This is a multi-part message in MIME format. ---------------0512010705712 Content-Type: text/plain; name="05-409.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-409.keywords" Skew-product, Schroedinger equation, Reducibility, Perturbation Series, Cancellations, Bryuno condition, Multiscale analysis, Renormalisation Group, Quantum Field Theory. ---------------0512010705712 Content-Type: application/postscript; name="skew5.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="skew5.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software %%Title: skew5.dvi %%Pages: 30 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX12 CMR9 CMBX9 CMMI6 CMSY9 CMMI9 CMR10 CMMI10 %%+ CMMIB10 CMSY10 CMMI7 EUFM10 CMR7 CMBX10 CMTI10 CMSY7 CMBX7 CMEX10 %%+ CMR5 CMMIB7 CMMI5 CMSY5 CMR6 CMBX6 CMSY6 CMMIB5 CMBX5 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: 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 23 /nu put dup 33 /omega 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 4FC57CB4420221C7B3EB8B891044B5FB0227009F0F6028D3F28545E63A3A6D58 CD5DAC837B4A93F1702AEA2768479E049E5ADB017236EF0139CD9AF0AD1D1D32 B111473BE1F09E3F718C9F57D5472C8E020FAC6F4FDC9B592A8281BE2812F020 F767F1C52D991A54A7A38E3A7BD66D12BD7BF228FDCB15836C4B3AE26C38555A 107AFCD506A2E066CB1EBAA60960578149FB26441D4CA4C5B0A15D4AACEE090C 7AD5D481F8F606E0C63F00D27181B64A44611CB4AB29A8719A0B67EF1389BB8A 8F1B33D9A1CB58E4A14F45E07242562414EEA2D8F73D071039E8EA829C855FC4 65D07B2E3F20CA79A33BC73E5A0DF5C2D4A3436578B1EC86890728AD43C75526 5FC58442DACDF340A7381338F035055011013DE32BED7531CB668ECABE0F55FD 2FEE4215850BA1E45A37DE6686A7098A0D0423C5ACEE1492F870EE55E0737857 653B18940DE79C487572EDE598D004F9A031F406F32DA9509D11231E292CF23B A58EF7F95B21AB44808E34F84CAB5316200BDA94D83C274E31800C5EC44AAFAA 881F73D8B956B02AA63D871097ED5A8F6128D42DE95ECC15C7C3A6328CFF1134 06768EF688CE49400381D190A0550445108782209D166E0315F3620EF87D7A45 6EF435E74EF7597B2232CB57F8681AE79DA9AF95D90EA4056BB333254248550E A7179B189BF29C8670EC0A2ADB2E3BD9687913137D47F4CCB24AC46F6486C2FB 49A3A293AD964F07558F3BD4743CE07DF6D56FFFDCF005921025CA3027795073 69EFCAA6B511AE636388F0B281AD29F126E47359F093C284F9C81CD0F49F4422 8E1DFABB3E48343DC9FBF1B8247E2E710A99BBB86459D6E8A814CC1982D0C8D3 017A657211E193F9D6EF88098BEE22382DA757D443516CDF03BD3C92F953839A B7DDFFA7CFA315B71A152DD692E803405549E477F1AFC621B869566FBD145B3B 177EA952802FF130F442F0684A3CAD8F950ADBB61B7074BE4AD68EAC4C157B1C 6DE013698F3B174099E922A38F3A03797046E95CCADB66779B82CE5F9EC0E991 DCE3C26EB7875A8A27449DA7E47C3E4161DC2EC748A8295D20 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 32 /psi 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: CMBX6 %!PS-AdobeFont-1.1: CMBX6 1.0 %%CreationDate: 1991 Aug 20 16:35:30 % 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 (CMBX6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX6 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{-49 -250 1367 753}readonly def /UniqueID 5000764 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 (1.0) 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 D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D5FC1B2109839E5B52DFBAF552B11EFFB6A16C F03FB920C15AE724EFDF0CCBF00A838D34440FF9FED532F44036AD22561184C5 283722DDFA7285E62754372D716D704AC0E00B2F6AB67154241C7449AA047833 94CEDB08E8C92907FE72A0B05AE36A7B9226ACD6E7890A0B528FDDE84A950FC6 801DE75CF2E739E9121149CCB8B1C87A106822648D84A3D3FBF295EE6C4BF403 BBE9A1C1F6DAEDD1E642ACC486E609703D7612BFFD10C324F5DC710811F7F614 3691B400E3773987424C0D2B0D8A736873C6371DDB2442F05E018A2B5CA9A4AA 17AABB95D09E5890CFFFED5AC01495D89A53D3CF44556D089203F8FC8E7F9119 8496FAB6BED78BAA8A4333579662139575C4339D7716630EA7A09F9CD8CEDD53 7DCD6C852325F0309810A7AB405E10071DF5A6D6E8D386C5EF58E25747EB3A0E 6EB47369D84838AECC135C9F78854670FBD36AFB4220EE46DFC45B338335AAEA C38924A21B5FADFBBBFC34CD4D897581D6439AE390F562FB03B5EABDD66FEBBD BFD3FE225545DE872A86E4E5AFA60FD2E145FB3806D6D125383C589F85F3A575 5990EBC081F912BB741EE1A97C321D2155325491E93EC43649B03034765BDC8F 3E08B294305AB69194AF78A653CCBBA39CB48A4FF7CF869393CFC754DB0EE245 B91A93F5B428D2C02F635708B95645A56D17CCD01DDFD96CEEE111582810F498 3C7D390E2C53EE2C43B092B010273757E1422C1C2E7E11ADB464CA84003FCCDE 832FE2EB754FF71CC10F37CBD490B1CA76A9C749960C20EDA3F20275E46460B2 746BA85423B9859E7BDF5C6BBBE5C71A7E6A1864D323639783DE2154C9C952D0 8FAA952B64342BCB41F4AD0DDB5605E507930787C481EE767A0827953F03A747 B6E082C9E598F0A7977DE6B1E5AA2FE5573B604BAD46370C34DF70EA34A49ECD D92F5A0307E7ACC154AB417557A22772E0019D5B1680C57D30441F1556478AAC F5FD1B756D09BCFBEEF77B79455CA056BA56E4DEE1CB1677885E6E809A8EBB28 D2B14354272B2934522B01332C20196BD81077C8B3F29F150280ECBFEA0EDB7A 17AA86268A1D1EA0DCAE4DE23A8B6960759282F92976D4E7CC5E9534D4A50455 E725E6AB50BC3F96BF686539A4D572573C6F3A0B985BD86BECF70C523FFEFCAF 2583392C4E358AC7F7988C790780095A9BF80F9B19E7E771A81A654C1E40346B A04BDE7E1953C179E2AA9DCB78182A92C144AFA4A2A0EFC9C26606E3D76D5C2F C9DDE5BF83F7B4B93F607B4B9E27EE2A3B911C041871391BFD7A454B67E3F84D C57982B394A6308A0C9516555410E08FB9A39051071E19A85863D99CE74A633A 183D91B54B4C63E7EA8895B34DF8BD19E97AAB1CDCFAF1E2ABE9357BC50C121F 4D244CC2EFB4E8AC4883372E8E284D85BC9FEE0244A1429C 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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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7093D5BAEA47480C2BEDD1748A6350EBB0F0752AAAB888ADB6113A2A0E81476C D8E5C14CD14A6F28B5D85A039AF8DDFE993A4DE7FA2E6F6B65E75EBA78610307 8225BCE0A6C05BD9A43373BDDD07E75AD5F634F4778E01081DF726FD818635D7 348347E59806C663A97F6012341A23473A3CED13CDB238949C93D16813E200FD 0FA4E78D3B9B7F8B158A55207734246D6264D6D4095173B8A9186F3CBD863B21 C5A2A70F60573ED69E1A7BCB9B87DDAC502B900113691E762457CBF2EA8DA8AE AC8BFB935B1A25F46AE9D5A98224E078CB54F2F7058B10597E89083D2A318E97 7A4F1A48BDB94424B5F8C8518ECBCF74E58DF7EFD77BDA22EC5290BB6375B9A5 4BBB10325716BA9791262D23B848C91384BFEB3007A032523FE3A8933A315BFE 1EA6A023E66DEF823F179D9740666823ECFAE8A28303EF088DA2B9527465E706 717369C17EB73043B1CA06F9774615BB255D131FD95E499EDBFA8374325CDB68 BDB974D744C9AB90E5F34BD82096281716329ED62D147B2FC450FF71F585E45A EA1ECF49A1C9236187681748F9B044544FCCBF56992ECF091F87BACE54BE1F55 54A23378D1E50E1FCA1E379B076D4F480A07F11E38F6C23EA48242AA1298C93F 287D95F250089194BF96C05AF2FF0CB1BCE5BF5F3BE32BB367B80E4230D8ADB0 2F850D8C236EE6EA30EB413E830460C271B6CD6CE93609ECFD7ECCE7D46EBF76 D9B182DAA7E0FD6B19F89D1B041F81A4A3AC2EC4516E39FD84205CCFDED000AF 6D51F3EC39C709C600669194775F6B37632642C20E88A511267D2A92CF3A53FF 6F4F2ABA98160383AE67004CFBC01DE5C88CD6F1AA9E35E15890EE8B17BAADC8 39AD0704859B1D8C57D48562608EEE3B087D8E4FF278716D2C7ABBFAA2F239A1 85384DE3A00A735651377CC8FC933DDD4FB690AA0CD21A72F403100FCEA5B0D9 47BFB346C67C0C0481E789CB72DAAC8706FAC16F5CFDD361E0170522CCAF04E9 3D139F4C351B8511920D7A4DCCC4855DC3357766A75506282CAECE4AC94F9073 C3E7A223CDED26082EB2A4B15EA782963D8C378A83CE29DAED12DFE2AF653AAC 7CDD7883ACCB8B3660E7760EB39C2805E7485EFB500D5B821558AD5A337A3855 B9E37219EBE0978676AA2F1D27C8C4BA26D0CD802F9200D66CC31DF540EA78CD 2D9799C7EB01276908A31ED8044FED0F14DD5B5D22B5F4A6B468CD9A2F5C1318 2A055D2D7369B80AAD8CEADC0BE441FAB040DC8B8A72D28C221B0041CE212C8C E7F6ABB7022E5A77A9ADDE8941870CB6F2E88A253B67E48091DD347528C238A3 6CC11E733B3ADB1519A43CA45A682A022D50F1C0B59D6F45F1D0BEC635C4C1E1 4B5BE5CB2A9E0190A5528202DBEE2F9E6A2328884AF0AEABA56BF379C566682C EB0280842A0F31F8D87B1D5CBB3A2A610EBB01C38429293AAA3A0E3A7D5D90F2 37E53C49DB16A18C6438425B3C8B9056C45740DB8A4FB23EEBD884BBDA879ED0 11C4490B657F76F530ECF8C780457A0AE7269924A6A34AC2F387330D1A8FC71B 677B19F3E46C2B635DA1190AFA7D65600962CD46C439B1BE8BC6B3B6EEB7A34C 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 12 /vextendsingle put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 26 /braceleftbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 40 /braceleftBigg put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 80 /summationtext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 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: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 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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1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueID 5000790 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6CC3F1E9AE32F234EB60FE7D E34995B1ACFF52428EA20C8ED4FD73E3935CEBD40E0EAD70C0887A451E1B1AC8 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) 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 (1.100) 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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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 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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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bop -29 169 a Fz(2)135 b(Preliminary)46 b(considerations)-29 368 y Fy(Assume)31 b Fx(\025)d Fu(2)h Fy([)p Fx(a;)14 b(b)p Fy(])27 b Fu(\032)h Fv(R)20 b Fu(n)g(f)p Fy(0)p Fu(g)p Fy(;)30 b(w)n(e)g(shall)g(see)h(later)e(that)i(the)g (condition)f(0)37 b Fx(=)-51 b Fu(2)28 b Fy([)p Fx(a;)14 b(b)p Fy(])30 b(can)g(b)r(e)h(relaxed)f(\(cf.)46 b(the)31 b(end)-29 468 y(of)h(Section)g(6\).)49 b(Let)32 b Fx(A)e Fu(2)g Fs(sl)p Fy(\(2)p Fx(;)14 b Fv(R)p Fy(\))32 b(with)g(imaginary)e (eigen)n(v)-5 b(alues.)48 b(P)n(ossibly)30 b(renaming)h Fx(a)g Fy(and)h Fx(b)f Fy(w)n(e)g(can)h(assume)-29 568 y(that)h(the)g(eigen)n(v)-5 b(alues)31 b(b)r(e)i Fu(\006)p Fx(i)p Fy(.)50 b(Let)33 b Fx(f)26 b Fy(:)31 b Fv(T)1360 538 y Ft(d)1430 568 y Fu(!)g Fs(sl)p Fy(\(2)p Fx(;)14 b Fv(R)p Fy(\))32 b(b)r(e)h(real-analytic,)f Fw(!)h Fu(2)f Fv(R)2716 538 y Ft(d)2787 568 y Fy(a)g(real)f(v)n(ector,)i(and)f Fx(")g Fy(a)g(real)-29 667 y(parameter.)96 785 y(Consider)27 b(the)h(ordinary)d(di\013eren)n(tial)j(equation)1493 957 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v 96 4767 a(De\014ne)914 4939 y Fs(M)83 b Fy(:=)f Fu(f)p Fx(G)23 b Fu(2)h Fy(SL\(2)p Fx(;)14 b Fv(C)p Fy(\))23 b(:)g Fx(G)1898 4951 y Fr(11)1991 4939 y Fy(=)g Fx(G)2144 4905 y Fo(\003)2144 4959 y Fr(22)2215 4939 y Fx(;)97 b(G)2400 4951 y Fr(12)2493 4939 y Fy(=)23 b Fx(G)2646 4905 y Fo(\003)2646 4959 y Fr(21)2717 4939 y Fu(g)13 b Fx(;)937 5063 y Fs(m)83 b Fy(:=)f Fu(f)p Fx(g)26 b Fu(2)d Fs(sl)p Fy(\(2)p Fx(;)14 b Fv(C)p Fy(\))23 b(:)g Fx(g)1812 5075 y Fr(11)1905 5063 y Fy(=)g Fx(g)2036 5029 y Fo(\003)2033 5084 y Fr(22)2103 5063 y Fx(;)97 b(g)2263 5075 y Fr(12)2356 5063 y Fy(=)22 b Fx(g)2486 5029 y Fo(\003)2483 5084 y Fr(21)2553 5063 y Fu(g)13 b Fx(:)949 b Fy(\(2.7\))-29 5235 y(It)28 b(is)g(easy)f(to)g(see)g(that) h Fs(M)g Fy(is)f(a)h(subgroup,)e(and)h Fs(m)h Fy(is)f(the)h(corresp)r (onding)e(Lie)h(algebra.)1840 5484 y(3)p eop end %%Page: 4 4 TeXDict begin 4 3 bop -29 169 a Fq(Lemma)32 b(2)41 b Fp(Consider)c(the)f(e)l(quation)49 b Fy(_)-37 b Fx(z)37 b Fy(=)c Fx(S)5 b(z)t Fp(,)36 b(with)g 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Fr(2)2348 1208 y Fy(=)g Fx(z)2487 1220 y Fr(21)2581 1208 y Fu(\000)23 b Fx(z)2712 1178 y Fo(\003)2708 1228 y Fr(12)2778 1208 y Fy(,)37 b(one)e(obtains)61 b(_)-49 b Fx(w)39 b Fy(=)c Fx(S)5 b(w)r Fy(.)61 b(If)-29 1307 y Fx(z)t Fy(\(0\))23 b Fu(2)g Fs(M)28 b Fy(then)g Fx(w)r Fy(\(0\))c(=)f(0,)k(so)g(that)h Fx(w)r Fy(\()p Fx(t)p Fy(\))c(=)f(0)k(for)g(all)g Fx(t)c Fu(2)h Fv(R)p Fy(.)36 b(Moreo)n(v)n(er,)25 b(if)j Fx(\016)s Fy(\()p Fx(t)p Fy(\))c(=)f(det)14 b Fx(z)t Fy(\()p Fx(t)p Fy(\),)27 b(one)g(\014nds)991 1468 y(_)978 1490 y Fx(\016)f Fy(=)d(\()p Fx(S)1212 1502 y Fr(11)1301 1490 y Fy(+)18 b Fx(S)1440 1456 y Fo(\003)1435 1511 y Fr(11)1505 1490 y Fy(\))c(\()p Fx(z)1622 1502 y Fr(11)1692 1490 y Fx(z)1731 1502 y Fr(22)1820 1490 y Fu(\000)k Fx(z)1942 1502 y Fr(12)2012 1490 y Fx(z)2051 1502 y Fr(21)2121 1490 y Fy(\))23 b(=)g(\()p Fx(S)2347 1502 y Fr(11)2436 1490 y Fy(+)18 b Fx(S)2575 1456 y Fo(\003)2570 1511 y Fr(11)2640 1490 y Fy(\))c Fx(\016)o(;)835 b Fy(\(2.9\))-29 1673 y(where)28 b Fx(S)263 1685 y Fr(11)351 1673 y Fy(+)18 b Fx(S)490 1643 y Fo(\003)485 1693 y Fr(11)579 1673 y Fy(=)k Fx(S)717 1685 y Fr(11)806 1673 y Fy(+)c Fx(S)940 1685 y Fr(22)1033 1673 y Fy(=)23 b(tr)13 b Fx(S)28 b Fy(=)23 b(0.)36 b(Hence)28 b Fx(\016)s Fy(\()p Fx(t)p Fy(\))c(=)e Fx(\016)s Fy(\(0\))i(=)e(1.)p 3704 1673 48 48 v 96 1837 a(Therefore)k(it)i(is)g(not)f(restrictiv)n(e) g(to)g(consider)g(the)h(di\013eren)n(tial)f(equation)1493 2020 y(_)-37 b Fx(x)23 b Fy(=)g(\()p Fx(\025A)c Fy(+)f Fx("f)9 b Fy(\()p Fw(!)s Fx(t)p Fy(\)\))14 b Fx(x;)1296 b Fy(\(2.10\))-29 2203 y(on)28 b Fs(M)p Fy(,)g(with)1220 2348 y Fx(A)23 b Fy(=)1393 2230 y Fm(\022)1460 2297 y Fx(i)115 b Fy(0)1454 2397 y(0)82 b Fu(\000)p Fx(i)1671 2230 y Fm(\023)1746 2348 y Fx(;)180 b(f)32 b Fu(2)23 b Fx(C)2165 2313 y Ft(!)2214 2348 y Fy(\()p Fv(T)2308 2313 y Ft(d)2347 2348 y Fx(;)14 b Fs(m)p Fy(\))p Fx(;)1036 b Fy(\(2.11\))-29 2547 y(and)28 b(this)g(w)n(e)f(shall)g(do)g (henceforth.)37 b(W)-7 b(rite)28 b Fx(\025)c Fy(=)e Fx(\025)1605 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b Fs(M)g Fy(is)g(a)g(group)e(and)i Fx(y)h Fu(2)e Fs(M)p Fy(,)j(then)e Fx(B)j Fu(2)c Fs(M)h Fy(if)g Fx(x)f Fu(2)g Fs(M)p Fy(.)51 b(More)31 b(generally)-7 b(,)32 b(det)14 b Fx(B)t Fy(\()p Fw(!)s Fx(t)p Fy(\))30 b(=)h(det)14 b Fx(B)t Fy(\(0\),)-29 4897 y(whic)n(h)30 b(means)f(that)i(det)14 b Fx(B)31 b Fy(=)26 b(1)19 b(+)h Fx(a)f Fy(+)h Fx(d)g Fy(+)g Fx(ad)g Fu(\000)f Fx(bc)29 b Fy(is)h(constan)n(t)f(along)g(the)h(\015o)n(w.)43 b(By)29 b(requiring)g(det)14 b Fx(B)31 b Fy(=)26 b(1)j(giv)n(es)-29 4997 y(\(2.16\).)p 3704 4997 V 96 5161 a(In)f(terms)f(of)h Fx(\014)t Fy(,)g(\(2.14\))f(b)r(ecomes)1840 5484 y(4)p eop end %%Page: 5 5 TeXDict begin 5 4 bop 1271 246 a Fy(_)1250 268 y Fx(\014)23 b Fy(+)18 b Fx(\025)1451 280 y Fr(0)1489 268 y Fy([)p Fx(\014)t(;)c(A)p Fy(])23 b(=)g(\()p Fx("f)k Fy(+)18 b Fx(\026A)p Fy(\))c(\()q(1)k(+)g Fx(\014)t Fy(\))c Fx(;)1067 b Fy(\(2.17\))-29 412 y(whic)n(h,)28 b(written)g(explicitly)g(for)f (the)h(corresp)r(onding)d(en)n(tries,)i(giv)n(es)1264 584 y(_)-34 b Fx(a)83 b Fy(=)g Fx("f)1608 596 y Fr(11)1696 584 y Fy(+)18 b Fx(i\026)g Fy(+)g Fx(")c Fy(\()p Fx(f)2085 596 y Fr(11)2155 584 y Fx(a)k Fy(+)g Fx(f)2341 596 y Fr(12)2411 584 y Fx(c)p Fy(\))h(+)f Fx(i\026)c(a;)975 686 y Fy(_)969 708 y Fx(b)k Fu(\000)g Fy(2)p Fx(i\025)1225 720 y Fr(0)1262 708 y Fx(b)82 b Fy(=)h Fx("f)1608 720 y Fr(12)1696 708 y Fy(+)18 b Fx(")c Fy(\()p Fx(f)1905 720 y Fr(11)1975 708 y Fx(b)k Fy(+)g Fx(f)2153 720 y Fr(12)2223 708 y Fx(d)p Fy(\))h(+)f Fx(i\026)c(b;)979 833 y Fy(_)-34 b Fx(c)19 b Fy(+)f(2)p Fx(i\025)1225 845 y Fr(0)1261 833 y Fx(c)83 b Fy(=)g Fx("f)1608 845 y Fr(21)1696 833 y Fy(+)18 b Fx(")c Fy(\()p Fx(f)1905 845 y Fr(21)1975 833 y Fx(a)k Fy(+)g Fx(f)2161 845 y Fr(22)2231 833 y Fx(c)p Fy(\))h Fu(\000)f Fx(i\026)c(c;)1278 936 y Fy(_)1254 957 y Fx(d)83 b Fy(=)g Fx("f)1608 969 y Fr(22)1696 957 y Fu(\000)18 b Fx(i\026)g Fy(+)g Fx(")c Fy(\()p Fx(f)2085 969 y Fr(21)2155 957 y Fx(b)k Fy(+)g Fx(f)2333 969 y Fr(22)2403 957 y Fx(d)p Fy(\))h Fu(\000)f Fx(i\026)c(d:)800 b Fy(\(2.18\))-29 1130 y(If)28 b(w)n(e)g(use)f(that)h Fx(d)23 b Fy(=)g Fx(a)698 1100 y Fo(\003)764 1130 y Fy(and)k Fx(b)c Fy(=)f Fx(c)1107 1100 y Fo(\003)1145 1130 y Fy(,)28 b(equations)f(\(2.18\))g(reduce)g(to)g(t)n(w)n(o)g(indep)r(enden)n(t)i (equations)1264 1302 y(_)-34 b Fx(a)83 b Fy(=)g Fx("f)1608 1314 y Fr(11)1696 1302 y Fy(+)18 b Fx(i\026)g Fy(+)g Fx(")c Fy(\()p Fx(f)2085 1314 y Fr(11)2155 1302 y Fx(a)k Fy(+)g Fx(f)2341 1314 y Fr(12)2411 1302 y Fx(c)p Fy(\))h(+)f Fx(i\026)c(a;)979 1426 y Fy(_)-34 b Fx(c)19 b Fy(+)f(2)p Fx(i\025)1225 1438 y Fr(0)1261 1426 y Fx(c)83 b Fy(=)g Fx("f)1608 1438 y Fr(21)1696 1426 y Fy(+)18 b Fx(")c Fy(\()p Fx(f)1905 1438 y Fr(21)1975 1426 y Fx(a)k Fy(+)g Fx(f)2161 1438 y Fr(22)2231 1426 y Fx(c)p Fy(\))h Fu(\000)f Fx(i\026)c(c;)986 b Fy(\(2.19\))-29 1599 y(whic)n(h)28 b(is)f(the)h(system)g(the)g(w)n(e)f(are)g(going)f(to)h(study)-7 b(.)96 1716 y(W)g(e)26 b(can)f(view)g(\(2.19\))g(as)g(a)g(system)g(of)g (ordinary)f(di\013eren)n(tial)h(equations)g(on)g Fv(C)2635 1686 y Fr(2)2672 1716 y Fy(.)36 b(Suc)n(h)26 b(a)f(system)g(admits)g(a) g(\014rst)-29 1816 y(in)n(tegral,)i(as)g(the)h(follo)n(wing)e(result)i (sho)n(ws.)-29 2006 y Fq(Lemma)k(4)41 b Fp(Given)31 b(the)f(system)f (\(2.19\),)j(the)e(function)1196 2178 y Fx(H)g Fy(=)23 b Fx(H)7 b Fy(\()p Fx(a;)14 b(c)p Fy(\))23 b(:=)f Fx(a)d Fy(+)f Fx(a)1963 2144 y Fo(\003)2019 2178 y Fy(+)2102 2111 y Fm(\000)2140 2178 y Fu(j)p Fx(a)p Fu(j)2230 2144 y Fr(2)2286 2178 y Fu(\000)g(j)p Fx(c)p Fu(j)2451 2144 y Fr(2)2488 2111 y Fm(\001)3539 2178 y Fy(\(2.20\))-29 2365 y Fp(is)31 b(a)f(c)l(onstant)f(of)h(motion,)h(that)f(is)1159 2344 y Fy(_)1128 2365 y Fx(H)g Fy(=)22 b(0)p Fp(.)-29 2555 y(Pr)l(o)l(of.)40 b Fy(Just)27 b(note)h(that)f(\(2.19\))g(is)g(a)g (rewriting)f(of)i(\(2.14\).)36 b(Lemma)27 b(3)g(sho)n(ws)f(that)h(det) 15 b Fx(B)31 b Fy(is)c(a)g(constan)n(t)g(of)g(motion.)-29 2655 y(In)h(terms)g(of)f Fx(a)h Fy(and)f Fx(c)p Fy(,)h(this)g(means)f 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Fr(21)2189 4741 y Fx(a)k Fy(+)g Fx(f)2375 4753 y Fr(22)2445 4741 y Fx(c)p Fy(\))2514 4766 y Fk(\027)2578 4741 y Fu(\000)g Fx(i\026)c(c)2790 4753 y Fk(\027)2836 4741 y Fx(;)721 b Fy(\(3.3\))-29 4914 y(and)28 b(for)f Fw(\027)i Fy(=)22 b Fq(0)1144 5086 y Fy(0)83 b(=)g Fx("f)1497 5098 y Fr(11)p Ft(;)p Fn(0)1642 5086 y Fy(+)18 b Fx(i\026)g Fy(+)g Fx(")c Fy(\()p Fx(f)2031 5098 y Fr(11)2101 5086 y Fx(a)k Fy(+)g Fx(f)2287 5098 y Fr(12)2357 5086 y Fx(c)p Fy(\))2426 5111 y Fn(0)2486 5086 y Fy(+)g Fx(i\026)c(a)2706 5098 y Fn(0)2747 5086 y Fx(;)939 5210 y Fy(2)p Fx(i\025)1058 5222 y Fr(0)1108 5210 y Fx(c)1144 5222 y Fn(0)1269 5210 y Fy(=)83 b Fx("f)1497 5222 y Fr(21)p Ft(;)p Fn(0)1642 5210 y Fy(+)18 b Fx(")c Fy(\()p Fx(f)1851 5222 y Fr(21)1921 5210 y Fx(a)k Fy(+)g Fx(f)2107 5222 y Fr(22)2177 5210 y Fx(c)p Fy(\))2245 5235 y Fn(0)2306 5210 y Fu(\000)g Fx(i\026)c(c)2518 5222 y Fn(0)2559 5210 y Fx(:)998 b Fy(\(3.4\))1840 5484 y(5)p eop end %%Page: 6 6 TeXDict begin 6 5 bop -29 169 a FG(3.1)113 b(Recursiv)m(e)37 b(equations)-29 340 y Fy(Assume)32 b Fx(\025)f Fu(6)p Fy(=)f(0.)50 b(W)-7 b(e)32 b(shall)g(see)f(that)i Fx(\026)d Fy(=)g Fx(O)r Fy(\()p Fx(")p Fy(\),)k(so)d(that)h(the)h(assumption)e (is)h(satis\014ed)g(for)f(all)h Fx(\025)f Fu(2)f Fy([)p Fx(a;)14 b(b)p Fy(])32 b(if)g Fx(")g Fy(is)-29 439 y(small)e(enough)f (and)g(0)35 b Fx(=)-51 b Fu(2)27 b Fy([)p Fx(a;)14 b(b)p Fy(].)43 b(In)29 b(fact)h(it)g(w)n(ould)g(b)r(e)g(enough)f(to)g (require)g(that)h(min)p Fu(fj)p Fx(a)p Fu(j)p Fx(;)14 b Fu(j)p Fx(b)p Fu(jg)28 b Fy(b)r(e)i(of)g(order)e Fu(j)p Fx(")p Fu(j)3567 409 y Ft(\033)3612 439 y Fy(;)j(cf.)-29 539 y(the)d(end)g(of)g(Section)f(6.)96 657 y(W)-7 b(e)28 b(can)f(write)h(a)f(formal)g(p)r(o)n(w)n(er)f(series)g(in)i Fx(")g Fy(for)f Fx(\014)t Fy(,)h(b)n(y)f(setting)805 884 y Fx(\014)g Fy(=)c Fx(\014)t Fy(\()p Fw(!)s Fx(t)p Fy(\))h(=)1314 781 y Fo(1)1287 805 y Fm(X)1286 984 y Ft(k)q Fr(=1)1421 884 y Fx(")1460 850 y Ft(k)1500 884 y Fx(\014)1551 850 y Fr(\()p Ft(k)q Fr(\))1645 884 y Fy(\()p Fw(!)r Fx(t)p Fy(\))p Fx(;)181 b(\014)2056 850 y Fr(\()p Ft(k)q Fr(\))2149 884 y Fy(\()p Fw( )s Fy(\))23 b(=)2414 805 y Fm(X)2390 989 y Fk(\027)t Fo(2)p Fj(Z)2523 973 y Fi(d)2572 884 y Fy(e)2609 850 y Ft(i)p Fk(\027)t Fo(\001)p Fk( )2750 884 y Fx(\014)2801 850 y Fr(\()p Ft(k)q Fr(\))2797 905 y Fk(\027)2894 884 y Fx(:)663 b Fy(\(3.5\))-29 1134 y(The)38 b(prop)r(erties)f Fx(a)i Fy(=)g Fx(d)783 1104 y Fo(\003)859 1134 y Fy(and)e Fx(b)j Fy(=)f Fx(c)1246 1104 y Fo(\003)1322 1134 y Fy(imply)e Fx(a)1608 1104 y Fo(\003)1608 1155 y Fk(\027)1694 1134 y Fy(=)i Fx(d)1841 1146 y Fo(\000)p Fk(\027)1977 1134 y Fy(and)e Fx(b)2184 1104 y Fo(\003)2184 1155 y Fk(\027)2269 1134 y Fy(=)j Fx(c)2410 1146 y Fo(\000)p Fk(\027)2507 1134 y Fy(.)67 b(In)38 b(the)g(same)f(w)n(a)n(y)f Fx(f)48 b Fu(2)40 b Fs(m)d Fy(yields)-29 1234 y Fx(f)21 1204 y Fo(\003)12 1254 y Fr(11)p Ft(;)p Fk(\027)167 1234 y Fy(=)23 b Fx(f)296 1246 y Fr(22)p Ft(;)p Fo(\000)p Fk(\027)479 1234 y Fy(,)28 b(hence)g Fx(f)802 1246 y Fr(11)p Ft(;)p Fk(\027)952 1234 y Fy(+)18 b Fx(f)1085 1204 y Fo(\003)1076 1254 y Fr(11)p Ft(;)p Fo(\000)p Fk(\027)1282 1234 y Fy(=)23 b(0,)k(and)h Fx(f)1674 1204 y Fo(\003)1665 1254 y Fr(12)p Ft(;)p Fk(\027)1819 1234 y Fy(=)23 b Fx(f)1948 1246 y Fr(21)p Ft(;)p Fo(\000)p Fk(\027)2131 1234 y Fy(.)96 1351 y(If)28 b(w)n(e)f(write)h(also)1591 1497 y Fx(\026)23 b Fy(=)1779 1393 y Fo(1)1752 1418 y Fm(X)1752 1597 y Ft(k)q Fr(=1)1886 1497 y Fx(")1925 1462 y Ft(k)1966 1497 y Fx(\026)2016 1462 y Fr(\()p Ft(k)q Fr(\))2108 1497 y Fx(;)1449 b Fy(\(3.6\))-29 1715 y(and)28 b(w)n(e)f(insert)g(\(3.5\))h (and)f(\(3.6\))g(in)n(to)h(\(3.3\))f(and)h(\(3.4\))f(w)n(e)g(\014nd) 1409 1924 y Fx(a)1453 1890 y Fr(\(1\))1453 1945 y Fk(\027)1625 1924 y Fy(=)82 b Fu(\000)p Fx(i)1878 1868 y(f)1919 1880 y Fr(11)p Ft(;)p Fk(\027)p 1876 1905 177 4 v 1876 1981 a Fw(!)21 b Fu(\001)d Fw(\027)2062 1924 y Fx(;)1417 2128 y(c)1453 2094 y Fr(\(1\))1453 2149 y Fk(\027)1625 2128 y Fy(=)82 b Fu(\000)p Fx(i)1992 2072 y(f)2033 2084 y 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Fk(\027)1311 3009 y Fm(\020)1361 3101 y Fx(f)1402 3113 y Fr(11)p Ft(;)p Fk(\027)1529 3121 y Fl(1)1566 3101 y Fx(a)1610 3067 y Fr(\()p Ft(k)q Fo(\000)p Fr(1\))1610 3122 y Fk(\027)1652 3130 y Fl(2)1806 3101 y Fy(+)f Fx(f)1930 3113 y Fr(12)p Ft(;)p Fk(\027)2057 3121 y Fl(1)2094 3101 y Fx(c)2130 3067 y Fr(\()p Ft(k)q Fo(\000)p Fr(1\))2130 3122 y Fk(\027)2172 3130 y Fl(2)2308 3009 y Fm(\021)2376 3101 y Fy(+)g Fx(i)2578 3022 y Fm(X)2502 3201 y Ft(k)2537 3209 y Fl(1)2569 3201 y Fr(+)p Ft(k)2655 3209 y Fl(2)2687 3201 y Fr(=)p Ft(k)2789 3101 y Fx(\026)2839 3067 y Fr(\()p Ft(k)2900 3075 y Fl(1)2933 3067 y Fr(\))2963 3101 y Fx(a)3007 3067 y Fr(\()p Ft(k)3068 3075 y Fl(2)3100 3067 y Fr(\))3007 3122 y Fk(\027)3130 2959 y Fm(!)3210 3101 y Fx(;)277 3396 y(c)313 3362 y Fr(\()p Ft(k)q Fr(\))313 3417 y Fk(\027)488 3396 y Fy(=)83 b Fu(\000)p Fx(i)921 3340 y Fy(1)p 740 3377 405 4 v 740 3453 a Fw(!)20 b Fu(\001)f Fw(\027)24 b Fy(+)18 b(2)p Fx(\025)1107 3465 y Fr(0)1168 3254 y Fm( )1320 3317 y(X)1234 3491 y 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3881 y Fy(=)82 b Fx(i)837 3739 y Fm( )987 3802 y(X)903 3978 y Fk(\027)944 3986 y Fl(1)977 3978 y Fr(+)p Fk(\027)1070 3986 y Fl(2)1102 3978 y Fr(=)p Fn(0)1205 3788 y Fm(\020)1254 3881 y Fx(f)1295 3893 y Fr(11)p Ft(;)p Fk(\027)1423 3901 y Fl(1)1459 3881 y Fx(a)1503 3846 y Fr(\()p Ft(k)q Fo(\000)p Fr(1\))1503 3901 y Fk(\027)1545 3909 y Fl(2)1699 3881 y Fy(+)18 b Fx(f)1823 3893 y Fr(12)p Ft(;)p Fk(\027)1951 3901 y Fl(1)1987 3881 y Fx(c)2023 3846 y Fr(\()p Ft(k)q Fo(\000)p Fr(1\))2023 3901 y Fk(\027)2065 3909 y Fl(2)2201 3788 y Fm(\021)2269 3881 y Fy(+)g Fx(i)2471 3802 y Fm(X)2395 3980 y Ft(k)2430 3988 y Fl(1)2462 3980 y Fr(+)p Ft(k)2548 3988 y Fl(2)2581 3980 y Fr(=)p Ft(k)2682 3881 y Fx(\026)2732 3846 y Fr(\()p Ft(k)2793 3854 y Fl(1)2826 3846 y Fr(\))2856 3881 y Fx(a)2900 3837 y Fr(\()p Ft(k)2961 3845 y Fl(2)2994 3837 y Fr(\))2900 3903 y Fn(0)3024 3739 y Fm(!)3103 3881 y Fx(;)435 4176 y(c)471 4132 y Fr(\()p Ft(k)q Fr(\))471 4198 y Fn(0)647 4176 y Fy(=)82 b Fu(\000)918 4119 y Fx(i)p 869 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n 2475 2400 75 75 0 360 DrawEllipse gs col7 0.00 shd ef gr gs col0 s gr % here ends figure; $F2psEnd rs showpage %%EndDocument @endspecial -29 2056 a FF(Figure)30 b(11:)41 b(Examples)29 b(of)g(renormalised)h(self-energy)f(clusters)g(b)r(elonging)h(to)f(the) f(same)h(equiv)l(alence)g(class:)41 b FA(T)e FF(is)29 b(a)g(renor-)-29 2147 y(malised)h(self-energy)f(cluster)f(of)h(the)f (second)h(t)n(yp)r(e,)f(while)h FA(T)1817 2115 y Fc(0)1867 2147 y FF(and)f FA(T)2075 2115 y Fc(00)2143 2147 y FF(are)h (renormalised)g(self-energy)g(clusters)g(of)g(the)f(\014rst)-29 2239 y(kind.)34 b(The)25 b(self-energy)h(v)l(alues)f(of)h FA(T)1082 2207 y Fc(0)1129 2239 y FF(and)e FA(T)1333 2207 y Fc(00)1398 2239 y FF(are)i(equal)f(to)g(eac)n(h)g(other:)34 b(in)25 b(fact)h(the)f(trees)g(con)n(taining)h(suc)n(h)e(renormalised) -29 2330 y(self-energy)i(clusters)g(can)f(b)r(e)g(obtained)g(from)h (eac)n(h)g(other)f(b)n(y)f(p)r(erm)n(uting)h(the)f(en)n(tering)i(lines) f(of)h FA(v)2945 2338 y Fg(0)2980 2330 y FF(.)35 b(The)25 b(external)g(lines)h FA(`)3696 2338 y Fg(in)-29 2421 y FF(and)h FA(`)154 2429 y Fg(out)272 2421 y FF(do)g(not)f(b)r(elong)i (to)f(the)f(self-energy)h(clusters,)h(and)f(ha)n(v)n(e)f(b)r(een)g(dra) n(wn)h(only)f(to)h(help)f(visualising)j(the)d(structure)g(of)-29 2513 y(the)g(self-energy)g(clusters.)-29 2776 y Fy(with)33 b(the)f(constan)n(t)f Fx(C)710 2788 y Fr(1)778 2776 y Fu(\024)f Fx(C)932 2788 y Fr(0)1002 2776 y Fy(to)i(b)r(e)g(de\014ned)g (later.)50 b(Set)32 b(also)f Fx(\037)p Fy(\()p Fx(x)p Fy(\))g(:=)f(1)21 b Fu(\000)g Fx( )s Fy(\()p Fx(x)p Fy(\),)34 b(and)e(de\014ne,)i(for)d(all)h Fx(n)e Fu(2)h Fv(Z)3673 2788 y Fr(+)3728 2776 y Fy(,)-29 2876 y Fx(\037)23 2888 y Ft(n)69 2876 y Fy(\()p Fx(x)p Fy(\))24 b(:=)e Fx(\037)p Fy(\()p Fx(\014)449 2846 y Fo(\000)p Fr(1)539 2876 y Fx(\015)587 2846 y Fo(\000)p Fr(1)582 2896 y Ft(n)676 2876 y Fx(x)p Fy(\))28 b(and)g Fx( )999 2888 y Ft(n)1044 2876 y Fy(\()p Fx(x)p Fy(\))c(:=)f Fx( )s Fy(\()p Fx(\014)1430 2846 y Fo(\000)p Fr(1)1519 2876 y Fx(\015)1567 2846 y Fo(\000)p Fr(1)1562 2896 y Ft(n)1656 2876 y Fx(x)p Fy(\),)29 b(with)f Fx(\014)f Fy(=)c(1)p Fx(=)p Fy(4.)96 2993 y(De\014ne)1138 3145 y(\001)1207 3157 y Fr(0)1245 3145 y Fy(\()p Fx(x)p Fy(\))h(=)1467 3028 y Fm(\022)1538 3089 y Fy(1)p 1538 3126 42 4 v 1538 3202 a(2)1604 3028 y Fm(\022)1696 3089 y Fy(1)p 1675 3126 85 4 v 1675 3202 a Fx(x)1722 3178 y Fr(2)1788 3145 y Fy(+)2049 3089 y(1)p 1881 3126 378 4 v 1881 3202 a(\()p Fx(x)19 b Fy(+)f(2)p Fx(\025)2152 3214 y Fr(0)2189 3202 y Fy(\))2221 3178 y Fr(2)2269 3028 y Fm(\023\023)2391 3045 y Fo(\000)p Fr(1)p Ft(=)p Fr(2)2561 3145 y Fx(;)996 b Fy(\(4.5\))-29 3369 y(and,)28 b(setting)g Fu(M)529 3326 y Fr([0])529 3391 y(1)603 3369 y Fy(\()p Fx(x)p Fy(\))c(:=)f(0)k(and)g Fu(M)1179 3326 y Fr([0])1179 3391 y(2)1254 3369 y Fy(\()p Fx(x)p Fy(\))d(:=)f Fx(\025)1548 3381 y Fr(0)1585 3369 y Fy(,)28 b(de\014ne)g(for)f Fx(n)c Fu(\025)g Fy(1)k(and)g Fx(j)h Fy(=)23 b(1)p Fx(;)14 b Fy(2)931 3619 y Fu(M)1031 3576 y Fr([)p Fo(\024)p Ft(n)p Fr(])1031 3642 y Ft(j)1165 3619 y Fy(\()p Fx(x)p Fy(\))85 b(=)1548 3515 y Ft(n)1508 3540 y Fm(X)1509 3716 y Ft(p)p Fr(=0)1642 3619 y Fu(M)1742 3576 y Fr([)p Ft(p)p Fr(])1742 3642 y Ft(j)1818 3619 y Fy(\()p Fx(x)p Fy(\))p Fx(;)983 3856 y Fu(M)1083 3813 y Fr([)p Ft(n)p Fr(])1083 3879 y Ft(j)1165 3856 y Fy(\()p Fx(x)p Fy(\))g(=)d Fx(\037)1560 3868 y Fr(0)1597 3856 y Fy(\(\001)1698 3868 y Fr(0)1736 3856 y Fy(\()p Fx(x)p Fy(\)\))14 b Fx(:)g(:)g(:)h(\037)2057 3868 y Ft(n)p Fo(\000)p Fr(1)2187 3856 y Fy(\(\001)2288 3868 y Fr(0)2326 3856 y Fy(\()p Fx(x)p Fy(\)\))p Fx(M)2559 3813 y Fr([)p Ft(n)p Fr(])2550 3879 y Ft(j)2642 3856 y Fy(\()p Fx(x)p Fy(\))p Fx(;)993 4060 y(M)1083 4017 y Fr([)p Ft(n)p Fr(])1074 4084 y Ft(j)1165 4060 y Fy(\()p Fx(x)p Fy(\))85 b(=)1525 4004 y Fx(i)p 1518 4041 42 4 v 1518 4117 a Fy(2)1611 3957 y Fo(1)1584 3982 y Fm(X)1583 4160 y Ft(k)q Fr(=1)1829 3982 y Fm(X)1718 4160 y Ft(T)9 b Fo(2S)1852 4169 y Fi(k)q(;j;n)p Fh(\000)p Fl(1)2074 4060 y Fu(V)2125 4072 y Ft(T)2177 4060 y Fy(\()p Fx(x)p Fy(\))p Fx(;)1269 b Fy(\(4.6\))-29 4338 y(where)26 b Fu(S)260 4350 y Ft(k)q(;j;n)435 4338 y Fy(is)f(the)i(set)e(of)h(all)g (renormalised)e(self-energy)g(clusters)i Fx(T)36 b Fy(on)26 b(scale)f Fx(n)h Fy(with)g Fu(j)p Fx(P)12 b Fy(\()p Fx(T)g Fy(\))p Fu(j)j(\000)f(j)p Fx(V)3249 4350 y Fr(2)3287 4338 y Fy(\()p Fx(T)e Fy(\))p Fu(j)23 b Fy(=)f Fx(k)29 b Fy(and)-29 4457 y(with)d(comp)r(onen)n(t)f(lab)r(el)h Fx(j)k Fy(asso)r(ciated)24 b(with)i(b)r(oth)g(external)e(lines.)36 b(F)-7 b(or)25 b Fx(n)e Fy(=)g(0)h(w)n(e)h(in)n(terpret)g Fu(M)3092 4414 y Fr([)p Fo(\024)p Fr(0])3092 4480 y Ft(j)3219 4457 y Fy(\()p Fx(x)p Fy(\))f(=)e Fu(M)3541 4414 y Fr([0])3541 4480 y Ft(j)3616 4457 y Fy(\()p Fx(x)p Fy(\).)-29 4557 y(One)28 b(has)893 4656 y(min)q Fu(fj)p Fx(x)p Fu(j)p Fx(;)14 b Fu(j)p Fx(x)19 b Fy(+)f(2)p Fx(\025)1466 4668 y Fr(0)1503 4656 y Fu(jg)23 b(\024)f Fy(\001)1747 4668 y Fr(0)1785 4656 y Fy(\()p Fx(x)p Fy(\))i Fu(\024)2007 4584 y(p)p 2077 4584 V 2077 4656 a Fy(2)13 b(min)p Fu(fj)p Fx(x)p Fu(j)p Fx(;)h Fu(j)p Fx(x)19 b Fy(+)f(2)p Fx(\025)2704 4668 y Fr(0)2741 4656 y Fu(jg)p Fx(:)751 b Fy(\(4.7\))96 4840 y(Then)31 b(the)g Fp(r)l(enormalise)l(d)i(pr)l(op)l(agator)42 b Fy(is)30 b(de\014ned)h(as)f Fx(g)1899 4810 y Fo(R)1896 4863 y Ft(`)1987 4840 y Fy(=)e Fx(g)2120 4852 y Ft(`)2182 4840 y Fy(if)j Fw(\027)2315 4852 y Ft(`)2374 4840 y Fy(=)d Fq(0)i Fy(and)h Fx(g)2753 4810 y Fo(R)2750 4863 y Ft(`)2841 4840 y Fy(=)d Fx(g)2977 4797 y Fr([)p Ft(n)3037 4806 y Fi(`)3065 4797 y Fr(])2974 4863 y Ft(j)3001 4872 y Fi(`)3088 4840 y Fy(\()p Fw(!)23 b Fu(\001)e Fw(\027)3300 4852 y Ft(`)3332 4840 y Fy(\))31 b(if)g Fw(\027)3528 4852 y Ft(`)3588 4840 y Fu(6)p Fy(=)c Fq(0)p Fy(,)-29 4940 y(with)947 5066 y Fx(g)990 5023 y Fr([)p Ft(n)p Fr(])987 5089 y Ft(j)1072 5066 y Fy(\()p Fx(x)p Fy(\))d(=)f Fu(\000)p Fx(i)1399 5010 y(\037)1451 5022 y Fr(0)1487 5010 y Fy(\(\001)1588 5022 y Fr(0)1626 5010 y Fy(\()p Fx(x)p Fy(\)\))14 b Fx(:)g(:)g(:)h(\037)1947 5022 y Ft(n)p Fo(\000)p Fr(1)2077 5010 y Fy(\(\001)2178 5022 y Fr(0)2216 5010 y Fy(\()p Fx(x)p Fy(\)\))p Fx( )2413 5022 y Ft(n)2459 5010 y Fy(\(\001)2560 5022 y Fr(0)2598 5010 y Fy(\()p Fx(x)p Fy(\)\))p 1399 5047 1345 4 v 1802 5144 a Fx(x)k Fy(+)f(2)p Fu(M)2093 5101 y Fr([)p Fo(\024)p Ft(n)p Fr(])2093 5167 y Ft(j)2227 5144 y Fy(\()p Fx(x)p Fy(\))2752 5066 y Fx(;)805 b Fy(\(4.8\))1820 5484 y(16)p eop end %%Page: 17 17 TeXDict begin 17 16 bop -29 169 a Fy(so)27 b(that)h(w)n(e)f(see)h(that) g Fx(g)733 139 y Fr([)p Ft(n)p Fr(])815 169 y Fy(\()p Fx(x)p Fy(\))c Fu(6)p Fy(=)f(0)k(implies)1247 326 y(1)p 1247 363 42 4 v 1247 439 a(2)1299 382 y Fx(\014)t(\015)1393 394 y Ft(n)1438 382 y Fx(C)1497 394 y Fr(1)1558 382 y Fu(\024)c Fy(\001)1715 394 y Fr(0)1752 382 y Fy(\()p Fx(x)p Fy(\))h Fu(\024)f Fx(\014)t(\015)2069 394 y Ft(n)p Fo(\000)p Fr(1)2199 382 y Fx(C)2258 394 y Fr(1)2296 382 y Fx(:)1261 b Fy(\(4.9\))-29 580 y(W)-7 b(e)28 b(asso)r(ciate)f(a)g (scale)g(lab)r(el)g Fx(n)982 592 y Ft(`)1042 580 y Fy(also)f(with)i (lines)g(with)g(v)-5 b(anishing)27 b(momen)n(tum,)h(b)n(y)f(setting)h Fx(n)3040 592 y Ft(`)3095 580 y Fy(=)22 b Fu(\000)p Fy(1.)96 714 y(Note)28 b(that)g Fu(M)577 671 y Fr([)p Fo(\024)p Ft(n)p Fr(])577 737 y Ft(j)711 714 y Fy(\()p Fx(x)p Fy(\))h(is)f (de\014ned)g(in)g(terms)f(of)h(propagators)d(on)i(scales)g Fx(n)2499 684 y Fo(0)2545 714 y Fx(<)c(n)p Fy(,)28 b(hence)f(in)h (terms)g(of)g Fu(M)3488 671 y Fr([)p Ft(n)3548 646 y Fh(0)3570 671 y Fr(])3488 738 y Ft(j)3518 722 y Fh(0)3593 714 y Fy(\()p Fx(x)3672 684 y Fo(0)3696 714 y Fy(\),)-29 845 y(with)35 b Fx(n)217 814 y Fo(0)274 845 y Fx(<)e(n)p Fy(:)50 b(this)34 b(means)g(that)g(\(4.6\))g(pro)n(vides)f(a)g (recursiv)n(e)f(de\014nition)j(of)f Fu(M)2655 801 y Fr([)p Fo(\024)p Ft(n)p Fr(])2655 868 y Ft(j)2789 845 y Fy(\()p Fx(x)p Fy(\),)j(hence)d(it)h(mak)n(es)d(sense.)-29 944 y(Note)g(also)f(that)h(self-energy)f(clusters)g(on)h(scale)f Fu(\000)p Fy(1)g(\(in)h(particular)f(those)g(consisting)g(of)h(a)g (single)f(no)r(de\))h(are)f(not)-29 1044 y(tak)n(en)c(in)n(to)h(accoun) n(t)f(in)g(\(4.6\);)h(this)g(will)f(b)r(e)h(motiv)-5 b(ated)28 b(b)n(y)g(Lemma)f(10)g(b)r(elo)n(w.)96 1161 y(De\014ne)h(the)g Fp(tr)l(e)l(e)h(value)35 b Fy(V)-7 b(al\()p Fx(\022)r Fy(\))28 b(as)1246 1361 y(V)-7 b(al\()p Fx(\022)r Fy(\))24 b(=)1583 1269 y Fm(\020)1695 1282 y(Y)1646 1464 y Ft(`)p Fo(2)p Ft(L)p Fr(\()p Ft(\022)r Fr(\))1863 1361 y Fx(g)1906 1327 y Fo(R)1903 1381 y Ft(`)1967 1269 y Fm(\021\020)2135 1282 y(Y)2080 1464 y Ft(v)r Fo(2)p Ft(P)9 b Fr(\()p Ft(\022)r Fr(\))2311 1361 y Fx(F)2364 1373 y Ft(v)2403 1269 y Fm(\021)2453 1361 y Fx(:)1063 b Fy(\(4.10\))-29 1640 y(Then,)29 b(if)g(\002)354 1610 y Fo(R)354 1663 y Ft(j;k)q(;)p Fk(\027)531 1640 y Fy(is)f(the)g(set)h (of)f(inequiv)-5 b(alen)n(t)28 b(renormalised)e(trees)i(with)g(lab)r (els)g Fx(j;)14 b Fw(\027)34 b Fy(asso)r(ciated)27 b(with)i(the)f(ro)r (ot)g(line)-29 1739 y(and)g(with)g Fu(j)p Fx(P)12 b Fy(\()p Fx(\022)r Fy(\))p Fu(j)19 b(\000)f(j)p Fx(V)711 1751 y Fr(2)749 1739 y Fy(\()p Fx(\022)r Fy(\))p Fu(j)24 b Fy(=)f Fx(k)s Fy(,)k(set)1473 1856 y Fx(u)1521 1813 y Fr([)p Ft(k)q Fr(])1521 1879 y Ft(j;)p Fk(\027)1637 1856 y Fy(=)1796 1777 y Fm(X)1724 1964 y Ft(\022)r Fo(2)p Fr(\002)1854 1944 y Fh(R)1854 1984 y Fi(j;k)q(;)p Fb(\027)2001 1856 y Fy(V)-7 b(al\()p Fx(\022)r Fy(\))p Fx(;)1290 b Fy(\(4.11\))-29 2140 y(with)29 b Fx(u)209 2097 y Fr([)p Ft(k)q Fr(])209 2162 y(3)p Ft(;)p Fn(0)326 2140 y Fy(:=)22 b Fx(\026)486 2110 y Fr([)p Ft(k)q Fr(])565 2140 y Fy(,)28 b(and)f(de\014ne)h(the)g(function)p 1485 2094 48 4 v 28 w Fx(u)p Fy(\()p Fx(t)p Fy(\))23 b(=)g(\()p Fx(u)1818 2152 y Fr(1)1855 2140 y Fy(\()p Fx(t)p Fy(\))p Fx(;)14 b(u)2034 2152 y Fr(2)2071 2140 y Fy(\()p Fx(t)p Fy(\)\))29 b(as)p 960 2348 V 960 2394 a Fx(u)1007 2406 y Ft(j)1042 2394 y Fy(\()p Fx(t)p Fy(\))24 b(=)1275 2290 y Fo(1)1248 2315 y Fm(X)1248 2494 y Ft(k)q Fr(=1)1382 2394 y Fx(")1421 2360 y Ft(k)1462 2394 y Fx(u)1510 2351 y Fr([)p Ft(k)q Fr(])1510 2417 y Ft(j)1588 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Fy(2)g(again)f(the)i(b)r(ound)g(follo)n(ws)f(inductiv) n(ely)-7 b(.)57 b(If)35 b Fx(p)f Fy(=)g(1)-29 5235 y(call)h Fx(\022)170 5247 y Fr(1)242 5235 y Fy(the)g(subtree)g(with)g(ro)r(ot)f (line)i Fx(`)1272 5247 y Fr(1)1308 5235 y Fy(,)h(and)e(call)f Fx(T)46 b Fy(the)36 b(set)f(of)f(p)r(oin)n(ts)h(and)g(lines)g(b)r(et)n (w)n(een)g Fx(`)3166 5247 y Fr(1)3237 5235 y Fy(and)g Fx(`)g Fy(\(that)g(is)1820 5484 y(18)p eop end %%Page: 19 19 TeXDict begin 19 18 bop -29 169 a Fy(whic)n(h)29 b(precede)f Fx(`)h Fy(but)g(not)g Fx(`)912 181 y Fr(1)949 169 y Fy(\).)40 b(Denote)29 b(b)n(y)g Fx(P)12 b Fy(\()p Fx(T)g Fy(\))28 b(the)h(set)g(of)g(p)r(oin)n(ts)f(in)h Fx(T)12 b Fy(,)28 b(and)h(de\014ne)g Fx(M)9 b Fy(\()p Fx(T)j Fy(\))24 b(:=)3252 106 y Fm(P)3339 194 y Ft(v)r Fo(2)p Ft(P)9 b Fr(\()p Ft(T)g Fr(\))3589 169 y Fu(j)p Fw(\027)3665 181 y Ft(v)3705 169 y Fu(j)p Fy(.)-29 279 y(Call)33 b Fw(\027)38 b Fy(and)33 b Fw(\027)459 249 y Fo(0)515 279 y Fy(the)g(momen)n(ta)f(asso)r(ciated) g(with)h Fx(`)f 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Fy(w)n(ould)34 b(b)r(e)g(a)g(renormalised)e(self-energy)h (cluster.)56 b(Therefore)32 b(b)n(y)i(the)h(second)-29 1094 y(Diophan)n(tine)28 b(conditions)f(\(4.14\),)g(one)g(obtains)g Fx(n)p Fy(\()p Fw(\027)d Fu(\000)19 b Fw(\027)1837 1064 y Fo(0)1860 1094 y Fy(\))24 b Fu(\025)e Fx(n)p Fy(,)28 b(so)f(that)h Fx(M)9 b Fy(\()p Fx(T)j Fy(\))22 b Fu(\025)h Fy(2)2753 1064 y Ft(n)p Fo(\000)p Fr(1)2910 1094 y Fy(also)k(in)g(suc)n (h)h(a)f(case.)96 1211 y(Hence,)h(b)n(y)f(the)h(inductiv)n(e)g(h)n(yp)r (othesis)227 1394 y Fx(N)294 1406 y Ft(n)339 1394 y Fy(\()p Fx(\022)r Fy(\))c Fu(\024)f Fy(1)18 b(+)699 1327 y Fm(\000)737 1394 y Fy(22)821 1360 y Fo(\000)p Ft(n)917 1394 y Fx(M)9 b Fy(\()p Fx(\022)1078 1406 y Fr(1)1115 1394 y Fy(\))19 b Fu(\000)f Fy(1)1291 1327 y Fm(\001)1351 1394 y Fu(\024)23 b Fy(1)18 b Fu(\000)g Fy(22)1666 1360 y Fo(\000)p Ft(n)1762 1394 y Fx(M)9 b Fy(\()p Fx(T)j Fy(\))18 b(+)2078 1327 y Fm(\000)2116 1394 y Fy(22)2200 1360 y Fo(\000)p Ft(n)2296 1394 y Fx(M)9 b Fy(\()p Fx(\022)r 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y Ft(v)2520 2119 y Fu(j)23 b Fx(>)g Fy(2)2696 2084 y Ft(n)2737 2092 y Fi(T)2782 2084 y Fo(\000)p Fr(1)2871 2119 y Fx(;)686 b Fy(\(5.5\))-29 2398 y Fp(with)31 b(the)f(same)g(c)l(onstant)f Fx(K)35 b Fp(as)30 b(in)g(\(5.2\).)-29 2599 y(Pr)l(o)l(of.)40 b Fy(W)-7 b(e)27 b(\014rst)f(pro)n(v)n(e)f(the)i(b)r(ound)g(on)f Fx(M)9 b Fy(\()p Fx(T)j Fy(\).)36 b(By)26 b(construction)g Fx(T)38 b Fy(m)n(ust)26 b(con)n(tain)g(at)h(least)f(a)g(line)h Fx(`)f Fy(on)g(scale)g Fx(n)3676 2611 y Ft(T)3728 2599 y Fy(,)-29 2698 y(so)j(that)g Fu(j)p Fx(x)326 2710 y Ft(`)378 2698 y Fy(+)19 b(2)p Fx(\032)547 2710 y Fr(0)584 2698 y Fy(\()p Fx(x)663 2710 y Ft(`)696 2698 y Fy(\))p Fu(j)26 b(\024)f Fx(\014)t(C)977 2710 y Fr(1)1015 2698 y Fx(\015)1058 2710 y Ft(n)1099 2718 y Fi(T)1145 2710 y Fo(\000)p Fr(1)1234 2698 y Fy(,)30 b(with)f Fx(x)1524 2710 y Ft(`)1582 2698 y Fy(=)d Fw(!)c Fu(\001)d Fw(\027)1851 2710 y Ft(`)1883 2698 y Fy(.)42 b(W)-7 b(rite)29 b Fw(\027)2234 2710 y Ft(`)2291 2698 y Fy(=)c Fw(\027)2435 2668 y Fr(0)2435 2722 y Ft(`)2463 2730 y Fl(0)2519 2698 y Fy(+)19 b Fx(\033)2650 2710 y Ft(`)2682 2698 y Fw(\027)6 b Fy(,)30 b(where)e Fw(\027)35 b Fy(is)29 b(the)h(momen)n(tum)-29 2798 y(asso)r(ciated)h (with)h(the)g(en)n(tering)f(line)h(of)g Fx(T)42 b Fy(and)32 b Fx(\033)1600 2810 y Ft(`)1662 2798 y Fy(=)e(0)p Fx(;)14 b Fy(1,)32 b(and)f(set)h Fx(x)e Fy(=)g Fw(!)24 b Fu(\001)d Fw(\027)37 b Fy(and)32 b Fx(x)2830 2768 y Fr(0)2830 2821 y Ft(`)2897 2798 y Fy(=)e Fw(!)24 b Fu(\001)d Fw(\027)3174 2768 y Fr(0)3174 2821 y Ft(`)3211 2798 y Fy(.)49 b(The)32 b(en)n(tering)-29 2897 y(line)j(of)f Fx(T)46 b Fy(has)34 b(scale)f(strictly)h(larger)f(than)h Fx(n)1475 2909 y Ft(T)1527 2897 y Fy(,)j(so)c(that)i Fu(j)p Fx(x)23 b Fy(+)g(2)p Fx(\032)2148 2909 y Fr(0)2185 2897 y Fy(\()p Fx(x)p Fy(\))p Fu(j)35 b(\024)f Fx(\014)t(C)2563 2909 y Fr(1)2601 2897 y Fx(\015)2644 2909 y Ft(n)2685 2917 y Fi(T)2731 2909 y Fo(\000)p Fr(1)2820 2897 y Fy(.)57 b(If)35 b Fx(M)9 b Fy(\()p Fx(T)j Fy(\))34 b Fu(\024)f Fy(2)3379 2867 y Ft(n)3420 2875 y Fi(T)3466 2867 y Fo(\000)p Fr(1)3590 2897 y Fy(then)-29 2997 y Fu(j)p Fw(\027)48 2967 y Fr(0)48 3021 y Ft(`)85 2997 y Fu(j)i(\024)e Fx(M)9 b Fy(\()p Fx(T)j Fy(\))34 b Fu(\024)g Fy(2)631 2967 y Ft(n)672 2975 y Fi(T)717 2967 y Fo(\000)p Fr(1)806 2997 y Fy(,)i(hence)f Fx(n)p Fy(\()p Fw(\027)1239 2967 y Fr(0)1239 3021 y Ft(`)1276 2997 y Fy(\))f Fu(\024)g Fx(n)1491 3009 y Ft(T)1566 2997 y Fu(\000)23 b Fy(1,)35 b(so)f(that)h Fu(j)p Fx(x)2120 2967 y Fr(0)2120 3021 y Ft(`)2180 2997 y Fy(+)23 b(2)p Fx(\032)2353 3009 y Fr(0)2389 2997 y Fy(\()p Fx(x)2468 2967 y Fr(0)2468 3021 y Ft(`)2506 2997 y Fy(\))p Fu(j)35 b Fx(>)f(C)2754 3009 y Fr(1)2791 2997 y Fx(\015)2834 3015 y Ft(n)p Fr(\()p Fk(\027)2943 2995 y Fl(0)2943 3035 y Fi(`)2976 3015 y Fr(\))3040 2997 y Fu(\025)g Fx(C)3198 3009 y Fr(1)3235 2997 y Fx(\015)3278 3009 y Ft(n)3319 3017 y Fi(T)3365 3009 y Fo(\000)p Fr(1)3455 2997 y Fy(,)i(b)n(y)e(the)-29 3106 y(Diophan)n(tine)28 b(conditions)f(\(4.14\).)36 b(Then)28 b(one)f(has)912 3289 y Fx(C)971 3301 y Fr(1)1008 3289 y Fx(\015)1051 3301 y Ft(n)1092 3309 y Fi(T)1138 3301 y Fo(\000)p Fr(1)1310 3289 y Fx(>)83 b Fu(j)p Fx(x)1528 3301 y Ft(`)1579 3289 y Fy(+)18 b(2)p Fx(\032)1747 3301 y Fr(0)1784 3289 y Fy(\()p Fx(x)1863 3301 y Ft(`)1895 3289 y Fy(\))p Fu(j)h Fy(+)f Fx(\033)2099 3301 y Ft(`)2131 3289 y Fu(j)p Fx(x)h Fy(+)f(2)p Fx(\032)2388 3301 y Fr(0)2425 3289 y Fy(\()p Fx(x)p Fy(\))p Fu(j)1310 3413 y(\025)83 b(j)p Fx(x)1528 3379 y Fr(0)1528 3434 y Ft(`)1584 3413 y Fy(+)18 b(2\()p Fx(\032)1784 3425 y Fr(0)1821 3413 y Fy(\()p Fx(x)1900 3425 y Ft(`)1933 3413 y Fy(\))h Fu(\000)f Fx(\033)2114 3425 y Ft(`)2146 3413 y Fx(\032)2189 3425 y Fr(0)2226 3413 y Fy(\()p Fx(x)p Fy(\))p Fu(j)24 b Fx(>)f(C)2531 3425 y Fr(1)2569 3413 y Fx(\015)2612 3425 y Ft(n)2653 3433 y Fi(T)2698 3425 y Fo(\000)p Fr(1)2788 3413 y Fx(;)769 b Fy(\(5.6\))-29 3596 y(whic)n(h)28 b(leads)f(to)g(a)h(con)n (tradiction.)96 3713 y(Next)35 b(w)n(e)g(pass)f(to)h(the)g(b)r(ound)g (on)g Fx(N)1336 3725 y Ft(n)1381 3713 y Fy(\()p Fx(T)12 b Fy(\).)58 b(Consider)34 b(a)g(subset)h Fx(G)2344 3725 y Fr(0)2417 3713 y Fy(of)g(the)g(lines)g(of)g(a)f(tree)h Fx(\022)i Fy(b)r(et)n(w)n(een)e(t)n(w)n(o)-29 3813 y(lines)g Fx(`)203 3825 y Fr(out)337 3813 y Fy(and)f Fx(`)540 3825 y Fr(in)634 3813 y Fy(Set)h Fx(G)f Fy(=)g Fx(G)1047 3825 y Fr(0)1108 3813 y Fu([)23 b(f)p Fx(`)1263 3825 y Fr(in)1322 3813 y Fu(g)f([)i(f)p Fx(`)1542 3825 y Fr(out)1641 3813 y Fu(g)p Fy(.)57 b(Let)34 b([)p Fx(n)1991 3825 y Fr(in)2051 3813 y Fy(])p Fx(;)14 b Fy([)p Fx(n)2184 3825 y Fr(out)2283 3813 y Fy(])35 b(b)r(e)g(the)g(scales)e(of)h(the)h(lines)f Fx(`)3331 3825 y Fr(out)3465 3813 y Fy(and)h Fx(`)3669 3825 y Fr(in)3728 3813 y Fy(,)-29 3913 y(resp)r(ectiv)n(ely)-7 b(,)28 b(and)g(supp)r(ose)g(that)h Fx(n)1147 3925 y Fr(in)1206 3913 y Fx(;)14 b(n)1293 3925 y Fr(out)1417 3913 y Fu(\025)24 b Fx(n)p Fy(,)29 b(while)f(all)g(lines)h(in)f Fx(G)2294 3925 y Fr(0)2360 3913 y Fy(\(if)h(an)n(y\))f(ha)n(v)n(e)f(scales)g Fx(n)3133 3882 y Fo(0)3181 3913 y Fu(\024)d Fx(n)3320 3925 y Ft(T)3391 3913 y Fu(\000)18 b Fy(1.)39 b(Note)-29 4012 y(that)33 b(in)f(general)e Fx(G)612 4024 y Fr(0)682 4012 y Fy(is)i(not)g(ev)n(en)f(a)h(cluster)g(unless)f Fx(n)1761 4024 y Fr(in)1821 4012 y Fx(;)14 b(n)1908 4024 y Fr(out)2038 4012 y Fu(\025)30 b Fx(n)2183 4024 y Ft(T)2235 4012 y Fy(.)50 b(Then)32 b(w)n(e)g(can)f(pro)n(v)n(e)f(that)j(if)f Fx(N)3372 4024 y Ft(n)3417 4012 y Fy(\()p Fx(G)3514 4024 y Fr(0)3552 4012 y Fy(\))f Fu(6)p Fy(=)e(0)-29 4112 y(then)c Fx(N)224 4124 y Ft(n)269 4112 y Fy(\()p Fx(G)366 4124 y Fr(0)403 4112 y Fy(\))f Fu(\024)e Fy(22)630 4082 y Fo(\000)p Ft(n)740 4050 y Fm(P)828 4137 y Ft(v)r Fo(2)p Ft(P)9 b Fr(\()p Ft(G)1037 4145 y Fl(0)1069 4137 y Fr(\))1113 4112 y Fu(j)p Fw(\027)1190 4124 y Ft(v)1229 4112 y Fu(j)i(\000)g Fy(1,)24 b(where)f Fx(P)12 b Fy(\()p Fx(G)1826 4124 y Fr(0)1864 4112 y Fy(\))24 b(is)g(the)g(set)g(of)f(p)r(oin)n(ts)h (preceding)f Fx(`)3006 4124 y Fr(out)3129 4112 y Fy(and)h(follo)n(wing) f Fx(`)3669 4124 y Fr(in)3728 4112 y Fy(.)96 4245 y(If)k Fx(G)243 4257 y Fr(0)306 4245 y Fy(has)f(no)g(lines)g(then)g(the)h(mo)r (de)f Fw(\027)1360 4257 y Fr(0)1423 4245 y Fy(of)g(the)h(\(only\))f(no) r(de)g(b)r(et)n(w)n(een)g Fx(`)2456 4257 y Fr(out)2582 4245 y Fy(and)g Fx(`)2777 4257 y Fr(in)2862 4245 y Fy(is)g(suc)n(h)g (that)h Fu(j)p Fw(\027)3385 4257 y Fr(0)3422 4245 y Fu(j)d(\025)e Fy(2)3598 4215 y Ft(n)p Fo(\000)p Fr(1)3728 4245 y Fy(,)-29 4345 y(b)n(y)i(the)h(second)e(Diophan)n(tine)h(conditions)g(\(4.14\),)g (and)g(the)g(statemen)n(t)g(is)g(true.)36 b(Hence)24 b(w)n(e)g(pro)r(ceed)f(inductiv)n(ely)h(on)-29 4444 y(the)j(n)n(um)n(b) r(er)e(of)h(lines)g(in)g Fx(G)854 4456 y Fr(0)891 4444 y Fy(.)37 b(If)26 b(no)f(line)h(of)g Fx(G)1458 4456 y Fr(0)1522 4444 y Fy(on)f(the)h(path)g Fu(P)7 b Fy(\()p Fx(G)p Fy(\))26 b(connecting)g(the)g(external)f(lines)h(of)f Fx(G)i Fy(has)e(scale)-29 4544 y Fx(n)30 b Fy(then)f(the)h(lines)f(in)h Fx(G)741 4556 y Fr(0)808 4544 y Fy(on)f(scale)f Fx(n)i Fy(\(if)g(an)n(y\))e(b)r(elong)h(to)h(trees)e(with)i(ro)r(ot)f(on)g Fu(P)7 b Fy(\()p Fx(G)p Fy(\),)30 b(and)f(the)h(statemen)n(t)f(follo)n (ws)-29 4643 y(from)j(the)f(b)r(ound)h(\(5.3\))g(for)e(trees)h(giv)n (en)g(in)h(the)g(pro)r(of)f(of)g(Lemma)g(11.)48 b(If)32 b(there)f(is)g(a)g(line)h Fx(`)d Fu(2)h(P)7 b Fy(\()p Fx(G)p Fy(\))32 b(on)f(scale)g Fx(n)p Fy(,)-29 4743 y(then)c(call)f Fx(G)375 4755 y Fr(1)439 4743 y Fy(and)h Fx(G)665 4755 y Fr(2)729 4743 y Fy(the)f(disjoin)n(t)h(subsets)f(of)g Fx(G)h Fy(suc)n(h)f(that)h Fx(G)2066 4755 y Fr(1)2119 4743 y Fu([)17 b Fx(G)2256 4755 y Fr(2)2309 4743 y Fu([)f(f)p Fx(`)p Fu(g)22 b Fy(=)h Fx(G)p Fy(.)37 b(Then)26 b Fx(G)3014 4755 y Fr(1)3068 4743 y Fu([)16 b(f)p Fx(`)p Fu(g)25 b Fy(and)h Fx(G)3508 4755 y Fr(2)3562 4743 y Fu([)16 b(f)p Fx(`)p Fu(g)-29 4843 y Fy(ha)n(v)n(e)25 b(the)h(same)f(structure) g(of)h Fx(G)g Fy(itself,)g(but)g(eac)n(h)f(has)g(less)g(lines.)37 b(Hence,)26 b(again)e(the)i(inductiv)n(e)g(assumption)f(yields)-29 4942 y(the)j(result.)96 5060 y(Therefore,)38 b(as)e(a)g(particular)g (case,)i(b)n(y)f(c)n(ho)r(osing)e Fx(G)1835 5072 y Fr(0)1911 5060 y Fy(=)j Fx(T)12 b Fy(,)38 b(with)f Fx(T)50 b Fu(2)38 b(S)2576 5072 y Ft(k)q(;j;n)2720 5080 y Fi(T)2767 5072 y Fo(\000)p Fr(1)2856 5060 y Fy(,)h(the)f(b)r(ound)f(for)f Fx(N)3539 5072 y Ft(n)3584 5060 y Fy(\()p Fx(G)3681 5072 y Fr(0)3719 5060 y Fy(\))-29 5159 y(implies)28 b(the)g(b)r(ound)g(on)g Fx(N)835 5171 y Ft(n)879 5159 y Fy(\()p Fx(T)12 b Fy(\))28 b(w)n(e)f(are)g(lo)r(oking)f(for.)p 3704 5159 V 1820 5484 a(19)p eop end %%Page: 20 20 TeXDict begin 20 19 bop -29 172 a Fq(Lemma)32 b(13)41 b Fp(Assume)29 b(that)h(the)g(pr)l(op)l(agators)h Fx(g)1553 129 y Fr([)p Ft(p)p Fr(])1550 196 y Ft(j)1629 172 y Fy(\()p Fx(x)p Fy(\))f Fp(c)l(an)g(b)l(e)g(uniformly)h(b)l(ounde)l(d)f(for)g (al)t(l)h Fy(0)23 b Fu(\024)f Fx(p)h Fu(\024)g Fx(n)18 b Fu(\000)g Fy(1)29 b Fp(as)1452 293 y Fm(\014)1452 343 y(\014)1452 392 y(\014)1480 388 y Fx(g)1523 345 y Fr([)p Ft(p)p Fr(])1520 411 y Ft(j)1598 388 y Fy(\()p Fx(x)p Fy(\))1709 293 y Fm(\014)1709 343 y(\014)1709 392 y(\014)1761 388 y Fu(\024)23 b Fx(K)1920 400 y Fr(1)1956 388 y Fx(C)2021 353 y Fo(\000)p Fr(1)2015 410 y(1)2111 388 y Fx(\015)2159 354 y Fo(\000)p Fr(1)2154 409 y Ft(p)2247 388 y Fx(;)1310 b Fy(\(5.7\))-29 586 y Fp(for)31 b(some)f(p)l(ositive)h(c)l(onstant)e Fx(K)1017 598 y Fr(1)1054 586 y Fp(.)39 b(Then)30 b(one)g(has)973 775 y Fu(jV)1047 787 y Ft(T)1100 775 y Fy(\()p Fw(!)21 b Fu(\001)e Fw(\027)5 b Fy(\))p Fu(j)24 b(\024)e(j)p Fx(")p Fu(j)1559 740 y Ft(k)1594 748 y Fi(T)1644 775 y Fx(D)1715 738 y Ft(k)1750 746 y Fi(T)1713 797 y Fr(1)1801 775 y Fx(C)1866 731 y Fo(\000)p Fr(\()p Ft(k)1979 739 y Fi(T)2025 731 y Fo(\000)p Fr(1\))1860 797 y(1)2140 775 y Fx(\015)2188 740 y Fo(\000)p Ft(k)2275 748 y Fi(T)2183 795 y Ft(m)2242 803 y Fl(0)2325 775 y Fy(e)2362 740 y Fo(\000)p Ft(\024)2453 748 y Fl(0)2485 740 y Ft(M)6 b Fr(\()p Ft(T)j Fr(\))p Ft(=)p Fr(2)2726 775 y Fx(;)831 b Fy(\(5.8\))-29 952 y Fp(for)31 b(a)f(suitable)g(c)l(onstant)f Fx(D)877 964 y Fr(1)914 952 y Fp(.)39 b(If)30 b(also)h(the)f (derivatives)i(of)e(the)g(pr)l(op)l(agators)i(ar)l(e)e(b)l(ounde)l(d)g (uniformly)h(as)1409 1054 y Fm(\014)1409 1104 y(\014)1409 1154 y(\014)1437 1149 y Fx(@)1481 1161 y Ft(x)1523 1149 y Fx(g)1566 1106 y Fr([)p Ft(p)p Fr(])1563 1172 y Ft(j)1641 1149 y Fy(\()p Fx(x)p Fy(\))1752 1054 y Fm(\014)1752 1104 y(\014)1752 1154 y(\014)1804 1149 y Fu(\024)22 b Fx(K)1962 1161 y Fr(2)1999 1149 y Fx(C)2064 1114 y Fo(\000)p Fr(2)2058 1171 y(1)2154 1149 y Fx(\015)2202 1115 y Fo(\000)p Fr(3)2197 1170 y Ft(p)2290 1149 y Fx(;)1267 b Fy(\(5.9\))-29 1347 y Fp(for)31 b(some)f(p)l(ositive)h(c)l(onstant)e Fx(K)1017 1359 y Fr(2)1054 1347 y Fp(,)h(one)g(has)h(also)877 1449 y Fm(\014)877 1499 y(\014)877 1549 y(\014)877 1598 y(\014)938 1513 y Fy(d)p 915 1550 94 4 v 915 1626 a(d)p Fx(x)1042 1569 y Fu(V)1093 1581 y Ft(T)1145 1569 y Fy(\()p Fx(x)p Fy(\))p Fu(j)1280 1594 y Ft(x)p Fr(=)p Fk(!)r Fo(\001)p Fk(\027)1484 1449 y Fm(\014)1484 1499 y(\014)1484 1549 y(\014)1484 1598 y(\014)1535 1569 y Fu(\024)22 b(j)p Fx(")p Fu(j)1707 1535 y Ft(k)1742 1543 y Fi(T)1793 1569 y Fx(D)1864 1532 y Ft(k)1899 1540 y Fi(T)1862 1591 y Fr(2)1949 1569 y Fx(C)2014 1532 y Fo(\000)p Ft(k)2101 1540 y Fi(T)2008 1591 y Fr(1)2152 1569 y Fx(\015)2200 1535 y Fo(\000)p Ft(k)2287 1543 y Fi(T)2332 1535 y Fo(\000)p Fr(2)2195 1590 y Ft(m)2254 1598 y Fl(0)2421 1569 y Fy(e)2458 1535 y Fo(\000)p Ft(\024)2549 1543 y Fl(0)2581 1535 y Ft(M)6 b Fr(\()p Ft(T)j Fr(\))p Ft(=)p Fr(2)2822 1569 y Fx(;)694 b Fy(\(5.10\))-29 1792 y Fp(for)31 b(a)f(suitable)g(c)l (onstant)f Fx(D)877 1804 y Fr(2)914 1792 y Fp(.)-29 1987 y(Pr)l(o)l(of.)39 b Fy(F)-7 b(or)24 b(an)n(y)f(renormalised)g (self-energy)f(cluster)i Fx(T)36 b Fy(consider)23 b(the)h(corresp)r (onding)f(self-energy)f(v)-5 b(alue)24 b(\(4.1\).)36 b(The)-29 2087 y(pro)r(duct)28 b(of)g(factors)e Fx(F)701 2099 y Ft(v)769 2087 y Fy(can)h(b)r(e)h(b)r(ounded)g(as)1234 2197 y Fm(Y)1172 2379 y Ft(v)r Fo(2)p Ft(P)9 b Fr(\()p Ft(T)g Fr(\))1417 2276 y Fx(F)1470 2288 y Ft(v)1533 2276 y Fu(\024)22 b Fx(F)1685 2239 y Ft(k)1720 2247 y Fi(T)1673 2298 y Fr(0)1955 2197 y Fm(Y)1784 2379 y Ft(v)r Fo(2)p Ft(V)1903 2387 y Fl(1)1936 2379 y Fr(\()p Ft(T)9 b Fr(\))p Fo([)p Ft(E)s Fr(\()p Ft(T)g Fr(\))2247 2276 y Fy(e)2284 2242 y Fo(\000)p Ft(\024)2375 2250 y Fl(0)2407 2242 y Fo(j)p Fk(\027)2468 2250 y Fi(v)2504 2242 y Fo(j)2528 2276 y Fx(;)988 b Fy(\(5.11\))-29 2545 y(while)28 b(the)g(pro)r(duct)g (of)f(propagators)e(can)i(b)r(e)h(b)r(ounded,)g(for)f(an)n(y)g Fx(m)2179 2557 y Fr(0)2239 2545 y Fu(2)d Fv(N)p Fy(,)k(as)803 2714 y Fm(Y)743 2896 y Ft(v)r Fo(2)p Ft(L)p Fr(\()p Ft(T)9 b Fr(\))982 2793 y Fx(g)1025 2758 y Fo(R)1022 2813 y Ft(`)1109 2793 y Fu(\024)23 b Fx(C)1262 2750 y Fo(\000)p Fr(\()p Ft(k)1375 2758 y Fi(T)1421 2750 y Fo(\000)p Fr(1\))1256 2815 y(1)1536 2793 y Fx(\015)1584 2758 y Fo(\000)p Ft(k)1671 2766 y Fi(T)1579 2813 y Ft(m)1638 2821 y Fl(0)1735 2793 y Fy(exp)1876 2651 y Fm( )1941 2793 y Fx(K)2132 2689 y Fo(1)2105 2714 y Fm(X)2032 2890 y Ft(n)p Fr(=)p Ft(m)2183 2898 y Fl(0)2215 2890 y Fr(+1)2345 2736 y Fy(1)p 2323 2774 87 4 v 2323 2850 a(2)2365 2826 y Ft(n)2433 2793 y Fy(log)2588 2736 y(1)p 2564 2774 89 4 v 2564 2850 a Fx(\015)2607 2862 y Ft(n)2663 2793 y Fx(M)9 b Fy(\()p Fx(T)j Fy(\))2878 2651 y Fm(!)2956 2793 y Fx(;)560 b Fy(\(5.12\))-29 3067 y(where)28 b(the)f(\014rst)h(b)r(ound)g(\(5.5\))f (of)h(Lemma)f(12)g(has)g(b)r(een)h(used.)37 b(If)28 b(w)n(e)f(c)n(ho)r (ose)f Fx(m)2589 3079 y Fr(0)2654 3067 y Fy(suc)n(h)h(that)1381 3311 y Fx(K)1572 3207 y Fo(1)1545 3232 y Fm(X)1472 3407 y Ft(n)p Fr(=)p Ft(m)1623 3415 y Fl(0)1655 3407 y Fr(+1)1785 3254 y Fy(1)p 1763 3291 87 4 v 1763 3367 a(2)1805 3343 y Ft(n)1873 3311 y Fy(log)2028 3254 y(1)p 2004 3291 89 4 v 2004 3367 a Fx(\015)2047 3379 y Ft(n)2125 3311 y Fu(\024)2223 3254 y Fx(\024)2271 3266 y Fr(0)p 2223 3291 86 4 v 2224 3367 a Fy(12)2318 3311 y Fx(;)1198 b Fy(\(5.13\))-29 3572 y(then)29 b(w)n(e)e(obtain)g(\(5.8\).)37 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b(Detac)n(h)-29 268 y(the)32 b(line)f Fx(`)313 280 y Fr(in)403 268 y Fy(from)f Fx(v)642 280 y Fr(in)733 268 y Fy(and)g(attac)n(h)h(it)g(to)f(the)i(no) r(de)e Fx(v)1737 280 y Fr(out)1838 268 y Fy(,)h(and)g(detac)n(h)f(the)h (line)g Fx(`)2666 280 y Fr(out)2797 268 y Fy(from)f Fx(v)3036 280 y Fr(out)3167 268 y Fy(and)h(attac)n(h)f(it)h(to)-29 368 y(the)i(no)r(de)g Fx(v)365 380 y Fr(in)425 368 y Fy(.)51 b(Consisten)n(tly)-7 b(,)33 b(orien)n(t)f(all)g(lines)g(along)g (the)g(path)h Fu(P)7 b Fy(\()p Fx(T)12 b Fy(\))32 b(b)r(et)n(w)n(een)g (the)h(external)f(lines)g(of)g Fx(T)44 b Fy(in)33 b(the)-29 468 y(opp)r(osite)c(direction,)h(i.e.)42 b(from)28 b Fx(v)1060 480 y Fr(out)1190 468 y Fy(to)h Fx(v)1333 480 y Fr(in)1393 468 y Fy(.)42 b(Finally)29 b(c)n(hange)f(the)i(mo)r(de)f (lab)r(els)g(of)g(all)g(no)r(des)g(along)f Fu(P)7 b Fy(\()p Fx(T)12 b Fy(\),)29 b(i.e.)42 b(of)-29 567 y(all)27 b(no)r(des)f Fx(v)g Fu(2)e Fx(V)18 b Fy(\()p Fu(P)7 b Fy(\()p Fx(T)12 b Fy(\)\),)27 b(if)g Fx(V)19 b Fy(\()p 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Ft(j)2031 1398 y Fy(\()p Fx(x)p Fy(\))c(=)f Fu(M)2354 1354 y Fr([)p Fo(\024)p Ft(n)p Fr(])p Fo(\003)2354 1421 y Ft(j)2522 1398 y Fy(\()p Fx(x)p Fy(\),)29 b(whic)n(h)f(pro)n(v)n(es)d (the)j(assertion.)p 3704 1398 48 48 v -29 1675 a Fq(Lemma)k(15)41 b Fp(Assume)h(that)g(the)h(pr)l(op)l(agators)h Fx(g)1604 1632 y Fr([)p Ft(p)p Fr(])1601 1698 y Ft(j)1680 1675 y Fy(\()p Fx(x)p Fy(\))f Fp(and)h(their)f(derivatives)h(c)l(an)f(b)l(e) g(uniformly)g(b)l(ounde)l(d)g(for)-29 1804 y(al)t(l)d Fy(0)e Fu(\024)g Fx(p)g Fu(\024)g Fx(n)25 b Fu(\000)f Fy(1)38 b Fp(as)g(in)h(\(5.7\))g(and)g(\(5.9\),)j(for)d(some)g(c)l (onstants)e Fx(K)2370 1816 y Fr(1)2445 1804 y Fp(and)i Fx(K)2686 1816 y Fr(2)2723 1804 y Fp(.)64 b(Then)39 b(one)f(has)h Fu(M)3459 1761 y Fr([)p Ft(n)p Fr(])3459 1826 y(1)3542 1804 y Fy(\(0\))f(=)-29 1921 y Fu(\000M)136 1878 y Fr([)p Ft(n)p Fr(])136 1943 y(2)218 1921 y Fy(\()p Fu(\000)p Fy(2)p Fx(\025)405 1933 y Fr(0)443 1921 y Fy(\))30 b Fp(for)g(al)t(l)h Fx(n)23 b Fu(\025)f Fy(1)p Fp(.)-29 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Fy(\))23 b(=)g Fu(V)1456 1679 y Ft(T)1508 1667 y Fy(\()p Fu(\000)p Fy(2)p Fx(\025)1695 1679 y Fr(0)1732 1667 y Fy(\))c(+)f(\()p Fu(V)1949 1679 y Ft(T)2001 1667 y Fy(\()p Fw(!)k Fu(\001)c Fw(\027)6 b Fy(\))19 b Fu(\000)f(V)2395 1679 y Ft(T)2447 1667 y Fy(\()p Fu(\000)p Fy(2)p Fx(\025)2634 1679 y Fr(0)2671 1667 y Fy(\)\))c Fx(;)767 b Fy(\(5.23\))-29 1817 y(where)532 1955 y Fu(V)583 1967 y Ft(T)635 1955 y Fy(\()p Fw(!)22 b Fu(\001)c Fw(\027)6 b Fy(\))19 b Fu(\000)f(V)1029 1967 y Ft(T)1081 1955 y Fy(\()p Fu(\000)p Fy(2)p Fx(\025)1268 1967 y Fr(0)1305 1955 y Fy(\))23 b(=)g(\()p Fw(!)e Fu(\001)e Fw(\027)24 b Fy(+)18 b(2)p Fx(\025)1848 1967 y Fr(0)1885 1955 y Fy(\))1931 1842 y Fm(Z)2014 1863 y Fr(1)1977 2031 y(0)2065 1955 y Fy(d)p Fx(s)2198 1899 y Fy(d)p 2174 1936 V 2174 2012 a(d)p Fx(x)2301 1955 y Fu(V)2352 1967 y Ft(T)2405 1955 y Fy(\()p Fx(x)p Fy(\))p Fu(j)2540 1980 y Ft(x)p Fr(=)p Fo(\000)p Fr(2)p Ft(\025)2753 1988 y Fl(0)2785 1980 y Fr(+)p Ft(s)p Fr(\()p Fk(!)s Fo(\001)p Fk(\027)s Fr(+2)p Ft(\025)3127 1988 y Fl(0)3160 1980 y Fr(\))3539 1955 y Fy(\(5.24\))-29 2151 y(can)28 b(b)r(e)g(b)r(ounded)g(b)n(y)f (using)g(\(5.10\),)g(while)819 2344 y Fx(i)p 812 2381 42 4 v 812 2457 a Fy(2)905 2296 y Fo(1)878 2321 y Fm(X)878 2500 y Ft(k)q Fr(=1)1125 2321 y Fm(X)1012 2499 y Ft(T)9 b Fo(2S)1146 2508 y Fi(k)q(;)p Fl(2)p Fi(;n)p Fh(\000)p Fl(1)1372 2400 y Fu(V)1423 2412 y Ft(T)1475 2400 y Fy(\()p Fu(\000)p Fy(2)p Fx(\025)1662 2412 y Fr(0)1699 2400 y Fy(\))24 b(=)e Fu(M)1942 2357 y Fr([)p Ft(n)p Fr(])1942 2422 y(2)2025 2400 y Fy(\()p Fu(\000)p Fy(2)p Fx(\025)2212 2412 y Fr(0)2249 2400 y Fy(\))h(=)g Fu(\000M)2557 2357 y Fr([)p Ft(n)p Fr(])2557 2422 y(1)2639 2400 y Fy(\(0\))g(=)g(0)p Fx(;)618 b Fy(\(5.25\))-29 2674 y(so)27 b(that)h(the)g(assertion)e(is)i (pro)n(v)n(ed)e(also)g(in)i(suc)n(h)f(a)h(case.)p 3704 2674 48 48 v -29 2933 a Fq(Lemma)k(17)41 b Fp(Assume)28 b(that)h(the)g(pr)l(op)l(agators)h Fx(g)1549 2890 y Fr([)p Ft(p)p Fr(])1546 2956 y Ft(j)1625 2933 y Fy(\()p Fx(x)p Fy(\))g Fp(ar)l(e)f(di\013er)l(entiable,)i(and)e(that,)h(to)l(gether)f (with)g(their)h(derivat-)-29 3033 y(ives,)g(they)f(c)l(an)f(b)l(e)g (uniformly)h(b)l(ounde)l(d)f(for)h(al)t(l)h Fy(0)22 b Fu(\024)h Fx(p)g Fu(\024)f Fx(n)15 b Fu(\000)f Fy(1)28 b Fp(as)g(in)g(\(5.7\))i(and)e(\(5.9\),)i(for)f(suitable)g(c)l (onstants)e Fx(K)3714 3045 y Fr(1)-29 3148 y Fp(and)k Fx(K)204 3160 y Fr(2)240 3148 y Fp(.)39 b(Then)30 b(for)h Fx(")f Fp(smal)t(l)g(enough)g Fu(M)1324 3105 y Fr([)p Fo(\024)p Ft(n)p Fr(])1324 3171 y Ft(j)1459 3148 y Fy(\()p Fx(x)p Fy(\))g Fp(is)h(di\013er)l(entiable)g(in)f Fx(x)p Fp(,)g(and)h(one)f(has)718 3274 y Fm(\014)718 3324 y(\014)718 3374 y(\014)745 3370 y Fu(M)845 3326 y Fr([)p Fo(\024)p Ft(n)p Fr(])845 3393 y Ft(j)980 3370 y Fy(\()p Fx(x)1059 3335 y Fo(0)1083 3370 y Fy(\))18 b Fu(\000)g(M)1316 3326 y Fr([)p Fo(\024)p Ft(n)p Fr(])1316 3393 y Ft(j)1451 3370 y Fy(\()p Fx(x)p Fy(\))h Fu(\000)f Fx(@)1708 3382 y Ft(x)1750 3370 y Fu(M)1850 3326 y Fr([)p Fo(\024)p Ft(n)p Fr(])1850 3393 y Ft(j)1985 3370 y Fy(\()p Fx(x)p Fy(\))c(\()q Fx(x)2190 3335 y Fo(0)2232 3370 y Fu(\000)k Fx(x)p Fy(\))2395 3274 y Fm(\014)2395 3324 y(\014)2395 3374 y(\014)2445 3370 y Fy(=)23 b Fx(o)p Fy(\()p Fx(")2644 3335 y Fr(2)2681 3370 y Fx(C)2746 3334 y Fo(\000)p Fr(2)2740 3392 y(1)2836 3370 y Fu(j)p Fx(x)2906 3335 y Fo(0)2948 3370 y Fu(\000)18 b Fx(x)p Fu(j)p Fy(\))p Fx(;)718 3457 y Fm(\014)718 3507 y(\014)718 3556 y(\014)745 3552 y Fx(@)789 3564 y Ft(x)831 3552 y Fu(M)931 3509 y Fr([)p Fo(\024)p Ft(n)p Fr(])931 3575 y Ft(j)1065 3552 y Fy(\()p Fx(x)p Fy(\))1176 3457 y Fm(\014)1176 3507 y(\014)1176 3556 y(\014)1228 3552 y Fu(\024)23 b Fx(B)1379 3564 y Fr(2)1416 3552 y Fu(j)p Fx(")p Fu(j)1501 3518 y Fr(2)1538 3552 y Fx(C)1603 3517 y Fo(\000)p Fr(2)1597 3574 y(1)1693 3552 y Fx(;)1823 b Fy(\(5.26\))-29 3755 y Fp(for)31 b(a)f(suitable)g(c) l(onstant)f Fx(B)871 3767 y Fr(2)909 3755 y Fp(.)-29 3967 y(Pr)l(o)l(of.)42 b Fy(By)28 b(writing)g Fu(M)748 3924 y Fr([)p Fo(\024)p Ft(n)p Fr(])748 3990 y Ft(j)882 3967 y Fy(\()p Fx(x)p Fy(\))h(according)e(to)h(\(4.6\),)g(one)g (\014nds)g(immediately)g(that)h(the)f(function)h(is)f(di\013eren)n (tiable)-29 4078 y(if)i(the)f(propagators)c(are)j(di\013eren)n(tiable,) h(and)f(that)h(the)g(deriv)-5 b(ativ)n(e)28 b(satis\014es)g(the)h(b)r (ound)g(in)g(\(5.19\).)39 b(The)29 b(factor)f Fx(")3714 4047 y Fr(2)-29 4177 y Fy(is)g(due)g(to)f(the)h(fact)g(that)g(a)f (self-energy)f(cluster)h Fx(T)39 b Fy(dep)r(ending)28 b(explicitly)g(on)f Fx(x)h Fy(has)f(at)h(least)f Fx(k)3072 4189 y Ft(T)3148 4177 y Fy(=)22 b(2.)p 3704 4177 V -29 4436 a Fq(Lemma)32 b(18)41 b Fp(Assume)32 b(that)g(the)h(pr)l(op)l (agators)h Fx(g)1564 4393 y Fr([)p Ft(p)p Fr(])1561 4460 y Ft(j)1639 4436 y Fy(\()p Fx(x)p Fy(\))g Fp(and)f(their)g(derivatives) h(c)l(an)f(b)l(e)f(uniformly)i(b)l(ounde)l(d)f(for)g(al)t(l)-29 4536 y Fy(0)23 b Fu(\024)g Fx(p)f Fu(\024)h Fx(n)18 b Fu(\000)g Fy(1)30 b Fp(as)g(in)g(\(5.7\))g(and)h(\(5.9\),)g(for)g(some) f(c)l(onstants)f Fx(K)2090 4548 y Fr(1)2156 4536 y Fp(and)h Fx(K)2388 4548 y Fr(2)2425 4536 y Fp(.)39 b(Then)30 b(for)h Fx(")e Fp(smal)t(l)i(enough)f(one)g(has)1370 4662 y Fm(\014)1370 4712 y(\014)1370 4762 y(\014)1397 4758 y Fx(x)19 b Fy(+)f(2)p Fu(M)1688 4715 y Fr([)p Fo(\024)p Ft(n)p Fr(])1688 4781 y Ft(j)1822 4758 y Fy(\()p Fx(x)p Fy(\))1933 4662 y Fm(\014)1933 4712 y(\014)1933 4762 y(\014)1985 4758 y Fu(\025)2083 4702 y Fy(1)p 2083 4739 42 4 v 2083 4815 a(2)2134 4758 y(\001)2203 4770 y Fr(0)2241 4758 y Fy(\()p Fx(x)p Fy(\))1187 b(\(5.27\))-29 4993 y Fp(as)31 b(far)f(as)g Fx(g)359 4950 y Fr([)p Ft(n)p Fr(])356 5016 y Ft(j)442 4993 y Fy(\()p Fx(x)p Fy(\))24 b Fu(6)p Fy(=)e(0)p Fp(.)1820 5484 y Fy(23)p eop end %%Page: 24 24 TeXDict begin 24 23 bop -29 172 a Fp(Pr)l(o)l(of.)46 b Fy(By)29 b(Lemma)g(16)g(one)g(has)g Fu(M)1181 129 y Fr([)p Fo(\024)p Ft(n)p Fr(])1181 195 y(1)1316 172 y Fy(\(0\))d(=)g(0)j(and)h Fu(M)1874 129 y Fr([)p Fo(\024)p Ft(n)p Fr(])1874 195 y(2)2008 172 y Fy(\()p Fu(\000)p Fy(2)p Fx(\025)2195 184 y Fr(0)2232 172 y Fy(\))d(=)f Fx(\025)2430 184 y Fr(0)2467 172 y Fy(.)43 b(Set)30 b Fx(j)5 b Fy(\()p Fx(x)p Fy(\))28 b(=)e(1)j(when)h Fx(\032)3280 184 y Fr(0)3317 172 y Fy(\()p Fx(x)p Fy(\))d(=)f(0)j(and)-29 272 y Fx(j)5 b Fy(\()p Fx(x)p Fy(\))24 b(=)f(2)k(when)h Fx(\032)562 284 y Fr(0)599 272 y Fy(\()p Fx(x)p Fy(\))c(=)f Fx(\025)870 284 y Fr(0)908 272 y Fy(,)k(so)g(that)h(one)f(can)h(write) 412 475 y Fx(x)19 b Fy(+)f(2)p Fu(M)703 432 y Fr([)p Fo(\024)p Ft(n)p Fr(])703 504 y Ft(j)s Fr(\()p Ft(x)p Fr(\))837 475 y Fy(\()p Fx(x)p Fy(\))84 b(=)f Fx(x)19 b Fy(+)f(2)p Fu(M)1471 432 y Fr([)p Fo(\024)p Ft(n)p Fr(])1471 504 y Ft(j)s Fr(\()p Ft(x)p Fr(\))1605 475 y Fy(\()p Fu(\000)p Fy(2)p Fx(\032)1787 487 y Fr(0)1823 475 y Fy(\()p Fx(x)p Fy(\)\))i(+)2069 383 y Fm(\020)2119 475 y Fy(2)p Fu(M)2261 432 y Fr([)p Fo(\024)p Ft(n)p Fr(])2261 504 y Ft(j)s Fr(\()p Ft(x)p Fr(\))2394 475 y Fy(\()p Fx(x)p Fy(\))g Fu(\000)e Fy(2)p Fu(M)2750 432 y Fr([)p Fo(\024)p Ft(n)p Fr(])2750 504 y Ft(j)s Fr(\()p Ft(x)p Fr(\))2884 475 y Fy(\()p Fu(\000)p Fy(2)p Fx(\032)3066 487 y Fr(0)3102 475 y Fy(\()p Fx(x)p Fy(\)\))3245 383 y Fm(\021)1032 658 y Fy(=)83 b Fx(x)19 b Fy(+)f(2)p Fx(\032)1414 670 y Fr(0)1450 658 y Fy(\()p Fx(x)p Fy(\))i(+)e(2)1720 566 y Fm(\020)1769 658 y Fu(M)1869 615 y Fr([)p Fo(\024)p Ft(n)p Fr(])1869 686 y Ft(j)s Fr(\()p Ft(x)p Fr(\))2003 658 y Fy(\()p Fx(x)p Fy(\))i Fu(\000)e(M)2317 615 y Fr([)p Fo(\024)p Ft(n)p Fr(])2317 686 y Ft(j)s Fr(\()p Ft(x)p Fr(\))2451 658 y Fy(\()p Fu(\000)p Fy(2)p Fx(\032)2633 670 y Fr(0)2670 658 y Fy(\))2702 566 y Fm(\021)2765 658 y Fx(;)751 b Fy(\(5.28\))-29 886 y(where)28 b Fu(jM)335 843 y Fr([)p Fo(\024)p Ft(n)p Fr(])335 914 y Ft(j)s Fr(\()p Ft(x)p Fr(\))469 886 y Fy(\()p Fx(x)p Fy(\))19 b Fu(\000)f(M)782 843 y Fr([)p Fo(\024)p Ft(n)p Fr(])782 914 y Ft(j)s Fr(\()p Ft(x)p Fr(\))917 886 y Fy(\()p Fu(\000)p Fy(2)p Fx(\032)1099 898 y Fr(0)1135 886 y Fy(\))p Fu(j)24 b(\024)e Fy(const.)p Fu(j)p Fx(")p Fu(j)1599 856 y Fr(2)1636 886 y Fx(C)1701 850 y Fo(\000)p Fr(2)1695 908 y(1)1791 886 y Fu(j)p Fx(x)c Fy(+)g(2)p Fx(\032)2047 898 y Fr(0)2084 886 y Fy(\()p Fx(x)p Fy(\))p Fu(j)p Fy(,)29 b(b)n(y)e(Lemma)h(17.)36 b(Then)27 b(b)n(y)h(\(4.7\))f(one)g(has)432 1049 y Fm(\014)432 1099 y(\014)432 1149 y(\014)460 1145 y Fx(x)19 b Fy(+)f(2)p Fu(M)751 1102 y Fr([)p Fo(\024)p Ft(n)p Fr(])751 1173 y Ft(j)s Fr(\()p Ft(x)p Fr(\))885 1145 y Fy(\()p Fx(x)p Fy(\))996 1049 y Fm(\014)996 1099 y(\014)996 1149 y(\014)1047 1145 y Fu(\025)1135 1077 y Fm(\000)1173 1145 y Fy(1)g Fu(\000)g Fy(const.)p Fu(j)p Fx(")p Fu(j)1614 1110 y Fr(2)1651 1145 y Fx(C)1716 1109 y Fo(\000)p Fr(2)1710 1167 y(1)1805 1077 y Fm(\001)1857 1145 y Fu(j)p Fx(x)h Fy(+)f(2)p Fx(\032)2114 1157 y Fr(0)2151 1145 y Fy(\()p Fx(x)p Fy(\))p Fu(j)24 b(\025)2407 1089 y Fy(1)18 b Fu(\000)g Fy(const.)o Fu(j)p Fx(")p Fu(j)2847 1058 y Fr(2)2884 1089 y Fx(C)2949 1053 y Fo(\000)p Fr(2)2943 1111 y(1)p 2407 1126 633 4 v 2667 1142 a Fu(p)p 2736 1142 42 4 v 69 x Fy(2)3049 1145 y(\001)3118 1157 y Fr(0)3155 1145 y Fy(\()p Fx(x)p Fy(\))p Fx(:)250 b Fy(\(5.29\))-29 1396 y(Since)28 b Fu(j)p Fx(x)19 b Fy(+)f(2)p Fu(M)502 1353 y Fr([)p Fo(\024)p Ft(n)p Fr(])502 1424 y(3)p Fo(\000)p Ft(j)s Fr(\()p Ft(x)p Fr(\))711 1396 y Fy(\()p Fx(x)p Fy(\))p Fu(j)24 b(\025)f Fy(\(1)18 b Fu(\000)g Fy(const.)p Fu(j)p Fx(")p Fu(j)1430 1366 y Fr(2)1467 1396 y Fx(C)1532 1360 y Fo(\000)p Fr(2)1526 1418 y(1)1621 1396 y Fy(\))p Fu(j)p Fx(x)h Fy(+)f(2)p Fu(M)1967 1353 y Fr([)p Fo(\024)p Ft(n)p Fr(])1967 1424 y Ft(j)s Fr(\()p Ft(x)p Fr(\))2101 1396 y Fy(\()p Fx(x)p Fy(\))p Fu(j)p Fy(,)29 b(the)f(b)r(ound)g(follo)n (ws.)p 3704 1396 48 48 v -29 1677 a Fq(Lemma)k(19)41 b Fp(The)31 b(pr)l(op)l(agators)g Fx(g)1105 1634 y Fr([)p Ft(n)p Fr(])1102 1700 y Ft(j)1188 1677 y Fy(\()p Fx(x)p Fy(\))f Fp(satisfy)h(the)f(b)l(ounds)g(\(5.7\))h(and)f(\(5.9\))h(for)f (al)t(l)h Fx(n)23 b Fu(\025)g Fy(0)p Fp(.)-29 1874 y(Pr)l(o)l(of.)60 b Fy(The)34 b(pro)r(of)g(can)h(b)r(e)g(p)r(erformed)f(b)n(y)g (induction.)59 b(F)-7 b(or)34 b Fx(n)g Fy(=)h(1)f(the)h(b)r(ounds)g (\(5.7\))f(and)h(\(5.9\))f(are)g(trivially)-29 1989 y(satis\014ed,)28 b(as)f Fu(M)516 1946 y Fr([0])516 2012 y(1)590 1989 y Fy(\()p Fx(x)p Fy(\))d(=)f(0)k(and)g Fu(M)1143 1946 y Fr([0])1143 2012 y(2)1218 1989 y Fy(\()p Fx(x)p Fy(\))d(=)f Fx(\025)1489 2001 y Fr(0)1526 1989 y Fy(,)28 b(b)r(ecause)f(of)h(the)g (Diophan)n(tine)f(conditions)h(\(4.14\).)96 2124 y(The)j(di\013erence)h (for)f Fx(n)e(>)g Fy(1)i(is)h(that)f(no)n(w)g(the)h(propagators)c(dep)r (end)k(also)f(on)g(the)h(functions)g Fu(M)3270 2081 y Fr([)p Ft(p)p Fr(])3270 2148 y Ft(j)3345 2124 y Fy(\()p Fx(x)p Fy(\),)i Fx(p)29 b(<)g(n)p Fy(,)-29 2224 y(app)r(earing)f(in)i (the)f(denominators)f(and)h(the)h(compact)e(supp)r(ort)h(functions.)42 b(Then)29 b(assume)g(\(5.7\))g(and)g(\(5.9\))g(for)f(all)-29 2340 y Fx(p)23 b(<)g(n)p Fy(.)37 b(Then)27 b(one)h(has)f Fu(j)p Fx(g)817 2297 y Fr([)p Ft(n)p Fr(])814 2363 y Ft(j)899 2340 y Fy(\()p Fx(x)p Fy(\))p Fu(j)d(\024)f Fy(const.)p Fx( )1412 2352 y Ft(n)1457 2340 y Fy(\(\001)1558 2352 y Fr(0)1596 2340 y Fy(\()p Fx(x)p Fy(\)\))p Fx(=)p Fy(\001)1850 2352 y Fr(0)1888 2340 y Fy(\()p Fx(x)p Fy(\))h Fu(\024)f Fy(const.)o Fx(C)2388 2304 y Fo(\000)p Fr(1)2382 2362 y(1)2478 2340 y Fx(\015)2526 2310 y Fo(\000)p Fr(1)2521 2360 y Ft(n)2614 2340 y Fy(,)28 b(b)n(y)f(Lemma)h(18.)36 b(Moreo)n(v)n(er)237 2611 y Fx(@)281 2623 y Ft(x)323 2611 y Fx(g)366 2568 y Fr([)p Ft(n)p Fr(])363 2634 y Ft(j)448 2611 y Fy(\()p Fx(x)p Fy(\))84 b(=)f Fu(\000)p Fx(i)899 2507 y Ft(n)p Fo(\000)p Fr(1)901 2532 y Fm(X)902 2708 y Ft(p)p Fr(=0)1038 2611 y Fx(\037)1090 2623 y Fr(0)1127 2611 y Fy(\(\001)1228 2623 y Fr(0)1266 2611 y Fy(\()p Fx(x)p Fy(\)\))14 b Fx(:)g(:)g(:)h(@)5 b(\037)1636 2623 y Ft(p)1674 2611 y Fy(\(\001)1775 2623 y Fr(0)1812 2611 y Fy(\()p Fx(x)p Fy(\)\))14 b Fx(:)g(:)g(:)h(\037)2133 2623 y Ft(n)p Fo(\000)p Fr(1)2263 2611 y Fy(\(\001)2364 2623 y Fr(0)2402 2611 y Fy(\()p Fx(x)p Fy(\)\))p Fx( )2599 2623 y Ft(n)2646 2611 y Fy(\(\001)2747 2623 y Fr(0)2784 2611 y Fy(\()p Fx(x)p Fy(\)\))3055 2555 y Fx(@)3099 2567 y Ft(x)3141 2555 y Fy(\001)3210 2567 y Fr(0)3247 2555 y Fy(\()p Fx(x)p Fy(\))p 2937 2592 537 4 v 2937 2688 a Fx(x)20 b Fy(+)e(2)p Fu(M)3229 2645 y Fr([)p Fo(\024)p Ft(n)p Fr(])3229 2712 y Ft(j)3363 2688 y Fy(\()p Fx(x)p Fy(\))643 2879 y Fu(\000)83 b Fx(i\037)872 2891 y Fr(0)909 2879 y Fy(\(\001)1010 2891 y Fr(0)1047 2879 y Fy(\()p Fx(x)p Fy(\)\))14 b Fx(:)g(:)g(:)h(\037)1368 2891 y Ft(n)p Fo(\000)p Fr(1)1498 2879 y Fy(\(\001)1599 2891 y Fr(0)1637 2879 y Fy(\()p Fx(x)p Fy(\)\))p Fx(@)5 b( )1883 2891 y Ft(n)1929 2879 y Fy(\(\001)2030 2891 y Fr(0)2068 2879 y Fy(\()p Fx(x)p Fy(\)\))2339 2823 y Fx(@)2383 2835 y Ft(x)2425 2823 y Fy(\001)2494 2835 y Fr(0)2531 2823 y Fy(\()p Fx(x)p Fy(\))p 2221 2860 V 2221 2957 a Fx(x)20 b Fy(+)e(2)p Fu(M)2513 2913 y Fr([)p Fo(\024)p Ft(n)p Fr(])2513 2980 y Ft(j)2647 2957 y Fy(\()p Fx(x)p Fy(\))643 3178 y(+)83 b Fx(i\037)872 3190 y Fr(0)909 3178 y Fy(\(\001)1010 3190 y Fr(0)1047 3178 y Fy(\()p Fx(x)p Fy(\)\))14 b Fx(:)g(:)g(:)h (\037)1368 3190 y Ft(n)p Fo(\000)p Fr(1)1498 3178 y Fy(\(\001)1599 3190 y Fr(0)1637 3178 y Fy(\()p Fx(x)p Fy(\)\))p Fx( )1834 3190 y Ft(n)1881 3178 y Fy(\(\001)1982 3144 y Fr([)p Ft(n)p Fr(])2065 3178 y Fy(\()p Fx(x)p Fy(\)\))2230 3111 y(1)j(+)g(2)p Fx(@)2459 3123 y Ft(x)2501 3111 y Fu(M)2601 3068 y Fr([)p Fo(\024)p Ft(n)p Fr(])2601 3134 y Ft(j)2735 3111 y Fy(\()p Fx(x)p Fy(\))p 2218 3159 639 4 v 2218 3255 a(\()p Fx(x)i Fy(+)e(2)p Fu(M)2542 3212 y Fr([)p Fo(\024)p Ft(n)p Fr(])2542 3279 y Ft(j)2676 3255 y Fy(\()p Fx(x)p Fy(\)\))2819 3231 y Fr(2)2868 3178 y Fx(;)648 b Fy(\(5.30\))-29 3435 y(where)28 b Fx(@)k Fy(denotes)27 b(deriv)-5 b(ativ)n(e)27 b(with)h(resp)r(ect)f(to)h(the)g(argumen)n(t.) 96 3552 y(One)f(c)n(hec)n(ks)g(immediately)g(that)h(for)f(all)h Fx(p)23 b Fu(\025)f Fy(0)466 3730 y Fx(@)5 b(\037)567 3742 y Ft(p)605 3730 y Fy(\()p Fx(x)p Fy(\))24 b Fu(\024)f Fy(const.)p Fx(C)1106 3695 y Fo(\000)p Fr(1)1100 3753 y(1)1195 3730 y Fx(\015)1243 3696 y Fo(\000)p Fr(1)1238 3751 y Ft(p)1332 3730 y Fx(;)180 b(@)5 b( )1638 3742 y Ft(p)1676 3730 y Fy(\()p Fx(x)p Fy(\))24 b Fu(\024)f Fy(const.)o Fx(C)2176 3695 y Fo(\000)p Fr(1)2170 3753 y(1)2266 3730 y Fx(\015)2314 3696 y Fo(\000)p Fr(1)2309 3751 y Ft(p)2402 3730 y Fx(;)180 b(@)2649 3742 y Ft(x)2691 3730 y Fy(\001)2760 3742 y Fr(0)2798 3730 y Fy(\()p Fx(x)p Fy(\))24 b Fu(\024)e Fy(const.)p Fx(;)283 b Fy(\(5.31\))-29 3936 y(so)27 b(that)h(the)g(deriv)-5 b(ativ)n(e)27 b Fx(@)821 3948 y Ft(x)863 3936 y Fu(M)963 3893 y Fr([)p Fo(\024)p Ft(n)p Fr(])963 3959 y Ft(j)1097 3936 y Fy(\()p Fx(x)p Fy(\))i(can)e(b)r(e)h(b)r(ounded)g(through)f(\(5.26\),)g(b)r (ecause)g(of)h(the)g(inductiv)n(e)f(h)n(yp)r(othesis.)96 4053 y(Hence,)h(b)n(y)f(using)g(once)g(more)g(\(4.9\))h(and)f(Lemma)g (18)g(to)g(b)r(ound)h(the)g(denominators,)f(w)n(e)g(obtain)g(from)h (\(5.30\))517 4215 y Fm(\014)517 4265 y(\014)517 4315 y(\014)545 4311 y Fx(@)589 4323 y Ft(x)631 4311 y Fx(g)674 4268 y Fr([)p Ft(n)734 4243 y Fh(0)756 4268 y Fr(])671 4334 y Ft(j)779 4311 y Fy(\()p Fx(x)p Fy(\))890 4215 y Fm(\014)890 4265 y(\014)890 4315 y(\014)941 4311 y Fu(\024)23 b Fy(const.)p Fx(C)1307 4275 y Fo(\000)p Fr(2)1301 4333 y(1)1410 4169 y Fm( )1476 4207 y Ft(n)p Fo(\000)p Fr(1)1479 4232 y Fm(X)1479 4408 y Ft(p)p Fr(=0)1615 4311 y Fx(\015)1663 4277 y Fo(\000)p Fr(1)1658 4331 y Ft(p)1752 4311 y Fx(\015)1800 4277 y Fo(\000)p Fr(1)1795 4331 y Ft(n)1907 4311 y Fy(+)18 b Fx(\015)2038 4277 y Fo(\000)p Fr(1)2033 4331 y Ft(n)2127 4311 y Fx(\015)2175 4277 y Fo(\000)p Fr(1)2170 4331 y Ft(n)2282 4311 y Fy(+)g Fx(\015)2413 4277 y Fo(\000)p Fr(2)2408 4331 y Ft(n)2502 4169 y Fm(!)2591 4311 y Fu(\024)k Fy(const.)p Fx(C)2956 4275 y Fo(\000)p Fr(2)2950 4333 y(1)3045 4311 y Fx(\015)3093 4277 y Fo(\000)p Fr(3)3088 4331 y Ft(n)3182 4311 y Fx(;)334 b Fy(\(5.32\))-29 4572 y(whic)n(h)28 b(pro)n(v)n(es)e(the)i(assertion.)p 3704 4572 48 48 v -29 4827 a Fq(Lemma)k(20)41 b Fp(L)l(et)33 b Fx(\025)e Fu(2)g Fy(\003)817 4797 y Fo(\003)817 4848 y Fr(0)855 4827 y Fp(.)51 b(Ther)l(e)34 b(exists)g Fx(")1444 4839 y Fr(0)1511 4827 y Fx(>)c Fy(0)j Fp(such)h(that)g(for)h Fu(j)p Fx(")p Fu(j)30 b Fx(<)g(")2434 4839 y Fr(0)2505 4827 y Fp(the)k(c)l(o)l(e\016cients)g Fx(u)3122 4784 y Fr([)p Ft(k)q Fr(])3122 4850 y Ft(j;)p Fk(\027)3215 4827 y Fp(,)h Fx(j)g Fy(=)30 b(1)p Fx(;)14 b Fy(2)p Fp(,)34 b(and)-29 4944 y Fx(\026)21 4913 y Fr([)p Ft(k)q Fr(])130 4944 y Fp(ar)l(e)c(b)l(ounde)l(d)g(by)1133 4968 y Fm(\014)1133 5018 y(\014)1133 5068 y(\014)1160 5064 y Fx(u)1208 5020 y Fr([)p Ft(k)q Fr(])1208 5087 y Ft(j;)p Fk(\027)1301 4968 y Fm(\014)1301 5018 y(\014)1301 5068 y(\014)1352 5064 y Fu(\024)22 b Fx(B)c Fy(e)1557 5029 y Fo(\000)p Ft(\024)p Fo(j)p Fk(\027)s Fo(j)1733 5064 y Fu(j)p Fx(")p Fu(j)1818 5029 y Ft(k)1859 5064 y Fx(;)2066 4968 y Fm(\014)2066 5018 y(\014)2066 5068 y(\014)2093 5064 y Fx(\026)2143 5029 y Fr([)p Ft(k)q Fr(])2222 4968 y Fm(\014)2222 5018 y(\014)2222 5068 y(\014)2272 5064 y Fu(\024)23 b Fx(B)18 b Fu(j)p Fx(")p Fu(j)2526 5029 y Ft(k)2567 5064 y Fx(;)949 b Fy(\(5.33\))-29 5235 y Fp(for)31 b(suitable)f Fx(k)s Fp(-indep)l(endent)g(c)l(onstants)f Fx(B)34 b Fp(and)c Fx(\024)p Fp(.)39 b(One)29 b(c)l(an)h(take)g Fx(")2222 5247 y Fr(0)2282 5235 y Fy(=)22 b Fx(O)r Fy(\()p Fx(C)2525 5247 y Fr(1)2564 5235 y Fx(\015)2607 5247 y Ft(m)2666 5255 y Fl(0)2702 5235 y Fy(\))p Fp(,)31 b(with)f Fx(m)3043 5247 y Fr(0)3110 5235 y Fp(dep)l(ending)h(on)f Fx(\024)3663 5247 y Fr(0)3700 5235 y Fp(.)1820 5484 y Fy(24)p eop end %%Page: 25 25 TeXDict begin 25 24 bop -29 169 a Fp(Pr)l(o)l(of.)39 b Fy(F)-7 b(or)23 b(an)n(y)g(tree)h Fx(\022)h Fu(2)e Fy(\002)895 139 y Fo(R)895 192 y Ft(k)q(;j;)p Fk(\027)1068 169 y Fy(the)i(v)-5 b(alue)23 b(V)-7 b(al\()p Fx(\022)r Fy(\))25 b(can)e(b)r(e)h(b)r(ounded)h(b)n(y)e(using)h(the)g(b)r(ounds)g (\(5.11\))f(for)g(the)h(factors)-29 268 y Fx(F)24 280 y Ft(v)94 268 y Fy(and)29 b(the)g(b)r(ounds)h(\(5.7\),)f(pro)n(v)n(ed)f (in)h(Lemma)g(19,)g(for)g(the)h(propagators.)38 b(Summation)30 b(o)n(v)n(er)d(the)j(F)-7 b(ourier)28 b(lab)r(els)-29 368 y(can)h(b)r(e)h(p)r(erformed)f(b)n(y)g(using)g(an)g(exp)r(onen)n (tial)g(deca)n(y)f(factor)h(e)2048 338 y Fo(\000)p Ft(\024)2139 346 y Fl(0)2171 338 y Ft(M)6 b Fr(\()p Ft(T)j Fr(\))p Ft(=)p Fr(4)2441 368 y Fy(whic)n(h)29 b(can)g(b)r(e)h(extracted)f(from) g(\(5.11\).)-29 468 y(Summation)35 b(o)n(v)n(er)d(the)j(other)e(lab)r (els)h(and)g(o)n(v)n(er)f(the)h(n)n(um)n(b)r(er)g(of)g(unlab)r(elled)h (trees)f(can)f(b)r(e)i(easily)e(b)r(ounded)i(as)e(a)-29 567 y(constan)n(t)27 b(to)h(the)g(p)r(o)n(w)n(er)e Fx(k)s Fy(.)p 3704 567 48 48 v -29 815 a Fq(Lemma)32 b(21)41 b Fp(The)31 b(function)p 943 769 48 4 v 29 w Fx(u)p Fy(\()p Fx(t)p Fy(\))f Fp(solves)h(\(3.3\))g(for)f(al)t(l)h Fw(\027)e Fu(6)p Fy(=)22 b Fq(0)p Fp(,)30 b(pr)l(ovide)l(d)i Fx(\026)23 b Fy(=)p 2568 769 51 4 v 23 w Fx(\026)p Fp(.)-29 1015 y(Pr)l(o)l(of.)40 b Fy(W)-7 b(e)28 b(write)p 885 1214 48 4 v 885 1260 a Fx(u)933 1272 y Ft(j)968 1260 y Fy(\()p Fx(t)p Fy(\))83 b(=)p 1293 1214 V 83 w Fx(u)1340 1272 y Ft(j;)p Fn(0)1448 1260 y Fy(+)1554 1181 y Fm(X)1531 1365 y Fk(\027)s Fo(2)p Fj(Z)1663 1348 y Fi(d)1712 1260 y Fx(e)1751 1226 y Ft(i)p Fk(!)s Fo(\001)p Fk(\027)p 1890 1214 V 1890 1260 a Fx(u)1937 1272 y Ft(j;)p Fk(\027)2030 1260 y Fx(;)p 2233 1214 V 180 w(u)2281 1272 y Ft(j;)p Fk(\027)2397 1260 y Fy(=)2514 1156 y Fo(1)2487 1181 y Fm(X)2484 1357 y Ft(n)p Fr(=0)p 2623 1214 V 2623 1260 a Fx(u)2671 1272 y Ft(j;)p Fk(\027)t Ft(;n)2825 1260 y Fx(;)p 861 1502 V 861 1548 a(u)908 1560 y Ft(j;)p Fk(\027)t Ft(;n)1145 1548 y Fy(=)1320 1444 y Fo(1)1293 1469 y Fm(X)1293 1648 y Ft(k)q Fr(=1)1427 1548 y Fx(")1466 1514 y Ft(k)1620 1469 y Fm(X)1521 1656 y Ft(\022)r Fo(2)p Fr(\002)1651 1636 y Fh(R)1651 1676 y Fi(k)q(;j;)p Fb(\027)r Fi(;n)1853 1548 y Fy(V)-7 b(al\()p Fx(\022)r Fy(\))p Fx(;)1438 b Fy(\(5.34\))-29 1854 y(where)28 b(\002)277 1823 y Fo(R)277 1877 y Ft(k)q(;j;)p Fk(\027)s Ft(;n)514 1854 y Fy(is)f(the)h(set)g(of)g (trees)f(in)g(\002)1325 1823 y Fo(R)1325 1877 y Ft(k)q(;j;)p Fk(\027)1502 1854 y Fy(with)h(ro)r(ot)f(line)h(on)f(scale)g Fx(n)p Fy(.)96 1971 y(An)h(imp)r(ortan)n(t)f(prop)r(ert)n(y)g(of)g(the) h(compact)f(supp)r(ort)h(functions)g(is)f(that)642 2216 y(1)22 b(=)824 2112 y Fo(1)797 2137 y Fm(X)794 2313 y Ft(n)p Fr(=0)933 2216 y Fy(\011)998 2228 y Ft(n)1043 2216 y Fy(\()p Fx(x)p Fy(\))p Fx(;)181 b Fy(\011)1423 2228 y Ft(n)1467 2216 y Fy(\()p Fx(x)p Fy(\))25 b(:=)d Fx(\037)1765 2228 y Fr(0)1802 2216 y Fy(\(\001)1903 2228 y Fr(0)1941 2216 y Fy(\()p Fx(x)p Fy(\)\))14 b Fx(:)g(:)g(:)h(\037)2262 2228 y Ft(n)p Fo(\000)p Fr(1)2392 2216 y Fy(\(\001)2493 2228 y Fr(0)2531 2216 y Fy(\()p Fx(x)p Fy(\)\))p Fx( )2728 2228 y Ft(n)2774 2216 y Fy(\(\001)2875 2228 y Fr(0)2913 2216 y Fy(\()p Fx(x)p Fy(\)\))p Fx(;)460 b Fy(\(5.35\))-29 2474 y(where)28 b(the)f(summand)h(for)f Fx(n)c Fy(=)g(0)k(is)g(mean)n (t)h(as)f Fx( )1581 2486 y Fr(0)1618 2474 y Fy(\(\001)1719 2486 y Fr(0)1757 2474 y Fy(\()p Fx(x)p Fy(\)\).)38 b(More)27 b(generally)f(one)h(has)g(for)g(all)g Fx(s)c Fu(\025)g Fy(1)587 2724 y(1)f(=)769 2620 y Fo(1)742 2645 y Fm(X)739 2819 y Ft(n)p Fr(=)p Ft(p)879 2724 y Fy(\011)944 2736 y Ft(p;n)1043 2724 y Fy(\()p Fx(x)p Fy(\))p Fx(;)181 b Fy(\011)1423 2736 y Ft(p;n)1521 2724 y Fy(\()p Fx(x)p Fy(\))25 b(:=)d Fx(\037)1819 2736 y Ft(p)1858 2724 y Fy(\(\001)1959 2736 y Fr(0)1996 2724 y Fy(\()p Fx(x)p Fy(\)\))14 b Fx(:)g(:)g(:)h(\037)2317 2736 y Ft(n)p Fo(\000)p Fr(1)2447 2724 y Fy(\(\001)2548 2736 y Fr(0)2586 2724 y Fy(\()p Fx(x)p Fy(\)\))p Fx( )2783 2736 y Ft(n)2830 2724 y Fy(\(\001)2931 2736 y Fr(0)2968 2724 y Fy(\()p Fx(x)p Fy(\)\))p Fx(;)405 b Fy(\(5.36\))-29 2992 y(where)28 b(again)e(the)i(summand)g(for)f Fx(n)c Fy(=)f Fx(p)28 b Fy(is)f(mean)n(t)g(as)g Fx( )1802 3004 y Ft(p)1841 2992 y Fy(\(\001)1942 3004 y Fr(0)1980 2992 y Fy(\()p Fx(x)p Fy(\)\).)96 3110 y(W)-7 b(e)28 b(can)f(rewrite)g(the)h(equation) f(\(3.3\))g(as)520 3292 y Fx(u)568 3304 y Ft(j;)p Fk(\027)684 3292 y Fy(=)c Fx(g)812 3304 y Ft(j)846 3292 y Fy(\()p Fx(x)p Fy(\)\010)1017 3304 y Ft(j;)p Fk(\027)1111 3292 y Fy(\()p Fx(u)p Fy(\))p Fx(;)180 b Fy(\010)1486 3304 y Ft(j)1544 3292 y Fy(=)23 b Fx("f)1712 3304 y Ft(j)s Fr(1)1798 3292 y Fy(+)18 b Fx(i\026)g Fy(+)g Fx("f)2141 3304 y Ft(j)s Fr(1)2208 3292 y Fx(u)2256 3304 y Fr(1)2311 3292 y Fy(+)h Fx("f)2475 3304 y Ft(j)s Fr(2)2542 3292 y Fx(u)2590 3304 y Fr(2)2645 3292 y Fy(+)f(\()p Fu(\000)p Fy(1\))2899 3258 y Ft(j)s Fr(+1)3018 3292 y Fx(i\026u)3145 3304 y Ft(j)3179 3292 y Fx(;)337 b Fy(\(5.37\))-29 3494 y(where)28 b Fx(x)23 b Fy(=)g Fw(!)e Fu(\001)d Fw(\027)6 b Fy(,)28 b Fx(g)637 3506 y Ft(j)671 3494 y Fy(\()p Fx(x)p Fy(\))c(=)f Fu(\000)p Fx(i)p Fy(\()p Fx(x)18 b Fy(+)g(2)p Fu(M)1310 3451 y Fr([0])1310 3517 y Ft(j)1384 3494 y Fy(\()p Fx(x)p Fy(\)\))1527 3464 y Fo(\000)p Fr(1)1618 3494 y Fy(,)28 b(with)g Fu(M)1958 3451 y Fr([0])1958 3516 y(1)2032 3494 y Fy(\()p Fx(x)p Fy(\))c(=)f(0)k(and)h Fu(M)2586 3451 y Fr([0])2586 3516 y(2)2660 3494 y Fy(\()p Fx(x)p Fy(\))c(=)f Fx(\025)2931 3506 y Fr(0)2968 3494 y Fy(.)96 3612 y(By)k(using)h(\(5.35\))e(w)n(e)i(can)f(write)694 3861 y Fx(g)734 3873 y Ft(j)768 3861 y Fy(\()p Fx(x)p Fy(\)\010)939 3873 y Ft(j;)p Fk(\027)1033 3861 y Fy(\()p 1065 3815 V Fx(u)p Fy(\))83 b(=)g Fx(g)1416 3873 y Ft(j)1450 3861 y Fy(\()p Fx(x)p Fy(\))1606 3757 y Fo(1)1579 3782 y Fm(X)1575 3958 y Ft(n)p Fr(=0)1715 3861 y Fy(\011)1780 3873 y Ft(n)1825 3861 y Fy(\()p Fx(x)p Fy(\)\010)1996 3873 y Ft(j;)p Fk(\027)2090 3861 y Fy(\()p 2122 3815 V Fx(u)p Fy(\))1228 4136 y(=)g Fx(g)1416 4148 y Ft(j)1450 4136 y Fy(\()p Fx(x)p Fy(\))1606 4033 y Fo(1)1579 4058 y Fm(X)1575 4233 y Ft(n)p Fr(=0)1715 4136 y Fy(\011)1780 4148 y Ft(n)1825 4136 y Fy(\()p Fx(x)p Fy(\))1950 4044 y Fm(\020)2000 4136 y Fx(g)2043 4093 y Fr([)p Ft(n)p Fr(])2040 4160 y Ft(j)2126 4136 y Fy(\()p Fx(x)p Fy(\))2237 4044 y Fm(\021)2287 4062 y Fo(\000)p Fr(1)2390 4044 y Fm(\020)2440 4136 y Fx(g)2483 4093 y Fr([)p Ft(n)p Fr(])2480 4160 y Ft(j)2565 4136 y Fy(\()p Fx(x)p Fy(\)\010)2736 4148 y Ft(j;)p Fk(\027)2830 4136 y Fy(\()p 2862 4091 V Fx(u)p Fy(\))2942 4044 y Fm(\021)3006 4136 y Fx(;)510 b Fy(\(5.38\))-29 4420 y(where)28 b(\011)277 4432 y Ft(n)321 4420 y Fy(\()p Fx(x)p Fy(\)\()p Fx(g)507 4377 y Fr([)p Ft(n)p Fr(])504 4443 y Ft(j)591 4420 y Fy(\()p Fx(x)p Fy(\)\))734 4390 y Fo(\000)p Fr(1)847 4420 y Fy(=)23 b Fx(i)p Fy(\()p Fx(x)c Fy(+)f(2)p Fu(M)1287 4377 y Fr([)p Fo(\024)p Ft(n)p Fr(])1287 4443 y Ft(j)1421 4420 y Fy(\()p Fx(x)p Fy(\)\),)29 b(and)1150 4683 y Fx(g)1193 4640 y Fr([)p Ft(n)p Fr(])1190 4706 y Ft(j)1276 4683 y Fy(\()p Fx(x)p Fy(\)\010)1447 4695 y Ft(j;)p Fk(\027)1541 4683 y Fy(\()p 1573 4637 V Fx(u)o Fy(\))24 b(=)1791 4579 y Fo(1)1764 4604 y Fm(X)1763 4782 y Ft(k)q Fr(=1)1898 4683 y Fx(")1937 4648 y Ft(k)2090 4604 y Fm(X)1991 4804 y Ft(\022)r Fo(2)p 2070 4756 51 3 v Fr(\002)2120 4770 y Fh(R)2120 4820 y Fi(k)q(;j;)p Fb(\027)t Fi(;n)2323 4683 y Fy(V)-7 b(al\()p Fx(\022)r Fy(\))p Fx(;)968 b Fy(\(5.39\))-29 5021 y(where)p 212 4954 65 4 v 28 w(\002)277 4968 y Fo(R)277 5041 y Ft(k)q(;j;)p Fk(\027)t Ft(;n)515 5021 y Fy(di\013ers)28 b(from)g(\002)1025 4990 y Fo(R)1025 5044 y Ft(k)q(;j;)p Fk(\027)t Ft(;n)1263 5021 y Fy(as)f(it)i(con)n(tains)e(also)g(trees)h (whic)n(h)g(can)f(ha)n(v)n(e)g(one)h(renormalised)e(self-energy)-29 5120 y(cluster)j Fx(T)41 b Fy(with)30 b(exiting)f(line)g(giv)n(en)g(b)n (y)g(the)h(ro)r(ot)e(line)i(of)f Fx(\022)r Fy(.)43 b(In)29 b(suc)n(h)g(a)g(case)g(if)h Fx(p)f Fy(is)g(the)h(line)f(of)h(the)f(en)n (tering)g(line)-29 5220 y(of)f Fx(T)12 b Fy(,)27 b(then)h Fx(p)23 b Fu(\025)f Fy(0)28 b(and)f(the)h(scale)f Fx(n)1141 5232 y Ft(T)1221 5220 y Fy(of)g Fx(T)39 b Fy(is)27 b(suc)n(h)h(that)f Fx(n)1903 5232 y Ft(T)1974 5220 y Fy(+)18 b(1)23 b Fu(\024)f Fy(min)q Fu(f)p Fx(n;)14 b(p)p Fu(g)p Fy(,)26 b(b)n(y)h(de\014nition)h (of)g(cluster.)1820 5484 y(25)p eop end %%Page: 26 26 TeXDict begin 26 25 bop 96 169 a Fy(Then)28 b(w)n(e)f(ha)n(v)n(e)947 259 y Fo(1)920 284 y Fm(X)917 460 y Ft(n)p Fr(=0)1056 363 y Fy(\011)1121 375 y 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b(in)i(the)g(third)g(and)f(fourth)h(lines.)96 2136 y(Then)g(the)g(sum)f(of)h(the)g(third)g(and)f(fourth)h(lines)f(in) h(\(5.40\))f(giv)n(es)555 2363 y Fu(\000)p Fy(2)p Fx(i)705 2221 y Fm( )798 2259 y Fo(1)772 2284 y Fm(X)769 2460 y Ft(n)p Fr(=1)908 2363 y Fy(\011)973 2375 y Ft(n)1018 2363 y Fy(\()p Fx(x)p Fy(\))1174 2259 y Fo(1)1147 2284 y Fm(X)1143 2459 y Ft(p)p Fr(=)p Ft(n)1323 2259 y(n)1284 2284 y Fm(X)1286 2460 y Ft(s)p Fr(=1)1417 2363 y Fx(M)1507 2320 y Fr([)p Ft(s)p Fr(])1498 2386 y Ft(j)1580 2363 y Fy(\()p Fx(x)p Fy(\))p 1705 2317 V 14 w Fx(u)1753 2375 y Ft(j;)p Fk(\027)t Ft(;p)1919 2363 y Fy(+)2031 2259 y Fo(1)2005 2284 y Fm(X)2002 2460 y Ft(n)p Fr(=2)2141 2363 y Fy(\011)2206 2375 y Ft(n)2251 2363 y Fy(\()p Fx(x)p Fy(\))2376 2259 y Ft(n)p Fo(\000)p Fr(1)2380 2284 y Fm(X)2380 2460 y Ft(p)p Fr(=1)2559 2255 y Ft(p)2516 2284 y Fm(X)2519 2460 y Ft(s)p Fr(=1)2650 2363 y Fx(M)2740 2320 y Fr([)p Ft(s)p Fr(])2731 2386 y Ft(j)2813 2363 y Fy(\()p Fx(x)p Fy(\))p 2938 2317 V 14 w Fx(u)2986 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2870 y Fm(X)1467 3046 y Ft(s)p Fr(=1)1599 2949 y Fx(M)1689 2906 y Fr([)p Ft(s)p Fr(])1680 2972 y Ft(j)1762 2949 y Fy(\()p Fx(x)p Fy(\))1916 2845 y Fo(1)1889 2870 y Fm(X)1887 3044 y Ft(n)p Fr(=)p Ft(s)2025 2949 y Fy(\011)2090 2961 y Ft(n)2134 2949 y Fy(\()p Fx(x)p Fy(\))p Fx(:)1271 b Fy(\(5.42\))-29 3180 y(If)28 b(w)n(e)g(de\014ne) 1005 3280 y(\004)1060 3292 y Ft(n)1106 3280 y Fy(\()p Fx(x)p Fy(\))c(:=)f Fx(\037)1404 3292 y Fr(0)1441 3280 y Fy(\(\001)1542 3292 y Fr(0)1580 3280 y Fy(\()p Fx(x)p Fy(\)\))14 b Fx(:)g(:)g(:)h(\037)1901 3292 y Ft(n)p Fo(\000)p Fr(1)2031 3280 y Fy(\(\001)2132 3292 y Fr(0)2169 3280 y Fy(\()p Fx(x)p Fy(\)\))p Fx(\037)2364 3292 y Ft(n)2411 3280 y Fy(\(\001)2512 3292 y Fr(0)2550 3280 y Fy(\()p Fx(x)p Fy(\)\))p Fx(;)823 b Fy(\(5.43\))-29 3409 y(then)29 b(in)e(\(5.42\))g(w)n(e)g(can)h(write)1150 3520 y Fo(1)1123 3545 y Fm(X)1122 3719 y Ft(n)p Fr(=)p Ft(s)1259 3624 y Fy(\011)1324 3636 y Ft(n)1369 3624 y Fy(\()p Fx(x)p Fy(\))c(=)e(\004)1646 3636 y Ft(s)p Fo(\000)p 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5220 y(1)2922 5198 y Fx(;)635 b Fy(\(6.7\))1820 5484 y(28)p eop end %%Page: 29 29 TeXDict begin 29 28 bop -29 169 a Fy(where)28 b(w)n(e)f(explicitly)g (used)h(that)g(the)g(\014rst)f(con)n(tribution)g(to)h Fx(\026)f Fy(dep)r(ending)h(on)g Fx(\025)2592 181 y Fr(0)2657 169 y Fy(has)f(size)g Fx(O)r Fy(\()p Fx(")3098 139 y Fr(2)3136 169 y Fx(C)3201 133 y Fo(\000)p Fr(1)3195 191 y(1)3291 169 y Fy(\).)96 286 y(Therefore)638 415 y(meas)o(\(\003)18 b Fu(n)g Fy(\003)1044 381 y Fo(\003)1082 415 y Fy(\))24 b(=)1225 302 y Fm(Z)1271 491 y Fr(\003)p Fo(n)p Fr(\003)1395 474 y Fh(\003)1449 415 y Fy(d)p Fx(\025)g Fu(\024)e(\000)p Fy(2)p Fx(A")1862 381 y Fr(2)1917 415 y Fy(+)2000 302 y Fm(Z)2046 491 y Fr(\003)2091 499 y Fl(0)2124 491 y Fo(n)p Fr(\003)2203 471 y Fh(\003)2203 509 y Fl(0)2256 415 y Fy(d)p Fx(\025)2350 427 y Fr(0)2402 295 y Fm(\014)2402 344 y(\014)2402 394 y(\014)2402 444 y(\014)2458 359 y Fy(d)p Fx(\025)p 2439 396 132 4 v 2439 472 a Fy(d)p Fx(\025)2533 484 y Fr(0)2581 295 y Fm(\014)2581 344 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y(are)f(b)r(ounded)h(prop)r(ortionally) d(to)j Fu(j)p Fx(")p Fu(j)1164 1261 y Fo(\000)p Ft(\033)1260 1291 y Fx(\015)1308 1255 y Fo(\000)p Fr(1)1303 1319 y Ft(n)p Fr(\()p Fk(\027)1412 1328 y Fi(`)1441 1319 y Fr(\))1495 1291 y Fy({)f(and)g(the)h(b)r(ounds)f(w)n(ere)g(obtained)g(b)n(y)g (using)g(that)h(one)f(has)g(at)g(w)n(orst)-29 1408 y(a)28 b(factor)e Fu(j)p Fx(")p Fu(j)363 1378 y Fo(\000)p Ft(\033)487 1408 y Fy(p)r(er)i(line.)37 b(Then)28 b(one)f(can)g(restrict)g(the)h (analysis)e(to)i([)p Fx(a;)14 b(b)p Fy(])k Fu(n)g Fy(\003)2486 1420 y Fr(1)2523 1408 y Fy(,)27 b(and)h(the)g(same)f(conclusions)f (hold.)-29 1762 y Fq(Ac)m(kno)m(wledgmen)m(ts.)37 b Fy(I'm)28 b(indebted)g(to)g(Gio)n(v)-5 b(anni)27 b(Galla)n(v)n(otti)f(for)h (useful)h(discussions.)-29 2054 y Fz(References)13 2253 y Fy([1])41 b(A.)h(Avila,)i(R.)e(Krik)n(orian,)g Fp(R)l(e)l(ducibility) h(or)g(non-uniform)f(hyp)l(erb)l(olicity)j(for)f(quasi-p)l(erio)l(dic)g (Schr\177)-42 b(odinger)142 2353 y(c)l(o)l(cycles)p Fy(,)29 b(Preprin)n(t,)d(to)i(app)r(ear)f(on)g(Ann.)h(Math.)13 2518 y([2])41 b(M.)36 b(Bartuccelli,)h(G.)f(Gen)n(tile,)i Fp(Lindste)l(dt)f(series)h(for)g(p)l(erturb)l(ations)f(of)h(iso)l(chr)l (onous)g(systems)p Fy(,)f(Rev.)f(Math.)142 2618 y(Ph)n(ys.)27 b Fq(14)g Fy(\(2002\),)f(no.)i(2,)f(121{171.)13 2783 y([3])41 b(F.)23 b(Bonetto,)h(G.)f(Galla)n(v)n(otti,)f(G.)h(Gen)n (tile,)h(V.)g(Mastropietro,)e Fp(Quasi-line)l(ar)j(\015ows)h(on)f (tori:)37 b(r)l(e)l(gularity)26 b(of)g(their)142 2883 y(line)l(arization)p Fy(,)k(Comm.)d(Math.)h(Ph)n(ys.)f Fq(192)f Fy(\(1998\),)h(no.)g(3,)g(707{736.)13 3048 y([4])41 b(Ch.-Q.)k(Cheng,)51 b Fp(Birkho\013-Kolmo)l(gor)l(ov-A)n(rnold-Moser)f (tori)d(in)g(c)l(onvex)f(Hamiltonian)i(systems)p Fy(,)i(Comm.)142 3148 y(Math.)28 b(Ph)n(ys)e Fq(177)h Fy(\(1996\),)g(no.)g(3,)g (529{559.)13 3313 y([5])41 b(E.I.)29 b(Dinaburg,)g(Ja.G.)f(Sina)-9 b(\025)-32 b(\020,)29 b Fp(The)j(one-dimensional)h(Schr\177)-42 b(odinger)32 b(e)l(quation)g(with)f(quasip)l(erio)l(dic)j(p)l(otential) p Fy(,)142 3413 y(F)-7 b(unk)n(cional.)32 b(Anal.)g(i)h(Prilo)n(\024) -39 b(zen.)30 b Fq(9)j Fy(\(1975\),)f(no.)g(4,)i(8{21;)f(English)e (translation:)46 b(F)-7 b(unctional)32 b(Anal.)h(Appl.)g Fq(9)142 3512 y Fy(\(1975\),)26 b(no.)i(4,)f(279{289)d(\(1976\).)13 3678 y([6])41 b(L.H.)e(Eliasson,)i Fp(Hamiltonian)g(systems)f(with)g (line)l(ar)h(normal)g(form)g(ne)l(ar)f(an)g(invariant)h(torus)p Fy(,)h(Nonlinear)142 3777 y(dynamics)27 b(\(Bologna,)f(1988\),)g (11{29,)g(W)-7 b(orld)27 b(Sci.)h(Publishing,)f(T)-7 b(eanec)n(k,)27 b(NJ,)g(1989.)13 3942 y([7])41 b(L.H.)31 b(Eliasson,)e Fp(Flo)l(quet)k(solutions)g(for)g(the)f Fy(1)p Fp(-dimensional)i(quasi-p)l(erio)l(dic)g(Schr\177)-42 b(odinger)34 b(e)l(quation)p Fy(,)e(Comm.)142 4042 y(Math.)c(Ph)n(ys.)e Fq(146)h Fy(\(1992\),)g(no.)g(3,)g(447{482.)13 4207 y([8])41 b(L.H.)32 b(Eliasson,)e Fp(R)l(e)l(ducibility)k(and)f(p)l(oint)h(sp)l 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e(nonr)l(esonant)f(harmonic)j(oscil)t(lators.)h("Wick)e(or-)142 5136 y(dering")f(of)f(the)h(p)l(erturb)l(ations)f(in)g(classic)l(al)h (me)l(chanics)g(and)g(invarianc)l(e)g(of)g(the)f(fr)l(e)l(quency)g(sp)l (e)l(ctrum)p Fy(,)e(Comm.)142 5235 y(Math.)k(Ph)n(ys.)e Fq(87)i Fy(\(1982/83\),)d(no.)i(3,)g(365{383.)1820 5484 y(29)p eop end %%Page: 30 30 TeXDict begin 30 29 bop -29 169 a Fy([11])41 b(G.)24 b(Galla)n(v)n(otti,)f(F.)h(Bonetto,)f(G.)h(Gen)n(tile,)h Fp(Asp)l(e)l(cts)g(of)i(er)l(go)l(dic,)h(qualitative)f(and)g(statistic) l(al)f(the)l(ory)h(of)f(motion)p Fy(,)142 268 y(T)-7 b(exts)28 b(and)f(Monographs)e(in)j(Ph)n(ysics,)f(Springer,)f(Berlin,)h (2004.)-29 430 y([12])41 b(G.)30 b(Galla)n(v)n(otti,)e(G.)i(Gen)n (tile,)g(A.)g(Giuliani,)g Fp(F)-6 b(r)l(actional)32 b(Lindste)l(dt)g (series)p Fy(,)e(Preprin)n(t,)f(to)g(app)r(ear)g(on)g(J.)g(Math.)142 530 y(Ph)n(ys.)-29 692 y([13])41 b(G.)34 b(Galla)n(v)n(otti,)h(G.)f (Gen)n(tile,)i Fp(Hyp)l(erb)l(olic)h(low-dimensional)h(invariant)e 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(ducible)g(quasi-p)l(erio)l(dic)h(c)l(o)l(cycles)f(on)f Fq(T)2656 2749 y Fr(1)2711 2786 y Fu(\002)19 b Fy(SU\(2\),)28 b(Ann.)h(of)f(Math.)g Fq(154)142 2886 y Fy(\(2001\),)e(no.)i(2,)f (269{326.)-29 3048 y([22])41 b(A.)28 b(Iserles,)f Fp(Exp)l(ansions)j (that)g(gr)l(ow)g(on)g(tr)l(e)l(es)p Fy(,)d(Notices)g(Amer.)h(Math.)g (So)r(c.)f Fq(49)g Fy(\(2002\),)g(no.)g(4,)g(430{440.)-29 3210 y([23])41 b(A.)28 b(Iserles,)e(S.P)-7 b(.)27 b(N\034rsett,)g Fp(On)h(the)i(solution)f(of)i(line)l(ar)f(di\013er)l(ential)g(e)l (quations)g(in)f(Lie)h(gr)l(oups)p Fy(,)e(R.)f(So)r(c.)g(Lond.)142 3309 y(Philos.)g(T)-7 b(rans.)27 b(Ser.)g(A)h(Math.)f(Ph)n(ys.)g(Eng.)g (Sci.)h Fq(357)f Fy(\(1999\),)f(no.)h(1754,)f(983{1019.)-29 3471 y([24])152 3450 y(\022)142 3471 y(A.)h(Jorba,)f(C.)g(Sim\023)-42 b(o,)27 b Fp(On)i(the)g(r)l(e)l(ducibility)h(of)g(line)l(ar)g(di\013er) l(ential)g(e)l(quations)f(with)h(quasip)l(erio)l(dic)h(c)l(o)l (e\016cients)p Fy(,)142 3571 y(J.)c(Di\013eren)n(tial)h(Equations)e Fq(98)i Fy(\(1992\),)e(no.)h(1,)h(111{124.)-29 3733 y([25])41 b(J.)27 b(Lop)r(es)h(Dias,)f Fp(A)j(normal)g(form)g(the)l(or)l(em)g (for)h(Brjuno)f(skew-systems)g(thr)l(ough)g(r)l(enormalization)p Fy(,)f(Preprin)n(t.)-29 3895 y([26])41 b(J.)35 b(Moser,)g(J.)g(P\177) -42 b(osc)n(hel,)35 b Fp(A)n(n)g(extension)h(of)i(a)e(r)l(esult)g(by)h (Dinabur)l(g)f(and)h(Sinai)g(on)f(quasip)l(erio)l(dic)j(p)l(otentials)p Fy(,)142 3995 y(Commen)n(t.)28 b(Math.)f(Helv.)h Fq(59)f Fy(\(1984\),)g(no.)g(1,)g(39{85.)-29 4157 y([27])41 b(L.)30 b(P)n(astur,)f(A.)h(Figotin,)g Fp(Sp)l(e)l(ctr)l(a)h(of)i(r)l(andom)f (and)g(almost-p)l(erio)l(dic)i(op)l(er)l(ators)p Fy(,)d(Grundlehren)f (der)f(Mathem-)142 4257 y(atisc)n(hen)e(Wissensc)n(haften)g(297,)g (Springer,)f(Berlin,)h(1992.)-29 4419 y([28])41 b(H.)20 b(R)r(\177)-44 b(ussmann,)21 b Fp(On)g(the)i(one-dimensional)h (Schr\177)-42 b(odinger)24 b(e)l(quation)e(with)h(a)g(quasip)l(erio)l (dic)i(p)l(otential)p Fy(,)d(Nonlinear)142 4518 y(dynamics)27 b(\(In)n(ternational)g(Conference,)g(New)h(Y)-7 b(ork,)27 b(1979\),)f(pp.)i(90{107,)d(Ann.)j(New)g(Y)-7 b(ork)27 b(Acad.)g(Sci.,)h(357,)142 4618 y(New)g(Y)-7 b(ork)27 b(Acad.)g(Sci.,)h(New)g(Y)-7 b(ork,)27 b(1980.)-29 4780 y([29])41 b(W.M.)34 b(Sc)n(hmidt,)h Fp(Diophantine)h(appr)l(oximation)p Fy(,)g(Lecture)d(Notes)g(in)g(Mathematics,)h(785,)f(Springer,)g (Berlin,)142 4880 y(1980.)-29 5042 y([30])41 b(Ju.)30 b(Xu,)i(Q.)e(Zheng,)h Fp(On)h(the)g(r)l(e)l(ducibility)i(of)f(line)l (ar)g(di\013er)l(ential)h(e)l(quations)e(with)h(quasip)l(erio)l(dic)i (c)l(o)l(e\016cients)142 5141 y(which)c(ar)l(e)g(de)l(gener)l(ate)p Fy(,)d(Pro)r(c.)e(Amer.)i(Math.)g(So)r(c.)f Fq(126)p Fy(\(1998\),)f(no.)h(5,)g(1445{1451.)1820 5484 y(30)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0512010705712--