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a0a2e1770c861a0415cffcd49dfb62b65e1abc7d6c1bc2640abfee00c11ee585 737bc0a7e7b0f0bc1d2b4b6e62456bb6be2db7a96308c75833908f08b0430c7a ec607de6afa959a1f8070954b566fc06b3d71845761ff8aa7bee20ef5ce4f26f 8bbd164da010d3e22dc5562a1a7e0510ba319305f93dd77a2e950105d35e7848 0f0f84cc80c4e34edb00af9215524ac882729631ee045b47c233f1eec1c99ba4 efd06b0802e67c9ef3930ebcf0e3376fb7bd73e1c6baff10621003b0137b329e aea825ee7c84d21ca546bcd4b0966524711c546aa733c0e162f5dfc9bcf2b954 2e6fdc323d5d4548cbe2810f8fb58c693c0410924d3b218b8b452b19b7a331fc 9981271a2c9fb3eee5428f9adf59f9f7d6b40b69c7a49af4861b651b1106419b 6512f0faec343870353d9dca17bcab94be34d7eb40aacf92a8d981383e3d691e 73d0554f7e198abdbae094b0dc67b6540964253a1d918b88d61193354e330025 96acbe71bb9590bcf30a050057f79f3502c2408ae0c7819971ba0cdb59bd8aed 2817154e3189db5f12a2a3523a255630d7adb82bcaa224ddd4ccda372a984265 c9 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % 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 (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /square put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueXX 5031981 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6f33457e3200f0935ea98f1a0cfaff49cfa2 d877ff805620cba84126dfd175a82e1b4c6ec58461fce97cdd749a012e2c4243 86b199a97c306b282350cbe5af90cf1d8e7371dcaedbd5fa3346dd3e558072c5 fd51a2edfee6890f62efd1b552699123f4f52c2817dadd5ece30b3bc178a3ef4 f45fd8ad0ee5bf30db5f9eee9ddefe249f3c349a886cf9130cba68ae559ad719 c439930ece1d52d38366116c390cbeabb8fdb36811cf3b8e2416825e8c880b24 6521ec6880ebe7f127c8741c756b29e1f89e3fd5d638a7bbade8c3a0fdbbf7ba ec37cd372afb2d9dd25d509726d20a05914e418f6b7e759c440beb61519df02a 06734ec1226890d17c51f25f775e6879fdc231bc4d7fcb4b33d04fa273bdb8e8 f124171714614f8a94e98a6c2d6bdc1254913999f9c181e42c0886556cdf48af 92a265c1eb06bb30fb2745840c17bd31c7b8bdfd37dfb9643a418a2eaae14094 6752e0a315207aef85c3c2630538907971ec1b05687208571a3504bb3369b5c8 9a43e5abe343d0599fd8d41a1bf01de49f5856bdd67d63a2b3ecea75d721a6ca 0c8751ab41e892e9cf674218dd44acfd40019bf59f4d69ea40d5cf9068a6ff96 9f8cc772b920baf17334833d732431a2969b9a30fe38bd44772cef653935a957 af61dd568f041847ccce3c587656cba7bbf46d8cefec936c6d34dbe51bc04929 0440354f3b29ed8e6b4f21ad53b5809779d67339f284563d54ed4d2a504b014d 974a31c0ceb57b364af519bad4e85c41d521a3ffc26dbac555b42dc191367152 1d60c30b95b5bbafb3c7d87a1c043ea5f0e6dab83a4de99a742e75f02837b923 c50eb478ac4131ee604e3d7b784bd8c93eb9430291d452c21be0495718546333 66a602a9f142a563fdae3c8d40de6a8aa581ff3172367ca325e6d23102c5c6a2 02cc99580287dbd76a05bb4cca591b12a9dd3d602d7957376ec30b16051a63c7 e26ddf009e21fdca5a31cc973f0b1f6967a66b30b973ddd9d9a3095b673ea8f8 5db8dbf3acb7bb2d85c04a0ea8853e4592e4f8e581adaadb78bc3ae58a2d4c80 3b09839eafc7bc3e7cdf784e852d6590ab1c4b7558e28505887c6858072f09dc 2caba91b6871920e5256b602a0de02b87e5eb456878d3634e8eb0e069fa2bb5c c10d5cd98401de1cf07ec9f7de62aaa20b583e2f7aad2fb7e727ebc43f712c74 e8eed1fa76133428ac71589b933cc37ad200c516045fc1a06d0086c541103b5d 4ec54dc77b 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSS12 %!PS-AdobeFont-1.1: CMSS12 1.0 %%CreationDate: 1991 Aug 20 17:33:47 % 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 (CMSS12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMSS12 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 77 /M put readonly def /FontBBox{-62 -251 978 758}readonly def /UniqueXX 5000804 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cddfaf2ed604ab4eafe1b34aa4 31dcb3f73c18fcc8e8014e34597896a814c48cab18fd9dc77f4ae3457ce2499b d24f4aa1976bde782344d6dc811f065f385688b7538ce76d0f5dcfb8fe458d13 888ad60eebdbb9cc6a025850c506e45069aa14b5e3c38665bfbe4a9286dde935 692b76af24bdc584e298eea841d45d9eb87ac61187def799331f88ab8ccaad72 e5bdd6ea94580161d4666a64d5b8f76c12736a38e956909f626892039a1ede98 f694485b5ea34576e5a2bdbb58b8c14ec6724b6a3678db4365afd61f276beecd c501c230d0827f4c8925b42f5f1be2de179d0e99954f2c258310e1d7dd1bcf50 00b2eeddb59ff8d39c999e25f71765f252cd6d490ec7fc64de93e2564f66aa27 2a795dbc634db4d437644b80248b159d9fe87167af12d23ec43245d4529a9abe 93b9a60847b5fd4651dc67da13d1bae2682c531c439ae77017dc6c262a84a3ec 3fa6157090d02388e68dcb5b17482fc97f5b4cc143cb206103a09e1753e7331e 8701a112af1c5622c4688ac0674569c19c8b9b027d73aee03d3e51eed237ecbe 597276e6a19bffb51bf5e169c00e4434159439456087af1371b571bcbd53384b 60fec89664d72ff71b6cd992abc6e5469218f853925df72d7ff171be74dcef6a ad23ddada04bd0e5a5fe71d65f0420a32823990d6f8277a73e5dfe48596fc627 3c908f77f31c7d0610621dee3e1286f8e2ccf3d6660a9d74ed93d1baab063d49 e61a1cdeed78bb46732abdc2484aabf7dd725d9fc6e3c0963a83396a250ba2ad 19432e151446f7d5db35cbb0ed6bc156049e60651ae834acb96b724d32a544a4 822c77575ef321ba3852a701573103f27174a5303b6245babec520efe0992b7e 4a14d4648a784602ea2b24a396f618e1fe4d7cecbef5c0e5f06bc8f6f9fe0e79 b9e8a4e7d81483bbbdbb344049529fe687df8dd4f24e268f8436808ed90790e1 24d71e706a9b1543b15e9a19186594bc1040f798e59535ba42c03e0eb6e9372c 17f9f054f0fa5e221494e6a936b12a0c342fdff00738f2c8a15e7f3700fb5f22 e4a82f51e041797ae5aac5973bb86f4a6c49c97fe6acbc282b9f60ab205d41c5 12cbc38b61c96ab24902bc87fe1dda0980727ea01edc9f39bfb7297d3712d846 0bd3d4950d2be0406595f7d2e1b784367c6825ec77982dfccbd18b3a98155899 7cf1b03f64123a0c6d08432bd83e1e056248efea3b73649250d2c08fa8163fc0 7aa47b808969aded4635894386d7c6a04b9e6698061961c5e5220a1899cebdd2 1402156e05a93ad59a0608d77b24 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTT12 %!PS-AdobeFont-1.1: CMTT12 1.0 %%CreationDate: 1991 Aug 20 16:45:46 % 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 (CMTT12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch true def end readonly def /FontName /CMTT12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 46 /period put dup 47 /slash put dup 49 /one put dup 50 /two put dup 58 /colon put dup 65 /A put dup 67 /C put dup 69 /E 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 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 122 /z put dup 126 /asciitilde put readonly def /FontBBox{-1 -234 524 695}readonly def /UniqueXX 5000833 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5b667eed85fade30ab2ec4e1c2f 4fe750cf9a69506458071de8896261c001b1c290937fa42a1fa87de406e9f86a 4c9c0f03f69fba719114f8ae51040a4c03a58b6f3de027f6726f32743dce7395 14c2c6f3c2de1aeddcfa8eed7da56dd68dbcde33c29ff7a694769049e4325ec9 61b2df16add2ad46f18781fbd63eec62c26f1db41e6a666a09886382dceba91d 8ff7cedd64ddc8a2f15fe34076a064c46febd1f4dc3707aede0ce8af161bcb8e 4ada202d0954baca5be71c28ab8bf32fd72029aa07f27d658a68f454a4bf386f 6314224d16c7f0a8624f6b86319a4e2fd178f596f3b20db032ea28c157a68a69 fad2bc88881658d2286c0d4ef061bc070c3145f2a37ebec03f51de7cda34cd97 7eeebd86b00f9758b3a496f41d9a74279ef3c6322dc54945f95062c5f37a1e51 cf7eaf45f2b3634b2d7d01ec3b298ff35b6616cd3020e096f6c6aa2bf04c4a93 61c19f3c67a28d936ffc8120f1e46980b29bf54047c5f1fa48e67611a74fa434 3384f802a4079b28b94681e119ab0c1692de5bbf426440ad1023506cb80732de c64d9061869cb34de924d2c26e404b7dbd9be55b9c0c5ae3095c15d278adbb49 12df1484ed2d73128e5d9c1f1a3b9ac19e2ce5366c96644f5b3d3eea589caedb 586fd9b8f4df32ec910e9b7434d5bbe57e03892eb03890567df5b811afa0da5e 929d456af362fda0789aa0cff464e238029294fde86cfc8d7d33d5c98b18c76c 319811e4d3d49560f2ac39103594ec513e0b1377c2c3e22979866c19b61448d5 a7201ca4b7538aa370f6347253b0853d4c5d575a7572f93cb73a598e2e26b11a d46a1126e44e7e1795dff01562316b6ccc6678663e66a73db285746df3f4405d d51714679581946ef723e22ddedd85ee8a03c1f444ce1e8a69db9dfa0b79efef da9bc49d9ab4eb085bab097e8919ad16461f271dcff326864fad6c71572376e8 c69b50b6569e4c1c7e48baef0b53da66a0ad4fe2b3978d094746797941c9d7f2 1986ca1dfdd3420922e73cffcb544576a8649fc31679aa8b8540b8555c18fd97 84f047d000db978e6bda8266a4855f9e7d39f27d886c1e77ecc76ba18781d4ff 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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 15 /bullet put dup 48 /prime put dup 49 /infinity 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 c43e28519169701856d797b0108d338a1ac7ca5e7ce82bfdaab14993f4291d24 3e7620ad5c7ba78ac94704719bcee0989af4528504a2b239d9315fb537f4d6ba 2fb3ed0c12619bf6762334d9a02a757f59bd6a3100cc8d5ed885ec5a36fa28e4 8ddc71f212c9fa837473c317b532920e0a6566a29f2d9c4033ef2e8e71c53751 6c39f2d93e6d9b09a438e6385d2ca42c13a36bb1a96862eae0ad3059f8e8da8d 3327c12e42f7981244331f6beb047c2ffa32f258b56004316db48757a704bae8 46830dcad23d158534f5aab0874f1a010b26169303de456293b185e93c42a92a 760e54116aaf315e6da77182f75bd9f971a6e986d5fc0be410058dec543fbff2 b4b880addf0076868d80a8c29a4a4da2e144ffbc7e5cf1c6edcb48199c4e9263 1c2a467a72f68cabb5100e1ea3a11962db424c84f3534e6bea6585c77dca4651 b9a1c114a0d7dbaa8a8ac6b4c3c92bba91da8cbfec9b78fd60c87e04cdefc7f5 c1ab0cc02b14159772e768ede96fc3dc854969dc2da68d8c1c9818e2aae9fc9c bb172abad6989c8d88cb851f08c7b06a3bb6c438adbd00a4e3cdc21932fd0e2a febdfdb5bfa1fb0f072c52c3c702e5b516d59cb6767566b5d9e3d404fe6aae07 6a20874630ce113041c28e601ff9a6a7432eda6ae8313757836391a87320b2c4 be9c52d1721899c129e9c12c03764c2494454f7447044eeb3fee09069d3c2906 89291d3884db00b214bf2429a7a17c18ae1f51a0a6b63c0992c516d9f8f775e7 6b1fe83db985fd3a35ffa492f9fd8ee77ea5d78864f190adf9f60a3a26b53f 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 11 /ff put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 61 /equal put dup 65 /A put dup 67 /C put dup 68 /D put dup 70 /F put dup 76 /L put dup 78 /N put dup 80 /P put dup 81 /Q put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 105 /i put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 120 /x put dup 121 /y 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 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ced170e675c08bc135a8c09b7e5e5f9983b119226b639fc40a9d687033c18d28 dbdd5835f251fc37d7ba25362821498c4bf0cbf54ce87d338c802dd117e35b2b 43c434d8b655816163dd1816a3e0043988efb452e76ce69a4c9fea193120d56e 3076fd7b90f3ac619e32ad90218b865e4d7b763ee6631b4826e3013d79642ec0 4459bd65b5450d5da7c6b6b6c142eca4a6da5ad1a75f99e1dc75694674c8685a 6fb94ec0d1f13577d17d41954aa5bb2d37ad28bec7855e7033f4592007681c06 e7ce3303a571a3835a60b5c0a6cc3be795c3b0e400c9a88f4c7d00b43cc517af cc38ea0fd7d08604af59c61178ec36f379f7969696128c20659df9dcb52073fb bd949601c43ae1265a10310558d993a21c1daf8528b468442c40ef3fc3fae633 a94185f63ac1626ed2b5a2aa12e22e8c99adbd50fe8234c3058939adbe9bada4 007ae62277ceeee315961552921b77f285f043bfd86e17c602326dbc38dd13f4 7901bc7021667af244a0345dea9ed4c4a5956b9b4d0da87f931b7dec952fd393 6cb9abb9dc32820959bd1b47dec46e15c6db96512d1f0d1884cda1c099e06627 4fc60f6ac2259489b86be76fc4eeca264b9e4fc16f9faf0d7fa4e0073b044e85 1a4f52183c91ff40c4a4074fde7a6a7f4464d544c59c7fedc9d2adfce14ebdcf 4883f904930f459a9a788991ae55cf5bcd943df2c1c9d9d75b086ef84cdb4c42 9927500bec2c09b608306ea47b33b47c7decb252a66e7179bc2dc9590f5c6bf0 92f0cb9e732db95fef0ad0a25bfdc35d1d68b3bb3341e44e020c6f3eae26403e 133ae7d5219fffeba2a1942699f096ac600045723f53713764409f9cbcfaee50 691ecca8cd8067c50a169b9d37b335019543ad1e7ebf2402581f0094cf9191c5 f50c88f394f9d464999f48f653fd7cf346e59e5f9336519cdd40d8c13dfc44c7 13585bb027e9f9ece788d5ab84c5e0a45edaeecb07dd8005c06bc77bf4458550 718b5564c6f4c02a8b88720f2186998c0639ae525aa4a41f877ffad8f9f501c1 f8630135c81835d9ca3059151bd71c07 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 17 /eta put dup 18 /theta put dup 61 /slash put dup 76 /L put dup 77 /M put dup 78 /N put dup 98 /b put dup 102 /f put dup 105 /i put dup 106 /j put dup 110 /n put dup 112 /p put dup 119 /w put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueXX 5087381 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a1e67ed7e41cc25acc04702f6 8ef703e3d45722c1a91fdef7100a48631a02a6f02a08c6b1f9b4df8310385b86 8632718fd87119a233f219d9411383b7fa9f3e4780d8c27e2e89e0cae883d664 c3eac57a3aef8988a2e9f0f8c7f53e0a80bdfc4620e21287d0390e1975398544 7f3ea66401024bea75e1b4c4437b7bb188f76f96b918ac7c6ad7e8ae7f21d8c2 790f08cccec904fe48ef39e597ed4d4237c1d1f596f5906b19ea308020f7a35c 168e327ec3246b1dfabe912f6b6daac09974876d3996e57d180261110db05f15 e3e8eebba3d90b5764c03df3033a1ed678ebc679569a2fb297378b25434c0f20 5313ecb8a952f07242d3ee731b0cdc086a4481178a3d65129c47c09b22e9c431 e11b3747b94c26a757c38d06001798c6a568303d541385244b967d3b1786edea f65bb53c4c2fe75e4b1b15c2c78d930b4296c80f08bad86012451edc8e9f0854 c3b390a16e27b11b3d45a9f72eff8baded2242dc928a61685d79e09681c97425 5b90a498614cf560fa5b1718981388268ba206a96989e6d0b5d485d9aca5594a e67dd7b34d8a369adb06647f8aff8814d6d9cdc04a4835918e557174c5bc0f3f bcea9907a04cf93c12727ec40db3f2f77596dca477862747435bdedacd9b2311 6cc97fa47ffdd7d897fb6bdd5572e35d34e7e1cb5e7273a4ffd86525323ace4a 84e1297028c2bd5469baa2e75d19360c2c9042139d5e7dd4390a6a3935424711 de21910126d750ae279916ceb71da3591d60dc62db333c5021e2c1cd61ade51e 93920049e7883cc4907b86ec4b734d2b94e73a93db52d853a90943a7b1e55264 0c18f2e833062bb526b171c5711e9b93d428fef64238c13b9c999dfe879b1bfb 4edb5a5a7992dd4c49556dea19c3fa03c9d8c7dcf51e2c12188fb5e53cd5c36c 94059ed4c7afe9a3eb9c33a9e58617544c9214d2226d554149ab320a6272a766 1aa6eacb8caa64cf105c713b1bf07e5e7545478ef9e515d934825021d5ee0790 2429093e9d1e89b8f94b8f872603ba626fb1e0d3690883acf6eb1f009d82fa27 e1285a189b394aec163a8b58503a7d6dc26fbfea2b2576d6e82fdc5c961c62a2 52fd40e7d8cf19de8a00625d44591b4d970c7ad55887711d277bad4eab0d28f9 ef62ee68c45eafb880e664f86c0eaa65eac91e32b80f2cf633bbffab6180a63c 03aa49afe0b2ebf6d140f97c40d0e60cbad1946b7929aac6197669d1a6e45289 8db0a7e939b8223973506683eb0789ca7836c24c2b12d08dfa433b07effb7555 2cdde9d44818829b95371e2ecab8e1daca1d7c8cd01a820df3262a6fd468384e 24a81ab1211d7455a566ca4628b88d26f0a52e361c84030e627e193395cce495 c0884b3c56066e1f6e603eaa5297e18eda26ec4e4fddeafa346270de07404f70 765de0015a705cc4a35b99e1bbfe7184c183d30c7a2fdfe6ea1e47bdf254305c a4ebf19e02be447ce8999c8b7e165a8e96dceaaa27363d183719d0cc09641062 c9dfb064b08c88a22e127848d9d67decdbde3b29b0466046b8f894c02aa8fece d7373546bd49cbf9e6cbaf6d3773c79fa95ae187b8d6f93edd7d7fe0c21d9e53 8b0448645fc046c92eab4da51e38cdff874a1a5f165636a7e9799ddc7faaaa3f 6a3c39d2c4c96d9d215d77163514044c6337b471055eed6a83d4a6183806fff2 9f626bd3abde83334aabd9c3e0897d302d5a138ef68de4172dd072209d3c18a9 a82c6db4ed512e02708f56a870591d478719f2a701fb5acf9461001827b3d1f5 96929a71f1a63f69ba41d1cf38dfaa22fe5b6b48488aa1c1d13f145f1ebc9c5c 3af3da3dbb0e3b86e9f0de41838acce73f796411bd07774099e3ef9d3ccd9045 a5ec38cccd48f0e7d2a1b4efa970cb43611ebe9c079c58807e5b7d4ca86c07c9 1f0d5108c2b3fe91f8daa5747fb875263f9c954b9b692ce840f12a605371abcf fcd7f01a34ea5cab3e6ad600ce1666166bcf501be5094662bcbea858bcb669ca 11162a8ef29436cdca7f964cd781fa38e3b9545b871afc9208db220a2eca2f76 1a31457cc49d8994405349f2c519862ec59c06a0d2315be16be7a34a7d8c2caa 2d5bb5af04235835b2232293ff15a0b13bf9760866f13d5866b190355b0dcf85 ade45a22a2ad9b079863cb7fa779c3431209010c00c313497d540c76a0397f04 4e47d9fd8f73d1179c98609584cf6c42553f6843085ff88e8f8c20cb04fe9238 9b6fa56d14cb7e12cb9880e980ca8841bd0570b0b1bf336a019981c6c1b94b8a 1c3b92c05409aa7d25484b3f52fc4804d23730f05f63255c7055d94bca7e7294 9aa16afc5d3dd642f0d374cdc67266f5212bb646b90e93c2e64a5d4432794680 42c5ef38c1e6baec70cb8b9ff93861a5ab33d03d28d1cc0f6f830cebdada06a9 c66293e51d6290bca5f5d764243e3695a98c0a6a41a203acdfecba64bded0488 6a7b3e7db3209d50d70aa9d453f4b2ec7198ccd67c70013004b3a93df05c6de1 e52bba4750ae1a2e451ce8541a8ca97fd56fae35728ce4129d385a51286dfa7b 39207983dd08f757af2b98f40cb6c102a6981939cd65b4081bc0c8426a27b3d0 2b9dca231681a8881877a44f75cfb31b3919d79482e81136f81db511b32200ff 63b8aff06c579b8181e2b99332ddeea6a59974002b9c5f46437dae7c754030e6 9c3973cf67e570a3f903bbcc1efec513ae1d374ade29b77e75869e92a4550601 adbdf9de95265b44df9f83bb1e53032288ebe13410a612513bfb0a63da869ca0 86fd0085612774c69fd75449a838a388925c41d8c63c1255b8460502ae274477 47c479158d686a6dd4b0b3f24f49e98b976d103a93ecc883af75e5fc92ee6898 84edc5e3d4f6a0a878c7b9032174abd719e1083a7bcbcaa48a71ba7785f96c1b c3a44c01cc3a53804dd50005ca098dca1ba4579b7106d1f9ca12a76ac0a24c2e be0d9e2cfec99fccd43cf1471b3031005f2a6a92b3f32b4b9d58f061cb5c11c3 6952d094449cae4a2f1f89a6020d6cb175aef2da6e59040063a8f0ac5308079d 9d22e7e0d2b4eb0192ca8ab312285e215ce529441d8624337fc813a326f246ce dbc9cb318ef67a06c69e1761f96312f2757d575ef0a288e8e97add5da6be25da 4ff08386d0f240f86e5ef9fcdacc2c4686b12546987f83a4036c3c839dd19e4f 3d4453223cf81a87a8c50a87d3d97543cfa3ddf485f03ff77b2b8b6c1768f322 6c5862b868dad1b1b9eb618e353c0ed3116beb29a5fcb02f73fdb7edd3ad879e e82b0bc4206f65fe5ade080637fcc0852dd5404d062ec5a955fb50c2a32a47d2 51ffe10c1ef885541ac38bcf411c6398a23d4a6affa243f36f1e59a1d61875ba c4ccb91c0e4aa08f180d121965e585ffc2807ca62485b9fd2a36991d60f30d03 68a12bb62b4132c5d7cac2af3e7a6b00276340375e00400d84e6a50c47053c56 70b6e964e4cbb4c5646b81a70c4d8882f4c027de6967605d7d1f9f90d77e9aa2 476697ab6ecefa592ea0cc986ae9b263f7a109cbaf989bd0134f7c55b3877837 637c2f63b28aee12394515116efb48e7e73ffaacb2c365e45f2713151b4f30fc f2de6c8404a670367f9875bc1297ce11d8be9fddb61e20cf5303961987aaf33e a0707bacb452b8b5267a3944e37956700187a153f8e7c00f393fdfbe4cbc8f92 3411951280d560bcf1b0441c6d8e16606c9e66e3454e24f28e762ad34a111ef0 0dd23cd7b31889cebf7872d1ff05fdd8f749efca20ed37a99396d60d1989002c f941b1e2b982a5289c4f0eb380ca08acb3c6acc9a3d092a9eef3eee647f5bb9c b01d987a33632a7b3a6d4c94f9a557f1ef401f6b41792933c10f7b20d5b54fd1 3b39dffdc497cce30e76a717e67be29da72cfc18fa27e25a9edd731691d4a823 2e51f57855717441de8bd4c3ab0014b56119da22a1e470a8da843b3d679aad3a 635f83ebc4ac3850189a2220c01c06c65dfb07bc977167dd1833b63c31100589 c66b2fa39a8218c78e1b971a9a64f58c507228d9bc03f78abb510315aed4a5e2 349ea3c84a988ee281a8802169bf977104a5f13a4bd14021342df8e620b719c1 f5a51fb4ea33d053463041380f64ed19e15e18e45c8b37651e123aeea809a787 e2a6fe758b3250f8f649e4cbb691c97a1ef258945b572da61ab755ae54bb5757 fbf9f4cfaf0eea079de185ec70956584bd390bff4ddece8ed8b8ebf3ffd2c15c 02a714566b24b918354476ec01608332d6f8c7f9413e0c57c9ed1731d9449d32 2336a291705cd54825a488ef65e5544526da5b61613ef378676e0216e4de6db5 ed962a3fcce0a80bac99800fb1fc9923932f1aaa571c7de981c8fe825621e004 e22a0f0bd28b9810535611af2168bb62731f2bd43dfcbc6075a6f42c76470a37 bed55a9a58162f0f932ec7266a72fe2ed9655a2c898243104efcc80d9c624fdb 4865b95ce82fc6893772923b2a247faa8aefa098d64875cd8111c83979ebec97 8eb4991de370620b68b527cfed03251a4657f254bda92a2427c38c0d4afc8c0e 3ec0f0aad8cc8278cd208dcfa705a939927c224daeafed8c97c01e9e6e1662b9 3a793c7503d79bcf904b70f155b86cdc73137d2468945e16f214705d277a29eb 4b9425c59a87c9043433fabf0a301efe75ddc15a21085d5bf95ea5145f629158 a099c03bb1d55a8f072c7acf2826896d66dd47aceb035fbbac945659f0467d48 ff2f999d3729ddb010a0b7678d9f8c3b3c02cc5b03f8df65f81e8d4f038b00e6 fcdd90347164eb7eb5a8d01d2100813cb5a4b3c293f16fffdae97ece546b48cd 298f1cc577afbaa34fa611f98aa5cf6842cd2dfa617748272f00c48527e61b36 bee1e838e1d533e6ff7c12361fac6282d97cd66eb888069cc79e39fb5ee9ba2b 87808ab71ac82596d2909e6869ad767a1beb23eafe313ad2e1f4d3df1497db78 44062dc89816805e09026747a01e620b00d21e75bf412e424a29e8c469a23e2c dee4fe5948fd361f22feb293775b2df19d2da21efa25756e69b3b4322baa3710 98ca74182e1cbd571732cb14814ce00cb230dbbbd444ad8b15f0eed2a1516d44 926f44de3fa54b7b67df5a2a846ea9f7317c9aba0e209f6347f3039d1a05680e a5fce34436cb7ce2dfab9a22fe8df262bf1d0273699ae73a55d6cf58a43c0860 60dd 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 19 /acute put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 59 /semicolon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K 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 87 /W put dup 89 /Y put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 127 /dieresis put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueXX 5000769 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5bfae5c1f476ee6f05eb1d2caeb 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17afb6daea46ad42336b57c864fe3d6f74c937c0a23d7bb5826903c7694310b8 e2250a4df013d161213ff3ea43c4737ddb67a2e2640229361954834b88dd7817 4daec031121edd7d94461edbb6da79ffa3954fe1973d37f88b02a69c33d90aaf 3169a9f1381ab77aeeb9904ceff967ae5af48332e8995ca72396b2f8001ef5cb 7d3a3be1ed04880269efaa 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 12 /vextendsingle put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 77 /circleplusdisplay put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 112 /radicalbig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueXX 5000774 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d5649664203ab24acb938d58d246707ffe7 d62f04bec4b70c21ef75beb2b812622b3c74e969d72d3cd11bd7106294a99caf 0b1629bc7d4de6b96ca82930831d64575f23f4ad06a0e45e315b1d392411be8d 6d73c998789ff258a07a3c8c2057325784514c845500bfd1a971310cfc11d41c 1a167dbd5ff012c60add4e87325f6e5299032a839de65fb1473a166aae1876a4 414a434f22c1d241591fb36f857df6fa930608750ffc0c54f44994662b1f00f1 400bf752ea8d83ffc4cb77a290bc2d99981ae59a191748ba5c7ba1a9d2583fd2 1398452b6ff5d83a059f7eadcd2ef744e9dd22bdf9c79d049bf06835e878c32b 7765c69bdd8ef4deb4ea7cfff4cf9354a4ddffa689de961d16772491c7afbd7f 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 19 /acute put dup 30 /OE put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 127 /dieresis put readonly def /FontBBox{-36 -251 1103 750}readonly def /UniqueXX 5000829 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd803271dc8bb1656323089e9e2bb50f2b95088 87d31dc36e555668d9578b338402f8c259f6813b0b71e6cbd95e19c20ced8495 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0ad793b35ff4eb248021f64b69667ed0464c1dacd92c1878c2e2e3e57065ec02 09c3c0cd424bd2fe537772347020ae96d89805ebf2e0c05a87a84b7befb6c8ad 03b235ad823a86a68e93b3811b7ef6958be25735eb5596e2c5bbb47d05ff956f 1c583f33723957a45a0918909406df1f01cb2ee57718a541ca130ac04616ad9e e4a2ad581e4042426a2eb08fefd8ba520eed32701ba71b037639414a9a36f51f faa85b41728326b8467c27f698d7ad3a56e3e3bf864db5c20699e48bccff5fe1 a46265633c3d09f896252c8a50717ca9bb49cbaefe901c751ff4b20f10b691dc 2f0520a83176a8af5868c2e5060ab807eb2980628d7bc6aaf2f20fc39ac7ce70 8f075dc31f2575b0c39eef4c9a86d5db0b47c41130838647eee11916c04831cc cf70d96871ac124084f17de24dc60f094213cfaeac3241818cce6d52d056873f 372c36cbb3bc2b1db0d738a0edcb2b8d76d7c1436492278723b34d9dc9980357 066623501e003373c5c71a96865acc14e9eb130d7aaa5d5c752ec51365c0018c a25af52ff953befa5ab0e2312ce2212d977ade28ca7b77a9e420be29eb1cacc1 046501cee7e0fc00a26842dc6f4efc4638ae9bccae175173da26b56d15d3382b 93c35a0ddf94697fb4cd487b99d0d7cbbac8cf0e8428bcff4431ab641dab18a5 be21e710ba76c1d2949bcdf5b37d2278a7563939f8f5a048213093666a1c7941 f9967d57b44e6095bfde63f46be7dfb9ea927b5fbe41bccf3aafde95393afc9b 97bc8a8cdd99270dd280048ce29968641517f19c943903e55203dd05ba74ae6a dc791d66766830913e2c005e7476744df60c28b804e32eefe1c81f4db2ea818b bd6e2aa7166d619237af4cc65bc52a66e26d9e9160ea797eb6bb38c995690156 31e4fd8e12b49d48661524a5544ab38305ab5fba296d6add6f39bcdbb1eb4875 f3ac4a128a9b32f83310a93260af0bd59f544ae116bfedfc40853bdc9d0944c2 6baa1767559f5c5372a2d9cf75689baeca726fec634fd5e41203ad7d25e10c3b 0462b36f133cfc3b82b1f084106b3f323a7e6e516c4345f7e03a1d7617447fe8 98bf1235a1ea4d10c820227dd1b558c6b9d79f4b27ceaab2a034550afb7ffb5a 4ef1da5dc0abd2c2de316d53130be37255f72644d2b05d8378b67ff2f22fc350 98fc82e2fccc6710d9bbd0abc25e87865dca43877bd9366db01db29dbf1f3312 f90ddbac727e7e860332c013f3568d584e77dba6e8dd3b7ae53df6177f44725b 5abd7f00d4bf795247ae9f249d9b8986f53cc379a4b1f6127361b730799cc393 b8f559830980de23d3f4d3b955dd85ca6180a18b1bf400f4b6e7c7ce92e49507 8956f2cdc415f0c1f804482d3caa95f87bc33687be45a8f5b9ddd98fd771d181 62d802972df197f47753171d9c8001cdcdd512dcd6ae05a62e7cdeb16f3995d1 a3fd07d4a41243735926d4785b46480b4d434855d1a2321663d420a30068a410 2fa56e1ad2bc3272 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 78 /N put dup 81 /Q put dup 82 /R put dup 84 /T put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueXX 5031982 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d56491bdecdcafdc988206c5a457a19270b 37d0ab776e03eaa7eb568eeab6b5e79dec03b0dcbf923a2aa8e4f4deda2cb043 858d8b430efef0ed914a80ffc9818aba0fb30dae3694e5b31df3855892d59b89 7f82ff79fce8444c7926ffc4ea008e63bc518936c9098afc2d1c14030a2c15bc c9285c1b57e80570d2f6301009e63d9ddeba44e3251c75ca9616a2c5de9ade66 1a15db37b40b67c1bd71f77343912c62f47dd24be0e03dd833ea60d2eb020407 eaf042fed30f0c2f01956c83d46d982b50dd979c2026ed5f71979dd98c9c1eee 25054000a5eed5ce9498b2cb59a7ee55e0538ab47d86e69b2ac79229c8c6bfc9 3a24e822a2f1c9964fe630344579a25f66646113f73d1315c447f070c575de76 266c6773b01137e4a039df5e4c2f199e8e3396ef9fd001eebe090e04384c4089 ec9f13a07ad8733c6af775f432f154ef50ec4477e08760c03c006e4d1eb1c7cd acb8955ca4b0d62c7de046ff58d431bb9883d28586552c51809e3ced4dfc2198 b97ad9e52ea33c91b9e335a80d9b77d3430e0487930f992459664abca5dd9177 150b0fb502fbb04ccaa94ffe2697ed07906f95fd98e7b325bfbf10ea6b39fd76 c1e5295022cb39e9cd471b7b0b3f9d72da99deb0615d35cc22f42dbf5cfb70dc 6279fd520d99f32201abf25e621cfe1d37961d9c0ad36ef3d9264c547638152a be30812b1188a10f6327c9e8714449de2ae636420f56a44cebd774f0b131c629 818455cc8dd8d40939f3ba8aa5abc54590c7286b172193c23adb1b4f9931a9f7 73b960b1f5b760438028478e4442dc93a567b07e20017c18fca2a1daa2649dee 5a6ec51fd6c3f4b27367c31e95a56ba4bbeda0493bc02306037e2ba69add9b94 373364db316691e73d6811cd76fbc92b0c67ca18dcdf05807f17d1b309456ea9 d113ea0fccd6eb73bd8dfdbd5905a5e38bbf61cfe58fb01ec95a4bdd9565433d 4a7abb8dca6f76c45b254635c3352c04b4ac72a87290d2af4efd3744481b5ea9 3d29504fb127e5885efe3dea4fee790f512861a5b404024f63ae4c4a6da49ef5 053060eea39bf2fa7a41665f3a61fe68d59b908f1e3b48cd21cd03a6638eae9f 62c61dbd191574b40e7414c359553deadfa82ccd746c8ea5f1f67299018da48c 9e943c0a3b6b1fa3792acdc8b2614566907a4ae582a5f25746b4583bf801df42 ce894904ed3e72499990f4a5f822a429bd91588d2ef55f69c6ba3ef60579bc78 f39a9d4075ee1a7b6103211d408e7f10a1d475b7598ef74c738340407f863a9d 410168e7488dba753aff94df055daf303a29bb6ed3ece67ce6533f782c59df2a ee42e16df33bdb7517ca41faf4af81923ef1e7064f4659c85b4d55ea28c7b26c 5289024b2261f65046189199d636ef6a069533180153fe16da003c57b8f5a457 e15f005ae48637da25c17bd8daf7d82d036328daa5aa3356b263c7a21061c752 ff22f499cdcb1c1df77631e2f599f404ce80da6747a57b104c57c885795ed3f9 49ebf0dad6ed0ebf52cdb4b308c12a26a4a08641f83ffa7a2849ee62bf3871cd df87556b4e8549bb0532e878440c02874ef410fe97e3e5a870ad1a573beafe8b 87208f07e2b8300ee69d7a644d71065dffdf8dfe00542f765d0fe3ffd8dcc8d0 cd883e5e6179758eff31ad81f43c059221c834b08ce0989212377ad7388f19ce 15e0222e1f82dc1167b482a29c847a21907529eacc7767fab3df96080b2af118 f58a043a0e2514433cb8904f3e11eec67bba04422c8f27eb77d26f10ab3e81d7 43a9fa9db69f8feed5e77d6918ea633b2d9d3d95f0060d25a1234df2d6087728 ea769e8372fccbb551df3610a72192244433381b90260b1087ee721876a23dde 7267a9cae5740e101218681b00e852a005d7541bdf09d48c77d09cbf809a0b2e 0643034b153268bf4b2a4bd4f6284facef07f80777482c523a331a4e65036153 e998b4511fe347b21be1483cba2ef847ad92163e3119d3aa725ad2e5aa9def2b 810c8925f649d2210c90d95af6fdc12a7f155b84369af1b5c4d480b2f232a260 0e71f4a37d90fbeb3a4408dfc8da33499cc9ad99e56c87f6266f106cfa9d5697 5ef1a5272f428430865520f433c7a7b52614d7d05130881c1a432324ab845166 8f4e43039639af9f820299f4a8739cfc159b95c82be93ccff106e9bc740940f0 2c9151c612950413a00feee691e7a4903b049e1770cdc3438939aac1dc7e38d0 59a1aed472e0c0868f2c299f07d1f11a78d9ba73d9d78f8aa52e189ac018dcca e32fb6876c6f38a7ac0ac739d18e7c9e473d73fe77e90cf89616535df5276049 892110e7c3aa3bcf0bd68468e3efeeeee5181571778165a545d8ed2c32913c4f 1765d9d89aed0d85f34fff2d49b19fdcd8df45599100f1a37e47dfde08de499a 51997e0d5c055e08d7bfec3a37eecad5faf408d7df8f9e2f96d1d829297c3f4d 3025b2d43e4ea1034e7d5d433d0b168b80c8f241054977be3af6d46638fda8e9 32d6e751ee91bde19bb767259fe0028b6f7d9994fe9f35d5efac72c80249db8a 28935c98b715191de37aee6fc073c4676dc0464ba9e963dd6cfad20bdbf29a26 3516780469e315420ed10435573e08307c94ed7923bedc3a7d978ebbc647a273 6b95ba4853fdc08c2ef712b468e70e96027993404bdc8c578bdbceca5a123a6d aba850d33ec257d0372c66f96add4be186c00ea82dd43024e5491ebaa93cc059 7b48705078f98adef7747a27d5932cfa6194e4a678b4836b64616d85c6baf9c7 dfffbb33fcc10594a9f516cfb5598dc46720a3ab00bade54da93188ed46127e5 c7e0588edf4fd1988425ea3ef0a3cc1d77d38667fc1b3597cacfb5192600b013 4d94a20e1d24ad69bbeff760d822d498e5b63e2aa0e4025984a8fdf4c891a124 cb213cee3d4e864010da0f4acfeec8617f8b8cead73d4b021ff05462ae39550a dc5f36dc9b 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 15 /bullet put dup 20 /lessequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 106 /bar put dup 112 /radical 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 b910bf03a427298b3c19530c7546a0c1d19509c200db96985216165bc2a1c458 e24670c9f68e76065209e2c047b6df32d1d005cc3a7969e60fb54112a57deed3 81b0d50ae434865f0ceb8861a75459e915fe3097fcc16958f253edc5de15a94d d8ac2c60c556cff50c95b73df6ca2b769c95fbb217a48efb1b2674203d146b66 a6ca35c68af79010eee771e9be20967849e49756015a96783f8a015a9790997c 8f270131e4d1682a403850fe51622e717fa14611ed060f58077de591068f8aac a97157ad36114d4d83c619a060c3afb625dd8686c0f46693a2006b07c6a500c8 6efb54fd24fb3e01f897e43439bdc3ce14f1fbebffdeadb4359a0c19f310b3cf 43de72cf476b2901c6b6c4bc3f4ab23a7675b4fc9c1923176ecb1853d95e2ba3 0cddab2bde50ebf2bcee415bfce1b945448d4df9fc6c17c13a45d35bd92b37a4 787e5feac849af0ec1301e35f4d558e74cddcec5781a6904202ee21a67fb93df d0ce1989305492497c5f049bc1c135856b12cccf19be7ae530089c25ab71c079 feb0b7d518927ca216062761021f95e2a85be74d571740e4f570df9144fdd871 a27f00945f2dd9ccff0c95e069f050f42e36ef9b29324179cff23b07dd76769c f708507b2f897ad13bcd2591d7c6d6957eb686fdfa79e5557a1f7a4d90f5134d 62966b7a6be14036d17393819efc240c78a22cd7b635f2968bc1e97dc4015ccd 6fb5434b9a5f2e9a810df438f51a6764153d5743a3318bc8cc05d1f72cf79c0c 77515bdddfc31f9a8476fbef41d053973a7f8c6cd25abbb6bc105e1e1fe25258 6a2067d836b2d0ec1328ad5cc0140a2aa9523448de9343abe0a65fa6da41ebb5 5645b7b55c450b4716b54b9e413b8b57f515f1db01e1e05dcc5398f6cd1a0597 b6bb81ea94bedb592f9543ffb057b9ff3d01e810471b3daa68cb8aa8d3381c33 d8598260d8203579a3f0c1894d8fcdf6bc52a52100428d19e4cd39b78f62e002 ded3ed39c06fa202cc11dc1a9f1c3324eeeb576d1c06ebe48b06e10519540d58 b52832f36f8e80dbe0cb8ef7097c5bb19ee42da7d86d4343541bead2d3fe7f09 17eff49a1d26d9690dfe404bffe1e7b316b44c47dbc99887938dd90ff1023b7b 52718f94db177650c0bd707cbc46e23a22b02d6e456eb6fefe2f5ffecedc108d 41d436cc4c51b20a8ae9b2276e32ff4d9ee49818161aa7ebc0be12bbfa46a943 46112fb69f79a18db9b1e4a6c928502eccd8065026dba34f788a7fb943429ec7 328489c407e449aaa51abd3ded9601a4d85f3a3ad337cd468410057b9bc622e9 86d9345b985d41dcda84e54772ade07bc61da9d9852c2253b4ee21550b00bbf3 4417e5232a6eb3f8e5f02786aedb7637244823c1615a5e00dba74fa102043502 3a4bd54e8704f021b2b4f870bdd0c0eb6824d2f041ece0ca22144c680519c97b f9be11108d33b2f17aebf0b2c970a2ae4dba124812d971b511a69e6e4fd10b67 9b72ad02061c72ace0fc76ca9e21aa8c549669df978d0db87e1f062572e7e39e d59addc542efa6d67adbfed30f034825c1fafd2392f96f429e65e281bb14f25a 4e833716a65e9050cebfe21ca9c0b4489b5b57cb9c998e8cf316fd5a99e8b70c ddcb048513fe85fbb6735192ae1b0630b6857b070b43a8e602da6127edcdd355 0a17fbc52bc26229399dbdfa557c94abd7409495921931c68d1372bb0af7f5f1 414735 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 17 /eta put dup 20 /kappa put dup 21 /lambda put dup 23 /nu put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 31 /chi put dup 33 /omega put dup 34 /epsilon put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 81 /Q put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 102 /f put dup 103 /g put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 110 /n put dup 112 /p put dup 115 /s put dup 116 /t put dup 120 /x put dup 121 /y 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 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 6 /plusminus put dup 8 /circleplus put dup 14 /openbullet put dup 15 /bullet put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 25 /approxequal put dup 26 /propersubset put dup 33 /arrowright put dup 35 /arrowdown put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 66 /B put dup 68 /D put dup 70 /F put dup 75 /K put dup 76 /L put dup 77 /M put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 112 /radical put dup 121 /dagger 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 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 86 /V put dup 87 /W 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 103 /g put dup 104 /h put dup 106 /j put dup 107 /k put dup 108 /l 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 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueXX 5087386 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 89 /Y put dup 90 /Z put readonly def /FontBBox{-56 -251 1102 750}readonly def /UniqueXX 5000799 def currentdict end currentfile eexec 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31d965181fd4ae94804b74a1c3f49c7806e69ad9853d22ac4b4b2db8c9954868 8db24295530ae34b2d5c72dfe31e1e28d7c05285d9db654151a9dc00fb78e6b7 2a59a792c8b011c5af61ac5c415c37f52e3c9437c34774627a79 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTT10 %!PS-AdobeFont-1.1: CMTT10 1.00B %%CreationDate: 1992 Apr 26 10:42:42 % 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 (CMTT10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch true def end readonly def /FontName /CMTT10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 46 /period put dup 64 /at put dup 97 /a put dup 100 /d put dup 101 /e put dup 104 /h put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put readonly def /FontBBox{-4 -235 731 800}readonly def /UniqueXX 5000832 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /asteriskmath put dup 121 /dagger put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueXX 5000817 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f73e9b697ddc0351763500e23c4d170ea8a2d2 75ad0b6b23097f02fa803c1f46f9ac10fe0e527c431b11b1b9a50e874896cca6 17fe3e255960846bc319ca68b9b930a45d5d0fc36b352c3443ab6f7793b38d2c 7fb9c409ab7980ead55e9f8f6b83c70088d1fa6d7f2214da8928e40b96334ad2 72d1b5db9d1355a579b26a8dac21f021e393d7d837a95deb8de92ed818a8b80b 98ee9cdd8a3c57ad7d6eb7a82f43eaa6ec520eefa7302e750edc4b495f70bba2 24a2182eec37f1a1c5dc8e20973007150237daa6bf03d498826777eeebe09014 ebfe05a8731e1a421cfc513d47d673e3d94d805f5d0668ffa265e0962cb5a75a 4a93d9b873912be368f3f300b584a7c5dcdc5e3f796d62bd15453c88ba7aa204 01cbb3dee5fa0a9d2b44dd122a565359edd6dab1c8481ee7c0ccad206bd3ea69 eb9028f90c198b6d86654c074b98cc73fc076a8d893563b78e999de97e8ac3a0 cb0b0bbdee41b05909141758571e9fd2fa666ffead0e00a2a2df28fced5bafc0 cebde1c84683c0582fffd2d4e45b1a8c6ae34ee3fccc062e93a0f3204dd663b1 0906213adfce1c6e22aa01eec403ee8def76ad79b48d997948b9a062a357fe3f b7067440a9c692a62dcf9bdd3e045f972d228a65f07d7c60f78a7df35ffe7bf5 82157f9f5f89ad65006863f8fc88109f95324b99f3d815b015fa83cee7a40fa8 1da4197d00be65b12a39e8ca1047ec8d7787a04ea06ac24f661025c03e32f52c b9c10247bcc6f7fac4346813a1e1f94c5f026733fc4b38ce5d13fbadcaed1b05 a6f5cee0ae6386cd8a1ce54347782a0876fdd994d9829ae3f421d9e495f85960 9d8fbc709c2519a6dad85e444bad9ee8f96b92c43ea6592b91968c9921c8813d baeeb41a80c44c203d920ee9ea2768e53e1b119ab37f8d4a8878895ac7e766a2 5ba6517116721c34b22aa5b3ec8d17ac1da99f6accdbdb3a0058010fb862b535 2ebb4eba91b4d2b2eb56474c5ad37cbcef260183d71ae3be0dab0068355c2032 98b9e8c5bdd1514a735839954eea2a01b0305a55da2f0599b90caac255d0ec38 de173b4fd18da3fc0a9778f52adec865fed8e0eb1e61183d4ebd963cac804a82 de5716377b779c4357870517f865fb3b19c3a1d8a17e23da08cf39eee9c3864d f58c6756231340ddcbaf8c335f 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 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 58 /colon put dup 65 /A put dup 76 /L put dup 80 /P put dup 83 /S put dup 84 /T put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 121 /y put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueXX 5000793 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306457c7e62c1d269f306bd3402e266de fc3b5e7d8a8d2f5bf0fe6ddd40d07391df4fad4a6018dce29a2b8f692b29f202 3a7c0e66de8ed85c14f1f8492167357f51a7e84cc5d92e0fee4d81cf7fbc8de5 2d2e7bb57142033993f9c08c315abade8dbc4a732e84e142d3bee51557910e12 cd8aa37c459a5e6b7f5269f59078aba3be4641a11ac48d0b625c8325b38ec08e 4c9e5e7fed976a5650d99d82114f449b9ca14c4ec957702295a39a93ef93f618 99b8ea06b092c3c1e503e6e436e0a9fa22576c8930ab3dc8c20f5d82b69cddf8 ff4dacfa9c54bed5a3aa3ea5b129fe96be632843b9b6bc91b615581a985db56b 1e01ca60ee69ca92cf5c0882ece62edad3e106d835348822400f0b66af658f2a 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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: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /Delta put dup 3 /Lambda put 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 24 /cedilla put dup 25 /germandbls put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 63 /question put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueXX 5000794 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204a520640fd7c4d05e35ea5486f a43d0e970787203855daccb1ed0b16325828b13b7c767a1a58c87902f0546ea5 404440fa6526975783c2d7a9572b8e5d5176bac598b8960609f49016aebf3763 89c1adfe6a1ffc0c53b3acd9d168ec3a278fd4f1d9b085dc2df701346c9d0887 3d230a161692b42a45bb5da343b4edd839dd60685a59aaa49fc65cd8e2ad2a9c 5cab953495172a1a9f40d5f38e922443a519dd96f667201a408bd1d89e35f4ab 98e2f68c3366efc14b5afe1018d9c845d808d4f8e07ea8168518a171ab4d6e21 70eeac099eb7ef46eb9781384da1fac3b3133bb718bc80082eaea88759745188 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500b954c9a9c6dff70366ad982cb370dc4dcbb7d04995f2794d3ebf6f9e39e16 1b25c546d1b47235093bd3d26b1b7888449f9190007665b103c0e130c52ff5fc 8d6266d09e57ca72d7 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 1 /Delta put dup 3 /Lambda put 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 57 /nine put dup 61 /equal put dup 67 /C put dup 78 /N put dup 81 /Q put dup 91 /bracketleft put dup 93 /bracketright put dup 101 /e put dup 103 /g put dup 108 /l put dup 111 /o put dup 112 /p put dup 120 /x put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueXX 5000791 def currentdict end currentfile eexec 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8fdd86b03a4796923f8f7d68dedaedf5dbbcfa64252e3b 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 69 /E put dup 77 /M put dup 82 /R put dup 83 /S put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 105 /i put dup 106 /j put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-33 -250 945 749}readonly def /UniqueXX 5000795 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 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b(the)g(articles)h([Shenk)m(er)35 b(1982)o(],)24 b([F)-8 b(eigen)m(baum)33 b(et)g(al.)g(1982)o(],)24 b([Rand)32 b(et)h(al.)g(1982)o(],)324 3897 y([Ostlund)g(et)g(al.)g(1983)o(],)28 b(it)f(w)m(as)h(found)g(n)m(umerically)h(that)e(cubic)h(critical)g (circle)g(maps)324 4017 y(exhibit)43 b(in)m(teresting)h(\\univ)m (ersal")f(prop)s(erties)g({)e(for)h(large)g(classes)h(of)f(circle)h (maps)324 4138 y(there)48 b(exist)h(n)m(um)m(b)s(ers)h(and)e(functions) g(that)g(are)f(the)h(same)h(for)e(all)h(functions)g(in)324 4258 y(the)f(class)i({)d(similar)i(to)f(the)g(F)-8 b(eigen)m (baum-Coullet-T)g(resser)50 b(univ)m(ersalit)m(y)g(of)d(uni-)324 4378 y(mo)s(dal)27 b(maps)g(of)g(the)g(in)m(terv)-5 b(al.)42 b(\(Lately)28 b(similar)f(studies)i(w)m(ere)f(carried)g(out)f(for)f (maps)324 4499 y(with)c(critical)h(p)s(oin)m(ts)f(of)g(higher)g (degrees)i([Dixon)32 b(et)h(al.)g(1997)o(],)24 b([Briggs)33 b(at)f(al.)g(1998]\).)324 4619 y(Shortly)g(after)f(the)h(initial)f(n)m 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b(in)e(some)g(sense)i (to)e(a)f(rotation,)j(i.e.,)324 2159 y(whether)f(there)g(exists)g(a)f (c)m(hange)g(of)g(v)-5 b(ariables)42 b Fr(y)47 b Fy(=)c Fr(h)p Fy(\()p Fr(x)p Fy(\))f(suc)m(h)i(that)d(in)h(the)h(new)324 2280 y(v)-5 b(ariables)33 b Fr(f)43 b Fy(\\lo)s(oks)33 b(lik)m(e")h(a)e(rotation.)43 b(The)33 b(de\014nition)h(is)f(the)g (follo)m(wing.)324 2495 y Fi(De\014nition)38 b(2.1)49 b Fl(Two)24 b(cir)-5 b(cle)24 b(maps)g Fr(f)36 b Fl(and)24 b Fr(g)29 b Fl(ar)-5 b(e)24 b Fy(top)s(ologically)e(\(resp.)h Fr(C)3138 2459 y Fp(k)3180 2495 y Fy(-,)h(smo)s(othly)-8 b(,)324 2615 y(analytically\))39 b(conjugate)h Fl(if)f(ther)-5 b(e)40 b(exists)f(a)g(home)-5 b(omorphism)38 b(\(r)-5 b(esp.)39 b Fr(C)3135 2579 y Fp(k)3177 2615 y Fl(,)i Fr(C)3325 2579 y Fo(1)3400 2615 y Fl(,)g(or)324 2736 y Fr(C)401 2699 y Fp(!)486 2736 y Fl(di\013e)-5 b(omorphism,)33 b(r)-5 b(esp.\))44 b Fr(h)28 b Fy(:)f Fm(T)h Fq(!)g Fm(T)35 b Fl(such)g(that)1594 2944 y Fr(f)k Fy(=)27 b Fr(h)1840 2903 y Fo(\000)p Fz(1)1957 2944 y Fq(\016)22 b Fr(g)j Fq(\016)d Fr(h)35 b(:)1146 b Fy(\(2\))324 3153 y Fl(The)34 b(map)g Fr(h)h Fl(is)g(c)-5 b(al)5 b(le)-5 b(d)34 b(the)h Fy(conjugacy)g Fl(b)-5 b(etwe)g(en)35 b Fr(f)45 b Fl(and)34 b Fr(g)t Fl(.)470 3368 y Fy(If)28 b Fr(f)39 b Fy(and)28 b Fr(g)j Fy(are)d(conjugate,)h(man)m(y)g(of)f(their)g(prop)s(erties)h (are)f(the)g(same)h({)f(e.g.,)h(the)324 3488 y(p)s(ossible)37 b(t)m(yp)s(es)h(of)e(the)h(orbits)f(of)g(the)h(p)s(oin)m(ts)g(of)f Fm(T)h Fy(under)g Fr(f)47 b Fy(and)36 b(under)h Fr(g)j Fy(are)c(the)324 3609 y(same,)d(the)g(rotation)f(n)m(um)m(b)s(er)i(of)e Fr(f)44 b Fy(is)33 b(equal)g(to)f(that)h(of)f Fr(g)t Fy(,)g(etc.)470 3729 y(Early)45 b(results)g(ab)s(out)e(the)h(existence) j(of)c(a)h(conjugacy)g(to)g(a)f(rotation)g(are)h(the)324 3849 y(follo)m(wing)33 b(theorems)h([P)m(oincar)m(\023)-46 b(e)33 b(1885)o(],)g([Denjo)m(y)g(1932)o(].)324 4065 y Fi(Theorem)38 b(2.1)f(\(P)m(oincar)n(\023)-54 b(e\))48 b Fl(Assume)e(that)h(the)f(r)-5 b(otation)46 b(numb)-5 b(er)45 b(of)h(the)g(cir)-5 b(cle)324 4185 y(map)37 b Fr(f)49 b Fl(is)38 b(irr)-5 b(ational)37 b(and)h(the)g(orbit)g(of)g (some)f(p)-5 b(oint)38 b Fr(x)c Fq(2)g Fm(T)k Fl(is)g(dense)f(in)h Fm(T)p Fl(.)54 b(Then)324 4305 y Fr(f)c Fl(is)40 b(top)-5 b(olo)g(gic)g(al)5 b(ly)38 b(c)-5 b(onjugate)40 b(to)g Fr(r)1689 4321 y Fp(\034)8 b Fz(\()p Fp(f)f Fz(\))1828 4305 y Fl(.)59 b(\(If)39 b(only)h(the)f(\014rst)h(c)-5 b(ondition)39 b(is)h(satis\014e)-5 b(d,)324 4426 y(then)30 b(ther)-5 b(e)31 b(exists)g(a)f(noninvertible)g(c)-5 b(ontinuous)30 b(map)g Fr(h)e Fy(:)g Fm(T)g Fq(!)f Fm(T)32 b Fl(such)e(that)h Fr(h)13 b Fq(\016)g Fr(f)39 b Fy(=)324 4546 y Fr(r)368 4562 y Fp(\034)8 b Fz(\()p Fp(f)f Fz(\))529 4546 y Fq(\016)22 b Fr(h)p Fl(.\))324 4761 y Fi(Theorem)38 b(2.2)f(\(Denjo)m(y\))49 b Fl(A)g(cir)-5 b(cle)48 b(di\013e)-5 b(omorphism)47 b(with)h(irr)-5 b(ational)49 b(r)-5 b(otation)324 4882 y(numb)g(er)44 b(and)g(derivative)g(of)h(b)-5 b(ounde)g(d)43 b(variation)h(is)h(top)-5 b(olo)g(gic)g(al)5 b(ly)44 b(c)-5 b(onjugate)44 b(to)h(a)324 5002 y(r)-5 b(otation.)p eop %%Page: 7 7 7 6 bop 324 241 a Ft(2)97 b(RIGOR)m(OUS)21 b(RESUL)-8 b(TS)23 b(ABOUT)f(CONJUGA)m(CIES)i(BETWEEN)g(CIR)m(CLE)f(MAPS)p Fy(7)470 548 y(Denjo)m(y's)30 b(theorem)f(implies)h(that)e(ev)m(ery)i Fr(C)2103 512 y Fz(2)2171 548 y Fy(circle)g(map)e(is)h(top)s (ologically)g(conju-)324 668 y(gate)24 b(to)g(a)g(rotation.)40 b(The)25 b Fr(C)1389 632 y Fz(2)1453 668 y Fy(condition)g(is)g(rather)f (sharp)h({)f(Herman)h([Herman)33 b(1979,)324 789 y(Sec.)28 b(X.3.19])f(constructed)i(examples)h(of)d Fr(C)1967 753 y Fz(2)p Fo(\000)p Fp(")2121 789 y Fy(circle)h(maps)h(that)e(are)g(not) g(conjugate)324 909 y(to)32 b(a)g(rotation,)g(ev)m(en)j(if)d(the)h (rotation)f(n)m(um)m(b)s(er)i(satis\014es)g(additional)f(restrictions.) 470 1029 y(In)j(Denjo)m(y's)g(theorem,)i(it)d(is)h(imp)s(ortan)m(t)g (that)f Fr(f)2344 993 y Fo(\000)p Fz(1)2474 1029 y Fy(is)h(di\013eren)m (tiable.)54 b(If)36 b Fr(f)3367 993 y Fo(\000)p Fz(1)3496 1029 y Fy(is)324 1150 y(not)42 b(di\013eren)m(tiable,)k(one)d(cannot)f (guaran)m(tee)h(the)f(existence)j(of)c(a)h(conjugacy)h(to)f(a)324 1270 y(rotation)f(ev)m(en)i(b)m(y)g(assuming)g(that)f Fr(f)52 b Fy(is)43 b Fr(C)2022 1234 y Fo(1)2096 1270 y Fy(.)72 b(The)43 b(article)f([Hall)33 b(1981)o(])42 b(con)m(tains)324 1391 y(an)c(example)h(of)f(a)g Fr(C)1133 1354 y Fo(1)1245 1391 y Fy(circle)i(map)e(with)h(rotation)e(n)m(um)m(b) s(er)i Fr(\032)g Fy(\(for)e(an)m(y)i(irrational)324 1511 y Fr(\032)28 b Fq(2)g Fy([0)p Fr(;)17 b Fy(1\)\))29 b(whic)m(h)j(has)f (no)f(dense)i(orbit)e(and)g(therefore)h(cannot)g(b)s(e)f(conjugate)g (to)g Fr(r)3494 1526 y Fp(\032)3535 1511 y Fy(.)324 1631 y(This)39 b(map)f(is)h(on)m(to,)g(has)g(no)f(p)s(erio)s(dic)g(orbits)g (and)g(has)h(no)f(more)g(than)g(t)m(w)m(o)h(p)s(oin)m(ts)324 1752 y(where)d(the)g(deriv)-5 b(ativ)m(e)37 b(of)e(the)g(map)h(v)-5 b(anishes)36 b(\(in)g(fact,)g(one)f(can)h(construct)g(suc)m(h)h(a)324 1872 y(map)c(with)g(one)g(critical)g(p)s(oin)m(t)f(only\).)470 1993 y(In)h(striking)g(con)m(trast)g(with)g(the)f Fr(C)1790 1956 y Fo(1)1897 1993 y Fy(Denjo)m(y)h(coun)m(terexamples)i(of)d([Hall) g(1981)o(],)324 2113 y(it)h(w)m(as)h(sho)m(wn)g([Y)-8 b(o)s(ccoz)33 b(1984a)o(])g(that)g(the)h(maps)f(exhibiting)i(the)e(b)s (eha)m(vior)h(of)f(those)324 2233 y(in)g([Hall)f(1981)o(])h(cannot)g(b) s(e)g(real)f(analytic.)44 b(More)33 b(precisely)-8 b(,)324 2462 y Fi(Theorem)38 b(2.3)f(\(Y)-9 b(o)s(ccoz\))47 b Fl(A)n(ny)39 b(r)-5 b(e)g(al)38 b(analytic)g(cir)-5 b(cle)38 b(map)g(with)g(no)g(p)-5 b(erio)g(dic)38 b(or-)324 2582 y(bits)d(is)f(top)-5 b(olo)g(gic)g(al)5 b(ly)34 b(c)-5 b(onjugate)35 b(to)g(a)f(r)-5 b(otation.)470 2810 y Fy(This)37 b(theorem)f(guaran)m(tees)h(that)e(an)m(y)i(t)m(w)m(o)f(real)g (analytic)g(circle)h(maps)f(with)g(ir-)324 2931 y(rational)29 b(rotation)g(n)m(um)m(b)s(er)i(are)f(top)s(ologically)g(conjugate)g (\(w)m(e)g(will)h(use)f(this)h(fact)e(in)324 3051 y(Section)k(4.3\).) 470 3171 y(The)h(ab)s(o)m(v)m(e)f(result)h(w)m(as)f(extended)i(in)d([) 1964 3146 y(\023)1961 3171 y(Swi)s(\030)-46 b(atek)34 b(1991)o(],)f([)2639 3146 y(\023)2636 3171 y(Swi)s(\030)-46 b(atek)34 b(1998)o(].)324 3460 y Fn(2.3)135 b(Smo)t(othness)45 b(of)g(the)h(Conjugacy)324 3645 y Fy(The)31 b(theorems)g(in)f(Section)h (2.2)f(do)g(not)g(assert)h(an)m(ything)f(ab)s(out)g(the)g(smo)s (othness)i(of)324 3765 y(the)j(conjugacy)g Fr(h)p Fy(.)49 b(Can)35 b(one)f(sa)m(y)i(more)e(ab)s(out)g(the)h(di\013eren)m (tiabilit)m(y)i(prop)s(erties)e(of)324 3886 y(the)e(conjugacies)i(in)e (the)h(case)g(of)f(smo)s(oth)g(or)g(analytic)h(maps?)46 b(The)34 b(answ)m(er)h(to)e(this)324 4006 y(question)h(in)m(v)m(olv)m (es)h(n)m(um)m(b)s(er-theoretic)g(prop)s(erties)e(of)f(the)h(rotation)f (n)m(um)m(b)s(er.)324 4266 y Fi(2.3.1)112 b(Arithmetic)37 b(Prop)s(erties)g(of)h(the)f(Rotation)g(Num)m(b)s(er)324 4451 y Fy(It)46 b(turns)g(out)g(that,)j(to)d(guaran)m(tee)g(that)g(a)g (smo)s(oth)g(circle)h(di\013eomorphism)g Fr(f)57 b Fy(is)324 4571 y(smo)s(othly)39 b(conjugate)h(to)e(a)h(rotation,)h(it)f(is)h(not) e(enough)i(that)f Fr(\034)11 b Fy(\()p Fr(f)g Fy(\))39 b(b)s(e)g(irrational,)324 4691 y(but)33 b(also)f(that)h(it)f(should)i (not)e(b)s(e)h(w)m(ell)h(appro)m(ximable)g(b)m(y)f(rationals.)p eop %%Page: 8 8 8 7 bop 324 241 a Ft(2)97 b(RIGOR)m(OUS)21 b(RESUL)-8 b(TS)23 b(ABOUT)f(CONJUGA)m(CIES)i(BETWEEN)g(CIR)m(CLE)f(MAPS)p Fy(8)324 548 y Fi(De\014nition)38 b(2.2)49 b Fl(A)37 b(numb)-5 b(er)36 b Fr(\032)h Fl(is)f(c)-5 b(al)5 b(le)-5 b(d)36 b Fy(Diophan)m(tine)h Fl(of)f(typ)-5 b(e)37 b Fy(\()p Fr(C)r(;)17 b(\027)6 b Fy(\))36 b Fl(\(or)g(simply)324 668 y(of)e(typ)-5 b(e)36 b Fr(\027)6 b Fl(\))35 b(for)f(p)-5 b(ositive)35 b Fr(C)41 b Fl(and)35 b Fr(\027)6 b Fl(,)35 b(if)g(for)f(any)h Fr(p=q)c Fq(2)d Fm(Q)1565 783 y Fk(\014)1565 843 y(\014)1565 903 y(\014)1565 962 y(\014)1598 928 y Fr(\032)22 b Fq(\000)1780 860 y Fr(p)p 1780 905 49 4 v 1781 996 a(q)1839 783 y Fk(\014)1839 843 y(\014)1839 903 y(\014)1839 962 y(\014)1900 928 y Fr(>)2093 860 y(C)p 2013 905 236 4 v 2013 996 a Fq(j)p Fr(q)t Fq(j)2116 967 y Fz(2+)p Fp(\027)2294 928 y Fr(:)1116 b Fy(\(3\))324 1181 y Fl(A)35 b(numb)-5 b(er)35 b(that)g(is)f(not)h(Diophantine)f(is)g (c)-5 b(al)5 b(le)-5 b(d)34 b(a)h Fy(Liouville)e(n)m(um)m(b)s(er)p Fl(.)324 1389 y Fy(It)39 b(is)h(easy)g(to)f(sho)m(w)i(that)e(there)h (are)f(no)g(n)m(um)m(b)s(ers)i(that)f(satisfy)g(\(3\))f(when)h Fr(\027)46 b(<)39 b Fy(0.)324 1509 y(Hence,)34 b(w)m(e)f(will)h(assume) g Fr(\027)g Fq(\025)28 b Fy(0.)470 1630 y(Eac)m(h)f(irrational)f(n)m (um)m(b)s(er)i Fr(\032)g Fq(2)g Fy(\(0)p Fr(;)17 b Fy(1\))25 b(can)i(b)s(e)f(written)h(in)g(the)g(con)m(tin)m(ued)h(fraction)324 1750 y(represen)m(tation)34 b(as)1044 1991 y Fr(\032)28 b Fy(=)f Fq(h)p Fr(\032)1314 2006 y Fz(0)1354 1991 y Fr(;)17 b(\032)1448 2006 y Fz(1)1487 1991 y Fr(;)g(\032)1581 2006 y Fz(2)1621 1991 y Fr(;)g(:)g(:)g(:)o Fq(i)28 b Fy(:=)2355 1924 y(1)p 1986 1968 786 4 v 1986 2112 a Fr(\032)2036 2127 y Fz(0)2098 2112 y Fy(+)2460 2044 y(1)p 2206 2089 556 4 v 2206 2232 a Fr(\032)2256 2247 y Fz(1)2318 2232 y Fy(+)2565 2165 y(1)p 2426 2209 327 4 v 2426 2301 a Fr(\032)2476 2316 y Fz(2)2538 2301 y Fy(+)22 b Fq(\001)17 b(\001)g(\001)2814 1991 y Fr(;)324 2479 y Fy(where)34 b(the)f(p)s(ositiv)m(e)h(in)m(tegers)f Fr(\032)1545 2494 y Fp(n)1625 2479 y Fy(are)g(called)g Fl(p)-5 b(artial)35 b(quotients)d Fy(of)h Fr(\032)p Fy(.)470 2599 y(In)42 b(this)h(represen)m(tation,)j(a)c(n)m(um)m(b)s(er)h(is)g(Diophan)m (tine)f(if)g(its)g(partial)g(quotien)m(ts)324 2720 y(increase)31 b(slo)m(wly)-8 b(.)45 b(In)30 b(this)h(sense,)h(the)f(\\most)f (irrational")f(n)m(um)m(b)s(er)j(b)s(et)m(w)m(een)g(0)e(and)g(1)324 2840 y(is)j(the)g Fl(golden)h(me)-5 b(an)1061 3107 y Fr(\015)33 b Fy(:=)28 b Fq(h)p Fy(1)p Fr(;)17 b Fy(1)p Fr(;)g Fy(1)p Fr(;)g(:)g(:)g(:)m Fq(i)28 b Fy(=)1886 2957 y Fq(p)p 1969 2957 49 4 v 83 x Fy(5)22 b Fq(\000)h Fy(1)p 1886 3084 303 4 v 2013 3175 a(2)2226 3107 y(=)52 b(lim)2330 3167 y Fp(n)p Fo(!1)2585 3040 y Fr(Q)2662 3055 y Fp(n)p 2540 3084 215 4 v 2540 3175 a Fr(Q)2617 3190 y Fp(n)p Fz(+1)2797 3107 y Fr(;)324 3356 y Fy(where)34 b Fr(Q)683 3371 y Fz(1)750 3356 y Fy(=)28 b(1,)k Fr(Q)1039 3371 y Fz(2)1106 3356 y Fy(=)c(2,)k Fr(Q)1395 3371 y Fp(n)p Fz(+1)1560 3356 y Fy(=)c Fr(Q)1741 3371 y Fp(n)1810 3356 y Fy(+)22 b Fr(Q)1985 3371 y Fp(n)p Fo(\000)p Fz(1)2155 3356 y Fy(are)33 b(the)g(Fib)s(onacci)f(n)m(um)m(b)s(ers.)470 3477 y(When)45 b(the)f(partial)f(quotien)m(ts)j(of)d Fr(\032)h Fy(are)g(b)s(ounded,)j Fr(\032)d Fy(is)g(called)h(a)e(n)m(um) m(b)s(er)i(of)324 3597 y Fl(b)-5 b(ounde)g(d)28 b Fy(\(or)g Fl(c)-5 b(onstant)p Fy(\))28 b(t)m(yp)s(e.)44 b(This)30 b(is)f(equiv)-5 b(alen)m(t)30 b(to)f(the)g(fact)g(that)f(\(3\))h(holds) g(with)324 3718 y Fr(\027)34 b Fy(=)27 b(0.)44 b(The)33 b Fl(c)-5 b(ombinatorial)34 b(typ)-5 b(e)33 b Fy(of)f Fr(\032)h Fy(is)g(b)m(y)g(de\014nition)h(sup)2651 3741 y Fp(n)2714 3718 y Fr(\032)2764 3733 y Fp(n)2811 3718 y Fy(.)324 3975 y Fi(2.3.2)112 b(Results)38 b(for)f(Non-Critical)h (Maps)324 4159 y Fy(The)47 b(question)h(ab)s(out)e(the)g(smo)s(othness) i(of)e(the)h(conjugacy)g(in)g(the)f(case)i(of)d(ana-)324 4280 y(lytic)i(circle)g(maps)g(close)g(to)e(a)h(rotation)g(w)m(as)h (\014rst)f(addressed)i(in)e([Arnold)33 b(1961)o(].)324 4400 y(There,)j(it)e(w)m(as)h(pro)m(v)m(ed)h(that)e(an)m(y)h(analytic)g (circle)h(map)e(whose)i(rotation)e(n)m(um)m(b)s(er)h Fr(\032)324 4521 y Fy(is)f(Diophan)m(tine)g(of)g(t)m(yp)s(e)h Fr(\027)h Fq(\025)30 b Fy(0)k(and)f(whic)m(h)j(is)e(close)h(to)e(a)h (rotation)f(is)h(analytically)324 4641 y(conjugate)26 b(to)f Fr(r)915 4656 y Fp(\032)955 4641 y Fy(.)41 b(This)27 b(result)g(w)m(as)f(extended)i(to)d(the)h(case)h(of)e(\014nite)h (di\013eren)m(tiabilit)m(y)324 4761 y(in)33 b([Moser)g(1966)o(].)470 4882 y(The)38 b(\014rst)g(theorem)g(without)f(the)h(h)m(yp)s(othesis)h (of)e(closeness)j(to)c(a)h(rotation)g(w)m(as)324 5002 y(pro)m(v)m(ed)23 b(in)f([Herman)34 b(1979)o(].)40 b(The)22 b(theorem)h(states)g(that)f(there)g(exists)i(a)d(set)i Fq(K)29 b(\032)f Fy([0)p Fr(;)17 b Fy(1])p eop %%Page: 9 9 9 8 bop 324 241 a Ft(2)97 b(RIGOR)m(OUS)21 b(RESUL)-8 b(TS)23 b(ABOUT)f(CONJUGA)m(CIES)i(BETWEEN)g(CIR)m(CLE)f(MAPS)p Fy(9)324 548 y(of)35 b(full)h(Leb)s(esgue)i(measure)f(suc)m(h)g(that)f (if)g Fr(f)44 b Fq(2)34 b Fr(C)2221 512 y Fp(k)2299 548 y Fy(for)h(3)f Fq(\024)f Fr(k)k Fq(\024)d Fr(!)39 b Fy(and)d Fr(\034)11 b Fy(\()p Fr(f)g Fy(\))33 b Fq(2)h(K)q Fy(,)324 668 y(then)45 b(the)h(conjugacy)f(is)g Fr(C)1393 632 y Fp(k)r Fo(\000)p Fz(2)p Fo(\000)p Fp(")1658 668 y Fy(for)g(an)m(y)g Fr(")j(>)h Fy(0.)80 b(The)46 b(set)f Fq(K)h Fy(is)g(c)m(haracterized) 324 789 y(in)h(terms)i(of)e(the)g(gro)m(wth)h(of)f(the)h(partial)f (quotien)m(ts)i(of)e(the)h(con)m(tin)m(ued)h(fraction)324 909 y(expansions)f(of)e(its)g(mem)m(b)s(ers;)55 b(all)46 b(n)m(um)m(b)s(ers)i(in)f Fq(K)g Fy(are)f(Diophan)m(tine)h(of)f(order)g Fr(\027)324 1029 y Fy(for)e(an)m(y)i Fr(\027)55 b(>)48 b Fy(0.)81 b(This)46 b(result)g(w)m(as)f(signi\014can)m(tly)i(impro)m (v)m(ed)g(in)e([Y)-8 b(o)s(ccoz)33 b(1984b)o(].)324 1150 y(Similar)49 b(theorems)h(w)m(ere)h(pro)m(v)m(ed)f(b)m(y)g(using)f (renormalization)h(group)f(tec)m(hniques)324 1270 y([Sinai)33 b(and)f(Khanin)h(1989)o(],)25 b(whic)m(h)f(lo)m(w)m(ered)g(the)f (minimal)g(regularit)m(y)g(required.)42 b(The)324 1391 y(most)49 b(comprehensiv)m(e)k(result)d(w)m(e)g(ha)m(v)m(e)g(found)f (in)h(the)f(literature)h(is)f(the)h(one)f(in)324 1511 y([Katznelson)33 b(and)g(Ornstein)h(1989)o(].)324 1739 y Fi(Theorem)k(2.4)49 b Fl(If)36 b Fr(f)47 b Fl(is)36 b(a)g Fr(C)1459 1703 y Fp(k)1539 1739 y Fl(cir)-5 b(cle)35 b(di\013e)-5 b(omorphism)35 b(whose)g(r)-5 b(otation)37 b(numb)-5 b(er)36 b(is)324 1860 y(Diophantine)44 b(of)i(or)-5 b(der)45 b Fr(\027)6 b Fl(,)49 b(and)d Fr(k)51 b(>)d(\027)37 b Fy(+)30 b(2)p Fl(,)48 b(then)e(the)g(home)-5 b(omorphism)43 b Fr(h)j Fl(that)324 1980 y(c)-5 b(onjugates)34 b Fr(f)46 b Fl(with)34 b(the)h(r)-5 b(otation)35 b Fr(r)1678 1996 y Fp(\034)8 b Fz(\()p Fp(f)f Fz(\))1852 1980 y Fl(is)35 b(of)f(class)g Fr(C)2382 1944 y Fp(k)r Fo(\000)p Fz(1)p Fo(\000)p Fp(\027)t Fo(\000)p Fp(")2732 1980 y Fl(for)g(any)h Fr(")27 b(>)h Fy(0)p Fl(.)324 2240 y Fi(2.3.3)112 b(Results)38 b(for)f(Critical)h(Maps)324 2425 y Fy(The)22 b(follo)m(wing)h(result)f (w)m(as)h(pro)m(v)m(ed)g(recen)m(tly)h(\([de)33 b(F)-8 b(aria)31 b(and)i(de)g(Melo)g(1999)o(],)24 b([de)34 b(F)-8 b(aria)31 b(and)i(de)g(Melo)g(2000)o(],)324 2545 y(see)g(also)g([de)g (Melo)g(1998],)f([de)h(F)-8 b(aria)32 b(1999)o(],)h([Y)-8 b(amp)s(olsky)34 b(1999)o(]\):)324 2773 y Fi(Theorem)k(2.5)49 b Fl(L)-5 b(et)29 b Fr(f)40 b Fl(and)29 b Fr(g)j Fl(b)-5 b(e)29 b(r)-5 b(e)g(al)29 b(analytic)g(critic)-5 b(al)28 b(cir)-5 b(cle)29 b(maps)f(with)h(the)g(same)324 2894 y(r)-5 b(otation)42 b(numb)-5 b(er)42 b Fr(\032)h Fl(of)f(b)-5 b(ounde)g(d)41 b(typ)-5 b(e.)68 b(Then)41 b Fr(f)54 b Fl(and)41 b Fr(g)46 b Fl(ar)-5 b(e)42 b Fr(C)2818 2858 y Fz(1+)p Fp(\013)3000 2894 y Fl(c)-5 b(onjugate)42 b(for)324 3014 y(some)36 b Fr(\013)d Fq(2)g Fy(\(0)p Fr(;)17 b Fy(1\))36 b Fl(dep)-5 b(ending)36 b(only)h(on)g(the)g(c)-5 b(ombinatorial)36 b(typ)-5 b(e,)39 b Fr(N)j Fy(:=)32 b(sup)3253 3038 y Fp(n)3317 3014 y Fr(\032)3367 3029 y Fp(n)3414 3014 y Fl(,)38 b(of)324 3134 y Fr(\032)p Fl(.)470 3363 y Fy(It)47 b(w)m(as)h(conjectured)h(in)e([de)33 b(Melo)h(1998)o(])47 b(that)g(the)g(H\177)-49 b(older)48 b(exp)s(onen)m(t)g(of)f(the)324 3483 y(conjugacy)41 b(b)s(et)m(w)m(een) j(t)m(w)m(o)d(critical)h(circle)g(maps)g(whose)g(critical)g(p)s(oin)m (ts)f(are)g(if)g(the)324 3604 y(same)32 b(order)g(and)f(whic)m(h)i(ha)m (v)m(e)g(the)f(same)h(rotation)d(n)m(um)m(b)s(er)j(of)e(b)s(ounded)i(t) m(yp)s(e)f(do)s(es)324 3724 y(not)k(dep)s(end)h(on)f Fr(N)47 b Fy(\(although)36 b(in)h(the)f(pro)s(of)g(of)f(Theorem)j(2.5)e Fr(\013)h Fy(do)s(es)f(dep)s(end)i(on)324 3844 y Fr(N)10 b Fy(\).)470 3965 y(F)-8 b(or)28 b(the)h(regularit)m(y)g(of)f(the)h (conjugacies)g(b)s(et)m(w)m(een)i(critical)e(maps)g(and)g(rotations,) 324 4085 y(whic)m(h)39 b(exist)h(b)s(ecause)f(of)f(Theorem)h(2.3,)g (the)g(sharp)s(est)g(result)g(w)m(e)g(are)f(a)m(w)m(are)h(of)e(is)324 4205 y(from)c([)585 4180 y(\023)582 4205 y(Swi)s(\030)-46 b(atek)34 b(1998)o(],)g(whose)h(Theorem)f(1.1)g(\(the)f(theorem)i(and)e (a)g(previous)i(pro)s(of)324 4326 y(are)d(credited)i(to)f(an)f (unpublished)j(man)m(uscript)f(of)e(M.)h(Herman\))g(implies:)324 4554 y Fi(Theorem)38 b(2.6)49 b Fl(L)-5 b(et)48 b Fr(f)59 b Fl(b)-5 b(e)47 b(an)h(analytic)f(critic)-5 b(al)48 b(cir)-5 b(cle)47 b(map)g(with)h(an)f(irr)-5 b(ational)324 4674 y(r)g(otation)42 b(numb)-5 b(er.)68 b(The)42 b(c)-5 b(onjugacy)42 b(b)-5 b(etwe)g(en)42 b(the)h(map)f(and)g(a)g(r)-5 b(otation)43 b(is)f(quasi-)324 4795 y(symmetric)34 b(if)h(and)f(only)h (if)g(the)f(r)-5 b(otation)35 b(numb)-5 b(er)35 b(is)f(of)h(c)-5 b(onstant)34 b(typ)-5 b(e.)p eop %%Page: 10 10 10 9 bop 324 241 a Ft(3)97 b(SOME)23 b(GENERAL)f(HEURISTIC)h(REMARKS)f (ON)g(RENORMALIZA)-8 b(TION)22 b(AND)g(CONJUGA)m(CIES)p Fy(10)470 548 y(W)-8 b(e)25 b(note)f(that)g(it)h(is)f(a)h(w)m(ell-kno)m (wn)h(fact)e(in)h(the)f(theory)h(of)f(quasi-conformal)h(maps)324 668 y(that)c(the)h(quasi-symmetric)j(maps)d(are)g(H\177)-49 b(older)22 b(\([V\177)-49 b(ais\177)g(al\177)g(a,)24 b(Sec.)f(18],)h([Ghering)32 b(and)h(P)m(alk)-5 b(a,)33 b(man)m(uscript)r(]\).)324 789 y(Hence,)f(w)m(e)g(can)e(conclude)i (that)e(the)h(conjugacy)g(b)s(et)m(w)m(een)i(a)d(critical)h(map)g(and)f (a)g(ro-)324 909 y(tation)37 b(is)h(H\177)-49 b(older.)58 b(Therefore,)39 b(the)f(conjugacies)g(b)s(et)m(w)m(een)i(critical)e (circle)g(maps)g(of)324 1029 y(the)33 b(same)g(rotation)f(n)m(um)m(b)s (er)i(ha)m(v)m(e)g(to)e(b)s(e)h(H\177)-49 b(older)33 b(with)g(exp)s(onen)m(t)h(1.)470 1150 y(It)25 b(is)f(not)h(di\016cult)g (to)f(sho)m(w)i(that,)g(when)f(the)g(critical)g(maps)g(ha)m(v)m(e)h (di\013eren)m(t)f(order,)324 1270 y(the)33 b(conjugacy)g(b)s(et)m(w)m (een)i(them)e(cannot)g(ha)m(v)m(e)h(H\177)-49 b(older)33 b(exp)s(onen)m(t)h(1.)470 1391 y(Putting)h(together)g(Theorem)h(2.6,)e (the)h(H\177)-49 b(older)35 b(regularit)m(y)h(of)e(quasi-symmetric)324 1511 y(maps)f(and)g(Theorem)h(2.5,)e(w)m(e)h(obtain:)324 1739 y Fi(Corollary)38 b(2.1)49 b Fl(The)29 b(c)-5 b(onjugacy)30 b(b)-5 b(etwe)g(en)29 b(critic)-5 b(al)30 b(cir)-5 b(cle)30 b(maps)f(with)h(golden)f(me)-5 b(an)324 1860 y(r)g(otation)37 b(numb)-5 b(er)38 b(is)f(H\177)-50 b(older.)53 b(The)37 b(H\177)-50 b(older)38 b(exp)-5 b(onent)37 b(dep)-5 b(ends)36 b(only)i(on)f(the)h(or)-5 b(der)324 1980 y(of)34 b(the)h(critic)-5 b(al)35 b(p)-5 b(oints)34 b(of)h(the)g(two)f(functions.)470 2208 y Fy(The)44 b(ab)s(o)m(v)m(e)f(results)i(seem)f(to)e(giv)m(e)i(v)m (ery)g(little)f(information)g(ab)s(out)f(what)h(the)324 2329 y(actual)27 b(v)-5 b(alues)27 b(of)f(these)i(regularities)g(are.) 42 b(In)27 b(this)g(pap)s(er,)h(w)m(e)g(will)f(dev)m(elop)i(metho)s(ds) 324 2449 y(that)41 b(allo)m(w)g(us)h(to)f(compute)h(these)g(n)m(um)m(b) s(ers)h(as)f(w)m(ell)g(as)f(to)g(explore)h(n)m(umerically)324 2570 y(some)33 b(geometric)h(prop)s(erties)f(of)f(the)h(conjugacies.) 324 2902 y Fs(3)161 b(Some)35 b(General)g(Heuristic)f(Remarks)j(on)e (Renor-)566 3085 y(malization)53 b(and)h(Conjugacies)324 3304 y Fy(A)34 b(unifying)h(p)s(oin)m(t)f(of)g(view)i(in)e(the)h(study) g(of)f(long)g(term)g(dynamics)i({)e(esp)s(ecially)i(in)324 3424 y(one-dimensional)i(systems)i({)d(has)h(b)s(een)g(pro)m(vided)h(b) m(y)f(the)g(scaling)g(and)g(renormal-)324 3545 y(ization)33 b(group)f(ideas.)470 3665 y(F)-8 b(orm)m(ulated)29 b(somewhat)h(lo)s (osely)-8 b(,)30 b(the)g(unifying)f(idea)g(of)f(renormalization)i (group)324 3786 y(can)39 b(b)s(e)h(form)m(ulated)g(as)f(sa)m(ying)i (that)e(\\highly)h(iterated)g(maps,)i(when)e(observ)m(ed)h(in)324 3906 y(small)35 b(scales,)i(ha)m(v)m(e)f(forms)f(that)g(are)g(largely)g (indep)s(enden)m(t)i(of)d(the)i(map")e(\(the)i(uni-)324 4026 y(v)m(ersal)e(prop)s(erties)f(can)g(b)s(e)g(di\013eren)m(t)h(in)e (sets)i(of)e(maps)i(of)e(p)s(ositiv)m(e)i(co)s(dimension\).)470 4147 y(In)i(this)h(section,)h(w)m(e)f(w)m(an)m(t)g(to)f(presen)m(t)i(a) d(heuristic)j(p)s(oin)m(t)e(of)g(view)h(of)f(the)g(rela-)324 4267 y(tions)30 b(of)f(the)i(renormalization)f(and)g(smo)s(othness)h (of)e(conjugacy)i(whic)m(h)g(seems)h(to)d(b)s(e)324 4387 y(applicable)k(to)g(a)f(wide)i(v)-5 b(ariet)m(y)33 b(of)f(mo)s(dels.) 470 4508 y(There)i(are)g(sev)m(eral)h(w)m(a)m(ys)f(to)f(mak)m(e)h(this) g(widely)h(applicable)f(heuristic)h(form)m(ula-)324 4628 y(tion,)30 b(precise)h(and)e(susceptible)j(of)d(theorems.)43 b(Nev)m(ertheless,)34 b(the)29 b(rigorous)g(analysis)324 4749 y(seems)44 b(to)e(dep)s(end)h(on)g(the)f(mo)s(dels)h(considered.) 75 b(Indeed,)46 b(in)d(Section)g(2)f(w)m(e)h(ha)m(v)m(e)324 4869 y(presen)m(ted)31 b(a)e(review)i(of)e(the)g(mathematical)h (literature)g(for)f(circle)h(maps)g(that)f(mak)m(es)p eop %%Page: 11 11 11 10 bop 324 241 a Ft(3)97 b(SOME)23 b(GENERAL)f(HEURISTIC)h(REMARKS)f (ON)g(RENORMALIZA)-8 b(TION)22 b(AND)g(CONJUGA)m(CIES)p Fy(11)324 548 y(precise)36 b(and)g(establishes)h(rigorously)e(part)g (of)g(the)g(general)g(renormalization)h(group)324 668 y(picture)d(in)g(this)g(case.)470 789 y(In)j(the)f(rest)h(of)f(the)h (pap)s(er)f(after)g(this)h(section,)h(w)m(e)f(will)g(presen)m(t)h(the)f (n)m(umerical)324 909 y(to)s(ols)30 b(that)h(w)m(e)h(need)f(to)g(v)m (erify)h(some)f(asp)s(ects)h(of)f(the)g(renormalization)g(group)g (ideas)324 1029 y(and)d(also)g(the)h(implemen)m(tation)h(for)d(the)i (circle)g(maps.)43 b(Since)29 b(in)f(this)h(case)g(w)m(e)g(ha)m(v)m(e)h (a)324 1150 y(go)s(o)s(d)e(mathematical)j(understanding,)h(w)m(e)f(can) f(asses)h(the)f(v)-5 b(alidit)m(y)31 b(of)e(the)i(metho)s(ds.)470 1270 y(Note)f(that)g(in)g(the)h(case)g(of)e(circle)j(maps,)f(b)s (esides)h(the)e(mathematically)i(rigorous)324 1391 y(literature,)49 b(one)d(also)g(has)g(a)f(v)m(ery)i(w)m(ell)g(dev)m(elop)s(ed)h 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563 y Fz(1)1552 548 y Fq(\016)22 b Fr(h)1680 507 y Fo(\000)p Fz(1)1680 572 y(2)1804 548 y Fy(=)28 b Fr(k)s Fy(,)34 b Fr(h)2079 507 y Fo(\000)p Fz(1)2079 572 y(1)2196 548 y Fq(\016)23 b Fr(h)2325 563 y Fz(2)2393 548 y Fy(=)29 b Fr(`)k Fy(are)g(v)m(ery)i(smo)s(oth)f(means)324 668 y(that)e Fr(h)591 683 y Fz(1)658 668 y Fy(=)c Fr(k)c Fq(\016)e Fr(h)965 683 y Fz(2)1004 668 y Fy(,)33 b Fr(h)1120 683 y Fz(1)1187 668 y Fy(=)27 b Fr(h)1346 683 y Fz(2)1408 668 y Fq(\016)21 b Fr(`)1520 632 y Fo(\000)p Fz(1)1614 668 y Fy(,)33 b(in)f(other)g(w)m(ords,)i(w)m(e)f(can)g(obtain)f Fr(h)3029 683 y Fz(1)3101 668 y Fy(from)g Fr(h)3387 683 y Fz(2)3459 668 y Fy(b)m(y)324 789 y(comp)s(osing)j(with)g(a)f(v)m(ery) i(smo)s(oth)e(map.)50 b(This)35 b(mak)m(es)h(precise)h(the)d(notion)h (that)f Fr(h)3522 804 y Fz(1)324 909 y Fy(and)e Fr(h)569 924 y Fz(2)640 909 y Fy(are)g(v)m(ery)h(similar.)44 b(Ev)m(en)34 b(if)d(eac)m(h)i(of)e(them)i(is)f(rather)g(rough,)g(the)g(roughness)324 1029 y(of)g(one)h(is)g(v)m(ery)h(precisely)h(comparable)e(to)f(that)h (of)f(the)h(other.)470 1150 y(W)-8 b(e)49 b(hop)s(e)g(that)f(the)h (presen)m(t)i(w)m(ork)e(could)g(serv)m(e)i(as)d(a)h(stim)m(ulus)h(for)e (further)324 1270 y(mathematical)37 b(in)m(v)m(estigations.)59 b(Our)36 b(n)m(umerics)j(are)e(precise)i(enough)e(that)f(w)m(e)i(can) 324 1391 y(ev)m(en)c(study)g(the)f(corrections)g(to)g(\(A3\).)43 b(W)-8 b(e)33 b(form)m(ulate)g(them)g(as)321 1589 y(\(A4\))49 b(The)33 b(con)m(v)m(ergence)i(to)e(self-similarit)m(y)h(is)f(exp)s (onen)m(tially)i(fast.)470 1787 y(Somewhat)c(more)f(precisely)i(\(but)e (still)g(v)m(ery)h(far)f(from)f(a)g(mathematically)j(rigor-)324 1907 y(ous)h(statemen)m(t\),)h(w)m(e)f(can)g(write)h(conjugacies)f(in)g (the)g(form)963 2139 y Fr(h)p Fy(\()p Fr(x)p Fy(\))28 b(=)1282 2044 y Fk(X)1442 2139 y Fr(\025)1499 2097 y Fp(n)1499 2163 y Fz(1)1563 2139 y Fr(H)1644 2154 y Fz(1)1683 2139 y Fy(\()p Fr(\013)1784 2097 y Fp(n)1830 2139 y Fr(x)p Fy(\))23 b(+)2044 2044 y 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a(2)p Fp(\031)1522 1610 y Fy(\()p Fr(K)d Fy(sin)18 b(2)p Fr(\031)t(x)10 b Fy(+)2062 1571 y Fz(9)p Fo(\000)p Fz(8)p Fp(K)p 2062 1587 190 4 v 2121 1645 a Fz(10)2278 1610 y Fy(sin)17 b(4)p Fr(\031)t(x)p Fy(+)2673 1571 y Fz(3)p Fp(K)5 b Fo(\000)p Fz(4)p 2673 1587 V 2733 1645 a(15)2889 1610 y Fy(sin)18 b(6)p Fr(\031)t(x)p Fy(\))3227 1530 y Fk(\003)3285 1610 y Fy(mo)s(d)e(1)22 b Fr(;)3437 1731 y Fy(\(7\))568 1851 y(where)33 b(the)g(co)s(e\016cien)m(ts)i(are)e (c)m(hosen)h(in)f(suc)m(h)h(a)e(w)m(a)m(y)i(that)e(for)g(ev)m(ery)j Fr(K)7 b Fy(,)1296 2059 y Fr(f)1355 2012 y Fz(Q)1344 2086 y Fp(K)q(;)k(!)1486 2059 y Fy(\()p Fr(x)p Fy(\))28 b(=)g Fr(!)d Fy(+)1943 2012 y Fz(8)p Fp(\031)2021 1988 y Fg(4)2056 2012 y Fz(\(3)p Fo(\000)p Fz(2)p Fp(K)5 b Fz(\))p 1943 2036 358 4 v 2104 2094 a(5)2311 2059 y Fr(x)2366 2023 y Fz(5)2428 2059 y Fy(+)22 b Fr(O)s Fy(\()p Fr(x)2697 2023 y Fz(7)2736 2059 y Fy(\))32 b Fr(:)324 2296 y Fy(Of)e(course,)i 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Fr(K)7 b Fy(\))324 4031 y(in)33 b(the)g(C)g(and)f(Q)h(cases.)470 4152 y(T)-8 b(o)22 b(double-c)m(hec)m(k)i(the)e(results,)k(w)m(e)d(also)e(used)i (the)f(GNU)g(MP)g(library)h([GMP)32 b(2000])324 4272 y({)i(a)g(public)h(domain)f(library)h(for)e(arbitrary)h(precision)i (arithmetic.)49 b(W)-8 b(e)35 b(wrote)f(sub-)324 4392 y(routines)42 b(for)e(high)i(precision)g(trigonometric)f(functions,)k (whic)m(h)d(w)m(e)g(did)g(b)m(y)g(using)324 4513 y(lo)s(cal)29 b(T)-8 b(a)m(ylor)29 b(series)i(expansion,)g(and)e(tested)i(them)e(b)m (y)h(using)g(the)g(follo)m(wing)f(elegan)m(t)324 4633 y(and)j(n)m(umerically)j(stable)f(metho)s(d.)470 4753 y(F)-8 b(or)32 b Fr(x)p Fy(,)h Fr(y)j Fy(p)s(ositiv)m(e,)e(de\014ne)f Fr(x)1568 4768 y Fz(1)1636 4753 y Fy(:=)28 b Fr(x)p Fy(,)33 b Fr(y)1930 4768 y Fz(1)1996 4753 y Fy(:=)28 b Fr(y)t Fy(,)k(and)1200 4984 y Fr(x)1255 4999 y Fp(n)p Fz(+1)1421 4984 y Fy(:=)1561 4916 y Fr(x)1616 4931 y Fp(n)1686 4916 y Fy(+)22 b Fr(y)1832 4931 y Fp(n)p 1561 4961 318 4 v 1695 5052 a Fy(2)1921 4984 y Fr(;)114 b(y)2110 4999 y Fp(n)p Fz(+1)2274 4984 y Fy(:=)2405 4917 y Fq(p)p 2488 4917 197 4 v 67 x Fr(x)2543 4999 y Fp(n)2591 4984 y Fr(y)2639 4999 y Fp(n)p eop %%Page: 15 15 15 14 bop 324 241 a Ft(4)97 b(COMPUTING)34 b(THE)g(CONJUGA)m(CIES)1374 b Fy(15)324 548 y(for)36 b Fr(n)g Fq(2)g Fm(N)p Fy(.)58 b(The)38 b(sequences)i Fq(f)p Fr(x)1587 563 y Fp(n)1635 548 y Fq(g)d Fy(and)g Fq(f)p Fr(y)2014 563 y Fp(n)2060 548 y Fq(g)g Fy(con)m(v)m(erge)i(quadratically)g(to)e(a)g(com-)324 668 y(mon)d(limit)g(called)g(the)g(arithmetic-geometric)h(mean)g(\(A)m (GM\))f(of)f Fr(x)h Fy(and)g Fr(y)t Fy(,)f Fd(M)o Fy(\()p Fr(x;)17 b(y)t Fy(\))324 789 y(\(see,)50 b(e.g.,)f([Borw)m(ein)34 b(and)e(Borw)m(ein)i(1987)o(]\).)82 b(The)47 b(A)m(GM)f(has)f(man)m(y)i (remark)-5 b(able)324 909 y(prop)s(erties,)33 b(e.g.,)1102 1117 y(2)p 1097 1161 59 4 v 1097 1252 a Fr(\031)1182 1049 y Fk(Z)1282 1075 y Fp(\031)r(=)p Fz(2)1237 1274 y(0)1826 1117 y Fr(d\020)p 1426 1161 902 4 v 1426 1181 a Fk(p)p 1525 1181 802 4 v 1525 1273 a Fr(x)1580 1244 y Fz(2)1637 1273 y Fy(cos)1768 1244 y Fz(2)1824 1273 y Fr(\020)c Fy(+)22 b Fr(y)2046 1244 y Fz(2)2101 1273 y Fy(sin)2221 1233 y Fz(2)2277 1273 y Fr(\020)2365 1184 y Fy(=)2609 1117 y(1)p 2478 1161 311 4 v 2478 1252 a Fd(M)p Fy(\()p Fr(x;)17 b(y)t Fy(\))324 1468 y(\(in)46 b(particular,)51 b(elliptic)c(in)m(tegrals)h(can)e(b)s(e)h(calculated)h (through)e(the)h(A)m(GM\).)g(T)-8 b(o)324 1588 y(calculate)46 b(precisely)i(trigonometric)e(functions,)j(e.g.,)h(sin,)f(one)d(can)f (use)i(the)f(fact)324 1709 y(that)1376 1829 y(arcsin)17 b Fr(x)28 b Fy(=)2112 1762 y Fr(x)p 1839 1806 602 4 v 1839 1908 a Fd(M)p Fy(\()1962 1826 y Fq(p)p 2044 1826 266 4 v 2044 1908 a Fy(1)22 b Fq(\000)h Fr(x)2270 1879 y Fz(2)2310 1908 y Fr(;)17 b Fy(1\))2483 1829 y Fr(;)324 2059 y Fy(and)47 b(calculate)h(sin)g(as)f(the)h(in)m(v)m(erse)h (function)f(b)m(y)g(using,)j(for)c(instance,)52 b(the)47 b(pro-)324 2180 y(gram)40 b Fe(zeroin)i Fy(\(whic)m(h)g(is)f 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b(ac)m(hiev)m(e)j(this,)f(w)m(e)324 3523 y(use)f(the)f(follo)m(wing)g(metho)s(d)g([Greene)g(1979)o(],)g ([Shenk)m(er)i(1982)o(].)44 b(First)33 b(w)m(e)h(determine)324 3644 y(the)f(phase-lo)s(c)m(king)h(in)m(terv)-5 b(als)1144 3874 y Fr(I)1195 3833 y Fo(\017)1187 3899 y Fp(K)q(;)10 b(n)1353 3874 y Fy(:=)1484 3764 y Fk(n)1550 3874 y Fr(!)31 b Fq(2)d Fy([0)p Fr(;)17 b Fy(1])1964 3760 y Fk(\014)1964 3820 y(\014)1964 3879 y(\014)2030 3874 y Fr(\034)11 b Fy(\()p Fr(f)2180 3833 y Fo(\017)2169 3899 y Fp(K)q(;)g(!)2311 3874 y Fy(\))28 b(=)2529 3831 y Fp(Q)2585 3839 y Fh(n)p 2490 3851 176 4 v 2490 3909 a Fp(Q)2546 3918 y Fh(n)p Fg(+1)2675 3764 y Fk(o)324 4110 y Fy(\()p Fr(n)g Fq(2)g Fm(N)p Fy(\).)42 b(It)29 b(is)g(guaran)m(teed)h(that)f(the)g(v)-5 b(alue)30 b(\012)2106 4074 y Fo(\017)2146 4110 y Fy(\()p Fr(K)7 b Fy(\))29 b(w)m(e)h(are)f(lo)s(oking)g(for)f(is)i(b)s(et)m(w)m (een)324 4231 y Fr(I)375 4195 y Fo(\017)367 4256 y Fp(K)q(;)10 b(n)543 4231 y Fy(and)37 b Fr(I)788 4195 y Fo(\017)780 4256 y 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Fy(\))g(is)324 668 y(eviden)m(tly)f(more)f(di\013eren)m(tiable)h(than)e(the)g(ones)h (b)s(et)m(w)m(een)h(maps)f(of)f(di\013eren)m(t)h(t)m(yp)s(es.)470 789 y(Another)26 b(observ)-5 b(ation)26 b(is)g(the)h(self-similar)f (structure)h(of)e(the)h(conjugacy)h(b)s(et)m(w)m(een)324 909 y(an)f(N)g(map)g(and)g(a)g(critical)g(\(C)h(or)e(Q\))h(map.)42 b(T)-8 b(o)26 b(illustrate)g(this,)j(in)d(Figure)g(4)f(w)m(e)i(sho)m(w) 324 1029 y(magni\014ed)33 b(regions)g(of)g(the)g(conjugacy)g(b)s(et)m (w)m(een)i Fr(f)2270 993 y Fz(N)2259 1054 y(0)p Fp(:)p Fz(8)2386 1029 y Fy(and)d Fr(f)2634 982 y Fz(Q)2623 1054 y(0)p Fp(:)p Fz(9)2718 1029 y Fy(.)43 b(The)34 b(self-similarit)m(y)324 1150 y(of)28 b(the)h(conjugacies)h(b)s(et)m(w)m(een)g(an)f(N)f(and)h(a) f(C)h(map)g(is)g(one)g(of)f(the)h(predictions)h(of)e(the)324 1270 y(theory)i(of)f(renormalization)i(for)e(C)h(maps;)i(w)m(e)e (observ)m(ed)i(a)e(self-similar)g(structure)h(in)324 1391 y(the)i(case)g(of)f(the)h(conjugacy)h(b)s(et)m(w)m(een)g(an)f(N)f 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4769 y Fp(n)1641 4835 y(y)1691 4810 y Fr(\022)s Fq(k)1789 4825 y Fp(L)1837 4807 y Ff(1)1902 4825 y Fz(\()p Fj(T)p Fz(\))2033 4810 y Fq(\024)g Fr(C)7 b Fq(j)p Fr(y)t Fq(j)2323 4769 y Fp(\013)3388 4810 y Fy(\(14\))324 5002 y Fl(for)34 b(al)5 b(l)35 b Fr(y)c Fq(2)d Fm(T)p Fl(.)p eop %%Page: 21 21 21 20 bop 324 241 a Ft(5)97 b(METHODS)34 b(F)m(OR)e(STUD)m(YING)h(THE)g (REGULARITY)742 b Fy(21)470 548 y(The)36 b(FD)d(metho)s(d)i(is)g (simple)h(and)f(con)m(v)m(enien)m(t)i(to)d(use)i(if)e(one)h(can)g (compute)g(the)324 668 y(v)-5 b(alues)34 b(of)e(the)h(function)h(in)f (p)s(oin)m(ts)g(that)g(are)g(arbitrarily)g(close.)45 b(Unfortunately)-8 b(,)34 b(w)m(e)324 789 y(kno)m(w)k(the)f(v)-5 b(alues)38 b(of)f(the)g(conjugacies)h(only)g(at)e(the)i(p)s(oin)m(ts)f (of)g(the)g(t)m(yp)s(e)h Fr(f)3238 753 y Fp(n)3285 789 y Fy(\(0\),)f(so)324 909 y(to)h(apply)i(Theorem)g(5.1,)g(w)m(e)g(ha)m (v)m(e)g(to)e(in)m(terp)s(olate.)63 b(The)40 b(in\015uence)g(of)f (large)f(gaps)324 1029 y(b)s(et)m(w)m(een)c(the)f(iterates)h Fr(f)1275 993 y Fp(n)1322 1029 y Fy(\(0\))e(is)h(discussed)i(in)e (Section)g(6.3.)324 1318 y Fn(5.3)135 b(F)-11 b(ourier)45 b(Metho)t(ds)f({)h(Littlew)l(o)t(o)t(d-P)l(aley)j(Theorem)324 1503 y Fy(The)38 b(trigonometric)g(system)h Fq(f)p Fr(e)1564 1467 y Fz(2)p Fp(\031)r(ik)r(x)1749 1503 y Fq(g)1799 1518 y Fp(k)r Fo(2)p Fj(Z)1970 1503 y Fy(is)f(an)f(orthonormal)g(basis) i(of)e Fr(L)3203 1467 y Fz(2)3242 1503 y Fy(\()p Fm(T)p Fr(;)17 b(dx)p Fy(\),)324 1623 y(hence,)34 b(according)f(to)f(the)h (Planc)m(herel's)i(theorem,)f(a)e(function)1540 1855 y Fr(\022)s Fy(\()p Fr(x)p Fy(\))c(=)1851 1760 y Fk(X)1858 1972 y Fp(k)r Fo(2)p Fj(Z)2019 1829 y Fy(^)2011 1855 y Fr(\022)2056 1870 y Fp(k)2116 1855 y Fr(e)2161 1814 y Fz(2)p Fp(\031)r(ik)r(x)3388 1855 y Fy(\(15\))324 2173 y(b)s(elongs)33 b(to)f Fr(L)859 2137 y Fz(2)899 2173 y Fy(\()p Fm(T)p Fy(\))h(if)f(and)h(only)g(if)1626 2315 y Fk(X)1633 2527 y Fp(k)r Fo(2)p Fj(Z)1786 2410 y Fq(j)1822 2384 y Fy(^)1814 2410 y Fr(\022)1859 2425 y Fp(k)1902 2410 y Fq(j)1930 2369 y Fz(2)1997 2410 y Fr(<)27 b Fq(1)33 b Fr(:)470 2718 y Fy(The)j(main)g(result)g(of)f(Littlew)m(o)s(o)s(d-P)m (aley)i(theory)f(is)g(that)f(similar)h(c)m(haracteriza-)324 2839 y(tion)k(of)f Fr(L)716 2802 y Fp(p)756 2839 y Fy(\()p Fm(T)p Fy(\))h(\(1)g Fr(<)g(p)g(<)g Fq(1)p Fy(\))f(can)h(b)s(e)g (obtained)g(b)m(y)h(grouping)f(the)g(terms)h(of)e(the)324 2959 y(F)-8 b(ourier)32 b(series)i(in)f(dy)m(adic)h(blo)s(c)m(ks.)45 b(De\014ne)33 b(the)g(decomp)s(osition)1664 3249 y Fr(\022)e Fy(=)1891 3124 y Fo(1)1854 3154 y Fk(X)1844 3366 y Fp(M)7 b Fz(=1)2025 3249 y Fq(L)2094 3264 y Fp(M)2173 3249 y Fr(\022)324 3562 y Fy(of)32 b Fr(\022)f Fq(2)d Fr(L)671 3526 y Fz(1)711 3562 y Fy(\()p Fm(T)p Fy(\))k(in)h Fl(dyadic)i(p)-5 b(artial)34 b(sums)1195 3799 y Fy(\()p Fq(L)1302 3814 y Fp(M)1381 3799 y Fr(\022)s Fy(\))17 b(\()p Fr(x)p Fy(\))27 b(:=)1954 3704 y Fk(X)1773 3923 y Fp(A)1826 3905 y Fh(M)5 b Ff(\000)p Fg(1)1972 3923 y Fo(\024j)p Fp(k)r Fo(j)p Fp()h Fy(1.)324 4381 y Fi(Remark)38 b(5.2)49 b Fy(Usually)41 b Fr(A)g Fy(is)g(tak)m(en)h(to)e(b)s(e)h(2,)h (since)g(the)f(precise)i(v)-5 b(alue)41 b(do)s(es)g(not)324 4501 y(mak)m(e)33 b(an)m(y)h(di\013erence)g(for)e(the)h(mathematical)g (treatmen)m(t.)470 4622 y(In)h(the)g(n)m(umerical)h(applications,)g(w)m (e)f(will)g(\014nd)g(it)g(con)m(v)m(enien)m(t)i(to)d(use)i(some)f Fr(A)p Fy('s)324 4742 y(di\013eren)m(t)42 b(from)f(2.)69 b(Nev)m(ertheless,)46 b(w)m(e)c(ha)m(v)m(e)h(not)e(in)m(tro)s(duced)h Fr(A)f Fy(in)g(the)h(notation,)324 4862 y(since)c(it)e(will)i(b)s(e)e (clear)h(from)g(the)g(con)m(text,)i(and)d(w)m(e)i(follo)m(w)f(the)g (standard)g(practice)324 4983 y(of)32 b(calling)h(the)g(decomp)s (osition)h(\\dy)m(adic".)1529 b Fc(\003)p eop %%Page: 22 22 22 21 bop 324 241 a Ft(5)97 b(METHODS)34 b(F)m(OR)e(STUD)m(YING)h(THE)g (REGULARITY)742 b Fy(22)470 548 y(The)34 b(dy)m(adic)g(blo)s(c)m(ks)f (can)g(b)s(e)g(written)g(as)1358 736 y Fq(L)1427 751 y Fp(M)1506 736 y Fr(\022)e Fy(=)c(\()p Fr(\036)1781 755 y Fp(A)1834 736 y Fh(M)35 b Fq(\000)23 b Fr(\036)2085 755 y Fp(A)2138 736 y Fh(M)5 b Ff(\000)p Fg(1)k Fy(\))22 b Fq(\003)g Fr(\022)35 b(;)324 924 y Fy(where)1485 1044 y Fr(\036)1543 1059 y Fp(N)1610 1044 y Fy(\()p Fr(x)p Fy(\))28 b(:=)1925 950 y Fk(X)1899 1165 y Fo(j)p Fp(k)r Fo(j)p Fp()g Fy(0,)1529 990 y Fr(P)1592 1005 y Fp(\025t)1662 990 y Fy(\()p Fr(\025x)p Fy(\))g(=)g Fr(\025)2039 949 y Fo(\000)p Fz(1)2133 990 y Fr(P)2196 1005 y Fp(t)2225 990 y Fy(\()p Fr(x)p Fy(\))1032 b(\(18\))470 1192 y(On)34 b(the)f(circle,)i(w)m(e)g (cannot)e(sp)s(eak)i(ab)s(out)d(scaling,)j(therefore)f(the)g(relation)f (\(18\))324 1312 y(do)s(es)38 b(not,)i(strictly)f(sp)s(eaking,)h(mak)m (e)f(sense)h(for)e(the)g(P)m(oisson)h(k)m(ernel)h(on)e(the)g(circle)324 1433 y(when)k Fr(\025)f Fy(is)h(not)f(an)g(in)m(teger.)69 b(Nev)m(ertheless,)47 b(for)41 b(small)g(scales,)k(the)d(circle)g(can)f (b)s(e)324 1553 y(iden)m(ti\014ed)50 b(with)f(the)f(real)g(line,)53 b(so)c(that)e(the)i(scalings)g(of)f(the)h(p)s(erio)s(dic)f(P)m(oisson) 324 1673 y(k)m(ernel)34 b(can)f(b)s(e)g(used)g(when)h(examining)g (asymptotic)g(features)f(in)g(small)g(scales.)113 b Fc(\003)470 1881 y Fy(Using)39 b(the)h(heuristic)g(argumen)m(t)f(that)g(the)g (partial)g(sums,)i Fr(\036)2819 1896 y Fp(n)2892 1881 y Fq(\003)27 b Fr(\022)s Fy(,)40 b(b)s(eha)m(v)m(e)g(lik)m(e)324 2001 y(the)32 b(Ab)s(el)g(means,)h Fr(P)1111 2026 y Fz(1)p Fo(\000)1215 1999 y Fg(1)p 1211 2011 39 3 v 1211 2052 a Fh(n)1283 2001 y Fq(\003)20 b Fr(\022)s Fy(,)32 b(one)g(easily)h (\014nds)g(that)e(the)h Fr(L)2582 1965 y Fp(p)2622 2001 y Fy(\()p Fm(T)p Fy(\))g(norms)g(of)f Fr(d)p Fy(\()p Fr(\022)s Fy(\))g(and)324 2156 y Fr(G)p Fy(\()p Fr(\022)s Fy(\))h(are)h(equiv)-5 b(alen)m(t)34 b(for)e(1)c Fr(<)f(p)h(<)f Fq(1)33 b Fy(if)1922 2130 y(^)1914 2156 y Fr(\022)1959 2171 y Fz(0)2027 2156 y Fy(=)27 b(0.)324 2363 y Fi(Theorem)38 b(5.2)f(\(Littlew)m(o)s(o)s(d-P)m(aley\))49 b Fl(If)40 b Fr(\022)g Fq(2)d Fr(L)2310 2327 y Fp(p)2350 2363 y Fy(\()p Fm(T)p Fy(\))p Fl(,)42 b Fy(1)37 b Fr(<)f(p)h(<)g Fq(1)p Fl(,)k(then)f(ther)-5 b(e)324 2483 y(exist)34 b(p)-5 b(ositive)35 b(c)-5 b(onstants)34 b Fr(A)1403 2498 y Fp(p)1478 2483 y Fl(and)g Fr(B)1741 2498 y Fp(p)1816 2483 y Fl(such)g(that)1083 2685 y Fr(A)1156 2700 y Fp(p)1212 2685 y Fq(k)p Fr(\022)s Fq(k)1360 2701 y Fp(L)1408 2682 y Fh(p)1444 2701 y Fz(\()p Fj(T)p Fz(\))1575 2685 y Fq(\024)28 b(k)p Fr(d)p Fy(\()p Fr(\022)s Fy(\))p Fq(k)1955 2701 y Fp(L)2003 2682 y Fh(p)2039 2701 y Fz(\()p Fj(T)p Fz(\))2170 2685 y Fq(\024)g Fr(B)2349 2700 y Fp(p)2405 2685 y Fq(k)p Fr(\022)s Fq(k)2553 2701 y Fp(L)2601 2682 y Fh(p)2637 2701 y Fz(\()p Fj(T)p Fz(\))2776 2685 y Fr(:)470 2892 y Fy(Theorem)46 b(5.2)e(has)h(man)m(y)g(imp)s(ortan)m(t)g (implications.)80 b(In)45 b(particular,)j(it)c(giv)m(es)324 3013 y(useful)36 b(c)m(haracterizations)g(of)e(Sob)s(olev,)j(H\177)-49 b(older,)35 b(Hardy)-8 b(,)36 b(Beso)m(v)h(spaces)f({)f(see,)h(e.g.,) 324 3133 y([Stein)d(1970)o(,)d(Ch.)g(5],)f([Hern\023)-49 b(andez)34 b(and)f(W)-8 b(eiss)34 b(1996)o(,)c(Ch.)f(6],)h([Mey)m(er)k (1990)o(,)c(Ch.)g(6],)324 3253 y([F)-8 b(razier)32 b(et)h(al.)g(1991)o (].)470 3374 y(In)39 b(our)f(n)m(umerical)h(explorations,)i(w)m(e)e (used)g(metho)s(ds)h(based)f(on)f(the)g(follo)m(wing)324 3494 y(t)m(w)m(o)49 b(corollaries)g(of)e(Theorem)j(5.2,)i(whic)m(h)d(w) m(e)h(will)f(call)f(\\discrete")i(\(DLP\))d(and)324 3615 y(\\con)m(tin)m(uous")34 b(\(CLP\))f(v)m(ersions)h(of)e(the)h(Littlew)m (o)s(o)s(d-P)m(aley)h(theorem.)324 3822 y Fi(Theorem)k(5.3)f(\(DLP\))48 b([Kran)m(tz)37 b(1983,)f(Theorem)f(5.9])e Fl(The)e(function)i Fr(\022)i Fl(\(15\))324 3942 y(is)h(of)h(class)f Fy(\003)851 3957 y Fp(\013)900 3942 y Fy(\()p Fm(T)p Fy(\))h Fl(\()p Fr(\013)32 b Fq(2)f Fm(R)1382 3957 y Fz(+)1441 3942 y Fl(\))37 b(if)f(and)h(only)f(if)h(ther)-5 b(e)36 b(exists)h(a)f Fr(C)j(>)31 b Fy(0)36 b Fl(such)h(that)g(for)324 4063 y(any)e Fr(M)j Fq(2)28 b Fm(N)1429 4183 y Fq(kL)1548 4198 y Fp(M)1626 4183 y Fr(\022)s Fq(k)1724 4199 y Fp(L)1772 4180 y Ff(1)1837 4199 y Fz(\()p Fj(T)p Fz(\))1968 4183 y Fq(\024)g Fr(C)c Fy(2)2216 4142 y Fo(\000)p Fp(\013M)2430 4183 y Fr(:)931 b Fy(\(19\))324 4390 y Fi(Theorem)38 b(5.4)f(\(CLP\))48 b([Stein)38 b(1970,)60 b(Ch.)c(5,)k(Lemma)d(5])49 b Fl(The)g(function)f Fr(\022)324 4511 y Fl(\(15\))36 b(is)h(of)g(class)g Fy(\003)1069 4526 y Fp(\013)1118 4511 y Fy(\()p Fm(T)p Fy(\))h Fl(\()p Fr(\013)32 b Fq(2)h Fm(R)1603 4526 y Fz(+)1662 4511 y Fl(\))k(if)g(and)g(only)g(if)g(for)g (e)-5 b(ach)36 b Fr(\021)g Fq(\025)d Fy(0)k Fl(ther)-5 b(e)37 b(exists)g(a)324 4631 y Fr(C)d(>)28 b Fy(0)35 b Fl(such)f(that)i(for)e(any)h Fr(t)28 b(>)f Fy(0)1210 4742 y Fk(\014)1210 4802 y(\014)1210 4862 y(\014)1210 4922 y(\014)1226 4742 y(\014)1226 4802 y(\014)1226 4862 y(\014)1226 4922 y(\014)1260 4746 y(\022)1360 4819 y Fr(@)p 1343 4864 93 4 v 1343 4955 a(@)5 b(t)1445 4746 y Fk(\023)1518 4769 y Fp(\021)1577 4887 y Fr(e)1622 4846 y Fo(\000)p Fp(t)1702 4793 y Fo(p)p 1761 4793 114 3 v 53 x(\000)p Fz(\001)1896 4887 y Fr(\022)1944 4742 y Fk(\014)1944 4802 y(\014)1944 4862 y(\014)1944 4922 y(\014)1960 4742 y(\014)1960 4802 y(\014)1960 4862 y(\014)1960 4922 y(\014)1994 4985 y Fp(L)2042 4967 y Ff(1)2107 4985 y Fz(\()p Fj(T)p Fz(\))2238 4887 y Fq(\024)28 b Fr(C)23 b(t)2471 4846 y Fp(\013)p Fo(\000)p Fp(\021)2649 4887 y Fr(:)712 b Fy(\(20\))p eop %%Page: 24 24 24 23 bop 324 241 a Ft(5)97 b(METHODS)34 b(F)m(OR)e(STUD)m(YING)h(THE)g (REGULARITY)742 b Fy(24)324 548 y Fn(5.4)135 b(W)-11 b(a)l(v)l(elet)47 b(Metho)t(ds)324 733 y Fy(Expansions)34 b(in)f(w)m(a)m(v)m(elet)i(bases)f(are)f(v)m(ery)h(w)m(ell)g(suited)f (to)g(studying)h(the)f(lo)s(cal)f(prop-)324 853 y(erties)40 b(of)f(functions)h(b)s(ecause)g(of)f(their)g(lo)s(calization)g(in)h (space.)64 b(W)-8 b(a)m(v)m(elet)41 b(metho)s(ds)324 973 y(are)27 b(esp)s(ecially)h(appropriate)f(for)g(analyzing)g (self-similar)h(functions)g(lik)m(e)g(some)g(of)e(the)324 1094 y(conjugacies)37 b(b)s(et)m(w)m(een)i(circle)f(maps)f(studied)h (in)f(this)g(pap)s(er.)55 b(Belo)m(w)38 b(w)m(e)g(in)m(tro)s(duce)324 1214 y(the)47 b(notations)g(and)g(collect)g(the)g(basic)h(theoretical)f (results)h(ab)s(out)f(regularit)m(y)g(of)324 1335 y(functions)31 b(expanded)i(in)d(w)m(a)m(v)m(elet)j(bases.)44 b(F)-8 b(or)30 b(more)h(details)g(w)m(e)h(refer)f(the)g(reader)g(to)324 1455 y([Mey)m(er)j(1990)o(],)24 b([Daub)s(ec)m(hies)34 b(1992],)24 b([Mallat)32 b(1998)o(],)24 b([Hern\023)-49 b(andez)35 b(and)d(W)-8 b(eiss)34 b(1996)o(],)324 1575 y([H\177)-49 b(ardle)33 b(et)g(al.)f(1998)o(],)h([Louis)g(et)g(al.)43 b(1997)o(].)470 1696 y(Let)e Fr(L)719 1660 y Fz(2)758 1696 y Fy(\()p Fm(T)p Fy(\))900 1714 y Fz(2)935 1695 y Fh(L)1027 1696 y Fy(b)s(e)f(the)h(\\discrete")g(v)m(ersion)h(of)e (the)h(space)g(of)f(square)i(in)m(tegrable)324 1816 y(circle)35 b(maps,)h(i.e.,)f(the)g(2)1270 1780 y Fp(L)1322 1816 y Fy(-dimensional)g(space)g(of)f(the)h(circle)g(maps)g(de\014ned)h(on)e (the)324 1936 y(grid)j Fr(x)584 1951 y Fp(`)653 1936 y Fy(=)e(2)813 1900 y Fo(\000)p Fp(L)920 1936 y Fr(`)p Fy(,)j Fr(`)e Fy(=)f(0,)j(1,)g Fr(:)17 b(:)g(:)p Fy(,)38 b(2)1671 1900 y Fp(L)1748 1936 y Fq(\000)26 b Fy(1.)57 b(W)-8 b(e)38 b(use)g(the)f(follo)m(wing)h(m)m(ultiresolution)324 2057 y(analysis)c(of)e Fr(L)868 2021 y Fz(2)908 2057 y Fy(\()p Fm(T)p Fy(\))1050 2076 y Fz(2)1085 2057 y Fh(L)9 b Fy(:)1106 2260 y Fr(V)1163 2275 y Fz(0)1230 2260 y Fq(\032)28 b Fr(V)1392 2275 y Fz(1)1459 2260 y Fq(\032)g(\001)17 b(\001)g(\001)26 b(\032)j Fr(V)1871 2275 y Fp(L)p Fo(\000)p Fz(1)2040 2260 y Fq(\032)g Fr(V)2203 2275 y Fp(L)2282 2260 y Fy(=)f Fr(L)2452 2219 y Fz(2)2492 2260 y Fy(\()p Fm(T)p Fy(\))2634 2279 y Fz(2)2669 2260 y Fh(L)41 b Fr(:)324 2464 y Fy(Let)33 b Fr(W)591 2479 y Fp(j)660 2464 y Fy(b)s(e)f(the)h (orthogonal)f(complemen)m(t)j(of)d Fr(V)2161 2479 y Fp(j)2230 2464 y Fy(in)h Fr(V)2401 2479 y Fp(j)t Fz(+1)2527 2464 y Fy(,)g(so)g(that)1336 2763 y Fr(L)1402 2722 y Fz(2)1442 2763 y Fy(\()p Fm(T)p Fy(\))1584 2781 y Fz(2)1619 2763 y Fh(L)k Fy(=)27 b Fr(V)1858 2778 y Fz(0)1920 2763 y Fq(\010)2019 2592 y Fk( )2104 2638 y Fp(L)p Fo(\000)p Fz(1)2098 2668 y Fk(M)2112 2878 y Fp(j)t Fz(=0)2265 2763 y Fr(W)2357 2778 y Fp(j)2394 2592 y Fk(!)2522 2763 y Fy(;)1449 3077 y(dim)17 b Fr(V)1685 3092 y Fp(j)1749 3077 y Fy(=)28 b(dim)17 b Fr(W)2124 3092 y Fp(j)2188 3077 y Fy(=)28 b(2)2341 3036 y Fp(j)2409 3077 y Fr(:)324 3258 y Fy(The)33 b(space)h Fr(W)877 3273 y Fp(j)946 3258 y Fy(is)f(spanned)h(b)m(y)g Fq(f)p Fr( )1673 3273 y Fp(j)t(k)1748 3258 y Fq(g)1798 3217 y Fz(2)1833 3193 y Fh(j)1865 3217 y Fo(\000)p Fz(1)1798 3286 y Fp(k)r Fz(=0)1960 3258 y Fy(,)e(where)1435 3461 y Fr( )1498 3476 y Fp(j)t(k)1573 3461 y Fy(\()p Fr(x)p Fy(\))c(=)g(2)1885 3420 y Fp(j)t(=)p Fz(2)1991 3461 y Fr( )t Fy(\(2)2145 3420 y Fp(j)2182 3461 y Fr(x)22 b Fq(\000)h Fr(k)s Fy(\))324 3679 y(and)41 b Fr( )k Fy(is)d(the)g(\\mother)g(w)m(a)m(v)m(elet".)71 b(Let)42 b Fr(\022)1996 3698 y Fz(2)2031 3679 y Fh(L)52 b Fy(:=)42 b Fq(f)p 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Fy(=)28 b(0,)k(1,)g Fr(:)17 b(:)g(:)p Fy(,)33 b Fr(L)p Fy(.)470 4521 y(Littlew)m(o)s(o)s(d-P)m(aley)50 b(theorem)f(can)g(b)s(e)g (generalized)h(to)f(bases)h(other)e(than)h(the)324 4641 y(trigonometric)34 b(one)h(b)m(y)g(observing)g(that)g(the)f(pro)s(ofs)g (do)g(not)g(use)h(the)g(explicit)h(form)324 4761 y(of)29 b Fr(\036)490 4776 y Fp(N)588 4761 y Fy(\(16\))g(and)h Fr(P)1041 4776 y Fp(s)1108 4761 y Fy(\(17\),)g(but)h(only)f(some)h(of)f (their)h(prop)s(erties,)g(so)g(that)f(the)g(results)324 4882 y(are)45 b(v)-5 b(alid)46 b(for)f(larger)g(function)h(classes.)84 b(In)46 b(particular,)j(the)d(follo)m(wing)g(theorem)324 5002 y(holds:)p eop %%Page: 25 25 25 24 bop 324 241 a Ft(6)97 b(NUMERICAL)34 b(IMPLEMENT)-8 b(A)g(TION)1422 b Fy(25)324 548 y Fi(Theorem)38 b(5.5)49 b([Hern)o(\023)-55 b(andez)38 b(and)g(W)-9 b(eiss)38 b(1996,)56 b(Th.)c(7.16])46 b Fl(If)g Fr( )53 b Fq(2)c Fy(\003)3340 563 y Fp(\013)3389 548 y Fy(\()p Fm(T)p Fy(\))p Fl(,)324 668 y(then)33 b(the)h(function)g Fr(\022)i Fl(is)e(of)f(class)g Fy(\003)1682 683 y Fp(\013)1732 668 y Fy(\()p Fm(T)p Fy(\))h Fl(if)f(and)g(only)h(if)f(ther)-5 b(e)34 b(exists)g(a)f Fr(C)i(>)27 b Fy(0)34 b Fl(such)324 789 y(that)h(for)g(any)g Fr(j)e Fq(2)28 b Fm(N)1362 1009 y Fy(sup)1265 1097 y Fz(0)p Fo(\024)p Fp(k)r Fo(\024)p Fz(2)1484 1078 y Fh(j)1516 1097 y Fo(\000)p Fz(1)1623 1009 y Fq(jh)p Fr(\022)s(;)17 b( )1845 1024 y Fp(j)t(k)1920 1009 y Fq(ij)27 b(\024)h Fr(C)c Fy(2)2262 968 y Fo(\000)p Fp(j)t Fz(\()p Fp(\013)p Fz(+)2487 940 y Fg(1)p 2486 952 31 3 v 2486 994 a(2)2527 968 y Fz(\))2593 1009 y Fr(:)768 b Fy(\(21\))470 1303 y(Another)33 b(form)m(ulation)g(whic)m(h) h(is)f(useful)h(for)e(n)m(umerical)i(computations)f(is)324 1531 y Fi(Theorem)38 b(5.6)49 b Fl(If)42 b Fr( )j Fq(2)e Fy(\003)1386 1546 y Fp(\013)1435 1531 y Fy(\()p Fm(T)p Fy(\))p Fl(,)h(then)f(the)f(function)g Fr(\022)k Fl(is)c(of)g(class)g Fy(\003)3071 1546 y Fp(\013)3120 1531 y Fy(\()p Fm(T)p Fy(\))h Fl(if)f(and)324 1652 y(only)34 b(if)h(ther)-5 b(e)35 b(exists)f(a)h Fr(C)g(>)27 b Fy(0)35 b Fl(such)g(that)g(for)g (any)f Fr(j)g Fq(2)28 b Fm(N)1384 1872 y Fq(k)p Fr(\022)d Fq(\000)e Fy(\005)1677 1887 y Fp(j)1713 1872 y Fr(\022)s Fq(k)1811 1904 y Fp(L)1859 1885 y Ff(1)1924 1904 y Fz(\()p Fj(T)p Fz(\))2055 1872 y Fq(\024)28 b Fr(C)c Fy(2)2303 1831 y Fo(\000)p Fp(j)t(\013)2474 1872 y Fr(:)887 b Fy(\(22\))470 2100 y(F)-8 b(or)23 b(more)i(subtle)g(results)g(on)f(applications)h(of) f(w)m(a)m(v)m(elets)i(to)e(studies)i(of)d(lo)s(cal)h(regu-)324 2221 y(larit)m(y)e(of)f(functions,)k(see)e([Ja\013ard)32 b(and)h(Mey)m(er)h(1996],)24 b([Holsc)m(hneider)35 b(and)d(Tc)m(hamitc) m(hian)j(1991)o(],)324 2341 y([Mey)m(er)f(1998)o(].)56 b(F)-8 b(or)36 b(the)h(study)h(of)e(conjugacies,)j(ho)m(w)m(ev)m(er,)h (the)d(lo)s(cal)g(regularit)m(y)g(is)324 2461 y(empirically)42 b(found)e(to)f(b)s(e)h(the)h(same)g(at)e(all)h(p)s(oin)m(ts.)67 b(A)40 b(heuristic)h(explanation)g(of)324 2582 y(this)33 b(phenomenon)i(often)e(rep)s(eated)g(in)g(the)h(literature)f(is)h(that) e(the)i(conjugacies)g(are)324 2702 y(obtained)i(b)m(y)g(iterating)g (\(11\))e(whic)m(h)j(\\spreads")g(the)f(regularit)m(y)-8 b(,)37 b(i.e.,)g(mak)m(es)g(it)e(the)324 2822 y(same)d(at)f(all)g(p)s (oin)m(ts.)44 b(A)31 b(di\016cult)m(y)i(with)f(this)g(argumen)m(t)g(is) f(that)g(the)h(critical)g(p)s(oin)m(ts)324 2943 y(can)41 b(c)m(hange)g(the)g(regularit)m(y)g(of)f(a)g(function.)68 b(In)41 b(this)g(pap)s(er)f(w)m(e)i(will)f(not)f(explore)324 3063 y(lo)s(cal)32 b(regularit)m(y)-8 b(.)324 3396 y Fs(6)161 b(Numerical)53 b(Implemen)l(tation)324 3615 y Fy(In)33 b(this)h(subsection)i(w)m(e)e(discuss)h(the)f(n)m(umerical)h (implemen)m(tation)g(of)e(the)g(theorems)324 3735 y(from)f(Section)i (5.)324 4024 y Fn(6.1)135 b(General)46 b(Remarks)324 4209 y Fy(The)30 b(n)m(umerical)h(implemen)m(tation)h(brings)e(us)g(a)g (w)m(ealth)g(of)g(issues)h(whic)m(h)g(w)m(e)g(will)f(try)324 4329 y(to)i(discuss.)470 4500 y(One)i(issue)g(is)g(that)f(the)h(c)m (haracterizations)h(men)m(tioned)f(ab)s(o)m(v)m(e)g(in)m(v)m(olv)m(e)i (inequal-)324 4620 y(ities)g(that)g(ha)m(v)m(e)h(to)e(b)s(e)h (satis\014ed)h(for)e(an)h(in\014nite)g(n)m(um)m(b)s(er)h(of)f(in)m (tegers.)54 b(Ob)m(viously)-8 b(,)324 4740 y(the)38 b(n)m(umerical)h (calculation)f(can)g(only)g(compute)g(the)g(F)-8 b(ourier)37 b(and)h(w)m(a)m(v)m(elet)i(trans-)324 4861 y(form)29 b(up)i(to)e(a)h(\014nite)g(order.)43 b(It)30 b(is)g(conceiv)-5 b(able)32 b(that)d(the)h(b)s(eha)m(vior)h(of)e(the)i(functions)324 4981 y(is)i(di\013eren)m(t)h(for)e(high)g(F)-8 b(ourier)33 b(mo)s(des)g(than)g(for)f(the)h(v)-5 b(alues)33 b(that)g(can)f(b)s(e)h (explored.)p eop %%Page: 26 26 26 25 bop 324 241 a Ft(6)97 b(NUMERICAL)34 b(IMPLEMENT)-8 b(A)g(TION)1422 b Fy(26)470 548 y(In)25 b(spite)h(of)f(the)g(ab)s(o)m (v)m(e)h(solipsistic)h(argumen)m(t,)g(there)f(are)f(go)s(o)s(d)e (reasons)j(\(a)f(renor-)324 668 y(malization)44 b(group)h (description\))g(that)f(strongly)h(suggest)g(that)g(the)f(functions)i (w)m(e)324 789 y(are)c(studying)i(are)f(asymptotically)i(self-similar,) h(so)d(that)f(the)h(the)g(study)h(of)f(a)f(\014-)324 909 y(nite)35 b(n)m(um)m(b)s(er)h(of)e(scales)h(predicts)h(accurately)g (the)f(b)s(eha)m(vior)g(at)f(all)g(scales.)51 b(Indeed,)324 1029 y(w)m(e)35 b(\014nd)f(empirically)h(that)f(the)g(upp)s(er)g(b)s (ounds)h(giving)f(the)g(regularit)m(y)g(b)s(ecome)h(ap-)324 1150 y(pro)m(ximately)j(iden)m(tities.)56 b(W)-8 b(e)37 b(see)g(that,)g(after)f(a)g(v)m(ery)i(short)e(transien)m(t,)i(the)f (upp)s(er)324 1270 y(b)s(ounds)g(b)s(ecome)h(iden)m(tities)h(up)f(to)e (a)h(small)g(p)s(erio)s(dic)h(error)e(whose)j(in)m(terpretation)324 1391 y(w)m(e)33 b(discuss)i(in)e(Section)g(6.5.)470 1511 y(Because)j(of)d(this)i(empirical)g(observ)-5 b(ation)35 b(and)f(the)g(renormalization)h(group)f(de-)324 1631 y(scription,)40 b(w)m(e)f(b)s(eliev)m(e)g(that)e(it)h(is)g(reasonable)h 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Fz(CC)1354 2814 y Fy(and)g Fr(\022)1592 2778 y Fz(QQ)1706 2814 y Fy(,)f(whose)i(smo)s(othness)g(is)f(more)g (than)g(one.)470 2935 y(In)45 b(these)h(t)m(w)m(o)f(cases,)k(as)c(w)m (ell)h(as)f(for)f(all)h(conjugacies)g(b)s(et)m(w)m(een)i Fr(f)55 b Fy(and)45 b Fr(g)j Fy(for)324 3055 y Fr(f)55 b Fy(b)s(eing)46 b(critical)f(\(C)h(or)e(Q\),)h(an)g(imp)s(ortan)m(t)g (issue)i(is)e(the)h(presence)h(of)d(big)h(gaps)324 3176 y(b)s(et)m(w)m(een)f(the)f(iterates)h Fr(f)1305 3139 y Fp(n)1351 3176 y Fy(\(0\))e(\(see)i(Section)f(4.3\).)73 b(This)44 b(is)f(due)g(to)g(the)f(fact)h(that)324 3296 y(w)m(e)g(p)s(erform)g(F)-8 b(ast)42 b(F)-8 b(ourier)42 b(T)-8 b(ransform)43 b(\(FFT\))f(or)g(Discrete)h(W)-8 b(a)m(v)m(elet)44 b(T)-8 b(ransform)324 3416 y(\(D)m(WT\))32 b(not)h(on)f(the)g(exact)i(v)-5 b(alues)33 b(of)f Fr(\022)j Fy(at)d(the)h(p)s(oin)m(ts)g Fr(x)2524 3431 y Fp(`)2589 3416 y Fy(\(13\),)f(but)h(on)f(the)h(v)-5 b(alues)324 3537 y(of)43 b(the)h(in)m(terp)s(olating)g(cubic)h(p)s(olynomials)g(at) 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b(it)f(is)h(w)m(ell)h(kno)m(wn,)h Fr(w)2197 1660 y Fp(a;b)2321 1645 y Fq(2)33 b Fy(\003)2488 1660 y Fo(\000)11 b Fz(log)2645 1677 y Fh(b)2689 1660 y Fp(a)2766 1645 y Fy(and)35 b(for)g(an)m(y)i Fr(\016)f(>)d Fy(0,)324 1765 y Fr(w)394 1780 y Fp(a;b)524 1765 y Fr(=)-60 b Fq(2)28 b Fy(\003)675 1780 y Fo(\000)11 b Fz(log)832 1797 y Fh(b)875 1780 y Fp(a)p Fz(+)p Fp(\016)1006 1765 y Fy(.)470 1885 y(T)-8 b(o)27 b(calibrate)g(our)g(n)m(umerical)h(metho) s(ds,)h(w)m(e)f(ha)m(v)m(e)g(generated)g(the)f Fr(w)3056 1900 y Fp(a;b)3174 1885 y Fy(functions)324 2006 y(at)33 b(p)s(oin)m(ts)h(obtained)f(b)m(y)i(iterating)e(the)h (di\013eomorphisms)h(w)m(e)f(are)f(studying.)47 b(Then,)324 2126 y(w)m(e)33 b(obtained)g(the)g(regularit)m(y)h(applying)f(the)g (metho)s(ds)h(outlined)f(ab)s(o)m(v)m(e.)470 2246 y(This)g(pro)s (cedure)g(ga)m(v)m(e)f(us)g(an)g(idea)g(of)f(the)h(sev)m(erit)m(y)j(of) c(the)h(problem)g(of)g(the)g(lac)m(k)324 2367 y(of)g(equidistribution)j (of)d(the)h(p)s(oin)m(ts)g(or)f(of)g(the)h(e\013ects)h(of)e(the)h(in)m (terp)s(olation)g(etc.)470 2487 y(W)-8 b(e)30 b(think)g(that)f(the)h (use)g(of)f(W)-8 b(eierstrass)31 b(function)f(to)f(calibrate)h(the)g (metho)s(ds)g(is)324 2608 y(appropriate)35 b(b)s(ecause)h(the)f(w)m (orking)h(h)m(yp)s(othesis)h(\(A2\))e(asserts)h(that)e(the)i(functions) 324 2728 y(w)m(e)e(are)f(studying)h(are)f(v)m(ery)i(similar)e(to)g(the) g(functions)h(\(23\).)44 b(Hence,)35 b(one)e(can)g(hop)s(e)324 2848 y(that)38 b(the)g(problems)i(of)d(in)m(terp)s(olation)i(and)f(lac) m(k)h(of)f(distribution)h(can)g(b)s(e)f(assessed)324 2969 y(b)m(y)33 b(testing)h(the)f(metho)s(ds)g(on)f(\(23\).)324 3258 y Fn(6.3)135 b(Finite)46 b(Di\013erences)g(Metho)t(d)324 3442 y Fy(W)-8 b(e)33 b(applied)g(Theorem)h(5.1)e(for)g Fr(y)f Fy(=)d(2)1783 3406 y Fo(\000)p Fp(j)1874 3442 y Fy(,)k(in)h(whic)m(h)h(case)g(\(14\))e(yields)1278 3662 y(log)1404 3686 y Fz(2)1460 3662 y Fq(kD)1590 3621 y Fp(n)1587 3689 y Fz(2)1622 3670 y Ff(\000)p Fh(j)1707 3662 y Fr(\022)s Fq(k)1805 3678 y Fp(L)1853 3659 y Ff(1)1918 3678 y Fz(\()p Fj(T)p Fz(\))2049 3662 y Fq(\024)c Fy(const)23 b Fq(\000)g Fr(\013)q(j)324 3882 y Fy(\(naturally)-8 b(,)40 b(one)f(can)g(consider)h(the)f(case)h(of)e(arbitrary)h Fr(y)t Fy('s\).)61 b(As)39 b(examples)i(of)d(the)324 4003 y(results)i(obtained)g(b)m(y)g(applying)f(this)h(metho)s(d,)h(w)m (e)f(sho)m(w)h(in)e(Figure)g(9)g(the)g(plot)g(of)324 4123 y(log)450 4147 y Fz(2)506 4123 y Fq(kD)636 4087 y Fz(1)633 4155 y(2)668 4136 y Ff(\000)p Fh(j)752 4123 y Fr(\022)s Fq(k)850 4139 y Fp(L)898 4120 y Ff(1)963 4139 y Fz(\()p Fj(T)p Fz(\))1094 4123 y Fy(as)28 b(a)f(function)i(of)e Fr(j)33 b Fy(for)27 b(four)h(conjugacies)g(of)f(t)m(yp)s(e)i(NC)f (\(x's\))g(and)324 4243 y(four)41 b(ones)i(of)f(t)m(yp)s(e)h(CQ)f (\(circles\);)48 b(to)42 b(calculate)h Fr(\022)2304 4207 y Fz(NC)2412 4243 y Fy(,)h(w)m(e)g(used)f(10)2967 4207 y Fz(7)3048 4243 y Fy(iterates)f(and)324 4364 y(2)373 4328 y Fz(22)486 4364 y Fy(in)m(terp)s(olated)d(v)-5 b(alues,)41 b(while)e(for)f Fr(\022)1827 4328 y Fz(CQ)1976 4364 y Fy(these)i(n)m(um)m(b)s(ers)g(w)m(ere)g(2)26 b Fq(\002)g Fy(10)3138 4328 y Fz(6)3216 4364 y Fy(and)38 b(2)3460 4328 y Fz(22)3535 4364 y Fy(,)324 4484 y(resp)s(ectiv)m(ely)-8 b(.)470 4605 y(In)33 b(the)f(fa)m(v)m(orable)i(case)f(\(NC\),)g(w)m(e)g (see)g(that)f(the)h(n)m(umerical)h(results)g(corresp)s(ond)324 4725 y(to)k(parallel)h(straigh)m(t)h(lines)g(that)e(co)m(v)m(er)j(the)e (whole)h(range)e(plotted.)63 b(On)39 b(the)g(other)324 4845 y(hand,)25 b(in)e(the)g(unfa)m(v)m(orable)h(case)g(\(CQ\),)f(the)g (n)m(umerical)h(results)g(presen)m(t)h(t)m(w)m(o)e(straigh)m(t)324 4966 y(lines)34 b(joined)e(b)m(y)i(a)e(break.)p eop %%Page: 28 28 28 27 bop 324 241 a Ft(6)97 b(NUMERICAL)34 b(IMPLEMENT)-8 b(A)g(TION)1422 b Fy(28)470 548 y(This)26 b(can)e(b)s(e)g(clearly)i (explained)g(b)s(ecause)f(the)g(graph)f(presen)m(ted)j(for)c(the)i(NC)g (case)324 668 y(includes)33 b(computations)e(of)g(in)g(whic)m(h)h(man)m (y)g(of)e(the)i(p)s(oin)m(ts)f(in)g(the)g(\014nite)h(di\013erence)324 789 y(op)s(erator)41 b(are)g(included)j(in)e(the)g(gaps.)71 b(Hence,)45 b(the)d(\014nite)h(di\013erence)g(op)s(erator)e(is)324 909 y(observing)34 b(the)f(regularit)m(y)g(of)f(the)h(in)m(terp)s (olating)g(spline.)470 1029 y(In)25 b(the)g(NC)g(case)g(the)g(gaps)f(b) s(et)m(w)m(een)j(the)e(iterates)g(did)g(not)f(exceed)j(1)p Fr(:)p Fy(5)5 b Fq(\002)g Fy(10)3291 993 y Fo(\000)p Fz(7)3386 1029 y Fy(.)40 b(A)m(t)324 1150 y(the)29 b(same)g(time,)h(in) f(the)f(CQ)h(case)g(the)g(maxim)m(um)h(gap)f(w)m(as)g(ab)s(out)f(2)14 b Fq(\002)g Fy(10)3130 1114 y Fo(\000)p Fz(4)3251 1150 y Fq(\031)28 b Fy(2)3405 1114 y Fo(\000)p Fz(12)3535 1150 y Fy(,)324 1270 y(whic)m(h)34 b(corresp)s(onds)g(quite)f(exactly)h (to)f(the)g(p)s(osition)f(of)g(the)h(break)h(in)f(the)g(graph.)470 1391 y(When)25 b(w)m(e)h(restrict)f(the)g(di\013erences)i(to)d(regions) 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b(uncertain)m(ties)i(are)324 3020 y(just)h(the)g(the)g(standard)g(errors)g(of)f(the)h(regression.)p 1104 3137 1678 4 v 1104 3140 V 1102 3261 4 121 v 1154 3225 a Fr(\021)p 1254 3261 V 103 w Fy(Range)g(of)f(log)1842 3249 y Fz(10)1933 3225 y Fr(t)p 2016 3261 V 209 w Fy(Regularit)m(y)p 2780 3261 V 1104 3264 1678 4 v 1104 3267 V 1102 3388 4 121 v 1155 3352 a(1)p 1254 3388 V 182 w([)p Fq(\000)p Fy(5)p Fr(:)p Fy(0)p Fr(;)17 b Fq(\000)p Fy(4)p Fr(:)p Fy(0])p 2016 3388 V 180 w(0)p Fr(:)p Fy(5247)k Fq(\006)i Fy(0)p Fr(:)p Fy(0009)p 2780 3388 V 1102 3509 V 1155 3472 a(2)p 1254 3509 V 182 w([)p Fq(\000)p Fy(3)p Fr(:)p Fy(5)p Fr(;)17 b Fq(\000)p Fy(2)p Fr(:)p Fy(5])p 2016 3509 V 180 w(0)p Fr(:)p Fy(5253)k Fq(\006)i Fy(0)p Fr(:)p Fy(0012)p 2780 3509 V 1102 3629 V 1155 3593 a(3)p 1254 3629 V 182 w([)p Fq(\000)p Fy(3)p Fr(:)p Fy(0)p Fr(;)17 b Fq(\000)p Fy(1)p Fr(:)p Fy(5])p 2016 3629 V 180 w(0)p Fr(:)p Fy(5244)k Fq(\006)i Fy(0)p Fr(:)p Fy(0008)p 2780 3629 V 1104 3632 1678 4 v 1104 3635 V 470 3825 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552 y Fp(n)1887 618 y Fz(1)1951 594 y Fy([)p Fr(P)2041 609 y Fp(\013)2086 590 y Fh(n)2128 609 y Fp(t)2180 594 y Fq(\003)22 b Fr(H)2333 609 y Fz(1)2372 594 y Fy(]\()p Fr(\013)2500 552 y Fp(n)2547 594 y Fr(x)p Fy(\))h(+)f Fr(\025)2818 552 y Fp(n)2818 618 y Fz(2)2881 594 y Fy([)p Fr(P)2971 609 y Fp(\013)3016 590 y Fh(n)3059 609 y Fp(t)3111 594 y Fq(\003)g Fr(H)3264 609 y Fz(2)3303 594 y Fy(]\()p Fr(\013)3431 552 y Fp(n)3478 594 y Fr(x)p Fy(\))g(+)g Fq(\001)17 b(\001)g(\001)3824 453 y Fk(\025)324 895 y Fy(If)37 b(w)m(e)h(tak)m(e)g(suprema)g(in)f Fr(x)h Fy(and)f(then)g (logarithms,)i(w)m(e)f(obtain)f(that)g(the)g(structure)324 1016 y(of)d(the)i(main)f(term)h(for)e(the)h(resulting)h(function)g (considered)h(as)e(a)g(function)g(of)g(log)17 b Fr(t)324 1136 y Fy(is)46 b(a)g(sum)h(of)f(a)g(linear)g(function)h(and)f(a)g (function)h(that)f(is)g(p)s(erio)s(dic.)85 b(The)47 b(slop)s(e)324 1256 y(of)d(the)g(linear)h(function)g(is,)j(of)c(course,)k(according)d (to)f(Theorem)i(5.4,)h(the)e(degree)324 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y(the)g(case)g(of)f(second)i(di\013erences)h(since)f(they)g(are)e(v)m (ery)i(susceptible)i(to)c(n)m(umerical)324 3062 y(errors.)55 b(Hence,)39 b(this)e(giv)m(es)h(us)f(con\014dence)h(on)e(the)h (reliabilit)m(y)h(of)e(the)g(metho)s(ds)i(w)m(e)324 3183 y(ha)m(v)m(e)c(used.)470 3303 y(W)-8 b(e)32 b(note)f(that)g(the)h (computation)f(of)g(\014rst)h(di\013erences)h(is)f(one)f(of)g(the)g (data)g(anal-)324 3423 y(ysis)e(features)f(included)i(in)e Fe(ACE/gr)p Fy(,)i(so)e(that)f(it)h(is)g(quite)h(feasible)f(to)g(carry) g(out)f(these)324 3544 y(explorations)33 b(in)g(an)g(in)m(teractiv)m(e) h(w)m(a)m(y)g(for)e(a)g(v)-5 b(ariet)m(y)34 b(of)e(functions.)324 3833 y Fn(6.6)135 b(Deca)l(y)46 b(of)f(W)-11 b(a)l(v)l(elet)47 b(Co)t(e\016cien)l(ts)324 4017 y Fy(Theorem)30 b(5.5)e(can)h(b)s(e)h (used)g(to)e(asses)i(the)g(regularit)m(y)f(of)g(the)g(functions)h(w)m (e)g(study)g(b)m(y)324 4138 y(examining)k(the)f(deca)m(y)h(of)e(the)h (co)s(e\016cien)m(ts)h(of)e(w)m(a)m(v)m(elet)j(transform.)470 4258 y(Nev)m(ertheless,)49 b(w)m(e)44 b(do)e(not)h(think)g(that)g(for)f (our)g(functions)i(it)f(is)g(necessary)i(to)324 4378 y(app)s(eal)32 b(to)h(Theorem)h(5.5.)470 4499 y(Note)28 b(that)f(the)h(w)m(orking)h(h)m(yp)s(othesis)h(\(A4\))d(giv)m(es)i(a)f (represen)m(tation)h(of)e(the)i(func-)324 4619 y(tion.)470 4740 y(It)k(is)g(not)f(di\016cult)i(to)e(sho)m(w)i(that,)f(for)f (functions)h(of)g(the)g(form)f(\(4\))g(in)h(the)g(w)m(ork-)324 4860 y(ing)39 b(h)m(yp)s(othesis)i(\(A4\),)g(the)f(degree)g(of)f (regularit)m(y)h(is)g(a)e(simple)j(ratio)e(b)s(et)m(w)m(een)i(the)324 4980 y(logarithms)33 b(of)f Fr(\025)g Fy(and)h(the)g(scaling)g(factor)f Fr(\013)h Fy(de\014ned)h(in)f(\(4\).)p eop %%Page: 31 31 31 30 bop 324 241 a Ft(6)97 b(NUMERICAL)34 b(IMPLEMENT)-8 b(A)g(TION)1422 b Fy(31)470 548 y(F)-8 b(or)30 b(functions)i(of)e(this) i(form,)f(the)g(logarithm)f(of)h(the)g(size)h(of)e(the)i(pro)5 b(jections)32 b(on)324 668 y(a)39 b(space)h Fr(V)736 683 y Fp(j)811 668 y Fy(should)h(deca)m(y)f(linearly)g(with)g Fr(j)45 b Fy(irresp)s(ectiv)m(e)d(of)d(what)g(is)h(the)f(w)m(a)m(v)m (elet)324 789 y(used.)59 b(In)38 b(particular,)h(one)f(do)s(es)g(not)f (need)i(to)e(use)h(w)m(a)m(v)m(elets)j(whic)m(h)e(are)e(smo)s(other)324 909 y(than)28 b(the)h(regularit)m(y)h(observ)m(ed)g(to)f(obtain)f(the)h (scaling)g(exp)s(onen)m(ts,)j(whic)m(h)e(also)f(giv)m(e)324 1029 y(the)k(regularit)m(y)-8 b(.)470 1150 y(In)41 b(our)g(n)m (umerical)h(studies)h(w)m(e)f(ha)m(v)m(e)g(used)g(Daub)s(ec)m(hies)g(w) m(a)m(v)m(elets)h(of)e(order)g(4,)324 1270 y(10,)33 b(20,)g(whic)m(h)h (w)m(e)h(will)e(denote)h(as)g(D4,)e(D10,)h(D20)f(resp)s(ectiv)m(ely)-8 b(.)48 b(It)34 b(is)f(kno)m(wn)i(that)324 1391 y(D4)p Fq(2)41 b Fy(\003)623 1406 y Fz(0)p Fp(:)p Fz(38)p Fp(:::)852 1391 y Fy(and)g(not)g(in)g(an)m(y)g(more)h(regular)f(space.)69 b(F)-8 b(or)40 b(large)h Fr(N)10 b Fy(,)43 b(D2)p Fr(N)52 b Fq(2)42 b Fy(\003)3479 1406 y Fp(l)3500 1417 y Fh(N)324 1511 y Fy(where)34 b Fr(l)635 1526 y 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b(w)m(e)f(can)f(tak)m(e)h Fr(a)d Fy(=)f(N,)k Fr(b)d Fy(=)f(C,)k Fr(c)c Fy(=)g(Q,)k(or)f Fr(a)d Fy(=)g(Q,)k Fr(b)d Fy(=)f(C,)k Fr(c)c Fy(=)g(N.)52 b(When)324 2192 y(w)m(e)34 b(try)f(to)f(carry)i(out)e(this)i(v)m(eri\014cation,)g(w)m (e)g(\014nd)g(out)e(that,)h(up)g(to)g(the)g(error)g(of)f(the)324 2312 y(calculation,)h(w)m(e)h(obtain)e(that)h(\(24\))e(b)s(ecomes)j(an) f(iden)m(tit)m(y)-8 b(.)470 2433 y(This)33 b(is)e(presumably)j(not)d(a) g(coincidence.)45 b(W)-8 b(e)32 b(b)s(eliev)m(e)h(that)e(it)g(is)h (again)f(a)g(man-)324 2553 y(ifestation)g(of)f(the)h(self-similarit)m (y)i(of)d(the)i(function)f(at)f(small)i(scales.)44 b(If)31 b(w)m(e)h(comp)s(ose)324 2673 y(t)m(w)m(o)i(functions)h(that)e(in)h (eac)m(h)h(small)f(scale)h(ha)m(v)m(e)g(oscillations)g(comparable)g(to) e(those)324 2794 y(allo)m(w)m(ed)38 b(b)m(y)g(the)f(H\177)-49 b(older)38 b(exp)s(onen)m(t,)h(the)f(resulting)g(function)f(will)h (also)f(ha)m(v)m(e)h(oscil-)324 2914 y(lations)h(that)f(are)h (comparable)g(to)g(the)g(pro)s(duct)g(of)f(the)h(H\177)-49 b(older)40 b(exp)s(onen)m(ts.)63 b(Note)324 3034 y(ho)m(w)m(ev)m(er)36 b(that)f(this)g(argumen)m(t)g(do)s(es)f(not)g(suggest)i(that)e(there)h (is)g(a)f(simple)h(relation)324 3155 y(b)s(et)m(w)m(een)f(the)f (regularit)m(y)h(of)e(a)g(function)h(and)g(its)g(in)m(v)m(erse.)470 3275 y(W)-8 b(e)35 b(note)f(that)g(the)g(equation)h(\(24\))f(can)g(b)s (e)g(describ)s(ed)i(as)f(sa)m(ying)g(that)f(the)g(reg-)324 3396 y(ularities)41 b(of)e(the)i(conjugacies)g(as)g(indexed)h(b)m(y)f (the)f(classes)i(form)e(a)g(m)m(ultiplicativ)m(e)324 3516 y(sup)s(erco)s(cycle.)46 b(W)-8 b(e)33 b(\014nd)g(empirically)h (it)f(is)g(a)f(co)s(cycle.)470 3636 y(In)40 b(the)f(follo)m(wing)h (section,)i(w)m(e)e(will)g(deriv)m(e)h(upp)s(er)f(b)s(ounds)f(for)g (the)h(regularit)m(y)324 3757 y(that)32 b(are)h(co)s(cycles.)324 4046 y Fn(8.2)135 b(Scalings)48 b(of)g(the)g(Recurrence)g(and)f(Upp)t (er)g(Bounds)g(on)630 4195 y(H\177)-67 b(older)46 b(Exp)t(onen)l(ts)f (of)g(Conjugacies)324 4380 y Fy(In)c(this)h(section,)j(w)m(e)d(discuss) h(the)f(relation)f(b)s(et)m(w)m(een)i(the)f(scaling)f(prop)s(erties)h (and)324 4500 y(smo)s(othness)34 b(of)e(the)h(conjugacy)-8 b(.)470 4621 y(Scalings)36 b(ha)m(v)m(e)h(b)s(een)f(studied)h(n)m (umerically)h(from)d(the)h(b)s(eginning)g(of)f(renormal-)324 4741 y(ization)e(theory)-8 b(.)44 b(Some)33 b(of)f(them)h(ha)m(v)m(e)h (b)s(een)f(prob)s(ed)g(to)f(hold.)470 4861 y(In)e(this)h(section,)g(w)m (e)g(will)f(rep)s(ort)g(some)g(rigorous)g(results)h(that)f(sho)m(w)g (that)g(if)f(cer-)324 4982 y(tain)i(scalings)h(hold,)f(then)g(there)h (are)f(b)s(ounds)g(for)g(the)g(regularit)m(y)g(of)g(the)g(conjugacy)-8 b(.)p eop %%Page: 35 35 35 34 bop 324 241 a Ft(8)97 b(SOME)33 b(BOUNDS)g(ON)f(THE)i(REGULARITY) f(OF)f(CONJUGA)m(CIES)208 b Fy(35)470 548 y(Since)31 b(these)f(scaling)g(relations)g({)f(h)m(yp)s(othesis)j(to)d(our)g (lemma)h({)f(are)g(n)m(umerically)324 668 y(accessible,)i(w)m(e)d(can)g (use)g(the)g(rigorous)g(results)g(to)g(obtain)f(n)m(umerical)i(upp)s (er)f(b)s(ounds.)470 789 y(One)39 b(of)g(the)g(\014rst)h(n)m(umerical)g (observ)-5 b(ations)40 b(that)f(w)m(ere)h(made)f(in)h(the)f(study)h(of) 324 909 y(the)33 b(golden)g(mean)g(rotation)f(n)m(um)m(b)s(er)i (critical)f(circle)h(maps)f(w)m(as)g(that)1564 1111 y(\()p Fr(f)1661 1070 y Fo(\017)1700 1111 y Fy(\))1738 1070 y Fp(Q)1794 1078 y Fh(n)1840 1111 y Fy(\(0\))27 b Fq(\031)h Fr(\013)2160 1070 y Fo(\000)p Fp(n)2159 1136 y Fo(\017)2295 1111 y Fr(;)1066 b Fy(\(25\))324 1313 y(where)30 b Fq(\017)g Fy(stands)g(for)f(N,)g(C,)h(Q,)g(and)f Fr(\013)1766 1328 y Fo(\017)1835 1313 y Fy(are)g(univ)m(ersal)j(constan)m(ts.)44 b(The)30 b(n)m(um)m(b)s(ers)h Fr(\013)3522 1328 y Fo(\017)324 1433 y Fy(pla)m(y)i(a)f(fundamen)m(tal)i(role)f(in)g(the)g(\014xed)g(p) s(oin)m(t)g(equations.)470 1554 y(F)-8 b(or)44 b(non-critical)h(maps,)k (b)s(ecause)d(of)e(Theorem)i(2.4,)i Fr(\013)2664 1569 y Fz(N)2766 1554 y Fy(is)d(the)g(same)h(as)f(for)324 1674 y(rotations)32 b(b)m(y)i(golden)f(mean,)g(and,)g(b)m(y)g(using)g (the)g(w)m(ell-kno)m(wn)i(relation)1320 1851 y Fr(Q)1397 1866 y Fp(n)p 1275 1896 215 4 v 1275 1987 a Fr(Q)1352 2002 y Fp(n)p Fz(+1)1527 1919 y Fy(=)28 b Fr(\015)f Fy(+)22 b Fr(C)7 b Fy(\()p Fq(\000)p Fr(\015)2055 1878 y Fz(2)2095 1919 y Fy(\))2133 1878 y Fp(n)2202 1919 y Fy(+)22 b Fr(o)p Fy(\()p Fr(\015)2441 1878 y Fz(2)p Fp(n)2523 1919 y Fy(\))32 b Fr(;)324 2178 y Fy(w)m(e)h(obtain)g Fr(\013)833 2193 y Fz(N)918 2178 y Fy(=)27 b Fr(\015)1077 2142 y Fo(\000)p Fz(1)1172 2178 y Fy(.)470 2298 y(W)-8 b(e)33 b(note)h(that)e(for)h(the) g(cubic)h(critical)g(case,)g(there)f(are)g(unpublished)j(computer)324 2419 y(assisted)23 b(pro)s(ofs)f(\([Mestel,)34 b(thesis)g(\(1984\))o (],)24 b([Lanford)32 b(and)h(de)g(la)f(Lla)m(v)m(e,)i(man)m(uscript)g (\(1984\))o(]\))324 2539 y(that)39 b(establish)i(the)f(existence)i(of)c (the)i Fr(\013)1902 2554 y Fz(C)1957 2539 y Fy(,)i(upp)s(er)d(and)h(lo) m(w)m(er)g(b)s(ounds)h(for)d(it,)j(and)324 2659 y(the)33 b(fact)f(that)h(\(25\))e(holds)j(for)e(maps)h(in)g(op)s(en)g(sets.)470 2780 y(Some)i(relation)f(b)s(et)m(w)m(een)h(the)g(scaling)f(prop)s (erties)h(of)f(the)g(returns)h(and)f(the)g(reg-)324 2900 y(ularit)m(y)f(of)f(the)h(conjugacy)g(is)h(giv)m(en)f(b)m(y)324 3088 y Fi(Lemma)39 b(8.1)49 b Fl(L)-5 b(et)1338 3290 y Fr(f)1397 3243 y Fp(Q)1453 3251 y Fh(n)1386 3314 y Fz(1)1499 3290 y Fy(\(0\))83 b(=)g Fr(C)1936 3305 y Fz(1)1975 3290 y Fr(\013)2038 3249 y Fo(\000)p Fp(n)2037 3315 y Fz(1)2162 3290 y Fy(+)22 b Fr(o)p Fy(\()p Fr(\013)2408 3249 y Fo(\000)p Fp(n)2407 3315 y Fz(1)2509 3290 y Fy(\))841 b(\(26\))1338 3435 y Fr(f)1397 3388 y Fp(Q)1453 3396 y Fh(n)1386 3460 y Fz(2)1499 3435 y Fy(\(0\))83 b(=)g Fr(C)1936 3450 y Fz(2)1975 3435 y Fr(\013)2038 3394 y Fo(\000)p Fp(n)2037 3460 y Fz(2)2162 3435 y Fy(+)22 b Fr(o)p Fy(\()p Fr(\013)2408 3394 y Fo(\000)p Fp(n)2407 3460 y Fz(2)2509 3435 y Fy(\))841 b(\(27\))324 3637 y Fl(and)1392 3758 y Fr(\033)31 b Fy(:=)d(log)17 b Fq(j)p Fr(\013)1841 3773 y Fz(2)1881 3758 y Fq(j)p Fr(=)g Fy(log)f Fq(j)p Fr(\013)2206 3773 y Fz(1)2245 3758 y Fq(j)39 b Fr(=)-60 b Fq(2)28 b Fm(N)p Fr(:)894 b Fy(\(28\))470 3924 y Fl(Then,)47 b(if)d Fr(h)h Fl(satis\014es)g Fr(h)29 b Fq(\016)h Fr(f)1562 3939 y Fz(1)1647 3924 y Fy(=)47 b Fr(f)1818 3939 y Fz(2)1887 3924 y Fq(\016)29 b Fr(h)p Fl(,)48 b Fr(h)p Fy(\(0\))e(=)g(0)p Fl(,)h(then,)g(for)e(every)g Fr(\016)50 b(>)c Fy(0)p Fl(,)324 4045 y Fr(h)39 b(=)-60 b Fq(2)28 b Fy(\003)570 4060 y Fp(\033)r Fz(+)p Fp(\016)705 4045 y Fl(.)324 4277 y(Pr)-5 b(o)g(of)21 b(.)43 b Fy(F)-8 b(or)32 b(an)m(y)h Fr(\037)28 b(>)f Fy(0)32 b(w)m(e)i(ha)m(v)m(e)787 4479 y Fr(h)22 b Fq(\016)g Fr(f)996 4432 y Fp(Q)1052 4440 y Fh(n)985 4503 y Fz(1)1098 4479 y Fy(\(0\))f Fq(\000)i Fr(h)p Fy(\(0\))p 787 4524 739 4 v 857 4542 a Fk(h)904 4652 y Fr(f)963 4605 y Fp(Q)1019 4613 y Fh(n)952 4677 y Fz(1)1065 4652 y Fy(\(0\))e Fq(\000)i Fy(0)1360 4542 y Fk(i)1407 4564 y Fp(\037)1563 4546 y Fy(=)1747 4479 y Fr(f)1806 4432 y Fp(Q)1862 4440 y Fh(n)1795 4503 y Fz(2)1908 4479 y Fy(\(0\))p 1676 4524 428 4 v 1676 4542 a Fk(h)1723 4652 y Fr(f)1782 4605 y Fp(Q)1838 4613 y Fh(n)1771 4677 y Fz(1)1884 4652 y Fy(\(0\))2009 4542 y Fk(i)2056 4564 y Fp(\037)2141 4546 y Fy(=)2306 4479 y Fr(C)2376 4494 y Fz(2)2416 4479 y Fr(\013)2479 4438 y Fo(\000)p Fp(n)2478 4503 y Fz(2)2602 4479 y Fy(+)f Fr(o)p Fy(\()p Fr(\013)2848 4438 y Fo(\000)p Fp(n)2847 4503 y Fz(2)2950 4479 y Fy(\))p 2255 4524 785 4 v 2255 4623 a Fr(C)2332 4576 y Fp(\037)2325 4647 y Fz(1)2380 4623 y Fr(\013)2443 4576 y Fo(\000)p Fp(\037n)2442 4647 y Fz(1)2610 4623 y Fy(+)g Fr(o)p Fy(\()p Fr(\013)2856 4576 y Fo(\000)p Fp(\037n)2855 4647 y Fz(1)3001 4623 y Fy(\))3082 4546 y Fr(:)279 b Fy(\(29\))470 4882 y(W)-8 b(e)41 b(argue)g(b)m(y)h(con)m(tradiction:)61 b(if)41 b Fr(h)h Fq(2)g Fy(\003)2084 4897 y Fp(\037)2172 4882 y Fy(for)f(some)g Fr(\037)h(>)g(\033)t Fy(,)h(w)m(e)f(use)g(\(29\))e (to)324 5002 y(pro)m(v)m(e)33 b(b)m(y)f(induction)h(that)e Fr(h)1419 4966 y Fz(\()p Fp(n)p Fz(\))1521 5002 y Fy(\(0\))c(=)h(0)j (for)g(all)h Fr(n)c Fq(\024)g Fr(\033)t Fy(,)k Fr(n)c Fq(2)g Fm(N)p Fy(.)43 b(Then)32 b(w)m(e)h(note)f(that)p eop %%Page: 36 36 36 35 bop 324 241 a Ft(8)97 b(SOME)33 b(BOUNDS)g(ON)f(THE)i(REGULARITY) f(OF)f(CONJUGA)m(CIES)208 b Fy(36)324 548 y Fr(h)40 b Fq(2)h Fy(\003)595 563 y Fp(\033)r Fz(+)p Fp(\016)769 548 y Fy(\(for)f(an)m(y)g Fr(\016)k(>)c Fy(0\))g(w)m(ould)h(imply)g (that)e(if)h(w)m(e)h(substitute)g Fr(\037)g Fy(=)f Fr(\033)31 b Fy(+)c Fr(\016)43 b Fy(in)324 668 y(\(29\),)d(the)g(left-hand)f(side) i(is)f(b)s(ounded)g(uniformly)g(in)g Fr(n)p Fy(.)64 b(A)m(t)39 b(the)h(same)g(time,)i(the)324 789 y(righ)m(t-hand)32 b(side)i(of)e(\(29\))g(is)h(un)m(b)s(ounded)h(in)f Fr(n)p Fy(.)1367 b Fc(\003)470 1042 y Fy(W)-8 b(e)33 b(emphasize)i(that)e (Lemma)g(8.1)g(do)s(es)g(not)g(conclude)h(an)m(ything)f(when)h Fr(\014)g Fq(2)29 b Fm(N)p Fy(;)324 1162 y(in)g(particular,)g(it)g(do)s (es)g(not)f(conclude)i(an)m(ything)g(in)f(the)g(cases)h(when)f Fr(f)2999 1177 y Fz(1)3068 1162 y Fy(and)f Fr(f)3301 1177 y Fz(2)3369 1162 y Fy(ha)m(v)m(e)324 1283 y(the)40 b(same)g(scaling)h(factor)e Fr(\013)q Fy(,)i(whic)m(h)g(happ)s(ens)g (when)g Fr(f)2481 1298 y Fz(1)2560 1283 y Fy(and)f Fr(f)2805 1298 y Fz(2)2884 1283 y Fy(are)g(in)g(the)g(same)324 1403 y(univ)m(ersalit)m(y)35 b(class.)470 1523 y(W)-8 b(e)39 b(ha)m(v)m(e)g(v)m(eri\014ed)i(the)d(relations)h(\(25\))f(for)g (the)g(maps)h(w)m(e)h(considered)g(and)e(ob-)324 1644 y(tained)33 b(v)-5 b(alues)33 b(of)f Fr(\013)i Fy(as)e(follo)m(ws:)1203 1864 y Fr(\013)1265 1879 y Fz(C)1348 1864 y Fy(=)27 b Fq(\000)p Fy(1)p Fr(:)p Fy(2886)32 b Fr(;)212 b(\013)2133 1879 y Fz(Q)2220 1864 y Fy(=)27 b Fq(\000)p Fy(1)p Fr(:)p Fy(194)32 b Fr(;)706 b Fy(\(30\))324 2084 y(whic)m(h)49 b(agree)f(with)h(the)f(v)-5 b(alues)49 b(rep)s(orted)g(in)f([Shenk)m (er)34 b(1982])48 b(\(for)f(the)h(C)h(case\),)324 2204 y(and)34 b(those)h(in)f(the)g(pap)s(ers)h([Hu)e(et)g(al.)f(1990)o(],)j ([Delb)s(ourgo)d(and)g(Kenn)m(y)i(1991)o(],)h(ev)m(en)324 2325 y(though)j(they)h(consider)h(functions)f(that)f(are)g(p)s(erio)s (dized)i(v)m(ersions)g(of)e(p)s(olynomials)324 2445 y(whic)m(h)c(are)e (not)h Fr(C)1016 2409 y Fz(1)1055 2445 y Fy(.)470 2565 y(T)-8 b(aking)48 b(the)h(v)-5 b(alues)48 b(in)g(\(30\))f(and)h(the)g (exact)h(v)-5 b(alue)48 b(for)f Fr(\013)2803 2580 y Fp(N)2870 2565 y Fy(,)52 b(w)m(e)d(obtain)e(the)324 2686 y(follo)m(wing)g(upp)s (er)g(b)s(ounds)h(for)e(the)h(regularit)m(y)h(of)e(the)h(conjugacies)h (b)s(et)m(w)m(een)h(the)324 2806 y(maps)33 b Fr(f)43 b Fy(and)33 b Fr(g)t Fy(.)p 1128 2945 1629 4 v 1128 2948 V 1126 3069 4 121 v 1178 3033 a Fq(#)60 b Fr(f)147 b(g)63 b Fq(!)p 1741 3069 V 174 w Fy(N)p 2063 3069 V 275 w(C)p 2434 3069 V 273 w(Q)p 2755 3069 V 1128 3072 1629 4 v 1128 3075 V 1126 3196 4 121 v 1178 3160 a(N)p 1741 3196 V 630 w(?)p 2063 3196 V 187 w(0)p Fr(:)p Fy(5269)p 2434 3196 V 99 w(0)p Fr(:)p 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(that)h(the)g(upp)s(er)g(b)s(ounds)h(that)e(w)m(ork)i(in)f(one)g(case)g (are)g(far)324 4361 y(o\013)28 b(in)g(another)h(seem)h(to)e(imply)h (that)f(there)h(are)g(di\013eren)m(t)g(mec)m(hanisms)i(limiting)e(the) 324 4481 y(regularit)m(y)-8 b(.)71 b(In)42 b(the)g(cases)h(CN,)f(QN,)g (QC,)g(these)h(mec)m(hanisms)h(cannot)e(b)s(e)g(just)g(a)324 4602 y(simple)34 b(scaling)f(argumen)m(t.)470 4722 y(W)-8 b(e)28 b(note)f(that)g(since)i(the)f Fr(\033)j Fy(de\014ned)e(in)e (\(28\))g(is)h(the)f(ratio)g(of)g(a)g(quan)m(tit)m(y)i(dep)s(end-)324 4842 y(ing)f(on)g(the)h(domain)g(and)f(a)g(quan)m(tit)m(y)i(dep)s (ending)f(on)f(the)h(range,)g(the)g(upp)s(er)g(b)s(ounds)324 4963 y(that)j(w)m(e)i(ha)m(v)m(e)g(deriv)m(ed)g(form)e(a)h(m)m (ultiplicativ)m(e)i(co)s(cycle.)p eop %%Page: 37 37 37 36 bop 324 241 a Ft(8)97 b(SOME)33 b(BOUNDS)g(ON)f(THE)i(REGULARITY) f(OF)f(CONJUGA)m(CIES)208 b Fy(37)470 548 y(F)-8 b(or)25 b(the)h(sak)m(e)h(of)e(completeness,)30 b(w)m(e)c(also)g(p)s(oin)m(t)g (out)f(that)g(there)i(is)f(a)f(v)m(ery)i(similar)324 668 y(argumen)m(t)44 b(that)f(giv)m(es)i(upp)s(er)f(b)s(ounds)h(for)e (the)h(regularit)m(y)g(of)f(the)h(conjugacy)g(for)324 789 y(the)i(maps)g(in)f(the)h(same)h(class.)83 b(The)46 b(follo)m(wing)g(set)g(of)f(ideas)i(w)m(as)f(found)g(useful)324 909 y(in)35 b([de)e(la)g(Lla)m(v)m(e)h(and)e(Sc)m(hafer,)i(man)m (uscript)g(\(1996\))o(].)51 b(There)37 b(it)e(is)g(sho)m(wn)i(that)e (this)324 1029 y(argumen)m(t)c(pro)s(duced)g(upp)s(er)g(b)s(ounds)g (that)f(w)m(ere)i(sharp)f(in)f(the)h(case)h(of)d(conjugacies)324 1150 y(of)j(limiting)h(sets)h(of)e(unimo)s(dal)h(maps.)324 1378 y Fi(Lemma)39 b(8.2)49 b Fl(Assume)35 b(that)1136 1719 y Fr(f)1195 1671 y Fp(Q)1251 1679 y Fh(n)1184 1743 y Fz(1)1297 1719 y Fy(\(0\))82 b(=)h Fr(C)1733 1734 y Fz(1)1772 1719 y Fr(\013)1835 1677 y Fo(\000)p Fp(n)1959 1719 y Fy(+)22 b Fr(D)2138 1734 y Fz(1)2187 1719 y Fy(~)-58 b Fr(\013)2241 1677 y Fo(\000)p Fp(n)2364 1719 y Fy(+)22 b Fr(o)p Fy(\()10 b(~)-59 b Fr(\013)2610 1677 y Fo(\000)p Fp(n)2712 1719 y Fy(\))1136 1864 y Fr(f)1195 1817 y Fp(Q)1251 1825 y Fh(n)1184 1888 y Fz(2)1297 1864 y Fy(\(0\))82 b(=)h Fr(C)1733 1879 y Fz(2)1772 1864 y Fr(\013)1835 1823 y Fo(\000)p Fp(n)1959 1864 y Fy(+)22 b Fr(D)2138 1879 y Fz(2)2187 1864 y Fy(~)-58 b Fr(\013)2241 1823 y Fo(\000)p Fp(n)2364 1864 y Fy(+)22 b Fr(o)p Fy(\()10 b(~)-59 b Fr(\013)2610 1823 y Fo(\000)p Fp(n)2712 1864 y Fy(\))324 2084 y Fl(with)34 b Fq(j)p Fr(\013)q Fq(j)27 b Fr(<)h Fq(j)10 b Fy(~)-59 b Fr(\013)p Fq(j)p Fl(,)35 b Fr(C)1038 2099 y Fz(1)1077 2084 y Fr(C)1147 2099 y Fz(2)1214 2084 y Fq(6)p Fy(=)27 b(0)p Fl(,)1585 2304 y Fr(D)1666 2319 y Fz(2)1733 2304 y Fq(6)p Fy(=)h(\()p Fr(C)1945 2319 y Fz(2)1984 2304 y Fr(=C)2103 2319 y Fz(1)2142 2304 y Fy(\))p Fr(D)2261 2319 y Fz(1)3388 2304 y Fy(\(31\))324 2524 y Fl(and)37 b Fr(\033)g Fy(:=)c(log)18 b Fq(j)10 b Fy(~)-59 b Fr(\013)p Fq(j)p Fr(=)17 b Fy(log)g Fq(j)p Fr(\013)q Fq(j)43 b Fr(=)-60 b Fq(2)33 b Fm(N)p Fl(.)53 b(Then,)38 b(if)f Fr(h)h Fl(satis\014es)f Fr(h)25 b Fq(\016)f Fr(f)2669 2539 y Fz(1)2741 2524 y Fy(=)33 b Fr(f)2898 2539 y Fz(2)2962 2524 y Fq(\016)24 b Fr(h)p Fl(,)38 b Fr(h)p Fy(\(0\))33 b(=)g(0)p Fl(,)324 2644 y(then)h Fr(h)40 b(=)-61 b Fq(2)28 b Fy(\003)786 2659 y Fp(\033)r Fz(+)p Fp(\016)956 2644 y Fl(for)35 b(any)g Fr(\016)d(>)27 b Fy(0)p Fl(.)324 2922 y(Pr)-5 b(o)g(of)21 b(.)40 b Fy(The)24 b(argumen)m(t)g(is)g(v)m(ery)h(similar)f(to)f(the)h(argumen)m(t)g(that) f(pro)m(v)m(ed)i(Lemma)f(8.1.)470 3043 y(Note)g(that)g(the)h (assumptions)h(imply)f(that)f Fr(\033)32 b(>)c Fy(1.)40 b(Hence)26 b(w)m(e)f(are)f(only)h(excluding)324 3163 y(regularities)33 b(higher)g(than)g Fr(C)1431 3127 y Fz(1)1470 3163 y Fy(.)470 3284 y(W)-8 b(e)33 b(argue)g(b)m(y)g(con)m (tradiction,)h(assuming)f(that)g Fr(h)28 b Fq(2)g Fy(\003)2541 3299 y Fp(\033)r Fz(+)p Fp(\016)2676 3284 y Fy(.)43 b(W)-8 b(e)33 b(note)g(that)1335 3504 y Fr(h)23 b Fq(\016)f Fr(f)1545 3456 y Fp(Q)1601 3464 y Fh(n)1534 3528 y Fz(1)1646 3504 y Fy(\(0\))g Fq(\000)h Fr(h)p Fy(\(0\))k(=)h Fr(f)2264 3456 y Fp(Q)2320 3464 y Fh(n)2253 3528 y Fz(2)2366 3504 y Fy(\(0\))k Fr(:)838 b Fy(\(32\))324 3724 y(Since)34 b Fr(h)27 b Fq(2)i Fy(\003)825 3739 y Fp(\033)r Fz(+)p Fp(\016)960 3724 y Fy(,)j(and)h Fr(\033)f(>)c Fy(1,)k(w)m(e)i(conclude) g(that)e Fr(h)2320 3687 y Fo(0)2344 3724 y Fy(\(0\))27 b(=)h Fr(C)2670 3739 y Fz(2)2709 3724 y Fr(=C)2828 3739 y Fz(1)2894 3724 y Fq(6)p Fy(=)g(0.)43 b(Hence)34 b Fr(h)f Fy(is)324 3844 y(in)m(v)m(ertible)i(in)d(a)h(neigh)m(b)s(orho)s(o)s(d)f (of)g(zero)h(and)g Fr(h)2125 3808 y Fo(\000)p Fz(1)2247 3844 y Fq(2)28 b Fy(\003)2409 3859 y Fp(\033)r Fz(+)p Fp(\016)2544 3844 y Fy(.)470 3964 y(W)-8 b(e)30 b(note)g(that)g Fr(h)p Fy(\()p Fr(t)p Fy(\))e(=)f Fr(P)14 b Fy(\()p Fr(t)p Fy(\))i(+)g Fr(R)q Fy(\()p Fr(t)p Fy(\))30 b(where)h Fr(P)14 b Fy(\()p Fr(t)p Fy(\))30 b(is)g(a)f(p)s(olynomial)h(of)g (degree)g([)p Fr(\033)t Fy(])324 4085 y(and)i Fq(j)p Fr(R)q Fy(\()p Fr(t)p Fy(\))p Fq(j)c(\024)g(j)p Fr(t)p Fq(j)979 4049 y Fp(\033)r Fz(+)p Fp(\016)1114 4085 y Fy(.)470 4205 y(Giv)m(en)34 b(the)f(assumptions)h(on)e(the)h(degrees)h (w)m(e)g(ha)m(v)m(e)g(made,)f(w)m(e)h(ha)m(v)m(e)424 4437 y Fr(P)14 b Fy(\()p Fr(C)609 4452 y Fz(1)648 4437 y Fr(\013)711 4396 y Fo(\000)p Fp(n)834 4437 y Fy(+)22 b Fr(D)1013 4452 y Fz(1)1063 4437 y Fy(~)-59 b Fr(\013)1116 4396 y Fo(\000)p Fp(n)1240 4437 y Fy(+)22 b Fr(o)p Fy(\()10 b(~)-59 b Fr(\013)1486 4396 y Fo(\000)p Fp(n)1587 4437 y Fy(\)\))27 b(=)h Fr(h)1850 4396 y Fo(0)1873 4437 y Fy(\(0\))10 b(~)-59 b Fr(\013)2061 4396 y Fo(\000)p Fp(n)2185 4437 y Fy(+)2283 4342 y Fk(X)2339 4552 y Fp(j)2443 4437 y Fr(C)2513 4452 y Fp(j)2550 4437 y Fy(\()p Fr(\013)2651 4396 y Fp(j)2687 4437 y Fy(\))2725 4396 y Fo(\000)p Fp(n)2849 4437 y Fy(+)22 b Fr(o)p Fy(\()p Fr(\013)3095 4396 y Fo(\000)p Fp(n)o Fz(\([)p Fp(\014)s Fz(]+1\))3423 4437 y Fy(\))470 4759 y(W)-8 b(e)33 b(also)g(note)f(that)h Fr(R)q Fy(\()p Fr(C)1445 4774 y Fz(1)1484 4759 y Fr(\013)1547 4723 y Fo(\000)p Fp(n)1671 4759 y Fy(+)22 b Fr(D)1850 4774 y Fz(1)1899 4759 y Fy(~)-59 b Fr(\013)1952 4723 y Fo(\000)p Fp(n)2076 4759 y Fy(+)22 b Fr(o)p Fy(\()10 b(~)-59 b Fr(\013)2322 4723 y Fo(\000)p Fp(n)2423 4759 y Fy(\)\))28 b(=)f Fr(o)p Fy(\()10 b(~)-59 b Fr(\013)2778 4723 y Fo(\000)p Fp(n)2880 4759 y Fy(\).)470 4880 y(Equating)33 b(the)g(co)s(e\016cien)m (ts)h(of)e(the)42 b(~)-59 b Fr(\013)1893 4844 y Fo(\000)p Fp(n)2027 4880 y Fy(in)32 b(\(32\),)g(w)m(e)h(obtain)f Fr(D)2900 4895 y Fz(2)2967 4880 y Fy(=)c(\()p Fr(C)3179 4895 y Fz(2)3218 4880 y Fr(=C)3337 4895 y Fz(1)3376 4880 y Fy(\))p Fr(D)3495 4895 y Fz(1)3535 4880 y Fy(,)324 5000 y(whic)m(h)47 b(con)m(tradicts)g(our)e(assumption)i(\(31\).)82 b(This)47 b(is)f(the)g(desired)h(con)m(tradiction)p eop %%Page: 38 38 38 37 bop 324 241 a Ft(9)97 b(CONCLUSION)2333 b Fy(38)324 548 y(with)27 b(the)g(assumption)i(that)d Fr(h)h Fy(admitted)h(a)e(T)-8 b(a)m(ylor)27 b(expansions)i(near)e(0)g(with)g(H\177)-49 b(older)324 668 y(b)s(ounds.)2827 b Fc(\003)470 887 y Fy(Note)31 b(that)g(in)g(con)m(trast)g(with)h(Lemma)f(8.1,)g(Lemma)h (8.2)e(con)m(tains)i(a)e(h)m(yp)s(othesis)324 1008 y(\(31\))44 b(whic)m(h)i(could)f(fail)g(for)f(a)h(\014nite)g(co)s(dimension)h(set)g (of)e(maps.)81 b(This)46 b(is)f(to)f(b)s(e)324 1128 y(exp)s(ected)k (since)g(the)f(other)g(h)m(yp)s(othesis)h(allo)m(ws)g(to)e(tak)m(e)h Fr(f)2653 1143 y Fz(1)2744 1128 y Fy(=)k Fr(f)2919 1143 y Fz(2)2959 1128 y Fy(,)f Fr(h)h Fy(=)h(Id.)85 b(In)324 1249 y(this)43 b(situation,)j(the)e(conjugacy)f Fr(h)g Fy(is)h(v)m(ery)g(regular.)75 b(The)44 b(elemen)m(tary)h(theory)e(of) 324 1369 y(renormalization)34 b(tells)h(us)f(that)g(\(31\))f(indeed)i (o)s(ccurs)f(in)g(a)g(set)g(whic)m(h)i(is)e(of)f(p)s(ositiv)m(e)324 1489 y(co)s(dimension.)470 1610 y(The)h(conjecture)g(3.1)e(amoun)m(ts)i (to)e(the)h(fact)g(that)f(the)h(upp)s(er)g(b)s(ounds)h(pro)s(duced)324 1730 y(b)m(y)f(applying)h(Lemma)f(8.2)f(are)h(sharp.)470 1851 y(A)m(t)41 b(the)g(momen)m(t,)j(unfortunately)-8 b(,)44 b(w)m(e)e(do)e(not)h(ha)m(v)m(e)h(accurate)f(enough)h(v)-5 b(alues)324 1971 y(for)37 b(the)48 b(~)-59 b Fr(\013)38 b Fy(for)g(cubic)g(critical)h(or)e(quin)m(tic)i(critical)g(maps)f(and,) h(hence,)i(cannot)d(giv)m(e)324 2091 y(concrete)27 b(v)-5 b(alues)27 b(for)e(the)h(upp)s(er)g(b)s(ounds)h(of)e(the)i(regularit)m (y)f(in)g(the)g(CC)h(or)f(QQ)f(cases.)324 2212 y(It)39 b(is)h(in)m(teresting)h(to)e(remark)h(that)g(in)f(the)h(NN)g(case,)i (these)e(b)s(ounds)h(are)e(far)g(from)324 2332 y(optimal.)324 2660 y Fs(9)161 b(Conclusion)324 2880 y Fy(W)-8 b(e)38 b(ha)m(v)m(e)h(studied)g(the)f(problem)h(of)e(the)h(smo)s(othness)i(of) d(maps)i(of)e(the)h(circle)h(with)324 3000 y(critical)33 b(p)s(oin)m(ts)g(and)g(golden)g(mean)g(rotation)f(n)m(um)m(b)s(er.)470 3120 y(The)39 b(\014rst)f(step)g(w)m(as)h(to)e(obtain)g(an)h(extremely) i(precise)f(calculation)f(of)f(the)h(pa-)324 3241 y(rameters)33 b(of)f(the)h(function)h(that)e(put)h(us)g(in)g(the)g(correct)g(univ)m (ersalit)m(y)i(class.)470 3361 y(The)43 b(most)g(imp)s(ortan)m(t)f (step)i(is)f(to)e(dev)m(elop)j(a)f(n)m(umerical)g(to)s(olkit)f(based)i (on)e(a)324 3481 y(wide)36 b(arra)m(y)f(of)f(metho)s(ds)i(from)f (harmonic)g(analysis)h(\(Littlew)m(o)s(o)s(d-P)m(aley)-8 b(,)37 b(w)m(a)m(v)m(elets\))324 3602 y(to)32 b(study)i(the)f (regularit)m(y)g(and)g(the)g(\014ne)g(scale)h(structure)g(of)e(these)i (functions.)470 3722 y(W)-8 b(e)40 b(ha)m(v)m(e)g(used)g(the)g(com)m (bination)g(of)f(the)g(metho)s(ds)h(applied)g(to)f(critical)g(circle) 324 3843 y(maps)33 b(to)f(asses)i(their)f(range)g(of)f(v)-5 b(alidit)m(y)d(.)470 3963 y(W)g(e)45 b(ha)m(v)m(e)h(found)f(indication) h(that)e(the)h(regularit)m(y)h(of)e(critical)i(circle)g(maps)f(is)324 4083 y(limited)33 b(b)m(y)h(considerations)g(other)f(than)f(just)h (scaling)h(of)e(recurrences.)324 4412 y Fs(10)161 b(Ac)l(kno)l (wledgmen)l(ts)324 4641 y Fy(W)-8 b(e)42 b(are)f(grateful)g(to)g(John)h (Gilb)s(ert,)1808 4616 y(\023)1796 4641 y(Alex)g(Haro,)i(Ricardo)d(P)m (\023)-46 b(erez-Marco,)45 b(Marek)324 4761 y(Ryc)m(hlik)34 b(and)f(John)g(V)-8 b(ano)32 b(for)g(fruitful)g(discussions.)470 4882 y(W)-8 b(e)34 b(w)m(ould)g(also)f(lik)m(e)i(to)d(thank)i(some)g(p) s(eople)g(w)m(e)g(do)f(not)g(kno)m(w)i(in)e(p)s(erson,)h(but)324 5002 y(who)47 b(generously)h(made)f(their)g(soft)m(w)m(are)g(pro)s (ducts)g(freely)h(a)m(v)-5 b(ailable.)85 b(W)-8 b(e)47 b(thank)p eop %%Page: 39 39 39 38 bop 324 241 a Ft(REFERENCES)2471 b Fy(39)324 548 y(P)m(aul)43 b(J.)g(T)-8 b(urner)44 b(for)e(the)h(plotting)f(to)s(ol)g Fe(ACE/gr)p Fy(,)47 b(Keith)c(Briggs)g(for)f(the)h(double-)324 668 y(double)35 b(routines,)g(and)f(the)h(team)f(dev)m(eloping)i(the)e (library)h(for)e(arbitrary)h(precision)324 789 y(arithmetic)k(GNU)f(MP) -8 b(.)39 b(Of)e(course,)j(w)m(e)e(ha)m(v)m(e)h(relied)f(on)g(the)g(w)m (ork)g(of)f(man)m(y)h(other)324 909 y(p)s(eople)32 b(included)h(in)f (the)g(references.)46 b(W)-8 b(e)32 b(ha)m(v)m(e)h(dev)m(elop)s(ed)g (and)f(run)g(our)f(programs)324 1029 y(on)43 b(the)h(computers)h(of)e (the)g(Departmen)m(t)h(of)f(Mathematics)i(of)e(the)h(Univ)m(ersit)m(y)i (of)324 1150 y(T)-8 b(exas)34 b(at)e(Austin.)470 1270 y(Our)h(researc)m(h)h(w)m(as)f(partially)g(supp)s(orted)h(b)m(y)f(NSF)g (and)f(T)-8 b(ARP)34 b(gran)m(ts.)324 1603 y Fs(References)324 1822 y Fy([Arnold)f(1961])48 b(V.)22 b(I.)g(Arnol'd,)j(\\Small)d (denominators.)h(I.)f(Mappings)h(of)e(the)i(circum-)524 1942 y(ference)34 b(on)m(to)e(itself)7 b(",)33 b Fl(Izv.)h(A)n(kad.)f (Nauk)i(SSSR)f(Ser.)g(Mat.)f Fi(25)f Fy(\(1961\),)g(21{86,)524 2063 y(correction:)50 b Fi(28)36 b Fy(\(1964\),)f(479{480.)f(English)i (translation:)50 b Fl(A)n(mer.)37 b(Math.)g(So)-5 b(c.)524 2183 y(T)e(r)i(ansl.)34 b(\(2\))e Fi(46)g Fy(\(1965\),)g(213{284.)324 2387 y([Berggren)h(et)g(al.)f(2000])48 b(L.)23 b(Berggren,)i(J.)d(Borw) m(ein,)k(and)d(P)-8 b(.)23 b(Borw)m(ein,)j Fl(Pi:)40 b(A)26 b(Sour)-5 b(c)g(e)524 2507 y(Bo)g(ok)p Fy(,)32 b(Second)i(ed.)f(Springer,)g(New)g(Y)-8 b(ork,)33 b(2000.)324 2710 y([Bohr)f(et)h(al.)g(1984])48 b(T.)34 b(Bohr,)h(P)-8 b(.)35 b(Bak,)g(and)f(M.)h(H.)g(Jensen,)h(\\T)-8 b(ransition)35 b(to)f(c)m(haos)524 2831 y(b)m(y)j(in)m(teraction)f(of)f(resonances)i (in)f(dissipativ)m(e)i(systems.)g(I)s(I.)e(Josephson)h(junc-)524 2951 y(tions,)29 b(c)m(harge-densit)m(y)h(w)m(a)m(v)m(es,)g(and)e (standard)f(maps",)i Fl(Phys.)h(R)-5 b(ev.)30 b(A)g(\(3\))d Fi(30)p Fy(:4)524 3072 y(\(1984\),)32 b(1970{1981.)324 3275 y([Borw)m(ein)i(and)e(Borw)m(ein)i(1984])48 b(J.)72 b(M.)h(Borw)m(ein)h(and)e(P)-8 b(.)73 b(B.)f(Borw)m(ein,)84 b(\\The)524 3395 y(arithmetic-geometric)47 b(mean)f(and)g(fast)g (computation)g(of)f(elemen)m(tary)j(func-)524 3516 y(tions",)33 b Fl(SIAM)i(R)-5 b(ev.)32 b Fi(26)p Fy(:3)h(\(1984\),)e(351{366.)324 3719 y([Borw)m(ein)j(and)e(Borw)m(ein)i(1987])48 b(J.)39 b(M.)g(Borw)m(ein)i(and)e(P)-8 b(.)39 b(B.)g(Borw)m(ein,)j Fl(Pi)e(and)h(the)524 3839 y(A)n(GM:)j(A)f(Study)g(in)g(A)n(nalytic)g (Numb)-5 b(er)43 b(The)-5 b(ory)42 b(and)h(Computational)f(Com-)524 3960 y(plexity)p Fy(,)33 b(Wiley)-8 b(,)34 b(New)f(Y)-8 b(ork,)33 b(1987.)324 4163 y([Bren)m(t)g(1976])48 b(R.)36 b(P)-8 b(.)35 b(Bren)m(t,)j(\\F)-8 b(ast)35 b(m)m(ultiple-precision)j (ev)-5 b(aluation)35 b(of)h(elemen)m(tary)524 4284 y(functions",)e Fl(J.)h(Asso)-5 b(c.)34 b(Comput.)h(Mach.)d Fi(23)p Fy(:2)g(\(1976\),)g (242{251.)324 4487 y([Briggs)h(2000])48 b(K.)180 b(M.)h(Briggs,)217 b(The)182 b(doubledouble)g(homepage.)524 4607 y Fe (http://www.labs.bt.com/p)q(eopl)q(e/b)q(rigg)q(sk2/)q(doub)q(led)q (oubl)q(e.ht)q(ml)p Fy(.)324 4811 y([Briggs)33 b(at)f(al.)g(1998])48 b(K.)j(M.)g(Briggs,)56 b(T.)51 b(M.)g(Dixon,)56 b(and)51 b(G.)f(Szek)m(eres,)58 b(\\Ana-)524 4931 y(lytic)25 b(solutions)h(of)d (the)i(Cvitano)m(vi)m(\023)-46 b(c-F)-8 b(eigen)m(baum)27 b(and)d(F)-8 b(eigen)m(baum-Kadano\013-)p eop %%Page: 40 40 40 39 bop 324 241 a Ft(REFERENCES)2471 b Fy(40)524 548 y(Shenk)m(er)52 b(equations",)j Fl(Internat.)50 b(J.)h(Bifur.)f(Chaos)g (Appl.)g(Sci.)g(Engr)-5 b(g.)50 b Fi(8)p Fy(:2)524 668 y(\(1998\),)32 b(347{357.)324 872 y([Cvitano)m(vi)m(\023)-46 b(c)34 b(1989])48 b(P)-8 b(.)49 b(Cvitano)m(vi)m(\023)-46 b(c)51 b(\(Ed.\),)i Fl(Universality)c(in)h(Chaos)p Fy(,)h(Second)f(ed.) 524 992 y(Adam)33 b(Hilger,)g(Bristol,)g(1989.)324 1196 y([Daub)s(ec)m(hies)h(1992])48 b(I.)24 b(Daub)s(ec)m(hies,)i Fl(T)-7 b(en)26 b(L)-5 b(e)g(ctur)g(es)26 b(on)h(Wavelets)p Fy(,)e(SIAM,)f(Philadel-)524 1316 y(phia,)33 b(P)-8 b(A,)33 b(1992.)324 1519 y([Denjo)m(y)g(1932])48 b(A.)73 b(Denjo)m(y)-8 b(,)83 b(\\Sur)73 b(les)h(courb)s(es)g(d)m(\023)-46 b(e\014nies)75 b(par)e(les)e(\023)-46 b(equations)524 1640 y(di\013)m(\023)g(eren)m (tielles)41 b(\022)-49 b(a)38 b(la)h(surface)g(du)f(tore",)i Fl(J.)g(Math.)h(Pur)-5 b(es)40 b(et)h(Appl.)f(\(9.)f(s)n(\023)-47 b(erie\))524 1760 y Fi(11)33 b Fy(\(1932\),)e(333{375.)324 1963 y([Dixon)h(et)h(al.)g(1997])48 b(T.)38 b(M.)h(Dixon,)g(B.)f(G.)g (Kenn)m(y)-8 b(,)40 b(and)f(K.)e(M.)i(Briggs,)g(\\On)f(the)524 2084 y(univ)m(ersalit)m(y)29 b(of)d(singular)h(circle)h(maps",)g Fl(Phys.)h(L)-5 b(ett.)30 b(A)d Fi(231)p Fy(:5{6)f(\(1997\),)h(359{)524 2204 y(366.)324 2408 y([Delb)s(ourgo)32 b(and)g(Kenn)m(y)i(1991])48 b(R.)27 b(Delb)s(ourgo)f(and)h(B.)h(G.)e(Kenn)m(y,)k(\\Relations)d(b)s (e-)524 2528 y(t)m(w)m(een)49 b(univ)m(ersal)g(scaling)f(constan)m(ts)h (for)e(the)g(circle)i(map)e(near)h(the)f(golden)524 2648 y(mean",)33 b Fl(J.)i(Math.)g(Phys.)e Fi(32)p Fy(:4)f(\(1991\),)g (1045{1051.)324 2852 y([Ec)m(kmann)i(and)f(Epstein)h(1986])48 b(J.-P)-8 b(.)46 b(Ec)m(kmann)j(and)d(H.)g(Epstein,)52 b(\\On)46 b(the)g(ex-)524 2972 y(istence)k(of)e(\014xed)i(p)s(oin)m(ts) f(of)e(the)i(comp)s(osition)g(op)s(erator)f(for)g(circle)h(maps",)524 3093 y Fl(Comm.)34 b(Math.)h(Phys.)e Fi(107)p Fy(:2)f(\(1986\),)g (213{231.)324 3296 y([Ec)m(kmann)i(and)f(Epstein)h(1987])48 b(J.-P)-8 b(.)37 b(Ec)m(kmann)h(and)e(H.)h(Epstein,)i(\\Fixed)e(p)s (oin)m(ts)524 3416 y(of)f(comp)s(osition)h(op)s(erators",,)h(in)e Fl(VIIIth)i(International)f(Congr)-5 b(ess)37 b(on)h(Math-)524 3537 y(ematic)-5 b(al)42 b(Physics)h(\(Marseil)5 b(le,)44 b(1986\))p Fy(,)e(W)-8 b(orld)42 b(Scien)m(ti\014c,)j(Singap)s(ore,)e (1987,)524 3657 y(pp.)33 b(517{530.)324 3861 y([Epstein)h(1986])48 b(H.)35 b(Epstein,)i(\\New)e(pro)s(ofs)g(of)f(the)i(existence)h(of)d (the)h(Feigen)m(baum)524 3981 y(functions",)f Fl(Comm.)f(Math.)j(Phys.) c Fi(106)p Fy(:3)h(\(1986\),)e(395{426.)324 4184 y([Epstein)j(1989])48 b(H.)27 b(Epstein,)i(\\Fixed)e(p)s(oin)m(ts)g(of)f(comp)s(osition)h(op) s(erators.)f(II",)h Fl(Non-)524 4305 y(line)-5 b(arity)33 b Fi(2)p Fy(:2)f(\(1989\),)g(305{310.)324 4508 y([F)-8 b(eigen)m(baum)33 b(et)g(al.)g(1982])48 b(M.)63 b(J.)g(F)-8 b(eigen)m(baum,)72 b(L.)62 b(P)-8 b(.)63 b(Kadano\013,)70 b(and)63 b(S.)g(J.)524 4628 y(Shenk)m(er,)56 b(\\Quasip)s(erio)s(dicit) m(y)50 b(in)g(dissipativ)m(e)i(systems:)79 b(a)49 b(renormalization)524 4749 y(group)33 b(analysis",)g Fl(Physic)-5 b(a)33 b Fi(5D)p Fy(:2{3)e(\(1982\),)h(370{386.)p eop %%Page: 41 41 41 40 bop 324 241 a Ft(REFERENCES)2471 b Fy(41)324 548 y([de)33 b(F)-8 b(aria)32 b(1999])48 b(E.)28 b(de)h(F)-8 b(aria,)29 b(\\Asymptotic)h(rigidit)m(y)f(of)f(scaling)i(ratios)e(for)g (critical)524 668 y(circle)46 b(mappings",)i Fl(Er)-5 b(go)g(dic)45 b(The)-5 b(ory)45 b(Dynam.)f(Systems)g Fi(19)p Fy(:4)g(\(1999\),)j(995{)524 789 y(1035.)324 989 y([de)33 b(F)-8 b(aria)32 b(and)g(de)h(Melo)g(1999])48 b(E.)24 b(de)f(F)-8 b(aria)22 b(and)i(W.)f(de)g(Melo,)j(\\Rigidit)m(y)d (of)g(critical)524 1110 y(circle)34 b(mappings.)g(I",)e Fl(J.)j(Eur.)g(Math.)g(So)-5 b(c.)35 b(\(JEMS\))d Fi(1)p Fy(:4)h(\(1999\),)e(339{392.)324 1310 y([de)i(F)-8 b(aria)32 b(and)g(de)h(Melo)g(2000])48 b(E.)24 b(de)f(F)-8 b(aria)22 b(and)i(W.)f(de)g(Melo,)j(\\Rigidit)m(y)d(of)g(critical)524 1431 y(circle)34 b(mappings.)g(I)s(I",)e Fl(J.)j(A)n(mer.)f(Math.)h(So) -5 b(c.)32 b Fi(13)p Fy(:2)h(\(2000\),)e(343{370.)324 1631 y([F)-8 b(orsythe)33 b(et)g(al.)f(1977])48 b(G.)g(E.)g(F)-8 b(orsythe,)52 b(M.)c(A.)g(Malcolm,)53 b(and)48 b(C.)g(B.)g(Moller,)524 1752 y Fl(Computer)37 b(Metho)-5 b(ds)37 b(for)g(Mathematic)-5 b(al)37 b(Computations)p Fy(,)e(Pren)m(tice-Hall,)j(En-)524 1872 y(glew)m(o)s(o)s(d)33 b(Cli\013s,)h(1977.)324 2073 y([F)-8 b(razier)32 b(et)h(al.)g(1991])48 b(M.)34 b(F)-8 b(razier,)35 b(B.)f(Ja)m(w)m(erth,)j(and)d(G.)g(W)-8 b(eiss,)36 b Fl(Littlewo)-5 b(o)g(d-Paley)524 2193 y(The)g(ory)35 b(and)f(the)h(Study)g(of)g(F)-7 b(unction)34 b(Sp)-5 b(ac)g(es)p Fy(,)31 b(AMS,)j(Pro)m(vidence,)h(RI,)d(1991.)324 2394 y([Ghering)g(and)h(P)m(alk)-5 b(a,)33 b(man)m(uscript])51 b(F.)24 b(W.)h(Ghering,)h(B.)f(P)m(alk)-5 b(a,)27 b Fl(Quasi-Conformal) 524 2514 y(Maps)p Fy(,)33 b(Man)m(uscript.)324 2715 y([Glass)g(1995])48 b(L.Glass,)53 b(\\Cardiac)d(arrh)m(ythmias)g(and)f(circle)i(maps)e({)g (a)g(classical)524 2835 y(problem",)c Fl(Internat.)e(J.)h(Bifur.)f (Chaos)g(Appl.)g(Sci.)g(Engr)-5 b(g.)41 b Fi(5)h Fy(\(1995\),)i(359{) 524 2956 y(371.)324 3156 y([GMP)33 b(2000])48 b(The)33 b(GMP)g(home)g(page.)g Fe(http://www.swox.com/gmp/)p Fy(.)324 3357 y([Greene)g(1979])48 b(J.)31 b(M.)g(Greene,)g(\\A)g (metho)s(d)g(for)f(determining)i(a)e(sto)s(c)m(hastic)i(transi-)524 3477 y(tion",)h Fl(J.)i(Math.)g(Phys.)d Fi(20)p Fy(:6)h(\(1979\),)e (1183{1201.)324 3678 y([Hall)h(1981])48 b(G.)29 b(R)g(Hall,)h(\\A)f Fr(C)1525 3642 y Fo(1)1628 3678 y Fy(Denjo)m(y)h(coun)m(terexample",)i Fl(Er)-5 b(go)g(dic)31 b(The)-5 b(ory)31 b(Dy-)524 3798 y(namic)-5 b(al)34 b(Systems)e Fi(1)p Fy(:3)h(\(1981\),)e(261{272.)324 3999 y([H\177)-49 b(ardle)33 b(et)g(al.)f(1998])48 b(W.)41 b(H\177)-49 b(ardle,)43 b(G.)d(Kerky)m(ac)m(harian,)k(D.)c(Picard,)j (and)e(A.)g(Tsy-)524 4119 y(bak)m(o)m(v,)31 b Fl(Wavelets,)g(Appr)-5 b(oximation,)30 b(and)h(Statistic)-5 b(al)30 b(Applic)-5 b(ations)p Fy(,)28 b(Springer-)524 4240 y(V)-8 b(erlag,)33 b(New)g(Y)-8 b(ork,)33 b(1998.)324 4440 y([Bai-Lin)f(Hao)g(1990])48 b(Bai-Lin)23 b(Hao)f(\(Ed.\),)j Fl(Chaos)g(II)p Fy(,)d(W)-8 b(orld)23 b(Scien)m(ti\014c,)k(Singap)s(ore,)524 4561 y(1990.)324 4761 y([Herman)33 b(1979])48 b(M.)75 b(R.)f(Herman,)86 b(\\Sur)74 b(la)h(conjugaison)f(di\013)m(\023)-46 b(eren)m(tiable)77 b(des)524 4882 y(di\013)m(\023)-46 b(eomorphismes)45 b(du)d(cercle)h(\022)-49 b(a)42 b(des)h(rotations",)h Fl(Inst.)e(Hautes)3088 4857 y(\023)3074 4882 y(Etudes)h(Sci.)524 5002 y(Publ.)35 b(Math.)e Fy(No.)f(49)h(\(1979\),)e(5{233.)p eop %%Page: 42 42 42 41 bop 324 241 a Ft(REFERENCES)2471 b Fy(42)324 548 y([Herman)33 b(1983])48 b(M.)32 b(R.)g(Herman,)h(\\Sur)f(les)h(di\013)m (\023)-46 b(eomorphismes)34 b(du)e(cercle)h(de)g(nom-)524 668 y(bre)e(de)f(rotation)g(de)g(t)m(yp)s(e)h(constan)m(t",)g(in)f Fl(Confer)-5 b(enc)g(e)31 b(on)h(Harmonic)g(A)n(nalysis)524 789 y(in)43 b(Honor)g(of)g(A)n(ntoni)f(Zygmund,)j(\(Chic)-5 b(ago,)43 b(Il)5 b(l.,)44 b(1981\))p Fy(,)f(W)-8 b(adsw)m(orth,)45 b(Bel-)524 909 y(mon)m(t,)33 b(CA,)h(1983,)d(pp.)i(708{725.)324 1112 y([Hern\023)-49 b(andez)34 b(and)f(W)-8 b(eiss)34 b(1996])48 b(E.)29 b(Hern\023)-49 b(andez)30 b(and)f(G.)f(W)-8 b(eiss,)31 b Fl(A)h(First)f(Course)g(on)524 1233 y(Wavelets)p Fy(,)i(CR)m(C)g(Press,)h(Bo)s(ca)f(Raton,)f(FL,)g(1996.)324 1436 y([Holsc)m(hneider)j(and)e(Tc)m(hamitc)m(hian)h(1991])48 b(M.)199 b(Holsc)m(hneider)h(and)e(Ph.)524 1557 y(Tc)m(hamitc)m(hian,) 69 b(\\P)m(oin)m(t)m(wise)62 b(analysis)e(of)f(Riemann's)i (\\nondi\013eren)m(tiable")524 1677 y(function",)33 b Fl(Invent.)h(Math.)f Fi(105)g Fy(\(1991\),)e(157{175.)324 1880 y([Hu)i(et)g(al.)f(1990])48 b(B.)40 b(Hu,)i(A.)d(V)-8 b(alinia,)42 b(and)e(O.)f(Piro,)j(\\Univ)m(ersalit)m(y)g(and)e(asymp-) 524 2001 y(totic)27 b(limits)h(of)e(the)i(scaling)f(exp)s(onen)m(ts)i (in)e(circle)h(maps",)h Fl(Phys.)g(L)-5 b(ett.)30 b(A)e Fi(144)p Fy(:1)524 2121 y(\(1990\),)k(7{10.)324 2325 y([Ignato)m(v)h(1995])48 b(A.)j(M.)h(Ignato)m(v,)57 b(\\T)-8 b(riv)m(elpiece-Gould)53 b(mo)s(des)g(in)e(a)g(corrugated)524 2445 y(plasma)33 b(slab",)g Fl(Phys.)i(R)-5 b(ev)35 b(E)d Fi(51)p Fy(:2)h(\(1995\),)f(1391{1399.)324 2648 y([Ja\013ard)g(and)h (Mey)m(er)h(1996])48 b(S.)g(Ja\013ard)f(and)g(Y.)h(Mey)m(er,)53 b(\\W)-8 b(a)m(v)m(elet)49 b(metho)s(ds)f(for)524 2769 y(p)s(oin)m(t)m(wise)e(regularit)m(y)f(and)f(lo)s(cal)f(oscillations)i (of)f(functions",)j Fl(Mem.)e(A)n(mer.)524 2889 y(Math.)35 b(So)-5 b(c.)32 b Fi(123)h Fy(\(1996\),)f(no.)g(587.)324 3093 y([Katok)g(and)h(Hasselblatt)h(1995])48 b(A.)37 b(Katok)g(and)h(B.)f(Hasselblatt,)j Fl(Intr)-5 b(o)g(duction)38 b(to)524 3213 y(the)32 b(Mo)-5 b(dern)31 b(The)-5 b(ory)30 b(of)h(Dynamic)-5 b(al)30 b(Systems)p Fy(,)g(Cam)m(bridge)g(Univ)m (ersit)m(y)h(Press,)524 3333 y(Cam)m(bridge,)j(1995.)324 3537 y([Katznelson)f(and)g(Ornstein)h(1989])48 b(Y.)24 b(Katznelson)i(and)e(D.)g(Ornstein,)j(\\The)f(di\013er-)524 3657 y(en)m(tiabilit)m(y)34 b(of)e(the)h(conjugation)f(o)f(f)h(certain) h(di\013eomorphisms)h(of)e(the)h(circle",)524 3778 y Fl(Er)-5 b(go)g(dic)35 b(The)-5 b(ory)34 b(Dynamic)-5 b(al)34 b(Systems)e Fi(9)p Fy(:4)g(\(1989\),)g(643{680.)324 3981 y([Kim)h(and)f(Ostlund)i(1989])48 b(S.)31 b(Kim)g(and)g(S.)g (Ostlund,)h(\\Univ)m(ersal)h(scaling)e(in)g(circle)524 4101 y(maps",)i Fl(Physic)-5 b(a)33 b Fi(39D)p Fy(:2{3)f(\(1989\),)f (365{392.)324 4305 y([Kran)m(tz)i(1983])48 b(S.)30 b(G.)f(Kran)m(tz,)i (\\Lipsc)m(hitz)h(spaces,)g(smo)s(othness)f(of)f(functions,)h(and)524 4425 y(appro)m(ximation)j(theory",)f Fl(Exp)-5 b(osition.)34 b(Math.)e Fi(1)p Fy(:3)h(\(1983\),)e(193{260.)324 4628 y([Lanford)h(1986])48 b(O.)36 b(E.)h(Lanford)g(I)s(I)s(I,)f (\\Renormalization)h(group)f(metho)s(ds)i(for)e(cir-)524 4749 y(cle)i(mappings",)g(in)f Fl(Statistic)-5 b(al)38 b(Me)-5 b(chanics)38 b(and)g(Field)f(The)-5 b(ory:)52 b(Mathemati-)524 4869 y(c)-5 b(al)41 b(Asp)-5 b(e)g(cts)41 b(\(Gr)-5 b(oningen,)41 b(1985\))p Fy(,)f(Lecture)g(Notes)g(in)f(Ph)m (ys.,)k(257,)d(Springer,)524 4990 y(Berlin,)34 b(1986,)d(pp.)i (176{189.)p eop %%Page: 43 43 43 42 bop 324 241 a Ft(REFERENCES)2471 b Fy(43)324 548 y([Lanford)32 b(1987])48 b(O.)32 b(E.)g(Lanford)f(I)s(I)s(I,)h (\\Renormalization)g(group)g(metho)s(ds)h(for)e(crit-)524 668 y(ical)48 b(circle)h(mappings)g(with)g(general)f(rotation)f(n)m(um) m(b)s(er",)53 b(in)48 b Fl(VIIIth)g(Inter-)524 789 y(national)e(Congr) -5 b(ess)45 b(on)h(Mathematic)-5 b(al)46 b(Physics)g(\(Marseil)5 b(le,)49 b(1986\))p Fy(,)f(W)-8 b(orld)524 909 y(Scien)m(ti\014c,)35 b(Singap)s(ore,)e(1987,)e(pp.)i(532{536.)324 1112 y([Lanford)f(1988])48 b(O.)36 b(E.)h(Lanford)g(I)s(I)s(I,)f(\\Renormalization)h(group)f (metho)s(ds)i(for)e(cir-)524 1233 y(cle)g(mappings",)g(in)e Fl(Nonline)-5 b(ar)37 b(Evolution)f(and)g(Chaotic)g(Phenomena)f (\(Noto,)524 1353 y(1987\))p Fy(,)47 b(NA)-8 b(TO)45 b(Adv.)g(Sci.)g(Inst.)h(Ser.)f(B)f(Ph)m(ys.,)50 b(176,)d(Plen)m(um,)i (New)c(Y)-8 b(ork,)524 1474 y(1988,)32 b(pp.)h(25{36.)324 1677 y([Lanford)f(and)h(de)g(la)f(Lla)m(v)m(e,)i(man)m(uscript)g (\(1984\)])48 b(O.)29 b(E.)h(Lanford)f(I)s(I)s(I)g(and)g(R.)g(de)h(la) 524 1797 y(Lla)m(v)m(e,)40 b(\\Solution)e(of)f(the)h(functional)g (equation)h(for)e(critical)h(circle)h(mappings)524 1918 y(with)34 b(golden)e(rotation)g(n)m(um)m(b)s(er",)i(Man)m(uscript)h (\(1984\).)324 2121 y([de)e(la)f(Lla)m(v)m(e)i(and)f(P)m(etro)m(v)h (1999])48 b(R.)33 b(de)h(la)g(Lla)m(v)m(e)g(and)g(N.)g(P)m(etro)m(v,)h (\\Theory)g(of)e(cir-)524 2242 y(cle)39 b(maps)f(and)g(the)g(problem)h (of)e(one-dimensional)j(optical)e(resonator)f(with)i(a)524 2362 y(p)s(erio)s(dically)34 b(mo)m(ving)f(w)m(all",)g Fl(Phys.)i(R)-5 b(ev.)35 b(E)g(\(3\))d Fi(59)p Fy(:6)g(\(1999\),)g (6637{6651.)324 2565 y([de)h(la)f(Lla)m(v)m(e)i(and)f(Sc)m(hafer,)g (man)m(uscript)i(\(1996\)])47 b(R.)77 b(de)h(la)e(Lla)m(v)m(e)i(and)g (R.)e(P)-8 b(.)524 2686 y(Sc)m(hafer,)46 b(\\Rigidit)m(y)c(prop)s (erties)h(of)f(one)g(dimensional)i(expanding)f(maps)f(and)524 2806 y(applications)34 b(to)e(renormalization",)h(Man)m(uscript,)h (1996.)324 3010 y([Louis)f(et)g(al.)f(1997])48 b(A.)37 b(K.)g(Louis,)i(P)-8 b(.)37 b(Maa\031,)h(and)f(A.)g(Rieder,)i Fl(Wavelets:)53 b(The)-5 b(ory)524 3130 y(and)35 b(Applic)-5 b(ations)p Fy(,)31 b(Wiley)-8 b(,)34 b(Chic)m(hester,)h(1997.)324 3333 y([Mallat)d(1998])48 b(S.)h(Mallat,)k Fl(A)d(Wavelet)g(T)-7 b(our)49 b(of)h(Signal)e(Pr)-5 b(o)g(c)g(essing)p Fy(,)52 b(Academic)524 3454 y(Press,)35 b(San)d(Diego,)g(CA,)i(1998.)324 3657 y([de)f(Melo)g(1998])48 b(W.)89 b(de)g(Melo,)103 b(\\Rigidit)m(y)89 b(and)f(renormalization)h(in)g(one-)524 3778 y(dimensional)52 b(dynamical)g(systems",)57 b(in)51 b Fl(Pr)-5 b(o)g(c)g(e)g(e)g(dings)50 b(of)h(the)h(International)524 3898 y(Congr)-5 b(ess)37 b(of)g(Mathematicians,)g(V)-7 b(ol.)37 b(II)f(\(Berlin,)h(1998\))p Fy(,)e Fl(Do)-5 b(c.)37 b(Math.)e Fi(1998)p Fy(,)524 4018 y Fl(Extr)-5 b(a)35 b(V)-7 b(ol.)35 b(II)p Fy(,)d(pp.)h(765{778)d(\(electronic\).) 324 4222 y([de)j(Melo)g(and)g(V)-8 b(an)32 b(Strien)h(1998])48 b(W.)79 b(de)g(Melo)g(and)g(S.)f(V)-8 b(an)79 b(Strien,)90 b Fl(One-)524 4342 y(Dimensional)33 b(Dynamics)p Fy(,)e(Springer,)j (Berlin,)f(1993.)324 4545 y([Mestel,)h(thesis)g(\(1984\)])48 b(B.)33 b(Mestel,)h(Univ.)f(of)g(W)-8 b(arwic)m(k)34 b(thesis,)g(\(1984\).)324 4749 y([Mey)m(er)g(1998])48 b(Y.)k(Mey)m(er,)59 b Fl(Wavelets,)e(Vibr)-5 b(ations)53 b(and)f(Sc)-5 b(alings)p Fy(,)55 b(AMS,)e(Pro)m(vi-)524 4869 y(dence,)34 b(RI,)f(1998.)p eop %%Page: 44 44 44 43 bop 324 241 a Ft(REFERENCES)2471 b Fy(44)324 548 y([Mey)m(er)34 b(1990])48 b(Y.)33 b(Mey)m(er,)h Fl(Ondelettes)h(et)g (Op)n(\023)-47 b(er)-5 b(ateurs.)33 b(I)p Fy(,)f(Hermann,)i(P)m(aris,)f (1990.)324 751 y([Moser)g(1966])48 b(J.)27 b(Moser,)j(\\A)c(rapid)i (con)m(v)m(ergen)m(t)h(iteration)e(metho)s(d)g(and)h(non-linear)524 872 y(di\013eren)m(tial)33 b(equations.)h(I)s(I",)e Fl(A)n(nn.)h (Scuola)h(Norm.)g(Sup.)g(Pisa)g(\(3\))e Fi(20)g Fy(\(1966\),)524 992 y(499{535.)324 1196 y([Moser)h(and)g(P\177)-49 b(osc)m(hel)34 b(1984])48 b(J.)28 b(Moser)h(and)f(J.)g(P\177)-49 b(osc)m(hel,)31 b(\\An)d(extension)i(of)d(a)h(result)524 1316 y(b)m(y)41 b(Dinaburg)e(and)g(Sinai)h(on)f(quasip)s(erio)s(dic)i(p)s(oten)m (tials",)g Fl(Comment.)f(Math.)524 1436 y(Helv.)33 b Fi(59)p Fy(:1)f(\(1984\),)g(39{85.)324 1640 y([Ostlund)h(et)g(al.)g (1983])48 b(S.)37 b(Ostlund,)j(D.)d(Rand,)i(J.)f(Sethna,)i(and)d(E.)h (Siggia,)h(\\Uni-)524 1760 y(v)m(ersal)33 b(prop)s(erties)g(of)e(the)h (transition)g(from)g(quasip)s(erio)s(dicit)m(y)i(to)d(c)m(haos)h(in)g (dis-)524 1880 y(sipativ)m(e)j(systems",)f Fl(Physic)-5 b(a)33 b Fi(8D)p Fy(:3)f(\(1983\),)f(303{342.)324 2084 y([Pin)m(to)i(and)g(Rand)f(1992])48 b(A.)42 b(A.)g(Pin)m(to)h(and)f(D.) f(A.)h(Rand,)j(\\Global)c(phase)h(space)524 2204 y(univ)m(ersalit)m(y) -8 b(,)31 b(smo)s(oth)d(conjugacies)g(and)g(renormalization.)g(I)s(I.)f (The)i Fr(C)3218 2168 y Fp(k)r Fz(+)p Fp(\013)3388 2204 y Fy(case)524 2325 y(using)42 b(rapid)f(con)m(v)m(ergence)j(of)c(Mark)m (o)m(v)i(families",)i Fl(Nonline)-5 b(arity)41 b Fi(5)p Fy(:1)g(\(1992\),)524 2445 y(49{79.)324 2648 y([P)m(oincar)m(\023)-46 b(e)34 b(1885])48 b(H.)60 b(P)m(oincar)m(\023)-46 b(e,)68 b(\\Sur)60 b(le)h(courb)s(es)g(d)m(\023)-46 b(e\014nies)62 b(par)e(les)f(\023)-46 b(equations)524 2769 y(di\013)m(\023)g(eren)m (tielles",)53 b Fl(J.)47 b(Math)h(Pur)-5 b(es)48 b(et)f(Appl.)h(\(4.)f (s)n(\023)-47 b(erie\))44 b Fi(1)i Fy(\(1885\),)j(167{244.)524 2889 y(Reprin)m(ted)25 b(in)e Fl(\036uvr)-5 b(es)26 b(de)f(Henri)i (Poinc)-5 b(ar)n(\023)-47 b(e)p Fy(,)23 b(T)-8 b(ome)23 b(I,)h(Gauthier-Villars,)h(P)m(aris,)524 3010 y(1928.)324 3213 y([P)m(ommeau)34 b(and)e(Manneville)j(1980])48 b(Y.)i(P)m(ommeau)h (and)f(P)-8 b(.)50 b(Manneville,)56 b(\\In)m(ter-)524 3333 y(mitten)m(t)j(transition)f(to)f(turbulence)j(in)d(dissipativ)m(e) j(dynamical)f(systems",)524 3454 y Fl(Comm.)34 b(Math.)h(Phys.)e Fi(74)p Fy(:2)f(\(1980\),)g(189{197.)324 3657 y([Press)i(et)f(al.)f (1992])48 b(W.)40 b(H.)g(Press,)j(S.)c(A.)h(T)-8 b(euk)m(olsky)g(,)44 b(W.)c(T.)g(V)-8 b(etterling,)42 b(and)d(B.)524 3778 y(P)-8 b(.)33 b(Flannery,)f Fl(Numeric)-5 b(al)35 b(R)-5 b(e)g(cip)g(es)33 b(in)h(C)p Fy(,)e(Second)h(ed.)g(Cam)m(bridge)g(Univ) m(ersit)m(y)524 3898 y(Press,)i(Cam)m(bridge,)f(1992.)d(Also)i(a)m(v)-5 b(ailable)33 b(at)f Fe(http://www.nr.com/)p Fy(.)324 4101 y([Rand)g(1987])48 b(D.)35 b(A.)g(Rand,)h(\\F)-8 b(ractal)35 b(bifurcation)g(sets,)i(renormalization)f(strange)524 4222 y(sets)f(and)e(their)h(univ)m(ersal)h(in)m(v)-5 b(arian)m(ts",)35 b Fl(Pr)-5 b(o)g(c.)35 b(R.)h(So)-5 b(c.)35 b(L)-5 b(ond.)33 b Fy(A)g Fi(413)h Fy(\(1987\),)524 4342 y(45{61.)324 4545 y([Rand)e(1988a])48 b(D.)35 b(A.)h(Rand,)g (\\Global)f(phase)i(space)f(univ)m(ersalit)m(y)-8 b(,)39 b(smo)s(oth)d(conju-)524 4666 y(gacies)d(and)f(renormalisation.)h(I.)f (The)h Fr(C)2087 4630 y Fz(1+)p Fp(\013)2259 4666 y Fy(case",)g Fl(Nonline)-5 b(arity)32 b Fi(1)p Fy(:1)g(\(1988\),)524 4786 y(181{202.)p eop %%Page: 45 45 45 44 bop 324 241 a Ft(REFERENCES)2471 b Fy(45)324 548 y([Rand)32 b(1988b])49 b(D.)29 b(Rand,)h(\\Univ)m(ersalit)m(y)i(and)e (renormalisation)g(in)g(dynamical)h(sys-)524 668 y(tems",)26 b(in)c Fl(New)k(Dir)-5 b(e)g(ctions)24 b(in)h(Dynamic)-5 b(al)25 b(Systems)p Fy(,)f(Cam)m(bridge)g(Univ.)f(Press,)524 789 y(Cam)m(bridge,)34 b(1988,)e(pp.)h(1{56.)324 986 y([Rand)f(1992])48 b(D.)28 b(A.)h(Rand,)h(\\Existence,)h(nonexistence)h (and)d(univ)m(ersal)h(breakdo)m(wn)524 1106 y(of)f(dissipativ)m(e)i (golden)e(in)m(v)-5 b(arian)m(t)30 b(tori.)f(I.)g(Golden)g(critical)g (circle)i(maps",)f Fl(Non-)524 1226 y(line)-5 b(arity)33 b Fi(5)p Fy(:3)f(\(1992\),)g(639{662.)324 1423 y([Rand)g(et)h(al.)g (1982])48 b(D.)34 b(Rand,)i(S.)g(Ostlund,)h(J.)e(Sethna,)i(and)e(E.)h (D.)e(Siggia,)i(\\Uni-)524 1544 y(v)m(ersal)e(transition)f(from)f (quasip)s(erio)s(dicit)m(y)i(to)e(c)m(haos)h(in)g(dissipativ)m(e)h (systems",)524 1664 y Fl(Phys.)h(R)-5 b(ev.)35 b(L)-5 b(ett.)33 b Fi(49)p Fy(:2)f(\(1982\),)g(132{135.)324 1861 y([Shenk)m(er)i(1982])48 b(S.)30 b(Shenk)m(er,)j(\\Scaling)d(b)s (eha)m(vior)g(in)g(a)g(map)g(of)f(a)g(circle)i(on)m(to)f(itself:)524 1982 y(empirical)k(results",)g Fl(Physic)-5 b(a)32 b Fi(5D)p Fy(:2{3)g(\(1982\),)g(405{411.)324 2179 y([Shraiman)h(1984])48 b(B.)29 b(I.)h(Shraiman,)g(\\T)-8 b(ransition)30 b(from)f(quasip)s (erio)s(dicit)m(y)j(to)d(c)m(haos:)524 2299 y(a)h(p)s(erturbativ)m(e)h (renormalization-group)e(approac)m(h",)i Fl(Phys.)h(R)-5 b(ev.)32 b(A)h(\(3\))c Fi(29)p Fy(:6)524 2419 y(\(1984\),)j(3464{3466.) 324 2616 y([Sinai)h(and)f(Khanin)h(1989])48 b(Y)-8 b(a.)44 b(G.)g(Sinai)g(and)h(K.)f(M.)g(Khanin,)k(\\Smo)s(othness)d(of)524 2737 y(conjugacies)j(of)f(di\013eomorphisms)i(of)d(the)h(circle)h(with) g(rotations",)i Fl(Usp)-5 b(ekhi)524 2857 y(Mat.)41 b(Nauk)d Fi(44)p Fy(:1\(265\))g(\(1989\),)g(57{82,)h(247.)e(English)j (translation:)55 b Fl(R)n(ussian)524 2977 y(Math.)35 b(Surveys)e Fi(44)p Fy(:1)f(\(1989\),)g(69{99.)324 3174 y([Stein)h(1970])48 b(E.)c(Stein,)i Fl(Singular)e(Inte)-5 b(gr)g(als)44 b(and)g(Di\013er)-5 b(entiability)44 b(Pr)-5 b(op)g(erties)44 b(of)524 3295 y(F)-7 b(unctions)p Fy(,)32 b(Princeton)h(Univ)m(ersit)m(y)j(Press,)e(Princeton,)g(1970.)324 3492 y([)354 3467 y(\023)351 3492 y(Swi)s(\030)-46 b(atek)34 b(1991])48 b(G.)1167 3467 y(\023)1165 3492 y(Swi)s(\030)-46 b(atek,)85 b(\\Circle)74 b(homeomorphisms)i(with)e(\015at)f(critical) 524 3612 y(p)s(oin)m(ts",)33 b Fl(F)-7 b(und.)34 b(Math.)f Fi(138)p Fy(:3)f(\(1991\),)g(205{217.)324 3809 y([)354 3784 y(\023)351 3809 y(Swi)s(\030)-46 b(atek)34 b(1998])48 b(G.)1130 3784 y(\023)1127 3809 y(Swi)s(\030)-46 b(atek,)38 b(\\On)e(critical)g(circle)h(homeomorphisms",)i Fl(Bol.)e(So)-5 b(c.)524 3930 y(Br)g(asil.)34 b(Mat.)i(\(N.S.\))c Fi(29)p Fy(:2)g(\(1998\),)g(329{351.)324 4127 y([T)-8 b(angerman)33 b(and)g(V)-8 b(eerman)33 b(1991])48 b(F.)h(M.)h(T)-8 b(angerman)51 b(and)e(J.)h(J.)g(P)-8 b(.)50 b(V)-8 b(eerman,)524 4247 y(\\Scalings)45 b(in)g(circle)g(maps.)h(II",)e Fl(Comm.)h(Math.)h (Phys.)e Fi(141)p Fy(:2)g(\(1991\))g(279{)524 4367 y(291.)324 4564 y([V\177)-49 b(ais\177)g(al\177)g(a])49 b(J.)37 b(V\177)-49 b(ais\177)g(al\177)g(a,)38 b Fl(L)-5 b(e)g(ctur)g(es)40 b(on)e Fr(n)p Fl(-Dimensional)f(Quasic)-5 b(onformal)37 b(Mappings)p Fy(,)524 4685 y(\(Lecture)d(Notes)f(in)g(Mathematics,)h(V) -8 b(ol.)33 b(229\))e(Springer,)j(Berlin,)f(1971.)324 4882 y([V)-8 b(eerman)33 b(and)g(T)-8 b(angerman)33 b(1990])48 b(J.)i(J.)g(P)-8 b(.)49 b(V)-8 b(eerman)51 b(and)f(F.)f(M.)h(T)-8 b(angerman,)524 5002 y(\\Scalings)34 b(in)e(circle)i(maps.)f(I",)g Fl(Comm.)g(Math.)i(Phys.)e Fi(134)p Fy(:1)f(\(1990\),)g(89{107.)p eop %%Page: 46 46 46 45 bop 324 241 a Ft(REFERENCES)2471 b Fy(46)324 548 y([Wic)m(k)m(erhauser)35 b(1994])48 b(M.)f(V.)f(Wic)m(k)m(erhauser,)53 b Fl(A)-5 b(dapte)g(d)47 b(Wavelet)g(A)n(nalysis)g(fr)-5 b(om)524 668 y(The)g(ory)29 b(to)g(Softwar)-5 b(e)p Fy(,)26 b(A)g(K)g(P)m(eters,)j(W)-8 b(ellesley)g(,)30 b(MA,)c(1994.)f(Soft)m(w) m(are)i(a)m(v)-5 b(ailable)524 789 y(at)33 b Fe (http://www.math.wustl.edu/~)q(vic)q(tor/)q Fy(.)324 992 y([Y)-8 b(amp)s(olsky)34 b(1999])48 b(M.)38 b(Y)-8 b(amp)s(olsky,)42 b(\\Complex)d(b)s(ounds)g(for)f(renormalization)h(of) 524 1112 y(critical)j(circle)g(maps",)i Fl(Er)-5 b(go)g(dic)42 b(The)-5 b(ory)42 b(Dynamic)-5 b(al)42 b(Systems)f Fi(19)p Fy(:1)g(\(1999\),)524 1233 y(227{257.)324 1436 y([Y)-8 b(o)s(ccoz)33 b(1984a])47 b(J.-C.)34 b(Y)-8 b(o)s(ccoz,)33 b(\\Il)g(n'y)h(a)f(pas)g(de)h(con)m(tre-exemple)h(de)f(Denjo)m(y)f(an-) 524 1557 y(alytique",)h Fl(C.)h(R.)f(A)-5 b(c)g(ad)35 b(Sc.)g(Paris)f(S)n(\023)-47 b(er.)34 b(I)g(Math.)f Fi(298)p Fy(:7)f(\(1984\),)g(141{144.)324 1760 y([Y)-8 b(o)s(ccoz)33 b(1984b])48 b(J.-C.)183 b(Y)-8 b(o)s(ccoz,)220 b(\\Conjugaison)182 b(di\013)m(\023)-46 b(eren)m(tiable)185 b(des)524 1880 y(di\013)m(\023)-46 b(eomorphismes)61 b(du)e(cercle)h(don)m(t)f(le)g (nom)m(bre)h(de)f(rotation)f(v)m(\023)-46 b(eri\014e)60 b(une)524 2001 y(condition)33 b(diophan)m(tienne",)g Fl(A)n(nn.)h(Sci.)2110 1976 y(\023)2095 2001 y(Ec)-5 b(ole)34 b(Norm.)g(Sup.)g(\(4\))d Fi(17)p Fy(:3)h(\(1984\),)524 2121 y(333{359.)p eop %%Page: 47 47 47 46 bop 324 241 a Ft(REFERENCES)2471 b Fy(47)324 548 y Fs(Figure)53 b(Captions)443 767 y Fy(1.)c(Densit)m(y)31 b(of)e(the)h(iterates)h(of)e(a)g(Q)h(map.)42 b(The)31 b(n)m(um)m(b)s(er)g(of)e(iterates)i(in)f(a)f(bin)h(\(in)568 887 y(thousands\))38 b(vs.)f(the)h(p)s(osition)f(of)f(the)h(bin,)i(for) d(four)h(million)g(iterates)h(of)e Fr(f)3478 840 y Fz(Q)3467 912 y(0)p Fp(:)p Fz(6)568 1008 y Fy(\(thin)d(line\))g(and)f(of)h Fr(f)1394 972 y Fz(N)1383 1032 y(0)p Fp(:)p Fz(5)1509 1008 y Fy(\(thic)m(k)h(line\),)g(in)e(256)g(bins.)443 1209 y(2.)49 b(Conjugacies)25 b Fr(\022)j Fy(b)s(et)m(w)m(een:)42 b Fr(f)1642 1173 y Fz(N)1631 1233 y(0)p Fp(:)p Fz(2)1750 1209 y Fy(and)24 b Fr(f)1990 1173 y Fz(C)1979 1233 y(0)p Fp(:)p Fz(3)2098 1209 y Fy(\(thin)h(solid)g(line\),)i Fr(f)2856 1173 y Fz(N)2845 1233 y(0)p Fp(:)p Fz(2)2964 1209 y Fy(and)d Fr(f)3204 1173 y Fz(C)3193 1233 y(0)p Fp(:)p Fz(6)3312 1209 y Fy(\(thic)m(k)568 1329 y(solid)33 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/languagelevel where {pop /gs_languagelevel languagelevel def} {/gs_languagelevel 1 def} ifelse gs_languagelevel 1 gt { <} image end } >> matrix makepattern /Pat0 exch def <} image end } >> matrix makepattern /Pat1 exch def <} image end } >> matrix makepattern /Pat2 exch def <} image end } >> matrix makepattern /Pat3 exch def <} image end } >> matrix makepattern /Pat4 exch def <} image end } >> matrix makepattern /Pat5 exch def <} image end } >> matrix makepattern /Pat6 exch def <} image end } >> matrix makepattern /Pat7 exch def <} image end } >> matrix makepattern /Pat8 exch def <} image end } >> matrix makepattern /Pat9 exch def <} image end } >> matrix makepattern /Pat10 exch def <} image end } >> matrix makepattern /Pat11 exch def <} image end } >> matrix makepattern /Pat12 exch def <} image end } >> matrix makepattern /Pat13 exch def <} image end } >> matrix makepattern /Pat14 exch def <} image end } >> matrix makepattern /Pat15 exch def }{ /Pat0 { 0.000000 setgray } def /Pat1 { 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sq o } bind def % Triangle symbols /t1 { moveto dup 0 exch rmoveto dup neg dup 2 mul rlineto 2 mul 0 rlineto closepath } bind def /dt1 { t1 s } bind def /ft1 { t1 f } bind def /ot1 { t1 o } bind def /t2 { moveto dup neg 0 rmoveto dup dup 2 mul exch neg rlineto 2 mul 0 exch rlineto closepath } bind def /dt2 { t2 s } bind def /ft2 { t2 f } bind def /ot2 { t2 o } bind def /t3 { moveto dup neg 0 exch rmoveto dup dup 2 mul rlineto neg 2 mul 0 rlineto closepath } bind def /dt3 { t3 s } bind def /ft3 { t3 f } bind def /ot3 { t3 o } bind def /t4 { moveto dup 0 rmoveto dup dup -2 mul exch rlineto -2 mul 0 exch rlineto closepath } bind def /dt4 { t4 s } bind def /ft4 { t4 f } bind def /ot4 { t4 o } bind def % Diamond symbol /di { moveto dup 0 exch rmoveto dup neg dup rlineto dup dup neg rlineto dup dup rlineto closepath } bind def /ddi { di s } bind def /fdi { di f } bind def /odi { di o } bind def % Plus symbol /pl { dup 0 rmoveto dup -2 mul 0 rlineto dup dup rmoveto -2 mul 0 exch rlineto } bind def /dpl { m pl s } bind def % x symbol /x { dup dup rmoveto dup -2 mul dup rlineto 2 mul dup 0 rmoveto dup neg exch rlineto } bind def /dx { m x s } bind def % Splat symbol /dsp { m dup pl dup 0 exch rmoveto 0.707 mul x s } bind def /RJ { stringwidth neg exch neg exch rmoveto } bind def /CS { stringwidth 2 div neg exch 2 div neg exch rmoveto } bind def 0.24 0.24 scale 1 setlinecap mark /ISOLatin1Encoding 8#000 1 8#054 {StandardEncoding exch get} for /minus 8#056 1 8#217 {StandardEncoding exch get} for /dotlessi 8#301 1 8#317 {StandardEncoding exch get} for /space /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen /registered /macron /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedilla /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis /ISOLatin1Encoding where not {256 array astore def} if cleartomark /makeISOEncoded { findfont /curfont exch def /newfont curfont maxlength dict def /ISOLatin1 (-ISOLatin1) def /curfontname curfont /FontName get dup length string cvs def /newfontname curfontname length ISOLatin1 length add string dup 0 curfontname putinterval dup curfontname length ISOLatin1 putinterval def curfont { exch dup /FID ne { dup /Encoding eq { exch pop ISOLatin1Encoding exch } if dup /FontName eq { exch pop 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1718 1030 l 1719 1030 l 1719 1029 l 1719 1028 l 1719 1027 l 1719 1026 l 1719 1025 l 1720 1025 l 1720 1024 l 1720 1023 l 1720 1022 l 1720 1021 l 1721 1021 l 1721 1020 l 1721 1019 l 1721 1018 l 1721 1017 l 1721 1018 l 1722 1019 l 1722 1018 l 1722 1017 l 1722 1016 l 1722 1015 l 1723 1015 l 1723 1014 l 1723 1013 l 1723 1012 l 1723 1013 l 1723 1012 l 1723 1013 l 1724 1013 l 1724 1012 l 1724 1011 l 1724 1012 l 1724 1013 l 1724 1014 l 1724 1016 l 1724 1017 l 1724 1018 l 1724 1017 l 1725 1018 l 1725 1017 l 1725 1018 l 1725 1017 l 1725 1016 l 1725 1015 l 1726 1015 l 1726 1014 l 1726 1013 l 1726 1012 l 1727 1011 l 1727 1010 l 1727 1009 l 1727 1008 l 1727 1007 l 1727 1006 l 1728 1006 l 1728 1005 l 1728 1004 l 1728 1003 l 1728 1002 l 1728 1001 l 1729 1001 l 1729 1000 l 1729 1001 l 1729 1000 l 1729 1001 l 1729 1000 l 1729 999 l 1729 998 l 1730 998 l 1730 997 l 1730 996 l 1730 995 l 1730 994 l 1730 993 l 1731 993 l 1731 992 l 1731 991 l 1731 990 l 1731 989 l 1731 988 l 1732 987 l 1732 986 l 1732 985 l 1732 984 l 1732 983 l 1732 982 l 1733 982 l 1733 981 l 1733 980 l 1733 979 l 1733 978 l 1733 977 l 1733 976 l 1734 976 l 1734 975 l 1734 974 l 1734 973 l 1734 972 l 1735 972 l 1735 971 l 1735 970 l 1735 969 l 1735 968 l 1735 967 l 1735 968 l 1735 967 l 1736 967 l 1736 966 l 1736 965 l 1736 964 l 1736 965 l 1737 965 l 1737 964 l 1737 963 l 1737 962 l 1737 961 l 1738 961 l 1738 960 l 1738 959 l 1738 958 l 1738 959 l 1738 958 l 1738 957 l 1739 957 l 1739 956 l 1739 955 l 1739 954 l 1739 953 l 1739 952 l 1740 952 l 1740 951 l 1740 950 l 1740 949 l 1740 948 l 1740 947 l 1741 947 l 1741 946 l 1741 945 l 1741 946 l 1741 945 l 1742 945 l 1742 944 l 1742 945 l 1742 944 l 1742 943 l 1742 942 l 1742 941 l 1742 940 l 1743 940 l 1743 941 l 1743 940 l 1743 941 l 1743 940 l 1743 939 l 1743 940 l 1744 940 l 1744 941 l 1744 942 l 1744 943 l 1744 945 l 1744 946 l 1744 947 l 1744 948 l 1744 947 l 1744 948 l 1744 947 l 1745 947 l 1745 946 l 1745 945 l 1745 944 l 1746 944 l 1746 943 l 1746 942 l 1746 941 l 1746 940 l 1747 940 l 1747 939 l 1747 938 l 1747 937 l 1747 936 l 1747 935 l 1748 935 l 1748 934 l 1748 933 l 1748 932 l 1748 933 l 1748 932 l 1749 932 l 1749 931 l 1749 930 l 1749 929 l 1749 928 l 1749 927 l 1750 926 l 1750 925 l 1750 924 l 1750 923 l 1750 922 l 1750 921 l 1751 921 l 1751 920 l 1751 919 l 1751 918 l 1751 917 l 1751 916 l 1752 915 l 1752 914 l 1752 913 l 1752 912 l 1752 911 l 1752 910 l 1753 910 l 1753 909 l 1753 908 l 1753 907 l 1753 906 l 1753 905 l 1754 905 l 1754 904 l 1754 903 l 1754 902 l 1754 901 l 1754 900 l 1755 900 l 1755 899 l 1755 898 l 1755 897 l 1755 896 l 1756 896 l 1756 897 l 1756 896 l 1756 895 l 1756 894 l 1756 893 l 1757 893 l 1757 892 l 1757 891 l 1757 890 l 1757 889 l 1757 888 l 1757 887 l 1758 887 l 1758 886 l 1758 885 l 1758 884 l 1758 883 l 1758 882 l 1759 882 l 1759 881 l 1759 880 l 1759 879 l 1759 878 l 1759 877 l 1759 876 l 1760 876 l 1760 875 l 1760 874 l 1760 873 l 1760 872 l 1760 871 l 1761 871 l 1761 870 l 1761 869 l 1761 868 l 1761 867 l 1761 866 l 1761 865 l 1762 865 l 1762 864 l 1762 863 l 1762 862 l 1762 861 l 1762 860 l 1762 859 l 1763 859 l 1763 858 l 1763 857 l 1763 856 l 1763 855 l 1763 854 l 1764 853 l 1764 852 l 1764 851 l 1764 850 l 1764 849 l 1764 848 l 1765 848 l 1765 847 l 1765 846 l 1765 845 l 1765 844 l 1765 843 l 1765 842 l 1766 842 l 1766 841 l 1766 840 l 1766 839 l 1766 838 l 1766 837 l 1767 836 l 1767 835 l 1767 834 l 1767 833 l 1767 832 l 1768 832 l 1768 831 l 1768 830 l 1768 829 l 1768 828 l 1768 827 l 1769 827 l 1769 826 l 1769 825 l 1769 824 l 1769 823 l 1769 822 l 1770 821 l 1770 820 l 1770 819 l 1770 818 l 1770 817 l 1770 816 l 1771 816 l 1771 815 l 1771 814 l 1771 813 l 1771 812 l 1771 811 l 1771 810 l 1772 810 l 1772 809 l 1772 808 l 1772 807 l 1772 806 l 1773 805 l 1773 804 l 1773 803 l 1773 802 l 1773 801 l 1773 800 l 1774 800 l 1774 799 l 1774 798 l 1774 797 l 1774 796 l 1775 796 l 1775 797 l 1775 798 l 1775 797 l 1776 797 l 1776 796 l 1776 795 l 1776 794 l 1776 793 l 1776 792 l 1777 792 l 1777 791 l 1777 790 l 1777 789 l 1777 788 l 1778 788 l 1778 787 l 1778 786 l 1778 785 l 1778 784 l 1778 783 l 1779 783 l 1779 782 l 1779 781 l 1779 780 l 1779 779 l 1779 778 l 1780 778 l 1780 777 l 1780 776 l s 1780 776 m 1780 776 m 1780 776 l 1780 775 l 1780 774 l 1781 774 l 1781 773 l 1781 772 l 1781 771 l 1781 770 l 1781 769 l 1781 768 l 1782 768 l 1782 767 l 1782 766 l 1782 765 l 1782 764 l 1782 763 l 1783 763 l 1783 762 l 1783 761 l 1783 760 l 1783 759 l 1783 758 l 1783 757 l 1784 757 l 1784 756 l 1784 755 l 1784 754 l 1784 753 l 1784 752 l 1785 752 l 1785 751 l 1785 750 l 1785 749 l 1785 748 l 1785 747 l 1786 747 l 1786 746 l 1786 745 l 1786 744 l 1786 743 l 1786 742 l 1787 742 l 1787 741 l 1787 740 l 1787 739 l 1787 740 l 1787 741 l 1788 741 l 1788 740 l 1788 741 l 1788 740 l 1788 739 l 1788 738 l 1788 737 l 1789 737 l 1789 736 l 1789 735 l 1789 734 l 1789 733 l 1790 733 l 1790 732 l 1790 731 l 1790 730 l 1790 729 l 1790 728 l 1791 728 l 1791 727 l 1791 726 l 1791 725 l 1791 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717 l 1804 716 l 1804 715 l 1804 714 l 1804 713 l 1805 713 l 1805 712 l 1805 711 l 1805 710 l 1805 709 l 1805 708 l 1806 708 l 1806 707 l 1806 706 l 1806 705 l 1806 706 l 1806 705 l 1807 705 l 1807 704 l 1807 703 l 1807 702 l 1807 701 l 1807 700 l 1808 700 l 1808 699 l 1808 698 l 1808 697 l 1808 698 l 1809 698 l 1809 697 l 1809 698 l 1809 700 l 1809 701 l 1809 700 l 1810 700 l 1810 699 l 1810 698 l 1810 697 l 1810 696 l 1811 696 l 1811 695 l 1811 694 l 1811 693 l 1811 692 l 1812 692 l 1812 691 l 1812 690 l 1812 689 l 1812 688 l 1812 687 l 1812 686 l 1813 686 l 1813 685 l 1813 684 l 1813 683 l 1813 682 l 1813 681 l 1814 681 l 1814 680 l 1814 679 l 1814 678 l 1814 677 l 1814 678 l 1814 677 l 1815 677 l 1815 676 l 1815 675 l 1815 674 l 1815 673 l 1815 672 l 1816 672 l 1816 671 l 1816 670 l 1816 669 l 1816 668 l 1816 667 l 1817 667 l 1817 666 l 1817 665 l 1817 664 l 1817 663 l 1817 662 l 1817 661 l 1818 661 l 1818 660 l 1818 659 l 1818 658 l 1818 657 l 1818 656 l 1818 655 l 1819 655 l 1819 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000001000000000000000000000000000000000000000020000000000000000000200000000000000000001000000000000000000188 % 000001000000000000000000002000000000000000000020000000000000000000200000000000000000001000000000000000000088 % 00000300000000000000000007c00000000000000000020000000000000000000020000000000000000001a000000000000000000050 %%EndImage %%EndPreview save countdictstack mark newpath /showpage {} def %%EndProlog %%Page 1 1 %!PS /maxrep 255 def % max repeat count % Initialize the strings for filling runs (lazily). /.ImageFill { maxrep string dup 0 1 maxrep 1 sub { 3 index put dup } for .ImageFills 4 2 roll put } bind def /.ImageFills [ 0 1 255 { /.ImageFill cvx 2 array astore cvx } for ] def % Initialize the procedure table for input dispatching. /.ImageProcs [ % Stack: 32 { { pop .ImageItem } } repeat 16 { { % 0x20-0x2f: (N-0x20) data bytes follow 32 sub 3 -1 roll add 3 index exch 0 exch getinterval 2 index exch readhexstring pop exch pop 0 exch dup } bind } repeat 16 { { % 0x30-0x3f: prefix hex digit (N-0x30) to next count 48 sub 3 -1 roll add 4 bitshift exch .ImageItem } bind } repeat 32 { { % 0x40-0x5f: repeat last data byte (N-0x40) times 64 sub 3 -1 roll add .ImageFills 2 index dup length 1 sub get get exec exch 0 exch getinterval 0 3 1 roll } bind } repeat 160 { { pop .ImageItem } } repeat ] readonly def % Read one item from a compressed image. % Stack contents: /.ImageItem { 2 index read pop dup .ImageProcs exch get exec } bind def % Read and print an entire compressed image. /.ImageRead % .ImageRead - { gsave [ 6 -2 roll exch 72 div 0 0 4 -1 roll -72 div 0 7 index ] { .ImageItem } 4 index 3 index mul 7 add 8 idiv string currentfile 0 () 9 4 roll image pop pop pop pop grestore showpage } def 300 300 2550 3300 1 .ImageRead !ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff 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?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff 2N#e03fff ?ON!ff 2N#800fff ?ON!ff 2M$fe0007ff ?ON!ff 2M$fc0003ff ?ON!ff 2M$f87e01ff ?ON!ff 2M$f9ff80ff ?ON!ff 2M$f3ffc0ff ?ON!ff 2M$e3ffc0ff ?ON!ff 2M%e7ffe07fff ?OM!ff 2M%e7ffe07fff ?OM!ff 2M%cffff07fff ?OM!ff 2M%cffff07fff ?OM!ff 2M%dffff07fff ?OM!ff 2O#f07fff ?OM!ff 2O#f07fff ?OM!ff 2O"f0ff ?ON!ff 2O"f0ff ?ON!ff 2O"e0ff ?ON!ff 2O"e1ff ?ON!ff 2O"e1ff ?ON!ff 2O"c3ff ?ON!ff 2O(c7ffffffffff8000 =@"3fff 3E!ff 2O(87ffffffffffbfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff 3E!ff 2O(8fffffffffffbfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff 3E!ff 2I.00000003ffff1fffffffffffbfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff 3E!ff 2I.00000003fffe3fffffffffffbfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff 3E!ff 2I.00000003fffe7fffffffffffbfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff 3E!ff 2I.00000003fffcffffffffffffbfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff 3E!ff 2N)f9ffffffffffffbfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff 3E!ff 2N)f3ffffffffffffbfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff 3E!ff 2N)e7ffffffffffffbfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff 3E!ff 2N)cfffffffffffffbfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff 3E!ff 2N)9fffffffffffffbfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 3E!ff 2N)3fffefffffffffbfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 3E!ff 2M*fe7fffcfffffffffbfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 3E!ff 2M*fcffff9fffffffffbfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 3E!ff 2M*f800001fffffffffbfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 3E!ff 2M*f000001fffffffffbfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 3E!ff 2M*e000003fffffffffbfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 3E!ff 2M*e000003fffffffffbfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 3E!ff 2M*c000007fffffffffbfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 3E!ff 3E"bfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E'bffffffffc0fff 32 { { pop .ImageItem } } repeat 16 { { % 0x20-0x2f: (N-0x20) data bytes follow 32 sub 3 -1 roll add 3 index exch 0 exch getinterval 2 index exch readhexstring pop exch pop 0 exch dup } bind } repeat 16 { { % 0x30-0x3f: prefix hex digit (N-0x30) to next count 48 sub 3 -1 roll add 4 bitshift exch .ImageItem } bind } repeat 32 { { % 0x40-0x5f: repeat last data byte (N-0x40) times 64 sub 3 -1 roll add .ImageFills 2 index dup length 1 sub get get exec exch 0 exch getinterval 0 3 1 roll } bind } repeat 160 { { pop .ImageItem } } repeat ] readonly def % Read one item from a compressed image. % Stack contents: /.ImageItem { 2 index read pop dup .ImageProcs exch get exec } bind def % Read and print an entire compressed image. /.ImageRead % .ImageRead - { gsave [ 6 -2 roll exch 72 div 0 0 4 -1 roll -72 div 0 7 index ] { .ImageItem } 4 index 3 index mul 7 add 8 idiv string currentfile 0 () 9 4 roll image pop pop pop pop grestore showpage } def 300 300 2550 3300 1 .ImageRead !ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff 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2I.00000003fffe7fffffffffffbfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff 3E!ff 2I.00000003fffcffffffffffffbfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff 3E!ff 2N)f9ffffffffffffbfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff 3E!ff 2N)f3ffffffffffffbfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff 3E!ff 2N)e7ffffffffffffbfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff 3E!ff 2N)cfffffffffffffbfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff S"bfff 3E!ff 2N)9fffffffffffffbfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 3E!ff 2N)3fffefffffffffbfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 3E!ff 2M*fe7fffcfffffffffbfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 3E!ff 2M*fcffff9fffffffffbfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 3E!ff 2M*f800001fffffffffbfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 3E!ff 2M*f000001fffffffffbfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 3E!ff 2M*e000003fffffffffbfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 3E!ff 2M*e000003fffffffffbfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 3E!ff 2M*c000007fffffffffbfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 3E!ff 3E"bfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 2H"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3E"bfff =@"bfff 3E!ff 3D#fe07ff =@"bfff 3E!ff 3D#fe07ff =@"bfff 3E!ff 3D(fc07fffffffc0fff 32 { { pop .ImageItem } } repeat 16 { { % 0x20-0x2f: (N-0x20) data bytes follow 32 sub 3 -1 roll add 3 index exch 0 exch getinterval 2 index exch readhexstring pop exch pop 0 exch dup } bind } repeat 16 { { % 0x30-0x3f: prefix hex digit (N-0x30) to next count 48 sub 3 -1 roll add 4 bitshift exch .ImageItem } bind } repeat 32 { { % 0x40-0x5f: repeat last data byte (N-0x40) times 64 sub 3 -1 roll add .ImageFills 2 index dup length 1 sub get get exec exch 0 exch getinterval 0 3 1 roll } bind } repeat 160 { { pop .ImageItem } } repeat ] readonly def % Read one item from a compressed image. % Stack contents: /.ImageItem { 2 index read pop dup .ImageProcs exch get exec } bind def % Read and print an entire compressed image. /.ImageRead % .ImageRead - { gsave [ 6 -2 roll exch 72 div 0 0 4 -1 roll -72 div 0 7 index ] { .ImageItem } 4 index 3 index mul 7 add 8 idiv string currentfile 0 () 9 4 roll image pop pop pop pop grestore showpage } def 300 300 2550 3300 1 .ImageRead !ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff 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roll add 3 index exch 0 exch getinterval 2 index exch readhexstring pop exch pop 0 exch dup } bind } repeat 16 { { % 0x30-0x3f: prefix hex digit (N-0x30) to next count 48 sub 3 -1 roll add 4 bitshift exch .ImageItem } bind } repeat 32 { { % 0x40-0x5f: repeat last data byte (N-0x40) times 64 sub 3 -1 roll add .ImageFills 2 index dup length 1 sub get get exec exch 0 exch getinterval 0 3 1 roll } bind } repeat 160 { { pop .ImageItem } } repeat ] readonly def % Read one item from a compressed image. % Stack contents: /.ImageItem { 2 index read pop dup .ImageProcs exch get exec } bind def % Read and print an entire compressed image. /.ImageRead % .ImageRead - { gsave [ 6 -2 roll exch 72 div 0 0 4 -1 roll -72 div 0 7 index ] { .ImageItem } 4 index 3 index mul 7 add 8 idiv string currentfile 0 () 9 4 roll image pop pop pop pop grestore showpage } def 300 300 2550 3300 1 .ImageRead !ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff ?O3O!ff 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findfont 81 scalefont setfont (h) show /Times-Roman-ISOLatin1 findfont 81 scalefont setfont ( = 2) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 81 scalefont setfont 1611 1241 m gsave 1611 1241 translate 0 rotate 0 -27 m /Symbol findfont 81 scalefont setfont (h) show /Times-Roman-ISOLatin1 findfont 81 scalefont setfont ( = 3) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont end showpage %%Trailer %%EndDocument @endspecial eop %%Page: 61 61 61 60 bop 324 241 a Ft(REFERENCES)2471 b Fy(61)324 4063 y @beginspecial 78 @llx 72 @lly 516 @urx 434 @ury 3885 @rwi @setspecial %%BeginDocument: CLP_1st_2nd.eps %!PS-Adobe-2.0 EPSF-1.2 %%BoundingBox: 78 72 516 434 %%Title: CLP_1st_2nd.eps %%Creator: xmgr %%CreationDate: Wed May 9 17:45:53 2001 %%EndComments 80 dict begin /languagelevel where {pop /gs_languagelevel languagelevel def} {/gs_languagelevel 1 def} ifelse gs_languagelevel 1 gt { <} image end } >> matrix 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875 translate 0 rotate 0 -20 m (-12) RJ (-12) show grestore newpath 398 1375 m gsave 398 1375 translate 0 rotate 0 -20 m (-7) RJ (-7) show grestore newpath 2198 414 m 2114 501 m 2114 501 l 2030 584 l 1946 671 l 1862 715 l 1778 792 l 1694 863 l 1610 905 l 1526 957 l 1442 995 l 1358 1117 l 1274 1186 l 1190 1211 l 1106 1300 l 1022 1334 l 938 1382 l 854 1497 l 770 1508 l 686 1569 l 602 1678 l 518 1741 l 434 1744 l /Times-Roman-ISOLatin1 findfont 14 scalefont setfont s 4 2114 501 fa 4 2030 584 fa 4 1946 671 fa 4 1862 715 fa 4 1778 792 fa 4 1694 863 fa 4 1610 905 fa 4 1526 957 fa 4 1442 995 fa 4 1358 1117 fa 4 1274 1186 fa 4 1190 1211 fa 4 1106 1300 fa 4 1022 1334 fa 4 938 1382 fa 4 854 1497 fa 4 770 1508 fa 4 686 1569 fa 4 602 1678 fa 4 518 1741 fa 4 434 1744 fa /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 2198 411 m 2114 491 m 2114 491 l 2030 583 l 1946 662 l 1862 705 l 1778 782 l 1694 853 l 1610 896 l 1526 956 l 1442 985 l 1358 1107 l 1274 1176 l 1190 1201 l 1106 1291 l 1022 1324 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4 938 1375 fa 4 854 1459 fa 4 770 1505 fa 4 686 1557 fa 4 602 1664 fa 4 518 1696 fa 4 434 1702 fa /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 2198 384 m 2114 428 m 2114 428 l 2030 527 l 1946 576 l 1862 632 l 1778 717 l 1694 780 l 1610 824 l 1526 919 l 1442 942 l 1358 1051 l 1274 1095 l 1190 1176 l 1106 1209 l 1022 1259 l 938 1350 l 854 1383 l 770 1469 l 686 1507 l 602 1605 l 518 1628 l 434 1631 l /Times-Roman-ISOLatin1 findfont 14 scalefont setfont s 4 2114 428 fa 4 2030 527 fa 4 1946 576 fa 4 1862 632 fa 4 1778 717 fa 4 1694 780 fa 4 1610 824 fa 4 1526 919 fa 4 1442 942 fa 4 1358 1051 fa 4 1274 1095 fa 4 1190 1176 fa 4 1106 1209 fa 4 1022 1259 fa 4 938 1350 fa 4 854 1383 fa 4 770 1469 fa 4 686 1507 fa 4 602 1605 fa 4 518 1628 fa 4 434 1631 fa /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 2198 414 m 2114 484 m 2114 484 l 2030 533 l 1946 622 l 1862 701 l 1778 780 l 1694 840 l 1610 867 l 1526 960 l 1442 1003 l 1358 1056 l 1274 1125 l 1190 1170 l 1106 1269 l 1022 1319 l 938 1384 l 854 1441 l 770 1528 l 686 1584 l 602 1623 l 518 1668 l 434 1671 l /Times-Roman-ISOLatin1 findfont 14 scalefont setfont s 4 2114 484 fa 4 2030 533 fa 4 1946 622 fa 4 1862 701 fa 4 1778 780 fa 4 1694 840 fa 4 1610 867 fa 4 1526 960 fa 4 1442 1003 fa 4 1358 1056 fa 4 1274 1125 fa 4 1190 1170 fa 4 1106 1269 fa 4 1022 1319 fa 4 938 1384 fa 4 854 1441 fa 4 770 1528 fa 4 686 1584 fa 4 602 1623 fa 4 518 1668 fa 4 434 1671 fa /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 2198 421 m 2114 483 m 2114 483 l 2030 538 l 1946 621 l 1862 693 l 1778 779 l 1694 830 l 1610 866 l 1526 951 l 1442 1003 l 1358 1055 l 1274 1132 l 1190 1177 l 1106 1268 l 1022 1309 l 938 1374 l 854 1441 l 770 1518 l 686 1574 l 602 1607 l 518 1679 l 434 1678 l /Times-Roman-ISOLatin1 findfont 14 scalefont setfont s 4 2114 483 fa 4 2030 538 fa 4 1946 621 fa 4 1862 693 fa 4 1778 779 fa 4 1694 830 fa 4 1610 866 fa 4 1526 951 fa 4 1442 1003 fa 4 1358 1055 fa 4 1274 1132 fa 4 1190 1177 fa 4 1106 1268 fa 4 1022 1309 fa 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938 1380 l 854 1443 l 770 1504 l 686 1555 l 602 1574 l 518 1702 l 434 1713 l /Times-Roman-ISOLatin1 findfont 14 scalefont setfont s 4 2114 427 fa 4 2030 548 fa 4 1946 613 fa 4 1862 683 fa 4 1778 745 fa 4 1694 815 fa 4 1610 836 fa 4 1526 913 fa 4 1442 1013 fa 4 1358 1029 fa 4 1274 1142 fa 4 1190 1187 fa 4 1106 1240 fa 4 1022 1309 fa 4 938 1380 fa 4 854 1443 fa 4 770 1504 fa 4 686 1555 fa 4 602 1574 fa 4 518 1702 fa 4 434 1713 fa /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 2198 601 m 2114 691 m 2114 691 l 2030 769 l 1946 842 l 1862 903 l 1778 962 l 1694 1014 l 1610 1066 l 1526 1122 l 1442 1178 l 1358 1226 l 1274 1279 l 1190 1336 l 1106 1387 l 1022 1438 l 938 1490 l 854 1546 l 770 1595 l 686 1646 l 602 1695 l 518 1745 l 434 1744 l /Times-Roman-ISOLatin1 findfont 14 scalefont setfont s 4 2114 691 fa 4 2030 769 fa 4 1946 842 fa 4 1862 903 fa 4 1778 962 fa 4 1694 1014 fa 4 1610 1066 fa 4 1526 1122 fa 4 1442 1178 fa 4 1358 1226 fa 4 1274 1279 fa 4 1190 1336 fa 4 1106 1387 fa 4 1022 1438 fa 4 938 1490 fa 4 854 1546 fa 4 770 1595 fa 4 686 1646 fa 4 602 1695 fa 4 518 1745 fa 4 434 1744 fa /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 2198 585 m 2114 679 m 2114 679 l 2030 738 l 1946 808 l 1862 868 l 1778 928 l 1694 980 l 1610 1032 l 1526 1087 l 1442 1143 l 1358 1192 l 1274 1245 l 1190 1301 l 1106 1353 l 1022 1403 l 938 1455 l 854 1512 l 770 1562 l 686 1607 l 602 1630 l 518 1665 l 434 1671 l /Times-Roman-ISOLatin1 findfont 14 scalefont setfont s 4 2114 679 fa 4 2030 738 fa 4 1946 808 fa 4 1862 868 fa 4 1778 928 fa 4 1694 980 fa 4 1610 1032 fa 4 1526 1087 fa 4 1442 1143 fa 4 1358 1192 fa 4 1274 1245 fa 4 1190 1301 fa 4 1106 1353 fa 4 1022 1403 fa 4 938 1455 fa 4 854 1512 fa 4 770 1562 fa 4 686 1607 fa 4 602 1630 fa 4 518 1665 fa 4 434 1671 fa /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 2198 593 m 2114 687 m 2114 687 l 2030 745 l 1946 814 l 1862 874 l 1778 934 l 1694 986 l 1610 1038 l 1526 1093 l 1442 1150 l 1358 1198 l 1274 1251 l 1190 1308 l 1106 1359 l 1022 1410 l 938 1461 l 854 1517 l 770 1565 l 686 1613 l 602 1652 l 518 1681 l 434 1683 l /Times-Roman-ISOLatin1 findfont 14 scalefont setfont s 4 2114 687 fa 4 2030 745 fa 4 1946 814 fa 4 1862 874 fa 4 1778 934 fa 4 1694 986 fa 4 1610 1038 fa 4 1526 1093 fa 4 1442 1150 fa 4 1358 1198 fa 4 1274 1251 fa 4 1190 1308 fa 4 1106 1359 fa 4 1022 1410 fa 4 938 1461 fa 4 854 1517 fa 4 770 1565 fa 4 686 1613 fa 4 602 1652 fa 4 518 1681 fa 4 434 1683 fa /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 2198 587 m 2114 681 m 2114 681 l 2030 746 l 1946 820 l 1862 880 l 1778 939 l 1694 992 l 1610 1043 l 1526 1099 l 1442 1155 l 1358 1203 l 1274 1256 l 1190 1313 l 1106 1364 l 1022 1415 l 938 1467 l 854 1523 l 770 1571 l 686 1619 l 602 1662 l 518 1698 l 434 1698 l /Times-Roman-ISOLatin1 findfont 14 scalefont setfont s 4 2114 681 fa 4 2030 746 fa 4 1946 820 fa 4 1862 880 fa 4 1778 939 fa 4 1694 992 fa 4 1610 1043 fa 4 1526 1099 fa 4 1442 1155 fa 4 1358 1203 fa 4 1274 1256 fa 4 1190 1313 fa 4 1106 1364 fa 4 1022 1415 fa 4 938 1467 fa 4 854 1523 fa 4 770 1571 fa 4 686 1619 fa 4 602 1662 fa 4 518 1698 fa 4 434 1698 fa /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 2198 624 m 2114 678 m 2114 678 l 2030 756 l 1946 836 l 1862 898 l 1778 959 l 1694 1009 l 1610 1062 l 1526 1118 l 1442 1172 l 1358 1221 l 1274 1274 l 1190 1331 l 1106 1382 l 1022 1433 l 938 1486 l 854 1542 l 770 1589 l 686 1640 l 602 1689 l 518 1735 l 434 1732 l /Times-Roman-ISOLatin1 findfont 14 scalefont setfont s 4 2114 678 fa 4 2030 756 fa 4 1946 836 fa 4 1862 898 fa 4 1778 959 fa 4 1694 1009 fa 4 1610 1062 fa 4 1526 1118 fa 4 1442 1172 fa 4 1358 1221 fa 4 1274 1274 fa 4 1190 1331 fa 4 1106 1382 fa 4 1022 1433 fa 4 938 1486 fa 4 854 1542 fa 4 770 1589 fa 4 686 1640 fa 4 602 1689 fa 4 518 1735 fa 4 434 1732 fa /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 2198 640 m 2114 673 m 2114 673 l 2030 756 l 1946 808 l 1862 864 l 1778 925 l 1694 975 l 1610 1027 l 1526 1084 l 1442 1137 l 1358 1187 l 1274 1240 l 1190 1296 l 1106 1347 l 1022 1399 l 938 1452 l 854 1507 l 770 1558 l 686 1603 l 602 1626 l 518 1679 l 434 1678 l /Times-Roman-ISOLatin1 findfont 14 scalefont setfont s 4 2114 673 fa 4 2030 756 fa 4 1946 808 fa 4 1862 864 fa 4 1778 925 fa 4 1694 975 fa 4 1610 1027 fa 4 1526 1084 fa 4 1442 1137 fa 4 1358 1187 fa 4 1274 1240 fa 4 1190 1296 fa 4 1106 1347 fa 4 1022 1399 fa 4 938 1452 fa 4 854 1507 fa 4 770 1558 fa 4 686 1603 fa 4 602 1626 fa 4 518 1679 fa 4 434 1678 fa /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 2198 648 m 2114 680 m 2114 680 l 2030 763 l 1946 815 l 1862 870 l 1778 931 l 1694 981 l 1610 1034 l 1526 1090 l 1442 1144 l 1358 1193 l 1274 1246 l 1190 1303 l 1106 1353 l 1022 1411 l 938 1458 l 854 1513 l 770 1560 l 686 1608 l 602 1646 l 518 1665 l 434 1665 l /Times-Roman-ISOLatin1 findfont 14 scalefont setfont s 4 2114 680 fa 4 2030 763 fa 4 1946 815 fa 4 1862 870 fa 4 1778 931 fa 4 1694 981 fa 4 1610 1034 fa 4 1526 1090 fa 4 1442 1144 fa 4 1358 1193 fa 4 1274 1246 fa 4 1190 1303 fa 4 1106 1353 fa 4 1022 1411 fa 4 938 1458 fa 4 854 1513 fa 4 770 1560 fa 4 686 1608 fa 4 602 1646 fa 4 518 1665 fa 4 434 1665 fa /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 2198 642 m 2114 674 m 2114 674 l 2030 757 l 1946 813 l 1862 875 l 1778 936 l 1694 986 l 1610 1039 l 1526 1096 l 1442 1149 l 1358 1199 l 1274 1251 l 1190 1308 l 1106 1359 l 1022 1410 l 938 1463 l 854 1518 l 770 1565 l 686 1614 l 602 1656 l 518 1684 l 434 1682 l /Times-Roman-ISOLatin1 findfont 14 scalefont setfont s 4 2114 674 fa 4 2030 757 fa 4 1946 813 fa 4 1862 875 fa 4 1778 936 fa 4 1694 986 fa 4 1610 1039 fa 4 1526 1096 fa 4 1442 1149 fa 4 1358 1199 fa 4 1274 1251 fa 4 1190 1308 fa 4 1106 1359 fa 4 1022 1410 fa 4 938 1463 fa 4 854 1518 fa 4 770 1565 fa 4 686 1614 fa 4 602 1656 fa 4 518 1684 fa 4 434 1682 fa /Times-Roman-ISOLatin1 findfont 60 scalefont setfont 434 375 m 434 1775 l 2114 1775 l 2114 375 l 434 375 l s /Times-Roman-ISOLatin1 findfont 81 scalefont setfont /Helvetica-ISOLatin1 findfont 81 scalefont setfont 1426 1245 m gsave 1426 1245 translate 0 rotate 0 -27 m /Times-Roman-ISOLatin1 findfont 81 scalefont setfont (NC) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont /Times-Roman-ISOLatin1 findfont 81 scalefont setfont /Helvetica-ISOLatin1 findfont 81 scalefont setfont 1090 1063 m gsave 1090 1063 translate 0 rotate 0 -27 m /Times-Roman-ISOLatin1 findfont 81 scalefont setfont (CN) show grestore newpath /Times-Roman-ISOLatin1 findfont 60 scalefont setfont end showpage %%Trailer %%EndDocument @endspecial eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF