Content-Type: multipart/mixed; boundary="-------------0212020727806" This is a multi-part message in MIME format. ---------------0212020727806 Content-Type: text/plain; name="02-500.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-500.keywords" Free Energy, First Order Phase Transition, Singularity, Non-analyticity, ---------------0212020727806 Content-Type: application/postscript; name="DEUXPHASES2.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="DEUXPHASES2.ps" %!PS-Adobe-1.0 statusdict begin { true setduplexmode } stopped {("Two-sided (duplex) printing is not supported on this PrintServer model.") print flush $error /newerror false put stop } if end %!PS-Adobe-2.0 %%Creator: dvips(k) 5.85 Copyright 1999 Radical Eye Software %%Title: DEUXPHASES.dvi %%Pages: 29 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX12 CMR12 CMCSC10 MSBM10 CMMI8 CMMI12 CMSY10 CMR8 %%+ CMTI12 CMSY8 CMR9 CMEX10 msbm8 CMMI6 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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 46 /period put dup 65 /A put dup 67 /C put dup 70 /F put dup 72 /H put dup 74 /J put dup 77 /M put dup 80 /P put dup 83 /S put dup 84 /T put dup 97 /a put dup 99 /c put dup 101 /e put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 121 /y put dup 122 /z put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueXX 5000828 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3ffd20326947b1065649b533eb5e9e3 a88a87fe987918a687950b26f647d6e08bf14d983ed1d4a8ca1e2da8d985f944 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1957a2f946055949715599443d06cfed4dc30788bf2392aad9b0445f0a39cf80 0ea046cf2b104350e94d9da90bb7ec317bad1a69b095889b803a3d48f8c32981 40d9ba6ff587517beff072577e417eea1bb66083daecba64bdec6e593fc7dfc0 1b99242f5a018db00bfc8d6b04504bf0e8258000de08c23ee08b7e2e6676c350 ed799d1d9ae9367bd28a701d47afca9ea3de618cadd20971f9f5b92ac6f77d96 720d0d3717df3c8dbec6884466f431c1298e6816c4102318599105a6e79595be fe94ddc1b01bab96695437f362eee272cd77175dd0078711a502c8f15cce4900 beb8ca406f5c13c028be9b778cbaf4aa21ae3acc1e6e6b4ae9e94afa840bde43 0e53d75008cf6aa70c4f899ebd739c32408a59f24d3ead4f9a2ef731c7f38dbe 5aba70f3a79443a98e4747bf05fe97c31e4a3f28d805364f08b4c6e1077cdbea 249dca4042c57e0738d58fb34f031e08069dce9a33d0ff71d919f952a868116c 1ae6e4fcb9466fdd66a6eb4ab7ba637aa89d13cf3da7ff4bd87be2c10edd6e9c d7c9202dbb9126ceded59b17033746ae1434193710b692782a351212fc96ee39 7b5742ba37afa4c47b1ecca06f9547fe3510b9a264633b9d7701442b4b644ce2 2603b485ffb89f17e9da9daeb7674bfde9cc7a3d88abfcee5fd41c78228488c4 10ee3cbd7ec553805cf04f1624713a3ba3529e5c50cf2c09706b08778ddd8cf7 a8e931be1a2a3552e60004302ef357f1fd33e9003474667d48da5cbb7c4342dc 38c2fb5c0508da45e94595151f30184181fd37af0a106b7f077cf62a64ec3f8d 9af0bca2188587737887b78f36616aa5de0dd3b759f9fb91a354c820dd4119f3 2102cf1f4d6c32e53dbb9afc36627ef0713e403f55c525fd591deac49aa5b293 bb42500de1cbc49b7a94edddfeb5bea5fb9c15e4b8d84fa1c8282df23916f70e 97951bc8cccd23f521ae1636c2c284a097b678ab9f49cf82e72ed8db81844ab7 5d22209cb9fd99e76705f33524cc31134c352185f30411fce66213a5621447e7 9f75ff937fb0c79debb3fb7e223c93e7ec69960ea4e9adc4f6675c05412d543b b943c023c8aaa08195dde117b849c3403b78035d725d2194cdc682885699a86b 95b9533efca5d58de2a9dbe1831179231b476621f6442e903c8365cacd97ed2d e1830dc310c7db01e157292a80030b1acbf3fef08dbc5d48134cac41a4c9baf5 6f100c6d9e5756e92687e2cc6e054df0c78a9455aadd94d52382841b15ebbd2b 75f600c5cb92912823a8ae7c88a1fbbd099995fe52906a77a20420c3e8b8695f ce259e01eab9cebc69aec43aefa9bdc74b798f09b54a89d6c1e933c00131bc1c 8e01879664eebc60c55bd02117e3a4297b7f82c32fb68c3d63232356501daf52 f1b2b107b14775fa146c9fdc04bca4ae321663d68c0c7f20750f037ed8fa6077 381e14f0a09f405590db4d145d0e98b23cc3e85b1a0719cd85f010070a9e4df6 2eec797ef52d452d92a58f21c5e64398a3c1865527b95b3e80bfe234eeec6cda cd227010e22b93620105df1448cece8ade68d0019b43c6413685c5867b842325 a381580bd51e3b44ef0d377dacda71699eac25c2d8ca7607a620e7a755e87067 a3b647c7fceedb0e106eaa4b062bcb67d996196864badf0157eea6557122c294 732b9f55bf191415bb48573d9864656dcf1108c82d1ed117e14c13e5c2a354a3 9e26dea6d544f9dd73380106a711809fb83e48e7d1a0bd9b06aebd7834c60de8 36fd235299662124fe3f3610476587098c421022369668f4e04d98687251572f 6ef3fdc03f112605be5dd38745e3b75ef4bf08bce859aeee02e2267600534d07 ce92e68000aabb42925d2c4d91ec69c618cd333335c63dc2eae080195c6a31db 2774440a0573f9f40114854ae055c9339097c9ce26de00466fd9b751b6f6edda 1c31ed1ddc 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTT12 %!PS-AdobeFont-1.1: CMTT12 1.0 %%CreationDate: 1991 Aug 20 16:45:46 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTT12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch true def end readonly def /FontName /CMTT12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 71 /G put dup 103 /g put readonly def /FontBBox{-1 -234 524 695}readonly def /UniqueXX 5000833 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5b667eed85fade30ab2ec4e1c2f 4fe750cf9a69506458071de8896261c001b1c290937fa42a1fa87de406e9f86a 4c9c0f03f69fba719114f8ae51040a4c03a58b6f3de027f6726f32743dce7395 14c2c6f3c2de1aeddcfa8eed7da56dd68dbcde33c29ff7a694769049e4325ec9 61b2df16add2ad46f18781fbd63eec62c26f1db41e6a666a09886382dceba91d 8ff7cedd64ddc8a2f15fe34076a064c46febd1f4dc3707aede0ce8af161bcb8e 4ada202d0954baca5be71c28ab8bf32fd72029aa07f27d658a68f454a4bf386f 6314224d16c7f0a8624f6b86319a4e2fd178f596f3b20db032ea28c157a68a69 fad2bc88881658d2286c0d4ef061bc070c3145f2a37ebec03f51de7cda34cd97 7eeebd86b00f9758b3a496f41d9a74279ef3c6322dc54945f95062c5f37a1e51 cf7eaf45f2b3634b2d7d01ec3b298ff35b6616cd3020e096f6c6aa2bf04c4a93 61c19f3c67a28d936ffc8120f1e46980b29bf54047c5f1fa48e67611a74fa434 3384f802a4079b28b94681e119ab0c1692de5bbf426440ad1023506cb80732de c64d9061869cb34de924d2c26e404b7dbd9be55b9c0c5ae3095c15d278adbb49 12df1484ed2d73128e5d9c1f1a3b9ac19e2ce5366c96644f5b3d3eea589caedb 586fd9b8f4df32ec910e9b7434d5bbe57e03892eb03890567df5b811afa0da5e 929d456af362fda0789aa0cff464e238029294fde86cfc8d7d33d5c98b18c76c 319811e4d3d49560f2ac39103594ec513e0b1377c2c3e22979866c19b61448d5 a7201ca4b7538aa370f6347253b0853d4c5d575a7572f93cb73a598e2e26b11a d46a1126e44e7e1795dff01562316b6ccc6678663e66a73db285746df3f4405d d51714679581946ef723f105e9623f044be53642047b0717be4512baa53a8fa1 56552670d9f870ecb50731caf55c931f28f8417c1fc70910e1e72da1c6344b66 9f066ec38fcc5e0a8eaba95d79a5573e8849215ea6462d150922562befa03305 50a224decd9ce7149d9eb4a13f5608168fee1350819b15aa70d06437c8c35d94 51edb2f6ffc31b5045db8afb71f2a07d6c0f13434993616fb51ae77a48d4ea94 9d1bfe7902b0b139022b7c2dc87263ff1babde951e25c1d20f9f2bf884e6cb85 1d9ea4e208852cf994fd0c8dc1d566d6307817d3ba6bc3291d79ebfa52725f6d 21ca08ad1038b2ad0919fd36e169af9df89aa0da28ac4c2de7452a99b284e502 57c9186f5aa1be2401ee95d53d49ad3e57512bcc90e0d0d65b8312b7c591d31f 036429d6d3b1102a3ca0bfca830b98af660d2abefe391b21dbaf59d1daf1992b 480910ada36f242c5a7cf31ecc3af0cbaaf15442133b19af0bc2b895359b4f49 3f0ef4d3a7b4a461754809bda1cac648dc242434d02119fd7acc43963009de96 7656dc63ec7438d1a31921f0915373e793ab750b694dfa8cf26992a1b8cc00a5 65beb0a57be3de75c7775906fe1733c26b99a4c26a9b3374055e3ff93d471b2b 0452a3dc8c6f9947eee2b92194811dd532cb3fdac5a47dab83a5d8130304bca1 356c8bcf62dfc73c5c3767959413da1528a0cb2bff2b48e2e4669c14279ec159 893f2afed35ea107c952576513db5119752ff1ff423534baf80b72cc18fdd7c9 b4f032ce4267a41dc9e5d9d9254e80a3f3284e7a9916f9dd39464dc297ef7f45 0a 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTT8 %!PS-AdobeFont-1.1: CMTT8 1.0 %%CreationDate: 1991 Aug 20 16:46:05 % 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 (CMTT8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch true def end readonly def /FontName /CMTT8 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 71 /G put dup 103 /g put readonly def /FontBBox{-5 -232 545 699}readonly def /UniqueXX 5000830 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be533de8dbe3ccd493ef4c9656d5d5b76d531 9e704539dd76b065f45e6d6f62ec499c9ab91923a364237ef75e1d62d52df3a9 43713cbc63b16c1880feedd1633766248fed9d060ff709951f2ec84ecdc06bbd 9cc4a865e1f275455525e5ac7e2e6289615e98cf7ce3763b6914d3a4c5b75b29 b8043dbd898ccd5be448064cf1af2f80f969a44b4f9b801c42f6d6a6d0c3ff42 e0b1cff748c1fc49b598eadde0c2802dd591c54b802ffd24cb7a7639295713d5 4e639514ac78460672c9c51742ae2014e4137ba4303051c06f686395aaef86c9 9d7f9a97009436c2ff55b770de3e80cef0d53f4df46aacecc21632c0c676e2f5 ca7bb101a239a0ff67ebd37a6b8c4c8246dc2c90b759dc03fb26bde8f134f552 23ef3d6496aa0c4a2d7b6398de6e6a0474ee1b0d78b2d865614609f1db92fc56 4ab5921078b4944e11a2162902f0e04f4b29448aaa22b4ea92fac48f8dc47535 d717dc7acd08b25d391c97c6c540ae0a98be5d6a8da564ab73e8b5d3b5e1989d 6e25c3a35ea44d596b5fd6b2373b30de7d59c1831245edb76d6a67816ab91a80 a27a6597c6dfd274db8b3e45cb01aedee59ba6efa1428e41ed557f051012a389 9b195ea44a4e4d0067398d9d40ea26195efbffd2cf785540bbeb8964f6391361 904fa5e187e4af3baeb4e8cbe7558fb7049c94a1769a2ff0b44f679bf6f14d8a 974bb8bdc735d5a9798c89d7d81176ba98dc22dddf6766acd52020e3ddcbc94b 71ad8a336dbe9262f302d8e8d5ced4481b19a586e1d73d0138ab0b732226185e 8cfb8408890ecb590f032a5633352c42df62a7dd1a3cb4fb12083709edc26146 f31161bae67a54aa5b41ca7af61b515c100b38a01b77f831245a013f804584b7 bb764dab019978bf8edc1dc7ed623a145b6c115b70835642d286fd3b306825c9 3535b8e4ce1b3787f162efa6afa9a2d728a8954cb2cc6faf123fd4f1270e5277 7eec5a271e926568f18851b1b456390ec0302b4c21dc0122a288625af00757e3 32e85011ed2e930d6755d217369d4c530ef240e808f997a8087fd963eb5a1b21 37230953697a3a3b1049904933e08d84e87db8b137ed9fcdc9fee61463bf1eec 1ca50b51c1a9e9e55e322ce22a4a7428709dc01c343f6ff728b2fefe75654743 fd74549a8970d8d1e7c0e9be982c82e3e03e1548c990e58422872119a78da706 800c54b305389589b558489e1f899bc33c3aaea868560c4d0e408937867d3bc9 6c4d1216cd081e7f9e72c5e087a61e56353b19d46de75471bbe0c14e38d50fb6 9d6c18b79a0ef25334a403c2bd03aec8dc151822ef4d0d5c77ca78d0e5bb1e21 47facb6db519aa47205a25f8c104081be7a899ab5fe07d319ba891848e7eb229 03d743358b1d6224150fcf447429d034e2e094550f1328c40aa5874db03eca02 74fb9f7a88d13a38096bcd1d74c6df52c07a8c671c1400fe136f53d7317805ee 71059fad62fb616dc02264444e7c6c11564f667523f4df23a7d617b52fb50005 cc36b755f3bce71c6e22f8fc1113b7de84ba110fa5d1d5c4345d7e5a1508bbeb 2eb73ad18b961a5fc52f52bde01304282b511260205e7bf858b7b4fcf0074888 b9b78af2f883a1f0cb2dfcb3695b9df6fd80361490c8057f4c4dd0c7391ef3e1 c013d2c929fd357282b0a26d4a404052346316ff2afd484a4091d0b5e4a60e 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY9 %!PS-AdobeFont-1.1: CMSY9 1.0 %%CreationDate: 1991 Aug 15 07:22:27 % 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 (CMSY9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY9 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 52 /triangle put readonly def /FontBBox{-30 -958 1146 777}readonly def /UniqueXX 5000819 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8 3750d597b842d8d96c423ba1273ddc63eb43f34fa90ea73a234af35fd9b24eb7 4f19626e9124df7acb92629cc87b2cf81c4a89b27447806db34393ddf0402958 e2daee0b4bf315f9aa0a2650d2e11ee289ffaec940d0af160fe7f29411c1c458 69d1ac3bcfe2ae4da72422549095f1fc9fec606289316f106e719a4ae2fd45ee 868fddf07a80e9a626bb43cdfdaa9dcd208b88fd99af20b27f9d6b5cda6a3fa8 8d5acfce8b5eebb38854378cecc3bfdac526514f1be9fb28ae628918ba942a3e 05b78c3aff808cf416a5ed01be280231b75a7aee0c66b0549713bef6b3cbc188 e625ba81a0a7f34771e69cf51959595a11c3e703d4cc8f9a29d4ff98a4581f1f 9dd4ec6a3ecefc689b9c4d0a36a436319f17daf6e464044f41951c290bb70240 7c74f272fb3353896b314ee620cb5ba6735564b3e5f92002538a87a6821fe03d 2fa0140b33408c278b45a971cca3be2d42e50fddc0b4995fb3e4e1f4f27ed44e 33ed9425c59a8790431d9b01a8edfed3aa105fcd7aac865b4c1510aa96d377cd 51bcd3d5ef548f7cd24814870051fba702f6bcabb3397b439e4be6da3d81b6af 07b3ed9e26aab276a673e8e9d3547c040e5aa5b309dded389911ad3c9fe583a7 a8365be7153f73d96cb7416f7a0350e941a11564c903e8df98d910688ec50849 b68f 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 12 /beta put dup 18 /theta put dup 22 /mu put dup 23 /nu put dup 64 /partialdiff put dup 68 /D put dup 82 /R put dup 86 /V put dup 114 /r put readonly def /FontBBox{-29 -250 1075 750}readonly def /UniqueXX 5087384 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 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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 107 /k put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueXX 5087382 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930f01bdaae649990604a33aa9eaffbe5b5489 e5c7c9ff9d9be01b08220832c41caac64816b53bbc087ae4b621d2b44b21ae5e 3f7fe4ddb05c675abfe30510eee4a7c874bb57b2ffe521a6875eddfdfd18c781 25bfca5a097aad361dd94df46f14026c25ea988194899b95001209cb7009cead 4b4ea03473ea120e7a7e341d4e3926796347fdb3a124c29660ac09f4648b313c b58e824884382f24ce08d2edc686923acace1201c16a07a42803288cd0b946f7 546510e95b1fa1cc96f0bf72aa823d8e83d7c68c752c82a54c752eed2b1e699d e9db1830272ffbf2f4996ccc2d6fe2ae272798989525ef3b67b0d09bffcef749 a805e5f76578222b9c4a8a09b13189a16ab746ddef7875f1ecf83e568f493d82 ff729baa1e0834dbec30a35d3c49c9b10c5e1d90c6e3c8fb737df5ceb3299d96 0fb79632f91f257753b4d2e34e3f54a26c7b950981fb7fe4dee3315db63e75b0 24b4d318baec8aac9cde186f6d65767df1dba35124287c2e805bc660a3a22772 0ae4fa097f1d75107441c0a0fbb9eec17d5516fa03b98756b7df7ac7eff9c7a2 364b9f691ed5cb692dfff9dd80c761e9c4d5d6061782b327302c053ce1cd09ba 0605ebf761cf9b730524e1994e7dfaa956799d522ec53d5acba27a35c4d28279 617771938ece58c5ab664aa00c1bc4efba2d15504247a68e3af0f9bd59a387b3 80db88cca083b06f5af70d768d1699826fb12ce12efc2b4c70923e052dbb964f 5fa957c3ce225c0fda64d9b5321ee1ade207c1cc93105096cc73118f4082594c 8ecddeb0d7bfde6f70bc62f10407c26a476ed88be7999d64a15912493829a9de 62c71f5a34fbf5a843e311873cd98817dc9f8a4aeb7e0738b5de10043f5cfea7 147f31c39019afacba581f8351d51c8da6e4e299427e120776055e8f3670eacb 6c0b673108232691a3c169f9be51e4f609834b56cbb6eb1c26763ae597330914 7280cd97c723f71fc2b6663a53660326d25a3344be83bde128181c44afe96e5a b865cc70715b4b3fbe0b5de17d05854bb80b9fa8cb06cc414b9da1619d9d9bed 999d3087c58c2cfc97f6f3afc485dab294dc3fbd4300a7d992369d0a9a119e60 a2bf2efd01539a19980f0b17474a3a5b52ea1a89d1cfa93c61ff493c58f4d31b 86f19a10616fdce79e8e2852c5ebc40264c299db337140c14de2ededeec685f5 37403f032345da9ab676ff2968340a4442574f1514989a7b6c44fb71f5e3b9ee 3863b9a0603134d2631afb49351ccb0cf05610e51a5bbac02bc1d5d77828ee0e 67f8c833c7e0c1300dfbb7ccec832d248d4eebdc0c05f58f7e48963d79fb4fb8 ad093a3472520a609a4fbc4aa37acf569e467faeb7f943b60d74ecc498e30fbc 3ee440207190bb7fe92aa867853cb82d2ac8fb673688bf394ca5ac5c7dbdbfab 7a965070cd37ed109ef029576031fbde49956e9dd80d1f2add6fdfd1cfe85bca b0 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /asteriskmath 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 4a93d9b872c2d3ac4391eedd46c6cae184e1af981f95ce21204dc4babbba38ac cc0a16964fcb3e98501c0392030d486c9aa8b35acadbea5e2bf9b4e732a90a25 6a3227e489f245b82b26f4e485b7567ffff5ee02e89d640e56b11d661791f511 df9cef5821b08b65b671db106348a8bdbbcd377bf82aef6020dfc0cac6fe1932 7140f3fc917d249129ad07ffdd9b6998cf46901131c08bc0e77fa38ee04143bc f4804222bee432c8d0d89ded7fc94b221bc7158766b997de21e51fa6e41af7fc 3c8329a272bbd2dedcf1bd2eb9d8664abc336bb5697c0c5cb95c7d93323c32bf 766f35e038d2d3b4e5c4e39d9923a1e36271d2197ca4a121765a994ecf4dffc5 9e92e783fbabe550dbbe3a6c21fca2652b83d3e550bdfa9b0044f0bb4af8f6d6 b145c7f1ad0e88f63a2385401cccb0e80f36a7454e6685a0199deb1d52a661ac 0244fd9d4de92e28930395a3b8c436bfc939fede4b97e8a1164a68d975534921 c4d36f2591 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 12 /beta put dup 28 /tau put dup 59 /comma put dup 62 /greater put dup 107 /k 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 d29c5e9effb7af3ef83400b0910c07873a8c56fa7b1eb1ba426043b00cc95dbe dc6e136cbbbcb3f67509964f9f281ebf81fe5b018122eaf66c4a838487e82e18 6e006093042848a903efb3469ab6b4049767aadb95c30408dfd58d8a10f4cb22 168decd9f3ee100f07b49aa44c92139b669cc312ba20192454eb2375be6284b0 26659d964b96ae82d4942e758027fcf23c25ed01115af27ce7f20efe2a822bb6 84004f20243a49c9e93301fc21b80815c033c3e2ba58ef53da2157d524b395f2 b37abca13bc6a2f42e824ab7e47106176b0d6db267fbb795ac7425582df2e3dc 55863468a9200742bd7b552c48f8cf58bc21343bd3b95abfa140f33f37c6f3f7 8b0d8a5154eb7c1f62ec598267f13e841a3e64172663935ac8b665d86540d316 ddece329c008049c5e74b27d59022c5515059bc3b89370b1bc6a169c888bb325 e0b74282d6f053a50da4024ed1e433271a32ab8c17d41c632b41cf9f3ecd5fe1 24daf7aea7ce8a63047b245822930d517df7baaafa69d2d17f7d93cebd45416d f3459365434123a76ccf883a4973bab19807e1f0c4bab03a45f7ed69ff2660ff 3963e4def11001eec4238c368a39d874fd30b3c14f4186ec7700fbce22abe468 eb32baf7b0164f8f21a9b5e706b91411faaa44467cb2180ab03de375c1ab93e9 76e11ab92fdd4a7280a6d1a8dc65c4c89d04c8ed6988045a2a3de9a4c7b1efd3 75b3a4105c3d10ef7781c955b819182c6c351e3c271a5e84789550ef09a61d28 477f55ed66adadcf1ba915c824f874089e4952cd96f14e59cd96e8532236573a 61cdc9a2eb56f67c2919258ce2f6f4d143b6de0e2552c21a75172018eb51f34e 3ca31f3e19dbfd9e2475cd2045cda8220809b1887b95c1baff3cb28f77313f06 15240c147ab0777bcc25a70001a622081430e462804e3d31a190b21030d44970 032868d54a4b0ae0b75cc4aed436cec660891faa939854eb24453b35b9f6a40c b4e9e8b6eecd8e3dea75e2f733278b95cf3dcf020a7846d4f964276f43944129 71b79e46a5019b8d531578fca7f282206a39059ab2fe277e687c0a4c496888a2 e1c36f0df5172d4aa5821229d73ff5e49a58f4d729e7a2bf973d0af0e747116f 3a2c066a9df69591b1139f6f58083407490e4e0ac76a94ef054994ad4eae98f1 9b4a56c27ba951caf62ba374b9bb6eba71bfdc125177f336bd0ded58af88c3ae c99aa47adb0f1b76eae9170470e9e4584da6da90c5b66d1cdafdfa52dfa2fdce d7f47f37eeb5feead1048c55ec391374075439e3c0aeb1cd5da6407dcb8d72e4 e3bb757a83ab1def425a332e99ba56dcf8bd06e669818400718f7f83da86c682 9b45f534bff32b5f9a3b8b9e3bf68c442c692a31afa73f01a12d9455f8cdbd32 52468683de02eab10c3d606338689cbeb2042f48afacb18bd4f422ff56f78bb6 11479a9171bbfc68b80c0159e79d49d343a4207f5545778d8096beff193e1b44 880b64622faaf735f77d84b28b0a60569c570393cddff734de8c69967002e310 382693be5e8296a57002e99f5dfef40d88d0c01f340a84173483493db06dc460 776f92d20bfbba238a0d87c20c74acce7e1736fb30d6ab66a7938e3f0349e40f b243359f12fba4beb15af83c25d72fd3d8793644a3b57f2344bce45ded865447 eca2878c298e51c083c0e2bd93a80f0a11d9e311dbb40fd5731c41899ead5977 a196870524dfd66b52cdfd867842fee181a872f1c5c9256ea2a26520d2dcfa1f 9d1155ff0da7cfd5a8a9455d07a447a12ba9ea8b92cf371815299d417a44e656 6997a55b0709d378a4fc5d80350762a0b61572e1a09688291289ca6d8f2a1ebb 6f7e1257d26c07900ac99131d5ebe914b3e129fae7da92c332f8a6a2d7b36043 fddfbdd9ba11af27dec519a247b503bcc823bf68ff89cc09f731a8793ee0c270 42b3c3b1f74167d5b15f3b4e4e9f94aff797cd382e9bbbf08d68a8f516a2eee4 fb4b5df1a7066d5eaf527ed093d38ee53f3c1f13f4d2bf3627a0337836e6732d 1801864fc682eebb37a06241ae46a9fffccbbeea08517b58fa3df47eba4eb4b1 04028facb79604658f77db1e4d4e901f58b317e45ab255019f5339b0e18b72f9 61edf71ddf37359f44185fa4876564d2ee91827f0d6ad7f61d28a376734df554 a3654c0b25f7350899 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 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 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 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 86 /V put dup 87 /W put dup 89 /Y put dup 90 /Z 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 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 dup 122 /z put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueXX 5000793 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306457c7e62c1d269f306bd3402e266de fc3b5e7d8a8d2f5bf0fe6ddd40d07391df4fad4a6018dce29a2b8f692b29f202 3a7c0e66de8ed85c14f1f8492167357f51a7e84cc5d92e0fee4d81cf7fbc8de5 2d2e7bb57142033993f9c08c315abade8dbc4a732e84e142d3bee51557910e12 cd8aa37c459a5e6b7f5269f59078aba3be4641a11ac48d0b625c8325b38ec08e 4c9e5e7fed976a5650d99d82114f449b9ca14c4ec957702295a39a93ef93f618 99b8ea06b092c3c1e503e6e436e0a9fa22576c8930ab3dc8c20f5d82b69cddf8 ff4dacfa9c54bed5a3aa3ea5b129fe96be632843b9b6bc91b615581a985db56b 1e01ca60ee69ca92cf5c0882ece62edad3e106d835348822400f0b66af658f2a e56ed08f8b0010571807009b73ab12a8cf14ca6c71f03c2a48c500f9d62266af 154a6375ff600d9bac3f05ce34142d6867a79581c533176bb2f3117336671e2e 44638a97167e2ea9644e31ea16c2ad2990ea33c54001e0c8156e6de8ab6a4d40 a7137ba275f39589fea2e2db8256adc103d6f9cc038037a47e8fd469c5f98a5e 3c15bd4ace40d340018b1cff7d1ed8abb0ac57b5b5a2c20a51957b96c453edb7 dae5affd91a46d938fe0a13363001d844ded4323f1ee6d30012aea19b024a552 315505535c85dc26bad31e09c50e6512802976d298c4e90d0044c362e6bf3ab3 62a454ee93de25ce54411090c29e9d75c80ce26a84404bd9de3aee0e3f921ac5 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65db26971cf49ddb5a4653cfae5b8d1d7e355dab5ad109e3126832a330771aec d409fbea902c8810be21ab5205a8482f57bda92dee7135a87e4c97638b5d3949 87e5885873abf946ee8ea303f3e4aea75836386aa2fb1942aaeac1cd2283117f 24347b8fa006f4824868c2377fe49acced5f15977ee6ff1cfc48f8f5c36d2960 9e5381788c044b2d2c0543016b2b6eeec8ef26339ffd56f07b328506651e9545 867b9c854adfaa49272c8dfe9cc4d8d3f304c28662c22a25fdee9974e6cf203a 7e68630fdb843bde30d1fb0916eece0614f22e638ee1a58222085baafa293f05 17e777a12ccb609481edb23997977cdc1c5fc0fd100751d76b39a227e0e047f4 9bd240f994e57100e09afcd6f25836fd9a50b1c173499423d19926ad0b32c1b0 f270815358e807230269b9edf87014e07ba58ee31157c9efb8ed6077943e4a0c 5b81e986cf4a52d4601b3690fffc982a7eea72ed33e4195c4dc3be1f9fe32033 02808bc84dea74be3046e2030bdde4b52c0ea6febf31db5a6c39fb9d4f0387fe d141aee8899694ba129e70b2b4f86e332a15c396b0a4f841f0546b016758e527 fccd903838684d3595d0bfd210036bd328d150b053a7babfd54d2702c28023d4 c94f7c121b1d1ecc1375083c0c8fa73795f9ab9131a7e394be10658837461be0 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01648b5f75be2421980cce17fd219bb423418c71859edf241ddf8aaf4f6198a0 b4f08f943766a10986070af392226bc97a0725de6afe818909b05bcd9323d6ab 94e17eb60f0bede0db822f13c16b003281b1fb54752fef3830a5cb8f1c4848fc ae5b9ecdbbb5058e98925e8b68a4474bef218d625358081e3290c83d8073d998 064fdbca335bd89bd20830e3c0c21da1530f0706759074eab453f7c6f0f78b23 40a5a9554a68f1a5c2e314e9c954646ffb04d2430a8610959583d0c66c5b3dd5 1e8d171d06a6995b3b40523596d5026658eb8a1f5ef67a58b1d992143c703c97 33f203b3810b6775e529a6a74c2518e5406a89210288f9ff3462d146271993f3 0225306ab0552d00aa68b7ad1d536b7d3b7eaf9839623cb8015538b68deb5f7d 0e90d9a4792e1990a5f466861dafa8c70ff9a74c7b763d21a29248380c13e7f5 e715195930905a8cc3be571e224f8cffee3b9cd31113b2bd9dff4f63c9ae73b6 4136c62895d25284cc8e8ea2d7e14ba18e63f0a0bcea6345983d73746ff99b36 125285971b3e999db91a8a6b388af9d92baecdea3473c8c62e19beee314d446a bce8993bb3b79a95176c276db158afe2cfad0e6e89e349c2c229a5b4de14f7ea 18cdb86072ea70ef8dad0997a6bbcf54bf22266fddbcc54f6cdd104d5a990002 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6fbc5aeec6061494d7822d5cda1a33cb8c2d26816aea0c159e46f8a6d1726094 bb1142c0623e7679c1d219a161a4cbed7fdbdbe49eac9d290508458070178f42 6dbca8bcb1cd492dd91b63dd3531e5cd6a7a8703d00513483b57ee413c7c9ac9 aa97cf43a901b3f707ef3540e085cd13dfdc02ad1a615f808f284d3f6e6e3637 0691501ba406f45889a9568ee735e29b441c346e26e436c22f7c20bdcea17981 212ec657899137c4f55bc5f847066518e6c4fb3048e55f903072010511b96475 7a7839b8b3cb1b0bc5afed29f55a51e8f57738154ec25b18a3c2dc778c7e0e99 c9bd411a9096f3727a77fe0fc354d40c7676ccfc1474738e4384d3e10836fcf0 a37018166b148c42d69ea09364ba8f1119d5be02e898e6e5a4b0e2c4e2dacab5 ec51371ebf15ce6eacff6b5960ab9e9f4becea7e6abb340c03222c5353ec27d1 bebd4cb215b410a65a80d2370cac2eb3a7d31ace6a7cadb73bd949d908c20866 dce5684bfc898522ad0773df1b1d5ffcfbf0bcb9d25ccafc400f6299b4b1f2e4 343fd634831e97a937ec2645496c770e9e647cdaaa81f960475e8c52d43e717e 5a908e0cbc59d3fe5f912897f271ef23c3b45c7f62b6bc88e60aae053deaf58d 34adcfabc8d84dead6bcabdd1a01fc7d7d77e05fa58d1c4f83ea1a927829ee31 8f488fc49bca2db2545686da3f4c0b86026484e0a8dbf548dc66ad88c75fda94 2ce87a4f75097b6657313c1a7e4c79b86c73afc02cf86a324a3f45925412a933 b192d0d3f4de782926c17e647fc776e93216d14ac543eac83e05856ff7ada9f7 1f695a29c5218d13f08bf210241adb41854ec405fb22eef28a3a28c2756ae169 0b064ccef4ba465a6440fe9498a3b158e99839d730620e3e182a8ae9626011a6 b8f7611b46c9e921b3cda6ff6e2a44e4512e37c8a8aec09b84a948f7b371789d 352758e5ee68a2be829bac69b45a069dadc583ab9995f3e52361655618c32cab 6c22dc117d4aa11059b6798170cd84f1ec18d03a9cfcb295cb29494985bfa685 efa0545047ba963faa3fe4000c762f288f503ae7930292f477de7dfc18174e5b c78ca3c219d46715ede492eaa096db4e12432cfb0b3bdc6180ec 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 49 /one put dup 50 /two 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 977f844d32b43b0e48caed5a4bdabcefe71695d69ca784dd32fbf3ff701f6b07 72c3661e0561ac614e9d741b232a0a42532bf9f91c1b6e4f7fb5df6a7d57a413 6087c7366279bea1da2e5b45cb2cb412ed06370ab7acfffde951760e5780f1a2 a5b0d3a36404e2cc8fe0d4974abb14ad10292e5dc1052bedb764929bbcd286d1 c08d9f7ccdccaa67dbc6be59a8ceda22dab94428ae070410145447d9aec13d02 017fb1f46e77ceedee642474b15cfcc31626213f04ab512e32081ba880b0f341 58eda42b8e5c337362cdc37858b7ce30fd20ff6407d3d2cc2fdf219725fe5cb9 cc52abff5ab3d8ffcda3bdac17a3a73171d598ff8f4ca694c7c42785193a055c d6041f1727a1decb1bc19834aa6afba9ced9ebedd351095c46f9635ea198b83f c2d35601180b315bf665497dcbc23afe3ed42c803a9d0e8f00021831ddc48526 30b565a2b60211b54517b353c56b506c1f8cd337af6ada68340a07236d8d47b2 4d8257058b300efd0373bab5c4ba73c24195b8c34f99b1357b47d5810e13b4e2 55102ce9702253480412830f50c112def6874aef7121f3802b0334983a62ee04 6736f791aa683776ea768fd0a090a28c4b8459 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 0 /Gamma put dup 3 /Lambda put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 57 /nine put dup 94 /circumflex 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 f4fab969bba2b44e47399fedf5caa1bcea216d7ba713d523dc2f54f7e2e63099 0959727536c2b31f5adc7f2f13c2ab97ba8d4d59f81bafa2cfefa9cc8af6070e 4653181bc31fa47f115efb87195f52fa39d813e094e0252ab89112b8e4150bb0 9907e311b57067c159cbbe5b994962ad32ac5947e0614817aa84827af882a709 936fe5002ad18215ca6b3bb5b85eff143e7757a36fd23ecb3ab376dea1dd8eb4 a9525eeae27447e252414bedd5471b7a2424a232dc6dbed858b41b41de0f533d b03f34603ec452c3fa4d86f6dc47aa491a2991ed351a69de9d0ea558a53a988a 03cd658d87117b28fd8cd553af1908f09b50bfb1d56c1cfe117269470a70278c 261d7785b8461f44577a691e4341b432f38241aa6194abe3bc5a226682402639 02b51da0c2989a75b501e6cfec648ef9af29599d102f1144eff7b59f206ac580 14d02d5831c8a54a1b88bf7fd7b83521b844e281a1c483110ba8eeebe82f60e1 47e6f2d32bd6fc1c2398d69e755014480b69dab449d58a96aad390fe9899283f cc5b9cd48e638010e87d6ebb1debd3409ba98eef9fa14e452d86016c4f04d361 170792d5cb1c8948c01428e54bcb82b16a82b99ccc1512bc7801e7b57087195f c245a7d78ce356e2f6cbb65e063ce7e772bcf0c7ca2302eb659acf56495ee07c dac9053b95d041cf2faa6c600c5d03d484ca5c7d33be449d0395eca3d69d18b7 0f5c20dcb3462008f22051b583a560e63b928a619ba2a693f8f52dc1265a47d4 377087e899bdc2bb5cef04308a1afcaff7daff0c8f1d3d2c5a4ac15495c47eed 4118c447e5a2672d8d7a0d9e52eb043a09df6a3245aa02022f23ee7dc0c25909 6d59440c82ff1363ee9309124fa4c3883d261e659c7499c5927b94a889ba26ef a7a19b8f8ec46044e64d284615787a72202ed3813ffef409ac79d207c5a2f6e5 8576277119cb92b51f107e7d3a369681a7504c49053635655c0e554ef4e7991f d5b072e665c48b22d42737d50690333214dd4da9cfa5c95e80dfcd66ae984172 a65ca2b326a8f8420c369fde949ea27edc328321a72c153eec58e0cd11316b54 e528149e65b14b0cd950c4a29bb56e0c95be7444fa998fd2857f5b150fc3bb6c 976b27c19d65d67e42892d3f60df449a460a111303a26ff5e36e7a50a9606581 b53c20e06889606ab48b4088d709b4342d3352e651bee26e7e6691cac424133c 5923ef165c7a1a7a22d95de915d8379dcdf29edc254502995a1e3defe439ce86 000f91ad8a28e883f4e2e8681e89f09e6331d526fcd038bb45cd37f65e825289 8f1e617bba151aae9b2b8e1a481ad05c6ad4210434d048dd56aa97b9a0585407 e37722768a60c0b442150896956d7e594acc14d73777536006eaea7d515bc972 7a64514b677778e1be68c57f72938c45e59b4b9a049f26fc5f9fdf2db4b28821 b5450b89b634e6af61cdbb5ee85f2c6366b344aa4224dbfcd4700713e08ad764 836c1daf0feaedc19da9d8df3b39849a7e97a871dd1bef1707fd6d9be92c2643 3bdec79bfa4ef595aa8930e8f3a6dc2499d1eece4c3270a8391267a2218dfa03 bcbd83233278dcce87912a40a05795439f0ac73cdff044c3d4a18167f8ecf490 14cb16d8577aa3f211356c5f3e949d67be7d7bf621ae54268b20be1d2723a108 f910daf9715a4123cd84089c6db261cec4709cc4270d8e22dbc812c3390f0b15 5f8b3d7847fe5fae4610eaea3fd13523a1c9b47e8c85ac31045a15d9734b5603 62153b2021fccd1bc00e38c85b85bfca1ec70bee832d60e9793ab3ce51d483cb 3b10a78ccd78c339a409bd86ebc32ffdbf341bb9eff284ed3365eca978bbe60c 998381f1a0b6583535a52357f6d06e83f3a02b99a4909c788e63db7cbaacb3bd 3108904cc96a108a145af0c1add394c77b2e9f7dbbe287fef48190685d1a787a 10b68a72af143e6c913b1099c6c87b9cfc6855a41313a82ef3caef47bfad032e d292615f694e2fe85125ea407c50a67a939d79793d0cd89e1debd8e6870ad703 b1d763cce31fc825f4b287a02355778d38c8fc7f8e735b1d506408931486ea9c 475360a2dfd2221fd0e7577203552f1bf5bf598df74fdcfe7bd3c5947880e6bd e4047e42a705002e4fbbbd1507361d7809cad6240a9a00490bd6b4e3e4fec7a7 107f8d5a3602379fb924a1d6eb6a6a8c2ed7768131cea269df6dc646996034f2 d09a01332ea7910dbc12c51b63eee5cb26681326141fbaa629a6c0dcc831590b e4aa61888dcd5b22fd79a7942d4722570e41f217adbdc14fc6944079f1819f89 1274ef02393fa9b00815a40ce6a9c01d70b995725b82e25a36f79dfd852a979f 3729e89ed0f8254fc25f40c044d024cc94501c3e839241d10151b9e789e0bcb3 3881efe891c5acdff9d16c15adf4869223cb7643fa926376b75df45cadc2e183 b51c9bc35aba4bdc202d46ec6fa9448b470c2a07867fb4c4dab6550e09d83065 bbed48e529a072540084d4b2d24d4515c1677af0cff904c9ec8a0c416d145994 21712d569fb0b439a6c986532cff6b19cc6355e2595d1a1ada9669a1ef22e075 fb925710eabb7201fbacd4f641dfa65049ce4ee6748e64826d8e2009391946f8 5d80323bb57f4173561e513a9373f131356ffb7df0aeb95df9765820bfc9744b 2febbbc7da97b7954cc7eb7a08a7c43eaa87fbd472614ecd88910f240a032412 552f994edd48655c860270ec7236ac83dbab252e640210047a26e2fa72c71582 c3ba0ec20d2ea7fe968a68e01723b2e47ae90d83701891e9721fd62b6cd7b33a da19a00f54bde1c379ad2dd3527365b8237655909cf1240d9544903f123a14d0 7512c568a95ddf7b37b7a82c2d6c5bf72878ff2df1ab95536514d7acf861eb71 c1e3a83179c3bf87631db43b10db2709433a8b1dcfb35ae82a3dd6cd830b2580 e431c10bdbf77fcefc1b7b6ec752c9f1689d355da3d8a491d667ec9d1d46c174 ba22481a353c53579959c455189d7ce27470ece2c282722357dcc8b012173645 aa6edecddf6ee9999da2a576a785bd84896c2998b1240311e69b4ef24ba73601 fa816e399033d4a7496b187c0c375125cbadaae5cc383dca44b8 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 48 /prime put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueXX 5000816 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6a18979f258b8c6ed3 ccafbc21aca420c9c83eea371adc20e038b4d7b8ac303004b0aa205f04135140 76407216032fdd22e6219da8f16b28ca12524deb7bca073cc5eba65c102a5e85 fd48e6d062cd4283ee570a7774597e5bf0e3400b6be72db0115f3cb12db70ce0 83722870cddfadee715f10f1fcaf20e06f3c54afe5ca238539bfe2b596116e83 f5371ff18fa5003d8543226cfd4025f9940365b392a858d27f078d3abcffe4a1 54e78c7692d1a32bf935967c64f01b24788ff8325d61145e2d4a489fd986fb77 38e6b254522c77ca2797a504a9ce4676a77ebacb026eca94dde5922c936f8e90 c43e2851905abd26842613d8fb644512275d43f312eea72f7ea3ba90c8fda6ce 7a9dacf52f55f0dd557f8d866ed6f3d3d5e6fe7e9ee79404fbe1a285bc519ecd 7ca162f0729044aff526bfe554ed59dd9beab0f842d55be08caada6136ac8e30 21c0b1f2869eae249565a126093d019e874f1d3e3de66d54ad31a98634d992d1 2087aa9caadb695d492b62edb22e89fddf96357f7dfe18336e102a892d42c1a8 924b450df9e62931117872ba4c4f60217f8dba3846b1df7f87862b42db5627bd 43eb0e86fd0a633d67f9c8d2e64df360c689be9b696536fd7d1886be07beb475 10befde02910376810d3894b6c3e16d0694dff896b2cd98f5e44f7f059776d8e 9bb6e3c852781ef8e6e0e0a9c8c78b899754aab4f9a8f9da4fd51bc78e39e7c9 a538ff680ef2c545bf9fcb4b038ed25337f8c02f345b1b7100eb6a13cad1d980 178c730a870f02d2c6c01f6606669510bec67c99986b891c142f8c2bf8df035e c436fa9b81563dcb484d18f85ae28341ea562614fad1bac1432d874695af6fb8 48171ebc83f63001e7c3a982d1fce641f1a622b9a8b7a28ad3d22df9c562ea5e 3ae864fa0a6413e60ef869d10970fd20c80b7568574276ef4c8c3d65cd96d424 100af3a3ad9c4b5f6c8dcd144af4a2da9f34a1b14c33f586a70f444a53fc565f 9082b4aa858c8bf637f8499fbd0dbc5576edd1ab85a4481adf4df8ffdb8ea103 9b99a919a11ed74d95ddcec7150896c48d0f80 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /beta put dup 18 /theta put dup 31 /chi put dup 59 /comma put dup 76 /L put dup 100 /d put dup 103 /g put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueXX 5087381 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a1e67ed7e41cc25acc04702f6 8ef703e3d45722c1a91fdef7100a48631a02a6f02a08c6b1f9b4df8310385b86 8632718fd87119a233f219d9411383b7fa9f3e4780d8c27e2e89e0cae883d664 c3eac57a3aef8988a2e9f0f8c7f53e0a80bdfc4620e21287d0390e1975398544 7f3ea66401024bea75e1b4c4437b7bb188f76f96b918ac7c6ad7e8ae7f21d8c2 790f08cccec904fe48ef39e597ed4d4237c1d1f596f5906b19ea308020f7a35c 168e327ec3246b1dfabe912f6b6daac09974876d3996e57d180261110db05f15 e3e8eebba3d90b5764c03df3033a1ed678ebc679569a2fb297378b25434c0f20 5313ecb8a952f07242d3ee731b0cdc086a4481178a3d65129c47c09b22e9c431 e11b3747b94c26a757c38d06001798c6a568303d541385244b967d3b1786edea f65bb53c4c2fe75e4b1b15c2c78d930b4296c80f08bad86012451edc8e9f0854 c3b390a16e27b11b3d45a9f72eff8baded2242dc928a61685d79e09681c97425 5b90a498614cf560fa5b1718981388268ba206a96989e6d0b5d485d9aca5594a e67dd7b34d8a369adb06647f8aff8814d6d9cdc04a4835918e557174c5bc0f3f bcea9907a04cf93c12727ec40db3f2f77596dca477862747435bdedacd9b2311 6cc97fa47ffdd7d897fb6bdd5572e35d34e7e1cb5e7273a4ffd86525323ace4a 84e1297028c2bd5469baa2e75d19360c2c9042139d5e7dd4390a6a3935424711 de21910126d750ae279916ceb71da3591d60dc62db333c5021e2c1cd61ade51e 939203f45921c242b0b0b4bb367ec770c0983ab09b18a5d04a08a29ba3cab62e a74cdb87265c5b342611777ad75307b0bbfbab54717352a814c13778ddea2fc6 b2acb9738345d6741f1ccc4fef7f64f299ecef962e5678e17ae357cb5d826a8e 121ff19ac4ca58f8e38722b7744bc63f592df2b3b3bc2faea58b9b699cee2ae8 70a80cc00d8d0431cf2e156ad4b43d690f970f3b404c03a202343e6b86ee4471 3a29a831036674e3b62a29651755bf7d22d522e3d8118593ea5239100909616d e5529d4f71284485b50873bfe0f7ed594c7b950699e6151f445b0461d5a9e559 c89fff5c2151b9705640983af01b4921192bb5821ee07a421f29ecf02755fa39 2a3e6944b44c5aed78be2b758977eb6b28ad719584be0ce34eef273cf824ec36 9bfa34ac93f564408a7f87dffc7cc5b5d5aa21ea8e95c9ac881f131be3ebea36 73265ed6dc591fd4875d8d693c85111e4a500d30128aac64af4a68bd9dc9cbb9 0ca39f8ef34179bce6719654589e3059357605eb7a72d3705887a84a22239caf 0e6c206ff0bd1fbcbf14d0876304c2b51c6998bb873d7050d137c3f116f54145 6cb294e9308b2c0fdb4fc95d8c529b8229cb7ed7ae81ad358eff7730baa8f5c7 3aa657efa1700fbcfba001d76d6d77f8b7f6726b1fae64d42ebda5126076e553 50707870cb8be355512aa0178c6792f46d95d7b8207c3bcf48c6f7d52bd30d90 f97c65f9459b2069cbd8bee799d45fa4ba5e7ad82c59e2cde1bf9a34496bbcd1 57e83763e01cb9ed415476cc1cf1a9d7d474452798978c272b0b7dab2d5d6a77 5501a6209207f3f5ea61145ab78760c27cc715db0d9532b2d3009ad54ee35ce3 7ad965f0e1bca9d0dce9b179b9d797dcc8bb8d580bb227598832a5a1c7bce07e fe38873ffe67ea9a26090b6a2967f5b5fd3661719c055a37b0ac3bd2dcf1815f e0221ea0282305f17afad9a57257a20955401d5e87bbe24e33bd9a6198487332 2c7f79e6a238e8415757c411db6bac2be1b560180884c836263d5f0690cc7c8a 5300f89b9e621f00860b3f75885aa9d1dc5a4142d528020c9bb949ebb3b55ed6 31e0680d823635cfb6f72eefba5ae6cb940d736d7986d7e108e7c3533daf34d2 cc8bab26fb34d1b5b63fbc4fd814afc2fd398eba5204960ea0bca5ce4400874d 51fda43125a7985fb6b00f59f5ba2c00d3148fc295afa576edab5c08b0a6412c 8d6bd1b240628c8c5728fd1029a8539d1f5d7510a8d8a518be3d0e6f2f9a0525 b44631c5edb71428459ea4729faf78000ebcb0319c8b4b520941004e84f838ec 54915add43154644dd56ce335bb97e31629084df3b8885b01413efbc0c35c1b7 2f83558aa8ad4b2d5d8d9c7ad7449b8ea56f40f5a12723006fba48fb8c2c86a9 7d6637853ded99eb40b5cd8d660b56d7b587a7fe489861554bc85ce709ce91ba c6939f4246a681444a18d71a2751f79828a7d009955804d6a7cd99e4bb2e700f 44fa42113294da85aeab80dcf157be6df7bea3d21ab10ed338992ee6ddf2a21f 92f27dc8f2170e8554edc0d346370f44fb17bd4d8d2b58f3562408f6ad0d053c 0357746571b9358e0b20598ea2ff5ca3acf9f1d29989c038af8039e98c6f089f fd0d02c8eb1661a8d1cb130565e0694d2ba53f4cfc661deca27b27d3a34f0d6d 83180942c28540812dfe5bdb2c232dc43b74a7c6ff583a918124e4e7d05c0c0b 7f0a353214478f5ee39730dc68bbdec7b498868e890eaa6f5736d48404d0b899 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96175c530edf821706596e291ef3ff9207a9ea3eee67cb3ed6bfd66e03bb84da b3b545276e1c6ee0ab57e3a5f40a738cdee9cfcfadfa49a27e7adf2bdcd6ad33 4aaf8823f08808cd05a7209ad185e3b292dff210de9d8a1ea8bb678d12792524 2c3969ef06d5d5b85f3a4ed30b738edb9073f22362174f639374010855878ca5 5aff4e675484fb685777c3442616f1c64fc6e0657807af9c40210ad6dde642de 68901729747af8426730acf1bdbb7ddc26ac6d9859964b81bf4fa7e7ed0001ec 94c8de8f2e7e8c5c0d3863224398912dd229b4155cf6541ade45de97b337b7b5 4275b64c5c01cb1710a7b058417c90efafef61bfb9ba4aedd886d658f64625be 42ded73f9f4eae7edddc8593e7f1293f27b277a9da6acd385d08b36a68cabca5 344ef3695b1e1739fb483b5a143ed353e64249839ecd6a752b804eedac0bd4a9 65925bceb079d9afef8d64ba9c45785fb8bb224ade00c981b4be2a3999bdfe08 868e466181deae56d416af977bd0642338c6c0f387a57ea64a6542bfc78550b1 54bfae3ba7e9df26c7ac4215b98af016f47926c8b7bde9e4c0541005f3ee5411 8fed20cd68fc632e01a80c3b920759596864dd2d5c9d8d67b77edb746341a797 e247f795141c01e344e2e9e7bafe5f0a9e0eb36f599faa6e44506229bea74cd7 aefa18d33662a605f337f7e2fc545f0fcc11b398f96811b2df7e06cb160998d6 12e9a9b962d0d0e017c715f8f9c0f5bfdd45f1a4461f34caaae034f2460d8191 bea0677a20f5afa245e4206754b433b3a98f822683538fd8d9d5db43a4e1c25b b38a92a74638ab473cec3c3169337d97d3802b1af3b4c57be1c90e38afa7f78b ae81bb14972983cddfbd8b28f477dbf982cede72b5604f78a9fa15e865d297ce 51a6516a90020911f1fc91743052727ecaed7eb43b7e01af544be4b981537940 9e35ee43274f27b687bfa0dbc149d810ddb200f149d2d9a27615bc6cb284053a 22ce30f47c0b255d0b62b596d13a47c5a1e1401c8d64372296b96149ce2be28e 969b5da41ef276c78a603f2535f7343785592aba6a7f183c694ffff7661df6cc 780d9dcf9f5070a695374097e017d428e9b8b9c8454420fe4c08937a2a1ce34f 819e969e39979334742683649d2f8385bab0510514ec002e1ea64163f4fd3584 38f80bad8d91ae569cbdae9e1dd5ee28e1196db138daf3f8db201e86cff4052d e47133578dbbac9f5c0891a2937412b684094f0efae966df78facf565eac0764 00264c0f608b5ea3a7483bb3a834e97aedde1bed2576aa58a864a31cec10811d cf6f42ca6e7709c93198c42191c1204dac8a44da87a46c930ee98d74ce599ebe 619aced7e5bd6697be793809e1984c13e38dd934525f4a9b77c1d213e7851c81 8b1741973b97aec305c00c4449200115b66d951a512eb01bd6576c9cc793c822 7183266a393a48f8e46556a73532d47b7abb1f309ca641c4351d459c64a5c71c 620986d8771c2d2d92f8fa5bef8ad914e94091586f37f2f64b14d9cbaaf91284 f5b34d169bc44d3398832593bdfcb055c940d9c82d2b6561e5b0d20bef8d7c9a bda9b601ebccf773af02ca6eff3466ba249c3426ad5b3faee6fbb742d5179a7c 3670a398f418b1f03e7bf3cf12e5826586db6ff843ec2c2dcbd1012ee2b14896 376ac209a50ad35016dfa44590aea1a90f846a736c9e464ae1618fadcfed30a7 c51387ffc887ac919250cfffb8ae540f19487860fdb49a8d175133ccf107d833 42fdfb2b1a2c0336d7c37fae55b4c40bd64ff81408ae1cb4e32d95b6940a98cb e4b1e83a1fea73ba2afa015de15a6484f9f0efada06ac107f0f22d9cbb873a39 a253fb4943105d4fdd4e1f4a6c866ab1fac3460b70660ce4020a3b636f89893b ac7dca5334d80e4e546914df6f095edff6e73be58e69ed9e675eaefbb13ac4dc 79701ec11fb56ff99338faabb26ac88f7aa5680e34ccb49515a0c300f57f31c4 5c8c71126b0551adae332458c71c22949d7f4b190c82d11873c4485e127d89df 0407703045436fbb61bdb09749664f9556af10014c6a7aa6bd2224bfa10bdea7 954dd4787906c0088a21e65cb4df772f552a605218d66e19935081c28b091515 5683429ec86b8848f40cef643a184df32ed37b6252eeecf176418970e921de14 3969f422830af7f259341ec50781b671488ce607aac56b50bec36a4015874a14 701d812865aee86885e595be6d427297f87ba6cdd3b40e6cb4c248253c552ffb 98 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: msbm8 %!FontType1-1.0: msbm8 1.2/19-Jan-95 %%VMusage: 32482 33913 % T1FMT-V2.0, Copyright (c) 1993,1994, Basil K. Malyshev. All rights reserved. 12 dict begin /FontInfo 13 dict dup begin /version (1.2/19-Jan-95) readonly def /Notice (Copyright \(C\) 1994, 1995, Basil K. Malyshev. All Rights Reserved.\012BaKoMa Fonts Collection, Level-C.) readonly def /FullName (msbm8) readonly def /FamilyName (msbm8) readonly def /Weight (Regular) readonly def /ItalicAngle 0 def /isFixedPitch false def /UnderlinePosition -133 def /UnderlineThickness 20 def /Descender -194 def /Ascender 694 def /CapHeight 683 def /XHeight 431 def end readonly def /FontName /msbm8 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 90 /Z put readonly def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /FontBBox [-59 -474 2478 974] readonly def /UniqueXX 4702307 def /StrokeWidth 0 def currentdict end currentfile eexec 9b9c0887db83fb1ecd8335b0bb39cef0af64f656fc6e5c230cc9d3a7346aac7a 06bd9a40393ca15d3773a21e06b9b4254d3050e90726bbb09120935a8d448cdb c799d90205a21291254fa633cc0c2ed88781ef21a5d45b72445c284fa44f8f39 cf566976075a92e9e2947a6fc93d879c29eb26809ecb4409cbc8666526cced92 65de661a2f8b0a16be45a9da17eeaf89b0e023de2b0373dcab997d60b7d7dc1b 4f7e650a2a9f13ca0543f3b080ae33d65f2802b8c6f032ed6f1f9ec1620e5d5e ac22d6b7a2c527d76fc035bf25a0299fd60a72ae7e0c045feb341646af23c463 ed2a5dc1919c6ff451154c679acff08a89ce1df2e249199fa1bda096e03fdf15 4361c76bdfa752edbddd62db3ab84a90e2bb9721b75b59a62e3ca79118748de0 4e0e6e26634604d253f71331cc6ba447a8a447031b1f313982407233d775612a 4a7724697c7493bcbf0472b29db0afd695aa19d59e0dac4be39cbf9c23c739cc cec6365429ea94ee6f803a5344c5f3f7cf9df9b4beedf4fbf49bebce757e1d8c 2a21c63d62f82f5dda542d838c67ec6f1bf39ae00a6d30401992944170e200db 493a28d6eede182c3d037a16efe0d9181729b170ccb04ec3ade7dabdc6b43e82 608979359b287adcb6d52fbdd08fc869310ca7729a48ad2b7624c662a80c6660 2f384ab69eaeae82818175fdf810bf18936efaaf360a316bb572f2ac44a3b67e a75d6dcc5a7df3de860cde7ef589080fae1f3203406b3feb87f4a8dc03273467 13194ee24d478e5c8d23d0c14046a09f3add6cd29bf26956c2fc04385053f00b c7a782a026a286876bc3bba3729426adccd4888f94d24ef5e0d8285622986082 510acbcbaedf777121f750fdc144725bb04ae1f2cfd6c2206ddbe5bbf143f5e7 a617f6f04e9881d6113e9a332ce952062cd29445c6eb8c4b09d040ab604caeee eea862d9df21ed360dff47ed25a9f4b1d4174a36899eb5f098056a566aa72224 f925f3a8cde994e3274718a340b54a4bf8e127da989452218800262b704f0eae 4d89590477c097b32e220e6773721abbff11d17212c3a6033c8922c7d1bc689f e56803d938115d577568ab6bf8b4be8c26679c8dccff62d788dacb0f4b545ebe f275e4c2e8eaff6be135d9c171aaaf1100230f843dec3067677ba2bce933ff49 1d37b8e609eaa5b4d65bd09bafa656428b785b277370c2c9036b166132fd5840 402c02172967f6f3df895b472295894ed9d80fdc3709e0c05a012af0f8c92854 7c8359b17adff1ff610e70f997d1406f3bfdb9ad9fee8c53729e1576cd193063 eaebe2d5a9910d2781bcdb80853428cacedebf7e578060a80f11e7e2ea295e6f 39682b2ef6db9cfe9d5a5a14c76d8b7770afea6d6624762f00377d453ac494c5 6209330b6b91cf8d4c6a1b5714c006d79b91e5f4a28bbd05c4ff672f2e965e01 ec28e8bcb4eafdb045a8b5d513204d9b28040d807bbe063cc9d935b5ffc5db1c ae5bed68ddeb78c3bc72a219646079cc9c879825e8bc9d7f3b1fbed40cf1f2f9 0d1f9faaf9754577ee4ab5fd30857da1da39007a6d1626a70368739813e418ce 45eb4987d2622ef9c4bdf3c5342bbcb454a68783d508bf 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 6 /ceilingleftbig put dup 7 /ceilingrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 12 /vextendsingle put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 22 /floorleftbigg put dup 23 /floorrightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 40 /braceleftBigg put dup 73 /contintegraldisplay put dup 80 /summationtext put dup 83 /uniontext put dup 88 /summationdisplay put dup 89 /productdisplay 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 dup 122 /bracehtipdownleft put dup 123 /bracehtipdownright put dup 124 /bracehtipupleft put dup 125 /bracehtipupright 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 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1117c17faa607d30805b0df2e3f9abc17c81a662cbc1313cf0cd5a1a21e7ff48 82a4c867f35857cf0ffae2adf53972903f59611b1530f88600cdbc4454609c00 e7b268d9c8173291502c3b9973f6b1ac8c846923f6e7aac7d40e3fd6eb5765a6 12c5f6bf675035bc7e084a905e4f2b5ce64e3318dd1c3562eb3284574f61de1e 7ecf600ff967451ce3f62f1729 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 0 /Gamma put dup 40 /parenleft put dup 41 /parenright put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 59 /semicolon put readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueXX 5000792 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 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86ceb6e0dbb5a4778ae8bc59c9b479b3a83bc84ae8337cc276bf94ed18d02c5f 16be35d7f3d345482780cbb8c4cae40086a6699c0936344724b9577eae74f685 c0c30ac30ebc6922fea81a77e06caabc1cdc225ba0dbe2a5936c6e33b7c6bc40 7ea46a83baa4092d9d6536e3ed7302d48277e5ddf78980ab830e00851358cf62 61eb930f0659cf9e1bbc463e2508308c95659fc33b498f8f2209b3fec54b32b3 e30e969fc0a3f431ac2cd08326b7686c8a6453ce83d5a5dce62108547f77c121 8469b54d31ccb0702cb8b38cfa89cace3549fa2058cd51f9f90a37fc0ceee2b0 e004fc08bbf24ed1f6b0b2bf6a21ebfc9a00594981fac1e2357f48491d74b4b8 0f638fb5a9cba1876768d6aca85ae19568509f8d729b1b25fb3a5cf83e5d0366 942ea580ab00e3120f2058e89328799ae698f0468f9f2093b8bd055ed66771ad a96aa791dea61c4ff86058aa700a1fb4a99c34a57242b1287bdea41a68059e15 1ba5825de6f01b21aa59264e816cb72a8be226652532ed083f7bce43f1cbb63d ba20726355baba489ff1a01f5d5d3d3a048b220e43d2bfb88e9dbe84f0d2f285 c1f1609a30b7051bfabb2a16f7c20db5d5215519f034a5512fcdc4716ff5b608 b6f36acd50451720545281501d2d1dbaa90bce05e79c0f5936d7e4cdf072dfce d2576ef65938281e6be1802bb0a193028215e6be8e856ed068fdcc384eaacd0c 6ddf845b3696f2cac5714b2c6442900b3cf6199386b2733281c357a5ef4ee872 8a0068a0e155c645218a673ace9620c09a1a15951411714ffe4cc3a0cfadbb11 ad76f07f83540943a5327f68420a7392a46c1b91671dc28e73e1f0532e3ddcb5 e0c2882ef5b8aede8b6feb0d5bc189588979ca7b87bcb7aaa7d0868e2cc2f723 7c2f9fdc13f862355b77624f64dcee76223f394f26fc22f9909cf5746fe8f226 4d5819c2fe66914efcf6f6116873a935064517c9633d9bbd38242b1258326270 a784973b084158f6ba85d1a898a45905b535faeb9124f9edc576dad6303f35cf da11fc6db26b9c2edbefa45a417f2aea141ebfdaba0cccc50d36057254357ca6 9ade000f740b5164c9f9555c9d943a5d5dc3cafbd1a996036368a7897400c6db c4d6aebb708b42a62952509d1ec8d0796ef446747128ac7b684eee90ea6a52ed dc4c456516b9094640fcaab6a4ae5ad4ec1cefd24ffd9127442e1716fb95aeb5 5c01b57cc25281aa8f1db861834b2df346ab7b67653be4dcae220e57fb454faa 44615b82b5601ac4789148fd6f857bd91faf70bf5c2f536d3bd67716dc096aad 5798a2cf9b43593b34360d2fa1f7552a5af95801be6d7b3510d2e44747a03364 d2decaeb78c93899fdba15b8905e1ea594d6830c18d75acab585793e70b45e12 10660efbccee8cd61f370d322f02f3906bc631899e2be544133fb50deeba94e3 20e8d1b879eb2131792d557ace6fc9d169afc428cf4b113814264cab116c322e 25729df1d51ac5bd4122c098caeca7fccae1d25e51d70527ec53be8da5abe7eb 393ca5246fc7986541f4d072d7a6f25db178174abce0846dee93977e2f0cfd48 bd23516b6798127cbcc7c6ca30968d86e118dd543e147d20bdf572c3e8442d1a 0f4ac1c0c4a50a737fba2c4c2c67eb62a4c9ec16bb3899c537acbb6f2426b37a cb8447a2de550aae78dc67ae7f11fe7aa25aea5ec3fe755beb0ef2d3bcb0f0ba 5e85851355064539921523e85e3b330fa24a5e4af2eeb543c7d0ad545c6571e5 f64177637024872e9c9415f9cf34d1d7a37480d5cdbde08d84d388629fe5c613 e6f6e9c436d00830a8e808157486692c6c790b8b84ad36f59deae81751da52b0 7ec3e0302853f2b53c49b1946f54a5bea5ea1fbfb20aa6f4f83da703a829350b 39a1706ed54686f9eefc0b8f5f898a5376ea0cad2ee3e0a5445b6d298e74cb61 67e00517231f75b848652500d94144fd888134de3770f3303e8bade91f12e4f3 7d6a73242fa0b6af83f28cfdc2201d805849e9b420283906392751cb69495c57 516293ca083ab9e5bf56de1454e73663a0cb2bf737d913e82e49bc4573597f7f 1357e389a03ff214739b67ec06a0dad7d9b0d43e246795dceb5db09f8e2d9b20 e651d45a1229717532121edf00edcef10cb64bcc5bbddf9f2a66e8b6fe613212 4482b3737446ac4ab64ceece6d8b5002ba7f2739540396517ec336c5678eb861 a69b5c152080196400362c13a2fc1b9185ce9df46108fb3f4ba62fe4188e3137 0f6634bfa47444ddd918cbad750b7c9b8d0a3c7a03442b78180630e08b020906 cf153c45ae94125a09b39ded96d9d17edceb730e01b7e8bcc2ca2ea83c0801a5 6dd2db778c6f8600d3299817e2eac56d3770d6795beb1f7dbdd47e3c96bbde29 ea2cacac802839d244e06a76eb58a3ad14532888e6cd2a265bea33f3c59b2960 7a037e6a42df325433408413f2dd673d550b1bac68f6edd47a2a5daf2f288472 e3fd9c15dbf034515a272c4cf95ed3cbf02b969264ce2f49c03a68f4878723ed 7df3dfa38111b75cddc13ceaf4f167b0bbae850c9efc607b39a741e0aaac5616 04f497ab0adde2d2d96c3bef6ffb5d8e64f9d2bfc017b52c94b2ead6598426dc a2aa41dd7588c773b1e2607e6e22428dc256644fe23ae2b181c3c42460adafcf 972d29883817c479a1671a467ca0e8029ccf4c2989a96f7caa158ed39794c2ce 5b64b87425342a76d57117c304b2b4282277564a5ec4c72812f1fb996f594979 0a11215a33ae1e1e0195718ff9af76711a152c7721d1bf05c7bf698ba4dc792d f88356af488ca8aa32c6d9680cdbad967068e15feb270d887ac6a9a2b391aae5 e9d750d409eb725ff3af4fe5097fd685e42f1b82c10b0528c79b20f8f8a0823a a63a180803a6c38b93f8510671389b7dc7dde330c60390 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 6 /plusminus put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 33 /arrowright put dup 34 /arrowup put dup 35 /arrowdown put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 54 /negationslash put dup 106 /bar put dup 107 /bardbl put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueXX 5000818 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b66566ca5ba42b1911a 5d7f1bf343015eece988b7a93bce0c7aa61344d48aed9c92c8698d4b7c9951c8 7d103f2414b39e1437f9d2e50c4ee5f218f2e6716926a79ea978f13b1f855345 191dd7d31d8f82c2e3343c7a5894d95bdc492c28226834efcb5c12fea36ac5cc 430e0aa604961e34888adf6c1f3954cbc2498e225d953cf5685852162346f474 5a2a7087d5d7ad486de16d2ca8e15cee26e012671ba3bdc7d95cc8c98bb774f5 08625e968aee27ff7d1a06e63bcfb5aa4876c3f8f13b30ccccee73c3caf4e70d 98e6ed2f422dbb4950bf789680e064150995941a9f4dd68a575949847a7d012b b910bf03a4204418590b6f69f7745418beab5769a67cc5218171ab5d3be26bfb 68c8e444efad985b272fb78bf83a8e9b92d921e3e11c30fc59d30c455d01ff0c b862caa635561c235dd6702e72882c0e3e6bb1ce5024aeea4ec333e6768f66c5 f2142d1769cac63c020d13662694cbc866d5da967e65d939acafbd38b601fe4b 2b0c8cdeca237a8b4a2439ba01a7bb03d84e40d6597499a8c3c239831dfc669d 23cd631552dcd42a8b3c042af1c6feca1952303fffca8723198e3f43dd0c4819 19fbc10e02dc3dbb343e16ff101136967f2c6ac3a9efc1b6df3145bde0a6cf45 63e348375a9da18a97db8d89ec1d1f0aa9149974fe8bdf5f2d3b91eb1a2d8d9f 5530d9db462896316de6fe49af90dffd8419ce68b624be9169eac5e40f159afe f677496cc0a9812d49a0dc71e2a6a4871767369a9fc337747763909401ef8fff d6d8233223c47745cceb8782484e11ecf353af8f20a3fe394a1c6b5ad248411d d4c64038ab00159fa4555e705bced40d81879d8466494fa79ef42cf795fa878a 60456658d92629beefafd479c2c9ec3a34358fc9948410cfde1b18543eb7d1f4 4119ead077cbd414a0e90d9c4641aeccd14ee14cebf738b51be59e42a7ce2c16 206470825edfe7c3c6ae00d575cda69f0f4cfb510b095caf4bc9c676cf2334d0 acfa3da5c01f97b3a9caef7dfe5840f54a2896abee388f1373976bb6d8861245 1cf44ebfb3b2a8746a85af15f3aea23d6c29ff12171a688de76d0512f71b6012 51877b188cd0cb1a3fe1d683f54a62e47822c6f9e7724cd3bcb7b402ee6373ac b53026d6df573ecd9095fb8a9b4187a87b963b0d70e8c184b8e40c5690db1928 eef5bb9793fe9dceebdc2381ea4969eadf64fea16e18caa94080336e20cf5f3f f63425c1db008b2a700bb9b1bf607f78be46d71c5aae006a663dcc44b968ad3f 2b7f669be9e6c6309451759c223b3d9a7386d7c62260d981b95b43ca7af08d4c 3d11efa25879c37321ace1f5491889c20382a418b6710b2604f1cb0f8ff31c09 24b8d6d991976051a2ab2627773cd06121d0d3473bb3a0688ac93d9866a7c74d 593713274946506ec46c985a2eca60a0d5abc512476e99e91e76289f9074ddae 61e9c623b0e41f84af36fdc9058acf98530d45a4309a1ef81531570b01f963b4 177928481133bdf51c906ff7546f013c927e32647d0219ca3f65321d05326c19 1127fda1942cdb6fc748ac70da5f9a53572af9f22249de8e0c8b5a90dcfd4079 b54896da2ab7e87addabb4cb74338f4f849fba9adc0743b38e60ff87aeb481c6 9e167b5ab76f8ca67eafa81d70f1eb933e6da816fc44034d3c9cc8e62cc15121 4084c3d0a648590c598501f89d06f309d9566abdeb46ad2f1f871a95f3726fbf f31e89087759b52c3c721c4187b5534acec13a76394c987411c5619864e863ab 2f948885677f25a558eb23e952cb5200070c917021d71f7782433ca7688bc331 3884f7695e4acb1d2ca18882b859e1e96fdd2ed1eee5c1380a3e54caaab7e635 01ac95c53c073c51e4cd665c1f058bb4e459857ea348c5676e12e2656db0fefd 290e1de836dc130ba8dcf2442d783d1f9223d46bd40ff2576eff1a6753b7cec5 db221c2d1e53442d5c62f2b439899ea6890460f4da8ebe536adca27356ec680c 4ffe81aa6dc5d83ac6477ad4ffba35107ea2922b62c938319087ab5f0b82c8a3 f032103ad5e683048b5bc8ea8bf3002b1c6f10fc0f44f7362cd0138a204b3452 5779e1cd5cecc6275161dc685dbd0c8cfc9e14baa91844f719e440045719b3cb c2f1dd5c1b4d2ef6a25a5d961ca7921711075d64b1679af4fac3c5c81ce0719d 20121184612ce0a8fb1a225d92146159a1a5c543ba1b7b2ef74d634fe45d2ca1 30c50b16cf709a9e0631f0939dd075a8368797e14f9b2fe4ac281e46c094cf8d 873a363f0fb46d07494953b87a4844b0b8de11ea7f0678d8254e01880abf9bae cc0e286316ba30ed15abaafc1e25552656beb6bff8af3e15a182e23c0f54561b 626c4dd2a41ab0db53b382c64a381f1e0de3016483176ef232ca634e02429357 488e04dab23f7199c4cc2e922741b1d2a5ab6dfe58f96c092b723249d6caa37f 4109bc4c68f3d46602ec938f626faadf33d126f875ba6479936cb8c8db6279ae 91d6fd4b32396fe99f013902454bc893ca6ab8e0364047b164c1dbaf6318dda5 3f9e34a2ed42a595ce2f42175921f8a2f653fef2137ceab829a88574cd73e567 ed930a7ce87875fd2d1a264b3a817ac5e91376c89590bb1f9086994fed2baeb0 dbff65b0a772a7a591a8a5ca286e581b635df48dc0884b010cc50b0deef0e0ea cda60dc8b06f6b1b407b86d3467f27272309dfc054fe254d4c3e00f0aa391210 a371ceac388035402ab131a5150e2500583741d7148edd2e0c2f995a3ab22a17 babeb8241d743a29ba4c61b37480dba73faf0084558ba8ff9949a78f8a1e6535 3c702b32b752f599fa62ac736475e557cdd232504dd1e00bffacbfe4812df517 fa32ecd9c87e21e8acae70f0bfe96381194e4be8423721bd90b0c07427cfa46f 5e5134e5abec0db769d997a44a3b9b32ec70e15eb4d3cce0f1a999cc20abde7a 90bb20867c21f487423b67e89e45e5a494a1b2757cae20334a8ee938e8e4b1fe 7b4da6e26b0ce44f553f16e762b2c54cadb0e22f119a2ba99444af6cf54b3311 1e602f9b99871253cc3d3730e75da1e7a62c4655b833fb4f436648d6ccfbf1be 0b180993461c4c97b5fa264433eff2efbb80ad75e7d898abe5aa2b68c75d6171 851d8923681594c7ac606657f71f94a28edbfd899ca6c854fba8082c5db11672 446328b768ef70ea3dc0b7567c08db763b61232a54c01067f46132cb5a4f7004 fc249354ef73f07140a1b09af28ea59cb4fe9a0ac88b5987be1d2647c41fba1c f630ac4b76a22ec11cb25c4c057d4257c2b0aa424f62425934fcd1a2fa8d44b8 384223af7947ea9bb6626ce2be425c00195caffa5cd7b0406bdaf4dc8190f9fa f73c681a8c6607729164aeb6cbd510adca8c3e75bb1829fed15368ff489af53c cae10625cb1ec68e007a45323827907d2852180bc6e81e73cd35a68fac867bbe d1c02cc44953d9e9ae4a8f1a7fabad3caa141a794a 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 14 /ffi put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 59 /semicolon put dup 65 /A put dup 66 /B put dup 67 /C put dup 70 /F put dup 73 /I put dup 76 /L put dup 77 /M put dup 80 /P put dup 83 /S put dup 84 /T put dup 85 /U put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 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{-36 -251 1103 750}readonly def /UniqueXX 5000829 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd803271dc8bb1656323089e9e2bb50f2b95088 87d31dc36e555668d9578b338402f8c259f6813b0b71e6cbd95e19c20ced8495 2559e47577e612a4d94de3c1b4c6c41a6a3fa9e2313457242de74c7e4c0d722b 3e224d18baa726369542682e9459b951b2ac638494880c42be4e2175a3c067e5 945bfc1a05d3de8c3e8ab97f80ab6000e1f78fc258b85d3c84b038d7d0f5be85 0d43fe03e26cdfbb41c36d14cb552e8447ca678efea1d275e85503123a2263b1 4077587683de69a8f5a88d666172668411d0d0fed3e037ee468627207505dbb3 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12f1ad5beb128152530a675695c732f54482493c29eaf5a567c7611b7ba0b01e 6805e989a958049b5199fb839250a9812247d54f47d8df2981a93b533d9d6875 350913dcc3fc7e13820dfbbef933b730f5a895824b998e40101d7e60b4d757d9 573b774d4418c3a5431af6b626b5a3e81edcfcbf9a31d0c44b156d36af8d662a 5e94c0f11f5b3c3a114bfc40c7bb3e18128898f5419f40e22c0d0df2863d6217 699446eaa3a19fa179e9492e9e488d47 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 0 /Gamma put dup 1 /Delta put dup 3 /Lambda put dup 10 /Omega put dup 33 /exclam put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 45 /hyphen 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 69 /E put dup 73 /I put dup 82 /R put dup 91 /bracketleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 99 /c put dup 101 /e put dup 104 /h put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 115 /s put dup 116 /t put dup 117 /u put dup 120 /x 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 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 6 /plusminus put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 25 /approxequal put dup 26 /propersubset put dup 27 /propersuperset put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 67 /C put dup 72 /H put dup 85 /U put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put dup 112 /radical put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueXX 5000820 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8 3750d597b842d8d96c423ba1273ddd32f3a54a912a443fcd44f7c3a6fe3956b0 aa1e784aaec6fce08dae0c76da9d0a3eba57b98a6233d9e9f0c3f00fcc6b2c6a 9ba23af389e6dfff4efec3de05d6276c6be417703ce508377f25960ef4ed83b4 9b01b873f3a639ce00f356229b6477a081933fef3bb80e2b9dffa7f75567b1fa 4d739b772f8d674e567534c6c5bbf1cf615372be20b18472f7aa58be8c216dbd df81cc0a86b6d8318ca68fe22c8af13b54d7576fe4ca5a7af9005ea5cc4edb79 c0ab668e4fec4b7f5a9eb5f0e4c088cd818ecc4feb4b40ec8bd2981bf2336074 b64c4301db2c1b0c919a7e5d6b6a4ee96c0f8df7f89c96b6d3ac2599d6ef2d6d 64d19915b8f9dce985f9e8969ba1dbaa694cfbf71699f23e716fcc7c79676fea de7949d94ab58be75f69c4904fdf1ead5821b364bb61146993daddde55d1a7e0 46296ef462744da3f3fa1c831a953f1bef348a6c04b4c84e0958f91b03bf694b 0194f083aa12489fc4020b2a19f50cb2ad76f6200af72d0ccf287e73ac1200cd 20c2cb367d813d854d858871a54a8a304ed20f821f885fb42815e1de4193ec3a 32a75b2e3d58d780f0745a7da66db84a1409818338ab4b83090166f550138b2d 58b4d0ee77bb44b4edb9ed06b356c3de6f66082153c7e1b28e23398ede337cf4 3cdc3529da0379b76c4a2452a0c0411e56c73b2e9120a2355264fb9c45db79b9 4212bca3e9907e4d80896a9878af2a1f5dc7b1b9c5c49f24ddb693f3d3b75223 a6502bdcfd9123075da5a4d26bbd2f03c586b83f10efe6a1146bed795da397f3 28375811bd8b6d3df1a8fcf43442b4763e3c2dd4954907ecdc4f6155ab32df9b 99b7480714ca63105ae02e38470c01956996c0d3f7946d36efceaef01b70e7c8 c374d97ab347ba8fe2729f498e42b59e201f868c5d37ac8e3640e42701800baf d32f4ea73548795d7be7a8350092e34ddda86cc643b14554dab1ebf3e108f3b0 8ad21511ce8a575da3f562b6bbe3c48bd02a844897de56a3737c761ea486a6a9 af8d47d6d5099803aa1d35469f8493161d85f9ba79bc5a488c45bae7b6894454 7bd7be71f86b5d2358e9b8c0fead25869b92da4628144ca2567e31860cb3be3b da9baffeb3c2de218a1e9df1926bf3eb608743aef34abd30232b3005acecc084 3b329ae38e2df4a06290f01caed186a0ab66023211e1af31bd84520ecc409175 26b336c3603ccbcdcb5a6e93bed6dbc7f8ae83d5d4eb5639201301ecdcd294bc a2240f82cfcf1dfc9c3a32c8a2916d1a2dfd901f8192fdf53e77813cbda559c2 0eecdd23f82cf6995979b8c0530622cfac44836edd60f5b76d8b4b56f0dd33cb e958ba41bbeee52989c696b690d1ecac7e8a9c9c15b0b20da4bea47070a74680 11361ef8b42b24345fa69f2c9a5943778b0f7226c49acb45323912fc47e926a0 0322cba73fe544840c914f2f2a2ca64bdcbf777b77e8cb68f1b613df41e0a82d 6ca53e2ee067b145cf9b2ce5ed0dc76bfd96275578ef5ccdfcc4188501c74072 804159cc6bb564768412dec7924d10e742bfa104919764b0dee5174e15c534ce 893c283c02cb7ffd7bed1cf1688c7502649cb00c711aff6ae894410059891d72 96392933b546d4724f4b048c5843e1039afb344773123fa2ccee98d1400d3d99 87e8d91d8843aa80f7791e818a649535312fb96a102eaa4c955a39ad30134e09 28800c2c5f8122efecc9cb9fcb3535b5902914f1440914464ee7d94bc1f16609 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3850e8e4aefac070ecc07189e147ad70fe5a3b26bcb0cf244c346aa0e427b6c6 18e3a67e1a922e48626f02a2b90bedc673bd6cc74aa6bc64e2252a3c361eccc8 7482c89de8ea4e234b41e548fdae261c9e81b00d224c6755a6b2fc39a689c9c6 6f69cf61f072d1732337336517ade88c2b3ee165ea76324442d242d850e10890 ea830cd07bfea69cec93fe5b2e175cab9caba45ca39882fc7f71bcb43593ecbf 8f583c0ad984da8d8d0dd436057ae5ba5cada567f4fea1d1aa4d6c0718369830 7f20b420cb05f407f53527ad34fe78ee846b19d81f8458befcf677ce9756dd12 3d5d19cb794bf48804e17bf1198d9defb1d41a5c144517d986fdb74e4a8b01dd ced112f3f923955ee6ec0b801fbc960e390aec9d4f4c828d7f9a63eb7fd4d16b 04d6d1a6bfa31b666f7cd81d8534f173f8a3e44c61f96f4468401bcc7ffd501d 6bb4eff0145958a4c322fa8849fe9389d88fd057b714c40a8b7e8d0f5a63590f 1efed903f6fcc026b93c0bb2063b14dd75457f47e6dece2ccaea9133f6365219 60b85a50f7f2dacaa20f6f3395c7b4e39883171394d76f17e042cd9e2722ee17 cf1304b32e84b5a9b730146ca74c88aaba89d9d60da4b493ecc049431f4d4398 00ec2b5297ab4342148bdafa4e6747c53028bc492881f34da84d1952156ea433 7126c2075e5f1c19d3556eb6b7f6f1ae180bf1f225d21501fe581d456a72f503 43090978a8781b372a22d8c6e414aab2443fa3bb349445a144b029515403406a 444f07289b3ee08e2afc5b0a0927e830315def11a007c92264266292de4ead7b 6bb91d1585dcd584ff47ea08a561a0ebf60e6ddeeabd31553dcea7cbc25af295 69764ca11a935946921e50c76c420b385489792ec193d4655ae283e99508de0b ec652bf7195b31124210cc571618d5406e22a813198da674cfd1ff428dc53f1b 7cb4e32308177493a92d0506cc5331a779977c9a08bf886a0c6b419abfcc76dd 6720fd370588601cf226fd036d7cfd19020a04f7fd0237ccfd4cd0a347920916 c371283532e6ed47cc0e019d84285ea5988cee4556abc6d612352234d40d0e8c a60bc29326d4b8929f14153d58cc06a9b190b77bbae3072d679b8703d1e7c9a7 b049472a2b000bf77cb0b4a3783722537773b9dd867f6e8bff0a3692e7a8c344 112ed005ad5ba89a27258ec9b75a21faf7ed16cc288ec387df32d9221aa2edb1 02c48e27560b5718fe609846141b0b158500756e31802ccc81563bb2b0508d3c 8f9642cc69aa865d74076359a409c9e5820105016b1c7f04a05f666f03a5db01 b1523231c51b15eaa8b2f5156b4243cb1ed0c1be19428da9fbcae1201ea4ebfc c58531dd865dffd8f47293deed6e3597349d4ccc6361c2d8ba08966986cf80e4 8918f074d89c89b52a45c1a0f43218fa165265ca69b10eb4fc9da76b0984cf1e 7dbc0870dc30028652c80c34240b6f9031b0df592431339b02f1fb24b501bbee 0d496b31b252685ec6e370b394a4f3bb8e8467be34d5ccf789af2364b81f9245 e86c66e97d964ec2afd0121e7dbac89cac18d40530f8222651db4b60f9598fe6 75f692c7357b8f669bbd84c67ec69e302779d875b12c23aeced11f2b9baa9449 59e664e13425df7da1c96cdc35ae198927ffdedca1c695aa0a8c534b72c3cbcf c9666d66607ae7c5b1588b2785a2c9b4474c713e9cda8da721e5c4eff9dc0501 a155ce4c5ae95f6c761c4199cf02121fae66f85f96038ffc059a55a18c85be00 b579d15b1b540d81873d21f708788747fe86933f770fe67eb36b06dfc82502a6 071f6d8f93b133e614657f81e2e80558a4e055bb116e27d38f100fbcd8e425d1 42781aae1f3ff96dec9c50bbf570de96407da25938c8d4ed394f35a62f9a43a5 ddaa4f75fe9c4c42eac722da98a60becdbac71deba66eb65716af027d2d68a6d 6a2d9ff9aaffbb2b5fd9b202a471398644da609922da0211a18641394f9ec102 d7faada5c0eaf91d4fe9c80ad8e22542738865becef32ec8219eaa744581292d bc110b60f07575e19f4f12a1d23ec12fe862613b528e579b4642ffb9741c8317 71b81298672d4fc07f1759eb4c87a37fbb63413dfc935b99533f31a57d0fe8b9 3d2363cfdc8c430846e11754d4df093ee698d334a0571e4da7d35a61de2a2d95 fde2a962e6ff1973b4849c4640f49c8b334339ef518b1410f47f2627a72ec6cd 58ff6e5399ecf1e15a98a89612e687abe31d0393b1d38aee1e44ce612e936ae3 b0938ddc46af47a838739b638c67704d3101b75317fc0bd6dd225fd11c0dfa19 f611412d1813c8bbfc8a8941db6d33a654f6bc986c62129ec75532d7158133a9 4e515be924eb33a14d740771e898bbd9994cec2418ffe075b160e5d6cc37bdcd abf5cafb35fd9c155bf03aea9d288c5538c8a0673b10b7b5a9dffe88f4161097 a071f20520a0fdf9b1041212c9d96b80354bd9c4ce2f9c1c08f406e657e6d178 37d2f96befdf24b510d14a6531d1db9f6095f15aed3364e7085a8878c76bdea8 148e557c10dac53cb2642fe8b02c5a59a042c46e815464dfc33b6e0a591496d9 d316b1718f69744bfae2af652e71bc45c8f086835c39fe3df6af05404f99f997 a82439b8b53cddbb130563b6bf7678300585d8c5cc2e4f45835a899de2f21719 2cc54fe7930ed3245640d53ff1c4c2729a5812e9bc6f56c374a8a01691dfb35d 2c9893ba9f113b62a4a4793222ace53df12c79778eae284de80ae6400740b4bf c99c 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 14 /delta put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 25 /pi put dup 26 /rho put dup 28 /tau put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 67 /C put dup 68 /D put dup 70 /F put dup 71 /G put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 82 /R put dup 83 /S put dup 85 /U put dup 86 /V put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d 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 112 /p 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 dup 122 /z put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueXX 5087386 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7bcb09b7064e91fa0ded228e42 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fae8012ff8322b2861ae120d47a78d1b165164060927060987ac380fcabfb533 44bd04b473fad6ed018c54648e7b071ee2072ea708fcd5e957d20a43eea3407f ad463d39dae460ed5fb3301e7a99 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 14 /delta put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 26 /rho put dup 28 /tau put dup 30 /phi put dup 32 /psi put dup 34 /epsilon put dup 39 /phi1 put dup 59 /comma put dup 64 /partialdiff put dup 67 /C put dup 68 /D put dup 76 /L put dup 84 /T put dup 86 /V put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d 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 112 /p put dup 113 /q put dup 114 /r put dup 116 /t put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueXX 5087383 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a196cf14c95a4e02f9cd2da8c db2546c6df52e524745992e18d9ff87aa25e4e1800bbe4ebb357c6ef55ed6d03 6d3a00c1ee8073266c21d2f0ac85d656abf61d7e5a4fa87da8ec3b5329e434d0 d2adab706b42a2e5331be5295399d803ccac03f631f01f39a022fcdf63486b68 7d15ef284a77def7fde4898543e7b5f7ec267756103e477f547cfb8d2311c4b0 09deff56085f5d419697af1846c8b88c1bbbae149f0f19ca3c8dafe19cec48fe 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 78 /N put dup 82 /R put dup 85 /U put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueXX 5031982 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d56491bdecdcafdc988206c5a457a19270b 37d0ab776e03eaa7eb568eeab6b5e79dec03b0dcbf923a2aa8e4f4deda2cb043 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981639fb58948b5caaab9c6436b17d5b35387df46380196ab129f47c96c3b665 5d1f5dd2fff5964fb59bd03cd2ca81ad1e1a0d3f76980300c89352d684890963 c74e259fe1c6af453a3ff8858da12c2330c69495055dc95e6a2b4e9083bb5fa8 c4aadb585f4e03e3b89671c8f71bfaaa25d2750f04cfd584b9fffd0c875882a2 784c1856f8daa276a9b52bdd807c9684cb2dbbca07fc8f9961523d2931b2a52e 7e47ba54c9ae34c8fea128e6e67621e737043b4b774d0f589fd4da11eb81672d 48d9da81db09de5feb3b7fe29d409b79086c417713c2d434c3d48a5d4ad29929 cc9f5050a906c81cb420ebe677af540b555261c802d1b1f359016114fac42007 353de4384b8ea37954d0427dac0de7e82fdf2d84962b05a71fcd674116c5a144 e500cbec2dfa5ca89476dfd83b699f633f9b37ef404d75e96c2631d4e444c3a1 541372269523d7b987b1dfa7e0fafdc56cd7b37a5ea316e091fb6259cab975d7 8a1d9026625135a147f55a1abbd772b2b0b21e7cf980cb077f03d466574f2223 3a2a5f5eddfb9140654bcfec5a611cfd206e1c81fff3e568bf58354e49492e12 5eac7f5e2252e05ae3123ac9bf678c76afa0e8e8aef4f42c24452d593dc71e47 a9bb62bb919e5b3220575ebfeb4b7b16aa6ce773037d0f1042846234ce754f13 7e57e1c1ebb45fdc5190865ff1f7e56a2358fc84fd5528614ea0b00c0bcb1486 34346703b8761a67f47c0ff5257d076a97f3eebb4e8106b2ef293276aca6e23b b999ad0e843e593f4d6e2e0eadff33e06a7e38219b3fe6fa44864fcc420f5ca4 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMCSC10 %!PS-AdobeFont-1.1: CMCSC10 1.0 %%CreationDate: 1991 Aug 18 17:46:49 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMCSC10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMCSC10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 46 /period put dup 49 /one put dup 70 /F put dup 73 /I put dup 80 /P put dup 82 /R put dup 84 /T 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 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 readonly def /FontBBox{14 -250 1077 750}readonly def /UniqueXX 5000772 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 58 /colon put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 76 /L put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 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{-53 -251 1139 750}readonly def /UniqueXX 5000769 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5bfae5c1f476ee6f05eb1d2caeb 269958b194521197b312fcced4867f3c8fbd030bd715d8ffda1dcd454b174e7a 1a97b59fe770e67702519d9d9b23d61ac08424d555242a8ca08c49aef300945d 99b999a79ce74804ae6bfde623f4463371442f6523a5f6ce19c839a708c02513 2e22c696c8ccade45680e5197189d0f98e7f0d5f955e353970b392cf530a68cc 56b0035ddfbf206c3074beeb0739dcbca272a6e629fb7aea2c5ba7bae50c7b4c a595df78200c352997ec3ee564df229fbb5473f5e8ccb1cc0153e9a7e299a8ea a29b69d1b622b1f0cffc58291248759607d91150cb0651120970dc9f743bebef 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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 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 19 /acute put dup 33 /exclam 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 63 /question put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 90 /Z put dup 91 /bracketleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueXX 5000794 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 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FA(\),)f(w)m(e)g(m)m(ust)h(c)m(ho)s(ose)f(carefully)g(the)g(t)m (w)m(o)g(sequences)i Fu(f)p Fv(b)2638 786 y Fw(q)2638 847 y(n)2685 822 y Fu(g)p FA(,)e Fv(q)f FA(=)d(1)p Fv(;)17 b FA(2.)48 b(In)34 b(order)0 938 y(to)29 b(ease)h(the)f(exp)s(osition)i (w)m(e)f(\014rst)g(describ)s(e)g(the)g(iterativ)m(e)g(pro)s(cedure)g (with)g(a)f(sp)s(eci\014c)i(c)m(hoice)0 1054 y(of)h Fu(f)p Fv(b)202 1018 y Fw(q)202 1079 y(n)249 1054 y Fu(g)p FA(,)h(based)g(on)g (the)g(isop)s(erimetric)h(inequalit)m(y)975 1235 y Fv(V)21 b FA(\(\000)1152 1194 y Fw(q)1190 1235 y FA(\))1238 1166 y Fn(d)p Fm(\000)p Fl(1)p 1238 1179 111 3 v 1277 1220 a Fn(d)1390 1235 y Fu(\024)28 b Fv(\037)1556 1194 y Fr(\000)p Ft(1)1651 1235 y Fu(k)p FA(\000)1762 1194 y Fw(q)1799 1235 y Fu(k)98 b(8)28 b FA(\000)2091 1194 y Fw(q)2145 1235 y Fv(;)34 b(q)d FA(=)d(1)p Fv(;)17 b FA(2)g Fv(:)-2570 b FA(\(2.2\))0 1401 y(Existence)34 b(of)c Fv(\037)h FA(in)h(\(2.2\))e (follo)m(ws)i(from)f(\(1.3\))o(,)h(\(1.4\))e(and)h(\(1.10\).)43 b(Then,)32 b(in)g(subsection)h(2.2,)0 1517 y(w)m(e)38 b(mak)m(e)g(another)f(c)m(hoice)i(for)d Fu(f)p Fv(b)1308 1481 y Fw(q)1308 1541 y(n)1355 1517 y Fu(g)p FA(.)57 b(This)38 b(iterativ)m(e)g(construction)g(is)g(giv)m(en)g(in)f(details) h(in)0 1633 y(the)33 b(pro)s(of)f(of)g(the)h(Prop)s(osition)g(2.1,)f (whic)m(h)i(is)f(the)g(main)g(result)h(of)e(subsection)i(2.1.)p Black 0 1809 a Fz(Prop)s(osition)k(2.1.)p Black 49 w Fs(L)-5 b(et)35 b FA(0)27 b Fv(<)h(")f(<)h(\032)35 b Fs(and)f FA(0)28 b Fv(<)f(\016)32 b(<)27 b FA(1)35 b Fs(so)g(that)g FA(\001)22 b Fu(\000)h FA(2)p Fv(\016)31 b(>)d FA(0)p Fs(.)44 b(Set)587 1969 y Fv(U)653 1984 y Ft(0)720 1969 y FA(:=)28 b(\()p Fu(\000)p Fv(C)1043 1928 y Fr(\000)p Ft(1)1036 1994 y(1)1138 1969 y Fv(";)17 b(C)1305 1928 y Fr(\000)p Ft(1)1298 1994 y(1)1398 1969 y Fv(")p FA(\))99 b Fs(and)h Fx(U)1908 1984 y Ft(0)1975 1969 y FA(:=)27 b Fu(f)p Fv(z)32 b Fu(2)d Fx(C)e FA(:)45 b(Re)p Fv(z)32 b Fu(2)c Fv(U)2850 1984 y Ft(0)2890 1969 y Fu(g)0 2129 y Fs(and)1235 2248 y Fv(\034)11 b FA(\()p Fv(\014)6 b FA(\))28 b(:=)f Fv(\014)6 b FA(\()p Fv(\032)22 b Fu(\000)h Fv(")p FA(\))f Fu(\000)g FA(3)p Fv(C)2178 2263 y Ft(0)2217 2248 y Fv(\016)f(:)0 2388 y Fs(Ther)-5 b(e)34 b(exists)h Fv(\014)602 2403 y Ft(0)669 2388 y Fu(2)28 b Fx(R)835 2351 y Ft(+)929 2388 y Fs(such)35 b(that)g(for)f(al)5 b(l)35 b Fv(\014)e Fu(\025)28 b Fv(\014)1893 2403 y Ft(0)1968 2388 y Fs(the)35 b(fol)5 b(lowing)33 b(holds.)0 2504 y(A\))51 b(Ther)-5 b(e)49 b(exists)h(a)h(c)-5 b(ontinuous)50 b(r)-5 b(e)g(al-value)g(d)49 b(function)i(on)f Fx(R)p Fs(,)k Fv(\027)63 b Fu(7!)56 b Fv(\026)2881 2468 y Fr(\003)2920 2504 y FA(\()p Fv(\027)6 b FA(;)17 b Fv(\014)6 b FA(\))p Fs(,)54 b(so)c(that)0 2620 y Fv(\026)59 2584 y Fr(\003)98 2620 y FA(\()p Fv(\027)6 b FA(;)17 b Fv(\014)6 b FA(\))22 b(+)g Fv(i\027)34 b Fu(2)28 b Fx(U)734 2635 y Ft(0)774 2620 y Fs(.)0 2736 y(B\))42 b(If)h Fv(\026)27 b FA(+)h Fv(i\027)49 b Fu(2)43 b Fx(U)763 2751 y Ft(0)845 2736 y Fs(and)g Fv(\026)f Fu(\024)h Fv(\026)1323 2700 y 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b(b)1458 2356 y Fw(q)1458 2427 y(n)p Fr(\000)p Ft(1)1618 2403 y Fu(\000)22 b Fv(b)1758 2362 y Fw(q)1758 2428 y(n)1834 2403 y Fv(>)2010 2336 y FA(2)p Fv(\016)2106 2299 y Fw(l)q Ft(\()p Fw(n)p Fr(\000)p Ft(1\))p 1947 2380 436 4 v 1947 2471 a Fv(\014)6 b FA(\(\001)22 b Fu(\000)h FA(2)p Fv(\016)t FA(\))2409 2403 y Fv(;)0 2609 y FA(so)41 b(that)f(Re)p Fv(F)539 2573 y Fw(n)p Fr(\000)p Ft(1)676 2609 y FA(\()p Fv(\026)773 2573 y Fr(\003)773 2633 y Fw(n)p Fr(\000)p Ft(1)937 2609 y Fu(\000)28 b Fv(b)1083 2573 y Ft(1)1083 2633 y Fw(n)p Fr(\000)p Ft(1)1249 2609 y FA(+)f Fv(i\027)6 b FA(\))42 b Fv(<)f FA(0)f(and)h(Re)p Fv(F)2115 2573 y Fw(n)p Fr(\000)p Ft(1)2252 2609 y FA(\()p Fv(\026)2349 2573 y Fr(\003)2349 2633 y Fw(n)p Fr(\000)p Ft(1)2513 2609 y FA(+)27 b Fv(b)2657 2573 y Ft(2)2657 2633 y Fw(n)p Fr(\000)p Ft(1)2823 2609 y FA(+)g Fv(i\027)6 b FA(\))42 b Fv(>)f FA(0.)67 b(This)0 2728 y(pro)m(v)m(es)35 b(the)e(existence)j(of)d Fv(\026)1057 2691 y Fr(\003)1057 2752 y Fw(n)1137 2728 y 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Ft(1)917 3298 y FA(.)54 b(The)37 b(implicit)g(function)g(theorem)f(implies)i (that)e Fv(\027)j Fu(7!)34 b Fv(\026)3161 3262 y Fr(\003)3161 3323 y Fw(n)3207 3298 y FA(\()p Fv(\027)6 b FA(;)17 b Fv(\014)6 b FA(\))36 b(is)0 3414 y(con)m(tin)m(uous)e(\(ev)m(en)g Fv(C)825 3378 y Fr(1)900 3414 y FA(\).)0 3671 y(B.)26 b(W)-8 b(e)26 b(pro)m(v)m(e)g(that)g Fv(!)804 3686 y Fw(n)850 3671 y FA(\(\000)949 3634 y Fw(q)987 3671 y FA(\))g(is)g Fv(\034)1184 3686 y Ft(1)1224 3671 y FA(-stable)f(for)g (all)h(con)m(tours)g(\000)2249 3634 y Fw(q)2287 3671 y FA(,)h Fv(q)32 b FA(=)27 b(1)p Fv(;)17 b FA(2.)41 b(Let)25 b(\000)2957 3634 y Fw(q)3021 3671 y FA(b)s(e)h(a)f(con)m(tour)0 3787 y(with)41 b Fv(V)21 b FA(\(\000)407 3751 y Fw(q)445 3787 y FA(\))41 b(=)g Fv(n)p FA(.)67 b(All)40 b(con)m(tours)h(con)m (tributing)g(to)f(\002)2127 3802 y Fw(m)2194 3787 y FA(\(In)m(t)2356 3802 y Fw(m)2440 3787 y FA(\000)2501 3751 y Fw(q)2539 3787 y FA(\))g(and)g(\002)2890 3802 y Fw(q)2928 3787 y FA(\(In)m(t)3091 3802 y Fw(m)3174 3787 y FA(\000)3235 3751 y Fw(q)3273 3787 y FA(\))g(ha)m(v)m(e)0 3903 y(v)m(olumes)33 b(smaller)f(than)f Fv(n)19 b Fu(\000)g FA(1,)31 b(so)h(that)e(for)h (these)h(con)m(tours)g Fv(!)t FA(\(\000\))27 b(=)g Fv(!)2723 3918 y Fw(n)p Fr(\000)p Ft(1)2860 3903 y FA(\(\000\).)43 b(If)31 b Fv(z)h Fu(2)c Fx(U)3406 3918 y Fw(n)p Fr(\000)p Ft(1)0 4019 y FA(\(use)33 b(\(1.8\),)g(\(1.4\))f(and)g(the)h (de\014nition)h(of)e Fv(U)1668 4034 y Ft(0)1708 4019 y FA(\),)g(then)131 4362 y Fu(j)p Fv(!)t FA(\(\000)323 4321 y Fw(q)360 4362 y FA(\))p Fu(j)27 b FA(=)h(exp)722 4282 y Fp(\002)786 4362 y Fu(\000)23 b Fv(\014)6 b FA(Re)o Fu(H)q FA(\()p Fv(')1248 4377 y Ft(\000)1292 4358 y Fn(q)1331 4362 y Fu(j)p Fv( )1422 4377 y Fw(q)1460 4362 y FA(\))1498 4282 y Fp(\003)1556 4248 y(\014)1556 4308 y(\014)1556 4367 y(\014)1659 4268 y(Y)1606 4480 y Fw(m)p Ft(:)p Fw(m)p Fr(6)p Ft(=)p Fw(q)1866 4295 y FA(\002)1942 4310 y Fw(m)2008 4295 y FA(\(In)m(t)2171 4310 y Fw(m)2254 4295 y FA(\000)2315 4259 y Fw(q)2353 4295 y FA(\))p 1866 4339 526 4 v 1880 4431 a(\002)1956 4446 y Fw(q)1994 4431 y FA(\(In)m(t)2156 4446 y Fw(m)2239 4431 y FA(\000)2300 4402 y Fw(q)2338 4431 y FA(\))2401 4248 y Fp(\014)2401 4308 y(\014)2401 4367 y(\014)0 4173 y FA(\(2.12\))453 4655 y Fu(\024)28 b FA(exp)724 4544 y Fp(h)793 4655 y Fu(\000)23 b Fv(\014)6 b Fu(k)p FA(\000)1065 4614 y Fw(q)1102 4655 y Fu(k)22 b FA(+)1272 4574 y Fp(\000)1318 4655 y Fv(\014)6 b(")21 b FA(+)h(2)p Fv(C)1663 4670 y Ft(0)1702 4655 y Fv(\016)1749 4574 y Fp(\001)1795 4655 y Fu(j)p FA(\000)1884 4614 y Fw(q)1921 4655 y Fu(j)g(\000)h Fv(\014)6 b FA(Re)2247 4574 y Fp(\000)2292 4655 y Fv(f)2351 4614 y Fw(n)p Fr(\000)p Ft(1)2340 4679 y Fw(m)2510 4655 y Fu(\000)23 b Fv(f)2669 4614 y Fw(n)p Fr(\000)p Ft(1)2658 4679 y Fw(q)2806 4574 y Fp(\001)2851 4655 y Fv(V)f FA(\(\000)3029 4614 y Fw(q)3067 4655 y FA(\))3105 4544 y Fp(i)453 4899 y FA(=)28 b(exp)722 4788 y Fp(h)791 4899 y Fu(\000)23 b Fv(\014)6 b Fu(k)p FA(\000)1063 4858 y Fw(q)1100 4899 y Fu(k)22 b FA(+)1270 4818 y Fp(\000)1316 4899 y Fv(\014)6 b(")21 b FA(+)h(2)p Fv(C)1661 4914 y Ft(0)1700 4899 y Fv(\016)1747 4818 y Fp(\001)1793 4899 y Fu(j)p FA(\000)1882 4858 y Fw(q)1920 4899 y Fu(j)g(\000)g Fv(\014)6 b FA(Re)2245 4818 y Fp(\000)2291 4899 y Fv(f)2350 4858 y Fw(n)p Fr(\000)p Ft(1)2339 4924 y Fw(m)2509 4899 y Fu(\000)22 b Fv(f)2667 4858 y Fw(n)p Fr(\000)p Ft(1)2656 4924 y Fw(q)2804 4818 y Fp(\001)2860 4832 y Fv(V)f FA(\(\000)3037 4795 y Fw(q)3075 4832 y FA(\))p 2860 4876 254 4 v 2887 4967 a Fu(k)p FA(\000)2998 4938 y Fw(q)3036 4967 y Fu(k)3123 4899 y(k)p FA(\000)3234 4858 y Fw(q)3272 4899 y Fu(k)3322 4788 y Fp(i)3385 4899 y Fv(:)0 5121 y FA(Let)36 b Fv(\026)d Fu(2)g Fv(U)435 5136 y Fw(n)p Fr(\000)p Ft(1)572 5121 y FA(\()p Fv(\027)6 b FA(;)17 b Fv(\014)6 b FA(\).)53 b(W)-8 b(e)36 b(pro)m(v)m(e)h(that)e Fv(b)1579 5085 y Fw(q)1579 5146 y(n)1662 5121 y FA(v)m(erify)i(the)f (follo)m(wing)g(conditions,)i(whic)m(h)f(imply)0 5237 y(the)c Fv(\034)210 5252 y Ft(1)250 5237 y FA(-stabilit)m(y)-8 b(.)564 5457 y Fu(\000)p FA(Re)757 5376 y Fp(\000)802 5457 y Fv(f)861 5416 y Fw(n)p Fr(\000)p Ft(1)850 5481 y(1)1020 5457 y Fu(\000)23 b Fv(f)1179 5416 y Fw(n)p Fr(\000)p Ft(1)1168 5481 y(2)1316 5376 y Fp(\001)1372 5389 y Fv(V)e FA(\(\000)1549 5353 y Ft(2)1588 5389 y FA(\))p 1372 5434 255 4 v 1399 5525 a Fu(k)p FA(\000)1510 5496 y Ft(2)1549 5525 y Fu(k)1664 5457 y(\024)28 b Fv(\022)1817 5416 y Fr(0)1938 5457 y FA(if)33 b Fv(\026)27 b Fu(\024)h Fv(\026)2278 5416 y Fr(\003)2278 5481 y Fw(n)2347 5457 y FA(+)22 b Fv(b)2486 5416 y Ft(2)2486 5481 y Fw(n)2566 5457 y FA(and)33 b Fv(V)21 b FA(\(\000)2933 5416 y Ft(2)2972 5457 y FA(\))28 b(=)g Fv(n)p FA(,)-3227 b(\(2.13\))564 5734 y Fu(\000)p FA(Re)757 5653 y Fp(\000)802 5734 y Fv(f)861 5693 y Fw(n)p Fr(\000)p Ft(1)850 5758 y(2)1020 5734 y Fu(\000)23 b Fv(f)1179 5693 y Fw(n)p Fr(\000)p Ft(1)1168 5758 y(1)1316 5653 y Fp(\001)1372 5666 y Fv(V)e FA(\(\000)1549 5630 y Ft(1)1588 5666 y FA(\))p 1372 5711 V 1399 5802 a Fu(k)p FA(\000)1510 5773 y Ft(1)1549 5802 y Fu(k)1664 5734 y(\024)28 b Fv(\022)1817 5693 y Fr(0)1938 5734 y FA(if)33 b Fv(\026)27 b Fu(\025)h Fv(\026)2278 5693 y Fr(\003)2278 5758 y Fw(n)2347 5734 y Fu(\000)23 b Fv(b)2488 5693 y Ft(1)2488 5758 y Fw(n)2567 5734 y FA(and)33 b Fv(V)22 b FA(\(\000)2935 5693 y Ft(1)2974 5734 y FA(\))28 b(=)f Fv(n)p FA(.)-3228 b(\(2.14\))p Black Black eop %%Page: 11 11 11 10 bop Black 3467 49 a Fq(11)p Black 0 241 a FA(F)-8 b(or)32 b(the)h(presen)m(t)h(c)m(hoice)g(of)e Fu(f)p Fv(b)1174 205 y Fw(q)1174 265 y(n)1221 241 y Fu(g)p FA(,)h(the)g(isop)s (erimetric)h(inequalit)m(y)i(\(2.2\))c(implies)1142 415 y Fv(V)22 b FA(\(\000)1320 379 y Fw(q)1358 415 y FA(\))p 1142 459 254 4 v 1169 551 a Fu(k)p FA(\000)1280 522 y Fw(q)1318 551 y Fu(k)1433 482 y(\024)1548 415 y Fv(V)g FA(\(\000)1726 379 y Fw(q)1764 415 y FA(\))1813 351 y Fl(1)p 1812 363 33 3 v 1812 404 a Fn(d)p 1548 459 310 4 v 1672 551 a Fv(\037)1965 482 y Fu(8)28 b Fv(q)k FA(=)27 b(1)p Fv(;)17 b FA(2)g Fv(;)0 698 y FA(and)33 b(therefore)429 808 y Fp(\014)429 868 y(\014)462 893 y FA(Re)577 812 y Fp(\000)623 893 y Fv(f)682 852 y Fw(n)p Fr(\000)p Ft(1)671 918 y Fw(m)841 893 y Fu(\000)23 b Fv(f)1000 852 y Fw(n)p Fr(\000)p Ft(1)989 918 y Fw(q)1136 812 y Fp(\001)1182 808 y(\014)1182 868 y(\014)1225 826 y Fv(V)f FA(\(\000)1403 790 y Fw(q)1441 826 y FA(\))p 1225 870 254 4 v 1253 961 a Fu(k)p FA(\000)1364 933 y Fw(q)1402 961 y Fu(k)1516 893 y FA(=)1620 779 y Fp(\014)1620 838 y(\014)1620 898 y(\014)1653 893 y FA(Re)1785 758 y Fp(Z)1884 784 y Fw(\026)1840 983 y(\026)1882 964 y Fm(\003)1882 1000 y Fn(n)1987 826 y Fv(d)p 1958 870 110 4 v 1958 961 a(d\026)2077 812 y Fp(\000)2123 893 y Fv(f)2182 852 y Fw(n)p Fr(\000)p Ft(1)2171 918 y Fw(m)2341 893 y Fu(\000)g Fv(f)2499 852 y Fw(n)p Fr(\000)p Ft(1)2488 918 y Fw(q)2636 812 y Fp(\001)2698 893 y Fv(d\026)2808 779 y Fp(\014)2808 838 y(\014)2808 898 y(\014)2851 826 y Fv(V)g FA(\(\000)3029 790 y Fw(q)3066 826 y FA(\))p 2851 870 254 4 v 2878 961 a Fu(k)p FA(\000)2989 933 y Fw(q)3027 961 y Fu(k)1516 1179 y(\024)29 b(j)p Fv(\026)21 b Fu(\000)i Fv(\026)1889 1138 y Fr(\003)1889 1204 y Fw(n)1935 1179 y Fu(j)p FA(\(\001)f(+)g(2)p Fv(\016)t FA(\))2346 1112 y Fv(V)g FA(\(\000)2524 1076 y Fw(q)2561 1112 y FA(\))p 2346 1156 V 2373 1247 a Fu(k)p FA(\000)2484 1219 y Fw(q)2522 1247 y Fu(k)2637 1179 y(\024)28 b Fv(\022)2790 1138 y Fr(0)2830 1179 y Fv(:)0 1400 y FA(Conditions)34 b(\(2.13\))e(and)g(\(2.14\))g(ensure)i(that)e(on)h Fx(U)1970 1415 y Fw(n)684 1593 y Fu(j)p Fv(!)t FA(\(\000)876 1552 y Fw(q)913 1593 y FA(\))p Fu(j)27 b(\024)h FA(exp)1277 1482 y Fp(h)1346 1593 y Fu(\000)22 b Fv(\014)1506 1512 y Fp(\000)1552 1593 y Fv(\032)p FA(\(1)g Fu(\000)g Fv(\022)1858 1552 y Fr(0)1882 1593 y FA(\))g Fu(\000)h Fv(")e Fu(\000)i FA(2)p Fv(\014)2319 1552 y Fr(\000)p Ft(1)2413 1593 y Fv(C)2483 1608 y Ft(0)2522 1593 y Fv(\016)2569 1512 y Fp(\001)2615 1593 y Fu(j)p FA(\000)2704 1552 y Fw(q)2741 1593 y Fu(j)2769 1482 y Fp(i)2833 1593 y Fv(:)0 1918 y FA(C.)33 b(W)-8 b(e)33 b(pro)m(v)m(e)h(that)e(on)h Fx(U)980 1933 y Fw(n)1208 2043 y Fp(\014)1208 2102 y(\014)1276 2060 y Fv(d)p 1251 2104 101 4 v 1251 2196 a(dz)1362 2127 y(!)1423 2142 y Fw(n)1469 2127 y FA(\(\000\))1606 2043 y Fp(\014)1606 2102 y(\014)1667 2127 y Fu(\024)28 b Fv(\014)6 b(C)1903 2142 y Ft(3)1942 2127 y FA(e)1985 2086 y Fr(\000)p Fw(\034)2071 2095 y Fl(2)2106 2086 y Ft(\()p Fw(\014)s Ft(\))p Fr(j)p Ft(\000)p Fr(j)2308 2127 y Fv(:)0 2324 y FA(Let)33 b Fv(V)21 b FA(\(\000)352 2287 y Fw(q)390 2324 y FA(\))28 b(=)f Fv(n)p FA(;)33 b(from)g(\(1.9\))568 2471 y Fv(d)p 543 2515 V 543 2606 a(dz)654 2538 y(!)715 2553 y Fw(n)761 2538 y FA(\(\000)860 2497 y Fw(q)898 2538 y FA(\))28 b(=)g Fv(!)1129 2553 y Fw(n)1175 2538 y FA(\(\000)1274 2497 y Fw(q)1312 2538 y FA(\))1350 2428 y Fp(\020)1432 2538 y Fu(\000)22 b Fv(\014)6 b(a)p FA(\()p Fv(')1745 2553 y Ft(\000)1789 2534 y Fn(q)1828 2538 y FA(\))22 b Fu(\000)g Fv(\014)2048 2457 y Fp(\000)2094 2538 y Fv(h)p FA(\()p Fv( )2251 2553 y Fw(m)2318 2538 y FA(\))g Fu(\000)g Fv(h)p FA(\()p Fv( )2634 2553 y Fw(q)2673 2538 y FA(\))2711 2457 y Fp(\001)2757 2538 y Fv(V)f FA(\(\000)2934 2497 y Fw(q)2972 2538 y FA(\))1056 2780 y(+)1189 2713 y Fv(d)p 1164 2758 V 1164 2849 a(dz)1274 2700 y Fp(\000)1337 2780 y FA(log)c(\002)1555 2795 y Fw(m)1622 2780 y FA(\(In)m(t)1784 2795 y Fw(m)1867 2780 y FA(\000)1928 2739 y Fw(q)1966 2780 y FA(\))23 b Fu(\000)f FA(log)c(\002)2345 2795 y Fw(q)2383 2780 y FA(\(In)m(t)2545 2795 y Fw(m)2628 2780 y FA(\000)2689 2739 y Fw(q)2727 2780 y FA(\))2765 2700 y Fp(\001)2811 2670 y(\021)2887 2780 y Fv(:)0 2977 y FA(By)33 b(Lemma)g(1.1)f(and)h(the)g(isop)s(erimetric)h(inequalit)m(y)i (\(1.3\))c(w)m(e)i(get)520 3107 y Fp(\014)520 3166 y(\014)588 3124 y Fv(d)p 563 3168 V 563 3260 a(dz)674 3191 y(!)735 3206 y Fw(n)781 3191 y FA(\(\000)880 3150 y Fw(q)918 3191 y FA(\))956 3107 y Fp(\014)956 3166 y(\014)1017 3191 y Fu(\024)28 b Fv(\014)6 b Fu(j)p Fv(!)1272 3206 y Fw(n)1318 3191 y FA(\(\000)1417 3150 y Fw(q)1455 3191 y FA(\))p Fu(j)1521 3111 y Fp(\000)1566 3191 y Fu(j)p FA(\000)1655 3150 y Fw(q)1693 3191 y Fu(j)p FA(\()p Fv(C)1829 3206 y Ft(1)1890 3191 y FA(+)22 b(2)p Fv(\016)t(C)2154 3206 y Ft(0)2193 3191 y FA(\))g(+)g Fv(V)g FA(\(\000)2529 3150 y Fw(q)2567 3191 y FA(\)\(\001)g(+)g(2)p Fv(\016)t FA(\))2978 3111 y Fp(\001)0 3191 y FA(\(2.15\))1017 3402 y Fu(\024)28 b Fv(\014)6 b(C)1253 3417 y Ft(3)1292 3402 y Fu(j)p Fv(!)1381 3417 y Fw(n)1427 3402 y FA(\(\000)1526 3361 y Fw(q)1564 3402 y FA(\))p Fu(jj)p FA(\000)1719 3361 y Fw(q)1757 3402 y Fu(j)1834 3332 y Fn(d)p 1794 3344 111 3 v 1794 3385 a(d)p Fm(\000)p Fl(1)1017 3570 y Fu(\024)28 b Fv(\014)6 b(C)1253 3585 y Ft(3)1292 3570 y FA(e)1335 3529 y Fr(\000)p Fw(\034)1421 3538 y Fl(2)1456 3529 y Ft(\()p Fw(\014)s Ft(\))p Fr(j)p Ft(\000)1617 3505 y Fn(q)1652 3529 y Fr(j)1693 3570 y Fv(:)0 3879 y FA(D.)23 b(W)-8 b(e)25 b(pro)m(v)m(e)g(that)f Fv(!)t FA(\(\000)906 3843 y Ft(2)944 3879 y FA(\)\()p Fv(z)t FA(\))h(is)f Fv(\034)11 b FA(\()p Fv(\014)6 b FA(\)-stable)24 b(for)g(an)m(y)g(con)m(tour)h(\000)2441 3843 y Ft(2)2504 3879 y FA(with)f(b)s(oundary)h(condition)0 3995 y Fv( )63 4010 y Ft(2)103 3995 y FA(,)31 b(if)g Fv(\026)d Fu(\024)g Fv(\026)500 3959 y Fr(\003)500 4020 y Fw(n)546 3995 y FA(\()p Fv(\027)6 b FA(;)17 b Fv(\014)6 b FA(\))19 b Fu(\000)h Fv(b)938 3959 y Ft(1)938 4020 y Fw(n)985 3995 y FA(.)43 b(Using)32 b(the)f(induction)h(h)m(yp)s(othesis)i(it)d(is)g (su\016cien)m(t)j(to)c(pro)m(v)m(e)j(this)0 4112 y(statemen)m(t)42 b(for)e Fv(z)45 b FA(=)c Fv(\026)27 b FA(+)g Fv(i\027)48 b Fu(2)41 b Fx(U)1322 4127 y Fw(n)p Fr(\000)p Ft(1)1499 4112 y FA(and)g Fv(\026)f Fu(\024)i Fv(\026)1974 4075 y Fr(\003)1974 4136 y Fw(n)2020 4112 y FA(\()p Fv(\027)6 b FA(;)17 b Fv(\014)6 b FA(\))27 b Fu(\000)h Fv(b)2428 4075 y Ft(1)2428 4136 y Fw(n)2476 4112 y FA(.)66 b(If)41 b Fv(z)k FA(=)c Fv(\026)27 b FA(+)g Fv(i\027)48 b Fu(2)41 b Fx(U)3379 4127 y Fw(n)p Fr(\000)p Ft(1)3516 4112 y FA(,)0 4228 y(then)46 b(all)g(con)m(tours)g(with)g(v)m(olume)h Fv(V)22 b FA(\(\000\))49 b Fu(\024)h Fv(n)32 b Fu(\000)f FA(1)45 b(are)h Fv(\034)2276 4243 y Ft(1)2316 4228 y FA(\()p Fv(\014)6 b FA(\)-stable,)48 b(and)e(for)f Fv(\026)k Fu(\024)i Fv(\026)3470 4192 y Fr(\003)3470 4252 y Fw(n)3516 4228 y FA(,)0 4345 y Fv(\026)29 b Fu(7!)h FA(Re\()p Fv(f)430 4304 y Fw(n)p Fr(\000)p Ft(1)419 4369 y(1)590 4345 y Fu(\000)23 b Fv(f)749 4304 y Fw(n)p Fr(\000)p Ft(1)738 4369 y(2)886 4345 y FA(\)\()p Fv(\026)f FA(+)h Fv(i\027)6 b FA(\))35 b(is)f(strictly)h(decreasing.)48 b(If)34 b Fv(\026)29 b Fu(\024)i Fv(\026)2603 4309 y Fr(\003)2603 4370 y Fw(n)2649 4345 y FA(\()p Fv(\027)6 b FA(;)17 b Fv(\014)6 b FA(\))23 b Fu(\000)g Fv(b)3048 4309 y Ft(1)3048 4370 y Fw(n)3096 4345 y FA(,)34 b(then)g(\(see)0 4461 y(\(2.8\))e(and)h(\(2.7\))o(\))279 4695 y Fv(\014)6 b FA(Re)o(\()p Fv(f)551 4654 y Fw(n)p Fr(\000)p Ft(1)540 4720 y(1)710 4695 y Fu(\000)23 b Fv(f)869 4654 y Fw(n)p Fr(\000)p Ft(1)858 4720 y(2)1006 4695 y FA(\)\()p Fv(\026)f FA(+)g Fv(i\027)6 b FA(\))28 b(=)f Fu(\000)p Fv(\014)1672 4560 y Fp(Z)1772 4586 y Fw(\026)1814 4562 y Fm(\003)1814 4603 y Fn(n)1727 4785 y Fw(\026)1917 4628 y Fv(d)p 1887 4672 110 4 v 1887 4764 a(d\026)2007 4695 y FA(Re\()p Fv(f)2219 4654 y Fw(n)p Fr(\000)p Ft(1)2208 4720 y(1)2378 4695 y Fu(\000)22 b Fv(f)2536 4654 y Fw(n)p Fr(\000)p Ft(1)2525 4720 y(2)2673 4695 y FA(\)\()p Fv(\026)g FA(+)g Fv(i\027)6 b FA(\))17 b Fv(d\026)1414 4991 y Fu(\025)28 b(\000)p Fv(\014)1674 4856 y Fp(Z)1773 4882 y Fw(\026)1815 4859 y Fm(\003)1815 4899 y Fn(n)1729 5081 y Fw(\026)1771 5062 y Fm(\003)1771 5098 y Fn(n)1814 5081 y Fr(\000)p Fw(b)1899 5062 y Fl(1)1899 5098 y Fn(n)2002 4924 y Fv(d)p 1972 4968 V 1972 5060 a(d\026)2092 4991 y FA(Re\()p Fv(f)2304 4950 y Fw(n)p Fr(\000)p Ft(1)2293 5016 y(1)2463 4991 y Fu(\000)22 b Fv(f)2621 4950 y Fw(n)p Fr(\000)p Ft(1)2610 5016 y(2)2758 4991 y FA(\)\()p Fv(\026)g FA(+)g Fv(i\027)6 b FA(\))17 b Fv(d\026)1414 5226 y Fu(\025)28 b Fv(\014)6 b(b)1621 5185 y Ft(1)1621 5251 y Fw(n)1668 5226 y FA(\(\001)22 b Fu(\000)h FA(2)p Fv(\016)t FA(\))k Fu(\025)i FA(2)p Fv(\016)2272 5185 y Fw(l)q Ft(\()p Fw(n)p Ft(\))2412 5226 y Fv(:)-2439 b FA(\(2.16\))0 5397 y(First)46 b(supp)s(ose)h(that)f Fv(V)22 b FA(\(\000)1034 5361 y Ft(2)1073 5397 y FA(\))51 b Fu(\024)g Fv(n)p FA(.)84 b(F)-8 b(rom)45 b(\(2.16\))g(and)h(\(2.12\)) g(it)g(follo)m(ws)g(that)g Fv(!)t FA(\(\000)3355 5361 y Ft(2)3394 5397 y FA(\))g(is)0 5514 y Fv(\014)6 b FA(\()p Fv(\032)22 b Fu(\000)h Fv(")e Fu(\000)i FA(2)p Fv(\014)548 5477 y Fr(\000)p Ft(1)642 5514 y Fv(C)712 5529 y Ft(0)751 5514 y Fv(\016)t FA(\)-stable,)33 b(in)g(particular)f Fv(\034)11 b FA(\()p Fv(\014)6 b FA(\)-stable.)44 b(Moreo)m(v)m(er,)34 b(if)f Fu(j)p FA(\003)p Fu(j)27 b(\024)h Fv(n)p FA(,)33 b(then)744 5619 y Fp(\014)744 5679 y(\014)744 5739 y(\014)794 5734 y FA(exp)959 5653 y Fp(\002)1023 5734 y Fu(\000)23 b Fv(\014)6 b(z)t FA(\()p Fv(h)p FA(\()p Fv( )1428 5749 y Ft(1)1468 5734 y FA(\))22 b Fu(\000)h Fv(h)p FA(\()p Fv( )1785 5749 y Ft(2)1825 5734 y FA(\)\))p Fu(j)p FA(\003)p Fu(j)2025 5653 y Fp(\003)2075 5666 y FA(\002)2151 5681 y Ft(1)2190 5666 y FA(\(\003\))p 2075 5711 259 4 v 2075 5802 a(\002)2151 5817 y Ft(2)2190 5802 y FA(\(\003\))2344 5619 y Fp(\014)2344 5679 y(\014)2344 5739 y(\014)2405 5734 y Fu(\024)28 b FA(e)2553 5693 y Ft(3)p Fw(\016)r(@)t Fr(j)p Ft(\003)p Fr(j)2772 5734 y Fv(:)-2799 b FA(\(2.17\))p Black Black eop %%Page: 12 12 12 11 bop Black 0 49 a Fq(12)p Black 0 241 a FA(Indeed,)34 b(all)f(con)m(tours)g(inside)h(\003)e(are)h Fv(\034)1455 256 y Ft(1)1495 241 y FA(\()p Fv(\014)6 b FA(\)-stable.)43 b(By)33 b(Lemma)g(1.1)407 352 y Fp(\014)407 412 y(\014)407 472 y(\014)440 467 y FA(e)483 425 y Fr(\000)p Fw(\014)s(z)s Ft(\()p Fw(h)p Ft(\()p Fw( )758 434 y Fl(1)794 425 y Ft(\))p Fr(\000)p Fw(h)p Ft(\()p Fw( )990 434 y Fl(2)1025 425 y Ft(\)\))p Fr(j)p Ft(\003)p Fr(j)1182 399 y FA(\002)1258 414 y Ft(1)1297 399 y FA(\(\003\))p 1182 444 259 4 v 1182 535 a(\002)1258 550 y Ft(2)1297 535 y FA(\(\003\))1451 352 y Fp(\014)1451 412 y(\014)1451 472 y(\014)1512 467 y Fu(\024)1617 382 y Fp(\014)1617 442 y(\014)1650 467 y FA(e)1693 425 y Fr(\000)p Fw(\014)s Ft(\()p Fw(z)s(h)p Ft(\()p Fw( )1968 434 y Fl(1)2003 425 y Ft(\))p Fr(\000)p Fw(z)s(h)p Ft(\()p Fw( )2235 434 y Fl(2)2270 425 y Ft(\)+)p Fw(g)2388 396 y Fn(n)p Fm(\000)p Fl(1)2386 447 y(1)2509 425 y Fr(\000)p Fw(g)2600 396 y Fn(n)p Fm(\000)p Fl(1)2598 447 y(2)2721 425 y Ft(\))p Fr(j)p Ft(\003)p Fr(j)2841 382 y Fp(\014)2841 442 y(\014)2890 467 y FA(e)2933 425 y Ft(2)p Fw(\016)r(@)t Fr(j)p Ft(\003)p Fr(j)1512 700 y FA(=)27 b(e)1658 659 y Fr(\000)p Fw(\014)s Ft(Re)q(\()p Fw(f)1908 629 y Fn(n)p Fm(\000)p Fl(1)1901 680 y(1)2029 659 y Ft(\()p Fw(z)s Ft(\))p Fr(\000)p Fw(f)2215 629 y Fn(n)p Fm(\000)p Fl(1)2208 680 y(2)2337 659 y Ft(\()p Fw(z)s Ft(\)\))p Fr(j)p Ft(\003)p Fr(j)2547 700 y FA(e)2590 659 y Ft(2)p Fw(\016)r(@)t Fr(j)p Ft(\003)p Fr(j)1512 868 y Fu(\024)h FA(e)1660 826 y Ft(2)p Fw(\016)r(@)t Fr(j)p Ft(\003)p Fr(j)1879 868 y Fv(:)0 1040 y FA(T)-8 b(o)29 b(pro)m(v)m(e)i(p)s(oin)m(t)e(D,)g(w)m(e)i(pro)m(v)m(e)f(b)m(y)g (induction)h(on)e Fu(j)p FA(\003)p Fu(j)f FA(that)h(\(2.17\))g(holds)h (for)f(an)m(y)h(\003.)42 b(Indeed,)0 1156 y(if)f(\(2.17\))33 b(is)i(true)f(and)g(if)g(w)m(e)h(set)g(\003)30 b(:=)g(In)m(t)1625 1171 y Ft(1)1664 1156 y FA(\000)1725 1120 y Ft(2)1765 1156 y FA(,)k(then)h(it)f(follo)m(ws)h(easily)g(from)f(the)g (de\014nition)0 1273 y(of)e Fv(!)t FA(\(\000)275 1236 y Ft(2)314 1273 y FA(\))g(and)h(from)f(\(1.9\))g(that)h Fv(!)t FA(\(\000)1413 1236 y Ft(2)1451 1273 y FA(\))g(is)g Fv(\034)11 b FA(\()p Fv(\014)6 b FA(\)-stable.)100 1439 y(The)35 b(argumen)m(t)g(to)e(pro)m(v)m(e)j(\(2.17\))d(is)i(due)g(to)e (Zahradnik)i([7].)48 b(The)35 b(statemen)m(t)h(is)e(true)h(for)0 1555 y Fu(j)p FA(\003)p Fu(j)j(\024)h Fv(n)p FA(.)63 b(Supp)s(ose)40 b(that)e(it)h(is)h(true)f(for)g Fu(j)p FA(\003)p Fu(j)e(\024)i Fv(m)p FA(,)i Fv(m)e(>)f(n)p FA(,)j(and)e(let)h Fu(j)p FA(\003)p Fu(j)d FA(=)i Fv(m)27 b FA(+)f(1.)62 b(The)0 1671 y(induction)34 b(h)m(yp)s(othesis)g (implies)g(that)f Fv(!)t FA(\(\000)1621 1635 y Ft(2)1659 1671 y FA(\)\()p Fv(z)t FA(\))g(is)h Fv(\034)11 b FA(\()p Fv(\014)6 b FA(\)-stable)32 b(if)h Fv(V)21 b FA(\(\000)2725 1635 y Ft(2)2765 1671 y FA(\))27 b Fu(\024)h Fv(m)p FA(.)44 b(Therefore)419 1783 y Fp(\014)419 1842 y(\014)419 1902 y(\014)452 1897 y FA(e)495 1856 y Fr(\000)p Fw(\014)s(z)s Ft(\()p Fw(h)p Ft(\()p Fw( )770 1865 y Fl(1)805 1856 y Ft(\))p Fr(\000)p Fw(h)p Ft(\()p Fw( )1001 1865 y Fl(2)1036 1856 y Ft(\)\))p Fr(j)p Ft(\003)p Fr(j)1193 1830 y FA(\002)1269 1845 y Ft(1)1309 1830 y FA(\(\003\))p 1193 1874 V 1193 1966 a(\002)1269 1981 y Ft(2)1309 1966 y FA(\(\003\))1462 1783 y Fp(\014)1462 1842 y(\014)1462 1902 y(\014)1523 1897 y Fu(\024)1628 1812 y Fp(\014)1628 1872 y(\014)1662 1897 y FA(e)1705 1856 y Fr(\000)p Fw(\014)s Ft(\()p Fw(z)s(h)p Ft(\()p Fw( )1980 1865 y Fl(1)2015 1856 y Ft(\))p Fr(\000)p Fw(z)s(h)p Ft(\()p Fw( )2247 1865 y Fl(2)2282 1856 y Ft(\))p Fr(\000)p Fw(g)2400 1833 y Fn(m)2398 1877 y Fl(2)2458 1856 y Ft(\))p Fr(j)p Ft(\003)p Fr(j)2578 1897 y FA(\002)2654 1912 y Ft(1)2694 1897 y FA(\(\003\))2838 1812 y Fp(\014)2838 1872 y(\014)2870 1897 y FA(e)2913 1856 y Fw(\016)r(@)t Fr(j)p Ft(\003)p Fr(j)3097 1897 y Fv(:)0 2124 y FA(F)-8 b(rom)32 b(\(1.5\))1276 2330 y(\002)1352 2345 y Ft(1)1392 2330 y FA(\(\003\))27 b(=)1667 2235 y Fp(X)1874 2206 y Fw(r)1827 2235 y Fp(Y)1829 2445 y Fw(j)t Ft(=1)1971 2330 y FA(\002\(\000)2146 2289 y Ft(1)2146 2355 y Fw(j)2185 2330 y FA(\))17 b Fv(;)0 2586 y FA(where)39 b(the)f(sum)g(is)g(o)m(v)m (er)h(all)f(families)g Fu(f)p FA(\000)1600 2550 y Ft(1)1600 2610 y(1)1639 2586 y Fv(;)17 b(:)g(:)g(:)33 b(;)17 b FA(\000)1936 2550 y Ft(1)1936 2610 y Fw(r)1975 2586 y Fu(g)37 b FA(of)g(compatible)i(external)g(con)m(tours)f(in)0 2702 y(\003.)69 b(W)-8 b(e)42 b(sa)m(y)g(that)f(an)g(external)i(con)m (tour)e(\000)1692 2666 y Ft(1)1692 2727 y Fw(j)1773 2702 y FA(is)g(large)h(if)f Fv(V)21 b FA(\(\000)2402 2666 y Ft(1)2402 2727 y Fw(j)2442 2702 y FA(\))42 b Fu(\025)h Fv(n)p FA(.)70 b(Supp)s(ose)42 b(that)f(the)0 2827 y(con)m(tours)33 b(\000)454 2791 y Ft(1)454 2851 y(1)494 2827 y Fv(;)17 b(:)g(:)g(:)f FA(\000)730 2791 y Ft(1)730 2851 y Fw(p)802 2827 y FA(are)32 b(large)h(and)g(all)f(other)h(con)m(tours)g(\000)2237 2791 y Ft(1)2237 2851 y Fw(p)p Ft(+1)2367 2827 y Fv(;)17 b(:)g(:)g(:)f FA(\000)2603 2791 y Ft(1)2603 2851 y Fw(r)2675 2827 y FA(not)32 b(large.)43 b(W)-8 b(e)33 b(set)1160 3090 y(Ext)1316 3043 y Fw(p)1316 3115 y Ft(1)1356 3090 y FA(\(\003\))27 b(:=)1658 3009 y Fp(\000)1763 2962 y Fw(p)1726 2996 y Fp(\\)1720 3206 y Fw(j)t Ft(=1)1859 3090 y FA(Ext)q(\000)2076 3049 y Ft(1)2076 3115 y Fw(j)2115 3009 y Fp(\001)2183 3090 y Fu(\\)c FA(\003)17 b Fv(:)0 3362 y FA(Summing)34 b(o)m(v)m(er)f(all)g(con)m(tours)g(whic)m(h)h(are) f(not)f(large,)h(and)f(using)j(\(1.6\),)d(w)m(e)i(get)204 3613 y(\002)280 3628 y Ft(1)319 3613 y FA(\(\003\))28 b(=)594 3518 y Fp(X)755 3613 y FA(\002)831 3572 y Fw(n)p Fr(\000)p Ft(1)831 3637 y(1)968 3532 y Fp(\000)1013 3613 y FA(Ext)1169 3566 y Fw(p)1169 3637 y Ft(1)1209 3613 y FA(\(\003\))1353 3532 y Fp(\001)1461 3484 y Fw(p)1415 3518 y Fp(Y)1417 3728 y Fw(j)t Ft(=1)1559 3613 y FA(exp)1724 3532 y Fp(\002)1788 3613 y Fu(\000)23 b Fv(\014)6 b Fu(H)q FA(\()p Fv(')2136 3633 y Ft(\000)2180 3611 y Fl(1)2180 3656 y Fn(j)2218 3613 y Fu(j)p Fv( )2309 3628 y Ft(1)2348 3613 y FA(\))2386 3532 y Fp(\003)2428 3613 y FA(\002)2504 3628 y Ft(1)2543 3613 y FA(\(In)m(t)2706 3628 y Ft(1)2745 3613 y FA(\000)2806 3572 y Ft(1)2806 3637 y Fw(j)2846 3613 y FA(\)\002)2960 3628 y Ft(2)2999 3613 y FA(\(In)m(t)3161 3628 y Ft(2)3201 3613 y FA(\000)3262 3572 y Ft(1)3262 3637 y Fw(j)3301 3613 y FA(\))491 3955 y(=)594 3861 y Fp(X)755 3955 y FA(\002)831 3914 y Fw(n)p Fr(\000)p Ft(1)831 3980 y(1)968 3875 y Fp(\000)1013 3955 y FA(Ext)1169 3908 y Fw(p)1169 3980 y Ft(1)1209 3955 y FA(\(\003\))1353 3875 y Fp(\001)1461 3827 y Fw(p)1415 3861 y Fp(Y)1417 4071 y Fw(j)t Ft(=1)1559 3955 y FA(e)1602 3883 y Fr(\000)p Fw(\014)s Fr(k)p Ft(\000)1779 3860 y Fl(1)1779 3905 y Fn(j)1814 3883 y Fr(k\000)p Fw(\014)s(z)s(a)p Ft(\()p Fw(')2093 3910 y Fl(\000)2131 3892 y(1)2131 3935 y Fn(j)2170 3883 y Ft(\)+)p Fw(\014)s(z)s Ft(\()p Fw(h)p Ft(\()p Fw( )2472 3892 y Fl(1)2507 3883 y Ft(\))p Fr(\000)p Fw(h)p Ft(\()p Fw( )2703 3892 y Fl(2)2738 3883 y Ft(\)\))p Fr(j)p Ft(In)n(t)2903 3892 y Fl(2)2937 3883 y Ft(\000)2981 3860 y Fl(1)2981 3905 y Fn(j)3016 3883 y Fr(j)627 4286 y Fu(\001)687 4210 y FA(\002)763 4225 y Ft(1)802 4210 y FA(\(In)m(t)965 4225 y Ft(1)1004 4210 y FA(\000)1065 4174 y Ft(1)1065 4235 y Fw(j)1104 4210 y FA(\))p 687 4263 456 4 v 687 4355 a(\002)763 4370 y Ft(2)802 4355 y FA(\(In)m(t)965 4370 y Ft(1)1004 4355 y FA(\000)1065 4320 y Ft(1)1065 4380 y Fw(j)1104 4355 y FA(\))1169 4286 y(\002)1245 4301 y Ft(2)1284 4286 y FA(\(In)m(t)1447 4301 y Ft(1)1486 4286 y FA(\000)1547 4245 y Ft(1)1547 4311 y Fw(j)1587 4286 y FA(\)\002)1701 4301 y Ft(2)1740 4286 y FA(\(In)m(t)1902 4301 y Ft(2)1942 4286 y FA(\000)2003 4245 y Ft(1)2003 4311 y Fw(j)2042 4286 y FA(\))17 b(;)0 4531 y(the)28 b(sums)h(are)e(o)m(v)m(er)i(all)e(families)h Fu(f)p FA(\000)1355 4495 y Ft(1)1355 4556 y(1)1395 4531 y Fv(;)17 b(:)g(:)g(:)e FA(\000)1630 4495 y Ft(1)1630 4556 y Fw(p)1670 4531 y Fu(g)27 b FA(of)g(compatible)i(external)f(large)g(con)m(tours)g(in)g (\003.)0 4651 y(All)33 b(con)m(tours)i(whic)m(h)f(are)g(not)f(large)g (are)g Fv(\034)1616 4666 y Ft(1)1656 4651 y FA(\()p Fv(\014)6 b FA(\)-stable,)33 b(and)g(w)m(e)h(use)h(the)e(cluster)i(expansion)0 4769 y(to)42 b(con)m(trol)g(\002)542 4728 y Fw(n)p Fr(\000)p Ft(1)542 4794 y(1)679 4689 y Fp(\000)725 4769 y FA(Ext)881 4722 y Fw(p)881 4794 y Ft(1)921 4769 y FA(\(\003\))1065 4689 y Fp(\001)1110 4769 y FA(,)i(\002)1257 4784 y Ft(2)1297 4769 y FA(\(In)m(t)1459 4784 y Ft(1)1499 4769 y FA(\000)1560 4733 y Ft(1)1560 4794 y Fw(j)1599 4769 y FA(\))e(and)h(\002)1955 4784 y Ft(2)1994 4769 y FA(\(In)m(t)2156 4784 y Ft(2)2196 4769 y FA(\000)2257 4733 y Ft(1)2257 4794 y Fw(j)2296 4769 y FA(\).)73 b(Notice)42 b(that)g Fv(@)5 b Fu(j)p FA(Ext)3211 4722 y Fw(p)3211 4794 y Ft(1)3250 4769 y FA(\(\003\))p Fu(j)44 b(\024)0 4894 y Fv(@)5 b Fu(j)p FA(\003)p Fu(j)22 b FA(+)300 4819 y Fp(P)405 4846 y Fw(p)405 4923 y(j)t Ft(=1)549 4894 y Fv(C)619 4909 y Ft(0)658 4894 y Fu(j)p FA(\000)747 4858 y Ft(1)747 4919 y Fw(j)786 4894 y Fu(j)p FA(.)43 b(By)33 b(Lemma)g(1.1)f(and)h(the)g(induction)h (h)m(yp)s(othesis,)298 5022 y Fp(\014)298 5082 y(\014)298 5142 y(\014)331 5137 y FA(e)374 5096 y Fr(\000)p Fw(\014)s(z)s Ft(\()p Fw(h)p Ft(\()p Fw( )649 5105 y Fl(1)684 5096 y Ft(\))p Fr(\000)p Fw(h)p Ft(\()p Fw( )880 5105 y Fl(2)915 5096 y Ft(\)\))p Fr(j)p Ft(\003)p Fr(j)1072 5070 y FA(\002)1148 5085 y Ft(1)1188 5070 y FA(\(\003\))p 1072 5114 259 4 v 1072 5205 a(\002)1148 5220 y Ft(2)1188 5205 y FA(\(\003\))1341 5022 y Fp(\014)1341 5082 y(\014)1341 5142 y(\014)1402 5137 y Fu(\024)28 b FA(e)1550 5096 y Ft(2)p Fw(\016)r(@)t Fr(j)p Ft(\003)p Fr(j)1769 5042 y Fp(X)1930 5137 y FA(e)1973 5096 y Fr(\000)p Fw(\014)s Ft(Re\()p Fw(f)2222 5066 y Fn(n)p Fm(\000)p Fl(1)2215 5117 y(1)2344 5096 y Fr(\000)p Fw(f)2440 5066 y Fn(n)p Fm(\000)p Fl(1)2433 5117 y(2)2561 5096 y Fr(\000)p Fw(g)2652 5072 y Fn(m)2650 5117 y Fl(2)2711 5096 y Ft(+)p Fw(g)2802 5066 y Fn(n)p Fm(\000)p Fl(1)2800 5117 y(2)2922 5096 y Ft(\))p Fr(j)p Ft(Ext)3082 5063 y Fn(p)3082 5117 y Fl(1)3118 5096 y Ft(\(\003\))p Fr(j)1084 5435 y Fu(\001)1180 5306 y Fw(p)1134 5340 y Fp(Y)1137 5550 y Fw(j)t Ft(=1)1278 5435 y FA(e)1321 5393 y Fr(\000)p Ft(\()p Fw(\014)s(\032)p Fr(\000)p Fw(\014)s(")p Ft(+6)p Fw(C)1753 5402 y Fl(0)1788 5393 y Fw(\016)r Ft(\))p Fr(j)p Ft(\000)1912 5369 y Fl(1)1912 5414 y Fn(j)1948 5393 y Fr(j)1972 5435 y FA(e)2015 5393 y Fr(\000)p Fw(\014)s Ft(Re\()p Fw(f)2264 5363 y Fn(n)p Fm(\000)p Fl(1)2257 5414 y(1)2386 5393 y Fr(\000)p Fw(f)2482 5363 y Fn(n)p Fm(\000)p Fl(1)2475 5414 y(2)2603 5393 y Fr(\000)p Fw(g)2694 5369 y Fn(m)2692 5414 y Fl(2)2752 5393 y Ft(+)p Fw(g)2843 5363 y Fn(n)p Fm(\000)p Fl(1)2841 5414 y(2)2964 5393 y Ft(\))p Fr(j)p Ft(\000)3055 5369 y Fl(1)3055 5414 y Fn(j)3090 5393 y Fr(j)3130 5435 y Fv(:)0 5701 y FA(W)-8 b(e)33 b(de\014ne)1240 5827 y(^)-54 b Fv(\034)12 b FA(\()p Fv(\014)6 b FA(\))27 b(:=)g Fv(\014)6 b FA(\()p Fv(\032)22 b Fu(\000)h Fv(")p FA(\))f Fu(\000)g FA(6)p Fv(C)2178 5842 y Ft(0)2217 5827 y Fv(\016)f(:)p Black Black eop %%Page: 13 13 13 12 bop Black 3467 49 a Fq(13)p Black 0 241 a FA(F)-8 b(rom)32 b(\(2.16\))g(and)h(Lemma)g(1.1)f(w)m(e)i(ha)m(v)m(e)1000 409 y Fv(\014)6 b FA(\()p Fv(f)1158 368 y Fw(n)p Fr(\000)p Ft(1)1147 434 y(1)1317 409 y Fu(\000)22 b Fv(f)1475 368 y Fw(n)p Fr(\000)p Ft(1)1464 434 y(2)1634 409 y Fu(\000)h Fv(g)1785 368 y Fw(m)1781 434 y Ft(2)1873 409 y FA(+)f Fv(g)2022 368 y Fw(n)p Fr(\000)p Ft(1)2018 434 y(2)2158 409 y FA(\))28 b Fu(\025)g Fv(\016)2376 368 y Fw(l)q Ft(\()p Fw(n)p Ft(\))2516 409 y Fv(:)0 569 y FA(Hence,)279 693 y Fp(\014)279 753 y(\014)279 813 y(\014)312 808 y FA(e)355 767 y Fr(\000)p Fw(\014)s(z)s Ft(\()p Fw(h)p Ft(\()p Fw( )630 776 y Fl(1)665 767 y Ft(\))p Fr(\000)p Fw(h)p Ft(\()p Fw( )861 776 y Fl(2)896 767 y Ft(\)\))p Fr(j)p Ft(\003)p Fr(j)1053 741 y FA(\002)1129 756 y Ft(1)1169 741 y FA(\(\003\))p 1053 785 259 4 v 1053 876 a(\002)1129 891 y Ft(2)1169 876 y FA(\(\003\))1322 693 y Fp(\014)1322 753 y(\014)1322 813 y(\014)1383 808 y Fu(\024)g FA(e)1531 767 y Ft(2)p Fw(\016)r(@)t Fr(j)p Ft(\003)p Fr(j)1750 713 y Fp(X)1911 808 y FA(e)1954 767 y Fr(\000)p Fw(\016)2042 743 y Fn(l)p Fl(\()p Fn(n)p Fl(\))2153 767 y Fr(j)p Ft(Ext)2286 734 y Fn(p)2286 788 y Fl(1)2322 767 y Ft(\(\003\))p Fr(j)2512 679 y Fw(p)2466 713 y Fp(Y)2468 923 y Fw(j)t Ft(=1)2610 808 y FA(e)2653 766 y Fr(\000)p Ft(\()p Fw(\016)2768 743 y Fn(l)p Fl(\()p Fn(n)p Fl(\))2880 766 y Ft(+)t(^)-39 b Fw(\034)8 b Ft(\()p Fw(\014)s Ft(\)\))p Fr(j)p Ft(\000)3162 743 y Fl(1)3162 788 y Fn(j)3197 766 y Fr(j)3238 808 y Fv(:)0 1068 y FA(W)-8 b(e)33 b(de\014ne)965 1254 y(^)-57 b Fv(!)s FA(\(\000\))28 b(:=)1317 1083 y Fp(\()1397 1187 y FA(e)1440 1151 y Fr(\000)p Ft(\()t(^)-39 b Fw(\034)8 b Ft(\()p Fw(\014)s Ft(\))p Fr(\000)p Fw(C)1763 1160 y Fl(0)1799 1151 y Fw(\016)r Ft(\))p Fr(j)p Ft(\000)p Fr(j)2045 1187 y FA(if)33 b Fu(j)p FA(\000)p Fu(j)27 b(\025)h Fv(l)r FA(\()p Fv(n)p FA(\);)1397 1326 y(0)599 b(otherwise.)0 1505 y(Let)188 1479 y(^)175 1505 y(\002\(\003\))32 b(b)s(e)h(de\014ned)h(b)m(y)g(\(1.11\),)e(replacing)i Fv(!)t FA(\(\000)1923 1468 y Fw(q)1960 1505 y FA(\))e(b)m(y)42 b(^)-57 b Fv(!)s FA(\(\000\),)32 b(and)h(let)1228 1713 y(^)-52 b Fv(g)31 b FA(:=)42 b(lim)1434 1784 y Ft(\003)p Fr(")p Fo(Z)1562 1765 y Fn(d)1615 1713 y Fu(\000)1770 1646 y FA(1)p 1702 1690 184 4 v 1702 1782 a Fv(\014)6 b Fu(j)p FA(\003)p Fu(j)1913 1713 y FA(log)2069 1688 y(^)2055 1713 y(\002\(\003\))17 b Fv(:)0 1928 y FA(W)-8 b(e)33 b(assume)h(that)e Fv(\014)771 1943 y Ft(0)843 1928 y FA(is)i(large)e(enough)h(so)g(that)f(for)g(all)h Fv(\014)g Fu(\025)28 b Fv(\014)2380 1943 y Ft(0)2420 1928 y FA(,)1487 2091 y Fv(K)7 b FA(e)1620 2050 y Fr(\000)t Ft(^)-39 b Fw(\034)8 b Ft(\()p Fw(\014)s Ft(\))1844 2091 y Fu(\024)28 b Fv(\016)21 b(;)-2040 b FA(\(2.18\))0 2268 y(where)37 b Fv(K)42 b FA(is)36 b(the)g(constan)m(t)h(of)e(Lemma)h (1.1.)52 b(Since)36 b Fv(\014)6 b Fu(j)s FA(^)-52 b Fv(g)s Fu(j)32 b(\024)h Fv(\016)2361 2232 y Fw(l)q Ft(\()p Fw(n)p Ft(\))2485 2268 y FA(,)j(putting)g(in)m(to)g(evidence)i(a)0 2387 y(factor)32 b(e)322 2351 y Fw(\014)6 b Ft(^)-38 b Fw(g)r Fr(j)p Ft(\003)p Fr(j)494 2387 y FA(,)32 b(w)m(e)i(get)386 2512 y Fp(\014)386 2571 y(\014)386 2631 y(\014)419 2626 y FA(e)462 2585 y Fr(\000)p Fw(\014)s(z)s Ft(\()p Fw(h)p Ft(\()p Fw( )737 2594 y Fl(1)772 2585 y Ft(\))p Fr(\000)p Fw(h)p Ft(\()p Fw( )968 2594 y Fl(2)1003 2585 y Ft(\)\))p Fr(j)p Ft(\003)p Fr(j)1161 2559 y FA(\002)1237 2574 y Ft(1)1276 2559 y FA(\(\003\))p 1161 2603 259 4 v 1161 2694 a(\002)1237 2709 y Ft(2)1276 2694 y FA(\(\003\))1429 2512 y Fp(\014)1429 2571 y(\014)1429 2631 y(\014)1490 2626 y Fu(\024)28 b FA(e)1638 2585 y Ft(2)p Fw(\016)r(@)t Fr(j)p Ft(\003)p Fr(j)p Ft(+)p Fw(\014)6 b Ft(^)-38 b Fw(g)s Fr(j)p Ft(\003)p Fr(j)2080 2531 y Fp(X)2286 2498 y Fw(p)2240 2531 y Fp(Y)2243 2741 y Fw(j)t Ft(=1)2384 2626 y FA(e)2427 2584 y Fr(\000)t Ft(^)f Fw(\034)8 b Ft(\()p Fw(\014)s Ft(\))p Fr(j)p Ft(\000)2682 2561 y Fl(1)2682 2606 y Fn(j)2718 2584 y Fr(j)2741 2626 y FA(e)2784 2584 y Fr(\000)p Fw(\014)e Ft(^)-38 b Fw(g)s Fr(j)p Ft(In)n(t)12 b(\000)3084 2561 y Fl(1)3084 2606 y Fn(j)3119 2584 y Fr(j)0 2626 y FA(\(2.19\))1490 2969 y Fu(\024)28 b FA(e)1638 2928 y Ft(2)p Fw(\016)r(@)t Fr(j)p Ft(\003)p Fr(j)p Ft(+)p Fw(\014)6 b Ft(^)-38 b Fw(g)s Fr(j)p Ft(\003)p Fr(j)2080 2874 y Fp(X)2286 2840 y Fw(p)2240 2874 y Fp(Y)2243 3084 y Fw(j)t Ft(=1)2384 2969 y FA(e)2427 2927 y Fr(\000)p Ft(\()t(^)f Fw(\034)9 b Ft(\()p Fw(\014)s Ft(\))p Fr(\000)p Fw(C)2751 2936 y Fl(0)2787 2927 y Fw(\016)r Ft(\))p Fr(j)p Ft(\000)2911 2903 y Fl(1)2911 2948 y Fn(j)2946 2927 y Fr(j)2983 2944 y FA(^)2970 2969 y(\002\(In)m(t)17 b(\000)3286 2928 y Ft(1)3286 2993 y Fw(j)3325 2969 y FA(\))g Fv(:)0 3251 y FA(In)44 b(the)g(last)g(line)h(of)50 b(\(2.19\))43 b(w)m(e)h(in)m(terpret)h(e)1738 3215 y Fr(\000)p Fw(\014)6 b Ft(^)-38 b Fw(g)s Fr(j)p Ft(In)n(t)12 b(\000)2038 3191 y Fl(1)2073 3215 y Fr(j)2140 3251 y FA(as)44 b(a)f(partition)h (function)g(\(up)g(to)g(a)0 3367 y(b)s(oundary)33 b(term\),)g(since)h (b)m(y)g(Lemma)f(1.1,)1136 3541 y(e)1179 3500 y Fr(\000)p Fw(\014)6 b Ft(^)-38 b Fw(g)r Fr(j)p Ft(In)n(t)13 b(\000)1479 3477 y Fl(1)1513 3500 y Fr(j)1565 3541 y Fu(\024)1683 3516 y FA(^)1670 3541 y(\002\(In)m(t)k(\000)1986 3500 y Ft(1)2025 3541 y FA(\))g(e)2123 3500 y Fw(C)2173 3509 y Fl(0)2208 3500 y Fw(\016)r Fr(j)p Ft(\000)2305 3477 y Fl(1)2340 3500 y Fr(j)2380 3541 y Fv(:)0 3701 y FA(W)-8 b(e)33 b(sum)g(o)m(v)m(er)h(external)g(con)m(tours)f(in)g(\(2.19\))f (and)g(get)612 3806 y Fp(\014)612 3866 y(\014)612 3926 y(\014)645 3921 y FA(e)688 3880 y Fr(\000)p Fw(\014)s(z)s Ft(\()p Fw(h)p Ft(\()p Fw( )963 3889 y Fl(1)998 3880 y Ft(\))p Fr(\000)p Fw(h)p Ft(\()p Fw( )1194 3889 y Fl(2)1230 3880 y Ft(\)\))p Fr(j)p Ft(\003)p Fr(j)1387 3853 y FA(\002)1463 3868 y Ft(1)1502 3853 y FA(\(\003\))p 1387 3898 V 1387 3989 a(\002)1463 4004 y Ft(2)1502 3989 y FA(\(\003\))1656 3806 y Fp(\014)1656 3866 y(\014)1656 3926 y(\014)1717 3921 y Fu(\024)c FA(e)1865 3880 y Ft(2)p Fw(\016)r(@)t Fr(j)p Ft(\003)p Fr(j)p Ft(+)p Fw(\014)6 b Ft(^)-38 b Fw(g)r Fr(j)p Ft(\003)p Fr(j)2303 3896 y FA(^)2289 3921 y(\002\(\003\))28 b Fu(\024)g FA(e)2685 3880 y Ft(3)p Fw(\016)r(@)t Fr(j)p Ft(\003)p Fr(j)2904 3921 y Fv(:)p 3473 4180 4 66 v 3477 4118 59 4 v 3477 4180 V 3535 4180 4 66 v 100 4399 a FA(It)37 b(is)h(not)f(di\016cult)h(to)f(pro)m(v)m(e)h (more)g(regularit)m(y)g(for)f(the)g(curv)m(e)i Fv(\027)j Fu(7!)35 b Fv(\026)2817 4363 y Fr(\003)2857 4399 y FA(\()p Fv(\027)6 b FA(;)17 b Fv(\014)6 b FA(\).)57 b(W)-8 b(e)37 b(need)0 4515 y(b)s(elo)m(w)c(only)g(the)g(follo)m(wing)g(result.)p Black 0 4739 a Fz(Lemma)39 b(2.1.)p Black 49 w Fs(L)-5 b(et)25 b FA(0)j Fv(<)f(\016)32 b(<)c FA(1)p Fs(.)41 b(If)24 b Fv(\014)31 b Fs(is)24 b(su\016ciently)h(lar)-5 b(ge,)27 b(then)d(for)h(al)5 b(l)24 b Fv(n)k Fu(\025)h FA(1)3052 4672 y Fv(d)p 3025 4716 106 4 v 3025 4808 a(d\027)3140 4739 y(\026)3199 4698 y Fr(\003)3199 4764 y Fw(n)3246 4739 y FA(\(0;)17 b Fv(\014)6 b FA(\))27 b(=)0 4885 y(0)p Fs(,)35 b(and)636 4978 y Fp(\014)636 5037 y(\014)707 4995 y Fv(d)758 4959 y Ft(2)p 680 5039 145 4 v 680 5131 a Fv(d\027)785 5102 y Ft(2)834 5062 y Fv(\026)893 5021 y Fr(\003)893 5087 y Fw(n)940 5062 y FA(\()p Fv(\027)6 b FA(;)17 b Fv(\014)6 b FA(\))1175 4978 y Fp(\014)1175 5037 y(\014)1235 5062 y Fu(\024)1452 4995 y FA(2)p Fv(\016)p 1351 5039 299 4 v 1351 5131 a FA(\001)22 b Fu(\000)h FA(2)p Fv(\016)1659 4952 y Fp(\020)q(\020)1890 4995 y FA(2)p Fv(\016)p 1788 5039 V 1788 5131 a FA(\001)g Fu(\000)f FA(2)p Fv(\016)2097 4952 y Fp(\021)2156 4974 y Ft(2)2218 5062 y FA(+)2428 4995 y(2)p Fv(\016)p 2326 5039 V 2326 5131 a FA(\001)g Fu(\000)h FA(2)p Fv(\016)2657 5062 y FA(+)f(1)2804 4952 y Fp(\021)2880 5062 y Fv(:)p Black 0 5333 a Fs(Pr)-5 b(o)g(of.)p Black 41 w FA(Let)38 b Fv(\016)j FA(b)s(e)d(as)f(in)h(the)g(pro)s(of)e(of)h(Prop)s(osition)h (2.1.)58 b(Because)39 b(the)f(free)g(energies)g Fv(f)3406 5292 y Fw(n)p Fr(\000)p Ft(1)3395 5358 y(1)0 5450 y FA(and)28 b Fv(f)244 5409 y Fw(n)p Fr(\000)p Ft(1)233 5475 y(2)408 5450 y FA(are)g(real)g(on)g(the)g(real)g(axis,)h(it)f(follo)m(ws)g (that)g Fv(\027)34 b Fu(7!)27 b Fv(\026)2334 5414 y Fr(\003)2334 5475 y Fw(n)2381 5450 y FA(\()p Fv(\027)6 b FA(;)17 b Fv(\014)6 b FA(\))27 b(is)h(ev)m(en,)j(and)d(therefore)37 5553 y Fv(d)p 10 5597 106 4 v 10 5688 a(d\027)125 5620 y(\026)184 5579 y Fr(\003)184 5645 y Fw(n)231 5620 y FA(\(0;)17 b Fv(\014)6 b FA(\))26 b(=)i(0.)43 b(By)33 b(de\014nition)h Fv(\026)1356 5584 y Fr(\003)1356 5645 y Fw(n)1402 5620 y FA(\()p Fv(\027)6 b FA(;)17 b Fv(\014)6 b FA(\))33 b(is)g(solution)g(of)676 5825 y(Re)791 5745 y Fp(\000)837 5825 y Fv(f)896 5784 y Fw(n)p Fr(\000)p Ft(1)885 5850 y(2)1032 5825 y FA(\()p Fv(\026)1129 5784 y Fr(\003)1129 5850 y Fw(n)1176 5825 y FA(\()p Fv(\027)6 b FA(;)17 b Fv(\014)6 b FA(\))22 b(+)g Fv(i\027)6 b FA(\))23 b Fu(\000)f Fv(f)1837 5784 y Fw(n)p Fr(\000)p Ft(1)1826 5850 y(1)1974 5825 y FA(\()p Fv(\026)2071 5784 y Fr(\003)2071 5850 y Fw(n)2118 5825 y FA(\()p Fv(\027)6 b FA(;)17 b Fv(\014)6 b FA(\))22 b(+)g Fv(i\027)6 b FA(\))2598 5745 y Fp(\001)2672 5825 y FA(=)27 b(0)17 b Fv(;)p Black Black eop %%Page: 14 14 14 13 bop Black 0 49 a Fq(14)p Black 0 241 a FA(whic)m(h)34 b(implies)g(that)590 450 y(\001)681 383 y Fv(d\026)791 346 y Fr(\003)791 407 y Fw(n)p 681 427 157 4 v 707 518 a Fv(d\027)876 450 y FA(=)1019 383 y Fv(d)p 989 427 110 4 v 989 518 a(d\026)1109 450 y FA(Re)1224 369 y Fp(\000)1269 450 y Fv(g)1320 409 y Fw(n)p Fr(\000)p Ft(1)1316 475 y(1)1479 450 y Fu(\000)23 b Fv(g)1630 409 y Fw(n)p Fr(\000)p Ft(1)1626 475 y(2)1766 369 y Fp(\001)1822 383 y Fv(d\026)1932 346 y Fr(\003)1932 407 y Fw(n)p 1822 427 157 4 v 1847 518 a Fv(d\027)2010 450 y FA(+)2145 383 y Fv(d)p 2118 427 106 4 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Fm(\000)p Fl(1)p 1534 1654 V 1573 1695 a Fn(d)p 1269 1750 390 4 v 1364 1841 a Fu(k)p FA(\000)1475 1812 y Ft(2)1514 1841 y Fu(k)1696 1773 y(\024)28 b Fv(C)c(;)33 b Fu(8)28 b FA(\000)2099 1732 y Ft(2)2166 1773 y FA(suc)m(h)34 b(that)28 b Fv(V)21 b FA(\(\000)2770 1732 y Ft(2)2809 1773 y FA(\))28 b Fu(\025)g Fv(n)3038 1662 y Fp(o)3121 1773 y Fv(:)0 2002 y(\037)61 2017 y Ft(2)101 2002 y FA(\()p Fv(n)p FA(\))k(is)h(a)g(b)s(ounded)g(increasing)h(sequence;)h(w)m(e)f (set)f Fv(\037)2089 2017 y Ft(2)2128 2002 y FA(\()p Fu(1)p FA(\))28 b(:=)f(lim)2598 2017 y Fw(n)2661 2002 y Fv(\037)2722 2017 y Ft(2)2762 2002 y FA(\()p Fv(n)p FA(\),)33 b(and)f(de\014ne)1277 2231 y Fv(R)1351 2246 y Ft(2)1390 2231 y FA(\()p Fv(n)p FA(\))c(:=)90 b(inf)1683 2292 y Fw(m)p Ft(:)p Fw(m)p Fr(\024)p Fw(n)1951 2163 y Fv(\037)2012 2178 y Ft(2)2052 2163 y FA(\()p Fv(m)p FA(\))p 1951 2208 262 4 v 2012 2316 a Fv(m)2108 2252 y Fl(1)p 2107 2264 33 3 v 2107 2305 a Fn(d)2240 2231 y Fv(:)0 2452 y FA(There)34 b(are)f(similar)h (de\014nitions)g(for)f Fv(\037)1449 2467 y Ft(1)1488 2452 y FA(\()p Fv(n)p FA(\))g(and)g Fv(R)1919 2467 y Ft(1)1959 2452 y FA(\()p Fv(n)p FA(\).)45 b(The)34 b(corresp)s(onding)g (isop)s(erimetric)0 2569 y(inequalities)h(con)m(trol)e(the)g (analyticit)m(y)h(prop)s(erties)f(of)f Fv(!)t FA(\(\000)2227 2532 y Ft(1)2266 2569 y FA(\))g(around)h Fv(\026)2726 2532 y Fr(\003)2765 2569 y FA(.)p Black 0 2750 a Fz(Lemma)39 b(2.2.)p Black 49 w Fs(F)-7 b(or)32 b(any)g Fv(\037)1045 2714 y Fr(0)1045 2775 y Fw(q)1111 2750 y Fv(<)27 b(\037)1275 2765 y Fw(q)1314 2750 y FA(\()p Fu(1)p FA(\))p Fs(,)32 b(ther)-5 b(e)32 b(exists)g Fv(N)10 b FA(\()p Fv(\037)2243 2714 y Fr(0)2243 2775 y Fw(q)2282 2750 y FA(\))32 b Fs(such)h(that)g (for)f(al)5 b(l)32 b Fv(n)c Fu(\025)g Fv(N)10 b FA(\()p Fv(\037)3437 2714 y Fr(0)3437 2775 y Fw(q)3475 2750 y FA(\))p Fs(,)1297 2922 y Fv(\037)1358 2886 y Fr(0)1358 2947 y Fw(q)p 1289 2976 115 4 v 1289 3084 a Fv(n)1358 3020 y Fl(1)p 1358 3032 33 3 v 1358 3073 a Fn(d)1441 2999 y Fu(\024)28 b Fv(R)1620 3014 y Fw(q)1659 2999 y FA(\()p Fv(n)p FA(\))g Fu(\024)1936 2931 y Fv(\037)1997 2946 y Fw(q)2035 2931 y FA(\()p Fu(1)p FA(\))p 1936 2976 275 4 v 2016 3084 a Fv(n)2085 3020 y Fl(1)p 2084 3032 33 3 v 2084 3073 a Fn(d)2237 2999 y Fv(:)0 3227 y Fs(F)-7 b(or)34 b Fv(q)d FA(=)d(1)p Fv(;)17 b FA(2)p Fs(,)34 b Fv(n)624 3191 y Fw(a)694 3227 y Fu(7!)27 b Fv(nR)953 3242 y Fw(q)991 3227 y FA(\()p Fv(n)p FA(\))35 b Fs(is)g(incr)-5 b(e)g(asing)33 b(in)i Fv(n)p Fs(,)g(pr)-5 b(ovide)g(d)34 b(that)h Fv(a)28 b Fu(\025)2746 3188 y Ft(1)p 2745 3204 37 4 v 2745 3262 a Fw(d)2791 3227 y Fs(.)p Black 0 3460 a(Pr)-5 b(o)g(of.)p Black 41 w FA(Let)33 b Fv(q)e FA(=)d(2)k(and)h (supp)s(ose)h(that)1167 3683 y Fv(R)1241 3698 y Ft(2)1281 3683 y FA(\()p Fv(n)p FA(\))28 b(=)1556 3615 y Fv(\037)1617 3630 y Ft(2)1657 3615 y FA(\()p Fv(m)p FA(\))p 1556 3660 262 4 v 1616 3768 a Fv(m)1712 3704 y Fl(1)p 1712 3716 33 3 v 1712 3757 a Fn(d)1925 3683 y FA(for)k Fv(m)c(<)g(n)p FA(.)0 3904 y(Then)34 b Fv(R)329 3919 y Ft(2)368 3904 y FA(\()p Fv(m)491 3868 y Fr(0)515 3904 y FA(\))28 b(=)f Fv(R)758 3919 y Ft(2)798 3904 y FA(\()p Fv(n)p FA(\))32 b(for)g(all)h Fv(m)28 b Fu(\024)g Fv(m)1552 3868 y Fr(0)1603 3904 y Fu(\024)h Fv(n)p FA(.)43 b(Let)33 b Fv(n)2070 3868 y Fr(0)2126 3904 y FA(b)s(e)g(the)g(largest)g Fv(n)28 b Fu(\025)g Fv(m)33 b FA(suc)m(h)h(that)1419 4133 y Fv(R)1493 4148 y Ft(2)1533 4133 y FA(\()p Fv(n)p FA(\))28 b(=)1808 4065 y Fv(\037)1869 4080 y Ft(2)1909 4065 y FA(\()p Fv(m)p FA(\))p 1808 4110 262 4 v 1869 4218 a Fv(m)1965 4154 y Fl(1)p 1964 4166 33 3 v 1964 4207 a Fn(d)2097 4133 y Fv(:)0 4353 y FA(W)-8 b(e)33 b(ha)m(v)m(e)h Fv(n)451 4316 y Fr(0)502 4353 y Fv(<)28 b Fu(1)p FA(,)k(otherwise)890 4576 y(0)27 b Fv(<)h(R)1144 4591 y Ft(2)1184 4576 y FA(\()p Fv(m)p FA(\))g(=)f Fv(R)1550 4591 y Ft(2)1590 4576 y FA(\()p Fv(n)p FA(\))h Fu(\024)1867 4508 y Fv(\037)1928 4523 y Ft(2)1967 4508 y FA(\()p Fu(1)p FA(\))p 1867 4553 277 4 v 1948 4661 a Fv(n)2017 4597 y Fl(1)p 2016 4609 33 3 v 2016 4650 a Fn(d)2250 4576 y Fu(8)g Fv(n)g Fu(\025)g Fv(m)17 b(;)0 4791 y FA(whic)m(h)34 b(is)f(imp)s(ossible.)45 b(Therefore,)34 b(either)760 5015 y Fv(R)834 5030 y Ft(2)873 5015 y FA(\()p Fv(n)969 4973 y Fr(0)993 5015 y FA(\))28 b(=)1172 4947 y Fv(\037)1233 4962 y Ft(2)1273 4947 y FA(\()p Fv(n)1369 4911 y Fr(0)1392 4947 y FA(\))p 1172 4992 258 4 v 1232 5102 a Fv(n)1290 5073 y Fr(0)1325 5036 y Fl(1)p 1324 5048 33 3 v 1324 5089 a Fn(d)1538 5015 y FA(or)97 b Fv(R)1796 5030 y Ft(2)1835 5015 y FA(\()p Fv(n)1931 4973 y Fr(0)1977 5015 y FA(+)22 b(1\))27 b(=)2303 4947 y Fv(\037)2364 4962 y Ft(2)2404 4947 y FA(\()p Fv(n)2500 4911 y Fr(0)2545 4947 y FA(+)22 b(1\))p 2303 4992 427 4 v 2325 5100 a(\()p Fv(n)2421 5071 y Fr(0)2467 5100 y FA(+)g(1\))2663 5036 y Fl(1)p 2661 5048 33 3 v 2661 5089 a Fn(d)2756 5015 y Fv(;)0 5261 y FA(and)33 b(for)f(all)g Fv(k)f Fu(\025)d Fv(n)719 5225 y Fr(0)765 5261 y FA(+)22 b(1,)32 b(since)i Fv(\037)1271 5276 y Ft(2)1311 5261 y FA(\()p Fv(m)p FA(\))e(is)i(increasing,)551 5489 y Fv(R)625 5504 y Ft(2)665 5489 y FA(\()p Fv(k)s FA(\))27 b(=)47 b(inf)926 5552 y Fw(m)p Fr(\024)p Fw(k)1108 5422 y Fv(\037)1169 5437 y Ft(2)1209 5422 y FA(\()p Fv(m)p FA(\))p 1108 5467 262 4 v 1169 5574 a Fv(m)1265 5511 y Fl(1)p 1264 5523 33 3 v 1264 5564 a Fn(d)1408 5489 y FA(=)106 b(inf)1511 5553 y Fw(n)1554 5534 y Fm(0)1576 5553 y Fr(\024)p Fw(m)p Fr(\024)p Fw(k)1814 5422 y Fv(\037)1875 5437 y Ft(2)1914 5422 y FA(\()p Fv(m)p FA(\))p 1814 5467 262 4 v 1874 5574 a Fv(m)1970 5511 y Fl(1)p 1969 5523 33 3 v 1969 5564 a Fn(d)2113 5489 y Fu(\025)h FA(inf)2218 5553 y Fw(n)2261 5534 y Fm(0)2284 5553 y Fr(\024)p Fw(m)p Fr(\024)p Fw(k)2521 5422 y Fv(\037)2582 5437 y Ft(2)2621 5422 y FA(\()p Fv(n)2717 5386 y Fr(0)2741 5422 y FA(\))p 2521 5467 258 4 v 2579 5574 a Fv(m)2675 5511 y Fl(1)p 2674 5523 33 3 v 2674 5564 a Fn(d)2816 5489 y FA(=)2930 5422 y Fv(\037)2991 5437 y Ft(2)3030 5422 y FA(\()p Fv(n)3126 5386 y Fr(0)3150 5422 y FA(\))p 2930 5467 258 4 v 3004 5574 a Fv(k)3069 5511 y Fl(1)p 3068 5523 33 3 v 3068 5564 a Fn(d)3214 5489 y Fv(:)-3241 b FA(\(2.22\))0 5711 y(Inequalit)m(y)37 b(\(2.22\))c(is)h(true)g(for)f(in\014nitely)j(man)m (y)e Fv(n)1937 5675 y Fr(0)1961 5711 y FA(;)g(since)h(there)g(exists)g Fv(m)f FA(suc)m(h)h(that)f Fv(\037)3397 5675 y Fr(0)3397 5735 y Ft(2)3466 5711 y Fu(\024)0 5827 y Fv(\037)61 5842 y Ft(2)101 5827 y FA(\()p Fv(m)p FA(\),)e(the)h(\014rst)g(statemen)m(t) h(is)g(pro)m(v)m(ed.)p Black Black eop %%Page: 17 17 17 16 bop Black 3467 49 a Fq(17)p Black 100 241 a FA(On)36 b(an)h(in)m(terv)-5 b(al)37 b(of)g(constancy)h(of)e Fv(R)1524 256 y Ft(2)1564 241 y FA(\()p Fv(n)p FA(\),)h Fv(n)e Fu(7!)g Fv(n)2048 205 y Fw(a)2090 241 y Fv(R)2164 256 y Ft(2)2203 241 y FA(\()p Fv(n)p FA(\))i(is)g(increasing.)57 b(On)37 b(the)g(other)0 357 y(hand,)c(if)f(on)h([)p Fv(m)608 372 y Ft(1)648 357 y Fv(;)17 b(m)777 372 y Ft(2)816 357 y FA(])1433 568 y Fv(R)1507 583 y Ft(2)1547 568 y FA(\()p Fv(n)p FA(\))27 b(=)1822 501 y Fv(\037)1883 516 y Ft(2)1923 501 y FA(\()p Fv(n)p FA(\))p 1822 546 235 4 v 1882 654 a Fv(n)1951 590 y Fl(1)p 1950 602 33 3 v 1950 643 a Fn(d)2083 568 y Fv(;)0 814 y FA(then)41 b Fv(n)h Fu(7!)g Fv(n)530 778 y Fw(a)572 814 y Fv(R)646 829 y Ft(2)685 814 y FA(\()p Fv(n)p FA(\))f(is)h(increasing)g(on)e([)p Fv(m)1686 829 y Ft(1)1726 814 y Fv(;)17 b(m)1855 829 y Ft(2)1895 814 y FA(])41 b(since)h Fv(n)g Fu(7!)f Fv(\037)2512 829 y Ft(2)2552 814 y FA(\()p Fv(n)p FA(\))g(and)f Fv(n)i Fu(7!)g Fv(n)3224 778 y Fw(a)p Fr(\000)3327 750 y Fl(1)p 3326 762 V 3326 804 a Fn(d)3413 814 y FA(are)0 931 y(increasing.)p 3473 1105 4 66 v 3477 1042 59 4 v 3477 1105 V 3535 1105 4 66 v 100 1373 a(The)29 b(next)h(prop)s(osition)f(giv)m(es)h(the)g (domains)f(of)g(analyticit)m(y)h(and)f(the)g(stabilit)m(y)h(prop)s (erties)0 1489 y(of)i(the)h(w)m(eigh)m(ts)h Fv(!)t FA(\(\000\))e (needed)i(for)e(estimating)i(the)f(deriv)-5 b(ativ)m(es)34 b(of)e(the)h(free)g(energy)-8 b(.)p Black 0 1684 a Fz(Prop)s(osition)38 b(2.3.)p Black 49 w Fs(L)-5 b(et)35 b FA(0)27 b Fv(<)h(\022)j(<)c FA(1)35 b Fs(and)f FA(0)28 b Fv(<)f(")h(<)f FA(1)35 b Fs(so)f(that)h Fv(\032)p FA(\(1)23 b Fu(\000)f Fv(\022)s FA(\))h Fu(\000)f Fv(")27 b(>)h FA(0)p Fs(.)0 1800 y(Ther)-5 b(e)37 b(exist)h FA(0)c Fv(<)f(\016)38 b(<)c FA(1)p Fs(,)k FA(0)c Fv(<)f(\022)1253 1764 y Fr(0)1310 1800 y Fv(<)h FA(1)k Fs(and)f Fv(\014)1754 1815 y Ft(0)1828 1800 y Fu(2)d Fx(R)2000 1764 y Ft(+)2059 1800 y Fs(,)39 b(such)f(that)g(for)g (al)5 b(l)38 b Fv(\014)h Fu(\025)34 b Fv(\014)3117 1815 y Ft(0)3195 1800 y Fv(!)t FA(\(\000)3359 1764 y Ft(2)3398 1800 y FA(\))k Fs(is)0 1916 y(analytic)d(and)f Fv(\034)600 1931 y Ft(1)640 1916 y FA(\()p Fv(\014)6 b FA(;)17 b Fv(\022)831 1880 y Fr(0)853 1916 y FA(\))p Fs(-stable)35 b(in)f(a)h(c)-5 b(omplex)34 b(neighb)-5 b(orho)g(o)g(d)33 b(of)644 2030 y Fp(\010)702 2111 y Fv(z)g Fu(2)28 b Fx(C)f FA(:)45 b(Re)p Fv(z)32 b Fu(\024)c Fv(\026)1401 2070 y Fr(\003)1441 2111 y FA(\(Im)p Fv(z)t FA(;)17 b Fv(\014)6 b FA(\))22 b(+)g Fv(\022)s FA(\001)2036 2070 y Fr(\000)p Ft(1)2131 2111 y Fv(R)2205 2126 y Ft(2)2245 2111 y FA(\()p Fv(V)f FA(\(\000)2460 2070 y Ft(2)2499 2111 y FA(\)\))2575 2030 y Fp(\011)2656 2111 y Fu(\\)h Fx(U)2816 2126 y Ft(0)2872 2111 y Fv(:)0 2304 y Fs(Mor)-5 b(e)g(over)1154 2401 y Fp(\014)1154 2460 y(\014)1222 2418 y Fv(d)p 1198 2462 101 4 v 1198 2554 a(dz)1308 2485 y(!)t FA(\(\000)1472 2444 y Ft(2)1511 2485 y FA(\))1549 2401 y Fp(\014)1549 2460 y(\014)1609 2485 y Fu(\024)29 b Fv(\014)6 b(C)1846 2500 y Ft(3)1884 2485 y FA(e)1927 2444 y Fr(\000)p Fw(\034)2013 2453 y Fl(2)2049 2444 y Ft(\()p Fw(\014)s Ft(;)p Fw(\022)2174 2421 y Fm(0)2196 2444 y Ft(\))p Fr(j)p Ft(\000)2287 2421 y Fl(2)2322 2444 y Fr(j)2362 2485 y Fv(:)0 2697 y Fs(Similar)34 b(pr)-5 b(op)g(erties)34 b(hold)g(for)h Fv(!)t FA(\(\000)1322 2661 y Ft(1)1361 2697 y FA(\))g Fs(in)f(a)h(c)-5 b(omplex)33 b(neighb)-5 b(orho)g(o)g(d)34 b(of)643 2811 y Fp(\010)701 2892 y Fv(z)f Fu(2)28 b Fx(C)g FA(:)44 b Fv(\026)1103 2851 y Fr(\003)1142 2892 y FA(\(Im)p Fv(z)t 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FA(\(\000)1174 5625 y Ft(2)1213 5661 y FA(\))1251 5625 y Fw(n)1298 5661 y FA(\()p Fv(z)t FA(\))p 955 5706 479 4 v 955 5797 a(\()p Fv(z)27 b Fu(\000)22 b Fv(\026)1223 5768 y Fr(\003)1262 5797 y FA(\))1300 5768 y Fw(k)r Ft(+1)1460 5729 y Fv(dz)32 b FA(=)27 b(Re)1807 5618 y Fp(n)1913 5661 y Fv(k)s FA(!)p 1883 5706 141 4 v 1883 5797 a(2)p Fv(\031)t(i)2050 5593 y Fp(I)2106 5819 y Fw(@)t(D)2205 5827 y Fn(r)2280 5661 y Fv(\036)2338 5676 y Ft(\003)2390 5661 y FA(\(\000)2489 5625 y Ft(2)2529 5661 y FA(\))2567 5625 y Fw(n)2614 5661 y FA(\()p Fv(z)t FA(\))p 2270 5706 479 4 v 2270 5797 a(\()p Fv(z)g Fu(\000)c Fv(\026)2539 5768 y Fr(\003)2578 5797 y FA(\))2616 5768 y Fw(k)r Ft(+1)2775 5729 y Fv(dz)2875 5618 y Fp(o)2959 5729 y Fv(:)-2986 b FA(\(2.26\))p Black Black eop %%Page: 20 20 20 19 bop Black 0 49 a Fq(20)p Black Black 0 241 a Fz(Remark)38 b(2.2.)p Black 49 w FA(F)-8 b(rom)32 b(Lemma)h(2.1,)g(there)g(exists)h Fv(C)2032 205 y Fr(0)2088 241 y FA(indep)s(enden)m(t)h(of)d Fv(\027)39 b FA(and)32 b Fv(n)p FA(,)h(so)g(that)1188 417 y Fv(\026)1247 376 y Fr(\003)1247 442 y Fw(n)1294 417 y FA(\()p Fv(\027)6 b FA(;)17 b Fv(\014)6 b FA(\))27 b Fu(\025)h Fv(\026)1720 376 y Fr(\003)1720 442 y Fw(n)1767 417 y FA(\(0;)17 b Fv(\014)6 b FA(\))21 b Fu(\000)h Fv(C)2194 376 y Fr(0)2218 417 y Fv(\027)2272 376 y Ft(2)2328 417 y Fv(:)0 594 y FA(This)34 b(implies)h(that)e(the)g(region)g Fu(f)p FA(Re)p Fv(z)g Fu(\024)c Fv(\026)1636 557 y Fr(\003)1698 594 y Fu(\000)22 b Fv(C)1874 557 y Fr(0)1898 594 y FA(\(Im)p Fv(z)t FA(\))2139 557 y Ft(2)2202 594 y FA(+)g Fv(\022)s FA(\001)2429 557 y Fr(\000)p Ft(1)2524 594 y Fv(R)2598 609 y Ft(2)2637 594 y FA(\()p Fv(V)g FA(\(\000)2853 557 y Ft(2)2892 594 y FA(\)\))p Fu(g)33 b FA(is)g(alw)m(a)m(ys)i(in)0 710 y(the)e(analyticit)m(y)h(region)f(of)f Fv(!)t FA(\(\000)1221 674 y Ft(2)1259 710 y FA(\),)h(whic)m(h)h(is)f(giv)m(en)h(in)f(Prop)s (osition)g(2.3.)43 b(Therefore,)33 b(if)1199 935 y Fv(C)1276 894 y Fr(0)1327 935 y Fu(\024)1842 867 y FA(1)p 1442 912 848 4 v 1442 1025 a(2)1491 944 y Fp(\000)1537 1025 y Fv(\022)s FA(\001)1666 996 y Fr(\000)p Ft(1)1761 1025 y Fv(R)1835 1040 y Ft(2)1874 1025 y FA(\()p Fv(V)22 b FA(\(\000)2090 996 y Ft(2)2129 1025 y FA(\)\))2205 944 y Fp(\001)2251 967 y Ft(2)2317 935 y Fv(;)0 1207 y FA(then)27 b(the)f(disc)i Fv(D)648 1222 y Fw(r)712 1207 y FA(of)e(cen)m(ter)h Fv(\026)1159 1171 y Fr(\003)1224 1207 y FA(and)g(radius)g Fv(r)j FA(=)e Fv(\022)s FA(\001)2002 1171 y Fr(\000)p Ft(1)2096 1207 y Fv(R)2170 1222 y Ft(2)2210 1207 y FA(\()p Fv(V)21 b FA(\(\000)2425 1171 y Ft(2)2465 1207 y FA(\)\))26 b(is)h(inside)g(the)g(analyticit)m(y)0 1324 y(region)33 b(of)f Fv(!)t FA(\(\000)568 1287 y Ft(2)606 1324 y FA(\).)44 b(This)34 b(happ)s(ens)f(as)g(so)s(on)f(as)h Fv(V)22 b FA(\(\000)1963 1287 y Ft(2)2002 1324 y FA(\))32 b(is)i(large)e (enough.)100 1562 y(Assuming)d(that)e(the)h(disc)g Fv(D)1183 1577 y Fw(r)1249 1562 y FA(is)g(inside)h(the)f(analyticit)m(y)h(region) e(of)g Fv(!)t FA(\(\000)2814 1526 y Ft(2)2853 1562 y FA(\),)h(w)m(e)h(decomp)s(ose)0 1678 y Fv(@)5 b(D)137 1693 y Fw(r)208 1678 y FA(in)m(to)509 1855 y Fv(@)g(D)649 1814 y Fw(g)646 1879 y(r)717 1855 y FA(:=)28 b Fv(@)5 b(D)985 1870 y Fw(r)1046 1855 y Fu(\\)22 b(f)p Fv(z)33 b FA(:)44 b(Re)p Fv(z)33 b Fu(\024)28 b Fv(\026)1690 1814 y Fr(\003)1729 1855 y FA(\(Im\()p Fv(z)t FA(\);)17 b Fv(\014)6 b FA(\))22 b Fu(\000)h Fv(\022)s FA(\001)2402 1814 y Fr(\000)p Ft(1)2497 1855 y Fv(R)2571 1870 y Ft(1)2610 1855 y FA(\()p Fv(V)f FA(\(\000)2826 1814 y Ft(2)2865 1855 y FA(\)\))p Fu(g)17 b Fv(;)0 2031 y FA(and)509 2167 y Fv(@)5 b(D)649 2126 y Fw(d)646 2191 y(r)717 2167 y FA(:=)28 b Fv(@)5 b(D)985 2182 y Fw(r)1046 2167 y Fu(\\)23 b(f)p Fv(z)32 b FA(:)44 b(Re)p Fv(z)33 b Fu(\025)28 b Fv(\026)1690 2126 y Fr(\003)1729 2167 y FA(\(Im\()p Fv(z)t FA(\);)17 b Fv(\014)6 b FA(\))22 b Fu(\000)h Fv(\022)s FA(\001)2402 2126 y Fr(\000)p Ft(1)2497 2167 y Fv(R)2571 2182 y Ft(1)2611 2167 y FA(\()p Fv(V)e FA(\(\000)2826 2126 y Ft(2)2865 2167 y FA(\)\))p Fu(g)c Fv(;)0 2323 y FA(and)33 b(write)g(\(2.26\))f(as)h(a)f(sum)h(of)f(t)m(w)m(o)i(in)m (tegrals)f Fv(I)1868 2276 y Fw(g)1860 2351 y(k)r(;n)1965 2323 y FA(\(\000)2064 2287 y Ft(2)2104 2323 y FA(\))f(and)h Fv(I)2415 2287 y Fw(d)2407 2349 y(k)r(;n)2512 2323 y FA(\(\000)2611 2287 y Ft(2)2650 2323 y FA(\))g(\(see)g(\014gure)g(1\),) 903 2578 y Fv(I)954 2531 y Fw(g)946 2606 y(k)r(;n)1051 2578 y FA(\(\000)1150 2537 y Ft(2)1190 2578 y FA(\))27 b(:=)h(Re)1501 2468 y Fp(n)1607 2511 y Fv(k)s FA(!)p 1577 2555 141 4 v 1577 2647 a(2)p Fv(\031)t(i)1745 2443 y Fp(I)1800 2668 y Fw(@)t(D)1901 2636 y Fn(g)1899 2679 y(r)1978 2511 y Fv(\036)2036 2526 y Ft(\003)2088 2511 y FA(\(\000)2187 2475 y Ft(2)2227 2511 y FA(\))2265 2475 y Fw(n)2312 2511 y FA(\()p Fv(z)t FA(\))p 1968 2555 479 4 v 1968 2647 a(\()p Fv(z)f Fu(\000)c Fv(\026)2237 2618 y Fr(\003)2276 2647 y FA(\))2314 2618 y Fw(k)r Ft(+1)2473 2578 y Fv(dz)2573 2468 y Fp(o)0 2578 y FA(\(2.27\))0 2817 y(and)881 3031 y Fv(I)932 2990 y Fw(d)924 3056 y(k)r(;n)1030 3031 y FA(\(\000)1129 2990 y Ft(2)1168 3031 y FA(\))k(:=)h(Re)1479 2921 y Fp(n)1586 2964 y Fv(k)s FA(!)p 1556 3008 141 4 v 1556 3100 a(2)p Fv(\031)t(i)1723 2896 y Fp(I)1779 3121 y Fw(@)t(D)1880 3102 y Fn(d)1878 3138 y(r)1956 2964 y Fv(\036)2014 2979 y Ft(\003)2067 2964 y FA(\(\000)2166 2928 y Ft(2)2205 2964 y FA(\))2243 2928 y Fw(n)2290 2964 y FA(\()p Fv(z)t FA(\))p 1946 3008 479 4 v 1946 3100 a(\()p Fv(z)f Fu(\000)c Fv(\026)2215 3071 y Fr(\003)2254 3100 y FA(\))2292 3071 y Fw(k)r Ft(+1)2451 3031 y Fv(dz)2551 2921 y Fp(o)2635 3031 y Fv(:)-2662 b FA(\(2.28\))0 3375 y(2.3.2.)49 b Fs(A)n(n)36 b(upp)-5 b(er)35 b(b)-5 b(ound)35 b(for)g Fv(I)1195 3328 y Fw(g)1187 3403 y(k)r(;n)1292 3375 y FA(\(\000)1391 3339 y Ft(2)1431 3375 y FA(\))p Fs(.)48 b Fv(I)1598 3328 y Fw(g)1590 3403 y(k)r(;n)1695 3375 y FA(\(\000)1794 3339 y Ft(2)1834 3375 y FA(\))33 b(is)h(not)f(the)g(main)h(con)m(tribution)g(to)f(\(2.26\))o(,)0 3493 y(so)44 b(that)g(it)g(is)g(su\016cien)m(t)i(to)d(get)h(an)g(upp)s 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FA(1)22 b(+)g(Re)17 b Fc(g)p FA(\(\000)1541 2056 y Ft(2)1581 2097 y FA(\))1619 2056 y Fr(0)1642 2097 y FA(\()p Fv(\020)8 b FA(\))1769 2016 y Fp(\001)1836 2097 y Fu(\000)22 b Fv(r)e FA(sin)d Fv(\013)q FA(Im)g Fc(g)p FA(\(\000)2481 2056 y Ft(2)2520 2097 y FA(\))2558 2056 y Fr(0)2582 2097 y FA(\()p Fv(\020)8 b FA(\))26 b(=)i Fv(k)19 b(:)0 2264 y FA(Since)34 b Fc(g)p FA(\(\000)405 2228 y Ft(2)444 2264 y FA(\))f(is)g(real)g(on)f(the)h(real)g(axis,)g Fv(\013)1583 2279 y Fw(k)r(;n)1716 2264 y FA(=)27 b(0)33 b(and)f Fv(r)2134 2279 y Fw(k)r(;n)2272 2264 y FA(is)h(solution)g(of) 1229 2426 y Fv(c)p FA(\()p Fv(n)p FA(\))p Fv(r)1452 2345 y Fp(\000)1497 2426 y FA(1)22 b(+)g Fc(g)p FA(\(\000)1816 2385 y Ft(2)1856 2426 y FA(\))1894 2385 y Fr(0)1917 2426 y FA(\()p Fv(r)s FA(\))2040 2345 y Fp(\001)2113 2426 y FA(=)28 b Fv(k)19 b(:)-2314 b FA(\(2.32\))p Black 0 2604 a Fz(Lemma)39 b(2.3.)p Black 49 w Fs(L)-5 b(et)36 b Fv(\013)851 2619 y Fw(i)907 2604 y Fu(\025)29 b Fv(\031)t(=)p FA(4)p Fs(,)35 b Fv(i)29 b 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Ft(0)1435 3313 y FA(\()p Fv(A)p FA(\))p Fv(;)17 b(c)p FA(\()p Fv(n)p FA(\)\(1)22 b Fu(\000)g FA(2)2061 3224 y Fu(p)p 2144 3224 74 4 v 89 x Fv(A)p FA(\))p Fv(R)2330 3233 y Fp(\003)2389 3313 y Fv(;)0 3472 y Fs(ther)-5 b(e)35 b(is)f(a)h(unique)g(solution)g FA(0)27 b Fv(<)h(r)1339 3487 y Fw(k)r(;n)1471 3472 y Fv(<)g(R)36 b Fs(of)55 b FA(\(2.32\))p Fs(.)44 b(Mor)-5 b(e)g(over,)413 3640 y FA(e)456 3604 y Fw(cr)519 3616 y Fn(k)q(;n)614 3604 y Ft(+)p Fw(c)p Ft(\()p Fw(n)p Ft(\))12 b Fd(g)p Ft(\(\000)915 3581 y Fl(2)950 3604 y Ft(\)\()p Fw(r)1036 3616 y Fn(k)q(;n)1133 3604 y Ft(\))p 413 3685 752 4 v 527 3789 a FA(10)625 3705 y Fp(p)p 725 3705 326 4 v 84 x Fv(c)p FA(\()p Fv(n)p FA(\))p Fv(r)945 3804 y Fw(k)r(;n)1202 3708 y Fu(\024)1346 3640 y FA(1)p 1317 3685 108 4 v 1317 3776 a(2)p Fv(\031)1451 3572 y Fp(Z)1551 3598 y Fw(\013)1596 3607 y Fl(2)1507 3798 y Fr(\000)p Fw(\013)1607 3807 y Fl(1)1662 3708 y FA(e)1705 3667 y Fw(c)p Ft(\()p Fw(n)p Ft(\))p Fw(r)i Ft(cos)f Fw(\013)p Ft(+)p Fw(c)p Ft(\()p Fw(n)p Ft(\)Re)f Fd(g)p Ft(\(\000)2415 3643 y Fl(2)2450 3667 y Ft(\))2499 3627 y Fp(\002)2557 3708 y FA(cos\()2742 3681 y Fp(e)2725 3708 y Fv( )t FA(\()p Fv(\013)q FA(\)\))2969 3627 y Fp(\003)3027 3708 y Fv(d\013)1202 4017 y Fu(\024)1317 3949 y FA(e)1360 3913 y Fw(c)p Ft(\()p Fw(n)p Ft(\))p Fw(r)1520 3925 y Fn(k)q(;n)1616 3913 y Ft(+)p Fw(c)p Ft(\()p Fw(n)p Ft(\))g Fd(g)p Ft(\(\000)1917 3890 y Fl(2)1952 3913 y Ft(\)\()p Fw(r)2038 3925 y Fn(k)q(;n)2134 3913 y Ft(\))p 1317 3994 849 4 v 1529 4014 a Fp(p)p 1629 4014 326 4 v 84 x Fv(c)p FA(\()p Fv(n)p FA(\))p Fv(r)1849 4113 y Fw(k)r(;n)2192 4017 y Fv(:)p Black 0 4303 a Fs(Pr)-5 b(o)g(of.)p Black 41 w FA(Existence)49 b(and)e(uniqueness)j(of)c Fv(r)1629 4318 y Fw(k)r(;n)1781 4303 y FA(is)h(a)f(consequence)k(of)c(the)h (monotonicit)m(y)h(of)0 4424 y Fv(r)31 b Fu(7!)d Fv(c)p FA(\()p Fv(n)p FA(\))p Fv(r)426 4343 y Fp(\000)472 4424 y FA(1)22 b(+)g Fc(g)p FA(\(\000)791 4388 y Ft(2)831 4424 y FA(\))869 4388 y Fr(0)892 4424 y FA(\()p Fv(r)s FA(\))1015 4343 y Fp(\001)1060 4424 y FA(.)45 b(The)34 b(last)f(part)g(of)g(Lemma)g(2.3)g(is)h(pro)m(v)m(en)g(in)f(app)s (endix)i(of)d([2].)0 4540 y(The)i(computation)f(is)g(relativ)m(ely)h (long,)f(but)f(standard.)p 3473 4703 4 66 v 3477 4640 59 4 v 3477 4703 V 3535 4703 4 66 v 100 4915 a(Setting)23 b Fv(c)p FA(\()p Fv(n)p FA(\))28 b(=)f Fv(n\014)6 b FA(\001)p Fv(V)22 b FA(\(\000)1111 4879 y Ft(2)1150 4915 y FA(\))h(and)g Fv(R)29 b FA(=)f Fv(\022)s FA(\001)1727 4879 y Fr(\000)p Ft(1)1821 4915 y Fv(R)1895 4930 y Ft(2)1935 4915 y FA(\()p Fv(V)22 b FA(\(\000)2151 4879 y Ft(2)2190 4915 y FA(\)\))h(in)g(Lemma)h (2.3)e(w)m(e)i(get)f(su\016cien)m(t)0 5031 y(conditions)30 b(for)e(the)i(existence)h(of)e(a)f(stationary)i(phase)f(p)s(oin)m(t)g (and)g(the)h(follo)m(wing)f(ev)-5 b(aluation)0 5148 y(of)32 b(the)h(in)m(tegral)g Fv(I)685 5111 y Fw(d)677 5174 y(k)r(;n)782 5148 y FA(\(\000)881 5111 y Ft(2)921 5148 y FA(\))f(b)m(y)h(that)g (metho)s(d.)44 b(Since)33 b Fv(r)2029 5163 y Fw(k)r(;n)2167 5148 y FA(is)g(solution)g(of)39 b(\(2.32\))o(,)33 b(w)m(e)h(ha)m(v)m(e) 471 5372 y Fv(k)25 b Fu(\000)752 5305 y Fv(k)s(A)p 657 5349 318 4 v 657 5440 a FA(\(1)c(+)h Fv(A)p FA(\))1012 5372 y(=)1258 5305 y Fv(k)p 1126 5349 V 1126 5440 a FA(\(1)g(+)g Fv(A)p FA(\))1481 5372 y Fu(\024)28 b Fv(c)p FA(\()p Fv(n)p FA(\))p Fv(r)1806 5387 y Fw(k)r(;n)1939 5372 y Fu(\024)2187 5305 y Fv(k)p 2054 5349 320 4 v 2054 5440 a FA(\(1)22 b Fu(\000)h Fv(A)p FA(\))2412 5372 y(=)k Fv(k)e FA(+)2795 5305 y Fv(k)s(A)p 2699 5349 V 2699 5440 a FA(\(1)d Fu(\000)h Fv(A)p FA(\))3045 5372 y Fv(;)0 5583 y FA(and)351 5770 y Fv(c)p FA(\()p Fv(n)p FA(\))p Fu(j)p Fc(g)p FA(\(\000)705 5729 y Ft(2)744 5770 y FA(\)\()p Fv(r)864 5785 y Fw(k)r(;n)969 5770 y FA(\))p Fu(j)28 b FA(=)f Fv(c)p FA(\()p Fv(n)p FA(\))1359 5656 y Fp(\014)1359 5716 y(\014)1359 5775 y(\014)1409 5635 y(Z)1508 5661 y Fw(r)1540 5673 y Fn(k)q(;n)1464 5860 y Ft(0)1656 5770 y Fc(g)p FA(\(\000)1806 5729 y Ft(2)1846 5770 y FA(\))1884 5729 y Fr(0)1907 5770 y FA(\()p Fv(\020)8 b FA(\))p Fv(d\020)2136 5656 y Fp(\014)2136 5716 y(\014)2136 5775 y(\014)2195 5770 y Fu(\024)28 b Fv(Ac)p FA(\()p Fv(n)p FA(\))p Fv(r)2593 5785 y Fw(k)r(;n)2726 5770 y Fu(\024)g Fv(k)2980 5703 y(A)p 2895 5747 244 4 v 2895 5839 a FA(1)22 b Fu(\000)h Fv(A)3165 5770 y(:)p Black Black eop %%Page: 23 23 23 22 bop Black 3467 49 a Fq(23)p Black 0 241 a FA(Therefore)34 b(Lemma)f(2.3)f(implies)565 324 y Fu(p)p 648 324 244 4 v 80 x FA(1)22 b Fu(\000)g Fv(A)p 565 449 327 4 v 611 553 a FA(10)709 469 y Fu(p)p 791 469 55 4 v 791 553 a Fv(k)901 472 y(c)943 430 y Fw(k)943 496 y Fr(\000)1019 472 y Fv(c)p FA(\()p Fv(n)p FA(\))1195 430 y Fw(k)1264 404 y Fv(k)s FA(!)17 b(e)1405 368 y Fw(k)p 1264 449 184 4 v 1308 540 a Fv(k)1362 511 y Fw(k)1475 472 y Fv(\036)1533 430 y Fr(\003)1533 496 y Ft(\003)1586 472 y FA(\(\000)1685 430 y Ft(2)1724 472 y FA(\))1762 430 y Fw(n)1837 472 y Fu(\024)28 b Fv(I)1993 430 y Fw(d)1985 496 y(k)r(;n)2090 472 y FA(\(\000)2189 430 y Ft(2)2228 472 y FA(\))-2249 b(\(2.33\))1837 757 y Fu(\024)1952 610 y(p)p 2035 610 243 4 v 80 x FA(1)22 b(+)g Fv(A)p 1952 735 326 4 v 2046 755 a Fu(p)p 2129 755 55 4 v 84 x Fv(k)2287 757 y(c)2329 716 y Fw(k)2329 782 y Ft(+)2404 757 y Fv(c)p FA(\()p Fv(n)p FA(\))2580 716 y Fw(k)2650 690 y Fv(k)s FA(!)17 b(e)2791 654 y Fw(k)p 2650 735 184 4 v 2693 826 a Fv(k)2747 797 y Fw(k)2860 757 y Fv(\036)2918 716 y Fr(\003)2918 782 y Ft(\003)2971 757 y FA(\(\000)3070 716 y Ft(2)3109 757 y FA(\))3147 716 y Fw(n)3211 757 y Fv(;)0 975 y FA(with)1035 1167 y Fv(c)1077 1182 y Fr(\006)1136 1167 y FA(\()p Fv(A)p FA(\))28 b(:=)g(\(1)21 b Fu(\006)i Fv(A)p FA(\))17 b(exp)1945 1056 y Fp(h)2015 1167 y Fu(\006)2205 1099 y FA(2)p Fv(A)p 2124 1144 284 4 v 2124 1235 a FA(1)22 b Fu(\000)h Fv(A)2368 1206 y Ft(2)2417 1056 y Fp(i)2481 1167 y Fv(:)-2508 b FA(\(2.34\))0 1374 y(If)33 b Fv(A)f FA(con)m(v)m(erges)j(to)d(0,)h (then)g Fv(c)1132 1389 y Fr(\006)1223 1374 y FA(con)m(v)m(erges)i(to)d (1.)44 b(W)-8 b(e)33 b(assume)h(that)e(\(see)i(\(2.30\))o(\))921 1601 y Fv(C)991 1616 y Ft(7)1047 1601 y FA(e)1090 1560 y Fr(\000)p Fw(\034)1176 1569 y Fl(2)1211 1560 y Ft(\()p Fw(\014)s Ft(;)p Fw(\022)1336 1536 y Fm(0)1359 1560 y Ft(\))1418 1601 y Fu(\024)1533 1533 y Fv(A)p 1533 1578 74 4 v 1545 1669 a FA(2)1714 1601 y(and)97 b Fv(C)2038 1616 y Ft(8)2137 1533 y Fu(j)p FA(\000)2226 1497 y Ft(2)2265 1533 y Fu(j)p 2088 1578 255 4 v 2088 1669 a Fv(V)21 b FA(\(\000)2265 1640 y Ft(2)2304 1669 y FA(\))2380 1601 y Fu(\024)2495 1533 y Fv(A)p 2495 1578 74 4 v 2507 1669 a FA(2)2595 1601 y Fv(:)-2622 b FA(\(2.35\))0 1853 y Fv(A)42 b FA(can)g(b)s(e)g(c)m(hosen)h(as)f(small)g(as)g(w)m(e)h(wish,) i(pro)m(vided)e(that)e Fv(\014)47 b FA(is)c(large)e(enough)h(and)3379 1806 y Fr(j)p Ft(\000)3443 1782 y Fl(2)3478 1806 y Fr(j)p 3343 1830 191 4 v 3343 1887 a Fw(V)16 b Ft(\(\000)3471 1868 y Fl(2)3506 1887 y Ft(\))0 1981 y FA(small)33 b(enough.)0 2219 y(2.4.)56 b Fz(Lo)m(w)m(er)37 b(b)s(ounds)h(on)f(the)g(deriv)-6 b(ativ)m(es)37 b(of)g(the)g(free)g(energy)g(at)g(\014nite)g(v)m(olume.) 0 2347 y FA(W)-8 b(e)42 b(estimate)g(the)g(deriv)-5 b(ativ)m(e)43 b(of)e([)p Fv(f)1415 2311 y Ft(2)1404 2372 y(\003)1457 2347 y FA(])1484 2296 y Ft(\()p Fw(k)r Ft(\))1484 2371 y Fw(\026)1526 2352 y Fm(\003)1623 2347 y FA(for)g(large)g(enough)h Fv(k)s FA(.)70 b(The)42 b(main)f(result)i(of)e(this)0 2463 y(subsection)35 b(is)e(Prop)s(osition)g(2.4.)100 2629 y(Let)f(0)c Fv(<)f(\022)k(<)d FA(1,)k Fv(A)c Fu(\024)g FA(1)p Fv(=)p FA(25,)k(and)h(set)1405 2781 y(^)1397 2807 y Fv(\022)e FA(:=)c Fv(\022)s FA(\(1)22 b Fu(\000)h FA(2)1909 2718 y Fu(p)p 1992 2718 74 4 v 89 x Fv(A)p FA(\))17 b Fv(:)0 2971 y FA(Let)33 b Fv(")221 2935 y Fr(0)271 2971 y Fv(>)28 b FA(0)k(and)h Fv(\037)707 2935 y Fr(0)707 2995 y Ft(2)779 2971 y FA(so)g(that)1339 3134 y(\(1)22 b(+)g Fv(")1592 3093 y Fr(0)1615 3134 y FA(\))p Fv(\037)1714 3093 y Fr(0)1714 3159 y Ft(2)1781 3134 y Fv(>)28 b(\037)1946 3149 y Ft(2)1985 3134 y FA(\()p Fu(1)p FA(\))17 b Fv(:)-2205 b FA(\(2.36\))0 3302 y(The)31 b(whole)g(analysis)h(dep)s(ends)f(on)g (the)f(parameters)h Fv(\022)j FA(and)c Fv(")2324 3266 y Fr(0)2347 3302 y FA(.)42 b(W)-8 b(e)31 b(\014x)g(the)f(v)-5 b(alues)32 b(of)d Fv(\022)s FA(,)i(and)0 3419 y Fv(")46 3382 y Fr(0)109 3419 y FA(b)m(y)42 b(the)f(follo)m(wing)g(conditions,)j (whic)m(h)e(are)e(needed)j(for)d(the)h(pro)s(of)f(of)g(Prop)s(osition)h (2.4.)0 3535 y(W)-8 b(e)33 b(c)m(ho)s(ose)g(0)28 b Fv(<)f(A)730 3550 y Ft(0)798 3535 y Fv(<)g FA(1)p Fv(=)p FA(25,)32 b Fv(\022)k FA(and)c Fv(")1472 3499 y Fr(0)1528 3535 y FA(so)h(that)508 3782 y(e)562 3714 y Fl(1)p 561 3726 33 3 v 561 3767 a Fn(d)879 3715 y FA(1)p 634 3760 539 4 v 634 3856 a Fv(\022)s FA(\(1)22 b Fu(\000)h FA(2)940 3779 y Fu(p)p 1023 3779 113 4 v 77 x Fv(A)1096 3871 y Ft(0)1135 3856 y FA(\))1211 3782 y Fv(<)1410 3715 y(d)p 1324 3760 222 4 v 1324 3851 a(d)f Fu(\000)h FA(1)1582 3715 y Fv(c)1624 3730 y Fr(\000)1683 3715 y FA(\()p Fv(A)1794 3730 y Ft(0)1834 3715 y FA(\))1882 3650 y Fn(d)p Fm(\000)p Fl(1)p 1881 3663 111 3 v 1921 3704 a Fn(d)p 1582 3760 424 4 v 1675 3851 a FA(1)f(+)g Fv(")1890 3822 y Fr(0)2113 3782 y FA(and)2378 3715 y(1)g Fu(\000)h FA(2)2598 3638 y Fu(p)p 2680 3638 113 4 v 2680 3715 a Fv(A)2753 3730 y Ft(0)p 2378 3760 415 4 v 2467 3851 a FA(1)f(+)g Fv(")2682 3822 y Fr(0)2915 3715 y Fv(d)p 2830 3760 222 4 v 2830 3851 a(d)f Fu(\000)i FA(1)3088 3782 y Fv(>)28 b FA(1)17 b Fv(:)-3285 b FA(\(2.37\))0 4006 y(This)34 b(is)f(p)s(ossible,)h (since)1634 4110 y Fv(d)p 1453 4154 414 4 v 1453 4262 a FA(\()p Fv(d)22 b Fu(\000)h FA(1\))17 b(e)1822 4199 y Fl(1)p 1820 4211 33 3 v 1820 4252 a Fn(d)1904 4177 y Fv(>)28 b FA(1)17 b Fv(:)0 4392 y FA(Indeed,)628 4599 y Fv(d)679 4488 y Fp(\020)739 4599 y FA(e)793 4531 y Fl(1)p 792 4543 V 792 4584 a Fn(d)860 4599 y Fu(\000)23 b FA(1)1009 4488 y Fp(\021)1096 4599 y FA(=)k Fv(d)1250 4488 y Fp(\020)1310 4599 y FA(e)1364 4531 y Fl(1)p 1363 4543 V 1363 4584 a Fn(d)1431 4599 y Fu(\000)c FA(1)f Fu(\000)1712 4532 y FA(1)p 1711 4576 51 4 v 1711 4667 a Fv(d)1794 4599 y FA(+)1903 4532 y(1)p 1902 4576 V 1902 4667 a Fv(d)1963 4488 y Fp(\021)2050 4599 y FA(=)2153 4504 y Fp(X)2159 4715 y Fw(n)p Fr(\025)p Ft(2)2342 4532 y FA(1)p 2324 4576 86 4 v 2324 4667 a Fv(n)p FA(!)2419 4488 y Fp(\020)2490 4532 y FA(1)p 2489 4576 51 4 v 2489 4667 a Fv(d)2549 4488 y Fp(\021)2609 4511 y Fw(n)p Fr(\000)p Ft(1)2768 4599 y FA(+)g(1)1096 4905 y(=)27 b(1)22 b(+)1368 4810 y Fp(X)1374 5021 y Fw(n)p Fr(\025)p Ft(1)1679 4837 y FA(1)p 1539 4882 331 4 v 1539 4973 a(\()p Fv(n)g FA(+)g(1\)!)1879 4794 y Fp(\020)1949 4837 y FA(1)p 1948 4882 51 4 v 1948 4973 a Fv(d)2009 4794 y Fp(\021)2068 4817 y Fw(n)1096 5211 y Fv(<)27 b FA(1)22 b Fu(\000)1405 5143 y FA(1)p 1380 5188 100 4 v 1380 5279 a(2)p Fv(d)1511 5211 y FA(+)1609 5116 y Fp(X)1615 5327 y Fw(n)p Fr(\025)p Ft(1)1798 5143 y FA(1)p 1780 5188 86 4 v 1780 5279 a Fv(n)p FA(!)1875 5100 y Fp(\020)1946 5143 y FA(1)p 1945 5188 51 4 v 1945 5279 a Fv(d)2005 5100 y Fp(\021)2065 5123 y Fw(n)2139 5211 y FA(=)28 b(e)2297 5142 y Fl(1)p 2296 5154 33 3 v 2296 5196 a Fn(d)2365 5211 y Fu(\000)2500 5143 y FA(1)p 2474 5188 100 4 v 2474 5279 a(2)p Fv(d)2600 5211 y(:)0 5478 y FA(Notice)41 b(that)f(conditions)i(\(2.37\))d(are)i(still)g(v)m (eri\014ed)h(with)f(the)g(same)h(v)-5 b(alues)41 b(of)f Fv(\022)k FA(and)c Fv(")3423 5442 y Fr(0)3486 5478 y FA(if)0 5595 y(w)m(e)d(replace)f Fv(A)553 5610 y Ft(0)628 5595 y FA(b)m(y)h(0)32 b Fv(<)h(A)g(<)f(A)1244 5610 y Ft(0)1284 5595 y FA(.)52 b(Giv)m(en)37 b Fv(\022)s FA(,)f(the)g(v)-5 b(alue)36 b(of)f Fv(\022)2348 5558 y Fr(0)2407 5595 y FA(is)h(\014xed)h(in)f(Prop)s(osition)g(2.3.)0 5711 y(>F)-8 b(rom)35 b(no)m(w)h(w)m(e)h(assume)g(that)e Fv(\014)41 b FA(is)36 b(so)f(large)h(that)f(all)g(results)i(of)e(subsections)j (2.1)d(and)h(2.2)0 5827 y(are)d(v)-5 b(alid.)43 b(The)34 b(v)-5 b(alue)33 b(of)f(0)27 b Fv(<)h(A)g(<)f(A)1457 5842 y Ft(0)1529 5827 y FA(is)33 b(\014xed)h(in)f(the)g(pro)s(of)f(of)g (Lemma)h(2.5.)p Black Black eop %%Page: 24 24 24 23 bop Black 0 49 a 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Fu(\024)29 b Fv(\022)s FA(\001)2019 5786 y Fr(\000)p Ft(1)2113 5827 y Fv(R)2187 5842 y Ft(2)2227 5827 y FA(\()p Fv(V)21 b FA(\(\000)2442 5786 y Ft(2)2482 5827 y FA(\)\))c Fv(:)p Black Black eop %%Page: 25 25 25 24 bop Black 3467 49 a Fq(25)p Black 0 241 a FA(W)-8 b(e)33 b(get)g(\(see)g(\(1.10\))o(\))485 436 y Fp(\014)485 496 y(\014)519 521 y FA([)p Fv(\036)604 536 y Ft(\003)657 521 y FA(\(\000)756 479 y Ft(2)795 521 y FA(\))833 479 y Fw(n)880 521 y FA(])907 470 y Ft(\()p Fw(k)r Ft(\))919 544 y Fw(\026)961 525 y Fm(\003)1004 436 y Fp(\014)1004 496 y(\014)1065 521 y Fu(\024)28 b Fv(k)s FA(!)1285 350 y Fp( )1374 453 y FA(\001)p Fv(V)21 b FA(\(\000)1632 417 y Ft(2)1672 453 y FA(\))1721 389 y Fl(1)p 1720 401 33 3 v 1720 442 a Fn(d)p 1374 498 393 4 v 1433 605 a Fv(\037)1494 620 y Ft(2)1534 605 y FA(\(1\))1667 579 y(^)1659 605 y Fv(\022)1776 350 y Fp(!)1855 372 y Fw(k)1931 521 y FA(e)1974 479 y Fr(\000)p Fw(n)p Ft([)p Fw(\034)2123 488 y Fl(1)2157 479 y Ft(\()p Fw(\014)s Ft(;)p Fw(\022)2282 456 y Fm(0)2304 479 y Ft(\))p Fr(\000)p Fw(C)2436 488 y Fl(5)2472 479 y Fw(\016)r Ft(])p Fr(j)p Ft(\000)2589 456 y Fl(2)2623 479 y Fr(j)1065 877 y Fu(\024)28 b Fv(k)s FA(!)1285 706 y Fp( )1374 809 y FA(\001)17 b(\()p Fv(C)1580 824 y Ft(2)1619 809 y Fv(\021)t FA(\))1758 738 y Fl(1)p 1718 750 111 3 v 1718 792 a Fn(d)p Fm(\000)p Fl(1)p 1374 854 470 4 v 1472 961 a Fv(\037)1533 976 y Ft(2)1572 961 y FA(\(1\))1705 935 y(^)1697 961 y Fv(\022)1853 706 y Fp(!)1932 729 y Fw(k)2008 877 y Fu(j)p FA(\000)2097 835 y Ft(2)2136 877 y Fu(j)2211 806 y Fn(k)p 2173 818 111 3 v 2173 859 a(d)p Fm(\000)p Fl(1)2298 877 y FA(e)2341 835 y Fr(\000)p Fw(n)p Ft([)p Fw(\034)2490 844 y Fl(1)2524 835 y Ft(\()p Fw(\014)s Ft(;)p Fw(\022)2649 812 y Fm(0)2672 835 y Ft(\))p Fr(\000)p Fw(C)2804 844 y Fl(5)2839 835 y Fw(\016)r Ft(])p Fr(j)p Ft(\000)2956 812 y Fl(2)2990 835 y Fr(j)3031 877 y Fv(:)0 1141 y FA(W)-8 b(e)33 b(sum)g(o)m(v)m(er)h Fv(n)f FA(and)g(o)m(v)m(er)g(\000)1133 1105 y Ft(2)1205 1141 y FA(using)g(the)g(inequalit)m(y)839 1258 y Fp(X)835 1468 y Fw(m)p Fr(\025)p Ft(1)1004 1352 y Fv(m)1089 1311 y Fw(p)1146 1352 y FA(e)1189 1311 y Fr(\000)p Fw(q)r(m)1372 1352 y Fu(\024)1506 1285 y FA(1)p 1487 1330 87 4 v 1487 1421 a Fv(q)1534 1392 y Fw(p)1600 1352 y FA(\000\()p Fv(p)22 b FA(+)g(1\))98 b(\()p Fv(p)27 b Fu(\025)h FA(2)17 b Fv(;)33 b(q)f Fu(\025)c FA(2\))17 b Fv(:)0 1625 y FA(There)34 b(exist)g Fv(C)583 1640 y Ft(11)690 1625 y FA(and)e Fv(C)949 1640 y Ft(12)1024 1625 y FA(\()p Fv(\022)1110 1589 y Fr(0)1133 1625 y FA(\))c Fv(>)f FA(0)33 b(so)g(that)486 1810 y Fp(X)377 2034 y Ft(\000)421 2011 y Fl(2)455 2034 y Ft(:In)n(t)13 b(\000)621 2011 y Fn(d)657 2034 y Fr(3)p Ft(0)275 2138 y Fw(V)i Ft(\(\000)402 2114 y Fl(2)438 2138 y Ft(\))475 2082 y Fn(d)p Fm(\000)p Fl(1)p 475 2094 111 3 v 514 2131 a Fn(d)600 2138 y Fr(\024)p Fw(\021)r Fr(k)p Ft(\000)771 2114 y Fl(2)806 2138 y Fr(k)335 2219 y Ft(\000)379 2195 y Fl(2)433 2219 y Ft(not)24 b Fw(k)r Ft(-small)858 1820 y Fp(\014)858 1880 y(\014)891 1905 y FA([)p Fv(u)974 1920 y Ft(\003)1027 1905 y FA(\(\000)1126 1864 y Ft(2)1165 1905 y FA(\)])1230 1854 y Ft(\()p Fw(k)r Ft(\))1242 1928 y Fw(\026)1284 1909 y Fm(\003)1328 1820 y Fp(\014)1328 1880 y(\014)1389 1905 y Fu(\024)k Fv(C)1564 1920 y Ft(11)1655 1734 y Fp( )1854 1837 y FA(\001)17 b(\()p Fv(C)2060 1852 y Ft(2)2099 1837 y Fv(\021)t FA(\))2238 1766 y Fl(1)p 2198 1778 V 2198 1820 a Fn(d)p Fm(\000)p Fl(1)p 1744 1882 689 4 v 1744 1997 a FA(\()p Fv(C)1852 2012 y Ft(12)1926 1997 y Fv(\014)6 b FA(\))2075 1926 y Fl(1)p 2035 1938 111 3 v 2035 1979 a Fn(d)p Fm(\000)p Fl(1)2159 1997 y Fv(\037)2220 2012 y Ft(2)2260 1997 y FA(\(1\))2393 1970 y(^)2385 1997 y Fv(\022)2443 1734 y Fp(!)2522 1757 y Fw(k)2581 1905 y Fv(k)s FA(!)17 b(\000)2740 1794 y Fp(\020)2893 1837 y Fv(k)p 2809 1882 222 4 v 2809 1973 a(d)22 b Fu(\000)g FA(1)3062 1905 y(+)g(1)3209 1794 y Fp(\021)1389 2460 y Fu(\024)28 b Fv(C)1564 2475 y Ft(11)1655 2290 y Fp( )1854 2393 y FA(\001)17 b(\()p Fv(C)2060 2408 y Ft(2)2099 2393 y Fv(\021)t FA(\))2238 2322 y Fl(1)p 2198 2334 111 3 v 2198 2375 a Fn(d)p Fm(\000)p Fl(1)p 1744 2437 689 4 v 1744 2552 a FA(\()p Fv(C)1852 2567 y Ft(12)1926 2552 y Fv(\014)6 b FA(\))2075 2482 y Fl(1)p 2035 2494 111 3 v 2035 2535 a Fn(d)p Fm(\000)p Fl(1)2159 2552 y Fv(\037)2220 2567 y Ft(2)2260 2552 y FA(\(1\))2393 2526 y(^)2385 2552 y Fv(\022)2443 2290 y Fp(!)2522 2312 y Fw(k)2581 2460 y Fv(k)s FA(!)33 b Fv(k)2797 2390 y Fn(k)p 2759 2402 V 2759 2443 a(d)p Fm(\000)p Fl(1)2900 2460 y Fv(:)0 2720 y FA(W)-8 b(e)33 b(c)m(ho)s(ose)g Fv(\021)k FA(so)32 b(small)i(that)e(\(see)i(\(2.39\))o(\))722 2909 y(\001)17 b(\()p Fv(C)928 2924 y Ft(2)967 2909 y Fv(\021)t FA(\))1107 2838 y Fl(1)p 1067 2850 V 1067 2891 a Fn(d)p Fm(\000)p Fl(1)p 612 2953 689 4 v 612 3068 a FA(\()p Fv(C)720 3083 y Ft(12)795 3068 y Fv(\014)6 b FA(\))943 2997 y Fl(1)p 903 3009 111 3 v 903 3051 a Fn(d)p Fm(\000)p Fl(1)1028 3068 y Fv(\037)1089 3083 y Ft(2)1128 3068 y FA(\(1\))1261 3042 y(^)1253 3068 y Fv(\022)1339 2976 y(<)1776 2909 y FA(\001)p 1452 2953 730 4 v 1452 3071 a Fv(\014)1563 3000 y Fl(1)p 1523 3012 111 3 v 1523 3053 a Fn(d)p Fm(\000)p Fl(1)1647 3071 y FA(\()1693 3044 y(^)1685 3071 y Fv(\022)s(\037)1794 3086 y Ft(2)1834 3071 y FA(\()p Fu(1)p FA(\)\))2097 3000 y Fn(d)p 2057 3012 V 2057 3053 a(d)p Fm(\000)p Fl(1)2220 2976 y Fv(<)2570 2909 y FA(\001)p 2333 2953 555 4 v 2333 3071 a Fv(\014)2444 3000 y Fl(1)p 2404 3012 111 3 v 2404 3053 a Fn(d)p Fm(\000)p Fl(1)2528 3071 y FA(\()2574 3044 y(^)2566 3071 y Fv(\022)s(\037)2675 3036 y Fr(0)2675 3095 y Ft(2)2715 3071 y FA(\))2802 3000 y Fn(d)p 2763 3012 V 2763 3053 a(d)p Fm(\000)p Fl(1)2914 2976 y Fv(:)-2941 b FA(\(2.40\))0 3337 y(2.4.2.)49 b Fs(Contribution)43 b(to)g FA([)p Fv(f)1073 3290 y Fw(q)1062 3364 y Ft(\003)1115 3337 y FA(])1142 3286 y Ft(\()p Fw(k)r Ft(\))1142 3361 y Fw(\026)1184 3342 y Fm(\003)1283 3337 y Fs(fr)-5 b(om)42 b(the)h Fv(k)s Fs(-lar)-5 b(ge)42 b(and)h(thin)f(c)-5 b(ontours.)49 b FA(F)-8 b(or)40 b Fv(k)s FA(-large)h(and)0 3483 y(thin)33 b(con)m(tours)g(w)m(e)h(get)f (lo)m(w)m(er)g(and)g(upp)s(er)g(b)s(ounds)g(for)f([)p Fv(\036)2202 3498 y Ft(\003)2255 3483 y FA(\(\000)2354 3446 y Ft(2)2394 3483 y FA(\))2432 3446 y Fw(n)2479 3483 y FA(])2506 3432 y Ft(\()p Fw(k)r Ft(\))2517 3506 y Fw(\026)2559 3487 y Fm(\003)2603 3483 y FA(.)44 b(There)33 b(are)g(t)m(w)m(o)g (cases.)0 3766 y(A.)g(Assume)h(that)f Fv(R)780 3781 y Ft(1)819 3766 y FA(\()p Fv(V)22 b FA(\(\000)1035 3730 y Ft(2)1074 3766 y FA(\)\))28 b Fu(\025)g Fv(R)1357 3781 y Ft(2)1396 3766 y FA(\()p Fv(V)22 b FA(\(\000)1612 3730 y Ft(2)1651 3766 y FA(\)\),)33 b(or)f(that)g Fv(V)22 b FA(\(\000)2295 3730 y Ft(2)2334 3766 y FA(\))33 b(is)g(so)f(large)h (that)1261 3920 y(^)1253 3947 y Fv(\022)s(\014)6 b(V)21 b FA(\(\000)1539 3906 y Ft(2)1579 3947 y FA(\))p Fv(R)1691 3962 y Ft(1)1730 3947 y FA(\()p Fv(V)h FA(\(\000)1946 3906 y Ft(2)1985 3947 y FA(\)\))28 b Fv(>)f(k)20 b(:)0 4115 y FA(F)-8 b(or)30 b(eac)m(h)i Fv(n)c Fu(\025)g FA(1)j(let)h Fv(c)p FA(\()p Fv(n)p FA(\))27 b(=)h Fv(n\014)6 b FA(\001)p Fv(V)22 b FA(\(\000)1487 4079 y Ft(2)1526 4115 y FA(\).)43 b(Under)32 b(these)g(conditions)g(w)m(e)g(can)g(apply)g(Lemma)0 4232 y(2.3)37 b(with)h(a)f(disc)h Fv(D)756 4247 y Fw(r)788 4259 y Fn(k)q(;n)924 4232 y FA(so)g(that)f Fv(@)5 b(D)1402 4247 y Fw(r)1434 4259 y Fn(k)q(;n)1570 4232 y FA(=)35 b Fv(@)5 b(D)1821 4195 y Fw(d)1818 4256 y(r)1850 4268 y Fn(k)q(;n)1950 4232 y FA(.)58 b(Indeed,)40 b(if)d Fv(R)2553 4247 y Ft(1)2592 4232 y FA(\()p Fv(V)22 b FA(\(\000)2808 4195 y Ft(2)2847 4232 y FA(\)\))36 b Fu(\025)g Fv(R)3146 4247 y Ft(2)3186 4232 y FA(\()p Fv(V)21 b FA(\(\000)3401 4195 y Ft(2)3440 4232 y FA(\)\),)0 4365 y(then)46 b(w)m(e)h(apply)f (Lemma)g(2.3)g(with)g Fv(R)51 b FA(=)e Fv(\022)s FA(\001)1819 4329 y Fr(\000)p Ft(1)1914 4365 y Fv(R)1988 4380 y Ft(2)2028 4365 y FA(\()p Fv(V)21 b FA(\(\000)2243 4329 y Ft(2)2283 4365 y FA(\)\),)48 b(and)e(in)g(the)g(other)g(case)g(w)m(e)0 4481 y(set)39 b Fv(R)h FA(=)e Fv(\022)s FA(\001)515 4445 y Fr(\000)p Ft(1)610 4481 y Fv(R)684 4496 y Ft(1)723 4481 y FA(\()p Fv(V)22 b FA(\(\000)939 4445 y Ft(2)978 4481 y FA(\)\).)62 b(In)39 b(b)s(oth)g(cases)h Fv(r)1803 4496 y Fw(k)r(;n)1946 4481 y Fv(<)e(R)q FA(,)j(whic)m(h)f(implies)g Fv(@)5 b(D)2962 4496 y Fw(r)2994 4508 y Fn(k)q(;n)3133 4481 y FA(=)38 b Fv(@)5 b(D)3387 4445 y Fw(d)3384 4506 y(r)3416 4518 y Fn(k)q(;n)3516 4481 y FA(.)0 4618 y(Therefore)34 b(w)m(e)f(get)g(for)f Fv(I)948 4582 y Fw(d)940 4644 y(k)r(;n)1045 4618 y FA(\(\000)1144 4582 y Ft(2)1183 4618 y FA(\))h(the)g(lo)m(w)m (er)h(and)e(upp)s(er)h(b)s(ounds)h(\(2.33\))o(.)p Black 0 4804 a Fz(Lemma)39 b(2.4.)p Black 49 w Fs(Ther)-5 b(e)35 b(exists)g(a)g(function)g Fv(D)s FA(\()p Fv(k)s FA(\))p Fs(,)g FA(lim)2051 4819 y Fw(k)r Fr(!1)2252 4804 y Fv(D)s FA(\()p Fv(k)s FA(\))28 b(=)g(0)p Fs(,)36 b(such)f(that)g(for)g Fv(\014)41 b Fs(suf-)0 4921 y(\014ciently)g(lar)-5 b(ge)41 b(and)g Fv(A)g Fs(su\016ciently)h(smal)5 b(l)41 b(the)g(fol)5 b(lowing)40 b(holds.)64 b(If)40 b Fv(k)j Fu(\025)e Fv(K)7 b FA(\()p Fv(A;)17 b(\021)t(;)g(\014)6 b FA(\))40 b Fs(and)0 5049 y Fv(R)74 5064 y Ft(1)114 5049 y FA(\()p Fv(V)21 b FA(\(\000)329 5013 y Ft(2)368 5049 y FA(\)\))28 b Fu(\025)g Fv(R)651 5064 y Ft(2)691 5049 y FA(\()p Fv(V)21 b FA(\(\000)906 5013 y Ft(2)946 5049 y FA(\)\))34 b Fs(or)1190 5023 y FA(^)1182 5049 y Fv(\022)s(\014)6 b(V)22 b FA(\(\000)1469 5013 y Ft(2)1508 5049 y FA(\))p Fv(R)1620 5064 y Ft(1)1659 5049 y FA(\()p Fv(V)g FA(\(\000)1875 5013 y Ft(2)1914 5049 y FA(\)\))28 b Fv(>)f(k)s Fs(,)35 b(then)445 5236 y FA(\(1)22 b Fu(\000)h Fv(D)s FA(\()p Fv(k)s FA(\)\))17 b([)p Fv(\036)1008 5251 y Ft(\003)1060 5236 y FA(\(\000)1159 5195 y Ft(2)1198 5236 y FA(\)])1263 5185 y Ft(\()p Fw(k)r Ft(\))1275 5259 y Fw(\026)1317 5240 y Fm(\003)1388 5236 y Fu(\024)29 b(\000)p FA([)p Fv(u)1654 5251 y Ft(\003)1707 5236 y FA(\(\000)1806 5195 y Ft(2)1845 5236 y FA(\)])1910 5185 y Ft(\()p Fw(k)r Ft(\))1922 5259 y Fw(\026)1964 5240 y Fm(\003)2035 5236 y Fu(\024)f FA(\(1)22 b(+)g Fv(D)s FA(\()p Fv(k)s FA(\)\))17 b([)p Fv(\036)2701 5251 y Ft(\003)2753 5236 y FA(\(\000)2852 5195 y Ft(2)2892 5236 y FA(\)])2957 5185 y Ft(\()p Fw(k)r Ft(\))2969 5259 y Fw(\026)3011 5240 y Fm(\003)3071 5236 y Fv(:)p Black 0 5467 a Fs(Pr)-5 b(o)g(of.)p Black 41 w FA(W)d(e)33 b(ha)m(v)m(e)380 5723 y Fu(\000)p FA([)p Fv(u)540 5738 y Ft(\003)593 5723 y FA(\(\000)692 5682 y Ft(2)731 5723 y FA(\)])796 5672 y Ft(\()p Fw(k)r Ft(\))808 5747 y Fw(\026)850 5728 y Fm(\003)922 5723 y FA(=)27 b([)p Fv(\036)1110 5738 y Ft(\003)1163 5723 y FA(\(\000)1262 5682 y Ft(2)1301 5723 y FA(\)])1366 5672 y Ft(\()p Fw(k)r Ft(\))1378 5747 y Fw(\026)1420 5728 y Fm(\003)1486 5723 y FA(+)22 b([)p Fv(\036)1669 5738 y Ft(\003)1722 5723 y FA(\(\000)1821 5682 y Ft(2)1860 5723 y FA(\)])1925 5672 y Ft(\()p Fw(k)r Ft(\))1937 5747 y Fw(\026)1979 5728 y Fm(\003)2040 5629 y Fp(X)2045 5839 y Fw(n)p Fr(\025)p Ft(2)2210 5656 y FA(\()p Fu(\000)p FA(1\))2412 5620 y Ft(\()p Fw(n)p Fr(\000)p Ft(1\))p 2210 5700 395 4 v 2378 5792 a Fv(n)2624 5648 y FA([)p Fv(\036)2709 5663 y Ft(\003)2762 5648 y FA(\(\000)2861 5612 y Ft(2)2900 5648 y FA(\))2938 5612 y Fw(n)2985 5648 y FA(])3012 5597 y Ft(\()p Fw(k)r Ft(\))3024 5672 y Fw(\026)3066 5653 y Fm(\003)p 2624 5700 486 4 v 2648 5813 a FA([)p Fv(\036)2733 5828 y Ft(\003)2785 5813 y FA(\(\000)2884 5784 y Ft(2)2924 5813 y FA(\)])2989 5762 y Ft(\()p Fw(k)r Ft(\))3001 5837 y Fw(\026)3043 5818 y Fm(\003)3136 5723 y Fv(:)p Black Black eop %%Page: 26 26 26 25 bop Black 0 49 a Fq(26)p Black 0 241 a FA(F)-8 b(rom)32 b(\(2.33\))g(there)h(exists)i(a)d(constan)m(t)h Fv(C)1601 256 y Ft(13)1676 241 y FA(,)912 445 y([)p Fv(\036)997 460 y Ft(\003)1050 445 y FA(\(\000)1149 409 y Ft(2)1188 445 y FA(\))1226 409 y Fw(n)1273 445 y FA(])1300 394 y Ft(\()p Fw(k)r Ft(\))1312 469 y Fw(\026)1354 450 y Fm(\003)p 912 497 486 4 v 936 610 a FA([)p Fv(\036)1021 625 y Ft(\003)1073 610 y FA(\(\000)1172 581 y Ft(2)1212 610 y FA(\)])1277 559 y Ft(\()p Fw(k)r Ft(\))1289 634 y Fw(\026)1331 615 y Fm(\003)1435 520 y Fu(\024)c Fv(C)1611 535 y Ft(13)1702 520 y Fv(\036)1760 479 y Fr(\003)1760 545 y Ft(\003)1813 520 y FA(\(\000)1912 479 y Ft(2)1951 520 y FA(\))1989 479 y Ft(\()p Fw(n)p Fr(\000)p Ft(1\))2197 410 y Fp(\020)2267 453 y Fv(c)2309 468 y Ft(+)p 2267 497 102 4 v 2267 589 a Fv(c)2309 604 y Fr(\000)2378 410 y Fp(\021)2437 432 y Fw(k)2497 520 y Fv(n)2555 479 y Fw(k)2614 520 y Fv(:)0 821 y FA(The)j(isop)s(erimetric)g(inequalit)m(y) i(\(2.21\),)d Fv(R)1614 836 y Ft(2)1654 821 y FA(\()p Fv(n)p FA(\))c Fu(\024)i Fv(\037)1982 836 y Ft(2)2021 821 y FA(\()p Fv(n)p FA(\))p Fv(n)2213 784 y Fr(\000)2279 757 y Fl(1)p 2278 769 33 3 v 2278 810 a Fn(d)2355 821 y FA(and)i(the)h(de\014nition)f(of)g Fv(k)s FA(-large)0 937 y(v)m(olume)j(con)m(tour)f(imply)549 1139 y Fv(\014)6 b Fu(k)p FA(\000)721 1098 y Ft(2)759 1139 y Fu(k)28 b(\025)g Fv(\014)6 b(\037)1064 1154 y Ft(2)1103 1139 y FA(\()p Fv(V)22 b FA(\(\000)1319 1098 y Ft(2)1358 1139 y FA(\)\))p Fv(V)f FA(\(\000)1611 1098 y Ft(2)1651 1139 y FA(\))1699 1070 y Fn(d)p Fm(\000)p Fl(1)p 1699 1083 111 3 v 1738 1124 a Fn(d)1851 1139 y Fu(\025)1964 1113 y FA(^)1956 1139 y Fv(\022)s(\014)6 b(R)2139 1154 y Ft(2)2179 1139 y FA(\()p Fv(V)21 b FA(\(\000)2394 1098 y Ft(2)2433 1139 y FA(\)\))p Fv(V)h FA(\(\000)2687 1098 y Ft(2)2726 1139 y FA(\))28 b Fu(\025)g Fv(k)20 b(:)0 1338 y FA(Let)27 b Fv(b)h FA(:=)g Fv(C)439 1353 y Ft(9)478 1338 y Fv(\016)j FA(\(see)d(\(2.31\))o(\);)h(w)m(e)e(ma)m(y)h(assume)1787 1299 y Ft(9)p 1769 1315 71 4 v 1769 1373 a(10)1860 1338 y Fu(\000)10 b Fv(b)29 b Fu(\025)2132 1299 y Ft(4)p 2132 1315 36 4 v 2132 1373 a(5)2204 1338 y FA(b)m(y)f(taking)f Fv(\014)32 b FA(large)27 b(enough.)42 b(Then)464 1533 y Fv(c)506 1497 y Fw(k)506 1558 y Ft(+)p 464 1581 102 4 v 464 1673 a Fv(c)506 1639 y Fw(k)506 1695 y Fr(\000)592 1509 y Fp(X)597 1720 y Fw(n)p Fr(\025)p Ft(2)752 1604 y Fv(n)810 1563 y Fw(k)r Fr(\000)p Ft(1)944 1604 y FA(e)987 1563 y Fr(\000)p Ft(\()p Fw(n)p Fr(\000)p Ft(1\)\(1)p Fr(\000)p Fw(b)p Ft(\))p Fw(k)1475 1604 y Fu(\024)1590 1533 y Fv(c)1632 1497 y Fw(k)1632 1558 y Ft(+)p 1590 1581 V 1590 1673 a Fv(c)1632 1639 y Fw(k)1632 1695 y Fr(\000)1718 1509 y Fp(X)1723 1720 y Fw(n)p Fr(\025)p Ft(2)1878 1604 y FA(e)1921 1563 y Fr(\000)2002 1535 y Fl(1)p 1987 1547 61 3 v 1987 1589 a(10)2058 1563 y Ft(\()p Fw(n)p Fr(\000)p Ft(1\))p Fw(k)2288 1604 y FA(e)2331 1558 y Fr(\000)p Fw(k)2425 1477 y Fp(\002)2466 1558 y Ft(\()2519 1531 y Fl(9)p 2504 1543 V 2504 1584 a(10)2575 1558 y Fr(\000)p Fw(b)p Ft(\)\()p Fw(n)p Fr(\000)p Ft(1\))p Fr(\000)p Ft(ln)12 b Fw(n)3043 1477 y Fp(\003)1475 1931 y Fu(\024)1590 1860 y Fv(c)1632 1824 y Fw(k)1632 1885 y Ft(+)p 1590 1908 102 4 v 1590 2000 a Fv(c)1632 1966 y Fw(k)1632 2022 y Fr(\000)1718 1836 y Fp(X)1723 2047 y Fw(n)p Fr(\025)p Ft(2)1878 1931 y FA(e)1921 1890 y Fr(\000)2002 1863 y Fl(1)p 1987 1875 61 3 v 1987 1916 a(10)2058 1890 y Ft(\()p Fw(n)p Fr(\000)p Ft(1\))p Fw(k)2288 1931 y FA(e)2331 1885 y Fr(\000)p Fw(k)2425 1804 y Fp(\002)2476 1858 y Fl(4)p 2476 1870 31 3 v 2476 1911 a(5)2517 1885 y Ft(\()p Fw(n)p Fr(\000)p Ft(1\))p Fr(\000)p Ft(ln)g Fw(n)2873 1804 y Fp(\003)1475 2240 y Fu(\024)1580 2129 y Fp(\020)1650 2172 y Fv(c)1692 2187 y Ft(+)p 1650 2217 102 4 v 1650 2308 a Fv(c)1692 2323 y Fr(\000)1761 2240 y FA(e)1804 2198 y Fr(\000)1884 2171 y Fl(1)p 1869 2183 61 3 v 1869 2224 a(10)1944 2129 y Fp(\021)2004 2151 y Fw(k)2079 2145 y Fp(X)2085 2355 y Fw(n)p Fr(\025)p Ft(1)2240 2240 y FA(e)2283 2198 y Fr(\000)2363 2171 y Fl(1)p 2348 2183 V 2348 2224 a(10)2419 2198 y Fw(nk)2521 2240 y Fv(:)0 2551 y FA(W)-8 b(e)33 b(c)m(ho)s(ose)g Fv(A)g FA(so)g(small)g(that)f Fv(c)1211 2566 y Ft(+)1270 2551 y FA(\()p Fv(A)p FA(\))p Fv(c)1461 2566 y Fr(\000)1520 2551 y FA(\()p Fv(A)p FA(\))1669 2502 y Fr(\000)p Ft(1)1780 2551 y FA(e)1823 2514 y Fr(\000)1904 2487 y Fl(1)p 1889 2499 V 1889 2540 a(10)1991 2551 y Fu(\024)c FA(1.)p 3473 2723 4 66 v 3477 2660 59 4 v 3477 2723 V 3535 2723 4 66 v 0 2981 a(B.)33 b(The)g(second)h(case)g(is)f (when)790 3141 y(^)782 3167 y Fv(\022)s(\014)6 b(V)21 b FA(\(\000)1068 3126 y Ft(2)1108 3167 y FA(\))p Fv(R)1220 3182 y Ft(1)1259 3167 y FA(\()p Fv(V)h FA(\(\000)1475 3126 y Ft(2)1514 3167 y FA(\)\))28 b Fu(\024)g Fv(k)j Fu(\024)1918 3141 y FA(^)1910 3167 y Fv(\022)s(\014)6 b(V)21 b FA(\(\000)2196 3126 y Ft(2)2235 3167 y FA(\))p Fv(R)2347 3182 y Ft(2)2387 3167 y FA(\()p Fv(V)g FA(\(\000)2602 3126 y Ft(2)2642 3167 y FA(\)\))c Fv(:)0 3353 y FA(Since)34 b(the)f(con)m(tours)g(are)g(also)f(thin,)867 3555 y Fv(\014)6 b Fu(k)p FA(\000)1039 3514 y Ft(2)1078 3555 y Fu(k)27 b(\024)h Fv(\021)1312 3514 y Fr(\000)p Ft(1)1414 3529 y FA(^)1406 3555 y Fv(\022)1454 3514 y Fr(\000)p Ft(1)1549 3555 y Fv(\037)1610 3570 y Ft(1)1649 3555 y FA(\(1\))1774 3514 y Fr(\000)p Ft(1)1868 3555 y Fv(\014)1936 3529 y FA(^)1929 3555 y Fv(\022)s(\037)2038 3570 y Ft(1)2078 3555 y FA(\(1\))p Fv(V)21 b FA(\(\000)2380 3514 y Ft(2)2419 3555 y FA(\))2467 3486 y Fn(d)p Fm(\000)p Fl(1)p 2467 3499 111 3 v 2506 3540 a Fn(d)1155 3727 y Fu(\024)28 b Fv(\021)1312 3686 y Fr(\000)p Ft(1)1414 3701 y FA(^)1406 3727 y Fv(\022)1454 3686 y Fr(\000)p Ft(1)1549 3727 y Fv(\037)1610 3742 y Ft(1)1649 3727 y FA(\(1\))1774 3686 y Fr(\000)p Ft(1)1868 3727 y Fv(\014)1936 3701 y FA(^)1929 3727 y Fv(\022)s(V)22 b FA(\(\000)2155 3686 y Ft(2)2194 3727 y FA(\))p Fv(R)2306 3742 y Ft(1)2346 3727 y FA(\()p Fv(V)f FA(\(\000)2561 3686 y Ft(2)2600 3727 y FA(\)\))1155 3899 y Fu(\024)28 b Fv(\021)1312 3858 y Fr(\000)p Ft(1)1414 3872 y FA(^)1406 3899 y Fv(\022)1454 3858 y Fr(\000)p Ft(1)1549 3899 y Fv(\037)1610 3914 y Ft(1)1649 3899 y FA(\(1\))1774 3858 y Fr(\000)p Ft(1)1868 3899 y Fv(k)j Fu(\021)d Fv(\025k)20 b(:)0 4095 y FA(W)-8 b(e)43 b(c)m(ho)s(ose)h Fv(R)i FA(=)f Fv(\014)6 b FA(\001)881 4059 y Fr(\000)p Ft(1)976 4095 y Fv(R)1050 4110 y Ft(2)1089 4095 y FA(\()p Fv(V)22 b FA(\(\000)1305 4059 y Ft(2)1344 4095 y FA(\)\))43 b(in)g(Lemma)g(2.3.)75 b(The)43 b(in)m(tegration)h(in)f(\(2.26\))f(is)h (de-)0 4211 y(comp)s(osed)f(in)m(to)g(t)m(w)m(o)g(parts)f(\(see)h (\014gure)g(1\).)69 b(W)-8 b(e)42 b(sho)m(w)g(that)f(the)h(con)m (tribution)g(from)f(the)0 4328 y(in)m(tegration)d(o)m(v)m(er)g Fv(@)5 b(D)854 4291 y Fw(g)851 4352 y(r)883 4364 y Fn(k)q(;n)1021 4328 y FA(is)38 b(negligible)h(for)d(large)i(enough)g Fv(\014)6 b FA(.)58 b(Since)38 b Fv(k)h Fu(\025)e Fv(K)7 b FA(\()p Fv(A;)17 b(\021)t(;)g(\014)6 b FA(\))36 b(and)0 4481 y(the)d(con)m(tours)g(v)m(erify)h Fv(V)22 b FA(\(\000)1010 4445 y Ft(2)1049 4481 y FA(\))1097 4416 y Fn(d)p Fm(\000)p Fl(1)p 1097 4429 V 1136 4471 a Fn(d)1249 4481 y Fv(>)28 b(\021)t Fu(k)p FA(\000)1516 4445 y Ft(2)1555 4481 y Fu(k)p FA(,)k(w)m(e)i(ha)m(v)m(e)580 4721 y Fv(n\014)6 b Fu(k)p FA(\000)810 4680 y Ft(2)849 4721 y Fu(k)p Fv(r)943 4736 y Fw(k)r(;n)1076 4721 y Fu(\024)1545 4654 y Fv(k)p 1191 4698 764 4 v 1191 4806 a FA(\001\(1)22 b Fu(\000)g Fv(A)p FA(\))p Fv(\021)t(V)g FA(\(\000)1821 4777 y Ft(2)1860 4806 y FA(\))1909 4742 y Fl(1)p 1908 4754 33 3 v 1908 4796 a Fn(d)1992 4721 y Fu(\024)2481 4654 y Fv(k)p 2107 4698 803 4 v 2107 4806 a FA(\001\(1)g Fu(\000)h Fv(A)2470 4821 y Ft(0)2509 4806 y FA(\))p Fv(\021)t(V)e FA(\(\000)2776 4777 y Ft(2)2816 4806 y FA(\))2865 4742 y Fl(1)p 2864 4754 33 3 v 2864 4796 a Fn(d)2936 4721 y Fv(:)0 4984 y FA(By)33 b(de\014nition)h(of)e Fv(K)7 b FA(\()p Fv(A;)17 b(\021)t(;)g(\014)6 b FA(\))31 b(\(see)j(\(2.38\))o(\))1264 5224 y Fv(n\014)6 b Fu(k)p FA(\000)1494 5183 y Ft(2)1533 5224 y Fu(k)p Fv(\032)1633 5183 y Fr(\000)p Ft(1)1727 5224 y Fv(C)1797 5239 y Ft(1)1836 5224 y Fv(r)1880 5239 y Fw(k)r(;n)2013 5224 y Fu(\024)2150 5157 y Fv(k)p 2128 5201 98 4 v 2128 5293 a FA(10)2252 5224 y Fv(:)0 5472 y FA(F)-8 b(rom)30 b(\(2.29\))g(with)h Fv(r)g FA(=)c Fv(r)976 5487 y Fw(k)r(;n)1112 5472 y FA(w)m(e)32 b(obtain)e(that)h (the)g(con)m(tribution)h(to)e Fu(j)p FA([)p Fv(u)2713 5487 y Ft(\003)2765 5472 y FA(\(\000)2864 5435 y Fw(q)2902 5472 y FA(\)])2967 5421 y Ft(\()p Fw(k)r Ft(\))2979 5495 y Fw(\026)3021 5476 y Fm(\003)3065 5472 y Fu(j)g FA(is)h(at)g(most)611 5723 y(\(1)22 b(+)g Fv(A)p FA(\))929 5682 y Fw(k)971 5643 y Fp(\000)1017 5723 y Fv(\014)6 b FA(\001)p Fv(V)21 b FA(\(\000)1336 5682 y Ft(2)1376 5723 y FA(\))1414 5643 y Fp(\001)1459 5665 y Fw(k)1519 5723 y FA(exp)1684 5643 y Fp(\000)1762 5656 y Fv(k)p 1740 5700 V 1740 5792 a FA(10)1847 5643 y Fp(\001)1911 5656 y Fv(k)s FA(!)p 1903 5700 97 4 v 1903 5792 a Fv(k)1957 5763 y Fw(k)2026 5629 y Fp(X)2032 5839 y Fw(n)p Fr(\025)p Ft(1)2187 5723 y Fv(n)2245 5682 y Fw(k)2288 5723 y FA(e)2331 5682 y Fr(\000)p Fw(n\014)s Fr(k)p Ft(\000)2551 5659 y Fl(2)2586 5682 y Fr(k)p Ft(\(1)p Fr(\000)p Fw(C)2788 5691 y Fl(6)2824 5682 y Fw(\016)r Ft(\))2905 5723 y Fv(:)p Black Black eop %%Page: 27 27 27 26 bop Black 3467 49 a Fq(27)p Black 0 241 a FA(As)29 b(in)f(the)h(pro)s(of)f(of)g(Lemma)h(2.4,)f(w)m(e)i(c)m(ho)s(ose)f Fv(\014)34 b FA(large)28 b(enough)h(so)f(that)h(w)m(e)g(can)g(assume)g (that)28 318 y Ft(9)p 10 334 71 4 v 10 391 a(10)113 357 y Fu(\000)22 b Fv(C)282 372 y Ft(6)321 357 y Fv(\016)32 b Fu(\025)511 318 y Ft(4)p 511 334 36 4 v 511 391 a(5)556 357 y FA(.)44 b(Then)251 484 y Fp(X)256 694 y Fw(n)p Fr(\025)p Ft(1)411 578 y Fv(n)469 537 y Fw(k)512 578 y FA(e)555 537 y Fr(\000)p Fw(n\014)s Fr(k)p Ft(\000)775 514 y Fl(2)810 537 y Fr(k)p Ft(\(1)p Fr(\000)p Fw(C)1012 546 y Fl(6)1048 537 y Fw(\016)r Ft(\))1141 578 y Fu(\024)28 b FA(e)1289 537 y Fr(\000)p Fw(\014)s Fr(k)p Ft(\000)1466 514 y Fl(2)1501 537 y Fr(k)p Ft(\(1)p Fr(\000)p Fw(C)1703 546 y Fl(6)1739 537 y Fw(\016)r Ft(\))1804 468 y Fp(\020)1864 578 y FA(1)22 b(+)2033 484 y Fp(X)2038 694 y Fw(n)p Fr(\025)p Ft(2)2193 578 y FA(e)2236 537 y Fr(\000)2317 510 y Fl(1)p 2301 522 61 3 v 2301 563 a(10)2372 537 y Ft(\()p Fw(n)p Fr(\000)p Ft(1\))p Fw(k)2603 578 y FA(e)2646 533 y Fr(\000)p Fw(k)2740 452 y Fp(\002)2791 506 y Fl(4)p 2791 518 31 3 v 2791 559 a(5)2831 533 y Ft(\()p Fw(n)p Fr(\000)p Ft(1\))p Fr(\000)p Ft(ln)13 b Fw(n)3188 452 y Fp(\003)3233 468 y(\021)1141 867 y Fu(\024)28 b FA(e)1289 826 y Fr(\000)p Fw(\014)s Fr(k)p Ft(\000)1466 803 y Fl(2)1501 826 y Fr(k)p Ft(\(1)p 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1567 y Fr(\000)p Fw(\014)s Fr(k)p Ft(\000)1428 1543 y Fl(2)1464 1567 y Fr(k)p Fw(C)1549 1576 y Fl(9)1584 1567 y Fw(\016)1649 1608 y Fu(\025)29 b FA(\(1)21 b Fu(\000)i Fv(D)s FA(\()p Fv(k)s FA(\)\))p Fv(c)2257 1567 y Fw(k)2257 1633 y Fr(\000)2316 1608 y FA(e)2359 1567 y Fw(k)2402 1608 y FA(e)2445 1567 y Fr(\000)p Fw(k)r(\025C)2630 1576 y Fl(9)2664 1567 y Fw(\016)2730 1608 y Fv(>)28 b FA(e)2887 1540 y Fl(2)p Fn(k)p 2887 1552 65 3 v 2904 1593 a Fl(3)0 1773 y FA(and)318 1937 y(\(1)22 b(+)g Fv(D)s FA(\()p Fv(k)s FA(\)\)\(1)f(+)h Fv(A)p FA(\))1094 1896 y Fw(k)1137 1937 y FA(e)1203 1869 y Fn(k)p 1190 1881 61 3 v 1190 1922 a Fl(10)1265 1937 y FA(e)1308 1896 y Fw(\014)s Fr(k)p Ft(\000)1430 1873 y Fl(2)1465 1896 y Fr(k)p Fw(C)1550 1905 y Fl(6)1585 1896 y Fw(\016)1651 1937 y Fu(\024)28 b FA(\(1)22 b(+)g Fv(D)s FA(\()p Fv(k)s FA(\)\)\(1)f(+)i Fv(A)p FA(\))2533 1896 y Fw(k)2575 1937 y FA(e)2642 1869 y Fn(k)p 2629 1881 V 2629 1922 a Fl(10)2704 1937 y FA(e)2747 1896 y Fw(\025k)r(C)2877 1905 y Fl(6)2911 1896 y Fw(\016)2977 1937 y Fv(<)28 b FA(e)3134 1869 y Fn(k)p 3134 1881 34 3 v 3136 1922 a Fl(3)3199 1937 y Fv(:)0 2120 y FA(If)33 b(these)i(inequalities)g(are)f(v)m(eri\014ed,)h(then)f(the)g(con)m (tribution)g(to)f Fu(\000)p FA([)p Fv(u)2623 2135 y Ft(\003)2676 2120 y FA(\(\000)2775 2083 y Fw(q)2813 2120 y FA(\)])2878 2069 y Ft(\()p Fw(k)r Ft(\))2890 2143 y Fw(\026)2932 2124 y Fm(\003)3009 2120 y FA(coming)g(from)0 2236 y(the)f(in)m (tegrations)g(o)m(v)m(er)h Fv(@)5 b(D)1048 2200 y Fw(g)1045 2260 y(r)1077 2272 y Fn(k)q(;n)1208 2236 y FA(is)32 b(negligible)h (with)f(resp)s(ect)h(to)e(that)g(coming)h(from)f(the)h(in)m(te-)0 2372 y(grations)g(o)m(v)m(er)i Fv(@)5 b(D)723 2336 y Fw(d)720 2397 y(r)752 2409 y Fn(k)q(;n)852 2372 y FA(.)44 b(T)-8 b(aking)33 b(in)m(to)g(accoun)m(t)g(\(2.33\))f(w)m(e)i(get)e (Lemma)h(2.5.)p Black 0 2562 a Fz(Lemma)39 b(2.5.)p Black 49 w Fs(Ther)-5 b(e)47 b(exists)g FA(0)k Fv(<)f(A)1491 2526 y Fr(0)1566 2562 y Fu(\024)h Fv(A)1767 2577 y Ft(0)1854 2562 y Fs(so)c(that)h(for)f(al)5 b(l)47 b Fv(\014)53 b Fs(su\016ciently)47 b(lar)-5 b(ge,)50 b(the)0 2678 y(fol)5 b(lowing)33 b(holds.)44 b(If)35 b Fv(k)30 b Fu(\025)f Fv(K)7 b FA(\()p Fv(A)1193 2642 y Fr(0)1216 2678 y Fv(;)17 b(\021)t(;)g(\014)6 b FA(\))34 b Fs(and)g FA(\000)1739 2642 y Ft(2)1813 2678 y Fs(is)h(a)f Fv(k)s Fs(-lar)-5 b(ge)35 b(and)f(thin)h(c)-5 b(ontour,)35 b(then)670 2892 y Fu(\000)p FA([)p Fv(u)830 2907 y Ft(\003)883 2892 y FA(\(\000)982 2851 y Ft(2)1021 2892 y FA(\)])1086 2841 y Ft(\()p Fw(k)r Ft(\))1098 2915 y Fw(\026)1140 2896 y Fm(\003)1212 2892 y Fu(\025)1351 2824 y FA(1)p 1327 2869 98 4 v 1327 2960 a(20)1434 2892 y(\(1)22 b Fu(\000)h Fv(D)s FA(\()p Fv(k)s FA(\)\))1895 2811 y Fp(\000)1940 2892 y Fv(\014)6 b FA(\001)p Fv(V)21 b FA(\(\000)2259 2851 y Ft(2)2299 2892 y FA(\))2337 2811 y Fp(\001)2382 2833 y Fw(k)2425 2892 y Fv(c)2467 2851 y Fw(k)2467 2916 y Fr(\000)2543 2892 y Fv(\036)2601 2851 y Fr(\003)2601 2916 y Ft(\003)2653 2892 y FA(\(\000)2752 2851 y Ft(2)2792 2892 y FA(\))c Fv(:)p Black 0 3104 a Fz(Prop)s(osition)38 b(2.4.)p Black 49 w Fs(Ther)-5 b(e)32 b(exists)g Fv(\014)1432 3068 y Fr(0)1488 3104 y Fs(so)h(that)g(for)g(al)5 b(l)32 b Fv(\014)i(>)27 b(\014)2352 3068 y Fr(0)2375 3104 y Fs(,)33 b(the)g(fol)5 b(lowing)32 b(holds.)43 b(Ther)-5 b(e)0 3220 y(exists)30 b(an)g(incr)-5 b(e)g(asing)29 b(diver)-5 b(ging)29 b(se)-5 b(quenc)g(e)29 b Fu(f)p Fv(k)1754 3235 y Fw(n)1801 3220 y Fu(g)h Fs(such)g(that)h(for)f(e)-5 b(ach)29 b Fv(k)2709 3235 y Fw(n)2786 3220 y Fs(ther)-5 b(e)30 b(exists)g FA(\003\()p Fv(L)3458 3235 y Fw(n)3505 3220 y FA(\))0 3336 y Fs(such)35 b(that)g(for)g(al)5 b(l)34 b FA(\003)27 b Fu(\033)i FA(\003\()p Fv(L)1089 3351 y Fw(n)1136 3336 y FA(\))911 3534 y Fu(\000)p FA([)p Fv(f)1074 3492 y Ft(2)1063 3558 y(\003)1117 3534 y FA(])1144 3483 y Ft(\()p Fw(k)1208 3491 y Fn(n)1250 3483 y Ft(\))1155 3557 y Fw(\026)1197 3538 y Fm(\003)1310 3534 y Fu(\025)f Fv(C)1492 3491 y Fw(k)1529 3499 y Fn(n)1485 3558 y Ft(14)1592 3534 y Fv(k)1643 3549 y 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b(con)m(tribution)g(of)f(the)h(small)f(con)m(tours)0 4173 y(to)g Fu(j)p FA([)p Fv(f)233 4137 y Ft(2)222 4198 y(\003)275 4173 y FA(])302 4122 y Ft(\()p Fw(k)r Ft(\))314 4197 y Fw(\026)356 4178 y Fm(\003)400 4173 y Fu(j)g FA(is)h(at)f(most) 427 4430 y Fv(C)497 4445 y Ft(10)588 4430 y FA(\001)669 4389 y Fw(k)728 4430 y Fv(\014)789 4386 y Fr(\000)892 4359 y Fn(k)p 854 4371 V 854 4412 a(d)p Fm(\000)p Fl(1)995 4430 y FA(\()1041 4404 y(^)1033 4430 y Fv(\022)s(\037)1142 4389 y Fr(0)1142 4455 y Ft(2)1182 4430 y FA(\))1220 4386 y Fr(\000)1307 4359 y Fn(k)q(d)p 1285 4371 V 1285 4412 a(d)p Fm(\000)p Fl(1)1426 4430 y Fv(k)s FA(!)17 b Fv(k)1626 4359 y Fn(k)p 1587 4371 V 1587 4412 a(d)p Fm(\000)p Fl(1)1740 4430 y Fu(\024)28 b Fv(C)1915 4445 y Ft(10)2006 4430 y FA(\001)2087 4389 y Fw(k)2147 4430 y Fv(\014)2208 4386 y Fr(\000)2310 4359 y Fn(k)p 2272 4371 V 2272 4412 a(d)p Fm(\000)p Fl(1)2396 4319 y Fp(\020)2491 4363 y FA(e)2545 4299 y Fl(1)p 2544 4311 33 3 v 2544 4352 a Fn(d)p 2466 4407 149 4 v 2474 4488 a FA(^)2466 4515 y Fv(\022)s(\037)2575 4480 y Fr(0)2575 4539 y Ft(2)2625 4319 y Fp(\021)2684 4342 y Fw(k)2772 4315 y Fn(d)p 2733 4327 111 3 v 2733 4368 a(d)p Fm(\000)p Fl(1)2857 4430 y Fv(k)s FA(!)2987 4359 y Fn(d)p 2948 4371 V 2948 4412 a(d)p Fm(\000)p Fl(1)3089 4430 y Fv(:)0 4670 y FA(The)j(con)m(tribution)h(of)d(the)i(fat)f(con)m (tours)h(is)g(m)m(uc)m(h)h(smaller)f(b)m(y)h(our)e(c)m(hoice)i(of)e Fv(\021)k FA(\(see)d(\(2.40\))o(\).)0 4804 y(The)47 b(con)m(tribution)g (to)e Fu(\000)p FA([)p Fv(f)1078 4768 y Ft(2)1067 4829 y(\003)1121 4804 y FA(])1148 4753 y Ft(\()p Fw(k)r Ft(\))1160 4828 y Fw(\026)1202 4809 y Fm(\003)1292 4804 y FA(of)g(eac)m(h)i(large) f(and)g(thin)g(con)m(tour)g(is)h(nonnegativ)m(e.)85 b(By)0 4924 y(assumption)31 b(\(2.36\))f(and)g(the)h(de\014nition)g(of)e(the)i (isop)s(erimetric)h(constan)m(t)f Fv(\037)2884 4939 y Ft(2)2923 4924 y FA(,)g(there)g(exists)h(a)0 5040 y(sequence)j(\000)465 5004 y Ft(2)465 5065 y Fw(n)512 5040 y FA(,)e Fv(n)28 b Fu(\025)g FA(1,)k(suc)m(h)i(that)785 5268 y(lim)760 5328 y Fw(n)p Fr(!1)961 5268 y Fu(k)p FA(\000)1072 5227 y Ft(2)1072 5292 y Fw(n)1119 5268 y Fu(k)27 b(!)g(1)98 b FA(and)f Fv(V)22 b FA(\(\000)1953 5227 y Ft(2)1953 5292 y Fw(n)2000 5268 y FA(\))2048 5199 y Fn(d)p Fm(\000)p Fl(1)p 2048 5212 V 2087 5253 a Fn(d)2200 5268 y Fu(\025)2418 5200 y(k)p FA(\000)2529 5164 y Ft(2)2529 5225 y Fw(n)2576 5200 y Fu(k)p 2315 5245 415 4 v 2315 5336 a FA(\(1)g(+)g Fv(")2568 5307 y Fr(0)2591 5336 y FA(\))p Fv(\037)2690 5302 y Fr(0)2690 5361 y Ft(2)2756 5268 y Fv(:)0 5523 y FA(Since)34 b Fv(x)310 5480 y Fw(k)398 5452 y Fn(d)p 359 5464 111 3 v 359 5506 a(d)p Fm(\000)p Fl(1)483 5523 y FA(e)526 5487 y Fr(\000)p Fw(x)658 5523 y FA(has)f(its)g(maxim)m(um)h (at)e Fv(x)c FA(=)g Fv(k)1838 5484 y Fw(d)p 1793 5500 127 4 v 1793 5558 a(d)p Fr(\000)p Ft(1)1929 5523 y FA(,)33 b(w)m(e)h(set)1301 5766 y Fv(k)1352 5781 y Fw(n)1426 5766 y FA(:=)1557 5625 y Fp(\026)1625 5698 y Fv(d)22 b Fu(\000)g FA(1)p 1625 5743 222 4 v 1710 5834 a Fv(d)1856 5766 y(\014)6 b Fu(k)p FA(\000)2028 5724 y Ft(2)2028 5790 y Fw(n)2074 5766 y Fu(k)2124 5625 y Fp(\027)2215 5766 y Fv(:)p Black Black eop %%Page: 28 28 28 27 bop Black 0 49 a Fq(28)p Black 0 241 a FA(F)-8 b(or)32 b(an)m(y)h Fv(n)p FA(,)g(\000)538 205 y Ft(2)538 265 y Fw(n)617 241 y FA(is)g(a)g(thin)g(and)f Fv(k)1243 256 y Fw(n)1290 241 y FA(-large)g(v)m(olume)i(con)m(tour,)f(since)h(b)m (y)h(\(2.37\))559 424 y Fv(\014)22 b FA(\(1)g Fu(\000)h FA(2)894 334 y Fu(p)p 976 334 97 4 v 976 424 a Fv(A)1049 395 y Fr(0)1073 424 y FA(\))p Fv(V)e FA(\(\000)1288 383 y Ft(2)1328 424 y FA(\))p Fv(R)1440 439 y Ft(2)1479 424 y FA(\()p Fv(V)h FA(\(\000)1695 383 y Ft(2)1734 424 y FA(\)\))28 b Fu(\025)g Fv(\014)22 b FA(\(1)g Fu(\000)g FA(2)2277 334 y Fu(p)p 2360 334 V 90 x Fv(A)2433 395 y Fr(0)2457 424 y FA(\))p Fv(V)f FA(\(\000)2672 383 y Ft(2)2711 424 y FA(\))2759 355 y Fn(d)p Fm(\000)p Fl(1)p 2759 368 111 3 v 2799 409 a Fn(d)2884 424 y Fv(\037)2945 383 y Fr(0)2945 448 y Ft(2)1838 660 y Fu(\025)1953 592 y FA(\(1)h Fu(\000)g FA(2)2210 508 y Fu(p)p 2293 508 97 4 v 84 x Fv(A)2366 564 y Fr(0)2389 592 y FA(\))p 1953 637 475 4 v 2071 728 a(1)g(+)g Fv(")2286 699 y Fr(0)2437 660 y Fv(\014)6 b Fu(k)p FA(\000)2609 619 y Ft(2)2609 684 y Fw(n)2656 660 y Fu(k)27 b(\025)h Fv(k)2889 675 y Fw(n)2953 660 y Fv(:)0 869 y FA(If)33 b(\003)27 b Fu(\033)h FA(\000)359 833 y Ft(2)359 894 y Fw(n)406 869 y FA(,)33 b(then)272 1090 y Fu(\000)p FA([)p Fv(u)432 1105 y Ft(\003)486 1090 y FA(\(\000)585 1049 y Ft(2)585 1115 y Fw(n)631 1090 y FA(\)])696 1039 y Ft(\()p Fw(k)760 1047 y Fn(n)803 1039 y Ft(\))708 1114 y Fw(\026)750 1095 y Fm(\003)862 1090 y Fu(\025)978 1023 y FA(1)21 b Fu(\000)i Fv(D)s FA(\()p Fv(k)s FA(\))p 978 1067 385 4 v 1121 1158 a(20)1388 1009 y Fp(\002)1430 1090 y Fv(\014)6 b FA(\001)p Fv(c)1614 1105 y Fr(\000)1673 1090 y Fv(V)21 b FA(\(\000)1850 1049 y Ft(2)1850 1115 y Fw(n)1897 1090 y FA(\))1935 1009 y Fp(\003)1977 1032 y Fw(k)2014 1040 y Fn(n)2077 1090 y Fv(\036)2135 1049 y Fr(\003)2135 1115 y Ft(\003)2187 1090 y FA(\(\000)2286 1049 y Ft(2)2286 1115 y Fw(n)2333 1090 y FA(\))862 1386 y Fu(\025)978 1319 y FA(1)g Fu(\000)i Fv(D)s FA(\()p Fv(k)s FA(\))p 978 1363 V 1121 1454 a(20)1372 1386 y(\001)1453 1345 y Fw(k)1490 1353 y Fn(n)1536 1386 y Fv(\014)1597 1342 y Fr(\000)1679 1313 y Fn(k)1712 1321 y(n)p 1662 1327 111 3 v 1662 1369 a(d)p Fm(\000)p Fl(1)1786 1276 y Fp(\020)2090 1319 y Fv(d)17 b(c)2210 1230 y Fn(d)p Fm(\000)p Fl(1)p 2209 1243 V 2248 1284 a Fn(d)2200 1340 y Fr(\000)p 1856 1363 712 4 v 1856 1454 a FA(\()p Fv(d)22 b Fu(\000)g FA(1\)\(1)g(+)g Fv(")2406 1426 y Fr(0)2429 1454 y FA(\))p Fv(\037)2528 1420 y Fr(0)2528 1479 y Ft(2)2577 1276 y Fp(\021)2648 1268 y Fn(dk)2713 1276 y(n)p 2647 1282 111 3 v 2647 1324 a(d)p Fm(\000)p Fl(1)2771 1386 y Fv(k)2837 1289 y Fn(k)2870 1297 y(n)2912 1289 y(d)p 2835 1303 V 2835 1345 a(d)p Fm(\000)p Fl(1)2822 1396 y Fw(n)2960 1386 y Fv(\036)3018 1345 y Fr(\003)3018 1411 y Ft(\003)3071 1386 y FA(\(\000)3170 1345 y Ft(2)3170 1411 y Fw(n)3216 1386 y FA(\))0 1608 y(and)33 b(\(see)g(\(2.31\)\))676 1824 y Fv(k)742 1727 y Fn(k)775 1735 y(n)817 1727 y(d)p 740 1741 V 740 1783 a(d)p Fm(\000)p Fl(1)727 1834 y Fw(n)864 1824 y Fv(\036)922 1783 y Fr(\003)922 1849 y Ft(\003)975 1824 y FA(\(\000)1074 1783 y Ft(2)1074 1849 y Fw(n)1121 1824 y FA(\))28 b Fu(\025)g Fv(k)1358 1727 y Fn(k)1391 1735 y(n)1433 1727 y(d)p 1356 1741 V 1356 1783 a(d)p Fm(\000)p Fl(1)1343 1834 y Fw(n)1497 1824 y FA(exp)1663 1743 y Fp(\002)1726 1824 y Fu(\000)1826 1743 y Fp(\000)1872 1824 y Fv(k)1923 1839 y Fw(n)2065 1757 y Fv(d)p 1979 1801 222 4 v 1979 1892 a(d)22 b Fu(\000)h FA(1)2233 1824 y(+)f(1)2380 1743 y Fp(\001)2425 1824 y FA(\(1)g(+)g Fv(C)2702 1839 y Ft(9)2741 1824 y Fv(\016)t FA(\))2826 1743 y Fp(\003)1187 2088 y Fu(\030)28 b Fv(k)1343 2103 y Fw(n)1390 2088 y FA(!)1466 2017 y Fn(d)p 1427 2029 111 3 v 1427 2070 a(d)p Fm(\000)p Fl(1)1551 2088 y FA(e)1594 2044 y Fr(\000)p Fw(C)1699 2053 y Fl(9)1734 2044 y Fw(\016)1817 2017 y Fn(d)p 1778 2029 V 1778 2070 a(d)p Fm(\000)p Fl(1)1898 2044 y Fw(k)1935 2052 y Fn(n)2083 2020 y FA(e)2126 1984 y Fr(\000)p Ft(1)p Fr(\000)p Fw(C)2321 1993 y Fl(9)2356 1984 y Fw(\016)p 1992 2065 495 4 v 1992 2188 a FA(\(2)p Fv(\031)t(k)2189 2203 y Fw(n)2235 2188 y FA(\))2361 2111 y Fn(d)p 2283 2123 189 3 v 2283 2165 a Fl(2\()p Fn(d)p Fm(\000)p Fl(1\))2512 2088 y Fv(:)0 2342 y FA(By)39 b(the)g(c)m(hoice)h (\(2.37\))e(of)g(the)g(parameters)i Fv(\022)i FA(and)c Fv(")2046 2306 y Fr(0)2069 2342 y FA(,)i(if)e Fv(\016)43 b FA(is)c(small)g(enough,)i(i.e.)61 b Fv(\014)44 b FA(large)0 2458 y(enough,)33 b(then)1266 2614 y(e)1320 2550 y Fl(1)p 1320 2562 33 3 v 1320 2603 a Fn(d)p 1055 2659 523 4 v 1055 2763 a Fv(\022)s FA(\(1)22 b Fu(\000)g FA(2)1360 2678 y Fu(p)p 1443 2678 97 4 v 85 x Fv(A)1516 2734 y Fr(0)1539 2763 y FA(\))1615 2681 y Fv(<)1814 2614 y(d)p 1728 2659 222 4 v 1728 2750 a(d)g Fu(\000)h FA(1)2017 2614 y Fv(c)2069 2526 y Fn(d)p Fm(\000)p Fl(1)p 2069 2539 111 3 v 2108 2580 a Fn(d)2059 2636 y Fr(\000)p 1986 2659 238 4 v 1986 2750 a FA(1)f(+)g Fv(")2201 2721 y Fr(0)2234 2681 y FA(e)2277 2640 y Fr(\000)p Fw(C)2382 2649 y Fl(9)2417 2640 y Fw(\016)2472 2681 y Fv(:)0 2896 y FA(Hence)30 b(the)f(con)m(tributions)i(of)d(the)h(small)h(and)f(fat)f (con)m(tours)i(are)f(negligible)h(for)e(large)h Fv(k)3305 2911 y Fw(n)3380 2896 y FA(\(see)0 3013 y(\(2.39\))i(and)g(\(2.40\))o (\).)43 b(Let)32 b(\003\()p Fv(L)1172 3028 y Fw(n)1219 3013 y FA(\))f(b)s(e)g(a)g(b)s(o)m(x)h(whic)m(h)h(con)m(tains)f(at)f (least)h Fu(j)p FA(\003\()p Fv(L)2889 3028 y Fw(n)2936 3013 y FA(\))p Fu(j)p Fv(=)p FA(4)e(translates)0 3129 y(of)d(\000)167 3093 y Ft(2)167 3153 y Fw(n)214 3129 y FA(.)42 b(F)-8 b(or)27 b(an)m(y)h(\003)g Fu(\033)g FA(\003\()p Fv(L)1005 3144 y Fw(n)1052 3129 y FA(\),)h(if)e Fv(k)1281 3144 y Fw(n)1356 3129 y FA(and)h Fv(\014)33 b FA(are)28 b(large)f(enough,)j(then)e(there)g(exists)i(a)d(constan)m (t)0 3245 y Fv(C)70 3260 y Ft(14)172 3245 y Fv(>)h FA(0,)k(indep)s (enden)m(t)j(of)d Fv(\014)6 b FA(,)32 b Fv(k)1219 3260 y Fw(n)1298 3245 y FA(and)h(\003)27 b Fu(\033)i FA(\003\()p Fv(L)1861 3260 y Fw(n)1908 3245 y FA(\),)j(suc)m(h)i(that)911 3440 y Fu(\000)p FA([)p Fv(f)1074 3399 y Ft(2)1063 3465 y(\003)1117 3440 y FA(])1144 3389 y Ft(\()p Fw(k)1208 3397 y Fn(n)1250 3389 y Ft(\))1155 3464 y Fw(\026)1197 3445 y Fm(\003)1310 3440 y Fu(\025)28 b Fv(C)1492 3397 y Fw(k)1529 3405 y Fn(n)1485 3464 y Ft(14)1592 3440 y Fv(k)1643 3455 y Fw(n)1689 3440 y FA(!)1766 3369 y Fn(d)p 1726 3381 111 3 v 1726 3422 a(d)p Fm(\000)p Fl(1)1868 3440 y FA(\001)1949 3399 y Fw(k)1986 3407 y Fn(n)2032 3440 y Fv(\014)2093 3396 y Fr(\000)2175 3367 y Fn(k)2208 3375 y(n)p 2158 3381 V 2158 3422 a(d)p Fm(\000)p Fl(1)2299 3440 y Fv(\037)2360 3399 y Fr(0)2360 3465 y Ft(2)2399 3388 y Fr(\000)2466 3359 y Fn(dk)2531 3367 y(n)p 2464 3373 V 2464 3414 a(d)p Fm(\000)p Fl(1)2605 3440 y Fv(:)p 3473 3651 4 66 v 3477 3588 59 4 v 3477 3651 V 3535 3651 4 66 v 0 3885 a FA(2.5.)56 b Fz(Lo)m(w)m(er)31 b(b)s(ounds)h(of)f(the)g (deriv)-6 b(ativ)m(es)31 b(of)g(the)g(free)g(energy)h(at)f(in\014nite)g (v)m(olume.)0 4001 y FA(W)-8 b(e)42 b(sho)m(w)g(that)g(w)m(e)g(can)g (in)m(terc)m(hange)h(the)f(thermo)s(dynamic)h(limit)f(and)g(the)g(op)s (eration)f(of)0 4117 y(taking)47 b(the)g(deriv)-5 b(ativ)m(es,)53 b(and)47 b(that)f(the)h(T)-8 b(a)m(ylor)48 b(series,)k(whic)m(h)c (exists,)53 b(has)47 b(a)f(radius)i(of)0 4233 y(con)m(v)m(ergence)31 b(equal)d(to)g(0.)42 b(These)30 b(statemen)m(ts)g(are)e(a)f (consequence)k(of)d(Lemmas)h(2.6)e(and)i(2.7.)p Black 0 4411 a Fz(Lemma)39 b(2.6.)p Black 49 w Fs(If)51 b Fv(\014)56 b Fs(is)51 b(su\016ciently)g(lar)-5 b(ge,)55 b(then)c(for)f(any)h Fv(k)61 b Fu(2)d Fx(N)51 b Fs(ther)-5 b(e)51 b(exists)g Fv(M)3367 4426 y Fw(k)3467 4411 y FA(=)0 4527 y Fv(M)94 4542 y Fw(k)137 4527 y FA(\()p Fv(\014)6 b FA(\))27 b Fv(<)h Fu(1)p Fs(,)34 b(such)h(that)g(for)g(al)5 b(l)34 b Fv(t)28 b Fu(2)g FA(\()p Fv(\026)1539 4491 y Fr(\003)1600 4527 y Fu(\000)23 b Fv(";)17 b(\026)1849 4491 y Fr(\003)1887 4527 y FA(])35 b Fs(and)g(for)f(al)5 b(l)35 b(\014nite)f FA(\003)p Fs(,)1455 4622 y Fp(\014)1455 4682 y(\014)1488 4707 y FA([)p Fv(f)1574 4666 y Ft(2)1563 4731 y(\003)1617 4707 y FA(])1644 4656 y Ft(\()p Fw(k)r Ft(\))1656 4729 y Fw(t)1741 4622 y Fp(\014)1741 4682 y(\014)1802 4707 y Fu(\024)28 b Fv(M)2001 4722 y Fw(k)2061 4707 y Fv(:)p Black 0 4938 a Fs(Pr)-5 b(o)g(of.)p Black 41 w Fv(!)t FA(\(\000)462 4902 y Ft(2)501 4938 y FA(\))48 b(is)g(analytic)g(and)g Fv(\034)1333 4953 y Ft(1)1373 4938 y FA(\()p Fv(\014)6 b(;)17 b(\022)1564 4902 y Fr(0)1587 4938 y FA(\)-stable)48 b(on)g(a)f(disc)i(of)f(radius)g Fv(\022)s FA(\001)2977 4902 y Fr(\000)p Ft(1)3072 4938 y Fv(R)3146 4953 y Ft(2)3186 4938 y FA(\()p Fv(V)21 b FA(\(\000)3401 4902 y Ft(2)3440 4938 y FA(\)\).)0 5054 y(F)-8 b(rom)32 b(Cauc)m(h)m(y)j(form)m(ula)1005 5149 y Fp(\014)1005 5209 y(\014)1039 5233 y FA([)p Fv(u)1122 5248 y Ft(\003)1174 5233 y FA(\(\000)1273 5192 y Ft(2)1313 5233 y FA(\)])1378 5183 y Ft(\()p Fw(k)r Ft(\))1389 5256 y Fw(t)1475 5149 y Fp(\014)1475 5209 y(\014)1536 5233 y Fu(\024)28 b Fv(k)s FA(!)17 b Fv(C)1816 5192 y Fw(k)1809 5258 y Ft(15)1883 5233 y Fu(j)p FA(\000)1972 5192 y Ft(2)2012 5233 y Fu(j)2087 5163 y Fn(k)p 2049 5175 111 3 v 2049 5216 a(d)p Fm(\000)p Fl(1)2190 5233 y FA(e)2233 5192 y Fr(\000)p Fw(\014)s(\024)p Fr(j)p Ft(\000)2436 5169 y Fl(2)2471 5192 y Fr(j)2511 5233 y Fv(;)0 5396 y FA(for)32 b(some)h(constan)m(ts)h Fv(C)895 5411 y Ft(15)1002 5396 y FA(and)f Fv(\024)28 b(>)f FA(0.)44 b(Therefore)237 5570 y Fu(j)p FA(\003)p Fu(j)17 b Fv(\014)454 5486 y Fp(\014)454 5545 y(\014)487 5570 y FA([)p Fv(f)573 5529 y Ft(2)562 5595 y(\003)615 5570 y FA(])642 5519 y Ft(\()p Fw(k)r Ft(\))654 5593 y Fw(t)740 5486 y Fp(\014)740 5545 y(\014)801 5570 y Fu(\024)925 5476 y Fp(X)906 5693 y Ft(\000)950 5674 y Fl(2)984 5693 y Fr(\032)p Ft(\003)1105 5486 y Fp(\014)1105 5545 y(\014)1138 5570 y FA([)p Fv(u)1221 5585 y Ft(\003)1274 5570 y FA(\(\000)1373 5529 y Ft(2)1412 5570 y FA(\)])1477 5519 y Ft(\()p Fw(k)r Ft(\))1489 5593 y Fw(t)1575 5486 y Fp(\014)1575 5545 y(\014)1635 5570 y Fu(\024)29 b Fv(k)s FA(!)17 b Fv(C)1916 5529 y Fw(k)1909 5595 y Ft(15)2019 5476 y Fp(X)2000 5693 y Ft(\000)2044 5674 y Fl(2)2078 5693 y Fr(\032)p Ft(\003)2199 5570 y Fu(j)p FA(\000)2288 5529 y Ft(2)2327 5570 y Fu(j)2403 5500 y Fn(k)p 2364 5512 V 2364 5553 a(d)p Fm(\000)p Fl(1)2505 5570 y FA(e)2548 5529 y Fr(\000)p Fw(\014)s(\024)p Fr(j)p Ft(\000)2751 5506 y Fl(2)2786 5529 y Fr(j)2837 5570 y Fu(\021)28 b(j)p FA(\003)p Fu(j)p Fv(\014)6 b(M)3221 5585 y Fw(k)3280 5570 y Fv(:)p 3473 5827 4 66 v 3477 5765 59 4 v 3477 5827 V 3535 5827 4 66 v Black Black eop %%Page: 29 29 29 28 bop Black 3467 49 a Fq(29)p Black Black 0 241 a Fz(Lemma)39 b(2.7.)p Black 1264 357 a FA(lim)1237 419 y Fw(L)p Fr(!1)1426 357 y FA([)p Fv(f)1512 316 y Ft(2)1501 382 y(\003\()p Fw(L)p Ft(\))1657 357 y FA(])1684 306 y Ft(\()p Fw(k)r Ft(\))1696 381 y Fw(\026)1738 362 y Fm(\003)1809 357 y FA(=)30 b(lim)1913 420 y Fw(t)p Fr(")p Fw(\026)2015 401 y Fm(\003)2052 357 y FA([)p Fv(f)11 b FA(])2165 306 y Ft(\()p Fw(k)r Ft(\))2177 379 y Fw(t)2279 357 y Fv(:)p Black 0 617 a Fs(Pr)-5 b(o)g(of.)p Black 41 w FA(W)d(e)33 b(compute)h(the)f(\014rst)g(deriv)-5 b(ativ)m(e)34 b(at)e(the)h(origin.)43 b(Let)33 b Fv(\021)f(>)27 b FA(0.)788 828 y Fv(A)p FA(\()p Fv(\021)t FA(\))h(:)g(=)1185 760 y Fv(f)11 b FA(\()p Fv(\026)1341 724 y Fr(\003)1380 760 y FA(\))22 b Fu(\000)h Fv(f)11 b FA(\()p Fv(\026)1696 724 y Fr(\003)1757 760 y Fu(\000)22 b Fv(\021)t FA(\))p 1185 805 761 4 v 1540 896 a Fv(\021)1072 1117 y FA(=)54 b(lim)1175 1179 y Fw(L)p Fr(!1)1391 1032 y Fv(f)1450 995 y Ft(2)1439 1061 y(\003\()p Fw(L)p Ft(\))1595 1032 y FA(\()p Fv(\026)1692 995 y Fr(\003)1731 1032 y FA(\))22 b Fu(\000)h Fv(f)1950 995 y Ft(2)1939 1061 y(\003\()p Fw(L)p Ft(\))2095 1032 y FA(\()p Fv(\026)2192 995 y Fr(\003)2253 1032 y Fu(\000)g Fv(\021)t FA(\))p 1391 1094 1052 4 v 1891 1185 a Fv(\021)1072 1429 y FA(=)54 b(lim)1175 1491 y Fw(L)p Fr(!1)1391 1343 y FA([)p Fv(f)1477 1306 y Ft(2)1466 1372 y(\003\()p Fw(L)p Ft(\))1622 1343 y FA(])1649 1292 y Ft(\(1\))1661 1366 y Fw(\026)1703 1347 y Fm(\003)1760 1343 y Fv(\021)26 b FA(+)1952 1303 y Ft(1)p 1942 1320 55 4 v 1942 1377 a(2!)2007 1343 y FA([)p Fv(f)2093 1306 y Ft(2)2082 1372 y(\003\()p Fw(L)p Ft(\))2238 1343 y FA(])2265 1292 y Ft(\(2\))2265 1374 y Fw(\026)2307 1355 y Fm(\003)2343 1374 y Fr(\000)p Fw(x)2438 1385 y Fn(L)2484 1374 y Ft(\()p Fw(\021)r Ft(\))2598 1343 y Fv(\021)2650 1306 y Ft(2)p 1391 1406 1299 4 v 2014 1498 a Fv(\021)1072 1686 y FA(=)54 b(lim)1175 1748 y Fw(L)p Fr(!1)1381 1575 y Fp(\020)1440 1686 y FA([)p Fv(f)1526 1645 y Ft(2)1515 1711 y(\003\()p Fw(L)p Ft(\))1672 1686 y FA(])1699 1635 y Ft(\(1\))1710 1710 y Fw(\026)1752 1691 y Fm(\003)1815 1686 y FA(+)1937 1619 y(1)p 1923 1663 76 4 v 1923 1754 a(2!)2009 1686 y([)p Fv(f)2095 1645 y Ft(2)2084 1711 y(\003\()p Fw(L)p Ft(\))2240 1686 y FA(])2267 1635 y Ft(\(2\))2267 1718 y Fw(\026)2309 1699 y Fm(\003)2346 1718 y Fr(\000)p Fw(x)2441 1729 y Fn(L)2487 1718 y Ft(\()p Fw(\021)r Ft(\))2600 1686 y Fv(\021)2652 1575 y Fp(\021)2728 1686 y Fv(:)0 1908 y FA(By)43 b(Lemma)g(2.6,)i Fu(j)p FA([)p Fv(f)832 1872 y Ft(2)821 1937 y(\003\()p Fw(L)p Ft(\))977 1908 y FA(])1004 1857 y Ft(\(2\))1004 1939 y Fw(\026)1046 1920 y Fm(\003)1082 1939 y Fr(\000)p Fw(x)1177 1950 y Fn(L)1223 1939 y Ft(\()p Fw(\021)r Ft(\))1320 1908 y Fu(j)g(\024)g Fv(M)1609 1923 y Ft(2)1649 1908 y FA(.)73 b(Therefore)44 b Fu(f)p Fv(A)p FA(\()p Fv(\021)t FA(\))p Fu(g)2502 1923 y Fw(\021)2585 1908 y FA(is)g(a)e(Cauc)m(h)m(y)i (sequence.)0 2033 y(Hence)34 b(the)f(follo)m(wing)g(limits)g(exist,)539 2244 y([)p Fv(f)11 b FA(])652 2193 y Ft(\(1\))664 2267 y Fw(\026)706 2249 y Fm(\003)774 2244 y FA(=)27 b(lim)891 2306 y Fw(\021)r Fr(#)p Ft(0)1039 2176 y Fv(f)11 b FA(\()p Fv(\026)1195 2140 y Fr(\003)1234 2176 y FA(\))22 b Fu(\000)h Fv(f)11 b FA(\()p Fv(\026)1550 2140 y Fr(\003)1611 2176 y Fu(\000)22 b Fv(\021)t FA(\))p 1039 2221 761 4 v 1394 2312 a Fv(\021)1838 2244 y FA(=)29 b(lim)1941 2306 y Fw(t)p Fr(")p Fw(\026)2043 2288 y Fm(\003)2081 2244 y FA([)p Fv(f)11 b FA(])2194 2193 y Ft(\(1\))2205 2266 y Fw(t)2315 2244 y FA(=)55 b(lim)2419 2306 y Fw(L)p Fr(!1)2608 2244 y FA([)p Fv(f)2694 2203 y Ft(2)2683 2268 y(\003\()p Fw(L)p Ft(\))2839 2244 y FA(])2866 2193 y Ft(\(1\))2878 2267 y Fw(\026)2920 2249 y Fm(\003)2977 2244 y Fv(:)0 2450 y FA(Same)33 b(pro)s(of)f(for)g(the)h(deriv)-5 b(ativ)m(es)34 b(of)f(an)m(y)g(order.)p 3473 2613 4 66 v 3477 2551 59 4 v 3477 2613 V 3535 2613 4 66 v 1481 2858 a Fy(References)p Black 125 3016 a Fj([1])p Black 41 w(Fisher)20 b(M.E.,)j(The)e(Theory)f (of)h(Condensation)f(and)h(the)h(Critical)e(P)n(oin)n(t,)i Fb(Physics)29 b Fa(3)p Fj(,)23 b(255-283)18 b(\(1967\).)p Black 125 3115 a([2])p Black 41 w(Isak)n(o)n(v)26 b(S.N.,)k (Nonanalytic)e(F)-7 b(eatures)29 b(of)f(the)i(First)e(Order)g(Phase)g (T)-7 b(ransition)28 b(in)h(the)g(Ising)f(Mo)r(del,)254 3215 y Fb(Commun.)i(Math.)h(Phys.)e Fa(95)p Fj(,)e(427-443)e(\(1984\).) p Black 125 3315 a([3])p Black 41 w(Isak)n(o)n(v)c(S.N.,)k(Phase)e (Diagrams)f(and)i(Singularit)n(y)f(at)g(the)h(P)n(oin)n(t)f(of)h(a)f (Phase)g(T)-7 b(ransition)23 b(of)g(the)i(First)254 3414 y(Kind)i(in)h(Lattice)g(Gas)f(Mo)r(dels,)g Fb(T)-6 b(e)l(or.)31 b(Mat.)f(Fiz.)f Fa(71)p Fj(,)e(426-440)e(\(1987\).)p Black 125 3514 a([4])p Black 41 w(Kunz)j(H.,)h(Souillard)f(B.,)h(Essen) n(tial)e(Singularit)n(y)g(and)i(Asymptotic)g(Beha)n(vior)d(of)j (Cluster)f(Size)g(Dis-)254 3613 y(tribution)f(in)h(P)n(ercolation)e (Problems,)g Fb(J.)k(Stat.)f(Phys.)g Fa(19)p Fj(,)e(77-106)e(\(1978\).) p Black 125 3713 a([5])p Black 41 w(P\014ster)f(C.-E.,)h(Large)e (Deviations)i(and)g(Phase)f(Separation)g(in)h(the)g(Tw)n(o)g (Dimensional)g(Ising)f(Mo)r(del,)254 3813 y Fb(Helv.)30 b(Phys.)h(A)l(cta)j Fa(64)p Fj(,)27 b(953-1054)d(\(1991\).)p Black 125 3912 a([6])p Black 41 w(Sinai)39 b(Y)-7 b(a.)40 b(G.,)j(Theory)c(of)h(Phase)f(T)-7 b(ransitions:)60 b(Rigorous)38 b(Results,)43 b(P)n(ergamon)37 b(Press,)k(Oxford)254 4012 y(\(1982\).)p Black 125 4112 a([7])p Black 41 w(Zahradnik)31 b(M.,)i(An)g(Alternate)g(V)-7 b(ersion)31 b(of)h(Pirogo)n(v-Sinai)e (Theory)-7 b(,)32 b Fb(Commun.)i(Math.)i(Phys.)e Fa(93)p Fj(,)254 4211 y(559-581)24 b(\(1984\).)p Black Black eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0212020727806--