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%%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 80 /P put dup 84 /T put dup 97 /a put dup 101 /e put dup 102 /f put dup 104 /h put dup 108 /l put dup 109 /m put dup 111 /o put dup 114 /r put readonly def /FontBBox{14 -250 1077 750}readonly def /UniqueXX 5000772 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 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50fed3ed6c43c700a6bdc67bc619f3e54ccff4b09aab91dbd9854944c08fc229 23313c7ea13e8a4aaf89a78ae6b6ad9cb77f03411f75c495111a97e38d60fa4a 3d4219c7b7cc9e5c2cb3a4fd8dc38c8271654884a5e7e36cb43bea92f67d135d b7fea968c64c61c1613710381d16e0d3004b60c42fb4984e4ece5cfe35de8c66 72cf84a071cbdaacd0bb8bd26c0635152252bf79dab20677d28f472c1f059e15 3177c41fffcbb34dd9ce4d7d63bad292a88b7558a0d065645818e6b111c8d4be 48a0811d94461390ba81eec83d2311327800494e90624dda63ff156516474365 da1a0ea29bc746ed8a05a005f5b878fce905d697fe54642a3e206aa0d20c1de5 51b531f9c5c05bdafe870b215697911a5df34593956856b3f70f26ae85575d16 dbcc5206a95716b3c8294f1f4696a541a948312bb7b6ed8664aa1781824f65ff fe01befe17326b47559b39a3813f3e81308c71b341b592003ee861dc94c50dbb f3bc1324a8b5c49241d81328bd3f289317c6bd215b36286b524efd9987aaa874 b7fea14dba31029a4189eb34c47080e618aa013b8fef93d5a5b6f6d666d33982 90cf83adbe873f5650dcc66a8403a13aed40023f9378b578b19f4bed3db06c99 c7d7236f97f3be 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 21 /lambda put dup 59 /comma put dup 66 /B put dup 68 /D put dup 70 /F put dup 73 /I put dup 77 /M put dup 97 /a put dup 100 /d put dup 105 /i put dup 106 /j put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 115 /s put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueXX 5087380 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311e252234d51741e18db3a68e8ad10242 29e5817a10e796a78d2c7f7c1f50961b9a57aa604c9f821dbf5a9295197bc666 31678d7d2c7e1f8f2151ce0c29efce46270570f4301c5dad1b38884732e53dad 05dfc36b020e726ca575f71f04abf2b49e1f1d6d2a08a477658ca9a48f1c8cce 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9b1e5f37a7a2350f29f179953fa1c122ccc23c8344ede2c5bb9a9a2591312753 efd300790e4c1b16b6b35bb60b54ef504ffbca4a26eb28091282c9b9a6c8f9e7 40c6f07a98ba543f864568e28cd92cfb20e00731d2bea9a6ed3447abef91bd38 bb85e213f3f154714d644928f982b668366adb9808c1cf4fe7bf1eb4c3f16bce cad8c20d191b1745ff2199132feccec4ca33430881572354cd8532f30dc7b821 1b7510a72116234e0da3575c70b2d01b5bf9d78ba19523493db47f97b7342993 2b6aae505989ea4d281aae75e64380435dbf241da975d1929ef306e83987c26a 23757b3c38119cb367b595cbfb2e52f9cdca86db5fd48196e38835c643ebe4c6 f7db3e6f9f8574a961f279cbb1872fd1459e5ae84a33667cc9fd2310f1c4b478 6d6ccaa89f847b703675566b927eaeed737909c0939ae69f676bd929cac10da1 97455ab7dc4ee3445e4d37a5a81ffcc798a464dc5c3940a0810e033c12209648 776b0d68de9969e2242389fdad3553b2866e7f24755de4840c821999db4520ae 9d3cc19f860b16510104967ccce8c6250971ae53ac7ca3b4ff3d6e4c3d9a4e15 aca15bb186a760b4fa6a0c4f80f226e121e399a454e767d93bba430f26108dcd 2b0858872ab8eb415c86935b47cd10d1ed39b0248873d2066af8dd76119bfcfc 7d2cfbdb556e9345250d454c7a83add13d3b296700adb67134e4e76501504b3a bf771e3f4f8f04b88091ed8ba63c7ec09902e359efca3b957303ec70f6313043 dc87bbef21b394b206ef74d2b31cb0f246fc94636877b5956dbf113549c39352 cd35d93dad0e9abfa212b7b0130d4e688345adda3d439d0b073245e91ab960ac ae6e46150c3524eaf46dc7500a6c2adc17813a8e5b1891f52bf8f5a90b133e9d fb42ceaed1d77ac56c9c433b0bef60f5abf1144adb70c9a45adf519bc1cd69df 8ca4cde5 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 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 readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueXX 5000788 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306457c7e62c1d269f306bd3402e266de 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76753f36f27c7995de6e03ac105ef2ddcf27c27e824c618d3e4f493a1a7ff128 fef9fb27b8c1522ff0608d38c3a77f4d71ad98976c0ab8a54659b4457a557304 92624f309276a19291dc5108a001db4b8f49b96392d7b3ba23f361c275de7ef0 a3005c44d70fc43489669a07b157fc8d7509e9bb55790a50bf6a0db24f076915 c09493b790c269d68b5775e5c9952b83d7056d3e13f82c85349affe99e357360 f39dedb0411c05b6a502d828f57d450c19672ca1fb5a1278356ef824e3210bc5 ce6a1433e71a1b9eca9f9e6785e5a9a104416b8ae12a78d836016b2db4fcac81 fbd198aa028686f94fa0445be58568f90ce8a35b0e8db690dde38ca7f253edca c47e3a28dff61c2c2276ad967bf223b57f917df0be4ef1279a987e3e564c7a6d a8343f69528d3d72dc5e5287f0e586607ce93cfa3993befb35db31bf8a8f32ba 8f6b779f71396f46e8f72ad761a887919c44109cffd64e1eee2f67e091d5b373 2cf6fdb23d00ef671739c46ced68f8cfffdf8c5b11eae4ce442f9f00f950caf6 604d7edb391f515a28e19f062543187c58d41f97c0caa3183726fd8aca382fe5 ead372cc422eae2b5d17929ae29d5dee92f412ae08255d0cdbe8c38c8e31bf52 0521c35de3da42f732adb7a7cfd2aba95e08beb4ee037d4179226780c457d512 588f7f5ddb4ddd07fb5f92f5f17c904fc87aa814c297743e689ea066c1dcc8cd 6faf51a94f75cddfda43418c3a366b555a48fa6e20caead68d48e8669b373e64 a3b57e1ddb39a8c5bd625e6265f14b168341bc970fab49936957b84596fa8ac0 ae0f3c642e6df7ba963cf8ef2791f50c9f6c77d537d48d0f758a8544ed9b15bc bf8538806ebecb839f6a40c1bb641f0bf346d87ef51355361bccfacbeea73223 35f24b2e47d12cebe3abdb1ba73c096399940e9990ef6c92158c204eec4c3ccf e7d04a77758c718693540ec447e821b163c5f035b37e724bc772e6e7f44b41de 0a4e523c9cde4031e4d779046b1987cf3dfb8ac75bbe8b574bd54f9d9ee784a2 4e791f4074f91113ddc72220ab29a9ef5488cc1a5d40eddea8d78bcf1308757b d049138331cc6ab1fd25f00ab5e508a1f975066044 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 82 /R put dup 90 /Z 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 075b3023be6386f3805aee05b47c7f370e8ad4101105f8595189305804679524 74901f675385d8b5de52af884bb4ea2f488c1806e0251a1257a2d4bc94024d03 2222a6c9214825745af37df86a77585d9e94af57ae0a20570a6d07bb413ea870 25766a86c7c31bca8fdb121c7c72d0e46b7ecb5b7e091d2a8bf4b308674d0a76 d7a11670c508e02405b50534e855c386667f6b10fbae6932a73f9dd50c1989da eee25f89a9e1579aeeece5d6183da722bf14b34f575e32ad16691a79baa8c603 6f0fcd9f400c9d428f22248ca9a305b7b72c7d79f6627e02d47dd7d74465f0af f3727a07daa0b6b4fe8648242e645f9964bd93beb6b682387c66541bf7041913 9d2d1e26eee14609d0c3676ad297eb0d50dc644420c1402b312fe6f28952ee9a 19781c7ebaaabed22e65327fa109bf598fd9978618b18614ee6b28d152b716c9 a6b0ed0fe7cdaa5dd925786243c62fdd70d3063dd98bf405fd55a8c1a0fd4627 102ea561971440e7761239551fee30481b9099126420d68d6a67b11519fd6ca8 81ee0fb1124cb44c5e8b622fedcb476f816a0d7fa9da1d39df61ed23e54d9b9d 376856ad92d36a01d5873e30cf616bf0c63de9e3fdc58e270b36f2fac904063e 1482b11776ef4a70365611e2af36a93251ec77b200ec8a9a97c63e5039c1919c c94c528592665bbf2abd65ecdf35b766d3d98a3869f1f5d04731e2246b178f65 ad4febba8c81b728d9386c41e2bb45ab93d78a7d7861499da9bbe07268279644 74100877949a7bea8670035a81471717d0d85538809eee9f56928e9d0c19d53f ce2dc5085759ca5bcfe94f0fcd0a41560c1693d021cb144c65215a3b5ea8e21f 717904ab0036bc8181d763f0cdeb47a2df78a1641ffa23a4367cec2910d5d505 73d95704c45e21f964a708579e0eb1b1a5 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX8 %!PS-AdobeFont-1.1: CMBX8 1.0 %%CreationDate: 1991 Aug 20 16:36:07 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 82 /R put dup 90 /Z put readonly def /FontBBox{-59 -250 1235 750}readonly def /UniqueXX 5000766 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be1f91da738298311c1360c8158d61cb73427 3c21bce59b0f2e1d38c2fbc0984ce6a04f5d3bc84e2601397499a981b200b307 771141805f1f5cd7b12befe020f7121ec103dc2baf17f0aca1d0740c92a8dd61 12fa9c0d30cdae4f2af757b9f713c127d6ca02ef315205b81eff629104b21462 17d2b0f823133bccc80c9e400d6840db5fdf828f845313290fedee040d34f309 8f1b5a6b8941c81d1e607dcb310b14169cb433329994b4d1ac1b491e7422d518 2a8faf7ed3cbc2e23c58a2e310b56da54f51360556d27d58783694c51aaf62a3 a2db91ac8df80303c35cfd503341005020a5cf501fc0b64ff4f497951df6cd50 f00fafb04725d23bdd36616c7c06cd9052305682f6b6e521882922377b4979a2 e6f031b91b2a86d8631e81de17f90158ca05045c3b823da8b2d2aefd37a32728 45bace7594ceaafaecbf6140212789b3937a24cdd8dee327c934a9b0e328fd83 a1b535d9bbea6f0204ce5ee74cc30769734d4fa3dd0cac3986510c99ba062868 9285146cbe64caa89cc2f18a61d924e7f312fa189e0ae4deb913ed1cdfeae61b 4e0fe25f2775a8b2edabadebab4eb2ecf47258c1a77101c810cccd3b06633907 f765fc80bb1a2527f8a59c52760fffabbbac1d65a11708a3bea4557f24536773 d17dfd0eeee4e22f406e04765cf582d6ca4414f2441b6ff463c56018e44aa2fd c9af3f0480da99856aaa97e56b4bc28d9a74e6d74e5faf2016475d2991256254 6b9edd4874e8a3acaa1630376769be1149a2f5bab8c18a7d7d3a1bfe32aa1e53 05e5cdddc7ae6ea0b7945065d40d736808683107ea60c6a28e64b890097100f5 4da60fd7d785a9159e668cfcb1710f359ccdbfc25ecada02943aceeeae0ec38b 4bb1cbb26f4f5e99aec1e96415e1e2d58c726d576aa4a9cdf17420c396f4f87a c3e7e966f8d06c0bc29bcb3fc35578dbc1d6579e74da662c7d863adc0fe771d5 125eebc715501555d086cfbeece3c47c10f5e9579d9c82d7677f4e306e7b886e fd23c023f05d37241d19e17ba46e6de1f67bc66b2d92e40b6ed4f1fb913ace3f d8570fbf237041f366abd1b44ae91d8b867686128f3f573ff586074d4c75f8bd 12c4b018b9b0215947422ba457e07660cae277bf7296d9f510e368f0b38265f0 f3e442b611898983f6853325adf0000e6d9330429ffa4c5a1c7d22073fee221c d57118aaa566f54d2fc82091c7a60cd19a2c4267811a27da88c03a7585b250e0 b1d2ae2714f6feace3d8f582bdf864596bdd6dc32bc8db31fa896219fefaef3d 41adde66b5d01e41a20d293ed287df477962f4a861382b9c64f627223c6835e4 6fa75f10dce43f5b09beb8d2a1642ee77e55d7982df84dbc712675fc296c34e8 41b15ef756111f8c4fd2b3e583106569e0051c589f29cdbad2b8d76b5a581246 b495ed429b88d11bb9a55ca081927c984883d698a0fb4b7b14defa29d2b64427 982c8151a4ab6740b3ac3f9cffa5002ae5a266f72d9c1ab2e429b9ce17d9962d 5d5ec9911af2fa6f49fe4abab501bea9be9df228e9ff4b4267eaf6e8572a5103 e9fea76eb14f4643527691f7a02ba07f32ae812f24641789aa966747d876ad7d 0e6b5118d02eec2dcf38e79cbb70ece4f01332aa020bb1916f4b679a70d87615 809cc9408a8638dc1229f20f9369de2d02e7de8aa00f1c7f7d6b59f9362b1708 c82a705773b1dc75bc48d8161d807666fa5f09d01501674feb6d7516ea199f2e 24f1bc0c08f9d326325d88215708fee0e127c53bf9c79ffe39027cb6156c38fe 0f17514f99f2067687647afd0b299a8ed0fe0d43ba9ec1c4df0bd067b76d267a 57581228527c9ce680cd519e441bac3f532a7c63d72bbd441e402c4c7a92f54a 039952e3 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 2 /multiply put dup 20 /lessequal put dup 25 /approxequal put dup 49 /infinity put dup 50 /element put dup 56 /universal put dup 57 /existential put dup 69 /E put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 107 /bardbl put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueXX 5000818 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a 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cb8eef27e12940ceff701950eb0cdd71dded48bf2cda30a2c29917d06ac949a9 e3fe8d0bbbc43126e0c10fa628fc4c2cab72a08a0be6f50dd1f35645cb369705 8743a62b16b63ff68bc22ef7d68b9345b29967f83936974834fe15c810e04e23 5e97641c846b274acf1f3b70369c3dfcc648479dc68792338091bc0e3c63d608 1a8091030d0d80d79bf3a4a6824da0886f1ba24800f252b37dd492fd35d2b0fa 408e94717b2f7659aee3f4873f61d6b22dcee7ac51bed4ec26ca1c619c753f75 ea8e15a0f5b7f53cc6455a26aa10138fd11ceb62fe790e75faa82b35f2ae8e37 f6881c22004a259c5da06aa8b3af64e5178c759cea0dcf357fe5d1bab3e51ee5 8a9410cbf8636faa64908c1cab6ef4b1282abd86a7336c7e1a042485a10f7a24 c74c747b9e9ac074e626b772482c218556f8d54dbaa0842eb692905e24828dc0 8958a7c143b31eeb5ebdfdf5e8869d8ab55d60b615f629639f7756dd36202ea0 1e84d2fe4aa1266406162ab8dba3c533fb769d7ef887424c551083147a4c704c e0e8166e38154c0a59dfae8ab0b347213cf32efef1c2ce0d39138ed90034c1fd 5142ccade06b3e6ae07244e519f8d4778bdc735588bcb1206378b8226f1c208f d8922fecad38910fc62a13be569bd4ea4bf004234e6156ab53aabf5518db8722 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8a8fcf5435ea8c0033699f3c9c47137f3d677b41df58a01f2d4b5b59cf908073 dff377f21d9e64540f400f10ea034b8657c451b3711209b5ff09607d7dff68ee 70801f57248779f8c7e70da54e75fae45f8ab185e3e2fae73ee298affb0a06a1 c4592817012a98b2155d24b5877ad6298bf0f9e100e97a2113bcb4d6479ab5fd 4625ee4136ada388ea3068da770727ce45a512b14a3766c63cf5f9c25156b8a2 0fba362432c300426fb0518215138a211a5dcbb866f356f634738a0f3bb43ac5 13a89ba3b4d630b553b6c69311e0ac51fb6da82908ac83b55cf7a78d7bc7bfac 40cc858a1a0d4ad8f3c57b29aba0ff3ceb5a5eedd89d3ccf8057c30e233db1e4 dbdf874907aca70237379263d7e61daa9820e62a27e5cb82d13f5397cf576333 aec4ed786c47acf548bf860b049165a900bf9e50678c74101e9ecf8f0ee5c501 6b5706d1a1ef5f3acca23681963b654fb0bc8cc8350419faa7813279a7368b49 d2f6ab763a37ec98f4977461303e0f2f0a1b02e33c70b53b6067d6931a344863 a79264e8f79d7741d2d4a907154266dd3e0d655f00b4079b753c805f4fd1290b 867c5bbba158be9010f86cf54067c25560d36c5bf03cdcad071efa470d7c9d56 b09f5cf587dd9fd56981f954d819c757568e366b8793d9 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 48 /prime put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueXX 5000816 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6a18979f258b8c6ed3 ccafbc21aca420c9c83eea371adc20e038b4d7b8ac303004b0aa205f04135140 76407216032fdd22e6219da8f16b28ca12524deb7bca073cc5eba65c102a5e85 fd48e6d062cd4283ee570a7774597e5bf0e3400b6be72db0115f3cb12db70ce0 83722870cddfadee715f10f1fcaf20e06f3c54afe5ca238539bfe2b596116e83 f5371ff18fa5003d8543226cfd4025f9940365b392a858d27f078d3abcffe4a1 54e78c7692d1a32bf935967c64f01b24788ff8325d61145e2d4a489fd986fb77 38e6b254522c77ca2797a504a9ce4676a77ebacb026eca94dde5922c936f8e90 c43e2851905abd26842613d8fb644512275d43f312eea72f7ea3ba90c8fda6ce 7a9dacf52f55f0dd557f8d866ed6f3d3d5e6fe7e9ee79404fbe1a285bc519ecd 7ca162f0729044aff526bfe554ed59dd9beab0f842d55be08caada6136ac8e30 21c0b1f2869eae249565a126093d019e874f1d3e3de66d54ad31a98634d992d1 2087aa9caadb695d492b62edb22e89fddf96357f7dfe18336e102a892d42c1a8 924b450df9e62931117872ba4c4f60217f8dba3846b1df7f87862b42db5627bd 43eb0e86fd0a633d67f9c8d2e64df360c689be9b696536fd7d1886be07beb475 10befde02910376810d3894b6c3e16d0694dff896b2cd98f5e44f7f059776d8e 9bb6e3c852781ef8e6e0e0a9c8c78b899754aab4f9a8f9da4fd51bc78e39e7c9 a538ff680ef2c545bf9fcb4b038ed25337f8c02f345b1b7100eb6a13cad1d980 178c730a870f02d2c6c01f6606669510bec67c99986b891c142f8c2bf8df035e c436fa9b81563dcb484d18f85ae28341ea562614fad1bac1432d874695af6fb8 48171ebc83f63001e7c3a982d1fce641f1a622b9a8b7a28ad3d22df9c562ea5e 3ae864fa0a6413e60ef869d10970fd20c80b7568574276ef4c8c3d65cd96d424 100af3a3ad9c4b5f6c8dcd144af4a2da9f34a1b14c33f586a70f444a53fc565f 9082b4aa858c8bf637f8499fbd0dbc5576edd1ab85a4481adf4df8ffdb8ea103 9b99a919a11ed74d95ddcec7150896c48d0f80 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 59 /comma put dup 100 /d put dup 105 /i put dup 107 /k put dup 108 /l put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueXX 5087381 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a1e67ed7e41cc25acc04702f6 8ef703e3d45722c1a91fdef7100a48631a02a6f02a08c6b1f9b4df8310385b86 8632718fd87119a233f219d9411383b7fa9f3e4780d8c27e2e89e0cae883d664 c3eac57a3aef8988a2e9f0f8c7f53e0a80bdfc4620e21287d0390e1975398544 7f3ea66401024bea75e1b4c4437b7bb188f76f96b918ac7c6ad7e8ae7f21d8c2 790f08cccec904fe48ef39e597ed4d4237c1d1f596f5906b19ea308020f7a35c 168e327ec3246b1dfabe912f6b6daac09974876d3996e57d180261110db05f15 e3e8eebba3d90b5764c03df3033a1ed678ebc679569a2fb297378b25434c0f20 5313ecb8a952f07242d3ee731b0cdc086a4481178a3d65129c47c09b22e9c431 e11b3747b94c26a757c38d06001798c6a568303d541385244b967d3b1786edea f65bb53c4c2fe75e4b1b15c2c78d930b4296c80f08bad86012451edc8e9f0854 c3b390a16e27b11b3d45a9f72eff8baded2242dc928a61685d79e09681c97425 5b90a498614cf560fa5b1718981388268ba206a96989e6d0b5d485d9aca5594a e67dd7b34d8a369adb06647f8aff8814d6d9cdc04a4835918e557174c5bc0f3f bcea9907a04cf93c12727ec40db3f2f77596dca477862747435bdedacd9b2311 6cc97fa47ffdd7d897fb6bdd5572e35d34e7e1cb5e7273a4ffd86525323ace4a 84e1297028c2bd5469baa2e75d19360c2c9042139d5e7dd4390a6a3935424711 de21910126d750ae279916ceb71da3591d60dc62db333c5021e2c1cd61ade51e 939205511ebfbac6b51a838f6e00d92a916021775f36c8730f20719142d48a8c 84e2d642cb86bb8828da3027536fd57b836084ceaa10a61941804e1de594f697 c34847da8de42b5534f06329cf43094d05a9ecbc5ba0a9e48cdf3660ae720976 5d56c31dfa759f774fea0cc6a76a1bbd12fcc81ca80e6db071d2c189cb63f87f 82ffa957a29350aef4457875bb7e8c1bed25f7f8823e14c2a089e4b654a747ca df52317413c57fbd33fccd94bb6e01eccc0c5a6fd06e0a46eecf20545ddcc1ae ff5a3ba28c40d959298ef3f938318cf9f156054343dba02ec950e6fd3f213ce2 b212d1cc022106effb10eef5f947ca58156fa7c0307260310ee9f2fd593b6f4f b3e2e423a90e7981bc79cb8493284502ca21e85375d61be2842995bb39b5b31d 4149a22f84b15fb3d9a170def318cbc2decdaca17f6d1ca77990182728657f2a 89d327f3a5a3d73f2cc464b0f2e4256e87968ecc6911dc76c87ef8d0b219e7fb 35e30271ec745ca9033752552324903f43687c17ad675d2015dc1e40474b6e37 f6d649dbff24268076f7a91d5798d4b90dcdf1a56677414d91f3e3709c42650c 0c43af18afe65fae8c46ee80a34d766d11d56f525be730ecf81cf836b1a152cb 59f9f21b74f1937637ce7e08a7bd9f6c42daba900abd8f82b376a7dd2f4f1e61 5fa8d30f0205d760da9c216d026dd14f42b57391576f19b5e642340f181fbca6 4d360184ad6c81903c57b959b00cd412dee40ab3e85b5542f454b1a0cdd5059f 7321a70bab719f6be2b4b91fa7f51783ecbe82bdb361709eaafac8a045519bee e4f5232a0341ec1c1b5a5e7806b37410471fea50d061950123ba1c6eb757ee4b 2c3c45ceb1918109c80b7d8d6184bf94b4cb30cc6221875592803f5c7672bab6 f3da07d666725d94157ac81e52cd3fbfaeaf1c93af82d146d0a8b1725a28ca94 8e88ff57592f57cf86432969324b402d0d320d2f6289feb63c92af0543204982 8311fcf247873b136c45f71f819326e737a346de2a8fb8439df748dc33bcd366 66575d8579a209dbc08386e9071a4cebd427c90f06dfadab60d62cabca5a9a2e a6dbbaa0bf3b4e83d62ceb64b7e84fec3f9667f2dd9178e5578d2b30250553f5 c4904dc76d546a61862d66d09eb2ad2de16213bc5316b2ca9f38d5b928d4a76a 1b8851054341afe5836e797837aaf2411c8bc7d46e576059470c952a93177673 fa45089ab79960e29d1ced2da9f28d5a66e7c635ae28f9acb65d64bd88451b1d dbf985b3d5a846dd633ccda83676f4bdabf220706254f7e0bf2126c9f25479cf efc87c78984451114831264eba5d4e68f54cce03cacaacef32ba60f9d793ee38 1647bae379e6b567331b05834091e3e9d130a76d7f96265e8dbe1b77f660be8e 5f11f09ad52d733b8a190dd46fc1df6cae3380d7f79ffcd49fa2c72b6f2cd57e 45a736983d9b71fd7b92163681413f120de160c06ea36fc30109fc1e9db1eab1 46c10820dcaff4e3a4103bc0c6235566d58b912b0c2c7d1e5b961e4d3f462483 99ab7f58e9d24267550cdfdfd464edf48d989edcbb71bd841102b82f72c7f964 d00b0a044105843b5301ad874019611bf3c5fea1da3acfd8f6d44ebad72d4dce cc683132d91c0e9e45f01246e95046c879d6b42eaa20e0da1fa93098599b787e b5c4463e694a7211f9a3a969e7cc4c9c97be75dbf0bb3653a448843c65a8482e 452e1ac0bae5efedab062d90c17ff953631f20e65ac4dc1885ec0785d724bab3 e9d85f01860a4f0bb8ad592680cf22f69305ca30e7f88091f0d9e945278abc9f b569a85d6008551708db7d00920a3957dc518e12e19f9197cff5583a4b0fca93 cc40c0bb2c08e55c00184958a4b3a9429b6d43885d400dd6ea5b9186ad17fc40 49cdcf533201a5f8590d52330389eaddb7c50229e4cabb30b0a5cc9acb92fc6d 8dd2c5db3ec137e15ecd57a544c4ec344590ec42b0f646a37dae9cb2db1aca8b dee87c86aec4836cf31b4b8f4a9dbbfc27a9754129c71755c83be2f8a854d284 f764a52fd121568f9033e22ae5558db05c3547d00e0154d282f95ebbf3738db6 1625e2d855d9a62d6a06e5f7def0b91968f74caf830d1e00a490d86951f0f372 3aea7a47c77aead51907073ddf60cc8378a77c76769a8ef4399944d10b4c2e95 ec7b4fc3b54b164f9f60b44a8f2befea7bdaf315bd83a61395dcefacd0204c7b 288a8f62308015cd03a42c25d6dbc8234df603bc3e64628cc8cb0bcb0ff0e1b9 7a57b42a63d0439a666265ee189c720f4f75f8ea0be0f4e4a1992d9cb0b31c4c 5a8e2fcefe9ea79de41b072d7aa2db38a172151d70548edad74e19c648fc924c 628f7b95cb874e9c49744bb30916b01110681127acae1ad62d68334bfcb13e95 be86800cf1847862ec1f40bcc58f5ec2ca171383b6cc16a61dd49cb2066e20b6 2bfccc5a87538f2e17eccfda20e8ecba00ac65a867a63e189297a615894b9a06 b47bb089fcf11e09b0c849bf318b97de9d42ccd57e8c4981cc413e783d13db0a d7ce49096e63ed109a08f12b7c3d41f8024343dbf913e546c6f3e835bf3d42dd bc97bbbbfce443537566af1a437105320285e7e971cf35a0de11d8fd7bd15b1b 9852097fef558a4936097734812702dcddda436bb7186254a2742ff54a30b1c3 837d029efa03b4a49a01a7a275831e85deb8e3a62082c2d7976888b2acca6931 dd2b1c97f234c1f30bfc07d5d9a13381a9ee83f6a66b6be61aaa6826ea79ba25 00b1e9861d60d98bf16407a108c2c4e1d9387b71bd067807f983614e5fd9e8fc ab9afecd01892cac88fb096b7f6d71a5b655 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 18 /theta put dup 22 /mu put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 62 /greater put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 73 /I put dup 78 /N put dup 80 /P put dup 82 /R put dup 88 /X put dup 99 /c put dup 100 /d put dup 102 /f put dup 103 /g put dup 105 /i put dup 107 /k put dup 108 /l put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 118 /v put dup 120 /x put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueXX 5087383 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a196cf14c95a4e02f9cd2da8c db2546c6df52e524745992e18d9ff87aa25e4e1800bbe4ebb357c6ef55ed6d03 6d3a00c1ee8073266c21d2f0ac85d656abf61d7e5a4fa87da8ec3b5329e434d0 d2adab706b42a2e5331be5295399d803ccac03f631f01f39a022fcdf63486b68 7d15ef284a77def7fde4898543e7b5f7ec267756103e477f547cfb8d2311c4b0 09deff56085f5d419697af1846c8b88c1bbbae149f0f19ca3c8dafe19cec48fe 6b38357246d8b5cef80b53bab4cbbbe8bc8318cc462a4158258f456e697f6058 86010493f252073ee4e4c9531f23485415ad6bc901bbaea659ef701f825f4b6d 5c9bed0dce7b63810f906b33ea4955531ca1ed1fc120cc56dc1e277a33222cc6 c3a97552607b865783faf0895cbeccd0df54f57bd162a88494768435eda71a39 69718c0c5912d3167bb0971546d733e735aa9bcd89f5aa154656031ab3f6fe41 94ec225ad03ab8b67a9da43ab365c4df17c23e79f685c3b6fd5b654e4a6f2603 6cec3a8641b0b57f93eb95f9e238832aeac2061f3153e0f375513c974f1c38f7 b59edcc94716f2862cf71c9a2be8430985da1cd813b1e1c1b9310564a2e9bdb8 c536db7caeb9c37a0693a01df0045080f8236ece54dcfa7b743562771f452398 100dd95b678b6f190cdef25df0e2b10423aef1d27dfd5b022a0f895423aebb50 34ea52f46b077f2022e28ecfa6137fdd4b13935ac19769600b2e0dc86b0a2359 4675106e26eea6b5033525c1d79e046ee941c9ad410ad3284092f9fb725d366e e88a7a4ceaa6caf2ff6a2c80749780481db49bf09d5b7dd77cb83ed6c74fe223 0076eb41e5e84c5f714b9cd7ae6b36396bb68c992ac812a85b87db809bcac7b8 acc373360dc771999d1cca5ea40f294db858063f000703aa1c64258fd16cfe8f c21c69a04519e8518fa932f6ce002c837116f0a2417fdeb71815c2bc13e17b29 a39379a6fb6d5f71621ef81c85fcc950c7cf8c92951e48a5c04f07528765b27e 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 3 /asteriskmath put dup 6 /plusminus put dup 20 /lessequal put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 63 /perpendicular put dup 76 /L put dup 106 /bar put dup 112 /radical 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 ffd94dea1743cb25be008581ad33816fa911ee4c456a63a46b720e66466b2054 570f54d3be508c4a8025d14b2f914d3ad23a9fae97660deb2e9d594ee7b9929b 864441b4b4c6ad4010439ae5f38d9aa4126678eb57c440eee2ad3f0fb0119361 ac73d692af5f4a57928f1a165dbb1e55ff9ffefc9e6ec9a14bea624fbb4ed2a8 82f58b9d46ba5ce28aad836434550d5b87db7ae70afef46b8ca2a26c9876b99c 1f502333c0c3fcef438d0218fa30aeff370f7d5d9ee8efc569de7e184ed8e6f5 454866bd03dd62e45308f557814d2557b8cfe14be9106a49ff0fb20d624aa093 3b8d2d850b0c3e580d27d70fd7bb946c8e30101998ded5e89d43a59b43d96496 1623555657bce3a0035c7bfa071a5aa62743581607d78ca2b082c43338c6a9f3 dcd24ef07dd46ee79246310ebe336ac6544ef12232c680631639e58bb0a7ec5d d389fd8949898415f195e526bf83d09945c468cfd42b1e448608d9c4a533061e 3bc2c2d9593cfbb9b435bb5d4037dc3cd66dbf2942253e35a4b9752518e79a94 2d9b4f4d8b74dee467fa2a97e632392286ed42965ebc265a77f0897a8b441866 644248f83a547c54e0815260c856bb0e39e0cdbc6654a5d1b442170dfcb92807 f737114a2b1bc768783bf2c53323275d41ae8f7b4f86fd14064e119c6281f7a7 6034c5fd0f6328ba101d483a0d48547f4f5d8765edd0bacf7c81d8bd95c81411 04473420f51c029973f2c5576f5ca68475c7ce7debe9f380f6052e328d2c695d 7a1f0d5336f42777fcdb6b1c69a8fe8ce9135fce9e1a66e944c73607bd6a1fe5 4ec439d2c4a1702c6d9c63eb1f49618a270c95ecbdd6e07c0772f15cde086477 8368b92674b1239adf925ab8118c3959e7d7ec23757518e9da772b5b1d5e2b09 d1f596f2fd94e2837103b3da59e22649e95fcbf19902d1d29886d1b5f61c03fb f803a02d3881ab4a25693ed4254aa14f448a3556bfbf880b762e9bda43f74113 a165ff6971bc6cf92cc87881e0d782d3bfdf65e9214044a8dbd4722bf5a97d37 3850c5fa8428bc8d3f27a4f4b4de109c553b3cf06a093e856675c68e70744851 90c3bf56b437009c21ab761a60562aaf156a6cb3a0b4d503491145316a3de58b a9269c92b216ad73fa72b10fdf0b1a4fe4fd97d5d7b87d0dad66692c8927c816 072d17c688990882fcb13d6ad9a3ae7251831b1e303bb26780d17728cd7f85f3 73b12b281be5f4a37f00c0457d28c8ca56ac15e0015c833f764904e3a5593b35 767f55bc9964878693ac8d68e34dd04ab395aba2f2ba716e2878306d6be0d1be e78d07953a7e3527de87e7b6ab8e58d392a5c7b4cb3b004c5277f2c33673a23a d8384fe08bf60099c25779ee679764e59db9537a085aba17aa604141ad97be42 fa1b49871ed0d7ed4cc862d76c05a337cc52d0cfb674372026f7cb142dec1208 6e3c74f0fccd50c2ca129e5ae097cd070e15de7d7ebb46b2186e2a8f9068fa29 5ac2ce55be1a6a3d0623480e6e30b93ba4c31bd53c9ce6abae0b7073ad7855c8 470ab7e70fa408edd1cf2e39fe64d85a8df2d4edcb7770cbb1d0b38863d31793 b3ef2e252aba63449669787fccc7c5f441e38e374da23c96bdb1554f00267b1b c568632224079acad4629cf6c3100d7a8056200d11cc81e7caa2a6b3c2195ab5 138c226fe174101c3c7d493255a5e7c6f3e327b3388279b6fe8289f99a71aa3a 7bea440963aa53658ea08940e319f816dbf424e0868d0fabfb14ea199a90b007 674549e6b06709ae13e76c6f5df86c45471beca1289cc7aa4bd4fb33d80584c1 d951b0d89a5f848f7a142f82da37227cd8e6a409a8e05f7e8a35a0dd496935db 62c7102b47f7c45ddabddc1cf24ef261209fffa1b689fb86b0b1247459b84a27 c8709ed99ca78c54d5188ebfbb266065effb47dd090c50b20fb509e4422b47a8 f41627407a996c53345df775ab148bfb64abfe35f8053cedc406ed3b66ff549c 106469f6a13583c2a6dd7f0df4b60502087a879dcf6bdf21ebb54a0f8d02f3c0 b80922b5604b6940ccfe3c720e882a6f76 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /vextendsingle put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 26 /braceleftbigg put dup 80 /summationtext put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 101 /tildewide put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 112 /radicalbig put dup 113 /radicalBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueXX 5000774 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d5649664203ab24acb938d58d246707ffe7 d62f04bec4b70c21ef75beb2b812622b3c74e969d72d3cd11bd7106294a99caf 0b1629bc7d4de6b96ca82930831d64575f23f4ad06a0e45e315b1d392411be8d 6d73c998789ff258a07a3c8c2057325784514c845500bfd1a971310cfc11d41c 1a167dbd5ff012c60add4e87325f6e5299032a839de65fb1473a166aae1876a4 414a434f22c1d241591fb36f857df6fa930608750ffc0c54f44994662b1f00f1 400bf752ea8d83ffc4cb77a290bc2d99981ae59a191748ba5c7ba1a9d2583fd2 1398452b6ff5d83a059f7eadcd2ef744e9dd22bdf9c79d049bf06835e878c32b 7765c69bdd8ef4deb4ea7cfff4cf9354a4ddffa689de961d16772491c7afbd7f ffde42400764c68e954ee5c455a5687959829bc3b319b2147deaab3628662c80 30c5e02fea09609abe4eaa12e217bc3af673f1bc36a7039eb13fcacb4218fe0f c5a3f9452d4edf46cc91db67b624d4f2d37502fb9f11af4da18ca40b61097f95 d44329375467ed13c5cb585ec53f62b83ef9502cc755af44bf32b87b8ae9f3f2 f8dbf72dab90acafbacd280db6aaffaefdff6d5eff26669bac56280a950560e3 d01714ba38659b0e4e28bf8cdc1f15ff7e3ea9d95a6afbbe3d8e6905aa6d531d 6b6dcab89e2036cc492c65f76ddcc3cfcff33fde3d9ffd10f7a90530fd5c7683 4329baa902e701ca42717476298c9ff511b1941244bb1dfd0e5cab92a43970cd 1bdb292eadd055c97972844fdcdb0aa06401e7b2632b9d134f96bce0064860f3 9859da7a9a82be0a840672a1331212752ae1b521b693b384a72dba8d37a86eaf a0341e7fc5f6ef7a8c566aacb91e8bf5005c18837894c4eacff41dba54e071ca 83fd01c81afd0f8ed01e8e7e1f83d1ca84a843297301ef120aba68e7b1f12ea7 e8103f84171005db4bbb2f2ccb2e814012be5cf2bf9b0852b4e9f8eff570ed00 75fe77a0607d8d50c4c7536b9a3834275f31173090f656c6801cd63c6b97d9d9 4c8180368bb78fde608e169c36c2d320d66ff0fcf4eeda8c41660cbb34ca7b5a 2baeb1473fec2027a7483a79bc080d0b1eac765d4dfbe55206f52aa7daa3ec2b ba6fb9c3cd2d82575f2b42ded5b2da17ccaa97d58ced89a0b1fc495d41860892 341fcb2089c6c912a1423545047ed8fa7e0c664d581447e67aa08b6bd857760c b4a1a1172e9100fb79f1c52b383b73c64620cb740fb98748e406c5171ace7ebd 832532187478970fe99a870da4bdee61ed47b67dfd75e602dc01c11a690e8a3c bb3dcb159c5e767c33f7b11918032db7f22bb793a48b3fe47c856d5b07a8258b e56cdf4110f80b45701af0189110610cf4a790f29a55e57353c6e654b4557367 1bb40a4e1e71cdac04a7142fe8b764e5e5355c8289f3c81dd5c32da2ac4a39c2 04b6df2a596786ecc06fae9b8d833c2cd0369dd07a5d41ab1b1e964973ded340 04cb22a9ba33557a2f1fa08156fec6365982d0972098679c11a726592f97c111 65616238232eba67e3a030b36b22259c1c069996fd4884b912ff945fec0f21a1 a239a20c239d0b2765db9ba167278cf5d4957f52f4e852271e6440c87a803cd5 10590cabe202fd54865da7435959f861eb8a7b1b7beae1abaf05085cc4b3ff2a 33e4b4a5eaff438df4924028c8c7803b19d14fc96e6544060aeacec8475ccb9e dd6f4f71364c359de311ea2e668d2e58ceb874a69363a7986e23bdfb8e420b38 43f64a612c11d00b83ee431450ff97766beff11078c0b627b5784551c2329d5d 975017690d71f9f37ae17451861e2e93fee4636a2ef92a30158923065ae170e3 5b336570ebb636e64b88c9ee84a226dc637d7b8bb934210c58cfccbba34193a7 d7f0ee4d4b281b66a53cfd992fc55598a71c960093a15badfc18baf74d7d7ceb 5e07ca088e0abe3f8da5a3201237d591b63052abcaf319efbed640bad8c14814 204c383f1cd9b309ce875330589a2bec4a4c20fdad2716af5fb5259667f9c0f4 3411b063b5133f7d088d871372ff3c059de128e745e4a995b195b3a92f2739f6 468db2aea0a33bba974fac119a74dc6cc44561dc8d752e556bdbb2b9635f340e 26e765e4766cb3e0d1c78f99e678e118de382cb4d7975dffb35bb3789d9f6cef 14cd3de5f818d2ae0ec6993e6eb0d5e58ec16e2859ebc41bf3a9350276928507 21109e0a530299dfeba66eb6815c18a9f359d1727df99c3b260a2362b715b2cb 9596a636b8980be1962291351ed7128f5069dc89ed6d009c20b35ebe6d59e22c a29c5e9eca0565092298b698217c5995061370d4fad1c278f4177df7464648f6 dfe649ddc0adbd3f5bdd7713dcb1384cd42d003f7db9d89633cf328694ffbf7f cee240b75fd19d7d81bb357d99d01dced0ff4f34f808ad31ad223988ce2ba6c6 bb1b1a276b3e1db2e3b3385cf597f1bb29236256562f2f6868f7216c5be76acf 931881386e19a2918fcfc4c59ddb3faa6ad6a6f3e2fd0085339d0692f126ee5c 5f93309114673303b03676b4009b28b473921d71f2b7fd3a35cbe84fefff17ae aff2d028a558e75c87388906aa6d51a79c6951f94bed21ab6491971bf44cce2d 0fef915abd7158a08cd3848177eabbb111c94851b101f79bccd56c974f24b71c b228b1fbebe52b53173c09e6bdd89f4e2b43f701d407be768db4f687365723de 02084cdbe50c3c69ac904f1ae8f933f8db0a14839eed4cc0e5a984bab34e0cd6 c70b6382bff361aa5620e9b7cbef36e504c55641bba321a1a29aa7d4d9adc3b1 815eb8061f87a53f8fdf5a84c20bd63e9e2a8e74e2f36e9a0fb2b04b8f74491e 5f5907e8d7fe7d04c4a33c711573498967cbcd4e1338cdeae94a0895e9c54d3d 758f2c514581c564c0a09b4b6eae28b0008f7fcc27a6cff459f5411ef102bbed 1640fb077796f9a2182a1adbb72513e30acf34e4223a9e63311908b08544bd36 29b1cd60535355441cd1b7611db7c8a32db713a8d64fa7191c30ab2d12c97ed5 7939047270490587c5f95cc2fc2bba572ae205ca6b6bbf060f9ae7202165a2db ddbda6cb5be25f90aa9eef281eefb5a8f1133d1acec82aed02240e49d9a6f59f 98986bfcdd9ad3c483d25fbd78268625d5f6a19713d38f19acdb7cdcaa103f27 976d1d862a6fbb427ea91907c554d6435b782947fb4a4c2bda75f6ec5fcb69ed c7e660142c9513a19a7b5035d728ecfa9a51da4946679caa770f1894dc71b5dd c3c77dbeca5ee08d9fc1c7f47d62ea053c50e05c503643707c6bfa0d7d140f45 7d5dae4e9673814113f24fef9ccdd52604cab7e810bf99f4eccd8f89f4441006 7dc5eca815416397b2ed700134fc663e1018ce20a170fd8e146e7824c5a0b963 9a14639a23546e585c4f4748cc5329eba252936f4faaedafc41de29be6da5b16 b2ed9c9ad77a2aa7d7c8c1dde42a3214849ba2defa4ca510351e861a9500e94c 2ded9513e3e1067fa2794ff6912e811c6db6cf976987bfbd47dc2be991af32fa 413fab306bf1bc4ea72f81397a108aca9131a63f2bf617e685c7636544e6b368 b98aa785c3fb2f7752ea664db8f4ab6cbbebd0dae1f4f5d821567d8483c9aad4 103be96f05c4c816b4fe46be9007e31e39fb25f26a8b8f8755a1ceae1c313c06 5ca2c91114bf7d12afbd01aabea4631b8e4607558d4bb4cc865b139d70e13e51 eee3b56887311747eb638d678877fee6ae9687ab732e0793b546707e8dfdd014 73a6f7d334604eea23cd4616723e99853c1661d80d042a3c0a52f0bb06c6fde6 e59931999bd4d3d44befb87c7ed127d16a3763736124faa051f92340ed6d4c81 44444bbb18d49da1b8889da1fdf67297f20f49931edde87b8b546375398c363c 1827ce25a0a081813d719ae69f814a57626304baea84efaff5b8e5c760484ea5 58d46f35697f22fa8e6638b4e95ac88471b117c259220399ba195e70ca585d63 0996f30b9a75f69d8a253906ce699175e5825b4dc9c813c8e995f189568320fb 485ec70082b10af5098fda75ea81a61b1fb60fee5a4787bbaffcd9f2ffa2589c 2d40559682808b5c43cc85f7c6694932b0a008fb247e916ff21d63b469d624b6 6adf71d32df0f35e2c1ae2b88dd263f62dc5e17f42ea8118d800d39890adf7f6 514b14bc4c91150abd89ce37f416ea6bcd0797b447fe76ca86131e2d653961f4 34b15b7c3bb1b1c3fd07afd45b6a2e0b683d3362a9245f31687ff2b123e8f5ff 094c466377de9a7145976ff586ce40e61d8c0d37082e00466cfe0755dd689977 e7ab496f1001bcc426de7be0ee5136021c825c7f8fe2c5c8691ad8401b554398 ccd126bfcb76de2325bcff4ca71030c9e05a1a918f042207f328739a79c064b3 d8e582252d4b54a9cb3afbaf1cd3f60b08c4e3c07c47a8a33af4abea55f68f15 39dea004335331bc00cf906f693174f0f13d18b5ff210737aad4eb228a71e934 3cf66af12321a8e570527ad69250e9b6e57f896b5505d3d73a9c49bde879ddeb 667e198a52c8cf5028b7d40a4a778ca3a4c7a34e3a02731875a80fb1b4afc88b 10b27994184f296b99b171ef815cbb40504f330564d501ba3a7e1ed1e1 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 5 /Pi put dup 11 /ff put dup 12 /fi put dup 34 /quotedblright 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 55 /seven put dup 56 /eight put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 68 /D put dup 69 /E put dup 70 /F put dup 73 /I put dup 77 /M put dup 78 /N put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 92 /quotedblleft put dup 95 /dotaccent put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueXX 5000791 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b094541f345ec367ae85ca9bd 24568a01d3b9f8095f7420e6c423c414b3dcce6da48dd1c89a56d078e0d0e2f2 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /Lambda put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueXX 5000789 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 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25e939b73aca2948bfa26663d40e56eec455ad24d2ddcd5cf6a184a28fdd16dc a52d2aba4efb2ccb7e185d92f733abc404cbce1327822b77681a3da839b5ba45 757391e6a32c8d931a965ada5d87c8ccb24ef757fb3a6f47fafef2778d8043cf b0dced5ed1487ffbf186b1208ce6d4768f9ab590a4049fc62a2f8b1e6c4d7465 2ff22ee70f91655b9857714a68f87e0b147d4bf433faeb9ffb9b0174d9ac25fe f8006005e9015bd5474ab929cb7194e757cccba8db1050785bf79bbd26446c04 57a2a3261689e43a0ad2123cbbabf213af6ae34a21f92723c3775adbafda865b 2048a7c6119924926ec118dbd2aeb95d01b5878ba97592b64086e51191906bc8 d2cec6b10b7d435b4e3a1cde13c405d47356236b66c748a2f4e09664b666c672 4baffff97d8b0d6fb3ac7bf06f37595fee5337b6d9208811cae6d40343c0dc88 74cfcf37d634b250134e2afc5a3421cde65eccf712ef756717d659 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 6 /plusminus put dup 48 /prime put dup 49 /infinity put dup 50 /element put readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueXX 5000815 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6784decd2fb36dec8f 7478da853762e00bc9c52ba86ef50d222ff142abc18e4fb95b86b3e57d9390ca 86e8de6347bf7f48f93322a372c5d7830521b14e8fa0771961b027c81365426d d95d16b950e216a90f28ce806244ff28dd969cb512a23e05045e7b2bcaf7d6f0 ca8b51494f2ec0681bdfde60420628d9bce98a0ca1ecca9140f452a88db19a30 ed46ae881b098c843a51d06f9a153489357fd24de6a0c36cb5a9eb9b3b64dd75 18f83018ae7cefa1c12fd205df6880a705e28457af7b869a6dd28f6b37d82cc3 e51cba348dcdb34efb119b8ce5a996f15492b0f5d34538850a2c1b6a509fe5fa 015eceaf28b2bfc953397c590d3d4ef8b94f890461dea021548475edd68bffe1 5140c6ab20345e73534abc5986619b5685c499861c31e6aeb0f53a94d7a4ef66 c2607e038fc84884015fbdc9781dea7d7e4deffd4403977095576bb653af7c21 5a20e74c247ca8458ea463cf068ce92d08cf11b26e6eda102d0241f6835ad887 e4257aa540ae9b1d6c65b85802420c890c015a59cda5ba24313056de064284cd 1d80c95e18c2573d05cf0168dc70faa5d5a6969cb7fbf2f2d632352ac3fa1568 7e078816960aa184f4d1372836ee0792991838ce5ea55bffc5b822411feeffa4 0196e40534b4f1106b01e53b40232104e1d69e6ab34384c60e1e10b6e8ed1c72 548dcab1c1e69c2162eb16333b482ecfc5315d8f477054c52de89d0059650c90 c7d06aa79b0402b4ee3fce2d27fe12199c5aff1e49eeaffd6e33cfc576afdb73 a35ef14e825724d6d8ead48901cae77d7f7ff1bc917105f0481be6df420ea1af 906b896747ebd06b67d4da7fe67b0d3a52cfe43665afd31aa5380ca94feda79a f6b0dc41912dc78b00ec4b2f0558e48ecbec97e8aa88d98c4fd779b085f1737f 45a29962384e0667f7390d535cd3146e848a2ab676dd5631eaaef59ef8fd9d6b 0c94649b0e3aa59c863c7488983bd0c6b03bd7b89ca9ef14d057b0fd0a10e8eb ed39fc15e706cd1ed115ada29f5be6681deb76a98f7273733a829a7dbae981d0 6c9222682a6b79121907f28e57e509f6cc60f193695618143f51fdf06abb7c5b 901b513275c06fa8e4124286cabb5d1eb4da6d9ca47e12e928cbd22851664ba4 dd7a3f6b76156dc920bb134e31fd08561233b2468dc867cce7e939d4af1879ef a433f5f8532a6d65357263867600141ecdd7227eb1a805234ad08598a979e99b eab2cede8f7f7f677ab4500ed78a25570d7490ce64dc9253af7bac119898b37c 547c2ccf87de2b08217974fd46f96b7812131c0465e5dd11f808424e4aa1e262 7b270c2099da7988f905044a5e48b328c85f5021c860d456fb0f0f2c0b766af4 953e3c7c0a756b28e4b90ffd30071beb634e969d05ce6ee50d78c04c231e265f a13e6314c45a16b6b4607292b97feedba1537bf1b74554aac408affa63792887 7c4cc4b3a45a1eaeb51d186d043b0cf5349b1df2d9b93ab9c1dd55401a01ab70 41b2962890cc6742f6cb7555c8a439d289c2210c97c6f9dc31cf4f988cf25598 378cfc1546e52979e77a4adb7f70db680465e88d2632664ac66f922fc6259f29 cadd4cfa2479041724598e0c70e5fba325d04b41cf7fb9169d3288540ab0bf15 ce5cfed746e12cca52c6de3eac91afd625898849296d111d01fe493cae469815 8f44bd6419412279b8c107ced0f85376f9fa24bf0b8694f0317508a26195f72f a5b74c86813fa5222ede0efc 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 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 16 /dotlessi put dup 18 /grave put dup 19 /acute put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 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 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueXX 5000828 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3ffd20326947b1065649b533eb5e9e3 a88a87fe987918a687950b26f647d6e08bf14d983ed1d4a8ca1e2da8d985f944 2eb916f5b6ed0697ac7c33e1e36a3bf460d34ce45f1631871097cb04f18e3889 4cf4ac1538eb19481311d24fe3be7beaa4a3730e8b4831fe59d6d9ce2e46116b 629c7ba2f9ce3ecba2f43bc162a5a077ca1b2882a42afdcec3f4b75b5d63e0bc 8e5dc95257766d8ea467ad9cbaf47be60f797580cced6884b3a68f70c91f4fdf 80fd00ed9139e7f480dc3a76af72ad9b434187730bdfaefe4cbfe5c7edcaaf24 9204fd703011932e5a3c27be468b7dab69daa18dbbb6335ab8ddfdc607961c7c 02582763fa069d43563ba17704029945cf42fcb19cf78c51df0ec4d851086d43 c1ab38e865ef36865fff3a08b01cbcb070ba4893f7482dc7819ad03d337e520e 7d8cd83bb8ac7aff4df36751f4f12aaaea2b6d7260c09a26e987c904003e0723 7dbcc4a8e4e85cd3259f40d3caf55c742cbb40ad0db1044c20a4f122a63f7b81 1d945bfa69662189ce4d089464d2ead9f6ed60b581b04114d5d45e97be27a576 35154e9b8464f2a0df18da855eb51b8f64d3054b0b2e84621f025a0776d16fed b988a43d2d64c32768dd106b9f541ae8e68ac0bb7399e16f4d4ba074b2152d0a 4e9b5bfc21bfa0d531e28165e518ffc054dd4ed0d74eca079d20e79670419f92 e03bcbfeecf3554d661596f6f29e12170804d15a250d917ffe7a8b6e8fa53047 1ed97ba56481589a65ca13619129f1e26affe6c22f0c38540f8ce37efb6e9e9b 60b51a5a9cb4454fd7c7f92f2f9437159687ab590f35b946a54491a7739c9269 e0f1fb48b129811c82edb3171cbd5105fe65906974db8a7da92f4f43dc7f6b36 6981146d86d23613a0b38d9d6191db9fea0172e982cb1cc50e382f69da1b377c 1be9407f88a2a761b56f398467a057f35d18c01cc377e006182c891ec80272e2 2e3a5c7123cd27eccdc87392a9fdf63bb40f02b7649cfc53ef933c6518744cb8 9cbbd437830ef5b1d58b00b67bbcf4426dc1e6ef583b0b4ad3aebd741e3ff0f7 29ae271f032019b1e7268043fd215be590f9cf24da14a90f1683afcba8006cef 5d5632c2481fbc1bd8224e8952debb7c4596a237e5505ac143180956ca1f603e ec70cdb4a33e1c4ca174436a38b06fd9d2b9196f02114b77a9a5f66595727faa ba323668573ae328f1f84e5a3e0e7634d559479c134207aad7a949da589970c8 da77d71a20a8ffe72beb0643b81044daf53e79809e17a14cce790765c281b4c5 cd8c35a9f87a34fe5981c199ffae7fe41503e38d5f437d0134d3a0a7bbda0350 4bbe59c8585052ac7c18ce6b421449d73b16961298c14548b796d7a2198080f3 45be1b247c5a2e454e47886dbaa56f166a4b875a9cc7b735bd0686a3570da5a5 ae911fcd2159fef835f73b4721f47d4734828e4c7e30997976d6c457941ec11a e59c8de8fdcd954c78ea7937031d7fc78b2773ff600cfe51766b2d67175e6d43 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 6 /plusminus put dup 7 /minusplus put dup 8 /circleplus put dup 14 /openbullet put dup 15 /bullet put dup 18 /reflexsubset put dup 19 /reflexsuperset put dup 20 /lessequal put dup 21 /greaterequal put dup 25 /approxequal put dup 26 /propersubset put dup 33 /arrowright put dup 41 /arrowdblright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 59 /emptyset 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 83 /S put dup 84 /T put dup 86 /V put dup 87 /W put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put dup 112 /radical put dup 114 /nabla put dup 120 /section 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 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4c44f8cd6089ee89a9f4cf35e020defad22e4208c3774f412d1eac005a3188e5 0fe43f9a5f8c68a0003400cdb457085cffbf0797c55c95f8dfa7454bec3a2089 3237ec2c1432e15d9a33d3735c232b361b8bbd97be8311e65daa20d9226feab2 f44d4619d3b6f85deeb6b2e03e19cb8a1b7f581ad31ce496f4ddeca99c1a0568 0ea5fce34436cb7ce2dfab9a22fe8df262bf1d0273699ae73a55d6cf58a43c08 6060dd 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 45 /hyphen 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 58 /colon put dup 65 /A put dup 68 /D put dup 69 /E put dup 73 /I put dup 82 /R put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def 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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 3 /Lambda put dup 5 /Pi put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put dup 99 /c put dup 100 /d 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 115 /s put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueXX 5000790 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 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14e873f4bac0ca279fe539cdd00f5af9277a4bd69e0a7b2d9ca6ae91d2006b89 f3b7e001daab09ee0587ab02bbd541468674dc71c09a1549d1c2542a8d9a893c b695d5d70a7dc1b8f85764bf1a1531c571d517f337592c479ed63d8cf4d5e0b4 2f1085c855adaef893605d6348b143753d83b9452c8fd7c50c90d9c6d0be4cdc 5922a89d0426b37352184d3c56919d23b15efcf1a18140549f41fc9ea3deabc4 8b57cf10d0f1e89bf91d44381757ef9f671cb1ceaed4bf60affaa5325e78d284 5e9319591c80463f4605ff1e87f4f383424e74acbbad91ad7a0caebb04f70e89 265d8a45786784e45fd5a8950f0ffa6882fcd79bff76ec64f1377ef62b22ea17 ff6ff1b4f618cd9426fb2608edc382f80c6c2e77f8162d1f6d661a9ed7bd3b9b 42e93cf71c9173a998b47b510e7d65362cca157d68b23fa9f2bbf79917bfc184 640ac335f9f2a9e03b94f2ceb32f5d9e46594d69d89adcbe399e71e12370c8ec ec4e7e46750458b63f13e765697585ba1554628ff3c1c2f9ab22c917ae8ad0ae 406ab09404e1edb16f48116b6bd8039aef65c71f88b27ad72e330d52b1b544fb f9dd16f95b86f7c100c14c688e754fd70d94687bce79e7aa6d5a01be27813ada fc07d54a361f46d2207ae56bf1fec018a1f7cd769d8c08b38346d1d12a566af8 1a78d2d1c375282e916e5c3d8630aab8e9264011d5e6dfa0f1f05507162b2ac4 8795123e95b5351f9eecb15a19fa9c11c8af525a7b3aa428faabd7e45469b91d 419dd4d52b730e796ab5423610468d 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 18 /theta put dup 21 /lambda put dup 22 /mu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 68 /D put dup 69 /E put dup 70 /F 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 87 /W put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put dup 122 /z put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueXX 5087382 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930f01bdaae649990604a33aa9eaffbe5b5489 e5c7c9ff9d9be01b08220832c41caac64816b53bbc087ae4b621d2b44b21ae5e 3f7fe4ddb05c675abfe30510eee4a7c874bb57b2ffe521a6875eddfdfd18c781 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939ad51df69d3166214f501c13dda53042d098b84e14bf 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 12 /beta put dup 13 /gamma put dup 14 /delta put dup 17 /eta put dup 18 /theta put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L 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be771b8b49b7e0250b5e6464ea5e0bb667ecc4231e1281aa07f7422c33664d6d d294156227417ee7ae486e1344a217b28d43a7dcabc482d42c248bee7e341610 7b0ab1d8cae8 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 0 /Gamma put dup 1 /Delta put dup 3 /Lambda put dup 5 /Pi put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 19 /acute put dup 22 /macron put dup 24 /cedilla put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk 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 63 /question put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueXX 5000793 def 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 58 /colon put dup 65 /A put dup 67 /C put dup 68 /D put dup 76 /L put dup 78 /N put dup 82 /R put dup 83 /S put dup 84 /T put dup 90 /Z 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 121 /y put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueXX 5000768 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306462c563d51ece66fcc9c831843ecae 1fefc1a232e2724f7baee428ae03aadb95c3035345c15e9922fe49e1f2cfc980 237316572dbc57064edac9b0db8913a5e2d45e97e19a91435ccf8adfc835b585 48e74b291446d689c7f2f8c4325e8356e974ae30c3e2977477baeaa33d141fb7 80f59351e84bfc88c87b3a4a1d25e168b9d134554f6a581378c7d2d6eca8ac09 045cc3a5e0ed86f147133f094029e3483bebc81cfde69942cf645ea20d0a2b64 5140ccc2f9cb4a3d969f4b0a79cd3329b440dfcff5285494406f3dd08d804c86 df191e57232f6f1f83df6430f214e03fa5412b1e13d83c60c52a698bc9a38159 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%%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 44 /comma put dup 66 /B put dup 76 /L put dup 77 /M put dup 80 /P put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueXX 5000794 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 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58.1154 /CMBX7 rf /Ff 165[50 7[61 14[59 67[{}3 66.4176 /CMBX8 rf /Fg 148[35 20 2[35 35 32[37 11[39 39 5[47 71 23[55 4[55 17[55 1[55{}13 66.4176 /CMSY8 rf /Fh 207[18 44[32 2[48{}3 49.8132 /CMSY6 rf /Fi 141[29 28 32 1[38 1[20 33 1[22 4[32 40[19 59[{}9 49.8132 /CMMI6 rf /Fj 133[33 1[40 1[34 2[33 32 32 36 1[43 1[21 37 1[24 1[34 34 1[36 31 10[58 5[53 1[45 1[56 4[31 3[52 58 50 53 3[55 1[55 20 20 18[46 16[42 3[33 18[{}31 66.4176 /CMMI8 rf /Fk 143[55 5[20 29[46 12[52 8[0 3[45 66 19 27[52 13[52 2[34 1[20 52{}13 58.1154 /CMSY7 rf /Fl 142[83 83 55 55 4[39 39 2[46 9[92 46 1[120 5[39 1[88 53[62 8[50 50 3[28 12[{}16 83.022 /CMEX10 rf /Fm 133[31 37 37 51 37 39 27 28 28 37 39 35 39 59 20 1[22 20 39 35 22 31 39 31 39 35 1[20 2[35 4[72 53 53 51 39 52 3[53 65 3[25 2[46 48 54 1[50 53 3[55 1[20 20 1[35 35 5[35 35 1[20 24 20 55 1[27 27 5[35 21[39 41 5[53 5[{}58 66.4176 /CMR8 rf /Fn 198[30 30 30 30 30 30 30 30 30 30 4[47 39[41 3[{}12 49.8132 /CMR6 rf /Fo 205[39 57 18 41[45 2[31 2[45{}6 41.511 /CMSY5 rf /Fp 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34 39 30 1[40 39 2[32 3[30 35 38 43 11[{}61 58.1154 /CMMI7 rf /Fu 133[39 41 47 59 40 48 30 39 37 37 42 40 50 73 25 43 34 29 48 40 41 39 43 36 36 44 6[57 2[78 48 57 49 51 63 66 53 63 67 81 57 71 46 36 69 65 53 61 69 59 63 62 44 1[65 42 65 23 23 18[54 4[39 52 54 52 49 1[36 47 43 47 36 41 50 48 48 1[39 41 2[37 43 47 53 11[{}77 83.022 /CMMI10 rf /Fv 128[42 3[42 37 44 44 60 44 46 32 33 33 44 46 42 46 69 23 44 25 23 46 42 25 37 46 37 46 42 1[23 42 23 42 23 51 62 1[85 62 62 60 46 61 1[57 65 62 76 52 65 43 30 62 65 54 57 63 60 59 62 1[39 1[65 1[23 23 42 42 42 42 42 42 42 42 42 42 1[23 28 23 65 42 32 32 23 4[42 9[37 1[42 2[42 4[69 46 46 48 60 65 60 2[62 1[58 1[69 52{}94 83.022 /CMR10 rf /Fw 134[50 3[53 37 38 39 50 53 48 53 80 27 50 1[27 53 48 29 44 53 42 53 46 6[58 5[66 53 72 3[75 1[57 7[73 69 1[72 6[27 1[48 48 48 48 48 48 48 48 2[27 33[53 12[{}41 83.022 /CMBX10 rf /Fx 138[54 38 38 38 1[54 49 54 81 27 2[27 54 2[43 54 43 1[49 16[66 2[89 61 9[69 21[27 44[{}20 99.6264 /CMR12 rf /Fy 136[117 85 90 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b(\014rst)g(sho)n(w)n(ed)f(up)h(this)g(instabilit)n(y)g (phenomenon)g(for)g(a)f(p)r(eculiar)h(Hamiltonian)g(in)g(the)g(famous) 236 3373 y(pap)r(er)27 b([2].)361 3473 y(As)41 b(suggested)f(b)n(y)g (normal)g(form)h(theory)f(near)g(simple)h(resonances,)i(the)e (Hamiltonian)g(mo)r(dels)g(whic)n(h)g(are)236 3572 y(usually)d(studied) h(ha)n(v)n(e)f(the)h(form)f Fu(H)7 b Fv(\()p Fu(I)g(;)14 b(';)g(p;)g(q)s Fv(\))42 b(=)f(\()p Fu(I)2050 3542 y Fs(2)2043 3593 y(1)2087 3572 y Fu(=)p Fv(2\))25 b(+)h Fu(!)i Fq(\001)e Fu(I)2484 3584 y Fs(2)2548 3572 y Fv(+)f(\()p Fu(p)2712 3542 y Fs(2)2749 3572 y Fu(=)p Fv(2\))g(+)h Fu(")p Fv(\(cos)13 b Fu(q)29 b Fq(\000)c Fv(1\))h(+)f Fu("\026f)9 b Fv(\()p Fu(I)e(;)14 b(';)g(p;)g(q)s Fv(\))236 3672 y(where)34 b Fu(")h Fv(and)f Fu(\026)h Fv(are)e(small)i (parameters,)f Fu(n)h Fv(:=)f Fu(n)1898 3684 y Fs(1)1958 3672 y Fv(+)23 b Fu(n)2096 3684 y Fs(2)2133 3672 y Fv(,)37 b(\()p Fu(I)2261 3684 y Fs(1)2299 3672 y Fu(;)14 b(I)2372 3684 y Fs(2)2409 3672 y Fu(;)g(p)p 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770 y(The)c(\014rst)f(pap)r(er)h(pro)n (ving)e(Arnold)i(di\013usion)g(in)g(presence)g(of)g(p)r(erturbations)f (not)h(preserving)e(the)i(unp)r(erturb)r(ed)236 870 y(tori)g(has)g(b)r (een)h([12)o(].)37 b(Extending)26 b(Arnold's)g(analysis,)f(it)i(is)f (pro)n(v)n(ed)f(in)i([12)o(])f(that,)h(if)g(the)g(p)r(erturbation)f(is) h(a)f(trigono-)236 969 y(metric)j(p)r(olynomial)f(in)h(the)g(angles)e Fu(')p Fv(,)i(then,)h(in)f(some)f(regions)f(of)h(the)h(phase)f(space,)h (the)g(\\densit)n(y")e(of)i(p)r(erturb)r(ed)236 1069 y(in)n(v)-5 b(arian)n(t)27 b(tori)g(is)g(high)h(enough)f(to)g(allo)n(w) g(the)h(construction)e(of)i(a)f(transition)g(c)n(hain.)361 1169 y(Regarding)h(the)i(shado)n(wing)e(problem,)i(geometrical)e(metho) r(d,)i(see)g(e.g.)42 b([12)o(],)31 b([15)o(],)f([13)o(],)g([14],)g(and) f(v)-5 b(ariational)236 1268 y(ones,)29 b(see)g(e.g.)40 b([9],)29 b(ha)n(v)n(e)f(b)r(een)h(applied,)h(in)f(the)g(last)g(y)n (ears,)e(in)j(order)d(to)i(pro)n(v)n(e)e(the)j(existence)e(of)h 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b(metho)r(d)h(is)g(not)236 2013 y(based)h(on)g(the)h(existence)f(of)g(a)g(transition)g(c)n(hain)f(of)i (tori:)47 b(w)n(e)33 b(a)n(v)n(oid)f(the)i(KAM)f(construction)g(of)g (the)h(p)r(erturb)r(ed)236 2113 y(h)n(yp)r(erb)r(olic)18 b(tori,)i(pro)n(ving)d(directly)h(the)h(existence)f(of)h(a)f(drifting)h (orbit)f(as)g(a)g(lo)r(cal)g(minim)n(um)h(of)f(an)h(action)f (functional.)236 2212 y(A)n(t)31 b(the)f(same)f(time)i(our)e(v)-5 b(ariational)28 b(approac)n(h)g(ac)n(hiev)n(es)h(the)h(optimal)g (di\013usion)g(time.)45 b(W)-7 b(e)30 b(also)f(pro)n(v)n(e)f(that)j (our)236 2312 y(di\013usion)21 b(time)f(estimate)g(is)g(the)h(optimal)f (one)g(as)f(a)h(consequence)f(of)h(a)g(general)f(stabilit)n(y)h (result,)h(pro)n(v)n(ed)e(via)g(classical)236 2411 y(p)r(erturbation)29 b(theory)-7 b(.)43 b(As)30 b(in)g([12)o(])g(w)n(e)f(deal)h(with)g(a)f (p)r(erturbation)g(whic)n(h)h(is)g(a)f(trigonometric)f(p)r(olynomial)h (in)h(the)236 2511 y(angles)c(and)g(our)f(di\013usion)i(orbits)f(will)g (not)h(connect)f(an)n(y)g(t)n(w)n(o)f(arbitrary)g(frequencies)g(of)i (the)g(action)e(space,)h(ev)n(en)g(if)236 2611 y(w)n(e)j(manage)e(to)i (connect)f(more)g(frequencies)g(than)h(in)g([12)o(],)g(pro)n(ving)e (the)i(drift)g(also)f(in)h(some)f(regions)f(of)h(the)h(phase)236 2710 y(space)20 b(where)h(transition)f(c)n(hains)g(migh)n(t)h(not)g (exist.)35 b(Clearly)19 b(if)j(the)f(p)r(erturbation)g(is)f(c)n(hosen)g (as)h(in)g(Arnold's)f(example)236 2810 y(w)n(e)28 b(can)h(drift)g(in)f (all)h(the)g(phase)f(space)f(with)i(no)g(restriction.)38 b(The)29 b(results)f(pro)n(v)n(ed)f(here)h(ha)n(v)n(e)g(b)r(een)g (announced)h(in)236 2910 y([3].)361 3009 y(In)h(this)h(pap)r(er)f(w)n (e)g(will)g(assume,)h(as)e(in)i(Arnold's)f(pap)r(er,)h(the)f(parameter) f Fu(\026)i Fv(to)f(b)r(e)h(small)f(enough)f(in)i(order)e(to)236 3109 y(v)-5 b(alidate)25 b(the)h(so)e(called)h(P)n(oincar)n(\023)-39 b(e-Melnik)n(o)n(v)20 b(appro)n(ximation,)k(when)i(the)f(\014rst)g (order)f(expansion)g(term)h(in)h Fu(\026)f Fv(for)f(the)236 3208 y(splitting,)34 b(the)f(so)e(called)h(P)n(oincar)n(\023)-39 b(e-Melnik)n(o)n(v)28 b(function,)34 b(is)e(the)h(dominan)n(t)f(one.)51 b(F)-7 b(or)31 b(this)i(reason,)f(through)g(this)236 3308 y(pap)r(er)f(w)n(e)g(will)g(\014x)g(the)h(\\Ly)n(apuno)n(v)d(exp)r (onen)n(t")h(of)i(the)f(p)r(endulum)h Fu(")d Fv(:=)g(1,)j(considering)d (the)j(so)e(called)h(\\a-priori)236 3408 y(unstable")h(case.)52 b(Actually)32 b(our)g(v)-5 b(ariational)31 b(shado)n(wing)h(tec)n (hnique)g(is)h(not)g(restricted)f(to)g(the)h(a-priori)e(unstable)236 3507 y(case,)23 b(but)g(w)n(ould)f(allo)n(w,)g(in)h(the)f(same)g (spirit)g(of)h([4)o(],)h([5)o(])f(and)f([6)o(],)i(once)e(a)g (\\splitting)g(condition")g(is)g(somew)n(a)n(y)e(pro)n(v)n(ed,)236 3607 y(to)28 b(get)f(di\013usion)h(orbits)f(with)h(the)g(b)r(est)g (di\013usion)g(time)g(\(in)g(terms)f(of)h(some)f(measure)g(of)g(the)h (splitting\).)361 3754 y(W)-7 b(e)28 b(will)g(consider)e(nearly)h(in)n (tegrable)f Fp(non-iso)l(chr)l(onous)i Fv(Hamiltonian)g(systems)f (de\014ned)h(b)n(y)1303 3985 y Fq(H)1373 3997 y Ft(\026)1441 3985 y Fv(=)1539 3929 y Fu(I)1582 3899 y Fs(2)p 1539 3966 81 4 v 1558 4042 a Fv(2)1647 3985 y(+)1740 3929 y Fu(p)1782 3899 y Fs(2)p 1740 3966 80 4 v 1759 4042 a Fv(2)1848 3985 y(+)18 b(\(cos)13 b Fu(q)21 b Fq(\000)d Fv(1\))h(+)f Fu(\026f)9 b Fv(\()p Fu(I)e(;)14 b(';)g(p;)g(q)s(;)g(t)p Fv(\))p Fu(;)896 b Fv(\(1.1\))236 4204 y(where)33 b(\()p Fu(';)14 b(q)s(;)g(t)p Fv(\))33 b Fq(2)g Fw(T)931 4174 y Ft(d)992 4204 y Fq(\002)22 b Fw(T)1145 4174 y Fs(1)1204 4204 y Fq(\002)g Fw(T)1357 4174 y Fs(1)1428 4204 y Fv(are)32 b(the)i(angle)e(v)-5 b(ariables,)33 b(\()p Fu(I)7 b(;)14 b(p)p Fv(\))33 b Fq(2)g Fw(R)2696 4174 y Ft(d)2756 4204 y Fq(\002)22 b Fw(R)2915 4174 y Fs(1)2985 4204 y Fv(are)32 b(the)i(action)e(v)-5 b(ariables)32 b(and)236 4303 y Fu(\026)23 b Fq(\025)g Fv(0)k(is)h(a)f(small)g(real)g(parameter.)35 b(The)28 b(Hamiltonian)f(system)g(asso)r(ciated)g(with)h Fq(H)3005 4315 y Ft(\026)3077 4303 y 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(will)h(b)r(e)h(a)e(c)n(hain)236 5013 y(of)h(suc)n(h)f(connecting)g (orbits.)361 5136 y(W)-7 b(e)39 b(\014rst)g(in)n(tro)r(duce)f(a)h(few)g (de\014nitions)g(and)g(notations.)71 b(F)-7 b(or)38 b Fu(\025)k Fv(:=)g(\()p Fu(\022)2791 5106 y Fs(+)2847 5136 y Fu(;)14 b(\022)2925 5106 y Fk(\000)2981 5136 y Fu(;)g(')3072 5106 y Fs(+)3127 5136 y Fu(;)g(')3218 5106 y Fk(\000)3275 5136 y Fv(\))42 b Fq(2)g Fw(R)3518 5106 y Fs(2)3581 5136 y Fq(\002)26 b Fw(R)3744 5106 y Fs(2)p Ft(d)3854 5136 y Fv(with)p 236 5189 1512 4 v 329 5257 a Fn(3)363 5281 y Fm(W)-6 b(e)27 b(will)e(dev)n(elop)j(all)d(the)j (computations)g(for)d Fj(f)7 b Fm(.)39 b(All)26 b(the)h(next)g(argumen) n(ts)g(remain)f(unc)n(hanged)j(if)c(the)i(p)r(erturbation)h(is)d Fj(f)g Fm(+)17 b Fj(\026)3963 5271 y Fl(e)3954 5281 y Fj(f)8 b Fm(,)236 5360 y(see)24 b(the)h(pro)r(of)e(of)h(Theorem)g(1.1.) 2105 6059 y Fv(5)p eop %%Page: 6 6 6 5 bop 236 311 a Fu(\022)277 281 y Fs(+)361 311 y Fu(<)31 b(\022)498 281 y Fk(\000)585 311 y Fv(w)n(e)g(de\014ne)g Fu(T)1003 323 y Ft(\025)1075 311 y Fv(:=)d Fu(\022)1232 281 y Fk(\000)1309 311 y Fq(\000)21 b Fu(\022)1436 281 y Fs(+)1522 311 y Fv(and)31 b(the)g(\\mean)f(frequency")g Fu(!)2572 323 y Ft(\025)2644 311 y Fq(2)f Fw(R)2800 281 y Ft(d)2870 311 y Fv(as)h Fu(!)3027 323 y Ft(\025)3099 311 y Fv(:=)3225 255 y Fu(')3279 225 y Fk(\000)3354 255 y Fq(\000)18 b Fu(')3491 225 y Fs(+)p 3225 292 322 4 v 3238 368 a Fu(\022)3279 344 y Fk(\000)3354 368 y Fq(\000)g Fu(\022)3478 344 y Fs(+)3557 311 y Fu(:)31 b Fv(The)g(\\small)236 454 y(denominator")26 b(of)i(a)f(frequency)g Fu(!)f Fq(2)d Fw(R)1530 424 y Ft(d)1596 454 y Fv(is)28 b(de\014ned)g(b)n(y)1420 637 y Fu(\014)t Fv(\()p Fu(!)s Fv(\))c(:=)e Fu(\014)1771 649 y Ft(N)1834 637 y Fv(\()p Fu(!)s Fv(\))i(:=)138 b(min)2087 694 y Fs(0)p Ft(<)p Fk(j)p Fs(\()p Ft(n;l)p Fs(\))p Fk(j\024)p Ft(N)2470 637 y Fq(j)p Fu(n)19 b Fq(\001)f Fu(!)j Fv(+)d Fu(l)r Fq(j)p Fu(:)1013 b Fv(\(2.3\))236 870 y Fu(\014)t Fv(\()p Fu(!)s Fv(\))38 b(measures)e(ho)n(w)g(close)h(the)g(frequency)g Fu(!)j Fv(lies)d(to)g(the)g(resonan)n(t)f(w)n(eb)h Fq(D)2834 882 y Ft(N)2934 870 y Fv(de\014ned)g(in)h(\(1.3\).)65 b(W)-7 b(e)37 b(use)g(the)236 970 y(abbreviation)28 b Fu(\014)767 982 y Ft(\025)840 970 y Fv(for)h Fu(\014)t Fv(\()p Fu(!)1104 982 y Ft(\025)1148 970 y Fv(\).)42 b(W)-7 b(e)30 b(shall)f(alw)n(a)n(ys)f(assume)g(through)h(this)g(pap)r (er)g(that)h Fu(!)i Fv(sta)n(ys)c(in)i(a)f(\014xed)g(b)r(ounded)236 1070 y(set)f(con)n(taining)e(the)i(curv)n(e)f Fu(\015)5 b Fv(.)361 1169 y(F)-7 b(or)26 b Fu(T)38 b Fv(large)25 b(enough,)h(there)h(exists)f(a)g(unique)h Fu(T)12 b Fv(-p)r(erio)r(dic) 26 b(solution)g Fu(Q)2668 1181 y Ft(T)2747 1169 y Fv(of)g(the)h(p)r (endulum)h(equation,)f(of)f(small)236 1269 y(p)r(ositiv)n(e)h(energy)g (with)h Fu(Q)1064 1281 y Ft(T)1116 1269 y Fv(\(0\))23 b(=)g Fq(\000)p Fu(\031)s Fv(,)k Fu(Q)1564 1281 y Ft(T)1616 1269 y Fv(\()p Fu(T)12 b Fv(\))23 b(=)g Fu(\031)s Fv(.)37 b(Moreo)n(v)n(er)25 b Fu(Q)2393 1281 y Ft(T)2472 1269 y Fv(satis\014es)i Fq(8)p Fu(t)c Fq(2)28 b Fv([0)p Fu(;)14 b(T)9 b(=)p Fv(2\))17 b Fq([)i Fv(\()p Fu(T)9 b(=)p Fv(2)p Fu(;)14 b(T)e Fv(],)934 1452 y Fq(j)p Fu(@)1001 1464 y Ft(T)1054 1452 y Fu(Q)1120 1464 y Ft(T)1172 1452 y Fv(\()p Fu(t)p Fv(\))p Fq(j)23 b(\024)g Fu(K)1471 1464 y Fs(1)1508 1452 y Fu(e)1547 1417 y Fk(\000)p Ft(K)1655 1425 y Fd(2)1686 1417 y Fs(\()p Ft(T)9 b Fk(\000)p Ft(t)p Fs(\))1951 1452 y Fu(;)97 b Fq(j)p Fu(@)2138 1464 y Ft(T)2190 1452 y Fv(\()p Fu(Q)2288 1464 y Ft(T)2340 1452 y Fv(\()p Fu(T)30 b Fq(\000)18 b(\001)p Fv(\)\)\()p Fu(t)p Fv(\))p Fq(j)24 b(\024)f Fu(K)2921 1464 y Fs(1)2958 1452 y Fu(e)2997 1417 y Fk(\000)p Ft(K)3105 1425 y Fd(2)3136 1417 y Fs(\()p Ft(T)9 b Fk(\000)p Ft(t)p Fs(\))236 1634 y Fv(and)479 1817 y Fq(j)p Fu(Q)568 1829 y Ft(T)620 1817 y Fv(\()p Fu(t)p Fv(\))19 b Fq(\000)f Fu(q)853 1829 y Fk(1)923 1817 y Fv(\()p Fu(t)p Fv(\))p Fq(j)h Fv(+)f Fq(j)1194 1796 y Fv(_)1165 1817 y Fu(Q)1231 1829 y Ft(T)1283 1817 y Fv(\()p Fu(t)p Fv(\))h Fq(\000)34 b Fv(_)-39 b Fu(q)1516 1829 y Fk(1)1587 1817 y Fv(\()p Fu(t)p Fv(\))p Fq(j)23 b(\024)g Fu(K)1886 1829 y Fs(1)1923 1817 y Fu(e)1962 1783 y Fk(\000)p Ft(K)2070 1791 y Fd(2)2102 1783 y Ft(T)2182 1817 y Fu(;)41 b Fq(j)2297 1796 y Fv(_)2269 1817 y Fu(Q)2335 1829 y Ft(T)2387 1817 y Fv(\()p Fu(t)p Fv(\))p Fq(j)24 b(\024)e Fu(K)2686 1829 y Fs(1)2737 1817 y Fv(max)p Fq(f)p Fu(e)2973 1783 y Fk(\000)p Ft(K)3081 1791 y Fd(2)3112 1783 y Ft(t)3141 1817 y Fu(;)14 b(e)3217 1783 y Fk(\000)p Ft(K)3325 1791 y Fd(2)3357 1783 y Fs(\()p Ft(T)9 b Fk(\000)p Ft(t)p Fs(\))3538 1817 y Fq(g)p Fu(;)242 b Fv(\(2.4\))236 1999 y(for)27 b(some)g(p)r(ositiv)n(e)g(constan)n(ts)g Fu(K)1317 2011 y Fs(1)1382 1999 y Fv(and)g Fu(K)1614 2011 y Fs(2)1651 1999 y Fv(,)g(where)h Fu(q)1979 2011 y Fk(1)2077 1999 y Fv(is)f(de\014ned)h(b)n(y)829 2182 y Fu(q)866 2194 y Fk(1)936 2182 y Fv(\()p Fu(t)p Fv(\))c(=)e Fu(q)1178 2194 y Fs(0)1216 2182 y Fv(\()p Fu(t)p Fv(\))d Fq(\000)f Fv(2)p Fu(\031)30 b Fv(if)35 b Fu(t)23 b Fq(2)g Fv([0)p Fu(;)14 b(T)9 b(=)p Fv(2\))p Fu(;)179 b(q)2260 2194 y Fk(1)2330 2182 y Fv(\()p Fu(t)p Fv(\))24 b(=)e Fu(q)2572 2194 y Fs(0)2609 2182 y Fv(\()p Fu(t)d Fq(\000)f Fu(T)12 b Fv(\))27 b(if)35 b Fu(t)23 b Fq(2)g Fv(\()p Fu(T)9 b(=)p Fv(2)p Fu(;)14 b(T)e Fv(])p Fu(:)236 2365 y Fw(Lemma)32 b(2.1)40 b Fp(Ther)l(e)46 b(exists)e Fu(\026)1284 2377 y Fs(2)1371 2365 y Fu(>)50 b Fv(0)44 b Fp(and)h(c)l(onstants)f Fu(C)2187 2377 y Fs(0)2224 2365 y Fu(;)14 b(C)2320 2377 y Fs(1)2358 2365 y Fu(;)p 2395 2319 36 4 v 14 w(c)o(;)g(c)2503 2377 y Fs(1)2590 2365 y Fu(>)50 b Fv(0)44 b Fp(such)h(that)f Fq(8)p Fv(0)50 b Fu(<)f(\026)h Fq(\024)g Fu(\026)3696 2377 y Fs(2)3733 2365 y Fp(,)f Fq(8)p Fu(\025)h Fv(=)236 2464 y(\()p Fu(\022)309 2434 y Fs(+)365 2464 y Fu(;)14 b(\022)443 2434 y Fk(\000)499 2464 y Fu(;)g(')590 2434 y Fs(+)645 2464 y Fu(;)g(')736 2434 y Fk(\000)793 2464 y Fv(\))44 b Fp(such)g(that)g Fu(C)1315 2476 y Fs(0)1352 2464 y Fu(\014)1403 2434 y Fs(2)1399 2488 y Ft(\025)1491 2464 y Fu(>)k(\026)c Fp(and)h Fu(C)1933 2476 y Fs(1)1970 2464 y Fq(j)14 b Fv(ln)g Fu(\026)p Fq(j)49 b(\024)f Fu(T)2374 2476 y Ft(\025)2466 2464 y Fq(\024)g Fu(C)2638 2476 y Fs(0)2676 2464 y Fu(\014)2723 2476 y Ft(\025)2766 2464 y Fu(=\026)43 b Fp(ther)l(e)h(exists)g(a)g(unique)f(solution)236 2564 y Fv(\()p Fu(')322 2576 y Ft(\026)367 2564 y Fv(\()p Fu(t)p Fv(\))p Fu(;)14 b(q)535 2576 y Ft(\026)580 2564 y Fv(\()p Fu(t)p Fv(\)\))39 b(:=)e(\()p Fu(')956 2576 y Ft(\026;\025)1060 2564 y Fv(\()p Fu(t)p Fv(\))p Fu(;)14 b(q)1228 2576 y Ft(\026;\025)1332 2564 y Fv(\()p Fu(t)p Fv(\)\))38 b Fp(of)h(\(2.1\),)i(de\014ne)l(d)d(for)h Fu(t)e Fq(2)h Fv(\()p Fu(\022)2507 2534 y Fs(+)2587 2564 y Fq(\000)23 b Fv(1)p Fu(;)14 b(\022)2795 2534 y Fk(\000)2875 2564 y Fv(+)24 b(1\))p Fp(,)40 b(satisfying)f Fu(')3534 2576 y Ft(\026)3579 2564 y Fv(\()p Fu(\022)3652 2534 y Fk(\006)3708 2564 y Fv(\))f(=)f Fu(')3934 2534 y Fk(\006)3990 2564 y Fp(,)236 2664 y Fu(q)273 2676 y Ft(\026)318 2664 y Fv(\()p Fu(\022)391 2634 y Fk(\006)447 2664 y Fv(\))24 b(=)e Fq(\007)p Fu(\031)33 b Fp(and)952 2848 y Fv(\()p Fu(i)p Fv(\))83 b Fq(j)p Fu(')1205 2860 y Ft(\026)1250 2848 y Fv(\()p Fu(t)p Fv(\))19 b Fq(\000)p 1446 2803 55 4 v 18 w Fu(')p Fv(\()p Fu(t)p Fv(\))p Fq(j)24 b(\024)p 1729 2803 36 4 v 23 w Fu(c\026)p Fv(\(1)18 b(+)g Fu(c)2026 2860 y Fs(1)2063 2848 y Fu(\026T)2174 2818 y Fs(2)2162 2872 y Ft(\025)2210 2848 y Fv(\))p Fu(=\014)2335 2818 y Fs(2)2331 2872 y Ft(\025)2375 2848 y Fu(;)183 b Fq(j)23 b Fv(_)-46 b Fu(')2658 2860 y Ft(\026)2703 2848 y Fv(\()p Fu(t)p Fv(\))19 b Fq(\000)f Fu(!)s Fq(j)23 b(\024)p 3088 2803 V 23 w Fu(c\026=\014)3263 2860 y Ft(\025)3306 2848 y Fu(;)923 2955 y Fv(\()p Fu(ii)p Fv(\))83 b Fq(j)p Fu(q)1188 2967 y Ft(\026)1233 2955 y Fv(\()p Fu(t)p Fv(\))19 b Fq(\000)f Fu(Q)1495 2967 y Ft(T)1534 2976 y Fc(\025)1577 2955 y Fv(\()p Fu(t)h Fq(\000)f Fu(\022)1782 2925 y Fs(+)1837 2955 y Fv(\))p Fq(j)23 b(\024)p 2003 2909 V 23 w Fu(c\026;)184 b Fq(j)15 b Fv(_)-38 b Fu(q)2356 2967 y Ft(\026)2400 2955 y Fv(\()p Fu(t)p Fv(\))19 b Fq(\000)2624 2934 y Fv(_)2596 2955 y Fu(Q)2662 2967 y Ft(T)2701 2976 y Fc(\025)2744 2955 y Fv(\()p Fu(t)g Fq(\000)f Fu(\022)2949 2925 y Fs(+)3004 2955 y Fv(\))p Fq(j)24 b(\024)p 3170 2909 V 22 w Fu(c\026;)3845 2902 y Fv(\(2.5\))236 3144 y Fp(wher)l(e)p 471 3098 55 4 v 31 w Fu(')p Fv(\()p Fu(t)p Fv(\))g(:=)e Fu(!)805 3156 y Ft(\025)848 3144 y Fv(\()p Fu(t)d Fq(\000)f Fu(\022)1053 3113 y Fs(+)1109 3144 y Fv(\))g(+)g Fu(')1296 3113 y Fs(+)1352 3144 y Fp(.)38 b(Mor)l(e)l(over)32 b Fu(')1835 3156 y Ft(\026;\025)1938 3144 y Fv(\()p Fu(t)p Fv(\))p Fp(,)54 b Fv(_)-46 b Fu(')2142 3156 y Ft(\026;\025)2246 3144 y Fv(\()p Fu(t)p Fv(\))p Fp(,)31 b Fu(q)2433 3156 y Ft(\026;\025)2537 3144 y Fv(\()p Fu(t)p Fv(\))f Fp(and)45 b Fv(_)-38 b Fu(q)2859 3156 y Ft(\026;\025)2963 3144 y Fv(\()p Fu(t)p Fv(\))30 b Fp(ar)l(e)g Fq(C)3277 3113 y Fs(1)3344 3144 y Fp(functions)f(of)i Fv(\()p Fu(t;)14 b(\025)p Fv(\))p Fp(.)361 3326 y Fv(The)27 b(pro)r(of)g(of)h(lemma)f (2.1)g(is)h(giv)n(en)f(in)g(the)h(App)r(endix.)236 3509 y Fw(Remark)k(2.1)41 b Fp(R)l(oughly,)30 b(the)g(me)l(aning)g(of)h(the) f(ab)l(ove)h(estimates)e(is)i(the)e(fol)t(lowing.)361 3609 y(1\))e(We)h(have)h(imp)l(ose)l(d)f Fu(C)1172 3621 y Fs(1)1210 3609 y Fq(j)14 b Fv(ln)g Fu(\026)p Fq(j)23 b Fu(<)f(T)1562 3621 y Ft(\025)1629 3609 y Fv(:=)g Fu(\022)1780 3578 y Fk(\000)1850 3609 y Fq(\000)13 b Fu(\022)1969 3578 y Fs(+)2052 3609 y Fp(so)27 b(that)h(by)g Fv(\(2)p Fu(:)p Fv(4\))p Fp(,)f(on)h(such)f(intervals)h(of)h(time,)f(the)g(p)l (erio)l(dic)236 3708 y(solution)i Fu(Q)616 3720 y Ft(T)655 3729 y Fc(\025)728 3708 y Fp(is)g Fu(O)r Fv(\()p Fu(\026)p Fv(\))h Fp(close)g(to)e(\\sep)l(ar)l(atric)l(es")i Fu(q)1903 3720 y Fk(1)2003 3708 y Fp(of)f(the)g(unp)l(erturb)l(e)l(d)f(p)l (endulum.)361 3808 y(2\))e(Estimate)f Fv(\()p Fu(ii)p Fv(\))h Fp(implies)h(that)e(for)i Fu(t)23 b Fq(\031)f Fv(\()p Fu(\022)1748 3778 y Fs(+)1815 3808 y Fv(+)11 b Fu(\022)1932 3778 y Fk(\000)1989 3808 y Fv(\))p Fu(=)p Fv(2)25 b Fp(the)i(p)l(erturb)l(e)l(d)f(solution)h Fu(q)2968 3820 y Ft(\026)3039 3808 y Fp(may)g(have)h Fu(O)r Fv(\()p Fu(\026)p Fv(\))g Fp(oscil)t(lations)236 3907 y(ar)l(ound)42 b(the)g(unstable)f(e)l(quilibrium)i(of)f(the)g(p)l(endulum)f Fu(q)48 b Fv(=)c(0)p Fp(,)h(mo)l(d)d Fv(2)p Fu(\031)s Fp(,)j(which)e(is)f(exactly)g(what)g(one)g(exp)l(e)l(cts)236 4007 y(p)l(erturbing)25 b(with)h(a)g(gener)l(al)g Fu(f)9 b Fp(.)36 b(On)25 b(the)g(c)l(ontr)l(ary)h(for)g(the)f(class)h(of)g(p)l (erturb)l(ations)f(c)l(onsider)l(e)l(d)h(in)g([2)q(])f(as)h Fu(f)9 b Fv(\()p Fu(';)14 b(q)s(;)g(t)p Fv(\))23 b(=)236 4107 y(\(1)g Fq(\000)g Fv(cos)13 b Fu(q)s Fv(\))p Fu(f)c Fv(\()p Fu(';)14 b(t)p Fv(\))38 b Fp(pr)l(eserving)f(al)t(l)h(the)f (invariant)g(tori,)j(estimate)c Fv(\()p Fu(ii)p Fv(\))h Fp(c)l(an)f(b)l(e)h(impr)l(ove)l(d,)j(getting)d Fv(max)o Fq(fj)p Fu(q)3812 4119 y Ft(\026)3857 4107 y Fv(\()p Fu(t)p Fv(\))p Fq(\000)236 4215 y Fu(Q)302 4227 y Ft(T)341 4236 y Fc(\025)384 4215 y Fv(\()p Fu(t)19 b Fq(\000)f Fu(\022)589 4185 y Fs(+)644 4215 y Fv(\))p Fq(j)p Fu(;)30 b Fq(j)16 b Fv(_)-39 b Fu(q)812 4227 y Ft(\026)857 4215 y Fv(\()p Fu(t)p Fv(\))19 b Fq(\000)1081 4194 y Fv(_)1053 4215 y Fu(Q)1119 4227 y Ft(T)1158 4236 y Fc(\025)1201 4215 y Fv(\()p Fu(t)g Fq(\000)f Fu(\022)1406 4185 y Fs(+)1461 4215 y Fv(\))p Fq(jg)23 b Fv(=)g Fu(O)r Fv(\()p Fu(\026)14 b Fv(max)p Fq(f)p Fv(exp\()p Fq(\000)p Fu(C)6 b Fq(j)p Fu(t)18 b Fq(\000)g Fu(\022)2511 4185 y Fs(+)2567 4215 y Fq(j)p Fv(\))p Fu(;)c Fv(exp\()p Fq(\000)p Fu(C)6 b Fq(j)p Fu(t)18 b Fq(\000)g Fu(\022)3143 4185 y Fk(\000)3200 4215 y Fq(j)p Fv(\))p Fq(g)p Fv(\))p Fp(.)361 4315 y(3\))27 b(F)-6 b(or)28 b Fu(\014)664 4327 y Ft(\025)731 4315 y Fq(\031)818 4263 y(p)p 887 4263 51 4 v 52 x Fu(\026)g Fp(estimate)f Fv(\()p Fu(i)p Fv(\))h Fp(b)l(e)l(c)l(omes)f(me)l (aningless:)38 b(for)28 b(a)g(me)l(an)f(fr)l(e)l(quency)h Fu(!)3050 4327 y Ft(\025)3120 4315 y Fp(such)g(that)f Fu(n)13 b Fq(\001)g Fu(!)3625 4327 y Ft(\025)3682 4315 y Fv(+)g Fu(l)24 b Fq(\031)3897 4263 y(p)p 3966 4263 V 52 x Fu(\026)236 4414 y Fp(for)j(some)f Fv(0)d Fu(<)f Fq(j)p Fv(\()p Fu(n;)14 b(l)r Fv(\))p Fq(j)23 b(\024)g Fu(N)34 b Fp(the)26 b(p)l(erturb)l(e)l(d)g(tr)l(ansition)g(orbits)g Fu(')2307 4426 y Ft(\026)2378 4414 y Fp(ar)l(e)g(no)g(mor)l(e)g(wel)t (l-appr)l(oximate)l(d)i(by)e(the)g(str)l(aight)236 4514 y(lines)p 431 4468 55 4 v 30 w Fu(')q Fv(\()p Fu(t)p Fv(\))d(:=)g Fu(')768 4484 y Fs(+)842 4514 y Fv(+)18 b Fu(!)977 4526 y Ft(\025)1020 4514 y Fv(\()p Fu(t)h Fq(\000)f Fu(\022)1225 4484 y Fs(+)1280 4514 y Fv(\))p Fp(.)236 4706 y Fw(Remark)32 b(2.2)41 b Fp(L)l(et)e(us)g(de\014ne)h Fq(D)1352 4666 y Ft(\014)1350 4731 y(N)1454 4706 y Fv(:=)h Fq(f)p Fu(!)j Fq(2)d Fw(R)1889 4676 y Ft(d)1968 4706 y Fq(j)e(j)p Fu(!)29 b Fq(\001)d Fu(n)f Fv(+)h Fu(l)r Fq(j)40 b Fu(>)h(\014)t(;)94 b Fq(8)40 b Fv(0)h Fu(<)f Fq(j)p Fv(\()p Fu(n;)14 b(l)r Fv(\))p Fq(j)42 b(\024)e Fu(N)9 b Fq(g)p Fp(.)69 b(In)39 b([12)q(])i(it)e(is)236 4819 y(pr)l(ove)l(d)34 b(that)f(hyp)l(erb)l(olic)i(invariant)f(tori)f Fq(T)1652 4789 y Ft(\026)1631 4840 y(!)1729 4819 y Fp(of)h(system)f(\() p Fq(S)2189 4831 y Ft(\026)2233 4819 y Fp(\))g(exist)g(for)g (Diophantine)h(fr)l(e)l(quencies)g Fu(!)d Fq(2)e(D)3767 4779 y Ft(\014)3805 4787 y Fd(1)3765 4844 y Ft(N)3818 4852 y Fd(1)3854 4819 y Fp(,)34 b(for)236 4919 y(some)c Fu(\014)495 4931 y Fs(1)555 4919 y Fv(=)23 b Fu(O)r Fv(\(1\))30 b Fp(and)g(some)g Fu(N)1284 4931 y Fs(1)1344 4919 y Fv(=)23 b Fu(O)r Fv(\()p Fu(dN)9 b Fv(\))24 b Fu(>)f(N)9 b Fp(,)30 b(namely)g(avoiding)i(mor)l(e)d(\\r)l(esonanc)l(es)h(with)g(the)g (trigonometric)236 5019 y(p)l(olynomial)g Fu(f)9 b Fp(")27 b(than)h(just)g Fu(N)9 b Fp(.)37 b(The)29 b(pr)l(esenc)l(e)f(of)h(such) f(\\r)l(esonant)f(hyp)l(erplanes)j Fu(E)2925 5031 y Ft(n;l)3011 5019 y Fp(")e(for)g Fu(N)k(<)23 b Fq(j)p Fv(\()p Fu(n;)14 b(l)r Fv(\))p Fq(j)23 b Fu(<)f(N)3800 5031 y Fs(1)3865 5019 y Fp(may)236 5118 y(b)l(e)29 b(r)l(e\015e)l(cte)l(d)f(in)g (estimate)h Fv(\()p Fu(i)p Fv(\))f Fp(by)h(the)f(term)g Fu(\026T)1761 5088 y Fs(2)1749 5142 y Ft(\025)1798 5118 y Fp(.)38 b(However)29 b(such)f(term,)h(for)g(our)f(purp)l(oses,)i(c)l (an)e(b)l(e)h(ignor)l(e)l(d.)39 b(F)-6 b(r)l(om)236 5218 y(this)26 b(p)l(oint)g(of)h(view)g(lemma)f(2.1)h(c)l(ould)f(p)l(erhaps) i(b)l(e)d(interpr)l(ete)l(d)h(as)g(the)g(\014rst)f(iter)l(ative)i(step) f(for)g(lo)l(oking)h(at)f(invariant)236 5317 y(hyp)l(erb)l(olic)32 b(tori)e(in)g(the)g(p)l(erturb)l(e)l(d)g(system)f(bifur)l(c)l(ating)i (fr)l(om)f(the)g(unp)l(erturb)l(e)l(d)f(one's)h(.)2105 6059 y Fv(6)p eop %%Page: 7 7 7 6 bop 361 272 a Fv(By)27 b(lemma)h(2.1,)e(for)h(0)c Fu(<)g(\026)g Fq(\024)f Fu(\026)1405 284 y Fs(2)1443 272 y Fv(,)27 b(w)n(e)h(can)f(de\014ne)h(on)f(the)h(set)901 487 y(\003)959 499 y Ft(\026)1026 487 y Fv(:=)1137 394 y Fl(n)1192 487 y Fu(\025)23 b Fv(=)g(\()p Fu(\022)1424 452 y Fs(+)1480 487 y Fu(;)14 b(\022)1558 452 y Fk(\000)1614 487 y Fu(;)g(')1705 452 y Fs(+)1760 487 y Fu(;)g(')1851 452 y Fk(\000)1908 487 y Fv(\))1967 391 y Fl(\014)1967 441 y(\014)1967 491 y(\014)2051 487 y Fu(C)2110 499 y Fs(0)2147 487 y Fu(\014)2198 452 y Fs(2)2194 507 y Ft(\025)2261 487 y Fu(>)22 b(\026;)42 b(C)2522 499 y Fs(1)2559 487 y Fq(j)14 b Fv(ln)g Fu(\026)p Fq(j)24 b(\024)e Fu(T)2912 499 y Ft(\025)2978 487 y Fq(\024)3076 430 y Fu(C)3135 442 y Fs(0)3172 430 y Fu(\014)3219 442 y Ft(\025)p 3076 468 188 4 v 3144 544 a Fu(\026)3273 394 y Fl(o)3328 487 y Fu(;)236 702 y Fv(the)28 b(Lagrangian)d(action)i(functional)h Fu(G)1518 714 y Ft(\026)1586 702 y Fv(:)23 b(\003)1690 714 y Ft(\026)1757 702 y Fq(!)g Fw(R)28 b Fv(as)848 956 y Fu(G)913 968 y Ft(\026)958 956 y Fv(\()p Fu(\025)p Fv(\))c(=)e Fu(G)1246 968 y Ft(\026)1291 956 y Fv(\()p Fu(\022)1364 921 y Fs(+)1420 956 y Fu(;)14 b(\022)1498 921 y Fk(\000)1554 956 y Fu(;)g(')1645 921 y Fs(+)1700 956 y Fu(;)g(')1791 921 y Fk(\000)1848 956 y Fv(\))23 b(:=)2014 843 y Fl(Z)2097 863 y Ft(\022)2131 838 y Fo(\000)2060 1031 y Ft(\022)2094 1015 y Fd(+)2197 956 y Fq(L)2254 968 y Ft(\026)2299 956 y Fv(\()p Fu(')2385 968 y Ft(\026)2430 956 y Fv(\()p Fu(t)p Fv(\))p Fu(;)37 b Fv(_)-46 b Fu(')2615 968 y Ft(\026)2660 956 y Fv(\()p Fu(t)p Fv(\))p Fu(;)14 b(q)2828 968 y Ft(\026)2873 956 y Fv(\()p Fu(t)p Fv(\))p Fu(;)30 b Fv(_)-39 b Fu(q)3041 968 y Ft(\026)3086 956 y Fv(\()p Fu(t)p Fv(\))p Fu(;)14 b(t)p Fv(\))29 b Fu(dt:)441 b Fv(\(2.6\))236 1174 y(W)-7 b(e)28 b(ha)n(v)n(e)236 1334 y Fw(Lemma)k(2.2)40 b Fu(G)803 1346 y Ft(\026)878 1334 y Fp(is)30 b(di\013er)l(entiable)h(and)g(\(with)f(the)g(abbr)l (eviations)h Fu(';)14 b(q)33 b Fp(for)e Fu(')2823 1346 y Ft(\026)2868 1334 y Fu(;)14 b(q)2942 1346 y Ft(\026)2986 1334 y Fp(\))439 1553 y Fq(r)508 1568 y Ft(')552 1552 y Fd(+)603 1553 y Fu(G)668 1565 y Ft(\026)713 1553 y Fv(\()p Fu(\025)p Fv(\))24 b(=)f Fq(\000)f Fv(_)-45 b Fu(')p Fv(\()p Fu(\022)1129 1518 y Fs(+)1184 1553 y Fv(\))p Fu(;)184 b(@)1467 1568 y Ft(\022)1501 1552 y Fd(+)1552 1553 y Fu(G)1617 1565 y Ft(\026)1662 1553 y Fv(\()p Fu(\025)p Fv(\))24 b(=)1895 1496 y(1)p 1895 1533 42 4 v 1895 1610 a(2)1947 1553 y Fq(j)e Fv(_)-45 b Fu(')p Fv(\()p Fu(\022)2097 1518 y Fs(+)2153 1553 y Fv(\))p Fq(j)2208 1518 y Fs(2)2264 1553 y Fv(+)2357 1496 y(1)p 2357 1533 V 2357 1610 a(2)2424 1553 y(_)-39 b Fu(q)2448 1518 y Fs(2)2485 1553 y Fv(\()p Fu(\022)2558 1518 y Fs(+)2614 1553 y Fv(\))19 b(+)f(cos)13 b Fu(q)s Fv(\()p Fu(\022)2986 1518 y Fs(+)3042 1553 y Fv(\))18 b Fq(\000)g Fv(1)g(+)g Fu(\026f)9 b Fv(\()p Fu(')3504 1518 y Fs(+)3560 1553 y Fu(;)14 b(\031)s(;)g(\022)3725 1518 y Fs(+)3780 1553 y Fv(\))416 1804 y Fq(r)485 1820 y Ft(')529 1803 y Fo(\000)582 1804 y Fu(G)647 1816 y Ft(\026)692 1804 y Fv(\()p Fu(\025)p Fv(\))24 b(=)45 b(_)-46 b Fu(')q Fv(\()p Fu(\022)1043 1770 y Fk(\000)1099 1804 y Fv(\))p Fu(;)99 b(@)1297 1820 y Ft(\022)1331 1803 y Fo(\000)1384 1804 y Fu(G)1449 1816 y Ft(\026)1494 1804 y Fv(\()p Fu(\025)p Fv(\))24 b(=)f Fq(\000)1783 1712 y Fl(\020)1842 1748 y Fv(1)p 1842 1785 V 1842 1861 a(2)1893 1804 y Fq(j)g Fv(_)-46 b Fu(')q Fv(\()p Fu(\022)2044 1770 y Fk(\000)2100 1804 y Fv(\))p Fq(j)2155 1770 y Fs(2)2211 1804 y Fv(+)2304 1748 y(1)p 2304 1785 V 2304 1861 a(2)2371 1804 y(_)-38 b Fu(q)2396 1770 y Fs(2)2433 1804 y Fv(\()p Fu(\022)2506 1770 y Fk(\000)2563 1804 y Fv(\))18 b(+)g(cos)13 b Fu(q)s Fv(\()p Fu(\022)2934 1770 y Fk(\000)2991 1804 y Fv(\))19 b Fq(\000)f Fv(1)g(+)g Fu(\026f)9 b Fv(\()p Fu(')3454 1770 y Fk(\000)3510 1804 y Fu(;)14 b(\031)s(;)g(\022)3675 1770 y Fk(\000)3731 1804 y Fv(\))3763 1712 y Fl(\021)3813 1804 y Fu(:)236 1998 y Fb(Pr)n(oof)p Fv(.)105 b(By)37 b(lemma)f(2.1)g(the)h(map)f(\()p Fu(\025;)14 b(t)p Fv(\))39 b Fq(7!)f Fv(\()p Fu(')1953 2010 y Ft(\026;\025)2057 1998 y Fv(\()p Fu(t)p Fv(\))p Fu(;)f Fv(_)-46 b Fu(')2242 2010 y Ft(\026;\025)2347 1998 y Fv(\()p Fu(t)p Fv(\))p Fu(;)14 b(q)2515 2010 y Ft(\026;\025)2619 1998 y Fv(\()p Fu(t)p Fv(\))p Fu(;)30 b Fv(_)-39 b Fu(q)2787 2010 y Ft(\026;\025)2891 1998 y Fv(\()p Fu(t)p Fv(\)\))38 b(is)e Fu(C)3212 1968 y Fs(1)3286 1998 y Fv(on)g(the)h(set)g Fq(f)p Fv(\()p Fu(\025;)14 b(t)p Fv(\))38 b Fq(2)236 2097 y Fv(\003)294 2109 y Ft(\026)357 2097 y Fq(\002)18 b Fw(R)27 b Fq(j)56 b Fu(\022)659 2067 y Fs(+)737 2097 y Fq(\024)23 b Fu(t)g Fq(\024)f Fu(\022)1006 2067 y Fk(\000)1063 2097 y Fq(g)p Fv(.)36 b(Hence)28 b Fu(G)1476 2109 y Ft(\026)1548 2097 y Fv(is)g(di\013eren)n(tiable)f(and)336 2351 y Fu(@)380 2367 y Ft(\022)414 2350 y Fd(+)465 2351 y Fu(G)530 2363 y Ft(\026)574 2351 y Fv(\()p Fu(\025)p Fv(\))84 b(=)f Fq(\000L)1040 2363 y Ft(\026)1084 2351 y Fv(\()p Fu(')1170 2317 y Fs(+)1226 2351 y Fu(;)36 b Fv(_)-45 b Fu(')p Fv(\()p Fu(\022)1390 2317 y Fs(+)1446 2351 y Fv(\))p Fu(;)14 b Fq(\000)p Fu(\031)s(;)29 b Fv(_)-38 b Fu(q)s Fv(\()p Fu(\022)1780 2317 y Fs(+)1836 2351 y Fv(\))p Fu(;)14 b(\022)1946 2317 y Fs(+)2001 2351 y Fv(\))19 b(+)2135 2238 y Fl(Z)2218 2259 y Ft(\022)2252 2234 y Fo(\000)2181 2427 y Ft(\022)2215 2410 y Fd(+)2341 2351 y Fv(_)-46 b Fu(')q Fv(\()p Fu(s)p Fv(\))19 b Fq(\001)f Fu(@)2580 2367 y Ft(\022)2614 2350 y Fd(+)2687 2351 y Fv(_)-45 b Fu(')p Fv(\()p Fu(s)p Fv(\))19 b(+)33 b(_)-38 b Fu(q)s Fv(\()p Fu(s)p Fv(\))p Fu(@)3111 2367 y Ft(\022)3145 2350 y Fd(+)3211 2351 y Fv(_)g Fu(q)s Fv(\()p Fu(s)p Fv(\))28 b Fu(ds)770 2605 y Fv(+)918 2492 y Fl(Z)1001 2513 y Ft(\022)1035 2488 y Fo(\000)964 2681 y Ft(\022)998 2664 y Fd(+)1102 2605 y Fv(sin)13 b Fu(q)s Fv(\()p Fu(s)p Fv(\))p Fu(@)1404 2621 y Ft(\022)1438 2604 y Fd(+)1489 2605 y Fu(q)s Fv(\()p Fu(s)p Fv(\))19 b Fq(\000)f Fu(\026@)1828 2617 y Ft(')1876 2605 y Fu(f)9 b Fv(\()p Fu(')p Fv(\()p Fu(s)p Fv(\))p Fu(;)14 b(q)s Fv(\()p Fu(s)p Fv(\))p Fu(;)g(s)p Fv(\))20 b Fq(\001)e Fu(@)2508 2621 y Ft(\022)2542 2604 y Fd(+)2593 2605 y Fu(')p Fv(\()p Fu(s)p Fv(\))h Fq(\000)f Fu(\026@)2946 2617 y Ft(q)2983 2605 y Fu(f)9 b Fv(\()p Fu(')p Fv(\()p Fu(s)p Fv(\))p Fu(;)14 b(q)s Fv(\()p Fu(s)p Fv(\))p Fu(;)g(s)p Fv(\))p Fu(@)3554 2621 y Ft(\022)3588 2604 y Fd(+)3639 2605 y Fu(q)s Fv(\()p Fu(s)p Fv(\))29 b Fu(ds:)236 2832 y Fv(In)n(tegrating)d(b)n(y)i(parts)f(and)g(using)g (that)h(\()p Fu(q)1621 2844 y Ft(\026;\025)1725 2832 y Fu(;)14 b(')1816 2844 y Ft(\026;\025)1920 2832 y Fv(\))28 b(satis\014es)f(\(2.1\))g(in)h(\()p Fu(\022)2654 2801 y Fs(+)2710 2832 y Fu(;)14 b(\022)2788 2801 y Fk(\000)2844 2832 y Fv(\),)28 b(w)n(e)f(obtain)682 3068 y Fu(@)726 3083 y Ft(\022)760 3067 y Fd(+)810 3068 y Fu(G)875 3080 y Ft(\026)920 3068 y Fv(\()p Fu(\025)p Fv(\))d(=)f Fq(\000L)1266 3080 y Ft(\026)1310 3068 y Fv(\()p Fu(')1396 3033 y Fs(+)1452 3068 y Fu(;)36 b Fv(_)-45 b Fu(')p Fv(\()p Fu(\022)1616 3033 y Fs(+)1672 3068 y Fv(\))p Fu(;)14 b Fq(\000)p Fu(\031)s(;)29 b Fv(_)-38 b Fu(q)s Fv(\()p Fu(\022)2006 3033 y Fs(+)2062 3068 y Fv(\))p Fu(;)14 b(\022)2172 3033 y Fs(+)2227 3068 y Fv(\))19 b(+)2361 2976 y Fl(h)2415 3068 y Fv(_)-38 b Fu(q)s Fv(\()p Fu(s)p Fv(\))p Fu(@)2587 3083 y Ft(\022)2621 3067 y Fd(+)2672 3068 y Fu(q)s Fv(\()p Fu(s)p Fv(\))19 b(+)41 b(_)-46 b Fu(')p Fv(\()p Fu(s)p Fv(\))19 b Fq(\001)g Fu(@)3179 3083 y Ft(\022)3213 3067 y Fd(+)3263 3068 y Fu(')p Fv(\()p Fu(s)p Fv(\))3420 2976 y Fl(i)3460 2993 y Ft(\022)3494 2968 y Fo(\000)3460 3126 y Ft(\022)3494 3109 y Fd(+)3547 3068 y Fu(:)236 3277 y Fv(No)n(w)35 b Fu(q)470 3289 y Ft(\026;\025)574 3277 y Fv(\()p Fu(\022)647 3247 y Fs(+)703 3277 y Fv(\))h(=)g Fq(\000)p Fu(\031)i Fv(for)d(all)g Fu(\025)h Fv(hence)51 b(_)-39 b Fu(q)s Fv(\()p Fu(\022)1715 3247 y Fs(+)1771 3277 y Fv(\))24 b(+)f Fu(@)1959 3292 y Ft(\022)1993 3276 y Fd(+)2044 3277 y Fu(q)s Fv(\()p Fu(\022)2157 3247 y Fs(+)2212 3277 y Fv(\))37 b(=)e(0.)60 b(Similarly)35 b(w)n(e)g(get)58 b(_)-45 b Fu(')p Fv(\()p Fu(\022)3271 3247 y Fs(+)3326 3277 y Fv(\))24 b(+)g Fu(@)3515 3292 y Ft(\022)3549 3276 y Fd(+)3599 3277 y Fu(')p Fv(\()p Fu(\022)3726 3247 y Fs(+)3782 3277 y Fv(\))37 b(=)e(0,)236 3376 y Fu(@)280 3392 y Ft(\022)314 3375 y Fd(+)365 3376 y Fu(q)s Fv(\()p Fu(\022)478 3346 y Fk(\000)534 3376 y Fv(\))24 b(=)e(0,)28 b Fu(@)814 3392 y Ft(\022)848 3375 y Fd(+)898 3376 y Fu(')p Fv(\()p Fu(\022)1025 3346 y Fk(\000)1082 3376 y Fv(\))23 b(=)g(0.)36 b(As)28 b(a)f(consequence)888 3591 y Fu(@)932 3607 y Ft(\022)966 3590 y Fd(+)1016 3591 y Fu(G)1081 3603 y Ft(\026)1126 3591 y Fv(\()p Fu(\025)p Fv(\))d(=)1359 3535 y(1)p 1359 3572 V 1359 3648 a(2)1411 3591 y Fq(j)f Fv(_)-46 b Fu(')p Fq(j)1511 3557 y Fs(2)1549 3591 y Fv(\()p Fu(\022)1622 3557 y Fs(+)1677 3591 y Fv(\))19 b(+)1821 3535 y(1)p 1821 3572 V 1821 3648 a(2)1888 3591 y(_)-39 b Fu(q)1912 3557 y Fs(2)1950 3591 y Fv(\()p Fu(\022)2023 3557 y Fs(+)2078 3591 y Fv(\))19 b(+)f(\(cos)13 b Fu(q)s Fv(\()p Fu(\022)2482 3557 y Fs(+)2538 3591 y Fv(\))19 b Fq(\000)f Fv(1\))g(+)g Fu(\026f)9 b Fv(\()p Fu(')3033 3557 y Fs(+)3089 3591 y Fu(;)14 b(\031)s(;)g(\022)3254 3557 y Fs(+)3309 3591 y Fv(\))p Fu(:)236 3791 y Fv(The)28 b(other)f(partial)g(deriv)-5 b(ativ)n(es)26 b(are)h(computed)h(in)f (the)h(same)f(w)n(a)n(y)-7 b(.)p 2468 3791 48 48 v 361 3937 a(F)g(or)27 b Fu(\014)g(>)c Fv(0)k(\014xed,)h(denoting)f Fu(\025)1354 3949 y Ft(i)1405 3937 y Fv(=)c(\()p Fu(\022)1564 3949 y Ft(i)1592 3937 y Fu(;)14 b(\022)1668 3949 y Ft(i)p Fs(+1)1779 3937 y Fu(;)g(')1870 3949 y Ft(i)1898 3937 y Fu(;)g(')1989 3949 y Ft(i)p Fs(+1)2101 3937 y Fv(\),)28 b(w)n(e)f(de\014ne)h(on)f(the)h(set)359 4138 y(\003)417 4150 y Ft(\026;k)541 4138 y Fv(:=)23 b(\003)710 4099 y Ft(\014)710 4163 y(\026;k)833 4138 y Fv(:=)944 4046 y Fl(n)999 4138 y Fu(\025)h Fv(=)e(\()p Fu(\022)1229 4150 y Fs(1)1267 4138 y Fu(;)14 b(:)g(:)g(:)g(;)g(\022)1491 4150 y Ft(k)1531 4138 y Fu(;)g(')1622 4150 y Fs(1)1660 4138 y Fu(;)g(:)g(:)g(:)f(;)h(')1898 4150 y Ft(k)1939 4138 y Fv(\))24 b Fq(2)f Fw(R)2145 4104 y Ft(k)2204 4138 y Fq(\002)18 b Fw(R)2359 4104 y Ft(k)q(d)2462 4043 y Fl(\014)2462 4093 y(\014)2462 4143 y(\014)2517 4138 y Fq(8)28 b Fv(1)22 b Fq(\024)h Fu(i)g Fq(\024)f Fu(k)g Fq(\000)c Fv(1)p Fu(;)68 b(\025)3211 4150 y Ft(i)3263 4138 y Fq(2)23 b Fv(\003)3399 4150 y Ft(\026)3471 4138 y Fu(;)41 b(\014)3582 4150 y Ft(\025)3621 4158 y Fc(i)3675 4138 y Fq(\025)23 b Fu(\014)3814 4046 y Fl(o)3870 4138 y Fu(;)236 4335 y Fv(the)28 b(reduced)f(action)h(functional)f Fq(F)1387 4347 y Ft(\026)1455 4335 y Fv(:)c(\003)1559 4347 y Ft(\026;k)1682 4335 y Fq(!)g Fw(R)28 b Fv(as)755 4596 y Fq(F)815 4608 y Ft(\026)859 4596 y Fv(\()p Fu(\025)p Fv(\))84 b(=)f Fu(!)1255 4608 y Ft(I)1292 4596 y Fu(')1346 4608 y Fs(1)1402 4596 y Fq(\000)1495 4539 y(j)p Fu(!)1570 4551 y Ft(I)1608 4539 y Fq(j)1631 4509 y Fs(2)p 1495 4577 174 4 v 1561 4653 a Fv(2)1678 4596 y Fu(\022)1717 4608 y Fs(1)1773 4596 y Fv(+)18 b Fu(\026)p Fv(\000)1958 4561 y Ft(u)2001 4596 y Fv(\()p Fu(!)2085 4608 y Ft(I)2124 4596 y Fu(;)c(\022)2200 4608 y Fs(1)2237 4596 y Fu(;)g(')2328 4608 y Fs(1)2365 4596 y Fv(\))19 b(+)f Fu(\026F)12 b Fv(\()p Fu(!)2698 4608 y Ft(I)2736 4596 y Fu(;)i(\022)2812 4608 y Fs(1)2849 4596 y Fu(;)g(')2940 4608 y Fs(1)2977 4596 y Fv(\))19 b(+)3111 4492 y Ft(k)q Fk(\000)p Fs(1)3112 4517 y Fl(X)3118 4694 y Ft(i)p Fs(=1)3247 4596 y Fu(G)3312 4608 y Ft(\026)3357 4596 y Fv(\()p Fu(\025)3437 4608 y Ft(i)3465 4596 y Fv(\))1055 4859 y Fq(\000)83 b 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b(unstable)g(and)f(the)h(stable)g(P)n(oincar)n(\023)-39 b(e-Melnik)n(o)n(v)22 b(primitiv)n(e,)28 b(and)1190 509 y Fu(F)12 b Fv(\()p Fu(!)s(;)i(\022)1418 521 y Fs(0)1455 509 y Fu(;)g(')1546 521 y Fs(0)1583 509 y Fv(\))24 b(:=)f Fq(\000)p Fu(f)1856 521 y Fs(0)p Ft(;)p Fs(0)1945 509 y Fu(\022)1984 521 y Fs(0)2039 509 y Fq(\000)2247 430 y Fl(X)2122 612 y Fs(0)p Ft(<)p Fk(j)p Fs(\()p Ft(n;l)p Fs(\))p Fk(j\024)p Ft(N)2505 509 y Fu(f)2546 521 y Ft(n;l)2642 453 y Fu(e)2681 423 y Fs(i\()p Ft(n)p Fk(\001)p Ft(')2831 431 y Fd(0)2862 423 y Fs(+)p Ft(l\022)2966 431 y Fd(0)2999 423 y Fs(\))p 2642 490 387 4 v 2645 566 a Fv(i\()p Fu(n)c Fq(\001)g Fu(!)i Fv(+)d Fu(l)r Fv(\))3039 509 y Fu(;)783 b Fv(\(2.9\))236 789 y Fu(f)277 801 y Ft(n;l)386 789 y Fv(:=)23 b Fu(f)538 801 y Ft(n;l)624 789 y Fv(\(0\))28 b(b)r(eing)f(the)h(F)-7 b(ourier)27 b(co)r(e\016cien)n(ts)g(of)h Fu(f)9 b Fv(\()p Fu(';)14 b Fv(0)p Fu(;)g(t)p Fv(\).)361 912 y(Critical)30 b(p)r(oin)n(ts)h(of)g(the)h(\\reduced)e(action)g (functional")h Fq(F)2263 924 y Ft(\026)2338 912 y Fv(giv)n(e)f(rise)g (to)h(di\013usion)h(orbits)e(whose)g(action)h(v)-5 b(ari-)236 1012 y(ables)32 b Fu(I)40 b Fv(go)32 b(from)g(a)h(small)f(neigh)n(b)r (ourho)r(o)r(d)g(of)g Fu(!)1863 1024 y Ft(I)1934 1012 y Fv(to)g(a)h(small)f(neigh)n(b)r(ourho)r(o)r(d)g(of)g Fu(!)3061 1024 y Ft(F)3116 1012 y Fv(,)i(as)e(stated)h(in)g(lemma)g (2.3)236 1128 y(b)r(elo)n(w.)j(The)27 b(\\b)r(oundary)e(terms")g Fu(!)1410 1140 y Ft(I)1448 1128 y Fu(')1502 1140 y Fs(1)1556 1128 y Fq(\000)1646 1088 y Fk(j)p Ft(!)1708 1096 y Fc(I)1741 1088 y Fk(j)1761 1063 y Fd(2)p 1646 1109 148 4 v 1703 1157 a Fs(2)1803 1128 y Fu(\022)1842 1140 y Fs(1)1895 1128 y Fv(+)16 b Fu(\026)p Fv(\000)2078 1098 y Ft(u)2121 1128 y Fv(\()p Fu(!)2205 1140 y Ft(I)2243 1128 y Fu(;)e(\022)2319 1140 y Fs(1)2357 1128 y Fu(;)g(')2448 1140 y Fs(1)2485 1128 y Fv(\))i(+)g Fu(\026F)c Fv(\()p Fu(!)2813 1140 y Ft(I)2851 1128 y Fu(;)i(\022)2927 1140 y Fs(1)2964 1128 y Fu(;)g(')3055 1140 y Fs(1)3092 1128 y Fv(\))27 b(and)f Fq(\000)p Fu(!)3428 1140 y Ft(F)3483 1128 y Fu(')3537 1140 y Ft(k)3594 1128 y Fv(+)3684 1088 y Fk(j)p Ft(!)3746 1096 y Fc(F)3793 1088 y Fk(j)3813 1063 y Fd(2)p 3684 1109 162 4 v 3748 1157 a Fs(2)3855 1128 y Fu(\022)3894 1140 y Ft(k)3951 1128 y Fv(+)236 1228 y Fu(\026)p Fv(\000)338 1198 y Ft(s)374 1228 y Fv(\()p Fu(!)458 1240 y Ft(F)513 1228 y Fu(;)14 b(\022)589 1240 y Ft(k)629 1228 y Fu(;)g(')720 1240 y Ft(k)762 1228 y Fv(\))f Fq(\000)g Fu(\026F)f Fv(\()p Fu(!)1084 1240 y Ft(F)1139 1228 y Fu(;)i(\022)1215 1240 y Ft(k)1256 1228 y Fu(;)g(')1347 1240 y Ft(k)1388 1228 y Fv(\))26 b(ha)n(v)n(e)e(b)r(een)h(added)g(also)f(to)h(enable)g(us)g (to)g(\014nd)g(critical)g(p)r(oin)n(ts)g(of)g Fq(F)3629 1240 y Ft(\026)3698 1228 y Fv(w.r.t.)36 b(all)236 1327 y(the)28 b(v)-5 b(ariables)27 b(\(including)h Fu(\022)1155 1339 y Fs(1)1192 1327 y Fu(;)14 b(')1283 1339 y Fs(1)1320 1327 y Fu(;)g(\022)1396 1339 y Ft(k)1437 1327 y Fu(;)g(')1528 1339 y Ft(k)1569 1327 y Fv(\).)361 1427 y(More)37 b(precisely)-7 b(,)39 b(for)e Fu(\025)k Fv(=)e(\()p Fu(\022)r(;)14 b(')p Fv(\))41 b Fq(2)f Fv(\003)1674 1439 y Ft(\026;k)1813 1427 y Fv(w)n(e)d(de\014ne)h(the)g(pseudo)f(di\013usion)h(solutions)f (\()p Fu(')3423 1439 y Ft(\026;\025)3527 1427 y Fu(;)14 b(q)3601 1439 y Ft(\026;\025)3705 1427 y Fv(\))38 b(on)f(the)236 1527 y(in)n(terv)-5 b(al)27 b([)p Fu(\022)598 1539 y Fs(1)636 1527 y Fu(;)14 b(\022)712 1539 y Ft(k)752 1527 y Fv(])28 b(b)n(y)896 1709 y(\()p Fu(')982 1721 y Ft(\026;\025)1087 1709 y Fv(\()p Fu(t)p Fv(\))p Fu(;)14 b(q)1255 1721 y Ft(\026;\025)1359 1709 y Fv(\()p Fu(t)p Fv(\)\))24 b(:=)e(\()p Fu(')1705 1721 y Ft(\026;\025)1804 1729 y Fc(i)1836 1709 y Fv(\()p Fu(t)p Fv(\))p Fu(;)14 b(q)2004 1721 y Ft(\026;\025)2103 1729 y Fc(i)2134 1709 y Fv(\()p Fu(t)p Fv(\))19 b(+)f(2)p Fu(\031)s Fv(\()p Fu(i)g Fq(\000)h Fv(1\)\))55 b(for)g Fu(t)23 b Fq(2)g Fv([)p Fu(\022)3094 1721 y Ft(i)3122 1709 y Fu(;)14 b(\022)3198 1721 y Ft(i)p Fs(+1)3309 1709 y Fv(])p Fu(;)236 1892 y Fv(where)35 b(\()p Fu(')570 1904 y Ft(\026;\025)669 1912 y Fc(i)700 1892 y Fv(\()p Fu(t)p Fv(\))p Fu(;)14 b(q)868 1904 y Ft(\026;\025)967 1912 y Fc(i)998 1892 y Fv(\()p Fu(t)p Fv(\)\))36 b(are)e(giv)n(en)g(b)n (y)h(lemma)f(2.1.)58 b(The)35 b(pseudo)f(di\013usion)h(solutions)g(\()p Fu(')3360 1904 y Ft(\026;\025)3464 1892 y Fu(;)14 b(q)3538 1904 y Ft(\026;\025)3641 1892 y Fv(\))35 b(are)f(then)236 1992 y(con)n(tin)n(uous)25 b(functions)i(whic)n(h)f(are)g(true)g (solutions)f(of)h(the)h(equations)f(of)g(motion)g(\(2.1\))g(on)g(eac)n (h)f(in)n(terv)-5 b(al)26 b(\()p Fu(\022)3745 2004 y Ft(i)3773 1992 y Fu(;)14 b(\022)3849 2004 y Ft(i)p Fs(+1)3960 1992 y Fv(\),)236 2091 y(but)28 b(the)g(time)h(deriv)-5 b(ativ)n(es)26 b(\()d(_)-46 b Fu(')1220 2103 y Ft(\026;\025)1324 2091 y Fu(;)30 b Fv(_)-39 b Fu(q)1398 2103 y Ft(\026;\025)1502 2091 y Fv(\))28 b(ma)n(y)f(undergo)f(a)h(jump)i(at)e(time)h Fu(\022)2673 2103 y Ft(i)2701 2091 y Fv(.)37 b(W)-7 b(e)28 b(ha)n(v)n(e)236 2288 y Fw(Lemma)k(2.3)40 b Fp(If)830 2266 y Fl(e)829 2288 y Fu(\025)31 b Fv(=)e(\()1038 2266 y Fl(e)1034 2288 y Fu(\022)s(;)c Fl(e)-57 b Fu(')p Fv(\))30 b Fq(2)h Fv(\003)1373 2300 y Ft(\026;k)1507 2288 y Fp(is)j(a)g(critic)l (al)h(p)l(oint)f(of)g Fq(F)2334 2300 y Ft(\026)2378 2288 y Fp(,)h(then)f Fv(\()p Fu(')2713 2324 y Ft(\026;)2770 2319 y Fl(e)2773 2324 y Ft(\025)2817 2288 y Fv(\()p Fu(t)p Fv(\))p Fu(;)14 b(q)2985 2324 y Ft(\026;)3042 2319 y Fl(e)3045 2324 y Ft(\025)3089 2288 y Fv(\()p Fu(t)p Fv(\)\))34 b Fp(is)g(a)g(solution)g(of)h(\(2.1\))236 2426 y(in)e(the)f(time)h (interval)g Fv(\()1016 2404 y Fl(e)1012 2426 y Fu(\022)1051 2438 y Fs(1)1088 2426 y Fu(;)1130 2404 y Fl(e)1125 2426 y Fu(\022)1164 2438 y Ft(k)1205 2426 y Fv(\))p Fp(.)47 b(Mor)l(e)l(over)57 b Fv(_)-46 b Fu(')1731 2438 y Ft(\026)1776 2426 y Fv(\()1812 2404 y Fl(e)1808 2426 y Fu(\022)1847 2438 y Fs(1)1884 2426 y Fv(\))29 b(=)i Fu(!)2093 2438 y Ft(I)2152 2426 y Fv(+)20 b Fu(O)r Fv(\()p Fu(\026)p Fv(\))p Fu(;)56 b Fv(_)-46 b Fu(')2526 2438 y Ft(\026)2571 2426 y Fv(\()2607 2404 y Fl(e)2603 2426 y Fu(\022)2642 2438 y Ft(k)2683 2426 y Fv(\))28 b(=)k Fu(!)2892 2438 y Ft(F)2968 2426 y Fv(+)20 b Fu(O)r Fv(\()p Fu(\026)p Fv(\))p Fu(;)33 b Fp(i.e.)48 b Fv(\()p Fu(')3535 2462 y Ft(\026;)3592 2457 y Fl(e)3595 2462 y Ft(\025)3639 2426 y Fu(;)14 b(q)3713 2462 y Ft(\026;)3770 2457 y Fl(e)3773 2462 y Ft(\025)3817 2426 y Fv(\))33 b Fp(is)f(a)236 2563 y(di\013usion)f(orbit)f(b)l(etwe)l(en)g Fu(!)1132 2575 y Ft(I)1199 2563 y Fp(and)g Fu(!)1412 2575 y Ft(F)1497 2563 y Fp(with)g(di\013usion)h(time)f Fu(T)2257 2575 y Ft(d)2318 2563 y Fv(=)23 b Fq(j)2433 2541 y Fl(e)2429 2563 y Fu(\022)2468 2575 y Ft(k)2527 2563 y Fq(\000)2614 2541 y Fl(e)2610 2563 y Fu(\022)2649 2575 y Fs(1)2686 2563 y Fq(j)p Fp(.)236 2757 y Fb(Pr)n(oof)p Fv(.)85 b(By)30 b(lemma)g(2.2)f(if)h Fq(r)1290 2769 y Ft(')1334 2777 y Fc(i)1365 2757 y Fq(F)1425 2769 y Ft(\026)1469 2757 y Fv(\()1502 2735 y Fl(e)1501 2757 y Fu(\025)q Fv(\))d(=)f(0,)31 b(then)f(for)f(2)e Fq(\024)f Fu(i)h Fq(\024)j Fu(k)22 b Fq(\000)e Fv(1)p Fu(;)52 b Fv(_)-46 b Fu(')2725 2793 y Ft(\026;)2782 2788 y Fl(e)2785 2793 y Ft(\025)2829 2757 y Fv(\()2865 2735 y Fl(e)2861 2757 y Fu(\022)2902 2722 y Fk(\000)2900 2780 y Ft(i)2959 2757 y Fv(\))27 b(=)52 b(_)-45 b Fu(')3167 2793 y Ft(\026;)3224 2788 y Fl(e)3227 2793 y Ft(\025)3271 2757 y Fv(\()3307 2735 y Fl(e)3303 2757 y Fu(\022)3344 2722 y Fs(+)3342 2780 y Ft(i)3399 2757 y Fv(\))31 b(and)52 b(_)-45 b Fu(')3680 2793 y Ft(\026;)3737 2788 y Fl(e)3740 2793 y Ft(\025)3783 2757 y Fv(\()3819 2735 y Fl(e)3815 2757 y Fu(\022)3854 2769 y Fs(1)3892 2757 y Fv(\))27 b(=)236 2895 y Fu(!)288 2907 y Ft(I)339 2895 y Fv(+)13 b Fu(O)r Fv(\()p Fu(\026)p Fv(\),)48 b(_)-46 b Fu(')698 2931 y Ft(\026;)755 2926 y Fl(e)758 2931 y Ft(\025)802 2895 y Fv(\()838 2873 y Fl(e)834 2895 y Fu(\022)873 2907 y Ft(k)914 2895 y Fv(\))24 b(=)e Fu(!)1109 2907 y Ft(F)1177 2895 y Fv(+)13 b Fu(O)r Fv(\()p Fu(\026)p Fv(\).)36 b(Moreo)n(v)n(er,)22 b(if)k Fq(r)2021 2907 y Ft(')2065 2915 y Fc(i)2095 2895 y Fq(F)2155 2907 y Ft(\026)2199 2895 y Fv(\()2232 2873 y Fl(e)2231 2895 y Fu(\025)q Fv(\))d(=)g(0)h(and)g Fu(@)2691 2907 y Ft(\022)2723 2915 y Fc(i)2754 2895 y Fq(F)2814 2907 y Ft(\026)2858 2895 y Fv(\()2891 2873 y Fl(e)2890 2895 y Fu(\025)q Fv(\))f(=)g(0)h(then)h(\(for)f(2)f Fq(\024)f Fu(i)h Fq(\024)h Fu(k)16 b Fq(\000)d Fv(2\),)252 3032 y(_)-39 b Fu(q)276 3002 y Fs(2)273 3080 y Ft(\026;)330 3075 y Fl(e)333 3080 y Ft(\025)377 3032 y Fv(\()413 3010 y Fl(e)409 3032 y Fu(\022)450 2996 y Fs(+)448 3055 y Ft(i)506 3032 y Fv(\))23 b(=)38 b(_)-38 b Fu(q)689 3002 y Fs(2)686 3080 y Ft(\026;)743 3075 y Fl(e)746 3080 y Ft(\025)789 3032 y Fv(\()825 3010 y Fl(e)821 3032 y Fu(\022)862 2996 y Fk(\000)860 3055 y Ft(i)919 3032 y Fv(\).)37 b(No)n(w,)27 b(b)n(y)g(lemma)g(2.1)f(and)h(\(2)p Fu(:)p Fv(4\),)43 b(_)-38 b Fu(q)2158 3068 y Ft(\026;)2215 3063 y Fl(e)2218 3068 y Ft(\025)2261 3032 y Fv(\()2297 3010 y Fl(e)2293 3032 y Fu(\022)2334 2996 y Fk(\006)2332 3055 y Ft(i)2391 3032 y Fv(\))23 b(=)38 b(_)-38 b Fu(q)2571 3044 y Fs(0)2608 3032 y Fv(\(0\))18 b(+)f Fu(O)r Fv(\()p Fu(\026)p Fv(\).)38 b(Hence)43 b(_)-38 b Fu(q)3338 3068 y Ft(\026;)3395 3063 y Fl(e)3398 3068 y Ft(\025)3442 3032 y Fv(\()3478 3010 y Fl(e)3474 3032 y Fu(\022)3515 2996 y Fs(+)3513 3055 y Ft(i)3570 3032 y Fv(\))24 b(=)38 b(_)-39 b Fu(q)3750 3068 y Ft(\026;)3807 3063 y Fl(e)3810 3068 y Ft(\025)3854 3032 y Fv(\()3890 3010 y Fl(e)3886 3032 y Fu(\022)3927 2996 y Fk(\000)3925 3055 y Ft(i)3983 3032 y Fv(\))236 3157 y(and)28 b(the)g(pro)r(of)f(is)g(complete.)p 1224 3157 48 48 v 236 3481 a Fr(3)135 b(The)44 b(appro)l(ximation)j(of)e (the)g(reduced)g(functional)236 3663 y Fv(In)40 b(order)f(to)h(pro)n(v) n(e)e(the)j(existence)e(of)h(critical)g(p)r(oin)n(ts)g(of)g(the)g (reduced)g(action)f(functional)i Fq(F)3420 3675 y Ft(\026)3504 3663 y Fv(thanks)f(to)f(the)236 3763 y(prop)r(erties)c(of)h(the)h(P)n (oincar)n(\023)-39 b(e-Melnik)n(o)n(v)31 b(primitiv)n(es)k(\000\()p Fu(!)s(;)14 b Fq(\001)p Fu(;)g Fq(\001)p Fv(\))37 b(w)n(e)e(need)h(an)g (appropriate)e(expression)h(of)h Fq(F)3804 3775 y Ft(\026)3848 3763 y Fv(,)i(see)236 3862 y(lemma)d(3.5.)59 b(W)-7 b(e)36 b(shall)f(express)f Fq(F)1410 3874 y Ft(\026)1489 3862 y Fv(as)h(the)g(sum)h(of)f(a)g(function)h(whose)e(de\014nition)i(con)n (tains)e(the)i(\000\()p Fu(!)s(;)14 b Fq(\001)p Fu(;)g Fq(\001)p Fv(\))35 b(\(for)236 3962 y(whic)n(h)e(w)n(e)f(can)h(pro)n(v) n(e)e(the)i(existence)f(of)h(critical)f(p)r(oin)n(ts\))h(and)g(of)g(a)f (remainder)g(whose)g(deriv)-5 b(ativ)n(es)31 b(are)h(so)g(small)236 4062 y(that)c(it)g(cannot)f(destro)n(y)g(the)h(critical)e(p)r(oin)n(ts) i(of)g(the)g(\014rst)f(function.)361 4161 y(The)g(\014rst)h(lemma)f (giv)n(es)g(an)g(appro)n(ximation)f(of)h Fu(G)2002 4173 y Ft(\026)2075 4161 y Fv(\(de\014ned)h(in)g(\(2)p Fu(:)p Fv(6\)\).)236 4344 y Fw(Lemma)k(3.1)40 b Fp(F)-6 b(or)30 b Fv(0)23 b Fu(<)f(\026)i Fq(\024)e Fu(\026)1256 4356 y Fs(3)1293 4344 y Fp(,)31 b(for)f Fu(\025)24 b Fq(2)f Fv(\003)1689 4356 y Ft(\026)1763 4344 y Fp(we)30 b(have)317 4604 y Fu(G)382 4616 y Ft(\026)427 4604 y Fv(\()p Fu(\025)p Fv(\))24 b(=)660 4548 y(1)p 660 4585 42 4 v 660 4661 a(2)722 4548 y Fq(j)p Fu(')799 4518 y Fk(\000)874 4548 y Fq(\000)18 b Fu(')1011 4518 y Fs(+)1066 4548 y Fq(j)1089 4518 y Fs(2)p 722 4585 405 4 v 744 4661 a Fv(\()p Fu(\022)817 4637 y Fk(\000)892 4661 y Fq(\000)g Fu(\022)1016 4637 y Fs(+)1072 4661 y Fv(\))1155 4604 y(+)g Fu(\026)p Fv(\000)1340 4570 y Ft(s)1375 4604 y Fv(\()p Fu(!)1459 4616 y Ft(\025)1503 4604 y Fu(;)c(\022)1581 4570 y Fs(+)1636 4604 y Fu(;)g(')1727 4570 y Fs(+)1782 4604 y Fv(\))19 b(+)f Fu(\026)p Fv(\000)2018 4570 y Ft(u)2061 4604 y Fv(\()p Fu(!)2145 4616 y Ft(\025)2189 4604 y Fu(;)c(\022)2267 4570 y Fk(\000)2323 4604 y Fu(;)g(')2414 4570 y Fk(\000)2470 4604 y Fv(\))19 b Fq(\000)f Fu(\026)2668 4491 y Fl(Z)2751 4512 y Ft(\022)2785 4487 y Fo(\000)2714 4680 y Ft(\022)2748 4663 y Fd(+)2852 4604 y Fu(f)9 b Fv(\()p 2934 4559 55 4 v Fu(')p Fv(\()p Fu(t)p Fv(\))p Fu(;)14 b Fv(0)p Fu(;)g(t)p Fv(\))30 b Fu(dt)18 b Fv(+)g Fu(R)3527 4616 y Fs(0)3564 4604 y Fv(\()p Fu(\026;)c(\025)p Fv(\))82 b(\(3.1\))236 4831 y Fp(wher)l(e)1493 4963 y Fq(r)1562 4975 y Ft(\025)1606 4963 y Fu(R)1669 4975 y Fs(0)1706 4963 y Fv(\()p Fu(\026;)14 b(\025)p Fv(\))24 b(=)f Fu(O)2082 4871 y Fl(\020)2142 4907 y Fu(\026)2192 4877 y Fs(2)2229 4907 y Fv(\(1)c(+)f Fu(\026T)2516 4877 y Fs(2)2504 4930 y Ft(\025)2552 4907 y Fv(\))p 2142 4944 443 4 v 2318 5020 a Fu(\014)2369 4991 y Fs(2)2365 5045 y Ft(\025)2594 4963 y Fu(T)2643 4975 y Ft(\025)2686 4871 y Fl(\021)2736 4963 y Fu(:)1086 b Fv(\(3.2\))236 5202 y Fb(Pr)n(oof)p Fv(.)87 b(By)30 b(lemma)h(2.1,)f(w)n(e)g(can)h(write)f Fu(')1720 5214 y Ft(\026;\025)1824 5202 y Fv(\()p Fu(t)p Fv(\))e(=)p 2039 5157 55 4 v 28 w Fu(')p Fv(\()p Fu(t)p Fv(\))21 b(+)f Fu(v)2333 5214 y Ft(\026;\025)2437 5202 y Fv(\()p Fu(t)p Fv(\),)32 b Fu(q)2623 5214 y Ft(\026;\025)2727 5202 y Fv(\()p Fu(t)p Fv(\))c(=)g Fu(Q)3008 5214 y Ft(T)3047 5223 y Fc(\025)3090 5202 y Fv(\()p Fu(t)20 b Fq(\000)g Fu(\022)3298 5172 y Fs(+)3354 5202 y Fv(\))g(+)g Fu(w)3550 5214 y Ft(\026;\025)3654 5202 y Fv(\()p Fu(t)p Fv(\),)32 b(where)236 5302 y Fu(v)276 5314 y Ft(\026;\025)380 5302 y Fv(\()p Fu(\022)453 5272 y Fs(+)509 5302 y Fv(\))43 b(=)f Fu(v)731 5314 y Ft(\026;\025)835 5302 y Fv(\()p Fu(\022)908 5272 y Fk(\000)965 5302 y Fv(\))h(=)g(0,)f Fq(jj)12 b Fv(_)-35 b Fu(v)1341 5314 y Ft(\026;\025)1445 5302 y Fq(jj)1491 5318 y Ft(L)1537 5301 y Fo(1)1597 5318 y Fs(\()p Ft(\022)1657 5301 y Fd(+)1703 5318 y Ft(;\022)1757 5301 y Fo(\000)1805 5318 y Fs(\))1878 5302 y Fv(=)43 b Fu(O)r Fv(\()p Fu(\026=\014)2222 5314 y Ft(\025)2266 5302 y Fv(\),)g Fq(jj)p Fu(v)2450 5314 y Ft(\026;\025)2554 5302 y Fq(jj)2600 5318 y Ft(L)2646 5301 y Fo(1)2706 5318 y Fs(\()p Ft(\022)2766 5301 y Fd(+)2812 5318 y Ft(;\022)2866 5301 y Fo(\000)2914 5318 y Fs(\))2987 5302 y Fv(=)g Fu(O)r Fv(\(\()p Fu(\026=\014)3367 5272 y Fs(2)3363 5326 y Ft(\025)3407 5302 y Fv(\)\(1)27 b(+)f Fu(\026T)3742 5272 y Fs(2)3730 5326 y Ft(\025)3778 5302 y Fv(\)\))40 b(and)236 5402 y Fu(w)295 5414 y Ft(\026;\025)399 5402 y Fv(\()p Fu(\022)472 5372 y Fs(+)528 5402 y Fv(\))23 b(=)g Fu(w)730 5414 y Ft(\026;\025)834 5402 y Fv(\()p Fu(\022)907 5372 y Fk(\000)964 5402 y Fv(\))g(=)g(0,)k Fq(jj)f Fv(_)-49 b Fu(w)1304 5414 y Ft(\026;\025)1408 5402 y Fq(jj)1454 5417 y Ft(L)1500 5401 y Fo(1)1560 5417 y Fs(\()p Ft(\022)1620 5401 y Fd(+)1666 5417 y Ft(;\022)1720 5401 y Fo(\000)1769 5417 y Fs(\))1817 5402 y Fv(+)18 b Fq(jj)p Fu(w)2005 5414 y Ft(\026;\025)2109 5402 y Fq(jj)2155 5417 y Ft(L)2201 5401 y Fo(1)2262 5417 y Fs(\()p Ft(\022)2322 5401 y Fd(+)2368 5417 y Ft(;\022)2422 5401 y Fo(\000)2470 5417 y Fs(\))2523 5402 y Fv(=)23 b Fu(O)r Fv(\()p Fu(\026)p Fv(\).)2105 6059 y(8)p eop %%Page: 9 9 9 8 bop 361 272 a Fv(In)33 b(the)g(follo)n(wing,)g(in)g(order)e(to)i(a) n(v)n(oid)e(cum)n(b)r(ersome)h(notation,)i(w)n(e)e(shall)g(use)h(the)g (abbreviations)e Fu(v)s(;)14 b(w)r(;)g(Q)33 b Fv(for)236 372 y Fu(v)276 384 y Ft(\026;\025)380 372 y Fu(;)14 b(w)476 384 y Ft(\026;\025)580 372 y Fu(;)g(Q)683 384 y Ft(T)722 393 y Fc(\025)765 372 y Fv(\()p Fq(\001)19 b(\000)f Fu(\022)963 341 y Fs(+)1018 372 y Fv(\),)28 b(the)g(dep)r(endency)g(w.r.t.)37 b Fu(\025)28 b Fv(and)g Fu(\026)f Fv(b)r(eing)h(implicit.)38 b(W)-7 b(e)28 b(ha)n(v)n(e)677 632 y Fu(G)742 644 y Ft(\026)787 632 y Fv(\()p Fu(\025)p Fv(\))84 b(=)1131 519 y Fl(Z)1214 539 y Ft(\022)1248 514 y Fo(\000)1177 708 y Ft(\022)1211 691 y Fd(+)1324 576 y Fv(1)p 1324 613 42 4 v 1324 689 a(2)1376 632 y Fq(j)1414 615 y Fv(_)p 1399 586 55 4 v 1399 632 a Fu(')p Fv(\()p Fu(t)p Fv(\))p Fq(j)1570 598 y Fs(2)1626 632 y Fv(+)1725 615 y(_)p 1709 586 V 1709 632 a Fu(')q Fv(\()p Fu(t)p Fv(\))19 b Fq(\001)31 b Fv(_)-36 b Fu(v)s Fv(\()p Fu(t)p Fv(\))19 b(+)2167 576 y(1)p 2167 613 42 4 v 2167 689 a(2)2219 632 y Fq(j)2299 605 y Fv(_)2242 632 y Fu(v)s Fv(\()p Fu(t)p Fv(\))q Fq(j)2403 598 y Fs(2)2459 632 y Fv(+)2551 576 y(1)p 2551 613 V 2551 689 a(2)2631 611 y(_)2603 632 y Fu(Q)2669 598 y Fs(2)2706 632 y Fv(\()p Fu(t)p Fv(\))g(+)2930 611 y(_)2902 632 y Fu(Q)o Fv(\()p Fu(t)p Fv(\))27 b(_)-50 b Fu(w)t Fv(\()p Fu(t)p Fv(\))19 b(+)3330 576 y(1)p 3330 613 V 3330 689 a(2)3407 632 y(_)-49 b Fu(w)3442 598 y Fs(2)3480 632 y Fv(\()p Fu(t)p Fv(\))983 853 y(+)1131 740 y Fl(Z)1214 760 y Ft(\022)1248 735 y Fo(\000)1177 928 y Ft(\022)1211 912 y Fd(+)1300 853 y Fv([1)18 b Fq(\000)g Fv(cos\()p Fu(Q)p Fv(\()p Fu(t)p Fv(\))h(+)f Fu(w)r Fv(\()p Fu(t)p Fv(\)\)])i Fq(\000)e Fu(\026f)9 b Fv(\()p 2317 807 55 4 v Fu(')p Fv(\()p Fu(t)p Fv(\))19 b(+)f Fu(v)s Fv(\()p Fu(t)p Fv(\))p Fu(;)c(Q)p Fv(\()p Fu(t)p Fv(\))19 b(+)f Fu(w)r Fv(\()p Fu(t)p Fv(\))p Fu(;)c(t)p Fv(\))29 b Fu(dt:)236 1165 y Fv(No)n(w)j(since)f Fu(v)s Fv(\()p Fu(\022)753 1135 y Fs(+)809 1165 y Fv(\))g(=)f Fu(v)s Fv(\()p Fu(\022)1083 1135 y Fk(\000)1139 1165 y Fv(\))h(=)f(0)h(and)h Fu(w)r Fv(\()p Fu(\022)1670 1135 y Fs(+)1726 1165 y Fv(\))f(=)f Fu(w)r Fv(\()p Fu(\022)2018 1135 y Fk(\000)2075 1165 y Fv(\))g(=)g(0,)2330 1052 y Fl(Z)2413 1072 y Ft(\022)2447 1047 y Fo(\000)2376 1241 y Ft(\022)2410 1224 y Fd(+)2529 1148 y Fv(_)p 2513 1119 V 2513 1165 a Fu(')q Fv(\()p Fu(t)p Fv(\))22 b Fq(\001)33 b Fv(_)-35 b Fu(v)s Fv(\()p Fu(t)p Fv(\))33 b Fu(dt)d Fv(=)3096 1052 y Fl(Z)3179 1072 y Ft(\022)3213 1047 y Fo(\000)3142 1241 y Ft(\022)3176 1224 y Fd(+)3280 1165 y Fu(!)3332 1177 y Ft(\025)3396 1165 y Fq(\001)k Fv(_)-36 b Fu(v)t Fv(\()p Fu(t)p Fv(\))32 b Fu(dt)f Fv(=)e(0)j(and)236 1281 y Fl(Z)319 1302 y Ft(\022)353 1277 y Fo(\000)282 1470 y Ft(\022)316 1453 y Fd(+)448 1373 y Fv(_)420 1394 y Fu(Q)o Fv(\()p Fu(t)p Fv(\))27 b(_)-50 b Fu(w)t Fv(\()p Fu(t)p Fv(\))31 b Fu(dt)d Fv(=)961 1281 y Fl(Z)1044 1302 y Ft(\022)1078 1277 y Fo(\000)1007 1470 y Ft(\022)1041 1453 y Fd(+)1144 1394 y Fq(\000)1228 1373 y Fv(\177)1209 1394 y Fu(Q)o Fv(\()p Fu(t)p Fv(\))p Fu(w)r Fv(\()p Fu(t)p Fv(\))33 b Fu(dt)28 b Fv(=)1750 1281 y Fl(Z)1833 1302 y Ft(\022)1867 1277 y Fo(\000)1796 1470 y Ft(\022)1830 1453 y Fd(+)1933 1394 y Fq(\000)p Fv(\(sin)14 b Fu(Q)p Fv(\()p Fu(t)p Fv(\)\))p Fu(w)r Fv(\()p Fu(t)p Fv(\))32 b Fu(dt:)f Fv(As)g(a)f(result,)h Fu(G)3173 1406 y Ft(\026)3218 1394 y Fv(\()p Fu(\025)p Fv(\))e(=)e Fu(G)3516 1364 y Fs(0)3516 1415 y Ft(\026)3561 1394 y Fv(\()p Fu(\025)p Fv(\))21 b(+)f Fu(R)3842 1406 y Fs(1)3880 1394 y Fv(\()p Fu(\025)p Fv(\),)236 1536 y(where)1129 1689 y Fu(G)1194 1655 y Fs(0)1194 1710 y Ft(\026)1239 1689 y Fv(\()p Fu(\025)p Fv(\))k(=)1462 1576 y Fl(Z)1545 1597 y Ft(\022)1579 1572 y Fo(\000)1509 1765 y Ft(\022)1543 1748 y Fd(+)1656 1633 y Fv(1)p 1656 1670 42 4 v 1656 1746 a(2)1707 1689 y Fq(j)1746 1673 y Fv(_)p 1730 1643 55 4 v 1730 1689 a Fu(')q Fq(j)1808 1655 y Fs(2)1864 1689 y Fv(+)1957 1633 y(1)p 1957 1670 42 4 v 1957 1746 a(2)2036 1668 y(_)2008 1689 y Fu(Q)2074 1655 y Fs(2)2129 1689 y Fv(+)18 b(\(1)h Fq(\000)f Fv(cos)13 b Fu(Q)p Fv(\))18 b Fq(\000)g Fu(\026f)9 b Fv(\()p 2844 1643 55 4 v Fu(';)14 b(Q;)g(t)p Fv(\))p Fu(;)382 1968 y(R)445 1980 y Fs(1)482 1968 y Fv(\()p Fu(\025)p Fv(\))24 b(=)706 1855 y Fl(Z)789 1876 y Ft(\022)823 1851 y Fo(\000)752 2044 y Ft(\022)786 2027 y Fd(+)899 1912 y Fv(1)p 899 1949 42 4 v 899 2025 a(2)951 1968 y Fq(j)12 b Fv(_)-35 b Fu(v)s Fq(j)1040 1934 y Fs(2)1096 1968 y Fv(+)1189 1912 y(1)p 1189 1949 V 1189 2025 a(2)1267 1968 y(_)-50 b Fu(w)1301 1934 y Fs(2)1358 1968 y Fv(+)18 b(\(cos)13 b Fu(Q)18 b Fq(\000)g Fv(cos)o(\()p Fu(Q)h Fv(+)f Fu(w)r Fv(\))h Fq(\000)f Fu(w)f Fv(sin)c Fu(Q)p Fv(\))19 b Fq(\000)f Fu(\026f)9 b Fv(\()p 2794 1922 55 4 v Fu(')18 b Fv(+)h Fu(v)s(;)14 b(Q)k Fv(+)g Fu(w)r(;)c(t)p Fv(\))19 b(+)f Fu(\026f)9 b Fv(\()p 3591 1922 V Fu(';)14 b(Q;)g(t)p Fv(\))p Fu(:)236 2197 y Fv(W)-7 b(e)27 b(shall)f(\014rst)g(pro)n(v)n(e) f(that)h Fq(jr)p Fu(R)1297 2209 y Fs(1)1335 2197 y Fq(j)d Fv(=)g Fu(O)1534 2105 y Fl(\020)1594 2155 y Ft(\026)1634 2130 y Fd(2)1667 2155 y Fs(\(1+)p Ft(\026T)1865 2130 y Fd(2)1856 2172 y Fc(\025)1898 2155 y Fs(\))p 1594 2178 330 4 v 1720 2228 a Ft(\014)1761 2208 y Fd(2)1758 2250 y Fc(\025)1933 2197 y Fu(T)1982 2209 y Ft(\025)2026 2105 y Fl(\021)2075 2197 y Fv(.)37 b(W)-7 b(e)26 b(ha)n(v)n(e)g Fu(@)2511 2213 y Ft(\022)2545 2196 y Fd(+)2595 2197 y Fu(R)2658 2209 y Fs(1)2718 2197 y Fv(=)d Fu(r)2843 2209 y Ft(s)2879 2197 y Fv(1)16 b(+)f Fu(r)3054 2209 y Fs(2)3108 2197 y Fv(+)g Fu(r)3225 2209 y Fs(3)3279 2197 y Fv(+)h Fu(r)3397 2209 y Fs(4)3450 2197 y Fv(+)g Fu(r)3568 2209 y Fs(5)3622 2197 y Fv(+)f Fu(r)3739 2209 y Fs(6)3803 2197 y Fv(where)711 2488 y Fu(r)748 2500 y Fs(1)809 2488 y Fv(:=)920 2375 y Fl(Z)1003 2395 y Ft(\022)1037 2370 y Fo(\000)966 2564 y Ft(\022)1000 2547 y Fd(+)1116 2488 y Fv(_)-36 b Fu(v)22 b Fq(\001)1232 2432 y Fu(d)p 1217 2469 74 4 v 1217 2545 a(dt)1300 2488 y Fv(\()p Fu(@)1376 2504 y Ft(\022)1410 2487 y Fd(+)1460 2488 y Fu(v)s Fv(\))d Fq(\000)f Fu(\026@)1731 2500 y Ft(')1779 2488 y Fu(f)9 b Fv(\()p 1861 2442 55 4 v Fu(')19 b Fv(+)f Fu(v)s(;)c(Q)k Fv(+)g Fu(w)r(;)c(t)p Fv(\))20 b Fq(\001)e Fv(\()p Fu(@)2561 2504 y Ft(\022)2595 2487 y Fd(+)2646 2488 y Fu(v)s Fv(\))p Fu(;)711 2709 y(r)748 2721 y Fs(2)809 2709 y Fv(:=)920 2596 y Fl(Z)1003 2616 y Ft(\022)1037 2591 y Fo(\000)966 2784 y Ft(\022)1000 2768 y Fd(+)1130 2709 y Fv(_)-50 b Fu(w)1190 2653 y(d)p 1175 2690 74 4 v 1175 2766 a(dt)1258 2709 y Fv(\()p Fu(@)1334 2724 y Ft(\022)1368 2708 y Fd(+)1419 2709 y Fu(w)r Fv(\))19 b(+)1614 2617 y Fl(h)1667 2709 y Fv(sin\()p Fu(Q)g Fv(+)f Fu(w)r Fv(\))h Fq(\000)f Fv(sin)c Fu(Q)k Fq(\000)g Fu(\026@)2541 2721 y Ft(q)2578 2709 y Fu(f)9 b Fv(\()p 2660 2663 55 4 v Fu(')18 b Fv(+)g Fu(v)s(;)c(Q)19 b Fv(+)f Fu(w)r(;)c(t)p Fv(\))3223 2617 y Fl(i)3263 2709 y Fv(\()p Fu(@)3339 2724 y Ft(\022)3373 2708 y Fd(+)3423 2709 y Fu(w)r Fv(\))p Fu(;)711 2930 y(r)748 2942 y Fs(3)809 2930 y Fv(:=)920 2817 y Fl(Z)1003 2837 y Ft(\022)1037 2812 y Fo(\000)966 3005 y Ft(\022)1000 2989 y Fd(+)1090 2930 y Fv(\()p Fq(\000)g Fv(sin)f Fu(Q)18 b Fv(+)g(sin\()p Fu(Q)g Fv(+)h Fu(w)r Fv(\))g Fq(\000)f Fu(w)e Fv(cos)d Fu(Q)p Fv(\))p Fu(@)2322 2945 y Ft(\022)2356 2929 y Fd(+)2407 2930 y Fu(Q;)711 3150 y(r)748 3162 y Fs(4)809 3150 y Fv(:=)23 b Fu(\026)984 3037 y Fl(Z)1067 3058 y Ft(\022)1101 3033 y Fo(\000)1030 3226 y Ft(\022)1064 3209 y Fd(+)1167 3058 y Fl(h)1207 3150 y Fu(@)1251 3162 y Ft(')1298 3150 y Fu(f)9 b Fv(\()p 1380 3105 V Fu(')q(;)14 b(Q;)g(t)p Fv(\))k Fq(\000)g Fu(@)1782 3162 y Ft(')1830 3150 y Fu(f)9 b Fv(\()p 1912 3105 V Fu(')18 b Fv(+)g Fu(v)s(;)c(Q)19 b Fv(+)f Fu(w)r(;)c(t)p Fv(\))2475 3058 y Fl(i)2533 3150 y Fq(\001)19 b Fu(@)2619 3166 y Ft(\022)2653 3150 y Fd(+)p 2703 3105 V 7 w Fu(';)711 3371 y(r)748 3383 y Fs(5)809 3371 y Fv(:=)k Fu(\026)984 3258 y Fl(Z)1067 3279 y Ft(\022)1101 3254 y Fo(\000)1030 3447 y Ft(\022)1064 3430 y Fd(+)1167 3279 y Fl(h)1207 3371 y Fu(@)1251 3383 y Ft(q)1287 3371 y Fu(f)9 b Fv(\()p 1369 3326 V Fu(';)14 b(Q;)g(t)p Fv(\))19 b Fq(\000)f Fu(@)1771 3383 y Ft(q)1807 3371 y Fu(f)9 b Fv(\()p 1889 3326 V Fu(')19 b Fv(+)f Fu(v)s(;)c(Q)k Fv(+)g Fu(w)r(;)c(t)p Fv(\))2452 3279 y Fl(i)2492 3371 y Fu(@)2536 3387 y Ft(\022)2570 3370 y Fd(+)2621 3371 y Fu(Q;)711 3557 y(r)748 3569 y Fs(6)809 3557 y Fv(:=)23 b Fq(\000)995 3500 y Fv(1)p 995 3538 42 4 v 995 3614 a(2)1046 3557 y Fq(j)12 b Fv(_)-35 b Fu(v)s Fv(\()p Fu(\022)1185 3522 y Fs(+)1241 3557 y Fv(\))p Fq(j)1296 3522 y Fs(2)1352 3557 y Fq(\000)1445 3500 y Fv(1)p 1445 3538 V 1445 3614 a(2)1523 3557 y(_)-50 b Fu(w)s Fv(\()p Fu(\022)1631 3522 y Fs(+)1687 3557 y Fv(\))1719 3522 y Fs(2)1756 3557 y Fu(:)236 3763 y Fv(No)n(w)27 b Fu(v)k Fv(and)d Fu(w)i Fv(satisfy)702 3873 y Fl(\032)1510 3940 y Fq(\000)s Fv(\177)-45 b Fu(v)s Fv(\()p Fu(t)p Fv(\))84 b(=)e Fu(\026@)2037 3952 y Ft(')2085 3940 y Fu(f)9 b Fv(\()p 2167 3894 55 4 v Fu(')p Fv(\()p Fu(t)p Fv(\))19 b(+)f Fu(v)s Fv(\()p Fu(t)p Fv(\))p Fu(;)c(Q)p Fv(\()p Fu(t)p Fv(\))20 b(+)e Fu(w)r Fv(\()p Fu(t)p Fv(\))p Fu(;)c(t)p Fv(\))806 4039 y Fq(\000)i Fv(\177)-58 b Fu(w)r Fv(\()p Fu(t)p Fv(\))19 b(+)f(sin\()p Fu(Q)p Fv(\()p Fu(t)p Fv(\))h(+)f Fu(w)r Fv(\()p Fu(t)p Fv(\)\))85 b(=)d Fu(\026@)2037 4051 y Ft(q)2074 4039 y Fu(f)9 b Fv(\()p 2156 3994 V Fu(')p Fv(\()p Fu(t)p Fv(\))19 b(+)f Fu(v)s Fv(\()p Fu(t)p Fv(\))p Fu(;)c(Q)p Fv(\()p Fu(t)p Fv(\))19 b(+)f Fu(w)r Fv(\()p Fu(t)p Fv(\))p Fu(;)c(t)p Fv(\))20 b(+)e(sin)c Fu(Q)p Fv(\()p Fu(t)p Fv(\))p Fu(:)236 4227 y Fv(Moreo)n(v)n(er,)34 b(deriving)f(w.r.t.)58 b Fu(\022)1253 4197 y Fs(+)1343 4227 y Fv(the)35 b(equalit)n(y)e Fu(v)s Fv(\()p Fu(\022)1931 4197 y Fs(+)1987 4227 y Fv(\))i(=)f(0)g(w)n(e)g (obtain)g(that)h(\()p Fu(@)2886 4242 y Ft(\022)2920 4226 y Fd(+)2971 4227 y Fu(v)s Fv(\)\()p Fu(\022)3119 4197 y Fs(+)3175 4227 y Fv(\))g(=)f Fq(\000)12 b Fv(_)-35 b Fu(v)s Fv(\()p Fu(\022)3522 4197 y Fs(+)3578 4227 y Fv(\).)57 b(Similarly)236 4326 y(\()p Fu(@)312 4342 y Ft(\022)346 4325 y Fd(+)397 4326 y Fu(w)r Fv(\)\()p Fu(\022)563 4296 y Fs(+)620 4326 y Fv(\))33 b(=)g Fq(\000)26 b Fv(_)-49 b Fu(w)r Fv(\()p Fu(\022)982 4296 y Fs(+)1038 4326 y Fv(\),)35 b(\()p Fu(@)1204 4342 y Ft(\022)1238 4325 y Fd(+)1289 4326 y Fu(v)s Fv(\)\()p Fu(\022)1437 4296 y Fk(\000)1494 4326 y Fv(\))f(=)e(0)i(and)f(\()p Fu(@)1976 4342 y Ft(\022)2010 4325 y Fd(+)2061 4326 y Fu(w)r Fv(\)\()p Fu(\022)2227 4296 y Fk(\000)2285 4326 y Fv(\))g(=)g(0.)55 b(Therefore)32 b(an)h(in)n(tegration)g(b)n(y)g(parts)g(giv)n(es)236 4426 y Fu(r)273 4438 y Fs(1)334 4426 y Fv(=)23 b Fq(j)12 b Fv(_)-35 b Fu(v)s Fv(\()p Fu(\022)561 4396 y Fs(+)617 4426 y Fv(\))p Fq(j)672 4396 y Fs(2)709 4426 y Fv(,)28 b Fu(r)797 4438 y Fs(2)858 4426 y Fv(=)49 b(_)-50 b Fu(w)s Fv(\()p Fu(\022)1080 4396 y Fs(+)1136 4426 y Fv(\))1168 4396 y Fs(2)1233 4426 y Fv(hence)27 b Fq(j)p Fu(r)1523 4438 y Fs(1)1561 4426 y Fq(j)19 b Fv(+)f Fq(j)p Fu(r)1746 4438 y Fs(2)1784 4426 y Fq(j)23 b Fv(=)f Fu(O)r Fv(\()p Fu(\026)2064 4396 y Fs(2)2102 4426 y Fu(=\014)2195 4396 y Fs(2)2233 4426 y Fv(\).)361 4526 y(By)32 b(the)h(prop)r(erties)e(of)i Fu(Q)1205 4538 y Ft(T)1257 4526 y Fv(,)g Fu(@)1357 4541 y Ft(\022)1391 4525 y Fd(+)1442 4526 y Fu(Q)f Fv(is)g(b)r(ounded)h(in)g (the)g(in)n(terv)-5 b(al)31 b([)p Fu(\022)2590 4495 y Fs(+)2646 4526 y Fu(;)14 b(\022)2724 4495 y Fk(\000)2780 4526 y Fv(])33 b(b)n(y)f(a)g(constan)n(t)g(indep)r(enden)n(t)h(of)f Fu(\025)p Fv(.)236 4625 y(Moreo)n(v)n(er)25 b Fq(\000)14 b Fv(sin)f Fu(Q)p Fv(\()p Fu(t)p Fv(\))19 b(+)f(sin\()p Fu(Q)p Fv(\()p Fu(t)p Fv(\))h(+)f Fu(w)r Fv(\()p Fu(t)p Fv(\)\))i Fq(\000)e Fu(w)r Fv(\()p Fu(t)p Fv(\))c(cos)g Fu(Q)p Fv(\()p Fu(t)p Fv(\))23 b(=)g Fu(O)r Fv(\()p Fu(w)r Fv(\()p Fu(t)p Fv(\))2561 4595 y Fs(2)2600 4625 y Fv(\).)37 b(Therefore)27 b Fu(r)3106 4637 y Fs(3)3166 4625 y Fv(=)c Fu(O)r Fv(\()p Fu(\026)3401 4595 y Fs(2)3439 4625 y Fu(T)12 b Fv(\).)361 4725 y(W)-7 b(e)28 b(ha)n(v)n(e)e(also,)h(for)g(some)g(p)r (ositiv)n(e)g(constan)n(t)g Fu(c)p Fv(,)770 4928 y Fq(j)p Fu(r)830 4940 y Fs(4)867 4928 y Fq(j)19 b Fv(+)f Fq(j)p Fu(r)1052 4940 y Fs(5)1090 4928 y Fq(j)23 b(\024)f Fu(c\026T)1370 4836 y Fl(h)1505 4928 y Fv(sup)1422 5002 y Ft(t)p Fk(2)p Fs([)p Ft(\022)1545 4985 y Fd(+)1591 5002 y Ft(;\022)1645 4985 y Fo(\000)1693 5002 y Fs(])1726 4928 y Fq(j)p Fu(@)1793 4943 y Ft(\022)1827 4927 y Fd(+)1878 4928 y Fu(Q)p Fv(\()p Fu(t)p Fv(\))p Fq(j)c Fv(+)g Fq(j)p Fu(@)2229 4943 y Ft(\022)2263 4927 y Fd(+)p 2314 4882 V 2314 4928 a Fu(')p Fv(\()p Fu(t)p Fv(\))p Fq(j)2485 4836 y Fl(i)q(h)2660 4928 y Fv(sup)2578 5002 y Ft(t)p Fk(2)p Fs([)p Ft(\022)2701 4985 y Fd(+)2747 5002 y Ft(;\022)2801 4985 y Fo(\000)2849 5002 y Fs(])2868 4928 y Fv(\()p Fq(j)p Fu(v)s Fv(\()p Fu(t)p Fv(\))p Fq(j)i Fv(+)e Fq(j)p Fu(w)r Fv(\()p Fu(t)p Fv(\))p Fq(j)p Fv(\))3419 4836 y Fl(i)3459 4928 y Fu(:)236 5212 y Fv(Since)27 b Fu(@)496 5227 y Ft(\022)530 5211 y Fd(+)p 581 5166 V 581 5212 a Fu(')g Fv(is)f(b)r(ounded)i(indep)r (enden)n(tly)f(of)g Fu(\025)p Fv(,)g(w)n(e)g(ha)n(v)n(e)e(b)n(y)i (lemma)f(2.1)g Fq(j)p Fu(r)2697 5224 y Fs(4)2735 5212 y Fq(j)17 b Fv(+)f Fq(j)p Fu(r)2916 5224 y Fs(5)2954 5212 y Fq(j)23 b Fv(=)g Fu(O)3153 5119 y Fl(\020)3213 5169 y Ft(\026)3253 5144 y Fd(2)3286 5169 y Fs(\(1+)p Ft(\026T)3484 5144 y Fd(2)3475 5186 y Fc(\025)3516 5169 y Fs(\))p 3213 5193 330 4 v 3339 5243 a Ft(\014)3380 5223 y Fd(2)3377 5265 y Fc(\025)3552 5212 y Fu(T)3601 5224 y Ft(\025)3644 5119 y Fl(\021)3694 5212 y Fv(.)37 b(Still)27 b(b)n(y)236 5341 y(lemma)h(2.1,)f Fu(r)698 5353 y Fs(6)758 5341 y Fv(=)c Fu(O)r Fv(\()p Fu(\026)993 5311 y Fs(2)1031 5341 y Fu(=\014)1124 5311 y Fs(2)1161 5341 y Fv(\).)37 b(The)28 b(estimate)g(of)f(the)h(other)f(deriv)-5 b(ativ)n(es)27 b(of)g Fu(R)2782 5353 y Fs(1)2847 5341 y Fv(is)h(obtained)f(in)h(the)g(same)f(w)n(a)n(y)-7 b(.)2105 6059 y(9)p eop %%Page: 10 10 10 9 bop 236 272 a Fv(inden)n(t)28 b(W)-7 b(e)28 b(no)n(w)f(dev)n(elop) g Fu(G)1173 242 y Fs(0)1173 293 y Ft(\026)1218 272 y Fv(\()p Fu(\025)p Fv(\))h(as)278 540 y Fu(G)343 506 y Fs(0)343 561 y Ft(\026)388 540 y Fv(\()p Fu(\025)p Fv(\))c(=)621 484 y(1)p 621 521 42 4 v 621 597 a(2)683 484 y Fq(j)p Fu(')760 454 y Fk(\000)835 484 y Fq(\000)18 b Fu(')972 454 y Fs(+)1027 484 y Fq(j)1050 454 y Fs(2)p 683 521 405 4 v 705 597 a Fv(\()p Fu(\022)778 573 y Fk(\000)853 597 y Fq(\000)g Fu(\022)977 573 y Fs(+)1033 597 y Fv(\))1116 540 y(+)g Fu(\026)p Fv(\000)1301 506 y Ft(s)1336 540 y Fv(\()p Fu(!)1420 552 y Ft(\025)1464 540 y Fu(;)c(\022)1542 506 y Fs(+)1597 540 y Fu(;)g(')1688 506 y Fs(+)1743 540 y Fv(\))19 b(+)f Fu(\026)p Fv(\000)1979 506 y Ft(u)2022 540 y Fv(\()p Fu(!)2106 552 y Ft(\025)2150 540 y Fu(;)c(\022)2228 506 y Fk(\000)2284 540 y Fu(;)g(')2375 506 y Fk(\000)2431 540 y Fv(\))19 b Fq(\000)f Fu(\026)2629 427 y Fl(Z)2712 448 y Ft(\022)2746 423 y Fo(\000)2675 616 y Ft(\022)2709 599 y Fd(+)2812 540 y Fu(f)9 b Fv(\()p 2894 495 55 4 v Fu(')q Fv(\()p Fu(t)p Fv(\))p Fu(;)14 b Fv(0)p Fu(;)g(t)p Fv(\))27 b Fu(dt)19 b Fv(+)f Fu(R)3486 552 y Fs(2)3523 540 y Fv(\()p Fu(\025)p Fv(\))i(+)e Fu(R)3801 552 y Fs(3)3838 540 y Fv(\()p Fu(\025)p Fv(\))p Fu(;)236 767 y Fv(where)698 921 y Fu(R)761 933 y Fs(2)798 921 y Fv(\()p Fu(\025)p Fv(\))24 b(=)1022 808 y Fl(Z)1105 828 y Ft(\022)1139 803 y Fo(\000)1068 996 y Ft(\022)1102 980 y Fd(+)1215 864 y Fv(1)p 1215 901 42 4 v 1215 977 a(2)1295 900 y(_)1267 921 y Fu(Q)1333 886 y Fs(2)1370 921 y Fv(\()p Fu(t)p Fv(\))19 b(+)f(\(1)g Fq(\000)g Fv(cos)13 b Fu(Q)p Fv(\()p Fu(t)p Fv(\)\))28 b Fu(dt)c Fv(=)2270 808 y Fl(Z)2353 828 y Ft(T)2392 837 y Fc(\025)2316 996 y Fs(0)2459 864 y Fv(1)p 2459 901 V 2459 977 a(2)2539 900 y(_)2511 921 y Fu(Q)2577 886 y Fs(2)2577 941 y Ft(T)2616 950 y Fc(\025)2659 921 y Fv(\()p Fu(t)p Fv(\))19 b(+)f(\(1)g Fq(\000)g Fv(cos)13 b Fu(Q)3221 933 y Ft(T)3260 942 y Fc(\025)3303 921 y Fv(\()p Fu(t)p Fv(\)\))29 b Fu(dt;)291 b Fv(\(3.3\))496 1200 y Fu(R)559 1212 y Fs(3)596 1200 y Fv(\()p Fu(\025)p Fv(\))24 b(=)820 1087 y Fl(Z)903 1107 y Ft(\022)937 1082 y Fo(\000)866 1275 y Ft(\022)900 1259 y Fd(+)1004 1200 y Fq(\000)p Fu(\026)1119 1107 y Fl(h)1157 1200 y Fv(\()p Fu(f)9 b Fv(\()p 1271 1154 55 4 v Fu(')q Fv(\()p Fu(t)p Fv(\))p Fu(;)14 b(Q)p Fv(\()p Fu(t)p Fv(\))p Fu(;)g(t)p Fv(\))19 b Fq(\000)f Fu(f)9 b Fv(\()p 1900 1154 V Fu(')p Fv(\()p Fu(t)p Fv(\))p Fu(;)14 b Fv(0)p Fu(;)g(t)p Fv(\))2226 1107 y Fl(i)2293 1200 y Fu(dt)19 b Fq(\000)f Fu(\026)p Fv(\000)2570 1165 y Ft(s)2605 1200 y Fv(\()p Fu(!)2689 1212 y Ft(\025)2733 1200 y Fu(;)c(\022)2811 1165 y Fs(+)2866 1200 y Fu(;)g(')2957 1165 y Fs(+)3013 1200 y Fv(\))k Fq(\000)g Fu(\026)p Fv(\000)3248 1165 y Ft(u)3292 1200 y Fv(\()p Fu(!)3376 1212 y Ft(\025)3419 1200 y Fu(;)c(\022)3497 1165 y Fk(\000)3553 1200 y Fu(;)g(')3644 1165 y Fk(\000)3701 1200 y Fv(\))p Fu(:)236 1396 y Fv(There)19 b(remains)f(to)h(pro)n(v)n (e)e(estimate)i(\(3.2\))g(for)g Fq(r)p Fu(R)1842 1408 y Fs(2)1898 1396 y Fv(and)g Fq(r)p Fu(R)2183 1408 y Fs(3)2221 1396 y Fv(.)34 b(By)18 b(\(3.3\))h Fu(@)2633 1411 y Ft(')2677 1395 y Fo(\006)2730 1396 y Fu(R)2793 1408 y Fs(2)2854 1396 y Fv(=)j(0)d(and)g Fu(@)3199 1411 y Ft(\022)3233 1395 y Fd(+)3283 1396 y Fu(R)3346 1408 y Fs(2)3383 1396 y Fv(\()p Fu(\025)p Fv(\))24 b(=)f Fq(\000)p Fu(@)3716 1411 y Ft(\022)3750 1395 y Fo(\000)3802 1396 y Fu(R)3865 1408 y Fs(2)3903 1396 y Fv(\()p Fu(\025)p Fv(\))236 1495 y(is)30 b(the)g(energy)e(of)i(the)f Fu(T)1024 1507 y Ft(\025)1068 1495 y Fv(-p)r(erio)r(dic)g(solution)g Fu(Q)1798 1507 y Ft(T)1837 1516 y Fc(\025)1909 1495 y Fv(of)h(the)g(p)r(endulum)h (equation.)42 b(No)n(w)29 b(this)h(energy)e(is)h Fu(O)r Fv(\()p Fu(e)3763 1465 y Fk(\000)p Ft(c)3845 1473 y Fd(2)3878 1465 y Ft(T)3917 1474 y Fc(\025)3960 1495 y Fv(\).)236 1595 y(Hence)f(\(pro)n(vided)f Fu(C)916 1607 y Fs(1)981 1595 y Fv(is)h(large)e(enough\))h Fq(jr)p Fu(R)1741 1607 y Fs(2)1779 1595 y Fv(\()p Fu(\025)p Fv(\))p Fq(j)d Fv(=)e Fu(O)r Fv(\()p Fu(\026)2172 1565 y Fs(2)2210 1595 y Fv(\).)361 1694 y(In)28 b(order)e(to)h(estimate)h(the)g(deriv)-5 b(ativ)n(es)26 b(of)i Fu(R)1830 1706 y Fs(3)1867 1694 y Fv(,)g(let)g(us)f(de\014ne)h Fu(g)s Fv(\()p Fu(';)14 b(q)s(;)g(t)p Fv(\))23 b(:=)g Fu(f)9 b Fv(\()p Fu(';)14 b(q)s(;)g(t)p Fv(\))19 b Fq(\000)f Fu(f)9 b Fv(\()p Fu(';)14 b Fv(0)p Fu(;)g(t)p Fv(\).)37 b(W)-7 b(e)28 b(ha)n(v)n(e)468 1952 y Fu(R)531 1964 y Fs(3)568 1952 y Fv(\()p Fu(\025)p Fv(\))c(=)792 1839 y Fl(Z)875 1859 y Ft(\022)909 1834 y Fo(\000)838 2027 y Ft(\022)872 2011 y Fd(+)975 1952 y Fq(\000)p Fu(\026g)s Fv(\()p 1165 1906 V Fu(')p Fv(\()p Fu(t)p Fv(\))p Fu(;)14 b(Q)p Fv(\()p Fu(t)p Fv(\))p Fu(;)g(t)p Fv(\))29 b Fu(dt)18 b Fq(\000)g Fu(\026)p Fv(\000)1914 1917 y Ft(s)1950 1952 y Fv(\()p Fu(!)2034 1964 y Ft(\025)2077 1952 y Fu(;)c(\022)2155 1917 y Fs(+)2211 1952 y Fu(;)g(')2302 1917 y Fs(+)2357 1952 y Fv(\))19 b Fq(\000)f Fu(\026)p Fv(\000)2593 1917 y Ft(u)2636 1952 y Fv(\()p Fu(!)2720 1964 y Ft(\025)2763 1952 y Fu(;)c(\022)2841 1917 y Fk(\000)2898 1952 y Fu(;)g(')2989 1917 y Fk(\000)3045 1952 y Fv(\))23 b(=)g Fu(\026)p Fv(\()p Fu(a)3314 1964 y Fs(3)3351 1952 y Fv(\()p Fu(\025)p Fv(\))d(+)e Fu(b)3602 1964 y Fs(3)3639 1952 y Fv(\()p Fu(\025)p Fv(\)\))236 2176 y(where)602 2404 y Fu(a)646 2416 y Fs(3)683 2404 y Fv(\()p Fu(\025)p Fv(\))25 b(:=)d Fq(\000)1009 2291 y Fl(Z)1092 2311 y Ft(T)1131 2320 y Fc(\025)1170 2311 y Ft(=)p Fs(2)1055 2479 y(0)1255 2404 y Fu(g)s Fv(\()p Fu(!)1382 2416 y Ft(\025)1425 2404 y Fu(t)c Fv(+)h Fu(')1611 2369 y Fs(+)1666 2404 y Fu(;)14 b(Q)1769 2416 y Ft(T)1808 2425 y Fc(\025)1851 2404 y Fv(\()p Fu(t)p Fv(\))p Fu(;)g(t)19 b Fv(+)f Fu(\022)2155 2369 y Fs(+)2210 2404 y Fv(\))28 b Fu(dt)19 b Fv(+)2445 2291 y Fl(Z)2528 2311 y Fk(1)2491 2479 y Fs(0)2612 2404 y Fu(g)s Fv(\()p Fu(!)2739 2416 y Ft(\025)2782 2404 y Fu(t)f Fv(+)h Fu(')2968 2369 y Fs(+)3023 2404 y Fu(;)14 b(q)3097 2416 y Fs(0)3134 2404 y Fv(\()p Fu(t)p Fv(\))p Fu(;)g(t)19 b Fv(+)f Fu(\022)3438 2369 y Fs(+)3493 2404 y Fv(\))28 b Fu(dt;)493 2701 y(b)529 2713 y Fs(3)566 2701 y Fv(\()p Fu(\025)p Fv(\))c(:=)e Fq(\000)891 2588 y Fl(Z)974 2608 y Fs(0)937 2776 y Fk(\000)p Ft(T)1028 2785 y Fc(\025)1067 2776 y Ft(=)p Fs(2)1152 2701 y Fu(g)s Fv(\()p Fu(!)1279 2713 y Ft(\025)1322 2701 y Fu(t)d Fv(+)f Fu(')1508 2666 y Fk(\000)1564 2701 y Fu(;)c(Q)1667 2713 y Ft(T)1706 2722 y Fc(\025)1749 2701 y Fv(\()p Fu(t)19 b Fv(+)f Fu(T)1962 2713 y Ft(\025)2005 2701 y Fv(\))p Fu(;)c(t)k Fv(+)g Fu(\022)2246 2666 y Fk(\000)2303 2701 y Fv(\))28 b Fu(dt)18 b Fv(+)2537 2588 y Fl(Z)2620 2608 y Fs(0)2584 2776 y Fk(\0001)2720 2701 y Fu(g)s Fv(\()p Fu(!)2847 2713 y Ft(\025)2890 2701 y Fu(t)g Fv(+)g Fu(')3075 2666 y Fk(\000)3131 2701 y Fu(;)c(q)3205 2713 y Fs(0)3243 2701 y Fv(\()p Fu(t)p Fv(\))p Fu(;)g(t)19 b Fv(+)f Fu(\022)3547 2666 y Fk(\000)3603 2701 y Fv(\))28 b Fu(dt:)236 2907 y Fv(W)-7 b(e)28 b(ha)n(v)n(e)236 3134 y Fu(a)280 3146 y Fs(3)317 3134 y Fv(\()p Fu(\025)p Fv(\))c(=)f Fq(\000)620 3021 y Fl(Z)702 3041 y Ft(T)741 3050 y Fc(\025)781 3041 y Ft(=)p Fs(2)666 3210 y(0)866 3042 y Fl(h)905 3134 y Fu(g)s Fv(\()p Fu(!)1032 3146 y Ft(\025)1075 3134 y Fu(t)7 b Fv(+)g Fu(')1238 3100 y Fs(+)1294 3134 y Fu(;)14 b(Q)1397 3146 y Ft(T)1436 3155 y Fc(\025)1479 3134 y Fv(\()p Fu(t)p Fv(\))p Fu(;)g(t)7 b Fv(+)g Fu(\022)1760 3100 y Fs(+)1815 3134 y Fv(\))g Fq(\000)g Fu(g)s Fv(\()p Fu(!)2053 3146 y Ft(\025)2097 3134 y Fu(t)g Fv(+)g Fu(')2260 3100 y Fs(+)2315 3134 y Fu(;)14 b(q)2389 3146 y Fs(0)2427 3134 y Fv(\()p Fu(t)p Fv(\))p Fu(;)g(t)7 b Fv(+)g Fu(\022)2708 3100 y Fs(+)2764 3134 y Fv(\))2796 3042 y Fl(i)2842 3134 y Fv(+)2914 3021 y Fl(Z)2997 3041 y Fk(1)2960 3210 y Ft(T)2999 3219 y Fc(\025)3039 3210 y Ft(=)p Fs(2)3124 3134 y Fu(g)s Fv(\()p Fu(!)3251 3146 y Ft(\025)3294 3134 y Fu(t)g Fv(+)g Fu(')3457 3100 y Fs(+)3512 3134 y Fu(;)14 b(q)3586 3146 y Fs(0)3623 3134 y Fv(\()p Fu(t)p Fv(\))p Fu(;)g(t)7 b Fv(+)g Fu(\022)3904 3100 y Fs(+)3960 3134 y Fv(\))p Fu(:)236 3385 y Fv(Recalling)32 b(that)h(sup)912 3406 y Ft(t)p Fk(2)p Fs(\(0)p Ft(;T)7 b(=)p Fs(2\))1218 3385 y Fq(j)p Fu(@)1285 3397 y Ft(T)1337 3385 y Fu(Q)1403 3397 y Ft(T)1455 3385 y Fv(\()p Fu(t)p Fv(\))p Fq(j)32 b Fv(=)f Fu(O)r Fv(\()p Fu(e)1836 3355 y Fk(\000)p Ft(c)1918 3363 y Fd(2)1951 3355 y Ft(T)2003 3385 y Fv(\),)j(sup)2217 3406 y Ft(t)p Fk(2)p Fs(\(0)p Ft(;T)7 b(=)p Fs(2\))2523 3385 y Fq(j)p Fu(Q)2612 3397 y Ft(T)2664 3385 y Fv(\()p Fu(t)p Fv(\))22 b Fq(\000)g Fu(q)2904 3397 y Fs(0)2941 3385 y Fv(\()p Fu(t)p Fv(\))p Fq(j)32 b Fv(=)f Fu(O)r Fv(\()p Fu(e)3322 3355 y Fk(\000)p Ft(c)3404 3363 y Fd(2)3436 3355 y Ft(T)3489 3385 y Fv(\),)j(it)f(is)f(easy)g(to)236 3498 y(see)d(that)g(the)g(deriv)-5 b(ativ)n(es)28 b(of)h(the)g(\014rst) f(in)n(tegral)g(are)g Fu(O)r Fv(\()p Fu(T)2114 3510 y Ft(\025)2158 3498 y Fu(e)2197 3468 y Fk(\000)p Ft(c)2279 3476 y Fd(2)2310 3468 y Ft(T)2349 3477 y Fc(\025)2393 3498 y Fv(\))d(=)g Fu(O)r Fv(\()p Fu(\026)p Fv(\))30 b(\(still)g(pro)n(vided)d Fu(C)3346 3510 y Fs(1)3413 3498 y Fv(is)i(large)e(enough\).)236 3598 y(Moreo)n(v)n(er,)19 b(using)h(that)g(\()p Fq(j)p Fu(g)s Fv(\()p Fu(!)1182 3610 y Ft(\025)1226 3598 y Fu(t)t Fv(+)t Fu(')1383 3568 y Fs(+)1437 3598 y Fu(;)14 b(q)1511 3610 y Fs(0)1548 3598 y Fv(\()p Fu(t)p Fv(\))p Fu(;)g(t)p Fv(\))p Fq(j)t Fv(+)t Fq(j)p Fu(@)1904 3610 y Ft(')1952 3598 y Fu(g)s Fv(\()p Fu(!)2079 3610 y Ft(\025)2122 3598 y Fu(t)t Fv(+)t Fu(')2279 3568 y Fs(+)2333 3598 y Fu(;)g(q)2407 3610 y Fs(0)2444 3598 y Fv(\()p Fu(t)p Fv(\))p Fu(;)g(t)p Fv(\))p Fq(j)t Fv(+)t Fq(j)p Fu(@)2800 3610 y Ft(t)2829 3598 y Fu(g)s Fv(\()p Fu(!)2956 3610 y Ft(\025)2999 3598 y Fu(t)t Fv(+)t Fu(')3156 3568 y Fs(+)3210 3598 y Fu(;)g(q)3284 3610 y Fs(0)3321 3598 y Fv(\()p Fu(t)p Fv(\))p Fu(;)g(t)p Fv(\))p Fq(j)p Fv(\))24 b(=)f Fu(O)r Fv(\()p Fu(q)3815 3610 y Fs(0)3853 3598 y Fv(\()p Fu(t)p Fv(\))t Fq(\000)236 3697 y Fv(2)p Fu(\031)s Fv(\))i(=)g Fu(O)r Fv(\()p Fu(e)611 3667 y Fk(\000)p Ft(c)693 3675 y Fd(2)726 3667 y Ft(t)755 3697 y Fv(\))k(for)f Fu(t)d Fq(2)h Fv(\()p Fu(T)1161 3709 y Ft(\025)1204 3697 y Fu(=)p Fv(2)p Fu(;)14 b Fv(+)p Fq(1)p Fv(\),)28 b(w)n(e)h(\014nd)g(that)g(the)g(deriv)-5 b(ativ)n(es)28 b(of)h(the)g(second)f(in)n(tegral)g(are)g Fu(O)r Fv(\()p Fu(\026)p Fv(\))i(as)e(w)n(ell.)236 3797 y(Hence)c Fq(jr)p Fu(a)615 3809 y Fs(3)653 3797 y Fv(\()p Fu(\025)p Fv(\))p Fq(j)g Fv(=)f Fu(O)r Fv(\()p Fu(\026)p Fv(\).)37 b(The)24 b(same)f(estimate)h(holds)g(for)f Fu(b)2212 3809 y Fs(3)2249 3797 y Fv(.)36 b(W)-7 b(e)25 b(then)f(conclude)g(that)g Fq(r)p Fu(R)3279 3809 y Fs(3)3317 3797 y Fv(\()p Fu(\025)p Fv(\))g(=)e Fu(O)r Fv(\()p Fu(\026)3687 3767 y Fs(2)3726 3797 y Fv(\),)j(whic)n(h)236 3897 y(completes)j(the)f (pro)r(of)g(of)h(lemma)g(3.1.)p 1508 3897 48 48 v 361 4046 a(In)g(section)f(6)g(w)n(e)g(will)h(lo)r(ok)f(for)g(a)g(critical)g (p)r(oin)n(t)h(of)f Fq(F)2114 4058 y Ft(\026)2186 4046 y Fv(in)h(the)g(set)278 4257 y Fu(E)g Fv(:=)478 4165 y Fl(n)533 4257 y Fu(\025)23 b Fv(=)g(\()p Fu(\022)763 4269 y Fs(1)801 4257 y Fu(;)14 b(:)g(:)g(:)f(;)h(\022)1024 4269 y Ft(k)1065 4257 y Fu(;)g(')1156 4269 y Fs(1)1193 4257 y Fu(;)g(:)g(:)g(:)g(;)g(')1432 4269 y Ft(k)1473 4257 y Fv(\))23 b Fq(2)h Fw(R)1679 4222 y Ft(k)1731 4257 y Fq(\002)12 b Fw(R)1880 4222 y Ft(k)q(d)1978 4161 y Fl(\014)1978 4211 y(\014)1978 4261 y(\014)2030 4257 y Fu(\022)2069 4269 y Ft(i)2120 4257 y Fv(=)p 2207 4189 42 4 v 22 w Fu(\022)2249 4269 y Ft(i)2288 4257 y Fv(+)g Fu(b)2401 4269 y Ft(i)2427 4257 y Fu(;)63 b(')2567 4269 y Ft(i)2618 4257 y Fv(=)p 2705 4211 55 4 v 22 w Fu(')2760 4277 y Ft(i)2799 4257 y Fv(+)12 b Fu(a)2920 4269 y Ft(i)2947 4257 y Fu(;)62 b Fq(j)p Fu(b)3091 4269 y Ft(i)3118 4257 y Fq(j)23 b(\024)g Fv(2)p Fu(\031)s(;)38 b Fq(j)p Fu(a)3472 4269 y Ft(i)3500 4257 y Fq(j)23 b(\024)f Fv(2)p Fu(\031)3725 4165 y Fl(o)3781 4257 y Fu(;)41 b Fv(\(3.4\))236 4476 y(where)25 b Fu(k)s(;)p 557 4431 V 14 w(')611 4497 y Ft(i)639 4476 y Fu(;)p 676 4409 42 4 v 14 w(\022)717 4488 y Ft(i)770 4476 y Fv(will)h(b)r(e)f(de\014ned)h(in)g(section)e(5.) 36 b(It)26 b(will)f(result)g(that)h Fu(E)i Fq(\032)23 b Fv(\003)2673 4488 y Ft(\026;k)2799 4476 y Fv(\(for)i(some)g Fu(\014)i(>)c Fv(0)i(dep)r(ending)g(on)g(the)236 4576 y(curv)n(e)i Fu(\015)5 b Fv(\).)37 b(In)27 b(particular,)g(for)g(all)g Fu(\025)d Fq(2)f Fu(E)1244 4802 y(C)1303 4814 y Fs(1)1341 4802 y Fq(j)14 b Fv(ln)f Fu(\026)p Fq(j)24 b(\024)e Fu(\022)1683 4814 y Ft(i)p Fs(+1)1813 4802 y Fq(\000)c Fu(\022)1935 4814 y Ft(i)1986 4802 y Fu(<)2084 4746 y(C)2143 4758 y Fs(0)2180 4746 y Fu(\014)2227 4758 y Ft(i)p 2084 4783 172 4 v 2144 4859 a Fu(\026)2265 4802 y(;)97 b Fq(8)p Fu(i)23 b Fv(=)f(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(k)21 b Fq(\000)d Fv(1)p Fu(;)837 b Fv(\(3.5\))236 5028 y(where)27 b Fu(\014)523 5040 y Ft(i)574 5028 y Fv(:=)c Fu(\014)732 5040 y Ft(\025)771 5048 y Fc(i)824 5028 y Fv(:=)g Fu(\014)t Fv(\()p Fu(!)1070 5040 y Ft(i)1098 5028 y Fv(\))28 b(and)g Fu(!)1372 5040 y Ft(i)1422 5028 y Fv(:=)23 b Fu(!)1585 5040 y Ft(\025)1624 5048 y Fc(i)1677 5028 y Fv(:=)g(\()p Fu(')1874 5040 y Ft(i)p Fs(+1)2005 5028 y Fq(\000)18 b Fu(')2142 5040 y Ft(i)2170 5028 y Fv(\))p Fu(=)p Fv(\()p Fu(\022)2315 5040 y Ft(i)p Fs(+1)2445 5028 y Fq(\000)g Fu(\022)2567 5040 y Ft(i)2594 5028 y Fv(\).)38 b(Moreo)n(v)n(er)25 b(w)n(e)i(will)h(assume)e(\(see)i(\(5.8\)\))615 5253 y Fq(j)p 638 5207 55 4 v Fu(!)693 5265 y Ft(i)p Fs(+1)823 5253 y Fq(\000)p 906 5207 V 18 w Fu(!)961 5265 y Ft(i)988 5253 y Fq(j)23 b(\024)g Fu(\032\026)166 b Fv(where)p 1649 5207 V 55 w Fu(!)1703 5265 y Ft(i)1754 5253 y Fv(:=)p 1875 5147 V 1875 5193 a Fu(')1929 5213 y Ft(i)p Fs(+1)2059 5193 y Fq(\000)p 2142 5147 V 18 w Fu(')2197 5213 y Ft(i)p 1875 5233 350 4 v 1888 5250 42 4 v 1888 5318 a Fu(\022)1929 5330 y Ft(i)p Fs(+1)2059 5318 y Fq(\000)p 2142 5250 V 18 w Fu(\022)2184 5330 y Ft(i)2262 5253 y Fv(\(1)23 b Fq(\024)f Fu(i)h Fq(\024)g Fu(k)e Fq(\000)d Fv(1\))p Fu(;)69 b(!)2951 5265 y Fs(0)3011 5253 y Fv(:=)23 b Fu(!)3174 5265 y Ft(I)3211 5253 y Fu(;)70 b(!)3356 5265 y Ft(k)3419 5253 y Fv(:=)23 b Fu(!)3582 5265 y Ft(F)3845 5253 y Fv(\(3.6\))2084 6059 y(10)p eop %%Page: 11 11 11 10 bop 236 272 a Fv(and)25 b Fu(\032)e(>)f Fv(0)i(is)h(a)f(small)g (constan)n(t)g(to)g(b)r(e)h(c)n(hosen)f(later)g(\(see)g(\(6.3\)\).)36 b(F)-7 b(or)24 b(the)h(time)g(b)r(eing,)h(assuming)d(\(3.5\))i(and)f (\(3.6\),)236 372 y(w)n(e)j(w)n(an)n(t)g(to)h(giv)n(e)f(a)g(suitable)g (expression)f(of)i Fq(F)1767 384 y Ft(\026)1839 372 y Fv(in)g Fu(E)5 b Fv(.)37 b(By)27 b(lemma)g(3.1,)g(for)g Fu(\025)d Fq(2)f Fu(E)5 b Fv(,)28 b(w)n(e)f(ha)n(v)n(e)632 617 y Fq(F)692 629 y Ft(\026)737 617 y Fv(\()p Fu(\025)p Fv(\))84 b(=)1080 513 y Ft(k)q Fk(\000)p Fs(1)1081 538 y Fl(X)1087 714 y Ft(i)p Fs(=1)1226 560 y Fv(1)p 1226 597 42 4 v 1226 673 a(2)1287 560 y Fq(j)p Fu(')1364 572 y Ft(i)p Fs(+1)1495 560 y Fq(\000)18 b Fu(')1632 572 y Ft(i)1660 560 y Fq(j)1683 530 y Fs(2)p 1287 597 433 4 v 1344 673 a Fu(\022)1383 685 y Ft(i)p Fs(+1)1513 673 y Fq(\000)g Fu(\022)1635 685 y Ft(i)1748 617 y Fv(+)g Fu(!)1883 629 y Ft(I)1921 617 y Fu(')1975 629 y Fs(1)2031 617 y Fq(\000)g Fu(!)2166 629 y Ft(F)2221 617 y Fu(')2275 629 y Ft(k)2335 617 y Fq(\000)2428 560 y(j)p Fu(!)2503 572 y Ft(I)2540 560 y Fq(j)2563 530 y Fs(2)p 2428 597 174 4 v 2493 673 a Fv(2)2611 617 y Fu(\022)2650 629 y Fs(1)2705 617 y Fv(+)2798 560 y Fq(j)p Fu(!)2873 572 y Ft(F)2928 560 y Fq(j)2951 530 y Fs(2)p 2798 597 191 4 v 2873 673 a Fv(2)2999 617 y Fu(\022)3038 629 y Ft(k)933 875 y Fv(+)1122 771 y Ft(k)1080 796 y Fl(X)1086 973 y Ft(i)p Fs(=1)1214 875 y Fu(\026)1264 783 y Fl(\020)1314 875 y Fv(\000)1366 841 y Ft(u)1409 875 y Fv(\()p Fu(!)1493 887 y Ft(i)p Fk(\000)p Fs(1)1605 875 y Fu(;)c(\022)1681 887 y Ft(i)1709 875 y Fu(;)g(')1800 887 y Ft(i)1828 875 y Fv(\))19 b(+)f(\000)2014 841 y Ft(s)2049 875 y Fv(\()p Fu(!)2133 887 y Ft(i)2161 875 y Fu(;)c(\022)2237 887 y Ft(i)2264 875 y Fu(;)g(')2355 887 y Ft(i)2383 875 y Fv(\))2415 783 y Fl(\021)2483 875 y Fv(+)k Fu(\026F)12 b Fv(\()p Fu(!)2765 887 y Ft(I)2803 875 y Fu(;)i(\022)2879 887 y Fs(1)2916 875 y Fu(;)g(')3007 887 y Fs(1)3045 875 y Fv(\))933 1134 y Fq(\000)1080 1030 y Ft(k)q Fk(\000)p Fs(1)1081 1055 y Fl(X)1087 1232 y Ft(i)p Fs(=1)1216 1134 y Fu(\026)1280 1021 y Fl(Z)1363 1041 y Ft(\022)1395 1049 y Fc(i)p Fd(+1)1326 1210 y Ft(\022)1358 1218 y Fc(i)1509 1134 y Fu(f)9 b Fv(\()p Fu(!)1643 1146 y Ft(i)1671 1134 y Fv(\()p Fu(t)18 b Fq(\000)g Fu(\022)1873 1146 y Ft(i)1901 1134 y Fv(\))h(+)f Fu(')2089 1146 y Ft(i)2117 1134 y Fu(;)c Fv(0)p Fu(;)g(t)p Fv(\))27 b Fu(dt)19 b Fq(\000)f Fu(\026F)12 b Fv(\()p Fu(!)2696 1146 y Ft(F)2751 1134 y Fu(;)i(\022)2827 1146 y Ft(k)2867 1134 y Fu(;)g(')2958 1146 y Ft(k)2999 1134 y Fv(\))19 b(+)3133 1030 y Ft(k)q Fk(\000)p Fs(1)3134 1055 y Fl(X)3140 1232 y Ft(i)p Fs(=1)3269 1134 y Fu(R)3332 1146 y Fs(0)3369 1134 y Fv(\()p Fu(\026;)14 b(\025)3536 1146 y Ft(i)3564 1134 y Fv(\))p Fu(;)3845 873 y Fv(\(3.7\))236 1371 y(where)36 b Fq(jr)577 1383 y Ft(\025)620 1371 y Fu(R)683 1383 y Fs(0)721 1371 y Fv(\()p Fu(\026;)14 b(\025)p Fv(\))p Fq(j)37 b Fv(satis\014es)e (\(3.2\).)61 b(W)-7 b(e)36 b(shall)g(write)g Fq(F)2184 1383 y Ft(\026)2264 1371 y Fv(in)g(an)g(appropriate)e(form)h(thanks)h (to)g(the)g(follo)n(wing)236 1471 y(lemmas.)h(The)27 b(\014rst)h(one)f(sa)n(ys)f(ho)n(w)h(close)g(the)h(\\mean)f (frequencies")f Fu(!)2535 1483 y Ft(i)2590 1471 y Fv(are)h(to)g(the)h (unp)r(erturb)r(ed)p 3454 1425 55 4 v 29 w Fu(!)3508 1483 y Ft(i)3536 1471 y Fv(.)236 1619 y Fw(Lemma)k(3.2)40 b Fp(L)l(et)30 b Fu(\025)23 b Fv(=)g(\()p Fu(\022)1112 1631 y Fs(1)1149 1619 y Fu(;)14 b(:)g(:)g(:)g(;)g(\022)1373 1631 y Ft(k)1414 1619 y Fu(;)g(')1505 1631 y Fs(1)1542 1619 y Fu(;)g(:)g(:)g(:)g(;)g(')1781 1631 y Ft(k)1822 1619 y Fv(\))30 b Fp(b)l(elong)g(to)g Fu(E)5 b Fp(.)38 b(Then)1409 1818 y Fq(j)p Fu(!)1484 1830 y Ft(i)1529 1818 y Fq(\000)p 1612 1772 V 18 w Fu(!)1667 1830 y Ft(i)1695 1818 y Fq(j)23 b Fv(=)f Fu(O)1893 1726 y Fl(\020)2092 1762 y Fv(1)p 1954 1799 319 4 v 1954 1875 a Fu(\022)1993 1887 y Ft(i)p Fs(+1)2123 1875 y Fq(\000)c Fu(\022)2245 1887 y Ft(i)2282 1726 y Fl(\021)2355 1818 y Fv(=)23 b Fu(O)2508 1726 y Fl(\020)2644 1762 y Fv(1)p 2568 1799 194 4 v 2568 1875 a Fq(j)14 b Fv(ln)g Fu(\026)p Fq(j)2771 1726 y Fl(\021)2820 1818 y Fu(:)1002 b Fv(\(3.8\))236 2021 y Fp(Mor)l(e)l(over)599 2181 y Fv(\000)651 2147 y Ft(u)695 2181 y Fv(\()p Fu(!)779 2193 y Ft(i)p Fk(\000)p Fs(1)891 2181 y Fu(;)14 b(\022)967 2193 y Ft(i)995 2181 y Fu(;)g(')1086 2193 y Ft(i)1114 2181 y Fv(\))k(+)g(\000)1299 2147 y Ft(s)1335 2181 y Fv(\()p Fu(!)1419 2193 y Ft(i)1446 2181 y Fu(;)c(\022)1522 2193 y Ft(i)1550 2181 y Fu(;)g(')1641 2193 y Ft(i)1669 2181 y Fv(\))23 b(=)g(\000\()p 1896 2135 55 4 v Fu(!)1950 2193 y Ft(i)1978 2181 y Fu(;)14 b(\022)2054 2193 y Ft(i)2082 2181 y Fu(;)g(')2173 2193 y Ft(i)2200 2181 y Fv(\))19 b(+)f Fu(R)2397 2193 y Fs(4)2434 2181 y Fv(\()p Fu(\025)2514 2193 y Ft(i)2543 2181 y Fv(\))p Fu(;)73 b Fv(where)59 b Fq(r)p Fu(R)3075 2193 y Fs(4)3136 2181 y Fv(=)22 b Fu(O)r Fv(\(1)p Fu(=)p Fq(j)14 b Fv(ln)g Fu(\026)p Fq(j)p Fv(\))p Fu(:)193 b Fv(\(3.9\))236 2341 y Fb(Pr)n(oof)p Fv(.)78 b(Set)26 b(\001)p Fu(\022)846 2353 y Ft(i)896 2341 y Fv(:=)d Fu(\022)1046 2353 y Ft(i)p Fs(+1)1172 2341 y Fq(\000)14 b Fu(\022)1290 2353 y Ft(i)1317 2341 y Fv(,)27 b(\001)p Fu(a)1480 2353 y Ft(i)1530 2341 y Fv(:=)c Fu(a)1685 2353 y Ft(i)p Fs(+1)1811 2341 y Fq(\000)14 b Fu(a)1934 2353 y Ft(i)1987 2341 y Fv(and)25 b(\001)p Fu(b)2251 2353 y Ft(i)2302 2341 y Fv(:=)d Fu(b)2448 2353 y Ft(i)p Fs(+1)2574 2341 y Fq(\000)14 b Fu(b)2689 2353 y Ft(i)2716 2341 y Fv(.)36 b(By)25 b(an)h(elemen)n(tary)e(computation)h (w)n(e)236 2441 y(get)j Fu(!)427 2453 y Ft(i)472 2441 y Fq(\000)p 555 2395 V 18 w Fu(!)610 2453 y Ft(i)661 2441 y Fv(=)22 b Fq(\000)p 813 2395 V Fu(!)868 2453 y Ft(i)895 2441 y Fv(\001)p Fu(b)1000 2453 y Ft(i)1028 2441 y Fu(=)p Fv(\001)p Fu(\022)1178 2453 y Ft(i)1223 2441 y Fv(+)c(\001)p Fu(a)1419 2453 y Ft(i)1447 2441 y Fu(=)p Fv(\001)p Fu(\022)1597 2453 y Ft(i)1624 2441 y Fu(:)28 b Fv(By)g(the)g(de\014nition)g(of)f Fu(E)33 b Fv(and)27 b(\(3.5\),)h(estimate)f(\(3.8\))h(follo)n(ws.)361 2540 y(F)-7 b(rom)29 b(the)h(de\014nition)g(of)f(\000)1245 2510 y Ft(u)1288 2540 y Fu(;)14 b Fv(\000)1377 2510 y Ft(s)1442 2540 y Fv(and)29 b(the)h(exp)r(onen)n(tial)f(deca)n(y)g(of)g Fu(q)2563 2552 y Fs(0)2630 2540 y Fv(it)h(results)f(that)h Fu(@)3207 2552 y Ft(!)3255 2540 y Fv(\000)3307 2510 y Ft(u;s)3431 2540 y Fv(is)f(b)r(ounded)h(b)n(y)f(a)236 2640 y(uniform)e(constan)n(t,)g(as)f(w)n(ell)h(as)f(its)i(partial)e (deriv)-5 b(ativ)n(es.)35 b(Hence)28 b(\(3.9\))e(is)h(a)g(straigh)n (tforw)n(ard)d(consequence)i(of)h(\(3.8\))236 2740 y(and)h(of)f (\(3.6\).)p 723 2740 48 48 v 236 2940 a Fw(Lemma)32 b(3.3)40 b Fp(F)-6 b(or)30 b Fv(0)23 b Fu(<)f(\026)i Fq(\024)e Fu(\026)1256 2952 y Fs(4)341 3178 y Fu(\026F)12 b Fv(\()p Fu(!)540 3190 y Ft(I)578 3178 y Fu(;)i(\022)654 3190 y Fs(1)691 3178 y Fu(;)g(')782 3190 y Fs(1)820 3178 y Fv(\))k Fq(\000)995 3074 y Ft(k)953 3099 y Fl(X)960 3276 y Ft(i)p Fs(=1)1087 3178 y Fu(\026)1151 3065 y Fl(Z)1234 3085 y Ft(\022)1266 3093 y Fc(i)p Fd(+1)1197 3253 y Ft(\022)1229 3261 y Fc(i)1381 3178 y Fu(f)9 b Fv(\()p Fu(!)1515 3190 y Ft(i)1542 3178 y Fv(\()p Fu(t)19 b Fq(\000)f Fu(\022)1745 3190 y Ft(i)1773 3178 y Fv(\))g(+)g Fu(')1960 3190 y Ft(i)1988 3178 y Fu(;)c Fv(0)p Fu(;)g(t)p Fv(\))29 b Fu(dt)19 b Fq(\000)f Fu(\026F)12 b Fv(\()p Fu(!)2569 3190 y Ft(F)2624 3178 y Fu(;)i(\022)2700 3190 y Ft(k)2741 3178 y Fu(;)g(')2832 3190 y Ft(k)2873 3178 y Fv(\))23 b(=)3058 3074 y Ft(k)3016 3099 y Fl(X)3022 3276 y Ft(i)p Fs(=1)3150 3178 y Fu(R)3214 3143 y Ft(i)3213 3198 y Fs(5)3250 3178 y Fv(\()p Fu(\026;)14 b(\025)3417 3190 y Ft(i)p Fk(\000)p Fs(1)3530 3178 y Fu(;)g(\025)3615 3190 y Ft(i)3643 3178 y Fv(\))p Fu(;)106 b Fv(\(3.10\))236 3420 y Fp(wher)l(e,)31 b(for)g(al)t(l)g Fu(i)806 3390 y Fs(4)371 3623 y Fq(r)p Fu(R)504 3589 y Ft(i)503 3644 y Fs(5)540 3623 y Fv(\()p Fu(\026;)14 b(\022)698 3635 y Ft(i)p Fk(\000)p Fs(1)811 3623 y Fu(;)g(')902 3635 y Ft(i)p Fk(\000)p Fs(1)1015 3623 y Fu(;)g(\022)1091 3635 y Ft(i)1118 3623 y Fu(;)g(')1209 3635 y Ft(i)1237 3623 y Fu(;)g(\022)1313 3635 y Ft(i)p Fs(+1)1425 3623 y Fu(;)g(')1516 3635 y Ft(i)p Fs(+1)1628 3623 y Fv(\))23 b(=)g Fu(O)1836 3531 y Fl(\020)2143 3567 y Fu(\026)p 1896 3604 544 4 v 1896 3680 a(\014)1947 3652 y Fs(2)1943 3704 y Ft(i)p Fk(\000)p Fs(1)2056 3680 y Fv(\()p Fu(\022)2127 3692 y Ft(i)2173 3680 y Fq(\000)18 b Fu(\022)2295 3692 y Ft(i)p Fk(\000)p Fs(1)2408 3680 y Fv(\))2468 3623 y(+)2772 3567 y Fu(\026)p 2561 3604 472 4 v 2561 3680 a(\014)2612 3652 y Fs(2)2608 3704 y Ft(i)2650 3680 y Fv(\()p Fu(\022)2721 3692 y Ft(i)p Fs(+1)2851 3680 y Fq(\000)g Fu(\022)2973 3692 y Ft(i)3001 3680 y Fv(\))3062 3623 y(+)3155 3567 y Fu(\026)p Fq(j)p Fu(\014)3275 3579 y Ft(i)3321 3567 y Fq(\000)g Fu(\014)3451 3579 y Ft(i)p Fk(\000)p Fs(1)3563 3567 y Fq(j)p 3155 3604 432 4 v 3253 3680 a Fu(\014)3300 3692 y Ft(i)p Fk(\000)p Fs(1)3413 3680 y Fu(\014)3460 3692 y Ft(i)3596 3531 y Fl(\021)3646 3623 y Fu(:)135 b Fv(\(3.11\))236 3837 y Fb(Pr)n(oof)p Fv(.)78 b(W)-7 b(e)28 b(ha)n(v)n(e)788 4038 y Fq(\000)867 3925 y Fl(Z)949 3946 y Ft(\022)981 3954 y Fc(i)p Fd(+1)912 4114 y Ft(\022)944 4122 y Fc(i)1096 4038 y Fu(f)9 b Fv(\()p Fu(')1232 4050 y Ft(i)1278 4038 y Fv(+)18 b Fu(!)1413 4050 y Ft(i)1441 4038 y Fv(\()p Fu(t)g Fq(\000)g Fu(\022)1643 4050 y Ft(i)1671 4038 y Fv(\))p Fu(;)c Fv(0)p Fu(;)g(t)p Fv(\))28 b Fu(dt)23 b Fv(=)g Fu(F)12 b Fv(\()p Fu(!)2242 4050 y Ft(i)2269 4038 y Fu(;)i(\022)2345 4050 y Ft(i)p Fs(+1)2457 4038 y Fu(;)g(')2548 4050 y Ft(i)p Fs(+1)2660 4038 y Fv(\))k Fq(\000)g Fu(F)12 b Fv(\()p Fu(!)2942 4050 y Ft(i)2970 4038 y Fu(;)i(\022)3046 4050 y Ft(i)3073 4038 y Fu(;)g(')3164 4050 y Ft(i)3192 4038 y Fv(\))954 4251 y(=)1042 4159 y Fl(\020)1091 4251 y Fu(F)e Fv(\()p Fu(!)1240 4263 y Ft(i)1268 4251 y Fu(;)i(\022)1344 4263 y Ft(i)p Fs(+1)1455 4251 y Fu(;)g(')1546 4263 y Ft(i)p Fs(+1)1658 4251 y Fv(\))19 b Fq(\000)f Fu(F)12 b Fv(\()p Fu(!)1941 4263 y Ft(i)p Fk(\000)p Fs(1)2053 4251 y Fu(;)i(\022)2129 4263 y Ft(i)2157 4251 y Fu(;)g(')2248 4263 y Ft(i)2276 4251 y Fv(\))2308 4159 y Fl(\021)2376 4251 y Fv(+)2459 4159 y Fl(\020)2509 4251 y Fu(F)e Fv(\()p Fu(!)2658 4263 y Ft(i)p Fk(\000)p Fs(1)2770 4251 y Fu(;)i(\022)2846 4263 y Ft(i)2874 4251 y Fu(;)g(')2965 4263 y Ft(i)2993 4251 y Fv(\))k Fq(\000)g Fu(F)12 b Fv(\()p Fu(!)3275 4263 y Ft(i)3303 4251 y Fu(;)i(\022)3379 4263 y Ft(i)3406 4251 y Fu(;)g(')3497 4263 y Ft(i)3525 4251 y Fv(\))3557 4159 y Fl(\021)3607 4251 y Fu(;)236 4436 y Fv(where)27 b Fu(F)12 b Fv(\()p Fu(!)s(;)i Fq(\001)p Fu(;)g Fq(\001)p Fv(\))28 b(is)g(de\014ned)f(in)h(\(2.9\).)37 b(W)-7 b(e)28 b(obtain)672 4678 y Fu(\026F)12 b Fv(\()p Fu(!)871 4690 y Ft(I)909 4678 y Fu(;)i(\022)985 4690 y Fs(1)1022 4678 y Fu(;)g(')1113 4690 y Fs(1)1150 4678 y Fv(\))19 b Fq(\000)1284 4574 y Ft(k)q Fk(\000)p Fs(1)1285 4599 y Fl(X)1291 4776 y Ft(i)p Fs(=1)1420 4678 y Fu(\026)1484 4565 y Fl(Z)1566 4585 y Ft(\022)1598 4593 y Fc(i)p Fd(+1)1530 4753 y Ft(\022)1562 4761 y Fc(i)1713 4678 y Fu(f)9 b Fv(\()p Fu(')1849 4690 y Ft(i)1896 4678 y Fv(+)18 b Fu(!)2031 4690 y Ft(i)2058 4678 y Fv(\()p Fu(t)h Fq(\000)f Fu(\022)2261 4690 y Ft(i)2288 4678 y Fv(\))p Fu(;)c Fv(0)p Fu(;)g(t)p Fv(\))28 b Fu(dt)18 b Fq(\000)g Fu(\026F)12 b Fv(\()p Fu(!)2899 4690 y Ft(F)2954 4678 y Fu(;)i(\022)3030 4690 y Ft(k)3071 4678 y Fu(;)g(')3162 4690 y Ft(k)3203 4678 y Fv(\))24 b(=)3388 4574 y Ft(k)3346 4599 y Fl(X)3352 4776 y Ft(i)p Fs(=1)3480 4678 y Fu(R)3544 4644 y Ft(i)3543 4698 y Fs(5)236 4910 y Fv(where)239 5070 y Fu(R)303 5036 y Ft(i)302 5091 y Fs(5)363 5070 y Fv(:=)e Fu(R)537 5036 y Ft(i)536 5091 y Fs(5)573 5070 y Fv(\()p Fu(\026;)14 b(\022)731 5082 y Ft(i)p Fk(\000)p Fs(1)844 5070 y Fu(;)g(')935 5082 y Ft(i)p Fk(\000)p Fs(1)1048 5070 y Fu(;)g(\022)1124 5082 y Ft(i)1152 5070 y Fu(;)g(')1243 5082 y Ft(i)1270 5070 y Fu(;)g(\022)1346 5082 y Ft(i)p Fs(+1)1458 5070 y Fu(;)g(')1549 5082 y Ft(i)p Fs(+1)1661 5070 y Fv(\))83 b(:=)g Fu(\026)1997 4978 y Fl(\020)2047 5070 y Fu(F)12 b Fv(\()p Fu(!)2196 5082 y Ft(i)p Fk(\000)p Fs(1)2308 5070 y Fu(;)i(\022)2384 5082 y Ft(i)2412 5070 y Fu(;)g(')2503 5082 y Ft(i)2530 5070 y Fv(\))19 b Fq(\000)f Fu(F)12 b Fv(\()p Fu(!)2813 5082 y Ft(i)2841 5070 y Fu(;)i(\022)2917 5082 y Ft(i)2944 5070 y Fu(;)g(')3035 5082 y Ft(i)3063 5070 y Fv(\))3095 4978 y Fl(\021)1788 5288 y Fv(=)94 b Fq(\000)p Fu(\026)2172 5209 y Fl(X)2048 5391 y Fs(0)p Ft(<)p Fk(j)p Fs(\()p Ft(n;l)p Fs(\))p Fk(j\024)p Ft(N)2403 5288 y Fu(f)2444 5300 y Ft(n;l)2540 5231 y Fu(e)2579 5201 y Fs(i\()p Ft(n)p Fk(\001)p Ft(')2729 5209 y Fc(i)2754 5201 y Fs(+)p Ft(l\022)2858 5209 y Fc(i)2884 5201 y Fs(\))p 2540 5268 375 4 v 2716 5344 a Fv(i)2924 5195 y Fl(\020)3196 5231 y Fv(1)p 2984 5268 467 4 v 2984 5344 a(\()p Fu(n)18 b Fq(\001)h Fu(!)3178 5356 y Ft(i)p Fk(\000)p Fs(1)3309 5344 y Fv(+)f Fu(l)r Fv(\))3479 5288 y Fq(\000)3742 5231 y Fv(1)p 3572 5268 382 4 v 3572 5344 a(\()p Fu(n)g Fq(\001)h Fu(!)3766 5356 y Ft(i)3812 5344 y Fv(+)f Fu(l)r Fv(\))3963 5195 y Fl(\021)p 236 5424 1512 4 v 329 5480 a Fn(4)363 5504 y Fm(In)24 b(the)h(cases)f Fj(i)c Fm(=)f(1)p Fj(;)12 b(i)20 b Fm(=)f Fj(k)25 b Fm(w)n(e)f(only)g(ha)n(v)n(e)h Fj(R)1590 5480 y Fn(1)1590 5526 y(5)1644 5504 y Fm(=)20 b Fj(R)1772 5480 y Fn(1)1772 5526 y(5)1807 5504 y Fm(\()p Fj(\026;)12 b(\022)1941 5513 y Fn(1)1976 5504 y Fj(;)f(')2053 5513 y Fn(1)2088 5504 y Fj(;)g(\022)2152 5513 y Fn(2)2187 5504 y Fj(;)g(')2264 5513 y Fn(2)2298 5504 y Fm(\))24 b(and)h Fj(R)2540 5480 y Fi(k)2540 5526 y Fn(5)2598 5504 y Fm(=)19 b Fj(R)2725 5480 y Fi(k)2725 5526 y Fn(5)2764 5504 y Fm(\()p Fj(\026;)12 b(\022)2898 5516 y Fi(k)q Fh(\000)p Fn(1)3015 5504 y Fj(;)f(')3092 5516 y Fi(k)q Fh(\000)p Fn(1)3209 5504 y Fj(;)g(\022)3273 5516 y Fi(k)3311 5504 y Fj(;)g(')3388 5516 y Fi(k)3426 5504 y Fm(\))p Fj(:)2084 6059 y Fv(11)p eop %%Page: 12 12 12 11 bop 236 272 a Fv(No)n(w)27 b(w)n(e)h(pro)n(v)n(e)d(\(3.11\).)37 b(Let)27 b(us)h(consider)e(for)i(example)f Fu(@)2119 284 y Ft(\022)2151 292 y Fc(i)2181 272 y Fu(R)2245 242 y Ft(i)2244 293 y Fs(5)2281 272 y Fv(.)37 b(W)-7 b(e)28 b(ha)n(v)n(e)790 465 y Fu(@)834 477 y Ft(\022)866 485 y Fc(i)896 465 y Fu(R)960 431 y Ft(i)959 486 y Fs(5)1079 465 y Fv(=)83 b Fu(\026@)1321 477 y Ft(\022)1353 485 y Fc(i)1383 373 y Fl(\020)1432 465 y Fu(F)12 b Fv(\()p Fu(!)1581 477 y Ft(i)p Fk(\000)p Fs(1)1694 465 y Fu(;)i(\022)1770 477 y Ft(i)1797 465 y Fu(;)g(')1888 477 y Ft(i)1916 465 y Fv(\))19 b Fq(\000)f Fu(F)12 b Fv(\()p Fu(!)2199 477 y Ft(i)2227 465 y Fu(;)i(\022)2303 477 y Ft(i)2330 465 y Fu(;)g(')2421 477 y Ft(i)2449 465 y Fv(\))2481 373 y Fl(\021)1079 657 y Fv(=)83 b Fu(\026)1277 565 y Fl(\020)1326 657 y Fu(@)1370 669 y Ft(!)1418 657 y Fu(F)12 b Fv(\()p Fu(!)1567 669 y Ft(i)p Fk(\000)p Fs(1)1680 657 y Fu(;)i(\022)1756 669 y Ft(i)1783 657 y Fu(;)g(')1874 669 y Ft(i)1902 657 y Fv(\))p Fu(:)2045 601 y Fq(\000)p Fu(!)2162 613 y Ft(i)p Fk(\000)p Fs(1)p 1968 638 385 4 v 1968 714 a Fv(\()p Fu(\022)2039 726 y Ft(i)2085 714 y Fq(\000)k Fu(\022)2207 726 y Ft(i)p Fk(\000)p Fs(1)2319 714 y Fv(\))2380 657 y Fq(\000)g Fu(@)2507 669 y Ft(!)2555 657 y Fu(F)12 b Fv(\()p Fu(!)2704 669 y Ft(i)2732 657 y Fu(;)i(\022)2808 669 y Ft(i)2835 657 y Fu(;)g(')2926 669 y Ft(i)2954 657 y Fv(\))p Fu(:)3172 601 y(!)3224 613 y Ft(i)p 3020 638 384 4 v 3020 714 a Fv(\()p Fu(\022)3091 726 y Ft(i)p Fs(+1)3221 714 y Fq(\000)k Fu(\022)3343 726 y Ft(i)3371 714 y Fv(\))3413 565 y Fl(\021)1079 877 y Fq(\000)83 b Fu(\026)1277 785 y Fl(\020)1465 799 y(X)1340 980 y Fs(0)p Ft(<)p Fk(j)p Fs(\()p Ft(n;l)p Fs(\))p Fk(j\024)p Ft(N)1723 877 y Fu(f)1764 889 y Ft(n;l)1850 877 y Fu(l)r(e)1916 843 y Fs(i\()p Ft(n)p Fk(\001)p Ft(')2066 851 y Fc(i)2090 843 y Fs(+)p Ft(l\022)2194 851 y Fc(i)2221 843 y Fs(\))2251 877 y Fv(\()2505 821 y(1)p 2293 858 467 4 v 2293 934 a(\()p Fu(n)19 b Fq(\001)f Fu(!)2487 946 y Ft(i)p Fk(\000)p Fs(1)2618 934 y Fv(+)g Fu(l)r Fv(\))2788 877 y Fq(\000)3051 821 y Fv(1)p 2881 858 382 4 v 2881 934 a(\()p Fu(n)g Fq(\001)h Fu(!)3075 946 y Ft(i)3121 934 y Fv(+)f Fu(l)r Fv(\))3272 877 y(\))3304 785 y Fl(\021)3354 877 y Fu(;)427 b Fv(\(3.12\))236 1138 y(where)1317 1276 y Fu(@)1361 1288 y Ft(!)1409 1276 y Fu(F)12 b Fv(\()p Fu(!)s(;)i(\022)1637 1288 y Fs(0)1674 1276 y Fu(;)g(')1765 1288 y Fs(0)1803 1276 y Fv(\))23 b(=)2070 1197 y Fl(X)1946 1379 y Fs(0)p Ft(<)p Fk(j)p Fs(\()p Ft(n;l)p Fs(\))p Fk(j\024)p Ft(N)2329 1276 y Fu(f)2370 1288 y Ft(n;l)2465 1220 y Fu(ne)2554 1190 y Fs(i\()p Ft(n)p Fk(\001)p Ft(')2704 1198 y Fd(0)2735 1190 y Fs(+)p Ft(l\022)2839 1198 y Fd(0)2872 1190 y Fs(\))p 2465 1257 437 4 v 2475 1333 a Fv(i\()p Fu(n)c Fq(\001)f Fu(!)j Fv(+)d Fu(l)r Fv(\))2855 1309 y Fs(2)2912 1276 y Fu(;)869 b Fv(\(3.13\))236 1520 y(Estimate)30 b(\(3.11\))f(follo)n (ws)g(immediately)i(from)f(\(3.12\))f(and)h(\(3.13\).)44 b(The)30 b(other)g(partial)f(deriv)-5 b(ativ)n(es)29 b(of)h Fu(R)3709 1490 y Ft(i)3708 1540 y Fs(5)3776 1520 y Fv(can)f(b)r(e)236 1619 y(estimated)f(similarly)-7 b(.)p 980 1619 48 48 v 361 1763 a(Finally)g(,)23 b(to)e(get)h(a)f (suitable)h(expression)e(of)i Fq(F)1793 1775 y Ft(\026)1837 1763 y Fv(,)h(w)n(e)e(\014nd)h(con)n(v)n(enien)n(t)f(to)h(in)n(tro)r (duce)f(co)r(ordinates)f(\()p Fu(b;)14 b(c)p Fv(\))23 b Fq(2)h Fw(R)3805 1733 y Fs(\(1+)p Ft(d)p Fs(\))p Ft(k)236 1863 y Fv(de\014ned)k(b)n(y)f(\(3.4\))h(and)1538 1962 y Fu(c)1574 1974 y Ft(i)1625 1962 y Fv(=)22 b Fu(a)1756 1974 y Ft(i)1802 1962 y Fq(\000)p 1885 1917 55 4 v 18 w Fu(!)1940 1974 y Ft(i)1967 1962 y Fu(b)2003 1974 y Ft(i)2031 1962 y Fu(;)180 b Fq(8)p Fu(i)22 b Fv(=)h(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(k)s(;)1090 b Fv(\(3.14\))236 2103 y(\(w)n(e)29 b(are)e(just)i(p)r(erforming)f(a)g(linear)g(c)n (hange)g(of)g(co)r(ordinates)f(adapted)h(to)h(the)g(direction)f(of)g (the)h(unp)r(erturb)r(ed)g(\015o)n(w)236 2203 y(at)f(eac)n(h)e Fu(i)p Fv(-transition)h(\()p Fu(b)1028 2215 y Ft(i)1055 2203 y Fu(;)14 b(a)1136 2215 y Ft(i)1164 2203 y Fv(\))23 b(=)g Fu(b)1343 2215 y Ft(i)1370 2203 y Fv(\(1)p Fu(;)p 1481 2157 V 14 w(!)1535 2215 y Ft(i)1563 2203 y Fv(\))c(+)f(\(0)p Fu(;)c(c)1844 2215 y Ft(i)1871 2203 y Fv(\)\).)236 2371 y Fw(Lemma)32 b(3.4)40 b Fp(We)30 b(have)519 2496 y Ft(k)q Fk(\000)p Fs(1)520 2521 y Fl(X)526 2698 y Ft(i)p Fs(=1)665 2544 y Fv(1)p 665 2581 42 4 v 665 2657 a(2)726 2544 y Fq(j)p Fu(')803 2556 y Ft(i)p Fs(+1)934 2544 y Fq(\000)18 b Fu(')1071 2556 y Ft(i)1099 2544 y Fq(j)1122 2514 y Fs(2)p 726 2581 433 4 v 751 2657 a Fv(\()p Fu(\022)822 2669 y Ft(i)p Fs(+1)952 2657 y Fq(\000)g Fu(\022)1074 2669 y Ft(i)1102 2657 y Fv(\))1187 2600 y(+)g Fu(!)1322 2612 y Ft(I)1360 2600 y Fu(')1414 2612 y Fs(1)1470 2600 y Fq(\000)g Fu(!)1605 2612 y Ft(F)1660 2600 y Fu(')1714 2612 y Ft(k)1774 2600 y Fq(\000)1867 2544 y(j)p Fu(!)1942 2556 y Ft(I)1980 2544 y Fq(j)2003 2514 y Fs(2)p 1867 2581 174 4 v 1933 2657 a Fv(2)2050 2600 y Fu(\022)2089 2612 y Fs(1)2144 2600 y Fv(+)2237 2544 y Fq(j)p Fu(!)2312 2556 y Ft(F)2367 2544 y Fq(j)2390 2514 y Fs(2)p 2237 2581 191 4 v 2312 2657 a Fv(2)2438 2600 y Fu(\022)2477 2612 y Ft(k)2601 2600 y Fv(=)2758 2544 y(1)p 2758 2581 42 4 v 2758 2657 a(2)2823 2496 y Ft(k)q Fk(\000)p Fs(1)2824 2521 y Fl(X)2830 2698 y Ft(i)p Fs(=1)3079 2544 y Fq(j)p Fu(c)3138 2556 y Ft(i)p Fs(+1)3268 2544 y Fq(\000)g Fu(c)3387 2556 y Ft(i)3415 2544 y Fq(j)3438 2514 y Fs(2)p 2969 2581 617 4 v 2969 2665 a Fv(\001)p 3038 2598 42 4 v Fu(\022)3079 2677 y Ft(i)3125 2665 y Fv(+)g(\()p Fu(b)3276 2677 y Ft(i)p Fs(+1)3407 2665 y Fq(\000)g Fu(b)3526 2677 y Ft(i)3553 2665 y Fv(\))3804 2600 y(\(3.15\))2601 2892 y(+)2790 2788 y Ft(k)2748 2813 y Fl(X)2754 2990 y Ft(i)p Fs(=1)2882 2892 y Fu(R)2946 2858 y Ft(i)2945 2913 y Fs(6)2982 2892 y Fv(\()p Fu(\026;)c(\022)3140 2904 y Ft(i)3168 2892 y Fu(;)g(')3259 2904 y Ft(i)3287 2892 y Fu(;)g(\022)3363 2904 y Ft(i)p Fs(+1)3474 2892 y Fu(;)g(')3565 2904 y Ft(i)p Fs(+1)3677 2892 y Fv(\))p Fu(;)236 3149 y Fp(wher)l(e)31 b Fv(\001)p 540 3082 V Fu(\022)581 3161 y Ft(i)632 3149 y Fv(:=)p 743 3082 V 23 w Fu(\022)784 3161 y Ft(i)p Fs(+1)914 3149 y Fq(\000)p 997 3082 V 18 w Fu(\022)1038 3161 y Ft(i)1096 3149 y Fp(and)1227 3119 y Fs(5)1107 3317 y Fq(r)p Fu(R)1240 3283 y Ft(i)1239 3338 y Fs(6)1276 3317 y Fv(\()p Fu(\026;)14 b(\022)1434 3329 y Ft(i)p Fk(\000)p Fs(1)1547 3317 y Fu(;)g(')1638 3329 y Ft(i)p Fk(\000)p Fs(1)1751 3317 y Fu(;)g(\022)1827 3329 y Ft(i)1854 3317 y Fu(;)g(')1945 3329 y Ft(i)1973 3317 y Fu(;)g(\022)2049 3329 y Ft(i)p Fs(+1)2161 3317 y Fu(;)g(')2252 3329 y Ft(i)p Fs(+1)2364 3317 y Fv(\))23 b(=)g Fu(O)r Fv(\(\001)p 2673 3272 55 4 v Fu(!)2728 3329 y Ft(i)2756 3317 y Fv(\))g(=)g Fu(O)r Fv(\()p Fu(\032\026)p Fv(\))p Fu(:)660 b Fv(\(3.16\))236 3486 y Fb(Pr)n(oof)p Fv(.)78 b(Let)28 b Fq(f)p Fu(\015)831 3498 y Ft(i)858 3486 y Fq(g)900 3498 y Ft(i)p Fs(=1)p Ft(;:::)o(;k)q Fk(\000)p Fs(1)1259 3486 y Fv(b)r(e)f(de\014ned)h(b)n(y) f Fu(')1826 3498 y Ft(i)p Fs(+1)1955 3486 y Fq(\000)18 b Fu(')2092 3498 y Ft(i)2143 3486 y Fv(=)p 2230 3440 V 22 w Fu(!)2285 3498 y Ft(i)2312 3486 y Fv(\()p Fu(\022)2383 3498 y Ft(i)p Fs(+1)2513 3486 y Fq(\000)f Fu(\022)2634 3498 y Ft(i)2662 3486 y Fv(\))h(+)f Fu(\015)2837 3498 y Ft(i)2865 3486 y Fv(.)37 b(W)-7 b(e)27 b(can)g(write)g Fu(!)3483 3498 y Ft(I)3521 3486 y Fu(')3575 3498 y Fs(1)3630 3486 y Fq(\000)17 b Fu(!)3764 3498 y Ft(F)3819 3486 y Fu(')3873 3498 y Ft(k)3941 3486 y Fv(as)613 3739 y Fu(!)665 3751 y Ft(I)703 3739 y Fu(')757 3751 y Fs(1)813 3739 y Fq(\000)h Fu(!)948 3751 y Ft(F)1003 3739 y Fu(')1057 3751 y Ft(k)1181 3739 y Fv(=)1329 3635 y Ft(k)q Fk(\000)p Fs(1)1329 3660 y Fl(X)1336 3837 y Ft(i)p Fs(=1)1464 3646 y Fl(\020)1514 3739 y Fv(\()p 1546 3693 V Fu(!)1601 3751 y Ft(i)p Fk(\000)p Fs(1)1732 3739 y Fq(\000)p 1815 3693 V 18 w Fu(!)1869 3751 y Ft(i)1897 3739 y Fv(\))p Fu(')1983 3751 y Ft(i)2030 3739 y Fq(\000)p 2113 3693 V 18 w Fu(!)2167 3751 y Ft(i)2195 3739 y Fv(\()p Fu(')2281 3751 y Ft(i)p Fs(+1)2412 3739 y Fq(\000)g Fu(')2549 3751 y Ft(i)2577 3739 y Fv(\))2609 3646 y Fl(\021)2677 3739 y Fv(+)g Fu(')2814 3751 y Ft(k)2855 3739 y Fv(\()p 2887 3693 V Fu(!)2942 3751 y Ft(k)q Fk(\000)p Fs(1)3086 3739 y Fq(\000)g Fu(!)3221 3751 y Ft(F)3276 3739 y Fv(\))1181 4031 y(=)1329 3927 y Ft(k)q Fk(\000)p Fs(1)1329 3952 y Fl(X)1336 4128 y Ft(i)p Fs(=1)1464 3938 y Fl(\020)1514 4031 y Fv(\()p 1546 3985 V Fu(!)1601 4043 y Ft(i)p Fk(\000)p Fs(1)1732 4031 y Fq(\000)p 1815 3985 V 18 w Fu(!)1869 4043 y Ft(i)1897 4031 y Fv(\))p Fu(')1983 4043 y Ft(i)2030 4031 y Fq(\000)g(j)p 2136 3985 V Fu(!)2190 4043 y Ft(i)2218 4031 y Fq(j)2241 3996 y Fs(2)2278 4031 y Fv(\()p Fu(\022)2349 4043 y Ft(i)p Fs(+1)2480 4031 y Fq(\000)g Fu(\022)2602 4043 y Ft(i)2629 4031 y Fv(\))h Fq(\000)p 2763 3985 V 18 w Fu(!)2818 4043 y Ft(i)2846 4031 y Fu(\015)2889 4043 y Ft(i)2916 3938 y Fl(\021)2984 4031 y Fv(+)f Fu(')3121 4043 y Ft(k)3162 4031 y Fv(\()p 3194 3985 V Fu(!)3249 4043 y Ft(k)q Fk(\000)p Fs(1)3394 4031 y Fq(\000)g Fu(!)3529 4043 y Ft(F)3583 4031 y Fv(\))p Fu(:)166 b Fv(\(3.17\))236 4271 y(W)-7 b(e)28 b(can)f(also)g(write)355 4501 y Fq(\000)430 4444 y(j)p Fu(!)505 4456 y Ft(I)543 4444 y Fq(j)566 4414 y Fs(2)p 430 4481 174 4 v 496 4557 a Fv(2)613 4501 y Fu(\022)652 4513 y Fs(1)708 4501 y Fv(+)801 4444 y Fq(j)p Fu(!)876 4456 y Ft(F)931 4444 y Fq(j)954 4414 y Fs(2)p 801 4481 191 4 v 875 4557 a Fv(2)1001 4501 y Fu(\022)1040 4513 y Ft(k)1104 4501 y Fv(=)1191 4397 y Ft(k)q Fk(\000)p Fs(1)1192 4422 y Fl(X)1199 4598 y Ft(i)p Fs(=1)1327 4408 y Fl(\020)1377 4501 y Fv(\()1419 4444 y Fq(j)p 1442 4399 55 4 v Fu(!)1497 4456 y Ft(i)1524 4444 y Fq(j)1547 4414 y Fs(2)p 1419 4481 166 4 v 1481 4557 a Fv(2)1613 4501 y Fq(\000)1706 4444 y(j)p 1729 4399 55 4 v Fu(!)1784 4456 y Ft(i)p Fk(\000)p Fs(1)1896 4444 y Fq(j)1919 4414 y Fs(2)p 1706 4481 251 4 v 1810 4557 a Fv(2)1967 4501 y(\))p Fu(\022)2038 4513 y Ft(i)2084 4501 y Fv(+)2177 4444 y Fq(j)p 2200 4399 55 4 v Fu(!)2255 4456 y Ft(i)2282 4444 y Fq(j)2305 4414 y Fs(2)p 2177 4481 166 4 v 2239 4557 a Fv(2)2352 4501 y(\()p Fu(\022)2423 4513 y Ft(i)p Fs(+1)2554 4501 y Fq(\000)18 b Fu(\022)2676 4513 y Ft(i)2704 4501 y Fv(\))2736 4408 y Fl(\021)2804 4501 y Fv(+)2887 4408 y Fl(\020)2946 4444 y Fq(j)p Fu(!)3021 4456 y Ft(F)3076 4444 y Fq(j)3099 4414 y Fs(2)p 2946 4481 191 4 v 3021 4557 a Fv(2)3165 4501 y Fq(\000)3258 4444 y(j)p 3281 4399 55 4 v Fu(!)3336 4456 y Ft(k)q Fk(\000)p Fs(1)3462 4444 y Fq(j)3485 4414 y Fs(2)p 3258 4481 264 4 v 3369 4557 a Fv(2)3532 4408 y Fl(\021)3581 4501 y Fu(\022)3620 4513 y Ft(k)3661 4501 y Fu(;)120 b Fv(\(3.18\))972 4732 y Ft(k)q Fk(\000)p Fs(1)973 4757 y Fl(X)979 4934 y Ft(i)p Fs(=1)1117 4780 y Fv(1)p 1117 4817 42 4 v 1117 4893 a(2)1179 4780 y Fq(j)p Fu(')1256 4792 y Ft(i)p Fs(+1)1386 4780 y Fq(\000)18 b Fu(')1523 4792 y Ft(i)1551 4780 y Fq(j)1574 4750 y Fs(2)p 1179 4817 433 4 v 1203 4893 a Fv(\()p Fu(\022)1274 4905 y Ft(i)p Fs(+1)1405 4893 y Fq(\000)g Fu(\022)1527 4905 y Ft(i)1554 4893 y Fv(\))1645 4836 y(=)1732 4732 y Ft(k)q Fk(\000)p Fs(1)1733 4757 y Fl(X)1739 4934 y Ft(i)p Fs(=1)1878 4780 y Fq(j)p 1901 4734 55 4 v Fu(!)1955 4792 y Ft(i)1983 4780 y Fq(j)2006 4750 y Fs(2)p 1878 4817 166 4 v 1940 4893 a Fv(2)2053 4836 y(\()p Fu(\022)2124 4848 y Ft(i)p Fs(+1)2255 4836 y Fq(\000)g Fu(\022)2377 4848 y Ft(i)2404 4836 y Fv(\))h(+)2548 4780 y(1)p 2548 4817 42 4 v 2548 4893 a(2)2724 4780 y Fq(j)p Fu(\015)2790 4792 y Ft(i)2818 4780 y Fq(j)2841 4750 y Fs(2)p 2609 4817 384 4 v 2609 4893 a Fv(\()p Fu(\022)2680 4905 y Ft(i)p Fs(+1)2811 4893 y Fq(\000)f Fu(\022)2933 4905 y Ft(i)2960 4893 y Fv(\))3021 4836 y(+)p 3104 4790 55 4 v 18 w Fu(!)3159 4848 y Ft(i)3187 4836 y Fu(\015)3230 4848 y Ft(i)3257 4836 y Fu(:)524 b Fv(\(3.19\))236 5054 y(Summing)28 b(\(3.17\),)f(\(3.18\))g(and)g(\(3.19\))g(w)n(e)g(get)739 5200 y Ft(k)q Fk(\000)p Fs(1)740 5225 y Fl(X)746 5402 y Ft(i)p Fs(=1)885 5248 y Fv(1)p 885 5285 42 4 v 885 5361 a(2)946 5248 y Fq(j)p Fu(')1023 5260 y Ft(i)p Fs(+1)1154 5248 y Fq(\000)18 b Fu(')1291 5260 y Ft(i)1319 5248 y Fq(j)1342 5218 y Fs(2)p 946 5285 433 4 v 971 5361 a Fv(\()p Fu(\022)1042 5373 y Ft(i)p Fs(+1)1172 5361 y Fq(\000)g Fu(\022)1294 5373 y Ft(i)1322 5361 y Fv(\))1407 5304 y(+)g Fu(!)1542 5316 y Ft(I)1580 5304 y Fu(')1634 5316 y Fs(1)1690 5304 y Fq(\000)g Fu(!)1825 5316 y Ft(F)1880 5304 y Fu(')1934 5316 y Ft(k)1994 5304 y Fq(\000)2087 5248 y(j)p Fu(!)2162 5260 y Ft(I)2199 5248 y Fq(j)2222 5218 y Fs(2)p 2087 5285 174 4 v 2152 5361 a Fv(2)2270 5304 y Fu(\022)2309 5316 y Fs(1)2364 5304 y Fv(+)2457 5248 y Fq(j)p Fu(!)2532 5260 y Ft(F)2587 5248 y Fq(j)2610 5218 y Fs(2)p 2457 5285 191 4 v 2532 5361 a Fv(2)2658 5304 y Fu(\022)2697 5316 y Ft(k)2760 5304 y Fv(=)2848 5200 y Ft(k)q Fk(\000)p Fs(1)2849 5225 y Fl(X)2855 5402 y Ft(i)p Fs(=1)2994 5248 y Fv(1)p 2994 5285 42 4 v 2994 5361 a(2)3170 5248 y Fq(j)p Fu(\015)3236 5260 y Ft(i)3263 5248 y Fq(j)3286 5218 y Fs(2)p 3055 5285 384 4 v 3055 5361 a Fv(\()p Fu(\022)3126 5373 y Ft(i)p Fs(+1)3256 5361 y Fq(\000)g Fu(\022)3378 5373 y Ft(i)3406 5361 y Fv(\))3448 5304 y(+)p 236 5424 1512 4 v 329 5480 a Fn(5)363 5504 y Fm(F)-6 b(or)24 b Fj(i)19 b Fm(=)h Fj(k)25 b Fm(w)n(e)f(ha)n(v)n (e)g Fj(R)989 5480 y Fi(k)989 5526 y Fn(6)1048 5504 y Fm(=)19 b Fj(R)1175 5480 y Fi(k)1175 5526 y Fn(6)1214 5504 y Fm(\()p Fj(\026;)12 b(\022)1348 5516 y Fi(k)1386 5504 y Fj(;)g(')1464 5516 y Fi(k)1502 5504 y Fm(\))p Fj(:)2084 6059 y Fv(12)p eop %%Page: 13 13 13 12 bop 516 238 a Ft(k)q Fk(\000)p Fs(1)517 263 y Fl(X)523 439 y Ft(i)p Fs(=1)652 249 y Fl(\020)711 285 y Fq(j)p 734 240 55 4 v Fu(!)789 297 y Ft(i)817 285 y Fq(j)840 255 y Fs(2)p 711 322 166 4 v 774 398 a Fv(2)906 341 y Fq(\000)999 285 y(j)p 1022 240 55 4 v Fu(!)1076 297 y Ft(i)p Fk(\000)p Fs(1)1189 285 y Fq(j)1212 255 y Fs(2)p 999 322 251 4 v 1103 398 a Fv(2)1259 249 y Fl(\021)1309 341 y Fu(\022)1348 353 y Ft(i)1394 341 y Fv(+)18 b(\()p 1509 296 55 4 v Fu(!)1564 353 y Ft(i)p Fk(\000)p Fs(1)1695 341 y Fq(\000)p 1778 296 V 18 w Fu(!)1833 353 y Ft(i)1860 341 y Fv(\))p Fu(')1946 353 y Ft(i)1993 341 y Fv(+)g Fu(')2130 353 y Ft(k)2171 341 y Fv(\()p 2203 296 V Fu(!)2258 353 y Ft(k)q Fk(\000)p Fs(1)2402 341 y Fq(\000)g Fu(!)2537 353 y Ft(F)2592 341 y Fv(\))h(+)2726 249 y Fl(\020)2785 285 y Fq(j)p Fu(!)2860 297 y Ft(F)2915 285 y Fq(j)2938 255 y Fs(2)p 2785 322 191 4 v 2860 398 a Fv(2)3004 341 y Fq(\000)3097 285 y(j)p 3120 240 55 4 v Fu(!)3175 297 y Ft(k)q Fk(\000)p Fs(1)3301 285 y Fq(j)3324 255 y Fs(2)p 3097 322 264 4 v 3208 398 a Fv(2)3371 249 y Fl(\021)3420 341 y Fu(\022)3459 353 y Ft(k)3500 341 y Fu(:)281 b Fv(\(3.20\))236 565 y(Substituting)p 713 519 55 4 v 30 w Fu(')767 585 y Ft(i)814 565 y Fv(+)18 b Fu(a)941 577 y Ft(i)997 565 y Fv(for)28 b Fu(')1179 577 y Ft(i)1235 565 y Fv(and)p 1398 497 42 4 v 29 w Fu(\022)1439 577 y Ft(i)1486 565 y Fv(+)18 b Fu(b)1605 577 y Ft(i)1661 565 y Fv(for)28 b Fu(\022)1828 577 y Ft(i)1855 565 y Fv(,)h(w)n(e)f(get)g Fu(\015)2212 577 y Ft(i)2264 565 y Fv(=)c(\()p Fu(a)2429 577 y Ft(i)p Fs(+1)2560 565 y Fq(\000)19 b Fu(a)2688 577 y Ft(i)2715 565 y Fv(\))h Fq(\000)p 2850 519 55 4 v 18 w Fu(!)2905 577 y Ft(i)2932 565 y Fv(\()p Fu(b)3000 577 y Ft(i)p Fs(+1)3131 565 y Fq(\000)f Fu(b)3251 577 y Ft(i)3278 565 y Fv(\).)40 b(Moreo)n(v)n(er)25 b(the)k(non)236 664 y(constan)n(t)e(terms)g(in)h(the)g(righ)n(t)f(handside)g(of)h (\(3.20\))f(\(i.e.)37 b(those)27 b(dep)r(ending)h(on)g Fu(a)2875 676 y Ft(i)2902 664 y Fu(;)14 b(b)2975 676 y Ft(i)3002 664 y Fv(\))28 b(are)f(the)h(\014rst)f(one)g(and)1105 800 y Ft(k)1064 825 y Fl(X)1070 1001 y Ft(i)p Fs(=1)1184 903 y Fv(\()p 1216 858 V Fu(!)1271 915 y Ft(i)p Fk(\000)p Fs(1)1402 903 y Fq(\000)p 1485 858 V 18 w Fu(!)1539 915 y Ft(i)1567 903 y Fv(\))p Fu(a)1643 915 y Ft(i)1689 903 y Fv(+)1772 811 y Fl(\020)1832 847 y Fq(j)p 1855 802 V Fu(!)1910 859 y Ft(i)1937 847 y Fq(j)1960 817 y Fs(2)p 1832 884 166 4 v 1894 960 a Fv(2)2026 903 y Fq(\000)2119 847 y(j)p 2142 802 55 4 v Fu(!)2197 859 y Ft(i)p Fk(\000)p Fs(1)2309 847 y Fq(j)2332 817 y Fs(2)p 2119 884 251 4 v 2223 960 a Fv(2)2380 811 y Fl(\021)2429 903 y Fu(b)2465 915 y Ft(i)2515 903 y Fv(=:)2668 800 y Ft(k)2626 825 y Fl(X)2632 1001 y Ft(i)p Fs(=1)2760 903 y Fu(R)2824 869 y Ft(i)2851 903 y Fv(\()p Fu(\026;)14 b(\022)3009 915 y Ft(i)3037 903 y Fu(;)g(')3128 915 y Ft(i)3156 903 y Fv(\))236 1144 y(with)23 b Fq(r)p Fu(R)553 1114 y Ft(i)581 1144 y Fv(\()p Fu(\026;)14 b(\022)739 1156 y Ft(i)766 1144 y Fu(;)g(')857 1156 y Ft(i)885 1144 y Fv(\))24 b(=)e Fu(O)r Fv(\(\001)p 1194 1098 55 4 v Fu(!)1250 1156 y Ft(i)1278 1144 y Fv(\))p Fu(:)h Fv(Finally)-7 b(,)23 b(expressing)e Fu(\015)2088 1156 y Ft(i)2138 1144 y Fv(in)i(terms)f(of) g(\()p Fu(b)2613 1156 y Ft(i)2641 1144 y Fu(;)14 b(c)2714 1156 y Ft(i)2741 1144 y Fv(\))23 b(w)n(e)f(get)h Fu(\015)3090 1156 y Ft(i)3140 1144 y Fv(=)g(\()p Fu(a)3304 1156 y Ft(i)p Fs(+1)3424 1144 y Fq(\000)8 b Fu(a)3541 1156 y Ft(i)3568 1144 y Fv(\))g Fq(\000)p 3681 1098 V 8 w Fu(!)3736 1156 y Ft(i)3763 1144 y Fv(\()p Fu(b)3831 1156 y Ft(i)p Fs(+1)3951 1144 y Fq(\000)236 1244 y Fu(b)272 1256 y Ft(i)299 1244 y Fv(\))24 b(=)e(\()p Fu(c)510 1256 y Ft(i)p Fs(+1)641 1244 y Fq(\000)c Fu(c)760 1256 y Ft(i)787 1244 y Fv(\))h(+)f Fu(b)957 1256 y Ft(i)p Fs(+1)1069 1244 y Fv(\001)p 1138 1198 V Fu(!)1192 1256 y Ft(i)1248 1244 y Fv(and)27 b(then)h(from)f(\(3.20\),)g(dev)n(eloping)g(the)h(square,)e (w)n(e)h(get)h(\(3.16\).)p 3428 1244 48 48 v 361 1383 a(F)-7 b(rom)37 b(\(3.7\))h(lemmas)f(3.2,)j(3.3)d(and)h(3.4)f(w)n(e)h (obtain)f(the)h(expression)f(of)h Fq(F)2868 1395 y Ft(\026)2950 1383 y Fv(in)g(the)g(new)g(co)r(ordinates)f(\()p Fu(b;)14 b(c)p Fv(\))236 1483 y(required)27 b(to)g(apply)h(the)g(v)-5 b(ariational)26 b(argumen)n(t)g(of)i(section)f(6.)236 1640 y Fw(Lemma)32 b(3.5)40 b Fp(Ther)l(e)31 b(exists)e Fu(\026)1254 1652 y Fs(5)1292 1640 y Fu(;)14 b(C)1388 1652 y Fs(2)1448 1640 y Fu(>)23 b Fv(0)29 b Fp(such)h(that)g Fq(8)f Fv(0)23 b Fu(<)g(\026)g Fq(\024)f Fu(\026)2404 1652 y Fs(5)2442 1640 y Fp(,)30 b(if)889 1797 y Fu(\014)936 1809 y Ft(i)986 1797 y Fq(\025)23 b Fu(C)1133 1809 y Fs(2)1184 1797 y Fv(max)29 b Fq(f)h Fu(\026)1490 1763 y Fs(1)p Ft(=)p Fs(2)1594 1797 y Fv(\()p Fu(\022)1665 1809 y Ft(i)p Fs(+1)1795 1797 y Fq(\000)18 b Fu(\022)1917 1809 y Ft(i)1945 1797 y Fv(\))1977 1763 y Fs(1)p Ft(=)p Fs(2)2082 1797 y Fu(;)43 b(\026)p Fv(\()p Fu(\022)2269 1809 y Ft(i)p Fs(+1)2400 1797 y Fq(\000)18 b Fu(\022)2522 1809 y Ft(i)2549 1797 y Fv(\))2581 1763 y Fs(3)p Ft(=)p Fs(2)2686 1797 y Fu(;)44 b Fv(\()p Fu(\022)2824 1809 y Ft(i)p Fs(+1)2954 1797 y Fq(\000)18 b Fu(\022)3076 1809 y Ft(i)3104 1797 y Fv(\))3136 1763 y Fk(\000)p Fs(1)p Ft(=)p Fs(2)3322 1797 y Fq(g)440 b Fv(\(3.21\))236 1954 y Fp(then)582 2172 y Fq(F)642 2184 y Ft(\026)686 2172 y Fv(\()p Fu(b;)14 b(c)p Fv(\))94 b(=)1122 2116 y(1)p 1122 2153 42 4 v 1122 2229 a(2)1188 2069 y Ft(k)q Fk(\000)p Fs(1)1189 2094 y Fl(X)1195 2270 y Ft(i)p Fs(=1)1443 2116 y Fq(j)p Fu(c)1502 2128 y Ft(i)p Fs(+1)1632 2116 y Fq(\000)18 b Fu(c)1751 2128 y Ft(i)1779 2116 y Fq(j)1802 2086 y Fs(2)p 1333 2153 617 4 v 1333 2238 a Fv(\001)p 1402 2170 42 4 v Fu(\022)1444 2250 y Ft(i)1490 2238 y Fv(+)g(\()p Fu(b)1641 2250 y Ft(i)p Fs(+1)1771 2238 y Fq(\000)g Fu(b)1890 2250 y Ft(i)1917 2238 y Fv(\))1978 2172 y(+)g Fu(\026)2166 2069 y Ft(k)2125 2094 y Fl(X)2131 2270 y Ft(i)p Fs(=1)2258 2172 y Fv(\000\()p 2342 2127 55 4 v Fu(!)2397 2184 y Ft(i)2425 2172 y Fu(;)p 2462 2105 42 4 v 14 w(\022)2503 2184 y Ft(i)2549 2172 y Fv(+)g Fu(b)2668 2184 y Ft(i)2696 2172 y Fu(;)p 2733 2127 55 4 v 14 w(')2787 2193 y Ft(i)2833 2172 y Fv(+)p 2916 2127 V 18 w Fu(!)2970 2184 y Ft(i)2998 2172 y Fu(b)3034 2184 y Ft(i)3080 2172 y Fv(+)g Fu(c)3199 2184 y Ft(i)3226 2172 y Fv(\))h(+)f Fu(R)3423 2184 y Fs(7)3460 2172 y Fv(\()p Fu(b;)c(c)p Fv(\))p Fu(;)148 b Fv(\(3.22\))585 2464 y Fu(R)648 2476 y Fs(7)686 2464 y Fv(\()p Fu(b;)14 b(c)p Fv(\))83 b(:=)1154 2361 y Ft(k)1112 2385 y Fl(X)1119 2562 y Ft(i)p Fs(=1)1246 2464 y Fu(R)1310 2430 y Ft(i)1309 2485 y Fs(7)1346 2464 y Fv(\()p Fu(\026;)14 b(b)1501 2476 y Ft(i)p Fk(\000)p Fs(1)1614 2464 y Fu(;)g(c)1687 2476 y Ft(i)p Fk(\000)p Fs(1)1799 2464 y Fu(;)g(b)1872 2476 y Ft(i)1900 2464 y Fu(;)g(c)1973 2476 y Ft(i)2000 2464 y Fu(;)g(b)2073 2476 y Ft(i)p Fs(+1)2184 2464 y Fu(;)g(c)2257 2476 y Ft(i)p Fs(+1)2369 2464 y Fv(\))p Fu(;)1380 b Fv(\(3.23\))236 2704 y Fp(wher)l(e)440 2674 y Fs(6)1857 2804 y Fq(jr)p Fu(R)2013 2769 y Ft(i)2012 2824 y Fs(7)2049 2804 y Fq(j)23 b(\024)g Fu(C)2242 2816 y Fs(2)2279 2804 y Fu(\032\026:)1409 b Fv(\(3.24\))236 2961 y Fb(Pr)n(oof)p Fv(.)78 b(It)28 b(is)g(easy)e(to)i(see)f(that)h (\(3.6\),)f(\(3.8\))h(and)f(\(3.21\))g(imply)h(\(pro)n(vided)f Fu(\026)h Fv(is)f(small)g(enough\))g(that)890 3105 y Fu(\014)937 3117 y Ft(i)p Fk(\000)p Fs(1)p 890 3142 160 4 v 949 3218 a Fv(2)1083 3161 y Fq(\024)c Fu(\014)1218 3173 y Ft(i)1268 3161 y Fq(\024)g Fv(2)p Fu(\014)1445 3173 y Ft(i)p Fk(\000)p Fs(1)1557 3161 y Fu(;)180 b Fq(j)p Fu(\014)1830 3173 y Ft(i)1876 3161 y Fq(\000)18 b Fu(\014)2006 3173 y Ft(i)p Fk(\000)p Fs(1)2119 3161 y Fq(j)23 b Fv(=)f Fu(O)2317 3069 y Fl(\020)2517 3105 y Fv(1)p 2378 3142 320 4 v 2378 3218 a Fu(\022)2417 3230 y Ft(i)2463 3218 y Fq(\000)c Fu(\022)2585 3230 y Ft(i)p Fk(\000)p Fs(1)2726 3161 y Fv(+)2957 3105 y(1)p 2819 3142 319 4 v 2819 3218 a Fu(\022)2858 3230 y Ft(i)p Fs(+1)2988 3218 y Fq(\000)g Fu(\022)3110 3230 y Ft(i)3166 3161 y Fv(+)g Fu(\026)3299 3069 y Fl(\021)3348 3161 y Fu(:)433 b Fv(\(3.25\))236 3366 y(Noting)28 b(that)g Fu(@)735 3378 y Ft(c)765 3386 y Fc(i)819 3366 y Fv(=)c Fu(@)952 3378 y Ft(')996 3386 y Fc(i)1054 3366 y Fv(and)k Fu(@)1260 3378 y Ft(b)1289 3386 y Fc(i)1343 3366 y Fv(=)p 1432 3320 55 4 v 24 w Fu(!)1486 3378 y Ft(i)1514 3366 y Fu(@)1558 3378 y Ft(')1602 3386 y Fc(i)1651 3366 y Fv(+)18 b Fu(@)1778 3378 y Ft(\022)1810 3386 y Fc(i)1840 3366 y Fv(,)29 b(estimate)f(\(3.24\))f(follo)n(ws)g (from)g(\(3.2\),)h(\(3.9\),)g(\(3.11\),)g(\(3.25\))f(and)236 3465 y(\(3.16\).)p 508 3465 48 48 v 236 3775 a Fr(4)135 b(Ergo)t(dization)46 b(times)236 3957 y Fv(In)26 b(order)d(to)i (de\014ne)p 889 3911 55 4 v 26 w Fu(')943 3977 y Ft(i)971 3957 y Fu(;)p 1008 3889 42 4 v 14 w(\022)1049 3969 y Ft(i)1102 3957 y Fv(\(1)e Fq(\024)f Fu(i)h Fq(\024)g Fu(k)s Fv(\))i(w)n(e)g(need)g(some)g(results,)g(stated)g(in)g(this)h (section,)f(on)g(the)h(ergo)r(dization)d(time)236 4057 y(of)28 b(the)g(torus)f Fw(T)753 4026 y Ft(l)802 4057 y Fv(:=)22 b Fw(R)984 4026 y Ft(l)1009 4057 y Fu(=)p Fw(Z)1109 4026 y Ft(l)1162 4057 y Fv(for)27 b(linear)g(\015o)n(ws)g(p)r (ossibly)g(resonan)n(t)f(but)i(only)g(at)f(a)g(\\su\016cien)n(tly)g (high)h(lev)n(el",)361 4180 y(Let)h(\012)d Fq(2)g Fw(R)750 4150 y Ft(l)775 4180 y Fv(;)k(it)f(is)g(w)n(ell)g(kno)n(wn)g(that,)h (if)f(\012)20 b Fq(\001)f Fu(p)26 b Fq(6)p Fv(=)f(0,)k Fq(8)p Fu(p)d Fq(2)g Fw(Z)2341 4150 y Ft(l)2386 4180 y Fq(n)19 b(f)p Fv(0)p Fq(g)p Fu(;)28 b Fv(then)i(the)f(tra)5 b(jectories)27 b(of)j(the)f(linear)f(\015o)n(w)236 4279 y Fq(f)p Fv(\012)p Fu(t)14 b Fv(+)h Fu(A)p Fq(g)566 4291 y Ft(t)p Fk(2)p Fe(R)721 4279 y Fv(are)25 b(dense)g(on)h Fw(T)1262 4249 y Ft(l)1314 4279 y Fv(for)f(an)n(y)g(initial)h(p)r(oin)n (t)g Fu(A)d Fq(2)g Fw(T)2276 4249 y Ft(l)2302 4279 y Fv(.)37 b(It)26 b(is)f(also)g(in)n(tuitiv)n(ely)h(clear)e(that)j(the)f (tra)5 b(jectories)236 4379 y(of)37 b(the)h(linear)f(\015o)n(w)f Fq(f)p Fv(\012)p Fu(t)25 b Fv(+)f Fu(A)p Fq(g)1266 4391 y Ft(t)p Fk(2)p Fe(R)1433 4379 y Fv(will)37 b(mak)n(e)g(an)g (arbitrarly)e(\014ne)j Fu(\016)s Fv(-net)f(\()p Fu(\016)42 b(>)d Fv(0\))e(if)h(\012)f(is)g(resonan)n(t)f(only)h(at)g(a)236 4479 y(su\016cien)n(tly)26 b(high)g(lev)n(el,)h(namely)e(if)i(\012)16 b Fq(\001)f Fu(p)23 b Fq(6)p Fv(=)g(0,)j Fq(8)p Fu(p)d Fq(2)g Fw(Z)2011 4449 y Ft(l)2063 4479 y Fv(with)k(0)c Fu(<)f Fq(j)p Fu(p)p Fq(j)h(\024)g Fu(M)9 b Fv(\()p Fu(\016)s Fv(\))26 b(for)g(some)g(large)e(enough)i Fu(M)9 b Fv(\()p Fu(\016)s Fv(\).)37 b(Let)236 4578 y(us)28 b(mak)n(e)f(more)f(precise)h (and)h(quan)n(titativ)n(e)e(these)i(considerations.)361 4678 y(F)-7 b(or)27 b(an)n(y)g(\012)c Fq(2)g Fw(R)900 4648 y Ft(l)953 4678 y Fv(de\014ne)28 b(the)g(ergo)r(dization)e(time)i Fu(T)12 b Fv(\(\012)p Fu(;)i(\016)s Fv(\))27 b(required)f(to)i(\014ll)g Fw(T)2900 4648 y Ft(l)2953 4678 y Fv(within)h Fu(\016)d(>)c Fv(0)28 b(as)1013 4860 y Fu(T)12 b Fv(\(\012)p Fu(;)i(\016)s Fv(\))23 b(=)g(inf)1500 4768 y Fl(n)1556 4860 y Fu(t)g Fq(2)g Fw(R)1759 4872 y Fs(+)1842 4765 y Fl(\014)1842 4814 y(\014)1842 4864 y(\014)1897 4860 y Fq(8)p Fu(x)h Fq(2)f Fw(R)2164 4826 y Ft(l)2189 4860 y Fu(;)42 b(d)p Fv(\()p Fu(x;)14 b(A)19 b Fv(+)f([0)p Fu(;)c(t)p Fv(]\012)k(+)g Fw(Z)2951 4826 y Ft(l)2977 4860 y Fv(\))23 b Fq(\024)g Fu(\016)3160 4768 y Fl(o)3215 4860 y Fu(;)236 5050 y Fv(where)29 b Fu(d)h Fv(is)g(the)g(Euclidean)f(distance)g(and)h Fu(A)g Fv(some)f(p)r(oin)n(t)h(of)g Fw(R)2347 5020 y Ft(l)2372 5050 y Fv(.)43 b Fu(T)12 b Fv(\(\012)p Fu(;)i(\016)s Fv(\))29 b(is)h(clearly)e(indep)r(enden)n(t)j(of)f(the)g(c)n(hoice)236 5150 y(of)e Fu(A)p Fv(.)37 b(Ab)r(o)n(v)n(e)27 b(and)g(in)h(what)g (follo)n(ws,)e(inf)21 b Fu(E)33 b Fv(is)28 b(equal)f(to)g(+)p Fq(1)g Fv(if)h Fu(E)33 b Fv(is)28 b(empt)n(y)-7 b(.)37 b(F)-7 b(or)27 b Fu(R)c(>)g Fv(0)k(let)1228 5327 y Fu(\013)p Fv(\(\012)p Fu(;)14 b(R)q Fv(\))23 b(=)g(inf)1732 5235 y Fl(n)1787 5327 y Fq(j)p Fu(p)18 b Fq(\001)h Fv(\012)p Fq(j)2023 5232 y Fl(\014)2023 5282 y(\014)2023 5332 y(\014)2078 5327 y Fu(p)k Fq(2)g Fw(Z)2279 5293 y Ft(l)2305 5327 y Fu(;)69 b(p)23 b Fq(6)p Fv(=)g(0)k Fu(;)42 b Fq(j)p Fu(p)p Fq(j)22 b(\024)h Fu(R)2946 5235 y Fl(o)3001 5327 y Fu(:)p 236 5424 1512 4 v 329 5480 a Fn(6)363 5504 y Fm(In)h(the)h(cases)f Fj(i)c Fm(=)f(1)p Fj(;)12 b(i)20 b Fm(=)f Fj(k)25 b Fm(w)n(e)f(ha)n(v)n(e)h Fj(R)1435 5480 y Fn(1)1435 5526 y(7)1489 5504 y Fm(=)20 b Fj(R)1617 5480 y Fn(1)1617 5526 y(7)1652 5504 y Fm(\()p Fj(\026;)12 b(\022)1786 5513 y Fn(1)1821 5504 y Fj(;)f(')1898 5513 y Fn(1)1933 5504 y Fj(;)g(\022)1997 5513 y Fn(2)2032 5504 y Fj(;)g(')2109 5513 y Fn(2)2144 5504 y Fm(\))24 b(and)g Fj(R)2385 5480 y Fi(k)2385 5526 y Fn(7)2443 5504 y Fm(=)c Fj(R)2571 5480 y Fi(k)2571 5526 y Fn(7)2609 5504 y Fm(\()p Fj(\026;)12 b(\022)2743 5516 y Fi(k)q Fh(\000)p Fn(1)2860 5504 y Fj(;)f(')2937 5516 y Fi(k)q Fh(\000)p Fn(1)3054 5504 y Fj(;)g(\022)3118 5516 y Fi(k)3156 5504 y Fj(;)h(')3234 5516 y Fi(k)3272 5504 y Fm(\))p Fj(:)2084 6059 y Fv(13)p eop %%Page: 14 14 14 13 bop 236 272 a Fw(Theorem)32 b(4.1)40 b Fq(8)p Fu(l)48 b Fq(2)e Fw(N)c Fp(ther)l(e)h(exists)f(a)h(p)l(ositive)h(c)l(onstant)d Fu(a)2386 284 y Ft(l)2454 272 y Fp(such)h(that,)47 b Fq(8)p Fv(\012)f Fq(2)g Fw(R)3192 242 y Ft(l)3217 272 y Fp(,)g Fq(8)p Fu(\016)j(>)d Fv(0)p Fp(,)g Fu(T)12 b Fv(\(\012)p Fu(;)i(\016)s Fv(\))45 b Fq(\024)236 372 y Fv(\()p Fu(\013)p Fv(\(\012)p Fu(;)14 b(a)494 384 y Ft(l)520 372 y Fu(=\016)s Fv(\)\))666 341 y Fk(\000)p Fs(1)756 372 y Fp(.)38 b(Mor)l(e)l(over)31 b Fu(T)12 b Fv(\(\012)p Fu(;)i(\016)s Fv(\))23 b Fq(\025)g Fv(\(1)p Fu(=)p Fv(4\))p Fu(\013)p Fv(\(\012)p Fu(;)14 b Fv(1)p Fu(=)p Fv(4)p Fu(\016)s Fv(\))2127 341 y Fk(\000)p Fs(1)2214 372 y Fp(.)361 536 y Fv(In)28 b(the)g(ab)r(o)n(v)n(e)e(Theorem)h Fu(\013)1247 506 y Fk(\000)p Fs(1)1364 536 y Fv(is)g(equal)g(to)h(0)f (if)h Fu(\013)c Fv(=)e(+)p Fq(1)27 b Fv(and)h(to)f(+)p Fq(1)h Fv(if)g Fu(\013)23 b Fv(=)g(0.)236 716 y Fw(Remark)32 b(4.1)41 b Fp(Assume)27 b(that)g Fv(\012)h Fp(is)g(a)g Fu(C)6 b Fp(-)p Fu(\034)38 b Fp(Diophantine)29 b(ve)l(ctor,)g(i.e.)39 b(ther)l(e)28 b(exist)f Fu(C)j(>)22 b Fv(0)28 b Fp(and)g Fu(\034)33 b Fq(\025)22 b Fu(l)16 b Fq(\000)e Fv(1)27 b Fp(such)h(that)236 816 y Fq(8)p Fu(k)f Fq(2)d Fw(Z)489 786 y Ft(l)545 816 y Fq(j)p Fu(k)d Fq(\001)e Fv(\012)p Fq(j)24 b(\025)f Fu(C)t(=)p Fq(j)p Fu(k)s Fq(j)1066 786 y Ft(\034)1107 816 y Fp(.)40 b(Then)31 b Fu(\013)p Fv(\(\012)p Fu(;)14 b(R)q Fv(\))24 b Fq(\025)f Fu(C)t(=R)1948 786 y Ft(\034)2019 816 y Fp(and)31 b(so)f Fu(T)12 b Fv(\(\012)p Fu(;)i(\016)s Fv(\))23 b Fq(\024)h Fu(a)2705 786 y Ft(\034)2705 839 y(l)2746 816 y Fu(=C)6 b(\016)2893 786 y Ft(\034)2934 816 y Fp(.)40 b(This)31 b(estimate)g(was)f(pr)l(ove)l(d)i(in)236 915 y(The)l(or)l(em)f(D)e(of)i([11)q(].)39 b(A)n(lso)30 b(The)l(or)l(em)h(B)f(of)h([11)q(])f(is)g(an)g(e)l(asy)h(c)l(onse)l (quenc)l(e)e(of)i(The)l(or)l(em)f(4.1.)361 1096 y Fv(Theorem)i(4.1)h (is)h(a)f(direct)g(consequence)g(of)g(more)g(general)f(statemen)n(ts,)j (see)e(Theorem)g(4.2)f(and)i(remark)e(4.2.)236 1195 y(Let)d(us)f(in)n (tro)r(duce)g(\014rst)h(some)e(notations.)39 b(Let)29 b(\003)f(b)r(e)h(a)f(lattice)h(of)f Fw(R)2486 1165 y Ft(l)2511 1195 y Fv(,)h Fp(i.e.)41 b Fv(a)28 b(discrete)g(subgroup)f (of)i Fw(R)3622 1165 y Ft(l)3675 1195 y Fv(suc)n(h)g(that)236 1295 y Fw(R)308 1265 y Ft(l)333 1295 y Fu(=)p Fv(\003)e(has)g(\014nite) h(v)n(olume.)37 b(F)-7 b(or)27 b(all)g(\012)c Fq(2)g Fw(R)1636 1265 y Ft(l)1689 1295 y Fv(w)n(e)k(de\014ne)1091 1500 y Fu(T)12 b Fv(\(\003)p Fu(;)i Fv(\012)p Fu(;)g(\016)s Fv(\))22 b(=)h(inf)1673 1408 y Fl(n)1728 1500 y Fu(t)g Fq(2)h Fw(R)1932 1512 y Fs(+)2014 1405 y Fl(\014)2014 1455 y(\014)2014 1504 y(\014)2069 1500 y Fq(8)p Fu(x)g Fq(2)f Fw(R)2336 1466 y Ft(l)2389 1500 y Fu(d)p Fv(\()p Fu(x;)14 b Fv([0)p Fu(;)g(t)p Fv(]\012)19 b(+)f(\003\))23 b Fq(\024)f Fu(\016)3105 1408 y Fl(o)236 1714 y Fv(\()p Fu(T)12 b Fv(\(\003)p Fu(;)i Fv(\012)p Fu(;)g(\016)s Fv(\))23 b(is)h(the)g(time)g(required)f(to)g(ha)n(v)n(e)g(a)g Fu(\016)s Fv(-net)g(of)h(the)g(torus)f Fw(R)2441 1684 y Ft(l)2466 1714 y Fu(=)p Fv(\003)g(endo)n(w)n(ed)g(with)h(the)g (metric)f(inherited)h(from)236 1813 y Fw(R)308 1783 y Ft(l)333 1813 y Fv(\).)38 b(F)-7 b(or)27 b Fu(R)c(>)g Fv(0,)k(let)807 2019 y(\003)865 1984 y Fk(\003)926 2019 y Fv(=)1013 1926 y Fl(n)1069 2019 y Fu(p)22 b Fq(2)i Fw(R)1284 1984 y Ft(l)1337 1923 y Fl(\014)1337 1973 y(\014)1337 2023 y(\014)1392 2019 y Fq(8)p Fu(\025)f Fq(2)h Fv(\003)p Fu(;)41 b(p)18 b Fq(\001)h Fu(\025)k Fq(2)h Fw(Z)2020 1926 y Fl(o)2158 2019 y Fv(and)83 b(\003)2433 1984 y Fk(\003)2433 2039 y Ft(R)2510 2019 y Fv(=)2598 1926 y Fl(n)2653 2019 y Fu(p)23 b Fq(2)h Fv(\003)2855 1984 y Fk(\003)2920 1923 y Fl(\014)2920 1973 y(\014)2920 2023 y(\014)2975 2019 y Fv(0)f Fu(<)g Fq(j)p Fu(p)p Fq(j)g(\024)f Fu(R)3390 1926 y Fl(o)236 2232 y Fv(\(\003)326 2202 y Fk(\003)392 2232 y Fv(is)27 b(a)h(lattice)f(of)h Fw(R)965 2202 y Ft(l)1018 2232 y Fv(whic)n(h)f(is)h(conjugated)f(to)g(\003\).)37 b(W)-7 b(e)28 b(de\014ne)1487 2437 y Fu(\013)p Fv(\(\003)p Fu(;)14 b Fv(\012)p Fu(;)g(R)q Fv(\))23 b(=)f(inf)2085 2345 y Fl(n)2141 2437 y Fq(j)p Fu(p)c Fq(\001)g Fv(\012)p Fq(j)2376 2342 y Fl(\014)2376 2392 y(\014)2376 2441 y(\014)2432 2437 y Fu(p)k Fq(2)i Fv(\003)2633 2403 y Fk(\003)2633 2458 y Ft(R)2687 2345 y Fl(o)2742 2437 y Fu(:)236 2638 y Fv(The)k(follo)n(wing)e(result)i(holds:)236 2802 y Fw(Theorem)k(4.2)40 b Fq(8)p Fu(l)d Fq(2)e Fw(N)h Fp(ther)l(e)g(exists) g(a)g(p)l(ositive)i(c)l(onstant)e Fu(a)2327 2814 y Ft(l)2388 2802 y Fp(such)g(that,)j(for)e(al)t(l)g(lattic)l(e)g Fv(\003)f Fp(of)h Fw(R)3574 2772 y Ft(l)3599 2802 y Fp(,)h Fq(8)p Fv(\012)d Fq(2)g Fw(R)3965 2772 y Ft(l)3990 2802 y Fp(,)236 2902 y Fq(8)p Fu(\016)26 b(>)d Fv(0)p Fp(,)30 b Fu(T)12 b Fv(\(\003)p Fu(;)i Fv(\012)p Fu(;)g(\016)s Fv(\))22 b Fq(\024)h Fv(\()p Fu(\013)p Fv(\(\003)p Fu(;)14 b Fv(\012)p Fu(;)g(a)1350 2914 y Ft(l)1375 2902 y Fu(=\016)s Fv(\)\))1521 2872 y Fk(\000)p Fs(1)1611 2902 y Fp(.)236 3082 y Fw(Remark)32 b(4.2)41 b Fp(It)30 b(is)h(fairly)h(obvious)g(that) e Fu(T)12 b Fv(\(\003)p Fu(;)i Fv(\012)p Fu(;)g(\016)s Fv(\))24 b Fq(\025)g Fv(\(1)p Fu(=)p Fv(4\))p Fu(\013)p Fv(\(\003)p Fu(;)14 b Fv(\012)p Fu(;)g Fv(1)p Fu(=)p Fv(4)p Fu(\016)s Fv(\))2764 3052 y Fk(\000)p Fs(1)2851 3082 y Fp(.)41 b(Inde)l(e)l(d,)31 b(assume)g(that)f Fv(\003)3724 3052 y Fk(\003)3724 3109 y Fs(1)p Ft(=)p Fs(4)p Ft(\016)3885 3082 y Fq(6)p Fv(=)24 b Fq(;)236 3202 y Fp(and)41 b(let)e Fu(p)j Fq(2)g Fv(\003)773 3172 y Fk(\003)773 3229 y Fs(1)p Ft(=)p Fs(4)p Ft(\016)949 3202 y Fp(b)l(e)e(such)g(that)g Fu(p)25 b Fq(\001)h Fv(\012)41 b(=)g Fu(\013)h Fv(:=)e Fu(\013)p Fv(\(\003)p Fu(;)14 b Fv(\012)p Fu(;)g Fv(1)p Fu(=)p Fv(4)p Fu(\016)s Fv(\))p Fp(.)68 b(L)l(et)39 b Fu(x)j Fq(2)g Fw(R)2965 3172 y Ft(l)3030 3202 y Fp(satisfy)f Fu(p)26 b Fq(\001)f Fu(x)42 b Fv(=)f(1)p Fu(=)p Fv(2)p Fp(.)68 b(Then)236 3314 y Fq(8)p Fu(t)23 b Fq(2)h Fv([0)p Fu(;)14 b Fv(1)p Fu(=)p Fv(4)p Fu(\013)p Fv(\))p Fp(,)29 b Fq(8)p Fu(\025)23 b Fq(2)h Fv(\003)p Fp(,)1033 3542 y Fq(j)p Fu(x)19 b Fq(\000)f Fv(\()p Fu(t)p Fv(\012)g(+)g Fu(\025)p Fv(\))p Fq(j)24 b(\025)1653 3486 y(j)p Fu(p)18 b Fq(\001)h Fv(\()p Fu(x)g Fq(\000)f Fu(t)p Fv(\012)g Fq(\000)g Fu(\025)p Fv(\))p Fq(j)p 1653 3523 602 4 v 1909 3599 a(j)p Fu(p)p Fq(j)2287 3542 y(\025)23 b Fv(4)p Fu(\016)s Fq(j)p Fu(p)17 b Fq(\001)i Fu(x)g Fq(\000)f Fu(tp)g Fq(\001)h Fv(\012)f Fq(\000)g Fu(p)g Fq(\001)h Fu(\025)p Fq(j)p Fu(;)236 3771 y Fp(and)30 b Fu(p)19 b Fq(\001)f Fu(x)h Fq(\000)f Fu(p)g Fq(\001)h Fu(\025)k Fq(2)h Fv(\(1)p Fu(=)p Fv(2\))17 b(+)h Fw(Z)p Fp(,)31 b(wher)l(e)l(as)f Fq(j)p Fu(tp)19 b Fq(\001)f Fv(\012)p Fq(j)23 b Fv(=)g Fu(t\013)g(<)g Fv(1)p Fu(=)p Fv(4)p Fp(.)37 b(Henc)l(e)30 b Fq(j)p Fu(x)19 b Fq(\000)f Fv(\()p Fu(t)p Fv(\012)g(+)g Fu(\025)p Fv(\))p Fq(j)24 b Fu(>)f(\016)s Fp(.)361 3951 y Fv(In)31 b(the)g(next)f(section)h(w)n(e)f(will)h(apply) f(Theorem)g(4.1)g(when)h(\012)d(=)g(\()p Fu(!)s(;)14 b Fv(1\))27 b Fq(2)i Fw(R)2873 3921 y Ft(d)p Fs(+1)2995 3951 y Fv(.)46 b(The)31 b(pro)r(of)f(of)h(Theorem)f(4.2)236 4051 y(is)j(giv)n(en)f(in)i(the)f(App)r(endix.)54 b(W)-7 b(e)34 b(could)f(giv)n(e)f(an)g(explicit)i(expression)d(of)i Fu(a)2747 4063 y Ft(l)2773 4051 y Fv(.)53 b(Ho)n(w)n(ev)n(er)31 b(it)j(is)f(not)g(useful)g(for)g(our)236 4151 y(purp)r(ose)27 b(and)h(the)g(constan)n(ts)e Fu(a)1264 4163 y Ft(l)1317 4151 y Fv(whic)n(h)i(can)f(b)r(e)h(deriv)n(ed)f(from)g(our)g(pro)r(of)g (are)g(certainly)f(far)h(from)h(b)r(eing)f(optimal.)236 4425 y Fr(5)135 b(The)44 b(unp)t(erturb)t(ed)g(pseudo-di\013usion)h (orbit)236 4607 y Fv(Consider)27 b(the)h(set)f Fu(Q)921 4619 y Ft(M)1022 4607 y Fv(of)h(\\non-ergo)r(dizing)d(frequencies")569 4807 y Fu(Q)635 4819 y Ft(M)731 4807 y Fv(:=)842 4715 y Fl(n)897 4807 y Fu(!)h Fq(2)e Fw(R)1126 4773 y Ft(d)1192 4712 y Fl(\014)1192 4762 y(\014)1192 4811 y(\014)1247 4807 y Fq(9)p Fv(\()p Fu(n;)14 b(l)r Fv(\))23 b Fq(2)g Fw(Z)1630 4773 y Ft(d)p Fs(+1)1781 4807 y Fv(with)28 b(0)23 b Fu(<)f Fq(j)p Fv(\()p Fu(n;)14 b(l)r Fv(\))p Fq(j)23 b(\024)g Fu(M)t(;)41 b Fv(and)28 b Fu(!)21 b Fq(\001)d Fu(n)h Fv(+)f Fu(l)24 b Fv(=)f(0)3213 4715 y Fl(o)3291 4807 y Fv(=)3425 4728 y Fl([)3378 4907 y Ft(h)p Fk(2)p Ft(S)3503 4915 y Fc(M)3579 4807 y Fu(E)3640 4819 y Ft(h)236 5083 y Fv(where)32 b Fu(S)532 5095 y Ft(M)635 5083 y Fv(:=)e Fq(f)p Fu(h)g Fv(=)g(\()p Fu(n;)14 b(l)r Fv(\))30 b Fq(2)g Fv(\()p Fw(Z)1351 5053 y Ft(d)1412 5083 y Fq(n)21 b(f)p Fv(0)p Fq(g)p Fv(\))f Fq(\002)h Fw(N)31 b Fq(j)h Fv(0)e Fu(<)g Fq(j)p Fu(h)p Fq(j)g(\024)g Fu(M)t(;)46 b(h)30 b Fq(6)p Fv(=)f Fu(j)5 b(h)2699 5053 y Fk(0)2723 5083 y Fu(;)14 b Fq(8)g Fu(j)35 b Fq(2)30 b Fw(Z)p Fu(;)14 b(h)3117 5053 y Fk(0)3171 5083 y Fq(2)30 b Fv(\()p Fw(Z)3346 5053 y Ft(d)3407 5083 y Fq(n)21 b(f)p Fv(0)p Fq(g)p Fv(\))f Fq(\002)h Fw(N)p Fq(g)31 b Fv(and)236 5183 y Fu(E)297 5195 y Ft(h)364 5183 y Fv(=)22 b Fu(E)512 5195 y Ft(n;l)622 5183 y Fv(:=)h Fq(f)p Fu(!)i Fq(2)f Fw(R)1003 5153 y Ft(d)1069 5183 y Fq(j)k Fv(\()p Fu(!)s(;)14 b Fv(1\))k Fq(\001)g Fu(h)23 b Fv(=)g Fu(!)e Fq(\001)e Fu(n)f Fv(+)g Fu(l)25 b Fv(=)d(0)p Fq(g)p Fu(:)27 b Fv(By)g(Theorem)g (4.1)g(\(or)g(Theorem)g(4.2,)g(with)h(\003)23 b(=)f(2)p Fu(\031)s Fw(Z)3837 5153 y Ft(d)p Fs(+1)3960 5183 y Fv(\),)236 5283 y(for)27 b Fu(\016)f(>)d Fv(0,)k(if)h Fu(!)j Fv(b)r(elongs)c(to) 1194 5483 y Fu(Q)1260 5449 y Ft(c)1260 5504 y(M)1356 5483 y Fv(=)1444 5391 y Fl(n)1499 5483 y Fu(!)f Fq(2)d Fw(R)1727 5449 y Ft(d)1793 5388 y Fl(\014)1793 5438 y(\014)1793 5487 y(\014)1848 5483 y Fu(!)f Fq(\001)c Fu(n)g Fv(+)g Fu(l)25 b Fq(6)p Fv(=)d(0)p Fu(;)42 b Fq(8)p Fv(0)22 b Fu(<)h Fq(j)p Fv(\()p Fu(n;)14 b(l)r Fv(\))p Fq(j)23 b(\024)f Fu(M)2980 5391 y Fl(o)3035 5483 y Fu(;)787 b Fv(\(5.1\))2084 6059 y(14)p eop %%Page: 15 15 15 14 bop 236 272 a Fv(with)28 b Fu(M)k Fv(=)23 b(8)p Fu(\031)s(a)762 284 y Ft(d)p Fs(+1)884 272 y Fu(=\016)s Fv(,)k(then)h(the)g(\015o)n(w)f(of)h(\()p Fu(!)s(;)14 b Fv(1\))27 b(pro)n(vides)f(a)i Fu(\016)s(=)p Fv(4-net)e(of)i(the)g (torus)f Fw(T)3049 242 y Ft(d)p Fs(+1)3172 272 y Fv(.)361 372 y(Moreo)n(v)n(er)e(if)j Fu(!)35 b(=)-52 b Fq(2)24 b Fu(Q)1024 384 y Ft(M)1125 372 y Fv(then)k(for)f(all)g(\()p Fu(n;)14 b(l)r Fv(\))23 b Fq(2)g Fw(Z)1893 341 y Ft(d)1933 372 y Fq(nf)p Fv(0)p Fq(g)16 b(\002)i Fw(Z)p Fv(,)1002 554 y Fq(j)p Fu(n)h Fq(\001)f Fu(!)j Fv(+)d Fu(l)r Fq(j)23 b Fv(=)g Fq(j)p Fu(n)p Fq(j)p Fv(dist\()p Fu(!)s(;)14 b(E)1867 566 y Ft(n;l)1953 554 y Fv(\))23 b Fq(\025)g Fv(dist\()p Fu(!)s(;)14 b(E)2415 566 y Ft(n;l)2502 554 y Fv(\))23 b Fq(\025)g Fv(dist\()p Fu(!)s(;)14 b(Q)2969 566 y Ft(M)3042 554 y Fv(\))23 b Fu(>)g Fv(0)p Fu(:)595 b Fv(\(5.2\))361 737 y(By)27 b(Theorem)g(4.1)g(\(or)g(Theorem)f(4.2\),) h(w)n(e)h(deduce)f(from)h(\(5)p Fu(:)p Fv(2\))f(the)h(estimate)1605 958 y Fu(T)12 b Fv(\(\()p Fu(!)s(;)i Fv(1\))p Fu(;)g(\016)s(=)p Fv(4\))21 b Fq(\024)2377 902 y Fv(2)p Fu(\031)p 2208 939 430 4 v 2208 1015 a Fv(dist\()p Fu(!)s(;)14 b(Q)2532 1027 y Ft(M)2605 1015 y Fv(\))3845 958 y(\(5.3\))236 1190 y(whic)n(h)28 b(measures)e(the)i(div)n(ergence)e(of)i(the)g(ergo)r (dization)e(time)i Fu(T)12 b Fv(\(\()p Fu(!)s(;)i Fv(1\))p Fu(;)g(\016)s Fv(\))27 b(as)g Fu(!)j Fv(approac)n(hes)25 b(the)j(set)g Fu(Q)3656 1202 y Ft(M)3729 1190 y Fv(.)236 1372 y Fw(De\014nition)j(5.1)41 b Fp(Given)24 b Fu(M)31 b(>)23 b Fv(0)p Fu(;)g Fp(a)h(c)l(onne)l(cte)l(d)f(c)l(omp)l(onent)g Fq(C)28 b Fp(of)c Fq(D)2441 1342 y Ft(c)2439 1395 y(N)2526 1372 y Fp(and)f Fu(!)2732 1384 y Ft(I)2770 1372 y Fu(;)14 b(!)2859 1384 y Ft(F)2937 1372 y Fq(2)23 b(C)5 b Fp(,)25 b(we)f(say)g(that)f(an)h(emb)l(e)l(dding)236 1472 y Fu(\015)k Fq(2)23 b Fu(C)450 1442 y Fs(2)488 1472 y Fv(\([0)p Fu(;)14 b(L)p Fv(])p Fu(;)g Fq(C)5 b Fv(\))29 b Fp(is)h(a)g Fu(Q)1076 1484 y Ft(M)1149 1472 y Fv(-admissible)d(connecting)g(curv)n(e)g(b)r (et)n(w)n(een)g Fu(!)2588 1484 y Ft(I)2654 1472 y Fv(and)g Fu(!)2867 1484 y Ft(F)2952 1472 y Fp(if)j(the)g(fol)t(lowing)i(pr)l(op) l(erties)f(ar)l(e)236 1571 y(satis\014e)l(d:)292 1737 y(\(a\))42 b Fu(\015)5 b Fv(\(0\))23 b(=)f Fu(!)760 1749 y Ft(I)798 1737 y Fu(;)30 b(\015)5 b Fv(\()p Fu(L)p Fv(\))22 b(=)h Fu(!)1182 1749 y Ft(F)1237 1737 y Fu(;)30 b Fq(j)12 b Fv(_)-35 b Fu(\015)t Fv(\()p Fu(s)p Fv(\))p Fq(j)24 b Fv(=)f(1)29 b Fq(8)14 b Fu(s)23 b Fq(2)g Fv(\(0)p Fu(;)14 b(L)p Fv(\))p Fu(;)296 1903 y Fp(\(b\))42 b Fq(8)p Fu(h)23 b Fv(=)f(\()p Fu(n;)14 b(l)r Fv(\))23 b Fq(2)h Fu(S)979 1915 y Ft(M)1052 1903 y Fp(,)30 b Fq(8)p Fu(s)23 b Fq(2)h Fv([0)p Fu(;)14 b(L)p Fv(])29 b Fp(such)g(that)h Fu(\015)5 b Fv(\()p Fu(s)p Fv(\))23 b Fq(2)h Fu(E)2177 1915 y Ft(h)2220 1903 y Fp(,)30 b Fu(n)18 b Fq(\001)31 b Fv(_)-35 b Fu(\015)5 b Fv(\()p Fu(s)p Fv(\))23 b Fq(6)p Fv(=)g(0)p Fp(.)361 2086 y Fv(Condition)29 b(\()p Fu(b)p Fv(\))h(means)g(that)g(for)f(all)g Fu(h)d Fq(2)h Fu(S)1772 2098 y Ft(M)1846 2086 y Fv(,)j Fu(\015)5 b Fv(\([0)p Fu(;)14 b(L)p Fv(]\))29 b(ma)n(y)g(in)n(tersect)h Fu(E)2803 2098 y Ft(h)2876 2086 y Fv(transv)n(ersally)d(only)-7 b(.)43 b(It)30 b(is)f(easy)g(to)236 2186 y(see)g(that)h(condition)g(\() p Fu(b)p Fv(\))g(implies)g(that)g Fq(I)6 b Fv(\()p Fu(\015)f Fv(\))26 b(=)g Fq(f)p Fu(s)g Fq(2)h Fv([0)p Fu(;)14 b(L)p Fv(])29 b Fq(j)h Fu(\015)5 b Fv(\()p Fu(s)p Fv(\))26 b Fq(2)h Fu(Q)2575 2198 y Ft(M)2648 2186 y Fq(g)j Fv(is)f(\014nite)h (and)g(that)g(there)f(exists)g Fu(\027)j(>)26 b Fv(0)236 2285 y(suc)n(h)h(that)h(for)f(all)h Fu(s)23 b Fq(2)g(I)6 b Fv(\()p Fu(\015)f Fv(\),)28 b(for)f(all)h Fu(h)23 b Fv(=)f(\()p Fu(n;)14 b(l)r Fv(\))23 b Fq(2)g Fu(S)1931 2297 y Ft(M)2033 2285 y Fv(suc)n(h)k(that)h Fu(\015)5 b Fv(\()p Fu(s)p Fv(\))23 b Fq(2)g Fu(E)2713 2297 y Ft(h)2757 2285 y Fv(,)k Fq(j)12 b Fv(_)-35 b Fu(\015)5 b Fv(\()p Fu(s)p Fv(\))19 b Fq(\001)g Fu(n)p Fq(j)p Fu(=)p Fq(j)p Fu(n)p Fq(j)j(\025)h Fu(\027)5 b Fv(.)361 2385 y(If)22 b(a)f(curv)n(e)f Fu(\013)i Fv(is)f(not)h(admissible)f(w)n(e)g(can)g (alw)n(a)n(ys)f(\014nd)i(\\close)e(to)h(it")h(an)f(admissible)g(one)g Fu(\015)5 b Fv(.)34 b(Indeed)22 b(the)g(follo)n(wing)236 2484 y(lemma)28 b(holds.)236 2667 y Fw(Lemma)k(5.1)40 b Fp(L)l(et)34 b Fu(M)k(>)30 b Fv(0)p Fp(,)35 b Fq(C)j Fp(b)l(e)c(a)g(c)l(onne)l(cte)l(d)g(c)l(omp)l(onent)f(of)i Fq(D)2431 2637 y Ft(c)2429 2690 y(N)2492 2667 y Fp(,)g Fu(!)2604 2679 y Ft(I)2642 2667 y Fu(;)14 b(!)2731 2679 y Ft(F)2816 2667 y Fq(2)31 b(C)38 b Fp(and)c(let)g Fu(\013)c Fq(2)h Fu(C)3504 2637 y Fs(2)3542 2667 y Fv(\([0)p Fu(;)14 b(L)3733 2679 y Fs(0)3769 2667 y Fv(])p Fu(;)g Fq(C)5 b Fv(\))34 b Fp(b)l(e)236 2767 y(an)g(emb)l(e)l(dding)h(with)g Fu(\013)p Fv(\(0\))c(=)g Fu(!)1285 2779 y Ft(I)1356 2767 y Fp(and)k Fu(\013)p Fv(\()p Fu(L)1664 2779 y Fs(0)1701 2767 y Fv(\))c(=)g Fu(!)1912 2779 y Ft(F)1966 2767 y Fp(.)52 b(Then,)36 b Fq(8)14 b Fu(\021)34 b(>)c Fv(0)p Fp(,)35 b(ther)l(e)f(exists)g(a)g(curve)g Fu(\015)5 b Fp(,)36 b Fu(Q)3544 2779 y Ft(M)3617 2767 y Fp(-admissible)236 2866 y(b)l(etwe)l(en)30 b Fu(!)592 2878 y Ft(I)659 2866 y Fp(and)g Fu(!)872 2878 y Ft(F)927 2866 y Fp(,)g(satisfying)i Fv(dist\()p Fu(\015)5 b Fv(\()p Fu(s)p Fv(\))p Fu(;)14 b(\013)p Fv(\([0)p Fu(;)g(L)1950 2878 y Fs(0)1987 2866 y Fv(]\)\))24 b Fu(<)e(\021)s(;)30 b Fq(8)p Fu(s)23 b Fq(2)h Fv([0)p Fu(;)14 b(L)p Fv(])p Fu(:)236 3049 y Fb(Pr)n(oof)p Fv(.)107 b(First)37 b(it)g(is)g(easy)f(to)h(see)g(that)g(there)g (exists)g(an)g(em)n(b)r(edding)g Fu(\013)2727 3061 y Fs(1)2803 3049 y Fv(:)i([0)p Fu(;)14 b(L)3024 3061 y Fs(1)3060 3049 y Fv(])39 b Fq(!)g(C)j Fv(suc)n(h)37 b(that)g Fu(\013)3769 3061 y Fs(1)3806 3049 y Fv(\(0\))i(=)236 3148 y Fu(!)288 3160 y Ft(I)326 3148 y Fu(;)14 b(\013)416 3160 y Fs(1)453 3148 y Fv(\()p Fu(L)542 3160 y Fs(1)579 3148 y Fv(\))37 b(=)f Fu(!)801 3160 y Ft(F)856 3148 y Fv(,)i(dist\()p Fu(\013)1136 3160 y Fs(1)1174 3148 y Fv(\()p Fu(s)p Fv(\))p Fu(;)14 b(\013)p Fv(\([0)p Fu(;)g(L)1558 3160 y Fs(0)1595 3148 y Fv(]\)\))37 b Fq(\024)f Fu(\021)s(=)p Fv(4)f(and)h Fq(8)f Fu(h)i Fv(=)f(\()p Fu(n;)14 b(l)r Fv(\))36 b Fq(2)h Fu(S)2777 3160 y Ft(M)2851 3148 y Fv(,)g Fu(!)2963 3160 y Ft(I)3047 3148 y Fu(=)-51 b Fq(2)36 b Fu(E)3190 3160 y Ft(h)3269 3148 y Fv(\(resp.)61 b Fu(!)3586 3160 y Ft(F)3687 3148 y Fu(=)-51 b Fq(2)36 b Fu(E)3830 3160 y Ft(h)3874 3148 y Fv(\))g(or)254 3248 y(_)-41 b Fu(\013)289 3260 y Fs(1)327 3248 y Fv(\(0\))18 b Fq(\001)h Fu(n)k Fq(6)p Fv(=)f(0)27 b(\(resp.)54 b(_)-40 b Fu(\013)1016 3260 y Fs(1)1053 3248 y Fv(\()p Fu(L)1142 3260 y Fs(1)1179 3248 y Fv(\))19 b Fq(\001)f Fu(n)23 b Fq(6)p Fv(=)g(0\).)361 3348 y(Let)36 b Fu(r)k(>)d Fv(0)p Fu(;)14 b(\027)817 3360 y Fs(1)891 3348 y Fu(>)36 b Fv(0)g(b)r(e)h(suc)n(h)e(that)i Fq(8)p Fu(s)g Fq(2)h Fv([0)p Fu(;)14 b(r)r Fv(])24 b Fq([)g Fv([)p Fu(L)2138 3360 y Fs(1)2199 3348 y Fq(\000)g Fu(r)n(;)14 b(L)2417 3360 y Fs(1)2454 3348 y Fv(],)38 b Fq(8)p Fu(h)f Fv(=)g(\()p Fu(n;)14 b(l)r Fv(\))37 b Fq(2)h Fu(S)3130 3360 y Ft(M)3203 3348 y Fv(,)h(dist\()p Fu(\013)3484 3360 y Fs(1)3522 3348 y Fv(\()p Fu(s)p Fv(\))p Fu(;)14 b(E)3723 3360 y Ft(h)3766 3348 y Fv(\))38 b Fq(\025)e Fu(\027)3978 3360 y Fs(1)236 3447 y Fv(or)h Fq(j)18 b Fv(_)-41 b Fu(\013)424 3459 y Fs(1)462 3447 y Fv(\()p Fu(s)p Fv(\))25 b Fq(\001)h Fu(n)p Fq(j)39 b(\025)h Fu(\027)897 3459 y Fs(1)934 3447 y Fv(.)68 b(Let)38 b Fu(\036)i Fv(:)g([0)p Fu(;)14 b(L)1495 3459 y Fs(1)1531 3447 y Fv(])40 b Fq(!)g Fv([0)p Fu(;)14 b Fv(1])37 b(b)r(e)h(a)g(smo)r(oth)f(function)i(suc)n (h)e(that)h Fu(\036)p Fv(\(0\))j(=)e Fu(\036)p Fv(\()p Fu(L)3588 3459 y Fs(1)3626 3447 y Fv(\))h(=)g(0)d(and)236 3547 y Fq(8)p Fu(s)23 b Fq(2)h Fv([)p Fu(r)n(;)14 b(L)575 3559 y Fs(1)630 3547 y Fq(\000)k Fu(r)r Fv(])29 b Fu(\036)p Fv(\()p Fu(s)p Fv(\))24 b(=)e(1.)361 3647 y(W)-7 b(e)34 b(shall)g(pro)n(v)n(e)e(that)j(for)e(all)h Fu(")f(>)h Fv(0)f(there)h(exists)g Fu(!)2136 3659 y Ft(")2205 3647 y Fq(2)g Fw(R)2366 3616 y Ft(d)2404 3647 y Fv(,)i Fq(j)p Fu(!)2538 3659 y Ft(")2573 3647 y Fq(j)e Fu(<)g(")p Fv(,)h(suc)n(h)f (that)g Fq(8)p Fu(h)g Fv(=)f(\()p Fu(n;)14 b(l)r Fv(\))34 b Fq(2)g Fu(S)3784 3659 y Ft(M)3858 3647 y Fv(,)h(for)236 3746 y(all)i Fu(s)j Fq(2)g Fv([)p Fu(r)n(;)14 b(L)687 3758 y Fs(1)749 3746 y Fq(\000)25 b Fu(r)r Fv(])38 b(suc)n(h)f(that)h Fu(\013)1379 3758 y Fs(1)1417 3746 y Fv(\()p Fu(s)p Fv(\))i Fq(2)g Fu(E)1716 3758 y Ft(h)1784 3746 y Fv(+)25 b Fu(!)1926 3758 y Ft(")1961 3746 y Fv(,)58 b(_)-41 b Fu(\013)2077 3758 y Fs(1)2115 3746 y Fv(\()p Fu(s)p Fv(\))25 b Fq(\001)g Fu(n)40 b Fq(6)p Fv(=)f(0.)66 b(F)-7 b(or)37 b Fu(h)j Fv(=)f(\()p Fu(n;)14 b(l)r Fv(\))39 b Fq(2)h Fu(S)3330 3758 y Ft(M)3404 3746 y Fv(,)g(let)e Fq(J)3653 3758 y Ft(h)3736 3746 y Fv(=)h Fq(f)p Fu(s)g Fq(2)236 3846 y Fv([)p Fu(r)n(;)14 b(L)388 3858 y Fs(1)445 3846 y Fq(\000)19 b Fu(r)r Fv(])31 b Fq(j)f Fu(n)19 b Fq(\001)38 b Fv(_)-41 b Fu(\013)840 3858 y Fs(1)877 3846 y Fv(\()p Fu(s)p Fv(\))27 b(=)f(0)p Fq(g)j Fv(and)h Fq(V)1426 3858 y Ft(h)1495 3846 y Fv(=)c Fq(f)p Fu(\013)1681 3858 y Fs(1)1718 3846 y Fv(\()p Fu(s)p Fv(\))20 b Fq(\000)g Fu(u)29 b Fq(j)h Fu(s)c Fq(2)h(J)2259 3858 y Ft(h)2302 3846 y Fu(;)73 b(u)26 b Fq(2)h Fu(E)2615 3858 y Ft(h)2659 3846 y Fq(g)p Fv(.)42 b(Let)30 b Fu( )2971 3858 y Ft(h)3041 3846 y Fv(:)c([)p Fu(r)n(;)14 b(L)3242 3858 y Fs(1)3299 3846 y Fq(\000)19 b Fu(r)r Fv(])i Fq(\002)e Fu(E)3611 3858 y Ft(h)3681 3846 y Fq(!)26 b Fw(R)3862 3816 y Ft(d)3930 3846 y Fv(b)r(e)236 3945 y(de\014ned)k(b)n(y)g Fu( )696 3957 y Ft(h)739 3945 y Fv(\()p Fu(s;)14 b(u)p Fv(\))27 b(=)f Fu(\013)1098 3957 y Fs(1)1135 3945 y Fv(\()p Fu(s)p Fv(\))21 b Fq(\000)e Fu(u)p Fv(.)43 b Fu(D)r( )1582 3957 y Ft(h)1625 3945 y Fv(\()p Fu(s;)14 b(u)p Fv(\))30 b(is)g(singular)e (i\013)i Fu(s)d Fq(2)g(J)2550 3957 y Ft(h)2593 3945 y Fv(.)44 b(Therefore)28 b Fq(V)3089 3957 y Ft(h)3162 3945 y Fv(is)i(the)g(set)g(of)f(the)h(critical)236 4045 y(v)-5 b(alues)32 b(of)g Fu( )641 4057 y Ft(h)717 4045 y Fv(and)g(b)n(y)f (Sard's)h(lemma,)h(meas\()p Fq(V)1817 4057 y Ft(h)1860 4045 y Fv(\))e(=)f(0.)50 b(Hence)32 b(for)g(all)g Fu(")e(>)h Fv(0)g(there)h(exists)g Fu(!)3377 4057 y Ft(")3443 4045 y Fq(2)f Fw(R)3601 4015 y Ft(d)3672 4045 y Fv(suc)n(h)h(that)236 4145 y Fq(j)p Fu(!)311 4157 y Ft(")346 4145 y Fq(j)24 b Fu(<)e(")p Fv(,)28 b Fu(!)622 4157 y Ft(")689 4145 y Fu(=)-51 b Fq(2)23 b(V)809 4157 y Ft(h)880 4145 y Fv(for)k(all)g Fu(h)c Fq(2)h Fu(S)1323 4157 y Ft(M)1396 4145 y Fv(.)37 b(Our)27 b(claim)g(follo)n(ws.)361 4244 y(No)n(w)g(w)n(e)g(can)h (de\014ne)g Fu(\013)1118 4256 y Fs(2)1179 4244 y Fv(:)23 b([0)p Fu(;)14 b(L)1384 4256 y Fs(1)1420 4244 y Fv(])24 b Fq(!)f(C)32 b Fv(b)n(y)c Fu(\013)1818 4256 y Fs(2)1855 4244 y Fv(\()p Fu(s)p Fv(\))c(=)f Fu(\013)2123 4256 y Fs(1)2160 4244 y Fv(\()p Fu(s)p Fv(\))d Fq(\000)e Fu(\036)p Fv(\()p Fu(s)p Fv(\))p Fu(!)2570 4256 y Ft(")2606 4244 y Fv(.)37 b(It)28 b(is)g(easy)f(to)h(c)n(hec)n(k)e(that,)j(pro)n(vided) d Fu(")i Fv(is)236 4344 y(small)h(enough,)g Fu(\013)819 4356 y Fs(2)885 4344 y Fv(is)g(an)g(em)n(b)r(edding)g(whic)n(h)g (satis\014es)g(condition)f(\()p Fu(b)p Fv(\).)42 b Fu(\015)34 b Fv(is)29 b(obtained)f(from)h Fu(\013)3340 4356 y Fs(2)3407 4344 y Fv(b)n(y)f(a)h(simple)g(time)236 4444 y(reparametrization.)p 943 4444 48 48 v 236 4605 a(If)40 b(\000\()p Fu(\013)p Fv(\()p Fu(s)p Fv(\))p Fu(;)14 b Fq(\001)p Fu(;)g Fq(\001)p Fv(\))40 b(p)r(ossesses,)h(for)e(eac)n(h)f Fu(s)p Fv(,)k(a)d (non-degenerate)f(lo)r(cal)g(minim)n(um)i(\()p Fu(\022)2930 4561 y Ft(\013)p Fs(\()p Ft(s)p Fs(\))2928 4627 y(0)3061 4605 y Fu(;)14 b(')3152 4561 y Ft(\013)p Fs(\()p Ft(s)p Fs(\))3152 4627 y(0)3283 4605 y Fv(\),)43 b(then,)f(b)n(y)d(the)h(Im-) 236 4704 y(plicit)c(F)-7 b(unction)36 b(Theorem,)h(along)d(an)n(y)g (curv)n(e)h Fu(\015)40 b Fv(su\016cien)n(tly)35 b(close)g(to)g Fu(\013)p Fv(,)j(\000\()p Fu(\015)5 b Fv(\()p Fu(s)p Fv(\))p Fu(;)14 b Fq(\001)p Fu(;)g Fq(\001)p Fv(\))36 b(p)r(ossesses)e(lo)r(cal)h(minima)236 4820 y(\()p Fu(\022)309 4777 y Ft(\015)t Fs(\()p Ft(s)p Fs(\))307 4842 y(0)436 4820 y Fu(;)14 b(')527 4777 y Ft(\015)t Fs(\()p Ft(s)p Fs(\))527 4842 y(0)653 4820 y Fv(\))28 b(suc)n(h)f(that)1243 4937 y Fu(D)1314 4903 y Fs(2)1312 4960 y(\()p Ft(\022)r(;')p Fs(\))1464 4937 y Fv(\000\()p Fu(\015)5 b Fv(\()p Fu(s)p Fv(\))p Fu(;)14 b(\022)1777 4894 y Ft(\015)t Fs(\()p Ft(s)p Fs(\))1775 4959 y(0)1904 4937 y Fu(;)g(')1995 4894 y Ft(\015)t Fs(\()p Ft(s)p Fs(\))1995 4959 y(0)2121 4937 y Fv(\))23 b Fu(>)g(\025)p Fv(Id)p Fu(;)180 b Fq(8)28 b Fu(s)23 b Fq(2)g Fv([0)p Fu(;)14 b(L)p Fv(])p Fu(;)835 b Fv(\(5.4\))236 5086 y(for)35 b(some)g(constan)n(t)f Fu(\025)j(>)e Fv(0)g(dep)r(ending)h(on)f Fu(\013)p Fv(.)61 b(Therefore,)36 b(b)n(y)f(the)h(ab)r(o)n(v)n(e)e(lemma,)j(it)f(is)f (enough)g(to)g(pro)n(v)n(e)e(the)236 5186 y(existence)28 b(of)f(drifting)h(orbits)f(along)f(admissible)h(curv)n(es)g Fu(\015)5 b Fv(.)36 b(Prop)r(ert)n(y)26 b(\(5)p Fu(:)p Fv(4\))h(will)h(b)r(e)g(used)g(in)g(lemma)f(6.1.)361 5286 y(Giv)n(en)g(a)g Fu(Q)736 5298 y Ft(M)810 5286 y Fv(-admissible)g(curv)n(e)f Fu(\015)5 b Fv(,)28 b(let)g(us)f(call)h Fu(s)1978 5256 y Fk(\003)1978 5306 y Fs(1)2016 5286 y Fu(;)14 b(:)g(:)g(:)f(;)h(s)2239 5256 y Fk(\003)2239 5306 y Ft(r)2305 5286 y Fv(the)28 b(elemen)n(ts)f(of)h Fq(I)6 b Fv(\()p Fu(\015)f Fv(\),)28 b(and)g Fu(!)3313 5256 y Fk(\003)3310 5306 y Fs(1)3374 5286 y Fv(=)22 b Fu(\015)5 b Fv(\()p Fu(s)3580 5256 y Fk(\003)3580 5306 y Fs(1)3618 5286 y Fv(\))p Fu(;)14 b(:)g(:)g(:)g(;)g(!)3890 5256 y Fk(\003)3887 5306 y Ft(r)3951 5286 y Fv(=)236 5404 y Fu(\015)5 b Fv(\()p Fu(s)355 5374 y Fk(\003)355 5425 y Ft(r)393 5404 y Fv(\))26 b(the)f(corresp)r(onding)e (frequencies.)35 b(Since,)26 b Fq(8)p Fu(m)d Fv(=)g(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(r)r Fv(,)26 b(\()p Fu(\022)2433 5359 y Ft(!)2477 5334 y Fo(\003)2475 5376 y Fc(m)2431 5426 y Fs(0)2535 5404 y Fu(;)14 b(')2626 5359 y Ft(!)2670 5334 y Fo(\003)2668 5376 y Fc(m)2626 5426 y Fs(0)2727 5404 y Fv(\))26 b(is)f(a)f(nondegenerate)g(lo)r(cal)g(minim)n(um)236 5504 y(of)32 b(\000\()p Fu(!)474 5474 y Fk(\003)471 5524 y Ft(m)534 5504 y Fu(;)14 b Fq(\001)p Fu(;)g Fq(\001)p Fv(\),)34 b(there)d(is)h(a)g(neigh)n(b)r(orho)r(o)r(d)f Fu(W)1725 5516 y Ft(m)1820 5504 y Fv(of)h Fu(!)1974 5474 y Fk(\003)1971 5524 y Ft(m)2066 5504 y Fv(suc)n(h)g(that,)h Fq(8)p Fu(!)g Fq(2)e Fu(W)2761 5516 y Ft(m)2824 5504 y Fv(,)j(\000\()p Fu(!)s(;)14 b Fq(\001)p Fv(\))32 b(admits)g(a)f (nondegenerate)2084 6059 y(15)p eop %%Page: 16 16 16 15 bop 236 272 a Fv(lo)r(cal)31 b(minim)n(um)g(\()p Fu(\022)883 242 y Ft(!)881 293 y Fs(0)932 272 y Fu(;)14 b(')1023 242 y Ft(!)1023 293 y Fs(0)1071 272 y Fv(\),)33 b(the)e(map)g Fu(!)g Fq(7!)e Fv(\()p Fu(\022)1761 242 y Ft(!)1759 293 y Fs(0)1810 272 y Fu(;)14 b(')1901 242 y Ft(!)1901 293 y Fs(0)1949 272 y Fv(\))32 b(b)r(eing)f(Lipsc)n (hitz-con)n(tin)n(uous)e(on)i Fu(W)3210 284 y Ft(m)3273 272 y Fv(.)48 b(Therefore)30 b(w)n(e)g(shall)236 372 y(assume)d(without)h(loss)f(of)g(generalit)n(y)f(that)i(for)f(all)h Fu(m)23 b Fv(=)f(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(r)r Fv(,)958 554 y Fq(8)p Fv(\()p Fu(!)s(;)g(!)1183 520 y Fk(0)1206 554 y Fv(\))24 b Fq(2)f Fv(\()p Fu(W)1450 566 y Ft(m)1532 554 y Fq(\\)c Fu(\015)5 b Fv(\([0)p Fu(;)14 b(L)p Fv(]\)\))1932 520 y Fs(2)2024 554 y Fq(j)p Fv(\()p Fu(\022)2120 520 y Ft(!)2118 575 y Fs(0)2169 554 y Fu(;)g(')2260 520 y Ft(!)2260 575 y Fs(0)2308 554 y Fv(\))19 b Fq(\000)f Fv(\()p Fu(\022)2515 520 y Ft(!)2559 495 y Fo(0)2513 575 y Fs(0)2586 554 y Fu(;)c(')2677 520 y Ft(!)2721 495 y Fo(0)2677 575 y Fs(0)2748 554 y Fv(\))p Fq(j)23 b(\024)g Fu(K)6 b Fq(j)p Fu(!)20 b Fq(\000)e Fu(!)3224 520 y Fk(0)3247 554 y Fq(j)p Fu(:)552 b Fv(\(5.5\))361 737 y(It)28 b(is)f(easy)g(to)g (pro)n(v)n(e)f(that,)i(if)g Fu(\015)33 b Fv(is)27 b(an)g(admissible)h (curv)n(e,)e(there)i(exists)f Fu(d)2727 749 y Fs(0)2787 737 y Fu(>)c Fv(0)k(suc)n(h)g(that)296 903 y(\(*\))42 b Fq(f)p Fu(s)27 b Fq(2)h Fv([0)p Fu(;)14 b(L)p Fv(])30 b Fq(j)g Fv(dist\()p Fu(\015)5 b Fv(\()p Fu(s)p Fv(\))p Fu(;)14 b(Q)1320 915 y Ft(M)1394 903 y Fv(\))28 b Fq(\024)g Fu(d)1590 915 y Fs(0)1627 903 y Fq(g)i Fv(is)g(the)h(union)g(of)f(a)g (\014nite)h(n)n(um)n(b)r(er)f(of)h(disjoin)n(t)f(in)n(terv)-5 b(als)30 b([)p Fu(S)3660 915 y Fs(1)3697 903 y Fu(;)14 b(S)3790 873 y Fk(0)3785 924 y Fs(1)3822 903 y Fv(])p Fu(;)g(:)g(:)g(:)g(;)444 1003 y Fv([)p Fu(S)518 1015 y Ft(r)555 1003 y Fu(;)g(S)648 972 y Fk(0)643 1023 y Ft(r)679 1003 y Fv(];)48 b(for)40 b(all)h Fu(m)k Fv(=)g(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(r)44 b Fv(eac)n(h)c(in)n(terv)-5 b(al)41 b([)p Fu(S)2165 1015 y Ft(m)2228 1003 y Fu(;)14 b(S)2321 972 y Fk(0)2316 1023 y Ft(m)2378 1003 y Fv(])42 b(in)n(tersects)e Fq(I)6 b Fv(\()p Fu(\015)f Fv(\))41 b(at)g(a)g(unique)g(p)r(oin)n(t)g Fu(s)3778 972 y Fk(\003)3778 1023 y Ft(m)3882 1003 y Fv(and)444 1102 y Fu(\015)5 b Fv(\([)p Fu(S)598 1114 y Ft(m)661 1102 y Fu(;)14 b(S)754 1072 y Fk(0)749 1123 y Ft(m)812 1102 y Fv(]\))48 b Fq(\032)f Fu(W)1105 1114 y Ft(m)1168 1102 y Fv(.)82 b(Moreo)n(v)n(er)39 b(\()p Fu(s)48 b Fq(7!)g Fv(dist\()p Fu(\015)5 b Fv(\()p Fu(s)p Fv(\))p Fu(;)14 b(Q)2322 1114 y Ft(M)2396 1102 y Fv(\)\))43 b(is)f(decreasing)f(on)h([)p Fu(S)3222 1114 y Ft(m)3285 1102 y Fu(;)14 b(s)3361 1072 y Fk(\003)3361 1123 y Ft(m)3424 1102 y Fv(\),)46 b(increasing)41 b(on)444 1202 y(\()p Fu(s)515 1172 y Fk(\003)515 1222 y Ft(m)578 1202 y Fu(;)14 b(S)671 1172 y Fk(0)666 1222 y Ft(m)729 1202 y Fv(],)28 b(and)f(dist\()p Fu(\015)5 b Fv(\()p Fu(s)p Fv(\))p Fu(;)14 b(Q)1384 1214 y Ft(M)1458 1202 y Fv(\))23 b Fq(\025)g Fv(\()p Fu(\027)5 b(=)p Fv(2\))p Fq(j)p Fu(s)18 b Fq(\000)g Fu(s)1997 1172 y Fk(\003)1997 1222 y Ft(m)2060 1202 y Fq(j)28 b Fv(for)f(all)g Fu(s)c Fq(2)h Fv([)p Fu(S)2568 1214 y Ft(m)2631 1202 y Fu(;)14 b(S)2724 1172 y Fk(0)2719 1222 y Ft(m)2782 1202 y Fv(].)236 1368 y(No)n(w)27 b(w)n(e)h(are)e(able)h(to)h(de\014ne)g(the)g(\\unp)r (erturb)r(ed)f(transition)g(c)n(hain":)36 b(for)27 b(some)g(small)g (constan)n(t)g Fu(\032)c(>)g Fv(0)k(whic)n(h)h(will)236 1468 y(b)r(e)g(sp)r(eci\014ed)g(later)f(w)n(e)g(c)n(ho)r(ose)g Fu(k)f Fq(2)d Fw(N)k Fv(and)h Fu(k)21 b Fv(+)d(1)27 b(\\in)n (termediate)g(frequencies")1535 1650 y Fu(!)1587 1662 y Ft(I)1648 1650 y Fv(=:)p 1758 1604 55 4 v 22 w Fu(!)1813 1662 y Fs(0)1850 1650 y Fu(;)p 1887 1604 V 14 w(!)1942 1662 y Fs(1)1979 1650 y Fu(;)14 b(:)g(:)g(:)g(;)p 2164 1604 V 14 w(!)2218 1662 y Ft(k)q Fk(\000)p Fs(1)2344 1650 y Fu(;)p 2381 1604 V 14 w(!)2436 1662 y Ft(k)2500 1650 y Fv(:=)22 b Fu(!)2662 1662 y Ft(F)236 1833 y Fv(with)p 425 1787 V 28 w Fu(!)480 1845 y Ft(i)531 1833 y Fv(:=)g Fu(\015)5 b Fv(\()p Fu(s)760 1845 y Ft(i)788 1833 y Fv(\))28 b(for)f(certain)g(0)22 b(=:)h Fu(s)1466 1845 y Fs(0)1526 1833 y Fu(<)g(s)1653 1845 y Fs(1)1713 1833 y Fu(<)g(:)14 b(:)g(:)23 b(<)f(s)2047 1845 y Ft(k)q Fk(\000)p Fs(1)2196 1833 y Fu(<)h(s)2323 1845 y Ft(k)2387 1833 y Fv(:=)f Fu(L)27 b Fv(v)n(erifying)1350 1981 y Fu(\032\026)p 1350 2018 93 4 v 1376 2094 a Fv(2)1476 2037 y Fq(\024)c Fu(s)1603 2049 y Ft(i)p Fs(+1)1733 2037 y Fq(\000)18 b Fu(s)1855 2049 y Ft(i)1905 2037 y Fq(\024)23 b Fu(\032\026;)180 b Fq(8)p Fu(i)23 b Fv(=)f(0)p Fu(;)14 b(:)g(:)g(:)f(;)h(k)21 b Fq(\000)d Fv(1)p Fu(:)933 b Fv(\(5.6\))236 2247 y(By)28 b(\(5.6\))f(there)g(results)g(that)1894 2333 y Fu(L)p 1875 2370 V 1875 2446 a(\032\026)2001 2389 y Fq(\024)c Fu(k)j Fq(\024)2255 2333 y Fv(2)p Fu(L)p 2255 2370 99 4 v 2258 2446 a(\032\026)2363 2389 y(;)1459 b Fv(\(5.7\))236 2583 y(moreo)n(v)n(er)25 b(it)j(follo)n(ws)f(from)g(\()p Fu(a)p Fv(\))h(that)1413 2765 y Fq(j)p 1436 2720 55 4 v Fu(!)1491 2777 y Ft(i)p Fs(+1)1621 2765 y Fq(\000)p 1704 2720 V 18 w Fu(!)1758 2777 y Ft(i)1786 2765 y Fq(j)23 b(\024)g Fu(\032\026;)180 b Fq(8)p Fu(i)23 b Fv(=)f(0)p Fu(;)14 b(:)g(:)g(:)f(;)h(k)21 b Fq(\000)d Fv(1)p Fu(:)1006 b Fv(\(5.8\))236 2948 y(This)31 b(condition)g(has)f(b)r(een)h(used)g(b) r(efore)g(in)g(lemma)g(3.4.)46 b(Giv)n(en)30 b Fu(k)k Fv(time)e(instan)n(ts)p 2968 2880 42 4 v 30 w Fu(\022)3010 2960 y Fs(1)3075 2948 y Fv(:=)d Fu(\022)p 3233 2876 44 4 v 3233 2911 a Ft(!)3277 2919 y Fd(1)3231 2970 y Fs(0)3342 2948 y Fu(<)p 3435 2880 42 4 v 28 w(\022)3476 2960 y Fs(2)3542 2948 y Fu(<)f(:)14 b(:)g(:)29 b(<)p 3854 2880 V 28 w(\022)3895 2960 y Ft(i)3951 2948 y Fu(<)236 3048 y(:)14 b(:)g(:)23 b(<)p 444 2980 V 23 w(\022)485 3060 y Ft(k)526 3048 y Fv(,)28 b(w)n(e)f(de\014ne)h(the)g Fq(f)p 1124 3002 55 4 v Fu(')1178 3068 y Ft(i)1205 3048 y Fq(g)1247 3060 y Ft(i)p Fs(=1)p Ft(;:::)o(;k)1521 3048 y Fv(b)n(y)g(the)g(iteration)f(form)n(ula)p 1379 3196 V 1379 3241 a Fu(')1434 3262 y Fs(1)1494 3241 y Fv(=)c Fu(')p 1636 3169 44 4 v -37 x Ft(!)1680 3212 y Fd(1)1636 3263 y Fs(0)1716 3241 y Fu(;)p 1919 3196 55 4 v 180 w(')1974 3262 y Ft(i)p Fs(+1)2108 3241 y Fv(=)p 2196 3196 V 23 w Fu(')2250 3262 y Ft(i)2296 3241 y Fv(+)p 2379 3196 V 18 w Fu(!)2434 3253 y Ft(i)2462 3241 y Fv(\()p 2494 3174 42 4 v Fu(\022)2535 3253 y Ft(i)p Fs(+1)2665 3241 y Fq(\000)p 2748 3174 V 18 w Fu(\022)2790 3253 y Ft(i)2817 3241 y Fv(\))p Fu(:)973 b Fv(\(5.9\))361 3434 y(The)25 b(c)n(hoice)f(of)i(the)f(instan)n(ts)g Fq(f)p 1358 3366 V Fu(\022)1399 3446 y Ft(i)1427 3434 y Fq(g)1469 3446 y Ft(i)p Fs(=1)p Ft(;:::)o(;k)1740 3434 y Fv(is)h(sp)r(eci\014ed)f(in)h (the)f(next)h(lemma:)35 b(the)26 b(main)f(request)g(is)g(that)g(\()p 3795 3366 V Fu(\022)3837 3446 y Ft(i)3865 3434 y Fu(;)p 3902 3388 55 4 v 14 w(')3956 3454 y Ft(i)3983 3434 y Fv(\))236 3545 y(m)n(ust)20 b(arriv)n(e)f Fu(\016)s Fv(-close)g(mo)r(d) h(2)p Fu(\031)s Fw(Z)1251 3515 y Ft(d)p Fs(+1)1374 3545 y Fv(,)h(to)f(the)h(lo)r(cal)e(minim)n(um)i(p)r(oin)n(t)f(\()p Fu(\022)p 2482 3473 44 4 v 2482 3508 a Ft(!)2527 3516 y Fc(i)2480 3567 y Fs(0)2557 3545 y Fu(;)14 b(')p 2648 3473 V -37 x Ft(!)2692 3516 y Fc(i)2648 3567 y Fs(0)2723 3545 y Fv(\))20 b(of)g(the)h(P)n(oincar)n(\023)-39 b(e-Melnik)n(o)n(v) 15 b(primitiv)n(e)236 3644 y(\000\()p 320 3599 55 4 v Fu(!)375 3656 y Ft(i)403 3644 y Fu(;)f Fq(\001)p Fu(;)g Fq(\001)p Fv(\),)38 b(see)d(\(5.11\)-\(5.12\).)60 b(F)-7 b(rom)36 b(\(5.3\))f(w)n(e)h(deriv)n(e)e(that)j(if)p 2384 3599 V 36 w Fu(!)2439 3656 y Ft(i)2502 3644 y Fv(is)f(1)p Fu(=)p Fq(j)14 b Fv(ln)f Fu(\026)p Fq(j)36 b Fv(far)f(from)h(the)g(set) g Fu(Q)3601 3656 y Ft(M)3710 3644 y Fv(of)g(\\non-)236 3744 y(ergo)r(dizing)29 b(frequencies")g(w)n(e)h(can)g(reac)n(h)f(this) i(goal)e(for)h(\\short")e(time)j(in)n(terv)-5 b(als)p 2908 3676 42 4 v 30 w Fu(\022)2949 3756 y Ft(i)p Fs(+1)3081 3744 y Fq(\000)p 3166 3676 V 20 w Fu(\022)3207 3756 y Ft(i)3262 3744 y Fq(\031)27 b(j)14 b Fv(ln)g Fu(\026)p Fq(j)p Fv(.)45 b(In)31 b(order)e(to)236 3844 y(cross)k(the)i(set)f Fu(Q)799 3856 y Ft(M)906 3844 y Fv(of)h(\\non-ergo)r(dizing)c (frequencies")i(w)n(e)h(need)h(to)f(use)g(longer)f(time)h(in)n(terv)-5 b(als)p 3468 3776 V 34 w Fu(\022)3509 3856 y Ft(i)p Fs(+1)3644 3844 y Fq(\000)p 3731 3776 V 22 w Fu(\022)3772 3856 y Ft(i)3834 3844 y Fq(\031)34 b Fv(1)p Fu(=)236 3943 y Fv(dist\()p Fu(Q)468 3955 y Ft(M)542 3943 y Fu(;)p 579 3897 55 4 v 14 w(!)634 3955 y Ft(i)661 3943 y Fv(\))k(if)816 3891 y Fq(p)p 885 3891 51 4 v 52 x Fu(\026=)p Fq(j)14 b Fv(ln)g Fu(\026)p Fq(j)38 b Fu(<)g Fv(dist)q(\()p Fu(Q)1544 3955 y Ft(M)1617 3943 y Fu(;)p 1654 3897 55 4 v 14 w(!)1709 3955 y Ft(i)1736 3943 y Fv(\))h Fu(<)g Fv(1)p Fu(=)p Fq(j)14 b Fv(ln)e Fu(\026)p Fq(j)p Fv(.)65 b(When)38 b(the)p 2678 3897 V 37 w Fu(!)2733 3955 y Ft(i)2798 3943 y Fv(are)d(\\close")h(\(less)g(than)3634 3891 y Fq(p)p 3704 3891 51 4 v 3704 3943 a Fu(\026=)p Fq(j)14 b Fv(ln)f Fu(\026)p Fq(j)p Fv(-)236 4054 y(distan)n(t\))40 b(to)f(the)h(set)g(of) g(non-ergo)r(dizing)d(h)n(yp)r(erplanes)i Fu(Q)2187 4066 y Ft(M)2300 4054 y Fv(w)n(e)g(c)n(ho)r(ose)f(again)p 2943 3986 42 4 v 39 w Fu(\022)2984 4066 y Ft(i)p Fs(+1)3122 4054 y Fq(\000)p 3213 3986 V 26 w Fu(\022)3255 4066 y Ft(i)3325 4054 y Fq(\031)43 b(j)14 b Fv(ln)g Fu(\026)p Fq(j)p Fv(.)73 b(W)-7 b(e)40 b(also)236 4165 y(estimate)28 b(in)g(\(5.13\))e(the)i(total)g(time)p 1436 4097 V 28 w Fu(\022)1477 4177 y Ft(k)1536 4165 y Fq(\000)p 1619 4097 V 18 w Fu(\022)1661 4177 y Fs(1)1721 4165 y Fv(=)1809 4103 y Fl(P)1896 4123 y Ft(k)1896 4190 y(i)p Fs(=1)p 2022 4097 V 2022 4165 a Fu(\022)2063 4177 y Ft(i)p Fs(+1)2193 4165 y Fq(\000)p 2276 4097 V 18 w Fu(\022)2318 4177 y Ft(i)2345 4165 y Fv(.)236 4360 y Fw(Lemma)k(5.2)40 b Fq(8)p Fu(\016)c(>)c Fv(0)i Fp(ther)l(e)h(exists)g Fu(\026)1525 4372 y Fs(6)1595 4360 y Fu(>)d Fv(0)i Fp(such)h(that)g Fq(8)p Fv(0)d Fu(<)g(\026)h Fq(\024)f Fu(\026)2584 4372 y Fs(6)2656 4360 y Fp(ther)l(e)j(exist)g Fq(f)p 3109 4292 V Fu(\022)3149 4372 y Ft(i)3177 4360 y Fq(g)3219 4372 y Ft(i)p Fs(=1)p Ft(;:::)o(;k)3500 4360 y Fp(with)p 3686 4292 V 36 w Fu(\022)3727 4372 y Fs(1)3797 4360 y Fv(=)d Fu(\022)p 3935 4288 44 4 v 3935 4323 a Ft(!)3979 4331 y Fd(1)3933 4382 y Fs(0)236 4459 y Fp(satisfying,)361 4637 y Fq(\017)41 b Fv(\()p Fu(i)p Fv(\))30 b Fp(if)60 b Fv(dist\()p 843 4591 55 4 v Fu(!)898 4649 y Ft(i)926 4637 y Fu(;)14 b(Q)1029 4649 y Ft(M)1102 4637 y Fv(\))23 b Fu(>)1286 4556 y Fk(p)p 1341 4556 41 3 v 36 x Ft(\026)p 1255 4618 158 4 v 1255 4665 a Fk(j)11 b Fs(ln)g Ft(\026)p Fk(j)1453 4637 y Fp(then)729 4881 y Fv(max)898 4789 y Fl(n)953 4881 y Fu(C)1012 4893 y Fs(1)1050 4881 y Fq(j)j Fv(ln)g Fu(\026)p Fq(j)p Fu(;)1472 4825 y Fv(2)p Fu(\031)p 1289 4862 458 4 v 1289 4938 a Fv(dist)q(\()p 1456 4892 55 4 v Fu(!)1511 4950 y Ft(i)1538 4938 y Fu(;)g(Q)1641 4950 y Ft(M)1714 4938 y Fv(\))1757 4789 y Fl(o)1835 4881 y Fu(<)p 1923 4813 42 4 v 23 w(\022)1964 4893 y Ft(i)p Fs(+1)2094 4881 y Fq(\000)p 2177 4813 V 18 w Fu(\022)2219 4893 y Ft(i)2269 4881 y Fu(<)23 b Fv(2)14 b(max)2581 4789 y Fl(n)2636 4881 y Fu(C)2695 4893 y Fs(1)2732 4881 y Fq(j)g Fv(ln)g Fu(\026)p Fq(j)p Fu(;)3155 4825 y Fv(2)p Fu(\031)p 2972 4862 458 4 v 2972 4938 a Fv(dist)q(\()p 3139 4892 55 4 v Fu(!)3194 4950 y Ft(i)3221 4938 y Fu(;)g(Q)3324 4950 y Ft(M)3397 4938 y Fv(\))3440 4789 y Fl(o)3495 4881 y Fu(;)286 b Fv(\(5.10\))444 5112 y Fp(wher)l(e)30 b Fu(M)i Fv(=)23 b(8)p Fu(\031)s(a)1015 5124 y Ft(d)p Fs(+1)1137 5112 y Fu(=\016)s Fp(;)361 5294 y Fq(\017)41 b Fv(\()p Fu(ii)p Fv(\))29 b Fp(if)i Fv(dist\()p 842 5249 V Fu(!)897 5306 y Ft(i)925 5294 y Fu(;)14 b(Q)1028 5306 y Ft(M)1101 5294 y Fv(\))23 b Fq(\024)1285 5213 y Fk(p)p 1340 5213 41 3 v 37 x Ft(\026)p 1254 5275 158 4 v 1254 5323 a Fk(j)11 b Fs(ln)g Ft(\026)p Fk(j)1451 5294 y Fp(then)30 b Fu(C)1695 5306 y Fs(1)1733 5294 y Fq(j)14 b Fv(ln)g Fu(\026)p Fq(j)23 b Fu(<)p 2036 5227 42 4 v 22 w(\022)2078 5306 y Ft(i)p Fs(+1)2208 5294 y Fq(\000)p 2291 5227 V 18 w Fu(\022)2332 5306 y Ft(i)2383 5294 y Fu(<)f Fv(2)p Fu(C)2571 5306 y Fs(1)2608 5294 y Fq(j)14 b Fv(ln)g Fu(\026)p Fq(j)p Fp(,)2084 6059 y Fv(16)p eop %%Page: 17 17 17 16 bop 236 272 a Fp(and)30 b(such)g(that)1130 372 y Fv(dist)1265 279 y Fl(\020)1314 372 y Fv(\()p 1346 304 42 4 v Fu(\022)1388 384 y Ft(i)1415 372 y Fu(;)p 1452 326 55 4 v 14 w(')1507 392 y Ft(i)1534 372 y Fv(\))p Fu(;)14 b Fv(\()p Fu(\022)p 1676 300 44 4 v 1676 335 a Ft(!)1721 343 y Fc(i)1674 394 y Fs(0)1751 372 y Fu(;)g(')p 1842 300 V -37 x Ft(!)1887 343 y Fc(i)1842 394 y Fs(0)1917 372 y Fv(\))19 b(+)f(2)p Fu(\031)s Fw(Z)2201 337 y Ft(d)p Fs(+1)2324 279 y Fl(\021)2396 372 y Fu(<)23 b(\016)o(;)98 b Fq(8)p Fu(i)23 b Fv(=)g(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(k)s(;)682 b Fv(\(5.11\))236 549 y Fp(wher)l(e)p 473 503 55 4 v 33 w Fu(')527 569 y Fs(1)564 549 y Fu(;)14 b(:)g(:)g(:)g(;)p 749 503 V 14 w(')803 569 y Ft(k)875 549 y Fp(ar)l(e)32 b(de\014ne)l(d)g(by)g Fv(\(5)p Fu(:)p Fv(9\))p Fp(.)44 b(Equivalently,)34 b Fq(8)p Fu(i)26 b Fv(=)g(1)p Fu(;)14 b(:)g(:)g(:)g(;)g(k)s Fp(,)32 b(ther)l(e)g(exist)f Fu(h)3122 561 y Ft(i)3176 549 y Fq(2)c Fw(Z)3316 519 y Ft(d)p Fs(+1)3470 549 y Fp(and)32 b Fu(\037)3685 561 y Ft(i)3740 549 y Fq(2)26 b Fw(R)3893 519 y Ft(d)p Fs(+1)236 648 y Fp(such)k(that)1215 748 y Fv(\()p 1247 680 42 4 v Fu(\022)1289 760 y Ft(i)1316 748 y Fu(;)p 1353 702 55 4 v 14 w(')1408 768 y Ft(i)1435 748 y Fu(;)14 b Fv(\))23 b(=)g(\()p Fu(\022)p 1688 676 44 4 v 1688 711 a Ft(!)1733 719 y Fc(i)1686 770 y Fs(0)1763 748 y Fu(;)14 b(')p 1854 676 V -37 x Ft(!)1898 719 y Fc(i)1854 770 y Fs(0)1929 748 y Fv(\))k(+)g(2)p Fu(\031)s(h)2202 760 y Ft(i)2248 748 y Fv(+)g Fu(\037)2383 760 y Ft(i)2496 748 y Fv(with)85 b Fq(j)p Fu(\037)2817 760 y Ft(i)2845 748 y Fq(j)23 b Fu(<)f(\016)o(:)767 b Fv(\(5.12\))236 892 y Fp(Mor)l(e)l(over)31 b(ther)l(e)f(exists)g(a)g(c)l(onstant)f Fu(K)6 b Fv(\()p Fu(\015)f Fv(\))29 b Fp(such)h(that)p 1727 1046 42 4 v 1727 1114 a Fu(\022)1769 1126 y Ft(k)1828 1114 y Fq(\000)p 1911 1046 V 18 w Fu(\022)1952 1126 y Fs(1)2012 1114 y Fq(\024)23 b Fu(K)6 b Fv(\()p Fu(\015)f Fv(\))2299 1058 y Fq(j)14 b Fv(ln)g Fu(\026)p Fq(j)p 2299 1095 194 4 v 2349 1171 a Fu(\032\026)2502 1114 y(:)1279 b Fv(\(5.13\))236 1347 y Fb(Pr)n(oof)p Fv(.)78 b(Let)28 b Fu(\026)796 1359 y Fs(6)856 1347 y Fu(>)23 b Fv(0)k(b)r(e)h(so)f (small)g(that)1625 1295 y Fq(p)p 1695 1295 88 4 v 1695 1347 a Fu(\026)1745 1359 y Fs(6)1782 1347 y Fu(=)p Fq(j)14 b Fv(ln)f Fu(\026)1993 1359 y Fs(6)2031 1347 y Fq(j)23 b Fu(<)f(d)2207 1359 y Fs(0)2272 1347 y Fv(and)2434 1276 y Fl(p)p 2517 1276 231 4 v 71 x Fq(j)14 b Fv(ln)g Fu(\026)2687 1359 y Fs(6)2724 1347 y Fq(j)23 b(\025)g Fv(32)2942 1283 y Fq(p)p 3010 1283 97 4 v 3010 1347 a Fu(C)3069 1359 y Fs(1)3107 1347 y Fu(=)p Fv(\()p Fu(\027)3227 1284 y Fq(p)p 3296 1284 83 4 v 63 x Fu(\016)s(\032)p Fv(\).)361 1462 y(Let)f(us)h(de\014ne)g(\()p 873 1394 42 4 v Fu(\022)914 1474 y Fs(1)951 1462 y Fu(;)p 988 1416 55 4 v 14 w(')1043 1482 y Fs(1)1080 1462 y Fv(\))g(:=)g(\()p Fu(\022)p 1319 1390 44 4 v 1319 1425 a Ft(!)1363 1433 y Fd(1)1317 1484 y Fs(0)1400 1462 y Fu(;)14 b(')p 1491 1390 V -37 x Ft(!)1535 1433 y Fd(1)1491 1484 y Fs(0)1572 1462 y Fv(\).)35 b(Assume)23 b(that)g(\()p 2172 1394 42 4 v Fu(\022)2213 1474 y Fs(1)2250 1462 y Fu(;)14 b(:)g(:)g(:)g(;)p 2435 1394 V 14 w(\022)2476 1474 y Ft(i)2504 1462 y Fv(\))23 b(has)f(b)r(een)h(de\014ned.)35 b(If)23 b(dist\()p 3453 1416 55 4 v Fu(!)3508 1474 y Ft(i)3536 1462 y Fu(;)14 b(Q)3639 1474 y Ft(M)3712 1462 y Fv(\))23 b Fu(>)3855 1410 y Fq(p)p 3924 1410 51 4 v 52 x Fu(\026=)236 1572 y Fq(j)14 b Fv(ln)g Fu(\026)p Fq(j)28 b Fv(then)g(b)n(y)f(\(5.3\))h(there)f(certainly)g(exists)g(\()p 1777 1505 42 4 v Fu(\022)1819 1584 y Ft(i)p Fs(+1)1930 1572 y Fu(;)p 1967 1527 55 4 v 14 w(')2022 1593 y Ft(i)p Fs(+1)2133 1572 y Fv(\))h(satisfying)f(\(5)p Fu(:)p Fv(9\),\(5)p Fu(:)p Fv(10\),)g(suc)n(h)g(that)1221 1777 y(dist)1355 1685 y Fl(\020)1405 1777 y Fv(\()p 1437 1710 42 4 v Fu(\022)1479 1789 y Ft(i)p Fs(+1)1590 1777 y Fu(;)p 1627 1732 55 4 v 14 w(')1681 1798 y Ft(i)p Fs(+1)1793 1777 y Fv(\))p Fu(;)14 b Fv(\()p Fu(\022)p 1935 1699 44 4 v 1935 1734 a Ft(!)1980 1742 y Fc(i)p Fd(+1)1933 1800 y Fs(0)2081 1777 y Fu(;)g(')p 2172 1699 V -43 x Ft(!)2216 1742 y Fc(i)p Fd(+1)2172 1800 y Fs(0)2318 1777 y Fv(\))k(+)g(2)p Fu(\031)s Fw(Z)2601 1743 y Ft(d)p Fs(+1)2724 1685 y Fl(\021)2797 1777 y Fu(<)23 b(\016)s(=)p Fv(4)p Fu(:)236 1976 y Fv(W)-7 b(e)25 b(no)n(w)f(consider)f(the)i(case)f(in)g(whic)n(h)p 1507 1930 55 4 v 25 w Fu(!)1561 1988 y Ft(i)1613 1976 y Fv(is)h(close)e(to)i(some)f(\\non-ergo)r(dizing")d(h)n(yp)r(erplanes) i(of)i Fu(Q)3447 1988 y Ft(M)3520 1976 y Fv(.)36 b(If)25 b(dist\()p 3825 1930 V Fu(!)3880 1988 y Ft(i)p Fk(\000)p Fs(1)3993 1976 y Fu(;)236 2076 y(Q)302 2088 y Ft(M)376 2076 y Fv(\))42 b Fu(>)556 2024 y Fq(p)p 625 2024 51 4 v 52 x Fu(\026=)p Fq(j)14 b Fv(ln)g Fu(\026)p Fq(j)39 b Fv(and)f(dist\()p 1287 2030 55 4 v Fu(!)1343 2088 y Ft(i)1370 2076 y Fu(;)14 b(Q)1473 2088 y Ft(M)1546 2076 y Fv(\))43 b Fq(\024)1727 2024 y(p)p 1796 2024 51 4 v 52 x Fu(\026=)p Fq(j)14 b Fv(ln)g Fu(\026)p Fq(j)39 b Fv(w)n(e)f(pro)r(ceed)g(as)h(follo)n(ws.)69 b(W)-7 b(e)40 b(ha)n(v)n(e)p 3380 2030 55 4 v 37 w Fu(!)3435 2088 y Ft(i)3504 2076 y Fv(=)i Fu(\015)5 b Fv(\()p Fu(s)3730 2088 y Ft(i)3757 2076 y Fv(\),)42 b(with)236 2175 y Fu(s)275 2187 y Ft(i)340 2175 y Fq(2)c Fv([)p Fu(S)507 2187 y Ft(q)543 2175 y Fu(;)14 b(S)636 2145 y Fk(0)631 2196 y Ft(q)668 2175 y Fv(])36 b(for)g(some)f Fu(q)s Fv(,)k(1)e Fq(\024)f Fu(q)41 b Fq(\024)c Fu(r)r Fv(.)63 b(Moreo)n(v)n(er,)36 b(b)n(y)g(prop)r(ert)n(y)f(\()p Fq(\003)p Fv(\))h(there)g(exists)g Fu(p)3180 2145 y Fk(\003)3255 2175 y Fq(2)h Fw(N)f Fv(suc)n(h)g(that)h Fq(f)p Fu(j)k Fq(2)236 2275 y(f)p Fv(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(k)s Fq(g)25 b(j)g Fu(s)704 2287 y Ft(j)762 2275 y Fq(2)f Fv([)p Fu(S)915 2287 y Ft(q)951 2275 y Fu(;)14 b(S)1044 2245 y Fk(0)1039 2295 y Ft(q)1076 2275 y Fv(])25 b(and)g(dist)q(\()p 1450 2229 V Fu(!)1505 2287 y Ft(j)1540 2275 y Fu(;)14 b(Q)1643 2287 y Ft(M)1716 2275 y Fv(\))23 b Fq(\024)1859 2223 y(p)p 1928 2223 51 4 v 52 x Fu(\026=)p Fq(j)14 b Fv(ln)f Fu(\026)p Fq(jg)23 b Fv(=)g Fq(f)p Fu(i;)14 b(:)g(:)g(:)e(;)i (i)g Fv(+)g Fu(p)2783 2245 y Fk(\003)2835 2275 y Fq(\000)f Fv(1)p Fq(g)p Fv(,)25 b(and)h Fu(s)3244 2287 y Ft(i)3294 2275 y Fq(\024)d Fu(s)3421 2245 y Fk(\003)3421 2295 y Ft(q)3482 2275 y Fq(\024)g Fu(s)3609 2287 y Ft(i)p Fs(+)p Ft(p)3717 2271 y Fo(\003)3752 2287 y Fk(\000)p Fs(1)3841 2275 y Fv(.)36 b(W)-7 b(e)236 2375 y(shall)27 b(use)h(the)g (abbreviations)e Fu(s)1271 2344 y Fk(\003)1337 2375 y Fv(for)h Fu(s)1503 2344 y Fk(\003)1503 2395 y Ft(q)1541 2375 y Fv(,)g(and)h Fu(!)1808 2344 y Fk(\003)1873 2375 y Fv(for)f Fu(!)2055 2344 y Fk(\003)2052 2395 y Ft(q)2093 2375 y Fv(.)37 b(W)-7 b(e)28 b(claim)g(that)1553 2622 y(1)22 b Fq(\024)h Fu(p)1747 2588 y Fk(\003)1808 2622 y Fq(\024)f Fu(p)h Fv(:=)2071 2530 y Fl(h)2319 2495 y Fq(p)p 2388 2495 41 4 v 71 x Fu(\016)p 2120 2603 508 4 v 2120 2691 a Fv(4)2162 2620 y Fl(p)p 2245 2620 383 4 v 71 x Fu(C)2304 2703 y Fs(1)2341 2691 y Fu(\032\026)p Fq(j)14 b Fv(ln)g Fu(\026)p Fq(j)2637 2530 y Fl(i)2676 2622 y Fu(:)1105 b Fv(\(5.14\))236 2861 y(In)28 b(fact,)g(b)n(y)f (\(5.6\))g(and)h(\()p Fq(\003)p Fv(\))490 3027 y Fu(\027)5 b(\032)p 490 3064 90 4 v 514 3140 a Fv(4)589 3083 y Fu(\026)p Fv(\()p Fu(p)713 3048 y Fk(\003)770 3083 y Fq(\000)18 b Fv(1\))23 b Fq(\024)1047 3027 y Fu(\027)p 1047 3064 47 4 v 1049 3140 a Fv(2)1103 3083 y([\()p Fu(s)1197 3095 y Ft(i)p Fs(+)p Ft(p)1305 3079 y Fo(\003)1341 3095 y Fk(\000)p Fs(1)1449 3083 y Fq(\000)18 b Fu(s)1571 3048 y Fk(\003)1609 3083 y Fv(\))h(+)f(\()p Fu(s)1814 3095 y Fk(\003)1870 3083 y Fq(\000)g Fu(s)1992 3095 y Ft(i)2020 3083 y Fv(\)])23 b Fq(\024)g Fv(dist\()p 2352 3037 55 4 v Fu(!)2407 3095 y Ft(i)p Fs(+)p Ft(p)2515 3079 y Fo(\003)2551 3095 y Fk(\000)p Fs(1)2640 3083 y Fu(;)14 b(Q)2743 3095 y Ft(M)2816 3083 y Fv(\))19 b(+)f(dist\()p 3116 3037 V Fu(!)3171 3095 y Ft(i)3198 3083 y Fu(;)c(Q)3301 3095 y Ft(M)3375 3083 y Fv(\))23 b Fq(\024)g Fv(2)3606 2971 y Fq(p)p 3675 2971 51 4 v 51 x Fu(\026)p 3570 3064 194 4 v 3570 3140 a Fq(j)14 b Fv(ln)f Fu(\026)p Fq(j)236 3310 y Fv(Hence)28 b Fu(p)525 3280 y Fk(\003)586 3310 y Fq(\024)23 b Fv(8\()p Fu(\027)5 b(\032)837 3259 y Fq(p)p 906 3259 51 4 v 51 x Fu(\026)p Fq(j)14 b Fv(ln)g Fu(\026)p Fq(j)p Fv(\))1181 3280 y Fk(\000)p Fs(1)1270 3310 y Fv(,)28 b(whic)n(h)g(implies)f(\(5)p Fu(:)p Fv(14\),)g(b)n(y)h(the)g(c)n(hoice) e(of)i Fu(\026)2753 3322 y Fs(6)2790 3310 y Fv(.)361 3421 y(No)n(w)34 b(w)n(e)h(can)f(de\014ne)i(the)p 1244 3353 42 4 v 35 w Fu(\022)1285 3433 y Ft(i)p Fs(+1)1397 3421 y Fu(;)14 b(:)g(:)g(:)g(;)p 1582 3353 V 14 w(\022)1623 3433 y Ft(i)p Fs(+)p Ft(p)1731 3417 y Fo(\003)1770 3421 y Fv(.)59 b(The)35 b(\015o)n(w)g(of)g(\()p Fu(!)2400 3391 y Fk(\003)2438 3421 y Fu(;)14 b Fv(1\),)37 b(as)d(an)n(y)g(linear) g(\015o)n(w)h(on)f(a)h(torus,)h(has)e(the)236 3521 y(follo)n(wing)26 b(prop)r(ert)n(y)h(:)36 b(there)27 b(exists)g Fu(T)1487 3490 y Fk(\003)1524 3521 y Fv(\()p Fu(!)1611 3490 y Fk(\003)1650 3521 y Fu(;)14 b(\016)s Fv(\))23 b Fu(>)f Fv(0)27 b(\(abbreviated)g(as) f Fu(T)2587 3490 y Fk(\003)2625 3521 y Fv(\))h(suc)n(h)g(that)h(an)n(y) e(time)i(in)n(terv)-5 b(al)27 b(of)g(length)236 3620 y Fu(T)297 3590 y Fk(\003)362 3620 y Fv(con)n(tains)g Fu(t)g Fv(satisfying)g(dist)q(\(\()p Fu(t!)1399 3590 y Fk(\003)1437 3620 y Fu(;)14 b(t)p Fv(\))p Fu(;)g Fv(2)p Fu(\031)s Fw(Z)1723 3590 y Ft(d)p Fs(+1)1846 3620 y Fv(\))23 b Fq(\024)g Fu(\016)s(=)p Fv(4.)361 3720 y(Therefore)j(\(pro)n(vided)h Fu(C)1170 3732 y Fs(1)1208 3720 y Fq(j)14 b Fv(ln)f Fu(\026)1377 3732 y Fs(6)1415 3720 y Fq(j)23 b Fu(>)f(T)1609 3690 y Fk(\003)1647 3720 y Fv(\))28 b(w)n(e)f(can)g(de\014ne)p 2221 3652 V 28 w Fu(\022)2262 3732 y Ft(i)p Fs(+1)2374 3720 y Fu(;)14 b(:)g(:)g(:)g(;)p 2559 3652 V 14 w(\022)2600 3732 y Ft(i)p Fs(+)p Ft(p)2708 3716 y Fo(\003)2775 3720 y Fv(suc)n(h)27 b(that)469 3922 y Fu(C)528 3934 y Fs(1)566 3922 y Fq(j)14 b Fv(ln)g Fu(\026)p Fq(j)23 b(\024)p 869 3854 V 22 w Fu(\022)911 3934 y Ft(i)p Fs(+)p Ft(j)s Fs(+1)1123 3922 y Fq(\000)p 1206 3854 V 18 w Fu(\022)1247 3934 y Ft(i)p Fs(+)p Ft(j)1379 3922 y Fq(\024)g Fv(2)p Fu(C)1568 3934 y Fs(1)1605 3922 y Fq(j)14 b Fv(ln)g Fu(\026)p Fq(j)p Fu(;)180 b Fv(dist)2135 3829 y Fl(\020)2185 3922 y Fv(\()p 2217 3854 V Fu(\022)2258 3934 y Ft(i)p Fs(+)p Ft(j)2368 3922 y Fu(;)25 b Fl(e)-57 b Fu(')2459 3934 y Ft(i)p Fs(+)p Ft(j)2568 3922 y Fv(\))p Fu(;)14 b Fv(\()p 2669 3854 V Fu(\022)2711 3934 y Ft(i)2739 3922 y Fu(;)p 2776 3876 55 4 v 14 w(')2830 3942 y Ft(i)2858 3922 y Fv(\))k(+)g(2)p Fu(\031)s Fw(Z)3141 3887 y Ft(d)p Fs(+1)3264 3829 y Fl(\021)3337 3922 y Fq(\024)23 b Fu(\016)s(=)p Fv(4)p Fu(;)232 b Fv(\(5.15\))236 4132 y(where)38 b Fl(e)-57 b Fu(')530 4144 y Ft(i)p Fs(+)p Ft(j)663 4132 y Fv(=)p 751 4086 V 23 w Fu(')805 4152 y Ft(i)851 4132 y Fv(+)18 b Fu(!)989 4102 y Fk(\003)1027 4132 y Fv(\()p 1059 4065 42 4 v Fu(\022)1101 4144 y Ft(i)p Fs(+)p Ft(j)1228 4132 y Fq(\000)p 1311 4065 V 18 w Fu(\022)1353 4144 y Ft(i)1380 4132 y Fv(\).)38 b(F)-7 b(or)27 b(1)22 b Fq(\024)h Fu(j)28 b Fq(\024)22 b Fu(p)1965 4102 y Fk(\003)2003 4132 y Fv(,)28 b(let)p 1415 4347 55 4 v 1415 4393 a Fu(')1470 4413 y Ft(i)p Fs(+)p Ft(j)1602 4393 y Fv(=)p 1690 4347 V 23 w Fu(')1744 4413 y Ft(i)1790 4393 y Fv(+)1918 4285 y Ft(j)1873 4314 y Fl(X)1875 4490 y Ft(q)r Fs(=1)p 2007 4347 V 2007 4393 a Fu(!)2062 4405 y Ft(i)p Fs(+)p Ft(q)r Fk(\000)p Fs(1)2258 4393 y Fv(\()p 2290 4325 42 4 v Fu(\022)2331 4405 y Ft(i)p Fs(+)p Ft(q)2461 4393 y Fq(\000)p 2544 4325 V 18 w Fu(\022)2585 4405 y Ft(i)p Fs(+)p Ft(q)r Fk(\000)p Fs(1)2781 4393 y Fv(\))p Fu(:)968 b Fv(\(5.16\))236 4666 y(W)-7 b(e)35 b(no)n(w)f(c)n(hec)n(k)g(that)g(for)g(all)h Fu(j)k Fv(=)34 b(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(p)1680 4636 y Fk(\003)1718 4666 y Fv(,)36 b(\()p 1809 4598 V Fu(\022)1851 4678 y Ft(i)p Fs(+)p Ft(j)1960 4666 y Fu(;)p 1997 4620 55 4 v 14 w(')2052 4686 y Ft(i)p Fs(+)p Ft(j)2161 4666 y Fv(\),)h(as)d(de\014ned)g(in)h(\(5.15\))f(and)g(\(5.16\),)i (satisfy)e(estimate)236 4766 y(\(5.11\),)27 b(namely)484 4964 y(dist)618 4976 y Ft(T)671 4872 y Fl(\020)720 4964 y Fv(\()p 752 4897 42 4 v Fu(\022)794 4976 y Ft(i)p Fs(+)p Ft(j)903 4964 y Fu(;)p 940 4919 55 4 v 14 w(')994 4985 y Ft(i)p Fs(+)p Ft(j)1104 4964 y Fv(\))p Fu(;)14 b Fv(\()p Fu(\022)p 1246 4884 44 4 v 1246 4919 a Ft(!)1290 4927 y Fc(i)p Fd(+)p Fc(j)1244 4986 y Fs(0)1390 4964 y Fu(;)g(')p 1481 4884 V -45 x Ft(!)1526 4927 y Fc(i)p Fd(+)p Fc(j)1481 4986 y Fs(0)1626 4964 y Fv(\))1658 4872 y Fl(\021)1730 4964 y Fv(:=)23 b(dist)1975 4872 y Fl(\020)2025 4964 y Fv(\()p 2057 4897 42 4 v Fu(\022)2099 4976 y Ft(i)p Fs(+)p Ft(j)2208 4964 y Fu(;)p 2245 4919 55 4 v 14 w(')2299 4985 y Ft(i)p Fs(+)p Ft(j)2409 4964 y Fv(\))p Fu(;)14 b Fv(\()p Fu(\022)p 2551 4884 44 4 v 2551 4919 a Ft(!)2595 4927 y Fc(i)p Fd(+)p Fc(j)2549 4986 y Fs(0)2695 4964 y Fu(;)g(')p 2786 4884 V -45 x Ft(!)2830 4927 y Fc(i)p Fd(+)p Fc(j)2786 4986 y Fs(0)2930 4964 y Fv(\))19 b(+)f(2)p Fu(\031)s Fw(Z)3214 4930 y Ft(d)p Fs(+1)3337 4872 y Fl(\021)3410 4964 y Fq(\024)k Fu(\016)o(:)248 b Fv(\(5.17\))236 5163 y(W)-7 b(e)28 b(ha)n(v)n(e)f(b)n(y)g(\(5.16\))g(that)342 5421 y(dist)476 5433 y Ft(T)528 5328 y Fl(\020)578 5421 y Fv(\()p 610 5353 42 4 v Fu(\022)651 5433 y Ft(i)p Fs(+)p Ft(j)761 5421 y Fu(;)p 798 5375 55 4 v 14 w(')852 5441 y Ft(i)p Fs(+)p Ft(j)962 5421 y Fv(\))p Fu(;)14 b Fv(\()p 1063 5353 42 4 v Fu(\022)1104 5433 y Ft(i)1132 5421 y Fu(;)p 1169 5375 55 4 v 14 w(')1223 5441 y Ft(i)1251 5421 y Fv(\))1283 5328 y Fl(\021)1416 5421 y Fq(\024)82 b Fv(dist)1697 5433 y Ft(T)1750 5328 y Fl(\020)1799 5421 y Fv(\()p 1831 5353 42 4 v Fu(\022)1873 5433 y Ft(i)p Fs(+)p Ft(j)1982 5421 y Fu(;)25 b Fl(e)-57 b Fu(')2073 5433 y Ft(i)p Fs(+)p Ft(j)2183 5421 y Fv(\))p Fu(;)14 b Fv(\()p 2284 5353 V Fu(\022)2326 5433 y Ft(i)2353 5421 y Fu(;)p 2390 5375 55 4 v 14 w(')2445 5441 y Ft(i)2472 5421 y Fv(\))2504 5328 y Fl(\021)2573 5421 y Fv(+)2656 5325 y Fl(\014)2656 5375 y(\014)2656 5425 y(\014)2742 5313 y Ft(j)2697 5342 y Fl(X)2699 5518 y Ft(q)r Fs(=1)2817 5421 y Fv(\()p 2849 5375 V Fu(!)2904 5433 y Ft(i)p Fs(+)p Ft(q)r Fk(\000)p Fs(1)3119 5421 y Fq(\000)k Fu(!)3257 5386 y Fk(\003)3294 5421 y Fv(\)\()p 3358 5353 42 4 v Fu(\022)3400 5433 y Ft(i)p Fs(+)p Ft(q)3530 5421 y Fq(\000)p 3613 5353 V 18 w Fu(\022)3654 5433 y Ft(i)p Fs(+)p Ft(q)r Fk(\000)p Fs(1)3850 5421 y Fv(\))3882 5325 y Fl(\014)3882 5375 y(\014)3882 5425 y(\014)2084 6059 y Fv(17)p eop %%Page: 18 18 18 17 bop 1416 349 a Fq(\024)82 b Fu(\016)s(=)p Fv(4)18 b(+)g(2)p Fu(C)1889 361 y Fs(1)1926 349 y Fq(j)c Fv(ln)g Fu(\026)p Fq(j)2158 242 y Ft(p)2192 217 y Fo(\003)2133 270 y Fl(X)2134 446 y Ft(q)r Fs(=1)2267 349 y Fq(j)p Fu(s)2329 361 y Ft(i)p Fs(+)p Ft(q)r Fk(\000)p Fs(1)2543 349 y Fq(\000)k Fu(s)2665 315 y Fk(\003)2703 349 y Fq(j)166 b Fv(\(b)n(y)57 b(\(5)p Fu(:)p Fv(15\))d(and)h(\()p Fu(a)p Fv(\)\))1416 561 y Fq(\024)82 b Fu(\016)s(=)p Fv(4)18 b(+)g(2)p Fu(C)1889 573 y Fs(1)1926 561 y Fq(j)c Fv(ln)g Fu(\026)p Fq(j)p Fu(p)2161 527 y Fk(\003)2199 561 y Fv(\()p Fu(s)2270 573 y Ft(i)p Fs(+)p Ft(p)2378 557 y Fo(\003)2413 573 y Fk(\000)p Fs(1)2521 561 y Fq(\000)k Fu(s)2643 573 y Ft(i)2671 561 y Fv(\))1416 686 y Fq(\024)82 b Fu(\016)s(=)p Fv(4)18 b(+)g(2)p Fu(C)1889 698 y Fs(1)1926 686 y Fq(j)c Fv(ln)g Fu(\026)p Fq(j)p Fu(p)2161 652 y Fs(2)2198 686 y Fu(\032\026)23 b Fq(\024)g Fv(3)p Fu(\016)s(=)p Fv(8)p Fu(;)236 869 y Fv(b)n(y)28 b(\(5.6\))f(and)g(\(5.14\).)36 b(Therefore,)27 b(b)n(y)g(\(5.5\),)551 1094 y(dist)685 1106 y Ft(T)738 1002 y Fl(\020)787 1094 y Fv(\()p 819 1027 42 4 v Fu(\022)861 1106 y Ft(i)p Fs(+)p Ft(j)970 1094 y Fu(;)p 1007 1049 55 4 v 14 w(')1062 1115 y Ft(i)p Fs(+)p Ft(j)1171 1094 y Fv(\))p Fu(;)14 b Fv(\()p Fu(\022)p 1313 1015 44 4 v 1313 1049 a Ft(!)1358 1057 y Fc(i)p Fd(+)p Fc(j)1311 1117 y Fs(0)1458 1094 y Fu(;)g(')p 1549 1015 V -45 x Ft(!)1593 1057 y Fc(i)p Fd(+)p Fc(j)1549 1117 y Fs(0)1693 1094 y Fv(\))1725 1002 y Fl(\021)1858 1094 y Fq(\024)2015 1038 y Fv(3)p Fu(\016)p 2015 1075 82 4 v 2035 1151 a Fv(8)2125 1094 y(+)k(dist)2342 1106 y Ft(T)2395 1002 y Fl(\020)2444 1094 y Fv(\()p 2476 1027 42 4 v Fu(\022)2518 1106 y Ft(i)2545 1094 y Fu(;)p 2582 1049 55 4 v 14 w(')2637 1115 y Ft(i)2664 1094 y Fv(\))p Fu(;)c Fv(\()p Fu(\022)p 2806 1023 44 4 v 2806 1058 a Ft(!)2851 1066 y Fc(i)2804 1117 y Fs(0)2881 1094 y Fu(;)g(')p 2972 1023 V -36 x Ft(!)3017 1066 y Fc(i)2972 1117 y Fs(0)3047 1094 y Fv(\))3079 1002 y Fl(\021)3147 1094 y Fv(+)k Fu(K)6 b Fq(j)p 3330 1049 55 4 v Fu(!)3384 1106 y Ft(i)p Fs(+)p Ft(j)3512 1094 y Fq(\000)p 3595 1049 V 18 w Fu(!)3650 1106 y Ft(i)3678 1094 y Fq(j)1858 1298 y(\024)2015 1242 y Fv(3)p Fu(\016)p 2015 1279 82 4 v 2035 1355 a Fv(8)2125 1298 y(+)2219 1242 y Fu(\016)p 2218 1279 42 4 v 2218 1355 a Fv(4)2288 1298 y(+)18 b Fu(K)6 b(\032\026p)22 b(<)h(\016)236 1509 y Fv(b)n(y)28 b(\(5)p Fu(:)p Fv(14\),)e(pro)n (vided)h Fu(\026)1006 1521 y Fs(6)1071 1509 y Fv(has)g(b)r(een)h(c)n (hosen)f(small)g(enough.)361 1609 y(There)g(remains)g(to)g(pro)n(v)n(e) f(\(5)p Fu(:)p Fv(13\).)36 b(By)27 b(\()p Fq(\003)p Fv(\))h(w)n(e)f (can)h(write)591 1839 y Fu(A)653 1851 y Ft(m)739 1839 y Fv(:=)850 1747 y Fl(n)905 1839 y Fu(s)23 b Fq(2)h Fv([)p Fu(S)1120 1851 y Ft(m)1183 1839 y Fu(;)14 b(S)1276 1805 y Fk(0)1271 1860 y Ft(m)1334 1839 y Fv(])1384 1744 y Fl(\014)1384 1794 y(\014)1384 1844 y(\014)1487 1727 y Fq(p)p 1556 1727 51 4 v 52 x Fu(\026)p 1450 1820 194 4 v 1450 1896 a Fq(j)g Fv(ln)g Fu(\026)p Fq(j)1676 1839 y(\024)22 b Fv(dist)q(\()p Fu(\015)5 b Fv(\()p Fu(s)p Fv(\))p Fu(;)14 b(Q)2184 1851 y Ft(M)2257 1839 y Fv(\))24 b Fq(\024)2555 1783 y Fv(1)p 2410 1820 332 4 v 2410 1896 a(2)p Fu(C)2511 1908 y Fs(1)2548 1896 y Fq(j)14 b Fv(ln)g Fu(\026)p Fq(j)2751 1747 y Fl(o)2830 1839 y Fv(=)22 b([)p Fu(U)2997 1851 y Ft(m)3060 1839 y Fu(;)14 b(V)3145 1851 y Ft(m)3208 1839 y Fv(])19 b Fq([)g Fv([)p Fu(V)3414 1805 y Fk(0)3395 1860 y Ft(m)3458 1839 y Fu(;)14 b(U)3561 1805 y Fk(0)3552 1860 y Ft(m)3615 1839 y Fv(])p Fu(;)236 2071 y Fv(with)34 b Fu(S)482 2083 y Ft(m)576 2071 y Fu(<)e(U)730 2083 y Ft(m)825 2071 y Fu(<)f(V)969 2083 y Ft(m)1064 2071 y Fu(<)h(s)1200 2041 y Fk(\003)1200 2091 y Ft(m)1295 2071 y Fu(<)f(V)1458 2041 y Fk(0)1439 2091 y Ft(m)1535 2071 y Fu(<)g(U)1697 2041 y Fk(0)1688 2091 y Ft(m)1783 2071 y Fu(<)g(S)1935 2041 y Fk(0)1930 2091 y Ft(m)2026 2071 y Fv(\(in)i(the)h(case)e(when)h Fu(!)2767 2041 y Fk(\003)2837 2071 y Fv(=)e Fu(!)2985 2083 y Ft(I)5 b(;F)3094 2071 y Fv(,)34 b Fu(A)3213 2083 y Ft(m)3309 2071 y Fv(is)f(just)g(an)g (in)n(terv)-5 b(al\).)236 2170 y(Moreo)n(v)n(er,)18 b(b)n(y)h(\()p Fu(a)p Fv(\),)i Fu(s)915 2140 y Fk(\003)915 2191 y Ft(m)980 2170 y Fq(\000)q Fu(V)1094 2182 y Ft(m)1157 2170 y Fu(;)14 b(V)1261 2140 y Fk(0)1242 2191 y Ft(m)1307 2170 y Fq(\000)q Fu(s)1412 2140 y Fk(\003)1412 2191 y Ft(m)1498 2170 y Fq(\025)1585 2119 y(p)p 1655 2119 51 4 v 1655 2170 a Fu(\026=)p Fq(j)g Fv(ln)f Fu(\026)p Fq(j)p Fv(.)34 b(De\014ne)20 b Fu(A)j Fv(:=)g Fq([)2496 2140 y Ft(r)2496 2191 y(m)p Fs(=1)2643 2170 y Fu(A)2705 2182 y Ft(m)2769 2170 y Fv(.)34 b(W)-7 b(e)19 b(ha)n(v)n(e)p 3143 2103 42 4 v 18 w Fu(\022)3184 2182 y Ft(k)3227 2170 y Fq(\000)p 3293 2103 V 1 w Fu(\022)3334 2182 y Fs(1)3394 2170 y Fv(=)k Fu(\033)3529 2182 y Fs(0)3568 2170 y Fv(+)3634 2108 y Fl(P)3721 2129 y Ft(r)3721 2195 y(m)p Fs(=1)3882 2170 y Fu(\033)3929 2182 y Ft(m)3993 2170 y Fv(,)236 2270 y(where)912 2370 y Fu(\033)959 2382 y Fs(0)1020 2370 y Fv(:=)1298 2291 y Fl(X)1131 2473 y Fs(1)p Fk(\024)p Ft(i)p Fk(\024)p Ft(k)q Fk(\000)p Fs(1)p Ft(;s)1463 2481 y Fc(i)1497 2473 y Ft(=)-41 b Fk(2)p Ft(A)1585 2370 y Fv(\()p 1617 2302 V Fu(\022)1658 2382 y Ft(i)p Fs(+1)1788 2370 y Fq(\000)p 1871 2302 V 18 w Fu(\022)1913 2382 y Ft(i)1940 2370 y Fv(\))p Fu(;)180 b(\033)2222 2382 y Ft(m)2309 2370 y Fv(:=)2614 2291 y Fl(X)2420 2470 y Fs(1)p Fk(\024)p Ft(i)p Fk(\024)p Ft(k)q Fk(\000)p Fs(1)p Ft(;s)2752 2478 y Fc(i)2779 2470 y Fk(2)p Ft(A)2874 2478 y Fc(m)2929 2370 y Fv(\()p 2961 2302 V Fu(\022)3002 2382 y Ft(i)p Fs(+1)3133 2370 y Fq(\000)p 3216 2302 V 18 w Fu(\022)3257 2382 y Ft(i)3285 2370 y Fv(\))p Fu(:)236 2625 y Fv(F)-7 b(or)32 b Fu(s)429 2637 y Ft(i)497 2625 y Fu(=)-51 b Fq(2)32 b Fu(A)p Fv(,)p 694 2557 V 34 w Fu(\022)735 2637 y Ft(i)p Fs(+1)869 2625 y Fq(\000)p 955 2557 V 21 w Fu(\022)996 2637 y Ft(i)1055 2625 y Fq(\024)f Fv(2)p Fu(C)1252 2637 y Fs(1)1290 2625 y Fq(j)14 b Fv(ln)f Fu(\026)p Fq(j)p Fv(,)34 b(hence)f Fu(\033)1822 2637 y Fs(0)1891 2625 y Fq(\024)e Fv(2)p Fu(C)2088 2637 y Fs(1)2125 2625 y Fu(k)s Fq(j)14 b Fv(ln)g Fu(\026)p Fq(j)31 b(\024)g Fv(4)p Fu(C)2592 2637 y Fs(1)2629 2625 y Fu(L)14 b Fv(ln)g Fu(\026=)p Fv(\()p Fu(\032\026)p Fv(\).)52 b(F)-7 b(or)32 b Fu(i)f Fq(2)g Fu(A)3469 2637 y Ft(m)3533 2625 y Fv(,)p 3590 2557 V 34 w Fu(\022)3631 2637 y Ft(i)p Fs(+1)3765 2625 y Fq(\000)p 3851 2557 V 21 w Fu(\022)3892 2637 y Ft(i)3951 2625 y Fq(\024)236 2724 y Fv(4)p Fu(\031)s Fv(\(dist)q(\()p 527 2679 55 4 v Fu(!)581 2736 y Ft(i)609 2724 y Fu(;)14 b(Q)712 2736 y Ft(M)785 2724 y Fv(\)\))849 2694 y Fk(\000)p Fs(1)962 2724 y Fq(\024)27 b Fv(8)p Fu(\031)s(=)p Fv(\()p Fu(\027)5 b Fq(j)p Fu(s)1328 2736 y Ft(i)1374 2724 y Fq(\000)18 b Fu(s)1496 2694 y Fk(\003)1496 2745 y Ft(m)1559 2724 y Fq(j)p Fv(\))28 b(b)n(y)g(\()p Fq(\003)p Fv(\),)f(and)h(hence,)g (using)f(that)h(b)n(y)f(\(5.6\))g Fu(s)3079 2736 y Ft(i)p Fs(+1)3214 2724 y Fq(\025)c Fu(s)3341 2736 y Ft(i)3387 2724 y Fv(+)18 b Fu(\032\026=)p Fv(2,)939 2946 y Fu(\033)986 2958 y Ft(m)1072 2946 y Fq(\024)1170 2890 y Fv(8)p Fu(\031)p 1170 2927 92 4 v 1193 3003 a(\027)1480 2867 y Fl(X)1286 3046 y Fs(1)p Fk(\024)p Ft(i)p Fk(\024)p Ft(k)q Fk(\000)p Fs(1)p Ft(;s)1618 3054 y Fc(i)1645 3046 y Fk(2)p Ft(A)1740 3054 y Fc(m)1956 2890 y Fv(1)p 1819 2927 317 4 v 1819 3003 a Fq(j)p Fu(s)1881 3015 y Ft(i)1927 3003 y Fq(\000)g Fu(s)2049 2979 y Fk(\003)2049 3024 y Ft(m)2112 3003 y Fq(j)2168 2946 y(\024)2268 2890 y Fv(16)p Fu(\031)p 2265 2927 140 4 v 2265 3003 a(\027)5 b(\032\026)2623 2867 y Fl(X)2428 3046 y Fs(1)p Fk(\024)p Ft(i)p Fk(\024)p Ft(k)q Fk(\000)p Fs(1)p Ft(;s)2760 3054 y Fc(i)2788 3046 y Fk(2)p Ft(A)2883 3054 y Fc(m)2961 2890 y Fu(s)3000 2902 y Ft(i)p Fs(+1)3130 2890 y Fq(\000)19 b Fu(s)3253 2902 y Ft(i)p 2961 2927 319 4 v 2962 3003 a Fq(j)p Fu(s)3024 3015 y Ft(i)3071 3003 y Fq(\000)f Fu(s)3193 2979 y Fk(\003)3193 3024 y Ft(m)3256 3003 y Fq(j)3290 2946 y Fu(:)236 3215 y Fv(Estimating)27 b(the)h(ab)r(o)n(v)n(e)e(sum)i(with)g(an)g(in)n (tegral)e(w)n(e)h(easily)g(get)772 3463 y Fu(\033)819 3475 y Ft(m)906 3463 y Fq(\024)1170 3406 y Fv(8)p Fu(\031)p 1003 3444 426 4 v 1003 3520 a(\027)5 b Fv(\()p Fu(s)1120 3496 y Fk(\003)1120 3540 y Ft(m)1202 3520 y Fq(\000)18 b Fu(V)1333 3532 y Ft(m)1397 3520 y Fv(\))1458 3463 y(+)1554 3406 y(16)p Fu(\031)p 1551 3444 140 4 v 1551 3520 a(\027)5 b(\032\026)1714 3350 y Fl(Z)1797 3370 y Ft(V)1836 3378 y Fc(m)1760 3538 y Ft(U)1805 3546 y Fc(m)1999 3406 y Fu(ds)p 1919 3444 243 4 v 1919 3520 a(s)1958 3496 y Fk(\003)1958 3540 y Ft(m)2039 3520 y Fq(\000)18 b Fu(s)2190 3463 y Fv(+)2450 3406 y(8)p Fu(\031)p 2283 3444 426 4 v 2283 3520 a(\027)5 b Fv(\()p Fu(V)2428 3496 y Fk(0)2409 3540 y Ft(m)2491 3520 y Fq(\000)18 b Fu(s)2613 3496 y Fk(\003)2613 3540 y Ft(m)2676 3520 y Fv(\))2737 3463 y(+)2833 3406 y(16)p Fu(\031)p 2830 3444 140 4 v 2830 3520 a(\027)5 b(\032\026)2993 3350 y Fl(Z)3076 3370 y Ft(U)3127 3345 y Fo(0)3121 3387 y Fc(m)3039 3538 y Ft(V)3093 3522 y Fo(0)3078 3555 y Fc(m)3284 3406 y Fu(ds)p 3204 3444 243 4 v 3204 3520 a(s)19 b Fq(\000)f Fu(s)3384 3496 y Fk(\003)3384 3540 y Ft(m)3457 3463 y Fu(:)236 3709 y Fv(\(5)p Fu(:)p Fv(13\))27 b(can)g(b)r(e)h(easily)f(deduced)h(b)n(y)f(the)h(b)r(ound)g (on)f Fu(s)1961 3679 y Fk(\003)1961 3729 y Ft(m)2043 3709 y Fq(\000)18 b Fu(V)2174 3721 y Ft(m)2237 3709 y Fu(;)c(V)2341 3679 y Fk(0)2322 3729 y Ft(m)2404 3709 y Fq(\000)k Fu(s)2526 3679 y Fk(\003)2526 3729 y Ft(m)2589 3709 y Fv(.)p 2649 3709 48 48 v 361 3858 a(In)27 b(the)h(next)g (section)f(w)n(e)g(will)g(pro)n(v)n(e)f(the)i(existence)f(of)g(a)g (di\013usion)h(orbit)f(\()p Fu(')2856 3870 y Ft(\026)2901 3858 y Fu(;)14 b(q)2975 3870 y Ft(\026)3019 3858 y Fv(\))28 b(close)f(to)g(the)h(\\unp)r(erturb)r(ed)236 3958 y(pseudo-di\013usion) g(orbit")f(\()p 1126 3912 55 4 v Fu(')q Fv(\()p Fu(t)p Fv(\))p Fu(;)p 1312 3912 41 4 v 14 w(q)s Fv(\()p Fu(t)p Fv(\)\))f(:)e(\()p 1583 3890 42 4 v Fu(\022)1624 3970 y Fs(1)1662 3958 y Fu(;)p 1699 3890 V 14 w(\022)1740 3970 y Ft(k)1781 3958 y Fv(\))g Fq(!)h Fw(R)2017 3928 y Ft(d)p Fs(+1)2167 3958 y Fv(de\014ned,)k(for)f Fu(t)c Fq(2)h Fv([)p 2762 3890 V Fu(\022)2803 3970 y Ft(i)2831 3958 y Fu(;)p 2868 3890 V 14 w(\022)2909 3970 y Ft(i)p Fs(+1)3021 3958 y Fv(],)j(as)p 3198 3912 55 4 v 28 w Fu(')p Fv(\()p Fu(t)p Fv(\))d(:=)p 3483 3912 V 24 w Fu(')3537 3978 y Ft(i)3584 3958 y Fv(+)p 3668 3912 V 19 w Fu(!)3722 3970 y Ft(i)3750 3958 y Fv(\()p Fu(t)19 b Fq(\000)p 3915 3890 42 4 v 19 w Fu(\022)3956 3970 y Ft(i)3983 3958 y Fv(\))236 4057 y(and)p 398 4012 41 4 v 28 w Fu(q)438 4078 y Fk(j)p Fs([)p Ft(\022)509 4086 y Fc(i)534 4078 y Ft(;\022)586 4086 y Fc(i)p Fd(+1)683 4078 y Fs(])729 4057 y Fv(:=)j Fu(Q)p 905 4035 34 4 v 29 x Ft(\022)938 4094 y Fc(i)p Fd(+1)1036 4086 y Fk(\000)p 1088 4035 V Ft(\022)1121 4094 y Fc(i)1151 4057 y Fv(\()p Fq(\001)d(\000)p 1308 3990 42 4 v 18 w Fu(\022)1349 4069 y Ft(i)1377 4057 y Fv(\))28 b(\(mo)r(d.)37 b(2)p Fu(\031)s Fv(\).)236 4332 y Fr(6)135 b(The)44 b(di\013usion)i(orbit)236 4514 y Fv(W)-7 b(e)27 b(need)f(the)g(follo)n(wing)f(prop)r(ert)n(y)g(of)h (the)h(Melnik)n(o)n(v)e(function)2315 4493 y Fl(e)2313 4514 y Fv(\000)o(\()p Fu(!)s(;)14 b Fq(\001)p Fu(;)g Fq(\001)p Fv(\))27 b(de\014ned)f(w.r.t.)37 b(to)26 b(the)g(v)-5 b(ariables)25 b(\()p Fu(b;)14 b(c)p Fv(\))26 b(b)n(y)1441 4686 y Fl(e)1438 4707 y Fv(\000\()p Fu(!)s(;)14 b(b;)g(c)p Fv(\))22 b(:=)h(\000\()p Fu(!)s(;)14 b(\022)2105 4672 y Ft(!)2103 4727 y Fs(0)2172 4707 y Fv(+)k Fu(b;)c(')2382 4672 y Ft(!)2382 4727 y Fs(0)2448 4707 y Fv(+)k Fu(b!)j Fv(+)d Fu(c)p Fv(\))p Fu(:)236 4889 y Fw(Lemma)32 b(6.1)40 b Fp(Assume)34 b(that)h Fv(\000\()p Fu(!)s(;)14 b Fq(\001)p Fu(;)g Fq(\001)p Fv(\))34 b Fp(p)l(ossesses)h(a)g(non-de)l(gener)l(ate) g(lo)l(c)l(al)g(minimum)g(in)f Fv(\()p Fu(\022)3319 4859 y Ft(!)3317 4910 y Fs(0)3368 4889 y Fu(;)14 b(')3459 4859 y Ft(!)3459 4910 y Fs(0)3507 4889 y Fv(\))p Fp(.)53 b(Then)36 b(ther)l(e)236 4989 y(exist)30 b Fu(r)25 b(>)e Fv(0)p Fp(,)p 677 4921 36 4 v 30 w Fu(b)f(>)h Fv(0)p Fp(,)30 b Fu(\027)961 5001 y Ft(j)1019 4989 y Fu(>)22 b Fv(0)30 b Fp(\()p Fu(j)e Fv(=)22 b(1)p Fu(;)14 b Fv(2)p Fp(\))29 b(dep)l(ending)i(only)f(on)g Fu(\015)k Fp(such)c(that)g Fq(8)p Fu(!)c Fv(=)c Fu(\015)5 b Fv(\()p Fu(s)p Fv(\))p Fp(,)31 b Fu(s)23 b Fq(2)g Fv([0)p Fu(;)14 b(L)p Fv(])361 5168 y Fq(\017)41 b Fv(\()p Fu(i)p Fv(\))30 b Fu(@)611 5180 y Ft(c)647 5147 y Fl(e)645 5168 y Fv(\000)o(\()p Fu(!)s(;)14 b(b;)g(c)p Fv(\))k Fq(\001)h Fu(c)k Fq(\025)g Fu(\027)1209 5180 y Fs(2)1269 5168 y Fu(>)g Fv(0)29 b Fw(or)h Fq(j)p Fu(@)1612 5180 y Ft(b)1648 5147 y Fl(e)1645 5168 y Fv(\000\()p Fu(!)s(;)14 b(b;)g(c)p Fv(\))p Fq(j)23 b(\025)f Fu(\027)2136 5180 y Fs(1)2197 5168 y Fu(>)g Fv(0)29 b Fp(for)i Fq(j)p Fu(c)p Fq(j)23 b Fv(=)g Fu(r)n(;)14 b Fq(j)p Fu(b)p Fq(j)23 b(\024)p 2945 5101 V 22 w Fu(b)p Fp(,)361 5345 y Fq(\017)41 b Fv(\()p Fu(ii)p Fv(\))29 b Fu(@)639 5357 y Ft(b)676 5324 y Fl(e)673 5345 y Fv(\000\()p Fu(!)s(;)14 b(b;)g(c)p Fv(\))k Fq(\002)g Fv(sign)o(\()p Fu(b)p Fv(\))23 b Fq(\025)g Fu(\027)1486 5357 y Fs(1)1546 5345 y Fu(>)g Fv(0)29 b Fp(for)i Fq(j)p Fu(c)p Fq(j)23 b(\024)f Fu(r)33 b Fp(and)d Fu(b)23 b Fv(=)g Fq(\006)p 2473 5277 V Fu(b)o Fp(.)2084 6059 y Fv(18)p eop %%Page: 19 19 19 18 bop 236 272 a Fb(Pr)n(oof)p Fv(.)104 b(W)-7 b(e)36 b(can)g(assume)f(that)h(\(5)p Fu(:)p Fv(4\))g(is)g(satis\014ed.)62 b(Since)36 b(\000\()p Fu(!)s(;)14 b Fq(\001)p Fu(;)g Fq(\001)p Fv(\))36 b(p)r(ossesses)e(a)i(non-degenerate)e(minim)n(um)236 382 y(in)e(\()p Fu(\022)410 352 y Ft(!)408 403 y Fs(0)458 382 y Fu(;)14 b(')549 352 y Ft(!)549 403 y Fs(0)598 382 y Fv(\),)688 361 y Fl(e)685 382 y Fv(\000\()p Fu(!)s(;)g(b;)g(c)p Fv(\))31 b(p)r(ossesses)f(in)i(\(0)p Fu(;)14 b Fv(0\))31 b(a)g(non)g(degenerate)f(minim)n(um.)49 b(Hence)31 b(w)n(e)g(write)3376 361 y Fl(e)3373 382 y Fv(\000\()p Fu(!)s(;)14 b(b;)g(c)p Fv(\),)32 b(up)g(to)f(a)236 492 y(constan)n(t,)26 b(as)697 471 y Fl(e)695 492 y Fv(\000)o(\()p Fu(!)s(;)14 b(b;)g(c)p Fv(\))23 b(=)g Fu(Q)1188 504 y Fs(2)1225 492 y Fv(\()p Fu(b;)14 b(c)p Fv(\))i(+)h Fu(Q)1562 504 y Fs(3)1598 492 y Fv(\()p Fu(b;)d(c)p Fv(\))27 b(where)f Fu(Q)2103 504 y Fs(2)2140 492 y Fv(\()p Fu(b;)14 b(c)p Fv(\))23 b(=:)k Fu(\014)2498 504 y Ft(!)2546 492 y Fu(b)2582 462 y Fs(2)2619 492 y Fu(=)p Fv(2)16 b(+)g(\()p Fu(\013)2885 504 y Ft(!)2950 492 y Fq(\001)h Fu(c)p Fv(\))p Fu(b)f Fv(+)h(\()p Fu(\015)3267 504 y Ft(!)3315 492 y Fu(c)g Fq(\001)f Fu(c)p Fv(\))p Fu(=)p Fv(2)26 b(is)h(a)f(p)r(ositiv)n(e)236 592 y(de\014nite)j(quadratic)e(form)g(\()p Fu(\014)1179 604 y Ft(!)1252 592 y Fq(2)d Fw(R)p Fu(;)14 b(\013)1493 604 y Ft(!)1564 592 y Fq(2)24 b Fw(R)1715 562 y Ft(d)1754 592 y Fu(;)14 b(\015)1834 604 y Ft(!)1905 592 y Fq(2)25 b Fv(Mat)o(\()p Fu(d)20 b Fq(\002)e Fu(d)p Fv(\)\))29 b(and)f Fu(Q)2676 604 y Fs(3)2736 592 y Fv(=)c Fu(O)r Fv(\()p Fq(j)p Fu(b)p Fq(j)3004 562 y Fs(3)3060 592 y Fv(+)19 b Fq(j)p Fu(c)p Fq(j)3226 562 y Fs(3)3263 592 y Fv(\).)39 b(More)27 b(precisely)-7 b(,)27 b(b)n(y)236 691 y(\(5)p Fu(:)p Fv(4\),)f(there)f(exists)h Fu(")c(>)h Fv(0)i(suc)n(h)g(that)h Fu(\014)1520 703 y Ft(!)1591 691 y Fu(>)d(")p Fv(,)j(and)f Fu(d)1969 703 y Ft(!)2018 691 y Fv(\()p Fu(c)p Fv(\))e(:=)g Fu(\014)2299 703 y Ft(!)2347 691 y Fv(\()p Fu(\015)2422 703 y Ft(!)2470 691 y Fu(c)15 b Fq(\001)f Fu(c)p Fv(\))h Fq(\000)f Fv(\()p Fu(\013)2805 703 y Ft(!)2868 691 y Fq(\001)g Fu(c)p Fv(\))2973 661 y Fs(2)3034 691 y Fu(>)22 b(")p Fq(j)p Fu(c)p Fq(j)3242 661 y Fs(2)3305 691 y Fv(for)j(all)g Fu(!)h Fq(2)d Fu(\015)5 b Fv(\([0)p Fu(;)14 b(L)p Fv(]\).)236 791 y(In)32 b(addition,)f(b)n(y)g (the)h(smo)r(othness)e(of)h(\000)g(and)g(the)h(fact)f(that)h Fu(!)f Fv(=)e Fu(\015)5 b Fv(\()p Fu(s)p Fv(\))31 b(liv)n(es)g(in)g(a)g (compact)f(subset)h(of)h Fw(R)3738 761 y Ft(d)3776 791 y Fv(,)g(there)236 891 y(exists)27 b(a)h(constan)n(t)e Fu(M)37 b Fv(suc)n(h)27 b(that,)h Fq(8)p Fu(!)e Fq(2)d Fu(\015)5 b Fv(\([0)p Fu(;)14 b(L)p Fv(]\),)27 b Fq(j)p Fu(\013)1999 903 y Ft(!)2047 891 y Fq(j)19 b Fv(+)f Fq(j)p Fu(\014)2242 903 y Ft(!)2290 891 y Fq(j)h Fv(+)f Fq(j)p Fu(\015)2481 903 y Ft(!)2529 891 y Fq(j)23 b(\024)f Fu(M)9 b Fv(,)28 b Fq(jr)p Fu(Q)2961 903 y Fs(3)2998 891 y Fv(\()p Fu(b;)14 b(c)p Fv(\))p Fq(j)23 b(\024)g Fu(M)9 b Fv(\()p Fu(b)3463 860 y Fs(2)3518 891 y Fv(+)18 b Fq(j)p Fu(c)p Fq(j)3683 860 y Fs(2)3720 891 y Fv(\).)361 990 y(W)-7 b(e)28 b(ha)n(v)n(e)e Fu(@)739 1002 y Ft(b)773 990 y Fu(Q)839 1002 y Fs(2)875 990 y Fv(\()p Fu(b;)14 b(c)p Fv(\))24 b(=)j Fu(\014)1211 1002 y Ft(!)1259 990 y Fu(b)18 b Fv(+)g Fu(\013)1449 1002 y Ft(!)1516 990 y Fq(\001)g Fu(c)28 b Fv(and)f Fu(@)1826 1002 y Ft(c)1860 990 y Fu(Q)1926 1002 y Fs(2)1963 990 y Fv(\()p Fu(b;)14 b(c)p Fv(\))k Fq(\001)h Fu(c)k Fv(=)k Fu(b\013)2436 1002 y Ft(!)2502 990 y Fq(\001)19 b Fu(c)f Fv(+)g(\()p Fu(\015)2756 1002 y Ft(!)2805 990 y Fu(c)g Fq(\001)g Fu(c)p Fv(\).)236 1090 y(Let)41 b(us)g(de\014ne)p 772 1044 47 4 v 42 w Fu(\027)818 1102 y Fs(1)901 1090 y Fv(:=)k(inf)1135 1105 y Ft(!)r Fk(2)p Ft(\015)t Fs(\([0)p Ft(;L)p Fs(]\))1468 1090 y Fu("=)p Fv(\(4)p Fq(j)p Fu(\013)1699 1102 y Ft(!)1746 1090 y Fq(j)p Fv(\))h Fu(>)f Fv(0)c(and)p 2215 1044 V 41 w Fu(\027)2261 1102 y Fs(2)2344 1090 y Fv(:=)k(inf)2578 1105 y Ft(!)r Fk(2)p Ft(\015)t Fs(\([0)p Ft(;L)p Fs(]\))2911 1090 y Fu("=)p Fv(\(4)p Fu(\014)3113 1102 y Ft(!)3160 1090 y Fv(\))h Fu(>)f Fv(0.)77 b(Then)41 b(consider)236 1202 y Fu(\027)277 1214 y Fs(1)346 1202 y Fv(:=)p 465 1156 V 31 w Fu(\027)511 1214 y Fs(1)548 1202 y Fu(r)r Fv(,)35 b Fu(\027)686 1214 y Fs(2)754 1202 y Fv(=)p 850 1156 V 31 w Fu(\027)896 1214 y Fs(2)933 1202 y Fu(r)972 1172 y Fs(2)1043 1202 y Fv(and)p 1209 1134 36 4 v 32 w Fu(b)c Fv(:=)g Fu(r)17 b Fv(sup)1574 1222 y Ft(!)r Fk(2)p Ft(\015)t Fs(\([0)p Ft(;L)p Fs(]\))1893 1202 y Fv(\(3)p 1967 1156 47 4 v Fu(\027)2013 1214 y Fs(1)2072 1202 y Fv(+)k Fq(j)p Fu(\013)2234 1214 y Ft(!)2282 1202 y Fq(j)p Fv(\))p Fu(=\014)2426 1214 y Ft(!)2474 1202 y Fv(,)34 b Fu(r)g Fq(2)d Fv(\(0)p Fu(;)14 b Fv(1].)51 b(W)-7 b(e)33 b(no)n(w)f(pro)n(v)n(e)f(that,)j(pro)n(vided)236 1301 y Fu(r)46 b(>)c Fv(0)d(has)g(b)r(een)h(c)n(hosen)f(su\016cien)n (tly)g(small,)k(conditions)c(\()p Fu(i)p Fv(\))g(and)h(\()p Fu(ii)p Fv(\))f(are)g(satis\014ed)g(with)h(the)g(ab)r(o)n(v)n(e)e(c)n (hoice)236 1411 y(of)i(the)g(constan)n(ts.)72 b(Indeed)40 b(if)g(\()p Fq(j)p Fu(\013)1412 1423 y Ft(!)1487 1411 y Fq(\001)27 b Fu(c)p Fq(j)g Fv(+)f(2)p 1756 1366 V Fu(\027)1802 1423 y Fs(1)1839 1411 y Fu(r)r Fv(\))p Fu(=\014)1999 1423 y Ft(!)2091 1411 y Fq(\024)42 b(j)p Fu(b)p Fq(j)h(\024)p 2431 1344 36 4 v 43 w Fu(b)d Fv(and)f Fq(j)p Fu(c)p Fq(j)44 b(\024)e Fu(r)h Fv(then)d Fu(@)3238 1423 y Ft(b)3275 1390 y Fl(e)3272 1411 y Fv(\000\()p Fu(!)s(;)14 b(b;)g(c)p Fv(\))26 b Fq(\001)g Fv(sign\()p Fu(b)p Fv(\))43 b Fq(\025)236 1511 y Fu(\014)283 1523 y Ft(!)331 1511 y Fq(j)p Fu(b)p Fq(j)20 b(\000)f(j)p Fu(\013)593 1523 y Ft(!)661 1511 y Fq(\001)h Fu(c)p Fq(j)f(\000)h(j)p Fu(@)934 1523 y Ft(b)967 1511 y Fu(Q)1033 1523 y Fs(3)1070 1511 y Fv(\()p Fu(b;)14 b(c)p Fv(\))p Fq(j)26 b(\025)j Fv(2)p 1428 1465 47 4 v Fu(\027)1474 1523 y Fs(1)1511 1511 y Fu(r)23 b Fq(\000)c Fu(O)r Fv(\()p Fu(r)1791 1481 y Fs(2)1830 1511 y Fv(\))26 b Fq(\025)g Fu(\027)2020 1523 y Fs(1)2087 1511 y Fv(for)j Fu(r)j Fv(su\016cien)n(tly)e(small.)42 b(In)29 b(particular)g(this)g(pro)n(v)n(es)f(\()p Fu(ii)p Fv(\).)236 1611 y(On)g(the)g(other)f(hand)g(if)h Fq(j)p Fu(b)p Fq(j)23 b Fu(<)g Fv(\()p Fq(j)p Fu(\013)1319 1623 y Ft(!)1386 1611 y Fq(\001)18 b Fu(c)p Fq(j)h Fv(+)f(2)p 1630 1565 V Fu(\027)1675 1623 y Fs(1)1713 1611 y Fu(r)r Fv(\))p Fu(=\014)1873 1623 y Ft(!)1949 1611 y Fv(and)27 b Fq(j)p Fu(c)p Fq(j)c Fv(=)g Fu(r)31 b Fv(then)571 1804 y Fu(@)615 1816 y Ft(c)652 1783 y Fl(e)649 1804 y Fv(\000\()p Fu(!)s(;)14 b(b;)g(c)p Fv(\))k Fq(\001)g Fu(c)83 b Fv(=)g Fu(b)p Fv(\()p Fu(\013)1413 1816 y Ft(!)1479 1804 y Fq(\001)19 b Fu(c)p Fv(\))g(+)f(\()p Fu(\015)1766 1816 y Ft(!)1814 1804 y Fu(c)g Fq(\001)h Fu(c)p Fv(\))g(+)f Fu(@)2124 1816 y Ft(c)2157 1804 y Fu(Q)2223 1816 y Fs(3)2260 1804 y Fv(\()p Fu(b;)c(c)p Fv(\))19 b Fq(\001)f Fu(c)23 b Fq(\025)g Fv(\()p Fu(\015)2715 1816 y Ft(!)2763 1804 y Fu(c)c Fq(\001)f Fu(c)p Fv(\))h Fq(\000)f(j)p Fu(b)p Fv(\()p Fu(\013)3173 1816 y Ft(!)3239 1804 y Fq(\001)h Fu(c)p Fv(\))p Fq(j)g Fv(+)f Fu(O)r Fv(\()p Fu(r)3610 1769 y Fs(3)3649 1804 y Fv(\))1144 1981 y Fq(\025)1302 1925 y Fu("r)1380 1895 y Fs(2)1436 1925 y Fv(+)g(\()p Fu(\013)1604 1937 y Ft(!)1671 1925 y Fq(\001)g Fu(c)p Fv(\))1780 1895 y Fs(2)1836 1925 y Fq(\000)g(j)p Fu(\013)1995 1937 y Ft(!)2062 1925 y Fq(\001)h Fu(c)p Fq(j)p Fv(\()p Fq(j)p Fu(\013)2271 1937 y Ft(!)2338 1925 y Fq(\001)f Fu(c)p Fq(j)h Fv(+)f(2)p 2582 1880 V Fu(\027)2627 1937 y Fs(1)2665 1925 y Fu(r)r Fv(\))p 1302 1962 1435 4 v 1972 2038 a Fu(\014)2019 2050 y Ft(!)2765 1981 y Fv(+)g Fu(O)r Fv(\()p Fu(r)2984 1947 y Fs(3)3023 1981 y Fv(\))1144 2206 y Fq(\025)1302 2150 y Fu(")g Fq(\000)g Fv(2)p 1484 2104 47 4 v Fu(\027)1530 2162 y Fs(1)1567 2150 y Fq(j)p Fu(\013)1643 2162 y Ft(!)1691 2150 y Fq(j)p 1302 2187 413 4 v 1461 2263 a Fu(\014)1508 2275 y Ft(!)1724 2206 y Fu(r)1763 2172 y Fs(2)1820 2206 y Fv(+)g Fu(O)r Fv(\()p Fu(r)2039 2172 y Fs(3)2078 2206 y Fv(\))23 b Fq(\025)2280 2150 y Fu(")p 2231 2187 137 4 v 2231 2263 a Fv(2)p Fu(\014)2320 2275 y Ft(!)2377 2206 y Fu(r)2416 2172 y Fs(2)2473 2206 y Fq(\000)18 b Fu(O)r Fv(\()p Fu(r)2692 2172 y Fs(3)2731 2206 y Fv(\))23 b Fq(\025)g Fv(2)p 2916 2160 47 4 v Fu(\027)2961 2218 y Fs(2)2999 2206 y Fu(r)3038 2172 y Fs(2)3094 2206 y Fv(+)18 b Fu(O)r Fv(\()p Fu(r)3313 2172 y Fs(3)3352 2206 y Fv(\))p Fu(:)236 2433 y Fv(Hence)28 b(\()p Fu(i)p Fv(\))g(is)f(satis\014ed)g(for)h Fu(r)i Fv(small)d(enough.)p 1736 2433 48 48 v 361 2593 a(The)21 b(partial)f(deriv)-5 b(ativ)n(es)21 b(of)1285 2572 y Fl(e)1282 2593 y Fv(\000)g(are)f(Lipsc) n(hitz-con)n(tin)n(uous)g(w.r.t.)35 b(\()p Fu(b;)14 b(c)p Fv(\))21 b(uniformly)g(in)h Fu(!)k Fq(2)d Fu(\015)5 b Fv(\([0)p Fu(;)14 b(L)p Fv(]\).)34 b(Therefore,)236 2692 y(b)n(y)28 b(lemma)f(6.1,)g(there)g(exists)g Fu(\016)g(>)22 b Fv(0)27 b(suc)n(h)h(that,)g Fq(8)p Fu(\021)e Fq(2)d Fw(R)28 b Fv(with)g Fq(j)p Fu(\021)s Fq(j)23 b(\024)g Fu(\016)s Fv(,)28 b Fq(8)p Fu(\030)f Fq(2)c Fw(R)2859 2662 y Ft(d)2925 2692 y Fv(with)28 b Fq(j)p Fu(\030)t Fq(j)c(\024)e Fu(\016)s Fv(,)28 b Fq(8)p Fu(!)e Fq(2)d Fu(\015)5 b Fv(\([0)p Fu(;)14 b(L)p Fv(]\),)305 2885 y Fu(@)349 2897 y Ft(c)386 2864 y Fl(e)383 2885 y Fv(\000\()p Fu(!)s(;)g(b)k Fv(+)g Fu(\021)s(;)c(c)19 b Fv(+)f Fu(\030)t Fv(\))g Fq(\001)h Fu(c)k Fq(\025)g Fv(3)p Fu(\027)1277 2897 y Fs(2)1313 2885 y Fu(=)p Fv(4)f Fu(>)h Fv(0)83 b Fw(or)g Fq(j)p Fu(@)1869 2897 y Ft(b)1905 2864 y Fl(e)1902 2885 y Fv(\000\()p Fu(!)s(;)14 b(b)k Fv(+)g Fu(\021)s(;)c(c)k Fv(+)g Fu(\030)t Fv(\))p Fq(j)24 b(\025)f Fv(3)p Fu(\027)2723 2897 y Fs(1)2759 2885 y Fu(=)p Fv(4)f Fu(>)h Fv(0)83 b(for)f Fq(j)p Fu(c)p Fq(j)23 b Fv(=)g Fu(r)n(;)14 b Fq(j)p Fu(b)p Fq(j)23 b(\024)p 3717 2818 36 4 v 22 w Fu(b;)69 b Fv(\(6.1\))874 3078 y Fu(@)918 3090 y Ft(b)954 3057 y Fl(e)951 3078 y Fv(\000\()p Fu(!)s(;)14 b(b)k Fv(+)g Fu(\021)s(;)c(c)19 b Fv(+)f Fu(\030)t Fv(\))h Fq(\002)f Fv(sign)o(\()p Fu(b)p Fv(\))23 b Fq(\025)g Fv(3)p Fu(\027)2094 3090 y Fs(1)2131 3078 y Fu(=)p Fv(4)f Fu(>)g Fv(0)83 b(for)f Fq(j)p Fu(c)p Fq(j)24 b(\024)e Fu(r)58 b Fv(and)d Fu(b)23 b Fv(=)g Fq(\006)p 3320 3011 V Fu(b)o(:)467 b Fv(\(6.2\))236 3228 y(Moreo)n(v)n(er)25 b(let)j(us)g(\014x)f Fu(\032)c(>)g Fv(0)k(suc)n(h)g(that)1626 3410 y Fu(\032)c Fq(\024)g Fv(min)p Fq(f)p Fu(\027)2001 3422 y Fs(1)2038 3410 y Fu(=)p Fv(2)p Fu(;)14 b(\027)2200 3422 y Fs(2)2236 3410 y Fu(=r)r Fq(g)p Fu(=)p Fv(\(6)p Fu(C)2534 3422 y Fs(2)2571 3410 y Fv(\))p Fu(;)1219 b Fv(\(6.3\))236 3593 y(where)33 b Fu(C)541 3605 y Fs(2)612 3593 y Fv(app)r(ears)e(in)j(\(3)p Fu(:)p Fv(24\).)52 b(These)33 b(are)f(the)i(p)r(ositiv)n(e)e(constan)n(ts)h(\()p Fu(\016)o(;)14 b(\032)p Fv(\))33 b(that)g(w)n(e)g(use)g(in)g(order)f (to)h(de\014ne,)h(for)236 3693 y(0)23 b Fu(<)f(\026)i(<)e(\026)599 3705 y Fs(6)636 3693 y Fv(,)p 715 3647 55 4 v 56 w Fu(!)769 3705 y Ft(i)797 3693 y Fv(,)p 848 3625 42 4 v 28 w Fu(\022)889 3705 y Ft(i)917 3693 y Fv(,)p 968 3647 55 4 v 28 w Fu(')1022 3713 y Ft(i)1077 3693 y Fv(b)n(y)27 b(lemma)h(5.2.)361 3792 y(Since)h Fu(\015)5 b Fv(\([0)p Fu(;)14 b(L)p Fv(]\))28 b(is)h(a)g(compact)f(subset)h(of)g Fq(D)1808 3762 y Ft(c)1806 3815 y(N)1869 3792 y Fv(,)h(inf)2023 3807 y Ft(s)p Fk(2)p Fs([0)p Ft(;L)p Fs(])2253 3792 y Fu(\014)t Fv(\()p Fu(\015)5 b Fv(\()p Fu(s)p Fv(\)\))26 b Fu(>)f Fv(0)k(and,)g(b)n(y)g(the)g(c)n (hoice)f(of)p 3497 3725 42 4 v 29 w Fu(\022)3538 3804 y Ft(i)3566 3792 y Fv(,)i(for)e Fu(\026)h Fv(small)236 3892 y(enough)21 b(\(3)p Fu(:)p Fv(21\))f(is)h(satis\014ed.)34 b(Therefore,)21 b(b)n(y)g(lemma)g(3.5)f(and)h(\(5.12\),)h(there)f (exists)f Fu(\026)2957 3904 y Fs(7)3018 3892 y Fu(>)i Fv(0)f(suc)n(h)g(that,)h Fq(8)p Fv(0)h Fu(<)g(\026)g Fq(\024)f Fu(\026)3955 3904 y Fs(7)3993 3892 y Fv(,)879 4157 y Fq(F)939 4169 y Ft(\026)983 4157 y Fv(\()p Fu(b;)14 b(c)p Fv(\))23 b(=)1277 4100 y(1)p 1277 4137 V 1277 4213 a(2)1342 4053 y Ft(k)q Fk(\000)p Fs(1)1343 4078 y Fl(X)1349 4254 y Ft(i)p Fs(=1)1598 4100 y Fq(j)p Fu(c)1657 4112 y Ft(i)p Fs(+1)1787 4100 y Fq(\000)18 b Fu(c)1906 4112 y Ft(i)1934 4100 y Fq(j)1957 4070 y Fs(2)p 1488 4137 617 4 v 1488 4222 a Fv(\001)p 1557 4154 42 4 v Fu(\022)1598 4234 y Ft(i)1644 4222 y Fv(+)g(\()p Fu(b)1795 4234 y Ft(i)p Fs(+1)1925 4222 y Fq(\000)g Fu(b)2044 4234 y Ft(i)2072 4222 y Fv(\))2132 4157 y(+)g Fu(\026)2321 4053 y Ft(k)2279 4078 y Fl(X)2285 4254 y Ft(i)p Fs(=1)2416 4136 y Fl(e)2413 4157 y Fv(\000\()p 2497 4111 55 4 v Fu(!)2552 4169 y Ft(i)2579 4157 y Fu(;)c(\021)2657 4169 y Ft(i)2704 4157 y Fv(+)k Fu(b)2823 4169 y Ft(i)2850 4157 y Fu(;)c(\030)2923 4169 y Ft(i)2969 4157 y Fv(+)k Fu(c)3088 4169 y Ft(i)3116 4157 y Fv(\))h(+)f Fu(R)3313 4169 y Fs(7)3350 4157 y Fu(;)472 b Fv(\(6.4\))236 4416 y(where)27 b Fq(j)p Fu(\021)540 4428 y Ft(i)568 4416 y Fq(j)c(\024)g Fu(\016)s Fv(,)28 b Fq(j)p Fu(\030)852 4428 y Ft(i)880 4416 y Fq(j)23 b(\024)g Fu(\016)s Fv(,)k Fu(R)1167 4428 y Fs(7)1232 4416 y Fv(is)h(giv)n(en)f (b)n(y)g(\(3)p Fu(:)p Fv(23\))g(and)g(satis\014es)g(\(3)p Fu(:)p Fv(24\).)361 4540 y(W)-7 b(e)28 b(minimize)g(the)g(functional)g Fq(F)1452 4552 y Ft(\026)1523 4540 y Fv(on)g(the)g(closure)e(of)700 4750 y Fu(W)35 b Fv(:=)924 4658 y Fl(n)979 4750 y Fv(\()p Fu(b;)14 b(c)p Fv(\))23 b(:=)g(\()p Fu(b)1354 4762 y Fs(1)1391 4750 y Fu(;)14 b(c)1464 4762 y Fs(1)1501 4750 y Fu(;)g(:)g(:)g(:)g(;)g(b)1722 4762 y Ft(k)1762 4750 y Fu(;)g(c)1835 4762 y Ft(k)1876 4750 y Fv(\))23 b Fq(2)h Fw(R)2082 4716 y Fs(\()p Ft(d)p Fs(+1\))p Ft(k)2320 4655 y Fl(\014)2320 4704 y(\014)2320 4754 y(\014)2376 4750 y Fq(j)p Fu(b)2435 4762 y Ft(i)2462 4750 y Fq(j)f Fu(<)p 2596 4683 36 4 v 23 w(b)o(;)42 b Fq(j)p Fu(c)2755 4762 y Ft(i)2783 4750 y Fq(j)23 b Fu(<)f(r)n(;)42 b Fq(8)p Fu(i)23 b Fv(=)f(1)p Fu(;)14 b(:)g(:)g(:)g(;)g(k)3474 4658 y Fl(o)3529 4750 y Fu(:)236 4977 y Fv(Since)p 460 4911 90 4 v 35 w Fu(W)47 b Fv(is)35 b(compact,)h Fq(F)1100 4989 y Ft(\026)1180 4977 y Fv(attains)e(its)h(minim)n(um)h(in)p 2070 4911 V 35 w Fu(W)12 b Fv(,)37 b(sa)n(y)c(at)i(\()2506 4955 y Fl(e)2511 4977 y Fu(b;)13 b Fl(e)-45 b Fu(c)p Fv(\).)59 b(By)35 b(lemma)g(2.3)f(the)h(existence)g(of)f(the)236 5091 y(di\013usion)20 b(orbit)g(will)g(b)r(e)g(pro)n(v)n(ed)f(once)g(w) n(e)h(sho)n(w)f(that)h(\()1969 5069 y Fl(e)1974 5091 y Fu(b;)13 b Fl(e)-45 b Fu(c)p Fv(\))23 b Fq(2)g Fu(W)12 b Fv(,)22 b(see)e(lemma)f(6.3.)34 b(Let)20 b(us)g(de\014ne)g(for)f Fu(i)k Fv(=)g(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(k)6 b Fq(\000)s Fv(1)1264 5308 y Fu(w)1323 5320 y Ft(i)1374 5308 y Fv(:=)23 b Fu(w)1544 5320 y Ft(i)1572 5308 y Fv(\()p Fu(b;)14 b(c)p Fv(\))23 b(:=)1892 5252 y Fu(c)1928 5264 y Ft(i)p Fs(+1)2058 5252 y Fq(\000)18 b Fu(c)2177 5264 y Ft(i)p 1889 5289 319 4 v 1889 5365 a Fu(\022)1928 5377 y Ft(i)p Fs(+1)2058 5365 y Fq(\000)g Fu(\022)2180 5377 y Ft(i)2241 5308 y Fv(=)2490 5252 y Fu(c)2526 5264 y Ft(i)p Fs(+1)2656 5252 y Fq(\000)g Fu(c)2775 5264 y Ft(i)p 2338 5289 617 4 v 2338 5373 a Fv(\001)p 2407 5305 42 4 v Fu(\022)2449 5385 y Ft(i)2495 5373 y Fv(+)g(\()p Fu(b)2646 5385 y Ft(i)p Fs(+1)2776 5373 y Fq(\000)g Fu(b)2895 5385 y Ft(i)2922 5373 y Fv(\))2965 5308 y Fu(;)2084 6059 y Fv(19)p eop %%Page: 20 20 20 19 bop 236 272 a Fv(and)28 b Fu(w)457 284 y Fs(0)517 272 y Fv(=)23 b Fu(w)664 284 y Ft(k)728 272 y Fv(=)g(0.)36 b(F)-7 b(rom)28 b(\(5.9\))f(and)g(\(3.14\),)g Fu(w)1817 284 y Ft(i)1873 272 y Fv(can)g(b)r(e)h(written)g(as)994 489 y Fu(w)1053 501 y Ft(i)1104 489 y Fv(=)1218 433 y Fu(')1272 445 y Ft(i)p Fs(+1)1403 433 y Fq(\000)18 b Fu(')1540 445 y Ft(i)p 1201 470 384 4 v 1201 546 a Fv(\()p Fu(\022)1272 558 y Ft(i)p Fs(+1)1403 546 y Fq(\000)g Fu(\022)1525 558 y Ft(i)1552 546 y Fv(\))1613 489 y Fq(\000)p 1696 443 55 4 v 18 w Fu(!)1751 501 y Ft(i)1797 489 y Fq(\000)1932 433 y Fv(\001)p 2001 387 V Fu(!)2056 445 y Ft(i)2084 433 y Fu(b)2120 445 y Ft(i)p Fs(+1)p 1890 470 384 4 v 1890 546 a Fv(\()p Fu(\022)1961 558 y Ft(i)p Fs(+1)2091 546 y Fq(\000)g Fu(\022)2213 558 y Ft(i)2241 546 y Fv(\))2306 489 y(=)2394 397 y Fl(\020)2443 489 y Fu(!)2495 501 y Ft(i)2541 489 y Fq(\000)p 2624 443 55 4 v 18 w Fu(!)2679 501 y Ft(i)2707 397 y Fl(\021)2775 489 y Fv(+)g Fu(O)2923 397 y Fl(\020)3054 433 y Fu(\026)p 2983 470 194 4 v 2983 546 a Fq(j)c Fv(ln)g Fu(\026)p Fq(j)3186 397 y Fl(\021)3235 489 y Fu(:)587 b Fv(\(6.5\))236 712 y(By)28 b(the)g(expression)e(of)h Fq(F)1064 724 y Ft(\026)1136 712 y Fv(in)h(\(6)p Fu(:)p Fv(4\))f(w)n(e)g(ha)n(v)n(e,)g (for)g(all)g Fu(i)c Fv(=)g(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(k)s Fv(,)1121 899 y Fu(@)1165 911 y Ft(c)1195 919 y Fc(i)1225 899 y Fq(F)1285 911 y Ft(\026)1330 899 y Fv(\()p Fu(b;)g(c)p Fv(\))23 b(=)f Fu(w)1672 911 y Ft(i)p Fk(\000)p Fs(1)1804 899 y Fq(\000)c Fu(w)1946 911 y Ft(i)1992 899 y Fv(+)g Fu(\026@)2169 911 y Ft(c)2206 878 y Fl(e)2203 899 y Fv(\000\()p 2287 853 55 4 v Fu(!)2342 911 y Ft(i)2370 899 y Fu(;)c(\021)2448 911 y Ft(i)2494 899 y Fv(+)k Fu(b)2613 911 y Ft(i)2640 899 y Fu(;)c(\030)2713 911 y Ft(i)2760 899 y Fv(+)k Fu(c)2879 911 y Ft(i)2906 899 y Fv(\))h(+)f Fu(R)3103 911 y Ft(i)3845 899 y Fv(\(6.6\))964 1115 y Fu(@)1008 1127 y Ft(b)1037 1135 y Fc(i)1068 1115 y Fq(F)1128 1127 y Ft(\026)1172 1115 y Fv(\()p Fu(b;)c(c)p Fv(\))23 b(=)1466 1059 y(1)p 1466 1096 42 4 v 1466 1172 a(2)1517 1023 y Fl(\020)1567 1115 y Fq(j)p Fu(w)1649 1127 y Ft(i)1677 1115 y Fq(j)1700 1080 y Fs(2)1756 1115 y Fq(\000)18 b(j)p Fu(w)1921 1127 y Ft(i)p Fk(\000)p Fs(1)2034 1115 y Fq(j)2057 1080 y Fs(2)2094 1023 y Fl(\021)2162 1115 y Fv(+)g Fu(\026@)2339 1127 y Ft(b)2376 1094 y Fl(e)2373 1115 y Fv(\000\()p 2457 1069 55 4 v Fu(!)2511 1127 y Ft(i)2539 1115 y Fu(;)c(\021)2617 1127 y Ft(i)2663 1115 y Fv(+)k Fu(b)2782 1127 y Ft(i)2810 1115 y Fu(;)c(\030)2883 1127 y Ft(i)2929 1115 y Fv(+)k Fu(c)3048 1127 y Ft(i)3076 1115 y Fv(\))g(+)g Fu(S)3260 1127 y Ft(i)3845 1115 y Fv(\(6.7\))236 1287 y(where)27 b Fu(R)539 1299 y Ft(i)590 1287 y Fv(:=)c Fu(@)745 1299 y Ft(c)775 1307 y Fc(i)805 1287 y Fu(R)868 1299 y Fs(7)905 1287 y Fv(,)28 b Fu(S)1007 1299 y Ft(i)1058 1287 y Fv(:=)22 b Fu(@)1212 1299 y Ft(b)1241 1307 y Fc(i)1272 1287 y Fu(R)1335 1299 y Fs(7)1400 1287 y Fv(satisfy)-7 b(,)27 b(b)n(y)h(\(3.24\))e(and)i(\(6)p Fu(:)p Fv(3\))1620 1486 y Fq(j)p Fu(R)1706 1498 y Ft(i)1734 1486 y Fq(j)p Fu(;)14 b Fq(j)p Fu(S)1868 1498 y Ft(i)1895 1486 y Fq(j)23 b(\024)2039 1429 y Fu(\026)p 2039 1466 51 4 v 2043 1543 a Fv(2)2113 1486 y(min)2265 1393 y Fl(n)2330 1429 y Fu(\027)2371 1441 y Fs(1)p 2330 1466 79 4 v 2349 1543 a Fv(2)2419 1486 y Fu(;)2465 1429 y(\027)2506 1441 y Fs(2)p 2465 1466 V 2485 1543 a Fu(r)2554 1393 y Fl(o)2609 1486 y Fu(:)1213 b Fv(\(6.8\))236 1693 y(By)22 b(\(6.6\)-\(6.7\),)g(a)f(w)n(a) n(y)f(to)i(see)f(critical)h(p)r(oin)n(ts)f(of)h Fq(F)1889 1705 y Ft(\026)1955 1693 y Fv(is)g(to)f(sho)n(w)g(that)h(the)g(terms)g Fu(w)2924 1705 y Ft(i)p Fk(\000)p Fs(1)3043 1693 y Fq(\000)7 b Fu(w)3174 1705 y Ft(i)3224 1693 y Fv(and)21 b Fq(j)p Fu(w)3461 1705 y Ft(i)3489 1693 y Fq(j)3512 1663 y Fs(2)3556 1693 y Fq(\000)7 b(j)p Fu(w)3710 1705 y Ft(i)p Fk(\000)p Fs(1)3823 1693 y Fq(j)3846 1663 y Fs(2)3905 1693 y Fv(are)236 1792 y(small)21 b(w.r.t)g(the)g Fu(O)r Fv(\()p Fu(\026)p Fv(\)-con)n(tribution)h(pro)n(vided)e(b)n(y)g(the)i(Melnik)n(o)n(v)e (function.)35 b(By)21 b(\(3.8\))g Fq(j)p Fu(!)3127 1804 y Ft(i)3159 1792 y Fq(\000)p 3229 1747 55 4 v 5 w Fu(!)3284 1804 y Ft(i)3312 1792 y Fq(j)i Fv(=)f Fu(O)r Fv(\(1)p Fu(=)p Fv(\()p Fu(\022)3697 1804 y Ft(i)p Fs(+1)3815 1792 y Fq(\000)5 b Fu(\022)3924 1804 y Ft(i)3951 1792 y Fv(\)\))236 1892 y(and)24 b(hence,)g(using)g(\(6.5\),)g(an)f (estimate)h(for)f(eac)n(h)g Fu(w)1881 1904 y Ft(i)1933 1892 y Fv(separately)f(is)h(giv)n(en)g(b)n(y)g Fu(w)2786 1904 y Ft(i)2838 1892 y Fv(=)f Fu(O)r Fv(\(1)p Fu(=)p Fq(j)p 3129 1824 42 4 v Fu(\022)3171 1904 y Ft(i)p Fs(+1)3293 1892 y Fq(\000)p 3369 1824 V 11 w Fu(\022)3409 1904 y Ft(i)3437 1892 y Fq(j)p Fv(\))11 b(+)g Fu(O)r Fv(\()p Fu(\026=)p Fq(j)j Fv(ln)f Fu(\026)p Fq(j)p Fv(\).)236 1992 y(Hence)26 b(eac)n(h)f Fq(j)p Fu(w)748 2004 y Ft(i)776 1992 y Fq(j)h Fv(is)f Fu(O)r Fv(\()p Fu(\026)p Fv(\)-small)h(if)g(the)g (time)g(to)g(mak)n(e)f(a)g(transition)g Fq(j)p 2513 1924 V Fu(\022)2554 2004 y Ft(i)p Fs(+1)2680 1992 y Fq(\000)p 2759 1924 V 14 w Fu(\022)2801 2004 y Ft(i)2828 1992 y Fq(j)e Fv(=)g Fu(O)r Fv(\(1)p Fu(=\026)p Fv(\),)j(as)f(in)h([7].)36 b(These)25 b(time)236 2091 y(in)n(terv)-5 b(als)27 b(are)g(to)r(o)g (large)f(to)h(obtain)h(the)g(appro)n(ximation)d(for)i(the)h(reduced)g (action)f(functional)g Fq(F)3390 2103 y Ft(\026)3462 2091 y Fv(giv)n(en)g(in)h(lemma)236 2191 y(3.5)i(and)h(\(6)p Fu(:)p Fv(4\).)46 b(Therefore)30 b(w)n(e)g(need)h(more)f(re\014ned)h (estimates:)43 b(the)31 b(pro)r(of)g(of)f(Theorem)g(1.1)g(\(and)h (Theorem)f(1.3\))236 2302 y(relies)d(on)g(the)h(follo)n(wing)f(crucial) g(prop)r(ert)n(y)f(for)42 b Fl(e)-61 b Fu(w)1851 2314 y Ft(i)1902 2302 y Fv(:=)23 b Fu(w)2072 2314 y Ft(i)2100 2302 y Fv(\()2127 2280 y Fl(e)2132 2302 y Fu(b;)13 b Fl(e)-45 b Fu(c)p Fv(\),)28 b(satis\014ed)f(b)n(y)g(the)h(minim)n(um)h (p)r(oin)n(t)e(\()3516 2280 y Fl(e)3521 2302 y Fu(b;)13 b Fl(e)-45 b Fu(c)p Fv(\).)236 2461 y Fw(Lemma)32 b(6.2)40 b Fp(We)30 b(have)h(\(for)g Fu(i)22 b Fv(=)h(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(k)s(;)p Fp(\))1175 2683 y Fu(i)p Fv(\))60 b Fq(j)14 b Fl(e)-60 b Fu(w)1378 2695 y Ft(i)1424 2683 y Fq(\000)33 b Fl(e)-61 b Fu(w)1566 2695 y Ft(i)p Fk(\000)p Fs(1)1679 2683 y Fq(j)23 b Fv(=)g Fu(O)r Fv(\()p Fu(\026)p Fv(\))p Fu(;)185 b(ii)p Fv(\))59 b Fq(j)15 b Fl(e)-61 b Fu(w)2431 2695 y Ft(i)2459 2683 y Fq(j)23 b Fv(=)g Fu(O)2658 2591 y Fl(\020)2796 2571 y Fq(p)p 2866 2571 51 4 v 2866 2622 a Fu(\026)p 2718 2664 277 4 v 2718 2680 a Fl(p)p 2801 2680 194 4 v 71 x Fq(j)14 b Fv(ln)g Fu(\026)p Fq(j)3004 2591 y Fl(\021)3054 2683 y Fu(:)768 b Fv(\(6.9\))236 2921 y Fb(Pr)n(oof)p Fv(.)82 b(Estimate)28 b(\(6)p Fu(:)p Fv(9\))19 b Fq(\000)g Fu(i)p Fv(\))29 b(is)f(a)h(straigh)n(tforw)n(ard) c(consequence)j(of)h(\(6.6\))f(and)h(\(6.8\))f(if)i Fq(j)p Fl(e)-46 b Fu(c)3322 2933 y Ft(i)3349 2921 y Fq(j)25 b Fu(<)g(r)r Fv(,)30 b(since)f(in)g(this)236 3032 y(case)e Fu(@)456 3044 y Ft(c)486 3052 y Fc(i)516 3032 y Fq(F)576 3044 y Ft(\026)620 3032 y Fv(\()647 3010 y Fl(e)652 3032 y Fu(b;)13 b Fl(e)-45 b Fu(c)p Fv(\))23 b(=)g(0.)37 b(W)-7 b(e)28 b(no)n(w)e(pro)n(v)n(e)g(that)i(\(6)p Fu(:)p Fv(9\))p Fu(i)p Fv(\))g(holds)f(also)f(if)i Fq(j)p Fl(e)-46 b Fu(c)2503 3044 y Ft(i)2531 3032 y Fq(j)23 b Fv(=)g Fu(r)30 b Fv(for)d(some)g Fu(i)p Fv(.)37 b(Indeed)28 b(if)g Fq(j)p Fl(e)-46 b Fu(c)3561 3044 y Ft(i)3588 3032 y Fq(j)24 b Fv(=)e Fu(r)31 b Fv(then)1438 3220 y Fu(@)1482 3232 y Ft(c)1512 3240 y Fc(i)1542 3220 y Fq(F)1602 3232 y Ft(\026)1646 3220 y Fv(\()1673 3198 y Fl(e)1678 3220 y Fu(b;)13 b Fl(e)-45 b Fu(c)p Fv(\))23 b(=)g Fu(\013)1983 3232 y Ft(\026)2027 3220 y Fl(e)-46 b Fu(c)2063 3232 y Ft(i)2174 3220 y Fv(for)27 b(some)82 b Fu(\013)2617 3232 y Ft(\026)2685 3220 y Fq(\024)23 b Fv(0)989 b(\(6.10\))236 3409 y(\(since)28 b(\()499 3387 y Fl(e)504 3409 y Fu(b;)13 b Fl(e)-45 b Fu(c)p Fv(\))28 b(is)f(a)g(minim)n(um)i(p)r(oin)n(t\))f (and)f(then)h(b)n(y)f(\(6.6\),\(6.10\))g(and)g(\(6.8\))h(w)n(e)f (deduce)1672 3582 y Fl(e)-61 b Fu(w)1716 3594 y Ft(i)p Fk(\000)p Fs(1)1848 3582 y Fq(\000)33 b Fl(e)-61 b Fu(w)1990 3594 y Ft(i)2041 3582 y Fv(=)23 b Fu(\013)2182 3594 y Ft(\026)2226 3582 y Fl(e)-46 b Fu(c)2262 3594 y Ft(i)2308 3582 y Fv(+)18 b Fu(O)r Fv(\()p Fu(\026)p Fv(\))p Fu(:)1211 b Fv(\(6.11\))236 3756 y(Let)20 b(us)f(decomp)r(ose)33 b Fl(e)-61 b Fu(w)943 3768 y Ft(i)p Fk(\000)p Fs(1)1075 3756 y Fv(and)34 b Fl(e)-61 b Fu(w)1287 3768 y Ft(i)1335 3756 y Fv(in)19 b(the)h(\\radial")d(and)i(\\tangen)n(t")e(directions)i (to)g(the)g(ball)g Fu(S)3199 3768 y Ft(i)3250 3756 y Fv(=)k Fq(fj)p Fu(b)3439 3768 y Ft(i)3466 3756 y Fq(j)g(\024)p 3599 3689 36 4 v 22 w Fu(b;)33 b Fq(j)p Fu(c)3750 3768 y Ft(i)3778 3756 y Fq(j)23 b(\024)f Fu(r)r Fq(g)p Fv(:)1490 3930 y Fl(e)-60 b Fu(w)1535 3942 y Ft(i)p Fk(\000)p Fs(1)1671 3930 y Fv(=)22 b Fu(a)1802 3942 y Ft(i)1830 3930 y Fl(e)-46 b Fu(c)1866 3942 y Ft(i)1912 3930 y Fv(+)18 b Fu(u)2043 3942 y Ft(i)2236 3930 y Fv(with)28 b Fu(u)2473 3942 y Ft(i)2519 3930 y Fq(\001)18 b Fl(e)-46 b Fu(c)2596 3942 y Ft(i)2647 3930 y Fv(=)23 b(0)1027 b(\(6.12\))1456 4104 y Fq(\000)14 b Fl(e)-60 b Fu(w)1580 4116 y Ft(i)1631 4104 y Fv(=)22 b Fu(a)1762 4070 y Fk(0)1762 4124 y Ft(i)1789 4104 y Fl(e)-45 b Fu(c)1826 4116 y Ft(i)1872 4104 y Fv(+)18 b Fu(u)2003 4070 y Fk(0)2003 4124 y Ft(i)2030 4104 y Fu(;)180 b Fv(with)28 b Fu(u)2470 4070 y Fk(0)2470 4124 y Ft(i)2516 4104 y Fq(\001)18 b Fl(e)-46 b Fu(c)2593 4116 y Ft(i)2644 4104 y Fv(=)22 b(0)p Fu(:)1008 b Fv(\(6.13\))236 4248 y(Since)28 b Fq(j)p Fl(e)-46 b Fu(c)512 4260 y Ft(i)p Fk(\000)p Fs(1)625 4248 y Fq(j)23 b(\024)f(j)p Fl(e)-46 b Fu(c)817 4260 y Ft(i)845 4248 y Fq(j)23 b Fv(=)g Fu(r)r Fv(,)28 b Fq(j)p Fl(e)-46 b Fu(c)1128 4260 y Ft(i)p Fs(+1)1240 4248 y Fq(j)23 b(\024)g(j)p Fl(e)-46 b Fu(c)1433 4260 y Ft(i)1460 4248 y Fq(j)23 b Fv(=)g Fu(r)r Fv(,)29 b(there)e(results)g (that)1268 4422 y Fu(a)1312 4434 y Ft(i)1339 4422 y Fu(r)1378 4388 y Fs(2)1439 4422 y Fv(=)38 b Fl(e)-61 b Fu(w)1586 4434 y Ft(i)p Fk(\000)p Fs(1)1717 4422 y Fq(\001)18 b Fl(e)-45 b Fu(c)1795 4434 y Ft(i)1846 4422 y Fq(\025)22 b Fv(0)83 b(and)g Fu(a)2319 4388 y Fk(0)2319 4442 y Ft(i)2346 4422 y Fu(r)2385 4388 y Fs(2)2446 4422 y Fv(=)23 b Fq(\000)14 b Fl(e)-60 b Fu(w)2658 4434 y Ft(i)2704 4422 y Fq(\001)18 b Fl(e)-46 b Fu(c)2781 4434 y Ft(i)2832 4422 y Fq(\025)23 b Fv(0)p Fu(;)819 b Fv(\(6.14\))236 4596 y(so)27 b(that)h Fu(a)562 4608 y Ft(i)590 4596 y Fu(;)14 b(a)671 4566 y Fk(0)671 4617 y Ft(i)721 4596 y Fq(\025)23 b Fv(0.)36 b(Summing)28 b(\(6.12\))f(and)g(\(6.13\))g(and)h(using)f(\(6.11\))g(w)n (e)g(obtain)1442 4770 y(\()p Fu(a)1518 4782 y Ft(i)1565 4770 y Fv(+)18 b Fu(a)1692 4735 y Fk(0)1692 4790 y Ft(i)1719 4770 y Fv(\))p Fl(e)-46 b Fu(c)1787 4782 y Ft(i)1834 4770 y Fv(+)18 b(\()p Fu(u)1997 4782 y Ft(i)2042 4770 y Fv(+)g Fu(u)2173 4735 y Fk(0)2173 4790 y Ft(i)2201 4770 y Fv(\))23 b(=)g Fu(O)r Fv(\()p Fu(\026)p Fv(\))c(+)f Fu(\013)2678 4782 y Ft(\026)2722 4770 y Fl(e)-45 b Fu(c)2759 4782 y Ft(i)2787 4770 y Fu(;)236 4944 y Fv(with)28 b Fu(a)469 4956 y Ft(i)497 4944 y Fu(;)14 b(a)578 4913 y Fk(0)578 4965 y Ft(i)605 4944 y Fu(;)g Fq(\000)p Fu(\013)760 4956 y Ft(\026)827 4944 y Fq(\025)23 b Fv(0.)37 b(This)27 b(implies)h(that)g Fu(\013)1721 4956 y Ft(\026)1788 4944 y Fv(=)23 b Fu(O)r Fv(\()p Fu(\026=r)r Fv(\))29 b(and)f(from)f (equation)g(\(6.11\))g(w)n(e)g(get)g(\(6)p Fu(:)p Fv(9\))p Fu(i)p Fv(\).)361 5043 y(W)-7 b(e)29 b(can)g(no)n(w)g(pro)n(v)n(e)e (\(6)p Fu(:)p Fv(9\))19 b Fq(\000)g Fu(ii)p Fv(\).)41 b(Let)30 b Fu(i)1667 5055 y Fs(0)1729 5043 y Fq(2)c(f)p Fv(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(k)22 b Fq(\000)d Fv(1)p Fq(g)28 b Fv(b)r(e)i(suc)n(h)e(that)i Fq(8)p Fv(1)25 b Fq(\024)g Fu(i)h Fq(\024)f Fu(k)d Fq(\000)d Fv(1,)29 b Fq(j)15 b Fl(e)-61 b Fu(w)3497 5055 y Ft(i)3520 5063 y Fd(0)3558 5043 y Fq(j)25 b(\025)h(j)14 b Fl(e)-60 b Fu(w)3779 5055 y Ft(i)3807 5043 y Fq(j)p Fv(.)41 b(F)-7 b(or)236 5152 y Fu(j)28 b Fq(2)c(f)p Fv(1)p Fu(;)14 b(:)g(:)g(:)e(;)i (k)21 b Fq(\000)d Fv(1)p Fq(g)p Fv(,)27 b Fu(j)h Fq(6)p Fv(=)23 b Fu(i)1104 5164 y Fs(0)1168 5152 y Fv(w)n(e)28 b(can)f(write)42 b Fl(e)-61 b Fu(w)1714 5164 y Ft(j)1773 5152 y Fv(=)37 b Fl(e)-61 b Fu(w)1919 5164 y Ft(i)1942 5172 y Fd(0)1998 5152 y Fv(+)18 b Fu(s)2120 5164 y Ft(j)2183 5152 y Fv(with)28 b Fu(s)2411 5164 y Ft(j)2469 5152 y Fv(=)2557 5090 y Fl(P)2644 5111 y Ft(j)s Fk(\000)p Fs(1)2644 5177 y Ft(i)p Fs(=)p Ft(i)2741 5185 y Fd(0)2779 5152 y Fv(\()15 b Fl(e)-61 b Fu(w)2870 5164 y Ft(i)p Fs(+1)3001 5152 y Fq(\000)32 b Fl(e)-60 b Fu(w)3143 5164 y Ft(i)3171 5152 y Fv(\))28 b(and)f(hence,)h(b)n(y)f(\(6)p Fu(:)p Fv(9\))p Fu(i)p Fv(\))1496 5423 y Fq(j)p Fu(s)1558 5435 y Ft(j)1593 5423 y Fq(j)d(\024)1734 5315 y Ft(j)s Fk(\000)p Fs(1)1732 5344 y Fl(X)1727 5521 y Ft(i)p Fs(=)p Ft(i)1824 5529 y Fd(0)1871 5423 y Fq(j)15 b Fl(e)-61 b Fu(w)1953 5435 y Ft(i)p Fs(+1)2084 5423 y Fq(\000)33 b Fl(e)-61 b Fu(w)2226 5435 y Ft(i)2254 5423 y Fq(j)23 b(\024)g Fu(C)6 b(\026)p Fq(j)p Fu(j)24 b Fq(\000)18 b Fu(i)2696 5435 y Fs(0)2732 5423 y Fq(j)1049 b Fv(\(6.15\))2084 6059 y(20)p eop %%Page: 21 21 21 20 bop 236 272 a Fv(for)27 b(some)g(constan)n(t)g Fu(C)i(>)23 b Fv(0.)37 b(Hence)996 512 y Fl(e)-45 b Fu(c)1033 524 y Ft(j)1086 512 y Fq(\000)18 b Fl(e)-46 b Fu(c)1205 524 y Ft(i)1228 532 y Fd(0)1288 512 y Fv(=)1383 404 y Ft(j)s Fk(\000)p Fs(1)1381 433 y Fl(X)1376 610 y Ft(i)p Fs(=)p Ft(i)1473 618 y Fd(0)1535 512 y Fl(e)-61 b Fu(w)1579 524 y Ft(i)1607 512 y Fv(\()1643 490 y Fl(e)1639 512 y Fu(\022)1678 524 y Ft(i)p Fs(+1)1809 512 y Fq(\000)1896 490 y Fl(e)1892 512 y Fu(\022)1931 524 y Ft(i)1958 512 y Fv(\))23 b(=)38 b Fl(e)-61 b Fu(w)2160 524 y Ft(i)2183 532 y Fd(0)2221 512 y Fv(\()2257 490 y Fl(e)2253 512 y Fu(\022)2292 524 y Ft(j)2345 512 y Fq(\000)2433 490 y Fl(e)2428 512 y Fu(\022)2467 524 y Ft(i)2490 532 y Fd(0)2527 512 y Fv(\))19 b(+)2668 404 y Ft(j)s Fk(\000)p Fs(1)2666 433 y Fl(X)2661 610 y Ft(i)p Fs(=)p Ft(i)2758 618 y Fd(0)2805 512 y Fu(s)2844 524 y Ft(i)2872 512 y Fv(\()2908 490 y Fl(e)2904 512 y Fu(\022)2943 524 y Ft(i)p Fs(+1)3073 512 y Fq(\000)3161 490 y Fl(e)3156 512 y Fu(\022)3195 524 y Ft(i)3223 512 y Fv(\))549 b(\(6.16\))236 771 y(and)28 b(then)g(b)n(y)f(\(6.15\))677 875 y Fl(\014)677 924 y(\014)677 974 y(\014)704 970 y(e)-46 b Fu(c)740 982 y Ft(j)794 970 y Fq(\000)17 b Fl(e)-45 b Fu(c)913 982 y Ft(i)936 990 y Fd(0)973 875 y Fl(\014)973 924 y(\014)973 974 y(\014)1023 970 y Fq(\025)23 b(j)15 b Fl(e)-61 b Fu(w)1193 982 y Ft(i)1216 990 y Fd(0)1254 970 y Fq(jj)1304 948 y Fl(e)1300 970 y Fu(\022)1339 982 y Ft(j)1392 970 y Fq(\000)1480 948 y Fl(e)1475 970 y Fu(\022)1514 982 y Ft(i)1537 990 y Fd(0)1574 970 y Fq(j)19 b(\000)f Fu(C)6 b(\026)p Fq(j)p Fu(j)24 b Fq(\000)18 b Fu(i)2007 982 y Fs(0)2043 970 y Fq(jj)2093 948 y Fl(e)2089 970 y Fu(\022)2128 982 y Ft(j)2182 970 y Fq(\000)2269 948 y Fl(e)2265 970 y Fu(\022)2304 982 y Ft(i)2327 990 y Fd(0)2364 970 y Fq(j)23 b Fv(=)2498 878 y Fl(\020)2547 970 y Fq(j)15 b Fl(e)-61 b Fu(w)2629 982 y Ft(i)2652 990 y Fd(0)2690 970 y Fq(j)18 b(\000)g Fu(C)6 b(\026)p Fq(j)p Fu(j)24 b Fq(\000)18 b Fu(i)3122 982 y Fs(0)3159 970 y Fq(j)3182 878 y Fl(\021)3232 970 y Fq(j)3259 948 y Fl(e)3255 970 y Fu(\022)3294 982 y Ft(j)3347 970 y Fq(\000)3435 948 y Fl(e)3430 970 y Fu(\022)3469 982 y Ft(i)3492 990 y Fd(0)3529 970 y Fq(j)p Fu(:)229 b Fv(\(6.17\))236 1188 y(Since)35 b Fq(j)487 1167 y Fl(e)483 1188 y Fu(\022)522 1200 y Ft(i)p Fs(+1)656 1188 y Fq(\000)748 1167 y Fl(e)744 1188 y Fu(\022)783 1200 y Ft(i)811 1188 y Fq(j)f Fu(>)g(C)1026 1200 y Fs(1)1063 1188 y Fq(j)14 b Fv(ln)g Fu(\026)p Fq(j)23 b Fv(+)g Fu(O)r Fv(\(1\))35 b(\(b)n(y)f(\(3.4\)\),)i Fq(8)p Fu(i)e Fv(=)g(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(k)26 b Fq(\000)c Fv(1,)36 b Fq(j)2707 1167 y Fl(e)2703 1188 y Fu(\022)2742 1200 y Ft(j)2800 1188 y Fq(\000)2892 1167 y Fl(e)2888 1188 y Fu(\022)2927 1200 y Ft(i)2950 1208 y Fd(0)2987 1188 y Fq(j)e Fu(>)g(C)3202 1200 y Fs(1)3239 1188 y Fq(j)p Fu(j)28 b Fq(\000)23 b Fu(i)3441 1200 y Fs(0)3478 1188 y Fq(j)g(\001)g(j)14 b Fv(ln)g Fu(\026)p Fq(j)p Fv(.)57 b(T)-7 b(ak)n(e)p 236 1233 39 4 v 236 1298 a Fu(j)40 b Fq(2)35 b(f)p Fv(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(k)25 b Fq(\000)e Fv(1)p Fq(g)34 b Fv(suc)n(h)g(that)h Fq(j)p 1346 1233 V Fu(j)28 b Fq(\000)23 b Fu(i)1525 1310 y Fs(0)1562 1298 y Fq(j)35 b Fv(=)f([\()1774 1246 y Fq(p)p 1843 1246 51 4 v 52 x Fu(\026)1893 1227 y Fl(p)p 1976 1227 194 4 v 71 x Fq(j)14 b Fv(ln)g Fu(\026)p Fq(j)p Fv(\))2201 1268 y Fk(\000)p Fs(1)2291 1298 y Fv(])23 b(+)g(1)34 b(\(suc)n(h)h(a)p 2804 1233 39 4 v 34 w Fu(j)k Fv(certainly)34 b(exists)g(since,)j(b)n(y)d(\(5.7\),)236 1398 y Fu(k)26 b Fq(\031)d Fv(1)p Fu(=\026)j Fv(for)i Fu(\026)f Fv(small\).)37 b(Then)28 b(w)n(e)f(obtain,)g(using)h(that)g Fq(j)p Fl(e)-46 b Fu(c)2117 1410 y Ft(i)2144 1398 y Fq(j)23 b(\024)g Fu(r)30 b Fv(for)d(all)h Fu(i)22 b Fv(=)h(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(k)s Fv(,)1133 1636 y(2)p Fu(r)25 b Fq(\025)1325 1541 y Fl(\014)1325 1591 y(\014)1325 1640 y(\014)1352 1636 y(e)-45 b Fu(c)1389 1648 y Ft(j)1442 1636 y Fq(\000)17 b Fl(e)-45 b Fu(c)1561 1648 y Ft(i)1584 1656 y Fd(0)1621 1541 y Fl(\014)1621 1591 y(\014)1621 1640 y(\014)1672 1636 y Fq(\025)1759 1544 y Fl(\020)1809 1636 y Fq(j)15 b Fl(e)-61 b Fu(w)1891 1648 y Ft(i)1914 1656 y Fd(0)1951 1636 y Fq(j)19 b(\000)f Fu(C)2230 1524 y Fq(p)p 2299 1524 51 4 v 52 x Fu(\026)p 2151 1617 277 4 v 2151 1634 a Fl(p)p 2234 1634 194 4 v 71 x Fq(j)c Fv(ln)g Fu(\026)p Fq(j)2456 1636 y(\000)k Fu(C)6 b(\026)2654 1544 y Fl(\021)2703 1636 y Fu(C)2762 1648 y Fs(1)2810 1508 y Fl(p)p 2893 1508 V 71 x Fq(j)14 b Fv(ln)g Fu(\026)p Fq(j)p 2810 1617 277 4 v 2888 1642 a(p)p 2958 1642 51 4 v 2958 1693 a Fu(\026)3096 1636 y(;)236 1935 y Fp(i.e.)39 b Fq(j)14 b Fl(e)-60 b Fu(w)470 1947 y Ft(i)493 1955 y Fd(0)530 1935 y Fq(j)23 b(\024)674 1875 y Fv(\(2)p Fu(r)e Fv(+)d Fu(C)6 b(C)1013 1887 y Fs(1)1051 1875 y Fv(\))1083 1823 y Fq(p)p 1152 1823 V 52 x Fu(\026)p 674 1916 529 4 v 752 2004 a(C)811 2016 y Fs(1)848 1933 y Fl(p)p 931 1933 194 4 v 71 x Fq(j)14 b Fv(ln)g Fu(\026)p Fq(j)1231 1935 y Fv(+)k Fu(C)6 b(\026)p Fv(.)37 b(W)-7 b(e)28 b(ha)n(v)n(e)e(th)n(us)i(pro)n(v)n(ed)e(the)i(imp)r(ortan)n(t)f (prop)r(ert)n(y)g(\(6)p Fu(:)p Fv(9\))18 b Fq(\000)g Fu(ii)p Fv(\).)p 3572 1935 48 48 v 236 2220 a Fw(Remark)32 b(6.1)41 b Fp(By)30 b(\(6.5\),)h Fv(\()t Fl(e)-50 b Fu(!)1208 2232 y Ft(i)1254 2220 y Fq(\000)p 1338 2174 55 4 v 19 w Fu(!)1392 2232 y Ft(i)1420 2220 y Fv(\))23 b(=)37 b Fl(e)-60 b Fu(w)1622 2232 y Ft(i)1668 2220 y Fv(+)18 b Fu(O)r Fv(\()p Fu(\026=)p Fq(j)c Fv(ln)h Fu(\026)p Fq(j)p Fv(\))p Fp(,)30 b(so)g(that,)h(by)f(\(5.8\),)h(\(6.9\))g (implies)1230 2442 y Fq(j)t Fl(e)-50 b Fu(!)1305 2454 y Ft(i)1351 2442 y Fq(\000)p 1434 2396 V 18 w Fu(!)1489 2454 y Ft(i)1516 2442 y Fq(j)23 b Fv(=)g Fu(O)1715 2350 y Fl(\020)1854 2330 y Fq(p)p 1923 2330 51 4 v 52 x Fu(\026)p 1775 2423 277 4 v 1775 2439 a Fl(p)p 1858 2439 194 4 v 71 x Fq(j)14 b Fv(ln)g Fu(\026)p Fq(j)2061 2350 y Fl(\021)2111 2442 y Fu(;)183 b Fq(j)t Fl(e)-50 b Fu(!)2392 2454 y Ft(i)p Fs(+1)2522 2442 y Fq(\000)23 b Fl(e)-51 b Fu(!)2657 2454 y Ft(i)2685 2442 y Fq(j)23 b Fv(=)f Fu(O)r Fv(\()p Fu(\026)p Fv(\))p Fu(:)784 b Fv(\(6.18\))236 2680 y Fp(Note)34 b(that,)i(fr)l(om)f(\(3.8\),)i(we)e(would)g(just)f(obtain)h Fq(j)t Fl(e)-50 b Fu(!)1947 2692 y Ft(i)1996 2680 y Fq(\000)p 2082 2635 55 4 v 21 w Fu(!)2137 2692 y Ft(i)2164 2680 y Fq(j)32 b Fv(=)f Fu(O)r Fv(\(1)p Fu(=)p Fq(j)14 b Fv(ln)g Fu(\026)p Fq(j)p Fv(\))p Fp(.)52 b(\(6.18\))36 b(c)l(an)e(b)l(e)h(se)l (en)f(as)g(an)h(a-priori)236 2791 y(estimate)30 b(satis\014e)l(d)g(by)h (the)e(minimum)h(p)l(oint)g Fv(\()1751 2769 y Fl(e)1747 2791 y Fu(\022)s(;)24 b Fl(e)-56 b Fu(')p Fv(\))p Fp(.)361 2965 y Fv(The)26 b(follo)n(wing)g(lemma)h(pro)n(v)n(es)d(the)j (existence)g(of)f(a)g(lo)r(cal)g(minim)n(um)i(of)e(the)h(reduced)g (action)f(functional)h(in)f(the)236 3065 y(in)n(terior)h(of)g Fu(W)40 b Fv(and)27 b(hence)h(of)f(a)h(true)f(di\013usion)h(orbit.)236 3239 y Fw(Lemma)k(6.3)40 b Fp(L)l(et)30 b Fv(\()909 3217 y Fl(e)914 3239 y Fu(b)o(;)13 b Fl(e)-45 b Fu(c)p Fv(\))30 b Fp(b)l(e)g(a)g(minimum)g(p)l(oint)g(of)g Fq(F)2001 3251 y Ft(\026)2075 3239 y Fp(over)p 2259 3172 90 4 v 31 w Fu(W)12 b Fp(.)38 b(Then)31 b Fv(\()2656 3217 y Fl(e)2661 3239 y Fu(b;)13 b Fl(e)-45 b Fu(c)p Fv(\))23 b Fq(2)g Fu(W)12 b Fp(,)30 b(namely)1476 3412 y Fq(j)p Fl(e)-46 b Fu(c)1535 3424 y Ft(i)1563 3412 y Fq(j)23 b Fu(<)g(r)172 b Fv(for)27 b(all)170 b Fu(i)22 b Fq(2)i(f)p Fv(1)p Fu(;)14 b(:)g(:)g(:)e(;)i(k)s Fq(g)1028 b Fv(\(6.19\))236 3586 y Fp(and)1467 3686 y Fq(j)1485 3664 y Fl(e)1490 3686 y Fu(b)1526 3698 y Ft(i)1553 3686 y Fq(j)23 b Fu(<)p 1687 3618 36 4 v 23 w(b)169 b Fv(for)28 b(all)169 b Fu(i)23 b Fq(2)g(f)p Fv(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(k)s Fq(g)p Fu(:)1018 b Fv(\(6.20\))236 3860 y Fb(Pr)n(oof)p Fv(.)89 b(By)31 b(\(6.9\))f(w)n(e)h(ha)n(v)n(e)f Fq(jj)15 b Fl(e)-61 b Fu(w)1369 3872 y Ft(i)p Fs(+1)1481 3860 y Fq(j)1504 3829 y Fs(2)1562 3860 y Fq(\000)21 b(j)15 b Fl(e)-61 b Fu(w)1730 3872 y Ft(i)1758 3860 y Fq(j)1781 3829 y Fs(2)1818 3860 y Fq(j)29 b(\024)i(j)15 b Fl(e)-61 b Fu(w)2048 3872 y Ft(i)p Fs(+1)2181 3860 y Fq(\000)35 b Fl(e)-61 b Fu(w)2325 3872 y Ft(i)2353 3860 y Fq(j)21 b(\001)g Fv(\()p Fq(j)15 b Fl(e)-61 b Fu(w)2555 3872 y Ft(i)p Fs(+1)2668 3860 y Fq(j)20 b Fv(+)h Fq(j)14 b Fl(e)-60 b Fu(w)2879 3872 y Ft(i)2907 3860 y Fq(j)p Fv(\))29 b(=)g Fu(O)r Fv(\()p Fu(\026)3232 3829 y Fs(3)p Ft(=)p Fs(2)3337 3860 y Fv(\),)j(and)f(hence,)h(from)236 3959 y(\(6.7\))c(w)n(e)f(deriv) n(e)1154 4070 y Fu(@)1198 4082 y Ft(b)1227 4090 y Fc(i)1258 4070 y Fq(F)1318 4082 y Ft(\026)1362 4070 y Fv(\()1389 4048 y Fl(e)1394 4070 y Fu(b;)13 b Fl(e)-45 b Fu(c)p Fv(\))23 b(=)g Fu(\026@)1740 4082 y Ft(b)1776 4049 y Fl(e)1773 4070 y Fv(\000\()p 1857 4024 55 4 v Fu(!)1912 4082 y Ft(i)1940 4070 y Fu(;)14 b(\021)2018 4082 y Ft(i)2064 4070 y Fv(+)2142 4048 y Fl(e)2147 4070 y Fu(b)2183 4082 y Ft(i)2210 4070 y Fu(;)g(\030)2283 4082 y Ft(i)2330 4070 y Fv(+)j Fl(e)-45 b Fu(c)2449 4082 y Ft(i)2476 4070 y Fv(\))19 b(+)f Fu(O)r Fv(\()p Fu(\026)2757 4036 y Fs(3)p Ft(=)p Fs(2)2862 4070 y Fv(\))h(+)f Fu(S)3047 4082 y Ft(i)3075 4070 y Fu(:)706 b Fv(\(6.21\))236 4229 y(Let)28 b(us)g(\014rst)f(assume)g(b)n(y)g(con)n(tradiction)f(that)i Fq(9)p Fu(i)g Fv(suc)n(h)f(that)h Fq(j)p Fl(e)-46 b Fu(c)2279 4241 y Ft(i)2306 4229 y Fq(j)24 b Fv(=)e Fu(r)31 b Fv(and)c Fq(j)2687 4207 y Fl(e)2692 4229 y Fu(b)2728 4241 y Ft(i)2755 4229 y Fq(j)c Fu(<)p 2889 4161 36 4 v 23 w(b)p Fv(.)37 b(In)27 b(this)h(case)f(w)n(e)g(claim)h(that)797 4414 y Fu(@)841 4426 y Ft(c)878 4393 y Fl(e)875 4414 y Fv(\000\()p 959 4368 55 4 v Fu(!)1014 4426 y Ft(i)1041 4414 y Fu(;)14 b(\021)1119 4426 y Ft(i)1166 4414 y Fv(+)1244 4392 y Fl(e)1249 4414 y Fu(b)1285 4426 y Ft(i)1312 4414 y Fu(;)g(\030)1385 4426 y Ft(i)1431 4414 y Fv(+)k Fl(e)-46 b Fu(c)1550 4426 y Ft(i)1578 4414 y Fv(\))19 b Fq(\001)f Fl(e)-46 b Fu(c)1706 4426 y Ft(i)1757 4414 y Fq(\024)22 b Fu(\027)1885 4426 y Fs(2)1923 4414 y Fu(=)p Fv(2)82 b Fw(and)h Fq(j)p Fu(@)2391 4426 y Ft(b)2428 4393 y Fl(e)2425 4414 y Fv(\000\()p 2509 4368 V Fu(!)2564 4426 y Ft(i)2591 4414 y Fu(;)14 b(\021)2669 4426 y Ft(i)2715 4414 y Fv(+)2793 4392 y Fl(e)2798 4414 y Fu(b)2834 4426 y Ft(i)2862 4414 y Fu(;)g(\030)2935 4426 y Ft(i)2981 4414 y Fv(+)j Fl(e)-45 b Fu(c)3100 4426 y Ft(i)3128 4414 y Fv(\))p Fq(j)23 b(\024)g Fu(\027)3335 4426 y Fs(1)3372 4414 y Fu(=)p Fv(2)348 b(\(6.22\))236 4599 y(con)n(tradicting)27 b(\(6)p Fu(:)p Fv(1\),)g(since)g Fq(j)p Fu(\021)1230 4611 y Ft(i)1258 4599 y Fq(j)p Fu(;)14 b Fq(j)p Fu(\030)1377 4611 y Ft(i)1405 4599 y Fq(j)23 b(\024)g Fu(\016)s Fv(.)37 b(Let)28 b(us)f(pro)n(v)n(e)f(\(6.22\).)36 b(Since)28 b(\()2634 4577 y Fl(e)2639 4599 y Fu(b;)13 b Fl(e)-45 b Fu(c)p Fv(\))28 b(is)f(a)g(minim)n(um)i(p)r(oin)n(t)414 4784 y Fu(@)458 4796 y Ft(c)488 4804 y Fc(i)518 4784 y Fq(F)578 4796 y Ft(\026)622 4784 y Fv(\()649 4762 y Fl(e)654 4784 y Fu(b;)13 b Fl(e)-45 b Fu(c)p Fv(\))19 b Fq(\001)f Fl(e)-46 b Fu(c)891 4796 y Ft(i)942 4784 y Fv(=)23 b(\()15 b Fl(e)-61 b Fu(w)1121 4796 y Ft(i)p Fk(\000)p Fs(1)1252 4784 y Fq(\000)33 b Fl(e)-61 b Fu(w)1394 4796 y Ft(i)1422 4784 y Fv(\))19 b Fq(\001)f Fl(e)-45 b Fu(c)1551 4796 y Ft(i)1597 4784 y Fv(+)18 b Fu(\026@)1774 4796 y Ft(c)1811 4763 y Fl(e)1808 4784 y Fv(\000\()p 1892 4738 V Fu(!)1946 4796 y Ft(i)1974 4784 y Fu(;)c(\021)2052 4796 y Ft(i)2098 4784 y Fv(+)2176 4762 y Fl(e)2181 4784 y Fu(b)2217 4796 y Ft(i)2245 4784 y Fu(;)g(\030)2318 4796 y Ft(i)2364 4784 y Fv(+)j Fl(e)-45 b Fu(c)2483 4796 y Ft(i)2511 4784 y Fv(\))18 b Fq(\001)g Fl(e)-45 b Fu(c)2639 4796 y Ft(i)2685 4784 y Fv(+)18 b Fu(R)2831 4796 y Ft(i)2877 4784 y Fq(\001)g Fl(e)-46 b Fu(c)2954 4796 y Ft(i)3005 4784 y Fv(=)23 b Fu(\013)3146 4796 y Ft(\026)3190 4784 y Fl(e)-46 b Fu(c)3226 4796 y Ft(i)3272 4784 y Fq(\001)18 b Fl(e)-45 b Fu(c)3350 4796 y Ft(i)3401 4784 y Fv(=)22 b Fu(\013)3541 4796 y Ft(\026)3586 4784 y Fu(r)3625 4750 y Fs(2)3686 4784 y Fq(\024)g Fv(0)p Fu(:)236 4972 y Fv(By)32 b(\(6.14\))f(and)h(\(6.8\))g(it)g(follo)n(ws)f(that)h Fu(@)1576 4984 y Ft(c)1613 4951 y Fl(e)1610 4972 y Fv(\000\()p 1694 4926 V Fu(!)1749 4984 y Ft(i)1777 4972 y Fu(;)14 b(\021)1855 4984 y Ft(i)1904 4972 y Fv(+)1984 4950 y Fl(e)1990 4972 y Fu(b)2026 4984 y Ft(i)2053 4972 y Fu(;)g(\030)2126 4984 y Ft(i)2175 4972 y Fv(+)21 b Fl(e)-46 b Fu(c)2297 4984 y Ft(i)2325 4972 y Fv(\))21 b Fq(\001)g Fl(e)-45 b Fu(c)2459 4984 y Ft(i)2517 4972 y Fq(\024)30 b Fu(\027)2653 4984 y Fs(2)2690 4972 y Fu(=)p Fv(2.)49 b(Moreo)n(v)n(er)29 b(since)j Fq(j)3441 4950 y Fl(e)3446 4972 y Fu(b)3482 4984 y Ft(i)3509 4972 y Fq(j)f Fu(<)p 3658 4904 36 4 v 30 w(b)g Fv(w)n(e)h(ha)n(v)n(e)236 5083 y Fu(@)280 5095 y Ft(b)309 5103 y Fc(i)340 5083 y Fq(F)400 5095 y Ft(\026)444 5083 y Fv(\()471 5061 y Fl(e)476 5083 y Fu(b;)13 b Fl(e)-45 b Fu(c)p Fv(\))26 b(=)g(0,)j(and)h(b)n(y)f (\(6.21\),)g(\(6.8\))g(it)h(follo)n(ws)e(that)i Fq(j)p Fu(@)2182 5095 y Ft(b)2218 5062 y Fl(e)2215 5083 y Fv(\000\()p 2299 5037 55 4 v Fu(!)2354 5095 y Ft(i)2382 5083 y Fu(;)14 b(\021)2460 5095 y Ft(i)2507 5083 y Fv(+)2586 5061 y Fl(e)2591 5083 y Fu(b)2627 5095 y Ft(i)2655 5083 y Fu(;)g(\030)2728 5095 y Ft(i)2775 5083 y Fv(+)19 b Fl(e)-46 b Fu(c)2895 5095 y Ft(i)2923 5083 y Fv(\))p Fq(j)26 b(\024)g Fu(\027)3136 5095 y Fs(1)3173 5083 y Fu(=)p Fv(2)j(\(pro)n(vided)f Fu(\026)i Fv(is)f(small)236 5182 y(enough\).)37 b(Estimate)27 b(\(6.22\))g(is)g(then)h(pro)n(v)n(ed.)36 b(As)27 b(a)h(result,)f(if)h (\(6)p Fu(:)p Fv(20\))f(holds,)g(so)g(do)r(es)g(\(6)p Fu(:)p Fv(19\).)361 5293 y(Let)k(us)g(\014nally)g(pro)n(v)n(e)e (\(6.20\).)47 b(If)32 b(b)n(y)e(con)n(tradiction)g Fq(9)p Fu(i)h Fv(with)h Fq(j)2419 5271 y Fl(e)2424 5293 y Fu(b)2460 5305 y Ft(i)2487 5293 y Fq(j)d Fv(=)p 2632 5226 36 4 v 28 w Fu(b)p Fv(,)j(b)n(y)f(\(6.21\),)g(\(6.8\))g(and)g(since)g(\() 3710 5271 y Fl(e)3715 5293 y Fu(b;)13 b Fl(e)-45 b Fu(c)p Fv(\))31 b(is)g(a)236 5404 y(minim)n(um)23 b(p)r(oin)n(t,)h(arguing)d 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Fv(=)3116 3307 y Fl(P)3204 3328 y Ft(k)q Fk(\000)p Fs(1)3204 3394 y Ft(j)s Fs(=1)3330 3369 y Fv(\()p 3362 3302 V Fu(\022)3403 3381 y Ft(j)s Fs(+1)3537 3369 y Fq(\000)p 3616 3302 V 14 w Fu(\022)3657 3381 y Ft(j)3692 3369 y Fv(\))d(=)g Fu(O)r Fv(\(1)p Fu(=)236 3481 y(\026)286 3451 y Fs(1+)p Ft(a)410 3481 y Fv(\))p Fu(:)p 493 3481 48 48 v 236 3796 a Fr(7)135 b(The)44 b(stabilit)l(y)j(result)f(and)f (the)g(optimal)h(time)236 3978 y Fv(In)31 b(this)g(section)f(w)n(e)g (will)h(pro)n(v)n(e,)e(via)h(classical)f(p)r(erturbation)h(theory)-7 b(,)31 b(stabilit)n(y)f(results)g(for)g(the)h(action)f(v)-5 b(ariables,)236 4077 y(implying,)26 b(in)g(particular,)e(Theorem)h (1.2.)35 b(W)-7 b(e)26 b(shall)e(use)i(the)f(follo)n(wing)g(notations:) 35 b(for)24 b Fu(l)h Fq(2)e Fw(N)p Fv(,)j Fu(A)d Fq(\032)g Fw(C)3591 4047 y Ft(l)3642 4077 y Fv(and)i Fu(r)h(>)c Fv(0,)236 4177 y(w)n(e)29 b(de\014ne)h Fu(A)664 4189 y Ft(r)726 4177 y Fv(:=)c Fq(f)p Fu(z)i Fq(2)f Fw(C)1100 4147 y Ft(l)1154 4177 y Fq(j)j Fv(dist\()p Fu(z)t(;)14 b(A)p Fv(\))26 b Fq(\024)f Fu(r)r Fq(g)30 b Fv(and)f Fw(T)2003 4147 y Ft(l)2003 4197 y(s)2065 4177 y Fv(:=)c Fq(f)p Fu(z)k Fq(2)d Fw(C)2438 4147 y Ft(l)2493 4177 y Fq(j)j(j)p Fv(Im)g Fu(z)2735 4189 y Ft(j)2770 4177 y Fq(j)d Fu(<)f(s;)43 b Fq(8)30 b Fv(1)25 b Fq(\024)h Fu(j)k Fq(\024)c Fu(l)r Fq(g)i Fv(\(though)n(t)h(of)g(as)236 4277 y(a)j(complex)f(neigh)n(b)r(orho)r(o)r(d)g(of)g Fw(T)1329 4246 y Ft(l)1355 4277 y Fv(\).)50 b(Giv)n(en)32 b(t)n(w)n(o)f(b)r(ounded)h(op)r(en)g(sets)g Fu(B)i Fq(\032)c Fw(C)2842 4246 y Fs(2)2879 4277 y Fu(;)i(D)g Fq(\032)e Fw(C)3199 4246 y Ft(l)3256 4277 y Fv(and)i Fu(f)9 b Fv(\()p Fu(I)e(;)14 b(';)g(p;)g(q)s Fv(\),)33 b(real)236 4376 y(analytic)24 b(function)h(with)f(holomorphic)f(extension)h(on)g Fu(D)2073 4388 y Ft(\033)2129 4376 y Fq(\002)12 b Fw(T)2272 4346 y Ft(l)2272 4397 y(s)p Fs(+)p Ft(\033)2411 4376 y Fq(\002)g Fu(B)2551 4388 y Ft(\033)2619 4376 y Fv(for)24 b(some)f Fu(\033)k(>)22 b Fv(0,)j(w)n(e)f(de\014ne)g(the)h(follo)n (wing)236 4492 y(norm)32 b Fq(k)p Fu(f)9 b Fq(k)592 4504 y Ft(B)s(;D)r(;s)805 4492 y Fv(=)900 4429 y Fl(P)987 4516 y Ft(k)q Fk(2)p Ft(Z)1117 4500 y Fc(l)1160 4492 y Fv(sup)1295 4495 y Fd(\()p Fc(p;q)q Fd(\))p Fo(2)p Fc(B)1341 4543 y(I)s Fo(2)p Fc(D)1531 4492 y Fq(j)1572 4470 y Fv(^)1554 4492 y Fu(f)1595 4504 y Ft(k)1635 4492 y Fv(\()p Fu(I)e(;)14 b(p;)g(q)s Fv(\))p Fq(j)p Fu(e)1960 4461 y Fk(j)p Ft(k)q Fk(j)p Ft(s)2118 4492 y Fv(where)2380 4470 y(^)2362 4492 y Fu(f)2403 4504 y Ft(k)2444 4492 y Fv(\()p Fu(I)7 b(;)14 b(p;)g(q)s Fv(\))32 b(denotes)g(the)g Fu(k)s Fv(-F)-7 b(ourier)31 b(co)r(e\016cien)n(t)h(of)236 4618 y(the)c(p)r(erio)r(dic)g(function)g Fu(')23 b Fq(!)g Fu(f)9 b Fv(\()p Fu(I)e(;)14 b(';)g(p;)g(q)s Fv(\).)361 4741 y(Let)27 b(us)h(consider)e(Hamiltonian)h Fq(H)1486 4753 y Ft(\026)1558 4741 y Fv(de\014ned)h(in)f(\(1.1\))h(and)f(assume)f (that)i Fu(f)9 b Fv(\()p Fu(I)e(;)14 b(';)g(p;)g(q)s(;)g(t)p Fv(\),)27 b(de\014ned)h(in)g(\(1.2\),)f(is)g(a)236 4840 y(real)j(analytic)f(function,)j(p)r(ossessing,)e(for)f(some)h Fu(r)n(;)p 1906 4795 40 4 v 14 w(r)s(;)15 b Fl(e)-47 b Fu(r)r(;)14 b(s)28 b(>)f Fv(0,)k(complex)f(analytic)f(exten)n(tion)h (on)g Fq(f)p Fu(I)k Fq(2)28 b Fw(R)3712 4810 y Ft(d)3781 4840 y Fq(j)i(j)p Fu(I)7 b Fq(j)28 b(\024)p 236 4894 V 236 4940 a Fu(r)s Fq(g)318 4952 y Ft(r)373 4940 y Fq(\002)18 b Fw(T)522 4910 y Ft(d)522 4961 y(s)579 4940 y Fq(\002)g(f)p Fu(p)23 b Fq(2)g Fw(R)k Fq(j)h(j)p Fu(p)p Fq(j)23 b(\024)h Fl(e)-47 b Fu(r)r Fq(g)1277 4952 y Ft(r)1332 4940 y Fq(\002)18 b Fw(T)1481 4952 y Ft(s)1536 4940 y Fq(\002)g Fw(T)1685 4952 y Ft(s)1720 4940 y Fv(.)361 5040 y(It)28 b(is)h(con)n(v)n(enien)n (t)e(to)h(write)g(Hamiltonian)g Fq(H)1810 5052 y Ft(\026)1883 5040 y Fv(in)g(autonomous)f(form.)39 b(F)-7 b(or)28 b(this)g(purp)r (ose)g(let)g(us)h(in)n(tro)r(duce)e(the)236 5139 y(new)35 b(action-angle)e(v)-5 b(ariables)33 b(\()p Fu(I)1306 5151 y Fs(0)1344 5139 y Fu(;)14 b(')1435 5151 y Fs(0)1472 5139 y Fv(\))35 b(with)g Fu(t)g Fv(=)f Fu(')1953 5151 y Fs(0)1991 5139 y Fv(,)i(that)f(will)g(still)g(b)r(e)f(denoted)h(b)n (y)f Fu(I)42 b Fv(:=)34 b(\()p Fu(I)3400 5151 y Fs(0)3438 5139 y Fu(;)14 b(I)3511 5151 y Fs(1)3549 5139 y Fu(;)g(:)g(:)g(:)f(;)h (I)3769 5151 y Ft(n)3815 5139 y Fv(\))35 b(and)236 5239 y Fu(')c Fv(:=)e(\()p Fu(')524 5251 y Fs(0)562 5239 y Fu(;)14 b(')653 5251 y Fs(1)691 5239 y Fu(;)g(:)g(:)g(:)f(;)h(')929 5251 y Ft(n)975 5239 y Fv(\).)50 b(De\014ning)32 b Fu(h)p Fv(\()p Fu(I)7 b Fv(\))31 b(:=)e Fu(I)1754 5251 y Fs(0)1813 5239 y Fv(+)21 b Fq(j)p Fu(I)7 b Fq(j)1988 5209 y Fs(2)2026 5239 y Fu(=)p Fv(2)31 b(and)g Fu(E)k Fv(:=)30 b Fu(E)5 b Fv(\()p Fu(p;)14 b(q)s Fv(\))31 b(:=)f Fu(p)2960 5209 y Fs(2)2997 5239 y Fu(=)p Fv(2)20 b(+)h(\(cos)13 b Fu(q)24 b Fq(\000)d Fv(1\))p Fu(;)32 b Fq(H)3690 5251 y Ft(\026)3767 5239 y Fv(is)f(then)236 5339 y(equiv)-5 b(alen)n(t)28 b(to)f(the)h(autonomous)e(Hamiltonian)1197 5504 y Fu(H)k Fv(:=)22 b Fu(H)7 b Fv(\()p Fu(I)g(;)14 b(';)g(p;)g(q)s Fv(\))24 b(:=)e Fu(h)p Fv(\()p Fu(I)7 b Fv(\))19 b(+)f Fu(E)5 b Fv(\()p Fu(p;)14 b(q)s Fv(\))19 b(+)f Fu(\026f)9 b Fv(\()p Fu(I)e(;)14 b(';)g(p;)g(q)s Fv(\))p Fu(:)790 b Fv(\(7.1\))2084 6059 y(22)p eop %%Page: 23 23 23 22 bop 236 272 a Fv(Clearly)-7 b(,)27 b(Hamiltonian)g Fu(H)35 b Fv(is)27 b(a)g(real)g(analytic)g(function,)h(with)g(complex)f (analytic)g(exten)n(tion)h(on)1070 374 y Fl(n)1126 466 y Fu(I)i Fq(2)23 b Fw(R)1342 432 y Ft(d)p Fs(+1)1492 371 y Fl(\014)1492 421 y(\014)1492 470 y(\014)1548 466 y Fq(j)p Fu(I)7 b Fq(j)23 b(\024)p 1747 421 40 4 v 22 w Fu(r)1787 374 y Fl(o)1843 524 y Ft(r)1898 466 y Fq(\002)18 b Fw(T)2047 432 y Ft(d)p Fs(+1)2047 487 y Ft(s)2188 466 y Fq(\002)2271 374 y Fl(n)2327 466 y Fu(p)23 b Fq(2)g Fw(R)2569 371 y Fl(\014)2569 421 y(\014)2569 470 y(\014)2625 466 y Fq(j)p Fu(p)p Fq(j)g(\024)h Fl(e)-48 b Fu(r)2863 374 y Fl(o)2918 524 y Ft(r)2974 466 y Fq(\002)18 b Fw(T)3123 478 y Ft(s)3159 466 y Fu(:)236 669 y Fv(In)40 b(the)f(sequel)g(w)n(e)g (will)h(denote)f(b)n(y)g Fu(z)t Fv(\()p Fu(t)p Fv(\))j(:=)d(\()p Fu(I)7 b Fv(\()p Fu(t)p Fv(\))p Fu(;)14 b(')p Fv(\()p Fu(t)p Fv(\))p Fu(;)g(p)p Fv(\()p Fu(t)p Fv(\))p Fu(;)g(q)s Fv(\()p Fu(t)p Fv(\)\))42 b(the)e(solution)e(of)i(the)f(Hamilton)h (equations)236 769 y(asso)r(ciated)26 b(to)i(Hamiltonian)f(\(7.1\))h (with)g(initial)g(condition)f Fu(z)t Fv(\(0\))22 b(=)28 b(\()p Fu(I)7 b Fv(\(0\))p Fu(;)14 b(')p Fv(\(0\))p Fu(;)g(p)p Fv(\(0\))p Fu(;)g(q)s Fv(\(0\)\))p Fu(:)361 892 y Fv(The)27 b(pro)r(of)g(of)h(the)g(stabilit)n(y)f(of)h(the)g(action)f(v)-5 b(ariables)26 b(is)i(divided)g(in)f(t)n(w)n(o)g(steps:)361 1051 y Fq(\017)41 b Fv(\(i\))28 b(\()p Fw(Stabilit)m(y)k(far)h(from)f (the)f(separatrices)i(of)e(the)h(p)s(endulum:)p Fv(\))j(pro)n(v)n(e)26 b(stabilit)n(y)i(in)g(the)g(region)551 1250 y Fq(E)595 1262 y Fs(1)655 1250 y Fv(:=)23 b Fq(E)817 1215 y Fs(+)810 1273 y(1)891 1250 y Fq([)c(E)1016 1215 y Fk(\000)1009 1273 y Fs(1)1095 1250 y Fv(:=)1206 1158 y Fl(n)1261 1250 y Fv(\()p Fu(I)7 b(;)14 b(';)g(p;)g(q)s Fv(\))28 b Fq(j)g Fu(E)5 b Fv(\()p Fu(p;)14 b(q)s Fv(\))23 b Fq(\025)g Fu(\026)2104 1216 y Ft(c)2134 1225 y Fc(d)2200 1158 y Fl(o)2274 1250 y Fq([)2348 1158 y Fl(n)2403 1250 y Fv(\()p Fu(I)7 b(;)14 b(';)g(p;)g(q)s Fv(\))28 b Fq(j)46 b(\000)18 b Fv(2)g(+)g Fu(\026)3130 1216 y Ft(c)3160 1225 y Fc(d)3222 1250 y Fq(\024)23 b Fu(E)5 b Fv(\()p Fu(p;)14 b(q)s Fv(\))23 b Fq(\024)g(\000)p Fu(\026)3785 1216 y Ft(c)3815 1225 y Fc(d)3853 1158 y Fl(o)444 1445 y Fv(in)h(whic)n(h)f(w)n(e)h(can)f (apply)h(the)g(Nekhoroshev)e(Theorem)h(obtaining)g(actually)g(stabilit) n(y)g(for)g(exp)r(onen)n(tially)h(long)444 1544 y(times,)361 1707 y Fq(\017)41 b Fv(\(ii\))28 b(\()p Fw(Stabilit)m(y)k(close)e(to)h (the)h(separatrices)f(of)h(the)f(p)s(endulum)f(and)i(to)f(the)h (elliptic)e(equilibrium)444 1807 y(p)s(oin)m(t:)p Fv(\))36 b(pro)n(v)n(e)26 b(stabilit)n(y)h(in)h(the)g(region)444 2006 y Fq(E)488 2018 y Fs(2)548 2006 y Fv(:=)23 b Fq(E)710 1970 y Fs(+)703 2028 y(2)772 2006 y Fq([)7 b(E)885 1970 y Fk(\000)878 2028 y Fs(2)965 2006 y Fv(:=)1076 1913 y Fl(n)1131 2006 y Fv(\()p Fu(I)g(;)14 b(';)g(p;)g(q)s Fv(\))22 b Fq(j)30 b(\000)7 b Fv(2)p Fu(\026)1724 1971 y Ft(c)1754 1980 y Fc(d)1815 2006 y Fq(\024)22 b Fu(E)5 b Fv(\()p Fu(p;)14 b(q)s Fv(\))24 b Fq(\024)e Fv(2)p Fu(\026)2354 1971 y Ft(c)2384 1980 y Fc(d)2422 1913 y Fl(o)2485 2006 y Fq([)2547 1913 y Fl(n)2603 2006 y Fv(\()p Fu(I)7 b(;)14 b(';)g(p;)g(q)s Fv(\))22 b Fq(j)29 b(\000)7 b Fv(2)23 b Fq(\024)f Fu(E)5 b Fv(\()p Fu(p;)14 b(q)s Fv(\))24 b Fq(\024)e(\000)p Fv(2)7 b(+)g(2)p Fu(\026)3893 1971 y Ft(c)3923 1980 y Fc(d)3960 1913 y Fl(o)444 2200 y Fv(in)28 b(whic)n(h)f(w)n(e)g(use)h(some)f Fa(ad)g(ho)r(c)h Fv(argumen)n(ts,)236 2359 y(where)f(0)c Fu(<)f(c)664 2371 y Ft(d)726 2359 y Fu(<)h Fv(1)k(is)h(a)f(p)r(ositiv)n(e)g(constan) n(t)g(that)h(will)f(b)r(e)h(c)n(hosen)f(later)g(on,)g(see)h(\(7.12\).) 361 2516 y(W)-7 b(e)28 b(\014rst)g(pro)n(v)n(e)f(\(i\).)39 b(In)28 b(the)h(regions)1554 2486 y Fs(7)1630 2495 y Fl(e)1618 2516 y Fq(E)1669 2481 y Fk(\006)1662 2538 y Fs(1)1749 2516 y Fv(:=)24 b(\005)1923 2528 y Ft(q)r(;p)2014 2516 y Fq(E)2065 2481 y Fk(\006)2058 2538 y Fs(1)2149 2516 y Fv(w)n(e)k(\014rst)g(write)g(the)h(p)r(endulum)g(Hamiltonian)f Fu(E)5 b Fv(\()p Fu(p;)14 b(q)s Fv(\))29 b(in)236 2629 y(action-angle)d(v)-5 b(ariables.)36 b(In)27 b(the)h(region)1548 2599 y Fs(8)1624 2608 y Fl(e)1612 2629 y Fq(E)1663 2594 y Fs(+)1656 2651 y(1)1737 2629 y Fq([)19 b(f)p Fu(p)j(>)h Fv(0)p Fq(g)j Fv(the)i(new)g(action)f(v)-5 b(ariable)27 b Fu(P)39 b Fv(is)28 b(de\014ned)f(b)n(y)h(the)g(form)n(ula)1303 2866 y Fu(P)35 b Fv(:=)22 b Fu(P)1566 2831 y Fs(+)1621 2866 y Fv(\()p Fu(E)5 b Fv(\))24 b(:=)1896 2741 y Fq(p)p 1965 2741 42 4 v 68 x Fv(2)p 1896 2847 111 4 v 1926 2923 a Fu(\031)2030 2753 y Fl(Z)2113 2773 y Ft(\031)2076 2941 y Fs(0)2172 2791 y Fl(p)p 2255 2791 558 4 v 75 x Fu(E)g Fv(+)18 b(\(1)g(+)g(cos)13 b Fu( )s Fv(\))h Fu(d )s(:)236 3105 y Fv(while)28 b(in)g(the)g(region)954 3084 y Fl(e)942 3105 y Fq(E)993 3069 y Fk(\000)986 3127 y Fs(1)1077 3105 y Fv(the)g(new)g(action)f(v)-5 b(ariable)26 b(is)1235 3346 y Fu(P)35 b Fv(:=)22 b Fu(P)1498 3312 y Fk(\000)1554 3346 y Fv(\()p Fu(E)5 b Fv(\))24 b(=)1806 3290 y(2)1848 3221 y Fq(p)p 1916 3221 42 4 v 1916 3290 a Fv(2)p 1806 3327 153 4 v 1857 3403 a Fu(\031)1982 3233 y Fl(Z)2065 3253 y Ft( )2109 3261 y Fd(0)2141 3253 y Fs(\()p Ft(E)s Fs(\))2028 3421 y(0)2263 3271 y Fl(p)p 2346 3271 558 4 v 75 x Fu(E)f Fv(+)18 b(\(1)h(+)f(cos)13 b Fu( )s Fv(\))h Fu(d )236 3566 y Fv(where)34 b Fu( )537 3578 y Fs(0)575 3566 y Fv(\()p Fu(E)5 b Fv(\))35 b(is)g(the)g(\014rst)f(p)r(ositiv)n(e) g(n)n(um)n(b)r(er)h(suc)n(h)f(that)h Fu(E)28 b Fv(+)23 b(\(1)g(+)f(cos)13 b Fu( )2704 3578 y Fs(0)2742 3566 y Fv(\()p Fu(E)5 b Fv(\)\))35 b(=)g(0.)57 b(W)-7 b(e)36 b(will)e(use)h(the)g(follo)n(w-)236 3666 y(ing)i(lemma,)j(pro)n(v)n(ed) c(in)i([10)o(],)i(regarding)35 b(the)j(analyticit)n(y)e(radii)h(of)g (these)h(action-angle)e(v)-5 b(ariables)36 b(close)g(to)h(the)236 3766 y(separatrices)26 b(of)h(the)h(p)r(endulum.)236 3925 y Fw(Lemma)k(7.1)40 b Fp(Ther)l(e)32 b(exist)e(intervals)h Fu(D)1582 3895 y Fk(\006)1662 3925 y Fq(\032)24 b Fw(R)p Fu(;)30 b Fp(symple)l(ctic)h(tr)l(ansformations)g Fu(\036)2925 3895 y Fk(\006)3006 3925 y Fv(=)24 b Fu(\036)3144 3895 y Fk(\006)3200 3925 y Fv(\()p Fu(P)r(;)14 b(Q)p Fv(\))31 b Fp(r)l(e)l(al)g(analytic)h(on)236 4024 y Fu(D)307 3994 y Fk(\006)369 4024 y Fq(\002)6 b Fw(T)24 b Fp(with)h(holomorphic)i (extension)d(on)g Fu(D)1717 3994 y Fk(\006)1715 4045 y Ft(r)1746 4053 y Fd(0)1788 4024 y Fq(\002)6 b Fw(T)1925 4036 y Ft(s)1956 4044 y Fd(0)2017 4024 y Fp(and)25 b(functions)f Fu(E)2594 3994 y Fk(\006)2674 4024 y Fp(r)l(e)l(al)g(analytic)i(on)e Fu(D)3319 3994 y Fk(\006)3399 4024 y Fp(with)g(holomorphic)236 4143 y(extension)30 b(on)f Fu(D)793 4112 y Fk(\006)791 4163 y Ft(r)822 4171 y Fd(0)888 4143 y Fp(such)h(that)g Fu(\036)1296 4112 y Fk(\006)1352 4143 y Fv(\()p Fu(D)1455 4112 y Fk(\006)1530 4143 y Fq(\002)18 b Fw(T)p Fv(\))24 b(=)1834 4122 y Fl(e)1823 4143 y Fq(E)1874 4107 y Fk(\006)1867 4165 y Fs(1)1960 4143 y Fp(and)1704 4317 y Fu(E)5 b Fv(\()p Fu(\036)1851 4282 y Fk(\006)1908 4317 y Fv(\()p Fu(P)r(;)14 b(Q)p Fv(\)\))24 b(=)e Fu(E)2339 4282 y Fk(\006)2396 4317 y Fv(\()p Fu(P)12 b Fv(\))p Fu(;)236 4491 y Fp(with)39 b Fu(r)462 4503 y Fs(0)538 4491 y Fv(=)g Fu(const\026)887 4461 y Ft(c)917 4470 y Fc(d)994 4491 y Fp(and)f Fu(s)1202 4503 y Fs(0)1278 4491 y Fv(=)g Fu(const=)p Fq(j)14 b Fv(ln)g Fu(\026)p Fq(j)p Fp(.)64 b(Mor)l(e)l(over,)42 b(for)e Fu(E)j Fp(b)l(ounde)l(d,)e(the)e(fol)t(lowing)i(estimates)d(on) h(the)236 4590 y(derivatives)32 b(hold)829 4560 y Fs(9)1446 4739 y Fu(dE)1555 4709 y Fk(\006)p 1446 4776 166 4 v 1475 4852 a Fu(dP)1621 4795 y Fv(\()p Fu(P)1718 4761 y Fk(\006)1774 4795 y Fv(\()p Fu(E)5 b Fv(\)\))84 b Fq(\031)f Fv(ln)2237 4758 y Fk(\000)p Fs(1)2326 4795 y Fv(\(1)18 b(+)2588 4739 y(1)p 2511 4776 196 4 v 2511 4793 a Fl(p)p 2594 4793 113 4 v 71 x Fq(j)p Fu(E)5 b Fq(j)2717 4795 y Fv(\))1096 b(\(7.2\))1334 5046 y Fq(\006)1409 4990 y Fu(d)1452 4960 y Fs(2)1489 4990 y Fu(E)1555 4960 y Fk(\006)p 1409 5027 203 4 v 1437 5103 a Fu(dP)1545 5079 y Fs(2)1621 5046 y Fv(\()p Fu(P)1718 5012 y Fk(\006)1774 5046 y Fv(\()p Fu(E)5 b Fv(\)\))84 b Fq(\031)2213 4990 y Fv(1)p 2178 5027 113 4 v 2178 5103 a Fq(j)p Fu(E)5 b Fq(j)2314 5046 y Fv(ln)2383 5009 y Fk(\000)p Fs(3)2472 5046 y Fv(\(1)18 b(+)2734 4990 y(1)p 2657 5027 196 4 v 2657 5044 a Fl(p)p 2740 5044 113 4 v 71 x Fq(j)p Fu(E)5 b Fq(j)2863 5046 y Fv(\))p Fu(:)927 b Fv(\(7.3\))p 236 5190 1512 4 v 329 5243 a Fn(7)363 5267 y Fm(\005)416 5275 y Fi(p;q)525 5267 y Fm(denotes)25 b(the)g(pro)t(jection)f(on)n(to) h(the)f(\()p Fj(p;)12 b(q)r Fm(\))23 b(v)l(ariables.)329 5322 y Fn(8)363 5346 y Fm(The)h(case)h(with)e Fj(p)d(<)f Fm(0)24 b(is)f(completely)i(analogous.)329 5402 y Fn(9)363 5425 y Fm(If)33 b Fj(f)7 b Fm(\()p Fj(x)p Fm(\))p Fj(;)12 b(g)r Fm(\()p Fj(x)p Fm(\))33 b(are)g(p)r(ositiv)n(e)g(function,)j (with)c(the)i(sym)n(b)r(ol)f Fj(f)42 b Fg(\031)34 b Fj(g)h Fm(w)n(e)e(mean)g(that)h Fg(9)12 b Fj(c)2817 5434 y Fn(1)2851 5425 y Fj(;)f(c)2913 5434 y Fn(2)2982 5425 y Fj(>)35 b Fm(0)e(suc)n(h)g(that)h Fj(c)3501 5434 y Fn(1)3535 5425 y Fj(g)r Fm(\()p Fj(x)p Fm(\))i Fg(\024)e Fj(f)7 b Fm(\()p Fj(x)p Fm(\))36 b Fg(\024)236 5504 y Fj(c)267 5513 y Fn(2)301 5504 y Fj(g)r Fm(\()p Fj(x)p Fm(\))p Fj(;)24 b Fg(8)12 b Fj(x:)2084 6059 y Fv(23)p eop %%Page: 24 24 24 23 bop 236 272 a Fv(After)28 b(this)g(c)n(hange)f(of)g(v)-5 b(ariables)26 b(Hamiltonian)i Fu(H)34 b Fv(b)r(ecomes)535 455 y Fu(H)611 420 y Fk(\006)690 455 y Fv(:=)22 b Fu(H)876 420 y Fk(\006)932 455 y Fv(\()p Fu(I)7 b(;)14 b(';)g(P)r(;)g(Q)p Fv(\))24 b(:=)f Fu(h)1508 420 y Fk(\006)1564 455 y Fv(\()p Fu(I)7 b(;)14 b(P)e Fv(\))18 b(+)g Fu(\026f)1974 420 y Fk(\006)2030 455 y Fv(\()p Fu(I)7 b(;)14 b(';)g(P)r(;)g(Q)p Fv(\))24 b(:=)f Fu(h)p Fv(\()p Fu(I)7 b Fv(\))19 b(+)f Fu(E)2881 420 y Fk(\006)2937 455 y Fv(\()p Fu(P)12 b Fv(\))19 b(+)f Fu(\026f)3268 420 y Fk(\006)3323 455 y Fv(\()p Fu(I)7 b(;)14 b(';)g(P)r(;)g(Q)p Fv(\))236 637 y(where)27 b Fu(f)526 607 y Fk(\006)582 637 y Fv(\()p Fu(I)7 b(;)14 b(';)g(P)r(;)g(Q)p Fv(\))24 b(:=)f Fu(f)9 b Fv(\()p Fu(I)e(;)14 b(';)g(\036)1412 607 y Fk(\006)1468 637 y Fv(\()p Fu(P)r(;)g(Q)p Fv(\)\))p Fu(:)361 784 y Fw(Stabilit)m(y)40 b(in)h(the)f(region)g Fq(E)1400 749 y Fs(+)1393 806 y(1)1455 784 y Fu(:)c Fv(In)f(the)h(region)e Fq(E)2084 749 y Fs(+)2077 806 y(1)2139 784 y Fu(;)i Fv(the)f(pro)r(of)g (of)g(the)h(stabilit)n(y)f(of)g(the)h(actions)e(v)-5 b(ariables)236 884 y(follo)n(ws)26 b(b)n(y)g(a)f(straigh)n(tforw)n(ard) f(application)h(of)i(the)f(Nekhoroshev)f(Theorem)g(as)h(pro)n(v)n(ed)f (in)h(Theorem)g(1)g(of)g([18)o(].)37 b(In)236 983 y(order)28 b(to)h(apply)g(suc)n(h)f(Theorem)g(w)n(e)h(need)g(some)g (de\014nitions.)41 b(F)-7 b(or)28 b Fu(l)r(;)14 b(m)25 b(>)g Fv(0,)k(a)g(function)g Fu(h)c Fv(:=)h Fu(h)p Fv(\()p Fu(J)8 b Fv(\))29 b(is)g(said)g(to)f(b)r(e)236 1083 y Fa(l,m-quasi-con)n(v)n(ex)d Fv(on)j Fu(A)23 b Fq(\032)g Fw(R)1227 1053 y Ft(d)p Fs(+1)1349 1083 y Fu(;)28 b Fv(if)g(at)f(ev)n (ery)g(p)r(oin)n(t)g Fu(J)k Fq(2)24 b Fu(A)k Fv(at)f(least)g(one)h(of)f (the)h(inequalities)1403 1266 y Fq(jh)p Fu(h)1506 1231 y Fk(0)1529 1266 y Fv(\()p Fu(J)8 b Fv(\))p Fu(;)14 b(\030)t Fq(ij)24 b Fu(>)f(l)r Fq(j)p Fu(\030)t Fq(j)k Fu(;)97 b Fq(h)p Fu(h)2231 1231 y Fk(00)2274 1266 y Fv(\()p Fu(J)8 b Fv(\))p Fu(\030)t(;)14 b(\030)t Fq(i)24 b(\025)f Fu(m)p Fq(j)p Fu(\030)t Fq(j)2812 1231 y Fs(2)236 1448 y Fv(holds)35 b(for)f(eac)n(h)g Fu(\030)40 b Fq(2)c Fw(R)1028 1418 y Ft(d)p Fs(+1)1150 1448 y Fu(:)f Fv(Using)g(the)g(previous)f(lemma)h (it)g(is)g(p)r(ossible)g(to)f(pro)n(v)n(e)g(that,)j(for)d(ev)n(ery)p 3626 1403 40 4 v 34 w Fu(r)k(>)d Fv(0)p Fu(;)f Fv(the)236 1548 y(Hamiltonian)d Fu(h)765 1518 y Fs(+)851 1548 y Fv(is)g Fa(l,m-quasi-con)n(v)n(ex)e Fv(in)i(the)g(set)g Fu(S)j Fv(:=)28 b Fu(D)2223 1518 y Fs(+)2221 1568 y Ft(r)2252 1576 y Fd(0)2309 1548 y Fq(\002)20 b(f)p Fu(I)36 b Fq(2)29 b Fw(R)2664 1518 y Ft(d)p Fs(+1)2817 1548 y Fq(j)j(j)p Fu(I)7 b Fq(j)29 b(\024)p 3083 1502 V 28 w Fu(r)s Fq(g)3165 1560 y Ft(r)3196 1568 y Fd(0)3263 1548 y Fv(with)i Fu(l)r(;)14 b(m)28 b Fv(=)h Fu(O)r Fv(\(1\))p Fu(:)j Fv(In)236 1648 y(the)c(previous)f(set)g(also)g(holds)998 1830 y Fq(k)p Fv(\()p Fu(h)1120 1796 y Fs(+)1175 1830 y Fv(\))1207 1796 y Fk(00)1250 1830 y Fq(k)22 b Fv(=:)h Fu(M)32 b Fv(=)22 b Fu(O)r Fv(\()p Fu(\026)1772 1796 y Fk(\000)p Ft(c)1854 1805 y Fc(d)1908 1830 y Fv(ln)1977 1793 y Fk(\000)p Fs(3)2066 1830 y Fv(\(1)p Fu(=\026)p Fv(\)\))28 b Fu(;)96 b Fq(k)p Fv(\()p Fu(h)2565 1796 y Fs(+)2620 1830 y Fv(\))2652 1796 y Fk(0)2676 1830 y Fq(k)22 b Fv(=:)h(\012)2911 1842 y Fs(0)2971 1830 y Fv(=)g Fu(O)r Fv(\(1\))p Fu(:)236 2023 y Fv(Putting)41 b Fu(")46 b Fv(:=)f Fu(\026)p Fq(k)p Fu(f)915 1993 y Fs(+)969 2023 y Fq(k)1011 2035 y Ft(S;s)1103 2043 y Fd(0)1185 2023 y Fv(=)g Fu(O)r Fv(\()p Fu(\026)p Fv(\))p Fu(;)e(")1579 2035 y Fs(0)1661 2023 y Fv(:=)j(2)1837 1993 y Fk(\000)p Fs(10)1958 2023 y Fu(r)1997 1993 y Fs(2)1995 2044 y(0)2035 2023 y Fu(m)p Fv(\()p Fu(m=)p Fv(11)p Fu(M)9 b Fv(\))2461 1993 y Fs(2\()p Ft(d)p Fs(+2\))2713 2023 y Fv(=)46 b Fu(O)r Fv(\()p Fu(\026)2971 1993 y Fs(2)p Ft(c)3034 2002 y Fc(d)3069 1993 y Fs(\()p Ft(d)p Fs(+3\))3258 2023 y Fv(ln)3327 1986 y Fs(6\()p Ft(d)p Fs(+2\))3535 2023 y Fv(\(1)p Fu(=\026)p Fv(\)\))p Fu(;)41 b(\013)46 b Fv(:=)236 2123 y(\(1)9 b Fq(\000)g Fv(2)p Fu(c)471 2135 y Ft(d)508 2123 y Fv(\()p Fu(d)g Fv(+)g(3\)\))p Fu(=)p Fv(2\()p Fu(d)g Fv(+)g(2\))21 b(w)n(e)i(obtain)f(that,)i(if)g (the)f(initial)g(data)f(\()p Fu(I)7 b Fv(\(0\))p Fu(;)14 b(')p Fv(\(0\))p Fu(;)g(p)p Fv(\(0\))p Fu(;)g(q)s Fv(\(0\)\))24 b Fq(2)f(E)3239 2087 y Fs(+)3232 2145 y(1)3295 2123 y Fu(;)f Fv(that)h(is)g Fu(P)12 b Fv(\(0\))23 b Fq(2)h Fu(D)3938 2093 y Fs(+)3993 2123 y Fu(;)236 2222 y Fv(then)687 2322 y Fq(j)p Fu(I)7 b Fv(\()p Fu(t)p Fv(\))19 b Fq(\000)f Fu(I)7 b Fv(\(0\))p Fq(j)23 b(\024)g Fu(const:\026)1500 2288 y Ft(\013)1561 2322 y Fv(ln)1630 2285 y Fk(\000)p Fs(3)1719 2322 y Fv(\(1)p Fu(=\026)p Fv(\))28 b Fu(;)97 b Fv(for)82 b Fq(j)p Fu(t)p Fq(j)23 b(\024)g Fu(const:)14 b Fv(exp)o(\()p Fu(const:\026)3092 2288 y Fk(\000)p Ft(\013)3206 2322 y Fv(ln)3275 2285 y Fs(2)3312 2322 y Fv(\(1)p Fu(=\026)p Fv(\)\))p Fu(:)280 b Fv(\(7.4\))236 2471 y(If)28 b Fu(c)355 2483 y Ft(d)417 2471 y Fu(<)23 b Fv(1)p Fu(=)p Fv(2\()p Fu(d)17 b Fv(+)h(3\))28 b(then)g Fu(\013)23 b(>)g Fv(0)k(and)g(w)n(e)h (obtain)f(stabilit)n(y)g(for)g(exp)r(onen)n(tially)g(long)g(times.)361 2618 y Fw(Stabilit)m(y)h(in)g(the)h(region)e Fq(E)1351 2583 y Fk(\000)1344 2640 y Fs(1)1408 2618 y Fu(:)d Fv(In)h(the)g (region)f Fq(E)1994 2583 y Fk(\000)1987 2640 y Fs(1)2075 2618 y Fv(w)n(e)g(cannot)g(use)h(the)g(Nekhoroshev)e(Theorem)g(as)h (pro)n(v)n(ed)g(in)236 2718 y([18)o(],)k(b)r(ecause)f Fu(E)789 2688 y Fk(\000)873 2718 y Fv(is)h(conca)n(v)n(e)d(and)j(so)f Fu(h)1575 2688 y Fk(\000)1658 2718 y Fv(is)h(not)g(quasi-con)n(v)n(ex.) 34 b(Ho)n(w)n(ev)n(er)26 b(w)n(e)h(can)g(still)h(apply)f(the)i (Nekhoroshev)236 2817 y(Theorem)h(in)h(its)g(original)e(and)i(more)f (general)f(form)h(as)g(pro)n(v)n(ed)g(in)h([16)o(])f(\(see)h(also)f ([17)o(]\);)j(in)e(fact)f(the)i(function)f Fu(h)3960 2787 y Fk(\000)236 2917 y Fv(pro)n(v)n(es)26 b(to)h(b)r(e)h Fa(steep)g Fv(\(see)g(De\014nition)g(1.7.C.)f(pag.)36 b(6)27 b(of)g([16]\).)236 3017 y(F)-7 b(or)36 b(simplicit)n(y)h(w)n(e)f (pro)n(v)n(e)g(the)h Fa(steepness)f Fv(of)h(the)g(function)g Fu(h)2307 2987 y Fk(\000)2400 3017 y Fv(in)g(the)g(case)f Fu(d)i Fv(=)g(1)e(only)-7 b(.)64 b(In)37 b(this)g(case)f Fu(h)3857 2987 y Fk(\000)3951 3017 y Fv(=)236 3116 y Fu(h)284 3086 y Fk(\000)340 3116 y Fv(\()p Fu(I)408 3128 y Fs(0)446 3116 y Fu(;)14 b(I)519 3128 y Fs(1)557 3116 y Fu(;)g(P)e Fv(\))23 b(=)f Fu(I)837 3128 y Fs(0)894 3116 y Fv(+)17 b Fu(I)1019 3086 y Fs(2)1012 3137 y(1)1057 3116 y Fu(=)p Fv(2)g(+)h Fu(E)1307 3086 y Fk(\000)1363 3116 y Fv(\()p Fu(P)12 b Fv(\))p Fu(:)28 b Fv(W)-7 b(e)28 b(need)g(more)f(informations)f(on)h(the)h(function)h Fu(E)3220 3086 y Fk(\000)3276 3116 y Fu(:)e Fv(In)h(the)g(follo)n (wing,)f(in)236 3216 y(order)f(to)i(simplify)g(the)g(notation,)f(w)n(e) g(will)h(forget)f(the)h(ap)r(ex)2225 3186 y Fk(\000)2309 3216 y Fv(writing,)f(for)g(example,)h Fu(E)g Fv(=)22 b Fu(E)3336 3186 y Fk(\000)3420 3216 y Fv(and)28 b Fu(P)34 b Fv(=)23 b Fu(P)3822 3186 y Fk(\000)3878 3216 y Fu(:)361 3316 y Fv(By)k(\(1.11\))g(of)g([16],)g(since)h Fq(r)p Fu(h)1326 3285 y Fk(\000)1405 3316 y Fq(6)p Fv(=)22 b(0,)28 b(a)f(su\016cien)n(t)h(condition)f(for)g Fu(h)2542 3285 y Fk(\000)2626 3316 y Fv(to)g(b)r(e)h(steep)g(is)f(that)h(the)g(system) 1628 3498 y Fu(\021)1669 3510 y Fs(1)1725 3498 y Fv(+)18 b Fu(I)7 b(\021)1892 3510 y Fs(2)1948 3498 y Fv(+)18 b Fu(E)2097 3464 y Fk(0)2121 3498 y Fv(\()p Fu(P)12 b Fv(\))p Fu(\021)2291 3510 y Fs(3)2411 3498 y Fv(:=)83 b(0)1826 3623 y Fu(\021)1870 3589 y Fs(2)1867 3643 y(2)1926 3623 y Fv(+)18 b Fu(E)2075 3589 y Fk(00)2118 3623 y Fv(\()p Fu(P)12 b Fv(\))p Fu(\021)2291 3589 y Fs(2)2288 3643 y(3)2411 3623 y Fv(:=)83 b(0)1990 3747 y Fu(E)2056 3713 y Fk(000)2118 3747 y Fv(\()p Fu(P)12 b Fv(\))p Fu(\021)2291 3713 y Fs(3)2288 3768 y(3)2411 3747 y Fv(:=)83 b(0)1221 b(\(7.5\))236 3930 y(has)27 b(no)h(real)e(solution)h(apart)g(from)g (the)h(trivial)f(one)g Fu(\021)1976 3942 y Fs(1)2037 3930 y Fv(=)c Fu(\021)2166 3942 y Fs(2)2226 3930 y Fv(=)g Fu(\021)2355 3942 y Fs(3)2415 3930 y Fv(=)g(0.)236 4030 y(Making)k(the)h(c)n(hange)e(of)i(v)-5 b(ariable)27 b Fu( )f Fv(=)c(arccos)n(\(1)d Fq(\000)1942 4009 y Fl(e)1925 4030 y Fu(E)k Fv(+)18 b Fu(\030)2149 4009 y Fl(e)2132 4030 y Fu(E)5 b Fv(\))p Fu(;)28 b Fv(where)2538 4009 y Fl(e)2522 4030 y Fu(E)g Fv(=)22 b Fu(E)i Fv(+)18 b(2)p Fu(;)27 b Fv(w)n(e)g(get)3191 3999 y Fs(10)642 4251 y Fv(_)614 4272 y Fu(P)12 b Fv(\()p Fu(E)5 b Fv(\))24 b(=)920 4159 y Fl(Z)1004 4179 y Fs(1)967 4347 y(0)1068 4272 y Fu(F)1121 4284 y Fs(1)1159 4272 y Fv(\()p Fu(\030)t Fv(;)14 b Fu(E)5 b Fv(\))14 b Fu(d\030)t(;)1602 4251 y Fv(\177)1584 4272 y Fu(P)e Fv(\()p Fu(E)5 b Fv(\))23 b(=)g(3)1932 4237 y Fk(\000)p Fs(1)p Ft(=)p Fs(2)2102 4159 y Fl(Z)2185 4179 y Fs(1)2148 4347 y(0)2250 4272 y Fu(F)2303 4284 y Fs(2)2340 4272 y Fv(\()p Fu(\030)t Fv(;)14 b Fu(E)5 b Fv(\))14 b Fu(d\030)t(;)2774 4206 y Ft(:::)2765 4280 y Fu(P)2830 4272 y Fv(\()p Fu(E)5 b Fv(\))24 b(=)3071 4159 y Fl(Z)3154 4179 y Fs(1)3117 4347 y(0)3219 4272 y Fu(F)3272 4284 y Fs(3)3310 4272 y Fv(\()p Fu(\030)t Fv(;)14 b Fu(E)5 b Fv(\))14 b Fu(d\030)t(;)208 b Fv(\(7.6\))236 4496 y(where)1456 4719 y Fu(F)1509 4731 y Fs(1)1546 4719 y Fv(\()p Fu(\030)t Fv(;)14 b Fu(E)5 b Fv(\))84 b(:=)2347 4594 y Fq(p)p 2416 4594 42 4 v 69 x Fv(2)p 2018 4700 769 4 v 2018 4824 a Fu(\031)2068 4761 y Fq(p)p 2137 4761 41 4 v 63 x Fu(\030)2177 4761 y Fq(p)p 2246 4761 184 4 v 63 x Fv(1)18 b Fq(\000)g Fu(\030)2430 4717 y Fl(q)p 2513 4717 274 4 v 2529 4803 a(e)2513 4824 y Fu(E)5 b(\030)22 b Fq(\000)c Fu(E)1456 5027 y(F)1509 5039 y Fs(2)1546 5027 y Fv(\()p Fu(\030)t Fv(;)c Fu(E)5 b Fv(\))84 b(:=)2158 4903 y Fq(p)p 2227 4903 42 4 v 68 x Fv(6)2269 4909 y Fq(p)p 2338 4909 184 4 v 62 x Fv(1)18 b Fq(\000)g Fu(\030)p 2018 5008 644 4 v 2018 5099 a Fv(2)p Fu(\031)2110 5036 y Fq(p)p 2179 5036 41 4 v 63 x Fu(\030)t Fv(\()2268 5078 y Fl(e)2251 5099 y Fu(E)5 b(\030)23 b Fq(\000)18 b Fu(E)5 b Fv(\))2557 5075 y Fs(3)p Ft(=)p Fs(2)1456 5285 y Fu(F)1509 5297 y Fs(3)1546 5285 y Fv(\()p Fu(\030)t Fv(;)14 b Fu(E)5 b Fv(\))84 b(:=)2087 5228 y(3)2129 5160 y Fq(p)p 2198 5160 42 4 v 68 x Fv(2\(1)18 b Fq(\000)g Fu(\030)t Fv(\))2487 5198 y Fs(3)p Ft(=)p Fs(2)p 2018 5266 644 4 v 2018 5356 a Fv(4)p Fu(\031)2110 5294 y Fq(p)p 2179 5294 41 4 v 62 x Fu(\030)t Fv(\()2268 5335 y Fl(e)2251 5356 y Fu(E)5 b(\030)23 b Fq(\000)18 b Fu(E)5 b Fv(\))2557 5332 y Fs(5)p Ft(=)p Fs(2)2671 5285 y Fu(:)1151 b Fv(\(7.7\))p 236 5397 1512 4 v 298 5451 a Fn(10)363 5474 y Fm(W)-6 b(e)25 b(will)d(denote)j(with)f(\\)14 b(_)f(")24 b(the)h(deriv)l(ativ)n(e)f (with)g(resp)r(ect)h(to)f Fj(E)t(;)e Fm(and)j(with)e(\\)2536 5451 y Fh(0)2582 5474 y Fm(")h(the)g(deriv)l(ativ)n(e)h(with)f(resp)r (ect)g(to)h Fj(P)r(:)2084 6059 y Fv(24)p eop %%Page: 25 25 25 24 bop 236 272 a Fv(F)-7 b(rom)27 b(the)h(equation)f Fu(E)5 b Fv(\()p Fu(P)12 b Fv(\()p Fu(E)5 b Fv(\)\))25 b(=)d Fu(E)5 b(;)28 b Fv(deriving)f(with)h(resp)r(ect)f(to)h Fu(E)5 b(;)28 b Fv(w)n(e)f(obtain)g(that)1220 454 y Fu(E)1286 419 y Fk(000)1347 454 y Fv(\()p Fu(P)12 b Fv(\()p Fu(E)5 b Fv(\)\))24 b(=)f Fq(\000)p Fv(\()1843 433 y(_)1815 454 y Fu(P)12 b Fv(\()p Fu(E)5 b Fv(\)\))2042 419 y Fk(\000)p Fs(5)2132 454 y Fv([)2183 433 y(_)2155 454 y Fu(P)12 b Fv(\()p Fu(E)5 b Fv(\))2359 388 y Ft(:::)2350 462 y Fu(P)2415 454 y Fv(\()p Fu(E)g Fv(\))19 b Fq(\000)f Fv(3\()2740 433 y(\177)2721 454 y Fu(P)12 b Fv(\()p Fu(E)5 b Fv(\)\))2948 419 y Fs(2)2986 454 y Fv(])p Fu(:)236 625 y Fv(W)-7 b(e)28 b(w)n(an)n(t)f(to)h(pro)n(v)n(e)e(that)1845 725 y Fu(E)1911 691 y Fk(000)1972 725 y Fv(\()p Fu(P)12 b Fv(\()p Fu(E)5 b Fv(\)\))24 b Fu(<)f Fv(0)p Fu(:)1437 b Fv(\(7.8\))236 878 y(for)30 b(ev)n(ery)g Fu(E)36 b Fv(with)31 b Fq(\000)p Fv(2)c Fu(<)h(E)33 b(<)28 b Fv(0)p Fu(:)i Fv(This)h(is)g(equiv)-5 b(alen)n(t)30 b(to)h(pro)n(v)n(e)e(that)2606 857 y(_)2579 878 y Fu(P)12 b Fv(\()p Fu(E)5 b Fv(\))2783 812 y Ft(:::)2774 886 y Fu(P)2839 878 y Fv(\()p Fu(E)g Fv(\))29 b Fu(>)f Fv(3\()3183 857 y(\177)3165 878 y Fu(P)11 b Fv(\()p Fu(E)5 b Fv(\)\))3391 848 y Fs(2)3429 878 y Fu(:)31 b Fv(Using)g(\(7.7\))f(w)n (e)236 977 y(see)i(that)h Fu(F)613 989 y Fs(1)650 977 y Fu(F)703 989 y Fs(3)772 977 y Fv(=)d Fu(F)932 947 y Fs(2)920 998 y(2)1002 977 y Fv(and)i(hence,)h(noting)f(that)h Fu(F)1928 989 y Fs(3)1966 977 y Fv(\()p Fu(\030)t Fv(;)14 b Fu(E)5 b Fv(\))33 b(is)f(not)g(prop)r(ortional)f(to)h Fu(F)3088 989 y Fs(1)3125 977 y Fv(\()p Fu(\030)t Fv(;)14 b Fu(E)5 b Fv(\))34 b(for)d(ev)n(ery)g Fu(E)38 b Fv(\014xed,)236 1077 y(w)n(e)25 b(conclude)g(that)872 1010 y Fl(R)941 1077 y Fu(F)994 1089 y Fs(1)1046 1010 y Fl(R)1115 1077 y Fu(F)1168 1089 y Fs(3)1229 1077 y Fu(>)d Fv(\()1348 1010 y Fl(R)1418 1077 y Fu(F)1471 1089 y Fs(2)1509 1077 y Fv(\))1541 1047 y Fs(2)1603 1077 y Fv(b)n(y)j(a)g(straigh)n(tforw)n (ard)d(application)i(of)i(Cauc)n(h)n(y-Sc)n(h)n(w)n(arz)21 b(inequalit)n(y)k(and)236 1177 y(\(7.8\))j(follo)n(ws)e(from)h (\(7.6\).)361 1276 y(By)c(\(7.8\))g(the)g(unique)h(solution)f(of)g(the) h(system)f(\(7.5\))g(is)g(the)g(trivial)g(one)g Fu(\021)2738 1288 y Fs(1)2799 1276 y Fv(=)f Fu(\021)2927 1288 y Fs(2)2988 1276 y Fv(=)g Fu(\021)3116 1288 y Fs(3)3177 1276 y Fv(=)h(0,)g(hence)h (the)f(function)236 1376 y Fu(h)284 1346 y Fk(\000)366 1376 y Fv(is)i(steep.)36 b(It)26 b(is)g(simple)g(to)f(pro)n(v)n(e)f (that)i(the)g(so)f(called)g Fa(steepness)g(co)r(e\016cien)n(ts)g Fv(and)h Fa(steepness)f(indices)h Fv(\(see)f(again)236 1476 y(De\014nition)31 b(1.7.C.)e(pag.)42 b(6)29 b(of)h([16)o(]\))g (can)g(b)r(e)g(tak)n(en)f(uniformly)g(for)h Fq(\000)p Fv(2)18 b(+)i Fu(\026)2689 1445 y Ft(c)2719 1454 y Fc(d)2784 1476 y Fq(\024)26 b Fu(E)32 b Fq(\024)26 b(\000)p Fu(\026)3174 1445 y Ft(c)3204 1454 y Fc(d)3269 1476 y Fv(:)k(that)g(is)f(they)h(do)g (not)236 1575 y(dep)r(end)e(on)g Fu(\026:)236 1675 y Fv(No)n(w)i(w)n(e)h(are)f(ready)f(to)i(apply)f(the)h(Nekhoroshev)e (Theorem)h(in)h(the)g(form)n(ulation)f(giv)n(en)g(in)h(Theorem)f(4.4)g (of)h([16)o(].)236 1774 y(In)d(order)e(to)i(use)f(the)h(notations)f(of) h([16)o(])f(w)n(e)h(need)f(the)h(follo)n(wing)f(substitutions)2838 1744 y Fs(11)2908 1774 y Fv(:)836 1946 y(\()p Fu(I)7 b(;)14 b(P)e Fv(\))23 b Fq(!)h Fu(I)7 b(;)69 b Fv(\()p Fu(';)14 b(Q)p Fv(\))23 b Fq(!)g Fu(';)70 b(H)1883 1912 y Fk(\000)1962 1946 y Fq(!)23 b Fu(H)r(;)69 b(h)2279 1912 y Fk(\000)2358 1946 y Fq(!)23 b Fu(H)2533 1958 y Fs(0)2571 1946 y Fu(;)69 b(\026f)2763 1912 y Fk(\000)2841 1946 y Fq(!)24 b Fu(H)3017 1958 y Fs(1)3054 1946 y Fu(;)69 b(r)3183 1958 y Fs(0)3244 1946 y Fq(!)23 b Fu(\032;)526 2117 y Fq(f)p Fu(I)29 b Fq(2)24 b Fw(R)784 2083 y Ft(d)p Fs(+1)962 2117 y Fq(j)55 b(j)p Fu(I)7 b Fq(j)23 b(\024)p 1240 2072 40 4 v 23 w Fu(r)s Fq(g)17 b(\002)i Fu(D)1494 2083 y Fk(\000)1573 2117 y Fq(!)k Fu(G;)180 b Fq(f)p Fu(I)30 b Fq(2)23 b Fw(R)2205 2083 y Ft(d)p Fs(+1)2355 2117 y Fq(j)28 b(j)p Fu(I)7 b Fq(j)23 b(\024)p 2606 2072 V 23 w Fu(r)r Fq(g)2687 2129 y Ft(r)2718 2137 y Fd(0)2773 2117 y Fq(\002)18 b Fw(T)2922 2083 y Ft(d)p Fs(+1)2922 2138 y Ft(s)2953 2146 y Fd(0)3063 2117 y Fq(\002)g Fu(D)3217 2083 y Fs(+)3215 2138 y Ft(r)3246 2146 y Fd(0)3301 2117 y Fq(\002)g Fw(T)3450 2129 y Ft(s)3481 2137 y Fd(0)3541 2117 y Fq(!)23 b Fu(F)r(:)236 2284 y Fv(De\014ning)28 b Fu(m)23 b Fv(:=)g(sup)899 2304 y Ft(F)968 2284 y Fq(k)1020 2250 y Ft(@)1059 2224 y Fd(2)1091 2250 y Ft(H)1145 2258 y Fd(0)p 1019 2265 159 4 v 1046 2312 a Ft(@)t(I)1119 2296 y Fd(2)1187 2284 y Fq(k)28 b Fv(and)f(remem)n(b)r(ering)g(\(7.3\)) g(and)g(the)h(de\014nition)g(of)g Fu(r)2920 2296 y Fs(0)2958 2284 y Fu(;)f Fv(w)n(e)h(ha)n(v)n(e)1325 2467 y Fu(m)23 b Fq(\024)f Fu(const:\026)1776 2433 y Fk(\000)p Ft(c)1858 2442 y Fc(d)1911 2467 y Fv(ln)1980 2430 y Fk(\000)p Fs(3)2069 2467 y Fv(\(1)p Fu(=\026)p Fv(\))p Fu(;)124 b(\032)23 b Fv(=)g Fu(const:\026)2836 2433 y Ft(c)2866 2442 y Fc(d)2904 2467 y Fu(:)918 b Fv(\(7.9\))236 2639 y(In)28 b(order)e(to)i(apply)f (the)h(Theorem)f(w)n(e)g(ha)n(v)n(e)f(only)i(to)f(v)n(erify)g(the)h (follo)n(wing)e(condition)1754 2810 y Fu(M)32 b Fv(:=)23 b(sup)2015 2880 y Ft(F)2117 2810 y Fq(j)p Fu(H)2209 2822 y Fs(1)2246 2810 y Fq(j)g Fu(<)g(M)2461 2822 y Fs(0)3804 2810 y Fv(\(7.10\))236 3031 y(where)k Fu(M)557 3043 y Fs(0)622 3031 y Fv(dep)r(ends)h(only)f(on)g(the)h(steepness)g(co)r (e\016cien)n(ts)f(and)g(steepness)g(indices)h(\(whic)n(h)g(are)e(indep) r(enden)n(t)j(of)e Fu(\026)p Fv(\))236 3130 y(and)j(on)g Fu(m)g Fv(and)g Fu(\032)h Fv(\(whic)n(h)f(dep)r(end)h(on)f Fu(\026)p Fv(\).)45 b(Moreo)n(v)n(er)27 b(w)n(e)j(use)g(the)h(fact)f (that)g(the)h(dep)r(endence)g(of)f Fu(M)3594 3142 y Fs(0)3661 3130 y Fv(on)g Fu(m)g Fv(and)236 3230 y Fu(\032)i Fv(is,)h(\\p)r (olynomial")d(\(although)i(it)g(is)f(quite)i(cum)n(b)r(ersome\):)44 b(that)33 b(is)e(there)h(exist)g(constan)n(t)e Fl(e)-45 b Fu(c)3324 3242 y Ft(d)3362 3230 y Fu(;)p 3399 3184 36 4 v 14 w(c)3435 3242 y Ft(d)3504 3230 y Fu(>)30 b Fv(0)h(suc)n(h)h(that)236 3349 y Fu(M)317 3361 y Fs(0)354 3349 y Fv(\()p Fu(m;)14 b(\032)p Fv(\))23 b Fq(\025)g Fu(const:m)973 3319 y Fk(\000)-5 b Fl(e)-41 b Ft(c)1055 3328 y Fc(d)1093 3349 y Fu(\032)p 1136 3284 30 4 v -30 x Ft(c)1166 3328 y Fc(d)1233 3349 y Fv(\(see)27 b Fq(x)p Fv(6.8)g(of)g([17)o(]\).)38 b(So)27 b(condition)g(\(7.10\))g(b)r (ecomes,)h(using)f(\(7.9\),)1559 3547 y Fu(\026)c Fq(\024)f Fu(const:\026)1987 3513 y Ft(c)2017 3522 y Fc(d)2052 3513 y Fs(\()-5 b Fl(e)-41 b Ft(c)2108 3522 y Fc(d)2142 3513 y Fs(+)p 2193 3478 V Ft(c)2223 3522 y Fc(d)2258 3513 y Fs(\))2302 3547 y Fv(ln)2371 3510 y Fs(3)-5 b Fl(e)-41 b Ft(c)2434 3519 y Fc(d)2473 3547 y Fv(\(1)p Fu(=\026)p Fv(\))p Fu(;)236 3719 y Fv(whic)n(h)28 b(is)f(v)n(eri\014ed) g(c)n(ho)r(osing)f Fu(c)1221 3731 y Ft(d)1283 3719 y Fu(<)d Fv(\()p Fl(e)-46 b Fu(c)1439 3731 y Ft(d)1496 3719 y Fv(+)p 1579 3673 36 4 v 18 w Fu(c)1615 3731 y Ft(d)1654 3719 y Fv(\))1686 3688 y Fk(\000)p Fs(1)1775 3719 y Fu(:)236 3818 y Fv(No)n(w)35 b(w)n(e)g(can)g(apply)g(the)h (Nekhoroshev)e(Theorem)g(as)h(form)n(ulated)g(in)h(Theorem)e(4.4)h(of)g ([16)o(],)j(obtaining)d(that)g(if)236 3918 y(\()p Fu(I)7 b Fv(\(0\))p Fu(;)14 b(')p Fv(\(0\))p Fu(;)g(p)p Fv(\(0\))p Fu(;)g(q)s Fv(\(0\)\))24 b Fq(2)f(E)1167 3882 y Fk(\000)1160 3940 y Fs(1)1251 3918 y Fv(then)402 4130 y Fq(j)p Fu(I)7 b Fv(\()p Fu(t)p Fv(\))19 b Fq(\000)f Fu(I)7 b Fv(\(0\))p Fq(j)23 b(\024)g Fv(d)p Fu(=)p Fv(2)f(:=)h Fu(M)1300 4095 y Ft(b)1333 4130 y Fu(=)p Fv(2)f(=)g Fu(O)r Fv(\()p Fu(\026)1673 4095 y Ft(b)1708 4130 y Fv(\))221 b Fq(8)28 b(j)p Fu(t)p Fq(j)23 b(\024)g Fu(T)34 b Fv(:=)2450 4074 y(1)p 2426 4111 90 4 v 2426 4187 a Fu(M)2539 4130 y Fv(exp)2680 4038 y Fl(\020)2763 4074 y Fv(1)p 2739 4111 V 2739 4187 a Fu(M)2839 4038 y Fl(\021)2888 4055 y Ft(a)2952 4130 y Fv(=)22 b Fu(O)3104 4038 y Fl(\020)3169 4074 y Fv(1)p 3165 4111 51 4 v 3165 4187 a Fu(\026)3238 4130 y Fv(exp)3379 4038 y Fl(\020)3443 4074 y Fv(1)p 3439 4111 V 3439 4187 a Fu(\026)3499 4038 y Fl(\021)3548 4055 y Ft(a)3588 4038 y Fl(\021)3804 4130 y Fv(\(7.11\))236 4341 y(where)27 b Fu(a;)14 b(b)23 b(>)f Fv(0)28 b(are)e(some)h(constan)n(ts)g(dep)r (ending)h(only)f(on)g(the)h(steepness)f(prop)r(erties)g(of)h Fu(H)3242 4353 y Fs(0)3279 4341 y Fv(.)37 b(Finally)-7 b(,)27 b(c)n(ho)r(osing)1493 4512 y Fu(c)1529 4524 y Ft(d)1590 4512 y Fu(<)c Fv(min)p Fq(f)p Fv(\(2)p Fu(d)18 b Fv(+)g(6\))2150 4478 y Fk(\000)p Fs(1)2239 4512 y Fu(;)28 b Fv(\()p Fl(e)-46 b Fu(c)2358 4524 y Ft(d)2416 4512 y Fv(+)p 2499 4467 36 4 v 18 w Fu(c)2534 4524 y Ft(d)2573 4512 y Fv(\))2605 4478 y Fk(\000)p Fs(1)2695 4512 y Fq(g)p Fu(;)1044 b Fv(\(7.12\))236 4684 y(w)n(e)27 b(ha)n(v)n(e)g(pro)n(v)n (ed)f(the)i(exp)r(onen)n(tial)f(stabilit)n(y)g(in)h(the)g(region)e Fq(E)2264 4696 y Fs(1)2302 4684 y Fu(:)236 4831 y Fw(Stabilit)m(y)j(in) g(the)g(region)f Fq(E)1229 4795 y Fs(+)1222 4853 y(2)1284 4831 y Fu(:)e Fv(In)f(the)h(follo)n(wing)e(w)n(e)g(will)i(denote)f Fu(I)2505 4801 y Fk(\003)2566 4831 y Fv(:=)e(\()p Fu(I)2745 4843 y Fs(1)2783 4831 y Fu(;)14 b(:)g(:)g(:)f(;)h(I)3003 4843 y Ft(d)3043 4831 y Fv(\))25 b(the)h(pro)5 b(jection)24 b(on)h(the)g(last)236 4930 y Fu(d)j Fv(co)r(ordinates.)36 b(W)-7 b(e)28 b(shall)f(pro)n(v)n(e)f(the)i(follo)n(wing)e(lemma)236 5087 y Fw(Lemma)32 b(7.2)40 b Fq(8)p Fu(\024)28 b(>)g Fv(0)p Fu(;)k Fq(9)p Fu(\024)1144 5099 y Fs(0)1181 5087 y Fu(;)14 b(\026)1268 5099 y Fs(8)1333 5087 y Fu(>)27 b Fv(0)32 b Fp(such)h(that)f Fq(8)h Fv(0)27 b Fu(<)h(\026)g Fq(\024)f Fu(\026)2324 5099 y Fs(8)2361 5087 y Fu(;)33 b Fp(if)g Fv(\()p Fu(I)7 b Fv(\()p Fu(t)p Fv(\))p Fu(;)14 b(')p Fv(\()p Fu(t)p Fv(\))p Fu(;)g(p)p Fv(\()p Fu(t)p Fv(\))p Fu(;)g(q)s Fv(\()p Fu(t)p Fv(\)\))31 b Fq(2)d(E)3395 5052 y Fs(+)3388 5110 y(2)3483 5087 y Fp(for)33 b Fv(0)27 b Fu(<)h(t)g Fq(\024)p 3930 5021 61 4 v 27 w Fu(T)11 b Fp(,)236 5187 y(then)1252 5305 y Fq(j)p Fu(I)1318 5271 y Fk(\003)1356 5305 y Fv(\()p Fu(t)p Fv(\))19 b Fq(\000)f Fu(I)1595 5271 y Fk(\003)1633 5305 y Fv(\(0\))p Fq(j)24 b(\024)1883 5249 y Fu(\024)p 1883 5286 48 4 v 1886 5362 a Fv(2)2170 5305 y Fq(8)30 b Fu(t)23 b Fq(\024)g Fv(min)p Fq(f)2577 5249 y Fu(\024)2625 5261 y Fs(0)p 2577 5286 86 4 v 2594 5362 a Fu(\026)2685 5305 y Fv(ln)2783 5249 y(1)p 2778 5286 51 4 v 2778 5362 a Fu(\026)2838 5305 y(;)p 2875 5238 61 4 v 14 w(T)11 b Fq(g)p Fu(:)p 236 5427 1512 4 v 298 5480 a Fn(11)363 5504 y Fm(W)-6 b(e)25 b(observ)n(e)f(that)h(w)n(e)f(do)g(not)g(need)h(to)f(in)n(tro)r(duce)h (the)f(\()p Fj(p;)12 b(q)r Fm(\))24 b(v)l(ariables)f(so)h(in)f(our)h (case)g Fj(C)h Fm(=)19 b(+)p Fg(1)p Fm(.)2084 6059 y Fv(25)p eop %%Page: 26 26 26 25 bop 236 272 a Fv(It)22 b(is)f(quite)h(ob)n(vious)e(that)i(for)f (initial)h(conditions)f(\()p Fu(I)7 b Fv(\(0\))p Fu(;)14 b(')p Fv(\(0\))p Fu(;)g(p)p Fv(\(0\))p Fu(;)g(q)s Fv(\(0\)\))23 b Fq(2)h(E)2746 236 y Fs(+)2739 294 y(2)2801 272 y Fv(,)f(Theorem)e (1.2)f(follo)n(ws)h(from)g(lemma)236 372 y(7.2)27 b(and)g(the)h(exp)r (onen)n(tial)f(stabilit)n(y)h(in)g(the)g(region)e Fq(E)1976 384 y Fs(1)2013 372 y Fv(.)361 471 y(In)31 b(order)g(to)g(pro)n(v)n(e)f (lemma)h(7.2)g(let)g(us)h(de\014ne,)h(for)e(some)f(\014xed)i(0)d Fu(<)g(\016)k(<)c(\031)s(=)p Fv(4,)j(the)g(follo)n(wing)e(t)n(w)n(o)h (regions)f(in)236 571 y(the)e(phase)f(space)g(:)37 b Fu(U)32 b Fv(:=)23 b Fq(f)p Fv(\()p Fu(I)7 b(;)14 b(';)g(p;)g(q)s Fv(\))p Fq(j)28 b(j)p Fu(q)s Fq(j)23 b(\024)g Fu(\016)31 b Fv(mo)r(d)c(2)p Fu(\031)s(;)42 b Fq(j)p Fu(E)5 b Fv(\()p Fu(p;)14 b(q)s Fv(\))p Fq(j)24 b(\024)e Fv(2)p Fu(\026)2645 541 y Ft(c)2675 550 y Fc(d)2714 571 y Fq(g)27 b Fv(and)g Fu(V)42 b Fv(:=)23 b Fq(f)p Fv(\()p Fu(I)7 b(;)14 b(';)g(p;)g(q)s Fv(\))p Fq(j)28 b(j)p Fu(q)s Fq(j)23 b Fu(>)g(\016)31 b Fv(mo)r(d)236 671 y(2)p Fu(\031)s(;)42 b Fq(j)p Fu(E)5 b Fv(\()p Fu(p;)14 b(q)s Fv(\))p Fq(j)23 b(\024)g Fv(2)p Fu(\026)891 640 y Ft(c)921 649 y Fc(d)959 671 y Fq(g)p Fv(.)36 b(W)-7 b(e)28 b(\014rst)g(note)f(that)1711 640 y Fs(12)302 836 y Fu(z)t Fv(\()p Fu(t)p Fv(\))22 b Fq(2)i Fu(V)66 b Fq(8)p Fu(t)730 848 y Fs(1)791 836 y Fu(<)22 b(t)h(<)g(t)1049 848 y Fs(2)1086 836 y Fu(;)38 b Fq(j)p Fu(q)s Fv(\()p Fu(t)1272 848 y Fs(1)1310 836 y Fv(\))p Fq(j)p Fu(;)14 b Fq(j)p Fu(q)s Fv(\()p Fu(t)1527 848 y Fs(2)1565 836 y Fv(\))p Fq(j)23 b Fv(=)g Fu(\016)j Fv(mo)r(d)e(2)p Fu(\031)50 b Fv(=)-14 b Fq(\))23 b Fu(t)2303 848 y Fs(2)2351 836 y Fq(\000)11 b Fu(t)2457 848 y Fs(1)2517 836 y Fu(<)22 b(c)2640 848 y Fs(1)2677 836 y Fu(;)38 b Fq(j)p Fu(I)7 b Fv(\()p Fu(t)2866 848 y Fs(2)2904 836 y Fv(\))k Fq(\000)g Fu(I)c Fv(\()p Fu(t)3128 848 y Fs(1)3165 836 y Fv(\))p Fq(j)23 b(\024)g Fu(c)3367 848 y Fs(2)3404 836 y Fv(\()p Fu(t)3466 848 y Fs(2)3514 836 y Fq(\000)11 b Fu(t)3620 848 y Fs(1)3657 836 y Fv(\))p Fu(\026:)42 b Fv(\(7.13\))236 1001 y(Indeed)31 b(in)g(this)g(case)f Fq(8)14 b Fu(t)1043 1013 y Fs(1)1109 1001 y Fu(<)28 b(t)g(<)g(t)1383 1013 y Fs(2)1420 1001 y Fv(,)k Fu(c)1511 1013 y Fs(3)1576 1001 y Fq(\024)c(j)16 b Fv(_)-39 b Fu(q)t Fv(\()p Fu(t)p Fv(\))p Fq(j)28 b(\024)g Fu(c)2007 1013 y Fs(4)2045 1001 y Fv(.)46 b(This)31 b(implies)g(that)g Fu(t)2805 1013 y Fs(2)2863 1001 y Fq(\000)20 b Fu(t)2978 1013 y Fs(1)3043 1001 y Fq(\024)28 b Fu(c)3172 1013 y Fs(1)3240 1001 y Fv(and)j(then,)h(in)n(tegrating)236 1101 y(the)c(equation)f(of)h (motion)1113 1080 y(_)1094 1101 y Fu(I)i Fv(=)23 b Fq(\000)p Fu(\026@)1407 1113 y Ft(')1454 1101 y Fu(f)37 b Fv(in)27 b(\()p Fu(t)1690 1113 y Fs(1)1728 1101 y Fu(;)14 b(t)1795 1113 y Fs(2)1832 1101 y Fv(\),)28 b(w)n(e)f(immediately)h(get)f (\(7.13\).)37 b(W)-7 b(e)28 b(also)e(claim)h(that)770 1266 y Fq(8)p Fu(t)846 1278 y Fs(1)906 1266 y Fu(<)c(t)g(<)g(t)1165 1278 y Fs(2)1202 1266 y Fu(;)41 b(z)t Fv(\()p Fu(t)p Fv(\))23 b Fq(2)h Fu(U)36 b Fv(and)28 b Fq(j)p Fu(q)s Fv(\()p Fu(t)1885 1278 y Fs(1)1922 1266 y Fv(\))p Fq(j)p Fu(;)14 b Fq(j)p Fu(q)s Fv(\()p Fu(t)2139 1278 y Fs(2)2177 1266 y Fv(\))p Fq(j)23 b Fv(=)g Fu(\016)31 b Fv(mo)r(d)d(2)p Fu(\031)53 b Fq(\))23 b Fu(t)2876 1278 y Fs(2)2932 1266 y Fq(\000)18 b Fu(t)3045 1278 y Fs(1)3105 1266 y Fq(\025)23 b Fu(c)3229 1278 y Fs(5)3266 1266 y Fq(j)14 b Fv(ln)g Fu(\026)p Fq(j)p Fu(:)322 b Fv(\(7.14\))236 1431 y(W)-7 b(e)33 b(denote)f(with)h Fu(t)880 1401 y Ft(i)880 1454 y(U)967 1431 y Fv(\(resp.)51 b Fu(t)1252 1401 y Ft(i)1252 1454 y(V)1309 1431 y Fv(\))33 b(the)f Fu(i)p Fv(-th)g(time)h(for)f (whic)n(h)g(the)g(orbits)g(en)n(ters)f(in)i(\(resp.)50 b(go)r(es)31 b(out)h(from\))g Fu(U)9 b Fv(,)33 b(so)236 1531 y(that)28 b Fu(t)446 1500 y Ft(i)446 1553 y(U)524 1531 y Fu(<)23 b(t)642 1500 y Ft(i)642 1553 y(V)723 1531 y Fu(<)f(t)840 1495 y Ft(i)p Fs(+1)840 1555 y Ft(U)975 1531 y Fu(<)h(t)1093 1495 y Ft(i)p Fs(+1)1093 1555 y Ft(V)1232 1531 y Fv(for)j(0)d Fq(\024)g Fu(i)f Fq(\024)h Fu(i)1679 1543 y Fs(0)1716 1531 y Fv(.)37 b(F)-7 b(rom)26 b(\(7.14\))h(it)g(follo)n(ws)g(that)g Fu(i)2796 1543 y Fs(0)2856 1531 y Fq(\024)c Fu(c)2980 1543 y Fs(6)3017 1531 y Fu(\024)3065 1543 y Fs(0)3102 1531 y Fu(=\026)k Fv(and,)g(from)g(\(7.13\),)g(that)236 1630 y(the)h(time)g Fu(T)617 1642 y Ft(V)702 1630 y Fv(sp)r(en)n(t)g(b)n(y)f(the)h(orbit)f (in)h(the)g(region)f Fu(V)46 b Fv(is)28 b(b)r(ounded)g(b)n(y)f Fu(c)2543 1642 y Fs(7)2580 1630 y Fu(\024)2628 1642 y Fs(0)2665 1630 y Fu(=\026:)361 1730 y Fv(In)22 b(order)f(to)h(pro)n(v)n (e)f(\(7.14\))g(w)n(e)h(use)g(the)h(follo)n(wing)e(normal)g(form)h (result)g(for)f(the)i(p)r(endulum)g(Hamiltonian)f Fu(E)5 b Fv(\()p Fu(p;)14 b(q)s Fv(\))236 1829 y(in)28 b(a)f(neigh)n(b)r(orho) r(o)r(d)g(of)g(its)h(h)n(yp)r(erb)r(olic)f(equilibrium)h(p)r(oin)n(t)g (\(see)f(e.g.)37 b([12)o(]\))236 2006 y Fw(Lemma)32 b(7.3)40 b Fp(Ther)l(e)h(exist)e Fu(R)q(;)1292 1984 y Fl(e)1291 2006 y Fu(\016)k(>)d Fv(0)p Fu(;)f Fp(an)g(analytic)i(function)e Fu(g)s(;)g Fp(with)h Fu(g)2706 1976 y Fk(0)2729 2006 y Fv(\(0\))g(=)g Fq(\000)p Fv(1)f Fp(and)g(an)h(analytic)g(c)l(anoni-) 236 2117 y(c)l(al)g(tr)l(ansformation)f Fv(\010)h(:)g Fu(B)k Fq(\000)-14 b(!)39 b(fj)p Fu(p)p Fq(j)h(\024)1667 2095 y Fl(e)1665 2117 y Fu(\016)s Fq(g)25 b(\002)f(fj)p Fu(q)s Fq(j)40 b(\024)f Fu(\016)j Fv(mo)r(d)d(2)p Fu(\031)s Fq(g)g Fp(wher)l(e)g Fu(B)44 b Fv(:=)c Fq(fj)p Fu(P)12 b Fq(j)p Fu(;)i Fq(j)p Fu(Q)p Fq(j)38 b(\024)i Fu(R)q Fq(g)p Fp(,)h(such)e(that)236 2217 y Fu(E)5 b Fv(\(\010\()p Fu(P)r(;)14 b(Q)p Fv(\)\))25 b(=)d Fu(g)s Fv(\()p Fu(P)12 b(Q)p Fv(\))p Fp(.)361 2382 y Fv(In)35 b(the)h(co)r(ordinates)f(\()p Fu(Q;)14 b(P)e Fv(\))35 b(the)h(lo)r(cal)f(stable)g(and)h(unstable)f (manifolds)h(are)e(resp.)60 b Fu(W)3306 2352 y Ft(s)3294 2405 y(loc)3419 2382 y Fv(=)36 b Fq(f)p Fu(P)47 b Fv(=)36 b(0)p Fq(g)f Fv(and)236 2481 y Fu(W)326 2451 y Ft(u)314 2505 y(loc)426 2481 y Fv(=)22 b Fq(f)p Fu(Q)g Fv(=)h(0)p Fq(g)k Fv(and)g(Hamiltonian)h(\(7.1\))f(writes)g(as)1194 2640 y Fl(e)1175 2661 y Fu(H)j Fv(:=)1404 2640 y Fl(e)1384 2661 y Fu(H)7 b Fv(\()p Fu(I)g(;)14 b(';)g(P)r(;)g(Q)p Fv(\))24 b(:=)f Fu(h)p Fv(\()p Fu(I)7 b Fv(\))19 b(+)f Fu(g)s Fv(\()p Fu(P)12 b(Q)p Fv(\))18 b(+)g Fu(\026)2649 2639 y Fl(e)2634 2661 y Fu(f)8 b Fv(\()p Fu(I)f(;)14 b(';)g(P)r(;)g(Q)p Fv(\))236 2837 y(where)492 2815 y Fl(e)476 2837 y Fu(f)9 b Fv(\()p Fu(I)e(;)14 b(';)g(P)r(;)g(Q)p Fv(\))24 b(:=)e Fu(f)9 b Fv(\()p Fu(I)e(;)14 b(';)g Fv(\010\()p Fu(P)r(;)g(Q)p Fv(\)\))p Fu(:)361 2937 y Fv(W)-7 b(e)21 b(are)g(no)n(w)f(able)h(to)g(pro)n(v)n(e)f(\(7.14\).)34 b(Certainly)20 b(there)h(exists)g(an)g(instan)n(t)g Fu(t)2749 2907 y Fk(\003)2749 2957 y Fs(1)2810 2937 y Fq(2)j Fv([)p Fu(t)2942 2949 y Fs(1)2979 2937 y Fu(;)14 b(t)3046 2949 y Fs(2)3083 2937 y Fv(\))22 b(for)e(whic)n(h)i(\()p Fu(p)p Fv(\()p Fu(t)3625 2907 y Fk(\003)3625 2957 y Fs(1)3663 2937 y Fv(\))p Fu(;)14 b(q)s Fv(\()p Fu(t)3834 2907 y Fk(\003)3834 2957 y Fs(1)3873 2937 y Fv(\)\))23 b Fq(2)236 3036 y Fv(\010\()p Fu(B)t Fv(\))32 b(but,)g Fq(8)p Fu(t)714 3048 y Fs(1)780 3036 y Fu(<)d(t)f(<)g(t)1055 3006 y Fk(\003)1055 3057 y Fs(1)1093 3036 y Fv(,)k(\()p Fu(p)p Fv(\()p Fu(t)p Fv(\))p Fu(;)14 b(q)s Fv(\()p Fu(t)p Fv(\)\))40 b Fu(=)-52 b Fq(2)29 b Fv(\010\()p Fu(B)t Fv(\).)48 b(It)31 b(follo)n(ws)f(that,)i (if)g(w)n(e)e(tak)n(e)h(the)g(represen)n(tan)n(t)e Fu(q)s Fv(\()p Fu(t)3587 3048 y Fs(1)3625 3036 y Fv(\))g Fq(2)g Fv([)p Fq(\000)p Fu(\016)o(;)14 b(\016)s Fv(],)236 3136 y(then)38 b Fu(p)p Fv(\()p Fu(t)539 3106 y Fk(\003)539 3157 y Fs(1)577 3136 y Fv(\))p Fu(q)s Fv(\()p Fu(t)711 3106 y Fk(\003)711 3157 y Fs(1)750 3136 y Fv(\))h Fu(<)g Fv(0)p Fu(:)e Fv(W)-7 b(e)38 b(will)f(denote)g(with)h Fu(Z)6 b Fv(\()p Fu(t)p Fv(\))39 b(:=)g(\()p Fu(I)7 b Fv(\()p Fu(t)p Fv(\))p Fu(;)14 b(')p Fv(\()p Fu(t)p Fv(\))p Fu(;)g(P)e Fv(\()p Fu(t)p Fv(\))p Fu(;)i(Q)p Fv(\()p Fu(t)p Fv(\)\))42 b(=)c(\()p Fu(I)7 b Fv(\()p Fu(t)p Fv(\))p Fu(;)14 b(')p Fv(\()p Fu(t)p Fv(\))p Fu(;)g Fv(\010)3520 3106 y Fk(\000)p Fs(1)3611 3136 y Fv(\()p Fu(p)p Fv(\()p Fu(t)p Fv(\))p Fu(;)g(q)s Fv(\()p Fu(t)p Fv(\)\)\))236 3246 y(the)33 b(corresp)r(onding)d(solution)i(of)g(the)h(Hamiltonian)f (system)g(asso)r(ciated)f(to)2776 3225 y Fl(e)2757 3246 y Fu(H)6 b Fv(.)52 b(F)-7 b(rom)31 b(the)i(fact)g(that)f Fq(j)p Fu(q)s Fv(\()p Fu(t)3755 3216 y Fk(\003)3755 3267 y Fs(1)3794 3246 y Fv(\))p Fq(j)f Fv(=)g Fu(\016)236 3346 y Fv(or)j(\()p Fu(p)p Fv(\()p Fu(t)481 3315 y Fk(\003)481 3366 y Fs(1)519 3346 y Fv(\))p Fu(;)14 b(q)s Fv(\()p Fu(t)690 3315 y Fk(\003)690 3366 y Fs(1)729 3346 y Fv(\)\))35 b Fq(2)f Fu(@)5 b Fv(\010\()p Fu(B)t Fv(\))35 b(and)f(that)h Fq(j)p Fu(g)s Fv(\()p Fu(P)12 b(Q)p Fv(\))p Fq(j)34 b(\024)g Fu(\026)2014 3315 y Ft(c)2044 3324 y Fc(d)2083 3346 y Fv(,)i Fu(p)p Fv(\()p Fu(t)2246 3315 y Fk(\003)2246 3366 y Fs(1)2284 3346 y Fv(\))p Fu(q)s Fv(\()p Fu(t)2418 3315 y Fk(\003)2418 3366 y Fs(1)2457 3346 y Fv(\))e Fu(<)g Fv(0,)i(it)f(follo)n(ws)e(that)i Fq(j)p Fu(P)12 b Fv(\()p Fu(t)3429 3315 y Fk(\003)3429 3366 y Fs(1)3467 3346 y Fv(\))p Fq(j)35 b(\024)f Fu(c)3692 3358 y Fs(8)3729 3346 y Fu(\026)3779 3315 y Ft(c)3809 3324 y Fc(d)3882 3346 y Fv(and)236 3445 y Fq(j)p Fu(Q)p Fv(\()p Fu(t)387 3415 y Fk(\003)387 3466 y Fs(1)425 3445 y Fv(\))p Fq(j)24 b(\025)e Fu(c)627 3457 y Fs(9)665 3445 y Fu(:)361 3545 y Fv(In)30 b(the)h(same)f(w)n(a)n(y)f(there)h(exists)g(an)g(instan)n(t) g Fu(t)1872 3515 y Fk(\003)1872 3565 y Fs(2)1941 3545 y Fv(with)g Fu(t)2162 3557 y Fs(1)2227 3545 y Fu(<)d(t)2349 3515 y Fk(\003)2349 3565 y Fs(1)2415 3545 y Fu(<)g(t)2537 3515 y Fk(\003)2537 3565 y Fs(2)2603 3545 y Fu(<)g(t)2725 3557 y Fs(2)2793 3545 y Fv(for)i(whic)n(h)i(\()p Fu(P)12 b Fv(\()p Fu(t)3322 3515 y Fk(\003)3322 3565 y Fs(2)3360 3545 y Fv(\))p Fu(;)i(Q)p Fv(\()p Fu(t)3557 3515 y Fk(\003)3557 3565 y Fs(2)3596 3545 y Fv(\)\))28 b Fq(2)g Fu(B)34 b Fv(but,)236 3644 y Fq(8)p Fu(t)k(>)f(t)482 3614 y Fk(\003)482 3665 y Fs(2)556 3644 y Fv(\()p Fu(P)12 b Fv(\()p Fu(t)p Fv(\))p Fu(;)i(Q)p Fv(\()p Fu(t)p Fv(\)\))47 b Fu(=)-51 b Fq(2)38 b Fu(B)t Fv(;)j(in)36 b(particular)f(it)h(results)g Fq(j)p Fu(P)12 b Fv(\()p Fu(t)2249 3614 y Fk(\003)2249 3665 y Fs(2)2287 3644 y Fv(\))p Fq(j)38 b(\025)f Fu(c)2518 3656 y Fs(10)2588 3644 y Fu(:)f Fv(W)-7 b(e)37 b(claim)f(that)g Fu(t)3247 3614 y Fk(\003)3247 3665 y Fs(2)3310 3644 y Fq(\000)23 b Fu(t)3428 3614 y Fk(\003)3428 3665 y Fs(1)3504 3644 y Fq(\025)37 b Fu(c)3642 3656 y Fs(11)3726 3644 y Fv(ln\(1)p Fu(=\026)p Fv(\).)236 3755 y(Indeed)f Fu(P)12 b Fv(\()p Fu(t)p Fv(\))36 b(satis\014es)f(the)h(Hamilton's)f(equation) 1979 3734 y(_)1951 3755 y Fu(P)12 b Fv(\()p Fu(t)p Fv(\))37 b(=)f Fq(\000)p Fu(g)2356 3725 y Fk(0)2379 3755 y Fv(\()p Fu(P)12 b Fv(\()p Fu(t)p Fv(\))p Fu(Q)p Fv(\()p Fu(t)p Fv(\)\))p Fu(P)g Fv(\()p Fu(t)p Fv(\))25 b Fq(\000)e Fu(\026@)3128 3767 y Ft(Q)3200 3734 y Fl(e)3185 3755 y Fu(f)8 b Fv(\()p Fu(I)f Fv(\()p Fu(t)p Fv(\))p Fu(;)14 b(')p Fv(\()p Fu(t)p Fv(\))p Fu(;)g(P)e Fv(\()p Fu(t)p Fv(\))p Fu(;)i(Q)p Fv(\()p Fu(t)p Fv(\)\))236 3855 y(with)33 b(initial)g(condition)g Fq(j)p Fu(P)12 b Fv(\()p Fu(t)1195 3825 y Fk(\003)1195 3876 y Fs(1)1233 3855 y Fv(\))p Fq(j)32 b(\024)f Fu(c)1452 3867 y Fs(8)1489 3855 y Fu(\026)1539 3825 y Ft(c)1569 3834 y Fc(d)1608 3855 y Fv(.)52 b(Since)32 b Fq(j)p Fu(P)12 b Fv(\()p Fu(t)2054 3825 y Fk(\003)2054 3876 y Fs(2)2093 3855 y Fv(\))p Fq(j)31 b(\025)g Fu(c)2311 3867 y Fs(10)2382 3855 y Fv(,)j(w)n(e)e(can)g(deriv)n(e)g(from)g(Gron)n (w)n(all's)f(lemma)i(that)236 3955 y Fu(t)266 3925 y Fk(\003)266 3975 y Fs(2)323 3955 y Fq(\000)18 b Fu(t)436 3925 y Fk(\003)436 3975 y Fs(1)497 3955 y Fq(\025)23 b Fu(c)621 3967 y Fs(11)705 3955 y Fv(ln\(1)p Fu(=\026)p Fv(\),)k(whic)n(h)h(implies)g(\(7.14\).)361 4054 y(By)f(the)g(follo)n (wing)g(normal-form)e(lemma)j(there)f(exists)g(a)g(close)f(to)h(the)h (iden)n(tit)n(y)f(symplectic)h(c)n(hange)e(of)h(co)r(ordi-)236 4154 y(nates)d(remo)n(ving)f(the)i(non-resonan)n(t)d(angles)h Fu(')i Fv(in)f(the)h(p)r(erturbation)f(up)g(to)g Fu(O)r Fv(\()p Fu(\026)2843 4124 y Fs(2)2882 4154 y Fv(\).)36 b(It)24 b(can)g(b)r(e)h(pro)n(v)n(ed)e(b)n(y)h(standard)236 4254 y(p)r(erturbation)j(theory)g(\(see)h(for)f(similar)g(lemmas)g (section)g Fq(x)p Fv(5)g(of)h([12)o(]\).)236 4406 y Fw(Lemma)k(7.4)40 b Fp(L)l(et)j Fu(\014)51 b(>)46 b Fv(0)p Fu(:)c Fp(Ther)l(e)i(exist)e Fu(R)q(;)14 b(\032)46 b(>)g Fv(0)d Fp(so)g(smal)t(l)g(that,)k (de\014ning)c Fu(\025)k Fv(:=)f(min)3313 4421 y Fk(j)p Ft(\030)r Fk(j\024)p Ft(R)3487 4404 y Fd(2)3538 4406 y Fq(j)p Fu(g)3604 4376 y Fk(0)3627 4406 y Fv(\()p Fu(\030)t Fv(\))p Fq(j)p Fp(,)h Fu(S)k Fv(:=)236 4514 y(max)391 4529 y Fk(j)p Ft(\030)r Fk(j\024)p Ft(R)565 4512 y Fd(2)615 4514 y Fq(j)p Fu(g)681 4484 y Fk(00)723 4514 y Fv(\()p Fu(\030)t Fv(\))p Fq(j)p Fp(,)33 b(then)e Fu(\025)26 b Fq(\025)g Fv(2)p Fu(S)5 b(R)1421 4484 y Fs(2)1488 4514 y Fp(and)32 b Fu(\032)f Fq(\024)g Fv(min)p Fq(f)p Fu(\025=)p Fv(4)p Fu(N)t(;)f(R)2321 4484 y Fs(2)2358 4514 y Fu(=)p Fv(8)p Fu(s;)h(\014)t(=)p Fv(2)p Fu(N)9 b Fp(,)31 b Fu(r)r Fq(g)p Fu(:)g Fp(L)l(et)g Fv(\003)g Fp(b)l(e)g(a)h(sublattic)l(e)f(of)i Fw(Z)3870 4484 y Ft(d)p Fs(+1)3993 4514 y Fu(:)236 4622 y Fp(L)l(et)h Fq(D)h(\032)c Fw(R)651 4592 y Ft(d)p Fs(+1)808 4622 y Fp(b)l(e)k(b)l(ounde)l(d)g(and)g Fu(\014)t Fp(-non-r)l(esonant)f (mo)l(d)h Fv(\003)p Fp(,)h(i.e.)55 b Fq(8)14 b Fu(I)39 b Fq(2)32 b(D)r Fp(,)37 b Fu(h)32 b Fq(2)g Fw(Z)3037 4592 y Ft(d)p Fs(+1)3182 4622 y Fq(n)22 b Fv(\003)p Fp(,)36 b Fq(j)p Fu(h)p Fq(j)c(\024)f Fu(N)44 b Fp(it)35 b(r)l(esults)236 4722 y Fq(j)p Fv(\(1)p Fu(;)14 b(I)413 4692 y Fk(\003)451 4722 y Fv(\))19 b Fq(\001)f Fu(h)p Fq(j)23 b(\025)g Fu(\014)t(:)30 b Fp(Supp)l(ose)g(that)1624 4833 y Fu(")23 b Fv(:=)g Fu(\026)p Fq(k)1904 4811 y Fl(e)1889 4833 y Fu(f)8 b Fq(k)1980 4845 y Ft(B)s(;D)r(;s)2186 4833 y Fq(\024)23 b Fv(2)2316 4799 y Fk(\000)p Fs(11)2438 4833 y Fu(\014)2485 4845 y Fk(\003)2523 4833 y Fu(\032s;)1176 b Fv(\(7.15\))236 4972 y Fp(wher)l(e)440 4942 y Fs(13)541 4972 y Fu(D)25 b Fv(:=)e Fq(D)810 4984 y Ft(\032)848 4972 y Fp(,)31 b Fu(\014)951 4984 y Fk(\003)1012 4972 y Fv(:=)22 b(min)q Fq(f)p Fu(\014)t(;)14 b(\025=)p Fv(2)p Fq(g)p Fu(:)28 b Fp(Then)j(ther)l(e)f(exists)f(an)h(analytic)h(c)l(anonic)l(al)g(tr)l (ansformation)1349 5145 y Fv(\011)23 b(:)p 1543 5079 72 4 v 83 w Fu(D)d Fq(\002)e Fw(T)1781 5110 y Ft(d)p Fs(+1)1781 5174 y Ft(s=)p Fs(4)1923 5145 y Fq(\002)p 2006 5079 68 4 v 18 w Fu(B)87 b Fq(\000)-14 b(!)83 b Fu(D)20 b Fq(\002)e Fw(T)2611 5115 y Ft(d)p Fs(+1)2611 5166 y Ft(s)2753 5145 y Fq(\002)g Fu(B)1606 5298 y Fv(\()p 1638 5232 44 4 v Fu(I)8 b(;)p 1719 5253 55 4 v 14 w(';)p 1810 5232 65 4 v 14 w(P)j(;)p 1911 5232 66 4 v 14 w(Q)p Fv(\))147 b Fq(7\000)-14 b(!)83 b Fv(\()p Fu(I)7 b(;)14 b(';)g(P)r(;)g(Q)p Fv(\))3804 5220 y(\(7.16\))p 236 5341 1512 4 v 298 5395 a Fn(12)363 5418 y Fm(In)24 b(the)h(follo)n(wing)e(w) n(e)h(will)e(use)i Fj(c)1261 5428 y Fi(i)1310 5418 y Fm(to)h(denote)g(some)f(p)r(ositiv)n(e)g(constan)n(t)i(indep)r(enden)n (t)g(on)e Fj(\026:)298 5480 y Fn(13)363 5504 y Fj(B)j Fm(and)d Fj(D)i Fm(are)e(though)n(t)h(as)f(complex)g(domains,)g(as)f (in)h(the)g(sequel)p 2208 5449 57 4 v 24 w Fj(B)j Fm(and)p 2425 5449 60 4 v 24 w Fj(D)r Fm(.)2084 6059 y Fv(26)p eop %%Page: 27 27 27 26 bop 236 272 a Fp(with)p 417 205 68 4 v 31 w Fu(B)27 b Fv(:=)22 b Fq(fj)p 682 205 65 4 v Fu(P)12 b Fq(j)p Fu(;)i Fq(j)p 830 205 66 4 v Fu(Q)p Fq(j)23 b(\024)f Fu(R)q(=)p Fv(8)p Fq(g)p Fp(,)p 1273 205 72 4 v 29 w Fu(D)j Fv(:=)d Fq(D)1541 287 y Ft(\032=)p Fs(4)1647 272 y Fu(;)30 b Fp(such)g(that)p 940 397 76 4 v 940 464 a Fu(H)g Fv(:=)p 1150 397 V 23 w Fu(H)7 b Fv(\()p 1258 397 44 4 v Fu(I)g(;)p 1338 418 55 4 v 14 w(';)p 1429 397 65 4 v 14 w(P)12 b(;)p 1531 397 66 4 v 14 w(Q)o Fv(\))24 b(:=)1782 443 y Fl(e)1762 464 y Fu(H)i Fq(\016)18 b Fv(\011)k(=)h Fu(h)p Fv(\()p 2172 397 44 4 v Fu(I)7 b Fv(\))19 b(+)p 2349 418 43 4 v 18 w Fu(g)r Fv(\()p 2423 397 44 4 v Fu(I)8 b(;)p 2504 418 55 4 v 14 w(';)p 2595 397 65 4 v 14 w(P)p 2660 397 66 4 v 12 w(Q)o Fv(\))19 b(+)p 2859 396 50 4 v 18 w Fu(f)9 b Fv(\()p 2941 397 44 4 v Fu(I)e(;)p 3021 418 55 4 v 14 w(';)p 3112 397 65 4 v 14 w(P)12 b(;)p 3214 397 66 4 v 14 w(Q)o Fv(\))236 637 y Fp(with)p 409 591 43 4 v 23 w Fu(g)r Fv(\()p 483 570 44 4 v Fu(I)c(;)p 564 591 55 4 v 14 w(';)14 b(\030)t Fv(\))23 b(:=)g Fu(g)s Fv(\()p Fu(\030)t Fv(\))r(+)r Fu(f)1127 607 y Fk(\003)1164 637 y Fv(\()p 1196 570 44 4 v Fu(I)7 b(;)p 1276 591 55 4 v 14 w(')q(;)14 b(\030)t Fv(\))p Fp(,)24 b Fu(f)1539 607 y Fk(\003)1577 637 y Fv(\()p 1609 570 44 4 v Fu(I)7 b(;)p 1689 591 55 4 v 14 w(';)14 b(\030)t Fv(\))23 b(=)1963 575 y Fl(P)2051 662 y Ft(h)p Fk(2)p Fs(\003)p Ft(;)29 b Fk(j)p Ft(h)p Fk(j\024)p Ft(N)2436 637 y Fu(f)2486 607 y Fk(\003)2477 660 y Ft(h)2524 637 y Fv(\()p 2556 570 44 4 v Fu(I)7 b(;)14 b(\030)t Fv(\))p Fu(e)2747 607 y Fs(i)p Ft(h)p Fk(\001)p 2825 572 V Ft(')2894 637 y Fp(and)23 b Fq(k)p Fu(f)3140 607 y Fk(\003)3177 637 y Fq(k)p 3219 615 53 4 v 28 x Ft(B)s(;)p 3292 615 57 4 v(D)r(;s=)p Fs(4)3493 637 y Fq(\024)g Fu(")p Fp(.)35 b(Mor)l(e)l(over)236 745 y(the)30 b(fol)t(lowing)i(estimates)e(hold)967 966 y Fq(j)p 990 900 44 4 v Fu(I)c Fq(\000)18 b Fu(I)7 b Fq(j)23 b(\024)1325 910 y Fv(2)1367 880 y Fs(4)1404 910 y Fu(")p 1322 947 125 4 v 1322 1023 a(\014)1369 1035 y Fk(\003)1407 1023 y Fu(s)1456 966 y(;)103 b Fq(j)p 1605 900 65 4 v Fu(P)30 b Fq(\000)18 b Fu(P)12 b Fq(j)p Fu(;)i Fq(j)p 1919 900 66 4 v Fu(Q)k Fq(\000)g Fu(Q)p Fq(j)23 b(\024)2311 910 y Fv(2)2353 880 y Fs(5)2390 910 y Fu(")p 2295 947 149 4 v 2295 1023 a(R)q(\014)2406 1035 y Fk(\003)2454 966 y Fu(;)103 b Fq(k)p 2622 899 50 4 v Fu(f)8 b Fq(k)p 2713 944 53 4 v 28 x Ft(B)s(;)p 2786 944 57 4 v(D)r(;s=)p Fs(4)2987 966 y Fq(\024)3091 910 y Fv(2)3133 880 y Fs(9)3169 910 y Fu(")3208 880 y Fs(2)p 3085 947 167 4 v 3085 1023 a Fu(\014)3132 1035 y Fk(\003)3170 1023 y Fu(\032s)3261 966 y(:)520 b Fv(\(7.17\))361 1191 y(Let)32 b Fq(L)g Fv(b)r(e)h(the)f(\(\014nite\))h(set)g(of)f(the)g (maximal)g(sublattices)f(\003)g(=)f Fq(h)p Fu(h)2553 1203 y Fs(1)2590 1191 y Fu(;)14 b(:)g(:)g(:)g(;)g(h)2823 1203 y Ft(s)2858 1191 y Fq(i)31 b(\032)f Fw(Z)3074 1161 y Ft(d)p Fs(+1)3229 1191 y Fv(for)h(some)h(indep)r(enden)n(t)236 1291 y Fu(h)284 1303 y Ft(i)357 1291 y Fq(2)46 b Fw(R)530 1261 y Ft(d)p Fs(+1)693 1291 y Fv(with)41 b Fq(j)p Fu(h)966 1303 y Ft(i)994 1291 y Fq(j)k(\024)g Fu(N)50 b Fv(for)40 b Fu(i)45 b Fv(=)g(1)p Fu(;)14 b(:)g(:)g(:)f(;)h(s)45 b Fq(\024)g Fu(d)p Fv(.)77 b(F)-7 b(or)40 b(\003)45 b Fq(2)g(L)d Fv(w)n(e)e(de\014ne)h(the)h(\003-resonan)n(t)c(frequencies) 236 1391 y Fu(R)300 1361 y Fs(\003)373 1391 y Fv(:=)22 b Fq(f)p Fu(I)568 1361 y Fk(\003)629 1391 y Fq(2)h Fw(R)779 1361 y Ft(d)836 1391 y Fq(j)c Fv(\(1)p Fu(;)14 b(I)1032 1361 y Fk(\003)1070 1391 y Fv(\))p Fq(\001)p Fu(h)23 b Fv(=)g(0)p Fu(;)32 b Fq(8)14 b Fu(h)23 b Fq(2)g Fv(\003)p Fq(g)18 b Fv(and)g(the)h(set)g(of)f(the)h Fu(s)p Fv(-order)d(resonan)n (t)h(frequencies)h Fu(Z)3413 1361 y Ft(s)3471 1391 y Fv(:=)23 b Fq([)3637 1403 y Fs(dim\003=)p Ft(s)3879 1391 y Fu(R)3943 1361 y Fs(\003)3993 1391 y Fv(.)361 1490 y(Setting)k Fu(h)694 1502 y Ft(i)744 1490 y Fv(=)c(\()p Fu(l)889 1502 y Ft(i)916 1490 y Fu(;)14 b(n)1003 1502 y Ft(i)1031 1490 y Fv(\))27 b(with)f Fu(l)1302 1502 y Ft(i)1353 1490 y Fq(2)d Fw(R)p Fv(,)k Fu(n)1603 1502 y Ft(i)1653 1490 y Fq(2)d Fw(R)1804 1460 y Ft(d)1842 1490 y Fv(,)j(w)n(e)f(remark)f(that)h(if)h Fu(R)2613 1460 y Fs(\003)2686 1490 y Fq(6)p Fv(=)22 b Fq(;)k Fv(then)h Fu(n)3079 1502 y Fs(1)3116 1490 y Fu(;)14 b(:)g(:)g(:)g(;)g(n)3351 1502 y Ft(s)3412 1490 y Fv(are)26 b(indep)r(enden)n(t.)236 1590 y(W)-7 b(e)32 b(also)e(de\014ne)i(the)f(\()p Fu(d)22 b Fq(\000)e Fu(s)p Fv(\)-dimensional)31 b(linear)f(subspace)h(\(asso)r (ciated)f(with)i(the)f(a\016ne)g(subspace)g Fu(R)3680 1560 y Fs(\003)3729 1590 y Fv(\))h Fu(L)3850 1560 y Fs(\003)3928 1590 y Fv(:=)236 1690 y Fq(\\)291 1659 y Ft(s)291 1711 y(i)p Fs(=1)403 1690 y Fu(n)453 1659 y Fk(?)453 1711 y Ft(i)532 1690 y Fq(\032)23 b Fw(R)692 1659 y Ft(d)758 1690 y Fv(and)k(w)n(e)g(denote)h(b)n(y)f(\005)1486 1659 y Fs(\003)1564 1690 y Fv(the)h(orthogonal)d(pro)5 b(jection)27 b(from)g Fw(R)2785 1659 y Ft(d)2851 1690 y Fv(on)n(to)g Fu(L)3095 1659 y Fs(\003)3144 1690 y Fu(:)361 1789 y Fv(Since)h Fq(L)f Fv(is)h(a)f(\014nite)h(set,)g Fu(\013)23 b Fv(:=)g(min)1506 1801 y Fs(\003)p Fk(2L)1660 1789 y Fv(min)1798 1806 y Ft(n)p Fk(2)p Fe(Z)1930 1789 y Fc(d)1965 1806 y Ft(;)p Fk(j)p Ft(n)p Fk(j\024)p Ft(N)s(;)p Fs(\005)2243 1789 y Fd(\003)2283 1806 y Ft(n)p Fk(6)p Fs(=0)2426 1789 y Fq(j)p Fv(\005)2511 1759 y Fs(\003)2561 1789 y Fu(n)p Fq(j)k Fv(is)h(strictly)f(p)r(ositiv)n(e.)361 1889 y(W)-7 b(e)31 b(no)n(w)e(p)r(erform)h(a)g(suitable)h(v)n(ersion)e(of)h(the)h (standard)f(\\co)n(v)n(ering)d(lemma")j(in)h(whic)n(h)f(the)h(whole)f (frequency)236 1988 y(space)e(is)h(co)n(v)n(ered)e(b)n(y)i(non-resonan) n(t)e(zones.)40 b(The)29 b(fundamen)n(tal)g(blo)r(c)n(ks)f(used)h(to)g (construct)f(this)i(co)n(v)n(ering)c(will)j(b)r(e)236 2088 y Fu(r)r Fv(-neigh)n(b)r(orho)r(o)r(ds)k(of)h(an)n(y)f Fu(R)1192 2058 y Fs(\003)1275 2088 y Fv(i.e.)56 b Fu(R)1501 2058 y Fs(\003)1500 2109 y Ft(r)1583 2088 y Fv(:=)33 b Fq(f)p Fu(I)1789 2058 y Fk(\003)1861 2088 y Fq(2)g Fw(R)2021 2058 y Ft(d)2094 2088 y Fq(j)h Fv(dist\()p Fu(I)2360 2058 y Fk(\003)2398 2088 y Fu(;)14 b(R)2499 2058 y Fs(\003)2548 2088 y Fv(\))34 b Fq(\024)f Fu(r)r Fq(g)h Fv(for)g(suitable)f Fu(r)k(>)c Fv(0)g(dep)r(ending)h(on)236 2188 y(dim)q(\003)p Fu(:)d Fv(Let)h Fu(r)677 2200 y Ft(d)746 2188 y Fu(>)d Fv(0)j(b)r(e)g(suc)n(h)f(that)h(\()p Fu(d)22 b Fv(+)e(1\))p Fu(r)1699 2200 y Ft(d)1768 2188 y Fu(<)29 b(c)1898 2200 y Fs(12)1969 2188 y Fu(\024;)i Fv(for)h(some)f Fu(c)2451 2200 y Fs(12)2553 2188 y Fv(su\016cien)n(tly)g(small)h(to)f (b)r(e)h(determined.)49 b(F)-7 b(or)236 2287 y(1)34 b Fq(\024)g Fu(s)g Fq(\024)g Fu(d)23 b Fq(\000)f Fv(1)34 b(w)n(e)g(can)g(de\014ne)h(recursiv)n(ely)d(n)n(um)n(b)r(ers)i Fu(r)2146 2299 y Ft(s)2216 2287 y Fv(su\016cien)n(tly)g(small)g(suc)n (h)g(that)h(0)f Fu(<)g(r)3455 2299 y Ft(s)3525 2287 y Fu(<)f(\013r)3713 2299 y Ft(s)p Fs(+1)3834 2287 y Fu(=)p Fv(2)p Fu(N)9 b Fv(,)236 2387 y(v)n(erifying)551 2357 y Fs(14)795 2487 y Fv(dim)q(\003)22 b(=)h(dim\003)1298 2452 y Fk(0)1344 2487 y Fv(=)g Fu(s;)69 b(R)1627 2452 y Fs(\003)1700 2487 y Fq(6)p Fv(=)22 b Fu(R)1851 2452 y Fs(\003)1896 2427 y Fo(0)2029 2487 y Fv(=)-14 b Fq(\))106 b Fu(R)2333 2452 y Fs(\003)2332 2509 y(\()p Ft(s)p Fs(+1\))p Ft(r)2530 2517 y Fc(s)2585 2487 y Fq(\\)18 b Fu(R)2722 2452 y Fs(\003)2767 2427 y Fo(0)2721 2509 y Fs(\()p Ft(s)p Fs(+1\))p Ft(r)2919 2517 y Fc(s)2979 2487 y Fq(\032)k([)3121 2452 y Ft(d)3121 2507 y(i)p Fs(=)p Ft(s)p Fs(+1)3316 2487 y Fu(Z)3379 2452 y Ft(i)3373 2507 y(r)3404 2515 y Fc(i)3434 2487 y Fu(:)347 b Fv(\(7.18\))236 2646 y(W)-7 b(e)26 b(also)f(de\014ne,)h(for)f(1)e Fq(\024)f Fu(s)h Fq(\024)g Fu(d)14 b Fq(\000)g Fv(1)p Fu(;)26 b(S)1513 2616 y Fs(0)1573 2646 y Fv(:=)c Fw(R)1755 2616 y Ft(d)1808 2646 y Fq(n)14 b Fv(\()p Fq([)1951 2616 y Ft(d)1951 2668 y(i)p Fs(=1)2063 2646 y Fu(Z)2126 2616 y Ft(i)2120 2667 y Fs(2)p Ft(r)2184 2675 y Fc(i)2214 2646 y Fv(\))26 b(and)g Fu(S)2488 2616 y Ft(s)2546 2646 y Fv(:=)c Fu(Z)2719 2616 y Ft(s)2713 2673 y Fs(\()p Ft(s)p Fs(+1\))p Ft(r)2911 2681 y Fc(s)2962 2646 y Fq(n)14 b Fv(\()p Fq([)3105 2616 y Ft(d)3105 2668 y(i)p Fs(=)p Ft(s)p Fs(+1)3299 2646 y Fu(Z)3362 2616 y Ft(i)3356 2673 y Fs(\()p Ft(s)p Fs(+2\))p Ft(r)3554 2681 y Fc(i)3585 2646 y Fv(\),)26 b(i.e.)36 b(the)26 b Fu(s)p Fv(-)236 2767 y(order)19 b(resonances)e(min)n(us)j (the)g(higher-order)e(ones.)33 b(W)-7 b(e)20 b(claim)g(that)g Fw(R)2490 2736 y Ft(d)2551 2767 y Fv(=)j Fu(S)2695 2736 y Fs(0)2735 2767 y Fq([)s Fu(:)14 b(:)g(:)r Fq([)s Fu(S)3006 2736 y Ft(d)p Fk(\000)p Fs(1)3132 2767 y Fq([)s Fu(Z)3253 2736 y Ft(d)3247 2793 y Fs(\()p Ft(d)p Fs(+1\))p Ft(r)3449 2802 y Fc(d)3507 2767 y Fv(is)20 b(the)g(co)n(v)n(ering)236 2887 y(that)28 b(w)n(e)f(need.)37 b(W)-7 b(e)28 b(also)f(de\014ne)h Fu(S)1370 2856 y Fs(0)1430 2887 y Fq(\032)22 b Fu(S)1573 2856 y Fs(0)1568 2907 y Fk(\003)1633 2887 y Fv(:=)h Fw(R)1816 2856 y Ft(d)1873 2887 y Fq(n)18 b Fv(\()p Fq([)2020 2856 y Ft(d)2020 2908 y(i)p Fs(=1)2132 2887 y Fu(Z)2195 2856 y Ft(i)2189 2907 y(r)2220 2915 y Fc(i)2250 2887 y Fv(\))28 b(and)f Fu(S)2527 2856 y Ft(s)2586 2887 y Fq(\032)22 b Fu(S)2729 2856 y Ft(s)2724 2907 y Fk(\003)2788 2887 y Fv(:=)g Fu(Z)2961 2856 y Ft(s)2955 2913 y Fs(\()p Ft(s)p Fs(+1\))p Ft(r)3153 2921 y Fc(s)3208 2887 y Fq(n)c Fv(\()p Fq([)3355 2856 y Ft(d)3355 2908 y(i)p Fs(=)p Ft(s)p Fs(+1)3549 2887 y Fu(Z)3612 2856 y Ft(i)3606 2913 y Fs(\()p Ft(s)p Fs(+1\))p Ft(r)3804 2921 y Fc(i)3834 2887 y Fv(\).)361 3006 y(If)35 b(the)g(orbit)g(lies)g(near)f(a)g(certain)h Fu(R)1580 2976 y Fs(\003)1664 3006 y Fv(\(but)h(far)e(a)n(w)n(a)n(y)f (from)i(higher)f(order)g(resonances\))f(then)j(the)f(follo)n(wing)236 3106 y(lemma)28 b(sa)n(ys)e(that)i(the)g(drift)g(of)f(the)h(actions)f Fu(I)1752 3075 y Fk(\003)1818 3106 y Fv(in)h(the)g(direction)f(whic)n (h)h(is)f(parallel)f(to)i Fu(R)3186 3075 y Fs(\003)3263 3106 y Fv(is)f(small.)236 3264 y Fw(Lemma)32 b(7.5)40 b Fp(Supp)l(ose)j(that)f Fu(I)1292 3234 y Fk(\003)1331 3264 y Fv(\(0\))k Fq(2)g Fu(S)1640 3234 y Ft(s)1675 3264 y Fp(,)g Fu(I)1789 3234 y Fk(\003)1827 3264 y Fv(\()p Fu(t)p Fv(\))h Fq(2)f Fu(S)2125 3234 y Ft(s)2120 3284 y Fk(\003)2203 3264 y Fp(and)d Fq(j)p Fu(I)2443 3234 y Fk(\003)2481 3264 y Fv(\()p Fu(t)p Fv(\))p Fq(j)k(\024)i Fv(\026)-46 b Fu(r)30 b Fv(+)e Fu(r)r(=)p Fv(2)p Fu(;)42 b Fq(8)14 b Fv(0)45 b Fq(\024)h Fu(t)g Fq(\024)f Fu(T)3608 3234 y Fk(\003)3688 3264 y Fp(for)e(some)236 3363 y Fu(T)297 3333 y Fk(\003)362 3363 y Fq(\024)27 b Fu(\024)502 3375 y Fs(0)540 3363 y Fq(j)14 b Fv(ln)f Fu(\026)p Fq(j)p Fu(=\026)32 b Fp(and)h Fv(0)27 b Fq(\024)h Fu(s)f Fq(\024)h Fu(d)20 b Fq(\000)g Fv(1)p Fu(:)32 b Fp(Then,)i(if)f Fu(s)28 b Fq(\025)f Fv(1)p Fp(,)33 b(ther)l(e)f(exists)g(a)h(sublattic) l(e)f Fv(\003)c Fq(\032)f Fw(Z)3288 3333 y Ft(d)p Fs(+1)3411 3363 y Fp(,)33 b Fv(dim)q(\003)27 b(=)g Fu(s)33 b Fp(such)236 3463 y(that)d Fu(I)449 3433 y Fk(\003)487 3463 y Fv(\()p Fu(t)p Fv(\))24 b Fq(2)f Fu(R)747 3433 y Fs(\003)746 3490 y(\()p Ft(s)p Fs(+1\))p Ft(r)944 3498 y Fc(s)999 3463 y Fq(n)18 b Fv(\()p Fq([)1146 3433 y Ft(d)1146 3485 y(i)p Fs(=)p Ft(s)p Fs(+1)1340 3463 y Fu(Z)1403 3433 y Ft(i)1397 3490 y Fs(\()p Ft(s)p Fs(+1\))p Ft(r)1595 3498 y Fc(i)1626 3463 y Fv(\))p Fp(,)30 b Fq(8)14 b Fv(0)23 b Fq(\024)f Fu(t)h Fq(\024)g Fu(T)2127 3433 y Fk(\003)2164 3463 y Fp(.)39 b(Mor)l(e)l(over)31 b(if)g Fu(\024)2722 3475 y Fs(0)2788 3463 y Fp(is)f(su\016ciently)h(smal)t(l)3487 3433 y Fs(15)1330 3660 y Fq(j)p Fv(\005)1415 3626 y Fs(\003)1465 3660 y Fv(\()p Fu(I)1540 3626 y Fk(\003)1578 3660 y Fv(\()p Fu(t)p Fv(\))19 b Fq(\000)f Fu(I)1817 3626 y Fk(\003)1856 3660 y Fv(\(0\)\))p Fq(j)23 b(\024)g Fu(r)2165 3672 y Fs(1)2202 3660 y Fu(=)p Fv(2)169 b Fq(8)30 b Fv(0)23 b Fq(\024)f Fu(t)h Fq(\024)g Fu(T)2885 3626 y Fk(\003)3804 3660 y Fv(\(7.19\))236 3833 y Fp(and)38 b(henc)l(e,)j(for)d Fu(s)f Fq(\025)g Fv(1)p Fu(;)g Fq(j)p Fu(I)1156 3803 y Fk(\003)1194 3833 y Fv(\()p Fu(t)p Fv(\))25 b Fq(\000)f Fu(I)1445 3803 y Fk(\003)1483 3833 y Fv(\(0\))p Fq(j)37 b(\024)g Fv(2\()p Fu(s)24 b Fv(+)g(1\))p Fu(r)2088 3845 y Ft(s)2147 3833 y Fv(+)g Fu(r)2273 3845 y Fs(1)2311 3833 y Fu(=)p Fv(2)p Fu(:)36 b Fp(In)i(p)l(articular)g(for)h Fu(I)3142 3803 y Fk(\003)3180 3833 y Fv(\(0\))e Fq(2)h Fu(S)3472 3803 y Fs(0)3546 3833 y Fp(we)g(have)h(that)236 3932 y Fq(j)p Fu(I)302 3902 y Fk(\003)340 3932 y Fv(\()p Fu(t)p Fv(\))19 b Fq(\000)f Fu(I)579 3902 y Fk(\003)618 3932 y Fv(\(0\))p Fq(j)23 b(\024)g Fu(r)895 3944 y Fs(1)932 3932 y Fu(=)p Fv(2)p Fu(;)29 b Fq(8)14 b Fv(0)23 b Fq(\024)f Fu(t)h Fq(\024)g Fu(T)1482 3902 y Fk(\003)1519 3932 y Fp(.)236 4090 y Fb(Pr)n(oof)p Fv(.)105 b(In)37 b(the)g(case)e Fu(s)j Fv(=)f(0)f(w)n(e)h(tak)n(e)e(\003)j(=)f Fq(f)p Fv(0)p Fq(g)p Fu(:)e Fv(The)i(existence)f(of)g(\003)h(is)f(trivial)g(b) r(ecause)g Fu(I)3479 4060 y Fk(\003)3517 4090 y Fv(\(0\))i Fq(2)g Fu(S)3810 4060 y Ft(s)3882 4090 y Fv(and)236 4190 y(hence)26 b Fu(I)508 4160 y Fk(\003)547 4190 y Fv(\(0\))d Fq(2)g Fu(R)818 4160 y Fs(\003)817 4217 y(\()p Ft(s)p Fs(+1\))p Ft(r)1015 4225 y Fc(s)1078 4190 y Fv(for)j(some)f(\003)e Fq(2)g(L)k Fv(with)g(dim\003)c(=)f Fu(s:)27 b Fv(The)f(fact)g(that)h Fu(I)2789 4160 y Fk(\003)2827 4190 y Fv(\()p Fu(t)p Fv(\))d Fq(2)f Fu(R)3087 4160 y Fs(\003)3086 4217 y(\()p Ft(s)p Fs(+1\))p Ft(r)3284 4225 y Fc(s)3336 4190 y Fq(n)16 b Fv(\()p Fq([)3481 4160 y Ft(d)3481 4212 y(i)p Fs(=)p Ft(s)p Fs(+1)3675 4190 y Fu(Z)3738 4160 y Ft(i)3732 4217 y Fs(\()p Ft(s)p Fs(+1\))p Ft(r)3930 4225 y Fc(i)3960 4190 y Fv(\),)236 4302 y Fq(8)e Fv(0)36 b Fq(\024)g Fu(t)h Fq(\024)f Fu(T)704 4272 y Fk(\003)741 4302 y Fv(,)i(follo)n(ws)d(from)g Fu(I)1330 4272 y Fk(\003)1368 4302 y Fv(\()p Fu(t)p Fv(\))i Fq(2)g Fu(S)1647 4272 y Ft(s)1642 4322 y Fk(\003)1682 4302 y Fv(,)h Fq(8)14 b Fv(0)36 b Fq(\024)g Fu(t)h Fq(\024)f Fu(T)2211 4272 y Fk(\003)2284 4302 y Fv(and)f(\(7.18\).)61 b(No)n(w)35 b(w)n(e)g(w)n(an)n(t)g(to)h(apply)f(lemma)h(7.4)236 4402 y(with)i Fu(\014)43 b Fv(:=)38 b Fu(\013r)741 4414 y Fs(1)779 4402 y Fu(=)p Fv(2)f(and)f Fq(D)42 b Fv(:=)c Fu(R)1366 4371 y Fs(\003)1365 4428 y(\()p Ft(s)p Fs(+1\))p Ft(r)1563 4436 y Fc(s)1624 4402 y Fq(n)24 b Fv(\()p Fq([)1777 4371 y Ft(d)1777 4423 y(i)p Fs(=)p Ft(s)p Fs(+1)1972 4402 y Fu(Z)2035 4371 y Ft(i)2029 4428 y Fs(\()p Ft(s)p Fs(+1\))p Ft(r)2227 4436 y Fc(i)2257 4402 y Fv(\).)66 b(W)-7 b(e)37 b(ha)n(v)n(e)f(to)h(v)n(erify)f(that)i Fq(D)h Fv(is)e Fu(\014)t Fv(-non-resonan)n(t)236 4513 y(mo)r(d)d(\003)p Fu(:)f Fv(Fix)g Fq(j)p Fu(h)768 4525 y Fs(0)805 4513 y Fq(j)f(\024)g Fu(N)9 b Fv(,)35 b Fu(h)1139 4525 y Fs(0)1208 4513 y Fv(=)d(\()p Fu(l)1362 4525 y Fs(0)1399 4513 y Fu(;)14 b(n)1486 4525 y Fs(0)1523 4513 y Fv(\))42 b Fu(=)-51 b Fq(2)32 b Fv(\003)h(\(resp.)53 b Fq(6)p Fv(=)32 b(0)h(for)g Fu(s)f Fv(=)g(0\).)53 b(W)-7 b(e)34 b(\014rst)f(estimate)g Fq(j)p Fu(l)3358 4525 y Fs(0)3417 4513 y Fv(+)22 b Fu(n)3554 4525 y Fs(0)3613 4513 y Fq(\001)g Fu(I)3701 4483 y Fk(\003)3694 4534 y Fs(0)3739 4513 y Fq(j)33 b Fv(for)g(all)236 4613 y Fu(I)279 4583 y Fk(\003)272 4634 y Fs(0)340 4613 y Fq(2)24 b(D)483 4625 y Fs(0)543 4613 y Fv(:=)f Fu(R)718 4583 y Fs(\003)778 4613 y Fq(n)11 b Fv(\()p Fq([)918 4583 y Ft(d)918 4635 y(i)p Fs(=)p Ft(s)p Fs(+1)1113 4613 y Fu(Z)1176 4583 y Ft(i)1170 4640 y Fs(\()p Ft(s)p Fs(+1\))p Ft(r)1368 4648 y Fc(i)1398 4613 y Fv(\))p Fu(:)24 b Fv(If)h(\003)1615 4583 y Fk(0)1661 4613 y Fv(:=)d(\003)11 b Fq(\010)g(h)p Fu(h)1996 4625 y Fs(0)2033 4613 y Fq(i)24 b Fv(and)g Fu(n)2297 4583 y Fk(\003)2297 4634 y Fs(0)2358 4613 y Fv(:=)f(\005)2531 4583 y Fs(\003)2581 4613 y Fu(n)2631 4625 y Fs(0)2692 4613 y Fv(w)n(e)g(ha)n(v)n(e)g(t)n(w)n(o)g(cases:)34 b Fu(n)3440 4583 y Fk(\003)3440 4634 y Fs(0)3501 4613 y Fq(6)p Fv(=)23 b(0)g(or)g Fu(n)3802 4583 y Fk(\003)3802 4634 y Fs(0)3864 4613 y Fv(=)f(0.)236 4741 y(In)g Fu(n)384 4711 y Fk(\003)384 4761 y Fs(0)445 4741 y Fq(6)p Fv(=)h(0)e(w)n(e)g (can)g(p)r(erform)g(the)h(follo)n(wing)f(decomp)r(osition:)33 b Fu(I)2267 4711 y Fk(\003)2260 4761 y Fs(0)2329 4741 y Fv(=)22 b Fu(I)2459 4711 y Fk(\003)2452 4761 y Fs(1)2504 4741 y Fv(+)6 b Fu(v)25 b Fv(with)d Fu(I)2866 4711 y Fk(\003)2859 4761 y Fs(1)2927 4741 y Fq(2)i Fu(R)3070 4711 y Fs(\003)3115 4686 y Fo(0)3141 4741 y Fv(,)f Fu(v)k Fq(2)c Fu(L)3389 4711 y Fs(\003)3460 4741 y Fv(and)e(moreo)n(v)n(er) 3948 4711 y Fs(16)236 4840 y Fu(v)33 b Fv(=)d Fq(\006j)p Fu(v)s Fq(j)p Fu(n)608 4810 y Fk(\003)608 4861 y Fs(0)646 4840 y Fu(=)p Fq(j)p Fu(n)761 4810 y Fk(\003)761 4861 y Fs(0)798 4840 y Fq(j)p Fu(:)i Fv(Since)g Fu(I)1140 4810 y Fk(\003)1133 4861 y Fs(0)1218 4840 y Fu(=)-52 b Fq(2)31 b Fv(\()p Fq([)1381 4810 y Ft(d)1381 4862 y(i)p Fs(=)p Ft(s)p Fs(+1)1575 4840 y Fu(Z)1638 4810 y Ft(i)1632 4867 y Fs(\()p Ft(s)p Fs(+1\))p Ft(r)1830 4875 y Fc(i)1861 4840 y Fv(\))h(then)g Fu(I)2161 4810 y Fk(\003)2154 4861 y Fs(0)2238 4840 y Fu(=)-51 b Fq(2)30 b Fu(Z)2377 4810 y Fs(\003)2422 4785 y Fo(0)2371 4867 y Fs(\()p Ft(s)p Fs(+1\))p Ft(r)2569 4875 y Fc(s)p Fd(+1)2708 4840 y Fv(and,)j(hence)e Fq(j)p Fu(v)s Fq(j)g(\025)e Fv(\()p Fu(s)22 b Fv(+)e(1\))p Fu(r)3635 4852 y Ft(s)p Fs(+1)3755 4840 y Fu(:)32 b Fv(Using)236 4953 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y Ft(s)p Fs(+1)3478 5126 y Fu(:)303 b Fv(\(7.20\))p 236 5175 1512 4 v 298 5229 a Fn(14)363 5252 y Fm(Assumption)25 b(\(7.18\))h(means)f(that,)h (in)e(order)h(to)g(go)g(from)f(a)h(neigh)n(b)r(orho)r(o)r(d)h(of)e(a)h (\()p Fj(d)17 b Fg(\000)f Fj(s)p Fm(\)-order)25 b(resonance)g(to)h(a)f (di\013eren)n(t)g(one,)g(w)n(e)236 5331 y(ha)n(v)n(e)g(to)f(pass)g (through)h(an)f(higher)f(order)h(dimensional)f(one.)298 5389 y Fn(15)363 5413 y Fm(In)h(the)h(case)f Fj(s)c Fm(=)f(0)24 b(\005)961 5389 y Fn(\003)1030 5413 y Fm(is)f(simply)g(the)h(iden)n (tit)n(y)h(on)f Ff(R)1868 5389 y Fi(d)1904 5413 y Fj(:)298 5480 y Fn(16)363 5504 y Fm(W)-6 b(e)25 b(observ)n(e)f(that)h(dist\()p Fj(I)1066 5480 y Fh(\003)1061 5526 y Fn(0)1102 5504 y Fj(;)12 b(R)1187 5480 y Fn(\003)1228 5460 y Fo(0)1255 5504 y Fm(\))20 b(=)g Fg(j)p Fj(v)r Fg(j)p Fj(:)2084 6059 y Fv(27)p eop %%Page: 28 28 28 27 bop 236 272 a Fv(No)n(w)32 b(w)n(e)g(consider)f(the)i(case)e(in)i (whic)n(h)f Fu(n)1607 242 y Fk(\003)1607 293 y Fs(0)1676 272 y 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Fu(T)1791 1898 y Fk(\003)1828 1929 y Fv(\))k Fu(=)-51 b Fq(2)23 b Fu(S)2018 1898 y Ft(s)2013 1949 y Fk(\003)2053 1929 y Fv(.)37 b(W)-7 b(e)28 b(will)g(pro)n(v)n(e)e(that)1678 2099 y Fu(I)1721 2065 y Fk(\003)1759 2099 y Fv(\()p Fu(T)1852 2065 y Fk(\003)1890 2099 y Fv(\))d Fq(2)g Fu(S)2079 2065 y Fs(0)2135 2099 y Fq([)c Fu(:)14 b(:)g(:)k Fq([)h Fu(S)2454 2065 y Ft(s)p Fk(\000)p Fs(1)3804 2099 y Fv(\(7.21\))236 2270 y(that)36 b(means)e(that)h(the)h(orbit)e(can)h(only)g(en)n(ter)f (in)h(zones)g(that)g(are)f(\\less")f(resonan)n(t.)58 b(In)35 b(fact)g(b)n(y)g(lemma)g(7.5)f(w)n(e)236 2369 y(see)i(that)g Fu(I)610 2339 y Fk(\003)648 2369 y Fv(\()p Fu(T)741 2339 y Fk(\003)779 2369 y Fv(\))46 b Fu(=)-51 b Fq(2)37 b([)995 2339 y Ft(d)995 2391 y(i)p Fs(=)p Ft(s)p Fs(+1)1190 2369 y Fu(Z)1253 2339 y Ft(i)1247 2396 y Fs(\()p Ft(s)p Fs(+1\))p Ft(r)1445 2404 y Fc(i)1475 2369 y Fv(,)h(moreo)n(v)n (er,)e(since)f Fu(I)2182 2339 y Fk(\003)2221 2369 y Fv(\()p Fu(T)2314 2339 y Fk(\003)2351 2369 y Fv(\))47 b 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Fv(\()p Fu(T)3508 2462 y Fk(\003)3545 2492 y Fv(\))d Fq(2)h([)3744 2456 y Ft(s)p Fk(\000)p Fs(1)3744 2515 y Ft(i)p Fs(=1)3865 2492 y Fu(Z)3928 2462 y Ft(i)3922 2512 y Fs(2)p Ft(r)3986 2520 y Fc(i)236 2614 y Fq(\022)23 b([)379 2579 y Ft(s)p Fk(\000)p Fs(1)379 2638 y Ft(i)p Fs(=1)500 2614 y Fu(Z)563 2584 y Ft(i)557 2641 y Fs(\()p Ft(i)p Fs(+1\))p Ft(r)747 2649 y Fc(i)777 2614 y Fv(.)36 b(If)24 b Fu(I)958 2584 y Fk(\003)996 2614 y Fv(\()p Fu(T)1089 2584 y Fk(\003)1127 2614 y Fv(\))f Fq(2)h Fu(S)1317 2584 y Fs(1)1377 2614 y Fv(w)n(e)g(ha)n(v)n(e)f(\014nished.)36 b(If)24 b Fu(I)2142 2584 y Fk(\003)2180 2614 y Fv(\()p Fu(T)2273 2584 y Fk(\003)2311 2614 y Fv(\))32 b Fu(=)-51 b Fq(2)23 b Fu(S)2500 2584 y Fs(1)2561 2614 y Fv(then)i Fu(I)2790 2584 y Fk(\003)2828 2614 y Fv(\()p Fu(T)2921 2584 y Fk(\003)2958 2614 y Fv(\))33 b Fu(=)-51 b Fq(2)23 b Fu(Z)3155 2584 y Fs(1)3149 2635 y(2)p Ft(r)3213 2643 y Fd(1)3260 2614 y Fq(n)11 b([)3368 2584 y Ft(d)3368 2636 y(i)p Fs(=2)3480 2614 y Fu(Z)3543 2584 y Ft(i)3537 2635 y Fs(3)p Ft(r)3601 2643 y Fc(i)3655 2614 y Fv(and)24 b(hence)236 2737 y Fu(I)279 2707 y Fk(\003)317 2737 y Fv(\()p Fu(T)410 2707 y Fk(\003)448 2737 y Fv(\))f Fq(2)h([)637 2701 y Ft(s)p Fk(\000)p Fs(1)637 2760 y Ft(i)p Fs(=2)757 2737 y Fu(Z)820 2707 y Ft(i)814 2764 y Fs(\()p Ft(i)p Fs(+1\))p Ft(r)1004 2772 y Fc(i)1035 2737 y Fu(:)k Fv(Iterating)e(this) i(pro)r(cedure)f(w)n(e)g(pro)n(v)n(e)f(\(7.21\).)361 2837 y(The)20 b(conclusion)g(is)h(that)g(if)g(the)g(order)e(of)i (resonance)e(c)n(hanges)g(along)g(the)i(orbit,)h(it)f(can)f(decrease)f (only)i(so)f(that)g(the)236 2936 y(orbit)26 b(ma)n(y)g(ev)n(en)n (tually)g(arriv)n(e)f(in)i(the)g(completely)f(non)h(resonan)n(t)e(zone) h Fu(S)2641 2906 y Fs(0)2705 2936 y Fv(where)g(there)g(is)h(stabilit)n (y)-7 b(.)36 b(Considering)236 3036 y(the)c(\\w)n(orst")d(case)h(i.e.) 47 b(when)31 b Fu(I)1289 3006 y Fk(\003)1328 3036 y Fv(\(0\))e Fq(2)g Fu(Z)1610 3006 y Ft(d)1604 3063 y Fs(\()p Ft(d)p Fs(+1\))p Ft(r)1806 3072 y Fc(d)1875 3036 y Fv(and)i(the)g(orbit)g (arriv)n(es)e(in)i Fu(S)2821 3006 y Fs(0)2858 3036 y Fv(,)h(summing)f(all)g(the)h(con)n(tributions)236 3135 y(from)27 b(lemma)h(7.5,)f(w)n(e)g(ha)n(v)n(e)f(that,)i(if)g Fu(c)1485 3147 y Fs(12)1583 3135 y Fv(is)g(su\016cien)n(tly)f(small,) 356 3383 y Fq(j)p Fu(I)422 3349 y Fk(\003)460 3383 y Fv(\()p Fu(t)p Fv(\))19 b Fq(\000)f Fu(I)699 3349 y Fk(\003)738 3383 y Fv(\(0\))p Fq(j)23 b(\024)f Fv(2\()p Fu(d)d Fv(+)f(1\))p Fu(r)1307 3395 y Ft(d)1364 3383 y Fv(+)1448 3279 y Ft(d)p Fk(\000)p Fs(1)1447 3304 y Fl(X)1450 3480 y Ft(s)p Fs(=1)1567 3383 y Fv(\(2\()p Fu(s)h Fv(+)f(1\))p Fu(r)1925 3395 y Ft(s)1979 3383 y Fv(+)g Fu(r)2099 3395 y Fs(1)2137 3383 y Fu(=)p Fv(2\))f(+)h Fu(r)2390 3395 y Fs(1)2428 3383 y Fu(=)p Fv(2)k(=)2665 3279 y Ft(d)2622 3304 y Fl(X)2624 3480 y Ft(s)p Fs(=1)2756 3383 y Fv(2\()p Fu(s)c Fv(+)g(1\))p Fu(r)3081 3395 y Ft(s)3135 3383 y Fv(+)g Fu(dr)3298 3395 y Fs(1)3336 3383 y Fu(=)p Fv(2)k Fq(\024)h Fu(\024=)p Fv(2)p Fu(:)119 b Fv(\(7.22\))236 3630 y(In)22 b(order)f(to)h(conclude) g(the)g(pro)r(of)f(of)h(the)h(lemma)e(w)n(e)h(ha)n(v)n(e)f(only)g(to)h (pro)n(v)n(e)f(that)h(if)g Fq(j)p Fu(I)2918 3600 y Fk(\003)2957 3630 y Fv(\(0\))p Fq(j)h(\024)p 3196 3584 V 22 w Fu(r)i Fv(then)e Fq(j)p Fu(I)3508 3600 y Fk(\003)3546 3630 y Fv(\()p Fu(t)p Fv(\))p Fq(j)h(\024)p 3774 3584 V 22 w Fu(r)10 b Fv(+)d Fu(r)r(=)p Fv(2)236 3730 y Fq(8)14 b Fv(0)23 b Fq(\024)f Fu(t)h Fq(\024)g Fu(\024)637 3742 y Fs(0)674 3730 y Fq(j)14 b Fv(ln)g Fu(\026)p Fq(j)p Fu(=\026)p Fv(.)37 b(This)27 b(is)h(an)f(immediate)h(consequence)f(of)g (\(7.22\))g(and)g(of)h(the)g(fact)g(that)f Fu(\024)c Fq(\024)g Fu(r)n(:)p 3603 3730 48 48 v 361 3922 a Fw(Stabilit)m(y)j(in) g(the)g(region)g Fq(E)1343 3886 y Fk(\000)1336 3944 y Fs(2)1399 3922 y Fu(:)d Fv(If,)h(for)e(all)h Fu(t)g Fq(\025)g Fv(0)45 b(\()p Fu(p)p Fv(\()p Fu(t)p Fv(\))p Fu(;)14 b(q)s Fv(\()p Fu(t)p Fv(\)\))24 b Fq(2)g(E)2533 3886 y Fk(\000)2526 3944 y Fs(2)2589 3922 y Fv(,)g(then)f(it)g(follo)n(ws)f (easily)g(that)h Fq(j)p Fu(p)p Fv(\()p Fu(t)p Fv(\))p Fq(j)p Fu(;)14 b Fq(j)p Fu(q)s Fv(\()p Fu(t)p Fv(\))9 b Fq(\000)236 4021 y Fu(\031)s Fq(j)50 b Fv(=)e Fu(O)r Fv(\()p Fu(\026)619 3991 y Ft(c)649 4000 y Fc(d)685 3991 y Ft(=)p Fs(2)756 4021 y Fv(\))p Fu(:)43 b Fv(Then,)48 b(de\014ning)43 b Fu(f)1484 4033 y Fs(1)1521 4021 y Fv(\()p Fu(I)7 b(;)14 b(')p Fv(\))50 b(:=)e Fu(f)9 b Fv(\()p Fu(I)e(;)14 b(';)g Fv(0)p Fu(;)g(\031)s Fv(\))43 b(and)g Fu(f)2580 4033 y Fs(2)2617 4021 y Fv(\()p Fu(I)7 b(;)14 b(';)g(t)p Fv(\))50 b(:=)f Fu(\026)3119 3991 y Fk(\000)p Ft(c)3201 4000 y Fc(d)3235 3991 y Ft(=)p Fs(2)3306 4021 y Fv([)p Fu(f)9 b Fv(\()p Fu(I)e(;)14 b(';)g(p)p Fv(\()p Fu(t)p Fv(\))p Fu(;)g(q)s Fv(\()p Fu(t)p Fv(\)\))30 b Fq(\000)236 4121 y Fu(f)277 4133 y Fs(1)314 4121 y Fv(\()p Fu(I)7 b(;)14 b(')p Fv(\)],)27 b(it)f(results)f(that)h Fq(j)p Fu(@)1173 4133 y Ft(I)1211 4121 y Fu(f)1252 4133 y Fs(2)1289 4121 y Fv(\()p Fu(I)7 b(;)14 b(')p Fv(;)g Fu(t)p Fv(\))p Fq(j)p Fu(;)g Fq(j)p Fu(@)1681 4133 y Ft(')1730 4121 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5338 y Ft(=)p Fs(4)3142 5372 y Fu(:)p 236 5421 1512 4 v 298 5480 a Fn(17)363 5504 y Fm(F)-6 b(or)23 b(brevit)n(y)g(w)n(e)g (pro)n(v)n(e)g(only)g(the)h(case)f(in)g(whic)n(h)g Fj(I)5 b Fm(\(0\))23 b(is)f(in)h(a)f(non-resonan)n(t)i(zone.)32 b(The)23 b(resonan)n(t)h(case)g(can)f(b)r(e)g(treated)h(as)f(in)f Fg(E)3945 5474 y Fn(+)3939 5526 y(2)3996 5504 y Fj(:)2084 6059 y Fv(28)p eop %%Page: 29 29 29 28 bop 236 272 a Fv(It)28 b(follo)n(ws)f(that,)h(if)g(\()p Fu(I)7 b Fv(\(0\))p Fu(;)14 b(')p Fv(\(0\))p Fu(;)g(p)p Fv(\(0\))p Fu(;)g(q)s Fv(\(0\)\))24 b Fq(2)f(E)1809 236 y Fk(\000)1802 294 y Fs(2)1865 272 y Fv(,)28 b(then)341 451 y Fq(j)p Fu(I)7 b Fv(\()p Fu(t)p Fv(\))19 b Fq(\000)f Fu(I)7 b Fv(\(0\))p Fq(j)24 b(\024)e(j)p Fu(I)7 b Fv(\()p Fu(t)p Fv(\))19 b Fq(\000)p 1148 385 44 4 v 18 w Fu(I)7 b Fv(\()p Fu(t)p Fv(\))p Fq(j)19 b Fv(+)f Fq(j)p 1433 385 V Fu(I)8 b Fv(\()p Fu(t)p Fv(\))19 b Fq(\000)p 1673 385 V 18 w Fu(I)7 b Fv(\(0\))p Fq(j)18 b Fv(+)g Fq(j)p 1969 385 V Fu(I)7 b Fv(\(0\))19 b Fq(\000)f Fu(I)7 b Fv(\(0\))p Fq(j)23 b(\024)g Fu(const:\026)2771 417 y Ft(c)2801 426 y Fc(d)2835 417 y Ft(=)p Fs(4)2906 451 y Fu(;)180 b Fq(8)28 b(j)p Fu(t)p Fq(j)23 b(\024)g Fu(const:\026)3638 417 y Fk(\000)p Fs(1)p Fk(\000)p Ft(c)3805 426 y Fc(d)3839 417 y Ft(=)p Fs(4)236 622 y Fv(\(if)29 b(at)e(some)g(instan)n(t)h Fu(t)f Fv(the)h(solution)f Fu(z)t Fv(\()p Fu(t)p Fv(\))h(escap)r(es)f (outside)g Fq(E)2244 587 y Fk(\000)2237 644 y Fs(2)2328 622 y Fv(it)h(is)g(exp)r(onen)n(tially)e(stable)i(in)g(time\).)361 769 y(Finally)-7 b(,)37 b(from)d(the)i(previous)e(steps,)j(w)n(e)e(can) f(conclude)h(that)g(there)g(exists)g Fu(\026)2934 781 y Fs(1)3007 769 y Fu(>)g Fv(0)g(suc)n(h)f(that)i(0)f Fu(<)g(\026)g Fq(\024)g Fu(\026)3978 781 y Fs(1)236 869 y Fv(Theorem)27 b(1.2)g(holds.)236 1141 y Fr(8)135 b(App)t(endix)236 1323 y Fb(Pr)n(oof)69 b(of)32 b(lemma)59 b Fv(2.1.)36 b(W)-7 b(e)28 b(shall)f(use)g(the)h(follo)n(wing)f(lemma:)236 1480 y Fw(Lemma)32 b(8.1)40 b Fp(Ther)l(e)28 b(exists)f Fu(T)1248 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Fk(\000)p Fs(1)p Ft(;T)9 b Fs(+1])1697 1992 y Fq(j)p Fu(h)p Fv(\()p Fu(t)p Fv(\))p Fq(j)19 b Fv(+)f Fq(j)2020 1970 y Fv(_)2010 1992 y Fu(h)p Fv(\()p Fu(t)p Fv(\))p Fq(j)23 b(\024)g Fu(C)115 b Fv(max)2365 2050 y Ft(t)p Fk(2)p Fs([)p Fk(\000)p Fs(1)p Ft(;T)9 b Fs(+1])2723 1992 y Fq(j)p Fu(f)g Fv(\()p Fu(t)p Fv(\))p Fq(j)932 b Fv(\(8.2\))236 2209 y Fp(for)31 b(some)f(p)l(ositive)h(c)l(onstant)e Fu(C)36 b Fp(indep)l(endent)31 b(of)f Fu(T)12 b Fp(.)236 2366 y Fb(Pr)n(oof)p Fv(.)78 b(W)-7 b(e)28 b(\014rst)g(note)f(that)h (the)g(homogeneous)e(problem)h(\(8.1\))g(\(i.e.)37 b Fu(f)32 b Fv(=)22 b(0\))28 b(admits)f(only)g(the)h(trivial)f(solution) 236 2466 y Fu(h)34 b Fv(=)f(0.)56 b(This)34 b(immediately)g(implies)g (the)h(uniqueness)f(of)f(the)i(solution)e(of)h(\(8.1\).)56 b(The)34 b(existence)g(result)g(follo)n(ws)236 2565 y(b)n(y)c(the)h (standard)e(theory)g(of)h(linear)f(second)h(order)f(di\013eren)n(tial)h (equations.)43 b(W)-7 b(e)31 b(no)n(w)e(pro)n(v)n(e)g(that)h(an)n(y)f (solution)h Fu(h)236 2665 y Fv(of)i(\(8)p Fu(:)p Fv(1\))f(satis\014es)g (\(8)p Fu(:)p Fv(2\).)49 b(It)33 b(is)e(enough)g(to)h(sho)n(w)f(that)h (max)2216 2680 y Ft(t)p Fk(2)p Fs([)p Fk(\000)p Fs(1)p Ft(;T)9 b Fs(+1])2579 2665 y Fq(j)p Fu(h)p Fv(\()p Fu(t)p Fv(\))p Fq(j)30 b(\024)g Fu(C)2957 2635 y Fk(0)2994 2665 y Fv(max)3149 2680 y Ft(t)p Fk(2)p Fs([)p Fk(\000)p Fs(1)p Ft(;T)9 b Fs(+1])3511 2665 y Fq(j)p Fu(f)g Fv(\()p Fu(t)p Fv(\))p Fq(j)p Fv(.)50 b(Indeed)236 2780 y(w)n(e)29 b(obtain)f(b)n(y)h (\(8)p Fu(:)p Fv(1\))f(that)h(max)1271 2795 y Ft(t)p Fk(2)p Fs([)p Fk(\000)p Fs(1)p Ft(;T)9 b Fs(+1])1633 2780 y Fq(j)p Fu(h)p Fv(\()p Fu(t)p Fv(\))p Fq(j)20 b Fv(+)e Fq(j)1948 2758 y Fv(\177)1947 2780 y Fu(h)p Fv(\()p Fu(t)p Fv(\))p Fq(j)26 b(\024)f Fv(\(2)p Fu(C)2367 2750 y Fk(0)2409 2780 y Fv(+)19 b(1\))14 b(max)2735 2795 y Ft(t)p Fk(2)p Fs([)p Fk(\000)p Fs(1)p Ft(;T)9 b Fs(+1])3097 2780 y Fq(j)p Fu(f)g Fv(\()p Fu(t)p Fv(\))p Fq(j)29 b Fv(and,)g(b)n(y)g(elemen)n(tary)236 2880 y(analysis,)e(this)g(implies)h (\(8)p Fu(:)p Fv(2\))g(for)f(an)g(appropriate)f(constan)n(t)h Fu(C)6 b Fv(.)361 2979 y(Arguing)27 b(b)n(y)g(con)n(tradiction,)f(w)n (e)h(assume)g(that)h(there)g(exist)f(sequences)g(\()p Fu(T)2781 2991 y Ft(n)2826 2979 y Fv(\))c Fq(!)g(1)p Fv(,)28 b(\()p Fu(f)3194 2991 y Ft(n)3239 2979 y Fv(\),)g(\()p Fu(h)3402 2991 y Ft(n)3448 2979 y Fv(\))g(suc)n(h)f(that)440 3150 y Fq(\000)506 3128 y Fv(\177)505 3150 y Fu(h)553 3162 y Ft(n)616 3150 y Fv(+)18 b(cos)13 b Fu(Q)890 3162 y Ft(T)929 3170 y Fc(n)973 3150 y Fv(\()p Fu(t)p Fv(\))p Fu(h)1115 3162 y Ft(n)1184 3150 y Fv(=)23 b Fu(f)1313 3162 y Ft(n)1357 3150 y Fu(;)70 b(h)1498 3162 y Ft(n)1543 3150 y Fv(\(0\))23 b(=)g Fu(h)1808 3162 y Ft(n)1853 3150 y Fv(\()p Fu(T)1934 3162 y Ft(n)1979 3150 y Fv(\))g(=)g(0)p Fu(;)68 b Fq(j)p Fu(h)2326 3162 y Ft(n)2371 3150 y Fq(j)2394 3162 y Ft(n)2463 3150 y Fv(:=)133 b(max)2573 3208 y Ft(t)p Fk(2)p Fs([)p Fk(\000)p Fs(1)p Ft(;T)2806 3216 y Fc(n)2847 3208 y Fs(+1])2963 3150 y Fq(j)p Fu(h)3034 3162 y Ft(n)3079 3150 y Fv(\()p Fu(t)p Fv(\))p Fq(j)24 b Fv(=)f(1)p Fu(;)68 b Fq(j)p Fu(f)3505 3162 y Ft(n)3550 3150 y Fq(j)3573 3162 y Ft(n)3642 3150 y Fq(!)23 b Fv(0)p Fu(:)236 3377 y Fv(By)29 b(the)g(Ascoli-Arzela)f(Theorem)g(there)h(exists)g Fu(h)c Fq(2)h Fu(C)2038 3347 y Fs(2)2075 3377 y Fv(\([)p Fq(\000)p Fv(1)p Fu(;)14 b Fq(1)p Fv(\))p Fu(;)g Fw(R)p Fv(\))29 b(suc)n(h)f(that,)i(up)g(to)e(a)h(subsequence,)g Fu(h)3789 3389 y Ft(n)3859 3377 y Fq(!)d Fu(h)236 3477 y Fv(in)j(the)f(top)r(ology)f(of)i Fu(C)980 3447 y Fs(2)1046 3477 y Fv(uniform)f(con)n(v)n(ergence)e(in)i([)p Fq(\000)p Fv(1)p Fu(;)14 b(M)9 b Fv(])27 b(for)h(all)g Fu(M)33 b(>)23 b Fv(0.)39 b(Since)28 b Fu(Q)3055 3489 y Ft(T)3094 3497 y Fc(n)3163 3477 y Fq(!)d Fu(q)3308 3489 y Fs(0)3364 3477 y Fq(\000)18 b Fv(2)p Fu(\031)31 b Fv(uniformly)e(in)236 3577 y(all)f(b)r(ounded)g(in)n(terv)-5 b(als)26 b(of)i([)p Fq(\000)p Fv(1)p Fu(;)14 b Fq(1)p Fv(\),)27 b(w)n(e)g(obtain)g(that) 1195 3748 y Fq(\000)1261 3726 y Fv(\177)1260 3748 y Fu(h)18 b Fv(+)g(cos)13 b Fu(q)1571 3760 y Fs(0)1608 3748 y Fv(\()p Fu(t)p Fv(\))p Fu(h)24 b Fv(=)f(0)p Fu(;)68 b(h)p Fv(\(0\))23 b(=)g(0)p Fu(;)149 b Fv(sup)2394 3821 y Ft(t)p Fk(2)p Fs([)p Fk(\000)p Fs(1)p Ft(;)p Fk(1)p Fs(\))2693 3748 y Fq(j)p Fu(h)p Fv(\()p Fu(t)p Fv(\))p Fq(j)24 b(\024)e Fv(1)p Fu(:)788 b Fv(\(8.3\))236 3986 y(No)n(w)39 b(the)g(solutions)f (of)h(the)h(linear)e(di\013eren)n(tial)h(equation)f(in)i(\(8)p Fu(:)p Fv(3\))e(ha)n(v)n(e)g(the)i(form)e Fu(h)k Fv(=)g Fu(K)3380 3998 y Fs(1)3417 3986 y Fu(\030)30 b Fv(+)c Fu(K)3645 3998 y Fs(2)3681 3986 y Fu( )s Fv(,)42 b(where)236 4085 y(\()p Fu(K)339 4097 y Fs(1)376 4085 y Fu(;)14 b(K)484 4097 y Fs(2)521 4085 y Fv(\))24 b Fq(2)h Fw(R)729 4055 y Fs(2)766 4085 y Fv(,)j Fu(\030)t Fv(\()p Fu(t)p Fv(\))d(=)39 b(_)-38 b Fu(q)1102 4097 y Fs(0)1139 4085 y Fv(\()p Fu(t)p Fv(\))25 b(=)1421 4053 y Fs(2)p 1356 4067 163 4 v 1356 4114 a(cosh)11 b Ft(t)1556 4085 y Fv(and)28 b Fu( )s Fv(\()p Fu(t)p Fv(\))d(=)1993 4053 y Fs(1)p 1993 4067 34 4 v 1993 4114 a(4)2036 4085 y Fv(\(sinh)14 b Fu(t)19 b Fv(+)2441 4053 y Ft(t)p 2372 4067 163 4 v 2372 4114 a Fs(cosh)11 b Ft(t)2544 4085 y Fv(\))29 b(satis\014es)2938 4063 y(_)2911 4085 y Fu( )s(\030)23 b Fq(\000)3129 4063 y Fv(_)3111 4085 y Fu(\030)t( )k Fv(=)d(1.)38 b(The)28 b(b)r(ound)h(on)f Fu(h)236 4185 y Fv(implies)e(that)g Fu(K)765 4197 y Fs(2)825 4185 y Fv(=)d(0)i(and)h Fu(h)p Fv(\(0\))d(=)g(0)i(implies)h(that)g Fu(K)2001 4197 y Fs(1)2061 4185 y Fv(=)d(0.)36 b(Hence)26 b Fu(h)d Fv(=)f(0.)36 b(In)26 b(the)g(same)g(w)n(a)n(y)e(w)n(e)i(can)f(pro)n(v)n(e)g(that)236 4285 y Fu(h)284 4297 y Ft(n)329 4285 y Fv(\()p Fq(\001)19 b(\000)f Fu(T)535 4297 y Ft(n)580 4285 y Fv(\))23 b Fq(!)g Fv(0)k(uniformly)h(in)g(ev)n(ery)e(b)r(ounded)i(subin)n(terv)-5 b(al)27 b(of)h(\()p Fq(\0001)p Fu(;)14 b Fv(1].)361 4384 y(No)n(w)28 b(let)h(us)f(\014x)p 898 4323 30 4 v 29 w Fu(t)g Fv(suc)n(h)g(that)h(for)f(all)g Fu(n)h Fv(large)e(enough,)h(for) g(all)g Fu(t)d Fq(2)g Fv([)p 2564 4323 V Fu(t;)14 b(T)2680 4396 y Ft(n)2743 4384 y Fq(\000)p 2827 4323 V 19 w Fu(t)p Fv(],)29 b(cos)13 b Fu(Q)3123 4396 y Ft(T)3162 4404 y Fc(n)3207 4384 y Fv(\()p Fu(t)p Fv(\))25 b Fq(\025)f Fv(1)p Fu(=)p Fv(2)j(\()p 3600 4323 V Fu(t)i Fv(do)r(es)f(exist)236 4484 y(b)r(ecause)d(of)f(\(2)p Fu(:)p Fv(4\)\).)36 b(By)25 b(the)g(previous)f(step,)h(for)g Fu(n)f Fv(large)g(enough,)h(there)f (exists)h(a)f(maxim)n(um)h(p)r(oin)n(t)g Fu(t)3524 4496 y Ft(n)3592 4484 y Fq(2)e Fv(\()p 3702 4423 V Fu(t)q(;)14 b(T)3819 4496 y Ft(n)3876 4484 y Fq(\000)p 3954 4423 V 13 w Fu(t)o Fv(\))236 4595 y(of)31 b Fu(h)382 4565 y Fs(2)382 4615 y Ft(n)427 4595 y Fv(\()p Fu(t)p Fv(\),)h(i.e.)47 b Fu(h)777 4565 y Fs(2)777 4615 y Ft(n)822 4595 y Fv(\()p Fu(t)884 4607 y Ft(n)930 4595 y Fv(\))28 b(=)g Fq(j)p Fu(h)1154 4607 y Ft(n)1199 4595 y Fq(j)1222 4565 y Fs(2)1222 4615 y Ft(n)1296 4595 y Fv(=)g(1.)46 b(Then)1788 4568 y(_)1720 4595 y(\()p Fu(h)1800 4571 y Fs(2)1800 4615 y Ft(n)1846 4595 y Fv(\)\()p Fu(t)1940 4607 y Ft(n)1985 4595 y Fv(\))29 b(=)f(2)p Fu(h)2229 4607 y Ft(n)2274 4595 y Fv(\()p Fu(t)2336 4607 y Ft(n)2381 4595 y Fv(\))2423 4573 y(_)2413 4595 y Fu(h)2461 4607 y Ft(n)2506 4595 y Fv(\()p Fu(t)2568 4607 y Ft(n)2614 4595 y Fv(\))h(=)f(0)i(and)3063 4568 y(\177)3005 4595 y(\()p Fu(h)3085 4571 y Fs(2)3085 4615 y Ft(n)3130 4595 y Fv(\)\()p Fu(t)3224 4607 y Ft(n)3270 4595 y Fv(\))e(=)h(2)3467 4573 y(\177)3466 4595 y Fu(h)3514 4607 y Ft(n)3558 4595 y Fv(\()p Fu(t)3620 4607 y Ft(n)3666 4595 y Fv(\))p Fu(h)3746 4607 y Ft(n)3791 4595 y Fv(\()p Fu(t)3853 4607 y Ft(n)3898 4595 y Fv(\))21 b(+)236 4694 y(2)288 4673 y(_)278 4694 y Fu(h)326 4664 y Fs(2)326 4715 y Ft(n)371 4694 y Fv(\()p Fu(t)p Fv(\))32 b Fq(\024)g Fv(0.)52 b(By)33 b(the)g(di\013eren)n(tial)g(equation)g(satis\014ed)f (b)n(y)h Fu(h)2254 4706 y Ft(n)2299 4694 y Fv(,)h(w)n(e)f(can)f(deriv)n (e)g(from)h(the)g(latter)g(inequalit)n(y)g(that)236 4794 y(cos)13 b Fu(Q)427 4806 y Ft(T)466 4814 y Fc(n)511 4794 y Fv(\()p Fu(t)573 4806 y Ft(n)618 4794 y Fv(\))p Fu(h)698 4764 y Fs(2)698 4815 y Ft(n)744 4794 y Fv(\()p Fu(t)806 4806 y Ft(n)851 4794 y Fv(\))28 b Fq(\024)f Fu(f)1044 4806 y Ft(n)1089 4794 y Fv(\()p Fu(t)1151 4806 y Ft(n)1196 4794 y Fv(\))p Fu(h)1276 4806 y Ft(n)1321 4794 y Fv(\()p Fu(t)1383 4806 y Ft(n)1429 4794 y Fv(\),)k(i.e.)45 b(cos)13 b Fu(Q)1857 4806 y Ft(T)1896 4814 y Fc(n)1940 4794 y Fv(\()p Fu(t)2002 4806 y Ft(n)2048 4794 y Fv(\))28 b Fq(\024)e Fu(f)2240 4806 y Ft(n)2285 4794 y Fv(\()p Fu(t)2347 4806 y Ft(n)2393 4794 y Fv(\),)31 b(whic)n(h,)g(for)e Fu(n)i Fv(large)d(enough,)j(con)n(tradicts)d(the)236 4894 y(prop)r(ert)n(y)f(of)p 670 4833 V 27 w Fu(t)h Fv(and)f(the)h (fact)g(that)g Fq(j)p Fu(f)1440 4906 y Ft(n)1485 4894 y Fq(j)1508 4906 y Ft(n)1576 4894 y Fq(!)23 b Fv(0.)p 1784 4894 48 48 v 361 5038 a(No)n(w)33 b(w)n(e)g(can)g(deal)g(with)h (the)f(existence)h(result)f(of)g(lemma)g(2.1.)54 b(Let)33 b Fu(T)44 b Fv(:=)32 b(\()p Fu(\022)2960 5008 y Fk(\000)3039 5038 y Fq(\000)22 b Fu(\022)3167 5008 y Fs(+)3223 5038 y Fv(\),)35 b Fu(!)g Fv(=)e(\()p Fu(')3584 5008 y Fk(\000)3663 5038 y Fq(\000)21 b Fu(')3803 5008 y Fs(+)3859 5038 y Fv(\))p Fu(=T)12 b Fv(,)p 236 5092 55 4 v 236 5138 a Fu(')q Fv(\()p Fu(t)p Fv(\))23 b(:=)g Fu(!)s Fv(\()p Fu(t)18 b Fq(\000)g Fu(\022)778 5108 y Fs(+)833 5138 y Fv(\))h(+)f Fu(')1021 5108 y Fs(+)1076 5138 y Fv(.)37 b(In)28 b(the)g(follo)n(wing)e(w)n(e)i(call)f Fu(c)2044 5150 y Ft(i)2099 5138 y Fv(constan)n(ts)f(dep)r(ending)i(only)f(on)h Fu(f)9 b Fv(.)36 b(W)-7 b(e)28 b(are)f(searc)n(hing)e(for)236 5238 y(solutions)i(\()p Fu(';)14 b(q)s Fv(\))29 b(of)e(\(2.1\))g(with)i Fu(')p Fv(\()p Fu(\022)1416 5208 y Fk(\006)1472 5238 y Fv(\))24 b(=)e Fu(')1669 5208 y Fk(\006)1726 5238 y Fv(,)28 b Fu(q)s Fv(\()p Fu(\022)1890 5208 y Fk(\006)1946 5238 y Fv(\))c(=)e Fq(\007)p Fu(\031)s Fv(,)28 b(in)g(the)g(follo)n (wing)e(form)1421 5341 y Fl(\032)1525 5408 y Fu(')p Fv(\()p Fu(t)p Fv(\))e(=)e Fu(!)s Fv(\()p Fu(t)d Fq(\000)f Fu(\022)2044 5378 y Fs(+)2099 5408 y Fv(\))h(+)f Fu(')2287 5378 y Fs(+)2361 5408 y Fv(+)g Fu(v)s Fv(\()p Fu(t)h Fq(\000)f Fu(\022)2692 5378 y Fs(+)2747 5408 y Fv(\))1525 5507 y Fu(q)s Fv(\()p Fu(t)p Fv(\))23 b(=)g Fu(Q)1836 5519 y Ft(T)1888 5507 y Fv(\()p Fu(t)c Fq(\000)f Fu(\022)2093 5477 y Fs(+)2148 5507 y Fv(\))h(+)f Fu(w)r Fv(\()p Fu(t)h Fq(\000)f Fu(\022)2548 5477 y Fs(+)2604 5507 y Fv(\))p Fu(:)2084 6059 y Fv(29)p eop %%Page: 30 30 30 29 bop 236 272 a Fv(Hence)28 b(w)n(e)f(need)h(to)f(\014nd)h(a)g (solution,)f(in)h(the)g(time)g(in)n(terv)-5 b(al)27 b Fu(I)j Fv(:=)23 b([)p Fq(\000)p Fv(1)p Fu(;)14 b(T)28 b Fv(+)18 b(1],)27 b(of)h(the)g(follo)n(wing)e(t)n(w)n(o)h(equations) 573 387 y Fl(\032)680 454 y Fv(\177)-45 b Fu(v)s Fv(\()p Fu(t)p Fv(\))23 b(=)g Fq(\000)p Fu(\026)p Fv([)p Fu(F)1116 466 y Ft(')1164 454 y Fv(\()p Fu(v)s(;)14 b(w)r Fv(\)]\()p Fu(t)p Fv(\))r Fu(;)1311 b(v)s Fv(\(0\))23 b(=)g Fu(v)s Fv(\()p Fu(T)12 b Fv(\))23 b(=)f(0)p Fu(;)677 553 y Fv([)p Fu(L)p Fv(\()p Fu(w)r Fv(\)]\()p Fu(t)p Fv(\))i(=)f([)p Fu(G)p Fv(\()p Fu(v)s(;)14 b(w)r Fv(\)]\()p Fu(t)p Fv(\))25 b(:=)e Fq(\000)p Fv([)p Fu(S)5 b Fv(\()p Fu(w)r Fv(\)]\()p Fu(t)p Fv(\))19 b(+)f Fu(\026)p Fv([)p Fu(F)2271 565 y Ft(q)2309 553 y Fv(\()p Fu(v)s(;)c(w)r Fv(\)]\()p Fu(t)p Fv(\))r Fu(;)166 b(w)r Fv(\(0\))23 b(=)g Fu(w)r Fv(\()p Fu(T)12 b Fv(\))23 b(=)g(0)p Fu(;)3845 504 y Fv(\(8.4\))236 732 y(where)796 810 y([)p Fu(F)872 822 y Ft(')920 810 y Fv(\()p Fu(v)s(;)14 b(w)r Fv(;)g Fu(\025;)g(\026)p Fv(\)]\()p Fu(t)p Fv(\))85 b(:=)e Fu(@)1714 822 y Ft(')1762 810 y Fu(f)9 b Fv(\()p Fu(!)s(t)18 b Fv(+)g Fu(')2084 780 y Fs(+)2158 810 y Fv(+)g Fu(v)s Fv(\()p Fu(t)p Fv(\))p Fu(;)c(Q)2481 822 y Ft(T)2533 810 y Fv(\()p Fu(t)p Fv(\))19 b(+)f Fu(w)r Fv(\()p Fu(t)p Fv(\))p Fu(;)c(t)20 b Fv(+)e Fu(\022)3095 780 y Fs(+)3150 810 y Fv(\))p Fu(;)796 910 y Fv([)p Fu(F)872 922 y Ft(q)909 910 y Fv(\()p Fu(v)s(;)c(w)r Fv(;)g Fu(\025;)g(\026)p Fv(\)]\()p Fu(t)p Fv(\))96 b(:=)83 b Fu(@)1714 922 y Ft(q)1751 910 y Fu(f)9 b Fv(\()p Fu(!)s(t)18 b Fv(+)g Fu(')2073 880 y Fs(+)2147 910 y Fv(+)g Fu(v)s Fv(\()p Fu(t)p Fv(\))p Fu(;)c(Q)2470 922 y Ft(T)2522 910 y Fv(\()p Fu(t)p Fv(\))19 b(+)f Fu(w)r Fv(\()p Fu(t)p Fv(\))p Fu(;)c(t)20 b Fv(+)e Fu(\022)3084 880 y Fs(+)3139 910 y Fv(\))p Fu(;)796 1010 y Fv([)p Fu(S)5 b Fv(\()p Fu(w)r Fv(\)]\()p Fu(t)p Fv(\))382 b(:=)83 b(sin\()p Fu(Q)1870 1022 y Ft(T)1922 1010 y Fv(\()p Fu(t)p Fv(\))19 b(+)f Fu(w)r Fv(\()p Fu(t)p Fv(\)\))i Fq(\000)e Fv(sin\()p Fu(Q)2608 1022 y Ft(T)2660 1010 y Fv(\()p Fu(t)p Fv(\)\))i Fq(\000)e Fv(cos)o(\()p Fu(Q)3098 1022 y Ft(T)3150 1010 y Fv(\()p Fu(t)p Fv(\)\))p Fu(w)r Fv(\()p Fu(t)p Fv(\))p Fu(;)796 1109 y Fv([)p Fu(L)p Fv(\()p Fu(w)r Fv(\)]\()p Fu(t)p Fv(\))381 b(:=)83 b Fq(\000)17 b Fv(\177)-59 b Fu(w)r Fv(\()p Fu(t)p Fv(\))19 b(+)f(cos)13 b Fu(Q)2183 1121 y Ft(T)2235 1109 y Fv(\()p Fu(t)p Fv(\))p Fu(w)r Fv(\()p Fu(t)p Fv(\))p Fu(:)236 1260 y Fv(W)-7 b(e)35 b(w)n(an)n(t)f(to)h(solv)n(e)f(\(8.4\))g(as)g(a)g(\014xed)h(p)r(oin)n (t)g(problem.)58 b(By)34 b(lemma)h(8.1,)h(the)f(second)f(equation)g(of) h(\(8.4\))f(can)g(b)r(e)236 1359 y(written)28 b Fu(w)e Fv(=)c Fu(K)29 b Fv(:=)23 b Fq(G)5 b Fv(\()p Fq(\000)p Fu(S)23 b Fv(+)18 b Fu(\026F)1321 1371 y Ft(q)1358 1359 y Fv(\).)37 b(Moreo)n(v)n(er)25 b(the)j(\014rst)f(equation)g(\(8.4\))h (can)f(b)r(e)h(written)997 1604 y Fu(v)s Fv(\()p Fu(t)p Fv(\))c(=)f Fu(J)8 b Fv(\()p Fu(t)p Fv(\))24 b(:=)e([)p Fu(J)8 b Fv(\()p Fu(v)s(;)14 b(w)r Fv(;)g Fu(\025;)g(\026)p Fv(\)]\()p Fu(t)p Fv(\))26 b(:=)p 2235 1537 55 4 v 22 w Fu(J)8 b Fv(\()p Fu(t)p Fv(\))19 b Fq(\000)p 2495 1481 V 2495 1548 a Fu(J)8 b Fv(\(0\)\()p Fu(T)30 b Fq(\000)18 b Fu(t)p Fv(\))h(+)p 3013 1481 V 18 w Fu(J)8 b Fv(\()p Fu(T)k Fv(\))p Fu(t)p 2495 1585 727 4 v 2828 1661 a(T)3232 1604 y(;)590 b Fv(\(8.5\))236 1814 y(where,)27 b(setting)h Fu(F)825 1826 y Ft(')873 1814 y Fv(\()p Fu(s)p Fv(\))c(=)e Fu(F)1140 1826 y Ft(')1189 1814 y Fv(\()p Fu(v)s Fv(\()p Fu(s)p Fv(\))p Fu(;)14 b(w)r Fv(\()p Fu(s)p Fv(\)\),)1303 2058 y([)p 1326 1991 55 4 v Fu(J)8 b Fv(\()p Fu(v)s(;)14 b(w)r Fv(;)g Fu(\025;)g(\026)p Fv(\)]\()p Fu(t)p Fv(\))25 b(:=)e Fq(\000)p Fu(\026)2139 1945 y Fl(Z)2221 1965 y Ft(t)2184 2133 y(T)7 b(=)p Fs(2)2316 1945 y Fl(Z)2399 1965 y Ft(x)2362 2133 y(T)g(=)p Fs(2)2493 2058 y Fu(F)2546 2070 y Ft(')2595 2058 y Fv(\()p Fu(s)p Fv(\))28 b Fu(ds)g(dx:)361 2310 y Fv(Let)38 b(us)f(consider)g(the)h(Banac)n(h)e(space)h Fu(Z)45 b Fv(=)40 b Fu(V)k Fq(\002)25 b Fu(W)51 b Fv(:=)40 b Fq(C)2356 2279 y Fs(1)2393 2310 y Fv(\()p Fu(I)7 b Fv(;)14 b Fw(R)2577 2279 y Ft(d)2615 2310 y Fv(\))26 b Fq(\002)e(C)2811 2279 y Fs(1)2848 2310 y Fv(\()p Fu(I)7 b Fv(;)14 b Fw(R)p Fv(\),)40 b(endo)n(w)n(ed)d(with)h(the)g(norm)236 2409 y Fq(k)p Fu(z)t Fq(k)22 b Fv(=)g Fq(k)p Fv(\()p Fu(v)s(;)14 b(w)r Fv(\))p Fq(k)24 b Fv(:=)e(max)p Fq(fk)p Fu(v)s Fq(k)1219 2421 y Ft(V)1275 2409 y Fu(;)28 b Fq(k)p Fu(w)r Fq(k)1471 2421 y Ft(W)1546 2409 y Fq(g)p Fv(,)f(de\014ned)h(b)n (y)644 2617 y Fq(k)p Fu(v)s Fq(k)771 2629 y Ft(V)851 2617 y Fv(:=)22 b(sup)972 2686 y Ft(t)p Fk(2)p Ft(I)1100 2525 y Fl(h)1139 2617 y Fq(j)p Fu(v)s Fv(\()p Fu(t)p Fv(\))p Fq(j)p Fv(\(1)d(+)g Fu(c)1535 2629 y Fs(1)1572 2617 y Fu(\026T)1683 2582 y Fs(2)1719 2617 y Fv(\))1751 2582 y Fk(\000)p Fs(1)1840 2617 y Fu(\014)1891 2582 y Fs(2)1947 2617 y Fv(+)f Fq(j)12 b Fv(_)-35 b Fu(v)t Fv(\()p Fu(t)p Fv(\))p Fq(j)p Fu(\014)2265 2525 y Fl(i)2305 2617 y Fu(;)180 b Fq(k)p Fu(w)r Fq(k)2653 2629 y Ft(W)2751 2617 y Fv(:=)23 b(sup)2872 2686 y Ft(t)p Fk(2)p Ft(I)3001 2525 y Fl(h)3040 2617 y Fq(j)p Fu(w)r Fv(\()p Fu(t)p Fv(\))p Fq(j)d Fv(+)e Fq(j)26 b Fv(_)-49 b Fu(w)r Fv(\()p Fu(t)p Fv(\))p Fq(j)3545 2525 y Fl(i)3585 2617 y Fu(:)237 b Fv(\(8.6\))236 2853 y(A)23 b(\014xed)g(p)r(oin)n(t)g(of)f(the)h(op)r (erator)e(\010)i(:)g Fu(Z)29 b Fq(!)23 b Fu(Z)28 b Fv(de\014ned)23 b Fq(8)p Fu(z)j Fq(2)e Fu(Z)k Fv(as)22 b(\010\()p Fu(z)t Fv(\))h(:=)g(\010\()p Fu(z)t Fv(;)14 b Fu(\025;)g(\026)p Fv(\))23 b(:=)f(\()p Fu(J)8 b Fv(\()p Fu(z)t Fv(\))p Fu(;)14 b(K)6 b Fv(\()p Fu(z)t Fv(\)\))23 b(is)f(a)g(solution)236 2952 y(of)j(\(8.4\).)35 b(W)-7 b(e)25 b(shall)g(pro)n(v)n(e)d(in)j(the) g(sequel)g(that)f(\010)h(is)f(a)h(con)n(traction)e(in)i(the)g(ball)2797 2922 y Fs(18)2891 2952 y Fu(D)g Fv(:=)e Fu(B)p 3159 2932 30 4 v 15 x Ft(c\026)p Fs(\()p Ft(Z)t Fs(\))3359 2952 y Fv(for)h(an)h(appropriate)236 3052 y(c)n(hoice)i(of)p 578 3006 36 4 v 28 w Fu(c)o(;)14 b(c)686 3064 y Fs(1)724 3052 y Fu(;)g(C)820 3064 y Fs(0)857 3052 y Fv(,)28 b(pro)n(vided)e Fu(\026)i Fv(is)f(small)h(enough.)361 3152 y(W)-7 b(e)28 b(ha)n(v)n(e)f Fq(j)p Fv([)p Fu(S)5 b Fv(\()p Fu(w)r Fv(\)]\()p Fu(t)p Fv(\))p Fq(j)26 b(\024)e Fu(w)1239 3121 y Fs(2)1277 3152 y Fv(\()p Fu(t)p Fv(\))p Fu(;)k Fv(so)g(that)g Fq(8)p Fu(t;)h Fq(j)p Fv([)p Fu(G)p Fv(\()p Fu(v)s(;)14 b(w)r Fv(\)]\()p Fu(t)p Fv(\))p Fq(j)26 b(\024)p 2404 3106 V 24 w Fu(c)2440 3120 y Fs(2)2477 3152 y Fu(\026)2527 3121 y Fs(2)2583 3152 y Fv(+)19 b Fu(c)2703 3164 y Fs(4)2740 3152 y Fu(\026:)28 b Fv(No)n(w,)g(c)n(ho)r(osing)f(\014rst)p 3564 3106 V 28 w Fu(c)h Fv(su\016cien)n(tly)236 3251 y(large)c(and)h(then)g Fu(\026)g Fv(su\016cien)n(tly)g(small,)h(w)n(e)e (can)h(conclude)g(using)f(\(8.2\))h(that,)h(if)g Fu(z)g Fq(2)d Fu(D)r Fv(,)j Fq(k)p Fu(K)6 b Fv(\()p Fu(z)t Fv(\))p Fq(k)3333 3263 y Ft(W)3430 3251 y Fq(\024)p 3518 3205 V 23 w Fu(c)o(\026=)p Fv(4)p Fu(:)25 b Fv(No)n(w)f(w)n(e)236 3351 y(study)k(the)g(b)r(eha)n(viour)e(of)i Fu(J)o(:)f Fv(Let)h(us)g(\014rst)f(consider)p 1934 3284 55 4 v 27 w Fu(J)8 b Fv(.)37 b(W)-7 b(e)28 b(de\014ne)1078 3533 y Fu(f)1119 3545 y Ft(nl)1185 3533 y Fv(\()p Fu(t)p Fv(\))23 b(:=)g Fu(f)1454 3545 y Ft(nl)1520 3533 y Fv(\()p Fu(Q)1618 3545 y Ft(T)1670 3533 y Fv(\()p Fu(t)p Fv(\))c(+)g Fu(w)r Fv(\()p Fu(t)p Fv(\)\))p Fu(;)98 b(g)2215 3545 y Ft(nl)2281 3533 y Fv(\()p Fu(t)p Fv(\))24 b(:=)e Fu(f)2559 3499 y Fk(0)2550 3554 y Ft(nl)2617 3533 y Fv(\()p Fu(Q)2715 3545 y Ft(T)2767 3533 y Fv(\()p Fu(t)p Fv(\))d(+)f Fu(w)r Fv(\()p Fu(t)p Fv(\)\))p Fu(;)1436 3716 y(\013)1489 3728 y Ft(nl)1579 3716 y Fv(:=)23 b Fu(n)18 b Fq(\001)g Fu(')1853 3682 y Fs(+)1927 3716 y Fv(+)g Fu(l)r(\022)2078 3682 y Fs(+)2133 3716 y Fu(;)97 b(\014)2300 3728 y Ft(nl)2390 3716 y Fv(:=)22 b Fu(n)d Fq(\001)f Fu(!)j Fv(+)d Fu(l)r(:)236 3866 y Fv(F)-7 b(or)27 b Fu(t)c Fq(2)h Fv([)p Fq(\000)p Fv(1)p Fu(;)14 b(T)28 b Fv(+)18 b(1],)28 b Fu(z)e Fq(2)d Fu(D)r Fv(,)28 b(w)n(e)f(w)n(an)n(t)g(to)h(estimate)951 4068 y(_)p 935 4039 V 935 4106 a Fu(J)8 b Fv(\()p Fu(t)p Fv(\))24 b(=)f Fq(\000)p Fu(\026)1324 3993 y Fl(Z)1406 4013 y Ft(t)1369 4182 y(T)7 b(=)p Fs(2)1501 4106 y Fu(F)1554 4118 y Ft(')1625 4106 y Fv(=)23 b Fq(\000)p Fu(\026)1923 4027 y Fl(X)1842 4209 y Fk(j)p Fs(\()p Ft(n;l)p Fs(\))p Fk(j\024)p Ft(N)2139 4106 y Fv(i)p Fu(ne)2251 4072 y Fs(i)p Ft(\013)2313 4081 y Fc(nl)2391 3993 y Fl(Z)2474 4013 y Ft(t)2437 4182 y(T)7 b(=)p Fs(2)2569 4106 y Fu(f)2610 4118 y Ft(nl)2676 4106 y Fv(\()p Fu(s)p Fv(\))p Fu(e)2818 4072 y Fs(i)p Ft(n)p Fk(\001)p Ft(v)r Fs(\()p Ft(s)p Fs(\))3020 4106 y Fu(e)3059 4072 y Fs(i)p Ft(\014)3116 4081 y Fc(nl)3176 4072 y Ft(s)3211 4106 y Fu(ds:)236 4381 y Fv(In)n(tegrating)26 b(b)n(y)i(parts,)f(w)n(e)g(obtain)586 4700 y Fq(\000)p Fv(i)p Fu(\014)721 4712 y Ft(nl)801 4587 y Fl(Z)884 4607 y Ft(t)847 4776 y(T)7 b(=)p Fs(2)978 4700 y Fu(f)1019 4712 y Ft(nl)1086 4700 y Fv(\()p Fu(s)p Fv(\))p Fu(e)1228 4666 y Fs(i)p Ft(n)p Fk(\001)p Ft(v)r Fs(\()p Ft(s)p Fs(\))1430 4700 y Fu(e)1469 4666 y Fs(i)p Ft(\014)1526 4675 y Fc(nl)1586 4666 y Ft(s)1621 4700 y Fu(ds)83 b Fv(=)g Fu(f)1975 4712 y Ft(nl)2041 4700 y Fv(\()p Fu(T)9 b(=)p Fv(2\))p Fu(e)2286 4666 y Fs(i)p Ft(n)p Fk(\001)p Ft(v)r Fs(\()p Ft(T)e(=)p Fs(2\))2569 4700 y Fu(e)2608 4666 y Fs(i)p Ft(\014)2665 4675 y Fc(nl)2725 4666 y Ft(T)g(=)p Fs(2)2861 4700 y Fq(\000)18 b Fu(f)2985 4712 y Ft(nl)3052 4700 y Fv(\()p Fu(t)p Fv(\))p Fu(e)3185 4666 y Fs(i)p Ft(n)p Fk(\001)p Ft(v)r Fs(\()p Ft(t)p Fs(\))3381 4700 y Fu(e)3420 4666 y Fs(i)p Ft(\014)3477 4675 y Fc(nl)3536 4666 y Ft(t)3845 4700 y Fv(\(8.7\))1786 4952 y(+)1934 4839 y Fl(Z)2017 4859 y Ft(t)1980 5027 y(T)7 b(=)p Fs(2)2112 4952 y Fu(g)2152 4964 y Ft(nl)2218 4952 y Fv(\()p Fu(s)p Fv(\))2349 4931 y(_)2321 4952 y Fu(Q)2387 4964 y Ft(T)2439 4952 y Fv(\()p Fu(s)p Fv(\))p Fu(e)2581 4917 y Fs(i)p Ft(n)p Fk(\001)p Ft(v)r Fs(\()p Ft(s)p Fs(\))2783 4952 y Fu(e)2822 4917 y Fs(i)p Ft(\014)2879 4926 y Fc(nl)2939 4917 y Ft(s)2975 4952 y Fu(ds)788 b Fv(\(8.8\))1786 5203 y(+)1934 5090 y Fl(Z)2017 5111 y Ft(t)1980 5279 y(T)7 b(=)p Fs(2)2098 5203 y Fv(\()p Fu(g)2170 5215 y Ft(nl)2236 5203 y Fv(\()p Fu(s)p Fv(\))27 b(_)-50 b Fu(w)s Fv(\()p Fu(s)p Fv(\))19 b(+)f Fu(f)2647 5215 y Ft(nl)2713 5203 y Fv(\()p Fu(s)p Fv(\)i)p Fu(n)h Fq(\001)31 b Fv(_)-35 b Fu(v)s Fv(\()p Fu(s)p Fv(\)\))p Fu(e)3167 5169 y Fs(i)p Ft(n)p Fk(\001)p Ft(v)r Fs(\()p Ft(s)p Fs(\))3370 5203 y Fu(e)3409 5169 y Fs(i)p Ft(\014)3466 5178 y Fc(nl)3525 5169 y Ft(s)3561 5203 y Fu(ds:)179 b Fv(\(8.9\))p 236 5313 1512 4 v 298 5367 a Fn(18)363 5390 y Fm(If)24 b Fj(X)k Fm(is)23 b(a)h(Banac)n(h)h(space)g(and)f Fj(r)d(>)f Fm(0)k(w)n(e)g(de\014ne)g Fj(B)1775 5398 y Fi(r)1810 5390 y Fm(\()p Fj(X)5 b Fm(\))21 b(:=)e Fg(f)p Fj(x)h Fg(2)f Fj(X)5 b Fm(;)12 b Fg(k)p Fj(x)p Fg(k)20 b(\024)f Fj(r)r Fg(g)p Fj(:)2084 6059 y Fv(30)p eop %%Page: 31 31 31 30 bop 361 272 a Fv(By)27 b(\(2)p Fu(:)p Fv(4\),)g(the)h(term)g (\(8.8\))f(is)h(b)r(ounded)g(b)n(y)f Fu(c)1826 284 y Fs(5)1877 272 y Fv(max)p Fq(f)p Fu(e)2113 242 y Fk(\000)p Ft(K)2221 250 y Fd(2)2252 242 y Ft(t)2281 272 y Fu(;)14 b(e)2357 242 y Fk(\000)p Ft(K)2465 250 y Fd(2)2497 242 y Fs(\()p Ft(T)9 b Fk(\000)p Ft(t)p Fs(\))2678 272 y Fq(g)p Fv(.)36 b(Hence,)28 b(for)f Fu(z)g Fq(2)c Fu(D)r Fv(,)319 392 y Fl(Z)402 413 y Ft(t)365 581 y(T)7 b(=)p Fs(2)496 505 y Fu(F)549 517 y Ft(')621 505 y Fv(=)22 b Fu(u)p Fv(\()p Fu(t)p Fv(\))d Fq(\000)f Fu(u)p Fv(\()p Fu(T)9 b(=)p Fv(2\))17 b(+)h Fu(R)q Fv(\()p Fu(t)p Fv(\))p Fu(;)97 b Fv(with)84 b Fq(j)p Fu(R)q Fv(\()p Fu(t)p Fv(\))p Fq(j)23 b(\024)2154 449 y Fu(c)2190 461 y Fs(6)p 2154 486 74 4 v 2165 562 a Fu(\014)2237 413 y Fl(h)2290 505 y Fv(max)2458 413 y Fl(n)2514 505 y Fu(e)2553 471 y Fk(\000)p Ft(K)2661 479 y Fd(2)2692 471 y Ft(t)2722 505 y Fu(;)14 b(e)2798 471 y Fk(\000)p Ft(K)2906 479 y Fd(2)2937 471 y Fs(\()p Ft(T)9 b Fk(\000)p Ft(t)p Fs(\))3119 413 y Fl(o)3192 505 y Fv(+)p 3275 459 36 4 v 18 w Fu(c)p Fv(\()p Fu(\026)19 b Fv(+)3506 449 y Fu(\026)p 3505 486 52 4 v 3505 562 a(\014)3566 505 y Fv(\))p Fu(T)3659 413 y Fl(i)3698 505 y Fu(;)83 b Fv(\(8.10\))236 753 y(where)27 b Fu(u)p Fv(\()p Fu(t)p Fv(\))c(=)729 691 y Fl(P)817 753 y Fv(\()p Fu(n=\014)988 765 y Ft(nl)1054 753 y Fv(\))p Fu(e)1125 723 y Fs(i)p Ft(\013)1187 732 y Fc(nl)1252 753 y Fu(f)1293 765 y Ft(nl)1359 753 y Fv(\()p Fu(t)p Fv(\))p Fu(e)1492 723 y Fs(i)p Ft(n)p Fk(\001)p Ft(v)r Fs(\()p Ft(t)p Fs(\))1688 753 y Fu(e)1727 723 y Fs(i)p Ft(\014)1784 732 y Fc(nl)1844 723 y Ft(t)1873 753 y Fv(.)361 853 y(So)k(w)n(e)g(can)g(write)p 963 786 55 4 v 28 w Fu(J)8 b Fv(\()p Fu(t)p Fv(\))24 b(=)e Fu(j)5 b Fv(\()p Fu(t)p Fv(\))19 b(+)f Fu(\026)p Fv(\()p Fu(t)h Fq(\000)f Fu(T)9 b(=)p Fv(2\))p Fu(u)p Fv(\()p Fu(T)g(=)p Fv(2\),)26 b(where)1381 1086 y Fu(j)5 b Fv(\()p Fu(t)p Fv(\))23 b(=)1625 973 y Fl(Z)1708 994 y Ft(t)1671 1162 y(T)7 b(=)p Fs(2)1802 1086 y Fq(\000)p Fu(\026u)p Fv(\()p Fu(s)p Fv(\))28 b Fu(ds)18 b Fv(+)2279 973 y Fl(Z)2362 994 y Ft(t)2325 1162 y(T)7 b(=)p Fs(2)2457 1086 y Fq(\000)p Fu(\026R)q Fv(\()p Fu(s)p Fv(\))27 b Fu(ds:)236 1328 y Fv(By)39 b(the)h(b)r(ound)f(of)g Fu(R)q Fv(\()p Fu(t)p Fv(\))h(giv)n(en)e(in)h(\(8.10\),)j(the)d(second)g(in)n(tegral)f(can)g (b)r(e)i(b)r(ounded)g(b)n(y)e Fu(c)3267 1340 y Fs(7)3305 1328 y Fv(\()p Fu(\026=\014)t Fv(\)[1)26 b(+)p 3694 1282 36 4 v 26 w Fu(c)o(T)3790 1298 y Fs(2)3827 1328 y Fu(\026=\014)t Fv(].)236 1428 y(In)n(tegrating)36 b(once)h(again)f(b)n(y)h(parts)f(as) h(ab)r(o)n(v)n(e,)h(w)n(e)f(\014nd)h(that)f(the)h(\014rst)f(in)n (tegral)f(is)h(b)r(ounded)h(b)n(y)f Fu(c)3580 1440 y Fs(8)3617 1428 y Fv(\()p Fu(\026=\014)3792 1397 y Fs(2)3829 1428 y Fv(\)[1)25 b(+)p 236 1481 V 236 1527 a Fu(c)p Fv(\()p Fu(\026T)9 b(=\014)t Fv(\)],)32 b(hence,)f(b)n(y)f(the)h (condition)f(imp)r(osed)g(on)g Fu(\026T)12 b Fv(,)30 b(it)h(can)f(b)r(e)h(b)r(ounded)g(b)n(y)f Fu(\026)p 2980 1481 V(c=)p Fv(8)p Fu(\014)3151 1497 y Fs(2)3187 1527 y Fv(,)h(pro)n(vided)f(that)g Fu(C)3827 1539 y Fs(0)3895 1527 y Fv(has)236 1627 y(b)r(een)e(c)n(hosen)f(small)g(enough)g(and)p 1363 1581 V 28 w Fu(c)g Fv(is)h(large)e(enough.)36 b(Hence)1587 1837 y Fq(j)p Fu(j)5 b Fv(\()p Fu(t)p Fv(\))p Fq(j)24 b(\024)1888 1781 y Fu(\026)p 1938 1735 V(c)p 1887 1818 89 4 v 1887 1894 a(\014)1938 1870 y Fs(2)1986 1745 y Fl(h)2035 1781 y Fu(c)2071 1793 y Fs(7)p 2035 1818 74 4 v 2053 1848 36 4 v 2053 1894 a Fu(c)2136 1837 y Fv(+)18 b Fu(c)2255 1849 y Fs(7)2293 1837 y Fu(\026T)2404 1803 y Fs(2)2458 1837 y Fv(+)2551 1781 y(1)p 2551 1818 42 4 v 2551 1894 a(8)2603 1745 y Fl(i)2642 1837 y Fu(:)236 2052 y Fv(In)28 b(addition)1312 2078 y Fl(\014)1312 2128 y(\014)1312 2178 y(\014)1364 2118 y Fu(d)p 1349 2155 74 4 v 1349 2231 a(dt)1432 2174 y(j)5 b Fv(\()p Fu(t)p Fv(\))1565 2078 y Fl(\014)1565 2128 y(\014)1565 2178 y(\014)1617 2174 y Fv(=)22 b Fu(\026)p Fq(j)p Fu(u)p Fv(\()p Fu(t)p Fv(\))d(+)f Fu(R)q Fv(\()p Fu(t)p Fv(\))p Fq(j)23 b(\024)g Fu(c)2349 2186 y Fs(10)2429 2118 y Fu(\026)p 2479 2072 36 4 v(c)p 2429 2155 86 4 v 2447 2231 a(\014)2525 2082 y Fl(\020)2585 2118 y Fv(1)p 2585 2155 42 4 v 2588 2185 36 4 v 2588 2231 a Fu(c)2655 2174 y Fv(+)2748 2118 y Fu(\026T)p 2748 2155 111 4 v 2777 2231 a(\014)2868 2082 y Fl(\021)2917 2174 y Fu(:)236 2363 y Fv(As)28 b(a)f(result)g Fq(jj)p Fu(j)5 b Fq(jj)790 2375 y Ft(V)871 2363 y Fq(\024)23 b Fu(\026)p 1009 2317 36 4 v(c=)p Fv(4,)k(pro)n(vided)p 1520 2317 V 26 w Fu(c)h Fv(and)f Fu(c)1781 2375 y Fs(1)1846 2363 y Fv(ha)n(v)n(e)f(b)r(een)j(c)n(hosen)d(large)g(enough,)h Fu(C)3070 2375 y Fs(0)3136 2363 y Fv(small)g(enough.)361 2463 y(No)n(w)p 553 2396 55 4 v 30 w Fu(J)8 b Fv(\()p Fu(t)p Fv(\))30 b(=)e Fu(j)5 b Fv(\()p Fu(t)p Fv(\))22 b(+)e Fu(at)h Fv(+)f Fu(b)p Fv(,)32 b(where)e Fu(a;)14 b(b)29 b Fq(2)g Fw(R)p Fv(,)j(so)e(that)i(w)n(e)e(ma)n(y)h(replace)p 2818 2396 V 30 w Fu(J)39 b Fv(with)32 b Fu(j)k Fv(in)31 b(\(8)p Fu(:)p Fv(5\).)47 b(Since)32 b Fq(j)p Fu(J)8 b Fv(\()p Fu(t)p Fv(\))p Fq(j)29 b(\024)236 2578 y(j)p Fu(j)5 b Fv(\()p Fu(t)p Fv(\))p Fq(j)11 b Fv(+)g(max)p Fq(fj)p Fu(j)5 b Fv(\(0\))p Fq(j)p Fu(;)14 b Fq(j)p Fu(j)5 b Fv(\()p Fu(T)12 b Fv(\))p Fq(jg)p Fv(\()p Fu(T)21 b Fv(+)11 b(2\))p Fu(=T)33 b Fv(and)24 b Fq(j)1765 2557 y Fv(_)1736 2578 y Fu(J)8 b Fv(\()p Fu(t)p Fv(\))p Fq(j)24 b(\024)f(j)p Fu(d)-9 b(j)5 b Fv(\()p Fu(t)p Fv(\))p Fu(=dt)p Fq(j)11 b Fv(+)g(\(1)p Fu(=T)h Fv(\))2657 2511 y Fl(R)2710 2531 y Ft(T)d Fs(+1)2694 2607 y(1)2860 2578 y Fq(j)p Fu(d)-9 b(j)5 b Fv(\()p Fu(s)p Fv(\))p Fu(=dt)p Fq(j)24 b Fu(ds)p Fv(,)h(w)n(e)f(obtain)f Fq(jj)p Fu(J)8 b Fq(jj)3870 2590 y Ft(V)3951 2578 y Fq(\024)236 2677 y Fv(3)p Fq(jj)p Fu(j)d Fq(jj)409 2689 y Ft(V)490 2677 y Fq(\024)22 b Fu(\026)p Fv(3)p 669 2632 36 4 v Fu(c=)p Fv(4.)36 b(W)-7 b(e)28 b(ha)n(v)n(e)e(\014nally)h(pro)n(v)n(ed)g(that)g(\010)h(maps)f Fu(D)j Fv(in)n(to)d(itself)h(\(in)h(fact)e(in)n(to)h Fu(B)3181 2692 y Fs(3)p 3214 2657 30 4 v Ft(c)o(\026=)p Fs(4)3355 2677 y Fv(\).)361 2777 y(No)n(w)21 b(w)n(e)g(m)n(ust)h(pro)n (v)n(e)e(that)h(\010)h(is)f(a)h(con)n(traction.)33 b(\010)21 b(is)h(di\013eren)n(tiable)f(and)h(for)f Fu(z)26 b Fv(=)d(\()p Fu(v)s(;)14 b(w)r Fv(\))24 b Fq(2)f Fu(D)r Fv(,)g(\()p Fu(D)r Fv(\010\()p Fu(z)t Fv(\)[)p Fu(h;)14 b(g)s Fv(]\)\()p Fu(t)p Fv(\))236 2877 y(=)23 b(\()p Fu(r)r Fv(\()p Fu(t)p Fv(\))p Fu(;)14 b(s)p Fv(\()p Fu(t)p Fv(\)\),)30 b Fu(r)g Fv(and)e Fu(s)23 b Fv(:)g([)p Fq(\000)p Fv(1)p Fu(;)14 b(T)28 b Fv(+)18 b(1])23 b Fq(!)g Fw(R)k Fv(b)r(eing)h(de\014ned)g(b)n (y)281 3052 y(\177)-45 b Fu(r)r Fv(\()p Fu(t)p Fv(\))24 b(=)f Fu(a)567 3064 y Fs(1)604 3052 y Fv(\()p Fu(t)p Fv(\))p Fu(:h)p Fv(\()p Fu(t)p Fv(\))18 b(+)e Fu(b)998 3064 y Fs(1)1035 3052 y Fv(\()p Fu(t)p Fv(\))p Fu(g)s Fv(\()p Fu(t)p Fv(\))p Fu(;)42 b(r)r Fv(\(0\))24 b(=)e Fu(r)r Fv(\()p Fu(T)12 b Fv(\))24 b(=)e(0)p Fu(;)67 b(L)p Fv(\()p Fu(s)p Fv(\)\()p Fu(t)p Fv(\))24 b(=)f Fu(a)2404 3064 y Fs(2)2441 3052 y Fv(\()p Fu(t)p Fv(\))p Fu(:h)p Fv(\()p Fu(t)p Fv(\))18 b(+)e Fu(b)2835 3064 y Fs(2)2872 3052 y Fv(\()p Fu(t)p Fv(\))p Fu(g)s Fv(\()p Fu(t)p Fv(\))p Fu(;)42 b(s)p Fv(\(0\))23 b(=)f Fu(s)p Fv(\()p Fu(T)12 b Fv(\))23 b(=)f(0)p Fu(;)42 b Fv(\(8.11\))236 3228 y(where)590 3382 y Fu(a)634 3394 y Fs(1)671 3382 y Fv(\()p Fu(t)p Fv(\))24 b(=)f Fq(\000)p Fu(\026@)1036 3394 y Ft('')1127 3382 y Fu(f)9 b Fv(\()p Fu(!)s(t)18 b Fv(+)g Fu(')1449 3351 y Fs(+)1523 3382 y Fv(+)g Fu(v)s Fv(\()p Fu(t)p Fv(\))p Fu(;)c(Q)1846 3394 y Ft(T)1898 3382 y Fv(\()p Fu(t)p Fv(\))19 b(+)f Fu(w)r Fv(\()p Fu(t)p Fv(\))p Fu(;)c(t)20 b Fv(+)e Fu(\022)2460 3351 y Fs(+)2515 3382 y Fv(\))p Fu(;)590 3481 y(b)626 3493 y Fs(1)663 3481 y Fv(\()p Fu(t)p Fv(\))24 b(=)e Fq(\000)p Fu(\026@)1027 3493 y Ft('q)1107 3481 y Fu(f)9 b Fv(\()p Fu(!)s(t)18 b Fv(+)g Fu(')1429 3451 y Fs(+)1503 3481 y Fv(+)g Fu(v)s Fv(\()p Fu(t)p Fv(\))p Fu(;)c(Q)1826 3493 y Ft(T)1879 3481 y Fv(\()p Fu(t)p Fv(\))19 b(+)f Fu(w)r Fv(\()p Fu(t)p Fv(\))p Fu(;)c(t)19 b Fv(+)f Fu(\022)2440 3451 y Fs(+)2496 3481 y Fv(\))p Fu(;)180 b(a)2775 3493 y Fs(2)2812 3481 y Fv(\()p Fu(t)p Fv(\))24 b(=)e Fq(\000)p Fu(b)3118 3493 y Fs(1)3155 3481 y Fv(\()p Fu(t)p Fv(\))p Fu(;)590 3581 y(b)626 3593 y Fs(2)663 3581 y Fv(\()p Fu(t)p Fv(\))i(=)e(cos)o(\()p Fu(Q)1077 3593 y Ft(T)1130 3581 y Fv(\()p Fu(t)p Fv(\))d(+)f Fu(w)r Fv(\()p Fu(t)p Fv(\)\))i Fq(\000)e Fv(cos)13 b Fu(Q)1807 3593 y Ft(T)1859 3581 y Fv(\()p Fu(t)p Fv(\))19 b(+)f Fu(\026@)2149 3593 y Ft(q)r(q)2218 3581 y Fu(f)9 b Fv(\()p Fu(!)s(t)18 b Fv(+)g Fu(')2540 3551 y Fs(+)2614 3581 y Fv(+)g Fu(v)s Fv(\()p Fu(t)p Fv(\))p Fu(;)c(Q)2937 3593 y Ft(T)2989 3581 y Fv(\()p Fu(t)p Fv(\))19 b(+)f Fu(w)r Fv(\()p Fu(t)p Fv(\))p Fu(;)c(t)20 b Fv(+)e Fu(\022)3551 3551 y Fs(+)3606 3581 y Fv(\))p Fu(:)236 3777 y Fv(By)28 b(the)g(same)f(argumen)n(ts)f(as)h(ab)r(o)n(v)n(e)f(\()p Fu(A;)14 b(B)t Fv(\))24 b Fq(2)f Fu(V)1839 3789 y Fs(1)1895 3777 y Fq(\002)18 b Fu(V)47 b Fv(\(where)27 b Fu(V)2393 3789 y Fs(1)2454 3777 y Fv(:=)c Fu(C)2630 3747 y Fs(1)2667 3777 y Fv(\()p Fu(I)7 b(;)14 b Fw(R)2851 3747 y Ft(d)2886 3722 y Fd(2)2922 3777 y Fv(\)\))28 b(de\014ned)g(b)n(y)890 3931 y(\177)868 3952 y Fu(A)p Fv(\()p Fu(t)p Fv(\))c(=)e Fu(a)1179 3964 y Fs(1)1216 3952 y Fv(\()p Fu(t)p Fv(\))p Fu(;)70 b(A)p Fv(\(0\))24 b(=)e Fu(A)p Fv(\()p Fu(T)12 b Fv(\))23 b(=)g(0)p Fu(;)2244 3931 y Fv(\177)2224 3952 y Fu(B)t Fv(\()p Fu(t)p Fv(\))h(=)f Fu(b)2533 3964 y Fs(1)2570 3952 y Fv(\()p Fu(t)p Fv(\))p Fu(;)69 b(B)t Fv(\(0\))24 b(=)e Fu(B)t Fv(\()p Fu(T)12 b Fv(\))23 b(=)g(0)236 4128 y(satisfy)k Fq(jj)p Fu(A)p Fq(jj)649 4140 y Ft(V)688 4148 y Fd(1)744 4128 y Fv(+)18 b Fq(jj)p Fu(B)t Fq(jj)986 4140 y Ft(V)1067 4128 y Fq(\024)23 b Fu(c)1191 4140 y Fs(11)p 1261 4082 36 4 v 1261 4128 a Fu(c\026)28 b Fv(\()p Fq(jj)g(jj)1527 4140 y Ft(V)1566 4148 y Fd(1)1630 4128 y Fv(b)r(eing)g(de\014ned)g(in)g(the)g(same)f(w)n(a)n(y)f(as)h Fq(jj)h(jj)2978 4140 y Ft(V)3036 4128 y Fv(\).)361 4227 y(Using)f(an)g(in)n(tegration)g(b)n(y)g(parts,)g(w)n(e)g(can)g(deriv)n (e)g(from)g(\(8)p Fu(:)p Fv(11\))g(and)h(the)g(b)r(ound)g(on)f Fq(jj)p Fu(A)p Fq(jj)3268 4239 y Ft(V)3307 4247 y Fd(1)3363 4227 y Fv(+)18 b Fq(jj)p Fu(B)t Fq(jj)3605 4239 y Ft(V)3690 4227 y Fv(that)932 4456 y Fq(j)13 b Fv(_)-36 b Fu(r)s Fv(\()p Fu(t)p Fv(\))p Fq(j)23 b(\024)g Fu(c)1259 4468 y Fs(12)p 1329 4410 V 1329 4456 a Fu(c)1376 4400 y(\026)p 1375 4437 52 4 v 1375 4513 a(\014)1436 4364 y Fl(h)q(\020)1535 4400 y Fv(1)18 b(+)g Fu(c)1714 4412 y Fs(1)1751 4400 y Fu(\026T)1862 4370 y Fs(2)p 1535 4437 364 4 v 1673 4513 a Fu(\014)1724 4489 y Fs(2)1909 4456 y Fq(jj)p Fu(h)p Fq(jj)2049 4468 y Ft(V)2125 4456 y Fv(+)g Fq(jj)p Fu(g)s Fq(jj)2343 4468 y Ft(W)2418 4364 y Fl(\021)2486 4456 y Fv(+)g Fu(T)2630 4364 y Fl(\020)2689 4400 y Fq(jj)p Fu(h)p Fq(jj)2829 4412 y Ft(V)p 2689 4437 198 4 v 2762 4513 a Fu(\014)2915 4456 y Fv(+)g Fq(jj)p Fu(g)s Fq(jj)3133 4468 y Ft(W)3208 4364 y Fl(\021i)3297 4456 y Fu(:)484 b Fv(\(8.12\))236 4676 y(Therefore,)27 b(for)g Fu(C)822 4688 y Fs(0)887 4676 y Fv(small)g(enough,)g Fq(j)p Fu(\014)18 b Fv(_)-37 b Fu(r)s Fv(\()p Fu(t)p Fv(\))p Fq(j)24 b(\024)f Fv(1)p Fu(=)p Fv(8)14 b(max)n Fq(fjj)p Fu(h)p Fq(jj)2231 4688 y Ft(V)2288 4676 y Fu(;)g Fq(jj)p Fu(g)s Fq(jj)2460 4688 y Ft(W)2535 4676 y Fq(g)p Fv(.)37 b(W)-7 b(e)28 b(deriv)n(e)e(also)h(from)g(\(8)p Fu(:)p Fv(12\))g(that)1121 4904 y Fq(j)p Fu(r)r Fv(\()p Fu(t)p Fv(\))p Fq(j)e(\024)e Fu(c)1449 4916 y Fs(13)p 1519 4859 36 4 v 1519 4904 a Fu(c)1555 4812 y Fl(h)1604 4848 y Fu(\026T)p 1604 4885 111 4 v 1615 4961 a(\014)1666 4937 y Fs(3)1742 4904 y Fv(+)1835 4848 y Fu(c)1871 4860 y Fs(1)1909 4848 y Fu(\026)1959 4818 y Fs(2)1996 4848 y Fu(T)2057 4818 y Fs(3)p 1835 4885 258 4 v 1920 4961 a Fu(\014)1971 4937 y Fs(3)2122 4904 y Fv(+)2215 4848 y Fu(\026T)2326 4818 y Fs(2)p 2215 4885 148 4 v 2244 4961 a Fu(\014)2295 4937 y Fs(2)2372 4812 y Fl(i)2425 4904 y Fv(max)o Fq(fjj)p Fu(h)p Fq(jj)2761 4916 y Ft(V)2819 4904 y Fu(;)14 b Fq(jj)p Fu(g)s Fq(jj)2991 4916 y Ft(W)3066 4904 y Fq(g)p Fu(;)236 5129 y Fv(whic)n(h)35 b(yields)g Fu(\014)773 5099 y Fs(2)811 5129 y Fv(\(1)23 b(+)g Fu(c)1032 5141 y Fs(1)1070 5129 y Fu(\026T)1181 5099 y Fs(2)1217 5129 y Fv(\))1249 5099 y Fk(\000)p Fs(1)1338 5129 y Fq(j)p Fu(r)r Fv(\()p Fu(t)p Fv(\))p Fq(j)38 b(\024)d Fu(c)1691 5141 y Fs(14)p 1761 5083 36 4 v 1761 5129 a Fu(c)p Fv(\()p Fu(\026T)9 b(=\014)28 b Fv(+)23 b(\(1)p Fu(=c)2294 5141 y Fs(1)2330 5129 y Fv(\)\))14 b(max)p Fq(fjj)p Fu(h)p Fq(jj)2745 5141 y Ft(V)2802 5129 y Fu(;)g Fq(jj)p Fu(g)s Fq(jj)2974 5141 y Ft(W)3050 5129 y Fq(g)35 b(\024)g Fv(max)p Fq(fjj)p Fu(h)p Fq(jj)3564 5141 y Ft(V)3621 5129 y Fu(;)14 b Fq(jj)p Fu(g)s Fq(jj)3793 5141 y Ft(W)3868 5129 y Fq(g)p Fu(=)p Fv(8,)236 5229 y(pro)n(vided)27 b Fu(C)637 5241 y Fs(0)702 5229 y Fv(is)h(small)f(enough)g(and)g Fu(c)1486 5241 y Fs(1)1523 5229 y Fu(=)p 1565 5183 V(c)g Fv(is)h(large)e(enough.)36 b(Finally)28 b Fq(jj)p Fu(r)r Fq(jj)2647 5241 y Ft(V)2729 5229 y Fq(\024)22 b Fv(max)p Fq(fjj)p Fu(h)p Fq(jj)3153 5241 y Ft(V)3210 5229 y Fu(;)14 b Fq(jj)p Fu(g)s Fq(jj)3382 5241 y Ft(W)3457 5229 y Fq(g)p Fu(=)p Fv(4.)361 5328 y(Using)27 b(the)h(prop)r(erties)f(of)g Fu(L)h Fv(and)f(the)h(fact)g(that)800 5504 y Fq(j)p Fu(a)867 5516 y Fs(2)904 5504 y Fv(\()p Fu(t)p Fv(\))p Fu(:h)p Fv(\()p Fu(t)p Fv(\))19 b(+)f Fu(b)1301 5516 y Fs(2)1338 5504 y Fv(\()p Fu(t)p Fv(\))p Fu(g)s Fv(\()p Fu(t)p Fv(\))p Fq(j)24 b(\024)f Fu(c)1740 5516 y Fs(15)1810 5504 y Fu(\026)p Fv(\(1)18 b(+)g Fu(c)2071 5516 y Fs(1)2108 5504 y Fu(\026T)2219 5470 y Fs(2)2256 5504 y Fv(\))p Fu(=\014)2381 5470 y Fs(2)2418 5504 y Fq(jj)p Fu(h)p Fq(jj)2558 5516 y Ft(V)2634 5504 y Fv(+)g Fu(c)2753 5516 y Fs(15)2824 5504 y Fv(\()p Fq(j)p Fu(w)r Fv(\()p Fu(t)p Fv(\))p Fq(j)i Fv(+)e Fu(\026)p Fv(\))p Fq(jj)p Fu(g)s Fq(jj)3377 5516 y Ft(W)2084 6059 y Fv(31)p eop %%Page: 32 32 32 31 bop 236 272 a Fv(w)n(e)36 b(easily)f(deriv)n(e)g Fq(jj)p Fu(s)p Fq(jj)988 284 y Ft(W)1101 272 y Fq(\024)i Fv(max)p Fq(fjj)p Fu(h)p Fq(jj)1540 284 y Ft(V)1597 272 y Fu(;)14 b Fq(jj)p Fu(g)s Fq(jj)1769 284 y Ft(W)1844 272 y Fq(g)p Fu(=)p Fv(4)35 b(\(again)g(pro)n(vided)g(that)i Fu(C)2865 284 y Fs(0)2902 272 y Fv(,)i(more)c(precisely)g Fu(C)3587 284 y Fs(0)3625 272 y Fu(c)3661 284 y Fs(1)3734 272 y Fv(is)h(small)236 372 y(enough\).)f(W)-7 b(e)24 b(ha)n(v)n(e)e(pro)n(v)n(ed)f(that)j(for)e(a)h(go)r(o)r(d)f(c)n(hoice)h (of)p 2070 326 36 4 v 23 w Fu(c;)14 b(c)2179 384 y Fs(1)2216 372 y Fu(;)g(C)2312 384 y Fs(0)2349 372 y Fv(,)24 b Fq(jj)p Fu(D)r Fv(\010\()p Fu(z)t Fv(\)[)p Fu(h;)14 b(g)s Fv(])p Fq(jj)23 b(\024)g(jj)p Fv(\()p Fu(h;)14 b(g)s Fv(\))p Fq(jj)p Fu(=)p Fv(2)22 b(for)g Fu(z)27 b Fq(2)c Fu(D)r Fv(.)36 b(Hence)236 471 y(\010)d(is)f(a)g(con)n(traction.)49 b(As)32 b(a)g(result,)i(it)e(has)g(a)g(unique)g(\014xed)h(p)r(oin)n(t)f Fu(z)2489 483 y Ft(\025)2565 471 y Fv(in)g Fu(D)j Fv(\(whic)n(h)d(in)h (fact)f(b)r(elongs)g(to)g Fu(B)3786 486 y Fs(3)p 3819 451 30 4 v Ft(c\026=)p Fs(4)3960 471 y Fv(\).)236 571 y(This)c(pro)n(v)n(es)d(existence.)361 671 y(No)n(w)g(there)h(remains)g (to)g(pro)n(v)n(e)e(that)j Fu(')1623 683 y Ft(\026;\025)1727 671 y Fv(\()p Fu(t)p Fv(\))p Fu(;)14 b(q)1895 683 y Ft(\026;\025)1999 671 y Fv(\()p Fu(t)p Fv(\))27 b(are)e Fu(C)2322 640 y Fs(1)2386 671 y Fv(functions)h(of)g(\()p Fu(\025;)14 b(t)p Fv(\).)38 b(Let)26 b(\()p Fu(\022)3295 635 y Fs(+)3293 693 y(0)3351 671 y Fu(;)14 b(\022)3429 635 y Fk(\000)3427 693 y Fs(0)3485 671 y Fv(\))26 b(b)r(e)h(\014xed)f(with)236 770 y Fu(T)285 782 y Fs(0)345 770 y Fv(:=)e Fu(\022)498 735 y Fk(\000)496 792 y Fs(0)572 770 y Fq(\000)19 b Fu(\022)697 735 y Fs(+)695 792 y(0)780 770 y Fv(and)28 b(let)g(\003)23 b(=)g Fq(f)p Fu(\025)28 b Fq(j)g(j)p Fu(\022)1464 740 y Fs(+)1538 770 y Fq(\000)18 b Fu(\022)1662 735 y Fs(+)1660 792 y(0)1717 770 y Fq(j)24 b(\024)f Fv(1)p Fu(=)p Fv(4)p Fu(;)68 b Fq(j)p Fu(\022)2133 740 y Fk(\000)2208 770 y Fq(\000)18 b Fu(\022)2332 735 y Fk(\000)2330 792 y Fs(0)2389 770 y Fq(j)23 b(\024)g Fv(1)p Fu(=)p Fv(4)p Fq(g)p Fv(.)36 b(F)-7 b(or)27 b Fu(\025)d Fq(2)g Fv(\003)j Fu(I)3171 782 y Fs(0)3232 770 y Fv(:=)d([)p Fq(\000)p Fv(1)p Fu(=)p Fv(2)p Fu(;)14 b(T)3644 782 y Fs(0)3697 770 y Fv(+)k(1)p Fu(=)p Fv(2])k Fq(\032)236 870 y Fv([)p Fq(\000)p Fv(1)p Fu(;)14 b(\022)444 840 y Fk(\000)518 870 y Fq(\000)k Fu(\022)642 840 y Fs(+)716 870 y Fv(+)g(1],)27 b(hence)h(the)g(restrictions)e Fu(v)1761 840 y Fs(0)1758 893 y Ft(\025)1830 870 y Fv(and)h Fu(w)2052 840 y Fs(0)2050 893 y Ft(\025)2122 870 y Fv(of)g Fu(v)2256 882 y Ft(\025)2328 870 y Fv(and)g Fu(w)2548 882 y Ft(\025)2620 870 y Fv(to)g Fu(I)2757 882 y Fs(0)2823 870 y Fv(are)f(w)n(ell)i(de\014ned.)361 969 y(Let)j Fu(V)561 981 y Fs(0)619 969 y Fq(\002)20 b Fu(W)782 981 y Fs(0)848 969 y Fv(:=)28 b Fu(C)1029 939 y Fs(1)1067 969 y Fv(\()p Fu(I)1135 981 y Fs(0)1173 969 y Fu(;)14 b Fw(R)1282 939 y Ft(n)1327 969 y Fv(\))21 b Fq(\002)f Fu(C)1530 939 y Fs(1)1567 969 y Fv(\()p Fu(I)1635 981 y Fs(0)1673 969 y Fu(;)14 b Fw(R)p Fv(\))31 b(b)r(e)g(endo)n(w)n (ed)f(with)i(the)f(norm)f Fq(jj)h(jj)2983 981 y Fs(0)3051 969 y Fv(as)g(de\014ned)g(in)g(\(8)p Fu(:)p Fv(6\).)46 b(De\014ne)236 1069 y(\011)36 b(:)g(\003)g Fq(!)h Fu(V)658 1081 y Fs(0)719 1069 y Fq(\002)23 b Fu(W)885 1081 y Fs(0)959 1069 y Fv(b)n(y)35 b(\011\()p Fu(\025)p Fv(\))i(=)f Fu(z)1440 1039 y Fs(0)1436 1093 y Ft(\025)1479 1069 y Fv(.)61 b(W)-7 b(e)35 b(shall)h(justify)g(brie\015y)f(that)h(\011)f(is)h(di\013eren)n (tiable)f(and)g(that)h Fq(jj)p Fu(D)r Fv(\011)p Fq(jj)g(\024)236 1169 y Fu(c)272 1181 y Fs(16)342 1169 y Fu(\026)p Fv(.)53 b Fu(z)511 1139 y Fs(0)507 1192 y Ft(\025)583 1169 y Fv(is)32 b(the)h(unique)g(solution)g(in)f Fu(B)p 1578 1147 V 13 x Ft(c\026)1685 1169 y Fv(of)h(\(8.4\))f(\(with)i Fu(T)43 b Fv(=)31 b Fu(\022)2444 1139 y Fk(\000)2522 1169 y Fq(\000)21 b Fu(\022)2649 1139 y Fs(+)2705 1169 y Fv(\),)34 b(whic)n(h)f(is)f(equiv)-5 b(alen)n(t)33 b(to)f(\()p Fu(v)3703 1181 y Ft(\025)3747 1169 y Fu(;)14 b(w)3843 1181 y Ft(\025)3887 1169 y Fv(\))32 b(=)236 1268 y(\010\()p Fu(z)367 1280 y Ft(\025)411 1268 y Fv(;)14 b Fu(\022)489 1238 y Fs(+)544 1268 y Fu(;)g(\022)622 1238 y Fk(\000)678 1268 y Fu(;)g(')769 1238 y Fs(+)824 1268 y Fu(;)g(')915 1238 y Fk(\000)972 1268 y Fu(;)g(\026)p Fv(\),)42 b(where)c(\010)j(:)h Fu(B)p 1636 1246 V 13 x Ft(c)o(\026)1736 1268 y Fq(\002)25 b Fv(\003)h Fq(\002)f Fv(\(0)p Fu(;)14 b(\026)2161 1280 y Fs(2)2198 1268 y Fv(\))42 b Fq(!)g Fu(V)2445 1280 y Fs(0)2508 1268 y Fq(\002)25 b Fu(W)2676 1280 y Fs(0)2753 1268 y Fv(is)38 b(smo)r(oth.)70 b(No)n(w,)41 b(b)n(y)e(the)g(previous)236 1368 y(step,)j Fq(jj)p Fu(D)564 1380 y Ft(z)602 1368 y Fv(\010)p Fq(jj)f(\024)g Fv(1)p Fu(=)p Fv(2)c(ev)n(erywhere,)i(so)f(that)h Fu(I)33 b Fq(\000)25 b Fu(D)2018 1380 y Ft(z)2056 1368 y Fv(\010)39 b(is)f(in)n(v)n(ertible.)69 b(Therefore,)40 b(b)n(y)e(the)h(Implicit)g (F)-7 b(unction)236 1468 y(Theorem,)30 b(\011)f(is)g Fu(C)857 1437 y Fs(1)895 1468 y Fv(.)43 b(This)30 b(pro)n(v)n(es)d (that)j(\()p Fu(\025;)14 b(t)p Fv(\))27 b Fq(7!)g Fu(')1963 1480 y Ft(\026;\025)2067 1468 y Fv(\()p Fu(t)p Fv(\))j(\(resp.)43 b(\()p Fu(\025;)14 b(t)p Fv(\))27 b Fq(7!)f Fu(q)2790 1480 y Ft(\026;\025)2894 1468 y Fv(\()p Fu(t)p Fv(\)\))31 b(and)e(\()p Fu(\025;)14 b(t)p Fv(\))27 b Fq(7!)49 b Fv(_)-45 b Fu(')3584 1480 y Ft(\026;\025)3688 1468 y Fv(\()p Fu(t)p Fv(\))30 b(\(resp.)236 1567 y(\()p Fu(\025;)14 b(t)p Fv(\))24 b Fq(7!)39 b Fv(_)-39 b Fu(q)582 1579 y Ft(\026;\025)686 1567 y Fv(\()p Fu(t)p Fv(\)\))24 b(ha)n(v)n(e)e(con) n(tin)n(uous)g(partial)g(deriv)-5 b(ativ)n(es)22 b(w.r.t.)35 b Fu(\025)24 b Fv(in)f(the)h(set)f Fq(f)p Fv(\()p Fu(\025;)14 b(t)p Fv(\))p Fq(j)9 b(\000)g Fv(1)p Fu(=)p Fv(2)g(+)g Fu(\022)3340 1537 y Fs(+)3418 1567 y Fu(<)22 b(t)h(<)g Fv(1)p Fu(=)p Fv(2)9 b(+)g Fu(\022)3896 1537 y Fk(\000)3951 1567 y Fq(g)p Fv(,)236 1667 y(and)26 b(b)n(y)f(the)h(standard)f(theory) f(of)i(di\013eren)n(tial)f(equations,)h(these)f(partial)g(deriv)-5 b(ativ)n(es)25 b(ha)n(v)n(e)f(con)n(tin)n(uous)h(extensions)236 1766 y(on)j Fq(f)p Fv(\()p Fu(\025;)14 b(t)p Fv(\))p Fq(j)k(\000)g Fv(1)g(+)g Fu(\022)881 1736 y Fs(+)960 1766 y Fu(<)23 b(t)g(<)f Fv(1)c(+)g Fu(\022)1372 1736 y Fk(\000)1429 1766 y Fq(g)p Fv(.)36 b(Finally)-7 b(,)28 b(b)n(y)f(\(2)p Fu(:)p Fv(1\),)41 b(\177)-56 b Fu(')2219 1778 y Ft(\026;\025)2351 1766 y Fv(and)34 b(\177)-49 b Fu(q)2549 1778 y Ft(\026;\025)2681 1766 y Fv(dep)r(end)28 b(con)n(tin)n(uously)e(on)i(\()p Fu(\025;)14 b(t)p Fv(\).)p 3807 1766 48 48 v 236 1912 a Fb(Pr)n(oof)70 b(of)33 b(Theorem)c Fv(4.2.)40 b(In)29 b(order)e(to)h(pro)n(v)n(e)g(Theorem)f(4.2)h(w)n(e)h (need)f(a)h(preliminary)e(lemma.)40 b(Observ)n(e)28 b(that)236 2012 y(\003)294 1981 y Fk(\003)294 2034 y Ft(R)381 2012 y Fv(is)k(a)g(\014nite)h(set)f(whic)n(h)h(is)f(symmetric)g(with)h(resp) r(ect)f(to)g(the)h(origin.)50 b(Hence,)34 b(if)f(it)g(is)f(not)h(empt)n (y)f(there)g(exists)236 2111 y Fu(p)23 b Fq(2)g Fv(\003)437 2081 y Fk(\003)437 2134 y Ft(R)519 2111 y Fv(suc)n(h)28 b(that)f Fu(p)19 b Fq(\001)f Fv(\012)23 b(=)g Fu(\013)p Fv(\(\003)p Fu(;)14 b Fv(\012)p Fu(;)g(R)q Fv(\).)236 2284 y Fw(Lemma)32 b(8.2)40 b Fp(Assume)25 b(that)h Fv(\003)1268 2254 y Fk(\003)1268 2307 y Ft(R)1346 2284 y Fq(6)p Fv(=)c Fq(;)k Fp(and)g(let)g Fu(p)d Fq(2)h Fv(\003)1973 2254 y Fk(\003)1973 2307 y Ft(R)2053 2284 y Fp(b)l(e)i(such)g(that)g Fu(p)10 b Fq(\001)g Fv(\012)23 b(=)g Fu(\013)g Fv(:=)g Fu(\013)p Fv(\(\003)p Fu(;)14 b Fv(\012)p Fu(;)g(R)q Fv(\))p Fp(.)38 b(Assume)25 b(mor)l(e)l(over)236 2384 y(that)30 b Fu(\013)23 b(>)g Fv(0)29 b Fp(and)i(de\014ne)e Fu(E)g Fv(:=)22 b([)p Fu(p)p Fv(])1332 2354 y Fk(?)1388 2384 y Fp(.)39 b(Then)30 b Fv(\003)1726 2396 y Fs(0)1786 2384 y Fv(:=)23 b(\003)18 b Fq(\\)h Fu(E)35 b Fp(is)30 b(a)g(lattic)l(e)g(of)h Fu(E)5 b Fp(.)38 b(In)30 b(addition)h(:)361 2593 y Fv(\()p Fu(i)p Fv(\))537 2537 y Fu(\013)p 494 2574 140 4 v 494 2650 a(\014)t Fq(j)p Fu(p)p Fq(j)666 2593 y(\024)775 2537 y Fv(2)p 763 2574 64 4 v 763 2650 a Fu(R)867 2593 y Fp(,)f(wher)l(e)g Fu(\014)e Fv(=)22 b(inf)7 b Fq(fj)p Fu(q)22 b Fq(\001)c Fv(\012)p Fq(j)p Fv(;)c Fu(q)26 b Fq(2)d Fv(\(\003)1935 2605 y Fs(0)1973 2593 y Fv(\))2005 2563 y Fk(\003)2005 2582 y(p)p 2059 2582 34 3 v 2059 2631 a Fs(3)q Ft(R=)p Fs(2)2214 2593 y Fq(g)p Fp(,)30 b Fv(\(\003)2401 2605 y Fs(0)2438 2593 y Fv(\))2470 2563 y Fk(\003)2532 2593 y Fv(=)22 b Fq(f)p Fu(q)k Fq(2)d Fu(E)35 b Fq(j)30 b(8)p Fu(x)23 b Fq(2)h Fv(\003)3204 2605 y Fs(0)3270 2593 y Fu(q)e Fq(\001)c Fu(x)24 b Fq(2)f Fw(Z)p Fq(g)p Fp(.)236 2784 y(In)30 b(p)l(articular)g Fu(\013)24 b Fq(\024)e Fv(2)p Fu(\014)t Fp(.)361 2955 y Fv(\()p Fu(ii)p Fv(\))29 b Fu(\013)p Fv(\(\003)p Fu(;)14 b Fv(\012)p Fu(;)789 2886 y Fq(p)p 859 2886 42 4 v 859 2955 a Fv(7)o Fu(R)q(=)p Fv(2\))22 b Fq(\024)h Fu(\014)t Fp(.)236 3128 y Fb(Pr)n(oof)p Fv(.)107 b(Since)37 b(\003)g(is)g(a)g(lattice,)i(it)f(is)f(not)g(con)n(tained)f (in)h Fu(E)5 b Fv(.)66 b(Hence)37 b Fu(p)25 b Fq(\001)g Fv(\003)36 b(is)h(a)g(non)g(trivial)f(subgroup)g(of)h Fw(Z)p Fv(,)236 3227 y Fu(p)19 b Fq(\001)h Fv(\003)25 b(=)f Fu(m)p Fw(Z)30 b Fv(for)e(some)g(in)n(teger)g Fu(m)d Fq(\025)g Fv(1,)k(whic)n(h)g(implies)g(that)g Fu(p=m)c Fq(2)g Fv(\003)2591 3197 y Fk(\003)2629 3227 y Fv(.)41 b(But)29 b Fu(p=m)19 b Fq(\001)g Fv(\012)25 b(=)g Fu(\013=m)k Fv(and)g Fq(j)p Fu(p=m)p Fq(j)24 b(\024)h Fu(R)q Fv(,)236 3327 y(hence)i(b)n(y)f(the)g(de\014nition)h(and)f(the)h(p)r(ositivit)n (y)f(of)g Fu(\013)p Fv(,)i Fu(m)23 b Fv(=)f(1.)36 b(As)27 b(a)f(result)g(there)g(exists)p 3099 3281 48 4 v 26 w Fu(x)d Fq(2)h Fv(\003)i(suc)n(h)g(that)g Fu(p)16 b Fq(\001)p 3793 3281 V 16 w Fu(x)24 b Fv(=)e(1.)236 3427 y(Ob)n(viously)28 b(\003)686 3439 y Fs(0)743 3427 y Fv(+)19 b Fw(Z)p 885 3381 V Fu(x)27 b Fq(\022)f Fv(\003.)42 b(On)30 b(the)g(other)f(hand)g (all)g Fu(x)e Fq(2)g Fv(\003)i(can)g(b)r(e)h(written)g(as)e Fu(x)f Fv(=)f(\()p Fu(x)20 b Fq(\001)g Fu(p)p Fv(\))p 3293 3381 V Fu(x)g Fv(+)g Fu(y)s Fv(,)30 b(where)f Fu(y)f Fq(2)f Fv(\003,)236 3526 y Fu(y)22 b Fq(\001)d Fu(p)25 b Fv(=)g(0,)k Fp(i.e.)42 b Fu(y)28 b Fq(2)d Fv(\003)954 3538 y Fs(0)991 3526 y Fv(.)40 b(So)29 b(the)g(rev)n(erse)e(inclusion)i (holds)f(and)h(w)n(e)f(ma)n(y)g(write)h(\003)c(=)f(\003)3071 3538 y Fs(0)3127 3526 y Fv(+)19 b Fw(Z)p 3269 3480 V Fu(x)q Fv(.)41 b(As)28 b(a)h(consequence)236 3626 y(\003)294 3638 y Fs(0)359 3626 y Fv(is)e(a)g(lattice)h(of)g Fu(E)k Fv(and)704 3799 y(\003)762 3764 y Fk(\003)823 3799 y Fv(=)22 b Fq(f)p Fu(r)k Fq(2)d Fw(R)1165 3764 y Ft(l)1218 3799 y Fq(j)28 b Fu(r)21 b Fq(\001)d Fv(\003)1426 3811 y Fs(0)1486 3799 y Fq(\032)23 b Fw(Z)28 b Fv(and)f Fu(r)22 b Fq(\001)p 1921 3753 V 18 w Fu(x)i Fq(2)f Fw(Z)p Fq(g)g Fv(=)g Fq(f)p Fu(q)e Fv(+)d Fu(ap)50 b Fv(:)h Fu(q)26 b Fq(2)d Fv(\003)2873 3764 y Fk(\003)2873 3819 y Fs(0)2939 3799 y Fu(;)69 b(a)23 b Fq(2)g Fw(Z)c Fq(\000)f Fu(q)k Fq(\001)p 3436 3753 V 18 w Fu(x)q Fq(g)p Fu(;)937 3972 y Fv(\003)995 3937 y Fk(\003)995 3992 y Ft(R)1072 3972 y Fv(=)h Fq(f)p Fu(q)e Fv(+)d Fu(ap)50 b Fv(:)h Fu(q)26 b Fq(2)e Fv(\003)1753 3937 y Fk(\003)1753 3992 y Fs(0)1818 3972 y Fu(;)69 b(a)23 b Fq(2)h Fw(Z)19 b Fq(\000)f Fu(q)j Fq(\001)p 2316 3926 V 19 w Fu(x)28 b(;)41 b Fv(0)23 b Fu(<)f Fq(j)p Fu(q)s Fq(j)2693 3937 y Fs(2)2749 3972 y Fv(+)c Fu(a)2876 3937 y Fs(2)2913 3972 y Fq(j)p Fu(p)p Fq(j)3001 3937 y Fs(2)3062 3972 y Fq(\024)k Fu(R)3213 3937 y Fs(2)3250 3972 y Fq(g)p Fu(:)236 4115 y Fv(If)28 b Fu(\014)g Fv(=)22 b(+)p Fq(1)28 b Fv(there)f(is)g(nothing)h(more)f (to)g(pro)n(v)n(e.)35 b(If)28 b Fu(\014)g(<)22 b Fv(+)p Fq(1)p Fv(,)28 b(let)g Fu(q)e Fq(2)d Fv(\(\003)2617 4127 y Fs(0)2654 4115 y Fv(\))2686 4085 y Fk(\003)2686 4104 y(p)p 2741 4104 34 3 v 49 x Fs(3)p Ft(R=)p Fs(2)2923 4115 y Fv(b)r(e)28 b(suc)n(h)g(that)g Fu(q)21 b Fq(\001)e Fv(\012)k(=)f Fu(\014)t Fv(.)38 b(Let)646 4327 y Fu(S)28 b Fv(=)23 b Fq(f)p Fu(a)f Fq(2)h Fw(R)51 b Fv(:)g Fu(q)21 b Fv(+)d Fu(ap)23 b Fq(2)g Fv(\003)1582 4292 y Fk(\003)1582 4347 y Ft(R)1636 4327 y Fq(g)g Fv(=)g Fq(f)p Fu(a)f Fq(2)h Fw(R)51 b Fv(:)g Fu(a)22 b Fq(2)i Fw(Z)19 b Fq(\000)f Fu(q)j Fq(\001)p 2577 4281 48 4 v 19 w Fu(x)28 b(;)41 b Fq(j)p Fu(a)p Fq(j)23 b(\024)g Fv(\()p Fu(R)3013 4292 y Fs(2)3069 4327 y Fq(\000)18 b(j)p Fu(q)s Fq(j)3238 4292 y Fs(2)3275 4327 y Fv(\))3307 4292 y Fs(1)p Ft(=)p Fs(2)3412 4327 y Fu(=)p Fq(j)p Fu(p)p Fq(jg)p Fu(:)236 4499 y Fv(Since)27 b Fq(j)p Fu(q)s Fq(j)538 4469 y Fs(2)598 4499 y Fq(\024)c Fv(3)p Fu(R)792 4469 y Fs(2)828 4499 y Fu(=)p Fv(4,)j Fu(S)i Fq(\023)22 b Fu(S)1183 4469 y Fk(0)1229 4499 y Fv(:=)h(\()p Fw(Z)16 b Fq(\000)g Fu(q)i Fq(\001)p 1621 4454 V 16 w Fu(x)q Fv(\))e Fq(\\)g Fv([)p Fq(\000)p Fu(R)q(=)p Fv(2)p Fq(j)p Fu(p)p Fq(j)p Fu(;)e(R)q(=)p Fv(2)p Fq(j)p Fu(p)p Fq(j)p Fv(].)33 b(Hence)26 b(b)n(y)g(the)h (de\014nition)f(of)h Fu(\013)p Fv(,)g(for)e(all)h Fu(a)d Fq(2)h Fu(S)3970 4469 y Fk(0)3993 4499 y Fv(,)236 4599 y Fq(j)p Fv(\()p Fu(q)e Fv(+)c Fu(ap)p Fv(\))g Fq(\001)h Fv(\012)p Fq(j)k Fv(=)g Fq(j)p Fu(\014)g Fv(+)18 b Fu(a\013)p Fq(j)23 b(\025)g Fu(\013)p Fv(,)28 b Fp(i.e.)38 b Fu(\014)t(=\013)33 b(=)-51 b Fq(2)23 b Fv(\()p Fq(\000)p Fv(1)18 b Fq(\000)g Fu(a;)c Fv(1)k Fq(\000)g Fu(a)p Fv(\).)361 4699 y(As)32 b Fq(j)p Fu(p)p Fq(j)d(\024)h Fu(R)q Fv(,)j(the)f(in)n(terv)-5 b(al)31 b([)p Fq(\000)p Fu(R)q(=)p Fv(2)p Fq(j)p Fu(p)p Fq(j)p Fu(;)14 b(R)q(=)p Fv(2)p Fq(j)p Fu(p)p Fq(j)p Fv(])28 b(has)k(length)f Fq(\025)f Fv(1)h(and)h(m)n(ust)g(in)n(tersect) f(\()p Fw(Z)22 b Fq(\000)e Fu(q)25 b Fq(\001)p 3515 4653 V 21 w Fu(x)p Fv(\).)50 b(Therefore)236 4798 y Fu(S)292 4768 y Fk(0)338 4798 y Fq(6)p Fv(=)23 b Fq(;)p Fv(,)j(more)g(precisely) g Fu(S)1119 4768 y Fk(0)1165 4798 y Fv(=)d Fq(f)p Fu(u;)14 b(u)h Fv(+)h(1)p Fu(;)e(:)g(:)g(:)f(;)h(u)i Fv(+)g Fu(K)6 b Fq(g)p Fv(,)26 b(for)g(some)g(in)n(teger)g Fu(K)i Fq(\025)23 b Fv(0,)j(where)g Fu(u)d Fv(=)g(inf)d Fu(S)3516 4768 y Fk(0)3539 4798 y Fv(.)37 b(As)27 b(a)f(result,)1094 5053 y Fu(\014)t(=\013)32 b(=)-51 b Fq(2)1372 4949 y Ft(K)1356 4974 y Fl([)1342 5152 y Ft(k)q Fs(=0)1462 5053 y Fv(\()p Fq(\000)p Fv(1)18 b Fq(\000)g Fu(u)g Fq(\000)g Fu(k)s(;)c Fv(1)k Fq(\000)g Fu(u)g Fq(\000)g Fu(k)s Fv(\))23 b(=)g(\()p Fq(\000)p Fv(1)18 b Fq(\000)g Fu(u)g Fq(\000)g Fu(K)q(;)c Fv(1)j Fq(\000)h Fu(u)p Fv(\))p Fu(:)236 5305 y Fv(No)n(w)23 b Fu(S)477 5275 y Fk(0)509 5305 y Fq(\\)9 b Fv([)p Fq(\000)p Fv(1)p Fu(=)p Fv(2)p Fu(;)14 b Fv(1)p Fu(=)p Fv(2])21 b Fq(6)p Fv(=)i Fq(;)p Fv(,)g(hence)g Fu(u)9 b Fv(+)g Fu(K)29 b Fq(\025)23 b(\000)p Fv(1)p Fu(=)p Fv(2)e(and)i Fq(\000)p Fv(1)9 b Fq(\000)g Fu(u)g Fq(\000)g Fu(K)28 b(<)23 b Fv(0.)35 b(As)23 b(a)f(consequence)h Fu(\014)t(=\013)g Fq(\025)g Fv(1)9 b Fq(\000)g Fu(u)p Fv(.)35 b(Since)236 5404 y([)p Fq(\000)p Fu(R)q(=)p Fv(2)p Fq(j)p Fu(p)p Fq(j)p Fu(;)14 b Fq(\000)p Fu(R)q(=)p Fv(2)p Fq(j)p Fu(p)p Fq(j)d Fv(+)g(1])18 b Fq(\022)23 b Fv([)p Fq(\000)p Fu(R)q(=)p Fv(2)p Fq(j)p Fu(p)p Fq(j)p Fu(;)14 b(R)q(=)p Fv(2)p Fq(j)p Fu(p)p Fq(j)p Fv(])21 b(in)n(tersects)i Fw(Z)11 b Fq(\000)g Fu(q)j Fq(\001)p 2391 5359 V 11 w Fu(x)o Fv(,)25 b Fu(u)d Fq(\024)h(\000)p Fu(R)q(=)p Fv(2)p Fq(j)p Fu(p)p Fq(j)11 b Fv(+)g(1.)33 b(Therefore)22 b Fu(\014)t(=\013)h Fq(\025)g Fu(R)q(=)p Fv(2)p Fq(j)p Fu(p)p Fq(j)p Fv(,)236 5504 y(whic)n(h)28 b(is)f(\()p Fu(i)p Fv(\).)37 b(In)28 b(particular,)e(since)i Fq(j)p Fu(p)p Fq(j)23 b(\024)f Fu(R)q Fv(,)28 b Fu(\013)23 b Fq(\024)g Fv(2)p Fu(\014)t Fv(.)2084 6059 y(32)p eop %%Page: 33 33 33 32 bop 361 272 a Fv(Finally)25 b(there)g(exists)f Fu(a)f Fq(2)h Fv([)p Fq(\000)p Fv(1)p Fu(;)14 b Fv(0\))f Fq(\\)g Fv(\()p Fw(Z)g Fq(\000)g Fu(q)k Fq(\001)p 1816 226 48 4 v 13 w Fu(x)q Fv(\);)26 b Fu(q)17 b Fv(+)c Fu(ap)23 b Fq(2)g Fv(\003)2322 242 y Fk(\003)2360 272 y Fv(,)j(and)e Fq(j)p Fu(q)17 b Fv(+)c Fu(ap)p Fq(j)2831 242 y Fs(2)2891 272 y Fv(=)23 b Fq(j)p Fu(q)s Fq(j)3065 242 y Fs(2)3115 272 y Fv(+)13 b Fu(a)3237 242 y Fs(2)3275 272 y Fq(j)p Fu(p)p Fq(j)3363 242 y Fs(2)3423 272 y Fq(\024)22 b Fv(3)p Fu(R)3616 242 y Fs(2)3653 272 y Fu(=)p Fv(4)13 b(+)g Fu(R)3892 242 y Fs(2)3951 272 y Fv(=)236 372 y(7)p Fu(R)342 341 y Fs(2)379 372 y Fu(=)p Fv(4.)36 b(Hence)27 b Fu(q)21 b Fv(+)c Fu(ap)23 b Fq(2)g Fv(\003)1153 341 y Fk(\003)1153 360 y(p)p 1207 360 34 3 v 1207 409 a Fs(7)p Ft(R=)p Fs(2)1362 372 y Fv(.)37 b(W)-7 b(e)27 b(ha)n(v)n(e)f Fq(j)p Fv(\()p Fu(q)21 b Fv(+)d Fu(ap)p Fv(\))f Fq(\001)h Fv(\012)p Fq(j)23 b Fv(=)g Fq(j)p Fu(\014)f Fv(+)17 b Fu(a\013)p Fq(j)23 b(\024)g Fu(\014)t Fv(,)28 b(b)r(ecause)f Fq(\000)p Fv(1)22 b Fq(\024)g Fu(a)h Fq(\024)g Fv(0)k(and)g Fu(\013)c Fq(\024)g Fv(2)p Fu(\014)t Fv(.)236 494 y(This)28 b(pro)n(v)n(es)d(\()p Fu(ii)p Fv(\).)p 864 494 48 48 v 361 629 a(No)n(w)31 b(w)n(e)g(turn)h(to)g(the)g(pro)r(of)f(of)h(Theorem)f(4.2)g(W)-7 b(e)32 b(\014rst)g(pro)n(v)n(e)e(that)i(the)g(statemen)n(t)g(is)g(true) f(for)h Fu(l)f Fv(=)f(1,)i(with)236 728 y Fu(a)280 740 y Fs(1)341 728 y Fv(=)24 b(1)p Fu(=)p Fv(2.)37 b(Here)28 b(\003)c(=)f Fu(\025)1031 740 y Fs(0)1069 728 y Fw(Z)29 b Fv(for)e(some)h Fu(\025)1540 740 y Fs(0)1602 728 y Fu(>)23 b Fv(0,)28 b(and)g(\003)2003 698 y Fk(\003)2065 728 y Fv(=)c(\()p Fu(\025)2234 740 y Fs(0)2272 728 y Fv(\))2304 698 y Fk(\000)p Fs(1)2393 728 y Fw(Z)p Fv(.)39 b(W)-7 b(e)29 b(can)f(assume)f(without)i(loss)e(of)h(generalit)n(y)236 828 y(that)g(\012)23 b Fu(>)g Fv(0.)36 b(If)28 b Fu(\025)819 840 y Fs(0)880 828 y Fu(<)23 b Fv(2)p Fu(\016)s Fv(,)k(then)h(for)f (all)g Fu(x)d Fq(2)f Fw(R)p Fv(,)28 b Fu(d)p Fv(\()p Fu(x;)14 b Fv(\003\))24 b Fu(<)e(\016)s Fv(.)37 b(Hence)28 b Fu(T)12 b Fv(\(\003)p Fu(;)i Fv(\012)p Fu(;)g(\016)s Fv(\))22 b(=)h(0)361 928 y(If)40 b Fu(\025)504 940 y Fs(0)586 928 y Fq(\025)j Fv(2)p Fu(\016)s Fv(,)g(then)d(it)g(is)g(easy) f(to)h(see)g(that)g Fu(T)12 b Fv(\(\003)p Fu(;)i Fv(\012)p Fu(;)g(\016)s Fv(\))43 b(=)g(\()p Fu(\025)2470 940 y Fs(0)2535 928 y Fq(\000)26 b Fv(2)p Fu(\016)s Fv(\))p Fu(=)p Fv(\012)43 b Fq(\024)g Fu(\025)3041 940 y Fs(0)3079 928 y Fu(=)p Fv(\012.)73 b(On)40 b(the)g(other)g(hand,)236 1027 y(1)p Fu(=\025)368 1039 y Fs(0)428 1027 y Fq(2)23 b Fv(\003)564 997 y Fk(\003)564 1054 y Fs(1)p Ft(=)p Fs(2)p Ft(\016)728 1027 y Fv(and)28 b Fu(\013)p Fv(\(\003)p Fu(;)14 b Fv(\012)p Fu(;)g Fv(1)p Fu(=)p Fv(\(2)p Fu(\016)s Fv(\)\))22 b(=)h(\012)p Fu(=\025)1689 1039 y Fs(0)1726 1027 y Fv(.)37 b(The)28 b(result)f(follo)n(ws.)361 1186 y(No)n(w)32 b(w)n(e)h(assume)f(that)i(the)f(statemen)n(t)g(holds)g (true)f(up)i(to)f(dimension)g Fu(l)23 b Fq(\000)f Fv(1)32 b(\()p Fu(l)i Fq(\025)e Fv(2\).)53 b(W)-7 b(e)33 b(shall)g(pro)n(v)n(e) e(it)j(in)236 1286 y(dimension)28 b Fu(l)r Fv(.)361 1386 y(Fix)g Fu(R)23 b(>)g Fv(0)k(and)g(de\014ne)h Fu(\016)1191 1398 y Ft(R)1269 1386 y Fv(=)22 b(\(4)p Fu(a)1474 1356 y Fs(2)1474 1409 y Ft(l)p Fk(\000)p Fs(1)1585 1386 y Fu(=)p Fv(3)17 b(+)h(4\))1843 1356 y Fs(1)p Ft(=)p Fs(2)1947 1386 y Fu(=R)q Fv(.)36 b(W)-7 b(e)28 b(claim)g(that:)236 1533 y(\()p Fu(a)p Fv(\))i Fp(If)h Fv(\003)520 1502 y Fk(\003)520 1555 y Ft(R)597 1533 y Fv(=)22 b Fq(;)30 b Fp(then)f Fu(T)12 b Fv(\(\003)p Fu(;)i Fv(\012)p Fu(;)g(\016)1262 1545 y Ft(R)1315 1533 y Fv(\))24 b(=)e(0)p Fp(.)236 1679 y Fv(\()p Fu(b)p Fv(\))30 b Fp(If)g Fv(\003)511 1649 y Fk(\003)511 1702 y Ft(R)588 1679 y Fq(6)p Fv(=)23 b Fq(;)p Fp(,)30 b(let)f Fu(p)23 b Fq(2)h Fv(\003)1091 1649 y Fk(\003)1091 1702 y Ft(R)1175 1679 y Fp(b)l(e)29 b(such)h(that)g Fu(p)18 b Fq(\001)h Fv(\012)k(=)f Fu(\013)i Fv(:=)e Fu(\013)p Fv(\(\003)p Fu(;)14 b Fv(\012)p Fu(;)g(R)q Fv(\))p Fp(,)31 b(and)f(de\014ne)g Fu(\014)k Fp(as)c(in)g(lemma)g(8.2.) 40 b(Then)1577 1826 y Fu(T)12 b Fv(\(\003)p Fu(;)i Fv(\012)p Fu(;)g(\016)1899 1838 y Ft(R)1952 1826 y Fv(\))23 b Fq(\024)g Fv(max)o Fq(f)p Fu(\013)2344 1791 y Fk(\000)p Fs(1)2433 1826 y Fu(;)14 b(\014)2521 1791 y Fk(\000)p Fs(1)2611 1826 y Fq(g)p Fu(:)361 2071 y Fv(P)n(ostp)r(oning)31 b(the)j(pro)r(of)e(of)h(\()p Fu(a)p Fv(\))h(and)e(\()p Fu(b)p Fv(\),)j(w)n(e)e(sho)n(w)f(ho)n(w)g(to)h(de\014ne)g Fu(a)2651 2083 y Ft(l)2677 2071 y Fv(.)53 b(In)33 b(the)g(case)g(\()p Fu(b)p Fv(\),)h(b)n(y)f(lemma)g(8.2)f(\(ii\),)236 2171 y Fu(T)12 b Fv(\(\003)p Fu(;)i Fv(\012)p Fu(;)g(\016)558 2183 y Ft(R)611 2171 y Fv(\))24 b Fq(\024)e Fu(\013)p Fv(\(\003)p Fu(;)14 b Fv(\012)p Fu(;)1031 2102 y Fq(p)p 1101 2102 42 4 v 1101 2171 a Fv(7)o Fu(R)q(=)p Fv(2\))1322 2141 y Fk(\000)p Fs(1)1410 2171 y Fv(.)37 b(This)26 b(estimate)g(ob)n (viously)e(holds)i(in)g(the)h(case)e(\()p Fu(a)p Fv(\))h(to)r(o.)36 b(Hence)27 b(for)e(all)h Fu(R)e(>)e Fv(0,)1225 2326 y Fu(T)12 b Fv(\(\003)p Fu(;)i Fv(\012)p Fu(;)g Fv(\(4)p Fu(a)1628 2292 y Fs(2)1628 2346 y Ft(l)p Fk(\000)p Fs(1)1737 2326 y Fu(=)p Fv(3)j(+)i(4\))1996 2292 y Fs(1)p Ft(=)p Fs(2)2100 2326 y Fu(=R)q Fv(\))j Fq(\024)h Fu(\013)p Fv(\(\003)p Fu(;)14 b Fv(\012)p Fu(;)2625 2253 y Fq(p)p 2694 2253 V 73 x Fv(7)p Fu(R)q(=)p Fv(2\))2916 2292 y Fk(\000)p Fs(1)3004 2326 y Fu(:)236 2483 y Fv(As)28 b(a)f(consequence,) g(the)h(statemen)n(t)f(of)h(Theorem)f(4.2)f(holds)i(with)g Fu(a)2478 2495 y Ft(l)2526 2483 y Fv(=)23 b(\()2646 2415 y Fq(p)p 2715 2415 V 68 x Fv(7\(4)p Fu(a)2875 2453 y Fs(2)2875 2507 y Ft(l)p Fk(\000)p Fs(1)2985 2483 y Fu(=)p Fv(3)17 b(+)h(4\))3243 2453 y Fs(1)p Ft(=)p Fs(2)3348 2483 y Fu(=)p Fv(2\).)361 2630 y(There)27 b(remains)h(to)g(pro)n(v)n(e) e(\()p Fu(a)p Fv(\))j(and)f(\()p Fu(b)p Fv(\).)38 b(First)28 b(assume)g(that)g(\003)2427 2600 y Fk(\003)2427 2653 y Ft(R)2505 2630 y Fv(=)c Fq(;)p Fv(.)38 b(Let)28 b Fu(p)c Fq(2)g Fv(\003)3049 2600 y Fk(\003)3106 2630 y Fq(n)18 b(f)p Fv(0)p Fq(g)27 b Fv(b)r(e)h(suc)n(h)g(that)h(for)e(all)236 2730 y Fu(p)278 2700 y Fk(0)324 2730 y Fq(2)d Fv(\003)461 2700 y Fk(\003)517 2730 y Fq(n)18 b(f)p Fv(0)p Fq(g)p Fv(,)26 b Fq(j)p Fu(p)p Fq(j)d(\024)g(j)p Fu(p)1016 2700 y Fk(0)1039 2730 y Fq(j)p Fv(.)37 b(Then)28 b Fq(j)p Fu(p)p Fq(j)23 b Fu(>)f(R)q Fv(.)37 b(Let)28 b Fu(E)5 b Fv(,)28 b(\003)1985 2742 y Fs(0)2049 2730 y Fv(b)r(e)g(de\014ned)g (from)f Fu(p)h Fv(as)f(in)h(lemma)f(8.2.)361 2830 y(Arguing)36 b(b)n(y)g(con)n(tradiction,)i(w)n(e)f(assume)f(that)h(\(\003)2062 2842 y Fs(0)2100 2830 y Fv(\))2132 2799 y Fk(\003)2132 2818 y(p)p 2186 2818 34 3 v 2186 2867 a Fs(3)p Ft(R=)p Fs(2)2380 2830 y Fq(6)p Fv(=)h Fq(;)p Fv(.)64 b(By)37 b(the)g(same)f(argumen)n(ts)g(as)g(previously)236 2957 y(there)c(exist)h Fu(q)h Fq(2)d Fv(\(\003)902 2969 y Fs(0)940 2957 y Fv(\))972 2927 y Fk(\003)972 2946 y(p)p 1026 2946 V 1026 2994 a Fs(3)q Ft(R=)p Fs(2)1214 2957 y Fv(and)h Fu(a)f Fq(2)g Fv([)p Fq(\000)p Fv(1)p Fu(=)p Fv(2)p Fu(;)14 b Fv(1)p Fu(=)p Fv(2])30 b(suc)n(h)i(that)g Fu(q)25 b Fv(+)c Fu(ap)31 b Fq(2)h Fv(\003)2757 2927 y Fk(\003)2794 2957 y Fv(.)52 b(But)33 b Fq(j)p Fu(q)24 b Fv(+)e Fu(ap)p Fq(j)3319 2927 y Fs(2)3387 2957 y Fv(=)30 b Fq(j)p Fu(q)s Fq(j)3568 2927 y Fs(2)3628 2957 y Fv(+)21 b Fu(a)3758 2927 y Fs(2)3795 2957 y Fq(j)p Fu(p)p Fq(j)3883 2927 y Fs(2)3951 2957 y Fq(\024)236 3085 y Fv(\(3)p Fu(=)p Fv(4\))p Fu(R)490 3055 y Fs(2)538 3085 y Fv(+)11 b Fq(j)p Fu(p)p Fq(j)702 3055 y Fs(2)739 3085 y Fu(=)p Fv(4)22 b Fu(<)g Fq(j)p Fu(p)p Fq(j)1020 3055 y Fs(2)1082 3085 y Fv(and)i(this)g(con)n(tradicts)f(the)h(de\014nition)h(of)f Fu(p)p Fv(.)35 b(Hence)25 b(\(\003)2852 3097 y Fs(0)2889 3085 y Fv(\))2921 3055 y Fk(\003)2921 3074 y(p)p 2976 3074 V 48 x Fs(3)p Ft(R=)p Fs(2)3153 3085 y Fv(=)e Fq(;)h Fv(and)f(b)n(y)h(the)h(iterativ)n(e)236 3220 y(h)n(yp)r(othesis,)i(all) h(p)r(oin)n(t)f(of)h Fu(E)33 b Fv(lies)27 b(at)h(a)f(distance)g(from)g (\003)2080 3232 y Fs(0)2145 3220 y Fv(less)g(than)h(2)p Fu(a)2578 3232 y Ft(l)p Fk(\000)p Fs(1)2688 3220 y Fu(=)2730 3152 y Fq(p)p 2798 3152 42 4 v 2798 3220 a Fv(3)p Fu(R)q Fv(.)361 3320 y(F)-7 b(rom)28 b(the)g(pro)r(of)g(of)g(lemma)g(8.2,)g (there)g(exists)p 1904 3274 48 4 v 27 w Fu(x)d Fq(2)f Fv(\003)k(suc)n(h)g(that)h Fu(p)18 b Fq(\001)p 2612 3274 V 19 w Fu(x)25 b Fv(=)e(1)28 b(and)g(\003)c(=)g(\003)3233 3332 y Fs(0)3288 3320 y Fv(+)19 b Fw(Z)p 3430 3274 V Fu(x)q Fv(.)38 b(Therefore)27 b(for)236 3420 y(all)k Fu(x)d Fq(2)h Fw(R)586 3389 y Ft(l)611 3420 y Fv(,)i(there)g(is)f Fu(x)1014 3389 y Fk(0)1066 3420 y Fq(2)f Fu(x)21 b Fv(+)f(\003)30 b(suc)n(h)h(that)g Fq(j)p Fu(x)1835 3389 y Fk(0)1879 3420 y Fq(\001)21 b Fu(p)p Fq(j)28 b(\024)f Fv(1)p Fu(=)p Fv(2.)45 b(This)31 b(implies)g(that)g Fu(d)p Fv(\()p Fu(x)3085 3389 y Fk(0)3109 3420 y Fu(;)14 b(E)5 b Fv(\))28 b Fq(\024)g Fv(1)p Fu(=)p Fv(\(2)p Fq(j)p Fu(p)p Fq(j)p Fv(\))f Fq(\024)h Fv(1)p Fu(=)p Fv(\(2)p Fu(R)q Fv(\))236 3519 y(and)g(hence)f(that)h Fu(d)p Fv(\()p Fu(x)930 3489 y Fk(0)954 3519 y Fu(;)14 b Fv(\003)1049 3531 y Fs(0)1086 3519 y Fv(\))24 b Fq(\024)e Fv(\(4)p Fu(a)1347 3489 y Fs(2)1347 3543 y Ft(l)p Fk(\000)p Fs(1)1458 3519 y Fu(=)p Fv(3)17 b(+)h(1)p Fu(=)p Fv(4\))1800 3489 y Fs(1)p Ft(=)p Fs(2)1903 3519 y Fu(=R)24 b Fq(\024)e Fu(\016)2156 3531 y Ft(R)2211 3519 y Fv(.)37 b(Hence)27 b(the)h(distance)g(from)f(an)n(y) g(p)r(oin)n(t)h(of)f Fw(R)3720 3489 y Ft(l)3773 3519 y Fv(to)h(\003)f(is)236 3619 y(not)h(greater)e(than)h Fu(\016)896 3631 y Ft(R)951 3619 y Fv(.)37 b(This)27 b(completes)h(the)g(pro)r(of)f(of)g(\()p Fu(a)p Fv(\).)361 3766 y(Next)f(assume)f(that)h(\003)1083 3735 y Fk(\003)1083 3788 y Ft(R)1161 3766 y Fq(6)p Fv(=)c Fq(;)k Fv(and)f(let)i Fu(p)e Fv(b)r(e)i(as)e(in)h(lemma)g(8.2.)35 b(De\014ne)27 b Fu(\013)f Fv(and)g Fu(\014)k Fv(in)c(the)h(same)e(w)n(a)n(y)g(as)g (in)h(lemma)236 3865 y(8.2.)42 b(Let)30 b Fu(x)d Fq(2)g Fw(R)787 3835 y Ft(l)812 3865 y Fv(.)43 b(Again)29 b(\003)d(=)g(\003) 1355 3877 y Fs(0)1412 3865 y Fv(+)19 b Fw(Z)p 1554 3820 V Fu(x)31 b Fv(for)e(some)p 1971 3820 V 29 w Fu(x)e Fq(2)g Fv(\003)i(suc)n(h)g(that)h Fu(p)20 b Fq(\001)p 2690 3820 V 20 w Fu(x)26 b Fv(=)g(1,)k(hence)g(there)f(exists)h Fu(x)3675 3835 y Fk(0)3725 3865 y Fq(2)c Fu(x)21 b Fv(+)e(\003)236 3965 y(suc)n(h)27 b(that)h Fu(p)19 b Fq(\001)f Fu(x)752 3935 y Fk(0)799 3965 y Fq(2)23 b Fv([0)p Fu(;)14 b Fv(1\).)37 b(W)-7 b(e)28 b(ha)n(v)n(e)1549 4132 y Fu(x)1596 4098 y Fk(0)1643 4132 y Fv(=)23 b Fu(y)e Fv(+)1918 4076 y Fu(w)p 1886 4113 126 4 v 1886 4189 a Fq(j)p Fu(p)p Fq(j)1974 4165 y Fs(2)2021 4132 y Fu(p)27 b(;)42 b Fv(\012)23 b(=)g Fu(U)k Fv(+)2539 4076 y Fu(\013)p 2503 4113 V 2503 4189 a Fq(j)p Fu(p)p Fq(j)2591 4165 y Fs(2)2638 4132 y Fu(p;)236 4335 y Fv(with)d Fu(y)s(;)14 b(U)32 b Fq(2)23 b Fu(E)28 b Fv(=)23 b([)p Fu(p)p Fv(])934 4305 y Fk(?)990 4335 y Fv(,)i Fu(w)g Fv(=)e Fu(p)11 b Fq(\001)g Fu(x)1344 4305 y Fk(0)1390 4335 y Fq(2)23 b Fv([0)p Fu(;)14 b Fv(1\).)35 b(W)-7 b(e)24 b(shall)g(assume)f(that)h Fu(\013)f(>)g Fv(0)g(\(if)i Fu(\013)e Fv(=)g(0,)h(there)f(is)h(nothing)g(to)f(pro)n (v)n(e\).)236 4435 y(Let)p 385 4374 30 4 v 28 w Fu(t)g Fv(=)g Fu(w)r(=\013)p Fv(,)28 b(and)f(consider)g(the)h(time)g(in)n (terv)-5 b(al)27 b(de\014ned)h(b)n(y)1116 4581 y Fu(J)k Fv(=)22 b([0)p Fu(;)14 b Fv(1)p Fu(=\014)t Fv(])54 b(if)p 1678 4520 V 35 w Fu(t)23 b(<)g Fv(1)p Fu(=\014)t(;)179 b(J)31 b Fv(=)23 b([)p 2344 4520 V Fu(t)18 b Fq(\000)g Fv(1)p Fu(=\014)t(;)p 2647 4520 V 14 w(t)o Fv(])56 b(if)p 2837 4520 V 34 w Fu(t)23 b Fq(\025)g Fv(1)p Fu(=\014)t(:)236 4727 y(J)33 b Fq(\032)25 b Fv([0)p Fu(;)14 b Fv(max)n Fq(f)p Fv(1)p Fu(=\014)t(;)g Fv(1)p Fu(=\013)p Fq(g)p Fv(],)27 b(and)i(it)g(is)g(enough)f(to)g(pro)n(v)n(e)f(that)i(there)g (exists)f Fu(t)d Fq(2)g Fu(J)37 b Fv(suc)n(h)28 b(that)h Fu(d)p Fv(\()p Fu(x)3406 4697 y Fk(0)3430 4727 y Fu(;)14 b(t)p Fv(\012)19 b(+)g(\003)3718 4739 y Fs(0)3755 4727 y Fv(\))25 b Fq(\024)f Fu(\016)3938 4739 y Ft(R)3993 4727 y Fv(.)236 4827 y(The)34 b(length)g(of)g Fu(J)42 b Fv(is)34 b(not)f(less)h(than)g(1)p Fu(=\014)t Fv(.)55 b(Hence)34 b(b)n(y)f(the)i(iterativ)n(e)e(h)n(yp)r(othesis,)i(there)e (exists)h Fu(t)f Fq(2)h Fu(J)42 b Fv(suc)n(h)34 b(that)236 4926 y Fu(d)p Fv(\()p Fu(y)s(;)14 b(tU)23 b Fv(+)13 b(\003)638 4938 y Fs(0)675 4926 y Fv(\))24 b Fq(\024)e Fv(2)p Fu(a)904 4938 y Ft(l)p Fk(\000)p Fs(1)1014 4926 y Fu(=)p Fv(\()1088 4858 y Fq(p)p 1157 4858 42 4 v 68 x Fv(3)p Fu(R)q Fv(\))j(\(notice)g (that)h(for)f(all)g Fu(q)h Fq(2)d Fv(\003)2209 4896 y Fk(\003)2209 4947 y Fs(0)2247 4926 y Fv(,)j Fu(q)17 b Fq(\001)d Fu(U)32 b Fv(=)22 b Fu(q)17 b Fq(\001)d Fv(\012,)26 b(so)f(that)g(the)h(linear)f(\015o)n(w)f(\()p Fu(tU)9 b Fv(\))26 b(creates)236 5026 y(a)h(2)p Fu(a)391 5038 y Ft(l)p Fk(\000)p Fs(1)501 5026 y Fu(=)p Fv(\()575 4957 y Fq(p)p 644 4957 V 69 x Fv(3)p Fu(R)q Fv(\)-net)g(of)h Fu(E)5 b(=)p Fv(\003)1213 5038 y Fs(0)1277 5026 y Fv(in)28 b(time)g Fu(\014)1614 4996 y Fk(\000)p Fs(1)1704 5026 y Fv(\).)37 b(W)-7 b(e)28 b(ha)n(v)n(e)900 5232 y Fu(d)p Fv(\()p Fu(x)1022 5198 y Fk(0)1047 5232 y Fu(;)14 b(t)p Fv(\012)k(+)g(\003)1333 5244 y Fs(0)1370 5232 y Fv(\))1402 5198 y Fs(2)1462 5232 y Fv(=)1550 5140 y Fl(\020)1610 5176 y Fv(\()p Fu(t)g Fq(\000)p 1773 5115 30 4 v 18 w Fu(t)p Fv(\))p Fu(\013)p 1610 5213 280 4 v 1705 5289 a Fq(j)p Fu(p)p Fq(j)1899 5140 y Fl(\021)1949 5157 y Fs(2)2004 5232 y Fv(+)g Fu(d)p Fv(\()p Fu(y)s(;)c(tU)28 b Fv(+)18 b(\003)2499 5244 y Fs(0)2535 5232 y Fv(\))2567 5198 y Fs(2)2628 5232 y Fq(\024)2716 5140 y Fl(\020)2818 5176 y Fu(\013)p 2775 5213 140 4 v 2775 5289 a(\014)t Fq(j)p Fu(p)p Fq(j)2924 5140 y Fl(\021)2974 5157 y Fs(2)3030 5232 y Fv(+)3123 5169 y(4)p Fu(a)3209 5139 y Fs(2)3209 5193 y Ft(l)p Fk(\000)p Fs(1)p 3123 5213 196 4 v 3149 5289 a Fv(3)p Fu(R)3255 5265 y Fs(2)3329 5232 y Fu(:)236 5438 y Fv(Hence,)28 b(b)n(y)f(lemma)h(8.2)f(\()p Fu(i)p Fv(\))g(,)h Fu(d)p Fv(\()p Fu(x)1316 5408 y Fk(0)1340 5438 y Fu(;)14 b(t)p Fv(\012)19 b(+)f(\003)1627 5450 y Fs(0)1664 5438 y Fv(\))23 b Fq(\024)g Fv(\(4)p Fu(a)1925 5408 y Fs(2)1925 5462 y Ft(l)p Fk(\000)p Fs(1)2035 5438 y Fu(=)p Fv(3)17 b(+)h(4\))2293 5408 y Fs(1)p Ft(=)p Fs(2)2397 5438 y Fu(=R)q Fv(.)37 b(This)27 b(completes)g(the)h(pro)r (of)f(of)h(\()p Fu(b)p Fv(\).)p 3750 5438 48 48 v 2084 6059 a(33)p eop %%Page: 34 34 34 33 bop 236 272 a Fr(References)278 454 y Fv([1])41 b(A.)25 b(Am)n(brosetti,)f(M.)h(Badiale:)34 b Fp(Homo)l(clinics:)39 b(Poinc)l(ar)n(\023)-40 b(e-Melnikov)29 b(typ)l(e)e(r)l(esults)f(via)i (a)f(variational)i(appr)l(o)l(ach)p Fv(,)407 553 y(Annales)e(I.)h(H.)g (P)-7 b(.)27 b(-)h(Analyse)f(nonlin.,)h(v)n(ol.)e(15,)h(n.2,)h(1998,)d (pp.)j(233-252.)278 719 y([2])41 b(V.)31 b(I.)f(Arnold:)43 b Fp(Instability)33 b(of)g(dynamic)l(al)h(systems)e(with)h(sever)l(al)g (de)l(gr)l(e)l(es)g(of)g(fr)l(e)l(e)l(dom)p Fv(,)f(So)n(v.)e(Math.)h (Dokl.)f(6,)407 819 y(1964,)c(pp.)i(581-585.)278 984 y([3])41 b(M.)29 b(Berti,)g(L.)g(Biasco,)e(P)-7 b(.)29 b(Bolle:)39 b Fp(Optimal)31 b(stability)h(and)f(instability)i(r)l (esults)d(of)h(a)g(class)h(of)g(ne)l(arly)f(inte)l(gr)l(able)407 1084 y(Hamiltonian)g(systems)p Fv(,)c(to)h(app)r(ear)e(on)i(Rend.)g (Mat.)f(Acc.)h(Naz.)f(Lincei.)278 1249 y([4])41 b(M.)e(Berti,)j(P)-7 b(.)38 b(Bolle:)59 b Fp(Di\013usion)40 b(time)h(and)f(splitting)h(of)g (sep)l(ar)l(atric)l(es)g(for)g(ne)l(arly)g(inte)l(gr)l(able)g(iso)l (chr)l(onous)407 1349 y(Hamiltonian)31 b(systems)p Fv(,)c(Rend.)h(Mat.) g(Acc.)g(Naz.)f(Lincei,)h(s.)f(9,)g(v)n(ol.)g(11,)g(fasc.)g(4,)g(2000,) f(pp.)i(235-243.)278 1514 y([5])41 b(M.)34 b(Berti,)i(P)-7 b(.)34 b(Bolle:)50 b Fp(A)35 b(functional)i(analysis)g(appr)l(o)l(ach)h (to)e(A)n(rnold)g(Di\013usion)p Fv(,)g(to)e(app)r(ear)g(on)g(Annales)g (de)407 1614 y(l'I.H.P)n(oincar)n(\023)-39 b(e,)25 b(analyse)h (non-lineaire.)278 1779 y([6])41 b(M.)21 b(Berti,)i(P)-7 b(.)21 b(Bolle:)33 b Fp(F)-6 b(ast)24 b(A)n(rnold's)h(di\013usion)f(in) h(systems)e(with)i(thr)l(e)l(e)f(time)g(sc)l(ales)p Fv(,)f(Discrete)f (and)f(Con)n(tin)n(uous)407 1879 y(Dynamical)27 b(Systems,)h(series)e (A,)i(V)-7 b(ol.)28 b(8,)f(n.3,)h(2002,)d(pp.795-811.)278 2044 y([7])41 b(U.)28 b(Bessi:)36 b Fp(A)n(n)29 b(appr)l(o)l(ach)j(to)d (A)n(rnold)h(di\013usion)g(thr)l(ough)g(the)g(c)l(alculus)g(of)g (variations)p Fv(,)f(Nonlinear)e(Analysis)g(T.)407 2144 y(M.)h(A.,)g(26,)f(1996,)e(pp.)j(1115-1135.)278 2310 y([8])41 b(U.)28 b(Bessi:)36 b Fp(A)n(rnold's)31 b(example)f(with)g (thr)l(e)l(e)g(r)l(otators)p Fv(,)e(Nonlinearit)n(y)-7 b(,)27 b(10,)g(pp.)h(763-781,)c(1997.)278 2475 y([9])41 b(U.)26 b(Bessi,)f(L.)h(Chierc)n(hia,)e(E.)i(V)-7 b(aldino)r(ci:)35 b Fp(Upp)l(er)29 b(Bounds)f(on)g(A)n(rnold)g(Di\013usion)g(Time)h(via)f (Mather)h(the)l(ory)p Fv(,)e(J.)407 2575 y(Math.)h(Pures)e(Appl.)i(v)n (ol.)f(80,)g(1,)g(2001,)f(pp.)i(105{129.)236 2740 y([10])41 b(L.)22 b(Biasco,)f(L.)h(Chierc)n(hia,)g Fp(On)i(the)h(stability)g(of)h (some)f(pr)l(op)l(erly{de)l(gener)l(ate)i(Hamiltonian)e(systems)p Fv(,)e(to)f(app)r(ear)407 2840 y(on)27 b(Discrete)h(and)f(Con)n(tin)n (uous)g(Dynamical)g(Systems,)g(series)g(A.)236 3005 y([11])41 b(J.)33 b(Bourgain,)h(F.)g(Golse,)h(B.)f(W)-7 b(enn)n(b)r(erg:)49 b Fp(On)34 b(the)i(distribution)g(of)g(fr)l(e)l(e)g(p)l(ath)g(lengths)g (for)g(p)l(erio)l(dic)h(L)l(or)l(entz)407 3105 y(gas)p Fv(,)28 b(Comm.)g(Math.)g(Ph)n(ys.,)e(v)n(ol.)h(190,)f(1998,)g(pp.)i (491-508.)236 3270 y([12])41 b(L.)29 b(Chierc)n(hia,)g(G.)h(Galla)n(v)n (otti:)39 b Fp(Drift)31 b(and)h(di\013usion)g(in)g(phase)g(sp)l(ac)l(e) p Fv(,)f(Annales)e(de)h(l'IHP)-7 b(,)29 b(section)g(Ph)n(ysique)407 3370 y(Th)n(\023)-39 b(eorique,)26 b(60,)g(1994,)g(pp.)i(1-144;)d(see)j (also)e(Erratum)h(in)g(V)-7 b(ol.)28 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4331 y(systems,)h Fv(Russian)f(Mathematical)g(Surv)n(ey)g Fw(32)p Fv(,)g(1977.)236 4496 y([17])41 b(N.)34 b(N.)h(Nekhoroshev:)48 b Fp(A)n(n)35 b(exp)l(onential)h(estimate)g(of)g(the)g(time)g(of)g (stability)h(of)g(ne)l(arly-inte)l(gr)l(able)f(Hamilto-)407 4596 y(nian)e(systems,)f Fv(I)r(I,)f(T)-7 b(r.)32 b(Semin.)g(P)n(etro)n (vsk.)d Fw(5)p Fv(,)k(pp.)f(5-50,)f(1979;)i(In:)45 b(Oleinik,)33 b(O.A.)e(\(ed\))i(T)-7 b(opics)31 b(in)h(Mo)r(dern)407 4695 y(Mathematics,)27 b(P)n(etro)n(vkii)f(Semin.,)i(no.)f(5.)h(New)f (Y)-7 b(ork:)37 b(Consultan)n(t)27 b(Bureau)f(1985.)236 4861 y([18])41 b(J.)35 b(P\177)-42 b(osc)n(hel:)51 b Fp(Nekhor)l(oshev)39 b(estimates)e(for)g(quasi{c)l(onvex)h(Hamiltonian) g(Systems)p Fv(,)f(Math.)e(Zeitsc)n(hrift)g Fw(213)p Fv(,)407 4960 y(187{216)24 b(\(1993\))236 5142 y Fp(Massimiliano)40 b(Berti)e(and)g(Luc)l(a)g(Biasc)l(o,)j(S.I.S.S.A.,)g(Via)d(Beirut)g (2-4,)i(34014,)i(T)-6 b(rieste,)40 b(Italy,)g(b)l(erti@sissa.it,)236 5241 y(biasc)l(o@sissa.it)32 b Fv(.)236 5388 y Fp(Philipp)l(e)45 b(Bol)t(le,)i(D)n(\023)-40 b(ep)l(artement)41 b(de)h(math)n(\023)-40 b(ematiques,)46 b(Universit)n(\023)-40 b(e)42 b(d'A)n(vignon,)47 b(33,)f(rue)41 b(L)l(ouis)h(Pasteur,)k(84000)236 5488 y(A)n(vignon,)31 b(F)-6 b(r)l(anc)l(e,)30 b(philipp)l(e.b)l(ol)t (le@univ-avignon.fr)2084 6059 y Fv(34)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF