Content-Type: multipart/mixed; boundary="-------------0207090328446" This is a multi-part message in MIME format. ---------------0207090328446 Content-Type: text/plain; name="02-299.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-299.comments" e-mail: benfatto@mat.uniroma2.it, alessandro.giuliani@roma1.infn.it, mastropi@mat.uniroma2.it ---------------0207090328446 Content-Type: text/plain; name="02-299.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-299.keywords" Fermion systems, Renormalization group ---------------0207090328446 Content-Type: application/postscript; name="bgimlat.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="bgimlat.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: bgimlat.dvi %%CreationDate: Tue Jul 09 10:16:36 2002 %%Pages: 60 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMR12 CMSY10 CMBX10 CMR10 CMMI10 CMBX12 CMMI12 %%+ CMMI8 CMTI12 CMSY7 CMSY8 CMR8 CMSY6 CMR6 MSBM10 CMBX8 CMEX10 CMSL12 %%+ CMBX6 CMMI6 EUFM10 EUFM7 CMCSC10 MSBM7 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips.exe -P pdf -G0 -t ppt bgimlat %DVIPSParameters: dpi=8000, compressed %DVIPSSource: TeX output 2002.07.09:1015 %%BeginProcSet: tex.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/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]{Ci}imagemask restore}B/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: alt-rule.pro %! % Patch by TVZ % Makes dvips files draw rules with stroke rather than fill. % Makes narrow rules more predictable at low resolutions % after distilling to PDF. % May have unknown consequences for very thick rules. % Tested only with dvips 5.85(k). 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (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 45 /hyphen put dup 80 /P put dup 102 /f put dup 111 /o put dup 114 /r put readonly def /FontBBox{14 -250 1077 750}readonly def /UniqueXX 5000772 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd800bcb325468c79ffacb9cd3037a112066957 7889d67fff57549064b985d67ede2fb03272083723541051ebe936eff01a4ab3 cd845d565dfdf25d9ddd2792db17f52f894bea484480f06c832140fdd6c3af0e fef0584c6b78b5e2bbb9a752f10581fd1c48cb498a822afb18cddebba34695b9 b418010368fdfb7bf758d50f3af03360b81c776325c33c9c81130b1e1be840cf 68c71ffd091dccc0df7b507fe8953dc3836af0da08ff9ff3e9cf7288d9eb8baf ffac546a0fd987ae8e0bcf7b759f4408e13524f478de5896d816cb0ad2904271 cbec82a5ca50f0598502b434ecd94531b4a65181b5b3afcb650904b15157737a a4386f1a7f1252caf2adbd5c8162bf3b065c01ca471d520c884ebbe1b29f9549 134e285dea2fe158c15a08a744cdda00bb6c62158262e24702803d9e1dc1d3a3 1383085631154b077fc20488ee3aa03f47ed2766644330b94133f2f9389b99b8 96d3707d1c5bb227fd3bbdf3a6ff572bf07164107c7a1c0710a5c7cfb67282f9 c238100fa138b75c553d7cbef17e2292a320d3d86a6210e6e3a1e590b10dac6e 177094d5dd163db623d7b344478752f6f3d7d5b3fa92e6bc72dc6fc343d2f946 9153e3d14c5a6d6b1d9ca52947b789cb4cf6196a02973aff0b097333f8d615c0 a1726ab7e31c378eab77150faa28cce535afe3ae5798402ca51cbd18dedc2c43 5281d0781351df30f4d11761579676449ae4a47a10f677cdd31ca9120fc2606a 9025d3cc09d58856a9bbfb17669f706ff8c7efe37a3656c9b20609cccb58b309 0595ca5666833f2bbf884aac1cd632fce69b4e0ea2e0a25b9c4d471f0e322e09 4cefda321359fdbe0a82656d3ecaeb68b7e0c7e733d85066ed7474c22b67169a 1bf1bdf7dfd0ce0aaeef78ee393ce6ea65edc1fdcdf3f744f714005d8221a156 4e75f39af0f7cac55a62a3c7c8b8ff93dbbf453ac91a10fc4c608e41b9e3f571 b108af8761c5b78d841b2ec35711b72e0e045c3bb32d4e7d0d2f24a61b8d0d5c 1af22db0688911a97a8e25dcc56f596f60f312dee53caed9fe495cbabb3a91c7 ada5007b6b4263586aca068ee0df14bd4b964d24a14345d3b18ade6f14df529d 8ccb99daf9fe3fa37d73a8eb09ca6ba792521c61e50afc6472676c7d4a6aab24 b27274aa9aa01d1e5ae0b05e9aab9b2b0032496f06a266ba0482e467abcefd07 0b1f6acc94bda5b4525ab57a1bf620775f744cdbc10be42db8552d3e9f3a8140 350d01a29407dd06d9b3634a9fc50c4e9b297070ab1ca1747d4fba3c374d577a c53b6679fad31c5a6d90131bced5b15a51845e753384b8796aeed1d68fa60a09 3d934d496d820d78b300d2dd315072c3ac9f57e9f4eac0db820b5416d689e120 1af6a73a186da042fb6101201ef5891a08502d69ca46dad8dcc06fe935f679a8 ad1d26e8cca2f9f883275663ffaadf9100ffc087d09e9bf778580d5ed2a6a311 b423f03177c72b9c44afb39de1d98eced0cdccb89e27bf0153c1bd3e2109e512 df2e486014cc260a2f24c8c61bfe205ca658b5dae0b114305986d4605391e966 93e2871aa778c2fe421c87404a0b180511471f8d55a57e43392876d3e5596a86 ef0e3e3a5ad57ea7c36231c1c24ff9dd384b2a478d0ece8eae7898013bd0c9ae db567712ffdc654ec583e2440e2229e35c5e311696d7f08f5d4f19b1d9e6954b 7f36a18602e2f8a8f2a2f115eff53431159481fa0c26cd4339271b41d105d449 5f58d347f6c142616e718bb8f59a8d40cf4bf4e336affa5cbcfde1f9b855a814 013d035da1819b290d1a680b09395f87f72c36e9a9841056fd6977c434b2a431 a26abe5824c4577f55c0f2c1e2001b518245ddea49b83a5f7c344ce68945478d b89dd7a18776cf9598ec5da86bc754c2eb9b774ee55c73a20b9c932388140185 861294d4c48bd55e50875d73acee73dce8b9d05c65477d4bc6cef94ef3b1bb7b 047cc870537cd106d62d7a5ba6cfb78a56e3022e3b8338f527e71f8bfc5d8412 f7bb7649a8a41c35fc57c3840371a74a01f4bbf870ce968a937a00385eddeb47 7b0c5eb080ccc6bd07d2cd3d8fbd9d6a02520e95a0be3a290078c29c5767fbe5 3504d9 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 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 16 /zeta put dup 17 /eta put dup 18 /theta put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon 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 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 81 /Q put dup 83 /S put dup 84 /T put dup 86 /V put dup 87 /W put dup 88 /X put dup 96 /lscript 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 120 /x put dup 121 /y put dup 122 /z put dup 126 /vector put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueXX 5087386 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 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a5387e9011eb2334c1f42c090f06a1125c207ea6324e87f46414050d88004e87 79bd39f94399c3f1de84b8aae5edc6c3611e8ac49460509e672000d0afd94568 4ab67ff5c6bf2301522775398e192532677826b8cc7821a3a6a2765ee6edd840 b30d07f445681d61f0af1154c46147dfea6f3500f327b50eb05b5007e9e2a5e1 50be9f8fb62b781a609cfc01a10c69b3c05cb9368d5080db5967cb19b03aeb8a 38d5229ecec9a0453df594e38df4004c27fbd2230371a01fa865070b1de53b95 3c9422ec87ff72882f150c22a5d364ca8c1f2723231101b6ac7f8506f5d25825 91b9aa77d893bfea886d0dc93a91c7fccb621382bd048902b63c0591b53cbd3d 2b95364c62b73012b107ad5a263511c3b10e763b41474928dac7821e52611748 8934e12a5deed19b7d1cbd85bf433b5131b0aacabf0b273b7b8e5e60754ee79b 844b11e2a3ba1009b34137b90d924fb6a5f07d6fa333cffd877eaecbb1b94614 74bfb68427cfd84051311e6bac3769b6a8af2a2a2a65ce4105d8c0af1984af41 60ff751b0ff71c69648820548486e6372e49d71614bb496a89df21f9ced68b8a e3ac3fd2a6432c25426ef99eb0ba213f6678784b62411e6df77d4a4c29ca8b36 f00ac044894320b95381e4ecacdf512a1404f7533b8633a8cad50b9186bf1462 d629aa76a88c44401715bf07cf4076c19cc08c476146bf1fef95335a2e499002 0448bc4d700577b86d51d465f88315b17f6d507ab24c2452cf7a1e24e9a65ff1 bfde7df80e6c2dd32c0f3ed78824e06c12ccbf07c7409e1deb3601a993b6641f 755f665c4aa2ed4e666b6786f25cc2724979fe67b507ed3fc6bb6f9daa7f1bcc 4b352ee664643888ac42c4b4da4632d7f9f55c2b8281be8e55d0578b5fa13de2 b5506726d10a14b45b9b2e8bf75d059f9c73858a2b5d566da78184e1e9d1a30c 514a4f303a104467d79cbbcb24b69229c16db4efb1195e39f33077687848eee2 6bdffe05b1fd0ad606abb3d705629e705468e9de2fcec12e0418585e8120cbd9 f4326eb23d9b45432658fa929e2a4c99a7b9d2b0bb49e2239513a798fd6a06f8 72263b411aa72ffff84e5920f36a214ccb7175aa7be3ad94e91f5cb19764fe67 f124d7ff186cc6721a6db8ecf0bf0df299cb2cbbc766de461b5fbceb9b67f17c a99f46f607680a56e2f0d9cc4f7bb99677c4911dec507f34edcea83a047d56bc 1c951964e93081773f16b949b84decceaaee71cc75dbef5ff66ef3e9f18b4b9c c07b98a68e6198a00b17e3913d1ef8248f88797185ed1d53933deed305841011 b9a4cb9ac5b352bde7860318ad52d234f83bb6dc7cf10242318145bd532c6d53 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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 100 /d put dup 104 /h 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 c555448d5f81a16259ded1e11bf4119d53e8ab07a802df900d3a2d5ccc1c6876 d29c5e9effb7af3ef83400b0910c07873a8c56fa7b1eb1ba426043b00cc95dbe dc6e136cbbbcb3f67509964f9f281ebf81fe5b018122eaf66c4a838487e82e18 6e006093042848a903efb3469ab6b4049767aadb95c30408dfd58d8a10f4cb22 168decd9f3ee100f07b49aa44c92139b669cc312ba20192454eb2375be6284b0 26659d964b96ae82d4942e758027fcf23c25ed01115af27ce7f20efe2a822bb6 84004f20243a49c9e93301fc21b80815c033c3e2ba58ef53da2157d524b395f2 b37abca13bc6a2f42e824ab7e47106176b0d6db267fbb795ac7425582df2e3dc 55863468a9200742bd7b552c48f8cf58bc21343bd3b95abfa140f33f37c6f3f7 8b0d8a5154eb7c1f62ec598267f13e841a3e64172663935ac8b665d86540d316 ddece329c008049c5e74b27d59022c5515059bc3b89370b1bc6a169c888bb325 e0b74282d6f053a50da4024ed1e433271a32ab8c17d41c632b41cf9f3ecd5fe1 24daf7aea7ce8a63047b245822930d517df7baaafa69d2d17f7d93cebd45416d f3459365434123a76ccf883a4973bab19807e1f0c4bab03a45f7ed69ff2660ff 3963e4def11001eec4238c368a39d874fd30b3c14f4186ec7700fbce22abe468 eb32baf7b0164f8f21a9b5e706b91411faaa44467cb2180ab03de375c1ab93e9 76e11ab92fdd4a7280a6d1a8dc65c4c89d04c8ed6988045a2a3de9a4c7b1efd3 75b6d05a48a30c24d13b30ca0bd0790ff38c7bf80a5de1d5dd01142136a15eb9 caa33c59ab318c9e54769c8dd10e3b3cc1d371692b60b0d9b9eea52d9397aa04 9b6e3b531744c1d78476883b9a642407088e2f5d63623c412efb284b3d4d3f1d f1faa09dd3fa7603853e6e2e662d83f27eaf92591a47435a38c3ac44787a4e54 49dd8b08b8b209e235d5e5fced559f9029b6a3e8bc3adc199a278c0de0e5e986 0254a78f9ad70707cfc7c6f2f0f575b3930455502d384a054434bdda189606ec 450547595731a4bc9f90d627144f84cee67f18888dec91eb3db0df15259bf07b 805f551c76d3c75aa4c1b81b550bb8c8628882c2335c5cd44fb67c29c38c4c51 b51a80c39a72e4026226fc02961e24ac812ca63edcb21b0062b51d4d02aaa774 d18142d20624c25eae49bcf9f9fa14c3aca36440523afff84c14541cc1ba399b 91696bb9cf60fcc2adda555daf3bc5f5387d82423015b91c9e51bb4be11a5f70 122b8da9d0c90643e96e9672a291613a89fa7b1814f0a7c56074cc149169a5a3 2100af9199349c7519c4d09f765c8be122acd8d5f5d166c70016a829575a0308 10ae168d5a1cadbe68ef74162178484e88e484ce1dff7c1a3d345aba2706aa34 6802993620198bfa722bd7e5338fa8b93bc12c9245e12653e7196f1d65340fca bffb886959f95b7e6036a35f333232236cd08f47e7fe2ef440275022c352f2db 9acfd79da37a086d1fa166897320b10101a18051a0cf4f2d399c3b8dfa2f5b22 4fb487e6fbddf96cf29ce2c361475ce00bce60b7bbc6c3fb76ce7a1edb1aa620 b099eb875f60c6b8f1ac3edd1cd752226afb 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: EUFM7 %!PS-AdobeFont-1.1: EUFM7 2.1 %%CreationDate: 1992 Nov 20 17:36:25 % Euler fonts were designed by Hermann Zapf. % 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 (EUFM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM7 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 83 /S put readonly def /FontBBox{0 -250 1193 750}readonly def /UniqueXX 5031992 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6f33457e320fe93acc64a4016412366d1ad2 8776cfdce37d3a89b95e5bb9373a99730cba9c11192589e22438b44f4ecc6a8d 1f72a7ea6a698d9580bcafcc2366c5964e5d3094b67d86eec82e16761e4b6f4d fd245d33c9b9c2e5efaf6b70fc1b26729242663b094e125b63b03099ae89aef3 960574bed8d38d9a5871585daeef1a4fee999d512520a302e184ec3fc9a08570 7c490d9d92ba9e23eb831c6f0d09095f97e570e19fbc20e2a04ae15de9836754 0071fa98cd0a687c158288903e7daecf04f31701e4e2f9e7f89f9444063548bf 107761142c46af966926ad8888abae91da9af232c3560a8f044ec566f2797b20 4c66aab3d13785980b5f6cd6edc225a2d939b5e3c6c2dcf7e85c911ec4be45fd ecf273f14f93bc789e2d567f6413fa3c0d46239dee0c7e98faaeb9643481b0de dc7821817e01bb2c4d14d6eb348422a86e7168ca0b0783312dd2c985805b1cf8 3048b9b127262460901ff6291c5c8540daaf8d3e370c1e1580a2eaee60acd2f2 20843291cf3bc3c8fbc397444ce5651559001ebd115ece976b74f40d75ac9a0e be0b5a33d5fb7a5ad3888af240cec5b189aa907095f26a83f21347a20dc6a274 070b50b813f007bd6ec6bf3dc7e6ca22e2b334770791a822f530efca26ae1d67 72f29c2372bcca17e57f13d88cdaa448255f9160bb173dbae40a205e7e4ea2e5 eae10ae90a9d46855cbc2e3a8fee8ae0a6d9c193b8223a89d6d64acf98a0f30d da3e845f80654229284286633e0a99290ca56f27900ecdf0e3e671a8bb4effd9 4ae4a57bdb1959d1e466e7071b2059e1e153939897702414815d1aaea56b2a71 7f464f4344355e9ce5c38746dd5439241f72ebbf2c6528bb5b7a31b72467edcd 835a91658b27b76237aa3a20ffad24b7ce6b10bcb41c58396f4216303115faa9 a11d8b1b8184a69389667ca81b651751b0c59cfc2d2d77bbf82eae48cca683be 05c25f9619b1afddbb02247fb70297589ca4fdf9afc308d21f832c72d2b47494 8f92c31ff9c0d8a3c6512ed96be70b4c914e66da59ef24b20185d5d64bc998db bdb99bf85291b70fba4a963e0e23de5fc9f0aa6b54cc5dcac9dfb4fe69f132f0 c6bcb0fb2874bb654bccc9a8e3670c5d3cd2da110356f3fe8b2579e05864fea4 190765e69f9c813c04b31d196057eb50930058bf984a9f83078892e3e773a899 3b668b248b70ff53b3fa0c051545ff00c9101cd43f6dc3ed20641b02d9c49e92 f7c4af1344fe3d43b1d51c05c7010fd1656a13c3654c6d6d0015194bffd2c66e a4aeda0ce73ad7f1f201c5f44c4f89b5180886693f7b87756113ec07fed05142 a0e02e740dec49134d1824e79af3b3c8f5656107106906b7892366e4624847c7 3cb0fd828b72cb2ba768669ebd9b12b8b755f70fd71f395fa8a504bfcc19b45a 64bf92d8fc9456acfd814ef18311f3f1af1fd208131fd74b371dcc2d92b045c6 70a4d722882011ff2f68a463ae7c84bc321b8c03e07f47c0fd0d807aa02ad5a1 2c1428aa156bd2a8f7b077b0e5106bf936669d437f049e557e941e349fc70cd9 2be9ff6443a7d6b9253b2157037b1d 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: EUFM10 %!PS-AdobeFont-1.1: EUFM10 2.1 %%CreationDate: 1992 Nov 20 17:36:20 % Euler fonts were designed by Hermann Zapf. % 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 (EUFM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM10 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 70 /F put dup 77 /M put dup 83 /S put readonly def /FontBBox{-26 -224 1055 741}readonly def /UniqueXX 5031986 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6f33457e320fe93acc64a4016412366d1ad2 8776cfdce37d3a89b95e5bb9373a99730cba9c11192589e22438b44f4ecc6a8d 1f72a7ea6a698d9580bcafcc2366c5964e5d3094b67d86eec82e16761e4b6f4d fd245d3c0b4188a0543ee88fa787fb10793268e415e389ff071cdd46bf6b260e 1e9c9ac037edba327c6379836655308ddf35e4a4488ad32d3ab301c67a2fa71e 8ae00daeead4271df276f5ce395a57e414f85237cf8f1be070309ed5c3078506 c9d38c2adeeac618d04c0be2ff891dcfc9ada7588efb91f739188a933e1d37bf 632cf042106df1bae0cdfb79c578e565014cead123bf856cc0c14983f5548a20 c0d45446e0dd2b017f528e46d394a88a2d5f86fc669ac7d489f1851c100b29a0 ee3ec82faf87eba944a57dc14ebc94f2445a8a21ee68eef1848d534bdd192ea1 181220351ff93c51e53fcd5ee6ec92bd644d121eac52bc163acb6948a7a4c256 7243c7113b1b5650aeeedb14ffd9b6ce33605907595230a06db01bfb510a460e 76598fb2bb67498f79f0f5b5c5fdb5515cf2257aad324ef1a877a296e2851d38 f8995123e527c4956247fc8ab910206a9ce8736563ff0d4cd5d00579d60bca51 650af348ac4d6422c4cd74d924b532d4a28ba95eff24aca4b3cd1d28e207c7f8 906f8dd99aa04c135fe46c1dc8ee89bd768df17240a70f1a96382defcc180b9a f1b6fa57a9b0a2440bae0797790f91733970b16e742e6341d95faa40c583dac5 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c861d310425acad463ca33e6d91c0140b6f0e74da6fb28cea2f3d239ca405045 569491a591c0e360fccb9ad956185f47d3fdcb879126a210a4466c69ba3bbb0d fddf06a57c276f4f937bb7f9e7be6e18b79b2ea738d2525236f5688b1275defa 2a9717e7fe0d233888d9908928e88ac17845f1872651acc3b3ee4be716ca4de0 969955d4874357c8df5587e5c215e2bdb43c2a2ebbcfc1dd6e5c29e61b75e968 944fdd444c2a599130d281379f129b8d4dfef87943eac6ac124fc54bb5f92edd 3206a43a89893d8981f58d5b2efbba5512f6fee8a08eba0f34358763aadede38 62c82eaf70392d59325931d53f241e8f74634c6421f4bb6f06314f6d573c5f58 9c0b2b65ba23a2cab503235bf076b6c08b668e8f3d4e6a470c63f778d4a678fa d6eb18c27a31968f776fc1777050bec058bc310255df37c31228c3764e7feccf 8a9f855ced3cc64fe52e5c3bd27a2051e2359509e3b83cb7f001d48bdab2803d 1cbc60e2d48cc7b5d8640501b046e5dddfbd42cb78bb3dac3d5d420a0c95e622 a2162b885b8c42d46f7497c5dc0fada75b78501ce11da2edf7c4703ab07dbf2d 29ae6b194e3ccbed30c6e018287be5cff486657451ab643e2b334fc5152f0e2b c9b2b3a9a2f70d3dcd40e3a17cc304f09e3732cf845d95881d9c88d1581c1f49 f176eb362ff48e6dd61e9f92b863d0012e519db2a6ab48bce4cf772900a58b81 14161e629c9260b342c3f0107578d77df797f91c06b8ad858190d331c393caba 1a9b052335466995ce8a45eb5994e3b2b6c418fe95fbaedd3996beb90af905e7 fc10c05c6b6d27e7fe6cb67811038457b78e4f7c0b9e5f19969050d985790819 c35ea28af548854dd81261326b005e 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 3 /Lambda put dup 10 /Omega put dup 22 /macron put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus 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 56 /eight put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put dup 101 /e put dup 102 /f put dup 105 /i put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 115 /s put dup 116 /t put dup 120 /x put dup 121 /y put dup 126 /tilde put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueXX 5000791 def currentdict end 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a38e8240f0a5ad40bca39050ebfe7f848d3ad45506565b81e033aeaaee140f9e e65fd5966e1c456e7c2413021b804c8a832c7dea4cd58341e6749f4a07bb4418 02ebfb8f9b48888b98c43bb47231 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 6 /plusminus put dup 10 /circlemultiply put dup 17 /equivalence put dup 18 /reflexsubset put dup 19 /reflexsuperset put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 30 /precedes put dup 33 /arrowright put dup 41 /arrowdblright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 56 /universal put dup 57 /existential put dup 59 /emptyset 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 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 18 /theta put dup 21 /lambda put dup 23 /nu put dup 27 /sigma put dup 28 /tau put dup 31 /chi put dup 33 /omega put dup 34 /epsilon put dup 58 /period put dup 59 /comma put dup 62 /greater put dup 67 /C put dup 70 /F put dup 76 /L put dup 77 /M put dup 79 /O put dup 80 /P put dup 84 /T put dup 86 /V put dup 99 /c 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 108 /l 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 120 /x put dup 121 /y put dup 126 /vector put 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ab2e3310f7b5939ba0a0cb04dc4a8c954974541fe934901312b28d4dc1160a45 2125a25e67d2d4a492dcc893d665e2948aad1ef961d8df8f22c2051236cf613c 8fbc11355549e890cab3b51734dde46536e589fcc2f0933aa6cd8079ef81b27c 0515ae73d7db6310178d7478ecac958625ffcae9f1bf13f8fcaaf5587a420da6 1a3dad9858e967a8e33da9b6796b 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX6 %!PS-AdobeFont-1.1: CMBX6 1.0 %%CreationDate: 1991 Aug 20 16:35:30 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 79 /O put dup 80 /P put dup 107 /k put dup 120 /x put readonly def /FontBBox{-49 -250 1367 753}readonly def /UniqueXX 5000764 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be1f91da738298311c1360c8158d61cb73427 3c21bce59b0f2e1d3c853547e19715d3d570cef600ee263180d3e7a38d50b375 8dd0cefd9101435d4abab6d6fc24a6e3b161ca0bc1bf52f71a10b608905028a2 2381239adcbc00060d6c1e7ef5829015a5afa1d1ab3473c01363d08e71452ff9 fc63cff9ce95e7afaf4e8a1722bd03fe5dd4048821cda268cc9ff1bb832da53f 174d9e4af1ba9d2b589b8b429ce7368ac70433b0ff5cc83a19f4048fe72e7708 d7a8660449ecec43435f71d329eca654dfd2d005e7e97113ef3fdea187a5b834 be9e2c464878232f477bedc0c5ab6629d49e7fcb218c5353d120c77a5988d359 3721bf0112b242344df6bb8204624de09d31e54700f4ae64872a0d8c6794f65d 5825f1dd4c2a95c1bd09358efed666b41467c451dacf553a7219a9b6f15c9e05 53d54ee3fc6fb3ab2600ff64cde6edc96e198f0655a6354613d9ee365ff117ea fbca446ea8a72885313152306723dfabb26561a0887f00184c5639975db79b5f cfb46295842fd49f43ca585f36c6322f3b062b09630dba1e9942d3bd7f998163 731692475cedfafc8c07dc96855dcadcaef131b0c7545c37f5b5e82f34fad7ae 9da5183f3bf5fc3a9534d0803e680198feac773a9b4cb8dd21574f71a5132e84 2bb805b079d9ef80aba443cc74ce606e48e3ebd33a3aed020ea919845dafc115 bcbc62c1bb2867d722037939b720ea1c2d87233d0c3463b13e48e2b08a45b825 4496aea4d0caf8afe2be8c52ea28d3ee8a9cd6410cfdfe1f7b2f2ee631bd20b0 64e2312cf06de8aa4e46e1286864fb0dad8b85ccc2020107ef45acd5a8e7fa08 1f1ca294726dc0d287f3e57b9104ca9106bcb78475e4ba3818be3d84d558ffcb 5ce3bba032ee1e079e6a53a6dc18df00cf15138e79b3ca2d53ec633ea62d5e38 84ef5c94e4d9d4a04fd82d259db796f9e00b7d8ee5cbd79238929e3cbb5e8bc7 de0a156cfb2569dbe99514796f7ae0d56ecdb50459b7651a198d5ddfaebf06fa 3fdf594369e5054c0ee102863dc972b187e9dcf97345523e0ac520f114e7c290 f7cb33de8dae2cf0f77d78fc81d7bd016e0796545606324834da9a29567ed471 530e5699e50b17b751859150b448a97edc62a30694a9cf59987a9f302f52aeb9 167068d565ec83d58ea0547b729351b17e62110f0ae7bcb5b78602b76b014ddc 33f735729832ffacce5b8b068adf0557a4f1fa69e5bc23811403cd2c927d4d3f 32ade58d0d156bb405a9abe51705637a80397002224e8654be7a6d62f2b2f219 326180f18a2b50e20a3a78fa171158bfced83bab69b24a668702c927637c35e9 cb3c33a33b8df29869544d33f758606e3a835dfa9c392031c205cc63de3c860a 8abf514274a5aca4779af343cee82d68ab909cb0592fe097ab4261ced2599746 cc14ef9bb2c098a4105cc3872acfbd1c2c342bb98b9f670f5f94ab1d835d3fc0 332cacb425afdd954d1139a23bda622a3e7093b4905dcf7792a083e1b964babd 2d38e00a2d0f0a55c92d2184b41bad8140714101ff31d03bfc2b796565bff754 2f79ab6f2d3f7de404041661caa9aa13710ca486119e2cad55e769560c5941f8 0374c216b8824e149b5a4fe095687d5285936c056fe4ea4581a8fd0cd64b6cba 48a1a548e9d05503c2df3c47fe74dbb8abead0acd8a793950a5de7265df67bc8 e6d6bc5c8a77f6fe9eb4b933ed3faa8297f6547ed029bcec1f8c2e737612af55 161ba418e67df3d60c7aa57b5b7a5ad97830dd4a222422447b6b736abd96f01e ce8f805fac8ca5e474b974f0278747c2cea078c1cbe94f32d7ed352ca50af9b9 9b68d58fbc5b970ca7b7a275e75ec1761a12504dbcfc20ac8e8dda8b8d395e81 0d654a2ea8847ceaad59499d4d953257de58e802a2935a01ed3a8d957cec0b6f c49d16c26b8535bf0518a842e6b9c14196c4fac5352ad8813e93a34b3c439c2d e7eb100637f7e0ebbdcca7ad56501a906759705e20abdb6aeef6ef79a2f6870a 25b8c5a13d5ba1850f53796fdc8cf04bb7a938da414146a68f65d2545a4f1d 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSL12 %!PS-AdobeFont-1.1: CMSL12 1.0 %%CreationDate: 1991 Aug 20 16:40:41 % 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 (CMSL12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -9.46 def /isFixedPitch false def end readonly def /FontName /CMSL12 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 68 /D put dup 71 /G 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 114 /r put dup 115 /s put readonly def /FontBBox{-56 -251 1102 750}readonly def /UniqueXX 5000799 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd803271dc8bb1656323089e9e2bb50f2b95088 806eb7b8c434269c8a30d52dd3a6e85ebcf727aecf5607b54ac3942d00365274 9fef6277965f241846e12c5fc12f452046f9c7df8d1d31e228d6011de7485f19 4de778c725bd819dd7c91f2cfd0a9994b0ccdf95f200c0247f2c39f74a6ec77a c20f4789830ad04647b0784bd6740b2c56e2e41da705d1f709b16b92abcf7e4b aa92321eb28f528123e04dd7a74742e176a447a518a4aaeb0a4d3d5c23003c2f 98ac15a3b49f60efe1fd02bfa0c4344ea0300cf33a57ae113a16baf4bc852b2c 0ae48195b9930218490abd06f6102cd3314b61286d6cf26ab735e56cd7cb2cc5 07dfa75f1b0eb40953064e80654dd9ba6aadbc499f64d92fa58245e6b42337b8 95ee240599a523542c9b75d02457f4206181f124211d7785b665c338cbf95a83 20beecde03ebc387ff69b8ea66cb977fc926cb9ccba1125834e9b45e838a9509 28da66519e8782afddd42da1a6fec0293d2c54bf37ee6f265ba589094d3170f4 74264806cb095940e14f8573c66f3695bfece5550daf89b3584c412fc8bc3666 48e689c3af3cc9671a951f0281aa81dce103dc01d718e2982306e185ffc8abef ec62b3cddd662b7ed2233dc6c11acff1dd39c8674eb818de0da9ade582824bf4 0a475de261046f32f0a1db2ba342df06062584ece7fdebf26e73d5a70c65a7ef 509eba5ba5fbc2d1c68161a01e13666ac624253e70fc8835bca2263da9a43277 bc7e6b1a6cfbd3f82e55d79cef1d4cc4744fdcc97d0e7453f9314af146e6642e 9645821305f50fe0e4f0f6085888adc6007d613f23e810e65fb05b53e1b98cb8 1ff9ef4b3af7b19f702447cc8270163f642f3ef1e195de5713b8cd78ceb42c76 b2fec84aaf2c337f7da8e54b86261c866a8437ce336ef50630a6bc0323615a9a 0d11209de96e3ad6645fb954bc4e9a3534f418e51641bbf97156e652a0d94c53 e42295cca604f6e5ed9492f672fc22a81fbc1ae139ca1a6b262507b023d8d640 f9ed1fc41ac0eed4b6921acba5909d0d70af511eadaf8495732de84760a362fa 9acc49340ef0666d5aabab453f23aaacf47e09232754768b6f589173852271ad 7930a0fad549551ead375ca82047af0f6d4337e354effe7c89be30c62fc63933 1bdb8f659bee0b1690255e52f44291d7bf315a62d2af06df7f8875019466272a 1771d3021bdb99a9db8ec8b77fd24fbff29534467e07a8b8b7f444ea9917fad2 49917a013338d463b218d1368eaf513d28515e7c59877f9389dfaa40fae09a69 c244caf5383918a8248cfcffd4ddeba28a2eeaabe5941caa13eca0afeece0a81 0d9a8ee9baeba6f7c3470880bdef634f625fe2837fb9f6d72a9e936ad8c4408e 69c5fd69920b07a9e74b12339b1084aaf5255b1827c5f35a7aa4097a62947f95 6d25911f7330c1cb1c6712a1baa2ccaef8df50c998f00e76c99f66b34733621e 2fe86f1086ec898d93eec69e02c3b19b92aa05a4dc799e9892e6bbf5fcf15c44 97ccf488826f3eeaaaad9fc6aceee3e73dbb81bff711bdaa9d62bd973ecbfce9 792eba88867853317f4885306f75671a278a3eca7abd8681028e874051e09f2f 3bc1628b7ae11261240582159aabdf19385157b16755aa856229e00f4ff607ec 7d33a15679fb74ad56404e27848fca8d8a3fab7f87cba5a7022e55f7122fb853 961a93c3c9e0fb925504c3e320466360110d57d235c656fa5b7ee2f33cbd07ec 0c4241bcb843e97372a02c182cb56338fccd284a66297ef1bad63515fda796fd fe6384e0737b649baddae5149a2d3f3a114489c529825f68ac4aa85f456c037d 76c0f8d9a7d6491ca13c158c1a4917fd9abe78bd0876a7164b7da40001f0568a fa6485da2e7efec90add76d16b149f6bd7530732930c93875dbe7c19b2ace724 be49b67a8ecb606ea26ab2bff755b3473735a7de949d473f7245ffc7510132d9 d64acd6ac28f632314956c9fee06f61cb7de7cddc83a6e2c66afe63ad07bf81e 7939037392ce75d67abc798c9f20b745ef69f377b0c6f541032477fbcbef8c7c 18fd250363c0818c865705feb7d2494a873bd045b304833038bfd5788df36515 3fd72dce8997ea83cbaba0a34591530285b18e02e7010abb7e306e7ae5e4e7aa dcd8e857ea764136bd417269321ab02f4954303450c6db438b29b6dc90549f42 063759375a060526d7c7bac43a17c7ce922804dfb5b93909c7c762521bf604da ef2206b784e7a9a88e1d9b72693d4c3db533dceff7af567d0566f3b9f25c9907 b2bd4adf67225e560bfd17498d01c3991d8f79be592517121840efc40ab34110 c74bdaf8c3bdf3c3e2a1714856db78167f5ffa2258ae5de0c1eb0f3ab0c51935 b8973cfbd69480ccbb894a1fbfcf51be525ea03eb14c2bfd44a5a739480ee412 864a774bf9915c7cedd9490d2fc03854a2970c7387ef794dbcc1a082477dd0da 0fc92005780337ac004ac1b72f52e47392a394f8e6ac2dbd2557a7c0a5db590d 8a03d91420a259c38b73e58f64a5065a78524f7bf50f580370ffcf18ae04b579 c6f177c25475950f1f1fae6df0ba8970894ad21eb621f350d90b079b38e947d6 02c3bdce4b8f70fd09981c56e9d5a6d788fc06842c943fcbd246c0f5c4a336f1 933ddf10d76d2239c28c5a4796dfea5e0821e9f07dc7a45fde16f2e874782741 b628f49d57eea2e8ed958d8422d2a52dac38d04b2f39193f9d6adba4f0730db9 75d8a556f43e1a8b54f096c2f22e5e48798ee0eb11109e2ed46674e711a0cd36 1cd9e924c6421f5689ff2ae9adc08623225ca2d9906d314b1944b645881a248d 0125996bd3ca74882752fb69dc725a9f918017ccd0cf8bdbea142a013fc52454 00ee51b494ee8882ff1811a28d92fe9d948e114fd5fb941ffbc4d21474b291e2 30678357905beb1aeb3f473e92fa9edf7e938b3ed1ddd7d08f9bde92965daa2a c161dfd0c2e27d015d809b528f53a70d08fcc12bbe5cd55a80857e0286649f0b 2c7af9471b2ead744dcae8d65a347f5632be9960cfdf404cb7c207bbcd75f2fb ae1a6e723c75877d089c71780887319383ce73099cf19b3304758c3b09cdcaf6 8b76eccb78a167ef95d604263778668165001cf69e76bddb8ef88e284afef7b3 83ef2af7f2568c947354d663f65f875aa8edd463150de22e84d277b979af1320 a58fe2fc3f186b81fc169775a9dc25654dbb3cfcc4853b039bb82d91e0336642 5739d5025cd5d2e8daaa36d25c9962056036cfe9705f1bf8ef2fd3441cd8cebc 01f4d4f54b16af25ad6a54edf0b0245ecce8190af5f9c09c9620ffc4672df9ee 56f451beac0bbd0791a162928e13fa5fbff208e755906f1df95745dc9f821134 526c394eb9f83b60d24b3830771abf270f4be558ecf1477edfe6616a839cf5b7 d65ee09d6766d9023cd1cfe219cf17c9a7f192abdbc02c91cec574935dff658b bf290c0e87010fc4cb744f71ee2a3fe5db45cc7246933267c0676aabfd10a26a e876c258d558d3bf642b65e99f989efef1d50f78766ca5d0d888f838b8e2d0f8 b3623bd4cd46072f75ae6854ed89d0f272871447b2196283b811e348a331c39a 04a29f64dbf993a785ea3d771a28f84fec6ae0dfae0e4790b60177ffa4b046b0 7e66e2c36d1cf4370aad5456f79fb752a18d66d31f74a1a539bc6467e2f81ddf af5c3f9f1be977ae5a074f0dc05876daa76ae536dd044279026c39eaef6ba51d 1bebdd6b8b80121dc441eed90890d6343aed2283f72912eef9c8b49978d82ad6 430c158cf683670e092ed36e5ddca44aeb7e994205f09c878e64e228db427775 5e36e188c1a7b51b126cb364eca407cc6ea9207df7404245adc89397653dddc5 bfdc6c3467316b57e375ccaad5a425a9d4d7ad48f290d34d66c2aedc66d57d33 f23c5865c73e032eee1e1689ad2ad16001a2f5b5567de4a09036796a13522491 45e733a40ea9a21b0349ec65df11b3e94bc943baeadd72802628bdb813c79f79 765005ab5159c76925381c2cddc97c64fec6493b29a8d29257755b9f545d96b0 c257b430d88d3d05b4bc1721772d8c63953796c151208a4e19cee49b6cd57994 dcf330b51bd3bbc8012bfc5821bd79bde4fd1749a36beb1e2606832f61cd7563 45ab9c61211804aeadbaa096e0a9e5e1becef1eff55e29fe517cc76fbf1cba35 58b75b7a373304f1af096d9f27cc9cb60e05dc4b0de613858a59c9bba1601bd0 ee2a86d89e8a9bc41d6217bb93fbbd669e6d303a9557c2c9d73ff41e209fb687 56ad91eb8402313303f0175fc35075e24b04e5b361c624b5bf8d4beb9be74032 db45371fd150f9170ab1bf1a7417de7154d2d8f4f0173c30e496ebe36141a3aa b9f870bbc3ad284b10c005c8968131a4196b103798b21099 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 12 /vextendsingle put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 34 /bracketleftBigg put dup 35 /bracketrightBigg put dup 40 /braceleftBigg put dup 41 /bracerightBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 50 /bracketlefttp put dup 51 /bracketrighttp put dup 52 /bracketleftbt put dup 53 /bracketrightbt put dup 54 /bracketleftex put dup 55 /bracketrightex put dup 56 /bracelefttp put dup 57 /bracerighttp put dup 58 /braceleftbt put dup 59 /bracerightbt put dup 60 /braceleftmid put dup 61 /bracerightmid put dup 62 /braceex put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 66 /parenleftex put dup 67 /parenrightex put dup 80 /summationtext put dup 81 /producttext put dup 82 /integraltext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 113 /radicalBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueXX 5000774 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 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34583873d4b49b72702dc9c0c771ef260595176639c9e101918851c997dd473f c7f4b78f3ee8087af23b8c759257682da71f859a1aa577bf3407d1405bd37067 f7165793e6bc5275f467f0f912eea1e5fe990c09ed39aa7d5d9665bdd270ac58 c1268be2f06c705a08ca9a8bf3a53874d15964b23ab3391efe08957c02a08963 086c52a26931dd88e59257f8bf46a8228ce3717d958865bced7db671b54b62a8 2ae0166dfef579f7a26c27f36e788227e59914249dfa92d818c94a8a1d5634cc 6376ad3ffaaa14a9c8cab80752bae702ba496ed1cd3b6239507cea5ba5101c18 7c317bf2a43090627414d37995e56d82a30d2d01785f129c6ff5f0dc83e981dc 181c7d1679639f0fa0e334f738ad151dde13de234737ae86cf4376b231af9571 a6bba3d1c798f2dc434ae4ef6223240187a44eeb9ceeb68cba3d0742000b5718 7dca30aa81c3ad38203d319087185b3ae789a3a5d6c9b5ee5a4c990b00dfb23c dffa170cb318d88001b0fc8f70bb8c88ae4bdb725fe7b8be33fea5623fd617c2 0bc8d4c4c7bb693fe73a5ee8ca48d8eb23e01a1e1078f561724565db1ec6f5de ee27c60d9f986b8e0297ba2bef28a9915d0c056ec13de658a7646229031f520b e64bf11e38feb01e4fc31b80603eb6f7dffa3192f5ed8f7bdf275ce2d0083463 fc757ca7e810ae2df530fb9432c65eb2a047ecdfa385f796a5f62d407ce7d0ff 05d15ad1802c4de3ef4dc173ec2eea68a492a087c03fc2e73642565557847a86 404ce284b0f0116742b0bfea8b671908d3d068a70989742588e4554d4b41d0e3 a289887e6dd6b64e578c0af66c71a0be945a793815783ac62529b65880ebe73d 6b51a634c4f68f08c32d0b8b3d8e361a4f024d0bd9cacf54de4ef380126f2945 f28f7df9c29a96b67dbe3af2b5cb99702c4a39b4cd1755af92fe15d77fda771d 4a3a3c3ca0787578514fbb80432d89271ad71aeccc487cb9714df132cc273d8f decbb88704c98cff5086866d6c8348d014618c7801dcb90cde4e1bc4abe79d2c 2543e668c6303c05fb4daf875f684a001db0c682607bd77d8936ca8e712dd9b6 e8c1a76a2757a86e516986f39b47c14746485f05fd4965669b4ba075cdeca6b2 719d7ede8bac28147e233e35846477899d9cf3353dc0dfa1a646305d1f705392 c1e6cd02f3fbd1fdd5164ad0062aa073873f0412832b841447cef4369f5b39aa 3dd735d0f1cf6a81d2637e18a509849d6e7f289dbffd8cffea502ccdc26b2874 9550e9ff94970e83ea8bac46f666c9960ede07b66714e96260e2f3349a8f80ee fbfdbbe018f4d8329b51c001a1b3748aa08ee3c7b413f92dbab258712cf7edc4 318fbe878db2769f55183d309ac623746683da08e03cc2ce60286bfad2f6f552 fbf219590cbf8c46ed1a2de4a8ead19d2c625fea2c4a8bf4a2ee728ca9421024 fd08a3a0e665f3ef0aa656a8112c17ac9a9fca39205d64c8f647adb654d74e4e eb64b9673616362d7806c0b3bd0b7117d80c4c7e0ea0436b0c5db46900380823 80 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX8 %!PS-AdobeFont-1.1: CMBX8 1.0 %%CreationDate: 1991 Aug 20 16:36:07 % 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 (CMBX8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX8 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 79 /O put dup 80 /P put dup 84 /T put dup 107 /k put dup 112 /p put dup 120 /x put dup 121 /y put readonly def /FontBBox{-59 -250 1235 750}readonly def /UniqueXX 5000766 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be1f91da738298311c1360c8158d61cb73427 3c21bce59b0f2e1d38c2fbc0984ce6a04f5d3bc84e2601397499a981b200b307 771141805f1f5cd7b12befe020f7121ec103dc2baf17f0aca1d0740c92a8dd61 12fa9c0d30cdae4f2af757b9f713c127d6ca02ef315205b81eff629104b21462 17d2b0f823133bccc80c9e400d6840db5fdf828f845313290fedee040d34f309 8f1b5a6b8941c81d1e607dcb310b14169cb433329994b4d1ac1b491e7422d518 2a8faf7ed3cbc2e23c58a2e310b56da54f51360556d27d58783694c51aaf62a3 a2db91ac8df80303c35cfd503341005020a5cf501fc0b64ff4f497951df6cd50 f00fafb04725d23bdd36616c7c06cd9052305682f6b6e521882922377b4979a2 e6f031b91b2a86d8631e81de17f90158ca05045c3b823da8b2d2aefd37a32728 45bace7594ceaafaecbf6140212789b3937a24cdd8dee327c934a9b0e328fd83 a1b535d9bbea6f0204ce5ee74cc30769734d4fa3dd0cac3986510c99ba062868 9285146cbe64caa89cc2f18a61d924e7f312fa189e0ae4deb913ed1cdfeae61b 4e0fe25f2775a8b2edabadebab4eb2ecf47258c1a77101c810cccd3b06633907 f765fc80bb1a2527f8a59c52760fffabbbac1d65a11708a3bea4557f24536773 d17dfd0eeee4e22f406e04765cf582d6ca4414f2441b6ff463c56018e44aa2fd c9af3f0480da99856aaa97e56b4bc28d9a74e6d74e5ee199932ba9548c9f5a24 4ad6d8509e006d51a96e77cf6decfe21ff0391441c8b95ace9dcb2d93694eb75 e76e0ef6fcdfb8a73c860e09dc49dcfea499450162b7373cccfe5e771eb05086 9552a2ff03a1c1c6a08405ec3ad93fa9c951b4aa746b4c4bf3f98a916ba7134e 93d777c339423c4fad91e1968b981744bda6f085de1cc19720308c550616f557 337f699747de79a218f2b7e3071958610108360d54fb5dcee09272bb7ffc60fe ec4fcaf59982e01e234ced0e740e6a1e62968a954359482482758f46ffc6a2be 9a934ba0ea1e1fe8571f834ebc616258e0df743ecf92a6620260266460260b2d af5b2498f29971a043409295b63ee35bd7a755a77f9fa9342725f7e764d30d91 2bd488d8243ab45ff5e6b5bd032269662e3e7148b3b392cb2482ea64c4c2a411 f6bc21db08e526ac8d5e90bc331a475fd27e96c47a7f595a69d1a7fa47ef01e9 1033274b12b1494803b79b7da498d560af4656593a3a37398503bfe6c2de6751 5dbf62e6a207549aa7685ef75e63e2eee1033e6dd84a09c59beaf7597f09cea7 f6f440b9945514b937a80fae8eeefbc361d45edd8c76d96a7c337284bce86395 d4c8da19187dc05c27db926ec8ec1bd6af267a489bfc61bb90bd8fbbf413c163 0c02b1e854a85481b9b5d34513efb224ee62b0438ad52e9d1c370007cc521098 06d9e44f42789d6c91399b979a1062457d109229bf53adeacceab63c9fd48969 d41acf58c048f74864cc8b203ff0ccb5f6eec11aafedae78bb11c7bd0105dcc4 5f4499780cffdda3df7bff74ab8d0b6503f4a3b48ff9a64ed105689d3d321353 74794c763102801823e9a0c3112c6ba4cd60d74bd658ba77acede3054b76a188 01fdc1cad60b8c81609dae432329a17c987b556a034efc4a0f15d8dd82bcd673 cb47d2791e5ef31ae61ec21ecdbaf9897d83a3ee36bfaff2ecc290d79e0e13e4 9460d0089363278a92da452470b04974f24e153bd5b4b868e2ed79de0cf5f3a0 7b22aa3eca81a480f420330b1cfc2ef64527107a3d7d64f0cdbd0fa8f703147a eb962e80e243fbeeff6a3b15f862bd8a1550a5989cec599bf7db3576d9689e5f 13ae0672167e25863dcad0d8ac7cf21d84045ef6fc75d33a8b1e20997e7317a7 fff58fa6c05124d3d3d7850d6940a310066b8942c58207ebdd5eefc5601b8f97 7255eec6b9e0aa68f36c2164f2ab8eb2a94a1a01f14622c5b9a62cc99f042035 733ed55354b70c03bd37a2f55a072a6d6cab569b6d4d3a55776158b85f56b099 9a6448111004f06a1badb819e62972f9c8fc4db3d0a05d66c9747c7eaf4ba47e c02a84678cd1cc97161a2119a44d48a5d26d78fcc35bc6daf167727417231a38 37c7873e35dacbea7018c6359c779def18caf4751649f354586b4fbff42c67a2 869443936f5d0b3e7ed81f66b0fab3184f3adb7ff7b419355b6905d93f7546a8 14b972ca06de54c50b592d6335dd4ea1e4348b2f6ddf89bf958bcf1d33e73b95 0491c9f7fead9041d1d52cfd3187900b9c3b541a59cea170af8e683cb50ecf5d 3eb1d3143241a1f4944a19f049fcea36171e6e82099d63dc6cc240ef59c21c7a 58d4fb1e5c39b8395ea0b93460a3256eb4893f4fdbd68f6d14b5f848b77dc815 e91ec70ab9e42b70ad399c23661af4fe94f03ed41888749d73e91f2a81376380 5154177a00386dbd3271569aa0829e771136e46519c3fea19581dcf3bad4e2ef 8c40b8397b81085a89a3bac9dc01be84aec57baac0a8657cfa9674f92ce389ca 8566ba1d0bd8d2c8a881e8c568b2481b1fe4eead7cbbadd60ccebb7d0ad116fb e4d101cbf5a623a7285628efd902b409e98adbe47f653933213088ffceee6a04 77d849251488fc209786aed3062c02ac49a8aa967c55bca0bff816885787cfb4 268c6d1ca54745293baa7b9d27a17c656b02 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 11 /ff put dup 12 /fi put dup 13 /fl put 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 65 /A put dup 66 /B put dup 68 /D put dup 71 /G put dup 73 /I put dup 76 /L put dup 77 /M put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S 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 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 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 a29b69d1b622b1f0cffc58291248759607d91150cb0651120970dc9f743bebef 44f6ae92facec57658999c8bf01f60611560043962c0695e1bb87d090e0fdaab 659d6557208a660be9aeef8ae55f71b8bc76394d792967e41bd8023b6e7f8c3d 9987d5e54c53bf2bbb32855b946473b44833bd78873df8e4c7e668ffff25aabb 8da4148f744b9d0524d6cb8281247b3052cc5d66f3f1bcf8a56a3fb446ea587b d8be30257652cee035628aeddfd8283b5ad21b8b1480ad24d898970ac8a1e33e 543ce4e3c48f9ee621c5ce3e8ae930f890011b630866c988606068d9e92396eb f51bb3db204505fc3d09d4701ac703a14d92bbfe27bd7411b363a88e33a8e9a8 025ac31f21e73dc52b333b918d0b1270c54873a490222d041a66e0192612be6d 2c88958a8f70c10390146ee5091df020ab97675e8e6acf98b28b539cd0a20d24 a42f535ae44b8f5e9290b38fc6b02e1bbbdf1aca96558c2dd5afbec53f03ab51 fbce958ed4a65b17fbc03f9defd4aea4a15229f3c320876235f2d8b69bda5f48 03a2798790d35b394c63295997dd487bc603cdd10d18f3e3fcc7f17d2a6afc77 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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 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 be30812eae30521c3c775adf1959595a43d8a9c4e12d9f9ece27a0c9b5f45e38 8e26a39cb0ad4e25ab8f21b25ad1b106ed22af7947ea8ae4dc17b9facbc5e22e 89a91b046ba4cddde69c1997fa7f4cd949ba3ac4b6e001475daf165b9432dc70 6e30aeead8cc342d94b20a4045a331175655d6aa279627efc0fb53a4ce66b68f a45b5765f0c1368c04a0907c59a0519e67505eba499580eea7d3467b18a5bc46 918b6ba40d310c9533800382fd7b2a7b383e1add9b19663ca0010aedda4bcb1b c0e384d6f24c49c213153391df88a0572ebae9f22f3028d6d99aee18c1ebebd3 460db69ad9f57178bdb95f1531110f2d8ba84dfb84fe950ebd7ff2c659919062 de4c311b47f75abc819f2c62aaba27bdcfbcc9b21423f2661e295c9a902197fd 80aca2b3b3a7ed580c8716929a800ca808712e3e75df7c469221c137c78b2463 10480d4828eef1bdc409c3d129b3312df2128513b39a04c102e4fd0353597319 5ef35c648a1a264a2ed2f0e42f494d0f517655b1db6c049bf006d4722f660873 25fb1b38d3587daa0dec6f393bfc0cfc0bcb3061f031f6977dcd3df30b10c115 96b70846df24886976d82f9b2ce92bd67e5f5095265250981b1b6a8f7ce19453 6beb108eab5e14257895c061813497609b30a1574945d58ceb0c4c36506f77d4 da430264c594a19258bdc989e186a458269aee6e1a6776ffe268d6ff38fcbafc 45e1377e9663b04ad9ecb9c367733d023926bd6a7daac4e2c1ac82745b7d0e7f c77a5a366b1af4edb925ef27265c08cd0232cac522582d8ddd428757c7a23ec8 5bdc839aab3729a7fff614a6b28bcaf86694b91c8cafab07f464d45485aade84 30f3dfc678233d57ab0408349fab2f61d274409803a2ef30fe923d24335af549 4b09fafd181bf764f1bb93077fa2783f4f0d01cc78b0c30b4cb324fe3cd216a1 6cc78f457cd79de2b419829fa22a0d97e72b12d272aea50dae4b08d393f2712c dbf9ae3d107d2865ea748b20d8058a30a888f782d9fd8db8ab63e5566cf634fc 5c504acb5297391b50cf1ef2051c9418578f287de7038433122929047420d862 fd440480509e0250ef40da0cd1ab3624130477cf7eecab47005d2fb245d9166e 4cb67dd49456dd4946d930dbed44de18cfc6e23dc1a81bda0b068f1fa04d4961 a044bff53fd1cc015c2b048e5d2d8c1f73ddb03b74f3b32f63c3b3843abe9307 c0026a33d9ff6cb8bc8abe4ac8f45601780797322e681de2c9797e645e7ef3de 6fa265b5b3d03cb1c1ae24af6caeb85844a4118398baef678cc98615d1212f3c 5f0144154989dc1d3cbadb5ae4fe401d4ad046800ba5b302770318ae9cdcf84d 99a79aa48eca67e9b3da97acd9bf0909c3b0d44022490ddeaec4897b7dead22d a63c4781a88fd3c00bcc6cfc379e129a91e6bd3b7a 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 10 /Omega put dup 22 /macron 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 58 /colon put dup 61 /equal put dup 101 /e put dup 110 /n put dup 111 /o put dup 112 /p put dup 116 /t put dup 120 /x put dup 126 /tilde put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueXX 5000789 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 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218dd41b196d1384f5222e4b4598f1f43da180161837e4156c185b765f6220bf 8b67f4c8eb680181188d81efb106949a4d161cef69d711d91184eba256c59576 77e97def742a75019ffb61bc64b29b2ba83608efbee4f01cc9909694f3b05551 8f0ce6824e2174e4878814e3b2ec3e8259446696667f5dcec8c2b039746b9fa3 7d16c653947ed0905750a3e6be1414cdf4b7db5adf9a59860d1433d1c7cc6729 49247f27542415c57adae3146432b1203d0657507fe33569ae131efe4420ad39 d8f17933eb9c5f1350801115058d83e22d0cbf3aff93d8254b92d990338d23e6 40310fb63207a1d9954089241483f5febd13e07fe363e13456791b78038628d7 11cbf908d1ab59aaa53adccf04eb346d9706e156f5845992100816063286bc63 f2859352b3209cfaec6ad1b0ac32252ccd926f659900185111f25be94609c9b5 ee255e54a2978cebad58af50d745d4d440b68977 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 6 /plusminus put dup 20 /lessequal put dup 21 /greaterequal put dup 30 /precedes put dup 48 /prime put dup 50 /element put dup 68 /D put dup 80 /P put dup 83 /S put dup 106 /bar put dup 110 /backslash put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueXX 5000816 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % 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 (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 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 6 /plusminus 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 59 /emptyset put dup 63 /perpendicular put dup 68 /D put dup 69 /E put dup 76 /L 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 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 110 /backslash put dup 112 /radical put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueXX 5000818 def currentdict end 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b0c130500b515824578ea51ae00a96e2b5a45b3414dd940552b279e9be955dc6 aeaf65d1bf187ecfe3f0e9271018071663600a04902cc23488009bac7f6696dc 423f8be4834c2278fde1326859850d17d9fcb81e9f160cbc2bedc78ef7f492ab 6a2e1cb59cf0ffe16e68 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 18 /grave put dup 34 /quotedblright 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 53 /five put dup 56 /eight put dup 61 /equal put dup 65 /A put dup 68 /D put dup 70 /F put dup 71 /G put dup 73 /I put dup 76 /L put dup 77 /M 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 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 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueXX 5000793 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306457c7e62c1d269f306bd3402e266de fc3b5e7d8a8d2f5bf0fe6ddd40d07391df4fad4a6018dce29a2b8f692b29f202 3a7c0e66de8ed85c14f1f8492167357f51a7e84cc5d92e0fee4d81cf7fbc8de5 2d2e7bb57142033993f9c08c315abade8dbc4a732e84e142d3bee51557910e12 cd8aa37c459a5e6b7f5269f59078aba3be4641a11ac48d0b625c8325b38ec08e 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246dd5e852818f822f98e607a8d42c1ea1de7674ccbb758698b0ce94ed2a43d8 fd5378183976ba740605d16e5def60d206276e0f90c56a1af04f5ff54daa71ba 6603e1a492f5d510fc8cfde9dca45988fda67f48b8443a7c1ec85c87fc4fa386 919f93f421c799417b6c28d959cff40e24b70b7a0cc7e3924b6e94b99e55700b c30ec57b65f2a069430758738f4cc30441ae45ffabc169ba215deb3b7e965b39 41e7c26a38f5b91335f7df844665a24f399ac51fd8b0c5746ccd5ddb092505a9 14bfd2a5175cc83ff39148420202bc64382ebda3a80b54b7446e51ecc530ecd5 ae1b31261bdbaf9c8ef9e07d8c0956555e0ae54bf962f672078a51cb7d0f1816 9d919822b000553ad626976f4707e310ab2ffb072738363f85c17043a4e71596 6f0f11ee412bcb651af9365544931f86f57960e2d5f911af93d94089561d899e 1d69a25da0177d5427e7f9e9d1f758affd2da0a53d71991121df9adcfd88eeda 5cc0dc4d6acac9932b4cb32270284e49fef06e13dc9632c9ee9963e8fff1f3da 27051195afe3eae74bf8e7d1f6b5cb82bb22b62e518606235961dd472f0b0e0e dc81bbc551db3c29536095b7474233d4b703c15fdf14405af0f342c5cb4e39e9 5a5a770f9a701c1a9d88e1340df50d89174e8fba9b886062f6d69493be9b1776 8deccd9ffac45c7c2a47fbd15e3060435604b76579ca4f7e3515844365d3f1d8 05572f1b0c01634df6f903cbc5ae138fb705537a2fea655e74046aba5a6fd76a 6b88e7ac61aacc5e5de7152f06dbf10e76ab4b11b82c1134f3f2ab6edcb26e82 66fe9bfef1fd2706d1c308deca5edcd7ef0fbba006f7b98485236c4d00623f8a 2fd5b875d616bfed1b2c542ead5cf04bfdc21b2555d43aff178ce2d2d0986bd1 feb683c09828876bb3fb75bf0db21bbee390fc8e70f0dd14eb92e9edabc4722a 69e94d524563097ceb7542ee0bec0d05ef75da9ee970b15f3a3e02f8f47db650 fb5fb46764f631f075ccef1246e2a6d66e1e81fed6864f5f7a586eab6829d6ea 76048ccffe51641439678a110c1fe45879b66dca18360c162d9427bf564d758e bac2366c2040cbaebafc0679ddcd6fa10ea451990ecf62cf99c93eaa682d47a7 7b4d6b6dca53b0b5bd93e31938a1b51f6bdf13863d53816122506eb578ec9bad a4725dd4d8dd3a779c44b30c13b5c2079e1e4afdefe24f82d2c9a463de9803d5 9fad16fc1b60ad055c4e34a92332fa293807951c810d391fcf9f52ceb7ee6318 354a6621112ccb8414eb408dc2ecc8f9b06b44da33d52b83873ecf4cda59ab5d 43232823ac3fd84353f635557e344ba93e9e0d8f125fb297cbcad5778e0242de 7e3cdddf195a9f22b7b8ca48fc1adcec480a20b64b6338c5825438eebb81d696 e94bc35f701227d32f189cd6e935235deeae3dfb481e2cd5bd2999b54e6544f7 d9a997f41dc6edfeb28245c33f872826cf8548d62812e52e66ec236a527ff6c8 f431b0f65c484e0863effa62e43bee1fe77b93ce64a30e918b7703cf6748efe5 a28fe29a666c327d38f598fd8e3e7ed9cb29d2aa9b0956bb59e7b96ae4ce6f6a a383912ec3634ac8ca65c99ae8f87ce13c772d4ceb81ed0d7e74dd3f6f7336b9 40082c2cc4b74a3458c42de73a01cd0936d0f94bc5d23cac9a49cf12bcedb1f1 dbfb1508183234451ae36e8d87a74ad8ddb223e6db086d72cb21242d60bababb 6eaccc9efcc3d3d095745119abc0745c45a724ceef691d0a1d551381effaaecd 1ce338cbbd9760caa73979e62638c2567e0b3faf8a9f3bbe2981c2d90319fed8 f349b7c37a6ab49aca56c3d967a322ac86ce2339b214b066ff8364e70e220dd5 4eb3ee4f278d088809d0512f3ee7fa183af7eb11d645815f493edff95e4f6a37 1ed24e6535cbddb931d09effc411a6f8af0c7ca7d366aa1603ba9b5a59301af1 5bb15596a80c3603b3e076912286748467108f1ada91164bbd646b82d9f20415 0926f2b82263ce95631531cce5afeb0f707b17fab5c6432ae2986641c32f32b6 53bb95bddbb30d714d7297cbbeeab2ef9c24c98024e03baca6dc13234f2be111 64046161f02df56dc700b07e60eb6407ac2deab09fb74ee6d58d658803f85280 1a4c575db2fc89804958513351d315d8502639b020bdb07f71f9f89d2db0e365 6f117273f43e652600cfc2f38c927d63b5e90c6ba9eeee8f011574eb8c8c3e42 d760634a7facf686960bc3fac9003781fddb672a76450a6381459a04c4989495 4eaa216c7b6652eea339b5c7fa5ea641a643b5b2cef1c259b2c42ea3a5ac8286 03abd8a1149121eabcc3fd4458a4eb56ef221b2832dd9efb805f72aedf3ad4fe 3f2582c405ceed2fabbfaf1273dd2aedae8d93f855fbf6e8dca560cc717b18ea c403609c877219cd58271a302dce801c3fe7b6f847a2a748022685fab7582c7b 798ce06cd58f83296403bd972087ee76bd96710fe289dec04cbd224734e90aa1 830ebb150027d459e632a38d2f433d906d255f78876d4c9cfc0fea06141446b6 76f2a89c50a41a68e0541e394d0d06be264e876196bd2a5c98cf27468454f18f b8edcc5bb99c0471caf31ec75a3936f0aaf8f0e5d01962c7772c47eb99af9b8e 9bbd0b69f6c3cb6f2a05d2fb9dd7a61b6c2fe7627d24f9074090bb061261a26a 575ae5f47a7a41c4ddebaaefd16c6f57514e37751c8c74df1c0d80a05dc53f76 bb852fe6ab4765b07da9d99729efd4747952a3e74cc5a8e122e6c1bbb31e15af 4b8ff0435cdf3f4388dc190b998897a59b44bec2e2951e0a3cb4fd306ef796ce b297da878658b875e10749aa58aced4d85052ca4306580badbabd5e75788779f a37aec7a4552b8c7cad711f1b3fae94981e64ed525dd62cc64bf8086f83b0f49 7dcbc58a0d87917839e35b72e2648e2fd0bf4f4d48fc4081e7420a2bf33a0220 22266e772af936623613ead05370a6648d30f5c4bc5457e8adb951b73bc7a3a3 353f24359e58380a21d0b537cfffd66a9f9293225b06aad15cef211b677b17b3 f1b19d54038404a4db4d2f77385d5161628b3b5a70ae8955fe42ce221d6b0673 ac3c10947f9588ad522fd88b6af7b1fa7902043b003b3f2690cb16856cd6ad29 037af859c49fbed2473c31513daf86da8105730feb6b11c66ae0999cbc886583 0bb3123af8cd80be12bd8680fbeecd71eac7e28bcbb9a1412d8ea69635f1b82c 7bf837be637c91eb40e1ed0f3aad655931dc324c04b75410632c5180c0818d19 571505f22f1dc951c66849573645764174a3298275d52cae11ec8e16cea75a98 934fcedcc50f8ab80f0b91ccd8b2cc2dc721cbba0362454f5947f3d07fce39c5 a5b71675558d7f16eb703ef118e03b7d5a0d268860ddbfccf1081f2dcc9cc0fd 72776e8619ad9046a628dc16433f68df2ae9aa1c369f733ad65f7a92df22a2f3 c1d01fcd50b48172bffc85fc7a5838ff65c8e31e9f83ac96100f2040b9a0c9cb 42a1295dcf2f2b82c80a5cbe2565168fd5adeaf9806571975c9201e93041d2d0 73348e50f36e7cac4c6db96b804fb5b2f9306ed060985ab0ca7b539b28354cbf ed543f15fd7a5742505fc94541216f645de0550ac2befcede604a55b910cb136 0c6d03bd5d2cbed2a8b4beb05c84059d8292d8327e3c14545a0740f024225d28 3ec9eb7c8dd9f4886d8e67bd3f1b16ee0a4cdb01c8d8e337cf6b66bd06da3b44 e0d5364c80d7983e399eef62dba2c2c92866f2bb252b0bd886eb69dde3a35b92 e448c96b80097a7937b862deca00f890c0a2a87583641a1b38b3faeae42bfa24 cb7567963248f3f9e1628dd02e381c44820f0d37f7511595ef6bfb5ad230018e 705d6dc49a510453ec3cb94625c24aae42dd6066d0b23dcce07a24f2008f9d46 d47e811e775f404c1b33deb1e04eea7aa02c0bc86c357eeda64bb45849c75068 6343bb316364e85db624c800a890500010944548096e9e5a577b08433bd35401 8746370873ad7e6e1613f347d5e0f6952539a9622fed10338188129cafd7814a 4e8dc4f74846782a2a9959c2482e0b38c47a6142004a1242739975e2eeda3a9c 64b9aa0e65b246cb629a8970dfa957325d4aa8874b26c39d277a438c6c6eeebd 3e7a5d855f6c8f70dc38bcab2907ce2fc4d54275751410db7905fd8ca0ce23a8 3dd35ff0af736d2e3a0983d3efb4fe49a747b3d2c07adc416aa45525a5bf0210 9d89df5ac1ccef373d892eb89e352fd1f77c439c09ec6617d0a02bb6d9f80532 bdf6ae85d19ea4edb499bb9224f8f26dbddf327867db804f859b3285ac50810e e10df241ca7be8f2b9a54ddf36c6ad768e96f395e2ce81ce8a0650d02a6fd88e 27b88a3bd40709a5526bd3ee3bc9c585ce7faf7b9a81a30af5b31ea95e83ff1c 0a67994530551f314b1d99034d3eaaea9556fa4f6fd051a79991e4c8d5caa9be 062e3b92490ede13736fa90beef624b48829395e53e2a9ef885efe6447776ecd 49bbd315d6d16d9b8ab03e774d23121a04e79bcb80c78bdd9c753260b23a82c6 c3a9e8ebc0b073cb12deb98870b6be8bb8d35b0dfa3df2e8185eef2a75bf998a 7f8e17f797f91c01c8ce9959a4a2fe480b0c64ce4946bff08b3392effba0716a 7a3f31d586633694ffb1c95d8224cc0b1e90004f8336c743e1845fe9a3361e84 ae82cbb0f081f333c845a856221a3a00f281b77d53dd43b16e1d91ca0fe1b008 4ac25a26b937642dfd6bd0 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 3 /asteriskmath put dup 121 /dagger put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueXX 5000817 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f73e9b697ddc0351763500e23c4d170ea8a2d2 75ad0b6b23097f02fa803c1f46f9ac10fe0e527c431b11b1b9a50e874896cca6 17fe3e255960846bc319ca68b9b930a45d5d0fc36b352c3443ab6f7793b38d2c 7fb9c409ab7980ead55e9f8f6b83c70088d1fa6d7f2214da8928e40b96334ad2 72d1b5db9d1355a579b26a8dac21f021e393d7d837a95deb8de92ed818a8b80b 98ee9cdd8a3c57ad7d6eb7a82f43eaa6ec520eefa7302e750edc4b495f70bba2 24a2182eec37f1a1c5dc8e20973007150237daa6bf03d498826777eeebe09014 ebfe05a8731e1a421cfc513d47d673e3d94d805f5d0668ffa265e0962cb5a75a 4a93d9b873912be368f3f300b584a7c5dcdc5e3f796d62bd15453c88ba7aa204 01cbb3dee5fa0a9d2b44dd122a565359edd6dab1c8481ee7c0ccad206bd3ea69 eb9028f90c198b6d86654c074b98cc73fc076a8d893563b78e999de97e8ac3a0 cb0b0bbdee41b05909141758571e9fd2fa666ffead0e00a2a2df28fced5bafc0 cebde1c84683c0582fffd2d4e45b1a8c6ae34ee3fccc062e93a0f3204dd663b1 0906213adfce1c6e22aa01eec403ee8def76ad79b48d997948b9a062a357fe3f b7067440a9c692a62dcf9bdd3e045f972d228a65f07d7c60f78a7df35ffe7bf5 82157f9f5f89ad65006863f8fc88109f95324b99f3d815b015fa83cee7a40fa8 1da4197d00be65b12a39e8ca1047ec8d7787a04ea06ac24f661025c03e32f52c b9c10247bcc6f7fac4346813a1e1f94c5f026733fc4b38ce5d13fbadcaed1b05 a6f5cee0ae6386cd8a1ce54347782a0876fdd994d9829ae3f421d9e495f85960 9d8fbc709c2519a6dad85e444bad9ee8f96b92c43ea6592b91968c9921c8813d baeeb41a80c44c203d920ee9ea2768e53e1b119ab37f8d4a8878895ac7e766a2 5ba6517116721c34b22aa5b3ec8d17ac1da99f6accdbdb3a0058010fb862b535 2ebb4eba91b4d2b2eb56474c5ad37cbcef260183d71ae3be0dab0068355c2032 98b9e8c5bdd1514a735839954eea2a01b0305a55da2f0599b90caac255d0ec38 de173b4fd18da3fc0a9778f52adec865fed8e0eb1e61183d4ebd963cac804a82 de5716377b779c4357870517f865fb3b19c3a1d8a17e23da08cf39eee9c3864d f58c6756231340ddcbaf8c335f 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 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 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 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 65 /A put dup 66 /B put dup 67 /C 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 76 /L put dup 77 /M put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 97 /a put dup 98 /b put 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c3440805f007167ece620b183abc3bd2aee26e2369838b05d3e9d2565141f940 45be1b6e793abe7caed35012af52e9a91629cc5542a0fd0edaba376e98e8ce78 e6b0a0ae81cda6ad12ddcd7ce044d48e5dec08c8f9c83d9a 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % 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 (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 17 /eta put dup 18 /theta put dup 21 /lambda put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 65 /A put dup 66 /B put dup 67 /C put dup 69 /E put dup 70 /F put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 83 /S put dup 84 /T put dup 86 /V put dup 88 /X 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 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 120 /x put dup 121 /y put dup 122 /z put dup 126 /vector put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueXX 5087383 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a196cf14c95a4e02f9cd2da8c 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /Lambda put dup 6 /Sigma put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 22 /macron put dup 33 /exclam put dup 34 /quotedblright 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 64 /at 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 89 /Y put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex 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{-34 -251 988 750}readonly def /UniqueXX 5000794 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 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922d0f244de900be9bb877f8e8b227e69048150e98fa57edcf997f45e352bc20 e0821857a8b3c7f3a128b66025646be60caba012e503369005b43d3f717786b0 ec281ec69244bd79d56f2f1ee7b3a63761f6b6b9f7639918e269331a29ef3927 e4160552cafee90044294fc869686214af7a21fffdc1300b5528b0cdeb3a9a28 49e6e03b9cf5c73e5024e262c178466e94d88e1a0de6c3ef7002d12c4866146a 8f01ef9f247c9715985b8650f38c8086ee8dbda8d407195285147616efc9304f ec1fc3fb62cc1cee015b9cc5667687494a268a76ba8a5a9984584a244e34b8cc 35ba2e7aa81f16bac0f02bdefdee18d6c89d60ae3a9f7f3d5bc26fe0431c692a e4b1c1f99865499269abcd3c892ae6a6d9b103146fb6b599de48c85fff5e4087 0272f8bba80a8a507e922c71156736d385d56fe1f0c9ff3c3a995af2d7b2d4aa bf762955fc3ed86610db73b8f4ffbf766bd7437de50267b74c45fc6956f73f89 708bf07b880a740c9d8b7e78ed767d5f0f964ec22fd9cf425b97abdbc9755962 e61d8714b142bf2e30458910b2239dd0ec5899a3b58ffdb146d5db3b5d688ede cd6c5d710c5d39dac1cb45224cd2f3961f94874907a1159d9bbd9213ac9f1179 b9f9cededec0c90e6cba4e13437a334b648e21e5b6c4818120a15ce1352d4dd2 981a2cfe6d24df2f7fed935df608e4c7b1f24b11a940e1292d06b0cc545c0f9e c4563125972137f2bef6941ce48e82eb7381d18149d18ee1e78d4b38007ccc33 02f9ab3e3a6d2211f546a2c459fca687db2f35a2847a51a0d52144e278019643 e98576cb42f1e61956849ccf1896af73b0d8d5d5d90802f3a5b1cec2b8420f91 969185be1eb8b91ad4159c28c84087ac60dae2ca1ad311f97707bae43ea7dfba f3df5f3c3801a450e4d61f02bb97f2e87230d5610f4c4027058360a5b5b94930 afaa9eb0129526a4a560df202aae2a53eb1bde3ce0e30d33867ae95c2e928184 41eee245cbac5753ec8d4752ef94284eb3c997dd1e563b9f7130e9f0a9644903 9ab25450aa2557b5b1d1518ab443e1f9d670da5d803dca91d3f1dd776523afa0 89bde3912a17c8c34a05c3e022ecbe8825cbe1d8c945e32e0af3857c0481d6ad b665ec4a988458d26d6c2a7aef39736b60a027f638a609ba3872be1b4ed304bd d5c719e32b0b1fc960abb9a75996c85ffe2a23804293b770bf1c74a816f20eb7 7cf68b014c4c2c7f3aab82a9ebf397c2fe76e02119151df715a98f40f5a83159 f6f0daa88a3f1683cc8c9337d496e34b806f57422b6383c775cd27a9a019406b ac09f4c273c0a8fcb45f9eb147d255f61c283667fc6d1b0a4bdf00c5ab0c1952 6a9d9e11ab07bf18c4fd11521dd80b711c4e118a11d4667c9ee16105e71007c8 4e8401ce2fc5ac0a42252a391ce197024a62a5bb91da729b0910af44c1458ca6 90fcf7eb6ea64b255103b1ee55ad3b33f5bf6dd8f68bf52d7f4be904fbcdcbe1 0d42e3fc5f3e6b184e5ff98abc23e0987005680fb79320f5eac1543168a77a31 8a4790e871904c820ed12dbfd0f49028b76b7309f8db4191d43b2613716f041f f3be188b04d16e57c4a2215c21b66ae5456a9779ada0ce5523589c5ed0345958 ca2964b6fd68c936677f4b8847c6cf8a11f03050011af3b670c19bb8ca8b02df 086988393d5e5c446b2864e1d83d0726541055a98e00cbe20d11bae3a4c52379 70f0e7b0e0d709348a12db63824d207957b28b637c566c637ce74ae8910c018a 85605da07a142c88894b8b1fa96c0a8f874806ddf0e74a6e1613897bcfc934da 8b401702eef7793fd95a52053b30c0d16e0cd2ca9bfc53e4ae5d778b954d67b0 948332d80ed97cbf038b151ce8fe14ed60f3e23ee9e8dc923428c0817894ef87 87882a11089e4b3c0af795adf18bb81d6ac9417f7963ba1f1e4f4faf667623d9 901c2e53ee21825ab8 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 65 /A put dup 97 /a put dup 98 /b put dup 99 /c put dup 114 /r put dup 115 /s put dup 116 /t put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueXX 5000768 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306462c563d51ece66fcc9c831843ecae 1fefc1a232e2724f7baee428ae03aadb95c3035345c15e9922fe49e1f2cfc980 237316572dbc57064edac9b0db8913a5e2d45e97e19a91435ccf8adfc835b585 48e74b291446d689c7f2f8c4325e8356e974ae30c3e2977477baeaa33d141fb7 80f59351e84bfc88c87b3a4a1d25e168b9d134554f6a581378c7d2d6eca8ac09 045cc3a5e0ed86f147133f094029e3483bebc81cfde69942cf645ea20d0a2b64 5140ccc2f9cb4a3d969f4b0a79cd3329b440dfcff5285494406f3dd08d804c86 df191e57232f6f1f83df6430f214e03fa5412b1e13d83c60c52a698bc9a38159 19a64162af7b83a9468fae9507c6cb55ea80df8c696dfbc3296c663902d77e89 a181502627281181ac8624f65be9de5fb32f056cc7dd7cbc7366a91717b5892a 34ef6b441871c6993da3a81afc9bf70ff3842a21a29c0e49da1ccba37dd6b920 2702db510b8b1109acf4c27ee8d676bdbd16abe189f34a9b61532eb777457698 cee0561838eaab2d3939c5093afbb215bac433d1918c739b62566906bb2187f3 378521fb52e70c18886e1aa15f53a1f8a169528cf2ddd54c203613ff88e619f2 6c9c50c2c3b8b8c868a88afab1dc8472e7b6f3f992a161723a3c0904865a7c42 3281601eee91f35222c4a71f343739f102eb43145a47e4fac3bd666e4c114a90 5950618ecb1bdc29f90f6ebed1d8325fe6b52e14a8870e9b4bb088e585c71b91 493c12845a101937e3473ac5e9ae24f7c358f06c8b42dee8ef2a1a7a9f536037 57517f2fb6da0ea305d9d20cc74d922ec4b95b8b299459b71825f6f67abee3a6 f5267c6283c7c62aa8d36663071f18e1a3f0b35dcd88d739caf8dbc344630a28 7f583f037768d861f37d7929f248146b5f31ed9f61ef0a9659b50f47ec328516 9fd999b19687e50303557945476fe9845e5c4780045d84d7df253fe61bf69607 fd277a265296b02b4e4b35884f985d29521ee7327887d983bb92dd84ff646661 2b348b5e558d3bc65047afcb202750ea6e9c18a769a4e945e0954787ee65948e 7e8e443a9f2b9f5408e0febf523423b2200166ade223570bd17a6f7dd0386f3d 2246ebb40648548e0a3ff7bab6637a16137c24b042010a3cce2ba9afb9317fcf 4a5b893c4915859ff2b53a77ff0334966f7a234ea8a4349c3e048acd6876f5f3 17b9c238f37ed4231b5e8b661ba52c32821de47f7a0f6fa9c7e0f4bf09bf98a1 8008087aa44eaf37894e83891f552626f1f46563b5188ad949521e1a398eeba3 b9b8f4ac3c24785d50bdf01e5c9b7e4ea1d6b7b6c8019d2d5f353cacf216e30b 71f09d9b95805210964ccbf8f2e30aa04ac3a3a57bde635c9d2be1f5c8f64ad6 19764c72aca8152a51003bc396bce21a56c29306bccc36a37d816de2568e7794 a8f80d857132a3dbaf3ef75409cace85e1b85a8233ef2757a507158bbba8f5f6 d9c08ab4ee39f33a1c1a41ff2b8e81320050b6c4c167596b93918824a65fe1b0 6e7dcf5d7e128a77e5509f3a3a2af6f71de243cccfa5788088d277175e59bd61 a3b5cdc10cfd232cc14682d3b37d3aca68df524af6f38511e5627286034a7c08 1c7596bfc3adecc7d1868cb9d645f8cb61041bccad92249142542934977000c1 bfbbaf570bf1a3fad2368d1e6480ed485642acd1d44258a885562397035dc979 b08877eadba7c99963e7d866be89270b10629964b5c88439c08a7721e2fd6a7d 2a03d9cf2d7fa713411824db51a9141a7c3ca37feeb758ab37376d75b9ad8534 6fe75968a1e7f6b9b549b7bbef5da78116a7d029e6e95db78593abf789ab985e 301f5945c1628f591b4867079526cd9c779b05cd30de4d9dbb820ce964cb3acf cf693b39b15335996a208373a9a1cd362e93d740535d4fe06aaef81a77034010 883c098f8fe19617d6a3cc3ccaa63742d3b251b8baa94aba7ebb7f3cad53c3c5 2c626e3918191af1e343757e0fc5f9fa620ce7e655482aaab7303710167e3041 148e3e6aae44ebe904d96a7dea9281fdbdf93b03ccd08a315df83531214f1726 dd6eff4d5f494262983201ee2afd6cc77eac76155fc59d9e42a5c45735a37af6 c3c9c4aa7c535717838a1ce8ca3424188a3ddc2bb7f58483ef08fe9b6ea090c8 bd5a639bb56a54bed34a443f0f1624f13fffd6b4f24024573a6cdc110f2a85b3 f669c706b0bcf205ee9ff8342b38525e9cbf1f1b772a71c7718b30bd59f46490 dd92c121c9fd0b99edf33c57611ce19ca35fe17079c90ce99afdfd9227184562 f4e2dc14aea6d0989a8f01f17c0983ae121614d8f578d34ead9c05a88007bc4f 90e135d97abf57c1e32fd94ed63ceeddab6c1d42f3238e6cf85a063d1239a1d9 c5de6f9e7954b873ccef98d8b44f1761e66da28b9759a24f181c9f7ca8ca190d 4d5a52b1286ad43b5f8caebe64283da13e6a9e66536a821ebc1b50e98073d79e 335c244c2b8efcd7c0b223983db1608d6f1837ae6f73d970e567960e7e03d1a6 5f7dcaeecd4f9afd0266592ed051f800ee3d5a8954460f3f57a9fb9716f4dca2 616d5e77e19918f2b1c290fa9d049267df21a4b33a06e29c758e0ecac9746be0 b7988da1634f2e7162c2c483f1c8d78eaefc94e907d3bcc79cc389979ab0dc15 bf1b2371fe907e9c6f46e26260566e38ed55175ce116e4da7024db03b6657b44 1543e575e850440d7613b96baa8320e2cde5ab834d1f333a9a749c3ab079546c 3429c90c61bfda4b758861c2e351866a18c4d14d3bee4d863edce298231a71dc 9e4c701ac2d01fead65c2244a81865874b1c22125e9f11720450c10366eb6eec 694ba3acd45ea6f9cda354cc938222a523494fa570ba76d359afd28f6ca82c35 dc9890f6e09346d93734ef8c82d7b62344fb8103a1fe6f84fb56f55e1969c4bb f61c916f5fc7aaf919d620ed2c25d2bac648ede51f981afe1d4ecf8263830258 75069803a3032b0fba0522 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR17 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 70 /F put dup 76 /L 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 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 119 /w put dup 121 /y put readonly def /FontBBox{-33 -250 945 749}readonly def /UniqueXX 5000795 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be356c29c3aec91ef17f5d31183b1e489f1fa 559a6693fbdd04d4f7e99886d8cb7cca13d4aa9ceb34a708af22e69ccef0e504 c3997075243585edf60b6447c8d01cffb04b6e3ddbaa7da5891286861322a795 e9d990fd56af2f5c2b35801a0b07874e4ac3170481d232e150443253c92a8b05 513d4a4154a99d91f629ccd30376500f5d16aaa9b203bad6350e4b19424c0e33 75f2949bc1927a6f6221252f8a96618e646aa1b0810753c98becc26b37837775 24bca72b818d314074c1c91f4e1f4148d3d703d12d179ee6970bd55d04aa9a9d 97f63f2c7dbd129ec2ed3ba5f8080270ea5854ea165dbcc061e2f47f1aaf0997 2ac9a689a2a9620b91b53f6b5092884372f5f7afca495f331138d85a97c5bb83 d8072dc88010fc89108b60bc06d4f93c722e8908d7ce3c1ea3c70e9e65edd769 7db17ce0e012ad37d4030f33d30a73fd655b7a2e572f3378ac933dd75082f2f1 68702be19bc60cef6bd5f0ada180d21408c04b7f7db9b6dbaefe09e809a80ddf a5366d10930d3e68ca53e4721ff525853f4c784f2c2c3aa0129a34d9df00debc 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19b789dc751dfa1ef940cd676223cb819f60a344d2e137da5207b5fbbc7a8e05 2e2bc6ccfaa552daf18cd62eec67e22062ea05e8c82eed0eef 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont TeXDict begin 39158280 55380996 1000 8000 8000 (bgimlat.dvi) @start /Fa 173[672 82[{}1 929.846 /MSBM7 rf /Fb 141[800 2[844 8[711 21[985 34[502 45[{}5 1328.35 /CMCSC10 rf /Fc 223[1015 32[{}1 1594.02 /CMMI12 rf /Fd 151[638 104[{}1 1106.96 /CMMI10 rf /Fe 172[888 83[{}1 929.846 /EUFM7 rf /Ff 172[1101 5[1394 6[812 70[{}3 1328.35 /EUFM10 rf /Fg 139[439 3[627 565 627 8[502 54[314 46[{}6 1062.68 /CMR8 rf /Fh 169[977 3[1351 5[1099 76[{}3 1594.02 /CMSY10 rf /Fi 129[424 4[420 464 1[415 1[317 388 385 379 427 1[511 726 267 439 340 296 482 1[399 391 1[371 12[485 1[486 3[518 620 1[771 562 5[515 2[587 4[640 2[252 252 23[390 518 1[510 2[387 476 3[413 1[489 2[392 5[464 534 11[{}40 664.176 /CMMI6 rf /Fj 135[486 12[486 26[622 686 79[{}4 664.176 /CMBX6 rf /Fk 140[513 506 3[723 1084 361 2[361 3[578 723 2[650 25[1021 2[993 55[723 12[{}12 1328.35 /CMSL12 rf /Fl 142[1107 1[738 738 4[523 523 13[615 1414 1599 5[523 1045 1168 12[969 969 969 969 1[984 984 984 984 984 984 984 738 738 738 738 738 738 969 969 6[892 892 4[646 646 876 876 4[830 830 4[584 584 815 815 661 661 3[369 12[{}45 1106.96 /CMEX10 rf /Fm 134[573 573 7[603 4[573 22[755 3[740 815 79[{}7 885.568 /CMBX8 rf /Fn 134[789 789 1[789 830 581 589 610 1[830 747 1[1245 1[789 2[830 2[682 3[726 12[1039 1[1115 1[1021 1122 2[898 18[747 747 747 747 747 747 747 747 747 747 1[415 498 45[{}31 1328.35 /CMBX12 rf /Fo 165[886 5[886 1[959 82[{}3 1328.35 /MSBM10 rf /Fp 129[406 5[427 3[320 3[449 406 449 8[363 39[621 2[234 4[406 406 406 406 406 406 4[621 1[320 320 17[406 11[578 10[{}20 664.176 /CMR6 rf /Fq 145[424 3[252 22[506 2[583 11[638 17[553 1[243 17[640 8[640 640 13[640 2[424 2[640{}13 664.176 /CMSY6 rf /Fr 129[470 4[497 497 3[366 371 2[523 470 523 784 3[261 2[288 418 7[261 1[261 29[732 1[261 261 1[470 2[470 470 470 470 470 470 4[732 1[366 366 17[470 11[680 6[653 3[{}30 885.568 /CMR8 rf /Fs 143[784 1[470 3[261 2[470 470 9[627 627 4[581 1[519 1[803 1[658 745 770 1[647 6[497 728 4[732 3[470 8[627 941 255 14[941 11[732 732 13[732 2[470 1[261 732{}28 885.568 /CMSY8 rf /Ft 135[590 5[1107 3[664 3[369 2[664 664 7[886 1[886 886 3[1312 814 1[723 804 1126 1[924 1058 1090 1595 916 2[723 1122 1[955 701 1025 699 872 1061 5[664 1[738 738 1[0 3[886 1328 7[1328 7[1328 2[1033 3[1033 4[1033 1033 1033 1033 1033 6[1033 3[1033 2[664 1033 369 1033{}48 1328.35 /CMSY10 rf /Fu 133[492 584 2[584 615 430 437 434 1[615 553 615 922 307 2[307 1[553 1[492 615 492 615 553 4[553 5[830 830 799 615 815 1[753 2[1015 692 2[400 2[723 1[846 2[830 8[553 2[553 1[553 1[553 553 1[307 369 307 2[430 430 5[553 15[553 5[615 12[{}45 1106.96 /CMR10 rf /Fv 134[406 117[454 3[{}2 774.872 /CMSY7 rf /Fw 133[531 631 598 863 598 697 432 531 548 598 664 664 731 1063 332 598 399 399 664 598 399 598 664 598 598 664 6[797 966 966 1298 2[930 731 948 1[881 996 1[1165 815 1[682 501 966 1005 848 881 1[930 914 966 5[399 399 664 664 664 664 664 1[664 664 664 664 1[399 465 399 2[531 531 27[731 797 11[{}64 1328.35 /CMTI12 rf /Fx 129[470 3[438 465 530 1[458 545 340 435 425 420 474 455 571 832 281 490 381 320 542 1[452 436 484 408 403 500 8[774 1[548 1[548 572 1[740 597 714 747 904 640 2[408 2[595 690 1[671 710 705 2[732 470 732 261 261 23[434 585 615 584 559 1[418 538 482 540 414 463 1[549 2[439 472 2[416 485 528 604 11[{}64 885.568 /CMMI8 rf /Fy 129[650 3[604 634 739 1[629 740 470 613 586 577 653 625 776 1138 387 676 538 444 749 623 641 603 676 560 553 683 546 7[1079 1228 758 1[762 800 1[1030 838 994 1042 1258 885 1107 719 574 1078 1026 842 963 1080 931 988 975 687 1[1012 650 1012 361 361 23[609 810 846 815 770 1[565 739 671 738 568 643 783 759 2[606 642 568 1[576 675 734 827 11[{}75 1328.35 /CMMI12 rf /Fz 128[650 650 2[650 578 686 686 939 686 723 506 513 506 686 723 650 723 1084 361 686 397 361 723 650 397 578 723 578 723 650 2[650 361 650 361 1[975 1[1336 975 975 939 723 957 1[885 1012 975 1192 813 1011 668 469 975 1021 849 885 993 939 921 975 3[1012 1[361 361 650 650 650 650 650 650 650 650 650 650 1[361 434 361 1012 1[506 506 361 4[650 361 10[650 7[1084 723 723 759 939 3[939 2[903 3[{}86 1328.35 /CMR12 rf /FA 133[797 946 946 1295 946 996 697 707 732 1[996 897 996 1494 498 946 1[498 996 897 548 818 996 797 996 872 12[1247 3[1225 2[1701 1077 4[1410 2[1374 1[1275 1354 9[897 897 897 897 897 897 897 2[498 4[697 697 27[996 1046 11[{}44 1594.02 /CMBX12 rf /FB 133[956 1136 1[1554 1136 1196 837 849 879 1[1196 1076 1196 1793 598 2[598 1196 1076 658 982 1196 956 1[1046 12[1497 1196 1605 1[1470 3[1293 2[801 1[1691 15[1076 1076 1076 1076 1076 1076 1076 2[598 717 31[1196 13[{}38 1912.83 /CMBX12 rf /FC 155[631 100[{}1 1212.12 /CMMI10 rf /FD 133[539 640 640 875 640 673 471 478 475 1[673 606 673 1010 337 640 1[337 673 606 370 539 673 539 673 606 9[1246 3[673 892 10[951 791 8[943 10[606 3[337 1[337 44[{}33 1212.12 /CMR10 rf /FE 139[542 550 574 14[620 774 678 31[1054 65[{}7 1212.12 /CMBX10 rf /FF 134[492 100[861 16[553 3[{}3 1106.96 /CMSY10 rf /FG 134[824 3[867 607 616 607 1[867 780 867 1301 434 2[434 1[780 477 694 867 1[867 780 10[1170 8[1430 2[802 2[1225 4[1105 1170 1214 5[434 4[780 2[780 780 780 1[434 520 434 2[607 607 40[{}34 1594.02 /CMR12 rf /FH 134[1113 1[1533 1[1173 813 825 813 1[1173 1053 1173 1773 573 2[573 1173 1[633 933 1173 933 1[1053 20[1322 5[1382 70[{}20 2295.84 /CMR17 rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 8000dpi TeXDict begin %%BeginPaperSize: a4 a4 %%EndPaperSize %%EndSetup %%Page: 1 1 1 0 bop 4206 13007 a FH(Lo)-60 b(w)693 b(temp)60 b(erature)692 b(analysis)h(of)g(t)-60 b(w)g(o)693 b(dimensional)4436 15774 y(F)-180 b(ermi)693 b(systems)g(with)f(symmetric)h(F)-180 b(ermi)693 b(surface)20969 19718 y FG(G.)520 b(Benfatto)29065 19139 y FF(\003)13181 21710 y FG(e-mail:)693 b(b)43 b (enfatto@mat.uniroma2.it,)21219 24483 y(A.)519 b(Giuliani)28817 23905 y FF(y)11113 26476 y FG(e-mail:)693 b (alessandro.giuliani@roma1.infn.it)19246 29249 y(V.)520 b(Mastropietro)30179 28670 y FF(\003)13264 31241 y FG(e-mail:)692 b(mastropi@mat.uniroma2.it)21083 34357 y(5)521 b(July)e(2002)22351 40617 y FE(Abstract)8501 42827 y FD(W)-101 b(e)564 b(pro)-34 b(v)g(e)565 b(the)f(con)-34 b(v)g(ergence)564 b(of)h(the)g(p)34 b(erturbativ)-34 b(e)564 b(expansion,)605 b(based)6683 44332 y(on)554 b(Renormalization)f(Group,)591 b(of)554 b(the)g(t)-34 b(w)g(o)555 b(p)34 b(oin)-34 b(t)554 b(Sc)-34 b(h)g(winger)554 b(function)h(of)6683 45838 y(a)563 b(system)f(of)h(w) -34 b(eakly)562 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Fz(of)k(the)e(Jellium)i(mo)36 b(del,)355 b(an)335 b(h)-36 b(yp)36 b(oth-)3432 70237 y(esis)360 b(whic)-36 b(h)360 b(is)h(indeed)e(quite)i(unrealistic)f (\(it)g(corresp)36 b(onds)360 b(to)g(completely)i(neglecting)3432 71842 y(the)433 b(e\013ect)g(of)h(the)f(lattice\).)24692 75163 y(2)p eop %%Page: 3 3 3 2 bop 5383 7638 a Fz(The)588 b(aim)h(of)h(this)e(pap)36 b(er)588 b(is)h(to)g(pro)-36 b(v)g(e)589 b(con)-36 b(v)g(ergence)589 b(of)g(the)f(p)36 b(erturbativ)-36 b(e)588 b(ex-)3432 9243 y(pansion)499 b(for)i(the)e(t)-36 b(w)g(o)500 b(p)36 b(oin)-36 b(t)500 b(Sc)-36 b(h)g(winger)499 b(function,)517 b(in)500 b(the)f(case)i(of)f(an)g(in)-36 b(teracting)3432 10848 y(system)543 b(of)g(fermions)g(in)g(a)g(lattice)g(or)g(in)g(the)f (con)-36 b(tin)g(uum.)905 b(Since)542 b(the)g(in)-36 b(teraction)3432 12453 y(mo)36 b(di\014es)428 b(the)f(F)-108 b(ermi)428 b(surface,)i(w)-36 b(e)429 b(write)f(the)g(disp)36 b(ersion)428 b(relation)h Fy(")38147 12652 y Fr(0)38672 12453 y Fz(\()39111 12103 y Fy(~)39178 12453 y(k)45 b Fz(\))428 b(of)h(the)f(free)3432 14193 y(mo)36 b(del)505 b(in)g(the)g(form)h Fy(")15022 14392 y Fr(0)15547 14193 y Fz(\()15986 13843 y Fy(~)16053 14193 y(k)45 b Fz(\))491 b(=)g Fy(")p Fz(\()20322 13843 y Fy(~)20389 14193 y(k)44 b Fz(\))344 b(+)g Fy(\016)50 b(")p Fz(\()24989 13843 y Fy(~)25056 14193 y(k)44 b Fz(\))505 b(and)g(try)g(to)h(c)-36 b(ho)36 b(ose)505 b(the)g Fw(c)-66 b(ounterterm)3432 15934 y Fy(\016)50 b(")p Fz(\()5106 15583 y Fy(~)5173 15934 y(k)44 b Fz(\),)550 b(whic)-36 b(h)526 b(b)36 b(ecomes)527 b(part)f(of)h(the)g(in)-36 b(teraction,)550 b(as)526 b(a)h(suitable)g(function)f(of)i(the)3432 17539 y(original)558 b(in)-36 b(teraction,)588 b(so)557 b(that)f(the)h(F)-108 b(ermi)556 b(surface)i(of)f(the)g(in)-36 b(teracting)556 b(system)i(is)3432 19144 y(the)464 b(set)h Fy(F)603 b Fz(=)422 b Ft(f)11241 18793 y Fy(~)11308 19144 y(k)467 b Fz(:)423 b Fy(")p Fz(\()14283 18793 y Fy(~)14350 19144 y(k)45 b Fz(\))422 b(=)g Fy(\026)p Ft(g)p Fz(.)672 b(W)-108 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51474 y Fx(i)21878 51275 y Ft(\024)f Fy(M)463 b Ft(\000)323 b Fz(1)476 b(and)e Fy(k)30799 51474 y Fr(0)31765 51275 y Fz(=)33349 50752 y Fr(2)p Fx(\031)p 33349 50965 1043 45 v 33584 51733 a(\014)34525 51275 y Fz(\()p Fy(n)35807 51474 y Fr(0)36656 51275 y Fz(+)38124 50752 y Fr(1)p 38124 50965 471 45 v 38124 51733 a(2)38727 51275 y Fz(\),)486 b Fy(n)40856 51474 y Fr(1)41382 51275 y Fy(;)221 b(n)42740 51474 y Fr(2)43267 51275 y Fy(;)g(n)44625 51474 y Fr(0)45591 51275 y Fz(=)3432 52880 y(0)p Fy(;)g Fz(1)p Fy(;)g(:)g(:)g(:)k(;)c(M) 467 b Ft(\000)327 b Fz(1.)719 b(With)481 b(eac)-36 b(h)480 b Fn(k)448 b Ft(2)h(D)517 b Fz(w)-36 b(e)480 b(asso)36 b(ciate)482 b(four)f(Grassmanian)f(v)-72 b(ariables)3698 54134 y(^)3432 54485 y Fy( )4326 54003 y Fx(")4278 54813 y Fm(k)p Fx(;\033)5736 54485 y Fz(,)457 b Fy(";)221 b(\033)450 b Ft(2)401 b(f)p Fz(+)p Fy(;)221 b Ft(\000g)p Fz(.)636 b(The)452 b(lattice)h(\003)22761 54684 y Fx(M)24267 54485 y Fz(is)g(in)-36 b(tro)36 b(duced)451 b(only)i(for)g(tec)-36 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y Fz(])590 b Ft(\024)24692 75163 y Fz(3)p eop %%Page: 4 4 4 3 bop 3432 7638 a Fy(n)4208 7837 y Fx(i)4953 7638 y Ft(\024)369 b Fz([)6849 7018 y Fr(\()p Fx(L)p Fs(\000)p Fr(1\))p 6849 7328 2575 45 v 7901 8096 a(2)9556 7638 y Fz(].)561 b(With)383 b(eac)-36 b(h)383 b Fn(k)368 b Ft(2)h(D)419 b Fz(w)-36 b(e)383 b(asso)36 b(ciate)384 b(four)f(Grassmanian)f(v)-72 b(ariables)44203 7287 y(^)43937 7638 y Fy( )44831 7156 y Fx(")44783 7966 y Fm(k)p Fx(;\033)46242 7638 y Fz(,)3432 9243 y Fy(";)221 b(\033)417 b Ft(2)368 b(f)p Fz(+)p Fy(;)221 b Ft(\000g)p Fz(.)5383 12453 y(All)513 b(the)f(mo)36 b(dels)513 b(are)g(de\014ned)e(b)-36 b(y)513 b(in)-36 b(tro)36 b(ducing)512 b(a)h(linear)g(functional)g Fy(P)181 b Fz(\()p Fy(d )48 b Fz(\))511 b(on)3432 14058 y(the)433 b(Grassmanian)g(algebra)h(generated)g(b)-36 b(y)433 b(the)g(v)-72 b(ariables)33680 13708 y(^)33414 14058 y Fy( )34308 13576 y Fx(")34260 14387 y Fm(k)p Fx(;\033)35718 14058 y Fz(,)434 b(suc)-36 b(h)432 b(that)13581 15789 y Fl(Z)14910 17351 y Fy(P)181 b Fz(\()p Fy(d )48 b Fz(\))18775 17000 y(^)18511 17351 y Fy( )19405 16801 y Fs(\000)19357 17685 y Fm(k)19930 17808 y Fp(1)20389 17685 y Fx(;\033)21188 17808 y Fp(1)21971 17000 y Fz(^)21705 17351 y Fy( )22599 16801 y Fr(+)22551 17685 y Fm(k)23124 17808 y Fp(2)23584 17685 y Fx(;\033)24383 17808 y Fp(2)25269 17351 y Fz(=)368 b Fy(L)27534 16802 y Fr(2)28060 17351 y Fy(\014)74 b(\016)29444 17550 y Fx(\033)29982 17673 y Fp(1)30443 17550 y Fx(;\033)31242 17673 y Fp(2)31758 17351 y Fy(\016)32334 17550 y Fm(k)32907 17673 y Fp(1)33368 17550 y Fx(;)p Fm(k)34202 17673 y Fp(2)34765 17351 y Fz(^)-696 b Fy(g)47 b Fz(\()p Fn(k)36684 17550 y Fr(1)37210 17351 y Fz(\))368 b Fy(;)5484 b Fz(\(1.1\))16229 20887 y(^)-696 b Fy(g)47 b Fz(\()p Fn(k)p Fz(\))369 b(=)23338 19653 y(\026)23042 19989 y Fy(C)24068 19439 y Fs(\000)p Fr(1)23973 20279 y(0)25326 19989 y Fz(\()25765 19638 y Fy(~)25832 19989 y(k)45 b Fz(\))p 20536 20577 9029 45 v 20536 22008 a Ft(\000)p Fy(ik)22689 22207 y Fr(0)23510 22008 y Fz(+)295 b Fy(")p Fz(\()25865 21657 y Fy(~)25932 22008 y(k)45 b Fz(\))295 b Ft(\000)g Fy(\026)30066 20887 y(;)3432 24867 y Fz(where)339 b Fy(")p Fz(\()8143 24516 y Fy(~)8210 24867 y(k)44 b Fz(\),)358 b(the)339 b Fw(disp)-66 b(ersion)377 b(r)-66 b(elation)338 b Fz(of)i(the)e(mo)36 b(del,)359 b(is)339 b(a)h(function)e(strictly)i(p)36 b(ositiv)-36 b(e)3432 26607 y(for)5344 26256 y Fy(~)5410 26607 y(k)415 b Ft(6)p Fz(=)368 b(0)426 b(and)f(equal)h(to)f(0)h(for) 19471 26256 y Fy(~)19538 26607 y(k)414 b Fz(=)368 b(0,)428 b Fy(\026)d Fz(is)h(the)e Fw(chemic)-66 b(al)456 b(p)-66 b(otential)424 b Fz(and)41583 26271 y(\026)41287 26607 y Fy(C)42313 26057 y Fs(\000)p Fr(1)42218 26897 y(0)43571 26607 y Fz(\()44010 26256 y Fy(~)44077 26607 y(k)45 b Fz(\))425 b(is)3432 28347 y(the)492 b Fw(ultr)-66 b(aviolet)519 b(cut-o\013)p Fz(.)757 b(In)493 b(the)g(case)h(of)g(lattice)g(mo)36 b(dels)493 b(w)-36 b(e)494 b(c)-36 b(ho)36 b(ose)39919 28011 y(\026)39622 28347 y Fy(C)40648 27797 y Fs(\000)p Fr(1)40553 28637 y(0)41906 28347 y Fz(\()42345 27997 y Fy(~)42412 28347 y(k)45 b Fz(\))470 b(=)g(1,)3432 30087 y(while)640 b(for)g(con)-36 b(tin)g(uum)638 b(mo)36 b(dels)640 b(the)g(function)28492 29751 y(\026)28196 30087 y Fy(C)29222 29537 y Fs(\000)p Fr(1)29127 30378 y(0)30480 30087 y Fz(\()30919 29737 y Fy(~)30986 30087 y(k)44 b Fz(\))640 b(is)g(de\014ned)e(as)41151 29751 y(\026)40855 30087 y Fy(C)41881 29537 y Fs(\000)p Fr(1)41786 30378 y(0)43139 30087 y Fz(\()43578 29737 y Fy(~)43645 30087 y(k)45 b Fz(\))719 b(=)3432 31858 y Fy(H)4833 30574 y Fl(\020)5494 31858 y Fy(")p Fz(\()6542 31508 y Fy(~)6609 31858 y(k)45 b Fz(\))295 b Ft(\000)g Fy(\026)10242 30574 y Fl(\021)11298 31858 y Fz(where)395 b Fy(H)103 b Fz(\()p Fy(t)p Fz(\))367 b Ft(2)i Fy(C)20328 31376 y Fs(1)21324 31858 y Fz(\()p Fo(R)p Fz(\))395 b(is)g(a)h(smo)36 b(oth)395 b(function)f(of)i(compact) f(supp)36 b(ort)3432 33630 y(suc)-36 b(h)432 b(that,)i(for)g(example,)g Fy(H)103 b Fz(\()p Fy(t)p Fz(\))368 b(=)h(1)433 b(for)i Fy(t)368 b(<)h Fz(1)434 b(and)f Fy(H)103 b Fz(\()p Fy(t)p Fz(\))367 b(=)i(0)434 b(for)g Fy(t)369 b(>)f Fz(2.)5383 35235 y(W)-108 b(e)433 b(in)-36 b(tro)36 b(duce)433 b(the)g(propagator) h(in)f(co)36 b(ordinate)434 b(space:)4096 38477 y Fy(g)4767 37929 y Fx(L;\014)6296 38477 y Fz(\()p Fn(x)179 b Ft(\000)g Fn(y)21 b Fz(\))368 b Ft(\021)906 b Fz(lim)12069 39267 y Fx(M)94 b Fs(!1)16087 37579 y Fz(1)p 15303 38167 2219 45 v 15303 39389 a Fy(L)16188 39005 y Fr(2)16713 39389 y Fy(\014)18053 37370 y Fl(X)17876 39830 y Fm(k)p Fs(2D)20272 38477 y Fy(e)20875 37929 y Fs(\000)p Fx(i)p Fm(k)p Fs(\001)p Fr(\()p Fm(x)p Fs(\000)p Fm(y)15 b Fr(\))25708 38477 y Fz(^)-696 b Fy(g)47 b Fz(\()p Fn(k)p Fz(\))368 b(=)906 b(lim)29882 39267 y Fx(M)94 b Fs(!1)32983 36915 y Fl(Z)34312 38477 y Fy(P)181 b Fz(\()p Fy(d )48 b Fz(\))p Fy( )38807 37929 y Fs(\000)38759 38806 y Fm(x)p Fx(;\033)40216 38477 y Fy( )41110 37929 y Fr(+)41062 38806 y Fm(y)15 b Fx(;\033)42904 38477 y Fy(;)664 b Fz(\(1.2\))3432 42149 y(where)433 b(the)g Fk(Grassmanian)g(\014eld)h Fy( )20969 41667 y Fx(")20921 42477 y Fm(x)21982 42149 y Fz(is)f(de\014ned)g(b)-36 b(y)17474 45462 y Fy( )18368 44913 y Fs(\006)18320 45790 y Fm(x)p Fx(;\033)20148 45462 y Fz(=)22445 44563 y(1)p 21661 45152 V 21661 46373 a Fy(L)22546 45989 y Fr(2)23072 46373 y Fy(\014)24411 44355 y Fl(X)24234 46815 y Fm(k)p Fs(2D)26675 45111 y Fz(^)26409 45462 y Fy( )27303 44912 y Fs(\006)27255 45796 y Fm(k)p Fx(;\033)28713 45462 y Fy(e)29316 44913 y Fs(\006)p Fx(i)p Fm(k)p Fs(\001)p Fm(x)32199 45462 y Fy(:)11369 b Fz(\(1.3\))5383 49207 y(The)475 b(\\Gaussian)h(measure")g Fy(P)181 b Fz(\()p Fy(d )48 b Fz(\))474 b(has)h(a)h(simple)g(represen)-36 b(tation)475 b(in)g(terms)g(of)3432 50813 y(the)433 b(\\Leb)36 b(esgue)433 b(Grassmanian)g(measure")17859 53537 y Fk(D)p Fy( )417 b Fz(=)22857 52430 y Fl(Y)21708 54736 y Fj(k)p Fq(2D)22 b Fi(;\033)i Fp(=)p Fq(\006)21628 55747 y Fi(C)22261 55426 y Fq(\000)p Fp(1)22215 56123 y(0)23362 55747 y(\()23645 55572 y Fi(~)23682 55747 y(k)13 b Fp(\))p Fi(>)p Fp(0)25854 53537 y Fy(d)26796 53186 y Fz(^)26530 53537 y Fy( )27424 52987 y Fr(+)27376 53871 y Fm(k)p Fx(;\033)28834 53537 y Fy(d)29776 53186 y Fz(^)29510 53537 y Fy( )30404 52987 y Fs(\000)30356 53871 y Fm(k)p Fx(;\033)31814 53537 y Fy(;)11754 b Fz(\(1.4\))3432 58386 y(de\014ned)363 b(as)i(the)g(linear) g(functional)g(on)g(the)f(Grassmanian)h(algebra,)380 b(suc)-36 b(h)364 b(that,)378 b(giv)-36 b(en)3432 59991 y(a)549 b(monomial)h Fy(Q)p Fz(\()12546 59641 y(^)12280 59991 y Fy( )13174 59509 y Fs(\000)13960 59991 y Fy(;)14809 59641 y Fz(^)14542 59991 y Fy( )15436 59509 y Fr(+)16223 59991 y Fz(\))f(in)g(the)g(v)-72 b(ariables)27039 59641 y(^)26773 59991 y Fy( )27667 59441 y Fs(\000)27619 60326 y Fm(k)p Fx(;\033)29077 59991 y Fy(;)29926 59641 y Fz(^)29659 59991 y Fy( )30553 59441 y Fr(+)30505 60326 y Fm(k)p Fx(;\033)31964 59991 y Fz(,)578 b(its)550 b(v)-72 b(alue)549 b(is)g(0,)579 b(except)549 b(in)3432 61882 y(the)452 b(case)i Fy(Q)p Fz(\()10266 61531 y(^)10000 61882 y Fy( 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Fl(X)22131 69237 y Fj(k)p Fq(2D)22 b Fi(;\033)i Fp(=)p Fq(\006)22229 70080 y Fp(\026)22051 70248 y Fi(C)22684 69927 y Fq(\000)p Fp(1)22638 70624 y(0)23785 70248 y(\()24068 70073 y Fi(~)24105 70248 y(k)13 b Fp(\))p Fi(>)p Fp(0)26573 67702 y Fz(\026)26277 68037 y Fy(C)27208 68236 y Fr(0)27733 68037 y Fz(\()28172 67687 y Fy(~)28239 68037 y(k)45 b Fz(\)\()p Ft(\000)p Fy(ik)32125 68236 y Fr(0)32946 68037 y Fz(+)295 b Fy(")p Fz(\()35301 67687 y Fy(~)35368 68037 y(k)44 b Fz(\))295 b Ft(\000)g Fy(\026)p Fz(\))39772 67687 y(^)39506 68037 y Fy( )40400 67488 y Fr(+)40352 68372 y Fm(k)p Fx(;\033)42076 67687 y Fz(^)41810 68037 y Fy( )42704 67488 y Fs(\000)42656 68372 y Fm(k)p Fx(;\033)44115 64717 y Fl(9)44115 65713 y(>)44115 66045 y(>)44115 66377 y(>)44115 66709 y(=)44115 68702 y(>)44115 69034 y(>)44115 69366 y(>)44115 69698 y(;)45689 68037 y Fy(;)43929 71842 y Fz(\(1.5\))24692 75163 y(4)p eop %%Page: 5 5 5 4 bop 3432 7638 a Fz(with)433 b Fy(N)507 b Fz(=)9324 6752 y Fl(Q)10370 7915 y Fm(k)p 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Fw(mar)-66 b(ginal)357 b Fz(in)-36 b(teractions)359 b(are)h(renormalized)3432 71474 y(at)541 b(an)-36 b(y)541 b(iteration)g(of)h(the)e (Renormalization)j(Group.)899 b(One)541 b(writes)38795 71138 y(^)38573 71474 y Fy(S)77 b Fz(\()p Fn(k)p Fz(\))541 b(in)f(terms)24692 75163 y(6)p eop %%Page: 7 7 7 6 bop 3432 7638 a Fz(of)513 b(a)f(set)g(of)h Fw(running)536 b(c)-66 b(oupling)537 b(functions)p Fz(,)530 b(whic)-36 b(h)512 b(ob)36 b(ey)513 b(recursiv)-36 b(e)513 b(equations,)533 b(the)3432 9243 y Fw(b)-66 b(eta)432 b(function)397 b Fz(of)i(the)g(mo)36 b(del.)567 b(W)-108 b(e)399 b(pro)-36 b(v)g(e)399 b(that)f(the)g(expansions)h(of)38001 8907 y(^)37779 9243 y Fy(S)77 b Fz(\()p Fn(k)p Fz(\))399 b(and)f(of)i(the) 3432 10848 y(b)36 b(eta)373 b(function)h(are)g(con)-36 b(v)g(ergen)g(t,)386 b(if)375 b(the)e(running)g(coupling)h(functions)f (are)h(small)h(in)f(a)3432 12453 y(suitable)431 b(norm;)h(the)f(con)-36 b(v)g(ergence)432 b(pro)36 b(of)432 b(is)g(based)f(on)g(the)g Fw(tr)-66 b(e)g(e)461 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Fr(1)35728 20614 y Fz(;)547 b Fy(c)37196 20813 y Fr(0)38230 20614 y Fz(is)509 b(expressed)f(in)3432 22219 y(terms)356 b(of)i(a)g(few)g (terms)f(of)h(\014rst)e(and)h(second)f(order,)373 b(so)357 b(m)-36 b(uc)g(h)356 b(closer)i(to)g(the)e(exp)36 b(ected)3432 23824 y(v)-72 b(alue)453 b(for)h(the)f(onset)g(of)g(sup)36 b(erconductivit)-36 b(y)-108 b(.)637 b(Our)452 b(non)h(p)36 b(erturbativ)-36 b(e)452 b(de\014nition)g(of)3432 25429 y(the)536 b(b)36 b(eta)538 b(function)f(is)g(in)-36 b(teresting)537 b(b)-36 b(y)537 b(itself,)565 b(as)537 b(it)h(could)f(b)36 b(e)537 b(used)f(to)i(detect)f(the)3432 27034 y(main)433 b(instabilities)i(of)f(the)f(mo)36 b(del)434 b(at)f(lo)-36 b(w)g(er)435 b(temp)36 b(eratures.)5383 28639 y(In)414 b(order)g(to)g(complete)h(our)f(program,)k(w)-36 b(e)415 b(should)f(pro)-36 b(v)g(e)414 b(that)488 b(^)-724 b Fy(\027)83 b Fz(\()38702 28289 y Fy(~)38769 28639 y(k)45 b(;)221 b(\025)p Fz(\))415 b(and)f Fy(")p Fz(\()45310 28289 y Fy(~)45377 28639 y(k)44 b Fz(\))3432 30244 y(can)303 b(b)36 b(e)304 b(c)-36 b(hosen)303 b(in)h(a)g(space)g(of)h(functions)e (with)h(the)f(same)i(di\013eren)-36 b(tiabilit)g(y)303 b(prop)36 b(erties)3432 31849 y(and)317 b(that)g(the)g(relation)h Fy(")15933 32048 y Fr(0)16459 31849 y Fz(\()16898 31499 y Fy(~)16965 31849 y(k)44 b Fz(\))369 b(=)g Fy(")p Fz(\()20989 31499 y Fy(~)21056 31849 y(k)44 b Fz(\))58 b(+)133 b(^)-725 b Fy(\027)84 b Fz(\()24576 31499 y Fy(~)24643 31849 y(k)45 b(;)221 b(\025)p Fz(\))318 b(can)f(b)36 b(e)318 b(solv)-36 b(ed)318 b(with)f(resp)36 b(ect)318 b(to)f Fy(")p Fz(\()44948 31499 y Fy(~)45015 31849 y(k)45 b Fz(\),)3432 33590 y(giv)-36 b(en)320 b Fy(")7323 33789 y Fr(0)7849 33590 y Fz(\()8288 33239 y Fy(~)8355 33590 y(k)44 b Fz(\))320 b(and)g Fy(\025)p Fz(.)540 b(This)320 b(w)-36 b(ould)320 b(imply)g(that)g(the)f(in)-36 b(tro)36 b(duction)319 b(of)i(the)e(coun)-36 b(terterm)3432 35195 y(is)424 b(only)g(a)g(tec)-36 b(hnical)424 b(tric)-36 b(k,)426 b(but)d(do)36 b(es)424 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b(in)f([DR)q(].)785 b(Ho)-36 b(w)g(ev)g(er,)520 b(ev)-36 b(en)502 b(in)h(this)f(case,)520 b(our)502 b(result)g(has)g(some)3432 65962 y(in)-36 b(terest,)531 b(since)511 b(w)-36 b(e)512 b(get)g(it)g(without)g(b)36 b(eing)511 b(in)-36 b(v)g(olv)g(ed)513 b(in)f(the)f(delicate)h(one)g (particle)3432 67567 y(irreducibilit)-36 b(y)433 b(analysis)i(of)g ([DR].)24692 75163 y(7)p eop %%Page: 8 8 8 7 bop 3432 7638 a FB(2)2151 b(Renormalization)717 b(Group)f(analysis) 3432 10946 y FA(2.1)1792 b(The)599 b(scale)f(decomp)50 b(osition)3432 13409 y Fz(It)404 b(is)h(con)-36 b(v)g(enien)g(t,)410 b(for)c(clarit)-36 b(y)405 b(reasons,)411 b(to)405 b(start)g(b)-36 b(y)404 b(studying)g(the)h("free)g(energy")g(of)3432 15014 y(the)433 b(mo)36 b(del,)434 b(de\014ned)e(as)14806 18318 y Fy(E)15769 18517 y Fx(L;\014)17668 18318 y Fz(=)368 b Ft(\000)20998 17419 y Fz(1)p 20214 18008 2219 45 v 20214 19229 a Fy(L)21099 18846 y Fr(2)21625 19229 y Fy(\014)22787 18318 y Fz(log)24688 16756 y Fl(Z)26017 18318 y Fy(P)181 b Fz(\()p Fy(d )29112 17770 y Fr(\()p Fs(\024)p Fr(1\))31100 18318 y Fz(\))p Fy(e)32209 17770 y Fs(\000V)33596 17457 y Fp(\(1\))34752 18318 y Fy(;)8816 b Fz(\(2.1\))14806 21143 y Fy(P)181 b Fz(\()p Fy(d )17901 20595 y Fr(\()p Fs(\024)p Fr(1\))19889 21143 y Fz(\))369 b Ft(\021)g Fy(P)181 b Fz(\()p Fy(d )48 b Fz(\))p Fy(;)2822 b Ft(V)29873 20595 y Fr(\(1\))31500 21143 y Ft(\021)369 b(V)404 b Fz(+)295 b Ft(N)565 b Fy(:)3432 24063 y Fz(Note)452 b(that)f(our)h(mo) 36 b(del)452 b(has)g(an)g(ultra)-36 b(violet)453 b(cut-o\013)e(in)h (the)33941 23713 y Fy(~)34008 24063 y(k)497 b Fz(momen)-36 b(tum,)456 b(but)451 b(the)3432 25669 y Fy(k)4108 25868 y Fr(0)5159 25669 y Fz(v)-72 b(ariable)526 b(is)f(un)-36 b(b)36 b(ounded)522 b(in)j(the)f(limit)i Fy(M)663 b Ft(!)525 b(1)p Fz(.)853 b(Hence,)547 b(it)525 b(is)g(con)-36 b(v)g(enien)g(t)525 b(to)3432 27274 y(decomp)36 b(ose)613 b(the)g(\014eld)g(as)h Fy( )18247 26792 y Fr(\()p Fs(\024)p Fr(1\))20911 27274 y Fz(=)675 b Fy( )23492 26792 y Fr(\(+1\))25898 27274 y Fz(+)418 b Fy( )28222 26792 y Fr(\()p Fs(\024)p Fr(0\))30210 27274 y Fz(,)659 b(where)613 b Fy( )36061 26792 y Fr(\(+1\))38664 27274 y Fz(and)g Fy( )42267 26792 y Fr(\()p Fs(\024)p Fr(0\))44869 27274 y Fz(are)3432 28879 y(indep)36 b(enden)-36 b(t)450 b(\014elds)i(whose)h(co)-36 b(v)-72 b(ariances)454 b(ha)-36 b(v)g(e)453 b(F)-108 b(ourier)452 b(transforms)g(with)h(supp) 36 b(ort,)3432 30484 y(resp)g(ectiv)-36 b(ely)-108 b(,)495 b(in)482 b(the)g Fw(ultr)-66 b(aviolet)509 b(r)-66 b(e)g(gion)480 b Fz(and)i(the)g Fw(infr)-66 b(ar)g(e)g(d)507 b(r)-66 b(e)g(gion)p Fz(,)493 b(de\014ned)481 b(in)i(the)3432 32089 y(follo)-36 b(wing)435 b(w)-36 b(a)g(y)-108 b(.)5383 33694 y(Item)530 b(1\))g(in)g(the)g(list)h(of)g(prop)36 b(erties)529 b(of)i(the)f(disp)36 b(ersion)530 b(relation)h(giv)-36 b(en)531 b(in)f Ft(x)p Fz(1.2)3432 35299 y(implies)434 b(that,)f(if)h Fy(H)13296 35498 y Fr(0)13822 35299 y Fz(\()p Fy(t)p Fz(\))f(is)h(a)f(smo)36 b(oth)434 b(function)f(of)i Fy(t)368 b Ft(2)h Fo(R)32352 34735 y Fr(1)32878 35299 y Fz(,)434 b(suc)-36 b(h)433 b(that)17482 38684 y Fy(H)18560 38883 y Fr(0)19085 38684 y Fz(\()p Fy(t)p Fz(\))369 b(=)22316 37068 y Fl(\032)23368 37918 y Fz(1)1301 b(if)434 b Fy(t)369 b(<)f(e)29333 38117 y Fr(0)29859 37918 y Fy(=\015)441 b(;)23368 39524 y Fz(0)1301 b(if)434 b Fy(t)369 b(>)f(e)29333 39723 y Fr(0)30228 39524 y Fy(;)43929 38684 y Fz(\(2.2\))3432 42370 y Fy(\015)440 b(>)369 b Fz(1)408 b(b)36 b(eing)408 b(a)g(parameter)f(to)h(b)36 b(e)408 b(\014xed)f(b)36 b(elo)-36 b(w,)414 b(then,)e(since)34923 42034 y(\026)34627 42370 y Fy(C)35653 41820 y Fs(\000)p Fr(1)35558 42660 y(0)36910 42370 y Fz(\()37349 42020 y Fy(~)37416 42370 y(k)45 b Fz(\))369 b(=)g(1)408 b(if)g Ft(j)p Fy(")p Fz(\()44034 42020 y Fy(~)44101 42370 y(k)45 b Fz(\))242 b Ft(\000)3432 43975 y Fy(\026)p Ft(j)368 b(\024)h Fy(e)6957 44174 y Fr(0)7483 43975 y Fz(,)10692 46560 y(\026)10396 46896 y Fy(C)11422 46347 y Fs(\000)p Fr(1)11327 47224 y(0)12679 46896 y Fz(\()13118 46545 y Fy(~)13185 46896 y(k)45 b Fz(\))1107 b(=)g Fy(C)18664 46347 y Fs(\000)p Fr(1)18569 47224 y(0)19921 46896 y Fz(\()p Fn(k)p Fz(\))295 b(+)g Fy(f)23965 47095 y Fr(1)24491 46896 y Fz(\()p Fn(k)p Fz(\))368 b Fy(;)10328 49331 y(C)11354 48783 y Fs(\000)p Fr(1)11259 49659 y(0)12612 49331 y Fz(\()p Fn(k)p Fz(\))1106 b(=)h Fy(H)18716 49530 y Fr(0)19463 47715 y Fl(\024)20047 47781 y(q)p 21154 47781 8845 45 v 1550 x Fy(k)21875 48874 y Fr(2)21830 49621 y(0)22696 49331 y Fz(+)295 b([)p Fy(")p Fz(\()25412 48981 y Fy(~)25479 49331 y(k)44 b Fz(\))295 b Ft(\000)h Fy(\026)p Fz(])29473 48947 y Fr(2)29998 47715 y Fl(\025)31173 49331 y Fy(;)12395 b Fz(\(2.3\))11445 52431 y Fy(f)12086 52630 y Fr(1)12612 52431 y Fz(\()p Fn(k)p Fz(\))1106 b(=)17934 52095 y(\026)17638 52431 y Fy(C)18664 51882 y Fs(\000)p Fr(1)18569 52759 y(0)19921 52431 y Fz(\()20360 52080 y Fy(~)20427 52431 y(k)45 b Fz(\))21875 50814 y Fl(\032)22706 52431 y Fz(1)295 b Ft(\000)g Fy(H)26057 52630 y Fr(0)26804 50814 y Fl(\024)27389 50880 y(q)p 28496 50880 V 1551 x Fy(k)29217 51973 y Fr(2)29172 52721 y(0)30038 52431 y Fz(+)f([)p Fy(")p Fz(\()32753 52080 y Fy(~)32820 52431 y(k)45 b Fz(\))295 b Ft(\000)h Fy(\026)p Fz(])36815 52047 y Fr(2)37340 50814 y Fl(\025\033)39345 52431 y Fy(:)3432 55864 y Fz(The)393 b(co)-36 b(v)-72 b(ariances)395 b Fy(g)13496 55382 y Fr(\(+1\))15878 55864 y Fz(and)e Fy(g)19038 55382 y Fr(\()p Fs(\024)p Fr(0\))21421 55864 y Fz(of)h(the)f(\014elds)g Fy( )29247 55382 y Fr(\(+1\))31629 55864 y Fz(and)g Fy( )35012 55382 y Fr(\()p Fs(\024)p Fr(0\))37394 55864 y Fz(are)h(de\014ned)e(as)i(in)3432 57469 y(\(1.8\),)434 b(with)g Fy(f)10504 57668 y Fr(1)11030 57469 y Fz(\()p Fn(k)p Fz(\))e(and)i Fy(C)16819 56919 y Fs(\000)p Fr(1)16724 57759 y(0)18076 57469 y Fz(\()p Fn(k)p Fz(\))f(in)g(place)h(of)26929 57133 y(\026)26633 57469 y Fy(C)27659 56919 y Fs(\000)p Fr(1)27564 57759 y(0)28916 57469 y Fz(\()29355 57119 y Fy(~)29422 57469 y(k)45 b Fz(\).)5383 59074 y(If)511 b(w)-36 b(e)510 b(p)36 b(erform)511 b(the)f(in)-36 b(tegration)510 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10436 y Fr(\()p Fx( )35688 9979 y Fp(\()p Fq(\024)p Fi(h)37130 10211 y(\014)37682 9979 y Fp(\))38057 10436 y Fr(\))38847 10984 y Fy(:)4071 b Fz(\(2.11\))5383 14335 y(If)379 b(w)-36 b(e)378 b(neglect)h(the)f(spin) g(indices)g(and)g(w)-36 b(e)379 b(put)f Fy(")29211 14534 y Fr(1)30105 14335 y Fz(=)369 b Fy(:)221 b(:)g(:)370 b Fz(=)f Fy(")35371 14534 y Fx(n)36366 14335 y Fz(=)g(+,)389 b Fy(")40118 14534 y Fx(n)p Fr(+1)42315 14335 y Fz(=)369 b Fy(:)221 b(:)g(:)370 b Fz(=)3432 15940 y Fy(")4041 16139 y Fr(2)p Fx(n)5506 15940 y Fz(=)f Ft(\000)p Fz(,)434 b(w)-36 b(e)433 b(can)h(write)g(the)f(e\013ectiv)-36 b(e)434 b(p)36 b(oten)-36 b(tials)434 b(in)f(the)g(form)6318 19579 y Ft(V)7241 19030 y Fr(\()p Fx(h)p Fr(\))8571 19579 y Fz(\()p Fy( )9971 19030 y Fr(\()p Fs(\024)p Fx(h)p Fr(\))12031 19579 y Fz(\))369 b(=)14703 18140 y Fs(1)14374 18472 y Fl(X)14286 20887 y Fx(n)p Fr(=1)16281 18017 y Fl(Z)17609 19579 y Fy(d)p Fn(x)19074 19778 y Fr(1)19600 19579 y Fy(:::d)p Fn(x)22148 19778 y Fr(2)p 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b(ernels,)3432 28084 y(as)557 b(w)-36 b(ell)558 b(as)f Fy(E)10596 28283 y Fx(L;\014)12126 28084 y Fz(,)588 b(ha)-36 b(v)g(e)558 b(w)-36 b(ell)557 b(de\014ned)f(limits)i(as)f Fy(M)696 b Fz(and)557 b Fy(L)g Fz(go)g(to)g(in\014nit)-36 b(y)-108 b(.)948 b(Suc)-36 b(h)3432 29689 y(result)638 b(is)g(ac)-36 b(hiev)g(ed)639 b(b)-36 b(y)638 b(studying)h(a)f (suitable)h(p)36 b(erturbativ)-36 b(e)637 b(expansion)i(of)g(these)3432 31294 y(quan)-36 b(tities)435 b(and)g(b)-36 b(y)435 b(pro)-36 b(ving)435 b(that)g(they)g(are)h(uniformly)f(\(in)g Fy(M)575 b Fz(and)434 b Fy(L)p Fz(\))h(con)-36 b(v)g(ergen)g(t)3432 32899 y(and,)577 b(in)549 b(the)f(case)i(of)f(the)g(k)-36 b(ernels,)578 b(that)549 b(they)f(ha)-36 b(v)g(e)550 b(fast)f(deca)-36 b(ying)550 b(prop)36 b(erties)548 b(in)3432 34504 y(the)604 b Fn(x)g Fz(v)-72 b(ariables;)691 b(see)605 b([BM])g(for)g(a)f(complete)h(analysis)h(of)f(this)f(t)-36 b(yp)36 b(e)605 b(in)f(the)g(one)3432 36109 y(dimensional)406 b(case.)569 b(Ho)-36 b(w)g(ev)g(er,)412 b(since)406 b(this)g(pro)36 b(cedure)404 b(is)i(cum)-36 b(b)36 b(ersome)405 b(and)g(di\016cult)3432 37714 y(to)529 b(describ)36 b(e)529 b(rigorously)h(without)f(making)h (obscure)f(the)f(main)h(ideas,)554 b(whic)-36 b(h)529 b(ha)-36 b(v)g(e)3432 39319 y(nothing)565 b(to)h(do)g(with)g(the)g (details)h(related)f(with)g(the)f(\014nite)h(v)-72 b(alues)566 b(of)h Fy(M)705 b Fz(and)566 b Fy(L)p Fz(,)3432 40924 y(w)-36 b(e)424 b(shall)h(discuss)f(in)h(the)e(follo)-36 b(wing)427 b(only)e(the)f(formal)h(limit)g(of)h(our)e(expansions)g(and) 3432 42529 y(w)-36 b(e)512 b(shall)h(pro)-36 b(v)g(e)512 b(that)g(the)g(k)-36 b(ernels)513 b(as)f(w)-36 b(ell)513 b(as)g(the)f(free)h(energy)f(constan)-36 b(ts)512 b Fy(E)43759 42728 y Fx(h)44869 42529 y Fz(are)3432 44134 y(w)-36 b(ell)504 b(de\014ned.)786 b(F)-108 b(or)503 b(similar)h(reasons,)522 b(w)-36 b(e)503 b(shall)h(also)h(consider)e Fy(k)36676 44333 y Fr(0)37706 44134 y Fz(as)g(a)h(con)-36 b(tin)g(uous)3432 45740 y(v)-72 b(ariable)458 b(and)f(w)-36 b(e)458 b(shall)g(tak)-36 b(e)458 b(in)-36 b(to)458 b(accoun)-36 b(t)457 b(the)g(essen)-36 b(tial)458 b(infrared)f(cut-o\013)g(related)3432 47345 y(with)468 b(the)f(\014nite)g(temp)36 b(erature)467 b(v)-72 b(alue,)478 b(b)-36 b(y)467 b(preserving)h(the)g(de\014nition)f (\(2.10\))i(of)g(the)3432 48950 y(cut-o\013)432 b(functions.)578 b(This)434 b(means,)g(in)f(particular)h(that,)f(from)h(no)-36 b(w)434 b(on)18921 51323 y(1)p 18137 51912 V 18137 53133 a Fy(L)19022 52749 y Fr(2)19548 53133 y Fy(\014)20887 51115 y Fl(X)20710 53575 y Fm(k)p Fs(2D)23032 52222 y Ft(!)26024 51323 y Fz(1)p 24863 51912 2974 45 v 24863 53133 a(\(2)p Fy(\031)48 b Fz(\))27311 52749 y Fr(3)28190 50660 y Fl(Z)28805 53175 y Fs(D)29836 52222 y Fy(d)p Fn(k)368 b Fy(:)11249 b Fz(\(2.13\))3432 55923 y(Moreo)-36 b(v)g(er,)503 b(w)-36 b(e)489 b(shall)h(supp)36 b(ose)488 b(that)h(the)f(space)h(co)36 b(ordinates)490 b(are)f(con)-36 b(tin)g(uous)488 b(v)-72 b(ari-)3432 57528 y(ables,)392 b(b)36 b(oth)380 b(in)h(the)g(con)-36 b(tin)g(uum)379 b(and)i(lattice)g(mo)36 b(dels.)562 b(This)381 b(means)g(that,)391 b(from)382 b(no)-36 b(w)3432 59134 y(on,)5596 58187 y Fl(R)6555 59134 y Fy(d)p Fn(x)429 b Fz(will)h(denote)f(the)f(in)-36 b(tegral)430 b(o)-36 b(v)g(er)430 b Fo(R)25789 58569 y Fr(3)26315 59134 y Fz(.)577 b(Finally)-108 b(,)431 b(w)-36 b(e)429 b(shall)h(still)g(use)e(the)h(sym-)3432 60739 y(b)36 b(ol)516 b Fy(L)6603 60257 y Fr(2)7129 60739 y Fy(\014)590 b Fz(to)516 b(denote)f(the)h(formally)h(in\014nite)f (space-time)f(v)-36 b(olume)517 b(in)f(the)f(extensiv)-36 b(e)3432 62344 y(quan)g(tities)433 b(lik)-36 b(e)435 b Fy(L)12706 61862 y Fr(2)13232 62344 y Fy(\014)74 b(E)15003 62543 y Fx(h)15600 62344 y Fz(.)3432 66169 y FA(2.2)1792 b(The)599 b(lo)50 b(calization)597 b(pro)50 b(cedure)3432 68632 y Fz(Let)416 b(us)g(no)-36 b(w)416 b(describ)36 b(e)416 b(our)h(expansion,)j(whic)-36 b(h)416 b(is)h(pro)36 b(duced)415 b(b)-36 b(y)417 b(using)f(an)g(inductiv)-36 b(e)3432 70237 y(pro)36 b(cedure.)601 b(First)442 b(of)g(all,)j(w)-36 b(e)442 b(de\014ne)f(an)g Ft(L)h Fz(op)36 b(erator)442 b(acting)g(on)g(the)f(k)-36 b(ernels)442 b(in)g(the)3432 71842 y(follo)-36 b(wing)435 b(w)-36 b(a)g(y:)24367 75163 y(10)p eop %%Page: 11 11 11 10 bop 5021 7653 a Fz(1.)651 b Ft(L)p Fy(W)9008 6974 y Fr(\()p Fx(h)p Fr(\))8827 7943 y(2)p Fx(n)10706 7653 y Fz(=)369 b(0)434 b(if)g Fy(n)370 b Ft(\025)f Fz(3.)5021 10365 y(2.)651 b(If)539 b Fy(n)546 b Fz(=)g(2)538 b(and)f(w)-36 b(e)538 b(put)f Fn(x)p 19297 10542 789 45 v 546 w Fz(=)545 b(\()p Fn(x)23484 10564 y Fr(1)24010 10365 y Fy(;)221 b(:)g(:)g(:)i(;)e Fn(x)27711 10564 y Fr(4)28238 10365 y Fz(\),)563 b Fn(x)30457 10564 y Fx(i)31379 10365 y Fz(=)546 b(\()p Fy(x)34182 10564 y Fx(i;)p Fr(0)35289 10365 y Fy(;)200 b(~)-629 b(x)36610 10564 y Fx(i)36987 10365 y Fz(\),)38487 10347 y(~)38418 10365 y Fn(x)39207 10564 y Fx(i)40128 10365 y Fz(=)546 b(\()81 b(~)-731 b Fy(x)42931 10564 y Fx(i;)p Fr(0)44038 10365 y Fy(;)200 b(~)-629 b(x)45359 10564 y Fx(i)45736 10365 y Fz(\),)6683 11970 y Fy(\016)50 b Fz(\()p Fy(x)p 7815 12147 740 45 v 277 x Fr(0)9080 11970 y Fz(\))369 b(=)f Fy(\016)50 b Fz(\()p Fy(x)13206 12169 y Fr(1)p Fx(;)p Fr(0)14759 11970 y Ft(\000)295 b Fy(x)16826 12169 y Fr(2)p Fx(;)p Fr(0)18084 11970 y Fz(\))p Fy(\016)50 b Fz(\()p Fy(x)20461 12169 y Fr(1)p Fx(;)p Fr(0)22014 11970 y Ft(\000)295 b Fy(x)24081 12169 y Fr(3)p Fx(;)p Fr(0)25339 11970 y Fz(\))p Fy(\016)50 b Fz(\()p Fy(x)27716 12169 y Fr(1)p Fx(;)p Fr(0)29268 11970 y Ft(\000)295 b Fy(x)31335 12169 y Fr(4)p Fx(;)p Fr(0)32593 11970 y Fz(\))15374 15342 y Ft(L)p Fy(W)17699 14664 y Fr(\()p Fx(h)p Fr(\))17518 15633 y(4)19028 15342 y Fz(\()p Fn(x)p 19534 15519 789 45 v Fz(\))369 b(=)f Fy(\016)50 b Fz(\()p Fy(x)p 23710 15519 740 45 v 277 x Fr(0)24975 15342 y Fz(\))25702 13780 y Fl(Z)27030 15342 y Fy(d)p Fz(\()44 b(~)-694 b Fy(x)p 28212 15519 V 277 x Fr(0)29477 15342 y Ft(n)81 b Fz(~)-731 b Fy(x)30880 15541 y Fr(1)p Fx(;)p Fr(0)32138 15342 y Fz(\))p Fy(W)34053 14664 y Fr(\()p Fx(h)p Fr(\))33872 15633 y(4)35382 15342 y Fz(\()35957 15324 y(~)35888 15342 y Fn(x)p 35888 15519 789 45 v -1 w Fz(\))369 b Fy(:)5367 b Fz(\(2.14\))6683 18602 y(Note)401 b(that,)407 b(b)36 b(ecause)400 b(of)h(translation)g(in)-36 b(v)-72 b(ariance,)408 b(this)400 b(de\014nition)g(is)h(indep)36 b(en-)6683 20207 y(den)-36 b(t)370 b(of)h(the)f(c)-36 b(hoice)371 b(of)g(the)f Fw(lo)-66 b(c)g(alization)405 b(p)-66 b(oint)p Fz(,)382 b(that)370 b(is)h(the)f(p)36 b(oin)-36 b(t)370 b(whose)h(time)6683 21812 y(co)36 b(ordinate)434 b(is)g(not)f(in)-36 b(tegrated)434 b(\()p Fn(x)24172 22011 y Fr(1)25130 21812 y Fz(in)g(\(2.14\)\).)5021 24705 y(3.)651 b(If)498 b Fy(n)477 b Fz(=)f(1)497 b(and)g(w)-36 b(e)497 b(put)f(\(b)-36 b(y)496 b(using)h(translation)g(in)-36 b(v)-72 b(ariance\))497 b Fy(W)39562 24026 y Fr(\()p Fx(h)p Fr(\))39381 24995 y(2)40891 24705 y Fz(\()p Fn(x)42186 24904 y Fr(1)42712 24705 y Fy(;)221 b Fn(x)44083 24904 y Fr(2)44609 24705 y Fz(\))476 b(=)7062 26155 y(~)6683 26491 y Fy(W)8092 25812 y Fr(\()p Fx(h)p 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Fz(\))433 b(is)h(de\014ned)e(inductiv)-36 b(ely)434 b(b)-36 b(y)434 b(the)f(relation)4382 61798 y Ft(V)5305 61250 y Fr(\()p Fx(h)p Fr(\))6635 61798 y Fz(\()p Fy(\034)76 b(;)221 b( )9258 61250 y Fr(\()p Fs(\024)p Fx(h)p Fr(\))11319 61798 y Fz(\))368 b(=)13707 60899 y(\()p Ft(\000)p Fz(1\))16402 60417 y Fx(s)p Fr(+1)p 13707 61488 4388 45 v 15414 62709 a Fy(s)p Fz(!)18227 61798 y Ft(E)19047 61250 y Fx(T)18928 62126 y(h)p Fr(+1)20728 61798 y Fz([)21262 61462 y(\026)21089 61798 y Ft(V)22012 61250 y Fr(\()p Fx(h)p Fr(+1\))24545 61798 y Fz(\()p Fy(\034)25616 61997 y Fr(1)26141 61798 y Fy(;)221 b( )27617 61250 y Fr(\()p Fs(\024)p Fx(h)p Fr(+1\))30881 61798 y Fz(\);)g Fy(:)g(:)g(:)i Fz(;)34473 61462 y(\026)34299 61798 y Ft(V)35222 61250 y Fr(\()p Fx(h)p Fr(+1\))37755 61798 y Fz(\()p Fy(\034)38826 61997 y Fx(s)39316 61798 y Fy(;)e( )40792 61250 y Fr(\()p Fs(\024)p Fx(h)p Fr(+1\))44055 61798 y Fz(\)])369 b Fy(;)43279 63816 y Fz(\(2.37\))3432 65421 y(and)6134 65086 y(\026)5961 65421 y Ft(V)6884 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Fr(\(1\))5612 71842 y Fz(\()p Fy( )7012 71360 y Fs(\024)p Fr(1)8269 71842 y Fz(\).)24367 75163 y(16)p eop %%Page: 17 17 17 16 bop 3865 7638 a Ft(E)4685 7156 y Fx(T)4566 7966 y(h)p Fr(+1)6800 7638 y Fz(denotes)433 b(the)g(truncated)f(exp)36 b(ectation)434 b(with)g(resp)36 b(ect)433 b(to)h(the)f(measure)3432 8358 y Fl(Q)4477 9520 y Fx(!)5371 9243 y Fy(P)181 b Fz(\()p Fy(d )8466 8761 y Fr(\()p Fx(h)p Fr(+1\))8418 9572 y Fx(!)10996 9243 y Fz(\),)434 b(that)f(is)11677 12177 y Ft(E)12497 11628 y Fx(T)12378 12505 y(h)p Fr(+1)14178 12177 y Fz(\()p Fy(X)15763 12376 y Fr(1)16289 12177 y Fz(;)221 b Fy(:)g(:)g(:)j Fz(;)d Fy(X)20281 12376 y Fx(p)20811 12177 y Fz(\))369 b Ft(\021)11677 15033 y(\021)15598 14134 y Fy(@)16357 13652 y Fx(p)p 13212 14723 6060 45 v 13212 15944 a Fy(@)72 b(\025)14730 16143 y Fr(1)15477 15944 y Fy(:)221 b(:)g(:)i(@)72 b(\025)18743 16143 y Fx(p)19759 15033 y Fz(log)21660 13470 y Fl(Z)22988 13926 y(Y)23387 16250 y Fx(!)24624 15033 y Fy(P)181 b Fz(\()p Fy(d )27719 14484 y 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y(where)433 b(w)-36 b(e)434 b(use)f(the)g(de\014nitions)g(\()p Fy(@)21078 46644 y Fr(0)22037 46445 y Fz(is)h(from)g(no)-36 b(w)433 b(on)h(the)f(time)h (deriv)-72 b(ativ)-36 b(e\))6727 49209 y(~)6461 49560 y Fy( )7355 48881 y Fr(\()p Fs(\024)p Fx(h)p Fr(\))7307 49884 y(\012)7987 49995 y Fi(v)9415 49560 y Fz(\()p Fy(P)10759 49759 y Fx(v)11305 49560 y Fz(\))368 b(=)13987 48453 y Fl(Y)13560 50913 y Fx(f)98 b Fs(2)p Fx(P)15334 51024 y Fi(v)16049 49560 y Fy(e)16652 49011 y Fx(i")p Fr(\()p Fx(f)g Fr(\))12 b Fx(~)-482 b(p)19162 49160 y Fi(F)19840 49011 y Fr(\()p Fx(\022)20645 49204 y Fi(h;!)24 b Fp(\()p Fi(f)76 b Fp(\))23090 49011 y Fr(\))-12 b Fx(~)-458 b(x)p Fr(\()p Fx(f)98 b Fr(\))25323 49560 y Fy(@)26082 48881 y Fx(m)p Fr(\()p Fx(f)g Fr(\))26010 49850 y(0)28251 49560 y Fy( )29145 48881 y Fr(\()p Fs(\024)p Fx(h)p Fr(\))p Fx(")p Fr(\()p Fx(f)g Fr(\))29097 49943 y Fm(x)p Fr(\()p Fx(f)g Fr(\))p Fx(;!)32 b Fr(\()p Fx(f)98 b Fr(\))34466 49560 y Fy(;)1522 b(h)369 b Ft(\024)g Fz(0)h Fy(;)3029 b 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59045 y Fy(\027)84 b Fz(\()-22 b Fy(~)-628 b(x)294 b Ft(\000)282 b Fy(~)-636 b(y)47 b Fz(\))p Fy(\016)j Fz(\()p Fy(x)19816 59244 y Fr(0)20637 59045 y Ft(\000)295 b Fy(y)22599 59244 y Fr(0)23125 59045 y Fz(\))2010 b(if)434 b Fy(v)27462 59244 y Fx(i)28271 59045 y Fz(is)g(of)g(t)-36 b(yp)36 b(e)434 b Fy(\027)84 b Fz(,)43279 60484 y(\(2.44\))3432 62089 y(and)319 b(w)-36 b(e)321 b(supp)36 b(ose)319 b(that)h(the)f(order)h(of)h(the)e(\(an)-36 b(ticomm)g(uting\))319 b(\014eld)h(v)-72 b(ariables)321 b(in)f(\(2.43\))3432 63694 y(is)482 b(suitable)g(c)-36 b(hosen)481 b(in)h(order)f(to)h(\014x)g(the)f(sign)i(as)f(in)f (\(2.41\).)725 b(Note)482 b(that)f(the)g(terms)3432 65299 y(with)559 b Fy(P)7358 65498 y Fx(v)7816 65621 y Fp(0)8916 65299 y Fz(=)583 b Ft(;)559 b Fz(in)h(the)e(r.h.s.)956 b(of)560 b(\(2.39\))g(con)-36 b(tribute)559 b(to)g Fy(L\014)35102 64964 y Fz(~)34799 65299 y Fy(E)35762 65498 y Fr(1)36288 65299 y Fz(,)591 b(while)560 b(the)e(others)3432 66904 y(con)-36 b(tribute)432 b(to)i Ft(V)12195 66422 y Fr(\(0\))13453 66904 y Fz(\()p Fy( )14853 66422 y Fr(\()p Fs(\024)p Fr(0\))16841 66904 y Fz(\).)5383 68510 y(W)-108 b(e)381 b(no)-36 b(w)381 b(write)h Ft(V)14422 68027 y Fr(\(0\))16061 68510 y Fz(as)f Ft(LV)19444 68027 y Fr(\(0\))20890 68510 y Fz(+)188 b Ft(RV)24139 68027 y Fr(\(0\))25397 68510 y Fz(,)392 b(with)381 b Ft(LV)30899 68027 y Fr(\(0\))32538 68510 y Fz(de\014ned)f(as)h(in)g Ft(x)p Fz(2.2)i(\(it)e(rep-)3432 70115 y(resen)-36 b(t,)396 b(in)388 b(the)f(usual)h(R)-36 b(G)387 b(language,)398 b(the)387 b(relev)-72 b(an)-36 b(t)388 b(and)f(marginal)i(con)-36 b(tributions)387 b(to)3432 71720 y Ft(V)4355 71238 y Fr(\(0\))5612 71720 y Fz(\()p Fy( )7012 71238 y Fr(\()p Fs(\024)p Fr(0\))9001 71720 y Fz(\)\),)419 b(and)c(w)-36 b(e)416 b(write)g(for)g Ft(RV)22525 71238 y Fr(\(0\))24199 71720 y Fz(a)f(decomp)36 b(osition)416 b(similar)h(to)f(the)f(previous)24367 75163 y(17)p eop %%Page: 18 18 18 17 bop 3432 7638 a Fz(one,)494 b(with)481 b Ft(RV)11297 7156 y 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y Fi(v)28723 12962 y Fp(0)29294 12453 y Fz(\))p Fy(K)31002 11774 y Fr(\(1\))30907 12778 y Fx(\034)50 b(;P)32233 12889 y Fi(v)32648 13088 y Fp(0)33219 12453 y Fz(\()p Fn(x)34514 12652 y Fx(v)34972 12775 y Fp(0)35488 12453 y Fz(\))369 b Fy(;)6555 b Fz(\(2.45\))3432 15178 y(where)433 b Fn(x)7978 15377 y Fx(P)8575 15488 y Fi(v)8990 15687 y Fp(0)9930 15178 y Fz(=)369 b Ft([)12197 15377 y Fx(f)98 b Fs(2)p Fx(P)13971 15488 y Fi(v)14386 15687 y Fp(0)14957 15178 y Fn(x)p Fz(\()p Fy(f)142 b Fz(\).)578 b(In)433 b(order)g(to)h(remem)-36 b(b)36 b(er)433 b(this)g(c)-36 b(hoice,)434 b(w)-36 b(e)434 b(write)10093 18688 y Ft(RV)12142 18140 y Fr(\(0\))13399 18688 y Fz(\()p Fy(\034)76 b(;)221 b(P)15966 18887 y Fx(v)16424 19010 y Fp(0)16942 18688 y Fy(;)g Fz(\012)18463 18887 y Fx(v)18921 19010 y Fp(0)19439 18688 y Fz(\))368 b(=)21694 17126 y Fl(Z)23022 18688 y Fy(d)p Fn(x)24487 18887 y Fx(v)24945 19010 y Fp(0)25728 18337 y Fz(~)25462 18688 y Fy( )26356 18009 y Fr(\()p Fs(\024)p Fr(0\))26308 19013 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31578 y Fz(the)g(in)-36 b(tersection)3432 33183 y(of)450 b Fy(P)5767 33382 y Fx(v)6762 33183 y Fz(and)f Fy(P)10145 33382 y Fx(v)10603 33517 y Fi(i)11010 33183 y Fz(;)458 b(this)449 b(de\014nition)f(implies)i(that)f Fy(P)28280 33382 y Fx(v)29222 33183 y Fz(=)395 b Ft([)31515 33382 y Fx(i)31891 33183 y Fy(Q)32921 33382 y Fx(v)33379 33517 y Fi(i)33785 33183 y Fz(.)626 b(The)450 b(subsets)e Fy(P)42817 33382 y Fx(v)43275 33517 y Fi(i)43683 33183 y Ft(n)p Fy(Q)45377 33382 y Fx(v)45835 33517 y Fi(i)46242 33183 y Fz(,)3432 34788 y(whose)474 b(union)f Ft(I)11685 34987 y Fx(v)12704 34788 y Fz(will)h(b)36 b(e)474 b(made,)484 b(b)-36 b(y)473 b(de\014nition,)483 b(of)474 b(the)f Fw(internal)501 b(\014elds)473 b Fz(of)h Fy(v)48 b Fz(,)484 b(ha)-36 b(v)g(e)3432 36393 y(to)433 b(b)36 b(e)434 b(non)f(empt)-36 b(y)-108 b(,)433 b(if)h Fy(s)15353 36592 y Fx(v)16267 36393 y Fy(>)369 b Fz(1.)5383 37998 y(Moreo)-36 b(v)g(er,)518 b(w)-36 b(e)501 b(asso)36 b(ciate)502 b(with)e(an)-36 b(y)501 b Fy(f)625 b 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66570 y Fi(v)26723 66769 y Fp(0)27737 66135 y Fz(\()p Fy(P)29081 66334 y Fx(v)29539 66457 y Fp(0)30056 66135 y Fz(\))p Fy(K)31764 65456 y Fr(\()p Fx(h)p Fr(+1\))31669 66462 y Fx(\034)50 b(;)p Fm(P)p Fx(;)p Fr(\012)34295 66135 y Fz(\()p Fn(x)35590 66334 y Fx(v)36048 66457 y Fp(0)36564 66135 y Fz(\))369 b Fy(;)5479 b Fz(\(2.48\))3432 69815 y(with)327 b Fy(K)7490 69136 y Fr(\()p Fx(h)p Fr(+1\))7395 70142 y Fx(\034)50 b(;)p Fm(P)p Fx(;)p Fr(\012)10022 69815 y Fz(\()p Fn(x)11317 70014 y Fx(v)11775 70137 y Fp(0)12291 69815 y Fz(\))327 b(de\014ned)f(inductiv)-36 b(ely)328 b(\(recall)g(that)f Fy(h)31405 70014 y Fx(v)31863 70137 y Fp(0)32749 69815 y Fz(=)368 b Fy(h)78 b Fz(+)g(1\))328 b(b)-36 b(y)327 b(the)g(equation,)24367 75163 y(18)p eop %%Page: 19 19 19 18 bop 3432 7638 a Fz(v)-72 b(alid)434 b(for)g(an)-36 b(y)434 b Fy(v)416 b Ft(2)369 b Fy(\034)582 b Fz(whic)-36 b(h)433 b(is)h(not)f(an)h(endp)36 b(oin)-36 b(t,)3476 10701 y Fy(K)4678 10022 y Fr(\()p Fx(h)5586 10133 y Fi(v)6080 10022 y Fr(\))4583 11028 y Fx(\034)50 b(;)p Fm(P)p Fx(;)p Fr(\012)7047 10701 y Fz(\()p Fn(x)8342 10900 y Fx(v)8887 10701 y Fz(\))369 b(=)11710 9802 y(1)p 11275 10391 1520 45 v 11275 11612 a Fy(s)11888 11811 y Fx(v)12433 11612 y Fz(!)13445 9250 y Fx(s)13880 9361 y Fi(v)13203 9594 y Fl(Y)13149 12024 y Fx(i)p Fr(=1)14671 10701 y Fz([)p Fy(K)16234 10152 y Fr(\()p Fx(h)17142 10263 y Fi(v)17637 10152 y Fr(+1\))16139 11029 y Fx(v)16597 11164 y Fi(i)19261 10701 y Fz(\()p Fn(x)20556 10900 y Fx(v)21014 11035 y Fi(i)21420 10701 y Fz(\)])738 b Ft(E)23845 10152 y Fx(T)23726 11029 y(h)24268 11140 y Fi(v)24818 10701 y Fz([)25445 10350 y(~)25179 10701 y Fy( )26073 10022 y Fr(\()p Fx(h)26981 10133 y Fi(v)27475 10022 y Fr(\))26025 11026 y(\012)26705 11149 y Fp(1)27896 10701 y Fz(\()p Fy(P)29240 10900 y Fx(v)29698 11023 y Fp(1)30215 10701 y Ft(n)p Fy(Q)31909 10900 y Fx(v)32367 11023 y Fp(1)32884 10701 y Fz(\))p Fy(;)221 b(:)g(:)g(:)j(;)36569 10350 y Fz(~)36303 10701 y Fy( )37197 10022 y Fr(\()p Fx(h)38105 10133 y Fi(v)38599 10022 y Fr(\))37149 11026 y(\012)37829 11137 y Fi(s)38217 11248 y(v)39020 10701 y Fz(\()p Fy(P)40364 10900 y Fx(v)40822 11011 y Fi(s)41210 11122 y(v)41816 10701 y Ft(n)p Fy(Q)43510 10900 y Fx(v)43968 11011 y Fi(s)44356 11122 y(v)44962 10701 y Fz(\)])369 b Fy(;)43279 13242 y Fz(\(2.49\))3432 15028 y(where)484 b(\012)8179 15227 y Fx(i)9012 15028 y Fz(=)456 b Ft(f)p Fy(!)48 b Fz(\()p Fy(f)142 b Fz(\))p Fy(;)221 b(f)598 b Ft(2)456 b Fy(P)17798 15227 y Fx(v)18256 15362 y Fi(i)18663 15028 y Ft(n)p Fy(Q)20357 15227 y Fx(v)20815 15362 y Fi(i)21223 15028 y Ft(g)485 b Fz(and)25218 14677 y(~)24952 15028 y Fy( )25846 14349 y Fr(\()p Fx(h)26754 14460 y Fi(v)27248 14349 y Fr(\))25798 15353 y(\012)26478 15488 y Fi(i)27669 15028 y Fz(\()p Fy(P)29013 15227 y Fx(v)29471 15362 y Fi(i)29878 15028 y Ft(n)p Fy(Q)31572 15227 y Fx(v)32030 15362 y Fi(i)32437 15028 y Fz(\))g(has)g(a)g (de\014nition)f(similar)3432 16633 y(to)433 b(\(2.42\),)i(if)f Fy(h)11081 16832 y Fx(v)11995 16633 y Ft(\024)369 b 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b(of)g(t)-36 b(yp)36 b(e)433 b Fy(\025)p Fz(,)14373 23361 y Fy(\015)15120 22879 y Fx(h)15662 22990 y Fi(v)16156 22879 y Fs(\000)p Fr(1)17414 23361 y Fy(\027)18057 23560 y Fx(h)18599 23671 y Fi(v)19093 23560 y Fs(\000)p Fr(1)20351 23361 y Fz(\()-22 b Fy(~)-628 b(x)295 b Ft(\000)281 b Fy(~)-636 b(y)48 b Fz(\))p Fy(\016)i Fz(\()p Fy(x)26278 23560 y Fr(0)27098 23361 y Ft(\000)296 b Fy(y)29061 23560 y Fr(0)29587 23361 y Fz(\))1300 b(if)435 b Fy(v)481 b Fz(is)434 b(of)g(t)-36 b(yp)36 b(e)433 b Fy(\027)84 b Fz(,)14373 24966 y Fy(z)14977 25165 y Fx(h)15519 25276 y Fi(v)16013 25165 y Fs(\000)p Fr(1)17271 24966 y Fz(\()-22 b Fy(~)-628 b(x)295 b Ft(\000)281 b Fy(~)-635 b(y)47 b Fz(\))p Fy(\016)j Fz(\()p Fy(x)23198 25165 y Fr(0)24019 24966 y Ft(\000)295 b Fy(y)25981 25165 y Fr(0)26507 24966 y Fz(\))4380 b(if)435 b Fy(v)481 b Fz(is)434 b(of)g(t)-36 b(yp)36 b(e)433 b Fy(z)59 b Fz(,)43279 23361 y(\(2.50\))3432 27211 y(where)425 b Fn(x)7970 27410 y Fx(v)8884 27211 y Fz(=)368 b(\()p Fn(x)11559 27410 y 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70237 y(expand)609 b(eac)-36 b(h)610 b(term)g(in)g(the)f(r.h.s.)1108 b(of)611 b(\(2.54\),)656 b(b)-36 b(y)610 b(a)g(pro)36 b(cedure)609 b(similar)i(to)f(that)3432 71842 y(explained)433 b(in)h([BM].)24367 75163 y(20)p eop %%Page: 21 21 21 20 bop 5383 7638 a Fz(W)-108 b(e)439 b(start)g(this)g(analysis)i(b) -36 b(y)439 b(supp)36 b(osing)439 b(that)g Ft(j)p Fy(P)30085 7837 y Fx(v)30543 7960 y Fp(0)31060 7638 y Ft(j)378 b Fy(>)h Fz(0)439 b(\(otherwise)h(there)e(is)i(no)3432 9243 y Ft(R)547 b Fz(op)36 b(eration)548 b(acting)g(on)g Fy(v)17602 9442 y Fr(0)18127 9243 y Fz(\))g(and)f(b)-36 b(y)548 b(considering)g(the)f(action)h(of)g Ft(R)g Fz(on)f(a)h(single) 3432 10848 y(con)-36 b(tribution)522 b(to)h(the)g(sum)g(in)g(the)f (r.h.s.)848 b(of)524 b(\(2.54\).)848 b(This)524 b(action)f(is)h Fw(trivial)p Fz(,)544 b(that)3432 12453 y(is)596 b Ft(R)644 b Fz(=)h Fy(I)104 b Fz(,)637 b(b)-36 b(y)595 b(de\014nition,)636 b(if)596 b Ft(j)p Fy(P)20879 12652 y Fx(v)21337 12775 y Fp(0)21854 12453 y Ft(j)645 b 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b Fy(m)p Fz(\()p Fy(f)142 b Fz(\))633 b(=)h(0)590 b(for)g(all)g Fy(f)777 b Ft(2)633 b Fy(P)27725 19073 y Fx(v)28183 19196 y Fp(0)28701 18874 y Fz(.)1045 b(If)591 b Fy(P)32402 19073 y Fx(v)32860 19196 y Fp(0)34011 18874 y Fz(=)634 b(\()p Fy(f)36804 19073 y Fr(1)37330 18874 y Fy(;)221 b(f)38553 19073 y Fr(2)39079 18874 y Fy(;)g(f)40302 19073 y Fr(3)40829 18874 y Fy(;)g(f)42052 19073 y Fr(4)42578 18874 y Fz(\),)629 b(with)3432 20479 y Fy(")p Fz(\()p Fy(f)5188 20678 y Fr(1)5713 20479 y Fz(\))369 b(=)f Fy(")p Fz(\()p Fy(f)9724 20678 y Fr(2)10250 20479 y Fz(\))h(=)f(+)h(=)f Ft(\000)p Fy(")p Fz(\()p Fy(f)18055 20678 y Fr(3)18581 20479 y Fz(\))h(=)f Ft(\000)p Fy(")p Fz(\()p Fy(f)23625 20678 y Fr(4)24151 20479 y Fz(\),)402 b(and)393 b(w)-36 b(e)393 b(put)g Fn(x)p Fz(\()p Fy(f)34064 20678 y Fx(i)34439 20479 y Fz(\))369 b(=)g Fn(x)37484 20678 y Fx(i)37859 20479 y Fz(,)38691 20461 y(~)38622 20479 y Fn(x)39411 20678 y Fx(i)40155 20479 y Fz(=)g(\()p Fy(x)42781 20678 y Fr(1)p Fx(;)p Fr(0)44038 20479 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28125 y(\001)12357 26841 y Fl(n)13095 28125 y Fz(\()p Fy(x)14340 28324 y Fr(2)p Fx(;)p Fr(0)15893 28125 y Ft(\000)295 b Fy(x)17960 28324 y Fr(1)p Fx(;)p Fr(0)19218 28125 y Fz(\))p Fy( )20618 27577 y Fr(\()p Fs(\024)p Fx(h)p Fr(\)+)20570 28454 y Fm(x)21143 28577 y Fp(1)21603 28454 y Fx(;!)22449 28577 y Fp(1)23410 28125 y Fz([)23934 27775 y(^)23771 28125 y Fy(@)24530 27577 y Fr(1)25056 28125 y Fz(\()p Fy(x)26301 28324 y Fr(1)p Fx(;)p Fr(0)27559 28125 y Fz(\))p Fy( )28959 27577 y Fr(\()p Fs(\024)p Fx(h)p Fr(\)+)28911 28454 y Fm(x)29484 28577 y Fp(2)29944 28454 y Fx(;!)30790 28577 y Fp(2)31751 28125 y Fz(])p Fy( )33006 27577 y Fr(\()p Fs(\024)p Fx(h)p Fr(\))p Fs(\000)32958 28454 y Fm(x)33531 28577 y Fp(3)33992 28454 y Fx(;!)34838 28577 y Fp(3)35798 28125 y Fy( )36692 27577 y Fr(\()p Fs(\024)p Fx(h)p Fr(\))p Fs(\000)36644 28454 y Fm(x)37217 28577 y Fp(4)37678 28454 y Fx(;!)38524 28577 y Fp(4)39485 28125 y Fz(+)11398 30576 y(+\()p Fy(x)13655 30775 y Fr(3)p Fx(;)p Fr(0)15207 30576 y Ft(\000)h 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Fr(0)33370 41745 y Fz(+)294 b Fy(s)p Fz(\()p Fy(y)36429 41944 y Fr(0)37250 41745 y Ft(\000)i Fy(x)39318 41944 y Fr(0)39843 41745 y Fz(\))369 b Fy(:)2200 b Fz(\(2.56\))3432 45133 y(Similar)379 b(expressions)g(are)g(obtained,) 389 b(if)380 b(the)e(lo)36 b(calization)380 b(p)36 b(oin)-36 b(t)379 b(\(see)f(commen)-36 b(t)379 b(after)3432 46738 y(\(2.14\)\))434 b(is)g(c)-36 b(hanged.)5383 48343 y(Let)453 b(us)h(no)-36 b(w)455 b(consider)f(the)g(case)h Ft(j)p Fy(P)23482 48542 y Fx(v)23940 48665 y Fp(0)24457 48343 y Ft(j)404 b Fz(=)g(2.)641 b(If)455 b(only)g(one)g(of)g(the)f(external) g(\014elds)3432 49949 y(of)g Fy(v)5562 50148 y Fr(0)6541 49949 y Fz(carries)g(a)g Fy(@)12478 50148 y Fr(0)13457 49949 y Fz(deriv)-72 b(ativ)-36 b(e,)460 b(the)453 b(action)i(of)f Ft(R)f Fz(w)-36 b(ould)454 b(not)f(b)36 b(e)454 b(trivial.)640 b(Ho)-36 b(w)g(ev)g(er,)3432 51554 y(w)g(e)565 b(can)g(limit)h(this)f (p)36 b(ossibilit)-36 b(y)567 b(to)e(the)g(con)-36 b(tribution)564 b(corresp)36 b(onding)565 b(to)g(the)g(tree)3432 53159 y(with)407 b Fy(n)370 b Fz(=)e(1,)414 b(whose)408 b(only)h(endp)36 b(oin)-36 b(t)407 b(is)h(of)g(t)-36 b(yp)36 b(e)408 b Fy(z)59 b Fz(,)414 b(whic)-36 b(h)407 b(giv)-36 b(es)409 b(no)f(con)-36 b(tribution)407 b(to)3432 54764 y Ft(RV)5481 54282 y Fr(\()p Fx(h)p Fr(\))6810 54764 y Fz(.)621 b(In)447 b(fact,)452 b(if)c(there)f(is)h(more)g(than)f(one)h(endp)36 b(oin)-36 b(t,)450 b(at)e(most)f(one)h(of)g(the)f(\014elds)3432 56369 y(of)425 b(an)-36 b(y)424 b(endp)36 b(oin)-36 b(t)423 b(of)j(t)-36 b(yp)36 b(e)424 b Fy(z)483 b Fz(can)425 b(b)36 b(elong)424 b(to)h Fy(P)27178 56568 y Fx(v)27636 56691 y Fp(0)28153 56369 y Fz(,)h(so)f(that)f(w)-36 b(e)425 b(can)f(use)g(the)g(freedom)3432 57974 y(in)508 b(the)f(c)-36 b(hoice)509 b(of)g(the)f(\014eld)g(whic)-36 b(h)508 b(carries)g(the)g (deriv)-72 b(ativ)-36 b(e)509 b(so)g(that)f Fy(m)p Fz(\()p Fy(f)142 b Fz(\))495 b(=)h(0)508 b(for)3432 59579 y(b)36 b(oth)433 b Fy(f)511 b Ft(2)368 b Fy(P)9747 59778 y Fx(v)10205 59901 y Fp(0)11156 59579 y Fz(\(see)433 b(remark)h(after)g(\(2.51\)\).) 5383 61184 y(Hence,)419 b(w)-36 b(e)416 b(ha)-36 b(v)g(e)415 b(to)h(discuss)f(only)h(the)f(case)h Fy(m)p Fz(\()p Fy(f)142 b Fz(\))368 b(=)h(0)415 b(for)h(b)36 b(oth)415 b Fy(f)511 b Ft(2)369 b Fy(P)42190 61383 y Fx(v)42648 61506 y Fp(0)43165 61184 y Fz(;)422 b(if)416 b(w)-36 b(e)3432 62789 y(put)432 b Fy(P)6654 62988 y Fx(v)7112 63111 y Fp(0)7998 62789 y Fz(=)369 b(\()p Fy(f)10526 62988 y Fr(1)11052 62789 y Fy(;)221 b(f)12275 62988 y Fr(2)12801 62789 y Fz(\),)434 b Fn(x)p Fz(\()p Fy(f)16038 62988 y Fx(i)16413 62789 y Fz(\))369 b(=)g Fn(x)19458 62988 y Fx(i)19833 62789 y Fz(,)434 b Fy(!)48 b Fz(\()p Fy(f)22633 62988 y Fx(i)23008 62789 y Fz(\))369 b(=)f Fy(!)26073 62988 y Fx(i)26449 62789 y Fz(,)441 b Fy(~)-657 b(p)27897 62988 y Fx(F)26 b(;i)29524 62789 y Fz(=)376 b Fy(~)-657 b(p)31558 62988 y Fx(F)32339 62789 y Fz(\()p Fy(\022)33451 62988 y Fx(h;!)34839 63123 y Fi(i)35246 62789 y Fz(\),)434 b(w)-36 b(e)434 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b Fx(;!)41721 68511 y Fp(2)43127 68060 y Fz(])369 b Fy(;)3432 71740 y Fz(where)433 b Fy(W)181 b Fz(\()p Fn(x)9893 71939 y Fr(1)10713 71740 y Ft(\000)296 b Fn(x)12831 71939 y Fr(2)13356 71740 y Fz(\))433 b(is)h(the)f(in)-36 b(tegral)434 b(of)h Fy(W)25467 71061 y Fr(\()p Fx(h)p Fr(\))25286 72067 y Fx(\034)50 b(;)p Fm(P)p Fx(;)p Fr(\012)p Fx(;T)28690 71740 y Fz(\()p Fn(x)29985 71939 y Fx(v)30443 72062 y Fp(0)30960 71740 y Fz(\))433 b(o)-36 b(v)g(er)434 b(the)f(v)-72 b(ariables)24367 75163 y(21)p eop %%Page: 22 22 22 21 bop 3432 7638 a Fn(x)4221 7837 y Fx(v)4679 7960 y Fp(0)5195 7638 y Ft(n)p Fz(\()p Fn(x)7154 7837 y Fr(1)7680 7638 y Fy(;)221 b Fn(x)9051 7837 y Fr(2)9577 7638 y Fz(\),)433 b(up)g(to)h(a)g(sign,)g(and)21164 7287 y(^)21001 7638 y Fy(@)21760 7156 y Fr(2)22286 7638 y Fz(\()p Fy(x)23531 7837 y Fr(0)24056 7638 y Fz(\))g(is)g(an)f(op)36 b(erator)434 b(de\014ned)e(b)-36 b(y)4258 10786 y(^)4096 11136 y Fy(@)4855 10588 y Fr(2)5380 11136 y Fz(\()p Fy(x)6625 11335 y Fr(0)7151 11136 y Fz(\))p Fy( )8551 10588 y Fr(\()p Fs(\024)p Fx(h)p Fr(\))p Fx(")8503 11464 y Fm(y)15 b Fx(;!)9937 11587 y Fp(2)11415 11136 y Fz(=)12796 9574 y Fl(Z)13903 9925 y Fr(1)13411 12089 y(0)14650 11136 y Fy(ds)p Fz(\(1)293 b Ft(\000)i Fy(s)p Fz(\))p Fy(@)20594 10588 y Fr(2)20522 11464 y(0)21119 11136 y Fy( )22013 10457 y Fr(\()p Fs(\024)p Fx(h)p Fr(\))p Fx(")21965 11520 y(\030)22379 11643 y Fp(0)22839 11520 y Fr(\()p Fx(s)p Fr(\))p Fx(;)m(~)-467 b(y)32 b(;!)26999 11136 y Fy(;)1522 b(\030)29450 11335 y Fr(0)29976 11136 y Fz(\()p Fy(s)p Fz(\))368 b(=)h Fy(x)34089 11335 y Fr(0)34909 11136 y Fz(+)293 b Fy(s)p Fz(\()p Fy(y)37967 11335 y Fr(0)38787 11136 y Ft(\000)h Fy(x)40853 11335 y Fr(0)41379 11136 y Fz(\))369 b Fy(:)664 b Fz(\(2.58\))3432 14746 y(Instead)416 b(of)i(\(2.57\),)j(one)c(could)g(also)h(use)e(a)i (similar)g(expression)f(with)g([)39262 14395 y(^)39099 14746 y Fy(@)39858 14264 y Fr(2)40384 14746 y Fz(\()p Fy(y)41524 14945 y Fr(0)42050 14746 y Fz(\))p Fy( )43450 14264 y Fr(\()p Fs(\024)p 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b(oint)p Fz(.)5383 21446 y(By)434 b(using)f(\(2.49\))i(and)e(\(2.52\),)h(w)-36 b(e)434 b(can)g(also)g(write)8023 24324 y Ft(RV)10072 23775 y Fr(\()p Fx(h)p Fr(\))11401 24324 y Fz(\()p Fy(\034)76 b(;)221 b Fn(P)p Fy(;)g Fz(\012)p Fy(;)g(T)181 b Fz(\))371 b(=)24193 b(\(2.59\))8023 26853 y(=)10185 25955 y(1)p 9536 26543 1949 45 v 9536 27764 a Fy(s)10149 27963 y Fx(v)10607 28086 y Fp(0)11123 27764 y Fz(!)12008 25746 y Fl(X)11839 28196 y Fx(\013)p Fs(2)p Fx(A)13997 25291 y Fl(Z)15325 26853 y Fy(d)p Fn(x)16790 27052 y Fx(v)17248 27175 y Fp(0)17986 25291 y Fl(Z)19314 26853 y Fy(dP)20828 27052 y Fx(T)21376 27163 y Fi(v)21791 27362 y Fp(0)22363 26853 y Fz(\()p Fn(t)p Fz(\))369 b Ft(R)p Fz([)26078 26503 y(~)25812 26853 y Fy( )26706 26174 y Fr(\()p Fs(\024)p Fx(h)p Fr(\))26658 27178 y(\012)27338 27289 y Fi(v)27753 27488 y Fp(0)28269 27178 y Fx(;\013)29190 26853 y Fz(\()p Fy(P)30534 27052 y Fx(v)30992 27175 y Fp(0)31509 26853 y Fz(\)])g(\()p Fy(y)33885 27052 y Fx(\013;)p Fr(0)35572 26853 y Ft(\000)295 b Fy(x)37639 27052 y Fx(\013;)p Fr(0)39031 26853 y Fz(\))39537 26305 y Fx(b)p Fr(\()p Fs(j)p Fx(P)41164 26416 y Fi(v)41579 26615 y Fp(0)42094 26305 y Fs(j)p Fr(\))43441 26853 y Ft(\001)8023 30948 y(\001)8761 29664 y Fl(h)9997 29841 y(Y)9505 32301 y Fx(l)11 b Fs(2)p Fx(T)10972 32412 y Fi(v)11387 32611 y Fp(0)12124 30948 y Fy(@)12883 30335 y Fx(m)13715 30491 y Fi(l)12811 31239 y Fr(0)14139 30948 y Fz(~)-696 b Fy(g)14764 30400 y Fr(\()p Fx(h)p Fr(+1\))14716 31277 y Fx(!)15301 31433 y Fi(l)17295 30948 y Fz(\()p Fn(x)18590 31147 y Fx(l)19231 30948 y Ft(\000)296 b Fn(y)21349 31147 y Fx(l)21695 30948 y Fz(\))p Fy(\016)22777 31312 y Fx(!)23394 30915 y Fq(\000)23362 31646 y Fi(l)24089 31312 y Fx(;!)24967 30915 y Fp(+)24935 31646 y Fi(l)25699 29664 y Fl(i)26443 30948 y Fz(det)221 b Fy(G)29497 30400 y Fx(h)p Fr(+1)p Fx(;T)32050 30511 y Fi(v)32465 30710 y Fp(0)33037 30948 y Fz(\()p Fn(t)p Fz(\))34930 29298 y Fx(s)35365 29409 y Fi(v)35780 29608 y Fp(0)34906 29841 y Fl(Y)34852 32272 y 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Fx(v)10099 38741 y Fp(0)10615 38419 y Fz(\)])476 b(=)13710 38069 y(~)13445 38419 y Fy( )14339 37740 y Fr(\()p Fs(\024)p Fx(h)p Fr(\))14291 38744 y(\012)14971 38855 y Fi(v)15386 39054 y Fp(0)16399 38419 y Fz(\()p Fy(P)17743 38618 y Fx(v)18201 38741 y Fp(0)18718 38419 y Fz(\),)512 b(except)496 b(if)g Ft(j)p Fy(P)26703 38618 y Fx(v)27161 38741 y Fp(0)27678 38419 y Ft(j)476 b Fz(=)e(4)497 b(or)f Ft(j)p Fy(P)34015 38618 y Fx(v)34473 38741 y Fp(0)34990 38419 y Ft(j)475 b Fz(=)g(2)496 b(and)f Fy(m)p Fz(\()p Fy(f)142 b Fz(\))475 b(=)g(0)3432 40024 y(for)534 b(b)36 b(oth)534 b Fy(f)682 b Ft(2)539 b Fy(P)12277 40223 y Fx(v)12735 40346 y Fp(0)13252 40024 y Fz(;)585 b(in)534 b(these)f(case,)560 b(its)534 b(expression)h(can)f (b)36 b(e)533 b(easily)j(deduced)d(from)3432 41629 y(\(2.55\))432 b(and)g(\(2.57\).)578 b(Finally)-108 b(,)433 b Fy(b)p Fz(\()p Fy(p)p Fz(\))f(is)g(an)g(in)-36 b(teger,)432 b(equal)g(to)g(1,)h(if)g Fy(p)368 b Fz(=)h(4,)433 b(equal)f(to)g(2,) 3432 43234 y(if)i Fy(p)369 b 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Fz(\()p Fn(x)27067 18108 y Fx(l)27709 17909 y Ft(\000)295 b Fn(y)29826 18108 y Fx(l)30173 17909 y Fz(\))p Fy(\016)31255 18273 y Fx(!)31872 17876 y Fq(\000)31840 18606 y Fi(l)32566 18273 y Fx(;!)33444 17876 y Fp(+)33412 18606 y Fi(l)34176 16625 y Fl(i)35363 17909 y Ft(\001)7508 22268 y(\001)591 b Fz(det)220 b Fy(G)11521 21719 y Fx(h)p Fr(+1)p Fx(;T)14074 21830 y Fi(v)14489 22029 y Fp(0)15062 22268 y Fz(\()p Fn(t)p Fz(\))16954 20617 y Fx(s)17389 20728 y Fi(v)17804 20927 y Fp(0)16930 21161 y Fl(Y)16876 23591 y Fx(i)p Fr(=1)18399 22268 y Fz([\()p Fy(x)20005 21589 y Fr(\()p Fx(i)p Fr(\))20005 22558 y(0)21408 22268 y Ft(\000)295 b Fy(y)23418 21589 y Fr(\()p Fx(i)p Fr(\))23370 22558 y(0)24526 22268 y Fz(\))25032 21719 y Fx(b)25435 21830 y Fi(v)25850 22041 y(i)26256 21719 y Fr(\()p Fx(\013)27226 21830 y Fi(v)27641 22029 y Fp(0)28158 21719 y Fr(\))28579 22268 y Fy(K)29781 21719 y Fr(\()p Fx(h)p Fr(+2\))29686 22596 y Fx(v)30144 22731 y Fi(i)32312 22268 y Fz(\()p Fn(x)33607 22467 y Fx(v)34065 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y(0)37733 36099 y Fz(\))38239 35617 y Fx(m)39127 36099 y Fz(,)564 b Fy(m)546 b Fz(=)g(2)367 b(+)f Fy(b)p Fz(,)3432 37704 y(whic)-36 b(h)506 b(m)-36 b(ultiplies)506 b Fy(K)14469 37222 y Fr(\()p Fx(h)p Fr(+2\))14374 38032 y Fx(v)14832 38155 y Fp(1)17001 37704 y Fz(\()p Fn(x)18296 37903 y Fx(v)18754 38026 y Fp(1)19270 37704 y Fz(\).)796 b(Hence,)525 b(if)507 b(w)-36 b(e)506 b(expand)g(this)g(factor,)526 b(b)-36 b(y)506 b(using)g(an)3432 39309 y(equation)411 b(similar)h(to)g(\(2.60\),)417 b(w)-36 b(e)411 b(get)g(terms)g(with)g (some)h(propagator)f(m)-36 b(ultiplied)411 b(b)-36 b(y)3432 40914 y(a)362 b(factor)g(\()p Fy(x)9338 41113 y Fx(l)11 b(;)p Fr(0)10565 40914 y Ft(\000)149 b Fy(y)12381 41113 y Fx(l)11 b(;)p Fr(0)13460 40914 y Fz(\))13966 40432 y Fx(b)14369 40588 y Fi(l)14745 40914 y Fz(,)377 b(with)362 b Fy(b)18927 41113 y Fx(l)19642 40914 y Fy(>)369 b Fz(2.)555 b(If)362 b(w)-36 b(e)362 b(further)f(iterate)h(this)f(pro)36 b(cedure,)376 b(w)-36 b(e)362 b(can)3432 42519 y(end)460 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b(v)g(e)24367 75163 y(23)p eop %%Page: 24 24 24 23 bop 3432 7638 a Fz(tak)-36 b(en)569 b Fn(x)p Fz(,)603 b(in)569 b(the)g(case)h(of)f(\(2.64\),)605 b(or)569 b Fn(y)21 b Fz(,)603 b(in)569 b(the)g(other)g(case,)604 b(as)569 b(the)g(lo)36 b(calization)3432 9243 y(p)g(oin)-36 b(t.)5383 10848 y(Ev)g(en)442 b(simpler)h(is)g(the)g(situation,)i(when) e Ft(j)p Fy(P)27209 11047 y Fx(v)27667 11170 y Fp(1)28184 10848 y Ft(j)384 b Fz(=)h(4.)606 b(In)443 b(fact,)j(if)e(w)-36 b(e)443 b(insert)f(\(2.21\))3432 12453 y(in)c(an)h(expression)h(lik)-36 b(e)15427 11506 y Fl(R)16386 12453 y Fy(d)p Fn(x)p 17062 12630 789 45 v Fy(F)181 b Fz(\()p Fn(x)p 19380 12630 V -1 w Fz(\)\()p Fy(y)21862 11971 y Fs(\003)21814 12782 y Fr(0)22686 12453 y Ft(\000)299 b Fy(x)24757 11971 y Fs(\003)24757 12782 y Fr(0)25283 12453 y Fz(\))p Ft(R)p Fy(W)28143 12652 y Fr(4)28668 12453 y Fz(\()p Fn(x)p 29174 12630 V Fz(\),)440 b Fy(y)31952 11971 y Fs(\003)32917 12453 y Fz(and)e Fy(x)36190 11971 y Fs(\003)37155 12453 y Fz(b)36 b(eing)439 b(t)-36 b(w)g(o)439 b(p)36 b(oin)-36 b(ts)3432 14058 y(of)434 b Fn(x)p 4913 14235 V Fz(,)f(w)-36 b(e)434 b(get)7275 15756 y Fl(Z)8603 17318 y Fy(d)p Fn(x)p 9279 17495 V Fy(F)181 b Fz(\()p Fn(x)p 11597 17495 V -1 w Fz(\)\()p Fy(y)14079 16769 y Fs(\003)14031 17646 y Fr(0)14899 17318 y Ft(\000)296 b Fy(x)16967 16769 y Fs(\003)16967 17646 y Fr(0)17493 17318 y Fz(\))p Ft(R)p Fy(W)20353 17517 y Fr(4)20878 17318 y Fz(\()p Fn(x)p 21384 17495 V -1 w Fz(\))369 b(=)24428 15756 y Fl(Z)25756 17318 y Fy(d)p Fn(x)p 26432 17495 V Fy(F)181 b Fz(\()p Fn(x)p 28750 17495 V -1 w Fz(\)\()p Fy(y)31232 16769 y Fs(\003)31184 17646 y Fr(0)32052 17318 y Ft(\000)296 b Fy(x)34120 16769 y Fs(\003)34120 17646 y Fr(0)34646 17318 y Fz(\))p Fy(W)36380 17517 y Fr(4)36905 17318 y Fz(\()p Fn(x)p 37411 17495 V Fz(\))369 b Fy(;)3843 b Fz(\(2.65\))3432 20577 y(so)434 b(that,)f(ev)-36 b(en)434 b(in)f(this)g(case,)i(the)e(p)36 b(o)-36 b(w)g(er)433 b(of)i(the)e(\\zero")h(can)g(not)f(increase.)5383 22182 y(There)474 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30208 y(happ)36 b(en)643 b(that)i(there)f(is)h(an)g(\\in)-36 b(terference")645 b(b)36 b(et)-36 b(w)g(een)645 b(the)f Ft(R)h Fz(op)36 b(erations)645 b(in)g(t)-36 b(w)g(o)3432 31813 y(di\013eren)g(t)403 b(v)-36 b(ertices,)412 b(whic)-36 b(h)404 b(w)-36 b(ould)405 b(mak)-36 b(e)405 b(more)g(in)-36 b(v)g(olv)g(ed)406 b(the)e(expansion.)569 b(Ho)-36 b(w)g(ev)g(er,)3432 33418 y(one)611 b(can)g(sho)-36 b(w,)657 b(b)-36 b(y)611 b(the)g(same)h(argumen)-36 b(ts)610 b(giv)-36 b(en)612 b(in)g Ft(x)p Fz(3.3)g(and)f Ft(x)p Fz(3.4)i(of)f([BM])g(in)3432 35023 y(the)485 b(one)i(dimensional)f(case,)501 b(that)486 b(these)g(problems)g(can)g(b)36 b(e)486 b(a)-36 b(v)g(oided)487 b(b)-36 b(y)486 b(using)h(the)3432 36628 y(freedom)589 b(in)g(the)g(c)-36 b(hoice)590 b(of)g(the)f(lo)36 b(calization)591 b(p)36 b(oin)-36 b(t)589 b(and,)629 b(mainly)-108 b(,)629 b(the)588 b(fact)i(that)3432 38233 y(some)538 b(regularization)h(op)36 b(erations)538 b(are)g(not)g(really)h(presen)-36 b(t.)891 b(Let)538 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y Fr(4)17945 50072 y Fz(\()p Fn(k)19240 50271 y Fr(1)19765 50072 y Fy(;)221 b Fn(k)21136 50271 y Fr(2)21663 50072 y Fy(;)g Fn(k)23034 50271 y Fr(3)23560 50072 y Fz(\))295 b Ft(\000)26068 49736 y Fz(^)25689 50072 y Fy(W)26917 50271 y Fr(4)27443 50072 y Fz(\()p Fn(k)28738 50271 y Fr(1)29263 50072 y Fy(;)221 b Fz(\(0)p Fy(;)31517 49721 y(~)31583 50072 y(k)32259 50271 y Fr(2)32787 50072 y Fz(\))p Fy(;)g Fn(k)34664 50271 y Fr(3)35190 50072 y Fz(\))35696 48787 y Fl(i)36809 50072 y Fy(:)6109 b Fz(\(2.66\))3432 53433 y(Ho)-36 b(w)g(ev)g(er,)531 b(if)10680 53082 y(\026)10397 53433 y Fy(f)653 b Fz(is)511 b(the)f(lab)36 b(el)512 b(of)f(the)f(\014eld)h Fy( )26279 52951 y Fr(\()p Fs(\024)p Fx(h)p Fr(\))p Fx(;)p Fr(+)26231 53761 y Fm(x)26804 53884 y Fp(2)29332 53433 y Fz(,)531 b(it)511 b(is)g(easy)g(to)g(see)g(that)43047 53097 y(^)42668 53433 y Fy(W)43896 53632 y Fr(4)44421 53433 y Fz(\()p Fn(k)45716 53632 y Fr(1)46242 53433 y Fy(;)3432 55292 y Fz(\(0)p Fy(;)5104 54942 y(~)5170 55292 y(k)5846 55491 y Fr(2)6373 55292 y Fz(\))p Fy(;)221 b Fn(k)8250 55491 y Fr(3)8776 55292 y Fz(\))400 b(=)g(0,)457 b(if)c(there)e(is)i(a)f(v) -36 b(ertex)502 b(\026)-700 b Fy(v)448 b(>)400 b(v)26618 55491 y Fr(0)27596 55292 y Fz(with)452 b(four)g(external)h(legs,)k(suc) -36 b(h)451 b(that)3432 56898 y Fy(f)510 b Ft(2)369 b Fy(P)6712 57097 y Fr(\026)-506 b Fx(v)7607 56898 y Fz(and)384 b Fy(f)527 b Fz(is)385 b(a\013ected)f(b)-36 b(y)385 b(the)f Ft(R)h Fz(op)36 b(eration)385 b(in)434 b(\026)-699 b Fy(v)47 b Fz(.)562 b(Hence,)395 b(in)385 b(this)f(case,)396 b(w)-36 b(e)385 b(can)3432 58503 y(substitute)432 b(the)h(\014rst)g (term)g(in)g(the)g(braces)h(of)g(\(2.55\))h(with)33484 57617 y Fl(Q)34530 58779 y Fx(i)35127 58503 y Fy( )36021 58021 y Fr(\()p Fs(\024)p Fx(h)p Fr(\))p Fx(;")38722 58156 y Fi(i)35973 58831 y Fm(x)36546 58966 y Fi(i)36896 58831 y Fx(;!)37742 58966 y Fi(i)39128 58503 y Fz(.)5383 60108 y(W)-108 b(e)382 b(refer)g(to)h Ft(x)p Fz(3.3)h(and)e Ft(x)p 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b(of)g(the)e(external)i(lines)f(b)36 b(elonging)3432 68283 y(to)462 b Fy(P)5888 68482 y Fx(v)6346 68605 y Fp(0)7325 68283 y Fz(can)g(b)36 b(e)462 b(a\013ected)g(from)h(one)f(op) 36 b(erator)27314 67933 y(^)27151 68283 y Fy(@)27910 67801 y Fx(q)28417 68283 y Fz(,)470 b(as)462 b(a)h(consequence)f(of)h (the)f(action)3432 69888 y(of)522 b Ft(R)f Fz(on)g Fy(v)9171 70087 y Fr(0)10218 69888 y Fz(or)h(some)g(other)f(v)-36 b(ertex)522 b Fy(v)566 b(>)518 b(v)26107 70087 y Fr(0)26632 69888 y Fz(.)842 b(Finally)-108 b(,)545 b(the)520 b(lines)i(in)-36 b(v)g(olv)g(ed)522 b(in)g(the)3432 71493 y(determinan)-36 b(ts)332 b(ma)-36 b(y)333 b(b)36 b(e)334 b(a\013ected)e(from)i(one)f (op)36 b(erator)30828 71143 y(^)30665 71493 y Fy(@)31424 71011 y Fx(q)31931 71493 y Fz(.)545 b(W)-108 b(e)333 b(in)-36 b(tro)36 b(duce)333 b(an)g(index)g Fy(\013)24367 75163 y Fz(24)p eop %%Page: 25 25 25 24 bop 3432 7638 a Fz(to)330 b(distinguish)f(these)g(di\013eren)-36 b(t)329 b(terms)g(and,)351 b(giv)-36 b(en)330 b Fy(\013)8 b Fz(,)352 b(w)-36 b(e)330 b(shall)g(denote)g(b)-36 b(y)41524 7287 y(^)41361 7638 y Fy(@)42120 7156 y Fx(q)42540 7267 y Fi(\013)43130 7156 y Fr(\()p Fx(f)98 b Fr(\))44797 7638 y Fz(the)3432 9243 y(di\013eren)-36 b(tial)496 b(op)36 b(erators)496 b(acting)h(on)f(the)g(external)h(lines)g(of)g Fy(P)34449 9442 y Fx(v)34907 9565 y Fp(0)35920 9243 y Fz(or)g(the)f(propagators)3432 10848 y(b)36 b(elonging)434 b(to)f Fy(T)181 b Fz(,)434 b(as)g(a)g(consequence)g(of)g(the)f (regularization)h(pro)36 b(cedure.)5383 12453 y(All)434 b(the)f(previous)h(considerations)f(imply)h(that)g Ft(RV)31913 11971 y Fr(\()p Fx(h)p Fr(\))33242 12453 y Fz(\()p Fy(\034)76 b(;)221 b Fn(P)p Fy(;)g Fz(\012)p Fy(;)g(T)181 b Fz(\))371 b(=)d(0,)435 b(if)3432 14058 y Ft(j)p Fy(P)4639 14257 y Fx(v)5097 14380 y Fp(0)5614 14058 y Ft(j)598 b Fz(=)g(4)568 b(and)g Fy(n)599 b Fz(=)e(1)569 b(\(that)e(is)i(there)f(is)g(only)h(an) f(endp)36 b(oin)-36 b(t)567 b(of)i(t)-36 b(yp)36 b(e)568 b Fy(\025)g Fz(and)g(no)3432 15663 y(in)-36 b(ternal)555 b(line)i(asso)36 b(ciated)557 b(with)f Fy(v)20968 15862 y Fr(0)21494 15663 y Fz(\))g(or)g Fy(P)25106 15862 y Fx(v)25564 15985 y Fp(0)26659 15663 y Fz(=)577 b(\()p Fy(f)29395 15862 y Fr(1)29921 15663 y Fy(;)221 b(f)31144 15862 y Fr(2)31671 15663 y Fz(\))556 b(and)f Fy(m)p Fz(\()p Fy(f)37669 15862 y Fr(1)38195 15663 y Fz(\))378 b(+)g Fy(m)p Fz(\()p Fy(f)42754 15862 y Fr(2)43280 15663 y Fz(\))577 b(=)h(1)3432 17269 y(\(since)g(this)g(can)h(happ)36 b(en)577 b(only)i(if)g Fy(n)616 b Fz(=)g(1)579 b(and)f(the)g(endp)36 b(oin)-36 b(t)577 b(is)i(of)g(t)-36 b(yp)36 b(e)578 b Fy(z)59 b Fz(,)616 b(as)579 b(a)3432 18874 y(consequence)444 b(of)i(the)e(freedom)h(in)f(the)h(c)-36 b(hoice)445 b(of)g(the)f (\014eld)h(carrying)g(the)f(deriv)-72 b(ativ)-36 b(e)3432 20479 y(in)405 b(the)f(endp)36 b(oin)-36 b(ts)404 b(of)i(t)-36 b(yp)36 b(e)405 b Fy(z)59 b Fz(\))406 b(or)f Fy(m)p Fz(\()p Fy(f)22808 20678 y Fr(1)23334 20479 y Fz(\))237 b(+)f Fy(m)p Fz(\()p Fy(f)27610 20678 y Fr(2)28136 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Fi(f)27367 68752 y Fz(:)28432 67645 y Fl(X)28098 70105 y Fx(f)98 b Fs(2)p Fx(P)29872 70216 y Fi(v)30587 68752 y Fy(")p Fz(\()p Fy(f)142 b Fz(\))32924 68402 y Fy(~)32991 68752 y(k)44 b Fz(\()p Fy(f)142 b Fz(\))369 b(=)f(0)37905 66472 y Fl(1)37905 68464 y(A)39465 68752 y Fy(;)3453 b Fz(\(2.76\))24367 75163 y(27)p eop %%Page: 28 28 28 27 bop 3432 7638 a Fz(where)477 b Fy(\037)p Fz(\()p Fy(condition)p Fz(\))g(is)h(the)f(function)g(=)443 b(1)478 b(when)f Fy(condition)h Fz(is)g(v)-36 b(eri\014ed,)488 b(and)477 b(=)444 b(0)3432 9243 y(in)503 b(the)g(opp)36 b(osite)503 b(case.)789 b(Moreo)-36 b(v)g(er,)522 b(giv)-36 b(en)504 b(a)f(set)h Fy(P)684 b Fz(of)504 b(\014eld)f(lab)36 b(els,)522 b(w)-36 b(e)504 b(denote)f(b)-36 b(y)3432 10848 y Ft(S)100 b Fz(\()p Fy(P)181 b Fz(\))433 b(the)g(sp)36 b(ecial)434 b(family)h(of)g(s-sectors)e(lab)36 b(elled)434 b(b)-36 b(y)434 b Fy(P)181 b Fz(,)434 b(de\014ned)e(as)16789 13782 y Ft(S)100 b Fz(\()p Fy(P)181 b Fz(\))369 b(=)f Ft(f)p Fy(S)22937 13988 y Fx(h)p Fr(\()p Fx(f)98 b Fr(\))p Fx(;!)32 b Fr(\()p Fx(f)98 b Fr(\))27410 13782 y Fy(;)655 b(f)511 b Ft(2)368 b Fy(P)181 b Ft(g)369 b Fy(:)10034 b Fz(\(2.77\))5383 16715 y(The)433 b(previous)h(considerations)g(imply) g(that)17104 20533 y Fy(E)18067 20732 y Fx(L;\014)19966 20533 y Ft(\024)22919 19094 y Fr(0)22355 19426 y Fl(X)21368 21886 y Fx(h)p Fr(=)p Fx(h)23184 22042 y Fi(\014)23738 21886 y Fs(\000)p Fr(1)25578 19094 y Fs(1)25249 19426 y Fl(X)25162 21842 y Fx(n)p Fr(=1)27156 20533 y Fy(J)27875 20732 y Fx(h;n)29305 20533 y Fz(\(0)p Fy(;)221 b Fz(0\))370 b Fy(;)10349 b Fz(\(2.78\))3432 24643 y(with)5765 28075 y Fy(J)6484 28274 y Fx(h;n)7913 28075 y Fz(\(2)p Fy(l)9456 28274 y Fr(0)9983 28075 y Fy(;)221 b(q)11142 28274 y Fr(0)11668 28075 y Fz(\))1107 b(=)16083 26968 y Fl(X)15399 29418 y Fx(\034)102 b Fs(2T)17065 29574 y Fi(h;n)21615 26968 y Fl(X)19178 29311 y Fj(P)p Fq(2P)20936 29422 y Fi(\034)21456 29311 y Fp(:)p Fq(j)p Fi(P)22460 29422 y(v)22875 29621 y Fp(0)23391 29311 y Fq(j)p Fp(=2)p 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b(if)g Fy(l)21138 40035 y Fr(0)22034 39836 y Fy(>)368 b Fz(0,)434 b(otherwise)g(it)g(coincides)g(with)40410 38950 y Fl(P)41578 40113 y Fr(\012)p Fs(2O)43630 40224 y Fi(\034)44204 39836 y Fz(.)6298 41441 y Fn(Remarks)527 b(-)984 b Fz(Note)458 b(that)f(w)-36 b(e)458 b(could)f(freely)i(insert)f([)33074 40555 y Fl(Q)34119 41718 y Fx(v)34886 41441 y Fy(\037)35701 41640 y Fx(v)36246 41441 y Fz(\()p Ft(S)100 b Fz(\()p Fy(P)39000 41640 y Fx(v)39546 41441 y Fz(\)\)])457 b(in)h(\(2.79\),) 3432 43046 y(b)36 b(ecause)522 b(of)h(the)f(constrain)-36 b(ts)522 b(follo)-36 b(wing)524 b(from)f(momen)-36 b(tum)521 b(conserv)-72 b(ation)523 b(and)f(the)3432 44651 y(compact)433 b(supp)36 b(ort)433 b(prop)36 b(erties)433 b(of)h(propagator's)g(F)-108 b(ourier)433 b(transform.)3878 46256 y(Note)446 b(also)h(that,)j(if)d Fy(l)14442 46455 y Fr(0)15358 46256 y Fz(=)390 b(0,)450 b(giv)-36 b(en)447 b Fy(\034)539 b Ft(2)390 b(T)24733 46455 y Fx(h;n)26163 46256 y Fz(,)450 b(the)c(n)-36 b(um)g(b)36 b(er)444 b(of)j(in)-36 b(ternal)446 b(lines)g(in)h(the)3432 47861 y(lo)-36 b(w)g(est)434 b(v)-36 b(ertex)434 b Fy(v)11908 48060 y Fr(0)12867 47861 y Fz(\(of)g(scale)h Fy(h)295 b Fz(+)f(1\))434 b(has)g(to)f(b)36 b(e)434 b(di\013eren)-36 b(t)432 b(from)i(zero.)5383 51072 y(Hence,)428 b(in)e(order)g(to)h(pro) -36 b(v)g(e)427 b(that)f(the)g(free)h(energy)g(and)f(the)g(e\013ectiv) -36 b(e)427 b(p)36 b(oten)-36 b(tials)3432 52677 y(are)575 b(w)-36 b(ell)576 b(de\014ned)d(\(in)i(the)f(limit)i Fy(L)609 b Ft(!)h(1)576 b Fz(and)e Fy(\014)649 b Fz(not)575 b(\\to)36 b(o)576 b(large"\),)611 b(w)-36 b(e)575 b(need)g(a)3432 54282 y(\\go)36 b(o)g(d")435 b(b)36 b(ound)432 b(of)i Fy(J)14122 54481 y Fx(h;n)15552 54282 y Fz(\(2)p Fy(l)17095 54481 y Fr(0)17621 54282 y Fy(;)221 b(q)18780 54481 y Fr(0)19307 54282 y Fz(\).)5383 55887 y(In)490 b(order)g(to)g(get)h (this)f(b)36 b(ound,)504 b(w)-36 b(e)490 b(shall)h(extend)f(the)g(pro) 36 b(cedure)490 b(used)f(in)i([BM])3432 57492 y(for)c(the)f(analysis)i 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71493 y(\001)g Fy(\015)11185 70852 y Fs(\000)p Fx(h)12459 70963 y Fi(v)13110 70077 y Fl(P)14279 71239 y Fi(l)p Fq(2)p Fi(T)15585 71350 y(v)16134 70963 y Fz([)16496 70852 y Fx(q)16916 70963 y Fi(\013)17505 70852 y Fr(\()p Fx(f)18421 70455 y Fp(+)18323 71185 y Fi(l)19097 70852 y Fr(\)+)p Fx(q)20615 70963 y Fi(\013)21205 70852 y Fr(\()p Fx(f)22121 70455 y Fq(\000)22023 71185 y Fi(l)22815 70852 y Fr(\)+)p Fx(m)p Fr(\()p Fx(f)25661 70455 y Fp(+)25563 71185 y Fi(l)26337 70852 y Fr(\)+)p Fx(m)p Fr(\()p Fx(f)29183 70455 y Fq(\000)29085 71185 y Fi(l)29878 70852 y Fr(\))30244 70963 y Fz(])31030 71493 y Fy(;)11888 b Fz(\(2.80\))24367 75163 y(28)p eop %%Page: 29 29 29 28 bop 3432 7638 a Fz(where,)433 b(if)i Fy(P)550 b Ft(\032)369 b Fy(I)12107 7837 y Fx(v)12565 7960 y Fp(0)13081 7638 y Fz(,)434 b(w)-36 b(e)434 b(de\014ne)e Fy(q)20125 7837 y Fx(\013)20785 7638 y Fz(\()p Fy(P)181 b Fz(\))368 b(=)24565 6752 y Fl(P)25733 7915 y Fx(f)98 b Fs(2)p Fx(P)27914 7638 y Fy(q)28491 7837 y Fx(\013)29151 7638 y 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b(oin)-36 b(t,)328 b(and)301 b(the)h(\014eld)g(with)g Fy(m)p Fz(\()p Fy(f)142 b Fz(\))368 b(=)h(1)302 b(b)36 b(elonging)24367 75163 y(31)p eop %%Page: 32 32 32 31 bop 3432 7638 a Fz(to)372 b(the)g(endp)36 b(oin)-36 b(t)372 b Fy(v)420 b Fz(is)372 b(con)-36 b(tracted)372 b(in)h(the)f(v)-36 b(ertex)373 b(immediately)g(preceding)f Fy(v)48 b Fz(,)385 b(whose)3432 9243 y(scale)434 b(is)g Fy(h)8603 9442 y Fx(v)9443 9243 y Ft(\000)295 b Fz(1,)435 b(w)-36 b(e)433 b(ha)-36 b(v)g(e)434 b(the)f(iden)-36 b(tit)g(y)11837 11268 y Fl(Y)10238 13658 y Fx(v)465 b Fr(not)434 b(e)p Fx(:)g Fr(p)p Fx(:)15073 12375 y Fy(\015)15820 11789 y Fx(h)16362 11900 y Fi(v)17013 11014 y Fl(P)18181 11288 y Fi(s)18569 11399 y(v)18181 12177 y(i)p Fp(=1)19560 11900 y Fz([)19921 11789 y Fx(m)p Fr(\()p Fx(P)21716 11900 y Fi(v)22131 12111 y(i)22538 11789 y Fs(n)p Fx(Q)23748 11900 y Fi(v)24163 12111 y(i)24570 11789 y Fr(\))24936 11900 y Fz(])25721 12375 y(=)27578 10936 y Fx(n)27156 11268 y Fl(Y)27102 13699 y Fx(i)p Fr(=1)28846 12375 y Fy(\015)29593 11827 y Fr(\()p Fx(h)30501 11962 y Fi(i)30852 11827 y Fs(\000)p Fr(1\))p Fx(\037)p Fr(\()p Fx(v)294 b Fr(is)262 b(of)331 b(t)-26 b(yp)26 b(e)261 b Fx(z)37 b Fr(\))39435 12375 y Fy(:)3483 b Fz(\(2.98\))5383 15806 y(Putting)405 b(together)h(the)g(previous)h(b)36 b(ounds)405 b(and)h(supp)36 b(osing)406 b(that)f(the)h(h)-36 b(yp)36 b(othesis)3432 17411 y(\(2.90\),)434 b(\(2.91\),)h(\(2.92\))g(of)f (Lemma)f(2.2)i(are)f(v)-36 b(eri\014ed,)433 b(w)-36 b(e)434 b(\014nd)e(that)6678 20024 y Fy(J)7397 20223 y Fx(h;n)8826 20024 y Fz(\(2)p Fy(l)10369 20223 y Fr(0)10896 20024 y Fy(;)221 b(q)12055 20223 y Fr(0)12581 20024 y Fz(\))369 b Ft(\024)g Fz(\()p Fy(c)p Ft(j)p Fy(\025)p Ft(j)p Fz(\))17927 19476 y Fx(n)19458 18917 y Fl(X)18774 21367 y Fx(\034)102 b Fs(2T)20440 21523 y Fi(h;n)24990 18917 y Fl(X)22553 21260 y Fj(P)p Fq(2P)24311 21371 y Fi(\034)24830 21260 y Fp(:)p Fq(j)p Fi(P)25834 21371 y(v)26249 21570 y Fp(0)26766 21260 y Fq(j)p Fp(=2)p Fi(l)28312 21459 y Fp(0)28773 21260 y Fi(;)22095 21703 y Fl(P)23263 22865 y Fi(f)76 b Fq(2)p Fi(P)24809 22976 y(v)25224 23175 y Fp(0)25938 22423 y Fi(q)26317 22534 y(\013)26907 22423 y Fp(\()p Fi(f)g Fp(\)=)p Fi(q)29022 22622 y Fp(0)30068 18917 y Fl(X)29837 21370 y Fx(T)131 b Fs(2)p Fm(T)32119 20024 y Fy(\015)32866 19476 y Fs(\000)p Fx(h)p Fr([)34534 19114 y Fp(1)p 34534 19277 406 45 v 34534 19823 a(2)35073 19476 y Fx(m)35905 19599 y Fp(4)36367 19476 y Fr(\()p Fx(v)37191 19599 y Fp(0)37652 19476 y Fr(\)+)38883 19114 y Fp(1)p 38882 19277 V 38882 19823 a(2)39421 19476 y Fx(\037)p Fr(\()p Fx(l)40652 19599 y Fp(0)41113 19476 y Fr(=0\)+)p Fx(q)43833 19599 y Fp(0)44295 19476 y Fr(])44907 20024 y Ft(\001)6678 25721 y(\001)7268 23773 y Fl(")8390 24282 y Fx(n)7968 24614 y Fl(Y)7914 27045 y Fx(i)p Fr(=1)9658 25721 y Fy(\015)10405 25173 y Fr(\()p Fx(h)11313 25308 y Fi(i)11664 25173 y Fs(\000)p Fr(1\))p Fx(\037)p Fr(\()p Fs(j)p Fx(P)g Fr(\()p Fx(v)32 b Fr(\))p Fs(j)p Fr(=2\))18278 23773 y Fl(#)20744 24614 y(Y)19145 27004 y Fx(v)465 b Fr(not)434 b(e)p Fx(:)g Fr(p)p Fx(:)23758 23773 y Fl(")24971 24823 y Fz(1)p 24537 25411 1520 45 v 24537 26633 a Fy(s)25150 26832 y Fx(v)25695 26633 y Fz(!)26189 25721 y Fy(\015)26936 25136 y Fx(h)27478 25247 y Fi(v)79 b Fz([)28467 24774 y Fp(3)p 28467 24937 406 45 v 28467 25483 a(4)29005 25247 y Fz(\()29511 24361 y Fl(P)30680 24634 y Fi(s)31068 24745 y(v)30680 25523 y(i)p Fp(=1)32215 25136 y Fs(j)p Fx(P)33073 25247 y Fi(v)33488 25458 y(i)33894 25136 y Fs(j\000j)p Fx(P)35745 25247 y Fi(v)36240 25136 y Fs(j)36501 25247 y Fz(\))37007 25136 y Fs(\000)37872 24774 y Fp(5)p 37872 24937 V 37872 25483 a(2)38410 25136 y Fr(\()p Fx(s)39211 25247 y Fi(v)39706 25136 y Fs(\000)p Fr(1\))41274 25247 y Fz(])41986 25721 y Ft(\001)6678 29722 y(\001)p Fy(\015)7794 29247 y Fz([)8155 29136 y Fs(\000)p Fx(z)37 b Fr(\()p Fx(v)32 b Fr(\))p Fs(\000)11449 28774 y Fp(1)p 11448 28937 V 11448 29484 a(2)11987 29136 y Fx(m)12819 29259 y Fp(4)13280 29136 y Fr(\()p Fx(v)g Fr(\)+)15367 28774 y Fp(1)p 15367 28937 V 15367 29484 a(2)15906 29136 y Fr(\()p Fs(j)p Fx(P)17130 29247 y Fi(v)17624 29136 y Fs(j\000)p Fr(3\))p Fx(\037)p Fr(\(4)p Fs(\024j)p Fx(P)22463 29247 y Fi(v)22958 29136 y Fs(j\024)p Fr(8\)+)25652 28774 y Fp(1)p 25652 28937 V 25652 29484 a(2)26191 29136 y Fr(\()p Fs(j)p Fx(P)27415 29247 y Fi(v)27909 29136 y Fs(j\000)p Fr(1\))p Fx(\037)p Fr(\()p Fs(j)p Fx(P)31546 29247 y Fi(v)32041 29136 y Fs(j\025)p Fr(10\))34340 29247 y Fz(])34758 27774 y Fl(#)35772 29722 y Fy(:)7146 b Fz(\(2.99\))3432 32990 y(On)423 b(the)h(other)g(hand,)i(if)f Fy(m)17135 33189 y Fr(2)17661 32990 y Fz(\()p Fy(v)48 b Fz(\))423 b(denotes)h(the)g(n)-36 b(um)g(b)36 b(er)423 b(of)i(endp)36 b(oin)-36 b(ts)424 b(of)h(t)-36 b(yp)36 b(e)424 b Fy(\027)508 b Fz(or)425 b Fy(z)3432 34595 y Fz(follo)-36 b(wing)435 b Fy(v)48 b Fz(,)434 b(w)-36 b(e)433 b(ha)-36 b(v)g(e,)435 b(if)483 b(~)-699 b Fy(v)481 b Fz(is)434 b(not)f(an)g(endp)36 b(oin)-36 b(t,)433 b(the)g(iden)-36 b(tities)10463 36681 y Fl(X)10504 38992 y Fi(v)24 b Fq(\025)38 b 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Fx(v)37642 51093 y Fz(!)38137 50182 y Fy(\015)38884 49633 y Fx(\016)33 b Fr(\()p Fs(j)p Fx(P)40557 49744 y Fi(v)41051 49633 y Fs(j)p Fr(\))42103 50182 y Fy(;)3432 54414 y Fz(where)12862 56851 y Fy(\016)50 b Fz(\()p Fy(p)p Fz(\))368 b(=)g Ft(\000)p Fy(\037)p Fz(\(2)i Ft(\024)f Fy(p)g Ft(\024)g Fz(4\))295 b(+)12862 59286 y(+)14095 57670 y Fl(\022)14909 59286 y Fz(1)h Ft(\000)17316 58387 y Fy(p)p 17316 58976 653 45 v 17317 60197 a Fz(4)18102 57670 y Fl(\023)19138 59286 y Fy(\037)p Fz(\(6)369 b Ft(\024)g Fy(p)g Ft(\024)g Fz(8\))295 b(+)28062 57670 y Fl(\022)28877 59286 y Fz(2)h Ft(\000)31284 58387 y Fy(p)p 31284 58976 V 31285 60197 a Fz(4)32069 57670 y Fl(\023)33106 59286 y Fy(\037)p Fz(\()p Fy(p)368 b Ft(\025)h Fz(10\))h Fy(;)3242 b Fz(\(2.102\))16226 62980 y Fy(\016)16802 63179 y Fr(ext)18138 62980 y Fz(\()p Fy(p)p Fz(\))368 b(=)21685 62081 y(5)p 21685 62670 651 45 v 21685 63891 a(2)22763 62980 y Ft(\000)24224 62081 y Fz(3)p 24224 62670 V 24224 63891 a(4)25007 62980 y Fy(p)295 b Ft(\000)27416 62081 y Fz(1)p 27416 62670 V 27416 63891 a(2)28200 62980 y Fy(\037)p Fz(\()p Fy(p)368 b Fz(=)h(0\))g Fy(:)8820 b Fz(\(2.103\))5383 65505 y(Since)433 b Fy(\016)50 b Fz(\()p Ft(j)p Fy(P)11118 65704 y Fx(v)11664 65505 y Ft(j)p Fz(\))370 b Fy(<)g Fz(0,)435 b(for)f(an)-36 b(y)435 b(v)-36 b(ertex)435 b Fy(v)48 b Fz(,)434 b(whic)-36 b(h)434 b(is)h(not)f(an)g(endp)36 b(oin)-36 b(t,)433 b(a)i(standard)3432 67110 y(argumen)-36 b(t,)433 b(see)g([BM)q(])h(or)f ([GM],)h(allo)-36 b(ws)435 b(to)f(sho)-36 b(w)434 b(that)14411 69032 y Fl(X)13727 71482 y Fx(\034)102 b Fs(2T)15393 71638 y Fi(h;n)18103 69032 y Fl(X)18592 71333 y Fj(P)17048 72085 y Fq(j)p Fi(P)17818 72196 y(v)18233 72395 y Fp(0)18750 72085 y Fq(j)p Fp(=2)p Fi(l)20296 72284 y Fp(0)21342 69032 y Fl(X)21111 71485 y Fx(T)131 b Fs(2)p Fm(T)24498 69032 y Fl(Y)24985 71163 y Fi(v)23526 71915 y Fp(not)433 b(e)p Fi(:)h Fp(p)p Fi(:)27805 69241 y Fz(1)p 27371 69829 1520 45 v 27371 71050 a Fy(s)27984 71249 y Fx(v)28529 71050 y Fz(!)29023 70139 y Fy(\015)29770 69591 y Fx(\016)33 b Fr(\()p Fs(j)p Fx(P)31443 69702 y Fi(v)31937 69591 y Fs(j)p Fr(\))32989 70139 y Ft(\024)369 b Fy(c)34951 69591 y Fx(n)35946 70139 y Fy(:)6322 b Fz(\(2.104\))24367 75163 y(32)p eop %%Page: 33 33 33 32 bop 3432 7638 a Fz(The)433 b(b)36 b(ounds)433 b(\(2.101\))h(and)f (\(2.104\))i(imply)f(the)f(follo)-36 b(wing)436 b(theorem.)3432 11572 y Fn(Theorem)499 b(2.1)651 b Fw(If)464 b(c)-66 b(onditions)463 b(\(2.90\),)i(\(2.91\),)g(\(2.92\))h(ar)-66 b(e)464 b(satis\014e)-66 b(d,)464 b(then)14846 13879 y Fy(J)15565 14078 y Fx(h;n)16994 13879 y Fz(\(2)p Fy(l)18537 14078 y Fr(0)19064 13879 y Fy(;)221 b(q)20223 14078 y Fr(0)20749 13879 y Fz(\))369 b Ft(\024)g Fz(\()p Fy(c)p Ft(j)p Fy(\025)p Ft(j)p Fz(\))26095 13330 y Fx(n)26720 13879 y Fy(\015)27467 13330 y Fx(h)p Fr([)p Fs(\000)p Fx(q)29422 13453 y Fp(0)29884 13330 y Fr(+)p Fx(\016)31032 13441 y Fp(ext)32198 13330 y Fr(\(2)p Fx(l)33315 13453 y Fp(0)33776 13330 y Fr(\)])34828 13879 y Fy(:)7440 b Fz(\(2.105\))5383 16207 y Fn(Remark)684 b(-)1281 b Fz(W)-108 b(e)596 b(will)h(pro)-36 b(v)g(e)595 b(in)h Ft(x)p Fz(4)g(that,)637 b(if)596 b Ft(j)p Fy(\025)p Ft(j)g Fz(is)g(small)g(enough)g(and)f Fy(C)45346 16406 y Fr(1)p Fx(;)p Fr(2)3432 17812 y Fz(log)240 b Fy(\014)74 b Ft(j)p Fy(\025)p Ft(j)369 b(\024)g Fz(1,)329 b(where)302 b Fy(C)15306 18011 y Fr(1)p Fx(;)p Fr(2)16865 17812 y Fz(is)h(a)f(constan)-36 b(t)302 b(dep)36 b(ending)301 b(only)i(on)f(\014rst)g(and)f(second)h(order)3432 19417 y(con)-36 b(tributions)349 b(of)j(p)36 b(erturbation)349 b(theory)-108 b(,)367 b(it)351 b(is)g(p)36 b(ossible)351 b(to)g(c)-36 b(ho)36 b(ose)425 b(~)-724 b Fy(\027)37921 19616 y Fr(1)38446 19417 y Fz(\()-22 b Fy(~)-628 b(x)p 38952 19594 740 45 v Fz(\))350 b(so)h(that)g(the)3432 21023 y(mo)36 b(di\014ed)559 b(running)f(coupling)i(functions)f (satisfy)i(the)f(h)-36 b(yp)36 b(otheses)559 b(of)h(Lemma)g(2.2,)3432 22628 y(\(2.90\),)579 b(\(2.91\))551 b(and)d(\(2.92\).)927 b(So,)579 b(in)549 b(that)g(case,)579 b(w)-36 b(e)550 b(see)g(from)g(Theorem)f(2.1)h(that)3432 24233 y(lim)5238 24432 y Fx(L)p Fs(!1)8037 24233 y Fy(E)9000 24432 y Fx(L;\014)10963 24233 y Fz(do)36 b(es)434 b(exist)g(and)f(is)h(of)g(order)f Fy(\025)p Fz(.)3432 28577 y FB(3)2151 b(Pro)60 b(of)717 b(of)g(Lemma)h(2.2)3432 31885 y FA(3.1)1792 b(The)599 b(sector)f(coun)-50 b(ting)599 b(Lemma)3432 34348 y Fz(In)433 b(order)g(to)h(presen)-36 b(t)433 b(the)g(pro)36 b(of)434 b(of)g(Lemma)g(2.2,)h(w)-36 b(e)434 b(need)e(to)i(in)-36 b(tro)36 b(duce)433 b(some)h(new)3432 35953 y(de\014nitions.)5021 38282 y(1.)651 b(Giv)-36 b(en)430 b(a)g(tree)g Fy(\034)578 b Fz(and)430 b Fn(P)368 b Ft(2)h(P)21361 38481 y Fx(\034)21936 38282 y Fz(,)431 b(w)-36 b(e)430 b(shall)h(call)g Fy(\037)p Fz(-v)-36 b(ertices)430 b(the)f(v)-36 b(ertices)430 b Fy(v)478 b Fz(of)431 b Fy(\034)148 b Fz(,)6683 39887 y(suc)-36 b(h)407 b(that)g Ft(I)13106 40086 y Fx(v)14059 39887 y Fz(\(the)g(set)h(of)g(in)-36 b(ternal)407 b(lines,)413 b(that)408 b(is)f(the)g(lines)h(con)-36 b(tracted)407 b(in)h Fy(v)48 b Fz(\))6683 41492 y(is)474 b(not)e(empt)-36 b(y)-108 b(.)697 b(W)-108 b(e)473 b(shall)g(also)h(call)g Fy(V)26063 41691 y Fx(\037)27176 41492 y Fz(the)f(family)i(of)e(all)h Fy(\037)p Fz(-v)-36 b(ertices,)484 b(whose)6683 43097 y(n)-36 b(um)g(b)36 b(er)432 b(is)i(of)g(order)g Fy(n)p Fz(.)5021 45630 y(2.)651 b(Giv)-36 b(en)476 b Fy(h)441 b Ft(\024)g Fz(0)477 b(and)e(a)h(set)g(of)h(\014eld)e(indices)h Fy(P)181 b Fz(,)487 b(w)-36 b(e)476 b(de\014ne)f Fn(O)37496 45829 y Fx(h)38094 45630 y Fz(\()p Fy(P)181 b Fz(\))441 b(=)g Ft(\012)43052 45829 y Fx(f)98 b Fs(2)p Fx(P)45011 45630 y Fy(O)46005 45829 y Fx(h)6683 47235 y Fz(and)433 b(w)-36 b(e)434 b(shall)g(call)h Fy(\033)p 16554 47412 787 45 v 417 w Fz(=)368 b Ft(f)p Fy(\033)20493 47434 y Fx(f)21468 47235 y Ft(2)g Fy(O)23716 47434 y Fx(h)24314 47235 y Fy(;)655 b(f)511 b Ft(2)369 b Fy(P)181 b Ft(g)433 b Fz(the)g(elemen)-36 b(ts)434 b(of)g Fn(O)40015 47434 y Fx(h)40613 47235 y Fz(\()p 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Ft(g)p Fz(,)505 b Fy(i)466 b Fz(=)f(1)p Fy(;)221 b Fz(2,)507 b(w)-36 b(e)490 b(shall)h(sa)-36 b(y)492 b(that)e Ft(S)39594 53425 y Fr(\(1\))41317 53907 y Ft(\036)467 b(S)43721 53425 y Fr(\(2\))44978 53907 y Fz(,)505 b(if)6683 55914 y Fy(S)7483 56426 y Fx(j)7915 55955 y Fp(\(1\))7864 56759 y Fi(f)9016 56426 y Fx(;\033)9847 55955 y Fp(\(1\))9815 56759 y Fi(f)11371 55914 y Ft(\032)370 b Fy(S)13574 56426 y Fx(j)14006 55955 y Fp(\(2\))13955 56759 y Fi(f)15106 56426 y Fx(;\033)15937 55955 y Fp(\(2\))15905 56759 y Fi(f)17093 55914 y Fz(,)434 b(for)g(an)-36 b(y)434 b Fy(f)511 b Ft(2)368 b Fy(P)181 b Fz(.)5383 58645 y(The)355 b(main)g(p)36 b(oin)-36 b(t)355 b(in)h(the)f(pro)36 b(of)356 b(is)f(the)g(follo)-36 b(wing)358 b(lemma,)372 b(whic)-36 b(h)355 b(is)g(an)h(extension)3432 60250 y(of)434 b(that)f(pro)-36 b(v)g(ed)433 b(in)h([FMR)-108 b(T])434 b(in)f(the)g(jellium)h(case;)h (see)f Ft(x)p Fz(7)g(for)g(a)g(pro)36 b(of.)3432 62578 y Fn(Lemma)499 b(3.1)650 b Fw(L)-66 b(et)476 b Fy(h)14290 62096 y Fs(0)14600 62578 y Fy(;)221 b(h;)g(L)478 b Fw(b)-66 b(e)475 b(inte)-66 b(gers)475 b(such)i(that)g Fy(h)30710 62096 y Fs(0)31410 62578 y Ft(\024)391 b Fy(h)f Ft(\024)h Fz(0)p Fw(.)632 b(L)-66 b(et)476 b Fy(v)524 b Fw(b)-66 b(e)476 b(a)g(vertex)3432 64183 y(of)446 b(a)i(tr)-66 b(e)g(e)445 b Fy(\034)148 b Fw(,)451 b(such)c(that)g Ft(j)p Fy(P)16888 64382 y Fx(v)17434 64183 y Ft(j)369 b Fz(=)f Fy(L)448 b Fw(and)f Fy(f)24032 64382 y Fr(1)25005 64183 y Fw(a)g(\014xe)-66 b(d)446 b(element)f(of)i Fy(P)36232 64382 y Fx(v)36778 64183 y Fw(.)591 b(Then,)450 b(given)445 b(the)3432 65788 y(se)-66 b(ctor)464 b(index)g Fy(\033)11394 65987 y Fx(f)11846 66110 y Fp(1)12731 65788 y Ft(2)369 b Fy(O)14980 65987 y Fx(h)15522 65735 y Fq(0)15876 65788 y Fw(,)464 b(and)h(a)g(set)g Fy(\033)p 22418 65965 V 417 w Ft(2)368 b Fn(O)25950 65987 y Fx(h)26548 65788 y Fz(\()p Fy(P)27892 65987 y Fx(v)28438 65788 y Ft(n)p Fy(f)29743 65987 y Fr(1)30269 65788 y Fz(\))p Fw(,)464 b(the)h(fol)66 b(lowing)464 b(b)-66 b(ound)465 b(holds:)7217 67888 y Fl(X)5112 70293 y Fi(\033)p 5112 70470 500 45 v 5612 70102 a Fq(0)5910 70293 y(2)p Fj(O)7149 70624 y Fi(h)7631 70433 y Fq(0)7985 70293 y Fp(\()p Fi(P)8823 70404 y(v)9317 70293 y Fq(n)p Fi(f)10140 70492 y Fp(1)10601 70293 y(\))4834 71400 y Fe(S)5722 71731 y Fi(h)6204 71540 y Fq(0)6557 71400 y Fp(\()p Fi(\033)p 6877 71577 V 7377 71209 a Fq(0)7675 71400 y Fp(\))p Fq(\036)p Fe(S)9522 71633 y Fi(h)10059 71400 y Fp(\()p Fi(\033)p 10379 71577 V 24 w Fp(\))11553 68995 y Fy(\037)12368 69194 y Fx(v)13134 67711 y Fl(\020)13796 68995 y Ft(f)p Fy(S)15260 69194 y Fx(h)15802 68943 y Fq(0)16100 69194 y Fx(;\033)16899 69350 y Fi(f)17298 69549 y Fp(1)17870 68995 y Ft(g)295 b([)g Ff(S)21111 69194 y Fx(h)21653 68943 y Fq(0)22007 68995 y Fz(\()p Fy(\033)p 22513 69172 787 45 v 23299 68447 a Fs(0)23610 68995 y Fz(\))24116 67711 y Fl(\021)25146 68995 y Ft(\024)26548 67047 y Fl(\()27661 68491 y Fy(c)28221 68009 y Fx(L)28916 68491 y Fy(\015)29796 67620 y Fi(h)p Fq(\000)p Fi(h)31400 67342 y Fq(0)p 29796 67801 1902 45 v 30544 68348 a Fp(2)31831 68000 y Fr(\()p Fx(L)p Fs(\000)p Fr(3\))35788 68491 y Fw(,)930 b(if)464 b Fy(L)369 b Ft(\025)g Fz(4)p Fw(,)27661 70096 y Fy(c)7567 b Fw(,)930 b(if)464 b Fy(L)369 b Fz(=)f(2)465 b Fw(.)43929 68995 y Fz(\(3.1\))24367 75163 y(33)p eop %%Page: 34 34 34 33 bop 3432 7638 a FA(3.2)1792 b(Pro)50 b(of)598 b(of)f(Lemma)i(2.2) 3432 10101 y Fz(First)433 b(of)h(all,)h(w)-36 b(e)434 b(note)f(that)16063 11927 y Fl(Y)16525 14252 y Fx(v)17698 13034 y Fy(\037)18513 13233 y Fx(v)19059 13034 y Fz(\()p Ft(S)100 b Fz(\()p Fy(P)21813 13233 y Fx(v)22358 13034 y Fz(\)\))368 b(=)25527 11927 y Fl(Y)25119 14378 y Fx(v)32 b Fs(2)p Fx(V)26784 14489 y Fi(\037)27571 13034 y Fy(\037)28386 13233 y Fx(v)28931 13034 y Fz(\()p Ft(S)100 b Fz(\()p Fy(P)31685 13233 y Fx(v)32230 13034 y Fz(\)\))369 b Fy(:)9957 b Fz(\(3.2\))5383 17163 y(Let)578 b(us)g(consider)h(\014rst)f(the)g (case)h Fy(P)23830 17362 y Fx(v)24288 17485 y Fp(0)25421 17163 y Ft(6)p Fz(=)616 b Ft(;)579 b Fz(and)f(let)628 b(~)-699 b Fy(v)33619 17362 y Fr(0)34723 17163 y Fz(b)36 b(e)579 b(the)f(\014rst)g Fy(\037)p Fz(-v)-36 b(ertex)3432 18769 y(follo)g(wing)340 b(the)d(ro)36 b(ot)339 b(\(p)36 b(ossibly)338 b(equal)h(to)f Fy(v)24529 18968 y Fr(0)25055 18769 y Fz(\);)370 b(note)338 b(that)f Fy(P)32683 18968 y Fr(~)-506 b Fx(v)33105 19091 y Fp(0)33991 18769 y Fz(=)369 b Fy(P)36210 18968 y Fx(v)36668 19091 y Fp(0)37523 18769 y Fz(and)338 b(that)f Fy(h)p Fz(\()p Fy(f)142 b Fz(\))368 b(=)3432 20374 y Fy(h)457 b Fz(for)h(an)-36 b(y)458 b Fy(f)551 b Ft(2)410 b Fy(P)12492 20573 y Fr(~)-506 b Fx(v)12914 20696 y Fp(0)13431 20374 y Fz(.)650 b(In)457 b(the)g(follo)-36 b(wing)460 b(it)e(will)g(also)h(v)-36 b(ery)458 b(imp)36 b(ortan)-36 b(t)457 b(to)h(remem)-36 b(b)36 b(er)3432 21979 y(that)405 b(\012)h(is)h(the)e(family)j(of)e (all)h(sector)f(indices)g Fy(!)48 b Fz(\()p Fy(f)142 b Fz(\))405 b(asso)36 b(ciated)407 b(with)f(the)g(\014eld)f(lab)36 b(els)3432 23584 y Fy(f)510 b Fz(and)368 b(that)g Fy(!)48 b Fz(\()p Fy(f)142 b Fz(\))368 b Ft(2)h Fy(O)15070 23790 y Fx(h)p Fr(\()p Fx(f)98 b Fr(\))16949 23584 y Fz(,)382 b Fy(h)p Fz(\()p Fy(f)142 b Fz(\))368 b(b)36 b(eing)368 b(the)g(scale)i(of)f(the)f(propagator)h(connected)f(to)3432 25189 y(the)444 b(corresp)36 b(onding)445 b(\014eld)g(v)-72 b(ariable,)450 b(see)445 b Ft(x)p Fz(2.4.)615 b(In)445 b(agreemen)-36 b(t)445 b(with)h(this)f(de\014nition,)3432 26794 y(if)4890 26458 y(\026)4745 26794 y(\012)556 b(is)f(a)g(subset)g (of)g(\012,)16473 25908 y Fl(P)17747 26847 y Fr(\026)17642 27071 y(\012)18932 26794 y Fz(will)h(denote)f(the)f(sum)h(o)-36 b(v)g(er)555 b Fy(!)48 b Fz(\()p Fy(f)142 b Fz(\))575 b Ft(2)g Fy(O)39645 27001 y Fx(h)p Fr(\()p Fx(f)98 b Fr(\))41524 26794 y Fz(,)586 b(for)556 b(an)-36 b(y)3432 28399 y Fy(f)510 b Ft(2)5983 28063 y Fz(\026)5838 28399 y(\012.)5383 30004 y(Let)423 b(us)g(call)h Fy(f)12377 30203 y Fr(0)13327 30004 y Fz(the)f(\014eld)g(whose)g(sector)h(index)g Fy(!)48 b Fz(\()p Fy(f)31446 30203 y Fr(0)31971 30004 y Fz(\))369 b Ft(2)f Fy(O)35094 30203 y Fx(h)36115 30004 y Fz(is)424 b(\014xed)f(in)h(the)f(sum)3432 31609 y(o)-36 b(v)g(er)434 b(\012.)578 b(W)-108 b(e)434 b(rewrite)g(the)f(sector)g (sum)g(in)h(the)f(l.h.s.)579 b(of)434 b(\(2.93\))h(as:)10000 33903 y Fs(\003)9436 34235 y Fl(X)9895 36685 y Fr(\012)11404 35342 y Fz(=)13355 33903 y Fs(\003)12790 34235 y Fl(X)12784 36685 y Fr(\012)13502 36828 y Fp(~)-444 b Fi(v)13879 37027 y Fp(0)15198 34235 y Fl(X)14617 36744 y Fr(\012)p Fs(n)p Fr(\012)16485 36887 y Fp(~)g Fi(v)16862 37086 y Fp(0)17747 35342 y Fz(=)22356 34235 y Fl(X)19128 36744 y Fx(\033)p 19128 36921 570 45 v 19735 37021 a Fp(~)g Fi(v)20112 37220 y Fp(0)20629 36744 y Fs(2)p Fm(O)22071 36900 y Fi(h)22591 37120 y Fp(~)g Fi(v)22968 37319 y Fp(0)23540 36744 y Fr(\()p Fx(P)24541 36887 y Fp(~)g Fi(v)24918 37086 y Fp(0)25434 36744 y Fs(n)p Fx(f)26356 36867 y Fp(0)26818 36744 y Fr(\))31800 33903 y Fs(\003)31235 34235 y Fl(X)31163 36534 y Fp(\012)31779 36754 y(~)g Fi(v)32156 36953 y Fp(0)32673 36534 y(:)27538 37729 y Fq(S)47 b Fp(\()p Fi(P)28967 37949 y Fp(~)-444 b Fi(v)29344 38148 y Fp(0)29860 37729 y Fq(n)p Fi(f)30683 37928 y Fp(0)31145 37729 y(\))p Fq(\036)p Fe(S)32992 37962 y Fi(h)33512 38181 y Fp(~)g Fi(v)33889 38380 y Fp(0)34461 37729 y(\()p Fi(\033)p 34781 37906 500 45 v 35318 38006 a Fp(~)g Fi(v)35695 38205 y Fp(0)36212 37729 y(\))37467 34235 y Fl(X)36886 36744 y Fr(\012)p Fs(n)p Fr(\012)38754 36887 y Fp(~)g Fi(v)39131 37086 y Fp(0)40238 35342 y Fy(:)3330 b Fz(\(3.3\))3432 40927 y(Then,)346 b(for)326 b(an)-36 b(y)325 b(\014xed)f Fy(\033)p 14363 41104 787 45 v 15186 41203 a Fr(~)-506 b Fx(v)15608 41326 y Fp(0)16494 40927 y Ft(2)368 b Fn(O)18870 41126 y Fx(h)19450 41269 y Fp(~)-444 b Fi(v)19827 41468 y Fp(0)20400 40927 y Fz(\()p Fy(P)21780 41126 y Fr(~)-506 b Fx(v)22202 41249 y Fp(0)22719 40927 y Ft(n)p Fy(f)24024 41126 y Fr(0)24550 40927 y Fz(\),)347 b(w)-36 b(e)325 b(b)36 b(ound)324 b(the)g(pro)36 b(duct)324 b(of)h Fy(\037)40559 41126 y Fx(v)41429 40927 y 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(sector)h(condition)g(b)-36 b(y)409 b(allo)-36 b(wing)411 b(the)3432 55114 y(external)456 b(\014elds)g(of)h Fy(v)48 b Fz(,)462 b(whic)-36 b(h)456 b(are)h(also)g(external)g(\014elds)f(of) 506 b(~)-699 b Fy(v)33932 55313 y Fr(0)34914 55114 y Fz(and)456 b(are)h(not)f(equal)h(to)3432 56719 y Fy(f)4073 56918 y Fr(0)4598 56719 y Fz(,)399 b(to)390 b(ha)-36 b(v)g(e)390 b(a)h(momen)-36 b(tum)389 b(v)-72 b(arying,)399 b(instead)390 b(than)f(in)h(the)f(original)j(sector,)399 b(of)390 b(scale)3432 58324 y Fy(h)p Fz(,)433 b(in)h(that)f(of)h(scale) g Fy(h)14691 58523 y Fr(~)-506 b Fx(v)15113 58646 y Fp(0)16063 58324 y Fz(con)-36 b(taining)434 b(it.)5383 59929 y(Let)449 b(us)f(no)-36 b(w)450 b(observ)-36 b(e)450 b(that)f(the)f(mo)36 b(di\014ed)449 b(running)f(coupling)i(functions)f(do)g(not)3432 61535 y(dep)36 b(end)457 b(on)i(\012)10780 61734 y Fr(~)-506 b Fx(v)11202 61857 y Fp(0)11719 61535 y Fz(,)466 b(if)460 b Fy(\033)p 13764 61712 V 14586 61811 a Fr(~)-505 b Fx(v)15009 61934 y Fp(0)15984 61535 y Fz(is)460 b(\014xed,)465 b(as)459 b(it)h(follo)-36 b(ws)461 b(from)e(de\014nition)f(\(2.74\);)473 b(hence)459 b(the)3432 63140 y(only)585 b(remaining)h(dep)36 b(endence)583 b(on)j(\012)22717 63339 y Fr(~)-506 b Fx(v)23139 63462 y Fp(0)24241 63140 y Fz(is)585 b(in)g Fy(\037)28220 63339 y Fr(~)-506 b Fx(v)28642 63462 y Fp(0)29159 63140 y Fz(\()p Ft(S)100 b Fz(\()p Fy(P)31949 63339 y Fr(~)-506 b Fx(v)32371 63462 y Fp(0)32888 63140 y Fz(\)\).)1033 b(It)585 b(follo)-36 b(ws,)625 b(b)-36 b(y)585 b(using)3432 64745 y(Lemma)433 b(3.1)i(for)f Ft(j)p Fy(P)13400 64944 y Fr(~)-506 b Fx(v)13822 65067 y Fp(0)14339 64745 y Ft(j)369 b(\024)g Fz(8)434 b(and)f(the)g(trivial)i(b)36 b(ound)18346 67039 y Fs(\003)17782 67371 y Fl(X)17827 69670 y Fp(\012)18443 69889 y(~)-444 b Fi(v)18820 70088 y Fp(0)14085 70865 y Fq(S)47 b Fp(\()p Fi(P)15514 71084 y Fp(~)-444 b Fi(v)15891 71283 y Fp(0)16407 70865 y Fq(n)p Fi(f)17230 71064 y Fp(0)17691 70865 y(\))p Fq(\036)p Fe(S)19538 71134 y Fi(h)20058 71354 y Fp(~)g Fi(v)20435 71553 y Fp(0)20952 71134 y(\()p Fi(\033)p 21272 71311 500 45 v 21810 71411 a Fp(~)g Fi(v)22187 71610 y Fp(0)22703 71134 y(\))23433 68478 y Fz(1)369 b Ft(\024)g Fy(c\015)27294 67567 y Fp(1)p 27294 67730 406 45 v 27294 68277 a(2)27832 67929 y Fr(\()p Fx(h)28778 68072 y Fp(~)-444 b Fi(v)29155 68271 y Fp(0)29672 67929 y Fs(\000)p Fx(h)p Fr(\)\()p Fs(j)p Fx(P)32574 68072 y Fp(~)g Fi(v)32951 68271 y Fp(0)33468 67929 y Fs(j\000)p Fr(1\))35722 68478 y Fy(;)7846 b Fz(\(3.6\))24367 75163 y(34)p eop %%Page: 35 35 35 34 bop 3432 7638 a Fz(for)434 b Ft(j)p Fy(P)6662 7837 y Fr(~)-506 b Fx(v)7084 7960 y Fp(0)7601 7638 y Ft(j)369 b(\025)g Fz(10,)434 b(that)g(w)-36 b(e)433 b(can)h(b)36 b(ound)432 b(the)h(sum)g(o)-36 b(v)g(er)434 b(\012)31714 7837 y Fr(~)-506 b Fx(v)32136 7960 y Fp(0)32654 7638 y Fz(,)433 b(for)h(an)-36 b(y)434 b Ff(S)38993 7837 y Fx(h)39573 7980 y Fp(~)-444 b Fi(v)39950 8179 y Fp(0)40522 7638 y Fz(\()p Fy(\033)p 41028 7815 787 45 v 41851 7915 a Fr(~)-506 b Fx(v)42273 8038 y Fp(0)42790 7638 y Fz(\),)433 b(as)16242 10142 y Fs(\003)15678 10474 y Fl(X)15722 12773 y Fp(\012)16338 12992 y(~)-444 b Fi(v)16715 13191 y Fp(0)11980 13968 y Fq(S)47 b Fp(\()p Fi(P)13409 14188 y Fp(~)-444 b Fi(v)13786 14387 y Fp(0)14303 13968 y Fq(n)p Fi(f)15126 14167 y Fp(0)15587 13968 y(\))p Fq(\036)p Fe(S)17434 14200 y Fi(h)17954 14420 y Fp(~)g Fi(v)18331 14619 y Fp(0)18903 13968 y(\()p Fi(\033)p 19223 14145 500 45 v 19761 14245 a Fp(~)g Fi(v)20138 14444 y Fp(0)20655 13968 y(\))21329 11581 y Fy(\037)22180 11780 y Fr(~)-506 b Fx(v)22602 11903 y Fp(0)23118 11581 y Fz(\()p Ft(S)100 b Fz(\()p Fy(P)25908 11780 y Fr(~)-506 b Fx(v)26330 11903 y Fp(0)26847 11581 y Fz(\)\))368 b Ft(\024)11848 16644 y(\024)h Fy(c\015)14557 16058 y Fr(\()p Fx(h)15503 16201 y Fp(~)-444 b Fi(v)15880 16400 y Fp(0)16396 16058 y Fs(\000)p Fx(h)p Fr(\))18036 16169 y Fz([)18530 15696 y Fp(1)p 18530 15859 406 45 v 18530 16406 a(2)19069 16058 y Fr(\()p Fs(j)p Fx(P)20331 16201 y Fp(~)g Fi(v)20708 16400 y Fp(0)21225 16058 y Fs(j\000)p Fr(3\))p Fx(\037)p Fr(\(4)p Fs(\024j)p Fx(P)26102 16201 y Fp(~)g Fi(v)26479 16400 y Fp(0)26996 16058 y Fs(j\024)p Fr(8\)+)29690 15696 y Fp(1)p 29690 15859 V 29690 16406 a(2)30229 16058 y Fr(\()p Fs(j)p Fx(P)31491 16201 y Fp(~)g Fi(v)31868 16400 y Fp(0)32385 16058 y Fs(j\000)p Fr(1\))p Fx(\037)p Fr(\()p Fs(j)p Fx(P)36060 16201 y Fp(~)g Fi(v)36437 16400 y Fp(0)36954 16058 y Fs(j\025)p Fr(10\))39253 16169 y Fz(])40040 16644 y Fy(:)3528 b Fz(\(3.7\))5383 19578 y(W)-108 b(e)357 b(are)g(th)-36 b(us)356 b(left)h(with)g(the)g(problem)f(of)i(b)36 b(ounding)356 b(a)h(sum)g(similar)h(to)f(the)f(initial)3432 21183 y(one,)544 b(but)521 b(with)g(all)i(the)e(external)i(sector)f (indices)f(on)h(scale)h Fy(h)35239 21382 y Fr(~)-506 b Fx(v)35661 21505 y Fp(0)36699 21183 y Fz(instead)522 b(of)g Fy(h)p Fz(.)843 b(W)-108 b(e)3432 22788 y(shall)520 b(do)g(that)g(b)-36 b(y)519 b(iterating)i(the)e(previous)h(pro)36 b(cedure,)541 b(in)520 b(a)h(w)-36 b(a)g(y)521 b(whic)-36 b(h)519 b(dep)36 b(ends)3432 24393 y(on)443 b(the)g(structure)f(of)i (the)f(tree)g Fy(\034)591 b Fz(and)443 b(of)h(the)f(graph)g Fy(T)181 b Fz(;)448 b(the)443 b(iteration)h(stops)f(at)h(the)3432 25998 y(endp)36 b(oin)-36 b(ts,)432 b(where)i(w)-36 b(e)434 b(can)f(use)h(the)f(h)-36 b(yp)36 b(otheses)433 b(\(2.90\),)h(\(2.91\)) h(and)e(\(2.92\).)5383 27603 y(T)-108 b(o)395 b(describ)36 b(e)395 b(this)f(inductiv)-36 b(e)395 b(pro)36 b(cedure,)402 b(w)-36 b(e)395 b(establish,)403 b(for)396 b(an)-36 b(y)395 b(v)-36 b(ertex)396 b Fy(v)416 b Ft(2)369 b Fy(V)45602 27802 y Fx(\037)46242 27603 y Fz(,)3432 29208 y(a)526 b(partial)h(ordering)f(of)g(the)g Fy(s)18634 29407 y Fx(v)19705 29208 y Fz(v)-36 b(ertices)527 b Fy(v)25131 29407 y Fr(1)25656 29208 y Fy(;)221 b(:)g(:)g(:)j(;)d(v)29198 29407 y Fx(s)29633 29518 y Fi(v)30710 29208 y Ft(2)526 b Fy(V)32880 29407 y Fx(\037)34046 29208 y Fz(immediately)h(follo)-36 b(wing)3432 30813 y Fy(v)526 b Fz(on)479 b Fy(\034)148 b Fz(,)490 b(b)-36 b(y)479 b(assigning)h(a)f(ro)36 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b(where)339 b Fy(v)30529 36752 y Fs(0)31178 37234 y Fz(is)h(the)f Fy(\037)p Fz(-v)-36 b(ertex)339 b(immediately)3432 38839 y(preceding)c Fy(v)416 b Ft(2)369 b Fy(V)12245 39038 y Fx(\037)12884 38839 y Fz(,)356 b(if)336 b Fy(f)15336 39038 y Fr(0)16231 38839 y Ft(62)369 b Fy(P)18324 39038 y Fx(v)18869 38839 y Fz(;)g(otherwise,)356 b(the)335 b(ro)36 b(ot)335 b(of)i Fy(T)32641 39038 y Fx(v)33522 38839 y Fz(is)f(the)f(v)-36 b(ertex)336 b(con)-36 b(taining)3432 40444 y(the)433 b(ro)36 b(ot)434 b(of)g Fy(T)10878 39962 y Fs(\003)11404 40444 y Fz(,)g(see)f(Fig.)579 b(2.)35559 45347 y(~)-699 b Fy(v)36139 45546 y Fr(0)21119 48156 y Fy(f)21760 48355 y Fr(0)32189 51679 y Fy(v)32818 51878 y Fr(1)28868 50370 y Fy(`)29414 50569 y Fr(1)27761 46495 y Fy(`)28307 46694 y Fr(3)24440 47805 y Fy(v)25069 48004 y Fr(2)32189 45038 y Fy(v)32818 45237 y Fr(3)13370 56642 y @beginspecial @setspecial %%BeginDocument: fig53.pst % x1 y1 x2 y2 cambio_coordinate /cambio_coordinate{ /y2 exch def /x2 exch def /y1 exch def /x1 exch def /dx x2 x1 sub def /dy y2 y1 sub def x1 y1 translate dy dx atan rotate dx 2 exp dy 2 exp add sqrt } 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 /cerchio { gsave % uso: x1 y1 r cerchio 0 360 newpath arc stroke grestore} def 60 0 translate .6 .6 scale 150 120 90 cerchio 100 120 20 cerchio 160 170 20 cerchio 160 70 20 cerchio 0 120 90 120 freccia 100 130 moveto 100 157 lineto 107 127 153 163 freccia 110 120 moveto 137 120 lineto 107 113 153 77 freccia 100 110 moveto 100 83 lineto % 167 177 moveto 210 220 lineto 160 160 moveto 160 133 lineto 167 163 moveto 187 143 lineto % 160 60 moveto 160 5 lineto 167 77 moveto 187 97 lineto 167 63 moveto 210 20 lineto stroke %%EndDocument @endspecial 3432 61125 a Fz(Figure)520 b(2:)753 b(A)520 b(p)36 b(ossible)521 b(cluster)f(structure)f(corresp)36 b(onding)520 b(to)h(a)g(tree)f Fy(\034)669 b Fz(of)521 b(the)f(ex-)3432 62731 y(pansion)453 b(for)g(the)g(e\013ectiv)-36 b(e)454 b(p)36 b(oten)-36 b(tials)453 b(suc)-36 b(h)452 b(that)h Fy(s)30000 62930 y Fr(~)-506 b Fx(v)30422 63053 y Fp(0)31341 62731 y Fz(=)402 b(3.)638 b(The)453 b(set)g Fy(T)39945 62930 y Fr(~)-506 b Fx(v)40367 63053 y Fp(0)41337 62731 y Fz(is)454 b(formed)3432 64336 y(b)-36 b(y)458 b(the)h(lines)g Fy(`)11070 64535 y Fr(1)12054 64336 y Fz(and)f Fy(`)15154 64535 y Fr(3)15680 64336 y Fz(.)654 b(The)459 b(lines)g(di\013eren)-36 b(t)457 b(from)j Fy(`)31187 64535 y Fr(1)32171 64336 y Fz(and)e Fy(`)35271 64535 y Fr(3)36256 64336 y Fz(and)g(not)g(b)36 b(elonging)3432 65941 y(to)433 b Fy(P)5895 66140 y Fr(~)-506 b Fx(v)6317 66263 y Fp(0)7268 65941 y Fz(ha)-36 b(v)g(e)434 b(to)f(b)36 b(e)434 b(con)-36 b(tracted)433 b(in)-36 b(to)433 b(the)g(Lesniewski)i (determinan)-36 b(ts.)5383 69160 y(The)472 b(l.h.s.)695 b(of)473 b(\(2.93\))g(is)g(b)36 b(ounded)471 b(b)-36 b(y)472 b(the)g(pro)36 b(duct)471 b(of)i(the)f(r.h.s.)695 b(of)473 b(\(3.7\))g(and)24367 75163 y(35)p eop %%Page: 36 36 36 35 bop 3432 7638 a Fz(the)433 b(follo)-36 b(wing)435 b(quan)-36 b(tit)g(y:)4578 9071 y Fl(2)4578 11019 y(6)4578 11728 y(4)6925 10577 y(Y)5316 13027 y Fx(v)32 b(>)k Fr(~)-506 b Fx(v)6996 13150 y Fp(0)7457 13027 y Fx(;v)32 b Fs(2)p Fx(V)9383 13138 y Fi(\037)10891 10577 y Fl(X)10644 13016 y Fr(~)10539 13239 y(\012)11219 13382 y Fi(v)24 b(;)38 b Fp(~)-444 b Fi(v)12325 13581 y Fp(0)13063 11684 y Fy(\037)13878 11883 y Fx(v)14645 10400 y Fl(\020)15617 11348 y Fz(~)15306 11684 y Ft(S)16110 11883 y Fx(v)32 b(;)k Fr(~)-506 b Fx(v)17319 12006 y Fp(0)17836 10400 y Fl(\021)18497 9071 y(3)18497 11019 y(7)18497 11728 y(5)19457 9403 y(2)19457 11396 y(4)20286 10577 y(Y)20195 13037 y Fx(l)11 b Fs(2)p Fx(T)22013 11684 y Fy(\016)22589 12048 y Fx(!)23206 11650 y Fp(+)23174 12381 y Fi(l)23882 12048 y Fx(;!)24760 11650 y Fq(\000)24728 12381 y Fi(l)25511 9403 y Fl(3)25511 11396 y(5)26470 10122 y(Z)28704 10577 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Fp(0)6550 23019 y Fz(b)36 b(elong)454 b(to)f Fy(O)13328 23218 y Fx(h)13908 23361 y Fp(~)-444 b Fi(v)14285 23560 y Fp(0)14858 23019 y Fz(,)458 b(since)453 b(in)h(this)f(case)g(the)g(\014elds)g(asso)36 b(ciated)454 b(with)g Fy(P)41393 23218 y Fx(v)41938 23019 y Ft(n)p Fy(P)43476 23218 y Fr(~)-506 b Fx(v)43898 23341 y Fp(0)44869 23019 y Fz(are)3432 24624 y(con)-36 b(tracted)433 b(on)g(scale)h Fy(h)15532 24823 y Fr(~)-506 b Fx(v)15954 24946 y Fp(0)16471 24624 y Fz(.)5383 26229 y(W)-108 b(e)483 b(no)-36 b(w)483 b(consider)g(the)f Fy(s)18484 26428 y Fr(~)-506 b Fx(v)18906 26551 y Fp(0)19906 26229 y Fy(\037)p Fz(-v)-36 b(ertices)483 b(immediately)h(follo)-36 b(wing)534 b(~)-699 b Fy(v)39534 26428 y Fr(0)40543 26229 y Fz(and)482 b(w)-36 b(e)484 b(re-)3432 27834 y(order)433 b(the)g(expression)h (\(3.8\))g(in)f(the)g(follo)-36 b(wing)436 b(w)-36 b(a)g(y:)5829 32285 y(\(3)p Fy(:)p Fz(8\))1107 b(=)11861 30602 y Fx(s)12334 30745 y Fp(~)-444 b Fi(v)12711 30944 y Fp(0)11837 31178 y Fl(Y)11728 33609 y 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Fs(\003)16715 71652 y Fz(is)g(an)f(endp)36 b(oin)-36 b(t.)577 b(In)432 b(this)g(case)32663 70767 y Fl(P)33831 71117 y Fs(\003)33936 71888 y Fr(~)33831 72112 y(\012)34511 72307 y Fi(v)34950 72116 y Fq(\003)35430 72307 y Fi(;)38 b Fp(~)-444 b Fi(v)36097 72506 y Fp(0)37038 71652 y Fz(=)38418 70767 y Fl(P)39587 71117 y Fs(\003)39587 71981 y Fx(\033)p 39587 72158 570 45 v 31 w Fs(2)p Fr(\012)41463 72137 y Fi(h)41945 72408 y(v)42384 72217 y Fq(\003)42919 72137 y(\000)p Fp(1)44507 71652 y Fz(and)24367 75163 y(36)p eop %%Page: 37 37 37 36 bop 3432 7638 a Fy(\037)4247 7837 y Fx(v)4737 7585 y Fq(\003)5271 7638 y Fz(\()6089 7302 y(~)5777 7638 y Ft(S)6581 7837 y Fx(v)7071 7585 y Fq(\003)7551 7837 y Fx(;)36 b Fr(~)-506 b Fx(v)8270 7960 y Fp(0)8787 7638 y Fz(\))369 b(=)g(1,)416 b(since)410 b(the)h(corresp)36 b(onding)410 b(constrain)-36 b(t)410 b(is)h(already)h(included)e(in)h (the)3432 9243 y(de\014nition)349 b(of)i(the)f(mo)36 b(di\014ed)350 b(coupling)g(functions,)367 b(so)351 b(that)e(the)h 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Fp(1)p 18322 18458 406 45 v 18322 19004 a(2)18861 18657 y Fr(\()p Fx(h)19769 18852 y Fi(v)20208 18661 y Fq(\003)20743 18657 y Fs(\000)p Fr(1\))p Fx(\037)p Fr(\()p Fs(j)p Fx(P)24119 18852 y Fi(v)24558 18661 y Fq(\003)25093 18657 y Fs(j)p Fr(=4\))26978 19205 y Fy(\015)27725 18657 y Fr(\()p Fx(h)28633 18852 y Fi(v)29072 18661 y Fq(\003)29607 18657 y Fs(\000)p Fr(1\))p Fx(\037)p Fr(\()p Fx(v)32615 18344 y Fq(\003)33357 18657 y Fr(is)261 b(of)332 b(t)-26 b(yp)26 b(e)261 b Fx(\027)58 b Fr(\))38715 19205 y Fy(:)4203 b Fz(\(3.13\))5727 21585 y(\(b\))343 b Fy(v)8482 21103 y Fs(\003)9351 21585 y Fz(is)i(not)f(an)g(end)f(p)36 b(oin)-36 b(t.)548 b(In)344 b(this)g(case,)363 b(b)-36 b(y)344 b(the)g(remark)g(follo)-36 b(wing)347 b(\(3.9\),)363 b(the)3432 23190 y(expression)441 b(in)h(the)f(r.h.s.)602 b(of)442 b(\(3.11\))h(has)e(exactly)i(the)e (same)h(structure)e(as)i(the)f(l.h.s.)3432 24795 y(of)447 b(\(2.93\),)452 b(whic)-36 b(h)447 b(w)-36 b(e)447 b(started)g(the)f (iteration)i(from;)455 b(one)447 b(has)g(only)g(to)h(substitute)495 b(~)-700 b Fy(v)46077 24994 y Fr(0)3432 26400 y Fz(with)359 b Fy(v)6997 25918 y Fs(\003)7522 26400 y Fz(,)374 b Fy(h)359 b Fz(with)h Fy(h)13039 26599 y Fr(~)-506 b Fx(v)13461 26722 y Fp(0)14336 26400 y Fz(and)359 b Fy(h)17576 26599 y Fr(~)-506 b Fx(v)17998 26722 y Fp(0)18874 26400 y Fz(with)359 b Fy(h)22511 26599 y Fx(v)23001 26347 y Fq(\003)23536 26400 y Fz(.)553 b(Hence)359 b(w)-36 b(e)360 b(can)f(b)36 b(ound)358 b(the)h(r.h.s.)553 b(of)360 b(\(3.11\))3432 28005 y(b)-36 b(y)433 b(extracting)h(a)g(factor)11091 30621 y Fy(c\015)12398 30035 y Fr(\()p Fx(h)13306 30230 y Fi(v)13745 30039 y Fq(\003)14280 30035 y Fs(\000)p Fx(h)15592 30178 y Fp(~)-444 b Fi(v)15969 30377 y Fp(0)16485 30035 y Fr(\))16851 30146 y Fz(\()17490 29673 y Fp(1)p 17490 29836 V 17490 30382 a(2)18029 30035 y Fr(\()p Fs(j)p Fx(P)19253 30230 y Fi(v)19692 30039 y Fq(\003)20226 30035 y Fs(j\000)p Fr(3\))p Fx(\037)p Fr(\(4)p Fs(\024j)p Fx(P)25065 30230 y Fi(v)25504 30039 y Fq(\003)26040 30035 y Fs(j\024)p Fr(8\)+)28734 29673 y Fp(1)p 28735 29836 V 28735 30382 a(2)29273 30035 y Fr(\()p Fs(j)p Fx(P)30497 30230 y Fi(v)30936 30039 y Fq(\003)31471 30035 y Fs(j\000)p Fr(1\))p Fx(\037)p Fr(\()p Fs(j)p Fx(P)35108 30230 y Fi(v)35547 30039 y Fq(\003)36083 30035 y Fs(j\025)p Fr(10\))38382 30146 y Fz(\))43279 30621 y(\(3.14\))3432 33000 y(and)445 b(w)-36 b(e)445 b(end)g(up)g(with)g(an)h(expression)f(similar)i(to)e(\(3.8\),) 450 b(the)445 b(line)g Fy(l)37832 32518 y Fs(\003)38804 33000 y Fz(acting)h(no)-36 b(w)446 b(as)3432 34605 y(an)351 b(external)i(\014eld,)368 b(since)352 b(there)f(is)h(only)h(one)e (sector)h(sum)g(asso)36 b(ciated)353 b(with)f(it,)368 b(thanks)3432 36211 y(to)433 b(the)g(factor)i Fy(\016)11559 36575 y Fx(!)12176 36177 y Fp(+)12144 36946 y Fi(l)12411 36755 y Fq(\003)12946 36575 y Fx(;!)13824 36177 y Fq(\000)13792 36946 y Fi(l)14059 36755 y Fq(\003)15083 36211 y Fz(presen)-36 b(t)432 b(in)i(the)f(r.h.s.)578 b(of)434 b(\(3.10\).)5383 39421 y(It)428 b(is)g(no)-36 b(w)428 b(completely)h(ob)-36 b(vious)429 b(that)e(w)-36 b(e)429 b(can)f(iterate)g(the)f(previous)i (pro)36 b(cedure,)3432 41026 y(for)510 b(eac)-36 b(h)510 b(lea)-36 b(v)g(e)511 b(of)g Fy(T)14146 41225 y Fr(~)-506 b Fx(v)14568 41348 y Fp(0)15085 41026 y Fz(,)529 b(ending)510 b(up)f(with)h(a)h(b)36 b(ound)508 b(of)j(the)f(l.h.s.)808 b(of)511 b(\(2.93\))g(of)g(the)3432 42631 y(form)9161 45841 y(\()p Fy(c)p Ft(j)p Fy(\025)p Ft(j)p Fz(\))12230 45293 y Fx(n)13076 43561 y Fl(2)13076 45553 y(4)14222 44734 y(Y)13814 47184 y Fx(v)32 b Fs(2)p Fx(V)15479 47295 y Fi(\037)16266 45841 y Fy(\015)17013 45255 y Fr(\()p Fx(h)17921 45366 y Fi(v)18415 45255 y Fs(\000)p Fx(h)19689 45510 y Fi(v)20128 45319 y Fq(0)20482 45255 y Fr(\))20848 45366 y Fz(\()21486 44893 y Fp(1)p 21486 45056 V 21486 45603 a(2)22025 45255 y Fr(\()p Fs(j)p Fx(P)23249 45366 y Fi(v)23743 45255 y Fs(j\000)p Fr(3\))p Fx(\037)p Fr(\(4)p Fs(\024j)p Fx(P)28582 45366 y Fi(v)29077 45255 y Fs(j\024)p Fr(8\)+)31771 44893 y Fp(1)p 31772 45056 V 31772 45603 a(2)32310 45255 y Fr(\()p Fs(j)p Fx(P)33534 45366 y Fi(v)34028 45255 y Fs(j\000)p Fr(1\))p Fx(\037)p Fr(\()p Fs(j)p Fx(P)37665 45366 y Fi(v)38160 45255 y Fs(j\025)p Fr(10\))40459 45366 y Fz(\))41022 43561 y Fl(3)41022 45553 y(5)42424 45841 y Ft(\001)9161 49970 y(\001)10120 48022 y Fl(")11242 48531 y Fx(n)10820 48863 y Fl(Y)10766 51293 y Fx(i)p Fr(=1)12510 49970 y Fy(\015)13257 49422 y Fs(\000)14122 49059 y Fp(1)p 14121 49222 V 14121 49769 a(2)14660 49422 y Fr(\()p Fx(h)15568 49557 y Fi(i)15919 49422 y Fs(\000)p Fr(1\))p Fx(\037)p Fr(\()p Fx(v)18927 49109 y Fq(\003)18895 49726 y Fi(i)19669 49422 y Fr(is)262 b(of)331 b(t)-26 b(yp)26 b(e)261 b Fx(\025)p Fr(\))24687 49970 y Fy(\015)25434 49422 y Fr(\()p Fx(h)26342 49557 y Fi(i)26693 49422 y Fs(\000)p Fr(1\))p Fx(\037)p Fr(\()p Fx(v)29701 49109 y Fq(\003)29669 49726 y Fi(i)30443 49422 y Fr(is)h(of)331 b(t)-26 b(yp)26 b(e)261 b Fx(\027)58 b Fr(\))35432 48022 y Fl(#)36668 49970 y Fy(;)6250 b Fz(\(3.15\))3432 53264 y(where)394 b Fy(v)7827 52782 y Fs(0)8531 53264 y Fz(is)h(the)f Fy(\037)p Fz(-v)-36 b(ertex)394 b(immediately)i(preceding)e Fy(v)441 b Fz(on)395 b Fy(\034)148 b Fz(,)403 b(if)395 b Fy(v)416 b(>)i Fz(~)-699 b Fy(v)38853 53463 y Fr(0)39378 53264 y Fz(,)403 b(or)394 b(the)g(ro)36 b(ot,)3432 54869 y(if)388 b Fy(v)416 b Fz(=)i(~)-699 b Fy(v)7633 55068 y Fr(0)8159 54869 y Fz(.)563 b(On)387 b(the)g(other)g(hand,)396 b(giv)-36 b(en)389 b Fy(v)416 b Ft(2)369 b Fy(V)26734 55068 y Fx(\037)27373 54869 y Fz(,)398 b Fy(P)29006 55068 y Fr(\026)-506 b Fx(v)29884 54869 y Fz(=)369 b Fy(P)32103 55068 y Fx(v)33036 54869 y Fz(if)389 b Fy(v)34860 54387 y Fs(0)35539 54869 y Fy(<)417 b Fz(\026)-699 b Fy(v)417 b Ft(\024)369 b Fy(v)48 b Fz(.)562 b(Moreo)-36 b(v)g(er,)8154 56618 y Fx(n)7732 56950 y Fl(Y)7678 59380 y Fx(i)p Fr(=1)9422 58057 y Fy(\015)10169 57508 y Fs(\000)11034 57146 y Fp(1)p 11034 57309 V 11034 57855 a(2)11573 57508 y Fr(\()p Fx(h)12481 57643 y Fi(i)12832 57508 y Fs(\000)p Fr(1\))p Fx(\037)p Fr(\()p Fx(v)15840 57196 y Fq(\003)15808 57813 y Fi(i)16581 57508 y Fr(is)262 b(of)332 b(t)-26 b(yp)26 b(e)260 b Fx(\025)p Fr(\))21968 58057 y Fz(=)369 b Fy(\015)24096 57508 y Fs(\000)24961 57146 y Fp(1)p 24961 57309 V 24961 57855 a(2)25499 57508 y Fx(hm)26873 57631 y Fp(4)27335 57508 y Fr(\()p Fx(v)28159 57631 y Fp(0)28620 57508 y Fr(\))30604 56950 y Fl(Y)29263 59339 y Fx(v)293 b Fr(not)262 b(e)p Fx(:)g Fr(p)p Fx(:)33581 58057 y Fy(\015)34328 57508 y Fs(\000)35193 57146 y Fp(1)p 35193 57309 V 35193 57855 a(2)35732 57508 y Fx(m)36564 57631 y Fp(4)37025 57508 y Fr(\()p Fx(v)32 b Fr(\))38671 58057 y Fy(;)4247 b Fz(\(3.16\))3432 61504 y(where)465 b Fy(m)8359 61703 y Fr(4)8885 61504 y Fz(\()p Fy(v)48 b Fz(\))464 b(is)i(the)f(n)-36 b(um)g(b)36 b(er)464 b(of)i(end)f(p)36 b(oin)-36 b(ts)465 b(of)i(t)-36 b(yp)36 b(e)465 b Fy(\025)g Fz(follo)-36 b(wing)468 b(v)-36 b(ertex)466 b Fy(v)513 b Fz(on)465 b Fy(\034)148 b Fz(.)3432 63109 y(It)433 b(follo)-36 b(ws)436 b(that)d(\(3.15\))h(can)g(b)36 b(e)433 b(written)g(in)h(the)f (form)10664 67592 y(\()p Fy(c)p Ft(j)p Fy(\025)p Ft(j)p Fz(\))13733 67044 y Fx(n)14358 67592 y Fy(\015)15105 67044 y Fs(\000)15970 66682 y Fp(1)p 15970 66845 V 15970 67391 a(2)16508 67044 y Fx(hm)17882 67167 y Fp(4)18344 67044 y Fr(\()p Fx(v)19168 67167 y Fp(0)19629 67044 y Fr(\))20272 65644 y Fl(")21394 66153 y Fx(n)20972 66485 y Fl(Y)20918 68915 y Fx(i)p Fr(=1)22662 67592 y Fy(\015)23409 67044 y Fr(\()p Fx(h)24317 67179 y Fi(i)24668 67044 y Fs(\000)p Fr(1\))p Fx(\037)p Fr(\()p Fx(v)27676 66731 y Fq(\003)27644 67348 y Fi(i)28417 67044 y Fr(is)262 b(of)332 b(t)-26 b(yp)26 b(e)260 b Fx(\027)58 b Fr(\))33407 65644 y Fl(#)34348 67592 y Ft(\001)10664 70940 y(\001)12595 69833 y Fl(Y)11254 72223 y Fx(v)293 b Fr(not)262 b(e)p Fx(:)g Fr(p)p Fx(:)15572 70940 y Fy(\015)16319 70465 y Fz([)16680 70355 y Fs(\000)17545 69992 y Fp(1)p 17545 70155 V 17545 70702 a(2)18084 70355 y Fx(m)18916 70478 y Fp(4)19377 70355 y Fr(\()p Fx(v)32 b Fr(\)+)21464 69992 y Fp(1)p 21464 70155 V 21464 70702 a(2)22003 70355 y Fr(\()p Fs(j)p Fx(P)23227 70466 y Fi(v)23721 70355 y Fs(j\000)p Fr(3\))p Fx(\037)p Fr(\(4)p Fs(\024j)p Fx(P)28560 70466 y Fi(v)29055 70355 y Fs(j\024)p Fr(8\)+)31749 69992 y Fp(1)p 31749 70155 V 31749 70702 a(2)32288 70355 y Fr(\()p Fs(j)p Fx(P)33512 70466 y Fi(v)34006 70355 y Fs(j\000)p Fr(1\))p Fx(\037)p Fr(\()p Fs(j)p Fx(P)37643 70466 y Fi(v)38138 70355 y Fs(j\025)p Fr(10\))40437 70466 y Fz(])41224 70940 y Fy(;)1694 b Fz(\(3.17\))24367 75163 y(37)p eop %%Page: 38 38 38 37 bop 3432 7638 a Fz(whic)-36 b(h)433 b(pro)-36 b(v)g(es)434 b(Lemma)f(2.2)h(in)g(the)f(case)h Ft(j)p Fy(P)25626 7837 y Fx(v)26084 7960 y Fp(0)26601 7638 y Ft(j)369 b Fy(>)g Fz(0.)5383 9243 y(The)471 b(case)g Fy(P)11722 9442 y Fx(v)12180 9565 y Fp(0)13130 9243 y Fz(=)433 b Ft(;)471 b Fz(is)g(treated)g(in)g(a)g(similar)h(w)-36 b(a)g(y)-108 b(.)692 b(The)471 b(only)h(real)f(di\013erence)g(is)3432 10848 y(that)377 b(one)h(has)f(to)h(sum)g(o)-36 b(v)g(er)378 b(all)g(sector)g(indices.)560 b(Ho)-36 b(w)g(ev)g(er,)390 b(since)378 b(the)f(set)h(of)g(in)-36 b(ternal)3432 12453 y(\014elds)511 b Ft(I)7564 12652 y Fx(v)8022 12775 y Fp(0)9051 12453 y Fz(is)h(necessarily)h(not)e(empt)-36 b(y)511 b(\(our)h(de\014nitions)f(imply)h(that,)531 b(in)512 b(this)f(case,)3481 14058 y(~)-699 b Fy(v)4061 14257 y Fr(0)5167 14058 y Fz(=)581 b Fy(v)7389 14257 y Fr(0)7914 14058 y Fz(\),)590 b(w)-36 b(e)558 b(can)g(c)-36 b(ho)36 b(ose)559 b(in)f(an)g(arbitrary)g(w)-36 b(a)g(y)559 b(one)f(\014eld)g Fy(f)36250 14257 y Fr(0)37356 14058 y Ft(2)581 b(I)39546 14257 y Fx(v)40004 14380 y Fp(0)41079 14058 y Fz(and)558 b(let)g(it)3432 15663 y(pla)-36 b(y)378 b(the)f(same)h(role)g(of)h(the) e(selected)g(external)i(\014eld)e(of)h Fy(v)32239 15862 y Fr(0)33143 15663 y Fz(in)f(the)g(previous)h(iterativ)-36 b(e)3432 17269 y(pro)36 b(cedure.)1065 b(Of)597 b(course,)637 b(the)596 b(\014rst)f(iteration)i(step,)637 b(whic)-36 b(h)596 b(pro)36 b(duced)595 b(b)36 b(efore)597 b(the)3432 18874 y(\\scale)327 b(jump")f(factor)g(in)g(the)g(r.h.s.)542 b(of)327 b(\(3.7\),)349 b(is)326 b(no)-36 b(w)326 b(missing,)348 b(but)325 b(this)h(is)h(irrelev)-72 b(an)-36 b(t,)3432 20479 y(since)433 b(that)g(factor)h(is)f(equal)h(to)g(1)f(if)h Ft(j)p Fy(P)22968 20678 y Fx(v)23426 20801 y Fp(0)23943 20479 y Ft(j)369 b Fz(=)g(0.)578 b(All)434 b(the)f(other)g(steps)g(are) g(absolutely)3432 22084 y(iden)-36 b(tical,)573 b(but,)f(at)545 b(the)g(end)f(of)i(the)e(iteration,)574 b(w)-36 b(e)545 b(end)g(up)f(with)h(the)f(sector)h(sum)3432 23689 y(related)535 b(with)g Fy(f)11574 23888 y Fr(0)12100 23689 y Fz(;)586 b(this)535 b(pro)36 b(duces)535 b(a)h(factor)g Fy(\015)27006 23199 y Fs(\000)27871 22837 y Fp(1)p 27870 23000 406 45 v 27870 23546 a(2)28409 23199 y Fx(h)28951 23310 y Fi(v)29366 23509 y Fp(0)30480 23689 y Fz(=)542 b Fy(\015)32781 23199 y Fs(\000)33646 22837 y Fp(1)p 33645 23000 V 33645 23546 a(2)34184 23199 y Fr(\()p Fx(h)p Fr(+1\))36716 23689 y Fz(.)883 b(This)536 b(completes)3432 25294 y(the)433 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67257 y Fy(;)34792 67239 y Fz(~)34722 67257 y Fn(x)35511 67456 y Fr(2)36037 67257 y Fz(\))369 b Fy(:)6656 b Fz(\(4.4\))3432 70237 y(Note)485 b(that)g(there)f(is)i(no)f(other)g(case)g(to)h (consider,)498 b(since,)g(as)486 b(a)f(consequence)h(of)g(the)3432 71842 y(freedom)435 b(in)h(the)f(c)-36 b(hoice)436 b(of)g(the)f (\014eld)g(carrying)h(the)f(deriv)-72 b(ativ)-36 b(e)436 b(in)g(the)f(endp)36 b(oin)-36 b(ts)434 b(of)24367 75163 y(38)p eop %%Page: 39 39 39 38 bop 3432 7638 a Fz(t)-36 b(yp)36 b(e)483 b Fy(z)59 b Fz(,)496 b(there)482 b(is)i(no)f(con)-36 b(tribution)482 b(to)h(the)g(e\013ectiv)-36 b(e)483 b(p)36 b(oten)-36 b(tial)483 b(with)g Fy(n)454 b Ft(\025)g Fz(2)483 b(and)g(a)3432 9243 y(deriv)-72 b(ativ)-36 b(e)432 b(acting)h(on)e(the)h(external)g (\014elds)f(of)i Fy(v)27793 9442 y Fr(0)28318 9243 y Fz(,)g(b)36 b(efore)432 b(the)g(application)g(of)h(the)e Ft(L)3432 10848 y Fz(op)36 b(erator.)5383 12453 y(Let)460 b(us)g(consider)g(\014rst)g(the)f(con)-36 b(tributions)460 b(to)h(the)e(r.h.s.)659 b(of)462 b(\(4.1\))f(coming)g(from)3432 14058 y(the)302 b(trees)i(with)f Fy(n)370 b Fz(=)e(1.)535 b(These)304 b(trees)f(ha)-36 b(v)g(e)304 b(only)g(t)-36 b(w)g(o)304 b(v)-36 b(ertices,)330 b Fy(v)35559 14257 y Fr(0)36388 14058 y Fz(\(of)304 b(scale)g Fy(h)29 b Fz(+)g(1\))303 b(and)3432 15663 y(the)331 b(endp)36 b(oin)-36 b(t)331 b Fy(v)11565 15181 y Fs(\003)12090 15663 y Fz(,)353 b(whose)332 b(scale)h(has)f(to)g(b)36 b(e)332 b(equal)g(to)h Fy(h)88 b Fz(+)g(2.)543 b(If)333 b(w)-36 b(e)332 b(imp)36 b(ose)332 b(the)g(further)3432 17269 y(condition)317 b(that)g Fy(P)12564 17468 y Fx(v)13054 17216 y Fq(\003)13958 17269 y Fz(=)369 b Fy(P)16177 17468 y Fx(v)16635 17591 y Fp(0)17152 17269 y Fz(,)341 b(the)317 b(sum)g(of)h(these)f(terms)g (is)h(equal)g(to)g Ft(LV)38520 16787 y Fr(\()p Fx(h)p Fr(+1\))41052 17269 y Fz(\()p Fy(\034)76 b(;)221 b( )43675 16787 y Fr(\()p Fs(\024)p Fx(h)p Fr(\))45736 17269 y Fz(\).)3432 18874 y(In)293 b(order)h(to)g(con)-36 b(trol)295 b(the)e(\015o)-36 b(w)294 b(of)h(the)f(running)f(coupling)h(functions,) 322 b(w)-36 b(e)294 b(need)f(a)i(\\go)36 b(o)g(d)3432 20479 y(b)g(ound")432 b(of)j(the)e(remaining)g(terms.)5383 22084 y(Let)483 b(us)h(consider)g(a)g(con)-36 b(tribution)483 b(to)i(the)e(r.h.s.)730 b(of)485 b(\(4.1\),)498 b(suc)-36 b(h)483 b(that)h Fy(n)455 b Ft(\025)g Fz(2)485 b(or)3432 23689 y Fy(n)369 b Fz(=)g(1)433 b(and)h Fy(P)10409 23888 y Fx(v)10899 23636 y Fq(\003)11803 23689 y Ft(6)p Fz(=)368 b Fy(P)14021 23888 y Fx(v)14479 24011 y Fp(0)14997 23689 y Fz(.)578 b(By)434 b(pro)36 b(ceeding)433 b(as)h(in)g Ft(x)p Fz(2.5,)h(it)f(is)g(easy)g(to)g(sho)-36 b(w)433 b(that)12020 26793 y Ft(L)p Fy(W)14345 26114 y Fr(\()p Fx(h)p Fr(\))14164 27177 y Fx(\034)50 b(;)p Fm(P)p Fx(;)p Fr(\012)p Fs(n)p Fr(\012)17724 27288 y Fi(v)18139 27487 y Fp(0)18655 27177 y Fx(;T)19651 26793 y Fz(\()p Fn(x)20946 26992 y Fx(v)21404 27115 y Fp(0)21920 26793 y Fz(\))369 b(=)24668 25686 y Fl(X)24175 28136 y Fx(\013)p Fs(2)p Fx(A)26111 28285 y Fi(T)26982 26793 y Fy(W)28391 26114 y Fr(\()p Fx(L)p Fr(\))28210 27177 y Fx(\034)50 b(;)p Fm(P)p Fx(;)p Fr(\012)p Fs(n)p Fr(\012)31770 27288 y Fi(v)32185 27487 y Fp(0)32701 27177 y Fx(;T)78 b(;\013)34509 26793 y Fz(\()p Fn(x)35804 26992 y Fx(v)36262 27115 y Fp(0)36779 26793 y Fz(\))369 b Fy(;)5914 b Fz(\(4.5\))3432 31178 y(where)296 b Fy(A)8027 31377 y Fx(T)9057 31178 y Fz(is)h(a)g(suitable)f(set)g(of)h(indices)g(and)f Fy(W)27058 30499 y Fr(\()p Fx(L)p Fr(\))26877 31562 y Fx(\034)50 b(;)p Fm(P)p Fx(;)p Fr(\012)p Fs(n)p Fr(\012)30437 31673 y Fi(v)30852 31872 y Fp(0)31368 31562 y Fx(;T)78 b(;\013)33177 31178 y Fz(\()p Fn(x)34472 31377 y Fx(v)34930 31500 y Fp(0)35446 31178 y Fz(\))296 b(can)g(b)36 b(e)297 b(represen)-36 b(ted)3432 32979 y(as)392 b(in)g(\(2.69\).)566 b(There)392 b(is)g(indeed)g(a)g(small)h(di\013erence,)400 b(b)36 b(ecause)392 b(of)h(the)f(delta)g(function)3432 34584 y(and)556 b(the)f(in)-36 b(tegral)557 b(app)36 b(earing)557 b(in)f(\(4.2\),)588 b(\(4.3\))557 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Fz(.)p 46160 66570 443 1329 v 24367 75163 a(39)p eop %%Page: 40 40 40 39 bop 3432 7638 a FA(4.2)1792 b(The)599 b(b)50 b(eta)598 b(function)3432 10101 y Fz(The)433 b(discussion)g(of)i Ft(x)p Fz(4.1)f(and)f(the)g(de\014nition)g(of)h(mo)36 b(di\014ed)433 b(running)f(coupling)i(func-)3432 11706 y(tions)f(\(MR)-36 b(CF)434 b(in)f(the)g(follo)-36 b(wing\))436 b(of)e Ft(x)p Fz(2.6)h(imply)f(that)7003 14289 y(~)6948 14640 y Fy(\025)7707 14839 y Fx(h;\033)p 8510 15016 570 45 v 9135 14640 a Fz(\()-22 b Fy(~)-628 b(x)p 9641 14817 740 45 v Fz(\))369 b(=)g(\()p Ff(F)13954 14862 y Fr(4)p Fx(;h;\033)p 15488 15039 570 45 v 16409 14640 a Ft(\003)295 b Fy(\025)18127 14839 y Fx(h)p Fr(+1)19927 14640 y Fz(\)\()-22 b Fy(~)-628 b(x)p 20939 14817 740 45 v Fz(\))295 b(+)f(\()p Ff(F)25103 14862 y Fr(4)p Fx(;h;\033)p 26637 15039 570 45 v 27559 14640 a Ft(\003)h Fy(\014)29326 14010 y Fr(4)p Fx(;)p Fr(0)29252 14974 y Fx(h)p Fr(+1)31052 14640 y Fz(\)\()32144 14622 y(~)32064 14640 y Fn(v)32853 14839 y Fx(h)p 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eop %%Page: 41 41 41 40 bop 3432 7638 a Fz(where,)320 b(ignoring)291 b(in)h(the)e (notation)i(the)e(spin)h(dep)36 b(endence)290 b(of)i Fy(v)48 b Fz(\()-22 b Fy(~)-628 b(x)o Fz(\),)320 b(see)292 b(\(1.8\),)42241 7287 y(~)42187 7638 y Fy(\025)42946 7837 y Fr(1)43472 7638 y Fz(\()-22 b Fy(~)-628 b(x)p 43978 7815 740 45 v Fz(\))368 b(=)3432 9243 y(\()p Ff(F)4749 9466 y Fr(4)p Fx(;)p Fr(1)p Fx(;)p Fr(0)p 6211 9643 471 45 v 7002 9243 a Ft(\003)263 b Fy(\025)8688 9442 y Fr(1)9214 9243 y Fz(\)\()-22 b Fy(~)-628 b(x)p 10226 9420 740 45 v Fz(\),)421 b(with)d Fy(\025)15959 9442 y Fr(1)16484 9243 y Fz(\()-22 b Fy(~)-628 b(x)p 16990 9420 V Fz(\))369 b(=)f Ft(\000)p Fy(\025v)48 b Fz(\()-22 b Fy(~)-628 b(x)23698 9442 y Fr(1)24487 9243 y Ft(\000)241 b Fy(~)-628 b(x)26522 9442 y Fr(2)27048 9243 y Fz(\))p Fy(\016)50 b Fz(\()-22 b Fy(~)-628 b(x)29425 9442 y Fr(3)30214 9243 y Ft(\000)241 b Fy(~)-628 b(x)32249 9442 y Fr(1)32775 9243 y Fz(\))p Fy(\016)50 b Fz(\()-22 b Fy(~)-628 b(x)35152 9442 y Fr(4)35940 9243 y Ft(\000)242 b Fy(~)-628 b(x)37976 9442 y Fr(2)38502 9243 y Fz(\),)421 b(and)491 b(~)-724 b Fy(\027)42946 9442 y Fr(1)43472 9243 y Fz(\()-22 b Fy(~)-628 b(x)p 43978 9420 V Fz(\))368 b(=)3432 10848 y(\()p Ff(F)4749 11071 y Fr(2)p Fx(;)p Fr(1)p Fx(;)p Fr(\(0)p Fx(;)p Fr(0\))8401 10848 y Ft(\003)198 b Fy(\027)9906 11047 y Fr(1)10432 10848 y Fz(\)\()-22 b Fy(~)-628 b(x)p 11444 11025 V -1 w Fz(\).)563 b(F)-108 b(urthermore)463 b(~)-729 b Fy(z)21701 11047 y Fr(1)22227 10848 y Fz(\()-22 b Fy(~)-628 b(x)p 22733 11025 V Fz(\))369 b(=)g Fy(z)26332 11047 y Fr(1)26858 10848 y Fz(\()-22 b Fy(~)-628 b(x)p 27364 11025 V -1 w Fz(\))369 b(=)g(0)386 b(and)g Fy(\027)34519 11047 y Fr(1)35044 10848 y Fz(\()-22 b Fy(~)-628 b(x)p 35550 11025 V Fz(\))386 b(m)-36 b(ust)386 b(b)36 b(e)386 b(suitably)3432 12453 y(c)-36 b(hosen.)5383 14058 y(W)-108 b(e)488 b(note)g(that)f(it)i (is)f(p)36 b(ossible)488 b(to)h(c)-36 b(ho)36 b(ose)488 b(the)g(functions)35234 13723 y(~)34940 14058 y Fy(F)35782 14257 y 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Fz(;)200 b Fy(~)-629 b(x)p 36697 47537 740 45 v Fz(\))369 b Fy(;)4607 b Fz(\(4.17\))3432 51276 y(where,)701 b(giv)-36 b(en)648 b Fy(\033)p 11428 51453 787 45 v 781 w Fz(=)733 b(\()p Fy(\033)15938 51475 y Fr(1)16464 51276 y Fy(;)221 b(\033)17785 51475 y Fr(2)18311 51276 y Fz(\))733 b Ft(2)g Fn(O)22291 51475 y Fx(h;)p Fr(2)23621 51276 y Fz(,)24683 50391 y Fl(P)25851 50741 y Fs(\003)25851 51605 y Fx(\033)p 25851 51782 570 45 v 26421 51882 a Fi(j)26855 51605 y Fs(2)p Fm(O)28297 51740 y Fi(j)-5 b(;)p Fp(2)30048 51276 y Fz(is)648 b(the)f(sum)g(restricted)g(to)h(the)3432 53118 y Fy(\033)p 3432 53295 787 45 v 4219 53394 a Fx(j)5074 53118 y Fz(=)369 b(\()p Fy(\033)7748 52636 y Fs(0)8058 53317 y Fr(1)8584 53118 y Fy(;)221 b(\033)9953 52636 y Fs(0)10264 53317 y Fr(2)10790 53118 y Fz(\))369 b Ft(2)f Fn(O)14041 53317 y Fx(j)o(;)p Fr(2)15642 53118 y Fz(suc)-36 b(h)433 b(that)g Fy(S)22194 53317 y Fx(h;\033)23535 53452 y Fi(i)24237 53118 y Ft(\\)294 b Fy(S)26217 53317 y Fx(j)o(;\033)27428 53065 y Fq(0)27726 53452 y Fi(i)28501 53118 y Ft(6)p Fz(=)369 b Ft(;)p Fz(,)434 b Fy(i)369 b Fz(=)f(1)p Fy(;)221 b Fz(2.)5383 54723 y(In)587 b(order)f(to)i(presen)-36 b(t)586 b(our)g(results,)626 b(w)-36 b(e)587 b(ha)-36 b(v)g(e)588 b(to)f(in)-36 b(tro)36 b(duce)586 b(a)i(few)g(other)e(def-) 3432 56328 y(initions.)1202 b(Giv)-36 b(en)642 b Fy(h)722 b Ft(\024)i Fz(1)641 b(and)h Fy(!)770 b Ft(2)723 b Fy(O)24608 56527 y Fx(h)25205 56328 y Fz(,)694 b(w)-36 b(e)642 b(denote)f(b)-36 b(y)642 b Fy(D)35877 56527 y Fx(h;\033)38028 56328 y Ft(2)722 b Fo(R)40596 55763 y Fr(2)41763 56328 y Fz(the)641 b(sup-)3432 58107 y(p)36 b(ort)518 b(of)8231 57772 y(~)7937 58107 y Fy(F)8779 58306 y Fx(h;\033)10207 58107 y Fz(\()10646 57757 y Fy(~)10713 58107 y(k)45 b Fz(\).)833 b(Moreo)-36 b(v)g(er,)540 b(if)520 b Fy(p)513 b Fz(=)g(2)p Fy(;)221 b Fz(4,)541 b(w)-36 b(e)519 b(call)g Ft(M)32126 58306 y Fx(h;p)33978 58107 y Fz(the)f(space)g(of)h(functions)3432 59712 y Fy(G)4458 59911 y Fx(\033)p 4458 60088 625 45 v 5082 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Ff(M)19715 22544 y Fx(p)20627 22345 y Fz(as)i(the)f(set)h(of)h(sequences)f Fy(G)369 b Fz(=)g Ft(f)p Fy(G)38092 22544 y Fx(h)39058 22345 y Ft(2)g(M)41908 22544 y Fx(h;p)43241 22345 y Fy(;)221 b(h)44572 22544 y Fx(\014)45570 22345 y Ft(\024)3432 23950 y Fy(h)368 b Ft(\024)h Fz(1)p Ft(g)p Fz(,)435 b(suc)-36 b(h)433 b(that)g(the)g(norm)18515 26710 y Ft(jj)p Fy(G)p Ft(jj)21017 26909 y Fx(p)21914 26710 y Fz(=)944 b(max)23295 27509 y Fx(h)23837 27665 y Fi(\014)24390 27509 y Fs(\024)p Fx(h)p Fs(\024)p Fr(1)27088 26710 y Ft(jj)p Fy(G)28852 26909 y Fx(h)29450 26710 y Ft(jj)30188 26909 y Fx(h;p)43279 26710 y Fz(\(4.20\))3432 30403 y(is)477 b(\014nite.)708 b(W)-108 b(e)477 b(w)-36 b(an)g(t)477 b(to)h(pro)-36 b(v)g(e)477 b(that)f(the)h(sequence)30064 30053 y(~)30010 30403 y Fy(\025)443 b Ft(\021)g(f)33406 30053 y Fz(~)33352 30403 y Fy(\025)34111 30602 y Fx(h)34708 30403 y Fy(;)221 b(h)36039 30602 y Fx(\014)37112 30403 y Ft(\024)443 b Fy(h)g Ft(\024)g Fz(1)p Ft(g)478 b Fz(is)f(w)-36 b(ell)3432 32008 y(de\014ned)444 b(as)i(an)g(elemen)-36 b(t)445 b(of)i Ff(M)19057 32207 y Fr(4)19583 32008 y Fz(,)i(while)d(the)f (sequences)520 b(~)-724 b Fy(\027)473 b Ft(\021)390 b(f)74 b Fz(~)-724 b Fy(\027)35815 32207 y Fx(h)36413 32008 y Fy(;)221 b(h)37744 32207 y Fx(\014)38763 32008 y Ft(\024)390 b Fy(h)f Ft(\024)h Fz(1)p Ft(g)447 b Fz(and)3510 33614 y(~)-728 b Fy(z)428 b Ft(\021)369 b(f)79 b Fz(~)-729 b Fy(z)7134 33813 y Fx(h)7732 33614 y Fy(;)221 b(h)9063 33813 y Fx(\014)10061 33614 y Ft(\024)369 b Fy(h)g Ft(\024)g Fz(1)p Ft(g)434 b Fz(are)g(t)-36 b(w)g(o)434 b(elemen)-36 b(ts)433 b(of)i Ff(M)28550 33813 y Fr(2)29075 33614 y Fz(.)5383 35219 y(W)-108 b(e)449 b(b)36 b(egin)449 b(our)g(analysis)h (b)-36 b(y)449 b(\\decoupling")h(equations)g(\(4.12\))g(and)e(\(4.14\)) j(from)3432 36824 y(\(4.13\),)582 b(that)552 b(is)g(w)-36 b(e)553 b(imagine)g(that,)581 b(in)552 b(the)f(r.h.s.)934 b(of)553 b(\(4.12\))g(and)e(\(4.14\),)657 b(~)-724 b Fy(\027)636 b Fz(is)552 b(an)3432 38429 y(arbitrary)622 b(elemen)-36 b(t)622 b(of)h Ff(M)17206 38628 y Fr(2)17732 38429 y Fz(,)669 b(acting)623 b(as)f(a)g(parameter.)1144 b(W)-108 b(e)622 b(w)-36 b(an)g(t)622 b(to)g(lo)36 b(ok)624 b(for)f(a)3432 40034 y(solution)433 b(\()8912 39683 y(~)8858 40034 y Fy(\025)p Fz(\()74 b(~)-724 b Fy(\027)83 b Fz(\))369 b Ft(2)g Ff(M)14373 40233 y Fr(4)14899 40034 y Fz(,)512 b(~)-729 b Fy(z)60 b Fz(\()74 b(~)-724 b Fy(\027)83 b Fz(\))369 b Ft(2)f Ff(M)21113 40233 y Fr(2)21638 40034 y Fz(\).)579 b(W)-108 b(e)433 b(shall)h(pro)-36 b(v)g(e)434 b(the)f(follo)-36 b(wing)435 b(lemma.)3432 44485 y Fn(Lemma)499 b(4.1)650 b Fw(Ther)-66 b(e)587 b(exist)g(p)-66 b(ositive)587 b(c)-66 b(onstants)587 b Fy(C)29862 44684 y Fr(1)30975 44485 y Fw(and)h Fy(C)34553 44684 y Fr(2)35078 44485 y Fw(,)618 b(dep)-66 b(ending)586 b(only)i(on)3432 46090 y(\014rst)495 b(and)g(se)-66 b(c)g(ond)495 b(or)-66 b(der)495 b(terms)f(in)h(our)h(exp)-66 b(ansion,)501 b(such)496 b(that,)503 b(given)494 b(two)h(p)-66 b(ositive)3432 47695 y(c)g(onstants)464 b Fy(C)10075 47894 y Fr(3)10969 47695 y Ft(\025)369 b Fy(C)13302 47894 y Fr(1)14292 47695 y Fw(and)465 b Fy(C)17747 47894 y Fr(4)18273 47695 y Fw(,)f(ther)-66 b(e)464 b(exists)g Fy(\025)26686 47894 y Fr(0)27677 47695 y Fw(so)h(that,)f(if)g Ft(j)p Fy(\025)p Ft(j)369 b(\024)g Fy(\025)37681 47894 y Fr(0)38207 47695 y Fw(,)16475 50456 y Fz(2)p Fy(C)18056 50655 y Fr(2)18582 50456 y Fy(C)19513 50655 y Fr(3)20260 50456 y Fz(max)q Ft(f)p Fz(1)p Fy(;)221 b(C)25603 49907 y Fs(\000)p Fr(1)25508 50784 y(4)26861 50456 y Ft(gj)p Fy(\025)p Ft(jj)p Fy(h)30140 50655 y Fx(\014)30769 50456 y Ft(j)369 b(\024)g Fz(1)9720 b(\(4.21\))3432 53216 y Fw(and)464 b Ft(jj)74 b Fz(~)-724 b Fy(\027)84 b Ft(jj)8158 53415 y Fr(2)8684 53216 y Fy(;)221 b Ft(jj)74 b Fz(~)-724 b Fy(\027)10731 52734 y Fs(0)11041 53216 y Ft(jj)11779 53415 y Fr(2)12674 53216 y Ft(\024)369 b Fy(C)15007 53415 y Fr(3)15533 53216 y Ft(j)p Fy(\025)p Ft(j)p Fw(,)464 b(then,)f(for)i Fy(h)24005 53415 y Fx(\014)25003 53216 y Ft(\024)369 b Fy(h)f Ft(\024)i Fz(1)p Fw(,)10757 56192 y Ft(jj)11549 55841 y Fz(~)11495 56192 y Fy(\025)p Fz(\()74 b(~)-724 b Fy(\027)83 b Fz(\))13992 56391 y Fx(h)14589 56192 y Ft(jj)15327 56391 y Fx(h;)p Fr(4)17026 56192 y Ft(\024)369 b Fz(2)p Fy(C)20009 56391 y Fr(1)20535 56192 y Ft(j)p Fy(\025)p Ft(j)p Fy(\015)22779 55643 y Fs(\000)23644 55281 y Fp(1)p 23643 55444 406 45 v 23643 55990 a(2)24182 55643 y Fx(h)26108 56192 y Fy(;)1550 b Ft(jj)79 b Fz(~)-729 b Fy(z)59 b Fz(\()74 b(~)-724 b Fy(\027)83 b Fz(\))31158 56391 y Fx(h)31756 56192 y Ft(jj)32494 56391 y Fx(h;)p Fr(2)34192 56192 y Ft(\024)369 b Fy(C)36525 56391 y Fr(1)37050 56192 y Ft(j)p Fy(\025)p Ft(j)g Fy(;)4002 b Fz(\(4.22\))10465 60323 y Ft(jj)11257 59972 y Fz(~)11203 60323 y Fy(\025)p Fz(\()74 b(~)-724 b Fy(\027)83 b Fz(\))13700 60522 y Fx(h)14593 60323 y Ft(\000)15975 59972 y Fz(~)15921 60323 y Fy(\025)p Fz(\()74 b(~)-724 b Fy(\027)17913 59774 y Fs(0)18223 60323 y Fz(\))18729 60522 y Fx(h)19326 60323 y Ft(jj)20064 60522 y Fx(h;)p Fr(4)22500 60323 y Ft(\024)1108 b Fy(C)25572 60522 y Fr(4)26097 60323 y Fy(\015)26844 59774 y Fs(\000)27709 59412 y Fp(1)p 27709 59575 V 27709 60121 a(2)28247 59774 y Fx(h)29066 60323 y Fz(max)29424 61122 y Fx(j)51 b(>h)31708 60323 y Ft(jj)74 b Fz(~)-724 b Fy(\027)33089 60522 y Fx(h)33982 60323 y Ft(\000)369 b Fz(~)-724 b Fy(\027)36037 59774 y Fs(0)35953 60651 y Fx(h)36550 60323 y Ft(jj)37288 60522 y Fx(h;)p Fr(2)38987 60323 y Fy(;)10656 62888 y Ft(jj)79 b Fz(~)-729 b Fy(z)59 b Fz(\()74 b(~)-724 b Fy(\027)84 b Fz(\))13796 63087 y Fx(h)14688 62888 y Ft(\000)374 b Fz(~)-728 b Fy(z)59 b Fz(\()74 b(~)-724 b Fy(\027)17913 62340 y Fs(0)18223 62888 y Fz(\))18729 63087 y Fx(h)19326 62888 y Ft(jj)20064 63087 y Fx(h;)p Fr(2)22500 62888 y Ft(\024)1108 b Fy(C)25572 63087 y Fr(4)26318 62888 y Fz(max)26676 63688 y Fx(j)51 b(>h)28960 62888 y Ft(jj)74 b Fz(~)-724 b Fy(\027)30341 63087 y Fx(h)31234 62888 y Ft(\000)369 b Fz(~)-724 b Fy(\027)33289 62340 y Fs(0)33205 63217 y Fx(h)33802 62888 y Ft(jj)34540 63087 y Fx(h;)p Fr(2)36239 62888 y Fy(:)6679 b Fz(\(4.23\))3432 66620 y Fb(Pr)-30 b(oof)503 b(-)935 b Fz(Note)434 b(that,)f(if)17335 66284 y(\026)17041 66620 y Fy(F)17883 66819 y Fx(h;!)19358 66620 y Fz(\()-22 b Fy(~)-628 b(x)p Fz(\))434 b(is)g(the)f(F)-108 b(ourier)432 b(transform)i(of)37333 66284 y(~)37038 66620 y Fy(F)37880 66819 y Fx(h;!)39356 66620 y Fz(\()39795 66269 y Fy(~)39862 66620 y(k)45 b Fz(\),)434 b(then)14744 69107 y Fl(\014)14744 69772 y(\014)14744 70436 y(\014)15407 70100 y Fz(\026)15113 70436 y Fy(F)15955 70635 y Fx(h;!)17431 70436 y Fz(\()-22 b Fy(~)-628 b(x)p Fz(\))19182 69107 y Fl(\014)19182 69772 y(\014)19182 70436 y(\014)368 b Ft(\024)26019 69537 y Fy(C)26950 69736 y Fx(N)27846 69537 y Fy(\015)28726 68685 y Fp(3)p 28726 68848 V 28726 69394 a(2)29265 69047 y Fx(h)p 21454 70126 12974 45 v 21454 71842 a Fz(1)296 b(+)23707 70558 y Fl(\020)24368 71842 y Fy(\015)25115 71458 y Fx(h)25712 71842 y Ft(j)p Fy(x)26820 71384 y Fs(0)26820 72132 y Fr(1)27346 71842 y Ft(j)f Fz(+)g Fy(\015)30197 70989 y Fi(h)p 30197 71152 482 45 v 30235 71699 a Fp(2)30867 71842 y Ft(j)p Fy(x)31975 71384 y Fs(0)31975 72132 y Fr(2)32501 71842 y Ft(j)32870 70558 y Fl(\021)33531 70864 y Fx(N)34929 70436 y Fy(;)7989 b Fz(\(4.24\))24367 75163 y(42)p eop %%Page: 43 43 43 42 bop 3432 7638 a Fz(so)445 b(that)7870 6691 y Fl(R)8830 7638 y Fy(d)-22 b(~)-628 b(x)10466 6310 y Fl(\014)10466 6974 y(\014)10466 7638 y(\014)11130 7302 y Fz(\026)10835 7638 y Fy(F)11677 7837 y Fx(h;!)13153 7638 y Fz(\()-22 b Fy(~)-628 b(x)p Fz(\))14904 6310 y Fl(\014)14904 6974 y(\014)14904 7638 y(\014)389 b Ft(\024)g Fy(c)17644 7837 y Fx(F)18871 7638 y Fz(for)446 b(some)f(constan)-36 b(t)445 b Fy(c)29958 7837 y Fx(F)31185 7638 y Fz(indep)36 b(enden)-36 b(t)443 b(of)j Fy(h)g Fz(and)e Fy(!)48 b Fz(.)614 b(It)3432 9243 y(follo)-36 b(ws)435 b(that)e(there)g(exists)h(a)g(constan)-36 b(t)433 b Fy(C)24696 9442 y Fr(1)25222 9243 y Fz(,)h(suc)-36 b(h)7200 12015 y Ft(jj)7992 11664 y Fz(~)7938 12015 y Fy(\025)8697 12214 y Fr(1)9223 12015 y Ft(jj)9961 12214 y Fr(1)p Fx(;)p Fr(4)11587 12015 y Ft(\024)369 b Fz(2)p Fy(C)14570 12214 y Fr(1)15096 12015 y Ft(j)p Fy(\025)p Ft(j)p Fy(\015)17340 11466 y Fs(\000)p Fr(1)p Fx(=)p Fr(2)20839 12015 y Fy(;)1522 b Ft(jj)p Ff(F)24272 12238 y Fr(4)p Fx(;h;\033)p 25806 12415 570 45 v 26728 12015 a Ft(\003)27741 11664 y Fz(~)27687 12015 y Fy(\025)28446 12214 y Fr(1)28971 12015 y Ft(jj)29709 12214 y Fx(h;)p Fr(4)31408 12015 y Ft(\024)369 b Fy(C)33741 12214 y Fr(1)34266 12015 y Ft(j)p Fy(\025)p Ft(j)p Fy(\015)36510 11466 y Fs(\000)p Fx(h=)p Fr(2)39149 12015 y Fy(;)3769 b Fz(\(4.25\))3432 14786 y(ha)-36 b(ving)434 b(used)e(also)j(Lemma)e(3.1)i(for)f(the)f (second)g(inequalit)-36 b(y)-108 b(.)5383 16391 y(W)g(e)564 b(shall)g(pro)-36 b(v)g(e)564 b(inductiv)-36 b(ely)565 b(that,)596 b(if)565 b Ft(jj)74 b Fz(~)-724 b Fy(\027)84 b Ft(jj)28243 16590 y Fr(2)29359 16391 y Ft(\024)592 b Fy(C)31915 16590 y Fr(3)32440 16391 y Ft(j)p Fy(\025)p Ft(j)p Fz(,)597 b(with)564 b Fy(C)38919 16590 y Fr(3)40035 16391 y Ft(\025)592 b Fy(C)42591 16590 y Fr(1)43116 16391 y Fz(,)597 b(then)3432 18115 y Ft(jj)4224 17764 y Fz(~)4170 18115 y Fy(\025)o Fz(\()74 b(~)-724 b Fy(\027)84 b Fz(\))6667 18314 y Fx(h)7264 18115 y Ft(jj)8002 18314 y Fx(h;)p Fr(4)9914 18115 y Ft(\024)584 b Fz(2)p Fy(C)13112 18314 y Fr(1)13638 18115 y Ft(j)p Fy(\025)p Ft(j)p Fy(\015)15882 17624 y Fs(\000)16747 17262 y Fp(1)p 16746 17425 406 45 v 16746 17972 a(2)17285 17624 y Fx(h)18442 18115 y Fz(and)558 b Ft(jj)79 b Fz(~)-729 b Fy(z)60 b Fz(\()74 b(~)-724 b Fy(\027)83 b Fz(\))24236 18314 y Fx(h)24833 18115 y Ft(jj)25571 18314 y Fx(h;)p Fr(2)27484 18115 y Ft(\024)583 b Fy(C)30031 18314 y Fr(1)30557 18115 y Ft(j)p Fy(\025)p Ft(j)p Fz(.)956 b(This)559 b(b)36 b(ound)558 b(is)i(satis\014ed)3432 19720 y(for)510 b Fy(h)499 b Fz(=)g(1,)530 b(b)-36 b(y)510 b(the)f(\014rst)h(inequalit)-36 b(y)511 b(of)g(\(4.25\))g(and)e(the)h(fact)g(that)589 b(~)-729 b Fy(z)39316 19919 y Fr(1)40341 19720 y Fz(=)499 b(0;)549 b(let)510 b(us)3432 21325 y(supp)36 b(ose)568 b(that)h(it)h(is)f(true)g(for)h(an)-36 b(y)570 b Fy(j)675 b(>)599 b(h)p Fz(.)986 b(Then,)603 b(b)-36 b(y)569 b(using)g(\(4.12\),) 605 b(\(4.14\),)g(the)3432 23111 y(second)518 b(inequalit)-36 b(y)520 b(of)f(\(4.25\),)541 b(Theorem)519 b(4.1)h(and)e(the)g(fact)h (that)f Fy(\014)38683 22432 y Fr(\(4)p Fx(;)p Fr(0\))38609 23416 y Fx(j)41191 23111 y Fz(and)h Fy(\014)44614 22432 y Fr(\(2)p Fx(;)p Fr(1\))44540 23416 y Fx(j)3432 24716 y Fz(do)433 b(not)g(ha)-36 b(v)g(e)434 b(\014rst)f(order)g(con)-36 b(tributions,)433 b(w)-36 b(e)434 b(\014nd)5720 28372 y Ft(jj)6512 28021 y Fz(~)6458 28372 y Fy(\025)o Fz(\()74 b(~)-724 b Fy(\027)84 b Fz(\))8955 28571 y Fx(h)9552 28372 y Ft(jj)10290 28571 y Fx(h;)p Fr(4)12726 28372 y Ft(\024)1107 b Fy(C)15797 28571 y Fr(1)16323 28372 y Ft(j)p Fy(\025)p Ft(j)p Fy(\015)18567 27823 y Fs(\000)19432 27461 y Fp(1)p 19431 27624 V 19431 28171 a(2)19970 27823 y Fx(h)20862 28372 y Fz(+)295 b Fy(\015)22916 27823 y Fs(\000)23781 27461 y Fp(1)p 23781 27624 V 23781 28171 a(2)24320 27823 y Fx(h)26357 26933 y Fr(1)25793 27265 y Fl(X)25139 29725 y Fx(j)51 b Fr(=)p Fx(h)p Fr(+1)28046 27088 y Fl(h)28569 28372 y Fy(C)29500 28571 y Fr(2)p Fx(;\025)30836 28372 y Fy(C)31767 28571 y Fr(1)32292 28372 y Fy(C)33223 28571 y Fr(3)33749 28372 y Ft(j)p Fy(\025)p Ft(j)35246 27823 y Fr(2)36066 28372 y Fz(+)37789 26933 y Fs(1)37460 27265 y Fl(X)37373 29680 y Fx(n)p Fr(=3)39146 28372 y Fz(\()p Fy(c)p Ft(j)p Fy(\025)p Ft(j)p Fz(\))42215 27823 y Fx(n)42841 27088 y Fl(i)43732 28372 y Fy(;)5815 32571 y Ft(jj)79 b Fz(~)-729 b Fy(z)59 b Fz(\()74 b(~)-724 b Fy(\027)84 b Fz(\))8955 32770 y Fx(h)9552 32571 y Ft(jj)10290 32770 y Fx(h;)p Fr(2)12726 32571 y Ft(\024)16085 31132 y Fr(1)15521 31464 y Fl(X)14866 33924 y Fx(j)51 b Fr(=)p Fx(h)p Fr(+1)17774 31287 y Fl(h)18297 32571 y Fy(C)19228 32770 y Fr(2)p Fx(;z)20490 32571 y Fy(C)21421 32770 y Fr(1)21946 32571 y Fy(C)22877 32770 y Fr(3)23403 32571 y Ft(j)p Fy(\025)p Ft(j)24900 32022 y Fr(2)25720 32571 y Fz(+)27443 31132 y Fs(1)27114 31464 y Fl(X)27027 33879 y Fx(n)p Fr(=3)28800 32571 y Fz(\()p Fy(c)p Ft(j)p Fy(\025)p Ft(j)p Fz(\))31869 32022 y Fx(n)32495 31287 y Fl(i)33386 32571 y Fy(:)9532 b Fz(\(4.26\))3432 36716 y(Hence,)364 b(if)347 b Fy(\025)f Fz(small)h(enough)f(and)g(2)p Ft(j)p Fy(\025)p Ft(jj)p Fy(h)23215 36915 y Fx(\014)23843 36716 y Ft(j)p Fy(C)25143 36915 y Fr(3)25890 36716 y Fz(max)q Ft(f)p Fy(C)29906 36915 y Fr(2)p Fx(;\025)31242 36716 y Fy(;)221 b(C)32755 36915 y Fr(2)p Fx(;z)34017 36716 y Ft(g)369 b(\024)g Fz(1,)c(then)345 b Ft(jj)41495 36365 y Fz(~)41441 36716 y Fy(\025)p Fz(\()74 b(~)-724 b Fy(\027)83 b Fz(\))43938 36915 y Fx(h)44536 36716 y Ft(jj)45274 36915 y Fx(h;)p Fr(4)3432 38478 y Ft(\024)369 b Fz(2)p Fy(C)6415 38677 y Fr(1)6940 38478 y Ft(j)p Fy(\025)p Ft(j)p Fy(\015)9184 37988 y Fs(\000)10049 37626 y Fp(1)p 10049 37789 V 10049 38335 a(2)10587 37988 y Fx(h)11619 38478 y Fz(and)433 b Ft(jj)79 b Fz(~)-729 b Fy(z)59 b Fz(\()74 b(~)-724 b Fy(\027)83 b Fz(\))17287 38677 y Fx(h)17885 38478 y Ft(jj)18623 38677 y Fx(h;)p Fr(2)20321 38478 y Ft(\024)369 b Fy(C)22654 38677 y Fr(1)23179 38478 y Ft(j)p Fy(\025)p Ft(j)p Fz(,)434 b(up)f(to)g Fy(h)369 b Fz(=)g Fy(h)32187 38677 y Fx(\014)32815 38478 y Fz(.)5383 40084 y(W)-108 b(e)332 b(still)i(ha)-36 b(v)g(e)332 b(to)h(pro)-36 b(v)g(e)333 b(that,)352 b(if)334 b Ft(jj)74 b Fz(~)-724 b Fy(\027)83 b Ft(jj)24139 40283 y Fr(2)24665 40084 y Fz(,)353 b Ft(jj)74 b Fz(~)-724 b Fy(\027)26844 39601 y Fs(0)27154 40084 y Ft(jj)27892 40283 y Fr(2)28787 40084 y Ft(\024)369 b Fy(C)31120 40283 y Fr(3)31645 40084 y Ft(j)p Fy(\025)p Ft(j)p Fz(,)353 b(then)332 b(the)g(b)36 b(ounds)331 b(\(4.23\))3432 41689 y(are)581 b(v)-36 b(eri\014ed.)1023 b(W)-108 b(e)581 b(shall)i(again)f(pro)36 b(ceed)581 b(b)-36 b(y)582 b(induction,)618 b(b)-36 b(y)582 b(using)f(that)42205 41338 y(~)42151 41689 y Fy(\025)p Fz(\()74 b(~)-724 b Fy(\027)83 b Fz(\))44648 41888 y Fr(1)45570 41689 y Ft(\000)3486 42943 y Fz(~)3432 43294 y Fy(\025)o Fz(\()74 b(~)-724 b Fy(\027)5423 42812 y Fs(0)5733 43294 y Fz(\))6239 43493 y Fr(1)7293 43294 y Fz(=)527 b(0,)551 b(since)13728 42943 y(~)13674 43294 y Fy(\025)14433 43493 y Fr(1)15486 43294 y Fz(is)527 b(indep)36 b(enden)-36 b(t)525 b(of)602 b(~)-725 b Fy(\027)84 b Fz(,)550 b(and)527 b(that)605 b(~)-729 b Fy(z)60 b Fz(\()74 b(~)-724 b Fy(\027)83 b Fz(\))35499 43493 y Fr(1)36552 43294 y Fz(=)528 b(0.)859 b(Then,)550 b(if)528 b(w)-36 b(e)3432 44899 y(supp)36 b(ose)433 b(that)g(the)g(b)36 b(ound)432 b(is)i(true)f(for)h(an)-36 b(y)434 b Fy(j)444 b(>)368 b(h)p Fz(,)434 b(w)-36 b(e)434 b(\014nd)4991 47670 y Ft(jj)5783 47320 y Fz(~)5729 47670 y Fy(\025)p Fz(\()74 b(~)-724 b Fy(\027)83 b Fz(\))8226 47869 y Fx(h)9119 47670 y Ft(\000)10502 47320 y Fz(~)10447 47670 y Fy(\025)p Fz(\()74 b(~)-724 b Fy(\027)12439 47122 y Fs(0)12749 47670 y Fz(\))13255 47869 y Fx(h)13852 47670 y Ft(jj)14590 47869 y Fx(h;)p Fr(4)17027 47670 y Ft(\024)1107 b Fy(\015)19914 47122 y Fs(\000)20779 46760 y Fp(1)p 20778 46923 V 20778 47469 a(2)21317 47122 y Fx(h)22136 47670 y Fz(max)22494 48470 y Fx(j)51 b(>h)24778 47670 y Ft(jj)74 b Fz(~)-724 b Fy(\027)26159 47869 y Fx(j)26941 47670 y Ft(\000)369 b Fz(~)-724 b Fy(\027)28996 47122 y Fs(0)28912 47999 y Fx(j)29399 47670 y Ft(jj)30137 47869 y Fx(j)o(;)p Fr(2)31598 47670 y Ft(\001)19167 51316 y(\001)20976 49877 y Fr(1)20412 50209 y Fl(X)19757 52669 y Fx(j)51 b Fr(=)p Fx(h)p Fr(+1)22886 49368 y Fl(")23828 50980 y Fz(~)23532 51316 y Fy(C)24463 51515 y Fr(2)p Fx(;\025)25799 51316 y Fy(C)26730 51515 y Fr(3)27476 51316 y Fz(max)q Ft(f)p Fz(1)p Fy(;)221 b(C)32724 51515 y Fr(4)33251 51316 y Ft(gj)p Fy(\025)p Ft(j)295 b Fz(+)37430 49877 y Fs(1)37101 50209 y Fl(X)37014 52625 y Fx(n)p Fr(=3)39008 51316 y Fy(c)39568 50768 y Fx(n)40194 51316 y Ft(j)p Fy(\025)p Ft(j)41691 50768 y Fx(n)p Fs(\000)p Fr(1)43520 49368 y Fl(#)5182 54435 y Ft(jj)79 b Fz(~)-729 b Fy(z)60 b Fz(\()74 b(~)-724 b Fy(\027)83 b Fz(\))8322 54634 y Fx(h)9214 54435 y Ft(\000)374 b Fz(~)-728 b Fy(z)59 b Fz(\()74 b(~)-724 b Fy(\027)12439 53887 y Fs(0)12749 54435 y Fz(\))13255 54634 y Fx(h)13852 54435 y Ft(jj)14590 54634 y Fx(h;)p Fr(2)17027 54435 y Ft(\024)1107 b Fz(max)19524 55235 y Fx(j)51 b(>h)21809 54435 y Ft(jj)74 b Fz(~)-724 b Fy(\027)23190 54634 y Fx(j)23971 54435 y Ft(\000)370 b Fz(~)-724 b Fy(\027)26027 53887 y Fs(0)25943 54763 y Fx(j)26429 54435 y Ft(jj)27167 54634 y Fx(j)o(;)p Fr(2)28629 54435 y Ft(\001)14281 b Fz(\(4.27\))19167 58081 y Ft(\001)20976 56642 y Fr(1)20412 56974 y Fl(X)19757 59434 y Fx(j)51 b Fr(=)p Fx(h)p Fr(+1)22886 56132 y Fl(")23828 57745 y Fz(~)23532 58081 y Fy(C)24463 58280 y Fr(2)p Fx(;z)25725 58081 y Fy(C)26656 58280 y Fr(3)27403 58081 y Fz(max)p Ft(f)p Fz(1)p Fy(;)221 b(C)32650 58280 y Fr(4)33177 58081 y Ft(gj)p Fy(\025)p Ft(j)295 b Fz(+)37356 56642 y Fs(1)37027 56974 y Fl(X)36940 59389 y Fx(n)p Fr(=3)38934 58081 y Fy(c)39494 57532 y Fx(n)40120 58081 y Ft(j)p Fy(\025)p Ft(j)41617 57532 y Fx(n)p Fs(\000)p Fr(1)43446 56132 y Fl(#)44682 58081 y Fy(:)3432 62211 y Fz(Hence,)582 b(if)553 b Fy(\025)f Fz(small)h(enough)f(and)g(2)p Ft(j)p Fy(\025)p Ft(jj)p Fy(h)24463 62410 y Fx(\014)25091 62211 y Ft(j)p Fy(C)26391 62410 y Fr(3)27138 62211 y Fz(max)q Ft(f)p Fz(1)p Fy(;)221 b(C)32481 61661 y Fs(\000)p Fr(1)32386 62501 y(4)33739 62211 y Ft(g)g Fz(max)r Ft(f)38006 61875 y Fz(~)37710 62211 y Fy(C)38641 62410 y Fr(2)p Fx(;\025)39976 62211 y Fy(;)40855 61875 y Fz(~)40558 62211 y Fy(C)41489 62410 y Fr(2)p Fx(;z)42752 62211 y Ft(g)571 b(\024)g Fz(1,)3432 63816 y(the)433 b(b)36 b(ound)432 b(is)i(v)-36 b(eri\014ed)433 b(up)g(to)h Fy(h)368 b Fz(=)h Fy(h)22258 64015 y Fx(\014)22887 63816 y Fz(.)p 46160 64038 443 1329 v 5383 68632 a(W)-108 b(e)542 b(w)-36 b(an)g(t)542 b(no)-36 b(w)542 b(to)h(sho)-36 b(w)542 b(that)g(there)f(is)i(indeed)e(a)i(solution)g(of)g(the)e(full)i(set)f (of)3432 70237 y(equations)434 b(\(4.12\)-\(4.14\),)h(satisfying)g (condition)f(\(4.16\).)24367 75163 y(43)p eop %%Page: 44 44 44 43 bop 3432 7638 a Fn(Theorem)499 b(4.2)651 b Fw(If)527 b Ft(j)p Fy(\025)p Ft(j)h Fw(is)f(smal)66 b(l)529 b(enough)f(and)g Fy(C)28750 7837 y Fr(1)p Fx(;)p Fr(2)30229 7638 y Fz(log)240 b Fy(\014)74 b Ft(j)p Fy(\025)p Ft(j)485 b(\024)h Fz(1)p Fw(,)544 b(wher)-66 b(e)528 b Fy(C)42696 7837 y Fr(1)p Fx(;)p Fr(2)44481 7638 y Fw(is)g(a)3432 9243 y(c)-66 b(onstant)589 b(dep)-66 b(ending)590 b(only)g(on)h(\014rst)g(and)g(se) -66 b(c)g(ond)590 b(or)-66 b(der)591 b(c)-66 b(ontributions)589 b(of)i(p)-66 b(ertur-)3432 10848 y(b)g(ation)610 b(the)-66 b(ory,)646 b(it)611 b(is)g(p)-66 b(ossible)611 b(to)h(cho)-66 b(ose)685 b Fz(~)-724 b Fy(\027)26440 11047 y Fr(1)26966 10848 y Fz(\()-22 b Fy(~)-628 b(x)p 27472 11025 740 45 v Fz(\))611 b Fw(so)h(that)f(the)g(MR)-33 b(CF)610 b(satisfy)h(the)3432 12453 y(hyp)-66 b(othesis)464 b(of)h(L)-66 b(emma)464 b(2.2,)g(\(2.90\),)h(\(2.91\))h(and)f(\(2.92\).)3432 15497 y Fb(Pr)-30 b(oof)543 b(-)1012 b Fz(In)470 b(order)g(to)h(pro)-36 b(v)g(e)471 b(the)f(Theorem,)480 b(it)471 b(is)g(su\016cien)-36 b(t)470 b(to)h(lo)36 b(ok)472 b(for)f(a)g(\014xed)3432 17103 y(p)36 b(oin)-36 b(t)519 b(of)h(the)g(op)36 b(erator)520 b Fn(T)c Fz(:)g Ff(M)19959 17302 y Fr(2)21000 17103 y Ft(!)g Ff(M)24238 17302 y Fr(2)24764 17103 y Fz(,)542 b(de\014ned)518 b(in)i(the)f(follo)-36 b(wing)522 b(w)-36 b(a)g(y)-108 b(,)542 b(if)594 b(~)-724 b Fy(\027)44744 16620 y Fs(0)45570 17103 y Ft(\021)3432 18708 y Fn(T)p Fz(\()74 b(~)-724 b Fy(\027)83 b Fz(\):)12396 22643 y(~)-725 b Fy(\027)13048 22095 y Fs(0)12964 22972 y Fx(h)13931 22643 y Fz(=)368 b Ft(\000)18025 21204 y Fx(h)17497 21537 y Fl(X)16566 23996 y Fx(j)51 b Fr(=)p Fx(h)18272 24152 y Fi(\014)18824 23996 y Fr(+1)20248 22643 y Fy(\015)20995 22095 y Fs(\000)p Fx(h)p Fr(+)p Fx(j)g Fs(\000)p Fr(1)12454 25898 y Fz(1)p 12454 26487 651 45 v 12454 27708 a(4)14970 25358 y Fs(\003)14406 25690 y Fl(X)13459 28143 y Fx(\033)p 13459 28320 570 45 v 14028 28419 a Fi(j)14462 28143 y Fs(2)p Fm(O)15904 28278 y Fi(j)-5 b(;)p Fp(2)16953 26797 y Fz(\()o Ff(F)18270 27020 y Fr(2)p Fx(;h;\033)p 19804 27197 V 20726 26797 a Ft(\003)295 b Ff(F)22497 27020 y Fr(2)p Fx(;j)o(;\033)p 23869 27197 V 24439 27297 a Fi(j)25223 26797 y Ft(\003)h Fy(\014)26991 26167 y Fr(2)p Fx(;)p Fr(0)26917 27102 y Fx(j)28248 25513 y Fl(\021)28909 26797 y Fz(\()29495 26779 y(~)29415 26797 y Fn(v)30204 26996 y Fx(j)30691 26797 y Fz(\()74 b(~)-724 b Fy(\027)83 b Fz(\))p Fy(;)221 b(::;)34396 26779 y Fz(~)34315 26797 y Fn(v)35104 26996 y Fr(1)35631 26797 y Fz(\()74 b(~)-724 b Fy(\027)83 b Fz(\);)200 b Fy(~)-629 b(x)p 37951 26974 740 45 v 1 w Fz(\))369 b Fy(;)3352 b Fz(\(4.28\))3432 31330 y(where)7269 31312 y(~)7189 31330 y Fn(v)7978 31529 y Fx(j)8465 31330 y Fz(\()74 b(~)-724 b Fy(\027)83 b Fz(\))369 b(=)f(\()12512 30980 y(~)12458 31330 y Fy(\025)p Fz(\()74 b(~)-724 b Fy(\027)83 b Fz(\))p Fy(;)296 b Fz(~)-725 b Fy(\027)84 b(;)301 b Fz(~)-730 b Fy(z)60 b Fz(\()74 b(~)-724 b Fy(\027)83 b Fz(\)\).)5383 32935 y(W)-108 b(e)511 b(w)-36 b(an)g(t)512 b(to)g(pro)-36 b(v)g(e)511 b(that)g(it)h(is)g(p)36 b(ossible)512 b(to)g(c)-36 b(ho)36 b(ose)512 b(the)f(constan)-36 b(t)511 b Fy(C)41284 33134 y Fx(\027)42362 32935 y Ft(\025)502 b Fz(1,)532 b(so)3432 34541 y(that,)409 b(if)404 b Fy(C)8680 34740 y Fr(1)9609 34541 y Fz(is)g(the)f(constan)-36 b(t)403 b(de\014ned)f(in)i(Lemma)f (4.1)i(and)e Ft(j)p Fy(\025)p Ft(j)g Fz(is)h(small)g(enough,)410 b(the)3432 36146 y(set)519 b Ft(F)648 b Fz(=)515 b Ft(f)74 b Fz(~)-724 b Fy(\027)599 b Ft(2)515 b Ff(M)13379 36345 y Fr(2)14421 36146 y Fz(:)h Ft(jj)74 b Fz(~)-724 b Fy(\027)83 b Ft(jj)17500 36345 y Fr(2)18542 36146 y Ft(\024)515 b Fz(2)p Fy(C)21671 36345 y Fr(1)22197 36146 y Fy(C)23128 36345 y Fx(\027)23703 36146 y Ft(j)p Fy(\025)p Ft(jg)520 b Fz(is)g(in)-36 b(v)-72 b(arian)-36 b(t)520 b(under)e Fn(T)i Fz(and)f(that)h Fn(T)g Fz(is)3432 37751 y(a)509 b(con)-36 b(traction)508 b(on)h(it.)803 b(This)509 b(is)g(su\016cien) -36 b(t)508 b(to)h(pro)-36 b(v)g(e)508 b(the)h(Theorem,)527 b(since)509 b Ff(M)43535 37950 y Fr(2)44570 37751 y Fz(is)g(a)3432 39356 y(Banac)-36 b(h)433 b(space,)h(as)g(one)f(can)h(easily)h(sho)-36 b(w.)5383 40961 y(By)380 b(using)g(Theorem)g(4.1)h(and)e(Lemma)h(4.1)h (\(with)f Fy(C)31652 41160 y Fr(3)32546 40961 y Fz(=)369 b(2)p Fy(C)35508 41160 y Fr(1)36033 40961 y Fy(C)36964 41160 y Fx(\027)37540 40961 y Fz(\),)390 b(w)-36 b(e)381 b(see)f(that,)390 b(if)3432 42566 y Ft(j)p Fy(\025)p Ft(j)445 b Fz(is)g(small)h(enough)f(and)g(4)p Fy(C)18722 42765 y Fr(2)19248 42566 y Fy(C)20179 42765 y Fr(1)20704 42566 y Fy(C)21635 42765 y Fx(\027)22432 42566 y Fz(max)p Ft(f)p Fz(1)p Fy(;)221 b(C)27774 42016 y Fs(\000)p Fr(1)27679 42856 y(4)29033 42566 y Ft(gj)p Fy(\025)p Ft(jj)p Fy(h)32312 42765 y Fx(\014)32941 42566 y Ft(j)389 b(\024)g Fz(1)445 b(\()p Fy(C)37653 42765 y Fr(4)38624 42566 y Fz(will)h(b)36 b(e)445 b(c)-36 b(hosen)3432 44171 y(later\),)7820 48107 y Ft(jj)74 b Fz(~)-724 b Fy(\027)9285 47559 y Fs(0)9201 48435 y Fx(h)9798 48107 y Ft(jj)10536 48306 y Fx(h;)p Fr(2)12235 48107 y Ft(\024)15096 46668 y Fx(h)14568 47000 y Fl(X)13637 49460 y Fx(j)51 b Fr(=)p Fx(h)15343 49616 y Fi(\014)15896 49460 y Fr(+1)17319 48107 y Fy(\015)18066 47559 y Fs(\000)p Fx(h)p Fr(+)p Fx(j)g Fs(\000)p Fr(1)21761 48107 y Fy(\015)22641 47142 y Fi(h)p Fq(\000)p Fi(j)p 22641 47359 1500 45 v 23188 47906 a Fp(2)24550 46159 y Fl(")25196 48107 y Fy(C)26127 48306 y Fr(1)p Fx(;\027)27434 48107 y Fy(C)28365 48306 y Fr(1)28891 48107 y Ft(j)p Fy(\025)p Ft(j)295 b Fz(+)32406 46668 y Fs(1)32077 47000 y Fl(X)31989 49416 y Fx(n)p Fr(=2)33984 48107 y Fy(c)34544 47559 y Fx(n)35170 48107 y Ft(j)p Fy(\025)p Ft(j)36667 47559 y Fx(n)37293 46159 y Fl(#)38529 48107 y Fy(;)4389 b Fz(\(4.29\))3432 52664 y(where)465 b Fy(\015)8101 51756 y Fi(h)p Fq(\000)p Fi(j)p 8101 51974 V 8648 52521 a Fp(2)10254 52664 y Fz(is,)474 b(up)464 b(to)i(a)f(constan)-36 b(t,)473 b(a)466 b(b)36 b(ound)464 b(for)i(the)f(n)-36 b(um)g(b)36 b(er)463 b(of)k(sectors)e Fy(\033)43080 52181 y Fs(0)43814 52664 y Ft(2)422 b Fy(O)46116 52863 y Fx(j)3432 54269 y Fz(with)317 b(non)g(empt)-36 b(y)317 b(in)-36 b(tersection)318 b(with)f(a)h(giv)-36 b(en)318 b Fy(\033)417 b Ft(2)368 b Fy(O)29912 54468 y Fx(h)30510 54269 y Fz(,)341 b Fy(h)369 b Ft(\025)g Fy(j)392 b Fz(and)317 b Fy(C)38006 54468 y Fr(1)p Fx(;\027)39631 54269 y Fz(is)h(a)g(constan)-36 b(t)3432 56007 y(dep)36 b(ending)377 b(on)h(the)g(\014rst)g(order)g (con)-36 b(tribution)378 b(\()p Fw(i.e.)463 b Fz(the)378 b Fw(tadp)-66 b(ole)p Fz(\).)559 b(So,)390 b(if)379 b Fy(C)41401 56206 y Fx(\027)42345 56007 y Ft(\025)44254 55356 y Fx(C)44925 55479 y Fp(1)p Fi(;\027)p 43880 55697 2590 45 v 43880 56465 a Fx(\015)52 b Fs(\000)45149 55910 y(p)p 45933 55910 537 40 v 555 x Fx(\015)3432 57862 y Fz(and)5961 56977 y Fl(P)7129 57327 y Fs(1)7129 58191 y Fx(n)p Fr(=2)9179 57862 y Fy(c)9739 57380 y Fx(n)10365 57862 y Ft(j)p Fy(\025)p Ft(j)11862 57380 y Fx(n)12857 57862 y Ft(\024)369 b Fy(C)15190 58061 y Fr(1)p Fx(;\027)16497 57862 y Fy(C)17428 58061 y Fr(1)17954 57862 y Ft(j)p Fy(\025)p Ft(j)p Fz(,)433 b(then)g Ft(jj)74 b Fz(~)-724 b Fy(\027)24673 57380 y Fs(0)24589 58191 y Fx(h)25186 57862 y Ft(jj)369 b(\024)g Fz(2)p Fy(C)29276 58061 y Fr(1)29802 57862 y Fy(C)30733 58061 y Fx(\027)31308 57862 y Ft(j)p Fy(\025)p Ft(j)p Fz(.)5383 59467 y(W)-108 b(e)514 b(then)g(sho)-36 b(w)515 b(that)f Fn(T)h Fz(is)g(a)g(con)-36 b(traction)514 b(on)h Ft(F)132 b Fz(.)821 b(In)515 b(fact,)536 b(giv)-36 b(en)589 b(~)-724 b Fy(\027)40979 59666 y Fr(1)41504 59467 y Fy(;)296 b Fz(~)-725 b Fy(\027)42729 59666 y Fr(2)43762 59467 y Ft(2)507 b(F)132 b Fz(,)3432 61072 y(b)-36 b(y)532 b(using)f(again)i(Theorem)g(4.1)g(and)e(Lemma)h(4.1,) 558 b(w)-36 b(e)533 b(see)f(that,)556 b(under)531 b(the)h(same)3432 62677 y(conditions)433 b(supp)36 b(osed)433 b(ab)36 b(o)-36 b(v)g(e,)14075 66540 y Ft(jj)74 b Fz(~)-724 b Fy(\027)15540 65991 y Fs(0)15456 66868 y Fr(1)p Fx(;h)17080 66540 y Ft(\000)369 b Fz(~)-724 b Fy(\027)19135 65991 y Fs(0)19051 66868 y Fr(2)p Fx(;h)20380 66540 y Ft(jj)369 b(\024)24349 65100 y Fx(h)23820 65433 y Fl(X)22889 67892 y Fx(j)51 b Fr(=)p Fx(h)24595 68048 y Fi(\014)25148 67892 y Fr(+1)26572 66540 y Fy(\015)27319 65991 y Fs(\000)p Fx(h)p Fr(+)p Fx(j)g Fs(\000)p Fr(1)31014 66540 y Fy(\015)31894 65574 y Fi(h)p Fq(\000)p Fi(j)p 31893 65792 1500 45 v 32440 66338 a Fp(2)33877 66540 y Ft(\001)14075 70724 y(\001)221 b Fz(max)15133 71500 y Fx(i)p Fs(\025)p Fx(j)17307 70724 y Ft(jj)74 b Fz(~)-724 b Fy(\027)18688 70923 y Fr(1)p Fx(;i)20090 70724 y Ft(\000)370 b Fz(~)-724 b Fy(\027)22062 70923 y Fr(2)p Fx(;i)23169 70724 y Ft(jj)24128 68776 y Fl(")24774 70724 y Fy(C)25705 70923 y Fr(1)p Fx(;\027)27012 70724 y Fy(C)27943 70923 y Fr(4)28764 70724 y Fz(+)30487 69285 y Fs(1)30158 69617 y Fl(X)30071 72033 y Fx(n)p Fr(=2)32065 70724 y Fy(c)32625 70175 y Fx(n)33251 70724 y Ft(j)p Fy(\025)p Ft(j)34748 70175 y Fx(n)p Fs(\000)p Fr(1)36576 68776 y Fl(#)37813 70724 y Fy(;)5105 b Fz(\(4.30\))24367 75163 y(44)p eop %%Page: 45 45 45 44 bop 3432 7638 a Fz(so)521 b(that,)542 b(if)9684 6752 y Fl(P)10852 7103 y Fs(1)10852 7966 y Fx(n)p Fr(=2)12902 7638 y Fy(c)13462 7156 y Fx(n)14088 7638 y Ft(j)p Fy(\025)p Ft(j)15585 7156 y Fx(n)p Fs(\000)p Fr(1)17931 7638 y Ft(\024)518 b Fy(C)20413 7837 y Fr(1)p Fx(;\027)21720 7638 y Fy(C)22651 7837 y Fr(4)23176 7638 y Fy(=)p Fz(2)k(and)e Fy(C)28545 7837 y Fr(4)29588 7638 y Fz(=)d(\(2)p Fy(C)33204 7837 y Fx(\027)33780 7638 y Fz(\))34286 7156 y Fs(\000)p Fr(1)35543 7638 y Fz(,)543 b(then)520 b Ft(jj)74 b Fz(~)-724 b Fy(\027)40962 7156 y Fs(0)40878 7966 y Fr(1)41758 7638 y Ft(\000)429 b Fz(~)-724 b Fy(\027)43873 7156 y Fs(0)43789 7966 y Fr(2)44314 7638 y Ft(jj)518 b(\024)3564 8720 y Fr(3)p 3564 8933 471 45 v 3564 9701 a(4)4168 9243 y Ft(jj)74 b Fz(~)-724 b Fy(\027)5549 9442 y Fr(1)6369 9243 y Ft(\000)370 b Fz(~)-724 b Fy(\027)8341 9442 y Fr(2)8866 9243 y Ft(jj)p Fz(,)434 b(if)g(8)p Fy(C)13172 9442 y Fr(2)13698 9243 y Fy(C)14629 9442 y Fr(1)15154 9243 y Fy(C)16180 8761 y Fr(2)16085 9572 y Fx(\027)16706 9243 y Ft(j)p Fy(\025)p Ft(jj)p Fy(h)19321 9442 y Fx(\014)19949 9243 y Ft(j)369 b(\024)g Fz(1.)p 46160 9465 443 1329 v 5383 12453 a Fn(Remark)420 b Fz(In)g([DR],)k(where)c(only)g(the)g(rotational)h(in)-36 b(v)-72 b(arian)-36 b(t)419 b(case)i(is)f(considered,)3432 14058 y(the)380 b(lo)36 b(calization)384 b(acts)d(only)h(on)f(the)f(k) -36 b(ernels)382 b(of)g(the)e(e\013ectiv)-36 b(e)382 b(p)36 b(oten)-36 b(tial)381 b(with)g Fy(n)370 b Fz(=)e(2.)3432 15663 y(The)485 b(consequence)g(of)h(this)e(c)-36 b(hoice)486 b(is)f(that)g(the)f(e\013ectiv)-36 b(e)486 b(p)36 b(oten)-36 b(tial)485 b(is)g(b)36 b(ounded)484 b(at)3432 17269 y(order)393 b Fy(n)h Fz(b)-36 b(y)394 b(\()p Fy(c)p Ft(j)p Fy(\025)p Ft(jj)p Fy(h)13406 17468 y Fx(\014)14034 17269 y Ft(j)p Fz(\))14909 16787 y Fx(n)15535 17269 y Fz(.)565 b(Hence)394 b(in)f([DR)q(])h(the)g(e\013ectiv)-36 b(e)394 b(p)36 b(oten)-36 b(tials)394 b(are)g(found)f(to)h(b)36 b(e)3432 19197 y(con)-36 b(v)g(ergen)g(t)522 b(only)h(for)f Fy(T)701 b Ft(\025)520 b Fy(O)36 b Fz(\()p Fy(e)20197 18541 y Fs(\000)21541 18179 y Fp(1)p 21062 18342 1365 45 v 21062 18888 a Fi(c)p Fq(j)p Fi(\025)p Fq(j)22615 19197 y Fz(\),)544 b Fy(c)522 b Fz(b)36 b(eing)522 b(a)h(\\bad")f(constan)-36 b(t,)544 b(so)523 b(that)f(suc)-36 b(h)3432 20802 y(v)-72 b(alue)428 b(is)h(v)-36 b(ery)429 b(far)f(from)h(the)f(true)f(critical) j(temp)36 b(erature,)428 b(whic)-36 b(h)428 b(is)h(supp)36 b(osed)427 b(to)h(b)36 b(e)3432 22407 y(driv)-36 b(en)373 b(b)-36 b(y)374 b(the)f(second)h(order)f(con)-36 b(tribution)373 b(to)h(the)g(e\013ectiv)-36 b(e)374 b(p)36 b(oten)-36 b(tial,)386 b(for)375 b Fy(\025)e Fz(small)3432 24012 y(enough.)559 b(Note)380 b(in)e(fact)i(that)f Fy(c)f Fz(dep)36 b(end)378 b(on)h(b)36 b(ounds)378 b(at)h(ev)-36 b(ery)380 b(order)e(in)h Fy(\025)p Fz(,)390 b(while)380 b Fy(C)45346 24211 y Fr(1)p Fx(;)p Fr(2)3432 25617 y Fz(only)434 b(dep)36 b(ends)432 b(on)i(a)f(few)i(lo)-36 b(w)g(er)434 b(orders.)3432 30039 y FB(5)2869 b(The)716 b(t)-60 b(w)g(o)718 b(p)60 b(oin)-60 b(t)716 b(Sc)-60 b(h)g(winger)716 b(function)3432 32959 y Fz(In)433 b(this)g(section)h (w)-36 b(e)434 b(pro)-36 b(v)g(e)433 b(Theorem)h(1.1.)5383 34564 y(The)294 b(Sc)-36 b(h)g(winger)294 b(functions)h(can)f(b)36 b(e)294 b(deriv)-36 b(ed)295 b(b)-36 b(y)294 b(the)g Fw(gener)-66 b(ating)335 b(function)292 b Fz(de\014ned)3432 36169 y(as)10897 37774 y Ft(W)109 b Fz(\()p Fy(\036)p Fz(\))369 b(=)g(log)17751 36212 y Fl(Z)19079 37774 y Fy(P)181 b Fz(\()p Fy(d )48 b Fz(\))p Fy(e)23283 37189 y Fs(\000V)74 b Fr(\()p Fx( )32 b Fr(\))p Fs(\000N)132 b Fr(\()p Fx( )32 b Fr(\)+)29794 36352 y Fl(R)30686 37189 y Fx(d)p Fm(x)31743 37300 y Fz([)32104 37189 y Fx(\036)32663 36791 y Fp(+)32663 37356 y Fj(x)33340 37189 y Fx( )33987 36791 y Fq(\000)33955 37356 y Fj(x)34681 37189 y Fr(+)p Fx( )36060 36791 y Fp(+)36028 37356 y Fj(x)36736 37189 y Fx(\036)37295 36791 y Fq(\000)37295 37356 y Fj(x)37990 37300 y Fz(])38776 37774 y Fy(;)4792 b Fz(\(5.1\))3432 40379 y(where)345 b(the)g(v)-72 b(ariables)347 b Fy(\036)15326 39897 y Fx(\033)15326 40707 y Fm(x)16299 40379 y Fz(are)f(de\014ned)f (to)h(b)36 b(e)345 b(Grassmanian)h(v)-72 b(ariables,)364 b(an)-36 b(ticomm)g(ut-)3432 41984 y(ing)435 b(with)g(themselv)-36 b(es)435 b(and)f Fy( )18508 41502 y Fx(\033)18460 42312 y Fm(x)19132 41984 y Fz(.)583 b(In)435 b(particular)g(the)f(t)-36 b(w)g(o)435 b(p)36 b(oin)-36 b(t)435 b(Sc)-36 b(h)g(winger)434 b(function)3432 43589 y(is)f(giv)-36 b(en)435 b(b)-36 b(y)15902 45951 y Fy(S)77 b Fz(\()p Fn(x)295 b Ft(\000)g Fn(y)21 b Fz(\))368 b(=)24701 45052 y Fy(@)25460 44570 y Fr(2)p 23028 45641 4632 45 v 23028 46862 a Fy(@)72 b(\036)24557 46478 y Fr(+)24557 47190 y Fm(x)25344 46862 y Fy(@)g(\036)26873 46478 y Fs(\000)26873 47190 y Fm(y)27792 45951 y Ft(W)109 b Fz(\()p Fy(\036)p Fz(\))30995 43958 y Fl(\014)30995 44622 y(\014)30995 45287 y(\014)30995 45951 y(\014)30995 46615 y(\014)31364 47418 y Fx(\036)p Fr(=0)33771 45951 y Fy(:)9797 b Fz(\(5.2\))5383 49370 y(W)-108 b(e)398 b(can)h(get)g(a)h(m)-36 b(ultiscale)399 b(expansion)g(for)h Ft(W)109 b Fz(\()p 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Ft(\025)369 b Fy(h)180 b Fz(+)g(1,)388 b(to)377 b(b)36 b(e)376 b(called)i(the)e Fw(external)412 b(pr)-66 b(op)g(agator)p Fz(.)3432 68632 y(Ho)-36 b(w)g(ev)g(er,)462 b(one)455 b(has)h(to)g(b)36 b(e)455 b(careful)i(in)e(the)g(c)-36 b(hoice)456 b(of)h(the)e(lo)36 b(calization)458 b(p)36 b(oin)-36 b(t)455 b(in)h(the)3432 70237 y(v)-36 b(ertices)431 b Fy(v)478 b Fz(suc)-36 b(h)431 b(that)15270 69886 y(\026)14987 70237 y Fy(f)511 b Ft(2)369 b Fy(P)18232 70436 y Fx(v)19208 70237 y Fz(and)431 b Ft(j)p Fy(P)22942 70436 y Fx(v)23487 70237 y Ft(j)369 b(\024)g Fz(4)432 b(\(so)f(that)f(the)h(action)g(of)h Ft(R)f Fz(in)g Fy(v)478 b Fz(is)431 b(not)3432 71842 y(trivial\);)406 b(w)-36 b(e)390 b(c)-36 b(ho)36 b(ose)390 b(it)g(as)h(that)e(one)h(whic)-36 b(h)390 b(connects)31215 71491 y(\026)30932 71842 y Fy(f)532 b Fz(with)390 b(the)f Fy(\036)h Fz(\014eld)g(\(hence)f(no)24367 75163 y(45)p eop %%Page: 46 46 46 45 bop 3432 7638 a Fz(deriv)-72 b(ativ)-36 b(e)500 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Ft(\024)g Fz(min)p Ft(f)p Fy(j)12573 64901 y Fx(x)13158 64702 y Fy(;)221 b(j)14278 64901 y Fx(y)14831 64702 y Ft(g)p Fz(,)5021 67414 y(2.)651 b Fy(f)7324 67613 y Fx(x)8343 67414 y Fz(and)433 b Fy(f)11513 67613 y Fx(y)12499 67414 y Fz(b)36 b(elong)434 b(to)f Fy(P)19117 67613 y Fr(\026)-506 b Fx(v)19627 67414 y Fz(.)24367 75163 y(46)p eop %%Page: 47 47 47 46 bop 3791 7638 a Fz(Let)360 b Ft(C)438 b Fz(b)36 b(e)359 b(the)g(path)h(on)f Fy(\034)508 b Fz(connecting)409 b(\026)-699 b Fy(v)407 b Fz(with)360 b Fy(v)27798 7837 y Fr(0)28324 7638 y Fz(.)554 b(Giv)-36 b(en)359 b Fy(v)417 b Ft(2)368 b(C)77 b Fz(,)376 b(w)-36 b(e)360 b(a)-36 b(v)g(oid)361 b(to)f(apply)3432 9243 y(there)620 b(the)h(lo)36 b(calization)623 b(pro)36 b(cedure,)668 b(b)36 b(ecause)621 b(the)g Ft(R)g Fz(op)36 b(eration,)669 b(no)621 b(matter)g(w)-36 b(e)3432 10848 y(c)g(ho)36 b(ose)315 b(the)g(lo)36 b(calization)317 b(p)36 b(oin)-36 b(t,)339 b(w)-36 b(ould)315 b(giv)-36 b(e)316 b(rise)f(to)g(terms)g(with)g(a)h(deriv)-72 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punto 240 60 punto 240 40 punto 240 50 punto 200 80 punto 260 90 punto 260 110 punto 220 130 punto 40 75 100 75 tlinea 100 75 140 90 tlinea 140 90 moveto 200 120 lineto 220 130 lineto 200 120 moveto 240 100 lineto 260 110 lineto 240 100 moveto 260 90 lineto 140 90 moveto 180 70 lineto 200 80 lineto 180 70 moveto 220 50 lineto 240 60 lineto 220 50 moveto 240 40 lineto 220 50 moveto 240 50 lineto 100 75 moveto 120 68 lineto 100 75 moveto 120 50 lineto 200 10 lineto 180 20 moveto 200 30 lineto 120 50 moveto 140 60 lineto stroke %%EndDocument @endspecial 8995 59204 a(Figure)434 b(3:)578 b(An)433 b(example)i(of)f(tree)f(con)-36 b(tributing)433 b(to)h Fy(S)77 b Fz(\()p Fn(x)295 b Ft(\000)g Fn(y)21 b Fz(\).)3862 62211 y(1\))430 b(There)g(are)g(t)-36 b(w)g(o)430 b(\014eld)g(lab)36 b(els,)432 b Fy(f)21290 62410 y Fx(x)22305 62211 y Fz(and)e Fy(f)25472 62410 y Fx(y)26024 62211 y Fz(,)h(t)-36 b(w)g(o)430 b(scale)h(lab)36 b(els)431 b Fy(j)36571 62410 y Fx(x)37525 62211 y Ft(\025)38940 61861 y 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y(external)339 b(deriv)-72 b(ativ)-36 b(e,)360 b(and)338 b(m)-36 b(ultiply)340 b(it)f(b)-36 b(y)339 b(a)h(factor)g Fy(\015)30432 64939 y Fs(\000)p Fx(j)31545 65050 y Fi(x)32063 64939 y Fs(\000)p Fx(j)33176 65050 y Fi(y)33731 65421 y Fz(,)358 b(whic)-36 b(h)339 b(comes)h(from)g(the) 3432 67026 y(external)485 b(propagators)h(\(the)f(deriv)-72 b(ativ)-36 b(es)486 b(p)36 b(ossibly)486 b(acting)g(on)f(them)g(are)h (absorb)36 b(ed)3432 68632 y(in)596 b(the)f(\\gain)i(factors")h Fy(\015)16940 68149 y Fs(\000)p Fr(\()p Fx(h)p Fs(\000)p Fx(h)19854 67837 y Fq(0)20152 68149 y Fr(\))20573 68632 y Fz(,)637 b(pro)36 b(duced)595 b(b)-36 b(y)596 b(the)g(lo)36 b(calization)598 b(pro)36 b(cedure,)637 b(so)3432 70237 y(that)505 b(they)g(do)h(not)f(giv)-36 b(e)507 b(an)-36 b(y)506 b(con)-36 b(tribution)504 b(to)i(the)f(\014nal)g(b)36 b(ound\).)793 b(There)506 b(are)g(t)-36 b(w)g(o)3432 71842 y(relev)-72 b(an)-36 b(t)433 b(di\013erences.)24367 75163 y(48)p eop %%Page: 49 49 49 48 bop 3724 7638 a Fz(1\))293 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Fr(1)7948 30660 y Ft(\002)296 b Fz([)p Ft(\000)p Fy(e)11274 30859 y Fr(0)11800 30660 y Fy(;)221 b(e)12985 30859 y Fr(0)13511 30660 y Fz(],)434 b(de\014ned)e(b)-36 b(y)13327 33408 y Fy(~)-658 b(p)369 b Fz(=)362 b Fy(~)-643 b(q)47 b Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))369 b(=)g Fy(u)p Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))-54 b Fy(~)-596 b(e)25117 33607 y Fx(r)25624 33408 y Fz(\()p Fy(\022)36 b Fz(\))1300 b Fy(;)1522 b Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))370 b Ft(2)e(A)h Fy(:)7214 b Fz(\(7.1\))3432 36157 y(Moreo)-36 b(v)g(er,)434 b(the)f(symmetry)h(prop)36 b(ert)-36 b(y)433 b(\(1.11\))h(implies)g(that)18597 38906 y Fy(~)-644 b(q)48 b Fz(\()p Fy(\022)331 b Fz(+)295 b Fy(\031)48 b(;)221 b(e)p Fz(\))368 b(=)h Ft(\000)-7 b Fy(~)-643 b(q)48 b Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))369 b Fy(;)12498 b Fz(\(7.2\))3432 41655 y(a)433 b(prop)36 b(ert)-36 b(y)434 b(that)f(will)h(ha)-36 b(v)g(e)434 b(an)g(imp)36 b(ortan)-36 b(t)433 b(role)h(in)f(the)g(follo)-36 b(wing.)5383 43260 y(Let)470 b(us)h(no)-36 b(w)471 b(in)-36 b(tro)36 b(duce)470 b(some)h(more)g(geometrical)i(de\014nitions,)479 b(whic)-36 b(h)471 b(w)-36 b(e)471 b(shall)3432 44865 y(need)490 b(in)g(the)h(follo)-36 b(wing.)752 b(F)-108 b(or)490 b(an)-36 b(y)491 b(\014xed)f Fy(e)p Fz(,)506 b(w)-36 b(e)491 b(can)f(lo)36 b(cally)493 b(de\014ne)d(the)g(arc)h(length)3432 46470 y Fy(s)p Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))498 b(on)g(\006\()p Fy(e)p Fz(\);)530 b(w)-36 b(e)498 b(shall)h(denote)e Fy(@)72 b(=@)g(s)498 b Fz(the)g(partial)g(deriv)-72 b(ativ)-36 b(e)499 b(with)f(resp)36 b(ect)498 b(to)3432 48075 y Fy(s)p Fz(,)641 b(at)600 b(\014xed)f Fy(e)p Fz(,)641 b(and)599 b(w)-36 b(e)600 b(shall)h(sometime)f(use)f(the)g(prime)g(to)h (denote)f(the)h(partial)3432 49680 y(deriv)-72 b(ativ)-36 b(e)611 b(with)f(resp)36 b(ect)610 b(to)g Fy(\022)36 b Fz(.)1109 b(If)577 b Fy(~)-616 b(\034)149 b Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))670 b(=)f Fy(@)80 b(~)-658 b(p)p Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))p Fy(=@)72 b(s)611 b Fz(is)f(the)g(unit)g(tangen)-36 b(t)3432 51285 y(v)g(ector)434 b(at)f(\006\()p Fy(e)p Fz(\))h(in)426 b Fy(~)-643 b(q)47 b Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\),)435 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)8895 54717 y Fy(s)9508 54169 y Fs(0)9818 54717 y Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))-35 b Fy(~)-615 b(\034)148 b Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))1108 b(=)19568 53818 y Fy(@)80 b(~)-658 b(p)p 19568 54407 1412 45 v 19573 55628 a(@)72 b(\022)21113 54717 y Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))369 b(=)g Fy(u)26442 54169 y Fs(0)26753 54717 y Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))-54 b Fy(~)-596 b(e)30195 54916 y Fx(r)30701 54717 y Fz(\()p Fy(\022)36 b Fz(\))295 b(+)g Fy(u)p Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))-54 b Fy(~)-596 b(e)38139 54916 y Fx(t)38534 54717 y Fz(\()p Fy(\022)36 b Fz(\))369 b Fy(;)12447 57559 y(s)13060 57011 y Fs(0)13370 57559 y Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))1108 b(=)19435 56160 y Fl(q)p 20542 56160 10124 45 v 1399 x Fy(u)21282 57176 y Fs(0)21593 57559 y Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))24432 57176 y Fr(2)25254 57559 y Fz(+)294 b Fy(u)p Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))30139 57176 y Fr(2)31035 57559 y Fy(;)12533 b Fz(\(7.3\))3432 60308 y(where)379 b Fy(~)-596 b(e)7792 60507 y Fx(t)8187 60308 y Fz(\()p Fy(\022)36 b Fz(\))369 b(=)f(\()p Ft(\000)221 b Fz(sin)h Fy(\022)36 b(;)221 b Fz(cos)i Fy(\022)36 b Fz(\).)5383 61913 y(Analogously)-108 b(,)356 b(if)297 b Fy(~)-611 b(n)p Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))336 b(is)g(the)e(outgoing)i(unit)f(normal)g(v)-36 b(ector)336 b(at)f(\006\()p Fy(e)p Fz(\))g(in)328 b Fy(~)-643 b(q)48 b Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))3432 63518 y(and)412 b(1)p Fy(=r)36 b Fz(\()p Fy(\022)g(;)221 b(e)p Fz(\))414 b(is)g(the)e(curv)-72 b(ature)412 b(\(whic)-36 b(h)413 b(satis\014es)g(the)f(con)-36 b(v)g(exit)g(y)414 b(condition)f(\(1.9\)\),)3432 65123 y(w)-36 b(e)433 b(ha)-36 b(v)g(e)10668 67872 y Fy(s)11281 67324 y Fs(0)11591 67872 y Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))-39 b Fy(~)-611 b(n)q Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))1108 b(=)e Fy(u)p Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))-54 b Fy(~)-596 b(e)25454 68071 y Fx(r)25961 67872 y Fz(\()p Fy(\022)36 b Fz(\))295 b Ft(\000)g Fy(u)29978 67324 y Fs(0)30289 67872 y Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))-54 b Fy(~)-596 b(e)33731 68071 y Fx(t)34126 67872 y Fz(\()p Fy(\022)36 b Fz(\))369 b Fy(;)13137 69700 y(@)13896 69218 y Fr(2)14429 69700 y Fy(~)-657 b(p)p 13137 70288 1938 45 v 13142 71510 a(@)72 b(\022)14543 71126 y Fr(2)15207 70598 y Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))1108 b(=)e Fy(s)21885 70050 y Fs(00)22451 70598 y Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))-35 b Fy(~)-615 b(\034)148 b Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))296 b Ft(\000)30599 69700 y Fy(s)31212 69218 y Fs(0)31522 69700 y Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))34361 69218 y Fr(2)p 30599 70288 4289 45 v 31012 71510 a Fy(r)36 b Fz(\()p Fy(\022)g(;)221 b(e)p Fz(\))34981 70598 y Fy(~)-611 b(n)q Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))369 b Fy(:)4563 b Fz(\(7.4\))24367 75163 y(52)p eop %%Page: 53 53 53 52 bop 3432 7638 a Fn(Lemma)499 b(7.1)650 b Fw(The)507 b(angle)e Fy(\013)8 b Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))508 b Fw(b)-66 b(etwe)g(en)465 b Fy(~)-611 b(n)q Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))506 b Fw(and)453 b Fy(~)-596 b(e)33676 7837 y Fx(r)34182 7638 y Fz(\(0\))506 b Fw(is)g(a)h(monotone)e(in-)3432 9243 y(cr)-66 b(e)g(asing)463 b(function)g(of)h Fy(\022)36 b Fw(,)465 b(such)h(that,)e(if)g Ft(jj)p Fy(\022)25194 9442 y Fr(1)26015 9243 y Ft(\000)295 b Fy(\022)27949 9442 y Fr(2)28475 9243 y Ft(jj)465 b Fw(denotes)g(the)f(distanc)-66 b(e)464 b(on)h Fo(T)44279 8679 y Fr(1)44805 9243 y Fw(.)10717 12177 y Fy(c)11277 12376 y Fr(1)11803 12177 y Ft(jj)p Fy(\022)13147 12376 y Fr(2)13968 12177 y Ft(\000)296 b Fy(\022)15903 12376 y Fr(1)16428 12177 y Ft(jj)369 b(\024)h(jj)p Fy(\013)8 b Fz(\()p Fy(\022)21623 12376 y Fr(2)22149 12177 y Fy(;)221 b(e)p Fz(\))296 b Ft(\000)f Fy(\013)8 b Fz(\()p Fy(\022)27411 12376 y Fr(1)27937 12177 y Fy(;)221 b(e)p Fz(\))p Ft(jj)370 b(\024)f Fy(c)32698 12376 y Fr(2)33223 12177 y Ft(jj)p Fy(\022)34567 12376 y Fr(2)35389 12177 y Ft(\000)295 b Fy(\022)37323 12376 y Fr(1)37849 12177 y Ft(jj)369 b Fz(;)4612 b(\(7.5\))3432 15110 y Fw(mor)-66 b(e)g(over,)462 b Fy(\013)8 b Fz(\()p Fy(\022)332 b Fz(+)295 b Fy(\031)48 b(;)221 b(e)p Fz(\))295 b Ft(\000)h Fy(\013)8 b Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))370 b(=)e Fy(\031)48 b Fw(.)3432 18154 y Fb(Pr)-30 b(oof)659 b(-)1238 b Fz(By)580 b(using)g(\(7.3\))g(and)f(\(7.4\))h(and)f(T)-108 b(a)-36 b(ylor)581 b(expansion,)616 b(one)580 b(can)g(easily)3432 19759 y(pro)-36 b(v)g(e)433 b(that,)h(if)g Fy(\013)12135 19958 y Fx(i)12880 19759 y Fz(=)368 b Fy(\013)8 b Fz(\()p Fy(\022)16207 19958 y Fx(i)16584 19759 y Fy(;)221 b(e)p Fz(\),)3678 23425 y(sin\()p Fy(\013)6608 23624 y Fr(2)7429 23425 y Ft(\000)295 b Fy(\013)9584 23624 y Fr(1)10110 23425 y Fz(\))1107 b(=)1067 b Fy(~)-611 b(n)q Fz(\()p Fy(\022)15730 23624 y Fr(2)16256 23425 y Fy(;)221 b(e)p Fz(\))295 b Ft(\001)261 b Fy(~)-616 b(\034)149 b Fz(\()p Fy(\022)20732 23624 y Fr(1)21257 23425 y Fy(;)221 b(e)p Fz(\))370 b(=)e(\()p Fy(\022)25810 23624 y Fr(2)26631 23425 y Ft(\000)295 b Fy(\022)28565 23624 y Fr(1)29091 23425 y Fz(\))29730 22527 y Fy(s)30343 22045 y Fs(0)30653 22527 y Fz(\()p Fy(\022)31765 22726 y Fr(1)32291 22527 y Fy(;)221 b(e)p Fz(\))p 29730 23115 4253 45 v 29880 24336 a Fy(r)36 b Fz(\()p Fy(\022)31614 24535 y Fr(1)32141 24336 y Fy(;)221 b(e)p Fz(\))34410 23425 y(+)295 b Fy(O)36 b Fz(\()p Fy(\022)37859 23624 y Fr(2)38680 23425 y Ft(\000)296 b Fy(\022)40615 23624 y Fr(1)41141 23425 y Fz(\))41647 22877 y Fr(2)42541 23425 y Fy(;)1027 b Fz(\(7.6\))3533 27063 y(cos)q(\()p Fy(\013)6608 27262 y Fr(2)7429 27063 y Ft(\000)295 b Fy(\013)9584 27262 y Fr(1)10110 27063 y Fz(\))1107 b(=)1067 b Fy(~)-611 b(n)q Fz(\()p Fy(\022)15730 27262 y Fr(2)16256 27063 y Fy(;)221 b(e)p Fz(\))295 b Ft(\001)256 b Fy(~)-611 b(n)q Fz(\()p Fy(\022)20795 27262 y Fr(1)21320 27063 y Fy(;)221 b(e)p Fz(\))370 b(=)e(1)296 b Ft(\000)27168 26164 y Fz(\()p Fy(\022)28280 26363 y Fr(2)29101 26164 y Ft(\000)f Fy(\022)31035 26363 y Fr(1)31561 26164 y Fz(\))32067 25682 y Fr(2)p 27168 26753 5425 45 v 29555 27974 a Fz(2)32859 26164 y Fy(s)33472 25682 y Fs(0)p Fr(2)34252 26164 y Fz(\()p Fy(\022)35364 26363 y Fr(1)35890 26164 y Fy(;)221 b(e)p Fz(\))p 32859 26753 4723 45 v 32982 27974 a Fy(r)33604 27590 y Fr(2)34129 27974 y Fz(\()p Fy(\022)35241 28173 y Fr(1)35767 27974 y Fy(;)g(e)p Fz(\))38009 27063 y(+)295 b Fy(O)36 b Fz(\()p Fy(\022)41458 27262 y Fr(2)42279 27063 y Ft(\000)296 b Fy(\022)44214 27262 y Fr(1)44740 27063 y Fz(\))45246 26514 y Fr(3)46140 27063 y Fy(;)3432 30741 y Fz(whic)-36 b(h)500 b(implies)i(\(7.5\))f(for)g Ft(j)p Fy(\022)17906 30940 y Fr(2)18773 30741 y Ft(\000)341 b Fy(\022)20753 30940 y Fr(1)21279 30741 y Ft(j)501 b Fz(small,)519 b(hence)500 b(ev)-36 b(en)501 b(for)h(an)-36 b(y)501 b(v)-72 b(alue)501 b(of)h Fy(\022)42869 30940 y Fr(2)43736 30741 y Ft(\000)341 b Fy(\022)45716 30940 y Fr(1)46242 30741 y Fz(,)3432 32346 y(together)333 b(with)h(the)g(monotonicit)-36 b(y)334 b(prop)36 b(ert)-36 b(y)-108 b(.)544 b(The)334 b(fact)h(that)e Fy(\013)8 b Fz(\()p Fy(\022)128 b Fz(+)92 b Fy(\031)48 b(;)221 b(e)p Fz(\))92 b Ft(\000)g Fy(\013)8 b Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))369 b(=)3432 33951 y Fy(\031)481 b Fz(is)433 b(a)h(trivial)h(consequence)f(of)g(\(7.2\).)p 46160 34173 443 1329 v 5383 38767 a(W)-108 b(e)641 b(denote)g(b)-36 b(y)649 b Fy(~)-658 b(p)14896 38966 y Fx(F)15678 38767 y Fz(\()p Fy(\022)36 b Fz(\))722 b(=)716 b Fy(~)-643 b(q)47 b Fz(\()p Fy(\022)36 b(;)221 b Fz(0\))643 b(the)e(generic)g(p)36 b(oin)-36 b(t)642 b(of)g(the)f(F)-108 b(ermi)641 b(surface)3432 40372 y(\006)4371 40571 y Fx(F)5736 40372 y Ft(\021)583 b Fz(\006\(0\).)956 b(Moreo)-36 b(v)g(er,)592 b(to)559 b(simplify)i(the)e(notation,)591 b(from)560 b(no)-36 b(w)559 b(on)h(w)-36 b(e)559 b(shall)h(in)3432 41977 y(general)493 b(suppress)f(the)g(v)-72 b(ariable)494 b Fy(e)f Fz(when)f(it)h(is)g(equal)h(to)f(0;)523 b(for)494 b(example,)508 b(w)-36 b(e)493 b(shall)3432 43582 y(put)440 b Fy(~)-658 b(p)6469 43781 y Fx(F)7250 43582 y Fz(\()p Fy(\022)36 b Fz(\))369 b(=)g Fy(u)p Fz(\()p Fy(\022)36 b Fz(\))-54 b Fy(~)-596 b(e)13651 43781 y Fx(r)14157 43582 y Fz(\()p Fy(\022)36 b Fz(\).)578 b(Let)433 b(us)h(consider)f(an) g(s-sector)h Fy(S)33133 43781 y Fx(h;!)34608 43582 y Fz(,)g(see)g(\(2.72\).)3432 46626 y Fn(Lemma)499 b(7.2)650 b Fw(If)472 b Fy(~)-658 b(p)369 b Fz(=)g Fy(\032)-54 b(~)-596 b(e)16327 46825 y Fx(r)16833 46626 y Fz(\()p Fy(\022)36 b Fz(\))368 b Ft(2)h Fy(S)20910 46825 y Fx(h;!)22386 46626 y Fw(,)464 b Fy(h)369 b Ft(\024)g Fz(0)p Fw(,)465 b Fy(!)416 b Ft(2)369 b Fy(O)30758 46825 y Fx(h)31355 46626 y Fw(,)465 b(then)13518 49560 y Ft(j)p Fy(\032)295 b Ft(\000)g Fy(u)p Fz(\()p Fy(\022)36 b Fz(\))p Ft(j)369 b(\024)g Fy(c\015)22022 49011 y Fx(h)23948 49560 y Fy(;)1550 b Ft(j)p Fy(\022)331 b Ft(\000)296 b Fy(\022)29100 49759 y Fx(h;!)30576 49560 y Ft(j)369 b(\024)g Fy(\031)48 b(\015)34249 49011 y Fx(h=)p Fr(2)36156 49560 y Fy(:)7412 b Fz(\(7.7\))3432 52604 y Fb(Pr)-30 b(oof)568 b(-)1062 b Fz(The)494 b(b)36 b(ound)494 b(on)g Fy(\022)531 b Fz(follo)-36 b(ws)497 b(directly)e(from)g(the)f(de\014nition)f(of)j Fy(S)42271 52803 y Fx(h;!)43746 52604 y Fz(.)762 b(On)3432 54209 y(the)433 b(other)g(hand,)g(the)g(iden)-36 b(tit)g(y)5084 57782 y Fy(")p Fz(\()7 b Fy(~)-657 b(p)p Fz(\))295 b Ft(\000)g Fy(\026)369 b Fz(=)f Fy(")p Fz(\()p Fy(\032)-54 b(~)-596 b(e)13902 57981 y Fx(r)14407 57782 y Fz(\()p Fy(\022)36 b Fz(\)\))295 b Ft(\000)h Fy(")p Fz(\()p Fy(u)p Fz(\()p Fy(\022)36 b Fz(\))-54 b Fy(~)-596 b(e)22303 57981 y Fx(r)22808 57782 y Fz(\()p Fy(\022)36 b Fz(\)\))369 b(=)26718 56220 y Fl(Z)27825 56570 y Fx(u)p Fr(\()p Fx(\022)26 b Fr(\))27333 58735 y Fx(\032)29843 57782 y Fy(d\032)31190 57233 y Fs(0)31816 57782 y Fy(~)-597 b(e)32472 57981 y Fx(r)32978 57782 y Fz(\()p Fy(\022)36 b Fz(\))34758 57446 y Fy(~)34632 57782 y Ft(r)p Fy(")221 b Fz(\()p Fy(\032)37746 57233 y Fs(0)38003 57782 y Fy(~)-597 b(e)38659 57981 y Fx(r)39165 57782 y Fz(\()p Fy(\022)36 b Fz(\)\))590 b Fy(;)1653 b Fz(\(7.8\))3432 61453 y(and)433 b(the)g(prop)36 b(ert)-36 b(y)433 b(\(1.10\))h(of)h Fy(")p Fz(\()7 b Fy(~)-657 b(p)o Fz(\),)433 b(easily)j(imply)e(the)f(b)36 b(ound)432 b(on)h Fy(\032)296 b Ft(\000)f Fy(u)p Fz(\()p Fy(\022)36 b(;)221 b Fz(0\).)p 46160 61675 V 5383 66269 a(The)353 b(follo)-36 b(wing)355 b(lemma)f(sho)-36 b(ws)354 b(that,)369 b(if)361 b Fy(~)-657 b(p)369 b Ft(2)f Fy(S)28457 66468 y Fx(h;!)29933 66269 y Fz(,)h(the)353 b(di\013erence)g(b)36 b(et)-36 b(w)g(een)360 b Fy(~)-658 b(p)354 b Fz(and)3439 67874 y Fy(~)-657 b(p)4085 68073 y Fx(F)4866 67874 y Fz(\()p Fy(\022)5978 68073 y Fx(h;!)7454 67874 y Fz(\))495 b(is)i(of)g(order)e Fy(\015)15575 67392 y Fx(h)16669 67874 y Fz(in)h(the)f(direction)h(normal)h(to)f(\006)33095 68073 y Fx(F)34373 67874 y Fz(in)g(the)f(p)36 b(oin)-36 b(t)503 b Fy(~)-657 b(p)42367 68073 y Fx(F)43148 67874 y Fz(\()p Fy(\022)44260 68073 y Fx(h;!)45736 67874 y Fz(\),)3432 69479 y(while)434 b(it)f(is)h(of)h(order)e Fy(\015)15061 68997 y Fx(h=)p Fr(2)17033 69479 y Fz(in)g(the)g(tangen) -36 b(t)433 b(direction.)24367 75163 y(53)p eop %%Page: 54 54 54 53 bop 3432 7638 a Fn(Lemma)499 b(7.3)650 b Fw(If)472 b Fy(~)-658 b(p)369 b Ft(2)g Fy(S)15727 7837 y Fx(h;!)17202 7638 y Fw(,)465 b Fy(h)369 b Ft(\024)g Fz(0)p Fw(,)464 b Fy(!)417 b Ft(2)369 b Fy(O)25575 7837 y Fx(h)26172 7638 y Fw(,)465 b(then)4103 10169 y Fy(~)-657 b(p)369 b Fz(=)376 b Fy(~)-658 b(p)7151 10368 y Fx(F)7932 10169 y Fz(\()p Fy(\022)9044 10368 y Fx(h;!)10520 10169 y Fz(\))188 b(+)g Fy(k)13090 10368 y Fr(1)13577 10169 y Fy(~)-610 b(n)p Fz(\()p Fy(\022)15505 10368 y Fx(h;!)16981 10169 y Fz(\))188 b(+)g Fy(k)19551 10368 y Fr(2)20043 10169 y Fy(~)-616 b(\034)148 b Fz(\()p Fy(\022)21902 10368 y Fx(h;!)23378 10169 y Fz(\))1328 b Fy(;)1550 b Ft(j)p Fy(k)28168 10368 y Fr(1)28695 10169 y Ft(j)369 b(\024)g Fy(c\015)32142 9620 y Fx(h)34067 10169 y Fy(;)1550 b Ft(j)p Fy(k)37023 10368 y Fr(2)37550 10169 y Ft(j)369 b(\024)g Fy(c\015)40997 9620 y Fx(h=)p Fr(2)42904 10169 y Fy(:)664 b Fz(\(7.9\))3432 12700 y Fw(Mor)-66 b(e)g(over,)11145 13994 y Fl(\014)11145 14659 y(\014)11145 15323 y(\014)11145 15987 y(\014)11145 16651 y(\014)12248 15088 y Fy(@)p 11647 15677 1962 45 v 11647 16898 a(@)72 b(k)13082 17097 y Fr(2)13741 15987 y Fy(")p Fz(\()7 b Fy(~)-657 b(p)15509 16186 y Fx(F)16290 15987 y Fz(\()p Fy(\022)17402 16186 y Fx(h;!)18877 15987 y Fz(\))295 b(+)g Fy(k)21661 16186 y Fr(1)22148 15987 y Fy(~)-611 b(n)q Fz(\()p Fy(\022)24076 16186 y Fx(h;!)25551 15987 y Fz(\))295 b(+)g Fy(k)28335 16186 y Fr(2)28827 15987 y Fy(~)-615 b(\034)148 b Fz(\()p Fy(\022)30687 16186 y Fx(h;!)32162 15987 y Fz(\)\))33174 13994 y Fl(\014)33174 14659 y(\014)33174 15323 y(\014)33174 15987 y(\014)33174 16651 y(\014)33912 15987 y Ft(\024)369 b Fy(c\015)36621 15439 y Fx(h=)p Fr(2)38528 15987 y Fy(:)4390 b Fz(\(7.10\))3432 19502 y Fb(Pr)-30 b(oof)735 b(-)1383 b Fz(If)658 b Fy(~)-657 b(p)736 b Fz(=)h Fy(\032)-54 b(~)-596 b(e)16135 19701 y Fx(r)16641 19502 y Fz(\()p Fy(\022)36 b Fz(\),)704 b(b)-36 b(y)650 b(Lemma)f(7.2)i Ft(j)7 b Fy(~)-657 b(p)442 b Ft(\000)450 b Fy(~)-657 b(p)32146 19701 y Fx(F)32927 19502 y Fz(\()p Fy(\022)36 b Fz(\))p Ft(j)737 b(\024)g Fy(c\015)38764 19020 y Fx(h)39362 19502 y Fz(.)1227 b(Hence,)704 b(to)3432 21107 y(pro)-36 b(v)g(e)390 b(\(7.9\),)399 b(it)391 b(is)f(su\016cien)-36 b(t)389 b(to)i(pro)-36 b(v)g(e)390 b(that)f Ft(j)p Fz([)7 b Fy(~)-657 b(p)27432 21306 y Fx(F)28214 21107 y Fz(\()p Fy(\022)36 b Fz(\))206 b Ft(\000)214 b Fy(~)-658 b(p)31966 21306 y Fx(F)32748 21107 y Fz(\()p Fy(\022)33860 21306 y Fx(h;!)35336 21107 y Fz(\)])-39 b Fy(~)-611 b(n)q Fz(\()p Fy(\022)38092 21306 y Fx(h;!)39567 21107 y Fz(\))p Ft(j)369 b(\024)g Fy(c\015)43520 20625 y Fx(h)44508 21107 y Fz(and)3432 22712 y Ft(j)p Fz([)7 b Fy(~)-657 b(p)4815 22911 y Fx(F)5596 22712 y Fz(\()p Fy(\022)36 b Fz(\))325 b Ft(\000)332 b Fy(~)-658 b(p)9585 22911 y Fx(F)10366 22712 y Fz(\()p Fy(\022)11478 22911 y Fx(h;!)12954 22712 y Fz(\)])-35 b Fy(~)-615 b(\034)148 b Fz(\()p Fy(\022)15646 22911 y Fx(h;!)17122 22712 y Fz(\))p Ft(j)442 b(\024)h Fy(c\015)21222 22230 y Fx(h=)p Fr(2)22760 22712 y Fz(.)707 b(These)477 b(b)36 b(ounds)475 b(immediately)j(follo)-36 b(ws)479 b(from)3432 24317 y(the)433 b(the)h(follo)-36 b(wing)437 b(ones,)e(whic)-36 b(h)434 b(can)g(b)36 b(e)434 b(easily)i(pro)-36 b(v)g(ed,)435 b(b)-36 b(y)434 b(using)g(\(7.3\),)i(\(7.4\))e(and)3432 25922 y(some)f(T)-108 b(a)-36 b(ylor)435 b(expansions:)13859 29133 y([)7 b Fy(~)-657 b(p)14873 29332 y Fx(F)15655 29133 y Fz(\()p Fy(\022)16767 29332 y Fr(1)17293 29133 y Fz(\))295 b Ft(\000)302 b Fy(~)-657 b(p)20075 29332 y Fx(F)20856 29133 y Fz(\()p Fy(\022)21968 29332 y Fr(2)22494 29133 y Fz(\)])295 b Ft(\001)256 b Fy(~)-611 b(n)q Fz(\()p Fy(\022)26209 29332 y Fr(2)26735 29133 y Fz(\))368 b(=)h Fy(O)36 b Fz(\()p Fy(\022)31132 29332 y Fr(1)31953 29133 y Ft(\000)295 b Fy(\022)33887 29332 y Fr(2)34413 29133 y Fz(\))34919 28584 y Fr(2)35814 29133 y Fy(;)7104 b Fz(\(7.11\))14154 32890 y([)7 b Fy(~)-657 b(p)15168 33089 y Fx(F)15949 32890 y Fz(\()p Fy(\022)17061 33089 y Fr(1)17587 32890 y Fz(\))295 b Ft(\000)303 b Fy(~)-658 b(p)20369 33089 y Fx(F)21151 32890 y Fz(\()p Fy(\022)22263 33089 y Fr(2)22789 32890 y Fz(\)])295 b Ft(\001)260 b Fy(~)-615 b(\034)148 b Fz(\()p Fy(\022)26440 33089 y Fr(2)26966 32890 y Fz(\))369 b(=)f Fy(O)36 b Fz(\()p Fy(\022)31363 33089 y Fr(1)32184 32890 y Ft(\000)296 b Fy(\022)34119 33089 y Fr(2)34645 32890 y Fz(\))369 b Fy(:)7398 b Fz(\(7.12\))3432 35042 y(It)539 b(is)i(su\016cien)-36 b(t)539 b(to)h(put)e(here)i Fy(\022)19647 35241 y Fr(1)20723 35042 y Fz(=)549 b Fy(\022)36 b Fz(,)567 b Fy(\022)24460 35241 y Fr(2)25536 35042 y Fz(=)549 b Fy(\022)27703 35241 y Fx(h;!)29719 35042 y Fz(and)539 b(to)h(recall)h(that)e Fy(\022)404 b Ft(\000)368 b Fy(\022)43566 35241 y Fx(h;!)45591 35042 y Fz(=)3432 36647 y Fy(O)36 b Fz(\()p Fy(\015)5715 36165 y Fx(h=)p Fr(2)7253 36647 y Fz(\).)5383 38252 y(Let)665 b(us)g(no)-36 b(w)665 b(observ)-36 b(e)666 b(that,)723 b(if)666 b(w)-36 b(e)666 b(deriv)-36 b(e)666 b(with)f(resp)36 b(ect)665 b(to)h Fy(\022)701 b Fz(the)665 b(iden)-36 b(tit)g(y)3432 39857 y Fy(")p Fz(\()p Fy(u)p Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))-54 b Fy(~)-596 b(e)8729 40056 y Fx(r)9235 39857 y Fz(\()p Fy(\022)36 b Fz(\)\))368 b(=)h Fy(e)p Fz(,)434 b(w)-36 b(e)433 b(get,)i(for)f(an)-36 b(y)441 b Fy(~)-658 b(p)369 b Ft(2)g(B)40 b Fz(,)7614 42508 y([)8101 42173 y Fy(~)7975 42508 y Ft(r)p Fy(")p Fz(\()7 b Fy(~)-657 b(p)o Fz(\))-35 b Fy(~)-615 b(\034)148 b Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\)])p Fy(s)15881 41960 y Fs(0)16192 42508 y Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))370 b(=)e(0)1670 b Ft(\))26225 42173 y Fy(~)26099 42508 y Ft(r)p Fy(")p Fz(\()7 b Fy(~)-657 b(p)o Fz(\))369 b(=)g Fy(a)p Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))-39 b Fy(~)-611 b(n)p Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))370 b Fy(;)4182 b Fz(\(7.13\))3432 45039 y Fy(a)p Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))324 b(b)36 b(eing)323 b(a)i(smo)36 b(oth)324 b(function,)345 b(strictly)325 b(p)36 b(ositiv)-36 b(e)325 b(b)-36 b(y)323 b(\(1.10\).)543 b(Hence,)346 b(if)332 b Fy(~)-657 b(p)368 b Ft(2)h Fy(S)44766 45238 y Fx(h;!)46242 45039 y Fz(,)3432 46644 y(b)-36 b(y)566 b(using)h(the)g(\014rst)f(line)h(of)h(\(7.6\),)601 b(\(7.13\))568 b(and)f(the)f(fact)i(that)e Ft(j)p Fy(")p Fz(\()7 b Fy(~)-657 b(p)p Fz(\))385 b Ft(\000)i Fy(\026)p Ft(j)595 b(\024)h Fy(c\015)45644 46162 y Fx(h)46242 46644 y Fz(,)3432 48249 y Ft(j)p Fy(\022)331 b Ft(\000)295 b Fy(\022)6672 48448 y Fx(h;!)8148 48249 y Ft(j)369 b(\024)g Fy(c\015)11595 47767 y Fx(h=)p Fr(2)13134 48249 y Fz(,)10114 50654 y Fy(@)72 b(")p Fz(\()7 b Fy(~)-657 b(p)o Fz(\))p 10114 51242 3032 45 v 10649 52463 a Fy(@)72 b(k)12084 52662 y Fr(2)14386 51552 y Fz(=)16630 51216 y Fy(~)16504 51552 y Ft(r)p Fy(")p Fz(\()7 b Fy(~)-657 b(p)o Fz(\))295 b Ft(\001)261 b Fy(~)-615 b(\034)148 b Fz(\()p Fy(\022)22669 51751 y Fx(h;!)24145 51552 y Fz(\))368 b(=)h Fy(a)p Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))-39 b Fy(~)-611 b(n)p Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))296 b Ft(\001)261 b Fy(~)-616 b(\034)148 b Fz(\()p Fy(\022)36322 51751 y Fx(h;!)37798 51552 y Fz(\))369 b(=)14386 54318 y(=)1106 b Fy(a)p Fz(\()p Fy(\022)36 b(;)221 b(e)p Fz(\))-39 b Fy(~)-611 b(n)q Fz(\()p Fy(\022)36 b Fz(\))295 b Ft(\001)260 b Fy(~)-615 b(\034)148 b Fz(\()p Fy(\022)25241 54517 y Fx(h;!)26717 54318 y Fz(\))295 b(+)g Fy(O)36 b Fz(\()p Fy(\015)31108 53770 y Fx(h)31705 54318 y Fz(\))369 b(=)g Fy(O)36 b Fz(\()p Fy(\015)36244 53770 y Fx(h=)p Fr(2)37782 54318 y Fz(\))369 b Fy(;)4261 b Fz(\(7.14\))3432 56849 y(whic)-36 b(h)433 b(pro)-36 b(v)g(es)434 b(\(7.10\).)p 46160 57070 443 1329 v 5383 61664 a(Giv)-36 b(en)348 b Fy(~)-657 b(p)368 b Ft(2)h Fy(S)12133 61863 y Fx(h;!)13609 61664 y Fz(,)359 b(w)-36 b(e)342 b(shall)g(also)g(consider)f(the)f(pro) 72 b(jection)342 b(on)f(the)g(F)-108 b(ermi)340 b(surface,)3432 63269 y(de\014ned)432 b(as)17061 64874 y Fy(~)-657 b(p)17707 65073 y Fs(?)18863 64874 y Fz(=)376 b Fy(~)-658 b(p)20896 65073 y Fx(F)21678 64874 y Fz(\()p Fy(\022)22790 65073 y Fs(?)23577 64874 y Fz(\))369 b(=)376 b Fy(~)-658 b(p)295 b Ft(\000)h Fy(x)182 b(~)-611 b(n)p Fz(\()p Fy(\022)30957 65073 y Fs(?)31745 64874 y Fz(\))368 b Fy(:)10299 b Fz(\(7.15\))3432 67026 y(Note)442 b(that)h(\(7.15\))g(has)g(to)f(b)36 b(e)443 b(though)-36 b(t)441 b(as)i(an)g(equation)g(for)g Fy(\022)35114 67225 y Fs(?)36344 67026 y Fz(and)f Fy(x)p Fz(,)j(giv)-36 b(en)451 b Fy(~)-658 b(p)443 b Fz(;)k(it)3432 68632 y(is)463 b(easy)g(to)g(pro)-36 b(v)g(e)463 b(that,)470 b(as)463 b(a)g(consequence)g(of)g(the)f(condition)h(\(1.9\),)471 b(this)463 b(equation)3432 70237 y(has)484 b(a)h(smo)36 b(oth)485 b(unique)f(solution,)498 b(if)485 b Fy(e)23054 70436 y Fr(0)24064 70237 y Fz(is)g(small)h(enough,)497 b(what)485 b(w)-36 b(e)484 b(shall)i(supp)36 b(ose)3432 71842 y(from)434 b(no)-36 b(w)433 b(on.)24367 75163 y(54)p eop %%Page: 55 55 55 54 bop 3432 7638 a Fn(Lemma)499 b(7.4)650 b Fw(If)556 b Fy(~)-657 b(p)525 b Fz(=)h Fy(\032)-54 b(~)-596 b(e)16725 7837 y Fx(r)17231 7638 y Fz(\()p Fy(\022)36 b Fz(\))525 b Ft(2)g Fy(S)21621 7837 y Fx(h;!)23097 7638 y Fw(,)570 b Fy(h)526 b Ft(\024)g Fz(0)p Fw(,)570 b(and)550 b Fy(x)f Fw(and)h Fy(\022)35631 7837 y Fs(?)36968 7638 y Fw(ar)-66 b(e)549 b(de\014ne)-66 b(d)548 b(as)i(in)3432 9243 y(\(7.15\),)465 b(then)f Ft(j)p Fy(x)p Ft(j)369 b(\024)g Fy(c\015)15193 8761 y Fx(h)16255 9243 y Fw(and)465 b Ft(j)p Fy(\022)19754 9442 y Fs(?)20837 9243 y Ft(\000)295 b Fy(\022)22771 9442 y Fx(h;!)24247 9243 y Ft(j)369 b(\024)g Fy(c\015)27694 8761 y Fx(h=)p Fr(2)29232 9243 y Fw(.)3432 11851 y Fb(Pr)-30 b(oof)503 b(-)935 b Fz(\(7.9\))434 b(and)f(\(7.15\))i(imply)f(that)9380 14403 y Fy(k)10056 14602 y Fr(1)11690 14403 y Fz(=)1106 b([)7 b Fy(~)-657 b(p)14822 14602 y Fx(F)15604 14403 y Fz(\()p Fy(\022)16716 14602 y Fs(?)17503 14403 y Fz(\))295 b Ft(\000)303 b Fy(~)-658 b(p)20285 14602 y Fx(F)21067 14403 y Fz(\()p Fy(\022)22179 14602 y Fx(h;!)23654 14403 y Fz(\)])296 b Ft(\001)256 b Fy(~)-611 b(n)p Fz(\()p Fy(\022)27369 14602 y Fx(h;!)28845 14403 y Fz(\))295 b(+)g Fy(x)-39 b(~)-611 b(n)p Fz(\()p Fy(\022)33580 14602 y Fs(?)34367 14403 y Fz(\))295 b Ft(\001)256 b Fy(~)-610 b(n)p Fz(\()p Fy(\022)37721 14602 y Fx(h;!)39197 14403 y Fz(\))368 b Fy(;)2847 b Fz(\(7.16\))9380 16340 y Fy(k)10056 16539 y Fr(2)11690 16340 y Fz(=)1106 b([)7 b Fy(~)-657 b(p)14822 16539 y Fx(F)15604 16340 y Fz(\()p Fy(\022)16716 16539 y Fs(?)17503 16340 y Fz(\))295 b Ft(\000)303 b Fy(~)-658 b(p)20285 16539 y Fx(F)21067 16340 y Fz(\()p Fy(\022)22179 16539 y Fx(h;!)23654 16340 y Fz(\)])296 b Ft(\001)260 b Fy(~)-615 b(\034)148 b Fz(\()p Fy(\022)27306 16539 y Fx(h;!)28782 16340 y Fz(\))295 b(+)g Fy(x)-39 b(~)-611 b(n)p Fz(\()p Fy(\022)33517 16539 y Fs(?)34304 16340 y Fz(\))295 b Ft(\001)261 b Fy(~)-616 b(\034)149 b Fz(\()p Fy(\022)37595 16539 y Fx(h;!)39070 16340 y Fz(\))369 b Fy(:)2973 b Fz(\(7.17\))3432 18892 y(By)514 b(using)g(the)f(\(7.11\),)535 b(\(7.12\))515 b(and)e(\(7.6\),)535 b(one)514 b(can)g(easily)h(complete)f(the)g(pro)36 b(of)514 b(of)3432 20497 y(the)433 b(lemma.)p 46160 20718 443 1329 v 3432 25897 a FA(7.2)1792 b(Pro)50 b(of)598 b(of)f(Lemma)i(2.1) 3432 28360 y Fz(The)471 b(b)36 b(ounds)469 b(on)i Fy(k)13223 28559 y Fr(1)14220 28360 y Fz(and)g Fy(k)17463 28559 y Fr(2)18460 28360 y Fz(in)g(\(7.9\))g(imply)h(that)29701 27413 y Fl(R)30660 28360 y Fy(d)p Fn(p)p Fy(F)33008 28559 y Fx(h;!)34484 28360 y Fz(\()p Fn(p)p Fz(\))433 b Ft(\024)f Fy(c\015)39531 27878 y Fr(5)p Fx(h=)p Fr(2)41540 28360 y Fz(.)690 b(On)471 b(the)3432 29965 y(other)359 b(hand,)375 b(if)361 b Fy(F)12270 30164 y Fx(h;!)13746 29965 y Fz(\()p Fn(p)p Fz(\))368 b Ft(6)p Fz(=)h(0,)376 b Ft(j)146 b(\000)g Fy(ip)21515 30164 y Fr(0)22185 29965 y Fz(+)g Fy(")p Fz(\()7 b Fy(~)-657 b(p)n Fz(\))146 b Ft(\000)g Fy(\026)p Ft(j)367 b(\025)j Fy(c\015)31169 29483 y Fx(h)31766 29965 y Fz(,)375 b(so)361 b(that)f(\(2.33\))h(implies)g(the)3432 31570 y(b)36 b(ound)496 b Ft(j)p Fy(g)8546 31088 y Fr(\()p Fx(h)p Fr(\))8498 31899 y Fx(!)9875 31570 y Fz(\()p Fn(x)p Fz(\))p Ft(j)478 b(\024)g Fy(c\015)15341 31088 y Fr(3)p Fx(h=)p Fr(2)17350 31570 y Fz(.)771 b(It)498 b(is)g(also)h(v)-36 b(ery)499 b(easy)g(to)f(pro)-36 b(v)g(e)498 b(that)f Ft(j)p Fy(@)39078 31088 y Fx(n)39750 31570 y Fz(^)-696 b Fy(g)40375 31088 y Fr(\()p Fx(h)p Fr(\))40327 31899 y Fx(!)41704 31570 y Fz(\()p Fn(p)p Fz(\))p Fy(=@)72 b(p)45608 31088 y Fx(n)45608 31899 y Fr(0)46234 31570 y Ft(j)3432 33175 y Fz(and)599 b Ft(j)p Fy(@)7255 32693 y Fx(n)7927 33175 y Fz(^)-696 b Fy(g)8552 32693 y Fr(\()p Fx(h)p Fr(\))8504 33504 y Fx(!)9881 33175 y Fz(\()p Fn(p)p Fz(\))p Fy(=@)72 b(k)13853 32693 y Fx(n)13808 33504 y Fr(1)14479 33175 y Ft(j)600 b Fz(are)h(b)36 b(ounded)598 b(b)-36 b(y)600 b Fy(c\015)26540 32693 y Fs(\000)p Fx(h)p Fr(\()p Fx(n)p Fr(+1\))30374 33175 y Fz(.)1078 b(Hence,)642 b(using)600 b(simple)g(in)-36 b(te-)3432 34780 y(gration)473 b(b)-36 b(y)473 b(parts)f(argumen)-36 b(ts,)483 b(one)473 b(can)g(sho)-36 b(w)473 b(that)g Ft(j)p Fy(x)31984 34298 y Fx(n)31984 35109 y Fr(0)32610 34780 y Fy(g)33281 34298 y Fr(\()p Fx(h)p Fr(\))33233 35109 y Fx(!)34610 34780 y Fz(\()p Fn(x)p Fz(\))p Ft(j)435 b(\024)h Fy(c\015)39991 34298 y Fx(h)p Fr(\(3)p Fx(=)p Fr(2)p Fs(\000)p Fx(n)p Fr(\))44508 34780 y Fz(and)3432 36386 y Ft(j)p Fy(x)4540 35904 y Fs(0)4540 36714 y Fr(1)5065 35762 y Fx(n)5692 36386 y Fy(g)6363 35904 y Fr(\()p Fx(h)p Fr(\))6315 36714 y Fx(!)7692 36386 y Fz(\()p Fn(x)p Fz(\))p Ft(j)368 b(\024)h Fy(c\015)12939 35904 y Fx(h)p Fr(\(3)p Fx(=)p Fr(2)p Fs(\000)p Fx(n)p Fr(\))16982 36386 y Fz(.)579 b(Moreo)-36 b(v)g(er,)434 b(it)g(is)g(easy)g(to)g(pro)-36 b(v)g(e)433 b(that)12472 39670 y Ft(j)p Fy(@)13600 39122 y Fx(n)14273 39670 y Fz(^)-696 b Fy(g)14898 39122 y Fr(\()p Fx(h)p Fr(\))14850 39998 y Fx(!)16226 39670 y Fz(\()p Fn(p)p Fz(\))p Fy(=@)72 b(k)20198 39122 y Fx(n)20153 39998 y Fr(2)20825 39670 y Ft(j)369 b(\024)g Fy(c\015)24272 39122 y Fs(\000)p Fx(h)25601 38386 y Fl(h)26124 39670 y Fy(\015)26871 39122 y Fs(\000)p Fx(h)28945 39670 y Fz(sup)28433 40748 y Fx(~)-482 b(p)p Fs(2)p Fx(S)30094 40904 y Fi(h;!)31647 39670 y Ft(j)32149 38771 y Fy(@)72 b(")p Fz(\()7 b Fy(~)-657 b(p)o Fz(\))p 32149 39360 3032 45 v 32684 40581 a Fy(@)72 b(k)34119 40780 y Fr(2)35314 39670 y Ft(j)35683 38386 y Fl(i)36206 38692 y Fx(n)37201 39670 y Fy(;)5717 b Fz(\(7.18\))3432 43347 y(whic)-36 b(h)329 b(implies)g(the)g(b)36 b(ound)328 b Ft(j)p Fy(x)18509 42865 y Fs(0)18509 43676 y Fr(2)19035 42724 y Fx(n)19662 43347 y Fy(g)20333 42865 y Fr(\()p Fx(h)p Fr(\))20285 43676 y Fx(!)21661 43347 y Fz(\()p Fn(x)p Fz(\))p Ft(j)369 b(\024)g Fy(c\015)26909 42865 y Fx(h)p Fr(\(3)p Fx(=)p Fr(2)p Fs(\000)p Fx(n=)p Fr(2\))31893 43347 y Fz(.)544 b(Finally)-108 b(,)351 b(b)-36 b(y)329 b(using)g(Lemma)3432 44952 y(7.3,)565 b(it)538 b(is)g(easy)g(to)g(pro) -36 b(v)g(e)538 b(that)f(the)h(previous)g(b)36 b(ounds)536 b(ha)-36 b(v)g(e)538 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47681 y Fx(m)38880 48163 y Fy(g)39551 47681 y Fr(\()p Fx(h)p Fr(\))39503 48491 y Fx(!)40880 48163 y Fz(\()p Fn(x)p Fz(\))p Fy(=@)72 b(x)44829 47681 y Fs(0)44829 48491 y Fr(2)45354 47539 y Fx(m)46242 48163 y Fz(.)5383 49768 y(The)403 b(b)36 b(ound)403 b(\(2.35\))h(is)g(a)g(simple)g(consequence)f(of)i(the)e (previous)h(considerations.)p 3432 51594 443 1329 v 3432 56773 a FA(7.3)1792 b(The)599 b(parallelogram)f(lemma)3432 59236 y Fz(Let)517 b(us)h(consider)g(the)g(map)g(F,)g(de\014ned)f(on)h (the)g(t)-36 b(w)g(o)518 b(dimensional)h(torus)f Fo(T)42647 58671 y Fr(2)43173 59236 y Fz(,)540 b(with)3432 60841 y(v)-72 b(alues)434 b(in)f Fo(R)9781 60276 y Fr(2)10307 60841 y Fz(,)h(suc)-36 b(h)433 b(that,)g(if)i(\()p Fy(\022)19520 61040 y Fr(1)20045 60841 y Fy(;)221 b(\022)21233 61040 y Fr(2)21760 60841 y Fz(\))369 b Ft(2)f Fo(T)24775 60276 y Fr(2)25734 60841 y Fz(and)28112 60490 y Fy(~)28263 60841 y(b)h Fz(=)g Fy(F)181 b Fz(\()p Fy(\022)32701 61040 y Fr(1)33226 60841 y Fy(;)221 b(\022)34414 61040 y Fr(2)34941 60841 y Fz(\),)434 b(then)18971 63177 y Fy(~)19122 63528 y(b)369 b Fz(=)376 b Fy(~)-657 b(p)22078 63727 y Fx(F)22859 63528 y Fz(\()p Fy(\022)23971 63727 y Fr(1)24497 63528 y Fz(\))295 b(+)302 b Fy(~)-658 b(p)27257 63727 y Fx(F)28039 63528 y Fz(\()p Fy(\022)29151 63727 y Fr(2)29676 63528 y Fz(\))369 b Fy(:)12367 b Fz(\(7.19\))3432 66080 y(The)673 b(di\013eren)-36 b(tial)673 b Fy(J)123 b Fz(\()p Fy(\022)15042 66279 y Fr(1)15568 66080 y Fy(;)221 b(\022)16756 66279 y Fr(2)17283 66080 y Fz(\))673 b(of)h(F)g(is)f(a)h(matrix,)735 b(whose)674 b(columns)f(coincide)h(with)3432 67685 y Fy(s)4045 67203 y Fs(0)4355 67685 y Fz(\()p Fy(\022)5467 67884 y Fr(1)5993 67685 y Fz(\))-35 b Fy(~)-615 b(\034)148 b Fz(\()p Fy(\022)8324 67884 y Fr(1)8849 67685 y Fz(\))489 b(and)f Fy(s)13041 67203 y Fs(0)13352 67685 y Fz(\()p Fy(\022)14464 67884 y Fr(2)14989 67685 y Fz(\))-35 b Fy(~)-615 b(\034)149 b Fz(\()p Fy(\022)17321 67884 y Fr(2)17846 67685 y Fz(\).)744 b(Then)489 b(Lemma)g(7.1)h(implies)f 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Ft(\000)33610 34122 y Fz(\026)33506 34473 y Fy(\022)34112 34672 y Fr(2)34638 34473 y Fz(,)g(so)g(that)g Fy(\036)368 b Fz(=)42466 34122 y(\026)42362 34473 y Fy(\022)42968 34672 y Fr(2)43785 34473 y Ft(\000)45214 34122 y Fz(\026)45110 34473 y Fy(\022)45716 34672 y Fr(1)46242 34473 y Fz(.)5383 36375 y(Let)562 b(us)g(de\014ne)g Fy(\016)14103 36574 y Fx(i)15067 36375 y Fz(=)588 b Fy(\022)17273 36574 y Fx(i)18032 36375 y Ft(\000)19553 36025 y Fz(\026)19448 36375 y Fy(\022)20054 36574 y Fx(i)20430 36375 y Fz(,)595 b Fy(\016)639 b Fz(=)24201 35004 y Fl(q)p 25308 35004 3906 45 v 1371 x Fy(\016)25934 35918 y Fr(2)25884 36665 y(1)26755 36375 y Fz(+)295 b Fy(\016)28688 35918 y Fr(2)28638 36665 y(2)29213 36375 y Fz(;)628 b(then)562 b(w)-36 b(e)562 b(can)h(write,)596 b(b)-36 b(y)562 b(using)3432 38361 y(\(7.19\),)590 b(\(7.3\))559 b(and)f(\(7.4\),)590 b(if)18382 38010 y Fy(~)18533 38361 y(b)380 b Fz(+)336 b Fy(~)-606 b(r)617 b Ft(2)581 b(D)595 b Fz(\(whic)-36 b(h)558 b(is)g(certainly)h (true,)589 b(if)515 b Fy(~)-606 b(r)595 b Fz(is)558 b(small)3432 39966 y(enough\),)12460 43039 y Fy(~)-606 b(r)1143 b Fz(=)16485 42140 y Fy(d)7 b(~)-657 b(p)17814 42339 y Fx(F)18595 42140 y Fz(\()19205 41790 y(\026)19101 42140 y Fy(\022)19707 42339 y Fr(1)20233 42140 y Fz(\))p 16485 42729 4254 45 v 17953 43950 a Fy(d\022)20871 43039 y(\016)21447 43238 y Fr(1)22268 43039 y Fz(+)23708 42140 y Fy(d)7 b(~)-657 b(p)25037 42339 y Fx(F)25818 42140 y Fz(\()26428 41790 y(\026)26324 42140 y Fy(\022)26930 42339 y Fr(2)27456 42140 y Fz(\))p 23708 42729 V 25176 43950 a Fy(d\022)28095 43039 y(\016)28671 43238 y Fr(2)29492 43039 y Fz(+)294 b Fy(O)36 b Fz(\()p Fy(\016)32960 42490 y Fr(2)33486 43039 y Fz(\))369 b(=)14233 45512 y(=)1107 b Fy(\016)16928 45711 y Fr(1)17454 45512 y Fy(s)18067 44964 y Fs(0)18377 45512 y Fz(\()18987 45161 y(\026)18883 45512 y Fy(\022)19489 45711 y Fr(1)20015 45512 y Fz(\))-35 b Fy(~)-615 b(\034)148 b Fz(\()21844 45161 y(\026)21740 45512 y Fy(\022)22346 45711 y Fr(1)22872 45512 y Fz(\))295 b(+)f Fy(\016)25555 45711 y Fr(2)26081 45512 y Fy(s)26694 44964 y Fs(0)27004 45512 y Fz(\()27614 45161 y(\026)27510 45512 y Fy(\022)28116 45711 y Fr(2)28642 45512 y Fz(\))-35 b Fy(~)-615 b(\034)148 b Fz(\()30471 45161 y(\026)30367 45512 y Fy(\022)30973 45711 y Fr(2)31499 45512 y Fz(\))295 b(+)g Fy(O)36 b Fz(\()p Fy(\016)35769 44964 y Fr(2)36294 45512 y Fz(\))369 b Fy(:)5749 b Fz(\(7.25\))3432 47748 y(Let)376 b(us)g(no)-36 b(w)377 b(put)f Fy(\016)12874 47947 y Fx(i)13619 47748 y Fz(=)369 b Fy(\021)48 b(x)16429 47947 y Fx(i)16804 47748 y Fz(,)388 b Fy(r)18139 47947 y Fr(1)19034 47748 y Fz(=)369 b Fy(\021)48 b(\036)58 b Fz(~)-708 b Fy(r)22461 47947 y Fr(1)22986 47748 y Fz(,)388 b Fy(r)24321 47947 y Fr(2)25216 47748 y Fz(=)369 b Fy(\021)106 b Fz(~)-708 b Fy(r)27873 47947 y Fr(2)28398 47748 y Fz(;)396 b(condition)377 b(\(7.23\))h(tak)-36 b(es)377 b(the)g(form)3432 49488 y Ft(j)58 b Fz(~)-708 b Fy(r)4387 49687 y Fr(1)4913 49488 y Ft(j)368 b(\024)i Fy(c)7613 49687 y Fr(1)8524 49488 y Fz(and)385 b Ft(j)58 b Fz(~)-708 b Fy(r)11960 49687 y Fr(2)12486 49488 y Ft(j)369 b(\024)g Fz(1.)563 b(Since,)395 b(b)-36 b(y)386 b(h)-36 b(yp)36 b(othesis,)395 b Fy(\021)417 b Ft(\024)369 b Fy(c)31431 49687 y Fr(2)31957 49488 y Fy(\036)p Fz(,)395 b(the)385 b(condition)41185 49138 y Fy(~)41336 49488 y(b)197 b Fz(+)154 b Fy(~)-606 b(r)405 b Ft(2)368 b(D)3432 51093 y Fz(is)505 b(satis\014ed,)524 b(together)506 b(with)f(\(7.24\),)525 b(if)506 b(and)f(only)i(if)f(the) f(follo)-36 b(wing)508 b(system)d(of)i(t)-36 b(w)g(o)3432 52698 y(equations)434 b(in)f(the)g(unkno)-36 b(wns)433 b Fy(x)19829 52897 y Fr(1)20355 52698 y Fy(;)221 b(x)21676 52897 y Fr(2)22636 52698 y Fz(has)434 b(a)g(unique)f(solution:)8670 55519 y Fy(x)9409 55718 y Fr(2)11042 55519 y Fz(=)19006 54621 y(~)-709 b Fy(r)19533 54820 y Fr(1)20059 54621 y Fy(\036)p 13293 55210 13190 45 v 13293 56443 a(s)13906 56059 y Fs(0)14216 56443 y Fz(\()14826 56093 y(\026)14722 56443 y Fy(\022)15328 56642 y Fr(2)15854 56443 y Fz(\))221 b(sin[)p Fy(\013)8 b Fz(\()19984 56093 y(\026)19880 56443 y Fy(\022)20486 56642 y Fr(1)21013 56443 y Fz(\))295 b Ft(\000)g Fy(\013)8 b Fz(\()24587 56093 y(\026)24483 56443 y Fy(\022)25089 56642 y Fr(2)25616 56443 y Fz(\)])26911 55519 y(+)295 b Fy(O)36 b Fz(\()p Fy(c)30314 55718 y Fr(2)30839 55519 y Fz(\))369 b Fy(;)8670 59217 y(x)9409 59416 y Fr(1)11042 59217 y Fz(=)14329 58319 y(~)-708 b Fy(r)14857 58518 y Fr(2)p 13293 58907 3067 45 v 13293 60141 a Fy(s)13906 59757 y Fs(0)14216 60141 y Fz(\()14826 59791 y(\026)14722 60141 y Fy(\022)15328 60340 y Fr(1)15854 60141 y Fz(\))16788 59217 y Ft(\000)18828 58319 y Fz(~)f Fy(r)19355 58518 y Fr(1)19881 58319 y Fy(\036)221 b Fz(cos)q([)p Fy(\013)8 b Fz(\()24420 57968 y(\026)24316 58319 y Fy(\022)24922 58518 y Fr(1)25449 58319 y Fz(\))295 b Ft(\000)g Fy(\013)8 b Fz(\()29023 57968 y(\026)28919 58319 y Fy(\022)29525 58518 y Fr(2)30052 58319 y Fz(\)])p 18249 58907 13190 45 v 18249 60141 a Fy(s)18862 59757 y Fs(0)19172 60141 y Fz(\()19782 59791 y(\026)19678 60141 y Fy(\022)20284 60340 y Fr(1)20810 60141 y Fz(\))221 b(sin[)p Fy(\013)8 b Fz(\()24940 59791 y(\026)24836 60141 y Fy(\022)25442 60340 y Fr(1)25969 60141 y Fz(\))295 b Ft(\000)g Fy(\013)8 b Fz(\()29543 59791 y(\026)29439 60141 y Fy(\022)30045 60340 y Fr(2)30572 60141 y Fz(\)])31867 59217 y(+)295 b Fy(O)36 b Fz(\()p Fy(\021)48 b Fz(\))294 b(+)h Fy(O)36 b Fz(\()p Fy(c)39603 59416 y Fr(2)40129 59217 y Fz(\))368 b Fy(;)1915 b Fz(\(7.26\))3432 62211 y(where)463 b Fy(\013)8 b Fz(\()p Fy(\022)36 b Fz(\))464 b(is)g(de\014ned)d(as)j(in)f(Lemma)h (7.1)g(and)f Fy(O)36 b Fz(\()p Fy(c)30646 62410 y Fr(2)31171 62211 y Fz(\),)471 b Fy(O)36 b Fz(\()p Fy(\021)48 b Fz(\))463 b(are)g(of)h(second)f(order)3432 63816 y(as)434 b(functions)f(of)h(the) f Fy(x)15096 64015 y Fx(i)15472 63816 y Fz('s.)5383 65421 y(By)513 b(using)g(Lemma)g(7.1,)534 b(w)-36 b(e)513 b(see)g(that)g(the) f(righ)-36 b(t)513 b(sides)g(of)g(\(7.26\))h(are)g(b)36 b(ounded)3432 67026 y(for)474 b Fy(\036)438 b Ft(!)g Fz(0.)700 b(Hence,)484 b(b)-36 b(y)474 b(Dini)h(Theorem,)484 b(\(7.26\))475 b(allo)-36 b(w)475 b(to)g(uniquely)f(determine)f Fy(x)46077 67225 y Fr(1)3432 68632 y Fz(and)c Fy(x)6736 68831 y Fr(2)7732 68632 y Fz(for)h(an)-36 b(y)470 b Fy(\036)431 b(>)g Fz(0,)479 b(giv)-36 b(en)427 b Fy(~)-606 b(r)36 b Fz(,)479 b(if)471 b Fy(\021)517 b Fz(and)470 b Fy(c)26791 68831 y Fr(2)27786 68632 y Fz(are)g(small)h(enough,)479 b(and)469 b Ft(j)p Fy(\016)41827 68831 y Fx(i)42203 68632 y Ft(j)431 b(\024)g Fy(c)45027 68831 y Fr(0)45552 68632 y Fy(\021)48 b Fz(,)3432 70237 y(with)433 b Fy(c)6954 70436 y Fr(0)7913 70237 y Fz(indep)36 b(enden)-36 b(t)432 b(of)i Fy(c)17325 70436 y Fr(2)17851 70237 y Fz(.)p 46160 70458 443 1329 v 24367 75163 a(56)p eop %%Page: 57 57 57 56 bop 3432 7638 a FA(7.4)1792 b(Pro)50 b(of)598 b(of)f(Lemma)i(3.1) e(\(sectors)i(coun)-50 b(ting)599 b(lemma\))3432 10101 y Fz(Let)469 b Fy(h)6547 9619 y Fs(0)6857 10101 y Fy(;)221 b(h;)g(L)472 b Fz(b)36 b(e)469 b(in)-36 b(tegers)470 b(suc)-36 b(h)470 b(that)f Fy(h)23356 9619 y Fs(0)24098 10101 y Ft(\024)431 b Fy(h)g Ft(\024)g Fz(0.)688 b(Giv)-36 b(en)470 b Fy(!)34518 10300 y Fr(1)35475 10101 y Ft(2)430 b Fy(O)37785 10300 y Fx(h)38327 10048 y Fq(0)39151 10101 y Fz(and)573 b(~)-753 b Fy(!)42527 10300 y Fx(i)43334 10101 y Ft(2)430 b Fy(O)45644 10300 y Fx(h)46242 10101 y Fz(,)3432 11706 y Fy(i)368 b Fz(=)h(2)p Fy(;)221 b(:)g(:)g(:)j(L)p Fz(,)420 b(let)d Fy(A)13109 11905 y Fx(h;h)14454 11653 y Fq(0)14809 11706 y Fz(\()p Fy(!)16125 11905 y Fr(1)16650 11706 y Fz(;)326 b(~)-755 b Fy(!)18042 11905 y Fr(2)18569 11706 y Fy(;)221 b(:)g(:)g(:)j(;)325 b Fz(~)-754 b Fy(!)22292 11905 y Fx(L)22987 11706 y Fz(\))417 b(b)36 b(e)417 b(the)f(set)h(of)h (the)e(sequences)h(\()p Fy(!)40792 11905 y Fr(2)41317 11706 y Fy(;)221 b(:)g(:)g(:)j(;)d(!)45040 11905 y Fx(L)45736 11706 y Fz(\),)3432 13311 y(suc)-36 b(h)441 b(that)884 b(i\))443 b Fy(S)11753 13510 y Fx(h)12295 13258 y Fq(0)12593 13510 y Fx(;!)13439 13645 y Fi(i)14230 13311 y Ft(\032)384 b Fy(S)16447 13510 y Fx(h;)73 b Fr(~)-543 b Fx(!)17835 13645 y Fi(i)18685 13311 y Fz(for)443 b Fy(i)383 b Fz(=)h(2)p Fy(;)221 b(:)g(:)g(:)j(;)d(L)p Fz(;)894 b(ii\))443 b(there)e(exists,) 446 b(for)d Fy(i)384 b Fz(=)f(1)p Fy(;)221 b(:)g(:)g(:)j(;)d(L)p Fz(,)3432 15091 y(a)433 b(v)-36 b(ector)8351 14740 y Fy(~)8417 15091 y(k)9138 14609 y Fr(\()p Fx(i)p Fr(\))10615 15091 y Ft(2)369 b Fy(S)12670 15290 y Fx(h)13212 15038 y Fq(0)13510 15290 y Fx(;!)14356 15425 y Fi(i)14763 15091 y Fz(,)434 b(so)g(that)19973 14205 y Fl(P)21141 14556 y Fx(L)21141 15419 y(i)p Fr(=1)22874 14740 y Fy(~)22941 15091 y(k)23662 14609 y Fr(\()p Fx(i)p Fr(\))25138 15091 y Fz(=)369 b(0.)3834 16696 y(If)402 b Fy(L)369 b Fz(=)g(2,)409 b(the)402 b(momen)-36 b(tum)401 b(conserv)-72 b(ation)402 b(ii\))h(and)e(the)h(symmetry)g(prop)36 b(ert)-36 b(y)402 b(\(1.11\))3432 18301 y(immediately)439 b(imply)f(that)g Ft(j)p Fy(A)18663 18500 y Fx(h;h)20008 18248 y Fq(0)20362 18301 y Fz(\()p Fy(!)21678 18500 y Fr(1)22204 18301 y Fz(;)325 b(~)-754 b Fy(!)23596 18500 y Fr(2)24122 18301 y Fy(;)221 b(:)g(:)g(:)j(;)325 b Fz(~)-754 b Fy(!)27845 18500 y Fx(L)28541 18301 y Fz(\))p Ft(j)376 b Fz(=)g(1.)592 b(Hence,)439 b(in)f(order)g(to)g(pro)-36 b(v)g(e)3432 19906 y(Lemma)433 b(3.1)i(it)e(is)h(su\016cien)-36 b(t)433 b(to)h(consider)f(the)g(case)h Fy(L)369 b Ft(\025)g Fz(4;)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b(to)g(pro)-36 b(v)g(e)433 b(that)14319 23180 y Ft(j)p Fy(A)15663 23379 y Fx(h;h)17008 23127 y Fq(0)17362 23180 y Fz(\()p Fy(!)18678 23379 y Fr(1)19204 23180 y Fz(;)325 b(~)-754 b Fy(!)20596 23379 y Fr(2)21122 23180 y Fy(;)221 b(:)g(:)g(:)j(;)325 b Fz(~)-754 b Fy(!)24845 23379 y Fx(L)25541 23180 y Fz(\))p Ft(j)369 b(\024)g Fy(c)28747 22631 y Fx(L)29442 23180 y Fy(\015)30322 22251 y Fi(h)p Fq(\000)p Fi(h)31926 21972 y Fq(0)p 30321 22432 1902 45 v 31069 22978 a Fp(2)32356 22631 y Fr(\()p Fx(L)p Fs(\000)p Fr(3\))35354 23180 y Fy(:)7564 b Fz(\(7.27\))5383 26113 y(Let)471 b Fy(\022)8357 26312 y Fx(i)9168 26113 y Ft(\021)435 b Fy(\022)11242 26312 y Fx(h)11784 26060 y Fq(0)12083 26312 y Fx(;!)12929 26447 y Fi(i)13336 26113 y Fz(,)482 b(so)473 b(that)f Fy(\022)19278 26312 y Fx(i)20126 26113 y Fz(is)h(the)f(cen)-36 b(ter)471 b(of)i(the)f Fy(\022)36 b Fz(-in)-36 b(terv)-72 b(al,)483 b(whic)-36 b(h)472 b(the)g(p)36 b(olar)3432 27718 y(angle)434 b(of)442 b Fy(~)-658 b(p)433 b Fz(has)h(to)g(b)36 b(elong)434 b(to,)g(if)441 b Fy(~)-657 b(p)369 b Ft(2)f Fy(S)23679 27917 y Fx(h)24221 27665 y Fq(0)24520 27917 y Fx(;!)25366 28052 y Fi(i)25773 27718 y Fz(.)578 b(F)-108 b(or)433 b(an)-36 b(y)434 b(pair)g(\()p Fy(i;)221 b(j)75 b Fz(\),)433 b(w)-36 b(e)434 b(de\014ne)14440 30651 y Fy(\036)15210 30850 y Fx(i;j)16647 30651 y Fz(=)369 b(min)p Ft(fjj)p Fy(\022)22204 30850 y Fx(i)22875 30651 y Ft(\000)295 b Fy(\022)24809 30850 y Fx(j)25296 30651 y Ft(jj)p Fy(;)221 b(\031)344 b Ft(\000)295 b(jj)p Fy(\022)30370 30850 y Fx(i)31041 30651 y Ft(\000)g Fy(\022)32975 30850 y Fx(j)33462 30651 y Ft(jjg)369 b Fy(:)7685 b Fz(\(7.28\))3432 33585 y(By)393 b(a)g(reordering)g(of)h(the)e(sectors,)402 b(whic)-36 b(h)392 b(is)i(unimp)36 b(ortan)-36 b(t)391 b(since)i(w)-36 b(e)394 b(are)f(lo)36 b(oking)394 b(for)3432 35190 y(a)433 b(b)36 b(ound)433 b(prop)36 b(ortional)434 b(to)f Fy(c)18154 34708 y Fx(L)18849 35190 y Fz(,)h(w)-36 b(e)434 b(can)g(get)f(the)g (condition)h(\(recall)g(that)f Fy(L)369 b Ft(\025)g Fz(4\):)14780 38123 y Fy(\036)g Ft(\021)g Fy(\036)18091 38322 y Fx(L)p Fs(\000)p Fr(1)p Fx(;L)21258 38123 y Ft(\025)h Fy(\036)23431 38322 y Fx(i;j)25800 38123 y Fy(;)1522 b Ft(8)p Fy(i;)221 b(j)444 b Ft(2)369 b Fz([2)p Fy(;)221 b(L)p Fz(])370 b Fy(:)8025 b Fz(\(7.29\))5383 41057 y(Note)433 b(that,)h(giv)-36 b(en)538 b(~)-754 b Fy(!)417 b Ft(2)368 b Fy(O)18576 41256 y Fx(h)19174 41057 y Fz(,)15878 44330 y Ft(j)p Fy(!)417 b Ft(2)368 b Fy(O)19722 44529 y Fx(h)20264 44278 y Fq(0)20987 44330 y Fz(:)h Fy(S)22517 44529 y Fx(h)23059 44278 y Fq(0)23358 44529 y Fx(;!)24660 44330 y Ft(\032)g Fy(S)26862 44529 y Fx(h;)73 b Fr(~)-543 b Fx(!)28338 44330 y Ft(j)369 b Fz(=)g Fy(\015)31337 43402 y Fi(h)p Fq(\000)p Fi(h)32941 43123 y Fq(0)p 31336 43583 V 32084 44129 a Fp(2)33795 44330 y Fy(:)9123 b Fz(\(7.30\))3432 47264 y(Hence,)433 b(giv)-36 b(en)434 b(an)-36 b(y)434 b(p)36 b(ositiv)-36 b(e)435 b(constan)-36 b(t)433 b Fy(c)24130 47463 y Fr(0)24655 47264 y Fz(,)h(if)g(w)-36 b(e)434 b(de\014ne)5969 50197 y Ft(A)7030 50396 y Fx(<)8186 50197 y Fz(=)368 b Ft(f)p Fz(\()p Fy(!)11546 50396 y Fr(2)12072 50197 y Fy(;)221 b(:)g(:)g(:)j(;)d(!)15795 50396 y Fx(L)16490 50197 y Fz(\))369 b Ft(2)g Fy(A)19595 50396 y Fx(h;h)20940 50144 y Fq(0)21294 50197 y Fz(\()p Fy(!)22610 50396 y Fr(1)23136 50197 y Fy(;)325 b Fz(~)-754 b Fy(!)24528 50396 y Fr(2)25054 50197 y Fy(;)221 b(:)g(:)g(:)j(;)325 b Fz(~)-754 b Fy(!)28777 50396 y Fx(L)29473 50197 y Fz(\))590 b(:)h Fy(\036)368 b Ft(\024)h Fy(Lc)35506 49649 y Fs(\000)p Fr(1)35506 50526 y(0)36764 50197 y Fy(\015)37511 49649 y Fx(h)38053 49336 y Fq(0)38351 49649 y Fx(=)p Fr(2)39348 50197 y Ft(g)g Fy(;)2537 b Fz(\(7.31\))3432 53131 y(w)-36 b(e)433 b(ha)-36 b(v)g(e:)17302 54736 y Ft(jA)18732 54935 y Fx(<)19519 54736 y Ft(j)369 b(\024)g Fy(\015)22539 53807 y Fi(h)p Fq(\000)p Fi(h)24143 53529 y Fq(0)p 22539 53988 V 23287 54535 a Fp(2)24573 54187 y Fr(\()p Fx(L)p Fs(\000)p Fr(3\))27203 54736 y Fz(\()p Fy(cLc)29714 54187 y Fs(\000)p Fr(1)29714 55064 y(0)30971 54736 y Fz(\))31477 54187 y Fr(2)32371 54736 y Fy(;)10547 b Fz(\(7.32\))3432 57061 y(where)378 b(\()p Fy(cLc)9645 56511 y Fs(\000)p Fr(1)9645 57351 y(0)10902 57061 y Fz(\))11408 56578 y Fr(2)12312 57061 y Fz(is)h(a)g(b)36 b(ound)377 b(on)h(the)g(n)-36 b(um)g(b)36 b(er)377 b(of)j(p)36 b(ossible)378 b(c)-36 b(hoices)379 b(of)h Fy(!)39987 57260 y Fx(L)p Fs(\000)p Fr(1)42263 57061 y Fz(and)e Fy(!)45547 57260 y Fx(L)46242 57061 y Fz(,)3432 58666 y(giv)-36 b(en)489 b Fy(!)7693 58865 y Fr(1)8218 58666 y Fy(;)221 b(:)g(:)g(:)j(;)d(!)11941 58865 y Fx(L)p Fs(\000)p Fr(2)13839 58666 y Fz(.)743 b(Hence,)502 b(in)488 b(order)g(to)g(pro)-36 b(v)g(e)489 b(\(7.27\),)503 b(it)488 b(is)h(su\016cien)-36 b(t)488 b(to)g(pro)-36 b(v)g(e)3432 60271 y(that,)433 b(if)h Fy(c)8363 60470 y Fr(0)9322 60271 y Fz(is)g(small)h(enough,)e(a)h (similar)g(b)36 b(ound)433 b(is)h(v)-72 b(alid)434 b(for)g(the)f(set) 5969 63204 y Ft(A)7030 63403 y Fx(>)8186 63204 y Fz(=)368 b Ft(f)p Fz(\()p Fy(!)11546 63403 y Fr(2)12072 63204 y Fy(;)221 b(:)g(:)g(:)j(;)d(!)15795 63403 y Fx(L)16490 63204 y Fz(\))369 b Ft(2)g Fy(A)19595 63403 y Fx(h;h)20940 63151 y Fq(0)21294 63204 y Fz(\()p Fy(!)22610 63403 y Fr(1)23136 63204 y Fy(;)325 b Fz(~)-754 b Fy(!)24528 63403 y Fr(2)25054 63204 y Fy(;)221 b(:)g(:)g(:)j(;)325 b Fz(~)-754 b Fy(!)28777 63403 y Fx(L)29473 63204 y Fz(\))590 b(:)h Fy(\036)368 b Ft(\025)h Fy(Lc)35506 62656 y Fs(\000)p Fr(1)35506 63533 y(0)36764 63204 y Fy(\015)37511 62656 y Fx(h)38053 62343 y Fq(0)38351 62656 y Fx(=)p Fr(2)39348 63204 y Ft(g)g Fy(:)2537 b Fz(\(7.33\))5383 66138 y(W)-108 b(e)433 b(ha)-36 b(v)g(e)18785 67743 y Ft(jA)20215 67942 y Fx(>)21002 67743 y Ft(j)369 b(\024)g Fy(m)24280 67942 y Fx(L)24975 67743 y Fy(\015)25855 66814 y Fi(h)p Fq(\000)p Fi(h)27459 66535 y Fq(0)p 25855 66995 V 26603 67541 a Fp(2)27890 67194 y Fr(\()p Fx(L)p Fs(\000)p Fr(3\))30888 67743 y Fy(;)12030 b Fz(\(7.34\))3432 70067 y(where)692 b Fy(m)8586 70266 y Fx(L)9974 70067 y Fz(is)h(a)g(b)36 b(ound)691 b(on)i(the)f(n)-36 b(um)g(b)36 b(er)691 b(of)i(c)-36 b(hoices)693 b(of)g Fy(!)35638 70266 y Fx(L)p Fs(\000)p Fr(1)38228 70067 y Fz(and)g Fy(!)41827 70266 y Fx(L)42522 70067 y Fz(,)757 b(giv)-36 b(en)3432 71672 y Fy(!)4242 71871 y Fr(1)4767 71672 y Fy(;)221 b(:)g(:)g(:)j(;)d(!)8490 71871 y Fx(L)p Fs(\000)p Fr(2)10388 71672 y Fz(.)24367 75163 y(57)p eop %%Page: 58 58 58 57 bop 5383 7638 a Fz(In)427 b(order)g(to)h(get)f Fy(m)15275 7837 y Fx(L)15970 7638 y Fz(,)i(w)-36 b(e)428 b(consider)g(a)f(particular)h(c)-36 b(hoice)428 b(of)g Fy(!)36943 7837 y Fr(2)37469 7638 y Fy(;)221 b(:)g(:)g(:)j(;)d(!)41192 7837 y Fx(L)p Fs(\000)p Fr(2)43459 7638 y Ft(2)368 b Fy(O)45707 7837 y Fx(h)46249 7585 y Fq(0)3432 9243 y Fz(and)530 b(w)-36 b(e)532 b(supp)36 b(ose)530 b(that)h(the)f(set)h Ft(E)654 b Fz(=)534 b Ft(f)p Fz(\()p Fy(!)25322 9442 y Fx(L)p Fs(\000)p Fr(1)27220 9243 y Fy(;)221 b(!)28612 9442 y Fx(L)29308 9243 y Fz(\))535 b(:)g(\()p Fy(!)32561 9442 y Fr(2)33086 9243 y Fy(;)221 b(:)g(:)g(:)j(;)531 b(!)37119 9442 y Fx(L)37815 9243 y Fz(\))j Ft(2)h(A)41337 9442 y Fx(>)42124 9243 y Ft(g)c Fz(is)g(not)3432 10848 y(empt)-36 b(y)-108 b(.)577 b(Moreo)-36 b(v)g(er,)435 b(w)-36 b(e)433 b(de\014ne)18413 13782 y Fy(\036)19183 13981 y Fr(0)20077 13782 y Fz(=)1980 b(max)21458 14630 y Fr(\()p Fx(!)22409 14779 y Fi(L)p Fq(\000)p Fp(1)24072 14630 y Fx(;!)24918 14779 y Fi(L)25535 14630 y Fr(\))p Fs(2E)27323 13782 y Fy(\036)28093 13981 y Fx(L)p Fs(\000)p Fr(1)p Fx(;L)31261 13782 y Fy(;)11657 b Fz(\(7.35\))5383 17417 y(By)568 b(de\014nition,)601 b(for)568 b(an)-36 b(y)568 b(c)-36 b(hoice)568 b(of)h(\()p Fy(!)25512 17616 y Fx(L)p Fs(\000)p Fr(1)27409 17417 y Fy(;)221 b(!)28801 17616 y Fx(L)29497 17417 y Fz(\))597 b Ft(2)g(E)119 b Fz(,)601 b(w)-36 b(e)569 b(can)e(\014nd)g Fy(L)h Fz(v)-36 b(ectors)3365 18807 y Fy(~)3432 19158 y(k)4153 18675 y Fr(\(1\))5410 19158 y Fy(;)221 b(:)g(:)g(:)j(;)8257 18807 y(~)8323 19158 y(k)9044 18675 y Fr(\()p Fx(L)p Fr(\))10472 19158 y Fz(,)434 b(suc)-36 b(h)432 b(that)16952 18807 y Fy(~)17019 19158 y(k)17740 18675 y Fr(\()p Fx(i)p Fr(\))19216 19158 y Ft(2)369 b Fy(S)21271 19357 y Fx(h)21813 19105 y Fq(0)22111 19357 y Fx(;!)22957 19492 y Fi(i)23798 19158 y Fz(and)22107 21684 y Fx(L)21628 22016 y Fl(X)21666 24446 y Fx(i)p Fr(=1)23381 22772 y Fy(~)23448 23123 y(k)24169 22574 y Fr(\()p Fx(i)p Fr(\))25646 23123 y Fz(=)f(0)i Fy(:)14872 b Fz(\(7.36\))3432 26919 y(Moreo)-36 b(v)g(er,)434 b(b)-36 b(y)433 b(Lemma)h(7.3,)h(for)f Fy(i)368 b Fz(=)h(1)p Fy(;)221 b(:)g(:)g(:)j(;)d(L)p Fz(,)435 b(w)-36 b(e)433 b(can)h(write)4589 29501 y Fy(~)4656 29852 y(k)5377 29304 y Fr(\()p Fx(i)p Fr(\))6854 29852 y Fz(=)376 b Fy(~)-658 b(p)8887 30051 y Fx(F)9669 29852 y Fz(\()p Fy(\022)10781 30051 y Fx(i)11156 29852 y Fz(\))295 b(+)g Fy(x)14003 30051 y Fx(i)14340 29852 y Fy(~)-611 b(n)p Fz(\()p Fy(\022)16267 30051 y Fx(i)16643 29852 y Fz(\))295 b(+)g Fy(y)19385 30051 y Fx(i)19726 29852 y Fy(~)-615 b(\034)148 b Fz(\()p Fy(\022)21586 30051 y Fx(i)21962 29852 y Fz(\))1300 b Fy(;)1522 b Ft(j)p Fy(x)26759 30051 y Fx(i)27135 29852 y Ft(j)369 b(\024)g Fy(c\015)30582 29304 y Fx(h)31124 28991 y Fq(0)32778 29852 y Fy(;)1523 b Ft(j)p Fy(y)35665 30051 y Fx(i)36041 29852 y Ft(j)369 b(\024)g Fy(c\015)39488 29304 y Fx(h)40030 28991 y Fq(0)40328 29304 y Fx(=)p Fr(2)41693 29852 y Fy(:)1225 b Fz(\(7.37\))3432 32785 y(Hence,)433 b(since)h Fy(\036)11615 32984 y Fr(0)12509 32785 y Ft(\025)370 b Fy(Lc)15357 32236 y Fs(\000)p Fr(1)15357 33076 y(0)16614 32785 y Fy(\015)17361 32303 y Fx(h)17903 31991 y Fq(0)18201 32303 y Fx(=)p Fr(2)19198 32785 y Fz(,)434 b(w)-36 b(e)433 b(get)7436 36073 y Ft(j)7738 35722 y Fy(~)7805 36073 y(k)8526 35524 y Fr(\()p Fx(i)p Fr(\))9929 36073 y Ft(^)302 b Fy(~)-657 b(p)11763 36272 y Fx(F)12544 36073 y Fz(\()p Fy(\022)13656 36272 y Fr(2)14182 36073 y Fz(\))p Ft(j)368 b Fz(=)h Ft(j)7 b Fy(~)-657 b(p)17828 36272 y Fx(F)18609 36073 y Fz(\()p Fy(\022)19721 36272 y Fx(i)20097 36073 y Fz(\))p Ft(j)221 b(j)7 b Fy(~)-657 b(p)22215 36272 y Fx(F)22996 36073 y Fz(\()p Fy(\022)24108 36272 y Fr(2)24634 36073 y Fz(\))p Ft(j)221 b Fz(sin)h Fy(\036)28319 36272 y Fx(i;)p Fr(2)29721 36073 y Fz(+)295 b Fy(O)36 b Fz(\()p Fy(\015)33444 35162 y Fi(h)33926 34884 y Fq(0)p 33444 35325 781 45 v 33631 35872 a Fp(2)34412 36073 y Fz(\))369 b Ft(\024)g Fy(c\036)38019 36272 y Fr(0)38913 36073 y Fy(;)4005 b Fz(\(7.38\))3432 39006 y(for)434 b Fy(i)368 b Fz(=)h(2)p Fy(;)221 b(:)g(:)g(:)j(;)d(L)p Fz(,)435 b(and,)e(b)-36 b(y)433 b(using)h(\(7.36\),)12916 42932 y Ft(j)13218 42582 y Fy(~)13285 42932 y(k)14006 42384 y Fr(\(1\))15559 42932 y Ft(^)302 b Fy(~)-657 b(p)17393 43131 y Fx(F)18174 42932 y Fz(\()p Fy(\022)19286 43131 y Fr(2)19812 42932 y Fz(\))p Ft(j)369 b Fz(=)22436 40940 y Fl(\014)22436 41604 y(\014)22436 42268 y(\014)22436 42932 y(\014)22436 43597 y(\014)23285 41493 y Fx(L)22805 41825 y Fl(X)22843 44256 y Fx(i)p Fr(=2)24559 42582 y Fy(~)24625 42932 y(k)25346 42384 y Fr(\()p Fx(i)p Fr(\))26749 42932 y Ft(^)302 b Fy(~)-657 b(p)28583 43131 y Fx(F)29364 42932 y Fz(\()p Fy(\022)30476 43131 y Fr(2)31002 42932 y Fz(\))31508 40940 y Fl(\014)31508 41604 y(\014)31508 42268 y(\014)31508 42932 y(\014)31508 43597 y(\014)32246 42932 y Ft(\024)369 b Fy(cL\036)35863 43131 y Fr(0)36757 42932 y Fy(;)6161 b Fz(\(7.39\))3432 46728 y(so)434 b(that)f Fy(\036)8617 46927 y Fr(1)p Fx(;)p Fr(2)10243 46728 y Ft(\024)369 b Fy(cL\036)13860 46927 y Fr(0)14385 46728 y Fz(.)5383 48333 y(Lemma)433 b(7.1,)i(\(7.37\))f(and)f(\(7.39\))i (easily)g(imply)f(that)9049 51051 y Fy(~)9116 51402 y(k)9837 50853 y Fr(\()p Fx(i)p Fr(\))11313 51402 y Fz(=)376 b Fy(~)-657 b(p)13347 51601 y Fx(F)14128 51402 y Fz(\()p Fy(\022)15240 51601 y Fx(i)15616 51402 y Fz(\))295 b(+)375 b(\026)-731 b Fy(x)18462 51601 y Fx(i)18799 51402 y Fy(~)-611 b(n)q Fz(\()p Fy(\022)20727 51601 y Fr(2)21252 51402 y Fz(\))295 b(+)383 b(\026)-738 b Fy(y)23994 51601 y Fx(i)24336 51402 y Fy(~)-616 b(\034)148 b Fz(\()p Fy(\022)26195 51601 y Fr(2)26721 51402 y Fz(\))p Fy(;)9116 54169 y Ft(j)81 b Fz(\026)-731 b Fy(x)10224 54368 y Fx(i)10599 54169 y Ft(j)369 b(\024)12739 52221 y Fl(\()13853 53487 y Fy(c\036)15183 53686 y Fr(0)15708 53487 y Fy(\015)16455 53005 y Fx(h)16997 52692 y Fq(0)17295 53005 y Fx(=)p Fr(2)20477 53487 y Fz(if)435 b Fy(i)368 b(>)h Fz(1)13853 55092 y Fy(cL\036)16068 55291 y Fr(0)16593 55092 y Fy(\015)17340 54610 y Fx(h)17882 54298 y Fq(0)18180 54610 y Fx(=)p Fr(2)20477 55092 y Fz(if)435 b Fy(i)368 b Fz(=)h(1)26168 54169 y Fy(;)1522 b Ft(j)88 b Fz(\026)-738 b Fy(y)29054 54368 y Fx(i)29430 54169 y Ft(j)369 b(\024)31570 52221 y Fl(\()32683 53487 y Fy(c\015)33990 53005 y Fx(h)34532 52692 y Fq(0)34831 53005 y Fx(=)p Fr(2)38013 53487 y Fz(if)434 b Fy(i)369 b(>)f Fz(1)32683 55092 y Fy(cL\015)34875 54610 y Fx(h)35417 54298 y Fq(0)35716 54610 y Fx(=)p Fr(2)38013 55092 y Fz(if)434 b Fy(i)369 b Fz(=)f(1)42772 54169 y Fy(:)146 b Fz(\(7.40\))5383 57859 y(Let)433 b(us)g(no)-36 b(w)434 b(de\014ne)7814 61453 y Fy(~)-564 b(a)369 b Fz(=)f Ft(\000)11586 60014 y Fx(L)p Fs(\000)p Fr(2)11708 60346 y Fl(X)11747 62776 y Fx(i)p Fr(=1)13658 61453 y Fy(~)-658 b(p)14303 61652 y Fx(F)15085 61453 y Fz(\()p Fy(\022)16197 61652 y Fx(i)16572 61453 y Fz(\))1301 b Fy(;)20111 61102 y(~)20262 61453 y(b)369 b Fz(=)22498 61102 y Fy(~)22565 61453 y(k)23286 60774 y Fr(\()p Fx(L)p Fs(\000)p Fr(1\))23241 61787 y Fs(?)26210 61453 y Fz(+)27450 61102 y Fy(~)27517 61453 y(k)28238 60774 y Fr(\()p Fx(L)p Fr(\))28193 61787 y Fs(?)30966 61453 y Fy(;)1478 b(~)-606 b(r)406 b Fz(=)35071 61102 y Fy(~)35221 61453 y(b)295 b Ft(\000)209 b Fy(~)-563 b(a)368 b(;)4469 b Fz(\(7.41\))3432 65623 y(where)7159 65273 y Fy(~)7226 65623 y(k)7902 65822 y Fs(?)9160 65623 y Fz(denotes)470 b(the)h(pro)72 b(jection)471 b(on)f(the)g(F)-108 b(ermi)470 b(surface,)481 b(see)471 b(\(7.15\).)690 b(By)471 b(using)3432 67228 y(Lemma)433 b(7.4,)i(the)e(momen)-36 b(tum)432 b(conserv)-72 b(ation)434 b(\(7.36\))h(and)e(\(7.40\),)i(w) -36 b(e)434 b(get)5478 70162 y Fy(~)-606 b(r)405 b Fz(=)369 b Fy(r)8480 70361 y Fr(1)8966 70162 y Fy(~)-611 b(n)q Fz(\()p Fy(\022)10894 70361 y Fr(2)11420 70162 y Fz(\))295 b(+)f Fy(r)14113 70361 y Fr(2)14605 70162 y Fy(~)-616 b(\034)149 b Fz(\()p Fy(\022)16465 70361 y Fr(2)16990 70162 y Fz(\))1301 b Fy(;)1522 b Ft(j)p Fy(r)21635 70361 y Fr(1)22161 70162 y Ft(j)369 b(\024)g Fy(cL\036)26516 70361 y Fr(0)27042 70162 y Fy(\015)27789 69613 y Fx(h)28331 69301 y Fq(0)28629 69613 y Fx(=)p Fr(2)30926 70162 y Fy(;)1522 b Ft(j)p Fy(r)33764 70361 y Fr(2)34290 70162 y Ft(j)369 b(\024)g Fy(cL\015)38622 69613 y Fx(h)39164 69301 y Fq(0)39462 69613 y Fx(=)p Fr(2)40828 70162 y Fy(:)2090 b Fz(\(7.42\))24367 75163 y(58)p eop %%Page: 59 59 59 58 bop 5383 7638 a Fz(Note)694 b(no)-36 b(w)695 b(that)e(the)h(v)-36 b(ector)609 b Fy(~)-564 b(a)694 b Fz(de\014ned)f(in)h(\(7.41\))h(is)g (\014xed,)759 b(if)695 b(the)f(indices)3432 9424 y Fy(!)4242 9623 y Fr(1)4767 9424 y Fy(;)221 b(:)g(:)g(:)j(;)d(!)8490 9623 y Fx(L)p Fs(\000)p Fr(2)10972 9424 y Fz(are)584 b(\014xed.)1030 b(Hence,)622 b(if)585 b(w)-36 b(e)584 b(put)590 b Fy(~)-657 b(p)28402 9623 y Fx(F)29183 9424 y Fz(\()29793 9073 y(\026)29689 9424 y Fy(\022)30295 9623 y Fx(i)30671 9424 y Fz(\))625 b(=)33372 9073 y Fy(~)33439 9424 y(k)34160 8745 y Fr(\()p Fx(i)p Fr(\))34115 9758 y Fs(?)35267 9424 y Fz(,)d Fy(i)j Fz(=)g Fy(L)397 b Ft(\000)h Fz(1)p Fy(;)221 b(L)p Fz(,)623 b Fy(m)45908 9623 y Fx(L)3432 11029 y Fz(can)433 b(b)36 b(e)433 b(calculated)i(b)-36 b(y)433 b(studying)g(the)g(p)36 b(ossible)434 b(solutions)g(of)g(the)f (equation)17182 13624 y Fy(~)-658 b(p)17827 13823 y Fx(F)18609 13624 y Fz(\()19219 13273 y(\026)19115 13624 y Fy(\022)19721 13823 y Fx(L)p Fs(\000)p Fr(1)21618 13624 y Fz(\))295 b(+)302 b Fy(~)-657 b(p)24379 13823 y Fx(F)25160 13624 y Fz(\()25770 13273 y(\026)25666 13624 y Fy(\022)26272 13823 y Fx(L)26967 13624 y Fz(\))369 b(=)283 b Fy(~)-564 b(a)295 b Fz(+)251 b Fy(~)-606 b(r)405 b(;)10419 b Fz(\(7.43\))3432 16399 y(as)310 b Fy(~)-606 b(r)391 b Fz(v)-72 b(aries)355 b(satisfying)h(\(7.42\).)553 b(Let)354 b(\()22317 16049 y(\026)22213 16399 y Fy(\022)22855 15721 y Fr(\(0\))22819 16724 y Fx(L)p Fs(\000)p Fr(1)24717 16399 y Fy(;)25404 16049 y Fz(\026)25299 16399 y Fy(\022)25941 15721 y Fr(\(0\))25905 16724 y Fx(L)27199 16399 y Fz(\))h(b)36 b(e)354 b(a)h(particular)f (solution)h(of)g(\(7.43\),)3432 18270 y(suc)-36 b(h)306 b(that)8865 17920 y Fy(~)8931 18270 y(k)9652 17788 y Fr(\()p Fx(i)p Fr(\))11129 18270 y Ft(2)368 b Fy(S)13183 18469 y Fx(h)13725 18217 y Fq(0)14024 18469 y Fx(;!)14870 18604 y Fi(i)15277 18270 y Fz(,)333 b Fy(i)369 b Fz(=)f Fy(L)38 b Ft(\000)g Fz(1)p Fy(;)221 b(L)p Fz(,)334 b(with)307 b Fy(\036)26576 18469 y Fx(L)p Fs(\000)p Fr(1)p Fx(;L)29744 18270 y Fz(=)369 b Fy(\036)31895 18469 y Fr(0)32420 18270 y Fz(,)333 b(and)307 b(put)37626 17920 y Fy(~)37776 18270 y(b)38329 18469 y Fr(0)39223 18270 y Fz(=)377 b Fy(~)-658 b(p)41257 18469 y Fx(F)42038 18270 y Fz(\()42648 17920 y(\026)42544 18270 y Fy(\022)43186 17591 y Fr(\(0\))43150 18595 y Fx(L)p Fs(\000)p Fr(1)45048 18270 y Fz(\))38 b(+)3439 20141 y Fy(~)-657 b(p)4085 20340 y Fx(F)4866 20141 y Fz(\()5476 19790 y(\026)5372 20141 y Fy(\022)6014 19462 y Fr(\(0\))5978 20466 y Fx(L)7272 20141 y Fz(\))657 b(=)572 b Fy(~)-564 b(a)410 b Fz(+)366 b Fy(~)-606 b(r)13206 20340 y Fr(0)13732 20141 y Fz(,)646 b(so)604 b(that)f(\(7.43\))h(can)f (b)36 b(e)603 b(written)g(as)611 b Fy(~)-657 b(p)35057 20340 y Fx(F)35838 20141 y Fz(\()36448 19790 y(\026)36344 20141 y Fy(\022)36950 20340 y Fx(L)p Fs(\000)p Fr(1)38847 20141 y Fz(\))411 b(+)418 b Fy(~)-658 b(p)41839 20340 y Fx(F)42621 20141 y Fz(\()43231 19790 y(\026)43127 20141 y Fy(\022)43733 20340 y Fx(L)44428 20141 y Fz(\))657 b(=)3281 21531 y Fy(~)3432 21881 y(b)3985 22080 y Fr(0)4884 21881 y Fz(+)374 b(\()-44 b Fy(~)-606 b(r)410 b Ft(\000)331 b Fy(~)-607 b(r)9765 22080 y Fr(0)10291 21881 y Fz(\).)926 b(The)550 b(de\014nition)e(of)i Fy(\036)23137 22080 y Fr(0)24212 21881 y Fz(implies)g(that)505 b Fy(~)-606 b(r)410 b Ft(\000)331 b Fy(~)-606 b(r)34667 22080 y Fr(0)35742 21881 y Fz(can)549 b(b)36 b(e)550 b(represen)-36 b(ted)3432 23667 y(as)480 b Fy(~)-606 b(r)394 b Ft(\000)313 b Fy(~)-606 b(r)8075 23866 y Fr(0)9125 23667 y Fz(=)523 b Fy(r)11282 23185 y Fs(0)11246 23995 y Fr(1)11733 23667 y Fy(~)-611 b(n)p Fz(\()13158 23316 y(\026)13054 23667 y Fy(\022)13696 22988 y Fr(\(0\))13660 23992 y Fx(L)14954 23667 y Fz(\))357 b(+)f Fy(r)17807 23185 y Fs(0)17771 23995 y Fr(2)18263 23667 y Fy(~)-616 b(\034)149 b Fz(\()19621 23316 y(\026)19517 23667 y Fy(\022)20159 22988 y Fr(\(0\))20123 23992 y Fx(L)21416 23667 y Fz(\),)548 b(with)524 b Ft(j)p Fy(r)26875 23185 y Fs(0)26839 23995 y Fr(1)27365 23667 y Ft(j)g(\024)g Fy(cL\036)32030 23866 y Fr(0)32555 23667 y Fy(\015)33302 23185 y Fx(h)33844 22872 y Fq(0)34142 23185 y Fx(=)p Fr(2)35663 23667 y Fz(and)g Ft(j)p Fy(r)39274 23185 y Fs(0)39238 23995 y Fr(2)39764 23667 y Ft(j)g(\024)f Fy(cL\015)44405 23185 y Fx(h)44947 22872 y Fq(0)45245 23185 y Fx(=)p Fr(2)46242 23667 y Fz(.)3432 25272 y(Hence)449 b(a)i(simple)f (application)g(of)h(Lemma)e(7.5)i(sho)-36 b(ws)450 b(that)g(the)f (solutions)h(of)h(\(7.43\))3432 26877 y(b)36 b(elong,)457 b(up)451 b(to)h(a)g(exc)-36 b(hange)453 b(b)36 b(et)-36 b(w)g(een)23305 26527 y(\026)23201 26877 y Fy(\022)23807 27076 y Fx(L)p Fs(\000)p Fr(1)26157 26877 y Fz(and)28809 26527 y(\026)28704 26877 y Fy(\022)29310 27076 y Fx(L)30006 26877 y Fz(,)457 b(to)452 b(a)g(connected)f(set)h(and)f(that)3432 28482 y Fy(m)4570 28681 y Fx(L)5633 28482 y Ft(\024)370 b Fy(cL)8481 28000 y Fr(2)9006 28482 y Fz(,)401 b(if)392 b Fy(c)11478 28681 y Fr(0)12373 28482 y Ft(\024)k Fz(\026)-677 b Fy(c)14335 28681 y Fr(2)14860 28482 y Fy(=c)16070 28681 y Fr(1)16596 28482 y Fz(,)401 b(where)391 b Fy(c)21633 28681 y Fr(1)22551 28482 y Fz(is)h(the)f(constan)-36 b(t)391 b Fy(c)h Fz(of)g(\(7.42\))h(and)418 b(\026)-677 b Fy(c)40373 28681 y Fr(2)41290 28482 y Fz(is)393 b(de\014ned)3432 30087 y(in)433 b(Lemma)g(7.5.)p 46160 30309 443 1329 v 4026 38022 a FA(Ac)-50 b(kno)g(wledgemen)g(ts)598 b Fz(-)c(W)-108 b(e)594 b(are)h(indebted)e(to)h(M.)h(Disertori)g(and)f (V.)h(Ri-)3432 39627 y(v)-72 b(asseau)434 b(for)g(their)f(enligh)-36 b(tening)433 b(explanation)i(of)f(their)f(w)-36 b(ork)434 b([DR)q(].)3432 44014 y FB(References)3432 46935 y Fz([A)-36 b(GD])1599 b(A.A.)382 b(Abrik)-36 b(oso)g(v,)393 b(L.P)-108 b(.)381 b(Gork)-36 b(o)g(v,)393 b(I.E.)381 b(Dzy)-36 b(aloshinski:)554 b Fw(Metho)-66 b(ds)417 b(of)f(quan-)8706 48540 y(tum)465 b(\014eld)g(the)-66 b(ory)464 b(in)g(statistic)-66 b(al)464 b(physics)p Fz(,)434 b(Do)-36 b(v)g(er,)434 b(New)g(Y)-108 b(ork)434 b(\(1963\).)3432 51155 y([BG])2610 b(G.)297 b(Benfatto,)325 b(G.)296 b(Galla)-36 b(v)g(otti:)511 b Fw(Perturb)-66 b(ation)338 b(the)-66 b(ory)339 b(of)f(the)h(F)-100 b(ermi)339 b(surfac)-66 b(e)8706 52760 y(in)614 b(a)h(quantum)g (liquid.)e(A)h(gener)-66 b(al)614 b(quasip)-66 b(article)613 b(formalism)h(and)h(one-)8706 54365 y(dimensional)464 b(systems)434 b Fz(J.)g(Stat.)f(Ph)-36 b(ys.)434 b Fn(59)p Fz(,)g(541{664)i(\(1990\).)3432 56981 y([BGPS])1002 b(G.)524 b(Benfatto,)547 b(G.)524 b(Galla)-36 b(v)g(otti,)548 b(A.)524 b(Pro)36 b(cacci,)548 b(B.)524 b(Scopp)36 b(ola:)760 b Fw(Beta)548 b(func-)8706 58586 y(tions)497 b(and)g(Schwinger)e (functions)h(for)g(a)i(many)e(fermions)g(system)g(in)g(one)8706 60191 y(dimension)p Fz(,)433 b(Comm.)h(Math.)f(Ph)-36 b(ys.)434 b Fn(160)p Fz(,)h(93{171)g(\(1994\).)3432 62806 y([BM])2439 b(G.)382 b(Benfatto,)393 b(V.)383 b(Mastropietro:)552 b Fw(R)-66 b(enormalization)416 b(gr)-66 b(oup,)426 b(hidden)417 b(sym-)8706 64411 y(metries)527 b(and)h(appr)-66 b(oximate)526 b(War)-66 b(d)528 b(identities)d(in)i(the)g(XYZ)g(mo)-66 b(del)p Fz(,)519 b(Rev.)8706 66016 y(Math.)434 b(Ph)-36 b(ys.)434 b Fn(13)p Fz(,)g(1323{1435)i(\(2001\).)3432 68632 y([BoM])1789 b(F.)610 b(Bonetto,)653 b(V.)609 b(Mastropietro:)929 b Fw(Beta)627 b(function)d(and)i(anomaly)g(of)g(the)8706 70237 y(F)-100 b(ermi)453 b(surfac)-66 b(e)452 b(for)g(a)h Fy(d)369 b Fz(=)g(1)453 b Fw(system)g(of)f(inter)-66 b(acting)450 b(fermions)h(in)h(a)h(p)-66 b(eri-)8706 71842 y(o)g(dic)465 b(p)-66 b(otential)p Fz(,)432 b(Comm.)i(Math.)g(Ph) -36 b(ys.)434 b Fn(174)p Fz(,)g(191{214)i(\(1995\).)24367 75163 y(59)p eop %%Page: 60 60 60 59 bop 3432 7638 a Fz([DR])2602 b(M.)426 b(Disertori,)h(V.)f(Riv)-72 b(asseau:)575 b Fw(Inter)-66 b(acting)454 b(F)-100 b(ermi)456 b(liquid)h(in)f(two)h(dimen-)8706 9243 y(sions)412 b(at)g(\014nite)e (temp)-66 b(er)g(atur)g(e)p Fz(,)385 b(I)376 b(and)g(I)36 b(I,)376 b(Comm.)h(Math.)e(Ph)-36 b(ys.)376 b Fn(215)p Fz(,)389 b(251{)8706 10848 y(290)435 b(and)e(291{341)j(\(2000\).)3432 13560 y([FMR)-108 b(T])723 b(J.)556 b(F)-108 b(eldman,)585 b(J.)556 b(Magnen,)586 b(V.)556 b(Riv)-72 b(asseau,)587 b(E.)555 b(T)-108 b(rub)36 b(o)-36 b(witz:)822 b Fw(A)-33 b(n)576 b(in\014nite)8706 15165 y(volume)484 b(exp)-66 b(ansion)483 b(for)h(many)g(fermions)e(Gr)-66 b(e)g(en)483 b(functions)p Fz(,)458 b(Helv.)e(Ph)-36 b(ys.)8706 16770 y(Acta)434 b Fn(65)p Fz(,)h(679{721)g(\(1992\).)3432 19482 y([FT])2764 b(J.)512 b(F)-108 b(eldman,)531 b(E.)511 b(T)-108 b(rub)36 b(o)-36 b(witz:)734 b(P)-36 b(erturbation)511 b(theory)g(for)h(man)-36 b(y)512 b(fermion)8706 21088 y(systems,)435 b(Helv.)f(Ph)-36 b(ys.)434 b(Acta)g Fn(63)p Fz(,)g(156{260)i(\(1990\).)3432 23800 y([FKT])1753 b(J.)517 b(F)-108 b(eldman,)537 b(H.)517 b(Kn\177)-650 b(orrer,)537 b(E.)516 b(T)-108 b(rub)36 b(o)-36 b(witz:)744 b Fw(A)541 b(two)g(dimensional)f(F)-100 b(ermi)8706 25405 y(liquid)p Fz(,)434 b(series)g(of)g(Preprin)-36 b(ts)432 b(\(2002\).)3432 28117 y([GM])2339 b(G.)484 b(Gen)-36 b(tile,)497 b(V.)484 b(Mastropietro:)680 b Fw(R)-66 b(enormalization)509 b(gr)-66 b(oup)511 b(for)g(fermions:)8706 29722 y(A)710 b(r)-66 b(eview)708 b(on)i(mathematic)-66 b(al)708 b(r)-66 b(esults)p Fz(,)768 b(Ph)-36 b(ys.)700 b(Rep)36 b(orts)700 b Fn(352)p Fz(,)768 b(273{437)8706 31327 y(\(2001\).)3432 34039 y([Le])3161 b(A.)434 b(Lesniewski:)579 b Fw(E\013e)-66 b(ctive)463 b(action)h(for)h(the)f(Y)-100 b(ukawa)35317 34238 y Fr(2)36308 34039 y Fw(quantum)465 b(\014eld)f(the-)8706 35644 y(ory)p Fz(,)434 b(Comm.)h(Math.)e(Ph)-36 b(ys.)434 b Fn(108)p Fz(,)g(437{467)i(\(1987\).)3432 38356 y([So])3179 b(J.)570 b(Soly)-36 b(om:)849 b Fw(The)589 b(F)-100 b(ermi)588 b(gas)h(mo)-66 b(del)589 b(of)g(one)f(dimensional)g(c)-66 b(onductors)p Fz(,)8706 39961 y(Adv.)434 b(Ph)-36 b(ys.)434 b Fn(28)p Fz(,)g(201{303)i(\(1958\).)3432 42673 y([VLSAR])650 b(C.M.)688 b(V)-108 b(arma,)750 b(P)-108 b(.B.)687 b(Littlew)-36 b(o)36 b(o)g(d,)751 b(S.)687 b(Sc)-36 b(hmitt-Rink,)749 b(E.)687 b(Abrahams,)8706 44278 y(A.E.)667 b(Ruc)-36 b(k)g(enstein:)1043 b Fw(Phenomenolo)-66 b(gy)678 b(of)g(the)g(normal)h (state)f(of)h(Cu-O)8706 45883 y(high-temp)-66 b(er)g(atur)g(e)463 b(sup)-66 b(er)g(c)g(onductors)p Fz(,)433 b(PRL)h Fn(63)p Fz(,)g(1996{1999)i(\(1989\).)24367 75163 y(60)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0207090328446--