Content-Type: multipart/mixed; boundary="-------------0304300929886" This is a multi-part message in MIME format. ---------------0304300929886 Content-Type: text/plain; name="03-203.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-203.keywords" Two-level systems, pure point spectrum, generalized Riccati equation, small divisors, quasi-periodic solutions, trees, multiscale analysis, resummation of divergent series, Cantor set ---------------0304300929886 Content-Type: application/postscript; name="riccati2.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="riccati2.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: riccati2.dvi %%Pages: 12 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX10 CMBX12 CMR10 CMCSC10 CMTI10 CMMI10 CMR7 CMSY10 %%+ CMMIB10 MSBM10 CMMI7 CMEX10 CMSY7 MSBM7 CMMI5 CMBX7 CMSY5 CMR5 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o riccati2.ps riccati2 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.04.30:0922 %%BeginProcSet: texc.pro %! 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1b773f8370649e808f2d86c099f2ef1ba545d61eb3a034110e01705ffab67ce1 f69dd0064e6226d2b912c3165df3022dc118f49ae7eb536a572bb9c638075719 b3dfe9435327b94f0b70aa0d4cb20d5fac4e87d42bdf8657a9141905609d6590 6de0a5ec9bcfbe96864c5c5b42f785c53f492732c0027c0bb7c73da6489b84bc 8378c9a5a669e2f9658eb608f507f3356d2158c18d76991819a924f0f3ed67d0 9a841fa8a6d2dc92fce5cb4db576334ae41535fa1b2ed542889835c17909bf9a f68d983fd5344f63484e8b75e104e289157bad243015732e4c71c1ce212f166a 86e2043e5cecc53f993a9721c7f51aff568936464ca4908aa565c68920d91158 bb5ca0d21d09fe04fdb2dc9c05b70f46e70c6f1a7e28d9312adcfbfd4ee9eccc 0da2479a922f8e94d8e6af7a8e84404a50427ade96a25c1b7d6f0d347c2c0188 832ae47a0c74a80a2c78f01078f243a9ab9f45de402ea31d811899d3061e905b c2c032197e10377aafc2d9ce65c2efdf16f05ae8ddeb88b32472620bf947a804 0b0151eeae090a54915b584219e2b18afab0c437aaa1f4a22c0668fb9ede9122 ecdfbc577ac63c1e7637cb2dcbbaf74976e1454994fc0b15d72148dc1ff7333f b6afe4564459cba807e8043918363269392bebbeb74e2ab3140f2656151d8e7c 20e09af525e36e6a012d35420446359e8b7ce259e27010954bfc24ca4e20b3f9 528603f1641427a9939181754491ce3c053dfcec89982422f5bbc03ad3f3bcfc 13c2855b60ae3d43848b4d534e473ef9f1cf7db2ae3c098a8834cfb4e2653eeb 3f5092e05695672da1f179a768a5784b59c7cb68ff72b6083140ff4b0e7cf4b6 9f334bcc40164054debcb9df4a3d5821869663fca125bbe8427f638a868ac634 f261a4d22e74c7a583f08d522e449a2726b7e15f195b2a540a1677c9bff3ca1d f642c58faa 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX7 %!PS-AdobeFont-1.1: CMBX7 1.0 %%CreationDate: 1991 Aug 20 16:35:49 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /zero put dup 120 /x put readonly def /FontBBox{-55 -250 1289 751}readonly def /UniqueXX 5000765 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5bfae5c1f476ee6f05eb1d2caeb 269958b194521197b1ee2d9de5f5346241e154574f14a4532b35968596ef7a20 102dad629475cf5d654585d4caac68ba7b2ff21b103a98af3e9ec01fae180804 26b73d62e93f0328ce7775e81898de587dd0fbcfe98934a58afb1aa4dd8c6eec d19373ddd360a5e85cdd4b752b2f2baa13b48a282214a05632d28e915ec90824 21ff8f2166e940c7122c583f235bfd098f8aac0f1dbb9a625b02bd889869664e f1856608899ba509d6eb8ece665afc34a459dd5b54dc7711f6f04a288214efe4 53095e239bf19cd36b7028d46c07a2b8c17d452d8563ea88a3bc5e5ea29a9ea9 a7e38481679f893f9e6a983ccb5512bed4aca5888793526701796b3fcac0fb62 f69c66d0688eede0fefd590fbce954b2cdd2abc9017390077a2cf6f6fa7d0db2 dcb4b0963e3dfbf02be1dca9cad104f3d29495d19337fbffdd745e1729e7c0a6 41fdd3bf42c30d715670b3ade44f481d795071f1b24f4f03dff59d91bf9d678b 4560da27ad826c9fc8f8cd9a333248cb2007b4ac0a9e2d5c044774124f1ee23c 0e3e005a4a16e1d0cba15797fc42fa75cb45b24fcf8a8cf41de8e06388b5f995 04454646c3246074f495564cfc8ad75a949cc477ecf03f0511b90b6a7456b730 cf09773b5544f6ec17a9a179714e47e2eb1142d8005bd228affaa375eed7fb1b 075b3023be6386f3805aee05b47c7f370e8ad410110429d2ee8801d0d2da9e68 a453a323c147716ca588fbf563175e5f47d1086a21b04102cba338a3580225ce 00dfb1784fbcad0fcd11434afcc2ec13337ee24775c2952be02e20572aaab17f d21287775e4a4d9fbf877a487b252b5c880463541489f5578c7472d39fc27320 2e4ec86f9935ab2ded14ab9ea36770d79a1a72161102412bc57c587426a39c77 b24a0746f71f62d109484272b91fedab5fd48f45011cbaa805f9cf27f83c270b b61433e2537fde0cd4eda9d15c7f03ecb26edd292c7055dbb33b67c45dfaa319 c613559388e83f43f8ebb575d07089101b621fbc5d1db97c2b65087a130b63c3 db3f5a0722b47b18ef88852954762e684620cd8968c9e8875a88c772d30460ae 836961ea4bd5ef4a1e2e2368070ecdd6ff1951a23187afc13cd99b8c34b592de 855b8f4df635b8102574e276d0bb562edf4a14988b2b44bb3513aac5c0f1e5e1 fb18fbc2c60db058cb989dfae35b2b313d1975d99fee5a298fc88fd14f36d63e ddc878cccdff56745b548ea71443b7a1f3f56e9a2f1e5d88e41938520e637f02 5dee37ddb4415cd44bea660c12675b135478f343f2052cfd611e6ac9cf33ef44 11841bda33b27e1764041b8f50856ec367128c769c0afe25a69c5d4fa7440e8e 54bb0870f17cf13a808397b737eb11ff918893786e1e79f7e31ce5be4db77fdd fa84bec5b4e459d766399192da022b26d69b3b415797932e9e1d26da208dfbd4 ba47bb1a588de231708d8403ca430626de0f1319af79ec441c243a19b1b19e99 c8d2737ce837cead357dcda86015fe827b3095a1d1a5ba6d0f9702252d388eb5 91abe715b8600ba5512220b51eb2fa5b588563493104ef2e9d22cf2dad70c0e6 5d99 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 23 /nu put dup 28 /tau put dup 59 /comma put dup 61 /slash put dup 96 /lscript put dup 100 /d put dup 105 /i put dup 106 /j put dup 107 /k put dup 110 /n put dup 112 /p put dup 118 /v put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueXX 5087380 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311e252234d51741e18db3a68e8ad10242 29e5817a10e796a78d2c7f7c1f50961b9a57aa604c9f821dbf5a9295197bc666 31678d7d2c7e1f8f2151ce0c29efce46270570f4301c5dad1b38884732e53dad 05dfc36b020e726ca575f71f04abf2b49e1f1d6d2a08a477658ca9a48f1c8cce 14382f0a42201de56e2821ee2b0e91a818b1b753d810ec2428dae5153b66af66 9442192024becf743a8f6264a58a0b3aa823510a4957d1bab3e1b83135362888 3e392f49926471a8021fdb48637baf0abcf31204b1b214f48c99f44af7782323 69d772ae4f49be53a1bf584747bf29b94f530d59a5be5370b8dd0691e32a21ba 9d0d592cf3bb50a99a3ecd18d6c4114cb4cfecc49705da90d7edb10e3a28f9eb a49347a09d31eec896f2a28e9607d99d42914aea16545c16e136bdd0b75fe486 61e24c7a9e9b233a76754c760822d0a88c575e08ae3e3f081d7bfb80ec4c03ac a08b65fb6474f71fc95c7c2462f2eb3754e647c29498288f2ced0bd3a2bca5c3 2f80a61fa667802d376cacd34de1f9572d1b3ab09d6f352750db4a768f869638 a87457889aa502850427b40e1294c5ba33e6b17532b2d13b5f50e71582f93daa 231e86597872a2a968127afb0f36257a76ce1eb24e6c4f29fce2aaf6532fb5cf df84da03f03a45b915209f67d398c7c4f7a3b0623750259087a478dca684b29e a8b7f52060bae9394369ed1cecc645d113cca3772a8dc36c6d916b5c5e65454c 0a188a33395fd3da67ad0a31ddab6a8434dd80f0bd0ea1a2a62eecc331d88b3e 369552daa1c575a98fe84383548626b65be7c24ae7eb61159f149872cfd87cf7 6c1a7c155c7186c7600e58cb61b8cb0c5dea98268a39e928bf6fc53985eaa763 2b28c1eb3d82e4d3fe225223f4266d2b6147e344aad2d4884a9308aadc7801af a606838925e3e50ee7aaf73f090baefcfd4e34b795cd92d83c944d91a4940f2f c955d727ea5bd76f0c8057b12621eceb0947b0a5dd10dbcbc52957dd978d7db0 c794660afdd4e946b3d7fc1ba60907fb3f756aab0406c0e7bce5a2238fb3e12e 07e2c0a7f465e90057642d704434ae0b57bd46c8ea91ab2a386e24c6deaf9cb0 8cdaa8fc4ec3fadc06bf28659cb878453db16f4b03925b2127b83be8fc240f33 23ea950b51d70c8fe861e30c346efa0a0125137598caca5478991e35f5bbbc19 4c787488ba339830bca15f8333431902e82b66a514a41c0bff92719a6a0d9587 4cff62c622fc61a0308a69f9fb7ef51bb274ce9e059c04b64e817abdb81b7303 78937b0b14edbd2e68a962c62758edecf7167dbe9a4a42869dde95625241c298 6746a7a51b8f0eeb0dbf07a3735df7a9c940c174c3fdf6d59d6a36f9475505ad 96a494bbc49e39f7391370759526217d2d52d466a8801f821608d6c83d50ae80 0a4d4837b579fc137dd3e9e4216d4b421dc5b0c3046132ea3bf37ed1af02620e 22831754938fc29a7abde270e532eaba622195dfbfa9ed53cef64e10fddd9eb6 0bf1e869b6bcdceb0e16894c4bf1c9ed5be3e7736757fee3b1a1cc5219f3e204 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27a57f515c2e16512d2df8437f5d6428034f29673c407a40d4c3faa05d6741cd a4b2539b9cc50abe0f92d8d215b54e40cf896b06c7daead1bb55ec51271cc0ab dfc0c5f8ddd8c82755031c40f159715007ca7fa960a58b573bb2bda1b57097be b996a6c457e14f632cfe05443f3a061c8b9c0b9ea470510e8970efaa4f8b0497 343b747af7c13f8458719b12b735a73e4e5295b0b58ed3656b8b4cd167bad4b0 b046148bc018ddd1acffa2da98d11205be625392150fe13e903e350ff65bf19b b13378ebc56221329efc7627bb25be671dbdfaefd3918da3e90f4d48d36376e7 0316ff7a555b0ddb2ef150bb1fb432871e6d7b7157b0eef4d4ae14f98e93519d cfd21bf264542127cde7792b6670db10a430a007b3640d6ead4748077124d032 14692c1b473ff7b1a37c887b10ce2c96af3f655b062b868fd2e57c331505ff5c 82b4726550b0fec1ed5da391a41c2b4361d650fa535fb332252b24ab5702f98d ef08f1fc5c90cf031432080d62c4cd85a0a829a5713ff7b385daad9ac9c7c7af 1a7ba60f0d628f05397ad4a5af0ade5c130f8d9911cf6a7e7a3c9525b3f755fa 8cb9e7408d1f8d8304d82d20909edf276becd4eff2059ff2f39724956a3a00c1 68843f881945f99ef8796196d70ed0a88f8829a58ada19428b2292913c1b1637 a0336d090f15461e12d419677603b6b51610006bc5a9a58fbd4bf25953dc9753 145461d39b253a610c2e47070755346f58c9e77656f10116243a642e252679e6 66bdd7bea7da851976038c815460e74117a687a207d1f4d8d6f4af57ded88cb2 ac24e111bdf06f0ebcc94e6ab5056663ee1141ce76564c567f49f7342a2bfaa2 e3665b947e5b1545022b730407d2afa6c71807baba6f697fced041f9566c1967 0f3b5c6dfd1d10c6eb2b8a20c4fdf6501d7320a292b70c1c6b4873eac5ee04e3 982bf6bc3f2bca251d2680719d0f8f39eb7d98eded3a0d0d3f620ff13fa99468 207a80ea35394a59e9d99c2c7518e5d666f121c71c44f9e44a02f4a1fa29c493 877a3c5b2815228175393f0df7e1823be091182f132c62f58adc66e9c3d0 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 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 554af524184702a02e534d1eb98dd3f7c5d51900409354dea37de4ac7cd42348 7c653dd50f6cda46266680119b22d86d17c4b63f13c00e0eac9bc346f92854e8 c2970254f55a231e056af8a18a32de3acaacfc722ce4dda7d5fd74cecc69b5d0 1dce2e5a86bd847bbdfebb0e9007c718727faa6637472e3b8629a9cd03a313ee 22c14b7894f13bd06dd32ba0d808628fa71320fe44a67b8792d1aeef6f8a07d9 f2420591a424b7f4afb559be14cdce92756b9b577787165b7e457d01873cbc72 4e6697a9ae81d712dcbc1638222177007037dbef6bf5ff1d976722a633a07ad2 3e9cf134f23b18fe315e877f941e35dd33870c3ed3d843e6e8673ae59215a0ae ea9a3c9db8b6a5df631b65b8481e1eabd0e9035dad89c59fee35158a6d5c411e 2f39658f385a4bc895af26e9b4d5173a36fcfea72083fafc712f213916e32c1d 946e26e88591b761f6cd59afd0a3974206f46c90207d83e18fd75f51be61ba8f 8ae7a83afd69a767d1 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 3 /asteriskmath put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 82 /R put dup 91 /union put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 110 /backslash put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueXX 5000817 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f73e9b697ddc0351763500e23c4d170ea8a2d2 75ad0b6b23097f02fa803c1f46f9ac10fe0e527c431b11b1b9a50e874896cca6 17fe3e255960846bc319ca68b9b930a45d5d0fc36b352c3443ab6f7793b38d2c 7fb9c409ab7980ead55e9f8f6b83c70088d1fa6d7f2214da8928e40b96334ad2 72d1b5db9d1355a579b26a8dac21f021e393d7d837a95deb8de92ed818a8b80b 98ee9cdd8a3c57ad7d6eb7a82f43eaa6ec520eefa7302e750edc4b495f70bba2 24a2182eec37f1a1c5dc8e20973007150237daa6bf03d498826777eeebe09014 ebfe05a8731e1a421cfc513d47d673e3d94d805f5d0668ffa265e0962cb5a75a 4a93d9b871209826fea9f7e131990ce2d23931207849afd4ced802911f76d55b 1869259753a830115a11c09c086a18358487e95b98e447e0e5d7654ec257cd1c aad9ebecac48d3cded9cab979e002f77eee52bf4c92ae1fcbf45638f02fc4ed4 613b39ccc78592011acd2218ad2262334402f743b19cd7f907156e2feecfd4d5 2a26d776ff3e84f8716eab52ef1d23f84bf0c981608f95789755ee447fa3c78d 11d70265320fe8efe76f9ae354752ebbc85c1658cd4b4623f00756f206831ebe fe44e57564b4db62b188bd8ddb3d837c3e6d0a5d4a0feb13633b12bc5f2d7837 164fa206dfe7c24156bb3385234b1c2857022cd0bf26f3e4b6e36bdcb98c0a29 43697cf4a8a49b4af0726775573340c0bc12f57bd8794a6d4afdf6ffa9879f8e ae63ae0f01deb728f9a0663b04712c16ce9732e6ec8d73a1ae12c30485f63135 6fb96c1341893fdd277a7ddd7443cf13ad97ff9bc6d6b6f852726a91ebe09757 ffd94dea1743cb25be008581ad33816fa911ee4c456a63a46b720e66467a179e 9ee277800ff5a189f4082279592181ff97e89e7d0d24faac850d8da20e41289d ef1da99fbe6db20dfa3e302b917f9d863378ceef467a39eed5186b921a159537 0abda89c40afbc45db074ac528fc29aebce68d09174e241acb5b0be9f825766a 1976339a5c2c0f8b2ffa8238ee9051a64ed4da3347f4f702ad3c68f18874c56d 52e1cefea1dbeec4fa0ec139b7dc9888c5b0128073301bc923ef3f739ad21a25 3d6ca0a7b92c7dc245707a7ed07e157fe810404a60817f3d5320d1b98cae39eb 51297a34a9e1051eaefea9b99aa60304ff093fbad8b828708be933a0ea2a3533 fa49e9e05a7d59072709e9c6df72b8f569fc20ec9260da70b85185ea068a6a78 f763a19252d08d6ed0ea1e40a20c819eefcc33bca4fc4ce93e0ec77d948bedbd 16ba4185e35058be8d6a75c2e86f5e151fa8dffc726c768a52599a9c923a0b54 12951aed296c8719993424b7194e7f697d2dd0ddc4ab61b2f823a6291363a557 4521403eb1299bf3b4bcdbffdb155369a71627259d73e8d3b781f0efa341a9d8 cb9cff30ea85b1c6e0a0cd5df9e6364b77516a83f091eb4809625f8d98dd6564 d5d1908c969a7e4dc1482d493add2702e887fcb494bcae28154229100d984ee5 d2bb284d2bbbb8564a32c89f5f30470c216e71bd5b04d0ad99f207b1c3f07bf0 90ef3aeecf5d28d93814979affde245eccfdd715d4e25a14e397418240fdbf63 ebb31a2eeed1b29c227042dfd454b6c03ca5b4e90ed6477c61402cb853bd0e26 041413ae8b604c85c3b778d9a4ea09d790169ada1b5eaccbc96d3f5c826dc45c 3dcf74ff882c4530a0d11910a34b5e806f2007f6f8be19789c74543e8cfde4d9 c5f634043739c7fb06518cbd6f58f7fd249db99a7a91bc02b48fbe89ce832b87 e64e788e182514080f55710712bce6e4422f177b5519cfde9054a9443e6dcadf ab6210aefe5957fc0c6d37000a2c968f421e1e41b026e1cd045b26ae2159037f 58b8acd059e7c44706f9a0349709c29cf7e96c0b1d1adfb31b2268b7269ced81 e77b8b0a34bceef98b315fa4b6bfac71053f8e47846e26339d37fe2f111c2071 73825a50a8cfb4e8ebb5e713e3ab8cf24a1bd02192875a92e38947ef27625821 76a014e443b2ec8d831c267b0123901711ea2651d3fb0d5fe083e63337a2a9e2 55d9bd656373ee31257de96aff168d84cfd7d30afa511ac41b35614ea224c1ba d579d27d0498c0ba50c8965181f81c0ae888c85ebf0987a3c7c53b839ec787a2 53650afbe606d83f34878a731d2b6dd85cc30f76c324024af7100a17f7c1d8f6 b418d7ceb70e1bbb68412fc25b63a3272a18fc048e9335e0f41bee8a8ad0c2bb c38330a7d809d2d7cc744c9422401ef1f4135fb7f73c026228697342707c83b6 7145ee30f670f66c040cf9366d42d549bec96a705a42737f608607aee370fca5 52f6d27b09b2d0c37690d7fae0c4e788de27f2cc3f24bbab832d49f17e55b4ef 7e342f89b3d63a0974b46e8f976fe868dd9865b6bcacb9d0194b3b7479438cdf 8936bfa51f2ac5f2aa344d48c10eb2464d4d1dfa4f052c6aa82d5d95ac7e77bb 0228f32eb4979dde0c7552582bde7a5c87ee353d544f16487e44969c0cdf871a 4957416c881a97311102679ac8b2764897230a6627b2c5b3c68aca48a41154a4 60a874874e7a7a2438200ade4bdf751f7fc4f0925aa4a76baf1d68a6ccda6ef2 f43d6a19db9c17da9976670a91db297741e708cc4ead3f2ab1a761329150a2e1 1250bd717328f3d39e4f8cc20867e1fe138590d487e6436c9552f16ebcbcdbd3 4e9328b61936ceca8161e3f47a1894da1eaae23dd9347e6efe4b87808534b2cd e164cebd2732af11754d96e73fcad2809feb46a5c3e688aa34071d75c3da9c07 28635ab9d12f54c086573cd8bb5c248147badba1b7224a65dc6b2403e94c0bdf e0a4c102fdddadc1bba558f21a356a8aa96f2999044b77ab299e91208ead4c0d 2d38dab204c1b549445dd17ad7e76b1f6d8f43923720c4b7c4ddad37edf275ee f8598c338e116a0cb522301f38b3c9e3e7947004255ddd5322bb0db585dd8444 32426d3a1ce7903d93788294bec8a136a5682d57d00d2ab9b048b85aa4dc93b0 f765ae6f7d9846f43575fdd67a8453f770eb20e7042f497d6839ac6e4fdd120e 3f 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMIB10 %!PS-AdobeFont-1.1: CMMIB10 1.100 %%CreationDate: 1996 Jul 23 07:54:00 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMIB10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMIB10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /beta put dup 18 /theta put dup 23 /nu put dup 32 /psi put dup 33 /omega put readonly def /FontBBox{-15 -250 1216 750}readonly def /UniqueXX 5087392 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d78041490e364babe4b85935b8c9c7758c2dbe253653c 892b1fc2c6a01103e0b8553f847bb38d8e4fcd563ebe51cb7d5f0877af29249d b61636618fb31eeb6cbb4ea4db687fd77ed1574da8c9f9d33b592ec0e38d798c 498a205cab252f7e2f1ddb82a38f331f917b8452af11b2b93e34c1ef55ee005c e8418dbcd0fb9d4d3fa8cb388ec91f16c8864913ad3fc8960d3c424650bdd12c 4db4049367faae97c660f00021f712e9eb6366b7ce74255ff62b72396ca43287 7c77fecdc53883427c90c1c3ff7bd46e472cea9bd72eef0e1be36abb3a50a7bf 6f5d460fa2fc81466014df9ee1e15365642c2de78706dfa740e035b4f28f290d 79b8ba16e5bca13fd9c3eb7eb54006ac2c1cf0bf363b3b69e923797a95b76515 073892c75cf733da7c07da8ed1a8d1ded902f784764c6df6fcd0fd28083d88c8 4c41aeaa6c2c2c1bb10219f4feaab75ee904f4314fc0b9de5e96054f2472ad86 42140315385400686cb07a0585d922ae4b1b25ae722edc6b4afe053560fc9ac9 20a1d2fb8d0921c52c9468d34dba9bed42e3a8308742d36186ed0a3b43f690a4 dca6bb2af8bd698fb303fd68df53edc5e01912cf007dd31ca7481ce844be1258 7002d4aafb204a8075fc6df86f22249d61dd840e7ef87362d72e35f4e8ece044 95ae41ee5c02d6aafaad37141935867a8770fbd9e9e894ea6cc5cf3770f35d98 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3831dfbbf1854b1862ef19ba58a34649e1b4c8c2426c7e256a22056686b98063 240a94a5974586e7a56604ce3623f73d54bb12b5614c701a2a06a6d39cf9d91b 5d3113c067c2cdb511e4ddc3cf5e61ac499bb5e4084586f1f05125e52bec1912 f3c60f266a14db30e39e57bb261f5f660464d6772165b28e30d7d746c76aa58c 3a77b23ecc89abfe8d5b3eecc4b8088abff07d204d9f3edfd2bbaebc7c1e95af c1abd6a4ec62c9f7f9942db985785ba3253b6ce6da9852a8de67203723416223 9724ad38788bee0f84519954e61181e0f29608978cbdbe49a9b93ab7df14e33d c2ac1a466b0a542523807d150a3a05f7dba2a845678c65f3b629154fc4333d01 3a1c36713e54dd4acbc05c154a76a857eac5fe373e1980b1f656f8795344e3da 404b80c79659449214ee95894857a37abfdf54049ace149556d5d2149e 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 12 /vextendsingle put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 80 /summationtext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 92 /intersectiondisplay put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueXX 5000774 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d5649664203ab24acb938d58d246707ffe7 d62f04bec4b70c21ef75beb2b812622b3c74e969d72d3cd11bd7106294a99caf 0b1629bc7d4de6b96ca82930831d64575f23f4ad06a0e45e315b1d392411be8d 6d73c998789ff258a07a3c8c2057325784514c845500bfd1a971310cfc11d41c 1a167dbd5ff012c60add4e87325f6e5299032a839de65fb1473a166aae1876a4 414a434f22c1d241591fb36f857df6fa930608750ffc0c54f44994662b1f00f1 400bf752ea8d83ffc4cb77a290bc2d99981ae59a191748ba5c7ba1a9d2583fd2 1398452b6ff5d83a059f7eadcd2ef744e9dd22bdf9c79d049bf06835e878c32b 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 18 /theta put dup 20 /kappa put dup 25 /pi put dup 27 /sigma put dup 28 /tau put dup 59 /comma put dup 61 /slash put dup 65 /A put dup 66 /B put dup 68 /D put dup 69 /E put dup 73 /I put dup 76 /L put dup 83 /S put dup 84 /T put dup 86 /V put dup 96 /lscript put dup 100 /d put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 115 /s put dup 116 /t put dup 118 /v put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueXX 5087382 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 82 /R put dup 84 /T put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueXX 5031982 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d56491bdecdcafdc988206c5a457a19270b 37d0ab776e03eaa7eb568eeab6b5e79dec03b0dcbf923a2aa8e4f4deda2cb043 858d8b430efef0ed914a80ffc9818aba0fb30dae3694e5b31df3855892d59b89 7f82ff79fce8444c7926ffc4ea008e63bc518936c9098afc2d1c14030a2c15bc c9285c1b57e80570d2f6301009e63d9ddeba44e3251c75ca9616a2c5de9ade66 1a15db37b40b67c1bd71f77343912c62f47dd24be0e03dd833ea60d2eb020407 eaf042fed30f0c2f01956c83d46d982b50dd979c2026ed5f71979dd98c9c1eee 25054000a5eed5ce9498b2cb59a7ee55e0538ab47d86e69b2ac79229c8c6bfc9 3a24e822a2f1c9964fe630344579a25f66646113f73d1315c447f070c575de76 266c6773b01137e4a039df5e4c2f199e8e3396ef9fd001eebe090e04384c4089 ec9f13a07ad8733c6af775f432f154ef50ec4477e08760c03c006e4d1eb1c7cd acb8955ca4b0d62c7de046ff58d431bb9883d28586552c51809e3ced4dfc2198 b97ad9e52ea33c91b9e335a80d9b77d3430e0487930f992459664abca5dd9177 150b0fb502fbb04ccaa94ffe2697ed07906f95fd98e7b325bfbf10ea6b39fd76 c1e5295022cb39e9cd471b7b0b3f9d72da99deb0615d35cc22f42dbf5cfb70dc 6279fd520d99f32201abf25e621cfe1d37961d9c0ad36ef3d9264c547638152a be30812eae30521c3c775adf1959595a43d8a9c4e12d9f9ece27a0c9b5f45e38 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6c9ab0f24f578ae7bfd9c710d508c6bc5c6ed1f98cf5fcb749383b2ccd67ea08 84a6778b189dbab132df907ca3dd83f6e4447868a9df74d0906259ca865d5256 4e369eecba79916280b0c3efef7a68399228c921a9465785ca73b1883b440627 9c4182d2977edf1aea32d6579e3633b687b01b4ffefb5917e2520a0304ff052a 96b006ee8d599c2fab96fd8d5299c2e4f834d0ce61e4d5ee0e06c64c8e1ce8d9 2064f55b0cb2ffe524bd575d63b5a7952aba9c227b2cb1fb50d826a0884b3f8c d8527091634c4265813a446b0753abc576ee589c1a4b5cd0700a2453e30ac167 39eee6c442410ef476d4f1dfa595353d85bebbbcbeb57322ae81074f73557dac 0a2e7681ae3668883e8a5750544d9deba1c1e50a52bf9ac3a03467b10532dbf2 729eb4146d898435c98e6e1f218826cdfb03996302da00cd907edf1482b38044 7b0eada6bee3b00243d59fc867bbb90eb707cc9bf616340cbcba075d63ee6d90 79bc3f5b6dff30241cf9603f0f5910078631120ce1c58434a314531587a4a5dd b953bc100d86b481cac72ec5e7a8618e1d4f25ea2e24d781eabbb63a749f4462 0942f0f6126da1e2be0558d1cfe50c2cace1e691d0aad9a49ec68ffba4ca6ba6 090a5bdf15de84ea3a94eff984762678d42a24083dbb4b66d3303a6e8bd1a1fb 18983be32b186e1c8945d252d18371a222736576a0afd744cc51a3dcdeabbb45 80fd8566e094a1296f7deaea904753165e78e39a5ae7451f91fe6ced6a9e1cc4 627db967cca76d8e020726e0a433426c7e24f2e6fb0b79f1a94a0cb13c3e74f1 b578b1a93758d9f7d2dac99bb340aa37ce889ef8f6dd28326ff8e00e4f2f60c6 f74688f8a76a085fd93016642674e24850a5b2793c1125ff30cb5cfe24a77f26 a720f00159e87f82974586c862e7528b576bbd89d37fb02bf3263c2b180c1a08 ab370d73913d41f358f74e95db0f570f5be76cc9477432830c22b219aa42ccdd a89cf6bc82391d7654b5c692a11c0dc783e3e65137ad7bbff112c2a1e8463e 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 50 /element put dup 54 /negationslash put dup 56 /universal put dup 59 /emptyset put dup 72 /H put dup 73 /I put dup 77 /M put dup 83 /S put dup 86 /V put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 110 /backslash put 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 b64c4301d00e489955b2ce2d9040bb11626f0633ded65ae053753f4af570b6da 256678ac3c38dbf0c3558b68ffd0eed303e91d8e6f146bfc1423c8e355d8d909 4db424413d8433175726ae6fb7d5792226a4d8896d0f0fd5c10d1ce906a2810f c0d86042ca52a9ed79c72f1680284d6a7539d80f52dc23c0d2359fcf778b040c 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4d6bafbc189961f6483cd2de08d1af40d8ad1eb724eb7097a4fd58dc1fe1290d 0366d7d2b4fdd15223fbe16df376866d194a430b384e6f251c298bf666acd5c3 065ac4a77d778bbc3d6b0596b2ea8bbb2e88296757ddfb5ffc3681d7a45e9996 ad0d90cfbc8e93871ba94014d97d4f8ffb81a098346f8e6cea7e6dea4952f22a e4366ebea9ea0e58a33e8507018bd984e91835ef6e707484e9490179d24d5014 4ee674878ed6fa736373cd96eabf986af76b51533614906faab8148e33f9b449 52a681936b189c4a050534bff672c075bb649f402efb86263d7155663d52fcce 21dd3d76baad04132e6e1a24e769a97f71c251ed3dedcd47b8a20dec74a309ca f9ccb5de5749214966c424bbd9dc0800563f609675d2a599917cadeb53f67ba4 cbcc0438a51cfaa2486bb48fcf875c93157f07c476c438f35101f747c25f1dd8 9d4e7cfebc9bb2490c0aced97faacef0e5a575e70c293bc850f3f7b13843d93e f8af58ddfa05c8662b86a41650a7756afc9f0f00a9efc025bf7d754c633991b0 1d89d47682ae7c04a79e93ae5beae8a6c1203f7e8a5609e65082156be2b30228 3af9e9457d58f3e2f5e205933634f96587f69fa506a4827ab41112410b3ce2f7 1f22775ea9d27c15d2b08f76497a080851eb5ced8760a1e42de4bba526c14e28 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5572f0383e976c5352e6316da7262dbcb46e9f1af28818600963f5775898d875 9aaa94184e26676b927df5a818d7bda5434f3d8fc1486c743205a6a22c335d7d 4066303bba8cf786039fefeab5fd0f230513b25b74d3bb84bfd1ad70488b832d 0c36e161bc52a09b24357b5ef33d3470b9a044b5ec90a2d35f6f346d54241c13 ddb90b9b998beac75543f54e8b4743aa4bfdee346b1fe99702d8b991656a68e6 a0a703aea083d31688b3bbb6ecd1a11b78c708b8e8c384a6bdf6b5c39f1e0d25 3e6cb70c88e242b58afc45b4484e79619b6f564f43d0644d9ab623a29d9e873b 0f3fe3aa325d9a0e08a84baf8a723d273e5efe7dd4f54d602a48711ebe2e5a9d cc86815b6119b4801a8d456db2f0ff78b9b8242f6b12c3a51af812582d48fbba 52e024d0ad77ccd6ebd8014d7eaf8c31b6dbbed27a9f42b5b9c3bed1928c6661 c4352ddfd7afa36311eb5be792d90cc12033655590d941307a2c7a647fa15b34 944289b05bd37ef5a6f02cb9795aa994d673becd27a8e98384da522c54db4f29 bbda3ee41abfc8328776c9747cb453c650e8d1cc3c8c44248bf42012e133d842 c1d39625013dd2d7d4d7592db934940d38a09237969241a60cc39dae5c9ddf0f cf77e7b31322b811888f411a65002d6ffcf559eadb2cd3ec85d96814c3ed1de4 27bb4b3f8cd85246a185f3eb1fb615dae79720cbf2115235bb419290733f803d 1fe5ea4ace7c04f64c513869601fec72fe736fdb25f2f30d511d2dbb91ace3dc 3fff03960f5cf2365317c5303bb779206fcd5caee645926f6c9c10a4eb54d26c d956a88305343ab6f4f81e82a2eb77566c8f01747495b4a99f7e48f3d3aced3d c171fff93696f781b44d75ba7169c88d44b95bc348b0e22a3779610075df7fa6 685128d015fb5f9c451c2a29c7a0de1c593c0c7091bbaa3e8a3b7e85982e7878 f9056e59430b1c9b89cd94c0486c57e56f38a4705423f0b08769c5f23a727144 14d7bbe5194cd1f86ca7cb86cb2c2c82caf57925dc4642a9a51756e168b3287e dc043fa5499af0e6883248c4f1b5b37697415aee74c9ac05e33cd9d0fb308e99 9a539adbb09c9a7d5126f079cf480b2f314f4f163dc639d9a27df5d2cd534cbf 350ba3e862167e6d3b767a4d2fadb8dc6554bf36333ed7b65131579f8f1b4591 84127d29863954b8291f8eb27b8e4d535e1f8530678eac1347dd4d4adba3872e 84ccdb4cda6bfa03fcfe80f0c96267e153e6c49abcae3a97bf66fe28a86dbb4f 3bd9440db8d390ddbfc45f7a821dbe4378321ad85b826e2cc7d19aee53312414 392dc0075895efdc70a24ffb4e7cd0d866cda97a440999d16c041fdc996f7f41 2e21 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 2 /Theta put dup 3 /Lambda put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueXX 5000790 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b448685940729e6e22f4be0e08c6505cba868f7950 93f556b64b808dd710eb936d3ac83e5783b5f7e56d363a59a026a5619a357d21 c4e77bea121eb24b09027d1aad93158edf912940e91cd2f17922b35885894644 7773730bb19af9e70453e1b78d5edc123b273a979891d6d5e624f4a388f3897e 13b045e34b4ee2f5e4fc5e996d3549640010c671de14d18b0b3cd300e57dac03 22e59bfbf7b29422230870f5897fcfaad4b50c7c1c58edcd119218163d6f6e56 25ccb882db7b6d09a1586508e8cb642a11c29a46915e4a96e282079cb21922c1 c2e360b487a45302fd22ec8c5fab63e54b5e844d4b17ca2fff37c69c366dd72a d02922c14c0932f65ed03e4219c117962edbad2dcdeaa9c10ce8af38a4ae52e2 b377245b0be19a77d6c936e7530cc4d0b78d0cc4a92698fa2870fa54f2d8503e 2d17b3d52fb2febb09f2b2af0c2a1892039ebe19a690098799a858e3d39631bd 6925a154d161df3918074ada6bd52baddd0adc3f07e2d9f15e27cbf7fe8b98c4 07205c811121fa91e059f2f99322fed63f359ac9da97aec383f067f23e5de331 51e80f0a88ab50fe8fdae4a5de93c1ae2fdca06150b37246140c0e87cb2325a6 0d2349162ae3ac93144eee1e665a1289105318fdfe86b6e76251cb25adc967d3 d0b97fe5e279e1161736ab22b4ca510b964342383a840defd38f96a7280e6ac1 34e48d740607ff2e7804164a16d47735864db847c97335e6d4215cb99911a1ec 015a3edaac1f28fedd56d2467130d07bae9416c15f0827d27c6c79f59054282a 418c12c157c91223a829947f47592f7cafd93ca182b25a73a9419127e3b12a9e 5167ac3963f2b019b338ac46d63880f94dda4b538835884d2a5538c85528d6bd 977f844d32b43b0e48caed5a4bdabcefe71695d69ca784df9b9a4adcf6669417 2fd3c6963e5fd0fbe96f80758aa2b4b2581bd4e0d85a98cd8717946c30324715 fae0c3babe98c1852708019e0a144d64bff9912b99fec79c9f3a276b9dd2a8e3 ec57dfa4de5817d511080c2cb7695014dbba456a298668b956464b8147db3e02 92fc7ae62376d7675dfccb8b947d6caf0fc3521b758a73eb5a7a51ebeb39e709 d44c2ea584c9029cccbee5d93104ccfd798bea4f1ee3c7e2f31c2b09d9d4b979 6143ffe66125de43805180037c24fc64337abe11c235f248766effe0da4b8b8b 1606829e9ea15b0f1f681e68c25f059371ad2778776ec1208b8786cc692189fd 250bb200234a10b96ccb05e78a7148857dddceb9caa51f4042b613b01c38fd74 751cd1bc858cc7e793f59b9afd5505b99d37f91c8cbfe24d0f8e66431d2b7058 2bf8d8650b524446850b638565422eb6e3311e19858b69c66f4167b694b35153 8f77bb58236b9e0b32a2f53fbd14cca266387dbd8bc15b2a01caefa2a26a07f3 6ecf44a7d6cee5252b8c494a3fde5939281b3a5b97b672c0b3c2ffdc5dd41cbf cdd55af8d4e5bdc8053de847bd5c55797bbbe871d3fc132e82052af0dbd783b1 28d03b40082d33a7ff7acbf4af501cb7da792cb2d7814e73b92ebb3e66fff809 c0b6be54d377c00a50e74c937c5de6e16af6697a1363db100014eee8c313b259 54a130b0d537abccaab31f4e9de36c16b1ba30bcf92ee81a4a88c2629b80b164 50c6b5279a5c26bc69a867ca0e3850b96ce5bcbd49a1d20f5fc273c78661b7a4 31d176fb7283b0dc9ddf221ef715a51bbf5233e52188dd71afb3e1ca75eb522e 5a1572368c085137f88529643c9816074f824c92381300eac732aa2c7e3bfb78 63d14a3add381c9303dc75d5d631c46e72e4df6368c6f435a4c9b02e429f1a7a f6449b556519a6c7dfc277ba124cbe0089b40bc4174807eeb3ce68e170958157 bd286755f8feb4c6265f8c4d4b9ed65bc6c3e6348782285c723caa9a8f575401 41d18d0ee1445773426cb4822a654477ad244018d70ee03dfe8597802e1930a8 941ea051cba983e0e55d60a10a6044306db32890a12bb9575b4f9cbeb54e818d 698d75572034eb018619c8827ec6d8df4cccaabea8f9112e1770c4e456cbca23 4e50efe58f25ee9ad665a7d19f185c520bef7b9e6aa5532dd5a0d3b75abaa5f8 d94eb0ac9d3846aaa23f7cad2300500dbde686bdaf03df628a0cdacee6fd535f d5322960b7c8cefb79e95863174b4ef397872c0b464d5374abc8b8e5442e8ebf 9f1c024d7a1871fcfe3c94b6422ec345b3f532feadf2403951e62d396500e035 a95f9937b1aaf463f5c7aaa287dd5676339d3da6859dc6c3fca42542f633ab0f d2b4513415fe2449a2f8ff68b9c009dbe0b554b7fa50946ec727ed75b3332686 445e76d95cdb66760034ad280c789dab2374456c260a38e7baa52178b31f5ac3 ee77841f84c55937511b3bd7fca7ce49c0c6f43e0ffde87b835c3cdf955357b5 8f560970f891bbe5fce30d5004294803e0ebbdad34dbab50539276883e55642b cc55d008e50af4ea8d2ea9a09f37aa4e707c7c9141d2e5da10c9f785bbba8938 9167c0d02e0e00c9a78128e3628cddddcd13a6b3c46050d6cc725a00787d18e1 2710aa594036af82f74be3827782b63269a9a197ea790adcff985fd2c59fcae7 ff2e932b4e6120a9f8524390ee85c667ef902fdd43eb28abbe43308ade1e6a23 90367b296c1614c2ec4f2ccef0e21071c2c9fe980004209c3f0e7abea1486776 6e01e3b693c4a4316c4b1c1bbcbd8b9a804f95eb11d5a0558ed11f2fc6857bd7 6f7598657f9b8962a2e7e6b1518b0bc7f3732b5c45cf57130e4cbfb081701603 0c60341ba41dfc48c771c917ea5fa0a1796f7750c6507a0ade51febb0bbf415d 7898736fd241e439e954fbb4794bc262c595d7ce77e5ee50392b3869d8dd6e28 2e4292cf9bd0fd5ac421b91aaf3afcfc47ab1a4171db744281ec5d36e62e87df f5d36c7e6b5fc8e1001ae80edb615abb1bfc2e8cefd9f53cf7b0ad5ef8ac8eb2 536e30a58eee3daa71d66c910485d972d240566a0fbaf591753fb9c1c520d115 e4779e0b6b41fe15c4d1df8712d367aa7661f772c91fcb1468b8cb901ac29a2d b55226b39350d3d3304af0e6f1c4c7f05e3e086b3cc462033841d535c3561e06 7532adb4a878e8ea72ee1352d654286d30a97110df3b14c69180241048862552 ee70e500d9677e0b1cf8b22c2e9cd5cf2fd441aaddb68f2331a4b016e8c5ef00 05eb269fe306f41d8e78039646d8b6f756f82b4b545bd5b88bb99ac6091d444e 1b6c7e577398dc42d89eed67c2d03d53f23e68abf0457e42aa3d278354506b7c 0fdc4d468dac3d9d1b631ebe480e54ea3f36d6cf440a6dfd98bfe6d74d11c494 73ecc734ba272a935f460a06cd12db8888daac308007d6e89b7d6652c1d97f96 e0bdfa9d4220d572d69f17c2f0f486bc656e86ed3c59b323022c8f438b73f99c a6843b09390922c57d022266b76ae116602dc8031f820eb43d84d368c6c05f5c 81a4406291f13068716c0de7f8f9557d787fc31a 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 18 /theta put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 25 /pi put dup 27 /sigma put dup 28 /tau put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 96 /lscript 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 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 115 /s put dup 116 /t put dup 117 /u put dup 120 /x put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueXX 5087385 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930da4f8d58abefd45db119b10eb409dd89792 3c6e705479464a4b310b58348c4b42393988fef4925cf984423aaf65fea9f0e6 4629953bcf50b919d968d99bd185f83112b2759cc411764e9bde677f57c5ee5a c555448d5f81a16259ded1e11bf4119d53e8ab07a802df900d3a2d5ccc1c6876 d29c5e9effb7af3ef83400b0910c07873a8c56fa7b1eb1ba426043b00cc95dbe 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031ee591906525fa45e30ab91c1fded43902ad035098c8f43f471b5978eed264 8c350f6044dc5decf18dcc4f7041f75e2ad29e2eb16351f55d13c3039d7ab814 294c742c186f97a56525415eb7108cae98dfa18e8ce64cff2afb446526b9b019 7be856ecd81182c45062983575781a318fa395f87be86c8787883d2e33371b98 536e46 0000000000000000000000000000000000000000000000000000000000000000 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 49 /one put dup 50 /two put dup 51 /three put dup 53 /five put dup 54 /six put dup 55 /seven put dup 58 /colon put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 127 /dieresis 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 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ebf8abb0dd0cb0b990d4ad5ab3ff69c0207d0ab15c5885521097cc21a6179b44 bec5b29bb1623c5e51aece84a1166a7a0d0b10ac88227167e6c7588093b2e933 bc0f94a2feffb35632db11ab6627d4a3d2003e87236d7a6dfb88c1f08077c252 f424ae9595f6f365a19c5f62f91fce878f3f7e9e7f9c49b10a449cf0807cd4fe 100dbf10a44f4f07fde5126c6ea6eac89b744f615d0d0080f79b27e18989036f ed00b63d569492d32b096c4a17c5a40e700d4a3f00163e7906e2d69a247f3fa2 520101ef057ba87e639d08fa377f8cd2fdf497d4e10d04c7f750860848aa32f1 287e68fc24ca6dcc90872236195a2f2fac2d92098e9981ef3df98840a0d66684 5999bd917cded778033ee15df367d5e502478fbe3315b267062901216f775f4d ef0b250c6e2febb803f299cc1d0610d3249c291ba9019ad7b71e2982508f1ee3 4378482b19ec369c2d50d161be9158a11e50196fc645b2d28ffe9d8489217aff 5fe4b31d367443303265c525d4a2a16fd69631d930a4e1f62c0520ff5ec88f92 861e1a1ba348c852c863c91d32e1297254209b27147606e7e6fa9ce8af59d3c3 ed8af2cc8d900a40c7a5f22d2b8eb0a33e3918df0196292aa7b49a3958c347d9 514e065666b8708bd887585d98e677c0b0eb3c2ef0954e58c4d6325bcce9aec2 5e60d8bff6b36523a0311a0dfc0e7abc5da5c63c2af784879d1400734d128a54 2c4226795bd37a84a5c19e348598a86b2e6f2ef33c59f6e53ec8e4ecc27786e8 b88a3a48eb448bffb3fad9180cec275534aa6014581062e8a30dd8c88f2ace6e 8504f07d6d985743b6a4e03aa6e2d817ae259ea71601fb75c98fee2dcd492e3e defe8bdd391dc8e332058d72f293bfc33d03d5a5cd6897cb52e444986149753b c7c5c4ef01063ea6133bc1c67a588a9c86d5fd700f956a64221c84c9646d1025 6eb709f3990c7871abf26dd0c409802788c713745bd2eb96c8d9743e21d94211 e0517dc7c10c60115c108865d621409d46294ddf788f50a4401dae727bdf8779 bc8e0610409c1a3226d48a3283b96259a4f0a758d1822ac7140ab4ba433996cb 163cf9d739be27281dcd8b9c2b7035c9279dd9f9253bae38cea3b55ae6f5ef41 6d7628b2b8f3d5adaf63ea4150aec1e5f897a6439c4c6b6ac827db0aae6360bf 5d199b441d929bfc5afb86401a7de93180431437d48a5ca928c7da1203848b23 6db01f3fe319c32a8f5364c9bb9618bf033e92af708c14f160589d120a0fb844 1f6c2181b5cdc6cc31a21e5094289f448d18c74bab930139267ee8b838c81f8f 525c9e747c1721bbc1a2318cb3c2c7b3ba1476d5cc21ff1c69d1a4f50ecf8481 39c916ae299041361849a9e03f4c43da6973efeb6fa0ca2ad1404d49024da5b2 886dd4514716e5550a6c219516c0fd0397c7c19719cf70cf4ba7942a4b457960 6c9a79010fec15991febd6a2b31fb40406df6c846a8bdb6f68aae0ece495a96b 7c480e8ca7d27432c610a6551de8e3776710dd5ab04b56b6e17c788d03769fbb 5c8e52e684c2e117a9cdf935545852341a2645ba1e67dad243afcaa17a110df2 7882f937b452878090c5a02215dd3bad0eb82e331f1235cb3da065d3a73330a5 fdab65147a42de960d3cb7c4b8c185db2e6ffccb49ac0da7f3939b893a811022 82a601cc626f696ceff91d6298e4efb7395390f0518797d1f1cdcd82d6768a71 4ef577de7b1c3901f8298718ee8c30 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 46 /period put dup 65 /A put dup 97 /a put dup 98 /b put dup 99 /c put dup 114 /r put dup 115 /s put dup 116 /t put readonly def /FontBBox{14 -250 1077 750}readonly def /UniqueXX 5000772 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd800bcb325468c79ffacb9cd3037a112066957 7889d67fff57549064b985d67ede2fb03272083723541051ebe936eff01a4ab3 cd845d565dfdf25d9ddd2792db17f52f894bea484480f06c832140fdd6c3af0e fef0584c6b78b5e2bbb9a752f10581fd1c48cb498a822afb18cddebba34695b9 b418010368fdfb7bf758d50f3af03360b81c776325c33c9c81130b1e1be840cf 68c71ffd091dccc0df7b507fe8953dc3836af0da08ff9ff3e9cf7288d9eb8baf ffac546a0fd987ae8e0bcf7b759f4408e13524f478de5896d816cb0ad2904271 cbec82a5ca50f0598502b434ecd94531b4a65181b5b3afcb650904b15157737a a4386f1a7f1252caf2adbd5c8162bf3b065c01ca471d520c884ebbe1b29f9549 134e285dea2fe158c15a08a744cdda00bb6c62158262e24702803d9e1dc1d3a3 1383085631154b077fc20488ee3aa03f47ed2766644330b94133f2f9389b99b8 96d3707d1c5bb227fd3bbdf3a6ff572bf07164107c7a1c0710a5c7cfb67282f9 c238100fa138b75c553d7cbef17e2292a320d3d86a6210e6e3a1e590b10dac6e 177094d5dd163db623d7b344478752f6f3d7d5b3fa92e6bc72dc6fc343d2f946 9153e3d14c5a6d6b1d9ca52947b789cb4cf6196a02973aff0b097333f8d615c0 a1726ab7e31c378eab77150faa28cce535afe3ae5798402ca51cbd18dedc2c43 5281d0781351df30f4d11761579676449ae4a47a10f677cdd31ca9120fc2606a 9025d3cc09d58856a9bbfb17669f706ff8c7efe37a3656c9b20609cccb58b309 0595ca5666833f2bbf884aac1cd632fce69b4e0ea2e0a25b9c4d471f0e322e09 4cefda321359fdbe0a82656d3ecaeb68b7e0c7e733d85066ed7474c22b67169a 1bf1bdf7dfd0ce0aaeef78ee393ce6ea65edc1fdcdf3f744f714005d8221a156 4e7afb117b9edb8b470bbad346f856507852a367877a199f0a199d704449538c ed349240ffa8898c77e5de3c7a36282ad76007e7c464dabdaede85a15b0354eb 06fdd3328991036e1081af3f436ad305ed24665d0725339e9dcc3c44b516a0e2 c91f8f4cc1ed7824d4efa363b503be97a3631fb4207727dfcf0ea73e8f19dc44 85386cf32533d52f57d10758dff2a1c6c8dc54a7d97cbf2e7a5070bc9541f569 f78b864fff16d4c8915767ab4d6053ee0c6b8fb4899c5e5fd460d1108c03eb9c f63f08d9123f6e6eb7a96c913b403ba3f5811bfadce7dd9a7b2cac73f903bb22 065c548bace04881b09ceebc928a285e726adcc0228d51b0573170bd07980246 75a11799e33afe4cb0025553ea2f8334600fd311366583fd9df70934b81e3b61 b7a7680b1d74ff3457f03090b0adc1df4e2261c23a57482c3d8873f4fa3dc381 3aac05724781413a2f215491c1f090f481f637c2a8e0b0ab4fd08ce012ebcd71 e7f4364bef1a7ab31e0ba8567aceee0980c6af355965b44cd6ccfbfcbf760a34 f8b105720cf50d32e1870e55f4708229e184f75e5f2f8f22a9d8cb911928b46c b5e1b22c00682399814dc61d7256c1e3f44ed42573d0affb65fb8e00789ab713 88811691dbea044284c251791774ee45f802d26d6e0cc16c2bee9cdaacaa9461 916338b625f2bbb7b2893abdfd194a5955ab3a6deb07de830d332e87234dd0ed a7f8a650547b4a5d58a08fe1329cdfd204e28b055fc6b30046786eb07b65f1c0 5cae6398d7ad3a56f892d6535be94d84b59a4bf7e120b1470319796fcc5eb190 3a4e26ea4f338b5a7be46db260302ea562d2e17d23a3265c329667d4a590997d 6e1ac98116a07aa735fdc8a8b8c8c316832abada3e5a399ec127ce5ca3ba6de9 d52ef98eda31c5a7ed0f0abe024ed95de9323b2ac8cb935d5075537bc9ef96d4 0516d10cefc86b5eaafdb99d07f59ac8e69f9156a5835ea37e3967bb1c9dac38 7ae08a4211a73612acf2e8db4550793c24b9235ffc92902a3d010629ed73ab04 4f0189ca190ad0a7dc55d71935d48be0f1310bac41945fc73e512815fc7ce822 8152ae94e4a0b807ea9af0f978b4cce77cbbc01b760c28796ff6cc778217a5fa 7c759438c99b0c6d1bd327818ba915dd8ffb6a5d5e5333b166e5f2d2b254a8df 5ce15b74a13988747fe688f0e2c33ec62dcc1e46012dc3c377b4e6b360b20199 01d900a26575712bd4350bc68fb576130c25333b5bc0fbb248949109a211ee8e a08238df32527c2d9213265aba880792e2b8b1dd40a8099a8b09bf49f3781b5b c8ac33b256183759f73dd7aee4986344963a09bcd40026d7126534e4823c82c7 d982e71d7e4c6a1f2186ec62653e3360ac5ae38480bec879abfe7e5b88d5ddf2 45144e423fb5630bf16a2a7c2ce0fedb067b645e2b93e6eb58255fa2227dad42 79efbfce1befd3658e982e059bcd8967948d054cdcdae474347db7834dde43d3 ccbcae558b7dc0e74302e8231942850a1c23bb7026cc683b36b00bb62448ff7a 1c07bbae8d062bc07e54a4c885a6e7c9f19b97248ad9cee1943ff698db1dc1db 4322fa3807c80f8526a0309f58c357f5aa5f7c97186b9db1d6025aa932abf298 8b81cc299278839e3c448f91d402a999d2b808a33177bdfa576bb0011030d5c8 9cc5af8f05420ca05a5214d63c782e8335c0a66949413994febebbce5f1d4c0a 0dfec20824a2ff5ed03bb05937bd67d4da66ccf1a3312a60f16c1a8e09c1aba8 7091a9641fd14ab95721fa030f498842784c7afb0e661f064b491ade93832e1e b03da8bbbc297a8c4ce6162ccdd14dc2b961a628be673b82fb49907c9d7a9767 8247dd523ae369394358be71a5537078fe6ce1fa5c91181e55a726d0b0c396d1 ee2f2c81b5267d398df8761a9cc933f824cf102ee564bc74d26eef96ddb65152 8f1dadf2c6f1401d01f9cae738db58c6d3e6097892dc032b074178fa498753 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /Lambda put dup 4 /Xi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 18 /grave put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 91 /bracketleft put dup 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 dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueXX 5000793 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306457c7e62c1d269f306bd3402e266de fc3b5e7d8a8d2f5bf0fe6ddd40d07391df4fad4a6018dce29a2b8f692b29f202 3a7c0e66de8ed85c14f1f8492167357f51a7e84cc5d92e0fee4d81cf7fbc8de5 2d2e7bb57142033993f9c08c315abade8dbc4a732e84e142d3bee51557910e12 cd8aa37c459a5e6b7f5269f59078aba3be4641a11ac48d0b625c8325b38ec08e 4c9e5e7fed976a5650d99d82114f449b9ca14c4ec957702295a39a93ef93f618 99b8ea06b092c3c1e503e6e436e0a9fa22576c8930ab3dc8c20f5d82b69cddf8 ff4dacfa9c54bed5a3aa3ea5b129fe96be632843b9b6bc91b615581a985db56b 1e01ca60ee69ca92cf5c0882ece62edad3e106d835348822400f0b66af658f2a e56ed08f8b0010571807009b73ab12a8cf14ca6c71f03c2a48c500f9d62266af 154a6375ff600d9bac3f05ce34142d6867a79581c533176bb2f3117336671e2e 44638a97167e2ea9644e31ea16c2ad2990ea33c54001e0c8156e6de8ab6a4d40 a7137ba275f39589fea2e2db8256adc103d6f9cc038037a47e8fd469c5f98a5e 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%%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 71 /G put dup 73 /I put dup 82 /R 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 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 121 /y put dup 122 /z put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueXX 5000769 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 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Fr(=)e(0)i(is)h(still) f(left)i(as)d(an)i(op)r(en)f(problem)g(in)118 414 y(literature.)189 520 y(In)37 b([1])f(the)i(problem)e(w)n(as)g(in)n(v)n(estigated)f(of)i (studying)g(quasi-p)r(erio)r(dic)e(solutions)h(of)h(the)g(corresp)r (onding)118 626 y(Sc)n(hr\177)-42 b(odinger)26 b(equation)1484 765 y Fo(i)1538 708 y(@)p 1523 745 79 4 v 1523 822 a(@)5 b(t)1611 765 y( )s Fr(\()p Fo(t)p Fr(\))24 b(=)f Fo(H)7 b Fr(\()p Fl(!)s Fo(t)p Fr(\))14 b Fo( )s Fr(\()p Fo(t)p Fr(\))p Fo(;)1196 b Fr(\(1)p Fo(:)p Fr(4\))118 931 y(for)27 b(small)h Fo(")p Fr(:)37 b(the)28 b(solutions)f(of)h(the)g(Sc)n(hr\177) -42 b(odinger)26 b(equation)h(\(1.4\))h(w)n(ere)f(sho)n(wn)g(to)h(b)r (e)g(expressible)f(in)h(terms)118 1037 y(of)g(an)n(y)e(particular)h (solution)g Fo(g)j Fr(of)e(the)g(generalized)e(Riccati)h(equation)1330 1176 y(d)p Fo(G)p 1330 1213 112 4 v 1347 1289 a Fr(d)p Fo(t)1470 1232 y Fm(\000)18 b Fo(iG)1647 1198 y Fn(2)1702 1232 y Fm(\000)g Fr(2)p Fo(if)9 b Fr(\()p Fo(t)p Fr(\))14 b Fo(G)k Fr(+)g Fo(i")2248 1198 y Fn(2)2308 1232 y Fr(=)k(0)p Fo(:)1031 b Fr(\(1)p Fo(:)p Fr(5\))118 1411 y(T)-7 b(o)30 b(come)g(bac)n(k)g(to)g(the)h(original)e(problem)h(ab)r(out)g(the)h(sp) r(ectrum)g(of)f(the)h(quasi-energy)e(op)r(erator)f(is)j(not)f(so)118 1517 y(immediate,)c(as)d(one)h(has)g(to)h(c)n(hec)n(k)e(some)h(prop)r (erties)f(of)i(the)g(solution)e(of)i(the)g(generalized)e(Riccati)h (equation,)118 1624 y(whic)n(h)k(are)e(not)i(ob)n(vious)e(\(see)i (Section)f(7)g(in)h([1];)g(w)n(e)f(shall)g(come)g(bac)n(k)g(to)g(this)h (in)g(Section)g(3\).)189 1730 y(In)e(particular)f(in)h([1])g(it)g(w)n (as)f(found)h(that)g(quasi-p)r(erio)r(dic)f(solutions)g(of)h(the)h (generalized)d(Riccati)i(equation)118 1836 y(exist)g(in)g(the)h(form)e (of)h(formal)g(p)r(o)n(w)n(er)e(series,)i(but)g(suc)n(h)g(series)f(w)n (ere)g(argued)g(to)h(b)r(e)g(in)h(general)d(div)n(ergen)n(t.)35 b(In)118 1943 y([7])27 b(it)g(w)n(as)f(pro)n(v)n(ed)f(that)i(quasi-p)r (erio)r(dic)f(solutions)g(exist)g(indeed,)i(and)e(that)h(they)g(are)f (not)h(analytic)f(in)h Fo(")p Fr(;)g(in)118 2049 y(fact)32 b(they)g(are)f(de\014ned)i(on)e(a)h(Can)n(tor)f(set)h(of)g(v)-5 b(alues)31 b(of)h(the)h(p)r(erturbativ)n(e)e(parameter)f Fo(")i Fr(cen)n(tered)g(around)118 2155 y(the)c(origin.)189 2262 y(F)-7 b(or)25 b(simplicit)n(y)g(w)n(e)g(assume)f(the)i(same)f (nondegeneracy)e(condition)i(of)g(the)h(external)f(\014eld)g(as)g(in)g ([7])g(\(whic)n(h)118 2368 y(corresp)r(onds)j(to)i(the)h(condition)f (of)g(case)f(\(1\))h(of)h(Theorem)e(2.2.)44 b(in)30 b([1]\),)h(but)g(v) n(ery)e(lik)n(ely)g(suc)n(h)h(a)g(condition)118 2474 y(can)d(b)r(e)h(easily)f(eliminated.)189 2580 y(The)35 b(problem)f(describ)r(ed)h(ab)r(o)n(v)n(e)e(is)i(sligh)n(tly)g (di\013eren)n(t)g(from)f(that)h(considered)f(in)h([4])g(and)g([10],)h (as)e(one)118 2687 y(\014xes)i(the)h(frequencies)f Fo(!)952 2699 y Fn(1)989 2687 y Fo(;)14 b(:)g(:)g(:)f(;)h(!)1225 2699 y Fj(d)1263 2687 y Fr(,)39 b(with)e Fo(d)h Fm(\025)f Fr(1,)h(and,)h(b)n(y)d(imp)r(osing)g(a)g(Diophan)n(tine)g(condition)g (on)g(the)118 2793 y(rotation)d(v)n(ector)g(\()p Fo(!)783 2805 y Fn(1)820 2793 y Fo(;)14 b(:)g(:)g(:)g(;)g(!)1057 2805 y Fj(d)1095 2793 y Fo(;)g(f)1173 2805 y Fd(0)1215 2793 y Fr(\))34 b(if)h Fo(f)1405 2805 y Fd(0)1480 2793 y Fm(6)p Fr(=)f(0)g(and)g(on)g(the)h(rotation)e(v)n(ector)g(\()p Fo(!)2760 2805 y Fn(1)2797 2793 y Fo(;)14 b(:)g(:)g(:)f(;)h(!)3033 2805 y Fj(d)3072 2793 y Fr(\))34 b(if)h Fo(f)3262 2805 y Fd(0)3338 2793 y Fr(=)e(0,)j(one)118 2899 y(\014nds)31 b(quasi-p)r(erio)r(dic)f(solutions)h(b)n(y)f(requesting)h(further)g (conditions)f(on)h(the)g(parameter)f Fo(")p Fr(:)44 b(therefore)30 b(one)118 3006 y(studies)g(the)g(dep)r(endence)h(on)e Fo(")h Fr(of)g(the)g(quasi-p)r(erio)r(dic)f(solutions.)43 b(But)30 b(of)g(course)e(one)i(can)f(also)g(\014x)h Fo(")g Fr(and)118 3112 y(\014nd)d(conditions)f(on)g(the)g(rotation)g(v)n (ector,)f(and)h(the)h(problem)f(can)f(b)r(e)i(solv)n(ed)e(essen)n (tially)h(in)g(the)h(same)f(w)n(a)n(y:)118 3218 y(for)36 b(details)g(w)n(e)g(refer)f(to)h(Section)h(2.)62 b(The)36 b(conclusions)f(are)h(summarized)f(in)i(the)f(follo)n(wing)g(result.)62 b(F)-7 b(or)118 3325 y(simplicit)n(y)24 b(w)n(e)f(state)g(the)h(result) f(only)h(when)f(one)g(has)g Fo(f)1900 3337 y Fd(0)1965 3325 y Fr(=)g(0,)h(as)e(this)i(is)g(the)g(case)e(whic)n(h)i(will)g(b)r (e)g(considered)118 3431 y(later,)h(but)h(the)g(pro)r(of)e(immediately) i(extends)f(to)g(an)n(y)g(analytic)f(function)i Fo(f)9 b Fr(\()p Fo(t)p Fr(\))26 b(in)f(\(1.5\),)h(without)f(requesting)118 3537 y(an)n(y)i(condition)g(on)h(its)f(a)n(v)n(erage.)118 3714 y Fc(Theorem)i(1.)35 b Fp(Consider)29 b(the)f(gener)l(alize)l(d)h (R)n(ic)l(c)l(ati)f(e)l(quation)g(\(1.5\),)i(with)e Fo(f)36 b Fp(a)28 b(r)l(e)l(al)g(analytic)h(quasi-p)l(erio)l(dic)118 3821 y(function)h(of)g(the)g(form)h(\(1.2\),)g(with)g Fo(f)1336 3833 y Fd(0)1400 3821 y Fr(=)23 b(0)29 b Fp(and)h Fl(!)j Fp(satisfying)e(the)f(Diophantine)h(c)l(ondition)1228 3983 y Fm(j)p Fl(!)21 b Fm(\001)d Fl(\027)6 b Fm(j)23 b Fo(>)g(C)1620 3995 y Fn(0)1657 3983 y Fm(j)p Fl(\027)6 b Fm(j)1757 3949 y Fg(\000)p Fj(\034)2020 3983 y Fm(8)p Fl(\027)29 b Fm(2)23 b Fk(Z)2281 3942 y Fj(d)2338 3983 y Fm(n)18 b(f)p Fc(0)p Fm(g)p Fo(;)938 b Fr(\(1)p Fo(:)p Fr(6\))118 4146 y Fp(with)30 b(Diophantine)i(c)l(onstants)d Fo(C)1195 4158 y Fn(0)1255 4146 y Fo(>)23 b Fr(0)29 b Fp(and)h Fo(\034)j(>)23 b(d)18 b Fm(\000)g Fr(1)p Fp(,)30 b(and)g(assume)g(that,)g(by)g(de\014ning)1072 4355 y Fo(Q)p Fr(\()p Fo(t)p Fr(\))24 b(=)e(exp)1484 4263 y Fi(\020)1534 4355 y Fr(2)p Fo(i)1619 4242 y Fi(Z)1700 4262 y Fj(t)1664 4430 y Fn(0)1744 4355 y Fr(d)p Fo(t)1820 4321 y Fg(0)1857 4355 y Fo(f)9 b Fr(\()p Fo(t)1969 4321 y Fg(0)1992 4355 y Fr(\))2024 4263 y Fi(\021)2097 4355 y Fr(=)2209 4276 y Fi(X)2184 4471 y Fh(\027)c Fg(2)p Ff(Z)2319 4443 y Fe(d)2367 4355 y Fo(e)2406 4321 y Fj(i)p Fh(\027)g Fg(\001)p Fh(!)r Fj(t)2571 4355 y Fo(Q)2637 4367 y Fh(\027)2684 4355 y Fo(;)784 b Fr(\(1)p Fo(:)p Fr(7\))118 4630 y Fp(one)30 b(has)h Fo(Q)490 4642 y Fd(0)554 4630 y Fm(6)p Fr(=)23 b(0)p Fp(.)38 b(Ther)l(e)31 b(exist)e(a)h(p)l (ositive)i(c)l(onstant)d Fo(")1921 4642 y Fn(0)1987 4630 y Fp(and,)i(for)g(any)f(b)l(al)t(l)g Fo(V)42 b Fm(\032)23 b Fk(R)2861 4588 y Fj(d)2929 4630 y Fp(of)31 b(L)l(eb)l(esgue)f(me)l (asur)l(e)118 4736 y(1,)h(a)f(p)l(ositive)h(c)l(onstant)e Fo(b)954 4748 y Fj(V)1041 4736 y Fp(and)h(a)g(set)g Fr(\012\()p Fo(V)19 b Fr(\))k Fm(\032)g Fo(V)48 b Fp(of)31 b(r)l(elative)g(L)l(eb)l (esgue)f(me)l(asur)l(e)1438 4884 y Fr(meas\(\012\()p Fo(V)19 b Fr(\)\))p 1438 4921 437 4 v 1501 4997 a(meas)o(\()p Fo(V)g Fr(\))1908 4940 y Fm(\025)j Fr(1)c Fm(\000)g Fo(b)2174 4952 y Fj(V)2232 4940 y Fo(C)2291 4952 y Fn(0)2328 4940 y Fo(;)1140 b Fr(\(1)p Fo(:)p Fr(8\))1869 5184 y(2)p eop %%Page: 3 3 3 2 bop 118 201 a Fp(such)30 b(that,)g(for)h(al)t(l)g Fm(j)p Fo(")p Fm(j)22 b Fo(<)h(")988 213 y Fn(0)1055 201 y Fp(and)30 b(for)h(al)t(l)f Fl(!)c Fm(2)e Fr(\012\()p Fo(V)19 b Fr(\))p Fp(,)30 b(\(1.5\))h(admits)f(a)h(p)l(articular)f (solution)g(of)h(the)f(form)1205 414 y Fo(g)s Fr(\()p Fo(t)p Fr(\))23 b Fm(\021)f Fo(g)s Fr(\()p Fo(t)p Fr(;)14 b Fo(";)g Fl(!)s Fr(\))23 b(=)1900 335 y Fi(X)1875 530 y Fh(\027)5 b Fg(2)p Ff(Z)2010 502 y Fe(d)2058 414 y Fo(e)2097 379 y Fj(i)p Fh(\027)g Fg(\001)p Fh(!)r Fj(t)2262 414 y Fo(g)2302 426 y Fh(\027)2349 414 y Fr(\()p Fo(";)14 b Fl(!)s Fr(\))p Fo(;)916 b Fr(\(1)p Fo(:)p Fr(9\))118 705 y Fp(analytic)31 b(and)f Fr(2)p Fo(\031)s Fp(-p)l(erio)l(dic)i(in)e Fl( )c Fr(=)c Fl(!)s Fo(t)p Fp(.)189 882 y Fr(Note)29 b(that)h(the)g(condition)f Fo(f)1125 894 y Fd(0)1193 882 y Fr(=)d(0)j(is)g(used)g(to)h(expand)f(the)h(function)g Fo(Q)p Fr(\()p Fo(t)p Fr(\))f(as)g(in)h(\(1.7\):)40 b(for)29 b Fo(f)3293 894 y Fd(0)3361 882 y Fm(6)p Fr(=)d(0,)j(the)118 988 y(rotation)e(v)n(ector)f(w)n(ould)h(b)r(e)h(\()p Fo(!)1125 1000 y Fn(1)1162 988 y Fo(;)14 b(:)g(:)g(:)g(;)g(!)1399 1000 y Fj(d)1437 988 y Fo(;)g(f)1515 1000 y Fd(0)1556 988 y Fr(\);)28 b(see)g([1])f(and)g([7].)189 1094 y(In)g(the)g (particular)f(case)g(in)h(whic)n(h)g(one)f(has)h Fo(f)1665 1106 y Fd(0)1729 1094 y Fr(=)c(0)j(\(whic)n(h)i(is)e(the)i(case)e(left) h(out)g(in)g(literature\),)g(w)n(e)f(sho)n(w)118 1201 y(in)35 b(Section)f(3)f(that)i(it)g(is)f(p)r(ossible)f(to)i(obtain)e (informations)h(ab)r(out)g(the)g(sp)r(ectrum)h(of)f(the)g(corresp)r (onding)118 1307 y(quasi-energy)22 b(op)r(erator:)33 b(w)n(e)24 b(\014nd)g(that)h(also)e(in)h(suc)n(h)g(a)f(case,)h(for)g(a) f(set)h(of)g(v)-5 b(alues)24 b Fl(!)i Fr(=)c(\()p Fo(!)3026 1319 y Fn(1)3064 1307 y Fo(;)14 b(:)g(:)g(:)f(;)h(!)3300 1319 y Fj(d)3339 1307 y Fr(\))24 b(of)g(large)118 1413 y(relativ)n(e)h(measure,)h(the)g(sp)r(ectrum)g(is)h(pure)e(p)r(oin)n (t.)37 b(More)25 b(precisely)h(the)g(follo)n(wing)f(statemen)n(t)h(can) g(b)r(e)h(made)118 1519 y(\(co)n(v)n(ering)f(also)g(the)i(cases)f(kno)n (wn)g(from)g(the)h(literature\).)118 1697 y Fc(Theorem)45 b(2.)70 b Fp(Consider)41 b(the)g(quasi-ener)l(gy)g(op)l(er)l(ator)g Fo(K)k Fp(in)c(\(1.3\),)j(with)d Fo(H)48 b Fr(=)42 b Fo(H)7 b Fr(\()p Fl(\022)s Fr(\))40 b Fp(given)h(by)f(\(1.1\))118 1803 y(and)31 b Fo(f)40 b Fp(a)31 b(r)l(e)l(al)g(analytic)h(quasi-p)l (erio)l(dic)h(function)e(of)g(the)g(form)h(\(1.2\),)g(with)g Fl(!)h Fp(satisfying)f(the)f(Diophantine)118 1909 y(c)l(ondition)37 b(\(1.6\),)j(and)c(assume)g(that,)j(by)d(de\014ning)h Fo(Q)p Fr(\()p Fo(t)p Fr(\))f Fp(as)g(in)h(\(1.7\),)i(one)d(has)h Fo(Q)2874 1921 y Fd(0)2950 1909 y Fm(6)p Fr(=)e(0)p Fp(.)57 b(Ther)l(e)37 b(exist)f(a)118 2016 y(p)l(ositive)30 b(c)l(onstant)e Fo(")785 2028 y Fn(0)850 2016 y Fp(and,)h(for)h(any)e(b)l(al)t(l)i Fo(V)42 b Fm(\032)22 b Fk(R)1718 1974 y Fj(d)1785 2016 y Fp(of)29 b(L)l(eb)l(esgue)g(me)l(asur)l(e)f(1,)h(a)g(p)l(ositive)h(c) l(onstant)d Fo(b)3373 2028 y Fj(V)3459 2016 y Fp(and)i(a)118 2122 y(set)h Fr(\012\()p Fo(V)19 b Fr(\))k Fm(\032)g Fo(V)48 b Fp(of)31 b(r)l(elative)g(L)l(eb)l(esgue)e(me)l(asur)l(e)1438 2309 y Fr(meas\(\012\()p Fo(V)19 b Fr(\)\))p 1438 2347 437 4 v 1501 2423 a(meas)o(\()p Fo(V)g Fr(\))1908 2366 y Fm(\025)j Fr(1)c Fm(\000)g Fo(b)2174 2378 y Fj(V)2232 2366 y Fo(C)2291 2378 y Fn(0)2328 2366 y Fo(;)1098 b Fr(\(1)p Fo(:)p Fr(10\))118 2610 y Fp(such)30 b(that,)g(for)h(al)t(l)g Fm(j)p Fo(")p Fm(j)22 b Fo(<)h(")988 2622 y Fn(0)1055 2610 y Fp(and)30 b(for)h(al)t(l)f Fl(!)c Fm(2)e Fr(\012\()p Fo(V)19 b Fr(\))p Fp(,)30 b(the)g(sp)l(e)l(ctrum)f(of)i Fo(K)k Fp(is)30 b(pur)l(e)g(p)l(oint.)189 2787 y Fr(F)-7 b(or)28 b Fo(f)380 2799 y Fd(0)447 2787 y Fm(6)p Fr(=)e(0)i(the)i(pro)r (of)e(can)h(b)r(e)h(found)f(in)g([10].)41 b(F)-7 b(or)28 b Fo(f)1957 2799 y Fd(0)2024 2787 y Fr(=)e(0)i(the)i(pro)r(of)e(is)h (carried)f(out)h(in)h(Section)f(3;)g(the)118 2894 y(set)f(\012\()p Fo(V)19 b Fr(\))28 b(and)f(the)h(constan)n(t)f Fo(b)1142 2906 y Fj(V)1227 2894 y Fr(can)g(b)r(e)h(tak)n(en)f(the)h(same)f(as)g (in)h(Theorem)f(1.)189 3071 y(Of)i(course)f(it)i(w)n(ould)f(b)r(e)h(in) n(teresting)e(also)g(to)i(in)n(v)n(estigate)e(systems)g(with)i (in\014nite)g(\(p)r(ossibly)f(degenerate\))118 3177 y(lev)n(els)i(in)h (a)f(quasi-p)r(erio)r(dic)g(\014eld.)50 b(P)n(artial)30 b(results)h(in)h(this)g(direction)f(ha)n(v)n(e)g(b)r(een)h(obtained,)g (for)g(instance,)118 3283 y(in)g([5])f(and)g(in)h([6],)g(where)f(the)g (case)g(of)g(w)n(eak)f(p)r(erio)r(dic)i(\014eld)f(has)g(b)r(een)h (studied.)48 b(W)-7 b(e)32 b(note)f(that)h(the)g(tec)n(h-)118 3390 y(nical)27 b(di\016culties)h(arising)e(in)h(studying)h(suc)n(h)f (problems)f(are)g(strongly)g(related)h(to)g(those)g(c)n(haracterizing)e (the)118 3496 y(problem)34 b(of)f(\014nding)i(p)r(erio)r(dic)e(and)h (quasi-p)r(erio)r(dic)f(solutions)g(for)h(Hamiltonian)f(systems:)50 b(the)34 b(case)f(of)h(a)118 3602 y(\014nite)26 b(n)n(um)n(b)r(er)f(of) h(lev)n(els)e(is)i(similar)e(to)i(the)f(case)g(of)g(elliptic)h(lo)n(w)n (er-dimensional)d(tori)i(for)g(\014nite-dimensional)118 3709 y(Hamiltonian)35 b(system)h(\(and)f(the)h(case)e(of)i(t)n(w)n(o)e (lev)n(els)h(is)g(analogous)f(to)h(the)h(case)e(in)i(whic)n(h)f(there)h (is)f(only)118 3815 y(one)j(normal)f(frequency\),)k(while)e(the)f (cases)g(of)g(systems)g(with)g(in\014nitely)h(man)n(y)f(lev)n(els)g(in) g(a)g(p)r(erio)r(dic)g(or)118 3921 y(quasi-p)r(erio)r(dic)26 b(\014eld)i(corresp)r(ond,)e(resp)r(ectiv)n(ely)-7 b(,)27 b(to)g(the)h(cases)e(of)h(p)r(erio)r(dic)h(or)e(quasi-p)r(erio)r(dic)h (solutions)f(in)118 4028 y(Hamiltonian)f(PDE)g(systems.)36 b(In)25 b(particular)f(it)i(is)f(w)n(ell)g(kno)n(wn)g(that,)h(for)e (the)i(latter)f(systems,)g(the)h(problem)118 4134 y(of)k(pro)n(ving)f (existence)h(of)g(quasi-p)r(erio)r(dic)f(solutions)g(is)i(more)e (di\016cult)i(than)f(that)h(of)f(pro)n(ving)f(existence)h(of)118 4240 y(p)r(erio)r(dic)e(solutions.)36 b(By)28 b(pursuing)f(further)h (suc)n(h)f(an)h(analogy)e(the)i(case)f(of)g(zero-a)n(v)n(erage)d (external)j(\014eld)h(w)n(e)118 4346 y(consider)g(in)g(this)h(pap)r(er) f(corresp)r(onds)f(to)i(the)g(case)e(of)i(a)f(system)g(with)h (degenerate)f(free)g(Hamiltonian,)h(and)118 4453 y(this)f(requires)e(p) r(erturbation)h(theory)g(in)h(presence)f(o)r(d)g(degeneracies,)f(as)h (already)f(p)r(oin)n(ted)i(out)g(in)g([10].)278 4701 y Fs(2.)50 b(Results)38 b(ab)s(out)g(the)f(solutions)i(of)e(the)h (generalized)g(Riccati)f(equation)118 4878 y Fr(T)-7 b(o)28 b(pro)n(v)n(e)f(Theorem)h(1)g(w)n(e)g(pro)r(ceed)g(as)g(in)h ([7],)f(where)g(w)n(e)g(studied)h(the)g(case)f(in)h(whic)n(h)f Fl(!)f Fr(=)e(\()p Fo(!)3233 4890 y Fn(1)3270 4878 y Fo(;)14 b(:)g(:)g(:)f(;)h(!)3506 4890 y Fj(d)3545 4878 y Fr(\))29 b(is)118 4984 y(\014xed)g(and)f Fo(")h Fr(is)f(a)h (parameter)e(to)i(b)r(e)g(v)-5 b(aried.)39 b(As)29 b(in)g([7])f(w)n(e)h (write)f(the)h(solution)g(of)f(the)h(generalized)f(Riccati)1869 5184 y(3)p eop %%Page: 4 4 4 3 bop 118 201 a Fr(equation)27 b(\(1.5\))g(as)g Fo(G)d Fr(=)e Fo(i"Qu)p Fr(,)27 b(where)1125 460 y Fo(u)22 b Fr(=)h Fo(u)p Fr(\()p Fo(t)p Fr(;)14 b Fo(";)g Fl(!)s Fr(\))22 b(=)1738 356 y Fg(1)1711 381 y Fi(X)1711 560 y Fj(k)q Fn(=0)1846 460 y Fo(")1885 426 y Fj(k)1963 381 y Fi(X)1939 576 y Fh(\027)5 b Fg(2)p Ff(Z)2073 549 y Fe(d)2122 460 y Fo(e)2161 426 y Fj(i)p Fh(!)r Fg(\001)p Fh(\027)g Fj(t)2326 460 y Fo(u)2374 417 y Fn([)p Fj(k)q Fn(])2374 474 y Fh(\027)2452 460 y Fr(\()p Fo(";)14 b Fl(!)s Fr(\))836 b(\(2)p Fo(:)p Fr(1\))118 752 y(admits)29 b(the)g(renormalized)e(tree)h(expansion)g(en)n(visaged)f(in)h(Section)h (6)f(in)h([7]:)39 b(w)n(e)28 b(refer)g(there)g(for)g(notations)118 859 y(and)e(details.)37 b(In)26 b(fact)h(note)f(that)g(the)h(tree)f (expansion)g(and)g(the)g(m)n(ultiscale)g(decomp)r(osition)g(are)g (carried)e(out)118 965 y(at)i(\014xed)g Fl(!)j Fr(and)d Fo(")p Fr(,)g(so)f(that)h(it)h(is)f(not)g(imp)r(ortan)n(t)f(if)i (either)f(w)n(e)f(are)g(supp)r(osing)h(that)g Fl(!)j Fr(is)d(\014xed)g(and)g Fo(")f Fr(v)-5 b(aries)118 1071 y(or)27 b(vice)g(v)n(ersa.)189 1177 y(As)20 b(w)n(e)g(w)n(an)n(t)g(to)g (consider)g Fl(!)j Fr(as)d(a)f(parameter)g(to)i(b)r(e)f(v)-5 b(aried)20 b(w)n(e)g(shall)g(write)g Fo(g)2645 1147 y Fn([)p Fj(n)p Fn(])2728 1177 y Fr(\()p Fl(!)s Fo(;)14 b Fl(\027)6 b Fr(;)14 b Fo(")p Fr(\))19 b(and)i Fo(M)3286 1147 y Fn([)p Fj(n)p Fn(])3368 1177 y Fr(\()p Fl(!)s Fo(;)14 b Fl(\027)6 b Fr(;)14 b Fo(")p Fr(\))118 1284 y(instead)28 b(of)f Fo(g)542 1254 y Fn([)p Fj(n)p Fn(])624 1284 y Fr(\()p Fl(!)22 b Fm(\001)c Fl(\027)6 b Fr(;)14 b Fo(")p Fr(\))28 b(and)f Fo(M)1220 1254 y Fn([)p Fj(n)p Fn(])1302 1284 y Fr(\()p Fl(!)22 b Fm(\001)c Fl(\027)6 b Fr(;)14 b Fo(")p Fr(\))27 b(as)g(in)h([7];)g(in)f(the)h(same)f(w)n(a) n(y)g(w)n(e)g(shall)g(write)h Fm(M)3229 1254 y Fn([)p Fj(n)p Fn(])3311 1284 y Fr(\()p Fl(!)s Fo(;)14 b Fl(\027)6 b Fr(;)14 b Fo(")p Fr(\).)189 1390 y(The)26 b(condition)g Fo(Q)787 1402 y Fd(0)851 1390 y Fm(6)p Fr(=)d(0)i(is)h(imp)r(osed)g(in) g(order)e(to)i(sho)n(w,)f(b)n(y)h(a)f(\014rst)h(order)f(analysis,)g (that)h(the)g(co)r(e\016cien)n(ts)118 1496 y Fo(u)166 1453 y Fn([)p Fj(k)q Fn(])166 1511 y Fh(\027)272 1496 y Fr(are)g(formally)h(w)n(ell)g(de\014ned;)h(again)f(w)n(e)g(refer)g (to)g([7])h(\(and)f([1]\))h(for)f(details.)189 1603 y(The)e(only)g (real)f(di\013erence)h(with)h(resp)r(ect)e(to)h([7],)h(in)f(the)h (renormalized)d(expansion,)i(is)g(in)g(the)g(Diophan)n(tine)118 1709 y(conditions.)37 b(F)-7 b(or)26 b(\014xed)i Fo(")g Fr(w)n(e)f(assume)g(that)h(one)f(has)594 1828 y Fi(\014)594 1877 y(\014)594 1927 y(\014)622 1923 y Fo(i)p Fl(!)20 b Fm(\001)f Fl(\027)24 b Fm(\000)18 b(M)1028 1889 y Fn([)p Fj(n)p Fn(])1110 1923 y Fr(\()p Fl(!)s Fo(;)c Fl(\027)6 b Fr(;)14 b Fo(")p Fr(\))1404 1828 y Fi(\014)1404 1877 y(\014)1404 1927 y(\014)1455 1923 y Fm(\025)1716 1867 y Fo(C)1775 1879 y Fn(0)p 1552 1904 424 4 v 1552 1981 a Fr(2)1594 1957 y Fn(\()p Fj(n)p Fn(+1\))p Fj(=)p Fn(2)1842 1981 y Fm(j)p Fo(\027)5 b Fm(j)1934 1957 y Fj(\034)2152 1923 y Fm(8)p Fl(\027)29 b Fm(2)23 b Fk(Z)2413 1882 y Fj(d)2470 1923 y Fm(n)18 b(f)p Fc(0)p Fm(g)26 b Fr(and)i Fm(8)p Fo(n)23 b Fm(\025)f(\000)p Fr(1)p Fo(;)305 b Fr(\(2)p Fo(:)p Fr(2\))118 2166 y(with)31 b(the)f(same)g(Diophan)n(tine)g (constan)n(ts)f Fo(C)1569 2178 y Fn(0)1637 2166 y Fr(and)h Fo(\034)40 b Fr(as)29 b(in)h(\(1.6\).)45 b(W)-7 b(e)30 b(call)g(\012)g(the)g(set)h(of)f Fl(!)i Fr(for)e(whic)n(h)g(the)118 2272 y(Diophan)n(tine)k(conditions)f(\(2.2\))g(are)f(satis\014ed;)37 b(for)32 b(an)n(y)h(subset)h Fo(V)52 b Fm(\032)32 b Fk(R)2517 2231 y Fj(d)2589 2272 y Fr(w)n(e)h(denote)h(with)g(\012\()p Fo(V)19 b Fr(\))34 b(the)f(set)118 2378 y(\012)19 b Fm(\\)f Fo(V)h Fr(.)189 2485 y(Note)27 b(that)f(the)h(set)g(\012)f(is)h(con)n (tained)f(inside)g(the)h(set)g Fo(D)1956 2497 y Fj(\034)1997 2485 y Fr(\()p Fo(C)2088 2497 y Fn(0)2126 2485 y Fr(\))g(of)f(Diophan)n (tine)g(v)n(ectors)f(with)i(Diophan)n(tine)118 2591 y(constan)n(ts)34 b Fo(C)552 2603 y Fn(0)625 2591 y Fr(and)h Fo(\034)9 b Fr(,)38 b(whic)n(h)d(corresp)r(onds)e(to)i(\(2.2\))g(for)g Fo(n)g Fr(=)h Fm(\000)p Fr(1)e(\(when)i Fm(M)2742 2561 y Fn([)p Fg(\000)p Fn(1])2868 2591 y Fr(\()p Fl(!)s Fo(;)14 b Fl(\027)6 b Fr(;)14 b Fo(")p Fr(\))35 b(is)g(iden)n(tically)118 2697 y(v)-5 b(anishing,)27 b(see)h([7]\).)189 2804 y(Supp)r(ose)g(that) f Fo(")733 2816 y Fn(1)798 2804 y Fr(is)h(suc)n(h)f(that)h(the)g (series)1073 2954 y Fg(1)1046 2979 y Fi(X)1045 3158 y Fj(k)q Fn(=0)1180 3058 y Fo(")1219 3024 y Fj(k)1298 2979 y Fi(X)1273 3174 y Fh(\027)5 b Fg(2)p Ff(Z)1408 3146 y Fe(d)1456 3058 y Fo(e)1495 3024 y Fj(i)p Fh(!)r Fg(\001)p Fh(\027)g Fj(t)p 1660 3012 48 4 v 1660 3058 a Fo(u)1708 3015 y Fn(\()p Fj(k)q Fn(\))1708 3072 y Fh(\027)1801 3058 y Fo(;)p 2004 3012 V 180 w(u)2051 3015 y Fn(\()p Fj(k)q Fn(\))2051 3072 y Fh(\027)2167 3058 y Fr(=)2303 2979 y Fi(X)2255 3166 y Fj(\022)r Fg(2)p Fn(\002)2385 3146 y Fb(R)2385 3188 y Fe(k)q(;\027)p 2485 2990 120 4 v 2485 3058 a Fr(V)-7 b(al\()p Fo(\022)r Fr(\))p Fo(;)758 b Fr(\(2)p Fo(:)p Fr(3\))118 3379 y(obtained)27 b(b)n(y)f(replacing)g (the)h(propagators)d Fo(g)1570 3349 y Fn([)p Fj(n)1630 3358 y Fe(`)1658 3349 y Fn(])1708 3379 y Fr(with)k Fo(G)p Fr(2)2004 3349 y Fj(n)2045 3358 y Fe(`)2073 3349 y Fn(+1)2162 3379 y Fo(C)2227 3344 y Fg(\000)p Fn(1)2221 3401 y(0)2343 3379 y Fr(\(for)f(some)f(constan)n(t)g Fo(G)p Fr(\),)i(con)n(v)n(erges) c(for)118 3486 y Fm(j)p Fo(")p Fm(j)f(\024)g Fo(")353 3498 y Fn(1)390 3486 y Fr(.)34 b(In)22 b(the)g(follo)n(wing)e (discussion)h(w)n(e)g(shall)g(consider)f(v)-5 b(alues)21 b(of)g Fo(")g Fr(suc)n(h)g(that)h Fm(j)p Fo(")p Fm(j)h Fo(<)g(")3009 3498 y Fn(0)3046 3486 y Fr(,)f(with)g Fo(")3313 3498 y Fn(0)3373 3486 y Fm(2)i Fr(\(0)p Fo(;)14 b(")3602 3498 y Fn(1)3638 3486 y Fr(])118 3592 y(small)27 b(enough)g(\(ho)n(w)h (small)f(can)g(b)r(e)h(deduced)g(from)f(the)h(discussion)f(itself)6 b(\).)189 3698 y(De\014ne)28 b(recursiv)n(ely)e(the)i(sets)f(\012)1225 3668 y Fn([)p Fj(n)p Fn(])1336 3698 y Fr(as)g(follo)n(ws.)35 b(Let)28 b(us)g(call)1620 3897 y(\012)1680 3863 y Fn([0])1778 3897 y Fr(=)22 b Fo(D)1934 3909 y Fj(\034)1976 3897 y Fr(\()p Fo(C)2067 3909 y Fn(0)2104 3897 y Fr(\))p Fo(;)1332 b Fr(\(2)p Fo(:)p Fr(4\))118 4097 y(and,)28 b(for)f Fo(n)c Fm(\025)f Fr(1,)690 4334 y(\012)750 4300 y Fn([)p Fj(n)p Fn(])856 4334 y Fr(=)944 4217 y Fi(\032)1006 4334 y Fl(!)k Fm(2)d Fr(\012)1230 4300 y Fn([)p Fj(n)p Fg(\000)p Fn(1])1444 4334 y Fr(:)1513 4239 y Fi(\014)1513 4289 y(\014)1513 4338 y(\014)1541 4334 y Fo(i)p Fl(!)e Fm(\001)d Fl(\027)24 b Fm(\000)18 b(M)1947 4300 y Fn([)p Fj(n)p Fg(\000)p Fn(1])2115 4334 y Fr(\()p Fl(!)s Fo(;)c Fl(\027)6 b Fr(;)14 b Fo(")p Fr(\))2409 4239 y Fi(\014)2409 4289 y(\014)2409 4338 y(\014)2459 4334 y Fo(>)2720 4278 y(C)2779 4290 y Fn(0)p 2556 4315 424 4 v 2556 4393 a Fr(2)2598 4369 y Fn(\()p Fj(n)p Fn(+1\))p Fj(=)p Fn(2)2846 4393 y Fm(j)p Fo(\027)5 b Fm(j)2938 4369 y Fj(\034)2990 4217 y Fi(\033)3066 4334 y Fr(;)402 b(\(2)p Fo(:)p Fr(5\))118 4572 y(\014nally)27 b(de\014ne)1432 4734 y(\012)c(=)1632 4630 y Fg(1)1619 4655 y Fi(\\)1602 4831 y Fj(n)p Fn(=0)1741 4734 y Fr(\012)1801 4700 y Fn([)p Fj(n)p Fn(])1907 4734 y Fr(=)52 b(lim)1995 4784 y Fj(n)p Fg(!1)2182 4734 y Fr(\012)2242 4700 y Fn([)p Fj(n)p Fn(])2325 4734 y Fo(;)1143 b Fr(\(2)p Fo(:)p Fr(6\))118 4984 y(and)28 b(for)f(an)n(y)f(op)r(en)i(subset)g Fo(V)42 b Fm(\032)22 b Fk(R)1261 4943 y Fj(d)1327 4984 y Fr(write)28 b(\012)1600 4954 y Fn([)p Fj(n)p Fn(])1683 4984 y Fr(\()p Fo(V)19 b Fr(\))k(=)g(\012)1985 4954 y Fn([)p Fj(n)p Fn(])2086 4984 y Fm(\\)c Fo(V)g Fr(.)1869 5184 y(4)p eop %%Page: 5 5 5 4 bop 189 201 a Fr(Then)28 b(Theorem)f(1)g(is)g(a)g(consequence)g(of) h(the)g(follo)n(wing)e(lemmata.)118 378 y Fc(Lemma)40 b(1.)57 b Fp(Assume)35 b(that)h(the)g(set)g Fr(\012)g Fp(has)h(non-zer)l(o)f(me)l(asur)l(e)g(and)g(that)h(for)g(al)t(l)g Fl(!)g Fm(2)e Fr(\012)h Fp(the)g(functions)118 485 y Fm(M)218 454 y Fn([)p Fj(p)p Fn(])294 485 y Fr(\()p Fl(!)s Fo(;)14 b Fc(x)p Fr(;)g Fo(")p Fr(\))30 b Fp(ar)l(e)g Fo(C)820 454 y Fn(1)887 485 y Fp(in)g Fc(x)g Fp(and)g(satisfy)h(the)f (b)l(ounds)943 611 y Fi(\014)943 661 y(\014)943 711 y(\014)971 706 y Fm(M)1071 672 y Fn([)p Fj(p)p Fn(])1147 706 y Fr(\()p Fl(!)s Fo(;)14 b Fc(x)p Fr(;)g Fo(")p Fr(\))1437 611 y Fi(\014)1437 661 y(\014)1437 711 y(\014)1487 706 y Fm(\024)23 b Fo(D)r Fm(j)p Fo(")p Fm(j)p Fo(;)1938 611 y Fi(\014)1938 661 y(\014)1938 711 y(\014)1965 706 y Fo(@)2009 718 y Fd(x)2053 706 y Fm(M)2153 672 y Fn([)p Fj(p)p Fn(])2229 706 y Fr(\()p Fl(!)s Fo(;)14 b Fc(x)p Fr(;)g Fo(")p Fr(\))2519 611 y Fi(\014)2519 661 y(\014)2519 711 y(\014)2570 706 y Fm(\024)22 b Fo(D)r Fm(j)p Fo(")p Fm(j)p Fo(;)655 b Fr(\(2)p Fo(:)p Fr(7\))118 928 y Fp(for)30 b(some)f(c)l(onstant)f Fo(D)j Fp(and)e(for)h(al)t(l)g Fo(p)22 b(<)h(n)p Fp(.)38 b(Then)30 b(for)f(any)g(r)l(enormalize)l(d)i (tr)l(e)l(e)d Fo(\022)j Fp(such)e(that)f Fr(V)-7 b(al\()p Fo(\022)r Fr(\))24 b Fm(6)p Fr(=)f(0)28 b Fp(the)118 1034 y(numb)l(er)h Fo(N)481 1046 y Fj(n)526 1034 y Fr(\()p Fo(\022)r Fr(\))i Fp(of)f(lines)g(on)g(sc)l(ale)h Fo(n)e Fp(satis\014es)h(the)g(b)l(ound)1339 1248 y Fo(N)1406 1260 y Fj(n)1451 1248 y Fr(\()p Fo(\022)r Fr(\))24 b Fm(\024)f Fo(c)14 b Fr(2)1760 1214 y Fg(\000)p Fj(n=)p Fn(\(2)p Fj(\034)1977 1222 y Fa(1)2008 1214 y Fn(\))2101 1169 y Fi(X)2052 1351 y Fj(v)r Fg(2)p Fj(B)s Fn(\()p Fj(\022)r Fn(\))2284 1248 y Fm(j)p Fl(\027)2355 1260 y Fj(v)2394 1248 y Fm(j)p Fo(;)1051 b Fr(\(2)p Fo(:)p Fr(8\))118 1537 y Fp(for)31 b(a)f(suitable)g(p)l(ositive)h(c)l(onstant) e Fo(c)p Fp(.)118 1714 y(Pr)l(o)l(of.)48 b Fr(One)30 b(pro)r(ceeds)f(as)h(in)h(the)g(pro)r(of)f(of)h(Lemma)f(1)g(of)h([7],)g (b)n(y)f(pro)n(ving)f(inductiv)n(ely)i(on)f(the)h(order)e Fo(k)34 b Fr(of)118 1820 y(the)28 b(renormalized)e(trees)h(the)h(b)r (ound)1157 2024 y Fo(N)1233 1990 y Fg(\003)1224 2045 y Fj(n)1271 2024 y Fr(\()p Fo(\022)r Fr(\))c Fm(\024)f Fr(max)o Fm(f)p Fr(0)p Fo(;)14 b Fr(2)p Fm(j)p Fl(\027)6 b Fr(\()p Fo(\022)r Fr(\))p Fm(j)p Fr(2)2052 1990 y Fn(\(3)p Fg(\000)p Fj(n)p Fn(\))p Fj(=)p Fn(\(2)p Fj(\034)2354 1998 y Fa(1)2385 1990 y Fn(\))2433 2024 y Fm(\000)18 b Fr(1)p Fm(g)p Fo(;)868 b Fr(\(2)p Fo(:)p Fr(9\))118 2228 y(where)23 b Fm(j)p Fl(\027)6 b Fr(\()p Fo(\022)r Fr(\))p Fm(j)23 b(\021)670 2165 y Fi(P)758 2252 y Fj(v)r Fg(2)p Fj(B)s Fn(\()p Fj(\022)r Fn(\))994 2228 y Fm(j)p Fl(\027)1065 2240 y Fj(v)1104 2228 y Fm(j)h Fr(and)e Fo(N)1383 2197 y Fg(\003)1374 2248 y Fj(n)1421 2228 y Fr(\()p Fo(\022)r Fr(\))i(is)f(the)h(n)n(um)n(b)r(er)f(of)g(lines)g(in) g Fo(L)p Fr(\()p Fo(\022)r Fr(\))h(on)e(scale)h Fo(n)2974 2197 y Fg(0)3020 2228 y Fm(\025)f Fo(n)p Fr(.)36 b(The)23 b(c)n(hanges)118 2334 y(with)28 b(resp)r(ect)g(to)f([7])g(are)g(ob)n (vious,)f(so)h(that)h(w)n(e)f(omit)h(details.)p 2164 2326 42 42 v 118 2511 a Fc(Lemma)34 b(2.)43 b Fp(F)-6 b(or)32 b Fl(!)d Fm(2)f Fr(\012)j Fp(and)i(for)f Fc(x)g Fp(such)g(that)g Fo(g)1783 2481 y Fn([)p Fj(n)p Fn(])1865 2511 y Fr(\()p Fl(!)s Fo(;)14 b Fc(x)p Fr(;)g Fo(")p Fr(\))27 b Fm(6)p Fr(=)g(0)p Fp(,)32 b(ther)l(e)g(exist)f(two)h(c)l (onstants)f Fo(D)j Fp(and)e Fo(D)3638 2481 y Fg(0)118 2617 y Fp(such)e(that)g(the)g(functions)f Fm(M)1075 2587 y Fn([)p Fj(j)s Fn(])1148 2617 y Fr(\()p Fl(!)s Fo(;)14 b Fc(x)p Fr(;)g Fo(")p Fr(\))29 b Fp(ar)l(e)h(smo)l(oth)h(functions)e (of)i Fc(x)f Fp(and)g(satisfy)h(the)f(b)l(ounds)954 2744 y Fi(\014)954 2793 y(\014)954 2843 y(\014)981 2839 y Fm(M)1081 2805 y Fn([)p Fj(j)s Fn(])1153 2839 y Fr(\()p Fl(!)s Fo(;)14 b Fc(x)p Fr(;)g Fo(")p Fr(\))1443 2744 y Fi(\014)1443 2793 y(\014)1443 2843 y(\014)1494 2839 y Fm(\024)23 b Fo(D)r Fm(j)p Fo(")p Fm(j)p Fo(;)1945 2744 y Fi(\014)1945 2793 y(\014)1945 2843 y(\014)1972 2839 y Fo(@)2016 2851 y Fd(x)2060 2839 y Fm(M)2160 2805 y Fn([)p Fj(j)s Fn(])2232 2839 y Fr(\()p Fl(!)s Fo(;)14 b Fc(x)p Fr(;)g Fo(")p Fr(\))2522 2744 y Fi(\014)2522 2793 y(\014)2522 2843 y(\014)2573 2839 y Fm(\024)23 b Fo(D)r Fm(j)p Fo(")p Fm(j)p Fo(;)954 2926 y Fi(\014)954 2976 y(\014)954 3026 y(\014)981 3022 y Fm(M)1081 2987 y Fn([)p Fj(j)s Fn(])1153 3022 y Fr(\()p Fl(!)s Fo(;)14 b Fc(x)p Fr(;)g Fo(")p Fr(\))19 b Fm(\000)f(M)1645 2987 y Fn([)p Fj(j)s Fg(\000)p Fn(1])1802 3022 y Fr(\()p 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Fm(V)954 3858 y Fj(T)1007 3846 y Fr(\()p Fl(!)s Fo(;)14 b Fc(x)p Fr(;)g Fo(")p Fr(\))23 b(=)1407 3754 y Fi(\020)1633 3767 y(Y)1471 3949 y Fj(v)r Fg(2)p Fj(E)s Fn(\()p Fj(T)9 b Fn(\))p Fg([)p Fj(V)14 b Fn(\()p Fj(T)9 b Fn(\))1915 3846 y Fo(F)1968 3858 y Fj(v)2008 3754 y Fi(\021)o(\020)2177 3767 y(Y)2121 3949 y Fj(`)p Fg(2)p Fj(L)p Fn(\()p Fj(T)g Fn(\))2353 3846 y Fo(g)2396 3812 y Fn([)p Fj(n)2456 3821 y Fe(`)2484 3812 y Fn(])2507 3846 y Fr(\()p Fl(!)s Fo(;)14 b Fl(\027)2687 3858 y Fj(`)2719 3846 y Fr(;)g Fo(")p Fr(\))2827 3754 y Fi(\021)3449 3846 y Fr(\(2)p Fo(:)p Fr(11\))118 4145 y(with)28 b Fl(\027)355 4157 y Fj(`)410 4145 y Fr(=)23 b Fl(\027)552 4115 y Fn(0)546 4168 y Fj(`)607 4145 y Fr(+)18 b Fc(x)28 b Fr(\(and)g Fl(\027)1016 4115 y Fn(0)1010 4168 y Fj(`)1080 4145 y Fr(giv)n(en)f(b)n(y)g(\(7.15\))g(in)h([7]\),)g(one)f(m)n(ust)h(ha)n(v)n (e)1492 4282 y Fi(X)1435 4464 y Fj(v)r Fg(2)p Fj(B)s Fn(\()p Fj(T)9 b Fn(\))1682 4361 y Fm(j)p Fl(\027)1753 4373 y Fj(v)1792 4361 y Fm(j)23 b Fo(>)g 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b(pro)r(of)f(mak)n(es)g(use)g(of)h(an)f (inductiv)n(e)h(application)e(of)i(Lemma)f(1)g(\(see)h([7])f(for)g (further)h(details\).)36 b(Then)27 b(the)118 307 y(t)n(w)n(o)g (inequalities)g(\(2.12\))g(and)h(\(2.13\))e(imply)-7 b(,)28 b(for)g(all)f Fo(T)34 b Fm(2)23 b(S)2061 277 y Fg(R)2055 331 y Fj(k)q(;j)2147 307 y Fr(,)949 493 y Fm(jV)1023 505 y Fj(T)1075 493 y Fr(\()p Fl(!)s Fo(;)14 b Fc(x)p Fr(;)g Fo(")p Fr(\))p Fm(j)23 b(\024)g(j)p Fo(")p Fm(j)1584 459 y Fj(k)1625 493 y Fo(A)1687 505 y Fn(1)1724 493 y Fo(A)1786 459 y Fj(k)1786 514 y Fn(2)1827 493 y Fo(e)1866 459 y Fg(\000)p Fj(A)1968 467 y Fa(3)2000 459 y Fn(2)2033 434 y Fe(j)r(=)p Fa(\(2)p Fe(\034)2168 446 y Fa(1)2202 434 y(\))2310 415 y Fi(Y)2247 596 y Fj(v)r Fg(2)p Fj(B)s Fn(\()p Fj(T)9 b Fn(\))2494 493 y Fo(e)2533 459 y Fg(\000)p Fj(\024)p Fg(j)p Fh(\027)2682 467 y Fe(v)2717 459 y Fg(j)p Fj(=)p Fn(2)2808 493 y Fo(;)618 b Fr(\(2)p Fo(:)p Fr(14\))118 738 y(for)27 b(suitable)h(constan)n(ts)e Fo(A)984 750 y Fn(1)1022 738 y Fr(,)i 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b(e)28 b(can)f(also)g(write)g(in)h(\(2.21\),)f(for)g Fl(!)s Fo(;)14 b Fl(!)1513 385 y Fg(0)1558 415 y Fm(2)24 b Fr(\012)1697 385 y Fn([)p Fj(n)p Fn(])1807 415 y Fr(and)k(for)f Fo(n)2146 427 y Fj(j)2204 415 y Fm(\024)22 b Fo(n)p Fr(,)221 625 y Fo(g)264 591 y Fn([)p Fj(n)324 599 y Fe(j)354 591 y Fn(])377 625 y Fr(\()p Fl(!)472 591 y Fg(0)495 625 y Fo(;)14 b Fl(\027)580 637 y Fj(j)615 625 y Fr(;)g Fo(")p Fr(\))k Fm(\000)g Fo(g)867 591 y Fn([)p Fj(n)927 599 y Fe(j)958 591 y Fn(])981 625 y Fr(\()p Fl(!)s Fo(;)c Fl(\027)1161 637 y Fj(j)1196 625 y Fr(;)g Fo(")p Fr(\))23 b(=)f Fo(@)1458 637 y Fh(!)1513 625 y Fo(g)1556 591 y Fn([)p Fj(n)1616 599 y Fe(j)1646 591 y Fn(])1669 625 y Fr(\()p Fl(!)s Fo(;)14 b Fl(\027)1849 637 y Fj(j)1884 625 y Fr(;)g Fo(")p Fr(\))g(\()p Fl(!)2101 591 y Fg(0)2142 625 y Fm(\000)k Fl(!)s Fr(\))g(+)g Fo(o)2475 558 y Fi(\000)2514 625 y Fr(2)2556 591 y Fn(2)p Fj(n)2630 599 y Fe(j)2664 625 y Fo(C)2729 589 y Fg(\000)p Fn(2)2723 647 y(0)2833 625 y Fm(j)p Fl(!)2919 591 y 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Fn(1)2070 1052 y Fr(2)2112 1017 y Fn(2)p Fj(n)2186 1025 y Fe(j)2221 1052 y Fo(C)2286 1016 y Fg(\000)p Fn(2)2280 1074 y(0)2389 1052 y Fm(j)p Fl(\027)2460 1064 y Fj(j)2495 1052 y Fm(j)14 b Fo(;)894 b Fr(\(2)p Fo(:)p Fr(26\))118 1234 y(for)27 b(some)g(constan)n(t)g Fo(G)853 1246 y Fn(1)891 1234 y Fr(.)189 1341 y(If)g(\011)336 1353 y Fj(n)377 1361 y Fe(j)412 1341 y Fr(\()p Fl(!)507 1311 y Fg(0)530 1341 y Fo(;)14 b Fl(\027)615 1353 y Fj(j)650 1341 y Fr(;)g Fo(")p Fr(\))23 b Fm(6)p Fr(=)g(0)j(this)i(follo)n(ws)e (from)g(\(2.21\))g(and)h(from)g(the)g(inductiv)n(e)g(h)n(yp)r(othesis,) g(b)n(y)g(using)f(\(2.16\),)118 1448 y(with)i Fo(p)23 b(<)g(n)k Fr(instead)h(of)f Fo(n)p Fr(,)h(and)f(\(2.24\))g(in)h(order)e (to)i(obtain)f(the)h(b)r(ound)626 1594 y Fi(\014)626 1644 y(\014)626 1694 y(\014)654 1690 y Fo(@)698 1702 y Fh(!)752 1690 y Fo(g)795 1656 y Fn([)p Fj(n)855 1664 y Fe(j)886 1656 y Fn(])909 1690 y Fr(\()p Fl(!)21 b Fm(\001)e Fl(\027)1112 1702 y Fj(j)1146 1690 y Fr(;)14 b Fo(")p Fr(\))1254 1594 y Fi(\014)1254 1644 y(\014)1254 1694 y(\014)1305 1690 y Fm(\024)1602 1634 y Fr(1)p 1403 1671 442 4 v 1403 1750 a Fo(C)1468 1721 y Fn(2)1462 1772 y(0)1505 1750 y Fr(2)1547 1724 y Fg(\000)p Fn(2\()p Fj(n)1699 1732 y Fe(j)1730 1724 y Fn(+1\))1868 1623 y Fi(\000)1906 1690 y Fr(1)k(+)g Fo(C)6 b Fm(j)p Fo(")p Fm(j)2199 1656 y Fn(2)2236 1623 y Fi(\001)2288 1690 y Fm(j)p Fl(\027)g Fm(j)1300 1951 y Fr(+)1590 1895 y(1)p 1393 1932 436 4 v 1393 2011 a Fo(C)1452 2023 y Fn(0)1490 2011 y Fr(2)1532 1986 y Fg(\000)p Fn(2\()p Fj(n)1684 1994 y Fe(j)1714 1986 y Fn(+1\))1876 1838 y Fj(n)1917 1846 y Fe(j)1852 1872 y Fi(X)1855 2048 y Fj(s)p Fn(=0)1986 1951 y Fr(2)2028 1917 y Fj(s)2063 1951 y Fo(C)2128 1916 y Fg(\000)p Fn(1)2122 1973 y(0)2218 1951 y Fo(K)19 b Fm(j)p Fl(\027)6 b Fm(j)23 b(\024)f Fo(D)r Fr(2)2631 1917 y Fn(2\()p Fj(n)2731 1925 y Fe(j)2762 1917 y Fn(+1\))2876 1951 y Fo(C)2941 1916 y Fg(\000)p Fn(2)2935 1973 y(0)3031 1951 y Fm(j)p Fl(\027)6 b Fm(j)p Fo(;)3449 1839 y Fr(\(2)p Fo(:)p 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y(\014)2579 3910 y(\014)2621 3906 y Fm(j)p Fl(!)2707 3872 y Fg(0)2748 3906 y Fm(\000)k Fl(!)s Fm(j)c(j)p Fl(\027)3002 3918 y Fj(j)3037 3906 y Fm(j)1277 4071 y(\024)23 b Fr(4)14 b(2)1463 4037 y Fn(2\()p Fj(n)1563 4045 y Fe(j)1593 4037 y Fn(+1\))1707 4071 y Fo(C)1772 4035 y Fg(\000)p Fn(2)1766 4093 y(0)1875 4071 y Fm(j)p Fl(!)1961 4037 y Fg(0)2002 4071 y Fm(\000)19 b Fl(!)s Fm(j)13 b(j)p Fl(\027)2256 4083 y Fj(j)2291 4071 y Fm(j)h Fo(;)3449 3976 y Fr(\(2)p Fo(:)p Fr(30\))118 4233 y(hence)28 b(\(2.25\))f(and)g(\(2.26\))g(follo)n(w)g(also)f(in)i (suc)n(h)f(a)g(case.)189 4340 y(T)-7 b(o)27 b(obtain)g(\(2.16\))g(w)n (e)h(ha)n(v)n(e)e(to)h(consider)352 4610 y Fm(M)452 4576 y Fn([)p Fj(n)p Fn(])534 4610 y Fr(\()p Fl(!)629 4576 y Fg(0)653 4610 y Fo(;)14 b Fl(\027)6 b Fr(;)14 b Fo(")p Fr(\))j Fm(\000)h(M)1052 4576 y Fn([)p Fj(n)p Fn(])1135 4610 y Fr(\()p Fl(!)s Fo(;)c Fl(\027)6 b Fr(;)14 b Fo(")p Fr(\))22 b(=)1578 4506 y Fj(n)1539 4531 y Fi(X)1540 4707 y Fj(p)p Fn(=0)1673 4610 y Fr(\004)1728 4622 y Fj(p)1767 4610 y Fr(\()p Fl(!)s Fo(;)14 b Fl(\027)6 b Fr(;)14 b Fo(")p Fr(\))2075 4518 y Fi(\020)2123 4610 y Fo(M)2213 4576 y Fn([)p Fj(p)p Fn(])2289 4610 y Fr(\()p Fl(!)2384 4576 y Fg(0)2407 4610 y Fo(;)g Fl(\027)6 b Fr(;)14 b Fo(")p Fr(\))k Fm(\000)g Fo(M)2797 4576 y Fn([)p Fj(p)p Fn(])2873 4610 y Fr(\()p Fl(!)s Fo(;)c Fl(\027)6 b Fr(;)14 b Fo(")p Fr(\))3167 4518 y Fi(\021)1447 4901 y Fr(+)1569 4797 y Fj(n)1530 4822 y Fi(X)1531 4998 y Fj(p)p Fn(=0)1706 4793 y Fj(p)1664 4822 y Fi(X)1666 4998 y Fj(s)p Fn(=0)1797 4901 y Fr(\004)1852 4913 y Fj(p;s)1942 4901 y Fr(\()p Fl(!)2037 4867 y Fg(0)2060 4901 y Fo(;)g Fl(!)s Fo(;)g Fl(\027)6 b Fr(;)14 b Fo(")p Fr(\))p Fo(M)2449 4867 y Fn([)p Fj(p)p Fn(])2524 4901 y Fr(\()p Fl(!)2619 4867 y Fg(0)2642 4901 y Fo(;)g Fl(\027)6 b Fr(;)14 b Fo(")p Fr(\))p Fo(:)3449 4766 y Fr(\(2)p Fo(:)p Fr(31\))1869 5184 y(8)p eop %%Page: 9 9 9 8 bop 118 201 a Fr(W)-7 b(e)41 b(can)e(insert)h(\(2.18\))g(in)n(to)f 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b(w)n(e)g(ha)n(v)n(e)f(to)i(consider)216 1001 y Fm(M)316 967 y Fn([0])390 1001 y Fr(\()p Fl(!)485 967 y Fg(0)509 1001 y Fo(;)14 b Fl(\027)6 b Fr(;)14 b Fo(")p Fr(\))k Fm(\000)g(M)909 967 y Fn([0])983 1001 y Fr(\()p Fl(!)s Fo(;)c Fl(\027)6 b Fr(;)14 b Fo(")p Fr(\))23 b(=)f Fo(\037)1439 1013 y Fn(0)1476 1001 y Fr(\()p Fm(j)p Fl(!)g Fm(\001)c Fl(\027)6 b Fm(j)p Fr(\))1763 909 y Fi(n)1860 897 y Fg(1)1833 922 y Fi(X)1833 1101 y Fj(k)q Fn(=1)2016 922 y Fi(X)1967 1109 y Fj(T)j Fg(2)p Fj(S)2104 1089 y Fb(R)2101 1131 y Fe(k)q(;)p Fa(0)2198 1001 y Fo(")2237 967 y Fj(k)2278 909 y Fi(\020)2503 922 y(Y)2341 1104 y Fj(v)r Fg(2)p Fj(E)s Fn(\()p Fj(T)g Fn(\))p Fg([)p Fj(V)14 b Fn(\()p Fj(T)9 b Fn(\))2785 1001 y Fo(F)2838 1013 y Fj(v)2878 909 y Fi(\021)562 1232 y Fg(j)p Fj(B)s Fn(\()p Fj(T)g Fn(\))p Fg(j)598 1264 y Fi(X)601 1441 y Fj(j)s Fn(=1)768 1251 y Fi(h\020)942 1264 y(Y)871 1446 y Fj(`)p Fg(2)p Fn(\003)989 1454 y Fe(j)1020 1446 y Fn(\()p Fj(T)g Fn(\))1134 1343 y Fo(g)1177 1309 y Fn([0])1251 1343 y Fr(\()p Fl(!)1346 1309 y Fg(0)1369 1343 y Fo(;)14 b Fl(\027)1454 1355 y Fj(`)1486 1343 y Fr(;)g Fo(")p Fr(\))1594 1251 y Fi(\021)f(\020)1707 1343 y Fo(g)1750 1309 y Fn([0])1824 1343 y Fr(\()p Fl(!)1919 1309 y Fg(0)1943 1343 y Fo(;)h Fl(\027)2028 1355 y Fj(`)2056 1363 y Fe(j)2090 1343 y Fr(;)g Fo(")p Fr(\))19 b Fm(\000)f Fo(g)2343 1309 y Fn([0])2417 1343 y Fr(\()p Fl(!)s Fo(;)c Fl(\027)2597 1355 y Fj(`)2625 1363 y Fe(j)2660 1343 y Fr(;)g Fo(")p Fr(\))2768 1251 y Fi(\021)548 1508 y(\020)850 1521 y(Y)611 1703 y Fj(`)p Fg(2)p Fn(\003\()p Fj(T)9 b Fn(\))p Fg(n)p Fn(\(\003)934 1711 y Fe(j)966 1703 y Fn(\()p Fj(T)g Fn(\))p Fg([)p Fj(`)1139 1711 y Fe(j)1169 1703 y Fn(\))1209 1600 y Fo(g)1252 1566 y Fn([0])1327 1600 y Fr(\()p Fl(!)s Fo(;)14 b Fl(\027)1507 1612 y Fj(`)1538 1600 y Fr(;)g Fo(")p Fr(\))1646 1508 y Fi(\021io)1809 1600 y Fr(+)k(\()p Fo(\037)1976 1612 y Fn(0)2013 1600 y Fr(\()p Fm(j)p Fl(!)2131 1566 y Fg(0)2173 1600 y Fm(\001)h Fl(\027)6 b Fm(j)p Fr(\))18 b Fm(\000)g Fo(\037)2477 1612 y Fn(0)2514 1600 y Fr(\()p Fm(j)p Fl(!)k Fm(\001)c Fl(\027)6 b Fm(j)p Fr(\)\))14 b Fo(M)2937 1566 y Fn([0])3012 1600 y Fr(\()p Fl(!)3107 1566 y Fg(0)3130 1600 y Fo(;)g Fl(\027)6 b Fr(;)14 b Fo(")p Fr(\))p Fo(;)3449 1316 y Fr(\(2)p Fo(:)p Fr(32\))118 1895 y(with)579 2024 y Fo(g)622 1990 y Fn([0])696 2024 y Fr(\()p Fl(!)791 1990 y Fg(0)815 2024 y Fo(;)g Fl(\027)900 2036 y Fj(`)928 2044 y Fe(j)962 2024 y Fr(;)g Fo(")p Fr(\))19 b Fm(\000)f Fo(g)1215 1990 y Fn([0])1289 2024 y Fr(\()p Fl(!)s Fo(;)c Fl(\027)1469 2036 y Fj(`)1497 2044 y Fe(j)1532 2024 y Fr(;)g Fo(")p Fr(\))23 b(=)1760 1968 y Fo( )1814 1980 y Fn(0)1852 1968 y Fr(\()p Fm(j)p Fl(!)1970 1938 y Fg(0)2011 1968 y Fm(\001)c Fl(\027)6 b Fm(j)p Fr(\))p 1760 2005 402 4 v 1818 2081 a Fo(i)p Fl(!)1910 2057 y Fg(0)1951 2081 y Fm(\001)19 b Fl(\027)2041 2093 y Fj(`)2069 2101 y Fe(j)2190 2024 y Fm(\000)2283 1968 y Fo( )2337 1980 y Fn(0)2375 1968 y Fr(\()p Fm(j)p Fl(!)i Fm(\001)e Fl(\027)6 b Fm(j)p Fr(\))p 2283 2005 379 4 v 2341 2081 a Fo(i)p Fl(!)21 b Fm(\001)e Fl(\027)2541 2093 y Fj(`)2569 2101 y Fe(j)934 2267 y Fr(=)k Fm(\000)1192 2205 y Fr(\()p Fl(!)1287 2175 y Fg(0)1328 2205 y Fm(\000)18 b Fl(!)s Fr(\))h Fm(\001)f Fl(\027)1614 2217 y Fj(`)1642 2225 y Fe(j)p 1096 2247 677 4 v 1096 2323 a Fr(\()p Fo(i)p Fl(!)1220 2299 y Fg(0)1262 2323 y Fm(\001)g Fl(\027)1351 2335 y Fj(`)1379 2343 y Fe(j)1414 2323 y Fr(\)\()p Fo(i)p Fl(!)j Fm(\001)e Fl(\027)1678 2335 y Fj(`)1706 2343 y Fe(j)1741 2323 y Fr(\))1797 2267 y Fo( )1851 2279 y Fn(0)1888 2267 y Fr(\()p Fm(j)p Fl(!)i Fm(\001)e Fl(\027)6 b Fm(j)p Fr(\))18 b(+)2286 2210 y Fo( )2340 2222 y Fn(0)2378 2210 y Fr(\()p Fm(j)p Fl(!)2496 2180 y Fg(0)2537 2210 y Fm(\001)h Fl(\027)6 b Fm(j)p Fr(\))18 b Fm(\000)g Fo( )2843 2222 y Fn(0)2881 2210 y Fr(\()p Fm(j)p Fl(!)j Fm(\001)e Fl(\027)6 b Fm(j)p Fr(\))p 2286 2247 882 4 v 2584 2323 a Fo(i)p Fl(!)2676 2299 y Fg(0)2717 2323 y Fm(\001)19 b Fl(\027)2807 2335 y Fj(`)2835 2343 y Fe(j)3178 2267 y Fo(;)3449 2150 y Fr(\(2)p Fo(:)p Fr(33\))118 2494 y(whic)n(h)32 b(is)g Fo(C)513 2464 y Fn(1)583 2494 y Fr(extendible)h(outside)f(\012)1338 2464 y Fn([0])1412 2494 y Fr(.)51 b(Then)32 b(the)h(formal)e(deriv)-5 b(ativ)n(e)32 b(of)g Fm(M)2705 2464 y Fn([0])2779 2494 y Fr(\()p Fl(!)s Fo(;)14 b Fl(\027)6 b Fr(;)14 b Fo(")p Fr(\))32 b(with)h(resp)r(ect)f(to)118 2601 y Fl(!)f Fr(pro)r(duces)d(a) g(sum)h(of)f(terms)g(whic)n(h)h(are)e(of)i(the)f(form)h(of)f(some)g (quan)n(tit)n(y)g(whic)n(h)g(can)g(b)r(e)h(b)r(ounded)g(b)n(y)f(the)118 2707 y(square)i(of)h(the)g(b)r(ound)g(holding)g(for)g Fm(M)1416 2677 y Fn([)p Fj(n)p Fn(])1498 2707 y Fr(\()p Fl(!)24 b Fm(\001)c Fl(\027)6 b Fr(;)14 b Fo(")p Fr(\))31 b(\(simply)g(b)r(ecause,)h(as)e(w)n(e)h(ha)n(v)n(e)f(seen)g(b)n(y)h (reasoning)e(as)118 2813 y(in)k(pro)n(ving)e(\(2.25\))g(and)h (\(2.26\),)h(the)g(di\013erence)g(of)f(propagators)d(\(2.33\))j(in)h (\(2.32\))e(can)h(b)r(e)h(b)r(ounded)g(b)n(y)f(a)118 2920 y(constan)n(t)g(times)h(the)g(pro)r(duct)g(of)f(the)h(b)r(ounds)g (of)g(the)g(single)f(propagators\))e(times)j Fm(j)p Fl(\027)2968 2932 y Fj(`)3000 2920 y Fm(j)p Fr(,)h(where)e Fo(`)h Fr(is)f(some)118 3026 y(line)h(in)f(\003\()p Fo(T)12 b Fr(\).)51 b(So)32 b(w)n(e)g(can)g(write)g Fl(\027)1307 3038 y Fj(`)1370 3026 y Fr(=)f Fl(\027)1520 2996 y Fn(0)1514 3049 y Fj(`)1578 3026 y Fr(+)21 b Fo(\033)1711 3038 y Fj(`)1744 3026 y Fl(\027)6 b Fr(,)33 b(with)g Fo(\033)2095 3038 y Fj(`)2158 3026 y Fm(2)f(f)p Fr(0)p Fo(;)14 b Fr(1)p Fm(g)p Fr(:)44 b(the)33 b(term)f(with)h Fl(\027)3116 2996 y Fn(0)3110 3049 y Fj(`)3186 3026 y Fr(can)f(b)r(e)g(easily)118 3132 y(estimated)27 b(since)f(the)h(pro)r(duct)f(of)h(no)r(de)f (factors)g(in)g(the)h(self-energy)e(v)-5 b(alue)27 b(can)f(b)r(e)h(b)r (ounded)g(b)n(y)f(a)g(quan)n(tit)n(y)118 3238 y(con)n(taining)g(an)g (exp)r(onen)n(tially)g(deca)n(ying)g(factor)g Fo(e)1763 3208 y Fg(\000)p Fj(\024)p Fg(j)p Fh(\027)1912 3217 y Fe(`)1937 3229 y Fa(0)1973 3208 y Fg(j)2023 3238 y Fr(\(b)r(ecause)h (of)g(the)g(analyticit)n(y)f(of)h Fo(f)9 b Fr(\()p Fo(t)p Fr(\)\),)27 b(while)g(the)118 3345 y(other)g(one)g(\(when)h(not)g(v)-5 b(anishing\))27 b(is)h(just)g(of)g(the)g(form)f(of)g(a)h(constan)n(t)e (times)i Fm(j)p Fo(")p Fm(j)2770 3315 y Fn(2)2835 3345 y Fr(times)g Fm(j)p Fl(\027)6 b Fm(j)p Fr(.)189 3452 y(T)-7 b(o)28 b(sho)n(w)g(that)i(the)f(b)r(ound)g(\(2.17\))f(holds)h (also)f(for)g Fo(n)d Fm(\025)g Fr(1,)k(w)n(e)f(can)h(reason)e(essen)n (tially)h(in)h(the)g(same)g(w)n(a)n(y)-7 b(,)118 3558 y(again)27 b(b)n(y)g(using)g(the)h(b)r(ounds)g(\(2.22\))f(and)g (\(2.25\))g(whic)n(h)g(follo)n(w)g(from)h(the)f(inductiv)n(e)h(h)n(yp)r (othesis.)189 3665 y(This)g(concludes)f(the)h(pro)r(of.)p 1145 3657 42 42 v 118 3843 a Fc(Lemma)i(4.)36 b Fp(F)-6 b(or)29 b(any)g(b)l(al)t(l)g Fo(V)42 b Fm(\032)23 b Fk(R)1272 3802 y Fj(d)1339 3843 y Fp(of)29 b(L)l(eb)l(esgue)g(me)l(asur)l(e)f(1,) i(ther)l(e)f(is)g(a)g(p)l(ositive)h(c)l(onstant)d Fo(b)3222 3855 y Fj(V)3308 3843 y Fp(such)i(that,)118 3949 y(for)i Fo(")290 3961 y Fn(0)356 3949 y Fp(smal)t(l)g(enough,)g(one)f(has)1399 4038 y Fr(meas)o(\(\012\()p Fo(V)20 b Fr(\)\))p 1399 4075 437 4 v 1461 4151 a(meas\()p Fo(V)f Fr(\))1868 4094 y Fm(\025)k Fr(\(1)18 b Fm(\000)g Fo(b)2167 4106 y Fj(V)2225 4094 y Fo(C)2284 4106 y Fn(0)2321 4094 y Fr(\))c Fo(;)1059 b Fr(\(2)p Fo(:)p Fr(34\))118 4298 y Fp(wher)l(e)31 b Fr(meas)e Fp(denotes)h(the)g(L)l(eb)l(esgue)f(me)l(asur)l(e.)118 4475 y(Pr)l(o)l(of.)39 b Fr(Fix)27 b(an)n(y)g(ball)h Fo(V)42 b Fm(\032)22 b Fk(R)1083 4434 y Fj(d)1122 4475 y Fr(.)37 b(De\014ne)1125 4605 y Fi(\032)1202 4678 y Fm(I)1253 4648 y Fn([0])1351 4678 y Fr(=)22 b Fm(;)p Fo(;)1202 4777 y Fm(I)1253 4747 y Fn([)p Fj(n)p Fn(])1359 4777 y Fr(=)g(\012)1506 4747 y Fn([)p Fj(n)p Fg(\000)p Fn(1])1674 4777 y Fr(\()p Fo(V)d Fr(\))g Fm(n)f Fr(\012)1944 4747 y Fn([)p Fj(n)p Fn(])2027 4777 y Fr(\()p Fo(V)h Fr(\))p Fo(;)83 b Fr(for)27 b Fo(n)c Fm(\025)g Fr(1;)3449 4722 y(\(2)p Fo(:)p Fr(35\))118 4984 y(note)28 b(that)f Fm(I)j(\021)23 b([)700 4954 y Fg(1)700 5005 y Fj(n)p Fn(=0)829 4984 y Fm(I)880 4954 y Fn([)p Fj(n)p Fn(])986 4984 y Fr(=)g Fo(D)1143 4996 y Fj(\034)1184 4984 y Fr(\()p Fo(C)1275 4996 y Fn(0)1313 4984 y Fr(\))c Fm(\\)g Fo(V)37 b Fm(n)18 b Fr(\012\()p Fo(V)h Fr(\).)1869 5184 y(9)p eop %%Page: 10 10 10 9 bop 189 201 a Fr(F)-7 b(or)27 b(all)g Fo(n)c Fm(\025)g Fr(1)k(and)g(for)g(all)h Fl(\027)g Fm(2)c Fk(Z)1302 160 y Fj(d)1359 201 y Fm(n)18 b(f)p Fc(0)p Fm(g)26 b Fr(de\014ne)570 435 y Fo(I)613 401 y Fn([)p Fj(n)p Fn(])696 435 y Fr(\()p Fl(\027)6 b Fr(\))23 b(=)925 318 y Fi(\032)987 435 y Fl(!)j Fm(2)d Fr(\012)1211 401 y Fn([)p Fj(n)p Fg(\000)p Fn(1])1379 435 y Fr(\()p Fo(V)c Fr(\))47 b(:)1626 339 y Fi(\014)1626 389 y(\014)1626 439 y(\014)1654 435 y Fo(i)p Fl(!)20 b Fm(\001)f Fl(\027)24 b Fm(\000)18 b(M)2060 401 y Fn([)p Fj(n)p Fg(\000)p Fn(1])2227 435 y Fr(\()p Fl(!)s Fo(;)c Fl(\027)6 b Fr(;)14 b Fo(")p Fr(\))2521 339 y Fi(\014)2521 389 y(\014)2521 439 y(\014)2571 435 y Fm(\024)2836 379 y Fo(C)2895 391 y Fn(0)p 2669 416 431 4 v 2669 493 a Fr(2)2711 469 y Fn(\()p Fj(n)p Fn(+1\))p Fj(=)p Fn(2)2959 493 y Fm(j)p Fl(\027)6 b Fm(j)3059 469 y Fj(\034)3110 318 y Fi(\033)3186 435 y Fo(:)240 b Fr(\(2)p Fo(:)p Fr(36\))189 669 y(F)-7 b(or)29 b(an)n(y)f Fl(\027)35 b Fr(set)30 b Fl(!)f Fr(=)c Fo(\013)p Fl(\027)6 b Fo(=)p Fm(j)p Fl(\027)g Fm(j)19 b Fr(+)g Fl(\014)s Fr(,)30 b(with)g Fl(\014)22 b Fm(\001)e Fl(\027)32 b Fr(=)25 b(0:)40 b Fo(\013)30 b Fr(is)f(the)h(comp)r(onen)n(t)f(of)h Fl(!)i Fr(along)c(the)i(direction)f Fl(\027)6 b Fr(,)118 775 y(while)28 b Fl(\014)i Fr(is)e(the)g(orthogonal)d(comp)r(onen)n(t,)i (so)g(that)h Fl(!)21 b Fm(\001)e Fl(\027)28 b Fr(=)23 b Fo(\013)p Fm(j)p Fl(\027)6 b Fm(j)p Fr(.)189 882 y(Therefore)30 b(one)g(has)h(to)g(exclude)g(from)f(the)i(set)f(\012)1821 852 y Fn([)p Fj(n)p Fg(\000)p Fn(1])1988 882 y Fr(\()p Fo(V)19 b Fr(\))32 b(all)f(the)g(v)-5 b(alues)30 b Fl(!)i Fr(=)c Fo(\013)p Fl(\027)6 b Fo(=)p Fm(j)p Fl(\027)g Fm(j)20 b Fr(+)g Fl(\014)34 b Fr(in)d Fo(I)3437 852 y Fn([)p Fj(n)p Fn(])3520 882 y Fr(\()p Fl(\027)6 b Fr(\),)118 988 y(whic)n(h)28 b(giv)n(es)e(a)h(set)h(of)f(measure)1018 1114 y Fi(Z)1064 1302 y Fj(I)1098 1286 y Fa([)p Fe(n)p Fa(])1172 1302 y Fn(\()p Fh(\027)5 b Fn(\))1285 1227 y Fr(d)p Fl(!)26 b Fr(=)1504 1114 y Fi(Z)1550 1302 y Fj(I)1584 1286 y Fa([)p Fe(n)p Fa(])1658 1302 y Fn(\()p Fh(\027)5 b Fn(\))1771 1227 y Fr(d)p Fl(\014)s Fr(d)p Fo(\013)23 b Fr(=)2085 1114 y Fi(Z)2182 1227 y Fr(d)p Fl(\014)2300 1114 y Fi(Z)2383 1134 y Fn(1)2346 1302 y Fg(\000)p Fn(1)2449 1227 y Fr(d)p Fo(t)2535 1171 y Fr(d)p Fo(\013)p Fr(\()p Fo(t)p Fr(\))p 2535 1208 195 4 v 2594 1284 a(d)p Fo(t)2739 1227 y(;)687 b Fr(\(2)p Fo(:)p Fr(37\))118 1472 y(if)28 b Fo(\013)p Fr(\()p Fo(t)p Fr(\))h(is)e(de\014ned)h(b)n(y) 936 1698 y Fo(i\013)p Fr(\()p Fo(t)p Fr(\))p Fm(j)p Fl(\027)6 b Fm(j)19 b(\000)f(M)1414 1664 y Fn([)p Fj(n)p Fn(])1496 1698 y Fr(\()p Fo(\013)p Fr(\()p Fo(t)p Fr(\))p Fl(\027)6 b Fo(=)p Fm(j)p Fl(\027)g Fm(j)19 b Fr(+)f Fl(\014)s Fo(;)c Fl(\027)6 b Fr(;)14 b Fo(")p Fr(\))22 b(=)h Fo(t)2547 1642 y(C)2606 1654 y Fn(0)p 2380 1679 431 4 v 2380 1757 a Fr(2)2422 1733 y Fn(\()p Fj(n)p Fn(+1\))p Fj(=)p Fn(2)2669 1757 y Fm(j)p Fl(\027)6 b Fm(j)2769 1733 y Fj(\034)2820 1698 y Fo(;)606 b Fr(\(2)p Fo(:)p Fr(38\))118 1944 y(where)27 b(the)h(Whitney)g(extension)g(of)f Fm(M)1404 1914 y Fn([)p Fj(n)p Fn(])1487 1944 y Fr(\()p Fl(!)s Fo(;)14 b Fl(\027)6 b Fr(;)14 b Fo(")p Fr(\))27 b(giv)n(en)f(b)n(y)i(Lemma)f(3)g(has)g(to)h (b)r(e)g(used.)189 2050 y(By)f(deriving)g(\(2.38\))g(with)h(resp)r(ect) f(to)h Fo(t)f Fr(w)n(e)h(obtain)725 2228 y(d)p Fo(\013)p Fr(\()p Fo(t)p Fr(\))p 725 2265 195 4 v 784 2341 a(d)p Fo(t)943 2284 y Fm(j)p Fl(\027)6 b Fm(j)1057 2167 y Fi(\022)1118 2284 y Fo(i)18 b Fm(\000)1300 2228 y Fl(\027)p 1258 2265 137 4 v 1258 2341 a Fm(j)p Fl(\027)6 b Fm(j)1358 2317 y Fn(2)1405 2284 y Fo(@)1449 2296 y Fh(!)1503 2284 y Fm(M)1603 2250 y Fn([)p Fj(n)p Fn(])1686 2284 y Fr(\()p Fo(\013)p Fr(\()p Fo(t)p Fr(\))p Fl(\027)g Fo(=)p Fm(j)p Fl(\027)g Fm(j)18 b Fr(+)g Fl(\014)s Fo(;)c Fl(\027)6 b Fr(;)14 b Fo(")p Fr(\))2419 2167 y Fi(\023)2503 2284 y Fr(=)2767 2228 y Fo(C)2826 2240 y Fn(0)p 2600 2265 431 4 v 2600 2342 a Fr(2)2642 2319 y Fn(\()p Fj(n)p Fn(+1\))p Fj(=)p Fn(2)2890 2342 y Fm(j)p Fl(\027)6 b Fm(j)2990 2319 y Fj(\034)3041 2284 y Fo(;)385 b Fr(\(2)p Fo(:)p Fr(39\))118 2518 y(so)27 b(that,)h(b)n(y)f(using)h(\(2.17\),)e(w)n(e)i (can)f(b)r(ound)916 2632 y Fi(\014)916 2681 y(\014)916 2731 y(\014)916 2781 y(\014)953 2696 y Fr(d)p Fo(\013)p Fr(\()p Fo(t)p Fr(\))p 953 2733 195 4 v 1012 2809 a(d)p Fo(t)1158 2632 y Fi(\014)1158 2681 y(\014)1158 2731 y(\014)1158 2781 y(\014)1208 2752 y Fm(\024)1450 2696 y Fr(1)p 1306 2733 331 4 v 1306 2809 a(1)18 b Fm(\000)g Fo(C)6 b Fm(j)p Fo(")p Fm(j)1599 2785 y Fn(2)1879 2696 y Fo(C)1938 2708 y Fn(0)p 1670 2733 516 4 v 1670 2811 a Fr(2)1712 2787 y Fn(\()p Fj(n)p Fn(+1\))p Fj(=)p Fn(2)1960 2811 y Fm(j)p Fl(\027)g Fm(j)2060 2787 y Fj(\034)h Fn(+1)2218 2752 y Fm(\024)2504 2696 y Fr(2)p Fo(C)2605 2708 y Fn(0)p 2316 2733 V 2316 2811 a Fr(2)2358 2787 y Fn(\()p Fj(n)p Fn(+1\))p Fj(=)p Fn(2)2605 2811 y Fm(j)p Fl(\027)f Fm(j)2705 2787 y Fj(\034)h Fn(+1)2841 2752 y Fo(;)585 b Fr(\(2)p Fo(:)p Fr(40\))118 2986 y(whic)n(h,)28 b(inserted)f(in)n(to)g (\(2.33\),)g(giv)n(es)1309 3100 y Fi(Z)1355 3288 y Fj(I)1389 3272 y Fa([)p Fe(n)p Fa(])1463 3288 y Fn(\()p Fh(\027)5 b Fn(\))1576 3213 y Fr(d)p Fl(!)26 b Fm(\024)c Fo(C)1854 3225 y Fj(V)2132 3157 y Fo(C)2191 3169 y Fn(0)p 1922 3194 V 1922 3271 a Fr(2)1964 3247 y Fn(\()p Fj(n)p Fn(+1\))p Fj(=)p Fn(2)2212 3271 y Fm(j)p Fl(\027)6 b Fm(j)2312 3247 y Fj(\034)h Fn(+1)2447 3213 y Fo(;)979 b Fr(\(2)p Fo(:)p Fr(41\))118 3453 y(for)27 b(some)g(constan)n(t)g Fo(C)847 3465 y Fj(V)905 3453 y Fr(.)189 3560 y(This)h(has)f(to)g(b)r (e)h(done)f(for)g(all)h Fl(\027)g Fm(2)c Fk(Z)1397 3518 y Fj(d)1454 3560 y Fm(n)18 b(f)p Fc(0)p Fm(g)p Fr(,)26 b(so)h(that)h(w)n(e)f(ha)n(v)n(e)g(to)g(exclude)h(from)f(\012)2946 3529 y Fn([)p Fj(n)p Fg(\000)p Fn(1])3114 3560 y Fr(\()p Fo(V)19 b Fr(\))28 b(a)f(set)1484 3760 y Fm(I)1535 3726 y Fn([)p Fj(n)p Fn(])1641 3760 y Fr(=)1837 3681 y Fi([)1729 3876 y Fh(\027)5 b Fg(2)p Ff(Z)1863 3848 y Fe(d)1898 3876 y Fg(nf)p Fd(0)p Fg(g)2051 3760 y Fo(I)2094 3726 y Fn([)p Fj(n)p Fn(])2177 3760 y Fr(\()p Fl(\027)h Fr(\))1154 b(\(2)p Fo(:)p Fr(42\))118 4049 y(of)28 b(measure)e(b)r(ounded)i(b)n(y) 302 4271 y(meas)o(\()p Fm(I)565 4237 y Fn([)p Fj(n)p Fn(])648 4271 y Fr(\))c Fm(\024)886 4192 y Fi(X)791 4387 y Fh(\027)5 b Fg(2)p Ff(Z)925 4359 y Fe(d)960 4387 y Fg(nf)p Fd(0)p Fg(g)1114 4271 y Fr(meas)o(\()p Fo(I)1369 4237 y Fn([)p Fj(n)p Fn(])1452 4271 y Fr(\()p Fl(\027)h Fr(\)\))24 b Fm(\024)e Fo(C)1772 4283 y Fj(V)1939 4192 y Fi(X)1844 4387 y Fh(\027)5 b Fg(2)p Ff(Z)1978 4359 y Fe(d)2013 4387 y Fg(nf)p Fd(0)p Fg(g)2386 4215 y Fo(C)2445 4227 y Fn(0)p 2177 4252 V 2177 4329 a Fr(2)2219 4305 y Fn(\()p Fj(n)p Fn(+1\))p Fj(=)p Fn(2)2466 4329 y Fm(j)p Fl(\027)h Fm(j)2566 4305 y Fj(\034)h Fn(+1)2725 4271 y Fm(\024)22 b Fo(B)2875 4283 y Fj(V)3040 4215 y Fo(C)3099 4227 y Fn(0)p 2943 4252 290 4 v 2943 4329 a Fr(2)2985 4305 y Fn(\()p Fj(n)p Fn(+1\))p Fj(=)p Fn(2)3243 4271 y Fo(;)183 b Fr(\(2)p Fo(:)p Fr(43\))118 4560 y(with)28 b Fo(B)370 4572 y Fj(V)456 4560 y Fr(implicitly)g(de\014ned,)g(as)f (the)h(sum)f(is)h(con)n(v)n(ergen)n(t)d(for)i Fo(\034)33 b(>)23 b(d)18 b Fm(\000)g Fr(1.)189 4666 y(W)-7 b(e)28 b(are)f(left)h(with)g(the)g(sum)g(o)n(v)n(er)d Fo(n)j Fr(to)f(p)r(erform,)h(but)g(this)g(can)f(b)r(e)h(done)f(trivially)-7 b(,)27 b(and)h(it)g(giv)n(es)868 4912 y(meas\()p Fm(I)6 b Fr(\))24 b Fm(\024)1305 4808 y Fg(1)1278 4833 y Fi(X)1275 5009 y Fj(n)p Fn(=0)1414 4912 y Fr(meas)o(\()p Fm(I)1677 4878 y Fn([)p Fj(n)p Fn(])1761 4912 y Fr(\))f Fm(\024)g Fo(B)1967 4924 y Fj(V)2024 4912 y Fo(C)2083 4924 y Fn(0)2164 4808 y Fg(1)2137 4833 y Fi(X)2135 5009 y Fj(n)p Fn(=0)2410 4856 y Fr(1)p 2284 4893 295 4 v 2284 4971 a(2)2326 4947 y Fn(\()p Fj(n)p Fn(+1\))p Fj(=\034)2611 4912 y Fm(\024)f Fo(b)2734 4924 y Fj(V)2792 4912 y Fo(C)2851 4924 y Fn(0)2888 4912 y Fo(;)538 b Fr(\(2)p Fo(:)p Fr(44\))1848 5184 y(10)p eop %%Page: 11 11 11 10 bop 118 201 a Fr(for)27 b(some)g(p)r(ositiv)n(e)g(constan)n(t)g Fo(b)1131 213 y Fj(V)1189 201 y Fr(.)p 1248 193 42 42 v 189 378 a(Lemma)k(1)g(and)h(Lemma)f(2)g(pro)n(v)n(e)f(that)i(there)f (exists)g Fo(")2001 390 y Fn(0)2068 378 y Fo(>)e Fr(0)i(suc)n(h)h(that) f(the)h(series)f(\(2.1\))g(con)n(v)n(erges)e(for)118 485 y(all)e Fm(j)p Fo(")p Fm(j)c Fo(<)g(")468 497 y Fn(0)532 485 y Fr(pro)n(vided)j(that)i(one)f(has)g Fl(!)e Fm(2)f Fr(\012.)37 b(In)27 b(fact)h(b)n(y)f(using)g(Lemma)g(1)g(one)f(pro)n(v) n(es)g(that)h(the)h(functions)118 591 y Fm(M)218 561 y Fn([)p Fj(n)p Fn(])301 591 y Fr(\()p Fl(!)s Fo(;)14 b Fl(\027)6 b Fr(;)14 b Fl(!)s Fr(\))25 b(admit)i(the)g(b)r(ounds)g (stated)f(in)h(Lemma)f(2.)36 b(This)26 b(in)h(turn)g(implies)f(that)h (the)g(b)r(ound)g(\(2.8\))f(on)118 697 y(the)34 b(n)n(um)n(b)r(er)f(of) g(lines)h(on)f(scale)f Fo(n)h Fr(holds)g(for)g(all)g Fo(n)g Fm(\025)f Fr(0,)i(so)f(that)h(one)f(can)g(reason)e(as)i(in)h ([7])f(to)g(conclude)118 813 y(that)g(the)g(co)r(e\016cien)n(ts)f Fo(u)924 770 y Fn([)p Fj(k)q Fn(])924 827 y Fh(\027)1035 813 y Fr(admit)g(the)h(b)r(ound)g Fo(C)1753 783 y Fj(k)1794 813 y Fo(e)1833 783 y Fg(\000)p Fj(\024)1924 758 y Fb(0)1946 783 y Fg(j)p Fh(\027)5 b Fg(j)2065 813 y Fr(for)32 b(some)g(p)r(ositiv) n(e)g(constan)n(ts)g Fo(C)39 b Fr(and)32 b Fo(\024)3407 783 y Fg(0)3430 813 y Fr(:)47 b(then)118 919 y(con)n(v)n(ergence)30 b(follo)n(ws.)49 b(Lemma)32 b(3)g(and)g(Lemma)g(4)f(pro)n(v)n(e)g(that) h(the)h(set)f(\012)g(is)g(not)g(empt)n(y:)46 b(more)31 b(precisely)118 1026 y(for)26 b(an)n(y)g(ball)h Fo(V)42 b Fm(\032)22 b Fk(R)803 984 y Fj(d)869 1026 y Fr(\(of)k(not)h(to)r(o)g (small)f(Leb)r(esgue)g(measure\))g(the)h(in)n(tersection)f(\012)17 b Fm(\\)g Fo(V)45 b Fr(has)26 b(large)g(relativ)n(e)118 1132 y(measure)h(pro)n(vided)f(that)i(the)g(Diophan)n(tine)g(constan)n (t)f Fo(C)1972 1144 y Fn(0)2037 1132 y Fr(is)g(small)h(enough.)423 1380 y Fs(3.)50 b(Results)38 b(ab)s(out)g(the)f(sp)s(ectrum)h(of)g(the) f(quasi-energy)j(op)s(erator)118 1557 y Fr(If)20 b(the)f(sp)r(ectrum)g (of)h(the)f(quasi-energy)e(op)r(erator)g(\(1.3\))i(is)g(pure)g(p)r(oin) n(t,)i(then)f(an)n(y)e(solution)h(of)g(the)g(Sc)n(hr\177)-42 b(odinger)118 1663 y(equation)27 b(\(1.4\))g(has)h(to)f(b)r(e)h(a)f (linear)g(com)n(bination)g(of)g(harmonics)1074 1860 y(exp)1215 1793 y Fi(\000)1253 1860 y Fo(i)p Fr(\()p Fl(!)21 b Fm(\001)e Fl(\027)24 b Fr(+)18 b Fo(\025)1640 1872 y Fj(m)1703 1860 y Fr(\))p Fo(t)1765 1793 y Fi(\001)1804 1860 y Fo(;)180 b(m)22 b Fr(=)h(1)p Fo(;)14 b Fr(2)p Fo(;)96 b Fl(\027)29 b Fm(2)23 b Fk(Z)2645 1819 y Fn(2)2682 1860 y Fo(;)786 b Fr(\(3)p Fo(:)p Fr(1\))118 2058 y(for)27 b(suitable)h Fo(\025)603 2070 y Fn(1)668 2058 y Fr(and)f Fo(\025)877 2070 y Fn(2)915 2058 y Fr(;)h(see)f([4].)189 2164 y(T)-7 b(o)36 b(deduce)g(\(3.1\))f(from)h(the)g(results)g(of)g(Section)g(2)f (requires)g(some)g(considerations,)h(as)g(an)n(y)f(particular)118 2270 y(quasi-p)r(erio)r(dic)29 b(solution)h Fo(g)s Fr(\()p Fo(t)p Fr(\))g(of)g(the)h(generalized)e(Riccati)h(equation)g(\(1.5\))g (and)g(the)g(solution)g Fo( )s Fr(\()p Fo(t)p Fr(\))h(of)f(the)118 2376 y(Sc)n(hr\177)-42 b(odinger)26 b(equation)h(\(1.4\))g(are)g (related)g(according)f(to)h(Theorem)g(2.1)g(of)g([1];)h(see)f(also)f ([2].)37 b(So)27 b(one)g(has)1314 2574 y Fo( )s Fr(\()p Fo(t)p Fr(\))d(=)f Fo(e)1616 2539 y Fj(i\031)r(\033)1718 2547 y Fa(2)1750 2539 y Fj(=)p Fn(4)1822 2574 y Fo(U)9 b Fr(\()p Fo(t)p Fr(\))p Fo(e)2021 2539 y Fg(\000)p Fj(i\031)r(\033) 2175 2547 y Fa(2)2208 2539 y Fj(=)p Fn(4)2279 2574 y Fo( )s Fr(\(0\))p Fo(;)1026 b Fr(\(3)p Fo(:)p Fr(2\))118 2771 y(with)28 b Fo( )s Fr(\(0\))23 b Fm(2)h Fk(C)637 2729 y Fn(2)702 2771 y Fr(and)j(the)h(unitary)f(transformation)f Fo(U)9 b Fr(\()p Fo(t)p Fr(\))28 b(giv)n(en)f(b)n(y)g(form)n(ula)g (\(2.3\))g(of)h([1]:)890 3036 y Fo(U)9 b Fr(\()p Fo(t)p Fr(\))24 b(=)1161 2894 y Fi( )1241 2961 y Fo(R)q Fr(\()p Fo(t)p Fr(\))14 b(\(1)k(+)g Fo(ig)s Fr(\(0\))p Fo(S)5 b Fr(\()p Fo(t)p Fr(\)\))234 b Fm(\000)p Fo(i"R)q Fr(\()p Fo(t)p Fr(\))p Fo(S)5 b Fr(\()p Fo(t)p Fr(\))1374 3086 y Fm(\000)p Fo(i")p 1507 3013 309 4 v(R)q Fr(\()p Fo(t)p Fr(\))p Fo(S)g Fr(\()p Fo(t)p Fr(\))p 2031 3013 159 4 v 216 w Fo(R)q Fr(\()p Fo(t)p Fr(\))2203 2994 y Fi(\020)2253 3086 y Fr(1)18 b Fm(\000)g Fo(i)p 2425 3013 299 4 v(g)s Fr(\(0\))p Fo(S)5 b Fr(\()p Fo(t)p Fr(\))2723 2994 y Fi(\021)2787 2894 y(!)2866 3036 y Fo(;)602 b Fr(\(3)p Fo(:)p Fr(3\))118 3296 y(where)691 3433 y Fo(R)q Fr(\()p Fo(t)p Fr(\))24 b(=)e(exp)1101 3341 y Fi(\020)1169 3433 y Fm(\000)c Fo(i)1295 3320 y Fi(Z)1378 3340 y Fj(t)1341 3509 y Fn(0)1421 3433 y Fr(d)p Fo(t)1497 3399 y Fg(0)1534 3433 y Fr(\()p Fo(f)9 b Fr(\()p Fo(t)1678 3399 y Fg(0)1701 3433 y Fr(\))19 b(+)f Fo(g)s Fr(\()p Fo(t)1940 3399 y Fg(0)1963 3433 y Fr(\)\))2042 3341 y Fi(\021)2091 3433 y Fo(;)180 b(S)5 b Fr(\()p Fo(t)p Fr(\))23 b(=)2555 3320 y Fi(Z)2638 3340 y Fj(t)2601 3509 y Fn(0)2681 3433 y Fr(d)p Fo(t)2757 3399 y Fg(0)2794 3433 y Fo(R)2858 3399 y Fg(\000)p Fn(2)2947 3433 y Fr(\()p Fo(t)3009 3399 y Fg(0)3033 3433 y Fr(\))p Fo(;)403 b Fr(\(3)p Fo(:)p Fr(4\))118 3633 y(and)21 b(the)g(bar)e(denotes)i(complex)f(conjugation)g(as)g(in)g (\(1.2\).)35 b(In)21 b(particular,)f(in)h(order)f(to)g(ha)n(v)n(e)f(a)i (quasi-p)r(erio)r(dic)118 3740 y(solution)j(one)g(needs)h(not)g(only)f (that)h Fo(g)s Fr(\()p Fo(t)p Fr(\))g(is)f(quasi-p)r(erio)r(dic,)g(but) h(also,)g(as)f(remark)n(ed)e(in)j([1],)g(that)g(it)g(satis\014es)118 3846 y(some)e(prop)r(erties,)g(whic)n(h)g(w)n(ould)g(b)r(e)g (di\016cult)h(to)g(pro)n(v)n(e)d(directly)i(from)g(the)g(p)r (erturbativ)n(e)g(expansion)f(of)h Fo(g)s Fr(\()p Fo(t)p Fr(\),)118 3952 y(but)28 b(whic)n(h)g(follo)n(w)f(automatically)f(from) h(the)h(unitarit)n(y)g(of)f Fo(U)9 b Fr(\()p Fo(t)p Fr(\),)28 b(namely)f(one)h(has)1359 4149 y Fo(g)1399 4161 y Fd(0)1463 4149 y Fm(2)23 b Fk(R)p Fo(;)1975 4082 y Fi(\000)2013 4149 y Fo(R)2077 4115 y Fg(\000)p Fn(2)2166 4082 y Fi(\001)2204 4183 y Fd(0)2269 4149 y Fr(=)f(0)p Fo(:)1070 b Fr(\(3)p Fo(:)p Fr(5\))118 4346 y(Ev)n(en)22 b(if)i(suc)n(h)e(prop)r(erties)g (are)g(satis\014ed,)i(in)f(general)f(it)h(is)g(not)g(ob)n(vious)e(that) j Fo( )s Fr(\()p Fo(t)p Fr(\))f(can)g(b)r(e)g(written)g(as)g(a)f (linear)118 4453 y(com)n(bination)31 b(of)h(the)h(functions)f(\(3.1\).) 50 b(Ho)n(w)n(ev)n(er)30 b(there)h(is)h(a)g(case)f(in)h(whic)n(h)g (this)g(can)g(b)r(e)g(easily)g(c)n(hec)n(k)n(ed,)118 4559 y(and)23 b(it)h(is)f(exactly)f(the)i(case)e(of)h(an)g(external)f (\014eld)i(with)f(zero)f(a)n(v)n(erage)f(whic)n(h)i(w)n(as)f(left)i (unsolv)n(ed)e(in)h(literature)118 4665 y(\(as)31 b(it)g(represen)n(ts) e(a)h(degenerate,)h(hence)g(more)f(complicated,)h(situation)f(in)h(the) h(direct)e(approac)n(h)f(of)i([10]\),)118 4772 y(so)c(w)n(e)g(think)h (that)g(it)g(can)f(b)r(e)h(of)g(in)n(terest)f(to)h(discuss)f(it)h (explicitly)-7 b(.)189 4878 y(If)38 b Fo(f)323 4890 y Fd(0)404 4878 y Fr(=)h(0)e(then)h(the)g(solution)f Fo(g)s Fr(\()p Fo(t)p Fr(\))g(of)h(the)g(generalized)e(Riccati)h(equation)g (\(1.5\))g(is)h(a)f(quasi-p)r(erio)r(dic)118 4984 y(function)e(dep)r (ending)h(on)e(time)h(through)f(the)i(quan)n(tit)n(y)e Fl(!)s Fo(t)g Fr(\(with)i(the)f(notations)f(in)h([1])g(and)f([7])h(one) f(has)1848 5184 y(11)p eop %%Page: 12 12 12 11 bop 118 201 a Fo(d)161 171 y Fg(0)208 201 y Fr(=)22 b Fo(d)28 b Fr(and)f Fo(!)p 527 214 55 4 v 26 w Fr(=)22 b Fl(!)s Fr(\),)28 b(so)e(that,)i(if)f(the)h(conditions)f(\(3.5\))g (are)f(satis\014ed,)h(it)h(is)f(straigh)n(tforw)n(ard)d(to)j(c)n(hec)n (k)f(that)118 307 y Fo( )s Fr(\()p Fo(t)p Fr(\))f(can)f(b)r(e)g (written)h(as)e(a)h(linear)f(com)n(bination)h(of)g(the)h(functions)f (\(3.1\))g(with)h Fo(\025)2715 319 y Fn(1)2775 307 y Fr(=)e Fm(\000)p Fo(g)2968 319 y Fd(0)3033 307 y Fr(and)h Fo(\025)3239 319 y Fn(2)3300 307 y Fr(=)f Fo(g)3428 319 y Fd(0)3469 307 y Fr(:)35 b(this)118 414 y(pro)n(v)n(es)c(that)i(the)h (sp)r(ectrum)f(of)g(the)g(quasi-energy)e(op)r(erator)g(is)i(pure)g(p)r (oin)n(t,)i(and)d(it)i(pro)n(vides)d(an)i(explicit)118 520 y(expansion)27 b(for)g(the)h(eigen)n(v)-5 b(alues.)118 768 y Fc(Ac)m(kno)m(wledgmen)m(ts.)37 b Fr(I'm)27 b(indebted)i(with)f (Gio)n(v)-5 b(anni)27 b(Galla)n(v)n(otti)f(for)h(useful)h(discussions.) 1628 1016 y Fs(References)118 1193 y Fr([1])78 b(J.)27 b(C.)f(A.)i(Barata,)d Fp(On)j(formal)i(quasi-p)l(erio)l(dic)i (solutions)d(of)h(the)f(Schr\177)-42 b(odinger)30 b(e)l(quation)f(for)h (a)f(two-level)284 1300 y(system)f(with)h(a)g(Hamiltonian)g(dep)l (ending)h(quasi-p)l(erio)l(dic)l(al)t(ly)h(on)e(time)p Fr(,)d(Rev.)36 b(Math.)h(Ph)n(ys.)e Fc(12)26 b Fr(\(2000\),)284 1406 y(no.)37 b(1,)27 b(25{64.)118 1576 y([2])78 b(J.)29 b(C.)g(A.)h(Barata,)e(W.F.)h(W)-7 b(reszinski,)29 b Fp(Str)l(ong-c)l (oupling)i(the)l(ory)h(of)g(two-level)g(atoms)f(in)h(p)l(erio)l(dic)h (\014elds)p Fr(,)284 1682 y(Ph)n(ys.)j(Rev.)h(Lett.)g Fc(84)27 b Fr(\(2000\),)g(no.)36 b(10,)27 b(2112{2115.)118 1856 y([3])78 b(J.)20 b(Bellissard,)g Fp(Stability)j(and)g(chaotic)i(b) l(ehavior)f(of)g(quantum)d(r)l(otators)p Fr(,)h(Sto)r(c)n(hastic)d(pro) r(cesses)g(in)h(classical)284 1963 y(and)36 b(quan)n(tum)f(systems)h (\(Ascona,)h(1985\),)f(Eds.)61 b(S.)36 b(Alb)r(ev)n(erio,)h(G.)f (Casati)f(and)h(D.)g(Merlini,)i(24{38,)284 2069 y(Lecture)27 b(Notes)h(in)g(Ph)n(ysics)e(262,)g(Springer,)h(Berlin,)g(1986.)118 2239 y([4])78 b(P)-7 b(.M.)25 b(Blekher,)g(H.R.)h(Jauslin,)f(J.L.)f (Leb)r(o)n(witz,)i Fp(Flo)l(quet)h(sp)l(e)l(ctrum)g(for)h(two-level)h (systems)e(in)g(quasip)l(eri-)284 2345 y(o)l(dic)k(time-dep)l(endent)f (\014elds)p Fr(,)e(J.)g(Statist.)37 b(Ph)n(ys.)f Fc(68)27 b Fr(\(1992\),)f(no.)37 b(1{2,)26 b(271{310.)118 2519 y([5])78 b(O.)33 b(Costin,)i(R.D.)f(Costin,)h(J.L.)e(Leb)r(o)n(witz,)i (A.)f(Rokhlenk)n(o,)g Fp(Evolution)i(of)g(a)f(mo)l(del)h(quantum)e (system)284 2626 y(under)k(time)g(p)l(erio)l(dic)j(for)l(cing:)57 b(c)l(onditions)39 b(for)g(c)l(omplete)g(ionization)p Fr(,)i(Comm.)64 b(Math.)h(Ph)n(ys.)e Fc(221)284 2732 y Fr(\(2001\),)27 b(no.)36 b(1,)27 b(1{26.)118 2906 y([6])78 b(P)-7 b(.)24 b(Duclos,)h(O.)g(Lev,)g(P)-7 b(.)24 b(Sto)n(vicek,)g(M.)h (Vittot,)h Fp(We)l(akly)i(r)l(e)l(gular)f(Flo)l(quet)g(Hamiltonians)h (with)f(pur)l(e)g(p)l(oint)284 3012 y(sp)l(e)l(ctrum)p Fr(,)g(Rev.)37 b(Math.)g(Ph)n(ys.)f Fc(14)p Fr(,)27 b(no.)37 b(6)27 b(\(2002\))f(531{568.)118 3187 y([7])78 b(G.)28 b(Gen)n(tile,)g Fp(Quasi-p)l(erio)l(dic)k(solutions)d(for)i(two-level)g (systems)p Fr(,)c(Preprin)n(t,)g(2002.)118 3357 y([8])78 b(H.R.)32 b(Jauslin,)f(J.L.)g(Leb)r(o)n(witz,)h Fp(Sp)l(e)l(ctr)l(al)h (and)g(stability)h(asp)l(e)l(cts)g(of)f(quantum)f(chaos)p Fr(,)i(Chaos)c Fc(1)h Fr(\(1991\),)284 3463 y(no.)37 b(1,)27 b(114{121.)118 3633 y([9])78 b(H.)28 b(M.)g(Nussenzv)n(eig,)f Fp(Intr)l(o)l(duction)i(to)g(Quantum)g(Optics)p Fr(,)f(Gordon)e(and)i (Breac)n(h,)e(New)i(Y)-7 b(ork,)26 b(1973.)118 3802 y([10])36 b(W.F.)d(W)-7 b(reszinski,)33 b(S.)f(Casmeridis,)g Fp(Mo)l(dels)j(of)g (two-level)g(atoms)f(in)g(quasip)l(erio)l(dic)i(external)e(\014elds)p Fr(,)g(J.)284 3909 y(Statist.)j(Ph)n(ys.)f Fc(90)27 b Fr(\(1998\),)g(no.)36 b(3-4,)27 b(1061{1068.)1848 5184 y(12)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0304300929886--