Content-Type: multipart/mixed; boundary="-------------0407091143653" This is a multi-part message in MIME format. ---------------0407091143653 Content-Type: text/plain; name="04-207.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-207.comments" MSC2000: 82B28; 82B44; 60K35. Number of pages 48 ---------------0407091143653 Content-Type: text/plain; name="04-207.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-207.keywords" Lattice systems, Cluster expansion, Disordered systems, Renormalization group. ---------------0407091143653 Content-Type: application/postscript; name="bcoastratto-preprint.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="bcoastratto-preprint.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software %%Title: bcoastratto-preprint.dvi %%Pages: 48 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMR12 CMR10 CMBX10 CMTI10 CMTT10 CMBX12 CMTI12 %%+ CMMI12 CMSY10 CMEX10 CMMI8 CMSY8 MSBM10 CMR8 CMR6 CMMI6 CMSY6 MSBM7 %%+ MSAM10 MSAM7 TeX-cmex7 LCIRCLE10 LINE10 TeX-cmex8 CMTT12 CMTI7 %%+ CMSL12 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips bcoastratto-preprint %DVIPSParameters: dpi=600, compressed %DVIPSSource: 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI7 %!PS-AdobeFont-1.1: CMTI7 1.0 %%CreationDate: 1991 Aug 18 21:07:18 % 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 (CMTI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI7 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 1268 750}readonly def /UniqueID 5000825 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D77639DF1232A4D6233A9CAF69B151DFD33F C0962CCA6FACCEA6B71BEEF7C056FBD376F2F0D0BD6BE0A0A8259139B28B99C6 25119B8C60FF7BA002476930DFDC2F6B1B5A80F1C0E544A22E3F0FB2FEAB64B6 A509B61E1AB96121FBF7F2BC27CDE5B4961048FC2169C8DB745FCC7AF0EED507 284038B15CE53E69AA22CA76C2A208F6AD8614CD500D479A5E0FD25E7BB14C7F C47C503E4CC38975CB0A8F7E109665FB69EBAE6EB68B2D6FA3C967D5BDE7830E 9CE6F13DFAEBC852DF1D7D0CDD33DD4D5CD6D94DD8AE0D48AB638013CF75A25C 04FDEA8775EC52A13AF40FA41F7354624D917EE620318C702237AF8C0E1FFA02 127F862D4DFD0A56774235A881AFBB8F7F4BCB33811C581CFF38462F669B7F97 1F97AC09373B8F9B7E653013AF8170613E8D7E17235A893BE296A0CD2096B71F 16778388993EDF1B003EBCE23DD02949CBFAAAB5D9F6A08178BFB8ED1043FC1B 0B90D9AFC27B19783740E8DDE0E5E01D116A8CB083C721FBB8EBF71018A9570E 78590CD831116BC0FDA4229F79A581874FC3A1F108A4FCA80CE27FA54A2A7505 5BDB3D52E2595512837732322FB5BBE459C0205EB38439E34A39544F0A1567C6 F29FBB7CF931708C8F92786CCBCBB443D54462B3B4B7BD38E90D325EBC31BA05 FAAB393C7EBDB9EC48E4ACC15A247D2348A62293A726FCA4250B4A407F64270F 923EB516CB8A3B5D19876C8755EB291357E4E4CC5903D1A536A635E267F7A971 B1C33AA7F58E1ADA931C450EA8367B18E20E683CCEEC1BC079E837DDAFFF0D16 23C4AED0DC04DD9458A630FD498794823FFA55705315F0687E7592A5DFC8B8D6 FE2F3C62D7A090886C9E92D1CA5329E9AAE5D6760B23949A9EBE069E84EE7709 F860EF08C04948E39A309B91E46FC622D3081F5B5ED0AA57C07915092B273F0F C3DFEFC85B408C198AABD700D23C14A32233B776731F63D7A2177A677816D8B8 82641414FE67D2590B6FC83ACCA65774D2EF6BE1EB58EB10B025ECAD93A3585B DDA793CE330D2E153BDF7954F0BB5FB6473AAF00498C21B2CCF8EF9548F6DDCB 5FAF601E62D0955F3C51765EF7E00D46BF570C16649C863EA61A24BA72A19584 0419781F044F1287686DE01A8611C154222177840C795CA82C3DBE543BA7BE93 41DAD66F10DD596D34E6F9C8D01765F534377EFE54A53A5F84A0AC17C8981910 EBAEA0F74130944E165D9B666C0BDA47B01BD6CD4B269C10B884D6674325E45D 0A4766FD0A954BF5AFF5B750F3A6EBCCC6A43F6FC4CF01DB884E0A5A34092D83 BE079F139597524762C6DD3EE86EE1FF8B48BA404616BB9BBC68EBFFA4C9E4E7 A6476A2EA73AADA413CF6868F76920E59529BCCFEF87B0ACB82129ECA6B11B10 E43867A29837903F9494C64F88C4B2A883A97D7243D9FC4678A8AD20C47FA8C9 5F0B476C228031F5459643D45AE9FC27143D17FC38EC6D63974B0C1DA46A6E8C A02808C213BE152B6D192FC317BE9E97D6E3976115106A5173809A51640D616E 946B51BA0E288BB66C5A0BDCDB253B0923CAB13505A03D01D229BF331D8FE369 8CEFE3B7BBB5AE12E833B94961B8DF24C7D6DD3B77DAB0E898B730264C6237EA 98B58A3367E36967730CEF3D4E87900EE2E4F0978DBD9DF73114091557F08F85 C053E1578FBE72B3976B5C5CA66115E7DF65B7A325A7B8B2A8B334427E588A20 735633EEDEA5745D606BCCA5BF5EDD2FB2009CFB9014147E2D6D367C32450879 8AFC6EB1BB68CB54B434F924CCEE2F676156D604835855479FD2D44C973EA6B3 05471B55A54A50212937873BA4FF5760815CCC954A479E26D20ED2F39345971C 502DDEE6741FC81A9A9FFED72ECE4BFC51F39E7E05F659ECAE44146A7347E9 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTT12 %!PS-AdobeFont-1.1: CMTT12 1.0 %%CreationDate: 1991 Aug 20 16:45:46 % 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 (CMTT12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch true def end readonly def /FontName /CMTT12 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{-1 -234 524 695}readonly def /UniqueID 5000833 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F0364CD5660FE13FF01BC20148F9C480BCD0E C81D5BFC66F04993DD73F0BE0AB13F53B1BA79FE5F618A4F672B16C06BE3251E 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2001 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.001) readonly def /Notice (Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved) readonly def /FullName (LINE10) readonly def /FamilyName (LaTeX) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /LINE10 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 /a0 put dup 18 /a18 put dup 64 /a64 put dup 73 /a73 put dup 82 /a82 put readonly def /FontBBox{-150 -150 1020 1020}readonly def /UniqueID 5011940 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: LCIRCLE10 %!PS-AdobeFont-1.1: LCIRCLE10 1.001 %%CreationDate: 1992 Oct 23 20:21:59 %%RevisionDate: 2001 Jun 05 20:21:59 % Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.001) readonly def /Notice (Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved) readonly def /FullName (LCIRCLE10) readonly def /FamilyName (LaTeX) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /LCIRCLE10 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 115 /a115 put readonly def /FontBBox{-2000 -2000 2040 2040}readonly def /UniqueID 5011942 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSAM7 %!PS-AdobeFont-1.1: MSAM7 2.1 %%CreationDate: 1992 Oct 17 08:30:29 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM7 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 22 /harpoonupright put readonly def /FontBBox{0 -576 1569 1076}readonly def /UniqueID 5032012 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: TeX-cmex7 %!PS-AdobeFont-1.0: TeX-cmex7 001.001 % Filtered by type1fix.pl 0.05 %%EndComments 13 dict dup begin /FontInfo 16 dict dup begin /Copyright (see\040copyright\040of\040original\040TeX\040font) def /FamilyName (TeX\040cmex7) def /FullName (TeX\040cmex7\040Regular) def /ItalicAngle 0 def /Notice (converted\040after\040April\0402001) def /UnderlinePosition -100 def /UnderlineThickness 50 def /Weight (Regular) def /isFixedPitch false def /version (001.001) def end readonly def /FontName /TeX-cmex7 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 100 /d put readonly def /FontBBox {-14 -2954 1627 771} readonly def /UniqueID 4314415 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 22 /harpoonupright put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueID 5031981 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 71 /G put dup 76 /L put dup 84 /T put dup 123 /kappa put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6A66A4000A13D5F68BFF326D 1D432B0D064B56C598F4338C319309181D78E1629A31ECA5DD8536379B03C383 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D5FC1B2109839E5B52DFB7605D7BA557CC35D6 49F6EB651B83771034BA0C39DB8D426A24543EF4529E2D939125B5157482688E 9045C2242F4AFA4C489D975C029177CD6497EACD181FF151A45F521A4C4043C2 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. 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American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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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 (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 12 /vextendsingle put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 40 /braceleftBigg put dup 41 /bracerightBigg put dup 56 /bracelefttp put dup 57 /bracerighttp put dup 58 /braceleftbt put dup 59 /bracerightbt put dup 60 /braceleftmid put dup 61 /bracerightmid put dup 62 /braceex put dup 78 /circlemultiplytext put dup 80 /summationtext put dup 81 /producttext put dup 83 /uniontext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 91 /uniondisplay put dup 98 /hatwide put dup 99 /hatwider put dup 100 /hatwidest put dup 101 /tildewide put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTT10 %!PS-AdobeFont-1.1: CMTT10 1.00B %%CreationDate: 1992 Apr 26 10:42:42 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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w(set)37 b Fq(G)475 3846 y Fs(j)544 3831 y Fx(:=)679 3757 y Ft(S)762 3861 y Fs(g)r Fr(2G)887 3871 y Fn(j)940 3831 y Fv(g)f Fu(\032)c Fq(L)p Fw(,)38 b Fq(B)1332 3846 y Fp(0)1404 3831 y Fx(:=)32 b Fq(L)24 b Fu(n)g Fq(G)1780 3846 y Fp(0)1857 3831 y Fw(and)37 b Fq(B)2115 3846 y Fs(j)2183 3831 y Fx(:=)32 b Fq(B)2384 3846 y Fs(j)t Fr(\000)p Fp(1)2535 3831 y Fu(n)24 b Fq(G)2686 3846 y Fs(j)2723 3831 y Fw(,)38 b(then)f(for)g(any) g Fv(g)e Fu(2)d(G)3596 3846 y Fs(j)3670 3831 y Fw(we)244 3976 y(have)i Fx(d)522 3991 y Fp(1)578 3976 y Fx(\()p Fv(g)t(;)17 b Fq(B)777 3991 y Fs(j)t Fr(\000)p Fp(1)925 3976 y Fu(n)22 b Fv(g)t Fx(\))27 b Fv(>)h(\015)1268 3991 y Fs(j)1339 3976 y Fw(for)34 b(any)h Fv(j)f Fu(\025)28 b Fx(1)p Fw(;)p Black 115 4203 a(6.)p Black 49 w(for)36 b(e)-5 b(ach)35 b Fv(x)30 b Fu(2)h Fq(L)36 b Fw(we)g(have)f Fv(k)1327 4218 y Fs(x)1400 4203 y Fx(:=)30 b(sup)1697 4123 y Ft(\010)1755 4203 y Fv(j)36 b Fu(\025)30 b Fx(1)g(:)46 b Fu(9)p Fv(g)34 b Fu(2)c(G)2381 4218 y Fs(j)2453 4203 y 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b(Algebra)f(of)h(the)f(expansion)0 574 y Fx(In)24 b(this)h(Section)f(w) m(e)h(in)m(tro)s(duce)g(the)f(algebra)g(of)f(the)h(graded)g(cluster)h (expansion)g(without)g(discussing)0 719 y(an)m(y)31 b(con)m(v)m (ergence)h(issue,)h(whic)m(h)e(will)g(b)s(e)f(dealt)h(up)s(on)f(in)g (Section)h(5.)42 b(W)-8 b(e)31 b(supp)s(ose)g(the)g(h)m(yp)s(otheses)0 863 y(of)h(Theorem)i(3.2)e(are)h(satis\014ed.)45 b(Moreo)m(v)m(er,)34 b(for)e(\003)27 b Fu(\032\032)i Fq(L)k Fx(w)m(e)g(de\014ne)h(the)f(set) 849 1095 y(\007)925 1110 y Fp(\003)1006 1095 y Fx(:=)28 b Fu(f)p Fv(X)35 b Fu(\032\032)28 b Fq(L)h Fx(:)44 b Fv(X)30 b Fu(\\)22 b Fx(\003)28 b Fu(6)p Fx(=)f Fu(;)33 b Fx(and)f Fv(X)e Fu(\\)23 b Fx(\()p 2559 1015 68 4 v(\003)o(\))2664 1053 y Fp(c)2728 1095 y Fx(=)k Fu(;g)648 b Fx(\(4.1\))0 1326 y(b)m(y)33 b(item)h(3)e(in)h(Condition)g(2.1)f(w)m(e)i(can)f (rewrite)g(\(2.4\))f(as)1189 1557 y Fv(Z)1256 1572 y Fp(\003)p Fr(\\)p Fl(G)1404 1581 y Fo(0)1459 1557 y Fx(\()p Fv(\033)t Fx(\))27 b(=)h(exp)1890 1447 y 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Fv(;)2752 5243 y Ft(b)2734 5268 y Fv(G)o Fx(\){compatible)g(and)g(that)0 5413 y(\010)70 5362 y Fp(\()p Fs(i;)p Fp(0\))70 5440 y Fs(G;s;)p Fp(\003)278 5413 y Fx(=)28 b(0)k(if)g(there)h(exists)h Fv(g)d Fu(2)d Fv(G)33 b Fx(suc)m(h)h(that)p 1783 5358 V 32 w Fv(g)d Fu(\032)d Fx(\003)2034 5377 y Fp(c)2069 5413 y Fx(.)44 b(W)-8 b(e)32 b(shall)h(lo)s(ok)g(at)f(\010)2936 5362 y Fp(\()p Fs(i;)p Fp(0\))2936 5440 y Fs(G;s;)p Fp(\003)3148 5413 y Fx(as)h(the)g(e\013ectiv)m(e)0 5557 y(in)m(teraction)g(at)g (scale)g Fv(i)g Fx(due)g(to)g(the)g(in)m(tegration)f(on)h(scale)g(0;)g (it)f(will)i(b)s(e)e(expanded)j(at)d(step)h Fv(i)p Fx(.)p Black 1841 5806 a(20)p Black eop end %%Page: 21 22 TeXDict begin 21 21 bop Black Black 146 407 a Fx(By)33 b(using)h(de\014nitions)g(\(4.5\))e(and)g(\(4.6\))g(w)m(e)i(ha)m(v)m(e) 609 541 y Ft(X)576 752 y Fs(X)5 b Fr(2)p Fp(\007)741 763 y Fo(\003)803 635 y Fv(V)860 650 y Fs(X)q(;)p Fp(\003)1020 635 y Fx(=)1166 541 y Ft(X)1136 741 y Fn(X)t Fm(2)p Fo(\007)1279 758 y(\003)1323 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Fs(j)2276 2618 y Fx(for)32 b Fv(j)h Fu(\025)28 b Fx(0.)44 b(F)-8 b(or)31 b Fv(i)d Fu(\025)h Fx(0)j(w)m(e)h(also)g(set)1036 2847 y(\003)1104 2862 y Fr(\025)p Fs(i)1215 2847 y Fx(:=)1346 2752 y Ft([)1345 2962 y Fs(j)t Fr(\025)p Fs(i)1473 2847 y Fx(\003)1541 2862 y Fs(j)1721 2847 y Fx(and)143 b(\001)2102 2862 y Fs(i)2158 2847 y Fx(:=)28 b(\003)2357 2862 y Fr(\025)p Fs(i)p Fp(+1)2552 2847 y Fu([)22 b Fx(\003)2708 2805 y Fp(c)0 3147 y Fx(Then,)38 b(giv)m(en)f Fv(\034)45 b Fu(2)34 b(S)7 b Fx(,)37 b(recalling)g(the)f(abuse)h(of)e(notation)h (men)m(tioned)i(at)d(the)h(end)h(of)e(Section)i(2.2,)0 3291 y(w)m(e)d(ha)m(v)m(e)372 3520 y Fv(Z)439 3535 y Fp(\003)492 3520 y Fx(\()p Fv(\034)11 b Fx(\))83 b(:=)1003 3425 y Ft(X)890 3637 y Fs(\021)r Fr(2S)5 b Fp(:)12 b Fs(\021)1089 3650 y Fo(\003)1130 3636 y(c)1166 3637 y Fp(=)p Fs(\034)1277 3520 y Fv(e)1322 3479 y Fs(H)1380 3490 y Fo(\003)1425 3479 y Fp(\()p Fs(\021)r Fp(\))1550 3520 y Fx(=)1786 3425 y Ft(X)1663 3629 y Fn(\021)1697 3615 y Fk({)1745 3629 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5196 y Fs(\031)r Fr(2)p Fp(\005)1669 5205 y Fo(0)1703 5196 y Fp(\()p Fs(k)r Fp(\))1866 5005 y Fs(k)1813 5035 y Ft(X)1829 5247 y Fs(l)q Fp(=1)1974 5129 y Fx(d)2028 5144 y Fp(1)2084 5049 y Ft(\000)2129 5129 y Fv(x)2184 5145 y Fs(\031)r Fp(\()p Fs(l)q Fr(\000)p Fp(1\))2399 5129 y Fv(;)17 b(x)2498 5145 y Fs(\031)r Fp(\()p Fs(l)q Fp(\))2622 5049 y Ft(\001)3579 5129 y Fx(\(5.2\))0 5413 y(whic)m(h)30 b(is)f(pro)m(v)m(en)i(as)e(follo)m(ws.) 43 b(By)29 b(induction)g(on)g(the)g(n)m(um)m(b)s(er)h(of)f(edges)g(in)g Fv(T)2912 5428 y Fs(X)3009 5413 y Fx(it)f(is)h(easy)h(to)f(pro)m(v)m (e,)0 5557 y(see)h(Fig.)e(1,)i(that)e(there)i(exists)g(a)f(path)g(\()p Fv(`)1554 5572 y Fp(0)1593 5557 y Fv(;)17 b(:)g(:)g(:)f(;)h(`)1853 5572 y Fs(M)7 b Fr(\000)p Fp(1)2022 5557 y Fx(\),)29 b(with)h Fv(`)2376 5572 y Fs(m)2470 5557 y Fu(2)e Fv(E)2636 5572 y Fs(X)2732 5557 y Fx(for)g(all)h Fv(m)f Fx(=)g(0)p Fv(;)17 b(:)g(:)g(:)e(;)i(M)25 b Fu(\000)14 b Fx(1,)p Black 1841 5806 a(24)p Black eop end %%Page: 25 26 TeXDict begin 25 25 bop Black Black Black 1079 847 a Ff(s)1295 631 y(s)1295 1063 y(s)1511 415 y(s)1511 847 y(s)1727 631 y(s)1727 1063 y(s)2374 631 y(s)2590 415 y(s)2590 847 y(s)-1544 b Fe(\000)1162 764 y(\000)1212 714 y(\000)1079 930 y(@)1162 1013 y(@)1212 1063 y(@)1295 631 y(\000)1378 548 y(\000)1428 498 y(\000)1295 714 y(@)1378 797 y(@)1428 847 y(@)1511 930 y(@)1594 1013 y(@)1644 1063 y(@)1511 498 y(@)1594 581 y(@)1644 631 y(@)647 b(\000)2457 548 y(\000)2507 498 y(\000)2374 714 y(@)2457 797 y(@)2507 847 y(@)p 1079 848 4 4 v 1079 850 V 1079 852 V 1079 853 V 1079 855 V 1079 857 V 1079 858 V 1079 860 V 1079 862 V 1080 863 V 1080 865 V 1080 866 V 1080 868 V 1080 870 V 1080 871 V 1080 873 V 1080 874 V 1080 876 V 1080 877 V 1080 879 V 1081 880 V 1081 882 V 1081 883 V 1081 885 V 1081 886 V 1081 888 V 1081 889 V 1082 891 V 1082 892 V 1082 894 V 1082 895 V 1082 897 V 1083 898 V 1083 900 V 1083 901 V 1083 903 V 1083 904 V 1084 906 V 1084 907 V 1084 908 V 1084 910 V 1085 911 V 1085 913 V 1085 914 V 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1006 V 1291 1007 V 1291 1009 V 1291 1010 V 1291 1012 V 1292 1013 V 1292 1014 V 1292 1016 V 1292 1017 V 1292 1019 V 1293 1020 V 1293 1022 V 1293 1023 V 1293 1025 V 1293 1026 V 1293 1028 V 1294 1029 V 1294 1031 V 1294 1032 V 1294 1034 V 1294 1035 V 1294 1037 V 1294 1039 V 1294 1040 V 1295 1042 V 1295 1043 V 1295 1045 V 1295 1046 V 1295 1048 V 1295 1050 V 1295 1051 V 1295 1053 V 1295 1054 V 1295 1056 V 1295 1058 V 1295 1059 V 1295 1061 V 1295 1063 V 1295 1064 V 1230 973 a(I)p 1079 848 V 1079 847 V 1079 845 V 1079 844 V 1079 842 V 1079 840 V 1079 839 V 1079 837 V 1079 835 V 1080 834 V 1080 832 V 1080 831 V 1080 829 V 1080 827 V 1080 826 V 1080 824 V 1080 823 V 1080 821 V 1080 820 V 1080 818 V 1081 817 V 1081 815 V 1081 814 V 1081 812 V 1081 810 V 1081 809 V 1081 807 V 1082 806 V 1082 804 V 1082 803 V 1082 802 V 1082 800 V 1083 799 V 1083 797 V 1083 796 V 1083 794 V 1083 793 V 1084 791 V 1084 790 V 1084 789 V 1084 787 V 1085 786 V 1085 784 V 1085 783 V 1085 782 V 1086 780 V 1086 779 V 1086 778 V 1087 776 V 1087 775 V 1087 773 V 1088 772 V 1088 771 V 1088 769 V 1089 768 V 1089 767 V 1089 766 V 1090 764 V 1090 763 V 1090 762 V 1091 760 V 1091 759 V 1092 758 V 1092 757 V 1092 755 V 1093 754 V 1093 753 V 1094 752 V 1094 750 V 1094 749 V 1095 748 V 1095 747 V 1096 745 V 1096 744 V 1097 743 V 1097 742 V 1098 741 V 1098 740 V 1099 738 V 1099 737 V 1100 736 V 1100 735 V 1101 734 V 1101 733 V 1102 732 V 1102 730 V 1103 729 V 1103 728 V 1104 727 V 1105 726 V 1105 725 V 1106 724 V 1106 723 V 1107 722 V 1107 721 V 1108 720 V 1109 719 V 1109 717 V 1110 716 V 1111 715 V 1111 714 V 1112 713 V 1112 712 V 1113 711 V 1114 710 V 1114 709 V 1115 708 V 1116 707 V 1117 706 V 1117 705 V 1118 704 V 1119 704 V 1119 703 V 1120 702 V 1121 701 V 1122 700 V 1122 699 V 1123 698 V 1124 697 V 1124 696 V 1125 695 V 1126 694 V 1127 693 V 1128 693 V 1128 692 V 1129 691 V 1130 690 V 1131 689 V 1132 688 V 1132 687 V 1133 687 V 1134 686 V 1135 685 V 1136 684 V 1137 683 V 1137 683 V 1138 682 V 1139 681 V 1140 680 V 1141 679 V 1142 679 V 1143 678 V 1144 677 V 1145 676 V 1145 676 V 1146 675 V 1147 674 V 1148 673 V 1149 673 V 1150 672 V 1151 671 V 1152 671 V 1153 670 V 1154 669 V 1155 669 V 1156 668 V 1157 667 V 1158 667 V 1159 666 V 1160 665 V 1161 665 V 1162 664 V 1163 663 V 1164 663 V 1165 662 V 1166 661 V 1167 661 V 1168 660 V 1169 660 V 1170 659 V 1172 658 V 1173 658 V 1174 657 V 1175 657 V 1176 656 V 1177 656 V 1178 655 V 1179 655 V 1180 654 V 1182 654 V 1183 653 V 1184 653 V 1185 652 V 1186 652 V 1187 651 V 1189 651 V 1190 650 V 1191 650 V 1192 649 V 1193 649 V 1195 648 V 1196 648 V 1197 647 V 1198 647 V 1199 647 V 1201 646 V 1202 646 V 1203 645 V 1204 645 V 1206 645 V 1207 644 V 1208 644 V 1210 643 V 1211 643 V 1212 643 V 1213 642 V 1215 642 V 1216 642 V 1217 641 V 1219 641 V 1220 641 V 1221 640 V 1223 640 V 1224 640 V 1225 639 V 1227 639 V 1228 639 V 1230 639 V 1231 638 V 1232 638 V 1234 638 V 1235 637 V 1237 637 V 1238 637 V 1239 637 V 1241 637 V 1242 636 V 1244 636 V 1245 636 V 1247 636 V 1248 635 V 1250 635 V 1251 635 V 1253 635 V 1254 635 V 1256 635 V 1257 634 V 1259 634 V 1260 634 V 1262 634 V 1263 634 V 1265 634 V 1266 634 V 1268 634 V 1269 633 V 1271 633 V 1273 633 V 1274 633 V 1276 633 V 1277 633 V 1279 633 V 1280 633 V 1282 633 V 1284 633 V 1285 633 V 1287 633 V 1289 633 V 1290 633 V 1292 633 V 1293 633 V 1295 633 V 1061 757 a(\022)p 1295 633 V 1295 634 V 1295 636 V 1295 638 V 1295 639 V 1295 641 V 1295 642 V 1295 644 V 1295 646 V 1295 647 V 1295 649 V 1296 651 V 1296 652 V 1296 654 V 1296 655 V 1296 657 V 1296 658 V 1296 660 V 1296 661 V 1296 663 V 1296 665 V 1297 666 V 1297 668 V 1297 669 V 1297 671 V 1297 672 V 1297 674 V 1297 675 V 1298 677 V 1298 678 V 1298 680 V 1298 681 V 1298 682 V 1299 684 V 1299 685 V 1299 687 V 1299 688 V 1300 690 V 1300 691 V 1300 693 V 1300 694 V 1301 695 V 1301 697 V 1301 698 V 1301 700 V 1302 701 V 1302 702 V 1302 704 V 1302 705 V 1303 706 V 1303 708 V 1303 709 V 1304 710 V 1304 712 V 1304 713 V 1305 714 V 1305 716 V 1306 717 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V 1366 809 V 1367 810 V 1368 811 V 1369 811 V 1370 812 V 1371 813 V 1372 813 V 1373 814 V 1374 815 V 1375 815 V 1376 816 V 1377 817 V 1378 817 V 1379 818 V 1380 818 V 1381 819 V 1382 820 V 1383 820 V 1384 821 V 1385 821 V 1386 822 V 1387 823 V 1388 823 V 1390 824 V 1391 824 V 1392 825 V 1393 825 V 1394 826 V 1395 826 V 1396 827 V 1397 828 V 1399 828 V 1400 829 V 1401 829 V 1402 830 V 1403 830 V 1404 831 V 1406 831 V 1407 831 V 1408 832 V 1409 832 V 1410 833 V 1412 833 V 1413 834 V 1414 834 V 1415 835 V 1417 835 V 1418 835 V 1419 836 V 1420 836 V 1422 837 V 1423 837 V 1424 837 V 1425 838 V 1427 838 V 1428 838 V 1429 839 V 1431 839 V 1432 840 V 1433 840 V 1435 840 V 1436 841 V 1437 841 V 1439 841 V 1440 841 V 1441 842 V 1443 842 V 1444 842 V 1445 843 V 1447 843 V 1448 843 V 1450 843 V 1451 844 V 1452 844 V 1454 844 V 1455 844 V 1457 845 V 1458 845 V 1460 845 V 1461 845 V 1463 845 V 1464 846 V 1466 846 V 1467 846 V 1468 846 V 1470 846 V 1471 846 V 1473 847 V 1474 847 V 1476 847 V 1478 847 V 1479 847 V 1481 847 V 1482 847 V 1484 848 V 1485 848 V 1487 848 V 1488 848 V 1490 848 V 1492 848 V 1493 848 V 1495 848 V 1496 848 V 1498 848 V 1499 848 V 1501 848 V 1503 848 V 1504 848 V 1506 848 V 1508 848 V 1509 848 V 1511 848 V 1277 804 a(R)p 1511 848 V 1511 850 V 1511 852 V 1511 853 V 1511 855 V 1511 857 V 1511 858 V 1511 860 V 1511 862 V 1511 863 V 1511 865 V 1511 866 V 1511 868 V 1512 870 V 1512 871 V 1512 873 V 1512 874 V 1512 876 V 1512 877 V 1512 879 V 1512 880 V 1512 882 V 1513 883 V 1513 885 V 1513 886 V 1513 888 V 1513 889 V 1513 891 V 1514 892 V 1514 894 V 1514 895 V 1514 897 V 1514 898 V 1514 900 V 1515 901 V 1515 903 V 1515 904 V 1515 906 V 1516 907 V 1516 908 V 1516 910 V 1516 911 V 1517 913 V 1517 914 V 1517 915 V 1517 917 V 1518 918 V 1518 919 V 1518 921 V 1519 922 V 1519 924 V 1519 925 V 1520 926 V 1520 928 V 1520 929 V 1521 930 V 1521 931 V 1521 933 V 1522 934 V 1522 935 V 1522 937 V 1523 938 V 1523 939 V 1524 940 V 1524 942 V 1524 943 V 1525 944 V 1525 945 V 1526 947 V 1526 948 V 1527 949 V 1527 950 V 1528 951 V 1528 953 V 1528 954 V 1529 955 V 1529 956 V 1530 957 V 1530 959 V 1531 960 V 1531 961 V 1532 962 V 1532 963 V 1533 964 V 1534 965 V 1534 967 V 1535 968 V 1535 969 V 1536 970 V 1536 971 V 1537 972 V 1537 973 V 1538 974 V 1539 975 V 1539 976 V 1540 977 V 1540 978 V 1541 980 V 1542 981 V 1542 982 V 1543 983 V 1544 984 V 1544 985 V 1545 986 V 1546 987 V 1546 988 V 1547 989 V 1548 990 V 1548 991 V 1549 992 V 1550 992 V 1550 993 V 1551 994 V 1552 995 V 1552 996 V 1553 997 V 1554 998 V 1555 999 V 1555 1000 V 1556 1001 V 1557 1002 V 1558 1003 V 1559 1004 V 1559 1004 V 1560 1005 V 1561 1006 V 1562 1007 V 1562 1008 V 1563 1009 V 1564 1010 V 1565 1010 V 1566 1011 V 1567 1012 V 1567 1013 V 1568 1014 V 1569 1014 V 1570 1015 V 1571 1016 V 1572 1017 V 1573 1018 V 1574 1018 V 1574 1019 V 1575 1020 V 1576 1021 V 1577 1021 V 1578 1022 V 1579 1023 V 1580 1024 V 1581 1024 V 1582 1025 V 1583 1026 V 1584 1026 V 1585 1027 V 1586 1028 V 1587 1028 V 1588 1029 V 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1537 849 V 1535 849 V 1534 849 V 1532 849 V 1531 849 V 1529 849 V 1527 849 V 1526 849 V 1524 849 V 1523 849 V 1521 849 V 1519 849 V 1518 849 V 1516 849 V 1514 848 V 1513 848 V 1511 848 V 1662 973 a(I)p 1511 848 V 1511 847 V 1511 845 V 1511 844 V 1511 842 V 1511 840 V 1511 839 V 1511 837 V 1511 835 V 1511 834 V 1511 832 V 1511 831 V 1511 829 V 1510 827 V 1510 826 V 1510 824 V 1510 823 V 1510 821 V 1510 820 V 1510 818 V 1510 817 V 1510 815 V 1509 814 V 1509 812 V 1509 810 V 1509 809 V 1509 807 V 1509 806 V 1509 804 V 1508 803 V 1508 802 V 1508 800 V 1508 799 V 1508 797 V 1507 796 V 1507 794 V 1507 793 V 1507 791 V 1506 790 V 1506 789 V 1506 787 V 1506 786 V 1505 784 V 1505 783 V 1505 782 V 1505 780 V 1504 779 V 1504 778 V 1504 776 V 1503 775 V 1503 773 V 1503 772 V 1502 771 V 1502 769 V 1502 768 V 1501 767 V 1501 766 V 1501 764 V 1500 763 V 1500 762 V 1500 760 V 1499 759 V 1499 758 V 1498 757 V 1498 755 V 1498 754 V 1497 753 V 1497 752 V 1496 750 V 1496 749 V 1495 748 V 1495 747 V 1494 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633 V 1303 633 V 1302 633 V 1300 633 V 1298 633 V 1297 633 V 1295 633 V 1446 757 a(I)p 1295 633 V 1295 631 V 1295 629 V 1295 628 V 1295 626 V 1295 624 V 1295 623 V 1295 621 V 1295 620 V 1295 618 V 1295 616 V 1296 615 V 1296 613 V 1296 612 V 1296 610 V 1296 608 V 1296 607 V 1296 605 V 1296 604 V 1296 602 V 1296 601 V 1297 599 V 1297 598 V 1297 596 V 1297 595 V 1297 593 V 1297 592 V 1297 590 V 1298 589 V 1298 587 V 1298 586 V 1298 584 V 1298 583 V 1299 581 V 1299 580 V 1299 578 V 1299 577 V 1300 576 V 1300 574 V 1300 573 V 1300 571 V 1301 570 V 1301 569 V 1301 567 V 1301 566 V 1302 564 V 1302 563 V 1302 562 V 1302 560 V 1303 559 V 1303 558 V 1303 556 V 1304 555 V 1304 554 V 1304 552 V 1305 551 V 1305 550 V 1306 548 V 1306 547 V 1306 546 V 1307 544 V 1307 543 V 1307 542 V 1308 541 V 1308 539 V 1309 538 V 1309 537 V 1309 536 V 1310 534 V 1310 533 V 1311 532 V 1311 531 V 1312 530 V 1312 528 V 1313 527 V 1313 526 V 1314 525 V 1314 524 V 1315 523 V 1315 521 V 1316 520 V 1316 519 V 1317 518 V 1317 517 V 1318 516 V 1318 515 V 1319 513 V 1319 512 V 1320 511 V 1320 510 V 1321 509 V 1322 508 V 1322 507 V 1323 506 V 1323 505 V 1324 504 V 1325 503 V 1325 502 V 1326 501 V 1326 500 V 1327 499 V 1328 498 V 1328 496 V 1329 495 V 1330 494 V 1330 493 V 1331 493 V 1332 492 V 1332 491 V 1333 490 V 1334 489 V 1334 488 V 1335 487 V 1336 486 V 1337 485 V 1337 484 V 1338 483 V 1339 482 V 1340 481 V 1340 480 V 1341 479 V 1342 478 V 1343 478 V 1343 477 V 1344 476 V 1345 475 V 1346 474 V 1347 473 V 1347 472 V 1348 472 V 1349 471 V 1350 470 V 1351 469 V 1352 468 V 1352 467 V 1353 467 V 1354 466 V 1355 465 V 1356 464 V 1357 464 V 1358 463 V 1359 462 V 1360 461 V 1360 460 V 1361 460 V 1362 459 V 1363 458 V 1364 458 V 1365 457 V 1366 456 V 1367 455 V 1368 455 V 1369 454 V 1370 453 V 1371 453 V 1372 452 V 1373 451 V 1374 451 V 1375 450 V 1376 449 V 1377 449 V 1378 448 V 1379 447 V 1380 447 V 1381 446 V 1382 446 V 1383 445 V 1384 444 V 1385 444 V 1386 443 V 1387 443 V 1388 442 V 1390 441 V 1391 441 V 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1514 417 V 1516 417 V 1518 417 V 1519 417 V 1521 417 V 1522 417 V 1524 417 V 1526 417 V 1527 417 V 1529 417 V 1530 417 V 1532 417 V 1534 417 V 1535 417 V 1537 418 V 1538 418 V 1540 418 V 1541 418 V 1543 418 V 1544 418 V 1546 418 V 1548 418 V 1549 419 V 1551 419 V 1552 419 V 1554 419 V 1555 419 V 1556 419 V 1558 420 V 1559 420 V 1561 420 V 1562 420 V 1564 420 V 1565 421 V 1567 421 V 1568 421 V 1569 421 V 1571 422 V 1572 422 V 1574 422 V 1575 422 V 1576 423 V 1578 423 V 1579 423 V 1581 424 V 1582 424 V 1583 424 V 1585 424 V 1586 425 V 1587 425 V 1589 425 V 1590 426 V 1591 426 V 1593 426 V 1594 427 V 1595 427 V 1597 428 V 1598 428 V 1599 428 V 1600 429 V 1602 429 V 1603 429 V 1604 430 V 1605 430 V 1607 431 V 1608 431 V 1609 432 V 1610 432 V 1612 432 V 1613 433 V 1614 433 V 1615 434 V 1616 434 V 1618 435 V 1619 435 V 1620 436 V 1621 436 V 1622 437 V 1623 437 V 1625 438 V 1626 438 V 1627 439 V 1628 439 V 1629 440 V 1630 440 V 1631 441 V 1632 441 V 1633 442 V 1635 443 V 1636 443 V 1637 444 V 1638 444 V 1639 445 V 1640 446 V 1641 446 V 1642 447 V 1643 447 V 1644 448 V 1645 449 V 1646 449 V 1647 450 V 1648 451 V 1649 451 V 1650 452 V 1651 453 V 1652 453 V 1653 454 V 1654 455 V 1655 455 V 1656 456 V 1657 457 V 1658 458 V 1659 458 V 1660 459 V 1661 460 V 1662 460 V 1662 461 V 1663 462 V 1664 463 V 1665 464 V 1666 464 V 1667 465 V 1668 466 V 1669 467 V 1670 467 V 1670 468 V 1671 469 V 1672 470 V 1673 471 V 1674 472 V 1675 472 V 1675 473 V 1676 474 V 1677 475 V 1678 476 V 1679 477 V 1679 478 V 1680 478 V 1681 479 V 1682 480 V 1682 481 V 1683 482 V 1684 483 V 1685 484 V 1685 485 V 1686 486 V 1687 487 V 1688 488 V 1688 489 V 1689 490 V 1690 491 V 1690 492 V 1691 492 V 1692 493 V 1692 494 V 1693 495 V 1694 496 V 1694 497 V 1695 498 V 1696 500 V 1696 501 V 1697 502 V 1697 503 V 1698 504 V 1699 505 V 1699 506 V 1700 507 V 1700 508 V 1701 509 V 1702 510 V 1702 511 V 1703 512 V 1703 513 V 1704 515 V 1704 516 V 1705 517 V 1705 518 V 1706 519 V 1706 520 V 1707 521 V 1707 523 V 1708 524 V 1708 525 V 1709 526 V 1709 527 V 1710 528 V 1710 530 V 1711 531 V 1711 532 V 1712 533 V 1712 534 V 1713 536 V 1713 537 V 1713 538 V 1714 539 V 1714 541 V 1715 542 V 1715 543 V 1715 544 V 1716 546 V 1716 547 V 1716 548 V 1717 550 V 1717 551 V 1718 552 V 1718 554 V 1718 555 V 1719 556 V 1719 558 V 1719 559 V 1720 560 V 1720 562 V 1720 563 V 1720 564 V 1721 566 V 1721 567 V 1721 568 V 1721 570 V 1722 571 V 1722 573 V 1722 574 V 1722 576 V 1723 577 V 1723 578 V 1723 580 V 1723 581 V 1724 583 V 1724 584 V 1724 586 V 1724 587 V 1724 589 V 1725 590 V 1725 592 V 1725 593 V 1725 595 V 1725 596 V 1725 598 V 1725 599 V 1726 601 V 1726 602 V 1726 604 V 1726 605 V 1726 607 V 1726 608 V 1726 610 V 1726 612 V 1726 613 V 1726 615 V 1727 616 V 1727 618 V 1727 620 V 1727 621 V 1727 623 V 1727 624 V 1727 626 V 1727 628 V 1727 629 V 1727 631 V 1727 633 V 1601 480 a(R)p 1727 633 V 1728 612 V 1731 594 V 1737 578 V 1744 564 V 1754 552 V 1766 542 V 1780 534 V 1796 529 V 1814 526 V 1835 525 V 1835 633 V 1856 633 V 1878 633 V 1900 633 V 1921 633 V 1943 633 V 1964 633 V 1986 633 V 2007 633 V 2029 633 V 2051 633 V 2072 633 V 2094 633 V 2115 633 V 2137 633 V 2159 633 V 2180 633 V 2202 633 V 2223 633 V 2245 633 V 2267 633 V 2267 525 V 2287 526 V 2305 529 V 2322 534 V 2336 542 V 2347 552 V 2357 564 V 2365 578 V 2370 594 V 2373 612 V 2374 633 V 2374 633 V 2374 634 V 2374 636 V 2374 638 V 2374 639 V 2375 641 V 2375 642 V 2375 644 V 2375 646 V 2375 647 V 2375 649 V 2375 651 V 2375 652 V 2375 654 V 2375 655 V 2375 657 V 2375 658 V 2375 660 V 2375 661 V 2376 663 V 2376 665 V 2376 666 V 2376 668 V 2376 669 V 2376 671 V 2376 672 V 2377 674 V 2377 675 V 2377 677 V 2377 678 V 2377 680 V 2378 681 V 2378 682 V 2378 684 V 2378 685 V 2378 687 V 2379 688 V 2379 690 V 2379 691 V 2379 693 V 2380 694 V 2380 695 V 2380 697 V 2380 698 V 2381 700 V 2381 701 V 2381 702 V 2381 704 V 2382 705 V 2382 706 V 2382 708 V 2383 709 V 2383 710 V 2383 712 V 2384 713 V 2384 714 V 2384 716 V 2385 717 V 2385 718 V 2386 719 V 2386 721 V 2386 722 V 2387 723 V 2387 725 V 2388 726 V 2388 727 V 2388 728 V 2389 730 V 2389 731 V 2390 732 V 2390 733 V 2391 734 V 2391 736 V 2391 737 V 2392 738 V 2392 739 V 2393 740 V 2393 742 V 2394 743 V 2394 744 V 2395 745 V 2395 746 V 2396 747 V 2396 748 V 2397 750 V 2397 751 V 2398 752 V 2399 753 V 2399 754 V 2400 755 V 2400 756 V 2401 757 V 2401 758 V 2402 759 V 2403 760 V 2403 762 V 2404 763 V 2404 764 V 2405 765 V 2406 766 V 2406 767 V 2407 768 V 2408 769 V 2408 770 V 2409 771 V 2410 772 V 2410 773 V 2411 774 V 2412 775 V 2412 776 V 2413 777 V 2414 778 V 2414 779 V 2415 779 V 2416 780 V 2417 781 V 2417 782 V 2418 783 V 2419 784 V 2420 785 V 2420 786 V 2421 787 V 2422 788 V 2423 789 V 2424 789 V 2424 790 V 2425 791 V 2426 792 V 2427 793 V 2428 794 V 2428 795 V 2429 795 V 2430 796 V 2431 797 V 2432 798 V 2433 799 V 2433 799 V 2434 800 V 2435 801 V 2436 802 V 2437 802 V 2438 803 V 2439 804 V 2440 805 V 2441 806 V 2442 806 V 2442 807 V 2443 808 V 2444 808 V 2445 809 V 2446 810 V 2447 811 V 2448 811 V 2449 812 V 2450 813 V 2451 813 V 2452 814 V 2453 815 V 2454 815 V 2455 816 V 2456 817 V 2457 817 V 2458 818 V 2459 818 V 2460 819 V 2461 820 V 2462 820 V 2463 821 V 2465 821 V 2466 822 V 2467 823 V 2468 823 V 2469 824 V 2470 824 V 2471 825 V 2472 825 V 2473 826 V 2474 826 V 2476 827 V 2477 828 V 2478 828 V 2479 829 V 2480 829 V 2481 830 V 2483 830 V 2484 831 V 2485 831 V 2486 831 V 2487 832 V 2488 832 V 2490 833 V 2491 833 V 2492 834 V 2493 834 V 2495 835 V 2496 835 V 2497 835 V 2498 836 V 2500 836 V 2501 837 V 2502 837 V 2503 837 V 2505 838 V 2506 838 V 2507 838 V 2509 839 V 2510 839 V 2511 840 V 2513 840 V 2514 840 V 2515 841 V 2517 841 V 2518 841 V 2519 841 V 2521 842 V 2522 842 V 2523 842 V 2525 843 V 2526 843 V 2528 843 V 2529 843 V 2530 844 V 2532 844 V 2533 844 V 2535 844 V 2536 845 V 2538 845 V 2539 845 V 2540 845 V 2542 845 V 2543 846 V 2545 846 V 2546 846 V 2548 846 V 2549 846 V 2551 846 V 2552 847 V 2554 847 V 2555 847 V 2557 847 V 2558 847 V 2560 847 V 2561 847 V 2563 848 V 2565 848 V 2566 848 V 2568 848 V 2569 848 V 2571 848 V 2572 848 V 2574 848 V 2576 848 V 2577 848 V 2579 848 V 2580 848 V 2582 848 V 2584 848 V 2585 848 V 2587 848 V 2589 848 V 2590 848 V 2356 804 a(R)p 2590 848 V 2590 847 V 2590 845 V 2590 844 V 2590 842 V 2590 840 V 2590 839 V 2590 837 V 2590 835 V 2590 834 V 2590 832 V 2590 831 V 2590 829 V 2590 827 V 2590 826 V 2590 824 V 2589 823 V 2589 821 V 2589 820 V 2589 818 V 2589 817 V 2589 815 V 2589 814 V 2589 812 V 2588 810 V 2588 809 V 2588 807 V 2588 806 V 2588 804 V 2588 803 V 2587 802 V 2587 800 V 2587 799 V 2587 797 V 2587 796 V 2586 794 V 2586 793 V 2586 791 V 2586 790 V 2585 789 V 2585 787 V 2585 786 V 2585 784 V 2584 783 V 2584 782 V 2584 780 V 2584 779 V 2583 778 V 2583 776 V 2583 775 V 2582 773 V 2582 772 V 2582 771 V 2581 769 V 2581 768 V 2581 767 V 2580 766 V 2580 764 V 2580 763 V 2579 762 V 2579 760 V 2578 759 V 2578 758 V 2578 757 V 2577 755 V 2577 754 V 2576 753 V 2576 752 V 2576 750 V 2575 749 V 2575 748 V 2574 747 V 2574 745 V 2573 744 V 2573 743 V 2572 742 V 2572 741 V 2571 740 V 2571 738 V 2570 737 V 2570 736 V 2569 735 V 2569 734 V 2568 733 V 2568 732 V 2567 730 V 2567 729 V 2566 728 V 2566 727 V 2565 726 V 2564 725 V 2564 724 V 2563 723 V 2563 722 V 2562 721 V 2561 720 V 2561 719 V 2560 717 V 2560 716 V 2559 715 V 2558 714 V 2558 713 V 2557 712 V 2556 711 V 2556 710 V 2555 709 V 2554 708 V 2554 707 V 2553 706 V 2552 705 V 2552 704 V 2551 704 V 2550 703 V 2550 702 V 2549 701 V 2548 700 V 2547 699 V 2547 698 V 2546 697 V 2545 696 V 2544 695 V 2544 694 V 2543 693 V 2542 693 V 2541 692 V 2540 691 V 2540 690 V 2539 689 V 2538 688 V 2537 687 V 2536 687 V 2536 686 V 2535 685 V 2534 684 V 2533 683 V 2532 683 V 2531 682 V 2530 681 V 2529 680 V 2529 679 V 2528 679 V 2527 678 V 2526 677 V 2525 676 V 2524 676 V 2523 675 V 2522 674 V 2521 673 V 2520 673 V 2519 672 V 2518 671 V 2518 671 V 2517 670 V 2516 669 V 2515 669 V 2514 668 V 2513 667 V 2512 667 V 2511 666 V 2510 665 V 2509 665 V 2508 664 V 2506 663 V 2505 663 V 2504 662 V 2503 661 V 2502 661 V 2501 660 V 2500 660 V 2499 659 V 2498 658 V 2497 658 V 2496 657 V 2495 657 V 2494 656 V 2493 656 V 2491 655 V 2490 655 V 2489 654 V 2488 654 V 2487 653 V 2486 653 V 2485 652 V 2483 652 V 2482 651 V 2481 651 V 2480 650 V 2479 650 V 2477 649 V 2476 649 V 2475 648 V 2474 648 V 2473 647 V 2471 647 V 2470 647 V 2469 646 V 2468 646 V 2466 645 V 2465 645 V 2464 645 V 2463 644 V 2461 644 V 2460 643 V 2459 643 V 2457 643 V 2456 642 V 2455 642 V 2453 642 V 2452 641 V 2451 641 V 2450 641 V 2448 640 V 2447 640 V 2445 640 V 2444 639 V 2443 639 V 2441 639 V 2440 639 V 2439 638 V 2437 638 V 2436 638 V 2434 637 V 2433 637 V 2432 637 V 2430 637 V 2429 637 V 2427 636 V 2426 636 V 2424 636 V 2423 636 V 2421 635 V 2420 635 V 2418 635 V 2417 635 V 2415 635 V 2414 635 V 2412 634 V 2411 634 V 2409 634 V 2408 634 V 2406 634 V 2405 634 V 2403 634 V 2402 634 V 2400 633 V 2399 633 V 2397 633 V 2396 633 V 2394 633 V 2392 633 V 2391 633 V 2389 633 V 2388 633 V 2386 633 V 2384 633 V 2383 633 V 2381 633 V 2379 633 V 2378 633 V 2376 633 V 2374 633 V 2526 757 a(I)p 2374 633 V 2374 631 V 2374 629 V 2374 628 V 2374 626 V 2375 624 V 2375 623 V 2375 621 V 2375 620 V 2375 618 V 2375 616 V 2375 615 V 2375 613 V 2375 612 V 2375 610 V 2375 608 V 2375 607 V 2375 605 V 2375 604 V 2376 602 V 2376 601 V 2376 599 V 2376 598 V 2376 596 V 2376 595 V 2376 593 V 2377 592 V 2377 590 V 2377 589 V 2377 587 V 2377 586 V 2378 584 V 2378 583 V 2378 581 V 2378 580 V 2378 578 V 2379 577 V 2379 576 V 2379 574 V 2379 573 V 2380 571 V 2380 570 V 2380 569 V 2380 567 V 2381 566 V 2381 564 V 2381 563 V 2381 562 V 2382 560 V 2382 559 V 2382 558 V 2383 556 V 2383 555 V 2383 554 V 2384 552 V 2384 551 V 2384 550 V 2385 548 V 2385 547 V 2386 546 V 2386 544 V 2386 543 V 2387 542 V 2387 541 V 2388 539 V 2388 538 V 2388 537 V 2389 536 V 2389 534 V 2390 533 V 2390 532 V 2391 531 V 2391 530 V 2391 528 V 2392 527 V 2392 526 V 2393 525 V 2393 524 V 2394 523 V 2394 521 V 2395 520 V 2395 519 V 2396 518 V 2396 517 V 2397 516 V 2397 515 V 2398 513 V 2399 512 V 2399 511 V 2400 510 V 2400 509 V 2401 508 V 2401 507 V 2402 506 V 2403 505 V 2403 504 V 2404 503 V 2404 502 V 2405 501 V 2406 500 V 2406 499 V 2407 498 V 2408 496 V 2408 495 V 2409 494 V 2410 493 V 2410 493 V 2411 492 V 2412 491 V 2412 490 V 2413 489 V 2414 488 V 2414 487 V 2415 486 V 2416 485 V 2417 484 V 2417 483 V 2418 482 V 2419 481 V 2420 480 V 2420 479 V 2421 478 V 2422 478 V 2423 477 V 2424 476 V 2424 475 V 2425 474 V 2426 473 V 2427 472 V 2428 472 V 2428 471 V 2429 470 V 2430 469 V 2431 468 V 2432 467 V 2433 467 V 2433 466 V 2434 465 V 2435 464 V 2436 464 V 2437 463 V 2438 462 V 2439 461 V 2440 460 V 2441 460 V 2442 459 V 2442 458 V 2443 458 V 2444 457 V 2445 456 V 2446 455 V 2447 455 V 2448 454 V 2449 453 V 2450 453 V 2451 452 V 2452 451 V 2453 451 V 2454 450 V 2455 449 V 2456 449 V 2457 448 V 2458 447 V 2459 447 V 2460 446 V 2461 446 V 2462 445 V 2463 444 V 2465 444 V 2466 443 V 2467 443 V 2468 442 V 2469 441 V 2470 441 V 2471 440 V 2472 440 V 2473 439 V 2474 439 V 2476 438 V 2477 438 V 2478 437 V 2479 437 V 2480 436 V 2481 436 V 2483 435 V 2484 435 V 2485 434 V 2486 434 V 2487 433 V 2488 433 V 2490 432 V 2491 432 V 2492 432 V 2493 431 V 2495 431 V 2496 430 V 2497 430 V 2498 429 V 2500 429 V 2501 429 V 2502 428 V 2503 428 V 2505 428 V 2506 427 V 2507 427 V 2509 426 V 2510 426 V 2511 426 V 2513 425 V 2514 425 V 2515 425 V 2517 424 V 2518 424 V 2519 424 V 2521 424 V 2522 423 V 2523 423 V 2525 423 V 2526 422 V 2528 422 V 2529 422 V 2530 422 V 2532 421 V 2533 421 V 2535 421 V 2536 421 V 2538 420 V 2539 420 V 2540 420 V 2542 420 V 2543 420 V 2545 419 V 2546 419 V 2548 419 V 2549 419 V 2551 419 V 2552 419 V 2554 418 V 2555 418 V 2557 418 V 2558 418 V 2560 418 V 2561 418 V 2563 418 V 2565 418 V 2566 417 V 2568 417 V 2569 417 V 2571 417 V 2572 417 V 2574 417 V 2576 417 V 2577 417 V 2579 417 V 2580 417 V 2582 417 V 2584 417 V 2585 417 V 2587 417 V 2589 417 V 2590 417 V 2356 541 a(\022)950 847 y Fv(x)1005 862 y Fp(0)1058 1052 y Fv(`)1099 1067 y Fp(0)1295 933 y Fv(`)1336 948 y Fp(1)1058 663 y Fv(`)1099 678 y Fp(2)1274 836 y Fv(`)1315 851 y Fp(3)1489 1052 y Fv(`)1530 1067 y Fp(4)1727 933 y Fv(`)1768 948 y Fp(5)1511 717 y Fv(`)1552 732 y Fp(6)1274 447 y Fv(`)1315 462 y Fp(7)1727 501 y Fv(`)1768 516 y Fp(8)1856 566 y Fv(`)1897 581 y Fp(9)2105 566 y Fv(`)2146 581 y Fs(M)7 b Fr(\000)p Fp(4)2353 890 y Fv(`)2394 905 y Fs(M)g Fr(\000)p Fp(3)2590 717 y Fv(`)2631 732 y Fs(M)g Fr(\000)p Fp(2)2342 394 y Fv(`)2383 409 y Fs(M)g Fr(\000)p Fp(1)0 1475 y Fx(Figure)37 b(1:)53 b(The)39 b(path)e Fv(`)f Fx(=)g Fu(f)p Fv(`)1159 1490 y Fp(0)1198 1475 y Fv(;)17 b(:)g(:)g(:)f(;)h(`)1458 1490 y Fs(M)7 b Fr(\000)p Fp(1)1627 1475 y Fu(g)37 b Fx(in)m(tro)s(duced)i(in)f(the)g(pro)s(of)e(of)h (Lemma)h(5.1.The)g(solid)0 1620 y(circles)c(represen)m(t)h(the)e(p)s (oin)m(ts)g Fu(f)p Fv(x)1280 1635 y Fp(0)1320 1620 y Fv(;)17 b(x)1419 1635 y Fp(1)1458 1620 y Fv(;)g(:)g(:)g(:)f(;)h(x)1732 1635 y Fs(k)1775 1620 y Fu(g)p Fx(.)p Black 0 1886 a(satisfying)36 b(the)g(follo)m(wing)g(prop)s(erties:)49 b Fv(`)1565 1901 y Fs(m)p Fr(\000)p Fp(1)1746 1886 y Fu(\\)24 b Fv(`)1877 1901 y Fs(m)1976 1886 y Fu(6)p Fx(=)32 b Fu(;)j Fx(for)g(all)g Fv(m)e Fx(=)f(1)p Fv(;)17 b(:)g(:)g(:)f(;)h(M)34 b Fu(\000)25 b Fx(1,)35 b Fv(x)3349 1901 y Fp(0)3421 1886 y Fu(2)e Fv(`)3561 1901 y Fp(0)3600 1886 y Fx(,)j(for)0 2031 y(eac)m(h)c Fv(v)g Fu(2)c Fv(V)448 2046 y Fs(X)546 2031 y Fx(there)33 b(exists)g Fv(m)28 b Fu(2)g(f)p Fx(0)p Fv(;)17 b(:)g(:)g(:)e(;)i(M)31 b Fu(\000)20 b Fx(1)p Fu(g)31 b Fx(suc)m(h)i(that)e Fv(`)2409 2046 y Fs(m)2503 2031 y Fu(3)d Fv(v)t Fx(,)k(and)f(eac)m(h)i Fv(e)28 b Fu(2)g Fv(E)3353 2046 y Fs(X)3452 2031 y Fx(app)s(ears)0 2175 y(in)33 b(the)g(path)f(at)h(most)g(t)m(wice.)45 b(The)33 b(b)s(ound)g(\(5.2\))f(then)h(follo)m(ws.)p 3709 2175 4 66 v 3713 2113 59 4 v 3713 2175 V 3771 2175 4 66 v 146 2320 a(W)-8 b(e)31 b(giv)m(e,)i(no)m(w,)f(a)e(recursiv)m(e)j (de\014nition)f(that)f(will)g(b)s(e)g(used)h(to)e(parameterize)i(the)g (exp)s(onen)m(tial)0 2464 y(deca)m(y)i(of)e(the)h(p)s(oten)m(tial)g(at) f(di\013eren)m(t)i(scales.)45 b(Recall)33 b(de\014nitions)h(\(4.17\))e (and)g(\(2.3\),)h(set)73 2656 y Fu(T)127 2671 y Fp(0)166 2656 y Fx(\()p Fv(G;)17 b(s)p Fx(\))28 b(:=)101 b(inf)567 2723 y Fs(x)p Fr(2)p Fs(y)689 2731 y Fn(s)722 2723 y Fp(\()p Fs(G)p Fp(\))932 2656 y Fx(inf)879 2702 y Fn(x)914 2710 y(m)972 2702 y Fm(2)p Fn(g)1044 2710 y(m)859 2758 y(m)p Fo(=1)p Fn(;:::;n)1150 2656 y Fq(T)17 b Fx(\()o Fu(f)p Fv(x;)g(x)1474 2671 y Fp(1)1514 2656 y Fv(;)g(:)g(:)g(:)f(;)h(x) 1788 2671 y Fs(n)1835 2656 y Fu(g)p Fx(\))115 b(for)33 b Fv(G)27 b Fx(=)h Fu(f)p Fv(g)2493 2671 y Fp(1)2532 2656 y Fv(;)17 b(:)g(:)g(:)e(;)i(g)2797 2671 y Fs(n)2844 2656 y Fu(g)27 b(\032\032)i(G)3163 2671 y Fr(\025)p Fp(1)3290 2656 y Fx(and)k Fv(s)27 b Fu(\025)i Fx(0)73 3027 y Fu(T)127 3042 y Fs(j)163 3027 y Fx(\()p Fv(G;)17 b(s)p Fx(\))28 b(:=)146 b(inf)564 3094 y Fs(R)p 564 3106 54 3 v 1 w Fr(2R)p 665 3106 61 3 v 19 x Fn(j)758 3094 y Fp(\()p Fs(G;s)p Fp(\))969 2933 y Ft(X)964 3144 y Fs(R)p Fr(2)p Fs(R)p 1064 3156 54 3 v 1228 2933 a Ft(X)1162 3148 y Fp(\()p Fs(H)q(;u)p Fp(\))p Fr(2)p Fs(R)1471 3027 y Fu(T)1525 3042 y Fs(j)t Fr(\000)p Fp(1)1652 3027 y Fx(\()p Fv(H)r(;)17 b(u)p Fx(\))127 b(for)33 b Fv(j)g Fu(\025)28 b Fx(1)p Fv(;)33 b(G)28 b Fu(\032\032)h(G)2822 3042 y Fr(\025)p Fs(j)t Fp(+1)3004 3027 y Fv(;)49 b Fx(and)32 b Fv(s)c Fu(\025)g Fx(0)3579 3258 y(\(5.3\))0 3402 y(As)34 b(usual)h(if)e Fv(R)p 492 3418 76 4 v 567 3427 a Fs(j)604 3402 y Fx(\()p Fv(G;)17 b(s)p Fx(\))29 b(=)g Fu(;)34 b Fx(w)m(e)g(understand)i Fu(T)1775 3417 y Fs(j)1811 3402 y Fx(\()p Fv(G;)17 b(s)p Fx(\))29 b(=)h(+)p Fu(1)p Fx(.)46 b(Note)34 b(that)f Fu(T)2941 3417 y Fp(0)2981 3402 y Fx(\()p Fv(G;)17 b Fx(0\))29 b(=)g Fu(T)d Fx(\()p Fv(G)p Fx(\),)34 b(see)0 3547 y(\(3.6\).)43 b(Finally)33 b(for)f(eac)m(h)i Fv(j)g Fu(\025)28 b Fx(0,)33 b Fv(G)27 b Fu(\032\032)i(G)1606 3562 y Fr(\025)p Fs(j)t Fp(+1)1821 3547 y Fx(and)j Fv(s)c Fu(\025)g Fx(0)33 b(w)m(e)h(set)2582 3522 y Ft(b)2567 3547 y Fu(T)2621 3562 y Fs(j)2658 3547 y Fx(\()p Fv(G;)17 b(s)p Fx(\))27 b(:=)h(inf)3177 3562 y Fp(0)p Fr(\024)p Fs(k)r Fr(\024)p Fs(j)3414 3547 y Fu(T)3468 3562 y Fs(k)3510 3547 y Fx(\()p Fv(G;)17 b(s)p Fx(\).)0 3691 y(In)31 b(order)f(to)g (clarify)g(the)h(recursiv)m(e)i(de\014nition)e(\(5.3\))e(w)m(e)j (consider)f(in)g(some)g(detail)f(the)h(case)g Fv(j)j Fx(=)27 b(1,)0 3836 y Fv(G)h Fx(=)f Fu(f)p Fv(g)305 3851 y Fp(1)344 3836 y Fv(;)17 b(g)435 3851 y Fp(2)474 3836 y Fu(g)27 b(\032)i(G)716 3851 y Fp(2)755 3836 y Fx(,)i(and)f Fv(s)e Fx(=)f(0.)43 b(Let)30 b Fv(R)p 1468 3852 V 1543 3794 a Fr(\003)1610 3836 y Fu(2)e(R)p 1704 3852 85 4 v 1789 3860 a Fp(1)1828 3836 y Fx(\()p Fu(f)p Fv(g)1963 3851 y Fp(1)2002 3836 y Fv(;)17 b(g)2093 3851 y Fp(2)2132 3836 y Fu(g)p Fv(;)g Fx(0\))29 b(b)s(e)i(a)f(minimizer)h(for)f(the)h (righ)m(t{hand)0 3980 y(side)j(of)e(\(5.3\).)43 b(Then)1110 4125 y Fu(T)1164 4140 y Fp(1)1204 4125 y Fx(\()p Fu(f)p Fv(g)1339 4140 y Fp(1)1378 4125 y Fv(;)17 b(g)1469 4140 y Fp(2)1508 4125 y Fu(g)p Fv(;)g Fx(0\))27 b(=)1842 4030 y Ft(X)1819 4244 y Fs(R)p Fr(2)p Fs(R)p 1919 4256 54 3 v 1973 4219 a Fm(\003)2092 4030 y Ft(X)2026 4246 y Fp(\()p Fs(H)q(;u)p Fp(\))p Fr(2)p Fs(R)2318 4125 y Fu(T)2372 4140 y Fp(0)2411 4125 y Fx(\()p Fv(H)r(;)17 b(u)p Fx(\))0 4391 y(W)-8 b(e)37 b(note)g(that)g(a)f(p)s(olymer)i Fv(R)e Fu(2)f Fv(R)p 1289 4407 76 4 v 1364 4350 a Fr(\003)1440 4391 y Fx(is)j(built)f(of)f(b)s(onds)i(\()p Fv(H)r(;)17 b(u)p Fx(\))36 b(connecting)i(on)e(1{gen)m(tle)i(atoms.)0 4536 y(Therefore,)47 b Fu(T)536 4551 y Fp(1)576 4536 y Fx(\()p Fu(f)p Fv(g)711 4551 y Fp(1)750 4536 y Fv(;)17 b(g)841 4551 y Fp(2)880 4536 y Fu(g)p Fv(;)g Fx(0\))42 b(can)i(b)s(e)g(strictly)g(smaller)h(than)e(d)2421 4551 y Fp(1)2461 4536 y Fx(\()p Fv(g)2546 4551 y Fp(1)2585 4536 y Fv(;)17 b(g)2676 4551 y Fp(2)2715 4536 y Fx(\))43 b(due)h(to)f(the)h(presence)h(of)0 4680 y(1{gen)m(tle)f(atoms)g(b)s(et) m(w)m(een)i Fv(g)1127 4695 y Fp(1)1209 4680 y Fx(and)e Fv(g)1457 4695 y Fp(2)1496 4680 y Fx(.)77 b(Ho)m(w)m(ev)m(er,)49 b(b)m(y)44 b(the)g(sparseness)i(conditions)f(4)e(and)h(5)f(of)0 4825 y(De\014nition)33 b(2.4,)f(w)m(e)i(ha)m(v)m(e)696 5046 y Fu(T)750 5061 y Fp(1)790 5046 y Fx(\()p Fu(f)p Fv(g)925 5061 y Fp(1)964 5046 y Fv(;)17 b(g)1055 5061 y Fp(2)1093 5046 y Fu(g)p Fv(;)g Fx(0\))27 b Fu(\025)1527 4978 y Fv(\015)1578 4993 y Fp(1)p 1417 5023 311 4 v 1417 5114 a Fx(\000)1478 5129 y Fp(1)1539 5114 y Fx(+)22 b Fv(\015)1688 5129 y Fp(1)1754 5046 y Fx(d)1808 5061 y Fp(1)1847 5046 y Fx(\()p Fv(g)1932 5061 y Fp(1)1971 5046 y Fv(;)17 b(g)2062 5061 y Fp(2)2101 5046 y Fx(\))28 b Fu(\025)2272 4935 y Ft(\020)2331 5046 y Fx(1)22 b Fu(\000)2512 4978 y Fx(\000)2573 4993 y Fp(1)p 2512 5023 101 4 v 2517 5114 a Fv(\015)2568 5129 y Fp(1)2622 4935 y Ft(\021)2698 5046 y Fx(d)2752 5061 y Fp(1)2792 5046 y Fx(\()p Fv(g)2877 5061 y Fp(1)2916 5046 y Fv(;)17 b(g)3007 5061 y Fp(2)3046 5046 y Fx(\))0 5268 y(Indeed,)43 b(the)d(maxim)m(um)i(n)m(um)m(b)s(er)f (of)e(1{gen)m(tle)h(atoms)g(that)g(can)g(b)s(e)g(arranged)f(b)s(et)m(w) m(een)j Fv(g)3544 5283 y Fp(1)3623 5268 y Fx(and)0 5413 y Fv(g)47 5428 y Fp(2)121 5413 y Fx(is)35 b(d)275 5428 y Fp(1)315 5413 y Fx(\()p Fv(g)400 5428 y Fp(1)439 5413 y Fv(;)17 b(g)530 5428 y Fp(2)569 5413 y Fx(\))p Fv(=)p Fx(\(\000)755 5428 y Fp(1)817 5413 y Fx(+)24 b Fv(\015)968 5428 y Fp(1)1007 5413 y Fx(\).)50 b(The)35 b(follo)m(wing)g(prop)s (osition)g(states)h(a)e(similar)i(b)s(ound)f(for)f(a)h(general)0 5557 y(situation.)p Black 1841 5806 a(25)p Black eop end %%Page: 26 27 TeXDict begin 26 26 bop Black Black Black 0 407 a Fy(Prop)s(osition)38 b(5.2.)p Black 41 w Fw(L)-5 b(et)36 b Fv(j)d Fu(\025)28 b Fx(0)p Fw(,)35 b Fv(G)28 b Fu(\032\032)g(G)1628 422 y Fr(\025)p Fs(j)t Fp(+1)1810 407 y Fw(,)35 b(and)f Fv(s)28 b Fu(\025)g Fx(0)p Fw(.)44 b(Then)544 716 y Fu(T)598 731 y Fs(j)635 716 y Fx(\()p Fv(G;)17 b(s)p Fx(\))27 b Fu(\025)1010 605 y Ft(\020)1070 716 y Fx(1)22 b Fu(\000)1296 587 y Fs(j)1240 621 y Ft(X)1248 834 y Fs(k)r Fp(=0)1411 649 y Fx(\000)1472 664 y Fs(k)p 1411 693 104 4 v 1416 784 a Fv(\015)1467 799 y Fs(k)1524 605 y Ft(\021)q(n)1650 716 y Fu(T)k Fx(\()p Fv(G)p Fx(\))c(+)g(1)-28 b(I)2059 731 y Fs(s)p Fr(\025)p Fp(1)2186 635 y Ft(\002)2228 716 y Fx(d)2282 731 y Fp(1)2321 716 y Fx(\()p Fu(Q)p Fx(\()2497 691 y Ft(b)2478 716 y Fv(G)p Fx(\))p Fv(;)17 b(y)2685 731 y Fs(s)2721 716 y Fx(\()p Fv(G)p Fx(\)\))22 b Fu(\000)h Fv(#)3091 731 y Fs(j)3128 635 y Ft(\003)3169 605 y(o)3579 716 y Fx(\(5.4\))0 1017 y Fw(wher)-5 b(e)34 b(we)h(understand)f Fx(0)p Fv(=\015)1073 1032 y Fp(0)1139 1017 y Fx(=)28 b(0)34 b Fw(even)h(if)f Fv(\015)1696 1032 y Fp(0)1763 1017 y Fx(=)28 b(0)p Fw(.)146 1237 y Fx(W)-8 b(e)36 b(remark)g(that)f (from)g(the)h(b)s(ound)f(\(5.4\))g(ab)s(o)m(v)m(e,)i(item)f(3)f(in)g (De\014nition)h(2.2,)f(and)g(\(2.9\))g(it)g(is)0 1381 y(straigh)m(tforw)m(ard)e(to)f(deduce)i(that)1085 1606 y Ft(b)1070 1631 y Fu(T)1124 1646 y Fs(j)1160 1631 y Fx(\()p Fv(G;)17 b(s)p Fx(\))28 b Fu(\025)1546 1564 y Fx(1)p 1546 1608 49 4 v 1546 1700 a(2)1604 1631 y Fu(T)e Fx(\()p Fv(G)p Fx(\))c(+)1967 1564 y(1)p 1967 1608 V 1967 1700 a(4)2026 1631 y(d)2080 1646 y Fp(1)2119 1631 y Fx(\()p Fu(Q)p Fx(\()2296 1606 y Ft(b)2276 1631 y Fv(G)q Fx(\))p Fv(;)17 b(y)2484 1646 y Fs(s)2520 1631 y Fx(\()p Fv(G)p Fx(\)\))868 b(\(5.5\))146 1866 y(T)-8 b(o)30 b(pro)m(v)m(e)g (Prop)s(osition)g(5.2)f(one)g(of)g(the)h(ingredien)m(ts)h(is)f(a)f (lemma)h(ab)s(out)f(one{side)i(pro)5 b(jections)0 2011 y(of)45 b(graphs)h(to)g(h)m(yp)s(er{planes.)85 b(In)46 b(order)g(to)g(state)g(it)f(w)m(e)i(need)g(a)f(few)g(more)g (de\014nitions.)85 b(Let)1 2155 y Ft(b)-56 b Fv(n)38 b Fu(2)g(f)p Fv(e)295 2170 y Fs(i)323 2155 y Fv(;)17 b Fu(\000)p Fv(e)489 2170 y Fs(i)518 2155 y Fv(;)33 b(i)38 b Fx(=)g(1)p Fv(;)17 b(:)g(:)g(:)f(;)h(d)p Fu(g)37 b Fx(b)s(e)i(a)f(co)s(ordinate)h(direction)g(and)g Fv(c)f Fu(2)g Fq(N)g 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Fr(\024)2423 2589 y Fu(\033)c Fv(\031)t Fx(.)146 2733 y(Giv)m(en)i(a)f(connected)i(graph)e(\()p Fv(V)5 b(;)17 b(E)6 b Fx(\))28 b Fu(\032\032)h Fx(\()p Fq(L)p Fv(;)17 b Fq(E)p Fx(\),)30 b(recall)g(the)g(de\014nition)g(ab)s (o)m(v)m(e)g(\(2.3\),)f(w)m(e)i(de\014ne)0 2877 y Fv(V)57 2892 y Fs(\031)r(;)p Fr(\024)224 2877 y Fx(:=)46 b Fv(V)51 b Fu(\\)30 b Fq(L)643 2892 y Fs(\031)r(;)p Fr(\024)765 2877 y Fx(,)46 b Fv(V)895 2892 y Fs(\031)r(;>)1063 2877 y Fx(:=)g Fv(V)51 b Fu(\\)30 b Fq(L)1482 2892 y Fs(\031)r(;>)1604 2877 y Fx(,)46 b Fv(E)1749 2892 y Fs(\031)r(;)p Fr(\024)1917 2877 y Fx(:=)g Fu(f)p Fv(e)g Fu(2)g Fv(E)6 b(;)34 b(e)46 b Fu(\032)g Fq(L)2738 2892 y Fs(\031)r(;)p Fr(\024)2860 2877 y Fu(g)p Fx(,)g(and)e Fv(E)3256 2892 y Fs(\031)r(;>)3424 2877 y Fx(:=)i Fu(f)p Fv(e)g Fu(2)0 3022 y Fv(E)6 b(;)33 b(e)23 b Fu(\\)g Fq(L)361 3037 y Fs(\031)r(;>)511 3022 y Fu(6)p Fx(=)28 b Fu(;g)p Fx(.)44 b(W)-8 b(e)34 b(note)f(that)g Fv(V)50 b Fx(=)28 b Fv(V)1652 3037 y Fs(\031)r(;)p Fr(\024)1796 3022 y Fu([)22 b Fv(V)1941 3037 y Fs(\031)r(;>)2096 3022 y Fx(and)33 b Fv(E)h Fx(=)28 b Fv(E)2568 3037 y Fs(\031)r(;)p Fr(\024)2712 3022 y Fu([)23 b Fv(E)2873 3037 y Fs(\031)r(;>)2995 3022 y Fx(.)45 b(W)-8 b(e)33 b(\014nally)h(de\014ne)0 3166 y Fv(E)78 3130 y Fr(?)72 3191 y Fs(\031)r(;>)221 3166 y Fx(:=)352 3086 y Ft(\010)410 3166 y Fu(f)p Fv(x;)17 b(y)t Fu(g)27 b(\032)h Fv(\031)t(;)33 b Fu(9)p Fv(k)e Fu(\025)d Fx(1)33 b(suc)m(h)h(that)e Fu(f)p Fv(x)22 b Fx(+)g Fv(k)5 b Ft(b)-57 b Fv(n)q(;)17 b(y)25 b Fx(+)d Fv(k)5 b Ft(b)-57 b Fv(n)p Fu(g)28 b(2)g Fv(E)2576 3181 y Fs(\031)r(;>)2697 3086 y Ft(\011)2756 3166 y Fx(.)p Black 0 3386 a Fy(Lemma)52 b(5.3.)p Black 47 w Fw(L)-5 b(et)46 b Fx(\()p Fv(V)5 b(;)17 b(E)6 b Fx(\))46 b Fu(\032)h Fx(\()p Fq(L)p Fv(;)17 b Fq(E)p Fx(\))45 b Fw(b)-5 b(e)45 b(a)f(c)-5 b(onne)g(cte)g(d)44 b(gr)-5 b(aph,)49 b Ft(b)-57 b Fv(n)47 b Fu(2)f(f)p Fv(e)2832 3401 y Fs(i)2861 3386 y Fv(;)17 b Fu(\000)p Fv(e)3027 3401 y Fs(i)3055 3386 y Fv(;)34 b(i)46 b Fx(=)g(1)p Fv(;)17 b(:)g(:)g(:)f(;)h(d)p Fu(g)44 b Fw(a)0 3531 y(c)-5 b(o)g(or)g(dinate)48 b(dir)-5 b(e)g(ction,)51 b(and)e Fv(c)k Fu(2)h Fq(N)p Fw(;)i(c)-5 b(onsider)47 b(the)i(hyp)-5 b(er{plane)48 b Fv(\031)2689 3546 y Fd(b)-41 b Fs(n;c)2838 3531 y Fu(\021)54 b Fv(\031)j Fu(\032)d Fq(L)p Fw(.)87 b(With)49 b(the)0 3675 y(de\014nitions)34 b(given)g(ab)-5 b(ove,)34 b(if)g Fv(V)1175 3690 y Fs(\031)r(;)p Fr(\024)1324 3675 y Fu(6)p Fx(=)28 b Fu(;)35 b Fw(then)p Black 115 3895 a(1.)p Black 49 w(the)g(b)-5 b(ound)1255 4039 y Fu(j)p Fv(E)6 b Fu(j)27 b(\025)i(j)p Fv(E)1622 4054 y Fs(\031)r(;)p Fr(\024)1765 4039 y Fu([)23 b Fv(E)1932 3998 y Fr(?)1926 4064 y Fs(\031)r(;>)2047 4039 y Fu(j)f Fx(+)66 b(sup)2195 4121 y Fs(v)r Fr(2)p Fs(V)2319 4129 y Fn(\031)r(;>)2446 4039 y Fx(d)2500 4054 y Fp(1)2539 4039 y Fx(\()p Fv(v)t(;)17 b(\031)t Fx(\))810 b(\(5.6\))244 4276 y Fw(holds,)46 b(wher)-5 b(e)44 b(we)g(understand)g(the)g(se)-5 b(c)g(ond)44 b(term)g(in)h(the)f(right{hand)g(side)f(e)-5 b(qual)45 b(to)f(zer)-5 b(o)244 4421 y(whenever)34 b Fv(V)726 4436 y Fs(\031)r(;>)875 4421 y Fx(=)27 b Fu(;)p Fw(;)p Black 115 4645 a(2.)p Black 49 w(the)35 b(p)-5 b(air)34 b Fx(\()p Fv(V)701 4660 y Fs(\031)r(;)p Fr(\024)823 4645 y Fv(;)17 b(E)939 4660 y Fs(\031)r(;)p Fr(\024)1082 4645 y Fu([)23 b Fv(E)1249 4609 y Fr(?)1243 4670 y Fs(\031)r(;>)1365 4645 y Fx(\))34 b Fw(is)h(a)g(c)-5 b(onne)g(cte)g(d)33 b(gr)-5 b(aph.)146 4865 y Fx(W)d(e)33 b(remark)f(that)g(this)g(lemma)h (dep)s(ends)g(on)f(the)g(use)h(of)e(the)i(distance)g(d)2917 4880 y Fp(1)2988 4865 y Fx(in)f(the)h(de\014nition)f(of)0 5010 y(the)h(edge)g(set)g Fq(E)p Fx(.)44 b(Indeed)34 b(it)f(w)m(ould)g(ha)m(v)m(e)h(b)s(een)g(false)f(if)f(w)m(e)i(had)e (used)i(the)f(distance)h(d)3301 5025 y Fr(1)3376 5010 y Fx(.)0 5178 y Fw(Pr)-5 b(o)g(of)36 b(of)g(L)-5 b(emma)35 b(5.3.)67 b Fx(Pro)s(of)34 b(of)f(the)i(item)f(1.)48 b(Let)34 b Fv(E)2104 5127 y Fr(k)2098 5197 y Fs(\031)r(;>)2250 5178 y Fx(:=)2383 5097 y Ft(\010)2441 5178 y Fu(f)p Fv(x;)17 b(y)t Fu(g)29 b(2)i Fv(E)2890 5193 y Fs(\031)r(;>)3011 5178 y Fv(;)j Fx(\()p Fv(y)26 b Fu(\000)d Fv(x)p Fx(\))h Fu(\001)g Ft(b)-56 b Fv(n)30 b Fu(6)p Fx(=)g(0)3695 5097 y Ft(\011)3753 5178 y Fx(;)0 5322 y(it)j(is)g(immediate)h(to)e(sho)m(w) h(that)1295 5557 y Fu(j)p Fv(E)6 b Fu(j)28 b(\025)g(j)p Fv(E)1662 5572 y Fs(\031)r(;)p Fr(\024)1805 5557 y Fu([)23 b Fv(E)1972 5516 y Fr(?)1966 5582 y Fs(\031)r(;>)2088 5557 y Fu(j)e Fx(+)h Fu(j)p Fv(E)2341 5506 y Fr(k)2335 5576 y Fs(\031)r(;>)2457 5557 y Fu(j)1094 b Fx(\(5.7\))p Black 1841 5806 a(26)p Black eop end %%Page: 27 28 TeXDict begin 27 27 bop Black Black 0 407 a Fx(If)34 b Fv(V)156 422 y Fs(\031)r(;>)308 407 y Fx(=)c Fu(;)j Fx(\(5.6\))h(trivially)h(follo)m(ws)g(from)e(\(5.7\).)48 b(Supp)s(ose,)35 b(no)m(w,)g Fv(V)2631 422 y Fs(\031)r(;>)2783 407 y Fu(6)p Fx(=)30 b Fu(;)p Fx(.)47 b(Pic)m(k)36 b Fv(v)e Fu(2)c Fv(V)3467 422 y Fs(\031)r(;>)3623 407 y Fx(and)0 551 y(let)37 b Fv(D)h Fx(:=)d(d)456 566 y Fp(1)495 551 y Fx(\()p Fv(\031)t(;)17 b(v)t Fx(\))35 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Ft(\014)1130 1058 y(\014)1163 1082 y Fu(ff)p Fv(x;)17 b(y)t Fu(g)27 b(2)h(f)p Fv(`)1676 1097 y Fp(1)1715 1082 y Fv(;)17 b(:)g(:)g(:)f(;)h (`)1975 1097 y Fs(h)2019 1082 y Fu(g)p Fv(;)33 b Fx(\()p Fv(y)25 b Fu(\000)e Fv(x)p Fx(\))g Fu(\001)g Ft(b)-57 b Fv(n)28 b Fu(6)p Fx(=)g(0)p Fu(g)2794 998 y Ft(\014)2794 1058 y(\014)2854 1082 y Fu(\025)g Fv(D)539 b Fx(\(5.8\))0 1325 y(The)34 b(inequalit)m(y)g(\(5.6\))e(follo)m(ws)h(from)g(\(5.7\))f (and)g(\(5.8\).)146 1469 y(Pro)s(of)j(of)h(the)g(item)g(2.)53 b(The)36 b(statemen)m(t)h(is)g(trivial)f(if)f Fu(j)p Fv(V)2287 1484 y Fs(\031)r(;)p Fr(\024)2408 1469 y Fu(j)e Fx(=)g(1.)52 b(Supp)s(ose,)38 b(no)m(w,)f Fu(j)p Fv(V)3439 1484 y Fs(\031)r(;)p Fr(\024)3560 1469 y Fu(j)c(\025)g Fx(2)0 1614 y(and)g(pic)m(k)h(t)m(w)m(o)g(distinct)g(v)m(ertexes)i Fv(v)t(;)17 b(w)30 b Fu(2)f Fv(V)1659 1629 y Fs(\031)r(;)p Fr(\024)1780 1614 y Fx(.)45 b(By)33 b(recalling)h(that)f(\()p Fv(V)5 b(;)17 b(E)6 b Fx(\))32 b(is)i(a)f(connected)h(graph)0 1758 y(w)m(e)25 b(ha)m(v)m(e)f(that)g(there)g(exists)h(a)e(connected)i (path)f(joining)f Fv(v)k Fx(to)c Fv(w)j Fx(namely)-8 b(,)27 b(there)d(exist)h Fv(`)3237 1773 y Fp(1)3276 1758 y Fv(;)17 b(:)g(:)g(:)f(;)h(`)3536 1773 y Fs(h)3608 1758 y Fu(2)28 b Fv(E)0 1903 y Fx(suc)m(h)34 b(that)e Fv(v)g Fu(2)c Fv(`)645 1918 y Fp(1)684 1903 y Fx(,)33 b Fv(w)d Fu(2)e Fv(`)979 1918 y Fs(h)1024 1903 y Fx(,)k(and)h Fv(`)1314 1918 y Fs(m)1403 1903 y Fu(\\)22 b Fv(`)1532 1918 y Fs(m)p Fp(+1)1717 1903 y Fu(6)p Fx(=)27 b Fu(;)33 b Fx(for)f Fv(m)c Fx(=)f(1)p Fv(;)17 b(:)g(:)g(:)f(;)h(h)22 b Fu(\000)g Fx(1.)146 2047 y(W)-8 b(e)37 b(let)g Fv(`)504 2011 y Fr(0)504 2072 y Fp(1)543 2047 y Fv(;)17 b(:)g(:)g(:)f(;)h(`)803 2011 y Fr(0)803 2073 y Fs(h)844 2055 y Fm(0)906 2047 y Fx(b)s(e)37 b(the)g(path)f(obtained)h(from)f Fv(`)2126 2062 y Fp(1)2165 2047 y Fv(;)17 b(:)g(:)g(:)f(;)h(`)2425 2062 y Fs(h)2506 2047 y Fx(b)m(y)37 b(remo)m(ving)h(all)e(the)h(edges)g (b)s(e-)0 2191 y(longing)30 b(to)f Fv(E)533 2141 y Fr(k)527 2211 y Fs(\031)r(;>)649 2191 y Fx(;)h(w)m(e)h(remark)g(that)e(the)h (path)g Fv(`)1817 2155 y Fr(0)1817 2216 y Fp(1)1857 2191 y Fv(;)17 b(:)g(:)g(:)e(;)i(`)2116 2155 y Fr(0)2116 2218 y Fs(h)2157 2199 y Fm(0)2213 2191 y Fx(is)30 b(not)g(necessarily)i (connected)f(and)f(that)0 2336 y(1)d Fu(\024)i Fv(h)238 2300 y Fr(0)289 2336 y Fu(\024)f Fv(h)p Fx(.)41 b(Let)25 b Fv(`)726 2300 y Fr(0)777 2336 y Fx(=)j Fu(f)p Fv(x)986 2300 y Fr(0)1009 2336 y Fv(;)17 b(y)1105 2300 y Fr(0)1127 2336 y Fu(g)25 b Fx(b)s(e)h(an)f(edge)g(of)g(suc)m(h)h(a)f(path;)j Fv(`)2351 2300 y Fr(0)2399 2336 y Fx(is)e(either)g(in)f Fv(E)2937 2351 y Fs(\031)r(;)p Fr(\024)3084 2336 y Fx(or)g(in)g Fv(E)3374 2351 y Fs(\031)r(;>)3502 2336 y Fu(n)7 b Fv(E)3637 2285 y Fr(k)3631 2355 y Fs(\031)r(;>)3753 2336 y Fx(.)0 2480 y(W)-8 b(e)32 b(set)324 2454 y(\026)317 2480 y Fv(`)358 2444 y Fr(0)409 2480 y Fx(:=)c Fv(`)581 2444 y Fr(0)635 2480 y Fx(in)k(the)f(former)h(case)g(and)1625 2454 y(\026)1618 2480 y Fv(`)1659 2444 y Fr(0)1710 2480 y Fx(:=)27 b Fu(f)p Fv(x)1945 2444 y Fr(0)1988 2480 y Fx(+)19 b(\()p Fv(c)h Fu(\000)f Fv(x)2334 2444 y Fr(0)2377 2480 y Fu(\001)i Ft(b)-57 b Fv(n)q Fx(\))q Ft(b)h Fv(n;)17 b(y)2675 2444 y Fr(0)2716 2480 y Fx(+)i(\()p Fv(c)g Fu(\000)h Fv(y)3059 2444 y Fr(0)3101 2480 y Fu(\001)g Ft(b)-56 b Fv(n)p Fx(\))q Ft(b)g Fv(n)p Fu(g)28 b(2)g Fv(E)3552 2444 y Fr(?)3546 2505 y Fs(\031)r(;>)3699 2480 y Fx(in)0 2625 y(the)33 b(latter.)146 2769 y(By)38 b(construction)874 2743 y(\026)867 2769 y Fv(`)908 2733 y Fr(0)908 2794 y Fp(1)948 2769 y Fv(;)17 b(:)g(:)g(:)e(;)1173 2743 y Fx(\026)1166 2769 y Fv(`)1207 2733 y Fr(0)1207 2796 y Fs(h)1248 2777 y Fm(0)1310 2769 y Fu(2)35 b Fv(E)1483 2784 y Fs(\031)r(;)p Fr(\024)1630 2769 y Fu([)26 b Fv(E)1800 2733 y Fr(?)1794 2794 y Fs(\031)r(;>)1916 2769 y Fx(.)57 b(Moreo)m(v)m(er)38 b(it)g(is)f(an)g(easy)h(task)g(to)f(pro)m(v)m(e)h(that)0 2914 y Fv(v)54 b Fu(2)225 2887 y Fx(\026)218 2914 y Fv(`)259 2878 y Fr(0)259 2938 y Fp(1)298 2914 y Fx(,)c Fv(w)i Fu(2)621 2887 y Fx(\026)614 2914 y Fv(`)655 2878 y Fr(0)655 2940 y Fs(h)696 2921 y Fm(0)722 2914 y Fx(,)d(and)d Fv(`)1042 2878 y Fr(0)1042 2938 y Fs(m)1140 2914 y Fu(\\)32 b Fv(`)1279 2878 y Fr(0)1279 2938 y Fs(m)p Fp(+1)1486 2914 y Fu(6)p Fx(=)50 b Fu(;)c Fx(for)f Fv(m)51 b Fx(=)f(1)p Fv(;)17 b(:)g(:)g(:)f(;)h(h)2456 2878 y Fr(0)2510 2914 y Fu(\000)32 b Fx(1.)83 b(The)47 b(pro)s(of)e(of)g(item)i(2)e(is)0 3058 y(completed.)p 3709 3058 4 66 v 3713 2996 59 4 v 3713 3058 V 3771 3058 4 66 v Black 0 3284 a Fy(Lemma)39 b(5.4.)p Black 42 w Fw(L)-5 b(et)35 b Fv(G)28 b Fu(\032\032)g(G)1127 3299 y Fr(\025)p Fp(1)1257 3284 y Fw(and)34 b Fv(s)28 b Fu(\025)g Fx(0)p Fw(.)44 b(Then)35 b(the)f(b)-5 b(ound)35 b(\(5.4\))f(holds)g(true)i(for)e Fv(j)g Fx(=)27 b(0)p Fw(.)0 3510 y(Pr)-5 b(o)g(of.)34 b Fx(The)i(statemen)m(t)g(is)f (trivial)g(in)g(the)g(case)g Fv(s)c Fx(=)g(0.)49 b(Let)35 b Fv(s)c Fu(\025)g Fx(1)k(and)f(lab)s(el)h(the)g(elemen)m(ts)i(of)d Fv(G)0 3655 y Fx(b)m(y)g(setting)g Fv(G)29 b Fx(=)f Fu(f)p Fv(g)764 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5124 a Fw(Pr)-5 b(o)g(of)39 b(of)h(Pr)-5 b(op)g(osition)39 b(5.2.)74 b Fx(W)-8 b(e)39 b(can)f(assume)h Fv(R)p 1843 5140 76 4 v 1918 5148 a Fs(j)1954 5124 y Fx(\()p Fv(G;)17 b(s)p Fx(\))36 b Fu(6)p Fx(=)h Fu(;)p Fx(,)i(otherwise)g Fu(T)2955 5139 y Fs(j)2992 5124 y Fx(\()p Fv(G;)17 b(s)p Fx(\))36 b(=)g(+)p Fu(1)p Fx(.)59 b(W)-8 b(e)0 5268 y(pro)m(v)m(e)34 b(\(5.4\))e(b)m(y)i(induction;)g(the)g(step)f Fv(j)i Fx(=)28 b(0)k(has)i(b)s(een)f(pro)m(v)m(en)h(in)g(the)f(Lemma)g(5.4.)44 b(W)-8 b(e)34 b(supp)s(ose)0 5413 y(\(5.4\))22 b(holds)i(for)e Fv(j)8 b Fu(\000)r Fx(1)23 b(and)g(w)m(e)h(sho)m(w)g(it)f(holds)g(true) g(for)f Fv(j)6 b Fx(.)41 b(T)-8 b(o)22 b(b)s(ound)h Fu(T)2613 5428 y Fs(j)2650 5413 y Fx(\()p Fv(G;)17 b(s)p Fx(\))22 b(w)m(e)i(let)f Fv(R)p 3180 5429 V 3255 5371 a Fr(\003)3322 5413 y Fu(2)29 b(R)p 3417 5429 85 4 v 24 x Fs(j)3538 5413 y Fx(\()p Fv(G;)17 b(s)p Fx(\))0 5557 y(b)s(e)43 b(a)g(minimizer)i(for)d(\(5.3\).)74 b(Note)44 b(that)e Fv(R)p 1630 5573 76 4 v 1705 5581 a Fs(j)1742 5557 y Fx(\()p Fv(G;)17 b(s)p Fx(\))42 b(is)i(not)f(a)g(\014nite)g(set)h(b)s (ecause)g(rep)s(etitions)g(of)p Black 1841 5806 a(27)p Black eop end %%Page: 28 29 TeXDict begin 28 28 bop Black Black 0 407 a Fx(the)42 b(some)g(b)s(ond)g(are)g(allo)m(w)m(ed.)71 b(Ho)m(w)m(ev)m(er)44 b(a)d(minimizer)j Fv(R)p 2226 423 76 4 v 2301 365 a Fr(\003)2382 407 y Fx(do)s(es)e(exist)g(b)s(ecause)h(without)f(suc)m(h)0 551 y(rep)s(etitions)34 b Fv(R)p 486 567 V 561 575 a Fs(j)597 551 y Fx(\()p Fv(G;)17 b(s)p Fx(\))32 b(w)m(ould)i(b)s(e)e (\014nite)i(and)e Fw(r)-5 b(ep)g(etita)35 b(juvant)p Fx(.)44 b(W)-8 b(e)33 b(ha)m(v)m(e)1189 784 y Fu(T)1243 799 y Fs(j)1280 784 y Fx(\()p Fv(G;)17 b(s)p Fx(\))27 b(=)1677 689 y Ft(X)1653 903 y Fs(R)p Fr(2)p Fs(R)p 1753 915 54 3 v 1808 878 a Fm(\003)1926 689 y Ft(X)1860 905 y Fp(\()p Fs(H)q(;u)p Fp(\))p Fr(2)p Fs(R)2152 784 y Fu(T)2206 799 y Fs(j)t Fr(\000)p Fp(1)2333 784 y Fx(\()p Fv(H)r(;)17 b(u)p Fx(\))987 b(\(5.9\))146 1103 y(W)-8 b(e)39 b(consider,)h(no)m(w,)g(the)e(case)h Fv(s)e Fx(=)f(0.)59 b(Let)38 b Fu(H)g(\021)f(H)q Fx(\()p Fv(R)p 2224 1119 76 4 v 2299 1061 a Fr(\003)2339 1103 y Fx(\))f(:=)h Fu(f)p Fv(H)44 b Fu(\032)37 b(G)2901 1118 y Fr(\025)p Fs(j)3030 1103 y Fx(:)53 b Fu(9)p Fv(R)38 b Fu(2)f Fv(R)p 3380 1119 V 3455 1061 a Fr(\003)3495 1103 y Fv(;)c Fu(9)p Fv(u)k Fu(\025)0 1247 y Fx(0)27 b(:)45 b(\()p Fv(H)r(;)17 b(u)p Fx(\))27 b Fu(2)h Fv(R)33 b Fx(and)g Fu(j)p Fv(H)8 b Fu(j)27 b(\025)h Fx(2)p Fu(g)p Fx(;)j(w)m(e)h(note)e(that)h Fu(H)h Fx(is)f(\014nite)h(and)e(not)h(empt)m(y)-8 b(.)44 b(F)-8 b(rom)30 b(\(5.9\))h(and)g(the)0 1392 y(inductiv)m(e)k(h)m(yp)s (othesis)f(w)m(e)g(ha)m(v)m(e)1145 1690 y Fu(T)1199 1705 y Fs(j)1236 1690 y Fx(\()p Fv(G;)17 b Fx(0\))27 b Fu(\025)1614 1579 y Ft(\020)1673 1690 y Fx(1)22 b Fu(\000)1855 1561 y Fs(j)t Fr(\000)p Fp(1)1844 1595 y Ft(X)1852 1808 y Fs(k)r Fp(=0)2014 1623 y Fx(\000)2075 1638 y Fs(k)p 2014 1667 104 4 v 2019 1758 a Fv(\015)2070 1773 y Fs(k)2128 1579 y Ft(\021)2217 1595 y(X)2204 1807 y Fs(H)5 b Fr(2H)2391 1690 y Fu(T)26 b Fx(\()p Fv(H)8 b Fx(\))0 1992 y(W)-8 b(e)32 b(also)g(remark)g(that)f(de\014nitions)i(\(3.4\))e(and)h (\(4.17\))f(imply)h(that)f(the)h(system)i Fu(H)e Fx(is)h Fv(j)6 b Fx({connected)0 2137 y(in)26 b(the)h(sense)h(sp)s(eci\014ed)g (just)e(ab)s(o)m(v)m(e)h(\(3.2\).)41 b(By)27 b(adding)f(and)g (subtracting)h(\000)2838 2152 y Fs(j)2875 2137 y Fv(=\015)2975 2152 y Fs(j)3037 2137 y Fx(and)f(b)m(y)h(remarking)0 2281 y(that)32 b Fu(j)p Fv(H)8 b Fu(j)27 b(\025)h Fx(2)k(implies)i Fu(T)26 b Fx(\()p Fv(H)8 b Fx(\))27 b Fu(\025)h Fv(\015)1328 2296 y Fs(j)1397 2281 y Fx(w)m(e)34 b(get)966 2588 y Fu(T)1020 2603 y Fs(j)1057 2588 y Fx(\()p Fv(G;)17 b Fx(0\))27 b Fu(\025)1435 2477 y Ft(\020)1494 2588 y Fx(1)22 b Fu(\000)1721 2459 y Fs(j)1665 2493 y Ft(X)1673 2705 y Fs(k)r Fp(=0)1835 2520 y Fx(\000)1896 2535 y Fs(k)p 1835 2565 V 1840 2656 a Fv(\015)1891 2671 y Fs(k)1949 2477 y Ft(\021)2038 2493 y(X)2025 2705 y Fs(H)5 b Fr(2H)2212 2588 y Fu(T)26 b 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y Fr(j)2524 3751 y Fx(the)39 b(elemen)m(ts)h(of)e Fu(H)3303 3766 y Fs(m)3370 3751 y Fx(;)j(for)d(eac)m(h)0 3896 y Fv(m)32 b Fx(=)f(0)p Fv(;)17 b(:)g(:)g(:)f(;)h(t)35 b Fx(and)g Fv(`)c Fx(=)h(1)p Fv(;)17 b(:)g(:)g(:)e(;)i Fu(jH)1313 3911 y Fs(m)1379 3896 y Fu(j)35 b Fx(w)m(e)h(let)f(\()p Fv(V)1826 3911 y Fs(m;`)1941 3896 y Fv(;)17 b(E)2057 3911 y Fs(m;`)2172 3896 y Fx(\))31 b Fu(\032)h Fx(\()p Fq(L)p Fv(;)17 b Fq(E)p Fx(\))36 b(b)s(e)f(a)f(connected)j(graph)d(suc) m(h)0 4040 y(that)1527 4185 y Fu(T)26 b Fx(\()p Fv(H)1726 4200 y Fs(m;`)1841 4185 y Fx(\))h(=)h Fu(j)p Fv(E)2110 4200 y Fs(m;`)2225 4185 y Fu(j)1278 b Fx(\(5.11\))0 4378 y(and)40 b(for)g(eac)m(h)h Fv(h)g Fu(2)g Fv(H)866 4393 y Fs(m;`)1021 4378 y Fx(w)m(e)g(ha)m(v)m(e)g(that)f Fv(V)1680 4393 y Fs(m;`)1822 4378 y Fu(\\)28 b Fv(h)41 b Fu(6)p Fx(=)f Fu(;)p Fx(.)66 b(W)-8 b(e)41 b(de\014ne,)i(no)m(w,)g(an)d (algorithm)g(that)0 4522 y(constructs)33 b(a)f(graph)f(\()p Fv(V)5 b(;)17 b(E)6 b Fx(\))28 b Fu(\032)g Fx(\()p Fq(L)p Fv(;)17 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Fv(V)d Fx(=)28 b Fv(V)43 b Fu([)23 b Fv(V)905 782 y Fs(m;`)1042 767 y Fu([)g Fv(W)64 b Fc(and)52 b Fv(E)34 b Fx(=)28 b Fv(E)g Fu([)23 b Fv(E)1963 782 y Fs(m;`)2100 767 y Fu([)f Fv(F)14 b Fc(;)p Black 93 983 a(7.)p Black 49 w(if)52 b Fv(`)27 b(<)h Fu(jH)682 998 y Fs(m)748 983 y Fu(j)51 b Fc(goto)h(3)g(else)g (goto)h(2;)p Black 93 1200 a(8.)p Black 49 w(exit;)0 1393 y Fx(By)28 b(recursion)g(it)f(is)h(easy)g(to)f(pro)m(v)m(e)i(that) e(this)h(algorithm)f(outputs)h(a)e(connected)j(graph)e(\()p Fv(V)5 b(;)17 b(E)6 b Fx(\))27 b(suc)m(h)0 1537 y(that)i(for)g(eac)m(h) h Fv(H)35 b Fu(2)28 b(H)j Fx(and)e Fv(h)f Fu(2)g Fv(H)37 b Fx(there)30 b(exists)h Fv(x)d Fu(2)g Fv(h)i Fx(suc)m(h)h(that)e Fv(x)f Fu(2)g Fv(V)22 b Fx(;)30 b(in)g(particular)f(for)g(eac)m(h)0 1682 y Fv(g)i Fu(2)d Fv(G)k Fx(there)g(exists)h Fv(x)28 b Fu(2)g Fv(g)35 b Fx(suc)m(h)e(that)f Fv(x)c Fu(2)g Fv(V)22 b Fx(,)31 b(hence)i Fu(j)p Fv(E)6 b Fu(j)28 b(\025)g(T)d Fx(\()p Fv(G)p Fx(\).)43 b(Moreo)m(v)m(er,)34 b(b)m(y)e(noticing)g (that)0 1826 y(the)h(graph)f(\()p Fv(W)m(;)17 b(F)d Fx(\))33 b(in)m(tro)s(duced)g(at)g(line)g(5)f(is)h(suc)m(h)h(that)f Fu(j)p Fv(F)14 b Fu(j)26 b(\024)i Fx(diam)2641 1841 y Fp(1)2680 1826 y Fx(\()p Fv(h)p Fx(\))g Fu(\024)g Fx(\000)3006 1841 y Fs(j)3075 1826 y Fx(w)m(e)34 b(ha)m(v)m(e)1162 2114 y Fu(j)p Fv(E)6 b Fu(j)27 b(\024)1491 1990 y Fs(t)1432 2020 y Ft(X)1428 2229 y Fs(m)p Fp(=0)1597 1982 y Fr(jH)1677 1990 y Fn(m)1735 1982 y Fr(j)1604 2020 y Ft(X)1616 2232 y Fs(`)p Fp(=1)1771 2114 y Fu(j)p Fv(E)1871 2129 y Fs(m;`)1986 2114 y Fu(j)22 b Fx(+)g(\()p Fu(jH)q(j)g(\000)g Fx(1\)\000)2582 2129 y Fs(j)3531 2114 y Fx(\(5.12\))0 2385 y(No)m(w,)33 b(b)m(y)h(using)f(\(5.10\){\(5.12\))e(w)m(e)i(get)309 2660 y Fu(T)363 2675 y Fs(j)400 2660 y Fx(\()p Fv(G;)17 b Fx(0\))27 b Fu(\025)778 2549 y Ft(\020)837 2660 y Fx(1)22 b Fu(\000)1064 2531 y Fs(j)1008 2565 y Ft(X)1016 2777 y Fs(k)r Fp(=0)1178 2593 y Fx(\000)1239 2608 y Fs(k)p 1178 2637 104 4 v 1183 2728 a Fv(\015)1234 2743 y Fs(k)1292 2549 y Ft(\021)1352 2579 y(\002)1393 2660 y Fu(j)p Fv(E)6 b Fu(j)22 b(\000)g Fx(\()p Fu(jH)q(j)g(\000)g Fx(1\)\000)2096 2675 y Fs(j)2133 2579 y Ft(\003)2196 2660 y Fx(+)g Fu(jH)q(j)p Fx(\000)2496 2675 y Fs(j)2560 2660 y Fu(\025)2665 2549 y Ft(\020)2724 2660 y Fx(1)g Fu(\000)2951 2531 y Fs(j)2895 2565 y Ft(X)2903 2777 y Fs(k)r Fp(=0)3065 2593 y Fx(\000)3126 2608 y Fs(k)p 3065 2637 V 3070 2728 a Fv(\015)3121 2743 y Fs(k)3179 2549 y Ft(\021)3239 2660 y Fu(T)j Fx(\()p Fv(G)p Fx(\))0 2930 y(whic)m(h)34 b(completes)g(the)f(inductiv)m(e)i (pro)s(of)d(of)g(\(5.4\))g(for)g Fv(s)c Fx(=)f(0.)146 3075 y(W)-8 b(e)33 b(consider,)g(no)m(w,)g(the)f(case)h Fv(s)27 b Fu(\025)h Fx(1.)43 b(Recalling)33 b(\(5.9\),)e(there)i (exists)g Fv(R)2895 3039 y Fr(0)2946 3075 y Fu(2)c Fv(R)p 3041 3091 76 4 v 3116 3033 a Fr(\003)3187 3075 y Fx(and)j(\()p Fv(H)3503 3039 y Fr(0)3525 3075 y Fv(;)17 b(u)3625 3039 y Fr(0)3648 3075 y Fx(\))28 b Fu(2)0 3219 y Fv(R)75 3183 y Fr(0)140 3219 y Fx(suc)m(h)43 b(that)e Fv(Y)646 3234 y Fs(u)687 3215 y Fm(0)713 3219 y Fx(\()p Fv(H)840 3183 y Fr(0)863 3219 y Fx(\))28 b Fu(\\)g Fv(y)1071 3234 y Fs(s)1108 3219 y Fx(\()p Fv(G)p Fx(\))42 b Fu(6)p Fx(=)h Fu(;)p Fx(.)70 b(Let)41 b Fu(H)1837 3183 y Fr(0)1903 3219 y Fu(\021)i(H)2108 3183 y Fr(0)2132 3219 y Fx(\()p Fv(R)p 2170 3235 V 2245 3177 a Fr(\003)2284 3219 y Fx(\))g(:=)g Fu(f)p Fv(H)50 b Fu(\032)43 b(G)2871 3234 y Fr(\025)p Fs(j)3005 3219 y Fx(:)60 b Fu(9)p Fv(R)44 b Fu(2)f Fv(R)p 3374 3235 V 3449 3177 a Fr(\003)3489 3219 y Fv(;)33 b Fu(9)p Fv(u)43 b Fu(\025)0 3364 y Fx(0)49 b(:)65 b(\()p Fv(H)r(;)17 b(u)p Fx(\))48 b Fu(2)h Fv(R)q(;)34 b Fx(\()p Fv(H)r(;)17 b(u)p Fx(\))48 b Fu(6)p Fx(=)h(\()p Fv(H)1307 3327 y Fr(0)1329 3364 y Fv(;)17 b(u)1429 3327 y Fr(0)1452 3364 y Fx(\))32 b(and)h Fu(j)p Fv(H)8 b Fu(j)48 b(\025)h Fx(2)p Fu(g)p Fx(.)80 b(Note)46 b(that,)i(as)d(in)g(the)g(previous)i (case,)0 3508 y Fu(j)p Fv(H)8 b Fu(j)32 b(\025)i Fx(2)i(implies)h Fu(T)26 b Fx(\()p Fv(H)8 b Fx(\))33 b Fu(\025)h Fv(\015)1147 3523 y 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Fr(\000)p Fp(1)1822 5409 y Fx(\()p Fv(H)r(;)17 b(u)p Fx(\))27 b Fu(\025)2223 5342 y Fx(1)p 2223 5386 49 4 v 2223 5478 a(2)2282 5409 y(d)2336 5424 y Fp(1)2375 5409 y Fx(\()p Fv(g)t(;)17 b(h)p Fx(\))929 b(\(5.22\))p Black 1841 5806 a(30)p Black eop end %%Page: 31 32 TeXDict begin 31 31 bop Black Black 0 407 a Fw(Pr)-5 b(o)g(of.)52 b Fx(The)27 b(Lemma)g(can)f(b)s(e)g(pro)m(v)m(en)i(b)m(y)f (using)g(\(5.4\),)g(the)f(simple)i(b)s(ound)e(1)9 b Fu(\000)2984 332 y Ft(P)3090 358 y Fr(1)3090 436 y Fp(0)3165 407 y Fx(\(\000)3264 422 y Fs(j)3300 407 y Fv(=\015)3400 422 y Fs(j)3436 407 y Fx(\))27 b Fu(\025)i Fx(1)p Fv(=)p Fx(2,)0 551 y(and)k(b)m(y)g(running)g(the)g(algorithm)g(in)m(tro)s (duced)g(in)g(the)g(pro)s(of)f(of)g(Lemma)h(5.2.)p 3709 551 4 66 v 3713 489 59 4 v 3713 551 V 3771 551 4 66 v Black 0 779 a Fy(Lemma)41 b(5.6.)p Black 42 w Fw(L)-5 b(et)36 b Fv(G)30 b Fu(\032\032)h(G)1135 794 y Fr(\025)p Fs(j)t Fp(+1)1317 779 y Fw(,)36 b Fv(s)29 b Fu(\025)i Fx(0)k Fw(and)h Fu(R)p 1841 795 85 4 v 24 x Fs(j)1962 779 y Fx(\()p Fv(G;)17 b(s)p Fx(\))35 b Fw(as)h(de\014ne)-5 b(d)35 b(in)g(\(4.17\).)48 b(Then,)35 b(for)h(e)-5 b(ach)0 923 y Fv(R)p 0 939 76 4 v 29 w Fu(2)28 b(R)p 197 939 85 4 v 24 x Fs(j)318 923 y Fx(\()p Fv(G;)17 b(s)p Fx(\))p Fw(,)1188 973 y Ft(X)1182 1184 y Fs(R)p Fr(2)p Fs(R)p 1282 1196 54 3 v 1441 973 a Ft(X)1375 1189 y Fp(\()p Fs(H)q(;u)p Fp(\))p Fr(2)p Fs(R)1667 1068 y Fu(j)p Fv(H)8 b Fu(j)27 b(\025)h(j)p Fv(G)p Fu(j)21 b Fx(+)2201 973 y Ft(X)2196 1184 y Fs(R)p Fr(2)p Fs(R)p 2296 1196 V 2367 1068 a Fu(j)p Fv(R)12 b Fj(\026)2534 1083 y Fs(j)2570 1068 y Fu(j)933 b Fx(\(5.23\))0 1381 y Fw(Pr)-5 b(o)g(of.)65 b Fx(The)33 b(Lemma)g(follo)m(ws)h(directly)g(from)e(the)h (de\014nition)h(of)e Fu(R)p 2489 1397 85 4 v 25 x Fs(j)2610 1381 y Fx(\()p Fv(G;)17 b(s)p Fx(\).)p 3709 1381 4 66 v 3713 1319 59 4 v 3713 1381 V 3771 1381 4 66 v 0 1626 a Fw(5.2.)79 b(Pr)-5 b(eliminary)34 b(lemmata)0 1795 y Fx(In)i(this)g(section)h(w)m(e)f(collect)h(some)f(tec)m(hnical)h(b)s (ounds)f(needed)h(to)f(pro)m(v)m(e)g(the)g(con)m(v)m(ergence)i(of)e (the)0 1939 y(m)m(ulti{scale)e(cluster)g(expansion.)p Black 0 2167 a Fy(Lemma)39 b(5.7.)p Black 42 w Fw(F)-7 b(or)34 b Fv(m)28 b(>)f Fx(0)35 b Fw(let)1392 2431 y Fv(K)7 b Fx(\()p Fv(m)p Fx(\))28 b(:=)1801 2320 y Ft(\020)1872 2363 y Fx(1)21 b(+)h Fv(e)2085 2327 y Fr(\000)p Fs(m=)p Fp(2)p 1871 2408 408 4 v 1871 2499 a Fx(1)g Fu(\000)g Fv(e)2086 2470 y Fr(\000)p Fs(m=)p Fp(2)2288 2320 y Ft(\021)2348 2343 y Fs(d)3531 2431 y Fx(\(5.24\))0 2685 y Fw(wher)-5 b(e)41 b(we)f(r)-5 b(e)g(c)g(al)5 b(l)41 b Fv(d)g Fw(is)g(the)g (dimension)e(of)i(the)h(lattic)-5 b(e)41 b Fq(L)p Fw(.)64 b(L)-5 b(et)41 b(also)g Fv(\015)5 b(;)17 b(L)39 b Fu(\025)h Fx(0)h Fw(b)-5 b(e)41 b(p)-5 b(ositive)41 b(r)-5 b(e)g(als;)0 2829 y(then)35 b(we)f(have)1170 2879 y Ft(X)1179 3079 y Fn(x)p Fm(2)p Fk(L)p Fo(:)1100 3136 y(d)1134 3151 y(1)1169 3136 y(\()p Fn(x;)p Fo(0\))p Fm(\025)p Fn(\015)1411 2973 y Fv(e)1456 2932 y Fr(\000)p Fs(m)12 b Fp(d)1624 2941 y Fo(1)1659 2932 y Fp(\()p Fs(x;B)1799 2943 y Fn(L)1845 2932 y Fp(\))1905 2973 y Fu(\024)28 b Fv(K)7 b Fx(\()p Fv(m)p Fx(\))17 b Fv(e)2323 2932 y Fr(\000)2388 2905 y Fn(m)p 2388 2917 55 3 v 2400 2958 a Fo(2)2452 2932 y Fp(\()p Fs(\015)t Fr(\000)p Fp(2)p Fs(L)p Fp(\))3531 2973 y Fx(\(5.25\))0 3292 y Fw(wher)-5 b(e)44 b(we)g(r)-5 b(e)g(c)g(al)5 b(l)44 b Fv(B)778 3307 y Fs(L)875 3292 y Fw(is)h(the)f(b)-5 b(al)5 b(l)45 b(of)f(r)-5 b(adius)44 b Fv(L)h Fw(c)-5 b(enter)g(e)g(d)45 b(at)f(the)h(origin)f(de\014ne)-5 b(d)44 b(at)g(the)h(end)f(of)0 3436 y(Se)-5 b(ction)34 b(2.1.)0 3664 y(Pr)-5 b(o)g(of.)65 b Fx(First)32 b(of)g(all)h(w)m(e)g (note)g(that)g(d)1432 3679 y Fp(1)1471 3664 y Fx(\()p Fv(x;)17 b Fx(0\))28 b Fu(\024)g Fv(L)22 b Fx(+)g(d)2068 3679 y Fp(1)2108 3664 y Fx(\()p Fv(x;)17 b(B)2319 3679 y Fs(L)2371 3664 y Fx(\).)44 b(Hence)419 3813 y Ft(X)427 4014 y Fn(x)p Fm(2)p Fk(L)p Fo(:)349 4070 y(d)383 4085 y(1)417 4070 y(\()p Fn(x;)p Fo(0\))p Fm(\025)p Fn(\015)659 3908 y Fv(e)704 3867 y Fr(\000)p Fs(m)13 b Fp(d)873 3876 y Fo(1)908 3867 y Fp(\()p Fs(x;B)1048 3878 y Fn(L)1094 3867 y Fp(\))1153 3908 y Fu(\024)1338 3813 y Ft(X)1347 4014 y Fn(x)p Fm(2)p Fk(L)p Fo(:)1268 4070 y(d)1302 4085 y(1)1337 4070 y(\()p Fn(x;)p Fo(0\))p Fm(\025)p Fn(\015)1579 3908 y Fv(e)1624 3867 y Fr(\000)p Fs(m)f Fp([d)1812 3876 y Fo(1)1847 3867 y Fp(\()p Fs(x;)p Fp(0\))p Fr(\000)p Fs(L)p Fp(])2150 3908 y Fu(\024)28 b Fv(e)2300 3867 y Fs(mL)p Fr(\000)p Fs(m\015)t(=)p Fp(2)2740 3813 y Ft(X)2748 4014 y Fn(x)p Fm(2)p Fk(L)p Fo(:)2670 4070 y(d)2704 4085 y(1)2738 4070 y(\()p Fn(x;)p Fo(0\))p Fm(\025)p Fn(\015)2981 3908 y Fv(e)3026 3867 y Fr(\000)p Fs(m)p Fp(d)3182 3876 y Fo(1)3217 3867 y Fp(\()p Fs(x;)p Fp(0\))p Fs(=)p Fp(2)0 4278 y Fx(Recalling)38 b(that)f(d)699 4293 y Fp(1)738 4278 y Fx(\()p Fv(x;)17 b Fx(0\))36 b(=)f Fu(j)p Fv(x)1192 4293 y Fp(1)1232 4278 y Fu(j)25 b Fx(+)g Fu(\001)17 b(\001)g(\001)23 b Fx(+)j Fu(j)p Fv(x)1712 4293 y Fs(d)1752 4278 y Fu(j)p Fx(,)38 b(where)h Fv(x)d Fx(=)f(\()p Fv(x)2427 4293 y Fp(1)2467 4278 y Fv(;)17 b(:)g(:)g(:)f(;)h(x)2741 4293 y Fs(d)2781 4278 y Fx(\),)39 b(and)e(using)h(the)g(b)s(ound)0 4422 y(ab)s(o)m(v)m(e)33 b(w)m(e)h(get)119 4618 y Ft(X)128 4818 y Fn(x)p Fm(2)p Fk(L)p Fo(:)49 4875 y(d)83 4890 y(1)117 4875 y(\()p Fn(x;)p Fo(0\))p Fm(\025)p Fn(\015)360 4713 y Fv(e)405 4671 y Fr(\000)p Fs(m)12 b Fp(d)573 4680 y Fo(1)608 4671 y Fp(\()p Fs(x;B)748 4682 y Fn(L)794 4671 y Fp(\))853 4713 y Fu(\024)29 b Fv(e)1004 4671 y Fr(\000)p Fs(m)p Fp(\()p Fs(\015)t Fr(\000)p Fp(2)p Fs(L)p Fp(\))p Fs(=)p Fp(2)1446 4618 y Ft(X)1452 4830 y Fs(x)p Fr(2)p Fl(L)1606 4713 y Fv(e)1651 4671 y Fr(\000)p Fs(m)p Fp(\()p Fr(j)p Fs(x)1855 4680 y Fo(1)1890 4671 y Fr(j)p Fp(+)p Fr(\001\001\001)n Fp(+)p Fr(j)p Fs(x)2138 4683 y Fn(d)2174 4671 y Fr(j)p Fp(\))p Fs(=)p Fp(2)2323 4713 y Fu(\024)f Fv(e)2473 4671 y Fr(\000)p Fs(m)p Fp(\()p Fs(\015)t Fr(\000)p Fp(2)p Fs(L)p Fp(\))p Fs(=)p Fp(2)2899 4602 y Ft(\020)2958 4713 y Fx(1)22 b(+)g(2)3229 4588 y Fr(1)3193 4618 y Ft(X)3200 4830 y Fs(k)r Fp(=1)3353 4713 y Fv(e)3398 4671 y Fr(\000)p Fs(m)13 b(k)r(=)p Fp(2)3641 4602 y Ft(\021)3700 4624 y Fs(d)0 5077 y Fx(and)33 b(the)g(Lemma)g (follo)m(ws)g(via)g(elemen)m(tary)i(computations.)p 3709 5077 4 66 v 3713 5014 59 4 v 3713 5077 V 3771 5077 4 66 v Black 0 5304 a Fy(Lemma)k(5.8.)p Black 42 w Fw(F)-7 b(or)34 b Fv(j)g Fu(\025)28 b Fx(1)34 b Fw(and)h Fv(m)28 b(>)f Fx(0)35 b Fw(let)1349 5548 y Fv(q)1392 5563 y Fs(j)1428 5548 y Fx(\()p Fv(m)p Fx(\))28 b(:=)g Fv(K)7 b Fx(\()p Fv(m=)p Fx(4\))17 b Fv(e)2159 5507 y Fr(\000)p Fs(m)12 b(\015)2324 5517 y Fn(j)2357 5507 y Fs(=)p Fp(8)3531 5548 y Fx(\(5.26\))p Black 1841 5806 a(31)p Black eop end %%Page: 32 33 TeXDict begin 32 32 bop Black Black 0 407 a Fw(wher)-5 b(e)34 b Fv(K)7 b Fx(\()p Fv(m)p Fx(\))35 b Fw(has)g(b)-5 b(e)g(en)34 b(de\014ne)-5 b(d)34 b(in)g(L)-5 b(emma)34 b(5.7.)45 b(Assume)34 b Fv(q)2354 422 y Fs(j)2391 407 y Fx(\()p Fv(m)p Fx(\))28 b Fv(<)g Fx(1)34 b Fw(and)h(set)266 696 y Fv(K)349 711 y Fs(j)386 696 y Fx(\()p Fv(m)p Fx(\))28 b(:=)f Fv(e)750 655 y Fr(\000)815 628 y Fn(m)p 815 640 55 3 v 827 681 a Fo(4)880 655 y Fs(#)922 665 y Fn(j)1054 629 y Fv(e)1099 592 y Fr(\000)p Fs(m=)p Fp(4)p 968 673 408 4 v 968 764 a Fx(1)22 b Fu(\000)h Fv(e)1184 736 y Fr(\000)p Fs(m=)p Fp(4)1408 696 y Fx(+)1506 585 y Ft(h)1553 696 y Fx(1)f(+)g Fv(e)1767 655 y Fr(\000)1832 628 y Fn(m)p 1832 640 55 3 v 1844 681 a Fo(4)1897 655 y Fs(#)1939 665 y Fn(j)2071 629 y Fv(e)2116 592 y Fr(\000)p Fs(m=)p Fp(4)p 1985 673 408 4 v 1985 764 a Fx(1)g Fu(\000)h Fv(e)2201 736 y Fr(\000)p Fs(m=)p Fp(4)2403 585 y Ft(i)2450 696 y Fx(\(\000)2549 711 y Fs(j)2608 696 y Fx(+)f(1\))2793 655 y Fs(d)2928 629 y Fv(q)2971 644 y Fs(j)3008 629 y Fx(\()p Fv(m)p Fx(\))p 2843 673 412 4 v 2843 764 a(1)g Fu(\000)g Fv(q)3056 779 y Fs(j)3093 764 y Fx(\()p Fv(m)p Fx(\))3531 696 y(\(5.27\))0 969 y Fw(Then)668 1141 y Fx(sup)663 1222 y Fs(g)r Fr(2G)788 1232 y Fn(j)914 1046 y Ft(X)847 1246 y Fn(G)p Fm(\032\032G)1027 1264 y(\025)p Fn(j)1108 1246 y 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b(")2322 4719 y Fr(j)p Fs(G)p Fr(j)2420 4760 y Fv(e)2465 4719 y Fr(\000)p Fs(m)2582 4729 y Fn(j)s Fm(\000)p Fo(1)2704 4703 y Fd(b)2694 4719 y Fr(T)2733 4729 y Fn(j)s Fm(\000)p Fo(1)2843 4719 y Fp(\()p Fs(G;s)p Fp(\))3531 4760 y Fx(\(5.50\))0 5093 y(where)31 b(w)m(e)g(used)h(that)e Fv(m)935 5108 y Fs(h)980 5093 y Fx(,)1052 5068 y Ft(b)1037 5093 y Fu(T)1091 5108 y Fs(h)1166 5093 y Fx(are)h(decreasing)g(in)f Fv(h)p Fx(.)43 b(Note)30 b(that)g(for)g Fv(g)h Fu(2)d(G)2853 5108 y Fr(\025)p Fs(j)2975 5093 y Fx(and)i Fv(s)d Fu(\025)i Fx(1)g(w)m(e)j(ha)m(v)m(e,)0 5237 y(b)m(y)h(recalling)h(the)f (inequalit)m(y)h(\(5.5\))e(and)h(de\014nition)g(\(2.9\),)g(that)754 5477 y Ft(b)740 5502 y Fu(T)794 5517 y Fs(h)839 5502 y Fx(\()p Fv(g)t(;)17 b(s)p Fx(\))26 b Fu(\025)1197 5435 y Fx(1)p 1197 5479 49 4 v 1197 5570 a(4)1256 5502 y(d)1310 5517 y Fp(1)1350 5502 y Fx(\()p Fu(Q)p Fx(\()p Fv(g)t Fx(\))p Fv(;)17 b(y)1688 5517 y Fs(s)1723 5502 y Fx(\()p Fv(g)t Fx(\)\))27 b Fv(>)2028 5435 y Fx(1)p 2028 5479 V 2028 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816 y Fv(h)g Fx(=)g(0)p Fv(;)17 b(:)g(:)g(:)e(;)i(j)28 b Fu(\000)23 b Fx(1)0 1060 y(W)-8 b(e)33 b(th)m(us)h(conclude)g(that)e (for)g(eac)m(h)h Fv(G)28 b Fu(\032\032)h(G)1710 1075 y Fr(\025)p Fs(j)1834 1060 y Fx(suc)m(h)34 b(that)e(\()p Fu(j)p Fv(G)p Fu(j)p Fv(;)17 b(s)p Fx(\))27 b Fu(6)p Fx(=)g(\(1)p Fv(;)17 b Fx(0\),)32 b Fv(j)i Fu(\025)28 b Fx(1,)k(w)m(e)i(ha)m(v)m(e)1157 1302 y Ft(b)1142 1327 y Fu(T)1196 1342 y Fs(h)1241 1327 y Fx(\()p Fv(G;)17 b(s)p Fx(\))27 b Fu(\025)1626 1260 y Fx(1)p 1626 1304 V 1626 1396 a(4)1685 1327 y Fv(\015)1736 1342 y Fs(j)1967 1327 y Fv(h)h Fx(=)f(0)p Fv(;)17 b(:)g(:)g(:)f(;)h(j)28 b Fu(\000)22 b Fx(1)893 b(\(5.51\))146 1604 y(Since)42 b(\010)479 1553 y Fp(\()p Fs(j)t Fp(\))479 1631 y Fs(G;s;)p Fp(\003)701 1604 y Fx(=)f(0)f(if)g(\()p Fu(j)p Fv(G)p Fu(j)p Fv(;)17 b(s)p Fx(\))40 b(=)h(\(1)p Fv(;)17 b Fx(0\),)42 b Fv(m)1832 1619 y Fs(j)t Fr(\000)p Fp(1)1959 1604 y Fv(\015)2010 1619 y Fs(j)2087 1604 y Fu(\025)g Fx(32)p Fv(j)k Fx(\(see)41 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2096 y Fp(\()p Fs(G;s)p Fp(\))911 2450 y Fu(\024)1139 2355 y Ft(Y)1071 2571 y Fp(\()p Fs(G;s)p Fp(\))p Fr(2)p Fs(R)1351 2450 y Fv(")1397 2409 y Fr(j)p Fs(G)p Fr(j)1511 2450 y Fv(e)1556 2409 y Fr(\000)p Fp(\()p Fs(m)1700 2419 y Fn(j)s Fm(\000)p Fo(1)1812 2409 y Fr(\000)p Fs(\016)1898 2419 y Fn(j)1931 2409 y Fp(\))1968 2392 y Fd(b)1958 2409 y Fr(T)1997 2419 y Fn(j)s Fm(\000)p Fo(1)2108 2409 y Fp(\()p Fs(G;s)p Fp(\))2291 2450 y Fx(sup)2303 2530 y Fs(j)t Fr(\025)p Fp(0)2454 2339 y Ft(h)2501 2450 y Fv(ej)h Fx(exp)2775 2369 y Ft(\010)2855 2450 y Fu(\000)g Fv(\016)2998 2465 y Fs(j)3044 2382 y Fx(1)p 3044 2427 V 3044 2518 a(4)3103 2450 y Fv(\015)3154 2465 y Fs(j)3190 2369 y Ft(\011)3248 2339 y(i)0 2782 y Fx(where)46 b(w)m(e)g(used)g(again)f(\(5.51\).)79 b(The)46 b(b)s(ound)f(\(5.49\))f(follo)m(ws)i(since)g(sup)2865 2805 y Fs(r)r Fr(\025)p Fp(0)2993 2782 y Fu(f)p Fv(e)17 b(r)i(e)3213 2746 y Fr(\000)p Fs(r)3306 2782 y Fu(g)49 b Fx(=)f(1)d(and)0 2926 y Fv(\016)43 2941 y Fs(j)80 2926 y Fv(\015)131 2941 y Fs(j)195 2926 y Fu(\025)28 b Fx(8)p Fv(j)33 b Fu(\025)28 b Fx(4)p Fv(j)6 b Fx(,)33 b(see)g(item)h(1)e (Lemma)h(5.9.)p 3709 2926 4 66 v 3713 2864 59 4 v 3713 2926 V 3771 2926 4 66 v Black 0 3154 a Fy(Lemma)39 b(5.14.)p Black 42 w Fw(F)-7 b(or)34 b Fv(R)29 b Fu(2)f(R)1132 3169 y Fs(j)1169 3154 y Fw(,)34 b(let)1074 3371 y Ft(e)1069 3398 y Fv(\020)1112 3413 y Fs(R)1197 3398 y Fx(:=)27 b Fv(")1373 3357 y Fr(j)p Fs(R)-11 b Fg(\026)1452 3367 y Fn(j)1485 3357 y Fr(j)1593 3303 y Ft(Y)1525 3519 y Fp(\()p Fs(G;s)p Fp(\))p Fr(2)p Fs(R)1804 3398 y Fx(exp)1970 3317 y Ft(\010)2050 3398 y Fu(\000)23 b Fv(\016)2193 3413 y Fs(j)2245 3373 y Ft(b)2230 3398 y Fu(T)2284 3413 y Fs(j)t Fr(\000)p Fp(1)2411 3398 y Fx(\()p Fv(G;)17 b(s)p Fx(\))2654 3317 y Ft(\011)3531 3398 y Fx(\(5.52\))0 3723 y Fw(Then)1300 3867 y Fx(sup)1294 3948 y Fs(g)r Fr(2G)1419 3958 y Fn(j)1483 3772 y Ft(X)1468 3981 y Fn(R)p Fm(2R)1609 3997 y Fn(j)1476 4048 y(R)-10 b Fi(\026)1528 4064 y Fn(j)1560 4048 y Fm(3)p Fn(g)1664 3841 y 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bop Black Black 146 407 a Fx(F)-8 b(rom)47 b(equation)g(\(3.8\))f(and)h(the)h(previous)g(de\014nitions)g (w)m(e)g(ha)m(v)m(e)g(that)f(the)g(iden)m(tit)m(y)i(\(2.11\))0 551 y(holds.)55 b(On)36 b(the)h(other)f(hand,)h(from)f(Condition)h (2.1,)g(the)g(\()p Fv(Y)2324 566 y Fp(0)2363 551 y Fx(\()p Fv(g)t Fx(\))p Fv(;)17 b Fu(;)p Fx(\){compatibilit)m(y)36 b(of)f Fv(Z)3450 566 y Fs(g)r(;)p Fp(\003)3559 551 y Fx(,)i(and)0 696 y(the)c(\(supp)18 b Fv(R)p 424 712 76 4 v 1 w(;)586 673 y Fq(\\)543 696 y Fv(R)p 543 712 V 12 w Fj(\026)682 711 y Fr(\025)p Fs(j)t Fp(+1)863 696 y Fx(\){compatibilit)m(y)34 b(of)e Fv(\020)1700 711 y Fs(R)p 1700 723 54 3 v(;)p Fp(\003)1858 696 y Fx(w)m(e)i(easily)f(get)g (that)f(item)i(2)e(holds)h(true.)146 840 y(No)m(w,)41 b(from)d(\(6.1\))f(and)i(\(6.3\))e(it)h(follo)m(ws)h(that)f(if)g(diam) 2264 855 y Fr(1)2338 840 y Fx(\()p Fv(X)8 b Fx(\))37 b Fv(>)g(\045)i Fx(and)f Fq(@)p Fv(g)j Fu(2)d(G)3243 855 y Fr(\025)p Fp(1)3376 840 y Fx(suc)m(h)h(that)0 985 y Fv(Y)57 1000 y Fp(0)96 985 y Fx(\()p Fv(g)t Fx(\))27 b(=)g Fv(X)38 b Fx(then)31 b(\011)768 1000 y Fs(X)q(;)p Fp(\003)927 985 y Fx(=)d(0.)42 b(Moreo)m(v)m(er,)32 b(recalling)f(item) f(6)g(in)g(De\014nition)g(2.4,)g(for)g(eac)m(h)g Fv(x)f Fu(2)f Fq(L)i Fx(w)m(e)0 1129 y(get)457 1243 y Ft(X)454 1454 y Fs(X)5 b Fr(3)p Fs(x)649 1337 y Fx(sup)621 1419 y Fp(\003)p Fr(\032\032)p Fl(L)840 1337 y Fu(k)p Fx(\011)966 1352 y Fs(X)q(;)p Fp(\003)1098 1337 y Fu(k)1148 1352 y Fr(1)1250 1337 y Fu(\024)1358 1243 y Ft(X)1355 1454 y Fs(X)g Fr(3)p Fs(x)1550 1337 y Fx(sup)1522 1419 y Fp(\003)p Fr(\032\032)p Fl(L)1741 1227 y Ft(h)1789 1337 y Fu(k)p Fx(\011)1915 1352 y Fs(X)q(;)p Fp(\003)p Fs(;)p Fp(0)2101 1337 y Fu(k)2151 1352 y Fr(1)2248 1337 y Fx(+)2380 1213 y Fs(k)2417 1221 y Fn(x)2346 1243 y Ft(X)2356 1453 y Fs(j)t Fp(=1)2558 1243 y Ft(X)2549 1443 y Fn(g)r Fm(2G)2660 1459 y Fn(j)2693 1443 y Fo(:)2516 1515 y Fn(Y)2552 1530 y Fo(0)2587 1515 y(\()p Fn(g)r Fo(\))p Fm(3)p Fn(x)2770 1337 y Fu(k)17 b Fx(log)g Fv(Z)3053 1286 y Fp(\()p Fs(j)t Fp(\))3046 1364 y Fs(g)r(;)p Fp(\003)3154 1337 y Fu(k)3204 1352 y Fr(1)3279 1227 y Ft(i)3579 1337 y Fx(\(6.4\))0 1665 y(By)31 b(exploiting)h(\(2.5\))e(in)g(Condition)h(2.1,)g(the)g (\014rst)g(term)g(on)g(the)f(righ)m(t{hand)h(side)h(of)e(\(6.4\))g(can) h(b)s(e)0 1810 y(easily)j(b)s(ounded)f(as)g(follo)m(ws)937 1933 y Ft(X)934 2144 y Fs(X)5 b Fr(3)p Fs(x)1129 2027 y Fx(sup)1101 2109 y Fp(\003)p Fr(\032\032)p Fl(L)1320 2027 y Fu(k)p Fx(\011)1446 2042 y Fs(X)q(;)p Fp(\003)p Fs(;)p Fp(0)1633 2027 y Fu(k)1683 2042 y Fr(1)1785 2027 y Fu(\024)1893 1933 y Ft(X)1890 2144 y Fs(X)g Fr(3)p Fs(x)2085 2027 y Fx(sup)2057 2109 y Fp(\003)p Fr(\032\032)p Fl(L)2277 2027 y Fu(k)p Fv(V)2384 2042 y Fs(X)q(;)p Fp(\003)2515 2027 y Fu(k)2565 2042 y Fr(1)2667 2027 y Fu(\024)29 b Fv(A)733 b Fx(\(6.5\))0 2324 y(T)-8 b(o)29 b(b)s(ound)f(the)h(second)h (term)f(on)f(the)h(righ)m(t{hand)g(side)g(of)f(\(6.4\))g(w)m(e)i(note)f (that)3014 2240 y Ft(\014)3014 2300 y(\014)3047 2324 y Fu(f)p Fv(g)i Fu(2)d(G)3328 2339 y Fs(j)3392 2324 y Fx(:)45 b Fv(Y)3521 2339 y Fp(0)3560 2324 y Fx(\()p Fv(g)t Fx(\))27 b Fu(3)0 2469 y Fv(x)p Fu(g)105 2384 y Ft(\014)105 2444 y(\014)182 2469 y Fu(\024)45 b Fx([\000)392 2484 y Fs(j)457 2469 y Fx(+)29 b(1)f(+)h(2)p Fv(#)850 2484 y Fs(j)886 2469 y Fx(])913 2433 y Fs(d)954 2469 y Fx(.)72 b(Hence)44 b(the)e(b)s(ound)g(\(2.12\),)i(whic)m(h)g(completes)g(the)e (pro)s(of)g(of)f(item)i(3,)0 2613 y(follo)m(ws)33 b(from)g(the)g(ab)s (o)m(v)m(e)g(inequalit)m(y)-8 b(,)34 b(\(3.9\),)f(\(6.4\),)f(and)g (\(6.5\).)146 2758 y(In)25 b(order)g(to)f(pro)m(v)m(e)i(item)f(4)g(let) g(us)g(\014rst)g(sho)m(w)h(that)e(for)g Fv(G)k Fu(\032\032)h(G)2529 2773 y Fs(j)2590 2758 y Fx(and)c Fv(s)i Fu(\025)h Fx(0,)e(if)f(\()p Fu(j)p Fv(G)p Fu(j)p Fv(;)17 b(s)p Fx(\))26 b Fu(6)p Fx(=)i(\(1)p Fv(;)17 b Fx(0\))0 2902 y(w)m(e)34 b(ha)m(v)m(e)1256 3022 y Ft(b)1241 3047 y Fu(T)1295 3062 y Fs(j)t Fr(\000)p Fp(1)1422 3047 y Fx(\()p Fv(G;)17 b(s)p Fx(\))27 b Fu(\025)1832 2979 y Fx(1)p 1808 3024 98 4 v 1808 3115 a(12)1915 3047 y(diam)2126 3062 y Fr(1)2201 2966 y Ft(\000)2247 3047 y Fv(Y)2304 3062 y Fs(s)2340 3047 y Fx(\()p Fv(G)p Fx(\))2493 2966 y Ft(\001)3579 3047 y Fx(\(6.6\))0 3234 y(It)46 b(is)h(in)m(teresting)g(to)f(remark)g(that)g(the)h(b)s(ound)f(\(6.6\))f (migh)m(t)i(fail)e(if)h(it)g(w)m(ere)h Fv(G)k Fu(\032\032)g(G)3485 3249 y Fr(\025)p Fs(j)3623 3234 y Fx(and)0 3378 y Fv(G)22 b Fu(\\)h(G)247 3393 y Fr(\025)p Fs(j)t Fp(+1)457 3378 y Fu(6)p Fx(=)k Fu(;)p Fx(.)44 b(If)33 b Fu(j)p Fv(G)p Fu(j)27 b Fx(=)h(1)k(then)i Fv(G)27 b Fx(=)h Fu(f)p Fv(g)t Fu(g)k Fx(for)g(some)h Fv(g)f Fu(2)c(G)2363 3393 y Fs(j)2400 3378 y Fx(;)33 b(b)m(y)g(recalling)h(\(2.9\),)e(\(5.5\))g(w)m(e)i(get,) 0 3523 y(since)g Fv(s)28 b Fu(\025)g Fx(1)k(and)h Fv(#)746 3538 y Fs(j)810 3523 y Fv(>)28 b Fx(3\000)1024 3538 y Fs(j)1060 3523 y Fx(,)89 3739 y Ft(b)74 3764 y Fu(T)128 3779 y Fs(j)t Fr(\000)p Fp(1)255 3764 y Fx(\()p Fu(f)p Fv(g)t Fu(g)p Fv(;)17 b(s)p Fx(\))26 b(=)712 3696 y(1)p 712 3741 49 4 v 712 3832 a(4)771 3764 y(d)825 3779 y Fp(1)881 3764 y Fx(\()p Fu(Q)p Fx(\()p Fv(g)t Fx(\))p Fv(;)17 b(y)1219 3779 y Fs(s)1254 3764 y Fx(\()p Fv(g)t Fx(\)\))27 b Fu(\025)1561 3696 y Fx(1)p 1561 3741 V 1561 3832 a(4)1620 3764 y(\()p Fv(#)1715 3779 y Fs(j)1774 3764 y Fx(+)22 b Fv(s)p Fx(\))28 b Fu(\025)2123 3696 y Fx(1)p 2099 3741 98 4 v 2099 3832 a(12)2206 3764 y(\(2)p Fv(#)2350 3779 y Fs(j)2409 3764 y Fx(+)22 b(2)p Fv(s)g Fx(+)g(\000)2783 3779 y Fs(j)2819 3764 y Fx(\))28 b Fu(\025)3025 3696 y Fx(1)p 3000 3741 V 3000 3832 a(12)3108 3764 y(diam)3319 3779 y Fr(1)3410 3764 y Fx(\()p Fv(Y)3505 3779 y Fs(s)3542 3764 y Fx(\()p Fv(g)t Fx(\)\))0 3981 y(Let,)36 b(no)m(w,)h Fu(j)p Fv(G)p Fu(j)31 b(\025)i Fx(2)i(and)g Fv(s)d Fx(=)g(0.)51 b(Recall)36 b Fv(#)1656 3996 y Fs(j)1725 3981 y Fv(<)c Fx(2)p Fv(\015)1933 3996 y Fs(j)2004 3981 y Fx(and)k Fu(T)25 b Fx(\()p Fv(G)p Fx(\))32 b Fu(\025)h Fv(\015)2622 3996 y Fs(j)2690 3981 y Fv(>)f Fx(2\000)2908 3996 y Fs(j)2944 3981 y Fx(.)52 b(By)36 b(applying)f(\(5.5\))0 4126 y(w)m(e)f(get)111 4333 y Ft(b)96 4358 y Fu(T)150 4373 y Fs(j)t Fr(\000)p Fp(1)277 4358 y Fx(\()p Fv(G;)17 b(s)p Fx(\))27 b Fu(\025)662 4291 y Fx(1)p 662 4335 49 4 v 662 4427 a(2)721 4358 y Fu(T)e Fx(\()p Fv(G)p Fx(\))j Fu(\025)1096 4291 y Fx(1)p 1096 4335 V 1096 4427 a(3)1155 4358 y Fv(\015)1206 4373 y Fs(j)1264 4358 y Fx(+)1372 4291 y(1)p 1372 4335 V 1372 4427 a(6)1431 4358 y Fu(T)d Fx(\()p Fv(G)p Fx(\))j Fu(\025)1806 4291 y Fx(1)p 1806 4335 V 1806 4427 a(6)1865 4358 y Fv(#)1922 4373 y Fs(j)1981 4358 y Fx(+)2113 4291 y(1)p 2089 4335 98 4 v 2089 4427 a(12)2196 4358 y(diam)2407 4373 y Fr(1)2482 4278 y Ft(\000)2528 4358 y Fu(Q)p Fx(\()2666 4333 y Ft(b)2647 4358 y Fv(G)p Fx(\))2762 4278 y Ft(\001)2835 4358 y Fu(\025)2975 4291 y Fx(1)p 2951 4335 V 2951 4427 a(12)3058 4358 y(diam)3269 4373 y Fr(1)3344 4278 y Ft(\000)3390 4358 y Fv(Y)3447 4373 y Fp(0)3486 4358 y Fx(\()3543 4333 y Ft(b)3524 4358 y Fv(G)p Fx(\))3639 4278 y Ft(\001)0 4576 y Fx(Finally)-8 b(,)33 b(in)g(the)g(case)g Fu(j)p Fv(G)p Fu(j)27 b(\025)h Fx(2)33 b(and)f Fv(s)c Fu(\025)g Fx(1)k(b)m(y)i(\(5.5\))15 4789 y Ft(b)0 4814 y Fu(T)54 4829 y Fs(j)t Fr(\000)p Fp(1)181 4814 y Fx(\()p Fv(G;)17 b(s)p Fx(\))27 b Fu(\025)566 4747 y Fx(1)p 566 4791 49 4 v 566 4882 a(6)625 4814 y Fu(T)f Fx(\()p Fv(G)p Fx(\))20 b(+)985 4747 y(1)p 985 4791 V 985 4882 a(6)1044 4814 y(\()p Fv(#)1139 4829 y Fs(j)1197 4814 y Fx(+)h Fv(s)p Fx(\))27 b Fu(\025)1545 4747 y Fx(1)p 1520 4791 98 4 v 1520 4882 a(12)1628 4814 y(diam)1839 4829 y Fr(1)1914 4733 y Ft(\000)1960 4814 y Fu(Q)p Fx(\()2098 4789 y Ft(b)2079 4814 y Fv(G)p Fx(\))2194 4733 y Ft(\001)2261 4814 y Fx(+)2392 4747 y(1)p 2367 4791 V 2367 4882 a(12)2475 4814 y(\(2)p Fv(#)2619 4829 y Fs(j)2677 4814 y Fx(+)20 b(2)p Fv(s)p Fx(\))28 b Fu(\025)3073 4747 y Fx(1)p 3049 4791 V 3049 4882 a(12)3156 4814 y(diam)3368 4829 y Fr(1)3442 4733 y Ft(\000)3488 4814 y Fv(Y)3545 4829 y Fs(s)3582 4814 y Fx(\()3638 4789 y Ft(b)3620 4814 y Fv(G)o Fx(\))3734 4733 y Ft(\001)146 5032 y Fx(F)-8 b(rom)37 b(\(6.6\))f(w)m(e)i(get)f(that,)h(giv)m(en)g Fv(X)43 b Fu(\032\032)36 b Fq(L)p Fx(,)i(for)e(an)m(y)i Fv(R)p 2249 5048 76 4 v 36 w Fu(2)e(R)p 2461 5048 85 4 v 24 x Fs(j)2618 5032 y Fx(suc)m(h)j(that)e(supp)17 b Fv(R)p 3276 5048 76 4 v 36 w Fx(=)35 b Fv(X)45 b Fx(and)0 5176 y Fv(R)p 0 5192 V 12 w Fj(\026)139 5191 y Fr(\025)p Fs(j)t Fp(+1)348 5176 y Fx(=)28 b Fu(;)k Fx(w)m(e)h(ha)m(v)m(e)1051 5322 y Ft(X)1046 5534 y Fs(R)p Fr(2)p Fs(R)p 1146 5546 54 3 v 1333 5322 a Ft(X)1281 5526 y Fo(\()p Fn(G;s)p Fo(\))p Fm(2)p Fn(R)p Fo(:)1227 5585 y(\()p Fm(j)p Fn(G)p Fm(j)p Fn(;s)p Fo(\))p Fm(6)p Fo(=\(1)p Fn(;)p Fo(0\))1624 5392 y Ft(b)1609 5417 y Fu(T)1663 5432 y Fs(j)t Fr(\000)p Fp(1)1790 5417 y Fx(\()p Fv(G;)17 b(s)p Fx(\))27 b Fu(\025)2200 5350 y Fx(1)p 2175 5394 98 4 v 2175 5485 a(12)2283 5417 y(diam)2494 5432 y Fr(1)2569 5417 y Fx(\()p Fv(X)8 b Fx(\))845 b(\(6.7\))p Black 1841 5806 a(42)p Black eop end %%Page: 43 44 TeXDict begin 43 43 bop Black Black 0 407 a Fx(F)-8 b(urthermore,)28 b(giv)m(en)g Fv(g)j Fu(2)d(G)1067 422 y Fs(j)1130 407 y Fx(and)f Fv(x)h Fu(2)g Fq(L)p Fx(,)g(for)e(an)m(y)h Fv(R)p 1933 423 76 4 v 29 w Fu(2)h(R)p 2130 423 85 4 v 24 x Fs(j)2277 407 y Fx(suc)m(h)g(that)e(supp)18 b Fv(R)p 2914 423 76 4 v 29 w Fu(3)28 b Fv(x)p Fx(,)g Fv(R)p 3221 423 V 12 w Fj(\026)3360 422 y Fs(j)3424 407 y Fu(3)g Fv(g)t Fx(,)f(and)0 551 y Fv(R)p 0 567 V 12 w Fj(\026)139 566 y Fr(\025)p Fs(j)t Fp(+1)348 551 y Fx(=)h Fu(;)22 b Fx(w)m(e)i(ha)m(v)m(e)g(that)e(the)h(left{hand)g(side)h(of)e(\(6.7\)) g(is)h(b)s(ounded)h(from)e(b)s(elo)m(w)i(b)m(y)f(d)3306 566 y Fr(1)3381 551 y Fx(\()p Fv(x;)17 b(g)t Fx(\))p Fv(=)p Fx(12.)0 696 y(Recalling)36 b(\(5.52\),)f(b)m(y)h(applying)g (Lemma)g(5.13,)f(and)h(noticing)f(that)h Fv(m)2734 711 y Fs(j)t Fr(\000)p Fp(1)2885 696 y Fu(\000)24 b Fx(2)p Fv(\016)3078 711 y Fs(j)3147 696 y Fu(\025)32 b Fv(m)3341 711 y Fs(j)3378 696 y Fx(,)k(w)m(e)h(ha)m(v)m(e)0 840 y(that)32 b(for)g(eac)m(h)i Fv(x)28 b Fu(2)g Fq(L)202 1002 y Ft(X)199 1213 y Fs(X)5 b Fr(3)p Fs(x)366 1096 y Fv(e)411 1055 y Fs(q)r(\013)p Fp(diam)643 1063 y Fm(1)708 1055 y Fp(\()p Fs(X)g Fp(\))875 1096 y Fx(sup)847 1178 y Fp(\003)p Fr(\032\032)p Fl(L)1067 1096 y Fu(k)p Fx(\010)1187 1111 y Fs(X)q(;)p Fp(\003)1319 1096 y Fu(k)1369 1111 y Fr(1)1471 1096 y Fu(\024)1579 1002 y Ft(X)1576 1213 y Fs(X)g Fr(3)p Fs(x)1743 1096 y Fv(e)1788 1055 y Fs(q)r(\013)p Fp(diam)2020 1063 y Fm(1)2085 1055 y Fp(\()p Fs(X)g Fp(\))2224 956 y Ft(\024)2321 1096 y Fx(sup)2293 1178 y Fp(\003)p Fr(\032\032)p Fl(L)2513 1096 y Fu(k)p Fx(\010)2633 1111 y Fs(X)q(;)p Fp(\003)p Fs(;)p Fp(0)2820 1096 y Fu(k)2870 1111 y Fr(1)449 1516 y Fx(+)547 1422 y Ft(X)558 1632 y Fs(j)t Fr(\025)p Fp(1)944 1422 y Ft(X)921 1622 y Fn(R)p 921 1632 47 3 v Fm(2R)p 1009 1632 53 3 v 1061 1638 a Fn(j)1094 1622 y Fo(:)718 1694 y(supp)10 b Fn(R)p 854 1704 47 3 v Fo(=)p Fn(X)q(;)f(R)p 1026 1704 V -10 w Fi(\026)1078 1712 y Fm(\025)p Fn(j)s Fo(+1)1236 1694 y(=)p Fm(;)1224 1432 y Ft(\014)1224 1491 y(\014)1258 1516 y 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b(.A.)28 b(P)m(earce,)i (\\Mathematical)f(prop)s(erties)f(of)f(p)s(osition{space)i (renormaliza-)201 551 y(tion)j(group)g(transformations.")h Fa(J.)g(Statist.)g(Ph)m(ys.)h Fy(20)p Fx(,)f(499{545)e(\(1979\).)p Black 0 779 a([27])p Black 49 w(K.)f(Haller,)h(T.)g(Kennedy)-8 b(,)32 b(\\Absence)h(of)d(renormalization)h(group)f(pathologies)h(near) f(the)h(crit-)201 923 y(ical)h(temp)s(erature.)i(Tw)m(o)f(examples.")i Fa(J.)e(Statist.)g(Ph)m(ys.)h Fy(85)p Fx(,)f(607{637)e(\(1996\).)p Black 0 1151 a([28])p Black 49 w(R.B.)42 b(Israel,)j(\\Banac)m(h)d (algebras)h(and)f(Kadano\013)g(transformations)g(in)g(random)h (\014elds.")g(J.)201 1295 y(F)-8 b(ritz,)28 b(J.L.)g(Leb)s(o)m(witz)g (and)g(D.)f(Szasz)i(editors)f(\(Esztergom)h(1979\),)e(V)-8 b(ol.)28 b(I)s(I,)f(593{608)f(\(North{)201 1440 y(Holland,)32 b(Amsterdam)i(1981\).)p Black 0 1667 a([29])p Black 49 w(R.)44 b(Kotec)m(k)q(\023)-50 b(y,)50 b(D.)44 b(Preiss,)49 b(\\Cluster)d(expansion)h(for)d(abstract)h(p)s(olymer)h(mo)s(dels.")f Fa(Comm.)201 1812 y(Math.)32 b(Ph)m(ys.)j Fy(103)p Fx(,)e(491{498)e (\(1986\).)p Black 0 2039 a([30])p Black 49 w(F.)j(Martinelli,)i(\\An)e (elemen)m(tary)j(approac)m(h)e(to)f(\014nite)h(size)h(conditions)g(for) e(the)h(exp)s(onen)m(tial)201 2183 y(deca)m(y)42 b(of)f(co)m(v)-5 b(ariance)41 b(in)h(lattice)f(spin)h(mo)s(dels.")g Fw(On)g(Dobrushin) -10 b('s)42 b(way.)g(F)-7 b(r)i(om)41 b(pr)-5 b(ob)g(ability)201 2328 y(the)g(ory)42 b(to)g(statistic)-5 b(al)42 b(physics)p Fx(,)g(169{181,)f(Amer.)g(Math.)f(So)s(c.)h(T)-8 b(rans.)41 b(Ser.)g(2,)h(198,)g Fw(A)n(mer.)201 2472 y(Math.)35 b(So)-5 b(c.,)34 b(Pr)-5 b(ovidenc)g(e,)33 b(RI.)f Fx(\(2000\))p Black 0 2700 a([31])p Black 49 w(F.)g(Martinelli,)h(priv)-5 b(ate)33 b(comm)m(unication.)p Black 0 2927 a([32])p Black 49 w(F.)e(Martinelli,)j(E.)e(Olivieri,)i(\\Approac)m(h)f(to)f (equilibrium)i(of)d(Glaub)s(er)h(dynamics)j(in)d(the)h(one)201 3072 y(phase)g(region)g(I.)f(The)i(attractiv)m(e)f(case.")h Fa(Comm)m(un.)g(Math.)f(Ph)m(ys.)h Fy(161)p Fx(,)f(447{486)e(\(1994\).) p Black 0 3299 a([33])p Black 49 w(F.)44 b(Martinelli,)49 b(E.)c(Olivieri,)k(\\Instabilit)m(y)d(of)e(renormalization)h(group)g (pathologies)g(under)201 3444 y(decimation.")33 b Fa(J.)g(Statist.)g (Ph)m(ys.)h Fy(79)p Fx(,)f(25{42)e(\(1995\).)p Black 0 3671 a([34])p Black 49 w(F.)d(Martinelli,)i(E.)f(Olivieri,)i(R.)d(Sc) m(honmann,)j(\\F)-8 b(or)28 b(2{)p Fv(D)i Fx(lattice)g(spin)f(systems)i (w)m(eak)f(mixing)201 3816 y(implies)k(strong)e(mixing.")i Fa(Comm)m(un.)g(Math.)f(Ph)m(ys.)h Fy(165)p Fx(,)f(33{47)e(\(1994\).)p Black 0 4043 a([35])p Black 49 w(C.)25 b(Maes,)j(F.)d(Redig,)i(S.)e (Shlosman,)j(A.)d(V)-8 b(an)25 b(Mo\013aert,)i(\\P)m(ercolation,)g (path)f(large)f(deviations)201 4188 y(and)32 b(w)m(eakly)j(Gibbs)e (states.")g Fa(Comm)m(un.)h(Math.)f(Ph)m(ys.)h Fy(209)p Fx(,)f(517{545)e(\(2000\).)p Black 0 4415 a([36])p Black 49 w(E.)36 b(Olivieri,)i(\\On)e(a)g(cluster)i(expansion)f(for)f (lattice)h(spin)g(systems:)53 b(a)36 b(\014nite)h(size)g(condition)201 4560 y(for)32 b(the)h(con)m(v)m(ergence.")i Fa(J.)d(Statist.)h(Ph)m (ys.)i Fy(50)p Fx(,)d(1179{1200)f(\(1988\).)p Black 0 4787 a([37])p Black 49 w(E.)c(Olivieri,)i(P)-8 b(.)28 b(Picco,)h(\\Cluster)f(expansion)g(for)f Fv(D)s Fx({dimensional)h (lattice)g(systems)h(and)e(\014nite)201 4931 y(v)m(olume)34 b(factorization)e(prop)s(erties.")h Fa(J.)g(Statist.)g(Ph)m(ys.)h Fy(59)p Fx(,)f(221{256)e(\(1990\).)p Black 0 5159 a([38])p Black 49 w(S.B.)56 b(Shlosman,)62 b(\\P)m(ath)56 b(large)f(deviation)i (and)e(other)h(t)m(ypical)h(prop)s(erties)f(of)g(the)g(lo)m(w{)201 5303 y(temp)s(erature)29 b(mo)s(dels,)g(with)g(applications)g(to)f(the) h(w)m(eakly)h(Gibbs)e(states.")h Fa(Mark)m(o)m(v)g(Pro)s(cess.)201 5448 y(Related)k(Fields)g Fy(6)p Fx(,)g(121{133)e(\(2000\).)p Black 1841 5806 a(46)p Black eop end %%Page: 47 48 TeXDict begin 47 47 bop Black Black Black 0 407 a Fx([39])p Black 49 w(R.H.)42 b(Sc)m(honmann,)47 b(S.B.)d(Shlosman,)i(\\Complete)e (analyticit)m(y)h(for)d(2D)g(Ising)i(completed.")201 551 y Fa(Comm.)33 b(Math.)g(Ph)m(ys.)h Fy(170)p Fx(,)f(453{482)e (\(1995\).)p Black 0 779 a([40])p Black 49 w(A.)j(Suto,)i(\\W)-8 b(eak)35 b(singularit)m(y)h(and)f(absence)i(of)d(metastabilit)m(y)i(in) f(random)g(Ising)h(ferromag-)201 923 y(nets.")d Fa(J.)g(Ph)m(ys.)h(A)f Fy(15)p Fx(,)f(L7494{L752)f(\(1982\).)p Black 1841 5806 a(47)p Black eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0407091143653--