Content-Type: multipart/mixed; boundary="-------------0107120818402" This is a multi-part message in MIME format. ---------------0107120818402 Content-Type: text/plain; name="01-263.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-263.keywords" Hyperbolic invariant tori, KAM theory, Resummation, Divergent series, Lindstedt series, Trees, Self-energy graphs ---------------0107120818402 Content-Type: application/postscript; name="hyptor.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="hyptor.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: bassi16.dvi %%Pages: 32 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX10 CMBX12 CMSY7 CMR10 CMCSC10 CMTI10 CMSL10 CMMI10 %%+ CMSY10 CMMIB10 MSBM10 CMMI7 CMR7 CMEX10 MSBM7 CMMI5 CMR5 CMBX7 CMR8 %%+ CMR6 CMSY5 CMR9 CMBX5 CMTT10 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o hyptor.ps bassi16 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.07.12:1305 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro %! 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX5 %!PS-AdobeFont-1.1: CMBX5 1.0 %%CreationDate: 1991 Aug 20 16:35:12 % 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 (CMBX5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /zero put readonly def /FontBBox{-40 -250 1483 756}readonly def /UniqueXX 5000763 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be28899b87700197ad7324570ad0fc53d9738 4a4e30a779b09ecbcd7b809dd648167d743332426b052cdfe33910579fd3bbab 2e760c399269571b5a80343c534ef49bd9740608f29379c5239eca7b4037cbb0 265c9efc841cb1a2ae57a601a16a789e5a588c3b39ca5e0db9cb4f7c193bba6a de9b9433619085040cd4068b0e0021b2b1f6518e2b6adbe1d48f33bebc15cb4f 8c70513fa15d916ea331a1c5576bf455e7ec5760567265bfdeafd91c6801be84 70032f036d9d61d27db018e2f35c85f5be645ad21502c177d36aa2b3851d4d66 2b5de0648c8c390ff33bf0d20cbb56a51969908fecce2ee79e9592862a6ff134 387379434a90905174ea81fd029b77146cc4a94ebefc2662eb7b19e7a5084d25 602fa58bd20b3c994f8f1e73704fc0812d92445e851ba513eab428e965b2b7c8 ea5b3f0db38245649ecff6daf3952ce3627e8ec8894ef187b86a8ef6aa3cb81c dac8b27b0dc31a2a064af93f6caa33e92d2b98e1df697a1c2fdb8d18fdf67f1f 8978bca712e32360481351d712d8ed45ecb97d23756b2d071ca1ebf2a665e121 5ce1848cd80c80600b9b7ee22ae0557a00ebcd6817a6a4e0a715a37850138a34 751bd5950b9cbef7c2043c4ee7e4c9dad028e9c175ddae33419f41e340a010b4 0c8eb0b52ba8530868095e3ba6a4e07608b36ef5297f51b545ea629b938046d4 ef48f1f2d594ddbd6bc10e711a5e6d58f871f6a199813f7f4e90c4f9e4341634 79ffcf4655ae7e1dc46198e3086347c5360f733c837de9060c14bf696a8e3e16 9e6051b88746cbfb6b11a522653121b3e5eff4711229e2cc5a3561351853992f fb82bc8c27dbb3dffd510200c2e03fa054f6f5b934877042f2a8b9e24fc0099d 6d1b717b2e37ac9642ce6f4f2de058ee70fe8b878dd769d7b99fb4a3034aecaa 28a9c0f18db4299115d70ebc128733281d70ad1083761ac21f3521a2ea6a7bce 2c7645b3178313e1afe70f17d86fa2b3e46673dd8396c7e4c47a42482ee5c462 8ad785137ff01cb153a8d79b5d735a739a0606ad80782692885b0d037eaf3082 7d229aec8487e33c1ca3ea53474a1064004688139395c51d054efa6505cb07b3 16946c0147b6e78414125fb82898627c1ef9ab945922eac8d059c0cadf05ff02 5745fc505b24843d14e4d5af3cca4a2867a7032c5d6ebf113cbe68021baf26b7 8d7f00683e4f613974f2e3cfa9e354cf3765013135d8b241c5e77cc331ac444e a594953b55eb8288c300b8c8dc1ce6e329fffe68a50cb2b1a43d2f3dc90a32b4 c9f73af809f743f9e66f676cfbc68c633a34954d78f8429c5559e7bb5b1def75 55b0186f85 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 49 /one put readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueXX 5000792 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b448685bdd20eb05d5e7c2126132e33a59a7170609 dcf4871a5d023c9ef57d3362d9f2d7a440bb69bf653364105f16f4d0f03582f9 aced3d05cc76489b16e3fa8a446094d30038b06ecceda269f2eab9d19a99c7f9 39f9548f206c5a457a19270b2b82c43b091dfc5573468eaa3e7a4a32f8042891 d85e4b180fcbcb3091d2800e54c87d84ce9cad6869b5aabbbe47f40c68799893 d22b765295e1e69e33aa048b7ed98ba480ceca91f3ebf8ef85fe9a3976909626 b95ac5940d53f9b02215d84a44837ba25ed15cce0d504f1d335065594f3bc824 5405407591cccb11cfd4645da60d960c0b93f187b0cf7b105543c0b70f89af5d 264b6c026e3ae646acf145950202ec73282111e3e601cf2bcde22ce3edf6db23 516481420f26552ff4472d749811f27768150450d0d0ebe3c79f999e99b5c0f2 2eaebb12d97782b1bd91b2a1f62a76412548ad53c0dd411d4a08c0f071c2c218 63d9adb75a4621803ecb84c2bb235b620b658984b2d8e0c4637e2811bc8f0d04 6c8935afc70141e1b2d9c23bdb251d304b3378faf8928bd09686aa0340fb0cb1 dc48c996ef91530ff078666fad227a3589f50b605267212d3a65ebc1019a8eec 9a0739a00279471a01e1505c17658c10030fae32f274fdc8c8774b0d5406c384 a1d17068dcb0c2575c562fdc5a2176609bad9a2c255e426a4325a4fc3053c7b3 2f35f7bbc9aa50135f30223360fa2fbf019294b3af98224c5c05d6038db08bc1 4103010094dc215ef9ccfb96b237851332c8ed81a9184cced4edbda26b381f0c 73a2bc4a92533b0e667e82ce49ddd8ac9e34d9d40fd4b5b9910eab3b1234bbf3 af6608d66eaa424c158eddb75fbe39462708da8f5c18d92fbdb2f534598a844a e076f96a741f8002af3b38d7e4bfa363f5f5d1193014f71b97d4b3e6b49bf0d4 e71db7a5d500ea99a03dba429d7f854bb2bfbe8d386f31f3e64d9f873e838ff6 f088be15e14457df901591c4ae08c940c57d80c5508812c08d85ff9b4690c387 29e0fe39ba911c3fc1ca7b667122b35c5dd48e1801edf133152e2e1d184a69dc e327973eee62fae0d31f16a5f6d54b0e9871de28d8257a33e8fd10ab8acdc9c2 434a8886ba5879de23a94181f440347a3be6ec483ce8799900b74d0dec88ae58 ecb318bfffe0e2033ee33902698a75586e5bab43c6cf00420d661e087323f602 5fb5f0ff98138222b931450307a71bf04ff477d750932a1090ea53e29a18802d f725548c57854b5d4bdfe254fbde50672ea1b2b2e875d7511231 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 22 /precedesequal put dup 48 /prime put dup 50 /element put dup 82 /R put dup 106 /bar put readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueXX 5000815 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6784decd2fb36dec8f 7478da853762e00bc9c52ba86ef50d222ff142abc18e4fb95b86b3e57d9390ca 86e8de6347bf7f48f93322a372c5d7830521b14e8fa0771961b027c81365426d d95d16b950e216a90f28ce806244ff28dd969cb512a23e05045e7b2bcaf7d6f0 ca8b51494f2ec0681bdfde60420628d9bce98a0ca1ecca9140f452a88db19a30 ed46ae881b098c843a51d06f9a153489357fd24de6a0c36cb5a9eb9b3b64dd75 18f83018ae7cefa1c12fd205df6880a705e28457af7b869a6dd28f6b37d82cc3 e51cba348dcdb34efb119b8ce5a996f15492b0f5d34538850a2c1b6a509fe5fa 015eceaf28bd5e74226f3617b02981c7005ca49478b9376227e59a4fd26c2a41 1967f340cd539ec09f3e17c244e8d9444c740b7e9af10974583713d57e21d228 786bf956a6b0e3541dab5e01438aa07a5d16a13737e2ceb30fde92c23c764523 c4327e980f71a031a8dd4af336af1dc2ce7ba746cabb132242d940262218552b 451185dc740a4efc7f310f1da9d5d79a5787494bbfb9c3f178731a72cd3d75a4 0138479fc26c17537f355041efa30062cb4414a90b336751b0d3b80b1ec9ab14 90c88cf89ebe70444850b43e38530e04ecda59cecd75e04a40a7ba6a03b643f5 69a9731c36c2b12d7208ee49867aff64382e7dfb152959c320b3721e7e24cb91 02785ee0f7bacfcf86b3bfab21c8966c53373fbabe2be067dc92c4490d976dc9 8bb4c2f1639566f7b3505bee1573e42cd2a0f1ae22ee376ee2109ab74b2e2703 2703df50db558b155025313c2db371f3de888f83c6aaac28f88fa2a7c0ee182a 9e0e5b8edd111e5bbf8f8524b595fe4b23700b3c0e037659215aab50c742b95b 85a19728f1e9310ae7eadd6576126816e6cfcbb46b9b6c47e5327a3e8fdc31ef 4cc175a0e5b7e3f3295b65907f8cc59b8a83ab443ab22617d389e6d3c02043d8 d4caeaaa6dc2d7bb560cc66bc3ed7c953617b46cd09c8d7c5e756ee8f591202c f96af28ecaa0074deadfd9bd747202f948a1932569b2aa58c92badbd5ae6dda1 c7a95654173ca2765074652343c10d84f12373335b1e6d78734cb6f9cbd3921c 9259f35bbf259ee885bb31ac569374ae7d7d785df60dae64884d66e8355379b0 f9bf5f052bf93030893db143a6296124428a12aa2fc58a6d8577df024311f114 f7d9be656732943ef22f849c3c0947fdc35af69d4c8e8c641322ff8a2e1ccfaa d0ed33a3886f8e351b8baabb6b2c6147caa63a406cb1426c3fb95362ad715dd2 49aa07673e2938ae17b8198d59b3fa47e2b6cc6776dd411f8339337bff7bbed6 1e7aa0948569ff6a094564bd6e3f7be26e106936110ac3a25b761e7748275424 ff2204162ca802682652347f3c2a0ea2ecee8cb19be0a3fd5583fb6e163e1862 9a4ff3a992d39bde15b5952113088d47a43e723623ed7b30e25a7aef5fca9826 dc9c33d1040807470ccd7d282497fb63aa3618050cec0066ae32d6ab402afd6a 02b2cc76efb5ef79a76647e7104ff326d28c10da7fa16653fea1b3b77930ce6d bdae4fda09c6832ecdc420b2e71fb922fe8a7d6e053735a24ed1c99aa72051d6 cd6e02be7dc6a7f3bdabc80b38eb49443f406ba7c58d39c6481eb00f54308f80 a29735aab4b09fc339b0f167a94be95383fbc52c8586ac8eb39ea9a049b3fc8c ca5c9a666cc5ae611d0d3462a6fc0826617b8a2a5c4dcc92d3338e30fd3b852d 4097a14c70b9adba9a2f52ccd96d7f485e4ec5283264afe3fbffa8cb5984c4e6 37d99a4cc4270d89e3fe954c4be0101e126bfba9c8308f969b35a95c8b114b74 2a01fa7ba623a2863be666cd968df874afe69c5ff13e72771483c25a1a59db33 6ab5c364d810623c3ce91f285d6c177a38161ccbd580fa2d9772fb1171e03f5c ae997d5cdd22567e279256e80e3bc0ab2cde7b27bad802f063978d078a985259 457077060cc00653e9e5b381ea9de8cd744a8b028928c3017cc70d4e4e8dbc09 fb2faed3586186ec24 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 61 /equal put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueXX 5000789 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 54 /six put dup 56 /eight put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 63 /question put dup 65 /A put dup 70 /F put dup 79 /O put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueXX 5000791 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b094541f345ec367ae85ca9bd 24568a01d3b9f8095f7420e6c423c414b3dcce6da48dd1c89a56d078e0d0e2f2 62a13640a06d17e44ee3866c3471fb58fedf5a3b77294517651c16bdd7267d39 a54e7171752dbde63ac19bb4b3021ce95eb5fe67390b09ae4d9ed4d704a67443 f55dce17acd996c1f5e023c9e5a18cbeecc3097f23763acb86cdd7cd13381ae7 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMCSC10 %!PS-AdobeFont-1.1: CMCSC10 1.0 %%CreationDate: 1991 Aug 18 17:46:49 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMCSC10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMCSC10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 65 /A put dup 67 /C put dup 70 /F put dup 76 /L put dup 80 /P put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put readonly def /FontBBox{14 -250 1077 750}readonly def /UniqueXX 5000772 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f 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a6104e3f1130505008d31e0b2aff7c25dd55b202f2b8aad930595bfc8ebec9ed 1a5ee83cd989a251384578f950ab14db529cfa1af6915811b3a813b070c30c1b cd3944cbe1f49ebc72e8bf744325 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX7 %!PS-AdobeFont-1.1: CMBX7 1.0 %%CreationDate: 1991 Aug 20 16:35:49 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /zero put readonly def /FontBBox{-55 -250 1289 751}readonly def /UniqueXX 5000765 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5bfae5c1f476ee6f05eb1d2caeb 269958b194521197b1ee2d9de5f5346241e154574f14a4532b35968596ef7a20 102dad629475cf5d654585d4caac68ba7b2ff21b103a98af3e9ec01fae180804 26b73d62e93f0328ce7775e81898de587dd0fbcfe98934a58afb1aa4dd8c6eec d19373ddd360a5e85cdd4b752b2f2baa13b48a282214a05632d28e915ec90824 21ff8f2166e940c7122c583f235bfd098f8aac0f1dbb9a625b02bd889869664e f1856608899ba509d6eb8ece665afc34a459dd5b54dc7711f6f04a288214efe4 53095e239bf19cd36b7028d46c07a2b8c17d452d8563ea88a3bc5e5ea29a9ea9 a7e38481679f893f9e6a983ccb5512bed4aca5888793526701796b3fcac0fb62 f69c66d0688eede0fefd590fbce954b2cdd2abc9017390077a2cf6f6fa7d0db2 dcb4b0963e3dfbf02be1dca9cad104f3d29495d19337fbffdd745e1729e7c0a6 41fdd3bf42c30d715670b3ade44f481d795071f1b24f4f03dff59d91bf9d678b 4560da27ad826c9fc8f8cd9a333248cb2007b4ac0a9e2d5c044774124f1ee23c 0e3e005a4a16e1d0cba15797fc42fa75cb45b24fcf8a8cf41de8e06388b5f995 04454646c3246074f495564cfc8ad75a949cc477ecf03f0511b90b6a7456b730 cf09773b5544f6ec17a9a179714e47e2eb1142d8005bd228affaa375eed7fb1b 075b3023be6386f3805aee05b47c7f370e8ad410110429d2ee8801d0d2da9e68 a453a323c147716ca588fbf563175e5f47d1086a21b04102cba338a3580225ce 00dfb1784fbcad0fcd11434afcc2ec13337ee24775c2952be02e20572aaab17f d21287775e4a4d9fbf877a487b252b5c880463541489f5578c7472d39fc27320 2e4ec86f9935ab2ded14ab9ea36770d79a1a72161102412bc57d6618a634fa8b 08a37e2c25c938ee57de5e7b3093a49da61f5a2331279c4381252aa1a2bce78d b5556e4a162507598240133a7aa9645868f385e78756325c58d9ec3e200bf548 8e82c747a98f3ff56d33baaff99db867ea2c09ad6728396f4f689471737c2311 2b1f75c27b8ad86fa7e9a8d0ad49effc5d48c8a74c403c0f641dfa32bd0457c4 9659662b175012a24d04420ce5d86a332f624758b3675bcc5a09e3feca67892e c5abc0979dc98bf0fb9efd09e50151c109175b1782c6086062991d3b1d50fb73 e7fdff340aadc51fc36032938131484da02c748d55fd9ffc703a2f135b8da724 a179223c35e04c8d236d4afe737f02f4243585e70e92f6822028df9884e36df0 b679c98b6e99431caa6f70e9491c9b9480f119b0a7985fb8602a0bd296f33a33 ce78017877e9 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 2 /Theta put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 97 /a put dup 100 /d put dup 101 /e put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 114 /r put dup 122 /z put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueXX 5000788 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306457c7e62c1d269f306bd3402e266de fc3b5e7d8a8d2f5bf0fe6ddd40d07391df4fad4a6018dce29a2b8f692b29f202 3a7c0e66de8ed85c14f1f8492167357f51a7e84cc5d92e0fee4a71813d2ffb26 445026f89b4787516ecd1afc78f8bd19e91e9ccc9402e8c36d2449c1ff850a8a f61165aac3fe931332dd28e261b91b05edd18f44ea7d58a8f35fe88493b64aad 6bfac3a0136215fc2f4ca8e91d70c5010e6f4013e6d63b44f6fb00afdbd7cdb4 5ec5b1d9736f45cefc8a0124b815987cebd81bbe0d44dedb2d5ef37923b3d551 abb6a1cefca7868fc7bf3814ef7d7b6b1ae6e869cb77aa29e3d90b12b0dc3ff6 ec945922b5899bbf2f12e92731486d2ef1230c528bf8d7e0ae09ad7632a38966 5963de49d1ef3d65bc483e4a577b927c940f5e121169ba52f6576c85793e5fd7 7f5863c488e55bdefd5d8b2514795533aabbc12e7f51816c7e1484f1c441aa9a 66a5dc77158e79cd6692d299b95e8058b35e771a6972d2b5eb1cd6d2bb8e835d 361a6eac0c90c906aea1cd75c4412a1339dc1439f93e59917311bf20af4f2bc8 bba5a4626f75f28a47d4595c37f4019df480d9385975159142618e22e964f019 d88125210e2ef21d65baf500adf802297b24eebbf4e93c617a613f856bd0d45f e17a9480980ef83950bdb8ebf208c895ebc4a7132f0765b7c19a332ebbf8a3c9 ab65063a42e19ca9d43dec3de7aba46b9a5d24fb93d615c0a1726ab7e31c378e ab77150858535b91362ab0e2b968dee1c539964126d937f9d4980ccefa92bc81 b054996a65a13995c9c9d819ff05511cfc89cbfe67447c940405001d61be633b b771eff383fc6977ec06af30f981504260de7b9ec97359974734f495ac073935 fb30e3700679092ef5d971b6b2da0e15960f2391fa6b63c33e7f56b4a9b86951 1d3686a5e8e613a69a0add8b8c73b127ddcd397a48d785079e786d24b30e3a3c 8f184c4c3ebe741567ce7f5a45f26d47643838a07d32951194800f224b875678 1a448a81cec4575af3c042b3d00abfe74cc175c3ce32b076d8ac0b109f5876df 767699d671ee73bb7778d1dd6dcc6e229cd3ba276361a7fd061a1a716fec3b1c 41c22196b0d4c6220b9eeaaaaa018be5ae4fb0eebdc4ba8bf2b3e7c1465b758d 698b5727c8b46bc70aa190006c23dcf67a434a189fc063a5446b3825cf455cb1 958870c35f6a931697fc8f0cec987f5edec1bcee6d6a98047348c79682a07436 9ed3b7d6ed74c17fe1f068616b298523e06a0b9f00727f3b6f736082a3940c3a 18d8f1a3ead472f8be28aa8c532ee972e48e28e3c445e8a2a529b5b9a8e5c203 972db04d2218b7fb4651f3f6d55b57b05f7b7b2acaf491187acbe91bc7978568 33146e29587975027d4c926e2204e5bf0bc61f83da5b13ed2e95bf6f576b4729 e4d7cb0968aefb5fc6f584a3cee1ae91dbc2a41e454e53b86f2dfdc349ba1a66 099840e963001fe57aaae881fddec3f114e8753371c8076337cd48cea7d147d5 831adf3cb2d4f1c0a042e4a70e225564f307c9ad882b53da6ca88963084ad223 7369b0410708e2e497a83ce49fdea75606923477f782f93984f9befe36630184 7403044069420b34828e759d29eb3dccb53ac19bdfad1dbbbddafd8730367e5b 1b5444d3194ce853fd9e850dfe362f6b762622e4d6832d90f1361e8564f96be3 514797514455b1c123eaf960596d52cb0726b7e1a15fcf62e4cdfe122ce87edd 3426c2fcbb9537f7b1c2b51251cd17d6c62112a9ec53e2c54b59761d9d114ebe d99b79f3213a19a342490ebfbab105a4fbea1ac407403e5a66b4cacd64212c46 8a2da0d9409978c328f23f0ab8dc986239cf6bba6b2c7c930df2835ef2ac7a13 cf7f5126f8758648e6bd75cdde48cdf1aa5574d53f59f76a937075df682176be 373a3ad52b162ed8346dcdcc5358937f3b46c41f027b31d6aefd14fbce452412 b714f0c836865d5152539786b66d227a0cf8d5fade610912fcbf1ce510955830 cc441028b8936cf1776f85817653dcbd0866afa897f5b9e2b6c1a106ba3ab079 da29651a86fbbc089027a736761b0b4b79f1a5828d2551d6a8eaf12a43f033c1 468f882b8e7fde42d06cd448c4dfaf1511b6709d14a0433ca6b3cf12314b9f6b b6b6190226ec55c4e0cf4bdfbca6f3d77cbeecb0a8fcdbfdfec60162f569b55b 86800b15db3214d84bcef40dd4e1eb56c009a31117176d5602f6ca1f231d2240 65fd32d7cc5fd86c04bc13a1a8240d4a337536dc3d9be046cabc323c7ea13d6c 997bd99b644ebe47b927aa15229939691d1db0fa1af4028620841ab2d008bf9c b7d30c2d2b8cbe604c2378c2f28cc4d6f1b076f60b8194959e89db85a3ea5c21 65ad7feed4cbe489109f7287ed0e86a092a977ca5726dbfcc389bbd437f0ec80 2154d08bb20c10d58829569c6d8a99edb7117dc05c5d65b200034857c514381d 111a1164426b633d7c408d9d71a4e8b99bdc7067e267c3e93ceda6268299c894 275f3688f821ab9c4cddd1db90c5c246d4efc5ce8f32a91cc96c432bef296f9a 9eb4849a52071eb96e8c84b9392608cba540b99b409776107468b21120048d6b d7e5a3533dc521cad3f0d182c8565359ffce375ffcda6698c9bf5211b2751833 17d6c7253eb3f1588cfffeec9bf404b00414c2e9079f5ea63b507f0ebc761d3a a7a7f6fb485996e483a84e345db7310b81b76b9dcbb984587adfa122ac76d0a7 e0d506f296b79892e0244a9cb299ede6d55628fb905da4efd116cb2d613c5ba3 feb54f68f82abc70656d42da28bf1190864b057d168823e774497d0e8163db7b f8a1f9b25ab8360b48747e5ca371a2ea4bc760fc2927d53c7ae168d23401aa55 27f27944e41763a875c6ae4f2ed4afe6d9dfadbf33632024b0a4b4b12366b975 d302e16e8c789a249ca83f5ed69c05110ae39b2e751a20315a25e65883cd30f0 72790e54a313297326e8470cf3c84de8de1830220b5429841a1e24604afdb371 da271a6f6e5b7b0ec26f8e7647d29cb323cc847ade411f98e61ab912e327539e 4e11065a8fd8b8e8bb47663e0e50e8f5bdd5e71480e67d53df414f7bfe4278c5 af6396ef553b2b6119084985cbd2208207c2c56aa094a437f08f67106703d68d 9311763b3b12bf83dedb868ab84d368c2dff60f4a606b8a6e07f35811e42ad7b 6e67372ac0fe3940091bd1515a466864a01cc29652991f866367bd59ba4c412f aa2552ba44e4898773dc3467b4543c162de912c1c98a2200bc0a1b12873ac715 8373b9dc9af24ab2a918ee1f8ee91dd66e8a1f33761753025a0e9c5c8f8bb535 921e77e6761b11e488bbf3902ad81ed3c1080ed194f8903f2c3943ba81f6022e 4a06fda296ec941e750f2a133f1bf2461211f150b1cc92cff8678b2215d62963 aec44a392f8ec8066c9400e64580d12a7c56824535edf485350585c6f5ae7682 35d85e09cb91dcb5c5c9229a74227c5d723d13c96a319b139e85041d774dab8f d901e97e4ec033ff59a37a242bbc49ff1c03bdedc219205e406344a9f7a96c95 2cbd784e9865c488556f5108256e77392e232cc1c1d5f4c17fa42d512e2cff05 223da509191b7827eee76b31e0e93e95744a7fea3764b5dd76dbb3af2263cafb 5ec2e250ce871c7696992f2629f6c99032a04ce68be4aff90ed1a511c8a7f8b6 4031081f133f92466c4be61eb9e1a05aad5781d081c9021f40201b8586ec1a5b 25910ae88aa5044e43b42f8a385449cfdb99a3b13197ebcc4a21aa451e091c83 375eb3becd75b0929dce35afdfa2e4eb6a6b4ca5dbfe9ac5ce66d0bd31ddffd8 163e0e438ff2fb206061719070279bab8f3f8dd35f3f90489b27ff2f43eaede8 831c289e0eb7002ea02e794808f8f4b9fe3b85332ccc7e703b70d59b02e3123f ec8d72f5c60ef1a15d72c15a2d122f024cd15a6b854e169d010dee81ff8fbf89 c41e35fdb50bf16000c09e682352f797870e761ff8adbceda35ee4da6b5c383f 2e62fd76d57d98bff06ff8fa52d9fc807e3272b7c7fd86ca7436e948e6e67ff7 cea298106cedb22f9c6e535c918780d5ce13352ae3cd9b71296d15f34354d23c 0985c0c3c91d25c2885c013d248d49c1cb9702021cbadbbea09b13cd1bd980b3 8fa57d2695dfd9fcf4171c2a04e6b45ea5caf3908989bfe341ca9ab3ddbaffb4 4f157d4b081698117b2597a833040c43d0a4f7deda816470c3b3f071d7ea7dde 6361e7d6411dc4c4e8e1fee55e98559d226a21c59b00d5363044c7cc90ee922c ddaf749271e2856fb355dd85833af88cd112d55c703e38c390ef184564bf5523 fdb5266b57b4406e7c9f45628ff2d0af4160431109496bdde0aa04f5caf767d4 eac90cd02418c845489d7a6869396ea5a8855944b20323d85a8325bab32ab4f1 e95bcf1c7ba4f1593374f3c7ef594d7e6b7678676107f550a3e20cae68a52ac1 52c37a197b225c063ce693ceafc2ad2b8502f9913fd1f995e43cd5068581011e 1e3a24752dd790490aa89671bc094ea3af360d33522ba39bff83324e3707aff7 a19d70a307a40dff3a941d8640d835787eb3a6e4cf28b724bab9b333b955e313 9ef48711ba405c33753d42505017f5328a1fec53e3120f790e981444e3a27226 c47e7a62cd96cafeb33368a609db996a27c3ee1652850f27706fb53cba3c7bc8 e9a6ab6c35b0e175ec4c169513180b7da4c102f1710b832865d7f21fd1fc8b3b dcaf9b164990041a3931b0ec6af8ec3e5195d0a119b3430281d6b24508e3e4c6 5cfb132bd6773a2b478355b0079b84c35d2cfe0ca202ad6988af54e7fd112602 d73b6d7d920360148acb2347a7c238dcfce955d56a8b9db8216a4c3f27ec275d e5566785634221bb2846e8acb4cba3ad0349a238938dfaf2655b78641888ff38 629a879ca9db82d49f0a56d4b9a8716a74101ff9939b974193324f35574ccb40 25fb5613110648093fbb61c451589aa30de40239bb0a6e31870869a61f143212 637d6fce4e8304ecbc1103552e1316a60d5bf1dba70cb0105803b017dd319f9a 742be205bb5c15bc458a15bd8b61aa847bba0363ca3f4ea249af5edd4d2ac772 40f1d85090b278de92bee2487581901fdcf3896361ec9d5cf2151b6948262f79 f0830b6b9958b6e1705a70c727b17c9844970a76148eaa4ead1a57d6564a9eb5 e8cc4b0f92042136f8b5b0faae83071fa00cf3dee5ccea5c6040003af394b295 c35d09c0f73c9ad1df9565d3c789e9771f9704bc33a9163de9cc09e1d8791b3b db9f9d43218fc9fe1718000d9df4721b6963621666b6b7a24f5395a13231d1a3 7cb87f5a80f9589e35e2725b99af2b84dd64c589bce7f926fa845237bd239745 f25db9a09a4ba0f29f7dfa2eeabb23e32cd276b51cebb2bde7c0c08ac616d90e 56d8f8b0d2e4e050179d45c95aacd9d63d375df583faca9ef3024bf59ed3f730 a5465e1aeb6549812e932e241145415d3579e78e7deef537434e5b2e9c27bda1 a3b02c09f396619c79a91c0c00390c3c667fc1f8383ff10aa844548b20601fbd a22f6e594de81f212bc0b0cf31e478debd004c37fccf46b7c023bbc775f55dc6 cf4f7033303905bcfc31ba41241f957575a8e4d84d04a574729abe8f56176480 84b4168dda190f0416ff573f51108ee738645ae78bbdc8a245b05affe452e76b 7a128d377b0b4e63148d4d0282aa137c28e40c490554ad776baefde6c858fce8 d2798d09b721f51d608f989af196a91dbb710378835cc1f6086559982ad4ae56 3876ecdc59ce5aa1fbd5b8480e8d905fc1a048a9176e791e2134c095b9712213 5c099e0686b543023202fb112d24e1fa44b299345859414816fd29d867067a0f c3cbcdc280bb8e5551e350d6c84878ac7bae4465c3346d40bf43a7e9d40764ae 6e78d2b95341db8127f71f79ace954a789708aba958ace9d539695f03cee1358 6e243f1f2629aa75a3286c5a39d2eadf63e99484cfa008f214f74b5a282d0e51 d693469412ebf63747c346962ec8dbfad170d5251f1f3c93ba929d7fda7f5c83 845f0c47b70f2f3e2f6f223c7e4fdd4583dbb18e15119cef85df4eedf7eae9 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 18 /theta put dup 20 /kappa put dup 23 /nu put dup 28 /tau put dup 34 /epsilon put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 68 /D put dup 77 /M put dup 84 /T put dup 86 /V put dup 96 /lscript put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 117 /u put dup 118 /v put dup 119 /w put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueXX 5087380 def currentdict end currentfile eexec 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945ea4e31489bb0fd2b152cdf94cf88972a86966b92e768771e865ce1c691d3a f06851b1506ab8970217083f195f4ed493b1785d75300dd5d6540c52fb37a6e6 958c1facc587a2dc39990eca143cfaa589a2b244460d0f705798755e3f2d2073 1d295acbddbe333920820d2ebba93f69af0fa8703c645198a7d5ae82f9f8f296 c2baea53711b93cab3f878066fadd77728aa84d7caea4ba24bd493f6c3ea58d1 7c19f3f50e16c2dae09ecb820e9edf31009a1932d54bc5f79d79234f27f93b0f 6a1a3af5b34108a93e906cc46a5e20e409687d339eb90c6105bea2077e172fac 5eec585f050a9de9c29f3680fac39ec4e1ea0e4e028439179d736b5648c3581e 1ee8859ceb3d7e2d75ba149abf720a7310927f243d43705cb3c1aaf1a39cb364 0a643278c8d0b48b87672c1a6e7cbd56edbb4f43df78cc385906ddacde4382c1 7b4d278a242e5f10555338280c22f6e2bf95c95f368e2416f368ab2e18d1cb3b d6eb5239f6acf37f4120c7c6cb1393eb7c64cbca32751c64459e12e23c261c94 ef4c17677ac723f048ff0fe7bb1c1380fd39e886eea80f41afd656dbec3386f1 cbc206dfde6c2796fc26127814b68467737c9e1e94d041800b7dd86d0e95e5c3 f5be85a44c1d94828c77d8ea02a47a788a3f62b0a604ec962c53850a1714f32c 72064d51f5bfeda5253589e67941a8d2770df7eaad18ed741c4c036dfc701647 d872ddde11a70c119473e58e7f5e4048ebdbb29a6fb6babda50ab9f1d2513af6 1e279f5f327eb38447e7d47928cc367d880dd68e4bc4fafdfe87bf629a58d055 f1d3ffcb8045ffc0298e9c347feedd71c87490f121f67d4068a27d6b5e76e30a aed0f138a79c82e86f4f877fc357e3d8113b3958f78328ef245ac1d1ff0c631b d1e2124ce51206ffc636bda197c76017fe86da4890c479b434dbbdee7bb4049d b869c4caf5fae4536877262ec3459cf45e64a03589d55c0c3905c032987a61f3 bdb82fe3cd9e88cf0c149939c07ceb7381c254a62771e47fb33a6ff3d5209ec5 b90cdb793edbe253417decfd2b7d4d3a01be7b608444253aee0c6942 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 78 /N put dup 90 /Z put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueXX 5032014 def currentdict end currentfile eexec b5e2457af4a6caaa4536453c498a618ecdce7b5f6f2d5944265309f833bd3d50 3df8bd14c071f404d2177500681a801856cb8c9c9cf0112c9bd17ccba9ab8944 1759a320b2078df0fafdd3e357994bd81740d1a7522ddfafb75d58f5def046f7 084d10fa2f702e570b09a7073d70efe37b48c355a7089772238b7b4538c4912f cf6703a14fa416057ea2e1b94455913dd271b42846ee10d78270a8118e7e4bcb 8deb048bb280c240d59409d47046c890e5f79e5dd4b0be65a28a77aeb53dd33e ac506a6da87bb55e329f433627e06c3c94b078838b908538d93f888e504e25e0 3a8ccf7d2c02122ab3d485525564295390d1154e15fee35edff4eb016fadd32c 26d1b4c09de87943780954cebc4965aa890fe1f80b03eb07103e13d468245c7b c26455f229e8859585c6eeafd9474acdb27290fd62950dc52f893c99a2a614d4 2a7e3618266f3fb0c68a01a30373e048385ffe7dd5844e208a55d1a9263ab2db cb9b19038724f95af00349840157be45cb337871b55a304b87ade64b10ff0eba 21aa0eaeef74aa01f049303a8b25de0acc26bb3a33fcf267f479a2a73f7dece3 2aa23083687322b417fe86e8f5afc2136d045ad6b392364811a17630a94ff097 038eedeedb8da4ebf8fa5a96ce27a2ebf771d14f4a11fb1f9f48cdd207209934 885ef9118463552984b0fd3cf15d55de967fbb051549dd65f60dfe07f299690e 58252f032bca78935b7d2df852ed688bd13dd171b91cf7bd46d5433aef9f50f1 41a2517c4912b837793f84ddcf1116f70f7e9b1e6ec752dbc9183d4791d92602 397bafe29ac6f879ceb72c673c8a1aa0135d6f92d0db7791f9a2f1a5b55347e0 3e63723efd258754f5f4a9ad35873a98d54b6c54b17a39892dc1d38660597448 fdba9258493a0db0d560c01d49f2d889a99eaa8a5b5919cbd33f6b3d97bf27b3 e0b94b4dcf3fe48b07398eec882c325567e28b9eb4d699111343b1910da9117d 74d6e9de75247ad908f8ce2cfc49e0faa22dbf4d1abe2c035b8f8eee2d1967e0 7cb9b5d8de5e0b1c3a86c7a08a580e51be75a8afdaba2cda08f87d352652a3c7 9bda6bcd32fd59e8079fee3fe7fa161ff626461e85ff8b9b60e2e6bc72884c6f dbaa5e8ad5912206cdd26b204cb34a5403c4fe9cc6d0a2e6805a98624c56baf0 3e673abaa926e941739ba23126dd78ca82499f376d15bfc7cc70f1586f5f10a1 fe99eed1a1f44ace1fc2bb8017599cf6ff3def1dcd7c683d21b5652c6c93b8fd f9fc70c3d5ade5f3509e03de2b93dc0c472061d17beec5a88b851b793206ce75 b32359f056a3c5e76af7a9627a6b2b293cd072e4904b8ea1ee2d0f2cee2175ca 61159ca39298a4f6af8a0ebb1b58b27e885e5708431c42838394cd45ffac06b8 154636f0b299cfec94493715c60025f42809234a89f13b3e1fbceadecd031fab 6a6283d5240402df432ecdc150513a5f917cc45341849958fe523cfb4f12c69b 2e88beef9104f76fd48daf72a990c050b4e5c2d83a06ea0d371d7c1af50e2d67 40cb2c9abc6d77dd70772fc9278e6d49da30c0212c2f94b8982c5712e4371957 2ba955ff6c60656c47616dc710c883796ac55106bf68c080194622842b1ad038 52341556036b02a3f7eb6f38b2309d6b6e99045f5bd4174518ba63d83666766c 93132f0f05f1f0fe8c7474a85a586aa3cd51ef9224707a0708a9966f537d0894 dcf9c38a780a51ba59a43c90bd24f1d6a2050e126c5fc34584ad719d39cbe7e1 143ef5e04ce291fabd56ec617cef4fdae6985b1b9578a354a47004f11f22070f 73ea58d77c881753362b0eba27d7ba8844f5f61ebe2db5bbeb5393a0ed2fa354 6983c159dd941027ea44d54767355c6ed7b16c94627eb33e2096be435cc31619 8a18f365e4150c0163b423490f452a5d417fa9ed181b136c268f5815afb45eb9 4e058d13a0627c4987680c1bc8330dbb569320979a83eed5cad9f45d88d50516 de0e929e0da1466cfcd7538a8d2ba843bafb1409a1b5cb2f705701e887c39ece 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMIB10 %!PS-AdobeFont-1.1: CMMIB10 1.100 %%CreationDate: 1996 Jul 23 07:54:00 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMIB10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMIB10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 23 /nu put dup 32 /psi put dup 33 /omega put readonly def /FontBBox{-15 -250 1216 750}readonly def /UniqueXX 5087392 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d78041490e364babe4b85935b8c9c7758c2dbe253653c 892b1fc2c6a01103e0b8553f847bb38d8e4fcd563ebe51cb7d5f0877af29249d b61636618fb31eeb6cbb4ea4db687fd77ed1574da8c9f9d33b592ec0e38d798c 498a205cab252f7e2f1ddb82a38f331f917b8452af11b2b93e34c1ef55ee005c e8418dbcd0fb9d4d3fa8cb388ec91f16c8864913ad3fc8960d3c424650bdd12c 4db4049367faae97c660f00021f712e9eb6366b7ce74255ff62b72396ca43287 7c77fecdc53883427c90c1c3ff7bd46e472cea9bd72eef0e1be36abb3a50a7bf 6f5d460fa2fc81466014df9ee1e15365642c2de78706dfa740e035b4f28f290d 79b8ba16e5bca13fd9c3eb7eb54006ac2c1cf0bf363b3b69e923797a95b76515 073892c75cf733da7c07da8ed1a8d1ded902f784764c6df6fcd0fd28083d88c8 4c41aeaa6c2c2c1bb10219f4feaab75ee904f4314fc0b9de5e96054f2472ad86 42140315385400686cb07a0585d922ae4b1b25ae722edc6b4afe053560fc9ac9 20a1d2fb8d0921c52c9468d34dba9bed42e3a8308742d36186ed0a3b43f690a4 dca6bb2af8bd698fb303fd68df53edc5e01912cf007dd31ca7481ce844be1258 7002d4aafb204a8075fc6df86f22249d61dd840e7ef87362d72e35f4e8ece044 95ae41ee5c02d6aafaad37141935867a8770fbd9e9e894ea6cc5cf3770f35d98 09be73c56fb053ebb622cd4a3826bc95c192bc2248e1a30e641799994debd48f c5a0ca09be0b2831672271e6a087154d1541899171bc12c8ae00dac586f5c0dc a9c8f4bc7910fe68531c2adf25f29002b74151ec41cfe904b25dc975f97ca163 59d6ea5787a3b0529b3a7a7f8b5b8606cf11dfa1029106881e56be3b79089194 3f00ab0020388dc3ce3ce7fe3efd542eae2b73e9eb4eedda137e3a20bcdb805d 68b88102ea106b3dd3e079dcd70bc4d8031934916155b5d5d061847dec176475 249d61f2177cee9fe6b3fe825b64c0f63e8a6179ae87f25df462500f5e4320d7 49e75b30da75db5011516e42a3b644f4c5111ceab91e4baabb859148ae93e166 c32bf01d6ec7c260b202044b1f4da6272d2a5f09bf554e4e841822cc61b50b7e f65a7a9fc68523b6aee99292be3fe7493053751edd12a7d5aaaffe365c7ffafe d241d32fe4416a8812c02d4c65e44bc6fb8aa5d227831e6b33da6d2ca34d1036 c562fd60fe8e3d8d3ea37a3a92ab2348865e16b68829d830d29acdbaae5017b8 59e9114514f190b8cf15d07718e02f2f513dd22a797a257a1332aacaec311db5 6dd679acf959e34199ce6558dfe8a6b3b3062df121282f35f08faaa214f84f6b 5b9dc1de483c5c5002ea7c53e0c3a420570bea66f651927f2dece97feee868b7 fbffe1cd36cb5c4ddd04e66f96f4c47ab0c48e34ed51026c8cc72246d1b72afc 739353f0481b4e0ff721384379e2d840e3d65d3fed0b210c7672ee5567d09f4e 0c279f8b2e04700cf35410641e6db8466a65b1065a1d5689c6a1b58c331dc2b2 cdea70a5ed5d262b0b4df3b63c03fb4287098363f8a6f5e833fe986cfff64565 0a959ac3a1ed516f5ed2c2644aee1de1180ccaa63e339fcfb4d940cb8f8cd18c 0d033ffb42635c1bbd7f1c317e76abf156dda52b8a992221a17ff0cd5c5d3b39 9f9378f59ec584aebfff65e17f28a50522321906045e14b40f32536666ff6b69 633a535f9c0c2e25689503242137ef5fc75fc9451811c93bdf3ebaa48ca934fe b979f1ff275a0a14abd759551c774cdb5b5fd59ed040c8263023448e42da92db 6483e4416a3e1924a7bd2268277fdfa09c3058c79d3adf6468eb2cb2d709efd1 ba0c3e25222968c949136f3c4dd01ca71d27133b007559a8ffccd8316cd11e84 26640e8a450f7e288a9e853efbd4944426cd00779b85abcda8ea9099ecb24618 ccafc6c7b048e15ae00fd2c1164aa66c43bd5543279f0f2d5df33cab8788dcdc 0aca26dd7fb8ab73facf5311ac66318ccd81cba4f769cc74418ea33be336f07e ecf26683735f5f42a8bc2a763cee5cd5fce347122520e947463ce6e5095da43d ca2cb1a122a13db3751bcd3e479682d99f4fef8395e718c30f9f249f44aea397 126d012814f58a0dec670f3c209e08f0eb7077cee17ccd81a767cd507a6c2e71 6eae62cd9bd9dd5d188df52273d90daa0a052903fca32173d80c6144d963651e 0c6a44f100e9994b98122a2f7148a5f1dbe67129f45cb0e24668d05d71b8954f 5d70c08b2d9ef198970379f6ac5fc3a20b7719f488328410d5f54287910b3372 0a317f3c288ecef77ab9c1e9c24e0262b0fa1a3d5edbecd2384c09a4fe00206d c65399f302b791d446c463c7a8f29ac8d3749daf78128a990593a7b088c4c17d aef6f38304a098fb1fb6f16ff7e77e220fb7abe7b5e1b110f087abdc7cb51d14 e5babc2385bc966cc8fa900320811f9d92764b443893834a2129cdcce8b12bf2 a70ca6f431fb3102db75464812758c8a98251ead9f8b2d7586a23584a615a064 c2ca53db2eb3ba0d6083148dfb9d0e68a409e9ea53997b561abc5856a0b822d0 07c21924342f 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 12 /vextendsingle put dup 13 /vextenddouble put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 26 /braceleftbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 34 /bracketleftBigg put dup 35 /bracketrightBigg put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 80 /summationtext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 91 /uniondisplay put dup 104 /bracketleftBig put dup 105 /bracketrightBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueXX 5000774 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d5649664203ab24acb938d58d246707ffe7 d62f04bec4b70c21ef75beb2b812622b3c74e969d72d3cd11bd7106294a99caf 0b1629bc7d4de6b96ca82930831d64575f23f4ad06a0e45e315b1d392411be8d 6d73c998789ff258a07a3c8c2057325784514c845500bfd1a971310cfc11d41c 1a167dbd5ff012c60add4e87325f6e5299032a839de65fb1473a166aae1876a4 414a434f22c1d241591fb36f857df6fa930608750ffc0c54f44994662b1f00f1 400bf752ea8d83ffc4cb77a290bc2d99981ae59a191748ba5c7ba1a9d2583fd2 1398452b6ff5d83a059f7eadcd2ef744e9dd22bdf9c79d049bf06835e878c32b 7765c69bdd8ef4deb4ea7cfff4cf9354a4ddffa689de961d16772491c7afbd7f ffde42400764c68e954ee5c455a5687959829bc3b319b2147deaab3628662c80 30c5e02fea09609abe4eaa12e217bc3af673f1bc36a7039eb13fcacb4218fe0f c5a3f9452d4edf46cc91db67b624d4f2d37502fb9f11af4da18ca40b61097f95 d44329375467ed13c5cb585ec53f62b83ef9502cc755af44bf32b87b8ae9f3f2 f8dbf72dab90acafbacd280db6aaffaefdff6d5eff26669bac56280a950560e3 d01714ba38659b0e4e28bf8cdc1f15ff7e3ea9d95a6afbbe3d8e6905aa6d531d 6b6dcab89e2036cc492c65f76ddcc3cfcff33fde3d9ffd10f7a90530fd5c7683 4329baa902e701ca42717476298c9ff511b1941244bb1dfd0e5cab92a43970cd 1bdb292eadd055c97972844fdcdb0aa06401e7b2632b9d134f96bce0064860f3 9859da7a9a82be0a840672a1331212752ae1b521b693b384a72dba8d37a86eaf a0341e7fc5f6ef7a8c566aacb91e8bf5005c18837894c4eacff41dba54e071ca 83fe3225b654c2758c0a6128e3272d1cea2184ad13521bd9deb2a021b0eea578 2af27bb20864fb39c5329b4da1f4c7854627fd1536eec6e58bb3ac4d9bcbcbf9 ee35a3f62fcedcb76df6cef6e5436f64ebd42bbc91fd9e02807e3614dbb3b77a 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d616a5b9dbe56f73a1edf164eb168e549043b4cc58fafc43f9dbb0902f93dbf2 aba54c87d2793d81ddc090408fc01d8c44bd915e83d6926f3b19dcd1e9ca3f5a 9c7bb53c00de81962e368459bfc7ba51d32ea51a697d8e854b2c5d4e28f89500 2e5350508ad0a7a3c97408dfe494135609a30364d8dd0d289688c2da7588d777 528ea493800b13bcf0f0a46f9be36e5c725bf6971d16d31882331dd9541b505c 3228165e56dcbee32db885df8f4fbf0db990818e332a69d9048f1bc842b2b99d 71dfdf7cec48e144b1179974c5b665150d2af0712aa74031cd45138f1ff5b5d3 5b5c013a777c5b8c6b110db2dc4615f125c80b802874a82e596b0f041106b408 5690a2d5b5c7af608e5bbe582cf31cee7c111a664c1a250e6e3a74e9cfddadb9 cfbba67288c23ca1472dd451e515a5c40db512cc1b8e5621e41be0a4afd0655e 9ec5d1b8132472e06639ece4094e23df95b6bc93e3cf1687d29b4da5f120105d fa3c65ff0c6fd3d0a2b6c1312180e30c6c6fdb89c656d94bdff06f6e6a8b3274 c6a50b643792619d0b912446a678749b7bc0c23c5b4cef0fa3b7d60de289eb78 184ced4e0d63cc1e36bd5ce0d305243fb19a3320ad0f174120c654421d66aecf 9ad6067c043b403b25d359b4e185774524a5f2d5c0e8f63cc448397a8f6ee3e6 6c80dc10fb02201a26d678c8d721f400d5c7c6af577dc503b728628d55fe9e9f 7b9cd9e258104a15379f6a07c2006e29b38b7efbf8e4530145d82b52921bad69 9dd794c358c539101b90747aaeaddb756a092f60bf67186c27655ab45c367430 7215478aea1869e5f868762a0ae1d796cde1b9a44a3700bdcefd061c85681728 3b868fbf27188e24f1435b508c6e306d21550c95907648b9bc408a5e39f70f07 53c7025b7c5f9ddc4fe0a55359ffd6246d247680c1d6fa3644dfe5b7f0260943 0e5f6824a8cc84ee329eb21d1059c5f8f9e65848c334b38597170f875187a214 3e7272db35268761ae24352c6eb0eae53db88747eb420d71ec2452c6cc38de5f cbed2ee2b6080f9c41f8e498a92e62dca9ad7e5237e9c4b6fcc89974489b02cf 53579ff0c325532a73243d6141d5473c5333fd732b21cb290a7efde13157ef99 4ad0698e8eb282dcde659116e5b922ed9d0140eb47eded829d1d0b38845cc685 2da89de070e75864ee2d56eab2ef4cb24263d80163b95c404ac346ad678ac281 7f3ea68a6c03782b8f797ad49b434153cbb2ff470df9f6dc26050c549283a962 b0e3e206171d9acf2f56724a972df020fbd1a314a899d51fc75265d8f74524ce 2dcf4f6060c4923711630596fa855d13259222de3e463cf556b90bf2b7be1d77 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e1ec03e16cf7354fe2d800bbb3b9f6e54297c551af986770e10a6abc00988883 08cf5d72342e12cef87a55272f687e75f8e873e56e9f8fc90e232c63935d0e65 8d9fbf2320796af0cb7dbb88a89c89b435e49d88a0d968db27fab9fbfad79a79 af4f100828f2a32d51ff183df52d0e8be979757540b2173712b530672672aa26 604ec00d9e114ed53645a20d69f7a5aeca4a7f9f7919ef04be02c3d1b041d3b0 20344e08f4eecd65fa51e3f3da7cd0eeb838a12584622ef330b532754dde6a21 926a7c1cdae4a25c082c73340c1b86cb30f84b5f6aaa53b4b1faba61da5ee331 50d0f790fe2744c0a9014cad3c6450a85895fd36c00d4651505e7168fa5ea540 4951cec3cd911a68120341f5ec3fef44e8b0cff4330df05025c2140e9a286008 e861a8cee033646811cee732b2a1780d79f0d92be094897fb099834fe6df4cd9 a9d75411068677403d51887fda24aafee7eecb766d9297d53ee9e41781f9433a 2e8d0ceaea7961aefc646cd58aeed7433cc127bf97dda977c86c6157cf90d18d 63de6e854421e13a6c69653e9b39a15c6515163fd138f1f99462e903bde83f13 fb947ed1487eef0f5100f6794889f5a721791b49e84dee4bc5884e8d9fec9b23 6dffc09fc6029cf15f25028fdf6edcd6594738dcdf795b921a633a8ba8eb5c4d bd19fff5b63b65b0594e9e24d6df1c79f275b9f94561ae97ed3d7af26b084888 a86173b7777983281765e31d20f3c2 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 2 /Theta put dup 3 /Lambda put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 116 /t put dup 120 /x put dup 122 /z put dup 126 /tilde put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueXX 5000790 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b448685940729e6e22f4be0e08c6505cba868f7950 93f556b64b808dd710eb936d3ac83e5783b5f7e56d363a59a026a5619a357d21 c4e77bea121eb24b09027d1aad93158edf912940e91cd2f17922b35885894644 7773730bb19af9e70453e1b78d5edc123b273a979891d6d5e624f4a388f3897e 13b045e34b4ee2f5e4fc5e996d3549640010c671de14d18b0b3cd300e57dac03 22e59bfbf7b29422230870f5897fcfaad4b50c7c1c58edcd119218163d6f6e56 25ccb882db7b6d09a1586508e8cb642a11c29a46915e4a96e282079cb21922c1 c2e360b487a45302fd22ec8c5fab63e54b5e844d4b17ca2fff37c69c366dd72a d02922c14c0932f65ed03e4219c117962edbad2dcdeaa9c10ce8af38a4ae52e2 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83758516049172740ed0c02d7f77e6d70a464947de048ee6c3a9deada0018ffd ef2242b6 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 17 /eta put dup 18 /theta put dup 20 /kappa put dup 24 /xi put dup 25 /pi put dup 27 /sigma put dup 28 /tau put dup 34 /epsilon put dup 39 /phi1 put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 68 /D put dup 73 /I put dup 75 /K put dup 76 /L put dup 78 /N put dup 79 /O put dup 84 /T put dup 86 /V put dup 88 /X put dup 96 /lscript put dup 99 /c put dup 100 /d put dup 102 /f put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 118 /v put dup 119 /w put dup 120 /x put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueXX 5087382 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930f01bdaae649990604a33aa9eaffbe5b5489 e5c7c9ff9d9be01b08220832c41caac64816b53bbc087ae4b621d2b44b21ae5e 3f7fe4ddb05c675abfe30510eee4a7c874bb57b2ffe521a6875eddfdfd18c781 25bfca5a097aad361dd94df46f14026c25ea988194899b95001209cb7009cead 4b4ea03473ea120e7a7e341d4e3926796347fdb3a124c29660ac09f4648b313c b58e824884382f24ce08d2edc686923acace1201c16a07a42803288cd0b946f7 546510e95b1fa1cc96f0bf72aa823d8e83d7c68c752c82a54c752eed2b1e699d e9db1830272ffbf2f4996ccc2d6fe2ae272798989525ef3b67b0d09bffcef749 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8eccbe94ec43d24e501486de733918380e5660e8a7c9ee04ec470eb02c770086 c092f18987c91621fcae15d8a0c265bfc41b9886d9a0d383801480bc9dc698cc 7a2b18b863a6fbabfd2a22f0f38e1c097d5dc63a5cf3bad8be1832303e84a1eb 0d0c2af7cc2b0c29b673272eed16844b8292e3f1702976eea6849eea1a0de32a efe5d5fb2adf4e326283e556fa60c40311383db0d93ffef9f34e0dc2c2cfb928 c5151217c9320ddbf849a6193a67124e81cc55f2bad931d85de23ecf155931e2 465d40689070378e95876a9d04e1ac03119ef8b3b7f3321f981116b426fc3879 eb0d518cb93d2843299a0ce5ebaec4fc4ede1ead69cfd4f781fd53effc3f81f4 59fbb4a884a8989565bb81f44564d7f1ec7f6cca0c4ae05786237d9d6811df3f 6a310b90ea3595aa10e691ed9e7f3c14738ee7f2767f6a354a6eb45277be7abc 870d07f1fa7679a236e70e33667b8d0a95ca7cf6ea27298710ee02fbac315f51 e2c3c08a2798a166380011d518b56d57a9b770ecbd8f11806db4f41e6c52b13f c4059d2b90a41fe96131e827a9f48d56a191fdfbd766375b50606b3b51773c03 743cbb278cd798399717d01bd48c8acf6d60fd9ccc9d6bfd74f6d4056c770928 cd4020596d7e4b1be2ea0293897d9618c0f9998f3488d1702066c6541e606165 9d39bbcc430450c2d8ea43b6244948a84246cda52b026845acef30229df4089e b502e1bd71f3c81eba5cf1d5e44bb23c1c45ad46f827ea9cf304feccec27a25d 2c8aad494432468d5b96d407f0f2ff89c1d5807adb3988a32e66a8ccf71380f5 303fd93bd8977598ba6d2c790ed15c01a1afbc5657949573fe1bc9b2db2b4792 5f6954709a56831809ac3965c2ef373fc0ac28e0feead49935af6bbca2953ae4 e11b9ab59fe3ef0444b68ebd5a7a0dd53d77fde4d81135495d56c5c0994ef057 5c7f45379604239626eb49722edef851a3fcb65b1a9769e2b523cb759500ce4d 2b939f39af09e123db96d43d0d50a1076ddd08cbeb06463434b97032432b146f 0989155463325ce2d613fb0ce9153386471e1f52402e43ac4b0e3649e763fa20 56aff3505deef5cd20fff442546b0fa495d7510e34b07e7c44bc9416f5729b4a 42f8c4ce8700daac3bd6776238cf7929467a8da773bc0046534b8835f05dd5c1 bf401a0306646d98dafe0f24827a2550f43f0fed1c3d68903ff4ff8ce3997a3a edffaa71626520ec2d18af2e3b7759ef355e7575be555247e11558754a1965cf 8fc08ccb39274924f8ae4bf80e9c8d57504a3c99949f1bd2ab283ec40473772d 49ac8dca7a2e832622b891926367ae69caca7512e0ef952f29bb51f5131d8472 d940221fe7fa6e23287c08b28c92cb603e468b9018a2e2227c89ea2c053ba658 e57fd2e630d26bba8d7f05b9b90828a27e890650cca75adcd0f2393d6cdfba3c f4f7742c94e2c82c93ba728c268f88f47efaabd3660657831f65668e9f3e36e0 e085768d27fe4600ac726b4c6c971ebaa2b36ed4c3793f21aadbdd6ae48bc9c2 dec473414c7c6b69e6f7ca31209397a04c6068c630c593f02c455efe8b9ea014 139b526f464ec956c98f8b1a1c5e6a44a9df6612b013db7c46f3206bad43975a 99983fb1e88803d77f4744e551cc00a590d864e5b7e56ae48be087f542955261 e0391ca54e2c973dfca6b105ce924bc73898314da17deab792ba15fcf7ded272 9f6fbe7cd0892c7c328e4e59c748759118355f35929ef38b65eecf63375171a0 e1fe4c3dbbfa260c6a5d5efa4fe99069e8c566cf874dcc233aa0f995ab3cfee1 4cee58afed143d6c7e727630534d50cfb251b7981f9470a2411780a40f1d2599 cdf29f6f8999bc1db03900070ad026d0ae4396690c0903487a70253cecc099d1 144a2c38c53687889e0fe191ecd3b8d6a247c9553fdbb40b4857acc69db4e4e8 eb5939d4f0c4dd8801df7d0d4f198cc8909b22c19ad7cf47c5b720956a6dcafd 7b7d694e98bb5ee900fb188dbfeca49d1c956733559cafdd1fc1456333a27324 f9d0a9b47c9156e8903fb431a4e6611f3bd2a8e481e5bdd802dce28f0f0b20c0 65104fe291237159268eb1e86c0256b99078a2da7c90f6f5f78eff4615c23aa2 86380658139c30bbe8b747b0a999ca061e92ceab90bdc0f9b3bd565712411384 5cff610ddbde47d44035c968296ad9d8fe5ccad9543fdc7716cbdf4addacc74f 0ae4f9558d1079fcf0c488bf63c3d0175b7f31576feea919c5175e906fc38eeb 32492e69a7d2a731ca3b1458d177718ccf1df1d1c52e7248ae6420bfb9cbde10 982ffd8c42030f0781726ce4c039cbc911073aaf8219707d9b66607645d6aef5 26aac8a4dd53915523d0a6912239ecbd1324c1093feb17c7dc11f95baf1b1c22 673e2b2ba54edbeb64a9f5635b308f8e0b292187a33ea5891d2a8956099d3141 f66e1001dca690fa7360f5949a3358aa573b79f819c52e9d95aa19d79de0cdbc f3c17a2ea56c5b4fe9c720768381529ff3c6d3d84a5cbac8d1f0f0cfecc25df8 08c34b4c75413441ccdb906480dac3e9934bd8249e7e9e5c04ddce7d601065b8 861d7e4d92ecaebf022c505285df5d740b06f14d9d5b377aea6522154775162e c3a2c5a9a831e365028843850c48e4fe2f6afad5fda91ea7a6919864699654cd 4cf63bd2cc8dd0adae22fab1a5f41cd8c4565cb7cc89fdd24808e86821a69403 d68f485f3ad970a8aa7acfb947b36f72336f4936d9cc6e462e7e4588ad2c26a6 c9f3823d69bc11a69706eebb48b8f207a0b2aba449ae4672ac8fe3fd1d064e33 09e508382736468f82f5df5c69a3f0d98ab22948f4447ff49ed37cd32a082867 802b8d68bfd159b8bd047829f6315ceb84e9ac8df0c8471da8e56b90ca203753 c4648b9e443a15028d12164c776727f1310c7e98b92f9b14e29023654807fb36 84299f86f230b906c9d43bb3f4cf1f543520fe1b635928170944cecb1ce9da29 15fdd0bd37ece24aa441391b9b01230845c2a4a500589cd3a5ee0716bfe52d22 24f5207c0895697f8813106be042eb281132cad1faf6dbfaff4de170956b5a2b 608bb74074ecd70e3ee1004c2d0d3fe1d0a73cad0629de4fcc5ac2f282d298fc e939b425204d687fb99b99b09d2819cd4eb49c5d31f67a9338cfd8ce2604d707 ed4cee875a6cca1d8cee40a7c0c8f332f147ce1d869d2649c53013828db5b338 d06c1ef781ee96d19a84840f94356dead459f9064ee34e4af2bbd0844c4468c6 6c3ba77456b8bb76c8d6737180739c3bf3e9763c73e82e047e7f4a451f6d96ab 5d6680b618b238305991c9e7d3c00d60c28760ea300cc86e43e8a0a00556b6d4 4d10c239aeefab9fffd520518292d64a04ff5c9f71fc8e459c7d7b268bfabf06 f3f341d5c7f5906687c3d20562b788c288de8ee6d894cab642fe8204d7666d1f b1f98d79656ea264d059795b4ee70d403d4ed2d06f8e2194fb5b5859156bf77d 8bd323257bd5e20b25ae446b39b57c4915789cd3aee45c246234beff3dbfe982 3a9035e8ab281aeb29c3d929e6b858a0238c4825 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 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 78 /N put dup 82 /R put dup 84 /T put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueXX 5031982 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d56491bdecdcafdc988206c5a457a19270b 37d0ab776e03eaa7eb568eeab6b5e79dec03b0dcbf923a2aa8e4f4deda2cb043 858d8b430efef0ed914a80ffc9818aba0fb30dae3694e5b31df3855892d59b89 7f82ff79fce8444c7926ffc4ea008e63bc518936c9098afc2d1c14030a2c15bc c9285c1b57e80570d2f6301009e63d9ddeba44e3251c75ca9616a2c5de9ade66 1a15db37b40b67c1bd71f77343912c62f47dd24be0e03dd833ea60d2eb020407 eaf042fed30f0c2f01956c83d46d982b50dd979c2026ed5f71979dd98c9c1eee 25054000a5eed5ce9498b2cb59a7ee55e0538ab47d86e69b2ac79229c8c6bfc9 3a24e822a2f1c9964fe630344579a25f66646113f73d1315c447f070c575de76 266c6773b01137e4a039df5e4c2f199e8e3396ef9fd001eebe090e04384c4089 ec9f13a07ad8733c6af775f432f154ef50ec4477e08760c03c006e4d1eb1c7cd acb8955ca4b0d62c7de046ff58d431bb9883d28586552c51809e3ced4dfc2198 b97ad9e52ea33c91b9e335a80d9b77d3430e0487930f992459664abca5dd9177 150b0fb502fbb04ccaa94ffe2697ed07906f95fd98e7b325bfbf10ea6b39fd76 c1e5295022cb39e9cd471b7b0b3f9d72da99deb0615d35cc22f42dbf5cfb70dc 6279fd520d99f32201abf25e621cfe1d37961d9c0ad36ef3d9264c547638152a be30812b1188a10f6327c9e8714449de2ae636420f56a44cebd774f0b131c629 818455cc8dd8d40939f3ba8aa5abc54590c7286b172193c23adb1b4f9931a9f7 73b960b1f5b760438028478e4442dc93a567b07e20017c18fca2a1daa2649dee 5a6ec51fd6c3f4b27367c31e95a56ba4bbeda0493bc02306037e2ba69add9b94 373364db316691e73d6811cd76fbc92b0c67ca18dcdf05807f17d1b309456ea9 d113ea0fccd6eb73bd8dfdbd5905a5e38bbf61cfe58fb01ec95a4bdd9565433d 4a7abb8dca6f76c458ba9cb7568b924f70eaf52ec0167e9a9af3b6a4e87e26a3 678a5b6b4605d1ae4039a653f2327262005c9d7f35bf1e86459bc12bfebd913f 866030f30fe1838d68ab298b6706d8ef667ea266915ea95b7ad35bde3daad0a2 bb92ee27dcac4334a7fab5a21da0a9a888af42de0c2b039ab679479535f05e71 7fbdaa74caa570a205c25f857a4844d415180397584dda6e850af7cbb099d0e1 b52863fdb4eb5705e3a68602139940df1a0758086545cd6d083fe79bfb5e72a5 d66b4f16e7f6aace05fb82f907fe91ff3bb494f1683ef2488d5873de036b6d05 2581c7f2177db6801cd44ce823d1bbc0ac738c9f77dc9594e515613432b13d2d f76187aa2fc9ee69560c592e758bbb72da44db9be903a1991e878ca5b006e122 824481cd2b1134ec4e3823179cc4d80ef3bdf884becfbfa8d1977f4b7ffdbd0d 9f095dfce04e4c8914e10062b7f8a65d77bbd1ac94ac47f127ae84d29783a8b9 9f33ea046883f67581e2795f293a83e07f5930049425d1a0d6eda155a4cdd242 ec6f1e0c709e6d4b54055ba725f2c23a99e7e3b8f955956563f5a4ec016e73ee d7f9c6b56f17d214a6c50fad7047db638b4a45b8f24b2b788a0b978cc3bb2e92 e7bd418a8f722166b4b50cc9516e5f9b6a378fd7c54f8ddaae55f4c037d3e674 9ac2948f7cdb769a1f749bee38d14095ee03a63a15f5adf7282368d3e285d1f1 c45865dc1d988705855c0c149f8f8f0df3a3769ebfd4b62ff527c94036b695b9 c8e6970e91ddd2a9338329f8475e1b68fe333b9b10a57b3c0e998fc72d682afb 33abf1060042aa4e757ca073aae7bf9576aa464c79699123debda6df39e29657 f920881c54416d13cb7403c0f9a87c7bf2bb63073ea9459cc419fffe831ac124 71b2f1b8a2fe131dcadaecf4d5d6f2a00ba32a3de55a333247d33a9473e6806f c7b60810bd48d4acde0958a87714e3bee90441795f967aeaa42249e528251f47 515b0639c9a6b6c8537f324441e006e3424d3465a6b6d2ade5a27b8dab73ab5b 83c650bd1fdfe981b2884ce1ad35106eb5919734d1c8aa4ea019c1f81ebc62c3 4ea69c894e23c3fcbdfc70083f21beacddc8fbf61f5b75918ab9f1d89006390b 7b91b67ba98d9cf28b678b474feb0e66040da221d0ae67e43b8d2718ec79d46e 3ff5849ed7ef9abce97bb0c85d2f9af0dfe7229bf28ab7417b3660839121271e 5d2a72920c3d074535809fb329e60085f446bcdda80a0f9ff58c59db7bd35f01 a219341bac1f4fdab886734c5e05bdacfff56beb0b9ddda4dd0b73eb373ea762 b8de820c0ef3d0bd69761fe737675d13994378a393b4c34baae6fd162b1cf26c 20784f89454d149ac5cf8bdc4bfcd211f9b1adc7c79e261549b21c8e4fbb8062 33cd258700d80e82b8f40cd0d42caaccd660ad1d0e36ed8d171b504dd15eef88 fd86be1c99504ca88617ac95d4ccb6a42996530f97812ab9c531bac778436724 a8bb9e2f432eeaa29d9702cb9b9bd71d0228a67f0334daaef8f0824a1c8b7a75 373a5ed98aec851097a84e08a49261a4 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 4 /divide put dup 6 /plusminus put dup 8 /circleplus put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 22 /precedesequal put dup 23 /followsequal put dup 26 /propersubset put dup 30 /precedes put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 56 /universal put dup 59 /emptyset put dup 65 /A put dup 70 /F put dup 72 /H put dup 76 /L put dup 80 /P put dup 84 /T put dup 86 /V put dup 91 /union put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put dup 120 /section put dup 121 /dagger put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueXX 5000820 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8 3750d597b842d8d96c423ba1273ddd32f3a54a912a443fcd44f7c3a6fe3956b0 aa1e784aaec6fce08dae0c76da9d0a3eba57b98a6233d9e9f0c3f00fcc6b2c6a 9ba23af389e6dfff4efec3de05d6276c6be417703ce508377f25960ef4ed83b4 9b01b873f3a639ce00f356229b6477a081933fef3bb80e2b9dffa7f75567b1fa 4d739b772f8d674e567534c6c5bbf1cf615372be20b18472f7aa58be8c216dbd df81cc0a86b6d8318ca68fe22c8af13b54d7576fe4ca5a7af9005ea5cc4edb79 c0ab668e4fec4b7f5a9eb5f0e4c088cd818ecc4feb4b40ec8bd2981bf2336074 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 66 /B put dup 71 /G put dup 72 /H put dup 86 /V put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 121 /y put readonly def /FontBBox{-301 -250 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 17 /eta put dup 18 /theta put dup 20 /kappa put dup 21 /lambda put dup 24 /xi put dup 25 /pi put dup 27 /sigma put dup 28 /tau put dup 33 /omega put dup 34 /epsilon put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 84 /T put dup 86 /V put dup 88 /X put dup 96 /lscript put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 118 /v put dup 119 /w put dup 120 /x put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueXX 5087385 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930da4f8d58abefd45db119b10eb409dd89792 3c6e705479464a4b310b58348c4b42393988fef4925cf984423aaf65fea9f0e6 4629953bcf50b919d968d99bd185f83112b2759cc411764e9bde677f57c5ee5a 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1b3c4515c53b33705978932056785a 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSL10 %!PS-AdobeFont-1.1: CMSL10 1.0 %%CreationDate: 1991 Aug 20 16:40:20 % 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 (CMSL10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -9.46 def /isFixedPitch false def end readonly def /FontName /CMSL10 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 44 /comma put dup 45 /hyphen put dup 67 /C put dup 71 /G put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 121 /y put readonly def /FontBBox{-62 -250 1123 750}readonly def /UniqueXX 5000798 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 19 /acute put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 63 /question put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 92 /quotedblleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueXX 5000828 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3ffd20326947b1065649b533eb5e9e3 a88a87fe987918a687950b26f647d6e08bf14d983ed1d4a8ca1e2da8d985f944 2eb916f5b6ed0697ac7c33e1e36a3bf460d34ce45f1631871097cb04f18e3889 4cf4ac1538eb19481311d24fe3be7beaa4a3730e8b4831fe59d6d9ce2e46116b 629c7ba2f9ce3ecba2f43bc162a5a077ca1b2882a42afdcec3f4b75b5d63e0bc 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 8 /Phi put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 18 /grave put dup 19 /acute put dup 20 /caron put dup 33 /exclam put dup 34 /quotedblright put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueXX 5000793 def currentdict end currentfile eexec 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 3 /asteriskmath put dup 20 /lessequal put dup 21 /greaterequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 70 /F put dup 82 /R put dup 91 /union put dup 106 /bar put dup 110 /backslash put dup 121 /dagger put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueXX 5000817 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a 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fbe6fc7af8e32748e4f611cacb35c03dc452743855831b12df434b424745d5e3 ca9521a42f8303dfcb9b0838cd56fc90513128cbddba278680b8ce3321b1fc77 22847cbdae8dcb16e77317fa10601e701a728b62e7ba1d2db92d869f071e0346 e0619d792122af166da426fb5bc2902765188450f293767fcd10dae6ba4368c9 2744c2dbd7bad3555c36700b25db522894d04a95ca3d51c202d666661ed19a95 66a8b21746594b9aead73c196f5496e29b 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 67 /C put dup 70 /F put dup 71 /G put dup 73 /I put dup 80 /P put dup 82 /R put dup 83 /S put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueXX 5000769 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5bfae5c1f476ee6f05eb1d2caeb 269958b194521197b312fcced4867f3c8fbd030bd715d8ffda1dcd454b174e7a 1a97b59fe770e67702519d9d9b23d61ac08424d555242a8ca08c49aef300945d 99b999a79ce74804ae6bfde623f4463371442f6523a5f6ce19c839a708c02513 2e22c696c8ccade45680e5197189d0f98e7f0d5f955e353970b392cf530a68cc 56b0035ddfbf206c3074beeb0739dcbca272a6e629fb7aea2c5ba7bae50c7b4c a595df78200c352997ec3ee564df229fbb5473f5e8ccb1cc0153e9a7e299a8ea 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2[65 65 65 65 2[65 8[65 1[65 1[65 42 65 23 65{}35 83.022 /CMSY10 rf /Fs 136[69 2[37 38 3[48 53 80 27 50 2[53 48 1[44 53 42 53 46 10[72 14[75 4[68 72 7[48 48 48 48 48 48 48 48 48 48 1[27 46[{}30 83.022 /CMBX10 rf /Ft 135[47 59 40 1[30 39 37 37 42 1[50 73 1[43 34 29 48 1[41 39 43 36 1[44 35 7[69 1[48 1[49 1[63 1[53 63 67 81 57 71 46 36 1[65 53 61 69 59 63 62 44 1[65 42 65 23 23 18[54 4[39 52 4[36 47 1[47 36 2[48 48 1[39 41 1[34 37 43 47 53 11[{}61 83.022 /CMMI10 rf /Fu 134[44 3[46 32 33 33 1[46 42 46 69 23 2[23 46 42 1[37 46 37 1[42 12[60 46 61 1[57 8[65 3[60 21[28 23 44[{}25 83.022 /CMSL10 rf /Fv 132[42 34 40 39 55 38 45 28 34 35 38 42 42 47 68 21 38 25 25 42 38 25 38 42 38 38 42 4[43 4[83 62 62 59 47 61 1[56 64 62 74 52 64 1[32 62 64 54 56 63 59 58 62 1[42 3[25 25 42 42 42 42 42 42 42 42 42 42 1[25 30 25 2[34 34 25 4[43 14[42 4[73 49 47 51 11[{}74 83.022 /CMTI10 rf /Fw 138[51 49 38 50 1[46 53 51 62 2[35 25 51 1[44 46 1[49 48 51 12[65 3[62 3[57 8[65 1[68 11[46 6[27 46[{}23 83.022 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y(shall)e(p)r(oin)n(t)h (out.)38 b(In)28 b(particular)e(the)i(analyticit)n(y)f(domain)h(is)f(m) n(uc)n(h)h(larger,)e(see)h(the)i(heart-lik)n(e)d(domain)h Ft(D)3506 3466 y Fn(0)0 3560 y Fx(in)k(Figure)e(1)h(b)r(elo)n(w)g (\(and)h(the)g(discussion)e(in)i(the)g(forthcoming)e(Section)i(5.9\).) 44 b(In)31 b(fact)g(w)n(e)f(think)g(that)h(our)0 3666 y(tec)n(hnique)26 b(can)g(lead)f(to)h(pro)n(v)n(e)e(existence)i(of)g (man)n(y)f(elliptic)i(in)n(v)-5 b(arian)n(t)25 b(tori,)h Fv(i.e.)31 b Fx(for)25 b(a)h(large)f(set)h(of)g(negativ)n(e)0 3773 y Ft(")p Fx('s,)j(and)g(to)g(understand)f(some)h(of)g(their)g (analyticit)n(y)f(prop)r(erties;)h(see)f(Section)h(6)g(for)f(further)h (remarks)f(and)0 3879 y(results.)1699 4182 y Fn(complex)1699 4247 y Fo(")p Fy(\000)p Fn(plane)1340 4447 y Fo(O)1190 4640 y @beginspecial @setspecial %%BeginDocument: fig1.ps /punto { gsave 2 0 360 newpath arc fill stroke grestore} def /origine1assexper2pilacon|P_2-P_1| { 4 2 roll 2 copy translate exch 4 1 roll sub 3 1 roll exch sub 2 copy atan rotate 2 copy exch 4 1 roll mul 3 1 roll mul add sqrt } def /punta0 {0 0 moveto dup dup 0 exch 2 div lineto 0 lineto 0 exch 2 div neg lineto 0 0 lineto fill stroke } def /dirpunta{% x1 y1 x2 y2 gsave origine1assexper2pilacon|P_2-P_1| 0 translate 7 punta0 grestore} def /P {3.1415926535898} def /R {15} def /h {0.01} def /gradi {180 P div} def /Xp {R P mul 5 add 0} def /Xm {R P mul neg 3 div 0} def /Yp {0 P mul R 2 mul } def /Ym {0 P mul R 2 mul neg} def 15 30 translate newpath R P mul 0 moveto 0 h P {dup dup dup neg P add R mul dup 4 2 roll gradi mul cos exch gradi mul sin 4 -1 roll mul 3 1 roll mul exch lineto} for 0.6 setgray fill stroke newpath R P mul 0 moveto 0 h P {dup dup dup neg P add R mul dup 4 2 roll gradi mul cos exch gradi mul sin 4 -1 roll mul 3 1 roll mul exch neg lineto} for 0.6 setgray fill stroke 0. setgray Ym Yp dirpunta Xm Xp dirpunta Ym moveto Yp lineto stroke Xm moveto Xp lineto stroke R P mul 0 moveto 0 h P {dup dup dup neg P add R mul dup 4 2 roll 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Fx(3)p eop %%Page: 4 4 4 3 bop 2010 28 a Fn(complex)2010 94 y Fo(")p Fy(\000)p Fn(plane)1377 777 y @beginspecial @setspecial %%BeginDocument: fig1.1.ps /origine1assexper2pilacon|P_2-P_1| { 4 2 roll 2 copy translate exch 4 1 roll sub 3 1 roll exch sub 2 copy atan rotate 2 copy exch 4 1 roll mul 3 1 roll mul add sqrt } def /punta0 {0 0 moveto dup dup 0 exch 2 div lineto 0 lineto 0 exch 2 div neg lineto 0 0 lineto fill stroke } def /dirpunta{% x1 y1 x2 y2 gsave origine1assexper2pilacon|P_2-P_1| 0 translate 7 punta0 grestore} def /puntatore {4 copy 4 copy 4 copy 4 copy dirpunta 4 2 roll moveto lineto stroke} def /punto { % x y punto gsave 3 0 360 newpath arc fill stroke grestore} def /R {100} def /h {1} def /p1 {20} def /p2 {23} def /p3 {30} def /p4 {34} def /p5 {40} def /p6 {45} def /p7 {60} def /p8 {66} def /p9 {80} def /p10 {88} def 0.5 0.5 scale 0 R translate R 0 moveto 0 h p1 {dup dup mul exch neg R add exch R div lineto} for R p1 p2 add 2 div sub 0 lineto p2 R sub neg p2 dup mul R div lineto p2 h p3 {dup dup mul exch neg R add exch R div lineto} for R p3 p4 add 2 div sub 0 lineto p4 R sub neg p4 dup mul R div lineto p4 h p5 {dup dup mul exch neg R add exch R div lineto} for R p5 p6 add 2 div sub 0 lineto p6 R sub neg p6 dup mul R div lineto p6 h p7 {dup dup mul exch neg R add exch R div lineto} for R p7 p8 add 2 div sub 0 lineto p8 R sub neg p8 dup mul R div lineto p8 h p9 {dup dup mul exch neg R add exch R div lineto} for R p9 p10 add 2 div sub 0 lineto p10 R sub neg p10 dup mul R div lineto p10 h R {dup dup mul exch neg R add exch R div lineto} for 0 100 moveto 105 85 180 50 180 0 curveto R 0 lineto 0.7 setgray fill stroke 1 -1 scale R 0 moveto 0 h p1 {dup dup mul exch neg R add exch R div lineto} for R p1 p2 add 2 div sub 0 lineto p2 R sub neg p2 dup mul R div lineto p2 h p3 {dup dup mul exch neg R add exch R div lineto} for R p3 p4 add 2 div sub 0 lineto p4 R sub neg p4 dup mul R div lineto p4 h p5 {dup dup mul exch neg R add exch R div lineto} for R p5 p6 add 2 div sub 0 lineto p6 R sub neg p6 dup mul R div lineto p6 h p7 {dup dup mul exch neg R add exch R div lineto} for R p7 p8 add 2 div sub 0 lineto p8 R sub neg p8 dup mul R div lineto p8 h p9 {dup dup mul exch neg R add exch R div lineto} for R p9 p10 add 2 div sub 0 lineto p10 R sub neg p10 dup mul R div lineto p10 h R {dup dup mul exch neg R add exch R div lineto} for 0 100 moveto 105 85 180 50 180 0 curveto R 0 lineto 1 -1 scale 0.7 setgray fill stroke 0 setgray R 0 moveto 0 h p1 {dup dup mul exch neg R add exch R div lineto} for R p1 p2 add 2 div sub 0 lineto p2 R sub neg p2 dup mul R div lineto p2 h p3 {dup dup mul exch neg R add exch R div lineto} for R p3 p4 add 2 div sub 0 lineto p4 R sub neg p4 dup mul R div lineto p4 h p5 {dup dup mul exch neg R add exch R div lineto} for R p5 p6 add 2 div sub 0 lineto p6 R sub neg p6 dup mul R div lineto p6 h p7 {dup dup mul exch neg R add exch R div lineto} for R p7 p8 add 2 div sub 0 lineto p8 R sub neg p8 dup mul R div lineto p8 h p9 {dup dup mul exch neg R add exch R div lineto} for R p9 p10 add 2 div sub 0 lineto p10 R sub neg p10 dup mul R div lineto p10 h R {dup dup mul exch neg R add exch R div lineto} for 0 100 moveto 105 85 180 50 180 0 curveto R 0 lineto stroke 1 -1 scale R 0 moveto 0 h p1 {dup dup mul exch neg R add exch R div lineto} for R p1 p2 add 2 div sub 0 lineto p2 R sub neg p2 dup mul R div lineto p2 h p3 {dup dup mul exch neg R add exch R div lineto} for R p3 p4 add 2 div sub 0 lineto p4 R sub neg p4 dup mul R div lineto p4 h p5 {dup dup mul exch neg R add exch R div lineto} for R p5 p6 add 2 div sub 0 lineto p6 R sub neg p6 dup mul R div lineto p6 h p7 {dup dup mul exch neg R add exch R div lineto} for R p7 p8 add 2 div sub 0 lineto p8 R sub neg p8 dup mul R div lineto p8 h p9 {dup dup mul exch neg R add exch R div lineto} for R p9 p10 add 2 div sub 0 lineto p10 R sub neg p10 dup mul R div lineto p10 h R {dup dup mul exch neg R add exch R div lineto} for 0 100 moveto 105 85 180 50 180 0 curveto R 0 lineto 1 -1 scale stroke R R neg R R puntatore 0 0 R 2 mul 0 puntatore %%EndDocument @endspecial 591 975 a Fg(Fig.)40 b(1)807 950 y Fd(0)829 975 y Fg(.)k Ff(The)26 b(domain)g Fo(D)1349 987 y Fe(0)1409 975 y Ff(of)f(Figure)h(1)f(can)i(b)r(e)f(further)f(extended?)40 b(the)26 b(conjec-)591 1049 y(ture)f(ab)r(o)n(v)n(e)h(asks)f(whether)h (the)f(extended)j(analyticit)n(y)e(domain)f(could)g(p)r(ossibly)g(b)r (e)591 1124 y(represen)n(ted)30 b(\(close)f(to)h(the)g(origin\))e(as)h (here:)42 b(the)30 b(domain)f(reac)n(hes)h(the)g(real)e(axis)591 1199 y(at)e(cusp)h(p)r(oin)n(ts)g(whic)n(h)f(are)g(in)g Fo(I)1496 1207 y(")1527 1215 y Fi(0)1591 1199 y Ff(and)h(corresp)r (ond,)g(in)e(the)j(complex)f Fo(")p Ff(-plane,)f(to)591 1274 y(the)h(elliptic)f(tori)g(whic)n(h)h(are)f(the)i(analytic)f(con)n (tin)n(uations)h(of)f(the)g(h)n(yp)r(erb)r(olic)g(tori.)591 1348 y(The)21 b(analytic)h(con)n(tin)n(uation)i(w)n(ould)d(b)r(e)h(con) n(tin)n(uous)h(thorough)f(the)g(real)f(axis)g(at)h(the)591 1423 y(p)r(oin)n(ts)i(of)f Fo(I)912 1431 y(")943 1439 y Fi(0)980 1423 y Ff(.)31 b(The)24 b(cusps)g(w)n(ould)g(b)r(e)g(at)g (least)h(quadratic.)623 1681 y Fz(2.)50 b(F)-9 b(ormal)39 b(solubilit)m(y)f(of)g(the)f(equations)i(of)e(motion)0 1858 y Fs(2.1.)e Fv(F)-6 b(ormal)31 b(exp)l(ansion)f(and)g(r)l(e)l (cursive)g(e)l(quations.)38 b Fx(W)-7 b(e)28 b(lo)r(ok)f(for)g(a)g (formal)g(expansion)567 2091 y Fs(a)p Fx(\()p Fq( )s Fx(;)14 b Ft(")p Fx(\))24 b(=)958 1987 y Fy(1)931 2012 y Fm(X)930 2191 y Fo(k)q Fn(=1)1065 2091 y Ft(")1104 2057 y Fo(k)1145 2091 y Fs(a)1191 2057 y Fn(\()p Fo(k)q Fn(\))1284 2091 y Fx(\()p Fq( )s Fx(\))f(=)1549 2012 y Fm(X)1525 2199 y Fl(\027)5 b Fy(2)p Fk(Z)1659 2171 y Fj(r)1707 2091 y Ft(e)1746 2057 y Fo(i)p Fl(\027)g Fy(\001)p Fl( )1888 2091 y Fs(a)1934 2103 y Fl(\027)1982 2091 y Fx(\()p Ft(")p Fx(\))23 b(=)2223 1987 y Fy(1)2196 2012 y Fm(X)2196 2191 y Fo(k)q Fn(=1)2330 2091 y Ft(")2369 2057 y Fo(k)2448 2012 y Fm(X)2424 2199 y Fl(\027)5 b Fy(2)p Fk(Z)2558 2171 y Fj(r)2605 2091 y Ft(e)2644 2057 y Fo(i)p Fl(\027)g Fy(\001)p Fl( )2801 2091 y Fs(a)2847 2048 y Fn(\()p Fo(k)q Fn(\))2847 2105 y Fl(\027)2940 2091 y Ft(;)561 2382 y Fs(b)p Fx(\()p Fq( )s Fx(;)14 b Ft(")p Fx(\))23 b(=)958 2278 y Fy(1)931 2303 y Fm(X)930 2481 y Fo(k)q Fn(=1)1065 2382 y Ft(")1104 2347 y Fo(k)1145 2382 y Fs(b)1198 2347 y Fn(\()p Fo(k)q Fn(\))1290 2382 y Fx(\()p Fq( )s Fx(\))h(=)1556 2303 y Fm(X)1532 2489 y Fl(\027)5 b Fy(2)p Fk(Z)1666 2462 y Fj(r)1713 2382 y Ft(e)1752 2347 y Fo(i)p Fl(\027)g Fy(\001)p Fl( )1895 2382 y Fs(b)1948 2394 y Fl(\027)1995 2382 y Fx(\()p Ft(")p Fx(\))23 b(=)2236 2278 y Fy(1)2209 2303 y Fm(X)2209 2481 y Fo(k)q Fn(=1)2344 2382 y Ft(")2383 2347 y Fo(k)2461 2303 y Fm(X)2437 2489 y Fl(\027)5 b Fy(2)p Fk(Z)2571 2462 y Fj(r)2618 2382 y Ft(e)2657 2347 y Fo(i)p Fl(\027)g Fy(\001)p Fl( )2814 2382 y Fs(b)2867 2338 y Fn(\()p Fo(k)q Fn(\))2867 2396 y Fl(\027)2960 2382 y Ft(;)3373 2249 y Fx(\(2)p Ft(:)p Fx(1\))0 2639 y(where)27 b(w)n(e)g(ha)n(v)n(e)g(not)g (explicitly)h(written)g(the)g(dep)r(endence)g(on)f Fq(\014)2110 2651 y Fn(0)2147 2639 y Fx(.)71 2746 y(Then)h(to)f(order)f Ft(k)31 b Fx(the)d(equations)f(of)g(motion)h(\(1.6\))f(b)r(ecome)1290 2933 y(\()q Fq(!)21 b Fr(\001)d Fq(\027)6 b Fx(\))1531 2891 y Fn(2)1582 2933 y Fs(a)1628 2890 y Fn(\()p Fo(k)q Fn(\))1628 2947 y Fl(\027)1744 2933 y Fx(=)23 b([)p Ft(@)1899 2945 y Fl(\013)1954 2933 y Ft(f)9 b Fx(])2026 2891 y Fn(\()p Fo(k)q Fy(\000)p Fn(1\))2026 2958 y Fl(\027)2218 2933 y Ft(;)1284 3089 y Fx(\()p Fq(!)21 b Fr(\001)e Fq(\027)6 b Fx(\))1524 3047 y Fn(2)1576 3089 y Fs(b)1629 3046 y Fn(\()p Fo(k)q Fn(\))1629 3103 y Fl(\027)1744 3089 y Fx(=)1832 3022 y Fm(\002)1867 3089 y Ft(@)1911 3107 y Fl(\014)1961 3089 y Ft(f)2011 3022 y Fm(\003)2045 3039 y Fn(\()p Fo(k)q Fy(\000)p Fn(1\))2045 3124 y Fl(\027)2237 3089 y Ft(;)3373 3006 y Fx(\(2)p Ft(:)p Fx(2\))0 3261 y(where,)27 b(giv)n(en)g(an)n(y)g(function)h Ft(F)39 b Fx(admitting)28 b(a)g(formal)e(expansion)1188 3494 y Ft(F)12 b Fx(\()p Fq( )s Fx(;)i Ft(")p Fx(\))23 b(=)1597 3391 y Fy(1)1570 3416 y Fm(X)1570 3594 y Fo(k)q Fn(=1)1704 3494 y Ft(")1743 3460 y Fo(k)1835 3416 y Fm(X)1812 3602 y Fl(\027)5 b Fy(2)p Fk(Z)1946 3574 y Fj(r)1993 3494 y Ft(e)2032 3460 y Fo(i)p Fl(\027)g Fy(\001)p Fl( )2175 3494 y Ft(F)2240 3451 y Fn(\()p Fo(k)q Fn(\))2228 3509 y Fl(\027)2332 3494 y Ft(;)1018 b Fx(\(2)p Ft(:)p Fx(3\))0 3781 y(w)n(e)27 b(denote)h(b)n(y)f([)p Ft(F)12 b Fx(])616 3738 y Fn(\()p Fo(k)q Fn(\))616 3795 y Fl(\027)737 3781 y Fx(the)27 b(co)r(e\016cien)n(t)h(with)g(T)-7 b(a)n(ylor)26 b(lab)r(el)h Ft(k)k Fx(and)c(F)-7 b(ourier)27 b(lab)r(el)h Fq(\027)6 b Fx(.)71 3888 y(W)-7 b(e)28 b(can)f(write)418 4119 y([)p Ft(@)485 4131 y Fl(\013)540 4119 y Ft(f)9 b Fx(])613 4078 y Fn(\()p Fo(k)q Fy(\000)p Fn(1\))613 4144 y Fl(\027)813 4119 y Fx(=)901 4041 y Fm(X)901 4219 y Fo(p)p Fy(\025)p Fn(0)1035 4041 y Fm(X)1036 4219 y Fo(q)r Fy(\025)p Fn(0)1190 4063 y Fx(1)p 1179 4100 65 4 v 1179 4176 a Ft(p)p Fx(!)1274 4063 y(1)p 1263 4100 64 4 v 1263 4176 a Ft(q)s Fx(!)1336 4041 y Fm(X)1456 4053 y Fy(\003)1508 4119 y Fx(\()q Ft(i)p Fq(\027)1618 4131 y Fn(0)1654 4119 y Fx(\))1687 4078 y Fo(p)p Fn(+1)1823 4119 y Ft(@)1872 4080 y Fo(q)1867 4150 y Fl(\014)1917 4119 y Ft(f)1958 4131 y Fl(\027)1996 4139 y Fi(0)2033 4119 y Fx(\()p Fq(\014)2120 4131 y Fn(0)2157 4119 y Fx(\))2189 4027 y Fm(\020)2293 4012 y Fo(p)2257 4041 y Fm(Y)2253 4217 y Fo(j)s Fn(=1)2381 4119 y Fs(a)2427 4075 y Fn(\()p Fo(k)2488 4083 y Fj(j)2520 4075 y Fn(\))2427 4134 y Fl(\027)2465 4142 y Fj(j)2550 4027 y Fm(\021\020)2704 4012 y Fo(p)p Fn(+)p Fo(q)2710 4041 y Fm(Y)2663 4217 y Fo(j)s Fn(=)p Fo(p)p Fn(+1)2877 4119 y Fs(b)2930 4075 y Fn(\()p Fo(k)2991 4083 y Fj(j)3022 4075 y Fn(\))2930 4134 y Fl(\027)2968 4142 y Fj(j)3052 4027 y Fm(\021)3102 4119 y Ft(;)248 b Fx(\(2)p Ft(:)p Fx(4\))0 4376 y(where)25 b(0)e Ft(<)g(k)434 4388 y Fo(j)492 4376 y Ft(<)f(k)29 b Fx(for)d(all)f Ft(j)j Fx(=)23 b(1)p Ft(;)14 b(:)g(:)g(:)f(;)h(p)h Fx(+)f Ft(q)29 b Fx(and)d(the)g Fr(\003)g Fx(denotes)f(that)i(the)f(sum)g(has)f(to)h (b)r(e)g(p)r(erformed)g(with)0 4482 y(the)i(constrain)n(ts)1119 4643 y(1)18 b(+)1263 4535 y Fo(p)p Fn(+)p Fo(q)1262 4564 y Fm(X)1265 4741 y Fo(j)s Fn(=1)1396 4643 y Ft(k)1439 4655 y Fo(j)1497 4643 y Fx(=)23 b Ft(k)s(;)180 b Fq(\027)1882 4655 y Fn(0)1937 4643 y Fx(+)2021 4535 y Fo(p)p Fn(+)p Fo(q)2020 4564 y Fm(X)2023 4741 y Fo(j)s Fn(=1)2154 4643 y Fq(\027)2202 4655 y Fo(j)2260 4643 y Fx(=)22 b Fq(\027)6 b Ft(;)949 b Fx(\(2)p Ft(:)p Fx(5\))0 4872 y(and,)27 b(analogously)-7 b(,)413 5053 y Fm(\002)448 5120 y Ft(@)492 5137 y Fl(\014)542 5120 y Ft(f)592 5053 y Fm(\003)626 5070 y Fn(\()p Fo(k)q Fy(\000)p Fn(1\))626 5154 y Fl(\027)827 5120 y Fx(=)915 5041 y Fm(X)915 5219 y Fo(p)p Fy(\025)p Fn(0)1049 5041 y Fm(X)1050 5219 y Fo(q)r Fy(\025)p Fn(0)1204 5064 y Fx(1)p 1192 5101 65 4 v 1192 5177 a Ft(p)p Fx(!)1288 5064 y(1)p 1277 5101 64 4 v 1277 5177 a Ft(q)s Fx(!)1350 5041 y Fm(X)1470 5053 y Fy(\003)1522 5120 y Fx(\()p Ft(i)p Fq(\027)1631 5132 y Fn(0)1668 5120 y Fx(\))1700 5078 y Fo(p)1753 5120 y Ft(@)1802 5080 y Fo(q)r Fn(+1)1797 5151 y Fl(\014)1922 5120 y Ft(f)1963 5132 y Fl(\027)2001 5140 y Fi(0)2038 5120 y Fx(\()p Fq(\014)2125 5132 y Fn(0)2162 5120 y Fx(\))2194 5028 y Fm(\020)2298 5012 y Fo(p)2262 5041 y Fm(Y)2258 5218 y Fo(j)s Fn(=1)2386 5120 y Fs(a)2432 5075 y Fn(\()p Fo(k)2493 5083 y Fj(j)2525 5075 y Fn(\))2432 5134 y Fl(\027)2470 5142 y Fj(j)2555 5028 y Fm(\021\020)2709 5012 y Fo(p)p Fn(+)p Fo(q)2715 5041 y Fm(Y)2668 5218 y Fo(j)s Fn(=)p Fo(p)p Fn(+1)2882 5120 y Fs(b)2935 5075 y Fn(\()p Fo(k)2996 5083 y Fj(j)3027 5075 y Fn(\))2935 5134 y Fl(\027)2973 5142 y Fj(j)3057 5028 y Fm(\021)3107 5120 y Ft(;)243 b Fx(\(2)p Ft(:)p Fx(6\))0 5376 y(with)28 b(the)g(same)f(meaning)g(of)h(the)g(sym)n(b)r(ols.)0 5554 y Fs(2.2.)33 b Fv(T)-6 b(r)l(e)l(e)21 b(formalism.)36 b Fx(By)19 b(iterating)f(\(2.2\),)j(\(2.4\))d(and)h(\(2.6\),)i(one)d (\014nds)h(that)g(one)g(can)f(represen)n(t)g(graphically)0 5669 y Fs(a)46 5626 y Fn(\()p Fo(k)q Fn(\))46 5683 y Fl(\027)170 5669 y Fx(and)30 b Fs(b)387 5626 y Fn(\()p Fo(k)q Fn(\))387 5683 y Fl(\027)511 5669 y Fx(in)g(terms)h(of)f Fv(tr)l(e)l(es.)46 b Fx(The)30 b(de\014nition)h(and)f(usage)g(of)g (graphical)f(to)r(ols)h(based)g(on)h(tree)f(graphs)0 5869 y Fi(12)p Fj(=lug)q(lio=)p Fi(2001;)f(13:05)1190 b Fx(4)p eop %%Page: 5 5 5 4 bop 0 83 a Fx(in)31 b(the)g(con)n(text)f(of)h(KAM)g(theory)f(has)g (b)r(een)h(adv)n(o)r(cated)f(recen)n(tly)g(in)h(the)g(literature)f(as)g (an)g(in)n(terpretation)0 189 y(of)e(the)g(w)n(ork)e([E];)h(see)g(for)g (instance)h([G1],)f([GG],)h([BGGM])g(and)g([BaG].)71 296 y(A)h(tree)f Ft(\022)j Fx(\(see)e(Figure)f(2)g(b)r(elo)n(w\))g(is)h (de\014ned)g(as)f(a)g(partially)g(ordered)f(set)i(of)f(p)r(oin)n(ts,)h (connected)f(b)n(y)h Fv(lines)p Fx(.)0 402 y(The)k(lines)g(are)e(orien) n(ted)h(to)n(w)n(ard)f(the)j Fv(r)l(o)l(ot)p Fx(,)g(whic)n(h)f(is)f (the)h(leftmost)h(p)r(oin)n(t;)h(the)e(line)g(en)n(tering)f(the)i(ro)r (ot)d(is)0 508 y(called)d(the)g Fv(r)l(o)l(ot)i(line)p Fx(.)38 b(If)28 b(a)g(line)g Ft(`)f Fx(connects)h(t)n(w)n(o)f(p)r(oin)n (ts)h Ft(v)1897 520 y Fn(1)1962 508 y Fx(and)g Ft(v)2164 520 y Fn(2)2229 508 y Fx(and)g(is)g(orien)n(ted)f(from)g Ft(v)3032 520 y Fn(2)3098 508 y Fx(to)h Ft(v)3240 520 y Fn(1)3305 508 y Fx(w)n(e)g(sa)n(y)0 615 y(that)k Ft(v)224 627 y Fn(2)292 615 y Fr(\036)e Ft(v)427 627 y Fn(1)497 615 y Fx(and)h(w)n(e)h(shall)f(write)h Ft(v)1247 584 y Fy(0)1244 635 y Fn(1)1312 615 y Fx(=)e Ft(v)1447 627 y Fn(2)1517 615 y Fx(and)h Ft(`)f Fx(=)g Ft(`)1877 627 y Fo(v)1910 635 y Fi(2)1947 615 y Fx(;)k(w)n(e)d(shall)h(sa)n(y)f(also) g(that)h Ft(`)g Fx(exits)f(from)h Ft(v)3340 627 y Fn(2)3410 615 y Fx(and)0 721 y(en)n(ters)27 b Ft(v)283 733 y Fn(1)320 721 y Fx(.)701 1507 y(ro)r(ot)1058 1526 y Fq(\027)12 b Fx(=)6 b Fq(\027)1237 1538 y Fo(`)1265 1546 y Fi(0)1141 1381 y Ft(`)1176 1393 y Fn(0)1357 1525 y Ft(v)1397 1537 y Fn(0)1323 1360 y Fq(\027)1371 1372 y Fo(v)1404 1380 y Fi(0)1996 1284 y Ft(v)2036 1296 y Fn(1)1946 1078 y Fq(\027)1994 1090 y Fo(v)2027 1098 y Fi(1)1705 1769 y Ft(v)1745 1781 y Fn(2)2253 1425 y Ft(v)2293 1437 y Fn(3)2533 1008 y Ft(v)2573 1020 y Fn(5)2533 1284 y Ft(v)2573 1296 y Fn(6)2533 1525 y Ft(v)2573 1537 y Fn(7)2533 2183 y Ft(v)2573 2195 y Fn(11)2533 1977 y Ft(v)2573 1989 y Fn(10)2325 1766 y Ft(v)2365 1778 y Fn(4)2533 1664 y Ft(v)2573 1676 y Fn(8)2533 1803 y Ft(v)2573 1815 y Fn(9)941 2079 y @beginspecial @setspecial %%BeginDocument: fig2.ps /punto { gsave 2 0 360 newpath arc fill stroke grestore} def /origine1assexper2pilacon|P_2-P_1| { 4 2 roll 2 copy translate exch 4 1 roll sub 3 1 roll exch sub 2 copy atan rotate 2 copy exch 4 1 roll mul 3 1 roll mul add sqrt } def /linea {gsave 4 2 roll moveto lineto stroke grestore} def /punta0 {0 0 moveto dup dup 0 exch 2 div lineto 0 lineto 0 exch 2 div neg lineto 0 0 lineto fill stroke } def /dirpunta{% x1 y1 x2 y2 gsave origine1assexper2pilacon|P_2-P_1| 0 translate 7 punta0 grestore} def /frecciac {0 index 3 index add 2 div 4 index 3 index add 2 div exch 4 2 roll 5 index 5 index linea dirpunta} def gsave 0.83333 0.83333 scale 0 90 punto 70 90 punto 120 60 punto 160 130 punto 200 110 punto 240 170 punto 240 130 punto 240 90 punto 240 0 punto 240 30 punto 200 60 punto 240 70 punto 240 50 punto 70 90 0 90 frecciac 120 60 70 90 frecciac 160 130 70 90 frecciac 200 110 160 130 frecciac 240 170 160 130 frecciac 240 130 200 110 frecciac 240 90 200 110 frecciac 240 0 120 60 frecciac 240 30 120 60 frecciac 200 60 120 60 frecciac 240 70 200 60 frecciac 240 50 200 60 frecciac grestore %%EndDocument @endspecial 591 2275 a Fg(Fig.)53 b(2.)k Ff(A)29 b(tree)h Fo(\022)h Ff(with)f Fe(12)g Ff(no)r(des;)j(one)d(has)g Fo(p)1976 2283 y Fj(v)2006 2295 y Fi(0)2043 2275 y Fe(=2)p Fo(;p)2174 2283 y Fj(v)2204 2295 y Fi(1)2240 2275 y Fe(=2)p Fo(;p)2371 2283 y Fj(v)2401 2295 y Fi(2)2438 2275 y Fe(=3)p Fo(;p)2569 2283 y Fj(v)2599 2295 y Fi(3)2635 2275 y Fe(=2)p Fo(;p)2766 2283 y Fj(v)2796 2295 y Fi(4)2833 2275 y Fe(=2)p Ff(.)591 2349 y(The)24 b(length)g(of)g(the)g(lines)f(should)h(b)r(e)g (the)h(same)f(but)g(it)f(is)g(dra)n(wn)h(of)g(arbitrary)f(size.)71 2630 y Fx(There)29 b(will)g(b)r(e)h(t)n(w)n(o)f(kinds)g(of)h(p)r(oin)n (ts:)40 b(the)30 b Fv(no)l(des)g Fx(and)g(the)f Fv(le)l(aves)p Fx(.)44 b(The)29 b(lea)n(v)n(es)f(can)h(only)g(b)r(e)h(endp)r(oin)n (ts,)0 2736 y Fv(i.e.)h Fx(they)h(ha)n(v)n(e)f(no)g(lines)h(en)n (tering)f(them,)j(but)e(an)g(endp)r(oin)n(t)g(can)f(b)r(e)i(either)e(a) h(no)r(de)g(or)f(a)g(leaf.)50 b(The)31 b(lines)0 2843 y(exiting)e(from)f(the)h(lea)n(v)n(es)f(pla)n(y)g(a)g(v)n(ery)g (di\013eren)n(t)h(r^)-42 b(ole)28 b(with)h(resp)r(ect)f(to)h(the)g (lines)g(exiting)g(from)f(the)i(no)r(des,)0 2949 y(as)24 b(w)n(e)h(shall)f(see)h(b)r(elo)n(w.)35 b(W)-7 b(e)26 b(shall)e(denote)h(b)n(y)g Ft(v)1558 2961 y Fn(0)1620 2949 y Fx(the)h(last)e(\()p Fv(i.e.)32 b Fx(leftmost\))25 b(no)r(de)g(of)g(the)h(tree,)f(and)g(b)n(y)f Ft(`)3366 2961 y Fn(0)3428 2949 y Fx(the)0 3055 y(ro)r(ot)j(line;)h(for)f(future) h(con)n(v)n(enience)e(w)n(e)h(shall)g(write)h Ft(v)1758 3025 y Fy(0)1755 3076 y Fn(0)1815 3055 y Fx(=)23 b Ft(r)30 b Fx(but)f Ft(r)h Fx(will)e(not)g(b)r(e)g(considered)e(a)h(no)r(de.)71 3162 y(W)-7 b(e)28 b(shall)f(denote)g(b)n(y)g Ft(V)19 b Fx(\()p Ft(\022)r Fx(\))29 b(the)f(set)f(of)h(no)r(des,)f(b)n(y)g Ft(L)p Fx(\()p Ft(\022)r Fx(\))h(the)g(set)g(of)f(lea)n(v)n(es)f(and)h (b)n(y)g(\003\()p Ft(\022)r Fx(\))i(the)e(set)h(of)f(lines.)71 3268 y(Fixed)h(an)n(y)f Ft(`)495 3280 y Fo(v)558 3268 y Fr(2)d Ft(\022)r Fx(,)29 b(w)n(e)f(shall)f(sa)n(y)g(that)i(the)f (subset)g(of)g Ft(\022)j Fx(con)n(taining)c Ft(`)2374 3280 y Fo(v)2441 3268 y Fx(as)g(w)n(ell)h(as)g(all)f(no)r(des)h Ft(w)f Fr(\026)c Ft(v)32 b Fx(and)0 3374 y(all)27 b(lines)h(connecting) f(them)h(is)g(a)f Fv(subtr)l(e)l(e)g Fx(of)g Ft(\022)j Fx(with)e(ro)r(ot)f Ft(v)1933 3344 y Fy(0)1957 3374 y Fx(:)37 b(of)27 b(course)f(a)i(subtree)f(is)g(a)h(tree.)71 3480 y(Giv)n(en)f(a)f(tree,)h(with)g(eac)n(h)g(no)r(de)g Ft(v)j Fx(w)n(e)d(asso)r(ciate)e(a)i Fv(mo)l(de)i(lab)l(el)f Fq(\027)2199 3492 y Fo(v)2262 3480 y Fr(2)23 b Fp(Z)2407 3432 y Fo(r)2443 3480 y Fx(,)28 b(and)e(to)h(eac)n(h)f(leaf)h Ft(v)k Fx(a)26 b Fv(le)l(af)k(lab)l(el)0 3587 y Ft(\024)48 3599 y Fo(v)110 3587 y Fr(2)24 b Fp(N)p Fx(.)37 b(The)27 b(quan)n(tit)n(y)1377 3693 y Ft(k)f Fx(=)d Fr(j)p Ft(V)c Fx(\()p Ft(\022)r Fx(\))p Fr(j)g Fx(+)1900 3614 y Fm(X)1854 3796 y Fo(v)r Fy(2)p Fo(L)p Fn(\()p Fo(\022)r Fn(\))2079 3693 y Ft(\024)2127 3705 y Fo(v)3373 3693 y Fx(\(2)p Ft(:)p Fx(7\))0 3934 y(is)27 b(called)h(the)g Fv(or)l(der)g Fx(of)g(the)g(tree)f Ft(\022)r Fx(.)71 4040 y(With)c(an)n(y)f(line)h Ft(`)g Fx(exiting)f(from)h(a)f(no)r(de)h Ft(v)j Fx(w)n(e)c(asso)r (ciate)g(t)n(w)n(o)g(lab)r(els)g Ft(\015)2314 4052 y Fo(`)2346 4040 y Fx(,)i Ft(\015)2441 4010 y Fy(0)2436 4064 y Fo(`)2491 4040 y Fx(assuming)d(the)j(sym)n(b)r(olic)e(v)-5 b(alues)0 4147 y Ft(\013;)14 b(\014)32 b Fx(and)c(a)f Fv(momentum)i(lab)l(el)f Fq(\027)1065 4159 y Fo(`)1120 4147 y Fr(2)c Fp(Z)1265 4098 y Fo(r)1302 4147 y Fx(,)k(whic)n(h)f(is)h (de\014ned)g(as)1383 4332 y Fq(\027)1431 4344 y Fo(`)1486 4332 y Fr(\021)22 b Fq(\027)1621 4344 y Fo(`)1649 4352 y Fj(v)1712 4332 y Fx(=)1851 4253 y Fm(X)1809 4423 y Fj(w)q Fd(2)p Fj(V)13 b Fi(\()p Fj(\022)q Fi(\))1851 4473 y Fj(w)q Fd(\026)p Fj(v)2036 4332 y Fq(\027)2084 4344 y Fo(w)2137 4332 y Ft(;)1213 b Fx(\(2)p Ft(:)p Fx(8\))0 4640 y(while)28 b(with)g(an)n(y)f(line)h Ft(`)f Fx(exiting)g(from)g(a)h (leaf)f Ft(v)k Fx(w)n(e)c(asso)r(ciate)f(only)i(the)g(lab)r(els)f Ft(\015)2620 4652 y Fo(`)2675 4640 y Fx(=)c Ft(\015)2811 4610 y Fy(0)2806 4664 y Fo(`)2861 4640 y Fx(=)f Ft(\014)t Fx(.)71 4747 y(W)-7 b(e)33 b(can)g(asso)r(ciate)e(with)i(eac)n(h)f(no)r (de)h(also)f(some)g(lab)r(els)h(dep)r(ending)g(on)g(the)g(en)n(tering)f (lines)h(and)f(on)h(the)0 4853 y(exiting)g(one:)46 b(the)33 b Fv(br)l(anching)j(lab)l(els)d Ft(p)1271 4865 y Fo(v)1343 4853 y Fx(and)g Ft(q)1547 4865 y Fo(v)1586 4853 y Fx(,)h(denoting)f(ho) n(w)f(man)n(y)g(lines)h Ft(`)f Fx(ha)n(ving)g(the)h(lab)r(el)f Ft(\015)3330 4865 y Fo(`)3394 4853 y Fx(=)f Ft(\013)0 4959 y Fx(and,)c(resp)r(ectiv)n(ely)-7 b(,)27 b Ft(\015)696 4971 y Fo(`)751 4959 y Fx(=)c Ft(\014)32 b Fx(en)n(ter)27 b Ft(v)s Fx(,)h(and)f(the)h(lab)r(el)g Ft(\016)1764 4971 y Fo(v)1803 4959 y Fx(,)g(de\014ned)g(as)1342 5183 y Ft(\016)1379 5195 y Fo(v)1442 5183 y Fx(=)1529 5066 y Fm(\032)1605 5131 y Fx(1)p Ft(;)83 b Fx(if)28 b Ft(\015)1872 5143 y Fo(`)1900 5151 y Fj(v)1963 5131 y Fx(=)22 b Ft(\014)32 b Fx(,)1605 5231 y(0)p Ft(;)83 b Fx(if)28 b Ft(\015)1872 5243 y Fo(`)1900 5251 y Fj(v)1963 5231 y Fx(=)22 b Ft(\013)28 b Fx(.)3373 5183 y(\(2)p Ft(:)p Fx(9\))71 5407 y(Then)g(with)g(eac)n(h) f(no)r(de)g Ft(v)k Fx(w)n(e)c(asso)r(ciate)f(a)i Fv(no)l(de)i(factor) 1010 5629 y Ft(F)1063 5641 y Fo(v)1126 5629 y Fx(=)1255 5573 y(1)p 1223 5610 105 4 v 1223 5686 a Ft(p)1265 5698 y Fo(v)1304 5686 y Fx(!)1376 5573 y(1)p 1347 5610 100 4 v 1347 5686 a Ft(q)1384 5698 y Fo(v)1424 5686 y Fx(!)1457 5537 y Fm(\020)1506 5629 y Ft(i)p Fq(\027)1583 5641 y Fo(v)1622 5537 y Fm(\021)1672 5554 y Fo(p)1706 5562 y Fj(v)1742 5554 y Fn(+\(1)p Fy(\000)p Fo(\016)1934 5562 y Fj(v)1970 5554 y Fn(\))2000 5629 y Ft(@)2049 5590 y Fo(q)2079 5598 y Fj(v)2114 5590 y Fn(+)p Fo(\016)2195 5598 y Fj(v)2044 5660 y Fl(\014)2235 5629 y Ft(f)2276 5641 y Fl(\027)2314 5649 y Fj(v)2354 5629 y Fx(\()p Fq(\014)2441 5641 y Fn(0)2478 5629 y Fx(\))p Ft(;)798 b Fx(\(2)p Ft(:)p Fx(10\))0 5869 y Fi(12)p Fj(=lug)q(lio=)p Fi(2001;)29 b(13:05)1190 b Fx(5)p eop %%Page: 6 6 6 5 bop 0 83 a Fx(whic)n(h)24 b(is)g(a)g(tensor)f(of)h(rank)g Ft(p)948 95 y Fo(v)999 83 y Fx(+)12 b Ft(q)1113 95 y Fo(v)1163 83 y Fx(+)g(1,)24 b(while)g(with)h(eac)n(h)e(leaf)h Ft(v)k Fx(w)n(e)c(asso)r(ciate)e(a)i Fv(le)l(af)k(factor)d Fx(\(to)f(b)r(e)h(de\014ned)0 189 y(recursiv)n(ely)-7 b(,)26 b(see)h(b)r(elo)n(w\))1565 319 y Ft(L)1622 331 y Fo(v)1684 319 y Fx(=)c Fs(b)1825 276 y Fn(\()p Fo(\024)1890 284 y Fj(v)1925 276 y Fn(\))1825 341 y Fh(0)1955 319 y Ft(;)1353 b Fx(\(2)p Ft(:)p Fx(11\))0 497 y(whic)n(h)36 b(is)f(a)g(tensor)g(of)h(rank)f(1)g(\()p Fv(i.e.)d Fx(a)j(v)n(ector\);) k(to)c(eac)n(h)g(line)h Ft(`)f Fx(exiting)h(from)f(a)h(no)r(de)f Ft(v)k Fx(w)n(e)d(asso)r(ciate)e(a)0 603 y Fv(pr)l(op)l(agator)1389 751 y Ft(G)1454 763 y Fo(`)1509 751 y Fr(\021)23 b Ft(\016)1634 763 y Fo(\015)1669 772 y Fj(`)1697 763 y Fo(;\015)1752 779 y Fj(`)1777 767 y Fd(0)1943 695 y Fc(1)-27 b Fx(1)p 1817 732 305 4 v 1817 808 a(\()p Fq(!)22 b Fr(\001)c Fq(\027)2020 820 y Fo(`)2052 808 y Fx(\))2084 784 y Fn(2)2132 751 y Ft(;)1176 b Fx(\(2)p Ft(:)p Fx(12\))0 971 y(whic)n(h)27 b(is)f(a)h(\(diagonal\))f Ft(d)17 b Fr(\002)f Ft(d)27 b Fx(matrix,)g(while)g(no)f(small)h(divisor)f(is)g(asso)r(ciated)g (with)h(the)g(lines)g(exiting)g(from)0 1078 y(the)h(lea)n(v)n(es.)35 b(F)-7 b(or)27 b(consistence)g(w)n(e)g(can)h(de\014ne)1417 1297 y Ft(G)1482 1309 y Fo(`)1537 1297 y Fr(\021)23 b Ft(\016)1662 1309 y Fo(\015)1697 1318 y Fj(`)1725 1309 y Fo(;\015)1780 1325 y Fj(`)1805 1313 y Fd(0)1849 1297 y Ft(\016)1886 1309 y Fo(\015)1921 1325 y Fj(`)1946 1313 y Fd(0)1973 1309 y Fo(;\014)2051 1297 y Fc(1)-27 b Fx(1)o Ft(;)1205 b Fx(\(2)p Ft(:)p Fx(13\))0 1517 y(for)27 b(lines)h(exiting)f (from)g(lea)n(v)n(es,)f(so)h(that)h(a)f(propagator)e Ft(G)1889 1529 y Fo(`)1949 1517 y Fx(is)j(in)f(fact)h(asso)r(ciated)e (with)j(eac)n(h)d(line.)0 1699 y Fs(2.3.)35 b Fv(R)l(emark.)j Fx(Note)27 b(that)h(w)n(e)g(can)f(write)g(\(2.12\))g(in)h(the)g(form) 1361 1953 y Ft(G)1426 1965 y Fo(`)1481 1953 y Fx(=)1569 1861 y Fm(\020)1632 1903 y Ft(G)1697 1915 y Fo(`;\013\013)1918 1903 y Ft(G)1983 1915 y Fo(`;\013\014)1633 2003 y Ft(G)1698 2015 y Fo(`;\014)s(\013)1920 2003 y Ft(G)1985 2015 y Fo(`;\014)s(\014)2133 1861 y Fm(\021)3331 1953 y Fx(\(2)p Ft(:)p Fx(14\))0 2203 y(where)i Ft(G)308 2215 y Fo(`;\013\013)446 2203 y Fx(,)h Ft(G)565 2215 y Fo(`;\013\014)701 2203 y Fx(,)g Ft(G)820 2215 y Fo(`;\014)s(\013)986 2203 y Fx(and)f Ft(G)1215 2215 y Fo(`;\014)s(\014)1379 2203 y Fx(are)f Ft(r)23 b Fr(\002)d Ft(r)r Fx(,)32 b Ft(r)23 b Fr(\002)d Ft(s)p Fx(,)31 b Ft(s)20 b Fr(\002)g Ft(r)33 b Fx(and)e Ft(s)20 b Fr(\002)g Ft(s)30 b Fx(matrices.)44 b(By)30 b(construction)0 2310 y(one)d(has)g Ft(G)365 2322 y Fo(`;\013\014)524 2310 y Fx(=)c Ft(G)677 2279 y Fo(T)677 2333 y(`;\014)s(\013)835 2310 y Fx(=)g(0,)k(and)936 2541 y Ft(G)1001 2553 y Fo(`)1056 2541 y Fx(=)c Ft(G)p Fx(\()p Fq(!)f Fr(\001)c Fq(\027)1412 2553 y Fo(`)1444 2541 y Fx(\))p Ft(;)180 b(G)1744 2506 y Fo(T)1797 2541 y Fx(\()p Fr(\000)p Ft(x)p Fx(\))24 b(=)e Ft(G)2149 2506 y Fy(y)2184 2541 y Fx(\()p Ft(x)p Fx(\))i(=)f Ft(G)p Fx(\()p Ft(x)p Fx(\);)725 b(\(2)p Ft(:)p Fx(15\))0 2760 y(here)27 b(and)h(henceforth)f Ft(T)39 b Fx(and)27 b Fr(y)h Fx(denote,)f(resp)r(ectiv)n(ely)-7 b(,)27 b(the)h(transp)r(osed) f(and)g(the)h(adjoin)n(t)f(of)h(a)f(matrix.)0 2942 y Fs(2.4.)44 b Fv(T)-6 b(r)l(e)l(e)32 b(values)h(and)f(r)l(e)l(duc)l(e)l (d)h(tr)l(e)l(e)e(values.)46 b Fx(Call)30 b(\002)1785 2954 y Fo(k)q(;)p Fl(\027)5 b Fo(;\015)1977 2942 y Fx(the)31 b(set)f(of)g(all)g(trees)g(of)h(order)e Ft(k)k Fx(with)e Fq(\027)3306 2954 y Fo(`)3334 2962 y Fi(0)3397 2942 y Fx(=)d Fq(\027)0 3048 y Fx(and)f Ft(\015)204 3060 y Fo(`)232 3068 y Fi(0)292 3048 y Fx(=)22 b Ft(\015)5 b Fx(,)28 b(if)g Ft(`)589 3060 y Fn(0)653 3048 y Fx(is)g(the)g(ro)r(ot)f(line.)37 b(Set)1349 3311 y Ft(d)1392 3323 y Fo(\015)1458 3311 y Fx(=)1546 3194 y Fm(\032)1622 3261 y Ft(r)n(;)87 b Fx(for)27 b Ft(\015)h Fx(=)22 b Ft(\013)28 b Fx(,)1622 3361 y Ft(s;)83 b Fx(for)27 b Ft(\015)h Fx(=)22 b Ft(\014)32 b Fx(;)3331 3311 y(\(2)p Ft(:)p Fx(16\))0 3600 y(w)n(e)27 b(can)g(de\014ne)h(an)g(application)f Fs(V)-8 b(al)9 b Fx(:)23 b(\002)1316 3612 y Fo(k)q(;)p Fl(\027)5 b Fo(;\015)1500 3600 y Fr(!)23 b Fp(R)1678 3552 y Fo(d)1713 3560 y Fj(\015)1755 3600 y Fx(,)28 b(de\014ned)g(as)934 3841 y Fs(V)-8 b(al)p Fx(\()p Ft(\022)r Fx(\))24 b(=)1287 3749 y Fm(\020)1407 3762 y(Y)1351 3944 y Fo(v)r Fy(2)p Fo(V)14 b Fn(\()p Fo(\022)r Fn(\))1583 3841 y Ft(F)1636 3853 y Fo(v)1676 3749 y Fm(\021\020)1842 3762 y(Y)1789 3944 y Fo(v)r Fy(2)p Fo(L)p Fn(\()p Fo(\022)r Fn(\))2014 3841 y Ft(L)2071 3853 y Fo(v)2110 3749 y Fm(\021)o(\020)2271 3762 y(Y)2223 3944 y Fo(`)p Fy(2)p Fn(\003\()p Fo(\022)r Fn(\))2440 3841 y Ft(G)2505 3853 y Fo(`)2537 3749 y Fm(\021)2587 3841 y Ft(;)721 b Fx(\(2)p Ft(:)p Fx(17\))0 4146 y(whic)n(h)28 b(is)f(called)g(the)h Fv(value)g Fx(of)g(the)g(tree)f Ft(\022)r Fx(.)71 4257 y(W)-7 b(e)28 b(can)f(de\014ne)h(also)875 4477 y Fs(V)-8 b(al)1012 4440 y Fy(0)1035 4477 y Fx(\()p Ft(\022)r Fx(\))24 b(=)1252 4385 y Fm(\020)1372 4398 y(Y)1315 4580 y Fo(v)r Fy(2)p Fo(V)15 b Fn(\()p Fo(\022)r Fn(\))1548 4477 y Ft(F)1601 4489 y Fo(v)1641 4385 y Fm(\021)o(\020)1806 4398 y(Y)1754 4580 y Fo(v)r Fy(2)p Fo(L)p Fn(\()p Fo(\022)r Fn(\))1978 4477 y Ft(L)2035 4489 y Fo(v)2074 4385 y Fm(\021\020)2283 4398 y(Y)2187 4580 y Fo(`)p Fy(2)p Fn(\003\()p Fo(\022)r Fn(\))p Fy(n)p Fo(`)2453 4588 y Fi(0)2499 4477 y Ft(G)2564 4489 y Fo(`)2596 4385 y Fm(\021)2645 4477 y Ft(;)663 b Fx(\(2)p Ft(:)p Fx(18\))0 4794 y(where,)27 b(as)g(usual,)g Ft(`)640 4806 y Fn(0)705 4794 y Fx(denotes)g(the)h(ro)r(ot)f(line;)h Fs(V)-8 b(al)1643 4757 y Fy(0)1666 4794 y Fx(\()p Ft(\022)r Fx(\))29 b(is)e(called)h(the)f Fv(r)l(e)l(duc)l(e)l(d)j(value)e Fx(of)g(the)g(tree)f Ft(\022)r Fx(.)71 4905 y(The)h(follo)n(wing)e (cancellation)h(is)g(pro)n(v)n(ed)f(in)i(App)r(endix)h(A1.)0 5087 y Fs(2.5.)43 b Fw(Lemma.)i Fv(Supp)l(ose)32 b(that)h(for)g(al)t(l) g(tr)l(e)l(es)f Ft(\022)d Fr(2)f Fx(\002)1703 5099 y Fo(k)q(;)p Fl(\027)5 b Fo(;\015)1897 5087 y Fv(the)32 b(set)g Fx(\003\()p Ft(\022)r Fx(\))21 b Fr(n)e Ft(`)2449 5099 y Fn(0)2518 5087 y Fv(do)l(es)33 b(not)f(c)l(ontain)g(any)h(lines) g Ft(`)0 5193 y Fv(with)d(momentum)f Fq(\027)661 5205 y Fo(`)716 5193 y Fx(=)23 b Fs(0)p Fv(.)38 b(Then)30 b Fs(V)-8 b(al)1269 5156 y Fy(0)1292 5193 y Fx(\()p Ft(\022)r Fx(\))30 b Fv(is)h(wel)t(l)f(de\014ne)l(d)g(and)1481 5344 y Fm(X)1404 5522 y Fo(\022)r Fy(2)p Fn(\002)1534 5531 y Fj(k)q(;)p Fb(0)p Fj(;\013)1691 5423 y Fs(V)-8 b(al)1828 5385 y Fy(0)1852 5423 y Fx(\()p Ft(\022)r Fx(\))23 b(=)g Fs(0)p Ft(:)1192 b Fx(\(2)p Ft(:)p Fx(19\))0 5869 y Fi(12)p Fj(=lug)q(lio=)p Fi(2001;)29 b(13:05)1190 b Fx(6)p eop %%Page: 7 7 7 6 bop 0 83 a Fs(2.6.)51 b Fv(Existenc)l(e)34 b(of)i(formal)g (solutions.)53 b Fx(The)33 b(follo)n(wing)f(result)g(states)h(the)g (existence)f(of)h(formal)f(solutions)0 189 y(to)d(\(1.6\))g(whic)n(h)h (are)e(conjugated)h(to)g(the)h(unp)r(erturb)r(ed)g(motion)f(\(1.5\),)h (pro)n(vided)e(the)i(v)-5 b(alue)29 b Fq(\014)3103 201 y Fn(0)3169 189 y Fx(is)h(suitably)0 296 y(\014xed.)0 473 y Fs(2.7.)35 b Fw(Lemma.)i Fv(One)29 b(c)l(an)h(write,)g(formal)t (ly,)j(for)d(al)t(l)h Fq(\027)e Fr(2)23 b Fp(Z)1923 425 y Fo(r)1978 473 y Fr(n)18 b(f)p Fs(0)p Fr(g)p Fv(,)1373 646 y Fs(a)1419 603 y Fn(\()p Fo(k)q Fn(\))1419 661 y Fl(\027)1535 646 y Fx(=)1700 568 y Fm(X)1623 746 y Fo(\022)r Fy(2)p Fn(\002)1753 755 y Fj(k)q(;\027)s(;\013)1911 646 y 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(other)g(lea)n(v)n(es)g(and)h(so)f(on.)36 b(If)27 b(eac)n(h)g(time)g(a) g(leaf)f(is)h(encoun)n(tered,)f(it)i(can)0 890 y(b)r(e)23 b(decomp)r(osed)f(in)n(to)g(trees,)h(at)g(the)g(end)g(w)n(e)f(ha)n(v)n (e)f(that)i(the)g(v)-5 b(alue)23 b(of)f(a)h(tree)f Ft(\022)j Fx(can)d(b)r(e)h(expressed)e(as)h(pro)r(duct)0 996 y(of)k(factors)e (whic)n(h)i(are)e(v)-5 b(alues)25 b(of)h(trees)f(without)h(lea)n(v)n (es,)e(that)i(w)n(e)g(can)f(call,)g(as)g(in)h([Ge],)h Fv(trimme)l(d)h(tr)l(e)l(es)p Fx(.)35 b(The)0 1102 y(sum)23 b(of)f(the)h(orders)e(of)h(all)h(the)f(so)g(obtained)h(trimmed)f(trees) g(is)h(equal)f(to)g Ft(k)s Fx(,)i(if)f(the)f(tree)h Ft(\022)i Fx(b)r(elonged)d(to)g(\002)3359 1114 y Fo(k)q(;)p Fl(\027)5 b Fo(;\015)3520 1102 y Fx(;)0 1209 y(moreo)n(v)n(er)30 b(for)j(all)f(trimmed)i(trees)e(the)h(order)f(equals)g(exactly)g(the)i (n)n(um)n(b)r(er)e(of)h(no)r(des,)h(as)e(it)i(follo)n(ws)e(from)0 1315 y(\(2.7\))27 b(b)n(y)h(using)f(that)h(a)f(trimmed)h(tree)f(has)g (no)h(lea)n(v)n(es.)0 1492 y Fs(3.2.)49 b Fv(Multi-sc)l(ale)34 b(de)l(c)l(omp)l(osition)i(and)e(clusters.)50 b Fx(Giv)n(en)32 b(a)g(v)n(ector)e Fq(!)j Fr(2)e Ft(D)2495 1504 y Fo(\034)2536 1492 y Fx(\()p Ft(C)2627 1504 y Fn(0)2665 1492 y Fx(\),)j(de\014ne)e Fq(!)3058 1504 y Fn(0)3125 1492 y Fx(=)e(2)3262 1462 y Fo(\034)3303 1492 y Ft(C)3368 1457 y Fy(\000)p Fn(1)3362 1514 y(0)3458 1492 y Fq(!)s Fx(.)0 1598 y(Then)e(there)f(exists)g(a)h (sequence)f Fr(f)p Ft(\015)1157 1610 y Fo(n)1201 1598 y Fr(g)1243 1618 y Fo(n)p Fy(2)p Fk(Z)1375 1626 y Fi(+)1426 1598 y Fx(,)h(with)g Ft(\015)1709 1610 y Fo(n)1777 1598 y Fr(2)c Fx([2)1921 1568 y Fo(n)p Fy(\000)p Fn(1)2050 1598 y Ft(;)14 b Fx(2)2129 1568 y Fo(n)2174 1598 y Fx(],)28 b(suc)n(h)f(that)887 1808 y Fr(j)q(j)p Fq(!)994 1820 y Fn(0)1049 1808 y Fr(\001)18 b Fq(\027)6 b Fr(j)19 b(\000)f Ft(\015)1312 1820 y Fo(p)1350 1808 y Fr(j)23 b(\025)g Fx(2)1526 1773 y Fo(n)p Fn(+1)1848 1808 y Fx(if)28 b(0)23 b Ft(<)f Fr(j)q Fq(\027)6 b Fr(j)22 b(\024)h Fx(2)2329 1773 y Fy(\000)p Fn(\()p Fo(n)p Fn(+3\))p Fo(=\034)2633 1808 y Ft(;)717 b Fx(\(3)p Ft(:)p Fx(3\))0 2002 y(for)29 b(all)g Ft(n)c Fr(\024)g Fx(0)k(and)g(for)g(all)g Ft(p)c Fr(\025)h Ft(n)p Fx(,)j(and)g Fr(j)p Fq(!)1397 2014 y Fn(0)1454 2002 y Fr(\001)19 b Fq(\027)6 b Fr(j)26 b(6)p Fx(=)f Ft(\015)1732 2014 y Fo(n)1806 2002 y Fx(for)k(all)g Fq(\027)i Fr(2)26 b Fp(Z)k Fx(and)f(for)g(all)f Ft(n)e Fr(\024)f Fx(0;)30 b(the)g(existence)e(of)0 2109 y(suc)n(h)f(a)g (sequence)h(\(dep)r(ending)g(on)f Fq(!)s Fx(\))h(is)f(pro)n(v)n(ed)f(b) n(y)i(Prop)r(osition)d(in)j(Section)g(3)f(of)h([GG].)71 2215 y(Giv)n(en)f(a)g(line)h Ft(`)g Fx(with)g(momen)n(tum)g Fq(\027)1273 2227 y Fo(`)1332 2215 y Fx(w)n(e)f(sa)n(y)g(that)h Ft(`)f Fx(has)g(scale)g(lab)r(el)g Ft(n)2437 2227 y Fo(`)2492 2215 y Fx(=)c(1)k(if)1523 2409 y Fr(j)p Fq(!)1606 2421 y Fn(0)1662 2409 y Fr(\001)18 b Fq(\027)1751 2421 y Fo(`)1783 2409 y Fr(j)23 b(\025)g Ft(\015)1960 2421 y Fn(0)1997 2409 y Ft(;)1353 b Fx(\(3)p Ft(:)p Fx(4\))0 2604 y(and)27 b(scale)g(lab)r(el)h Ft(n)611 2616 y Fo(`)666 2604 y Fx(=)22 b Ft(n)h Fr(2)h Fp(Z)18 b Fr(n)g Fp(Z)1116 2616 y Fn(+)1199 2604 y Fx(if)1377 2799 y Ft(\015)1420 2811 y Fo(n)p Fy(\000)p Fn(1)1574 2799 y Fr(\024)k(j)p Fq(!)1744 2811 y Fn(0)1800 2799 y Fr(\001)c Fq(\027)1889 2811 y Fo(`)1921 2799 y Fr(j)23 b Ft(<)g(\015)2098 2811 y Fo(n)2143 2799 y Ft(:)1207 b Fx(\(3)p Ft(:)p Fx(5\))71 2993 y(Once)24 b(the)h(scale)f(lab)r(els)h(ha)n(v)n(e)e(b)r(een)j(assigned)d(to)i(the) g(lines)g(one)f(has)g(a)h(natural)f(decomp)r(osition)g(of)h(the)g(tree) 0 3100 y(in)n(to)33 b(clusters.)54 b(A)34 b Fv(cluster)g Ft(T)44 b Fx(on)34 b(scale)e Ft(n)i Fx(is)f(a)g(maximal)g(set)h(of)f (no)r(des)h(and)f(lines)h(connecting)f(them)h(suc)n(h)0 3206 y(that)e(all)f(the)h(lines)g(ha)n(v)n(e)e(scales)h Ft(n)1125 3176 y Fy(0)1178 3206 y Fr(\025)e Ft(n)j Fx(and)f(there)h(is) f(at)h(least)f(one)g(line)h(with)g(scale)f Ft(n)p Fx(;)j(if)e(a)f (cluster)g Ft(T)3433 3176 y Fy(0)3487 3206 y Fx(is)0 3312 y(con)n(tained)d(inside)i(a)e(cluster)h Ft(T)40 b Fx(w)n(e)29 b(shall)g(sa)n(y)e(that)j Ft(T)1751 3282 y Fy(0)1802 3312 y Fx(is)f(a)g(sub)r(cluster)g(of)g Ft(T)12 b Fx(.)40 b(The)30 b Ft(m)2821 3324 y Fo(T)2898 3312 y Fr(\025)25 b Fx(0)k(lines)g(en)n(tering)0 3419 y(the)d(cluster)g Ft(T)36 b Fx(and)26 b(the)g(p)r(ossible)g(exiting)f(line)h(\(unique)h (if)f(existing)f(at)h(all\))g(are)f(called)g(the)h Fv(external)i(lines) e Fx(of)0 3525 y(the)h(cluster)f Ft(T)12 b Fx(;)26 b(giv)n(en)g(a)g (cluster)g Ft(T)38 b Fx(on)26 b(scale)g Ft(n)p Fx(,)h(w)n(e)f(shall)g (denote)g(b)n(y)h Ft(n)2314 3537 y Fo(T)2389 3525 y Fx(=)22 b Ft(n)27 b Fx(the)g(scale)e(of)i(the)g(cluster.)36 b(W)-7 b(e)0 3631 y(call)27 b Fr(T)22 b Fx(\()p Ft(\022)r Fx(\))28 b(the)g(set)g(of)f(all)h(clusters)e(in)i(a)g(tree)f Ft(\022)r Fx(.)71 3738 y(Giv)n(en)j(a)f(cluster)h Ft(T)38 b Fr(2)28 b(T)21 b Fx(\()p Ft(\022)r Fx(\))31 b(call)f Ft(V)18 b Fx(\()p Ft(T)12 b Fx(\),)31 b Ft(L)p Fx(\()p Ft(T)12 b Fx(\))29 b(and)h(\003\()p Ft(T)12 b Fx(\))29 b(the)i(set)f(of)g(no)r (des,)g(the)h(set)f(of)g(lea)n(v)n(es)e(and)i(the)0 3844 y(set)e(of)f(lines)h(of)f Ft(T)12 b Fx(,)27 b(resp)r(ectiv)n(ely)-7 b(.)36 b(Let)28 b(us)f(de\014ne)h(also)1487 4039 y Fq(\027)1535 4051 y Fo(T)1611 4039 y Fx(=)1755 3960 y Fm(X)1698 4142 y Fo(v)r Fy(2)p Fo(V)15 b Fn(\()p Fo(T)9 b Fn(\))1946 4039 y Fq(\027)1994 4051 y Fo(v)2033 4039 y Ft(;)1317 b Fx(\(3)p Ft(:)p Fx(6\))0 4323 y(and)31 b(denote)h(b)n(y)f Fr(T)601 4335 y Fh(0)643 4323 y Fx(\()p Ft(\022)r Fx(\))i(the)f(set)g (of)f(all)h(clusters)f Ft(T)42 b Fx(with)33 b Fq(\027)1918 4335 y Fo(T)2000 4323 y Fx(=)c Fs(0)p Fx(.)49 b(Giv)n(en)31 b(a)h(cluster)f Ft(T)43 b Fx(call)31 b Ft(T)3101 4335 y Fn(0)3169 4323 y Fx(the)h(subset)0 4430 y(of)e Ft(T)42 b Fx(obtained)30 b(from)g Ft(T)42 b Fx(b)n(y)30 b(eliminating)g(all)g (the)h(no)r(des)f(and)g(lines)h(of)f(the)h(sub)r(clusters)f Ft(T)2968 4400 y Fy(0)3018 4430 y Fr(\032)d Ft(T)42 b Fx(suc)n(h)30 b(that)0 4536 y Fq(\027)48 4548 y Fo(T)96 4532 y Fd(0)146 4536 y Fx(=)22 b Fs(0)p Fx(,)28 b(and)f(denote)h(b)n(y) f Ft(V)19 b Fx(\()p Ft(T)1024 4548 y Fn(0)1061 4536 y Fx(\))28 b(and)f(\003\()p Ft(T)1421 4548 y Fn(0)1458 4536 y Fx(\))h(the)g(set)g(of)f(no)r(des)h(and)f(lines,)h(resp)r(ectiv) n(ely)-7 b(,)26 b(in)i Ft(T)3107 4548 y Fn(0)3144 4536 y Fx(.)0 4713 y Fs(3.3.)35 b Fv(Self-ener)l(gy)c(gr)l(aphs.)38 b Fx(W)-7 b(e)28 b(call)f Fv(self-ener)l(gy)k(gr)l(aphs)e Fx(of)e(a)h(tree)f Ft(\022)j Fx(the)e(clusters)f Ft(T)34 b Fr(2)23 b(T)f Fx(\()p Ft(\022)r Fx(\))28 b(suc)n(h)f(that)0 4819 y(\(1\))h Ft(T)38 b Fx(has)28 b(only)f(one)g(en)n(tering)g(line)h Ft(`)1217 4789 y Fn(2)1217 4842 y Fo(T)1296 4819 y Fx(and)g(one)f (exiting)g(line)h Ft(`)2076 4789 y Fn(1)2076 4842 y Fo(T)2128 4819 y Fx(,)0 4926 y(\(2\))g Ft(T)34 b Fr(2)23 b(T)340 4938 y Fh(0)382 4926 y Fx(\()p Ft(\022)r Fx(\),)29 b Fv(i.e.)1408 5042 y Fq(\027)1456 5054 y Fo(T)1531 5042 y Fr(\021)1676 4963 y Fm(X)1619 5145 y Fo(v)r Fy(2)p Fo(V)15 b Fn(\()p Fo(T)9 b Fn(\))1866 5042 y Fq(\027)1914 5054 y Fo(v)1977 5042 y Fx(=)22 b Fs(0)p Ft(;)1238 b Fx(\(3)p Ft(:)p Fx(7\))0 5292 y(\(3\))28 b(one)f(has)1398 5329 y Fm(X)1330 5511 y Fo(v)r Fy(2)p Fo(V)14 b Fn(\()p Fo(T)1528 5519 y Fi(0)1561 5511 y Fn(\))1601 5408 y Fr(j)p Fq(\027)1672 5420 y Fo(v)1711 5408 y Fr(j)23 b(\024)g Fx(2)1887 5374 y Fy(\000)p Fn(\()p Fo(n)p Fn(+3\))p Fo(=\034)2191 5408 y Ft(;)1159 b Fx(\(3)p Ft(:)p Fx(8\))0 5663 y(where)27 b Ft(n)290 5681 y Fo(`)318 5661 y Fi(2)318 5703 y Fj(T)391 5663 y Fx(=)22 b Ft(n)28 b Fx(is)f(the)h(scale)f(of)h(the)g(line)g Ft(`)1411 5633 y Fn(2)1411 5686 y Fo(T)1462 5663 y Fx(.)0 5869 y Fi(12)p Fj(=lug)q(lio=)p Fi(2001;)h(13:05)1190 b Fx(8)p eop %%Page: 9 9 9 8 bop 71 83 a Fx(W)-7 b(e)32 b(sa)n(y)f(that)h(the)g(line)g Ft(`)893 53 y Fn(1)893 106 y Fo(T)977 83 y Fx(exiting)f(a)h (self-energy)e(graph)h Ft(T)43 b Fx(is)31 b(a)h Fv(self-ener)l(gy)i (line)p Fx(;)h(w)n(e)c(call)h Fv(normal)i(line)0 189 y Fx(an)n(y)27 b(line)h(of)f(the)h(tree)f(whic)n(h)h(is)f(not)h(a)f (self-energy)f(line.)71 301 y(Giv)n(en)31 b(a)g(self-energy)f(graph)g Ft(T)40 b Fr(2)30 b(T)21 b Fx(\()p Ft(\022)r Fx(\))32 b(w)n(e)f(sa)n(y)f(that)i(a)f(self-energy)e(graph)i Ft(T)2664 271 y Fy(0)2715 301 y Fr(2)f(T)21 b Fx(\()p Ft(\022)r Fx(\))32 b(con)n(tained)f(in)g Ft(T)0 407 y Fx(is)f Fv(maximal)h Fx(if)f(there)g(are)f(no)g(other)h(self-energy)e(graphs)h(in)n(ternal)g (to)h Ft(T)41 b Fx(and)30 b(con)n(taining)f Ft(T)3011 377 y Fy(0)3033 407 y Fx(.)44 b(W)-7 b(e)30 b(sa)n(y)f(that)0 513 y(a)h(self-energy)f(graph)g Ft(T)41 b Fx(has)30 b Fv(height)h Ft(D)1278 525 y Fo(T)1358 513 y Fx(=)c(0)j(if)h(it)f(do)r (es)g(not)g(con)n(tain)g(an)n(y)f(other)h(self-energy)f(graphs,)h(and)0 620 y(that)25 b(it)f(has)g(heigh)n(t)g Ft(D)718 632 y Fo(T)793 620 y Fx(=)f Ft(D)i Fr(2)e Fp(Z)1120 632 y Fn(+)1175 620 y Fx(,)i(recursiv)n(ely)-7 b(,)23 b(if)i(it)g(con)n(tains)e (maximal)h(self-energy)f(graphs)g(with)i(heigh)n(t)0 726 y Ft(D)20 b Fr(\000)f Fx(1.)1564 1632 y Ft(v)1604 1644 y Fn(1)2021 1540 y Ft(v)2061 1552 y Fn(2)1680 1358 y Ft(v)1720 1370 y Fn(3)2220 1482 y Ft(v)2260 1494 y Fn(5)2062 1159 y Ft(v)2102 1171 y Fn(6)2104 1416 y Ft(v)2144 1428 y Fn(4)1190 1498 y Ft(`)1225 1467 y Fn(1)1225 1520 y Fo(T)1813 1877 y Ft(T)2187 1243 y(T)2248 1213 y Fy(0)1938 1385 y Ft(T)1999 1354 y Fy(00)2685 1680 y Ft(`)2720 1650 y Fn(2)2720 1703 y Fo(T)1107 2094 y @beginspecial @setspecial %%BeginDocument: fig3.ps /origine1assexper2pilacon|P_2-P_1| { % P1 P2 ori.... 4 2 roll 2 copy translate exch 4 1 roll sub 3 1 roll exch sub 2 copy atan rotate 2 copy exch 4 1 roll mul 3 1 roll mul add sqrt } def /ellisse0 {% Rx Ry ellisse0 exch dup 0 moveto 0 1 360 { 3 1 roll dup 3 2 roll dup 5 1 roll exch 5 2 roll 3 1 roll dup 4 1 roll sin mul 3 1 roll exch cos mul exch lineto} for stroke pop pop} def /ellisse {% Rx Ry (assi) x y (centro) X Y (direzione asse) gsave origine1assexper2pilacon|P_2-P_1| pop ellisse0 grestore} def /punto { gsave 1.5 0 360 newpath arc fill stroke grestore} def /puntone { gsave 4 0 360 newpath arc fill stroke grestore} def /punta0 {0 0 moveto dup dup 0 exch 2 div lineto 0 lineto 0 exch 2 div neg lineto 0 0 lineto fill stroke } def /dirpunta{% x1 y1 x2 y2 gsave origine1assexper2pilacon|P_2-P_1| 0 translate 7 punta0 grestore} def /linea {gsave 4 2 roll moveto lineto stroke grestore} def /frecciac {0 index 3 index add 2 div 4 index 3 index add 2 div exch 4 2 roll 5 index 5 index linea dirpunta} def gsave 1 0.8 scale /R {50} def /H {80} def /V {2 5 div R mul 3 5 div R mul} def /D {10 0} def D translate /P0 {R -1.5 div H} def /P1 {R 3 mul 6 div H} def /P2 {R 1.3 mul H} def /P3 {P1 V 3 -1 roll add 3 1 roll add exch} def /P4 {P2 V 3 -1 roll add 3 1 roll add exch} def /P5 {P4 2 R mul 5 div 0.2 R mul add 0 3 -1 roll add 3 1 roll add 10 sub exch } def /P6 {P4 2 R mul 5 div 2 R mul 2 div 3 -1 roll add 3 1 roll add -30 add exch 15 sub exch 2 5 div R mul add exch} def /P7 {3 R mul R 2 div add H 30 sub} def R 2 div 0 translate P0 P2 linea P1 P0 R 4 div 3 -1 roll add exch frecciac P2 P1 frecciac P3 P1 frecciac P4 P2 frecciac P5 P4 frecciac P6 P4 frecciac P7 P2 frecciac P0 puntone P1 punto P2 punto P3 punto P4 punto P5 punto P6 punto P7 puntone 1.5 R mul 1.7 R mul P2 R 5 div add P2 R 5 div add exch 10 add exch ellisse R 2 div R 4 div P4 P4 exch 10 add exch ellisse R 0.7 mul R 0.9 mul P4 P4 exch 10 add exch ellisse grestore %%EndDocument @endspecial 591 2302 a Fg(Fig.)52 b(3.)k Ff(Three)29 b(self-energy)g(graphs)h Fo(T)5 b(;)11 b(T)1818 2277 y Fy(0)1842 2302 y Fo(;)g(T)1921 2277 y Fy(00)1993 2302 y Ff(of)29 b(heigh)n(t)h(2,)g(1,)h(0,)f(resp)r(ectiv)n(ely)-6 b(.)591 2376 y(The)22 b(blac)n(k)g(balls)g(represen)n(t)g(the)h (remaining)f(parts)g(of)g(the)g(trees;)h(the)g(lab)r(els)e(are)h(not) 591 2451 y(explicitly)30 b(sho)n(wn.)51 b(The)31 b(self-energy)f(graph) h Fo(T)1927 2426 y Fy(0)1981 2451 y Ff(is)e(maximal)h(in)g Fo(T)9 b Ff(.)51 b(The)31 b(ellipses)591 2526 y(are)24 b(visual)g(aids)h(to)g(iden)n(tify)f(the)i(clusters.)33 b(The)25 b(scales)g(\(not)h(mark)n(ed\))f(of)f(the)i(lines)591 2600 y(increase)f(as)g(one)h(crosses)f(in)n(w)n(ard)g(the)g(ellipses)g (b)r(oundaries:)34 b(recall,)25 b(ho)n(w)n(ev)n(er,)h(that)591 2675 y(the)g(scale)g(lab)r(els)f(are)g Fy(\024)d Ff(0.)37 b(The)26 b(cluster)f Fo(T)1785 2687 y Fe(0)1820 2675 y Ff(,)g(de\014ned)i(after)f(\(3.6\),)g(consists)g(in)f(this)591 2750 y(case)h(of)f(the)h(no)r(des)g Fo(v)1180 2762 y Fe(1)1240 2750 y Ff(and)g Fo(v)1412 2762 y Fe(3)1447 2750 y Ff(,)g(and)g(of)f(the)h(lines)f Fo(`)2028 2758 y(v)2061 2766 y Fi(3)2123 2750 y Ff(and)h Fo(`)2290 2758 y(v)2323 2766 y Fi(2)2359 2750 y Ff(.)37 b(The)25 b(mo)r(de)h(lab)r (els)591 2825 y(of)e(the)h(pairs)e(of)h(no)r(des)h(\()p Fo(v)1307 2837 y Fe(1)1343 2825 y Fo(;)11 b(v)1407 2837 y Fe(3)1441 2825 y Ff(\),)25 b(\()p Fo(v)1573 2837 y Fe(2)1608 2825 y Fo(;)11 b(v)1672 2837 y Fe(6)1707 2825 y Ff(\))25 b(and)f(\()p Fo(v)1956 2837 y Fe(4)1992 2825 y Fo(;)11 b(v)2056 2837 y Fe(5)2091 2825 y Ff(\))24 b(add)h(up)g(to)g Fh(0)p Ff(.)32 b(An)24 b(example)591 2899 y(of)d(a)h(simpler)e (self-energy)i(graph)g(could)g(b)r(e)g(obtained)h(b)n(y)f(deleting)h (the)f(line)f Fo(`)2724 2907 y(v)2757 2915 y Fi(3)2815 2899 y Ff(and)591 2974 y(assuming)i(that)i(the)g(mo)r(de)f(lab)r(el)f (of)h Fo(v)1646 2986 y Fe(1)1704 2974 y Ff(v)l(anishes.)71 3289 y Fx(Giv)n(en)j(a)g(line)h Ft(`)23 b Fr(2)g Fx(\003\()p Ft(T)812 3301 y Fn(0)849 3289 y Fx(\))28 b(with)g(momen)n(tum)g Fq(\027)1582 3301 y Fo(`)1614 3289 y Fx(,)g(its)f Fv(r)l(e)l(duc)l(e)l (d)j(momentum)d Fq(\027)2557 3259 y Fn(0)2551 3313 y Fo(`)2621 3289 y Fx(is)h(de\014ned)g(as)1261 3523 y Fq(\027)1315 3489 y Fn(0)1309 3543 y Fo(`)1374 3523 y Fx(=)1532 3444 y Fm(X)1472 3614 y Fj(w)q Fd(2)p Fj(V)12 b Fi(\()p Fj(T)1657 3626 y Fi(0)1690 3614 y(\))1532 3666 y Fj(w)q Fd(\026)p Fj(v)1736 3523 y Fq(\027)1784 3535 y Fo(w)1837 3523 y Ft(;)180 b(`)23 b Fr(\021)g Ft(`)2221 3535 y Fo(v)2260 3523 y Ft(;)1090 b Fx(\(3)p Ft(:)p Fx(9\))0 3874 y(and)27 b(it)h(can)g(b)r(e)g(giv)n(en)e(a)i(scale)e Ft(n)1044 3844 y Fn(0)1044 3898 y Fo(`)1109 3874 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Fn(1)466 5267 y Fo(T)518 5244 y Fx(,)g(hence,)g(b)n(y)f(construction,)g(the)h (same)f(scale)f Ft(n)2036 5262 y Fo(`)2064 5242 y Fi(2)2064 5284 y Fj(T)2137 5244 y Fx(=)d Ft(n)2275 5262 y Fo(`)2303 5242 y Fi(1)2303 5284 y Fj(T)2353 5244 y Fx(.)0 5350 y(\(3\))h(The)f(notion)g(of)g(self-energy)f(lines)i(has)e(b)r(een)i(in) n(tro)r(duced)f(b)n(y)g(Eliasson)f(who)h(named)g(them)h(\\resonances",) 0 5457 y([E].)g(W)-7 b(e)25 b(c)n(hange)e(the)i(name)f(here)g(not)g (only)g(to)g(a)n(v)n(oid)f(confusion)h(with)h(the)f(notion)g(of)h(mec)n (hanical)e(resonance)0 5563 y(\(whic)n(h)33 b(is)g(related)f(to)h(a)g (rational)f(relation)g(b)r(et)n(w)n(een)h(frequencies)f(of)h(a)g(quasi) f(p)r(erio)r(dic)h(motion\))g(but)g(also)0 5669 y(b)r(ecause)24 b(the)i(\\tree)e(expansions")f(that)i(w)n(e)f(use)h(here)f(\(also)g (basically)g(due)h(to)g(Eliasson\))e(can)i(b)r(e)g(in)n(terpreted,)0 5869 y Fi(12)p Fj(=lug)q(lio=)p Fi(2001;)k(13:05)1190 b Fx(9)p eop %%Page: 10 10 10 9 bop 0 83 a Fx([GGM],)31 b(as)e(F)-7 b(eynman)31 b(graphs)d(of)i(a)g(suitable)g(\014eld)h(theory)-7 b(.)44 b(As)30 b(suc)n(h)g(they)g(corresp)r(ond)f(to)h(classes)e(of)j(self-)0 189 y(energy)38 b(graphs:)59 b(w)n(e)39 b(use)g(here)f(the)i(corresp)r (ondence)d(to)j(p)r(erform)e(resummation)h(op)r(erations)f(t)n(ypical)g (of)0 296 y(renormalization)25 b(theory)-7 b(.)0 474 y Fs(3.5.)34 b Fv(V)-6 b(alue)26 b(of)h(a)f(self-ener)l(gy)h(gr)l(aph.) 37 b Fx(Giv)n(en)24 b(a)f(self-energy)f(graph)h Ft(T)34 b Fx(and)24 b(denoting)f(b)n(y)g Fr(j)p Ft(V)c Fx(\()p Ft(T)12 b Fx(\))p Fr(j)24 b Fx(the)g(n)n(um)n(b)r(er)0 580 y(of)k(no)r(des)f(in)h Ft(T)12 b Fx(,)27 b(de\014ne)g(the)h Fv(self-ener)l(gy)j(value)d Fx(as)447 804 y Fr(V)498 816 y Fo(T)550 804 y Fx(\()p Fq(!)21 b Fr(\001)e Fq(\027)6 b Fx(\))23 b(=)f Ft(")940 770 y Fy(j)p Fo(V)14 b Fn(\()p Fo(T)9 b Fn(\))p Fy(j)1137 712 y Fm(\020)1264 725 y(Y)1201 907 y Fo(v)r Fy(2)p Fo(V)14 b Fn(\()p Fo(T)9 b Fn(\))1448 804 y Ft(F)1501 816 y Fo(v)1541 712 y Fm(\021)o(\020)1714 725 y(Y)1654 907 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y(in)n(terpretation)d(of)g(Bryuno)g([B])h(and)f(P\177)-42 b(osc)n(hel)35 b([P].)i(This)f(is)h(pro)n(v)n(ed)e(as)h(in)h([G1])f(or) g([BaG],)g(for)h(insatnce;)0 3548 y(ho)n(w)n(ev)n(e,)26 b(for)h(completeness,)g(a)g(pro)r(of)g(is)h(also)e(giv)n(en)h(in)h(App) r(endix)h(A3.)0 3726 y Fs(3.6.)35 b Fw(Lemma.)i Fv(F)-6 b(or)30 b(any)g(tr)l(e)l(e)f Ft(\022)c Fr(2)f Fx(\002)1209 3738 y Fo(\024;)p Fl(\027)5 b Fo(;\015)1402 3726 y Fv(one)30 b(has)1386 3937 y Ft(N)1462 3902 y Fy(\003)1453 3957 y Fo(n)1500 3937 y Fx(\()p Ft(\022)r Fx(\))24 b Fr(\024)f Ft(c)14 b(M)9 b Fx(\()p Ft(\022)r Fx(\))14 b(2)2018 3902 y Fo(n=\034)2134 3937 y Ft(;)1174 b Fx(\(3)p Ft(:)p Fx(17\))0 4147 y Fv(for)31 b(some)f(c)l(onstant)f Ft(c)p Fv(.)0 4325 y Fs(3.7.)35 b Fv(L)l(o)l(c)l(alization)d(op)l(er)l(ators.)38 b Fx(F)-7 b(or)27 b(an)n(y)g(self-energy)f(graph)h Ft(T)38 b Fx(w)n(e)27 b(de\014ne)1092 4535 y Fr(LV)1200 4547 y Fo(T)1253 4535 y Fx(\()p Fq(!)21 b Fr(\001)e Fq(\027)6 b Fx(\))23 b Fr(\021)f(V)1655 4547 y Fo(T)1708 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(to)h Ft(x)h Fx(=)g Fq(!)27 b Fr(\001)e Fq(\027)6 b Fx(,)39 b(and)d(thence)0 5064 y(replacing)26 b Fq(\027)403 5076 y Fo(`)463 5064 y Fx(with)i Fq(\027)706 5034 y Fn(0)700 5088 y Fo(`)771 5064 y Fx(in)f(the)h(argumen)n(t)f(of)g(eac)n(h)g (propagator)e Ft(G)2153 5076 y Fo(`)2185 5064 y Fx(.)71 5172 y(W)-7 b(e)23 b(shall)g(call)g Fr(L)g Fx(the)g Fv(lo)l(c)l (alization)28 b(op)l(er)l(ator)c Fx(and)f Fr(LV)1777 5184 y Fo(T)1829 5172 y Fx(\()p Fq(!)12 b Fr(\001)d Fq(\027)d Fx(\))24 b(the)g Fv(lo)l(c)l(alize)l(d)j(p)l(art)c Fx(of)g(the)h (self-energy)d(v)-5 b(alue.)0 5350 y Fs(3.8.)67 b Fv(F)-6 b(amilies)41 b(of)g(self-ener)l(gy)f(gr)l(aphs.)71 b Fx(Giv)n(en)38 b(a)g(tree)g Ft(\022)j Fx(con)n(taining)c(a)h (self-energy)f(graph)g Ft(T)12 b Fx(,)40 b(w)n(e)e(can)0 5457 y(consider)33 b(all)h(trees)f(obtained)h(b)n(y)f(c)n(hanging)g (the)h(lo)r(cation)g(of)g(the)g(no)r(des)g(in)g Ft(T)2595 5469 y Fn(0)2666 5457 y Fx(\(note)g(that)g Ft(T)3124 5469 y Fn(0)3195 5457 y Fx(is)g(de\014ned)0 5563 y(after)g(\(3.6\)\))f (whic)n(h)h(the)h(external)e(lines)g(of)h Ft(T)45 b Fx(are)33 b(attac)n(hed)g(to:)50 b(w)n(e)33 b(denote)h(b)n(y)g Fr(F)2771 5575 y Fo(T)2810 5583 y Fi(0)2846 5563 y Fx(\()p Ft(\022)r Fx(\))h(the)f(set)g(of)g(trees)0 5669 y(so)f(obtained,)j(and) e(call)f(it)i Fv(the)h(self-ener)l(gy)g(family)g Fx(asso)r(ciated)d (with)i(the)f(self-energy)f(graph)g Ft(T)12 b Fx(.)55 b(And)35 b(w)n(e)0 5869 y Fi(12)p Fj(=lug)q(lio=)p Fi(2001;)29 b(13:05)1169 b Fx(10)p eop %%Page: 11 11 11 10 bop 0 83 a Fx(shall)28 b(refer)g(to)g(the)h(op)r(eration)e(of)i (detac)n(hing)f(and)g(reattac)n(hing)f(the)i(external)f(lines,)g(b)n(y) g(sa)n(ying)f(that)i(w)n(e)f(are)0 189 y Fv(shifting)h Fx(suc)n(h)e(lines.)71 296 y(Of)f(course)f(shifting)i(the)f(external)g (lines)g(of)g(a)g(self-energy)e(graph)h(pro)r(duces)h(a)g(c)n(hange)f (of)h(the)g(propagators)0 402 y(of)33 b(the)h(trees.)53 b(In)34 b(particular)e(since)h(all)g(arro)n(ws)e(ha)n(v)n(e)h(to)h(p)r (oin)n(t)g(to)n(w)n(ard)f(the)i(ro)r(ot,)g(some)f(lines)g(can)g(rev)n (ert)0 508 y(their)28 b(arro)n(ws.)71 615 y(Moreo)n(v)n(er)e(the)k (momen)n(tum)g(can)f(c)n(hange,)g(as)f(a)h(rev)n(ersal)e(of)j(the)f (arro)n(w)e(implies)j(a)f(c)n(hange)f(of)h(the)h(partial)0 721 y(ordering)h(of)i(the)g(no)r(des)g(inside)f(the)h(self-energy)f (graph)f(and)i(a)f(shifting)i(of)e(the)h(en)n(tering)f(line)h(can)g (add)f(or)0 827 y(subtract)23 b(the)g(con)n(tribution)g(of)g(the)g (momen)n(tum)g(\015o)n(wing)g(through)f(it.)36 b(More)22 b(precisely)-7 b(,)23 b(if)h(the)f(external)f(lines)0 933 y(of)g(a)f(self-energy)g(graph)g Ft(T)33 b Fx(are)21 b(detac)n(hed)g(then)i(reattac)n(hed)e(to)g(some)h(other)f(no)r(des)h (in)g Ft(V)d Fx(\()p Ft(T)12 b Fx(\),)23 b(the)f(momen)n(tum)0 1040 y(\015o)n(wing)h(through)h(the)g(line)g Ft(`)f Fr(2)g Fx(\003\()p Ft(T)12 b Fx(\))24 b(can)g(b)r(e)g(c)n(hanged)f(in)n(to)h Fr(\006)p Fq(\027)2085 1010 y Fn(0)2079 1063 y Fo(`)2133 1040 y Fx(+)11 b Ft(\033)s Fq(\027)6 b Fx(,)25 b(with)g Ft(\033)h Fr(2)d(f)p Fx(0)p Ft(;)14 b Fx(1)p Fr(g)p Fx(:)34 b(if)24 b(w)n(e)g(call)g Ft(V)3348 1052 y Fn(1)3410 1040 y Fx(and)0 1146 y Ft(V)48 1158 y Fn(2)115 1146 y Fx(the)29 b(t)n(w)n(o)f(disjoin)n(t)h(sets)f(in)n(to)h(whic)n(h)g Ft(`)f Fx(divides)h Ft(T)40 b Fx(\(see)29 b(Figure)f(2\),)h(suc)n(h)f (that)i(the)f(arro)n(w)d(sup)r(erp)r(osed)j(on)0 1252 y Ft(`)h Fx(is)f(directed)h(from)g Ft(V)718 1264 y Fn(2)785 1252 y Fx(to)g Ft(V)937 1264 y Fn(1)1005 1252 y Fx(\(b)r(efore)f(detac) n(hing)h(the)g(external)f(lines\),)i(then)f(the)g(sign)g(is)f(+)h(if)g (the)h(exiting)0 1359 y(line)k(is)g(reattac)n(hed)e(to)i(a)f(no)r(de)h (inside)g Ft(V)1354 1371 y Fn(1)1426 1359 y Fx(and)g(it)g(is)g Fr(\000)f Fx(otherwise,)i(while)f Ft(\033)j Fx(=)d(1)f(if)h(the)h(en)n (tering)e(line)g(is)0 1465 y(reattac)n(hed)25 b(to)g(a)h(no)r(de)g (inside)g Ft(V)1055 1477 y Fn(2)1119 1465 y Fx(when)g(the)g(sign)f(is)h (+)g(and)f(to)h(a)g(no)r(de)f(inside)h Ft(V)2624 1477 y Fn(1)2688 1465 y Fx(when)g(the)g(sign)g(is)g Fr(\000)p Fx(,)g(and)0 1571 y Ft(\033)g Fx(=)d(0)k(otherwise.)1647 1998 y Ft(V)1695 2010 y Fn(1)2104 2620 y Ft(V)2152 2632 y Fn(2)1323 2060 y Ft(v)1363 2072 y Fn(1)1564 1827 y Ft(v)1604 1839 y Fn(2)1821 2558 y Ft(v)1861 2570 y Fn(3)1913 2317 y Ft(v)1953 2329 y Fn(4)2120 2450 y Ft(v)2160 2462 y Fn(5)1896 2666 y Ft(v)1936 2678 y Fn(6)1024 1926 y Ft(`)1059 1895 y Fn(1)1059 1948 y Fo(T)2477 2424 y Ft(`)2512 2394 y Fn(2)2512 2447 y Fo(T)1606 2414 y Ft(`)2104 1956 y(T)941 2896 y @beginspecial @setspecial %%BeginDocument: fig4.ps /fx [ 0.00 1.11 2.22 3.33 4.44 5.56 6.67 7.78 8.89 10.00 11.11 12.22 13.33 14.44 15.56 16.67 17.78 18.89 20.00 21.11 22.22 23.33 24.44 25.56 26.67 27.78 28.89 30.00 31.11 32.22 33.33 34.44 35.56 36.67 37.78 38.89 40.00 41.11 42.22 43.33 44.44 45.56 46.67 47.78 48.89 50.00 51.11 52.22 53.33 54.44 55.56 56.67 57.78 58.89 60.00 61.11 62.22 63.33 64.44 65.56 66.67 67.78 68.89 70.00 71.11 72.22 73.33 74.44 75.56 76.67 77.78 78.89 80.00 81.11 82.22 83.33 84.44 85.56 86.67 87.78 88.89 90.00 91.11 92.22 93.33 94.44 95.56 96.67 97.78 98.89 100.00 ] def /fy [ 0.00 2.07 3.98 5.58 6.75 7.39 7.46 6.96 5.91 4.41 2.57 0.52 -1.56 -3.52 -5.21 -6.50 -7.28 -7.50 -7.14 -6.22 -4.82 -3.05 -1.04 1.04 3.05 4.82 6.22 7.14 7.50 7.28 6.50 5.21 3.52 1.56 -0.52 -2.57 -4.41 -5.91 -6.96 -7.46 -7.39 -6.75 -5.58 -3.98 -2.07 0.00 2.07 3.98 5.58 6.75 7.39 7.46 6.96 5.91 4.41 2.57 0.52 -1.56 -3.52 -5.21 -6.50 -7.28 -7.50 -7.14 -6.22 -4.82 -3.05 -1.04 1.04 3.05 4.82 6.22 7.14 7.50 7.28 6.50 5.21 3.52 1.56 -0.52 -2.57 -4.41 -5.91 -6.96 -7.46 -7.39 -6.75 -5.58 -3.98 -2.07 0.00 ] def /cambio_coordinate{ % st: x1 y1 x2 y2 4 copy exch pop exch sub % x1 dy 6 1 roll exch pop sub % dy x1 y1 dx 4 1 roll translate % sposta l'origine; st: dx dy 2 copy exch atan rotate % rotazione di arctan(dy/dx); st: dx dy 2 exp exch 2 exp add sqrt % sqrt(dx^2 +dy^2) } def /nx fx length 1 sub def /normonda { %st: l 100 div dup scale fx 0 get fy 0 get moveto 1 3 nx { dup dup 1 add exch 2 add 3 1 roll exch dup fx exch get 4 1 roll fy exch get 3 1 roll dup fx exch get 3 1 roll fy exch get exch dup fx exch get exch fy exch get curveto } for } def /onda { gsave % uso: x1 y1 x2 y2 onda cambio_coordinate % st: l normonda stroke grestore } def /freccia { gsave % uso: x1 y1 x2 y2 freccia cambio_coordinate % st: l dup 0 0 moveto 0 lineto % st: l 2 div 0 translate 15 rotate 0 0 moveto -5 0 lineto -30 rotate 0 0 moveto -5 0 lineto stroke grestore } def /punto { gsave % uso: x1 y1 punto 2 0 360 newpath arc fill stroke grestore} def /puntone { gsave % uso: x1 y1 punto 4 0 360 newpath arc fill stroke grestore} def /cerchio { gsave % uso: x1 y1 r cerchio 0 360 newpath arc stroke grestore} def /tlinea { gsave % uso: x1 y1 x2 y2 tlinea moveto [4 4] 2 setdash lineto stroke grestore} def /normarco { gsave newpath % l normarco dup dup 2 div exch 3 sqrt -2 div mul % l x0 y0 3 copy 3 -1 roll add % l x0 y0 x0 y1 5 2 roll 3 -1 roll 60 120 arc % disegna l'arco; st: x0 y1 translate 15 rotate 0 0 moveto -5 0 lineto -30 rotate 0 0 moveto -5 0 lineto stroke grestore } def /cfreccia { gsave % uso: x1 y1 x2 y2 cfreccia cambio_coordinate % st: l normarco grestore } def /cfrecciaspe { gsave % uso: x1 y1 x2 y2 cfrecciaspe cambio_coordinate 1 -1 scale % st: l normarco grestore } def /normarcofin { gsave newpath % uso: l normarcofin dup dup dup 2 div exch 3 sqrt -2 div mul % l l x0 y0 3 -1 roll 60 120 arc % disegna l'arco; st: l 0 translate -15 rotate 0 0 moveto -5 0 lineto -30 rotate 0 0 moveto -5 0 lineto stroke grestore } def /cfrecciafin { gsave % uso: x1 y1 x2 y2 cfrecciafin cambio_coordinate % st: l normarcofin grestore } def /normzigzag { % st: l 100 div dup scale 0 0 moveto 4 { 6.25 7.5 rlineto 12.5 -15 rlineto 6.25 7.5 rlineto} repeat } def /zigzag { gsave % uso: x1 y1 x2 y2 zigzag cambio_coordinate % st: l normzigzag stroke grestore } def /slinea { gsave % uso: x1 y1 x2 y2 n slinea setlinewidth 4 2 roll moveto lineto stroke grestore} def /sfreccia { gsave % uso: x1 y1 x2 y2 n sfreccia setlinewidth freccia grestore} def 0 110 puntone 50 110 punto 70 130 punto 110 50 punto 130 30 punto 130 70 punto 140 50 punto 200 70 puntone 70 110 30 cerchio 130 50 30 cerchio 100 80 80 cerchio 50 110 0 110 freccia 70 130 50 110 freccia 110 50 50 110 freccia 130 30 110 50 freccia 130 70 110 50 freccia 140 50 110 50 freccia 200 70 130 70 freccia stroke grestore %%EndDocument @endspecial 591 3092 a Fg(Fig.)57 b(4.)k Ff(The)31 b(sets)h Fo(V)1244 3100 y Fi(1)1307 3092 y Ff(and)f Fo(V)1490 3100 y Fi(2)1553 3092 y Ff(in)g(a)g(self-energy)f(graph)i Fo(T)9 b Ff(;)34 b(note)e(that,)h(ev)n(en)f(if)591 3167 y(they)g(are)f(dra)n(wn)g(lik)n(e)g(circles,)i(the)f(sets)f Fo(V)1795 3175 y Fi(1)1859 3167 y Ff(and)h Fo(V)2043 3175 y Fi(2)2106 3167 y Ff(are)f(not)i(clusters.)53 b(One)32 b(has)591 3241 y Fl(\027)634 3216 y Fi(0)629 3258 y Fj(`)666 3241 y Fe(=)p Fl(\027)751 3249 y Fj(v)781 3261 y Fi(3)817 3241 y Fe(+)p Fl(\027)902 3249 y Fj(v)932 3261 y Fi(4)969 3241 y Fe(+)p Fl(\027)1054 3249 y Fj(v)1084 3261 y Fi(5)1120 3241 y Fe(+)p Fl(\027)1205 3249 y Fj(v)1235 3261 y Fi(6)1301 3241 y Ff(and)e Fl(\027)5 b Fe(=)p Fl(\027)1572 3262 y Fj(`)1597 3248 y Fi(2)1597 3290 y Fj(T)1646 3241 y Ff(;)32 b(of)d(course)h Fl(\027)2044 3262 y Fj(`)2069 3248 y Fi(1)2069 3290 y Fj(T)2119 3241 y Fe(=)p Fl(\027)2204 3262 y Fj(`)2229 3248 y Fi(2)2229 3290 y Fj(T)2308 3241 y Ff(and)g Fl(\027)2494 3216 y Fi(0)2489 3258 y Fj(`)2526 3241 y Fe(=)p Fy(\000)p Fe(\()p Fl(\027)2687 3249 y Fj(v)2717 3261 y Fi(1)2754 3241 y Fe(+)p Fl(\027)2839 3249 y Fj(v)2869 3261 y Fi(2)2905 3241 y Fe(\))591 3344 y Ff(b)n(y)25 b(de\014nition)h(of)f(rself-energy)f(graph.)35 b(The)26 b(blac)n(k)f(balls)g(represen)n(t)g(the)h(remaining)591 3419 y(parts)d(of)h(the)g(trees.)32 b(The)24 b(lab)r(els)f(are)h(not)g (explicitly)g(sho)n(wn.)71 3609 y Fx(Referring)34 b(to)h(\(3.13\))f (for)h(the)g(notion)g(of)g(t)n(yp)r(e)g(of)g(self-energy)f(graph)g(one) h(sho)n(ws)f(the)h(existence)g(of)g(the)0 3715 y(follo)n(wing)27 b(cancellations)f(\(the)i(pro)r(of)f(is)h(in)g(App)r(endix)g(A4\).)0 3892 y Fs(3.9.)35 b Fw(Lemma.)i Fv(Given)30 b(a)g(tr)l(e)l(e)g Ft(\022)r Fv(,)g(for)h(any)f(self-ener)l(gy)g(gr)l(aph)h Ft(T)j Fr(2)24 b(T)d Fx(\()p Ft(\022)r Fx(\))31 b Fv(one)f(has)761 4141 y Fm(X)669 4322 y Fo(\022)703 4306 y Fd(0)724 4322 y Fy(2F)817 4330 y Fj(T)852 4342 y Fi(0)888 4322 y Fn(\()p Fo(\022)r Fn(\))987 4219 y Fr(LV)1095 4231 y Fo(T)1147 4219 y Fx(\()p Fq(!)22 b Fr(\001)c Fq(\027)6 b Fx(\))23 b(=)1499 3999 y Fm(8)1499 4074 y(>)1499 4099 y(>)1499 4124 y(<)1499 4273 y(>)1499 4298 y(>)1499 4323 y(:)1587 4044 y Fx(0)p Ft(;)554 b Fv(if)31 b Ft(T)41 b Fv(is)30 b(of)g(typ)l(e)g(1)h(,)1587 4143 y Fx(\()p Fq(!)21 b Fr(\001)d Fq(\027)6 b Fx(\))p Ft(B)1894 4113 y Fy(0)1890 4170 y(F)1938 4178 y Fj(T)1973 4190 y Fi(0)2009 4170 y Fn(\()p Fo(\022)r Fn(\))2098 4143 y Ft(;)85 b Fv(if)31 b Ft(T)41 b Fv(is)30 b(of)g(typ)l(e)g(2)h(,)1587 4262 y Fx(\()p Fq(!)21 b Fr(\001)d Fq(\027)6 b Fx(\))p Ft(B)1894 4232 y Fy(00)1890 4288 y(F)1938 4296 y Fj(T)1973 4308 y Fi(0)2009 4288 y Fn(\()p Fo(\022)r Fn(\))2098 4262 y Ft(;)85 b Fv(if)31 b Ft(T)41 b Fv(is)30 b(of)g(typ)l(e)g(3)h(,)1587 4375 y Ft(A)1649 4390 y Fy(F)1697 4398 y Fj(T)1732 4410 y Fi(0)1768 4390 y Fn(\()p Fo(\022)r Fn(\))1857 4375 y Ft(;)326 b Fv(if)31 b Ft(T)41 b Fv(is)30 b(of)g(typ)l(e)g(4)h(,)3331 4219 y Fx(\(3)p Ft(:)p Fx(19\))0 4547 y Fv(wher)l(e)e Fq(\027)f Fx(=)23 b Fq(\027)445 4565 y Fo(`)473 4545 y Fi(2)473 4586 y Fj(T)523 4547 y Fv(,)28 b(the)g(sum)g(is)g(over)g (the)h(self-ener)l(gy)f(family)i(asso)l(ciate)l(d)f(with)g Ft(T)12 b Fv(,)27 b(and)i Ft(A)2851 4562 y Fy(F)2899 4570 y Fj(T)2934 4582 y Fi(0)2970 4562 y Fn(\()p Fo(\022)r Fn(\))3059 4547 y Fv(,)g Ft(B)3180 4516 y Fy(0)3176 4573 y(F)3224 4581 y Fj(T)3259 4593 y Fi(0)3294 4573 y Fn(\()p Fo(\022)r Fn(\))3412 4547 y Fv(and)0 4665 y Ft(B)67 4635 y Fy(00)63 4692 y(F)111 4700 y Fj(T)146 4712 y Fi(0)182 4692 y Fn(\()p Fo(\022)r Fn(\))303 4665 y Fv(ar)l(e)j(matric)l(es)g Ft(s)20 b Fr(\002)g Ft(s)p Fv(,)32 b Ft(r)23 b Fr(\002)c Ft(s)32 b Fv(and)g Ft(s)20 b Fr(\002)g Ft(r)r Fv(,)33 b(r)l(esp)l(e)l(ctively,)h(dep)l(ending)f(only)g(on)f(the)f(self-ener)l (gy)i(gr)l(aph)0 4771 y Ft(T)12 b Fv(;)29 b(in)h(p)l(articular)h(they)f (ar)l(e)g(indep)l(endent)h(of)f(the)g(quantity)g Fq(!)21 b Fr(\001)d Fq(\027)6 b Fv(.)0 4948 y Fs(3.10.)39 b Fv(First)32 b(step)f(towar)l(d)h(the)f(r)l(esummation)g(of)h(self-ener)l(gy)f(gr)l (aphs.)43 b Fx(Let)29 b Ft(\022)j Fx(b)r(e)d(a)g(tree)f Ft(\022)g Fr(2)e Fx(\002)3121 4960 y Fl(\027)5 b Fo(;k)q(;\015)3311 4948 y Fx(with)30 b(a)0 5055 y(self-energy)c(graph)g Ft(T)12 b Fx(.)36 b(De\014ne)28 b Ft(\022)1062 5067 y Fn(0)1122 5055 y Fx(=)23 b Ft(\022)d Fr(n)d Ft(T)39 b Fx(as)26 b(the)i(set)f(of)g(no)r(des)g(and)h(lines)f(of)g Ft(\022)j Fx(outside)d Ft(T)38 b Fx(\(of)28 b(course)e Ft(\022)3423 5067 y Fn(0)3487 5055 y Fx(is)0 5161 y(not)e(a)g(tree\).) 35 b(Consider)24 b(simoultaneously)e(all)i(trees)g(suc)n(h)g(that)g (the)g(structure)g Ft(\022)2585 5173 y Fn(0)2646 5161 y Fx(outside)g(of)g(the)h(self-energy)0 5267 y(graph)32 b(is)i(the)f(same,)i(while)e(the)h(self-energy)e(graph)g(itself)i(can)f (b)r(e)h(arbitrary)-7 b(,)33 b Fv(i.e.)e Ft(T)44 b Fx(can)33 b(b)r(e)h(replaced)f(b)n(y)0 5373 y(an)n(y)27 b(other)f(self-energy)g (graph)g Ft(T)1080 5343 y Fy(0)1130 5373 y Fx(with)i Ft(k)1362 5385 y Fo(T)1410 5369 y Fd(0)1460 5373 y Fr(\025)23 b Fx(1.)36 b(This)27 b(allo)n(ws)f(us)i(to)f(de\014ne)g(as)g(a)g (formal)g(p)r(o)n(w)n(er)f(series)g(the)0 5480 y(matrix)1191 5586 y Ft(M)9 b Fx(\()p Fq(!)21 b Fr(\001)e Fq(\027)6 b Fx(;)14 b Ft(")p Fx(\))22 b(=)1780 5507 y Fm(X)1708 5686 y Fo(\022)r Fn(=)p Fo(\022)1825 5694 y Fi(0)1856 5686 y Fy([)p Fo(T)1949 5669 y Fd(0)1986 5586 y Fr(V)2037 5598 y Fo(T)2085 5582 y Fd(0)2111 5586 y Fx(\()p Fq(!)g Fr(\001)c Fq(\027)6 b Fx(\))979 b(\(3)p Ft(:)p Fx(20\))0 5869 y Fi(12)p Fj(=lug)q(lio=)p Fi(2001;)29 b(13:05)1169 b Fx(11)p eop %%Page: 12 12 12 11 bop 0 83 a Fx(where)25 b(the)i(sum)f(is)g(o)n(v)n(er)e(all)i (trees)f Ft(\022)j Fx(suc)n(h)e(that)g Ft(\022)18 b Fr(n)d Ft(T)37 b Fx(is)26 b(\014xed)g(to)g(b)r(e)g Ft(\022)2285 95 y Fn(0)2348 83 y Fx(and)g(the)g(mo)r(de)h(lab)r(els)e(of)h(the)h(no) r(des)0 189 y Ft(v)f Fr(2)e Ft(V)19 b Fx(\()p Ft(T)12 b Fx(\))27 b(ha)n(v)n(e)f(to)i(satisfy)f(the)h(conditions)f(\(1\))19 b Fr(\004)f Fx(\(3\))27 b(in)h Fr(x)p Fx(3.3)f(de\014ning)h(the)g (self-energy)e(graphs.)71 300 y(The)35 b(follo)n(wing)g(prop)r(ert)n(y) f(holds)h(\(the)h(pro)r(of)f(is)h(in)f(App)r(endix)i(A5\))e(as)g(an)g (algebraic)f(iden)n(tit)n(y)h(b)r(et)n(w)n(een)0 407 y(formal)27 b(p)r(o)n(w)n(er)f(series.)0 589 y Fs(3.11.)34 b Fw(Lemma.)i Fv(The)28 b(fol)t(lowing)i(two)d(pr)l(op)l(erties)i (hold:)39 b(\(1\))28 b Fx(\()p Ft(M)9 b Fx(\()p Ft(x)p Fx(;)14 b Ft(")p Fx(\)\))2312 547 y Fo(T)2388 589 y Fx(=)22 b Ft(M)9 b Fx(\()p Fr(\000)p Ft(x)p Fx(;)14 b Ft(")p Fx(\))p Fv(,)28 b(and)g(\(2\))g Fx(\()p Ft(M)9 b Fx(\()p Ft(x)p Fx(;)14 b Ft(")p Fx(\)\))3509 547 y Fy(y)0 695 y Fx(=)29 b Ft(M)9 b Fx(\()p Ft(x)p Fx(;)14 b Ft(")p Fx(\))p Fv(;)31 b(the)f(latter)f(me)l(ans)h(that)g(the)g(matrix)f Ft(M)9 b Fx(\()p Ft(x)p Fx(;)14 b Ft(")p Fx(\))30 b Fv(is)g (self-adjoint.)0 877 y Fs(3.12.)35 b Fv(R)l(emarks.)h Fx(\(1\))26 b(The)g(function)h Ft(M)9 b Fx(\()p Fq(!)17 b Fr(\001)e Fq(\027)6 b Fx(;)14 b Ft(")p Fx(\))25 b(dep)r(ends)i(on)e Ft(")h Fx(but,)h(b)n(y)e(construction,)h(it)g(is)g(indep)r(enden)n(t)0 983 y(of)i Ft(\022)134 995 y Fn(0)171 983 y Fx(:)37 b(hence)27 b(w)n(e)h(can)f(rewrite)g(\(3.20\))f(as)1252 1213 y Ft(M)9 b Fx(\()p Fq(!)21 b Fr(\001)d Fq(\027)6 b Fx(;)14 b Ft(")p Fx(\))23 b(=)1768 1134 y Fm(X)1793 1312 y Fo(T)1841 1296 y Fd(0)1902 1213 y Fr(V)1953 1225 y Fo(T)2001 1209 y Fd(0)2028 1213 y Fx(\()p Fq(!)e Fr(\001)e Fq(\027)6 b Fx(\))p Ft(;)1039 b Fx(\(3)p Ft(:)p Fx(21\))0 1500 y(where)24 b(the)h(sum)g(is)g(o)n(v)n(er)e(all)h(self-energy)f(graphs)h(of)g (order)g Ft(k)i Fr(\025)c Fx(1)j(with)g(external)f(lines)g(with)i (momen)n(tum)f Fq(\027)6 b Fx(.)0 1607 y(\(2\))34 b(In)h(\(3.20\))e(or) h(\(3.21\),)h(if)g Ft(\015)1003 1619 y Fo(n)p Fy(\000)p Fn(1)1167 1607 y Fr(\024)f(j)p Fq(!)1349 1619 y Fn(0)1408 1607 y Fr(\001)23 b Fq(\027)6 b Fr(j)34 b Ft(<)g(\015)1707 1619 y Fo(n)1752 1607 y Fx(,)i(the)f(sum)f(has)g(to)g(b)r(e)g (restricted)g(to)g(the)h(self-energy)0 1713 y(graphs)26 b Ft(T)329 1683 y Fy(0)379 1713 y Fx(on)i(scale)e Ft(n)743 1725 y Fo(T)791 1709 y Fd(0)841 1713 y Fr(\025)d Ft(n)18 b Fx(+)g(3.)36 b(W)-7 b(riting,)28 b(for)f(an)n(y)g(line)h Ft(`)22 b Fr(2)i Ft(T)2147 1683 y Fy(0)2135 1734 y Fn(0)2172 1713 y Fx(,)j Fq(\027)2270 1725 y Fo(`)2330 1713 y Fx(as)g(in)h (\(3.12\))e(one)i(has)675 1896 y Fm(\014)675 1946 y(\014)703 1967 y Fq(!)763 1979 y Fn(0)818 1967 y Fr(\001)19 b Fq(\027)914 1933 y Fn(0)908 1987 y Fo(`)950 1896 y Fm(\014)950 1946 y(\014)1001 1967 y Ft(>)k Fx(2)1131 1933 y Fo(\034)1172 1967 y Ft(C)1237 1931 y Fy(\000)p Fn(1)1231 1989 y(0)1326 1967 y Ft(C)1385 1979 y Fn(0)1437 1896 y Fm(\014)1437 1946 y(\014)1464 1967 y Fq(\027)1518 1933 y Fn(0)1512 1987 y Fo(`)1555 1896 y Fm(\014)1555 1946 y(\014)1583 1916 y Fy(\000)p Fo(\034)1699 1967 y Fr(\025)g Fx(2)1829 1933 y Fo(\034)1870 1875 y Fm(\020)2002 1888 y(X)1933 2070 y Fo(v)r Fy(2)p Fo(V)15 b Fn(\()p Fo(T)2132 2078 y Fi(0)2164 2070 y Fn(\))2204 1967 y Fr(j)p Fq(\027)2275 1979 y Fo(v)2315 1967 y Fr(j)2338 1875 y Fm(\021)2387 1892 y Fy(\000)p Fo(\034)2504 1967 y Fr(\025)22 b Fx(2)2633 1933 y Fo(\034)2674 1967 y Fx(2)2716 1933 y Fo(n)p Fn(+3)2845 1967 y Ft(;)463 b Fx(\(3)p Ft(:)p Fx(22\))0 2277 y(while)28 b Fr(j)p Fq(!)300 2289 y Fn(0)355 2277 y Fr(\001)19 b Fq(\027)6 b Fr(j)22 b Ft(<)h Fx(2)626 2247 y Fo(n)671 2277 y Fx(,)k(so)g(that,)h(b)n(y)g(using)f(\(3.11\),)g(one)g(obtains) 1202 2497 y Fr(j)p Fq(!)1285 2509 y Fn(0)1340 2497 y Fr(\001)19 b Fq(\027)1430 2509 y Fo(`)1461 2497 y Fr(j)24 b Ft(>)e Fx(2)1637 2463 y Fo(\034)1678 2497 y Fx(2)1720 2463 y Fo(n)p Fn(+3)1867 2497 y Fr(\000)c Fx(2)1992 2463 y Fo(n)2060 2497 y Ft(>)23 b Fx(2)2190 2463 y Fo(n)p Fn(+2)2319 2497 y Ft(;)989 b Fx(\(3)p Ft(:)p Fx(23\))0 2717 y(whic)n(h)28 b(implies)f Ft(n)569 2729 y Fo(`)624 2717 y Fr(\025)c Ft(n)18 b Fx(+)g(3.)0 2824 y(\(3\))28 b(The)f(matrix)g Ft(M)9 b Fx(\()p Fq(!)21 b Fr(\001)e Fq(\027)6 b Fx(;)14 b Ft(")p Fx(\))27 b(can)g(b)r(e)h(written)g(as)922 3079 y Ft(M)9 b Fx(\()p Fq(!)21 b Fr(\001)e Fq(\027)6 b Fx(;)14 b Ft(")p Fx(\))22 b(=)1439 2987 y Fm(\020)1502 3029 y Ft(M)1583 3041 y Fo(\013\013)1673 3029 y Fx(\()p Fq(!)g Fr(\001)c Fq(\027)6 b Fx(;)14 b Ft(")p Fx(\))83 b Ft(M)2154 3041 y Fo(\013\014)2241 3029 y Fx(\()p Fq(!)22 b Fr(\001)c Fq(\027)6 b Fx(;)14 b Ft(")p Fx(\))1504 3128 y Ft(M)1585 3140 y Fo(\014)s(\013)1672 3128 y Fx(\()p Fq(!)21 b Fr(\001)e Fq(\027)6 b Fx(;)14 b Ft(")p Fx(\))85 b Ft(M)2155 3140 y Fo(\014)s(\014)2240 3128 y Fx(\()p Fq(!)21 b Fr(\001)e Fq(\027)6 b Fx(;)14 b Ft(")p Fx(\))2572 2987 y Fm(\021)3331 3079 y Fx(\(3)p Ft(:)p Fx(24\))0 3334 y(where)25 b Ft(M)319 3346 y Fo(\013\013)409 3334 y Fx(\()p 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Fo(\013\014)1255 3786 y Fx(\()p Fq(!)21 b Fr(\001)e Fq(\027)6 b Fx(;)14 b Ft(")p Fx(\))22 b(=)h Ft(O)r Fx(\()p Ft(")1818 3751 y Fn(2)1856 3786 y Fx(\()p Fq(!)e Fr(\001)e Fq(\027)6 b Fx(\)\))p Ft(;)1089 3917 y(M)1170 3929 y Fo(\014)s(\014)1255 3917 y Fx(\()p Fq(!)21 b Fr(\001)e Fq(\027)6 b Fx(;)14 b Ft(")p Fx(\))22 b(=)h Ft(O)r Fx(\()p Ft(")p Fx(\))c(+)f Ft(O)r Fx(\()p Ft(")2088 3883 y Fn(2)2126 3917 y Fx(\()p Fq(!)k Fr(\001)c Fq(\027)6 b Fx(\))2367 3883 y Fn(2)2404 3917 y Fx(\))p Ft(:)3331 3785 y Fx(\(3)p Ft(:)p Fx(25\))0 4136 y(The)29 b(prop)r(ortionalit)n(y)d(of)j Ft(M)924 4148 y Fo(\013\013)1014 4136 y Fx(\()p Fq(!)22 b Fr(\001)d Fq(\027)6 b Fx(;)14 b Ft(")p Fx(\))28 b(to)h(\()p Fq(!)22 b Fr(\001)d Fq(\027)6 b Fx(\))1705 4106 y Fn(2)1771 4136 y Fx(and)28 b(of)h Ft(M)2110 4148 y Fo(\013\014)2197 4136 y Fx(\()p Fq(!)22 b Fr(\001)d Fq(\027)6 b Fx(;)14 b Ft(")p Fx(\))29 b(to)f Fq(!)22 b Fr(\001)d Fq(\027)34 b Fx(is)29 b(a)f(consequence)f(of)0 4242 y(the)h(lemma)f(3.9.)37 b(First)27 b(orders)f(computations)h(already)f(giv)n(e,)h(for)g (instance,)151 4452 y Ft(M)232 4464 y Fo(\014)s(\014)317 4452 y Fx(\()p Fq(!)21 b Fr(\001)e Fq(\027)6 b Fx(;)14 b Ft(")p Fx(\))22 b(=)h Ft("@)832 4418 y Fn(2)827 4477 y Fl(\014)877 4452 y Ft(f)918 4464 y Fh(0)959 4452 y Fx(\()p Fq(\014)1046 4464 y Fn(0)1084 4452 y Fx(\))18 b(+)g Ft(O)r Fx(\()p Ft(")1353 4418 y Fn(2)1391 4452 y Fx(\))h(+)f Ft(O)r Fx(\()p Ft(")1661 4418 y Fn(2)1699 4452 y Fx(\()p Fq(!)j Fr(\001)e Fq(\027)6 b Fx(\))1940 4418 y Fn(2)1977 4452 y Fx(\))p Ft(;)148 4633 y(M)229 4645 y Fo(\013\014)317 4633 y Fx(\()p Fq(!)21 b Fr(\001)e Fq(\027)6 b Fx(;)14 b Ft(")p Fx(\))22 b(=)h Fr(\000)p Fx(2)p Ft(")890 4599 y Fn(2)926 4633 y Ft(i)14 b Fx(\()p Fq(!)21 b Fr(\001)d Fq(\027)6 b Fx(\))1463 4554 y Fm(X)1403 4720 y Fj(\027)1433 4732 y Fi(1)1465 4720 y(+)p Fj(\027)1538 4732 y Fi(2)1570 4720 y(=)p Fb(0)1233 4789 y Fd(j)p Fj(\027)1282 4801 y Fi(1)1314 4789 y Fd(j)p Fi(+)p Fd(j)p Fj(\027)1425 4801 y Fi(2)1457 4789 y Fd(j)p Fj(<)p Fi(2)1549 4774 y Fd(\000)p Fi(\()p Fj(n)p Fi(+3\))p Fj(=\034)1982 4577 y Fx(1)p 1848 4614 310 4 v 1848 4690 a(\()p Fq(!)21 b Fr(\001)e Fq(\027)2051 4702 y Fn(2)2088 4690 y Fx(\))2120 4666 y Fn(3)799 4846 y Fm(h)838 4938 y Fq(\027)886 4950 y Fn(1)923 4938 y Ft(@)967 4955 y Fl(\014)1018 4938 y Ft(f)1059 4950 y Fl(\027)1097 4958 y Fi(1)1133 4938 y Fx(\()p Fq(\014)1220 4950 y Fn(0)1258 4938 y Fx(\))p Ft(@)1339 4904 y Fn(2)1334 4963 y Fl(\014)1384 4938 y Ft(f)1425 4950 y Fl(\027)1463 4958 y Fi(2)1500 4938 y Fx(\()p Fq(\014)1587 4950 y Fn(0)1624 4938 y Fx(\))g Fr(\000)f Fq(\027)1812 4904 y Fn(2)1806 4958 y(1)1849 4938 y Ft(f)1890 4950 y Fl(\027)1928 4958 y Fi(1)1964 4938 y Fx(\()p Fq(\014)2051 4950 y Fn(0)2088 4938 y Fx(\))p Fq(\027)2168 4950 y Fn(2)2206 4938 y Ft(@)2250 4955 y Fl(\014)2300 4938 y Ft(f)2341 4950 y Fl(\027)2379 4958 y Fi(2)2416 4938 y Fx(\()p Fq(\014)2503 4950 y Fn(0)2540 4938 y Fx(\))2572 4846 y Fm(i)2630 4938 y Fx(+)g Ft(O)r Fx(\()p Ft(")2849 4904 y Fn(3)2887 4938 y Fx(\()p Fq(!)j Fr(\001)e Fq(\027)6 b Fx(\)\))p Ft(;)3331 4707 y Fx(\(3)p Ft(:)p Fx(26\))0 5176 y(where)21 b Ft(\015)277 5188 y Fo(n)p Fy(\000)p Fn(1)430 5176 y Fr(\024)i(j)p Fq(!)601 5188 y Fn(0)644 5176 y Fr(\001)6 b Fq(\027)g Fr(j)22 b Ft(<)h(\015)903 5188 y Fo(n)948 5176 y Fx(.)35 b(Therefore)20 b Ft(M)1457 5188 y Fo(\014)s(\014)1542 5176 y Fx(\()p Fq(!)9 b Fr(\001)d Fq(\027)g Fx(;)14 b Ft(")p Fx(\))23 b Fr(6)p Fx(=)f(0)f(b)n(y)g(h)n(yp)r(othesis)g(\(see)g(\(1.8\)\),)i (and)e Ft(M)3164 5188 y Fo(\013\014)3252 5176 y Fx(\()p Fq(!)9 b Fr(\001)d Fq(\027)g Fx(;)14 b Ft(")p Fx(\))0 5283 y(is)27 b(generically)g(non)n(v)-5 b(anishing.)0 5389 y(\(4\))30 b(The)h(lemma)f(3.11)f(implies,)i(that,)g(b)n(y)f (de\014ning)g(the)h(matrices)e Ft(B)2254 5359 y Fy(0)2250 5416 y(F)2298 5424 y Fj(T)2333 5436 y Fi(0)2369 5416 y Fn(\()p Fo(\022)r Fn(\))2488 5389 y Fx(and)i Ft(B)2720 5359 y Fy(00)2716 5416 y(F)2764 5424 y Fj(T)2799 5436 y Fi(0)2834 5416 y Fn(\()p Fo(\022)r Fn(\))2954 5389 y Fx(as)e(in)i(the)f(lemma)0 5495 y(3.9,)d(one)g(has)1312 5649 y Ft(B)1379 5614 y Fy(0)1375 5671 y(F)1423 5679 y Fj(T)1458 5691 y Fi(0)1493 5671 y Fn(\()p Fo(\022)r Fn(\))1606 5649 y Fx(=)c Fr(\000)1773 5556 y Fm(\020)1822 5649 y Ft(B)1889 5614 y Fy(00)1885 5671 y(F)1933 5679 y Fj(T)1968 5691 y Fi(0)2003 5671 y Fn(\()p Fo(\022)r Fn(\))2093 5556 y Fm(\021)2142 5574 y Fo(T)2208 5649 y Ft(:)1100 b Fx(\(3)p Ft(:)p Fx(27\))0 5869 y Fi(12)p Fj(=lug)q(lio=)p Fi(2001;)29 b(13:05)1169 b Fx(12)p eop %%Page: 13 13 13 12 bop 0 221 a Fs(3.13.)45 b Fv(Changing)34 b(sc)l(ales.)48 b Fx(When)32 b(shifting)f(the)h(lines)e(external)h(to)f(the)i (self-energy)d(graphs,)i(the)g(momen)n(ta)0 327 y(of)e(the)g(in)n (ternal)f(lines)g(can)g(c)n(hange.)39 b(As)29 b(a)f(consequence)g(in)h (principle)f(also)g(the)h(scale)f(lab)r(els)g(could)g(c)n(hange;)0 433 y(ho)n(w)n(ev)n(er)d(this)j(do)r(es)g(not)f(happ)r(en,)h(as)f(the)h (follo)n(wing)f(result)g(sho)n(ws;)f(for)i(the)g(pro)r(of)f(see)g(App)r (endix)h(A6.)0 612 y Fs(3.14.)33 b Fw(Lemma.)i Fv(F)-6 b(or)25 b(al)t(l)g(lines)g Ft(`)e Fr(2)h Fx(\003\()p Ft(\022)r Fx(\))h Fv(one)g(has)g Ft(n)1694 624 y Fo(`)1749 612 y Fx(=)d Ft(n)1886 582 y Fn(0)1886 635 y Fo(`)1924 612 y Fv(;)k(in)f(p)l(articular)h(this)f(implies)h(that,)g(when)f (shifting)0 718 y(the)i(lines)h(external)f(to)g(the)h(self-ener)l(gy)g (gr)l(aphs)g(of)g(a)g(tr)l(e)l(e)f Ft(\022)r Fv(,)h(the)g(sc)l(ale)f (lab)l(els)i Ft(n)2559 730 y Fo(`)2618 718 y Fv(of)f(al)t(l)g(line)g Ft(`)22 b Fr(2)i Fx(\003\()p Ft(\022)r Fx(\))k Fv(do)g(not)0 824 y(change.)151 1072 y Fz(4.)50 b(Resummations)39 b(of)f(self-energy) h(graphs:)51 b(renormalized)38 b(propagators)0 1251 y Fs(4.1.)j Fv(R)l(enormalize)l(d)33 b(tr)l(e)l(es.)42 b Fx(So)29 b(far)h(w)n(e)f(considered)f(formal)h(p)r(o)n(w)n(er)f (expansions)h(in)g Ft(")p Fx(.)43 b(By)29 b(in)n(tro)r(ducing)g(the)0 1357 y(function)f Fs(h)23 b Fx(=)g(\()p Fs(h)574 1369 y Fo(\013)622 1357 y Ft(;)14 b Fs(h)712 1369 y Fo(\014)756 1357 y Fx(\))24 b(=)e(\()p Fs(a)p 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Ft(")1215 1682 y Fo(k)1251 1657 y Fd(0)p 1278 1649 V 1278 1716 a Fs(h)1331 1663 y Fn([)p Fo(k)q(;k)1442 1638 y Fd(0)1466 1663 y Fn(])1331 1737 y Fl(\027)1488 1716 y Fx(\()p Ft(")p Fx(\))p Ft(;)p 1795 1649 V 181 w Fs(h)1848 1663 y Fn([)p Fo(k)q(;k)1959 1638 y Fd(0)1982 1663 y Fn(])1848 1737 y Fo(\015)t Fl(\027)2005 1716 y Fx(\()p Ft(")p Fx(\))g(=)2305 1638 y Fm(X)2219 1824 y Fo(\022)r Fy(2)p Fn(\002)2349 1804 y Fd(R)2349 1853 y Fj(k)2381 1841 y Fd(0)2403 1853 y Fj(;\027)s(;\015)p 2525 1649 138 4 v 2525 1716 a Fs(V)-8 b(al)2662 1663 y Fn([)p Fo(k)q Fn(])2741 1716 y Fx(\()p Ft(\022)r Fx(\))p Ft(;)3373 1620 y Fx(\(4)p Ft(:)p Fx(8\))0 1978 y(where)38 b(the)i(last)e(form)n(ula)g(holds)h(for)f Fq(\027)48 b Fr(6)p Fx(=)41 b Fs(0)p Fx(,)h(b)r(ecause)c(for)h Fq(\027)47 b Fx(=)42 b Fs(0)p Fx(,)g(one)c(has)h(\(4.6\))f(for)h Ft(\015)3058 1990 y Fo(`)3131 1978 y Fx(=)j Ft(\014)t Fx(,)g(while)p 0 2041 54 4 v 0 2109 a Fs(h)53 2055 y Fn([)p Fo(k)q(;k)164 2030 y Fd(0)187 2055 y Fn(])53 2129 y Fo(\015)t Fh(0)233 2109 y Fr(\021)23 b Fx(0.)0 2286 y Fs(4.2.)53 b Fv(R)l(emark.)i Fx(Note)33 b(that)h(if)g(w)n(e)f(expand) g(the)h(quan)n(tit)n(y)f Ft(M)2034 2256 y Fn([)p Fo(k)q Fn(])2112 2286 y Fx(\()p Fq(!)25 b Fr(\001)e Fq(\027)6 b Fx(;)14 b Ft(")p Fx(\))33 b(in)h(p)r(o)n(w)n(ers)e(of)h Ft(")p Fx(,)i(b)n(y)e(expanding)0 2411 y(the)e(propagators)p 606 2344 66 4 v 27 w Ft(G)671 2358 y Fn([)p Fo(k)q Fy(\000)p Fn(1])671 2432 y Fo(`)835 2411 y Fx(,)g(w)n(e)f(reconstruct)f(the)h (sum)h(of)f(the)h(v)-5 b(alues)29 b(of)i(all)f(self-energy)f(graphs)f (con)n(taining)0 2517 y(only)i(self-energy)f(graphs)g(with)i(heigh)n(t) f Ft(D)g Fr(\024)d Ft(k)s Fx(.)46 b(Therefore)29 b(if)i(w)n(e)f(expand) g Ft(M)2586 2487 y Fn([)p Fo(k)q Fn(+1])2748 2517 y Fx(\()p Fq(!)23 b Fr(\001)e Fq(\027)6 b Fx(;)14 b Ft(")p Fx(\))30 b(in)g(p)r(o)n(w)n(ers)f(of)0 2624 y Ft(")j Fx(w)n(e)h(obtain)f(the)h (same)f(terms)g(as)g(if)h(expanding)f Ft(M)1741 2594 y Fn([)p Fo(k)q Fn(])1819 2624 y Fx(\()p Fq(!)25 b Fr(\001)d Fq(\027)6 b Fx(;)14 b Ft(")p Fx(\),)33 b(plus)g(the)g(sum)f(of)h(the)g (v)-5 b(alues)32 b(of)h(all)f(the)0 2730 y(self-energy)d(graphs)f(con)n (taining)h(also)g(self-energies)g(graphs)g(with)h(heigh)n(t)g Ft(k)23 b Fx(+)d(1,)30 b(whic)n(h)g(are)f(absen)n(t)h(in)g(the)0 2836 y(self-energy)i(graphs)h(con)n(tributing)g(to)g Ft(M)1368 2806 y Fn([)p Fo(k)q Fn(])1446 2836 y Fx(\()p Fq(!)26 b Fr(\001)d Fq(\027)6 b Fx(;)14 b Ft(")p Fx(\).)54 b(Suc)n(h)34 b(a)g(result)f(will)h(b)r(e)g(used)g(in)g(App)r(endix)h (A7)e(in)0 2943 y(order)26 b(to)i(pro)n(v)n(e)e(the)i(follo)n(wing)e (result.)0 3135 y Fs(4.3.)38 b Fw(Lemma.)g Fv(The)32 b(p)l(ower)f(series)g(de\014ning)g(the)g(functions)p 1991 3068 54 4 v 30 w Fs(h)2044 3082 y Fn([)p Fo(k)q Fn(])2044 3156 y Fl(\027)2123 3135 y Fx(\()p Ft(")p Fx(\))p Fv(,)g(trunc)l(ate)l(d)f(at)g(or)l(der)h Ft(k)3012 3105 y Fy(0)3060 3135 y Fr(\024)24 b Ft(k)s Fv(,)31 b(c)l(oincide)0 3256 y(with)f(the)g(functions)g Fs(h)732 3212 y Fn(\()p Fy(\024)p Fo(k)846 3187 y Fd(0)869 3212 y Fn(\))732 3270 y Fl(\027)899 3256 y Fx(\()p Ft(")p Fx(\))g Fv(given)g(by)h(\(4.2\).) 311 3504 y Fz(5.)50 b(Con)m(v)m(ergence)38 b(of)g(the)f(renormalized)i (p)s(erturbativ)m(e)f(expansion)0 3681 y Fs(5.1.)46 b Fv(Domains)33 b(of)h(analyticity)h(and)e(norms.)48 b Fx(Consider)30 b(in)i(the)f(complex)g Ft(")p Fx(-plane)f(the)i(domain)e Ft(D)3256 3693 y Fo(")3287 3701 y Fi(0)3324 3681 y Fx(\()p Ft(')p Fx(\))i(in)0 3787 y(Figure)f(5)g(b)r(elo)n(w:)45 b(if)32 b Ft(')g Fx(denotes)g(the)g(half-op)r(ening)f(of)h(the)g (sector)f Ft(D)2272 3799 y Fo(")2303 3807 y Fi(0)2339 3787 y Fx(\()p Ft(')p Fx(\),)j(then)e(the)g(radius)f(of)h(the)g(circle) 0 3893 y(delimiting)c Ft(D)461 3905 y Fo(")492 3913 y Fi(0)529 3893 y Fx(\()p Ft(')p Fx(\))g(will)g(b)r(e)g(of)g(the)g(form)f Ft(")p Fx(\()p Ft(')p Fx(\))d(=)e(\()p Ft(\031)g Fr(\000)c Ft(')p Fx(\))p Ft(")1956 3905 y Fn(0)2021 3893 y Fx(\(see)28 b(b)r(elo)n(w\).)71 4000 y(W)-7 b(e)30 b(shall)f(de\014ne)h Fr(k)19 b(\001)h(k)29 b Fx(an)g(algebraic)f(matrix)h(norm)g(\()p Fv(i.e.)i Fx(a)f(norm)f(whic)n(h)g(v)n(eri\014es)f Fr(k)p Ft(AB)t Fr(k)e(\024)g(k)p Ft(A)p Fr(k)14 b(k)p Ft(B)t Fr(k)28 b Fx(for)0 4106 y(all)f(matrices)g Ft(A)h Fx(and)f Ft(B)t Fx(\);)i(for)e(instance)g Fr(k)18 b(\001)g(k)28 b Fx(can)f(b)r(e)h(the)g(uniform)f(norm.)71 4231 y(The)37 b(propagators)p 718 4165 66 4 v 34 w Ft(G)784 4179 y Fn([)p Fo(k)q Fn(])784 4252 y Fo(`)899 4231 y Fx(in)h(\(4.4\))f (satisfy)g(in)n(teresting)f Ft(k)s Fx(-indep)r(enden)n(t)i(b)r(ounds)f (describ)r(ed)g(and)g(pro)n(v)n(ed)0 4338 y(b)r(elo)n(w.)1954 4596 y Ft(")p Fx(-plane)1738 4817 y Ft(D)1807 4829 y Fo(")1838 4837 y Fi(0)1875 4817 y Fx(\()p Ft(')p Fx(\))1431 4997 y Fr(\000)p Ft(x)1543 4966 y Fn(2)1357 5236 y @beginspecial @setspecial %%BeginDocument: fig5.ps /origine1assexper2 { 4 2 roll 2 copy translate exch 4 1 roll sub 3 1 roll exch sub 2 copy atan rotate 2 copy exch 4 1 roll mul 3 1 roll mul add sqrt } def /punta0{0 0 moveto dup dup 0 exch 2 div lineto 0 lineto 0 exch 2 div neg lineto 0 0 lineto fill stroke } def /dirpunta{ gsave origine1assexper2 0 translate 7 punta0 grestore} def /punto { % x y punto gsave 2 0 360 newpath arc fill stroke grestore} def /R {40} def /O {R R} def /A {R 2 mul R 4 div add} def 0 R moveto A R lineto stroke 0 R A R dirpunta R 0 moveto R A lineto stroke R 0 R A dirpunta R 2 div R punto O moveto R R 45 cos mul neg add R R 45 sin mul neg add lineto stroke O R -135 135 arc stroke O moveto R R 45 cos mul neg add R R 45 sin mul add lineto stroke %%EndDocument @endspecial 591 5432 a Fg(Fig.)d(5.)j Ff(The)25 b(domain)f Fo(D)1311 5440 y Fj(")1339 5452 y Fi(0)1376 5432 y Fe(\()p Fo(')p Fe(\))f Ff(in)g(the)i(complex)g Fo(")p Ff(-plane:)32 b(the)24 b(half-op)r(ening)g(angle)591 5507 y(of)k(the)i(sector)f(is)f Fo('<\031)j Ff(but,)f(otherwise,)g(arbitrary)-6 b(,)30 b(and)f(the)h(radius)e(of)h(the)g(circle)591 5582 y(delimiting)23 b Fo(D)978 5590 y(")1009 5598 y Fi(0)1046 5582 y Ff(\()p Fo(')p Ff(\))h(is)f(giv)n(en)h(b)n(y)g Fo(")p Ff(\()p Fo(')p Ff(\))c(=)f(\()p Fo(\031)e Fy(\000)e Fo(')p Ff(\))p Fo(")1996 5594 y Fe(0)2031 5582 y Ff(.)0 5869 y Fi(12)p Fj(=lug)q(lio=)p Fi(2001;)29 b(13:05)1169 b Fx(14)p eop %%Page: 15 15 15 14 bop 0 83 a Fs(5.2.)65 b Fw(Pr)n(oposition.)h Fv(L)l(et)38 b Ft(D)1032 95 y Fo(")1063 103 y Fi(0)1100 83 y Fx(\()p Ft(')p Fx(\))i Fv(b)l(e)f(obtaine)l(d)h(fr)l(om)f(the)g(disk)h(of)g (diameter)g Ft(")2740 95 y Fn(0)2816 83 y Ft(>)f Fx(0)g Fv(in)g(the)g(c)l(omplex)0 189 y Ft(")p Fv(-plane)28 b(by)g(taking)f(out)g(a)h(se)l(ctor)f(of)i(half-op)l(ening)g Ft(\031)17 b Fr(\000)c Ft(')27 b Fv(ar)l(ound)h(the)f(ne)l(gative)h(r)l (e)l(al)g(axis.)38 b(Assume)27 b(that)g(the)0 310 y(pr)l(op)l(agators)p 446 243 66 4 v 32 w Ft(G)511 257 y Fn([)p Fo(k)q Fn(])511 331 y Fo(`)612 310 y Fr(\021)p 700 243 V 23 w Ft(G)765 257 y Fn([)p Fo(k)q Fn(])844 310 y Fx(\()p Fq(!)21 b Fr(\001)e Fq(\027)1047 322 y Fo(`)1078 310 y Fx(;)14 b Ft(")p Fx(\))30 b Fv(satisfy)741 447 y Fm(\020)p 791 473 V 92 x Ft(G)856 486 y Fn([)p Fo(k)q Fn(])934 539 y Fx(\()p Ft(x)p Fx(;)14 b Ft(")p Fx(\))1121 447 y Fm(\021)1172 463 y Fo(T)1247 539 y Fx(=)p 1335 473 V 23 w Ft(G)1400 486 y Fn([)p Fo(k)q Fn(])1478 539 y Fx(\()p Fr(\000)p Ft(x)p Fx(;)g Ft(")p Fx(\))p Ft(;)1937 469 y Fm(\015)1937 519 y(\015)p 1983 473 V 20 x Ft(G)2049 486 y Fn([)p Fo(k)q Fn(])2127 539 y Fx(\()p Ft(x)p Fx(;)g Ft(")p Fx(\))2314 469 y Fm(\015)2314 519 y(\015)2384 539 y Ft(<)2564 483 y Fx(2)p 2481 520 207 4 v 2481 596 a Ft(\031)22 b Fr(\000)c Ft(')2729 483 y Fx(1)p 2707 520 85 4 v 2707 596 a Ft(x)2754 572 y Fn(2)3373 539 y Fx(\(5)p Ft(:)p Fx(1\))0 767 y Fv(for)28 b(al)t(l)f Fr(j)p Ft(")p Fr(j)c Ft(<)g Fx(\()p Ft(\031)15 b Fr(\000)d Ft(')p Fx(\))p Ft(")737 779 y Fn(0)774 767 y Fv(,)28 b(if)g Ft(")944 779 y Fn(0)1008 767 y Fv(is)f(smal)t(l)g(enough.)38 b(Then)28 b(ther)l(e)f(is)g(a)g(c)l (onstant)f Ft(B)2590 779 y Fo(f)2660 767 y Fv(such)g(that,)i(summing)e (over)0 897 y(al)t(l)31 b(r)l(enormalize)l(d)g(tr)l(e)l(es)e Ft(\022)j Fv(with)e Fr(j)p Ft(V)19 b Fx(\()p Ft(\022)r Fx(\))p Fr(j)24 b Fx(=)f Ft(V)49 b Fv(no)l(des)30 b(the)g(values)g Fr(j)p 2119 829 138 4 v Fs(V)-8 b(al)2256 843 y Fn([)p Fo(k)q Fn(])2334 897 y Fx(\()p Ft(\022)r Fx(\))p Fr(j)p Fv(,)32 b(one)e(has)p 931 1082 54 4 v 931 1150 a Fs(h)984 1096 y Fn([)p Fo(k)q(;V)15 b Fn(])984 1170 y Fo(\015)t Fl(\027)1159 1150 y Fr(\024)1324 1071 y Fm(X)1246 1258 y Fo(\022)r Fy(2)p Fn(\002)1376 1238 y Fd(R)1376 1279 y Fj(V)s(;\027)s(;\015)1535 1054 y Fm(\014)1535 1104 y(\014)1535 1154 y(\014)p 1563 1082 138 4 v -4 x Fs(V)-8 b(al)1700 1096 y Fn([)p Fo(k)q Fn(])1778 1150 y Fx(\()p Ft(\022)r Fx(\))1883 1054 y Fm(\014)1883 1104 y(\014)1883 1154 y(\014)1935 1150 y Fr(\024)2023 1033 y Fm(\022)2101 1094 y Fr(j)p Ft(")p Fr(j)p Ft(B)2249 1106 y Fo(f)p 2094 1131 207 4 v 2094 1207 a Ft(\031)21 b Fr(\000)d Ft(')2310 1033 y Fm(\023)2371 1050 y Fo(V)2442 1150 y Ft(;)1023 1390 y Fm(X)955 1571 y Fj(\022)q Fd(2)p Fi(\002)1066 1557 y Fd(R)1066 1598 y Fj(V)s(;\027)s(;\015)982 1647 y(M)5 b Fi(\()p Fj(\022)q Fi(\)=)p Fj(s)1236 1373 y Fm(\014)1236 1423 y(\014)1236 1473 y(\014)p 1263 1401 138 4 v 1263 1469 a Fs(V)-8 b(al)1401 1415 y Fn([)p Fo(k)q Fn(])1479 1469 y Fx(\()p Ft(\022)r Fx(\))1584 1373 y Fm(\014)1584 1423 y(\014)1584 1473 y(\014)1636 1469 y Fr(\024)1723 1351 y Fm(\022)1802 1412 y Fr(j)p Ft(")p Fr(j)p Ft(B)1950 1424 y Fo(f)p 1794 1449 207 4 v 1794 1525 a Ft(\031)22 b Fr(\000)c Ft(')2010 1351 y Fm(\023)2071 1369 y Fo(V)2143 1469 y Ft(e)2182 1434 y Fy(\000)p Fo(\024s=)p Fn(8)2375 1469 y Ft(;)945 1743 y Fm(\014)945 1793 y(\014)945 1843 y(\014)972 1839 y Fs(b)1025 1796 y Fn([)p Fo(k)q(;V)d Fn(])1025 1861 y Fh(0)1177 1743 y Fm(\014)1177 1793 y(\014)1177 1843 y(\014)1228 1839 y Fr(\024)1315 1743 y Fm(\015)1315 1793 y(\015)1315 1843 y(\015)1362 1747 y(\020)1411 1839 y Ft(@)1460 1804 y Fn(2)1455 1864 y Fl(\014)1505 1839 y Ft(f)1546 1851 y Fh(0)1588 1839 y Fx(\()p Fq(\014)1675 1851 y Fn(0)1712 1839 y Fx(\))1744 1804 y Fy(\000)p Fn(1)1834 1747 y Fm(\021)1883 1743 y(\015)1883 1793 y(\015)1883 1843 y(\015)1943 1722 y(\022)2064 1783 y Ft(B)2127 1795 y Fo(f)p 2014 1820 V 2014 1896 a Ft(\031)22 b Fr(\000)c Ft(')2230 1722 y Fm(\023)2291 1739 y Fo(V)d Fn(+1)2447 1839 y Fr(j)p Ft(")p Fr(j)2532 1804 y Fo(V)2589 1839 y Ft(;)3373 1485 y Fx(\(5)p Ft(:)p Fx(2\))0 2068 y Fv(for)31 b(al)t(l)f Ft(s)23 b(>)g Fx(0)p Fv(.)0 2245 y Fs(5.3.)49 b Fv(R)l(emark.)j Fx(Note)32 b(that,)i(although)d(the)i(propagators)c (are)i(no)h(longer)f(diagonal,)h(they)h(still)f(satisfy)g(the)0 2351 y(same)24 b(prop)r(ert)n(y)f(as)h(\(2.15\),)g(whic)n(h)h(is)f(the) h(crucial)f(one)g(whic)n(h)g(is)g(used)h(in)g(order)e(to)h(pro)n(v)n(e) f(b)r(oth)i(the)g(lemmata)0 2458 y(2.5)k(and)h(2.7)f(ab)r(out)h(the)g (formal)f(solubilit)n(y)h(of)f(the)i(equations)e(of)h(motion)f(and)h (the)g(lemmata)g(3.9)f(and)h(3.11)0 2564 y(ab)r(out)e(the)g(formal)e (cancellations)h(b)r(et)n(w)n(een)g(tree)h(v)-5 b(alues.)0 2741 y Fs(5.4.)34 b Fv(Pr)l(o)l(of)26 b(of)h(the)e(pr)l(op)l(osition)j (5.2.)37 b Fx(W)-7 b(e)23 b(can)g(consider)f(\014rst)g(trees)h(without) g(lea)n(v)n(es,)f(so)h(that)g(the)g(tree)g(v)-5 b(alues)0 2848 y(are)27 b(giv)n(en)f(b)n(y)i(\(4.3\))f(with)h Ft(L)p Fx(\()p Ft(\022)r Fx(\))c(=)e Fr(;)p Fx(.)71 2973 y(The)28 b(h)n(yp)r(othesis)f(\(5.1\))g(implies)h(that)g(for)f(all)g (propagators)p 2011 2906 66 4 v 25 w Ft(G)2076 2920 y Fn([)p Fo(k)q Fn(])2076 2993 y Fo(`)2182 2973 y Fx(one)g(has)933 3132 y Fm(\015)933 3182 y(\015)933 3231 y(\015)p 979 3160 V -4 x Ft(G)1044 3174 y Fn([)p Fo(k)q Fn(])1044 3248 y Fo(`)1123 3132 y Fm(\015)1123 3182 y(\015)1123 3231 y(\015)1192 3227 y Fr(\024)c Ft(C)1339 3239 y Fn(1)1376 3227 y Fx(2)1418 3193 y Fy(\000)p Fn(2)p Fo(n)1544 3202 y Fj(`)1577 3227 y Ft(;)180 b(C)1839 3239 y Fn(1)1899 3227 y Fx(=)2079 3171 y(2)p 1997 3208 207 4 v 1997 3284 a Ft(\031)22 b Fr(\000)c Ft(')2227 3110 y Fm(\022)2298 3171 y Fx(2)2340 3141 y Fo(\034)7 b Fn(+2)p 2298 3208 168 4 v 2333 3284 a Ft(C)2392 3296 y Fn(0)2475 3110 y Fm(\023)2536 3124 y Fn(2)2587 3227 y Ft(:)763 b 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b Ft(E)2106 5457 y Fy(\000)p Fn(1)2101 5511 y Fo(n)2214 5491 y Fr(\000)k Fx(1)1057 5649 y Fr(\024)k Fx(1)c(+)1287 5556 y Fm(\020)1337 5649 y Ft(M)9 b Fx(\()p Ft(\022)r Fx(\))19 b Fr(\000)f Ft(E)1695 5661 y Fo(n)1740 5556 y Fm(\021)1790 5649 y Ft(E)1856 5614 y Fy(\000)p Fn(1)1851 5669 y Fo(n)1964 5649 y Fr(\000)g Fx(1)k Fr(\024)h Ft(M)9 b Fx(\()p Ft(\022)r Fx(\))14 b Ft(E)2474 5614 y Fy(\000)p Fn(1)2469 5669 y Fo(n)2582 5649 y Fr(\000)k Fx(1)p Ft(;)3310 5582 y Fx(\()p Ft(A)p Fx(3)p Ft(:)p Fx(9\))0 5869 y Fi(12)p Fj(=lug)q(lio=)p Fi(2001;)29 b(13:05)1169 b Fx(23)p eop %%Page: 24 24 24 23 bop 0 83 a Fx(hence)28 b(the)g(b)r(ound)g(is)f(pro)n(v)n(ed)f (also)h(if)h(the)g(ro)r(ot)f(line)g(is)h(normal)e(and)i(on)f(scale)g Ft(n)p Fx(.)0 260 y Fs(A3.3.)44 b Fv(R)l(emark.)h Fx(The)31 b(same)f(argumen)n(t)f(pro)n(v)n(es)f(the)j(b)r(ound)g(\(5.4\))f(for)g (renormalized)e(trees,)j(b)n(y)f(using)g(the)0 366 y(observ)-5 b(ation)26 b(that)i(there)f(are)g(no)g(self-energy)g(lines)g(in)h(the)g 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y(a)g(w)n(a)n(y)e(that)j Fr(j)p Fq(!)541 201 y Fn(0)596 189 y Fr(\001)644 188 y Fx(~)638 189 y Fq(\027)686 201 y Fo(`)717 189 y Fr(j)g Fx(di\013ers)e(from)1252 119 y Fm(\014)1252 169 y(\014)1280 189 y Fq(!)1340 201 y Fn(0)1395 189 y Fr(\001)19 b Fq(\027)1491 159 y Fn(0)1485 213 y Fo(`)1527 119 y Fm(\014)1527 169 y(\014)1596 189 y Fx(b)n(y)41 b(a)f(quan)n(tit)n(y)h(b)r(ounded)g(b)n (y)g Ft(\015)2677 201 y Fo(n)2750 189 y Fx(+)27 b Ft(\015)2885 201 y Fo(n)2926 209 y Fi(2)2990 189 y Fx(+)g Ft(:)14 b(:)g(:)27 b Fx(+)g Ft(\015)3341 201 y Fo(n)3382 209 y Fj(D)3479 189 y Fr(\024)0 296 y Fx(2)42 265 y Fo(n)105 296 y Fx(+)18 b(2)230 265 y Fo(n)271 273 y Fi(2)326 296 y Fx(+)g Ft(:)c(:)g(:)k Fx(+)g(2)649 265 y Fo(n)690 273 y Fj(D)769 296 y Ft(<)k Fx(2)898 265 y Fo(n)p Fn(+1)1027 296 y Fx(.)71 403 y(As)199 333 y Fm(\014)199 383 y(\014)227 403 y Fq(\027)281 373 y Fn(0)275 427 y Fo(`)318 333 y Fm(\014)318 383 y(\014)377 403 y Ft(<)32 b Fx(2)516 373 y Fy(\000)p Fn(\()p Fo(n)p Fn(+3\))p Fo(=\034)820 403 y Fx(,)i(b)n(y)f(de\014nition)h(of)f(self-energy)e(graph,)j(w)n(e)e (can)h(apply)g(\(3.2\))f(and)h(conclude)g(that)0 510 y Fr(j)p Fq(!)83 522 y Fn(0)132 510 y Fr(\001)12 b Fq(\027)221 480 y Fn(0)215 533 y Fo(`)257 510 y Fr(j)25 b Fx(has)e(to)h(b)r(e)h (con)n(tained)f(inside)g(an)g(in)n(terv)-5 b(al)24 b([)p Ft(\015)1737 522 y Fo(p)p Fy(\000)p Fn(1)1860 510 y Ft(;)14 b(\015)1940 522 y Fo(p)1978 510 y Fx(],)25 b(with)g Ft(p)e Fx(=)g Ft(n)2438 480 y Fn(0)2438 533 y Fo(`)2498 510 y Fr(\025)f Ft(n)12 b Fx(+)g(3)23 b(\(see)h(the)h(remark)e(3.12,)0 616 y(\(2\)\),)j(at)g(a)f(distance)g(at)h(least)f(2)1009 586 y Fo(n)p Fn(+1)1163 616 y Fx(from)g(the)h(extremes:)35 b(therefore)25 b(the)h(quan)n(tit)n(y)f Fr(j)p Fq(!)2779 628 y Fn(0)2834 616 y Fr(\001)2882 615 y Fx(~)2876 616 y Fq(\027)2924 628 y Fo(`)2955 616 y Fr(j)h Fx(still)g(falls)f(inside)0 722 y(the)j(same)f(in)n(terv)-5 b(al)27 b([)p Ft(\015)717 734 y Fo(p)p Fy(\000)p Fn(1)840 722 y Ft(;)14 b(\015)920 734 y Fo(p)959 722 y Fx(].)37 b(In)27 b(particular)g(this)h(implies)f (the)h(iden)n(tit)n(y)1524 933 y Ft(n)1574 945 y Fo(`)1629 933 y Fx(=)f(~)-46 b Ft(n)1767 945 y Fo(`)1821 933 y Fx(=)23 b Ft(n)1959 899 y Fn(0)1959 953 y Fo(`)1996 933 y Ft(;)1291 b Fx(\()p Ft(A)p Fx(6)p Ft(:)p Fx(2\))0 1144 y(if)32 b(~)-46 b Ft(n)126 1156 y Fo(`)186 1144 y Fx(is)27 b(de\014ned)h(as)f(the)h(in)n(teger)f(suc)n(h)g(that)h Ft(\015)1489 1156 y Fn(~)-37 b Fo(n)1526 1165 y Fj(`)1555 1156 y Fy(\000)p Fn(1)1667 1144 y Fr(\024)23 b(j)p Fq(!)1838 1156 y Fn(0)1893 1144 y Fr(\001)1940 1143 y Fx(~)1934 1144 y Fq(\027)1982 1156 y Fo(`)2014 1144 y Fr(j)g Ft(<)g(\015)2195 1156 y Fn(~)-37 b Fo(n)2232 1165 y Fj(`)2265 1144 y Fx(.)835 1392 y Fz(App)s(endix)39 b(A7.)50 b(Pro)s(of)37 b(of)g(the)h(lemma)h (4.3)0 1570 y Fs(A7.1.)d Fv(Set-up.)g Fx(W)-7 b(e)28 b(ha)n(v)n(e)e(to)i(pro)n(v)n(e)e(that,)i(for)f(all)g Ft(k)1700 1540 y Fy(0)1746 1570 y Fr(\024)c Ft(k)s Fx(,)28 b(one)f(has)1371 1796 y([)p 1394 1729 54 4 v Fs(h)1447 1743 y Fn([)p Fo(k)q Fn(])1526 1796 y Fx(\()p Fq( )s Fx(;)14 b Ft(")p Fx(\)])1755 1753 y Fn(\()p Fo(k)1817 1728 y Fd(0)1840 1753 y Fn(\))1755 1811 y Fl(\027)1893 1796 y Fx(=)23 b Fs(h)2034 1753 y Fn(\()p Fo(k)2096 1728 y Fd(0)2119 1753 y Fn(\))2034 1811 y Fl(\027)2149 1796 y Ft(;)1138 b Fx(\()p Ft(A)p Fx(7)p Ft(:)p Fx(1\))0 2027 y(whic)n(h)30 b(yields)g(that)g(the)h(functions)f Fs(h)g Fx(and)p 1411 1959 V 30 w Fs(h)1464 1973 y Fn(\()p Fy(\024)p Fo(k)q Fn(\))1639 2027 y Fx(admit)g(the)g(same)g(p)r(o)n(w)n(er)f (series)g(expansions)f(up)j(to)f(order)0 2133 y Ft(k)s Fx(.)71 2241 y(Of)f(course)f(b)r(oth)i Fs(h)697 2198 y Fn(\()p Fo(k)759 2173 y Fd(0)782 2198 y Fn(\))697 2255 y Fl(\027)841 2241 y Fx(and)f([)p 1027 2174 V Fs(h)1080 2187 y Fn([)p Fo(k)q Fn(])1159 2241 y Fx(\()p Fq( )s Fx(;)14 b Ft(")p Fx(\)])1388 2198 y Fn(\()p Fo(k)1450 2173 y Fd(0)1473 2198 y Fn(\))1388 2255 y Fl(\027)1533 2241 y Fx(are)28 b(giv)n(en)g(b)n(y)h(sums)g(of)g(sev)n(eral)f(con)n (tributions;)h(in)g(the)h(same)0 2347 y(w)n(a)n(y)20 b([)p Ft(M)275 2317 y Fn([)p Fo(k)q Fn(])353 2347 y Fx(\()p Fq(!)9 b Fr(\001)d Fq(\027)g Fx(;)14 b Ft(")p Fx(\)])668 2317 y Fn(\()p Fo(k)730 2292 y Fd(0)752 2317 y Fn(\))804 2347 y Fx(can)21 b(b)r(e)g(expressed)g(as)f(sum)i(of)f(sev)n(eral)e (terms.)35 b(W)-7 b(e)21 b(shall)g(pro)n(v)n(e)f(that,)j(giv)n(en)d(an) n(y)h(tree)0 2468 y(v)-5 b(alue)28 b Fs(V)-8 b(al)q Fx(\()p Ft(\022)r Fx(\))29 b(con)n(tributing)f(to)h Fs(h)1117 2425 y Fn(\()p Fo(k)1179 2400 y Fd(0)1202 2425 y Fn(\))1117 2482 y Fl(\027)1261 2468 y Fx(and)f(an)n(y)g(self-energy)f(v)-5 b(alue)29 b Fr(V)2260 2480 y Fo(T)2312 2468 y Fx(\()p Ft(x)p Fx(\))g(corresp)r(onding)e(to)i(a)f(self-energy)0 2594 y(graph)f Ft(T)39 b Fx(of)28 b(order)f Ft(k)683 2564 y Fy(0)706 2594 y Fx(,)i(one)f(can)f(\014nd)i(the)f(same)g(terms)g (con)n(tributing,)g(resp)r(ectiv)n(ely)-7 b(,)27 b(to)h([)p 2905 2527 V Fs(h)2958 2540 y Fn([)p Fo(k)q Fn(])3037 2594 y Fx(\()p Fq( )s Fx(;)14 b Ft(")p Fx(\)])3266 2551 y Fn(\()p Fo(k)3328 2526 y Fd(0)3351 2551 y 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2987 y Fn(\()p Fo(k)446 2962 y Fd(0)469 2987 y Fn(\))384 3044 y Fl(\027)537 3030 y Fx(is)37 b(the)h(co)r (e\016cien)n(t)g(to)f(order)f Ft(k)1565 3000 y Fy(0)1626 3030 y Fx(that)i(one)f(obtains)g(b)n(y)h(dev)n(elopping)p 2871 2963 V 36 w Fs(h)2924 2976 y Fn(\()p Fo(k)q Fn(\))3017 3030 y Fx(\()p Fq( )s Fx(;)14 b Ft(")p Fx(\))38 b(in)g(p)r(o)n(w-)0 3136 y(ers)27 b(of)g Ft(")p Fx(.)0 3243 y(\(2\))34 b(The)g(con)n (tributions)f(to)h([)p 960 3175 V Fs(h)1013 3189 y Fn([)p Fo(k)q Fn(])1092 3243 y Fx(\()p Fq( )s Fx(;)14 b Ft(")p Fx(\)])1321 3200 y Fn(\()p Fo(k)1383 3175 y Fd(0)1406 3200 y Fn(\))1321 3257 y Fl(\027)1470 3243 y Fx(and)34 b(to)g([)p Ft(M)1859 3213 y Fn([)p Fo(k)q Fn(])1937 3243 y Fx(\()p Ft(x)p Fx(;)14 b Ft(")p Fx(\)])2147 3213 y Fn(\()p Fo(k)2209 3188 y Fd(0)2233 3213 y Fn(\))2297 3243 y Fx(can)33 b(arise)g(only)h(from)f(trees)h(of)g(order)0 3369 y Ft(k)46 3339 y Fy(00)120 3369 y Fr(\024)e Ft(k)263 3339 y Fy(0)286 3369 y Fx(.)53 b(F)-7 b(urthermore)32 b([)p 872 3302 V Fs(h)925 3315 y Fn([)p Fo(k)q Fn(])1004 3369 y Fx(\()p Fq( )s Fx(;)14 b Ft(")p Fx(\)])1233 3326 y Fn(\()p Fo(k)1295 3301 y Fd(0)1318 3326 y Fn(\))1233 3383 y Fl(\027)1380 3369 y Fx(=)32 b([)p 1500 3302 V Fs(h)1553 3315 y Fn([)p Fo(k)q Fn(])1631 3369 y Fx(\()p Fq( )s Fx(;)14 b Ft(")p Fx(\)])1860 3326 y Fn(\()p Fo(k)1922 3301 y Fd(0)1946 3326 y Fn(\))1860 3383 y Fl(\027)2009 3369 y Fx(and)33 b([)p Ft(M)2289 3339 y Fn([)p Fo(k)q Fn(])2367 3369 y Fx(\()p Ft(x)p Fx(;)14 b Ft(")p Fx(\)])2577 3339 y Fn(\()p Fo(k)2639 3314 y Fd(0)2663 3339 y Fn(\))2725 3369 y Fx(=)31 b([)p Ft(M)2934 3339 y Fn([)p Fo(k)q Fn(])3012 3369 y Fx(\()p Ft(x)p Fx(;)14 b Ft(")p Fx(\)])3222 3339 y Fn(\()p Fo(k)3284 3314 y Fd(0)3308 3339 y Fn(\))3338 3369 y Fx(:)48 b(this)0 3475 y(symply)33 b(follo)n(ws)f(from)h(the)h (remark)e(4.2)g(and)h(the)h(trivial)f(observ)-5 b(ation)32 b(that,)j(for)e Ft(k)i(>)d(k)2930 3445 y Fy(0)2953 3475 y Fx(,)j(the)f(self-energy)0 3602 y(graphs)29 b([trees])g(whose)h(v)-5 b(alues)29 b(con)n(tribute)h(to)g Ft(M)1611 3572 y Fn([)p Fo(k)q Fn(])1689 3602 y Fx(\()p Ft(x)p Fx(;)14 b Ft(")p Fx(\))30 b([)p 1929 3534 V Fs(h)1982 3548 y Fn([)p Fo(k)q Fn(])2061 3602 y Fx(\()p Fq( )s Fx(;)14 b Ft(")p Fx(\)])30 b(but)h(not)f(to)g Ft(M)2819 3572 y Fn([)p Fo(k)2874 3547 y Fd(0)2896 3572 y Fn(])2919 3602 y Fx(\()p Ft(x)p Fx(;)14 b Ft(")p Fx(\))31 b([)p 3160 3534 V Fs(h)3213 3548 y Fn([)p Fo(k)3268 3523 y Fd(0)3291 3548 y Fn(])3314 3602 y Fx(\()p Fq( )s Fx(;)14 b Ft(")p Fx(\)])0 3708 y(are)24 b(those)h(con)n(taining)g(also)f(self-energy)g(graphs)g(with)i (heigh)n(t)f Ft(D)g(>)d(k)2255 3678 y Fy(0)2279 3708 y Fx(:)35 b(suc)n(h)25 b(con)n(tributions)g(are)f(of)i(order)e(at)0 3830 y(least)j Ft(D)j Fx(in)e Ft(")p Fx(,)f(so)g(that)h(they)g(can)f (not)h(con)n(tribute)f(to)g([)p Ft(M)1861 3800 y Fn([)p Fo(k)q Fn(])1939 3830 y Fx(\()p Ft(x)p Fx(;)14 b Ft(")p Fx(\)])2149 3800 y Fn(\()p Fo(k)2211 3775 y Fd(0)2235 3800 y Fn(\))2293 3830 y Fx([[)p 2339 3762 V Fs(h)2392 3776 y Fn([)p Fo(k)q 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Fn(2)2173 5669 y Fo(i)2210 5649 y Fc(1)-27 b Fx(1)18 b Fr(\000)g Ft(M)2454 5614 y Fn([)p Fo(k)q Fy(\000)p Fn(1])2617 5649 y Fx(\()p Ft(x)2696 5661 y Fo(i)2725 5649 y Fx(;)c Ft(")p Fx(\))2833 5556 y Fm(i)2872 5574 y Fy(\000)p Fn(1)2975 5649 y Ft(;)312 b Fx(\()p Ft(A)p Fx(7)p Ft(:)p Fx(3\))0 5869 y Fi(12)p Fj(=lug)q(lio=)p Fi(2001;)29 b(13:05)1169 b Fx(26)p eop %%Page: 27 27 27 26 bop 0 83 a Fx(and)27 b(it)h(can)g(b)r(e)g(expanded)f(in)h(p)r(o)n (w)n(ers)e(of)h Ft(M)1437 53 y Fn([)p Fo(k)q Fy(\000)p Fn(1])1601 83 y Fx(\()p Ft(x)1680 95 y Fo(i)1708 83 y Fx(;)14 b Ft(")p Fx(\))27 b(as)p 153 269 66 4 v 153 336 a Ft(G)218 283 y Fn([)p Fo(k)q Fy(\000)p Fn(1])382 336 y Fx(\()p Ft(x)461 348 y Fo(i)489 336 y Fx(;)14 b Ft(")p Fx(\))23 b(=)739 280 y(1)p 717 317 85 4 v 717 393 a Ft(x)764 364 y Fn(2)764 416 y Fo(i)826 219 y Fm(\022)887 336 y Fc(1)-27 b Fx(1)18 b(+)g Ft(M)1131 301 y Fn([)p Fo(k)q Fy(\000)p Fn(1])1294 336 y Fx(\()p Ft(x)1373 348 y Fo(i)1401 336 y Fx(;)c Ft(")p Fx(\))1541 280 y(1)p 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5361 y Fn(1)729 5349 y Fx(,)28 b(with)g Ft(\021)e Fx(=)d Ft(")1163 5319 y Fn(2)1200 5349 y Ft(x)p Fx(.)71 5456 y(De\014ne)685 5542 y Ft(B)t Fx(\()p Ft(x)p Fx(;)14 b Ft(")p Fx(\))24 b(=)e Ft(e)1089 5508 y Fo(\021)r(X)1188 5542 y Ft(A)p Fx(\()p Ft(x)p Fx(;)14 b Ft(")p Fx(\))p Ft(e)1476 5508 y Fy(\000)p Fo(\021)r(X)1651 5542 y Fx(=)23 b Ft(e)1778 5508 y Fo(\021)r(X)1877 5542 y Fx(\003)1935 5554 y Fn(1)1972 5542 y Ft(e)2011 5508 y Fy(\000)p Fo(\021)r(X)2180 5542 y Fx(+)18 b Ft(")2302 5508 y Fn(2)2339 5542 y Ft(x)2386 5508 y Fn(2)2424 5542 y Ft(e)2463 5508 y Fo(\021)r(X)2562 5542 y Fx(\001)2631 5554 y Fn(2)2668 5542 y Ft(e)2707 5508 y Fy(\000)p Fo(\021)r(X)963 5673 y Fr(\021)k Ft(B)1113 5685 y Fn(0)1151 5673 y Fx(\()p Ft(x)p Fx(;)14 b Ft(")p Fx(\))19 b(+)f Ft(")1479 5639 y Fn(2)1516 5673 y Ft(x)1563 5639 y Fn(2)1600 5673 y Ft(e)1639 5639 y Fo(\021)r(X)1738 5673 y Fx(\001)1807 5685 y Fn(2)1845 5673 y Ft(e)1884 5639 y Fy(\000)p Fo(\021)r(X)2035 5673 y Fx(;)3310 5606 y(\()p Ft(A)p Fx(8)p Ft(:)p Fx(5\))0 5869 y Fi(12)p Fj(=lug)q(lio=)p Fi(2001;)29 b(13:05)1169 b Fx(27)p eop %%Page: 28 28 28 27 bop 0 83 a Fx(of)28 b(course)e Ft(B)t Fx(\()p Ft(x)p Fx(;)14 b Ft(")p Fx(\))28 b(has)f(the)h(same)f(eigen)n(v)-5 b(alues)27 b(as)g Ft(A)p Fx(\()p Ft(x)p Fx(;)14 b Ft(")p Fx(\).)0 260 y Fs(A8.2.)57 b Fv(Blo)l(ck-diagonalization.)63 b Fx(Consider)34 b Ft(B)1549 272 y Fn(0)1586 260 y Fx(\()p Ft(x)p Fx(;)14 b Ft(")p Fx(\):)52 b(w)n(e)34 b(shall)h(\014x)f(the)i (matrix)e Ft(X)41 b Fx(in)35 b(suc)n(h)f(a)h(w)n(a)n(y)e(that)0 366 y Ft(B)63 378 y Fn(0)100 366 y Fx(\()p Ft(x)p Fx(;)14 b Ft(")p Fx(\))29 b(is)e(blo)r(c)n(k-diagonal)e(up)j(to)g(order)e Ft(\021)1433 336 y Fn(2)1470 366 y Fx(.)71 473 y(So)h(w)n(e)g(lo)r(ok)g (for)g Ft(X)34 b Fx(suc)n(h)28 b(that)541 596 y Fm(\000)579 663 y Fc(1)-27 b Fx(1)18 b(+)g Ft(\021)s(X)25 b Fx(+)18 b Ft(O)r Fx(\()p Ft(\021)1095 629 y Fn(2)1134 663 y Fx(\))1166 596 y Fm(\001)1218 663 y Fx(\(\003)g(+)g Ft(\021)s Fx(\001)1522 675 y Fn(1)1560 663 y Fx(\))1606 596 y Fm(\000)1644 663 y Fc(1)-27 b Fx(1)18 b Fr(\000)g Ft(\021)s(X)25 b Fx(+)18 b Ft(O)r Fx(\()p Ft(\021)2160 629 y Fn(2)2198 663 y Fx(\))2230 596 y Fm(\001)2292 663 y Fx(=)k(\003)c(+)g Ft(\021)s(J)2628 675 y Fn(1)2684 663 y Fx(+)g Ft(O)r Fx(\()p Ft(\021)2908 629 y Fn(2)2947 663 y Fx(\))p Ft(;)308 b Fx(\()p Ft(A)p Fx(8)p Ft(:)p Fx(6\))0 854 y(with)993 983 y Ft(J)1039 995 y Fn(1)1099 983 y Fx(=)1187 866 y Fm(\022)1262 933 y Ft(J)1308 945 y Fn(1)p Fo(;\013\013)1534 933 y Ft(J)1580 945 y Fn(1)p Fo(;\013\014)1263 1033 y Ft(J)1309 1045 y Fn(1)p Fo(;\014)s(\013)1536 1033 y Ft(J)1582 1045 y Fn(1)p Fo(;\014)s(\014)1735 866 y Fm(\023)1819 983 y Fx(=)1907 866 y Fm(\022)1982 933 y Ft(J)2028 945 y Fn(1)p Fo(;\013\013)2325 933 y Fx(0)2056 1033 y(0)156 b Ft(J)2300 1045 y Fn(1)p Fo(;\014)s(\014)2452 866 y Fm(\023)2527 983 y Ft(:)760 b Fx(\()p Ft(A)p Fx(8)p Ft(:)p Fx(7\))0 1184 y(By)27 b(expanding)g(to)h(\014rst)f(order)f(\(A8.6\))i(w)n(e)f (obtain)1454 1374 y([)p Ft(X)r(;)14 b Fx(\003])j(+)h(\001)1835 1386 y Fn(1)1896 1374 y Fx(=)k Ft(J)2029 1386 y Fn(1)2067 1374 y Ft(;)1220 b Fx(\()p Ft(A)p Fx(8)p Ft(:)p Fx(8\))0 1565 y(while)28 b(imp)r(osing)f(\(A8.7\))h(giv)n(es)1222 1683 y Ft(X)1291 1695 y Fo(\013\014)1401 1683 y Fx(=)23 b Fr(\000)p Ft(B)t Fx(\()p Ft(")p Fx(\))1738 1616 y Fm(\000)1776 1683 y Fx(\003)1834 1695 y Fo(\014)s(\014)1937 1683 y Fr(\000)18 b Ft(x)2067 1649 y Fn(2)2105 1683 y Fc(1)-27 b Fx(1)2158 1616 y Fm(\001)2196 1632 y Fy(\000)p Fn(1)2299 1683 y Ft(;)1222 1834 y(X)1291 1846 y Fo(\014)s(\013)1401 1834 y Fx(=)23 b Fr(\000)1568 1767 y Fm(\000)1605 1834 y Fx(\003)1663 1846 y Fo(\014)s(\014)1767 1834 y Fr(\000)18 b Ft(x)1897 1799 y Fn(2)1935 1834 y Fc(1)-27 b Fx(1)1987 1767 y Fm(\001)2025 1783 y Fy(\000)p Fn(1)2128 1834 y Ft(B)t Fx(\()p Ft(")p Fx(\))p Ft(;)3310 1749 y Fx(\()p Ft(A)p Fx(8)p Ft(:)p Fx(9\))0 1980 y(where)943 2013 y Fm(\015)943 2063 y(\015)943 2113 y(\015)989 2042 y(\000)1027 2109 y Fx(\003)1085 2121 y Fo(\014)s(\014)1188 2109 y Fr(\000)18 b Ft(x)1318 2075 y Fn(2)1356 2109 y Fc(1)-27 b Fx(1)1409 2042 y Fm(\001)1447 2058 y Fy(\000)p Fn(1)1536 2013 y Fm(\015)1536 2063 y(\015)1536 2113 y(\015)1605 2109 y Fx(=)1703 2053 y(1)p 1703 2090 42 4 v 1704 2166 a Ft(")1768 1988 y Fm(\015)1768 2038 y(\015)1768 2088 y(\015)1768 2138 y(\015)1814 2017 y(\020)1864 2109 y Ft(@)1913 2075 y Fn(2)1908 2134 y Fl(\014)1958 2109 y Ft(f)1999 2121 y Fh(0)2040 2109 y Fx(\()p Fq(\014)2127 2121 y Fn(0)2165 2109 y Fx(\))2197 2017 y Fm(\021)2246 2034 y Fy(\000)p Fn(1)2336 1988 y Fm(\015)2336 2038 y(\015)2336 2088 y(\015)2336 2138 y(\015)2405 2109 y Fr(\024)2502 2053 y Ft(C)p 2502 2090 66 4 v 2515 2166 a(")2578 2109 y(;)668 b Fx(\()p Ft(A)p Fx(8)p Ft(:)p Fx(10\))0 2305 y(for)27 b(some)g(constan)n(t)g Ft(C)6 b Fx(.)37 b(F)-7 b(urthermore,)27 b(b)n(y)g(c)n(ho)r(osing)f Ft(X)1821 2317 y Fo(\013\013)1935 2305 y Fx(=)c(0)27 b(and)h Ft(X)2322 2317 y Fo(\014)s(\014)2430 2305 y Fx(=)23 b(0,)k(one)g(obtains)g Ft(J)3099 2317 y Fn(1)3160 2305 y Fr(\021)22 b Fx(0.)71 2411 y(Then)28 b(it)g(follo)n(ws)e(that)i(one)f(has)859 2602 y Ft(B)t Fx(\()p Ft(x)p Fx(;)14 b Ft(")p Fx(\))24 b(=)f(\003)18 b(+)g Ft(O)r Fx(\()p Ft(\021)1525 2568 y Fn(2)1563 2602 y Ft(X)1639 2568 y Fn(2)1676 2602 y Fx(\))g(+)g Ft(O)r Fx(\()p Ft(")1945 2568 y Fn(2)1983 2602 y Ft(x)2030 2568 y Fn(2)2068 2602 y Fx(\))23 b(=)g(\003)18 b(+)g Ft(O)r Fx(\()p Ft(")2506 2568 y Fn(2)2544 2602 y Ft(x)2591 2568 y Fn(2)2629 2602 y Fx(\))p Ft(;)585 b Fx(\()p Ft(A)p Fx(8)p Ft(:)p Fx(11\))0 2793 y(so)27 b(that)1263 2899 y Ft(B)1330 2865 y Fy(\000)p Fn(1)1420 2899 y Fx(\()p Ft(x)p Fx(;)14 b Ft(")p Fx(\))23 b(=)g(\003)1776 2865 y Fy(\000)p Fn(1)1883 2899 y Fx(+)18 b Ft(O)r Fx(\()p Ft(")2102 2865 y Fn(2)2140 2899 y Ft(x)2187 2865 y Fn(2)2225 2899 y Fx(\))p Ft(;)989 b Fx(\()p Ft(A)p Fx(8)p Ft(:)p Fx(12\))0 3057 y(where)27 b(the)h(eigen)n(v)-5 b(alues)26 b(of)i(\003)965 3027 y Fy(\000)p Fn(1)1081 3057 y Fx(are)f(of)h(the)g (form)f(either)g(1)p Ft(=x)2020 3027 y Fn(2)2085 3057 y Fx(or)f(1)p Ft(=)p Fx(\()p Ft(x)2349 3027 y Fn(2)2405 3057 y Fx(+)18 b Ft(a)2532 3069 y Fo(j)2567 3057 y Ft(")p Fx(\).)71 3163 y(Therefore)26 b(one)h(has)991 3185 y Fm(\015)991 3235 y(\015)991 3285 y(\015)1037 3281 y Ft(G)1102 3247 y Fn([)p Fo(k)q Fn(])1181 3281 y Fx(\()p Ft(x)p Fx(;)14 b Ft(")p Fx(\))1368 3185 y Fm(\015)1368 3235 y(\015)1368 3285 y(\015)1438 3281 y Fr(\024)1525 3210 y Fm(\015)1525 3260 y(\015)1571 3281 y Ft(e)1610 3247 y Fo(\021)r(X)1709 3210 y Fm(\015)1709 3260 y(\015)1769 3210 y(\015)1769 3260 y(\015)1816 3281 y Ft(B)1883 3247 y Fy(\000)p Fn(1)1972 3210 y Fm(\015)1972 3260 y(\015)2032 3210 y(\015)2032 3260 y(\015)2078 3281 y Ft(e)2117 3247 y Fy(\000)p Fo(\021)r(X)2268 3210 y Fm(\015)2268 3260 y(\015)2337 3281 y Fr(\024)2456 3225 y Fx(2)p 2435 3262 85 4 v 2435 3338 a Ft(x)2482 3314 y Fn(2)2529 3281 y Ft(;)717 b Fx(\()p Ft(A)p Fx(8)p Ft(:)p Fx(13\))0 3455 y(whic)n(h)28 b(pro)n(v)n(es)d(the)j(lemma)g(5.4.)0 3632 y Fs(A8.3.)62 b Fv(Bounds)38 b(in)g(the)g(c)l(omplex)g(plane.)64 b Fx(The)37 b(ab)r(o)n(v)n(e)e(analysis)g(applies)h(also)f(for)g (complex)h(v)-5 b(alues)36 b(of)h Ft(")p Fx(.)0 3739 y(Consider)26 b(the)h(domain)f Ft(D)851 3751 y Fn(0)915 3739 y Fx(represen)n(ted)f(in)i(Fig.)37 b(5,)26 b(with)h(half-op)r (ening)f(angle)g Ft(')e(<)e(\031)s Fx(,)28 b(and)e(set)h Ft(\013)c Fx(=)g Ft(\031)d Fr(\000)c Ft(')p Fx(.)0 3860 y(F)-7 b(or)27 b Ft(")c Fr(2)g Ft(D)358 3872 y Fn(0)423 3860 y Fx(one)k(has)g(that)h(the)g(norms)f(of)p 1390 3793 66 4 v 27 w Ft(G)1456 3807 y Fn([)p Fo(k)q Fn(])1534 3860 y Fx(\()p Ft(x)p Fx(;)14 b Ft(")p Fx(\)\))1753 3829 y Fy(\000)p Fn(1)1871 3860 y Fx(are)26 b(b)r(ounded)i(from)g(b)r(elo)n (w)f(b)n(y)1569 4041 y Ft(x)1616 4010 y Fn(2)p 1569 4078 85 4 v 1591 4154 a Fx(2)1678 4097 y(\()p Ft(\031)22 b Fr(\000)c Ft(\013)p Fx(\))c Ft(;)1285 b Fx(\()p Ft(A)p Fx(8)p Ft(:)p Fx(14\))0 4304 y(when)28 b Ft(\013)g Fx(is)f(close)g(to)h (0.)835 4623 y Fz(App)s(endix)39 b(A9.)50 b(Pro)s(of)37 b(of)g(the)h(lemma)h(5.8)0 4800 y Fs(A9.1.)33 b Fv(Set-up.)h Fx(Both)21 b Ft(M)834 4770 y Fn([)p Fo(k)q Fn(])912 4800 y Fx(\()p Fq(!)7 b Fr(\001)t Fq(\027)f Fx(;)14 b Ft(")p Fx(\))21 b(and)f Ft(M)1465 4770 y Fn([)p Fo(k)q Fn(+1])1627 4800 y Fx(\()p Fq(!)7 b Fr(\001)t Fq(\027)f Fx(;)14 b Ft(")p Fx(\))21 b(can)g(b)r(e)g(expressed)e(b)n(y)h(\(4.5\):)33 b(the)21 b(only)f(di\013erence)0 4926 y(is)27 b(that)h(one)g(has)f(to)g (use)h(the)g(propagators)p 1409 4859 66 4 v 24 w Ft(G)1474 4873 y Fn([)p Fo(k)q Fn(])1474 4946 y Fo(`)1580 4926 y Fx(for)g Ft(M)1798 4895 y Fn([)p Fo(k)q Fn(+1])1960 4926 y Fx(\()p Fq(!)21 b Fr(\001)e Fq(\027)6 b Fx(;)14 b Ft(")p Fx(\).)71 5032 y(This)20 b(means)g(that)g(there)g(is)g(a)g (corresp)r(ondence)e(1-to-1)h(b)r(et)n(w)n(een)h(the)g(graphs)f(con)n (tributing)h(to)g Ft(M)3179 5002 y Fn([)p Fo(k)q Fn(])3257 5032 y Fx(\()p Fq(!)7 b Fr(\001)t Fq(\027)f Fx(;)14 b Ft(")p Fx(\))0 5138 y(and)27 b(those)h(con)n(tributing)f(to)g Ft(M)1043 5108 y Fn([)p Fo(k)q Fn(+1])1205 5138 y Fx(\()p Fq(!)22 b Fr(\001)c Fq(\027)6 b Fx(;)14 b Ft(")p Fx(\),)27 b(so)g(that)h(w)n(e)f(can)h(write)357 5339 y Ft(M)447 5305 y Fn([)p Fo(k)q Fn(])525 5339 y Fx(\()p Fq(!)22 b Fr(\001)c Fq(\027)6 b Fx(;)14 b Ft(")p Fx(\))k Fr(\000)g Ft(M)1033 5305 y Fn([)p Fo(k)q Fn(+1])1195 5339 y Fx(\()p Fq(!)k Fr(\001)c Fq(\027)6 b Fx(;)14 b Ft(")p Fx(\))23 b(=)1782 5260 y Fm(X)1622 5439 y Fn(renormalized)16 b Fo(T)2075 5339 y Fr(V)2133 5296 y Fn([)p Fo(k)q(;k)q Fn(+1])2126 5363 y Fo(T)2352 5339 y Fx(\()p Fq(!)21 b Fr(\001)e Fq(\027)6 b Fx(\))p Ft(;)357 5577 y Fr(V)415 5533 y Fn([)p Fo(k)q(;k)q Fn(+1])408 5601 y Fo(T)634 5577 y Fx(\()p Fq(!)21 b Fr(\001)e Fq(\027)6 b Fx(\))23 b(=)f Ft(")1024 5542 y Fy(j)p Fo(V)14 b Fn(\()p Fo(T)9 b Fn(\))p Fy(j)1235 5484 y Fm(\020)1362 5498 y(Y)1298 5680 y Fo(v)r Fy(2)p Fo(V)15 b Fn(\()p Fo(T)9 b Fn(\))1546 5577 y Ft(F)1599 5589 y Fo(v)1639 5484 y Fm(\021)o(h\020)1847 5498 y(Y)1791 5680 y Fo(`)p Fy(2)p Fn(\003\()p Fo(T)g Fn(\))p 2023 5510 V 2023 5577 a Ft(G)2088 5524 y Fn([)p Fo(k)q Fy(\000)p Fn(1])2088 5597 y Fo(`)2251 5484 y Fm(\021)2319 5577 y Fr(\000)2402 5484 y Fm(\020)2522 5498 y(Y)2466 5680 y Fo(`)p Fy(2)p Fn(\003\()p Fo(T)g Fn(\))p 2698 5510 V 2698 5577 a Ft(G)2763 5524 y Fn([)p Fo(k)q Fn(])2763 5597 y Fo(`)2841 5484 y Fm(\021i)2930 5577 y Ft(:)3310 5498 y Fx(\()p Ft(A)p Fx(9)p Ft(:)p Fx(1\))0 5869 y Fi(12)p Fj(=lug)q(lio=)p Fi(2001;)29 b(13:05)1169 b Fx(28)p eop %%Page: 29 29 29 28 bop 0 87 a Fx(F)-7 b(or)36 b(eac)n(h)g(renormalized)g (self-energy)f Ft(T)48 b Fx(w)n(e)37 b(can)f(write)h Fr(V)1948 44 y Fn([)p Fo(k)q(;k)q Fn(+1])1941 111 y Fo(T)2166 87 y Fx(\()p Fq(!)28 b Fr(\001)d Fq(\027)6 b Fx(\))37 b(as)f(sum)h(of)g Ft(V)57 b Fx(=)38 b Fr(j)p Ft(V)19 b Fx(\()p Ft(T)12 b Fx(\))p Fr(j)37 b Fx(terms)0 216 y(corresp)r(onding)29 b(to)i(trees)f(whose)g(lines)h(ha)n(v)n(e)f(all)g (the)i(propagators)27 b(of)k(the)h(form)e(either)p 2888 149 66 4 v 31 w Ft(G)2953 163 y Fn([)p Fo(k)q Fy(\000)p Fn(1])2953 236 y Fo(`)3148 216 y Fx(or)p 3253 149 V 30 w Ft(G)3318 163 y Fn([)p Fo(k)q Fn(])3318 236 y Fo(`)3396 216 y Fx(,)i(up)0 341 y(to)27 b(one)h(whic)n(h)f(has)g(a)g(new)h (propagator)d(giv)n(en)i(b)n(y)g(the)h(di\013erence)p 2151 274 V 28 w Ft(G)2216 288 y Fn([)p Fo(k)q Fy(\000)p Fn(1])2216 362 y Fo(`)2398 341 y Fr(\000)p 2481 274 V 18 w Ft(G)2546 288 y Fn([)p Fo(k)q Fn(])2546 362 y Fo(`)2624 341 y Fx(.)0 518 y Fs(A9.2.)39 b Fv(R)l(emark.)h Fx(Note)29 b(that)g(the)g(scales)e(of)i(all)f(lines)h(are)e(uniquely)i(\014xed)f (b)n(y)h(the)g(momen)n(ta,)f(so)g(that)h(b)r(oth)0 639 y(propagators)p 458 572 V 25 w Ft(G)523 586 y Fn([)p Fo(k)q Fy(\000)p Fn(1])523 660 y Fo(`)714 639 y Fx(and)p 875 572 V 27 w Ft(G)941 586 y Fn([)p Fo(k)q Fn(])941 660 y Fo(`)1047 639 y Fx(admit)f(the)g(same)f(b)r(ounds)g(\(see)h (\(5.6\)\).)0 816 y Fs(A9.3.)35 b Fv(Bounds.)i Fx(W)-7 b(e)25 b(can)f(order)f(the)i(lines)g(in)g(\003\()p Ft(T)12 b Fx(\))24 b(and)g(construct)h(a)f(set)h(of)f Ft(V)44 b Fx(subsets)24 b(\003)2957 828 y Fn(1)2994 816 y 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y(Moser's)c(the)l(or)l(em)p Fx(,)g(Ann.)72 b(Scuola.)f(Norm.)g (Sup.)h(Pisa)38 b(Cl.)71 b(Sci.)h(\(4\))39 b Fs(26)g Fx(\(1998\),)i(no.)71 b(3,)348 1490 y(545{593.)0 1644 y([B])243 b(A.D.)32 b(Bryuno:)42 b Fv(A)n(nalytic)33 b(form)h(of)g(di\013er)l(ential)g(e)l(quations.)47 b(I,)34 b(II.)f(\(R)n(ussian\))p Fx(,)e(T)-7 b(rudy)30 b(Mosk)n(o)n(v.)348 1750 y(Mat.)37 b(Ob)l(\024)-37 b(s)n(\024)e(c.)34 b Fs(25)28 b Fx(\(1971\),)e(119{262;)e(ibid.)38 b Fs(26)27 b Fx(\(1972\),)f (199{239.)0 1924 y([BKS])132 b(J.)19 b(Bricmon)n(t,)h(A.)f(Kupiainen,)h (A.)g(Sc)n(henk)n(el:)32 b Fv(R)l(enormalization)22 b(gr)l(oup)g(for)h (the)f(Melnikov)h(pr)l(oblem)348 2031 y(for)31 b(PDE's)p Fx(,)d(Comm.)37 b(Math.)g(Ph)n(ys.,)27 b(in)h(press.)0 2200 y([CGM])101 b(E.)32 b(Caliceti,)g(V.)h(Grecc)n(hi,)f(M.)g(Maioli:) 45 b Fv(The)34 b(distributional)h(Bor)l(el)g(summability)g(and)f(the)g (lar)l(ge)348 2307 y(c)l(oupling)d Fx(\010)733 2277 y Fn(4)800 2307 y Fv(lattic)l(e)f(\014elds)p Fx(,)e(Comm)n(un.)37 b(Math.)g(Ph)n(ys.)f Fs(104)27 b Fx(\(1986\),)f(163{174.)0 2481 y([CG])177 b(L.)29 b(Chierc)n(hia,)f(G.)h(Galla)n(v)n(otti:)39 b Fv(Drift)31 b(and)g(di\013usion)h(in)f(phase)h(sp)l(ac)l(e)p Fx(,)e(Ann.)41 b(Inst.)g(H.)30 b(P)n(oincar)n(\022)-39 b(e)348 2587 y(Ph)n(ys.)35 b(Th)n(\023)-39 b(eor.)33 b Fs(60)23 b Fx(\(1994\),)h(no.)35 b(1,)24 b(1{144;)e Fv(Err)l(atum)k(for)h(\\Drift)f(and)g(di\013usion)h(in)f(phase)i(sp)l (ac)l(e",)348 2694 y(by)j(L.)g(Chier)l(chia)i(and)e(G.)g(Gal)t (lavotti,)i(A)n(nn.)39 b(Inst.)h(H.)30 b(Poinc)l(ar)n(\023)-40 b(e)32 b(Phys.)42 b(Th)n(\023)-40 b(eor.)42 b(60)31 b(\(1994\),)i(1,) 348 2800 y(1994)p Fx(,)d(Ann.)37 b(Inst.)g(H.)28 b(P)n(oincar)n(\023) -39 b(e)25 b(Ph)n(ys.)36 b(Th)n(\023)-39 b(eor.)34 b Fs(68)28 b Fx(\(1998\),)e(135.)0 2974 y([E])245 b(Eliasson,)20 b(L.H.:)33 b Fv(A)n(bsolutely)22 b(c)l(onver)l(gent)g(series)h(exp)l (ansions)f(for)h(quasi-p)l(erio)l(dic)i(motions)p Fx(,)c(Math.)348 3081 y(Ph)n(ys.)36 b(Electron.)g(J.)27 b Fs(2)h Fx(\(1996\),)e(P)n(ap)r (er)g(4,)h Fa()p Fx(.)0 3255 y([G1])195 b(G.)28 b(Galla)n(v)n(otti:)36 b Fv(Twistless)30 b(KAM)g(tori)p Fx(,)e(Comm.)37 b(Math.)g(Ph)n(ys.)f Fs(164)27 b Fx(\(1994\),)f(145{156.)0 3429 y([G2])195 b(G.)26 b(Galla)n(v)n (otti:)35 b Fv(Twistless)28 b(KAM)g(tori,)h(quasi\015at)g(homo)l (clinic)h(interse)l(ctions,)e(and)h(other)f(c)l(anc)l(el-)348 3536 y(lations)34 b(in)f(the)h(p)l(erturb)l(ation)f(series)h(of)f(c)l (ertain)h(c)l(ompletely)g(inte)l(gr)l(able)g(Hamiltonian)g(systems.)348 3642 y(A)c(r)l(eview)p Fx(,)e(Rev.)37 b(Math.)g(Ph)n(ys.)f Fs(6)27 b Fx(\(1994\),)g(no.)36 b(3,)28 b(343{411.)0 3816 y([GG])172 b(G.)29 b(Galla)n(v)n(otti,)e(G.)i(Gen)n(tile:)38 b Fv(Major)l(ant)32 b(series)f(c)l(onver)l(genc)l(e)g(for)g(twistless)f (KAM)h(tori)p Fx(,)e(Ergo)r(dic)348 3922 y(Theory)e(Dynam.)37 b(Systems)27 b Fs(15)g Fx(\(1995\),)g(857{869.)0 4097 y([Ge])200 b(G.)35 b(Gen)n(tile:)52 b Fv(Whisker)l(e)l(d)38 b(tori)f(with)g(pr)l(e\014xe)l(d)f(fr)l(e)l(quencies)h(and)g(Lyapunov)h (sp)l(e)l(ctrum)p Fx(,)e(Dynam.)348 4203 y(Stabilit)n(y)28 b(of)f(Systems)h Fs(10)f Fx(\(1995\),)f(no.)37 b(3,)27 b(269{308.)0 4377 y([GGM])96 b(G.)34 b(Gen)n(tile,)h(G.)e(Galla)n(v)n (otti,)h(V.)f(Mastropietro:)47 b Fv(Field)36 b(the)l(ory)g(and)f(KAM)g (tori)p Fx(,)g(Math.)54 b(Ph)n(ys.)348 4484 y(Electron.)36 b(J.)27 b Fs(1)h Fx(\(1995\),)e(P)n(ap)r(er)g(1,)i Fa()p Fx(.)0 4658 y([GM])161 b(G.)30 b(Gen)n(tile,)h(V.)f (Mastropietro:)39 b Fv(Metho)l(ds)33 b(for)g(the)f(analysis)h(of)g(the) f(Lindste)l(dt)f(series)i(for)f(KAM)348 4764 y(tori)27 b(and)f(r)l(enormalizability)j(in)d(Classic)l(al)i(me)l(chanics.)38 b(A)26 b(r)l(eview)h(with)f(some)h(applic)l(ations)p Fx(,)g(Rev.)348 4871 y(Math.)37 b(Ph)n(ys.)f Fs(8)27 b Fx(\(1996\),)g(no.)36 b(3,)28 b(393{444.)0 5045 y([Gr])204 b(S.M.)30 b(Gra\013:)41 b Fv(On)31 b(the)h(c)l(onservation)h(for)f(hyp) l(erb)l(olic)i(invariant)f(tori)f(for)h(Hamiltonian)g(systems)p Fx(,)348 5151 y(J.)28 b(Di\013eren)n(tial)f(Equations)g Fs(15)g Fx(\(1974\),)f(1{69.)0 5869 y Fi(12)p Fj(=lug)q(lio=)p Fi(2001;)j(13:05)1169 b Fx(31)p eop %%Page: 32 32 32 31 bop 0 83 a Fx([Me])189 b(V.K.)27 b(Mel'nik)n(o)n(v:)36 b Fv(On)29 b(some)g(c)l(ases)h(of)g(c)l(onservation)g(of)h(c)l (onditional)t(ly)g(p)l(erio)l(dic)h(motions)e(under)348 189 y(a)25 b(smal)t(l)h(change)f(of)h(the)e(Hamiltonian)i(function)p Fx(,)d(So)n(viet)e(Math.)35 b(Dokl.)g Fs(6)47 b Fx(\(1965\),)22 b(1592{1596;)e Fv(A)348 296 y(family)32 b(of)e(c)l(onditional)t(ly)j(p) l(erio)l(dic)e(solutions)f(of)h(a)f(Hamiltonian)h(systems)p Fx(,)c(So)n(viet)g(Math.)38 b(Dokl.)348 402 y Fs(9)59 b Fx(\(1968\),)27 b(882{886.)0 576 y([Mo])184 b(J.)35 b(Moser:)49 b Fv(Conver)l(gent)36 b(series)h(exp)l(ansions)g(for)g (quasi)f(p)l(erio)l(dic)j(motions)p Fx(,)d(Math.)58 b(Ann.)g Fs(169)348 683 y Fx(\(1967\),)27 b(136{176.)0 857 y([P])245 b(J.)26 b(P\177)-42 b(osc)n(hel:)34 b(In)n(v)-5 b(arian)n(t)25 b(manifolds)h(of)g(complex)f(analytic)g(mappings)h(near)f(\014xed)h(p)r (oin)n(ts,)g(in)g Fu(Crit-)348 963 y(ical)f(Phenomena,)g(Random)g (Systems,)h(Gauge)e(Theories)p Fx(,)h(Les)g(Houc)n(hes,)g(Session)g (XLI)r(I)r(I)h(\(1984\),)348 1070 y(V)-7 b(ol.)37 b(I)r(I,)28 b(949{964,)c(Ed.)37 b(K.)27 b(Osterw)n(alder)f(&)h(R.)h(Stora,)f(North) g(Holland,)g(Amsterdam,)h(1986.)0 1239 y([R])241 b(H.)31 b(R)r(\177)-44 b(ussmann:)43 b Fv(Invariant)33 b(tori)g(in)f(the)h(p)l (erturb)l(ation)f(the)l(ory)h(of)h(we)l(akly)g(non{de)l(gener)l(ate)f (inte-)348 1346 y(gr)l(able)c(Hamiltonian)g(systems)p Fx(,)d(Preprin)n(t-Rehie)e(des)h(F)-7 b(ac)n(h)n(b)r(ereic)n(hs)24 b(Mathematik,)i(nr.)36 b(14,)25 b(1{89,)348 1452 y(1998.)0 1605 y([XY])178 b(J.)26 b(Xu,)h(J.)f(Y)-7 b(ou:)37 b Fv(Persistenc)l(e)29 b(of)g(L)l(ower)g(Dimensional)h(T)-6 b(ori)29 b(Under)g(the)f(First)h(Melnikov's)i(Non-)348 1712 y(r)l(esonanc)l(e)f(Condition)p Fx(,)f(Nanjing)f(Univ)n(ersit)n(y) f(Preprin)n(t,)f(1{21,)g(2001.)0 5869 y Fi(12)p Fj(=lug)q(lio=)p Fi(2001;)j(13:05)1169 b Fx(32)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0107120818402--