Content-Type: multipart/mixed; boundary="-------------0006010513224" This is a multi-part message in MIME format. ---------------0006010513224 Content-Type: text/plain; name="00-252.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-252.keywords" Generalized standard maps, perturbation theory, resonances, invariant tori, scaling properties, genericity ---------------0006010513224 Content-Type: application/postscript; name="bg4.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="bg4.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: bg4.dvi %%Pages: 12 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX10 CMR9 CMCSC10 CMMI9 CMMI6 CMTI9 CMSY6 CMR10 %%+ CMTI10 CMR8 CMMI10 CMMI8 CMEX10 CMSY8 CMSY10 MSBM7 CMR6 MSBM10 CMBX9 %%+ CMTT9 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o bg4.ps bg4 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.06.01:1157 %%BeginProcSet: texc.pro %! 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1.0 %%CreationDate: 1991 Aug 20 16:36:25 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 55 /seven put dup 56 /eight put readonly def /FontBBox{-58 -250 1195 750}readonly def /UniqueXX 5000767 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5bfae5c1f476ee6f05eb1d2caeb 269958b194521197b312fcced4867f3c8fbd030bd715d8ffda1dcd454b174e7a 1a97b59fe770e67702519d9d9b23d61ac08424d55fd4dda249cff0b56b9f3aff e9d0de215c02a52a6bc77155ff6b8cba5cd6646ba331254ac58ace650a967d3b 272331b87b6df06d5ab9d80fabe9f9ceac10139b61244814dd9fc295ed42d1b5 cd11c2e4beab318b20f51ca2c2930657e5343ab7bcf0e8870a0d12de4ffacb6c b3fdbdda481c2fcb84408d3d902e9a32070b2af6cd9317a33a42a43857c114b3 f4c3005cef9401f1c2bad3e69150d7145b79f95c9cfaf7a335b277e6435ab374 f6a3e78e124ac1e4615511f743ad65c5d778403a840310ac4902985f107fd33c 0049623e4c496cd353d2ae2a4804110c6a420a38ece5d5b235c5e35886a0987d 284e6110527b653b1b2c68c636ef423536180acadb954843eff4d9a82568a713 b682c574e30f793f093aef755c650e0e7175c2e3138695212d9fea7bd4b5805c a90985b134c6bc15a19afae0bfb3066363ca3e2568c2ac4559a69fc32feb443b 61243b4508b7524d974636855c83c8149d489b68fe18bd497116a953431251df bd09fa1d2ef66c85b997ab2f6b7c6d9d096cb43931379ff9782fa00c5f3cbd06 04265d16891ffc4f66e4a0307765d384c32aa83a4e4f89ac79ef8e76eed08450 e3466fbaf54f85324f0a4f04bd00ccc22d32e4132c51749bea28f112c2051857 423ea203df902ae80224d75efeb59094bdc5f4b6784ebeb01c38c34b695e681e 57a93ae18b657d13770300bde28833c732010ded4c8e79078d17acc3d4b521b1 6f2cac338c4f72e2e07c667227ef142d0273e5b501f24b02e9cad75fe9d18741 b36bbb29c9cd3d46964e00d8ba318912e69a6aa7ccf823c12a94ea675daff7bc 2d93f346e5fb9504d60dd1711f9b276e7b3dcbd16cadec80733eab3bde7de8b8 1f226935d9fb502d58686337d153a05899fe5f7bf0f9a4be3549c92db0b609ce 3c5a74e03bbde9ee97944d53bdfccf4afe9ee22af4854f2320577b8109dd3bde ffa074c4af676714b6147127b1bf83655a21fd4c917bc5acffc93ceb7eb4ce41 4e863a41546048cc2e8cb23029cbc4ab37f98bdb184f146e3cf8dce229c7dffc daf81fa54c41a032e2c6827d07f16efee1b0a2de84e205ef00eb648bbf4f26ac 0936f59e4f04dc31b8d6a16bc92f5f254c23d25f58e56424aaf63d11ab2dcb1c 2d2960ff199d486cad10fef0ac74def46ac5fdfbc0062e14c6da55a03f2c39e7 4e34179bf4a38d5b380fdda19cee03271873a5b510e61f53b39a3ffe1ca16b92 7ac8bce0d18ef7770b6e7d8ecd85979eab597d60ee5dbb70d82f053b97f30837 16057e480a2636a0131cecc580f3434cf5905099570220e5b38669b9147724f2 6f54e7d38a7231334f9b806b25494930c01c885383d79467a97fd430052a65a5 7674ebe390e4665c1aa03a0d84b22c27c75499094565bdfa3f08d9fd06cc1fe4 88146fed13390767fa2972a6ccd93ea5b9ab1fef1fba7b542fea690b76d29c2d b38551ea025bc60c8fe2a8fea1ca8c7347f20b9af9 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 67 /C put dup 78 /N put dup 82 /R put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueXX 5031982 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d56491bdecdcafdc988206c5a457a19270b 37d0ab776e03eaa7eb568eeab6b5e79dec03b0dcbf923a2aa8e4f4deda2cb043 858d8b430efef0ed914a80ffc9818aba0fb30dae3694e5b31df3855892d59b89 7f82ff79fce8444c7926ffc4ea008e63bc518936c9098afc2d1c14030a2c15bc c9285c1b57e80570d2f6301009e63d9ddeba44e3251c75ca9616a2c5de9ade66 1a15db37b40b67c1bd71f77343912c62f47dd24be0e03dd833ea60d2eb020407 eaf042fed30f0c2f01956c83d46d982b50dd979c2026ed5f71979dd98c9c1eee 25054000a5eed5ce9498b2cb59a7ee55e0538ab47d86e69b2ac79229c8c6bfc9 3a24e822a2f1c9964fe630344579a25f66646113f73d1315c447f070c575de76 266c6773b01137e4a039df5e4c2f199e8e3396ef9fd001eebe090e04384c4089 ec9f13a07ad8733c6af775f432f154ef50ec4477e08760c03c006e4d1eb1c7cd acb8955ca4b0d62c7de046ff58d431bb9883d28586552c51809e3ced4dfc2198 b97ad9e52ea33c91b9e335a80d9b77d3430e0487930f992459664abca5dd9177 150b0fb502fbb04ccaa94ffe2697ed07906f95fd98e7b325bfbf10ea6b39fd76 c1e5295022cb39e9cd471b7b0b3f9d72da99deb0615d35cc22f42dbf5cfb70dc 6279fd520d99f32201abf25e621cfe1d37961d9c0ad36ef3d9264c547638152a be30812b1188a10f6327c9e8714449de2ae636420f56a44cebd774f0b131c629 818455cc8dd8d40939f3ba8aa5abc54590c7286b172193c23adb1b4f9931a9f7 73b960b1f5b760438028478e4442dc93a567b07e20017c18fca2a1daa2649dee 5a6ec51fd6c3f4b27367c31e95a56ba4bbeda0493bc02306037e2ba69add9b94 373364db316691e73d6811cd76fbc92b0c67ca18dcdf05807f17d1b309456ea9 d113ea0fccd6eb73bd8dfdbd5905a5e38bbf61cfe58fb01ec95a4bdd9565433d 4a7abb8dca6f76c459484ad29c3480b588fae3d782340da9ccf5a3d15ebc4b0e 37e0e981f80deda1e9edc94ebc4269488bc7d09c3feaf61f454abe05a5c41289 f919b23cbd7cc570de9de9854d6be3f91530fd17c83115c02f6b34beeff7942c fae26f720a6defb2716a35ceb107fb9fdb87b6bcc3481ae05e08ca8eadc508b0 4b7ff96b6d32f54dec7e8b464057722370f5903fbc264c89eb31ed7e18c1a047 a7c173d8e80fe2fe10bfd4ef52781a5ac7d618463633c73d8e414e9129e2f05e abf6f3f2e9ff1a5e48863c810ba563fc8c1a1f1945422414ba0eb4086eaf68bf f0eabc6258f2d9015cef22a54c9417a293521eaa35917ff95937ae1b1a0dd3f9 98835ce49a704000529dd9cb93afebbebb69155c0b158bc0aadb786f1cde5f35 26cb4dc0baefde30ea5e277dc17e6037df3fe03283cc889a55498040985817ef 6af3b4d335e85314aa81a6fab86184ecb0f9cd53ab6b52673c6e0ce1afed5e83 50f55d705184b3f2e1f1ef51bb8b217e9f5c34314b677b375e7099d51e0b79eb ad84d113b155df64e6aaa5e1a80427862cc1c4c7ae5b6bd3dffa0f5312c6fc5a 66c6015faf233d08e090f7bd1907844c8a55b0d078afe863ab50469b5101ee45 6132a08a5901e40e3834f9c0bdafd9c29884f077783a3c1a1e9a6742f73f0d1b 599f84733fda79fbcc7a2ab64f3e722b6ab3a5d085d19b53bd3663ed7f80b2ae 650101063e7033b1c5c29ffa251605d6f4f7bb4a086068823a7bf37b091002e9 6ffb935b164d92835ebe08648d7312a5b26b82b9fcef182eb69fbdc7bcfbc509 a441d6e9f815949503bad5deee7a641293d740e6c942a4f4477f16fb6cd73296 e755c5db4b7246db87cfef1cbcfc76da11e1ffc136eec32a4bbae4e17e88aa30 b5b14695fc99a342d3bac179ff423edeea8dd6ae38c20120922daa7fc9ef1a4a 372067795107cf12b8ae6297a37fa4ac7056a97fb38b6f8ffe8212fb8399dca1 f7b6c29e821a92905579eae68c8c9343a7f5c1c9233f26d15ea56b3be0e9c6ee 717053a5eb4508632e8f93eba56eb5548999b000e17c300f1e7b589bd69398bd 6e0a084ca920a9b4e22a97d976c93edc285eb39638af274b5334ac1ac52a27e7 888b666b2ca668d7a54016260725afc68b239a54a918500667a57e96feac5d05 a293f9fb751d306a36f2e409f028fd267a7229f4188213e80449643c264741e0 194732d842ad35cfd947de0a43ef9fb9d9577ffffdbae496c1078aec6f050067 4a0123525ce76ba027b9e1c62b0690c7e2eba30ba3454f6492 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 43 /plus put dup 48 /zero put dup 49 /one put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueXX 5000789 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b031f3fc6c13d7da7277a94ba 018e9998b3dd888011a5d7c4204989f30f908b95533bda845746b673ab71ea57 65a0d14f4350707e47c8276305b28513cbe1bb0dbd269a53719bda46e536685d df78ca0146b6b93e760256b74d939d4e35b5e77238f04c92298dfdd188feea30 e053eefbcbb52f2011772b3aae39f5805597bbc1e8bb75a446ce014030f4f2f0 f49f9e962ee4a1024a746fa92a3628db5270732b54e43fe5ecfa524f127e5fcc 788e77e66098336ad67fe4cccaf0253272d5df79864bf4b734cb9a5859d557d8 bc11b8e00221ebc12e97de4b1f466ead83a4c894709363bca9040410a52d592e 34ee40cc7e5efa920546b981aa659513a24b1b85c221a1875b62d0b89e57a368 321b8043a5b094e0379760a443d632892b14ad6d19dacc8c78093243ad67e6a3 08e56e6b68412ee690b10dac6e17708754a00d51fc957b500eb80175716eef4b 2ca1ef867614659bee3f2b7319e97b6fdf1efc847bf3cee3156f72f21751da8e 5fb6898919e6799820d3de0642d756e09d6fae4ff08dd3deda3173bff4bb11f7 9109c97ddc05897af709ea199a90fcee8ce4c7a3c15b18170c41c04de2d3fba8 f34296a95b8e1e8de3739b17273f8f2c85e914615e8eac5e8bd2387ba3b1edf4 7968f06e2067d836d0f9f3e085cdfd2de06a62c81d786b304326f7002e83160a 36598589228b4dddddc43c85e1d126f8fe81b828028e26317af5894aaccf4f69 6301e1a9fc45935d8a414957f08febebbc3a72ada80f101e47447d019ade56e9 f4fab969bba2b44e47399fedf5caa1bcea216d7ba713d523d98f2e44ef37ad46 282d7a587974734c2b1e24d85e2f81581a8e36a6bcfbfe9dd0920ea5cc7255dc 2040d88b37bf8afa1d04f1ec06a739f608b5fa14bac8a50bb50a12d6574847a7 cf42db3fb99524d4ac40eb874d5d228b557be3c5eede986279a04e0d5fff1fb6 72e8393e9db5a96a98000702512357b8978176cc69e6c3cf9747115422e29b0f f86d6537e14ff074d3c212d8108ddcae70e19acbe1c7ea84bae91afbc6a79342 61a28fea71d96bcbc88fcff79b2178ef1f7e42e2b3e40512eae1e0c8789e1ee1 d2f0e33258da1052b483312ba035b30f42b48be9aa4c26cea2eaa920de93c57a b79a9e6bd5a9b9e86ea6e9c91ef191af606eb00e6854dbe0a12ae21e4822f384 000397bc6fb739496cc0a9813243665cc1354c554737c8c4b4a722ec2bd73918 e1f88864096a6496f7f40ad49d987b85a4e626ddfeeec3826eafc8fe8abaf455 6ee9a76f4797fda6013268c49f508faf0730a052698e5493bbd84bc238e944b5 aaef4d4c9cebc9017ba687d22e28f99e7db6bf7cc3591c9e5fdf6e5fed0dbae7 8552682bee7eba72c930ada06b8dcd3be8af83250107bc9429e3247c9bf6797b 9bf02fbce576fc791a850a725f4f046595c9a5b0efaebc9f88586b1b774730a4 383077c785aff17ad6fc708e90e381076a558eaf78777b9b39ee2f6b4aa81c87 f0dd8b6eb6e39b09de7dd1a4466c6f51ba6daaf1d21e464dd2641b2f 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 84 /T put dup 90 /Z put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueXX 5032014 def currentdict end currentfile eexec b5e2457af4a6caaa4536453c498a618ecdce7b5f6f2d5944265309f833bd3d50 3df8bd14c071f404d2177500681a801856cb8c9c9cf0112c9bd17ccba9ab8944 1759a320b2078df0fafdd3e357994bd81740d1a7522ddfafb75d58f5def046f7 084d10fa2f702e570b09a7073d70efe37b48c355a7089772238b7b4538c4912f cf6703a14fa416057ea2e1b94455913dd271b42846ee10d78270a8118e7e4bcb 8deb048bb280c240d59409d47046c890e5f79e5dd4b0be65a28a77aeb53dd33e ac506a6da87bb55e329f433627e06c3c94b078838b908538d93f888e504e25e0 3a8ccf7d2c02122ab3d485525564295390d1154e15fee35edff4eb016fadd32c 26d1b4c09de87943780954cebc4965aa890fe1f80b03eb07103e13d468245c7b c26455f229e8859585c6eeafd9474acdb27290fd62950dc52f893c99a2a614d4 2a7e3618266f3fb0c68a01a30373e048385ffe7dd5844e208a55d1a9263ab2db cb9b19038724f95af00349840157be45cb337871b55a304b87ade64b10ff0eba 21aa0eaeef74aa01f049303a8b25de0acc26bb3a33fcf267f479a2a73f7dece3 2aa23083687322b417fe86e8f5afc2136d045ad6b392364811a17630a94ff097 038eedeedb8da4ebf8fa5a96ce27a2ebf771d14f4a11fb1f9f48cdd207209934 885ef9118463552984b0fd3cf15d55de967fbb051549dd65f60dfe07f299690e 58252f032bca78935b7d2df852ed688bd13dd171b91cf7bd46d5433aef9f50f1 41a2517c4912b837793f84ddcf1116f70f7e9b1e6ec752dbc9183d4791d92602 397bafe29ac6f879ceb72c673c8a1aa0135d6f92d0db7791f9a2f1a5b55347e0 3e63723efd258754f5f4a9ad35873a98d54b6c54b17a39892dc1d38660597448 fdba9258493a0db0d560c01d49f2d889a99eaa8a5b5919cbd33f6b3d97bf27b3 e0b94b48d5d21d6c29aee4faca71bd775bb9defa37ff627f1e4926f10d59313e 77d81a511027ef9a8ed5b7fa9b5bd67758cd13bab6ac18411c974a65d0c90085 67bf5860d0930d7f43e557ee45e76ecb2806d21f4732bb282ac8b591f2b8775d 4b3ce2a37ebb371cb74a3cc513bdd70a0115de4760c21f034f5080d769f7ff83 554af524184702a02f6172e26f9b180f2de3cccd3ed216085ce7348f331d2233 39ef6c753f77c8dabf1994d5897b2068199fa1a14809457125164e02de42e133 4a3f7844380d2809c61c28923383a204528f1d29b3d1de283e1748ff804a05ac f1b3e69755ad508141a595e909c67768e6a4d09bb74bdfdd56eb5ca89bec6614 be8d5f5b9e67389928a781744b56e7285906a6081610f52361d4fe44349ddd8e 3611b7ee6b9d4b26c5dd6ad9e5c03600887dad5caa1320de2e38736011566ccd b05f2b034e47ba619773a2298220690891e9078892a314f42129f8b1500a9e1c af1ce45d1c4c38b5f9a3bfa7c25ac04095298eab4c0d9d880b97836bddb85e3c 1f458b2b333db6bb1a5f47859579badab924ee413400c0d6443b2202f26a65db a33226bd873d914616c0f205d653b18107cc0e6be04304e16cbefab4a738075a 0cb6f82ed9f052463cd6cf50577bce0fa4495f02c05dd6fc92120034bd89aed3 8aec5ee05c4c624d766622545c75bf2adcba1498faa2b681d923fadc299c13bc fd9dcc04f9bc9a6562616d777afab6d965d06a37b94907162bf57dcc340ab0fc 3f257338922c0b086605700f18efd63bd71404f983d1e39b13c61c5fd87847b3 fb0aa8eabd00cea3fc18a7dcc27be13ca2ebc78f9a3cd814939e037decd3714b 111672f62fdf4b07fbe09f0bee715569411eccc576421b78eab6f42541c1bfdb 9b6914ddd9e7795d68b3393d6323479f1c29deda2570f156ad99709dc8c024ea 7b6dcce0067ba9a16af82b3bbe7fa40707ef0a9a7042a6 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 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal 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 70 /F put dup 83 /S put dup 84 /T put dup 98 /floorleft put dup 99 /floorright put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueXX 5000820 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8 3750d597b842d8d96c423ba1273ddd32f3a54a912a443fcd44f7c3a6fe3956b0 aa1e784aaec6fce08dae0c76da9d0a3eba57b98a6233d9e9f0c3f00fcc6b2c6a 9ba23af389e6dfff4efec3de05d6276c6be417703ce508377f25960ef4ed83b4 9b01b873f3a639ce00f356229b6477a081933fef3bb80e2b9dffa7f75567b1fa 4d739b772f8d674e567534c6c5bbf1cf615372be20b18472f7aa58be8c216dbd df81cc0a86b6d8318ca68fe22c8af13b54d7576fe4ca5a7af9005ea5cc4edb79 c0ab668e4fec4b7f5a9eb5f0e4c088cd818ecc4feb4b40ec8bd2981bf2336074 b64c4301d624e41de0d56224b0e6eeff70187e66984b00b02f0c4ff369dc3377 7ff1caca8d2880942f5dad83094ca76f9e4f70d036418d21983a65085b998ff6 2ac3a8c84db1d44dd08776fdf140dd318c750d8d54dcfa0ddefde7cfb1d8851f 6e0a647fde1b27d27b435938fa15e8618153e3e63566dd6f16ed40eb03a555fc 2255854e5ed30e3eb4b08154172b0952b790283c93007f59b09df08283e5a1b1 56fce942139b6075cf127ad9502a5634ec8b666a640c43f5f5c33759be1760a8 6c651b8451ca0c1f8eabd7d984fb5a46b10043348a14d3f7095411dc8e9e3713 1bb6f19bcbe6eda148f366a6e698213911db836fc9228c8920ba08d96bfc1f0a bdccb87e6a30053c591d278000802af12f295674f9d1c766b48c3c4a4dec4721 2be38f8e768791f7ba05c4a74a0b00b9cb2071c71a6dddb00373b3d59655771e b050ca9e936c60c9875c93502bc9454402aa7f2e5afb875ba68b5fe7606aeb4d 3333b060f1af71177f4109b0bd3a66e823519a56547416616ee2f30d079ba3c6 06d1f1f1c21a12128a9b618095c7afe5e55d0bfdff591e985cc39da6e5baaf1d 5d36b180cbd937e10cd53b2cb3d9fe282ea8a870662a268df1561fe636d7acd1 f028f0f2231b25b583c31232d930ec0b07832f9e19bf76795db00931db990e87 04f00f98f87283739fc23e329995153f3c6fe82ff1baa198c75f234d417b3191 be8a7d88f3fb3ebd7d3c5351b4d13a165e3fcf73062b92298ff63e729051012b 60982c66bee898eeac9f8e7515d374740a37497a5da40bff1acfc20f6b36f2fd 858249b1c9504fa13a54952ef8952d60f9e3828bb44e57145a5bfcd4c365f82f 2f54ece4431a4964a9112e63a0987297e2dbcff6aa6e39386085f8e7ad49c06f a8405d41c3c1e422f9495e491a44d0f0ac5ae277100b10a1db5f414cc8344cd5 a471917be8d9a7b3033f44047caafaf7eb357d89b5bb315ef23ecacf8af3ed93 67469cd92ae23a439bd9fd2a1ed5002acb7b185bc955b9dd41b6bf94640ee6e0 435be4dae15e86c0c527e01f067331fb1d4dc4a0f3b64bf2d37d63f2572398a3 d7b578807f1fb42e5b93fc73c320040dbef39eba4dda7fcc0e02d1271126195d f4e4155436e0e09a8cb4c6f443fd1f803cd7462c26629ec1257e104ad43cedf3 371f408793b866f26ce9e673f14623d0bb15590057fbf648812d2ccd51a41cdd 98b617951906ebe18d8757eb5307bf22d0b09ff3b7c85e2c24285a9c20e7bee2 13f0ee98dd41681d0b94e22b65ad03663968480c2c97202a1d40f418b34537c3 20eb8f8957b9483eee38d0d8dcf12d35d8f035ff7c67cc8bac3734180e1d63eb 9750d3ca7f75a51913be0dd2a5ddfbc0294072ab8135785f0e39a67f653cfbc1 76c104bd4163570f22b84c207b301f46963d6cc7f096ad0e38385b39fc9f56e0 f2f55cdfad0061d9dc8090812202e173b3036cf604782b95aae1a99e4a66bcad 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90b3490350e744411b36569cd568d1f7118ed228d70dd77645ebc33841b0c4d6 52e90d5116e8f2374301e8df645f150bd17d9c8f6b339719f68f0235e067f098 bbea6fcb6990ae3edd2ecb65853b1fd3c206bea58b98afdab36ff1a59d323adc da81a3161b5ee3735f7b55c74187eff79ffe4ca0d9e4301ad801c615bb879926 4fdaf6fb8f72a1a36220088897512e2cfeffffd18635557e74aaa4f3b7abe2b2 eee78902b4419ae31c9021e464020b1ae64da18534b5befe4d68fdb4b88e1a12 fe0ab95cf4a52ae4e146d7f1bdbf5338bc7b6e5dcd48f6b3eeb26a71fd9f6d9e 5ecb96d44151b6b09b061786467d6bed8301c604ab19df8d7c8574e7a93b64eb 692427e16bd15570bddd2bce2b111f7fba2a5da14c36b42dd4e74ac4948b8bdf 5787a24d814510062d346231a861c287b41c7877c4c06968e643d4dd8808e569 98ac3f5e5be2658ed4cade9e8156a1beffb9f3a81fd4bc521724aefec5da7ec1 0e26e17d29e14e4bc81b48becfc19b8a9683383be7944e2c5381f80f12c19607 c73c2fbd9c6ab81a22d5afc2c02ae54d8dac3cf67a5cb7a719f8f3e2d4e1cd16 fde4941c0af517ef11e032716690f0caf873ca13154f7c21c6ad9cb40b51ba98 ab8cc6a2deb0a8d056001472f6ee4a1920eaf3ebd9002c0d975eab9f0e4cbfb9 753bf104db737e25c0308db764bfb182e69207ede734116d4b4397e409fe8efd cf422edd2f5b5556e6f00cab4be6def6931313bc0008e78447b82bd72897aefa 08cf2d7671458fbd2e3306a4e7cafe7b59092cc6dfc8c714aead588614dcd112 02a46c4fb989ada29f34485fb8523b430fe6344817058116af57adfc97c5b56d 9026995dc30369837bd10ce0f26b31f1fe3f7425f2154fd101cbcc1cc60ec808 b01f0537f9198a9b1ff1bb7206419b7e7cbacc6a662863ed547038b7d755d91e e1aba5e47a49fc1e4acf6734d6bbb4696b85f74e7088e9bc8199a7afb9f4ae6e bcbb91f6a56a90751470f9383b7328b03aba48348d1c84e1269e9dd0a40016c8 04e02781c3ce8ef418a0253d80101a15fbccba2db11ab385845ee438aefb5854 9eedb4b962c53f0cfa36ec69ef75c3b58c55013aac3531c80b63ed0330dfcffd a899c710f776fc4ea54bcf66701547cce95e97c846878f2282600dbd3184b988 b0f1282ef1d514971cd1f4d9a4e377c18144a845042746177518ed44f2dcb045 1515a5d83f5386a7ab03a4c64b489628f5ca032897c521cc9a2f3b14b5a8a74e 2d103a0c2c03766ee36925f4992a2967d9e20c1c5fdf0eee4057a9f9423bdafa 28bc6f4ad802ffa0d745e62b38857ea1eeba042ce8130f61aba881dc48c26df7 9d54863537e6719283694ea5984b1e363e4a4ac5f006a30b09e0c66c132f288e 03f0168c9423c96eb7fc2158e6e31af5ce364da7194db3981c52ea5be63db739 044672a500eb4a00ff1d3aa0115b43d73cbfda3a8f6cd3267f30d16405b78fe2 361819ccb87b2bc74dab3b9a41a402aa3e28a65e87e0c9cc8f496f40dcde15b5 d709ce35392a298082e998faec3b426272d71c751345f3395bcf1e9dd1aebf70 41b332f6dda07e1ae7566a83ebc19f52d7c92d4ee2d3ca643120b4f045977734 1c199e307d6e56ad2b0a19fc3b27304a1e30307a1a4490d2cf3f1cb37af56988 c045fc8cd6e24859fada30d3b938e744d5a8c3e258d95272848e3b900fd6a3f1 c276269da02a70b05606ed992a73880aa00880993f2c5dea5b909f480d76f6d9 a0df119151a37138452854e7a5141ecc0d07315f173aa02e98424051de1233d8 95470d5c60dcf11e1fee6adfc144755382bfe07948becd872172be37f1a33c55 5db9578f6c4398da74d38b1bdb660297bf9c463e7f599217f5cbda87d7da0465 838a44dbe2207d00216ac7d64a1e50771aa5cb8481b596073826765040135013 d621767a1528ffd1671d50ca55c58bc8cc4a51138521b57a27795d30f9ec082b b0f15cf25c37b871417abdba493f57cf22b0b536999d27bf91856e0c89635e3f 627497df7bd8c1c4d365b151756d73bf9dbbdb4b6f420df797afa5072ee4abbc e38847d2503940af770e4ab05a2fe9f5aa380aea4cddc892b8f87ccfb8855f83 0f11e24727840bc8fa56aca775e435e28115eb8bf0f6691a2dbdbeea9213b7a4 5d2e5321c1b1763c772542b1960274719ef2635d4cfcddbf6f93d9075e23323f 0b746bdef95fb3c47731e2afeaed6dee16c411ec87a0a0bbf029e8c564d0367f a07ee21f6ed56cb3f728d29aa946857427c03128b8047a25b25866284ee5d461 364fdb5a342e1456b5ff2110cb2df4ea1e068ee37cd719242b4321d8e1122147 23437981ae9b4009849dc87ebc90617135752c808f8c90b7a6492122c3600d9e 3e132978dba232bc1465e51fbe17af 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 3 /asteriskmath put dup 20 /lessequal put dup 21 /greaterequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 83 /S put dup 84 /T put dup 106 /bar put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueXX 5000818 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b66566ca5ba42b1911a 5d7f1bf343015eece988b7a93bce0c7aa61344d48aed9c92c8698d4b7c9951c8 7d103f2414b39e1437f9d2e50c4ee5f218f2e6716926a79ea978f13b1f855345 191dd7d31d8f82c2e3343c7a5894d95bdc492c28226834efcb5c12fea36ac5cc 430e0aa604961e34888adf6c1f3954cbc2498e225d953cf5685852162346f474 5a2a7087d5d7ad486de16d2ca8e15cee26e012671ba3bdc7d95cc8c98bb774f5 08625e968aee27ff7d1a06e63bcfb5aa4876c3f8f13b30ccccee73c3caf4e70d 98e6ed2f422dbb4950bf789680e064150995941a9f4dd68a575949847a7d012b b910bf03a4254c8f517e6ea430fac1155de17dc54e402e9f22beb39a64dbf0f4 ac415a28a2342100ebae2bbdd0e51225bd1825c5e0a2b0a5b4be4818bf1211e0 c4d56495f952a0a94dcc6ef1db7da90c9dd357202bfcee62fb05105a08c954c7 100cb2b7a78db25b19482b52485c73718f0978bc8677f96e9bfdb43d37c62a7b b7333146c0addce183cdbef937bb40a242d332f0ae19c0fa20cc606cb17c0b96 7e05c35df373cab82b25de3e37bd957b8048e4d5efba59f32c289fb7fd6d3287 480e90d492c9c499ab0c112c3a62a032c101804dec6cfef5e10fc123d012b1d6 b9ea3915527110c20952124ac0a11ecc640958e247cb334d4ed812ec8d07471d 2aef40981fcc48dd1043b12b3a6393270c7d12a6ddf55bb6b71126164799cb7d 087a6e4ca827def0602b5c734240e765ab27230d71fafbe2a41eb460c6d9af6a 7ab19d140d9a1f9ba6bb01e5dc4456261e2d3e9f2d1430fb5a3c91b19ac74398 873a030fcd3e85dbbaa4ac654ccfca0a80648a77100b61a758e4b2327574d2bb 9af439f05f5ce3b87c948790c52483a42ef890f01d0b7accc0513101bc2942d4 eda23d215c140d77eb214426b5bbbd16fac1028ddece66ab4f8de390f24215bc 01be7dad61180f2e590aedff2fe01bc8fee75ec3cc83745244c999f5ff66eac0 f02f16aa835a25d8405051f44e43d0cdecc9cfc5c25b5ad8b054701aec9c41c7 cc6ab33e89515f5c2663025f5e4c80da8691576511b9e4f6aa64b3ac823e430e a8ddef8f831fef3e6a14bfcad069200ebed9a23ceec3b98c1b9f0caeab2db8d6 ebda3917ce553d06fe96532176ba3b9651af1d598b2c149ba449971d149e7e48 0282215c71575466b22fb01133e5944636c9cee162ad35fb20c2a626908aeb17 31fa096d3ab002f2133a3c50a032bbe076fa71501e521914d767bbb2585a2502 0f799983ee71734a6a7345b6b5244fc0ca2bf93ec125031425b8ae961fbd231c ec9fbafa22250279728a700b046770292358adff308b5d50377e3ae29225db77 3b4c788120a9729b8b3ee33f23fd66d741b33a7bb3a7667afb4ac9efdd9de6a3 c9d8a36b35d9365c3f0c3493f6e49859dcf9fa5f7f534a019a8405437f3b4dfa 140d2aef952323f26ade493a8228d4f8f7324dd475dc8618e8acf76e5bb31f50 3694c67f49760f3f707931007a73e71c25262a99c103a358c27759421b7e9dff 329caceec6ace18a52f5842a9afadba7586e625c413fb5b1308ac99141d5f557 bffebcc27a4d14a3ca8c2f9a1b1d844211e13bf0bf6150a0895f29c39a2f25d3 f1b558682df36c19d3841c8eb9f2f24d3d1e7f876aac31cc7f879aae8a4d81ab b22a2c5764092ed3820cc0a4f6bb46e3c1c8f6cc468789bf33ffc597247d9de4 f09caacdd9658b430e1f0c99587e4576a43837305f08eb40ada72c54040d5146 a36e9f253a5e9e6227382ab7db9f5e902932c346cc6c7945e695a87246514769 f74d961d18e500c67c02d0d046968e3974277ae06d909384b792d502b16e8387 c1214332935e206c5f90796510fa7af4e03fb520f030fab7b45f41fce5b96a5a 8ff2ac879f747552b29e3768c1d802dbe6b3d6fae3da743f264c24c9f2c1884a 89d5b48e43a49da1af5dae5da40d8b17b3bfd83ca4e60680d7de3cbafbccca5b 153d6dbb0a74cf8b37da776615825e65cf8da621c6656d7fc516179a3ae4e36c a0e524eeaba21b7c3f2302610119b26e4e26d22d1d6ac8c6722a2990acaf63e9 70657ceed26fa87a95d944366e5b0c8f9d22fc070cd850f97c13c60eb109ebf7 853285def81282d47a6a6c01966d01e82c9eb30908db1d54cf6b6e5e4f763197 3b6f41264d898840512128ba49c8542f19912a1bee47b17cd68f2e516e80cd64 bd7a9b6a746bce11c814495089c55953efcd4b39900e84bf16ba9b3ee4401197 11498debca32941aeaa9cd9f5e46d62f9fd3a4470b7a184c547aafc202221e66 aed61a1d4622e0fd2db7a18bda25dd6d2083043b9c3af6d9950d132bd5c0ab61 976fb1ea1c581ff19fa28ef8f04dca6316c89d24dc09cf631a63e5e1d453bc98 496fd0efab6fc792888427ec03faec68e1b42816d720df27eea1c76b09f02892 7dcafe109fd1fc499111a0bee746633fe4e1947d0dea0178c16fa2e01a29fb71 7bac322e421982c80a4fffbaebe0b273bd9e51fbccc1cfc789f82779a5c509ed 2f3623b2305e713a460316a2d756cce9d6981c5f2e2d4668a1090d1225a6cec3 7a243fb04e0d00be16617ff09f18f0f6638ebd9cac9af61753681bdda3c911fc d453d41d28c842578ee3cfab45b351ccc87a5d79a8cae0fe9423ea07b9c8023c e05871230d4d6fbe431cb3406b1f059bfa3ae133ec9db0a86e4e746ed83baa64 1c7eb63150bb144e9326a7138615d0c0b4c4994f9af6667890f2fab76d07f007 15df5259552db24fd6983f7d4667dfb0c17674844341229e9516f4e9d86e3eb4 a4f5115befcab40289945f41f22ca52770c047943157fe2a50bdf13b62c8b588 991f788184f70a23b9ae 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 32 /parenleftBigg put dup 33 /parenrightBigg put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 80 /summationtext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 106 /floorleftBig put dup 107 /floorrightBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueXX 5000774 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d5649664203ab24acb938d58d246707ffe7 d62f04bec4b70c21ef75beb2b812622b3c74e969d72d3cd11bd7106294a99caf 0b1629bc7d4de6b96ca82930831d64575f23f4ad06a0e45e315b1d392411be8d 6d73c998789ff258a07a3c8c2057325784514c845500bfd1a971310cfc11d41c 1a167dbd5ff012c60add4e87325f6e5299032a839de65fb1473a166aae1876a4 414a434f22c1d241591fb36f857df6fa930608750ffc0c54f44994662b1f00f1 400bf752ea8d83ffc4cb77a290bc2d99981ae59a191748ba5c7ba1a9d2583fd2 1398452b6ff5d83a059f7eadcd2ef744e9dd22bdf9c79d049bf06835e878c32b 7765c69bdd8ef4deb4ea7cfff4cf9354a4ddffa689de961d16772491c7afbd7f ffde42400764c68e954ee5c455a5687959829bc3b319b2147deaab3628662c80 30c5e02fea09609abe4eaa12e217bc3af673f1bc36a7039eb13fcacb4218fe0f c5a3f9452d4edf46cc91db67b624d4f2d37502fb9f11af4da18ca40b61097f95 d44329375467ed13c5cb585ec53f62b83ef9502cc755af44bf32b87b8ae9f3f2 f8dbf72dab90acafbacd280db6aaffaefdff6d5eff26669bac56280a950560e3 d01714ba38659b0e4e28bf8cdc1f15ff7e3ea9d95a6afbbe3d8e6905aa6d531d 6b6dcab89e2036cc492c65f76ddcc3cfcff33fde3d9ffd10f7a90530fd5c7683 4329baa902e701ca42717476298c9ff511b1941244bb1dfd0e5cab92a43970cd 1bdb292eadd055c97972844fdcdb0aa06401e7b2632b9d134f96bce0064860f3 9859da7a9a82be0a840672a1331212752ae1b521b693b384a72dba8d37a86eaf a0341e7fc5f6ef7a8c566aacb91e8bf5005c18837894c4eacff41dba54e071ca 83fd0e43f31438730a3e275309aaa4287a2fa1c3c4118ec3f7f5660ffa38a476 c3076f10b250be2e1409a08ea579fd006546a25d5210f623c45f774d08e55c9b 553d3f46628395ab6075df29b01e75e41c9d811b76dd28be0b9bde4a8d93a674 0fed645a78fa3a54f2a875e5c26f8c8d3dc1745a1caa1eb2635be7c5c54913a1 1e64344531c08ae4e8c6130fa509222b0ed7592be0a1468fcedc84183dabf7e0 9612d0cda7b59f5689fd36ad64e264aec2c9b9c7b7b911bcf9a34ed42a83a4fd 1c7d24666d3a5a8226e14e5a149da7e0dc28b861253ca54674cb594f19d835da 43b44ce395ee2532ee563229282a7b3320b689acdb47b8ac371e716a5e727a59 f5a5600938eccc5e4d50b98e8f4062f7d3fc5b5dd64e0fed2418195bdf3153d1 ffe53db7150b5c7540978f9271a97c775bdac66622dfc0f2c8cac25f02cf6bf9 37b4f63897f9624fef25f164a76f6fdfdd7ba5f54af53ffbc7ca8879a1362d2c 609edb3a164e10403c7dd9188fd84440f533e92c21ad6d4b6e71f02274b5982c e4282821d12bb8379aecfb75ff145a48364232c3f35c2da8f51bbd1edfcd7a77 3c01d48c2fbb07e59cbe3f3bc7b9454f49dc118ac32286e9a47b81e6f20f73f1 563d38c65282c155eb068f7b2b7504e2e201e4c5ddafd5ecb31fde9e815c5c42 fac171b09fc2812c2df6a7f55ee92e696a63823c0eebd0cdbada12ea89179c3a 9ea54ef254a25123ad8af0209abd8379998a9cc6d0582a71edc63796c5b41fd4 4645b2d122e553a15a3c68d5a8ea9083c8028157746b035c222d05c075889da6 816569fc06ad3a085d2bf5e9b23345b3204e1b2d55a4ec12c4f4aebe5c714708 644aa2e516c5369cf8e8f375a78c41e42e32e3a882aa9b113c5ed5a5edd575ba ad4ca16ab3a77b4d89a2f02e1fc61e71f983b02b887520f166aa51a7adf6c2d2 958de847b44d7a40f73128d8be8ba0c4f2522561212bd021a97e157a987cf631 b1c785a08c3590530504c9f4f9a2023b09b829b655191e2d3f2c4e652d75dfe4 7356c2fc7a285823891f312f48895537bdecf496ff565db2ad78a641e51bcaec 7e3a74e26884e715cd9a442872174b608450d8b2b35b86d45539622454f5f504 ab520dd2264be206f041c382e5a7c2968eeb250489a80b03152ff14de3630fbd b7d3c3a1d9bfca9684db5d48a93fd2c42f7537a3811182fbe4babb8acf28dcd3 f0623a6dc489127c8617627ad8af1266bd2bd838949c99ba3f2caa85d84fdc1d 7b85f7bea1a82500b1423bd6e88dadc3be7a38b744de2731030b92642b363291 c0fb3c44da4706f1678edeecef0c9633bb7524cbc60a43d53ef06d997b2e4a0f 5787f4e3fa19b0d713518e4e0ff784378d3eb6106ad1ef9f3674b9efefd906a7 598180a9e7fdacf7178ca3eb0363f11bd9a0cf05a70c985fc475c733fd6eec60 26ffba1631e994566b613ae2c1822ee4035ec55c6772a261de7a8dc095c62aeb 48155204019153f7f72b8e23815f3d30bb1f7d62c94953365026408ca26027e2 6f0de91a64ef635a021dc8f0b15950a3ea2f3410770e3019f765fdc3583719ee a4b90821973b57f476adbe005a4efa53fcc105868b4b8292909c1b4b6375997f d5ccf3a46a4e14f69deff4606fdb40e2d1938a0924c5f913049ae8158a7e3c39 08d590075c684a2f4fc9ff1480318ce3231313f417a9c6852cd415257039cf34 faae383b88e2afc752ac1ea6dc866ffc9d84f4ad4a0a037beb5bfc2ea9fd2068 19597609ae5985532214d4e5333e9512049051355d834fdab2fca71ffe45ea79 8ffd863140b6da22beae31b27c8115f993661e9ad6cf46620b895cab2f3e8b97 2a187c1ac5cf77cabb509a2aa51690bab620d03cd6b4cf9282de2e64ca7c76fd 354d22010581b72464f215029cb4890c9b1fab597f7c29597132a76a689f7690 ca4d59b3088dca6db97336f12619e79beab688c9bec795fca207799e6f27a185 689cf320df769ce501eee266f983442bdce0d3c28a9e44644b3df5f91645eae0 8cbe72ec6c09a0aa63b8a6d2b374179bc8e868319efe466908bcbbb38cd476d9 2a14013dc9a45e225a258728d48ff5bb41c2dd226710b9e7876f34406adfc1b2 39220767c5c1ba606fd8caed015a6f83a6ea681ad136530fc344b09da229267d 234c04d4b230b03d2b531db46c4aec8b7cb778390f2943b9660d37f9edd65cb8 18ffba1f2bdf555269d3574079e82a348f06d82e7dbc4241f5b17d9e162182b2 d62d1e8233f4088b4a7abf529ea820629af2d6469933a9db00f514e7be0a0c6c d207787113a29228c34bbb4ce002b356c89284a305ebc34d3dfb94f8fafead47 09aa75bc41291aef8a8f10ce75c9813ddfe6be0f8d54d44575fd440932ce2c50 3a4fad9d353277f4922723fca87d1d4d529b73f0b91f13ffc3d17e668446ca43 15f9fe3a9b3a30b6e908f15ca83546d52a93cebd9c559aad08d3de5ec54c3586 91e6b324ddd82494ba369aff5e3f1ae5d6ef7a56215bba45118e4a33cb5ddb7e 05f5f546ffe00c0b05c122780da8031c01566550d72dbdd5b19d6ef9694b66fa 5daaf6346e194bd51392b0161b3d5ef0ca1284ea38b5eb950fc2ff13bad11c0f 122184ce0c8b6747b2a7cd82ed98f2406e74330207b273d872051a0cdadf9bb7 aa5678c7307881ee3c4edffa805dc18306829b6cf86c34e6085217d58af527e5 cf0f5d6952b90d291057026757a335e8e92dd8631d260232a05793ca7b0d74d2 a6cb9c5110af1297a37ce460405da45359631d080f5665003e6f79669c6efc4a 9a5cedaf8d4587817f13332704a08a9f53b2d0d0528e2cb06f862f55f24243a7 9845385ef106cbd8913c9755a730d76aa762c7694ac0491e45c937e2ef18da2e 4b725bf05c6c3da6f46f9ed52d4f576336c3c066a5e7db1c5716b689f330ee6e b9bb879ea72c74952679972d12747c74d60c4ccb84ec49d0c0a96d3b6099eb15 529fd1b7686e031ae55663d88b09c6981f3b8a1303f9ea736476eab7dd31c4b8 4fce1ae459913c0758990b06fd3d68fb96dee133a074e5c9a8c82a9881d4b101 0053d01605e6944d849c48dee02efcf44962de701e863a36508f796883e3616c d07ab4d1c77bf989d1065cb8d145657ad0dbe2dad5a365b35eb04aa2064d8599 72cda54ec6bb157b6b147501a40a3b0106940e74d74b4feea99471c307f32224 feb93c6c76e86d5cbfd4aefdaaa7a8418aec4a8121b5f1683957f5c5231cb8fa defd1750b98106d7f01c29d74652f3c6fe50d44dbcd99bea6a82be4908a802de 740b7bc46793b6d1fd83dbfbd90091300b52a55a3adbd3177cb6c4be1c4a314a ac7005145f6e789b1c354dd32404cbb6431a325d03b60c261bcab8189ecdf87e 3c57733c2c5c4fcf4a24ec00e3ed8720774fc77f5b9857fe89438b72b8a61d4a 245dc68e713294f8a1e123c22ad9fa207094be81724b72f07f9d49616af6e721 8297f5629c9193c7dd1ca71bd5b9df79 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 23 /nu put dup 24 /xi put dup 26 /rho put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 59 /comma put dup 61 /slash put dup 77 /M put dup 78 /N put dup 81 /Q put dup 82 /R put dup 96 /lscript put dup 99 /c put dup 102 /f 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 117 /u put dup 120 /x 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 db2546c6df52e524745992e18d9ff87aa25e4e1800bbe4ebb357c6ef55ed6d03 6d3a00c1ee8073266c21d2f0ac85d656abf61d7e5a4fa87da8ec3b5329e434d0 d2adab706b42a2e5331be5295399d803ccac03f631f01f39a022fcdf63486b68 7d15ef284a77def7fde4898543e7b5f7ec267756103e477f547cfb8d2311c4b0 09deff56085f5d419697af1846c8b88c1bbbae149f0f19ca3c8dafe19cec48fe 6b38357246d8b5cef80b53bab4cbbbe8bc8318cc462a4158258f456e697f6058 86010493f252073ee4e4c9531f23485415ad6bc901bbaea659ef701f825f4b6d 5c9bed0dce7b63810f906b33ea4955531ca1ed1fc120cc56dc1e277a33222cc6 c3a97552607b865783faf0895cbeccd0df54f57bd162a88494768435eda71a39 69718c0c5912d3167bb0971546d733e735aa9bcd89f5aa154656031ab3f6fe41 94ec225ad03ab8b67a9da43ab365c4df17c23e79f685c3b6fd5b654e4a6f2603 6cec3a8641b0b57f93eb95f9e238832aeac2061f3153e0f375513c974f1c38f7 b59edcc94716f2862cf71c9a2be8430985da1cd813b1e1c1b9310564a2e9bdb8 c536db7caeb9c37a0693a01df0045080f8236ece54dcfa7b743562771f452398 100dd95b678b6f190cdef25df0e2b10423aef1d27dfd5b022a0f895423aebb50 34ea52f46b077f2022e28ecfa6137fdd4b13935ac19769600b2e0dc86b0a2359 4675106e26eea6b5033525c1d79e046ee941c9ad410ad3284092f9fb725d366e e88a7a4ceaa6caf2ff6a2c80749780481db49bf09d5b7dd77cb83ed6c74fe223 0076eb41e5e84c5f714b9cd7ae6b36396bb68c992ac812a85b87db809bcac7b8 acc0c8f1a5dcbc17b40e84b5f4cfdc313a519a12f649e244f362d6dabbb86e63 deb75d3139363cdf5ff72ac93a939a6b1f5c215b00a8d67d271320bf16afd82a 40dd3ca0009444b7604f6617fe970d3fc0a5cbd84fb8004876a4c17fd96b6f90 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 16 /zeta put dup 17 /eta put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 30 /phi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 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 68 /D put dup 70 /F put dup 73 /I put dup 75 /K put dup 77 /M put dup 78 /N put dup 80 /P put dup 81 /Q put dup 82 /R put dup 84 /T put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 101 /e put dup 102 /f put dup 103 /g put dup 105 /i put 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%%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 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 71 /G put dup 73 /I put dup 76 /L 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 109 /m put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueXX 5000791 def currentdict end currentfile eexec 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 58 /colon put dup 59 /semicolon put dup 65 /A put dup 67 /C put dup 68 /D put dup 70 /F put dup 73 /I put dup 76 /L put dup 80 /P put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 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 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{-163 -250 1146 969}readonly def /UniqueXX 5000828 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3ffd20326947b1065649b533eb5e9e3 a88a87fe987918a687950b26f647d6e08bf14d983ed1d4a8ca1e2da8d985f944 2eb916f5b6ed0697ac7c33e1e36a3bf460d34ce45f1631871097cb04f18e3889 4cf4ac1538eb19481311d24fe3be7beaa4a3730e8b4831fe59d6d9ce2e46116b 629c7ba2f9ce3ecba2f43bc162a5a077ca1b2882a42afdcec3f4b75b5d63e0bc 8e5dc95257766d8ea467ad9cbaf47be60f797580cced6884b3a68f70c91f4fdf 80fd00ed9139e7f480dc3a76af72ad9b434187730bdfaefe4cbfe5c7edcaaf24 9204fd703011932e5a3c27be468b7dab69daa18dbbb6335ab8ddfdc607961c7c 02582763fa069d43563ba17704029945cf42fcb19cf78c51df0ec4d851086d43 c1ab38e865ef36865fff3a08b01cbcb070ba4893f7482dc7819ad03d337e520e 7d8cd83bb8ac7aff4df36751f4f12aaaea2b6d7260c09a26e987c904003e0723 7dbcc4a8e4e85cd3259f40d3caf55c742cbb40ad0db1044c20a4f122a63f7b81 1d945bfa69662189ce4d089464d2ead9f6ed60b581b04114d5d45e97be27a576 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8e1dd3dd6378a208d816f3be79 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMCSC10 %!PS-AdobeFont-1.1: CMCSC10 1.0 %%CreationDate: 1991 Aug 18 17:46:49 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMCSC10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMCSC10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 18 /grave put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 46 /period put dup 48 /zero put dup 49 /one put dup 51 /three put dup 52 /four put dup 54 /six put dup 65 /A put dup 66 /B put dup 67 /C 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 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 121 /y put dup 122 /z 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 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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 8 /Phi put dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 22 /macron put dup 33 /exclam put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 72 /H put dup 73 /I put dup 76 /L 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 86 /V put dup 87 /W put dup 91 /bracketleft 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 126 /tilde put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueXX 5000793 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 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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 48 /prime put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueXX 5000816 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6a18979f258b8c6ed3 ccafbc21aca420c9c83eea371adc20e038b4d7b8ac303004b0aa205f04135140 76407216032fdd22e6219da8f16b28ca12524deb7bca073cc5eba65c102a5e85 fd48e6d062cd4283ee570a7774597e5bf0e3400b6be72db0115f3cb12db70ce0 83722870cddfadee715f10f1fcaf20e06f3c54afe5ca238539bfe2b596116e83 f5371ff18fa5003d8543226cfd4025f9940365b392a858d27f078d3abcffe4a1 54e78c7692d1a32bf935967c64f01b24788ff8325d61145e2d4a489fd986fb77 38e6b254522c77ca2797a504a9ce4676a77ebacb026eca94dde5922c936f8e90 c43e2851905abd26842613d8fb644512275d43f312eea72f7ea3ba90c8fda6ce 7a9dacf52f55f0dd557f8d866ed6f3d3d5e6fe7e9ee79404fbe1a285bc519ecd 7ca162f0729044aff526bfe554ed59dd9beab0f842d55be08caada6136ac8e30 21c0b1f2869eae249565a126093d019e874f1d3e3de66d54ad31a98634d992d1 2087aa9caadb695d492b62edb22e89fddf96357f7dfe18336e102a892d42c1a8 924b450df9e62931117872ba4c4f60217f8dba3846b1df7f87862b42db5627bd 43eb0e86fd0a633d67f9c8d2e64df360c689be9b696536fd7d1886be07beb475 10befde02910376810d3894b6c3e16d0694dff896b2cd98f5e44f7f059776d8e 9bb6e3c852781ef8e6e0e0a9c8c78b899754aab4f9a8f9da4fd51bc78e39e7c9 a538ff680ef2c545bf9fcb4b038ed25337f8c02f345b1b7100eb6a13cad1d980 178c730a870f02d2c6c01f6606669510bec67c99986b891c142f8c2bf8df035e c436fa9b81563dcb484d18f85ae28341ea562614fad1bac1432d874695af6fb8 48171ebc83f63001e7c3a982d1fce641f1a622b9a8b7a28ad3d22df9c562ea5e 3ae864fa0a6413e60ef869d10970fd20c80b7568574276ef4c8c3d65cd96d424 100af3a3ad9c4b5f6c8dcd144af4a2da9f34a1b14c33f586a70f444a53fc565f 9082b4aa858c8bf637f8499fbd0dbc5576edd1ab85a4481adf4df8ffdb8ea103 9b99a919a11ed74d95ddcec7150896c48d0f80 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI9 %!PS-AdobeFont-1.1: CMTI9 1.0 %%CreationDate: 1991 Aug 18 21:08:07 % 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 (CMTI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 45 /hyphen put dup 46 /period put dup 58 /colon put dup 66 /B put dup 67 /C put dup 69 /E put dup 70 /F put dup 71 /G 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 97 /a 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 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 121 /y put dup 122 /z put readonly def /FontBBox{-35 -250 1148 750}readonly def /UniqueXX 5000827 def currentdict end currentfile eexec 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 23 /nu put dup 59 /comma put dup 107 /k put dup 117 /u put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueXX 5087381 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a1e67ed7e41cc25acc04702f6 8ef703e3d45722c1a91fdef7100a48631a02a6f02a08c6b1f9b4df8310385b86 8632718fd87119a233f219d9411383b7fa9f3e4780d8c27e2e89e0cae883d664 c3eac57a3aef8988a2e9f0f8c7f53e0a80bdfc4620e21287d0390e1975398544 7f3ea66401024bea75e1b4c4437b7bb188f76f96b918ac7c6ad7e8ae7f21d8c2 790f08cccec904fe48ef39e597ed4d4237c1d1f596f5906b19ea308020f7a35c 168e327ec3246b1dfabe912f6b6daac09974876d3996e57d180261110db05f15 e3e8eebba3d90b5764c03df3033a1ed678ebc679569a2fb297378b25434c0f20 5313ecb8a952f07242d3ee731b0cdc086a4481178a3d65129c47c09b22e9c431 e11b3747b94c26a757c38d06001798c6a568303d541385244b967d3b1786edea f65bb53c4c2fe75e4b1b15c2c78d930b4296c80f08bad86012451edc8e9f0854 c3b390a16e27b11b3d45a9f72eff8baded2242dc928a61685d79e09681c97425 5b90a498614cf560fa5b1718981388268ba206a96989e6d0b5d485d9aca5594a e67dd7b34d8a369adb06647f8aff8814d6d9cdc04a4835918e557174c5bc0f3f bcea9907a04cf93c12727ec40db3f2f77596dca477862747435bdedacd9b2311 6cc97fa47ffdd7d897fb6bdd5572e35d34e7e1cb5e7273a4ffd86525323ace4a 84e1297028c2bd5469baa2e75d19360c2c9042139d5e7dd4390a6a3935424711 de21910126d750ae279916ceb71da3591d60dc62db333c5021e2c1cd61ade51e 93964fa67d27fae86cba2166fd5bcdea975949a3e06aa8514ac1b65fef5314da 5e87cd51cb66480ee274cf436cc23935583f9ae66f748da022a14413e14fd4ff 891113dd490ccf165c756ecc71c8c09fd9ffec1e3b357a6e45095f36ccdedd0c 9d2aa5d5f2ad574f526316698ee74768688e4a93089158f75c99315f65426fda 5cb1da24552f6310215841528cf8fed30e7ede474706f34d2642af03e0770359 70c6956aa40fecbbb4eeac557aa60604cee8f013b1453a2e6b0f01db58bcc74f b17c38d7af52291322ce0417801d139adcb0488c4946d5cbbf2ae733a43ec35c f37f424114caafa5f07c4c6977042ef761e4dac81ec1408f284fa62ff9d5e93b c94692a876cd3286dc592001535e6faeb38279535cb0a2a67e6aa51863f9dd58 71a1716b4da5af1bd9376507f5ac4fc7fc09940fb7075cb35030ace6466f080f a3754669d67d1521c7ce019e3c80585fbccde38babb7f037b4a348b8a1679477 0f6a109392680b208b6c2431f2deea8460e410609b351e449ac237ba20fb81b3 bc40ac5f8de0944a1dbf9ae2c3017c0b9f991cf08a9e27981f2a3acaf1eeacbc f8fa12e64dd15e9e70f87872824128fff4787af267dec2cfbb6fd4a9cd94afdd c487f5bae23656491fdbd1e39a823d1d36268d9e7a529db268d3acf449275625 7a54488f2b6004828640274daca4d04359ac917235be1b75e89db6eb1f02362b 2d7389c179399cf53b2a49d32bde3cd4a0127126d12586d075ad287a000c7842 6b5ef4d46e4f9c1ac0355d7ffb2a0d2146df46d4459a79640b8f72682848d413 c9a9ef4f74d45bb55e80cfd66249691cb89a69f562c68e5fc6aec051a995202e e27d8535736edef05df727930c1fb5885ae59e42beaaaa01d6c54e98bd225b9e 1817982c3df91e213c0a184dd3a5d8959a9a65cd91c2f648d854470c407af20d baf370876a621660266eb6f4b6c3632b5291721cbedfd67cb2288a126c5bcbfb 2a10c1903ad5e19f28fe095999750fe0e9613001b47b206a6131087412c333cc ecafb3cbe597c2a0c90ebe3149115f85d7e80b5025828d2a3dd86c4a5b97cd4a 8ddb3f49c7b76c81250ddd2d1ab4e0b2c33598f61ac6ef38bda17409fa125556 2660da97d345475dec9ef63d65d553d1e6a5471f6dfc967605a6b7fea6ec2f48 199e7813d0aee28969762a3fd8c8108c91de5b15caa1542da6218a7c5b31bb87 5dee42ea4753b83240b0a173140c2f0f333353ee27aa6a06de5c2c982ca39680 ba871a440e15c4a7243c074916999b8006b8b38feb1f0e 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI9 %!PS-AdobeFont-1.1: CMMI9 1.100 %%CreationDate: 1996 Jul 23 07:53:55 % 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 (CMMI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 23 /nu put dup 102 /f put readonly def /FontBBox{-29 -250 1075 750}readonly def /UniqueXX 5087384 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930da4f8d58abefd45db119b10eb409dd89792 3c6e705479464a4b33ae3d31bfe98efe259f07f7950237bbaee4f7b64ffea83a 757fa717d50c3298392891bfd60e34a056b0d6021da3fd9b8b01bf78b0b23dc4 ea3b0605150d20b27ff9ea5f2524661019d982a2e47cd7c21ee5ca9ed9227821 f8d07119397de26838c11d7b35bcedc43e011626e300f8249ed846d7b5ccbd89 02550cda17ddf113df658fb13f8162681013766863784efac128e01ef997e1a4 30312afac8f3d948edecd7d5090ab5c864b91d7f6e80256f0e1fc99f1102b74c 61cc456e2e1bedf0e627d5f70f738f963a13666405c51ebf23ca077e97570a2f 1118536b92e585cf1dbd5fbf36a0a5d197538c05304e93a15f3c931168d91d42 483ef428b1a3f1b7e548fd1d23b80f5523c8b8bf450b02804fb689f915c86693 f7dfb1c43f681627c7f027021aa3a8179148f1819cd94b8890a9ac63dc1e8a68 f254b64f6f173c3d02e3ebc037da05074017c8b113e94f4e2d26b467ef9da07e afd7ea7c86e025af52c5ceeb835951e67e02a38daf1e9e698bd21996c1071e84 ff7fbe77f6ddf9d06fa92b16ecebcbe0370b5cd9d450d0f2906a105cadf93ed0 0abc7fcf2c996e7c15e452006e35390d99b9cfc59925f1444c20d9c3e1fc2910 32cf9fa9d580eac1bc956334da391b870a757c3bee6aa3e005b20988d8587876 3ced20c6edf2ac2b22890e37ff7aedeb0986d85ae05bc84339bf2116df48c5d1 cc888bb33d409b06dd70ffd0618b2080442aac439aeb39c20a87ad74ef10629c f1604db8611e8e4224c5b21ab0c7ebddc77fe9a283b3a8c6a2a9508f6836e3fc 3eae1366b4da4f44cb7a77fd81d278c17775d56cb28b70f18d894bfd1ec1827f fb177e1d46b0dc5afee78aaca1c43d03b7be6c8768b3cf192c3ba9b6e6caa4bf 7f6069600697446f33e6d043b4570e8f3138246334c7bcf8b7c1e30718797224 d6ce4372de5d47e346042f84050abc1ccbe402f9b5dc3d7f9de95238cad5ea06 60247b75a49f49f1b12237640dadc75bd047b028438449cbec75c2d1247d24f9 ef67a8a44c373c521ea8a6e976503a60e607347322b9a0a92a65d9ed6c619aad 4c8cb102e39aef5575d6a3a10c14786732d6bba702eececa4539d478b8dc65bd ecfdedd2405695f4a54c4b43c25fe7ddc73752cd6bc416c11e893a504e5a4567 3a6df03b88fc5d8f086876d7a740bf2762606d437039501d6e2e20b3f3396646 82283f3744233439f891ba2a6406d363cf1b1e6b7118f807a4e24bc357c14788 1e80dfa974a8cbe31bf2376b15c38d847abcd7abc54610f9e27bf9de80564349 8852ec7e5f73152139a56538c7b5b463c2d0b1ac1ed4556eabee75e33d19ed02 07f0bdbc6b35dc760f216a1467b534af95f1a73616dfdf86745a28bac0c42541 9dadeacbd2d58c82a9288cb3faa30ef02f0d5ce4ba9ab4726ba5e51413cc473b 22bc40c75b199f09cc15a1e5d960f3f0f492cb42ffbd7993bd7a2c035d2816c8 120fee5db81077711b08f686e7b222abeaa984b6a1f9a65b33122182b7728a15 4ea986e68ba07dd67518657f93173685ea7ce47ad342967e 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 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 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 61 /equal put dup 65 /A put dup 66 /B put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 84 /T put dup 85 /U put dup 87 /W put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueXX 5000792 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b448685bdd20eb05d5e7c2126132e33a59a7170609 dcf4871a5d023c9ef57d3362d9f2d7a440bb69bf653364105f16f4d0f03582f9 aced3d05cc76489b16e3fa8a446094d30038b06ecceda269f2eab9d19a99c7f9 39f9548f206c5a457a19270b2b82c43b091dfc5573468eaa3e7a4a32f8042891 d85e4b180fcbcb3091d2800e54c87d84ce9cad6869b5aabbbe47f40c68799893 d22b765295e1e69e33aa048b7ed98ba480ceca91f3ebf8ef85fe9a3976909626 b95ac5940d53f9b02215d84a44837ba25ed15cce0d504f1d335065594f3bc824 5405407591cccb11cfd4645da60d960c0b93f187b0cf7b105543c0b70f89af5d 264b6c026e3ae646acf145950202ec73282111e3e601cf2bcde22ce3edf6db23 516481420f26552ff4472d749811f27768150450d0d0ebe3c79f999e99b5c0f2 2eaebb12d97782b1bd91b2a1f62a76412548ad53c0dd411d4a08c0f071c2c218 63d9adb75a4621803ecb84c2bb235b620b658984b2d8e0c4637e2811bc8f0d04 6c8935afc70141e1b2d9c23bdb251d304b3378faf8928bd09686aa0340fb0cb1 dc48c996ef91530ff078666fad227a3589f50b605267212d3a65ebc1019a8eec 9a0739a00279471a01e1505c17658c10030fae32f274fdc8c8774b0d5406c384 a1d17068dcb0c2575c562fdc5a2176609bad9a2c255e426a4325a4fc3053c7b3 2f35f7bbc9aa50135f30223360fa2fbf019294b2e1d1405e078341097090ddba 1561f22f8738fe105c0e3b84532bed145d51428c9cb1807e655408807dc69161 1109a98af82680d6dccc987cb312835dec4ea0645f47a15b6707d654ae54846b 3d3301d61150fdea8ae2da33e79976812c7764050a3e86a2a819ffd3c64c38ec 16b2d0bec3922847fde375ffdd82f8f8bcdfa39c53068da393b35277ba7af682 9e48b1607f22eeefd7d0eea93ba1e2fb9b5ecc6c6fe1075db980999f0be7a3e0 e8021f1657fb5c18e996009eb94ced22846c21e3afc8acf7d8f9f4402a00df48 d9c0eb196f6272b4fd1df76563ff2e3b22bbe0179354d8b721265cd299c99b33 706b9a040fad5a36f7e09d255799da13276bf08e803a43fd16734c0d547a20b8 7f9ac43c2bcc227108f7cc47745fabd1ccb3a347ec13b854626c54bb3e8417fc 62eb327ed7891cfa42d117179cf1768cc4549ec36601adcc6f55d095808ceb0c 3a44dfd8728cd53d7a3d1406f1dc3eb2ef10fd99da850405be87a0ff1edeb7c4 f6abd7a0e02003945ecce903d79c34f8232c48d7d78e534c8ad3d402fc30a32e 14a9fa6ac2a07666c4768108b68676ca4223d762f6f57c1752eefabee59cfc9c 32693bd5756129c633f46af45905802f7fdf9f06ab9b382f646e4cf0ddfd31c9 dd7f95082b5e028125e502198faae514771e3069d7b419f3e78c58dbccb65a75 6be1e32306abc8be436c44e67049fd775a309a110c1344864f9dce8bc5db9361 d38004150fcb8fabb81e6ff8d8bc617b43a164f7ca9a5fe4ae35c6284fc7d512 947f1b3486226a9998c75d0cdbff68c099c80385cb4f3515a76b6448ac4b90e4 e3f496278f5709db3935f2f1040ae889b63d53414336845596991ae9fc9c4795 846ad63132669e2ab3e1d9400b9cf685ad35e9eb184acbac69f9828d67c67fa4 49849a68796229579f518ae9613e057643532cb40e49eabf6776655a28d8ce3b 6912d5f7ff928be89b6fd03c6c3df7f7abade3f2a4abff84111459be4d09872a 0b1ac5b1a4c5774ca4142644a8a6f6ff75aea1b8956178235f0b2e4e98b5e36a 3ce4aaa49aa12836bc22dbb7f0af7d3cd989d3a66e21b3ce7121bc29066706f1 4df396cc01b3ada02f36c04fa4d6e5147d2e6429133e65821b8d050b662966b1 0acf35883e9048fd3a556fc2a57d2d0370ced3e06edf17f2ce697fcc9472ed0a 1322b0bc19f91019e9a006a874ffc92dc3a18b467ed4932fb2e4123cfc40eb5c e372491c8a9d42436fdb1868ba640bdb1095e1845fbf94d86843d1b0aced831e bebeed35c7e1cdfae2ab99cf246addfa6545584c139e8e7190606ec1ed319672 4a484bd898fa4d2cf8b941aa47f175e3ac7d443c68a20f2ef3ee033372e2bbeb 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two 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 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 90 /Z put dup 97 /a put dup 101 /e put dup 104 /h put dup 105 /i 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 readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueXX 5000768 def currentdict end currentfile eexec 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b(the)h(semistandard)e(map)h(the)g(parameter)h Fj(")f Fn(in)g(\(2.1\))i(is)e(not)h(necessary)f(\(and)270 2525 y(in)d(fact)h(it)g(is)f(usually)g(replaced)g(b)m(y)g Fj(i)p Fn(\).)62 b(If)36 b(one)i(considers)f(the)g(conjugating)h (function)270 2661 y(as)32 b(a)h(function)e(of)i Fj(z)f Fn(=)c Fj(e)1139 2628 y Fi(i\013)1212 2661 y Fn(,)33 b Fl(i.e.)h Fj(u)1482 2675 y Fk(0)1521 2661 y Fn(\()p Fj(\013;)15 b(i;)g(!)s Fn(\))30 b Ff(\021)e Fn(\010\()p Fj(z)t(;)15 b(!)s Fn(\),)33 b(then)f(\010\()p Fj(z)t(;)15 b(!)s Fn(\))33 b(is)f(expanded)f(in)270 2797 y(p)s(o)m(w)m(ers)24 b(of)h Fj(z)t Fn(,)h(and)e(the)h(analyticit)m(y)i(domain)d(in)h Fj(z)j Fn(is)d(studied:)37 b(this)24 b(leads)h(to)g(a)g(condition)270 2933 y Ff(j)p Fj(z)t Ff(j)h Fj(<)f(K)7 b Fn(,)30 b(for)g(some)h (constan)m(t)h Fj(K)7 b Fn(,)30 b(whic)m(h)g(is)g(equiv)-5 b(alen)m(t)32 b(to)f(require)f(that)h(the)g(mo)s(dulus)270 3069 y(of)36 b(the)g(imaginary)h(part)e(of)h Fj(\013)g Fn(can)h(not)f(b)s(e)f(to)s(o)h(large.)59 b(Instead,)37 b(w)m(e)f(can)g(\014x)g Fj(\013)g Fn(to)h(b)s(e)270 3205 y(real)d(and)f(imp)s(ose)f(suc)m(h)h(a)h(condition)g(on)f(a)h (auxiliary)f(parameter)h Fj(")g Fn(\(whic)m(h)f(pla)m(ys)h(the)270 3341 y(r^)-45 b(ole)37 b(of)f Fj(e)604 3308 y Fi(i)12 b Fk(Im)g Fi(\013)786 3341 y Fn(\).)58 b(W)-8 b(e)38 b(prefer)d(to)i(adopt)f(this)g(second)h(con)m(v)m(en)m(tion)h(also)f (in)f(the)h(case)g(of)270 3477 y(the)31 b(semistandard)g(map)g(in)g (order)f(to)i(mak)m(e)g(more)f(transparen)m(t)h(the)f(connection)h (with)270 3613 y(the)f(standard)e(map.)370 3749 y(The)22 b Fl(Lindste)-5 b(dt)28 b(series)j Fn(for)23 b(the)h(problem)e(is)i (the)f(formal)h(expansion)f(of)h Fj(u)2899 3763 y Fk(0)2961 3749 y Fn(as)g(a)g(T)-8 b(a)m(ylor)270 3885 y(series)31 b(in)f Fj(")g Fn(and)g(a)h(F)-8 b(ourier)31 b(series)g(in)f Fj(\013)p Fn(:)741 4069 y Fj(u)793 4083 y Fk(0)833 4069 y Fn(\()p Fj(\013;)15 b(";)g(!)s Fn(\))27 b(=)1266 3983 y Fh(X)1268 4180 y Fi(k)r Fg(\025)p Fk(1)1438 4069 y Fj(")1480 4031 y Fi(k)1523 4069 y Fj(u)1575 4021 y Fk(\()p Fi(k)r Fk(\))1575 4095 y(0)1673 4069 y Fn(\()p Fj(\013;)15 b(!)s Fn(\))26 b(=)2023 3983 y Fh(X)2025 4180 y Fi(k)r Fg(\025)p Fk(1)2170 4069 y Fj(")2212 4031 y Fi(k)2270 3983 y Fh(X)2270 4180 y Fi(\027)t Fg(\025)p Fi(k)2418 4069 y Fj(e)2460 4031 y Fi(i\027)t(\013)2598 4069 y Fj(u)2650 4021 y Fk(\()p Fi(k)r Fk(\))2650 4095 y(0)p Fi(;\027)2748 4069 y Fn(\()p Fj(!)s Fn(\))p Fj(;)255 b Fn(\(2.11\))270 4319 y(where)44 b(w)m(e)g(used)f(that)i(the)f(F)-8 b(ourier)45 b(lab)s(el)f Fj(\027)50 b Fn(has)44 b(to)g(b)s(e)g(p)s(ositiv)m(e)h (and)e(not)h(smaller)270 4455 y(than)h(the)h(T)-8 b(a)m(ylor)46 b(lab)s(el)g Fj(k)i Fn(\(b)m(y)e(construction\).)86 b(In)45 b(the)h(follo)m(wing)g(w)m(e)g(shall)g(write)270 4591 y Fj(u)322 4544 y Fk(\()p Fi(k)r Fk(\))322 4617 y(0)p Fi(;\027)420 4591 y Fn(\()p Fj(!)s Fn(\))26 b(=)f Fj(u)724 4544 y Fk(\()p Fi(k)r Fk(\))724 4617 y(0)p Fi(;\027)822 4591 y Fn(.)370 4727 y(The)30 b Fl(r)-5 b(adius)34 b(of)f(c)-5 b(onver)g(genc)g(e)38 b Fn(of)31 b(the)f(Lindstedt)g(series)h(for)g (generalized)h(semistand-)270 4863 y(ard)e(maps)g(\(2.1\))i(is)e (de\014ned)f(as)857 5106 y(\026)854 5118 y(\026)-53 b Fj(\032)893 5133 y Fk(0)p Fi(;f)993 5118 y Fn(\()p Fj(!)s Fn(\))26 b Ff(\021)f Fj(\032)1292 5132 y Fk(0)1331 5118 y Fn(\()p Fj(!)s Fn(\))h(=)38 b(inf)1583 5179 y Fi(\013)p Fg(2)p Fe(T)1720 4962 y Fh( )1792 5118 y Fn(lim)15 b(sup)1841 5197 y Fi(k)r Fg(!1)2070 4986 y Fh(\014)2070 5040 y(\014)2070 5095 y(\014)2070 5149 y(\014)2100 5118 y Fj(u)2152 5070 y Fk(\()p Fi(k)r Fk(\))2152 5144 y(0)2250 5118 y Fn(\()p Fj(\013;)g(!)s Fn(\))2478 4986 y Fh(\014)2478 5040 y(\014)2478 5095 y(\014)2478 5149 y(\014)2510 5012 y Fk(1)p Fi(=k)2623 4962 y Fh(!)2695 4985 y Fg(\000)p Fk(1)2789 5118 y Fj(:)344 b Fn(\(2.12\))p eop %%Page: 4 4 4 3 bop 390 46 a Fk(4)739 b(ALBER)-6 b(TO)30 b(BERRETTI)h(AND)f(GUIDO)h (GENTILE)490 241 y Fn(W)-8 b(e)44 b(are)f(in)m(terested)h(in)e(the)h(b) s(eha)m(viour)g(of)g(the)g(radius)f(of)h(con)m(v)m(ergence)i Fj(\032)3224 255 y Fk(0)3263 241 y Fn(\()p Fj(!)s Fn(\))f(of)390 377 y(the)38 b(Lindstedt)f(series)h(\(2.11\))h(as)f(the)g (complexi\014ed)g(rotation)h(n)m(um)m(b)s(er)d Fj(!)k Fn(tends)d(to)i(a)390 513 y Fl(r)-5 b(esonant)54 b Fn(v)-5 b(alue)45 b Fj(p=q)s Fn(,)i(with)d(gcd\()p Fj(p;)15 b(q)s Fn(\))49 b(=)e(1.)82 b(W)-8 b(e)45 b(then)e(consider)h(the)g Fj(q)j Fn(sequences)401 626 y(~)390 649 y Fj(I)430 663 y Fi(c)465 649 y Fn(\()p Fj(f)10 b Fn(\))43 b(=)h Ff(f)p Fj(f)838 664 y Fi(l)q(q)r Fk(+)p Fi(c)983 649 y Ff(g)1028 664 y Fi(l)q Fg(2)p Fe(Z)1141 673 y Fd(+)1197 649 y Fn(,)g Fj(c)g Fn(=)g(1)p Fj(;)15 b(:)g(:)g(:)i(;)e(q)45 b Fn(\(recall)e(that)f Fj(f)51 b Fn(is)41 b(assumed)g(to)h(ha)m(v)m(e)h(not)f(the)390 785 y(F)-8 b(ourier)25 b(comp)s(onen)m(t)g(with)f Fj(\027)31 b Fn(=)25 b(0;)i(see)e(\(2.1\))r(\).)39 b(F)-8 b(or)26 b(eac)m(h)f(sequence)2793 762 y(~)2782 785 y Fj(I)2822 799 y Fi(c)2857 785 y Fn(\()p Fj(f)10 b Fn(\),)26 b Fj(c)f Fn(=)g(1)p Fj(;)15 b(:)g(:)g(:)i(;)e(q)s Fn(,)390 921 y(let)1405 1057 y Fj(I)1445 1071 y Fi(c)1480 1057 y Fn(\()p Fj(f)10 b Fn(\))25 b(=)g Ff(f)p Fj(f)1816 1071 y Fi(\027)1884 1057 y Ff(2)1981 1034 y Fn(~)1970 1057 y Fj(I)2010 1071 y Fi(c)2045 1057 y Fn(\()p Fj(f)10 b Fn(\))p Ff(j)p Fj(f)2240 1071 y Fi(\027)2308 1057 y Ff(6)p Fn(=)25 b(0)p Ff(g)390 1216 y Fn(b)s(e)30 b(the)g(set)h(of)g(nonzero)f(v)-5 b(alues)31 b(of)g(the)f(sequence)2173 1194 y(~)2162 1216 y Fj(I)2202 1230 y Fi(c)2237 1216 y Fn(\()p Fj(f)10 b Fn(\).)490 1352 y(W)-8 b(e)31 b(de\014ne)f(the)g(follo)m(wing)i(set)f (of)g(in)m(tegers:)1218 1536 y Fj(A)1286 1555 y Fi(p=q)1395 1536 y Fn(\()p Fj(f)10 b Fn(\))25 b(=)g Ff(f)p Fj(c)h Ff(2)f(f)p Fn(1)p Fj(;)15 b(:)g(:)g(:)j(;)d(q)s Ff(gj)p Fj(I)2284 1550 y Fi(c)2319 1536 y Fn(\()p Fj(f)10 b Fn(\))26 b Ff(6)p Fn(=)f Ff(;g)p Fj(:)597 b Fn(\(2.13\))390 1720 y(Of)30 b(course)g Fj(A)865 1738 y Fi(p=q)974 1720 y Fn(\()p Fj(f)10 b Fn(\))26 b(=)f Ff(f)p Fj(c)1305 1734 y Fk(1)1345 1720 y Fj(;)15 b(:)g(:)g(:)h(;)f(c)1585 1734 y Fi(M)1665 1720 y Ff(g)p Fn(,)31 b(where)f(1)c Ff(\024)f Fj(c)2235 1734 y Fk(1)2300 1720 y Fj(<)g Ff(\001)15 b(\001)g(\001)26 b Fj(<)f(c)2662 1734 y Fi(M)2767 1720 y Ff(\024)g Fj(q)s Fn(,)30 b(and)g Fj(M)35 b Ff(\024)25 b Fj(q)s Fn(.)490 1856 y(W)-8 b(e)30 b(de\014ne)f Fj(M)40 b Fn(in)m(tegers)30 b Fj(r)1411 1870 y Fk(1)1451 1856 y Fj(;)15 b(:)g(:)g(:)h(;)f(r)1693 1870 y Fi(M)1773 1856 y Fn(,)30 b(with)f Fj(r)2075 1870 y Fi(i)2128 1856 y Ff(\025)c Fn(0,)31 b(and)e(an)g(in)m(teger)i Fj(R)26 b(>)f Fn(0)30 b(as)f(those)390 1992 y(in)m(tegers)j(whic)m(h)e (satisfy)h(the)f(follo)m(wing)i(conditions:)1449 2175 y Fj(r)1490 2189 y Fk(1)1530 2175 y Fj(c)1569 2189 y Fk(1)1629 2175 y Fn(+)20 b Ff(\001)15 b(\001)g(\001)21 b Fn(+)f Fj(r)1978 2189 y Fi(M)2057 2175 y Fj(c)2096 2189 y Fi(M)2201 2175 y Fn(=)25 b Fj(R)q(q)s(;)796 b Fn(\(2.14a\))1449 2345 y Fj(r)1490 2359 y Fk(1)1550 2345 y Fn(+)20 b Ff(\001)15 b(\001)g(\001)21 b Fn(+)f Fj(r)1899 2359 y Fi(M)2003 2345 y Fn(=)25 b Fj(r)2140 2359 y Fk(0)2205 2345 y Ff(\025)g Fn(1)p Fj(;)856 b Fn(\(2.14b\))1449 2515 y Fj(r)1490 2529 y Fk(0)1560 2515 y Fn(is)31 b(minimal)p Fj(;)1237 b Fn(\(2.14c\))390 2699 y(where)35 b Ff(f)p Fj(c)742 2713 y Fk(1)782 2699 y Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(c)1038 2713 y Fi(M)1118 2699 y Ff(g)34 b Fn(=)g Fj(A)1370 2717 y Fi(p=q)1479 2699 y Fn(\()p Fj(f)10 b Fn(\);)39 b(note)d(that,)i(when)c(considering)i(generalized)h(semi-)390 2835 y(standard)f(maps,)i(it)g(is)f(not)g(necessary)h(to)f(in)m(tro)s (duce)g(the)g(set)h Ff(A)p Fn(\()p Fj(f)10 b Fn(\))36 b(\(see)i(the)f(lemma)390 2971 y(1)e(in)e([3)q(]\),)j(as)f(it)f(is)g (straigh)m(tforw)m(ard)h(to)g(con)m(vince)h(himself)e(b)m(y)g(using)g (that)g(all)h(F)-8 b(ourier)390 3107 y(lab)s(els)31 b(are)f(p)s(ositiv) m(e.)490 3243 y(The)25 b(Diophan)m(tine)i(problem)f(\(2.14\))i(has)e (alw)m(a)m(ys)i(a)e(\014nite,)h(nonzero)g(n)m(um)m(b)s(er)d(of)j(solu-) 390 3379 y(tions)g(and)f Fj(r)823 3393 y Fk(0)887 3379 y Ff(\024)f Fj(q)s Fn(,)i(as)g(it)g(can)g(b)s(e)f(pro)m(v)m(ed)g(b)m(y) h(reasoning)f(analogous)i(to)f(the)g(corresp)s(ond-)390 3515 y(ing)g(Lemma)g(2)h(in)e([3)q(].)40 b(In)26 b(particular)h(if)g Fj(q)h Ff(2)d Fj(A)2024 3533 y Fi(p=q)2133 3515 y Fn(\()p Fj(f)10 b Fn(\))27 b(then)g(\(2.14\))i(admits)e(the)g(solution)390 3651 y Fj(r)431 3665 y Fi(M)535 3651 y Fn(=)e(1,)31 b Fj(r)773 3665 y Fi(i)827 3651 y Fn(=)25 b(0)30 b Ff(8)p Fj(i)25 b(<)g(M)10 b Fn(,)31 b Fj(c)1394 3665 y Fi(M)1499 3651 y Fn(=)25 b(1,)31 b Fj(R)26 b Fn(=)f(1)30 b(and)g Fj(r)2180 3665 y Fk(0)2245 3651 y Fn(=)25 b(1.)490 3787 y(De\014ne)32 b Fj(r)817 3754 y Fg(\003)856 3787 y Fn(\()p Fj(f)10 b Fn(\))28 b Ff(\021)f Fj(r)1151 3754 y Fg(\003)1222 3787 y Fn(as)32 b Fj(r)1379 3754 y Fg(\003)1446 3787 y Fn(=)c Fj(r)1586 3801 y Fk(0)1625 3787 y Fn(.)46 b(W)-8 b(e)33 b(in)m(tro)s(duce)f(the)g(notation)h Fj(r)2827 3754 y Fg(\003)2898 3787 y Fn(simply)e(to)i(mak)m(e)390 3923 y(easier)j(the)g(comparison)g(with)f([3)q(])h(\(where)f (distinction)h(had)f(to)i(b)s(e)e(made)g(b)s(et)m(w)m(een)i Fj(r)3471 3937 y Fk(0)390 4059 y Fn(and)30 b Fj(r)611 4026 y Fg(\003)650 4059 y Fn(;)g(see)h(\(1.18\))i(in)d([3)q(]\).)490 4195 y(W)-8 b(e)31 b(pro)m(v)m(e)g(the)g(follo)m(wing)h(result.)390 4385 y Fu(Theorem.)46 b Fl(L)-5 b(et)27 b Fj(f)35 b Fl(b)-5 b(e)27 b(any)g(function)g(in)g Ff(B)1897 4399 y Fi(\032)1937 4385 y Fl(,)h(with)f Fj(\032)e(>)g Fn(0)p Fl(.)41 b(Consider)27 b(the)h(c)-5 b(one)27 b Ff(C)3243 4403 y Fi(p=q)r(;\014)3439 4385 y Fn(=)390 4521 y Ff(f)p Fj(z)34 b Ff(2)28 b Fc(C)h Fn(:)58 b Ff(j)15 b Fn(Im)g Fj(z)t Ff(j)30 b Fj(>)e Fn(0)p Fj(;)j Ff(j)15 b Fn(Re)h Fj(z)26 b Ff(\000)21 b Fj(p=q)s Ff(j)30 b(\024)e Fj(\014)5 b Ff(j)15 b Fn(Im)h Fj(z)t Ff(j)p Fl(,)35 b Fj(\014)f Ff(\025)29 b Fn(0)p Ff(g)p Fl(;)36 b(let)f Fj(!)d Ff(2)c(C)2905 4539 y Fi(p=q)r(;\014)3077 4521 y Fl(.)48 b(Then)35 b(the)390 4657 y Fn(rescaled)c(conjugating)h (function)1125 4862 y(\026)-51 b Fj(u)1171 4881 y Fk(0)p Fi(;p=q)1335 4862 y Fn(\()p Fj(\013;)15 b(";)g(!)s Fn(\))27 b(=)e Fj(u)1820 4734 y Fh(\022)1887 4862 y Fj(\013;)15 b(")2027 4761 y Fh(\020)2083 4862 y Fj(!)23 b Ff(\000)2263 4801 y Fj(p)p 2263 4841 46 4 v 2264 4925 a(q)2319 4761 y Fh(\021)2373 4784 y Fk(2)p Fi(=r)2477 4761 y Fo(\003)2514 4784 y Fk(\()p Fi(f)7 b Fk(\))2614 4862 y Fj(;)15 b(!)2714 4734 y Fh(\023)3278 4862 y Fn(\(2.15\))390 5068 y Fl(extends)45 b(to)f(a)g(function)g(c)-5 b(ontinuous)45 b(in)f Fj(!)j Fl(in)d(the)g(closur)-5 b(e)45 b(of)f(the)g(c)-5 b(one)45 b Ff(C)3151 5086 y Fi(p=q)r(;\014)3366 5068 y Fl(and)390 5204 y(analytic)36 b(in)f Fj(!)i Fl(in)e(the)g(interior)h(of)f Ff(C)1705 5222 y Fi(p=q)r(;\014)1876 5204 y Fl(,)g(for)g(any)h Fj(\014)e Ff(\025)29 b Fn(0)p Fl(,)35 b(analytic)h(in)f Fj(")g Fl(for)g Ff(j)p Fj(")p Ff(j)30 b Fj(<)f(a)p eop %%Page: 5 5 5 4 bop 1425 46 a Fk(SCALING)31 b(PR)n(OPER)-6 b(TIES)1121 b(5)270 241 y Fl(and)29 b(analytic)g(in)e Fj(\013)i Fl(for)f Ff(j)15 b Fn(Im)g Fj(\013)p Ff(j)26 b Fj(<)f(b)p Fl(,)j(with)h Fj(a)p Fl(,)g Fj(b)e Fl(two)i(p)-5 b(ositive)29 b(c)-5 b(onstants.)42 b(In)28 b(p)-5 b(articular,)270 377 y(the)33 b(fol)5 b(lowing)34 b(limit)f(exists:)1161 548 y Fn(\026)1159 560 y(\026)-51 b Fj(u)1205 579 y Fk(0)p Fi(;p=q)1369 560 y Fn(\()p Fj(\013;)15 b(")p Fn(\))27 b(=)73 b(lim)1702 625 y Fi(!)r Fg(!)p Fi(p=q)1944 560 y Fn(\026)-50 b Fj(u)1991 579 y Fk(0)p Fi(;p=q)2154 560 y Fn(\()p Fj(\013;)15 b(";)g(!)s Fn(\))p Fj(;)669 b Fn(\(2.16\))270 770 y Fl(it)44 b(is)h(indep)-5 b(endent)46 b(fr)-5 b(om)45 b(the)g(non-tangential)h(p)-5 b(ath)46 b(chosen)f(either)g(in)g(the)f(c)-5 b(omplex)270 906 y(upp)g(er)36 b(half)f(plane)h Fn(Im)14 b Fj(!)32 b(>)c Fn(0)35 b Fl(or)g(in)g(the)g(c)-5 b(omplex)36 b(lower)g(half)f (plane)h Fn(Im)14 b Fj(!)32 b(<)c Fn(0)p Fl(,)35 b(and)h(it)270 1042 y(is)d(analytic)g(for)h Ff(j)p Fj(")p Ff(j)26 b Fj(<)1082 1030 y Fn(\026)1079 1042 y(\026)-53 b Fj(\032)1118 1057 y Fk(0)p Fi(;f)1218 1042 y Fn(\()p Fj(p=q)s Fn(\))p Fl(,)33 b(for)g(some)1873 1030 y Fn(\026)1870 1042 y(\026)-53 b Fj(\032)1909 1057 y Fk(0)p Fi(;f)2009 1042 y Fn(\()p Fj(p=q)s Fn(\))26 b Fj(>)f Fn(0)p Fl(.)42 b(De\014ning)562 1190 y Fn(\026)559 1203 y(\026)539 1226 y Fj(R)609 1240 y Fk(0)649 1226 y Fn(\()p Fj(p=q)s Fn(\))26 b(=)52 b(inf)976 1287 y Fi(f)7 b Fg(2S)1107 1296 y Fd(1)1167 1214 y Fn(\026)1165 1226 y(\026)-54 b Fj(\032)1203 1241 y Fk(0)p Fi(;f)1304 1226 y Fn(\()p Fj(p=q)s Fn(\))p Fj(;)1758 1190 y Fn(\026)1755 1203 y(\026)1736 1226 y Fj(R)1805 1240 y Fk(0)p Fi(;r)1898 1226 y Fn(\()p Fj(p=q)s Fn(\))26 b(=)97 b(inf)2260 1287 y Fi(f)7 b Fg(2S)2391 1296 y Fd(1)2425 1287 y Fi(;)2225 1359 y(r)2259 1335 y Fo(\003)2295 1359 y Fk(\()p Fi(f)g Fk(\)=)p Fi(r)2506 1214 y Fn(\026)2503 1226 y(\026)-53 b Fj(\032)2542 1241 y Fk(0)p Fi(;f)2642 1226 y Fn(\()p Fj(p=q)s Fn(\))p Fj(;)286 b Fn(\(2.17\))270 1527 y Fl(ther)-5 b(e)40 b(exist)g(\014nite)f(strictly)h(p)-5 b(ositive)40 b(c)-5 b(onstants)2039 1492 y Fn(\026)2036 1504 y(\026)2016 1527 y Fj(C)2087 1541 y Fi(r)2125 1527 y Fj(;)2188 1492 y Fn(\026)2186 1504 y(\026)2165 1527 y Fj(C)2237 1541 y Fk(0)2276 1527 y Fj(;)2340 1492 y Fn(\026)2338 1504 y(\026)2316 1527 y Fj(D)2394 1541 y Fi(r)2432 1527 y Fj(;)2497 1492 y Fn(\026)2494 1504 y(\026)2472 1527 y Fj(D)2551 1541 y Fk(0)2590 1527 y Fl(,)41 b(dep)-5 b(ending)40 b(only)g(on)270 1663 y Fj(p=q)s Fl(,)32 b(such)h(that)h(one)f(has)1423 1811 y Fn(\026)1420 1824 y(\026)1399 1847 y Fj(C)1471 1861 y Fk(0)1536 1847 y Ff(\024)1654 1811 y Fn(\026)1651 1824 y(\026)1632 1847 y Fj(R)1701 1861 y Fk(0)1741 1847 y Fn(\()p Fj(p=q)s Fn(\))26 b Ff(\024)2091 1811 y Fn(\026)2089 1824 y(\026)2068 1847 y Fj(D)2145 1861 y Fk(0)2185 1847 y Fj(;)917 b Fn(\(2.18a\))1423 1986 y(\026)1420 1998 y(\026)1399 2021 y Fj(C)1471 2035 y Fi(r)1534 2021 y Ff(\024)1652 1986 y Fn(\026)1650 1998 y(\026)1630 2021 y Fj(R)1700 2035 y Fk(0)p Fi(;r)1793 2021 y Fn(\()p Fj(p=q)s Fn(\))26 b Ff(\024)2143 1986 y Fn(\026)2141 1998 y(\026)2120 2021 y Fj(D)2197 2035 y Fi(r)2235 2021 y Fj(;)862 b Fn(\(2.18b\))270 2205 y Fl(for)33 b(al)5 b(l)33 b(inte)-5 b(ger)33 b Fj(r)28 b Ff(\024)d Fj(q)s Fl(.)370 2409 y Fn(The)31 b(pro)s(of)g(of)i(the)f (Theorem)f(ab)s(o)m(v)m(e)j(essen)m(tially)g(follo)m(ws)f(from)e(the)i (analysis)f(in)g([3],)270 2545 y(up)21 b(to)i(the)g(last)g(assertion)g (on)f(the)h(\014niteness)f(of)g(the)h(quan)m(tities)2513 2509 y(\026)2510 2522 y(\026)2491 2545 y Fj(R)2560 2559 y Fk(0)2600 2545 y Fn(\()p Fj(p=q)s Fn(\))g(and)3019 2509 y(\026)3016 2522 y(\026)2996 2545 y Fj(R)3066 2559 y Fk(0)p Fi(;r)3159 2545 y Fn(\()p Fj(p=q)s Fn(\):)270 2681 y(in)34 b(fact)h(a)f(constructiv)m(e)i(b)s(ound)c(from)i(ab)s(o)m (v)m(e)h(requires)f(some)g(extra)h(w)m(ork,)h(whic)m(h)e(can)270 2817 y(b)s(e)i(b)s(orro)m(w)m(ed)g(from)g(the)g(analysis)h(p)s (ermormed)e(in)h([4)q(].)59 b(F)-8 b(urthermore,)38 b(as)f(in)f([4],)j (the)270 2953 y(result)23 b(whic)m(h)h(can)g(b)s(e)e(pro)m(v)m(ed)i (for)g(generalized)h(semistandard)e(maps)g(giv)m(es)i(information)270 3089 y(also)39 b(on)f(the)g(radius)f(of)h(con)m(v)m(ergence)j(of)d(the) g(limit)h(functions)f(of)g(the)g(corresp)s(onding)270 3225 y(generalized)32 b(standard)d(maps;)i(see)g(sect.)41 b(5.)370 3361 y(In)23 b(the)h(follo)m(wing)h(w)m(e)f(shall)g(consider)g (explicitly)h(the)f(case)h Fj(!)10 b Ff(\000)d Fj(p=q)28 b Fn(=)d Fj(i\021)s Fn(,)h(with)e Fj(\021)k Ff(2)d Fc(R)p Fn(:)270 3497 y(the)31 b(analysis)f(can)h(b)s(e)f(easily)h(extended)g (to)g(the)f(case)i Fj(\020)g Ff(2)24 b(C)2361 3515 y Fi(p=q)r(;\014)2563 3497 y Fn(as)31 b(in)f([1])h(and)f([3].)1365 3708 y(3.)51 b Fm(The)34 b(lo)n(wer)f(bound)370 3898 y Fn(W)-8 b(e)36 b(can)g(consider)g(a)g(tree)g(expansion)f(for)h(the)f (conjugating)i(function)e(as)h(in)f([3)q(],)i(to)270 4034 y(whic)m(h)26 b(w)m(e)h(refer)g(\(see)g(also)h([1]\):)40 b(the)26 b(only)h(di\013erence)g(is)g(that)g(no)m(w)f Fj(\027)2677 4048 y Fi(u)2747 4034 y Ff(\025)f Fn(1)i(for)g(all)g Fj(u)e Ff(2)g Fj(#)270 4170 y Fn(and)30 b(\(as)h(a)f(consequence\))i Fj(\027)f Ff(\025)25 b Fj(k)s Fn(.)41 b(So)30 b(w)m(e)h(ha)m(v)m(e)1429 4354 y Fj(u)1481 4306 y Fk(\()p Fi(k)r Fk(\))1481 4380 y(0)p Fi(;\027)1604 4354 y Fn(=)1744 4267 y Fh(X)1700 4465 y Fi(#)p Fg(2T)1828 4477 y Fq(k)q(;\027)1935 4354 y Fn(V)-8 b(al)q(\()p Fj(#)p Fn(\))p Fj(;)988 b Fn(\(3.1\))270 4614 y(where)1086 4798 y(V)-8 b(al)q(\()p Fj(#)p Fn(\))26 b(=)f Fj(i)1494 4761 y Fi(k)r Fg(\000)p Fk(1)1627 4697 y Fh(h)1677 4712 y(Y)1670 4909 y Fi(u)p Fg(2)p Fi(#)1825 4737 y Fj(f)1870 4751 y Fi(\027)1905 4759 y Fq(u)1949 4737 y Fj(\027)2000 4704 y Fi(m)2062 4712 y Fq(u)1994 4759 y Fi(u)p 1825 4777 283 4 v 1891 4860 a Fj(m)1971 4874 y Fi(u)2016 4860 y Fn(!)2117 4697 y Fh(ih)2204 4712 y(Y)2203 4909 y Fi(`)p Fg(2)p Fi(#)2336 4798 y Fj(g)s Fn(\()p Fj(\027)2462 4813 y Fi(`)2496 4798 y Fn(\))2531 4697 y Fh(i)3203 4798 y Fn(\(3.2\))p eop %%Page: 6 6 6 5 bop 390 46 a Fk(6)739 b(ALBER)-6 b(TO)30 b(BERRETTI)h(AND)f(GUIDO)h (GENTILE)390 241 y Fn(where)f(the)g(same)h(notations)h(as)e(in)g([3)q (])h(are)f(used.)40 b(In)30 b(particular)1287 439 y Fj(g)s Fn(\()p Fj(\027)1413 454 y Fi(`)1447 439 y Fn(\))25 b(=)1691 378 y(1)p 1613 419 201 4 v 1613 502 a Fj(\015)5 b Fn(\()p Fj(\027)1745 517 y Fi(`)1779 502 y Fn(\))1849 439 y(=)2256 378 y(1)p 1955 419 648 4 v 1955 502 a(2\(cos)17 b(2)p Fj(\031)s(!)s(\027)2378 517 y Fi(`)2431 502 y Ff(\000)j Fn(1\))3323 439 y(\(3.3\))390 644 y(is)28 b(the)g Fl(pr)-5 b(op)g(agator)42 b Fn(of)29 b(the)f(line)g Fj(`)p Fn(.)40 b(The)28 b(line)g Fj(`)g Fn(is)g(said)g(to)h(b)s(e)f(on)g(scale)h Fj(n)c Fn(=)g(0)j(if)g Fj(\027)3256 659 y Fi(`)3315 644 y Ff(2)c Fj(q)s Fc(N)390 780 y Fn(and)30 b(on)g(scale)i Fj(n)25 b Fn(=)f(1)31 b(otherwise.)490 916 y(The)k(main)g(di\013erence) h(\(and)f(simpli\014cation\))i(with)e(resp)s(ect)g(to)i([3])f(is)f (that)i(no)m(w)e(no)390 1052 y(resonances)c(are)g(p)s(ossible)f(as)h Fj(\027)1492 1066 y Fi(u)1563 1052 y Ff(\025)25 b Fn(1)31 b(for)f(all)i Fj(u)25 b Ff(2)h Fj(#)p Fn(.)41 b(So)30 b(the)h(analogue)h(of)f(Prop)s(osition)390 1188 y(1)37 b(of)h([3])g(holds,)g(and)e(the)i(pro)s(of)e(is)h(the)g(same)h(\(w)m(e) g(refer)e(to)i([3)q(])f(for)g(the)g(de\014nition)g(of)390 1324 y(clusters\).)390 1528 y Fu(Prop)s(osition)f(1.)46 b Fl(A)32 b(cluster)h(has)h(at)f(le)-5 b(ast)34 b Fj(r)1999 1495 y Fg(\003)2070 1528 y Fl(no)-5 b(des.)490 1732 y Fn(Then)32 b(a)j(result)e(analogous)j(to)e(Lemma)g(4)g(in)g([3)q(])g (is)g(easily)h(obtained)f(\(of)g(course)g(for)390 1867 y(the)f(n)m(um)m(b)s(er)f(of)h(all)g(lines)g(on)g(scale)h Fj(n)29 b Fn(=)g(0)k(as)g(no)g(resonan)m(t)h(line)f(is)g(p)s (ossible\),)g(that)h(is,)390 2003 y(b)m(y)c(denoting)h(with)f Fj(N)1170 2017 y Fk(0)1210 2003 y Fn(\()p Fj(#)p Fn(\))g(the)h(n)m(um)m (b)s(er)e(of)h(lines)h(in)f Fj(#)g Fn(on)g(scale)i Fj(n)25 b Fn(=)g(0,)31 b(one)g(has)1659 2207 y Fj(N)1732 2221 y Fk(0)1771 2207 y Fn(\()p Fj(#)p Fn(\))26 b Ff(\024)2017 2106 y Fh(j)2091 2145 y Fj(k)p 2075 2186 83 4 v 2075 2269 a(r)2119 2243 y Fg(\003)2168 2106 y Fh(k)2216 2207 y Fj(:)1082 b Fn(\(3.4\))390 2390 y(if)30 b Fj(k)k Fn(is)c(the)h(n)m (um)m(b)s(er)e(of)h(no)s(des)g(of)g Fj(#)p Fn(.)490 2526 y(By)g(de\014ning)936 2710 y Fj(R)1005 2724 y Fk(0)1045 2710 y Fn(\()p Fj(!)s Fn(\))c(=)52 b(inf)1296 2771 y Fi(f)7 b Fg(2S)1427 2780 y Fd(1)1477 2710 y Fj(\032)1524 2725 y Fk(0)p Fi(;f)1625 2710 y Fn(\()p Fj(!)s Fn(\))p Fj(;)197 b(R)2046 2724 y Fk(0)p Fi(;r)2139 2710 y Fn(\()p Fj(!)s Fn(\))26 b(=)97 b(inf)2426 2771 y Fi(f)7 b Fg(2S)2557 2780 y Fd(1)2591 2771 y Fi(;)2391 2843 y(r)2425 2819 y Fo(\003)2461 2843 y Fk(\()p Fi(f)g Fk(\)=)p Fi(r)2661 2710 y Fj(\032)2708 2725 y Fk(0)p Fi(;f)2808 2710 y Fn(\()p Fj(!)s Fn(\))p Fj(:)360 b Fn(\(3.5\))390 2988 y(then)28 b(one)g(immediately)h(obtains)f(that)g(there)g(exist)h(constan)m(ts)g Fj(C)2692 3002 y Fk(0)2731 2988 y Fj(;)15 b(C)2836 3002 y Fi(r)2902 2988 y Fn(suc)m(h)28 b(that,)h(for)f Fj(\021)390 3124 y Fn(small,)1213 3307 y Fj(R)1282 3321 y Fk(0)p Fi(;r)1375 3307 y Fn(\()p Fj(!)s Fn(\))e Ff(\025)f Fj(C)1692 3321 y Fi(r)1730 3307 y Fj(\021)1778 3270 y Fk(2)p Fi(=r)1887 3307 y Fj(;)197 b(R)2178 3321 y Fk(0)2217 3307 y Fn(\()p Fj(!)s Fn(\))26 b Ff(\025)f Fj(C)2534 3321 y Fk(0)2574 3307 y Fj(\021)2622 3270 y Fk(2)2661 3307 y Fj(:)637 b Fn(\(3.6\))390 3491 y(With)34 b(resp)s(ect)g(to)h([3)q(])f(there)g (is)g(an)f(o)m(v)m(ersimpli\014cation)j(due)e(to)g(the)g(absence)h(of)f (reson-)390 3627 y(ances:)40 b(so)29 b(no)g(cancellation)i(mec)m (hanism)e(has)g(to)g(b)s(e)g(discussed)e(in)i(order)f(to)i(obtain)f (the)390 3763 y(righ)m(t)f(b)s(ound)d(from)i(b)s(elo)m(w)g(of)h(the)f (radius)f(of)i(con)m(v)m(ergence.)42 b(Note)28 b(also)g(that)g(the)g (formal)390 3899 y(solubilit)m(y)g(requires)f(no)g(discussion)g(as)h Fj(\027)j Ff(\025)25 b Fj(k)30 b Fn(yields)d Fj(\027)2321 3914 y Fi(`)2380 3899 y Ff(6)p Fn(=)d(0)k(for)f(an)m(y)h(line)g Fj(`)f Fn(in)g(an)m(y)g(tree)390 4035 y Fj(#)p Fn(.)490 4171 y(If)598 4159 y(\026)595 4171 y(\026)-53 b Fj(\032)634 4186 y Fk(0)p Fi(;f)734 4171 y Fn(\()p Fj(p=q)s Fn(\))37 b(denotes)g(the)g(radius)f(of)g(con)m(v)m(ergence)j(for)e(the)g (rescaled)g(function)3402 4159 y(\026)3399 4171 y(\026)-51 b Fj(u)3445 4185 y Fk(0)3485 4171 y Fn(,)390 4307 y Fl(i.e.)1151 4526 y Fn(\026)1148 4538 y(\026)e Fj(\032)1187 4553 y Fk(0)p Fi(;f)1287 4538 y Fn(\()p Fj(p=q)s Fn(\))26 b(=)38 b(inf)1614 4599 y Fi(\013)p Fg(2)p Fe(T)1751 4383 y Fh( )1823 4538 y Fn(lim)15 b(sup)1872 4617 y Fi(k)r Fg(!1)2101 4406 y Fh(\014)2101 4461 y(\014)2101 4515 y(\014)2101 4570 y(\014)2140 4526 y Fn(\026)2137 4538 y(\026)-51 b Fj(u)2183 4490 y Fk(\()p Fi(k)r Fk(\))2183 4564 y(0)2281 4538 y Fn(\()p Fj(\013)p Fn(\))2409 4406 y Fh(\014)2409 4461 y(\014)2409 4515 y(\014)2409 4570 y(\014)2440 4433 y Fk(1)p Fi(=k)2554 4383 y Fh(!)2625 4405 y Fg(\000)p Fk(1)2720 4538 y Fj(;)578 b Fn(\(3.7\))390 4776 y(w)m(e)31 b(can)g(de\014ne,)e(analogously)j(to)f(\(3.5\))r(,)807 4924 y(\026)805 4937 y(\026)785 4959 y Fj(R)855 4973 y Fk(0)894 4959 y Fn(\()p Fj(p=q)s Fn(\))26 b(=)53 b(inf)1221 5021 y Fi(f)7 b Fg(2S)1352 5030 y Fd(1)1413 4947 y Fn(\026)1410 4959 y(\026)-53 b Fj(\032)1449 4974 y Fk(0)p Fi(;f)1549 4959 y Fn(\()p Fj(p=q)s Fn(\))p Fj(;)2000 4924 y Fn(\026)1997 4937 y(\026)1977 4959 y Fj(R)2047 4973 y Fk(0)p Fi(;r)2140 4959 y Fn(\()p Fj(p=q)s Fn(\))26 b(=)97 b(inf)2502 5021 y Fi(f)7 b Fg(2S)2633 5030 y Fd(1)2667 5021 y Fi(;)2467 5092 y(r)2501 5069 y Fo(\003)2537 5092 y Fk(\()p Fi(f)g Fk(\)=)p Fi(r)2747 4947 y Fn(\026)2745 4959 y(\026)-53 b Fj(\032)2784 4974 y Fk(0)p Fi(;f)2884 4959 y Fn(\()p Fj(p=q)s Fn(\))p Fj(:)209 b Fn(\(3.8\))p eop %%Page: 7 7 7 6 bop 1425 46 a Fk(SCALING)31 b(PR)n(OPER)-6 b(TIES)1121 b(7)270 245 y Fn(so)40 b(that)h(the)f(b)s(ounds)e(\(3.6\))k(imply)-8 b(,)43 b(for)d(suitable)g(constan)m(ts)2534 209 y(\026)2532 222 y(\026)2511 245 y Fj(C)2583 259 y Fk(0)2622 245 y Fj(;)2686 209 y Fn(\026)2683 222 y(\026)2662 245 y Fj(C)2734 259 y Fi(r)2812 245 y Fn(\(dep)s(ending)f(on)270 381 y Fj(p=q)s Fn(\),)1148 529 y(\026)1145 541 y(\026)1126 564 y Fj(R)1195 578 y Fk(0)p Fi(;r)1288 564 y Fn(\()p Fj(p=q)s Fn(\))26 b Ff(\025)1638 529 y Fn(\026)1636 541 y(\026)1615 564 y Fj(C)1687 578 y Fi(r)1725 564 y Fj(;)1969 529 y Fn(\026)1967 541 y(\026)1947 564 y Fj(R)2017 578 y Fk(0)2056 564 y Fn(\()p Fj(p=q)s Fn(\))g Ff(\025)2406 529 y Fn(\026)2404 541 y(\026)2383 564 y Fj(C)2454 578 y Fk(0)2494 564 y Fj(;)684 b Fn(\(3.9\))270 752 y(whic)m(h)30 b(giv)m(e)i(a)f(b)s(ound)d(from)i(b)s(elo)m(w)h(on)f(the)g(quan)m (tities)2270 717 y(\026)2268 729 y(\026)2248 752 y Fj(R)2318 766 y Fk(0)p Fi(;r)2411 752 y Fn(\()p Fj(p=q)s Fn(\))h(and)2846 717 y(\026)2843 729 y(\026)2823 752 y Fj(R)2893 766 y Fk(0)2933 752 y Fn(\()p Fj(p=q)s Fn(\).)1376 964 y(4.)50 b Fm(The)35 b(upper)e(bound)370 1154 y Fn(F)-8 b(or)31 b(the)f(rescaled)i(function)1398 1142 y(\026)1395 1154 y(\026)-50 b Fj(u)1442 1168 y Fk(0)1506 1154 y Fn(=)1611 1142 y(\026)1608 1154 y(\026)f Fj(u)1654 1172 y Fk(0)p Fi(;p=q)1818 1154 y Fn(,)31 b(one)f(has)874 1419 y(\026)872 1431 y(\026)-51 b Fj(u)918 1383 y Fk(\()p Fi(k)r Fk(\))918 1457 y(0)p Fi(;\027)1041 1431 y Fn(=)1137 1245 y Fh(8)1137 1327 y(<)1137 1490 y(:)1218 1291 y(P)1314 1318 y Fg(0)1314 1386 y Fi(#)p Fg(2T)1442 1398 y Fq(k)q(;\027)1553 1359 y Fn(V)-8 b(al)1684 1323 y Fg(0)1707 1359 y Fn(\()p Fj(#)p Fn(\))p Fj(;)92 b Fn(if)30 b Fj(k)f Ff(2)24 b Fj(r)2236 1326 y Fg(\003)2275 1359 y Fc(N)p Fj(;)106 b(\027)31 b Ff(2)25 b Fj(q)s Fc(N)p Fj(;)1218 1523 y Fn(0)685 b(otherwise,)3203 1431 y(\(4.1\))270 1707 y(where)532 1639 y Fh(P)628 1665 y Fg(0)681 1707 y Fn(means)29 b(that)h(only)g(trees)g Fj(#)f Fn(with)g Fj(N)1932 1721 y Fk(0)1972 1707 y Fn(\()p Fj(#)p Fn(\))d(=)e Fj(k)s(=r)2356 1674 y Fg(\003)2426 1707 y Fn(ha)m(v)m(e)30 b(to)h(b)s(e)d(summed)h(o)m(v)m(er)270 1843 y(and)i(V)-8 b(al)579 1806 y Fg(0)602 1843 y Fn(\()p Fj(#)p Fn(\))32 b(di\013ers)f(from)g(V)-8 b(al)r(\()p Fj(#)p Fn(\))31 b(as,)i(for)e Fj(\027)37 b Fn(m)m(ultiple)32 b(of)g Fj(q)s Fn(,)f Fj(\015)5 b Fn(\()p Fj(\027)h Fn(\))32 b(has)f(to)i(b)s(e)d(replaced)270 1979 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y(sidered)d(class)h(of)g(functions)e Ff(S)1349 4367 y Fk(\()p Fi(N)7 b Fk(\))1342 4441 y(1)1471 4415 y Fn(.)41 b(So)30 b(the)h(b)s(ounds)d(from)i(ab)s(o)m(v)m(e)1217 4563 y(\026)1214 4576 y(\026)1195 4599 y Fj(R)1264 4613 y Fk(0)p Fi(;r)1357 4599 y Fn(\()p Fj(!)s Fn(\))c Ff(\024)1633 4563 y Fn(\026)1630 4576 y(\026)1609 4599 y Fj(D)1687 4613 y Fi(r)1725 4599 y Fj(;)1969 4563 y Fn(\026)1967 4576 y(\026)1947 4599 y Fj(R)2017 4613 y Fk(0)2056 4599 y Fn(\()p Fj(!)s Fn(\))g Ff(\024)2332 4563 y Fn(\026)2329 4576 y(\026)2308 4599 y Fj(D)2386 4613 y Fk(0)2425 4599 y Fj(;)753 b Fn(\(4.4\))270 4787 y(immediately)43 b(follo)m(w,)i(for)c (suitable)h(constan)m(ts)2038 4751 y(\026)2035 4764 y(\026)2014 4787 y Fj(D)2092 4801 y Fi(r)2171 4787 y Fn(and)2382 4751 y(\026)2380 4764 y(\026)2359 4787 y Fj(D)2436 4801 y Fk(0)2476 4787 y Fn(,)i(and)d(the)h(pro)s(of)e(of)i(the)270 4923 y(theorem)31 b(is)f(complete.)p eop %%Page: 8 8 8 7 bop 390 46 a Fk(8)739 b(ALBER)-6 b(TO)30 b(BERRETTI)h(AND)f(GUIDO)h (GENTILE)490 241 y Fn(No)m(w)g(w)m(e)g(pro)m(v)m(e)g(\(4.3\))r(.)40 b(T)-8 b(ak)m(e)32 b(an)m(y)f(function)f(in)g Ff(S)2237 255 y Fk(1)2307 241 y Fn(of)g(the)h(form)1247 488 y Fj(f)10 b Fn(\()p Fj(x)p Fn(\))25 b(=)1579 374 y Fi(N)1545 401 y Fh(X)1546 595 y Fi(\027)t Fk(=1)1692 488 y Fj(f)1737 502 y Fi(\027)1794 488 y Fj(e)1836 450 y Fi(i\027)t(x)1944 488 y Fj(;)197 b(f)2211 502 y Fi(\027)2279 488 y Fj(>)25 b(F)38 b(>)25 b Fn(0)p Fj(;)686 b Fn(\(4.5\))390 758 y(for)30 b(some)h(\014xed)f(constan)m(t)h Fj(F)13 b Fn(:)41 b(this)30 b(de\014nes)g(the)g(class)h Ff(S)2387 710 y Fk(\()p Fi(N)7 b Fk(\))2380 784 y(1)2509 758 y Fn(.)490 894 y(The)31 b(propagators)i Fj(g)s Fn(\()p Fj(\027)1307 909 y Fi(`)1341 894 y Fn(\))f(in)g(\(3.2\))i(are)e(all)h(negativ)m(e,)i (so)e(that)f(the)h(c)m(hoice)h(\(4.5\))g(and)390 1030 y(the)e(de\014nition)g(of)g(V)-8 b(al)1191 993 y Fg(0)1214 1030 y Fn(\()p Fj(#)p Fn(\))32 b(yield)h(that,)g(for)f(an)m(y)g Fj(#)c Ff(2)g(T)2351 1045 y Fi(k)r(;\027)2452 1030 y Fn(,)k(V)-8 b(al)2641 993 y Fg(0)2664 1030 y Fn(\()p Fj(#)p Fn(\))32 b(is)g(giv)m(en)i(b)m(y)d(\()p Ff(\000)p Fn(1\))3466 997 y Fi(k)390 1166 y Fn(times)42 b Fj(i)675 1133 y Fi(k)r Fg(\000)p Fk(1)849 1166 y Fn(times)g(a)g(p)s(ositiv)m(e)h (quan)m(tit)m(y)f Fj(P)13 b Fn(\()p Fj(#)p Fn(\).)75 b(Therefore)41 b(all)h(trees)g(in)f Ff(T)3163 1181 y Fi(k)r(;\027)3305 1166 y Fn(ha)m(v)m(e,)390 1302 y(apart)31 b(from)g(a)g(comm)m(un)g(factor)h([)p Ff(\000)p Fn(\()p Ff(\000)p Fj(i)p Fn(\))1821 1269 y Fi(k)r Fg(\000)p Fk(1)1954 1302 y Fn(],)g(a)f(v)-5 b(alue)32 b Fj(P)13 b Fn(\()p Fj(#)p Fn(\))31 b(strictly)h(p)s(ositiv)m(e:)43 b(suc)m(h)31 b(a)390 1438 y(prop)s(ert)m(y)f(can)g(b)s(e)g(fruitfully)g(used)f(in)h (order)g(to)h(pro)m(v)m(e)h(the)e(follo)m(wing)i(result.)390 1642 y Fu(Lemma.)45 b Fl(L)-5 b(et)976 1630 y Fn(\026)973 1642 y(\026)-50 b Fj(u)1020 1656 y Fk(0)1102 1642 y Fl(b)-5 b(e)43 b(the)g(limit)g(function)50 b Fn(\(2.16\))45 b Fl(for)f Fj(f)53 b Ff(2)43 b(S)2681 1594 y Fk(\()p Fi(N)7 b Fk(\))2674 1668 y(1)2803 1642 y Fl(,)45 b(i.e.)e(of)g(the)g(form)390 1778 y Fn(\(4.5\))r Fl(,)e(and)g(let)f Fj(r)1009 1745 y Fg(\003)1087 1778 y Fn(=)e Fj(r)1237 1792 y Fk(0)1316 1778 y Fl(and)j Fj(R)f Fl(b)-5 b(e)40 b(the)h(inte)-5 b(gers)40 b(app)-5 b(e)g(aring)42 b(in)e(the)g(solution)h(of)g(the)390 1914 y(Diophantine)34 b(pr)-5 b(oblem)42 b Fn(\(2.14\))s Fl(.)h(Then)33 b(ther)-5 b(e)34 b(exists)f(a)h(universal)f(p)-5 b(ositive)34 b(c)-5 b(onstant)35 b Fj(D)390 2050 y Fl(such)e(that)1647 2129 y Fh(\014)1647 2184 y(\014)1647 2238 y(\014)1685 2222 y Fn(\026)1683 2234 y(\026)-51 b Fj(u)1729 2186 y Fk(\()p Fi(k)r Fk(\))1729 2260 y(0)p Fi(;\027)1827 2129 y Fh(\014)1827 2184 y(\014)1827 2238 y(\014)1883 2234 y Ff(\025)25 b Fj(D)2057 2196 y Fi(k)2099 2234 y Fj(F)2170 2196 y Fi(k)2213 2234 y Fj(;)1085 b Fn(\(4.6\))390 2446 y Fl(for)33 b(any)g Fj(k)c Fn(=)c Fj(nr)980 2413 y Fg(\003)1018 2446 y Fl(,)33 b(with)g Fj(n)25 b Ff(2)g Fc(N)p Fl(,)32 b(and)i(for)f(any)g Fj(\027)e Fn(=)25 b Fj(nR)q(q)s Fl(,)31 b(while)2705 2434 y Fn(\026)2703 2446 y(\026)-51 b Fj(u)2749 2398 y Fk(\()p Fi(k)r Fk(\))2749 2472 y(0)p Fi(;\027)2872 2446 y Fn(=)25 b(0)33 b Fl(otherwise.)390 2650 y(Pr)-5 b(o)g(of.)43 b Fn(Set)32 b Fj(r)873 2617 y Fg(\003)938 2650 y Ff(\021)27 b Fj(r)34 b Fn(and)d Fj(R)q(q)e Ff(\021)e Fj(Q)p Fn(.)44 b(The)31 b(mo)s(dulus)e(of)j(eac)m (h)h(propagator)f(with)f Fj(\027)3253 2665 y Fi(`)3323 2650 y Fj(=)-55 b Ff(2)26 b Fj(q)s Fc(N)390 2786 y Fn(can)c(b)s(e)f(b)s (ounded)f(from)h(ab)s(o)m(v)m(e)i(with)f(4,)i(while)d(the)h (propagators)h(with)e Fj(\027)2886 2801 y Fi(`)2944 2786 y Fn(=)k Fj(n)3095 2801 y Fi(`)3128 2786 y Fj(q)s Fn(,)e Fj(n)3275 2801 y Fi(`)3333 2786 y Ff(2)i Fc(N)p Fn(,)390 2922 y(are)31 b(giv)m(en)g(b)m(y)f Fj(g)s Fn(\()p Fj(\027)1031 2937 y Fi(`)1065 2922 y Fn(\))c(=)f(\(2)p Fj(\031)s(iq)s Fn(\))1467 2889 y Fg(\000)p Fk(2)1563 2922 y Fj(n)1618 2884 y Fg(\000)p Fk(2)1618 2952 y Fi(`)1711 2922 y Fn(.)490 3058 y(Note)35 b(that)f(an)m(y)h Fj(#)e Fn(con)m(tributing)i(to)1821 3046 y(\026)1818 3058 y(\026)-51 b Fj(u)1865 3010 y Fk(\()p Fi(nr)r Fk(\))1865 3084 y(0)p Fi(;\027)2034 3058 y Fn(m)m(ust)34 b(ha)m(v)m(e)h Fj(nr)h Fn(no)s(des,)e Fj(n)g Fn(lines)g(on)g(scale)390 3194 y Fj(n)28 b Fn(=)g(0,)33 b(hence)g Fj(n)f Fn(clusters)g(with)g Fj(r)j Fn(no)s(des:)43 b(eac)m(h)34 b(cluster)e(con)m(tributes)h Fj(Q)f Fn(to)h Fj(\027)6 b Fn(,)33 b(so)f(that)390 3330 y(one)i(m)m(ust)g(ha)m(v)m(e)g(for)g(the)g(ro)s(ot)g(line)g(a)g(momen)m (tum)f Fj(\027)k Fn(=)30 b Fj(r)s(Q)j Fn(in)h(order)f(that)3128 3318 y(\026)3125 3330 y(\026)-51 b Fj(u)3171 3282 y Fk(\()p Fi(nr)r Fk(\))3171 3356 y(0)p Fi(;\027)3338 3330 y Ff(6)p Fn(=)30 b(0.)390 3466 y(This)g(suggests)h(us)e(to)i(write)1358 3638 y(\026)1356 3650 y(\026)-51 b Fj(u)1402 3602 y Fk(\()p Fi(k)r Fk(\))1402 3676 y(0)p Fi(;\027)1525 3650 y Fn(=)25 b Ff(j)q Fj(g)s Fn(\()p Fj(\027)6 b Fn(\))p Ff(j)1855 3549 y Fh(h)1898 3650 y Fn(\()p Ff(\000)p Fn(1\))2084 3612 y Fi(k)2127 3650 y Fj(i)2158 3612 y Fi(k)r Fg(\000)p Fk(1)2292 3549 y Fh(i)2350 3650 y Fj( )2412 3612 y Fk(\()p Fi(k)r Fk(\))2409 3672 y Fi(\027)2510 3650 y Fj(;)788 b Fn(\(4.7\))390 3863 y(so)27 b(de\014ning)e(the)i(quan)m(tit)m(y)h Fj( )1412 3815 y Fk(\()p Fi(k)r Fk(\))1409 3874 y Fi(\027)1535 3863 y Fj(>)d Fn(0.)40 b(W)-8 b(e)27 b(can)g(pro)m(v)m(e)g(that)g Fj( )2552 3815 y Fk(\()p Fi(k)r Fk(\))2549 3874 y Fi(\027)2677 3863 y Fn(satis\014es)g(the)f(recursiv)m(e)390 3999 y(b)s(ound)1329 4182 y Fj( )1391 4134 y Fk(\()p Fi(nr)r Fk(\))1388 4211 y Fi(nQ)1553 4182 y Ff(\025)f Fj(n)1704 4145 y Fk(2)1743 4182 y Fn(8)1788 4145 y Fk(1)p Fg(\000)p Fi(n)1940 4182 y Ff(j)q Fj(g)s Fn(\()p Fj(Q)p Fn(\))p Ff(j)2180 4140 y Fi(n)p Fg(\000)p Fk(1)2332 4182 y Fj( )2394 4134 y Fk(\()p Fi(r)r Fk(\))p Fi(n)2391 4211 y(Q)2530 4182 y Fj(;)768 b Fn(\(4.8\))390 4366 y(for)33 b(an)m(y)g Fj(n)c Ff(2)g Fc(N)p Fn(.)48 b(The)32 b(pro)s(of)g(of)h(suc)m(h)g(an)f (assertion)i(is)f(b)m(y)g(induction)f(on)h Fj(n)p Fn(.)48 b(The)32 b(case)390 4502 y 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b(recall)37 b(\(1.18\))h(in)e([3)q(])390 4094 y(and,)g(noting)g(that)g Fj(N)46 b Fn(giv)m(en)36 b(b)m(y)f(the)h(second)f(line)h(of)43 b(\(5.2\))37 b(satis\014es)f Fj(N)44 b(<)33 b(q)s Fn(,)k(use)e(the)390 4230 y(second)f(assertion)g(in)g(Prop)s(osition)f(1)h(of)g([3)q(].)51 b(In)33 b(the)h(latter)h(case)f Fj(N)41 b(<)31 b(q)36 b Fn(implies)e(also)390 4366 y(that)d(one)g(has)f Ff(A)p Fn(\()p Fj(f)10 b Fn(\))25 b(=)g Fj(A)1303 4384 y Fi(p=q)1412 4366 y Fn(\()p Fj(f)10 b Fn(\))30 b(for)g(the)h(generalized)h(standard) d(maps.)490 4502 y(In)i(order)h(to)h(ha)m(v)m(e)h(the)e(same)h (solution)g(for)f(b)s(oth)g(the)g(Diophan)m(tine)i(problems,)e(it)h(is) 390 4638 y(enough)38 b(to)h(sho)m(w)f(that)g(there)h(exists)f(a)h (solution)g(of)f(\(1.17\))i(in)e([3)q(])g(suc)m(h)g(that)g Fj(r)3292 4605 y Fg(0)3289 4663 y Fi(i)3356 4638 y Fn(=)g(0)390 4774 y Ff(8)p Fj(i)31 b Fn(=)g(1)p Fj(;)15 b(:)g(:)g(:)32 b(;)15 b(M)10 b Fn(.)52 b(This)33 b(is)h(easily)h(obtained)f(for)g (functions)f(of)h(the)g(form)g(\(5.1\))q(,)h(simply)390 4910 y(sho)m(wing)21 b(that,)j(in)d(suc)m(h)f(a)i(case,)i(a)d(solution) h(\(with)f Fj(R)26 b Fn(=)f(1\))d(of)f(the)g(Diophan)m(tine)i(problem) 390 5046 y(\(2.14\))39 b(can)f(b)s(e)f(explicitly)h(constructed)g(b)m (y)f(the)g(pro)s(cedure)f(describ)s(ed)g(b)s(elo)m(w,)k(hence)390 5182 y(realizing)32 b(that)f(it)g(solv)m(es)g(also)h(the)e(Diophan)m (tine)i(problem)d(\(1.17\))k(in)d([3)q(].)p eop %%Page: 11 11 11 10 bop 1425 46 a Fk(SCALING)31 b(PR)n(OPER)-6 b(TIES)1085 b(11)370 241 y Fn(F)-8 b(or)32 b(\014xed)e Fj(q)k Fn(and)d Fj(r)s Fn(,)g(let)h Fj(N)41 b Fn(b)s(e)31 b(suc)m(h)g(that)h(\(5.2\))h (is)e(satis\014ed.)43 b(Note)33 b(that)f(then)f(b)s(oth)270 377 y(for)e(the)g(function)g(\(4.5\))j(in)c(the)i(case)g(of)g (generalized)h(semistandard)d(maps)h(and)g(for)g(the)270 513 y(function)42 b(\(5.1\))h(in)f(the)g(case)h(of)f(generalized)h (standard)e(maps,)k(one)d(has)g Fj(A)3041 531 y Fi(p=q)3150 513 y Fn(\()p Fj(f)10 b Fn(\))44 b(=)270 649 y Ff(f)p Fn(1)p Fj(;)15 b Fn(2)p Fj(;)g Fn(3)p Fj(;)g(:)g(:)g(:)34 b(;)15 b(N)10 b Ff(g)32 b Fn(and)d Fj(M)36 b Fn(=)25 b Fj(N)10 b Fn(.)40 b(Set)1458 832 y Fj(r)1499 846 y Fi(M)1603 832 y Ff(\021)25 b Fj(r)1740 846 y Fi(N)1833 832 y Fn(=)g Ff(b)p Fj(q)s(=)-5 b(N)10 b Ff(c)p Fj(:)1002 b Fn(\(5.3\))370 1016 y(If)39 b Fj(q)s(=r)45 b Ff(2)c Fc(N)f Fn(then)g(for)g Fj(N)52 b Fn(=)41 b Fj(q)s(=r)i Fn(one)e(has)f Fj(q)s(=)-5 b(N)52 b Fn(=)41 b Fj(r)s Fn(,)h(so)f(that)g(in)f(\(5.3\))i(one)e(has)270 1152 y Fj(r)311 1166 y Fi(N)404 1152 y Fn(=)25 b Fj(r)s Fn(.)41 b(Then)30 b(if)h(one)f(c)m(ho)s(oses)i Fj(r)1464 1166 y Fi(N)1557 1152 y Fn(=)25 b Fj(r)33 b Fn(and)d Fj(r)1945 1166 y Fi(i)1999 1152 y Fn(=)c(0)31 b Ff(8)p Fj(i)25 b(<)h(N)10 b Fn(,)30 b(one)h(obtains)g(a)g(solution)270 1288 y(of)g(the)f(Diophan)m(tine)i(problem)d(\(2.14\))k(with)d Fj(r)1911 1302 y Fk(0)1976 1288 y Fn(=)25 b Fj(r)2113 1302 y Fi(N)2205 1288 y Fn(=)g Fj(r)s Fn(,)30 b(so)h(that)g Fj(r)2753 1255 y Fg(\003)2817 1288 y Fn(=)25 b Fj(r)s Fn(.)370 1424 y(If)i Fj(q)s(=r)38 b(=)-55 b Ff(2)25 b Fc(N)p Fn(,)j(then)g Fj(N)37 b Fn(is)28 b(c)m(hosen)h(as)f(in)f(the)h (second)g(line)g(of)35 b(\(5.2\))r(.)40 b(Note)29 b(that)g(in)e(suc)m (h)270 1560 y(a)35 b(case)g Fj(r)587 1574 y Fi(N)687 1560 y Fj(<)c(q)s(=)-5 b(N)45 b Fn(and)34 b(there)g(exists)h Fj(c)e Ff(2)e Fj(A)1895 1578 y Fi(p=q)2004 1560 y Fn(\()p Fj(f)10 b Fn(\))23 b Ff(n)h(f)p Fj(N)10 b Ff(g)35 b Fn(suc)m(h)f(that)h Fj(c)d Fn(=)g Fj(q)26 b Ff(\000)d Fj(r)3215 1574 y Fi(N)3282 1560 y Fj(N)10 b Fn(.)270 1696 y(Hence)34 b Fj(r)584 1710 y Fi(N)651 1696 y Fj(N)39 b Fn(=)29 b Fj(q)c Ff(\000)c Fj(c)34 b Fn(satis\014es)f Fj(q)24 b Ff(\000)e Fj(N)32 b Fn(+)21 b(1)30 b Ff(\024)f Fj(r)2002 1710 y Fi(N)2069 1696 y Fj(N)40 b Ff(\024)29 b Fj(q)24 b Ff(\000)e Fn(1,)34 b(whic)m(h)f(can)g(b)s(e)f(written)270 1832 y(as)i Fj(q)25 b Fn(+)e(1)31 b Ff(\024)g Fn(\()p Fj(r)799 1846 y Fi(N)887 1832 y Fn(+)20 b(1\))c Fj(N)41 b Ff(\024)30 b Fj(q)c Ff(\000)c Fn(1)h(+)f Fj(N)10 b Fn(,)35 b(implying)f Fj(r)2174 1846 y Fi(N)2263 1832 y Fn(+)23 b(1)31 b(=)g Fj(r)s Fn(.)50 b(Then,)34 b(b)m(y)g(c)m(ho)s(osing)270 1968 y Fj(r)311 1982 y Fi(N)407 1968 y Fn(as)29 b(giv)m(en)h(b)m(y)h(\(5.3\))q(,)f Fj(r)1162 1982 y Fi(c)1222 1968 y Fn(=)25 b(1)k(and)g Fj(r)1609 1982 y Fi(i)1662 1968 y Fn(=)c(0)k(for)g(all)h(other)f Fj(i)p Fn(,)h(one)f(obtains)h(a)f(solution)h(of)270 2104 y(the)h(Diophan)m(tine)g(problem)f(\(2.14\))i(with)e Fj(r)1807 2118 y Fk(0)1872 2104 y Fn(=)25 b Fj(r)2009 2118 y Fi(N)2097 2104 y Fn(+)19 b(1)26 b(=)f Fj(r)s Fn(,)30 b(so)h(that)g Fj(r)2806 2071 y Fg(\003)2870 2104 y Fn(=)25 b Fj(r)s Fn(.)370 2240 y(So)c(w)m(e)h(are)g(left)g(with)f(c)m(hec)m (king)i(that)g(an)m(y)e(solution)h(of)g(the)g(form)f(w)m(e)h(ha)m(v)m (e)g(found)f(in)g(the)270 2376 y(case)26 b(of)e(the)h(generalized)i (semistandard)d(map)g(is)h(also)g(a)g(solution)g(of)g(the)g(corresp)s (onding)270 2512 y(Diophan)m(tine)40 b(problem)e(\(1.17\))i(of)f([3)q (])f(for)h(the)f(generalized)j(standard)c(map.)65 b(This)38 b(is)270 2648 y(quite)31 b(ob)m(vious)g(in)f(the)g(case)i Fj(q)s(=r)c Ff(2)d Fc(N)p Fn(.)370 2784 y(In)e(the)h(other)g(cases)h (one)f(can)g(reason)g(as)g(follo)m(ws.)40 b(Supp)s(ose)22 b(that)i(a)g(solution)h(of)f(\(1.17\))270 2920 y(of)k([3])g(con)m (tains)g(a)g(negativ)m(e)i(harmonic,)e(sa)m(y)g Fj(c)1891 2934 y Fi(i)1919 2920 y Fn(:)40 b(then)27 b Fj(R)q(q)g Fn(=)e Fj(r)2463 2934 y Fk(1)2517 2920 y Fn(+)14 b(2)p Fj(r)2688 2934 y Fk(2)2742 2920 y Fn(+)g Fj(:)h(:)g(:)g Fn(+)f Fj(N)c(r)3156 2934 y Fi(N)3238 2920 y Ff(\000)k Fj(c)3362 2934 y Fi(i)270 3056 y Fn(and)29 b Fj(r)f Fn(=)d Fj(r)652 3070 y Fk(1)711 3056 y Fn(+)19 b Fj(r)842 3070 y Fk(2)901 3056 y Fn(+)g Fj(:)c(:)g(:)20 b Fn(+)f Fj(r)1247 3070 y Fi(N)1334 3056 y Fn(+)g Fj(r)1468 3023 y Fg(0)1465 3081 y Fi(i)1493 3056 y Fn(.)41 b(As)30 b(\(1.17c\))i(implies)e Fj(r)2344 3070 y Fi(N)2437 3056 y Ff(\025)25 b Fn(1,)30 b(w)m(e)h(can)f(write)g Fj(R)q(q)j Fn(as)270 3192 y Fj(R)q(q)28 b Fn(=)d Fj(r)546 3206 y Fk(1)603 3192 y Fn(+)18 b(2)p Fj(r)778 3206 y Fk(2)836 3192 y Fn(+)g Fj(:)d(:)g(:)k Fn(+)f Fj(N)10 b Fn(\()p Fj(r)1297 3206 y Fi(N)1383 3192 y Ff(\000)18 b Fn(1\))h(+)f(\()p Fj(N)28 b Ff(\000)18 b Fj(c)1924 3206 y Fi(i)1952 3192 y Fn(\),)31 b(with)e(1)c Ff(\024)g Fj(N)j Ff(\000)18 b Fj(c)2644 3206 y Fi(i)2698 3192 y Ff(\021)25 b Fj(c)2833 3206 y Fi(j)2895 3192 y Ff(\024)g Fj(N)j Ff(\000)18 b Fn(1,)30 b(for)270 3328 y(some)23 b Fj(j)5 b Fn(,)26 b(so)d(that)h(w)m(e)f(can)g(lo)m(w)m(er)i (the)e(v)-5 b(alue)23 b(of)g Fj(r)j Fn(\(without)d(c)m(hanging)h Fj(q)s Fn(\),)h(b)m(y)e(c)m(ho)s(osing)h Fj(r)3323 3342 y Fi(N)270 3464 y Fn(dimin)m(uished)30 b(b)m(y)h(one)h(unit)e(and)h Fj(r)1485 3478 y Fi(i)1540 3464 y Fn(=)26 b(0,)32 b(pro)m(vided)f(that) h Fj(r)2353 3478 y Fi(j)2421 3464 y Fn(is)f(increased)g(b)m(y)h(one)f (unit.)270 3600 y(One)d(can)g(exclude)g Fl(a)j(fortiori)39 b Fn(the)28 b(presence)g(of)g(more)g(than)g(one)g(negativ)m(e)i (harmonic)e(in)270 3736 y(the)i(solution)g(of)f(the)h(Diophan)m(tine)g (problem)f(\(1.17\))j(of)d([3)q(])h(\(for)f(the)h(class)g(of)f (functions)270 3872 y(w)m(e)i(are)h(considering\).)43 b(The)30 b(condition)i(that)f Fj(r)1930 3886 y Fk(0)2000 3872 y Fn(to)h(b)s(e)e(minimal)h(requires)g Fj(R)c Fn(=)f(1)31 b(as)g(in)270 4008 y(the)24 b(case)h(of)f(the)g(generalized)h (semistandard)e(map,)i(so)f(that)h(the)f(pro)s(of)f(is)h(complete.)p 3324 4008 4 62 v 3328 3950 55 4 v 3328 4008 V 3382 4008 4 62 v 370 4193 a(By)g(denoting)h(with)1083 4181 y(\026)1081 4193 y(\026)-51 b Fj(u)24 b Fn(the)h(quan)m(tit)m(y)g(analogous)h(to) 2186 4181 y(\026)2184 4193 y(\026)-51 b Fj(u)2230 4207 y Fk(0)2293 4193 y Fn(for)24 b(the)h(generalized)h(standard)270 4329 y(map)k(\(see)h([3)q(]\),)g(w)m(e)g(ha)m(v)m(e)1143 4547 y(\026)1141 4559 y(\026)-51 b Fj(u)1187 4573 y Fk(0)1226 4559 y Fn(\()p Fj(\013;)15 b(")p Fn(\))28 b(=)1591 4446 y Fg(1)1561 4473 y Fh(X)1560 4667 y Fi(n)p Fk(=1)1708 4559 y Fj(")1750 4522 y Fi(nr)1827 4498 y Fo(\003)1867 4559 y Fj(e)1909 4522 y Fi(inRq)r(\013)2116 4547 y Fn(\026)2119 4559 y(\026)-51 b Fj(u)2165 4511 y Fk(\()p Fi(nr)2269 4488 y Fo(\003)2305 4511 y Fk(\))2165 4588 y(0)p Fi(;nRq)2354 4559 y Fj(;)779 b Fn(\(5.4a\))1143 4858 y(\026)1141 4870 y(\026)-51 b Fj(u)p Fn(\()p Fj(\013;)15 b(")p Fn(\))27 b(=)1551 4756 y Fg(1)1521 4784 y Fh(X)1520 4978 y Fi(n)p Fk(=1)1668 4870 y Fj(")1710 4833 y Fi(nr)1787 4809 y Fo(\003)1909 4784 y Fh(X)1899 4981 y Fi(\027)t Fg(2)p Fe(Z)p Fi(;)1843 5053 y Fg(j)p Fi(\027)t Fg(j\024)p Fi(nRq)2121 4870 y Fj(e)2163 4833 y Fi(i\027)t(\013)2279 4858 y Fn(\026)2282 4870 y(\026)-51 b Fj(u)2328 4832 y Fk(\()p Fi(nr)2432 4809 y Fo(\003)2468 4832 y Fk(\))2328 4893 y Fi(\027)2500 4870 y Fj(:)628 b Fn(\(5.4b\))p eop %%Page: 12 12 12 11 bop 390 46 a Fk(12)704 b(ALBER)-6 b(TO)30 b(BERRETTI)h(AND)f (GUIDO)h(GENTILE)390 255 y Fn(Note)h(that)815 243 y(\026)812 255 y(\026)-50 b Fj(u)859 208 y Fk(\()p Fi(nr)963 184 y Fo(\003)999 208 y Fk(\))859 284 y(0)p Fi(;nRq)1073 255 y Fn(=)1178 243 y(\026)1175 255 y(\026)f Fj(u)1221 208 y Fk(\()p Fi(nr)1325 184 y Fo(\003)1361 208 y Fk(\))1221 284 y Fi(nRq)1393 255 y Fn(,)30 b(b)m(y)h(construction.)490 391 y(It)g(follo)m(ws)h(from)f(Lemma)h(3.1,)g(Prop)s(osition)g(3.3)g (\(and)f(the)g(remark)g(follo)m(wing)i(it\))f(in)390 527 y([4)q(])d(that)g(the)g(radius)e(of)i(con)m(v)m(ergence)1750 515 y(\026)1748 527 y(\026)-54 b Fj(\032)1786 542 y Fi(f)1832 527 y Fn(\()p Fj(p=q)s Fn(\))29 b(of)g(the)f(series)h(\(5.4b\))h(can)f (b)s(e)f(de\014ned)g(as)1141 775 y(\026)1139 787 y(\026)-54 b Fj(\032)1177 802 y Fi(f)1223 787 y Fn(\()p Fj(p=q)s Fn(\))26 b(=)1550 631 y Fh( )1622 787 y Fn(lim)15 b(sup)1669 859 y Fi(n)p Fg(!1)1962 787 y Fn(max)1915 852 y Fg(j)p Fi(\027)t Fg(j\024)p Fi(nRq)2178 655 y Fh(\014)2178 709 y(\014)2178 764 y(\014)2178 818 y(\014)2217 775 y Fn(\026)2215 787 y(\026)-51 b Fj(u)2261 749 y Fk(\()p Fi(nr)r Fk(\))2261 809 y Fi(\027)2396 655 y Fh(\014)2396 709 y(\014)2396 764 y(\014)2396 818 y(\014)2427 681 y Fk(1)p Fi(=nr)2578 631 y Fh(!)2650 654 y Fg(\000)p Fk(1)2744 787 y Fj(:)554 b Fn(\(5.5\))390 1024 y(so)42 b(that)f(the)h(radius)e(of)i(con)m(v)m (ergence)h(of)f(the)f(series)h(\(5.4b\))h(cannot)f(b)s(e)e(larger)i (than)390 1160 y(the)e(radius)e(of)i(con)m(v)m(ergence)h(of)f(the)f (series)h(\(5.4a\))r(.)68 b(On)39 b(the)g(other)h(hand)e(the)i(latter) 390 1296 y(is)33 b(b)s(ounded)e(from)h(ab)s(o)m(v)m(e)j(as)e(in)f (\(4.3\))r(:)46 b(in)33 b(this)g(w)m(a)m(y)h(a)f(complete)h(b)s(ound,)e (from)h(ab)s(o)m(v)m(e)390 1432 y(and)i(from)g(b)s(elo)m(w,)i(is)f (obtained)g(for)f(the)h(radius)f(of)h(con)m(v)m(ergence)i(of)d(the)h (rescaled)h(lin-)390 1568 y(earizations)i(for)d(generalized)i(standard) e(maps)h(with)f Fj(f)10 b Fn(\()p Fj(x)p Fn(\))37 b(giv)m(en)g(b)m(y)h (\(5.1\))r(,)g(therefore)390 1704 y(completing)32 b(the)e(w)m(ork)h(of) f([3)q(])g(to)i(the)e(same)h(lev)m(el)h(as)f(in)f([1])h(for)f(the)h (standard)e(map.)1685 1916 y Fm(References)428 2085 y Ft([1])40 b(Berretti,)g(A.)d(and)f(Gen)n(tile,)41 b(G.,)80 b Fp(Sc)l(aling)38 b(Pr)l(op)l(erties)i(for)d(the)i(R)l(adius)f(of)f (Conver)l(genc)l(e)j(of)e(a)548 2200 y(Lindste)l(dt)28 b(Series:)35 b(the)27 b(Standar)l(d)h(Map)p Ft(,)c(J.)h(Math.)g(Pures.) g(Appl.)e(\(9\))h Fb(78)h Ft(\(1999\),)h(no.)e(2,)h(159{176)428 2315 y([2])40 b(Berretti,)27 b(A.)e(and)g(Gen)n(tile,)i(G.,)54 b Fp(Bryuno)29 b(F)-6 b(unction)28 b(and)g(the)g(Standar)l(d)i(Map)p Ft(,)c(Preprin)n(t)f(\(1998\).)428 2430 y([3])40 b(Berretti,)g(A.)d (and)f(Gen)n(tile,)41 b(G.,)80 b Fp(Sc)l(aling)38 b(Pr)l(op)l(erties)i (for)d(the)i(R)l(adius)f(of)f(Conver)l(genc)l(e)j(of)e(a)548 2545 y(Lindste)l(dt)29 b(Series:)36 b(Gener)l(alize)l(d)29 b(Standar)l(d)g(Maps)p Ft(,)e(to)f(app)r(ear)g(in)f(J.)i(Math.)f (Pures.)g(Appl.)428 2660 y([4])40 b(Da)n(vie,)25 b(A.)f(M.,)i Fp(The)g(critic)l(al)h(function)g(for)f(the)i(semistandar)l(d)g(map)p Ft(,)d(Nonlinearit)n(y)h Fb(7)e Ft(\(1994\),)j(no.)548 2775 y(1,)f(219{229)490 3227 y Fs(Alber)-5 b(to)39 b(Berretti,)h(Dip)-5 b(ar)g(timento)37 b(di)h(Ma)-5 b(tema)g(tica,)40 b(I)r(I)f(Universit) 2892 3221 y(\022)2890 3227 y(a)e(di)h(R)n(oma)g(\(Tor)390 3342 y(Ver)n(ga)-5 b(t)g(a\),)27 b(Via)h(della)g(Ricer)n(ca)g (Scientifica,)e(00133)k(R)n(oma,)d(It)-5 b(al)g(y)490 3457 y Fp(E-mail)27 b(addr)l(ess)6 b Ft(:)36 b Fa (berretti@mat.uniroma2.it)490 3653 y Fs(Guido)f(Gentile,)h(Dip)-5 b(ar)g(timento)34 b(di)h(Ma)-5 b(tema)g(tica,)37 b(Universit)2642 3647 y(\022)2640 3653 y(a)d(di)h(R)n(oma)g(Tre,)i(Lar)n(go)390 3768 y(San)28 b(Leonardo)g(Murialdo)g(1,)g(00146)i(R)n(oma,)e(It)-5 b(al)g(y)490 3883 y Fp(E-mail)27 b(addr)l(ess)6 b Ft(:)36 b Fa(gentile@matrm3.mat.uniroma3.)q(it)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0006010513224--