Content-Type: multipart/mixed; boundary="-------------0206170231461" This is a multi-part message in MIME format. ---------------0206170231461 Content-Type: text/plain; name="02-267.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-267.comments" This considerably improves upon the results in our preprint mp_arc 02-1. ---------------0206170231461 Content-Type: text/plain; name="02-267.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-267.keywords" quantum transport ---------------0206170231461 Content-Type: application/postscript; name="polymer.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="polymer.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: polymer.dvi %%Pages: 24 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMR12 CMR10 CMBX10 CMBX12 CMMI12 CMSY10 CMR8 %%+ CMMI8 CMSL12 CMEX10 CMSY8 CMR6 MSBM7 MSBM10 CMSY6 CMMI6 CMTI12 %%+ LASY10 CMTT12 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips polymer %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.06.17:0924 %%BeginProcSet: texc.pro %! 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e32ba19a00db8a22415fc720b852adf3ce9c92559edaa3bcc22b093bafbf4f8e d4c3ff5b5662ae30ffc107360d942700d72364b654a98f4f7558edb12a98ca0b 3d6de6a384675ca475eb836f1ff35b28a3ffa4aa3cbec43d804d673478a3ad0e 78cac90d4c7d8b9e3d9ac9f2eed4f422fe8c3f8b462eab86f2c139afc0973416 655263dfc146e4d0451dd19888ca1cc99422c4ecac11655fe1f8fa9220b8 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 12 /fi put dup 14 /ffi put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 58 /colon put dup 65 /A put dup 70 /F put dup 72 /H put dup 73 /I 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 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 readonly def /FontBBox{-36 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891625a1ba52efb7a0e22af38d9ab395e9c99417caa8bafd7276c20989ef7105 ef51506624782151e7c7f554987b5fb55cf8eff406692bbc157b15d1b448bc38 c64cff1b2cf4e1ee07e30425185282a7382ca34b37acc4d26e1752f93fa5191e 20 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 14 /delta put dup 15 /epsilon1 put dup 21 /lambda put dup 25 /pi put dup 33 /omega put dup 59 /comma put dup 67 /C put dup 69 /E put dup 82 /R put dup 84 /T put dup 99 /c put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 114 /r put dup 115 /s put dup 116 /t put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueXX 5087381 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a1e67ed7e41cc25acc04702f6 8ef703e3d45722c1a91fdef7100a48631a02a6f02a08c6b1f9b4df8310385b86 8632718fd87119a233f219d9411383b7fa9f3e4780d8c27e2e89e0cae883d664 c3eac57a3aef8988a2e9f0f8c7f53e0a80bdfc4620e21287d0390e1975398544 7f3ea66401024bea75e1b4c4437b7bb188f76f96b918ac7c6ad7e8ae7f21d8c2 790f08cccec904fe48ef39e597ed4d4237c1d1f596f5906b19ea308020f7a35c 168e327ec3246b1dfabe912f6b6daac09974876d3996e57d180261110db05f15 e3e8eebba3d90b5764c03df3033a1ed678ebc679569a2fb297378b25434c0f20 5313ecb8a952f07242d3ee731b0cdc086a4481178a3d65129c47c09b22e9c431 e11b3747b94c26a757c38d06001798c6a568303d541385244b967d3b1786edea f65bb53c4c2fe75e4b1b15c2c78d930b4296c80f08bad86012451edc8e9f0854 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98249c46403c56afb17533ed4928892a8969a439f1db113d9f06435902335bf1 f6d2e2d39a5fab024f42480a38c4977097cedf672795307577c424adb16f701b 1584c684554ab01ce1a3990fb0b0c545df1c79e6092948fef7389ede21dc386d 880bd62c26492e13919d348c433ab012379c8c47a809ee8ab436f0a04e874295 ba5d4a2831ac1fa68c95b18316e7e14c2d93260e41bcdcfa2a820a960464c16d 57b271e30e4b102193b4ab4d111d3e7c07f532fdbd00bf4e010362c90ebe2664 4a86e38c32d9adf133205d9e866a50147fb3a0bee4763ee47a8a8587cc845cef 42019de3ca885fad905f0b5c362d189cd89244ebb74f384916032c096bbccb75 22f6e0f18009f726471039ba1651c8ccb0e42cae9be825f2e57f8088142cc183 ab23816e403f8d5dc2ddbc19f8c2b7417a1b8b67fe6d9e250eb37be225fd0406 4786f2b9958227d0dc391cb5a021f66ff4abc7c48dc342fdb1 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 6 /plusminus put dup 48 /prime put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueXX 5000816 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6a18979f258b8c6ed3 ccafbc21aca420c9c83eea371adc20e038b4d7b8ac303004b0aa205f04135140 76407216032fdd22e6219da8f16b28ca12524deb7bca073cc5eba65c102a5e85 fd48e6d062cd4283ee570a7774597e5bf0e3400b6be72db0115f3cb12db70ce0 83722870cddfadee715f10f1fcaf20e06f3c54afe5ca238539bfe2b596116e83 f5371ff18fa5003d8543226cfd4025f9940365b392a858d27f078d3abcffe4a1 54e78c7692d1a32bf935967c64f01b24788ff8325d61145e2d4a489fd986fb77 38e6b254522c77ca2797a504a9ce4676a77ebacb026eca94dde5922c936f8e90 c43e2851905abd26842613d8fb644512275d43f312eea72f7ea3ba90c8fda6ce 7a9dacf52f55f0dd557f8d866ed6f3d3d5e6fe7e9ee79404fbe1a285bc519ecd 7ca162f0729044aff526bfe554ed59dd9beab0f842d55be08caada6136ac9f94 4cd96ff1c2f4927d87fea4496a1d612d58eba570eef0007c665fd143e9030923 517ac31aaf6912012671dd87686f2aaab25a2adc7501663d8a1ef80cc7da7dbe 13fb5333a4618ad8b4a6c2cc8fa3d616c65848c95ac10893c62198f642f2d2f3 503204e5d5aaef2602d4953fd4b77b402232f603d83de73bb3c2d50e2416481d 92e8147eef905c2812d6f99a347d02a88fab1c99dfc7a239e35262b5a40dbdef a0811fec7b3bd9cf2fcafcc714d3a5af1169f908d3df07e8705961d038e1881a 4f69eb69c9df1c78540f42b37de0ede2647d115acac4dd8dd54302911ed1dfd8 1ade02edfee46f8e3c5de5159938e4f5076a66243838a28e42b5f757e5fa2223 c5b56414540510a396366fe857023eeefa3b4c970609ea0ee08ba4dc6a1e57fd 6f3a1ee5257880b5d85e32bad58be98f232d603ec894e282326baec250e7e9fd acc49ef9b1dc997b3fbdc6ff862e02b1a2baf7c00b39aa7d82002d7256d9ae 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 78 /N put dup 82 /R put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueXX 5031982 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d56491bdecdcafdc988206c5a457a19270b 37d0ab776e03eaa7eb568eeab6b5e79dec03b0dcbf923a2aa8e4f4deda2cb043 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818455cc8dd8d40939f3ba8aa5abc54590c7286b172193c23adb1b4f9931a9f7 73b960b1f5b760438028478e4442dc93a567b07e20017c18fca2a1daa2649dee 5a6ec51fd6c3f4b27367c31e95a56ba4bbeda0493bc02306037e2ba69add9b94 373364db316691e73d6811cd76fbc92b0c67ca18dcdf05807f17d1b309456ea9 d113ea0fccd6eb73bd8dfdbd5905a5e38bbf61cfe58fb01ec95a4bdd9565433d 4a7abb8dca6f76c458ba9cb7568b924f70eaf52ec0167e9a9af3b6a4e5d3fd4d 62b615142bc52f2bee5557f7483d9377ec4d123091a300b3a6ae990d5ba252ab 95e88d1ac5df908a57a9896b043e7a4cf1efce44489d118414cccec128cebd1c c8c0f3e48c52461da294731af5936aee45de7f41d43e2762f740a2479405c947 e94092c2b06bc2a6483af211e9afe88a7c8995973b084f63152c2a446b291ee5 a63630e14f08283a6a6f922785e61b9cae0ce7aed3a6f99c1fa52bbc3e370bdb 0ca8bdff4bc0b21fd84de65439f4908c35dffe10d6356ed0851a66ee3d33066c c7912ef1b9f929375ec464e5c98bbfa5d5ab59d4f9d191336426a4fbccd95ef3 bf73184faef62d83670a072ad2d80d40f452cdae68c3f649ce52c9d71ce34365 e658685bd459cd7e1d24fde49e45db47d70b3afd7d67925dcb41ff49f2c46ad2 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 82 /R put dup 90 /Z put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueXX 5032014 def currentdict end currentfile eexec b5e2457af4a6caaa4536453c498a618ecdce7b5f6f2d5944265309f833bd3d50 3df8bd14c071f404d2177500681a801856cb8c9c9cf0112c9bd17ccba9ab8944 1759a320b2078df0fafdd3e357994bd81740d1a7522ddfafb75d58f5def046f7 084d10fa2f702e570b09a7073d70efe37b48c355a7089772238b7b4538c4912f cf6703a14fa416057ea2e1b94455913dd271b42846ee10d78270a8118e7e4bcb 8deb048bb280c240d59409d47046c890e5f79e5dd4b0be65a28a77aeb53dd33e ac506a6da87bb55e329f433627e06c3c94b078838b908538d93f888e504e25e0 3a8ccf7d2c02122ab3d485525564295390d1154e15fee35edff4eb016fadd32c 26d1b4c09de87943780954cebc4965aa890fe1f80b03eb07103e13d468245c7b c26455f229e8859585c6eeafd9474acdb27290fd62950dc52f893c99a2a614d4 2a7e3618266f3fb0c68a01a30373e048385ffe7dd5844e208a55d1a9263ab2db cb9b19038724f95af00349840157be45cb337871b55a304b87ade64b10ff0eba 21aa0eaeef74aa01f049303a8b25de0acc26bb3a33fcf267f479a2a73f7dece3 2aa23083687322b417fe86e8f5afc2136d045ad6b392364811a17630a94ff097 038eedeedb8da4ebf8fa5a96ce27a2ebf771d14f4a11fb1f9f48cdd207209934 885ef9118463552984b0fd3cf15d55de967fbb051549dd65f60dfe07f299690e 58252f032bca78935b7d2df852ed688bd13dd171b91cf7bd46d5433aef9f50f1 41a2517c4912b837793f84ddcf1116f70f7e9b1e6ec752dbc9183d4791d92602 397bafe29ac6f879ceb72c673c8a1aa0135d6f92d0db7791f9a2f1a5b55347e0 3e63723efd258754f5f4a9ad35873a98d54b6c54b17a39892dc1d38660597448 fdba9258493a0db0d560c01d49f2d889a99eaa8a5b5919cbd33f6b3d97bf27b3 e0b94b48d5d21d6c29aee4faca71bd775bb9defa37ff627f1e4926f10d59313e 77d81a511027ef9a8ed5b7fa9b5bd67758cd13bab6ac18411c974a65d0c90085 67bf5860d0930d7f43e557ee45e76ecb2806d21f4732bb282ac8b591f2b8775d 4b3ce2a37ebb371cb74a3cc513bdd70a0115de4760c21f034f5080d769f7ff83 554af524184702a02f6172e26f9b180f2de3cccd3ed216085ce7348f35fa4f29 b4c4140f140f1ad1e334c1fb6faffe0c0762b63994a9a169dffc2e9269b8051f 0faf67d7a974b6116623c357030becd3cc0c8e36acf741f64e172f71b782eb39 a4dc049ea1a1fdb3cfabd30941caf1b0bb5b4b8531456062df25cb115b547462 d18a1c756af1aeab302c58a72e63f9a4cf74772025b0347a905351e53f13b2df 0eaec2a56b4731d85573578108a244db23a2670010492b3d18340cd368bf114b b47ef637da9946f2c666011736b0d17dfbb358d63cda81d4390c92e60f5a23c5 ff9b8bcd9c21ec7ca83c3cd4cf1f24c6a267f830d3088b3a8cc4188ab4e148a5 0494ad203f83d05895ea054c36b348c8fc74e7bc0e1e0a9cfb71faf2edd626b3 95593d36eda154f393694b1a5b63a593e24b50bab02fc91db9ead80cabc9ec4f 91b68dc9087600dfd4b28f71ea264021bf83a392973764380057d5d4db096e1f 56af3403c9de4334082ed3afe942197d67c64b63de9fad01d7447f2186fe767b 8e9d6c769f384b65e7926d8e1104a5715724c1224a5bbfdbc0a34d2687e73207 4dbab7d9eba92667ad8609e7e6868ed5ddcc002087b772262c4d38fda852482c 4534ccd2827af975554b15036b525d95a12eb6b5215080f4a563effcb001aa4c ed12fbd933cbc662a08e69a4de5d77b575a0fd4a51e78ace687906355673d4aa e7b59cc92ec5f785a6e035f21b2a9ff3c697251e3fed777ede4dbc9eeb0ab1e0 99d8e56d2b589d84c75aa5ae310bfd7c081fd896a96a60697ec15011e1b6cab4 e0da744f109820a5e16da164938965c638f1383a68961623dea4ae78f9a66312 23f4f84a046da695eb214fe79294029b177948904c7fd46d858b53ca4cb08928 0b9498b837d2e3576c5d5138c5008995c2ad8e59379df6c1efa16cc48608d970 3091f3316e 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 48 /zero put dup 49 /one put dup 50 /two put dup 52 /four put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueXX 5000789 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b031f3fc6c13d7da7277a94ba 018e9998b3dd888011a5d7c4204989f30f908b95533bda845746b673ab71ea57 65a0d14f4350707e47c8276305b28513cbe1bb0dbd269a53719bda46e536685d df78ca0146b6b93e760256b74d939d4e35b5e77238f04c92298dfdd188feea30 e053eefbcbb52f2011772b3aae39f5805597bbc1e8bb75a446ce014030f4f2f0 f49f9e962ee4a1024a746fa92a3628db5270732b54e43fe5ecfa524f127e5fcc 788e77e66098336ad67fe4cccaf0253272d5df79864bf4b734cb9a5859d557d8 bc11b8e00221ebc12e97de4b1f466ead83a4c894709363bca9040410a52d592e 34ee40cc7e5efa920546b981aa659513a24b1b85c221a1875b62d0b89e57a368 321b8043a5b094e0379760a443d632892b14ad6d19dacc8c78093243ad67e6a3 08e56e6b68412ee690b10dac6e17708754a00d51fc957b500eb80175716eef4b 2ca1ef867614659bee3f2b7319e97b6fdf1efc847bf3cee3156f72f21751da8e 5fb6898919e6799820d3de0642d756e09d6fae4ff08dd3deda3173bff4bb11f7 9109c97ddc05897af709ea199a90fcee8ce4c7a3c15b18170c41c04de2d3fba8 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f01ac2798bc42fc890bc8aa468700c6d2c6003e34efd493f3638be41fff79f8f aafca93029eda8a6d35a17d8bee75b7c6862ffee9a45a86246971f34c605f3df 220f2ee9bf2bf76f823eaba6399b75837f27e46b4d70b1b705c3fbd07d69294d df9fed9e3847c31bad52d234e310bf00ee31b9c6d7124af751a20de67aa7da33 7b673c9e640c7a94a0c08a5840a71657561732d4b2ee06b94afe39aece8059dc 0c8a69ed6ed1de32b862f556093efab6b73482e0eff983ccb978bba22be44cac 0765f7158b98b5d0e4a5aa4bac2eb3f7bfce601087cb0f3971664402bbb89b76 01e8 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 6 /plusminus put dup 20 /lessequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 83 /S put dup 106 /bar put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueXX 5000818 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b66566ca5ba42b1911a 5d7f1bf343015eece988b7a93bce0c7aa61344d48aed9c92c8698d4b7c9951c8 7d103f2414b39e1437f9d2e50c4ee5f218f2e6716926a79ea978f13b1f855345 191dd7d31d8f82c2e3343c7a5894d95bdc492c28226834efcb5c12fea36ac5cc 430e0aa604961e34888adf6c1f3954cbc2498e225d953cf5685852162346f474 5a2a7087d5d7ad486de16d2ca8e15cee26e012671ba3bdc7d95cc8c98bb774f5 08625e968aee27ff7d1a06e63bcfb5aa4876c3f8f13b30ccccee73c3caf4e70d 98e6ed2f422dbb4950bf789680e064150995941a9f4dd68a575949847a7d012b b910bf03a4261be7bd48981fe5d74f36597276d570bdc1ca4d7a29ec280a13e7 cf740be69a5d79aff1ac325360e35ace9b7338a0d639d1a12ea484c43a1ae98c 3b6388532dbf33a3d9d0269db68b7a40451f68e14ea905164d8cfdf7e01d1a3d 7d97a280fd7b1664708aa91fdec822b75cbca7e846cc17cf2c03e485367838a8 44339ad3194ef665a11f1e1a246d1b7d6e005d5fbe263fc4b151e730ab3dc8f3 1b19da178175dbbfacb789821ab7c3fa6d28c9cd518bd5874253ec2a04729533 a3d4cacd2e697e38807d1245e753873a16e5254326c439e988498e60fe374897 d383c6ffa145fd7617fe093cf56c9552dd4dc003ced67336a1e10389b649144d 962a3cf7c16f4a43338bcf35cfc5b54f87cc0caa974179179c34ed5eee919f83 df67b2c7d175a6d085e8421d6ccacbb0f4cd6c8d559dab50997048f5b1a471da 1ff0f75d7d139afd61742807ce6a1fc6729936a057cd8173e74801be42825b65 9305fe04c79a211e887b0fb6ed5390bde19d453e565632ce56dda290a03ab241 aadbe56c880165eef59873eef2c9ffd220d65eff2e3322a8068e2643e1e1c008 0da0805be372a695a845a90ba4ccce269b9bb1c534375a3d3c725bd2d6bb8e80 769a3d99ff88727b238e1e320c4fb0352431b9f137f2c224efefd8a0dc089942 846519708dee04107fa15be0c81576a63040a77fd93f50fb7f6fe1bc1bb4b2eb aebda86e6b6b4fa1b5a8f1b01d5ad26464d5a488184518447a5a424369b614e2 adbd6754929a72ffc0c395ef9eaadfee8daf8d04af4f1c1c7c926e9f8aabdcbd f9df2e2c6fe7c29422a1a7e3614d850c5300f6752eabecff1e30409ea8534151 d1ddb4c1c0b0e755fe2348ef33d51a4af834461006ad6745775feae4a414c42e a506758f4aea0207e3d38120f8c38b3d1ab41fc6b5930a4eb6e399d77257daa4 216509d9449c5e179ae856b03830af9c3a2c6a074e54252298958880c78e7529 86cb616ea6ae1941d9858de754544ee913f03e626a08a01a965f6c5c99bfac18 cd578af9bb0063a04776f74fcc19442c06a5d0c6d7e1ce97039759285ad88337 7f8d27c73c4ed1c281ab5f27964bd8e6cf2ff03af21f88db25e3fd03dcaa17d2 afdfaca8c2e83b36c24adb73e01205655892d3203a6646c183f19510c0aec22b a391a4a0d8e39a42b7f676cb5d16f004147546ba9d76999483b35babc804ee57 934d552a3c49c0391d6f6d090f0e26953200c5b5fbb546fbe6682b6b127033ac de9d7945545cab266ee6b17ee80b8aa740b2cc43a84a2bed43b96839c76c9030 2897f391d7b724cae3ef1d4c8217333dd8675d0f186ce1a8b739c763330c1278 f1b0640072ba179b2fcb55256966ad273c0bf9c5a93155640fbb06a1ad16c26e 7f991772a27542fc848ea5c3db589d22f199add838dde80299798b14e13c6804 637fd189b44451abae4fd533ba1d44332e2f7eac07615e4b02796f310eeb8668 63b72e94639f09776dadbd85f9f6ef643dac4e1ac5a5cca1525cb4520335ed40 911e42dc8c669ce2874cd6344a8e0e18d41b4c0b331816401b48afd900211552 08d10967e704712d599f613bf882760717cbdc9768d6a78966ca7e7b0695c2fe 7b10f3db05b6c59d24e334610a6c79220a6a130e98114e2af932f6896fa823bb 32722ffd3e73867587002678e02cbcc292dbee229c19681311d18fd41a089c9b 62f632ed27be025060430f7ab4d5b9bd98dd4dc0e16df781749164bf86e386a2 b701fcebaf99113f357893ad8dfb7e7c3a50e21e4e95373fc2bb36180cb63d4f 70bda126318970e370c87d1063bbbe4c48c4dac15335df41924728bbc306cb91 6267985288976473b656f694a2e07804b35fc6210c3b00d238cdb4de4aa84256 afaf64f27f6a5ed0bde7cfd49dbe572e3a0518bd64c723538bc347e55859a67e 31176e8db552b87cb84cd7cc3569c3b1901a2cca6257cdcb74b435079d16c8e8 b6f4e3a330c3e5227ab77a3141f65011c58fd01b5f6d0de8bfcff7adea1bf171 119a95bf221ce2c8f8ede8995cfd011e16bdbda7941965cf0d8339a934297c14 f52f258bc691f9126b94762ec18ce663701ecd812185654f77a0d22934444655 084a0bbe3de51cfe26cf8f013273f80796ba41af573788b45b1c31669146a936 24a1b2216f85629b283997ee6b7ef42dde6c8d03c930acf19f596e23440633b6 051f47 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 10 /angbracketleftbig put dup 11 /angbracketrightbig put dup 12 /vextendsingle put dup 13 /vextenddouble put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 28 /angbracketleftbigg put dup 29 /angbracketrightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 34 /bracketleftBigg put dup 35 /bracketrightBigg put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 80 /summationtext put dup 82 /integraltext put dup 83 /uniontext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueXX 5000774 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d5649664203ab24acb938d58d246707ffe7 d62f04bec4b70c21ef75beb2b812622b3c74e969d72d3cd11bd7106294a99caf 0b1629bc7d4de6b96ca82930831d64575f23f4ad06a0e45e315b1d392411be8d 6d73c998789ff258a07a3c8c2057325784514c845500bfd1a971310cfc11d41c 1a167dbd5ff012c60add4e87325f6e5299032a839de65fb1473a166aae1876a4 414a434f22c1d241591fb36f857df6fa930608750ffc0c54f44994662b1f00f1 400bf752ea8d83ffc4cb77a290bc2d99981ae59a191748ba5c7ba1a9d2583fd2 1398452b6ff5d83a059f7eadcd2ef744e9dd22bdf9c79d049bf06835e878c32b 7765c69bdd8ef4deb4ea7cfff4cf9354a4ddffa689de961d16772491c7afbd7f ffde42400764c68e954ee5c455a5687959829bc3b319b2147deaab3628662c80 30c5e02fea09609abe4eaa12e217bc3af673f1bc36a7039eb13fcacb4218fe0f c5a3f9452d4edf46cc91db67b624d4f2d37502fb9f11af4da18ca40b61097f95 d44329375467ed13c5cb585ec53f62b83ef9502cc755af44bf32b87b8ae9f3f2 f8dbf72dab90acafbacd280db6aaffaefdff6d5eff26669bac56280a950560e3 d01714ba38659b0e4e28bf8cdc1f15ff7e3ea9d95a6afbbe3d8e6905aa6d531d 6b6dcab89e2036cc492c65f76ddcc3cfcff33fde3d9ffd10f7a90530fd5c7683 4329baa902e701ca42717476298c9ff511b1941244bb1dfd0e5cab92a43970cd 1bdb292eadd055c97972844fdcdb0aa06401e7b2632b9d134f96bce0064860f3 9859da7a9a82be0a840672a1331212752ae1b521b693b384a72dba8d37a86eaf a0341e7fc5f6ef7a8c566aacb91e8bf5005c18837894c4eacff41dba54e071ca 83fe3c35a5e4206f60b2b4f567b948f7ac54e3ed16b58e137958bb0c141e411b c1c47810add8abfba9f5ef95f96d064cbea94293bf05dc6979c6001553e168fc 75fc6cffc105d3eecca34913cebaad61f3a0949cc0c6768d5795ac7405b64118 0c28dd9d2fbba6debe3de789753f299b268de623c58dd73f912c935c525fabd5 a8a45699842cb930cc3a9fc92d950210de61d71d4457d71b3788acf2afe5ecce 8ac9e714628742820e52127239253ac17b853d8d03e0530d0f53ca9d2e0f7830 daa082f471eb27eb9f2b2f4b06aeccc1b80c0895128e0724fdfe44d7a47e72a1 eca3df3411b50056b6811b41d513597d04f5c2ef0794c38971ae5a662330983d d76e5bd67dbb1836cb7dd48bdb38b59309a82edfa5e1aa1a4ab74694c6d1975b 7c73d3c5be222139b25c7b2ad2f252004a7f1f092028da2845ef8fec8e10ef4f a24b6989b7fd9306c7490a51cc3e70b318b68e92018275763f89b51652e0e2c6 dad5bd2b4b5ed3ee82410d9d2b0006431088c9dcbfbc818d96974b056f4065ac 01b8d94db75373774585092f926a0d075a89a3df284913fda4251e2a0bc05092 9db8a911cabbf8b8c3226014bd25de3fa341fb48b0146a9c529879ca6f109d08 c90a961181c9dd46e92860c9fc21742cf122a5c681aea23a96240e993f3e3b0a 75b0cab9d4cf08d085d144f937d54dad40c32662c160d89294f6ea196a0b5138 9f0caa78cfce549cb30c2abdd0ba0cef41e290147e49e0cd4ab364b3cf3d8e5c 4987589a52f3156af2ae9c5650c3211aede0f3c2fd78d0936cfa482d9e1e1a98 6c5e9863cec478cfa950bff940499c36018455bc522ae149415e947a858e43a7 7425b7f46bfd909829aacb845fec379d7611c906d6e26c9b3fcb7f089c9eb1c5 d273a642334e21d7a79326457012d4feead90eefad736f63d93b7b492fc07030 1440709ae7898312133c5cff038a5148f94a0218bfe60007c525a25fd91424d0 7768355aeb29f6e2a3f63b34faf8cb61b849cd7659e60252ea95e6f5d483f7f1 36fb36d43555d2db8eb98863aaf0b96d8307e25844b19cb10e2834aafadc1126 71fe91a14cb6d15554cec48f1155ddb16ad57b0dd88e948201e18a9934582505 975db4270dae814f3c4e422d26e1e234321c29ea94045801104c3c18274020c0 db0d24def9240f23339120f64e0c7cf2bf754e304a72fb1b2497abebaf919f9e de100d4e11ac98262aa3eaaaa85680ac7a195f27ffaca15206427c5b0430475d 5d2c898bb7f295dc9b7ab76dcb318130b65c86de8d9eb319892f9595f88ea0c5 7ec99b282c5660d0989e2213339cbb44a728e2c4af217b9ac268c774057739e9 ba98e0d8fc6b2da6ca37eeb1ab0910a935f3b69a5588df1eaf6facf156af7d50 bd892350c1afa618c6706a36729b30cd98534f56e0b49f5cda65103cc961b160 3883cda239b2942dd8662c117fa17e71318707d752bb02e4f4de5615e227763e 96f2e3e6117b892c3f29424a3859f075fa7b6cb7a86ee704635c01205b0259d7 98901b1139b24713f048188bafedbc8b1013d719280f232e1af3ef1bf8701c42 90dba910696841f94d825f4575b12f9f7d49e1f787608884a510e862e1bd2b70 218ded2cdef1e1bc0dafc5ffcc45afbf574dfc8e8bbd764589eec6bc5005dbd2 8ca7cc608ab8fd4d002ad1cfb5b88aa993f5c7e90fc941a7d639e1c96df12066 bb975c2747e9ffb6c1d81d4ff59bda83edf7095944ef560843b84a62b6f451da e9825ce9b6d5de9c5a23c4e3aeaa8eec3c8a6d20f0328bf8e6a784274b126562 fd20fdb7e847dd3f1bb64e049c445f8502a4bd9e73cca03263a175a2d4af315b 60faacabcdd2f3609e87fb9c632583e8c6ecffd34fb23f85cccd522c1804e069 3c0bfc32a8f8ec802a1760e0784c5520e6e714e8590d41a6d1dd12bb3b2210be d3e79c368867c06faa2b0865c3fdd85505dc3de7a4619dbaecf697e002793000 ed24a18de6a0251374350be1f662e18d289bb615b87715dc25d11b6eb401638e c872d53bfe47fab537e26aefb092006f85b9b1afd38fdd6c310a40f971259578 797cd772fe104f13a59eb90722a95047fe4938b0d982d391c01359d8b661f546 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5876e87caf62458a205962b507c46bcabcd76e34ae91c2c28e688fa9633c5c81 c92ee0349a3f3419655e6718d6d2085c98e1d47d0d5a8aa9a71c9f385ef3454a df166ff6fbdcb7716c4490444c45d759d44c903b1bab934f276950ead3632a75 4c7fcd8d47bd77f5dc24f0ca3eb6ca6fd8cbdd6259d14699429ca832afa6004a f2fbe2e20de52792137287ae0f8f74e3a45f7b55ec7cfec9a2399657cec3bbc7 c777ab875a453bade88661d4ac7943ef2c1b63dd5e85996bb4041a901e73523d 0ef5a68e929ce173e203805a2e2b904456cf3c420893065fe5386ab22314ab5e 23608b0413afe83da0e144377003fcaeeb10d713248d12c2facf771d58624ce4 403a1f73fe474425c166f5a1f840deeb9cccdb9a1c9d164e4dca51e5eb07a6d2 5c6d459cec7856874b53e91bf58fd6762a67097e68acb6293a3f4cfa807e6bc8 c152b3b59397b0a55f381ae003d82c022ea8f16db3061e442b39950a41c9308f 410b8407224de8c27d9573f6fd948d00605d07cf638fbb06926418656ead5f52 eeee6b030aacdd77905c3d31c05461676f01ab8917099460a5ea93848ecdb892 a82bf78bc46bf478cf568188ee958f96b460b09e1a92870782b9d9174e3c6070 7ce8c0e3bb491f0721bae386866b64ce2f4f72c0be66e0690c43487bcbe653d8 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSL12 %!PS-AdobeFont-1.1: CMSL12 1.0 %%CreationDate: 1991 Aug 20 16:40:41 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSL12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -9.46 def /isFixedPitch false def end readonly def /FontName /CMSL12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 71 /G put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 85 /U put dup 86 /V put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 127 /dieresis put readonly def /FontBBox{-56 -251 1102 750}readonly def /UniqueXX 5000799 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd803271dc8bb1656323089e9e2bb50f2b95088 806eb7b8c434269c8a30d52dd3a6e85ebcf727aecf5607b54ac3942d00365274 9fef6277965f241846e12c5fc12f452046f9c7df8d1d31e228d6011de7485f19 4de778c725bd819dd7c91f2cfd0a9994b0ccdf95f200c0247f2c39f74a6ec77a c20f4789830ad04647b0784bd6740b2c56e2e41da705d1f709b16b92abcf7e4b aa92321eb28f528123e04dd7a74742e176a447a518a4aaeb0a4d3d5c23003c2f 98ac15a3b49f60efe1fd02bfa0c4344ea0300cf33a57ae113a16baf4bc852b2c 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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 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 20 /kappa put dup 21 /lambda put dup 25 /pi put dup 33 /omega put dup 59 /comma put dup 61 /slash put dup 62 /greater put dup 67 /C put dup 69 /E put dup 72 /H put dup 73 /I put dup 76 /L put dup 78 /N put dup 82 /R put dup 83 /S put dup 84 /T put dup 89 /Y put dup 90 /Z put dup 99 /c 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 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 118 /v put dup 120 /x put dup 122 /z put dup 123 /dotlessi put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueXX 5087383 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a196cf14c95a4e02f9cd2da8c db2546c6df52e524745992e18d9ff87aa25e4e1800bbe4ebb357c6ef55ed6d03 6d3a00c1ee8073266c21d2f0ac85d656abf61d7e5a4fa87da8ec3b5329e434d0 d2adab706b42a2e5331be5295399d803ccac03f631f01f39a022fcdf63486b68 7d15ef284a77def7fde4898543e7b5f7ec267756103e477f547cfb8d2311c4b0 09deff56085f5d419697af1846c8b88c1bbbae149f0f19ca3c8dafe19cec48fe 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199f558428e7df6067e47af92d01d4cff8c19068a6ff969f8cc152786391337a 66f5b847fd9b3a7468476ea1f8ac43797590554cbca38af6697314249958e856 6571b9c79ecc 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 10 /Omega put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put dup 94 /circumflex put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueXX 5000791 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 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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 10 /circlemultiply put dup 20 /lessequal put dup 21 /greaterequal put dup 25 /approxequal put dup 26 /propersubset put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 57 /existential put dup 60 /Rfractur put dup 61 /Ifractur put dup 69 /E put dup 78 /N put dup 79 /O put dup 83 /S put dup 84 /T put dup 91 /union put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put dup 112 /radical put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueXX 5000820 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8 3750d597b842d8d96c423ba1273ddd32f3a54a912a443fcd44f7c3a6fe3956b0 aa1e784aaec6fce08dae0c76da9d0a3eba57b98a6233d9e9f0c3f00fcc6b2c6a 9ba23af389e6dfff4efec3de05d6276c6be417703ce508377f25960ef4ed83b4 9b01b873f3a639ce00f356229b6477a081933fef3bb80e2b9dffa7f75567b1fa 4d739b772f8d674e567534c6c5bbf1cf615372be20b18472f7aa58be8c216dbd df81cc0a86b6d8318ca68fe22c8af13b54d7576fe4ca5a7af9005ea5cc4edb79 c0ab668e4fec4b7f5a9eb5f0e4c088cd818ecc4feb4b40ec8bd2981bf2336074 b64c4300276df7b05bc5105fdf3c2d2569e28b3a1e8323541e56e0a3e974ec21 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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 16 /zeta put dup 17 /eta put dup 18 /theta put dup 20 /kappa put dup 21 /lambda put dup 23 /nu put dup 25 /pi put dup 26 /rho put dup 31 /chi 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 64 /partialdiff put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 71 /G put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 88 /X put dup 89 /Y put dup 90 /Z 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 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put dup 122 /z put dup 123 /dotlessi put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueXX 5087386 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7bcb09b7064e91fa0ded228e42 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 5 /Pi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 18 /grave put dup 19 /acute put dup 35 /numbersign 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 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e1d1bb22e56789237c6f77ef32ae0a1a21558f507f309b0a464d40673592fa1b ab9968a5363764e3865febb4c6f553846a3def3600d74d70a0f35e47e8efac62 ccb756b66d5aeb6db4c242b25c2ab63c75e6c8a87b4cd50f8142dfeead627eb7 6fc6689479ccf481f1124a0d10fff33849c07f440c51fe4a7dddf56f07dcb1b3 c17167b5a774cdb67cd5a8cd5417e13d1ff70f0a6e02b0827eb04f836220be6f 6ac9741139b508074520d09e6c1a57da6430e98c656213eb000dfd7541a93593 333fede47977afe4d20c2f9aba315cb11ea849338273bb15b0dfffee82c1fddd 4753f336eebbe1baaac3fb760c5d6d09cc7fd7bbf2c70796f3f59c6d26fd6d77 069901df50bc86e44a2926c8d7ee7569abb55c99d0720112fb6a7489295341ae 3b2345467869e716036a673473b4e8d5bd9262e6a2ddbf82bd68e7ac1aa121ec 29c5e81352ecac05dc64d14214c277f49c24997fa5cf071da5e284a89b1e98de 261f0cc2a5335e0cf5dd0f5698e54cdb238676431d945cf4cd4a1da23a3fda72 32a6c02577535ac02cbcd8cff61bd3c068590ea2d308fa7bb8ec442a9bcf232f 6e67d4de94f4fafe98016758a7ac95ab92ee8f8c08f0d45fb81a6c63442162dd 565e14329a1d13ec06c1cad7236af517a1327db1ca1dfc62d856d2de5c47e619 392c5bfcb6e99e04971d0c70a25bb5526d2d28da0d3b6a0740cdcca4dd5c97ba 1cdf1ae7aa03deedca0cf7f6e56a5aa3a72a2ba00ae49e64a73d3d303c5cd239 f20f089055af6a20c54e2af0408208960a3db4cf85cce9b50b23bf2427e398f2 6e237a0dfe2856cd647fff90296af37f7e5879a3e82f803a027117cbfbc4c0e1 6298b9494a4bec09f76d8a7e8982af6d3f2434db164808ea781c5df0ffb35574 9032a29bee0a5d0585d0253b63f704e389a2c287485070657ce030dfb8b36ab2 9fef8a0b7bd156364c830d2bb1ef0cae92ac7c450a75f588a9bc9b5589f82e22 8ddc98c069dc03b2947d1831453a5c6517e0faaa8c5e9a6cd2078eecc00b7749 cc7701e99f75621822ce6926be9ba1cac70d6baf5f6f1a7d39282d1ae73fef78 da0a46593f60662c06948e64b0bcab08329a008ed76b0f734142cc778fc42dc8 a5eb12a2a28e3c6173e2749b2c93764e325237d96a5db7aaaf0df4e9be718612 1b270d3fe06448da674a4d769af68383174b0eb4e20e38bcbce477caf9fc6163 84d42fe82c89820a41d6ffadf0a8721da47f4a25f2b8e4e2e9cca03e25b0ebb9 5e190d685728e58bd285f6d017e4d3d58015a5163254bae14d0aa3e7a21db5c6 569f5af2636eaf41ffa26a13ab5c75d8ae7eab6594ecdae269c14ed0f5c5691d a51d76f44f4b1f8580a16446f7d6ab0010393017c8f66cc7d13400a25ffb32bd ebeb50d126f925725805c459e1a49c6bb61f0531719c85eda459899bc78fe51f b56846300f9057547940ffdf5344bdf4471a080896e17c516ac36f8de6330f05 85a7d0c987fc28331dfddb164d22a522c92ad16679266ea896d7dd26b9ba9cdd 4dac77248ca9ba16a031bbba14011022136f18cf1ba62af513150912d176772a ff1d77e8dffb9242dcdbd61cc2bfaccb9feba7e86defbe865591193336720632 b0ea9c7d135fea2f03fc41cbb8c59e415bcaa09d00a0d0796c200009b60cd055 4b32c57f920c8580d1e952c598fd8578721027ad1bf4064c963f53d800178387 faccca12be73dbd87ada0fe129c50e0143bcbecb569e7322070a645bd215a604 52f08a20d9c33ad8697c87621e10d5b7ce9ce0d594066686a8f8e6d5faf211c8 13c202081141d958916dddfeac819e9936ff5b181f2a2282ec26513f425c46ce 79ac877d1de2c7018a27977ea28cc99b969bda20e54b17a3b40b7d9946618384 330a314f7ffd8776c5643ae9104097b7918bb075b6c70d5ef563eb3644a4ce28 b218bf0c18f3eaca65f8f9a2fb67dbb95078a06eea912c9c15383a9ac4a87ef1 b94db96de9fd9e19707da3c8f2a315eb99c4251a02feaf7ac8d0d9f90f294068 53e8926c7889c239a20a908f69180f7e16cf59572fca18bdb884bb79765af63d a47553674f258c3506dd4df0d7176caa528ebb461096210238580663083ca1cf 7eae17ba062073cfcf72659a81fb996aca85bfde662bde72edbc544a29952185 f751127e0b854d289404bf2f02643d94dd4a3250d4e4798f0002c920ef0a42fa cb3a866ac315fa3a534bb88307f457ed791c9301effa2fd6d6a7758b59effd1a 053b1958139b6bcc3c6614759ea899766ffec7105a868878ba6b05a2918bed43 50c0c2c15b44adbd2126fb3523010e5b06a5f4b2a759682e434035b4e0d3a356 9e2485319be4ddea0c69b78b13a6c25c9c5e61a6c81236f28f377c462a30e596 b06f787dd63fc7418981e9a19c618c31e35c392edd065bae3edc60899b4ddb9a 921406999e82a1f6a1c8a0fac5804b684ff9c9b44aa1ec31bd8fd5036e68273e 7367870ddb73cb65c7bb1180677f219d1e0fe39acea29d4a00349e18c40f633e bff917df78104ad62fe407db2d1e3414e273e80a5d3cf2985495e05b4aeac84b b651ae2bf9fb13f1e5139d57b7aab356082c2bc143cbfab15af01e7b4b16e2fa 0cf9087081cb2f297a56fb8479521a10c21f638e9dcc0a6ceeff447f5f95e0cb de65e18ee3bb31249651052e756adce43a822ce55bfd621517b20b09519d1c48 3ba62c18cc3c55 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 65 /A put dup 97 /a put dup 98 /b put dup 99 /c put dup 114 /r put dup 115 /s put dup 116 /t put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueXX 5000768 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306462c563d51ece66fcc9c831843ecae 1fefc1a232e2724f7baee428ae03aadb95c3035345c15e9922fe49e1f2cfc980 237316572dbc57064edac9b0db8913a5e2d45e97e19a91435ccf8adfc835b585 48e74b291446d689c7f2f8c4325e8356e974ae30c3e2977477baeaa33d141fb7 80f59351e84bfc88c87b3a4a1d25e168b9d134554f6a581378c7d2d6eca8ac09 045cc3a5e0ed86f147133f094029e3483bebc81cfde69942cf645ea20d0a2b64 5140ccc2f9cb4a3d969f4b0a79cd3329b440dfcff5285494406f3dd08d804c86 df191e57232f6f1f83df6430f214e03fa5412b1e13d83c60c52a698bc9a38159 19a64162af7b83a9468fae9507c6cb55ea80df8c696dfbc3296c663902d77e89 a181502627281181ac8624f65be9de5fb32f056cc7dd7cbc7366a91717b5892a 34ef6b441871c6993da3a81afc9bf70ff3842a21a29c0e49da1ccba37dd6b920 2702db510b8b1109acf4c27ee8d676bdbd16abe189f34a9b61532eb777457698 cee0561838eaab2d3939c5093afbb215bac433d1918c739b62566906bb2187f3 378521fb52e70c18886e1aa15f53a1f8a169528cf2ddd54c203613ff88e619f2 6c9c50c2c3b8b8c868a88afab1dc8472e7b6f3f992a161723a3c0904865a7c42 3281601eee91f35222c4a71f343739f102eb43145a47e4fac3bd666e4c114a90 5950618ecb1bdc29f90f6ebed1d8325fe6b52e14a8870e9b4bb088e585c71b91 493c12845a101937e3473ac5e9ae24f7c358f06c8b42dee8ef2a1a7a9f536037 57517f2fb6da0ea305d9d20cc74d922ec4b95b8b299459b71825f6f67abee3a6 f5267c6283c7c62aa8d36663071f18e1a3f0b35dcd88d739caf8dbc344630a28 7f583f037768d861f37d7929f248146b5f31ed9f61ef0a9659b50f47ec328516 9fd999b19687e50303557945476fe9845e5c4780045d84d7df253fe61bf69607 fd277a265296b02b4e4b35884f985d29521ee7327887d983bb92dd84ff646661 2b348b5e558d3bc65047afcb202750ea6e9c18a769a4e945e0954787ee65948e 7e8e443a9f2b9f5408e0febf523423b2200166ade223570bd17a6f7dd0386f3d 2246ebb40648548e0a3ff7bab6637a16137c24b042010a3cce2ba9afb9317fcf 4a5b893c4915859ff2b53a77ff0334966f7a234ea8a4349c3e048acd6876f5f3 17b9c238f37ed4231b5e8b661ba52c32821de47f7a0f6fa9c7e0f4bf09bf98a1 8008087aa44eaf37894e83891f552626f1f46563b5188ad949521e1a398eeba3 b9b8f4ac3c24785d50bdf01e5c9b7e4ea1d6b7b6c8019d2d5f353cacf216e30b 71f09d9b95805210964ccbf8f2e30aa04ac3a3a57bde635c9d2be1f5c8f64ad6 19764c72aca8152a51003bc396bce21a56c29306bccc36a37d816de2568e7794 a8f80d857132a3dbaf3ef75409cace85e1b85a8233ef2757a507158bbba8f5f6 d9c08ab4ee39f33a1c1a41ff2b8e81320050b6c4c167596b93918824a65fe1b0 6e7dcf5d7e128a77e5509f3a3a2af6f71de243cccfa5788088d277175e59bd61 a3b5cdc10cfd232cc14682d3b37d3aca68df524af6f38511e5627286034a7c08 1c7596bfc3adecc7d1868cb9d645f8cb61041bccad92249142542934977000c1 bfbbaf570bf1a3fad2368d1e6480ed485642acd1d44258a885562397035dc979 b08877eadba7c99963e7d866be89270b10629964b5c88439c08a7721e2fd6a7d 2a03d9cf2d7fa713411824db51a9141a7c3ca37feeb758ab37376d75b9ad8534 6fe75968a1e7f6b9b549b7bbef5da78116a7d029e6e95db78593abf789ab985e 301f5945c1628f591b4867079526cd9c779b05cd30de4d9dbb820ce964cb3acf cf693b39b15335996a208373a9a1cd362e93d740535d4fe06aaef81a77034010 883c098f8fe19617d6a3cc3ccaa63742d3b251b8baa94aba7ebb7f3cad53c3c5 2c626e3918191af1e343757e0fc5f9fa620ce7e655482aaab7303710167e3041 148e3e6aae44ebe904d96a7dea9281fdbdf93b03ccd08a315df83531214f1726 dd6eff4d5f494262983201ee2afd6cc77eac76155fc59d9e42a5c45735a37af6 c3c9c4aa7c535717838a1ce8ca3424188a3ddc2bb7f58483ef08fe9b6ea090c8 bd5a639bb56a54bed34a443f0f1624f13fffd6b4f24024573a6cdc110f2a85b3 f669c706b0bcf205ee9ff8342b38525e9cbf1f1b772a71c7718b30bd59f46490 dd92c121c9fd0b99edf33c57611ce19ca35fe17079c90ce99afdfd9227184562 f4e2dc14aea6d0989a8f01f17c0983ae121614d8f578d34ead9c05a88007bc4f 90e135d97abf57c1e32fd94ed63ceeddab6c1d42f3238e6cf85a063d1239a1d9 c5de6f9e7954b873ccef98d8b44f1761e66da28b9759a24f181c9f7ca8ca190d 4d5a52b1286ad43b5f8caebe64283da13e6a9e66536a821ebc1b50e98073d79e 335c244c2b8efcd7c0b223983db1608d6f1837ae6f73d970e567960e7e03d1a6 5f7dcaeecd4f9afd0266592ed051f800ee3d5a8954460f3f57a9fb9716f4dca2 616d5e77e19918f2b1c290fa9d049267df21a4b33a06e29c758e0ecac9746be0 b7988da1634f2e7162c2c483f1c8d78eaefc94e907d3bcc79cc389979ab0dc15 bf1b2371fe907e9c6f46e26260566e38ed55175ce116e4da7024db03b6657b44 1543e575e850440d7613b96baa8320e2cde5ab834d1f333a9a749c3ab079546c 3429c90c61bfda4b758861c2e351866a18c4d14d3bee4d863edce298231a71dc 9e4c701ac2d01fead65c2244a81865874b1c22125e9f11720450c10366eb6eec 694ba3acd45ea6f9cda354cc938222a523494fa570ba76d359afd28f6ca82c35 dc9890f6e09346d93734ef8c82d7b62344fb8103a1fe6f84fb56f55e1969c4bb f61c916f5fc7aaf919d620ed2c25d2bac648ede51f981afe1d4ecf8263830258 75069803a3032b0fba0522 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: 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 11 /ff put dup 12 /fi put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 57 /nine put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 71 /G put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 83 /S 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 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR17 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 68 /D put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 121 /y put dup 122 /z put readonly def /FontBBox{-33 -250 945 749}readonly def /UniqueXX 5000795 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 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Fk(\000)p Fn({H)d(t)2063 1035 y Fp(j)p Fr(0)p Fp(i)54 b(\025)i Fq(C)2436 1050 y Fn(\013)2502 1035 y Fq(T)2573 994 y Fn(q)r Fk(\000)2672 967 y Fj(1)p 2672 979 31 3 v 2672 1020 a(2)2712 994 y Fk(\000)p Fn(\013)2849 1035 y Fr(,)1015 b(\(1\))-24 1308 y(where)39 b Fp(j)p Fr(0)p Fp(i)d Fr(is)h(the)h(state)f(lo)s(calized)h (at)f(the)g(origin.)57 b(This)38 b(rigorously)g(con\014rms)g(the)f (\014ndings)i(of)d([D)m(WP)q(].)-24 1428 y(Moreo)m(v)m(er,)e(it)f(is)g (sho)m(wn)g(that)g(for)e Fm(ev)m(ery)k Fr(con\014guration)d(the)h (l.h.s.)44 b(is)33 b(greater)g(or)f(equal)h(than)f Fq(C)24 b(T)3739 1392 y Fn(q)r Fk(\000)p Fo(1)3899 1428 y Fr(for)-24 1548 y(some)33 b Fq(C)i(>)28 b Fr(0.)43 b(Note)32 b(that)g(\(1\))g (implies)i(that)e(the)g(conductivit)m(y)j(is)e(in\014nite)g(either)g (at)f(\014nite)h(temp)s(erature)-24 1669 y(or)g(if)f(the)h(critical)g (energy)h(is)f(at)f(the)h(F)-8 b(ermi)33 b(lev)m(el)h([SB].)123 1836 y(This)k(should)h(b)s(e)e(confron)m(ted)h(with)g(the)g(fact)g 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y(and)e(m)m(uc)m(h)g(researc)m(h)h(since)g(then) f(w)m(as)g(dev)m(oted)h(to)d(the)i(program)f(of)g(pro)m(ving)h (dynamical)g(lo)s(calization)f(\(i.e.)-24 3569 y(b)s(oundedness)35 b(in)e(time)g(of)g(the)g(left-hand)f(side)i(of)e(\(1\)\))g(in)h(mo)s (dels)h(of)e(ph)m(ysical)i(in)m(terest)g(with)g(previously)-24 3689 y(established)j(sp)s(ectral)e(lo)s(calization,)g(b)m(y)g(means)g (of)f(upgrading)h(the)f(pro)s(of)g(of)g(pure)g(p)s(oin)m(t)h(sp)s (ectrum)h(\(see)-24 3810 y([GK])i(and)h(references)i(therein,)g(also)e ([BJ]\).)61 b(The)40 b(success)h(of)d(this)h(program)f(ma)m(y)i(ha)m(v) m(e)g(raised)f(doubts)-24 3930 y(as)f(to)g(the)g(v)-5 b(alidit)m(y)39 b(of)e(the)h(distinction)h(b)s(et)m(w)m(een)h(sp)s (ectral)f(and)e(dynamical)i(lo)s(calization)f(in)g(ph)m(ysically)-24 4051 y(relev)-5 b(an)m(t)40 b(con)m(texts.)65 b(This)40 b(pap)s(er)f(demonstrates)h(that)f(the)h(distinction)g(should)g(indeed) g(b)s(e)f(made)h(as)f(it)-24 4171 y(sho)m(ws)c(that)d(exp)s(onen)m 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b(p)s(olymer)g(transfer)g(matrices)g Fq(T)2979 4550 y Fn(E)2965 4611 y Fk(\006)3082 4587 y Fr(at)f(energy)h Fq(E)52 b Fp(2)46 b Fg(R)e Fr(are)-24 4707 y(in)m(tro)s(duced)34 b(b)m(y)124 5062 y Fq(T)195 5020 y Fn(E)181 5086 y Fk(\006)309 5062 y Fr(=)56 b Fq(T)501 5086 y Fo(^)-38 b Fn(v)532 5095 y Ff(\006)584 5086 y Fo(\()p Fn(L)659 5095 y Ff(\006)712 5086 y Fk(\000)p Fo(1\))p Fk(\000)p Fn(E)t(;)961 5074 y Fo(^)960 5086 y Fn(t)985 5095 y Ff(\006)1037 5086 y Fo(\()p Fn(L)1112 5095 y Ff(\006)1164 5086 y Fk(\000)p Fo(1\))1303 5062 y Fq(:)17 b(:)g(:)f(T)1494 5086 y Fo(^)-38 b Fn(v)1525 5095 y Ff(\006)1578 5086 y Fo(\(0\))p Fk(\000)p Fn(E)t(;)1799 5074 y Fo(^)1798 5086 y Fn(t)1823 5095 y Ff(\006)1875 5086 y Fo(\(0\))2002 5062 y Fr(,)228 b(where)172 b Fq(T)2734 5077 y Fn(v)r(;t)2876 5062 y Fr(=)3017 4994 y(1)p 3017 5039 49 4 v 3024 5130 a Fq(t)3109 4921 y Fl(\022)3223 5001 y Fq(v)87 b Fp(\000)p Fq(t)3469 4964 y Fo(2)3224 5121 y Fr(1)136 b(0)3551 4921 y Fl(\023)3673 5062 y Fr(.)191 b(\(3\))-24 5315 y(The)34 b(transfer)f(matrices)h(o)m(v)m(er)f(sev)m (eral)i(p)s(olymers)e(are)g(then)1972 5564 y(3)p eop %%Page: 4 4 4 3 bop 942 359 a Fq(T)1013 318 y Fn(E)999 384 y(!)1072 359 y Fr(\()p Fq(k)s(;)17 b(m)p Fr(\))28 b(=)g Fq(T)1534 318 y Fn(E)1520 384 y(!)1564 396 y Fe(k)q Ff(\000)p Fj(1)1706 359 y Fp(\001)22 b Fq(T)1827 318 y Fn(E)1813 384 y(!)1857 396 y Fe(k)q Ff(\000)p Fj(2)2000 359 y Fp(\001)g Fq(:)17 b(:)g(:)k Fp(\001)h Fq(T)2307 318 y Fn(E)2293 384 y(!)2337 392 y Fe(m)2433 359 y Fr(,)227 b Fq(k)31 b(>)d(m)33 b Fr(,)873 b(\(4\))-24 560 y(and)38 b Fq(T)242 524 y Fn(E)228 585 y(!)301 560 y Fr(\()p Fq(k)s(;)17 b(m)p Fr(\))36 b(=)g Fq(T)779 524 y Fn(E)765 585 y(!)838 560 y Fr(\()p Fq(m;)17 b(k)s Fr(\))1097 524 y Fk(\000)p Fo(1)1229 560 y Fr(if)37 b Fq(k)i(<)c(m)p Fr(,)k Fq(T)1746 524 y Fn(E)1732 585 y(!)1806 560 y Fr(\()p Fq(m;)17 b(m)p Fr(\))36 b(=)f Fi(1)p Fr(.)58 b(The)38 b(Ly)m(apuno)m(v)h(exp)s(onen)m(t)g(at)e (energy)i Fq(E)6 b Fr(,)-24 681 y(also)36 b(called)h(in)m(v)m(erse)h (lo)s(calization)d(length,)i(is)g(then)f(almost)g(surely)h(de\014ned)g (b)m(y)g(\(some)f(more)g(details)h(are)-24 801 y(giv)m(en)d(in)f (Sections)h(3.4)e(and)h(4.1\))1142 1129 y Fq(\015)5 b Fr(\()p Fq(E)h Fr(\))55 b(=)77 b(lim)1538 1192 y Fn(k)r Fk(!1)1857 1062 y Fr(1)p 1744 1106 274 4 v 1744 1198 a Fq(k)20 b Fp(h)p Fq(L)1920 1213 y Fk(\006)1979 1198 y Fp(i)2044 1129 y Fr(log)2187 1049 y Fl(\000)2233 1045 y(\015)2233 1105 y(\015)2288 1129 y Fq(T)2359 1088 y Fn(E)2345 1154 y(!)2418 1129 y Fr(\()p Fq(k)s(;)d Fr(0\))2641 1045 y Fl(\015)2641 1105 y(\015)2696 1049 y(\001)2791 1129 y Fr(,)1073 b(\(5\))-24 1374 y(where)36 b Fp(h)p Fq(c)341 1389 y Fk(\006)400 1374 y Fp(i)30 b Fr(=)g Fq(p)624 1389 y Fo(+)683 1374 y Fq(c)725 1389 y Fo(+)807 1374 y Fr(+)23 b Fq(p)955 1389 y Fk(\000)1014 1374 y Fq(c)1056 1389 y Fk(\000)1150 1374 y Fr(for)33 b(an)m(y)i(complex)h(n)m(um)m(b)s (ers)g Fq(c)2307 1389 y Fk(\006)2366 1374 y Fr(.)48 b(V)-8 b(anishing)35 b(of)f(the)g(Ly)m(apuno)m(v)i(exp)s(onen)m(t)-24 1494 y(is)e(considered)h(an)d(indicator)i(for)e(p)s(ossible)i(delo)s (calization.)45 b(F)-8 b(or)32 b(a)h(p)s(olymer)h(c)m(hain,)g(this)f (happ)s(ens)h(in)f(the)-24 1615 y(follo)m(wing)g(situation:)-24 1807 y Fi(De\014nition)38 b(1)49 b Fd(A)n(n)41 b(ener)-5 b(gy)41 b Fq(E)1156 1822 y Fn(c)1231 1807 y Fp(2)f Fg(R)h Fd(is)g(c)-5 b(al)5 b(le)-5 b(d)41 b(critic)-5 b(al)41 b(for)g(the)g(r)-5 b(andom)41 b(family)f Fr(\()p Fq(H)3284 1822 y Fn(!)3335 1807 y Fr(\))3373 1822 y Fn(!)r Fk(2)p Fo(\012)3562 1807 y Fd(of)h(p)-5 b(olymer)-24 1927 y(Hamiltonians)38 b(if)g(the)g(p)-5 b(olymer)37 b(tr)-5 b(ansfer)38 b(matric)-5 b(es)38 b Fq(T)2050 1885 y Fn(E)2102 1893 y Fe(c)2036 1949 y Fk(\006)2176 1927 y Fd(ar)-5 b(e)38 b(el)5 b(liptic)37 b Fr(\()p Fd(i.e.)54 b Fp(j)p Fr(T)-8 b(r\()p Fq(T)3133 1885 y Fn(E)3185 1893 y Fe(c)3119 1949 y Fk(\006)3221 1927 y Fr(\))p Fp(j)33 b Fq(<)h Fr(2\))k Fd(or)g(e)-5 b(qual)38 b(to)-24 2048 y Fp(\006)p Fi(1)e 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y(p)s(o)m(w)m(ers)44 b(of)e Fq(\017)p Fr(.)74 b(Since)43 b Fq(M)10 b(T)1009 5152 y Fn(E)1061 5160 y Fe(c)995 5216 y Fk(\006)1098 5195 y Fq(M)1202 5159 y Fk(\000)p Fo(1)1339 5195 y Fr(are)43 b(rotations,)h(this)g (implies)g(that)e Fp(k)p Fq(M)10 b(T)2951 5152 y Fn(E)3003 5160 y Fe(c)3035 5152 y Fo(+)p Fn(\017)2937 5216 y Fk(\006)3122 5195 y Fq(M)3226 5159 y Fk(\000)p Fo(1)3321 5195 y Fp(k)45 b(\024)g Fr(1)28 b(+)h Fq(c)p Fp(j)p Fq(\017)p Fp(j)42 b Fr(for)-24 5315 y Fp(j)p Fq(\017)p Fp(j)28 b(\024)g Fq(\017)243 5330 y Fo(0)315 5315 y Fr(and)33 b(one)g(deduces)h(the)f (follo)m(wing:)1972 5564 y(4)p eop %%Page: 5 5 5 4 bop -24 171 a Fi(Prop)s(osition)38 b(1)49 b Fd(F)-7 b(or)47 b Fq(\017)921 186 y Fo(0)1014 171 y Fq(>)53 b Fr(0)48 b Fd(ther)-5 b(e)49 b(exists)f(a)h(c)-5 b(onstant)48 b Fq(C)60 b(<)53 b Fp(1)48 b Fd(such)g(that)h(for)f(al)5 b(l)49 b Fp(j)p Fq(\017)p Fp(j)j(\024)i Fq(\017)3773 186 y Fo(0)3861 171 y Fd(and)-24 291 y Fq(m;)17 b(k)31 b Fp(2)d Fg(Z)p Fd(,)1305 403 y Fl(\015)1305 463 y(\015)1360 488 y Fq(T)1431 447 y Fn(E)1483 455 y Fe(c)1515 447 y Fo(+)p Fn(\017)1417 513 y(!)1602 488 y Fr(\()p Fq(k)s(;)17 b(m)p Fr(\))1861 403 y Fl(\015)1861 463 y(\015)1972 488 y Fp(\024)56 b Fq(C)24 b(e)2244 447 y Fn(C)5 b Fk(j)p Fn(\017)p Fk(j)10 b(j)p Fn(k)r Fk(\000)p Fn(m)p Fk(j)2612 488 y Fd(.)1249 b Fr(\(8\))-24 685 y Fd(In)35 b(p)-5 b(articular,)34 b Fp(j)p Fq(\015)5 b Fr(\()p Fq(E)773 700 y Fn(c)830 685 y Fr(+)22 b Fq(\017)p Fr(\))p Fp(j)28 b(\024)44 b Fq(C)1259 649 y Fk(0)1299 685 y Fp(j)p Fq(\017)p Fp(j)35 b Fd(for)g Fq(C)1662 649 y Fk(0)1712 685 y Fq(>)28 b Fr(0)p Fd(.)123 887 y Fr(Note)45 b(that)g(the)h(b)s(ound)f(in)g (\(8\))g(do)s(es)h(not)f(dep)s(end)h(on)f(the)h(con\014guration.)81 b(T)-8 b(o)45 b(study)h(a)f(p)s(ossible)-24 1007 y(spreading)38 b(of)f(w)m(a)m(v)m(e)h(pac)m(k)m(ets)h(due)f(to)f(the)g(div)m(ergence)i (of)d(the)i(lo)s(calization)f(length,)h(one)f(b)s(est)h(considers)-24 1128 y(the)d(momen)m(ts)g(of)e(the)h(asso)s(ciated)h(probabilit)m(y)f (distribution,)i(notably)e(the)g(time-a)m(v)m(eraged)h(momen)m(ts)g(of) -24 1248 y(the)e(p)s(osition)g(op)s(erator)f Fq(X)40 b Fr(on)33 b Fq(`)1207 1212 y Fo(2)1246 1248 y Fr(\()p Fg(Z)p Fr(\):)738 1591 y Fq(M)832 1606 y Fn(!)r(;q)936 1591 y Fr(\()p Fq(T)14 b Fr(\))55 b(=)1269 1456 y Fl(Z)1369 1482 y Fk(1)1325 1681 y Fo(0)1470 1524 y Fq(dt)p 1470 1568 86 4 v 1478 1659 a(T)1583 1591 y(e)1628 1550 y Fk(\000)1703 1523 y Fe(t)p 1693 1535 45 3 v 1693 1576 a(T)1779 1591 y Fp(h)p Fr(0)p Fp(j)p Fq(e)1940 1550 y Fn({H)2021 1558 y Fe(!)2065 1550 y Fn(t)2095 1591 y Fp(j)p Fq(X)8 b Fp(j)2240 1550 y Fn(q)2277 1591 y Fq(e)2322 1550 y Fk(\000)p Fn({H)2458 1558 y Fe(!)2503 1550 y Fn(t)2533 1591 y Fp(j)p Fr(0)p Fp(i)32 b Fr(,)227 b Fq(q)32 b(>)27 b Fr(0)33 b(,)669 b(\(9\))-24 1835 y(The)32 b(exp)s(onen)m(tial)g(time)f(a)m(v)m(erage)g (ma)m(y)h(b)s(e)e(replaced)i(b)m(y)f(a)g(Cesaro)g(mean)g(without)g(c)m (hanging)g(the)g(asymp-)-24 1956 y(totics)k(\()p Fm(e.g.)50 b Fr([GSB]\).)g(Prop)s(osition)35 b(1)f(will)h(lead)g(more)g(or)f(less) i(directly)g(to)e(the)i(follo)m(wing)e(deterministic)-24 2076 y(lo)m(w)m(er)g(b)s(ound)f(on)f(transp)s(ort.)-24 2278 y Fi(Theorem)38 b(1)49 b Fd(Ther)-5 b(e)34 b(exists)h(a)g(c)-5 b(onstant)34 b Fq(C)42 b Fd(such)35 b(that)g(for)g(every)f(c)-5 b(on\014gur)g(ation)34 b Fq(!)39 b Fd(and)34 b(for)h Fq(q)c Fp(\025)d Fr(0)1528 2595 y Fq(M)1622 2610 y Fn(!)r(;q)1727 2595 y Fr(\()p Fq(T)14 b Fr(\))55 b Fp(\025)h Fq(C)23 b(T)2226 2554 y Fn(q)r Fk(\000)p Fo(1)2389 2595 y Fd(.)1423 b Fr(\(10\))-24 2797 y Fi(Remark)38 b(3)49 b Fr(It)41 b(is)g(imp)s(ortan)m(t)f(that)h(the)g(initial)f(condition)i(in)e(\(9\)) g(is)i Fp(j)p Fr(0)p Fp(i)d Fr(and)i(not)f(an)h(arbitrary)f(state)-24 2918 y Fq( )32 b Fp(2)c Fq(`)206 2881 y Fo(2)246 2918 y Fr(\()p Fg(Z)p Fr(\).)43 b(In)33 b(fact,)g Fq( )j Fr(could)d(b)s(e)g (an)g(eigenstate)g(of)f Fq(H)2075 2933 y Fn(!)2158 2918 y Fr(and)h(hence)h(not)e(lead)h(to)f(an)m(y)i(di\013usion.)123 3167 y(In)25 b(order)g(to)g(study)h(the)f(b)s(eha)m(vior)h(of)e(the)i (Ly)m(apuno)m(v)g(exp)s(onen)m(t)h(in)e(the)g(vicinit)m(y)i(of)e(the)g (critical)h(energy)-8 b(,)-24 3287 y(that)37 b(is,)h(go)e(b)s(ey)m(ond) i(the)f(trivial)g(upp)s(er)g(b)s(ound)g Fp(j)p Fq(\015)5 b Fr(\()p Fq(E)2047 3302 y Fn(c)2106 3287 y Fr(+)25 b Fq(\017)p Fr(\))p Fp(j)34 b(\024)52 b Fq(C)2552 3251 y Fk(0)2592 3287 y Fp(j)p Fq(\017)p Fp(j)p Fr(,)37 b(let)g(us)g (de\014ne)h(the)f(transmission)-24 3408 y(and)c(re\015ection)h(co)s (e\016cien)m(ts)g Fq(a)1133 3371 y Fn(\017)1133 3432 y Fk(\006)1225 3408 y Fr(and)f Fq(b)1456 3371 y Fn(\017)1456 3432 y Fk(\006)1548 3408 y Fr(b)m(y)805 3758 y Fq(M)10 b(T)980 3717 y Fn(E)1032 3725 y Fe(c)1064 3717 y Fo(+)p Fn(\017)966 3783 y Fk(\006)1151 3758 y Fq(M)1255 3717 y Fk(\000)p Fo(1)1367 3758 y Fq(v)59 b Fr(=)c Fq(a)1655 3717 y Fn(\017)1655 3783 y Fk(\006)1715 3758 y Fq(v)26 b Fr(+)c Fq(b)1927 3717 y Fn(\017)1927 3783 y Fk(\006)p 1986 3703 51 4 v 1986 3758 a Fq(v)36 b Fr(,)228 b Fq(v)59 b Fr(=)2613 3691 y(1)p 2571 3735 132 4 v 2571 3755 a Fp(p)p 2654 3755 49 4 v 83 x Fr(2)2730 3618 y Fl(\022)2874 3697 y Fr(1)2844 3817 y Fp(\000)p Fq({)2995 3618 y Fl(\023)3117 3758 y Fr(.)698 b(\(11\))-24 4008 y(Both)33 b(are)f(p)s(olynomials)i (in)f Fq(\017)p Fr(.)44 b(As)33 b Fq(v)j Fr(is)d(an)g(eigen)m(v)m (ector)h(of)e(all)h(rotations,)f(one)h(has)1405 4325 y Fq(a)1456 4284 y Fo(0)1456 4350 y Fk(\006)1570 4325 y Fr(=)55 b Fq(e)1746 4284 y Fn({\021)1804 4293 y Ff(\006)1894 4325 y Fr(,)228 b Fq(b)2190 4284 y Fo(0)2190 4350 y Fk(\006)2305 4325 y Fr(=)55 b(0)32 b(.)1298 b(\(12\))-24 4522 y(F)-8 b(urthermore)41 b(let)f(us)g(set)h Fq(e)1034 4484 y Fn({\021)1094 4460 y Fe(\017)1092 4504 y Ff(\006)1189 4522 y Fr(=)f Fq(a)1356 4486 y Fn(\017)1356 4547 y Fk(\006)1416 4522 y Fq(=)p Fp(j)p Fq(a)1544 4486 y Fn(\017)1544 4547 y Fk(\006)1602 4522 y Fp(j)g Fr(so)g(that)g Fq(\021)2068 4486 y Fo(0)2064 4547 y Fk(\006)2163 4522 y Fr(=)g Fq(\021)2327 4537 y Fk(\006)2386 4522 y Fr(.)66 b(Av)m(erages)41 b(will)g(alw)m(a)m (ys)g(b)s(e)g(denoted)f(as)-24 4642 y Fp(h)p Fq(c)57 4657 y Fk(\006)116 4642 y Fp(i)28 b Fr(=)f Fq(p)335 4657 y Fo(+)394 4642 y Fq(c)436 4657 y Fo(+)517 4642 y Fr(+)22 b Fq(p)664 4657 y Fk(\000)723 4642 y Fq(c)765 4657 y Fk(\000)824 4642 y Fr(.)-24 4844 y Fi(Theorem)38 b(2)49 b Fd(Supp)-5 b(ose)47 b(that)i Fp(h)p Fq(e)1237 4808 y Fo(2)p Fn({\021)1330 4817 y Ff(\006)1387 4844 y Fp(i)i(6)p Fr(=)h(1)c Fd(and)f Fp(h)p Fq(e)1988 4808 y Fo(4)p Fn({\021)2081 4817 y Ff(\006)2138 4844 y Fp(i)52 b(6)p Fr(=)g(1)p Fd(.)83 b(Then)48 b(the)g(Lyapunov)f(exp)-5 b(onent)47 b(of)h(a)-24 4965 y(r)-5 b(andom)34 b(p)-5 b(olymer)35 b(chain)f(satis\014es)669 5218 y Fq(\015)5 b Fr(\()p Fq(E)835 5233 y Fn(c)892 5218 y Fr(+)22 b Fq(\017)p Fr(\))55 b(=)1263 5151 y(2)17 b Fq(p)1378 5166 y Fo(+)1437 5151 y Fq(p)1486 5166 y Fk(\000)p 1263 5195 282 4 v 1303 5287 a Fp(h)p Fq(L)1408 5302 y Fk(\006)1467 5287 y Fp(i)1592 5148 y(j)p Fq(b)1661 5112 y Fn(\017)1661 5172 y Fo(+)1754 5148 y Fr(sin)q(\()p Fq(\021)1964 5112 y Fn(\017)1960 5172 y Fk(\000)2019 5148 y Fr(\))22 b Fp(\000)g Fq(b)2219 5112 y Fn(\017)2219 5172 y Fk(\000)2312 5148 y Fr(sin)q(\()p Fq(\021)2522 5112 y Fn(\017)2518 5172 y Fo(+)2577 5148 y Fr(\))p Fp(j)2643 5112 y Fo(2)p 1592 5195 1090 4 v 1868 5290 a Fp(j)p Fr(1)g Fp(\000)g(h)p Fq(e)2150 5252 y Fo(2)p Fn({\021)2245 5229 y Fe(\017)2243 5273 y Ff(\006)2300 5290 y Fp(ij)2367 5261 y Fo(2)2714 5218 y Fr(+)g Fp(O)2911 5138 y Fl(\000)2956 5218 y Fp(j)p Fq(b)3025 5177 y Fn(\017)3025 5243 y Fk(\006)3085 5218 y Fp(j)3113 5177 y Fo(3)3152 5138 y Fl(\001)3249 5218 y Fd(.)563 b Fr(\(13\))1972 5564 y(5)p eop %%Page: 6 6 6 5 bop -24 171 a Fi(De\014nition)38 b(2)49 b Fd(A)43 b(critic)-5 b(al)43 b(ener)-5 b(gy)42 b Fq(E)1442 186 y Fn(c)1519 171 y Fp(2)h Fg(R)f Fd(of)h(a)f(p)-5 b(olymer)42 b(Hamiltonian)g(is)h(said)f(to)h(b)-5 b(e)42 b(of)g(or)-5 b(der)43 b Fq(r)i Fd(if)-24 291 y Fp(j)p Fq(b)45 255 y Fn(\017)45 316 y Fk(\006)105 291 y Fp(j)27 b Fr(=)g Fp(O)s Fr(\()p Fq(\017)422 255 y Fn(r)461 291 y Fr(\))p Fd(,)35 b(but)g(not)g Fp(j)p Fq(b)969 255 y Fn(\017)969 316 y Fk(\006)1028 291 y Fp(j)27 b Fr(=)h Fp(O)s Fr(\()p Fq(\017)1346 255 y Fn(r)r Fo(+1)1474 291 y Fr(\))35 b Fd(for)g(b)-5 b(oth)35 b(p)-5 b(olymers.)-24 519 y Fi(Remark)38 b(4)49 b Fr(If)41 b Fq(E)684 534 y Fn(c)760 519 y Fr(is)g(a)g(critical) g(energy)h(of)e(order)h Fq(r)s Fr(,)i(then)e Fq(\015)5 b Fr(\()p Fq(E)2504 534 y Fn(c)2567 519 y Fr(+)28 b Fq(\017)p Fr(\))42 b(=)f Fq(C)24 b(\017)3040 483 y Fo(2)p Fn(r)3141 519 y Fr(+)k Fp(O)s Fr(\()p Fq(\017)3404 483 y Fo(2)p Fn(r)r Fo(+1)3568 519 y Fr(\))41 b(for)f(some)-24 640 y(non-negativ)m(e)46 b(constan)m(t)f Fq(C)7 b Fr(.)78 b(Since)46 b Fq(b)1458 604 y Fn(\017)1458 664 y Fk(\006)1565 640 y Fr(=)i Fp(O)s Fr(\()p Fq(\017)p Fr(\),)f(the)e(order)f(of)g(ev)m (ery)i(critical)f(energy)h(is)e(as)h(least)g(1,)-24 760 y(i.e.)38 b(the)g(Ly)m(apuno)m(v)h(exp)s(onen)m(t)g(v)-5 b(anishes)39 b(at)e(least)h(quadratically)-8 b(.)58 b(Generically)39 b(the)f(order)f(of)g(a)g(critical)-24 880 y(energy)f(is)e(1)g(\(this)h (is)g(the)g(case)g(in)f(the)h(dimer)g(mo)s(del\).)49 b(In)34 b(the)h(latter)f(case,)i(more)f(explicit)g(form)m(ulas)g(for) -24 1001 y(the)28 b(co)s(e\016cien)m(t)g(in)f(\(13\))f(in)m(v)m(oking)i (only)f(the)h(v)-5 b(alues)27 b(of)f Fq(T)2111 958 y Fn(E)2163 966 y Fe(c)2097 1023 y Fk(\006)2226 1001 y Fr(and)h Fq(@)2461 1016 y Fn(E)2521 1001 y Fq(T)2592 958 y Fn(E)2644 966 y Fe(c)2578 1023 y Fk(\006)2706 1001 y Fr(at)f(the)h(critical)h(energy)f(can)g(easily)-24 1121 y(b)s(e)36 b(written)h(out.)52 b(A)36 b(comparison)g(of)f(\(13\))g (with)i(a)e(random)h(phase)g(appro)m(ximation)h(is)f(made)g(in)g (Section)-24 1242 y(4.7.)52 b(Finally)-8 b(,)37 b(let)e(us)h(note)g (that)f(\(13\))g(also)g(pro)m(v)m(es)j(p)s(ositivit)m(y)f(of)e(the)g (Ly)m(apuno)m(v)i(exp)s(onen)m(t)g(close)g(to)e Fq(E)3981 1257 y Fn(c)-24 1362 y Fr(whenev)m(er)h(the)d(n)m(umerator)g(do)s(es)g (not)f(v)-5 b(anish.)-24 1590 y Fi(Remark)38 b(5)49 b 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b(IDS,)f(its)g(de\014nition)-24 2300 y(is)41 b(recalled)f(in)g(Section)h(3.4)e(b)s(elo)m(w\).)65 b(Let)40 b Fp(N)1717 2315 y Fk(\006)1815 2300 y Fr(and)g Fp(N)2109 2264 y Fk(0)2094 2325 y(\006)2192 2300 y Fr(denote)h(the)f (absolutely)h(con)m(tin)m(uous)g(IDS)f(and)-24 2420 y(their)d (densities)h(asso)s(ciated)f(to)f(the)g(mo)s(dels)h(with)g Fq(L)1999 2435 y Fk(\006)2058 2420 y Fr(-p)s(erio)s(dic)f(mo)s(dels)h (comp)s(osed)g(of)f(only)g(one)h(of)e(the)-24 2541 y(p)s(olymers.)45 b(By)33 b(de\014nition)h(of)e(a)g(critical)h(energy)-8 b(,)34 b Fp(N)1964 2505 y Fk(0)1949 2565 y(\006)2008 2541 y Fr(\()p Fq(E)2118 2556 y Fn(c)2153 2541 y Fr(\))27 b Fq(>)h Fr(0.)-24 2769 y Fi(Theorem)38 b(3)49 b Fd(Supp)-5 b(ose)34 b(that)i Fp(h)p Fq(e)1211 2733 y Fo(2)p Fn({\021)1304 2742 y Ff(\006)1361 2769 y Fp(i)27 b(6)p Fr(=)h(1)p Fd(.)44 b(Then)34 b(the)h Fr(IDS)g Fd(of)f(a)h(r)-5 b(andom)34 b(p)-5 b(olymer)34 b(chain)g(satis\014es)794 3143 y Fp(N)15 b Fr(\()p Fq(E)1001 3158 y Fn(c)1057 3143 y Fr(+)22 b 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Fq(T)2198 4047 y Fn(v)r Fo(\()p Fn(l)q Fo(\))p Fk(\000)p Fn(z)s(;t)p Fo(\()p Fn(l)q Fo(\))2561 4031 y Fr(.)-24 4303 y(It)33 b(allo)m(ws)h(to)e(rewrite)h(the)g(\(formal\))f (eigenfunction)i(equation)g(\(19\))e(as)1004 4536 y Fl(\022)1119 4615 y Fq(t)p Fr(\()p Fq(n)p Fr(\))17 b Fq(u)1361 4579 y Fn(z)1400 4615 y Fr(\()p Fq(n)p Fr(\))1127 4735 y Fq(u)1183 4699 y Fn(z)1222 4735 y Fr(\()p Fq(n)22 b Fp(\000)h Fr(1\))1576 4536 y Fl(\023)1705 4676 y Fr(=)55 b Fp(T)1916 4635 y Fn(z)1955 4676 y Fr(\()p Fq(n;)17 b(k)s Fr(\))2231 4536 y Fl(\022)2346 4615 y Fq(t)p Fr(\()p Fq(k)s Fr(\))g Fq(u)2584 4579 y Fn(z)2623 4615 y Fr(\()p Fq(k)s Fr(\))2352 4735 y Fq(u)2408 4699 y Fn(z)2447 4735 y Fr(\()p Fq(k)25 b Fp(\000)e Fr(1\))2795 4536 y Fl(\023)2917 4676 y Fr(.)898 b(\(20\))-24 4954 y(Note)38 b(that)g(the)g(transfer)g(matrices)h (satisfy)g(the)f(transitivit)m(y)i(relation)e Fp(T)2807 4918 y Fn(z)2847 4954 y Fr(\()p Fq(n;)17 b(k)s Fr(\))36 b(=)h Fp(T)3307 4918 y Fn(z)3347 4954 y Fr(\()p Fq(n;)17 b(m)p Fr(\))p Fp(T)3690 4918 y Fn(z)3730 4954 y Fr(\()p Fq(m;)g(k)s Fr(\).)-24 5075 y(A)33 b(direct)g(inductiv)m(e)i(argumen)m (t)e(then)h(sho)m(ws)g(that,)e(for)g Fq(\020)j Fp(2)28 b Fg(C)p Fr(,)1972 5564 y(8)p eop %%Page: 9 9 9 8 bop 474 444 a Fp(T)554 403 y Fn(z)s Fo(+)p Fn(\020)684 444 y Fr(\()p Fq(n;)17 b(k)s Fr(\))55 b(=)h Fp(T)1182 403 y Fn(z)1222 444 y Fr(\()p Fq(n;)17 b(k)s Fr(\))22 b Fp(\000)h Fq(\020)1664 320 y Fn(n)p Fk(\000)p Fo(1)1659 350 y Fl(X)1673 562 y Fn(l)q Fo(=)p Fn(k)1836 444 y Fp(T)1916 403 y Fn(z)s Fo(+)p Fn(\020)2046 444 y Fr(\()p Fq(n;)17 b(l)24 b Fr(+)e(1\))2508 377 y(1)p 2462 421 143 4 v 2462 513 a Fq(t)p Fr(\()p Fq(l)r Fr(\))2647 304 y Fl(\022)2762 383 y Fr(1)83 b(0)2762 504 y(0)g(0)2984 304 y Fl(\023)3091 444 y Fp(T)3170 403 y Fn(z)3210 444 y Fr(\()p Fq(l)r(;)17 b(k)s Fr(\))32 b(.)368 b(\(21\))-24 743 y(T)-8 b(aking)44 b(the)f(norm)h(of)e(\(21\),)j(estimating)f(the)g(r.h.s.)75 b(and)44 b(consequen)m(tly)i(taking)d(the)g(suprem)m(um)j(o)m(v)m(er) -24 864 y(0)34 b Fp(\024)h Fq(k)i Fp(\024)e Fq(n)f Fp(\024)h Fq(m)i Fr(leads)g(to)f(the)h(follo)m(wing)g(p)s(erturbativ)m(e)h (result)f(whic)m(h)h(in)e(a)h(sligh)m(tly)g(di\013eren)m(t)h(form)e(is) -24 984 y(giv)m(en)e(in)f([Sim2])g(\(see)h(also)e([DT].\))-24 1199 y Fi(Lemma)39 b(1)49 b Fd(Supp)-5 b(ose)1017 1447 y Fr(sup)918 1529 y Fo(0)p Fk(\024)p Fn(k)r Fk(\024)p Fn(n)p Fk(\024)p Fn(m)1306 1447 y Fp(kT)1436 1406 y Fn(z)1475 1447 y Fr(\()p Fq(n;)17 b(k)s Fr(\))p Fp(k)55 b(\024)h Fq(C)206 b(D)58 b Fr(=)142 b(sup)2492 1529 y Fo(0)p Fk(\024)p Fn(l)q Fk(\024)p Fn(m)p Fk(\000)p Fo(1)2872 1379 y Fr(1)p 2838 1424 117 4 v 2838 1515 a Fp(j)p Fq(t)2901 1530 y Fn(l)2927 1515 y Fp(j)2999 1447 y Fd(.)-24 1721 y(Then,)34 b(as)h(long)f(as)h Fq(C)7 b(D)s Fp(j)p Fq(\020)h Fp(j)p Fq(m)26 b(<)h Fr(1)p Fd(,)1210 1985 y Fr(sup)1112 2067 y Fo(0)p Fk(\024)p Fn(k)r Fk(\024)p Fn(n)p Fk(\024)p Fn(m)1500 1985 y Fp(kT)1629 1944 y Fn(z)s Fo(+)p Fn(\020)1759 1985 y Fr(\()p Fq(n;)17 b(k)s Fr(\))p Fp(k)55 b(\024)2462 1918 y Fq(C)p 2239 1962 522 4 v 2239 2054 a Fr(1)22 b Fp(\000)h Fq(C)7 b(D)s Fp(j)p Fq(\020)h Fp(j)p Fq(m)2806 1985 y Fd(.)-24 2318 y Fc(3.2)136 b(F)-11 b(ree)45 b(Pr)t(\177)-71 b(ufer)44 b(v)-7 b(ariables)-24 2502 y Fr(Let)46 b(no)m(w)h Fq(E)56 b Fp(2)50 b Fg(R)c Fr(and)g Fq(u)1002 2466 y Fn(E)1106 2502 y Fr(b)s(e)g(giv)m(en)h(b)m(y)g(\(19\).)82 b(The)46 b(free)h(Pr)s(\177)-51 b(ufer)45 b(phases)i Fq(\022)3052 2466 y Fo(0)p Fn(;E)3167 2502 y Fr(\()p Fq(n)p Fr(\))e(and)h(amplitudes)-24 2623 y Fq(R)51 2587 y Fo(0)p Fn(;E)166 2623 y Fr(\()p Fq(n)p Fr(\))28 b Fq(>)f Fr(0)33 b(are)f(no)m(w)h(de\014ned)h(b)m(y)979 2985 y Fq(R)1054 2944 y Fo(0)p Fn(;E)1168 2985 y Fr(\()p Fq(n)p Fr(\))1336 2844 y Fl(\022)1451 2924 y Fr(cos\()p Fq(\022)1667 2888 y Fo(0)p Fn(;E)1782 2924 y Fr(\()p Fq(n)p Fr(\)\))1456 3044 y(sin)q(\()p Fq(\022)1662 3008 y Fo(0)p Fn(;E)1776 3044 y Fr(\()p Fq(n)p Fr(\)\))1995 2844 y Fl(\023)2124 2985 y Fr(=)2255 2844 y Fl(\022)2371 2924 y Fq(t)p Fr(\()p Fq(n)p Fr(\))p Fq(u)2596 2888 y Fn(E)2655 2924 y Fr(\()p Fq(n)p Fr(\))2370 3044 y Fq(u)2426 3008 y Fn(E)2485 3044 y Fr(\()p Fq(n)22 b Fp(\000)h Fr(1\))2831 2844 y Fl(\023)2954 2985 y Fr(,)861 b(\(22\))-24 3246 y(the)33 b(ab)s(o)m(v)m(e)h(initial)f (conditions)g(as)g(w)m(ell)h(as)1229 3506 y Fp(\000)1316 3439 y Fq(\031)p 1316 3483 59 4 v 1321 3575 a Fr(2)1413 3506 y Fq(<)28 b(\022)1565 3465 y Fo(0)p Fn(;E)1679 3506 y Fr(\()p Fq(n)23 b Fr(+)f(1\))g Fp(\000)g Fq(\022)2152 3465 y Fo(0)p Fn(;E)2267 3506 y Fr(\()p Fq(n)p Fr(\))28 b Fq(<)2542 3439 y Fr(3)p Fq(\031)p 2542 3483 108 4 v 2572 3575 a Fr(2)2692 3506 y(.)-24 3711 y(Note)33 b(that)f(the)h Fq(\022)639 3675 y Fo(0)679 3711 y Fr(-dep)s(endence)i(of)d(the)h(Pr)s (\177)-51 b(ufer)32 b(v)-5 b(ariables)33 b(is)g(suppressed.)-24 3926 y Fi(Lemma)39 b(2)811 4235 y Fq(R)886 4194 y Fo(0)p Fn(;E)1000 4235 y Fr(\()p Fq(n)p Fr(\))1134 4194 y Fo(2)1190 4235 y Fq(@)1241 4250 y Fn(E)1318 4235 y Fq(\022)1366 4194 y Fo(0)p Fn(;E)1481 4235 y Fr(\()p Fq(n)p Fr(\))55 b(=)1801 4031 y Fl(8)1801 4121 y(<)1801 4300 y(:)1972 4039 y(P)2077 4066 y Fn(n)p Fk(\000)p Fo(1)2077 4143 y Fn(l)q Fo(=0)2231 4114 y Fq(u)2287 4078 y Fn(E)2346 4114 y Fr(\()p Fq(l)r Fr(\))2453 4078 y Fo(2)2621 4114 y Fd(if)90 b Fq(n)28 b(>)f Fr(0)35 b Fd(,)1931 4362 y Fp(\000)2025 4287 y Fl(P)2131 4313 y Fk(\000)p Fo(1)2131 4391 y Fn(l)q Fo(=)p Fn(n)2271 4362 y Fq(u)2327 4326 y Fn(E)2386 4362 y Fr(\()p Fq(l)r Fr(\))2493 4326 y Fo(2)2616 4362 y Fd(if)90 b Fq(n)28 b(<)f Fr(0)35 b Fd(.)3842 4235 y Fr(\(23\))-24 4583 y Fi(Pro)s(of:)44 b Fr(F)-8 b(rom)32 b(the)h(recurrence)h(relation)f(\(19\))f(and)g(the)h(de\014nition)h(of) e Fq(\022)2732 4547 y Fo(0)p Fn(;E)2847 4583 y Fr(\()p Fq(n)p Fr(\))g(one)h(gets)684 4823 y(cot\()p Fq(\022)900 4782 y Fo(0)p Fn(;E)1014 4823 y Fr(\()p Fq(n)p Fr(\)\))56 b(=)f Fp(\000)p Fq(t)1485 4782 y Fo(2)1525 4823 y Fr(\()p Fq(n)22 b Fp(\000)h Fr(1\))17 b(tan)o(\()p Fq(\022)2073 4782 y Fo(0)p Fn(;E)2188 4823 y Fr(\()p Fq(n)22 b Fp(\000)h Fr(1\)\))e(+)h Fq(v)t Fr(\()p Fq(n)g Fp(\000)h Fr(1\))f Fp(\000)h Fq(E)38 b Fr(.)-24 4993 y(Di\013eren)m(tiation)33 b(leads)h(to)545 5272 y Fq(@)596 5287 y Fn(E)656 5272 y Fq(\022)704 5230 y Fo(0)p Fn(;E)818 5272 y Fr(\()p Fq(n)p Fr(\))56 b(=)1149 5204 y Fq(t)1184 5168 y Fo(2)1224 5204 y Fr(\()p Fq(n)22 b Fp(\000)g Fr(1\))17 b(sin)1665 5164 y Fo(2)1704 5204 y Fr(\()p Fq(\022)1790 5168 y Fo(0)p Fn(;E)1905 5204 y Fr(\()p Fq(n)p Fr(\)\))p 1149 5249 928 4 v 1256 5340 a(cos)1387 5311 y Fo(2)1426 5340 y Fr(\()p Fq(\022)1512 5311 y Fo(0)p Fn(;E)1627 5340 y Fr(\()p Fq(n)22 b Fp(\000)h Fr(1\)\))2114 5272 y Fq(@)2165 5287 y Fn(E)2225 5272 y Fq(\022)2273 5230 y Fo(0)p Fn(;E)2388 5272 y Fr(\()p Fq(n)f Fp(\000)h Fr(1\))f(+)g(sin)2933 5230 y Fo(2)2972 5272 y Fr(\()p Fq(\022)3058 5230 y Fo(0)p Fn(;E)3173 5272 y Fr(\()p Fq(n)p Fr(\)\))32 b(.)1972 5564 y(9)p eop %%Page: 10 10 10 9 bop -24 171 a Fr(Multiplying)34 b(with)g Fq(R)805 134 y Fo(0)p Fn(;E)919 171 y Fr(\()p Fq(n)p Fr(\))1053 134 y Fo(2)1125 171 y Fr(and)f(using)g(the)g(de\014nition)h(of)e Fq(R)2358 134 y Fo(0)p Fn(;E)2472 171 y Fr(\()p Fq(n)p Fr(\))h(and)g Fq(\022)2877 134 y Fo(0)p Fn(;E)2991 171 y Fr(\()p Fq(n)p Fr(\))g(giv)m(es)644 511 y Fq(R)719 470 y Fo(0)p Fn(;E)834 511 y Fr(\()p Fq(n)p Fr(\))968 470 y Fo(2)1007 511 y Fq(@)1058 526 y Fn(E)1118 511 y Fq(\022)1166 470 y Fo(0)p Fn(;E)1281 511 y Fr(\()p Fq(n)p Fr(\))56 b(=)f Fq(R)1677 470 y Fo(0)p Fn(;E)1791 511 y Fr(\()p Fq(n)23 b Fp(\000)f Fr(1\))2096 470 y Fo(2)2135 511 y Fq(@)2186 526 y Fn(E)2246 511 y Fq(\022)2294 470 y Fo(0)p Fn(;E)2409 511 y Fr(\()p Fq(n)g Fp(\000)h Fr(1\))f(+)g Fq(u)2890 470 y Fn(E)2949 511 y Fr(\()p Fq(n)g Fp(\000)h Fr(1\))3254 470 y Fo(2)3321 511 y Fq(:)494 b Fr(\(24\))-24 731 y(The)41 b(ab)s(o)m(v)m(e)g(deduction)g(of)e(\(24\))h(has)g(used)h (that)e Fq(u)1942 695 y Fn(E)2002 731 y Fr(\()p Fq(n)27 b Fp(\000)g Fr(1\))40 b Fp(6)p Fr(=)g(0.)65 b(If)40 b Fq(u)2774 695 y Fn(E)2833 731 y Fr(\()p Fq(n)27 b Fp(\000)h Fr(1\))40 b(=)g(0,)h(then)g(one)f(ma)m(y)-24 851 y(deduce)35 b(\(24\))c(in)i(a)g(similar)g(w)m(a)m(y)h(from)636 1133 y(tan\()p Fq(\022)863 1092 y Fo(0)p Fn(;E)977 1133 y Fr(\()p Fq(n)p Fr(\)\))56 b(=)1958 1066 y(cot\()p Fq(\022)2174 1030 y Fo(0)p Fn(;E)2289 1066 y Fr(\()p Fq(n)22 b Fp(\000)h Fr(1\)\))p 1346 1110 1898 4 v 1346 1201 a Fp(\000)p Fq(t)1458 1173 y Fo(2)1498 1201 y Fr(\()p Fq(n)f Fp(\000)h Fr(1\))f(+)g(\()p Fq(v)t Fr(\()p Fq(n)g Fp(\000)g Fr(1\))g Fp(\000)h Fq(E)6 b Fr(\))17 b(cot)o(\()p Fq(\022)2786 1173 y Fo(0)p Fn(;E)2901 1201 y Fr(\()p Fq(n)22 b Fp(\000)h Fr(1\)\))3286 1133 y(.)-24 1409 y(The)34 b(lemma)f(no)m(w)g(follo)m(ws)h(b)m(y)f (iterating)g(\(24\).)2194 b Fb(2)123 1577 y Fr(Note)32 b(in)g(particular)g(that)f(\(23\))g(implies)i(that)f Fq(@)1927 1592 y Fn(E)1987 1577 y Fq(\022)2035 1541 y Fo(0)p Fn(;E)2150 1577 y Fr(\()p Fq(n)p Fr(\))f(is)i(strictly)g(p)s (ositiv)m(e)g(for)e Fq(n)d Fp(\025)g Fr(2)k(and)f(strictly)-24 1697 y(negativ)m(e)g(for)e Fq(n)f Fp(\024)g(\000)p Fr(2.)43 b(F)-8 b(urthermore,)30 b(it)g(follo)m(ws)g(from)f(elemen)m(tary)i (considerations)g(for)e(transfer)h(matri-)-24 1817 y(ces)k(that)e (there)i(are)e(constan)m(ts)i Fq(C)1259 1832 y Fo(1)1331 1817 y Fr(and)f Fq(C)1591 1832 y Fo(2)1662 1817 y Fr(suc)m(h)h(that) 1146 2158 y(0)56 b Fq(<)f(C)1452 2173 y Fo(1)1546 2158 y Fp(\024)1679 2073 y Fl(\014)1679 2133 y(\014)1712 2158 y Fq(@)1763 2173 y Fn(E)1840 2158 y Fq(\022)1888 2117 y Fo(0)p Fn(;E)2003 2158 y Fr(\()p Fq(n)p Fr(\))2137 2073 y Fl(\014)2137 2133 y(\014)2225 2158 y Fp(\024)h Fq(C)2428 2173 y Fo(2)2523 2158 y Fq(<)f Fp(1)32 b Fr(,)1029 b(\(25\))-24 2378 y(where)50 b Fq(C)344 2393 y Fo(1)431 2378 y Fr(and)e Fq(C)706 2393 y Fo(2)793 2378 y Fr(can)h(b)s(e)f (uniformly)h(b)s(ounded)g(a)m(w)m(a)m(y)g(from)f(0)g(and)g Fp(1)g Fr(as)g(long)g(as)g Fp(j)p Fq(n)p Fp(j)54 b(\025)h Fr(2)48 b(and)-24 2498 y(the)35 b(quan)m(tities)h Fp(j)p Fq(n)p Fp(j)p Fr(,)e Fq(E)40 b Fr(and)34 b(max)1257 2514 y Fk(j)p Fn(k)r Fk(j\024j)p Fn(n)p Fk(j)1476 2498 y Fp(fj)p Fq(v)t Fr(\()p 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Fq(k)30 b Fp(!)e(1)p Fr(.)39 b(F)-8 b(or)22 b Fq(!)31 b Fp(2)3482 5290 y Fr(~)3471 5315 y(\012)3541 5330 y Fo(0)3604 5315 y Fr(it)22 b(is)h(easily)1947 5564 y(13)p eop %%Page: 14 14 14 13 bop -24 171 a Fr(seen)35 b(that)e(lim)536 186 y Fn(N)7 b Fk(!1)785 131 y Fo(1)p 771 148 64 4 v 771 205 a Fn(N)861 171 y Fr(log)q(\()p Fp(kT)1154 134 y Fn(E)1129 200 y Fo(\()p Fn(!)r(;)p Fo(0\))1289 171 y Fr(\()p Fq(N)e(;)17 b Fr(0\))p Fp(k)p Fr(\))27 b(=)i Fq(\015)1812 186 y Fo(0)1851 171 y Fr(\()p Fq(E)6 b Fr(\))p Fq(=)p Fp(h)p Fq(L)2159 186 y Fk(\006)2218 171 y Fp(i)p Fr(.)44 b(Since)34 b Fi(P)p Fp(f)p Fr(\()p Fq(!)t(;)17 b Fr(0\))27 b Fp(j)h Fq(!)j Fp(2)3225 145 y Fr(~)3214 171 y(\012)3284 186 y Fo(0)3324 171 y Fp(g)d Fr(=)h(1)p Fq(=)p Fp(h)p Fq(L)3710 186 y Fk(\006)3768 171 y Fp(i)g Fq(>)f Fr(0,)-24 300 y(one)33 b(concludes)i(from)d(\(34\))g(that)1499 660 y Fq(\015)5 b Fr(\()p Fq(E)h Fr(\))55 b(=)1983 592 y(1)p 1906 637 203 4 v 1906 728 a Fp(h)p Fq(L)2011 743 y Fk(\006)2070 728 y Fp(i)2146 660 y Fq(\015)2197 675 y Fo(0)2236 660 y Fr(\()p Fq(E)6 b Fr(\))33 b(.)1392 b(\(36\))-24 930 y(While)42 b Fq(\015)320 945 y Fo(0)400 930 y Fr(is)f(not)f(de\014ned)j (through)d(a)h(co)m(v)-5 b(arian)m(t)41 b(op)s(erator,)h(it)f(follo)m (ws)g(b)m(y)h(the)f(same)g(argumen)m(t)h(as)e(in)-24 1050 y(Lemma)34 b(3)e(that)g(for)g(an)m(y)i(con)m(tin)m(uous)g(measure) g Fq(\027)39 b Fr(on)32 b([0)p Fq(;)17 b(\031)t Fr(\))803 1414 y Fq(\015)854 1429 y Fo(0)893 1414 y Fr(\()p Fq(E)6 b Fr(\))55 b(=)77 b(lim)1233 1477 y Fn(k)r Fk(!1)1442 1347 y Fr(1)p 1440 1391 55 4 v 1440 1483 a Fq(k)1520 1279 y Fl(Z)1636 1414 y Fq(d\027)6 b Fr(\()p Fq(\022)s Fr(\))p Fi(E)1939 1429 y Fo(0)1995 1414 y Fr(log)2138 1334 y Fl(\000)2184 1414 y Fp(k)p Fq(M)k(T)2409 1373 y Fn(E)2395 1439 y(!)2468 1414 y Fr(\()p Fq(k)s(;)17 b Fr(0\))p Fq(M)2795 1373 y Fk(\000)p Fo(1)2890 1414 y Fq(e)2935 1429 y Fn(\022)2974 1414 y Fp(k)3024 1334 y Fl(\001)3119 1414 y Fr(.)696 b(\(37\))-24 1677 y Fi(Coun)m (terexample:)53 b Fr(The)37 b(con)m(tin)m(uit)m(y)i(condition)d(on)h Fq(\027)42 b Fr(in)37 b(Lemma)g(3)f(cannot)g(b)s(e)h(w)m(eak)m(ened)i (as)d(sho)m(ws)-24 1798 y(the)k(follo)m(wing)f(example.)64 b(Consider)40 b(the)f(p)s(olymer)h(mo)s(del)f(with)g Fq(L)2571 1813 y Fk(\006)2669 1798 y Fr(=)f(3,)j Fq(t)p Fr(\()p Fq(l)r Fr(\))d(=)g(1)h(for)f(all)h Fq(l)i Fp(2)d Fg(Z)i Fr(and)-20 1918 y(^)-53 b Fq(v)23 1933 y Fo(+)130 1918 y Fr(=)46 b(\()300 1879 y Fo(1)p 300 1895 36 4 v 300 1952 a(2)346 1918 y Fq(;)17 b Fr(2)p Fq(;)g Fr(0\))42 b(and)48 b(^)-53 b Fq(v)860 1933 y Fk(\000)967 1918 y Fr(=)46 b(\()p Fp(\000)1214 1879 y Fo(1)p 1214 1895 V 1214 1952 a(2)1260 1918 y Fq(;)17 b Fp(\000)p Fr(2)p Fq(;)g Fr(0\))p Fq(;)43 b Fr(and)h(c)m(ho)s(ose)h Fq(M)58 b Fr(=)47 b Fi(1)p Fr(.)77 b(F)-8 b(or)44 b Fq(E)53 b Fr(=)47 b(0)c(it)h(is)g(easily)i(seen)f(that)-24 2038 y Fq(T)47 2002 y Fo(0)33 2063 y Fk(\006)93 2038 y Fq(e)138 2054 y Fn(\031)r(=)p Fo(2)286 2038 y Fr(=)30 b Fp(\007)479 1999 y Fo(1)p 479 2016 V 479 2073 a(2)525 2038 y Fq(e)570 2054 y Fn(\031)r(=)p Fo(2)722 2038 y Fr(and)k(th)m(us)1141 1999 y Fo(1)p 1139 2016 39 4 v 1139 2073 a Fn(k)1204 2038 y Fr(log)18 b Fp(k)p Fq(T)1468 2002 y Fo(0)1454 2063 y Fn(!)1507 2038 y Fr(\()p Fq(k)s(;)f Fr(0\))p Fq(e)1775 2054 y Fn(\031)r(=)p Fo(2)1892 2038 y Fp(k)30 b Fr(=)g Fp(\000)2165 1999 y Fo(1)p 2165 2016 36 4 v 2165 2073 a(2)2245 2038 y Fr(for)j(all)i Fq(!)i Fr(and)d Fq(k)s Fr(,)h(while)g Fq(\015)3245 2053 y Fo(0)3284 2038 y Fr(\(0\))30 b(=)3555 1999 y Fo(1)p 3555 2016 V 3555 2073 a(2)3600 2038 y Fr(.)49 b(Hence)35 b(a)-24 2159 y(measure)f(ha)m(ving)e(an)g (atom)f(at)h Fq(\022)f Fr(=)1360 2120 y Fn(\031)p 1360 2136 43 4 v 1364 2193 a Fo(2)1445 2159 y Fr(will)h(not)g(satisfy)h (\(37\).)43 b(This)33 b(also)f(pro)m(vides)h(a)f(coun)m(terexample)i (to)-24 2279 y(Lemma)i(3)e(with)h(\(\012)p Fq(;)17 b(T)8 b(;)17 b Fg(Z)p Fq(;)g Fi(P)p Fr(\))35 b(as)g(ab)s(o)m(v)m(e.)50 b(F)-8 b(or)34 b(this)i(one)e(uses)j(that)d(the)h(ev)m(en)m(t)i Fp(f)p Fq(!)19 b Fp(j)e Fq(v)3222 2294 y Fn(!)3272 2279 y Fr(\(0\))31 b(=)k(^)-53 b Fq(v)3582 2294 y Fn(!)3626 2303 y Fj(0)3665 2279 y Fr(\(0\))p Fp(g)34 b Fr(has)-24 2400 y(probabilit)m(y)g(1)p Fq(=)p Fr(3)e(in)h(\012.)-24 2688 y Fc(4.2)136 b(P)l(olymer)46 b(phase)f(shifts)-24 2873 y Fr(F)-8 b(or)23 b Fq(M)35 b Fr(giv)m(en)25 b(b)m(y)g(\(7\),)g (let)f(the)g(p)s(olymer)h(action)f(m)m(ultipliers)i Fq(\032)2294 2837 y Fn(\017)2294 2898 y Fk(\006)2353 2873 y Fr(\()p Fq(\022)s Fr(\))e(and)g(the)g(p)s(olymer)h(phase)f(shifts)h Fp(S)3784 2888 y Fn(\017;)p Fk(\006)3892 2873 y Fr(\()p Fq(\022)s Fr(\))-24 2994 y(b)s(e)31 b(the)f Fq(M)10 b Fr(-mo)s(di\014ed)32 b(Pr)s(\177)-51 b(ufer)29 b(amplitude)j(and)e (phase)h(for)f(the)g Fq(L)2398 3009 y Fk(\006)2458 2994 y Fr(-p)s(erio)s(dic)g(p)s(olymers)h(with)g(initial)g(phase)-24 3114 y Fq(\022)40 b Fr(at)c(0)g(and)g(ev)-5 b(aluated)37 b(at)f Fq(L)1091 3129 y Fk(\006)1187 3114 y Fr(\(i.e.)h(o)m(v)m(er)h(a) e(single)h(p)s(olymer)g(\()2381 3096 y(^)2380 3114 y Fq(t)2415 3129 y Fo(+)2474 3114 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4451 y Fr(,)h(iterate)g(\(38\))f(in)g(order) h(to)f(deduce)793 4859 y(log)935 4778 y Fl(\000)981 4774 y(\015)981 4834 y(\015)1036 4859 y Fq(M)10 b(T)1211 4818 y Fn(E)1263 4826 y Fe(c)1296 4818 y Fo(+)p Fn(\017)1197 4883 y(!)1383 4859 y Fr(\()p Fq(N)5 b(;)17 b Fr(0\))p Fq(M)1739 4818 y Fk(\000)p Fo(1)1834 4859 y Fq(e)1879 4874 y Fn(\022)1918 4774 y Fl(\015)1918 4834 y(\015)1973 4778 y(\001)2074 4859 y Fr(=)2206 4734 y Fn(N)7 b Fk(\000)p Fo(1)2210 4764 y Fl(X)2226 4976 y Fn(l)q Fo(=0)2375 4859 y Fr(log)2518 4778 y Fl(\000)2564 4859 y Fq(\032)2614 4818 y Fn(\017)2614 4883 y(!)2658 4895 y Fe(l)2686 4859 y Fr(\()p Fp(S)2791 4818 y Fn(l)2784 4883 y(\017;!)2883 4859 y Fr(\()p Fq(\022)s Fr(\)\))3045 4778 y Fl(\001)3140 4859 y Fr(,)675 b(\(39\))-24 5169 y(whic)m(h)34 b(com)m(bined)h(with)e (\(36\))f(and)h(\(37\))e(giv)m(es)1947 5564 y(14)p eop %%Page: 15 15 15 14 bop 570 438 a Fq(\015)5 b Fr(\()p Fq(E)736 453 y Fn(c)793 438 y Fr(+)22 b Fq(\017)p Fr(\))55 b(=)1269 371 y(1)p 1192 415 203 4 v 1192 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Fo(\()p Fn(!)r(;l)q Fo(\))3261 895 y Fr(\()p Fq(n)3357 911 y Fo(\()p Fn(!)r(;l)q Fo(\))p Fn(;k)3562 895 y Fr(\))i(=)k(^)-52 b Fq(v)3778 910 y Fn(!)3822 922 y Fe(k)3864 895 y Fr(\(0\))p Fq(;)-24 1016 y Fr(for)34 b(an)m(y)g(c)m(hoice)h(of)i(^)-52 b Fq(v)s(:)34 b Fr(F)-8 b(or)33 b Fq(N)41 b Fp(2)30 b Fg(N)p Fr(,)k(let)g Fq(n)1551 1031 y Fo(\()p Fn(!)r(;l)q Fo(\))p Fn(;k)1790 1016 y Fr(b)s(e)g(the)g(p)s(olymer)h(no)s(de)f(closest)h(to)f Fq(N)10 b Fr(.)47 b(Since)35 b Fp(S)3628 979 y Fn(k)3621 1040 y(\017;!)3720 1016 y Fr(\()p Fq(\022)s Fr(\))23 b Fp(\000)h Fq(\022)-24 1136 y Fr(is)32 b(a)f(rotation)f(n)m(um)m(b)s (er)i(for)f(a)f(matrix)i(whic)m(h)g(arises)g(from)f Fq(H)2273 1151 y Fo(\()p Fn(!)r(;l)q Fo(\))p Fn(;N)2533 1136 y Fr(b)m(y)h(a)f(p)s(erturbation)g(of)g(rank)g(b)s(ounded)-24 1256 y(b)m(y)41 b Fq(C)24 b Fr(max)p Fp(f)p Fq(L)510 1271 y Fk(\000)569 1256 y Fq(;)17 b(L)679 1271 y Fo(+)738 1256 y Fp(g)p Fr(,)42 b(it)e(follo)m(ws)g(that)g Fp(j)p Fq(\022)1584 1214 y Fn(E)1636 1222 y 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Fp(\000)h(S)3012 2121 y Fn(l)3005 2187 y(\017;!)3104 2162 y Fr(\()p Fq(\022)s Fr(\))3228 2081 y Fl(\001)3323 2162 y Fr(.)492 b(\(41\))-24 2517 y Fc(4.3)136 b(Calculation)46 b(of)f(phase)h(shifts)f(and)g(action)g(m) l(ultipliers)-24 2702 y Fr(The)29 b(aim)e(of)g(this)h(paragraph)e(is)i (to)f(calculate)h(the)g(p)s(olymer)g(phase)g(shifts)g(and)g(action)f(m) m(ultipliers)i(needed)-24 2822 y(in)h(\(41\))f(and)g(\(40\))g(in)g (terms)i(of)e(the)g(transmission)i(and)f(re\015ection)g(co)s(e\016cien) m(ts)i(de\014ned)f(in)e(\(11\).)42 b(Because)-24 2943 y(det)q(\()p Fq(M)10 b(T)325 2900 y Fn(E)377 2908 y Fe(c)409 2900 y Fo(+)p Fn(\017)311 2964 y Fk(\006)497 2943 y Fq(M)601 2906 y Fk(\000)p Fo(1)696 2943 y Fr(\))27 b(=)h(1,)k(these)i(co)s (e\016cien)m(ts)h(satisfy)1566 3183 y Fp(j)p Fq(a)1645 3142 y Fn(\017)1645 3208 y Fk(\006)1704 3183 y Fp(j)1732 3142 y Fo(2)1793 3183 y Fp(\000)23 b(j)p Fq(b)1962 3142 y Fn(\017)1962 3208 y Fk(\006)2021 3183 y Fp(j)2049 3142 y Fo(2)2143 3183 y Fr(=)56 b(1)32 b(.)-24 3349 y(A)h(further)g(short)g (calculation)g(sho)m(ws)h(that)1123 3669 y Fq(\032)1173 3628 y Fn(\017)1173 3693 y Fk(\006)1233 3669 y Fr(\()p Fq(\022)s Fr(\))1357 3628 y Fo(2)1452 3669 y Fr(=)55 b(1)22 b(+)g(2)17 b Fp(<)p Fq(e)1952 3588 y Fl(\000)1997 3669 y Fq(a)2048 3628 y Fn(\017)2048 3693 y Fk(\006)2107 3669 y Fq(b)2148 3628 y Fn(\017)2148 3693 y Fk(\006)2224 3669 y Fq(e)2269 3628 y Fo(2)p Fn({\022)2367 3588 y Fl(\001)2434 3669 y Fr(+)22 b(2)p Fp(j)p Fq(b)2650 3628 y Fn(\017)2650 3693 y Fk(\006)2709 3669 y Fp(j)2737 3628 y Fo(2)2809 3669 y Fr(,)1006 b(\(42\))-24 3868 y(and)1309 4157 y Fq(e)1354 4115 y Fn({)p Fo(\()p Fk(S)1447 4124 y Fe(\017;)p Ff(\006)1544 4115 y Fo(\()p Fn(\022)r Fo(\))13 b Fk(\000)f Fn(\022)r Fo(\))1834 4157 y Fr(=)2009 4086 y Fq(a)2060 4050 y Fn(\017)2060 4111 y Fk(\006)2141 4086 y Fr(+)p 2239 4005 101 4 v 22 w Fq(b)2280 4052 y Fn(\017)2280 4108 y Fk(\006)2340 4086 y Fq(e)2385 4050 y Fk(\000)p Fo(2)p Fn({\022)p 1976 4134 595 4 v 1976 4150 a Fl(\014)1976 4210 y(\014)2009 4235 y Fq(a)2060 4200 y Fn(\017)2060 4256 y Fk(\006)2141 4235 y Fr(+)p 2239 4154 101 4 v 22 w Fq(b)2280 4200 y Fn(\017)2280 4256 y Fk(\006)2340 4235 y Fq(e)2385 4206 y Fk(\000)p Fo(2)p Fn({\022)2537 4150 y Fl(\014)2537 4210 y(\014)2613 4157 y Fr(.)123 4440 y(No)m(w)33 b(using)g(the)g(phase)h Fq(\021)1092 4403 y Fn(\017)1088 4464 y Fk(\006)1179 4440 y Fr(of)e Fq(a)1341 4403 y Fn(\017)1341 4464 y Fk(\006)1401 4440 y Fr(,)1479 4759 y Fq(a)1530 4718 y Fn(\017)1530 4784 y Fk(\006)1645 4759 y Fr(=)55 b Fq(e)1821 4718 y Fn({\021)1881 4695 y Fe(\017)1879 4739 y Ff(\006)1958 4759 y Fr(+)22 b Fp(O)s Fr(\()p Fp(j)p Fq(b)2245 4718 y Fn(\017)2245 4784 y Fk(\006)2305 4759 y Fp(j)2333 4718 y Fo(2)2372 4759 y Fr(\))32 b(.)1373 b(\(43\))-24 4959 y(This)34 b(leads)g(to)e(the)h(follo)m(wing)g (expansions:)471 5279 y(log)q(\()p Fq(\032)685 5238 y Fn(\017)685 5304 y Fk(\006)744 5279 y Fr(\()p Fq(\022)s Fr(\))868 5238 y Fo(2)908 5279 y Fr(\))55 b(=)g(2)17 b Fp(<)p Fq(e)1332 5198 y Fl(\000)1377 5279 y Fq(a)1428 5238 y Fn(\017)1428 5304 y Fk(\006)1487 5279 y Fq(b)1528 5238 y Fn(\017)1528 5304 y Fk(\006)1588 5279 y Fq(e)1633 5238 y Fo(2)p Fn({\022)1730 5198 y Fl(\001)1798 5279 y Fr(+)22 b Fp(j)p Fq(b)1965 5238 y Fn(\017)1965 5304 y Fk(\006)2024 5279 y Fp(j)2052 5238 y Fo(2)2114 5279 y Fp(\000)g(<)p Fq(e)2347 5198 y Fl(\000)2393 5279 y Fr(\()p Fq(a)2482 5238 y Fn(\017)2482 5304 y Fk(\006)2541 5279 y Fq(b)2582 5238 y Fn(\017)2582 5304 y Fk(\006)2641 5279 y Fr(\))2679 5238 y Fo(2)2719 5279 y Fq(e)2764 5238 y Fo(4)p Fn({\022)2861 5198 y Fl(\001)2929 5279 y Fr(+)g Fp(O)3126 5198 y Fl(\000)3172 5279 y Fp(j)p Fq(b)3241 5238 y Fn(\017)3241 5304 y Fk(\006)3300 5279 y Fp(j)3328 5238 y Fo(3)3367 5198 y Fl(\001)3462 5279 y Fr(,)353 b(\(44\))1947 5564 y(15)p eop %%Page: 16 16 16 15 bop -24 171 a Fr(and)665 459 y Fq(e)710 418 y Fo(2)p Fn({)p Fo(\()p Fk(S)838 427 y Fe(\017;)p Ff(\006)936 418 y Fo(\()p Fn(\022)r Fo(\))p Fk(\000)p Fn(\022)r Fo(\))1203 459 y Fr(=)55 b Fq(e)1379 418 y Fo(2)p Fn({\021)1474 394 y Fe(\017)1472 438 y Ff(\006)1552 459 y Fr(+)p 1650 378 101 4 v 22 w Fq(b)1691 425 y Fn(\017)1691 481 y Fk(\006)1750 459 y Fq(e)1795 418 y Fn({\021)1855 394 y Fe(\017)1853 438 y Ff(\006)1927 459 y Fq(e)1972 418 y Fk(\000)p Fo(2)p Fn({\022)2147 459 y Fp(\000)22 b Fq(b)2287 418 y Fn(\017)2287 484 y Fk(\006)2347 459 y Fq(e)2392 418 y Fo(3)p Fn({\021)2487 394 y Fe(\017)2485 438 y Ff(\006)2559 459 y Fq(e)2604 418 y Fo(2)p Fn({\022)2724 459 y Fr(+)g Fp(O)2920 378 y Fl(\000)2966 459 y Fp(j)p Fq(b)3035 418 y Fn(\017)3035 484 y Fk(\006)3094 459 y Fp(j)3122 418 y Fo(2)3161 378 y Fl(\001)3256 459 y Fr(.)559 b(\(45\))-24 740 y Fc(4.4)136 b(Oscillatory)46 b(sums)-24 925 y Fi(Prop)s(osition)38 b(2)49 b Fd(L)-5 b(et)35 b Fq(c)897 940 y Fk(\006)984 925 y Fp(2)28 b Fg(C)p Fd(,)34 b Fq(j)g Fr(=)28 b(1)p Fq(;)17 b Fr(2)p Fd(,)34 b(and)g(set)623 1233 y Fq(I)674 1186 y Fn(j)666 1260 y(N)734 1233 y Fr(\()p Fq(\022)s(;)17 b(\017)p Fr(\))55 b(=)g Fi(E)1201 1248 y Fo(0)1240 1233 y Fr(\()p Fq(I)1329 1186 y Fn(j)1321 1260 y(!)r(;N)1454 1233 y Fr(\()p Fq(\022)s(;)17 b(\017)p Fr(\)\))35 b Fd(,)234 b Fq(I)2049 1186 y Fn(j)2041 1260 y(!)r(;N)2175 1233 y Fr(\()p Fq(\022)s(;)17 b(\017)p Fr(\))55 b(=)2568 1109 y Fn(N)7 b Fk(\000)p Fo(1)2573 1139 y Fl(X)2589 1351 y Fn(l)q Fo(=0)2766 1233 y Fq(c)2808 1248 y Fn(!)2852 1260 y Fe(l)2897 1233 y Fq(e)2942 1192 y Fo(2)p Fn({j)t Fk(S)3081 1169 y Fe(l)3076 1209 y(\017;!)3165 1192 y Fo(\()p Fn(\022)r Fo(\))3294 1233 y Fd(.)-24 1492 y(L)-5 b(et)37 b Fq(\017)f Fd(b)-5 b(e)36 b(su\016ciently)g(smal)5 b(l.)48 b(If)35 b Fp(h)p Fq(e)1325 1456 y Fo(2)p Fn({j)t(\021)1451 1465 y Ff(\006)1508 1492 y Fp(i)29 b(6)p Fr(=)h(1)p Fd(,)36 b(then)g Fq(I)2066 1445 y Fn(j)2058 1519 y(N)2125 1492 y Fr(\()p Fq(\022)s(;)17 b(\017)p Fr(\))30 b(=)g Fp(O)s Fr(\()p Fq(N)10 b Fp(j)p Fq(b)2745 1456 y Fn(\017)2745 1517 y Fk(\006)2805 1492 y Fp(j)p Fq(;)33 b Fr(1\))p Fd(.)48 b(If)35 b Fp(h)p Fq(e)3245 1456 y Fo(2)p Fn({j)t(\021)3371 1465 y Ff(\006)3428 1492 y Fp(i)29 b(6)p Fr(=)h(1)36 b Fd(for)g(b)-5 b(oth)-24 1612 y Fq(j)34 b Fr(=)28 b(1)p Fq(;)17 b Fr(2)p Fd(,)961 1898 y Fq(I)1012 1857 y Fo(1)1004 1922 y Fn(N)1071 1898 y Fr(\()p Fq(\022)s(;)g(\017)p Fr(\))56 b(=)f Fq(N)38 b Fp(h)p Fq(c)1662 1913 y Fk(\006)1721 1898 y Fp(i)1850 1830 y(h)p 1889 1749 V Fq(b)1930 1796 y Fn(\017)1930 1852 y Fk(\006)1989 1830 y Fq(e)2034 1792 y Fn({\021)2094 1769 y Fe(\017)2092 1812 y Ff(\006)2149 1830 y Fp(i)p 1797 1875 444 4 v 1797 1970 a Fr(1)22 b Fp(\000)h(h)p Fq(e)2052 1931 y Fo(2)p Fn({\021)2147 1908 y Fe(\017)2145 1952 y Ff(\006)2202 1970 y Fp(i)2273 1898 y Fr(+)f Fp(O)s Fr(\()p Fq(N)10 b Fp(j)p Fq(b)2648 1857 y Fn(\017)2648 1922 y Fk(\006)2707 1898 y Fp(j)2735 1857 y Fo(2)2775 1898 y Fq(;)33 b Fr(1\))h Fd(.)-24 2133 y Fi(Pro)s(of:)44 b Fr(Since)33 b Fp(S)644 2097 y Fn(l)q Fo(+1)637 2158 y Fn(\017;!)761 2133 y Fr(\()p Fq(\022)s Fr(\))28 b(=)g Fp(S)1077 2148 y Fn(\017;!)1170 2160 y Fe(l)1197 2133 y Fr(\()p Fp(S)1302 2097 y Fn(l)1295 2158 y(\017;!)1394 2133 y Fr(\()p Fq(\022)s Fr(\)\))33 b(and)f Fp(S)1845 2097 y Fn(l)1838 2158 y(\017;!)1937 2133 y Fr(\()p Fq(\022)s Fr(\))h(is)g(indep)s(enden)m(t)i(of)d Fq(!)2917 2148 y Fn(l)2942 2133 y Fr(,)h(one)g(gets)117 2490 y Fq(I)168 2448 y Fo(1)160 2514 y Fn(N)227 2490 y Fr(\()p Fq(\022)s(;)17 b(\017)p Fr(\))28 b(=)f Fp(h)p Fq(e)649 2448 y Fo(2)p Fn({\021)744 2425 y Fe(\017)742 2469 y Ff(\006)799 2490 y Fp(i)17 b Fq(I)906 2448 y Fo(1)898 2514 y Fn(N)7 b Fk(\000)p Fo(1)1055 2490 y Fr(\()p Fq(\022)s(;)17 b(\017)p Fr(\))23 b(+)f Fp(h)p Fq(c)1464 2505 y Fk(\006)1522 2490 y Fp(i)17 b Fq(e)1623 2448 y Fo(2)p Fn({\022)1743 2490 y Fr(+)22 b Fp(h)p Fq(c)1922 2505 y Fk(\006)1980 2490 y Fp(i)2036 2365 y Fn(N)7 b Fk(\000)p Fo(1)2040 2395 y Fl(X)2056 2607 y Fn(l)q Fo(=1)2206 2490 y Fi(E)2280 2505 y Fo(0)2335 2379 y Fl(\020)2395 2490 y Fq(e)2440 2448 y Fo(2)p Fn({)p Fk(S)2541 2456 y Fe(\017;!)2625 2474 y(l)2653 2448 y Fo(\()p Fk(S)2728 2425 y Fe(l)2723 2465 y(\017;!)2813 2448 y Fo(\()p Fn(\022)r Fo(\)\))2957 2490 y Fp(\000)22 b Fq(e)3101 2445 y Fo(2)p Fn({)p Fo(\()p Fn(\021)3223 2421 y Fe(\017)3221 2461 y(!)3260 2479 y(l)3290 2445 y Fo(+)p Fk(S)3393 2421 y Fe(l)3388 2461 y(\017;!)3477 2445 y Fo(\()p Fn(\022)r Fo(\)\))3599 2379 y Fl(\021)3675 2490 y Fq(:)140 b Fr(\(46\))-24 2768 y(Equation)37 b(\(45\))e(sho)m(ws) i(that)f Fq(e)1153 2731 y Fo(2)p Fn({)p Fk(S)1254 2739 y Fe(\017;!)1338 2757 y(l)1366 2731 y Fo(\()p Fn(\022)r Fo(\))1485 2768 y Fp(\000)24 b Fq(e)1631 2723 y Fo(2)p Fn({)p Fo(\()p Fn(\021)1753 2699 y Fe(\017)1751 2739 y(!)1790 2757 y(l)1820 2723 y Fo(+)p Fn(\022)r Fo(\))1975 2768 y Fr(=)33 b Fp(O)s Fr(\()p Fp(j)p Fq(b)2273 2731 y Fn(\017)2273 2792 y Fk(\006)2332 2768 y Fp(j)p Fr(\).)52 b(As)37 b Fq(I)2676 2731 y Fo(1)2668 2793 y Fn(N)2735 2768 y Fr(\()p Fq(\022)s(;)17 b(\017)p Fr(\))33 b(=)g Fq(I)3135 2731 y Fo(1)3127 2793 y Fn(N)7 b Fk(\000)p Fo(1)3285 2768 y Fr(\()p Fq(\022)s(;)17 b(\017)p Fr(\))24 b(+)g Fp(O)s Fr(\(1\))36 b(and)-24 2888 y Fp(h)p Fq(e)60 2850 y Fo(2)p Fn({\021)155 2827 y Fe(\017)153 2870 y Ff(\006)211 2888 y Fp(i)48 b(6)p Fr(=)h(1)c(b)m(y)h(h)m(yp)s(othesis,)k (one)c(can)f(solv)m(e)i(for)d Fq(I)2035 2852 y Fo(1)2027 2913 y Fn(N)2095 2888 y Fr(\()p Fq(\022)s(;)17 b(\017)p Fr(\))45 b(whic)m(h)h(directly)h(implies)g(that)d Fq(I)3624 2852 y Fo(1)3616 2913 y Fn(N)3684 2888 y Fr(\()p Fq(\022)s(;)17 b(\017)p Fr(\))49 b(=)-24 3008 y Fp(O)s Fr(\()p Fq(N)10 b Fp(j)p Fq(b)253 2972 y Fn(\017)253 3033 y Fk(\006)313 3008 y Fp(j)17 b Fr(1\).)62 b(Along)39 b(the)h(same)g(lines,)i Fq(I)1560 2972 y Fo(2)1552 3033 y Fn(N)1619 3008 y Fr(\()p Fq(\022)s(;)17 b(\017)p Fr(\))39 b(=)g Fp(O)s Fr(\()p Fq(N)10 b Fp(j)p Fq(b)2257 2972 y Fn(\017)2257 3033 y Fk(\006)2316 3008 y Fp(j)p Fq(;)33 b Fr(1\).)63 b(No)m(w)40 b(insert)g(the)f(expansion)i(\(45\))d(in)-24 3129 y(\(46\).)k(Due)27 b(to)h(the)g(ab)s(o)m(v)m(e,)i(the)e(oscillatory)h(terms)f(in)g(that)g (form)m(ula)g(are)g(then)g(of)g(order)f Fp(O)s Fr(\()p Fq(N)10 b Fp(j)p Fq(b)3567 3093 y Fn(\017)3567 3153 y Fk(\006)3627 3129 y Fp(j)3655 3093 y Fo(2)3694 3129 y Fr(\).)42 b(Th)m(us)-24 3249 y(only)33 b(the)g(non-oscillatory)g(term)g (on)g(the)g(r.h.s.)44 b(of)32 b(\(45\))g(giv)m(es)i(a)f(con)m (tribution)g(to)f(the)h(leading)g(order.)67 b Fb(2)-24 3530 y Fc(4.5)136 b(Asymptotics)45 b(of)g(the)g(IDS)-24 3715 y Fi(Pro)s(of)40 b Fr(of)g(Theorem)h(3:)58 b(F)-8 b(orm)m(ula)40 b(\(45\))f(leads)i(to)e Fp(S)2002 3730 y Fn(\017;)p Fk(\006)2110 3715 y Fr(\()p Fq(\022)s Fr(\))27 b Fp(\000)h Fq(\022)43 b Fr(=)d Fq(\021)2618 3730 y Fk(\006)2705 3715 y Fr(+)27 b Fq(\017d)2898 3730 y Fk(\006)2984 3715 y Fp(\000)h Fq(\017)17 b Fp(=)p Fq(m)p Fr(\()p Fq(c)3382 3730 y Fk(\006)3441 3715 y Fq(e)3486 3679 y Fo(2)p Fn({\022)3584 3715 y Fr(\))27 b(+)g Fp(O)s Fr(\()p Fq(\017)3911 3679 y Fo(2)3951 3715 y Fr(\),)-24 3835 y(where)34 b Fq(d)309 3850 y Fk(\006)396 3835 y Fr(=)j(\()p Fq(@)598 3850 y Fn(\017)631 3835 y Fq(\021)683 3799 y Fn(\017)679 3860 y Fk(\006)738 3835 y Fr(\))776 3751 y Fl(\014)776 3810 y(\014)810 3874 y Fn(\017)p Fo(=0)965 3835 y Fr(and)c Fq(c)1197 3850 y Fk(\006)1283 3835 y Fr(=)38 b(\()p Fq(@)1486 3850 y Fn(\017)1519 3835 y Fq(b)1560 3799 y Fn(\017)1560 3860 y Fk(\006)1619 3835 y Fr(\))1657 3751 y Fl(\014)1657 3810 y(\014)1691 3874 y Fn(\017)p Fo(=0)1830 3835 y Fq(e)1875 3799 y Fn({\021)1933 3808 y Ff(\006)1990 3835 y Fr(.)44 b(Inserting)33 b(this)g(in)g(\(41\))f(yields)194 4192 y Fp(N)15 b Fr(\()p Fq(E)401 4207 y Fn(c)458 4192 y Fr(+)22 b Fq(\017)p Fr(\))28 b(=)881 4125 y(1)p 774 4169 262 4 v 774 4261 a Fq(\031)t Fp(h)p Fq(L)938 4276 y Fk(\006)997 4261 y Fp(i)1063 4022 y Fl( )1141 4192 y Fp(h)p Fq(\021)1228 4207 y Fk(\006)1287 4192 y Fp(i)22 b Fr(+)g Fq(\017)17 b Fp(h)p Fq(d)1592 4207 y Fk(\006)1651 4192 y Fp(i)22 b(\000)g Fq(\017)39 b Fr(lim)1867 4255 y Fn(k)r Fk(!1)2076 4125 y Fr(1)p 2073 4169 55 4 v 2073 4261 a Fq(k)2165 4192 y Fp(=)p Fq(m)28 b Fi(E)2424 4207 y Fo(0)2480 4022 y Fl( )2566 4068 y Fn(k)r Fk(\000)p Fo(1)2559 4098 y Fl(X)2575 4310 y Fn(l)q Fo(=0)2719 4192 y Fq(c)2761 4207 y Fn(!)2805 4219 y Fe(l)2833 4192 y Fq(e)2878 4151 y Fo(2)p Fn({S)2983 4128 y Fe(l)2979 4168 y(\017;!)3070 4151 y Fo(\()p Fn(\022)r Fo(\))3164 4022 y Fl(!)3265 4192 y Fr(+)22 b Fp(O)s Fr(\()p Fq(\017)3522 4151 y Fo(2)3561 4192 y Fr(\))3599 4022 y Fl(!)3727 4192 y Fr(.)-24 4450 y(By)33 b(Prop)s(osition)g(2,)g(the)g(exp)s(ectation)g(of)f(the)h (oscillatory)h(sum)f(is)g(of)f(order)h Fp(O)s Fr(\()p Fq(k)s Fp(j)p Fq(b)3118 4414 y Fn(\017)3118 4475 y Fk(\006)3177 4450 y Fp(j)p Fq(;)17 b Fr(1\))32 b(and)h(th)m(us)1012 4758 y Fp(N)15 b Fr(\()p Fq(E)1219 4773 y Fn(c)1276 4758 y Fr(+)22 b Fq(\017)p Fr(\))28 b(=)1699 4691 y(1)p 1592 4735 262 4 v 1592 4826 a Fq(\031)t Fp(h)p Fq(L)1756 4841 y Fk(\006)1815 4826 y Fp(i)1881 4677 y Fl(\000)1926 4758 y Fp(h)p Fq(\021)2013 4773 y Fk(\006)2072 4758 y Fp(i)22 b Fr(+)g Fq(\017)p Fp(h)p Fq(d)2360 4773 y Fk(\006)2419 4758 y Fp(i)g Fr(+)g Fp(O)s Fr(\()p Fq(\017)2737 4717 y Fo(2)2777 4758 y Fr(\))2815 4677 y Fl(\001)2909 4758 y Fr(.)906 b(\(47\))-24 4989 y(Setting)43 b Fq(p)371 5004 y Fo(+)473 4989 y Fr(=)h(1)e(and)g Fq(p)932 5004 y Fo(+)1035 4989 y Fr(=)h(0)f(yields)h(that)f(in)g(particular)h Fp(N)2414 5004 y Fk(\006)2472 4989 y Fr(\()p Fq(E)2582 5004 y Fn(c)2646 4989 y Fr(+)28 b Fq(\017)p Fr(\))44 b(=)3055 4950 y Fo(1)p 3001 4966 144 4 v 3001 5024 a Fn(\031)r(L)3092 5033 y Ff(\006)3154 4989 y Fr(\()p Fq(\021)3240 5004 y Fk(\006)3327 4989 y Fr(+)29 b Fq(\017d)3522 5004 y Fk(\006)3610 4989 y Fr(+)f Fp(O)s Fr(\()p Fq(\017)3873 4953 y Fo(2)3913 4989 y Fr(\)\),)-24 5123 y(allo)m(wing)33 b(to)f(iden)m(tify)i Fp(N)914 5138 y Fk(\006)973 5123 y Fr(\()p Fq(E)1083 5138 y Fn(c)1117 5123 y Fr(\))28 b(=)g Fq(\021)1335 5138 y Fk(\006)1394 5123 y Fq(=\031)t(L)1568 5138 y Fk(\006)1659 5123 y Fr(and)k Fp(N)1945 5086 y Fk(0)1930 5147 y(\006)1989 5123 y Fr(\()p Fq(E)2099 5138 y Fn(c)2134 5123 y Fr(\))c(=)f Fq(d)2354 5138 y Fk(\006)2413 5123 y Fq(=\031)t(L)2587 5138 y Fk(\006)2646 5123 y Fr(.)43 b(Using)33 b(this)g(to)f(insert)h(for)f Fq(\021)3767 5138 y Fk(\006)3859 5123 y Fr(and)-24 5243 y Fq(d)27 5258 y Fk(\006)119 5243 y Fr(in)g(\(47\))g(completes)j(the)e(pro)s(of.) 2634 b Fb(2)1947 5564 y Fr(16)p eop %%Page: 17 17 17 16 bop -24 171 a Fc(4.6)136 b(Asymptotics)45 b(of)g(the)g(Ly)l (apuno)l(v)g(exp)t(onen)l(t)-24 355 y Fi(Pro)s(of)32 b Fr(of)g(Theorem)i(2:)43 b(Replacing)33 b(\(44\))e(in)m(to)i(\(40\))e (sho)m(ws)j(that)e(2)p Fp(h)p Fq(L)2626 370 y Fk(\006)2685 355 y Fp(i)17 b Fq(\015)5 b Fr(\()p Fq(E)2907 370 y Fn(c)2963 355 y Fr(+)22 b Fq(\017)p Fr(\))32 b(is,)h(up)g(to)f(corrections)-24 476 y(of)g(order)h Fp(O)s Fr(\()p Fp(j)p Fq(b)531 440 y Fn(\017)531 500 y Fk(\006)590 476 y Fp(j)618 440 y Fo(3)658 476 y Fr(\),)f(equal)h(to)g(the)g Fq(\027)6 b Fr(-a)m(v)m(erage)33 b(of)-24 756 y Fp(hj)p Fq(b)84 715 y Fn(\017)84 780 y Fk(\006)143 756 y Fp(j)171 715 y Fo(2)210 756 y Fp(i)r Fr(+)r(2)17 b Fp(<)p Fq(e)529 585 y Fl( )607 756 y Fp(h)p Fq(a)697 715 y Fn(\017)697 780 y Fk(\006)756 756 y Fq(b)797 715 y Fn(\017)797 780 y Fk(\006)856 756 y Fp(i)51 b Fr(lim)912 818 y Fn(N)7 b Fk(!1)1162 688 y Fr(1)p 1142 733 89 4 v 1142 824 a Fq(N)1257 631 y Fn(N)g Fk(\000)p Fo(1)1262 661 y Fl(X)1278 873 y Fn(l)q Fo(=0)1427 756 y Fi(E)1501 771 y Fo(0)1557 645 y Fl(\020)1616 756 y Fq(e)1661 715 y Fo(2)p Fn({)p Fk(S)1767 691 y Fe(l)1762 731 y(\017;!)1853 715 y Fo(\()p Fn(\022)r Fo(\))1946 645 y Fl(\021)2006 585 y(!)2087 756 y Fp(\000)r(<)p Fq(e)2300 585 y Fl( )2379 756 y Fp(h)p Fr(\()p Fq(a)2507 715 y Fn(\017)2507 780 y Fk(\006)2565 756 y Fq(b)2606 715 y Fn(\017)2606 780 y Fk(\006)2666 756 y Fr(\))2704 715 y Fo(2)2743 756 y Fp(i)51 b Fr(lim)2799 818 y Fn(N)7 b Fk(!1)3049 688 y Fr(1)p 3029 733 V 3029 824 a Fq(N)3145 631 y Fn(N)g Fk(\000)p Fo(1)3149 661 y Fl(X)3165 873 y Fn(l)q Fo(=0)3314 756 y Fi(E)3388 771 y Fo(0)3444 645 y Fl(\020)3504 756 y Fq(e)3549 715 y Fo(4)p Fn({)p Fk(S)3655 691 y Fe(l)3650 731 y(\017;!)3740 715 y Fo(\()p Fn(\022)r Fo(\))3834 645 y Fl(\021)3893 585 y(!)3989 756 y Fq(:)-24 1031 y Fr(By)40 b(Prop)s(osition)f(2,)h (the)f(\014rst)h(oscillatory)f(sum)h(has)f(a)g(con)m(tribution)g(of)g (the)g(order)g Fp(O)s Fr(\()p Fp(j)p Fq(b)3424 995 y Fn(\017)3424 1055 y Fk(\006)3483 1031 y Fp(j)3511 995 y Fo(2)3550 1031 y Fr(\))g(\(whic)m(h)h(is)-24 1151 y(giv)m(en)d (there\))f(while)g(the)g(second)g(oscillatory)g(sum)g(is)g(of)f(order)g Fp(O)s Fr(\()p Fp(j)p Fq(b)2611 1115 y Fn(\017)2611 1176 y Fk(\006)2670 1151 y Fp(j)2698 1115 y Fo(3)2737 1151 y Fr(\))g(and)h(can)f(hence)i(b)s(e)e(neglected.)-24 1271 y(Therefore)f(one)f(obtains)453 1635 y Fq(\015)5 b Fr(\()p Fq(E)619 1650 y Fn(c)676 1635 y Fr(+)22 b Fq(\017)p Fr(\))55 b(=)1125 1567 y(1)p 1048 1612 203 4 v 1048 1703 a Fp(h)p Fq(L)1153 1718 y Fk(\006)1212 1703 y Fp(i)1294 1464 y Fl(")1362 1567 y Fr(1)p 1362 1612 49 4 v 1362 1703 a(2)1448 1635 y Fp(hj)p Fq(b)1556 1594 y Fn(\017)1556 1659 y Fk(\006)1615 1635 y Fp(j)1643 1594 y Fo(2)1682 1635 y Fp(i)22 b Fr(+)g Fp(<)p Fq(e)1975 1464 y Fl( )2064 1564 y Fp(h)p Fq(b)2144 1528 y Fn(\017)2144 1589 y Fk(\006)2203 1564 y Fq(e)2248 1526 y Fn({\021)2308 1503 y Fe(\017)2306 1547 y Ff(\006)2363 1564 y Fp(i)27 b(h)p 2468 1483 101 4 v Fq(b)2509 1530 y Fn(\017)2509 1586 y Fk(\006)2569 1564 y Fq(e)2614 1526 y Fn({\021)2674 1503 y Fe(\017)2672 1547 y Ff(\006)2729 1564 y Fp(i)p 2064 1612 704 4 v 2194 1707 a Fr(1)22 b Fp(\000)g(h)p Fq(e)2448 1668 y Fo(2)p Fn({\021)2543 1645 y Fe(\017)2541 1689 y Ff(\006)2599 1707 y Fp(i)2777 1464 y Fl(!#)2937 1635 y Fr(+)g Fp(O)3133 1554 y Fl(\000)3179 1635 y Fp(j)p Fq(b)3248 1594 y Fn(\017)3248 1659 y Fk(\006)3307 1635 y Fp(j)3335 1594 y Fo(3)3374 1554 y Fl(\001)3469 1635 y Fr(.)346 b(\(48\))-24 1897 y(It)31 b(can)f(b)s(e)h(directly)h(v)m(eri\014ed)g(that)e(the)h(giv)m (en)g(leading)g(order)f(term)h(v)-5 b(anishes)32 b(if)e(either)h Fq(p)3314 1912 y Fo(+)3401 1897 y Fr(=)d(0)i(or)g Fq(p)3750 1912 y Fo(+)3836 1897 y Fr(=)e(1,)-24 2017 y(whic)m(h)e(also)e(follo)m (ws)h(from)f(the)h(fact)f(that)g(in)g(this)h(case)g Fq(H)2081 2032 y Fn(!)2156 2017 y Fr(is)f(a)g(p)s(erio)s(dic)h(Jacobi)f(matrix,)j (whose)e(Ly)m(apuno)m(v)-24 2138 y(exp)s(onen)m(t)43 b(v)-5 b(anishes)43 b(in)f(the)g(in)m(terior)g(of)f(its)h(sp)s(ectral)g (bands.)71 b(Next)42 b(rewrite)g(\(48\))f(as)h(a)f(fraction)g(with)-24 2258 y(common)30 b(denominator)f Fp(j)p Fr(1)15 b Fp(\000)g(h)p Fq(e)1196 2220 y Fo(2)p Fn({\021)1291 2197 y Fe(\017)1289 2241 y Ff(\006)1345 2258 y Fp(ij)1412 2222 y Fo(2)1451 2258 y Fr(.)42 b(Since)30 b Fq(p)1820 2273 y Fk(\000)1907 2258 y Fr(=)e(1)15 b Fp(\000)g Fq(p)2216 2273 y Fo(+)2274 2258 y Fr(,)29 b(the)h(n)m(umerator)f(is)h(a)e(p)s(olynomial)h(of)g (degree)-24 2379 y(at)36 b(most)g(3)g(in)g Fq(p)592 2394 y Fo(+)687 2379 y Fr(v)-5 b(anishing)36 b(at)g Fq(p)1293 2394 y Fo(+)1385 2379 y Fr(=)d(0)j(and)g Fq(p)1821 2394 y Fo(+)1913 2379 y Fr(=)d(1.)54 b(Elemen)m(tary)37 b(but)g(length)m(y)g (algebra)e(sho)m(ws)j(that)-24 2499 y(moreo)m(v)m(er)h(its)e(third)h (deriv)-5 b(ativ)m(e)38 b(v)-5 b(anishes)39 b(iden)m(tically)-8 b(.)59 b(Calculating)38 b(the)f(\014rst)h(order)f(deriv)-5 b(ativ)m(e)39 b(allo)m(ws)-24 2619 y(to)33 b(conclude.)3451 b Fb(2)-24 2902 y Fc(4.7)136 b(Commen)l(ts)-24 3087 y Fr(A)52 b(random)f(phase)h(appro)m(ximation)g(consists)h(in)f(supp)s (osing)g(that)f(the)h(incoming)g(phases)g Fp(S)3667 3051 y Fn(l)3660 3112 y(\017;!)3759 3087 y Fr(\()p Fq(\022)s Fr(\))f(in)-24 3208 y(eac)m(h)c(summand)g(of)e(\(40\))g(and)h(\(41\))f (is)h(completely)i(random,)h(that)c(is)i(distributed)g(according)f(to)f (the)-24 3328 y(Leb)s(esgue)39 b(measure.)60 b(It)38 b(can)g(easily)h(b)s(e)f(c)m(hec)m(k)m(ed)j(that)c(one)h(actually)g (obtains)g(the)g(correct)h(answ)m(ers)g(for)-24 3448 y(the)f(deriv)-5 b(ativ)m(es)39 b(of)d(b)s(oth)h(the)g(IDS)g(and)g(the) h(Ly)m(apuno)m(v)g(exp)s(onen)m(t)h(at)d(the)i(critical)f(energy)h (within)g(this)-24 3569 y(appro)m(ximation.)77 b(Ho)m(w)m(ev)m(er,)48 b(the)c(lo)m(w)m(est)h(order)f(non-v)-5 b(anishing)43 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1817 y Fk(\000)p Fo(1)2243 1853 y Fr(,)44 b(one)e(deduces)h(that)f(for)f(all)g Fq(!)47 b Fp(2)c Fr(\012)3684 1817 y Fo(0)3684 1879 y Fn(N)3752 1853 y Fr(\()p Fq(\013)q(;)17 b(\016)t Fr(\))3982 1817 y Fn(c)-24 1974 y Fr(with)42 b Fp(j)p Fq(\016)t Fp(j)e Fq(<)i(N)556 1938 y Fk(\000)p Fn(\013)p Fk(\000)p Fo(1)p Fn(=)p Fo(2)821 1974 y Fr(,)h(norms)e(of)f(the)h(transfer-matrices)h Fq(T)2318 1938 y Fn(E)2370 1946 y Fe(c)2402 1938 y Fo(+)p Fn(\016)2304 1998 y(!)2494 1974 y Fr(\()p Fq(k)s(;)17 b(m)p Fr(\))41 b(are)g(uniformly)g(b)s(ounded)g(b)m(y)h(a)-24 2094 y(constan)m(t,)34 b(not)e(dep)s(enden)m(t)j(on)d Fq(\016)n(:)123 2215 y Fr(No)m(w)h(let)g(\012)556 2230 y Fn(N)623 2215 y Fr(\()p Fq(\013)q(;)17 b(\016)t Fr(\))27 b(=)h Fp(f)p Fr(\()p Fq(!)t(;)17 b(l)r Fr(\))27 b Fp(2)h Fr(\012)17 b Fp(j)27 b Fq(!)k Fp(2)d Fr(\012)1769 2178 y Fo(0)1769 2240 y Fn(N)1837 2215 y Fr(\()p Fq(\013)q(;)17 b(\016)t Fr(\))p Fp(g)9 b Fr(.)43 b(It)33 b(follo)m(ws)g(that)843 2479 y Fi(P)p Fr(\(\012)1028 2494 y 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3049 y Fr(,)-24 3291 y(for)34 b(all)h(\()p Fq(!)t(;)17 b(l)r Fr(\))30 b Fp(2)h Fr(\012)678 3306 y Fn(N)746 3291 y Fr(\()p Fq(\013)q(;)17 b(\016)t Fr(\))976 3255 y Fn(c)1009 3291 y Fr(,)35 b Fp(j)p Fq(\016)t Fp(j)30 b(\024)i Fq(N)1401 3255 y Fk(\000)p Fn(\013)p Fk(\000)p Fo(1)p Fn(=)p Fo(2)1701 3291 y Fr(and)i(0)d Fp(\024)g Fq(m)g Fp(\024)h Fq(k)i Fp(\024)d Fq(N)45 b Fr(\(in)34 b(fact)h(this)g(holds)g(for)f Fq(m;)17 b(k)37 b Fr(up)e(to)-24 3412 y(the)30 b Fq(N)10 b Fr(-th)29 b(p)s(olymer)i(no)s(de\).)42 b(Set)30 b Fq(\017)e Fr(=)f Fq(N)1493 3376 y Fk(\000)p Fn(\013)p Fk(\000)p Fo(1)p Fn(=)p Fo(2)1759 3412 y Fr(.)42 b(The)31 b(theorem)f(then)g(follo)m(ws) g(from)f(Lemma)h(1,)f(b)m(y)h(taking)-24 3532 y(\012)46 3547 y Fn(N)114 3532 y Fr(\()p Fq(\013)q Fr(\))d(=)384 3457 y Fl(S)467 3484 y Fn(N)467 3561 y(k)r Fo(=)p Fk(\000)p Fn(N)699 3532 y Fr(\012)769 3547 y Fn(N)837 3532 y Fr(\()p Fq(\013)q(;)17 b(k)s(\017=)-5 b(N)10 b Fr(\).)2669 b Fb(2)-24 3821 y Fc(5.1)136 b(Correlation)47 b(b)t(ounds:)59 b(Pro)t(of)45 b(of)g(Theorem)g(6)-24 4006 y Fi(Lemma)39 b(4)49 b Fd(L)-5 b(et)38 b Fq(\024)c Fr(=)e Fp(jh)p Fq(e)959 3969 y Fo(2)p Fn({\021)1052 3978 y Ff(\006)1109 4006 y Fp(ij)h Fq(<)g Fr(1)p Fd(.)53 b(Then)37 b(ther)-5 b(e)37 b(exists)h(a)f(c)-5 b(enter)g(e)g(d)38 b(c)-5 b(omplex)36 b(r)-5 b(andom)37 b(variable)g Fq(X)8 b Fr(\()p Fq(!)t Fr(\))-24 4126 y Fd(dep)-5 b(ending)34 b(on)g Fq(!)630 4141 y Fo(1)669 4126 y Fq(;)17 b(:)g(:)g(:)33 b(;)17 b(!)966 4141 y Fn(r)1038 4126 y Fd(such)35 b(that)1245 4367 y Fq(e)1290 4323 y Fo(2)p Fn({)p Fk(S)1396 4300 y Fe(r)1391 4346 y(\016)n(;!)1482 4323 y Fo(\()p Fn(\022)r Fo(\))1631 4367 y Fr(=)55 b Fq(X)8 b Fr(\()p Fq(!)t Fr(\))p Fq(e)2037 4326 y Fo(2)p Fn({\022)2156 4367 y Fr(+)22 b Fp(O)s Fr(\()p Fq(r)s(\016)n(;)17 b(\024)2562 4326 y Fn(r)2600 4367 y Fr(\))35 b Fd(.)-24 4541 y(Mor)-5 b(e)g(over,)35 b Fp(j)p Fq(X)8 b Fr(\()p Fq(!)t Fr(\))p Fp(j)33 b Fd(is)i(uniformly)f(b)-5 b(ounde)g(d)34 b(by)h Fr(2)p Fd(.)-24 4769 y Fi(Pro)s(of:)77 b Fr(Let)49 b(us)h(set)g Fq(\024)914 4784 y Fn(r)952 4769 y Fr(\()p Fq(!)t Fr(\))55 b(=)h(exp)q(\(2)p Fq({)1564 4695 y Fl(P)1669 4721 y Fn(r)1669 4798 y(m)p Fo(=1)1842 4769 y Fq(\021)1890 4784 y Fn(!)1934 4792 y Fe(m)1997 4769 y Fr(\).)93 b(Note)50 b(that)f Fp(j)p Fi(E)2738 4784 y Fo(0)2777 4769 y Fr(\()p Fq(\024)2871 4784 y Fn(r)2909 4769 y Fr(\()p Fq(!)t Fr(\)\))p Fp(j)54 b Fr(=)i Fq(\024)3358 4733 y Fn(r)3396 4769 y Fr(.)94 b(Iteration)49 b(of)-24 4890 y Fq(e)21 4854 y Fo(2)p Fn({)p Fk(S)122 4866 y Fe(\016)n(;)p Ff(\006)221 4854 y Fo(\()p Fn(\022)r Fo(\))342 4890 y Fr(=)28 b Fq(e)491 4854 y Fo(2)p Fn({)p Fo(\()p Fn(\021)611 4863 y Ff(\006)664 4854 y Fo(+)p Fn(\022)r Fo(\))808 4890 y Fr(+)22 b Fp(O)s Fr(\()p Fq(\016)t Fr(\))32 b(and)h(cen)m(tering)h(the)f(random)g(v)-5 b(ariable)32 b Fq(\024)2700 4905 y Fn(r)2739 4890 y Fr(\()p Fq(!)t Fr(\))f(sho)m(ws)994 5131 y Fq(e)1039 5087 y Fo(2)p Fn({)p Fk(S)1145 5064 y Fe(r)1140 5110 y(\016)n(;!)1231 5087 y Fo(\()p Fn(\022)r 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Fr(\()p Fq(\022)s(;)17 b(\016)t Fr(\))55 b(=)1288 575 y Fn(k)r Fk(\000)p Fo(1)1280 605 y Fl(X)1296 817 y Fn(l)q Fo(=0)1440 700 y Fq(c)1482 715 y Fn(!)1526 727 y Fe(l)p Fj(+)p Fe(r)r Fj(+1)1736 700 y Fq(X)8 b Fr(\()p Fq(T)1934 659 y Fn(l)1920 725 y Fo(0)1977 700 y Fq(!)t Fr(\))17 b Fq(e)2142 656 y Fo(2)p Fn({)p Fk(S)2248 633 y Fe(l)2243 679 y(\016)n(;!)2332 656 y Fo(\()p Fn(\022)r Fo(\))2448 700 y Fr(+)22 b Fp(O)s Fr(\()p Fq(r)s(k)s(\016)n(;)34 b(k)s(\024)2979 659 y Fn(r)3017 700 y Fq(;)f(r)s Fr(\))f(.)621 b(\(53\))-24 1018 y(\(Recall)24 b(here)f(that)g Fq(T)773 981 y Fn(l)759 1042 y Fo(0)799 1018 y Fq(!)j Fr(is)e(the)f Fq(l)r Fr(-fold)g(shift)g (of)f Fq(!)t Fr(\).)40 b(Under)24 b(the)f(h)m(yp)s(othesis)i(of)e(the)g (theorem)h(and)f(b)s(ecause)h(of)-24 1138 y(the)h(c)m(hoice)h(of)e Fq(r)s Fr(,)i(the)f(error)g(term)g(in)g(\(53\))f(is)h Fp(O)s Fr(\()p Fq(N)1839 1102 y Fo(1)p Fn(=)p Fo(2)1966 1138 y Fr(log)18 b Fq(N)10 b Fr(\).)41 b(Th)m(us)26 b(it)f(is)g (su\016cien)m(t)h(to)f(pro)m(v)m(e)h(probabilistic)-24 1258 y(estimates)35 b(of)d(the)h(app)s(earing)f(sum,)i(whic)m(h)g(will) f(b)s(e)g(denoted)g(b)m(y)h Fq(Z)2538 1273 y Fn(k)2580 1258 y Fr(\()p Fq(!)t(;)17 b(\016)t Fr(\).)123 1379 y(In)26 b(order)h(to)f(decouple)i(the)f(correlations,)h(divide)g Fp(f)p Fr(0)p Fq(;)17 b(:)g(:)g(:)e(;)i(k)c Fp(\000)d Fr(1)p Fp(g)26 b Fr(in)g(2)p Fq(R)i Fr(pieces)g Fq(I)3159 1394 y Fo(0)3198 1379 y Fq(;)17 b(:)g(:)g(:)33 b(;)17 b(I)3477 1394 y Fo(2)p Fn(R)p Fk(\000)p Fo(1)3686 1379 y Fr(of)26 b(equal)-24 1499 y(length)32 b([)p Fq(k)354 1463 y Fn(\013)404 1499 y Fr(],)g(where)g Fq(R)d Fr(=)f([)p Fq(k)s(=)p Fr(\(2[)p Fq(k)1275 1463 y Fn(\013)1324 1499 y Fr(]\)],)k(i.e.)g Fq(I)1674 1514 y Fn(s)1738 1499 y Fr(=)c Fp(f)p Fq(s)p Fr([)p Fq(k)2019 1463 y Fn(\013)2068 1499 y Fr(])p Fq(;)17 b Fr(\()p Fq(s)i Fr(+)h(1\)[)p Fq(k)2506 1463 y Fn(\013)2555 1499 y Fr(])g Fp(\000)g Fr(1)p Fp(g)p Fr(,)31 b Fq(s)d Fr(=)g(0)p Fq(;)17 b(:)g(:)g(:)32 b(;)17 b Fr(2)p Fq(R)j Fp(\000)g Fr(1.)43 b(Here)32 b([)p Fq(x)p Fr(])-24 1619 y(denotes)d(the)f(largest)g(in)m(teger)g(smaller)h (or)e(equal)h(to)f Fq(x)p Fr(.)43 b(This)28 b(excludes)i Fq(ck)2747 1583 y Fn(\013)2824 1619 y Fr(terms)f(whic)m(h)g(in)e(the)h (follo)m(wing)-24 1740 y(can)33 b(b)s(e)g(absorb)s(ed)g(in)g(the)g (error.)43 b(Set)33 b(for)f Fq(j)i Fr(=)28 b(0)p Fq(;)17 b Fr(1:)406 2048 y Fq(Z)480 2001 y Fn(j)473 2075 y(R)531 2048 y Fr(\()p Fq(!)t(;)g(\016)t Fr(\))54 b(=)948 1924 y Fn(R)p Fk(\000)p Fo(1)948 1954 y Fl(X)959 2163 y Fn(s)p Fo(=0)1109 2048 y Fq(Y)1166 2063 y Fo(2)p Fn(s)p Fo(+)p Fn(j)1325 2048 y Fr(\()p Fq(!)t(;)17 b(\016)t Fr(\))31 b(,)228 b Fq(Y)1900 2063 y Fn(s)1936 2048 y Fr(\()p Fq(!)t(;)17 b(\016)t Fr(\))54 b(=)2353 1954 y Fl(X)2359 2166 y Fn(l)q Fk(2)p Fn(I)2459 2174 y Fe(s)2514 2048 y Fq(c)2556 2063 y Fn(!)2600 2075 y Fe(l)p Fj(+)p Fe(r)r Fj(+1)2810 2048 y Fq(X)8 b Fr(\()p Fq(T)3008 2007 y Fn(l)2994 2073 y Fo(0)3033 2048 y Fq(!)t Fr(\))17 b Fq(e)3198 2005 y Fo(2)p Fn({)p Fk(S)3304 1981 y Fe(l)3299 2028 y(\016)n(;!)3389 2005 y Fo(\()p Fn(\022)r Fo(\))3515 2048 y Fr(.)-24 2316 y(Th)m(us)1075 2656 y Fq(Z)1142 2671 y Fn(k)1184 2656 y Fr(\()p Fq(!)t(;)g(\016)t Fr(\))55 b(=)g Fq(Z)1676 2615 y Fo(0)1669 2681 y Fn(R)1726 2656 y Fr(\()p Fq(!)t(;)17 b(\016)t Fr(\))k(+)h Fq(Z)2151 2615 y Fo(1)2144 2681 y Fn(R)2202 2656 y Fr(\()p Fq(!)t(;)17 b(\016)t Fr(\))k(+)h Fp(O)s Fr(\()p Fq(k)2727 2615 y Fn(\013)2776 2656 y Fr(\))33 b(.)968 b(\(54\))-24 2876 y(The)35 b(random)f(v)-5 b(ariable)34 b Fq(Y)961 2891 y Fn(s)997 2876 y Fr(\()p Fq(!)t(;)17 b(\016)t Fr(\))33 b(satis\014es)i(uniformly)g Fp(j)p Fq(Y)2153 2891 y Fn(s)2189 2876 y Fr(\()p Fq(!)t(;)17 b(\016)t Fr(\))p Fp(j)28 b(\024)i Fq(c)2626 2891 y Fo(1)2682 2876 y Fq(k)2736 2840 y Fn(\013)2786 2876 y Fq(:)j Fr(If)h Fi(E)3019 2891 y Fn(s)3089 2876 y Fr(denotes)h(the)f(a)m(v)m(eraging) -24 2996 y(pro)s(cedure)25 b(\(conditional)f(exp)s(ectation\))h(o)m(v)m (er)g(all)e(random)h(v)-5 b(ariables)25 b Fq(!)2640 3011 y 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Fo(0)684 3830 y Fp(j)17 b Fq(Z)796 3845 y Fn(k)838 3830 y Fr(\()p Fq(!)t(;)g(\016)t Fr(\))26 b Fq(>)i(\025)p Fp(g)p Fr(\))55 b Fp(\024)g Fq(e)1577 3788 y Fk(\000)p Fn(\014)s(\025)1749 3830 y Fi(E)1823 3845 y Fo(0)1862 3830 y Fr(\()p Fq(e)1945 3788 y Fn(\014)s(Z)2036 3800 y Fe(k)2074 3788 y Fo(\()p Fn(!)r(;\016)r Fo(\))2232 3830 y Fr(\))h Fp(\024)f Fq(e)2503 3788 y Fk(\000)p Fn(\014)s(\025)p Fo(+)p Fn(C)5 b(\014)s(k)2834 3765 y Fe(\013)2927 3830 y Fr(max)2929 3890 y Fn(j)t Fo(=0)p Fn(;)p Fo(1)3153 3830 y Fi(E)3227 3845 y Fo(0)3266 3830 y Fr(\()p Fq(e)3349 3788 y Fo(2)p Fn(\014)s(Z)3480 3756 y Fe(j)3475 3812 y(R)3525 3788 y Fo(\()p Fn(!)r(;\016)r Fo(\))3684 3830 y Fr(\))32 b(.)123 4103 y(No)m(w)41 b(if)g Fp(\000)p Fr(1)g Fp(\024)h Fq(Y)63 b Fp(\024)42 b Fr(1,)h(b)m(y)f(con)m(v)m(exit) m(y)h(2)17 b Fq(e)1791 4067 y Fn(\014)s(Y)1936 4103 y Fp(\024)42 b Fr(\(1)28 b Fp(\000)g Fq(Y)21 b Fr(\))p Fq(e)2436 4067 y Fk(\000)p Fn(\014)2566 4103 y Fr(+)28 b(\(1)f(+)h Fq(Y)21 b Fr(\))p Fq(e)3049 4067 y Fn(\014)3096 4103 y Fr(.)68 b(Th)m(us)42 b(if)f Fq(Y)62 b Fr(is)41 b(a)g(real)-24 4224 y(cen)m(tered)35 b(random)e(v)-5 b(ariable,)1191 4564 y Fi(E)p Fr(\()p Fq(e)1348 4523 y Fn(\014)s(Y)16 b Fo(\()p Fn(!)r Fo(\))1553 4564 y Fr(\))55 b Fp(\024)h Fr(\()p Fq(e)1862 4523 y Fk(\000)p Fn(\014)1986 4564 y Fr(+)22 b Fq(e)2129 4523 y Fn(\014)2177 4564 y Fr(\))p Fq(=)p Fr(2)55 b Fp(\024)g Fq(e)2545 4523 y Fn(\014)2588 4499 y Fj(2)2623 4523 y Fn(=)p Fo(2)2730 4564 y Fr(.)1085 b(\(55\))-24 4805 y(one)30 b(ma)m(y)g(assume)h(that)e([)p Fq(k)983 4769 y Fn(\013)1033 4805 y Fr(])e Fq(>)h(r)18 b Fp(\000)e Fr(1)29 b(and)h Fq(k)g Fp(\025)f Fq(N)1896 4740 y Fj(1+)p Fe(\013)p 1896 4753 118 3 v 1939 4794 a Fj(2)2057 4805 y Fr(\(otherwise)h(it)g(is)g(trivially)g(true)g(that)f Fp(j)p Fq(I)3570 4769 y Fo(1)3562 4831 y Fn(!)r(;k)3670 4805 y Fr(\()p Fq(\022)s(;)17 b(\016)t Fr(\))p Fp(j)27 b Fq(<)-24 4943 y(N)64 4907 y Fn(\013)p Fo(+1)p Fn(=)p Fo(2)275 4943 y Fr(\).)65 b(Th)m(us)41 b Fq(Z)733 4896 y Fn(j)726 4970 y(R)p Fk(\000)p Fo(1)874 4943 y Fr(\()p Fq(!)t(;)17 b(\016)t Fr(\))39 b(do)s(es)h(not)f(dep)s(end)i(on)f(the)g Fq(!)2277 4958 y Fn(l)2343 4943 y Fr(with)g Fq(l)i Fp(\025)f Fr(\(2\()p Fq(R)27 b Fp(\000)h Fr(1\))f(+)f Fq(j)6 b Fr(\)[)p Fq(k)3473 4907 y Fn(\013)3523 4943 y Fr(])27 b Fp(\000)h Fr(1)39 b(and)h(a)-24 5063 y(rescaled)34 b(v)m(ersion)h(of)d(\(55\))g(can)g(b)s(e)h(iterativ)m(ely)i(applied)e (to)f(the)h(conditional)g(exp)s(ectations,)i(leading)e(to)1947 5564 y(20)p eop %%Page: 21 21 21 20 bop 546 372 a Fi(E)620 387 y Fo(0)659 372 y Fr(\()p Fq(e)742 331 y Fo(2)p Fn(\014)s(Z)873 298 y Fe(j)868 354 y(R)919 331 y Fo(\()p Fn(!)r(;\016)r Fo(\))1077 372 y Fr(\))83 b Fp(\024)h Fi(E)1433 387 y Fo(0)1488 261 y Fl(\020)1548 372 y Fi(E)1622 387 y Fo(\(2\()p Fn(R)p Fk(\000)p Fo(1\)+)p Fn(j)t Fo(\)[)p Fn(k)2055 368 y Fe(\013)2100 387 y Fo(])p Fk(\000)p Fo(1)2214 372 y Fr(\()p Fq(e)2297 331 y Fo(2)p Fn(\014)s(Y)2416 345 y Fj(2\()p Fe(R)p Ff(\000)p Fj(1\)+)p Fe(j)2698 331 y Fo(\()p 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Fq(G)1450 5229 y Fn(z)1450 5295 y(!)1500 5270 y Fr(\()p Fq(n)p Fr(\))56 b(=)1821 5130 y Fl(\034)1896 5270 y Fq(n)1971 5126 y Fl(\014)1971 5186 y(\014)1971 5245 y(\014)1971 5305 y(\014)2141 5203 y Fr(1)p 2014 5247 303 4 v 2014 5339 a Fq(H)2095 5354 y Fn(!)2167 5339 y Fp(\000)23 b Fq(z)2326 5126 y Fl(\014)2326 5186 y(\014)2326 5245 y(\014)2326 5305 y(\014)2376 5270 y Fr(0)2425 5130 y Fl(\035)2549 5270 y Fr(.)1947 5564 y(21)p eop %%Page: 22 22 22 21 bop -24 171 a Fr(Note)33 b(that)374 505 y Fp(\000)p Fq(t)486 520 y Fn(!)537 505 y Fr(\()p Fq(n)23 b Fr(+)f(1\))17 b Fq(G)935 464 y Fn(z)935 530 y(!)984 505 y Fr(\()p Fq(n)23 b Fr(+)f(1\))f(+)h(\()p Fq(v)1492 520 y Fn(!)1543 505 y Fr(\()p Fq(n)p Fr(\))g Fp(\000)h Fq(E)1871 520 y Fn(c)1928 505 y Fp(\000)g Fq(z)t Fr(\))17 b Fq(G)2209 464 y Fn(z)2209 530 y(!)2259 505 y Fr(\()p Fq(n)p Fr(\))22 b Fp(\000)h Fq(t)2550 520 y Fn(!)2601 505 y Fr(\()p Fq(n)p Fr(\))17 b Fq(G)2829 464 y Fn(z)2829 530 y(!)2879 505 y Fr(\()p Fq(n)22 b Fp(\000)h Fr(1\))55 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Fp(j)284 3400 y Fo(2)351 3436 y Fp(\025)c Fq(C)526 3451 y Fo(6)593 3436 y Fq(>)f Fr(0,)33 b(then)g(it)g(follo)m(ws)g(from)f(\(58\))g(and)h Fp(k)p Fr(\()p Fp(T)2238 3400 y Fn(z)2213 3461 y(!)2278 3436 y Fr(\()p Fq(n;)17 b Fr(1\)\))2543 3400 y Fk(\000)p Fo(1)2637 3436 y Fp(k)27 b Fr(=)h Fp(kT)2947 3400 y Fn(z)2922 3461 y(!)2987 3436 y Fr(\()p Fq(n;)17 b Fr(1\))p Fp(k)32 b Fr(that)977 3700 y(max)1176 3620 y Fl(\010)1234 3700 y Fp(j)p Fq(G)1339 3659 y Fn(z)1339 3725 y(!)1389 3700 y Fr(\()p Fq(n)p Fr(\))p Fp(j)1551 3659 y Fo(2)1590 3700 y Fq(;)17 b Fp(j)p Fq(G)1739 3659 y Fn(z)1739 3725 y(!)1789 3700 y Fr(\()p Fq(n)22 b Fp(\000)h Fr(1\))p Fp(j)2122 3659 y Fo(2)2160 3620 y Fl(\011)2274 3700 y Fp(\025)2605 3633 y Fq(C)2675 3648 y Fo(7)p 2417 3677 486 4 v 2417 3769 a Fp(kT)2546 3740 y Fn(z)2521 3793 y(!)2586 3769 y Fr(\()p Fq(n;)17 b Fr(1\))p Fp(k)2863 3740 y Fo(2)2944 3700 y Fr(.)-24 3928 y(According)41 b(to)e(the)h(ab)s(o)m(v)m(e,)i(as)e (long)g(as)f Fq(\016)44 b Fp(2)c Fr([)p Fp(\000)p Fq(\017;)17 b(\017)p Fr(])41 b(the)f(transfer)g(matrices)h(are)e(b)s(ounded)i(from) e(ab)s(o)m(v)m(e)-24 4048 y(b)m(y)d Fq(C)41 b Fr(as)35 b(long)g(as)f Fq(n)e Fp(\024)f Fr([)p Fq(c)949 4063 y Fo(1)1006 4048 y Fq(T)14 b Fr(],)35 b(in)f(whic)m(h)j(case)e(at)f (least)i(ev)m(ery)g(second)g Fp(j)p Fq(G)2804 4012 y Fn(z)2804 4073 y(!)2854 4048 y Fr(\()p Fq(n)p Fr(\))p Fp(j)3016 4012 y Fo(2)3090 4048 y Fr(is)f(bigger)g(than)f Fq(C)3784 4063 y Fo(7)3823 4048 y Fq(=C)3949 4012 y Fo(2)3989 4048 y Fr(.)-24 4169 y(Replacing)g(this)f(in)m(to)g(\(59\),)604 4433 y Fq(M)698 4448 y Fn(!)r(;q)802 4433 y Fr(\()p Fq(T)14 b Fr(\))44 b Fp(\025)1190 4366 y Fr(1)p 1125 4410 179 4 v 1125 4501 a(2)p Fq(\031)t(T)1452 4339 y Fl(X)1346 4554 y Fo(0)p Fk(\024)p Fn(n)p Fk(\024)p Fo([)p Fn(c)1585 4563 y Fj(1)1631 4554 y Fn(T)10 b Fo(])1718 4433 y Fq(n)1776 4392 y Fn(q)1859 4298 y Fl(Z)1914 4523 y Fo([)p Fk(\000)p Fn(\017;\017)p Fo(])2022 4433 y Fq(d\016)2161 4366 y(C)2231 4381 y Fo(7)p 2158 4410 117 4 v 2158 4501 a Fq(C)2235 4473 y Fo(2)2328 4433 y Fp(\025)45 b Fq(C)2520 4448 y Fo(8)2576 4433 y Fq(T)2647 4392 y Fn(q)2685 4433 y Fq(\017)f Fr(=)g Fq(C)2958 4448 y Fo(8)3014 4433 y Fq(T)3085 4392 y Fn(q)r Fk(\000)3184 4365 y Fj(1)p 3184 4377 31 3 v 3184 4418 a(2)3224 4392 y Fk(\000)p Fn(\013)3361 4433 y Fr(,)-24 4720 y(for)g(some)h(constan)m(t)g Fq(C)868 4735 y Fo(8)955 4720 y Fq(>)j Fr(0.)78 b(If,)47 b(on)d(the)h(other)f (hand)h Fp(j)p Fq(G)2326 4684 y Fn(z)2326 4744 y(!)2376 4720 y Fr(\()p Fp(\000)p Fr(1\))p Fp(j)2606 4684 y Fo(2)2693 4720 y Fp(\025)j Fq(C)2888 4735 y Fo(6)2974 4720 y Fq(>)g Fr(0,)f(then)e(one)f(gets)h(this)-24 4840 y(estimate)29 b(in)f(the)h(same)f(w)m(a)m(y)-8 b(,)30 b(but)e(based)h(on)f(\(57\))f (instead)i(of)e(\(58\).)41 b(This)29 b(uses)h(the)e(fact)g(that)f(the)h (analysis)-24 4960 y(of)38 b(Section)g(5)g(can)g(also)g(b)s(e)f (carried)i(out)e(for)h Fp(T)1791 4924 y Fn(E)1843 4932 y Fe(c)1875 4924 y Fo(+)p Fn(\016)1766 4985 y(!)1968 4960 y Fr(\()p Fq(n;)17 b Fr(0\))37 b(with)h(negativ)m(e)h Fq(n)p Fr(.)59 b(A)38 b(Borel-Can)m(telli)h(lemma)-24 5081 y(sho)m(ws)c(that)d(a.s.)h Fq(\014)703 5045 y Fk(\000)697 5105 y Fn(!)r(;q)829 5081 y Fp(\025)28 b Fr(1)22 b Fp(\000)g Fr(\()1152 5042 y Fo(1)p 1152 5058 36 4 v 1152 5115 a(2)1220 5081 y Fp(\000)g Fq(\013)q Fr(\))p Fq(=q)t Fr(.)43 b(Since)34 b Fq(\013)28 b(>)f Fr(0)33 b(is)g(arbitrary)-8 b(,)32 b(this)i(\014nishes)g(the)f(pro)s(of.)343 b Fb(2)-24 5248 y Fi(Pro)s(of)33 b Fr(of)f(Theorem)i(1.)43 b(F)-8 b(ollo)m(w)32 b(the)h(ab)s(o)m(v)m(e)h(argumen)m(t)f(b)m(y)h(using)f (the)g(deterministic)i(Prop)s(osition)e(1.)84 b Fb(2)1947 5564 y Fr(22)p eop %%Page: 23 23 23 22 bop -24 171 a Fs(References)-24 390 y Fr([BT])50 b(J.-M.)36 b(Barbaroux,)g(S.)g(Tc)m(heremc)m(han)m(tsev,)41 b Fm(Univ)m(ersal)c(lo)m(w)m(er)g(b)s(ounds)f(for)f(quan)m(tum)i (di\013usion)p Fr(,)h(J.)177 510 y(F)-8 b(unct.)33 b(Anal.)g Fi(168)p Fr(,)f(327-354)f(\(1999\).)-24 710 y([BG])49 b(S.)40 b(de)g(Bi)m(\022)-46 b(evre,)43 b(F.)d(Germinet,)i Fm(Dynamical)f(Lo)s(calization)e(for)g(the)i(Random)f(Dimer)f(Sc)m (hr\177)-49 b(odinger)177 830 y(Op)s(erator)p Fr(,)32 b(J.)h(Stat.)f(Ph)m(ys.)j Fi(98)p Fr(,)d(1135-1148)f(\(2000\).)-24 1030 y([BL])49 b(P)-8 b(.)36 b(Bougerol,)h(J.)f(Lacroix,)h Fm(Pro)s(ducts)f(of)f(Random)h(Matrices)h(with)f(Applications)h(to)e (Sc)m(hr\177)-49 b(odinger)177 1151 y(Op)s(erators)p Fr(,)32 b(\(Birkh\177)-49 b(auser,)35 b(Boston,)d(1985\).)-24 1351 y([Bo)m(v])50 b(A.)31 b(Bo)m(vier,)h Fm(P)m(erturbation)g(theory)g (for)e(the)h(random)g(dimer)h(mo)s(del)p Fr(,)g(J.)f(Ph)m(ys.)h(A:)f (Math.)h(Gen.)f Fi(25)p Fr(,)177 1471 y(1021-1029)f(\(1992\).)-24 1671 y([BJ])50 b(J.)42 b(Bourgain,)j(S.)d(Jitomirsk)-5 b(a)m(y)m(a,)46 b Fm(Anderson)e(lo)s(calization)e(for)g(the)g(band)h (mo)s(del)p Fr(,)i(in:)63 b(Geometric)177 1791 y(asp)s(ects)31 b(of)f(functional)h(analysis,)h(Lecture)g(Notes)f(in)f(Math.)h Fi(1745)p Fr(,)g(67-79,)e(Springer,)j(Berlin,)g(2000.)-24 1991 y([CK])50 b(M.)45 b(Campanino,)k(A.)c(Klein,)k Fm(Anomalies)e(in)e (the)g(one-dimensional)i(Anderson)f(mo)s(del)g(at)e(w)m(eak)177 2112 y(disorder)p Fr(,)33 b(Comm)m(un.)i(Math.)d(Ph)m(ys.)j Fi(130)p Fr(,)e(441-456)d(\(1990\).)-24 2312 y([CKM])50 b(R.)c(Carmona,)k(A.)c(Klein,)51 b(F.)45 b(Martinelli,)51 b Fm(Anderson)c(lo)s(calization)g(for)e(Bernoulli)i(and)g(other)177 2432 y(singular)33 b(p)s(oten)m(tials)p Fr(,)g(Comm.)h(Math.)f(Ph)m (ys.)h Fi(108)p Fr(,)f(41-66)e(\(1987\).)-24 2632 y([CS])50 b(V.)44 b(Ch)m(ulaevsky)-8 b(,)50 b(T.)45 b(Sp)s(encer,)j Fm(P)m(ositiv)m(e)e(Ly)m(apuno)m(v)g(Exp)s(onen)m(ts)g(for)d (Deterministic)j(P)m(oten)m(tials)p Fr(,)177 2752 y(Comm)m(un.)34 b(Math.)f(Ph)m(ys.)h Fi(168)p Fr(,)f(455-466)e(\(1995\).)-24 2952 y([DSS1])49 b(D.)31 b(Damanik,)h(R.)f(Sims,)i(G.)e(Stolz,)h Fm(Lo)s(calization)g(of)f(one)h(dimensional,)h(con)m(tin)m(uum,)g (Bernoulli-)177 3073 y(Anderson)h(mo)s(dels)p Fr(,)f(to)f(app)s(ear)h (in)g(Duk)m(e)g(Math.)g(J..)-24 3273 y([DSS2])49 b(D.)32 b(Damanik,)h(R.)f(Sims,)i(G.)e(Stolz,)h(in)g(preparation.)-24 3473 y([DT])49 b(D.)32 b(Damanik,)h(S.)g(Tc)m(heremc)m(han)m(tsev,)k (in)c(preparation.)-24 3673 y([RJLS])49 b(R.)c(del)g(Rio,)i(S.)d (Jitomirsk)-5 b(a)m(y)m(a,)49 b(Y.)c(Last,)i(B.)e(Simon,)j Fm(Op)s(erators)c(with)h(singular)g(con)m(tin)m(uous)177 3793 y(sp)s(ectrum:)g(IV.)32 b(Hausdor\013)h(dimension,)h(rank-one)f(p) s(erturbations)g(and)f(lo)s(calization)p Fr(,)g(J.)h(d'Analyse)177 3913 y(Math.)g Fi(69)p Fr(,)f(153-200)f(\(1996\).)-24 4113 y([D)m(WP])50 b(D.)42 b(H.)g(Dunlap,)j(H.-L.)d(W)-8 b(u,)46 b(P)-8 b(.)42 b(W.)h(Phillips,)k Fm(Absence)d(of)e(Lo)s (calization)g(in)h(Random-Dimer)177 4234 y(Mo)s(del)p Fr(,)33 b(Ph)m(ys.)h(Rev.)g(Lett.)e Fi(65)p Fr(,)h(88-91)e(\(1990\).) -24 4434 y([F)-8 b(ur])49 b(H.)30 b(F)-8 b(ursten)m(b)s(erg,)31 b Fm(Noncomm)m(uting)f(random)g(pro)s(ducts)p Fr(,)h(T)-8 b(rans.)30 b(Amer.)g(Math.)g(So)s(c.)f Fi(108)p Fr(,)i(377-428)177 4554 y(\(1963\).)-24 4754 y([GK])49 b(F.)25 b(Germinet,)j(A.)d(Klein,)j Fm(Bo)s(otstrap)d(Multiscale)i(Analysis)h(and)d(Lo)s(calization)h(in)f (Random)h(Media)p Fr(,)177 4874 y(Comm)m(un.)34 b(Math.)f(Ph)m(ys.)h Fi(222)f Fr(415-448)e(\(2001\).)-24 5074 y([Gua])49 b(I.)24 b(Guarneri,)i Fm(Sp)s(ectral)f(prop)s(erties)g(of)f(quan)m(tum)h (di\013usion)g(on)f(discrete)i(lattices)p Fr(,)h(Europh)m(ys.)f(Lett.,) 177 5195 y Fi(10)p Fr(,)f(95-100,)f(\(1989\);)i Fm(On)d(an)h(estimate)g (concerning)h(quan)m(tum)g(di\013usion)f(in)g(the)g(presence)i(of)d(a)g (fractal)177 5315 y(sp)s(ectrum)34 b Fr(,)e(Europh)m(ys.)j(Lett.,)e Fi(21)p Fr(,)g(729-733,)d(\(1993\).)1947 5564 y(23)p eop %%Page: 24 24 24 23 bop -24 171 a Fr([GSB])49 b(I.)d(Guarneri,)j(H.)c(Sc)m(h)m (ulz-Baldes,)52 b Fm(In)m(termitten)m(t)47 b(lo)m(w)m(er)g(b)s(ound)e (on)h(quan)m(tum)h(di\013usion)p Fr(,)i(Lett.)177 291 y(Math.)33 b(Ph)m(ys.)h Fi(49)p Fr(,)f(317-324)e(\(1999\).)-24 494 y([JSS])50 b(S.)34 b(Jitomirsk)-5 b(a)m(y)m(a,)36 b(H.)e(Sc)m(h)m(ulz-Baldes,)j(G.)c(Stolz,)i(Delo)s(calization)f(in)g(p) s(olymer)g(mo)s(dels.)h(Preprin)m(t.)177 615 y Fa(mp-arc)53 b(02-1)p Fr(.)-24 818 y([KS])c(V.)33 b(Kostrykin,)h(R.)e(Sc)m(hrader,)h Fm(Global)f(b)s(ounds)h(for)f(the)h(Ly)m(apuno)m(v)h(exp)s(onen)m(t)g (and)e(the)h(in)m(tegrated)177 939 y(densit)m(y)h(of)f(states)g(of)f (random)h(Sc)m(hr\177)-49 b(odinger)34 b(op)s(erators)e(in)h(one)g (dimension)p Fr(,)h(J.)f(Ph)m(ys.)h(A)f Fi(33)p Fr(,)g(8231{)177 1059 y(8240)f(\(2000\).)-24 1262 y([PTB])50 b(A.)d(P)m(arisini,)52 b(L.)47 b(T)-8 b(arricone,)51 b(V.)c(Bellani,)k(G.)46 b(P)m(arra)m(vicini,)53 b(E.)47 b(Diez,)j(F)d(Dominguez-Adame,)177 1383 y(R.)c(Hey)-8 b(,)46 b Fm(Electronic)f(structure)f(and)g(v)m (ertical)g(transp)s(ort)f(in)g(random)g(dimer)h Fr(GaAs-Al)3601 1398 y Fn(x)3645 1383 y Fr(Ga)3771 1398 y Fo(1)p Fk(\000)p Fn(x)3904 1383 y Fr(As)177 1503 y Fm(sup)s(erlattices)p Fr(,)34 b(Ph)m(ys.)h(Rev.)e Fi(B)k(63)p Fr(,)c(1653218)e(\(2001\).)-24 1706 y([PF])49 b(L.)g(P)m(astur,)k(A.)c(Figotin,)j Fm(Sp)s(ectra)c(of)g (Random)h(and)f(Almost-P)m(erio)s(dic)i(Op)s(erators)p Fr(,)i(\(Springer,)177 1827 y(Berlin,)33 b(1992\).)-24 2030 y([PW])50 b(P)-8 b(.)25 b(Phillips,)j(H.-L.)d(W)-8 b(u,)27 b Fm(Lo)s(calization)d(and)h(Its)h(Absence:)41 b(A)25 b(New)h(Metallic)g(State)f(for)g(Conducting)177 2151 y(P)m(olymers)p Fr(,)35 b(Science)f Fi(252)p Fr(,)f(1805-1812)d (\(1992\).)-24 2354 y([SB])50 b(H.)40 b(Sc)m(h)m(ulz-Baldes,)45 b(J.)c(Bellissard,)j Fm(Anomalous)d(transp)s(ort:)59 b(a)40 b(mathematical)h(framew)m(ork)p Fr(,)j(Rev.)177 2474 y(Math.)33 b(Ph)m(ys.)h Fi(10)p Fr(,)f(1-46)e(\(1998\).)-24 2678 y([Sim])50 b(B.)33 b(Simon,)g Fm(Absence)h(of)e(ballistic)i (motion)p Fr(,)f(Comm.)g(Math.)g(Ph)m(ys.)i Fi(134)p Fr(,)d(209-212)f(\(1990\).)-24 2881 y([Sim2])50 b(B.)22 b(Simon,)j Fm(Bounded)e(eigenfunctions)h(and)f(absolutely)h(con)m(tin)m (uous)g(sp)s(ectra)f(for)e(one-dimensional)177 3002 y(Sc)m(hro)s (edinger)34 b(op)s(erators)p Fr(,)e(Pro)s(c.)h(Amer.)g(Math.)g(So)s(c.) g Fi(124)p Fr(,)g(3361{3369)d(\(1996\))-24 3205 y([SVW])50 b(C.)k(Sh)m(ubin,)60 b(R.)53 b(V)-8 b(akilian,)59 b(T.)54 b(W)-8 b(ol\013,)58 b Fm(Some)c(harmonic)g(analysis)h(questions)g (suggested)g(b)m(y)177 3325 y(Anderson-Bernoulli)34 b(mo)s(dels)p Fr(,)g(Geom.)e(F)-8 b(unct.)33 b(Anal.)65 b Fi(8)33 b Fr(932-964)d(\(1998\).)1947 5564 y(24)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0206170231461--