Content-Type: multipart/mixed; boundary="-------------9909291753689" This is a multi-part message in MIME format. ---------------9909291753689 Content-Type: text/plain; name="99-364.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-364.keywords" Hausdorff dimension, singular spectrum, subordinacy, quasiperiodic operators ---------------9909291753689 Content-Type: application/postscript; name="finalitff.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="finalitff.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.66a Copyright 1986-97 Radical Eye Software (www.radicaleye.com) %%Title: finalitff.dvi %%Pages: 18 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: CMR17 CMR12 CMR10 CMBX10 CMBX12 CMMI12 CMR8 msbm10 %%+ CMSY10 CMSY8 CMMI8 CMR6 CMEX10 CMTI12 CMR7 CMSY6 CMMI6 %%EndComments %DVIPSCommandLine: G:\PUBLIC~1\TEX\BIN\WIN32\DVIPS.EXE -Palltype1 %+ finalitff %DVIPSParameters: dpi=300, compressed %DVIPSSource: TeX output 1999.09.29:2325 %%BeginProcSet: texc.pro %! 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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 6 /plusminus put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6a18979f258b8c6ed3 ccafbc21aca420c9c83eea371adc20e038b4d7b8ac303004b0aa205f04135140 76407216032fdd22e6219da8f16b28ca12524deb7bca073cc5eba65c102a5e85 fd48e6d062cd4283ee570a7774597e5bf0e3400b6be72db0115f3cb12db70ce0 83722870cddfadee715f10f1fcaf20e06f3c54afe5ca238539bfe2b596116e83 f5371ff18fa5003d8543226cfd4025f9940365b392a858d27f078d3abcffe4a1 54e78c7692d1a32bf935967c64f01b24788ff8325d61145e2d4a489fd986fb77 38e6b254522c77ca2797a504a9ce4676a77ebacb026eca94dde5922c936f8e90 c43e2851973f31a3280c08220536dd2c2de1ffd15fc739091fb7cd5fb9bba2be a5ab0785843300a48f080377b45809107ea6c4e3b0bf4e928ef65d468a19dc1d 34a8be67669f93aa7ca73e2eda716c26be7f59bd96bede57fab460fb9288d0cd a12af19bc08615a5a40900c8683a1585ebc1604fb5514ce4b177e3833864af0a 4d3fae75ba71f5583e98352a7b0ba633f68a02ef3af2d812fe8f9d8305885a51 2719e96704eac14c4c27581e74ec63f74201be2994462054054c50e9105b9c75 a1e8abc0cfbb3066a084a35012ad7673fd90d311735c150526373f90f04140a9 306648562c2391b7380172fceeccd5b7ea82eae324248a7d3011d3134b29ea78 9f72ed9068c632503c3e695d8957b332f5f3a31e945c0a8fc12476690fa822e2 0d307d55650e77e5c019de363c5d4a0ffbc562dba38412dab372e1f2b868bfa0 47e40122b14a6139290948e144d006f4819fde1f9c1960049cba67e6b32fb867 00dfa9 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 45 /hyphen put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueID 5000790 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b448685940729e6e22f4be0e08c6505cba868f7950 93f556b64b808dd710eb936d3ac83e5783b5f7e56d363a59a026a5619a357d21 c4e77bea121eb24b09027d1aad93158edf912940e91cd2f17922b35885894644 7773730bb19af9e70453e1b78d5edc123b273a979891d6d5e624f4a388f3897e 13b045e34b4ee2f5e4fc5e996d3549640010c671de14d18b0b3cd300e57dac03 22e59bfbf7b29422230870f5897fcfaad4b50c7c1c58edcd119218163d6f6e56 25ccb882db7b6d09a1586508e8cb642a11c29a46915e4a96e282079cb21922c1 c2e360b487a45302fd22ec8c5fab63e54b5e844d4b17ca2fff37c69c366dd72a d02922c14c0932f65ed03e4219c117962edbad2dcdeaa9c10ce8af38a4ae52e2 b377245b0be19a77d6c936e7530cc4d0b78d0cc4a92698fa2870fa54f2d8503e 2d17b3d52fb2febb09f2b2af0c2a1892039ebe19a690098799a858e3d39631bd 6925a154d161df3918074ada6bd52baddd0adc3f07e2d9f15e27cbf7fe8b98c4 07205c811121fa91e059f2f99322fed63f359ac9da97aec383f067f23e5de331 51e80f0a88ab50fe8fdae4a5de93c1ae2fdca06150b37246140c0e87cb2325a6 0d2349162ae3ac93144eee1e665a1289105318fdfe86b6e76251cb25adc967d3 d0b97fe5e279e1161736ab22b4ca510b964342383a840defd38f96a7280e6ac1 34e48d740607ff2e7804164a16d47735864db847c97335e6d4215cb99911a1ec 015a3edaac1f28fedd56d2467130d07bae9416c15f0827d27c6c79f59054282a 418c12c157c91223a829947f47592f7cafd93ca182b25a73a9419127e3b12a9e 5167ac3963f2b019b338ac46d63880f94dda4b538835884d2a5538c85528d6bd 977f844d32b43b0e48caed5a4bdabcefe71695d69ca784dc64b133e0a1ad75f6 c55499b6157e9c9fe0a4117324c0c1e3257db52ac86166cd79bbf36356a12da4 87188618e4abe886348d2dcda4453b3b2a92fd6016e5696e30722dbc572a20f0 517ae2ba03360bfb7a9dc56d71647be00d09ad1e3c0e3736f7c607bb71d32aac 6d3fb4dcde28cf5ba10d4f15fc99198fc63f77ef8064e4dc427da233505b2305 ec1d88f7f6c169c9ac5ecc74c5dab6bfd5e68c27bdb1fea70f1a1f921875d859 15e823d278f3dd6b877151dee01baa7a479d3585e42f154001eaa59bade29c89 b24e9de1eb 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 65 /A put dup 66 /B put dup 70 /F put dup 72 /H put dup 76 /L put dup 77 /M put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 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 -251 1103 750}readonly def /UniqueID 5000829 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd803271dc8bb1656323089e9e2bb50f2b95088 87d31dc36e555668d9578b338402f8c259f6813b0b71e6cbd95e19c20ced8495 2559e47577e612a4d94de3c1b4c6c41a6a3fa9e2313457242de74c7e4c0d722b 3e224d18baa726369542682e9459b951b2ac638494880c42be4e2175a3c067e5 945bfc1a05d3de8c3e8ab97f80ab6000e1f78fc258b85d3c84b038d7d0f5be85 0d43fe03e26cdfbb41c36d14cb552e8447ca678efea1d275e85503123a2263b1 4077587683de69a8f5a88d666172668411d0d0fed3e037ee468627207505dbb3 59cfdc5d7f526fca5416d741073284f5dd2471e09f5916f4bff1a90eea4d1283 7731f6a36e5ea695f4efc3bf5669b7f60ab606df5819149c0c8a4817f06dc172 0543467645daeffe4457293e0c0407c4ec7857558d0452de2ac773f141783edf 81b30215ae8d2d062aeb3cb69eb0d62bf04780997c9c027a22896f456b784378 7f2ccdd4804375321e7946d3d254521249a4d28a360f4d3163912a688c16e807 15d0440bd27bb78fcbc89bc28028aab2f594f8b641e45b9c599180b69663eef0 03465a2425aa4f8b29dade493002fb335b23eb145108ef3ab828755d3753ea8d f0683265497b1cb5cb484c13f7a4e003fa6ab3ed5e24fcc6125cee4e33936122 3cc907af3c64e5aaad882c356e673e6ca2509a71920e99549d8797e9aea189f7 fe18bbf718017cd2e32b4652acf12f16e785462980f5895afc917ea83e9c3eab 6297f1de23785f79172538b39984ab261f14ee296a42cca3159e63fb30583af1 911646257a99b4162635fb79e25404242cccc3bf224d4c279e731333107ab2cb 2598dc582dd185669c1fad35d2be0428389a4b175db159508c08a77e842fd400 bbb81291cbda2d137c403246950ae040b7e5722802c24e598ba999a46fe58d7b 4af2fb6af1ec42ba29f609a1f68dd87ccea8ef2cb9372be548bc3dcd7ee944c0 dfe223d67cc908b91f2a7f54775bd43c2db6dda17ffe9a20a2e1508ffb84fdd4 c483b45c92f6990a551aee39d2d1c1984d706499fb148c30686fad06b7c11b96 0219c84da21da8e5162d18146c7ac4ce567606fab37c3b37bcb263b3a65c49d3 16ff0616098dd8a0fbe65ed98546695a57476ca677e10b4293fa8a46c076998d a0ced21d13e6e8649e321b4b040f8d69dd4dbe7d289b3ae82b2ebb3539f931a2 cf8c7716202a7fa4ad297113852d6ff97184ad54d3f4595ed9ee70052c3fd62a c5a038ea7abaedc205f098ad3dafdc34029a9c9adae7898fa6bde431c6416568 2fa4829a14f8d7f59d314672577abcbf79c169ed0f167d0039a7ddc538e49d8e d81e552b6c3ee20e105ff571bfd02691750b091ade267b7af5a1ed4f3c1a7aaf 4f11fa87f2d284398a9356608fba122b677f334b648e740e9bb6bd8812901e42 874cc681a4e429b9c4e205ab0a206973e0c96b988ebb6997a47f04e3c382f214 c02b827ede2a9fd50095f424be9b1ba68cbcd187f46f5a90f84afd84e076dd77 a2202e5f99397d0e36378feda05ae6861e508fae6158f284559f80ab739679d7 67a4f6d901db7b2939a8ce80a73808aeb1eeaf62c8121a2c0b2e5679e4cad7e2 6932b6c72f0c7724f10efb7cf54af09f75d10965fe1ba4ef97ba6f9dbf396666 38ae48fe23a6f14b5ac15ba27edd78a2d8a3af750f1b97d01f281846584b23b8 43d58c86a62f8a4c6e9da3edd1c1adb7efd65111368be1eae4e9416b684fd908 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6c49d430932f9ee14112e747f8e5788306d1b590a0c2f01febe67253bd917c9b d74bb8e9fd27a201f9f1a0a09ca8f28b99fd59ea9623b22292cd89dc12be81e4 336eed57236b2fed998092d19e1b1f9c9fd2ecc725c8b9488eb661cea110dbba 8345e7494f5dc62c020fc94ffe65f6caba0f2d386faaeec5a40c488b08024091 9ab7bda7c8b29bcd178696 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 67 /C put dup 69 /E put dup 70 /F put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 87 /W put dup 89 /Y put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 121 /y put dup 122 /z put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueID 5000769 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5bfae5c1f476ee6f05eb1d2caeb 269958b194521197b312fcced4867f3c8fbd030bd715d8ffda1dcd454b174e7a 1a97b59fe770e67702519d9d9b23d61ac08424d555242a8ca08c49aef300945d 99b999a79ce74804ae6bfde623f4463371442f6523a5f6ce19c839a708c02513 2e22c696c8ccade45680e5197189d0f98e7f0d5f955e353970b392cf530a68cc 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338dc4e74ba6ebb1324c936b18e7e3e547987121c5d553e6403c2367297acb74 f7c7aa054b98821ab67fe1a1da61257568226a64a0571ca773b350291e89f35f 3f2dfa2a4e2097b833698af6a04a578115632a0f295dea0f8fdef5581d8a3ed7 ff7cebcceb8fc64cb3b04a67b2603392230521e4d5060d1d1fb0a9691aff344b 42f59d9115b887d7fa21245933a7ca30c3d29f5aa8918f728baeb71a0a2c2919 4f8d4d29ee0f6dff37ccebfbcc0a9f612422b2943e4a5ef6ee74cd88ee36bc10 975436094faedc5cf8e9ec70b77a46451812fdcb9490df3519e3fc126acef4f9 8bca6111467336ef797ab6f4c7e1a0ec442cffb80d073508de37f634ac514a15 27e63f2cf4f515b910c051a2dc8af95e75d4338c6072156240622829e28ea8e5 ea85a4af24e6d1dc0ec28d4716f62305917d9f77b2e67f4fe37c591126f64633 72ccec7ec24a3f0629450162d3a6d95789f8df68614e937374324196d8997248 04182b7c9e3be4af345e5ad6907bb670ff0ba56b3a962dae566b2a8b4312c63c ec0706809965c30de7fdd1183b1e2d7af52df09d56cc49b7dba1de897a5b13fa 408c31d2944bcded6e5bfc49bd4a6c99997ad4b0882ed30dfe14238c1a28b041 1dcacecef73ded9e9f3d2c07cba6ace625db040a02d1bc08d0573c17d4c21328 c6ca111adc71ef728b8eaa9d44caf67fade519fc8206a7af71892d885344a4e3 57011244a388184ef45d1580d9d39a71f901fb0b85081169e76ae99d5ddb7439 1534613e94534880f037087adaaa5ed736a354a6dc943778ec0d25da3508eccf 12834929b5be828c66beb5a2dbcdfd32fda7d737ed9d523796b31d42cd5582d0 68644349d896e91b6e6873305cf9016a17b936225e01be4c3b85d665b233592e e6b6edd2311b2bf02b300eb883c78307aeb574c5b80a81973b0f88f2a8f02bd5 dedf51bbfeadc8c84678e9e4a4028f9f55c8f5e4b70f1aa33fbb79edbe04b82e 446b74c26f76c5b4fad20351392402fa51b03f64d12a29fa2be3b3abce0acdc5 dbfe8317ad8e1d940c3d42bc92c575cffdbc55fea8b11c32835752781098f5e3 a387e39725a0571485b9a1db46b33a9ce115281dae38629e20d84f423ac70996 36fa999dd488e34f60abc0a46f95a06dd00b401896e505a0f9dba50a4948d017 fc5cb213c34206d7fc62c5fe94902e7203203be78663c32d5c3f932e601e3d1c 813db52cce67f35c3595c3a01f15636f74a2677c8e4ce3e80ad5437eb9c12946 a4ec77ad89d131a9ec8fc05474f4ad8a948cecb1d935fb690059b36382e40580 6e47b09857dc725a69e8189b6194db530cae1667811452a34a70a35d52578f17 a1f407d757e7e0a5394c6403405ed9fb6d98daf85275d39def01f97da3351dc3 313cf0cd4c24c56086c68ae9ef4c92ccc9884f5e6d71c26266e85c291b7bb82e 6703fc2a17abc7b4928a773e1be666446dca3a4bbdfe315d4a4adadda4ac1fca 34b3a78695a39c40ef3e49081025bcd47d455399877edc75c1ca45956090328f 27cef0d88637c1a7ac9bc011942e06beda20757076e9409428aec503b0f78d62 577484b805a609beb4a4aabeb842e2494d6986cc5c2ce46906f91f77820d3d5f bc912710edff41506857fb19bb63d7f267e3fbbd2967cf0663b56585beb401f6 46757ddddbde3d1dbddd63e63a041b8e5b9342e590622d868921af6efbae65f1 823f7393f8c0e5ff55649247cec4f81d6c4ce40f48d5d2932fd9e95c6b55fa2d 48161743fb04fdd2aa478fb5d15756422b6cd9daf5e58dffd6d90bc90334fd98 803b31b0bdd4dedb40129d0f7b0b10f2941608b5188171389652d9a068447e3d 92e96634847c17d1319e95adc809bfb386b88d6da2ee6743b6634bfc7fe1713a e79b5669e89c78ee5646cb09efc57aff696d8d146c82686c86f616063842722c 3fbf23b27b18fad1c1551a9ebeb0fc6e4e955e88e3ca54310dd6cffee567d76b ac9e4f76464d9113ab25292a805b82236d1d0df64d32586afd71346e146f1a4d da89c6f9803aeaf5de86fdb35da1ac14a4ee90070da42f24f1b464ac073d8332 5669c20d367a4070656001bd3e32365116e42d2bd6c7cf8703e7781acebe80e6 a0d141527c663dbf5f751057918341eb2b863bb8fc2bb6be2ffaf2915adc98c5 bfa21bad7716a55d535a161615adf7388bb7247de928b67973d762d332e03e36 31eefae720837fc92ff37bd4aa2dc0b41f9d60bf025f4cb65e7e2546123835c2 7f613362 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 18 /parenleftbigg put dup 19 /parenrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 34 /bracketleftBigg put dup 35 /bracketrightBigg put dup 50 /bracketlefttp put dup 51 /bracketrighttp put dup 52 /bracketleftbt put dup 53 /bracketrightbt put dup 80 /summationtext put dup 82 /integraltext put dup 83 /uniontext put dup 88 /summationdisplay put dup 90 /integraldisplay put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 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 83fd0e43f31438730a3e275309aaa4287a2fa1c3c417a33a47968a838191810a 71c285c9a4d631967aef3f2154f8e9deeea19e47e77eec513c1aff4bc89083ec bf7303cb3f38764683e8a93090744def9e35b2035cc46a70ca1d18423ea0b362 147c79416517d52dd72ecc3317dd32bbb2ce08af7816025b6e759b7cc5280d50 46ec226631368ac2fe45b518805d1e6b2b47e453169564e79557b32e0d117050 f71ab4e614615048717aecae77332eb143fea30577ed765f9d991ed34f56d504 1e04b1b29b05753c502b0cbce8e061f08085bfe4a87a851a1ab0ac11a4dc51be bbf61818257eaacb3a2762325ac2b7ccd0c19189daff00bef48de95e8bd2d28e 9184fedc852da2e46dc4645a93dcc1f51004cb2d775544eb4933e898ffbbb0f2 372c8c06319ffab27eb0509038943c8fdff2c885b99f7a1069eb166308ef4147 8e593b291f1e7ef0872d0ec0242de4a70bbd0d715e768c0f39f81352cd702950 10b1398c42a9a286240632b2a67228f544a4b33898883a452c1e372a88a5051e d544d7e30529ee97a5504ae9e2d9a6dffc867b20d1e752ee206175fff521657a d7cc70b076328d308df8fd81b0c871e654874a6595f3cde0cf49a214b8abc134 d87ccecc74a6fba46c49311e3e53fdb6786057e15d029fdbefb62da508d7a3d5 63341f90ca4b3d09d62752ebd21e3959abfcf12619be38dc36b63e698809016f acfcf4d799bd75e88080ebca40b5a5d7302e23baca6a82d5a871b05bf154c7df cea5db4a732a5315fcd5abc2aa3776d12085e84e1513093a57bb3befb97ed025 fbeca644ca2dda965e2308f972c148ea4b094896ac198d73755d6783e8dde028 2276f80e7e990eea09afff268a34a664382fe0d887d009a65d45e55d7b5ba85d c839ae4c2386dd207579a403f99de5ab5766e7a8feeee475cb9f174c44c78d07 4b59f9978db64c0764fb6ee22443e7aebd068882bff077b7c0be4d2c19f885d3 503c6787406d0695f33f2c9d330049108f84e9c186127b8a980e789ce544d2d6 d6ed95851f721bd38b2be8f6eca2a9eae9a49dbad16636ce0256b79febf92336 a0b2520f888308a205574af8b1bd4dccb83c741bc0d37d5bfbd05b94dd2e32d3 fd25704ae6cceb882f6e067a0c360d06e79a8b87eea7acab8496283187d58ce3 4d85de010d3650a779587a6af38f4f639a2e001d74297ec19e9f7b1a6836a12e 8aa731fcfa36c7a47297226e3a3ec2ab5df9123d17fa137cda71c0df48cf3479 65cac790507736c4bde0eb9fedbde817cde43c583efdfd648f98af23f444debd 91ca6eb59263d5f281d95ed6fc28fc51f290d2823e13c19459a031bb0e084afd fcdfe85a9b0e7f7f7118228a0be5d70c7c692c27a75a29276c99473eeb1ec848 a887469059dd19ab147e6bf5ad9e1c84e6027a59d6942b6d6b192ab634adeff6 98999de22fef68ca5029ab1681b1edb74e4f90c69b02d67e70a82864983024d7 461f77ffa75b2d48239ec30ee2314ac226efedf3586ab0b1edb9853820912463 65c51e224b1e9bfa7a4108d80b1cd947ec461c14ae7ca424e2f4801fc7cdd9fd 5b28d7e9206e20122c7b761686f63b9f8d46563d644c8496b14e8025afc20523 d251bd3f7ffb51ecf62e65ebccf01192fbefdf23073b1be2b5e17cf67adb3986 7963ab3db0d093000421d0d1d890ef3d51c4377f53db87f49c817e4457b15711 1c9df44c62e1027695d6e9242d1d315458a3b1d3eae09b0fa5464d8e36d76051 f7abf3e687395adcb828d7767ada195d450fc412481b10508a7bbd64b91f2597 7bc208d6ae34c5967a8263ff6e00616fbbcb31964e4a2305caed32c052cbe851 4e67c2a08e9762d312145ae42b62697041f1b3f17b39781034fce6fb4e28afc1 00d12f42366fa0bcebf6b5f1863a76dd39d65201ed8e1f1e867890d9b5edf706 80be4ef84d7dc8afb0cb6ac62478a6816765dcf52da64ff0df26e655dbc43567 c8f61d913a1683e92badb77d55228eba54832dbb8c90d5f421d392ecee2662c4 b3a03e83ee155c66f7a98c15f88e8fd4fddb0aa1b19e42b7b6e844ca217a1010 32bd41d9eb7942cbd91db6c9fafcf418cf20c6e3db0be0aee922eeec870b6364 5bad82c6107de47f4137c705328b320f8e7f5016956491c10aaff25191fa8e88 9a265a36fcc2690abdf6e50f6045eaab997e646b246987224252904891720070 81017d587cbf98249e54f3212705cf3cef66757f155fff82c7afc4e0907d6fcc 62a160d6e016270a0c0bdfe5f51cb623d38aac2a0cb03b050735fa09d293b943 b33dd34b9acb00cee27609e9d19a999a37f02e17d9f42d5c8e1c504ef8657ff3 e94dc2ebc4c6a5d3bd8208001cb33f951d0e21777921b6f7dcb4f38a44aec2e9 bde3d17a24423f923695ea3a57249e0016dd660d8ac5cc4f718481a4481fc692 c082c9e635886dbfc9b718561470fe84912d14782e136f09bce0ebfbe2a8f077 413eaef4f4f23991323a52d0408cf0f2c78c774c7bde4f1c9f5214c3e682417e 1482db2aa104ea5ff511a9e82b55a2145fada11055a6cac590e01187efc07d4c 01808341a86abde0e086895bc25426702c52bec2456a654857156498610e47cd 6c98dfc1d32b3cdb5d7dcc1f1186f7b8671b372407ed5be06d9c530bd3e00409 0afe47e267d0f41b88e7b0a87676cc7a1284c8b3d440065e66eee35389dfc498 4b17f4bdbcd28e636f9a08dfb9886c706adffd3103854182a5e460bdd5aa73ae 554080346c18d74c7ad3de650b7a9e0c080410f3754ff21101d023329e425e01 973a8b32de41d0693fd8de30f1bb4c4ba9b14876ff48135c26542135fa083f2c 6b1f31f2393d5876154af37a4dc80913cf9a7651e7018a9385166eb14d8ffa8d e8a6cdca15b18f224be41017e6cd18e1e9f052009707a062d68babe7d7f3c0ea 9a52a867d95a23504174627c14817ce956ae100d436f01cc8140b5cdf7e9e55d 64cb817e37b24761d71a51ca30f423ca8587626594ec7330fec92df4b037b7dd 01630edc260ee463f8f7a3a96a91e47a3bfd207811323d10e81e9f897d34d4db 1afdaaa24069d239c88a44a76757c154a703244e9433696c677d73463cfb5667 bc99ccb8363dcffab8820c07b37a8c45ea083cac79e658dda9a12f2a5400f874 0a51f7a339bf9c42ccd383d8241756a611f81bbb9ba4ffdabb364b9bb5cf58ed 4171568cb5c082727d62459107d499181d7e6d22b5c98b3b980f100665681ffa dd4fbfe35c63979513ce2c4da1f22029a83a1f32a11ca08fc3c5122c7a47554e ab0805599b8d2bb1a00c0e7945ba625d69811b26f710b5d9129e62861325f924 6f1a5c9e1a351b66307689f22263b985381a692ae44c49624e5cd1b314ca43a2 f7efc4a23694cc9ddaed2c890a959e478dc51eb043b56875f9110748a692fa5b 06a61574ef58fc9655dac7af7d8969fc50fd0cb03080c9979514cc61951e3b6f 3067021e22ac7a3326a4ec3db457e9e378d2666e5d567a4ea1665e7b4bdfddd0 4de85e59e24621873266a1c317e672978613879668b33f9a48943ab70dfc97d3 2ee98b6bdfbea8f519c8ede888e6cf6a8839a4f5973e33ed255a7042574b7e91 442c56df024a3f23fb8f4f6ca58033285095e85c627e9b8714ecc2a656a6fbeb 2251e07f9286a9c13b8691135be5e81fedecbe7b23cabfc287f365b95ebdf9aa 2954c63ae904cf7e85ee611e6424cc96d1018811dc3c0b9785a4e5 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 49 /one put dup 50 /two put dup 52 /four put dup 56 /eight put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b031f3fc6c13d7da7277a94ba 018e9998b3dd888011a5d7c4204989f30f908b95533bda845746b673ab71ea57 65a0d14f4350707e47c8276305b28513cbe1bb0dbd269a53719bda46e536685d df78ca0146b6b93e760256b74d939d4e35b5e77238f04c92298dfdd188feea30 e053eefbcbb52f2011772b3aae39f5805597bbc1e8bb75a446ce014030f4f2f0 f49f9e962ee4a1024a746fa92a3628db5270732b54e43fe5ecfa524f127e5fcc 788e77e66098336ad67fe4cccaf0253272d5df79864bf4b734cb9a5859d557d8 bc11b8e00221ebc12e97de4b1f466ead83a4c894709363bca9040410a52d592e 34ee40cc7e5efa920546b981aa659513a24b1b85c221a1875b62d0b89e57a368 321b8043a5b094e0379760a443d632892b14ad6d19dacc8c78093243ad67e6a3 08e56e6b68412ee690b10dac6e17708754a00d51fc957b500eb80175716eef4b 2ca1ef867614659bee3f2b7319e97b6fdf1efc847bf3cee3156f72f21751da8e 5fb6898919e6799820d3de0642d756e09d6fae4ff08dd3deda3173bff4bb11f7 9109c97ddc05897af709ea199a90fcee8ce4c7a3c15b18170c41c04de2d3fba8 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b34c83cfc3685f22160a1abb4c5fe7cc0d25a2b1cb7228afe9ce1a40485a6190 d0445d1c7be7195e0d8132ccc5a5d4028aa716df4b2df8cec5a8e48e3c5dfb47 10fdf016d7dbfa3c3fed88142eab5f359337427e27ed6e6f77fa396290ae8bd3 67c962a2f7ffbbda02ff6fb8e910f4e3b392ea97cf2bff7ac6dc0cfab1044d27 d13382751be8be6bf7bbee6209e00d3ab8744606c3cf7a4927aa5a0fa7ebf5ac c22ddac7c00833d44bb49cf8e5736a13c0191d71ce93cb06a5e05136ddd6fbe4 854442eccfe11dd5a2774cd63a7ba52fdf5b87235d03fd9f4a3482b196d6adf5 6bb3b91439eb2e3055ee89f31bbd1b3998b81ca81aa8683452121aaf805b9a51 03b46bff2adf2f228f9f33fb15fcc0b851a0f94b275387eae94387ea508ce19c c92f9fe88ce1a2a31169aca703d79c9bcfe2555c3630ab35c6dcd1b796567627 6cdca88ddae8ddf17fef2f4ec13f0882163046e14250af7b1bdabd3d33db2822 1946ae74450c5b7a530ce729e8ac36f45c15305932755673592c8e7d41920257 d52d1080e85bb90c32cdc9b7a7ab43600c30aacffe46129246380cea99fe190d 34e784459e2e939774992572b1622c6237b01f58598c16841a5f0202b6e3395f bb630c36fa0f13af4c3bab495e1bde18fde49dcabc87f7cabeec8b59b39b8209 e9e87cd126d9feb429e1f19af734d20b614bd22b362a31d683840e1117ab9ad7 215a721de2864745a68c235612490f40fb9de185de8bf8d379a9ea64fd9a0809 e3bccc3b6d1855d3e58fc4a2020997b9533f325913ccc8534cafaa2a151f7a9f fbb686e59407ab3b3b3c6421e554bf28a0bbbf79d56a7b6a38375ad988469c4d eaed43654d84550870b74fd4767ee55efa0f5805c482b190a86c593f2b442cc1 96c58af60cdb3684c57ec366f5df613bf9daebb1fee2a5e7385991ce1a4a6107 7dffd524e2ee4ad393b5dec4837158110e320e7537f5a453e4b117c7f17d6271 7f088c4527ebc58657b87f6e925ffacf3703a2126fcf14220ed3190545ba2853 8757a09182e5e18f26c4d2eed14c 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 18 /theta put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 25 /pi put dup 59 /comma put dup 61 /slash put dup 65 /A put dup 67 /C put dup 69 /E put dup 70 /F put dup 76 /L put dup 84 /T put dup 88 /X put dup 96 /lscript put dup 97 /a put dup 99 /c put dup 100 /d put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 115 /s put dup 120 /x put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueID 5087383 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a196cf14c95a4e02f9cd2da8c db2546c6df52e524745992e18d9ff87aa25e4e1800bbe4ebb357c6ef55ed6d03 6d3a00c1ee8073266c21d2f0ac85d656abf61d7e5a4fa87da8ec3b5329e434d0 d2adab706b42a2e5331be5295399d803ccac03f631f01f39a022fcdf63486b68 7d15ef284a77def7fde4898543e7b5f7ec267756103e477f547cfb8d2311c4b0 09deff56085f5d419697af1846c8b88c1bbbae149f0f19ca3c8dafe19cec48fe 6b38357246d8b5cef80b53bab4cbbbe8bc8318cc462a4158258f456e697f6058 86010493f252073ee4e4c9531f23485415ad6bc901bbaea659ef701f825f4b6d 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121b5d0500f132763e7bfe357e296be83b972b1faff4e1ef17997a7896 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 6 /plusminus put dup 7 /minusplus put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 110 /backslash put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b66566ca5ba42b1911a 5d7f1bf343015eece988b7a93bce0c7aa61344d48aed9c92c8698d4b7c9951c8 7d103f2414b39e1437f9d2e50c4ee5f218f2e6716926a79ea978f13b1f855345 191dd7d31d8f82c2e3343c7a5894d95bdc492c28226834efcb5c12fea36ac5cc 430e0aa604961e34888adf6c1f3954cbc2498e225d953cf5685852162346f474 5a2a7087d5d7ad486de16d2ca8e15cee26e012671ba3bdc7d95cc8c98bb774f5 08625e968aee27ff7d1a06e63bcfb5aa4876c3f8f13b30ccccee73c3caf4e70d 98e6ed2f422dbb4950bf789680e064150995941a9f4dd68a575949847a7d012b b910bf03ad78e7fe3ede25ebd7b90e10c8ffdb9b6b4b460b11e84d5d1ed2c307 f30f1dd56f09206083aeb1e5c4efa99c7550086966a11485846744c64511501b 824efac74c4f7f82a5899bd4661ec1e8799ed0d5c8127cc83617f26d23f456af d2f610990fef9efc8ae3c6c1670cbec8b93f2f63d58ed5cc18dc56d2a0ea6f91 53148ae01822852f080547d9b6bcd35a2addaa663e75fed73e97dfa161ff5620 a0a41b4b712ddf19568250a761b3da02ab40c43a5d4c319886b509e965730ab0 c037fb7bbd5cd6e99d2062f915f9937f93e7f0e7fc4f247a2d0a1dafb56f0848 59b3b8b49ac2f1b35ee269c3d8a9ca369499997c1a3fac9338d25a3701598319 78dd459e78116e0d63315590d4afdfacb28c9a57d2d1dd912e3ee4cbe200f075 9ceed67e165686e2c4022c38b2ea6dc09d7554cfbcaed974461bf0cbe5857b85 032c16a9bd00cebe96cc7576a9e5598e678144a44b3dfe282e8f11d435a781c4 7a442b4dd14d65b2b5675cd91d1839bac4b542237be735801cfe9a9642c300c9 c3f6c7f23a820f6233e8afd9d3fb40642e0c7ae6c27e8d4eeb5b107e181bafe4 d2d2e4ebb5a510f473dd95be0a1a4b68cc62e5094f542838b01f77f34b175e5f 824bd74293c808309e4fef84a655493c729bd9e945778f4c24e50be27e8d1a5b 108f0e7ab669adec345064399fd774ccb75959d756add8fb7f3af223088550d5 ec48ca8bcd3ca35412d9692ca80708d975158f8a2248294e15abad864b5eda4f cea89fc09b767ae180f79eb08382f6b5959027a511583fd2eb160ed1d31e0c6a 809f7fdd8bcaeb6c71e23d26c6d66ba8d4fc58792fe3e22ffa9f03661cc713a8 5940cf5e70ae41741f039814bb61aa30ef507facb75494cde3e7aec8a3e9e274 3f154983c7a9b4208f9a4faa25a7ced82817596ca172264e4ddf879ef9d4aaa5 c49b396cf42d96ebfcbf54cba5aba04b7da2242f63ead2dd8016d30daa12295b 7655b3df01f9c173ea3797fdb97069195cd1627895e5f59a62874c18ffa1610a 179716e9778be8325f333249301dd93358eedc3d3dab2a86d99b269ff0104fbc 9ae5888a58a00f7eaa76fb1d4e8e57d03152d8298b3b8bc1fb5279bacfc8ab9d 537464129a5f5e099d59dec7cfe779cd96b49124df18a264d991484288fe2433 71c371d2de076d3c73c322097b1fa7123b729bffd1e8d7ec384ec7f72dcd25be 681f4d9ce3d00560bd42a4a4c58e8bd6d0f9d54a82ba20548eda3a4813371c64 459e12bbc457074f45da4beaaea770c4ecc37fc340326eabe4358fd84b50a65b a716817447938dd8d04ff593a6f7711c151862261c9faa81b4bf25712d109262 7caf52ac3177c0eb54fbc6b41f68a98f8997bbe95ba840c54d585afc14f705c2 4dee6b279b9e0405cb0202da9374b1d65e15ba2ad9c7cbb82033418541ec 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 6 /plusminus put dup 7 /minusplus put dup 17 /equivalence put dup 18 /reflexsubset put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 26 /propersubset put dup 33 /arrowright put dup 44 /arrowdblboth put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 59 /emptyset put dup 61 /Ifractur put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put dup 112 /radical put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8 3750d597b842d8d96c423ba1273ddd32f3a54a912a443fcd44f7c3a6fe3956b0 aa1e784aaec6fce08dae0c76da9d0a3eba57b98a6233d9e9f0c3f00fcc6b2c6a 9ba23af389e6dfff4efec3de05d6276c6be417703ce508377f25960ef4ed83b4 9b01b873f3a639ce00f356229b6477a081933fef3bb80e2b9dffa7f75567b1fa 4d739b772f8d674e567534c6c5bbf1cf615372be20b18472f7aa58be8c216dbd df81cc0a86b6d8318ca68fe22c8af13b54d7576fe4ca5a7af9005ea5cc4edb79 c0ab668e4fec4b7f5a9eb5f0e4c088cd818ecc4feb4b40ec8bd2981bf2336074 b64c4301d447e8310ba4f0c31d35df76280fc0dc4b3edd7b831516d589875797 e3e0482553c0510cf3ce38ebb476550ccba8cae0ab847e67cfa8e4673aa7c58c 9aba857d2bdcf38e2148a399733289d4d57dd86dbbd7b814e94091586e2c389d 03794c1776d79d11a37fb1843e5b8bfc1832349469b1f44b6564cf10bedf7eea 7e145de55ef2b5241efbd5a6d6620fd808681a509b8d096756d3bb19f717d404 fb1034f864950e769f33459f1d31dd719cb5f0eb05f65524ce7c86ad84c0ea33 19dfc8e11de81e77f42eb758ffde19d119511e54c606db3f6030d166700a063f 1c3e7587717f1ade6446a190086ea058c01522e6555374773f7dedca6c980515 90b26ca8fb400ff8c7f6b10a4d869b9e74abb825c90f154b463d974bfab76a07 b7e3f83d8f0dbb3ece8eb93e17be65b179e3dce2c7a868c281c5a2ab01c57989 95a9d94744b520ed9dceba6fe9c818f686a48bd24afd2996281cb21a4471e099 13b734f3cf8e7f72b5e7898d96effc5255720963c89c53c2248459fabd52ce54 6deb5932b742a5d61b1cbd94a5b3b6790c96a298a423d26fd2f38f8bbd2f7a5a f73ad068d29e5d7b17ffc1528a777a6eb3635c6d79a1a194152e95ea88afe10a ecd603e0783ed0e1a9d36a0c14905260a280c7d46a4e6f0191424b825b4e7001 7140dc0704fd2ec869c030a0de2bf2f49371693f7833eac092b961b5ea03842c 4c9a63671c1c30b7b8246116bce33dcb63a629e271fe6fe93354e2b90c7492f7 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: msbm10 %!FontType1-1.0: msbm10 1.2/19-Jan-95 %%VMusage: 30885 32362 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.2/19-Jan-95) readonly def /Notice (Copyright \(C\) 1994, 1995, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-C.) readonly def /FullName (msbm10) readonly def /FamilyName (msbm10) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /XHeight 431 def /CapHeight 683 def /Ascender 694 def /Descender -194 def end readonly def /FontName /msbm10 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 78 /N put dup 82 /R put dup 90 /Z put dup dup 161 10 getinterval 0 exch putinterval dup dup 173 23 getinterval 10 exch putinterval dup dup 127 exch 196 get put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-56 -420 2332 920] readonly def /UniqueID 4702309 def /StrokeWidth 0 def currentdict end currentfile eexec 9b9c0887db83fb1ecd8335b0bb39cef0af64f656fc6e5c230cc9d3a7346aac7a 06bd9a40393ca15d3773a21e06b9b4254d3050e90726bbb09120935a8d448cdb 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f62b1c8e12748788823fd59b96fa534af23b5e56456c931684e32f9318354271 95714b27f97c7dc4f22439c65f359a708018052bbc37b8a3e1ad8f60554ce7d5 fbdbbae3eb65c870a7ddeba1dfb59ff91d5e1763a9b1a37da8df0aa078bd965a 14cda00b74c268f5237eaaaeb4c067f2dad301d2ef5444f54a7c930f584bf793 76884422ca4225e269f7ae6168222b6516b643ac9c2c4dce6aceb8febd8700e1 73b03b3774f240a6f00daa55dede85ae84c908c0953177122b59500ea8c239b9 110407fc3414bd50bcbf1c156ac3c8db3efc9a25db2c7e15c58e4e2ed70781a2 a3ce82a94916e12f10fa370a266ccd519ba4ca5f6326fd43ecc1a37da4f1fcea 3dfc62640157c826914fd0ad02e7cd86c1631baaa59b2c507dc9ae4591f28b75 115fad4cd3864f67807b7adadb99c780f75ea8ec765980e3687fda0201172310 f1bdee1e779a72d76d241f60a819cd67164d762751ab822fe480d3943052c19d 8e69c275d5487fee887178047d62bf6711c3edfbd9bb5ad30781abd60d689921 8f69aa1e02506935b021e059c8109cfd6a76bb4dab8361f6b18303abb9e2c6c9 0a6540c73c05c5477ee2dfdd5cb08ff438983efbfc982b848d8554a3ba30206a c43623031187285cd78dc6811ab6e0eb288950805127326192602d496f34c3fd 79101f11ab759043279c38af695100ff79845133771c78af2e60efbdecf241b4 4888fbb4f455d202f2370745c1a8338224e2a93128c61b4a67c19528975751d9 81a260ba67fe749ed727abbacf7d0af02eb71455e07620498e3968b9a88980f3 3ecc4cf504fb405fab2da5157052f9cb97049d5b517abba9d16e95294826a7dc adaa36858dc2e6d2d6340222ed1475659d8b3c0a7f35ff66d4705a00c888f469 8517e086087c61b2bdaf55224cc19b6d5c4bd285492ac77cb46233b9d4c9419e 0b0dd300caa89868b830843fe57fa87ab7785c64b160f98e6aa4f54a9cc4a18b 4dd5fb9d522200f44a0050817e 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 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 52 /four put dup 56 /eight put dup 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put dup 99 /c put dup 101 /e put dup 111 /o put dup 114 /r put dup 115 /s put dup 118 /v put dup 126 /tilde put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b094541f345ec367ae85ca9bd 24568a01d3b9f8095f7420e6c423c414b3dcce6da48dd1c89a56d078e0d0e2f2 62a13640a06d17e44ee3866c3471fb58fedf5a3b77294517651c16bdd7267d39 a54e7171752dbde63ac19bb4b3021ce95eb5fe67390b09ae4d9ed4d704a67443 f55dce17acd996c1f5e023c9e5a18cbeecc3097f23763acb86cdd7cd13381ae7 4e48495ec7fa520539d87f8a8dcb3c826275469b6800876a457e7d1e5be867c7 b1ccad69742a8c9b0ad943482bf2a4ad0aed40baeb69a0233bad36b4ca2d2da7 322956c70375d152653500b2f22d2ab6990cadde2da14b4917f7515e64bc3d96 bf775258fc7dae4e42a4c9b6da8eddec4a800c8aadc8d75e48cae52137e05c03 677f5d6a82fa46d9f2fc7f56d62e5c605a1b7898b8d1401c2cac1a0122a2c8a7 aae09607f2c5f29293a09b9959399283be89051452898238b777db9830ff4318 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74e1d1933c12c5143298d98956d1d6df8dd36a4671923103617fc5b07f649cb1 f5369305553fb726a35bc419e9fa6b63753bec8720f56fecdd2997059ca84b73 6d1b7f8467fb284cd398e037e7e111ad4cbc970a274cfcebe3767d68a22c26d7 f0bc4f396395824f3918cef6b4ab3471d4c33e810457a0fa82c3504b5440d49a a98014af1da05a1cb7d46666ccf08ca34d97ca15e09beea8208143bf0654387c 67c26d23b9d293eac3888de78c18cb14b3fae1ee93dc2ef96ed18dc512bf0564 42bae2f03b7e22e0e9e3d831077e9725bac37ba9155608a3a069cee1cef83f8c 365c85d55a2cd8ecf73ee0922ee498a8bab4aefb1154b146231f268fc76eee10 76445b70984297ef9625e66742d5bd5f39a41d7e0e06ed719a10dd4185ee8aef 81f1488c12960e6f4fc35d0bbbde6725a8cf8983e77aa86bb36e0e0d9bf6c724 b17f73d91dbdaa19463634357166479a4862fc2cb42c29e79b50a5dc2f343da9 bf4a996fc3d249b081e6ef5beebfa135c2a6412ffe829766d46525d8bdca53b5 48d86736bea4b7b77e2e7f5886df29335d7c570f23279bcd75f8d7928bf191ef ca1e18de7ee29f84427a308d24d40848b25241bb954cbf98d4d53855e0ef8bb7 dd68730d719bb3f1a2397dc508ee12d9e9b61fab77069ad8f294d7ebfe4e3591 6a994717f88b50a8899debcd37a4dace167f339902a5385a4e540e83c49c75fb ab33e9d89cb9777180eb9635edf3273827823224a51764b779507b64a99c6676 1e31bf5615cd299266cdd0ef9908a2cc0a57bfb086dc94adbb990c8ec05a5f8f 06f2e932accbaa9d8563a1319dfaff071c86ff6848ee170a12056d3737879e11 7e9bc1b75fa62c839566fc0ad79356582580ffc58944c229861c18222812bc62 e4418c0124cf01012b5a31270ddd99ab63f7ded1f02e75c810b9acdd88e30060 35866e5229774377286a3a123df9fbf5b5d82558aa1ce7fe24b25c881eafd379 4f462d82645df46257fc44447c004aaa5b8a2a627ec17f92a00c71126a369a9e 36f4abc3790f8c75891578212fd40ae5a6b8507f4c041c61afdd13d750eb5f51 3846de07db8a7539b555dd95fad3503e0e77df13b42ea7fce6d9c80aaa09deb0 1c8b7e1b1b93e0fc865de57090fdf83097bf3eee7fa5269613c2a1bfdc6ce267 a2c0ef7eb06c3db004bd06f7c04a24d74ca47aea5433eae3fb251853830b2649 39 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 18 /theta put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 25 /pi put dup 32 /psi put dup 33 /omega put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 72 /H put dup 76 /L put dup 77 /M put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 88 /X put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 115 /s put dup 117 /u put dup 120 /x put dup 122 /z put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueID 5087386 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7bcb09b7064e91fa0ded228e42 09aae407a7aaca60b1076299ac4abd23ef02f108765f0e3d91f92f3afbfded37 2fcf6e4b1416901517da8f2fb3c9fe7a87bcbe6fd36cd5b5823fdb74229036a6 3c3346a1093e6b1036902c1bf42fc317c80abf04020a47b344c36de42f05c490 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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 1 /Delta put dup 8 /Phi put dup 9 /Psi put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 33 /exclam put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 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 88 /X 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 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 125 /hungarumlaut put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueID 5000794 def currentdict end currentfile eexec 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442fc578c75f6b45bbdf6283d574a095cf75d6e2627fc7aa02a10258ad1d0597 036939bcdf571395385ad5e66ff1f452d1f0a4eaae06efe6126bccbdeb9ebc79 5ef68dd0068067858060e3b3e0f5dfe1a5651f375a8460f32731b72d5daf0b2c a98481ea6fc5b746ca79eaec886abd15fb371753090f158d25736d4aa52ea52b 42286e7e6b12f52335bee8f1eb97efb5a781b5813f0e6e542bff6e8cfd58d6d5 7be411e6c7a876164c3ef19805e84583188053fb72b6a1294805f1ffc9925d15 b555e6c8f411d5da617315c4c869222cda17729489118af6a5c637d676d380d7 fbcec270393653cf31484b993d30e9de0e050bc02070e88574bd25b6e61ffd44 8f3bdb0ca10bed9179f91bc1951a31a1b6d02a80eb42a35e90c01440f6e0087b 408af9c8ac58cbb61cebe80dfe0560ac460dfb9015612350c4b14dc9e8e6164e 09a1a847c66f2a31938be3822a7f720e7690e138c94ae33f2b3bbfb605595f20 8c413b53088168d28ff9c0b570a57c4af34b202225c122747a7189beeef11b56 44485dfa654b239804aac45bca481ce3455d7d816bc8c228cdb3a43c05d2af73 f4cee1724ea06c1f99d11587b94489947dba2bb9e4cfcb25d1816bf43af05ded 26f2ee54962c639b537ab9a9f99b18a6aa5886dc86c94cb14c3e8e77c48de3df 2eaf9dab8f8fecba2b0ded0fc3cbb8530e44fe5a42b7b84a207e123e077f8621 f58cf0de98a1901097d08e3eb21a2ca57d53565ddeb91bb31a4e94c293881d5c 0f47d41819ea5b96c5937fac31bdc4d8cf2c31570e45ea8cd5bf4df9f00ecfe2 39eb8f2f6f1ef21b30e6ab2bdb7843be31c453a0447e16fdb91beaf394a69f04 ce7306d40e24d61817ed6ca15c7b629b8b8b9e4949e9bee251f3a3c75a6825d6 eb8c2c2043f796310b0469b271ca2cea32c886fdfdff430916b011106309af74 24aaf799c780afd8dc47bd99bf7d3aac67e2bcf7b326d0d4afdf5ffec5d6a1a0 c8c989ff720aaf97baca7c8bbfe12d76041f7b6035d8621e5278a78b772d8696 01b806f6d5262c3926d3c40c502e441ef6ab92d3de6ffdd24ef152142c270dba e13037dc58946cbabe28d8e9613e02a505551da1e6e3137f82bb39abf9aef75c 8cc00534e057b5ea4aeec370ec077682d63c8fe443d7c270d2d48884dec8c6a8 e080883b46f2ba8d99cc978bc260cc0bf9f84446f35576ea5ecb082ed09968ac 12a128eb131a55a6d559b0bef2a636f9da2784d43656ec1ff8b2c59cff818e79 5717fc8ca9ecc437001a1a007cc395346f5f5e6a1dd9d01af4bc4398c6c34ae6 41c8a6bc7104a5020d0f4181389e79109be786f948105342a89033f0e06ab68e d5b9539769a2d37af2be744e1acbf285ccd61a8689c367865a6df16600bd7ed9 3db5f7c6bcdc65fbc20731edfd07ef265384c996fbc4870df5cc7a125f87275d bc7e84d9effc2698ec7f74895c6301c5fe0832d7e3560d71765bf9b03d58812e 76d5f9960a0e7b166582347586f1186550be58d1db187c59e59e455af9615381 f347744e6c952c122419866adc9fd31596030834ff000778495bd3741b4f5dd2 ba5ae2d41323ac8bd5dba21fd1080b71b41ba867fc9f783baab63045260f9155 b5e3cf70d7bcaef797c883ab7fd9848bfc2ff9f7e40f72d934036ac6d1b28839 1ddd2185d3daeb100501886f76f1f90436751586ba55abab6010e029bc730ae1 8cc25b6b6da461f957a8e199603d41b29af33bf8fe8b0039aedd009a3f030ba8 6330ebbabcbfc72370885654b417218dd1d5980c33d729efc549629f0e198245 8075289ecea463bd29af2aa86ceaa7c0d2c835de050b28ac7943caef5852367f 8b4eadda85eb95b71ee8f2374349d6c475c5a40f301932bdb5767700d9b55a05 7f195130a47c57add210ea17e4621145be3e72508aed49a7ea762fa4e4d399f9 bf9306e7c3c5cae2cfb2db098c8b09123f789c74d556b871db644d00ba874ce4 3bae5329b8721052784205eb68b7da506025679fe8a64e19760001ebccb7bbdb 417457b5760755180cfafa398d93c5818a2db45644e266c1a81cdcbb8cc39546 34aaeb9d9388 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 65 /A put dup 97 /a put dup 98 /b put dup 99 /c put dup 114 /r put dup 115 /s put dup 116 /t put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306462c563d51ece66fcc9c831843ecae 1fefc1a232e2724f7baee428ae03aadb95c3035345c15e9922fe49e1f2cfc980 237316572dbc57064edac9b0db8913a5e2d45e97e19a91435ccf8adfc835b585 48e74b291446d689c7f2f8c4325e8356e974ae30c3e2977477baeaa33d141fb7 80f59351e84bfc88c87b3a4a1d25e168b9d134554f6a581378c7d2d6eca8ac09 045cc3a5e0ed86f147133f094029e3483bebc81cfde69942cf645ea20d0a2b64 5140ccc2f9cb4a3d969f4b0a79cd3329b440dfcff5285494406f3dd08d804c86 df191e57232f6f1f83df6430f214e03fa5412b1e13d83c60c52a698bc9a38159 19a64162af7b83a9468fae9507c6cb55ea80df8c696dfbc3296c663902d77e89 a181502627281181ac8624f65be9de5fb32f056cc7dd7cbc7366a91717b5892a 34ef6b441871c6993da3a81afc9bf70ff3842a21a29c0e49da1ccba37dd6b920 2702db510b8b1109acf4c27ee8d676bdbd16abe189f34a9b61532eb777457698 cee0561838eaab2d3939c5093afbb215bac433d1918c739b62566906bb2187f3 378521fb52e70c18886e1aa15f53a1f8a169528cf2ddd54c203613ff88e619f2 6c9c50c2c3b8b8c868a88afab1dc8472e7b6f3f992a161723a3c0904865a7c42 3281601eee91f35222c4a71f343739f102eb43145a47e4fac3bd666e4c114a90 5950618ecb1bdc29f90f6ebed1d8325fe6b52e14a8870e9b4bb088e585c71b91 493c12845a101937e3473ac5e9ae24f7c358f06c8b42dee8ef2a1a7a9f536037 57517f2fb6da0ea305d9d20cc74d922ec4b95b8b299459b71825f6f67abee3a6 f5267c6283c7c62aa8d36663071f18e1a3f0b35dcd88d739caf8dbc344630a28 7f583f037768d861f37d7929f248146b5f31ed9f61ef0a9659b50f47ec328516 9fd999b19687e50303557945476fe9845e5c4780045d84d7df253fe61bf69607 fd277a265296b02b4e4b35884f985d29521ee7327887d983bb92dd84ff646661 2b348b5e558d3bc65047afcb202750ea6e9c18a769a4e945e0954787ee65948e 7e8e443a9f2b9f5408e0febf523423b2200166ade223570bd17a6f7dd0386f3d 2246ebb40648548e0a3ff7bab6637a16137c24b042010a3cce2ba9afb9317fcf 4a5b893c4915859ff2b53a77ff0334966f7a234ea8a4349c3e048acd6876f5f3 17b9c238f37ed4231b5e8b661ba52c32821de47f7a0f6fa9c7e0f4bf09bf98a1 8008087aa44eaf37894e83891f552626f1f46563b5188ad949521e1a398eeba3 b9b8f4ac3c24785d50bdf01e5c9b7e4ea1d6b7b6c8019d2d5f353cacf216e30b 71f09d9b95805210964ccbf8f2e30aa04ac3a3a57bde635c9d2be1f5c8f64ad6 19764c72aca8152a51003bc396bce21a56c29306bccc36a37d816de2568e7794 a8f80d857132a3dbaf3ef75409cace85e1b85a8233ef2757a507158bbba8f5f6 d9c08ab4ee39f33a1c1a41ff2b8e81320050b6c4c167596b93918824a65fe1b0 6e7dcf5d7e128a77e5509f3a3a2af6f71de243cccfa5788088d277175e59bd61 a3b5cdc10cfd232cc14682d3b37d3aca68df524af6f38511e5627286034a7c08 1c7596bfc3adecc7d1868cb9d645f8cb61041bccad92249142542934977000c1 bfbbaf570bf1a3fad2368d1e6480ed485642acd1d44258a885562397035dc979 b08877eadba7c99963e7d866be89270b10629964b5c88439c08a7721e2fd6a7d 2a03d9cf2d7fa713411824db51a9141a7c3ca37feeb758ab37376d75b9ad8534 6fe75968a1e7f6b9b549b7bbef5da78116a7d029e6e95db78593abf789ab985e 301f5945c1628f591b4867079526cd9c779b05cd30de4d9dbb820ce964cb3acf cf693b39b15335996a208373a9a1cd362e93d740535d4fe06aaef81a77034010 883c098f8fe19617d6a3cc3ccaa63742d3b251b8baa94aba7ebb7f3cad53c3c5 2c626e3918191af1e343757e0fc5f9fa620ce7e655482aaab7303710167e3041 148e3e6aae44ebe904d96a7dea9281fdbdf93b03ccd08a315df83531214f1726 dd6eff4d5f494262983201ee2afd6cc77eac76155fc59d9e42a5c45735a37af6 c3c9c4aa7c535717838a1ce8ca3424188a3ddc2bb7f58483ef08fe9b6ea090c8 bd5a639bb56a54bed34a443f0f1624f13fffd6b4f24024573a6cdc110f2a85b3 f669c706b0bcf205ee9ff8342b38525e9cbf1f1b772a71c7718b30bd59f46490 dd92c121c9fd0b99edf33c57611ce19ca35fe17079c90ce99afdfd9227184562 f4e2dc14aea6d0989a8f01f17c0983ae121614d8f578d34ead9c05a88007bc4f 90e135d97abf57c1e32fd94ed63ceeddab6c1d42f3238e6cf85a063d1239a1d9 c5de6f9e7954b873ccef98d8b44f1761e66da28b9759a24f181c9f7ca8ca190d 4d5a52b1286ad43b5f8caebe64283da13e6a9e66536a821ebc1b50e98073d79e 335c244c2b8efcd7c0b223983db1608d6f1837ae6f73d970e567960e7e03d1a6 5f7dcaeecd4f9afd0266592ed051f800ee3d5a8954460f3f57a9fb9716f4dca2 616d5e77e19918f2b1c290fa9d049267df21a4b33a06e29c758e0ecac9746be0 b7988da1634f2e7162c2c483f1c8d78eaefc94e907d3bcc79cc389979ab0dc15 bf1b2371fe907e9c6f46e26260566e38ed55175ce116e4da7024db03b6657b44 1543e575e850440d7613b96baa8320e2cde5ab834d1f333a9a749c3ab079546c 3429c90c61bfda4b758861c2e351866a18c4d14d3bee4d863edce298231a71dc 9e4c701ac2d01fead65c2244a81865874b1c22125e9f11720450c10366eb6eec 694ba3acd45ea6f9cda354cc938222a523494fa570ba76d359afd28f6ca82c35 dc9890f6e09346d93734ef8c82d7b62344fb8103a1fe6f84fb56f55e1969c4bb f61c916f5fc7aaf919d620ed2c25d2bac648ede51f981afe1d4ecf8263830258 75069803a3032b0fba0522 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: 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 44 /comma put dup 48 /zero put dup 49 /one put dup 50 /two put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 57 /nine put dup 67 /C put dup 68 /D put dup 72 /H put dup 73 /I put dup 74 /J put dup 77 /M put dup 80 /P put dup 84 /T put dup 85 /U put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 121 /y put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306457c7e62c1d269f306bd3402e266de fc3b5e7d8a8d2f5bf0fe6ddd40d07391df4fad4a6018dce29a2b8f692b29f202 3a7c0e66de8ed85c14f1f8492167357f51a7e84cc5d92e0fee4d81cf7fbc8de5 2d2e7bb57142033993f9c08c315abade8dbc4a732e84e142d3bee51557910e12 cd8aa37c459a5e6b7f5269f59078aba3be4641a11ac48d0b625c8325b38ec08e 4c9e5e7fed976a5650d99d82114f449b9ca14c4ec957702295a39a93ef93f618 99b8ea06b092c3c1e503e6e436e0a9fa22576c8930ab3dc8c20f5d82b69cddf8 ff4dacfa9c54bed5a3aa3ea5b129fe96be632843b9b6bc91b615581a985db56b 1e01ca60ee69ca92cf5c0882ece62edad3e106d835348822400f0b66af658f2a 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%%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR17 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 46 /period put dup 73 /I put dup 76 /L put dup 79 /O put dup 80 /P put dup 83 /S put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 103 /g put dup 105 /i put dup 108 /l put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 121 /y put readonly def /FontBBox{-33 -250 945 749}readonly def /UniqueID 5000795 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 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1104 y(It)19 b(is)h(called)e Fm(\013)p Fo(-singular)j(if)e(it)g(is)g(supp)q(orted)i(on)f(some)f(set) h Fm(S)i Fo(with)e Fm(h)1342 1086 y Fh(\013)1367 1104 y Fo(\()p Fm(S)s Fo(\))f(=)h(0.)31 b(W)l(e)20 b(sa)o(y)f(that)h Fm(\026)g Fo(is)0 1164 y(zero-dimensional)c(if)f(it)h(is)g Fm(\013)p Fo(-singular)h(for)g(ev)o(ery)d Fm(\013)h(>)e Fo(0.)90 1242 y(Giv)o(en)i(a)i(\(p)q(ositiv)o(e,)e(\014nite\))h (measure)f Fm(\026)i Fo(and)g Fm(\013)d Fj(2)g Fo([0)p Fm(;)8 b Fo(1],)16 b(w)o(e)f(de\014ne)616 1377 y Fm(D)657 1357 y Fh(\013)656 1389 y(\026)683 1377 y Fo(\()p Fm(x)p Fo(\))e Fj(\021)h Fo(lim)8 b(sup)856 1415 y Fh(\017)p Fi(!)p Fl(0)978 1343 y Fm(\026)p Fo(\(\()p Fm(x)j Fj(\000)g Fm(\017;)k(x)c Fo(+)g Fm(\017)p Fo(\)\))p 978 1365 352 2 v 1100 1411 a(\(2)p Fm(\017)p Fo(\))1182 1397 y Fh(\013)1850 1377 y Fo(\(2.2\))0 1513 y(and)g Fm(T)118 1520 y Fi(1)169 1513 y Fj(\021)j(f)p Fm(x)8 b Fj(j)g Fm(D)346 1495 y Fh(\013)345 1525 y(\026)371 1513 y Fo(\()p Fm(x)p Fo(\))14 b(=)g Fj(1g)p Fo(.)19 b(The)11 b(restriction)f Fm(\026)p Fo(\()p Fm(T)1011 1520 y Fi(1)1048 1513 y Fj(\\)e(\001)g Fo(\))14 b Fj(\021)g Fm(\026)1226 1520 y Fh(\013s)1278 1513 y Fo(is)d Fm(\013)p Fo(-singular,)h(and)g Fm(\026)p Fo(\(\()p Fk(R)f Fj(n)g Fm(T)1832 1520 y Fi(1)1869 1513 y Fo(\))g Fj(\\)19 b(\001)8 b Fo(\))0 1573 y Fj(\021)27 b Fm(\026)95 1580 y Fh(\013c)160 1573 y Fo(is)d Fm(\013)p Fo(-con)o(tin)o(uous.)45 b(Th)o(us,)26 b(eac)o(h)e(measure)g(decomp)q (oses)g(uniquely)f(in)o(to)h(an)h Fm(\013)p Fo(-con)o(tin)o(uous)0 1633 y(part)20 b(and)g(an)g Fm(\013)p Fo(-singular)g(part:)28 b Fm(\026)20 b Fo(=)f Fm(\026)779 1640 y Fh(\013c)833 1633 y Fo(+)13 b Fm(\026)913 1640 y Fh(\013s)954 1633 y Fo(.)31 b(Moreo)o(v)o(er,)18 b(an)i Fm(\013)p Fo(-singular)h(measure) e(m)o(ust)g(ha)o(v)o(e)0 1693 y Fm(D)41 1675 y Fh(\013)40 1706 y(\026)67 1693 y Fo(\()p Fm(x)p Fo(\))e(=)h Fj(1)h Fo(a.e.)f(\(with)g(resp)q(ect)h(to)g(it\))f(and)i(an)f Fm(\013)p Fo(-con)o(tin)o(uous)g(measure)g(m)o(ust)f(ha)o(v)o(e)g Fm(D)1735 1675 y Fh(\013)1734 1706 y(\026)1760 1693 y Fo(\()p Fm(x)p Fo(\))g Fm(<)g Fj(1)0 1753 y Fo(a.e..)24 b(It)16 b(is)i(imp)q(ortan)o(t)f(to)h(note)f(that)h Fm(D)765 1735 y Fh(\013)764 1766 y(\026)790 1753 y Fo(\()p Fm(x)p Fo(\))f(is)g(de\014ned)g(with)h(a)f(lim)8 b(sup.)24 b(The)17 b(corresp)q(onding)i(limit)0 1814 y(need)d(not)h(exist.)90 1892 y(The)h(study)h(of)g(the)f(sp)q(ectral)g(measure)g Fm(\026)h Fo(of)g(an)g(op)q(erator)h(of)e(the)g(form)h(\(1.1\))g(is)f (related)f(to)i(the)0 1952 y(study)g(of)h(the)e(W)l(eyl-Titc)o(hmarsh)g Fm(m)p Fo(-function.)29 b(W)l(e)19 b(will)f(brie\015y)g(list)g(the)h (necessary)g(related)f(facts.)0 2012 y(More)e(details)g(can)g(b)q(e)h (found,)f(for)g(example,)f(in)h([5].)90 2090 y(Let)11 b Fk(Z)205 2069 y Fl(+)249 2090 y Fo(=)j Fj(f)p Fo(1)p Fm(;)8 b Fo(2)p Fm(;)g Fo(3)p Fm(;)g(:)g(:)g(:)o Fj(g)p Fm(;)j Fo(and)h Fk(Z)694 2069 y Fi(\000)737 2090 y Fo(=)i Fj(f)p Fm(:)8 b(:)g(:)f(;)h Fj(\000)p Fo(2)p Fm(;)g Fj(\000)p Fo(1)p Fm(;)g Fo(0)p Fj(g)p Fm(:)j 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b(no)o(w)g(recall)f(the)h(main)g(construction)g (of)h([17].)k(Giv)o(en)15 b(an)i(op)q(erator)0 1949 y Fm(H)j Fo(of)c(the)f(form)g(de\014ned)h(b)o(y)f(\(1.1\),)g(w)o(e)g (\014x)h Fm(E)g Fj(2)f Fk(R)o Fo(,)h(and)g(let)f Fm(u)1173 1928 y Fi(\006)1173 1961 y Fl(1)p Fh(;\014)1239 1949 y Fo(b)q(e)h Fm(u)1333 1928 y Fi(\006)1333 1961 y Fl(1)p Fh(;\014)r(;E)1421 1949 y Fm(;)f Fo(the)g(normalized)g(solution)0 2013 y(of)h(\(1.3\),)g(de\014ned)g(on,)f(corresp)q(ondingly)l(,)h Fk(Z)828 1992 y Fl(+)874 2013 y Fo(or)g Fk(Z)966 1992 y Fi(\000)996 2013 y Fm(;)f Fo(and)i(ob)q(eying)f(the)f(b)q(oundary)i (conditions)f(\(2.3\).)0 2073 y(Let)g Fm(u)115 2052 y Fi(\006)115 2086 y Fl(2)p Fh(;\014)182 2073 y Fo(b)q(e)h Fm(u)277 2052 y Fi(\006)277 2086 y Fl(2)p Fh(;\014)r(;E)365 2073 y Fm(;)f Fo(the)g(solution)g(ob)q(eying)h(the)f(orthogonal)i(b)q (oundary)f(conditions.)90 2155 y(F)l(or)h(an)o(y)g(function)f Fm(u)f Fo(:)g Fk(Z)572 2134 y Fl(+)618 2155 y Fj(!)g Fk(R)p Fo(,)i(w)o(e)f(denote)h(b)o(y)f Fj(k)p 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Fo(of)g(op)q(erators)h(acting)f(on)h Fk(Z)1151 1682 y Fl(+)1180 1703 y Fm(;)f Fo(it)f(will)g(b)q(e)h(con)o (v)o(enien)o(t)e(for)i(us)g(to)h(\014nd)0 1763 y(an)e(expression)f(for) g Fm(M)5 b Fo(\()p Fm(z)r Fo(\))17 b(that)f(uses)h(only)f Fm(m)868 1745 y Fl(+)897 1763 y Fm(:)90 1841 y Fo(W)l(e)11 b(observ)o(e)f(that)h(the)g(space)g Fm(`)661 1823 y Fl(2)681 1841 y Fo(\()p Fk(Z)733 1820 y Fi(\000)763 1841 y Fo(\))g(is)f (naturally)h(iden)o(ti\014ed)e(with)i Fm(`)1374 1823 y Fl(2)1394 1841 y Fo(\()p Fk(Z)1446 1820 y Fl(+)1476 1841 y Fo(\))g(b)o(y)f(a)h(unitary)g(op)q(erator)0 1901 y Fm(U)26 b Fo(:)20 b Fm(`)113 1883 y Fl(2)133 1901 y Fo(\()p Fk(Z)p Fo(\))h Fj(!)f Fm(`)315 1883 y Fl(2)336 1901 y Fo(\()p Fk(Z)p Fo(\),)g(de\014ned)g(b)o(y)g(\()p Fm(U)5 b( )r Fo(\)\()p Fm(n)p Fo(\))21 b(=)f Fm( )r Fo(\()p Fj(\000)p Fm(n)13 b Fo(+)h(1\))p Fm(;)29 b(n)20 b Fj(2)h Fk(Z)p Fm(:)f Fo(Clearly)l(,)g Fm(U)1605 1883 y Fl(2)1645 1901 y Fo(is)g(the)g(iden)o(tit)o(y)0 1962 y(op)q(erator.)i(F)l(or)17 b(an)o(y)f(op)q(erator)h Fm(H)j Fo(on)d Fm(`)739 1944 y Fl(2)759 1962 y Fo(\()p Fk(Z)p Fo(\),)f(w)o(e)f(de\014ne)h(an)g(op)q (erator)1348 1949 y(~)1336 1962 y Fm(H)k Fo(on)d Fm(`)1484 1944 y Fl(2)1504 1962 y Fo(\()p Fk(Z)p Fo(\))f(b)o(y)1671 1949 y(~)1659 1962 y Fm(H)i Fo(=)13 b Fm(U)5 b(H)t(U)1888 1944 y Fi(\000)p Fl(1)1936 1962 y Fm(:)0 2022 y Fo(F)l(or)18 b(op)q(erators)h(of)f(the)f(form)h(\(1.1\),)709 2009 y(~)697 2022 y Fm(H)k Fo(will)16 b(b)q(e)i Fm(H)k Fo(with)17 b(a)i(p)q(oten)o(tial)e(re\015ected)f(with)i(resp)q(ect)f(to)h(the)0 2082 y(p)q(oin)o(t)j(1)p Fm(=)p Fo(2)p Fm(:)h Fo(Let)30 b(~)-33 b Fm(m)375 2064 y Fl(+)404 2082 y Fm(;)39 b Fo(~)-33 b Fm(m)491 2061 y Fl(+)491 2095 y Fh(\014)520 2082 y Fm(;)33 b Fo(~)-27 b Fm(u)592 2061 y Fi(\006)592 2095 y Fh(\014)621 2082 y Fm(;)671 2069 y Fo(~)665 2082 y Fm(L)698 2061 y Fi(\006)698 2095 y Fh(\014)728 2082 y Fm(;)42 b Fo(denote,)22 b(corresp)q(ondingly)l(,)g Fm(m)1372 2064 y Fl(+)1401 2082 y Fm(;)30 b(m)1488 2061 y Fl(+)1488 2095 y Fh(\014)1517 2082 y Fm(;)f(u)1588 2061 y Fi(\006)1588 2095 y Fh(\014)1639 2082 y Fo(and)21 b Fm(L)1771 2061 y Fi(\006)1771 2095 y Fh(\014)1822 2082 y Fo(of)g(the)0 2151 y(op)q(erator)209 2138 y(~)196 2151 y Fm(H)5 b(:)16 b Fo(Then)g(it)g(is)g(immediate)e(from)i(\(2.6\))h(that)839 2283 y Fm(m)882 2262 y Fi(\000)925 2283 y Fo(=)d Fj(\000)1044 2249 y Fo(1)p 1021 2271 73 2 v 1030 2317 a(~)-33 b Fm(m)1064 2303 y Fl(+)1098 2283 y Fm(:)738 b Fo(\(3.5\))0 2409 y(Using)23 b(\(2.5\),\(2.7\),)h(and)g(\(3.5\),)g(w)o(e)e(obtain)i(b)o (y)e(an)i(elemen)o(tary)d(computation)i(that)g(for)g(an)o(y)g Fm(\014)28 b Fj(2)0 2469 y Fo(\()p Fj(\000)p Fm(\031)r(=)p Fo(2)p Fm(;)8 b(\031)r(=)p Fo(2],)661 2526 y Fm(m)704 2505 y Fl(+)704 2539 y Fh(\014)733 2526 y Fo(\()p Fm(z)r Fo(\))h(~)-33 b Fm(m)839 2505 y Fl(+)839 2540 y Fh(\031)q(=)p Fl(2)p Fi(\000)p Fh(\014)947 2526 y Fo(\()p Fm(z)r Fo(\))10 b Fj(\000)h Fo(1)p 661 2555 434 2 v 674 2601 a Fm(m)717 2581 y Fl(+)717 2614 y Fh(\014)746 2601 y Fo(\()p Fm(z)r Fo(\))g(+)20 b(~)-33 b Fm(m)912 2581 y Fl(+)912 2616 y Fh(\031)q(=)p Fl(2)p Fi(\000)p Fh(\014)1019 2601 y Fo(\()p Fm(z)r Fo(\))1113 2567 y(=)14 b Fm(M)5 b Fo(\()p Fm(z)r Fo(\))p Fm(:)556 b Fo(\(3.6\))963 2828 y(7)p eop %%Page: 8 9 8 8 bop 0 203 a Fo(Also,)14 b(w)o(e)h(will)e(use)i(the)g(fact)f(that)h (the)g(op)q(erator)h Fm(U;)e Fo(applied)g(formally)l(,)g(sends)h (sequences)f Fm(u)f Fo(:)h Fk(Z)1807 182 y Fi(\006)1850 203 y Fj(!)g Fk(R)0 264 y Fo(in)o(to)i(sequences)f(\()p Fm(U)5 b(u)p Fo(\))14 b(:)g Fk(Z)499 242 y Fi(\007)542 264 y Fj(!)g Fk(R)p Fm(:)i Fo(It)f(follo)o(ws)i(immediately)c(from)j (the)g(de\014nition)g(that)707 367 y Fm(U)5 b(u)773 346 y Fi(\000)773 379 y Fh(i;\014)832 367 y Fo(=)17 b(~)-27 b Fm(u)912 346 y Fl(+)912 381 y Fh(i;\031)q(=)p Fl(2)p Fi(\000)p Fh(\014)1041 367 y Fm(;)22 b(i)13 b Fo(=)h(1)p Fm(;)8 b Fo(2)p Fm(:)607 b Fo(\(3.7\))0 470 y(It)16 b(is)g(also)h (clear)e(from)h(\(2.10\),\(2.11\),\(2.12\))i(that)846 560 y(~)840 573 y Fm(L)873 552 y Fl(+)873 587 y Fh(\031)q(=)p Fl(2)p Fi(\000)p Fh(\014)995 573 y Fo(=)c Fm(L)1080 552 y Fi(\000)1080 585 y Fh(\014)1850 573 y Fo(\(3.8\))0 676 y(and)813 736 y Fj(k)p Fm(u)p Fj(k)891 743 y Fh(L)930 736 y Fo(=)g Fj(k)p Fm(U)5 b(u)p Fj(k)1098 743 y Fh(L)1124 736 y Fm(:)712 b Fo(\(3.9\))0 849 y Fe(LEMMA)19 b(5)24 b Fd(Supp)n(ose)13 b(that)f(ther)n(e)h(exists)g(a)g Fm(\014)j Fj(2)e Fo(\()p Fj(\000)p Fm(\031)r(=)p Fo(2)p Fm(;)8 b(\031)r(=)p Fo(2])k Fd(and)g(a)h(se)n(quenc)n(e)h Fm(\017)1541 856 y Fh(j)1573 849 y Fj(!)f Fo(0)p Fd(,)h(such)f(that)g(for)0 910 y(every)i Fm(E)i Fd(in)e(some)f(Bor)n(el)g(set)h Fm(A)p Fd(,)f(we)i(have)e(that)h Fm(\017)917 889 y Fl(1)p Fi(\000)p Fh(\013)917 921 y(j)986 910 y Fm(m)1029 889 y Fl(+)1029 922 y Fh(\014)1059 910 y Fo(\()p Fm(E)7 b Fo(+)t Fm(i\017)1200 917 y Fh(j)1217 910 y Fo(\))14 b Fj(!)f(1)i Fd(and)f Fm(\017)1489 889 y Fl(1)p Fi(\000)p Fh(\013)1489 921 y(j)1568 910 y Fo(~)-33 b Fm(m)1602 889 y Fl(+)1602 924 y Fh(\031)q(=)p Fl(2)p Fi(\000)p Fh(\014)1709 910 y Fo(\()p Fm(E)7 b Fo(+)t Fm(i\017)1850 917 y Fh(j)1867 910 y Fo(\))14 b Fj(!)0 970 y(1)j Fd(as)h Fm(j)e Fj(!)e(1)p Fm(:)j Fd(Then)h(the)g(r)n(estriction)f Fm(\026)p Fo(\()p Fm(A)11 b Fj(\\)20 b(\001)8 b Fo(\))17 b Fd(is)h Fm(\013)p Fd(-singular.)0 1084 y Fe(Pro)r(of.)j Fo(F)l(rom)15 b(\(3.6\))h(w)o(e)f(immediately)d(obtain)k(that)g(under)f (the)h(conditions)f(of)h(the)f(Lemma)g(w)o(e)g(ha)o(v)o(e)0 1144 y Fm(\017)20 1123 y Fl(1)p Fi(\000)p Fh(\013)20 1155 y(j)90 1144 y Fm(M)5 b Fo(\()p Fm(E)14 b Fo(+)d Fm(i\017)297 1151 y Fh(j)315 1144 y Fo(\))i Fj(!)h(1)i Fo(as)h Fm(j)g Fj(!)c(1)p Fm(:)j Fo(Th)o(us,)g(\(2.9\))h(yields)e(the)h (result.)p 1366 1148 21 30 v 90 1268 a(The)i(rest)f(of)h(the)f(pro)q (of)i(uses)f(k)o(ey)e(ingredien)o(ts)h(from)g(the)h(pro)q(ofs)g(of)g (Corollaries)g(4.1)g(and)g(4.2)g(of)0 1328 y([17].)j(F)l(rom)16 b(\(3.3\),\(3.4\))g(w)o(e)g(see)g(that)h(for)f(an)o(y)g Fm(\014)s Fo(,)g(w)o(e)f(ha)o(v)o(e)115 1431 y Fj(k)p Fo(\010)175 1410 y Fi(\006)175 1443 y Fh(n)205 1431 y Fo(\()p Fm(E)s Fo(\))p Fj(k)307 1410 y Fl(2)340 1431 y Fj(\024)f(j)p Fm(u)435 1410 y Fi(\006)435 1443 y Fl(1)p Fh(;\014)485 1431 y Fo(\()p Fj(\006)p Fo(\()p Fm(n)d Fo(+)g(1\)\))p Fj(j)727 1410 y Fl(2)758 1431 y Fo(+)g Fj(j)p Fm(u)849 1410 y Fi(\006)849 1443 y Fl(1)p Fh(;\014)900 1431 y Fo(\()p Fj(\006)p Fo(\()p Fm(n)p Fo(\)\))p Fj(j)1058 1410 y Fl(2)1088 1431 y Fo(+)g Fj(j)p Fm(u)1179 1410 y Fi(\006)1179 1443 y Fl(2)p Fh(;\014)1230 1431 y Fo(\()p Fj(\006)p Fo(\()p Fm(n)g Fo(+)g(1\)\))p Fj(j)1472 1410 y Fl(2)1503 1431 y Fo(+)g Fj(j)p Fm(u)1594 1410 y Fi(\006)1594 1443 y Fl(2)p Fh(;\014)1644 1431 y Fo(\()p Fj(\006)p Fo(\()p Fm(n)p Fo(\)\))p Fj(j)1802 1410 y Fl(2)1821 1431 y Fm(;)0 1540 y Fo(and)16 b(therefore)e(w)o(e)h(ha)o(v)o(e)f Fj(k)p Fm(u)532 1519 y Fi(\006)532 1552 y Fl(1)p Fh(;\014)583 1540 y Fj(k)608 1522 y Fl(2)608 1552 y Fh(L)643 1540 y Fo(+)9 b Fj(k)p Fm(u)743 1519 y Fi(\006)743 1552 y Fl(2)p Fh(;\014)793 1540 y Fj(k)818 1522 y Fl(2)818 1552 y Fh(L)858 1540 y Fj(\025)916 1520 y Fl(1)p 916 1528 18 2 v 916 1557 a(2)947 1506 y Ff(P)990 1520 y Fh(L)990 1552 y(n)p Fl(=1)1067 1540 y Fj(k)p Fo(\010)1127 1522 y Fi(\006)1127 1552 y Fh(n)1157 1540 y Fo(\()p Fm(E)s Fo(\))p Fj(k)1259 1522 y Fl(2)1278 1540 y Fm(:)15 b Fo(Th)o(us)h(,)f (under)g(the)g(conditions)g(of)0 1600 y(Lemma)h(4,)g(for)h(an)o(y)f Fm(L)e Fj(\025)g Fm(j)518 1582 y Fi(\006)515 1612 y Fh(n)563 1600 y Fo(and)j(an)o(y)f Fm(\014)g Fj(2)e Fo(\()p Fj(\000)p Fm(\031)r(=)p Fo(2)p Fm(;)8 b(\031)r(=)p Fo(2],)16 b(w)o(e)g(obtain)708 1714 y Fj(k)p Fm(u)761 1694 y Fi(\006)761 1727 y Fl(1)p Fh(;\014)812 1714 y Fj(k)837 1694 y Fl(2)837 1727 y Fh(L)874 1714 y Fo(+)11 b Fj(k)p Fm(u)976 1694 y Fi(\006)976 1727 y Fl(2)p Fh(;\014)1027 1714 y Fj(k)1052 1694 y Fl(2)1052 1727 y Fh(L)1091 1714 y Fj(\025)1149 1681 y Fm(a)p 1149 1703 26 2 v 1150 1749 a Fo(2)1180 1714 y Fm(k)1207 1694 y Fh(\016)1205 1727 y(n)1228 1714 y Fm(:)583 b Fo(\(3.10\))0 1833 y(In)17 b(particular,)f(inequalit)o(y)f(\(3.10\))i(holds)g(for)g (an)o(y)g Fm(L)e Fj(\025)g Fm(ck)1113 1840 y Fh(n)1137 1833 y Fm(:)h Fo(Since)g(for)h(a.e.)f Fm(E)k Fo(w.r.t.)c Fm(\026)p Fo(,)g(there)h(exists)0 1893 y(a)i(phase)g Fm(\014)s Fo(\()p Fm(E)s Fo(\))e Fj(2)h Fo(\()p Fj(\000)p Fm(\031)r(=)p Fo(2)p Fm(;)8 b(\031)r(=)p Fo(2])19 b(suc)o(h)f(that)h (\(2.14\))g(holds)g(for)g(b)q(oth)h Fm(u)1342 1872 y Fi(\000)1342 1907 y Fl(1)p Fh(;\014)r Fl(\()p Fh(E)r Fl(\))1466 1893 y Fo(and)f Fm(u)1591 1872 y Fl(+)1591 1907 y(1)p Fh(;\014)r Fl(\()p Fh(E)r Fl(\))1697 1893 y Fm(;)g Fo(w)o(e)f(deduce)0 1953 y(that)f(for)f(suc)o(h)g Fm(E)329 1935 y Fi(0)341 1953 y Fo(s)g(and)h(large)f Fm(n)p Fo('s,)792 2028 y Fj(k)p Fm(u)845 2007 y Fi(\006)845 2041 y Fl(2)p Fh(;\014)896 2028 y Fj(k)921 2035 y Fh(ck)954 2039 y Fa(n)992 2028 y Fj(\025)d Fm(a)1070 2035 y Fl(1)1090 2028 y Fm(k)1122 1984 y Fa(\016)p 1122 1990 16 2 v 1122 2011 a Fg(2)1115 2032 y Fh(n)1144 2028 y Fm(;)667 b Fo(\(3.11\))0 2135 y(for)17 b(some)g Fm(a)224 2142 y Fl(1)259 2135 y Fm(>)e Fo(0)p Fm(:)i Fo(T)l(ak)o(e)g Fm(\017)506 2142 y Fh(n)545 2135 y Fo(=)e Fm(k)625 2105 y Fi(\000)660 2091 y Fg(1)p 657 2097 21 2 v 657 2118 a Fa(\013)623 2139 y Fh(n)684 2135 y Fm(:)i Fo(Then,)g(b)o(y)g(\(2.12\),\(2.14\),)g (for)h(a.e.)e Fm(E)k Fo(w.r.t.)c Fm(\026)i Fo(and)f(large)g Fm(n)p Fo(,)g(w)o(e)0 2195 y(get)501 2250 y Fj(k)p Fm(u)554 2229 y Fi(\006)554 2264 y Fl(2)p Fh(;\014)r Fl(\()p Fh(E)r Fl(\))660 2250 y Fj(k)685 2263 y Fh(L)709 2249 y Fb(\006)709 2277 y Fa(\014)q Fg(\()p Fa(E)q Fg(\))778 2263 y Fl(\()p Fh(\017)806 2267 y Fa(n)827 2263 y Fl(\))p 501 2290 342 2 v 501 2336 a Fj(k)p Fm(u)554 2315 y Fi(\006)554 2350 y Fl(1)p Fh(;\014)r Fl(\()p Fh(E)r Fl(\))660 2336 y Fj(k)685 2350 y Fh(L)709 2335 y Fb(\006)709 2363 y Fa(\014)q Fg(\()p Fa(E)q Fg(\))778 2350 y Fl(\()p Fh(\017)806 2354 y Fa(n)827 2350 y Fl(\))861 2302 y Fm(>)1163 2268 y(c)1184 2275 y Fl(1)p 918 2290 531 2 v 918 2338 a Fo(2)p Fm(\017)962 2345 y Fh(n)986 2338 y Fm(L)1019 2317 y Fi(\006)1019 2352 y Fh(\014)r Fl(\()p Fh(E)r Fl(\))1098 2338 y Fo(\()p Fm(\017)1137 2345 y Fh(n)1160 2338 y Fo(\))8 b(ln)1228 2317 y Fl(2)1256 2338 y Fm(L)1289 2317 y Fi(\006)1289 2352 y Fh(\014)r Fl(\()p Fh(E)r Fl(\))1368 2338 y Fo(\()p Fm(\017)1407 2345 y Fh(n)1430 2338 y Fo(\))1825 2302 y(\(3.12\))0 2439 y(for)18 b(some)g(p)q(ositiv)o(e)g(constan)o(t)g Fm(c)602 2446 y Fl(1)622 2439 y Fo(.)26 b(If)18 b(it)f(so)i(happ)q(ens) g(that)f Fm(L)1158 2418 y Fi(\006)1158 2453 y Fh(\014)r Fl(\()p Fh(E)r Fl(\))1237 2439 y Fo(\()p Fm(\017)1276 2446 y Fh(n)1300 2439 y Fo(\))e Fm(<)h(ck)1436 2446 y Fh(n)1460 2439 y Fo(,)h(for)g(either)f(the)h(+)g(or)g Fj(\000)0 2499 y Fo(case,)e(then)g(\(3.12\))h(implies)d(that)j(for)g (some)f Fm(c)858 2506 y Fl(2)891 2499 y Fm(>)e Fo(0,)688 2600 y Fj(k)p Fm(u)741 2580 y Fi(\006)741 2615 y Fl(2)p Fh(;\014)r Fl(\()p Fh(E)r Fl(\))846 2600 y Fj(k)871 2614 y Fh(L)895 2599 y Fb(\006)895 2628 y Fa(\014)q Fg(\()p Fa(E)q Fg(\))964 2614 y Fl(\()p Fh(\017)992 2618 y Fa(n)1013 2614 y Fl(\))p 688 2641 342 2 v 688 2686 a Fj(k)p Fm(u)741 2666 y Fi(\006)741 2701 y Fl(1)p Fh(;\014)r Fl(\()p Fh(E)r Fl(\))846 2686 y Fj(k)871 2700 y Fh(L)895 2685 y Fb(\006)895 2714 y Fa(\014)q Fg(\()p Fa(E)q Fg(\))964 2700 y Fl(\()p Fh(\017)992 2704 y Fa(n)1013 2700 y Fl(\))1048 2652 y Fm(>)f(c)1120 2659 y Fl(2)1155 2619 y Fm(k)1187 2574 y Fg(1)p Fb(\000)p Fa(\013)p 1187 2581 60 2 v 1206 2601 a(\013)1180 2622 y Fh(n)p 1145 2641 118 2 v 1145 2689 a Fo(ln)1186 2667 y Fl(2)1214 2689 y Fm(k)1239 2696 y Fh(n)1825 2652 y Fo(\(3.13\))963 2828 y(8)p eop %%Page: 9 10 9 9 bop 0 203 a Fo(for)17 b(the)f(corresp)q(onding)i(case.)23 b(Otherwise)16 b(\(namely)l(,)f(if)h Fm(L)1104 183 y Fi(\006)1104 218 y Fh(\014)r Fl(\()p Fh(E)r Fl(\))1183 203 y Fo(\()p Fm(\017)1222 210 y Fh(n)1245 203 y Fo(\))f Fj(\025)f Fm(ck)1378 210 y Fh(n)1402 203 y Fo(\),)i(\(2.12\))i(and)f (\(3.11\))g(imply)0 264 y(that)g(for)f Fm(E)h Fj(2)d Fm(A)p Fo(,)568 318 y Fj(k)p Fm(u)621 298 y Fi(\006)621 333 y Fl(2)p Fh(;\014)r Fl(\()p Fh(E)r Fl(\))727 318 y Fj(k)752 332 y Fh(L)776 317 y Fb(\006)776 346 y Fa(\014)q Fg(\()p Fa(E)q Fg(\))844 332 y Fl(\()p Fh(\017)872 336 y Fa(n)893 332 y Fl(\))p 568 359 342 2 v 568 404 a Fj(k)p Fm(u)621 384 y Fi(\006)621 419 y Fl(1)p Fh(;\014)r Fl(\()p Fh(E)r Fl(\))727 404 y Fj(k)752 418 y Fh(L)776 403 y Fb(\006)776 432 y Fa(\014)q Fg(\()p Fa(E)q Fg(\))844 418 y Fl(\()p Fh(\017)872 422 y Fa(n)893 418 y Fl(\))928 370 y Fj(\025)f Fo(2)p Fm(a)1030 350 y Fl(2)1030 382 y(1)1050 370 y Fm(k)1077 340 y Fh(\016)q Fi(\000)1129 326 y Fg(1)p 1126 332 21 2 v 1126 353 a Fa(\013)1075 374 y Fh(n)1167 370 y Fo(=)h(2)p Fm(a)1269 350 y Fl(2)1269 382 y(1)1289 370 y Fm(k)1321 326 y Fg(1)p Fb(\000)p Fa(\013)p 1321 333 60 2 v 1341 353 a(\013)1314 374 y Fh(n)1825 370 y Fo(\(3.14\))0 507 y(for)j(this)g(case.)22 b(Th)o(us,)17 b(w)o(e)g(see)f(that)h(in)g(either)e(case,)i(for)g(a)g(set)g(of)g Fm(E)s Fo('s)f(of)h(full)f Fm(\026)p Fo(\()p Fm(A)c Fj(\\)19 b(\001)8 b Fo(\))17 b(measure,)f(w)o(e)0 567 y(ha)o(v)o(e)f(p)q(ositiv) o(e)h(constan)o(ts)h Fm(C)t Fo(\()p Fm(E)s Fo(\),)e(suc)o(h)h(that)h (for)f(large)h Fm(n)p Fo(,)643 675 y Fj(k)p Fm(u)696 655 y Fi(\006)696 690 y Fl(2)p Fh(;\014)r Fl(\()p Fh(E)r Fl(\))802 675 y Fj(k)827 689 y Fh(L)851 674 y Fb(\006)851 703 y Fa(\014)q Fg(\()p Fa(E)q Fg(\))920 689 y Fl(\()p Fh(\017)948 693 y Fa(n)969 689 y Fl(\))p 643 716 342 2 v 643 762 a Fj(k)p Fm(u)696 741 y Fi(\006)696 776 y Fl(1)p Fh(;\014)r Fl(\()p Fh(E)r Fl(\))802 762 y Fj(k)827 775 y Fh(L)851 760 y Fb(\006)851 789 y Fa(\014)q Fg(\()p Fa(E)q Fg(\))920 775 y Fl(\()p Fh(\017)948 779 y Fa(n)969 775 y Fl(\))1003 727 y Fm(>)d(C)t Fo(\()p Fm(E)s Fo(\))1185 694 y Fm(k)1217 649 y Fg(1)p Fb(\000)p Fa(\013)p 1217 656 60 2 v 1237 677 a(\013)1210 697 y Fh(n)p 1176 716 118 2 v 1176 764 a Fo(ln)1216 742 y Fl(2)1244 764 y Fm(k)1269 771 y Fh(n)1298 727 y Fm(:)513 b Fo(\(3.15\))0 887 y(W)l(e)16 b(th)o(us)g(obtain)h(b)o(y)f(\(2.13\),)g(for)h(an)o(y)f Fm(\017)d(>)h Fo(0)p Fm(;)181 1046 y(\017)201 1025 y Fl(1)p Fi(\000)p Fh(\013)p Fi(\000)p Fh(\017)201 1058 y(n)313 1046 y Fm(m)356 1025 y Fl(+)356 1058 y Fh(\014)385 1046 y Fo(\()p Fm(E)s Fo(\)\()p Fm(E)g Fo(+)d Fm(i\017)617 1053 y Fh(n)640 1046 y Fo(\))i Fj(\030)h Fm(\017)745 1025 y Fl(1)p Fi(\000)p Fh(\013)p Fi(\000)p Fh(\017)745 1058 y(n)861 994 y Fj(k)p Fm(u)914 973 y Fl(+)914 1008 y(2)p Fh(;\014)r Fl(\()p Fh(E)r Fl(\))1020 994 y Fj(k)1045 1007 y Fh(L)1069 992 y Fg(+)1069 1021 y Fa(\014)q Fg(\()p Fa(E)q Fg(\))1137 1007 y Fl(\()p Fh(\017)1165 1011 y Fa(n)1187 1007 y Fl(\))p 861 1034 342 2 v 861 1080 a Fj(k)p Fm(u)914 1059 y Fl(+)914 1094 y(1)p Fh(;\014)r Fl(\()p Fh(E)r Fl(\))1020 1080 y Fj(k)1045 1094 y Fh(L)1069 1079 y Fg(+)1069 1107 y Fa(\014)q Fg(\()p Fa(E)q Fg(\))1137 1094 y Fl(\()p Fh(\017)1165 1098 y Fa(n)1187 1094 y Fl(\))1221 1046 y Fj(\025)g Fm(C)t Fo(\()p Fm(E)s Fo(\))1423 1012 y Fm(k)1459 968 y Fa(\017)p 1455 974 21 2 v 1455 995 a(\013)1448 1016 y Fh(n)p 1395 1034 118 2 v 1395 1082 a Fo(ln)1435 1061 y Fl(2)1463 1082 y Fm(k)1488 1089 y Fh(n)1531 1046 y Fj(!)f(1)181 b Fo(\(3.16\))0 1189 y(as)17 b Fm(n)d Fj(!)g(1)p Fm(:)90 1267 y Fo(Similarly)l(,)g(using)i(in)g (addition)h(\(3.7\),\(3.8\),\(3.9\),)e(w)o(e)h(get)17 1426 y Fm(\017)37 1406 y Fl(1)p Fi(\000)p Fh(\013)p Fi(\000)p Fh(\017)37 1439 y(n)157 1426 y Fo(~)-33 b Fm(m)191 1406 y Fl(+)191 1441 y Fh(\031)q(=)p Fl(2)p Fi(\000)p Fh(\014)r Fl(\()p Fh(E)r Fl(\))354 1426 y Fo(\()p Fm(E)14 b Fo(+)d Fm(i\017)509 1433 y Fh(n)532 1426 y Fo(\))i Fj(\030)h Fm(\017)637 1406 y Fl(1)p Fi(\000)p Fh(\013)p Fi(\000)p Fh(\017)637 1439 y(n)753 1374 y Fj(k)s Fo(~)-27 b Fm(u)806 1354 y Fl(+)806 1389 y(2)p Fh(;\031)q(=)p Fl(2)p Fi(\000)p Fh(\014)r Fl(\()p Fh(E)r Fl(\))996 1374 y Fj(k)1025 1379 y Fl(~)1021 1388 y Fh(L)1045 1373 y Fg(+)1045 1402 y Fa(\031)q(=)p Fg(2)p Fb(\000)p Fa(\014)q Fg(\()p Fa(E)q Fg(\))1188 1388 y Fl(\()p Fh(\017)1216 1392 y Fa(n)1237 1388 y Fl(\))p 753 1415 500 2 v 753 1461 a Fj(k)s Fo(~)g Fm(u)806 1440 y Fl(+)806 1475 y(1)p Fh(;\031)q(=)p Fl(2)p Fi(\000)p Fh(\014)r Fl(\()p Fh(E)r Fl(\))996 1461 y Fj(k)1025 1466 y Fl(~)1021 1474 y Fh(L)1045 1459 y Fg(+)1045 1488 y Fa(\031)q(=)p Fg(2)p Fb(\000)p Fa(\014)q Fg(\()p Fa(E)q Fg(\))1188 1474 y Fl(\()p Fh(\017)1216 1478 y Fa(n)1237 1474 y Fl(\))1272 1426 y Fo(=)13 b Fm(\017)1343 1406 y Fl(1)p Fi(\000)p Fh(\013)p Fi(\000)p Fh(\017)1343 1439 y(n)1460 1374 y Fj(k)p Fm(u)1513 1354 y Fi(\000)1513 1389 y Fl(2)p Fh(;\014)r Fl(\()p Fh(E)r Fl(\))1618 1374 y Fj(k)1643 1388 y Fh(L)1667 1373 y Fb(\000)1667 1402 y Fa(\014)q Fg(\()p Fa(E)q Fg(\))1736 1388 y Fl(\()p Fh(\017)1764 1392 y Fa(n)1785 1388 y Fl(\))p 1460 1415 342 2 v 1460 1461 a Fj(k)p Fm(u)1513 1440 y Fi(\000)1513 1475 y Fl(1)p Fh(;\014)r Fl(\()p Fh(E)r Fl(\))1618 1461 y Fj(k)1643 1474 y Fh(L)1667 1459 y Fb(\000)1667 1488 y Fa(\014)q Fg(\()p Fa(E)q Fg(\))1736 1474 y Fl(\()p Fh(\017)1764 1478 y Fa(n)1785 1474 y Fl(\))1820 1426 y Fj(!)g(1)1825 1536 y Fo(\(3.17\))0 1596 y(as)k Fm(n)d Fj(!)g(1)p Fm(:)h Fo(Therefore,)h(Lemma)g(5)g(implies)f(Lemma)h(4.)p 1114 1600 21 30 v 90 1724 a(F)l(or)g(the)g(zero-dimensionalit)o(y)f (our)i(main)e(to)q(ol)i(will)f(b)q(e)g(the)g(follo)o(wing)0 1844 y Fe(COR)n(OLLAR)-5 b(Y)20 b(6)k Fd(L)n(et)13 b Fm(H)18 b Fd(b)n(e)c(an)g(op)n(er)n(ator)e(of)h(the)h(form)f(\(1.1\).) 21 b(Supp)n(ose)14 b(that)f(for)g(every)h Fm(E)j Fd(in)d(some)0 1904 y(Bor)n(el)j(set)f Fm(A)p Fd(,)h(ther)n(e)f(exist)h Fm(c)d(>)g Fo(0)j Fd(and)f(se)n(quenc)n(es)i Fm(k)967 1911 y Fh(n)1005 1904 y Fj(!)13 b(1)p Fm(;)22 b(j)1177 1886 y Fi(\006)1174 1916 y Fh(n)1220 1904 y Fj(\024)14 b Fm(ck)1319 1911 y Fh(n)1343 1904 y Fm(;)i Fd(such)g(that)h Fj(k)p Fo(\010)1641 1883 y Fi(\006)1641 1924 y Fh(j)1657 1909 y Fb(\006)1655 1930 y Fa(n)1685 1904 y Fj(k)d(\025)g Fm(e)1800 1886 y Fh(ak)1838 1870 y Fa(\015)1837 1892 y(n)1860 1904 y Fm(;)i Fd(for)0 1972 y(some)i Fm(a;)8 b(\015)16 b(>)e Fo(0)p Fm(:)j Fd(Then)h(the)g(r)n(estriction)f Fm(\026)p Fo(\()p Fm(A)11 b Fj(\\)20 b(\001)8 b Fo(\))17 b Fd(is)h(zer)n(o-dimensional.)0 2092 y Fe(Remark.)28 b Fo(This)19 b(Corollary)g(is)g(still)e(more)i(general)g(than)g(needed) f(for)h(our)h(curren)o(t)e(application)g(\(in)0 2152 y(particular,)e(w)o(e)g(will)g(only)h(use)g(it)f(with)g Fm(\015)i Fo(=)d(1\).)23 b(Ho)o(w)o(ev)o(er,)15 b(this)h(more)h (general)f(form)o(ulation)h(ma)o(y)f(b)q(e)0 2212 y(useful)g(in)g (certain)f(other)i(situations.)90 2326 y Fe(Pro)r(of.)29 b Fo(Clearly)l(,)18 b Fj(k)p Fo(\010)513 2306 y Fi(\006)513 2346 y Fh(j)529 2332 y Fb(\006)527 2352 y Fa(n)558 2326 y Fj(k)g(\025)g Fm(e)681 2308 y Fh(ak)719 2292 y Fa(\015)718 2314 y(n)760 2326 y Fo(implies)938 2269 y Fh(j)954 2255 y Fb(\006)952 2275 y Fa(n)929 2285 y Ff(X)930 2376 y Fh(i)p Fl(=1)997 2326 y Fj(k)p Fo(\010)1057 2306 y Fi(\006)1057 2338 y Fh(j)1087 2326 y Fj(k)1112 2306 y Fl(2)1150 2326 y Fm(>)g(ak)1259 2306 y Fh(\014)1257 2338 y(n)1301 2326 y Fo(for)i(an)o(y)e Fm(\014)j(<)d Fj(1)p Fm(:)h Fo(Therefore,)g(b)o(y)0 2421 y(Lemma)d(4,)g Fm(\026)p Fo(\()p Fm(A)11 b Fj(\\)20 b(\001)8 b Fo(\))16 b(is)g Fm(\013)p Fo(-singular)h(for)g(an)o(y)f Fm(\013)e(>)g Fo(0)p Fm(:)p 1042 2425 V 963 2828 a Fo(9)p eop %%Page: 10 11 10 10 bop 0 203 a Fn(4)104 b(Quasip)r(erio)r(dic)24 b(op)r(erators)g (and)f(zero-dimensional)h(sp)r(ec-)121 295 y(trum.)37 b(Pro)r(of)27 b(of)g(Theorem)g(1)0 422 y Fo(As)14 b(a)h(w)o(arm)o(up,)g (w)o(e)f(\014rst)g(pro)o(v)o(e)g(a)h(half-line)f(v)o(ersion)g(of)h (Theorem)f(1.)21 b(Let)15 b(us)f(consider)h Fm(H)1693 402 y Fl(+)1689 435 y Fh(\014)1722 422 y Fm(;)g Fo(op)q(erators)0 487 y(on)i Fm(`)88 469 y Fl(2)108 487 y Fo(\()p Fk(Z)160 465 y Fl(+)190 487 y Fo(\),)e(de\014ned)h(b)o(y)g(\(1.1\))h(along)g (with)f(a)g(phase)h(b)q(oundary)h(condition)e(\(2.3\).)21 b(In)16 b(case)g(of)h(ergo)q(dic)0 547 y Fm(V)11 b Fo(\()p Fm(n)p Fo(\),)21 b(the)e(Ly)o(apuno)o(v)h(exp)q(onen)o(t)g Fm(\015)s Fo(\()p Fm(E)s Fo(\))g(is)g(de\014ned)f(as)i(b)q(efore)f(and) g(is,)g(clearly)l(,)f(not)h(dep)q(enden)o(t)g(on)0 607 y Fm(\014)s(;)8 b(\022)q(:)0 718 y Fe(THEOREM)18 b(7)25 b Fd(L)n(et)20 b Fm(V)33 b Fd(b)n(e)21 b(as)g(in)g(The)n(or)n(em)f(1.) 33 b(Supp)n(ose)21 b(that)g Fm(\015)s Fo(\()p Fm(E)s Fo(\))g Fm(>)f Fo(0)i Fd(for)e(every)i Fm(E)i Fd(in)d(some)0 779 y(Bor)n(el)e(set)h Fm(A;)e Fd(then)i(for)f(every)g(b)n(oundary)g (phase)g Fm(\014)j Fd(and)d(every)h Fm(\022)q Fd(,)f(the)h(r)n (estriction)f Fm(\026)p Fo(\()p Fm(A)12 b Fj(\\)21 b(\001)8 b Fo(\))p Fd(,)20 b(of)f(the)0 839 y(c)n(orr)n(esp)n(onding)e(sp)n(e)n (ctr)n(al)f(me)n(asur)n(e)h(of)g(the)h(op)n(er)n(ator)e Fm(H)1044 818 y Fl(+)1040 851 y Fh(\014)1074 839 y Fd(,)h(is)h(zer)n (o-dimensional.)0 950 y Fe(Pro)r(of.)j Fo(Recall)15 b(that)i(the)f(upp) q(er)g(Ly)o(apuno)o(v)h(exp)q(onen)o(t)p 1085 924 29 2 v 16 w Fm(\015)s Fo(\()p Fm(E)s Fo(\))f(is)g(de\014ned)g(b)o(y)p 629 1024 V 629 1051 a Fm(\015)s Fo(\()p Fm(E)s Fo(\))d Fj(\021)h Fo(lim)8 b(sup)829 1085 y Fh(n)p Fi(!1)949 1051 y Fo(\(1)p Fm(=n)p Fo(\))g(ln)h Fj(k)p Fo(\010)1182 1058 y Fh(n)1206 1051 y Fo(\()p Fm(E)s Fo(\))p Fj(k)p Fm(:)0 1171 y Fo(Note)j(that,)g(unlik)o(e)f Fm(\015)s Fo(\()p Fm(E)s Fo(\),)h(the)g(upp)q(er)g(Ly)o(apuno)o(v)g(exp)q(onen)o (t)g(of)g(an)h(ergo)q(dic)f(op)q(erator)h Fm(H)1654 1178 y Fh(\022)1686 1171 y Fo(is)f(dep)q(enden)o(t)0 1231 y(on)17 b Fm(\022)q(:)f Fo(W)l(e)g(will)f(use)h(the)g(follo)o(wing)0 1343 y Fe(THEOREM)i(8)25 b(\(Corollary)15 b(4.3)h(of)h([17)o(]\))e Fd(Supp)n(ose)h(that)p 1189 1316 V 16 w Fm(\015)s Fo(\()p Fm(E)s Fo(\))e Fm(>)f Fo(0)j Fd(for)f(every)i Fm(E)h Fd(in)e(some)g(Bor)n(el)0 1403 y(set)i Fm(A)p Fd(,)f(then)h(for)f (every)h(b)n(oundary)f(phase,)g(the)h(r)n(estriction)g Fm(\026)p Fo(\()p Fm(A)11 b Fj(\\)19 b(\001)8 b Fo(\))18 b Fd(is)f(zer)n(o-dimensional.)0 1514 y Fo(Therefore,)h(w)o(e)g(need)g (to)h(pro)o(v)o(e)f(that)h(under)f(the)g(conditions)h(of)g(Theorem)f (7,)h(for)g(ev)o(ery)e Fm(\022)q(;)h Fo(w)o(e)g(ha)o(v)o(e)p 0 1548 V 0 1574 a Fm(\015)s Fo(\()p Fm(E)s(;)8 b(\022)q Fo(\))14 b Fm(>)g Fo(0)p Fm(:)f Fo(Note)f(that)i(b)q(oundedness)g(of)f Fm(V)e Fo(\()p Fm(n)p Fo(\))i(immediately)e(implies)g(the)i(follo)o (wing)g(a)g(priori)g(upp)q(er)0 1635 y(b)q(ound:)789 1695 y Fj(k)p Fo(\010)849 1702 y Fh(n)873 1695 y Fo(\()p Fm(\022)q(;)8 b(E)s Fo(\))p Fj(k)14 b(\024)f Fm(e)1110 1674 y Fh(nC)1850 1695 y Fo(\(4.1\))0 1782 y(for)20 b(some)f Fm(C)k(<)c Fj(1)p Fm(:)g Fo(Put)g Fm(c)p Fo(\()p Fm(E)s Fo(\))g(=)715 1759 y Fh(\015)r Fl(\()p Fh(E)r Fl(\))p 678 1771 148 2 v 678 1800 a(2)p Fh(C)r Fi(\000)p Fh(\015)r Fl(\()p Fh(E)r Fl(\))831 1782 y Fm(:)g Fo(Let)h Fm(A)992 1789 y Fh(k)1032 1782 y Fo(=)f Fj(f)p Fm(\022)h Fj(2)f Fo([0)p Fm(;)8 b Fo(1\))20 b(:)e Fj(k)p Fo(\010)1424 1789 y Fh(k)1446 1782 y Fo(\()p Fm(\022)q(;)8 b(E)s Fo(\))p Fj(k)19 b Fm(>)g(e)1693 1764 y Fh(k)q(\015)r Fl(\()p Fh(E)r Fl(\))p Fh(=)p Fl(2)1824 1782 y Fj(g)p Fm(:)g Fo(W)l(e)0 1849 y(obtain,)d(using)h(\(1.6\),)f(that)163 1973 y Fm(k)r(\015)s Fo(\()p Fm(E)s Fo(\))e Fj(\024)361 1914 y Ff(Z)403 1928 y Fl(1)385 2009 y(0)431 1973 y Fo(ln)8 b Fj(k)p Fo(\010)540 1980 y Fh(k)561 1973 y Fo(\()p Fm(\022)q(;)g(E)s Fo(\))p Fj(k)p Fm(d\022)16 b Fo(=)824 1914 y Ff(Z)847 2009 y Fh(A)873 2015 y Fa(k)903 1973 y Fo(+)949 1914 y Ff(Z)973 2009 y Fl([0)p Fh(;)p Fl(1\))p Fi(n)p Fh(A)1087 2015 y Fa(k)1120 1973 y Fj(\024)e(j)p Fm(A)1224 1980 y Fh(k)1244 1973 y Fj(j)p Fm(k)r(C)h Fo(+)c(\(1)g Fj(\000)g(j)p Fm(A)1539 1980 y Fh(k)1560 1973 y Fj(j)p Fo(\))p Fm(k)r(\015)s Fo(\()p Fm(E)s Fo(\))p Fm(=)p Fo(2)p Fm(;)0 2099 y Fo(where)18 b Fj(j)13 b(\001)f(j)18 b Fo(stands)i(for)f(the)f(Leb)q(esgue)h (measure.)28 b(W)l(e,)18 b(therefore,)g(obtain)h Fj(j)p Fm(A)1501 2106 y Fh(k)1522 2099 y Fj(j)e(\025)h Fm(c)p Fo(\()p Fm(E)s Fo(\))p Fm(:)g Fo(Since)f Fm(f)24 b Fo(is)0 2159 y(a)17 b(trigonometric)f(p)q(olynomial)h(of)g(degree)g Fm(p;)f Fj(k)p Fo(\010)921 2166 y Fh(k)943 2159 y Fj(k)968 2141 y Fl(2)1004 2159 y Fo(is)h(a)g(trigonometric)f(p)q(olynomial)h(of) g(degree)g(2)p Fm(k)r(p;)0 2219 y Fo(and)j(the)f(set)h Fm(A)302 2226 y Fh(k)342 2219 y Fo(consists)g(of)g(no)g(more)f(than)h (4)p Fm(k)r(p)g Fo(in)o(terv)m(als.)30 b(Therefore,)20 b(there)e(exists)h(a)h(segmen)o(t,)0 2279 y(\001)41 2286 y Fh(k)76 2279 y Fj(\032)13 b Fm(A)165 2286 y Fh(k)186 2279 y Fm(;)j Fo(with)g Fj(j)p Fo(\001)382 2286 y Fh(k)403 2279 y Fj(j)e(\025)488 2256 y Fh(c)p Fl(\()p Fh(E)r Fl(\))p 488 2268 71 2 v 496 2297 a(4)p Fh(k)q(p)564 2279 y Fm(:)90 2363 y Fo(W)l(e)j(will)f(no)o(w)i(sho)o(w)g(that)g(for)g(ev)o(ery)d Fm(\022)q Fo(,)j(there)e(exists)h(a)h(sequence)e Fm(j)1387 2370 y Fh(n)1427 2363 y Fj(!)f(1)i Fo(suc)o(h)g(that)h Fj(k)p Fo(\010)1837 2370 y Fh(j)1851 2374 y Fa(n)1875 2363 y Fj(k)f Fo(is)0 2423 y(exp)q(onen)o(tially)g(large.)26 b(This)19 b(sequence)e(will)g(b)q(e)h(related)f(to)i(a)f(con)o(tin)o (ued)f(fractions)h(expansion)h(of)f Fm(\013:)0 2483 y Fo(Let)c Fm(p)109 2490 y Fh(n)133 2483 y Fm(=q)179 2490 y Fh(n)216 2483 y Fo(b)q(e)g(the)g(sequence)e(of)i(con)o(tin)o(ued)f (fraction)h(appro)o(ximan)o(ts)f(of)h Fm(\013:)g Fo(W)l(e)f(will)g(use) h(the)f(follo)o(wing)0 2543 y(elemen)o(tary)k(prop)q(erties)h(of)h(con) o(tin)o(ued)e(fractions,)i(that)g(can)g(b)q(e)f(found,)h(for)g (example,)e(in)i([19].)27 b(\(W)l(e)0 2603 y(assume)17 b(here)e Fm(\013)f(>)g Fo(0.)22 b(Otherwise,)15 b(the)h(signs)h(in)f (\(4.3\))g(will)f(c)o(hange.\))361 2704 y Fj(j)p Fm(q)397 2711 y Fh(n)420 2704 y Fm(\013)c Fj(\000)g Fm(p)536 2711 y Fh(n)560 2704 y Fj(j)i Fm(<)h Fj(j)p Fm(k)r(\013)e Fj(\000)e Fm(p)p Fj(j)p Fm(;)22 b(k)16 b Fo(=)e(1)p Fm(;)8 b(:)g(:)g(:)g(;)g(q)1095 2711 y Fh(n)p Fl(+1)1174 2704 y Fj(\000)j Fo(1)p Fm(;)22 b(k)16 b Fj(6)p Fo(=)d Fm(q)1398 2711 y Fh(n)1422 2704 y Fm(;)21 b(p)15 b Fj(2)f Fk(Z)p Fo(;)260 b(\(4.2\))951 2828 y(10)p eop %%Page: 11 12 11 11 bop 422 203 a Fm(q)444 210 y Fh(n)467 203 y Fm(\013)12 b Fj(\000)f Fm(p)584 210 y Fh(n)621 203 y Fm(>)j Fo(0)g(for)h Fm(n)f Fo(o)q(dd)j(and)d Fm(q)1037 210 y Fh(n)1060 203 y Fm(\013)e Fj(\000)e Fm(p)1176 210 y Fh(n)1214 203 y Fm(<)k Fo(0)g(for)g Fm(n)g Fo(ev)o(en)o(;)322 b(\(4.3\))779 313 y Fj(j)p Fm(q)815 320 y Fh(n)838 313 y Fm(\013)11 b Fj(\000)g Fm(p)954 320 y Fh(n)978 313 y Fj(j)j Fm(<)1095 279 y Fo(1)p 1062 301 91 2 v 1062 347 a Fm(q)1084 354 y Fh(n)p Fl(+1)1158 313 y Fm(:)678 b Fo(\(4.4\))0 426 y(Let)16 b(\001)e Fj(\032)f Fo([0)p Fm(;)8 b Fo(1\))17 b(b)q(e)f(an)h(arbitrary)f(segmen)o(t.)21 b(W)l(e)16 b(pro)o(v)o(e)f(b)q(elo)o(w)h(the)g(follo)o(wing)g(simple)0 546 y Fe(LEMMA)j(9)24 b Fd(L)n(et)h Fm(n)k Fj(2)g Fk(N)d Fd(b)n(e)g(such)g(that)f Fj(j)p Fo(\001)p Fj(j)k Fm(>)1037 527 y Fl(1)p 1027 534 37 2 v 1027 563 a Fh(q)1043 567 y Fa(n)1069 546 y Fd(.)47 b(Then,)28 b(for)d(any)h Fm(\022)q(;)f Fd(ther)n(e)g(exists)i(a)e Fm(k)j Fd(in)0 615 y Fj(f)p Fo(0)p Fm(;)8 b Fo(1)p Fm(;)g(:)g(:)g(:)g(;)g(q)227 622 y Fh(n)261 615 y Fo(+)j Fm(q)332 622 y Fh(n)p Fi(\000)p Fl(1)411 615 y Fj(\000)g Fo(1)p Fj(g)18 b Fd(such)g(that)f Fm(T)774 597 y Fh(k)767 627 y(\013)795 615 y Fm(\022)e Fj(2)f Fo(\001)p Fm(:)0 735 y Fo(W)l(e)i(will)f(no)o(w)i(\014nish)f (the)g(pro)q(of)h(of)g(Theorem)f(7.)90 813 y(Set)768 892 y Fm(k)793 899 y Fh(n)831 892 y Fo(=)883 819 y Ff(")912 858 y Fm(c)p Fo(\()p Fm(E)s Fo(\))p Fm(q)1032 865 y Fh(n)p 912 880 144 2 v 959 926 a Fo(4)p Fm(p)1060 819 y Ff(#)1095 892 y Fo(+)11 b(1)p Fm(:)668 b Fo(\(4.5\))0 1013 y(Then,)12 b(b)o(y)g(Lemma)g(9,)g(for)g(ev)o(ery)f Fm(\022)i Fo(there)e(exists)g (a)i Fm(j)i Fo(in)c Fj(f)p Fo(0)p Fm(;)d Fo(1)p Fm(;)g(:)g(:)g(:)g(;)g (q)1253 1020 y Fh(n)1278 1013 y Fo(+)r Fm(q)1340 1020 y Fh(n)p Fi(\000)p Fl(1)1411 1013 y Fj(\000)r Fo(1)p Fj(g)k Fo(suc)o(h)f(that)i Fm(T)1756 995 y Fh(j)1749 1025 y(\013)1773 1013 y Fm(\022)i Fj(2)f Fm(A)1895 1020 y Fh(k)1913 1024 y Fa(n)1936 1013 y Fm(;)0 1073 y Fo(and)i(so)g Fj(k)p Fo(\010)213 1080 y Fh(k)231 1084 y Fa(n)255 1073 y Fo(\()p Fm(T)310 1055 y Fh(j)303 1085 y(\013)327 1073 y Fm(\022)q Fo(\))p Fj(k)e Fm(>)g(e)484 1055 y Fh(k)502 1059 y Fa(n)523 1055 y Fh(\015)r Fl(\()p Fh(E)r Fl(\))p Fh(=)p Fl(2)636 1073 y Fm(:)h Fo(Since)f(\010)826 1080 y Fh(j)r Fl(+)p Fh(k)887 1084 y Fa(n)911 1073 y Fo(\()p 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Fh(j)1306 1275 y Fo(\()p Fm(\022)q Fo(\))p Fj(k)g(\025)g Fm(e)1488 1238 y Fa(k)1504 1242 y(n)1525 1238 y(\015)r Fg(\()p Fa(E)q Fg(\))p 1487 1247 V 1531 1268 a(4)1597 1215 y Ff(\033)1637 1275 y Fm(:)199 b Fo(\(4.6\))0 1404 y(By)16 b(\(4.1\))g(w)o(e)g(ha)o(v)o(e)831 1492 y Fm(j)851 1499 y Fh(n)889 1492 y Fj(\025)946 1458 y Fm(k)971 1465 y Fh(n)995 1458 y Fm(\015)s Fo(\()p Fm(E)s Fo(\))p 946 1480 154 2 v 992 1526 a(2)p Fm(C)1105 1492 y(;)731 b Fo(\(4.7\))0 1592 y(so)22 b Fm(j)85 1599 y Fh(n)131 1592 y Fj(!)h(1)e Fo(as)h Fm(n)g Fj(!)h(1)p Fm(:)d Fo(T)l(o)i(\014nish)g(the)f(pro)q(of)h(of)g(Theorem)f(7)h(it)f (no)o(w)g(remains)g(to)h(notice)f(that)0 1666 y (\(4.5\),\(4.6\),\(4.7\))16 b(yield)f Fj(k)p Fo(\010)522 1673 y Fh(j)536 1677 y Fa(n)560 1666 y Fo(\()p Fm(\022)q Fo(\))p Fj(k)f(\025)f Fm(e)736 1641 y Fh(j)750 1645 y Fa(n)790 1624 y(c)p Fg(\()p Fa(E)q Fg(\))p Fa(\015)r Fg(\()p Fa(E)q Fg(\))p 776 1634 156 2 v 776 1654 a(4\(8)p Fa(p)p Fg(+)p Fa(c)p Fg(\()p Fa(E)q Fg(\)\))939 1666 y Fm(;)i Fo(whic)o(h)h(implies)p 1274 1639 29 2 v 15 w Fm(\015)s Fo(\()p Fm(E)s(;)8 b(\022)q Fo(\))13 b Fj(\025)1512 1642 y Fh(c)p Fl(\()p Fh(E)r Fl(\))p Fh(\015)r Fl(\()p Fh(E)r Fl(\))p 1496 1654 179 2 v 1496 1683 a(4\(8)p Fh(p)p Fl(+)p Fh(c)p Fl(\()p Fh(E)r Fl(\)\))1679 1666 y Fm(:)p 1734 1670 21 30 v 90 1806 a Fe(Pro)r(of)25 b(of)h(Lemma)e(9.)39 b Fo(Let)22 b Fm(k)730 1788 y Fl(+\()p Fi(\000)p Fl(\))837 1806 y Fo(de\014ne)g(suc)o(h)g Fm(s)h Fj(2)h(f)p Fo(0)p Fm(;)8 b(:)g(:)g(:)g(;)g(q)1384 1813 y Fh(n)1422 1806 y Fo(+)15 b Fm(q)1497 1813 y Fh(n)p Fi(\000)p Fl(1)1580 1806 y Fj(\000)g Fo(1)p Fj(g)23 b Fo(that)f Fm(T)1853 1788 y Fh(s)1846 1818 y(\013)1871 1806 y Fm(\022)h Fo(is)0 1866 y(the)c(nearest)h(neigh)o(b)q(or)f(to)h(the)f(righ)o(t)g(\(left\)) f(of)i Fm(T)954 1848 y Fh(k)947 1879 y(\013)975 1866 y Fm(\022)q(:)f Fo(T)l(o)h(pro)o(v)o(e)e(the)i(lemma)e(w)o(e)h(sho)o(w) h(that)g(for)f(an)o(y)0 1926 y Fm(k)d Fj(2)e(f)p Fo(0)p Fm(;)8 b(:)g(:)g(:)g(;)g(q)269 1933 y Fh(n)303 1926 y Fo(+)j Fm(q)374 1933 y Fh(n)p Fi(\000)p Fl(1)453 1926 y Fj(\000)g Fo(1)p Fj(g)p Fo(,)16 b(w)o(e)g(ha)o(v)o(e)654 2059 y(dist\()p Fm(T)788 2038 y Fh(k)781 2071 y(\013)809 2059 y Fm(\022)q(;)8 b(T)891 2038 y Fh(k)910 2026 y Fg(+\()p Fb(\000)p Fg(\))884 2071 y Fh(\013)985 2059 y Fm(\022)q Fo(\))14 b Fm(<)1109 2025 y Fo(1)p 1098 2047 46 2 v 1098 2093 a Fm(q)1120 2100 y Fh(n)1162 2059 y Fm(<)g Fj(j)p Fo(\001)p Fj(j)p Fm(:)553 b Fo(\(4.8\))0 2191 y(T)l(o)17 b(this)f(end)g(w)o(e)g(consider)g(three)f(cases.)90 2269 y(First,)g(supp)q(ose)i Fm(k)e(<)f(q)519 2276 y Fh(n)p Fi(\000)p Fl(1)587 2269 y Fm(:)h Fo(Without)h(loss)g(of)g(generalit)o (y)e(w)o(e)h(assume)h(that)g Fm(n)g Fo(is)f(ev)o(en)f(\(otherwise)0 2330 y(the)i(+)g(and)h Fj(\000)f Fo(will)f(c)o(hange)h(places\).)21 b(Then,)16 b(b)o(y)f(\(4.2\))i(and)f(\(4.3\),)g Fm(k)1299 2311 y Fl(+)1343 2330 y Fo(=)e Fm(k)e Fo(+)f Fm(q)1503 2337 y Fh(n)1543 2330 y Fo(and)16 b Fm(k)1664 2311 y Fi(\000)1708 2330 y Fo(=)e Fm(k)e Fo(+)f Fm(q)1868 2337 y Fh(n)p Fi(\000)p Fl(1)1936 2330 y Fm(:)0 2390 y Fo(Therefore,)23 b(b)q(ecause)g(of)g(\(4.4\),)h(dist\()p Fm(T)763 2372 y Fh(k)756 2402 y(\013)784 2390 y Fm(\022)q(;)8 b(T)866 2372 y Fh(k)885 2360 y Fg(+)859 2402 y Fh(\013)912 2390 y Fm(\022)q Fo(\))25 b(=)f Fj(j)p Fm(q)1078 2397 y Fh(n)1101 2390 y Fm(\013)16 b Fj(\000)f Fm(p)1226 2397 y Fh(n)1250 2390 y Fj(j)25 b Fm(<)1386 2370 y Fl(1)p 1357 2378 76 2 v 1357 2407 a Fh(q)1373 2412 y Fa(n)p Fg(+1)1460 2390 y Fo(and)e(dist\()p Fm(T)1695 2372 y Fh(k)1688 2402 y(\013)1716 2390 y Fm(\022)q(;)8 b(T)1798 2372 y Fh(k)1817 2360 y Fb(\000)1791 2402 y Fh(\013)1844 2390 y Fm(\022)q Fo(\))25 b(=)0 2460 y Fj(j)p Fm(q)36 2467 y Fh(n)p Fi(\000)p Fl(1)104 2460 y Fm(\013)11 b Fj(\000)g Fm(p)220 2467 y Fh(n)p Fi(\000)p Fl(1)289 2460 y Fj(j)j Fm(<)383 2441 y Fl(1)p 374 2448 37 2 v 374 2477 a Fh(q)390 2481 y Fa(n)416 2460 y Fm(;)i Fo(and)g(\(4.8\))h(holds.)90 2544 y(Second,)g(assume)g Fm(q)461 2551 y Fh(n)p Fi(\000)p Fl(1)545 2544 y Fj(\024)d Fm(k)k(<)d(q)716 2551 y Fh(n)739 2544 y Fm(:)h Fo(Then,)h(similarly)l (,)e Fm(k)1148 2526 y Fl(+)1193 2544 y Fo(=)g Fm(k)f Fj(\000)d Fm(q)1357 2551 y Fh(n)p Fi(\000)p Fl(1)1442 2544 y Fo(and)18 b Fm(k)1565 2526 y Fi(\000)1609 2544 y Fo(=)e Fm(k)d Fo(+)f Fm(q)1773 2551 y Fh(n)p Fi(\000)p Fl(1)1841 2544 y Fm(;)k Fo(and)0 2604 y(\(4.8\))h(holds)f(in)g(the)g (same)g(w)o(a)o(y)l(.)90 2682 y(Finally)l(,)e(for)j Fm(k)f Fj(\025)d Fm(q)455 2689 y Fh(n)494 2682 y Fo(w)o(e)j(ha)o(v)o(e)f Fm(k)705 2664 y Fl(+)748 2682 y Fo(=)f Fm(k)f Fj(\000)d Fm(q)909 2689 y Fh(n)p Fi(\000)p Fl(1)994 2682 y Fo(and)16 b Fm(k)1115 2664 y Fi(\000)1159 2682 y Fo(=)d Fm(k)g Fj(\000)d Fm(q)1319 2689 y Fh(n)1343 2682 y Fm(;)15 b Fo(whic)o(h)h(also)g(implies)f(\(4.8\).)p 1929 2686 21 30 v 951 2828 a(11)p eop %%Page: 12 13 12 12 bop 90 203 a Fe(Pro)r(of)16 b(of)g(Theorem)e(1.)20 b Fo(In)14 b(the)f(quasip)q(erio)q(dic)h(case,)f Fm(V)f Fo(\()p Fm(n)p Fo(\))i(=)f Fm(f)5 b Fo(\()p Fm(T)1406 185 y Fh(n)1399 216 y(\013)1430 203 y Fm(\022)q Fo(\))p Fm(;)13 b Fo(w)o(e)g(ha)o(v)o(e)g(the)g(follo)o(wing)0 264 y(relations)j(for)h(the)f(left)f(and)i(righ)o(t)f (transfer-matrices:)403 362 y(\010)438 341 y Fl(+)p Fh(;\013)438 374 y(n)500 362 y Fo(\()p Fm(\022)q(;)8 b(E)s Fo(\))14 b(=)g Fm(M)5 b Fo(\()p Fm(T)796 341 y Fh(n)789 374 y(\013)819 362 y Fm(\022)q(;)j(E)s Fo(\))p Fm(M)d Fo(\()p Fm(T)1030 341 y Fh(n)p Fi(\000)p Fl(1)1023 374 y Fh(\013)1098 362 y Fm(\022)q(;)j(E)s Fo(\))22 b Fj(\001)8 b(\001)g(\001)23 b Fm(M)5 b Fo(\()p Fm(T)1405 369 y Fh(\013)1429 362 y Fm(\022)q(;)j(E)s Fo(\))p Fm(;)0 460 y Fo(where)16 b Fm(M)5 b Fo(\()p Fm(\022)q(;)j(E)s Fo(\))16 b(is)h(giv)o(en)e(b)o(y)h (\(1.4\))g(with)g Fm(f)5 b Fo(\()p Fm(\022)q Fo(\))17 b(as)g(in)f(Theorem)g(1,)g(and)642 558 y(\010)677 537 y Fi(\000)p Fh(;\013)677 570 y(n)739 558 y Fo(\()p Fm(\022)q(;)8 b(E)s Fo(\))14 b(=)g(\010)963 537 y Fl(+)p Fh(;)p Fl(\()p Fi(\000)p Fh(\013)p Fl(\))963 570 y Fh(n)1080 558 y Fo(\()p Fm(\022)e Fo(+)f Fm(\013;)d(E)s Fo(\))p Fm(:)542 b Fo(\(4.9\))90 674 y(Fix)22 b Fm(\017)i(>)g Fo(0)p Fm(:)f Fo(Note)f(that)h(the)f (denominators)h Fm(q)1009 681 y Fh(n)1054 674 y Fo(are)g(the)f(same)h (for)f Fm(\013)h Fo(and)g Fj(\000)p Fm(\013:)f Fo(Then,)i(from)0 734 y(\(4.6\),\(4.7\),)15 b(and\(4.9\),)g(w)o(e)f(obtain)h(for)g Fm(n)g Fo(su\016cien)o(tly)d(large,)i(that)h(the)g(conditions)f(of)h (Corollary)g(6)g(are)0 799 y(satis\014ed)i(with)g Fm(A)d Fo(=)h Fj(f)p Fm(E)i Fo(:)23 b Fm(\015)s Fo(\()p Fm(E)s Fo(\))15 b Fm(>)f Fo(0)p Fj(g)p Fm(;)23 b(a)15 b Fo(=)892 776 y Fh(c)p Fl(\()p Fh(E)r Fl(\))p Fh(\015)r Fl(\()p Fh(E)r Fl(\))p 875 787 179 2 v 875 816 a(4\(8)p Fh(p)p Fl(+)p Fh(c)p Fl(\()p Fh(E)r Fl(\)\))1059 799 y Fo(\(1)c Fj(\000)h Fm(\017)p Fo(\))p Fm(;)22 b(\015)c Fo(=)c(1)p Fm(;)23 b(c)15 b Fo(=)1506 778 y Fl(8)p Fh(p)p 1489 787 71 2 v 1489 816 a(c)p Fl(\()p Fh(E)r Fl(\))1576 799 y Fo(+)c(1)p Fm(;)17 b Fo(the)f(sequence)0 867 y Fm(k)25 874 y Fh(n)65 867 y Fo(de\014ned)g(b)o(y)g(\(4.5\),)g(and)h(the)f (sequences)f Fm(j)854 849 y Fi(\006)851 879 y 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Fm(\013)h Fo(is)0 1400 y(some)i(p)q(ositiv)o(e)f(constan)o(t)h(dep)q (ending)g(only)f(on)h Fm(\025)p Fo(\).)23 b(Second,)17 b(w)o(e)f(will)g(use)g(the)h(symmetry)e(prop)q(erties)0 1461 y(of)i(the)g(p)q(oten)o(tial)g(to)g(conclude)f(a)h(corresp)q (onding)h(result)f(for)g(the)f(whole-line)g Fm(m)p Fo(-function.)23 b(T)l(o)18 b(pro)o(v)o(e)0 1521 y(all)e(that,)g(w)o(e)g(use)g(a)g(n)o (um)o(b)q(er)g(of)g(facts)h(that)f(where)g(established)g(b)o(y)f(S)q (\177)-26 b(ut})i(o)18 b(in)e([27],)g(as)g(w)o(ell)f(as)i(a)g(result)0 1581 y(of)g(Io)q(c)o(h)o(um)e(and)i(T)l(estard)g([13].)k(W)l(e)16 b(start)g(with)0 1691 y Fe(PR)n(OPOSITION)j(10)25 b Fd(L)n(et)20 b Fm(\025)f Fj(2)h Fk(R)g Fd(and)g(let)i Fm(H)950 1698 y Fh(\025)993 1691 y Fd(b)n(e)e(the)h(c)n(orr)n(esp)n(onding)f(Fib)n (onac)n(ci)g(Hamiltonian.)0 1751 y(Then)h(ther)n(e)f(exists)h(an)k Fo(~)-28 b Fm(\013)19 b(>)g Fo(0)h Fd(such)h(that)f(for)g(any)g Fm(E)j Fd(in)e(the)f(sp)n(e)n(ctrum)g(of)g Fm(H)1520 1758 y Fh(\025)1543 1751 y Fd(,)h(any)f(solution)h Fm(u)f Fd(of)0 1811 y Fm(H)40 1818 y Fh(\025)63 1811 y Fm(u)14 b Fo(=)f Fm(E)s(u)18 b Fd(ob)n(eys)f Fo(lim)8 b(inf)389 1841 y Fh(L)p Fi(!1)512 1811 y Fm(L)545 1790 y Fi(\000)s Fl(~)-21 b Fh(\013)598 1811 y Fj(k)p Fm(u)p Fj(k)676 1818 y Fh(L)715 1811 y Fm(>)14 b Fo(0)p Fd(.)0 1936 y Fe(Pro)r(of.)20 b Fo(Let)12 b Fj(f)p Fm(F)312 1943 y Fh(n)335 1936 y Fj(g)360 1917 y Fi(1)360 1948 y Fh(n)p Fl(=1)441 1936 y Fo(b)q(e)g(the)g(Fib)q(onacci)g(n)o(um)o(b)q(ers,)g (de\014ned)g(b)o(y)f Fm(F)1268 1943 y Fl(0)1302 1936 y Fo(=)i(1,)g Fm(F)1436 1943 y Fl(1)1469 1936 y Fo(=)h(1,)f Fm(F)1604 1943 y Fh(n)p Fl(+1)1686 1936 y Fo(=)h Fm(F)1770 1943 y Fh(n)1796 1936 y Fo(+)s Fm(F)1869 1943 y Fh(n)p Fi(\000)p Fl(1)1936 1936 y Fo(.)0 1996 y(As)j(sho)o(wn)i(b)o(y)e(S)q (\177)-26 b(ut})i(o)19 b(\(Prop)q(osition)f(1)g(of)g([27]\),)f(the)h (Fib)q(onacci)f(p)q(oten)o(tial)g(ob)q(eys)h Fm(V)11 b Fo(\()p Fm(F)1658 2003 y Fh(n)1693 1996 y Fo(+)h Fm(`)p Fo(\))k(=)g Fm(V)c Fo(\()p Fm(`)p 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b(Lemma)h(1)h(of)f([27)q(],)f (w)o(e)h(ha)o(v)o(e)f(for)i(an)o(y)f(2)11 b Fj(\002)g Fo(2)17 b(matrix)e Fm(B)k Fo(with)d(det)8 b Fm(B)17 b Fo(=)d(1,)582 2433 y(max)o Fj(fj)p Fo(T)l(r)9 b Fm(B)s Fj(j)h(\001)h(k)p Fm(B)s Fo(\011)p Fj(k)p Fm(;)d Fj(k)p Fm(B)1074 2412 y Fl(2)1092 2433 y Fo(\011)p Fj(kg)14 b(\025)1251 2399 y Fo(1)p 1251 2421 25 2 v 1251 2467 a(2)1281 2433 y Fj(k)p Fo(\011)p Fj(k)481 b Fo(\(5.2\))0 2546 y(for)17 b(an)o(y)f(2-v)o(ector)g(\011.)21 b(Th)o(us,)16 b(if)g(w)o(e)g(ha)o(v)o(e)g(a)g(b)q(ound)i(on)f Fj(j)p Fo(T)l(r)8 b Fm(B)s Fj(j)16 b Fo(of)g(the)g(form)g Fj(j)p Fo(T)l(r)9 b Fm(B)s Fj(j)k(\024)h Fm(C)20 b Fo(with)c Fm(C)h(>)d Fo(1,)0 2606 y(w)o(e)i(get)649 2686 y Fj(k)p Fm(B)s Fo(\011)p Fj(k)777 2665 y Fl(2)807 2686 y Fo(+)11 b Fj(k)p Fm(B)921 2665 y Fl(2)940 2686 y Fo(\011)p Fj(k)1003 2665 y Fl(2)1036 2686 y Fm(>)1122 2652 y Fo(1)p 1093 2674 83 2 v 1093 2720 a(4)p Fm(C)1156 2706 y Fl(2)1181 2686 y Fj(k)p Fo(\011)p Fj(k)1269 2665 y Fl(2)1288 2686 y Fm(:)548 b Fo(\(5.3\))951 2828 y(12)p eop %%Page: 13 14 13 13 bop 0 203 a Fo(F)l(or)18 b(ev)o(ery)e Fm(m)g(<)g(k)r Fo(,)h(let)g(\010)497 210 y Fh(m;k)578 203 y Fo(denote)g(the)g(2)c Fj(\002)f Fo(2)18 b(transfer)g(matrix)e(whic)o(h)h(tak)o(es)h(\()p Fm(u)p Fo(\()p Fm(m)11 b Fo(+)h(1\))p Fm(;)c(u)p Fo(\()p Fm(m)p Fo(\)\))1923 185 y Fh(T)0 264 y Fo(to)18 b(\()p Fm(u)p Fo(\()p Fm(k)c Fo(+)e(1\))p Fm(;)c(u)p Fo(\()p Fm(k)r Fo(\)\))393 246 y Fh(T)421 264 y Fo(.)26 b(Let)18 b(1)f Fj(\024)g Fm(`)g Fj(\024)g Fm(F)772 271 y Fh(n)p Fi(\000)p Fl(2)840 264 y Fo(.)26 b(By)17 b(\(5.1\),)i(w)o(e)e(ha)o(v)o (e)g(that)i(\010)1421 271 y Fh(`;F)1467 275 y Fa(n)1489 271 y Fl(+)p Fh(`)1549 264 y Fo(=)e(\010)1639 271 y Fh(F)1661 275 y Fa(n)1683 271 y Fl(+)p Fh(`;)p Fl(2)p Fh(F)1774 275 y Fa(n)1796 271 y Fl(+)p Fh(`)1840 264 y Fo(,)g(and)0 324 y(moreo)o(v)o(er,)g(w)o(e)h(ha)o(v)o(e)g(T)l(r)8 b(\010)508 331 y Fh(`;F)554 335 y Fa(n)576 331 y Fl(+)p Fh(`)637 324 y Fo(=)18 b(T)l(r)8 b(\010)786 331 y Fl(0)p Fh(;F)836 335 y Fa(n)859 324 y Fo(.)28 b(As)18 b(sho)o(wn)h(in)f(the)g (pro)q(of)h(of)g(Lemma)f(2)h(of)f([27)q(],)g(there)0 384 y(exists)i(a)i(constan)o(t)f Fm(C)421 391 y Fh(\025)444 384 y Fo(,)h(dep)q(ending)f(only)g(on)g Fm(\025)p Fo(,)i(suc)o(h)d (that)i Fj(j)p Fo(T)l(r)8 b(\010)1300 391 y Fl(0)p Fh(;F)1350 395 y Fa(n)1373 384 y Fj(j)22 b Fm(<)g(C)1504 391 y Fh(\025)1548 384 y Fo(for)f(ev)o(ery)f Fm(E)k Fo(in)c(the)0 444 y(sp)q(ectrum)c(of)h Fm(H)309 451 y Fh(\025)348 444 y Fo(and)g(ev)o(ery)d Fm(n)p Fo(.)22 b(Th)o(us,)16 b(\(5.3\))g(implies)25 576 y Fj(j)p Fm(u)p Fo(\()p Fm(F)118 583 y Fh(n)142 576 y Fo(+)q Fm(`)q Fo(+)q(1\))p Fj(j)298 556 y Fl(2)321 576 y Fo(+)q Fj(j)p Fm(u)p Fo(\()p Fm(F)453 583 y Fh(n)477 576 y Fo(+)q Fm(`)p Fo(\))p Fj(j)569 556 y Fl(2)590 576 y Fo(+)q Fj(j)p Fm(u)p Fo(\(2)p Fm(F)746 583 y Fh(n)771 576 y Fo(+)q Fm(`)q Fo(+)q(1\))p Fj(j)927 556 y Fl(2)950 576 y Fo(+)q Fj(j)p Fm(u)p Fo(\(2)p Fm(F)1106 583 y Fh(n)1130 576 y Fo(+)q Fm(`)p Fo(\))p Fj(j)1222 556 y Fl(2)1256 576 y Fm(>)1342 543 y Fo(1)p 1313 565 83 2 v 1313 611 a(4)p Fm(C)1376 593 y Fl(2)1372 623 y Fh(\025)1401 576 y Fo(\()p Fj(j)p Fm(u)p Fo(\()p Fm(`)q Fo(+)q(1\))p Fj(j)1598 556 y Fl(2)1620 576 y Fo(+)q Fj(j)p Fm(u)p Fo(\()p Fm(`)p Fo(\))p Fj(j)1773 556 y Fl(2)1792 576 y Fo(\))p Fm(;)25 b Fo(\(5.4\))0 712 y(from)16 b(whic)o(h)g(w)o(e)g(easily)f(deduce)h (that)627 862 y Fj(k)p Fm(u)p Fj(k)705 869 y Fh(F)727 874 y Fa(n)p Fg(+2)803 862 y Fm(>)854 789 y Ff( )887 862 y Fo(1)c(+)1006 828 y(1)p 977 850 V 977 896 a(4)p Fm(C)1040 879 y Fl(2)1036 909 y Fh(\025)1064 789 y Ff(!)1097 800 y Fl(1)p Fh(=)p Fl(2)1161 862 y Fj(k)p Fm(u)p Fj(k)1239 869 y Fh(F)1261 874 y Fa(n)p Fb(\000)p Fg(2)1850 862 y Fo(\(5.5\))0 1003 y(for)17 b(large)f Fm(n)p Fo(.)21 b(Since)15 b(the)h Fm(F)501 1010 y Fh(n)525 1003 y Fo('s)g(gro)o(w)g (exp)q(onen)o(tially)f(in)h Fm(n)p Fo(,)g(\(5.5\))h(pro)o(v)o(es)e(the) h(required)f(result.)p 1830 1007 21 30 v 90 1131 a(W)l(e)h(can)g(no)o (w)h(get)f(the)g(required)f(estimate)h(for)g Fm(m)1034 1110 y Fl(+)1034 1144 y Fh(\014)1063 1131 y Fo(.)0 1251 y Fe(PR)n(OPOSITION)j(11)25 b Fd(L)n(et)20 b Fm(\025)f Fj(2)h Fk(R)g Fd(and)g(let)i Fm(H)950 1258 y Fh(\025)993 1251 y Fd(b)n(e)e(the)h(c)n(orr)n(esp)n(onding)f(Fib)n(onac)n(ci)g (Hamiltonian.)0 1311 y(Then)f(ther)n(e)g(exists)g(an)g Fm(\013)d(>)g Fo(0)p Fd(,)j(which)g(dep)n(ends)g(only)g(on)g Fm(\025)p Fd(,)g(such)g(that)g(for)f(any)g Fm(E)k Fd(in)d(the)g(sp)n(e) n(ctrum)0 1371 y(of)e Fm(H)97 1378 y Fh(\025)138 1371 y Fd(and)g(every)h Fm(\014)s Fd(,)f Fo(lim)8 b(sup)574 1383 y Fh(\017)p Fi(!)p Fl(0)651 1371 y Fm(\017)671 1353 y Fl(1)p Fi(\000)p Fh(\013)741 1371 y Fj(j)p Fm(m)798 1350 y Fl(+)798 1384 y Fh(\014)827 1371 y Fo(\()p Fm(E)14 b Fo(+)d Fm(i\017)p Fo(\))p Fj(j)i Fm(<)g Fj(1)p Fd(.)0 1491 y Fe(Pro)r(of.)21 b Fo(By)16 b(com)o(bining)f(\(2.12\))i(and)g (\(2.13\),)f(w)o(e)g(get)22 1647 y Fm(\017)42 1627 y Fl(1)p Fi(\000)p Fh(\013)112 1647 y Fj(j)p Fm(m)169 1626 y Fl(+)169 1660 y Fh(\014)198 1647 y Fo(\()p Fm(E)e Fo(+)d Fm(i\017)p Fo(\))p Fj(j)i(\030)h Fo(\()p Fj(k)p Fm(u)524 1626 y Fl(+)524 1660 y(1)p Fh(;\014)574 1647 y Fj(k)599 1661 y Fh(L)623 1646 y Fg(+)623 1673 y Fa(\014)649 1661 y Fl(\()p Fh(\017)p Fl(\))692 1647 y Fj(k)p Fm(u)745 1626 y Fl(+)745 1660 y(2)p Fh(;\014)796 1647 y Fj(k)821 1661 y Fh(L)845 1646 y Fg(+)845 1673 y Fa(\014)870 1661 y Fl(\()p Fh(\017)p Fl(\))914 1647 y Fo(\))933 1627 y Fh(\013)p Fi(\000)p Fl(1)1008 1598 y Fj(k)p Fm(u)1061 1577 y Fl(+)1061 1610 y(2)p Fh(;\014)1112 1598 y Fj(k)1137 1611 y Fh(L)1161 1597 y Fg(+)1161 1624 y Fa(\014)1186 1611 y Fl(\()p Fh(\017)p Fl(\))p 1008 1635 222 2 v 1008 1681 a Fj(k)p Fm(u)1061 1661 y Fl(+)1061 1694 y(1)p Fh(;\014)1112 1681 y Fj(k)1137 1695 y Fh(L)1161 1680 y Fg(+)1161 1707 y Fa(\014)1186 1695 y Fl(\()p Fh(\017)p Fl(\))1248 1647 y Fo(=)g(\()p Fj(k)p Fm(u)1372 1626 y Fl(+)1372 1660 y(2)p Fh(;\014)1423 1647 y Fj(k)1448 1661 y Fh(L)1472 1646 y Fg(+)1472 1673 y Fa(\014)1497 1661 y Fl(\()p Fh(\017)p Fl(\))1541 1647 y Fo(\))1560 1627 y Fh(\013)1585 1647 y Fo(\()p Fj(k)p Fm(u)1657 1626 y Fl(+)1657 1660 y(1)p Fh(;\014)1707 1647 y Fj(k)1732 1661 y Fh(L)1756 1646 y Fg(+)1756 1673 y Fa(\014)1782 1661 y Fl(\()p Fh(\017)p Fl(\))1825 1647 y Fo(\))1844 1627 y Fh(\013)p Fi(\000)p Fl(2)1914 1647 y Fm(:)1850 1754 y Fo(\(5.6\))0 1814 y(Io)q(c)o(h)o(um) 21 b(and)h(T)l(estard)g([13])f(pro)o(v)o(ed)g(an)h(estimate)e(that)i (is)f(analogus)i(to)f(our)g(Prop)q(osition)h(10,)g(but)0 1874 y(b)q(ounds)17 b(the)f(gro)o(wth)g(of)g(solutions)h(from)e(ab)q(o) o(v)o(e.)21 b(Their)16 b(result)f(implies)f(that)j(for)f(ev)o(ery)e Fm(\025)g Fj(2)g Fk(R)p Fo(,)i(there)0 1934 y(exists)d(a)h(\014nite)f (p)q(ositiv)o(e)g(constan)o(t)19 b(~)-29 b Fm(\013)695 1941 y Fl(1)714 1934 y Fo(,)14 b(suc)o(h)f(that)h(for)g(ev)o(ery)e Fm(E)17 b Fo(in)c(the)g(sp)q(ectrum)h(of)g Fm(H)1642 1941 y Fh(\025)1665 1934 y Fo(,)f(an)o(y)h(solution)0 1995 y Fm(u)h Fo(of)h Fm(H)138 2002 y Fh(\025)161 1995 y Fm(u)e Fo(=)f Fm(E)s(u)i Fo(ob)q(eys)h(lim)8 b(sup)621 2006 y Fh(L)p Fi(!1)726 1995 y Fm(L)759 1976 y Fi(\000)s Fl(~)-21 b Fh(\013)809 1981 y Fg(1)828 1995 y Fj(k)p Fm(u)p Fj(k)906 2002 y Fh(L)946 1995 y Fm(<)14 b Fj(1)p Fo(.)20 b(This)c(means)f(that)h(for)g(appropriate)g(energies)0 2055 y Fm(E)s Fo(,)e(the)f(solution)g Fm(u)357 2034 y Fl(+)357 2067 y(2)p Fh(;\014)421 2055 y Fo(of)h(\(5.6\))g(ob)q(eys)f Fj(k)p Fm(u)773 2034 y Fl(+)773 2067 y(2)p Fh(;\014)824 2055 y Fj(k)849 2068 y Fh(L)873 2053 y Fg(+)873 2081 y Fa(\014)898 2068 y Fl(\()p Fh(\017)p Fl(\))956 2055 y Fm(<)h(C)1043 2062 y Fl(2)1062 2055 y Fo(\()p Fm(L)1114 2034 y Fl(+)1114 2067 y Fh(\014)1144 2055 y Fo(\()p Fm(\017)p Fo(\)\))1224 2037 y Fl(~)-21 b Fh(\013)1244 2042 y Fg(1)1276 2055 y Fo(for)14 b(some)f(constan)o(t)h Fm(C)1696 2062 y Fl(2)1715 2055 y Fo(.)21 b(Similarly)l(,)0 2131 y(Prop)q(osition)15 b(10)g(implies)e(that)i Fj(k)p Fm(u)644 2110 y Fl(+)644 2143 y(1)p Fh(;\014)694 2131 y Fj(k)719 2144 y Fh(L)743 2129 y Fg(+)743 2157 y Fa(\014)769 2144 y Fl(\()p Fh(\017)p Fl(\))826 2131 y Fm(>)f(C)913 2138 y Fl(1)933 2131 y Fo(\()p Fm(L)985 2110 y Fl(+)985 2143 y Fh(\014)1014 2131 y Fo(\()p Fm(\017)p Fo(\)\))1094 2112 y Fl(~)-21 b Fh(\013)1130 2131 y Fo(for)14 b(some)g(constan)o(t)h Fm(C)1552 2138 y Fl(1)1572 2131 y Fo(.)20 b(Th)o(us,)15 b(for)f Fm(\013)h Fj(\024)e Fo(1,)0 2209 y(the)21 b(r.h.s.)f(of)i (\(5.6\))f(is)g(b)q(ounded)i(from)e(ab)q(o)o(v)o(e)g(b)o(y)g Fm(C)1032 2191 y Fh(\013)1028 2221 y Fl(2)1056 2209 y Fm(C)1095 2188 y Fh(\013)p Fi(\000)p Fl(2)1091 2220 y(1)1164 2209 y Fo(\()p Fm(L)1216 2188 y Fl(+)1216 2221 y Fh(\014)1246 2209 y Fo(\()p Fm(\017)p Fo(\)\))1323 2191 y Fh(\013)p Fl(\()s(~)-21 b Fh(\013)1383 2196 y Fg(1)1399 2191 y Fl(+)s(~)g Fh(\013)p Fl(\))p Fi(\000)p Fl(2)s(~)g Fh(\013)1532 2209 y Fo(,)23 b(whic)o(h)d(is)h(b)q(ounded)0 2269 y(from)16 b(ab)q(o)o(v)o(e)g(\(as)h Fm(\017)d Fj(!)f Fo(0\))k(if)f(w)o(e)g(tak)o (e)f Fm(\013)f Fj(\024)g Fo(2)5 b(~)-29 b Fm(\013)q(=)p Fo(\()5 b(~)-29 b Fm(\013)939 2276 y Fl(1)970 2269 y Fo(+)16 b(~)-29 b Fm(\013)p Fo(\).)p 1129 2273 21 30 v 90 2397 a Fe(Pro)r(of)14 b(of)h(Theorem)e(3.)20 b Fo(By)12 b(Prop)q(osition)h(1)g(of)g([27],)g(the)f(Fib)q(onacci)g(p)q(oten)o (tial)g(ob)q(eys)h Fm(V)e Fo(\()p Fj(\000)p Fm(n)p Fo(\))j(=)0 2457 y Fm(V)d Fo(\()p Fm(n)f Fj(\000)g Fo(1\))15 b(for)h(an)o(y)g Fm(n)e Fj(\025)f Fo(2.)22 b(Let)15 b Fm(z)h Fo(=)e Fm(E)f Fo(+)c Fm(i\017)p Fo(,)15 b(where)g Fm(E)i Fj(2)d Fk(R)h Fo(and)i Fm(\017)c(>)h Fo(0,)i(then)f(the)g(uniqueness)g(of)h(the)0 2517 y(solutions)k(^)-27 b Fm(u)232 2499 y Fi(\006)232 2530 y Fh(z)277 2517 y Fo(\(in)o(tro)q(duced)16 b(in)g(Section)g(2\))g (along)h(with)g(this)f(symmetry)l(,)e(implies)h(that)795 2622 y(^)-28 b Fm(u)819 2604 y Fl(+)819 2634 y Fh(z)849 2622 y Fo(\(1\))p 791 2644 120 2 v 794 2690 a(^)h Fm(u)819 2675 y Fl(+)819 2702 y Fh(z)849 2690 y Fo(\(0\))930 2656 y(=)990 2622 y(^)g Fm(u)1015 2604 y Fi(\000)1015 2634 y Fh(z)1044 2622 y Fo(\()p Fj(\000)p Fo(2\))p 987 2644 159 2 v 990 2690 a(^)g Fm(u)1015 2675 y Fi(\000)1015 2702 y Fh(z)1044 2690 y Fo(\()p Fj(\000)p Fo(1\))1150 2656 y Fm(:)686 b Fo(\(5.7\))951 2828 y(13)p eop %%Page: 14 15 14 14 bop 0 203 a Fo(F)l(or)16 b(an)o(y)h(solution)f Fm(u)g Fo(of)h Fm(H)504 210 y Fh(\025)527 203 y Fm(u)c Fo(=)h Fm(z)r(u)p Fo(,)i(w)o(e)f(clearly)g(ha)o(v)o(e)h(for)g(ev)o(ery) f Fm(n)p Fo(,)624 304 y Fm(u)p Fo(\()p Fm(n)c Fo(+)g(1\))p 624 326 180 2 v 667 372 a Fm(u)p Fo(\()p Fm(n)p Fo(\))822 337 y(=)j Fm(z)f Fj(\000)e Fm(V)g Fo(\()p Fm(n)p Fo(\))g Fj(\000)1132 304 y Fm(u)p Fo(\()p Fm(n)g Fj(\000)g Fo(1\))p 1132 326 181 2 v 1175 372 a Fm(u)p Fo(\()p Fm(n)p Fo(\))1317 337 y Fm(;)519 b Fo(\(5.8\))0 472 y(and)17 b(th)o(us)466 525 y(^)-27 b Fm(u)491 507 y Fi(\000)491 538 y Fh(z)521 525 y Fo(\()p Fj(\000)p Fo(2\))p 463 548 159 2 v 466 593 a(^)g Fm(u)491 579 y Fi(\000)491 606 y Fh(z)521 593 y Fo(\()p Fj(\000)p Fo(1\))640 559 y(=)14 b Fm(z)f Fj(\000)e Fm(V)g Fo(\()p Fj(\000)p Fo(1\))g Fj(\000)979 486 y Ff( )1012 559 y Fm(z)i Fj(\000)e Fm(V)g Fo(\(0\))h Fj(\000)1268 525 y Fo(^)-27 b Fm(u)1293 507 y Fi(\000)1293 538 y Fh(z)1323 525 y Fo(\(1\))p 1265 548 120 2 v 1268 593 a(^)g Fm(u)1293 579 y Fi(\000)1293 606 y Fh(z)1323 593 y Fo(\(0\))1390 486 y Ff(!)1423 498 y Fi(\000)p Fl(1)1478 559 y Fm(:)358 b Fo(\(5.9\))0 677 y(By)16 b(using)g(\(2.6\))h(and)g(the)f(fact)g(that) g Fm(V)c Fo(\(0\))i(=)g(0)i(and)h Fm(V)11 b Fo(\()p Fj(\000)p Fo(1\))j(=)g Fm(\025)p Fo(,)i(\(5.7\))h(and)g(\(5.9\))f(imply)608 787 y Fj(\000)p Fm(m)690 766 y Fl(+)719 787 y Fo(\()p Fm(z)r Fo(\))e(=)f Fm(z)g Fj(\000)e Fm(\025)h Fj(\000)f Fo(\()p Fm(z)h Fj(\000)f Fm(m)1170 766 y Fi(\000)1199 787 y Fo(\()p Fm(z)r Fo(\)\))1281 766 y Fi(\000)p Fl(1)1328 787 y Fm(;)483 b Fo(\(5.10\))0 896 y(whic)o(h)16 b(can)g(b)q(e)g (rearanged)h(to)g(yield)627 1005 y Fm(m)670 985 y Fi(\000)700 1005 y Fo(\()p Fm(z)r Fo(\))c(=)h Fm(z)f Fj(\000)e Fo(\()p Fm(z)i Fj(\000)e Fm(\025)g Fj(\000)g Fm(m)1151 985 y Fl(+)1180 1005 y Fo(\()p Fm(z)r Fo(\)\))1262 985 y Fi(\000)p Fl(1)1309 1005 y Fm(:)502 b Fo(\(5.11\))0 1115 y(Since)18 b(w)o(e)h(already)f(kno)o(w)h(that)h Fm(H)658 1122 y Fh(\025)700 1115 y Fo(has)f(purely)f(singular)i(sp)q(ectrum,)e(its)h (sp)q(ectral)g(measure)g(is)f(sup-)0 1175 y(p)q(orted)g(on)g(the)g(set) f(of)h(energies)f Fm(E)k Fo(for)d(whic)o(h)f Fj(j)p Fm(M)5 b Fo(\()p Fm(E)15 b Fo(+)d Fm(i)p Fo(0\))p Fj(j)k Fo(=)h Fj(1)p Fo(.)25 b(Moreo)o(v)o(er,)16 b(it)h(had)i(b)q(een)e(sho)o(wn)0 1235 y(b)o(y)e(Gilb)q(ert)h([9])f(that)i(the)e(sp)q(ectral)h(measure)g (is)g(supp)q(orted)g(on)h(the)f(set)f(of)i Fm(E)s Fo('s)e(for)i(whic)o (h)e Fm(H)1767 1242 y Fh(\025)1790 1235 y Fm(u)e Fo(=)h Fm(E)s(u)0 1295 y Fo(has)k(solutions)g(that)g(are)f(sub)q(ordinate)h(b) q(oth)g(to)g(left)e(and)i(to)g(the)f(righ)o(t.)24 b(This)17 b(implies)f(that)i(for)f(suc)o(h)0 1355 y(energies,)c Fm(m)239 1337 y Fi(\006)269 1355 y Fo(\()p Fm(E)c Fo(+)e Fm(i)p Fo(0\))14 b(exist)g(and)g(are)h(either)e(real)h(or)g(in\014nit)o (y)f(\(where)h(w)o(e)f(iden)o(tify)g Fm(m)1636 1337 y Fi(\006)1665 1355 y Fo(\()p Fm(E)d Fo(+)d Fm(i)p Fo(0\))13 b(=)h Fj(1)0 1416 y Fo(with)22 b Fj(j)p Fm(m)174 1397 y Fi(\006)203 1416 y Fo(\()p Fm(E)c Fo(+)c Fm(i)p Fo(0\))p Fj(j)24 b Fo(=)f Fj(1)p Fo(,)f(namely)f(with)h(lim)933 1423 y Fh(\017)p Fi(!)p Fl(0)1010 1416 y Fj(j)p Fm(m)1067 1397 y Fi(\006)1096 1416 y Fo(\()p Fm(E)c Fo(+)d Fm(i\017)p Fo(\))p Fj(j)22 b Fo(=)i Fj(1)p Fo(\).)37 b(That)23 b(is,)g(the)e(sp)q (ectral)0 1476 y(measure)d(of)g Fm(H)289 1483 y Fh(\025)330 1476 y Fo(is)g(supp)q(orted)h(on)f(a)h(set)e Fm(S)k Fo(that)e(is)e(con) o(tained)h(in)g(its)f(sp)q(ectrum)h(and)h(suc)o(h)e(that)i(for)0 1536 y(an)o(y)j Fm(E)27 b Fj(2)d Fm(S)s Fo(,)f Fj(j)p Fm(M)5 b Fo(\()p Fm(E)18 b Fo(+)d Fm(i)p Fo(0\))p Fj(j)24 b Fo(=)g Fj(1)e Fo(and)h Fm(m)856 1518 y Fi(\006)885 1536 y Fo(\()p Fm(E)18 b Fo(+)d Fm(i)p Fo(0\))22 b(b)q(oth)h(exist)f (and)g(are)h(in)e Fk(R)15 b Fj([)h(f1g)p Fo(.)38 b(F)l(rom)0 1596 y(\(2.7\))21 b(along)h(with)f(the)f(fact)h(that)g Fm(m)714 1578 y Fi(\006)744 1596 y Fo(\()p Fm(z)r Fo(\))f(ha)o(v)o(e)g (p)q(ositiv)o(e)g(imaginary)h(parts,)h(w)o(e)f(see)f(that)h(for)h(eac)o (h)0 1656 y Fm(E)17 b Fj(2)d Fm(S)s Fo(,)i(w)o(e)f(m)o(ust)h(ha)o(v)o (e)g(either)f Fj(j)p Fm(m)663 1638 y Fl(+)692 1656 y Fo(\()p Fm(E)f Fo(+)d Fm(i)p Fo(0\))p Fj(j)i Fo(=)h Fj(j)p Fm(m)1006 1638 y Fi(\000)1035 1656 y Fo(\()p Fm(E)g Fo(+)d Fm(i)p Fo(0\))p Fj(j)j Fo(=)g Fj(1)i Fo(or)g(that)h Fm(m)1567 1638 y Fi(\006)1596 1656 y Fo(\()p Fm(E)d Fo(+)d Fm(i)p Fo(0\))16 b(are)g(real)0 1716 y(and)k Fm(m)141 1698 y Fl(+)170 1716 y Fo(\()p Fm(E)d Fo(+)c Fm(i)p Fo(0\))19 b(=)h Fj(\000)p Fm(m)512 1698 y Fi(\000)541 1716 y Fo(\()p Fm(E)c Fo(+)d Fm(i)p Fo(0\).)31 b(But,)20 b(b)o(y)f(\(5.11\))h(w)o(e)f (see)g(that)h(if)f Fj(j)p Fm(m)1469 1698 y Fl(+)1498 1716 y Fo(\()p Fm(E)e Fo(+)c Fm(i)p Fo(0\))p Fj(j)19 b Fo(=)g Fj(1)p Fo(,)h(then)0 1777 y Fm(m)43 1759 y Fi(\000)72 1777 y Fo(\()p Fm(E)d Fo(+)c Fm(i)p Fo(0\))20 b(is)g(\014nite,)f(and)i (so)f(for)g(eac)o(h)f Fm(E)k Fj(2)d Fm(S)j Fo(w)o(e)d(m)o(ust)f(ha)o(v) o(e)g(that)h Fm(m)1456 1759 y Fi(\006)1485 1777 y Fo(\()p Fm(E)d Fo(+)c Fm(i)p Fo(0\))20 b(are)g(real)f(and)0 1837 y Fm(m)43 1819 y Fl(+)72 1837 y Fo(\()p Fm(E)d Fo(+)d Fm(i)p Fo(0\))18 b(=)g Fj(\000)p Fm(m)410 1819 y Fi(\000)439 1837 y Fo(\()p Fm(E)e Fo(+)d Fm(i)p Fo(0\).)29 b(Let)19 b Fm(E)i Fj(2)e Fm(S)s Fo(,)g(then)f(b)o(y)h(in)o(v)o(erting)e(\(2.5\)) i(to)g(yield)f Fm(m)1665 1816 y Fl(+)1665 1849 y Fh(\014)1713 1837 y Fo(in)g(terms)h(of)0 1897 y Fm(m)43 1879 y Fl(+)72 1897 y Fo(,)d(w)o(e)g(obtain)760 1983 y Fm(m)803 1962 y Fl(+)803 1995 y Fh(\014)846 1983 y Fo(=)903 1949 y Fm(m)946 1931 y Fl(+)983 1949 y Fo(cot)9 b Fm(\014)k Fo(+)e(1)p 903 1971 269 2 v 919 2017 a(cot)d Fm(\014)14 b Fj(\000)c Fm(m)1126 2003 y Fl(+)1176 1983 y Fm(;)635 b Fo(\(5.12\))0 2098 y(from)18 b(whic)o(h)g(w)o(e)g(see)g(that)h Fj(j)p Fm(m)579 2078 y Fl(+)579 2111 y Fh(\014)608 2098 y Fo(\()p Fm(E)c Fo(+)d Fm(i)p Fo(0\))p Fj(j)18 b Fo(=)f Fj(1)h Fo(exactly)f(when)i(cot)8 b Fm(\014)20 b Fo(=)d Fm(m)1462 2080 y Fl(+)1491 2098 y Fo(\()p Fm(E)f Fo(+)c Fm(i)p Fo(0\),)19 b(and)g(in)f(that)0 2158 y(case,)23 b Fj(j)p Fm(m)181 2138 y Fl(+)181 2171 y Fh(\014)210 2158 y Fo(\()p Fm(E)c Fo(+)c Fm(i\017)p Fo(\))p Fj(j)23 b(\030)h(j)p Fo(\(cot)8 b Fm(\014)18 b Fj(\000)d Fm(m)742 2140 y Fl(+)771 2158 y Fo(\()p Fm(E)j Fo(+)d Fm(i\017)p Fo(\)\))972 2140 y Fi(\000)p Fl(1)1019 2158 y Fj(j)22 b Fo(as)h Fm(\017)h Fj(!)g Fo(0.)39 b(Th)o(us,)24 b(Prop)q(osition)g (11)f(implies)0 2219 y(that)e(for)g(small)f Fm(\017)p Fo(,)h Fj(j)8 b Fo(cot)g Fm(\014)16 b Fj(\000)e Fm(m)611 2200 y Fl(+)640 2219 y Fo(\()p Fm(E)j Fo(+)d Fm(i\017)p Fo(\))p Fj(j)20 b Fo(is)g(m)o(uc)o(h)g(larger)g(than)h Fm(\017)p Fo(.)34 b(By)20 b(di\013eren)o(tiating)f(the)i(r.h.s)0 2279 y(of)e(\(5.11\))h(with)f(resp)q(ect)g(to)g Fm(m)591 2261 y Fl(+)640 2279 y Fo(\(at)g(the)g(p)q(oin)o(t)g(where)g Fm(z)i Fo(=)d Fm(E)k Fo(and)e Fm(m)1381 2261 y Fl(+)1429 2279 y Fo(=)f(cot)8 b Fm(\014)s Fo(\),)18 b(w)o(e)h(see)g(that)g(as)0 2339 y Fm(\017)14 b Fj(!)h Fo(0,)i Fj(j)p Fm(m)211 2321 y Fl(+)240 2339 y Fo(\()p Fm(E)d Fo(+)e Fm(i\017)p Fo(\))e(+)i Fm(m)518 2321 y Fi(\000)547 2339 y Fo(\()p Fm(E)i Fo(+)e Fm(i\017)p Fo(\))p Fj(j)i(\030)g(j)8 b Fo(cot)h Fm(\014)k Fj(\000)e Fm(m)1033 2321 y Fl(+)1063 2339 y Fo(\()p Fm(E)j Fo(+)d Fm(i\017)p Fo(\))p Fj(j)16 b Fo(and)i(th)o(us,)e(b)o(y)g (\(2.7\),)h(w)o(e)f(also)i(ha)o(v)o(e)0 2399 y Fj(j)p Fm(M)5 b Fo(\()p Fm(E)14 b Fo(+)d Fm(i\017)p Fo(\))p Fj(j)i(\030)h(j)p Fm(m)377 2378 y Fl(+)377 2412 y Fh(\014)406 2399 y Fo(\()p Fm(E)g Fo(+)d Fm(i\017)p Fo(\))p Fj(j)p Fm(:)k Fo(By)g(Prop)q(osition)j(11,)e(this)h(implies)d(that)667 2513 y(lim)8 b(sup)708 2551 y Fh(\017)p Fi(!)p Fl(0)825 2513 y Fm(\017)845 2493 y Fl(1)p Fi(\000)p Fh(\013)914 2513 y Fj(j)p Fm(M)d Fo(\()p Fm(E)15 b Fo(+)c Fm(i\017)p Fo(\))p Fj(j)i Fm(<)g Fj(1)542 b Fo(\(5.13\))0 2643 y(for)21 b(a.e.)e Fm(E)24 b Fo(w.r.t.)19 b(the)h(sp)q(ectral)g(measure)g(of)h Fm(H)935 2650 y Fh(\025)958 2643 y Fo(,)g(and)g(th)o(us,)g Fm(H)1258 2650 y Fh(\025)1301 2643 y Fo(has)g(purely)f Fm(\013)p Fo(-con)o(tin)o(uous)h(sp)q(ec-)0 2704 y(trum.)p 160 2708 21 30 v 951 2828 a(14)p eop %%Page: 15 16 15 15 bop 0 203 a Fe(Remark.)23 b Fo(The)17 b(pro)q(of)h(of)g(Theorem)e (3)i(uses)f(the)g(symmetry)f(of)i(the)e(p)q(oten)o(tial)h(to)h(sho)o(w) g(that)f(for)h(an)0 264 y(appropriate)e Fm(\014)s Fo(,)f Fj(j)p Fm(m)379 243 y Fl(+)379 276 y Fh(\014)408 264 y Fo(\()p Fm(E)e Fo(+)d Fm(i\017)p Fo(\))p Fj(j)j(\030)h(j)p Fm(M)5 b Fo(\()p Fm(E)13 b Fo(+)d Fm(i\017)p Fo(\))p Fj(j)15 b Fo(as)h Fm(\017)e Fj(!)f Fo(0,)j(and)g(th)o(us)g(that)g(the)g (con)o(tin)o(uit)o(y)e(result)h(for)0 324 y(the)f(half-line)g(problem)g (\(Prop)q(osition)h(11\))g(translates)g(to)g(a)g(corresp)q(onding)g (con)o(tin)o(uit)o(y)e(result)h(for)g(the)0 384 y(whole-line)h (problem.)21 b(The)16 b(crucial)f(fact)i(to)f(sho)o(wing)h(this)f(is)g (\(5.11\),)h(whic)o(h)e(has)i(the)f(form)743 494 y Fm(m)786 473 y Fi(\000)816 494 y Fo(\()p Fm(z)r Fo(\))d(=)h Fm(f)5 b Fo(\()p Fm(z)r(;)j(m)1082 473 y Fl(+)1111 494 y Fo(\()p Fm(z)r Fo(\)\))p Fm(;)618 b Fo(\(5.14\))0 604 y(where)13 b Fm(f)18 b Fo(is)13 b(a)g(rational)h(function)e(of)i(t)o(w)o(o)f(v)m (ariables)f(with)h(real)g(co)q(e\016cien)o(ts.)19 b(By)12 b(using)h(\(5.8\))h(and)f(\(2.6\),)0 664 y(one)j(can)g(sho)o(w)h(that)g (a)f(relation)g(of)g(the)g(form)g(\(5.14\))h(w)o(ould)f(hold)g(whenev)o (er)f Fm(V)c Fo(\()p Fm(k)i Fj(\000)d Fm(n)p Fo(\))k(=)g Fm(V)d Fo(\()p Fm(m)f Fo(+)h Fm(n)p Fo(\))0 724 y(for)17 b(some)g Fm(k)r(;)8 b(m)14 b Fj(2)g Fk(Z)j Fo(and)g(all)g Fm(n)d Fj(2)h Fk(N)p Fo(,)h(and)h(the)g(implication)e Fj(j)p Fm(m)1214 704 y Fl(+)1214 737 y Fh(\014)1243 724 y Fo(\()p Fm(E)f Fo(+)e Fm(i\017)p Fo(\))p Fj(j)h(\030)i(j)p Fm(M)5 b Fo(\()p Fm(E)14 b Fo(+)d Fm(i\017)p Fo(\))p Fj(j)16 b Fo(can)h(th)o(us)0 785 y(b)q(e)j(extended)g(to)h(an)o(y)f (suc)o(h)g(case.)33 b(That)21 b(is,)g(if)f Fm(V)11 b Fo(\()p Fm(k)16 b Fj(\000)d Fm(n)p Fo(\))21 b(=)g Fm(V)11 b Fo(\()p Fm(m)j Fo(+)f Fm(n)p Fo(\))21 b(for)f(some)h Fm(k)r(;)8 b(m)20 b Fj(2)h Fk(Z)f Fo(and)0 845 y(all)f Fm(n)f Fj(2)h Fk(N)p Fo(,)h(and)f(if)g(there)g(exists)f(an)i(0)f Fm(<)f(\013)h Fj(\024)g Fo(1)g(suc)o(h)g(that)h(for)f(an)o(y)g Fm(E)j Fo(in)d(some)g(Borel)f(set)h Fm(S)j Fo(and)0 905 y(ev)o(ery)c Fm(\014)s Fo(,)h(lim)8 b(sup)345 917 y Fh(\017)p Fi(!)p Fl(0)423 905 y Fm(\017)443 887 y Fl(1)p Fi(\000)p Fh(\013)512 905 y Fj(j)p Fm(m)569 884 y Fl(+)569 918 y Fh(\014)598 905 y Fo(\()p Fm(E)16 b Fo(+)d Fm(i\017)p Fo(\))p Fj(j)18 b Fm(<)h Fj(1)p Fo(,)h(then)f(the)g(restriction)f(of)i (the)f(sp)q(ectral)g(measure)g(of)0 965 y(the)d(whole-line)f(problem)h (to)h Fm(S)i Fo(is)d Fm(\013)p Fo(-con)o(tin)o(uous.)0 1150 y Fn(References)24 1277 y Fo([1])24 b(J.)d(Avron)f(and)i(B.)e (Simon,)h(Singular)g(con)o(tin)o(uous)g(sp)q(ectrum)g(for)g(a)g(class)g (of)g(almost)g(p)q(erio)q(dic)100 1337 y(Jacobi)c(matrices.)e(Bull.)f (AMS)i Fe(6)p Fo(,)g(81-85)i(\(1982\).)24 1439 y([2])24 b(Bellissard,)k(J.,)g(Io)q(c)o(h)o(um,)g(B.,)f(Scopp)q(ola,)j(E.,)e(T)l (estard,)i(D.:)41 b(Sp)q(ectral)26 b(prop)q(erties)h(of)g(one-)100 1499 y(dimensional)16 b(quasi-crystals.)g(Comm)o(un.)f(Math.)h(Ph)o (ys.)g Fe(125)p Fo(,)g(527{543)j(\(1989\))24 1601 y([3])24 b(Berezanskii,)16 b(Y.:)23 b(Expansions)c(in)e(Eigenfunctions)h(of)g (Selfadjoin)o(t)f(Op)q(erators.)h(T)l(ransl.)f(Math.)100 1661 y(Monogr.,)g(V)l(ol.)e(17.)h(Pro)o(vidence,)f(RI:)g(Am.)g(Math.)h (So)q(c.)g(1968.)24 1763 y([4])24 b(Bourgain,)15 b(J.,)f(Goldstein,)g (M.:)20 b(On)14 b(nonp)q(erturbativ)o(e)h(lo)q(calization)f(with)g (quasip)q(erio)q(dic)g(p)q(oten-)100 1823 y(tial.)i(Preprin)o(t.)24 1925 y([5])24 b(Carmona,)17 b(R.,)e(Lacroix,)h(J.:)21 b(Sp)q(ectral)16 b(Theory)g(of)h(Random)f(Sc)o(hr\177)-24 b(odinger)16 b(Op)q(erators.)h(Boston,)100 1985 y(MA:)e(Birkhauser)h (1990.)24 2087 y([6])24 b(Cornfeld,)16 b(I.,)f(F)l(omin,)g(S.,)h (Sinai,)f(Y)l(a.:)21 b(Ergo)q(dic)c(Theory)l(.)f(New)f(Y)l(ork:)21 b(Springer)16 b(1982)24 2188 y([7])24 b(Cycon,)16 b(H.L.,)g(F)l(ro)q (ese,)g(R.G.,)f(Kirsc)o(h,)g(W.,)h(Simon,)g(B.:)k(Sc)o(hr\177)-24 b(odinger)17 b(Op)q(erators.)g(Berlin,)d(Hei-)100 2248 y(delb)q(erg,)i(New)g(Y)l(ork:)k(Springer)d(1987)24 2350 y([8])24 b(del)d(Rio,)g(R.,)h(Jitomirsk)m(a)o(y)o(a,)f(S.,)g(Last,)i (Y.,)e(Simon,)h(B.:)30 b(Op)q(erators)22 b(with)f(singular)h(con)o(tin) o(u-)100 2410 y(ous)f(sp)q(ectrum,)g(IV.)e(Hausdor\013)j(dimensions,)e (rank-one)h(p)q(erturbations,)h(and)f(lo)q(calization.)e(J.)100 2471 y(d'Analyse)c(Math.)h Fe(69)p Fo(,)g(153{200)j(\(1996\).)24 2572 y([9])24 b(Gilb)q(ert,)19 b(D.J.:)26 b(On)19 b(sub)q(ordinacy)h (and)g(analysis)f(of)g(the)g(sp)q(ectrum)g(of)g(Sc)o(hr\177)-24 b(odinger)19 b(op)q(erators)100 2632 y(with)d(t)o(w)o(o)h(singular)f (endp)q(oin)o(ts.)g(Pro)q(c.)g(Ro)o(y)l(.)g(So)q(c.)g(Edin)o(burgh)h (Sect.)e(A)h Fe(112)p Fo(,)g(213{229)j(\(1989\))951 2828 y(15)p eop %%Page: 16 17 16 16 bop 0 203 a Fo([10])24 b(Gilb)q(ert,)k(D.J.,)f(P)o(earson,)h(D.:) 41 b(On)26 b(sub)q(ordinacy)g(and)h(analysis)f(of)g(the)g(sp)q(ectrum)f (of)i(one-)100 264 y(dimensional)16 b(Sc)o(hr\177)-24 b(odinger)16 b(op)q(erators.)h(J.)f(Math.)g(Anal.)f Fe(128)p Fo(,)h(30{56)i(\(1987\))0 362 y([11])24 b(Gordon,)g(A.,)f(Jitomirsk)m (a)o(y)o(a,)f(S.,)g(Last,)i(Y.,)f(Simon,)f(B.:)32 b(Dualit)o(y)21 b(and)i(singular)f(con)o(tin)o(uous)100 422 y(sp)q(ectrum)16 b(in)g(the)g(almost)h(Mathieu)e(equation.)h(Acta)g(Math.)g Fe(178)p Fo(,)g(169-183)j(\(1997\).)0 521 y([12])24 b(Herman,)d(M.:)29 b(Une)19 b(metho)q(de)i(p)q(our)g(minorer)f(les)g(exp)q(osan)o(ts)h(de) f(Ly)o(apuno)o(v)h(et)f(quelques)f(ex-)100 581 y(emples)e(mon)o(tran)o (t)i(le)e(caractere)h(lo)q(cal)g(d'un)g(theoreme)f(d'Arnold)h(et)g(de)g (moser)g(sur)h(le)e(tore)h(en)100 641 y(dimension)e(2.)g(Comm)o(un.)g (Math.)g(helv.)e Fe(58)p Fo(,)i(453-502)j(\(1983\).)0 740 y([13])24 b(Io)q(c)o(h)o(um,)17 b(B.,)g(T)l(estard,)h(D.:)24 b(P)o(o)o(w)o(er)17 b(la)o(w)h(gro)o(wth)h(for)f(the)f(resistance)g(in) h(the)f(Fib)q(onacci)h(mo)q(del.)100 800 y(J.)e(Stat.)g(Ph)o(ys.)g Fe(65)p Fo(,)g(715{723)j(\(1991\).)0 898 y([14])24 b(Jitomirsk)m(a)o(y) o(a,)13 b(S.:)19 b(Almost)13 b(ev)o(erything)f(ab)q(out)j(the)e(almost) h(Mathieu)f(op)q(erator)h(I)q(I.)f(Pro)q(ceedings)100 958 y(of)k(the)f(XI)g(In)o(ternational)g(Congress)i(of)e(Mathematical)g (Ph)o(ysics,)f(P)o(aris)i(1994,)h(In)o(t.)d(Press,)h(373{)100 1019 y(382)i(\(1995\).)0 1117 y([15])24 b(Jitomirsk)m(a)o(y)o(a,)14 b(S.:)20 b(Metal)15 b(insulator)g(transition)g(for)g(the)g(almost)g (Mathieu)f(op)q(erator.)i(Annals)f(of)100 1177 y(Math.,)h(to)g(app)q (ear.)0 1276 y([16])24 b(Jitomirsk)m(a)o(y)o(a,)g(S.,)g(Last,)h(Y.:)34 b(Dimensional)23 b(Hausdor\013)h(prop)q(erties)f(of)h(singular)f(con)o (tin)o(uous)100 1336 y(sp)q(ectra.)16 b(Ph)o(ys.)g(Rev.)f(Lett.)h Fe(76)p Fo(,)g(1765{1769)k(\(1996\).)0 1434 y([17])k(Jitomirsk)m(a)o(y) o(a,)d(S.,)g(Last,)h(Y.:)29 b(P)o(o)o(w)o(er-La)o(w)21 b(sub)q(ordinacy)g(and)g(singular)g(sp)q(ectra,)g(I.)f(Half)g(line)100 1495 y(op)q(erators.)e(Acta)d(Math.,)h(to)h(app)q(ear.)0 1593 y([18])24 b(Jitomirsk)m(a)o(y)o(a,)13 b(S.,)f(Simon,)h(B.:)19 b(Op)q(erators)14 b(with)f(Singular)g(Con)o(tin)o(uous)g(Sp)q(ectrum,)g (I)q(I)q(I.)f(Almost)100 1653 y(P)o(erio)q(dic)k(Sc)o(hro)q(dinger)g (Op)q(erators.)h(Comm.)e(Math.)h(Ph)o(ys.)g Fe(165)p Fo(,)g(201-206)j(\(1994\).)0 1752 y([19])24 b(Khin)o(tc)o(hin,)14 b(A.Y)l(a.:)20 b(Con)o(tin)o(ued)c(F)l(ractions.)g(Groningen:)22 b(No)q(ordho\013)c(1963.)0 1850 y([20])24 b(Last,)15 b(Y.:)k(A)14 b(relation)f(b)q(et)o(w)o(een)g(absolutely)h(con)o(tin)o (uous)g(sp)q(ectrum)g(of)g(ergo)q(dic)g(Jacobi)g(matrices)100 1910 y(and)j(the)f(sp)q(ectra)h(of)f(p)q(erio)q(dic)g(appro)o(ximan)o (ts.)g(Comm)o(un.)f(Math.)h(Ph)o(ys.)g Fe(151)p Fo(,)g(183{192)j (\(1993\))0 2009 y([21])24 b(Last,)f(Y.:)29 b(Zero)21 b(measure)f(sp)q(ectrum)h(for)g(the)f(almost)h(Mathieu)g(op)q(erator.)g (Comm)o(un.)f(Math.)100 2069 y(Ph)o(ys.)c Fe(164)p Fo(,)g(421{432)j (\(1994\))0 2167 y([22])24 b(Last)h(Y.:)35 b(Quan)o(tum)23 b(dynamics)g(and)h(decomp)q(ositions)g(of)g(singular)g(con)o(tin)o (uous)f(sp)q(ectra.)g(J.)100 2228 y(F)l(unct.)16 b(Anal.)f Fe(142)p Fo(,)h(406{445)j(\(1996\))0 2326 y([23])24 b(Last,)16 b(Y.,)e(Simon,)h(B.:)k(Eigenfunctions,)c(transfer)g(matrices,)f(and)i (absolutely)f(con)o(tin)o(uous)g(sp)q(ec-)100 2386 y(trum)h(of)h (one-dimensional)f(Sc)o(hr\177)-24 b(odinger)16 b(op)q(erators.)h(In)o (v)o(en)o(t.)d(Math.)i Fe(135)p Fo(,)g(329{367)j(\(1999\))0 2485 y([24])24 b(Sorets,)d(E.,)f(Sp)q(encer,)g(T.:)28 b(P)o(ositiv)o(e)18 b(Ly)o(apuno)o(v)j(exp)q(onen)o(ts)e(for)h(Sc)o (hr\177)-24 b(odinger)20 b(op)q(erators)h(with)100 2545 y(quasi-p)q(erio)q(dic)c(p)q(oten)o(tials.)e(Comm)o(un.)h(Math.)g(Ph)o (ys.)f Fe(142)p Fo(,)h(543-566)j(\(1991\))0 2643 y([25])24 b(Ra)o(ymond,)19 b(L.:)25 b(A)18 b(constructiv)o(e)f(gap)j(lab)q (elling)d(for)i(the)f(discrete)f(Sc)o(hr\177)-24 b(odinger)18 b(op)q(erator)i(on)f(a)100 2704 y(quasip)q(erio)q(dic)d(c)o(hain.)g (Preprin)o(t)951 2828 y(16)p eop %%Page: 17 18 17 17 bop 0 203 a Fo([26])24 b(Simon,)16 b(B.:)k(Sc)o(hr\177)-24 b(odinger)16 b(semigroups,)g(Bull.)f(AMS)g Fe(7)p Fo(,)h(447-526)j (\(1982\).)0 305 y([27])24 b(S)q(\177)-26 b(ut})i(o,)15 b(A.:)k(The)13 b(sp)q(ectrum)g(of)g(a)h(quasip)q(erio)q(dic)f(Sc)o (hr\177)-24 b(odinger)12 b(op)q(erator.)i(Comm)o(un.)f(Math.)g(Ph)o (ys.)100 365 y Fe(111)p Fo(,)j(409{415)j(\(1987\))0 467 y([28])24 b(S)q(\177)-26 b(ut})i(o,)20 b(A.:)j(Singular)18 b(con)o(tin)o(uous)f(sp)q(ectrum)h(on)g(a)g(Can)o(tor)h(set)e(of)h (zero)g(Leb)q(esgue)g(measure)g(for)100 527 y(the)e(Fib)q(onacci)g (Hamiltonian.)f(J.)h(Stat.)g(Ph)o(ys.)g Fe(56)p Fo(,)g(525{531)j (\(1989\))951 2828 y(17)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9909291753689--