Content-Type: multipart/mixed; boundary="-------------0306060905764" This is a multi-part message in MIME format. ---------------0306060905764 Content-Type: text/plain; name="03-267.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-267.comments" A Morame, Universit de Nantes, 44322 Nantes Cedex 3, France. email: morame@math.univ-nantes.fr F Truc, Institut Fourier, 38402 Saint Martin d'H res Cedex , France. email: Francoise.Truc@ujf-grenoble.fr Mathematical Subject Classification: 35P20 ---------------0306060905764 Content-Type: text/plain; name="03-267.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-267.keywords" Neumann problem, magnetic field, spectral theory, eigenvalues. ---------------0306060905764 Content-Type: application/postscript; name="prep607.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="prep607.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86d Copyright 1999 Radical Eye Software %%Title: prep607.dvi %%Pages: 16 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMR12 CMBX9 CMR9 CMMI9 CMBX12 CMR10 CMMI10 CMR7 %%+ CMSY10 CMMI7 CMSL8 CMR8 CMEX10 CMSY7 CMMI5 CMR5 CMBX10 CMTI10 CMSY5 %%+ CMMI12 CMR6 CMSY6 CMTI8 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o prep607.ps prep607.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.06.06:1559 %%BeginProcSet: texc.pro %! 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 14 /openbullet 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 c43e28519671e8def84a1526a8b89450ef2bd624857da91e78f75dfbf7b8bcdc dc20960373fe601e8ea49e3d1cff1deb5c14e11975cd39edb706b363607acacc 3355ca449db8fc533b240f899da7a68418f9f0c3a4f0e82a6dff680742c4f0b5 bca221fcbd0bcc643f93787e2cac3f808ed2b8f4815cfb0881d58bb8f79f759b 5fe30bc3492d4e40dbf69e0178d4d70ee5f7def0ac821a51130fcde7d902aa74 87a0be32cb54fb38d9c49056cc4ffebaa3f33f23805d1807a65411fa8db1da30 a77af5891ba4d5bf30e66a5397cf175a574844f8e8255321fe74d501fd3918ef 43c34d2a7697b83b09e600d41f1910557c9247e272c7a293078bbf572b2ec5b4 e95a 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 50 /two put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueXX 5000789 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b031f3fc6c13d7da7277a94ba 018e9998b3dd888011a5d7c4204989f30f908b95533bda845746b673ab71ea57 65a0d14f4350707e47c8276305b28513cbe1bb0dbd269a53719bda46e536685d df78ca0146b6b93e760256b74d939d4e35b5e77238f04c92298dfdd188feea30 e053eefbcbb52f2011772b3aae39f5805597bbc1e8bb75a446ce014030f4f2f0 f49f9e962ee4a1024a746fa92a3628db5270732b54e43fe5ecfa524f127e5fcc 788e77e66098336ad67fe4cccaf0253272d5df79864bf4b734cb9a5859d557d8 bc11b8e00221ebc12e97de4b1f466ead83a4c894709363bca9040410a52d592e 34ee40cc7e5efa920546b981aa659513a24b1b85c221a1875b62d0b89e57a368 321b8043a5b094e0379760a443d632892b14ad6d19dacc8c78093243ad67e6a3 08e56e6b68412ee690b10dac6e17708754a00d51fc957b500eb80175716eef4b 2ca1ef867614659bee3f2b7319e97b6fdf1efc847bf3cee3156f72f21751da8e 5fb6898919e6799820d3de0642d756e09d6fae4ff08dd3deda3173bff4bb11f7 9109c97ddc05897af709ea199a90fcee8ce4c7a3c15b18170c41c04de2d3fba8 f34296a95b8e1e8de3739b17273f8f2c85e914615e8eac5e8bd2387ba3b1edf4 7968f06e2067d836d0f9f3e085cdfd2de06a62c81d786b304326f7002e83160a 36598589228b4dddddc43c85e1d126f8fe81b828028e26317af5894aaccf4f69 6301e1a9fc45935d8a414957f08febebbc3a72ada80f101e47447d019ade56e9 f4fab969bba2b44e47399fedf5caa1bcea216d7ba713d523dfa618d367b001af bfc9543651fadf6389c92ef4a113933bfb4ce45b9ec1da149da8266d7ae816ac 591763c1ce0975ca2c6fad4dbf00da2d5ea9005dfe188787b84606f607dcb299 dbd62d54c0d9a3f63151b4605487b20b6612bacb429274d9c41584d6b0364b8e 3d9974f806134b12b859a7d761a1cf69b88c3b8fcd7df349ecea60c16041cf86 b3759ca18300e771a59922287a619d7d1e85a6c05fd807d62fb128a85222d66a 5ffdfc23a68bf4b49d6cb703a4b61c83a49b8e965c46262f03314b3b201179ba 4523b7f14bc486d8c369a8681b7a2837f5f82810d48ab9c8063598298c56340e 20e031c2603386b7564b3ff929be77050ce216cf7ccca14fd4916821b5e568bc 46c3ae7c6c63e4bb5f9cff87417005f6403af598ac4fbf73ff557e9124485569 1327cc0942e6061e12c0f107926d130a 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 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 48 /zero put dup 50 /two put dup 51 /three put dup 54 /six put dup 65 /A put dup 67 /C put dup 69 /E put dup 70 /F put dup 74 /J put dup 77 /M put dup 79 /O put dup 82 /R put dup 84 /T put dup 85 /U put dup 97 /a put dup 98 /b put dup 100 /d put dup 101 /e put dup 105 /i put dup 109 /m put dup 110 /n put dup 111 /o put dup 114 /r put dup 115 /s put dup 117 /u 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 f266b31d0c7e791d633f16ed94e05517a32163e74cbd96dc5db066f2a5313d6d 7f09465608cb404613e0f058073af000e5bdfafbbf9da0c0d5c02a3447f2cdc1 f56481901e34265c26d331ac3f85bdcbc7c51ef9eddf2b9c189c5253f8e74282 d6f053d7d183ae17ea175e5ebc401e67d21a9fea139cbbf3bb1358c713a75843 b39a69aa2b1a7b975f53ff22d1dc3bdf7acd17afa1da21568310c68241b31d3d 03bf404bd30baf9cc19c75b958bce7155f03cb0cfe2cae0327aad4b603284a51 7c26de75b8c6f6aaa9debe4aec08ca4b646d787bab1e702054272909211e814c 1d2c33152bda973de70ec02436551ff9be604d16ffa29b93380b5a04a493f00a c1d3c429bd6c92508652f7ab7347e9dab047409e478b59e22a73993a96ad13f7 3a58f110ca20a1d65e0577f470498b6faed25a690fcb92f9f19bac5ac431c893 607dfdc528e7d7ccd1a953ffda264220271f146682ae3af7b7d80807576dcb76 0e5c25929b3a4ac45cc6fa965b8ba72eb5f411d089e080b28c17e3345b0fdcf0 cb2c8db62d0f8d69d94b8ce6a8269f995fa377426d67ce1423f015d712f2d7f2 2faf422346ce9a406a660568dfc043e861cfc829dd546a5638952aa9e634a1b7 a81e005b676e28bc937f228ad5bdd35537c0e597bcfa92913055b7696901f99f 6ecd856712745a329880c0a5720b36dc391ead1e3825cd968dab6f9b7817abe6 20df90a298eb6eaaac1a7716e26a622c13f0fc9134a7002a353ef8a5a91247ce 0d2ebac87ce6c3adc323e6a64a960b99d81ef07b37a61043261f28b4e7401492 57c6646d29ac096891f7fcf1fe8221d669aff5241b4a0073810dd24d4e588f04 9158073dd6357d1c0ee4b813a4980b39c8e46585d3a16076d732f59d37cacb5b 98e91e49841b3f013a33fce02f14ca91b5553264012989e7232ef5cc397834e4 a0bb368dd9610100928b86ecaf45ed8881c4c0fd989f0a0d80c1b7dbe243f114 08d2a9f58e84e72b38d9c7ffafaef9514a9c1270d434d5fb5e6af4131158758d 63dc0ca3c665831fba40230123c53423ef85c119a9919c62146ea9153142dfa6 2ccc331a2c2b47da369ee6d8e2a8da38347718c13e91750ec5afc7b9669d86af d7364270f9a46a0fa5bbef7a72e8e9dc88914f47a2a2c4433494991294364bc4 5d77f2f0212f71e671f4f7f06af55eaebe28df4f7452d45d050df2ab6a27e4f9 d87c5c8b271dc2e018afd4459e5a5e23a4ef06dd484deca6650eb33ad1f87ea7 0d4cf717b0407ae27bb19e4c61142e4e3dd2fef1707b26070288a3a5d6ad0eb6 7871052aabce7999056a3df080b162f01bc3cff22d91bfcfb4cf4f8db336587f 86a620d35d4cd19d5f6f8c02503551b0d05a2915428d79a6f7bc26f25cfb4a94 19fe3f5da1d29eb897f98760b6e2846947bb2edd27e253b4fb18e132046ba6f5 9860db092dea9137f29399200b168b3aad9373278373c84e24e8c3405326bb58 a5835fdaa020ead79ed35b4c5f0a255cd94b4f82b69dc2436f447ade15b1a012 a10ca29546e95080e8c819984f2e5aea0b490ce79e425e8c3443459b638080fe 4d6e2dfd2c010f57e396bbcab3131148759e08278419c06b5695cc655b47b7de e1625101021c4dcf9525df640181e3117670bcf93334a96c5be0fe1de6745c12 c44481d0b3b117d6e849537f3c7255d69ce8ea2d6e5ba008241937df18a45219 7c216850e72184693556eae3219b2c96e1e04bb2f631c7f66572ce6c4727ef53 cad10e361f6c736d6b22d829333b56a52921b32c8ff56fb290bf4242e3b0bd55 9627f154b4d30bc02450a5b047536f459e36aacc03b7f86166a548d6a9498c2f 7332c0bd205dc380746d47311818302cab89ea08f134f25db2aab939ca18a8dc 7d4fb5e0cd8141ac049c561d3f7e339695d0617f70ac3f7fd77a8bdc11abe68c 0bd423bab250f15a8042cfd9b2f4d227c7293b63b8f391633822afa54c63b3e9 31950daee5fb6460eafaf310eae580163c027c45036a7dddb176d1de2b64e432 a76c3a88f0dd68bc27c9114e2cc40e7a5e0a6e3dff2cd8724b0080970b122621 d41eccb1cceb0c65954f2e65572e7dcb9183d5f0ca737d089e5e8ea6ef61bb63 5f193a7d7efb46f16c702d21a75dcc79e265e35d261dc1ebf962c337801fc891 48babeda7e0c5cb2dbc2e8e7e4477e31c73ff9ad5149355f87a821bfaa7f1e8b 62b8b0276300a9fb6322b55306c9cb8a8635bfcf304804c5ede17a2ed9ced0ee 96bf6fd819088429579da51f2cd459ea1b2d3ec23ef2168ea584a149960688ce 19e13326872837f94ae42daa33c568ae63b171c6da5d10c681b3762e039f94f7 1634aaf4feb497ae4edda9b788aa5c0bee46ca6d2a249b99ffaefd63b7fa6ec2 f46eb2d5cb3aa039ccc1bc20b57a47da11a4bf702ba4bbf63e761c0495e84e50 27f1b85db6928cced1551b95f96e9f300d26aace160253679435135381bd4325 72350f5d44d9d6fe96d481f93433895be59639710b2a69c35f456a5197dd9cf1 05f218ee7f6dc55c0564fcf253a6d2cebacdca193b11651c617b6552b93074af b48dea7cfe93523096574d553e2533ac8e11bd4372a3a1df85fc90d4d3eb80e0 67c30ed47268298d8c8c106c40517d101f280d04d49e530ba81439f4c2ee93e0 10e7f92deab4211a8b5548558a8aed6540b294ba273ab0493fe7311f16ec1f66 cdeb3ab848ae51bf6c7ad4a0972435a51973fe653e8d6d148200bee6425ac889 92219e42fcafa6f7d9369a4415e2654f886ec4ccefea0cd340114d50cc148316 99e7434049f970c10db6575feaf0cb1f01c15fefea790b3d4173c085652efd90 59e63280b3af32867a97a858bb3663d675d9a92323b7e7b97034bd6fc43da831 c38f5bb7fe6f13bfae3289efa75f8f278cbf1c2e247d2d07167d23f2ce1242e8 1b032f879a0c70147727f70c658466961d031d1fcd34bb446e2044ebbbe18a06 94de9287c1d8233e00b12eccc6260ca99ba2edde52dc8377a23d9c5e13962253 daf88fdfacd43c63d66f2b7748215d49431173c042cd0aa496f4d6ba08627835 848af1abd1043224d91a4b30232702e386da936aac4699b5372017d3d68b6b74 980c44255b153886abc61d634672332a802ed5a57f842e1e49d73b38da0a6664 169522e991a3bb343db42830112637837cac81b5dc13467fce644cf0f0a660cd ea6c8052848b5c0a472d7c8695edcd8d276630d2d85a15444753a9f83143e1ca a7b8f81a7ee7be898e10f78ffb3cab8e8049e4e226adffddcc198a147fbc606b b94d74fc5369c4b41b111047dabdc2b058e046914a6a8bdb0e87ef38a2a9817d a6916c5e062d7e2a5562147fe8cb44d7a2c59925746e320fc5772ea906842c86 daa7fb38eb8c150d36c6aa8ad42c6892e818cfe3eaa96bcd7665164a692227d5 fd4ce866c82fc8fbbb593fcac0468e2dd77bd1ebfdb974ed8a12a8ab490dca05 3b7a7ff3879249a652e41501b3366a77b389eedb8ca79c5dbd09da77eb41c4c5 4c10d38ad67326c4c35d56c5c09972f7f21e789e562633073b5465e4731f3c50 372cf5861a9a4ec62abd206bf98af185550ae6ffdc5a6440d613792444e56dc2 4f6500f33c11a45bb4df81e68c147b1f0120d7f89836324f13d6ee498e0a6a40 8c22bda86afc26d9bbcd9101b8449fdcf523dcf62707e6f8f73fe754dde6aaf3 f0941506ae75021303979c737ad7f41c03c3ed6f6f960913540bf2a3bcd6a954 b3e07bf088fd1dcaa3d8559e222b9fffa18c0183646d661874a71873c3ed9166 49068e3a96ca04d93775230da25db8a4918b6d5a496a7ac039e2621e204095e6 d6006e72138ca09b7306436ea16178fdefc1a2281f36091053f86f32c1d1fe06 d403ea16c6aa6660308e9dcf32d55aa3c63c9f4c19268627651b07c8f042b963 da2bd706b21e6e49d0bde12cbfd1c08995757dc5e39f9689a2d6e649a847b4a6 8dfb2da555e8f3e4fe3bc5159a4a4b4ada53e3bb5a5dd3e42369221d08614a9b 8a15572498a3c375ea76f3cbd06ec610b3a93ec71be1285bf698d2dcb2998b8e 5c995b738b4ce314bfa37e5884681ae5f12e2be2b3b4c5f4ddff7b893cac31ad bcd68c777c1e05019a1dc8c3d42adf1d8ab9bba5f0aaf7ba350564e643db6dcc 7ba69b514e10bb38b016e3e34d6f22d6381e71d55c1d244e03a193d0e22b6bdc 89c2ae1dedf6920dcf72320a29d80888f5bc0c9ad5ee12df6cf1d233a2d68afb 5565eab367942ebe2014292c1a6821284752bbca61e5bcff28d6aff96cfb9e4b 5c37228382ebd1a366773ed4ea221dc6ca8112bf864987ca19d6d2898314db8e 51b4a939f97ab83995c9c7abbcf440e1777fd187d585cf50d7b652eb17fcc335 4128d5b1c412c6f871f77b2e7fa3d418366298bb18923a8a78b89e9bd0aae51a e8e02103217ff751d78f5d3991f98fb73080ad29da1e83f35e9266bcf6abcbba d9a18edaf0682ad9637aa085c24f7d92ce4b3c6af34df4d58325540168d68734 4d237d3c0cf1521bdd94ead26378ba906edde9546ae513cc50730b18f0eba831 5278dd9efc2c1ea83501c3eeba85544b335089cc735f8cd04a5edbfb530af2b0 90df6dd9fc4e5d0c648b00e06479dfb3cce22a949baad6a1e2573481cd04ad7f 2f41d6c32be860cf65cf9b7630ba3cbe7ad7223dc0b582d0bf7aee59a4e1dd06 be7a83c286ee0c4f332ed64091d3b07b86bd13b5f7225a73600fd0bb7ba481ca 672d095f624175418f04f1fa6acf1ae742178a40df47f052f8f628992270a326 0d5c16666e4142659e266bd9eaa7caa91c8a2f406958dc032e9ac633aa5cfda9 28ec6f0305ee68faa15b5cbeeb35bb07c8ff6040a66abbc060a59f1534c1d9b3 8831438c08f6dcd2a0c952ad92ef2c341214fb2fa4f4d1fb9f527979ea925d4e 41a8330e55806e4f82d9df4fc250ca1019361e5371d37af06306be147eee38aa 20165d1d3f0fb6f81724e19fbad3ea8f9425dad95ed33faaf0f77f07286751d4 fdc01a82e0425d5dc72e679f64c5e1fd10ce1c3040831acf076f6eebc94e9285 7fe560fd4936aa70642b2384a71590da742442ba0249bb4b6e17537e0c1e45be 41689fdc4064d4be18d8eb5fdc4c795f52ef2ef12fbd457eb55a52a6e39660bb 894e4069056a13e82e0a0a814b172fa6d90c07cdd41c0a52abca192199f0032a 9c413ceceff13b0fecf5455f9415d4fe19b54551f231df915c7e661ccdc38c5c a66eaa410b07fc59d9b86d9e6191c6c784fb4d0129b7c0e6010078aa531c5fc1 07819b16d82362b362c7b8ad5cec242f814d14ca12731f9bec6244dc717951ca 42a39c33e02d3faf3d45cb7ec707f040312c94ae11a3559597c84e45f835a615 8591826e2c2e6d5480bf6e1d231d6d885a2c08e4083358689d5412bb22a05077 0cef250c1b689e78b60ddcaec8c79612b6e13bece476e162703865fbc599b85d 410edd4cc67bfedb7e25b671c4276c6fa2b839e9f81cd5206e72db0498c96877 5bb08681bb9b1709623a74e5c4c26487d55eb19d15318b73624393e793a2d8af b19754c8bbf91a3fb8eff5ed8e926ec0cb3de18edd7a191194602c1c56f44565 980726775f216727898220ab8420291299cd9a60f8f890410014de2ffeb8faf0 9caec8784450a1fc07445dfd83b609d73a264a8b1aa2793b8a8c0fca58f4c8a3 c61dca189eb5e88b7a56a18b5b25994ce7c000a8b4b4e94d449e2d26167bb7b3 88ddf6f9e347f3b0cc3aa7b357d17b2befeeb528a65e6850fa4638ba5495914c 6cac539d87e8cf1734a28a1e8c793f8c0156929a152f11de50a2dd85a0435062 7fda04f0f40885596a2360c23cd210f406ea5aaba33a5245b28fecdb608dd871 1365193d91cfe4ae372cf918a438388544e7c52c3bd154affe8a2b51cc9953de 9654398b2cdc60a8b5757c1ce416fa5b8dd93ab8b312fb057b1f83d184802863 f83d8870746eb8cb4564076277f398125b6a9ec938e68a4c21805affc7f391d9 61f698308b42de2f062056c79cfe1930b7fa1325e1548507b5ee96e7b68548de 96790cbdbdbcfcd571a6d94254626184b6a1e4473e3ed077cc37e9585afca571 3d78f43df920f48e4c3f3de87a33e67e8bc3f9ab06db57fceff7df369c6685df b05f5f7cfd2d8afa30b14587e23e1176571f63cb67aa694c3100e34614fc6d55 59c525d498fc021f767154bda5eab30cfd04794beda6d3bd361b6863930f9490 9460375d759b7db65ad3b9d1a332b67d89b2fd281549ce58df97df80000b86dc dfb4b82b2fe3409612c71279d53228123faef537da65a7a9cf3bd89295954584 8afb4d5bc593988d045da0cab683576a0beb713d0589a4d1f30b9d41fef9be65 007056b6e2476ba3bebb73eb723a33cea690c31a64c3afcb9cbbd3ca2a859cf0 814c09e059f46d8c0218ba13c347e9bee5bd3d7379a213e6dff6101e006be2f7 097f98ec58ae1686f703c9dba24a69602f75d900fda56c5705d7211f614c68e8 ee027091e4386330ed863d13aa14116a8dae429ace656c417e3a08c433aa4567 2e17aac2e2a96f51bd5d7aed1131bc37aaf4bd90368dacf2e7006dd738f11472 59f34c631070df8d5ac01f31f4ca0f9c484e944a489e2de6202f2b709eb512db aadd91e82a93bb762f1d6f716af92d537ccc95bde73b25377c62a6de359f2dd7 2cb2f91b8337eb46a2ad8fa77b1bd6a283a5a84ad8fc6db42de4dccebda2779a a4e42e5819e67928289cd534621f9b6f011e5423359cf7c4819e17d92681f1c8 6a15bbb1c5fde8251b5c1423829eadc50f6fd553e5c0f81cc0c1c0916cadd609 0ff04904757a3f802fa22d829a53c37517d9477a9a21675e3e06d81d0cf4a486 aa520c0592a854d3bb3db4744e8d2bfd24666004eb3f6913cd78298d67fcd207 21d63a9e776d60d84c1fbe9e6eae66da159d44c30ce19a91933a49e61833421c 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 18 /theta put dup 22 /mu put dup 24 /xi put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueXX 5087386 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7bcb09b7064e91fa0ded228e42 09aae407a7aaca60b1076299ac4abd23ef02f108765f0e3d91f92f3afbfded37 2fcf6e4b1416901517da8f2fb3c9fe7a87bcbe6fd36cd5b5823fdb74229036a6 3c3346a1093e6b1036902c1bf42fc317c80abf04020a47b344c36de42f05c490 a0ff44ab6d5249e9f552a8707bb7661e242644814001c8430ebd5e5f0b944ceb 666ee64359d663e355b2f17093a964139d17287f6ca6a024767eba4fe4873855 babe2f07b91560f68300b06dfe27264c163195d446980c35bca0b48f7806626a e72636593a05ba403ce1c0f8b2cea3ecd586e90ac17d034ba4af708304f23131 3459fbbbfb97d4834d0395754ab3f22d6495d2144087d448616fa1ce27bc50d3 46543287e3860d99b433624119bb9920a2113604c0e260fd275ba55e0fd19c83 e19addc3baa1f32f6b7284038845ccee71a3311ddb17b84975f7a984bed7c6ec 2a06e5b335a763d081c6273f86a46632fd9141a27902074fc860df3a2eb59b89 774c767022dbb577e30da128bd7706a43af886d0c256b50fa968ef06776aac0b a5387e9011eb2334c1f42c090f06a1125c207ea6324e87f46414050d88022b83 5a65936ed3c2784d902d6a64517029ec3b54bec09a243ffea3ed713493049355 c918bc7aace9ae959f7e424b3672bffe59ef3fc66e5e1551c4988bba50810cfa db733fd611e15d137e5ead7c33c4366ae713f484bb0c079fbeae748c7fd8cc83 9b9653f139ef25d7774a8d71de6f1d593cb3d8d395baddbf3a1b2a3b891d41dc 178c5d392138e7847ee7dff0bd930a88335b42475aa761092ba6d7e0199dc847 9e690bea12ce7b4dd8d09e8c800bbe7cce2852abeb732dd1fd2b4f5ed8b179d2 344e69b0bc3105ab3c0c28106ee9cca60576eff94081f04a7fd008ef15f9b1bb b00ebb979b02b5d6190dab3934c06962e0af308fa791276d712f72f92768b163 79f0400a92274402cd3160baf324cc1561636b154319757c994f397ab40acedb 953bf2d87de5c3a03268070ef7fbff1e783c160471fc4a81c3c02372039e750a 8ec8936886a9b33530526fe163874bba66f5f7412911cb063bc09bc32794b3b9 ea3f7a983121c537a2b12d782364d5363f93056f8e9257cb43e385f98caf9520 6af52376a0b83cdfa0b65deaf8484a5669d6443403c6ca4fa377688c9483d188 0b9b86b5b3693351b632efe97eeec61e7c75009b54bd59c3890a1440f5877e52 15ef278df5e98e72f6269e284bab5a38cba4e1ed69b6763584ac969eb311db36 291b81bcd1e08289f1ec45562e7f831a857a645d50dce194385c2d6354b20878 5ac0166d99133ee308cfb8c5324d2c46a716c9c59423ecddfb482f227c6b2dbe 563d5c463dda30edaecf8127ff417a98e7385d5ef8922cf9882751142f3bd4e8 831dbeff8ee2c0a4968c1390aab3fe3ae4bc30e0d226a44cb286d59e51891dfc 5792ed74bb9687e1198894d3062d18cf89bf63cdfc3e39fbe064d4f4beae38bd 2ace7ec2017affacf1bca5fc8e7c2b3097996f369158ff6d8b5b6115731294a4 bb5590f75a2954ead0392d73a00bba5c2043ba91a22ffca810393626c1213775 ef8c8020e80e767a492c84af093fcce4950e75da50c5d57ba518dd053b566585 b578b80a2e6c2db022e8a85cf3441dd0fd4c515ed7f96e85b355bca49382cedc ec2dc3f26f0522dcc8a9a212500471ec8e01a97910d64d2dfc068d99605c7057 149434d3f0d9491fed47d4cc2dd0987c2df4c634d256dc97f193563d15ecbb76 244d62cfacbab3010933a16a3fd7a08bb6848702f78a6cf6276e4aa08505655b 0eab28fd6f3c7a617332fa656f6636aa0015def94435775aa61185a8550ef51b 6b421adc4fe70d7b11aa909b333f02c6f9d8ac20dbca377aaff438113fb7eae1 93d345fb07fdd688d9df5cb7b46a04505b05c175a76ebf550b10b85ce59742d7 dde01ba35b9cd47b10362b8c911737e1158c9152039c13cecdcf49b0f0994c8d e3f1a970ed0e9a67f630424cc2a5d018562b54296f68ea05dcfbfc6e90b9cbe5 5cde0c60d82ecfbe54e8e49145e9e436220eda0dbbc6930e448d585a54035fd0 b2d9dca26c35b5a50d7dcb47a6621406e1296b640e3e84c6d0ea91 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueXX 5000815 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6784decd2fb36dec8f 7478da853762e00bc9c52ba86ef50d222ff142abc18e4fb95b86b3e57d9390ca 86e8de6347bf7f48f93322a372c5d7830521b14e8fa0771961b027c81365426d d95d16b950e216a90f28ce806244ff28dd969cb512a23e05045e7b2bcaf7d6f0 ca8b51494f2ec0681bdfde60420628d9bce98a0ca1ecca9140f452a88db19a30 ed46ae881b098c843a51d06f9a153489357fd24de6a0c36cb5a9eb9b3b64dd75 18f83018ae7cefa1c12fd205df6880a705e28457af7b869a6dd28f6b37d82cc3 e51cba348dcdb34efb119b8ce5a996f15492b0f5d34538850a2c1b6a509fe5fa 015eceaf28b787e54f5f78aa5a89e3ed380237c85d8bf04e23701563b175d822 5c5d7d2174a8a83c21e412e1b77f7cc76c8c403d66e4c99542e26b931f9e92c4 2ee9e882d00aa046f70c982e6121e012e7bb8cf181815ba51aa769280a1df1da a08840b0c10deb9e0010080c236be2fbb1c88c49e1b0cd30809a855491a5de4f 5c15f5557a5202280a0ad1e52ebf42d16363add70d9b0538498c3267186d228d f03095909d4de4b8e027df58ad507b4f60ce048f65f3695263233c29591190ec 745f1725a1ff1b01004d56cdf15af9911ea15351ce0123d1bcad 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 46 /period put dup 49 /one put dup 50 /two put dup 52 /four put dup 65 /A put dup 70 /F put dup 73 /I put dup 77 /M put dup 78 /N put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-163 -250 1146 969}readonly def 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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 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 65 /A put dup 67 /C put dup 76 /L put dup 79 /O put dup 80 /P put dup 82 /R put dup 84 /T 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 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 114 /r put dup 115 /s put dup 116 /t put dup 118 /v put dup 119 /w put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueXX 5000768 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 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 99 /c put dup 105 /i put dup 110 /n put dup 111 /o put dup 115 /s put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueXX 5000788 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 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f13ec048ad1f0bc6494eb6cca257833105e37be6f18e27753d8c92fb0c089441 6da9716879b4cd674ff777bc062a29a14dd33ee7b72c66f009fbd15434cd01d8 99449a4fad3318aabc346bf958629f1f5f0ccf0b6c31f4e51d63e67f84cf1cee 08b630a90d5f6297da204b95675657a3c0debcb5969d85b971802bf998e5f061 e9f69399f55cabc9dd28130297665eb14e7bb77e4145f0a77cad49cc630118fb bc903b0f5478e763f476be664afb6fbb4cefa55e111fd4200256da96aff9e434 b129f17cdf25060d449008fe3a42571d12e199969573b3d3aa5ae534e07ae860 e199953a223fdd4151555c5610d5e9e89cf08ea37b4593c68ad0375b9a9c6aea 80ca7d41682bae9fd29417995dd10a878a5528a5f5291734595e5c2abb9b59e7 727b4cd638d20b56ee48d7b4a53ca12101c98b2d88eb70467cf32d295e598362 a60a96138b35bb9a2becf5837ab1 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 18 /theta put dup 25 /pi put dup 59 /comma put dup 61 /slash put dup 73 /I put dup 76 /L put dup 82 /R put dup 106 /j put dup 115 /s put dup 121 /y put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueXX 5087380 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311e252234d51741e18db3a68e8ad10242 29e5817a10e796a78d2c7f7c1f50961b9a57aa604c9f821dbf5a9295197bc666 31678d7d2c7e1f8f2151ce0c29efce46270570f4301c5dad1b38884732e53dad 05dfc36b020e726ca575f71f04abf2b49e1f1d6d2a08a477658ca9a48f1c8cce 14382f0a42201de56e2821ee2b0e91a818b1b753d810ec2428dae5153b66af66 9442192024becf743a8f6264a58a0b3aa823510a4957d1bab3e1b83135362888 3e392f49926471a8021fdb48637baf0abcf31204b1b214f48c99f44af7782323 69d772ae4f49be53a1bf584747bf29b94f530d59a5be5370b8dd0691e32a21ba 9d0d592cf3bb50a99a3ecd18d6c4114cb4cfecc49705da90d7edb10e3a28f9eb a49347a09d31eec896f2a28e9607d99d42914aea16545c16e136bdd0b75fe486 61e24c7a9e9b233a76754c760822d0a88c575e08ae3e3f081d7bfb80ec4c03ac a08b65fb6474f71fc95c7c2462f2eb3754e647c29498288f2ced0bd3a2bca5c3 2f80a61fa667802d376cacd34de1f9572d1b3ab09d6f352750db4a768f869638 a87457889aa502850427b40e1294c5ba33e6b17532b2d13b5f50e71582f93daa 231e86597872a2a968127afb0f36257a76ce1eb24e6c4f29fce2aaf6532fb5cf df84da03f03a45b915209f67d398c7c4f7a3b0623750259087a478dca684b29e a8b7f52060bae9394369ed1cecc645d113cca3772a8dc36c6d916b5c5e65454c 0a188a33395fd3da67ad0a31ddab6a8434dd80f0bd0ea1a2a62eecc331d88b3e 369552daa1c575a98fe84383548626b65be7c24ae7eb61159f149872cfd87cf7 6c1a7c155c7186c7600e58cb61b8cb0c5dea98268a39e928bf6fc53985eaa763 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 6 /plusminus put dup 21 /greaterequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 63 /perpendicular put dup 106 /bar put dup 107 /bardbl put dup 112 /radical put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueXX 5000817 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f73e9b697ddc0351763500e23c4d170ea8a2d2 75ad0b6b23097f02fa803c1f46f9ac10fe0e527c431b11b1b9a50e874896cca6 17fe3e255960846bc319ca68b9b930a45d5d0fc36b352c3443ab6f7793b38d2c 7fb9c409ab7980ead55e9f8f6b83c70088d1fa6d7f2214da8928e40b96334ad2 72d1b5db9d1355a579b26a8dac21f021e393d7d837a95deb8de92ed818a8b80b 98ee9cdd8a3c57ad7d6eb7a82f43eaa6ec520eefa7302e750edc4b495f70bba2 24a2182eec37f1a1c5dc8e20973007150237daa6bf03d498826777eeebe09014 ebfe05a8731e1a421cfc513d47d673e3d94d805f5d0668ffa265e0962cb5a75a 4a93d9b87126ae4175aaa7a9a5a5e8bda6e3a2f916f54e217750f7a8d2b8a6a7 926bca7b3e3ec4a41de1f07491614f24a1972cefd10ff5a96c0a0c43395fd7d6 ee18a644127b768e803e6a3a8c589e947d5d987c6a2d3d62b48d8e6c6e0cc913 4325a29a5357f29f1f860a1b286bc4f06462a294a4c28c13068b4745bf45efb5 c8ebe503309bdec50392b4b6c50525126a1640e4100be4c98febca52390b64c2 3676fd9a19a79c3791e284f4f05fd0587cbe751937dfb994ecc2ef5e1f83be20 2854ed0c9ceac76c04e0ca2c254a528307934e6a052666325035680e451f0cbc 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7a44618ba8ff6be6bf662151ac7f4ca641e215f0e1b63fc9451f5ddf3e01d36a cbe1d31c1ab2225d7281e1d59ae90a262bcbaefe302879a738e9ca613a9eb058 ae6733cc53e8d68a38238ff1c3fd6a42ac74358139a86488b99771ed3efcf20c 3a3ba09fd75fc3104b76d4d26ab327de625fa1d2db46dd5a096e2b5df8ce79d2 2d65915fc7a77711e315db48cc165ab7228cc197e8a100fa3521633ff6a8efcc 574718b379de53b843fc412e5fd587ef5303c69f15f7358158dcd88f4cc1e1e2 8ef5a7f9bf1ef061811a5e24f08223b6a2eb0b4bc51b58bd3bc5844c9c26bbb2 83d4a4eb350df75726769ee6739d3e80dd5a0688b04defec85ad0c9c618ddf50 b8fdc58ed96ce55eec7e9cb5aa284deaaea6bcff474eeaf3dba2e49498a26ba0 f6da6926e02db7905005cbb86aa3d8c77b2abf1fab410a760661330e9ec8224e 495226e61ec997ad1c71708044b1c3486f753fc618d75411053a935b253cda65 24e481df7fc19bdf57dddd80e9dc3049e0a8fd2c10d6151a1b338033d40598d9 4437a411f28aa1efa57aad48ab77d0e598d066cb325ded0321c8c226ba3c6a34 36ceed473a1b0f319513d562864e71963c88df3492daa70e7d489fe06073edbf 08746321197ad4510104ac4d470287abab1c2c785e47c9ccec63aaff0d3f0b01 02d3fd2be338d0f2508acab4db3ee7f8201931362253ffff134a4a3ea7c8aac4 8c355cb913998c6701258b571881ce52bbaa3beb9258b7e1b58607c2936dd003 a1405d14970a902eb7e3288111fa9c0e70175ddb826e92baf5124976e2abd6f6 1cd6309e88fc1b329008ea179dcf27881bd2dcd2e87f564330453640a130ea9e 1be8d5c9c639ee532f7089bc7cc787ef946939d9418082319672ac07961557a6 1890982c84822b4bc4dd2b9f79dc3c8e916ebcac4315bc84bb6e3a5535ed7021 af8fbb3c0c89e3659897f1b38dc085491a587319beb730f83c6aeea4e7138c60 b0792b720a2ca696d3ad1d041d568df912cd79a6c2a583f287fb6c8bbeb6f173 0fbc8053fc42b8195d6f1ea70fa2b9014de5e589747dcfe211 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 2 /bracketleftbig put dup 3 /bracketrightbig put dup 12 /vextendsingle put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 80 /summationtext put dup 82 /integraltext put dup 84 /intersectiontext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 92 /intersectiondisplay put dup 101 /tildewide put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 112 /radicalbig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueXX 5000774 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d5649664203ab24acb938d58d246707ffe7 d62f04bec4b70c21ef75beb2b812622b3c74e969d72d3cd11bd7106294a99caf 0b1629bc7d4de6b96ca82930831d64575f23f4ad06a0e45e315b1d392411be8d 6d73c998789ff258a07a3c8c2057325784514c845500bfd1a971310cfc11d41c 1a167dbd5ff012c60add4e87325f6e5299032a839de65fb1473a166aae1876a4 414a434f22c1d241591fb36f857df6fa930608750ffc0c54f44994662b1f00f1 400bf752ea8d83ffc4cb77a290bc2d99981ae59a191748ba5c7ba1a9d2583fd2 1398452b6ff5d83a059f7eadcd2ef744e9dd22bdf9c79d049bf06835e878c32b 7765c69bdd8ef4deb4ea7cfff4cf9354a4ddffa689de961d16772491c7afbd7f ffde42400764c68e954ee5c455a5687959829bc3b319b2147deaab3628662c80 30c5e02fea09609abe4eaa12e217bc3af673f1bc36a7039eb13fcacb4218fe0f c5a3f9452d4edf46cc91db67b624d4f2d37502fb9f11af4da18ca40b61097f95 d44329375467ed13c5cb585ec53f62b83ef9502cc755af44bf32b87b8ae9f3f2 f8dbf72dab90acafbacd280db6aaffaefdff6d5eff26669bac56280a950560e3 d01714ba38659b0e4e28bf8cdc1f15ff7e3ea9d95a6afbbe3d8e6905aa6d531d 6b6dcab89e2036cc492c65f76ddcc3cfcff33fde3d9ffd10f7a90530fd5c7683 4329baa902e701ca42717476298c9ff511b1941244bb1dfd0e5cab92a43970cd 1bdb292eadd055c97972844fdcdb0aa06401e7b2632b9d134f96bce0064860f3 9859da7a9a82be0a840672a1331212752ae1b521b693b384a72dba8d37a86eaf a0341e7fc5f6ef7a8c566aacb91e8bf5005c18837894c4eacff41dba54e071ca 83fd0f12cbb9773faea8074d8bd9ea37f030d986e99295d3788aae84f6276773 06b79e1c1f6bba324dec82590b78028d387c52cc492805c6892afc44a317ce2d 92d656dc235b11fb9674ed35df330a3eb6d7ba06569674b3873a35ee2192b331 077fd4129780824feecea5b12767f61065c55580a2f49bfbd808777d2ae0ae48 0ec1ae2f6b2245be1219d40fc9a0f1ccbdb92446b6e37512a69a05819f34905b 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69ccb2679d531e695116040fe49b99e3974552447f004e6bbc9910cbe4e49ba4 9778bd245800a8127394c6d10264844c52a7e5cbdac63065e88c40b4c968ab23 9138c42b5b4303a72fa4b7e4ed0a19938788dd57b652cffaa1bff4dac71846ca 3c169fff429ff6d69191bf729d7fc2db0128520af4cd0eeb727aa2e6bca9baa9 a4a9fb2085cdfe350c697a25d739a5a8305bc1559b81a96d6c4405059d8df6fb c21de8023f2888e0a7f6f600ea0ea65ffe24ff730afb2d380e24178f1cfbc7de 5a987ca0b79ea2c77d643f35ee0574ea5371527cb818f8f1d6d16b73c5622a23 55df7845e8047a3bde1e4406a8e2c24b4aac49d301528ae1cebbe5167c1668df 05c6945f28a3be939e79acb53fcc8fa37358e19ee412c4c1a2940a7b235c0b3c 032f05c41d245d885da47fb17d6bc62023b50a6c65dd807fc78da523bac78d75 899b93a9041f2fbb0a1149adc5cba8bbaa3985a4c9f74ebf33797523e2bd8577 8d7a0ac38a6d3a185037278f13eea40302b743b32e2fee67b023f26af57dd7fe 6d1efeef2a18253b817b6cb7de963eaf7362fc3bfea772d0989366c20767ab0b 9c889042fde429ff5bba9d63b2a9c093272f06dd4edc69c95c1f1d6f32ce391b f1581e7d389b1391ef5e327840c20ba2a7ad4c3f3ea1a71ed50077ea0682fa38 8c4c73d5e64deec2f197327cffeebf24c4f2788a35d3b3e11326f7911d485070 516be3e8b66527cc77574af376b2d15c2cce6e9c73a83695d425389975a84415 3bf08c1876712275ad0d0f8d9e3c07ab5abae96c26156a1bc472325f233f9ba9 69606284642d7d31739cfe7aa800bc5ebfc918867cda9003a419e3a96e8adf59 d9d1d8ca14ccfee1c2521c4d96680410706b3f3b800efc3f42218acec4a5260c 8021ecf2a6d98c39469b10b0ad5612cf7db841010df550a79c211120262edfa5 5076e927b0be5f3e5ed36b05a3999deeb033431f104831b4f9fb406fabc2be3c 2ebb4b7ce0ba15fdc0cd205eec2126d378540814fa46ee3ce0572daa1e86f391 2df2b8a6741772cc2fbb56ddf62a5b0a453dbea319edbc60cfb55dda9df9a590 2ab858048b38e4345627e7cd708bcdc775fff103ea46832d7eec97b351076105 f0a9c4acdf48007aef92c833a232316a7a27c68367f0a325aace030cf0f4121f 9ab6231304c34114264383e03e45165bb29fd1e3c01949b50f5c016230cb131a faaac8bb0ffbad2c529194d53520341bf998a92afe075b210c462c08939c3c4f bec4c8388b4c1a8e918e85cdb30e891fb91cb50cfd2282c4d3b36d501112c8b3 4dd29f045c6d56a039895ebb8a4568ad56931e35d133098f1b2f5cd31f322b30 40ba5cec75a662cda816928753a84977a8758638feec5dc5dac9e19fc52984cd ff153be40853a2ed8f1151ba01d7ecee41c118bb8e3b01bca183bc6e9187f580 ed0483c9148d730d1c56a3b014e30149230bd12449e670500085481dbfd0ac31 389deea249efeb7dec1c7a1075818115180b2f20aea1250c638896ebed41b152 fcb928ce3b79cfb3a6bb683fe294f7ac5b3a2e737ae41831bb4ed57964219e17 9a441174ead333252e96cd18f11891a63f6114aa777c09d424e42719b6fb464a 1da53c6d0dde55dceafedb0094966e65b1a7aa09360f63677977398a09a6c9bc 20b7ca2b480cb72eadd3c140e05744f5a130015f5763cfe42d0489dec17ba4bb 9736868bd49bd5a1b88b492b4ea85b3be084b3b9d9379da887eae981cfab40b4 8e104be50f22e0566a2cde313065572172edea1c1494cc27f4097d6a0042790c 5f2f66ff7ad8a2ded5d4622aa155a2acabaaa6f2bb39abd612b4686b14b354b1 7df2a0bb254f64418fe8a403beeb4470b3c8fc5d31717d47a24c87007718c35e 0827608156bc5269a120b0b8bd2b165a5b08b8cd756c21eae30d08cbd258ab3d a88e7c5827a54b788e3d24113558aebd2d0effa32e8073257d37166775f30c00 95e9c735b0a209667190fec08392f86b0415a455ac641d2f4655f68ed7124fcd 1e5d3bc088505719d801ee2ea25a5a76be450b7c739afd5346c4badc5e5b3bcc e8029bc735ed7d4f2b68dbcafae8f308dae8649915f123ee2c985e61913df064 6ef8cce27bf2303b5f37196f580034f87acba8809aea6d8092060a3f2c326e5b 1a455b59363578111b507941eb69663682660be58156626a9881b6b6337a15d6 6fe310104f887e6ba5fbda2609ed9051e6134bb6fef754c083fe2a62b38083ae 65f4ab87433de68b51fd30b34832d258b6ee33f2a380f7bc0ae128ec4de7dba6 afc736b21cc0f574036e7fe55a70725e3aade59e28bd889f0f7234d8e6760760 a84a9d82f609fe0fe6a1e93e3bb9c157ad8834a29e1f5e00a42a4d8860494609 ea136735d7620ac29f5f0d14397cc804ce8a5afcf17e68f38eb22fe15249e059 d87badeec8b4c59760094be8b973843d0be76d075807aa28d97eb0ff7df331fb 18175e0a972e940dcd7eed1990d90bfd502e98f337f00759a4bb9dc95766af32 3560a0a2e804aa25f7f08f88f12e672e8489c80c69074ddb158e8df95dbe6bab d1401e660cbc1fbdafb12eaa2717d4e19e7b2d6d5a0901057bb9142b6ca6eabd 200bd8fe1bfc19ffe3b7e8d71e9593f3a5770cbd18dd73024edc27e0cad6a285 a0d75905bab842821ac96ce122c2ef4bcfb9e45df7088d36f265e6f19838e6a2 4190ef17e3c697952b030d2d85a6b0da43338cbff134efcb494813af2717f5db 7ae9e7b27f68247b167754acbc34d33a7514981a060b9345500827b3b8db38e5 4fad1cc58240b092ae57d96557cc6ade3d7f6070abc69c3aca68ae5a0fd0bff8 44769fbb3cfb43f590ad3eae0a59d9a37b3864cfb37b865ff22dcce264d543ad 308903773e44e9330586798dc28f3b3b747ae7a0a0c0db3a283a562905db4f2c cc0dd9a793c9f436 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 11 /ff put dup 12 /fi put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 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 91 /bracketleft put dup 92 /quotedblleft 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{-36 -250 1070 750}readonly def /UniqueXX 5000791 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b094541f345ec367ae85ca9bd 24568a01d3b9f8095f7420e6c423c414b3dcce6da48dd1c89a56d078e0d0e2f2 62a13640a06d17e44ee3866c3471fb58fedf5a3b77294517651c16bdd7267d39 a54e7171752dbde63ac19bb4b3021ce95eb5fe67390b09ae4d9ed4d704a67443 f55dce17acd996c1f5e023c9e5a18cbeecc3097f23763acb86cdd7cd13381ae7 4e48495ec7fa520539d87f8a8dcb3c826275469b6800876a457e7d1e5be867c7 b1ccad69742a8c9b0ad943482bf2a4ad0aed40baeb69a0233bad36b4ca2d2da7 322956c70375d152653500b2f22d2ab6990cadde2da14b4917f7515e64bc3d96 bf775258fc7dae4e42a4c9b6da8eddec4a800c8aadc8d75e48cae52137e05c03 677f5d6a82fa46d9f2fc7f56d62e5c605a1b7898b8d1401c2cac1a0122a2c8a7 aae09607f2c5f29293a09b9959399283be89051452898238b777db9830ff4318 a298b221c4a820c700ca964fc99e6b1d9eb0bafc39be9aa9dffa2fa326b2a466 1e67261bb3dbc85844e76f9313ab533ee16bcb478166de5502b68e6ec1ea6110 f53206cf1cfea419571b0243244b32ea38b828a01e62b03bcf476a32c2341cec b8a0fa2a3f3f339315d5f3aa7704408f6617d0278d5c892bb12e35c9c7c11c20 a6ddf5832b2ff46bff9cde62a884dd607933864f6e1e9e2899faa6f2c9712eb5 22f0e723416ff6662065e9e132ecfba92a0c51ae000c87515df4d95aa141a0ee 5e16a5f3f833aec3b0951bd68b348f65df291829a67ee1ca79e14902eac1b912 6016755a1dcc184c6087667bc8aa7cd79507d23799c678826d169c172a7567b8 b783af25752af5a7ae470e8076782899d4879c8369eae52c461c6a0dd23db998 512cdb7c384e237be3f07171734b67ed4c7f340b2221f89f2aaaf458015581ac 258dd463c6d9917b0e2abe82511ccfe7df9740fac1de449b4cd3e62b858f1351 2c134e8085bce9c34131260342916a07626d3609dda2d8bcaeba914c8b0a5c18 614daf1000c841f4dfb3dbea7bf60728dd2bbf583ceecd7f1850f9280a4808b1 41bf2567f534cb1a5e5b4b0c4ca14fd8469adafbc0cf15dffec9d4bc7eaf375a 79929a900c13e429b14775648ad7d620def97b639236eca3a540fe53844cb538 42b2867b157d12c16726c55ac79052a8942d0b25ea0cb343832d6727d02e6e9f 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSL8 %!PS-AdobeFont-1.1: CMSL8 1.0 %%CreationDate: 1991 Aug 20 16:41:02 % 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 (CMSL8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -9.46 def /isFixedPitch false def end readonly def /FontName /CMSL8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 67 /C put dup 75 /K put dup 77 /M put dup 83 /S put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 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 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 121 /y put readonly def /FontBBox{-57 -250 1183 750}readonly def /UniqueXX 5000796 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 15 /epsilon1 put dup 17 /eta put dup 18 /theta put dup 21 /lambda put dup 22 /mu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 28 /tau put dup 59 /comma put dup 61 /slash put dup 64 /partialdiff put dup 67 /C put dup 68 /D put dup 70 /F put dup 73 /I put dup 74 /J put dup 76 /L put dup 78 /N put dup 79 /O put dup 82 /R put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 105 /i put dup 106 /j put dup 107 /k put dup 110 /n put dup 115 /s put dup 116 /t put dup 120 /x put dup 121 /y put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueXX 5087382 def currentdict end 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6e00fc4720aa128fbcccd05c6f601288c67c0a1e192285de84014f9d599f10da a497f554927949662cc8c1c850719fee6950cc20d941d801cafa90881a8cff1f ba7ee3691043b34e4d5cc466129dd42b42abfa85fc2f5f30af6f2b0beb0b177f 2ad351df8435e69a6b489aef50894cadc5c0 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 2 /multiply put dup 6 /plusminus put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 26 /propersubset put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 56 /universal put dup 57 /existential put dup 82 /R put dup 92 /intersection put dup 94 /logicaland 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 readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueXX 5000820 def currentdict end currentfile eexec 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 8 /Phi 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 61 /equal put dup 91 /bracketleft put dup 99 /c put dup 105 /i put dup 110 /n put dup 111 /o put dup 115 /s put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueXX 5000790 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b448685940729e6e22f4be0e08c6505cba868f7950 93f556b64b808dd710eb936d3ac83e5783b5f7e56d363a59a026a5619a357d21 c4e77bea121eb24b09027d1aad93158edf912940e91cd2f17922b35885894644 7773730bb19af9e70453e1b78d5edc123b273a979891d6d5e624f4a388f3897e 13b045e34b4ee2f5e4fc5e996d3549640010c671de14d18b0b3cd300e57dac03 22e59bfbf7b29422230870f5897fcfaad4b50c7c1c58edcd119218163d6f6e56 25ccb882db7b6d09a1586508e8cb642a11c29a46915e4a96e282079cb21922c1 c2e360b487a45302fd22ec8c5fab63e54b5e844d4b17ca2fff37c69c366dd72a d02922c14c0932f65ed03e4219c117962edbad2dcdeaa9c10ce8af38a4ae52e2 b377245b0be19a77d6c936e7530cc4d0b78d0cc4a92698fa2870fa54f2d8503e 2d17b3d52fb2febb09f2b2af0c2a1892039ebe19a690098799a858e3d39631bd 6925a154d161df3918074ada6bd52baddd0adc3f07e2d9f15e27cbf7fe8b98c4 07205c811121fa91e059f2f99322fed63f359ac9da97aec383f067f23e5de331 51e80f0a88ab50fe8fdae4a5de93c1ae2fdca06150b37246140c0e87cb2325a6 0d2349162ae3ac93144eee1e665a1289105318fdfe86b6e76251cb25adc967d3 d0b97fe5e279e1161736ab22b4ca510b964342383a840defd38f96a7280e6ac1 34e48d740607ff2e7804164a16d47735864db847c97335e6d4215cb99911a1ec 015a3edaac1f28fedd56d2467130d07bae9416c15f0827d27c6c79f59054282a 418c12c157c91223a829947f47592f7cafd93ca182b25a73a9419127e3b12a9e 5167ac3963f2b019b338ac46d63880f94dda4b538835884d2a5538c85528d6bd 977f844d32b43b0e48caed5a4bdabcefe71695d69ca784df97a0f787b40bcc99 9f4a584bc3d3e5aa2f816b45b006976ca7e928d140c84e76b3bdd2bfcd4fc1a6 6df9c9535c75d6db43c926ab3cfe08213a239b1a165fea71fc672c3c87e6f755 0f0aaa697788714fc790e6bdb3abce68221b68654d80bba25ccec0e1b0885672 185a0be14a36270b2fbfa08d5fc801c7c7631d4d21237081c92902f851a32143 89ac9dcce11ab9490cc69508a081097c9bdf4c0b19dbdfd9985d69b66a1b090f fc51ade0cacc7f9960f4555de6650f271bead9a6dd2f1e538589ae89c2f40e0b c1616198ae975d444df5d2e761da5a5824176ae96ad77e074a4403b66bbab012 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d8d86bcf76b50d3d9b3daa48005e4957d82d280471359d06e9041ce24f557a4c c82b4f2228071b2d429db0f9687b6176f2fbb7ab360f699530302b7f00ba974f 876ff875e826e27adae4b9ced08eb8e5ae1defe8f4f134ba45ca3411f2fa86af dad44eb93a2720ef4f89b7cfc8bbd7c105ed09fdd06bc3e3abe11bc800e40b48 24c8cffcd1b9841211d919d57fe96fa6a303e660acb53831bf6e23efd1f0d111 080e6f829e45208ff6c578e1695f3f00644d08bb38c93273dbe9a99621d40b68 fefffc20448c64368b110ef081526e2a463716edbfb268eb2b17a449c333ed62 60f9d2d46eccb698bfd27665f62dc30f22d3ab8a294716546303c6a3ec4f5baa 2935a1b82e2e1fe0738ba3c01cc05ee9a32e3afa53dd21066040076991f5ec3b ae56b8a6412b6a94657c9d7bb468c702dccc3fda37aa1b290ada7027f8bde27a c08a601ba8e83b5e9c39b9e8ede73a33a53c117e948b7655eb625be5f6f15f18 b7f53e189fe66c11db12ee51a0ed56eee40da96c308a935a873b87bd6abf3121 8e37a65c54938af87b1513d9976a49f27f4229a7c161387ee13c0c97403307f5 262513a62880999b5034de8ace9da3ce33d091cb2614b81f600b6b0aead05417 9f4db73a64830a2be9d66a27afe8e69868989897bafa662d690b652567749e41 8ab1e01e2935992035c08a6751e588a4c4dd336c90616fd739f0296d50ef0c5d abe884f9951d9781bc39ca95b402a95aef5bb035cf576f869e5328c8489f661f 27bb91988420edecde200ffff2818fd267a46a1582b2d852ee63a1df9ff60266 22a7d6aabab0b1b05421196cc9b996e3714c6c06bba5413edc76c394ef2e3c5d 6e2abc7b9718a65f93c23da94102090eb32d8732904e7f6861a1e048c54a5362 3cac7868fb90f47f1137301939455cc8b87915c20d40ba43447489952172b06e b3d8a79b8e3df90e34b976d21df541be719bca5b798ebac5e58363860fa33479 9049156d4c81380ec673793435d4e7429b75b7e30a7941a5f334fe5a03d8b369 de790e1edff2656e9e1022997efd701aea3105c9b70970058cc12543a12063ec b55bd70a7130af4e526571b00b2cff2b0f9ac54c026b67e2f9cbd0566acd4144 25a54122300dd8688321c9532d0778520d194c9900a6ace7a27f8fe87f5901a5 7240dd4cece281971d3e61c468a026a14b0184885596e41036237e53014a5d55 ba970eff9c44e78375e9baa9ecacb7390c71aff94c921d095d70bc36a322bf8e 8a8cdda82a839fee026c28f47d106f80 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 14 /delta put dup 15 /epsilon1 put dup 17 /eta put dup 18 /theta put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 31 /chi put dup 32 /psi put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 84 /T put dup 86 /V put dup 87 /W put dup 93 /sharp 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 113 /q 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{-32 -250 1048 750}readonly def /UniqueXX 5087385 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930da4f8d58abefd45db119b10eb409dd89792 3c6e705479464a4b310b58348c4b42393988fef4925cf984423aaf65fea9f0e6 4629953bcf50b919d968d99bd185f83112b2759cc411764e9bde677f57c5ee5a c555448d5f81a16259ded1e11bf4119d53e8ab07a802df900d3a2d5ccc1c6876 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c52d0c133119eca4fae72f7553d276c817003234a98337d7c037c0f1bce8a804 57113bcd97515c002d20f836fa086a3277727873ec2b64a92bedb01193491134 b6692a08de4a201a3f11dbe070d797f7629ad487d2b8dbf89a0631956c93864a 402144c9d2a406198630e4fe9e12cfd6a28ae1fe5f57e7ae86626e6ee9cb45ba 7ef6b6ddbfc5e6f551d66863c577461ce3730b94206136e284ce18eedf249ba5 3dc731ec4796cf2e7043302767542a8b87dfb8dba23be0303858595c808c9ea5 b73e8630bd5fb352d23bebd471bf5cef31e955ca919e87e4ebf5008c7ea97b07 3c7ca7259eb84eaf3a4aac02fca5c4250790a9aacd5f76ab6825b4bea3d9a3bf c443c9d54b30218f9ed9b5a3afca4ea753108530e45f5c9546500d267ba45eab 4eb75aed449ad7b67267690b727a4473c80881f98dbf0ca65f0f5fc248fb1ca5 d74d2b6e4937317073f2f9f1a773702364e83587247fb2f15a6568b14143ddec f8b74ae7ef36cdc1e1a7bcf200880aef55634f14f314df2620be2d7da0ee38a8 d5a8edd4dc74eabdf3216663ae19a8077963d07c6157b82e727b856284b42153 dc2aabf92a6da38a64c9a7b28add2c5a7e87f54cd631d1efd0520c12a1229d99 ad86bd535647c2e7fbf143472a68c637fe2e27b2bae6cdc10195b9a7118152d1 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Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 3 /Lambda put dup 5 /Pi put dup 8 /Phi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 62 /questiondown put dup 65 /A put dup 66 /B 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 88 /X 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 readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueXX 5000793 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 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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 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 70 /F put dup 73 /I put dup 82 /R put dup 83 /S put dup 84 /T put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueXX 5000769 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 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39cf89ce1613b78162e3c3d61412428f4873f94d50e84d60e3fafb823416398e 3c9fba53509006331c64254aceda4ff60b6eef152dc5c6936aeadf61be434436 dade138a9dae69534880ec4d9fee725bc34f27bd22e0bbb895a9a2733679fa24 0a476841d8caca5f5dcfc8be10981cda8f1b82f633220900649e70a158ffb5d8 bc327965767fcaeff77b6d95995127d7a9ca61a7c2d3bc36df7747269f510e99 e4e780afefc4f23de8018768233bbfbf545011ff023f9c68fdae60215d02675f d6d3759bd3f6b30cbbdca22be0523672ca3617584cb5501fd1db9060a946a410 533a2f0bccdb19aae1a7befa137185dee9d958f078cfe0c177b8c0e9c025db86 bd3b25c47c704c637b6cfd3d074be9dbf48c2bda8aa344b02e0c79499dbabb6c 62583d7a7545e70ff7bc435ee483a37ddc197fba48eb7a02148ba819f90d140b 939b860a5fb3b366cb79ca4118a6b5f320f82803cad384d39ed174d869a595a8 287c8dc5781fc981bab1858c6a0763c63d65ed102e67b9d88ceb102e0594ed89 be6b121fc3bc0353945923cefbf2ad9e551743ca8ecb24460a88652f1745bd72 5daf64abb665d10fea311b9d884ecb100a111ddbaaa68d3293466cf6b301926e 8d9e9ace397cb15fefce3b881ae1a93fb352fb6b647ee01cbcf82979aa91be9c 5575bc80c0cddea1687f06890266f3ba933b96cceb447ae13b9b19673fb9c5da a215b11d5a8c7feb5764ae3e0ccad9724a2e5753cd69b123107b132c71b07c8f b1f09eb2e286c755510a762e8a37193f78d4a6f5382144cc997a876354d9e5db e949d58581775852030b01cbdd4b6331359490cae05af34d6769f302dae943c7 04d82ec69c1d4454e4b3010733c3395bdb16e1e6e76b7c12552f9c1c7f94f8cb 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI9 %!PS-AdobeFont-1.1: CMMI9 1.100 %%CreationDate: 1996 Jul 23 07:53:55 % 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 (CMMI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI9 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 98 /b put readonly def /FontBBox{-29 -250 1075 750}readonly def /UniqueXX 5087384 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930da4f8d58abefd45db119b10eb409dd89792 3c6e705479464a4b33ae3d31bfe98efe259f07f7950237bbaee4f7b64ffea83a 757fa717d50c3298392891bfd60e34a056b0d6021da3fd9b8b01bf78b0b23dc4 ea3b0605150d20b27ff9ea5f2524661019d982a2e47cd7c21ee5ca9ed9227821 f8d07119397de26838c11d7b35bcedc43e011626e300f8249ed846d7b5ccbd89 02550cda17ddf113df658fb13f8162681013766863784efac128e01ef997e1a4 30312afac8f3d948edecd7d5090ab5c864b91d7f6e80256f0e1fc99f1102b74c 61cc456e2e1bedf0e627d5f70f738f963a13666405c51ebf23ca077e97570a2f 1118536b92e585cf1dbd5fbf36a0a5d197538c05304e93a15f3c931168d91d42 483ef428b1a3f1b7e548fd1d23b80f5523c8b8bf450b02804fb689f915c86693 f7dfb1c43f681627c7f027021aa3a8179148f1819cd94b8890a9ac63dc1e8a68 f254b64f6f173c3d02e3ebc037da05074017c8b113e94f4e2d26b467ef9da07e afd7ea7c86e025af52c5ceeb835951e67e02a38daf1e9e698bd21996c1071e84 ff7fbe77f6ddf9d06fa92b16ecebcbe0370b5cd9d450d0f2906a105cadf93ed0 0abc7fcf2c996e7c15e452006e35390d99b9cfc59925f1444c20d9c3e1fc2910 32cf9fa9d580eac1bc956334da391b870a757c3bee6aa3e005b20988d8587876 3ced20c6edf2ac2b22890e37ff7aedeb0986d85ae05bc84339bf2116df48c5d1 cc888bb33d409b06dd70ffd0618b2080442aac439aeb39c20a87ad74ef10629c f1604db8611e8e4224c5b21ab0c7ebddc77fe9a283b3a8c6a2a9508f6836e3fc 3eae1366b4da4f44cb7a77fd81d278c17775d56cb28b70f18d894bfd1ec1827f fb16b0d4c1a4ce155da066810170fea53e0d24874d6e6e740121ad15e7d286fc 3b9037c64549d8f100b16ec1dfbcc62f29ee53565baec00e73aa39888db2e0c0 002e87fc0abc4f2a9470894ff1f08223ecf1c24ba2df2fec514f5dba03ed1685 deec0ec531e9a826d90a62f741d821aff244622ac44f9c2aa39189de3fd1da22 4ab8beb30033ebb6aaec08c1327935da67bcceac0d5bdd7ec408ca942118390a de5dcb46f25facf956e62343fc4be59feecd2ec12189d999d0ddbb4073c529c1 1aac43a76716c35f9cfaa0d1f9d4ff3c883e67620a626866d2dc734ae3c900d8 121e2ba60657e1cfd5efa6ba410fc8bffa7e1d86a8c5e111794326bc51d99862 1632edc64f77b0365d9b63b7a7d912d949c72d4d9afa7b40394b759916e7833c 523507a069609de0c1b92c565e593e5139e335dd5d3828489c7e8ad7aaa86932 85a9c21a5de92e2401053a54adccca 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % 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 (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 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 51 /three put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 83 /S put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 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 readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueXX 5000792 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b448685bdd20eb05d5e7c2126132e33a59a7170609 dcf4871a5d023c9ef57d3362d9f2d7a440bb69bf653364105f16f4d0f03582f9 aced3d05cc76489b16e3fa8a446094d30038b06ecceda269f2eab9d19a99c7f9 39f9548f206c5a457a19270b2b82c43b091dfc5573468eaa3e7a4a32f8042891 d85e4b180fcbcb3091d2800e54c87d84ce9cad6869b5aabbbe47f40c68799893 d22b765295e1e69e33aa048b7ed98ba480ceca91f3ebf8ef85fe9a3976909626 b95ac5940d53f9b02215d84a44837ba25ed15cce0d504f1d335065594f3bc824 5405407591cccb11cfd4645da60d960c0b93f187b0cf7b105543c0b70f89af5d 264b6c026e3ae646acf145950202ec73282111e3e601cf2bcde22ce3edf6db23 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fbb1f00df8f6e08dac86b222eaf602f2b3e6fc055d70d7c0781d572b6b1812e2 633ca72aef39db3ce1765d0871143457756808f498c24c36b122dc0097430e23 4e8b0c69a7ce06e638c41cbf9121000ed4e0ecd452ab2f42aedae730d9d6dbb6 3510007d044558211a319eeaca93f68b9e 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX9 %!PS-AdobeFont-1.1: CMBX9 1.0 %%CreationDate: 1991 Aug 20 16:36:25 % 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 (CMBX9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX9 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{-58 -250 1195 750}readonly def /UniqueXX 5000767 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5bfae5c1f476ee6f05eb1d2caeb 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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 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 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 readonly def /FontBBox{-33 -250 945 749}readonly def /UniqueXX 5000795 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont TeXDict begin 39158280 55380996 1000 600 600 (prep607.dvi) @start /Fa 138[38 24 29 30 1[36 36 40 1[18 2[22 36 33 1[33 1[33 1[36 13[40 4[53 64 30[22 46[{}18 66.4176 /CMTI8 rf /Fb 241[32 14[{}1 49.8132 /CMSY6 rf /Fc 205[30 50[{}1 49.8132 /CMR6 rf /Fd 214[46 46 40[{}2 119.552 /CMR12 rf /Fe 231[51 1[70 3[55 18[{}3 119.552 /CMMI12 rf /Ff 255[45{}1 41.511 /CMSY5 rf /Fg 134[40 39 55 38 45 28 34 35 38 42 42 47 68 21 2[25 42 38 25 38 42 38 38 42 12[59 5[62 74 3[32 2[54 4[62 12[42 1[42 42 2[25 1[25 2[34 34 27[47 12[{}37 83.022 /CMTI10 rf /Fh 136[69 50 1[37 38 39 2[48 53 80 27 50 1[27 53 48 29 44 53 42 1[46 12[66 1[72 1[65 72 2[57 8[69 1[72 12[48 48 48 48 2[27 46[{}30 83.022 /CMBX10 rf /Fi 140[23 3[28 31 4[17 5[25 46[28 28 28 28 28 4[43 1[22 22 40[{}13 41.511 /CMR5 rf /Fj 134[30 5[28 8[24 23[43 5[40 2[26 11[31 1[19 33[35 6[28 18[{}10 41.511 /CMMI5 rf /Fk 143[55 4[34 20 42[52 12[45 66 19 14[66 11[52 14[52 5[52{}11 58.1154 /CMSY7 rf /Fl 143[83 55 55 4[39 39 2[46 8[92 92 46 1[120 3[69 1[39 1[88 62[50 50 3[28 8[35 35 2[{}18 83.022 /CMEX10 rf /Fm 128[35 3[35 31 37 37 51 37 39 27 28 28 37 39 35 39 59 20 37 22 20 39 35 22 31 39 31 39 35 3[20 35 20 1[53 53 72 53 53 51 39 52 55 48 55 53 65 44 55 36 25 53 55 46 48 54 51 50 53 55 5[20 35 35 35 35 35 35 35 35 35 35 1[20 24 20 2[27 27 20 4[35 21[39 41 11[{}77 66.4176 /CMR8 rf /Fn 134[37 1[51 1[39 27 28 28 2[35 39 59 20 1[22 20 39 2[31 39 31 39 35 13[39 5[65 1[55 7[51 54[39 12[{}23 66.4176 /CMSL8 rf /Fo 134[34 38 3[25 31 4[41 2[35 27 23 2[32 31 35 30 29 36 14[50 2[50 53 1[46 1[37 29 2[42 1[54 48 2[35 2[34 1[20 30[31 1[34 39 30 1[40 39 2[32 34 1[28 15[{}35 58.1154 /CMMI7 rf /Fp 143[69 4[42 23 32 32 42 42 7[55 1[55 9[70 24[46 46 1[0 3[55 83 15[83 6[65 1[65 2[65 65 13[65 3[65 1[65{}23 83.022 /CMSY10 rf /Fq 140[26 3[33 37 4[19 5[30 7[19 29[51 8[33 33 33 33 33 4[51 1[26 26 29[47 1[47 8[{}17 58.1154 /CMR7 rf /Fr 134[41 47 2[48 30 39 37 37 6[34 29 48 40 41 39 43 36 36 44 3[32 5[78 48 1[49 1[63 66 53 63 67 1[57 1[46 36 69 1[53 61 69 59 63 62 44 1[65 42 65 23 23 18[54 6[54 52 2[36 47 43 47 36 41 50 48 2[39 41 1[34 37 14[{}57 83.022 /CMMI10 rf /Fs 132[42 37 44 44 60 44 46 32 33 33 44 46 42 46 69 23 44 25 23 46 42 25 37 46 37 46 42 3[23 1[23 2[62 85 62 62 60 46 61 1[57 1[62 76 52 65 1[30 62 65 54 57 63 1[59 62 2[39 65 1[23 23 42 42 42 42 42 42 42 42 42 42 1[23 28 23 65 1[32 32 23 26[46 48 60 1[60 2[62 1[58 2[52{}77 83.022 /CMR10 rf /Ft 138[75 52 53 55 1[75 67 75 112 37 2[37 75 1[41 61 75 60 1[65 12[94 75 100 8[50 2[85 17[67 67 67 67 49[{}25 119.552 /CMBX12 rf /Fu 157[33 98[{}1 74.7198 /CMMI9 rf /Fv 134[41 41 55 41 43 30 30 30 1[43 38 43 64 21 41 1[21 43 38 23 34 43 34 43 38 9[79 3[43 2[52 1[58 70 48 2[28 58 20[38 4[21 26 21 2[30 30 27[43 45 11[{}39 74.7198 /CMR9 rf /Fw 139[34 35 36 14[39 49 43 31[67 65[{}7 74.7198 /CMBX9 rf /Fx 138[54 1[38 38 2[49 54 81 3[27 3[43 54 1[54 49 11[73 70 1[72 2[76 1[89 2[50 3[64 66 1[70 1[73 10[49 2[49 49 1[49 3[27 44[{}26 99.6264 /CMR12 rf /Fy 168[137 1[99 96 73 98 1[90 103 99 122 83 103 1[47 99 104 86 90 101 96 94 99 65[{}20 143.462 /CMR17 rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%BeginPaperSize: a4 a4 %%EndPaperSize %%EndSetup %%Page: 1 1 1 0 bop 635 872 a Fy(REMARKS)44 b(ON)g(THE)g(SPECTR)l(UM)f(OF)h(THE)625 1054 y(NEUMANN)h(PR)l(OBLEM)f(WITH)f(MA)l(GNETIC)1040 1237 y(FIELD)h(IN)f(THE)h(HALF)g(SP)-11 b(A)l(CE)983 1478 y Fx(Ab)s(deremane)34 b(MORAME)g(and)f(F)-8 b(ranoise)32 b(TR)m(UC)1678 1672 y(June)h(6,)g(2003)1778 2189 y Fw(Abstract)837 2319 y Fv(W)-6 b(e)30 b(consider)g(a)g(Sc)n(hrdinger)f(op)r(erator)h (with)g(a)g(constan)n(t)g(magnetic)g(\014eld)f(in)722 2410 y(a)f(half)h(3-dimensional)g(space,)g(with)f(Neumann)f(t)n(yp)r(e) g(b)r(oundary)g(conditions.)42 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b Fs(=)38 b Fr(I)-18 b(R)2528 3609 y Fq(+)2608 3597 y Fp(\002)24 b Fr(I)-18 b(R)2779 3561 y Fq(2)2816 3597 y Fs(,)39 b(the)f(Neumann)515 3697 y(realization)26 b(of)h(the)h(op)r (erator)e(with)i(magnetic)g(\014eld)1139 3879 y Fr(H)i Fs(=)23 b(\()p Fr(D)1427 3891 y Fo(t)1475 3879 y Fp(\000)18 b Fr(A)1620 3891 y Fq(1)1657 3879 y Fs(\))1689 3845 y Fq(2)1745 3879 y Fs(+)g(\()p Fr(D)1929 3891 y Fo(x)1989 3879 y Fp(\000)g Fr(A)2134 3891 y Fq(2)2172 3879 y Fs(\))2204 3845 y Fq(2)2260 3879 y Fs(+)g(\()p Fr(D)2444 3891 y Fo(y)2502 3879 y Fp(\000)g Fr(A)2647 3891 y Fq(3)2685 3879 y Fs(\))2717 3845 y Fq(2)515 4062 y Fs(where)27 b Fr(D)824 4074 y Fo(s)882 4062 y Fs(=)c Fp(\000)p Fr(i)p Fs(\()1121 4029 y Fo(@)p 1105 4043 71 4 v 1105 4091 a(@)t(s)1185 4062 y Fs(\).)639 4162 y(W)-7 b(e)36 b(will)f(assume)g(that)g(the)h (magnetic)f(\014eld)g Fr(B)40 b Fs(=)35 b Fr(dA)p Fs(,)j(seen)d(as)g(a) f(3-dimensional)515 4261 y(v)n(ector)i(\014eld,)41 b(is)d(not)g(tangen) n(t)f(to)h(the)g(b)r(oundary)g Fr(@)5 b(E)g Fs(,)40 b(and)e(denote)g(b) n(y)f Fr(\022)k Fs(the)d(angle)515 4361 y(b)r(et)n(w)n(een)27 b Fr(B)32 b Fs(and)c(the)g(plane)f Fr(t)c Fs(=)g(0)k(and)g(b)n(y)h Fr(b)f Fs(the)h(norm)f(of)g Fr(B)t Fs(.)639 4461 y(This)h(implies)g (that)g(a)f(suitable)g(c)n(hoice)g(for)g(the)h(gauge)e Fr(A)i Fs(is)g(the)g(1-form)1502 4643 y Fr(A)23 b Fs(=)g Fr(b)p Fs(\()p Fr(x)14 b Fs(sin)g Fr(\022)21 b Fp(\000)d Fr(t)c Fs(cos)e Fr(\022)r Fs(\))p Fr(dy)p 515 4696 1146 4 v 500 4773 a Fn(Keyw)n(ords)p Fm(:)23 b(Neumann)i(problem,)e (magnetic)i(\014eld,)e(sp)r(ectral)h(theory)-6 b(,)25 b(eigen)n(v)l(alues.)515 4852 y Fn(Mathematics)g(Sub)t(ject)f (Classi\014cation)p Fm(:)32 b(35P20.)1926 5255 y Fs(1)p eop %%Page: 2 2 2 1 bop 515 523 a Fs(\(so)27 b(that)h Fr(A)891 535 y Fq(1)951 523 y Fs(=)23 b Fr(A)1101 535 y Fq(2)1162 523 y Fs(=)f(0\),)28 b(since)f(the)h(condition)g Fr(B)f Fs(=)22 b Fr(dA)29 b Fs(leads)e(to)g(the)h(2-form)1014 727 y Fr(B)f Fs(=)c Fr(b)14 b Fs(sin)f Fr(\022)30 b(dx)19 b Fp(^)g Fr(dy)i Fp(\000)d Fr(b)c Fs(cos)f Fr(\022)30 b(dt)18 b Fp(^)h Fr(dy)s(;)69 b Fs(\()p Fr(\022)49 b Fp(2)2583 660 y Fl(\002)2618 727 y Fs(0)p Fr(;)2706 671 y(\031)p 2706 708 51 4 v 2710 784 a Fs(2)2767 660 y Fl(\003)2801 727 y Fs(\))24 b Fr(:)515 933 y Fs(No)n(w)j(the)h(op)r(erator)e Fr(H)34 b Fs(can)27 b(b)r(e)h(written)g(as)1156 1116 y Fr(H)1225 1128 y Fo(\022)1286 1116 y Fs(=)23 b Fr(D)1445 1081 y Fq(2)1443 1136 y Fo(t)1500 1116 y Fs(+)18 b Fr(D)1654 1081 y Fq(2)1652 1136 y Fo(x)1712 1116 y Fs(+)g(\()p Fr(D)1896 1128 y Fo(y)1955 1116 y Fp(\000)g Fr(b)p Fs(\()p Fr(x)c Fs(sin)g Fr(\022)20 b Fp(\000)e Fr(t)c Fs(cos)f Fr(\022)r Fs(\)\))2699 1081 y Fq(2)3208 1116 y Fs(\(1.1\))515 1298 y(When)55 b Fr(\022)26 b Fs(=)c(0)p Fr(;)27 b Fs(it)h(is)g(easy)e (to)h(see)g(that)h(the)g(sp)r(ectrum)g(of)f(the)h(Neumann)g(op)r (erator)53 b Fr(H)3342 1310 y Fq(0)515 1398 y Fs(is)27 b(absolutely)g(con)n(tin)n(uous.)36 b(More)27 b(precisely)f(one)i(has)f (:)1382 1581 y Fr(\033)s Fs(\()p Fr(H)1533 1593 y Fq(0)1571 1581 y Fs(\))c(=)g Fr(\033)1761 1593 y Fo(ac)1831 1581 y Fs(\()p Fr(H)1932 1593 y Fq(0)1969 1581 y Fs(\))h(=)e([)p Fr(b\026)2221 1593 y Fq(0)2258 1581 y Fr(;)14 b Fs(+)p Fp(1)p Fs([)23 b(;)696 b(\(1.2\))1661 1763 y Fr(\026)1711 1775 y Fq(0)1771 1763 y Fs(=)40 b(inf)1859 1817 y Fo(\030)r Fk(2)p Fo(I)-20 b(R)2032 1763 y Fr(\026)p Fs(\()p Fr(\030)t Fs(\))24 b Fr(;)975 b Fs(\(1.3\))515 1957 y(where)54 b Fr(\026)p Fs(\()p Fr(\030)t Fs(\))29 b(denotes)e(the)h(\014rst)f (eigen)n(v)-5 b(alue)27 b(of)h(the)g(Neumann)g(op)r(erator)667 2057 y Fr(Q)733 2069 y Fo(\030)792 2057 y Fs(=)23 b Fr(D)951 2027 y Fq(2)949 2078 y Fo(t)1006 2057 y Fs(+)18 b(\()p Fr(t)h Fp(\000)f Fr(\030)t Fs(\))1325 2027 y Fq(2)1414 2057 y Fs(on)27 b Fr(L)1586 2027 y Fq(2)1623 2057 y Fs(\()p Fr(I)-18 b(R)1737 2069 y Fq(+)1792 2057 y Fs(\))46 b(:)733 2285 y Fr(\026)p Fs(\()p Fr(\030)t Fs(\))23 b(=)g(inf)44 b Fr(\033)s Fs(\()p Fr(Q)1284 2297 y Fo(\030)1321 2285 y Fs(\))23 b(=)175 b(inf)1464 2342 y Fk(k)p Fo(f)7 b Fk(k)1571 2361 y Fj(L)1611 2349 y Fi(2)1643 2361 y(\()p Fj(I)-22 b(R)1712 2374 y Fi(+)1759 2361 y(\))1786 2342 y Fq(=1)1907 2172 y Fl(Z)1953 2361 y Fo(I)i(R)2012 2369 y Fi(+)2063 2218 y Fl(\002)2098 2285 y Fp(j)p Fr(D)2190 2297 y Fo(t)2219 2285 y Fr(f)9 b Fp(j)2292 2251 y Fq(2)2347 2285 y Fs(+)18 b(\()p Fr(t)h Fp(\000)f Fr(\030)t Fs(\))2666 2251 y Fq(2)2704 2285 y Fp(j)p Fr(f)9 b Fp(j)2800 2251 y Fq(2)2836 2218 y Fl(\003)2871 2285 y Fr(dt)23 b(:)218 b Fs(\(1.4\))515 2539 y(It)19 b(is)h(also)e(easy)g(to)i(see)f(that,)i (if)39 b Fr(\022)26 b Fs(=)1669 2506 y Fo(\031)p 1669 2520 41 4 v 1673 2568 a Fq(2)1720 2539 y Fr(;)20 b Fs(the)f(sp)r (ectrum)h(of)39 b Fr(H)2437 2529 y Fj(\031)p 2437 2538 36 3 v 2441 2572 a Fi(2)2506 2539 y Fs(is)19 b(absolutely)g(con)n(tin)n (uous:)1499 2722 y Fr(\033)s Fs(\()p Fr(H)1660 2712 y Fj(\031)p 1661 2721 V 1665 2754 a Fi(2)1711 2722 y Fs(\))k(=)g Fr(\033)1901 2734 y Fo(ac)1994 2722 y Fs(=)g([)p Fr(b;)14 b Fs(+)p Fp(1)p Fs([)22 b Fr(:)814 b Fs(\(1.5\))639 2916 y(When)63 b Fr(\022)e Fp(2)1101 2849 y Fl(\003)1135 2916 y Fs(0)p Fr(;)1224 2883 y Fo(\031)p 1224 2897 41 4 v 1228 2945 a Fq(2)1274 2916 y Fs([,)33 b(the)f(sp)r(ectrum)f(of)63 b Fr(H)2065 2928 y Fo(\022)2133 2916 y Fs(is)32 b(no)f(longer)f (absolutely)g(con)n(tin)n(uous)515 3016 y(as)d(pro)n(v)n(ed)f(b)n(y)h (K.)g(Lu)h(and)f(X-B.)h(P)n(an)e([LuP)n(a)o(],)i(\(see)g(also)e([HeMo2) o(])i(\).)515 3115 y(W)-7 b(e)28 b(are)e(precisely)h(in)n(terested)g (in)h(that)g(case:)1737 3315 y(0)22 b Fr(<)h(\022)i(<)2051 3259 y(\031)p 2051 3296 51 4 v 2055 3372 a Fs(2)2134 3315 y Fr(:)1051 b Fs(\(1.6\))515 3521 y(First,)27 b(w)n(e)h(observ)n (e)d(that)1541 3632 y Fr(\033)s Fs(\()p Fr(H)1692 3644 y Fo(\022)1730 3632 y Fs(\))e(=)1898 3553 y Fl([)1873 3731 y Fo(\034)7 b Fk(2)p Fo(I)-20 b(R)2028 3632 y Fr(\033)s Fs(\()p Fr(H)2179 3644 y Fo(\022)r(;\034)2275 3632 y Fs(\))23 b Fr(;)855 b Fs(\(1.7\))515 3860 y(where)27 b Fr(H)824 3872 y Fo(\022)r(;\034)946 3860 y Fs(denotes)g(the)h (Neumann)g(realization)e(in)i(the)g(half)g(plane)639 3960 y Fr(F)35 b Fs(=)23 b Fr(I)-18 b(R)897 3972 y Fq(+)970 3960 y Fp(\002)18 b Fr(I)-18 b(R)29 b Fs(of)e(the)h(op)r(erator)1136 4142 y Fr(H)1205 4154 y Fo(\022)r(;\034)1323 4142 y Fs(=)23 b Fr(D)1482 4108 y Fq(2)1480 4163 y Fo(t)1537 4142 y Fs(+)18 b Fr(D)1691 4108 y Fq(2)1689 4163 y Fo(x)1749 4142 y Fs(+)g(\()p Fr(\034)29 b Fp(\000)18 b Fr(b)p Fs(\()p Fr(x)c Fs(sin)g Fr(\022)20 b Fp(\000)e Fr(t)c Fs(cos)f Fr(\022)r Fs(\)\))2673 4108 y Fq(2)2734 4142 y Fr(:)451 b Fs(\(1.8\))515 4325 y(F)-7 b(urthermore)26 b(using)i(for)f(an)n(y)f Fr(\034)38 b Fs(the)28 b(c)n(hange)e(of)i(co)r(ordinates)e Fr(x)d Fp(!)g Fr(x)c Fp(\000)2924 4292 y Fo(\034)p 2859 4306 168 4 v 2859 4353 a(b)12 b Fq(sin)f Fo(\022)3036 4325 y Fs(,)515 4424 y(w)n(e)27 b(see)g(that)56 b Fr(\033)s Fs(\()p Fr(H)1130 4436 y Fo(\022)r(;\034)1225 4424 y Fs(\))23 b(=)g Fr(\033)s Fs(\()p Fr(H)1519 4436 y Fo(\022)r(;)p Fq(0)1610 4424 y Fs(\),)515 4524 y(and)k(then)h(the)g(sp)r(ectrum)g(of) 55 b Fr(H)1562 4536 y Fo(\022)1627 4524 y Fs(is)28 b(essen)n(tial)f (and)g(giv)n(en)g(b)n(y)g(:)1180 4723 y Fr(\033)s Fs(\()p Fr(H)1331 4735 y Fo(\022)1369 4723 y Fs(\))c(=)g Fr(\033)1559 4735 y Fo(ess)1658 4723 y Fs(\()p Fr(H)1759 4735 y Fo(\022)1797 4723 y Fs(\))g(=)g Fr(\033)s Fs(\()p Fr(H)2091 4735 y Fo(\022)r(;)p Fq(0)2182 4723 y Fs(\))g(=)g Fr(b)18 b Fp(\002)g Fr(\033)s Fs(\()p Fr(P)2597 4735 y Fo(\022)2635 4723 y Fs(\))24 b Fr(;)494 b Fs(\(1.9\))515 4873 y(if)49 b Fr(P)665 4885 y Fo(\022)726 4873 y Fs(=)22 b Fr(D)884 4843 y Fq(2)882 4893 y Fo(t)933 4873 y Fs(+)12 b Fr(D)1081 4843 y Fq(2)1079 4893 y Fo(x)1131 4873 y Fs(+)g(\()p Fr(t)i Fs(cos)f Fr(\022)h Fp(\000)e Fr(x)i Fs(sin)f Fr(\022)r Fs(\))1788 4843 y Fq(2)1850 4873 y Fs(is)24 b(the)h(Neumann)f(op)r (erator)f(on)g(the)i(half)f(plane)515 4972 y Fr(F)35 b Fs(=)22 b Fr(I)-18 b(R)772 4984 y Fq(+)846 4972 y Fp(\002)18 b Fr(I)-18 b(R)24 b(:)1926 5255 y Fs(2)p eop %%Page: 3 3 3 2 bop 639 523 a Fs(In)28 b([LuP)n(a)o(],)g(\(see)f(also)g([HeMo2)o(]) h(\),)g(it)g(w)n(as)e(pro)n(v)n(ed)g(that)1269 706 y(inf)44 b Fr(\033)s Fs(\()p Fr(P)1542 718 y Fo(\022)1581 706 y Fs(\))23 b(=)g Fr(\027)5 b Fs(\()p Fr(\022)r Fs(\))24 b Fr(<)f Fs(1)f(=)h(inf)44 b Fr(\033)2324 718 y Fo(ess)2423 706 y Fs(\()p Fr(P)2508 718 y Fo(\022)2546 706 y Fs(\))23 b(;)543 b(\(1.10\))515 888 y(so)27 b(there)g(exists)g(a)g(coun)n(table) g(set)h(of)g(eigen)n(v)-5 b(alues)26 b(of)55 b Fr(P)2332 900 y Fo(\022)2370 888 y Fr(;)42 b Fs(\()p Fr(\027)2508 900 y Fo(j)2543 888 y Fs(\()p Fr(\022)r Fs(\)\))2680 900 y Fo(j)s Fk(2)p Fo(I)2795 888 y Fr(;)f Fs(\()p Fr(I)31 b Fp(\032)22 b Fr(I)-18 b(N)9 b Fs(\))p Fr(;)515 988 y Fs(in)23 b([)p Fr(\027)5 b Fs(\()p Fr(\022)r Fs(\))p Fr(;)14 b Fs(1[)24 b(.)36 b(Eac)n(h)22 b(eigen)n(v)-5 b(alue)22 b(is)h(of)g(\014nite)h(m)n(ultiplicit)n(y)-7 b(,)24 b(so)f(w)n(e)g(will)g(assume)f(that)i(eac)n(h)515 1088 y(one)33 b(is)h(rep)r(eated)g(according)e(to)i(its)g(m)n (ultiplicit)n(y)-7 b(.)57 b(The)34 b(asso)r(ciated)f(orthonormalized) 515 1187 y(sequence)27 b(of)g(eigenfunctions)h(will)g(b)r(e)g(denoted)f (b)n(y)55 b(\()p Fr( )2308 1199 y Fo(\022)r(;j)2397 1187 y Fs(\))2429 1199 y Fo(j)s Fk(2)p Fo(I)2566 1187 y Fs(:)1060 1370 y Fr(\027)5 b Fs(\()p Fr(\022)r Fs(\))24 b(=)f Fr(\027)1364 1382 y Fq(1)1401 1370 y Fs(\()p Fr(\022)r Fs(\))h Fp(\024)f Fr(\027)1659 1382 y Fq(2)1696 1370 y Fs(\()p Fr(\022)r Fs(\))h Fp(\024)e Fr(:)14 b(:)g(:)g(\027)2064 1382 y Fo(j)2099 1370 y Fs(\()p Fr(\022)r Fs(\))24 b Fp(\024)f Fr(\027)2357 1382 y Fo(j)s Fq(+1)2476 1370 y Fs(\()p Fr(\022)r Fs(\))h Fp(\024)e Fr(:)14 b(:)g(:)23 b(<)g Fs(1)p Fr(;)202 b Fs(\(1.11\))979 1494 y Fr(P)1032 1506 y Fo(\022)1070 1494 y Fr( )1124 1506 y Fo(\022)r(;j)1235 1494 y Fs(=)23 b Fr(\027)1364 1506 y Fo(j)1399 1494 y Fs(\()p Fr(\022)r Fs(\))p Fr( )1558 1506 y Fo(\022)r(;j)1647 1494 y Fr(;)834 1619 y Fp(h)p Fr( )920 1631 y Fo(\022)r(;j)1009 1619 y Fp(j)p Fr( )1086 1631 y Fo(\022)r(;k)1180 1619 y Fp(i)g Fs(=)g Fr(\016)1360 1631 y Fo(j)s(k)1431 1619 y Fr(;)717 1769 y(E)778 1744 y Fl(\003)813 1811 y Fk(\0001)p Fo(;)p Fq(1[)1007 1769 y Fs(\()p Fr(P)1092 1781 y Fo(\022)1130 1769 y Fs(\))p Fr(f)32 b Fs(=)1323 1690 y Fl(X)1367 1867 y Fo(j)1443 1769 y Fp(h)p Fr( )1529 1781 y Fo(\022)r(;j)1617 1769 y Fp(j)p Fr(f)9 b Fp(i)p Fr( )1776 1781 y Fo(\022)r(;j)1887 1769 y Fs(;)515 2045 y(\()p Fp(h)p Fr(g)32 b Fp(j)p Fr(f)9 b Fp(i)29 b Fs(=)879 1978 y Fl(R)918 2074 y Fo(F)p 987 1999 43 4 v 987 2045 a Fr(g)s(f)9 b(dtdx)32 b Fs(and)62 b Fr(E)1532 2057 y Fo(J)1579 2045 y Fs(\()p Fr(P)1664 2057 y Fo(\022)1702 2045 y Fs(\))32 b(denotes)f(the)g(sp)r(ectral)g (pro)5 b(jection)30 b(of)h Fr(P)3081 2057 y Fo(\022)3151 2045 y Fs(on)g Fr(J)8 b Fs(\))p Fr(:)515 2145 y Fs(So)713 2327 y Fr(\033)s Fs(\()p Fr(H)864 2339 y Fo(\022)902 2327 y Fs(\))p Fp(\\)1017 2260 y Fl(\003)1071 2327 y Fp(\000)18 b(1)p Fr(;)c(b)p Fs([=)22 b Fp(f)p Fr(b\027)1539 2339 y Fq(1)1575 2327 y Fs(\()p Fr(\022)r Fs(\))p Fr(;)38 b(b\027)1818 2339 y Fq(2)1855 2327 y Fs(\()p Fr(\022)r Fs(\))p Fr(;)14 b(:)g(:)g(:)g(;)37 b(b\027)2245 2339 y Fo(j)2280 2327 y Fs(\()p Fr(\022)r Fs(\))p Fr(;)h(b\027)2523 2339 y Fo(j)s Fq(+1)2641 2327 y Fs(\()p Fr(\022)r Fs(\))p Fr(;)14 b(:)g(:)g(:)q Fp(g)23 b Fr(;)198 b Fs(\(1.12\))515 2510 y(\(eac)n(h)27 b Fr(b\027)811 2522 y Fo(j)845 2510 y Fs(\()p Fr(\022)r Fs(\))i(is)e(an)h(eigen)n(v)-5 b(alue)26 b(of)i(in\014nite)g(m)n(ultiplicit)n(y)g(of)g Fr(H)2559 2522 y Fo(\022)2620 2510 y Fs(\))p Fr(:)639 2610 y Fs(F)-7 b(or)29 b(an)n(y)f Fr(d)e Fp(\024)f Fs(1)j(let)i(us)f(denote)g(b)n(y)58 b Fr(N)9 b Fs(\()p Fr(d;)39 b(P)2088 2622 y Fo(\022)2126 2610 y Fs(\))30 b(the)f(n)n(um)n(b)r(er)g(of)g(eigen)n(v)-5 b(alues)28 b(of)58 b Fr(P)3341 2622 y Fo(\022)515 2709 y Fs(in)612 2642 y Fl(\003)665 2709 y Fp(\000)18 b(1)p Fr(;)c(d)p Fs([)23 b(:)1079 2892 y Fr(N)9 b Fs(\()p Fr(d;)37 b(P)1343 2904 y Fo(\022)1381 2892 y Fs(\))24 b(=)e Fr(T)12 b(r)r Fs(\()p Fr(E)1717 2867 y Fl(\003)1752 2934 y Fk(\0001)p Fo(;d)p Fq([)1948 2892 y Fs(\()p Fr(P)2033 2904 y Fo(\022)2071 2892 y Fs(\))23 b(=)g Fr(])p Fp(f)p Fr(j)5 b Fs(;)41 b Fr(\027)2432 2904 y Fo(j)2467 2892 y Fs(\()p Fr(\022)r Fs(\))24 b Fr(<)f(d)p Fp(g)f Fr(:)353 b Fs(\(1.13\))639 3125 y(The)25 b(aim)f(of)h(this)g(w)n(ork)e(is)i(\014rst)f(to)g(pro)n (v)n(e)f(that)i(for)f(an)n(y)g Fr(\022)49 b Fp(2)2605 3057 y Fl(\003)2640 3125 y Fs(0)p Fr(;)2728 3092 y Fo(\031)p 2728 3106 41 4 v 2732 3153 a Fq(2)2779 3125 y Fs([)23 b Fr(;)i Fs(the)g(n)n(um)n(b)r(er)f(of)515 3233 y(eigen)n(v)-5 b(alues)22 b(of)h Fr(P)1083 3245 y Fo(\022)1144 3233 y Fs(in)1236 3165 y Fl(\003)1281 3233 y Fp(\000)10 b(1)p Fr(;)36 b Fs(1[)23 b(is)g(\014nite.)36 b(This)23 b(is)g(the)h(purp)r (ose)e(of)h(section)g(2.)35 b(Another)515 3332 y(in)n(teresting)22 b(question)h(is)f(to)h(get)g(the)g(asymptotic)g(b)r(eha)n(viour)e(of)46 b Fr(N)9 b Fs(\()p Fr(d;)37 b(P)2872 3344 y Fo(\022)2910 3332 y Fs(\))24 b(as)e Fr(\022)j Fs(go)r(es)d(to)515 3432 y(zero,)i(when)48 b Fr(d)23 b(<)g Fs(1)g Fr(:)i Fs(This)g(is)f(done)h(in)g(section)f(4.)36 b(Section)25 b(3)f(is)h(dev)n(oted)f(to)h(a)g(surv)n(ey)e(of)515 3532 y(preliminary)g(results)h(ab)r(out)g(the)g(function)h Fr(\026)p Fs(\()p Fr(\030)t Fs(\))g(de\014ned)g(in)g(\(1.4\))f(,)h (whic)n(h)f(are)f(required)515 3631 y(in)28 b(the)g(computation)f(of)g (the)h(asymptotics)f(in)h(section)f(4.)515 3906 y Ft(2)134 b(Finiteness)46 b(of)f(the)h(discrete)f(sp)t(ectrum)515 4088 y Fs(The)27 b(purp)r(ose)g(of)h(this)g(section)f(is)h(to)f(pro)n (v)n(e)f(the)i(follo)n(wing)e(Theorem.)515 4254 y Fh(Theorem)31 b(2.1)41 b Fg(Ther)l(e)30 b(exists)g(a)g(c)l(onstant)f Fr(C)g Fp(\025)23 b Fs(1)29 b Fg(such)h(that,)g(for)h(any)f Fr(\022)48 b Fp(2)3042 4186 y Fl(\003)3076 4254 y Fs(0)p Fr(;)3165 4221 y Fo(\031)p 3165 4235 V 3169 4282 a Fq(2)3215 4254 y Fs([)24 b Fr(;)1610 4474 y(N)9 b Fs(\(1)p Fr(;)43 b(P)1879 4486 y Fo(\022)1917 4474 y Fs(\))24 b Fp(\024)2116 4418 y Fr(C)p 2070 4455 158 4 v 2070 4531 a Fs(sin)14 b Fr(\022)2260 4474 y(:)925 b Fs(\(2.1\))515 4664 y Fh(Pro)s(of)515 4846 y(Con)m(v)m(en)m(tion)32 b(2.2)40 b Fr(\022)49 b Fp(2)1349 4779 y Fl(\003)1384 4846 y Fs(0)p Fr(;)1472 4814 y Fo(\031)p 1472 4828 41 4 v 1476 4875 a Fq(2)1523 4846 y Fs([)30 b Fg(is)g(\014xe)l(d.)1926 5255 y Fs(3)p eop %%Page: 4 4 4 3 bop 515 523 a Fh(Con)m(v)m(en)m(tion)32 b(2.3)40 b Fg(F)-6 b(r)l(om)30 b(now)g(on,)g(any)g(c)l(onstant)f(dep)l(ending)j (only)e(on)59 b Fr(\022)515 623 y Fg(wil)t(l)31 b(b)l(e)f(denote)l(d)g (invariably)62 b Fr(C)1551 635 y Fo(\022)1612 623 y Fr(:)639 722 y Fg(If)30 b(the)g(c)l(onstant)f(do)l(es)i(not)e(dep)l(end)i(on)60 b Fr(\022)r(;)30 b Fg(it)f(wil)t(l)i(b)l(e)f(denote)l(d)h(invariably)61 b Fr(C)q(:)639 893 y Fs(Let)28 b(us)g(denote)f(b)n(y)g Fr(q)1314 905 y Fo(\022)1380 893 y Fs(the)h(quadratic)e(form)h(asso)r (ciated)g(to)g Fr(P)2640 905 y Fo(\022)2733 893 y Fs(:)890 1101 y Fr(q)927 1113 y Fo(\022)965 1101 y Fs(\()p Fr(u)p Fs(\))c(=)1187 988 y Fl(Z)1234 1177 y Fo(F)1289 1034 y Fl(\002)1323 1101 y Fs([)p Fr(D)1415 1113 y Fo(t)1444 1101 y Fr(u)p Fp(j)1515 1067 y Fq(2)1571 1101 y Fs(+)18 b Fp(j)p Fr(D)1746 1113 y Fo(x)1787 1101 y Fr(u)p Fp(j)1858 1067 y Fq(2)1914 1101 y Fs(+)g(\()p Fr(t)c Fs(cos)f Fr(\022)20 b Fp(\000)e Fr(x)c Fs(sin)g Fr(\022)r Fs(\))2590 1067 y Fq(2)2628 1101 y Fp(j)p Fr(u)p Fp(j)2722 1067 y Fq(2)2759 1034 y Fl(\003)2794 1101 y Fr(dtdx)24 b(;)204 b Fs(\(2.2\))515 1323 y Fp(8)p Fr(u)22 b Fp(2)i Fr(H)786 1293 y Fq(1)823 1323 y Fs(\()p Fr(F)12 b Fs(\))966 1261 y Fl(T)1049 1323 y Fr(L)1106 1293 y Fq(2)1143 1323 y Fs(\()p Fr(F)g Fs(;)37 b(\()p Fr(t)14 b Fs(cos)f Fr(\022)21 b Fp(\000)d Fr(x)c Fs(sin)g Fr(\022)r Fs(\))1894 1293 y Fq(2)1932 1323 y Fr(dtdx)p Fs(\))24 b(;)69 b(\()p Fr(F)36 b Fs(=)22 b Fr(I)-18 b(R)2533 1335 y Fq(+)p Fo(;t)2651 1323 y Fp(\002)18 b Fr(I)-18 b(R)2816 1335 y Fo(x)2881 1323 y Fs(\))p Fr(:)639 1423 y Fs(There)27 b(exists)h(a)f(partition)g(of)g(unit)n(y)h(\()p Fr(\037)1919 1435 y Fq(0)1957 1423 y Fs(\()p Fr(t)p Fs(\))p Fr(;)37 b(\037)2163 1435 y Fq(1)2200 1423 y Fs(\()p Fr(t)p Fs(\)\))29 b(satisfying)55 b(:)1532 1593 y Fr(\037)1584 1605 y Fq(0)1621 1593 y Fs(\()p Fr(t)p Fs(\))24 b(=)f(1)k(if)34 b Fr(t)23 b(<)g Fs(1)p Fr(;)1532 1718 y(\037)1584 1730 y Fq(0)1621 1718 y Fs(\()p Fr(t)p Fs(\))h(=)f(0)k(if)34 b Fr(t)23 b(>)g Fs(2)p Fr(;)1530 1842 y(\037)1582 1808 y Fq(2)1582 1863 y(0)1619 1842 y Fs(\()p Fr(t)p Fs(\))c(+)f Fr(\037)1867 1808 y Fq(2)1867 1863 y(1)1905 1842 y Fs(\()p Fr(t)p Fs(\))23 b(=)g(1)f Fr(:)1011 b Fs(\(2.3\))639 2013 y(Let)56 b Fr(R)23 b(>)g Fs(1)k(b)r(e)h(\014xed.)37 b(W)-7 b(e)28 b(consider)f(the)h(follo)n(wing)e(co)n(v)n(ering)g(of)h Fr(F)40 b Fs(:)1173 2183 y Fr(O)1236 2195 y Fq(0)p Fo(;R)1427 2183 y Fs(=)82 b Fp(f)p Fs(\()p Fr(t;)14 b(x)p Fs(\))24 b Fp(2)f Fr(I)-18 b(R)1978 2195 y Fq(+)2052 2183 y Fp(\002)18 b Fr(I)-18 b(R;)42 b Fs(0)23 b Fr(<)f(t)h(<)g Fs(2)p Fr(R)q Fp(g)1173 2308 y Fr(O)1236 2320 y Fq(1)p Fo(;R)1427 2308 y Fs(=)82 b Fp(f)p Fs(\()p Fr(t;)14 b(x)p Fs(\))24 b Fp(2)f Fr(I)-18 b(R)1978 2320 y Fq(+)2052 2308 y Fp(\002)18 b Fr(I)-18 b(R;)42 b(R)24 b(<)e(t)p Fp(g)681 b Fs(\(2.4\))639 2478 y(W)-7 b(e)28 b(de\014ne)g(the)g(partition)f(of)h(unit)n(y)g(\()p Fr(\037)1907 2490 y Fq(0)p Fo(;R)2014 2478 y Fs(\()p Fr(t)p Fs(\))p Fr(;)38 b(\037)2221 2490 y Fq(1)p Fo(;R)2328 2478 y Fs(\()p Fr(t)p Fs(\)\))28 b(b)n(y)55 b(:)1627 2688 y Fr(\037)1679 2700 y Fo(j;R)1780 2688 y Fs(\()p Fr(t)p Fs(\))24 b(=)f Fr(\037)2038 2700 y Fo(j)2072 2688 y Fs(\()2132 2632 y Fr(t)p 2114 2669 64 4 v 2114 2745 a(R)2188 2688 y Fs(\))h Fr(:)941 b Fs(\(2.5\))639 2882 y(Let)28 b(us)g(recall)e(that)1260 3052 y Fr(q)1297 3064 y Fo(\022)1335 3052 y Fs(\()p Fr(u)p Fs(\))d(=)1557 2974 y Fl(X)1602 3150 y Fo(j)1691 3052 y Fr(q)1728 3064 y Fo(\022)1766 3052 y Fs(\()p Fr(\037)1850 3064 y Fo(j;R)1951 3052 y Fr(u)p Fs(\))c Fp(\000)2133 2974 y Fl(X)2177 3150 y Fo(j)2266 3052 y Fp(k)p Fr(\037)2360 3018 y Fk(0)2360 3073 y Fo(j;R)2461 3052 y Fr(u)p Fp(k)2551 3018 y Fq(2)2611 3052 y Fr(:)574 b Fs(\(2.6\))515 3307 y(W)-7 b(e)28 b(de\014ne)g(the)g (follo)n(wing)e(quadratic)h(forms:)556 3515 y Fr(q)593 3527 y Fo(\022)r(;)p Fq(0)684 3515 y Fs(\()p Fr(u)p Fs(\))c(=)906 3402 y Fl(Z)953 3591 y Fo(O)1003 3599 y Fi(0)p Fj(;R)1102 3448 y Fl(\002)1136 3515 y Fr(D)1205 3527 y Fo(t)1234 3515 y Fr(u)p Fp(j)1305 3481 y Fq(2)1356 3515 y Fs(+)13 b Fp(j)p Fr(D)1526 3527 y Fo(x)1568 3515 y Fr(u)p Fp(j)1639 3481 y Fq(2)1690 3515 y Fs(+)g(\()23 b(\()p Fr(t)14 b Fs(cos)g Fr(\022)i Fp(\000)d Fr(x)h Fs(sin)g Fr(\022)r Fs(\))2408 3481 y Fq(2)2460 3515 y Fp(\000)g Fr(V)2587 3527 y Fo(R)2641 3515 y Fs(\()p Fr(t)p Fs(\))24 b(\))p Fp(j)p Fr(u)p Fp(j)2885 3481 y Fq(2)2922 3448 y Fl(\003)2957 3515 y Fr(dtdx)g(;)41 b Fs(\(2.7\))515 3754 y Fp(8)p Fr(u)p Fp(2)p Fr(H)740 3724 y Fq(1)777 3754 y Fs(\()p Fr(O)872 3766 y Fq(0)p Fo(;R)980 3754 y Fs(\))1026 3691 y Fl(T)1109 3754 y Fr(L)1166 3724 y Fq(2)1203 3754 y Fs(\()p Fr(O)1298 3766 y Fq(0)p Fo(;R)1405 3754 y Fs(;)14 b Fr(x)1489 3724 y Fq(2)1527 3754 y Fr(dtdx)p Fs(\))p Fr(;)36 b(u=)p Fp(f)p Fr(t)22 b Fs(=)h(2)p Fr(R)q Fp(g)e Fs(=)i(0,)f(with)g Fr(V)2628 3766 y Fo(R)2683 3754 y Fs(\()p Fr(t)p Fs(\))i(=)2888 3691 y Fl(P)2976 3779 y Fo(j)3025 3754 y Fp(j)p Fr(\037)3100 3724 y Fk(0)3100 3777 y Fo(j;R)3201 3754 y Fs(\()p Fr(t)p Fs(\))p Fp(j)3318 3724 y Fq(2)3356 3754 y Fr(;)515 3853 y Fs(and)556 4045 y Fr(q)593 4057 y Fo(\022)r(;)p Fq(1)684 4045 y Fs(\()p Fr(u)p Fs(\))f(=)906 3932 y Fl(Z)953 4121 y Fo(O)1003 4129 y Fi(1)p Fj(;R)1102 3978 y Fl(\002)1136 4045 y Fr(D)1205 4057 y Fo(t)1234 4045 y Fr(u)p Fp(j)1305 4011 y Fq(2)1356 4045 y Fs(+)13 b Fp(j)p Fr(D)1526 4057 y Fo(x)1568 4045 y Fr(u)p Fp(j)1639 4011 y Fq(2)1690 4045 y Fs(+)g(\()23 b(\()p Fr(t)14 b Fs(cos)g Fr(\022)i Fp(\000)d Fr(x)h Fs(sin)g Fr(\022)r Fs(\))2408 4011 y Fq(2)2460 4045 y Fp(\000)g Fr(V)2587 4057 y Fo(R)2641 4045 y Fs(\()p Fr(t)p Fs(\))24 b(\))p Fp(j)p Fr(u)p Fp(j)2885 4011 y Fq(2)2922 3978 y Fl(\003)2957 4045 y Fr(dtdx)g(;)41 b Fs(\(2.8\))515 4304 y Fp(8)23 b Fr(u)46 b Fp(2)g Fr(H)855 4269 y Fq(1)892 4304 y Fs(\()p Fr(O)987 4316 y Fq(1)p Fo(;R)1095 4304 y Fs(\))1141 4225 y Fl(\\)1247 4304 y Fr(L)1304 4269 y Fq(2)1341 4304 y Fs(\()p Fr(O)1436 4316 y Fq(1)p Fo(;R)1544 4304 y Fs(;)14 b(\()p Fr(t)g Fs(cos)f Fr(\022)20 b Fp(\000)f Fr(x)14 b Fs(sin)g Fr(\022)r Fs(\))2175 4269 y Fq(2)2212 4304 y Fr(dtdx)p Fs(\))p Fr(;)43 b(u=)p Fp(f)p Fr(t)22 b Fs(=)g Fr(R)q Fp(g)h Fs(=)f(0)p Fr(:)639 4415 y Fs(By)28 b(min-max)f(principle,)g(w)n(e)h(ha)n(v)n(e)1283 4586 y Fr(N)9 b Fs(\(1)p Fr(;)41 b(q)1534 4598 y Fo(\022)1572 4586 y Fs(\))23 b Fp(\024)g Fr(N)9 b Fs(\(1)p Fr(;)41 b(q)1966 4598 y Fo(\022)r(;)p Fq(0)2057 4586 y Fs(\))18 b(+)g Fr(N)9 b Fs(\(1)p Fr(;)42 b(q)2442 4598 y Fo(\022)r(;)p Fq(1)2532 4586 y Fs(\))23 b Fr(:)598 b Fs(\(2.9\))515 4756 y(This)27 b(estimate)h(remains)f(if)h(w)n(e)f(c)n(hange)f Fr(O)1882 4768 y Fq(1)p Fo(;R)2018 4756 y Fs(in)n(to)h Fr(I)-18 b(R)2268 4720 y Fq(2)2333 4756 y Fs(in)28 b(the)g (de\014nition)f(of)h Fr(q)3073 4768 y Fo(\022)r(;)p Fq(1)3187 4756 y Fs(:)556 4972 y Fr(q)593 4984 y Fo(\022)r(;)p Fq(1)684 4972 y Fs(\()p Fr(u)p Fs(\))23 b(=)906 4859 y Fl(Z)953 5048 y Fo(I)-20 b(R)1012 5029 y Fi(2)1049 4905 y Fl(\002)1083 4972 y Fr(D)1152 4984 y Fo(t)1181 4972 y Fr(u)p Fp(j)1252 4938 y Fq(2)1304 4972 y Fs(+)15 b Fp(j)p Fr(D)1476 4984 y Fo(x)1518 4972 y Fr(u)p Fp(j)1589 4938 y Fq(2)1641 4972 y Fs(+)g(\()23 b(\()p Fr(t)14 b Fs(cos)f Fr(\022)18 b Fp(\000)d Fr(x)f Fs(sin)g Fr(\022)r Fs(\))2364 4938 y Fq(2)2417 4972 y Fp(\000)h Fr(V)2545 4984 y Fo(R)2600 4972 y Fs(\()p Fr(t)p Fs(\))24 b(\))p Fp(j)p Fr(u)p Fp(j)2844 4938 y Fq(2)2881 4905 y Fl(\003)2915 4972 y Fr(dtdx)g(;)42 b Fs(\(2.10\))1926 5255 y(4)p eop %%Page: 5 5 5 4 bop 515 527 a Fp(8)23 b Fr(u)46 b Fp(2)g Fr(H)855 493 y Fq(1)892 527 y Fs(\()p Fr(I)-18 b(R)1006 491 y Fq(2)1043 527 y Fs(\))1089 448 y Fl(\\)1196 527 y Fr(L)1253 493 y Fq(2)1289 527 y Fs(\()p Fr(I)g(R)1403 491 y Fq(2)1441 527 y Fs(;)14 b(\()p Fr(t)g Fs(cos)f Fr(\022)20 b Fp(\000)e Fr(x)c Fs(sin)g Fr(\022)r Fs(\))2071 493 y Fq(2)2109 527 y Fr(dtdx)p Fs(\))24 b Fr(:)639 639 y Fs(As)k(the)g(op)r(erator)e Fr(P)1293 651 y Fo(\022)r(;)p Fq(0)1383 639 y Fr(;)i Fs(asso)r(ciated)e(to)i Fr(q)1968 651 y Fo(\022)r(;)p Fq(0)2086 639 y Fs(has)f(compact)g(resolv)n(en)n(t,)f(and)571 867 y Fr(q)608 879 y Fo(\022)r(;)p Fq(0)698 867 y Fs(\()p Fr(u)p Fs(\))e Fp(\025)921 754 y Fl(Z)967 943 y Fo(O)1017 951 y Fi(0)p Fj(;R)1116 775 y Fl(h)1155 867 y Fr(D)1224 879 y Fo(t)1253 867 y Fr(u)p Fp(j)1324 833 y Fq(2)1380 867 y Fs(+)18 b Fp(j)p Fr(D)1555 879 y Fo(x)1596 867 y Fr(u)p Fp(j)1667 833 y Fq(2)1723 867 y Fs(+)1806 775 y Fl(\020)1865 811 y Fs(1)p 1865 848 42 4 v 1865 924 a(2)1917 867 y Fr(x)1964 833 y Fq(2)2015 867 y Fs(sin)2117 832 y Fq(2)2168 867 y Fr(\022)j Fp(\000)d Fs(4)p Fr(R)2417 833 y Fq(2)2467 867 y Fs(cos)2578 833 y Fq(2)2629 867 y Fr(\022)j Fp(\000)2800 811 y Fr(C)p 2782 848 101 4 v 2782 924 a(R)2846 900 y Fq(2)2893 775 y Fl(\021)2943 867 y Fp(j)p Fr(u)p Fp(j)3037 833 y Fq(2)3074 775 y Fl(i)3113 867 y Fr(dtdx)j(;)515 1108 y Fs(w)n(e)j(get)g(easily)1276 1245 y Fr(N)9 b Fs(\(1)p Fr(;)41 b(q)1527 1257 y Fo(\022)r(;)p Fq(0)1617 1245 y Fs(\))24 b Fp(\024)1784 1189 y Fr(C)6 b(R)p 1770 1226 158 4 v 1770 1302 a Fs(sin)14 b Fr(\022)1937 1178 y Fl(\002)1972 1245 y Fs(1)k(+)g Fr(R)2179 1211 y Fq(2)2229 1245 y Fs(cos)2341 1211 y Fq(2)2392 1245 y Fr(\022)2433 1178 y Fl(\003)2468 1195 y Fq(3)p Fo(=)p Fq(2)2595 1245 y Fr(:)549 b Fs(\(2.11\))639 1418 y(Using)28 b(the)g(orthonormal)d(c)n(hange)i(of)g(co)r(ordinates)f(:)639 1518 y(\()p Fr(t;)14 b(x)p Fs(\))24 b Fp(!)f Fs(\()p Fr(s;)14 b(y)s Fs(\))28 b(with)g Fr(s)23 b Fs(=)g Fr(t)14 b Fs(cos)f Fr(\022)20 b Fp(\000)e Fr(x)c Fs(sin)g Fr(\022)r Fs(,)28 b(and)g Fr(y)e Fs(=)c Fr(t)14 b Fs(sin)g Fr(\022)21 b Fs(+)d Fr(x)c Fs(cos)f Fr(\022)r Fs(,)515 1617 y(w)n(e)27 b(can)g(tak)n(e)g(for)g Fr(q)1133 1629 y Fo(\022)r(;)p Fq(1)1251 1617 y Fs(the)h(follo)n(wing)f(expression)f(:)784 1921 y Fr(q)821 1933 y Fo(\022)r(;)p Fq(1)911 1921 y Fs(\()p Fr(u)p Fs(\))d(=)1134 1808 y Fl(Z)1180 1997 y Fo(I)-20 b(R)1239 1978 y Fi(2)1276 1854 y Fl(\002)1311 1921 y Fr(D)1380 1933 y Fo(s)1415 1921 y Fr(u)p Fp(j)1486 1887 y Fq(2)1541 1921 y Fs(+)18 b Fp(j)p Fr(D)1716 1933 y Fo(y)1756 1921 y Fr(u)p Fp(j)1827 1887 y Fq(2)1882 1921 y Fs(+)g(\()p Fr(s)2036 1887 y Fq(2)2092 1921 y Fp(\000)g Fr(V)2223 1933 y Fo(R)2278 1921 y Fs(\()p Fr(s;)c(y)s Fs(\))23 b(\))p Fp(j)p Fr(u)p Fp(j)2611 1887 y Fq(2)2648 1854 y Fl(\003)2683 1921 y Fr(dsdy)j(;)269 b Fs(\(2.12\))515 2175 y Fp(8)23 b Fr(u)46 b Fp(2)g Fr(H)855 2141 y Fq(1)892 2175 y Fs(\()p Fr(I)-18 b(R)1006 2139 y Fq(2)1043 2175 y Fs(\))1089 2096 y Fl(\\)1196 2175 y Fr(L)1253 2141 y Fq(2)1289 2175 y Fs(\()p Fr(I)g(R)1403 2139 y Fq(2)1441 2175 y Fs(;)37 b Fr(s)1540 2141 y Fq(2)1577 2175 y Fr(dsdy)s Fs(\))23 b Fr(;)28 b Fs(with)1181 2397 y Fr(V)1229 2409 y Fo(R)1284 2397 y Fs(\()p Fr(s;)14 b(y)s Fs(\))23 b(=)1618 2341 y(1)p 1588 2378 101 4 v 1588 2454 a Fr(R)1652 2430 y Fq(2)1713 2318 y Fl(X)1758 2495 y Fo(j)1833 2302 y Fl(\014)1833 2351 y(\014)1833 2401 y(\014)1861 2397 y Fr(\037)1913 2363 y Fk(0)1913 2418 y Fo(j)1948 2305 y Fl(\020)2007 2341 y Fr(s)14 b Fs(cos)f Fr(\022)21 b Fs(+)d Fr(y)e Fs(sin)e Fr(\022)p 2007 2378 536 4 v 2243 2454 a(R)2552 2305 y Fl(\021)2602 2302 y(\014)2602 2351 y(\014)2602 2401 y(\014)2630 2322 y Fq(2)2690 2397 y Fr(:)454 b Fs(\(2.13\))639 2630 y(Let)28 b(us)g(consider)e(the)i(orthogonal)d(pro)5 b(jections)1150 2852 y(\005)1212 2864 y Fq(1)1249 2852 y Fs(\()p Fr(u)p Fs(\)\()p Fr(s;)14 b(y)s Fs(\))84 b(=)e Fr(e)1815 2817 y Fk(\000)p Fo(s)1898 2792 y Fi(2)1931 2817 y Fo(=)p Fq(2)2016 2739 y Fl(Z)2062 2927 y Fo(I)-20 b(R)2139 2852 y Fr(u)p Fs(\()p Fr(\034)5 b(;)14 b(y)s Fs(\))p Fr(e)2412 2817 y Fk(\000)p Fo(\034)2502 2792 y Fi(2)2533 2817 y Fo(=)p Fq(2)2630 2796 y Fr(d\034)p 2614 2833 120 4 v 2614 2849 a Fp(p)p 2684 2849 51 4 v 2684 2909 a Fr(\031)1403 3024 y Fs(\003)1461 3036 y Fq(1)1498 3024 y Fr(u)83 b Fs(=)f Fr(u)18 b Fp(\000)g Fs(\005)1987 3036 y Fq(1)2025 3024 y Fr(u)k(;)1049 b Fs(\(2.14\))515 3207 y(so)27 b(that,)h(for)f(an)n(y)g Fr(u)45 b Fp(2)i Fr(L)1356 3177 y Fq(2)1392 3207 y Fs(\()p Fr(I)-18 b(R)1506 3171 y Fq(2)1544 3207 y Fs(\),)28 b(w)n(e)f(get)g(:)37 b Fp(k)p Fr(u)p Fp(k)2079 3177 y Fq(2)2138 3207 y Fs(=)23 b Fp(k)p Fs(\005)2330 3219 y Fq(1)2367 3207 y Fr(u)p Fp(k)2457 3177 y Fq(2)2512 3207 y Fs(+)18 b Fp(k)p Fs(\003)2695 3219 y Fq(1)2731 3207 y Fr(u)p Fp(k)2821 3177 y Fq(2)2885 3207 y Fs(.)639 3337 y(W)-7 b(riting)28 b(:)92 b(\005)1121 3349 y Fq(1)1159 3337 y Fr(u)p Fs(\()p Fr(s;)14 b(y)s Fs(\))22 b(=)1511 3305 y Fo(e)1542 3280 y Ff(\000)p Fj(s)1615 3263 y Fi(2)1648 3280 y Fj(=)p Fi(2)p 1511 3319 200 4 v 1545 3369 a Fo(\031)1586 3353 y Fi(1)p Fj(=)p Fi(4)1721 3337 y Fr( )s Fs(\()p Fr(y)s Fs(\))55 b(and)1346 3574 y Fr(W)1424 3586 y Fo(R)1479 3574 y Fs(\()p Fr(y)s Fs(\))23 b(=)1747 3517 y(1)p 1708 3555 120 4 v 1708 3571 a Fp(p)p 1777 3571 51 4 v 60 x Fr(\031)1851 3461 y Fl(Z)1897 3649 y Fo(I)-20 b(R)1974 3574 y Fr(e)2013 3539 y Fk(\000)p Fo(s)2096 3514 y Fi(2)2133 3574 y Fr(V)2181 3586 y Fo(R)2236 3574 y Fs(\()p Fr(s;)14 b(y)s Fs(\))p Fr(ds)23 b(;)515 3800 y Fs(w)n(e)k(obtain)g(that)h(:)1096 4004 y Fr(q)1133 4016 y Fo(\022)r(;)p Fq(1)1224 4004 y Fs(\(\005)1318 4016 y Fq(1)1355 4004 y Fr(u)p Fs(\))23 b(=)1546 3891 y Fl(Z)1592 4080 y Fo(I)-20 b(R)1656 3937 y Fl(\002)1690 4004 y Fp(j)p Fr(D)1782 4016 y Fo(y)1822 4004 y Fr( )s Fp(j)1902 3970 y Fq(2)1958 4004 y Fs(+)18 b(\(1)g Fp(\000)g Fr(W)2294 4016 y Fo(R)2349 4004 y Fs(\()p Fr(y)s Fs(\)\))p Fp(j)p Fr( )s Fp(j)2592 3970 y Fq(2)2630 3937 y Fl(\003)2664 4004 y Fr(dy)26 b(:)370 b Fs(\(2.15\))515 4229 y(W)-7 b(e)28 b(ha)n(v)n(e)e(also)994 4433 y Fr(q)1031 4445 y Fo(\022)r(;)p Fq(1)1122 4433 y Fs(\(\003)1212 4445 y Fq(1)1249 4433 y Fr(u)p Fs(\))d Fp(\025)1439 4320 y Fl(Z)1485 4509 y Fo(I)-20 b(R)1545 4490 y Fi(2)1581 4341 y Fl(h)1621 4433 y Fp(j)p Fr(D)1713 4445 y Fo(y)1752 4433 y Fs(\003)1810 4445 y Fq(1)1847 4433 y Fr(u)p Fp(j)1918 4399 y Fq(2)1973 4433 y Fs(+)2056 4341 y Fl(\020)2106 4433 y Fs(3)18 b Fp(\000)2277 4377 y Fr(C)p 2259 4414 101 4 v 2259 4490 a(R)2323 4466 y Fq(2)2370 4341 y Fl(\021)2419 4433 y Fp(j)p Fs(\003)2500 4445 y Fq(1)2537 4433 y Fr(u)p Fp(j)2608 4399 y Fq(2)2645 4341 y Fl(i)2684 4433 y Fr(dsdy)27 b(:)267 b Fs(\(2.16\))515 4657 y(But)660 4861 y Fr(q)697 4873 y Fo(\022)r(;)p Fq(1)787 4861 y Fs(\()p Fr(u)p Fs(\))24 b(=)e Fr(q)1047 4873 y Fo(\022)r(;)p Fq(1)1138 4861 y Fs(\(\005)1232 4873 y Fq(1)1269 4861 y Fr(u)p Fs(\))d(+)f Fr(q)1488 4873 y Fo(\022)r(;)p Fq(1)1578 4861 y Fs(\(\003)1668 4873 y Fq(1)1705 4861 y Fr(u)p Fs(\))h Fp(\000)f Fs(2)p Fp(R)p Fr(e)2052 4748 y Fl(Z)2097 4937 y Fo(I)-20 b(R)2157 4918 y Fi(2)2207 4861 y Fr(V)2255 4873 y Fo(R)2310 4861 y Fs(\()p Fr(s;)14 b(y)s Fs(\)\005)2556 4873 y Fq(1)2593 4861 y Fr(u:)p 2664 4794 143 4 v Fs(\003)2722 4873 y Fq(1)2759 4861 y Fr(u)o(dsdy)26 b Fs(;)146 b(\(2.17\))1926 5255 y(5)p eop %%Page: 6 6 6 5 bop 639 523 a Fs(so,)27 b(for)g(an)n(y)55 b Fr(\017)46 b Fp(2)1234 456 y Fl(\003)1269 523 y Fs(0)p Fr(;)14 b Fs(1)1390 456 y Fl(\003)1446 523 y Fr(;)556 717 y(q)593 729 y Fo(\022)r(;)p Fq(1)684 717 y Fs(\()p Fr(u)p Fs(\))23 b Fp(\025)f Fr(q)943 729 y Fo(\022)r(;)p Fq(1)1034 717 y Fs(\(\005)1128 729 y Fq(1)1166 717 y Fr(u)p Fs(\))8 b Fp(\000)1335 660 y Fs(1)p 1335 698 42 4 v 1339 774 a Fr(\017)1400 604 y Fl(Z)1447 792 y Fo(I)-20 b(R)1506 773 y Fi(2)1556 717 y Fr(V)1623 682 y Fq(2)1604 737 y Fo(R)1660 717 y Fs(\()p Fr(s;)14 b(y)s Fs(\))p Fp(j)p Fs(\005)1929 729 y Fq(1)1967 717 y Fr(u)p Fp(j)2038 682 y Fq(2)2075 717 y Fr(dsdy)33 b Fs(+)d Fr(q)2406 729 y Fo(\022)r(;)p Fq(1)2496 717 y Fs(\(\003)2586 729 y Fq(1)2623 717 y Fr(u)p Fs(\))8 b Fp(\000)g Fr(\017)p Fp(k)p Fs(\003)2918 729 y Fq(1)2953 717 y Fr(u)p Fp(k)3043 682 y Fq(2)3102 717 y Fr(:)42 b Fs(\(2.18\))515 906 y(Thanks)27 b(to)g(\(2.16\),)g(w)n (e)g(can)h(tak)n(e)e Fr(\017)d Fs(=)g(1)k(and)55 b Fr(R)28 b Fs(large)f(enough)g(suc)n(h)g(that)1371 1049 y Fr(q)1408 1061 y Fo(\022)r(;)p Fq(1)1498 1049 y Fs(\(\003)1588 1061 y Fq(1)1626 1049 y Fr(u)p Fs(\))18 b Fp(\000)g Fr(\017)p Fp(k)p Fs(\003)1941 1061 y Fq(1)1977 1049 y Fr(u)p Fp(k)2067 1015 y Fq(2)2126 1049 y Fr(>)23 b Fp(k)p Fs(\003)2314 1061 y Fq(1)2350 1049 y Fr(u)p Fp(k)2440 1015 y Fq(2)2500 1049 y Fr(;)644 b Fs(\(2.19\))515 1192 y(for)27 b(example)55 b Fr(R)28 b Fs(satisfying)f(:)37 b(3)18 b Fp(\000)1684 1159 y Fo(C)p 1669 1173 83 4 v 1669 1220 a(R)1719 1204 y Fi(2)1780 1192 y Fp(\000)g Fr(\017)23 b(>)f Fs(1)h Fr(:)515 1291 y Fs(Then,)28 b(b)n(y)f(\(2.15\))g(\(2.17\))g(and)g (\(2.19\),)g(w)n(e)g(get)h(that)1473 1434 y Fr(N)9 b Fs(\(1)p Fr(;)36 b(q)1719 1446 y Fo(\022)r(;)p Fq(1)1810 1434 y Fs(\))23 b Fp(\024)g Fr(N)9 b Fs(\(0)p Fr(;)36 b(q)2199 1446 y Fo(\022)r(;)p Fq(1)p Fo(;)p Fq(0)2343 1434 y Fs(\))23 b Fr(;)746 b Fs(\(2.20\))515 1577 y(if)1192 1698 y Fr(q)1229 1710 y Fo(\022)r(;)p Fq(1)p Fo(;)p Fq(0)1372 1698 y Fs(\()p Fr( )s Fs(\))24 b(=)1605 1585 y Fl(Z)1651 1774 y Fo(I)-20 b(R)1714 1631 y Fl(\002)1749 1698 y Fp(j)p Fr(D)1841 1710 y Fo(y)1881 1698 y Fr( )s Fp(j)1961 1664 y Fq(2)2017 1698 y Fp(\000)18 b Fr(W)2178 1710 y Fo(R;)p Fq(1)2285 1698 y Fs(\()p Fr(y)s Fs(\))p Fp(j)p Fr( )s Fp(j)2496 1664 y Fq(2)2534 1631 y Fl(\003)2568 1698 y Fr(dy)26 b(;)466 b Fs(\(2.21\))515 1876 y Fp(8)23 b Fr( )49 b Fp(2)e Fr(H)865 1846 y Fq(1)902 1876 y Fs(\()p Fr(I)-18 b(R)q Fs(\))23 b Fr(;)28 b Fs(with)1090 2061 y Fr(W)1168 2073 y Fo(R;)p Fq(1)1276 2061 y Fs(\()p Fr(y)s Fs(\))23 b(=)1544 2005 y(1)p 1505 2042 120 4 v 1505 2058 a Fp(p)p 1574 2058 51 4 v 60 x Fr(\031)1648 1948 y Fl(Z)1694 2137 y Fo(I)-20 b(R)1771 2061 y Fr(e)1810 2027 y Fk(\000)p Fo(s)1893 2002 y Fi(2)1930 1994 y Fl(\002)1964 2061 y Fr(V)2012 2073 y Fo(R)2067 2061 y Fs(\()p Fr(s;)14 b(y)s Fs(\))19 b(+)f Fr(V)2420 2027 y Fq(2)2401 2081 y Fo(R)2457 2061 y Fs(\()p Fr(s;)c(y)s Fs(\))2641 1994 y Fl(\003)2676 2061 y Fr(ds)23 b(:)515 2252 y Fs(>F)-7 b(rom)27 b(\(2.3\))g(and)h(the) g(form)n(ula)e(\(2.13\),)h(the)h(follo)n(wing)f(b)r(ound)h(holds)f(:) 1263 2508 y(0)c Fp(\024)f Fr(W)1493 2520 y Fo(R;)p Fq(1)1601 2508 y Fs(\()p Fr(y)s Fs(\))h Fp(\024)1848 2452 y Fr(C)p 1830 2489 101 4 v 1830 2565 a(R)1894 2541 y Fq(2)1955 2395 y Fl(Z)2048 2389 y Fi(2)p Fj(R)p Ff(\000)p Fj(y)12 b Fi(sin)d Fj(\022)p 2048 2403 269 3 v 2124 2436 a Fi(cos)h Fj(\022)2011 2557 y(R)p Ff(\000)p Fj(y)h Fi(sin)f Fj(\022)p 2011 2571 241 3 v 2073 2604 a Fi(cos)g Fj(\022)2344 2508 y Fr(e)2383 2474 y Fk(\000)p Fo(s)2466 2449 y Fi(2)2502 2508 y Fr(ds)24 b(;)515 2700 y Fs(\(w)n(e)30 b(used)h(the)g(fact)g (that,)h(for)e(an)n(y)g(\014xed)h Fr(y)s(;)44 b(V)2038 2712 y Fo(R)2093 2700 y Fs(\()p Fr(s;)14 b(y)s Fs(\))29 b(=)f(0)i(for)g Fr(s)h Fs(outside)g(the)g(in)n(terv)-5 b(al)515 2800 y(de\014ned)28 b(b)n(y)f Fr(R)d(<)e(s)14 b Fs(cos)f Fr(\022)21 b Fs(+)d Fr(y)e Fs(sin)e Fr(\022)26 b(<)c Fs(2)p Fr(R)q Fs(\),)27 b(so)795 2943 y(0)22 b Fp(\024)h Fr(W)1025 2955 y Fo(R;)p Fq(1)1133 2943 y Fs(\()p Fr(y)s Fs(\))1926 b(\(2.22\))859 3119 y Fp(\024)975 3062 y Fr(C)p 957 3099 101 4 v 957 3176 a(R)1021 3152 y Fq(2)1068 3119 y Fr(\037)1120 3093 y Fl(\002)1217 3138 y Fj(R)p 1164 3147 149 3 v 1164 3181 a Fi(2)11 b(sin)e Fj(\022)1323 3161 y Fo(;)1400 3138 y Fi(3)p Fj(R)p 1380 3147 111 3 v 1380 3181 a Fi(sin)h Fj(\022)1501 3093 y Fl(\003)1539 3119 y Fs(\()p Fr(y)s Fs(\))19 b(+)1848 3062 y Fr(C)p 1759 3099 244 4 v 1759 3176 a(R)c Fs(cos)e Fr(\022)2013 3051 y Fl(\002)2047 3119 y Fr(e)2086 3081 y Fk(\000)2148 3052 y Fi(\()p Fj(y)f Fi(sin)e Fj(\022)q Ff(\000)p Fj(R)p Fi(\))2432 3035 y(2)p 2148 3068 318 3 v 2232 3110 a(cos)2309 3098 y(2)2351 3110 y Fj(\022)2498 3119 y Fs(+)18 b Fr(e)2620 3081 y Fk(\000)2682 3052 y Fi(\()p Fj(y)12 b Fi(sin)e Fj(\022)q Ff(\000)p Fi(2)p Fj(R)p Fi(\))2994 3035 y(2)p 2682 3068 346 3 v 2780 3110 a(cos)2857 3098 y(2)2899 3110 y Fj(\022)3041 3051 y Fl(\003)3076 3119 y Fr(:)515 3351 y Fs(As)20 b(the)g(op)r(erators)d(on)j Fr(L)1289 3321 y Fq(2)1326 3351 y Fs(\()p Fr(I)-18 b(R)q Fs(\))p Fr(;)1558 3319 y Fq(1)p 1558 3333 34 4 v 1558 3380 a(3)1601 3351 y Fr(D)1672 3321 y Fq(2)1670 3372 y Fo(y)1713 3351 y Fp(\000)1862 3319 y Fo(C)p 1790 3333 196 4 v 1790 3380 a(R)12 b Fq(cos)f Fo(\022)1995 3351 y Fr(e)2034 3314 y Fk(\000)2096 3285 y Fi(\()p Fj(y)h Fi(sin)e Fj(\022)q Ff(\000)p Fj(R)p Fi(\))2380 3268 y(2)p 2096 3301 318 3 v 2180 3342 a(cos)2257 3330 y(2)2299 3342 y Fj(\022)2427 3351 y Fs(,)2482 3319 y Fq(1)p 2482 3333 34 4 v 2482 3380 a(3)2525 3351 y Fr(D)2596 3321 y Fq(2)2594 3372 y Fo(y)2636 3351 y Fp(\000)2785 3319 y Fo(C)p 2713 3333 196 4 v 2713 3380 a(R)i Fq(cos)f Fo(\022)2919 3351 y Fr(e)2958 3314 y Fk(\000)3019 3285 y Fi(\()p Fj(y)i Fi(sin)c Fj(\022)q Ff(\000)p Fi(2)p Fj(R)p Fi(\))3331 3268 y(2)p 3019 3301 346 3 v 3118 3342 a(cos)3194 3330 y(2)3237 3342 y Fj(\022)515 3476 y Fs(and)712 3443 y Fq(sin)794 3418 y Fi(2)838 3443 y Fo(\022)p 687 3457 211 4 v 687 3505 a Fq(3)i(cos)820 3488 y Fi(4)864 3505 y Fo(\022)907 3476 y Fr(D)978 3446 y Fq(2)976 3497 y Fo(y)1035 3476 y Fp(\000)1200 3443 y Fo(C)p 1128 3457 196 4 v 1128 3505 a(R)h Fq(cos)f Fo(\022)1333 3476 y Fr(e)1372 3446 y Fk(\000)p Fo(y)1460 3421 y Fi(2)1524 3476 y Fs(ha)n(v)n(e)27 b(the)i(same)e(sp)r (ectrum,)i(w)n(e)e(get)h(from)g(\(2.20\){\(2.22\))515 3576 y(that)1167 3675 y Fr(N)9 b Fs(\(1)p Fr(;)37 b(q)1414 3687 y Fo(\022)r(;)p Fq(1)1504 3675 y Fs(\))24 b Fp(\024)e Fs(2)p Fr(N)9 b Fs(\(0)p Fr(;)36 b(q)1935 3687 y Fo(\022)r(;)p Fq(1)p Fo(;)p Fq(1)2078 3675 y Fs(\))42 b(+)f Fr(N)9 b Fs(\(0)p Fr(;)37 b(q)2505 3687 y Fo(\022)r(;)p Fq(1)p Fo(;)p Fq(2)2648 3675 y Fs(\))23 b Fr(;)441 b Fs(\(2.23\))515 3801 y(if)982 3938 y Fr(q)1019 3950 y Fo(\022)r(;)p Fq(1)p Fo(;)p Fq(1)1162 3938 y Fs(\()p Fr( )s Fs(\))24 b(=)1394 3825 y Fl(Z)1441 4013 y Fo(I)-20 b(R)1504 3845 y Fl(h)1586 3881 y Fs(sin)1687 3847 y Fq(2)1739 3881 y Fr(\022)p 1553 3919 259 4 v 1553 3995 a Fs(3)14 b(cos)1720 3971 y Fq(4)1771 3995 y Fr(\022)1822 3938 y Fp(j)p Fr(D)1914 3950 y Fo(y)1954 3938 y Fr( )s Fp(j)2034 3903 y Fq(2)2090 3938 y Fp(\000)2272 3881 y Fr(C)p 2183 3919 244 4 v 2183 3995 a(R)g Fs(cos)f Fr(\022)2436 3938 y(e)2475 3903 y Fk(\000)p Fo(y)2563 3878 y Fi(2)2599 3938 y Fp(j)p Fr( )s Fp(j)2702 3903 y Fq(2)2740 3845 y Fl(i)2779 3938 y Fr(dy)26 b(;)255 b Fs(\(2.24\))515 4105 y(and)953 4226 y Fr(q)990 4238 y Fo(\022)r(;)p Fq(1)p Fo(;)p Fq(2)1134 4226 y Fs(\()p Fr( )s Fs(\))23 b(=)1366 4113 y Fl(Z)1412 4302 y Fo(I)-20 b(R)1476 4134 y Fl(h)1525 4170 y Fs(1)p 1525 4207 42 4 v 1525 4283 a(3)1576 4226 y Fp(j)p Fr(D)1668 4238 y Fo(y)1708 4226 y Fr( )s Fp(j)1788 4192 y Fq(2)1844 4226 y Fp(\000)1955 4170 y Fr(C)p 1937 4207 101 4 v 1937 4283 a(R)2001 4259 y Fq(2)2048 4226 y Fr(\037)2100 4201 y Fl(\002)2197 4246 y Fj(R)p 2144 4255 149 3 v 2144 4288 a Fi(2)11 b(sin)e Fj(\022)2303 4268 y Fo(;)2380 4246 y Fi(3)p Fj(R)p 2360 4255 111 3 v 2360 4288 a Fi(sin)h Fj(\022)2481 4201 y Fl(\003)2519 4226 y Fs(\()p Fr(y)s Fs(\))p Fp(j)p Fr( )s Fp(j)2730 4192 y Fq(2)2768 4134 y Fl(i)2807 4226 y Fr(dy)26 b(;)227 b Fs(\(2.25\))515 4407 y Fp(8)23 b Fr( )49 b Fp(2)e Fr(H)865 4377 y Fq(1)902 4407 y Fs(\()p Fr(I)-18 b(R)q Fs(\))23 b Fr(;)28 b Fs(for)f(some)54 b Fr(R)24 b(>)f Fs(1)k(indep)r(enden)n(t)i(of)e Fr(\022)e Fp(2)2412 4340 y Fl(\003)2447 4407 y Fs(0)p Fr(;)2536 4374 y Fo(\031)p 2536 4388 41 4 v 2540 4436 a Fq(2)2587 4340 y Fl(\002)2644 4407 y Fr(:)639 4533 y Fs(It)j(is)g(w)n(ell)f(kno)n (wn)g(that)h Fr(N)9 b Fs(\(0)p Fr(;)36 b(q)1670 4545 y Fo(\022)r(;)p Fq(1)p Fo(;)p Fq(1)1814 4533 y Fs(\))23 b Fp(\024)g Fr(C)2032 4500 y Fq(cos)2121 4475 y Fi(3)p Fj(=)p Fi(2)2223 4500 y Fo(\022)p 2032 4514 225 4 v 2081 4562 a Fq(sin)11 b Fo(\022)515 4642 y Fs(and)27 b Fr(N)9 b Fs(\(0)p Fr(;)37 b(q)923 4654 y Fo(\022)r(;)p Fq(1)p Fo(;)p Fq(2)1066 4642 y Fs(\))23 b Fp(\024)1256 4610 y Fo(C)p 1219 4624 127 4 v 1219 4671 a Fq(sin)11 b Fo(\022)1355 4642 y Fr(;)28 b Fs(so)1595 4836 y Fr(N)9 b Fs(\(1)p Fr(;)37 b(q)1842 4848 y Fo(\022)r(;)p Fq(1)1932 4836 y Fs(\))24 b Fp(\024)2131 4779 y Fr(C)p 2085 4816 158 4 v 2085 4892 a Fs(sin)14 b Fr(\022)2275 4836 y(:)869 b Fs(\(2.26\))515 5006 y(W)-7 b(e)28 b(conclude)f(from)g(\(2.9\),)h (\(2.11\))f(and)g(\(2.26\))g(that)h(the)g(estimate)f(\(2.1\))h(is)f(v) -5 b(alid.)1926 5255 y(6)p eop %%Page: 7 7 7 6 bop 515 530 a Ft(3)134 b(Some)46 b(prop)t(erties)f(of)g Fe(\026)p Fd(\()p Fe(\030)6 b Fd(\))515 712 y Fs(The)27 b(prop)r(erties)g(of)h(the)g(\014rst)f(eigen)n(v)-5 b(alue,)54 b Fr(\026)p Fs(\()p Fr(\030)t Fs(\))p Fr(;)29 b Fs(of)e(the)h(Neumann)g (op)r(erator)e(on)639 811 y Fr(L)696 781 y Fq(2)733 811 y Fs(\()p Fr(I)-18 b(R)847 823 y Fq(+)902 811 y Fs(\))24 b Fr(;)41 b(Q)1088 823 y Fo(\030)1147 811 y Fs(=)23 b Fr(D)1306 781 y Fq(2)1304 832 y Fo(t)1362 811 y Fs(+)18 b(\()p Fr(t)g Fp(\000)g Fr(\030)t Fs(\))1680 781 y Fq(2)1741 811 y Fr(;)515 911 y Fs(can)27 b(b)r(e)h(found)g(in)g([DaHe],)g(or)e (in)i([BeSt],)g(or)f(in)h([HeMo1)o(].)515 1010 y(The)f(main)h(one)f(is) 1168 1162 y Fr(\026)47 b Fp(2)f Fr(C)1431 1132 y Fk(1)1502 1162 y Fs(\()p Fr(I)-18 b(R)q Fs(\))p Fr(;)1168 1261 y(\026)1218 1231 y Fk(0)1242 1261 y Fs(\()p Fr(\030)t Fs(\))23 b Fp(6)p Fs(=)g(0)55 b(if)35 b Fr(\030)27 b Fp(6)p Fs(=)22 b Fr(\030)1823 1273 y Fq(0)1861 1261 y Fr(;)1168 1361 y(\026)p Fs(\()p Fr(\030)t Fs(\))i Fr(>)f Fs(1)p Fr(;)69 b Fs(if)62 b Fr(\030)27 b(<)c Fs(0)p Fr(;)1168 1461 y(\026)p Fs(\()p Fr(\030)t Fs(\))h Fr(<)f Fs(1)p Fr(;)69 b Fs(if)62 b Fr(\030)27 b(>)c Fs(0)p Fr(;)1168 1560 y Fs(lim)1284 1572 y Fo(\030)r Fk(!\0001)1518 1560 y Fr(\026)p Fs(\()p Fr(\030)t Fs(\))h(=)f(+)p Fp(1)p Fr(;)68 b Fs(lim)2139 1572 y Fo(\030)r Fk(!)p Fq(+)p Fk(1)2372 1560 y Fr(\026)p Fs(\()p Fr(\030)t Fs(\))24 b(=)f(1)f(;)3208 1362 y(\(3.1\))515 1739 y(\()p Fr(\030)583 1751 y Fq(0)644 1739 y Fr(>)g Fs(0)27 b(is)h(suc)n(h)f(that)56 b Fr(\030)1319 1709 y Fq(2)1315 1760 y(0)1379 1739 y Fs(=)23 b Fr(\026)p Fs(\()p Fr(\030)1585 1751 y Fq(0)1623 1739 y Fs(\))g(\))p Fr(:)515 1839 y Fs(Let)k Fr(')717 1851 y Fo(\030)782 1839 y Fs(b)r(e)h(a)f(normalized)g(eigenfunction)g (asso)r(ciated)g(to)55 b Fr(\026)p Fs(\()p Fr(\030)t Fs(\))23 b(:)1086 2012 y Fp(k)p Fr(')1182 2024 y Fo(\030)1218 2012 y Fp(k)1260 2027 y Fo(L)1306 2010 y Fi(2)1337 2027 y Fq(\()p Fo(I)-20 b(R)1423 2035 y Fi(+)1470 2027 y Fq(\))1523 2012 y Fs(=)22 b(1)p Fr(;)69 b(')1798 1978 y Fk(0)1798 2032 y Fo(\030)1835 2012 y Fs(\(0\))23 b(=)g(0)p Fr(;)96 b(Q)2279 2024 y Fo(\030)2315 2012 y Fr(')2369 2024 y Fo(\030)2429 2012 y Fs(=)22 b Fr(\026)p Fs(\()p Fr(\030)t Fs(\))p Fr(')2724 2024 y Fo(\030)2785 2012 y Fr(;)400 b Fs(\(3.2\))515 2185 y(then)1449 2285 y Fr(\026)1499 2250 y Fk(0)1523 2285 y Fs(\()p Fr(\030)t Fs(\))23 b(=)g Fp(\000)p Fs(\()p Fr(\026)p Fs(\()p Fr(\030)t Fs(\))c Fp(\000)f Fr(\030)2131 2250 y Fq(2)2168 2285 y Fs(\))p Fr(')2254 2250 y Fq(2)2254 2305 y Fo(\030)2292 2285 y Fs(\(0\))23 b Fr(:)764 b Fs(\(3.3\))515 2428 y(It)28 b(is)f(easy)g(to)g(see)g(that)h Fr(')1340 2440 y Fo(\030)1405 2428 y Fs(is)f(exp)r(onen)n(tially)g(decreasing.)35 b(More)27 b(precisely)g(w)n(e)g(ha)n(v)n(e)515 2601 y Fh(Lemma)k(3.1)70 b Fg(F)-6 b(or)30 b(any)60 b Fr(\030)27 b(>)c Fs(1)29 b Fg(and)60 b Fr(\021)50 b Fp(2)1972 2534 y Fl(\003)2007 2601 y Fs(0)p Fr(;)14 b Fs(1)2128 2534 y Fl(\002)2184 2601 y Fr(;)738 2712 y Fl(Z)785 2900 y Fo(I)-20 b(R)844 2909 y Fi(+)895 2757 y Fl(\002)930 2825 y Fr(\021)s Fs(\()p Fr(t)18 b Fp(\000)g Fr(\030)t Fs(\))1209 2790 y Fq(2)1266 2825 y Fp(\000)g Fr(\026)p Fs(\()p Fr(\030)t Fs(\))1503 2757 y Fl(\003)1538 2858 y Fq(+)1593 2825 y Fr(e)1632 2790 y Fq(\(1)p Fk(\000)p Fo(\021)r Fq(\))1805 2765 y Fi(1)p Fj(=)p Fi(2)1896 2790 y Fq(\()p Fo(t)p Fk(\000)p Fo(\030)r Fq(\))2057 2765 y Fi(2)2094 2825 y Fp(j)p Fr(')2171 2837 y Fo(\030)2207 2825 y Fs(\()p Fr(t)p Fs(\))p Fp(j)2324 2790 y Fq(2)2362 2825 y Fr(dt)24 b Fp(\024)e Fr(\026)p Fs(\()p Fr(\030)t Fs(\))p Fr(e)2739 2790 y Fo(\026)p Fq(\()p Fo(\030)r Fq(\))p Fo(=\021)2962 2825 y Fs(;)223 b(\(3.4\))515 3055 y Fg(c)l(onse)l(quently,)30 b(ther)l(e)f(exists)59 b Fr(C)1540 3067 y Fq(0)1601 3055 y Fr(>)23 b Fs(1)29 b Fg(such)h(that)1515 3239 y Fs(1)18 b Fp(\000)g Fr(C)1717 3251 y Fq(0)1754 3239 y Fr(e)1793 3204 y Fk(\000j)p Fo(\030)r Fk(j)1917 3179 y Fi(2)1949 3204 y Fo(=C)2031 3212 y Fi(0)2090 3239 y Fp(\024)23 b Fr(\026)p Fs(\()p Fr(\030)t Fs(\))h Fr(:)829 b Fs(\(3.5\))515 3412 y Fg(Mor)l(e)30 b(over,)h(if)61 b Fr(\026)1100 3424 y Fq(2)1137 3412 y Fs(\()p Fr(\030)t Fs(\))30 b Fg(is)g(the)g(se)l(c)l(ond)g(eigenvalue)h(of)61 b Fr(Q)2356 3424 y Fo(\030)2415 3412 y Fr(;)29 b Fg(then)h(we)g(have)h (also)1519 3603 y Fs(3)18 b Fp(\000)g Fr(C)1721 3615 y Fq(0)1759 3603 y Fr(e)1798 3569 y Fk(\000j)p Fo(\030)r Fk(j)1922 3544 y Fi(2)1954 3569 y Fo(=C)2036 3577 y Fi(0)2095 3603 y Fp(\024)k Fr(\026)2232 3615 y Fq(2)2270 3603 y Fs(\()p Fr(\030)t Fs(\))834 b(\(3.6\))515 3776 y Fg(and)30 b(then)1313 3898 y Fp(k)1385 3842 y Fr(@)p 1365 3879 89 4 v 1365 3955 a(@)5 b(\030)1463 3898 y(')1517 3910 y Fo(\030)1554 3898 y Fs(\()23 b Fr(:)g Fs(\))p Fp(k)1729 3913 y Fo(L)1775 3897 y Fi(2)1807 3913 y Fq(\()p Fo(I)-20 b(R)1893 3921 y Fi(+)1939 3913 y Fq(\))1992 3898 y Fp(\024)23 b Fr(C)2139 3910 y Fq(0)2177 3898 y Fr(;)73 b Fp(8)23 b Fr(\030)k(>)c Fs(1)g Fr(:)627 b Fs(\(3.7\))515 4115 y Fh(Pro)s(of)50 b Fs(W)-7 b(e)25 b(pro)r(ceed)f(as)h(in)g([He1])g(to)g (get)f(\(3.4\).)36 b(F)-7 b(or)25 b(an)n(y)f(Lipsc)n(hitz)h(and)f(real) g(function)515 4215 y(\010)p Fr(;)j Fs(with)i(compact)e(supp)r(ort,)891 4388 y Fp(k)p Fr(D)1002 4400 y Fo(t)1030 4388 y Fs(\()p Fr(e)1101 4353 y Fq(\010)1153 4388 y Fr(')1207 4400 y Fo(\030)1244 4388 y Fs(\))p Fp(k)1318 4353 y Fq(2)1318 4410 y Fo(L)1364 4394 y Fi(2)1395 4410 y Fq(\()p Fo(I)-20 b(R)1481 4418 y Fi(+)1527 4410 y Fq(\))1581 4388 y Fs(=)22 b Fp(h)1700 4321 y Fl(\002)1735 4388 y Fr(\026)p Fs(\()p Fr(\030)t Fs(\))d Fp(\000)f Fs(\()p Fr(t)h Fp(\000)f Fr(\030)t Fs(\))2227 4353 y Fq(2)2283 4388 y Fs(+)g(\(\010)2458 4353 y Fk(0)2482 4388 y Fs(\))2514 4353 y Fq(2)2551 4321 y Fl(\003)2586 4388 y Fr(e)2625 4353 y Fq(\010)2676 4388 y Fr(')2730 4400 y Fo(\030)2767 4388 y Fp(j)p Fr(e)2829 4353 y Fq(\010)2880 4388 y Fr(')2934 4400 y Fo(\030)2971 4388 y Fp(i)515 4561 y Fs(so)515 4734 y Fp(h)547 4666 y Fl(\002)582 4734 y Fs(\()p Fr(t)9 b Fp(\000)g Fr(\030)t Fs(\))799 4699 y Fq(2)844 4734 y Fp(\000)g Fr(\026)p Fs(\()p Fr(\030)t Fs(\))g Fp(\000)g Fs(\(\010)1247 4699 y Fk(0)1269 4734 y Fs(\))1301 4699 y Fq(2)1339 4666 y Fl(\003)1373 4767 y Fq(+)1428 4734 y Fr(e)1467 4699 y Fq(\010)1519 4734 y Fr(')1573 4746 y Fo(\030)1609 4734 y Fp(j)p Fr(e)1671 4699 y Fq(\010)1722 4734 y Fr(')1776 4746 y Fo(\030)1813 4734 y Fp(i)47 b(\024)e(h)2034 4666 y Fl(\002)2069 4734 y Fr(\026)p Fs(\()p Fr(\030)t Fs(\))9 b Fp(\000)g Fs(\()p Fr(t)g Fp(\000)g Fr(\030)t Fs(\))2523 4699 y Fq(2)2568 4734 y Fs(+)g(\(\010)2734 4699 y Fk(0)2757 4734 y Fs(\))2789 4699 y Fq(2)2826 4666 y Fl(\003)2861 4767 y Fq(+)2916 4734 y Fr(e)2955 4699 y Fq(\010)3006 4734 y Fr(')3060 4746 y Fo(\030)3097 4734 y Fp(j)p Fr(e)3159 4699 y Fq(\010)3210 4734 y Fr(')3264 4746 y Fo(\030)3301 4734 y Fp(i)23 b Fr(:)515 4907 y Fs(This)30 b(estimate)g(is)f(still)i (v)-5 b(alid)30 b(for)f(\010)h(with)h(non)f(compact)f(supp)r(ort,)h (pro)n(vided)f(that)i(the)515 5006 y(righ)n(t)c(hand)g(side)h(of)f(the) h(inequalit)n(y)f(is)h(\014nite;)g(so)f(w)n(e)g(can)g(tak)n(e)1926 5255 y(7)p eop %%Page: 8 8 8 7 bop 667 523 a Fs(\010\()p Fr(t)p Fs(\))24 b(=)e(\(1)c Fp(\000)g Fr(\021)s Fs(\))1183 493 y Fq(1)p Fo(=)p Fq(2)1288 523 y Fs(\()p Fr(t)h Fp(\000)f Fr(\030)t Fs(\))1524 493 y Fq(2)1562 523 y Fr(=)p Fs(2)26 b(to)i(get)f(\(3.4\).)515 623 y(No)n(w,)g(let)55 b Fr(\037)28 b Fs(b)r(e)g(a)f(smo)r(oth)h (cut-o\013)f(function)h(on)g Fr(I)-18 b(R)24 b Fs(:)1467 800 y Fr(\037)46 b Fp(2)g Fr(C)1731 770 y Fk(1)1802 800 y Fs(\()p Fr(I)-18 b(R)q Fs(\))p Fr(;)1467 899 y(\037)p Fs(\()p Fr(t)p Fs(\))23 b(=)g(1)p Fr(;)69 b Fs(if)80 b Fp(\000)18 b Fs(1)23 b Fr(<)g(t)g(<)f Fs(1)p Fr(;)1467 999 y(\037)p Fs(\()p Fr(t)p Fs(\))h(=)g(0)p Fr(;)69 b Fs(if)62 b Fp(j)p Fr(t)p Fp(j)23 b Fr(>)g Fs(2)p Fr(;)1467 1099 y Fs(0)f Fp(\024)h Fr(\037)g Fp(\024)g Fs(1)f Fr(:)3208 950 y Fs(\(3.8\))515 1297 y(If)28 b Fr(\030)f(>)c Fs(1)p Fr(;)k Fs(w)n(e)g(de\014ne)h(the)g(function)67 b Fl(e)-57 b Fr(')1753 1309 y Fq(1)p Fo(;\030)1842 1297 y Fs(\()p Fr(t)p Fs(\))24 b(=)e Fr(\037)p Fs(\(4)2183 1260 y Fo(t)p Fk(\000)p Fo(\030)p 2183 1278 110 4 v 2222 1326 a(\030)2302 1297 y Fs(\))p Fr(')2388 1309 y Fo(\030)2425 1297 y Fs(\()p Fr(t)p Fs(\))p Fr(:)515 1397 y Fs(So)767 1602 y Fp(k)p Fr(D)878 1614 y Fo(t)918 1602 y Fl(e)-57 b Fr(')961 1614 y Fq(1)p Fo(;\030)1050 1602 y Fp(k)1092 1568 y Fq(2)1148 1602 y Fs(+)18 b Fp(k)p Fs(\()p Fr(t)g Fp(\000)g Fr(\030)t Fs(\))11 b Fl(e)-57 b Fr(')1562 1614 y Fq(1)p Fo(;\030)1652 1602 y Fp(k)1694 1568 y Fq(2)1754 1602 y Fs(=)22 b Fr(\026)p Fs(\()p Fr(\030)t Fs(\)\))p Fp(k)11 b Fl(e)-57 b Fr(')2123 1614 y Fq(1)p Fo(;\030)2213 1602 y Fp(k)2255 1568 y Fq(2)2310 1602 y Fs(+)2403 1546 y(16)p 2403 1583 84 4 v 2406 1659 a Fr(\030)2446 1635 y Fq(2)2496 1602 y Fp(k)p Fr(\037)2590 1568 y Fk(0)2613 1602 y Fs(\(4)2697 1546 y Fr(t)18 b Fp(\000)g Fr(\030)p 2697 1583 172 4 v 2763 1659 a(\030)2878 1602 y Fs(\))11 b Fl(e)-57 b Fr(')2964 1614 y Fo(\030)3001 1602 y Fp(k)3043 1568 y Fq(2)3103 1602 y Fr(:)515 1834 y Fs(As)35 b Fl(e)-57 b Fr(')688 1846 y Fq(1)p Fo(;\030)801 1834 y Fs(is)24 b(of)f(compact)h(supp)r(ort)f(and)h(the)g(\014rst)f (eigen)n(v)-5 b(alue)23 b(of)h Fr(D)2622 1804 y Fq(2)2620 1855 y Fo(t)2670 1834 y Fs(+)11 b(\()p Fr(t)g Fp(\000)g Fr(\030)t Fs(\))2967 1804 y Fq(2)3027 1834 y Fs(on)24 b Fr(L)3196 1804 y Fq(2)3233 1834 y Fs(\()p Fr(I)-18 b(R)q Fs(\))515 1934 y(is)55 b(1)22 b Fr(;)28 b Fs(then)1117 2072 y Fp(k)11 b Fl(e)-57 b Fr(')1213 2084 y Fq(1)p Fo(;\030)1302 2072 y Fp(k)1344 2038 y Fq(2)1404 2072 y Fp(\024)23 b Fr(\026)p Fs(\()p Fr(\030)t Fs(\)\))p Fp(k)11 b Fl(e)-57 b Fr(')1774 2084 y Fq(1)p Fo(;\030)1864 2072 y Fp(k)1906 2038 y Fq(2)1961 2072 y Fs(+)2054 2016 y(16)p 2054 2053 84 4 v 2057 2129 a Fr(\030)2097 2105 y Fq(2)2147 2072 y Fp(k)p Fr(\037)2241 2038 y Fk(0)2263 2072 y Fs(\(4)2347 2016 y Fr(t)19 b Fp(\000)f Fr(\030)p 2347 2053 172 4 v 2413 2129 a(\030)2529 2072 y Fs(\))p Fr(')2615 2084 y Fo(\030)2652 2072 y Fp(k)2694 2038 y Fq(2)2754 2072 y Fs(;)515 2265 y(then)53 b(w)n(e)f(use)g(the)h(estimate)f(\(3.4\))g (to)g(see)g(that,)59 b(for)52 b(some)g(constan)n(t)g Fr(C)70 b(>)64 b Fs(1,)515 2376 y(1)22 b Fp(\024)h Fr(\026)p Fs(\()p Fr(\030)t Fs(\))c(+)f Fr(C)6 b(e)1027 2346 y Fk(\000)p Fo(\030)1111 2321 y Fi(2)1144 2346 y Fo(=C)1233 2376 y Fs(:)37 b(the)28 b(estimate)g(\(3.5\))f(follo)n(ws.)639 2476 y(If)h Fr(\026)772 2488 y Fq(2)809 2476 y Fs(\()p Fr(\030)t Fs(\))h(is)e(the)h(second)f(eigen)n(v)-5 b(alue)26 b(of)56 b Fr(Q)2021 2488 y Fo(\030)2084 2476 y Fs(and)f Fr(')2327 2488 y Fq(2)p Fo(;\030)2444 2476 y Fs(the)28 b(asso)r(ciated)e(normalized)515 2576 y(eigenfunction,)h(then)i(w)n(e)e (ha)n(v)n(e)f(in)i(the)g(same)f(w)n(a)n(y)-7 b(,)27 b(for)g(an)n(y)f (real)h(function)h(\010)p Fr(;)598 2783 y Fp(h)630 2716 y Fl(\002)665 2783 y Fs(\()p Fr(t)18 b Fp(\000)g Fr(\030)t Fs(\))900 2749 y Fq(2)956 2783 y Fp(\000)h Fr(\026)1090 2795 y Fq(2)1127 2783 y Fs(\()p Fr(\030)t Fs(\))g Fp(\000)f Fs(\(\010)1425 2749 y Fk(0)1449 2783 y Fs(\))1481 2749 y Fq(2)1518 2716 y Fl(\003)1553 2816 y Fq(+)1635 2783 y Fr(e)1674 2749 y Fq(\010)1726 2783 y Fr(')1780 2795 y Fq(2)p Fo(;\030)1869 2783 y Fp(j)p Fr(e)1931 2749 y Fq(\010)1982 2783 y Fr(')2036 2795 y Fq(2)p Fo(;\030)2126 2783 y Fp(i)1625 2931 y(\024)23 b(h)1745 2863 y Fl(\002)1779 2931 y Fr(\026)1829 2943 y Fq(2)1867 2931 y Fs(\()p Fr(\030)t Fs(\))c Fp(\000)f Fs(\()p Fr(t)h Fp(\000)f Fr(\030)t Fs(\))2309 2896 y Fq(2)2365 2931 y Fs(+)g(\(\010)2540 2896 y Fk(0)2563 2931 y Fs(\))2595 2896 y Fq(2)2633 2863 y Fl(\003)2667 2964 y Fq(+)2750 2931 y Fr(e)2789 2896 y Fq(\010)2840 2931 y Fr(')2894 2943 y Fq(2)p Fo(;\030)2984 2931 y Fp(j)p Fr(e)3046 2896 y Fq(\010)3097 2931 y Fr(')3151 2943 y Fq(2)p Fo(;\030)3241 2931 y Fp(i)p Fs(;)515 3139 y(so)27 b Fr(')671 3151 y Fq(2)p Fo(;\030)788 3139 y Fs(is)g(exp)r(onen)n(tially)g(decreasing)f(as)h Fr(')1941 3151 y Fo(\030)1978 3139 y Fr(;)h Fs(and)f(if)67 b Fl(e)-57 b Fr(')2348 3151 y Fq(2)p Fo(;\030)2437 3139 y Fs(\()p Fr(t)p Fs(\))24 b(=)e Fr(\037)p Fs(\(4)2778 3102 y Fo(t)p Fk(\000)p Fo(\030)p 2778 3120 110 4 v 2817 3168 a(\030)2897 3139 y Fs(\))p Fr(')2983 3151 y Fq(2)p Fo(;\030)3073 3139 y Fs(\()p Fr(t)p Fs(\))p Fr(;)515 3239 y Fs(then)1108 3339 y Fp(k)p Fr(D)1219 3351 y Fo(t)1259 3339 y Fl(e)-57 b Fr(')1302 3351 y Fq(2)p Fo(;\030)1391 3339 y Fp(k)1433 3304 y Fq(2)1488 3339 y Fs(+)18 b Fp(k)p Fs(\()p Fr(t)h Fp(\000)f Fr(\030)t Fs(\))11 b Fl(e)-57 b Fr(')1903 3351 y Fq(2)p Fo(;\030)1992 3339 y Fp(k)2034 3304 y Fq(2)2094 3339 y Fp(\024)23 b Fr(\026)2232 3351 y Fq(2)2269 3339 y Fs(\()p Fr(\030)t Fs(\))c(+)f Fr(C)6 b(e)2579 3304 y Fk(\000)p Fo(\030)2663 3279 y Fi(2)2696 3304 y Fo(=C)515 3488 y Fs(and)1503 3578 y Fp(j)23 b(k)11 b Fl(e)-57 b Fr(')1645 3590 y Fq(2)p Fo(;\030)1734 3578 y Fp(k)18 b(\000)g Fs(1)p Fp(j)23 b(\024)f Fr(C)6 b(e)2156 3548 y Fk(\000)p Fo(\030)2240 3523 y Fi(2)2273 3548 y Fo(=C)2363 3578 y Fr(;)1503 3690 y Fp(jh)11 b Fl(e)-57 b Fr(')1612 3702 y Fq(1)p Fo(;\030)1702 3690 y Fp(j)11 b Fl(e)-57 b Fr(')1779 3702 y Fq(2)p Fo(;\030)1868 3690 y Fp(ij)24 b(\024)e Fr(C)6 b(e)2138 3660 y Fk(\000)p Fo(\030)2222 3635 y Fi(2)2255 3660 y Fo(=C)2368 3690 y Fr(:)3208 3629 y Fs(\(3.9\))515 3860 y(Those)24 b(estimates,)i(\(3.5\))e(and)h(the)h (min-max)f(principle)g(sho)n(w)f(that)h Fr(\026)2754 3872 y Fq(2)2791 3860 y Fs(\()p Fr(\030)t Fs(\))13 b(+)g Fr(C)6 b(e)3090 3830 y Fk(\000)p Fo(\030)3174 3805 y Fi(2)3208 3830 y Fo(=C)3323 3860 y Fs(is)515 3960 y(greater)26 b(then)i(the)g(second)f(eigen)n(v)-5 b(alue)26 b(of)i Fr(D)1959 3930 y Fq(2)1957 3980 y Fo(t)2015 3960 y Fs(+)18 b(\()p Fr(t)g Fp(\000)g Fr(\030)t Fs(\))2333 3930 y Fq(2)2399 3960 y Fs(on)27 b Fr(L)2571 3930 y Fq(2)2608 3960 y Fs(\()p Fr(I)-18 b(R)q Fs(\))p Fr(;)515 4071 y Fs(so)27 b(3)22 b Fp(\024)h Fr(\026)819 4083 y Fq(2)856 4071 y Fs(\()p Fr(\030)t Fs(\))c(+)f Fr(C)6 b(e)1166 4041 y Fk(\000)p Fo(\030)1250 4016 y Fi(2)1283 4041 y Fo(=C)1396 4071 y Fs(:)27 b(this)h(ends)g(the)g(pro)r(of)f(of)g(the)h(Lemma.)639 4289 y(W)-7 b(e)28 b(recall)f(that)h(some)f(elemen)n(tary)f(tec)n (hnique)i(of)f(p)r(erturbation)h(sho)n(ws)e(that)1427 4454 y Fr(@)p 1407 4491 89 4 v 1407 4567 a(@)5 b(\030)1506 4510 y(')1560 4522 y Fo(\030)1597 4510 y Fs(\()p Fr(t)p Fs(\))23 b(=)g(2)1844 4443 y Fl(\002)1878 4510 y Fr(Q)1944 4522 y Fo(\030)1998 4510 y Fp(\000)18 b Fr(\026)p Fs(\()p Fr(\030)t Fs(\))2235 4443 y Fl(\003)2271 4460 y Fk(\000)p Fq(1)2360 4510 y Fr( )2414 4522 y Fo(\030)2473 4510 y Fr(;)671 b Fs(\(3.10\))515 4732 y(with)1169 4831 y Fr( )1223 4843 y Fo(\030)1260 4831 y Fs(\()p Fr(t)p Fs(\))23 b(=)g(\()p Fr(t)c Fp(\000)f Fr(\030)t Fs(\))p Fr(')1755 4843 y Fo(\030)1792 4831 y Fs(\()p Fr(t)p Fs(\))h Fp(\000)f(h)p Fs(\()p Fr(t)h Fp(\000)f Fr(\030)t Fs(\))p Fr(')2310 4843 y Fo(\030)2347 4831 y Fp(j)p Fr(')2424 4843 y Fo(\030)2461 4831 y Fp(i)p Fr(')2547 4843 y Fo(\030)2584 4831 y Fs(\()p Fr(t)p Fs(\))23 b Fr(:)515 4981 y Fs(As)k Fp(k)p Fr(D)750 4951 y Fq(2)748 5001 y Fo(t)787 4981 y Fr(')841 4993 y Fo(\030)878 4981 y Fp(k)18 b Fs(+)g Fp(k)p Fs(\()p Fr(t)g Fp(\000)g Fr(\030)t Fs(\))p Fr(')1352 4993 y Fo(\030)1389 4981 y Fp(k)23 b Fs(=)f Fr(\026)p Fs(\()p Fr(\030)t Fs(\))p Fp(k)p Fr(')1791 4993 y Fo(\030)1828 4981 y Fp(k)h Fs(=)g Fr(\026)p Fs(\()p Fr(\030)t Fs(\))g Fr(;)28 b Fs(w)n(e)f(get)h(easily)e(\(3.7\))i(from)f (\(3.10\).)1926 5255 y(8)p eop %%Page: 9 9 9 8 bop 515 523 a Fh(Remark)32 b(3.2)40 b Fg(As)29 b Fr(\030)34 b Fg(go)l(es)d(to)e Fp(\0001)p Fr(;)h Fg(the)g(fol)t(lowing) i(asymptotic)f(exp)l(ansions)f(hold)1241 706 y Fr(\026)p Fs(\()p Fr(\030)t Fs(\))24 b(=)e Fr(\030)1546 671 y Fq(2)1625 706 y Fs(+)41 b(\()p Fp(\000)p Fs(2)p Fr(\030)t Fs(\))1942 671 y Fq(2)p Fo(=)p Fq(3)2046 706 y Fr(\032)2089 718 y Fq(1)2168 706 y Fs(+)g Fh(O)p Fs(\()p Fp(j)p Fr(\030)t Fp(j)2464 671 y Fk(\000)p Fq(2)p Fo(=)p Fq(3)2621 706 y Fs(\))514 b(\(3.11\))515 888 y Fg(and)1222 988 y Fr(\026)1272 1000 y Fq(2)1309 988 y Fs(\()p Fr(\030)t Fs(\))24 b(=)f Fr(\030)1565 954 y Fq(2)1644 988 y Fs(+)41 b(\()p Fp(\000)p Fs(2)p Fr(\030)t Fs(\))1961 954 y Fq(2)p Fo(=)p Fq(3)2065 988 y Fr(\032)2108 1000 y Fq(2)2187 988 y Fs(+)g Fh(O)p Fs(\()p Fp(j)p Fr(\030)t Fp(j)2483 954 y Fk(\000)p Fq(2)p Fo(=)p Fq(3)2639 988 y Fs(\))496 b(\(3.12\))515 1137 y Fg(if)38 b Fs(\()p Fr(\032)678 1149 y Fo(j)713 1137 y Fs(\))g Fg(is)g(the)g(incr)l(e)l(asing)g(se)l(quenc)l(e)f(of)i (eigenvalues)f(of)h(the)f(Neumann)e(op)l(er)l(ator)j(on)515 1237 y Fr(L)572 1207 y Fq(2)608 1237 y Fs(\()p Fr(I)-18 b(R)723 1249 y Fq(+)778 1237 y Fs(\))30 b Fg(asso)l(ciate)l(d)h(to)f Fr(D)1399 1207 y Fq(2)1397 1258 y Fo(t)1454 1237 y Fs(+)18 b Fr(t)23 b(:)515 1420 y Fs(As)31 b(a)f(matter)g(of)h(fact,)h(if)f Fr(\030)h(<)c Fp(\000)p Fs(1)p Fr(;)i Fs(w)n(e)g(ha)n(v)n(e)g(easily)91 b Fr(\033)s Fs(\()p Fr(Q)2438 1432 y Fo(\030)2475 1420 y Fs(\))29 b(=)e Fr(\030)2668 1390 y Fq(2)2706 1352 y Fl(\002)2740 1420 y Fs(1)49 b(+)f Fr(\033)s Fs(\()p Fr(Q)3092 1432 y Fo(\030)r(;)p Fq(2)3182 1420 y Fs(\))3214 1352 y Fl(\003)3277 1420 y Fr(;)31 b Fs(if)515 1519 y Fr(Q)581 1531 y Fo(\030)r(;)p Fq(2)702 1519 y Fs(is)g(the)i(Neumann)f(op)r (erator)e(on)i Fr(L)1833 1489 y Fq(2)1870 1519 y Fs(\()p Fr(I)-18 b(R)1984 1531 y Fq(+)2039 1519 y Fs(\))32 b(asso)r(ciated)f (to)h Fr(h)2657 1489 y Fq(2)2694 1519 y Fr(D)2765 1489 y Fq(2)2763 1540 y Fo(t)2823 1519 y Fs(+)21 b(2)p Fr(t)g Fs(+)g Fr(t)3118 1489 y Fq(2)3218 1519 y Fs(with)515 1619 y Fr(h)i Fs(=)f Fr(\030)713 1589 y Fk(\000)p Fq(2)803 1619 y Fr(:)639 1719 y Fs(Using)35 b(semi-classical)e(metho)r(d)i(as)f (in)h([Si1])f(or)g(in)h([HeSj1])g(\(or)f(in)h([He1]\),)i(w)n(e)d(get) 515 1818 y(easily)20 b(that)i(the)g Fr(N)9 b Fs(-\014rst)21 b(eigen)n(v)-5 b(alues)20 b(of)i Fr(Q)1895 1830 y Fo(\030)r(;)p Fq(2)2005 1818 y Fs(are)e(equal)h(to)h(\(2)p Fr(h)p Fs(\))2600 1788 y Fq(2)p Fo(=)p Fq(3)2704 1818 y Fr(\032)2747 1830 y Fq(1)2784 1818 y Fr(;)37 b(:)14 b(:)g(:)g(;)37 b Fs(\(2)p Fr(h)p Fs(\))3169 1788 y Fq(2)p Fo(=)p Fq(3)3273 1818 y Fr(\032)3316 1830 y Fo(N)515 1918 y Fs(mo)r(dulo)27 b(a)g Fh(O)p Fs(\()p Fr(h)1033 1888 y Fq(4)p Fo(=)p Fq(3)1138 1918 y Fs(\))c Fr(:)515 2192 y Ft(4)134 b(The)45 b(case)g(of)h(small)g Fe(\022)515 2374 y Fs(W)-7 b(e)27 b(are)e(still)i(in)n(v)n(estigating)e (the)i(sp)r(ectrum)g(of)g(the)g(op)r(erator)d Fr(P)2567 2386 y Fo(\022)2605 2374 y Fs(,)j(de\014ned)g(in)g(the)g(in)n(tro-)515 2474 y(duction)h(as)e(follo)n(wing)h(:)1294 2673 y Fr(P)1347 2685 y Fo(\022)1408 2673 y Fs(=)c Fr(D)1567 2639 y Fq(2)1565 2694 y Fo(t)1622 2673 y Fs(+)18 b Fr(D)1776 2639 y Fq(2)1774 2694 y Fo(x)1834 2673 y Fs(+)g(\()p Fr(t)c Fs(cos)f Fr(\022)21 b Fp(\000)d Fr(x)c Fs(sin)g Fr(\022)r Fs(\))2511 2639 y Fq(2)2577 2673 y Fr(:)515 2823 y Fs(P)n(erforming)19 b(the)j(scaling)f(\()p Fr(t;)14 b(x)p Fs(\))24 b Fp(!)f Fs(\()p Fr(t)1713 2752 y Fp(p)p 1782 2752 167 4 v 71 x Fs(cos)13 b Fr(\022)r(;)h Fp(\000)2071 2790 y Fo(xsin\022)p 2060 2804 189 4 v 2060 2812 a Fk(p)p 2114 2812 134 3 v 2114 2862 a Fq(cos)e Fo(\022)2258 2823 y Fs(\),)23 b(w)n(e)e(observ)n(e)f(that)i(this)g(op)r(erator)515 2934 y(has)27 b(the)h(same)f(sp)r(ectrum)h(as)e(:)1270 3170 y Fr(P)1323 3182 y Fo(\022)1384 3170 y Fs(=)d(cos)13 b Fr(\022)1638 3102 y Fl(\002)1672 3170 y Fr(D)1743 3135 y Fq(2)1741 3190 y Fo(t)1799 3170 y Fs(+)18 b(\()p Fr(t)h Fp(\000)f Fr(x)p Fs(\))2125 3135 y Fq(2)2163 3102 y Fl(\003)2216 3170 y Fs(+)2309 3113 y(sin)2411 3079 y Fq(2)2462 3113 y Fr(\022)p 2309 3151 195 4 v 2323 3227 a Fs(cos)13 b Fr(\022)2513 3170 y(D)2584 3135 y Fq(2)2582 3190 y Fo(x)515 3380 y Fs(\(w)n(e)27 b(k)n(eep)g(on)h(the)g(same)f(notation)g(for)g (simpli\014cation\).)639 3480 y(It)38 b(has)f(b)r(een)h(pro)n(v)n(ed)e ([HeMo2)o(])i(that)f(for)g(small)g(v)-5 b(alues)37 b(of)h Fr(\022)j(>)e Fs(0)e(the)h(follo)n(wing)515 3579 y(asymptotics)27 b(hold)g(:)1462 3691 y(inf)44 b Fr(\033)s Fs(\()p Fr(P)1735 3703 y Fo(\022)1773 3691 y Fs(\))24 b Fp(\030)e Fr(\026)1966 3703 y Fq(0)2022 3691 y Fs(+)2105 3612 y Fl(X)2107 3790 y Fo(j)s Fk(\025)p Fq(1)2239 3691 y Fr(c)2275 3703 y Fo(j)2310 3691 y Fr(\022)2351 3657 y Fo(j)2409 3691 y Fr(:)639 3926 y Fs(Therefore)27 b(let)h(us)f(consider)g(a)g(set)1297 4108 y Fr(I)1333 4120 y Fo(d)1395 4108 y Fs(=)1510 4041 y Fl(\003)1563 4108 y Fp(\000)18 b(1)p Fr(;)37 b(d)1846 4041 y Fl(\002)1936 4108 y Fs(with)56 b Fr(d)23 b Fp(2)2288 4041 y Fl(\003)2337 4108 y Fr(\026)2387 4120 y Fq(0)2424 4108 y Fr(;)14 b Fs(1)2517 4041 y Fl(\002)2574 4108 y Fr(:)515 4291 y Fs(The)27 b(goal)g(of)g(this)h(section)f(is)h(to)f(get) h(information)f(ab)r(out)1388 4474 y Fr(N)9 b Fs(\()p Fr(d;)14 b(P)1629 4486 y Fo(\022)1667 4474 y Fs(\))23 b(=)g Fr(]\033)s Fs(\()p Fr(P)1977 4486 y Fo(\022)2015 4474 y Fs(\))p Fp(\\)2102 4406 y Fl(\003)2156 4474 y Fp(\000)18 b(1)p Fr(;)37 b(d)2425 4406 y Fl(\002)2483 4474 y Fr(;)702 b Fs(\(4.1\))515 4656 y(whic)n(h)27 b(denotes)h(the)g (n)n(um)n(b)r(er)f(of)g(eigen)n(v)-5 b(alues)27 b(of)g Fr(P)2169 4668 y Fo(\022)2235 4656 y Fs(included)h(in)g(the)g(set)f Fr(I)2972 4668 y Fo(d)3012 4656 y Fs(.)639 4756 y(Here)k(ma)n(y)f(b)r (e)h(one)g(can)f(apply)h(the)g(tec)n(hnique)g(of)f(Balazard-Konlein)e ([Bal])j(to)f(get)515 4856 y(the)37 b(asymptotics)f(of)g Fr(N)9 b Fs(\()p Fr(d;)14 b(P)1480 4868 y Fo(\022)1518 4856 y Fs(\))39 b Fr(;)e Fs(but)g(the)g(result)g(will)g(b)r(e)g(rough,) h(compared)d(to)i(our)515 4955 y(result)28 b(in)g(Theorem)f(4.2:)38 b(our)27 b(remainder)g(is)h(an)g Fh(O)p Fs(\(1\))52 b(and)28 b(the)h(result)f(of)g([Bal)o(])h(w)n(ould)1926 5255 y(9)p eop %%Page: 10 10 10 9 bop 515 523 a Fs(giv)n(e)26 b Fh(O)p Fs(\(sin)891 488 y Fk(\000)p Fo(\032)996 523 y Fr(\022)r Fs(\))51 b(with)28 b Fr(\032)23 b(>)g Fs(1)p Fr(=)p Fs(2)e Fr(:)515 623 y Fs(More)26 b(o)n(v)n(er)g(the)i(assumptions)f(in)h([Bal)o(])g (are)f(not)g(satis\014ed)g(in)h(our)f(case.)639 722 y(F)-7 b(or)27 b(a)g(\014xed)h Fr(a)23 b(>)g Fs(1)k(let)h(us)f(consider)g(the) h(follo)n(wing)e(sets)1261 902 y Fr(J)1307 914 y Fq(0)1367 902 y Fs(=)1455 835 y Fl(\002)1489 902 y Fp(\000)p Fr(a;)14 b Fs(+)p Fp(1)1783 835 y Fl(\002)1817 902 y Fr(;)152 b(J)2038 914 y Fq(1)2099 902 y Fs(=)2164 835 y Fl(\003)2216 902 y Fp(\000)18 b(1)p Fr(;)c Fp(\000)2494 846 y Fr(a)p 2494 883 44 4 v 2495 959 a Fs(2)2548 835 y Fl(\003)2610 902 y Fr(;)515 1088 y Fs(and)27 b(a)g(partition)h(of)f(unit)n(y)h(:)866 1263 y Fr(\037)918 1228 y Fq(2)918 1283 y(0)955 1263 y Fs(\()p Fr(x)p Fs(\))20 b(+)e Fr(\037)1221 1228 y Fq(2)1221 1283 y(1)1258 1263 y Fs(\()p Fr(x)p Fs(\))24 b(=)f(1)p Fr(;)96 b Fs(supp)r(ort)27 b(\()p Fr(\037)2033 1275 y Fo(j)2069 1263 y Fs(\))c Fp(\032)g Fr(J)2258 1275 y Fo(j)2293 1263 y Fr(;)2385 1184 y Fl(X)2429 1361 y Fo(j)2519 1263 y Fp(j)p Fr(\037)2594 1228 y Fk(0)2594 1283 y Fo(j)2629 1263 y Fs(\()p Fr(x)p Fs(\))p Fp(j)2763 1228 y Fq(2)2824 1263 y Fr(<)g(C)34 b(:)515 1509 y Fs(F)-7 b(or)27 b Fr(j)h Fs(=)22 b(0)p Fr(;)14 b Fs(1)27 b(let)h(us)f(denote)h(b)n(y)f(\012)1630 1521 y Fo(j)1693 1509 y Fs(the)h(domains)f Fr(I)-18 b(R)2246 1521 y Fq(+)2319 1509 y Fp(\002)18 b Fr(J)2448 1521 y Fo(j)2511 1509 y Fs(:)639 1609 y(\012)699 1621 y Fq(0)760 1609 y Fs(=)k Fr(I)-18 b(R)929 1621 y Fq(+)984 1609 y Fp(\002)1049 1542 y Fl(\003)1102 1609 y Fp(\000)18 b Fr(a;)c Fs(+)p Fp(1)1414 1542 y Fl(\002)1475 1609 y Fs(and)28 b(\012)1697 1621 y Fq(1)1757 1609 y Fs(=)23 b Fr(I)-18 b(R)1926 1621 y Fq(+)1982 1609 y Fp(\002)2047 1542 y Fl(\003)2099 1609 y Fp(\000)18 b(1)p Fr(;)c Fp(\000)2377 1576 y Fo(a)p 2377 1590 37 4 v 2378 1637 a Fq(2)2423 1542 y Fl(\002)2457 1609 y Fs(.)515 1725 y(W)-7 b(e)26 b(tak)n(e)f(no)n(w)g(the)h(realization)e(of)i(the)g(op)r(erators)e Fr(P)2215 1685 y Fo(j)2203 1750 y(\022)2276 1725 y Fs(on)h(eac)n(h)g (domain)g(\012)2927 1737 y Fo(j)2962 1725 y Fs(,)h(asso)r(ciated)515 1824 y(to)35 b(the)g(quadratic)f(form)h Fr(q)1394 1836 y Fq(\012)1441 1844 y Fj(j)1476 1824 y Fs(,)i(with)f(Neumann)f (conditions)f(on)h(\000)2695 1836 y Fo(N)2793 1824 y Fs(=)g Fp(f)p Fs(0)p Fp(g)22 b(\002)h Fr(J)3175 1836 y Fo(j)3245 1824 y Fs(and)515 1924 y(Diric)n(hlet)k(conditions)h(on)f (\000)1422 1936 y Fo(D)1505 1924 y Fs(=)c Fr(I)-18 b(R)1674 1936 y Fq(+)1748 1924 y Fp(\002)18 b Fr(@)5 b(J)1926 1936 y Fo(j)1961 1924 y Fs(.)639 2023 y(The)28 b(quadratic)e(forms)h (are)g(de\014ned)h(as)f(follo)n(ws)g(:)598 2264 y Fr(q)635 2276 y Fq(\012)682 2284 y Fj(j)717 2264 y Fs(\()p Fr(u)p Fs(\))d(=)940 2151 y Fl(Z)986 2340 y Fq(\012)1033 2348 y Fj(j)1069 2264 y Fp(f)p Fs(cos)12 b Fr(\022)1276 2197 y Fl(\002)1311 2264 y Fp(j)p Fr(D)1403 2276 y Fo(t)1432 2264 y Fr(u)p Fp(j)1503 2230 y Fq(2)1559 2264 y Fs(+)18 b(\()p Fr(t)g Fp(\000)g Fr(x)p Fs(\))1884 2230 y Fq(2)1922 2264 y Fp(j)p Fr(u)p Fp(j)2016 2230 y Fq(2)2053 2197 y Fl(\003)2106 2264 y Fs(+)2199 2208 y(sin)2301 2173 y Fq(2)2352 2208 y Fr(\022)p 2199 2245 195 4 v 2213 2321 a Fs(cos)13 b Fr(\022)2403 2264 y Fp(j)p Fr(D)2495 2276 y Fo(x)2537 2264 y Fr(u)p Fp(j)2608 2230 y Fq(2)2233 2539 y Fp(\000)2326 2483 y Fs(sin)2427 2448 y Fq(2)2479 2483 y Fr(\022)p 2326 2520 V 2340 2596 a Fs(cos)g Fr(\022)2587 2435 y Fq(1)2544 2460 y Fl(X)2546 2637 y Fo(j)s Fq(=0)2677 2539 y Fp(j)p Fr(\037)2752 2505 y Fk(0)2752 2560 y Fo(j)2787 2539 y Fs(\()p Fr(x)p Fs(\))p Fp(j)2921 2505 y Fq(2)2960 2539 y Fp(j)p Fr(u)p Fp(j)3054 2505 y Fq(2)3091 2539 y Fp(g)p Fr(dtdx)639 2786 y Fs(Let)42 b(us)f(\014rst)h(explain)f(wh)n (y)g Fr(q)1636 2798 y Fq(\012)1683 2806 y Fi(1)1762 2786 y Fs(will)g(not)h(giv)n(e)e(an)n(y)h(con)n(tribution)g(to)g(the)h(term) 515 2886 y Fr(N)9 b Fs(\()p Fr(d;)14 b(P)756 2898 y Fo(\022)794 2886 y Fs(\))23 b Fr(:)639 2985 y Fs(According)d(to)h(section)g(3)f(w)n (e)h(kno)n(w)f(that)h Fr(\026)p Fs(\()p Fr(x)p Fs(\),)j(the)d(\014rst)g (eigen)n(v)-5 b(alue)20 b(of)h Fr(D)3012 2955 y Fq(2)3010 3006 y Fo(t)3055 2985 y Fs(+)5 b(\()p Fr(t)g Fp(\000)g Fr(x)p Fs(\))3341 2955 y Fq(2)515 3085 y Fs(is)27 b(decreasing)f(on)h Fr(J)1161 3097 y Fq(1)1222 3085 y Fr(;)h Fs(so)f(w)n(e)g(ha)n(v)n(e)f (:)1127 3321 y Fr(q)1164 3333 y Fq(\012)1211 3341 y Fi(1)1248 3321 y Fs(\()p Fr(u)p Fs(\))d Fp(\025)1471 3254 y Fl(\002)1505 3321 y Fr(\026)p Fs(\()p Fp(\000)1662 3265 y Fr(a)p 1662 3302 44 4 v 1663 3378 a Fs(2)1716 3321 y(\))14 b(cos)f Fr(\022)21 b Fp(\000)d Fr(C)2133 3265 y Fs(sin)2235 3230 y Fq(2)2286 3265 y Fr(\022)p 2133 3302 195 4 v 2147 3378 a Fs(cos)13 b Fr(\022)2337 3254 y Fl(\003)2372 3321 y Fp(k)p Fr(u)p Fp(k)2504 3287 y Fq(2)2504 3344 y Fo(L)2550 3327 y Fi(2)2580 3344 y Fq(\(\012)2653 3352 y Fi(1)2686 3344 y Fq(\))2744 3321 y Fr(:)515 3487 y Fs(But)28 b Fr(\026)p Fs(\()p Fp(\000)837 3454 y Fo(a)p 836 3468 37 4 v 837 3515 a Fq(2)882 3487 y Fs(\))c Fr(>)e Fs(1,)28 b(so)f(for)g(small)g Fr(\022)j Fs(the)e(preceding)f(minoring)f(ensues) 1251 3658 y Fr(q)1288 3670 y Fq(\012)1335 3678 y Fi(1)1372 3658 y Fs(\()p Fr(u)p Fs(\))47 b Fp(\025)e(k)p Fr(u)p Fp(k)1773 3624 y Fq(2)1773 3681 y Fo(L)1819 3664 y Fi(2)1850 3681 y Fq(\(\012)1923 3689 y Fi(1)1956 3681 y Fq(\))2009 3658 y Fr(;)69 b Fs(if)62 b(0)23 b Fr(<)f(\022)k(<)c(\022)2554 3670 y Fq(0)2619 3658 y Fr(;)566 b Fs(\(4.2\))515 3838 y(for)27 b(some)54 b Fr(\022)916 3850 y Fq(0)1004 3838 y Fp(2)1087 3770 y Fl(\003)1122 3838 y Fs(0)p Fr(;)1210 3805 y Fo(\031)p 1210 3819 41 4 v 1214 3866 a Fq(4)1261 3770 y Fl(\002)1324 3838 y Fr(:)639 3937 y Fs(In)26 b(order)f(to)h (study)g(the)g(form)f Fr(q)1656 3949 y Fq(\012)1703 3957 y Fi(0)1740 3937 y Fs(,)i(it)f(is)g(con)n(v)n(enien)n(t)e(to)i(use)g (the)g(normalized)f(eigen-)515 4037 y(function)j Fr(')894 4049 y Fo(x)936 4037 y Fs(,)g(asso)r(ciated)e(to)i Fr(\026)p Fs(\()p Fr(x)p Fs(\),)g(in)g(the)g(follo)n(wing)f(w)n(a)n(y)-7 b(.)639 4136 y(Let)28 b(us)g(denote)f(b)n(y)g(\005)1339 4148 y Fq(0)1377 4136 y Fs(\()p Fr(u)p Fs(\))h(the)g(orthogonal)d(pro)5 b(jection)27 b(on)g(the)h(set)1353 4299 y Fr(F)1406 4311 y Fq(0)1467 4299 y Fs(=)23 b Fp(f)p Fr(')1651 4311 y Fo(x)1692 4299 y Fs(\()p Fr(t)p Fs(\))p Fr( )s Fs(\()p Fr(x)p Fs(\);)71 b Fr( )26 b Fp(2)e Fr(L)2264 4265 y Fq(2)2300 4299 y Fs(\()p Fr(J)2378 4311 y Fq(0)2416 4299 y Fs(\))p Fp(g)j Fr(;)668 b Fs(\(4.3\))515 4462 y(de\014ned)28 b(b)n(y)1291 4599 y(\005)1353 4611 y Fq(0)1391 4599 y Fs(\()p Fr(u)p Fs(\))23 b(=)g Fr(')1668 4611 y Fo(x)1710 4599 y Fs(\()p Fr(t)p Fs(\)\()1836 4486 y Fl(Z)1883 4675 y Fo(I)-20 b(R)1942 4683 y Fi(+)2007 4599 y Fr(u)p Fs(\()p Fr(s;)14 b(x)p Fs(\))p Fr(')2296 4611 y Fo(x)2338 4599 y Fs(\()p Fr(s)p Fs(\))p Fr(ds)p Fs(\))24 b Fr(;)606 b Fs(\(4.4\))515 4807 y(and)31 b(b)n(y)63 b Fr(F)884 4819 y Fq(1)951 4807 y Fs(=)29 b(\()p Fr(F)1130 4819 y Fq(0)1168 4807 y Fs(\))1200 4777 y Fk(?)1288 4807 y Fs(the)j(orthogonal)d(set)i(of)h Fr(F)2139 4819 y Fq(0)2208 4807 y Fs(.)48 b(The)32 b(corresp)r(onding)d(orthogonal)515 4907 y(pro)5 b(jection)26 b(is)1697 5006 y(\005)1759 5018 y Fq(1)1819 5006 y Fs(=)d Fr(I)i Fp(\000)18 b Fs(\005)2113 5018 y Fq(0)2174 5006 y Fr(:)1905 5255 y Fs(10)p eop %%Page: 11 11 11 10 bop 515 523 a Fs(A)28 b(direct)f(computation)g(giv)n(es)g(:)1432 698 y Fr(@)1476 710 y Fo(x)1518 698 y Fs(\(\005)1612 710 y Fq(0)1650 698 y Fr(u)p Fs(\))c(=)g(\005)1903 710 y Fq(0)1940 698 y Fs(\()p Fr(@)2016 710 y Fo(x)2058 698 y Fr(u)p Fs(\))18 b(+)g Fr(R)q Fs(\()p Fr(u)p Fs(\))23 b Fr(;)515 873 y Fs(where)k Fr(R)h Fs(is)g(de\014ned)g(b)n(y)581 1086 y Fr(R)q Fs(\()p Fr(u)p Fs(\))23 b(=)g Fr(')922 1098 y Fo(x)964 1086 y Fs(\()p Fr(t)p Fs(\))1058 994 y Fl(\020)1108 973 y(Z)1154 1162 y Fo(I)-20 b(R)1214 1170 y Fi(+)1278 1086 y Fr(u)p Fs(\()p Fr(s;)14 b(x)p Fs(\))p Fr(@)1557 1098 y Fo(x)1600 1086 y Fr(')1654 1098 y Fo(x)1696 1086 y Fs(\()p Fr(s)p Fs(\))p Fr(ds)1881 994 y Fl(\021)1949 1086 y Fs(+)k Fr(@)2076 1098 y Fo(x)2118 1086 y Fr(')2172 1098 y Fo(x)2214 1086 y Fs(\()p Fr(t)p Fs(\))2308 994 y Fl(\020)2359 973 y(Z)2405 1162 y Fo(I)-20 b(R)2464 1170 y Fi(+)2529 1086 y Fr(u)p Fs(\()p Fr(s;)14 b(x)p Fs(\))p Fr(')2818 1098 y Fo(x)2860 1086 y Fs(\()p Fr(s)p Fs(\))p Fr(ds)3045 994 y Fl(\021)3119 1086 y Fr(:)66 b Fs(\(4.5\))639 1318 y(The)28 b(additional)f(fact)h(that)1455 1493 y Fr(@)1499 1505 y Fo(x)1541 1493 y Fs(\(\005)1635 1505 y Fq(1)1673 1493 y Fr(u)p Fs(\))23 b(=)g(\005)1926 1505 y Fq(1)1963 1493 y Fs(\()p Fr(@)2039 1505 y Fo(x)2081 1493 y Fr(u)p Fs(\))c Fp(\000)f Fr(R)q Fs(\()p Fr(u)p Fs(\))515 1668 y(yields)27 b(the)h(follo)n(wing)f(b)r(ounds)g(:)598 1903 y(\(1)18 b Fp(\000)g Fr(\017)p Fs(\))839 1835 y Fl(\002)874 1903 y Fp(k)p Fr(@)960 1915 y Fo(x)1001 1903 y Fs(\(\005)1095 1915 y Fq(0)1133 1903 y Fr(u)p Fs(\))p Fp(k)1255 1868 y Fq(2)1255 1925 y Fo(L)1301 1909 y Fi(2)1332 1925 y Fq(\(\012)1405 1933 y Fi(0)1438 1925 y Fq(\))1486 1903 y Fs(+)g Fp(k)p Fr(@)1655 1915 y Fo(x)1697 1903 y Fs(\(\005)1791 1915 y Fq(1)1829 1903 y Fr(u)p Fs(\))p Fp(k)1951 1868 y Fq(2)1951 1925 y Fo(L)1997 1909 y Fi(2)2028 1925 y Fq(\(\012)2101 1933 y Fi(0)2134 1925 y Fq(\))2164 1835 y Fl(\003)2217 1903 y Fs(+)g(2\(1)f Fp(\000)2526 1847 y Fs(1)p 2526 1884 42 4 v 2530 1960 a Fr(\017)2578 1903 y Fs(\))p Fp(k)p Fr(R)q Fs(\()p Fr(u)p Fs(\))p Fp(k)2870 1868 y Fq(2)2870 1925 y Fo(L)2916 1909 y Fi(2)2947 1925 y Fq(\(\012)3020 1933 y Fi(0)3053 1925 y Fq(\))1699 2065 y Fp(\024)23 b(k)p Fr(@)1873 2077 y Fo(x)1914 2065 y Fr(u)p Fp(k)2004 2030 y Fq(2)2004 2087 y Fo(L)2050 2071 y Fi(2)2081 2087 y Fq(\(\012)2154 2095 y Fi(0)2187 2087 y Fq(\))677 2245 y Fp(\024)g Fs(\(1)18 b(+)g Fr(\017)p Fs(\))1006 2177 y Fl(\002)1041 2245 y Fp(k)p Fr(@)1127 2257 y Fo(x)1168 2245 y Fs(\(\005)1262 2257 y Fq(0)1300 2245 y Fr(u)p Fs(\))p Fp(k)1422 2210 y Fq(2)1422 2267 y Fo(L)1468 2251 y Fi(2)1499 2267 y Fq(\(\012)1572 2275 y Fi(0)1605 2267 y Fq(\))1653 2245 y Fs(+)g Fp(k)p Fr(@)1822 2257 y Fo(x)1864 2245 y Fs(\(\005)1958 2257 y Fq(1)1996 2245 y Fr(u)p Fs(\))p Fp(k)2118 2210 y Fq(2)2118 2267 y Fo(L)2164 2251 y Fi(2)2195 2267 y Fq(\(\012)2268 2275 y Fi(0)2301 2267 y Fq(\))2331 2177 y Fl(\003)2384 2245 y Fs(+)g(2\(1)g(+)2694 2188 y(1)p 2694 2225 V 2698 2302 a Fr(\017)2745 2245 y Fs(\))p Fp(k)p Fr(R)q Fs(\()p Fr(u)p Fs(\))p Fp(k)3037 2210 y Fq(2)3037 2267 y Fo(L)3083 2251 y Fi(2)3114 2267 y Fq(\(\012)3187 2275 y Fi(0)3220 2267 y Fq(\))3273 2245 y Fr(:)639 2448 y Fs(Using)28 b(the)g(result)f (\(3.10\))g(in)h(the)g(Lemma)f(3.1,)g(w)n(e)g(get)g(the)h(follo)n(wing) f(b)r(ound)h(:)515 2623 y Fh(Lemma)j(4.1)739 2798 y Fp(9)23 b Fr(C)867 2810 y Fq(0)928 2798 y Fr(>)f Fs(0)p Fr(;)43 b(s:t:)60 b Fp(8)23 b Fr(u)46 b Fp(2)g Fr(L)1619 2763 y Fq(2)1656 2798 y Fs(\(\012)1748 2810 y Fq(0)1785 2798 y Fs(\))p Fr(;)74 b Fp(k)p Fr(R)q Fs(\()p Fr(u)p Fs(\))p Fp(k)2174 2813 y Fo(L)2220 2796 y Fi(2)2251 2813 y Fq(\(\012)2324 2821 y Fi(0)2356 2813 y Fq(\))2410 2798 y Fp(\024)22 b Fr(C)2556 2810 y Fq(0)2594 2798 y Fp(k)p Fr(u)p Fp(k)2726 2813 y Fo(L)2772 2796 y Fi(2)2802 2813 y Fq(\(\012)2875 2821 y Fi(0)2908 2813 y Fq(\))2961 2798 y Fr(:)224 b Fs(\(4.6\))639 2973 y(Let)28 b(us)g(sk)n(etc)n(h)e(the)i(pro)r(of)f(of) h(the)g(lemma)f(4.1.)36 b(T)-7 b(aking)27 b(norms)g(in)h(\(4.5\))f(w)n (e)g(ha)n(v)n(e)g(:)516 3190 y Fp(k)p Fr(R)q Fs(\()p Fr(u)p Fs(\))p Fp(k)776 3156 y Fq(2)776 3213 y Fo(L)822 3196 y Fi(2)852 3213 y Fq(\(\012)925 3221 y Fi(0)958 3213 y Fq(\))1011 3190 y Fp(\024)c Fs(2)1141 3077 y Fl(Z)1186 3266 y Fq(\012)1233 3274 y Fi(0)1270 3190 y Fp(j)p Fr(u)p Fs(\()p Fr(s;)14 b(x)p Fs(\)\()p Fr(@)1604 3202 y Fo(x)1647 3190 y Fr(')1701 3202 y Fo(x)1743 3190 y Fs(\()p Fr(s)p Fs(\)\))p Fp(j)1901 3156 y Fq(2)1939 3190 y Fr(dsdx)19 b Fs(+)f(2)27 b(sup)2269 3260 y Fo(x)p Fk(2)p Fo(J)2389 3268 y Fi(0)2434 3190 y Fp(k)p Fr(@)2520 3202 y Fo(x)2562 3190 y Fr(')2616 3202 y Fo(x)2658 3190 y Fs(\()p Fr(t)p Fs(\))p Fp(k)2794 3156 y Fq(2)2794 3213 y Fo(L)2840 3196 y Fi(2)2872 3213 y Fq(\()p Fo(I)-20 b(R)2957 3221 y Fi(+)3004 3213 y Fq(\))3034 3190 y Fp(k)p Fr(u)p Fp(k)3166 3156 y Fq(2)3166 3213 y Fo(L)3212 3196 y Fi(2)3242 3213 y Fq(\(\012)3315 3221 y Fi(0)3348 3213 y Fq(\))1011 3381 y Fp(\024)23 b Fs(4)j(sup)1155 3451 y Fo(x)p Fk(2)p Fo(J)1275 3459 y Fi(0)1320 3381 y Fp(k)p Fr(@)1406 3393 y Fo(x)1447 3381 y Fr(')1501 3393 y Fo(x)1543 3381 y Fs(\()p Fr(t)p Fs(\))p Fp(k)1679 3347 y Fq(2)1679 3404 y Fo(L)1725 3387 y Fi(2)1757 3404 y Fq(\()p Fo(I)-20 b(R)1842 3412 y Fi(+)1889 3404 y Fq(\))1919 3381 y Fp(k)p Fr(u)p Fp(k)2051 3347 y Fq(2)2051 3404 y Fo(L)2097 3387 y Fi(2)2128 3404 y Fq(\(\012)2201 3412 y Fi(0)2233 3404 y Fq(\))2319 3381 y Fr(:)639 3623 y Fs(The)31 b(lemma)g(will)g(then)h(b)r(e)f(pro)n(v)n (ed)e(if)j(w)n(e)e(sho)n(w)g(that)76 b(sup)2455 3693 y Fo(x)p Fk(2)p Fo(J)2575 3701 y Fi(0)2620 3623 y Fp(k)p Fr(@)2706 3635 y Fo(x)2747 3623 y Fr(')2801 3635 y Fo(x)2843 3623 y Fs(\()p Fr(t)p Fs(\))p Fp(k)2979 3589 y Fq(2)2979 3646 y Fo(L)3025 3629 y Fi(2)3057 3646 y Fq(\()p Fo(I)-20 b(R)3143 3654 y Fi(+)3190 3646 y Fq(\))3220 3623 y Fp(k)61 b Fs(is)515 3775 y(\014nite.)639 3875 y(Going)27 b(bac)n(k)g(to)g(the)h (relation)f(\(3.10\))1415 4037 y Fr(@)p 1392 4074 97 4 v 1392 4150 a(@)5 b(x)1498 4093 y(')1552 4105 y Fo(x)1594 4093 y Fs(\()p Fr(t)p Fs(\))24 b(=)e(2)1841 4026 y Fl(\002)1875 4093 y Fr(Q)1941 4105 y Fo(x)2001 4093 y Fp(\000)c Fr(\026)p Fs(\()p Fr(x)p Fs(\))2245 4026 y Fl(\003)2281 4043 y Fk(\000)p Fq(1)2370 4093 y Fr( )2424 4105 y Fo(x)2489 4093 y Fr(;)696 b Fs(\(4.7\))515 4292 y(with)1148 4391 y Fr( )1202 4403 y Fo(x)1244 4391 y Fs(\()p Fr(t)p Fs(\))24 b(=)f(\()p Fr(t)18 b Fp(\000)g Fr(x)p Fs(\))p Fr(')1746 4403 y Fo(x)1789 4391 y Fs(\()p Fr(t)p Fs(\))h Fp(\000)f(h)p Fs(\()p Fr(t)h Fp(\000)f Fr(x)p Fs(\))p Fr(')2314 4403 y Fo(x)2357 4391 y Fp(j)p Fr(')2434 4403 y Fo(x)2476 4391 y Fp(i)p Fr(')2562 4403 y Fo(x)2605 4391 y Fs(\()p Fr(t)p Fs(\))23 b Fr(;)515 4536 y Fs(and)k(using)g(:)779 4711 y Fp(k)p Fr(D)892 4677 y Fq(2)890 4732 y Fo(t)929 4711 y Fr(')983 4723 y Fo(x)1025 4711 y Fp(k)1067 4677 y Fq(2)1067 4737 y Fo(L)1113 4720 y Fi(2)1145 4737 y Fq(\()p Fo(I)-20 b(R)1230 4717 y Fi(2)1230 4756 y(+)1277 4737 y Fq(\))1325 4711 y Fs(+)18 b Fp(k)p Fs(\()p Fr(t)h Fp(\000)f Fr(x)p Fs(\))p Fr(')1747 4723 y Fo(x)1790 4711 y Fp(k)1832 4677 y Fq(2)1832 4737 y Fo(L)1878 4720 y Fi(2)1909 4737 y Fq(\()p Fo(I)-20 b(R)1994 4717 y Fi(2)1994 4756 y(+)2041 4737 y Fq(\))2094 4711 y Fs(=)23 b Fr(\026)p Fs(\()p Fr(x)p Fs(\))p Fp(k)p Fr(')2439 4723 y Fo(x)2482 4711 y Fp(k)2524 4677 y Fq(2)2524 4737 y Fo(L)2570 4720 y Fi(2)2601 4737 y Fq(\()p Fo(I)-20 b(R)2686 4717 y Fi(2)2686 4756 y(+)2733 4737 y Fq(\))2786 4711 y Fs(=)23 b Fr(\026)p Fs(\()p Fr(x)p Fs(\))56 b Fr(;)515 4904 y Fs(w)n(e)27 b(get)g(that)h(:)1438 5004 y Fp(k)p Fs(\()p Fr(t)18 b Fp(\000)g Fr(x)p Fs(\))p Fr(')1776 5016 y Fo(x)1819 5004 y Fp(k)1861 5022 y Fo(L)1907 5005 y Fi(2)1938 5022 y Fq(\()p Fo(I)-20 b(R)2023 5002 y Fi(2)2023 5041 y(+)2070 5022 y Fq(\))2123 5004 y Fp(\024)2211 4929 y Fl(p)p 2294 4929 163 4 v 75 x Fr(\026)p Fs(\()p Fr(x)p Fs(\))1905 5255 y(11)p eop %%Page: 12 12 12 11 bop 515 523 a Fs(and)27 b(then)1341 623 y Fp(jh)p Fs(\()p Fr(t)19 b Fp(\000)f Fr(x)p Fs(\))p Fr(')1693 635 y Fo(x)1736 623 y Fp(j)p Fr(')1813 635 y Fo(x)1855 623 y Fp(i)1887 641 y Fo(L)1933 624 y Fi(2)1965 641 y Fq(\()p Fo(I)-20 b(R)2051 621 y Fi(2)2051 660 y(+)2098 641 y Fq(\))2128 623 y Fp(j)23 b(\024)2261 548 y Fl(p)p 2344 548 163 4 v 75 x Fr(\026)p Fs(\()p Fr(x)p Fs(\))i Fr(;)515 772 y Fs(so)1499 872 y Fp(k)p Fr( )1595 884 y Fo(x)1637 872 y Fp(k)1679 890 y Fo(L)1725 873 y Fi(2)1756 890 y Fq(\()p Fo(I)-20 b(R)1842 870 y Fi(2)1842 909 y(+)1888 890 y Fq(\))1942 872 y Fp(\024)22 b Fs(2)2071 797 y Fl(p)p 2154 797 V 75 x Fr(\026)p Fs(\()p Fr(x)p Fs(\))56 b Fr(:)639 1032 y Fs(Since)30 b Fr( )912 1044 y Fo(x)984 1032 y Fs(liv)n(es)f(on)h(the)g(orthogonal)e(space)h(of)h Fr(')2227 1044 y Fo(x)2296 1032 y Fr(;)g Fs(let)g(us)g(consider)f(the)h(norm)f Fr(N)3337 1044 y Fo(x)515 1143 y Fs(of)e(the)h(restriction)f(of)1244 1076 y Fl(\002)1279 1143 y Fr(Q)1345 1155 y Fo(x)1405 1143 y Fp(\000)18 b Fr(\026)p Fs(\()p Fr(x)p Fs(\))1649 1076 y Fl(\003)1685 1093 y Fk(\000)p Fq(1)1801 1143 y Fs(to)28 b(this)g(orthogonal)d(space.)36 b(It)28 b(is)g(giv)n(en)e(b)n (y)i(:)1241 1374 y Fr(N)1308 1386 y Fo(x)1373 1374 y Fs(=)1700 1317 y(1)p 1471 1354 501 4 v 1471 1430 a Fr(\026)1521 1442 y Fq(2)1558 1430 y Fs(\()p Fr(x)p Fs(\))19 b Fp(\000)f Fr(\026)1821 1442 y Fq(1)1859 1430 y Fs(\()p Fr(x)p Fs(\))2004 1374 y(=)2312 1317 y(1)p 2101 1354 463 4 v 2101 1430 a Fr(\026)2151 1442 y Fq(2)2188 1430 y Fs(\()p Fr(x)p Fs(\))i Fp(\000)e Fr(\026)p Fs(\()p Fr(x)p Fs(\))2629 1374 y Fr(;)515 1605 y Fs(where)27 b(\()p Fr(\026)837 1617 y Fo(j)872 1605 y Fs(\()p Fr(x)p Fs(\)\))1015 1617 y Fo(j)1079 1605 y Fs(is)h(the)g(increasing)e(sequence)h(of)g(the)h (eigen)n(v)-5 b(alues)27 b(of)g Fr(Q)2865 1617 y Fo(x)2930 1605 y Fr(:)639 1704 y Fs(According)j(to)h(\(3.1\))f(and)h(\(3.6\))f(,) i Fr(\026)p Fs(\()p Fr(x)p Fs(\))g(and)e Fr(N)2188 1716 y Fo(x)2260 1704 y Fs(are)g(uniformly)h(b)r(ounded)g(on)f Fr(J)3290 1716 y Fq(0)3356 1704 y Fr(;)515 1804 y Fs(so)d(there)g (exists)g Fr(c)1094 1816 y Fq(0)1155 1804 y Fr(>)22 b Fs(0)27 b(suc)n(h)h(that)1044 2043 y(sup)1031 2113 y Fo(x)p Fk(2)p Fo(J)1151 2121 y Fi(0)1196 2043 y Fp(k)p Fr(@)1282 2055 y Fo(x)1323 2043 y Fr(')1377 2055 y Fo(x)1419 2043 y Fs(\()p Fr(t)p Fs(\))p Fp(k)1555 2058 y Fo(L)1601 2042 y Fi(2)1633 2058 y Fq(\()p Fo(I)-20 b(R)1719 2066 y Fi(+)1766 2058 y Fq(\))1819 2043 y Fp(\024)22 b Fs(2)27 b(sup)1962 2113 y Fo(x)p Fk(2)p Fo(J)2082 2121 y Fi(0)2246 1914 y Fl(p)p 2329 1914 163 4 v 71 x Fr(\026)p Fs(\()p Fr(x)p Fs(\))p 2137 2024 463 4 v 2137 2100 a Fr(\026)2187 2112 y Fq(2)2224 2100 y Fs(\()p Fr(x)p Fs(\))20 b Fp(\000)e Fr(\026)p Fs(\()p Fr(x)p Fs(\))2633 2043 y Fp(\024)k Fr(C)2779 2055 y Fq(0)2840 2043 y Fr(;)515 2278 y Fs(so)27 b(the)h(Lemma)f(4.1)g(follo)n(ws.)639 2496 y(>F)-7 b(rom)25 b(Lemma)f(4.1,)g(w)n(e)h(see)f(that)h(w)n(e)f(can)g(\014nd)h(a)f (constan)n(t)g Fr(C)2657 2508 y Fq(1)2718 2496 y Fr(>)f Fs(0)p Fr(;)h Fs(suc)n(h)g(that,)i(for)515 2596 y(an)n(y)h Fr(\017)22 b Fp(2)783 2529 y Fl(\003)818 2596 y Fs(0)p Fr(;)14 b Fs(1)939 2529 y Fl(\002)973 2596 y Fr(;)865 2829 y Fs(\(1)19 b Fp(\000)f Fr(\017)p Fs(\))1107 2762 y Fl(\002)1141 2829 y Fp(k)p Fr(@)1227 2841 y Fo(x)1269 2829 y Fs(\(\005)1363 2841 y Fq(0)1400 2829 y Fr(u)p Fs(\))p Fp(k)1522 2795 y Fq(2)1522 2852 y Fo(L)1568 2835 y Fi(2)1600 2852 y Fq(\(\012)1673 2860 y Fi(0)1706 2852 y Fq(\))1754 2829 y Fs(+)g Fp(k)p Fr(@)1923 2841 y Fo(x)1964 2829 y Fs(\(\005)2058 2841 y Fq(1)2096 2829 y Fr(u)p Fs(\))p Fp(k)2218 2795 y Fq(2)2218 2852 y Fo(L)2264 2835 y Fi(2)2296 2852 y Fq(\(\012)2369 2860 y Fi(0)2401 2852 y Fq(\))2431 2762 y Fl(\003)2484 2829 y Fp(\000)2577 2773 y Fr(C)2636 2785 y Fq(1)p 2577 2810 97 4 v 2609 2886 a Fr(\017)2684 2829 y Fp(k)p Fr(u)p Fp(k)2816 2795 y Fq(2)2816 2852 y Fo(L)2862 2835 y Fi(2)2892 2852 y Fq(\(\012)2965 2860 y Fi(0)2998 2852 y Fq(\))1688 3049 y Fp(\024)23 b(k)p Fr(@)1862 3061 y Fo(x)1903 3049 y Fr(u)p Fp(k)1993 3015 y Fq(2)1993 3072 y Fo(L)2039 3055 y Fi(2)2070 3072 y Fq(\(\012)2143 3080 y Fi(0)2176 3072 y Fq(\))822 3257 y Fp(\024)f Fs(\(1)d(+)f Fr(\017)p Fs(\))1151 3190 y Fl(\002)1185 3257 y Fp(k)p Fr(@)1271 3269 y Fo(x)1313 3257 y Fs(\(\005)1407 3269 y Fq(0)1444 3257 y Fr(u)p Fs(\))p Fp(k)1566 3223 y Fq(2)1566 3280 y Fo(L)1612 3263 y Fi(2)1644 3280 y Fq(\(\012)1717 3288 y Fi(0)1749 3280 y Fq(\))1798 3257 y Fs(+)g Fp(k)p Fr(@)1967 3269 y Fo(x)2008 3257 y Fs(\(\005)2102 3269 y Fq(1)2140 3257 y Fr(u)p Fs(\))p Fp(k)2262 3223 y Fq(2)2262 3280 y Fo(L)2308 3263 y Fi(2)2339 3280 y Fq(\(\012)2412 3288 y Fi(0)2445 3280 y Fq(\))2475 3190 y Fl(\003)2528 3257 y Fs(+)2621 3201 y Fr(C)2680 3213 y Fq(1)p 2621 3238 V 2653 3314 a Fr(\017)2728 3257 y Fp(k)p Fr(u)p Fp(k)2860 3223 y Fq(2)2860 3280 y Fo(L)2906 3263 y Fi(2)2936 3280 y Fq(\(\012)3009 3288 y Fi(0)3042 3280 y Fq(\))515 3430 y Fs(>F)-7 b(rom)27 b(that)h(w)n(e)f(obtain)h(the)f(corresp)r(onding)f(b)r(ounds)i(on)f (the)h(quadratic)f(form)g Fr(q)3161 3442 y Fq(\012)3208 3450 y Fi(0)3300 3430 y Fs(:)909 3629 y Fr(q)949 3589 y Fo(\017;)p Fk(\000)946 3654 y Fq(\012)993 3662 y Fi(0)1052 3629 y Fs(\(\005)1146 3641 y Fq(0)1184 3629 y Fr(u)p Fs(\))18 b(+)g Fr(q)1405 3589 y Fo(\017;)p Fk(\000)1402 3654 y Fq(\012)1449 3662 y Fi(0)1509 3629 y Fs(\(\005)1603 3641 y Fq(1)1641 3629 y Fr(u)p Fs(\))23 b Fp(\024)f Fr(q)1868 3641 y Fq(\012)1915 3649 y Fi(0)1952 3629 y Fs(\()p Fr(u)p Fs(\))h Fp(\024)g Fr(q)2215 3589 y Fo(\017;)p Fq(+)2212 3654 y(\012)2259 3662 y Fi(0)2318 3629 y Fs(\(\005)2412 3641 y Fq(0)2449 3629 y Fr(u)p Fs(\))c(+)f Fr(q)2671 3589 y Fo(\017;)p Fq(+)2668 3654 y(\012)2715 3662 y Fi(0)2773 3629 y Fs(\(\005)2867 3641 y Fq(1)2905 3629 y Fr(u)p Fs(\))223 b(\(4.8\))515 3779 y(where)27 b(w)n(e)g(used)h(the)g(natural) e(notations)h(:)674 3999 y Fr(q)714 3959 y Fo(\017;)p Fk(\000)711 4023 y Fq(\012)758 4031 y Fi(0)817 3999 y Fs(\()p Fr(u)p Fs(\))c(=)1040 3886 y Fl(Z)1086 4074 y Fq(\012)1133 4082 y Fi(0)1170 3906 y Fl(n)1225 3999 y Fs(cos)13 b Fr(\022)1391 3931 y Fl(\002)1426 3999 y Fp(j)p Fr(D)1518 4011 y Fo(t)1547 3999 y Fr(u)p Fp(j)1618 3964 y Fq(2)1673 3999 y Fs(+)18 b(\()p Fr(t)h Fp(\000)f Fr(x)p Fs(\))1999 3964 y Fq(2)2037 3999 y Fp(j)p Fr(u)p Fp(j)2131 3964 y Fq(2)2168 3931 y Fl(\003)2221 3999 y Fs(+)g(\(1)g Fp(\000)g Fr(\017)p Fs(\))2555 3942 y(sin)2657 3907 y Fq(2)2708 3942 y Fr(\022)p 2555 3979 195 4 v 2569 4055 a Fs(cos)13 b Fr(\022)2760 3999 y Fp(j)p Fr(D)2852 4011 y Fo(x)2893 3999 y Fr(u)p Fp(j)2964 3964 y Fq(2)3001 3906 y Fl(o)3056 3999 y Fr(dtdx)1673 4244 y Fp(\000)1766 4188 y Fs(sin)1868 4153 y Fq(2)1919 4188 y Fr(\022)p 1766 4225 V 1780 4301 a Fs(cos)g Fr(\022)1971 4152 y Fl(h)2010 4244 y Fr(C)25 b Fs(+)2187 4188 y Fr(C)2246 4200 y Fq(1)p 2187 4225 97 4 v 2218 4301 a Fr(\017)2293 4152 y Fl(i)2332 4244 y Fp(k)p Fr(u)p Fp(k)2464 4210 y Fq(2)2464 4267 y Fo(L)2510 4250 y Fi(2)2541 4267 y Fq(\(\012)2614 4275 y Fi(0)2647 4267 y Fq(\))515 4450 y Fs(and)674 4670 y Fr(q)714 4630 y Fo(\017;)p Fq(+)711 4694 y(\012)758 4702 y Fi(0)817 4670 y Fs(\()p Fr(u)p Fs(\))e(=)1039 4557 y Fl(Z)1085 4746 y Fq(\012)1132 4754 y Fi(0)1169 4578 y Fl(n)1225 4670 y Fs(cos)13 b Fr(\022)1391 4603 y Fl(\002)1426 4670 y Fp(j)p Fr(D)1518 4682 y Fo(t)1547 4670 y Fr(u)p Fp(j)1618 4636 y Fq(2)1673 4670 y Fs(+)18 b(\()p Fr(t)h Fp(\000)f Fr(x)p Fs(\))1999 4636 y Fq(2)2037 4670 y Fp(j)p Fr(u)p Fp(j)2131 4636 y Fq(2)2168 4603 y Fl(\003)2221 4670 y Fs(+)g(\(1)g(+)g Fr(\017)p Fs(\))2555 4614 y(sin)2657 4579 y Fq(2)2708 4614 y Fr(\022)p 2555 4651 195 4 v 2569 4727 a Fs(cos)13 b Fr(\022)2759 4670 y Fp(j)p Fr(D)2851 4682 y Fo(x)2893 4670 y Fr(u)p Fp(j)2964 4636 y Fq(2)3001 4578 y Fl(o)3056 4670 y Fr(dtdx)1673 4916 y Fs(+)1766 4860 y(sin)1868 4825 y Fq(2)1919 4860 y Fr(\022)p 1766 4897 V 1780 4973 a Fs(cos)g Fr(\022)1980 4860 y(C)2039 4872 y Fq(1)p 1980 4897 97 4 v 2012 4973 a Fr(\017)2087 4916 y Fp(k)p Fr(u)p Fp(k)2219 4881 y Fq(2)2219 4938 y Fo(L)2265 4922 y Fi(2)2295 4938 y Fq(\(\012)2368 4946 y Fi(0)2401 4938 y Fq(\))1905 5255 y Fs(12)p eop %%Page: 13 13 13 12 bop 639 523 a Fs(W)-7 b(riting)1717 661 y Fr(h)23 b Fs(=)1925 605 y(sin)13 b Fr(\022)p 1885 642 236 4 v 1885 659 a Fp(p)p 1955 659 167 4 v 1955 730 a Fs(cos)g Fr(\022)2154 661 y(;)1031 b Fs(\(4.9\))515 855 y(taking)27 b(in)n(to)g(accoun)n(t)g(\(4.3\),)g(w)n(e)g(de\014ne)1462 1075 y Fr(W)12 b Fs(\()p Fr(x)p Fs(\))24 b(=)1775 962 y Fl(Z)1821 1151 y Fo(I)-20 b(R)1880 1159 y Fi(+)1945 1075 y Fp(j)2002 1019 y Fr(@)p 1978 1056 97 4 v 1978 1132 a(@)5 b(x)2084 1075 y(')2138 1087 y Fo(x)2180 1075 y Fs(\()p Fr(t)p Fs(\))p Fp(j)2297 1041 y Fq(2)2335 1075 y Fr(dt;)515 1313 y Fs(and)27 b(w)n(e)g(get,)h(using)f(\(4.4\))g(that)h (:)813 1490 y Fr(q)853 1450 y Fo(\017;)p Fk(\000)850 1514 y Fq(\012)897 1522 y Fi(0)956 1490 y Fs(\(\005)1050 1502 y Fq(0)1088 1490 y Fr(u)p Fs(\))23 b(=)g Fr(q)1319 1455 y Fo(\017;)p Fk(\000)1422 1490 y Fs(\()p Fr( )s Fs(\))1191 1669 y(=)1279 1556 y Fl(Z)1325 1744 y Fo(J)1362 1752 y Fi(0)1398 1669 y Fp(f)1440 1601 y Fl(\002)1474 1669 y Fr(\026)p Fs(\()p Fr(x)p Fs(\))43 b(cos)13 b Fr(\022)21 b Fs(+)d(\(1)g Fp(\000)g Fr(\017)p Fs(\))p Fr(h)2235 1634 y Fq(2)2272 1669 y Fr(W)12 b Fs(\()p Fr(x)p Fs(\))2473 1601 y Fl(\003)2508 1669 y Fp(j)p Fr( )s Fs(\()p Fr(x)p Fs(\))p Fp(j)2722 1634 y Fq(2)1353 1899 y Fs(+)18 b(\(1)g Fp(\000)g Fr(\017)p Fs(\))p Fr(h)1725 1864 y Fq(2)1762 1899 y Fp(j)p Fr(D)1854 1911 y Fo(x)1895 1899 y Fr( )s Fs(\()p Fr(x)p Fs(\))p Fp(j)2086 1864 y Fq(2)2143 1899 y Fp(\000)g Fr(h)2274 1864 y Fq(2)2311 1806 y Fl(h)2351 1899 y Fr(C)24 b Fs(+)2527 1842 y Fr(C)2586 1854 y Fq(1)p 2527 1880 V 2559 1956 a Fr(\017)2634 1806 y Fl(i)2673 1899 y Fp(j)p Fr( )s Fs(\()p Fr(x)p Fs(\))p Fp(j)2887 1864 y Fq(2)2926 1899 y Fp(g)p Fr(dx:)639 2098 y Fs(In)k(the)g(same)f (w)n(a)n(y)f(w)n(e)i(ha)n(v)n(e)e(:)973 2275 y Fr(q)1013 2235 y Fo(\017;)p Fq(+)1010 2299 y(\012)1057 2307 y Fi(0)1116 2275 y Fs(\(\005)1210 2287 y Fq(0)1248 2275 y Fr(u)p Fs(\))d(=)f Fr(q)1478 2241 y Fo(\017;)p Fq(+)1581 2275 y Fs(\()p Fr( )s Fs(\))1351 2454 y(=)1438 2341 y Fl(Z)1484 2529 y Fo(J)1521 2537 y Fi(0)1558 2454 y Fp(f)1600 2386 y Fl(\002)1634 2454 y Fr(\026)p Fs(\()p Fr(x)p Fs(\))42 b(cos)13 b Fr(\022)21 b Fs(+)d(\(1)g(+)g Fr(\017)p Fs(\))p Fr(h)2394 2419 y Fq(2)2431 2454 y Fr(W)12 b Fs(\()p Fr(x)p Fs(\))2632 2386 y Fl(\003)2668 2454 y Fp(j)p Fr( )s Fs(\()p Fr(x)p Fs(\))p Fp(j)2882 2419 y Fq(2)1429 2684 y Fs(+)18 b(\(1)g(+)g Fr(\017)p Fs(\))p Fr(h)1801 2650 y Fq(2)1838 2684 y Fp(j)p Fr(D)1930 2696 y Fo(x)1972 2684 y Fr( )s Fs(\()p Fr(x)p Fs(\))p Fp(j)2163 2650 y Fq(2)2220 2684 y Fs(+)g Fr(h)2351 2650 y Fq(2)2398 2628 y Fr(C)2457 2640 y Fq(1)p 2398 2665 V 2429 2741 a Fr(\017)2504 2684 y Fp(j)p Fr( )s Fs(\()p Fr(x)p Fs(\))p Fp(j)2718 2650 y Fq(2)2757 2684 y Fp(g)p Fr(dx:)515 2884 y Fs(No)n(w)27 b(w)n(e)g(ha)n(v)n(e)g(to)h(deal)f(with)h(the)h(terms)e(in)n(v)n (olving)f(the)j(second)e(pro)5 b(jection)26 b(\005)3082 2896 y Fq(1)3120 2884 y Fr(u)d(:)28 b Fs(But)515 2983 y(the)c(de\014nition)g(of)g(\005)1172 2995 y Fq(1)1210 2983 y Fr(u)e(;)i Fs(the)h(min-max)e(principle)h(and)g(the)g(estimate) 48 b(1)11 b Fp(\000)g Fs(cos)i Fr(\022)25 b Fp(\024)e Fr(h)3253 2953 y Fq(2)3290 2983 y Fr(C)515 3083 y Fs(giv)n(e)j(the)i (follo)n(wing)f(lo)n(w)n(er)f(b)r(ound)i(:)1008 3304 y Fr(q)1048 3264 y Fo(\017;)p Fk(\006)1045 3328 y Fq(\012)1092 3336 y Fi(0)1151 3304 y Fs(\(\005)1245 3316 y Fq(1)1283 3304 y Fr(u)p Fs(\))23 b Fp(\025)1474 3236 y Fl(\002)1534 3304 y Fs(inf)1508 3357 y Fo(x)p Fk(2)p Fo(J)1628 3365 y Fi(0)1674 3304 y Fr(\026)1724 3316 y Fq(2)1761 3304 y Fs(\()p Fr(x)p Fs(\))c Fp(\000)f Fr(h)2022 3269 y Fq(2)2060 3211 y Fl(\020)2109 3304 y Fr(C)25 b Fs(+)2286 3247 y Fr(C)2345 3259 y Fq(1)p 2286 3285 V 2317 3361 a Fr(\017)2392 3211 y Fl(\021)2442 3304 y Fp(k)p Fs(\005)2546 3316 y Fq(1)2583 3304 y Fr(u)p Fp(k)2673 3319 y Fo(L)2719 3302 y Fi(2)2750 3319 y Fq(\(\012)2823 3327 y Fi(0)2856 3319 y Fq(\))515 3490 y Fs(where)18 b Fr(\026)796 3502 y Fq(2)833 3490 y Fs(\()p Fr(x)p Fs(\))i(denotes)e(the)i(second)e(eigen)n(v)-5 b(alue)17 b(de\014ned)i(in)g(the)h(Lemma)e(3.1.)33 b(This)19 b(eigen-)515 3590 y(v)-5 b(alue)28 b(has)h(to)f(b)r(e)i(greater)d(than) i(the)g(\014rst)f(eigen)n(v)-5 b(alue)28 b(of)h(the)g(corresp)r(onding) e(Diric)n(hlet)515 3690 y(problem,)g(so)1748 3889 y Fr(\026)1798 3901 y Fq(2)1835 3889 y Fs(\()p Fr(x)p Fs(\))d Fr(>)f Fs(1)g Fr(:)515 4035 y Fs(If)28 b(w)n(e)f(tak)n(e)g(for)g(example)g Fr(\017)c Fs(=)g Fr(h)f(;)28 b Fs(w)n(e)f(get)h(that)1505 4211 y Fr(N)9 b Fs(\()p Fr(d;)14 b(q)1733 4171 y Fo(\017;)p Fk(\006)1730 4236 y Fq(\012)1777 4244 y Fi(0)1836 4211 y Fs(\))24 b(=)e Fr(N)9 b Fs(\()p Fr(d;)14 b(q)2207 4177 y Fo(\017;)p Fk(\006)2311 4211 y Fs(\))23 b Fr(:)515 4388 y Fs(Let)k(us)h(tak)n(e)f(an)g(extension)32 b Fl(e)-51 b Fr(\026)p Fs(\()p Fr(x)p Fs(\))29 b(of)f Fr(\026)p Fs(\()p Fr(x)p Fs(\))g(outside)g(of)1234 4564 y Fr(J)1280 4576 y Fq(0)p Fo(;d)1395 4564 y Fs(=)22 b Fp(f)p Fr(x)h Fp(2)h Fr(J)1719 4576 y Fq(0)1756 4564 y Fr(;)42 b(\026)p Fs(\()p Fr(x)p Fs(\))24 b Fr(<)e(d)d Fs(+)f(\(1)g Fp(\000)g Fr(d)p Fs(\))p Fr(=)p Fs(2)p Fp(g)k Fr(;)508 b Fs(\(4.10\))515 4741 y(suc)n(h)27 b(that)1415 4819 y Fl(e)-51 b Fr(\026)p Fs(\()p Fr(x)p Fs(\))25 b(=)d Fr(\026)p Fs(\()p Fr(x)p Fs(\))p Fr(;)43 b Fs(if)34 b Fr(x)47 b Fp(2)f Fr(J)2233 4831 y Fq(0)p Fo(;d)1415 4918 y Fl(e)-51 b Fr(\026)p Fs(\()p Fr(x)p Fs(\))25 b Fp(\025)d Fs(\(1)c(+)g Fr(d)p Fs(\))p Fr(=)p Fs(2)23 b Fr(;)42 b Fp(8)23 b Fr(x)33 b(=)-52 b Fp(2)47 b Fr(J)2392 4930 y Fq(0)p Fo(;d)1415 5018 y Fl(e)-51 b Fr(\026)p Fs(\()p Fr(x)p Fs(\))25 b(=)d(1)27 b(if)35 b Fp(j)p Fr(x)p Fp(j)24 b Fr(>)e(C)2098 5030 y Fo(d)3167 4919 y Fs(\(4.11\))1905 5255 y(13)p eop %%Page: 14 14 14 13 bop 515 523 a Fs(for)27 b(some)g(constan)n(t)g Fr(C)1244 535 y Fo(d)1306 523 y Fr(>)22 b Fs(0)p Fr(:)28 b Fs(Let)f(us)h(de\014ne)985 740 y Fr(q)1025 700 y Fo(\017;)p Fk(\006)1022 762 y Fq(0)1128 740 y Fs(\()p Fr( )s Fs(\))c(=)1360 627 y Fl(Z)1407 815 y Fo(I)-20 b(R)1470 673 y Fl(\002)1509 740 y(e)-50 b Fr(\026)p Fs(\()p Fr(x)p Fs(\))28 b Fp(j)p Fr( )s Fs(\()p Fr(x)p Fs(\))p Fp(j)1908 706 y Fq(2)1965 740 y Fs(+)18 b(\(1)h Fp(\006)f Fr(\017)p Fs(\))p Fr(h)2338 706 y Fq(2)2375 740 y Fp(j)p Fr(D)2467 752 y Fo(x)2508 740 y Fr( )s Fs(\()p Fr(x)p Fs(\))p Fp(j)2699 706 y Fq(2)2738 673 y Fl(\003)2772 740 y Fr(dx)24 b(:)515 973 y Fs(The)k(exp)r(onen)n (tial)g(lo)r(calization)f(of)i(eigenfunctions)f(of)g Fr(q)2347 943 y Fo(\017;)p Fk(\006)2479 973 y Fs(asso)r(ciated)f(to)h (eigen)n(v)-5 b(alues)515 1073 y(in)617 1005 y Fl(\003)674 1073 y Fp(\000)22 b(1)p Fr(;)14 b(d)924 1005 y Fl(\002)959 1073 y Fr(;)33 b Fs(\(see)g([He1]\),)i(and)e(the)h(p)r(olynomial)f(b)r (ound)g(of)h Fr(N)9 b Fs(\()p Fr(d;)14 b(q)2805 1042 y Fo(\017;)p Fk(\006)2908 1073 y Fs(\))34 b(and)f Fr(N)9 b Fs(\()p Fr(d)22 b Fs(+)515 1186 y Fr(hC)q(;)14 b(q)700 1146 y Fo(\017;)p Fk(\006)697 1208 y Fq(0)804 1186 y Fs(\))28 b(obtained)f(b)n(y)g(Theorem)g(2.1,)g(sho)n(w)f(as)h(in)h ([He1])g(that)1057 1368 y Fr(N)9 b Fs(\()p Fr(d)19 b Fp(\000)f Fr(hC)q(;)c(q)1495 1329 y Fo(\017;)p Fk(\006)1492 1391 y Fq(0)1599 1368 y Fs(\))23 b Fp(\024)g Fr(N)9 b Fs(\()p Fr(d;)14 b(q)1970 1334 y Fo(\017;)p Fk(\006)2073 1368 y Fs(\))24 b Fp(\024)e Fr(N)9 b Fs(\()p Fr(d)19 b Fs(+)f Fr(hC)q(;)c(q)2654 1329 y Fo(\017;)p Fk(\006)2651 1391 y Fq(0)2758 1368 y Fs(\))23 b Fr(;)331 b Fs(\(4.12\))515 1551 y(\(w)n(e)27 b(ha)n(v)n(e)g(used)g(that)h Fr(W)12 b Fs(\()p Fr(x)p Fs(\))29 b(is)e(b)r(ounded)h(in)g Fr(J)2025 1563 y Fq(0)2062 1551 y Fr(;)g Fs(thanks)f(to)h(\(3.7\)\).)639 1651 y(Applying)39 b(a)f(classical)f(estimate)h(of)h Fr(N)9 b Fs(\()p Fr(d;)14 b(q)2097 1611 y Fo(\017;)p Fk(\006)2094 1673 y Fq(0)2200 1651 y Fs(\))p Fr(;)39 b Fs(\(see)g(for)e(example)h(D.)h(Rob)r(ert's)515 1750 y(b)r(o)r(ok)31 b([Ro]\),)j(\(Theorem)d(V-11,)h(page)f(263\),)h(w)n(e)g (ha)n(v)n(e)e(that,)k(for)d(an)n(y)g Fr(\025)g(<)f Fs(\(1)21 b(+)g Fr(d)p Fs(\))p Fr(=)p Fs(2)30 b Fr(;)515 1850 y Fs(there)d(exists)g Fr(C)1015 1862 y Fo(\025)1082 1850 y Fr(>)c Fs(0)k(suc)n(h)g(that,)h(for)f(an)n(y)55 b Fr(h)46 b Fp(2)2113 1783 y Fl(\003)2148 1850 y Fs(0)p Fr(;)14 b Fs(1)2269 1783 y Fl(\002)2330 1850 y Fs(and)27 b(an)n(y)55 b Fr(\017)46 b Fp(2)2834 1783 y Fl(\003)2869 1850 y Fs(0)p Fr(;)14 b Fs(1)p Fr(=)p Fs(2)3074 1783 y Fl(\002)3106 1850 y Fr(;)967 1970 y Fl(\014)967 2020 y(\014)967 2070 y(\014)967 2120 y(\014)995 2091 y Fr(N)9 b Fs(\()p Fr(\025;)14 b(q)1228 2051 y Fo(\017;)p Fk(\006)1225 2113 y Fq(0)1331 2091 y Fs(\))42 b Fp(\000)1693 2035 y Fs(1)p 1521 2072 386 4 v 1521 2153 a(2)p Fr(\031)s(h)1661 2088 y Fp(p)p 1730 2088 177 4 v 65 x Fs(1)18 b Fp(\006)g Fr(\017)1954 1978 y Fl(Z)2000 2166 y Fo(I)-20 b(R)2063 2023 y Fl(\002)2098 2091 y Fr(\025)19 b Fp(\000)f Fr(\026)p Fs(\()p Fr(x)p Fs(\))2409 2023 y Fl(\003)2444 2041 y Fq(1)p Fo(=)p Fq(2)2444 2124 y(+)2549 2091 y Fr(dx)2639 1970 y Fl(\014)2639 2020 y(\014)2639 2070 y(\014)2639 2120 y(\014)2690 2091 y Fp(\024)23 b Fr(C)2837 2103 y Fo(\025)2904 2091 y Fr(:)240 b Fs(\(4.13\))515 2323 y(T)-7 b(aking)61 b Fr(\017)29 b Fs(=)f Fr(h)h(;)i Fs(w)n(e)g(get)f(from)h(\(4.8\),)h(\(4.9\),)g (\(4.10\),)f(\(4.11\),)g(\(4.12\))f(and)h(\(4.13\))g(with)515 2423 y Fr(\025)23 b Fs(=)g Fr(d)c Fp(\006)f Fr(hC)29 b(;)e Fs(that)h(there)g(exists)f Fr(C)1686 2435 y Fo(d)1748 2423 y Fr(>)c Fs(0)p Fr(;)k Fs(dep)r(ending)h(only)f(on)55 b Fr(d;)28 b Fs(suc)n(h)1047 2530 y Fl(\014)1047 2580 y(\014)1047 2630 y(\014)1047 2680 y(\014)1075 2651 y Fr(N)9 b Fs(\()p Fr(d;)14 b(q)1300 2663 y Fq(\012)1347 2671 y Fi(0)1384 2651 y Fs(\))42 b Fp(\000)1684 2594 y Fs(1)p 1574 2632 263 4 v 1574 2708 a(2)p Fr(\031)17 b Fs(sin)c Fr(\022)1883 2538 y Fl(Z)1930 2726 y Fo(I)-20 b(R)1993 2583 y Fl(\002)2028 2651 y Fr(d)18 b Fp(\000)g Fr(\026)p Fs(\()p Fr(x)p Fs(\))2333 2583 y Fl(\003)2369 2601 y Fq(1)p Fo(=)p Fq(2)2369 2684 y(+)2473 2651 y Fr(dx)2563 2530 y Fl(\014)2563 2580 y(\014)2563 2630 y(\014)2563 2680 y(\014)2615 2651 y Fp(\024)k Fr(C)2761 2663 y Fo(d)2823 2651 y Fr(:)321 b Fs(\(4.14\))639 2878 y(W)-7 b(e)28 b(get)g(easily)e(from)i(the)g(ab)r(o)n(v)n(e)e(discussion)h(the)h (follo)n(wing)e(theorem)h(:)515 3061 y Fh(Theorem)k(4.2)41 b Fg(F)-6 b(or)30 b(any)59 b Fr(d)47 b Fp(2)1595 2994 y Fl(\003)1630 3061 y Fr(\026)1680 3073 y Fq(0)1717 3061 y Fr(;)d Fs(1)1826 2994 y Fl(\002)1860 3061 y Fr(;)515 3161 y Fg(ther)l(e)30 b(exists)f Fr(C)1008 3173 y Fo(d)1070 3161 y Fr(>)23 b Fs(0)29 b Fg(such)h(that)1062 3265 y Fl(\014)1062 3314 y(\014)1062 3364 y(\014)1062 3414 y(\014)1090 3385 y Fr(N)9 b Fs(\()p Fr(d;)14 b(P)1331 3397 y Fo(\022)1369 3385 y Fs(\))42 b Fp(\000)1669 3329 y Fs(1)p 1559 3366 V 1559 3442 a(2)p Fr(\031)16 b Fs(sin)e Fr(\022)1868 3272 y Fl(Z)1914 3461 y Fo(I)-20 b(R)1978 3318 y Fl(\002)2013 3385 y Fr(d)18 b Fp(\000)g Fr(\026)p Fs(\()p Fr(x)p Fs(\))2318 3318 y Fl(\003)2354 3335 y Fq(1)p Fo(=)p Fq(2)2354 3418 y(+)2458 3385 y Fr(dx)2548 3265 y Fl(\014)2548 3314 y(\014)2548 3364 y(\014)2548 3414 y(\014)2600 3385 y Fp(\024)k Fr(C)2746 3397 y Fo(d)2808 3385 y Fr(:)336 b Fs(\(4.15\))515 3617 y Fh(Remark)32 b(4.3)40 b Fg(The)25 b(c)l(ondition)f Fr(\022)i(<)c(\022)1752 3629 y Fq(0)1813 3617 y Fg(\(4.2\))j(c)l(an)f (b)l(e)f(r)l(emove)l(d)i(sinc)l(e)f Fr(N)9 b Fs(\()p Fr(d;)14 b(P)3018 3629 y Fo(\022)3056 3617 y Fs(\))23 b Fg(is)h(\014nite)515 3717 y(for)30 b(\014xe)l(d)g Fr(\022)i Fg(ac)l(c)l(or)l(ding)f(to)e(The)l(or)l(em)i(2.1.)515 4099 y 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