Content-Type: multipart/mixed; boundary="-------------0205130553254" This is a multi-part message in MIME format. ---------------0205130553254 Content-Type: text/plain; name="02-225.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-225.comments" Revised (in full accordance with comments from a referee of a regular journal) version of the mp_arc preprint 01-383. ---------------0205130553254 Content-Type: text/plain; name="02-225.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-225.keywords" renormalization group, self-avoiding walk, fractals, Sierpinski gasket ---------------0205130553254 Content-Type: application/postscript; name="revMPARC.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="revMPARC.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.78 p1.4c Copyright 1996-99 ASCII Corp.(www-ptex@ascii.co.jp) %%based on dvipsk 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: revMPARC.dvi %%Pages: 37 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMMI12 CMR12 CMSY8 CMBX10 CMR10 CMBX9 CMR9 CMMI9 %%+ CMSY6 CMR8 CMBX12 MSBM10 CMMI7 CMMI10 CMSY7 CMMI5 MSAM10 CMSY10 CMR7 %%+ CMTI10 CMEX10 CMR5 CMSY5 LASY10 MSBM7 MSAM7 %%EndComments %DVIPSCommandLine: dvipsk -D600 -t a4size -P dl revMPARC.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.05.13:1924 %%BeginProcSet: texc.pro %! 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%%BeginFont: MSAM7 %!PS-AdobeFont-1.1: MSAM7 2.1 %%CreationDate: 1992 Oct 17 08:30:29 % 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 (MSAM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM7 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 53 /lessdblequal put dup 61 /greaterdblequal put readonly def /FontBBox{0 -576 1569 1076}readonly def /UniqueID 5032012 def currentdict end currentfile eexec bd2bc8312da7a34353963e088820ad4a2494871a5c866f8f44672a854e9e8a27 6fd3b21943b2867c3b30802d749780e1ebc10edad1e60864715c2b0d510d631b b3ea24b6ff392735adc334e1acc208f5ccbeb37a64e819267c12c01f65ce2874 139c00fa924304665e8e3b6caa150bdfd78b46291153c060657197929952dd84 82addc6c49f5d25d846cf29a9c369fdc1ce1c946d51ee4e7d9ccbb5bdfe9c7c3 a6580882b53120964199c2746ef2c9ea3ced8ce05368fe658f58459c2a6886f1 7c789209b30f9b93e74f5e019ac17509073328a8682797d746830f479a8e37cf ef4dd3d0a969af6601ed88c3f5da2b0415c21021eb3087080a055b3c606a43b6 97e4bb7b3990c55c4be557a4be0894a58e95c57ee60f448ec6f015ce7685030c d95b811e80f5dab0906d041dc606245ba655b7afb3416127adfca35b103b24e9 06ca962e00d3a07f24e9376a8cee5718c202eae8a11a4a53f41cad7b34fcb2a4 11b47a0de489195ee9ea72a2718bc6ea2b7074bc3a3ded23916fc900d0f1d17f 67359d48512d59703e5aea38c7cf6baf0326f9ece5db36d6864e7b55e36d0749 cf9bea28196bcc01ab02a3be5d1fdc611d1bc5bf26fceaf7472309b1f719d14e 7c313341089d6123d2647668f5f341150b5c4be4256bf275fd7af002614c84f5 8419b0d0a919f7ec918c9560ba8791f5689c72ef8ef24640989707df578af06b 79a3c5ef18d734209a3274c8242ee11d88885c3ea85c05609481d285d414210a 93ce5b8a05e4c9843b04b2c2da77035f113906e55b088c32b1eb27a45078e91d 994bee0b677775ec9cc4c8e285d39bdd188614c75929665b47210bf4bcf2bd27 38006d36aa619e4b023ce04c02c73305f72e2a74aac6789d7da24b495be899f5 888866192df45563080f7c64839aa1671946e1c7149c9ee6454c783e4edcf813 2a6166378f3b8b5d0b5c3a45aa38cccaf71bb5ce98e672daaf10b25cc169afd6 c5ddde5817983aa96909c1312dd6bd5e6faaabba55a9569259bb567e1471322c 39d916a7d96deb7cbc8e67d2d0bcebfadaaacf5a6b47e60f850121d48e3dd352 eb25c98058b68c3a64ace274aa38ecdbf9d97642a3c5f8f421a6780a70b780b6 1de05255c654b662cd1fd2e8b98382732594698ba1234d2011b0d3f91f9dc811 82045d7c1add6bacf9eab3a5b55d28d1f915f44a283090ca21902f34cbe47d60 64d466f0ab176baa369f844c34a60ff5e4139355b4db7182e603381f88a6e248 617d8f3d79c0dc817a7e45d799d4f8f58a53d84fb71f75f651350e9aaeaed3f9 971c26f60cdd3b3154fed96095ae1fe9135280cc819c3fca7e4a3d46c5c80ee6 c8b85afbb7a63c5872e55859e5fdfc4ba745aa1c1630255c7d02de35ff578db3 788f0ea642f1df5c3fca7414ca35fa7d4c8d181496270a4412c2f33ddf932005 ff02570ceef16aa39fdbf358ef9d11e7c55554bf0cbbd51e96ef0e1683e9d6ac 313ad0387042fc491ca42818bbc2011f2a56a9793a78ebbcb3e57eb74d0bd147 c6f6b11b670defc778978128941643025e15e364772a18479185186914f22973 4260cbe04d44ef222d8560d4a1bb26aa558470d8515761e06f1dbb4b5fce96d7 877ee8c09a4aa4b658d3fc327009c4a3b718b095fe125d43ae88651d8ef9fc28 0efa8a1d0b1146fecfbfce2bdce8cacb3768d27d27b45dfbb31cd41540d9ada1 c8266d006c5f8bfcc214fcc3b875b185c21404b433991f7b5f80ffe97a94c218 ed2826a55c16763cf3d00a96eec13eac28dca261076b94afcd1f4f542ea3c2ba 8b6529ecc1bc6593a5d3c70fa61b05e6ffd579fb67bc9bebb78d9a8722bc1d12 17fc34e9300df057ed718e221722dfa0e5f134da23c93786c127ca33582bf599 3992f26ad3a8 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 82 /R put dup 90 /Z put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 def currentdict end currentfile eexec b5e2457af4a6caaa4536453c498a618ecdce7b5f6f2d5944265309f833bd3d50 3df8bd14c071f404d2177500681a801856cb8c9c9cf0112c9bd17ccba9ab8944 1759a320b2078df0fafdd3e357994bd81740d1a7522ddfafb75d58f5def046f7 084d10fa2f702e570b09a7073d70efe37b48c355a7089772238b7b4538c4912f cf6703a14fa416057ea2e1b94455913dd271b42846ee10d78270a8118e7e4bcb 8deb048bb280c240d59409d47046c890e5f79e5dd4b0be65a28a77aeb53dd33e ac506a6da87bb55e329f433627e06c3c94b078838b908538d93f888e504e25e0 3a8ccf7d2c02122ab3d485525564295390d1154e15fee35edff4eb016fadd32c 26d1b4c09de87943780954cebc4965aa890fe1f80b03eb07103e13d468245c7b c26455f229e8859585c6eeafd9474acdb27290fd62950dc52f893c99a2a614d4 2a7e3618266f3fb0c68a01a30373e048385ffe7dd5844e208a55d1a9263ab2db cb9b19038724f95af00349840157be45cb337871b55a304b87ade64b10ff0eba 21aa0eaeef74aa01f049303a8b25de0acc26bb3a33fcf267f479a2a73f7dece3 2aa23083687322b417fe86e8f5afc2136d045ad6b392364811a17630a94ff097 038eedeedb8da4ebf8fa5a96ce27a2ebf771d14f4a11fb1f9f48cdd207209934 885ef9118463552984b0fd3cf15d55de967fbb051549dd65f60dfe07f299690e 58252f032bca78935b7d2df852ed688bd13dd171b91cf7bd46d5433aef9f50f1 41a2517c4912b837793f84ddcf1116f70f7e9b1e6ec752dbc9183d4791d92602 397bafe29ac6f879ceb72c673c8a1aa0135d6f92d0db7791f9a2f1a5b55347e0 3e63723efd258754f5f4a9ad35873a98d54b6c54b17a39892dc1d38660597448 fdba9258493a0db0d560c01d49f2d889a99eaa8a5b5919cbd33f6b3d97bf27b3 e0b94b48d5d21d6c29aee4faca71bd775bb9defa37ff627f1e4926f10d59313e 77d81a511027ef9a8ed5b7fa9b5bd67758cd13bab6ac18411c974a65d0c90085 67bf5860d0930d7f43e557ee45e76ecb2806d21f4732bb282ac8b591f2b8775d 4b3ce2a37ebb371cb74a3cc513bdd70a0115de4760c21f034f5080d769f7ff83 554af524184702a02f6172e26f9b180f2de3cccd3ed216085ce7348f35fa4f29 b4c4140f140f1ad1e334c1fb6faffe0c0762b63994a9a169dffc2e9269b8051f 0faf67d7a974b6116623c357030becd3cc0c8e36acf741f64e172f71b782eb39 a4dc049ea1a1fdb3cfabd30941caf1b0bb5b4b8531456062df25cb115b547462 d18a1c756af1aeab302c58a72e63f9a4cf74772025b0347a905351e53f13b2df 0eaec2a56b4731d85573578108a244db23a2670010492b3d18340cd368bf114b b47ef637da9946f2c666011736b0d17dfbb358d63cda81d4390c92e60f5a23c5 ff9b8bcd9c21ec7ca83c3cd4cf1f24c6a267f830d3088b3a8cc4188ab4e148a5 0494ad203f83d05895ea054c36b348c8fc74e7bc0e1e0a9cfb71faf2edd626b3 95593d36eda154f393694b1a5b63a593e24b50bab02fc91db9ead80cabc9ec4f 91b68dc9087600dfd4b28f71ea264021bf83a392973764380057d5d4db096e1f 56af3403c9de4334082ed3afe942197d67c64b63de9fad01d7447f2186fe767b 8e9d6c769f384b65e7926d8e1104a5715724c1224a5bbfdbc0a34d2687e73207 4dbab7d9eba92667ad8609e7e6868ed5ddcc002087b772262c4d38fda852482c 4534ccd2827af975554b15036b525d95a12eb6b5215080f4a563effcb001aa4c ed12fbd933cbc662a08e69a4de5d77b575a0fd4a51e78ace687906355673d4aa e7b59cc92ec5f785a6e035f21b2a9ff3c697251e3fed777ede4dbc9eeb0ab1e0 99d8e56d2b589d84c75aa5ae310bfd7c081fd896a96a60697ec15011e1b6cab4 e0da744f109820a5e16da164938965c638f1383a68961623dea4ae78f9a66312 23f4f84a046da695eb214fe79294029b177948904c7fd46d858b53ca4cb08928 0b9498b837d2e3576c5d5138c5008995c2ad8e59379df6c1efa16cc48608d970 3091f3316e 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: LASY10 %!PS-AdobeFont-1.1: LASY10 1.00 %%CreationDate: 1992 Oct 23 20:19:17 % 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 (LASY10) readonly def /FamilyName (LaTeX) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /LASY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 50 /a50 put dup 51 /a51 put readonly def /FontBBox{-19 -192 944 683}readonly def /UniqueID 5011949 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192e3f6620e5383410 9c674a944e5ff85290c767088ede1e6835bf2aaf7a0e947193d98442808fc3dd 148069ab44a71076493a05cbf836aeb3dd5c78829d1482b3c40c2f4cf5fdd0be c3c3c1e08cd5d21db9363551e5ea23a95a928a532e09d2b59879067059a5974e 4912c690ee71858bbf009a75913237a9ab9eb9d8d683ea3fe066fb2ecd3df7b5 c7cb950bdefac7931a3501dd1dd59c10d45f8f91ff07c7c41361e1fafb605c38 0431bdf3b1e44d21d351cf80d8dc6fbb5131fb8ad1a0e9a9a5fdea6432ec1d59 bee32029965fb5c141768d6f2c624b2dae5c9ede58cb4f522099a96749094f15 07e9a6b977ebc86c877b8c4a0758b69dca966be63e606c8efde1f65b295997e5 1205bef471d087f51dc9ff816ef9f6a4a23226f9305feba8da8baa909e515d85 b60901cd8e958e0c94d2a64c1d9510e66d37d18fdf08c5408032c63ae7a667bd 1ca377a9eab6b0bfd434ccc71b779a6cbab2183016bddbc05f171719794bf19d 0118b9e708296d876e7ebd011b0d49e51b56e10bae4b5f2d07b16e0c83df55ab 42cf85da87350d752cd4522682b02f82c49d08a2e0d47d58374443a63831e432 c48f945cc5e42d807ba86cf3e7b3b09f636c17360c5c9bd7717f7e4909e8bb4a 7efb7a580cc9357613467b64fd0161dc026602cc382bdc1779ce8ebc0587f172 654aff1ed908166f3755ccdc64a183e50dc8ff8b9d8ea9ab556615ce390b4660 1749955a3ac5f7c061d6b9948677ca68373ee7c6738abdd0856c90a91d904c91 914f3ba09a3a8b54fa319b1ba9bbd7441b69f0431716a4156816329b7e09b57f 495591d0d9c9049cd115e5c135fd9e340626baed2c37beabeb735bc14c38b852 9ebbb0a729bdbb359ed4ce7245d6d8add06d35a8177fd5ce1f05b7e64cdd2ae6 9c75b7f88e025c13440a564e79937f34792696e3125329a1c13e6aa86bc94c2e 6e5e152b639c9d2315cffd4f9f01b99aaaefddefafb475bf665e6a414c15c704 a1983104a46d4b430b75e2c4a5aaf1cb81fda5082141ffeec1fee27d5f0a5c65 3c1c3832aa680dbf18530bf1e023ad12b6864e8fc8c23397466a162b2d8933b3 c4bbc60032e215fa094b4cc3cbe328352a42aafbfcbc4bbac391e5574bfabe29 7b2325cc1f23c41583fbaa1ad3bea5f6b8b9c3ac3be4779969c41d9614030f51 3e8adf2e0af205b2779d14359fa32c026545914ad1213d984d28dad7c64c5285 5f68ecaf9c35d8cc74c941c63cc2a511c8f6c7840914f13f7a83d74908bd29a4 1f3d5fe916b62ca9bcd7e4c1fc81a761aded9a8d83195614644905d7fa1dd2c7 c9e2fa995182fa6f3494db799525e3b656450eaceaddae50c747dbb656650334 b07fc39b2605be76fc 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 8 /circleplus put dup 48 /prime put dup 73 /I put dup 75 /K put readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueID 5000815 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6784decd2fb36dec8f 7478da853762e00bc9c52ba86ef50d222ff142abc18e4fb95b86b3e57d9390ca 86e8de6347bf7f48f93322a372c5d7830521b14e8fa0771961b027c81365426d d95d16b950e216a90f28ce806244ff28dd969cb512a23e05045e7b2bcaf7d6f0 ca8b51494f2ec0681bdfde60420628d9bce98a0ca1ecca9140f452a88db19a30 ed46ae881b098c843a51d06f9a153489357fd24de6a0c36cb5a9eb9b3b64dd75 18f83018ae7cefa1c12fd205df6880a705e28457af7b869a6dd28f6b37d82cc3 e51cba348dcdb34efb119b8ce5a996f15492b0f5d34538850a2c1b6a509fe5fa 015eceaf28b2bfc953397c590d3d4ef8b94f890461dea021548475edd68bffe1 5140c6ab20345e73534abc5986619b5685c499861c31e6aeb0f53a94d7a4ef66 c2607e038fc84884015fbdc9781dea7d7e4deffd4403977095576bb653af7c30 24a733445d13c6d7d60992dc02087b4be10153465f69e8c150e3f94cb0981d12 de4f552962c9217ea3d6dbb152b9c3864abfc2cfa9eee9a59e977d4ffafed60d 470757c567549630383b07bcaf2b3902b5b90672f6907af9559ab5a160648557 a0ef77f1b866a6fe43c3813be7853e7c9a1f430c50afd3d4b150e6c309574f76 6279d2533f16da4eee46b3ef424590c39e92ff70bbef6ebbbba5f05cecba4de5 d40f7d238e12a94e9d9e97126d1a64465f6507b08401b418415ee8cfe7b5c838 bba325338e66209b9fd51e6a94cf8bf63aea7b8f185212bd9845e92301269763 fa974f238f6e461f633a48bafb566c11bf28bf34b8422a78c5ca3731685bed35 71258e3a53fe92e3620b04c596627d42b83568e01b5f5383c65d55ac103d47b7 a67b165e355744a978b7d4a85e568bb34c38b63ed8512a94a123f0db4af96fd6 dba1c4bdd4372b9d4a62a2e5973967e6b615d4527629ebd6c14ae308c5550940 c3bc1102e5a406e450662935669367c7e6de3235556ad688b13a34f644a67956 745fe20499b049177e61208113b1dbf4a1c8a06c6098c81941b436c357eaf425 9708fbd4ac901a954830f29d1dffde59a19f17333d598bf8addd8955c66b7027 2866b19387a01af9096678e95815721ab9d1f8c67a3901ca45abdb117e342ec7 fdfe90495755bf773e95f7bb2fed558adcd5ed905ed1d7b929886e622ac6aa4c 33be2cc35b78c4be17f6532f5e2ded2e5b8ae95a46ac06fc9389cde7045d5794 5091a1224ca5b8f3674f7d911ae2bafb91e6fd83bf302cf8347138a265b7c41c ca65e7044256c725a05660f8bad29f5ba3abd80ecaac2f105f2e1e3e25d7c772 dc19b7d3b8cbd362e6d844dedb3fefa580bd5f1a2a9782329b5fb6a1cdef09fa ab87cfc61413b56aaeaedcb8fac042ff60b6929f60f04374b7c7fafff423f789 fa301b62149bcf1882cf0278d8480f88111d564e344d9b4af0929a68d076e77c f04276343d8fc0258129c0b6ed829270bbcc8c216f36209a5058b004a8616e00 1be73e102df3ddb8f6bb0d7485133e4fbc736d706b13558d18a26f1d149cd8fe 4543ee3e87744d4aba68accb6cdffaba29514e36c672e6db0ff682d3309046e5 8657b797b999d5a88aaccf566b622858bcd47004b64e1041df7e8a410ead0b69 fe3a2640b8a224d49e4344f8f057c4d9fccfdec9e861dc7626ab517933ddd136 5a10758cfe63d0cb6d9088c7b8e9b3f3f9a01ed0a38b271548b02aa6 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueID 5000788 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306457c7e62c1d269f306bd3402e266de fc3b5e7d8a8d2f5bf0fe6ddd40d07391df4fad4a6018dce29a2b8f692b29f202 3a7c0e66de8ed85c14f1f8492167357f51a7e84cc5d92e0fee4a71813d2ffb26 445026f89b4787516ecd1afc78f8bd19e91e9ccc9402e8c36d2449c1ff850a8a f61165aac3fe931332dd28e261b91b05edd18f44ea7d58a8f35fe88493b64aad 6bfac3a0136215fc2f4ca8e91d70c5010e6f4013e6d63b44f6fb00afdbd7cdb4 5ec5b1d9736f45cefc8a0124b815987cebd81bbe0d44dedb2d5ef37923b3d551 abb6a1cefca7868fc7bf3814ef7d7b6b1ae6e869cb77aa29e3d90b12b0dc3ff6 ec945922b5899bbf2f12e92731486d2ef1230c528bf8d7e0ae09ad7632a38966 5963de49d1ef3d65bc483e4a577b927c940f5e121169ba52f6576c85793e5fd7 7f5863c488e55bdefd5d8b2514795533aabbc12e7f51816c7e1484f1c441aa9a 66a5dc77158e79cd6692d299b95e8058b35e771a6972d2b5eb1cd6d2bb8e835d 361a6eac0c90c906aea1cd75c4412a1339dc1439f93e59917311bf20af4f2bc8 bba5a4626f75f28a47d4595c37f4019df480d9385975159142618e22e964f019 d88125210e2ef21d65baf500adf802297b24eebbf4e93c617a613f856bd0d45f 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df897f33135eb78e66a25376dda3ddfa51d2f055bfcc7360f98d990492490529 635cfc5a0a7163 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 12 /vextendsingle put dup 13 /vextenddouble put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 26 /braceleftbigg put dup 40 /braceleftBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 66 /parenleftex put dup 67 /parenrightex put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 112 /radicalbig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 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fe747b009902e41e176b9cb425bce3aecbfb9b31228fdd8c6cfe1ab61459c47a 22f761452ce0a756ef37ec1b50d31f24ced1db93636917715a751cbaf70cc1ea 71f9f526b393a79a13b39a7e949dd381c21dd1746c0437200c2c1ee915171620 7eeb564e42b3a312d404041ef7560c9a7599cf4c55885b7adc 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 100 /d put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueID 5087386 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7bcb09b7064e91fa0ded228e42 09aae407a7aaca60b1076299ac4abd23ef02f108765f0e3d91f92f3afbfded37 2fcf6e4b1416901517da8f2fb3c9fe7a87bcbe6fd36cd5b5823fdb74229036a6 3c3346a1093e6b1036902c1bf42fc317c80abf04020a47b344c36de42f05c490 a0ff44ab6d5249e9f552a8707bb7661e242644814001c8430ebd5e5f0b944ceb 666ee64359d663e355b2f17093a964139d17287f6ca6a024767eba4fe4873855 babe2f07b91560f68300b06dfe27264c163195d446980c35bca0b48f7806626a e72636593a05ba403ce1c0f8b2cea3ecd586e90ac17d034ba4af708304f23131 3459fbbbfb97d4834d0395754ab3f22d6495d2144087d448616fa1ce27bc50d3 46543287e3860d99b433624119bb9920a2113604c0e260fd275ba55e0fd19c83 e19addc3baa1f32f6b7284038845ccee71a3311ddb17b84975f7a984bed7c6ec 2a06e5b335a763d081c6273f86a46632fd9141a27902074fc860df3a2eb59b89 774c767022dbb577e30da128bd7706a43af886d0c256b50fa968ef06776aac0b a5387e9011eb2334c1f42c090f06a1125c207ea6324e87f46414050d88070346 7b9a6e4d2045feac295d15ca55b900da594770ed9843365bbc81bff2190b731b 8790385f7c313b4f64683aededa87b8c04da713754a05b7ad08b814b94b7be0a 5f20abfa50f767a51aa084b330955da5231cf6660a76c929e56afc63797a999f 757018dd00ca061ace2eae761c8e56838535e4f0dde8afaf899c418727ddfd13 eaf2262f80d0e55d748a8911d5e9446dd6e2e7bb4ee37fadc6ab1858bc51e23a c993d8400ce3e83eb9c880240d2292a3df09eb6338c544e1dde19a0dcd10902f c6f0fa008e22d1e54f7522f3dc475f03ff71bb68da3ec1b35c69dd04bb61ad13 9beb14b69f362657fae35f272040a2acdc00e40d81e99f4b5d359f704c2733e7 1441ed88fa0d864e6753d053f3436ebc2c7f11a0a7c536d94e7da1e741a91a20 49d35b39dc5c23a88a752ce93ff729a2d556171733fabe0c5388f9fde5716afb 06b36bfef2145f4532b49be2d1620188a130d47be95a1d0612a1c3feaeb05e0f 20905288e1be273bee6c4b6b26af107c7fa69ce0e6a2244b6148d6b0a6ffea75 06ff3d49164adfa9a7c793802dca3a0d441e43cc9e35b08fdee65ea37ab91c68 d563400b3c33768e52e980773708141c28fcd9373063749b100b542a6ce6be59 df441708fc9d0d48b777aa2bff598614cebcefc19b294eec5c7ac0654c7791c8 0acc907ef38a6ec2f549b54529d787e44960877929872e099dc3433a0698adb2 b75bd0 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 19 /acute put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3ffd20326947b1065649b533eb5e9e3 a88a87fe987918a687950b26f647d6e08bf14d983ed1d4a8ca1e2da8d985f944 2eb916f5b6ed0697ac7c33e1e36a3bf460d34ce45f1631871097cb04f18e3889 4cf4ac1538eb19481311d24fe3be7beaa4a3730e8b4831fe59d6d9ce2e46116b 629c7ba2f9ce3ecba2f43bc162a5a077ca1b2882a42afdcec3f4b75b5d63e0bc 8e5dc95257766d8ea467ad9cbaf47be60f797580cced6884b3a68f70c91f4fdf 80fd00ed9139e7f480dc3a76af72ad9b434187730bdfaefe4cbfe5c7edcaaf24 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%%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 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 59 /semicolon put dup 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put dup 103 /g put dup 108 /l put dup 111 /o put dup 126 /tilde put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueID 5000790 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 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6563578772262d3a962d2b1ecca516d9365cb60c8d4bb503577cbcdd19f2eeea 53bdfaf83aaa0118661b1c31bbb18a27e0450a70f8f1eb962d37331f56d2e0c7 9cab218b421ce52e001de61a70920c999ecf6f5004 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 8 /circleplus put dup 14 /openbullet put dup 15 /bullet put dup 16 /equivasymptotic put dup 24 /similar put dup 26 /propersubset put dup 27 /propersuperset put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 59 /emptyset put dup 68 /D put dup 73 /I put dup 74 /J put dup 75 /K put dup 84 /T put dup 91 /union put dup 92 /intersection put dup 94 /logicaland 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 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67a256a41ba3fe0b840205286f7e64b659fb45fb15a3536c0e72da49b435cb27 cb5932958644aba5136a85a743328fc77117c467b05ad7e58ca34e1e3e704345 d698cfd1a628ba2e0c7d3743d5fdf967753205d2bae376fb41dcfd2278b17a50 6046d7e81452132970e55f6bf9cf1fb7cc2bfe2af776a879db4d6273636e26f2 6570f82cf3927fa2718bb928a56e2cc97c96ad444a9e7f88d460b55c56838b82 a5ccba1eb398801544e9a6c8a0b985aedfb83678a9599a5fbbb1f65d9d5b7ed8 9e46b00f34327775b639e63977b8903ff17fcfda019cdcd47611530912cfe3f1 5d731d76219dfc0502ada6c2d3265748f3692dc82e914da60295e8247df28fb5 2bccceab0ae77a8c7c5e36232f867d3e58047a31f7c2c7027b4ea29a085c5ba7 7fbf30d6d6daa5f8e6ab16c5103306bf48efe321f3037264acee3b506b009817 a2e2fc534ad2d65943cdab3d41fdc4a928642ced9fd46817bd4f5c121815e1cc a140ddd74e0a10aedb8d21309862068082a36339c128358774d4a4fbb50e6525 2b7ce954c5264fd6a678006bc060cd95245a02477ea4e2760694a12d2295818c 60ee87019db6a0756188dc196a327d5132469d387b97ebc3bcdec02899b86a70 c290dbd9411a97c62154d5b12b3318df09b19ab3bdf87b805b8b0f093619524a b5d49982b1d0df712a6eb4dc9587193172159f9e926cdf65955f91e3bc0f9d5e 8a2db70a83d47f4c3291cbc3ebfdb199dadf58d2542720bb9c27e19cbc637ddb 27dc8ca4d42b3d16bd37df76c2aabb179c674b90bb4185da 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 53 /lessdblequal put dup 61 /greaterdblequal put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueID 5031981 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6f33457e3200f0935ea98f1a0cfaff49cfa2 d877ff805620cba84126dfd175a82e1b4c6ec58461fce97cdd749a012e2c4243 86b199a97c306b282350cbe5af90cf1d8e7371dcaedbd5fa3346dd3e558072c5 fd51a2edfee6890f62efd1b552699123f4f52c2817dadd5ece30b3bc178a3ef4 f45fd8ad0ee5bf30db5f9eee9ddefe249f3c349a886cf9130cba68ae559ad719 c439930ece1d52d38366116c390cbeabb8fdb36811cf3b8e2416825e8c880b24 6521ec6880ebe7f127c8741c756b29e1f89e3fd5d638a7bbade8c3a0fdbbf7ba ec37cd372afb2d9dd25d509726d20a05914e418f6b7e759c440beb61519df02a 06734ec1226890d17c51f25f775e6879fdc231bc4d7fcb4b33d04fa273bdb8e8 f124171714614f8a94e98a6c2d6bdc1254913999f9c181e42c0886556cdf48af 92a265c1eb06bb30fb2745840c17bd31c7b8bdfd37dfb9643a418a2eaae14094 6752e0a315207aef85c3c2630538907971ec1b05687208571a3504bb3369b5c8 9a43e5abe343d0599fd8d41a1bf01de49f5856bdd67d63a2b3ecea75d721a6ca 0c8751ab41e892e9cf674218dd44acfd40019bf59f4d69ea40d5cf9068a6ff96 9f8cc772b920baf17334833d732431a2969b9a30fe38bd44772cef653935a957 af61dd568f041847ccce3c587656cba7bbf46d8cefec936c6d34dbe51bc04929 0440354f3b29ed8e6b4f21ad53b5809779d67339f284563d54ed4d2a504b014d 974a31c0ceb57b364af519bad4e85c41d521a3ffc26dbac555b42dc191367152 1d60c30b95b5bbafb3c7d87a1c043ea5f0e6dab83a4de99a742e75f02837b923 c50eb478ac4131ee604e3d7b784bd8c93eb9430291d452c21be0495718546333 66a602a9f142a563fdae3c8d40de6a8aa4f3118e6b59bd7a2c35420557fc465c b2c852522b60e76b1d2cb7df1e79b9db221da87a532f8c1175ee3ed93b795315 f1222e1f8cfd2d65a501c7c087666903230375c74103242386156e2671f1532c 9744ffc1af623cc4c6d385331bd15bb0621094b3455a3f384c033672292b8dc6 16f318cbd3b9187dbd0d79b4de471e9767145968bcabe586bbcd1c3ee2a62dc2 07fbc0195399830e5846677a9815a6f5e194ef5ace37c4bf49efaf4d318cf805 ff239f71478d6fb93d503c79c2acb5645401701374a4b5168b9a089ff963e0cc 03abcb186f51cfff07505c71cdc58b36a9d9f6b33425ca7ccf2343cbd066ec71 c5c4e4c6aca7e3f0ce0acf59fdb58c569c0f59b8a8af07844f6332582022e25e ee6690f5eaf479154e3ab88255e084bd402754bf62f0a1397950e8697ea4ea78 ff5fc589f40bc449942e6c3bdb33388939448904a08481e98678c51faf7c18b6 e69521c3d3f6ff4061f92d339c055ef3f54690a3b6cb4a9e095438086199aad9 9a7234b542c8c34e80a48358896ff25950c1ebbef7bdcf5259a207e0b67840d0 da60409e22cddebd0c458c4a835ee4c7ff2ca76c372c66ef7210f0fd3b35fbb5 6ac2c82e03ac73ae1842347d70bc23d377bc6de7f2538c1fb8c09c9b25b1a577 007b522a4ed63c5be6776144deefe5390e1f399b01c62c2cac58dbbc9a87d6b5 563b8400d9300bc78f6b50a6a34ce4a9a9919d2f252c107ed84136c1e1818c93 648e9d 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 59 /comma put dup 61 /slash put dup 73 /I put dup 74 /J put dup 75 /K put dup 96 /lscript put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 114 /r put dup 115 /s put dup 119 /w put dup 120 /x put dup 126 /vector put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueID 5087380 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311e252234d51741e18db3a68e8ad10242 29e5817a10e796a78d2c7f7c1f50961b9a57aa604c9f821dbf5a9295197bc666 31678d7d2c7e1f8f2151ce0c29efce46270570f4301c5dad1b38884732e53dad 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 3 /asteriskmath put dup 8 /circleplus put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 73 /I put dup 75 /K put dup 106 /bar put dup 110 /backslash put dup 112 /radical put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a 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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 11 /alpha put dup 12 /beta put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 19 /iota put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 30 /phi put dup 31 /chi put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 93 /sharp put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 126 /vector put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 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876bf2d506222638967406286ccf32756f08eab80e35161a712aa1468f7a3655 477755ae349c5e192c6838147df78c850f56eb8d8f308804b94ba08ee715e479 1de7e995905451e40e32a6bd1dfb262b6be47b2c2991a06374b2b40e643bed28 abb1e684367edc1e3182da1b86702a08aec3b4560da2c8b1d6d75670b48d3e04 059d34ac0f1014a1ad1e4b 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 14 /delta put dup 15 /epsilon1 put dup 21 /lambda put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 59 /comma put dup 61 /slash put dup 65 /A put dup 67 /C put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 82 /R put dup 87 /W put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 105 /i put dup 106 /j put dup 107 /k 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 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 78 /N put dup 82 /R put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d56491bdecdcafdc988206c5a457a19270b 37d0ab776e03eaa7eb568eeab6b5e79dec03b0dcbf923a2aa8e4f4deda2cb043 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 19 /acute put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 73 /I put dup 77 /M put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 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{-53 -251 1139 750}readonly def /UniqueID 5000769 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5bfae5c1f476ee6f05eb1d2caeb 269958b194521197b312fcced4867f3c8fbd030bd715d8ffda1dcd454b174e7a 1a97b59fe770e67702519d9d9b23d61ac08424d555242a8ca08c49aef300945d 99b999a79ce74804ae6bfde623f4463371442f6523a5f6ce19c839a708c02513 2e22c696c8ccade45680e5197189d0f98e7f0d5f955e353970b392cf530a68cc 56b0035ddfbf206c3074beeb0739dcbca272a6e629fb7aea2c5ba7bae50c7b4c a595df78200c352997ec3ee564df229fbb5473f5e8ccb1cc0153e9a7e299a8ea 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4fab37474eb6603be696ab13bf99da14c932a00deced16db660e6a46c5c22df1 af988e5147f9e648f26451f95837da9fa9af27da311da17bbd9704f56b92edd4 16c3fbcae8550b3035239132ed29ad68744a5937e51c68b30ddface5cb92a5da 1a7262f05a3825289be69bafd79528a2245a48b553eef75a759e9f64362bb663 4fc4abceda6332ab04159ec98d687e9eb3d0922a51a718af4c51f789b33c5bfe 73142b819ba1ef5442ff5085c3e59946cfdb6d9f88347827c7ed2e3c2d197955 1bbe7a3bc55db3264f3622ca2c14ac6274676c10d48352910800d8031c3203a5 2987c464ef97b4357eb1bb325c967167de01533a4bff3cb5328f20beb287cf1b 57d9d072fee3588a88cb4bd9f2e7591fbc17a2e9ab04f31b7ddebb1e38ed5546 625e8be53340a6337e1ed8e3dcfeb76fc245f51b0a025a5e8fe2f09d28bead3a a218fdbece91766de04e46197a30305fbd45717fb3f0e123f4f0759afa196fed 3f94b1cec006aa6b618da8e33249a6bec3e76f18c3d0b7328a1a85b7620ca5b8 54f22021c8425d18f6c4bdb0d1d4b4f3428bfaefe93bb36a1abba1d5817f25f7 4dfdd41ce3f640e5a4fb670c8431df6ac946f099c5165d1b4a6632ae179ef831 7640974dd9d68a283dc6918641a0ab77468bb49cd9f5e47fb76d1bc6a8c03393 a260d163d5af273912c3535fb57ba1f9bbb6702195154378c52e29737aa5ba20 ce4246dce9f007b970ee54c01e2ba6b1f08ca36706c17b1a610d8247a1134759 e4b723557d90aeed0905b8b954ffeb504b7aaa60cb635e341996f894190ae5dc 08ab2705bedfbd98e1b811dddd390f01d0e9bb3790c78231bd23e43a5c8d2a15 732b6b7222c344e0af0da4e633 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 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 54 /six put dup 56 /eight put dup 58 /colon put dup 64 /at put dup 65 /A put dup 67 /C put dup 68 /D put dup 71 /G put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 89 /Y put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 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 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 121 /y put dup 123 /endash put dup 126 /tilde put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueID 5000791 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b094541f345ec367ae85ca9bd 24568a01d3b9f8095f7420e6c423c414b3dcce6da48dd1c89a56d078e0d0e2f2 62a13640a06d17e44ee3866c3471fb58fedf5a3b77294517651c16bdd7267d39 a54e7171752dbde63ac19bb4b3021ce95eb5fe67390b09ae4d9ed4d704a67443 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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 3 /asteriskmath put dup 121 /dagger put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6a18979f258b8c6ed3 ccafbc21aca420c9c83eea371adc20e038b4d7b8ac303004b0aa205f04135140 76407216032fdd22e6219da8f16b28ca12524deb7bca073cc5eba65c102a5e85 fd48e6d062cd4283ee570a7774597e5bf0e3400b6be72db0115f3cb12db70ce0 83722870cddfadee715f10f1fcaf20e06f3c54afe5ca238539bfe2b596116e83 f5371ff18fa5003d8543226cfd4025f9940365b392a858d27f078d3abcffe4a1 54e78c7692d1a32bf935967c64f01b24788ff8325d61145e2d4a489fd986fb77 38e6b254522c77ca2797a504a9ce4676a77ebacb026eca94dde5922c936f8e90 c43e2851973f31a3280c08220536dd2c2de1ffd15fc7390913427ab78eb6d43b 8063223d65fd28ff3d1247f681b2b45681509b57ddc243e6cb03cc2ca12bab7a e851b881f9bdd89fcffa4f8091bd827649a5b36353db09f98c8c02ce4d952a03 b627a8e3a463bbab6dd9b6d37bef0d0b80a8eccadd52569152625253d39d9d51 a7d9f6b731bed1f2b2446ed069bc913fdf127bb1f1ead88b0e31d4b96c31d371 5135ef273e1f19fde6a48695c0ec0dc75a9be460b057704adf3d59cf65ed666b 0cfac98391d5df064100a0afe65339786504e6475faa102489d0cf4e53b6e7db 2f1d9adf662bcea1d4e141c924a2f2ee310ca66093cca2448d5341c3840bf846 8d0e9da55154ce2023c25c6f8c7ed1556c12f310e49fec8673ba71e1f1de4301 90121e072fcde7bb807d107d94d98a449e8bf022c1ef0e5f28e54eb9b58f9f44 464b9d859869fb43bf35b2e311bddc4e4f7356eaae39a9e42013328bca7445e7 487fbd4932181016896d0bbbd85ec2b9021c012aa8aa7cfa8ed89156711e39e9 b4a2a56188096b990a6008f8ba10b193ea09f323fdd1be6f86448cca98982a67 aef978431afd96e43a57d64a20507305a174f6624c12e251014af09501714210 e9d11dabe913e5b93e591ca0a220d551f431087a39bdbf12ebff76fa99774777 e5e83e755355e8d643dbd062c6eec55e99ca5f9d67639125e04da9e5da65a1ad 0dfab673b971bd1bd9f222657a9754e3bdba89177c15791f330a8675ba6cfb25 7a7b281226ad29d2fffa9b783143aa32f8c0b532fe59ce0cbeb85d7b0381701f 32006cf6f3cabc941288cfa0eed982d44484c865489743 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 100 /d put dup 110 /n put readonly def /FontBBox{-29 -250 1075 750}readonly def /UniqueID 5087384 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930da4f8d58abefd45db119b10eb409dd89792 3c6e705479464a4b33ae3d31bfe98efe259f07f7950237bbaee4f7b64ffea83a 757fa717d50c3298392891bfd60e34a056b0d6021da3fd9b8b01bf78b0b23dc4 ea3b0605150d20b27ff9ea5f2524661019d982a2e47cd7c21ee5ca9ed9227821 f8d07119397de26838c11d7b35bcedc43e011626e300f8249ed846d7b5ccbd89 02550cda17ddf113df658fb13f8162681013766863784efac128e01ef997e1a4 30312afac8f3d948edecd7d5090ab5c864b91d7f6e80256f0e1fc99f1102b74c 61cc456e2e1bedf0e627d5f70f738f963a13666405c51ebf23ca077e97570a2f 1118536b92e585cf1dbd5fbf36a0a5d197538c05304e93a15f3c931168d91d42 483ef428b1a3f1b7e548fd1d23b80f5523c8b8bf450b02804fb689f915c86693 f7dfb1c43f681627c7f027021aa3a8179148f1819cd94b8890a9ac63dc1e8a68 f254b64f6f173c3d02e3ebc037da05074017c8b113e94f4e2d26b467ef9da07e afd7ea7c86e025af52c5ceeb835951e67e02a38daf1e9e698bd21996c1071e84 ff7fbe77f6ddf9d06fa92b16ecebcbe0370b5cd9d450d0f2906a105cadf93ed0 0abc7fcf2c996e7c15e452006e35390d99b9cfc59925f1444c20d9c3e1fc2910 32cf9fa9d580eac1bc956334da391b870a757c3bee6aa3e005b20988d8587876 3ced20c6edf2ac2b22890e37ff7aedeb0986d85ae05bc84339bf2116df48c5d1 cc888bb33d409b06dd70ffd0618b2080442aac439aeb39c20a87ad74ef10629c f1604db8611e8e4224c5b21ab0c7ebddc77fe9a283b3a8c6a2a9508f6836e3fc 3eae1366b4da4f44cb7a77fd81d278c17775d56cb28b70f18d894bfd1ec1827f fb177e1d46b0dc5afee78aaca1c43d03b7be6c8768b9b0ea6f8c098cd4c05faf ba081c028de9af6e4978006cf6c3b98049092b8025778dd23741c3b080f956f6 38af18661f286db6c84da3d7f7cc8ebc68194a64db43e7dbf6e1af1fb7bd9851 652b967c27c32778f3f5026f6b28218c435b5b23889a5ca5cf2dfb202701185d 692de2e8360b575fd1f2226fd9ca346bd9e997ca0dc1ebfec48d1bbd5d103ff0 79d022dfb5eaf9932437d0d227705c57efa1aa9b4133e5756ba99738cf8bf3e9 5d6b3deef003ee1534d9bcf8ec26b58944144d5cbce367fc7dffa14d5e0a2997 2ce358b5ddad4035858fe5e3356af24783ee5c856fa5b58769a27102ace70d2f 1e52b35a69ee68db6374869956f223434b31762d4ccd9c9471408b46b9ddfa33 a327c9a9f465afc726dc66e75274575cd715661d47e5f105c804c5785ea39d49 b00d1393df55071589176472ec147dbb5eeb6a60f7713164a809501d3aecbf8c d2f1affc32457b696b4b8ef1a4b6aa16feea27fbd01b500c8dc914bfe91336e4 0f343abfed140ddba5d25f9eb10cc9e9b82cb58094b1ffe22df7849f5a801c62 fdf7b0318fee15165c1606dae5a38eefac1230e5e4ae68965305876f5047ca85 bb6ecd08791f97d11797135a0416441e5d45e0ab650eb184526d991bfd4ce64e b5ea47f1bfd30a6475d0a1d4d4fa6cffd8525e01ca438454cd24004420946baf 26b223e30b61b4729e2425f70fd1705a57d73980fd8e84e0dd286ea41adb5a7a 5603b8af6d6c17c26a87a1683daf51bcf15813ff8cac196f34103bdc4616dd81 0585601eabe8fe9d58289fd020ded82195d49315af76674d7eee0c8543ee11e5 6c7281a1b6d0aa941cf5d52736bb0166bdf8627cc1d0 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 12 /fi put dup 19 /acute put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 52 /four put dup 78 /N put dup 83 /S put dup 87 /W put dup 96 /quoteleft 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{-39 -250 1036 750}readonly def /UniqueID 5000792 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b448685bdd20eb05d5e7c2126132e33a59a7170609 dcf4871a5d023c9ef57d3362d9f2d7a440bb69bf653364105f16f4d0f03582f9 aced3d05cc76489b16e3fa8a446094d30038b06ecceda269f2eab9d19a99c7f9 39f9548f206c5a457a19270b2b82c43b091dfc5573468eaa3e7a4a32f8042891 d85e4b180fcbcb3091d2800e54c87d84ce9cad6869b5aabbbe47f40c68799893 d22b765295e1e69e33aa048b7ed98ba480ceca91f3ebf8ef85fe9a3976909626 b95ac5940d53f9b02215d84a44837ba25ed15cce0d504f1d335065594f3bc824 5405407591cccb11cfd4645da60d960c0b93f187b0cf7b105543c0b70f89af5d 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX9 %!PS-AdobeFont-1.1: CMBX9 1.0 %%CreationDate: 1991 Aug 20 16:36:25 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 65 /A put dup 97 /a put dup 98 /b put dup 99 /c put dup 114 /r put dup 115 /s put dup 116 /t put readonly def /FontBBox{-58 -250 1195 750}readonly def /UniqueID 5000767 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5bfae5c1f476ee6f05eb1d2caeb 269958b194521197b312fcced4867f3c8fbd030bd715d8ffda1dcd454b174e7a 1a97b59fe770e67702519d9d9b23d61ac08424d55fd4dda249cff0b56b9f3aff e9d0de215c02a52a6bc77155ff6b8cba5cd6646ba331254ac58ace650a967d3b 272331b87b6df06d5ab9d80fabe9f9ceac10139b61244814dd9fc295ed42d1b5 cd11c2e4beab318b20f51ca2c2930657e5343ab7bcf0e8870a0d12de4ffacb6c b3fdbdda481c2fcb84408d3d902e9a32070b2af6cd9317a33a42a43857c114b3 f4c3005cef9401f1c2bad3e69150d7145b79f95c9cfaf7a335b277e6435ab374 f6a3e78e124ac1e4615511f743ad65c5d778403a840310ac4902985f107fd33c 0049623e4c496cd353d2ae2a4804110c6a420a38ece5d5b235c5e35886a0987d 284e6110527b653b1b2c68c636ef423536180acadb954843eff4d9a82568a713 b682c574e30f793f093aef755c650e0e7175c2e3138695212d9fea7bd4b5805c a90985b134c6bc15a19afae0bfb3066363ca3e2568c2ac4559a69fc32feb443b 61243b4508b7524d974636855c83c8149d489b68fe18bd497116a953431251df bd09fa1d2ef66c85b997ab2f6b7c6d9d096cb43931379ff9782fa00c5f3cbd06 04265d16891ffc4f66e4a0307765d384c32aa83a4e4f89ac79ef8e76eed08450 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738ef5bc71c155a02d8f22fb937ad5e0859fba27465ba0bd41e06844010f6520 dfcbbe65acebf937f7099d1f5e62e666ed88886f674cf7d3bdd829db734e1351 c5d0dbae95c05d882faa8c6a290bef91d4defebccf65591e5bc4d79edfbd8f25 2261ce5f5b80a3d5b05dd71dd25011d2641c32e793dd1384483a3e297531e05a 5b7d7909b40b8a7b0a10fd9066c696df7cfdc39556da7fe10ca91652f048c399 945acee3e3453bf7ead5b3af2042175b239496f80d501f1b74a0a14cdbee6a87 4399b170911ed99b0a0b37d5c8e6688e78486976affcc8e74bfe52782f3fe838 0de0be783e217274095a39cd23999ff209da182ee2a846ba6843ef561135237f 34205db4e4bfc08f357f206aaa37175bd03c30c3dbb7f955109d225a93b4ae47 88ceede6a57e38371db28e19a163556d4afa3a0d5cdc029984dec1e4be531aec bcb6ffc0985a0e3491b438b1603c0e63b49ffd9e0c1bb1a7a6a5e1640baa2297 9fd7c136b1cba76e92be354c4df0682496c15bbdfa0296c99a18ba60a3aa3ad9 701071fe4990e15a230a23d456724d59966096c9b96c0e90e660157789c04675 314f2f098963e8f78aa888db00167a021af4a4b7243c1104b3a2480df077ecc5 7d41a49fa79d927709c777008adb58f5ed959f25ab6e125a2fb15af35eff32ad 42cc36ea6428cbcb074eb0d6c09de44f5dcd4baf74a680c3d1f26231bd956b20 385d23b2c68a1619fe2441af6eabbf203c339af755063771d42b4c9fcfdf1859 84b013ce91b074112a866b5a42a31a50a163940470d04cdd3239c3145e6953c4 a023ff958542a63cb6494edf4e8827113fd1835608aeda65faf500450eebd22f 19ec5c3032dd2161b6d92551d07fc4f104ad2a85e455da9032b156adf5bf74ee cb875b4d2e9e93400a24c4908539d8fccc0c2488a6f1fd9f38293d24d9c5d0bc 64f0d077fc97496444436cc5a53e7e396c16aeee0ed40a0652fbd7f3759904e0 6e80612d8cb804f9a72a3a76641173415dc07fc79bf8ced1eea521e0a1adeacc 939906f1a234e36c526e75261273c5b3ccac65480f8dfa9c35b8a6232f7ccdfb 073a8e6b113ad610c0419e9a57b5528ad42e2b578377fa4effbf855d553b7b2f f7cff7ebcb05925f8150930c42355e5124b8a8e20ebd0bdd2815e9966f03d143 5117c5a138e8287e1e2e5c07af239d2ed22d9913bc5d8e69af710d5612cec7fd 9c 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 1 /Delta put dup 3 /Lambda put dup 4 /Xi put dup 8 /Phi put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 19 /acute put dup 22 /macron 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 64 /at put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 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2beaa8ada983f6b82f0655e8c11b49292041c38538d51b527d8aec842ec217c1 d01d600e14d9bbdd9e9d2dbc97f478b1a1a8a2b170c9c909f6d8afe5539abf8f cc99abd4ac1b10862dfbec55910ef46994b38c8b64b95842e94d90b51d07ec52 599a5921aa3df4165bc2c6ae29b324097c70eeff5306a63e5b1460ac2ac36394 000ba03e567c460a52a21c2ed7caf29ff170bf9a7c0c8e40689356a39d50d19e 0953277c7ac44d399b29b0564ecadc7aae148fabc6aa1c89dc65e8744ba49ab7 04337d097c86533cbdc679945413b4d8a06dfd1e55cf0a99ddc98fdea1b97f4d 4d4616056573b7a32cf533718b423f476d8d22f031a5ceb7f6bcc526a769812e 4ba6205b29bb57aa822da1c1a44d97302fb21264d19de070d5264ce84f654474 7a2ae8e53af2fbc4bf3b828f3b58875500d626cc2052112aa7363095855929bf 016b6bae4f62f0279779a7e18c2d5f1db13faaa4130ac5a91265de566e668ebc bb22c2222feba91926187bdd71694c7a688003d01930bfb1558fbfcf54f8a018 51bdad9a1fabda5216092cc491fefb567e0affc5494fbaf9882cc81bc1b03407 70deb4e99a20dd40b76320b10bec8fe3f7b7505d8ac1f4ecf2c9272e598ed1e1 60dfd7d09cb04da7c6ac02ee116b95a8b94baa4e42b9aa6a28333ccde3d19f2a ecd1f7c835c2016baf17704372863a4399c5167c3c45b7860c6571eb9ce7cd78 f280fc6916d5e08795fb4a0c44ceb59f19e0beb351752a6595f39e5193d44a8b c81ebb9cd9f95081813877856af64492fa1a1f1a0da368070652732879290b5e bca9d5f5d75b45da6706206df844cbe8a0e79053d4fe4eddd9bf2850740ad998 f340367eb0ebe0628807 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 13 /fl put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 67 /C put dup 73 /I put dup 75 /K put dup 76 /L put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T 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 121 /y put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306462c563d51ece66fcc9c831843ecae 1fefc1a232e2724f7baee428ae03aadb95c3035345c15e9922fe49e1f2cfc980 237316572dbc57064edac9b0db8913a5e2d45e97e19a91435ccf8adfc835b585 48e74b291446d689c7f2f8c4325e8356e974ae30c3e2977477baeaa33d141fb7 80f59351e84bfc88c87b3a4a1d25e168b9d134554f6a581378c7d2d6eca8ac09 045cc3a5e0ed86f147133f094029e3483bebc81cfde69942cf645ea20d0a2b64 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68d4a9489ee65deaa8005f08b2117b15684cd03de9d72439f76550a00acaf5ad 178be0d654ba3d1474cc822acb0d4704500c3ea9c55ac786d5b0a0c13b150cc1 80b6ead0108dee8be99ed51e9bbefcd9a43ba63229789265ac01eddc569475df 041900e90eb243dc7b43d0e0f8992cd1aed199e6bac7d2fa9ec6a6a9224dcd7b 9d82ec883a4df0929e0f1ec25b7878c03fbf909643ac3f9f95804cfce6151b79 6f57b68d4c93464b1fbdcda30d3722a4041a802df7e03cf06e41178eaa855364 9e7efe605116cfbe68b1f1ba0fe8fdc11352d2bf0ab476f50ad9e9940b1d8d3c 56555f98ca896c6461200f6fe2f78784a5262151ce441e476d331773b621c3ed 97e3aa2bb6608761173cfaed4d10c555020f2b92406e179f46de9458f3999907 215c21dc217ed66b9bd9cac513ab843ff625dbb5187efc88c87b5e292890e55a 2d0be11d60efd4ee6277de142bf37d3c97303c37f95ec334943ed479623aa106 2561 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 3 /asteriskmath put dup 121 /dagger put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b66566ca5ba42b1911a 5d7f1bf343015eece988b7a93bce0c7aa61344d48aed9c92c8698d4b7c9951c8 7d103f2414b39e1437f9d2e50c4ee5f218f2e6716926a79ea978f13b1f855345 191dd7d31d8f82c2e3343c7a5894d95bdc492c28226834efcb5c12fea36ac5cc 430e0aa604961e34888adf6c1f3954cbc2498e225d953cf5685852162346f474 5a2a7087d5d7ad486de16d2ca8e15cee26e012671ba3bdc7d95cc8c98bb774f5 08625e968aee27ff7d1a06e63bcfb5aa4876c3f8f13b30ccccee73c3caf4e70d 98e6ed2f422dbb4950bf789680e064150995941a9f4dd68a575949847a7d012b b910bf03a69374e341e8036ff92c949f3dc6e86aaa7a2b79532b36ca1496c973 3f5e789fa3754f83951039e765b6ef8d37389a18ca2655d971ea0ab2f6131886 1d71a694f55fc3186d85d232f23b49634341f3d4279a84def32013aabbb89fa4 20ba8b5a82ec05c02340bfc3e91a5db9a027ddccd503a463a6bf851d16d578a6 b0a76a6efb49266813508e17c5b74b2890d4c8dad59c3a09170f03dc0265a344 3356687a0d87b14590aafdc0739693bb5e81d1a98faa93a01a1b550d6ff8f53f 06b1c4158d18ebfd24cf56f2a597d93da619a4034396e94a01a3c6b9f05cc22f 2737de6e9fa943644df9df52bcc4ee3550ee1c9f00383af205a2016ce22bc5b8 d7b7d6b290f789a119fad68df399a69485b9f13b16d7f2d811d2020c5368a913 9692a10e32c342ca6be46c7ee4e69643553793e9857e831e2099b529de302d0f 356d81a7feda3dc35d0abe1ec537106ca1cb9815912006e0ce4d2a640088be29 70c324895e9779a66ad59251ffeb638e4ca38e68ce7f9b8d716d4467b410c9fc 1fac82102e92daa1a05d0969faa31cb84a8ea418f4c13b088d63f1778e21f250 cd9b6d86a35cb901126de5836f52582bad67a6f7fcfebdf846a4459a3f5fe5a5 ab34c12fc1d5dc9537c48acdf8cc0103c5c21f2ffdc123193256168cb17364c6 270f2ce1b885f784db6ae5b22bcaa967f9e0e4d94c9677cc83877e021c941a37 ac1c7c18f9ed222c2cce8dd259d45d94ce767001a82114f83eea5768d5d1b78a e6da4abf31cd8ee3d400945ec1dd1f6cfe8a626cebeb12be0ad52ba7051ccd1d 03676fb5 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR17 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 19 /acute put dup 45 /hyphen put dup 82 /R put dup 83 /S put dup 97 /a 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 121 /y put dup 122 /z put readonly def /FontBBox{-33 -250 945 749}readonly def /UniqueID 5000795 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be356c29c3aec91ef17f5d31183b1e489f1fa 559a6693fbdd04d4f7e99886d8cb7cca13d4aa9ceb34a708af22e69ccef0e504 c3997075243585edf60b6447c8d01cffb04b6e3ddbaa7da5891286861322a795 e9d990fd56af2f5c2b35801a0b07874e4ac3170481d232e150443253c92a8b05 513d4a4154a99d91f629ccd30376500f5d16aaa9b203bad6350e4b19424c0e33 75f2949bc1927a6f6221252f8a96618e646aa1b0810753c98becc26b37837775 24bca72b818d314074c1c91f4e1f4148d3d703d12d179ee6970bd55d04aa9a9d 97f63f2c7dbd129ec2ed3ba5f8080270ea5854ea165dbcc061e2f47f1aaf0997 2ac9a689a2a9620b91b53f6b5092884372f5f7afca495f331138d85a97c5bb83 d8072dc88010fc89108b60bc06d4f93c722e8908d7ce3c1ea3c70e9e65edd769 7db17ce0e012ad37d4030f33d30a73fd655b7a2e572f3378ac933dd75082f2f1 68702be19bc60cef6bd5f0ada180d21408c04b7f7db9b6dbaefe09e809a80ddf 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92a1f51aad3e778a72912bc35879b8012e69620b6e47ec0e6ee51b3ba6e9fcca 0fc376112d1f22a25cea7fbf8335c53926177dabd342941c84e22c5f328d379e f85ef5ba20b2da378c67ba97d0631356afe994caa0baf331d3c5497f31bd0fc7 00157da442d9833cc14162eadc3bb465bb9b954b3e30301989d73b4ab9bc3bbd c0a532b93aee1e35f8ea7492e2fcbf9872c4ac543e387978a264be6bb652cd2b cae9bc7fd50ec733e56533d6a5b77010342d745dc5ff2d6be337de547e987641 fc47a4b9198227fa0cf7aa9d02a345fade6af5b901106116c175e0c7795923ef 2467f4bd71d27cd0205891af70a4657b142d26e81e4d9892fa41b445f3251187 3b36c427c2230bda12a107269c75e1a2172f0b2e48a4c435f76f009b75fff097 04aafaa321fea278c01531a8bdc29c6bb36381844ff567f1f7f9626f2a6ee872 d65dcf5cafb520d9df0b0f7f9062e668bf9eb0a481a13c34bda5f10ba1c0e11e de3dbea999f1818973b2f8f367abb0fe4aaa2023e6f7f3eca0a0e4e8800985b9 e6f6e3d1b4acffe0c90cd2b8ba899575050aa5ea661736fa350ec9baa6cad40a a3554df6116692ef38e8ee222634bdec7c7e306dc80434156f75af675b1a2998 941601cd64172ed691b4ede2e3900107b0d608784b3415e97db25b86edc05713 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont TeXDict begin 39158280 55380996 1000 600 600 (revMPARC.dvi) @start /Fa 203[ 59 52[{ } 1 119.552 /CMR12 rf /Fb 194[ 52 7[ 52 53[{ } 2 58.1154 /MSAM7 rf /Fc 165[ 39 7[ 42 82[{ } 2 58.1154 /MSBM7 rf /Fd 204[ 66 62 50[{ } 2 83.022 /LASY10 rf /Fe 180[ 44 1[ 34 24[ 18 39[ 45 6[ 19 45{ } 6 41.511 /CMSY5 rf /Ff 203[ 28 28 28 28 28 4[ 43 1[ 22 22 40[{ } 8 41.511 /CMR5 rf /Fg 143[ 83 20[ 92 46 106 120 20[ 73 73 73 73 3[ 74 1[ 74 1[ 74 6[ 73 73 7[ 67 13[ 62 4[ 44 44 61 61 4[ 46 28 12[{ } 22 83.022 /CMEX10 rf /Fh 155[ 51 100[{ } 1 99.6264 /CMMI12 rf /Fi 132[ 42 34 40 39 55 38 45 28 34 35 38 42 42 47 68 21 38 25 25 42 38 25 38 42 38 38 42 9[ 83 2[ 59 47 61 1[ 56 1[ 62 74 52 1[ 44 32 62 64 54 56 63 59 58 62 6[ 25 42 1[ 42 42 42 42 42 42 42 42 1[ 25 30 25 2[ 34 34 25 19[ 42 4[ 73 1[ 47 51 11[{ } 65 83.022 /CMTI10 rf /Fj 129[ 33 14[ 33 2[ 19 4[ 33 9[ 19 1[ 19 29[ 51 1[ 19 1[ 33 33 33 33 33 33 33 33 33 33 4[ 51 1[ 26 26 40[{ } 21 58.1154 /CMR7 rf /Fk 143[ 69 1[ 42 2[ 42 23 2[ 42 42 7[ 55 1[ 55 55 6[ 45 8[ 63 56 45 4[ 64 8[ 42 4[ 0 3[ 55 83 15[ 83 5[ 65 65 1[ 65 7[ 65 42 42 5[ 65 4[ 42 65 23 65{ } 30 83.022 /CMSY10 rf /Fl 194[ 65 7[ 65 53[{ } 2 83.022 /MSAM10 rf /Fm 129[ 31 5[ 33 42 3[ 28 28 3[ 37 51 1[ 31 24 22 3[ 28 31 27 1[ 32 25 20[ 47 32 26 11[ 31 1[ 19 59[{ } 20 41.511 /CMMI5 rf /Fn 143[ 55 1[ 34 3[ 20 30[ 51 1[ 38 18[ 0 3[ 45 66 19 14[ 66 24[ 52 4[ 34 1[ 20 52{ } 14 58.1154 /CMSY7 rf /Fo 129[ 42 3[ 39 41 47 59 40 48 30 39 37 37 42 1[ 50 73 1[ 43 34 29 48 40 41 39 43 36 36 44 35 2[ 32 2[ 57 48 69 78 48 57 49 51 63 66 53 63 67 81 57 71 46 36 69 65 53 61 69 59 63 62 44 1[ 65 42 65 23 23 18[ 54 7[ 52 49 3[ 43 47 36 41 50 48 1[ 29 39 41 36 34 37 1[ 47 53 11[{ } 76 83.022 /CMMI10 rf /Fp 129[ 34 5[ 38 48 33 39 25 31 31 1[ 34 33 41 59 1[ 35 27 23 3[ 31 35 30 29 36 28 8[ 62 4[ 50 2[ 50 53 63 46 56 37 29 5[ 48 1[ 50 3[ 34 1[ 20 33[ 39 30 33 1[ 39 5[ 28 30 1[ 38 43 11[{ } 42 58.1154 /CMMI7 rf /Fq 165[ 55 7[ 60 3[ 60 10[ 60 67[{ } 4 83.022 /MSBM10 rf /Fr 133[ 50 59 59 81 59 62 44 44 46 59 62 56 62 93 31 59 34 31 62 56 34 51 62 50 62 54 13[ 62 84 1[ 77 84 5[ 42 2[ 70 74 86 81 1[ 85 11[ 56 56 56 56 56 2[ 31 37 25[ 56 6[ 62 12[{ } 45 99.6264 /CMBX12 rf /Fs 133[ 60 4[ 75 52 53 55 1[ 75 67 75 112 37 71 1[ 37 75 67 41 61 75 60 1[ 65 13[ 75 100 1[ 92 2[ 128 3[ 50 19[ 67 67 67 67 67 2[ 37 45 25[ 67 19[{ } 32 119.552 /CMBX12 rf /Ft 129[ 35 2[ 35 1[ 37 1[ 51 37 39 27 28 28 1[ 39 35 39 59 20 37 22 20 39 35 22 31 39 31 1[ 35 7[ 53 3[ 53 51 39 52 1[ 48 1[ 53 65 44 1[ 36 25 1[ 55 2[ 54 51 1[ 53 55 5[ 20 1[ 35 1[ 35 1[ 35 35 35 35 35 35 20 24 20 44[{ } 52 66.4176 /CMR8 rf /Fu 134[ 29 117[ 32 3[{ } 2 49.8132 /CMSY6 rf /Fv 145[ 46 9[ 40 100[{ } 2 74.7198 /CMMI9 rf /Fw 133[ 34 41 41 55 41 43 30 30 30 41 43 38 43 64 21 41 23 21 43 38 23 34 43 34 43 38 21 8[ 79 3[ 43 4[ 58 25[ 38 5[ 21 26 21 2[ 30 30 21 19[ 38 6[ 43 12[{ } 39 74.7198 /CMR9 rf /Fx 139[ 34 35 36 14[ 39 49 43 31[ 67 65[{ } 7 74.7198 /CMBX9 rf /Fy 128[ 42 42 1[ 83 42 37 44 44 60 44 46 32 33 33 44 46 42 46 69 23 44 25 23 46 42 25 37 46 37 46 42 23 1[ 42 23 1[ 23 1[ 62 1[ 85 1[ 62 60 46 61 1[ 57 65 62 76 52 65 43 30 62 65 54 57 63 60 59 62 65 2[ 65 1[ 23 23 42 42 42 42 42 42 42 42 42 42 1[ 23 28 23 65 1[ 32 32 23 5[ 23 10[ 42 2[ 42 4[ 69 46 46 48 2[ 60 3[ 55 58 1[ 69 1[{ } 88 83.022 /CMR10 rf /Fz 134[ 50 1[ 69 50 53 37 38 39 1[ 53 48 53 80 27 50 1[ 27 53 48 29 44 53 42 53 46 3[ 27 1[ 27 6[ 66 53 72 1[ 65 3[ 57 75 1[ 36 5[ 69 1[ 72 6[ 27 48 48 48 48 48 48 48 48 48 48 1[ 27 1[ 27 2[ 37 37 26[ 53 13[{ } 49 83.022 /CMBX10 rf /FA 134[ 31 117[ 35 3[{ } 2 66.4176 /CMSY8 rf /FB 134[ 51 3[ 54 38 38 38 2[ 49 5[ 27 54 2[ 43 3[ 49 11[ 73 70 54 72 2[ 76 5[ 35 73 3[ 75 2[ 73 11[ 49 1[ 49 49 49 49 49 47[{ } 25 99.6264 /CMR12 rf /FC 155[ 73 100[{ } 1 143.462 /CMMI12 rf /FD 133[ 58 70 2[ 70 73 51 52 51 1[ 73 66 73 111 36 70 1[ 36 73 66 40 58 73 2[ 66 13[ 73 98 36[ 43 25[ 66 19[{ } 24 143.462 /CMR17 rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: a4 %%EndSetup %%Page: 1 1 1 0 bop FD 71 432 a(Renormalization) p 1058 432 a(group) p 1430 432 a(analysis) p 1923 432 a(of) p 2071 432 a(the) p 2297 432 a(self-a) l(v) l(oiding) p 3046 432 a(paths) p 3404 432 a(on) p 3587 432 a(the) p FC 995 614 a(d) p FD(-dimensional) p 1834 614 a(Sierpi) t(\023) p 2161 614 a(nski) p 2435 614 a(gask) l(ets) p FB 1506 855 a(T) p 1568 855 a(etsuy) m(a) p 1872 855 a(HA) p 2010 855 a(TTORI) p FA 2333 819 a(\003) p FB 1574 972 a(T) p 1636 972 a(oshiro) p 1924 972 a(TSUD) m(A) p FA 2266 936 a(y) p FB 1676 1173 a(2002/05/13) p Fz 0 1597 a(Running) p 390 1597 a(head.) p Fy 696 1597 a(Renormalization) p 1320 1597 a(group) p 1555 1597 a(of) p 1650 1597 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2490 a(lengths) p 3571 2490 a(of) 208 2582 y(`canonical) p 569 2582 a(ensem) n(ble',) p 943 2582 a(the) p 1078 2582 a(connectivit) n(y) p 1514 2582 a(constan) n(t) p 1827 2582 a(\(exp) r(onen) n(tial) p 2272 2582 a(gro) n(wth) p 2534 2582 a(of) p 2624 2582 a(path) p 2807 2582 a(n) n(um) n(b) r(ers) p 3120 2582 a(with) p 3298 2582 a(resp) r(ect) p 3564 2582 a(to) 208 2673 y(the) p 340 2673 a(length\),) p 626 2673 a(and) p 775 2673 a(the) p 907 2673 a(exp) r(onen) n(t) p 1238 2673 a(for) p 1356 2673 a(mean) p 1560 2673 a(square) p 1802 2673 a(displacemen) n(t.) 323 2764 y(W) p 396 2764 a(e) p 447 2764 a(apply) p 650 2764 a(the) p 774 2764 a(so) p 860 2764 a(de\014ned) p 1116 2764 a(framew) n(ork) p 1486 2764 a(to) p 1571 2764 a(pro) n(v) n(e) p 1770 2764 a(these) p 1959 2764 a(asymptotic) p 2345 2764 a(b) r(eha) n(viors) p 2681 2764 a(of) p 2760 2764 a(the) p 2884 2764 a(restricted) p 3218 2764 a(self-a) n(v) n(oiding) 208 2856 y(w) n(alks) p 417 2856 a(on) p 524 2856 a(the) p 656 2856 a(4-dimensional) p 1143 2856 a(pre-Sierpi) r(\023) p 1467 2856 a(nski) p 1628 2856 a(gask) n(et.) p Fz 0 3072 a(Key) p 201 3072 a(w) m(ords.) p Fy 554 3072 a(renormalization) p 1150 3072 a(group,) p 1408 3072 a(self-a) n(v) n(oiding) p 1884 3072 a(w) n(alk,) p 2101 3072 a(fractals,) p 2417 3072 a(Sierpinski) p 2799 3072 a(gask) n(et) p 0 5583 1536 4 v Fu 91 5636 a(\003) p Ft 127 5659 a(Graduate) p 496 5659 a(Sc) n(ho) r(ol) p 775 5659 a(of) p 910 5659 a(Mathematics,) p 1420 5659 a(Nago) n(y) n(a) p 1726 5659 a(Univ) n(ersit) n(y) p 2036 5659 a(,) p 2148 5659 a(Chikusa-ku,) p 2609 5659 a(Nago) n(y) n(a) p 2915 5659 a(464-8602,) p 3297 5659 a(JAP) p 3428 5659 a(AN,) p 3632 5659 a(e-mail:) 0 5738 y(hattori@math.nago) n(y) n(a-u.ac.jp,) p 933 5738 a(URL:) p 1125 5738 a(h) n(ttp://www.math.nago) n(y) n (a-u.ac.jp/~) p 2225 5738 a(hattori/) p Fu 94 5797 a(y) p Ft 127 5820 a(Dai-ic) n(hi) p 391 5820 a(Mutual) p 640 5820 a(Life) p 780 5820 a(Insurance,) p 1119 5820 a(1{13{1) p 1354 5820 a(Y) p 1401 5820 a(uraku-c) n(ho,) p 1750 5820 a(Chiy) n(o) r(da-ku,) p 2150 5820 a(T) p 2195 5820 a(oky) n(o) p 2362 5820 a(100-8411,) p 2676 5820 a(JAP) p 2807 5820 a(AN) p Fy 1899 6120 a(1) p 90 rotate dyy eop %%Page: 2 2 2 1 bop Fs 0 83 a(1) p 202 83 a(In) l(tro) t(duction) p 987 83 a(and) p 1246 83 a(main) p 1581 83 a(results.) p Fr 0 281 a(1.1) p 255 281 a(Self-a) m(v) m(oiding) p 915 281 a(w) m(alk) p 1175 281 a(on) p 1331 281 a(the) p 1525 281 a(Sierpi) s(\023) p 1808 281 a(nski) p 2043 281 a(gask) m(et.) p Fy 0 435 a(Self-a) n(v) n(oiding) p 496 435 a(w) n(alks) p 730 435 a(on) p 853 435 a(h) n(yp) r(ercubic) p 1282 435 a(lattices) p Fq 1576 435 a(Z) p Fp 1631 405 a(n) p Fy 1712 435 a(ha) n(v) n(e) p 1911 435 a(b) r(een) p 2114 435 a(studied) p 2412 435 a(mathematically) p 3008 435 a(for) p 3142 435 a(half) p 3313 435 a(a) p 3390 435 a(cen) n(tury) p 3656 435 a(,) p 3715 435 a(but) 0 534 y(compared) p 387 534 a(to) p 496 534 a(random) p 809 534 a(w) n(alks) p 1044 534 a(\(and) p 1246 534 a(di\013usion) p 1589 534 a(pro) r(cesses,) p 1982 534 a(their) p 2188 534 a(con) n(tin) n(uum) p 2607 534 a(limits\),) p 2904 534 a(amazingly) p 3306 534 a(little) p 3513 534 a(is) p 3605 534 a(kno) n(wn) 0 634 y([11) o(].) 125 734 y(F) p 172 734 a(or) p 272 734 a(random) p 575 734 a(w) n(alks,) p 823 734 a(nice) p 992 734 a(prop) r(erties) p 1381 734 a(suc) n(h) p 1567 734 a(as) p 1667 734 a(Mark) n(o) n(v) p 1968 734 a(prop) r(erties) p 2357 734 a(enabled) p 2660 734 a(deep) p 2852 734 a(and) p 3012 734 a(accurate) p 3342 734 a(studies,) p 3642 734 a(man) n(y) 0 833 y(of) p 100 833 a(whic) n(h) p 344 833 a(are) p 488 833 a(e\013ectiv) n(e) p 814 833 a(for) p 947 833 a(spaces) p 1207 833 a(with) p 1402 833 a(an) n(y) p 1565 833 a(dimension) p Fo 1963 833 a(n) p Fy(.) p 2090 833 a(On) p 2234 833 a(the) p 2383 833 a(other) p 2606 833 a(hand,) p 2844 833 a(self-a) n(v) n (oiding) p 3325 833 a(w) n(alks) p 3557 833 a(seem) p 3766 833 a(to) 0 933 y(ha) n(v) n(e) p 192 933 a(little) p 391 933 a(suc) n(h) p 579 933 a(strong) p 835 933 a(general) p 1122 933 a(metho) r(ds.) p 1491 933 a(In) p 1596 933 a(fact,) p 1784 933 a(their) p 1983 933 a(b) r(eha) n(viors) p 2356 933 a(are) p 2495 933 a(exp) r(ected) p 2842 933 a(to) p 2944 933 a(v) p 2983 933 a(ary) p 3130 933 a(drastically) p 3535 933 a(with) p 3725 933 a(the) 0 1032 y(dimension) p Fo 393 1032 a(n) p Fy 470 1032 a(for) p 597 1032 a(small) p Fo 814 1032 a(n) p Fy(,) p 915 1032 a(so) p 1017 1032 a(that) p 1197 1032 a(e\013ectiv) n(e) p 1517 1032 a(metho) r(ds) p 1852 1032 a(p) r(ossibly) p 2171 1032 a(v) p 2210 1032 a(ary) p 2356 1032 a(for) p 2483 1032 a(di\013eren) n(t) p 2811 1032 a(spaces.) 125 1132 y(T) p 178 1132 a(urning) p 440 1132 a(our) p 587 1132 a(atten) n(tion) p 943 1132 a(to) p 1044 1132 a(the) p 1186 1132 a(2-) p 1282 1132 a(and) p 1442 1132 a(3-dimensional) p 1968 1132 a(Sierpi) r(\023) p 2176 1132 a(nski) p 2348 1132 a(gask) n(ets,) p 2657 1132 a(there) p 2869 1132 a(are) p 3006 1132 a(w) n(orks) p 3241 1132 a(on) p 3356 1132 a(the) p 3498 1132 a(restricted) 0 1232 y(self-a) n(v) n(oiding) p 472 1232 a(w) n(alks) p 696 1232 a(\(a) p 794 1232 a(subset) p 1046 1232 a(of) p 1137 1232 a(self-a) n(v) n(oiding) p 1609 1232 a(w) n(alks,) p 1856 1232 a(to) p 1955 1232 a(b) r(e) p 2065 1232 a(de\014ned) p 2348 1232 a(in) p 2441 1232 a(Section) p 2729 1232 a(3.1\)) p 2892 1232 a(in) p 2985 1232 a([1],) p 3121 1232 a(and) p 3280 1232 a(mathematically) 0 1331 y(rigorous) p 325 1331 a(studies) p 608 1331 a(for) p 741 1331 a(the) p 889 1331 a(full) p 1040 1331 a(self-a) n(v) n(oiding) p 1521 1331 a(w) n(alk) p 1720 1331 a(\(including) p 2118 1331 a(a) p 2193 1331 a(pro) r(of) p 2415 1331 a(that) p 2601 1331 a(the) p 2750 1331 a(restriced) p 3093 1331 a(self-a) n(v) n(oiding) p 3574 1331 a(w) n(alk) p 3773 1331 a(of) 0 1431 y([1]) p 121 1431 a(are) p 265 1431 a(in) p 367 1431 a(the) p 516 1431 a(same) p 729 1431 a(univ) n(ersalit) n(y) p 1183 1431 a(classes) p 1452 1431 a(with) p 1647 1431 a(the) p 1795 1431 a(full) p 1946 1431 a(self-a) n(v) n(oiding) p 2427 1431 a(w) n(alk\),) p 2683 1431 a(with) p 2877 1431 a(further) p 3162 1431 a(precise) p 3441 1431 a(asymptotic) 0 1531 y(results,) p 287 1531 a(exist) p 482 1531 a(for) p 609 1531 a(b) r(oth) p 804 1531 a(the) p 947 1531 a(2-dimensional) p 1472 1531 a(Sierpi) r(\023) p 1680 1531 a(nski) p 1853 1531 a(gask) n(et) p 2106 1531 a([5,) p 2221 1531 a(4) o(,) p 2313 1531 a(7) o(]) p 2404 1531 a(and) p 2565 1531 a(the) p 2707 1531 a(3-dimensional) p 3233 1531 a(Sierpi) r(\023) p 3441 1531 a(nski) p 3614 1531 a(gask) n(et) 0 1630 y(\(4-simplex) p 404 1630 a(lattice\)) p 690 1630 a([6].) 125 1730 y(In) p 222 1730 a(the) p 358 1730 a(direction) p 697 1730 a(of) p 785 1730 a(generalization) p 1309 1730 a(to) p Fo 1403 1730 a(d) p Fy(-dimensional) p 1925 1730 a(Sierpi) r(\023) p 2133 1730 a(nski) p 2299 1730 a(gask) n(ets,) p 2603 1730 a(there) p 2809 1730 a(is) p 2886 1730 a(a) p 2948 1730 a(w) n(ork) p 3145 1730 a([9) o(]) p 3253 1730 a(on) p 3362 1730 a(the) p 3498 1730 a(restricted) 0 1829 y(mo) r(del) p 247 1829 a(for) p Fo 375 1829 a(d) p Fy 442 1829 a(=) p 531 1829 a(4) p Fo(;) p Fy 610 1829 a(5,) p 702 1829 a(follo) n(wing) p 1053 1829 a(the) p 1197 1829 a(lines) p 1387 1829 a(of) p 1482 1829 a([1]) p 1598 1829 a(with) p 1788 1829 a(a) p 1858 1829 a(propsal) p 2149 1829 a(of) p 2245 1829 a(an) p 2360 1829 a(appro) n(ximation) p 2915 1829 a(metho) r(d) p 3217 1829 a(for) p 3345 1829 a(general) p Fo 3632 1829 a(d) p Fy(.) p 3737 1829 a(\() p Fo(d) p Fy(-) 0 1929 y(dimensional) p 468 1929 a(Sierpi) r(\023) p 676 1929 a(nski) p 859 1929 a(gask) n(et) p 1124 1929 a(is) p 1218 1929 a(the) p Fo 1372 1929 a(d) p Fy 1441 1929 a(+) p 1531 1929 a(1-simplex) p 1913 1929 a(lattice) p 2177 1929 a(in) p 2285 1929 a([9) o(].\)) p 2496 1929 a(Ho) n(w) n(ev) n (er,) p 2867 1929 a(studies) p 3155 1929 a(in) p 3263 1929 a(the) p 3416 1929 a(direction) p 3773 1929 a(of) 0 2029 y(extending) p 387 2029 a(the) p 536 2029 a(rigorous) p 861 2029 a(renormalization) p 1462 2029 a(group) p 1704 2029 a(analysis) p 2022 2029 a(to) p Fo 2129 2029 a(d) p Fy(-dimensional) p 2663 2029 a(cases) p 2878 2029 a(ha) n(v) n(e) p 3075 2029 a(not) p 3229 2029 a(app) r(eared,) p 3617 2029 a(to) p 3725 2029 a(the) 0 2128 y(authors') p 324 2128 a(kno) n(wledge.) 125 2228 y(A) p 217 2228 a(main) p 428 2228 a(ob) p 521 2228 a(ject) p 682 2228 a(of) p 780 2228 a(this) p 944 2228 a(pap) r(er) p 1180 2228 a(to) p 1285 2228 a(prop) r(ose) p 1595 2228 a(a) p 1667 2228 a(general) p 1956 2228 a(and) p 2121 2228 a(mathematically) p 2712 2228 a(rigorous) p 3034 2228 a(renormalization) p 3632 2228 a(group) 0 2328 y(form) n(ulation) p 450 2328 a(of) p 547 2328 a(the) p 693 2328 a(self-a) n(v) n(oiding) p 1171 2328 a(w) n(alks) p 1400 2328 a(on) p Fo 1518 2328 a(d) p Fy(SG) p 1703 2328 a(for) p 1832 2328 a(all) p Fo 1950 2328 a(d) p Fy(,) p 2047 2328 a(from) p 2246 2328 a(whic) n(h) p 2486 2328 a(one) p 2641 2328 a(can) p 2796 2328 a(deriv) n(e) p 3043 2328 a(asymptotic) p 3473 2328 a(b) r(eha) n(viors.) 0 2427 y(As) p 131 2427 a(an) p 254 2427 a(application) p 691 2427 a(w) n(e) p 821 2427 a(pro) n(v) n(e) p 1053 2427 a(asymptotic) p 1488 2427 a(b) r(eha) n(viors,) p 1893 2427 a(suc) n(h) p 2088 2427 a(as) p 2198 2427 a(the) p 2349 2427 a(exp) r(onen) n(t) p 2715 2427 a(for) p 2850 2427 a(mean) p 3079 2427 a(square) p 3348 2427 a(displacemen) n(t,) 0 2527 y(of) p 97 2527 a(the) p 242 2527 a(restricted) p 614 2527 a(mo) r(del) p 863 2527 a(of) p 959 2527 a(self-a) n(v) n(oiding) p 1437 2527 a(w) n(alks) p 1665 2527 a(on) p 1783 2527 a(4SG.) p 1988 2527 a(\(The) p 2194 2527 a(restricted) p 2566 2527 a(mo) r(del) p 2814 2527 a(considers) p 3173 2527 a(those) p 3392 2527 a(self-a) n(v) n(oiding) 0 2626 y(w) n(alks) p 226 2626 a(whic) n(h) p 464 2626 a(do) r(es) p 651 2626 a(not) p 799 2626 a(tak) n(e) p 979 2626 a(2) p 1048 2626 a(or) p 1150 2626 a(more) p 1357 2626 a(steps) p 1566 2626 a(in) p 1663 2626 a(ro) n(w) p 1822 2626 a(in) p 1919 2626 a(eac) n(h) p 2106 2626 a(unit) p 2281 2626 a(simplices) p 2632 2626 a(\(Section) p 2955 2626 a(3.1\).\)) 125 2726 y(W) p 203 2726 a(e) p 270 2726 a(emphasize) p 669 2726 a(that) p 851 2726 a(a) p 922 2726 a(rigorous) p 1244 2726 a(renormalization) p 1841 2726 a(group) p 2079 2726 a(analysis) p 2393 2726 a(is) p 2479 2726 a(non-trivial) p 2889 2726 a(for) p 3018 2726 a(the) p 3164 2726 a(self-a) n(v) n(oiding) p 3641 2726 a(w) n(alks) 0 2826 y(on) p Fo 113 2826 a(d) p Fy(SG.) p 316 2826 a(Though) p 622 2826 a(it) p 703 2826 a(is) p 784 2826 a(easy) p 964 2826 a(to) p 1063 2826 a(write) p 1273 2826 a(do) n(wn) p 1490 2826 a(the) p 1631 2826 a(renormalization) p 2224 2826 a(group) p 2457 2826 a(recursion) p 2810 2826 a(equations) p 3180 2826 a(for) p 3305 2826 a(small) p Fo 3519 2826 a(d) p Fy(,) p 3611 2826 a(it) p 3692 2826 a(is) p 3773 2826 a(of) 0 2925 y(course) p 256 2925 a(another) p 563 2925 a(thing) p 782 2925 a(to) p 885 2925 a(analyze) p 1185 2925 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n(ets) p 2298 3224 a(are) p 2430 3224 a(app) r(ealing,) p 2823 3224 a(but) p 2969 3224 a(b) r(ecause) p 3269 3224 a(\(as) p 3396 3224 a(w) n(e) p 3512 3224 a(will) p 3662 3224 a(sho) n(w) 0 3324 y(in) p 101 3324 a(this) p 268 3324 a(pap) r(er\)) p 538 3324 a(w) n(e) p 664 3324 a(can) p 821 3324 a(form) n(ulate) p 1199 3324 a(and) p 1365 3324 a(pro) n(v) n(e) p 1593 3324 a(with) p 1787 3324 a(mathematical) p 2313 3324 a(rigor) p 2516 3324 a(that) p 2700 3324 a(an) p 2820 3324 a(appropriate) p 3272 3324 a(renormalization) 0 3424 y(group) p 247 3424 a(form) n(ulation) p 706 3424 a(con) n(tains) p 1043 3424 a(full) p 1199 3424 a(imformation) p 1684 3424 a(of) p 1789 3424 a(asymptotic) p 2228 3424 a(b) r(eha) n(viors) p 2611 3424 a(of) p 2717 3424 a(self-a) n(v) n (oiding) p 3204 3424 a(w) n(alks.) p 3497 3424 a(Since) p 3725 3424 a(the) 0 3523 y(renormalization) p 597 3523 a(group) p 835 3523 a(analysis) p 1149 3523 a(con) n(tains) p 1477 3523 a(full) p 1624 3523 a(information) p 2076 3523 a(on) p 2193 3523 a(asymptotic) p 2622 3523 a(b) r(eha) n(viors,) p 3019 3523 a(the) p 3164 3523 a(authors) p 3467 3523 a(think) p 3688 3523 a(that) 0 3623 y(it) p 80 3623 a(is) p 160 3623 a(to) r(o) p 302 3623 a(imp) r(ortan) n(t) p 690 3623 a(not) p 835 3623 a(to) p 933 3623 a(analyze) p 1227 3623 a(them) p 1436 3623 a(with) p 1621 3623 a(mathematical) p 2139 3623 a(rigor) p 2335 3623 a(and) p 2493 3623 a(in) p 2586 3623 a(generalit) n(y) p 2966 3623 a(\(as) p 3096 3623 a(w) n(e) p 3215 3623 a(do) p 3327 3623 a(in) p 3421 3623 a(this) p 3579 3623 a(pap) r(er\).) p Fr 0 3855 a(1.2) p 255 3855 a(Renormalization) p 1099 3855 a(group) p 1419 3855 a(approac) m(h.) p Fy 0 4008 a(General) p 313 4008 a(`philosoph) n(y') p 778 4008 a(of) p 876 4008 a(the) p 1023 4008 a(renormalization) p 1622 4008 a(group) p 1861 4008 a(\(R) n(G\)) p 2080 4008 a(in) p 2181 4008 a(ph) n(ysics) p 2471 4008 a(\(and) p 2668 4008 a(the) p 2815 4008 a(previous) p 3149 4008 a(rigorous) p 3471 4008 a(studies) p 3752 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a(the) p 585 4473 a(\014xed) p 784 4473 a(p) r(oin) n(ts) p 1032 4473 a(and) p 1192 4473 a(the) p 1333 4473 a(tra) p 1445 4473 a(jectories) p 1769 4473 a(of) p 1862 4473 a(the) p 2003 4473 a(R) n(G) p 2153 4473 a(\015o) n(ws,) p 2381 4473 a(suc) n(h) p 2566 4473 a(as) p 2667 4473 a(uniqueness) p 3083 4473 a(of) p 3176 4473 a(certain) p 3452 4473 a(\014xed) p 3651 4473 a(p) r(oin) n(t) 208 4573 y(and) p 369 4573 a(con) n(v) n(ergence) p 828 4573 a(of) p 923 4573 a(critical) p 1200 4573 a(tra) p 1312 4573 a(jectories.) 55 4739 y(\(ii\)) p 208 4739 a(Deriv) n(e) p 470 4739 a(asymptotic) p 897 4739 a(b) r(eha) n(viors) p 1269 4739 a(of) p 1363 4739 a(the) p 1506 4739 a(self-a) n(v) n(oiding) p 1982 4739 a(paths) p 2208 4739 a(from) p 2404 4739 a(the) p 2547 4739 a(prop) r(erties) p 2938 4739 a(of) p 3033 4739 a(R) n(G) p 3184 4739 a(\015o) n(ws.) 125 4915 y(The) p 298 4915 a(R) n(G) p 452 4915 a(is) p 538 4915 a(a) p 610 4915 a(dynamical) p 1012 4915 a(system) p 1290 4915 a(determined) p 1726 4915 a(b) n(y) p 1844 4915 a(a) p 1916 4915 a(recursion) p 2274 4915 a(map) p Fo 2464 4894 a(~) p Fy 2462 4915 a(\010,) p 2576 4915 a(whic) n(h) p 2816 4915 a(will) p 2975 4915 a(b) r(e) p 3091 4915 a(de\014ned) p 3379 4915 a(in) p 3479 4915 a(\(10\),) p 3681 4915 a(on) p 3798 4915 a(a) 0 5014 y(\014nite) p 216 5014 a(dimensional) p 677 5014 a(Eulidean) p 1028 5014 a(space) p 1254 5014 a(\(the) p 1433 5014 a(parameter) p 1834 5014 a(space) p Fq 2059 5014 a(R) p Fn 2119 4980 a(I) p Fm 2157 4989 a(d) p Fy 2228 5014 a(de\014ned) p 2517 5014 a(in) p 2618 5014 a(\(3\)\).) p 2828 5014 a(F) p 2875 5014 a(or) p 2981 5014 a(general) p 3271 5014 a(case) p 3451 5014 a(of) p 3549 5014 a(ph) n(ysical) 0 5114 y(in) n(terest,) p 325 5114 a(w) n(e) p 451 5114 a(should) p 717 5114 a(consider) p 1044 5114 a(in\014nite) p 1328 5114 a(dimensional) p 1789 5114 a(parameter) p 2189 5114 a(space,) p 2438 5114 a(but) p 2593 5114 a(the) p 2739 5114 a(so) p 2845 5114 a(called) p 3083 5114 a(\014nite) p 3298 5114 a(rami\014edness) p 3773 5114 a(of) p Fo 0 5214 a(d) p Fy(SG) p 189 5214 a(implies) p 478 5214 a(that) p 666 5214 a(the) p 816 5214 a(R) n(G) p 975 5214 a(in) p 1079 5214 a(the) p 1229 5214 a(presen) n(t) p 1525 5214 a(study) p 1761 5214 a(is) p 1852 5214 a(\014nite) p 2071 5214 a(dimensional.) p 2582 5214 a(The) p 2760 5214 a(R) n(G) p 2919 5214 a(map) p 3111 5214 a(is) p 3201 5214 a(a) p 3278 5214 a(resp) r(onse) p 3621 5214 a(in) p 3725 5214 a(the) 0 5313 y(parameter) p 399 5313 a(space) p 624 5313 a(to) p 727 5313 a(the) p 873 5313 a(`scale) p 1097 5313 a(transformation') p 1688 5313 a(\(smo) r(othing) p 2127 5313 a(out) p 2277 5313 a(or) p 2381 5313 a(putting) p 2678 5313 a(in) p 2777 5313 a(\014ner) p 2969 5313 a(structures) p 3360 5313 a(to) p 3464 5313 a(the) p 3609 5313 a(paths\)) 0 5413 y(on) p 109 5413 a(the) p 246 5413 a(space) p 461 5413 a(of) p 550 5413 a(paths.) p 806 5413 a(\(The) p 1003 5413 a(transformation) p 1563 5413 a(suitable) p 1866 5413 a(for) p 1987 5413 a(paths) p 2207 5413 a(on) p Fo 2316 5413 a(d) p Fy(SG) p 2492 5413 a(is) p 2569 5413 a(a) p 2632 5413 a(decimation,) p 3075 5413 a(whic) n(h) p 3306 5413 a(will) p 3456 5413 a(b) r(e) p 3563 5413 a(implicit) 0 5513 y(in) p 97 5513 a(the) p 240 5513 a(pro) r(of) p 457 5513 a(of) p 551 5513 a(Prop) r(osition) p 998 5513 a(4.\)) 125 5622 y(The) p 295 5622 a(quan) n(tities) p 678 5622 a(w) n(e) p 801 5622 a(need) p 994 5622 a(to) p 1096 5622 a(extract) p 1380 5622 a(from) p 1576 5622 a(the) p 1719 5622 a(R) n(G) p 1871 5622 a(map) p Fo 2058 5601 a(~) p Fy 2055 5622 a(\010) p 2143 5622 a(are) p 2281 5622 a(the) p 2424 5622 a(follo) n(wing.) 78 5788 y(\(i\)) p 208 5788 a(The) p 378 5788 a(largest) p 646 5788 a(eigen) n(v) p 868 5788 a(alue) p Fo 1043 5788 a(\025) p Fy 1119 5788 a(of) p 1214 5788 a(the) p 1357 5788 a(di\013eren) n(tial) p 1772 5788 a(map) p 1957 5788 a(of) p Fo 2054 5767 a(~) p Fy 2051 5788 a(\010) p 2139 5788 a(at) p 2240 5788 a(a) p 2309 5788 a(self-a) n(v) n(oiding) p 2785 5788 a(\014xed) p 2986 5788 a(p) r(oin) n(t) p Fo 3201 5788 a(~) p 3202 5788 a(x) p Fp 3249 5800 a(c) p Fy 3297 5788 a(.) 1899 6120 y(2) p 90 rotate dyy eop %%Page: 3 3 3 2 bop Fy 55 83 a(\(ii\)) p 208 83 a(The) p 384 83 a(critical) p 667 83 a(p) r(oin) n(t) p Fo 889 83 a(\014) p Fp 936 95 a(c) p Fy 970 83 a(,) p 1028 83 a(whic) n(h) p 1271 83 a(is) p 1360 83 a(the) p 1509 83 a(in) n(tersection) p 1960 83 a(p) r(oin) n(t) p 2183 83 a(of) p 2283 83 a(the) p 2432 83 a(critical) p 2714 83 a(surface) p 3000 83 a(\(the) p 3181 83 a(set) p 3316 83 a(of) p 3416 83 a(p) r(oin) n(ts) p 3671 83 a(from) 208 183 y(whic) n(h) p 451 183 a(the) p 600 183 a(tra) p 712 183 a(jectories) p 1043 183 a(of) p 1143 183 a(R) n(G) p 1300 183 a(con) n(v) n(erge) p 1646 183 a(to) p 1753 183 a(the) p 1902 183 a(self-a) n(v) n(oiding) p 2383 183 a(\014xed) p 2590 183 a(p) r(oin) n(t) p Fo 2811 183 a(~) p 2812 183 a(x) p Fp 2859 195 a(c) p Fy 2894 183 a(\)) p 2960 183 a(and) p 3127 183 a(the) p 3276 183 a(canonical) p 3646 183 a(curv) n(e) 208 282 y(\(the) p 383 282 a(curv) n(e) p 604 282 a(de\014ned) p 890 282 a(b) n(y) p 1006 282 a(\(17\)\).) 0 456 y(W) p 78 456 a(e) p 144 456 a(giv) n(e) p 315 456 a(the) p 458 456 a(precise) p 732 456 a(de\014nitions) p 1134 456 a(of) p Fo 1229 456 a(\025) p Fy 1306 456 a(and) p Fo 1468 456 a(\014) p Fp 1515 468 a(c) p Fy 1577 456 a(and) p 1739 456 a(also) p 1906 456 a(the) p 2049 456 a(assumptions) p 2522 456 a(on) p 2638 456 a(the) p 2781 456 a(R) n(G) p 2933 456 a(map) p Fo 3121 435 a(~) p Fy 3118 456 a(\010) p 3206 456 a(at) p 3308 456 a(\(FP1\)) p 3553 456 a({) p 3623 456 a(\(FP4\)) 0 556 y(and) p 155 556 a(\() p Fo(C) p 252 556 a(S) p Fy 308 556 a(1\)) p 404 556 a(in) p 495 556 a(Section) p 780 556 a(3.1.) p 944 556 a(\(T) p 1029 556 a(o) p 1092 556 a(state) p 1290 556 a(them) p 1496 556 a(rigorously) p 1850 556 a(,) p 1893 556 a(w) n(e) p 2010 556 a(need) p 2197 556 a(to) p 2293 556 a(prepare) p 2587 556 a(tec) n(hnically) p 2996 556 a(cum) n(b) r(ersome) p 3466 556 a(de\014nitions) 0 655 y(in) p 97 655 a(Section) p 387 655 a(2) p 457 655 a(starting) p 766 655 a(from) p 963 655 a(the) p 1106 655 a(de\014nition) p 1475 655 a(of) p Fo 1569 655 a(d) p Fy(SG.\)) 125 755 y(In) p 232 755 a(this) p 397 755 a(pap) r(er) p 634 755 a(w) n(e) p 760 755 a(will) p 920 755 a(pro) n(v) n(e) p 1148 755 a(the) p 1294 755 a(follo) n(wing.) p 1688 755 a(Fix) p Fo 1840 755 a(d) p Fl 1912 755 a(=) p Fy 2006 755 a(2.) p 2118 755 a(F) p 2165 755 a(or) p 2271 755 a(eac) n(h) p Fo 2461 755 a(k) p Fk 2536 755 a(2) p Fq 2620 755 a(Z) p Fj 2675 767 a(+) p Fy 2731 755 a(,) p 2786 755 a(let) p Fo 2910 755 a(N) p Fy 2986 755 a(\() p Fo(k) p Fy 3064 755 a(\)) p 3127 755 a(b) r(e) p 3244 755 a(the) p 3390 755 a(n) n(um) n(b) r(er) p 3696 755 a(of) p Fo 3794 755 a(k) p Fy 0 855 a(step) p 178 855 a(self-a) n(v) n(oiding) p 655 855 a(paths) p 884 855 a(on) p Fo 1001 855 a(d) p Fy(SG) p 1185 855 a(starting) p 1497 855 a(from) p 1695 855 a(the) p 1840 855 a(origin) p Fo 2078 855 a(O) p Fy 2143 855 a(,) p 2196 855 a(and) p 2360 855 a(let) p Fo 2482 855 a(E) p Fp 2543 867 a(k) p Fy 2584 855 a([) p Fk(\001) p Fy(]) p 2683 855 a(b) r(e) p 2798 855 a(the) p 2943 855 a(exp) r(ectation) p 3392 855 a(with) p 3583 855 a(resp) r(ect) 0 954 y(to) p 101 954 a(the) p 244 954 a(uniform) p 556 954 a(distribution) p 1009 954 a(\(a) n(v) n(eraging) p 1413 954 a(with) p 1602 954 a(equal) p 1821 954 a(w) n(eigh) n(t\)) p 2116 954 a(on) p 2231 954 a(suc) n(h) p 2418 954 a(paths.) p Fz 0 1118 a(Theorem) p 405 1118 a(1) p 485 1118 a(\(Theorem) p 928 1118 a(10) p 1055 1118 a(and) p 1239 1118 a(Theorem) p 1645 1118 a(11\)) p Fi 1819 1118 a(If) p 1906 1118 a(ther) l(e) p 2113 1118 a(exists) p 2340 1118 a(a) p 2412 1118 a(critic) l(al) p 2691 1118 a(p) l(oint) p Fo 2901 1118 a(\014) p Fp 2948 1130 a(c) p Fi 3012 1118 a(then) 73 1322 y(\(i\)) p Fy 234 1322 a(lim) p Fp 208 1376 a(k) p Fn 244 1376 a(!1) p Fy 403 1265 a(1) p 400 1303 46 4 v Fo 400 1379 a(k) p Fy 484 1322 a(log) p Fo 605 1322 a(N) p Fy 681 1322 a(\() p Fo(k) p Fy 759 1322 a(\)) p 814 1322 a(=) p Fo 902 1322 a(\014) p Fp 949 1334 a(c) p Fi 997 1322 a(.) 47 1572 y(\(ii\)) p Fy 234 1572 a(lim) p Fp 208 1626 a(k) p Fn 244 1626 a(!1) p Fy 463 1516 a(1) p 400 1553 167 4 v 400 1629 a(log) p Fo 521 1629 a(k) p Fy 591 1572 a(log) p Fo 712 1572 a(E) p Fp 773 1584 a(k) p Fy 814 1572 a([) p Fk(j) p Fo(w) p Fy 921 1572 a(\() p Fo(k) p Fy 999 1572 a(\)) p Fk(j) p Fp 1054 1537 a(s) p 1098 1537 a(d) p Fm 1133 1545 a(w) p Fy 1184 1572 a(]) p 1230 1572 a(=) p Fo 1318 1572 a(s;) p 1453 1572 a(s) p Fl 1515 1572 a(=) p Fy 1603 1572 a(0) p Fi(,) p 1699 1572 a(wher) l(e) p Fk 1934 1572 a(j) p 1975 1572 a(\001) p 2017 1572 a(j) p Fi 2070 1572 a(denotes) p 2369 1572 a(the) p 2507 1572 a(Euclide) l(an) p 2888 1572 a(length) p 3132 1572 a(and) p Fo 3293 1572 a(d) p Fp 3336 1584 a(w) p Fy 3413 1572 a(=) 3510 1516 y(log) p Fo 3632 1516 a(\025) p 3510 1553 170 4 v Fy 3513 1629 a(log) p 3635 1629 a(2) p Fi 3704 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a(the) p 575 2386 a(notions) p 857 2386 a(suc) n(h) p 1036 2386 a(as) p 1129 2386 a(\014xed) p 1321 2386 a(p) r(oin) n(ts) p 1562 2386 a(and) p 1714 2386 a(critical) p 1982 2386 a(p) r(oin) n(ts) p 2223 2386 a(are) p 2353 2386 a(not) p 2492 2386 a(new) p 2654 2386 a(from) p 2841 2386 a(the) p 2976 2386 a(view) p 3158 2386 a(p) r(oin) n(t) p 3366 2386 a(of) p 3452 2386 a(philosoph) n(y) 0 2485 y(of) p 92 2485 a(R) n(G.) p 263 2485 a(What) p 493 2485 a(is) p 574 2485 a(new) p 741 2485 a(here) p 919 2485 a(is) p 999 2485 a(that) p 1176 2485 a(w) n(e) p 1295 2485 a(prop) r(ose) p 1600 2485 a(a) p 1666 2485 a(mathematically) p 2251 2485 a(w) n(ell-de\014ned) p 2702 2485 a(form) n(ulation) p 3147 2485 a(\(Section) p 3467 2485 a(3.1\)) p 3630 2485 a(whic) n(h) 0 2585 y(are) p 140 2585 a(su\016cien) n(t) p 491 2585 a(\(Section) p 815 2585 a(3.2\)) p 983 2585 a(to) p 1086 2585 a(pro) n(v) n(e) p 1312 2585 a(asymptotic) p 1741 2585 a(b) r(eha) n(viors) p 2114 2585 a(of) p 2211 2585 a(self-a) n(v) n(oiding) p 2688 2585 a(w) n(alks) p 2916 2585 a(on) p Fo 3033 2585 a(d) p Fy(SG) p 3217 2585 a(with) p 3408 2585 a(all) p Fo 3525 2585 a(d) p Fy(,) p 3621 2585 a(giving) 0 2685 y(a) p 75 2685 a(mathematical) p 602 2685 a(evidence) p 942 2685 a(that) p 1127 2685 a(the) p 1276 2685 a(dynamics) p 1649 2685 a(of) p 1749 2685 a(R) n(G) p 1906 2685 a(con) n(tains) p 2238 2685 a(information) p 2693 2685 a(on) p 2814 2685 a(the) p 2963 2685 a(asymptotic) p 3395 2685 a(b) r(eha) n(viors) p 3773 2685 a(of) 0 2784 y(sto) r(c) n(hastic) p 384 2784 a(pro) r(cesses.) 125 2884 y(As) p 252 2884 a(an) p 371 2884 a(application) p 804 2884 a(of) p 903 2884 a(the) p 1051 2884 a(form) n(ulation,) p 1527 2884 a(w) n(e) p 1653 2884 a(pro) n(v) n(e) p 1882 2884 a(in) p 1983 2884 a(Section) p 2278 2884 a(5) p 2351 2884 a(that) p 2536 2884 a(the) p 2683 2884 a(assumptions) p 3159 2884 a(on) p 3279 2884 a(the) p 3426 2884 a(R) n(G) p 3582 2884 a(map) p 3771 2884 a(in) 0 2984 y(Section) p 291 2984 a(3.1) p 424 2984 a(are) p 563 2984 a(satis\014ed) p 882 2984 a(for) p 1009 2984 a(the) p 1152 2984 a(restricted) p 1522 2984 a(mo) r(del) p 1769 2984 a(of) p 1863 2984 a(self-a) n(v) n(oiding) p 2339 2984 a(paths) p 2565 2984 a(on) p 2681 2984 a(4SG.) p Fz 0 3164 a(Theorem) p 405 3164 a(2) p 485 3164 a(\(Theorem) p 928 3164 a(31) p 1055 3164 a(and) p 1239 3164 a(Theorem) p 1645 3164 a(33\)) p Fi 1819 3164 a(The) p 2001 3164 a(self-avoiding) p 2493 3164 a(\014xe) l(d) p 2697 3164 a(p) l(oint) p Fo 2919 3164 a(~) p 2920 3164 a(x) p Fp 2967 3176 a(c) p Fi 3044 3164 a(and) p 3217 3164 a(the) p 3368 3164 a(critic) l(al) p 3660 3164 a(p) l(oint) p Fo 0 3263 a(\014) p Fp 47 3275 a(c;r) r(es) p Fi 226 3263 a(of) p 323 3263 a(the) p 461 3263 a(r) l(estricte) l(d) p 824 3263 a(mo) l(del) p 1062 3263 a(on) p 1180 3263 a(the) p Fy 1318 3263 a(4) p Fi 1390 3263 a(dimensional) p 1852 3263 a(pr) l(e-Sierpi) r(\023) p 2204 3263 a(nski) p 2380 3263 a(gasket) p 2629 3263 a(\() p Fy(4) p Fi(SG\)) p 2879 3263 a(exists.) 125 3363 y(In) p 233 3363 a(p) l(articular,) p 638 3363 a(the) p 776 3363 a(numb) l(er) p Fo 1072 3363 a(N) p Fp 1139 3375 a(r) r(es) p Fy 1238 3363 a(\() p Fo(k) p Fy 1316 3363 a(\)) p Fi 1378 3363 a(of) p 1476 3363 a(r) l(estricte) l(d) p 1839 3363 a(self-avoiding) p 2318 3363 a(p) l(aths) p 2533 3363 a(of) p 2630 3363 a(length) p Fo 2874 3363 a(k) p Fi 2950 3363 a(starting) p 3256 3363 a(fr) l(om) p Fy 3453 3363 a(0) p Fi 3524 3363 a(satis\014es) p Fy 1438 3582 a(lim) p Fp 1411 3637 a(k) p Fn 1447 3637 a(!1) p Fy 1606 3526 a(1) p 1604 3563 46 4 v Fo 1604 3639 a(k) p Fy 1688 3582 a(log) p Fo 1809 3582 a(N) p Fp 1876 3594 a(r) r(es) p Fy 1975 3582 a(\() p Fo(k) p Fy 2053 3582 a(\)) p 2108 3582 a(=) p Fo 2196 3582 a(\014) p Fp 2243 3594 a(c;r) r(es) p Fo 2406 3582 a(:) p Fi 0 3848 a(and) p 163 3848 a(the) p 302 3848 a(exp) l(onent) p 650 3848 a(for) p 784 3848 a(me) l(an) p 1007 3848 a(squar) l(e) p 1266 3848 a(displac) l(ement) p 1758 3848 a(for) p 1892 3848 a(the) p 2031 3848 a(r) l(estricte) l(d) p 2396 3848 a(mo) l(del) p 2635 3848 a(is) p Fo 2726 3848 a(d) p Fp 2769 3860 a(w) p Fy 2848 3848 a(=) 2949 3792 y(log) p Fo 3070 3792 a(\025) p 2949 3829 170 4 v Fy 2952 3905 a(log) p 3073 3905 a(2) 3154 3848 y(=) p 3245 3848 a(1) p Fo(:) p Fy(6657696) p Fk 3618 3848 a(\001) p 3655 3848 a(\001) p 3692 3848 a(\001) p Fi 3711 3848 a(,) p 3768 3848 a(in) 0 3987 y(the) p 138 3987 a(sense) p 359 3987 a(that) p Fy 1159 4105 a(lim) p Fp 1133 4159 a(k) p Fn 1169 4159 a(!1) p Fy 1388 4049 a(1) p 1325 4086 167 4 v 1325 4162 a(log) p Fo 1446 4162 a(k) p Fy 1516 4105 a(log) p Fo 1637 4105 a(E) p Fp 1698 4117 a(r) r(es;k) p Fy 1854 4105 a([) p Fk(j) p Fo(w) p Fy 1961 4105 a(\() p Fo(k) p Fy 2039 4105 a(\)) p Fk(j) p Fp 2094 4070 a(s) p 2138 4070 a(d) p Fm 2173 4078 a(w) p Fy 2224 4105 a(]) p 2270 4105 a(=) p Fo 2358 4105 a(s;) p 2493 4105 a(s) p Fl 2555 4105 a(=) p Fy 2643 4105 a(0) p Fo(;) p Fi 0 4292 a(wher) l(e) p Fo 227 4292 a(E) p Fp 288 4304 a(r) r(es;k) p Fi 467 4292 a(is) p 548 4292 a(the) p 679 4292 a(exp) l(e) l(ctation) p 1100 4292 a(with) p 1273 4292 a(r) l(esp) l(e) l(ct) p 1536 4292 a(to) p 1629 4292 a(the) p 1759 4292 a(pr) l(ob) l(ability) p 2152 4292 a(me) l(asur) l(e) p 2466 4292 a(with) p 2639 4292 a(e) l(qual) p 2841 4292 a(weight) p 3091 4292 a(on) p 3202 4292 a(length) p Fo 3439 4292 a(k) p Fi 3507 4292 a(r) l(estricte) l(d) 0 4392 y(self-avoiding) p 479 4392 a(p) l(aths) p 694 4392 a(starting) p 1000 4392 a(at) p Fo 1100 4392 a(O) p Fi 1165 4392 a(.) p Fz 125 4596 a(Ac) m(kno) m (wledgemen) m(ts.) p Fy 1004 4596 a(T.) p 1123 4596 a(Hattori) p 1424 4596 a(thanks) p 1702 4596 a(Prof.) p 1916 4596 a(S.) p 2021 4596 a(Kusuok) p 2292 4596 a(a) p 2368 4596 a(and) p 2538 4596 a(Prof.) p 2752 4596 a(K.) p 2875 4596 a(Hattori) p 3176 4596 a(for) p 3311 4596 a(collab) r(orations,) 0 4696 y(and) p 164 4696 a(Prof.) p 374 4696 a(T.) p 487 4696 a(Hara) p 696 4696 a(for) p 825 4696 a(encouragemen) n(ts.) p 1469 4696 a(The) p 1642 4696 a(authors) p 1946 4696 a(thank) p 2186 4696 a(the) p 2332 4696 a(mem) n(b) r(ers) p 2686 4696 a(in) p 2786 4696 a(Departmen) n(t) p 3251 4696 a(of) p 3348 4696 a(Mathematics,) 0 4796 y(Rikky) n(o) p 282 4796 a(Univ) n(erist) n(y) p 648 4796 a(,) p 697 4796 a(where) p 937 4796 a(part) p 1116 4796 a(of) p 1210 4796 a(this) p 1371 4796 a(w) n(ork) p 1574 4796 a(w) n(as) p 1733 4796 a(done,) p 1954 4796 a(and) p 2115 4796 a(the) p 2257 4796 a(referee) p 2522 4796 a(of) p 2616 4796 a(a) p 2685 4796 a(journal) p 2968 4796 a(for) p 3095 4796 a(detailed) p 3408 4796 a(c) n(hec) n(k) p 3631 4796 a(of) p 3725 4796 a(the) 0 4895 y(pro) r(ofs.) 125 4995 y(The) p 295 4995 a(researc) n(h) p 616 4995 a(of) p 711 4995 a(T.) p 821 4995 a(Hattori) p 1114 4995 a(is) p 1197 4995 a(supp) r(orted) p 1588 4995 a(in) p 1684 4995 a(part) p 1864 4995 a(b) n(y) p 1979 4995 a(a) p 2048 4995 a(Gran) n(t-in-Aid) p 2547 4995 a(for) p 2674 4995 a(Scien) n(ti\014c) p 3027 4995 a(Researc) n(h) p 3377 4995 a(\(C\)) p 3529 4995 a(from) p 3725 4995 a(the) 0 5095 y(Ministry) p 337 5095 a(of) p 432 5095 a(Education,) p 853 5095 a(Culture,) p 1181 5095 a(Sp) r(orts,) p 1465 5095 a(Science) p 1755 5095 a(and) p 1917 5095 a(T) p 1970 5095 a(ec) n(hnology) p 2320 5095 a(.) p Fs 0 5369 a(2) p 202 5369 a(Renormalization) p 1214 5369 a(group.) p Fr 0 5567 a(2.1) p 255 5567 a(Self-a) m(v) m(oiding) p 915 5567 a(paths) p 1219 5567 a(on) p 1375 5567 a(the) p Fh 1569 5567 a(d) p Fr(-dimensional) p 2274 5567 a(pre-Sierpi) s(\023) p 2753 5567 a(nski) p 2989 5567 a(gask) m(et.) p Fy 0 5720 a(Let) p Fo 148 5720 a(d) p Fl 214 5720 a(=) p Fy 302 5720 a(2) p 370 5720 a(b) r(e) p 483 5720 a(an) p 597 5720 a(in) n(teger.) p 904 5720 a(W) p 982 5720 a(e) p 1046 5720 a(de\014ne) p 1285 5720 a(a) p Fo 1353 5720 a(d) p Fy(-dimensional) p 1881 5720 a(pre-Sierpi) r(\023) p 2233 5720 a(nski) p 2404 5720 a(gask) n(et) p 2658 5720 a(\(pre-) p Fo(d) p Fy(SG\)) p 3047 5720 a(as) p 3148 5720 a(follo) n(ws.) p 3453 5720 a(Consider) p 3798 5720 a(a) p Fo 0 5820 a(d) p Fy(-simplex) p 376 5820 a(of) p 473 5820 a(a) p 544 5820 a(unit) p 722 5820 a(side) p 890 5820 a(length) p 1146 5820 a(em) n(b) r(edded) p 1541 5820 a(in) p Fq 1640 5820 a(R) p Fp 1700 5790 a(d) p Fy 1738 5820 a(,) p 1792 5820 a(and) p 1956 5820 a(let) p Fo 2078 5820 a(G) p Fj 2143 5832 a(0) p Fy 2207 5820 a(=) p Fk 2299 5820 a(f) p Fo(v) p Fj 2381 5832 a(0) p Fo 2418 5820 a(;) p 2455 5820 a(v) p Fj 2495 5832 a(1) p Fo 2532 5820 a(;) p 2569 5820 a(v) p Fj 2609 5832 a(2) p Fo 2646 5820 a(;) p Fk 2683 5820 a(\001) p 2720 5820 a(\001) p 2757 5820 a(\001) p Fo 2794 5820 a(;) p 2831 5820 a(v) p Fp 2871 5832 a(d) p Fk 2910 5820 a(g) p Fy 2981 5820 a(b) r(e) p 3096 5820 a(the) p 3242 5820 a(set) p 3374 5820 a(of) p 3470 5820 a(v) n(ertices) p 3773 5820 a(of) 1899 6120 y(3) p 90 rotate dyy eop %%Page: 4 4 4 3 bop Fy 0 83 a(the) p Fo 143 83 a(d) p Fy(-simplex,) p 540 83 a(where) p Fo 781 83 a(v) p Fj 821 95 a(0) p Fy 882 83 a(=) p Fo 970 83 a(O) p Fy 1060 83 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1291 y(W) p 203 1291 a(e) p 268 1291 a(call) p Fo 1186 1437 a(F) p Fy 1274 1437 a(=) p 1362 1437 a(\() p Fo(G;) p 1496 1437 a(B) p Fy 1563 1437 a(\);) p Fo 1688 1437 a(G) p Fy 1776 1437 a(=) p Fn 1893 1333 a(1) p Fg 1880 1358 a([) p Fp 1864 1534 a(n) p Fj(=0) p Fo 2003 1437 a(G) p Fp 2068 1449 a(n) p Fo 2127 1437 a(;) p 2192 1437 a(B) p Fy 2282 1437 a(=) p Fn 2399 1333 a(1) p Fg 2386 1358 a([) p Fp 2370 1534 a(n) p Fj(=0) p Fo 2509 1437 a(B) p Fp 2572 1449 a(n) p Fo 2631 1437 a(;) p Fy 3734 1437 a(\(2\)) 0 1669 y(the) p Fo 139 1669 a(d) p Fy(-dimensional) p 663 1669 a(pre-Sierpi) r(\023) p 1015 1669 a(nski) p 1183 1669 a(gask) n(et) p 1433 1669 a(\(pre-) p Fo(d) p Fy(SG\).) p 1842 1669 a(W) p 1920 1669 a(e) p 1981 1669 a(iden) n(tify) p 2279 1669 a(\() p Fo(v) s(;) p 2391 1669 a(v) p Fn 2434 1639 a(0) p Fy 2458 1669 a(\)) p Fk 2513 1669 a(2) p Fo 2592 1669 a(B) p Fy 2682 1669 a(with) p 2867 1669 a(line) p 3020 1669 a(segmen) n(ts) p 3370 1603 110 4 v Fo 3370 1669 a(v) s(v) p Fn 3456 1645 a(0) p Fy 3503 1669 a(whenev) n(er) 0 1769 y(it) p 83 1769 a(w) n(ould) p 325 1769 a(b) r(e) p 438 1769 a(natural) p 729 1769 a(to) p 830 1769 a(do) p 946 1769 a(so.) 125 1868 y(Denote) p 407 1868 a(the) p 547 1868 a(set) p 674 1868 a(of) p 767 1868 a(non-negativ) n(e) p 1253 1868 a(in) n(tegers) p 1558 1868 a(b) n(y) p Fq 1671 1868 a(Z) p Fj 1726 1880 a(+) p Fy 1781 1868 a(,) p 1830 1868 a(and) p 1989 1868 a(for) p Fo 2114 1868 a(w) p Fy 2198 1868 a(:) p Fq 2270 1868 a(Z) p Fj 2325 1880 a(+) p Fk 2403 1868 a(!) p Fo 2509 1868 a(G) p Fy(,) p 2623 1868 a(denote) p 2888 1868 a(b) n(y) p Fo 3001 1868 a(L) p Fy(\() p Fo(w) p Fy 3151 1868 a(\)) p Fk 3207 1868 a(2) p Fq 3285 1868 a(Z) p Fj 3340 1880 a(+) p Fk 3409 1868 a([) p 3477 1868 a(f1g) p Fy 3669 1868 a(\(`the) 0 1968 y(length) p 254 1968 a(of) p Fo 348 1968 a(w) p Fy 409 1968 a('\)) p 493 1968 a(the) p 636 1968 a(smallest) p 955 1968 a(in) n(teger) p 1230 1968 a(satisfying) p Fo 1414 2144 a(w) p Fy 1475 2144 a(\() p Fo(i) p Fy(\)) p 1592 2144 a(=) p Fo 1680 2144 a(w) p Fy 1741 2144 a(\() p Fo(L) p Fy(\() p Fo(w) p Fy 1923 2144 a(\)\)) p Fo(;) p 2081 2144 a(i) p Fl 2132 2144 a(=) p Fo 2220 2144 a(L) p Fy(\() p Fo(w) p Fy 2370 2144 a(\)) p Fo(:) p Fy 0 2320 a(De\014ne) p 257 2320 a(the) p 400 2320 a(set) p 530 2320 a(of) p 624 2320 a(self-a) n(v) n(oiding) p 1100 2320 a(paths) p Fo 1326 2320 a(W) p Fj 1404 2332 a(0) p Fy 1470 2320 a(to) p 1571 2320 a(b) r(e) p 1684 2320 a(the) p 1827 2320 a(set) p 1957 2320 a(of) p 2051 2320 a(maps) p Fo 2268 2320 a(w) p Fy 2353 2320 a(:) p Fq 2427 2320 a(Z) p Fj 2482 2332 a(+) p Fk 2561 2320 a(!) p Fo 2667 2320 a(G) p Fy(,) p 2783 2320 a(suc) n(h) p 2970 2320 a(that) p Fo 1135 2493 a(w) p Fy 1196 2493 a(\() p Fo(i) p Fj 1257 2505 a(1) p Fy 1295 2493 a(\)) p Fk 1350 2493 a(6) p Fy(=) p Fo 1438 2493 a(w) p Fy 1499 2493 a(\() p Fo(i) p Fj 1560 2505 a(2) p Fy 1598 2493 a(\)) p Fo(;) p Fy 1994 2493 a(0) p Fl 2058 2493 a(5) p Fo 2146 2493 a(i) p Fj 2175 2505 a(1) p Fo 2235 2493 a(<) p 2322 2493 a(i) p Fj 2351 2505 a(2) p Fl 2411 2493 a(5) p Fo 2499 2493 a(L) p Fy(\() p Fo(w) p Fy 2649 2493 a(\)) p Fo(;) p Fk 1135 2593 a(j) p Fo(w) p Fy 1219 2593 a(\() p Fo(i) p Fy(\)) p Fk 1332 2593 a(\000) p Fo 1415 2593 a(w) p Fy 1476 2593 a(\() p Fo(i) p Fy 1556 2593 a(+) p 1639 2593 a(1\)) p Fk(j) p Fy 1758 2593 a(=) p 1846 2593 a(1) p Fo(;) p Fy 1994 2593 a(0) p Fl 2058 2593 a(5) p Fo 2146 2593 a(i) p Fl 2198 2593 a(5) p Fo 2285 2593 a(L) p Fy(\() p Fo(w) p Fy 2435 2593 a(\)) p Fk 2486 2593 a(\000) p Fy 2569 2593 a(1) p Fo(;) p 1135 2629 453 4 v 1135 2702 a(w) p Fy 1196 2702 a(\() p Fo(i) p Fy(\)) p Fo(w) p Fy 1350 2702 a(\() p Fo(i) p Fy 1431 2702 a(+) p 1514 2702 a(1\)) p Fk 1611 2702 a(2) p Fo 1689 2702 a(B) t(;) p Fy 1994 2702 a(0) p Fl 2058 2702 a(5) p Fo 2146 2702 a(i) p Fl 2198 2702 a(5) p Fo 2285 2702 a(L) p Fy(\() p Fo(w) p Fy 2435 2702 a(\)) p Fk 2486 2702 a(\000) p Fy 2569 2702 a(1) p Fo 2625 2702 a(:) p Fr 0 2924 a(2.2) p 255 2924 a(Ov) m(erview) p 752 2924 a(of) p 880 2924 a(tec) m(hnical) p 1350 2924 a(de\014nitions.) p Fy 0 3077 a(W) p 78 3077 a(e) p 142 3077 a(need) p 334 3077 a(to) p 434 3077 a(prepare) p 733 3077 a(sev) n(eral) p 1004 3077 a(basic) p 1210 3077 a(de\014nitions) p 1611 3077 a(in) p 1706 3077 a(Section) p 1995 3077 a(2.3,) p 2151 3077 a(Section) p 2440 3077 a(2.4,) p 2596 3077 a(and) p 2755 3077 a(Section) p 3045 3077 a(2.5) p 3177 3077 a(b) r(efore) p 3425 3077 a(in) n(tro) r (ducing) 0 3177 y(the) p 143 3177 a(main) p 351 3177 a(notions) p 642 3177 a(in) p 738 3177 a(Section) p 1029 3177 a(3.1.) p 1195 3177 a(Here) p 1391 3177 a(w) n(e) p 1514 3177 a(will) p 1670 3177 a(brie\015y) p 1927 3177 a(explain) p 2215 3177 a(the) p 2358 3177 a(basic) p 2566 3177 a(de\014nitions.) p Fz 0 3338 a(Section) p 335 3338 a(2.3.) p Fy 526 3338 a(W) p 604 3338 a(e) p 667 3338 a(\014rst) p 837 3338 a(classify) p 1122 3338 a(ho) n(w) p 1294 3338 a(a) p 1361 3338 a(self-a) n(v) n(oiding) p 1835 3338 a(path) p 2028 3338 a(in) n(tersects) p 2394 3338 a(a) p 2462 3338 a(unit) p Fo 2635 3338 a(d) p Fy(-simplex.) p 3041 3338 a(A) p 3129 3338 a(path) p 3321 3338 a(whic) n(h) p 3557 3338 a(en) n(ters) p 3798 3338 a(a) 208 3437 y(simplex) p 507 3437 a(mo) n(v) n(es) p 751 3437 a(within) p 1006 3437 a(the) p 1146 3437 a(simplex) p 1446 3437 a(for) p 1569 3437 a(at) p 1668 3437 a(most) p Fo 1868 3437 a(d) p Fy 1936 3437 a(steps) p 2141 3437 a(\(b) r(ecause) p 2478 3437 a(it) p 2557 3437 a(ma) n(y) p 2734 3437 a(not) p 2879 3437 a(hit) p 3005 3437 a(the) p 3144 3437 a(same) p 3349 3437 a(v) n(ertex) p 3598 3437 a(t) n(wice\).) 208 3537 y(If) p 285 3537 a(a) p 348 3537 a(path) p 536 3537 a(tak) n(es) p Fo 742 3537 a(i) p Fj 771 3549 a(1) p Fy 830 3537 a(steps) p 1032 3537 a(in) p 1123 3537 a(the) p 1260 3537 a(simplex) p 1557 3537 a(and) p 1712 3537 a(go) r(es) p 1889 3537 a(in) n(to) p 2051 3537 a(an) p 2161 3537 a(adjacen) n(t) p 2487 3537 a(one,) p 2657 3537 a(and) p 2813 3537 a(nev) n(er) p 3029 3537 a(returns) p 3310 3537 a(to) p 3405 3537 a(the) p 3542 3537 a(simplex,) 208 3637 y(w) n(e) p 334 3637 a(lab) r(el) p 539 3637 a(the) p 686 3637 a(in) n(tersection) p 1136 3637 a(of) p 1234 3637 a(the) p 1382 3637 a(path) p 1579 3637 a(and) p 1745 3637 a(the) p 1892 3637 a(simplex) p 2199 3637 a(b) n(y) p 2318 3637 a(the) p 2465 3637 a(index) p 2693 3637 a(\() p Fo(i) p Fj 2754 3649 a(1) p Fo 2791 3637 a(;) p Fy 2828 3637 a(0) p Fo(;) p Fk 2907 3637 a(\001) p 2944 3637 a(\001) p 2981 3637 a(\001) p Fo 3017 3637 a(;) p Fy 3054 3637 a(0\).) p 3200 3637 a(Alternativ) n(ely) p 3669 3637 a(,) p 3725 3637 a(the) 208 3736 y(path) p 411 3736 a(ma) n(y) p 600 3736 a(return) p 864 3736 a(to) p 975 3736 a(the) p 1127 3736 a(simplex) p 1439 3736 a(a) p 1518 3736 a(n) n(um) n(b) r(er) p 1830 3736 a(of) p 1934 3736 a(times,) p 2190 3736 a(and) p 2361 3736 a(for) p 2498 3736 a(eac) n(h) p 2694 3736 a(return) p 2958 3736 a(the) p 3110 3736 a(in) n(tersection) p 3565 3736 a(ma) n(y) p 3755 3736 a(b) r(e) 208 3836 y(lab) r(elled) p 513 3836 a(b) n(y) p 626 3836 a(ho) n(w) p 798 3836 a(man) n(y) p 1022 3836 a(steps) p 1229 3836 a(the) p 1370 3836 a(path) p 1562 3836 a(tak) n(es) p 1773 3836 a(in) p 1868 3836 a(the) p 2009 3836 a(simplex.) p 2344 3836 a(Th) n(us) p 2552 3836 a(if) p 2627 3836 a(a) p 2694 3836 a(path) p 2886 3836 a(sp) r(ends) p 3155 3836 a(3) p 3223 3836 a(steps) p 3430 3836 a(for) p 3555 3836 a(the) p 3696 3836 a(\014rst) 208 3935 y(in) n(tersection) p 655 3935 a(and) p 818 3935 a(1) p 888 3935 a(step) p 1066 3935 a(for) p 1194 3935 a(the) p 1339 3935 a(second) p 1608 3935 a(in) n(tersection) p 2055 3935 a(with) p 2246 3935 a(a) p 2317 3935 a(simplex,) p 2644 3935 a(then) p 2835 3935 a(w) n(e) p 2958 3935 a(lab) r(el) p 3161 3935 a(the) p 3305 3935 a(in) n(tersection) p 3752 3935 a(b) n(y) 208 4035 y(the) p 352 4035 a(index) p 577 4035 a(\(1) p Fo(;) p Fy 688 4035 a(3) p Fo(;) p Fy 767 4035 a(0) p Fo(;) p Fk 846 4035 a(\001) p 883 4035 a(\001) p 920 4035 a(\001) p Fo 955 4035 a(;) p Fy 992 4035 a(0\).) p 1130 4035 a(\(F) p 1209 4035 a(or) p 1312 4035 a(our) p 1462 4035 a(purp) r(ose) p 1775 4035 a(w) n(e) p 1898 4035 a(ma) n(y) p 2080 4035 a(iden) n(tify) p 2383 4035 a(\(1) p Fo(;) p Fy 2494 4035 a(3) p Fo(;) p Fy 2573 4035 a(0) p Fo(;) p Fk 2652 4035 a(\001) p 2689 4035 a(\001) p 2726 4035 a(\001) p Fo 2761 4035 a(;) p Fy 2798 4035 a(0\)) p 2901 4035 a(and) p 3064 4035 a(\(3) p Fo(;) p Fy 3175 4035 a(1) p Fo(;) p Fy 3254 4035 a(0) p Fo(;) p Fk 3333 4035 a(\001) p 3370 4035 a(\001) p 3407 4035 a(\001) p Fo 3442 4035 a(;) p Fy 3479 4035 a(0\);) p 3606 4035 a(w) n(e) p 3729 4035 a(are) 208 4135 y(free) p 365 4135 a(to) p 465 4135 a(rearrange) p 833 4135 a(a) p 901 4135 a(sequence) p 1244 4135 a(in) p 1339 4135 a(an) p 1453 4135 a(index) p 1675 4135 a(in) p 1770 4135 a(the) p 1912 4135 a(ascending) p 2289 4135 a(order.\)) p 2570 4135 a(W) p 2648 4135 a(e) p 2711 4135 a(denote) p 2977 4135 a(the) p 3119 4135 a(set) p 3247 4135 a(of) p 3340 4135 a(the) p 3481 4135 a(indices) p 3752 4135 a(b) n(y) p Fk 208 4234 a(I) p Fp 253 4246 a(d) p Fy 291 4234 a(.) 208 4366 y(Eac) n(h) p 408 4366 a(index) p 626 4366 a(corresp) r(onds) p 1076 4366 a(to) p 1171 4366 a(a) p 1234 4366 a(comp) r(onen) n(t) p 1653 4366 a(in) p 1744 4366 a(the) p 1881 4366 a(parameter) p 2272 4366 a(space) p 2488 4366 a(on) p 2597 4366 a(whic) n(h) p 2829 4366 a(the) p 2966 4366 a(R) n(G) p 3111 4366 a(map) p 3290 4366 a(acts.) p 3491 4366 a(Therefore) 208 4477 y(for) p 340 4477 a(eac) n(h) p 532 4477 a(index) p Fo 761 4477 a(I) p Fk 835 4477 a(2) p 922 4477 a(I) p Fp 967 4489 a(d) p Fy 1006 4477 a(,) p 1063 4477 a(w) n(e) p 1191 4477 a(need) p 1390 4477 a(a) p 1464 4477 a(set) p 1599 4477 a(of) p 1698 4477 a(self-a) n(v) n(oiding) p 2179 4477 a(paths) p Fo 2411 4477 a(W) p Fj 2501 4434 a(\() p Fp(n) p Fj(\)) p Fp 2489 4501 a(I) p Fy 2631 4477 a(on) p Fo 2751 4477 a(G) p Fp 2816 4489 a(n) p Fy 2894 4477 a(lab) r(elled) p 3206 4477 a(b) n(y) p Fo 3327 4477 a(I) p Fy 3403 4477 a(whic) n(h) p 3645 4477 a(has) p 3798 4477 a(a) 208 4577 y(similar) p 484 4577 a(structure) p 844 4577 a(as) p 949 4577 a(the) p 1096 4577 a(in) n(tersection) p 1545 4577 a(of) p 1643 4577 a(a) p 1716 4577 a(path) p 1914 4577 a(and) p 2079 4577 a(a) p 2151 4577 a(unit) p 2330 4577 a(simplex) p 2637 4577 a(lab) r(elled) p 2947 4577 a(b) n(y) p Fo 3066 4577 a(I) p Fy 3109 4577 a(.) p 3180 4577 a(F) p 3227 4577 a(or) p 3332 4577 a(an) p 3451 4577 a(index) p 3679 4577 a(with) 208 4688 y(more) p 422 4688 a(than) p 622 4688 a(one) p 780 4688 a(non-zero) p 1124 4688 a(en) n(tries,) p 1421 4688 a(suc) n(h) p 1614 4688 a(as) p Fo 1723 4688 a(I) p Fy 1800 4688 a(=) p 1898 4688 a(\(1) p Fo(;) p Fy 2009 4688 a(3) p Fo(;) p Fy 2088 4688 a(0) p Fo(;) p Fk 2167 4688 a(\001) p 2204 4688 a(\001) p 2241 4688 a(\001) p Fo 2276 4688 a(;) p Fy 2313 4688 a(0\),) p 2445 4688 a(the) p 2595 4688 a(set) p Fo 2731 4688 a(W) p Fj 2821 4645 a(\() p Fp(n) p Fj(\)) p Fp 2809 4712 a(I) p Fy 2952 4688 a(is) p 3042 4688 a(de\014ned) p 3334 4688 a(to) p 3442 4688 a(b) r(e) p 3562 4688 a(a) p 3637 4688 a(set) p 3773 4688 a(of) 208 4807 y(collection) p 580 4807 a(of) p 679 4807 a(self-) p 856 4807 a(and) p 1021 4807 a(m) n(utually-a) n(v) n(oiding) p 1706 4807 a(paths) p 1936 4807 a(on) p Fo 2055 4807 a(G) p Fp 2120 4819 a(n) p Fy 2166 4807 a(.) p 2237 4807 a(F) p 2284 4807 a(or) p 2390 4807 a(example,) p Fo 2743 4807 a(W) p Fj 2833 4764 a(\() p Fp(n) p Fj(\)) 2821 4835 y(\(1) p Fp(;) p Fj(3) p Fp(;) p Fj(0) p Fp(;) p Fn(\001\001\001) p Fp 3065 4835 a(;) p Fj(0\)) p Fy 3180 4807 a(is) p 3267 4807 a(a) p 3340 4807 a(set) p 3474 4807 a(of) p 3572 4807 a(disjoin) n(t) 208 4916 y(pairs) p 417 4916 a(of) p 518 4916 a(self-a) n(v) n(oiding) p 1000 4916 a(paths) p 1232 4916 a(on) p Fo 1354 4916 a(G) p Fp 1419 4928 a(n) p Fy 1464 4916 a(,) p 1523 4916 a(suc) n(h) p 1716 4916 a(that) p 1902 4916 a(one) p 2061 4916 a(path) p 2261 4916 a(starts) p 2499 4916 a(and) p 2666 4916 a(ends) p 2862 4916 a(at) p 2970 4916 a(outmost) p 3299 4916 a(v) n(ertices) p 3606 4916 a(of) p Fo 3706 4916 a(G) p Fp 3771 4928 a(n) p Fy 3817 4916 a(,) 208 5015 y(but) p 366 5015 a(hits) p 534 5015 a(no) p 655 5015 a(other) p 879 5015 a(outmost) p 1208 5015 a(v) n(ertices,) p 1539 5015 a(while) p 1762 5015 a(the) p 1911 5015 a(other) p 2135 5015 a(path) p 2334 5015 a(hits) p 2503 5015 a(t) n(w) n(o) p 2666 5015 a(outmost) p 2995 5015 a(v) n(ertices) p 3302 5015 a(other) p 3525 5015 a(than) p 3725 5015 a(the) 208 5115 y(endp) r(oin) n(ts.) p Fz 0 5278 a(Section) p 335 5278 a(2.4.) p Fy 526 5278 a(The) p 706 5278 a(R) n(G) p 868 5278 a(in) p 975 5278 a(our) p 1133 5278 a(study) p 1372 5278 a(is) p 1466 5278 a(the) p 1619 5278 a(recursion) p 1986 5278 a(map) p 2180 5278 a(in) p Fo 2287 5278 a(n) p Fy 2375 5278 a(of) p 2480 5278 a(the) p 2633 5278 a(join) n(t) p 2837 5278 a(generating) p 3253 5278 a(functions) p Fo 3639 5257 a(~) 3621 5278 y(X) p Fp 3690 5290 a(n) p Fy 3775 5278 a(=) 208 5390 y(\() p Fo(X) p Fp 309 5402 a(n;I) p Fy 407 5390 a(\() p Fo 438 5390 a(~) p 439 5390 a(x) p Fy 487 5390 a(\)) p Fo(;) p 578 5390 a(I) p Fk 644 5390 a(2) p 722 5390 a(I) p Fp 767 5402 a(d) p Fy 806 5390 a(\)) p 860 5390 a(of) p Fo 948 5390 a(s) p Fp 987 5402 a(J) p Fo 1033 5390 a(;) p 1092 5390 a(J) p Fk 1169 5390 a(2) p 1247 5390 a(I) p Fp 1292 5402 a(d) p Fy 1331 5390 a(,) p 1377 5390 a(for) p Fo 1498 5390 a(W) p Fj 1588 5347 a(\() p Fp(n) p Fj(\)) p Fp 1576 5414 a(I) p Fy 1685 5390 a(,) p 1730 5390 a(where) p Fo 1964 5390 a(s) p Fp 2003 5402 a(J) p Fy 2071 5390 a(is) p 2148 5390 a(the) p 2285 5390 a(n) n(um) n(b) r(er) p 2581 5390 a(of) p 2669 5390 a(unit) p 2838 5390 a(simplices) p 3183 5390 a(whose) p 3422 5390 a(in) n(tersection) 208 5489 y(with) p 397 5489 a(the) p 540 5489 a(path) p 733 5489 a(is) p 817 5489 a(of) p 911 5489 a(t) n(yp) r(e) p Fo 1098 5489 a(J) p Fy 1152 5489 a(.) 208 5621 y(A) p 295 5621 a(similarit) n(y) p 663 5621 a(of) p 755 5621 a(\014nite) p 965 5621 a(gask) n(ets) p Fo 1250 5621 a(G) p Fp 1315 5633 a(n) p Fy 1386 5621 a(among) p 1651 5621 a(di\013eren) n(t) p Fo 1977 5621 a(n) p Fy(s) p 2085 5621 a(implies) p 2364 5621 a(a) p 2431 5621 a(recursion) p 2785 5621 a(relation) p 3088 5621 a(to) p 3187 5621 a(hold) p 3369 5621 a(for) p 3494 5621 a(all) p Fo 3607 5621 a(n) p Fy(,) p 3706 5621 a(and) 208 5720 y(this) p 373 5720 a(is) p 460 5720 a(our) p 612 5720 a(R) n(G.) p 790 5720 a(In) p 898 5720 a(this) p 1063 5720 a(w) n(a) n(y) p 1235 5720 a(w) n(e) p 1361 5720 a(arriv) n(e) p 1601 5720 a(at) p 1706 5720 a(a) p 1779 5720 a(mathematically) p 2371 5720 a(w) n(ell-de\014ned) p 2829 5720 a(notion) p 3091 5720 a(of) p 3189 5720 a(`a) p 3285 5720 a(resp) r(onse) p 3624 5720 a(in) p 3725 5720 a(the) 208 5820 y(parameter) p 605 5820 a(space) p 827 5820 a(of) p 921 5820 a(the) p 1064 5820 a(scale) p 1263 5820 a(transformation) p 1829 5820 a(in) p 1926 5820 a(the) p 2069 5820 a(path) p 2262 5820 a(space'.) 1899 6120 y(4) p 90 rotate dyy eop %%Page: 5 5 5 4 bop Fz 0 83 a(Section) p 335 83 a(2.5.) p Fy 526 83 a(A) p 620 83 a(study) p 852 83 a(in) p 953 83 a(3SG) p 1138 83 a(sho) n(ws) p 1381 83 a([6) o(]) p 1500 83 a(that) p 1684 83 a(in) p 1785 83 a(general) p 2075 83 a(there) p 2292 83 a(are) p 2435 83 a(more) p 2647 83 a(than) p 2844 83 a(one) p 3001 83 a(non-trivial) p 3413 83 a(\014xed) p 3618 83 a(p) r(oin) n(ts) 208 183 y(of) p 305 183 a(the) p 452 183 a(R) n(G.) p 630 183 a(Therefore) p 1009 183 a(w) n(e) p 1135 183 a(ha) n(v) n(e) p 1330 183 a(to) p 1434 183 a(kno) n(w) p 1655 183 a(whic) n(h) p 1895 183 a(\014xed) p 2099 183 a(p) r(oin) n(t) p 2319 183 a(is) p 2406 183 a(relev) p 2575 183 a(an) n(t) p 2723 183 a(for) p 2854 183 a(the) p 3000 183 a(asymptotic) p 3431 183 a(b) r(eha) n(vior) p 3773 183 a(of) 208 282 y(the) p 352 282 a(self-a) n(v) n(oiding) p 829 282 a(paths.) p 1091 282 a(It) p 1183 282 a(turns) p 1402 282 a(out) p 1551 282 a(that) p 1732 282 a(the) p 1876 282 a(condition) p 2242 282 a(that) p 2423 282 a(the) p 2567 282 a(\014xed) p 2769 282 a(p) r(oin) n(t) p 2987 282 a(is) p 3072 282 a(in) p 3170 282 a(a) p 3241 282 a(certain) p 3519 282 a(in) n(v) p 3625 282 a(arian) n(t) 208 382 y(set) p 339 382 a(of) p 436 382 a(the) p 582 382 a(R) n(G) p 735 382 a(ensures) p 1029 382 a(our) p 1180 382 a(pro) r(of) p 1399 382 a(to) p 1503 382 a(w) n(ork.) p 1745 382 a(T) p 1798 382 a(o) p 1869 382 a(form) n(ulate) p 2245 382 a(the) p 2391 382 a(condition) p 2757 382 a(\(see) p 2926 382 a(\(FP4\)\),) p 3229 382 a(w) n(e) p 3353 382 a(in) n(tro) r(duce) p 3725 382 a(the) 208 482 y(in) n(v) p 314 482 a(arian) n(t) p 556 482 a(set) p 686 482 a(\004) p Fp 741 494 a(d) p Fy 780 482 a(.) 0 627 y(W) p 78 627 a(e) p 153 627 a(note) p 348 627 a(that) p 538 627 a(it) p 631 627 a(w) n(ould) p 884 627 a(also) p 1061 627 a(b) r(e) p 1184 627 a(useful) p 1432 627 a(for) p 1570 627 a(in) n(tuitiv) n(e) p 1910 627 a(understanding) p 2465 627 a(to) p 2577 627 a(lo) r(ok) p 2767 627 a(at) p 2878 627 a(the) p 3032 627 a(case) p 3218 627 a(of) p 3323 627 a(3SG,) p 3536 627 a(whic) n(h) p 3784 627 a(is) 0 727 y(explicitly) p 360 727 a(giv) n(en) p 577 727 a(in) p 673 727 a([6],) p Fr 0 954 a(2.3) p 255 954 a(Classi\014cation) p 938 954 a(of) p 1066 954 a(self-a) m(v) m(oiding) p 1708 954 a(paths.) p Fy 0 1108 a(Denote) p 289 1108 a(b) n(y) p Fk 409 1108 a(T) p Fp 454 1120 a(b) p Fy 520 1108 a(the) p 667 1108 a(family) p 925 1108 a(of) p 1024 1108 a(all) p 1144 1108 a(the) p 1291 1108 a(translations) p 1749 1108 a(of) p Fo 1848 1108 a(B) p Fj 1911 1120 a(0) p Fy 1980 1108 a(that) p 2164 1108 a(are) p 2307 1108 a(subsets) p 2599 1108 a(of) p Fo 2698 1108 a(B) p Fy 2765 1108 a(.) p 2838 1108 a(Namely) p 3108 1108 a(,) p Fk 3164 1108 a(T) p Fp 3209 1120 a(b) p Fy 3275 1108 a(con) n(tains) p 3605 1108 a(all) p 3725 1108 a(the) 0 1207 y(unit) p Fo 175 1207 a(d) p Fy(-simplices) p 598 1207 a(whic) n(h) p 835 1207 a(comp) r(ose) p 1170 1207 a(the) p 1313 1207 a(pre-) p Fo(d) p Fy(SG.) p 1661 1207 a(\(with) p 1883 1207 a(eac) n(h) p 2070 1207 a(simplex) p 2372 1207 a(regarded) p 2714 1207 a(as) p 2816 1207 a(a) p 2885 1207 a(collection) p 3254 1207 a(of) p 3349 1207 a(b) r(onds\).) p 3656 1207 a(Put) p Fk 725 1368 a(I) p Fp 770 1380 a(d) p Fy 832 1368 a(=) p Fk 919 1368 a(f) p Fy(\() p Fo(i) p Fj 1022 1380 a(1) p Fo 1059 1368 a(;) p 1096 1368 a(i) p Fj 1125 1380 a(2) p Fo 1162 1368 a(;) p Fk 1199 1368 a(\001) p 1236 1368 a(\001) p 1273 1368 a(\001) p Fo 1309 1368 a(;) p 1346 1368 a(i) p Fp 1375 1380 a(k) p Fy 1416 1368 a(\)) p Fk 1471 1368 a(2) p Fq 1549 1368 a(Z) p Fp 1604 1334 a(k) p Fj 1604 1389 a(+) p Fk 1683 1368 a(j) p Fo 1729 1368 a(k) p Fy 1798 1368 a(=) p 1886 1368 a(1) p Fo(;) p Fy 1965 1368 a(2) p Fo(;) p Fy 2044 1368 a(3) p Fo(;) p Fk 2123 1368 a(\001) p 2160 1368 a(\001) p 2197 1368 a(\001) p Fo 2232 1368 a(;) p Fy 2296 1368 a(0) p Fo 2361 1368 a(<) p 2448 1368 a(i) p Fj 2477 1380 a(1) p Fl 2537 1368 a(5) p Fo 2625 1368 a(i) p Fj 2654 1380 a(2) p Fl 2714 1368 a(5) p Fk 2801 1368 a(\001) p 2838 1368 a(\001) p 2875 1368 a(\001) p Fl 2921 1368 a(5) p Fo 3009 1368 a(i) p Fp 3038 1380 a(k) p Fo 3092 1368 a(;) 1692 1473 y(i) p Fj 1721 1485 a(1) p Fy 1776 1473 a(+) p Fo 1859 1473 a(i) p Fj 1888 1485 a(2) p Fy 1944 1473 a(+) p Fk 2027 1473 a(\001) p 2064 1473 a(\001) p 2101 1473 a(\001) p Fy 2142 1473 a(+) p Fo 2225 1473 a(i) p Fp 2254 1485 a(k) p Fy 2313 1473 a(+) p Fo 2396 1473 a(k) p Fl 2465 1473 a(5) p Fo 2553 1473 a(d) p Fy 2614 1473 a(+) p 2697 1473 a(1) p Fk(g) p Fo(;) p Fy 3734 1419 a(\(3\)) 0 1626 y(and) p 161 1626 a(denote) p 429 1626 a(the) p 572 1626 a(n) n(um) n(b) r(er) p 874 1626 a(of) p 969 1626 a(elemen) n(ts) p 1308 1626 a(of) p Fk 1403 1626 a(I) p Fp 1448 1638 a(d) p Fy 1514 1626 a(b) n(y) p Fo 1630 1626 a(f) p Fp 1671 1638 a(d) p Fy 1732 1626 a(=) p Fo 1820 1626 a(]) p Fk(I) p Fp 1897 1638 a(d) p Fy 1936 1626 a(.) p Fz 0 1771 a(Prop) s(osition) p 516 1771 a(3) p Fi 605 1771 a(L) l(et) p Fo 749 1771 a(w) p Fk 833 1771 a(2) p Fo 912 1771 a(W) p Fj 990 1783 a(0) p Fi 1057 1771 a(and) p Fy 1218 1771 a(\001) p Fk 1311 1771 a(2) p 1389 1771 a(T) p Fp 1434 1783 a(b) p Fi 1482 1771 a(,) p 1537 1771 a(and) p 1698 1771 a(c) l(onsider) p 2026 1771 a(the) p 2164 1771 a(set) p 2293 1771 a(of) p 2391 1771 a(b) l(onds) p Fo 1082 1928 a(A) p Fy 1167 1928 a(=) p Fk 1255 1928 a(f) p 1297 1856 453 4 v Fo(w) p Fy 1358 1928 a(\() p Fo(i) p Fy(\)) p Fo(w) p Fy 1512 1928 a(\() p Fo(i) p Fy 1592 1928 a(+) p 1675 1928 a(1\)) p Fk 1772 1928 a(2) p Fy 1850 1928 a(\001) p Fk 1943 1928 a(j) p Fo 1989 1928 a(i) p Fy 2040 1928 a(=) p 2128 1928 a(0) p Fo(;) p Fy 2207 1928 a(1) p Fo(;) p Fy 2286 1928 a(2) p Fo(;) p Fk 2365 1928 a(\001) p 2402 1928 a(\001) p 2439 1928 a(\001) p Fo 2474 1928 a(;) p 2511 1928 a(L) p Fy(\() p Fo(w) p Fy 2661 1928 a(\)) p Fk(g) p Fo(:) p Fi 0 2085 a(If) p Fo 87 2085 a(A) p Fi 179 2085 a(is) p 268 2085 a(not) p 415 2085 a(empty,) p 686 2085 a(then) p 871 2085 a(ther) l(e) p 1078 2085 a(exists) p Fo 1305 2085 a(I) p Fy 1371 2085 a(=) p 1459 2085 a(\() p Fo(i) p Fj 1520 2097 a(1) p Fo 1557 2085 a(;) p 1594 2085 a(i) p Fj 1623 2097 a(2) p Fo 1660 2085 a(;) p Fk 1697 2085 a(\001) p 1734 2085 a(\001) p 1771 2085 a(\001) p Fo 1807 2085 a(;) p 1844 2085 a(i) p Fp 1873 2097 a(k) p Fy 1914 2085 a(\)) p Fk 1969 2085 a(2) p 2047 2085 a(I) p Fp 2092 2097 a(d) p Fi 2161 2085 a(such) p 2350 2085 a(that) p Fo 2519 2085 a(A) p Fi 2611 2085 a(is) p 2701 2085 a(c) l(ongruent) p 3084 2085 a(to) p Fy 663 2242 a(\001) p Fp 732 2254 a(I) p Fy 793 2242 a(=) p Fk 881 2242 a(f) p 923 2175 631 4 v Fo(O) r(v) p Fj 1028 2254 a(1) p Fo 1065 2242 a(v) p Fj 1105 2254 a(2) p Fk 1157 2242 a(\001) p 1194 2242 a(\001) p 1231 2242 a(\001) p Fo 1267 2242 a(v) p Fp 1307 2254 a(i) p Ff 1330 2262 a(1) p Fn 1364 2254 a(\000) p Fj(1) p Fo 1453 2242 a(v) p Fp 1493 2254 a(i) p Ff 1516 2262 a(1) p Fo 1553 2242 a(;) p 1590 2195 517 4 v 1590 2242 a(v) p Fp 1630 2254 a(i) p Ff 1653 2262 a(1) p Fj 1686 2254 a(+1) p Fk 1788 2242 a(\001) p 1825 2242 a(\001) p 1862 2242 a(\001) p Fo 1899 2242 a(v) p Fp 1939 2254 a(i) p Ff 1962 2262 a(1) p Fj 1995 2254 a(+) p Fp(i) p Ff 2069 2262 a(2) p Fo 2106 2242 a(;) p Fk 2143 2242 a(\001) p 2180 2242 a(\001) p 2217 2242 a(\001) p Fo 2254 2242 a(;) p 2291 2195 823 4 v 2291 2242 a(v) p Fp 2331 2254 a(i) p Ff 2354 2262 a(1) p Fj 2387 2254 a(+) p Fn(\001\001\001) p Fp 2497 2254 a(i) p Fm 2520 2263 a(k) p Fe 2552 2263 a(\000) p Ff(1) p Fj 2630 2254 a(+1) p Fk 2732 2242 a(\001) p 2769 2242 a(\001) p 2806 2242 a(\001) p Fo 2842 2242 a(v) p Fp 2882 2254 a(i) p Ff 2905 2262 a(1) p Fj 2939 2254 a(+) p Fn(\001\001\001) p Fp 3049 2254 a(i) p Fm 3072 2263 a(k) p Fk 3113 2242 a(g) p Fo(;) p Fy 3734 2242 a(\(4\)) p Fi 0 2398 a(wher) l(e) p 234 2398 a(we) p 357 2398 a(use) l(d) p 542 2398 a(an) p 661 2398 a(abbr) l(eviation) p 1127 2398 a(such) p 1315 2398 a(as) p 1032 2489 631 4 v Fo 1032 2555 a(O) r(v) p Fj 1137 2567 a(1) p Fo 1176 2555 a(v) p Fj 1216 2567 a(2) p Fk 1267 2555 a(\001) p 1304 2555 a(\001) p 1341 2555 a(\001) p Fo 1378 2555 a(v) p Fp 1418 2567 a(i) p Ff 1441 2575 a(1) p Fn 1474 2567 a(\000) p Fj(1) p Fo 1563 2555 a(v) p Fp 1603 2567 a(i) p Ff 1626 2575 a(1) p Fy 1686 2555 a(=) p 1774 2489 144 4 v Fo 1774 2555 a(O) r(v) p Fj 1879 2567 a(1) p Fo 1917 2555 a(;) p 1954 2510 155 4 v 1954 2555 a(v) p Fj 1994 2567 a(1) p Fo 2031 2555 a(v) p Fj 2071 2567 a(2) p Fo 2109 2555 a(;) p 2146 2510 V 2146 2555 a(v) p Fj 2186 2567 a(2) p Fo 2223 2555 a(v) p Fj 2263 2567 a(3) p Fo 2301 2555 a(;) p Fk 2338 2555 a(\001) p 2375 2555 a(\001) p 2412 2555 a(\001) p Fo 2448 2555 a(;) p 2485 2510 286 4 v 2485 2555 a(v) p Fp 2525 2567 a(i) p Ff 2548 2575 a(1) p Fn 2581 2567 a(\000) p Fj(1) p Fo 2670 2555 a(v) p Fp 2710 2567 a(i) p Ff 2733 2575 a(1) p Fo 2785 2555 a(:) p Fi -42 2712 a(Example.) p Fk 457 2712 a(\017) p 540 2712 a(I) p Fj 585 2724 a(2) p Fy 646 2712 a(=) p Fk 733 2712 a(f) p Fy(\(1\)) p Fo(;) p Fy 918 2712 a(\(2\)) p Fk(g) p Fy(:) p Fo 1125 2712 a(A) p Fk 1211 2712 a(6) p Fy(=) p Fk 1298 2712 a(;) p Fy 1367 2712 a(is) p 1451 2712 a(congruen) n(t) p 1836 2712 a(to) p 1938 2712 a(either) p Fk 2173 2712 a(f) p 2215 2645 144 4 v Fo(O) r(v) p Fj 2320 2724 a(1) p Fk 2358 2712 a(g) p Fy 2427 2712 a(or) p Fk 2529 2712 a(f) p 2571 2645 221 4 v Fo(O) r(v) p Fj 2676 2724 a(1) p Fo 2713 2712 a(v) p Fj 2753 2724 a(2) p Fk 2791 2712 a(g) p Fy(.) p Fk 125 2868 a(\017) p 208 2868 a(I) p Fj 253 2880 a(3) p Fy 332 2868 a(=) p Fk 439 2868 a(f) p Fy(\(1\)) p Fo(;) p Fy 624 2868 a(\(2\)) p Fo(;) p Fy 767 2868 a(\(3\)) p Fo(;) p Fy 910 2868 a(\(1) p Fo(;) p Fy 1021 2868 a(1\)) p Fk(g) p Fy(:) p 1218 2868 a(There) p 1469 2868 a(is) p 1564 2868 a(a) p 1645 2868 a(p) r(ossibilit) n(y) p 2052 2868 a(that) p 2243 2868 a(a) p 2323 2868 a(path) p 2529 2868 a(en) n(ters) p 2783 2868 a(a) p 2863 2868 a(unit) p 3050 2868 a(tetrahedron) p 3514 2868 a(t) n(wice,) p 3766 2868 a(as) p Fk 208 2967 a(f) p 250 2901 144 4 v Fo(O) r(v) p Fj 355 2979 a(1) p Fo 392 2967 a(;) p 429 2922 155 4 v 429 2967 a(v) p Fj 469 2979 a(2) p Fo 507 2967 a(v) p Fj 547 2979 a(3) p Fk 584 2967 a(g) p Fy(.) p Fk 125 3123 a(\017) p 208 3123 a(I) p Fj 253 3135 a(4) p Fy 313 3123 a(=) p Fk 401 3123 a(f) p Fy(\(1\)) p Fo(;) p Fy 586 3123 a(\(2\)) p Fo(;) p Fy 729 3123 a(\(3\)) p Fo(;) p Fy 872 3123 a(\(4\)) p Fo(;) p Fy 1015 3123 a(\(1) p Fo(;) p Fy 1126 3123 a(1\)) p Fo(;) p Fy 1237 3123 a(\(1) p Fo(;) p Fy 1348 3123 a(2\)) p Fk(g) p Fy(.) 125 3223 y(Corresp) r(ondingly) p 716 3223 a(,) p Fo 764 3223 a(f) p Fj 805 3235 a(2) p Fy 865 3223 a(=) p 953 3223 a(2,) p Fo 1045 3223 a(f) p Fj 1086 3235 a(3) p Fy 1146 3223 a(=) p 1234 3223 a(4,) p Fo 1326 3223 a(f) p Fj 1367 3235 a(4) p Fy 1427 3223 a(=) p 1515 3223 a(6.) p Fd 3774 3223 a(3) p Fi -42 3516 a(Pr) l(o) l(of) p 181 3516 a(of) p 279 3516 a(Pr) l(op) l(osition) p 720 3516 a(3.) p Fy 830 3516 a(If) p Fo 914 3516 a(A) p Fk 1000 3516 a(6) p Fy(=) p Fk 1089 3516 a(;) p Fy(,) p 1182 3516 a(namely) p 1436 3516 a(,) p 1488 3516 a(if) p 1565 3516 a(the) p 1709 3516 a(path) p Fo 1903 3516 a(w) p Fy 1993 3516 a(en) n(ters) p 2237 3516 a(the) p 2381 3516 a(unit) p Fo 2557 3516 a(d) p Fy(-simplex) p 2931 3516 a(sp) r(eci\014ed) p 3267 3516 a(b) n(y) p 3383 3516 a(\001,) p 3504 3516 a(then) p Fo 3693 3516 a(A) p Fy 3784 3516 a(is) 0 3615 y(comp) r(osed) p 373 3615 a(of) p 459 3615 a(one) p 602 3615 a(or) p 696 3615 a(more) p 895 3615 a(connected) p 1274 3615 a(clusters.) p 1604 3615 a(That) p 1803 3615 a(is,) p Fo 1903 3615 a(w) p Fy 1984 3615 a(ma) n(y) p 2156 3615 a(pass) p 2328 3615 a(through) p 2633 3615 a(\001) p 2722 3615 a(and) p 2875 3615 a(ma) n(y) p 3046 3615 a(come) p 3250 3615 a(bac) n(k) p 3435 3615 a(and) p 3588 3615 a(reen) n(ter) 0 3715 y(\001.) p 130 3715 a(Since) p Fo 348 3715 a(w) p Fy 437 3715 a(is) p 521 3715 a(self-a) n(v) n(oiding,) p 1020 3715 a(the) p 1164 3715 a(second) p 1432 3715 a(passage) p 1733 3715 a(do) r(es) p 1921 3715 a(not) p 2069 3715 a(in) n(tersect) p 2404 3715 a(with) p 2594 3715 a(the) p 2737 3715 a(\014rst) p 2909 3715 a(one.) p 3095 3715 a(Th) n(us) p 3306 3715 a(w) n(e) p 3428 3715 a(can) p 3581 3715 a(classify) p Fo 0 3815 a(A) p Fy 92 3815 a(b) n(y) p 209 3815 a(the) p 354 3815 a(size) p 513 3815 a(of) p 609 3815 a(the) p 754 3815 a(connected) p 1143 3815 a(segmen) n(ts.) p 1535 3815 a(One) p 1712 3815 a(ma) n(y) p 1894 3815 a(rearrange) p 2265 3815 a(the) p 2410 3815 a(segmen) n(ts) p 2766 3815 a(in) p 2865 3815 a(an) p 2982 3815 a(increasing) p 3372 3815 a(order) p 3591 3815 a(of) p 3687 3815 a(size,) 0 3914 y(hence) p 230 3914 a(eac) n(h) p 417 3914 a(class) p 611 3914 a(is) p 695 3914 a(determined) p 1128 3914 a(b) n(y) p 1243 3914 a(an) p 1358 3914 a(increasing) p 1746 3914 a(\014nite) p 1958 3914 a(sequence) p 2302 3914 a(of) p 2396 3914 a(p) r(ositiv) n(e) p 2703 3914 a(in) n(tegers,) p Fo 3033 3914 a(i) p Fj 3062 3926 a(1) p Fl 3122 3914 a(5) p Fo 3210 3914 a(i) p Fj 3239 3926 a(2) p Fl 3299 3914 a(5) p Fo 3386 3914 a(i) p Fj 3415 3926 a(3) p Fl 3475 3914 a(5) p Fk 3563 3914 a(\001) p 3600 3914 a(\001) p 3637 3914 a(\001) p Fl 3683 3914 a(5) p Fo 3771 3914 a(i) p Fp 3800 3926 a(k) p Fy 0 4014 a(for) p 128 4014 a(some) p Fo 337 4014 a(k) p Fl 408 4014 a(=) p Fy 498 4014 a(1) p 553 4014 a(.) p 616 4014 a(The) p 788 4014 a(meaning) p 1121 4014 a(of) p 1217 4014 a(the) p 1361 4014 a(conditions) p 1759 4014 a(in) p 1857 4014 a(the) p 2001 4014 a(de\014nition) p 2371 4014 a(of) p Fk 2467 4014 a(I) p Fp 2512 4026 a(d) p Fy 2580 4014 a(should) p 2844 4014 a(no) n(w) p 3018 4014 a(b) r(e) p 3132 4014 a(ob) n(vious.) p 3468 4014 a(Since) p 3686 4014 a(\001) p 3784 4014 a(is) 0 4113 y(a) p 69 4113 a(translation) p 490 4113 a(of) p Fo 584 4113 a(B) p Fj 647 4125 a(0) p Fy 712 4113 a(whic) n(h) p 950 4113 a(is) p 1033 4113 a(the) p 1176 4113 a(set) p 1306 4113 a(of) p 1400 4113 a(b) r(onds) p 1643 4113 a(in) p 1740 4113 a(the) p 1883 4113 a(unit) p Fo 2058 4113 a(d) p Fy(-simplex) p Fo 2431 4113 a(O) r(v) p Fj 2536 4125 a(1) p Fo 2575 4113 a(v) p Fj 2615 4125 a(2) p Fk 2666 4113 a(\001) p 2703 4113 a(\001) p 2740 4113 a(\001) p Fo 2777 4113 a(v) p Fp 2817 4125 a(d) p Fy 2869 4113 a(,) p 2920 4113 a(the) p 3063 4113 a(statemen) n(t) p 3449 4113 a(follo) n(ws.) p Fd 3778 4113 a(2) p Fy 125 4349 a(In) p 237 4349 a(analogy) p 553 4349 a(with) p 751 4349 a(Prop) r(osition) p 1207 4349 a(3) p 1285 4349 a(w) n(e) p 1416 4349 a(can) p 1577 4349 a(classify) p 1873 4349 a(the) p 2025 4349 a(set) p 2164 4349 a(of) p 2267 4349 a(self-a) n(v) n(oiding) p 2752 4349 a(paths) p 2987 4349 a(on) p Fo 3112 4349 a(F) p Fp 3165 4361 a(n) p Fy 3247 4349 a(b) n(y) p Fk 3371 4349 a(I) p Fp 3416 4361 a(d) p Fy 3469 4349 a(,) p 3531 4349 a(and) p 3701 4349 a(also) 0 4449 y(generalize) p 383 4449 a(to) p 485 4449 a(t) n(w) n(o) p 641 4449 a(or) p 743 4449 a(more) p 951 4449 a(self-a) n(v) n(oiding) p 1426 4449 a(paths.) 125 4549 y(F) p 172 4549 a(or) p Fo 274 4549 a(n) p Fk 346 4549 a(2) p Fq 425 4549 a(Z) p Fj 480 4561 a(+) p Fy 563 4549 a(and) p Fo 724 4549 a(u;) p 809 4549 a(v) p Fk 875 4549 a(2) p Fo 954 4549 a(G) p Fp 1019 4561 a(n) p Fy 1064 4549 a(,) p 1115 4549 a(de\014ne) p Fo 1355 4549 a(W) p Fj 1445 4518 a(\() p Fp(n;u;v) p Fj 1626 4518 a(\)) p Fy 1683 4549 a(b) n(y) p Fo 658 4705 a(W) p Fj 748 4671 a(\() p Fp(n;u;v) p Fj 929 4671 a(\)) p Fy 982 4705 a(=) p Fk 1069 4705 a(f) p Fo(w) p Fk 1196 4705 a(2) p Fo 1274 4705 a(W) p Fj 1352 4717 a(0) p Fk 1413 4705 a(j) p Fo 1459 4705 a(w) p Fy 1520 4705 a(\(0\)) p 1650 4705 a(=) p Fo 1737 4705 a(u;) p 1849 4705 a(w) p Fy 1910 4705 a(\() p Fo(L) p Fy(\() p Fo(w) p Fy 2092 4705 a(\)\)) p 2181 4705 a(=) p Fo 2269 4705 a(v) s(;) p 2432 4705 a(w) p Fy 2493 4705 a(\() p Fo(i) p Fy(\)) p Fk 2610 4705 a(2) p Fo 2688 4705 a(G) p Fp 2753 4717 a(n) p Fo 2813 4705 a(;) p 2877 4705 a(i) p Fk 2929 4705 a(2) p Fq 3007 4705 a(Z) p Fj 3062 4717 a(+) p Fk 3118 4705 a(g) p Fo(:) p Fy 0 4878 a(F) p 47 4878 a(or) p Fo 149 4878 a(n) p Fk 222 4878 a(2) p Fq 300 4878 a(Z) p Fj 355 4890 a(+) p Fy 438 4878 a(and) p Fo 600 4878 a(I) p Fy 666 4878 a(=) p 754 4878 a(\() p Fo(i) p Fj 815 4890 a(1) p Fo 852 4878 a(;) p 889 4878 a(i) p Fj 918 4890 a(2) p Fo 954 4878 a(;) p Fk 991 4878 a(\001) p 1028 4878 a(\001) p 1065 4878 a(\001) p Fo 1102 4878 a(;) p 1139 4878 a(i) p Fp 1168 4890 a(k) p Fy 1208 4878 a(\)) p Fk 1264 4878 a(2) p 1342 4878 a(I) p Fp 1387 4890 a(d) p Fy 1426 4878 a(,) p 1477 4878 a(de\014ne) p Fo 1717 4878 a(W) p Fj 1807 4835 a(\() p Fp(n) p Fj(\)) p Fp 1795 4903 a(I) p Fy 1931 4878 a(b) n(y) p Fo 184 5055 a(W) p Fj 274 5012 a(\() p Fp(n) p Fj(\)) p Fp 262 5079 a(I) p Fy 395 5055 a(=) p Fk 482 5055 a(f) p Fy(\() p Fo(w) p Fj 615 5067 a(1) p Fo 653 5055 a(;) p 690 5055 a(w) p Fj 749 5067 a(2) p Fo 786 5055 a(;) p Fk 823 5055 a(\001) p 860 5055 a(\001) p 897 5055 a(\001) p Fo 934 5055 a(;) p 971 5055 a(w) p Fp 1030 5067 a(k) p Fy 1071 5055 a(\)) p Fk 1126 5055 a(2) p Fo 1205 5055 a(W) p Fj 1295 5021 a(\() p Fp(n;O) r(;v) p Fm 1487 5029 a(n;i) p Ff 1565 5041 a(1) p Fj 1601 5021 a(\)) p Fk 1649 5055 a(\002) p Fo 1733 5055 a(W) p Fj 1823 5021 a(\() p Fp(n;v) p Fm 1943 5029 a(n;i) p Ff 2021 5041 a(1) 2053 5029 y(+1) p Fp 2128 5021 a(;v) p Fm 2181 5029 a(n;i) p Ff 2259 5041 a(1) 2291 5029 y(+) p Fm(i) p Ff 2356 5041 a(2) 2388 5029 y(+1) p Fj 2463 5021 a(\)) p Fk 2511 5055 a(\002) p Fo 2594 5055 a(W) p Fj 2684 5021 a(\() p Fp(n;v) p Fm 2804 5029 a(n;i) p Ff 2882 5041 a(1) 2914 5029 y(+) p Fm(i) p Ff 2979 5041 a(2) 3012 5029 y(+2) p Fp 3087 5021 a(;v) p Fm 3140 5029 a(n;i) p Ff 3218 5041 a(1) 3250 5029 y(+) p Fm(i) p Ff 3315 5041 a(2) 3347 5029 y(+) p Fm(i) p Ff 3412 5041 a(3) 3444 5029 y(+2) p Fj 3519 5021 a(\)) p Fk 478 5172 a(\002) p 561 5172 a(\001) p 598 5172 a(\001) p 635 5172 a(\001) p 676 5172 a(\002) p Fo 759 5172 a(W) p Fj 849 5132 a(\() p Fp(n;v) p Fm 969 5141 a(n;i) p Ff 1047 5153 a(1) 1079 5141 y(+) p Fm(i) p Ff 1144 5153 a(2) 1176 5141 y(+) p Fe(\001\001\001) p Ff(+) p Fm(i) 1341 5156 y(k) p Fe 1373 5156 a(\000) p Ff(1) 1450 5141 y(+) p Fm(k) p Fe 1525 5141 a(\000) p Ff(1) p Fp 1602 5132 a(;v) p Fm 1655 5141 a(n;i) p Ff 1733 5153 a(1) 1765 5141 y(+) p Fm(i) p Ff 1830 5153 a(2) 1863 5141 y(+) p Fe(\001\001\001) p Ff 1962 5141 a(+) p Fm(i) 2027 5156 y(k) p Ff 2064 5141 a(+) p Fm(k) p Fe 2139 5141 a(\000) p Ff(1) p Fj 2216 5132 a(\)) p Fk 2269 5172 a(j) p Fy 510 5288 a(if) p Fo 586 5288 a(i) p Fk 638 5288 a(6) p Fy(=) p Fo 725 5288 a(j) p Fy 792 5288 a(then) p Fo 981 5288 a(w) p Fp 1040 5300 a(i) p Fy 1096 5288 a(and) p Fo 1257 5288 a(w) p Fp 1316 5300 a(j) p Fy 1379 5288 a(do) p 1495 5288 a(not) p 1642 5288 a(hit) p 1771 5288 a(common) p 2103 5288 a(p) r(oin) n(ts,) p 2376 5288 a(and) p 2537 5288 a(for) p 2665 5288 a(eac) n(h) p Fo 2851 5288 a(j) 510 5405 y(w) p Fp 569 5417 a(j) p Fy 632 5405 a(hits) p 794 5405 a(p) r(oin) n(ts) p Fo 1043 5405 a(v) p Fp 1083 5417 a(n;i) p Ff 1167 5425 a(1) p Fj 1200 5417 a(+) p Fp(i) p Ff 1274 5425 a(2) p Fj 1307 5417 a(+) p Fn(\001\001\001) p Fj 1417 5417 a(+) p Fp(i) p Fm 1491 5425 a(j) p Fe 1517 5425 a(\000) p Ff(1) p Fj 1596 5417 a(+) p Fp(j) p Fn 1677 5417 a(\000) p Fj(1) p Fo 1767 5405 a(;) p 1827 5405 a(v) p Fp 1867 5417 a(n;i) p Ff 1951 5425 a(1) p Fj 1984 5417 a(+) p Fp(i) p Ff 2058 5425 a(2) p Fj 2091 5417 a(+) p Fn(\001\001\001) p Fj 2201 5417 a(+) p Fp(i) p Fm 2275 5425 a(j) p Fe 2301 5425 a(\000) p Ff(1) p Fj 2379 5417 a(+) p Fp(j) p Fo 2465 5405 a(;) p 2525 5405 a(v) p Fp 2565 5417 a(n;i) p Ff 2649 5425 a(1) p Fj 2682 5417 a(+) p Fp(i) p Ff 2756 5425 a(2) p Fj 2789 5417 a(+) p Fn(\001\001\001) p Fj 2899 5417 a(+) p Fp(i) p Fm 2973 5425 a(j) p Fe 2999 5425 a(\000) p Ff(1) p Fj 3078 5417 a(+) p Fp(j) p Fj 3159 5417 a(+1) p Fo 3248 5405 a(;) p Fk 496 5521 a(\001) p 533 5521 a(\001) p 570 5521 a(\001) p Fo 630 5521 a(v) p Fp 670 5533 a(n;i) p Ff 754 5541 a(1) p Fj 787 5533 a(+) p Fp(i) p Ff 861 5541 a(2) p Fj 894 5533 a(+) p Fn(\001\001\001) p Fj 1004 5533 a(+) p Fp(i) p Fm 1078 5541 a(j) p Fe 1104 5541 a(\000) p Ff(1) p Fj 1182 5533 a(+) p Fp(i) p Fm 1256 5541 a(j) p Fj 1288 5533 a(+) p Fp(j) p Fn 1369 5533 a(\000) p Fj(1) p Fo 1459 5521 a(;) p Fy 1523 5521 a(in) p 1620 5521 a(this) p 1782 5521 a(order,) 510 5638 y(but) p 662 5638 a(hits) p 824 5638 a(no) p 939 5638 a(other) p 1156 5638 a(p) r(oin) n(ts) p 1406 5638 a(in) p Fk 1503 5638 a(f) p Fo(v) p Fp 1585 5650 a(n;`) p Fk 1700 5638 a(j) p Fo 1746 5638 a(`) p Fy 1804 5638 a(=) p 1892 5638 a(0) p Fo(;) p Fy 1971 5638 a(1) p Fo(;) p Fy 2050 5638 a(2) p Fo(;) p Fk 2129 5638 a(\001) p 2166 5638 a(\001) p 2203 5638 a(\001) p Fo 2238 5638 a(;) p 2275 5638 a(d) p Fk(g) o(g) p Fo(:) p Fy 3734 5339 a(\(5\)) 0 5820 y(Ob) n(viously) p 357 5820 a(,) p Fo 406 5820 a(k) p Fy 480 5820 a(is) p 563 5820 a(equal) p 782 5820 a(to) p 884 5820 a(the) p 1027 5820 a(n) n(um) n(b) r(er) p 1329 5820 a(of) p 1424 5820 a(path) p 1617 5820 a(segmen) n(ts) p 1971 5820 a(that) p 2151 5820 a(form) p 2347 5820 a(an) p 2463 5820 a(elemen) n(t) p 2769 5820 a(in) p Fo 2866 5820 a(W) p Fj 2956 5777 a(\() p Fp(n) p Fj(\)) p Fp 2944 5844 a(I) p Fy 3067 5820 a(.) 1899 6120 y(5) p 90 rotate dyy eop %%Page: 6 6 6 5 bop Fi -42 87 a(Example.) p Fy 332 87 a(F) p 379 87 a(or) p Fo 481 87 a(d) p Fy 548 87 a(=) p 635 87 a(4,) p 728 87 a(there) p 940 87 a(are) p Fo 1079 87 a(f) p Fj 1120 99 a(4) p Fy 1180 87 a(=) p 1267 87 a(6) p 1336 87 a(t) n(yp) r(es) p 1556 87 a(of) p 1650 87 a(sets) p Fo 1813 87 a(W) p Fj 1903 44 a(\() p Fp(n) p Fj(\)) p Fp 1891 111 a(I) p Fy 2014 87 a(,) p 2064 87 a(whic) n(h) p 2302 87 a(are) p Fk 0 253 a(f) p Fy(\(1\)) p Fk(g) p Fz(:) p Fy 300 253 a(Set) p 443 253 a(of) p 537 253 a(paths) p 764 253 a(from) p Fo 960 253 a(O) p Fy 1053 253 a(to) p Fo 1155 253 a(v) p Fp 1195 265 a(n;) p Fj(1) p Fy 1321 253 a(whic) n(h) p 1558 253 a(do) p 1673 253 a(not) p 1821 253 a(hit) p Fo 1950 253 a(v) p Fp 1990 265 a(n;) p Fj(2) p Fy 2088 253 a(,) p Fo 2139 253 a(v) p Fp 2179 265 a(n;) p Fj(3) p Fy 2277 253 a(,) p Fo 2328 253 a(v) p Fp 2368 265 a(n;) p Fj(4) p Fy 2480 253 a(.) p Fk 0 419 a(f) p Fy(\(2\)) p Fk(g) p Fz(:) p Fy 300 419 a(Set) p 443 419 a(of) p 537 419 a(paths) p 764 419 a(from) p Fo 960 419 a(O) p Fy 1053 419 a(to) p Fo 1155 419 a(v) p Fp 1195 431 a(n;) p Fj(2) p Fy 1321 419 a(passing) p 1612 419 a(through) p Fo 1926 419 a(v) p Fp 1966 431 a(n;) p Fj(1) p Fy 2092 419 a(whic) n(h) p 2329 419 a(do) p 2445 419 a(not) p 2592 419 a(hit) p Fo 2721 419 a(v) p Fp 2761 431 a(n;) p Fj(3) p Fy 2860 419 a(,) p Fo 2910 419 a(v) p Fp 2950 431 a(n;) p Fj(4) p Fy 3062 419 a(.) p Fk 0 585 a(f) p Fy(\(3\)) p Fk(g) p Fz(:) p Fy 300 585 a(Set) p 443 585 a(of) p 537 585 a(paths) p 764 585 a(from) p Fo 960 585 a(O) p Fy 1053 585 a(to) p Fo 1155 585 a(v) p Fp 1195 597 a(n;) p Fj(3) p Fy 1321 585 a(passing) p 1612 585 a(through) p Fo 1926 585 a(v) p Fp 1966 597 a(n;) p Fj(1) p Fy 2092 585 a(and) p Fo 2253 585 a(v) p Fp 2293 597 a(n;) p Fj(2) p Fy 2419 585 a(in) p 2516 585 a(this) p 2678 585 a(order) p 2895 585 a(and) p 3057 585 a(a) n(v) n(oiding) p Fo 3386 585 a(v) p Fp 3426 597 a(n;) p Fj(4) p Fy 3538 585 a(.) p Fk 0 751 a(f) p Fy(\(4\)) p Fk(g) p Fz(:) p Fy 300 751 a(Set) p 443 751 a(of) p 537 751 a(paths) p 764 751 a(from) p Fo 960 751 a(O) p Fy 1053 751 a(to) p Fo 1155 751 a(v) p Fp 1195 763 a(n;) p Fj(4) p Fy 1321 751 a(passing) p 1612 751 a(through) p Fo 1926 751 a(v) p Fp 1966 763 a(n;) p Fj(1) p Fy 2064 751 a(,) p Fo 2115 751 a(v) p Fp 2155 763 a(n;) p Fj(2) p Fy 2253 751 a(,) p 2304 751 a(and) p Fo 2465 751 a(v) p Fp 2505 763 a(n;) p Fj(3) p Fy 2631 751 a(in) p 2728 751 a(this) p 2890 751 a(order.) p Fk 0 917 a(f) p Fy(\(1) p Fo(;) p Fy 153 917 a(1\)) p Fk(g) p Fz(:) p Fy 378 917 a(Set) p 519 917 a(of) p 611 917 a(pair) p 780 917 a(of) p 873 917 a(\(self-) p 1076 917 a(and) p 1236 917 a(m) n(utually-a) n(v) n(oiding\)) p 1946 917 a(paths,) p 2194 917 a(one) p 2344 917 a(from) p Fo 2538 917 a(O) p Fy 2630 917 a(to) p Fo 2729 917 a(v) p Fp 2769 929 a(n;) p Fj(1) p Fy 2893 917 a(and) p 3052 917 a(the) p 3193 917 a(other) p 3408 917 a(from) p Fo 3602 917 a(v) p Fp 3642 929 a(n;) p Fj(2) p Fy 3766 917 a(to) p Fo 208 1017 a(v) p Fp 248 1029 a(n;) p Fj(3) p Fy 373 1017 a(neither) p 655 1017 a(hitting) p Fo 927 1017 a(v) p Fp 967 1029 a(n;) p Fj(4) p Fy 1079 1017 a(.) p Fk 0 1183 a(f) p Fy(\(1) p Fo(;) p Fy 153 1183 a(2\)) p Fk(g) p Fz(:) p Fy 378 1183 a(Set) p 521 1183 a(of) p 616 1183 a(pair) p 786 1183 a(of) p 881 1183 a(paths,) p 1130 1183 a(one) p 1283 1183 a(from) p Fo 1479 1183 a(O) p Fy 1572 1183 a(to) p Fo 1674 1183 a(v) p Fp 1714 1195 a(n;) p Fj(1) p Fy 1840 1183 a(and) p 2001 1183 a(the) p 2144 1183 a(other) p 2361 1183 a(from) p Fo 2557 1183 a(v) p Fp 2597 1195 a(n;) p Fj(2) p Fy 2723 1183 a(to) p Fo 2825 1183 a(v) p Fp 2865 1195 a(n;) p Fj(4) p Fy 2991 1183 a(via) p Fo 3127 1183 a(v) p Fp 3167 1195 a(n;) p Fj(3) p Fy 3279 1183 a(.) p Fd 3774 1349 a(3) p Fy 125 1614 a(F) p 172 1614 a(or) p Fo 285 1614 a(w) p Fk 389 1614 a(2) p Fg 526 1535 a([) p Fp 487 1714 a(n) p Fn(2) p Fc(Z) p Ff 612 1722 a(+) p Fg 701 1535 a([) p Fp 672 1714 a(I) p Fn 706 1714 a(2I) p Fm 789 1723 a(d) p Fo 837 1614 a(W) p Fj 927 1571 a(\() p Fp(n) p Fj(\)) p Fp 915 1639 a(I) p Fy 1063 1614 a(denote) p 1342 1614 a(b) n(y) p 1486 1614 a(^) p Fo 1469 1614 a(w) p Fy 1570 1614 a(the) p 1724 1614 a(set) p 1866 1614 a(of) p 1972 1614 a(b) r(onds) p 2226 1614 a(whic) n(h) p Fo 2475 1614 a(w) p Fy 2576 1614 a(passes.) p 2893 1614 a(Namely) p 3163 1614 a(,) p 3228 1614 a(for) p Fo 3367 1614 a(n) p Fk 3459 1614 a(2) p Fq 3557 1614 a(Z) p Fj 3612 1626 a(+) p Fy 3706 1614 a(and) p Fo 0 1832 a(I) p Fy 66 1832 a(=) p 154 1832 a(\() p Fo(i) p Fj 215 1844 a(1) p Fo 252 1832 a(;) p 289 1832 a(i) p Fj 318 1844 a(2) p Fo 355 1832 a(;) p Fk 392 1832 a(\001) p 429 1832 a(\001) p 466 1832 a(\001) p Fo 502 1832 a(;) p 539 1832 a(i) p Fp 568 1844 a(k) p Fy 609 1832 a(\)) p Fk 664 1832 a(2) p 742 1832 a(I) p Fp 787 1844 a(d) p Fy 840 1832 a(,) p 891 1832 a(and) p 1052 1832 a(for) p Fo 1179 1832 a(w) p Fy 1264 1832 a(=) p 1352 1832 a(\() p Fo(w) p Fj 1443 1844 a(1) p Fo 1481 1832 a(;) p 1518 1832 a(w) p Fj 1577 1844 a(2) p Fo 1614 1832 a(;) p Fk 1651 1832 a(\001) p 1688 1832 a(\001) p 1725 1832 a(\001) p Fo 1762 1832 a(;) p 1799 1832 a(w) p Fp 1858 1844 a(k) p Fy 1899 1832 a(\)) p Fk 1954 1832 a(2) p Fo 2033 1832 a(W) p Fj 2123 1789 a(\() p Fp(n) p Fj(\)) p Fp 2111 1857 a(I) p Fy 2234 1832 a(,) 697 2027 y(^) p Fo 680 2027 a(w) p Fy 765 2027 a(=) p Fk 853 2027 a(f) p 895 1955 519 4 v Fo(w) p Fp 954 2039 a(j) p Fy 989 2027 a(\() p Fo(i) p Fy(\)) p Fo(w) p Fp 1141 2039 a(j) p Fy 1176 2027 a(\() p Fo(i) p Fy 1255 2027 a(+) p 1339 2027 a(1\)) p Fk 1435 2027 a(2) p Fo 1514 2027 a(B) p Fk 1604 2027 a(j) p Fo 1650 2027 a(i) p Fy 1702 2027 a(=) p 1789 2027 a(0) p Fo(;) p Fy 1868 2027 a(1) p Fo(;) p Fy 1947 2027 a(2) p Fo(;) p Fk 2026 2027 a(\001) p 2063 2027 a(\001) p 2100 2027 a(\001) p Fo 2135 2027 a(;) p 2172 2027 a(L) p Fy(\() p Fo(w) p Fp 2320 2039 a(j) p Fy 2355 2027 a(\)) p Fk 2406 2027 a(\000) p Fy 2489 2027 a(1) p Fo(;) p 2595 2027 a(j) p Fy 2657 2027 a(=) p 2745 2027 a(1) p Fo(;) p Fy 2824 2027 a(2) p Fo(;) p Fk 2903 2027 a(\001) p 2940 2027 a(\001) p 2977 2027 a(\001) p Fo 3012 2027 a(;) p 3049 2027 a(k) p Fk 3095 2027 a(g) p Fo(:) p Fy 0 2210 a(Also) p 187 2210 a(de\014ne) p Fo 427 2210 a(S) p Fp 478 2222 a(I) p Fy 516 2210 a(\() p Fo(w) p Fy 609 2210 a(\),) p Fo 693 2210 a(I) p Fk 759 2210 a(2) p 838 2210 a(I) p Fp 883 2222 a(d) p Fy 921 2210 a(,) p 972 2210 a(b) n(y) p 1053 2210 a(,) p Fo 925 2392 a(S) p Fp 976 2404 a(I) p Fy 1014 2392 a(\() p Fo(w) p Fy 1107 2392 a(\)) p 1163 2392 a(=) p Fk 1251 2392 a(f) p Fy(\001) p Fk 1385 2392 a(2) p 1463 2392 a(T) p Fp 1508 2404 a(b) p Fk 1565 2392 a(j) p Fy 1628 2392 a(^) p Fo 1611 2392 a(w) p Fk 1691 2392 a(\\) p Fy 1765 2392 a(\001) p 1889 2392 a(is) p 1973 2392 a(congruen) n(t) p 2358 2392 a(to) p 2487 2392 a(\001) p Fp 2556 2404 a(I) p Fy 2650 2392 a(of) p 2744 2392 a(\(4\)) p Fk(g) p Fo(;) p Fy 3734 2392 a(\(6\)) 0 2575 y(and) p 159 2575 a(denote) p 423 2575 a(b) n(y) p Fo 536 2575 a(s) p Fp 575 2587 a(I) p Fy 613 2575 a(\() p Fo(w) p Fy 706 2575 a(\)) p 762 2575 a(=) p Fo 850 2575 a(]S) p Fp 933 2587 a(I) p Fy 971 2575 a(\() p Fo(w) p Fy 1064 2575 a(\),) p 1146 2575 a(the) p 1286 2575 a(cardinalit) n(y) p 1698 2575 a(of) p Fo 1790 2575 a(S) p Fp 1841 2587 a(I) p Fy 1879 2575 a(\() p Fo(w) p Fy 1972 2575 a(\).) p Fo 2064 2575 a(s) p Fp 2103 2587 a(I) p Fy 2141 2575 a(\() p Fo(w) p Fy 2234 2575 a(\)) p 2293 2575 a(is) p 2373 2575 a(the) p 2513 2575 a(n) n(um) n(b) r(er) p 2813 2575 a(of) p 2905 2575 a(unit) p Fo 3077 2575 a(d) p Fy(-simplices) p 3497 2575 a(in) p Fo 3591 2575 a(F) p Fy 3680 2575 a(suc) n(h) 0 2675 y(that) p 176 2675 a(the) p 316 2675 a(tra) p 428 2675 a(jectory) p 700 2675 a(of) p 791 2675 a(the) p 931 2675 a(path) p 1121 2675 a(\(or) p 1251 2675 a(the) p 1391 2675 a(paths\)) p Fo 1646 2675 a(w) p Fy 1732 2675 a(is) p 1812 2675 a(congruen) n(t) p 2193 2675 a(to) p 2291 2675 a(\001) p Fp 2360 2687 a(I) p Fy 2412 2675 a(.) p 2471 2675 a(It) p 2558 2675 a(is) p 2638 2675 a(a) p 2703 2675 a(generalized) p 3129 2675 a(notion) p 3384 2675 a(of) p 3475 2675 a(the) p 3614 2675 a(length) 0 2774 y(of) p 95 2774 a(the) p 238 2774 a(path) p 431 2774 a(in) p 528 2774 a(the) p 671 2774 a(sense) p 884 2774 a(that) p Fp 344 2930 a(k) p Fg 302 2955 a(X) p Fp 309 3132 a(i) p Fj(=1) p Fo 436 3034 a(L) p Fy(\() p Fo(w) p Fp 584 3046 a(i) p Fy 612 3034 a(\)) p 667 3034 a(=) p Fg 775 2955 a(X) p Fp 755 3133 a(J) p Fn 797 3133 a(2I) p Fm 880 3142 a(d) p Fk 928 3034 a(j) p Fo(J) p Fk 1005 3034 a(j) p Fo(s) p Fp 1067 3046 a(J) p Fy 1114 3034 a(\() p Fo(w) p Fy 1207 3034 a(\)) p Fo(;) p 1332 3034 a(w) p Fy 1417 3034 a(=) p 1504 3034 a(\() p Fo(w) p Fj 1595 3046 a(1) p Fo 1633 3034 a(;) p 1670 3034 a(w) p Fj 1729 3046 a(2) p Fo 1767 3034 a(;) p Fk 1804 3034 a(\001) p 1841 3034 a(\001) p 1878 3034 a(\001) p Fo 1915 3034 a(;) p 1952 3034 a(w) p Fp 2011 3046 a(k) p Fy 2052 3034 a(\)) p Fk 2107 3034 a(2) p Fo 2186 3034 a(W) p Fj 2276 2991 a(\() p Fp(n) p Fj(\)) p Fp 2264 3058 a(I) p Fo 2386 3034 a(;) p 2451 3034 a(I) p Fy 2517 3034 a(=) p 2605 3034 a(\() p Fo(i) p Fj 2666 3046 a(1) p Fo 2703 3034 a(;) p Fk 2740 3034 a(\001) p 2777 3034 a(\001) p 2814 3034 a(\001) p Fo 2850 3034 a(;) p 2887 3034 a(i) p Fp 2916 3046 a(k) p Fy 2957 3034 a(\)) p Fk 3012 3034 a(2) p 3090 3034 a(I) p Fp 3135 3046 a(d) p Fo 3188 3034 a(;) p 3253 3034 a(n) p Fk 3326 3034 a(2) p Fq 3404 3034 a(Z) p Fj 3459 3046 a(+) p Fo 3515 3034 a(;) p Fy 3734 3034 a(\(7\)) 0 3304 y(where,) p 263 3304 a(for) p Fo 390 3304 a(J) p Fk 467 3304 a(2) p 546 3304 a(I) p Fp 591 3316 a(d) p Fy 657 3304 a(w) n(e) p 780 3304 a(de\014ne) p Fk 1019 3304 a(j) p Fo(J) p Fk 1096 3304 a(j) p Fy(,) p 1170 3304 a(the) p 1313 3304 a(length) p 1567 3304 a(of) p Fo 1661 3304 a(J) p Fy 1715 3304 a(,) p 1766 3304 a(b) n(y) p Fk 1176 3487 a(j) p Fo(J) p Fk 1253 3487 a(j) p Fy 1299 3487 a(=) p Fo 1387 3487 a(j) p Fj 1421 3499 a(1) p Fy 1477 3487 a(+) p Fk 1560 3487 a(\001) p 1597 3487 a(\001) p 1634 3487 a(\001) p Fy 1675 3487 a(+) p Fo 1758 3487 a(j) p Fp 1792 3499 a(`) p Fo 1838 3487 a(;) p Fy 1958 3487 a(if) p Fo 2090 3487 a(J) p Fy 2167 3487 a(=) p 2254 3487 a(\() p Fo(j) p Fj 2320 3499 a(1) p Fo 2358 3487 a(;) p Fk 2395 3487 a(\001) p 2432 3487 a(\001) p 2469 3487 a(\001) p Fo 2506 3487 a(;) p 2543 3487 a(j) p Fp 2577 3499 a(`) p Fy 2609 3487 a(\)) p Fo(:) p Fy 3734 3487 a(\(8\)) p Fr 0 3719 a(2.4) p 255 3719 a(P) m(arameter) p 806 3719 a(space) p 1105 3719 a(and) p 1322 3719 a(the) p 1516 3719 a(renormalization) p 2322 3719 a(group.) p Fy 0 3872 a(Assumptions) p 495 3872 a(of) p 592 3872 a(the) p 738 3872 a(main) p 948 3872 a(results) p 1215 3872 a(are) p 1356 3872 a(stated) p 1609 3872 a(in) p 1708 3872 a(terms) p 1942 3872 a(of) p 2039 3872 a(the) p 2185 3872 a(\015o) n(ws) p 2393 3872 a(of) p 2490 3872 a(the) p 2636 3872 a(asso) r(ciated) p 3034 3872 a(renormalization) p 3632 3872 a(group) 0 3972 y(\(R) n(G\),) p 236 3972 a(whic) n(h) p 471 3972 a(is) p 551 3972 a(a) p 617 3972 a(map) p 799 3972 a(\(discrete-time) p 1322 3972 a(dynamical) p 1718 3972 a(system\)) p 2022 3972 a(in) p 2116 3972 a(a) p 2182 3972 a(parameter) p 2576 3972 a(space) p 2795 3972 a(of) p 2887 3972 a(v) p 2926 3972 a(ariables) p 3228 3972 a(in) p 3322 3972 a(the) p 3462 3972 a(generating) 0 4072 y(function) p 326 4072 a(of) p 422 4072 a(generalized) p 852 4072 a(path) p 1047 4072 a(length) p 1301 4072 a(\() p Fo(s) p Fp 1372 4084 a(J) p Fo 1419 4072 a(;) p 1481 4072 a(J) p Fk 1559 4072 a(2) p 1640 4072 a(I) p Fp 1685 4084 a(d) p Fy 1724 4072 a(\).) p 1819 4072 a(The) p 1991 4072 a(dynamical) p 2391 4072 a(system) p 2667 4072 a(is) p 2751 4072 a(deriv) n(ed) p 3043 4072 a(as) p 3146 4072 a(the) p 3290 4072 a(resp) r(onse) p 3627 4072 a(in) p 3725 4072 a(the) 0 4171 y(parameter) p 401 4171 a(space) p 627 4171 a(to) p 733 4171 a(the) p 880 4171 a(c) n(hange) p 1159 4171 a(in) p Fo 1260 4171 a(n) p Fy(.) p 1383 4171 a(A) p 1477 4171 a(graphical) p 1841 4171 a(prop) r(ert) n(y) p 2185 4171 a(of) p Fo 2284 4171 a(d) p Fy(SG) p 2470 4171 a(called) p 2709 4171 a(\014nite) p 2926 4171 a(rami\014deness) p 3402 4171 a(implies) p 3688 4171 a(that) 0 4271 y(the) p 143 4271 a(R) n(G) p 295 4271 a(is) p 378 4271 a(a) p 447 4271 a(\014nite) p 659 4271 a(dimensional) p 1117 4271 a(dynamical) p 1516 4271 a(system.) 125 4370 y(De\014ne) p 382 4370 a(the) p 525 4370 a(generating) p 931 4370 a(function) p Fo 1313 4532 a(~) 1296 4553 y(X) p Fp 1365 4565 a(n) p Fy 1433 4553 a(=) p 1520 4553 a(\() p Fo(X) p Fp 1621 4565 a(n;I) p Fo 1720 4553 a(;) p 1785 4553 a(I) p Fk 1851 4553 a(2) p 1929 4553 a(I) p Fp 1974 4565 a(d) p Fy 2013 4553 a(\)) p 2068 4553 a(:) p Fq 2142 4553 a(C) p Fn 2202 4519 a(I) p Fm 2240 4528 a(d) p Fk 2302 4553 a(!) p Fq 2408 4553 a(C) p Fn 2468 4519 a(I) p Fm 2506 4528 a(d) p Fy 0 4755 a(of) p 95 4755 a(\() p Fo(s) p Fp 166 4767 a(J) p Fo 226 4755 a(;) p 291 4755 a(J) p Fk 368 4755 a(2) p 446 4755 a(I) p Fp 491 4767 a(d) p Fy 530 4755 a(\)) p 590 4755 a(for) p 717 4755 a(a) p 786 4755 a(family) p 1040 4755 a(of) p 1134 4755 a(paths) p 1361 4755 a(sets) p 1523 4755 a(\() p Fo(W) p Fj 1645 4712 a(\() p Fp(n) p Fj(\)) p Fp 1633 4779 a(I) p Fo 1756 4755 a(;) p 1821 4755 a(I) p Fk 1887 4755 a(2) p 1965 4755 a(I) p Fp 2010 4767 a(d) p Fy 2049 4755 a(\),) p 2132 4755 a(b) n(y) p 2213 4755 a(,) p Fo 614 4958 a(X) p Fp 683 4970 a(n;I) p Fy 782 4958 a(\() p Fo 813 4958 a(~) p 814 4958 a(x) p Fy(\)) p 917 4958 a(=) p Fg 1070 4879 a(X) p Fp 1004 5079 a(w) p Fn 1054 5079 a(2) p Fp(W) p Ff 1170 5048 a(\() p Fm(n) p Ff(\)) p Fm 1161 5098 a(I) p Fg 1296 4879 a(Y) p Fp 1270 5057 a(J) p Fn 1312 5057 a(2I) p Fm 1395 5066 a(d) p Fo 1443 4958 a(x) p Fp 1490 4915 a(s) p Fm 1521 4923 a(J) p Fj 1562 4915 a(\() p Fp(w) p Fj 1638 4915 a(\)) p Fp 1490 4982 a(J) p Fo 1681 4958 a(;) p 1772 4958 a(~) p 1773 4958 a(x) p Fy 1844 4958 a(=) p 1932 4958 a(\() p Fo(x) p Fp 2011 4970 a(J) p Fo 2058 4958 a(;) p 2122 4958 a(J) p Fk 2199 4958 a(2) p 2278 4958 a(I) p Fp 2323 4970 a(d) p Fy 2362 4958 a(\)) p Fk 2417 4958 a(2) p Fq 2495 4958 a(C) p Fn 2555 4923 a(I) p Fm 2593 4932 a(d) p Fo 2632 4958 a(;) p 2697 4958 a(n) p Fy 2769 4958 a(=) p 2857 4958 a(0) p Fo(;) p Fy 2936 4958 a(1) p Fo(;) p Fy 3015 4958 a(2) p Fo(;) p Fk 3094 4958 a(\001) p 3131 4958 a(\001) p 3168 4958 a(\001) p Fo 3203 4958 a(:) p Fy 3734 4958 a(\(9\)) 0 5272 y(The) p 171 5272 a(righ) n(t) p 372 5272 a(hand) p 579 5272 a(side) p 746 5272 a(is) p 829 5272 a(a) p 898 5272 a(\014nite) p 1110 5272 a(summation,) p 1563 5272 a(so) p Fo 1665 5272 a(X) p Fp 1734 5284 a(n;I) p Fy 1860 5272 a(is) p 1944 5272 a(de\014ned) p 2230 5272 a(on) p Fq 2345 5272 a(C) p Fn 2405 5238 a(I) p Fm 2443 5247 a(d) p Fy 2481 5272 a(.) 125 5372 y(The) p 295 5372 a(starting) p 605 5372 a(p) r(oin) n(t) p 822 5372 a(of) p 916 5372 a(our) p 1064 5372 a(analysis) p 1376 5372 a(is) p 1460 5372 a(the) p 1603 5372 a(follo) n(wing.) p Fz 0 5538 a(Prop) s(osition) p 516 5538 a(4) p Fo 623 5517 a(~) 605 5538 y(X) p Fp 674 5550 a(n) p Fy 742 5538 a(=) p 830 5538 a(\() p Fo(X) p Fp 931 5550 a(n;I) p Fo 1030 5538 a(;) p 1097 5538 a(I) p Fk 1163 5538 a(2) p 1241 5538 a(I) p Fp 1286 5550 a(d) p Fy 1325 5538 a(\)) p Fi(,) p Fo 1412 5538 a(n) p Fy 1485 5538 a(=) p 1573 5538 a(0) p Fo(;) p Fy 1652 5538 a(1) p Fo(;) p Fy 1731 5538 a(2) p Fo(;) p Fk 1810 5538 a(\001) p 1847 5538 a(\001) p 1884 5538 a(\001) p Fi 1905 5538 a(,) p 1960 5538 a(satisfy) p 2219 5538 a(the) p 2357 5538 a(fol) t(lowing) p 2709 5538 a(r) l(e) l(cursion) p 3070 5538 a(r) l(elations.) p Fo 1562 5712 a(~) 1544 5733 y(X) p Fj 1613 5745 a(0) p Fy 1650 5733 a(\() p Fo 1681 5733 a(~) p 1682 5733 a(x) p Fy 1730 5733 a(\)) p 1785 5733 a(=) p Fo 1872 5733 a(~) p 1873 5733 a(x) q(;) p 1986 5733 a(~) p 1987 5733 a(x) p Fk 2058 5733 a(2) p Fq 2136 5733 a(C) p Fn 2196 5699 a(I) p Fm 2234 5708 a(d) p Fo 2273 5733 a(;) p Fy 1899 6120 a(6) p 90 rotate dyy eop %%Page: 7 7 7 6 bop Fi 0 83 a(and) p Fo 1638 162 a(~) 1621 183 y(X) p Fp 1690 195 a(n) p Fj(+1) p Fy 1842 183 a(=) p Fo 1933 162 a(~) p Fy 1930 183 a(\010) p Fk 2008 183 a(\016) p Fo 2086 162 a(~) 2068 183 y(X) p Fp 2137 195 a(n) p Fo 2196 183 a(;) p Fy 3692 183 a(\(10\)) p Fi 0 332 a(wher) l(e) p Fo 1482 411 a(~) p Fy 1479 432 a(\010) p 1562 432 a(=) p 1650 432 a(\(\010) p Fp 1742 444 a(I) p Fo 1780 432 a(;) p 1847 432 a(I) p Fk 1913 432 a(2) p 1991 432 a(I) p Fp 2036 444 a(d) p Fy 2075 432 a(\)) p 2130 432 a(=) p Fo 2235 411 a(~) 2218 432 y(X) p Fj 2287 444 a(1) p Fo 2338 432 a(;) p Fi 0 581 a(is) p 89 581 a(a) p Fo 161 581 a(f) p Fp 202 593 a(d) p Fi 269 581 a(dimensional) p 731 581 a(ve) l(ctor) p 976 581 a(value) l(d) p 1228 581 a(function) p 1555 581 a(whose) p 1796 581 a(c) l(omp) l(onents) p 2244 581 a(ar) l(e) p 2384 581 a(p) l(olynomials) p 2836 581 a(in) p Fo 2937 581 a(f) p Fp 2978 593 a(d) p Fi 3046 581 a(variables) p 3391 581 a(with) p 3571 581 a(p) l(ositive) 0 681 y(inte) l(ger) p 275 681 a(c) l(o) l(e\016cients.) p 732 681 a(In) p 840 681 a(p) l(articular,) p Fq 1246 681 a(R) p Fn 1306 643 a(I) p Fm 1344 652 a(d) p Fj 1306 701 a(+) p Fi 1412 681 a(is) p 1501 681 a(an) p 1620 681 a(invariant) p 1979 681 a(set) p 2109 681 a(of) p Fo 2209 660 a(~) p Fy 2206 681 a(\010) p Fi(.) 125 780 y(The) p 302 780 a(de) l(gr) l(e) l(e) p 556 780 a(of) p 661 780 a(e) l(ach) p 855 780 a(term) p 1061 780 a(in) p 1170 780 a(the) p 1315 780 a(p) l(olynomials) p 1774 780 a(ar) l(e) p 1922 780 a(no) p 2049 780 a(less) p 2213 780 a(than) p Fy 2409 780 a(2) p Fi 2487 780 a(and) p 2656 780 a(no) p 2782 780 a(gr) l(e) l(ater) p 3065 780 a(than) p Fo 3261 780 a(d) p Fy 3328 780 a(+) p 3417 780 a(1) p Fi(,) p 3522 780 a(and) p Fy 3691 780 a(\010) p Fj 3751 795 a(\(1\)) p Fi 0 889 a(c) l(ontains) p 329 889 a(terms) p Fo 561 889 a(x) p Fj 608 904 a(\(1\)) 698 859 y(2) p Fi 765 889 a(and) p Fo 926 889 a(x) p Fj 973 904 a(\(1\)) p Fp 1063 859 a(d) p Fj(+1) p Fi 1199 889 a(.) -42 1071 y(Pr) l(o) l(of.) p Fy 219 1071 a(Let) p Fo 371 1071 a(n) p Fk 449 1071 a(2) p Fq 532 1071 a(Z) p Fj 587 1083 a(+) p Fy 643 1071 a(.) p Fo 712 1071 a(F) p Fj 765 1083 a(1) p Fy 834 1071 a(is) p 920 1071 a(comp) r(osed) p 1304 1071 a(of) p Fo 1402 1071 a(d) p Fy 1466 1071 a(+) p 1551 1071 a(1) p Fo 1623 1071 a(d) p Fy(-simplices) p 2049 1071 a(congruen) n(t) p 2437 1071 a(to) p Fo 2542 1071 a(F) p Fj 2595 1083 a(0) p Fy 2646 1071 a(.) p 2716 1071 a(Similarly) p 3035 1071 a(,) p Fo 3089 1071 a(F) p Fp 3142 1083 a(n) p Fj(+1) p Fy 3302 1071 a(is) p 3389 1071 a(comp) r(osed) p 3773 1071 a(of) p Fo 0 1171 a(d) p Fy 62 1171 a(+) p 145 1171 a(1) p Fo 214 1171 a(d) p Fy(-simplices) p Fo 636 1171 a(F) p Fp 689 1183 a(n) p Fy 749 1171 a(.) p 809 1171 a(The) p 979 1171 a(similarit) n(y) p 1349 1171 a(of) p 1443 1171 a(the) p 1586 1171 a(t) n(w) n(o) p 1743 1171 a(comp) r(ositions) p 2240 1171 a(leads) p 2448 1171 a(to) p 2550 1171 a(a) p 2619 1171 a(natural) p 2910 1171 a(map) p Fo 1378 1365 a(\031) p Fy 1451 1365 a(:) p Fo 1525 1365 a(W) p Fj 1615 1322 a(\() p Fp(n) p Fj(+1\)) p Fp 1603 1390 a(I) p Fk 1819 1365 a(!) p Fo 1925 1365 a(W) p Fj 2015 1322 a(\(1\)) p Fp 2003 1390 a(I) p Fo 2118 1365 a(;) p 2211 1365 a(I) p Fk 2277 1365 a(2) p 2355 1365 a(I) p Fp 2400 1377 a(d) p Fo 2439 1365 a(:) p Fy 0 1548 a(F) p 47 1548 a(or) p 161 1548 a(eac) n(h) p Fo 359 1548 a(X) p Fp 428 1560 a(n) p Fj(+1) p Fp(;I) p Fy 625 1548 a(,) p 690 1548 a(classify) p 989 1548 a(the) p 1144 1548 a(summation) p 1585 1548 a(in) p 1694 1548 a(the) p 1848 1548 a(righ) n(t) p 2061 1548 a(hand) p 2280 1548 a(side) p 2459 1548 a(of) p 2565 1548 a(\(9\)) p 2711 1548 a(\(with) p Fo 2944 1548 a(n) p Fy 3020 1548 a(+) p 3111 1548 a(1) p 3192 1548 a(in) p 3300 1548 a(place) p 3524 1548 a(of) p Fo 3631 1548 a(n) p Fy(\)) p 3752 1548 a(b) n(y) p Fo 0 1667 a(\031) p Fy 50 1667 a(\() p Fo(w) p Fy 143 1667 a(\)) p Fk 200 1667 a(2) p Fo 278 1667 a(W) p Fj 368 1623 a(\(1\)) p Fp 356 1691 a(I) p Fy 485 1667 a(to) p 586 1667 a(\014nd) p Fo 225 1869 a(X) p Fp 294 1881 a(n) p Fj(+1) p Fp(;I) p Fy 477 1869 a(\() p Fo 508 1869 a(~) p 509 1869 a(x) p Fy 557 1869 a(\)) p 612 1869 a(=) p Fg 801 1791 a(X) p Fp 700 1991 a(w) p Fn 750 1991 a(2) p Fp(W) p Ff 866 1960 a(\() p Fm(n) p Ff(+1\)) p Fm 857 2010 a(I) p Fg 1063 1791 a(Y) p Fp 1036 1969 a(J) p Fn 1078 1969 a(2I) p Fm 1161 1978 a(d) p Fo 1209 1869 a(x) p Fp 1256 1826 a(s) p Fm 1287 1834 a(J) p Fj 1328 1826 a(\() p Fp(w) p Fj 1404 1826 a(\)) p Fp 1256 1894 a(J) p Fy 1457 1869 a(=) p Fg 1617 1791 a(X) p Fp 1545 1991 a(w) p Fe 1595 1974 a(0) p Fn 1617 1991 a(2) p Fp(W) p Ff 1733 1960 a(\(1\)) p Fm 1724 2010 a(I) p Fg 2077 1791 a(X) p Fp 1824 1991 a(w) p Fn 1874 1991 a(2) p Fp(W) p Ff 1990 1960 a(\() p Fm(n) p Ff(+1\)) p Fm 1981 2010 a(I) p Fj 2146 1991 a(;) p Fp 2184 1991 a(\031) p Fj 2225 1991 a(\() p Fp(w) p Fj 2301 1991 a(\)=) p Fp(w) p Fe 2428 1974 a(0) p Fg 2490 1791 a(Y) p Fp 2463 1969 a(J) p Fn 2505 1969 a(2I) p Fm 2588 1978 a(d) p Fo 2636 1869 a(x) p Fp 2683 1826 a(s) p Fm 2714 1834 a(J) p Fj 2755 1826 a(\() p Fp(w) p Fj 2831 1826 a(\)) p Fp 2683 1894 a(J) p Fy 225 2241 a(=) p Fg 386 2162 a(X) p Fp 313 2362 a(w) p Fe 363 2346 a(0) p Fn 385 2362 a(2) p Fp(W) p Ff 501 2332 a(\(1\)) p Fm 492 2382 a(I) p Fg 626 2162 a(Y) p Fp 592 2340 a(I) p Fe 626 2324 a(0) p Fn 649 2340 a(2I) p Fm 732 2349 a(d) p Fg 780 2049 a(0) 780 2195 y(B) 780 2249 y(@) 939 2162 y(X) p Fp 853 2362 a(w) p Fe 903 2346 a(00) p Fn 943 2362 a(2) p Fp(W) p Ff 1059 2332 a(\() p Fm(n) p Ff(\)) p Fm 1050 2389 a(I) p Fe 1079 2377 a(0) p Fg 1185 2162 a(Y) p Fp 1159 2340 a(J) p Fn 1201 2340 a(2I) p Fm 1284 2349 a(d) p Fo 1332 2241 a(x) p Fp 1379 2198 a(s) p Fm 1410 2206 a(J) p Fj 1451 2198 a(\() p Fp(w) p Fe 1527 2173 a(00) p Fj 1567 2198 a(\)) p Fp 1379 2266 a(J) p Fg 1597 2049 a(1) 1597 2195 y(C) 1597 2249 y(A) p Fp 1670 2067 a(s) p Fm 1701 2082 a(I) p Fe 1730 2070 a(0) p Fj 1757 2067 a(\() p Fp(w) p Fe 1833 2042 a(0) p Fj 1855 2067 a(\)) p Fy 1908 2241 a(=) p Fg 2068 2162 a(X) p Fp 1996 2362 a(w) p Fe 2046 2346 a(0) p Fn 2068 2362 a(2) p Fp(W) p Ff 2184 2332 a(\(1\)) p Fm 2175 2382 a(I) p Fg 2309 2162 a(Y) p Fp 2275 2340 a(I) p Fe 2309 2324 a(0) p Fn 2331 2340 a(2I) p Fm 2414 2349 a(d) p Fy 2449 2241 a(\() p Fo(X) p Fp 2550 2253 a(n;I) p Fe 2645 2237 a(0) p Fy 2671 2241 a(\() p Fo 2702 2241 a(~) p 2703 2241 a(x) p Fy 2751 2241 a(\)\)) p Fp 2815 2207 a(s) p Fm 2846 2222 a(I) p Fe 2875 2210 a(0) p Fj 2902 2207 a(\() p Fp(w) p Fe 2978 2182 a(0) p Fj 3000 2207 a(\)) p Fy 3054 2241 a(=) p Fo 3141 2241 a(X) p Fj 3210 2253 a(1) p Fp(;I) p Fy 3301 2241 a(\() p Fo 3351 2220 a(~) 3333 2241 y(X) p Fp 3402 2253 a(n) p Fy 3447 2241 a(\() p Fo 3478 2241 a(~) p 3479 2241 a(x) p Fy 3527 2241 a(\)\)) p Fo(:) p Fy 125 2576 a(By) p 260 2576 a(de\014nition) p 635 2576 a(\(9\),) p 798 2576 a(eac) n(h) p 990 2576 a(term) p 1194 2576 a(in) p 1296 2576 a(\010) p Fp 1356 2588 a(I) p Fy 1426 2576 a(=) p Fo 1523 2576 a(X) p Fj 1592 2588 a(1) p Fp(;I) p Fy 1716 2576 a(has) p 1869 2576 a(a) p 1944 2576 a(form) p Fg 2172 2497 a(Y) p Fp 2145 2675 a(J) p Fn 2187 2675 a(2I) p Fm 2270 2684 a(d) p Fo 2318 2576 a(x) p Fp 2365 2532 a(s) p Fm 2396 2540 a(J) p Fj 2437 2532 a(\() p Fp(w) p Fj 2513 2532 a(\)) p Fp 2365 2600 a(J) p Fy 2543 2576 a(,) p 2600 2576 a(hence) p 2836 2576 a(its) p 2957 2576 a(degree) p Fg 3241 2497 a(X) p Fp 3221 2675 a(J) p Fn 3263 2675 a(2I) p Fm 3346 2684 a(d) p Fo 3395 2576 a(s) p Fp 3434 2588 a(J) p Fy 3480 2576 a(\() p Fo(w) p Fy 3573 2576 a(\)) p 3639 2576 a(is,) p 3752 2576 a(b) n(y) 0 2762 y(de\014nition) p 375 2762 a(\(6\),) p 539 2762 a(the) p 689 2762 a(n) n(um) n(b) r(er) p 997 2762 a(of) p 1098 2762 a(unit) p 1279 2762 a(simplices) p 1636 2762 a(in) p Fo 1739 2762 a(F) p Fj 1792 2774 a(1) p Fy 1864 2762 a(that) p 2050 2762 a(a) p 2125 2762 a(path) p Fo 2325 2762 a(w) p Fy 2420 2762 a(passes) p 2677 2762 a(through.) p 3041 2762 a(This) p 3237 2762 a(is) p 3326 2762 a(b) r(ounded) p 3671 2762 a(from) 0 2862 y(ab) r(o) n(v) n(e) p 237 2862 a(b) n(y) p 354 2862 a(the) p 498 2862 a(total) p 698 2862 a(n) n(um) n(b) r(er) p 1002 2862 a(of) p 1098 2862 a(unit) p 1275 2862 a(simplices) p 1628 2862 a(in) p Fo 1726 2862 a(F) p Fj 1779 2874 a(1) p Fy 1817 2862 a(,) p 1870 2862 a(whic) n(h) p 2109 2862 a(is) p Fo 2194 2862 a(d) p Fy 2257 2862 a(+) p 2341 2862 a(1,) p 2435 2862 a(and) p 2598 2862 a(from) p 2796 2862 a(b) r(elo) n(w) p 3033 2862 a(b) n(y) p 3149 2862 a(2,) p 3244 2862 a(b) r(ecause) p 3552 2862 a(an) n(y) p 3711 2862 a(t) n(w) n(o) 0 2962 y(extreme) p 316 2962 a(\(outmost\)) p 704 2962 a(v) n(ertices) p 1005 2962 a(of) p Fo 1099 2962 a(F) p Fj 1152 2974 a(1) p Fy 1217 2962 a(is) p 1301 2962 a(apart) p 1523 2962 a(b) n(y) p 1638 2962 a(length) p 1892 2962 a(2.) 125 3061 y(P) n(ositivit) n(y) p 500 3061 a(of) p 595 3061 a(co) r(e\016cien) n(ts) p 1016 3061 a(of) p Fo 1111 3061 a(X) p Fj 1180 3073 a(1) p Fp(;I) p Fy 1299 3061 a(are) p 1438 3061 a(ob) n(vious.) p 1772 3061 a(Existence) p 2145 3061 a(of) p 2240 3061 a(terms) p Fo 2472 3061 a(x) p Fj 2519 3031 a(2) 2519 3088 y(\(1\)) p Fy 2637 3061 a(and) p Fo 2799 3061 a(x) p Fp 2846 3026 a(d) p Fj(+1) 2846 3090 y(\(1\)) p Fy 2997 3061 a(in) p 3095 3061 a(\010) p Fj 3155 3076 a(\(1\)) p Fy 3268 3061 a(=) p Fo 3356 3061 a(X) p Fj 3425 3076 a(1) p Fp(;) p Fj(\(1\)) p Fy 3595 3061 a(follo) n(ws) 0 3199 y(from) p 196 3199 a(the) p 339 3199 a(paths) p 566 3132 327 4 v Fo 566 3199 a(O) r(v) p Fj 671 3211 a(0) p Fp(;) p Fj(1) p Fo 762 3199 a(v) p Fj 802 3211 a(1) p Fp(;) p Fj(1) p Fy 920 3199 a(and) p 1081 3127 1993 4 v Fo 1081 3199 a(O) r(v) p Fj 1186 3211 a(0) p Fp(;d) p Fy 1278 3199 a(\() p Fo(v) p Fj 1350 3211 a(0) p Fp(;d) p Fy 1461 3199 a(+) p Fo 1544 3199 a(v) p Fj 1584 3211 a(0) p Fp(;d) p Fn(\000) p Fj(1) p Fy 1761 3199 a(\)\() p Fo(v) p Fj 1865 3211 a(0) p Fp(;d) p Fn(\000) p Fj(1) p Fy 2060 3199 a(+) p Fo 2144 3199 a(v) p Fj 2184 3211 a(0) p Fp(;d) p Fn(\000) p Fj(2) p Fy 2360 3199 a(\)) p Fk 2406 3199 a(\001) p 2443 3199 a(\001) p 2480 3199 a(\001) p Fy 2517 3199 a(\() p Fo(v) p Fj 2589 3211 a(0) p Fp(;) p Fj(2) p Fy 2698 3199 a(+) p Fo 2781 3199 a(v) p Fj 2821 3211 a(0) p Fp(;) p Fj(1) p Fy 2911 3199 a(\)) p Fo(v) p Fj 2983 3211 a(1) p Fp(;) p Fj(1) p Fy 3102 3199 a(in) p Fo 3199 3199 a(W) p Fj 3289 3156 a(\(1\)) 3277 3227 y(\(1\)) p Fy 3391 3199 a(.) p Fd 3778 3199 a(2) p Fy 125 3491 a(Large) p Fo 356 3491 a(n) p Fy 432 3491 a(means) p 685 3491 a(that) p 864 3491 a(the) p 1006 3491 a(endp) r(oin) n(ts) p 1384 3491 a(of) p 1477 3491 a(the) p 1619 3491 a(paths) p 1845 3491 a(are) p 1982 3491 a(far) p 2108 3491 a(apart,) p 2352 3491 a(hence) p 2582 3491 a(it) p 2664 3491 a(corresp) r(onds) p 3119 3491 a(to) p 3219 3491 a(large) p 3421 3491 a(path) p 3614 3491 a(length) p Fo 0 3590 a(L) p Fy(.) p 130 3590 a(In) n(tuitiv) n(ely) p 538 3590 a(sp) r(eaking) p 884 3590 a(Prop) r(osition) p 1335 3590 a(4) p 1409 3590 a(therefore) p 1762 3590 a(giv) n(es) p 1970 3590 a(a) p 2044 3590 a(resp) r(onse) p 2383 3590 a(to) p 2489 3590 a(the) p 2637 3590 a(c) n(hange) p 2916 3590 a(in) p 3017 3590 a(the) p 3164 3590 a(length) p 3422 3590 a(scale) p 3626 3590 a(of) p 3725 3590 a(the) 0 3690 y(system) p 283 3690 a(in) p 387 3690 a(consideration,) p 927 3690 a(the) p 1077 3690 a(sets) p 1247 3690 a(of) p 1348 3690 a(self-a) n(v) n(oiding) p 1831 3690 a(paths,) p 2090 3690 a(in) p 2194 3690 a(terms) p 2432 3690 a(of) p 2534 3690 a(the) p 2684 3690 a(parameter) p 3088 3690 a(space) p 3318 3690 a(of) p 3419 3690 a(v) p 3458 3690 a(ariables) p 3771 3690 a(in) 0 3789 y(the) p 149 3789 a(generating) p 561 3789 a(functions) p 925 3789 a(of) p Fo 1025 3789 a(s) p Fp 1064 3801 a(I) p Fy 1116 3789 a(,) p 1174 3789 a(the) p 1323 3789 a(generalized) p 1758 3789 a(path) p 1958 3789 a(length.) p 2262 3789 a(Global) p 2536 3789 a(prop) r(erties) p 2932 3789 a(of) p 3033 3789 a(the) p 3182 3789 a(tra) p 3294 3789 a(jectories) p 3624 3789 a(of) p 3725 3789 a(the) 0 3889 y(map) p Fo 189 3868 a(~) p Fy 186 3889 a(\010) p 276 3889 a(therefore) p 626 3889 a(is) p 712 3889 a(exp) r(ected) p 1060 3889 a(to) p 1163 3889 a(giv) n(e) p 1336 3889 a(\(and) p 1531 3889 a(w) n(e) p 1655 3889 a(will) p 1814 3889 a(sho) n(w) p 2022 3889 a(that) p 2203 3889 a(it) p 2288 3889 a(do) r(es\)) p 2510 3889 a(large) p 2715 3889 a(length) p 2970 3889 a(asymptotic) p 3399 3889 a(b) r(eha) n(viors) p 3773 3889 a(of) 0 3989 y(self-a) n(v) n(oiding) p 476 3989 a(paths) p 702 3989 a(on) p Fo 817 3989 a(d) p Fy(SG.) 125 4088 y(In) p 232 4088 a(analogy) p 542 4088 a(to) p 646 4088 a(the) p 793 4088 a(\(mathematically) p 1416 4088 a(misleading\)) p 1863 4088 a(terminology) p 2325 4088 a(in) p 2425 4088 a(ph) n(ysics) p 2716 4088 a(literature,) p 3108 4088 a(w) n(e) p 3233 4088 a(call) p 3389 4088 a(the) p 3535 4088 a(discrete-) 0 4198 y(time) p 189 4198 a(dynamical) p 588 4198 a(system) p 863 4198 a(on) p Fq 979 4198 a(R) p Fn 1039 4160 a(I) p Fm 1077 4169 a(d) p Fj 1039 4218 a(+) p Fy 1143 4198 a(de\014ned) p 1429 4198 a(b) n(y) p 1544 4198 a(the) p 1687 4198 a(map) p Fo 1875 4177 a(~) p Fy 1872 4198 a(\010,) p 1982 4198 a(the) p 2125 4198 a(renormalization) p 2721 4198 a(group) p 2956 4198 a(\(R) n(G\).) p Fr 0 4430 a(2.5) p 255 4430 a(In) m(v) p 409 4430 a(arian) m(t) p 735 4430 a(sets.) p Fy 0 4583 a(If) p 83 4583 a(there) p 296 4583 a(is) p 379 4583 a(a) p 449 4583 a(subset) p 703 4583 a(of) p Fq 798 4583 a(R) p Fn 858 4546 a(I) p Fm 896 4555 a(d) p Fj 858 4604 a(+) p Fy 962 4583 a(whic) n(h) p 1200 4583 a(is) p 1284 4583 a(an) p 1399 4583 a(in) n(v) p 1505 4583 a(arian) n(t) p 1748 4583 a(set) p 1877 4583 a(of) p 1972 4583 a(the) p 2115 4583 a(R) n(G) p 2267 4583 a(map) p Fo 2454 4562 a(~) p Fy 2452 4583 a(\010,) p 2562 4583 a(then) p 2752 4583 a(the) p 2895 4583 a(recursion) p 3251 4583 a(\(10\)) p 3426 4583 a(is) p 3510 4583 a(naturally) 0 4683 y(regarded) p 342 4683 a(as) p 444 4683 a(a) p 513 4683 a(recursion) p 869 4683 a(equation) p 1208 4683 a(on) p 1323 4683 a(the) p 1466 4683 a(subset.) 125 4783 y(F) p 172 4783 a(or) p Fo 270 4783 a(I) p Fy 336 4783 a(=) p 423 4783 a(\() p Fo(i) p Fj 484 4795 a(1) p Fo 521 4783 a(;) p Fk 558 4783 a(\001) p 595 4783 a(\001) p 632 4783 a(\001) p Fo 669 4783 a(;) p 706 4783 a(i) p Fp 735 4795 a(k) p Fy 775 4783 a(\)) p 831 4783 a(and) p Fo 989 4783 a(J) p Fy 1066 4783 a(=) p 1154 4783 a(\() p Fo(j) p Fj 1220 4795 a(1) p Fo 1257 4783 a(;) p Fk 1294 4783 a(\001) p 1331 4783 a(\001) p 1368 4783 a(\001) p Fo 1405 4783 a(;) p 1442 4783 a(j) p Fp 1476 4795 a(`) p Fy 1508 4783 a(\)) p 1564 4783 a(in) p Fk 1657 4783 a(I) p Fp 1702 4795 a(d) p Fy 1755 4783 a(,) p 1802 4783 a(denote) p 2066 4783 a(b) n(y) p Fo 2177 4783 a(I) p Fk 2231 4783 a(\010) p Fo 2307 4783 a(J) p Fy 2384 4783 a(the) p 2523 4783 a(rearrangemen) n(t) p 3071 4783 a(of) p Fo 3161 4783 a(i) p Fj 3190 4795 a(1) p Fo 3227 4783 a(;) p Fk 3264 4783 a(\001) p 3301 4783 a(\001) p 3338 4783 a(\001) p Fo 3375 4783 a(;) p 3412 4783 a(i) p Fp 3441 4795 a(k) p Fo 3481 4783 a(;) p 3518 4783 a(j) p Fj 3552 4795 a(1) p Fo 3589 4783 a(;) p Fk 3626 4783 a(\001) p 3663 4783 a(\001) p 3700 4783 a(\001) p Fo 3737 4783 a(;) p 3774 4783 a(j) p Fp 3808 4795 a(`) p Fy 0 4882 a(in) p 97 4882 a(non-decreasing) p 660 4882 a(order,) p 901 4882 a(and) p 1062 4882 a(de\014ne) p 1302 4882 a(\004) p Fp 1357 4894 a(d) p Fk 1419 4882 a(\032) p Fq 1507 4882 a(R) p Fn 1567 4845 a(I) p Fm 1605 4854 a(d) p Fj 1567 4903 a(+) p Fy 1671 4882 a(b) n(y) 613 5078 y(\004) p Fp 668 5090 a(d) p Fy 730 5078 a(=) p Fk 818 5078 a(f) p Fo 859 5078 a(~) p 860 5078 a(x) p Fk 930 5078 a(2) p Fq 1008 5078 a(R) p Fn 1068 5041 a(I) p Fm 1106 5050 a(d) p Fj 1068 5099 a(+) p Fk 1168 5078 a(j) p Fo 1214 5078 a(x) p Fp 1261 5090 a(I) p Fn 1295 5090 a(\010) p Fp(J) p Fl 1417 5078 a(5) p Fo 1504 5078 a(x) p Fp 1551 5090 a(I) p Fo 1590 5078 a(x) p Fp 1637 5090 a(J) p Fy 1739 5078 a(for) p 1866 5078 a(all) p Fo 2009 5078 a(I) p 2052 5078 a(;) p 2089 5078 a(J) p Fk 2166 5078 a(2) p 2244 5078 a(I) p Fp 2289 5090 a(d) p Fy 2384 5078 a(suc) n(h) p 2571 5078 a(that) p Fo 2779 5078 a(I) p Fk 2840 5078 a(\010) p Fo 2923 5078 a(J) p Fk 3000 5078 a(2) p 3078 5078 a(I) p Fp 3123 5090 a(d) p Fk 3162 5078 a(g) p Fo(:) p Fy 3692 5078 a(\(11\)) p Fi -42 5274 a(Example.) p Fy 332 5274 a(\004) p Fj 387 5286 a(3) p Fy 448 5274 a(=) p Fk 536 5274 a(f) p Fo 577 5274 a(~) p 578 5274 a(x) p Fk 648 5274 a(2) p Fq 726 5274 a(R) p Fn 786 5237 a(I) p Ff 824 5245 a(3) p Fj 786 5294 a(+) p Fk 883 5274 a(j) p Fo 929 5274 a(x) p Fj 976 5289 a(\(11\)) p Fl 1122 5274 a(5) p Fo 1210 5274 a(x) p Fj 1257 5244 a(2) 1257 5300 y(\(1\)) p Fk 1346 5274 a(g) p Fy 1402 5274 a(,) p 1452 5274 a(\004) p Fj 1507 5286 a(4) p Fy 1568 5274 a(=) p Fk 1656 5274 a(f) p Fo 1697 5274 a(~) p 1698 5274 a(x) p Fk 1768 5274 a(2) p Fq 1846 5274 a(R) p Fn 1906 5237 a(I) p Ff 1944 5245 a(4) p Fj 1906 5294 a(+) p Fk 2003 5274 a(j) p Fo 2049 5274 a(x) p Fj 2096 5289 a(\(11\)) p Fl 2242 5274 a(5) p Fo 2330 5274 a(x) p Fj 2377 5244 a(2) 2377 5300 y(\(1\)) p Fo 2466 5274 a(;) p 2531 5274 a(x) p Fj 2578 5289 a(\(12\)) p Fl 2724 5274 a(5) p Fo 2811 5274 a(x) p Fj 2858 5289 a(\(1\)) p Fo 2948 5274 a(x) p Fj 2995 5289 a(\(2\)) p Fk 3085 5274 a(g) p Fy 3140 5274 a(.) p Fd 3774 5274 a(3) p Fz 0 5653 a(Prop) s(osition) p 516 5653 a(5) p Fy 605 5653 a(\004) p Fp 660 5665 a(d) p Fi 729 5653 a(is) p 818 5653 a(an) p 937 5653 a(invariant) p 1297 5653 a(set) p 1426 5653 a(of) p Fo 1527 5632 a(~) p Fy 1524 5653 a(\010) p Fi(.) p Fy 1899 6120 a(7) p 90 rotate dyy eop %%Page: 8 8 8 7 bop Fi -42 87 a(Pr) l(o) l(of.) p Fy 219 87 a(Let) p Fo 370 87 a(I) p 413 87 a(;) p 450 87 a(J) o(;) p 532 87 a(I) p Fk 595 87 a(\010) p Fo 679 87 a(J) p Fk 761 87 a(2) p 843 87 a(I) p Fp 888 99 a(d) p Fy 941 87 a(.) p 1008 87 a(Note) p 1212 87 a(that) p 1394 87 a(there) p 1609 87 a(is) p 1695 87 a(a) p 1767 87 a(natural) p 2060 87 a(one-to-one) p 2468 87 a(in) n(to) p 2639 87 a(map) p Fo 2826 87 a(W) p Fj 2916 44 a(\(1\)) p Fp 2904 111 a(I) p Fn 2938 111 a(\010) p Fp(J) p Fk 3064 87 a(!) p Fo 3174 87 a(W) p Fj 3264 44 a(\(1\)) p Fp 3252 111 a(I) p Fk 3374 87 a(\002) p Fo 3458 87 a(W) p Fj 3548 44 a(\(1\)) p Fp 3536 111 a(J) p Fy 3651 87 a(.) p 3719 87 a(F) p 3766 87 a(or) p Fo 0 213 a(w) p Fk 86 213 a(2) p Fo 166 213 a(W) p Fj 256 170 a(\(1\)) p Fp 244 237 a(I) p Fn 278 237 a(\010) p Fp(J) p Fy 390 213 a(,) p 442 213 a(let) p 563 213 a(\() p Fo(w) p Fj 654 225 a(1) p Fo 692 213 a(;) p 729 213 a(w) p Fj 788 225 a(2) p Fy 825 213 a(\)) p Fk 882 213 a(2) p Fo 961 213 a(W) p Fj 1051 170 a(\(1\)) p Fp 1039 237 a(I) p Fk 1159 213 a(\002) p Fo 1243 213 a(W) p Fj 1333 170 a(\(1\)) p Fp 1321 237 a(J) p Fy 1450 213 a(b) r(e) p 1564 213 a(the) p 1708 213 a(corresp) r(onding) p 2242 213 a(pair.) p 2448 213 a(Then,) p 2689 213 a(for) p 2817 213 a(eac) n(h) p 3004 213 a(\001) p Fk 3098 213 a(2) p 3177 213 a(T) p Fp 3222 225 a(b) p Fy 3270 213 a(,) p 3338 213 a(^) p Fo 3321 213 a(w) p Fk 3402 213 a(\\) p Fy 3476 213 a(\001) p 3574 213 a(ma) n(y) p 3755 213 a(b) r(e) 0 313 y(regarded) p 337 313 a(as) p 435 313 a(a) p 499 313 a(comp) r(osition) p 959 313 a(of) p 1066 313 a(^) p Fo 1049 313 a(w) p Fj 1108 325 a(1) p Fk 1155 313 a(\\) p Fy 1219 313 a(\001) p 1312 313 a(and) p 1486 313 a(^) p Fo 1469 313 a(w) p Fj 1528 325 a(2) p Fk 1575 313 a(\\) p Fy 1639 313 a(\001,) p 1757 313 a(hence) p 1983 313 a(if) p 2071 313 a(^) p Fo 2054 313 a(w) p Fk 2125 313 a(\\) p Fy 2189 313 a(\001) p 2283 313 a(is) p 2362 313 a(congruen) n(t) p 2742 313 a(to) p 2839 313 a(\001) p Fp 2908 325 a(K) p Fy 2996 313 a(of) p 3086 313 a(\(4\)) p 3215 313 a(for) p 3338 313 a(some) p Fo 3541 313 a(K) p Fk 3641 313 a(2) p 3719 313 a(I) p Fp 3764 325 a(d) p Fy 3817 313 a(,) 0 412 y(then) p 197 412 a(there) p 418 412 a(exists) p Fo 655 412 a(K) p Fj 726 424 a(1) p Fo 763 412 a(;) p 800 412 a(K) p Fj 871 424 a(2) p Fk 945 412 a(2) p 1037 412 a(I) p Fp 1082 424 a(d) p Fy 1157 412 a(\(allo) n(wing) p 1522 412 a(an) p 1646 412 a(empt) n(yset\)) p 2042 412 a(suc) n(h) p 2238 412 a(that) p Fo 2426 412 a(K) p Fy 2539 412 a(=) p Fo 2640 412 a(K) p Fj 2711 424 a(1) p Fk 2772 412 a(\010) p Fo 2860 412 a(K) p Fj 2931 424 a(2) p Fy 3004 412 a(and) p 3174 412 a(suc) n(h) p 3369 412 a(that) p 3574 412 a(^) p Fo 3557 412 a(w) p Fp 3616 424 a(i) p Fk 3668 412 a(\\) p Fy 3748 412 a(\001,) p Fo 0 512 a(i) p Fy 52 512 a(=) p 139 512 a(1) p Fo(;) p Fy 218 512 a(2,) p 310 512 a(is) p 393 512 a(congruen) n(t) p 779 512 a(to) p 880 512 a(\001) p Fp 949 524 a(K) p Fm 1005 532 a(i) p Fy 1036 512 a(,) p Fo 1086 512 a(i) p Fy 1138 512 a(=) p 1226 512 a(1) p Fo(;) p Fy 1305 512 a(2,) p 1396 512 a(resp) r(ectiv) n(ely) p 1815 512 a(.) p 1874 512 a(Note) p 2075 512 a(also) p 2242 512 a(that) p Fo 2420 512 a(~) p 2421 512 a(x) p Fk 2492 512 a(2) p Fy 2570 512 a(\004) p Fp 2625 524 a(d) p Fy 2692 512 a(implies) p Fo 2974 512 a(x) p Fp 3021 524 a(K) p Ff 3077 532 a(1) p Fn 3110 524 a(\010) p Fp(K) p Ff 3218 532 a(2) p Fl 3277 512 a(5) p Fo 3365 512 a(x) p Fp 3412 524 a(K) p Ff 3468 532 a(1) p Fo 3504 512 a(x) p Fp 3551 524 a(K) p Ff 3607 532 a(2) p Fy 3658 512 a(.) 125 612 y(Therefore) p 501 612 a(b) n(y) p 616 612 a(de\014nition) p 985 612 a(\(Prop) r(osition) p 1464 612 a(4) p 1534 612 a(and) p 1695 612 a(\(9\)\),) 82 808 y(\010) p Fp 142 820 a(I) p Fn 176 820 a(\010) p Fp(J) p Fy 274 808 a(\() p Fo 305 808 a(~) p 306 808 a(x) p Fy(\)) p 409 808 a(=) p Fg 572 729 a(X) p Fp 496 929 a(w) p Fn 546 929 a(2) p Fp(W) p Ff 662 899 a(\(1\)) p Fm 653 949 a(I) p Fe 682 949 a(\010) p Fm(J) p Fg 817 729 a(Y) p Fp 781 907 a(K) p Fn 841 907 a(2I) p Fm 924 916 a(d) p Fo 972 808 a(x) p Fp 1019 820 a(K) 1084 774 y(s) p Fm 1115 782 a(K) p Fj 1169 774 a(\() p Fp(w) p Fj 1245 774 a(\)) p Fl 1298 808 a(5) p Fg 1463 729 a(X) p Fp 1386 929 a(w) p Ff 1434 937 a(1) p Fn 1466 929 a(2) p Fp(W) p Ff 1582 899 a(\(1\)) p Fm 1573 949 a(I) p Fg 1750 729 a(X) p Fp 1673 929 a(w) p Ff 1721 937 a(2) p Fn 1754 929 a(2) p Fp(W) p Ff 1870 899 a(\(1\)) p Fm 1861 949 a(J) p Fg 2011 729 a(Y) p Fp 1961 907 a(K) p Ff 2017 915 a(1) p Fn 2049 907 a(2I) p Fm 2132 916 a(d) p Fo 2180 808 a(x) p Fp 2227 820 a(K) p Ff 2283 828 a(1) p Fp 2320 774 a(s) p Fm 2351 782 a(K) p Ff 2398 794 a(1) p Fj 2435 774 a(\() p Fp(w) p Ff 2509 782 a(1) p Fj 2542 774 a(\)) p Fg 2635 729 a(Y) p Fp 2586 907 a(K) p Ff 2642 915 a(2) p Fn 2674 907 a(2I) p Fm 2757 916 a(d) p Fo 2805 808 a(x) p Fp 2852 820 a(K) p Ff 2908 828 a(2) p Fp 2945 774 a(s) p Fm 2976 782 a(K) p Ff 3023 794 a(2) p Fj 3060 774 a(\() p Fp(w) p Ff 3134 782 a(2) p Fj 3166 774 a(\)) p Fy 3219 808 a(=) p 3307 808 a(\010) p Fp 3367 820 a(I) p Fy 3405 808 a(\() p Fo 3436 808 a(~) p 3437 808 a(x) p Fy 3485 808 a(\)\010) p Fp 3577 820 a(J) p Fy 3623 808 a(\() p Fo 3654 808 a(~) p 3655 808 a(x) p Fy 3703 808 a(\)) p Fo(:) p Fd 3778 1089 a(2) p Fy 125 1366 a(In) p 228 1366 a(the) p 371 1366 a(follo) n(wing,) p 745 1366 a(for) p Fk 872 1366 a(K) p 959 1366 a(\032) p 1047 1366 a(I) p Fp 1092 1378 a(d) p Fy 1145 1366 a(,) p 1196 1366 a(w) n(e) p 1318 1366 a(use) p 1461 1366 a(a) p 1530 1366 a(\(somewhat) p 1951 1366 a(irregular\)) p 2320 1366 a(notation) p Fq 1160 1546 a(R) p Fn 1220 1512 a(K) p Fj 1220 1567 a(+) p Fy 1298 1546 a(=) p Fk 1386 1546 a(f) p Fo 1427 1546 a(~) p 1428 1546 a(x) p Fk 1498 1546 a(2) p Fq 1576 1546 a(R) p Fn 1636 1509 a(I) p Fm 1674 1518 a(d) p Fj 1636 1567 a(+) p Fk 1736 1546 a(j) p Fo 1782 1546 a(x) p Fp 1829 1558 a(J) p Fy 1899 1546 a(=) p 1987 1546 a(0) p Fo(;) p 2093 1546 a(J) p Fk 2170 1546 a(62) p 2248 1546 a(K) q(g) p 2405 1546 a(\032) p Fq 2520 1546 a(R) p Fn 2580 1509 a(I) p Fm 2618 1518 a(d) p Fj 2580 1567 a(+) p Fo 2657 1546 a(:) p Fy 3692 1546 a(\(12\)) 0 1740 y(W) p 78 1740 a(e) p 143 1740 a(also) p 309 1740 a(write) p Fq 522 1740 a(C) p Fn 582 1709 a(K) p Fk 660 1740 a(\032) p Fq 747 1740 a(C) p Fn 807 1709 a(I) p Fm 845 1718 a(d) p Fy 884 1740 a(,) p Fq 935 1740 a(Z) p Fn 990 1709 a(K) p Fj 990 1760 a(+) p Fk 1068 1740 a(\032) p Fq 1156 1740 a(Z) p Fn 1211 1702 a(I) p Fm 1249 1711 a(d) p Fj 1211 1760 a(+) p Fy 1288 1740 a(,) p 1339 1740 a(etc.) 125 1839 y(De\014ne) p Fk 1357 1939 a(K) p Fp 1420 1951 a(r) r(es) p Fy 1543 1939 a(=) p Fk 1630 1939 a(f) p Fy(\(1\)) p Fo(;) p Fy 1815 1939 a(\(11\)) p Fo(;) p Fk 2000 1939 a(\001) p 2037 1939 a(\001) p 2074 1939 a(\001) p Fo 2110 1939 a(;) p Fy 2147 1939 a(\(1) p Fk 2235 1939 a(\001) p 2272 1939 a(\001) p 2309 1939 a(\001) p Fy 2345 1939 a(1\)) p Fk(g) p Fo(:) p Fy 3692 1939 a(\(13\)) 0 2087 y(The) p 167 2087 a(indices) p 436 2087 a(in) p Fk 529 2087 a(K) p Fp 592 2099 a(r) r(es) p Fy 715 2087 a(corresp) r(ond) p 1134 2087 a(to) p 1232 2087 a(those) p 1445 2087 a(paths) p 1668 2087 a(whic) n(h) p 1902 2087 a(go) p 2009 2087 a(out) p 2152 2087 a(of) p 2243 2087 a(a) p 2308 2087 a(simplex) p 2607 2087 a(after) p 2800 2087 a(single) p 3027 2087 a(step) p 3199 2087 a(passage) p 3496 2087 a(eac) n(h) p 3679 2087 a(time) 0 2186 y(they) p 187 2186 a(en) n(ter) p 397 2186 a(the) p 540 2186 a(simplex.) p Fz 0 2350 a(Prop) s(osition) p 516 2350 a(6) p Fq 605 2350 a(R) p Fn 665 2313 a(K) p Fm 716 2321 a(r) q(es) p Fj 665 2371 a(+) p Fi 839 2350 a(is) p 928 2350 a(an) p 1047 2350 a(invariant) p 1407 2350 a(subset) p 1653 2350 a(of) p Fo 1753 2329 a(~) p Fy 1751 2350 a(\010) p Fi(.) -42 2530 y(Pr) l(o) l(of.) p Fy 219 2530 a(This) p 408 2530 a(is) p 492 2530 a(pro) n(v) n(ed) p 762 2530 a(b) n(y) p 877 2530 a(generalizing) p 1334 2530 a(the) p 1477 2530 a(argumen) n(ts) p 1881 2530 a(in) p 1978 2530 a(the) p 2121 2530 a(pro) r(of) p 2338 2530 a(of) p 2433 2530 a([6) o(,) p 2548 2530 a(Prop) r(osition) p 2995 2530 a(2.1) p 3129 2530 a(\(4\)\(5\)].) p Fd 3778 2530 a(2) p Fs 0 2981 a(3) p 202 2981 a(Main) p 552 2981 a(results.) p Fr 0 3180 a(3.1) p 255 3180 a(Fixed) p 567 3180 a(p) s(oin) m(t) p 859 3180 a(and) p 1076 3180 a(critical) p 1450 3180 a(tra) p 1600 3180 a(jectory) p 1931 3180 a(.) p Fy 0 3333 a(Based) p 237 3333 a(on) p 345 3333 a(exp) r(eriences) p 777 3333 a(with) p Fo 960 3333 a(d) p Fy(SG) p 1135 3333 a(for) p Fo 1255 3333 a(d) p Fy 1322 3333 a(=) p 1409 3333 a(2) p Fo(;) p Fy 1488 3333 a(3) p Fo(;) p Fy 1567 3333 a(4,) p 1653 3333 a(w) n(e) p 1768 3333 a(de\014ne) p 2001 3333 a(notions) p 2286 3333 a(whic) n(h) p 2516 3333 a(are) p 2648 3333 a(relev) p 2817 3333 a(an) n(t) p 2955 3333 a(for) p 3075 3333 a(asymptotic) p 3496 3333 a(b) r(eha) n(viors) 0 3433 y(of) p 95 3433 a(self-a) n(v) n(oiding) p 570 3433 a(paths) p 797 3433 a(on) p Fo 912 3433 a(d) p Fy(SG.) 125 3532 y(Denote) p 409 3532 a(the) p 552 3532 a(Jacobi) p 812 3532 a(matrix) p 1082 3532 a(of) p Fo 1179 3511 a(~) p Fy 1176 3532 a(\010) p 1264 3532 a(in) p 1361 3532 a(Prop) r(osition) p 1808 3532 a(4) p 1877 3532 a(b) n(y) p Fk 1992 3532 a(J) p Fy 2087 3532 a(=) p 2175 3532 a(\() p Fk(J) p Fp 2263 3544 a(I) p 2297 3544 a(J) p Fy 2343 3532 a(\):) p Fk 1202 3755 a(J) p Fp 1258 3767 a(I) p 1292 3767 a(J) p Fy 1338 3755 a(\() p Fo 1369 3755 a(~) p 1370 3755 a(x) p Fy 1418 3755 a(\)) p 1473 3755 a(=) p Fo 1571 3699 a(@) p Fy 1620 3699 a(\010) p Fp 1680 3711 a(I) p 1571 3736 147 4 v Fo 1573 3812 a(@) p 1622 3812 a(x) p Fp 1669 3824 a(J) p Fy 1728 3755 a(\() p Fo 1759 3755 a(~) p 1760 3755 a(x) p Fy(\)) p Fo(;) p 1932 3755 a(I) p 1975 3755 a(;) p 2012 3755 a(J) p Fk 2089 3755 a(2) p 2167 3755 a(I) p Fp 2212 3767 a(d) p Fo 2265 3755 a(;) p 2329 3755 a(~) p 2330 3755 a(x) p Fk 2400 3755 a(2) p Fq 2479 3755 a(C) p Fn 2539 3721 a(I) p Fm 2577 3730 a(d) p Fo 2615 3755 a(:) p Fy 3692 3755 a(\(14\)) 125 3990 y(W) p 203 3990 a(e) p 268 3990 a(sa) n(y) p 411 3990 a(that) p Fo 590 3990 a(~) p 591 3990 a(x) p Fp 638 4002 a(c) p Fk 695 3990 a(2) p Fq 774 3990 a(R) p Fn 834 3953 a(I) p Fm 872 3962 a(d) p Fj 834 4011 a(+) p Fy 938 3990 a(is) p 1021 3990 a(a) p Fi 1090 3990 a(self-avoiding) p 1570 3990 a(\014xe) l(d) p 1761 3990 a(p) l(oint) p Fy 1949 3990 a(,) p 2000 3990 a(if) p 2076 3990 a(the) p 2219 3990 a(follo) n(wing) p 2570 3990 a(hold.) -51 4171 y(\(FP1\)) p Fo 210 4150 a(~) p Fy 208 4171 a(\010\() p Fo 299 4171 a(~) p 300 4171 a(x) p Fp 347 4183 a(c) p Fy 381 4171 a(\)) p 436 4171 a(=) p Fo 523 4171 a(~) p 524 4171 a(x) p Fp 571 4183 a(c) p Fy 619 4171 a(.) -51 4336 y(\(FP2\)) p Fk 208 4336 a(J) p Fy 279 4336 a(\() p Fo 310 4336 a(~) p 311 4336 a(x) p Fp 358 4348 a(c) p Fy 393 4336 a(\)) p 450 4336 a(in) p 545 4336 a(\(14\)) p 717 4336 a(is) p 798 4336 a(diagonalizable) p 1336 4336 a(b) n(y) p 1449 4336 a(an) p 1561 4336 a(in) n(v) n(ertible) p 1926 4336 a(matrix.) p 2227 4336 a(The) p 2396 4336 a(eigen) n(v) p 2618 4336 a(alue) p Fo 2790 4336 a(\025) p Fy 2863 4336 a(whic) n(h) p 3099 4336 a(is) p 3180 4336 a(largest) p 3445 4336 a(in) p 3540 4336 a(absolute) 208 4435 y(v) p 247 4435 a(alue) p 426 4435 a(satis\014es) p Fo 736 4435 a(\025) p 814 4435 a(>) p Fy 908 4435 a(1) p 981 4435 a(with) p 1174 4435 a(m) n(ultiplicit) n(y) p 1623 4435 a(1,) p 1720 4435 a(and) p 1885 4435 a(all) p 2004 4435 a(the) p 2151 4435 a(other) p 2372 4435 a(eigen) n(v) p 2594 4435 a(alues) p 2805 4435 a(ha) n(v) n(e) p 3000 4435 a(absolute) p 3332 4435 a(v) p 3371 4435 a(alues) p 3583 4435 a(strictly) 208 4535 y(less) p 361 4535 a(than) p 554 4535 a(1) p 610 4535 a(.) 208 4667 y(Denote) p 492 4667 a(b) n(y) p Fo 604 4667 a(~) p 608 4667 a(v) p Fp 648 4679 a(L) p Fy 721 4667 a(=) p 808 4667 a(\() p Fo(v) p Fp 880 4679 a(L;I) p Fo 984 4667 a(;) p 1049 4667 a(I) p Fk 1115 4667 a(2) p 1193 4667 a(I) p Fp 1238 4679 a(d) p Fy 1277 4667 a(\)) p 1337 4667 a(a) p 1406 4667 a(left) p 1552 4667 a(eigen) n(v) n(ector) p 1983 4667 a(of) p Fk 2078 4667 a(J) p Fy 2149 4667 a(\() p Fo 2180 4667 a(~) p 2181 4667 a(x) p Fp 2228 4679 a(c) p Fy 2263 4667 a(\)) p 2323 4667 a(corresp) r(onding) p 2856 4667 a(to) p Fo 2958 4667 a(\025) p Fy(;) p Fg 1397 4788 a(X) p Fp 1382 4966 a(I) p Fn 1416 4966 a(2I) p Fm 1499 4975 a(d) p Fo 1547 4867 a(v) p Fp 1587 4879 a(L;I) p Fk 1690 4867 a(J) p Fp 1746 4879 a(I) p 1780 4879 a(J) p Fy 1827 4867 a(\() p Fo 1858 4867 a(~) p 1859 4867 a(x) p Fp 1906 4879 a(c) p Fy 1940 4867 a(\)) p 1996 4867 a(=) p Fo 2083 4867 a(\025v) p Fp 2171 4879 a(L;J) p Fo 2297 4867 a(;) p 2390 4867 a(J) p Fk 2467 4867 a(2) p 2545 4867 a(I) p Fp 2590 4879 a(d) p Fo 2643 4867 a(;) p Fy 208 5134 a(whic) n(h) p 452 5134 a(w) n(e) p 580 5134 a(c) n(hose) p 806 5134 a(to) p 914 5134 a(ha) n(v) n(e) p 1112 5134 a(non-negativ) n(e) p 1608 5134 a(comp) r(onen) n(ts) p 2071 5134 a(\(p) r(ossible,) p 2447 5134 a(thanks) p 2724 5134 a(to) p 2832 5134 a(F) p 2879 5134 a(rob) r(enius') p 3244 5134 a(theorem\).) p 3651 5134 a(Then) p Fo 208 5233 a(v) p Fp 248 5245 a(L;J) p Fo 382 5233 a(>) p Fy 470 5233 a(0,) p Fo 562 5233 a(J) p Fk 639 5233 a(2) p 718 5233 a(I) p Fp 763 5245 a(d) p Fy 816 5233 a(.) 208 5366 y(Similarly) p 527 5366 a(,) p 577 5366 a(denote) p 844 5366 a(b) n(y) p Fo 956 5366 a(~) p 960 5366 a(v) p Fp 1000 5378 a(R) p Fy 1082 5366 a(a) p 1151 5366 a(righ) n(t) p 1352 5366 a(eigen) n(v) n(ector) p 1784 5366 a(corresp) r(onding) p 2317 5366 a(to) p Fo 2419 5366 a(\025) p Fy 2495 5366 a(with) p 2684 5366 a(non-negativ) n(e) p 3173 5366 a(comp) r(onen) n (ts;) p Fg 1398 5479 a(X) p Fp 1378 5657 a(J) p Fn 1420 5657 a(2I) p Fm 1503 5666 a(d) p Fk 1551 5558 a(J) p Fp 1607 5570 a(I) p 1641 5570 a(J) p Fy 1688 5558 a(\() p Fo 1719 5558 a(~) p 1720 5558 a(x) p Fp 1767 5570 a(c) p Fy 1802 5558 a(\)) p Fo(v) p Fp 1874 5570 a(R;J) p Fy 2014 5558 a(=) p Fo 2101 5558 a(\025v) p Fp 2189 5570 a(R;I) p Fo 2312 5558 a(;) p 2404 5558 a(I) p Fk 2470 5558 a(2) p 2549 5558 a(I) p Fp 2594 5570 a(d) p Fo 2646 5558 a(:) p Fy 208 5820 a(Then) p Fo 424 5820 a(v) p Fp 464 5835 a(R;) p Fj(\(1\)) p Fo 647 5820 a(>) p Fy 734 5820 a(0) p 790 5820 a(.) 1899 6120 y(8) p 90 rotate dyy eop %%Page: 9 9 9 8 bop Fy -51 83 a(\(FP3\)) p 208 83 a(F) p 255 83 a(or) p 358 83 a(all) p Fo 474 83 a(I) p Fk 542 83 a(2) p 623 83 a(I) p Fp 668 95 a(d) p Fy 736 83 a(suc) n(h) p 924 83 a(that) p Fo 1105 83 a(x) p Fp 1152 95 a(c;I) p Fk 1265 83 a(6) p Fy(=) p 1355 83 a(0,) p 1448 83 a(there) p 1662 83 a(exists) p Fo 1892 83 a(m) p Fk 1990 83 a(2) p Fq 2071 83 a(Z) p Fn 2126 46 a(I) p Fm 2164 55 a(d) p Fj 2126 104 a(+) p Fy 2203 83 a(,) p 2255 83 a(satisfying) p Fo 2626 83 a(m) p Fj 2699 98 a(\(1\)) p Fo 2813 83 a(>) p Fy 2903 83 a(0) p 2973 83 a(and) p Fo 3136 83 a(m) p Fp 3209 95 a(J) p Fy 3280 83 a(=) p 3370 83 a(0) p 3440 83 a(if) p Fo 3518 83 a(x) p Fp 3565 95 a(c;J) p Fy 3686 83 a(=) p 3775 83 a(0,) 208 208 y(suc) n(h) p 395 208 a(that) p 575 208 a(there) p 787 208 a(is) p 871 208 a(a) p 940 208 a(term) p Fg 1165 129 a(Y) p Fp 1138 307 a(J) p Fn 1180 307 a(2I) p Fm 1263 316 a(d) p Fo 1312 208 a(x) p Fp 1359 220 a(J) 1405 174 y(m) p Fm 1464 182 a(J) p Fy 1536 208 a(in) p 1633 208 a(\010) p Fp 1693 220 a(I) p Fy 1745 208 a(.) -51 476 y(\(FP4\)) p Fo 206 476 a(~) p 208 476 a(x) p Fp 255 488 a(c) p Fk 312 476 a(2) p Fy 390 476 a(\004) p Fp 445 488 a(d) p Fk 503 476 a(n) p 563 476 a(f) p Fo 599 458 a(~) p Fy 605 476 a(0) p Fk 646 476 a(g) p Fy(.) 125 642 y(Assume) p 441 642 a(that) p 630 642 a(there) p 852 642 a(exists) p 1090 642 a(a) p 1168 642 a(self-a) n(v) n(oiding) p 1653 642 a(\014xed) p 1863 642 a(p) r(oin) n(t) p Fo 2088 642 a(~) p 2089 642 a(x) p Fp 2136 654 a(c) p Fy 2184 642 a(.) p 2271 642 a(W) p 2349 642 a(e) p 2423 642 a(sa) n(y) p 2576 642 a(that) p Fo 2764 642 a(~) p 2765 642 a(x) p Fk 2851 642 a(2) p Fq 2944 642 a(R) p Fn 3004 605 a(I) p Fm 3042 614 a(d) p Fj 3004 663 a(+) p Fy 3118 642 a(is) p Fi 3210 642 a(in) p 3320 642 a(the) p 3467 642 a(domain) p 3772 642 a(of) 0 742 y(attr) l(action) p 381 742 a(of) p Fo 477 742 a(~) p 478 742 a(x) p Fp 525 754 a(c) p Fy 560 742 a(,) p 610 742 a(if) p 687 742 a(the) p 830 742 a(follo) n(wing) p 1180 742 a(hold.) -63 908 y(\(D) n(A1\)) p 237 908 a(lim) p Fp 208 958 a(n) p Fn(!1) p Fo 412 887 a(~) 395 908 y(X) p Fp 464 920 a(n) p Fy 509 908 a(\() p Fo 540 908 a(~) p 541 908 a(x) p Fy 589 908 a(\)) p 644 908 a(=) p Fo 730 908 a(~) p 731 908 a(x) p Fp 778 920 a(c) p Fy 827 908 a(.) -63 1103 y(\(D) n(A2\)) p 208 1103 a(If) p Fo 291 1103 a(x) p Fp 338 1115 a(c;I) p Fk 449 1103 a(6) p Fy(=) p 536 1103 a(0) p 605 1103 a(then) p Fo 794 1103 a(x) p Fp 841 1115 a(I) p Fk 903 1103 a(6) p Fy(=) p 991 1103 a(0) p 1046 1103 a(.) 0 1279 y(W) p 78 1279 a(e) p 143 1279 a(denote) p 410 1279 a(b) n(y) p Fk 526 1279 a(D) p Fi 592 1279 a(om) p Fy 709 1279 a(\() p Fo 740 1279 a(~) p 741 1279 a(x) p Fp 788 1291 a(c) p Fy 822 1279 a(\)) p 882 1279 a(the) p 1025 1279 a(set) p 1155 1279 a(of) p Fo 1248 1279 a(~) p 1250 1279 a(x) p Fk 1320 1279 a(2) p Fq 1398 1279 a(R) p Fn 1458 1241 a(I) p Fm 1496 1250 a(d) p Fj 1458 1299 a(+) p Fy 1563 1279 a(whic) n(h) p 1800 1279 a(are) p 1939 1279 a(in) p 2036 1279 a(the) p 2179 1279 a(domain) p 2474 1279 a(of) p 2568 1279 a(attraction) p 2956 1279 a(of) p Fo 3049 1279 a(~) p 3051 1279 a(x) p Fp 3098 1291 a(c) p Fy 3146 1279 a(.) p Fi -42 1445 a(Example.) p Fy 332 1445 a(If) p Fo 420 1445 a(~) p 421 1445 a(x) p Fp 468 1457 a(c) p Fy 535 1445 a(is) p 624 1445 a(a) p 699 1445 a(self-a) n(v) n(oiding) p 1180 1445 a(\014xed) p 1386 1445 a(p) r(oin) n(t,) p 1633 1445 a(then) p Fo 1826 1445 a(~) p 1827 1445 a(x) p Fp 1874 1457 a(c) p Fk 1940 1445 a(2) p 2028 1445 a(D) p Fi 2094 1445 a(om) p Fy 2211 1445 a(\() p Fo 2242 1445 a(~) p 2243 1445 a(x) p Fp 2290 1457 a(c) p Fy 2325 1445 a(\),) p 2414 1445 a(i.e.,) p 2578 1445 a(a) p 2653 1445 a(self-a) n(v) n(oiding) p 3134 1445 a(\014xed) p 3340 1445 a(p) r(oin) n(t) p 3562 1445 a(satis\014es) 0 1544 y(\(D) n(A1\)) p 257 1544 a({) p 326 1544 a(\(D) n(A2\).) p Fd 3774 1644 a(3) p Fy 125 1959 a(Let) p Fk 273 1959 a(K) p 360 1959 a(\032) p 448 1959 a(I) p Fp 493 1971 a(d) p Fy 545 1959 a(.) p 605 1959 a(Instead) p 898 1959 a(of) p 992 1959 a(\(5\),) p 1148 1959 a(w) n(e) p 1269 1959 a(ma) n(y) p 1449 1959 a(consider) p 1771 1959 a(a) p 1840 1959 a(set) p 1969 1959 a(of) p 2063 1959 a(w) n(alks) p Fo 2288 1959 a(W) p Fj 2378 1916 a(\() p Fp(n) p Fj(\)) p Fn 2366 1984 a(K) p Fp(;I) p Fy 2502 1959 a(b) n(y) p 2617 1959 a(restricting) p 3014 1959 a(to) p 3115 1959 a(those) p 3331 1959 a(paths) p 3557 1959 a(in) p Fo 3653 1959 a(W) p Fj 3743 1916 a(\() p Fp(n) p Fj(\)) p Fp 3731 1984 a(I) p Fy 0 2059 a(whic) n(h) p 238 2059 a(satisfy) p Fo 497 2059 a(s) p Fp 536 2071 a(J) p Fy 582 2059 a(\() p Fo(w) p Fy 675 2059 a(\)) p 731 2059 a(=) p 819 2059 a(0) p 888 2059 a(if) p Fo 964 2059 a(J) p Fk 1041 2059 a(62) p 1120 2059 a(K) p Fy 1184 2059 a(:) p Fo 1184 2258 a(W) p Fj 1274 2215 a(\() p Fp(n) p Fj(\)) p Fn 1262 2282 a(K) p Fp(;I) p Fy 1394 2258 a(=) p Fk 1482 2258 a(f) p Fo(w) p Fk 1608 2258 a(2) p Fo 1687 2258 a(W) p Fj 1777 2215 a(\() p Fp(n) p Fj(\)) p Fp 1765 2282 a(I) p Fk 1897 2258 a(j) p Fo 1943 2258 a(s) p Fp 1982 2270 a(J) p Fy 2028 2258 a(\() p Fo(w) p Fy 2121 2258 a(\)) p 2177 2258 a(=) p 2265 2258 a(0) p Fo(;) p 2371 2258 a(J) p Fk 2448 2258 a(62) p 2527 2258 a(K) q(g) p Fo(:) p Fy 3692 2258 a(\(15\)) 0 2471 y(If) p Fk 81 2471 a(K) p Fy 169 2471 a(=) p Fk 256 2471 a(I) p Fp 301 2483 a(d) p Fy 354 2471 a(,) p 404 2471 a(then) p 591 2471 a(w) n(e) p 711 2471 a(are) p 848 2471 a(dealing) p 1133 2471 a(with) p 1320 2471 a(the) p 1461 2471 a(original) p 1759 2471 a(\(full\)) p 1968 2471 a(mo) r(del;) p Fo 2236 2471 a(W) p Fj 2326 2428 a(\() p Fp(n) p Fj(\)) p Fn 2314 2496 a(I) p Fm 2352 2505 a(d) p Fp 2398 2496 a(;I) p Fy 2479 2471 a(=) p Fo 2567 2471 a(W) p Fj 2657 2428 a(\() p Fp(n) p Fj(\)) p Fp 2645 2496 a(I) p Fy 2754 2471 a(.) p 2813 2471 a(W) p 2891 2471 a(e) p 2955 2471 a(de\014ne) p 3193 2471 a(the) p 3334 2471 a(corresp) r(onding) 0 2571 y(generating) p 406 2571 a(functions) p 764 2571 a(b) n(y) p Fo 456 2766 a(X) p Fn 525 2778 a(K) p Fp(;n;I) p Fy 694 2766 a(\() p Fo 725 2766 a(~) p 726 2766 a(x) p Fy 774 2766 a(\)) p 829 2766 a(=) p Fg 983 2687 a(X) p Fp 917 2887 a(w) p Fn 967 2887 a(2) p Fp(W) p Ff 1083 2856 a(\() p Fm(n) p Ff(\)) p Fe 1074 2906 a(K) p Fm(;I) p Fg 1199 2687 a(Y) p Fp 1184 2865 a(J) p Fn 1226 2865 a(2K) p Fo 1335 2766 a(x) p Fp 1382 2723 a(s) p Fm 1413 2731 a(J) p Fj 1454 2723 a(\() p Fp(w) p Fj 1530 2723 a(\)) p Fp 1382 2790 a(J) p Fo 1574 2766 a(;) p 1665 2766 a(~) p 1666 2766 a(x) p Fy 1736 2766 a(=) p 1824 2766 a(\() p Fo(x) p Fp 1903 2778 a(J) p Fo 1950 2766 a(;) p 2015 2766 a(J) p Fk 2092 2766 a(2) p 2170 2766 a(K) p Fy 2234 2766 a(\)) p Fk 2290 2766 a(2) p Fq 2368 2766 a(C) p Fn 2428 2731 a(K) p Fo 2483 2766 a(;) p 2548 2766 a(n) p Fy 2621 2766 a(=) p 2708 2766 a(0) p Fo(;) p Fy 2787 2766 a(1) p Fo(;) p Fy 2866 2766 a(2) p Fo(;) p Fk 2945 2766 a(\001) p 2982 2766 a(\001) p 3019 2766 a(\001) p Fo 3054 2766 a(;) p 3119 2766 a(I) p Fk 3185 2766 a(2) p 3263 2766 a(I) p Fp 3308 2778 a(d) p Fo 3361 2766 a(:) p Fy 3692 2766 a(\(16\)) 125 3094 y(If) p Fq 215 3094 a(R) p Fn 275 3064 a(K) p Fj 275 3115 a(+) p Fy 364 3094 a(is) p 455 3094 a(an) p 577 3094 a(in) n(v) p 683 3094 a(arian) n(t) p 932 3094 a(subset) p 1194 3094 a(of) p Fq 1295 3094 a(R) p Fn 1355 3057 a(I) p Fm 1393 3066 a(d) p Fj 1355 3115 a(+) p Fy 1432 3094 a(,) p 1491 3094 a(then) p Fo 1705 3073 a(~) 1687 3094 y(X) p Fn 1756 3106 a(K) p Fp(;n) p Fy 1906 3094 a(satisfy) p 2173 3094 a(\(10\),) p 2380 3094 a(with) p 2576 3094 a(a) p 2652 3094 a(con) n(v) n(en) n(tion) p 3074 3094 a(that) p 3261 3094 a(the) p 3411 3094 a(comp) r(onen) n(ts) 0 3194 y(corresp) r(onding) p 534 3194 a(to) p Fo 635 3194 a(J) p Fk 712 3194 a(62) p 791 3194 a(K) p Fy 883 3194 a(are) p 1021 3194 a(0) p 1077 3194 a(.) 125 3294 y(F) p 172 3294 a(or) p Fk 274 3294 a(K) p 361 3294 a(\032) p 449 3294 a(I) p Fp 494 3306 a(d) p Fy 560 3294 a(and) p Fo 722 3294 a(\014) p Fk 796 3294 a(2) p Fq 875 3294 a(R) p Fy(,) p 985 3294 a(de\014ne) p Fo 1224 3294 a(~) p 1225 3294 a(x) p Fp 1272 3306 a(can;) p Fn(K) p Fy 1454 3294 a(\() p Fo(\014) p Fy 1537 3294 a(\)) p 1593 3294 a(=) p 1680 3294 a(\() p Fo(x) p Fp 1759 3306 a(can;) p Fn(K) p Fp(;I) p Fy 1995 3294 a(\() p Fo(\014) p Fy 2078 3294 a(\)) p Fo(;) p 2176 3294 a(I) p Fk 2242 3294 a(2) p 2320 3294 a(I) p Fp 2365 3306 a(d) p Fy 2404 3294 a(\)) p 2464 3294 a(b) n(y) p Fo 1314 3533 a(x) p Fp 1361 3545 a(can;) p Fn(K) p Fp(;I) p Fy 1597 3533 a(\() p Fo(\014) p Fy 1680 3533 a(\)) p 1736 3533 a(=) p Fg 1824 3416 a(\032) p 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1881 3891 a(K) p Fp(;n) p Fy 2019 3879 a(=) p 2107 3879 a(\() p Fo(Z) p Fn 2196 3891 a(K) p Fp(;n;I) p Fo 2379 3879 a(;) p 2439 3879 a(I) p Fk 2505 3879 a(2) p 2583 3879 a(I) p Fp 2628 3891 a(d) p Fy 2667 3879 a(\),) p 2750 3879 a(with) p Fo 970 4081 a(Z) p Fn 1027 4093 a(K) p Fp(;n;I) p Fy 1196 4081 a(\() p Fo(\014) p Fy 1279 4081 a(\)) p 1335 4081 a(=) p Fg 1489 4002 a(X) p Fp 1422 4202 a(w) p Fn 1472 4202 a(2) p Fp(W) p Ff 1588 4172 a(\() p Fm(n) p Ff(\)) p Fe 1579 4222 a(K) p Fm(;I) p Fo 1689 4081 a(e) p Fn 1728 4047 a(\000) p Fp(\014) s(L) p Fj(\() p Fp(w) p Fj 1943 4047 a(\)) p Fo 1971 4081 a(;) p 2064 4081 a(\014) p Fk 2138 4081 a(2) p Fq 2216 4081 a(R) p Fo(;) p 2341 4081 a(n) p Fy 2414 4081 a(=) p 2501 4081 a(0) p Fo(;) p Fy 2580 4081 a(1) p Fo(;) p Fy 2659 4081 a(2) p Fo(;) p Fk 2738 4081 a(\001) p 2775 4081 a(\001) p 2812 4081 a(\001) p Fo 2847 4081 a(:) p Fy 3692 4081 a(\(18\)) 0 4392 y(With) p 214 4392 a(\(7\),) p 371 4392 a(w) n(e) p 494 4392 a(see) p 628 4392 a(that) p 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n(e) p 617 4840 a(also) p 783 4840 a(use) p 926 4840 a(an) p 1042 4840 a(abbreviation) p Fo 1523 4840 a(~) p 1524 4840 a(x) p Fp 1571 4852 a(can) p Fy 1682 4840 a(\() p Fo(\014) p Fy 1765 4840 a(\)) p 1821 4840 a(=) p Fo 1908 4840 a(~) p 1909 4840 a(x) p Fp 1956 4852 a(can;) p Fn(I) p Fm 2121 4861 a(d) p Fy 2159 4840 a(\() p Fo(\014) p Fy 2242 4840 a(\)) p 2303 4840 a(and) p Fo 2476 4819 a(~) 2464 4840 y(Z) p Fp 2521 4852 a(n) p Fy 2566 4840 a(\() p Fo(\014) p Fy 2649 4840 a(\)) p 2705 4840 a(=) p Fo 2804 4819 a(~) 2793 4840 y(Z) p Fn 2850 4852 a(I) p Fm 2888 4861 a(d) p Fp 2933 4852 a(;n) p Fy 2998 4840 a(\() p Fo(\014) p Fy 3081 4840 a(\).) p 3174 4840 a(Hence) p Fo 697 5043 a(Z) p Fp 754 5055 a(n;I) p Fy 853 5043 a(\() p Fo(\014) p Fy 936 5043 a(\)) p 992 5043 a(=) p Fo 1079 5043 a(X) p Fp 1148 5055 a(n;I) p Fy 1247 5043 a(\() p Fo 1278 5043 a(~) p 1279 5043 a(x) p Fp 1326 5055 a(can) p Fy 1437 5043 a(\() p Fo(\014) p Fy 1520 5043 a(\)\)) p 1609 5043 a(=) p Fg 1762 4964 a(X) p Fp 1696 5164 a(w) p Fn 1746 5164 a(2) p Fp(W) p Ff 1862 5133 a(\() p Fm(n) p Ff(\)) p Fm 1853 5183 a(I) p Fo 1962 5043 a(e) p Fn 2001 5008 a(\000) p Fp(\014) s(L) p Fj(\() p Fp(w) p Fj 2216 5008 a(\)) p Fo 2244 5043 a(;) p 2336 5043 a(\014) p Fk 2411 5043 a(2) p Fq 2489 5043 a(R) p Fo(;) p 2613 5043 a(n) p Fy 2686 5043 a(=) p 2774 5043 a(0) p Fo(;) p Fy 2853 5043 a(1) p Fo(;) p Fy 2932 5043 a(2) p Fo(;) p Fk 3011 5043 a(\001) p 3048 5043 a(\001) p 3085 5043 a(\001) p Fo 3120 5043 a(:) p Fy 3692 5043 a(\(20\)) 125 5345 y(In) p 226 5345 a(the) p 366 5345 a(follo) n(wing,) p 738 5345 a(the) p 879 5345 a(set) p 1006 5345 a(of) p 1098 5345 a(paths) p 1322 5345 a(in) p 1416 5345 a(\(15\)) p 1589 5345 a(with) p Fk 1776 5345 a(K) p Fy 1863 5345 a(=) p Fk 1951 5345 a(K) p Fp 2014 5357 a(r) r(es) p Fy 2139 5345 a(will) p 2294 5345 a(b) r(e) p 2404 5345 a(called) p 2637 5345 a(the) p 2778 5345 a(restricted) p 3145 5345 a(self-a) n(v) n(oiding) p 3618 5345 a(paths,) 0 5445 y(the) p 143 5445 a(corresp) r(onding) p 677 5445 a(generating) p 1083 5445 a(function) p 1408 5445 a(\(16\),) p 1606 5445 a(the) p 1749 5445 a(generating) p 2155 5445 a(function) p 2481 5445 a(for) p 2608 5445 a(the) p 2751 5445 a(restricted) p 3120 5445 a(mo) r(del,) p 3390 5445 a(and) p 3552 5445 a(so) p 3654 5445 a(on.) 125 5544 y(W) p 203 5544 a(e) p 268 5544 a(need) p 461 5544 a(the) p 604 5544 a(follo) n(wing) p 955 5544 a(additional) p 1347 5544 a(de\014nitions) p 1749 5544 a(for) p 1876 5544 a(Theorem) p 2226 5544 a(10.) -46 5710 y(\(CS1\)) p 208 5710 a(W) p 286 5710 a(e) p 347 5710 a(sa) n(y) p 487 5710 a(that) p Fo 663 5710 a(\014) p Fp 710 5722 a(c) p Fk 767 5710 a(2) p Fq 845 5710 a(R) p Fy 929 5710 a(is) p 1009 5710 a(a) p 1074 5710 a(critical) p 1348 5710 a(p) r(oin) n(t) p 1561 5710 a(of) p 1652 5710 a(the) p 1791 5710 a(full) p 1933 5710 a(mo) r(del) p 2176 5710 a(if) p Fo 2247 5710 a(~) p 2248 5710 a(x) p Fp 2295 5722 a(can) p Fy 2407 5710 a(\() p Fo(\014) p Fp 2486 5722 a(c) p Fy 2520 5710 a(\)) p Fk 2575 5710 a(2) p 2653 5710 a(D) p Fi 2719 5710 a(om) p Fy 2836 5710 a(\() p Fo 2867 5710 a(~) p 2868 5710 a(x) p Fp 2915 5722 a(c) p Fy 2950 5710 a(\)) p 3006 5710 a(for) p 3130 5710 a(a) p 3195 5710 a(self-a) n(v) n(oiding) p 3667 5710 a(\014xed) 208 5810 y(p) r(oin) n(t) p Fo 423 5810 a(~) p 424 5810 a(x) p Fp 471 5822 a(c) p Fy 519 5810 a(.) 1899 6120 y(9) p 90 rotate dyy eop %%Page: 10 10 10 9 bop Fy -46 83 a(\(CS2\)) p 208 83 a(W) p 286 83 a(e) p 350 83 a(sa) n(y) p 494 83 a(that) p Fo 674 83 a(\014) p Fp 721 95 a(c;r) r(es) p Fk 893 83 a(2) p Fq 971 83 a(R) p Fy 1059 83 a(is) p 1142 83 a(a) p 1211 83 a(critical) p 1488 83 a(p) r(oin) n(t) p 1705 83 a(of) p 1799 83 a(the) p 1942 83 a(restricted) p 2312 83 a(mo) r(del) p 2559 83 a(if) p Fo 2634 83 a(~) p 2635 83 a(x) p Fp 2682 95 a(can;) p Fn(K) p Fm 2860 103 a(r) q(es) p Fy 2953 83 a(\() p Fo(\014) p Fp 3032 95 a(c;r) r(es) p Fy 3181 83 a(\)) p Fk 3237 83 a(2) p 3315 83 a(D) p Fi 3381 83 a(om) p Fy 3498 83 a(\() p Fo 3529 83 a(~) p 3530 83 a(x) p Fp 3577 95 a(c) p Fy 3612 83 a(\)) p 3671 83 a(for) p 3798 83 a(a) 208 183 y(self-a) n(v) n(oiding) p 683 183 a(\014xed) p 884 183 a(p) r(oin) n(t) p Fo 1099 183 a(~) p 1101 183 a(x) p Fp 1148 195 a(c) p Fy 1196 183 a(.) 125 349 y(Note) p 325 349 a(that) p 505 349 a(b) n(y) p 620 349 a(de\014nition) p 989 349 a(\(11\),) p Fo 1061 531 a(~) p 1062 531 a(x) p Fp 1109 543 a(can) p Fy 1220 531 a(\() p Fo(\014) p Fp 1299 543 a(c) p Fy 1333 531 a(\)) p Fk 1389 531 a(2) p Fy 1467 531 a(\004) p Fp 1522 543 a(d) p Fo 1575 531 a(;) p Fy 1723 531 a(and) p Fo 1966 531 a(~) p 1967 531 a(x) p Fp 2014 543 a(can;) p Fn(K) p Fm 2192 551 a(r) q(es) p Fy 2285 531 a(\() p Fo(\014) p Fp 2364 543 a(c;r) r(es) p Fy 2513 531 a(\)) p Fk 2569 531 a(2) p Fy 2647 531 a(\004) p Fp 2702 543 a(d) p Fo 2755 531 a(:) p Fy 3692 531 a(\(21\)) p Fi -42 727 a(R) l(emark.) p Fy 382 727 a(\(i\)) p 511 727 a(The) p 690 727 a(b) r(oundary) p Fo 1072 727 a(@) p 1121 727 a(D) p Fy 1228 727 a(of) p 1331 727 a(the) p 1483 727 a(set) p Fo 1621 727 a(D) p Fk 1729 727 a(\032) p Fy 1831 727 a(\004) p Fp 1886 739 a(d) p Fy 1961 727 a(de\014ned) p 2255 727 a(in) p 2361 727 a(\(31\)) p 2545 727 a(is) p 2636 727 a(a) p 2714 727 a(b) r(ounded) p 3062 727 a(closed) p 3315 727 a(non-empt) n(y) p Fo 3741 706 a(~) p Fy 3739 727 a(\010) o({) 208 826 y(in) n(v) p 314 826 a(arian) n(t) p 551 826 a(subset) p 801 826 a(of) p Fq 891 826 a(R) p Fn 951 792 a(I) p Fm 989 801 a(d) p Fy 1050 826 a(\(whic) n(h) p 1315 826 a(are) p 1449 826 a(easy) p 1627 826 a(consequences) p 2123 826 a(of) p 2213 826 a(Theorem) p 2559 826 a(15\).) p 2733 826 a(Hence,) p 2999 826 a(the) p 3137 826 a(\014xed) p 3333 826 a(p) r(oin) n(t) p 3545 826 a(theorem) 208 936 y(implies) p 489 936 a(that) p 669 936 a(there) p 882 936 a(exists) p 1111 936 a(a) p 1180 936 a(\014xed) p 1381 936 a(p) r(oin) n(t) p 1597 936 a(of) p Fo 1695 915 a(~) p Fy 1692 936 a(\010) p 1780 936 a(whic) n(h) p 2017 936 a(satis\014es) p 2323 936 a(\(FP1\)) p 2567 936 a(and) p 2729 936 a(\(FP4\).) 208 1069 y(The) p 376 1069 a(other) p 590 1069 a(conditions) p 985 1069 a(on) p 1098 1069 a(the) p 1239 1069 a(self-a) n(v) n(oiding) p 1712 1069 a(\014xed) p 1910 1069 a(p) r(oin) n(t) p 2124 1069 a(\(FP2\)) p 2366 1069 a(and) p 2525 1069 a(\(FP3\)) p 2767 1069 a(dep) r(end) p 3053 1069 a(more) p 3259 1069 a(on) p 3372 1069 a(the) p 3512 1069 a(details) p 3773 1069 a(of) 208 1168 y(the) p 352 1168 a(self-a) n(v) n(oiding) p 829 1168 a(paths) p 1057 1168 a(on) p Fo 1173 1168 a(d) p Fy(SG.) p 1380 1168 a(Ho) n(w) n(ev) n(er,) p 1740 1168 a(these) p 1954 1168 a(conditions) p 2352 1168 a(deal) p 2529 1168 a(with) p 2719 1168 a(conditions) p 3118 1168 a(of) p 3214 1168 a(P) n(erron-F) p 3535 1168 a(rob) r(enius) 208 1268 y(t) n(yp) r(e) p 402 1268 a(and) p 572 1268 a(irreducibilit) n(y) p 1032 1268 a(,) p 1092 1268 a(whic) n(h) p 1337 1268 a(are) p 1484 1268 a(`soft') p 1698 1268 a(conditions,) p 2128 1268 a(hence) p 2367 1268 a(w) n(e) p 2497 1268 a(exp) r(ect) p 2768 1268 a(them) p 2988 1268 a(to) p 3097 1268 a(hold.) p 3338 1268 a(\(These) p 3619 1268 a(condi-) 208 1367 y(tions) p 415 1367 a(are) p 557 1367 a(used) p 750 1367 a(in) p 851 1367 a(the) p 998 1367 a(pro) r(ofs) p 1251 1367 a(of) p 1350 1367 a(Prop) r(osition) p 1800 1367 a(12,) p 1939 1367 a(Prop) r(osition) p 2389 1367 a(14,) p 2528 1367 a(Prop) r(osition) p 2979 1367 a(17,) p 3117 1367 a(Prop) r(osition) p 3568 1367 a(20,) p 3706 1367 a(and) 208 1467 y(Lemma) p 504 1467 a(22.\)) 55 1633 y(\(ii\)) p 208 1633 a(What) p 439 1633 a(ma) n(y) p 617 1633 a(b) r(e) p 729 1633 a(more) p 935 1633 a(di\016cult) p 1238 1633 a(is) p 1320 1633 a(the) p 1461 1633 a(condition) p 1824 1633 a(\(CS1\),) p 2085 1633 a(whic) n(h) p 2321 1633 a(states) p 2556 1633 a(existence) p 2908 1633 a(of) p 3001 1633 a(a) p 3068 1633 a(tra) p 3180 1633 a(jectory) p 3455 1633 a(con) n(v) n(erging) 208 1733 y(to) p 311 1733 a(a) p 383 1733 a(\014xed) p 586 1733 a(p) r(oin) n(t.) p 842 1733 a(This) p 1034 1733 a(essen) n(tially) p 1434 1733 a(suggests) p 1761 1733 a(that) p 1943 1733 a(a) p 2015 1733 a(b) r(ounded) p 2357 1733 a(tra) p 2469 1733 a(jectory) p 2747 1733 a(necessary) p 3117 1733 a(con) n(v) n(erges) p 3492 1733 a(to) p 3595 1733 a(a) p 3667 1733 a(\014xed) 208 1832 y(p) r(o) r(oin) n(t) p 468 1832 a(\(at) p 601 1832 a(least) p 795 1832 a(in) p 892 1832 a(the) p 1035 1832 a(domain) p 1329 1832 a(\004) p Fp 1384 1844 a(d) p Fy 1424 1832 a(\),) p 1506 1832 a(that) p 1686 1832 a(the) p 1829 1832 a(renormalization) p 2424 1832 a(group) p 2659 1832 a(dynamical) p 3058 1832 a(system) p 3333 1832 a(is) p 3416 1832 a(free) p 3575 1832 a(of) p 3669 1832 a(limit) 208 1932 y(cycles) p 447 1932 a(m) n(uc) n(h) p 670 1932 a(less) p 825 1932 a(an) n(y) p 983 1932 a(c) n(haotic) p 1268 1932 a(b) r(eha) n(viors.) p 1677 1932 a(There) p 1919 1932 a(are) p 2059 1932 a(of) p 2155 1932 a(course) p 2411 1932 a(man) n(y) p 2638 1932 a(discrete) p 2945 1932 a(dynamical) p 3345 1932 a(systems,) p 3679 1932 a(ev) n(en) 208 2032 y(on) p 323 2032 a(one-dimensional) p 932 2032 a(space,) p 1177 2032 a(whic) n(h) p 1415 2032 a(exhibit) p 1694 2032 a(c) n(haotic) p 1977 2032 a(b) r(eha) n(viors,) p 2372 2032 a(hence) p 2603 2032 a(this) p 2765 2032 a(condition) p 3129 2032 a(is) p 3213 2032 a(far) p 3340 2032 a(from) p 3536 2032 a(trivial.) 208 2164 y(On) p 350 2164 a(the) p 496 2164 a(other) p 717 2164 a(hand,) p 952 2164 a(it) p 1039 2164 a(is) p 1126 2164 a(pro) n(v) n(ed) p 1400 2164 a(in) p 1500 2164 a([5]) p 1619 2164 a(and) p 1784 2164 a([6]) p 1903 2164 a(that) p 2087 2164 a(for) p Fo 2217 2164 a(d) p Fy 2290 2164 a(=) p 2383 2164 a(2) p 2456 2164 a(and) p Fo 2621 2164 a(d) p Fy 2694 2164 a(=) p 2787 2164 a(3,) p 2884 2164 a(all) p 3003 2164 a(the) p 3150 2164 a(conditions) p 3550 2164 a(\(FP1\)) p 3798 2164 a({) 208 2264 y(\(FP4\)) p 448 2264 a(and) p 606 2264 a(\(CS1\)) p 842 2264 a(are) p 977 2264 a(satis\014ed.) p 1328 2264 a(W) p 1406 2264 a(e) p 1467 2264 a(also) p 1630 2264 a(pro) n(v) n(e) p 1850 2264 a(in) p 1943 2264 a(Section) p 2230 2264 a(5) p 2296 2264 a(that) p 2472 2264 a(\(FP1\)) p 2713 2264 a({) p 2779 2264 a(\(FP4\)) p 3019 2264 a(and) p 3177 2264 a(\(CS2\)) p 3414 2264 a(are) p 3548 2264 a(satis\014ed) 208 2364 y(for) p 344 2364 a(the) p 497 2364 a(restricted) p 876 2364 a(mo) r(del) p 1133 2364 a(on) p 1257 2364 a(4SG.) p 1471 2364 a(Based) p 1724 2364 a(on) p 1849 2364 a(these) p 2071 2364 a(results,) p 2370 2364 a(w) n(e) p 2502 2364 a(conjecture) p 2911 2364 a(that) p 3100 2364 a(these) p 3322 2364 a(conditions) p 3729 2364 a(are) 208 2463 y(satis\014ed) p 527 2463 a(\(hence) p 789 2463 a(the) p 932 2463 a(results) p 1196 2463 a(ab) r(out) p 1434 2463 a(the) p 1577 2463 a(asymptotic) p 2004 2463 a(b) r(eha) n(viors) p 2376 2463 a(of) p 2470 2463 a(the) p 2613 2463 a(self-a) n(v) n(oiding) p 3088 2463 a(w) n(alks) p 3315 2463 a(hold\)) p 3532 2463 a(for) p 3658 2463 a(all) p Fo 3774 2463 a(d) p Fy(.) p Fd 3774 2563 a(3) p Fr 0 2978 a(3.2) p 255 2978 a(Asymptotic) p 860 2978 a(b) s(eha) m(viors.) p Fy 0 3131 a(Here) p 196 3131 a(w) n(e) p 318 3131 a(will) p 475 3131 a(state) p 679 3131 a(main) p 886 3131 a(consequences) p 1388 3131 a(of) p 1482 3131 a(assumptions) p 1954 3131 a(on) p 2069 3131 a(R) n(G) p 2221 3131 a(form) n(ulated) p 2641 3131 a(in) p 2738 3131 a(Section) p 3028 3131 a(3.1.) 125 3231 y(First) p 327 3231 a(w) n(e) p 449 3231 a(note) p 634 3231 a(the) p 777 3231 a(follo) n(wing) p 1127 3231 a(c) n(haracterization) p 1734 3231 a(of) p 1829 3231 a(a) p 1898 3231 a(critical) p 2175 3231 a(p) r(oin) n(t) p Fo 2392 3231 a(\014) p Fp 2439 3243 a(c) p Fy 2486 3231 a(.) p Fz 0 3397 a(Theorem) p 405 3397 a(7) p Fi 495 3397 a(If) p Fo 580 3397 a(\014) p Fp 627 3409 a(c) p Fk 684 3397 a(2) p Fq 762 3397 a(R) p Fi 850 3397 a(is) p 938 3397 a(a) p 1008 3397 a(critic) l(al) p 1286 3397 a(p) l(oint) p 1494 3397 a(of) p 1590 3397 a(the) p 1726 3397 a(ful) t(l) p 1871 3397 a(mo) l(del) p 2107 3397 a(and) p Fo 2265 3397 a(~) p 2266 3397 a(x) p Fp 2313 3409 a(c) p Fi 2376 3397 a(the) p 2512 3397 a(c) l(orr) l(esp) l (onding) p 3034 3397 a(\014xe) l(d) p 3224 3397 a(p) l(oint) p 3432 3397 a(\(implicit) p 3768 3397 a(in) 0 3497 y(the) p 138 3497 a(de\014nition) p 509 3497 a(\(CS1\)\),) p 815 3497 a(then) p 999 3497 a(for) p Fo 1132 3497 a(I) p Fk 1198 3497 a(2) p 1276 3497 a(I) p Fp 1321 3509 a(d) p Fi 1374 3497 a(,) p Fy 1319 3729 a(lim) p Fp 1290 3779 a(n) p Fn(!1) p Fo 1478 3729 a(Z) p Fp 1535 3741 a(n;I) p Fy 1633 3729 a(\() p Fo(\014) p Fy 1716 3729 a(\)) p 1772 3729 a(=) p Fg 1860 3612 a(\032) p Fy 1964 3678 a(0) p Fo(;) p 2218 3678 a(\014) p 2293 3678 a(>) p 2380 3678 a(\014) p Fp 2427 3690 a(c) p Fo 2475 3678 a(;) 1964 3778 y(x) p Fp 2011 3790 a(c;I) p Fo 2112 3778 a(;) p 2218 3778 a(\014) p Fy 2293 3778 a(=) p Fo 2380 3778 a(\014) p Fp 2427 3790 a(c) p Fo 2475 3778 a(:) p Fi 0 3956 a(Mor) l(e) l(over,) p Fy 1332 4133 a(lim) p Fp 1303 4183 a(n) p Fn(!1) p Fp 1533 4029 a(d) p Fg 1491 4054 a(X) p Fp 1497 4231 a(i) p Fj(=1) p Fo 1624 4133 a(Z) p Fp 1681 4148 a(n;) p Fj(\() p Fp(i) p Fj(\)) p Fy 1821 4133 a(\() p Fo(\014) p Fy 1904 4133 a(\)) p 1960 4133 a(=) p Fk 2048 4133 a(1) p Fo(;) p 2257 4133 a(\014) p 2331 4133 a(<) p 2419 4133 a(\014) p Fp 2466 4145 a(c) p Fo 2514 4133 a(:) p Fi 125 4359 a(In) p 233 4359 a(p) l(articular,) p 638 4359 a(critic) l(al) p 917 4359 a(p) l(oint) p 1127 4359 a(\(if) p 1242 4359 a(exists\)) p 1503 4359 a(is) p 1592 4359 a(unique.) 125 4459 y(Similar) p 418 4459 a(r) l(esult) p 644 4459 a(holds) p 857 4459 a(also) p 1026 4459 a(for) p 1159 4459 a(the) p 1297 4459 a(r) l(estricte) l(d) p 1660 4459 a(mo) l(del) p 1897 4459 a(\(CS2\).) p Fy 0 4625 a(Since) p 211 4625 a(the) p 348 4625 a(critical) p 619 4625 a(p) r(oin) n(t) p 830 4625 a(\(if) p 933 4625 a(exists\)) p 1189 4625 a(is) p 1266 4625 a(unique,) p 1554 4625 a(there) p 1761 4625 a(is) p 1839 4625 a(one) p 1985 4625 a(and) p 2141 4625 a(only) p 2317 4625 a(one) p 2463 4625 a(self-a) n(v) n(oiding) p 2933 4625 a(\014xed) p 3128 4625 a(p) r(oin) n(t) p 3339 4625 a(that) p 3513 4625 a(is) p 3591 4625 a(related) 0 4725 y(b) n(y) p 115 4725 a(\(CS1\)) p 355 4725 a(to) p 457 4725 a(the) p 600 4725 a(critical) p 877 4725 a(p) r(oin) n(t.) 125 4824 y(Though) p 436 4824 a(it) p 521 4824 a(is) p 606 4824 a(not) p 756 4824 a(trivial) p 1005 4824 a(to) p 1109 4824 a(pro) n(v) n(e) p 1335 4824 a(the) p 1480 4824 a(uniqueness) p 1901 4824 a(of) p 1997 4824 a(self-a) n(v) n (oiding) p 2475 4824 a(\014xed) p 2678 4824 a(p) r(oin) n(t,) p 2920 4824 a(w) n(e) p 3044 4824 a(therefore) p 3395 4824 a(can) p 3550 4824 a(\(and) p 3745 4824 a(w) n(e) 0 4924 y(will,) p 190 4924 a(in) p 295 4924 a(the) p 446 4924 a(pro) r(of) p 671 4924 a(of) p 773 4924 a(Theorem) p 1132 4924 a(10\)) p 1283 4924 a(talk) p 1459 4924 a(ab) r(out) p 1705 4924 a(the) p 1856 4924 a(unique) p 2134 4924 a(self-a) n(v) n (oiding) p 2617 4924 a(\014xed) p 2826 4924 a(p) r(oin) n(t) p 3051 4924 a(that) p 3239 4924 a(is) p 3330 4924 a(related) p 3615 4924 a(to) p 3725 4924 a(the) 0 5024 y(critical) p 277 5024 a(p) r(oin) n(t,) p 517 5024 a(under) p 752 5024 a(the) p 895 5024 a(asumption) p 1302 5024 a(that) p 1481 5024 a(the) p 1624 5024 a(critical) p 1901 5024 a(p) r(oin) n(t) p 2118 5024 a(exists.) 125 5223 y(T) p 178 5223 a(o) p 246 5223 a(state) p 448 5223 a(the) p 590 5223 a(next) p 775 5223 a(Theorem,) p 1148 5223 a(w) n(e) p 1270 5223 a(note) p 1453 5223 a(a) p 1521 5223 a(relation) p 1825 5223 a(b) r(et) n(w) n(een) p 2144 5223 a(0) p 2212 5223 a(comp) r(onen) n(ts) p 2668 5223 a(of) p 2761 5223 a(a) p 2830 5223 a(\014xed) p 3029 5223 a(p) r(oin) n(t) p 3245 5223 a(and) p 3405 5223 a(an) p 3519 5223 a(in) n(v) p 3625 5223 a(arian) n(t) 0 5322 y(subset) p 256 5322 a(of) p Fo 354 5302 a(~) p Fy 351 5322 a(\010.) p 474 5322 a(In) p 578 5322 a(the) p 722 5322 a(kno) n(wn) p 986 5322 a(case) p 1163 5322 a(of) p Fo 1258 5322 a(d) p Fy 1326 5322 a(=) p 1415 5322 a(2) p 1485 5322 a(and) p Fo 1648 5322 a(d) p Fy 1715 5322 a(=) p 1804 5322 a(3,) p 1898 5322 a(the) p 2042 5322 a(self-a) n(v) n(oiding) p 2518 5322 a(\014xed) p 2720 5322 a(p) r(oin) n(ts) p 2970 5322 a(ha) n(v) n(e) p 3162 5322 a(0) p 3233 5322 a(comp) r(onen) n(ts.) p 3725 5322 a(W) p 3803 5322 a(e) 0 5422 y(will) p 157 5422 a(write) p Fk 1451 5522 a(K) p Fp 1513 5535 a(~) p 1514 5535 a(x) p Fm 1552 5543 a(c) p Fy 1610 5522 a(=) p Fk 1698 5522 a(f) p Fo(I) p Fk 1806 5522 a(2) p 1884 5522 a(I) p Fp 1929 5534 a(d) p Fk 1991 5522 a(j) p Fo 2037 5522 a(x) p Fp 2084 5534 a(c;I) p Fk 2195 5522 a(6) p Fy(=) p 2283 5522 a(0) p Fk(g) p Fo(;) p Fy 3692 5522 a(\(22\)) 0 5671 y(and,) p 184 5671 a(as) p 286 5671 a(in) p 383 5671 a(\(12\),) p Fq 1235 5787 a(R) p Fn 1295 5740 a(K) p Fm 1345 5750 a(~) p 1346 5750 a(x) 1379 5758 y(c) p Fj 1295 5807 a(+) p Fy 1441 5787 a(=) p Fk 1528 5787 a(f) p Fo 1569 5787 a(~) p 1570 5787 a(x) p Fk 1640 5787 a(2) p Fq 1719 5787 a(R) p Fn 1779 5749 a(I) p Fm 1817 5758 a(d) p Fj 1779 5807 a(+) p Fk 1878 5787 a(j) p Fo 1925 5787 a(x) p Fp 1972 5799 a(J) p Fy 2041 5787 a(=) p 2129 5787 a(0) p Fo(;) p 2235 5787 a(J) p Fk 2312 5787 a(62) p 2391 5787 a(K) p Fp 2453 5800 a(~) p 2454 5800 a(x) p Fm 2492 5808 a(c) p Fk 2527 5787 a(g) p Fo 2583 5787 a(:) p Fy 1878 6120 a(10) p 90 rotate dyy eop %%Page: 11 11 11 10 bop Fz 0 87 a(Prop) s(osition) p 516 87 a(8) p Fq 605 87 a(R) p Fn 665 40 a(K) p Fm 715 50 a(~) p 716 50 a(x) 749 58 y(c) p Fj 665 107 a(+) p Fi 818 87 a(is) p 907 87 a(an) p 1026 87 a(invariant) p 1386 87 a(subset) p 1632 87 a(of) p Fo 1732 66 a(~) p Fy 1729 87 a(\010) p Fi(.) -42 269 y(Pr) l(o) l(of.) p Fy 219 269 a(If) p Fo 301 269 a(x) p Fp 348 281 a(c;I) p Fy 459 269 a(=) p 547 269 a(0) p 616 269 a(then) p 804 269 a(\010) p Fp 864 281 a(I) p Fy 902 269 a(\() p Fo 933 269 a(~) p 934 269 a(x) p Fp 981 281 a(c) p Fy 1016 269 a(\)) p 1071 269 a(=) p Fo 1159 269 a(x) p Fp 1206 281 a(c;I) p Fy 1317 269 a(=) p 1404 269 a(0) p 1460 269 a(.) p 1520 269 a(On) p 1657 269 a(the) p 1800 269 a(other) p 2016 269 a(hand) p 2224 269 a(\010) p Fp 2284 281 a(I) p Fy 2349 269 a(is) p 2432 269 a(a) p 2501 269 a(p) r(olynomial) p 2929 269 a(with) p 3118 269 a(p) r(ositiv) n(e) p 3424 269 a(co) r(e\016cien) n(ts.) 0 369 y(Therefore,) p 398 369 a(eac) n(h) p 582 369 a(term) p 778 369 a(in) p 873 369 a(\010) p Fp 933 381 a(I) p Fy 971 369 a(\() p Fo 1002 369 a(~) p 1003 369 a(x) p Fy 1051 369 a(\)) p 1108 369 a(con) n(tains) p 1431 369 a(one) p 1581 369 a(of) p Fo 1673 369 a(x) p Fp 1720 381 a(J) p Fy 1767 369 a('s) p 1848 369 a(suc) n(h) p 2033 369 a(that) p Fo 2211 369 a(x) p Fp 2258 381 a(c;J) p Fy 2377 369 a(=) p 2465 369 a(0) p 2520 369 a(.) p 2579 369 a(In) p 2681 369 a(other) p 2895 369 a(w) n(ords,) p 3155 369 a(eac) n(h) p 3339 369 a(term) p 3535 369 a(in) p 3630 369 a(\010) p Fp 3690 381 a(I) p Fy 3728 369 a(\() p Fo 3759 369 a(~) p 3760 369 a(x) p Fy 3808 369 a(\)) 0 468 y(con) n(tains) p Fo 326 468 a(x) p Fp 373 480 a(J) p Fy 447 468 a(suc) n(h) p 634 468 a(that) p Fo 814 468 a(J) p Fk 891 468 a(62) p 970 468 a(K) p Fp 1032 481 a(~) p 1033 481 a(x) p Fm 1071 489 a(c) p Fy 1120 468 a(.) 125 585 y(Therefore,) p 524 585 a(if) p Fo 599 585 a(~) p 600 585 a(x) p Fk 671 585 a(2) p Fq 749 585 a(R) p Fn 809 538 a(K) p Fm 859 548 a(~) p 860 548 a(x) 893 556 y(c) p Fj 809 606 a(+) p Fy 946 585 a(,) p 996 585 a(then) p 1186 585 a(\010) p Fp 1246 597 a(I) p Fy 1284 585 a(\() p Fo 1315 585 a(~) p 1316 585 a(x) p Fy(\)) p 1419 585 a(=) p 1506 585 a(0) p 1576 585 a(for) p 1703 585 a(those) p Fo 1920 585 a(I) p Fy 1991 585 a(satisfying) p Fo 2360 585 a(x) p Fp 2407 597 a(c;I) p Fy 2518 585 a(=) p 2606 585 a(0,) p 2698 585 a(or) p 2800 585 a(equiv) p 2989 585 a(alen) n(tly) p 3227 585 a(,) p Fo 3277 585 a(I) p Fk 3343 585 a(62) p 3422 585 a(K) p Fp 3484 598 a(~) p 3485 598 a(x) p Fm 3523 606 a(c) p Fy 3572 585 a(.) p Fd 3778 585 a(2) p Fi -42 950 a(R) l(emark.) p Fy 303 950 a(F) p 350 950 a(or) p Fo 456 950 a(d) p Fy 528 950 a(=) p 621 950 a(2) p 694 950 a(and) p Fo 858 950 a(d) p Fy 930 950 a(=) p 1024 950 a(3,) p 1120 950 a(the) p 1266 950 a(results) p 1534 950 a(in) p 1634 950 a([5]) p 1753 950 a(and) p 1918 950 a([6) o(]) p 2036 950 a(resp) r(ectiv) n(ely) p 2492 950 a(pro) n(v) n(es) p 2752 950 a(\(b) n(y) p 2903 950 a(explicit) p 3200 950 a(calculations\)) p 3688 950 a(that) 0 1050 y(the) p 143 1050 a(self-a) n(v) n(oiding) p 619 1050 a(\014xed) p 819 1050 a(p) r(oin) n(t) p Fo 1035 1050 a(~) p 1036 1050 a(x) p Fp 1083 1062 a(c) p Fy 1145 1050 a(is) p 1228 1050 a(unique) p 1498 1050 a(and) p 1660 1050 a(that) p Fk 1839 1050 a(K) p Fp 1902 1062 a(r) r(es) p Fy 2025 1050 a(=) p Fk 2113 1050 a(K) p Fp 2175 1063 a(~) p 2176 1063 a(x) p Fm 2214 1071 a(c) p Fy 2263 1050 a(.) p Fd 3774 1050 a(3) p Fy 125 1349 a(F) p 172 1349 a(or) p Fo 274 1349 a(I) p Fk 340 1349 a(2) p 418 1349 a(I) p Fp 463 1361 a(d) p Fy 502 1349 a(,) p Fo 553 1349 a(n) p Fk 626 1349 a(2) p Fq 704 1349 a(Z) p Fj 759 1361 a(+) p Fy 814 1349 a(,) p 865 1349 a(and) p Fo 1025 1349 a(~) p 1027 1349 a(x) p Fk 1097 1349 a(2) p Fq 1176 1349 a(R) p Fn 1236 1311 a(I) p Fm 1274 1320 a(d) p Fj 1236 1370 a(+) p Fy 1326 1349 a(,) p 1377 1349 a(de\014ne) p 1616 1349 a(a) p 1686 1349 a(probabilit) n(y) p 2110 1349 a(measure) p Fo 2434 1349 a(\026) p Fp 2483 1362 a(~) p 2484 1362 a(x) o(;n;I) p Fy 2668 1349 a(on) p 2783 1349 a(the) p 2926 1349 a(\014nite) p 3138 1349 a(set) p Fo 3268 1349 a(W) p Fj 3358 1306 a(\() p Fp(n) p Fj(\)) p Fp 3346 1373 a(I) p Fy 3482 1349 a(b) n(y) p Fo 1078 1577 a(\026) p Fp 1127 1590 a(~) p 1128 1590 a(x) o(;n;I) p Fy 1284 1577 a([) p Fk(f) p Fo(w) p Fk 1410 1577 a(g) p Fy(]) p 1498 1577 a(=) 1715 1521 y(1) p 1595 1558 280 4 v Fo 1595 1634 a(X) p Fp 1664 1646 a(n;I) p Fy 1763 1634 a(\() p Fo 1794 1634 a(~) p 1795 1634 a(x) p Fy 1843 1634 a(\)) p Fg 1926 1499 a(Y) p Fp 1899 1677 a(J) p Fn 1941 1677 a(2I) p Fm 2024 1686 a(d) p Fo 2072 1577 a(x) p Fp 2119 1534 a(s) p Fm 2150 1542 a(J) p Fj 2191 1534 a(\() p Fp(w) p Fj 2267 1534 a(\)) p Fp 2119 1602 a(J) p Fo 2297 1577 a(;) p 2389 1577 a(w) p Fk 2474 1577 a(2) p Fo 2552 1577 a(W) p Fj 2642 1534 a(\() p Fp(n) p Fj(\)) p Fp 2630 1602 a(I) p Fo 2739 1577 a(;) p Fy 3692 1577 a(\(23\)) 0 1847 y(whenev) n(er) p Fo 365 1847 a(X) p Fp 434 1859 a(n;I) p Fy 532 1847 a(\() p Fo 563 1847 a(~) p 564 1847 a(x) p Fy 612 1847 a(\)) p Fk 667 1847 a(6) p Fy(=) p 755 1847 a(0) p 810 1847 a(.) 125 1947 y(Note) p 328 1947 a(that) p 511 1947 a(if) p Fo 590 1947 a(x) p Fp 637 1959 a(c;I) p Fk 752 1947 a(6) p Fy(=) p 845 1947 a(0) p 916 1947 a(and) p Fo 1080 1947 a(~) p 1081 1947 a(x) p Fk 1156 1947 a(2) p 1239 1947 a(D) p Fi 1305 1947 a(om) p Fy 1422 1947 a(\() p Fo 1453 1947 a(~) p 1454 1947 a(x) p Fp 1501 1959 a(c) p Fy 1536 1947 a(\),) p 1622 1947 a(then) p 1814 1947 a(\(D) n(A1\)) p 2074 1947 a(implies) p 2359 1947 a(that) p Fo 2541 1947 a(X) p Fp 2610 1959 a(n;I) p Fy 2709 1947 a(\() p Fo 2740 1947 a(~) p 2741 1947 a(x) p Fy 2789 1947 a(\)) p Fo 2849 1947 a(>) p Fy 2941 1947 a(0) p 3013 1947 a(for) p 3143 1947 a(su\016cien) n(tly) p 3561 1947 a(large) p Fo 3767 1947 a(n) p Fy(,) 0 2047 y(hence) p 231 2047 a(if) p Fo 306 2047 a(~) p 307 2047 a(x) p Fk 377 2047 a(2) p 456 2047 a(D) p Fi 522 2047 a(om) p Fy 639 2047 a(\() p Fo 670 2047 a(~) p 671 2047 a(x) p Fp 718 2059 a(c) p Fy 752 2047 a(\)) p 812 2047 a(then) p Fo 1001 2047 a(\026) p Fp 1050 2060 a(~) p 1051 2060 a(x;n;I) p Fy 1235 2047 a(is) p 1319 2047 a(w) n(ell) p 1487 2047 a(de\014ned.) p Fz 0 2213 a(Theorem) p 405 2213 a(9) p Fi 495 2213 a(L) l(et) p Fo 637 2213 a(~) p 638 2213 a(x) p Fp 685 2225 a(c) p Fi 749 2213 a(b) l(e) p 851 2213 a(a) p 923 2213 a(self-avoiding) p 1402 2213 a(\014xe) l(d) p 1594 2213 a(p) l(oint) p 1804 2213 a(and) p Fo 1964 2213 a(~) p 1965 2213 a(x) p Fk 2035 2213 a(2) p 2114 2213 a(D) p Fi 2180 2213 a(om) p Fy 2297 2213 a(\() p Fo 2328 2213 a(~) p 2329 2213 a(x) p Fp 2376 2225 a(c) p Fy 2411 2213 a(\)) p Fi(.) p 2506 2213 a(Then) p 2723 2213 a(the) p 2861 2213 a(fol) t(lowing) p 3213 2213 a(hold.) 73 2379 y(\(i\)) p 208 2379 a(Ther) l(e) p 446 2379 a(exists) p Fo 674 2379 a(f) p Fp 715 2391 a(d) p Fk 771 2379 a(\002) p Fo 854 2379 a(f) p Fp 895 2391 a(d) p Fi 963 2379 a(matrix) p Fy 1230 2379 a(\003\() p Fo 1319 2379 a(~) p 1320 2379 a(x) p Fy(\)) p Fi 1429 2379 a(whose) p 1671 2379 a(elements) p 2013 2379 a(ar) l(e) p 2154 2379 a(non-ne) l(gative) p 2640 2379 a(such) p 2829 2379 a(that) p Fy 1599 2561 a(\003\() p Fo 1688 2561 a(~) p 1689 2561 a(x) p Fy(\)) p 1792 2561 a(=) p 1908 2561 a(lim) p Fp 1879 2611 a(n) p Fn(!1) p Fo 2067 2561 a(\025) p Fn 2115 2527 a(\000) p Fp(n) p Fk 2212 2561 a(J) p Fp 2268 2573 a(n) p Fy 2314 2561 a(\() p Fo 2345 2561 a(~) p 2346 2561 a(x) p Fy(\)) p Fo(;) p Fy 3692 2561 a(\(24\)) p Fi 208 2773 a(wher) l(e) p Fk 442 2773 a(J) p Fp 498 2785 a(n) p Fi 573 2773 a(is) p 662 2773 a(as) p 768 2773 a(in) p 870 2773 a(\(14\).) 47 2939 y(\(ii\)) p 208 2939 a(F) p 256 2939 a(or) p Fo 362 2939 a(I) p Fk 428 2939 a(2) p 506 2939 a(K) p Fp 568 2952 a(~) p 569 2952 a(x) p Fm 607 2960 a(c) p Fi 657 2939 a(,) p 712 2939 a(the) p 849 2939 a(joint) p 1046 2939 a(distribution) p 1490 2939 a(of) p 1587 2939 a(sc) l(ale) l(d) p 1824 2939 a(gener) l(alize) l(d) p 2244 2939 a(p) l(ath) p 2424 2939 a(lengths) p Fy 2702 2939 a(\() p Fo(\025) p Fn 2782 2909 a(\000) p Fp(n) p Fo 2879 2939 a(s) p Fp 2918 2951 a(J) p Fo 2979 2939 a(;) p 3045 2939 a(J) p Fk 3122 2939 a(2) p 3200 2939 a(K) p Fp 3262 2952 a(~) p 3263 2952 a(x) p Fm 3301 2960 a(c) p Fy 3336 2939 a(\)) p Fi 3398 2939 a(under) p Fo 3634 2939 a(\026) p Fp 3683 2952 a(~) p 3684 2952 a(x) o(;n;I) p Fi 208 3039 a(c) l(onver) l(ges) p 577 3039 a(we) l(akly) p 837 3039 a(to) p 935 3039 a(a) p 1006 3039 a(Bor) l(el) p 1226 3039 a(pr) l(ob) l(ability) p 1625 3039 a(me) l(asur) l(e) p Fo 1945 3039 a(p) p Fn 1987 3004 a(\003) p Fp 1986 3060 a(~) p 1987 3060 a(x;I) p Fi 2111 3039 a(on) p Fq 2229 3039 a(R) p Fn 2289 3004 a(I) p Fm 2327 3013 a(d) p Fi 2394 3039 a(as) p Fo 2499 3039 a(n) p Fk 2572 3039 a(!) p 2678 3039 a(1) p Fi(.) p 2825 3039 a(Her) l(e,) p Fo 3048 3039 a(\025) p Fi 3125 3039 a(is) p 3213 3039 a(as) p 3318 3039 a(in) p 3419 3039 a(\(FP2\).) p Fo 3703 3039 a(p) p Fn 3745 3004 a(\003) p Fp 3744 3060 a(~) p 3745 3060 a(x) o(;I) p Fi 208 3156 a(is) p 297 3156 a(supp) l(orte) l(d) p 667 3156 a(on) p Fq 786 3156 a(R) p Fn 846 3119 a(I) p Fm 884 3128 a(d) p Fj 846 3177 a(+) p Fi 922 3156 a(.) 208 3289 y(The) p 377 3289 a(gener) l(ating) p 779 3289 a(function) p Fo 1106 3289 a(') p Fn 1160 3255 a(\003) p Fp 1160 3310 a(I) p Fy 1222 3289 a(=) p Fo 1309 3289 a(') p Fn 1363 3255 a(\003) p Fp 1362 3310 a(~) p 1363 3310 a(x;I) p Fi 1459 3289 a(,) p 1514 3289 a(as) p 1620 3289 a(a) p 1692 3289 a(function) p 2019 3289 a(of) p Fy 2117 3289 a(\() p Fo(t) p Fp 2179 3301 a(J) p Fo 2225 3289 a(;) p 2292 3289 a(J) p Fk 2369 3289 a(2) p 2447 3289 a(K) p Fp 2509 3302 a(~) p 2510 3302 a(x) p Fm 2548 3310 a(c) p Fy 2584 3289 a(\)) p Fi(,) p 2671 3289 a(de\014ne) l(d) p 2951 3289 a(by) p Fo 1331 3530 a(') p Fn 1385 3496 a(\003) p Fp 1384 3551 a(~) p 1385 3551 a(x;I) p Fy 1481 3530 a(\() p Fo 1508 3515 a(~) 1513 3530 y(t) p Fy(\)) p 1598 3530 a(=) p Fg 1686 3417 a(Z) p Fn 1769 3438 a(1) p Fj 1732 3606 a(0) p Fo 1853 3530 a(e) p Fp 1888 3485 a(~) 1892 3496 y(t) p Fn(\001) p Fp 1939 3480 a(~) 1937 3496 y(\030) p Fo 1973 3530 a(p) p Fn 2015 3496 a(\003) p Fp 2014 3551 a(~) p 2015 3551 a(x) o(;I) p Fy 2110 3530 a([) p Fo(d) 2178 3508 y(~) 2176 3530 y(\030) p Fy 2216 3530 a(]) p Fo(;) 2360 3515 y(~) 2366 3530 y(t) p Fk 2419 3530 a(2) p Fq 2497 3530 a(C) p Fn 2557 3496 a(K) p Fm 2607 3506 a(~) p 2608 3506 a(x) 2641 3514 y(c) p Fo 2694 3530 a(;) p Fi 208 3772 a(is) p 297 3772 a(an) p 415 3772 a(entir) l(e) p 652 3772 a(function) p 979 3772 a(in) p Fo 1075 3756 a(~) 1081 3772 y(t) p Fi(.) 22 3938 y(\(iii\)) p 208 3938 a(The) p 377 3938 a(set) p 507 3938 a(of) p 604 3938 a(functions) p Fo 965 3938 a(') p Fn 1019 3903 a(\003) p Fp 1019 3958 a(I) p Fy 1081 3938 a(=) p Fo 1168 3938 a(') p Fn 1222 3903 a(\003) p Fp 1221 3958 a(~) p 1222 3958 a(x;I) p Fi 1332 3938 a(,) p Fo 1387 3938 a(I) p Fk 1453 3938 a(2) p 1532 3938 a(K) p Fp 1594 3951 a(~) p 1595 3951 a(x) p Fm 1633 3959 a(c) p Fi 1682 3938 a(,) p 1737 3938 a(ar) l(e) p 1878 3938 a(uniquely) p 2207 3938 a(determine) l(d) p 2635 3938 a(by) p Fo 1079 4194 a(x) p Fp 1126 4206 a(c;I) p Fo 1237 4138 a(@) p 1286 4138 a(') p Fn 1340 4108 a(\003) p Fp 1340 4161 a(I) p 1237 4175 142 4 v Fo 1245 4251 a(@) p 1294 4251 a(t) p Fp 1324 4263 a(J) p Fy 1388 4194 a(\() p Fo 1414 4176 a(~) p Fy 1420 4194 a(0\)) p 1518 4194 a(=) p 1605 4194 a(\003) p Fp 1663 4206 a(I) p 1697 4206 a(J) p Fy 1743 4194 a(\() p Fo 1774 4194 a(~) p 1775 4194 a(x) p Fy 1823 4194 a(\)) p Fo 1869 4194 a(x) p Fp 1916 4206 a(J) p Fo 1977 4194 a(;) p Fi 2103 4194 a(if) p Fo 2213 4194 a(I) p 2256 4194 a(;) p 2307 4194 a(J) p Fk 2384 4194 a(2) p 2462 4194 a(K) p Fp 2524 4207 a(~) p 2525 4207 a(x) p Fm 2563 4215 a(c) p Fo 2612 4194 a(;) 1079 4352 y(x) p Fp 1126 4364 a(c;I) p Fo 1214 4352 a(') p Fn 1268 4317 a(\003) p Fp 1268 4372 a(I) p Fy 1306 4352 a(\() p Fo(\025) 1381 4336 y(~) 1386 4352 y(t) p Fy 1417 4352 a(\)) p 1472 4352 a(=) p 1560 4352 a(\010) p Fp 1620 4364 a(I) p Fy 1658 4352 a(\() p Fo 1689 4352 a(~) p 1690 4352 a(x) p Fp 1737 4364 a(c) p Fo 1792 4352 a(~) p 1785 4352 a(') p Fn 1839 4317 a(\003) p Fy 1878 4352 a(\() p Fo 1905 4336 a(~) 1910 4352 y(t) p Fy(\)\)) p Fo(;) 2096 4336 y(~) 2101 4352 y(t) p Fk 2154 4352 a(2) p Fq 2232 4352 a(C) p Fn 2292 4317 a(K) p Fm 2342 4327 a(~) p 2343 4327 a(x) 2376 4335 y(c) p Fo 2415 4352 a(;) p Fi 2541 4352 a(if) p Fo 2651 4352 a(I) p Fk 2717 4352 a(2) p 2796 4352 a(K) p Fp 2858 4365 a(~) p 2859 4365 a(x) p Fm 2897 4373 a(c) p Fo 2946 4352 a(;) p Fy 3692 4242 a(\(25\)) p Fi 208 4553 a(wher) l(e) p 442 4553 a(we) p 565 4553 a(de\014ne) p Fo 807 4553 a(') p Fn 861 4519 a(\003) p Fp 861 4574 a(J) p Fy 931 4553 a(=) p 1018 4553 a(0) p Fi 1090 4553 a(for) p Fo 1222 4553 a(J) p Fk 1299 4553 a(62) p 1378 4553 a(K) p Fp 1440 4566 a(~) p 1441 4566 a(x) p Fm 1479 4574 a(c) p Fi 1528 4553 a(,) p 1583 4553 a(and) p 1744 4553 a(in) p 1846 4553 a(the) p 1984 4553 a(variable) p 2295 4553 a(for) p Fo 2430 4532 a(~) p Fy 2427 4553 a(\010) p Fi 2517 4553 a(we) p 2640 4553 a(use) l(d) p 2825 4553 a(an) p 2944 4553 a(\(irr) l(e) l(gular\)) p 3348 4553 a(notation) p Fy 1451 4736 a(\() p Fo 1482 4736 a(~) p 1483 4736 a(x) p 1552 4736 a(~) p 1545 4736 a(') p Fn 1599 4701 a(\003) p Fy 1637 4736 a(\() p Fo 1664 4720 a(~) 1669 4736 y(t) p Fy(\)\)) p Fp 1763 4748 a(J) p Fy 1833 4736 a(=) p Fo 1921 4736 a(x) p Fp 1968 4748 a(J) p Fo 2029 4736 a(') p Fn 2083 4701 a(\003) p Fp 2083 4756 a(J) p Fy 2129 4736 a(\() p Fo 2156 4720 a(~) 2161 4736 y(t) p Fy 2192 4736 a(\)) p Fo(;) p 2320 4736 a(J) p Fk 2397 4736 a(2) p 2476 4736 a(I) p Fp 2521 4748 a(d) p Fo 2573 4736 a(:) p Fi 35 4968 a(\(iv\)) p 208 4968 a(If) p Fo 294 4968 a(~) p 295 4968 a(x) p Fk 365 4968 a(2) p 444 4968 a(D) p Fi 510 4968 a(om) p Fy 627 4968 a(\() p Fo 658 4968 a(~) p 659 4968 a(x) p Fp 706 4980 a(c) p Fy 740 4968 a(\)) p Fk 791 4968 a(\\) p Fy 865 4968 a(\004) p Fp 920 4980 a(d) p Fi 988 4968 a(and) p Fo 1150 4968 a(I) p Fk 1216 4968 a(2) p 1294 4968 a(K) p Fp 1356 4981 a(~) p 1357 4981 a(x) p Fm 1395 4989 a(c) p Fi 1444 4968 a(,) p 1500 4968 a(then) p 1684 4968 a(the) p 1822 4968 a(distribution) p 2266 4968 a(of) p Fo 2364 4968 a(\025) p Fn 2412 4933 a(\000) p Fp(n) p Fo 2509 4968 a(L) p Fy(\() p Fo(w) p Fy 2659 4968 a(\)) p Fi(,) p 2747 4968 a(the) p 2885 4968 a(sc) l(ale) l(d) p 3123 4968 a(length) p 3367 4968 a(of) p Fo 3464 4968 a(w) p Fk 3549 4968 a(2) p Fo 3628 4968 a(W) p Fj 3718 4925 a(\() p Fp(n) p Fj(\)) p Fp 3706 4992 a(I) p Fi 3815 4968 a(,) 208 5067 y(under) p Fo 444 5067 a(\026) p Fp 493 5080 a(~) p 494 5080 a(x;n;I) p Fi 680 5067 a(c) l(onver) l(ges) p 1050 5067 a(we) l(akly) p 1311 5067 a(to) p 1411 5067 a(a) p 1483 5067 a(Bor) l(el) p 1704 5067 a(pr) l(ob) l(ability) p 2104 5067 a(me) l(asur) l(e) p Fy 2432 5067 a(\026) p Fo 2425 5067 a(p) p Fn 2467 5033 a(\003) p Fp 2466 5088 a(~) p 2467 5088 a(x;I) p Fi 2577 5067 a(,) p 2632 5067 a(which) p 2865 5067 a(has) p 3014 5067 a(a) p Fo 3086 5067 a(C) p Fn 3151 5037 a(1) p Fi 3251 5067 a(density) p Fy 3543 5067 a(\026) p Fo 3535 5067 a(\032) p Fn 3578 5033 a(\003) p Fp 3577 5088 a(~) p 3578 5088 a(x;I) p Fi 3688 5067 a(.) 208 5200 y(In) p 316 5200 a(p) l(articular,) p Fy 729 5200 a(\026) p Fo 721 5200 a(\032) p Fn 764 5166 a(\003) p Fp 763 5223 a(~) p 764 5223 a(x;) p Fj(\(1\)) p Fy 911 5200 a(\() p Fo(\030) p Fy 983 5200 a(\)) p Fo 1038 5200 a(>) p Fy 1126 5200 a(0) p Fi(,) p Fo 1223 5200 a(\030) p 1286 5200 a(>) p Fy 1374 5200 a(0) p Fi 1429 5200 a(.) -42 5383 y(R) l(emark.) p Fy 382 5383 a(\(i\)) p 511 5383 a(If) p Fo 594 5383 a(x) p Fp 641 5395 a(c;I) p Fy 753 5383 a(=) p 841 5383 a(0,) p 934 5383 a(then) p 1124 5383 a(the) p 1267 5383 a(righ) n(t) p 1468 5383 a(hand) p 1676 5383 a(sides) p 1876 5383 a(in) p 1973 5383 a(\(25\)) p 2149 5383 a(are) p 2288 5383 a(0) p 2357 5383 a(b) r(ecause) p 2665 5383 a(of) p 2760 5383 a(Prop) r(osition) p 3207 5383 a(8) p 3276 5383 a(and) p 3438 5383 a(\(24\),) p 3637 5383 a(hence) 208 5482 y(the) p 351 5482 a(equations) p 722 5482 a(in) p 819 5482 a(\(25\)) p 994 5482 a(are) p 1133 5482 a(trivially) p 1447 5482 a(correct) p 1724 5482 a(for) p 1851 5482 a(if) p Fo 1927 5482 a(I) p Fk 1993 5482 a(62) p 2072 5482 a(K) p Fp 2134 5495 a(~) p 2135 5495 a(x) p Fm 2173 5503 a(c) p Fy 2222 5482 a(.) 55 5648 y(\(ii\)) p 208 5648 a(Existence) p 580 5648 a(and) p 742 5648 a(p) r(ositivit) n(y) p 1111 5648 a(of) p 1206 5648 a(densit) n(y) p 1492 5648 a(is) p 1576 5648 a(used) p 1765 5648 a(in) p 1862 5648 a(the) p 2005 5648 a(pro) r(of) p 2222 5648 a(of) p 2317 5648 a(Theorem) p 2668 5648 a(10.) p Fd 3774 5748 a(3) p Fy 1878 6120 a(11) p 90 rotate dyy eop %%Page: 12 12 12 11 bop Fy 125 83 a(W) p 203 83 a(e) p 268 83 a(mo) n(v) n(e) p 482 83 a(on) p 597 83 a(to) p 699 83 a(the) p 842 83 a(results) p 1106 83 a(on) p 1221 83 a(paths) p 1448 83 a(with) p 1637 83 a(step) p 1812 83 a(n) n(um) n(b) r(ers) p 2148 83 a(\014xed,) p 2371 83 a(instead) p 2658 83 a(of) p 2752 83 a(paths) p 2979 83 a(with) p 3168 83 a(endp) r(oin) n(ts) p 3546 83 a(\014xed.) 125 183 y(W) p 203 183 a(e) p 264 183 a(denote) p 528 183 a(the) p 668 183 a(self-a) n(v) n(oiding) p 1140 183 a(paths) p 1363 183 a(starting) p 1669 183 a(from) p 1862 183 a(origin) p Fo 2094 183 a(O) p Fy 2184 183 a(b) n(y) p Fo 2296 183 a(W) p Fj 2386 153 a(\(0\)) p Fy 2475 183 a(:) p Fo 2534 183 a(W) p Fj 2624 153 a(\(0\)) p Fy 2736 183 a(=) p Fk 2823 183 a(f) p Fo(w) p Fk 2950 183 a(2) p Fo 3028 183 a(W) p Fj 3106 195 a(0) p Fk 3167 183 a(j) p Fo 3213 183 a(w) p Fy 3274 183 a(\(0\)) p 3404 183 a(=) p Fo 3491 183 a(O) p Fk 3556 183 a(g) p Fy(.) p 3657 183 a(Also,) 0 298 y(w) n(e) p 122 298 a(de\014ne,) p 385 298 a(in) p 482 298 a(analogy) p 789 298 a(with) p 978 298 a(\(15\),) p Fo 1176 298 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a(approac) n(hes) p 2421 3955 a(to) p 2527 3955 a(asymptotic) p 2959 3955 a(b) r(eha) n(viors,) p 3359 3955 a(our) p 3512 3955 a(emphasis) 208 4055 y(here) p 385 4055 a(is) p 465 4055 a(on) p 577 4055 a(the) p 717 4055 a(precise) p 987 4055 a(mathematical) p 1505 4055 a(statemen) n(ts) p 1920 4055 a(and) p 2079 4055 a(rigorous) p 2395 4055 a(pro) r(ofs) p 2641 4055 a(that) p 2818 4055 a(\014t) p 2921 4055 a(to) p 3019 4055 a(suc) n(h) p 3204 4055 a(in) n(tuitiv) n(e) p 3530 4055 a(pictures.) 55 4219 y(\(ii\)) p 208 4219 a(As) p 332 4219 a(ma) n(y) p 513 4219 a(b) r(e) p 628 4219 a(seen) p 810 4219 a(from) p 1008 4219 a(the) p 1153 4219 a(fact) p 1318 4219 a(that) p Fo 1500 4219 a(\025) p Fy 1578 4219 a(is) p 1663 4219 a(de\014ned) p 1950 4219 a(in) p 2049 4219 a(\(FP2\)) p 2295 4219 a(as) p 2399 4219 a(the) p 2543 4219 a(largest) p 2813 4219 a(eigen) n(v) p 3035 4219 a(alue) p 3211 4219 a(of) p 3308 4219 a(the) p 3452 4219 a(di\013eren) n(tial) 208 4328 y(map) p 404 4328 a(of) p Fo 514 4307 a(~) p Fy 511 4328 a(\010) p 611 4328 a(at) p Fo 724 4328 a(~) p 725 4328 a(x) p Fp 772 4340 a(c) p Fy 846 4328 a(while) p Fo 1075 4328 a(\014) p Fp 1122 4340 a(c) p Fy 1196 4328 a(is) p 1291 4328 a(de\014ned) p 1590 4328 a(in) p 1699 4328 a(\(CS1\)) p 1951 4328 a(as) p 2065 4328 a(the) p 2220 4328 a(in) n(tersection) p 2678 4328 a(of) p 2785 4328 a(the) p 2940 4328 a(canonical) p 3317 4328 a(curv) n(e) p 3551 4328 a(and) p 3725 4328 a(the) 208 4428 y(critical) p 485 4428 a(surface,) p 789 4428 a(these) p 1003 4428 a(t) n(w) n(o) p 1160 4428 a(quan) n(tities) p 1544 4428 a(ha) n(v) n(e) p 1737 4428 a(no) p 1853 4428 a(direct) p 2089 4428 a(relations.) p 2461 4428 a(In) p 2566 4428 a(fact,) p 2754 4428 a(in) p 2851 4428 a(the) p 2995 4428 a(common) p 3328 4428 a(wisdom) p 3629 4428 a(of) p 3725 4428 a(the) 208 4527 y(renormalization) p 796 4527 a(group) p 1025 4527 a(ideas,) p 1250 4527 a(the) p 1387 4527 a(exp) r(onen) n(t) p 1737 4527 a(for) p 1858 4527 a(mean) p 2072 4527 a(square) p 2327 4527 a(displacemen) n(t) p 2816 4527 a(is) p 2893 4527 a(considered) p 3293 4527 a(to) p 3387 4527 a(b) r(e) p 3493 4527 a(univ) n(ersal,) 208 4627 y(i.e.,) p 368 4627 a(indep) r(enden) n(t) p 842 4627 a(of) p 940 4627 a(details) p 1206 4627 a(of) p 1304 4627 a(the) p 1450 4627 a(system,) p 1753 4627 a(\(in) p 1885 4627 a(fact) p 2052 4627 a(the) p 2198 4627 a(full) p 2346 4627 a(mo) r(del) p 2596 4627 a(and) p 2761 4627 a(the) p 2907 4627 a(restricted) p 3280 4627 a(mo) r(del) p 3530 4627 a(ha) n(v) n(e) p 3725 4627 a(the) 208 4727 y(same) p Fo 415 4727 a(\025) p Fy(\),) p 546 4727 a(while) p 763 4727 a(the) p 905 4727 a(connectiv) n(e) p 1311 4727 a(constan) n(t) p 1645 4727 a(dep) r(ends) p 1966 4727 a(on) p 2081 4727 a(the) p 2223 4727 a(details) p 2486 4727 a(of) p 2580 4727 a(the) p 2723 4727 a(system) p 2998 4727 a(\(the) p 3172 4727 a(full) p 3317 4727 a(mo) r(del) p 3564 4727 a(and) p 3725 4727 a(the) 208 4826 y(restricted) p 583 4826 a(mo) r(del) p 835 4826 a(ha) n(v) n(e) p 1032 4826 a(di\013eren) n(t) p 1365 4826 a(v) p 1404 4826 a(alues) p 1617 4826 a(of) p Fo 1717 4826 a(\014) p Fp 1764 4838 a(c) p Fy 1798 4826 a(\).) p 1906 4826 a(This) p 2101 4826 a(is) p 2190 4826 a(related) p 2472 4826 a(to) p 2579 4826 a(the) p 2727 4826 a(fact) p 2896 4826 a(that) p Fo 3081 4826 a(\025) p Fy 3163 4826 a(is) p 3252 4826 a(a) p 3326 4826 a(solution) p 3646 4826 a(to) p 3752 4826 a(an) 208 4926 y(algebraic) p 554 4926 a(equation) p 888 4926 a(and) p 1045 4926 a(can) p 1193 4926 a(b) r(e) p 1301 4926 a(calculated) p 1689 4926 a(explicitly) p 2044 4926 a(to) p 2141 4926 a(arbitrary) p 2490 4926 a(precision,) p 2856 4926 a(while) p Fo 3068 4926 a(\014) p Fp 3115 4938 a(c) p Fy 3172 4926 a(has) p 3315 4926 a(no) p 3426 4926 a(suc) n(h) p 3609 4926 a(simple) 208 5026 y(algebraically) p 690 5026 a(closed) p 935 5026 a(form) n(ula) p 1239 5026 a(and) p 1401 5026 a(is) p 1484 5026 a(di\016cult) p 1789 5026 a(to) p 1890 5026 a(calculate) p 2236 5026 a(explicitly) p 2561 5026 a(.) 32 5190 y(\(iii\)) p 208 5190 a(It) p 292 5190 a(ma) n(y) p 467 5190 a(b) r(e) p 575 5190 a(w) n(orth) n(while) p 994 5190 a(to) p 1090 5190 a(note) p 1269 5190 a(that) p 1444 5190 a(the) p 1582 5190 a(self-a) n(v) n(oiding) p 2052 5190 a(w) n(alks) p 2273 5190 a(on) p 2383 5190 a(h) n(yp) r(ercubic) p 2800 5190 a(lattice) p Fq 3049 5190 a(Z) p Fp 3104 5160 a(n) p Fy 3172 5190 a(in) p 3263 5190 a(high) p 3442 5190 a(dimensions) 208 5290 y(\() p Fo(n) p 313 5290 a(>) p Fy 400 5290 a(4\)) p 499 5290 a(are) p 636 5290 a(pro) n(v) n(ed) p 903 5290 a(to) p 1002 5290 a(b) r(e) p 1113 5290 a(in) p 1208 5290 a(the) p 1348 5290 a(same) p 1554 5290 a(univ) n(ersalit) n(y) p 1999 5290 a(class) p 2192 5290 a(as) p 2291 5290 a(the) p 2432 5290 a(random) p 2734 5290 a(w) n(alks) p 2958 5290 a(|) p 3066 5290 a(i.e.,) p 3221 5290 a(they) p 3406 5290 a(ha) n(v) n(e) p 3595 5290 a(similar) 208 5389 y(asymptotic) p 635 5389 a(b) r(eha) n(viors) p 1008 5389 a(|) p 1120 5389 a(b) n(y) p 1236 5389 a(the) p 1380 5389 a(lace) p 1547 5389 a(expansion) p 1933 5389 a(metho) r(ds) p 2269 5389 a([2) o(,) p 2385 5389 a(3) o(].) p 2512 5389 a(In) p 2616 5389 a(a) p 2686 5389 a(sense,) p 2923 5389 a(the) p 3067 5389 a(self-a) n(v) n(oiding) p 3544 5389 a(w) n(alks) p 3771 5389 a(in) 208 5489 y(high) p 392 5489 a(dimensional) p 849 5489 a(spaces) p 1104 5489 a(ma) n(y) p 1284 5489 a(b) r(e) p 1397 5489 a(seen) p 1577 5489 a(as) p 1679 5489 a(\(non-trivial\)) p 2152 5489 a(p) r(erturbations) p 2672 5489 a(to) p 2773 5489 a(the) p 2916 5489 a(random) p 3221 5489 a(w) n(alks.) 208 5621 y(Ho) n(w) n(ev) n(er,) p 561 5621 a(it) p 638 5621 a(is) p 716 5621 a(also) p 877 5621 a(b) r(eliev) n(ed) p 1192 5621 a(\(and) p 1380 5621 a(trivially) p 1689 5621 a(true) p 1859 5621 a(for) p Fo 1980 5621 a(n) p Fy 2053 5621 a(=) p 2141 5621 a(1!\),) p 2284 5621 a(that) p 2458 5621 a(for) p Fo 2580 5621 a(n) p 2652 5621 a(<) p Fy 2740 5621 a(4) p 2804 5621 a(the) p 2941 5621 a(asymptotic) p 3363 5621 a(b) r(eha) n(viors) p 3729 5621 a(are) 208 5720 y(v) n(ery) p 391 5720 a(di\013eren) n(t,) p 743 5720 a(hence) p 974 5720 a(the) p 1118 5720 a(problem) p 1442 5720 a(remains) p 1753 5720 a(in) p 1850 5720 a(lo) n(w) n(er) p 2068 5720 a(dimensional) p 2526 5720 a(spaces.) p 2816 5720 a(W) p 2894 5720 a(e) p 2960 5720 a(note) p 3145 5720 a(that) p Fo 3326 5720 a(d) p Fy(SG) p 3509 5720 a(are,) p 3671 5720 a(from) 208 5820 y(the) p 356 5820 a(renormalization) p 956 5820 a(group) p 1197 5820 a(p) r(oin) n(t) p 1419 5820 a(of) p 1519 5820 a(view,) p 1740 5820 a(spaces) p 1999 5820 a(`b) r(et) n(w) n(een) p Fq 2348 5820 a(Z) p Fj 2403 5790 a(1) p Fy 2474 5820 a(and) p Fq 2640 5820 a(Z) p Fj 2695 5790 a(2) p Fy 2747 5820 a('.) p 2845 5820 a(W) p 2923 5820 a(e) p 2993 5820 a(also) p 3165 5820 a(p) r(oin) n(t) p 3387 5820 a(out) p 3540 5820 a(that) p 3725 5820 a(the) 1878 6120 y(12) p 90 rotate dyy eop %%Page: 13 13 13 12 bop Fy 208 83 a(lace) p 372 83 a(expansion) p 757 83 a(metho) r(d) p 1058 83 a(hea) n(vily) p 1340 83 a(uses) p 1515 83 a(translational) p 1999 83 a(in) n(v) p 2105 83 a(ariance) p 2390 83 a(of) p Fq 2483 83 a(Z) p Fp 2538 53 a(n) p Fy 2584 83 a(,) p 2633 83 a(while) p 2849 83 a(fractals) p 3141 83 a(lac) n(k) p 3311 83 a(the) p 3452 83 a(in) n(v) p 3558 83 a(ariance.) 208 183 y(In) p 307 183 a(fact,) p 491 183 a(the) p 630 183 a(random) p 930 183 a(w) n(alks) p 1153 183 a(and) p 1310 183 a(the) p 1449 183 a(self-a) n(v) n(oiding) p 1921 183 a(w) n(alks) p 2143 183 a(are) p 2278 183 a(kno) n(wn) p 2536 183 a(to) p 2634 183 a(b) r(e) p 2743 183 a(in) p 2836 183 a(di\013eren) n(t) p 3159 183 a(univ) n(ersalit) n(y) p 3603 183 a(classes) 208 282 y(on) p 328 282 a(2SG) p 514 282 a(and) p 681 282 a(3SG;) p 889 282 a(the) p 1038 282 a(v) p 1077 282 a(alues) p 1290 282 a(of) p 1390 282 a(exp) r(onen) n(ts) p 1785 282 a(for) p 1918 282 a(mean) p 2144 282 a(square) p 2411 282 a(displacemen) n(t) p 2912 282 a(of) p 3012 282 a(the) p 3160 282 a(self-a) n(v) n(oiding) p 3641 282 a(w) n(alks) 208 382 y(and) p 369 382 a(the) p 512 382 a(random) p 817 382 a(w) n(alks) p 1043 382 a(ha) n(v) n(e) p 1234 382 a(no) p 1350 382 a(explicit) p 1643 382 a(simple) p 1901 382 a(relations) p 2239 382 a([8) o(].) p Fd 3774 482 a(3) p Fs 0 922 a(4) p 202 922 a(Pro) t(ofs.) p Fr 0 1120 a(4.1) p 255 1120 a(Phase) p 581 1120 a(structure.) p Fy 0 1273 a(Here) p 196 1273 a(w) n(e) p 318 1273 a(will) p 475 1273 a(pro) n(v) n(e) p 699 1273 a(Theorem) p 1050 1273 a(7.) 125 1373 y(W) p 203 1373 a(e) p 268 1373 a(\014rst) p 439 1373 a(note) p 623 1373 a(some) p 831 1373 a(simple) p 1090 1373 a(consequences) p 1591 1373 a(of) p 1686 1373 a(the) p 1829 1373 a(de\014nitions.) p Fz 0 1532 a(Prop) s(osition) p 516 1532 a(12) p Fi 653 1532 a(L) l(et) p Fo 795 1532 a(~) p 796 1532 a(x) p Fp 843 1544 a(c) p Fi 907 1532 a(b) l(e) p 1009 1532 a(a) p 1081 1532 a(self-avoiding) p 1561 1532 a(\014xe) l(d) p 1752 1532 a(p) l(oint) p 1962 1532 a(and) p Fo 2123 1532 a(\025) p Fi 2202 1532 a(b) l(e) p 2303 1532 a(as) p 2410 1532 a(in) p 2511 1532 a(\(FP2\).) p 2796 1532 a(Then) p Fy 3012 1532 a(2) p Fo 3077 1532 a(<) p 3164 1532 a(\025) p 3236 1532 a(<) p 3324 1532 a(d) p Fy 3385 1532 a(+) p 3468 1532 a(1) p Fi(.) -42 1706 y(Pr) l(o) l(of.) p Fy 219 1706 a(Prop) r(osition) p 666 1706 a(4) p 735 1706 a(and) p 896 1706 a(\(14\)) p 1071 1706 a(imply) 811 1897 y(2\010) p Fp 913 1909 a(I) p Fy 951 1897 a(\() p Fo 982 1897 a(~) p 983 1897 a(x) p Fy 1031 1897 a(\)) p Fo 1086 1897 a(<) p Fg 1193 1818 a(X) p Fp 1174 1996 a(J) p Fn 1216 1996 a(2I) p Fm 1299 2005 a(d) p Fk 1347 1897 a(J) p Fp 1403 1909 a(I) p 1437 1909 a(J) p Fy 1483 1897 a(\() p Fo 1514 1897 a(~) p 1515 1897 a(x) p Fy 1563 1897 a(\)) p Fo(x) p Fp 1642 1909 a(J) p Fo 1712 1897 a(<) p Fy 1800 1897 a(\() p Fo(d) p Fy 1894 1897 a(+) p 1977 1897 a(1\)\010) p Fp 2111 1909 a(I) p Fy 2149 1897 a(\() p Fo 2180 1897 a(~) p 2181 1897 a(x) p Fy(\)) p Fo(;) p 2352 1897 a(~) p 2353 1897 a(x) p Fk 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3953 a(\() p Fo 2278 3932 a(~) 2261 3953 y(X) p Fj 2330 3965 a(1) p Fy 2367 3953 a(\() p Fo 2398 3953 a(~) p 2399 3953 a(x) p Fy 2447 3953 a(\)\)) p Fk 2530 3953 a(\001) p 2571 3953 a(J) p Fy 2643 3953 a(\() p Fo 2674 3953 a(~) p 2675 3953 a(x) p Fy 2723 3953 a(\)) p Fo(;) p 2847 3953 a(n) p Fk 2920 3953 a(2) p Fq 2998 3953 a(Z) p Fj 3053 3965 a(+) p Fo 3109 3953 a(;) p Fy 3692 3953 a(\(29\)) 0 4127 y(whic) n(h) p 238 4127 a(further) p 517 4127 a(implies) p 799 4127 a(with) p 988 4127 a(\(27\),) p Fk 1421 4227 a(J) p Fp 1477 4239 a(n) p Fy 1523 4227 a(\() p Fo 1554 4227 a(~) p 1555 4227 a(x) p Fp 1602 4239 a(c) p Fy 1636 4227 a(\)) p 1692 4227 a(=) p Fk 1779 4227 a(J) p Fy 1851 4227 a(\() p Fo 1882 4227 a(~) p 1883 4227 a(x) p Fp 1930 4239 a(c) p Fy 1964 4227 a(\)) p Fp 1996 4193 a(n) p Fo 2042 4227 a(;) p 2134 4227 a(n) p Fk 2207 4227 a(2) p Fq 2285 4227 a(Z) p Fj 2340 4239 a(+) p Fo 2396 4227 a(:) p Fy 3692 4227 a(\(30\)) p Fz 0 4401 a(Prop) s(osition) p 516 4401 a(13) p Fy 653 4401 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297 5621 a(\(i\)) p Fo 425 5621 a(~) p 426 5621 a(x) p Fp 473 5633 a(c) p Fk 535 5621 a(6) p Fy(=) p Fo 621 5603 a(~) p Fy 627 5621 a(0) o(,) p 722 5621 a(b) n(y) p 840 5621 a(\(FP4\).) p 1125 5621 a(Hence) p 1374 5621 a(there) p 1589 5621 a(exists) p Fo 1821 5621 a(I) p Fk 1892 5621 a(2) p 1974 5621 a(I) p Fp 2019 5633 a(d) p Fy 2089 5621 a(suc) n(h) p 2279 5621 a(that) p Fo 2461 5621 a(x) p Fp 2508 5633 a(c;I) p Fo 2623 5621 a(>) p Fy 2715 5621 a(0) p 2771 5621 a(.) p 2839 5621 a(With) p 3056 5621 a(\(FP4\),) p 3326 5621 a(w) n(e) p 3451 5621 a(further) p 3733 5621 a(see) 208 5720 y(that) p 387 5720 a(there) p 598 5720 a(exists) p 827 5720 a(1) p Fl 891 5720 a(5) p Fo 979 5720 a(i) p Fl 1031 5720 a(5) p Fo 1118 5720 a(d) p Fy 1188 5720 a(suc) n(h) p 1375 5720 a(that) p Fo 1554 5720 a(x) p Fp 1601 5735 a(c;) p Fj(\() p Fp(i) p Fj(\)) p Fo 1753 5720 a(>) p Fy 1841 5720 a(0) p 1896 5720 a(.) p 1956 5720 a(Prop) r(osition) p 2402 5720 a(13) p 2512 5720 a(then) p 2701 5720 a(implies) p 2982 5720 a(that) p Fo 3161 5720 a(x) p Fp 3208 5735 a(c;) p Fj(\(1\)) p Fy 3370 5720 a(=) p 3457 5720 a(\010) p Fj 3517 5735 a(\(1\)) p Fy 3607 5720 a(\() p Fo 3638 5720 a(~) p 3639 5720 a(x) p Fp 3686 5732 a(c) p Fy 3720 5720 a(\)) p Fl 3775 5720 a(=) p Fo 208 5820 a(x) p Fj 255 5790 a(2) p Fp 255 5847 a(c;) p Fj(\() p Fp(i) p Fj(\)) p Fo 407 5820 a(>) p Fy 495 5820 a(0) p 550 5820 a(.) 1878 6120 y(13) p 90 rotate dyy eop %%Page: 14 14 14 13 bop Fy 55 110 a(\(ii\)) p 208 110 a(\(FP3\)) p 444 110 a(and) p Fo 597 110 a(I) p Fk 663 110 a(2) p 741 110 a(K) p Fp 803 123 a(~) p 804 123 a(x) p Fm 842 131 a(c) p Fy 896 110 a(imply) p Fk 1121 110 a(J) p Fy 1193 110 a(\() p Fo 1224 110 a(~) p 1225 110 a(x) p Fp 1272 122 a(c) p Fy 1306 110 a(\)) p Fp 1338 125 a(I) p 1372 125 a(;) p Fj(\(1\)) p Fo 1504 110 a(>) p Fy 1592 110 a(0) p 1647 110 a(.) p 1704 110 a(With) p 1910 110 a(\(FP2\),) p 2171 110 a(this) p 2325 110 a(further) p 2596 110 a(implies) p Fo 2869 110 a(v) p Fp 2909 122 a(R;I) p Fl 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a(d) p Fy 3673 4028 a(,) p 3729 4028 a(are) 208 4173 y(p) r(olynomials) p 665 4173 a(with) p 850 4173 a(p) r(ositiv) n(e) p 1153 4173 a(co) r(e\016cien) n (ts) p 1569 4173 a(and) p 1726 4173 a(eac) n(h) p 1909 4173 a(term) p 2103 4173 a(of) p 2193 4173 a(degree) p 2448 4173 a(no) p 2559 4173 a(less) p 2708 4173 a(than) p 2897 4173 a(2) p 2962 4173 a(\(Prop) r(osition) p 3437 4173 a(4\),) p 3558 4173 a(w) n(e) p 3676 4173 a(ha) n(v) n(e) 208 4272 y(max) p Fp 209 4326 a(I) p Fn 243 4326 a(2I) p Fm 326 4335 a(d) p Fy 376 4272 a(\010) p Fp 436 4284 a(I) p Fy 474 4272 a(\() p Fo 505 4272 a(~) p 506 4272 a(x) p Fn 554 4238 a(0) p Fy 577 4272 a(\)) p Fl 633 4272 a(5) p Fo 722 4272 a(r) p Fj 761 4238 a(2) p Fy 813 4272 a(max) p Fp 815 4326 a(I) p Fn 849 4326 a(2I) p Fm 932 4335 a(d) p Fy 981 4272 a(\010) p Fp 1041 4284 a(I) p Fy 1079 4272 a(\() p Fo 1110 4272 a(~) p 1111 4272 a(x) p Fy 1159 4272 a(\)) p 1220 4272 a(if) p 1296 4272 a(0) p Fl 1362 4272 a(5) p Fo 1451 4272 a(r) p Fl 1514 4272 a(5) p Fy 1603 4272 a(1) p 1658 4272 a(.) p 1720 4272 a(By) p 1851 4272 a(induction,) p 2244 4272 a(max) p Fp 2246 4326 a(I) p Fn 2280 4326 a(2I) p Fm 2363 4335 a(d) p Fo 2412 4272 a(X) p Fp 2481 4284 a(n;I) p Fy 2580 4272 a(\() p Fo 2611 4272 a(~) p 2612 4272 a(x) p Fn 2660 4238 a(0) p Fy 2683 4272 a(\)) p Fl 2739 4272 a(5) p Fo 2828 4272 a(r) p Fj 2867 4238 a(2) p Fm 2900 4213 a(n) p Fy 2960 4272 a(max) p Fp 2961 4326 a(I) p Fn 2995 4326 a(2I) p Fm 3078 4335 a(d) p Fo 3128 4272 a(X) p Fp 3197 4284 a(n;I) p Fy 3296 4272 a(\() p Fo 3327 4272 a(~) p 3328 4272 a(x) p Fy(\),) p Fo 3459 4272 a(n) p Fk 3533 4272 a(2) p Fq 3612 4272 a(Z) p Fj 3667 4284 a(+) p Fy 3723 4272 a(.) p 3785 4272 a(If) p Fo 206 4428 a(~) p 208 4428 a(x) p Fk 278 4428 a(2) p Fo 356 4428 a(D) p Fk 442 4428 a(n) p 498 4428 a(f) p Fo 534 4410 a(~) p Fy 540 4428 a(0) p Fk 581 4428 a(g) p Fy 648 4428 a(and) p Fo 806 4428 a(~) p 807 4428 a(x) p Fn 855 4398 a(0) p Fk 901 4428 a(2) p Fy 979 4428 a(\004) p Fp 1034 4440 a(d) p Fy 1087 4428 a(,) p 1136 4428 a(and) p 1296 4428 a(if) p 1370 4428 a(for) p 1495 4428 a(eac) n(h) p Fo 1679 4428 a(I) p Fk 1745 4428 a(2) p 1824 4428 a(I) p Fp 1869 4440 a(d) p Fy 1933 4428 a(either) p Fo 2167 4428 a(x) p Fn 2214 4398 a(0) p Fp 2214 4451 a(I) p Fo 2275 4428 a(<) p 2363 4428 a(x) p Fp 2410 4440 a(I) p Fy 2474 4428 a(or) p Fo 2574 4428 a(x) p Fn 2621 4398 a(0) p Fp 2621 4451 a(I) p Fy 2682 4428 a(=) p Fo 2770 4428 a(x) p Fp 2817 4440 a(I) p Fy 2879 4428 a(=) p 2966 4428 a(0) p 3033 4428 a(holds,) p 3272 4428 a(then) p 3459 4428 a(0) p Fl 3523 4428 a(5) p Fo 3611 4428 a(r) p 3674 4428 a(<) p Fy 3762 4428 a(1) p 3817 4428 a(.) 208 4528 y(Since) p Fk 424 4528 a(f) p Fo(X) p Fp 535 4540 a(n;I) p Fy 633 4528 a(\() p Fo 664 4528 a(~) p 665 4528 a(x) p Fy 713 4528 a(\)) p Fk(g) p Fy 815 4528 a(is) p 898 4528 a(b) r(ounded,) p 1260 4528 a(w) n(e) p 1382 4528 a(ha) n(v) n(e) p Fo 1573 4528 a(~) p 1574 4528 a(x) p Fn 1621 4498 a(0) p Fk 1668 4528 a(2) p Fy 1765 4507 a(~) p Fo 1746 4528 a(D) p Fy 1817 4528 a(.) 32 4714 y(\(iii\)) p 208 4714 a(F) p 255 4714 a(or) p Fo 358 4714 a(\017) p 416 4714 a(>) p Fy 506 4714 a(0) p 576 4714 a(let) p Fo 697 4714 a(D) p Fp 766 4726 a(\017) p Fy 823 4714 a(=) p Fk 912 4714 a(f) p Fo 953 4714 a(~) p 954 4714 a(x) p Fk 1026 4714 a(2) p Fy 1106 4714 a(\004) p Fp 1161 4726 a(d) p Fk 1225 4714 a(j) p Fg 1289 4635 a(X) p Fp 1273 4813 a(I) p Fn 1307 4813 a(2I) p Fm 1390 4822 a(d) p Fo 1438 4714 a(x) p Fp 1485 4726 a(I) p Fo 1549 4714 a(<) p 1638 4714 a(\017) p Fk(g) p Fy(.) p 1777 4714 a(Since) p 1994 4714 a(\010) p Fp 2054 4726 a(I) p Fy 2121 4714 a(is) p 2206 4714 a(a) p 2276 4714 a(p) r(olynomial) p 2706 4714 a(with) p 2897 4714 a(eac) n(h) p 3084 4714 a(term) p 3284 4714 a(of) p 3380 4714 a(order) p 3598 4714 a(no) p 3715 4714 a(less) 208 4900 y(than) p 401 4900 a(2) p 470 4900 a(\(Prop) r(osition) p 950 4900 a(4\),) p 1074 4900 a(there) p 1287 4900 a(exists) p 1516 4900 a(a) p 1585 4900 a(constan) n(t) p Fo 1920 4900 a(M) p 2033 4900 a(>) p Fy 2120 4900 a(0) p 2189 4900 a(suc) n(h) p 2377 4900 a(that) p Fg 1205 5021 a(X) p Fp 1190 5199 a(I) p Fn 1224 5199 a(2I) p Fm 1307 5208 a(d) p Fy 1354 5100 a(\010) p Fp 1414 5112 a(I) p Fy 1453 5100 a(\() p Fo 1484 5100 a(~) p 1485 5100 a(x) p Fy(\)) p Fl 1588 5100 a(5) p Fo 1675 5100 a(M) p 1765 5100 a(\017) p Fg 1828 5021 a(X) p Fp 1813 5199 a(I) p Fn 1847 5199 a(2I) p Fm 1930 5208 a(d) p Fo 1977 5100 a(x) p Fp 2024 5112 a(I) p Fo 2077 5100 a(;) p 2168 5100 a(~) p 2169 5100 a(x) p Fk 2239 5100 a(2) p Fo 2318 5100 a(D) p Fp 2387 5112 a(\017) p Fo 2418 5100 a(;) p Fy 2483 5100 a(0) p Fo 2548 5100 a(<) p 2635 5100 a(\017) p 2692 5100 a(<) p Fy 2780 5100 a(1) p Fo 2836 5100 a(:) p Fy 208 5411 a(Let) p Fo 356 5411 a(\017) p Fy 413 5411 a(=) p Fj 545 5378 a(1) p 511 5392 103 4 v 511 5439 a(2) p Fp(M) p Fk 642 5411 a(^) p Fj 725 5378 a(1) p 725 5392 34 4 v 725 5439 a(2) p Fy 782 5411 a(.) p 842 5411 a(Then) p 1059 5411 a(b) n(y) p 1174 5411 a(induction) p 1543 5411 a(in) p Fo 1640 5411 a(n) p Fy 1718 5411 a(with) p Fg 1431 5539 a(X) p Fp 1416 5717 a(I) p Fn 1450 5717 a(2I) p Fm 1533 5726 a(d) p Fo 1581 5618 a(X) p Fp 1650 5630 a(n) p Fj(+1) p Fp(;I) p Fy 1832 5618 a(\() p Fo 1863 5618 a(~) p 1864 5618 a(x) p Fy 1912 5618 a(\)) p 1968 5618 a(=) p Fg 2071 5539 a(X) p Fp 2055 5717 a(I) p Fn 2089 5717 a(2I) p Fm 2172 5726 a(d) p Fy 2220 5618 a(\010) p Fp 2280 5630 a(I) p Fy 2318 5618 a(\() p Fo 2368 5597 a(~) 2350 5618 y(X) p Fp 2419 5630 a(n) p Fy 2464 5618 a(\() p Fo 2495 5618 a(~) p 2496 5618 a(x) p Fy 2544 5618 a(\)\)) p Fo(;) p Fy 1878 6120 a(14) p 90 rotate dyy eop %%Page: 15 15 15 14 bop Fy 208 83 a(w) n(e) p 330 83 a(ha) n(v) n(e,) p 544 83 a(for) p Fo 670 83 a(~) p 671 83 a(x) p Fk 742 83 a(2) p Fo 820 83 a(D) p Fp 889 95 a(\017) p Fy 935 83 a(,) p Fo 1003 62 a(~) 985 83 y(X) p Fp 1054 95 a(n) p Fy 1099 83 a(\() p Fo 1130 83 a(~) p 1131 83 a(x) p Fy 1179 83 a(\)) p Fk 1234 83 a(2) p Fo 1313 83 a(D) p Fp 1382 95 a(\017) p Fy 1413 83 a(,) p Fo 1464 83 a(n) p Fk 1537 83 a(2) p Fq 1615 83 a(Z) p Fj 1670 95 a(+) p Fy 1726 83 a(.) p 1786 83 a(Then) p 2003 83 a(w) n(e) p 2125 83 a(further) p 2404 83 a(ha) n(v) n(e) p Fg 925 221 a(X) p Fp 909 399 a(I) p Fn 943 399 a(2I) p Fm 1026 408 a(d) p Fo 1074 300 a(X) p Fp 1143 312 a(n) p Fj(+1) p Fp(;I) p Fy 1326 300 a(\() p Fo 1357 300 a(~) p 1358 300 a(x) p Fy 1406 300 a(\)) p Fl 1461 300 a(5) p Fo 1549 300 a(M) p 1639 300 a(\017) p Fg 1701 221 a(X) p Fp 1687 399 a(I) p Fn 1721 399 a(2I) p Fm 1804 408 a(d) p Fo 1851 300 a(X) p Fp 1920 312 a(n;I) p Fy 2018 300 a(\() p Fo 2049 300 a(~) p 2050 300 a(x) p Fy 2098 300 a(\)) p Fl 2154 300 a(5) p Fy 2251 243 a(1) p 2251 281 42 4 v 2251 357 a(2) p Fg 2332 221 a(X) p Fp 2316 399 a(I) p Fn 2350 399 a(2I) p Fm 2433 408 a(d) p Fo 2481 300 a(X) p Fp 2550 312 a(n;I) p Fy 2649 300 a(\() p Fo 2680 300 a(~) p 2681 300 a(x) p Fy 2729 300 a(\)) p Fo(;) p 2853 300 a(n) p Fk 2926 300 a(2) p Fq 3005 300 a(Z) p Fj 3060 312 a(+) p Fo 3115 300 a(;) p Fy 208 579 a(whic) n(h) p 444 579 a(implies) p Fo 723 579 a(~) p 724 579 a(x) p Fk 795 579 a(2) p Fy 892 558 a(~) p Fo 873 579 a(D) p Fy 944 579 a(.) p 1003 579 a(Hence) p Fo 1249 579 a(D) p Fp 1318 591 a(\017) p Fk 1372 579 a(\032) p Fy 1479 558 a(~) p Fo 1460 579 a(D) p Fy 1557 579 a(for) p Fo 1683 579 a(\017) p Fy 1740 579 a(=) p Fj 1872 546 a(1) p 1837 560 103 4 v 1837 607 a(2) p Fp(M) p Fk 1966 579 a(^) p Fj 2046 546 a(1) p 2046 560 34 4 v 2046 607 a(2) p Fy 2103 579 a(.) p 2163 579 a(On) p 2300 579 a(the) p 2441 579 a(other) p 2657 579 a(hand,) p 2886 579 a(the) p 3028 579 a(de\014nition) p 3395 579 a(of) 3508 558 y(~) p Fo 3489 579 a(D) p Fy 3586 579 a(implies) 208 696 y(that) p 384 696 a(if) p Fo 455 696 a(~) p 456 696 a(x) p Fk 526 696 a(2) p Fy 624 675 a(~) p Fo 605 696 a(D) p Fy 699 696 a(then) p Fo 902 675 a(~) 885 696 y(X) p Fp 954 708 a(n) p Fy 999 696 a(\() p Fo 1030 696 a(~) p 1031 696 a(x) p Fy(\)) p Fk 1134 696 a(2) p Fo 1212 696 a(D) p Fp 1281 708 a(\017) p Fy 1336 696 a(for) p 1459 696 a(su\016cien) n (tly) p 1871 696 a(large) p Fo 2070 696 a(n) p Fy(.) p Fo 2196 675 a(~) 2179 696 y(X) p Fp 2248 708 a(n) p Fy 2317 696 a(is) p 2396 696 a(a) p 2461 696 a(con) n(tin) n(uous) p 2873 696 a(function) p 3194 696 a(and) p Fo 3352 696 a(D) p Fp 3421 708 a(\017) p Fy 3476 696 a(is) p 3556 696 a(an) p 3667 696 a(op) r(en) 208 805 y(set,) p 365 805 a(hence) p 600 805 a(there) p 817 805 a(exists) p 1050 805 a(an) p 1170 805 a(op) r(en) p 1375 805 a(neigh) n(b) r(orho) r (o) r(d) p Fo 1900 805 a(U) p Fy 1998 805 a(of) p Fo 2095 805 a(~) p 2096 805 a(x) p Fy 2176 805 a(suc) n(h) p 2367 805 a(that) p Fo 2569 784 a(~) 2551 805 y(X) p Fp 2620 817 a(n) p Fy 2665 805 a(\() p Fo(U) p Fy 2763 805 a(\)) p Fk 2826 805 a(\032) p Fo 2920 805 a(D) p Fp 2989 817 a(\017) p Fy 3021 805 a(.) p 3093 805 a(Hence) p Fo 3344 805 a(U) p Fk 3440 805 a(\032) p Fy 3554 784 a(~) p Fo 3535 805 a(D) p Fy 3606 805 a(.) p 3678 805 a(This) 208 905 y(implies) p 489 905 a(that) 689 884 y(~) p Fo 669 905 a(D) p Fy 768 905 a(is) p 851 905 a(an) p 967 905 a(op) r(en) p 1167 905 a(subset) p 1422 905 a(of) p Fo 1517 905 a(D) p Fy 1588 905 a(.) p 1648 905 a(Since) p Fo 1864 905 a(D) p Fp 1935 875 a(o) p Fy 2000 905 a(is) p 2083 905 a(the) p 2226 905 a(largest) p 2495 905 a(op) r(en) p 2695 905 a(set) p 2825 905 a(of) p Fo 2919 905 a(D) p Fy 2990 905 a(,) p 3041 905 a(this) p 3203 905 a(implies) 3504 884 y(~) p Fo 3485 905 a(D) p Fk 3579 905 a(\032) p Fo 3667 905 a(D) p Fp 3738 875 a(o) p Fy 3775 905 a(.) 208 1037 y(T) p 261 1037 a(o) p 330 1037 a(pro) n(v) n(e) 574 1016 y(~) p Fo 555 1037 a(D) p Fk 650 1037 a(\033) p Fo 738 1037 a(D) p Fp 809 1006 a(o) p Fy 846 1037 a(,) p 897 1037 a(assume) p Fo 1184 1037 a(~) p 1185 1037 a(x) p Fn 1232 1006 a(0) p Fk 1280 1037 a(2) p Fo 1359 1037 a(D) p Fp 1430 1006 a(o) p Fy 1467 1037 a(.) p 1528 1037 a(Since) p Fo 1746 1037 a(D) p Fp 1817 1006 a(o) p Fy 1882 1037 a(is) p 1966 1037 a(an) p 2081 1037 a(op) r(en) p 2283 1037 a(set,) p 2436 1037 a(there) p 2649 1037 a(exists) p Fo 2877 1037 a(~) p 2879 1037 a(x) p Fk 2950 1037 a(2) p Fo 3029 1037 a(D) p Fp 3100 1006 a(o) p Fy 3165 1037 a(suc) n(h) p 3353 1037 a(that) p Fo 3533 1037 a(x) p Fn 3580 1006 a(0) p Fp 3580 1059 a(I) p Fo 3643 1037 a(<) p 3731 1037 a(x) p Fp 3778 1049 a(I) p Fy 3817 1037 a(,) p Fo 208 1136 a(I) p Fk 274 1136 a(2) p 352 1136 a(I) p Fp 397 1148 a(d) p Fy 450 1136 a(.) p 510 1136 a(Then,) p 750 1136 a(the) p 893 1136 a(ab) r(o) n(v) n(e) p 1128 1136 a(claim) p 1349 1136 a(imples) p Fo 1607 1136 a(~) p 1608 1136 a(x) p Fn 1655 1106 a(0) p Fk 1702 1136 a(2) p Fy 1800 1115 a(~) p Fo 1780 1136 a(D) p Fy 1851 1136 a(.) 35 1300 y(\(iv\)) p 208 1300 a(By) p 340 1300 a(\(33\),) p 542 1300 a(\(32\),) p 743 1300 a(and) p 907 1300 a(\(34\),) p 1108 1300 a(w) n(e) p 1233 1300 a(see) p 1369 1300 a(that) p Fo 1552 1300 a(D) p Fy 1623 1300 a(,) p Fo 1676 1300 a(D) p Fp 1747 1270 a(o) p Fy 1784 1300 a(,) p 1838 1300 a(and) p Fo 2002 1300 a(D) p Fp 2073 1270 a(c) p Fy 2137 1300 a(are) p 2278 1300 a(in) n(v) p 2384 1300 a(arian) n(t) p 2629 1300 a(sets) p 2794 1300 a(of) p Fo 2893 1279 a(~) p Fy 2891 1300 a(\010) o(.) p 3018 1300 a(Since) p Fo 3237 1300 a(D) p Fy 3335 1300 a(=) p 3427 1234 72 4 v Fo 3427 1300 a(D) p Fy 3498 1300 a(,) p Fo 3551 1300 a(@) p 3600 1300 a(D) p Fy 3701 1300 a(also) 208 1400 y(is) p 297 1400 a(in) n(v) p 403 1400 a(arian) n(t.) p 695 1400 a(Ob) n(viously) p 1052 1400 a(,) p Fo 1107 1400 a(~) p 1109 1400 a(x) p Fy 1189 1400 a(with) p 1384 1400 a(small) p 1607 1400 a(enough) p 1899 1400 a(comp) r(onen) n(ts) p 2362 1400 a(is) p 2451 1400 a(in) p Fo 2554 1400 a(D) p Fp 2625 1370 a(o) p Fy 2695 1400 a(and) p 2863 1400 a(that) p 3048 1400 a(with) p 3243 1400 a(su\016cien) n(tly) p 3664 1400 a(large) 208 1499 y(comp) r(onen) n(ts) p 665 1499 a(is) p 748 1499 a(in) p Fo 845 1499 a(D) p Fp 916 1469 a(c) p Fy 950 1499 a(,) p 1001 1499 a(hence) p 1231 1499 a(the) p 1374 1499 a(b) r(oundary) p Fo 1748 1499 a(@) p 1797 1499 a(D) p Fy 1895 1499 a(is) p 1979 1499 a(also) p 2145 1499 a(non-empt) n(y) p 2526 1499 a(.) p Fd 3778 1599 a(2) p Fi -42 1950 a(Pr) l(o) l(of) p 181 1950 a(of) p 279 1950 a(The) l(or) l(em) p 624 1950 a(7.) p Fy 733 1950 a(The) p 904 1950 a(case) p Fo 1080 1950 a(\014) p Fy 1154 1950 a(=) p Fo 1242 1950 a(\014) p Fp 1289 1962 a(c) p Fy 1350 1950 a(holds) p 1567 1950 a(b) n(y) p 1683 1950 a(the) p 1826 1950 a(de\014nition) p 2195 1950 a(\(CS1\).) 125 2059 y(By) p 258 2059 a(the) p 403 2059 a(de\014nitions) p 808 2059 a(\(CS1\)) p 1050 2059 a(and) p 1215 2059 a(\(FP4\),) p 1515 2059 a(lim) p Fp 1486 2109 a(n) p Fn(!1) p Fo 1690 2038 a(~) 1673 2059 y(X) p Fp 1742 2071 a(n) p Fy 1787 2059 a(\() p Fo 1818 2059 a(~) p 1819 2059 a(x) p Fp 1866 2071 a(can) p Fy 1978 2059 a(\() p Fo(\014) p Fp 2057 2071 a(c) p Fy 2091 2059 a(\)\)) p 2183 2059 a(=) p Fo 2274 2059 a(~) p 2275 2059 a(x) p Fp 2322 2071 a(c) p Fk 2384 2059 a(6) p Fy(=) p Fo 2470 2042 a(~) p Fy 2476 2059 a(0) p 2532 2059 a(,) p 2586 2059 a(whic) n(h,) p 2850 2059 a(with) p 3042 2059 a(\(31\)) p 3220 2059 a(and) p 3384 2059 a(\(33\),) p 3586 2059 a(implies) p Fo -1 2202 a(~) p 0 2202 a(x) p Fp 47 2214 a(can) p Fy 158 2202 a(\() p Fo(\014) p Fp 237 2214 a(c) p Fy 271 2202 a(\)) p Fk 340 2202 a(2) p Fo 432 2202 a(@) p 481 2202 a(D) p Fy 552 2202 a(.) p 636 2202 a(Monotonicit) n(y) p 1153 2202 a(prop) r(ert) n(y) p 1501 2202 a(in) p 1605 2202 a(Theorem) p 1964 2202 a(15) p 2083 2202 a(then) p 2280 2202 a(implies) p Fo 2569 2202 a(~) p 2570 2202 a(x) p Fp 2617 2214 a(can) p Fy 2728 2202 a(\() p Fo(\014) p Fy 2811 2202 a(\)) p Fk 2881 2202 a(2) p Fy 2992 2181 a(~) p Fo 2972 2202 a(D) p Fy 3079 2202 a(if) p Fo 3163 2202 a(\014) p 3251 2202 a(>) p 3352 2202 a(\014) p Fp 3399 2214 a(c) p Fy 3446 2202 a(,) p 3507 2202 a(hence,) p 3771 2202 a(in) 0 2312 y(particular,) p 435 2312 a(lim) p Fp 406 2361 a(n) p Fn(!1) p Fo 605 2291 a(~) 594 2312 y(Z) p Fp 651 2324 a(n) p Fy 695 2312 a(\() p Fo(\014) p Fy 778 2312 a(\)) p 834 2312 a(=) p 922 2312 a(0) p 977 2312 a(.) 125 2451 y(Finally) p 373 2451 a(,) p 437 2451 a(if) p Fo 524 2451 a(\014) p 616 2451 a(<) p 722 2451 a(\014) p Fp 769 2463 a(c) p Fy 841 2451 a(and) p Fo 1012 2451 a(~) p 1013 2451 a(x) p Fp 1060 2463 a(can) p Fy 1171 2451 a(\() p Fo(\014) p Fy 1254 2451 a(\)) p Fk 1328 2451 a(2) p Fo 1425 2451 a(D) p Fy 1537 2451 a(=) p 1642 2385 V Fo 1642 2451 a(D) p Fy 1713 2451 a(,) p 1778 2451 a(then) p 1978 2451 a(the) p 2131 2451 a(monotonicit) n(y) p 2645 2451 a(prop) r(ert) n(y) p 2995 2451 a(in) p 3103 2451 a(Theorem) p 3464 2451 a(15) p 3586 2451 a(implies) p Fo -1 2551 a(~) p 0 2551 a(x) p Fp 47 2563 a(can) p Fy 158 2551 a(\() p Fo(\014) p Fp 237 2563 a(c) p Fy 271 2551 a(\)) p Fk 335 2551 a(2) p Fy 440 2530 a(~) p Fo 421 2551 a(D) p Fy 523 2551 a(=) p Fo 618 2551 a(D) p Fp 689 2521 a(o) p Fy 726 2551 a(,) p 783 2551 a(whic) n(h) p 1025 2551 a(con) n(tradicts) p Fo 1456 2551 a(~) p 1457 2551 a(x) p Fp 1504 2563 a(can) p Fy 1615 2551 a(\() p Fo(\014) p Fp 1694 2563 a(c) p Fy 1728 2551 a(\)) p Fk 1791 2551 a(2) p Fo 1877 2551 a(@) p 1926 2551 a(D) p Fy 1997 2551 a(.) p 2071 2551 a(Hence) p Fo 2322 2551 a(~) p 2323 2551 a(x) p Fp 2370 2563 a(can) p Fy 2481 2551 a(\() p Fo(\014) p Fy 2564 2551 a(\)) p Fk 2628 2551 a(2) p Fo 2714 2551 a(D) p Fp 2785 2521 a(c) p Fy 2851 2551 a(and) p 3017 2551 a(in) p 3119 2551 a(particular,) p 3531 2551 a(with) p 3725 2551 a(the) 0 2732 y(same) p 208 2732 a(argumen) n(t) p 580 2732 a(as) p 681 2732 a(that) p 861 2732 a(led) p 995 2732 a(to) p 1097 2732 a(\(34\),) p 1295 2732 a(w) n(e) p 1417 2732 a(ha) n(v) n(e) p 1638 2732 a(lim) p Fp 1609 2782 a(n) p Fn(!1) p Fp 1838 2629 a(d) p Fg 1796 2654 a(X) p Fp 1802 2830 a(i) p Fj(=1) p Fo 1930 2732 a(Z) p Fp 1987 2747 a(n;) p Fj(\() p Fp(i) p Fj(\)) p Fy 2127 2732 a(\() p Fo(\014) p Fy 2210 2732 a(\)) p 2265 2732 a(=) p Fk 2353 2732 a(1) p Fy 2450 2732 a(.) 125 2909 y(The) p 295 2909 a(case) p 471 2909 a(of) p 565 2909 a(restricted) p 935 2909 a(mo) r(del) p 1182 2909 a(is) p 1266 2909 a(similarly) p 1605 2909 a(pro) n(v) n(ed,) p 1898 2909 a(if) p 1975 2909 a(w) n(e) p 2097 2909 a(note) p 2281 2909 a(\(CS2\)) p 2521 2909 a(in) p 2618 2909 a(place) p 2830 2909 a(of) p 2925 2909 a(\(CS1\).) p Fd 3778 3009 a(2) p Fy 125 3282 a(W) p 203 3282 a(e) p 268 3282 a(will) p 424 3282 a(use) p 568 3282 a(the) p 711 3282 a(follo) n(wing) p 1061 3282 a(in) p 1158 3282 a(the) p 1301 3282 a(pro) r(of) p 1518 3282 a(of) p 1613 3282 a(Theorem) p 1963 3282 a(10.) p Fz 0 3444 a(Prop) s(osition) p 516 3444 a(16) p Fi 653 3444 a(If) p Fo 739 3444 a(~) p 740 3444 a(x) p Fk 811 3444 a(2) p Fo 889 3444 a(D) p Fp 960 3414 a(o) p Fi 1027 3444 a(then) p 1241 3376 116 4 v Fy 1241 3444 a(lim) p Fp 1212 3494 a(n) p Fn(!1) p Fy 1399 3444 a(max) p Fp 1400 3497 a(I) p Fn 1434 3497 a(2I) p Fm 1517 3506 a(d) p Fy 1567 3444 a(2) p Fn 1609 3410 a(\000) p Fp(n) p Fy 1720 3444 a(log) p Fo 1841 3444 a(X) p Fp 1910 3456 a(n;I) p Fy 2008 3444 a(\() p Fo 2039 3444 a(~) p 2040 3444 a(x) p Fy 2088 3444 a(\)) p Fo 2143 3444 a(<) p Fy 2231 3444 a(0) p Fi 2287 3444 a(.) -42 3658 y(Pr) l(o) l(of.) p Fy 219 3658 a(W) p 297 3658 a(rite) p Fo 451 3658 a(Y) p Fp 499 3670 a(n) p Fy 570 3658 a(=) p 660 3658 a(max) p Fp 662 3712 a(I) p Fn 696 3712 a(2I) p Fm 779 3721 a(d) p Fy 828 3658 a(log) p Fo 950 3658 a(X) p Fp 1019 3670 a(n;I) p Fy 1131 3658 a(,) p Fo 1184 3658 a(n) p Fk 1259 3658 a(2) p Fq 1340 3658 a(Z) p Fj 1395 3670 a(+) p Fy 1450 3658 a(.) p 1515 3658 a(Since) p 1733 3658 a(\010) p Fp 1793 3670 a(I) p Fy 1860 3658 a(are) p 2001 3658 a(p) r(olynomials) p 2464 3658 a(of) p 2560 3658 a(p) r(ositiv) n(e) p 2868 3658 a(co) r(e\016cien) n(ts) p 3290 3658 a(with) p 3481 3658 a(eac) n(h) p 3669 3658 a(term) 0 3800 y(of) p 95 3800 a(degree) p 353 3800 a(no) p 468 3800 a(less) p 622 3800 a(than) p 815 3800 a(2) p 884 3800 a(\(Prop) r(osition) p 1364 3800 a(4\),) p 1488 3800 a(there) p 1701 3800 a(exists) p 1930 3800 a(a) p 1999 3800 a(p) r(olynomial) p Fo 2428 3800 a(P) p Fj 2481 3812 a(1) p Fy 2546 3800 a(with) p 2735 3800 a(p) r(ositiv) n(e) p 3042 3800 a(co) r(e\016cien) n(ts) p 3463 3800 a(suc) n(h) p 3650 3800 a(that) p Fo 846 3977 a(Y) p Fp 894 3989 a(n) p Fj(+1) p Fy 1023 3977 a(\() p Fo 1054 3977 a(~) p 1055 3977 a(x) p Fy 1103 3977 a(\)) p Fl 1158 3977 a(5) p Fy 1246 3977 a(2) p Fo(Y) p Fp 1336 3989 a(n) p Fy 1381 3977 a(\() p Fo 1412 3977 a(~) p 1413 3977 a(x) p Fy(\)) p 1511 3977 a(+) p 1594 3977 a(log) p Fo 1715 3977 a(P) p Fj 1768 3989 a(1) p Fy 1806 3977 a(\(exp\() p Fo(Y) p Fp 2045 3989 a(n) p Fy 2091 3977 a(\() p Fo 2122 3977 a(~) p 2123 3977 a(x) p Fy(\)\)\)) p Fo(;) p 2359 3977 a(n) p Fk 2432 3977 a(2) p Fq 2511 3977 a(Z) p Fj 2566 3989 a(+) p Fo 2621 3977 a(;) p 2685 3977 a(~) p 2686 3977 a(x) p Fk 2756 3977 a(2) p Fq 2835 3977 a(R) p Fn 2895 3940 a(I) p Fm 2933 3949 a(d) p Fj 2895 3998 a(+) p Fo 2971 3977 a(:) p Fy 125 4171 a(If) p Fo 209 4171 a(~) p 210 4171 a(x) p Fk 285 4171 a(2) p Fo 367 4171 a(D) p Fp 438 4141 a(o) p Fy 475 4171 a(,) p 529 4171 a(Theorem) p 882 4171 a(15) p 995 4171 a(implies) p 1279 4171 a(that) p Fo 1479 4150 a(~) 1462 4171 y(X) p Fp 1531 4183 a(n) p Fy 1576 4171 a(\() p Fo 1607 4171 a(~) p 1608 4171 a(x) p Fy(\)) p Fk 1715 4171 a(2) p Fo 1797 4171 a(D) p Fp 1868 4141 a(o) p Fy 1905 4171 a(,) p Fo 1959 4171 a(n) p Fk 2036 4171 a(2) p Fq 2118 4171 a(Z) p Fj 2173 4183 a(+) p Fy 2229 4171 a(.) p 2296 4171 a(Hence,) p Fk 2569 4171 a(f) p Fo 2628 4150 a(~) 2611 4171 y(X) p Fp 2680 4183 a(n) p Fy 2724 4171 a(\() p Fo 2755 4171 a(~) p 2756 4171 a(x) p Fy 2804 4171 a(\)) p Fk 2864 4171 a(j) p Fo 2914 4171 a(n) p Fk 2991 4171 a(2) p Fq 3073 4171 a(Z) p Fj 3128 4183 a(+) p Fk 3184 4171 a(g) p Fy 3255 4171 a(is) p 3341 4171 a(b) r(ounded,) p 3706 4171 a(and) 0 4271 y(there) p 212 4271 a(exists) p Fo 442 4271 a(M) p Fk 554 4271 a(2) p Fq 633 4271 a(R) p Fy(,) p 743 4271 a(a) p 813 4271 a(constan) n(t) p 1147 4271 a(in) p Fo 1244 4271 a(n) p Fy(,) p 1345 4271 a(suc) n(h) p 1532 4271 a(that) p Fo 1315 4448 a(Y) p Fp 1363 4460 a(n) p Fj(+1) p Fy 1492 4448 a(\() p Fo 1523 4448 a(~) p 1524 4448 a(x) p Fy 1572 4448 a(\)) p Fl 1627 4448 a(5) p Fy 1715 4448 a(2) p Fo(Y) p Fp 1805 4460 a(n) p Fy 1850 4448 a(\() p Fo 1881 4448 a(~) p 1882 4448 a(x) p Fy(\)) p 1980 4448 a(+) p Fo 2063 4448 a(M) t(;) p 2240 4448 a(n) p Fk 2313 4448 a(2) p Fq 2392 4448 a(Z) p Fj 2447 4460 a(+) p Fo 2502 4448 a(:) p Fy 3692 4448 a(\(35\)) 0 4626 y(In) p 104 4626 a(particular,) p 510 4626 a(2) p Fn 552 4596 a(\000) p Fp(n) p Fy 649 4626 a(\() p Fo(Y) p Fp 729 4638 a(n) p Fy 774 4626 a(\() p Fo 805 4626 a(~) p 806 4626 a(x) p Fy 854 4626 a(\)) p 905 4626 a(+) p Fo 988 4626 a(M) p Fy 1078 4626 a(\)) p 1137 4626 a(is) p 1221 4626 a(decreasing,) p 1646 4626 a(hence) p 947 4736 V 947 4803 a(lim) p Fp 918 4853 a(n) p Fn(!1) p Fy 1105 4803 a(2) p Fn 1147 4769 a(\000) p Fp(n) p Fy 1244 4803 a(\() p Fo(Y) p Fp 1324 4815 a(n) p Fy 1370 4803 a(\() p Fo 1401 4803 a(~) p 1402 4803 a(x) p Fy(\)) p 1500 4803 a(+) p Fo 1583 4803 a(M) p Fy 1673 4803 a(\)) p Fl 1728 4803 a(5) p Fy 1816 4803 a(2) p Fn 1858 4769 a(\000) p Fp(n) p Ff 1951 4777 a(0) p Fy 1987 4803 a(\() p Fo(Y) p Fp 2067 4815 a(n) p Ff 2108 4823 a(0) p Fy 2145 4803 a(\() p Fo 2176 4803 a(~) p 2177 4803 a(x) p Fy(\)) p 2275 4803 a(+) p Fo 2358 4803 a(M) p Fy 2448 4803 a(\)) p Fo(;) p 2600 4803 a(n) p Fj 2650 4815 a(0) p Fk 2710 4803 a(2) p Fq 2788 4803 a(Z) p Fj 2843 4815 a(+) p Fo 2899 4803 a(:) p Fy 0 5007 a(Also) p Fo 186 5007 a(~) p 187 5007 a(x) p Fk 258 5007 a(2) p Fo 336 5007 a(D) p Fp 407 4977 a(o) p Fy 472 5007 a(implies) 1612 5107 y(lim) p Fp 1583 5156 a(n) p Fn(!1) p Fo 1770 5107 a(Y) p Fp 1818 5119 a(n) p Fy 1864 5107 a(\() p Fo 1895 5107 a(~) p 1896 5107 a(x) p Fy(\)) p 1999 5107 a(=) p Fk 2086 5107 a(\0001) p Fo(:) p Fy 0 5277 a(Hence) p 1147 5309 V 1147 5376 a(lim) p Fp 1118 5426 a(n) p Fn(!1) p Fy 1305 5376 a(2) p Fn 1347 5342 a(\000) p Fp(n) p Fo 1444 5376 a(Y) p Fp 1492 5388 a(n) p Fy 1537 5376 a(\() p Fo 1568 5376 a(~) p 1569 5376 a(x) p Fy 1617 5376 a(\)) p 1672 5376 a(=) p 1789 5309 V 1789 5376 a(lim) p Fp 1760 5426 a(n) p Fn(!1) p Fy 1947 5376 a(2) p Fn 1989 5342 a(\000) p Fp(n) p Fy 2086 5376 a(\() p Fo(Y) p Fp 2166 5388 a(n) p Fy 2211 5376 a(\() p Fo 2242 5376 a(~) p 2243 5376 a(x) p Fy 2291 5376 a(\)) p 2342 5376 a(+) p Fo 2425 5376 a(M) p Fy 2515 5376 a(\)) p Fo 2570 5376 a(<) p Fy 2657 5376 a(0) p Fo(;) p Fy 0 5547 a(whic) n(h) p 238 5547 a(implies) p 519 5547 a(the) p 662 5547 a(statemen) n(t.) p Fd 3778 5647 a(2) p Fy 1878 6120 a(15) p 90 rotate dyy eop %%Page: 16 16 16 15 bop Fr 0 83 a(4.2) p 255 83 a(Distribution) p 892 83 a(of) p 1020 83 a(path) p 1280 83 a(length.) p Fy 0 236 a(Here) p 196 236 a(w) n(e) p 318 236 a(will) p 475 236 a(pro) n(v) n(e) p 699 236 a(Theorem) p 1050 236 a(9.) p Fz 0 402 a(Prop) s(osition) p 516 402 a(17) p Fi 653 402 a(L) l(et) p Fo 799 402 a(~) p 800 402 a(x) p Fp 847 414 a(c) p Fi 914 402 a(b) l(e) p 1020 402 a(a) p 1095 402 a(self-avoiding) p 1578 402 a(\014xe) l(d) p 1773 402 a(p) l(oint) p 1987 402 a(and) p Fo 2150 402 a(~) p 2152 402 a(x) p Fk 2228 402 a(2) p 2313 402 a(D) p Fi 2379 402 a(om) p Fy 2496 402 a(\() p Fo 2527 402 a(~) p 2528 402 a(x) p Fp 2575 414 a(c) p Fy 2610 402 a(\)) p Fi 2675 402 a(then) p 2863 402 a(ther) l(e) p 3074 402 a(exists) p Fo 3305 402 a(f) p Fp 3346 414 a(d) p Fk 3405 402 a(\002) p Fo 3491 402 a(f) p Fp 3532 414 a(d) p Fi 3603 402 a(matrix) p Fy 0 502 a(\003\() p Fo 89 502 a(~) p 90 502 a(x) p Fy(\)) p Fi 199 502 a(such) p 388 502 a(that) p 558 502 a(\(24\)) p 740 502 a(holds.) p 987 502 a(The) p 1156 502 a(elements) p 1498 502 a(of) p 1595 502 a(the) p 1733 502 a(matrix) p 2000 502 a(ar) l(e) p 2141 502 a(non-ne) l(gative,) p 2652 502 a(and) p Fy 1481 685 a(\003\() p Fo 1570 685 a(~) p 1571 685 a(x) p Fy(\)) p Fp 1650 700 a(I) p 1684 700 a(;) p Fj(\(1\)) p Fo 1816 685 a(>) p Fy 1904 685 a(0) p Fo(;) p 2042 685 a(I) p Fk 2108 685 a(2) p 2186 685 a(K) p Fp 2248 698 a(~) p 2249 698 a(x) p Fm 2287 706 a(c) p Fo 2336 685 a(:) p Fy 3692 685 a(\(36\)) p Fi -42 867 a(Pr) l(o) l(of.) p Fy 219 867 a(First) p 421 867 a(note) p 606 867 a(that) p 785 867 a(\(FP2\)) p 1030 867 a(and) p 1191 867 a(\(30\)) p 1367 867 a(imply) p 1600 867 a(an) p 1715 867 a(existence) p 2068 867 a(of) p 2163 867 a(the) p 2306 867 a(limit) p 2504 867 a(in) p 2601 867 a(\(24\)) p 2776 867 a(for) p Fo 2902 867 a(~) p 2903 867 a(x) p Fy 2974 867 a(=) p Fo 3060 867 a(~) p 3061 867 a(x) p Fp 3108 879 a(c) p Fy 3156 867 a(.) 125 967 y(Next) p 327 967 a(recall) p 549 967 a(the) p 692 967 a(follo) n(wing) p 1043 967 a(t) n(w) n(o) p 1199 967 a(results.) p Fz 0 1133 a(Lemma) p 339 1133 a(18) p 466 1133 a(\([6) o(,) p 636 1133 a(Lemma) p 974 1133 a(4.2],) p 1181 1133 a([5) o(,) p 1314 1133 a(Lemma) p 1652 1133 a(\(3.1\)]\)) p Fi 1954 1133 a(L) l(et) p Fo 2092 1133 a(N) p Fi 2192 1133 a(b) l(e) p 2289 1133 a(a) p 2357 1133 a(p) l(ositive) p 2651 1133 a(inte) l(ger,) p 2947 1133 a(and) p Fo 3103 1133 a(A) p Fi 3190 1133 a(and) p Fo 3347 1133 a(A) p Fp 3409 1145 a(n) p Fi 3454 1133 a(,) p Fo 3505 1133 a(n) p Fk 3578 1133 a(2) p Fq 3657 1133 a(N) p Fi(,) p 3768 1133 a(b) l(e) p Fo 0 1233 a(N) p Fk 84 1233 a(\002) p Fo 158 1233 a(N) p Fi 258 1233 a(matric) l(es.) p 625 1233 a(Assume) p 930 1233 a(that) p 1096 1233 a(ther) l(e) p 1298 1233 a(is) p 1382 1233 a(an) p 1497 1233 a(invertible) p Fo 1856 1233 a(N) p Fk 1940 1233 a(\002) p Fo 2014 1233 a(N) p Fi 2114 1233 a(matrix) p Fo 2376 1233 a(Q) p Fi 2467 1233 a(such) p 2652 1233 a(that) p Fo 2817 1233 a(Q) p Fn 2883 1202 a(\000) p Fj(1) p Fo 2971 1233 a(AQ) p Fi 3125 1233 a(is) p 3209 1233 a(a) p 3277 1233 a(diagonal) p 3603 1233 a(matrix) 0 1394 y(whose) p 244 1394 a(eigenvalue) p Fo 648 1394 a(\025) p Fi 728 1394 a(that) p 900 1394 a(is) p 991 1394 a(lar) l(gest) p 1256 1394 a(in) p 1360 1394 a(absolute) p 1681 1394 a(value,) p 1923 1394 a(is) p 2015 1394 a(p) l(ositive.) p 2355 1394 a(Assume) p 2667 1394 a(further) p 2947 1394 a(that) p Fn 3149 1290 a(1) p Fg 3122 1315 a(X) p Fp 3119 1491 a(n) p Fj(=1) p Fk 3258 1394 a(k) p Fo(A) p Fp 3362 1406 a(n) p Fk 3426 1394 a(\000) p Fo 3509 1394 a(A) p Fk(k) p Fo 3640 1394 a(<) p Fk 3732 1394 a(1) p Fi(.) 0 1565 y(Then) p 216 1565 a(ther) l(e) p 423 1565 a(exists) p 651 1565 a(a) p 723 1565 a(matrix) p Fo 989 1565 a(R) p Fi 1083 1565 a(such) p 1272 1565 a(that) p Fy 1025 1748 a(lim) p Fp 987 1798 a(m) p Fn(!1) p 1221 1680 116 4 v Fy 1221 1748 a(lim) p Fp 1192 1798 a(n) p Fn(!1) p Fg 1380 1677 a(\015) 1380 1727 y(\015) p Fo 1426 1748 a(\025) p Fn 1474 1714 a(\000) p Fp(n) p Fo 1571 1748 a(A) p Fp 1633 1760 a(n) p Fj(+) p Fp(m) p Fo 1789 1748 a(A) p Fp 1851 1760 a(n) p Fj(+) p Fp(m) p Fn(\000) p Fj(1) p Fk 2105 1748 a(\001) p 2142 1748 a(\001) p 2179 1748 a(\001) p Fo 2215 1748 a(A) p Fp 2277 1760 a(m) p Fj(+1) p Fk 2457 1748 a(\000) p Fo 2554 1748 a(R) p Fg 2618 1677 a(\015) 2618 1727 y(\015) p Fy 2687 1748 a(=) p 2774 1748 a(0) p Fo 2830 1748 a(:) 0 1965 y(Q) p Fn 66 1935 a(\000) p Fj(1) p Fo 155 1965 a(R) q(Q) p Fi 323 1965 a(is) p 422 1965 a(a) p 504 1965 a(diagonal) p 844 1965 a(matrix) p 1121 1965 a(whose) p 1372 1965 a(diagonal) p 1713 1965 a(elements) p 2064 1965 a(satisfy) p Fy 2332 1965 a(\() p Fo(Q) p Fn 2430 1935 a(\000) p Fj(1) p Fo 2520 1965 a(R) q(Q) p Fy(\)) p Fp 2682 1977 a(ii) p Fy 2773 1965 a(=) p 2878 1965 a(1) p Fi 2959 1965 a(if) p Fy 3049 1965 a(\() p Fo(Q) p Fn 3147 1935 a(\000) p Fj(1) p Fo 3236 1965 a(AQ) p Fy(\)) p Fp 3396 1977 a(ii) p Fy 3488 1965 a(=) p Fo 3593 1965 a(\025) p Fi(,) p 3708 1965 a(and) p Fy 0 2065 a(\() p Fo(Q) p Fn 98 2034 a(\000) p Fj(1) p Fo 187 2065 a(R) q(Q) p Fy(\)) p Fp 349 2077 a(ii) p Fy 423 2065 a(=) p 510 2065 a(0) p Fi 582 2065 a(otherwise.) p 984 2065 a(Mor) l(e) l(over) p Fy 1378 2065 a(lim) p Fp 1349 2114 a(n) p Fn(!1) p Fo 1536 2065 a(\025) p Fn 1584 2030 a(\000) p Fp(n) p Fo 1682 2065 a(A) p Fp 1744 2077 a(n) p Fo 1789 2065 a(A) p Fp 1851 2077 a(n) p Fn(\000) p Fj(1) p Fk 1995 2065 a(\001) p 2032 2065 a(\001) p 2069 2065 a(\001) p Fo 2106 2065 a(A) p Fj 2168 2077 a(1) p Fi 2235 2065 a(exists.) 125 2193 y(\() p Fk(k) p Fo 200 2193 a(A) p Fk(k) p Fi 333 2193 a(is) p 423 2193 a(the) p 560 2193 a(standar) l(d) p 899 2193 a(op) l(er) l(ator) p 1226 2193 a(norm) p 1447 2193 a(de\014ne) l(d) p 1727 2193 a(as) p 1833 2193 a(the) p 1971 2193 a(squar) l(e) p 2229 2193 a(r) l(o) l(ot) p 2398 2193 a(of) p 2495 2193 a(lar) l(gest) p 2757 2193 a(eigenvalue) p 3158 2193 a(of) p Fo 3256 2193 a(A) p Fn 3318 2163 a(\003) p Fo 3356 2193 a(A) p Fi(.\)) p Fz 0 2376 a(Lemma) p 339 2376 a(19) p 466 2376 a(\([6) o(,) p 636 2376 a(Prop) s(osition) p 1152 2376 a(3.8]\)) p Fi 1379 2376 a(If) p Fo 1462 2376 a(~) p 1463 2376 a(x) p Fp 1510 2388 a(c) p Fi 1572 2376 a(is) p 1658 2376 a(a) p 1728 2376 a(self-avoiding) p 2205 2376 a(\014xe) l(d) p 2393 2376 a(p) l(oint) p 2601 2376 a(and) p Fo 2758 2376 a(~) p 2759 2376 a(x) p Fk 2830 2376 a(2) p 2908 2376 a(D) p Fi 2974 2376 a(om) p Fy 3091 2376 a(\() p Fo 3122 2376 a(~) p 3123 2376 a(x) p Fp 3170 2388 a(c) p Fy 3205 2376 a(\)) p Fi(,) p 3290 2376 a(then) p 3472 2376 a(ther) l(e) p 3676 2376 a(exist) 0 2476 y(p) l(ositive) p 299 2476 a(c) l(onstants) p Fo 664 2476 a(\032) p Fi 737 2476 a(and) p Fo 898 2476 a(C) p Fi 963 2476 a(,) p 1018 2476 a(satisfying) p Fo 1388 2476 a(\032) p 1454 2476 a(<) p Fy 1541 2476 a(1) p Fi(,) p 1638 2476 a(such) p 1827 2476 a(that) p Fk 1162 2658 a(j) p Fo(X) p Fp 1254 2670 a(n;I) p Fy 1353 2658 a(\() p Fo 1384 2658 a(~) p 1385 2658 a(x) p Fy(\)) p Fk 1483 2658 a(\000) p Fo 1566 2658 a(x) p Fp 1613 2670 a(c;I) p Fk 1701 2658 a(j) p Fl 1747 2658 a(5) p Fo 1835 2658 a(C) p 1900 2658 a(\032) p Fp 1943 2624 a(n) p Fo 1988 2658 a(;) p 2084 2658 a(n) p Fk 2157 2658 a(2) p Fq 2236 2658 a(Z) p Fj 2291 2670 a(+) p Fo 2346 2658 a(;) p 2413 2658 a(I) p Fk 2479 2658 a(2) p 2557 2658 a(I) p Fp 2602 2670 a(d) p Fo 2655 2658 a(:) p Fi -42 2841 a(R) l(emark.) p Fy 382 2841 a(\(i\)) p 511 2841 a(The) p 683 2841 a(pro) r(of) p 901 2841 a(in) p 998 2841 a([6,) p 1115 2841 a(Prop) r(osition) p 1563 2841 a(3.8]) p 1720 2841 a(is) p 1805 2841 a(for) p Fo 1933 2841 a(d) p Fy 2001 2841 a(=) p 2090 2841 a(3,) p 2183 2841 a(but) p 2337 2841 a(the) p 2480 2841 a(pro) r(of) p 2698 2841 a(is) p 2783 2841 a(directly) p 3086 2841 a(applicable) p 3479 2841 a(to) p 3581 2841 a(general) 208 2940 y(cases.) p 448 2940 a(In) p 552 2940 a(fact,) p 739 2940 a(the) p 882 2940 a(pro) r(of) p 1099 2940 a(is) p 1182 2940 a(an) p 1298 2940 a(application) p 1727 2940 a(of) p 1821 2940 a(a) p 1890 2940 a(general) p 2177 2940 a(theory) p 2437 2940 a(for) p 2564 2940 a(di\013eomorphisms) p 3181 2940 a([10) o(].) 55 3107 y(\(ii\)) p 208 3107 a(Though) p 518 3107 a(w) n(e) p 641 3107 a(do) p 757 3107 a(not) p 906 3107 a(use) p 1051 3107 a(this,) p 1237 3107 a(the) p 1381 3107 a(pro) r(of) p 1599 3107 a(in) p 1697 3107 a([6,) p 1814 3107 a(Prop) r(osition) p 2262 3107 a(3.8]) p 2420 3107 a(implies) p 2703 3107 a(that) p 2883 3107 a(Lemma) p 3181 3107 a(19) p 3293 3107 a(holds) p 3511 3107 a(for) p 3639 3107 a(an) n(y) p Fo 3797 3107 a(\032) p Fy 208 3206 a(satisfying) 1587 3306 y(max) p Fk(f) p Fo(\025) p Fn 1832 3271 a(\000) p Fj(1) p Fo 1921 3306 a(;) p Fk 1958 3306 a(j) p Fo(\025) p Fj 2029 3318 a(2) p Fk 2067 3306 a(jg) p Fo 2154 3306 a(<) p 2242 3306 a(\032) p 2308 3306 a(<) p Fy 2396 3306 a(1) p Fo(;) p Fy 208 3455 a(where) p Fo 448 3455 a(\025) p Fj 496 3467 a(2) p Fy 561 3455 a(is) p 644 3455 a(the) p 787 3455 a(eigen) n(v) p 1009 3455 a(alue) p 1184 3455 a(of) p Fk 1279 3455 a(J) p Fy 1350 3455 a(\() p Fo 1381 3455 a(~) p 1382 3455 a(x) p Fp 1429 3467 a(c) p Fy 1464 3455 a(\)) p 1524 3455 a(whic) n(h) p 1761 3455 a(is) p 1845 3455 a(second) p 2113 3455 a(largest) p 2381 3455 a(in) p 2478 3455 a(absolute) p 2806 3455 a(v) p 2845 3455 a(alue.) p Fd 3774 3555 a(3) p Fy 125 3837 a(Let) p 273 3837 a(us) p 380 3837 a(pro) r(ceed) p 687 3837 a(with) p 876 3837 a(the) p 1019 3837 a(pro) r(of) p 1236 3837 a(of) p 1330 3837 a(existence) p 1684 3837 a(of) p 1778 3837 a(the) p 1921 3837 a(limit) p 2120 3837 a(in) p 2216 3837 a(Prop) r(osition) p 2663 3837 a(17.) p 2806 3837 a(By) p 2937 3837 a(the) p 3080 3837 a(mean-v) p 3341 3837 a(alue) p 3516 3837 a(theorem,) p Fk 830 4058 a(J) p Fp 886 4070 a(I) p 920 4070 a(J) p Fy 967 4058 a(\() p Fo 998 4058 a(~) p 999 4058 a(x) p Fy(\)) p Fk 1097 4058 a(\000) p 1180 4058 a(J) p Fp 1236 4070 a(I) p 1270 4070 a(J) p Fy 1317 4058 a(\() p Fo 1348 4058 a(~) p 1349 4058 a(x) p Fp 1396 4070 a(c) p Fy 1430 4058 a(\)) p 1486 4058 a(=) p Fg 1602 3980 a(X) p Fp 1573 4158 a(K) p Fn 1633 4158 a(2I) p Fm 1716 4167 a(d) p Fo 1774 4002 a(@) p Fk 1823 4002 a(J) p Fp 1879 4014 a(I) p 1913 4014 a(J) p 1774 4039 186 4 v Fo 1787 4115 a(@) p 1836 4115 a(x) p Fp 1883 4127 a(K) p Fy 1969 4058 a(\() p Fo 2000 4058 a(~) p 2001 4058 a(u) p Fy(\)) p 2095 4058 a(\() p Fo(x) p Fp 2174 4070 a(K) p Fk 2258 4058 a(\000) p Fo 2341 4058 a(x) p Fp 2388 4070 a(c;K) p Fy 2501 4058 a(\)) p Fo(;) p 2654 4058 a(I) p 2697 4058 a(;) p 2734 4058 a(J) 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1983 4438 a(x) p Fp 2030 4450 a(c) p Fy 2092 4438 a(for) p 2219 4438 a(su\016cien) n(tly) p 2635 4438 a(large) p Fo 2838 4438 a(m) p Fy(.) p 2971 4438 a(Therefore) p Fo 3364 4417 a(~) 3347 4438 y(X) p Fp 3416 4450 a(m) p Fy 3479 4438 a(\() p Fo 3510 4438 a(~) p 3511 4438 a(x) p Fy(\)) p 3618 4438 a(is) p 3702 4438 a(in) p 3798 4438 a(a) 0 4537 y(b) r(ounded) p 339 4537 a(domain.) p 666 4537 a(In) p 770 4537 a(particular,) p 1177 4537 a(there) p 1389 4537 a(exists) p 1618 4537 a(a) p 1687 4537 a(p) r(ositiv) n(e) p 1995 4537 a(constan) n(t) p Fo 2329 4537 a(M) p Fy 2447 4537 a(suc) n(h) p 2634 4537 a(that) p Fg 515 4645 a(\014) 515 4695 y(\014) 515 4744 y(\014) 515 4794 y(\014) p Fo 553 4709 a(@) p Fk 602 4709 a(J) p Fp 658 4721 a(I) p 692 4721 a(J) p 553 4746 V Fo 565 4822 a(@) p 614 4822 a(x) p Fp 661 4834 a(K) p Fy 748 4765 a(\() p Fo 779 4765 a(~) p 780 4765 a(u) p Fy(\)) p Fg 860 4645 a(\014) 860 4695 y(\014) 860 4744 y(\014) 860 4794 y(\014) p Fl 911 4765 a(5) p Fo 999 4765 a(M) t(;) p 1203 4765 a(I) p 1246 4765 a(;) p 1283 4765 a(J) o(;) p 1365 4765 a(K) p Fk 1465 4765 a(2) p 1543 4765 a(I) p Fp 1588 4777 a(d) p Fo 1641 4765 a(;) p 1704 4765 a(~) p 1705 4765 a(u) p Fy 1776 4765 a(=) p Fo 1862 4765 a(~) p 1864 4765 a(x) p Fp 1911 4777 a(c) p Fy 1963 4765 a(+) p 2046 4765 a(\() p Fo 2096 4744 a(~) 2078 4765 y(X) p Fp 2147 4777 a(m) p Fy 2210 4765 a(\() p Fo 2241 4765 a(~) p 2242 4765 a(x) p Fy 2290 4765 a(\)) p Fk 2341 4765 a(\000) p Fo 2423 4765 a(~) p 2424 4765 a(x) p Fp 2471 4777 a(c) p Fy 2505 4765 a(\)) p Fo(\022) r(;) p Fy 2643 4765 a(0) p Fo 2708 4765 a(<) p 2796 4765 a(\022) p 2860 4765 a(<) p Fy 2947 4765 a(1) p Fo 3003 4765 a(;) p 3067 4765 a(m) p Fk 3163 4765 a(2) p Fq 3242 4765 a(N) p Fo(:) p Fy 0 4998 a(This) p 190 4998 a(with) p 379 4998 a(\(37\)) p 554 4998 a(and) p 715 4998 a(Lemma) p 1012 4998 a(19) p 1122 4998 a(implies) p Fg 1186 5110 a(\015) 1186 5159 y(\015) 1186 5209 y(\015) p Fk 1232 5205 a(J) p Fy 1303 5205 a(\() p Fo 1353 5184 a(~) 1335 5205 y(X) p Fp 1404 5217 a(m) p Fy 1467 5205 a(\() p Fo 1498 5205 a(~) p 1499 5205 a(x) p Fy 1547 5205 a(\)\)) p Fk 1630 5205 a(\000) p 1713 5205 a(J) p Fy 1785 5205 a(\() p Fo 1816 5205 a(~) p 1817 5205 a(x) p Fp 1864 5217 a(c) p Fy 1898 5205 a(\)) p Fg 1930 5110 a(\015) 1930 5159 y(\015) 1930 5209 y(\015) p Fl 2000 5205 a(5) p Fo 2087 5205 a(C) p Fn 2152 5171 a(0) p Fo 2176 5205 a(\032) p Fp 2219 5171 a(m) p Fo 2282 5205 a(;) p 2347 5205 a(m) p Fk 2443 5205 a(2) p Fq 2521 5205 a(Z) p Fj 2576 5217 a(+) p Fo 2631 5205 a(;) p Fy 0 5413 a(for) p 127 5413 a(some) p Fo 335 5413 a(C) p Fn 400 5383 a(0) p Fo 447 5413 a(>) p Fy 534 5413 a(0) p 604 5413 a(and) p 765 5413 a(0) p Fo 830 5413 a(<) p 917 5413 a(\032) p 983 5413 a(<) p Fy 1071 5413 a(1) p 1126 5413 a(.) p 1186 5413 a(Therefore) p 1562 5413 a(w) n(e) p 1685 5413 a(see) p 1819 5413 a(that) p 1999 5413 a(the) p 2142 5413 a(assumptions) p 2614 5413 a(in) p 2710 5413 a(Lemma) p 3007 5413 a(18) p 3118 5413 a(are) p 3256 5413 a(satis\014ed) p 3575 5413 a(with) p Fo 1630 5596 a(A) p Fp 1692 5608 a(n) p Fy 1760 5596 a(=) p Fk 1848 5596 a(J) p Fy 1920 5596 a(\() p Fo 1969 5575 a(~) 1952 5596 y(X) p Fp 2021 5608 a(n) p Fy 2066 5596 a(\() p Fo 2097 5596 a(~) p 2098 5596 a(x) p Fy 2146 5596 a(\)\)) p 3692 5596 a(\(38\)) 0 5778 y(and) p Fo 161 5778 a(A) p Fy 247 5778 a(=) p Fk 334 5778 a(J) p Fy 406 5778 a(\() p Fo 437 5778 a(~) p 438 5778 a(x) p Fp 485 5790 a(c) p Fy 520 5778 a(\).) p 612 5778 a(Hence,) p 882 5778 a(the) p 1025 5778 a(limit) p 1223 5778 a(\(24\)) p 1398 5778 a(exists.) 1878 6120 y(16) p 90 rotate dyy eop %%Page: 17 17 17 16 bop Fy 125 83 a(T) p 178 83 a(o) p 254 83 a(pro) n(v) n(e) p 486 83 a(\(36\),) p 694 83 a(let) p Fo 821 83 a(i) p Fy 885 83 a(b) r(e) p 1006 83 a(an) p 1128 83 a(index) p 1360 83 a(satisfying) p 1737 83 a(\() p Fo(Q) p Fn 1835 53 a(\000) p Fj(1) p Fo 1924 83 a(AQ) p Fy(\)) p Fp 2084 95 a(ii) p Fy 2171 83 a(=) p Fo 2272 83 a(\025) p Fy(.) p 2403 83 a(Then) p 2627 83 a(the) p 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a(n) p Fy 1631 482 a(giv) n(en) p 1855 482 a(b) n(y) p 1977 482 a(\(38\)) p 2160 482 a(has) p 2315 482 a(p) r(ositiv) n(e) p 2629 482 a(\() p Fo(I) p 2704 482 a(;) p 2741 482 a(J) p Fy 2795 482 a(\)) p 2863 482 a(elemen) n(t) p 3177 482 a(for) p 3311 482 a(all) p Fo 3434 482 a(J) p Fk 3523 482 a(2) p 3613 482 a(I) p Fp 3658 494 a(d) p Fy 3711 482 a(,) p 3771 482 a(in) 0 616 y(particular,) p 405 616 a(for) p Fo 530 616 a(J) p Fy 607 616 a(=) p 694 616 a(\(1\).) p 860 616 a(Also,) p 1068 616 a(since) p 1270 616 a(\010) p Fj 1330 631 a(\(1\)) p Fy 1419 616 a(\() p Fo 1450 616 a(~) p 1451 616 a(x) p Fy(\)) p 1556 616 a(con) n(tains) p 1880 616 a(a) p 1947 616 a(term) p Fo 2143 616 a(x) p Fj 2190 631 a(\(1\)) 2280 585 y(2) p Fy 2342 616 a(\(Prop) r(osition) p 2820 616 a(4\),) p Fo 2942 616 a(A) p Fp 3004 628 a(n) p Fy 3075 616 a(has) p 3221 616 a(p) r(ositiv) n(e) p 3526 616 a(\(\(1\)) p Fo(;) p Fy 3701 616 a(\(1\)\)) 0 715 y(elemen) n(t,) p 330 715 a(whic) n(h) p 568 715 a(further) p 847 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1176 a(2) p 754 1176 a(I) p Fp 799 1188 a(d) p Fy 838 1176 a(\)) p Fk 893 1176 a(2) p Fq 972 1176 a(C) p Fn 1032 1141 a(I) p Fm 1070 1150 a(d) p Fy 1136 1176 a(and) p Fo 1292 1160 a(~) 1297 1176 y(t) p Fy 1350 1176 a(=) p 1438 1176 a(\() p Fo(t) p Fp 1500 1188 a(I) p Fo 1538 1176 a(;) p 1598 1176 a(I) p Fk 1664 1176 a(2) p 1743 1176 a(I) p Fp 1788 1188 a(d) p Fy 1827 1176 a(\)) p Fk 1882 1176 a(2) p Fq 1960 1176 a(C) p Fn 2020 1141 a(I) p Fm 2058 1150 a(d) p Fy 2097 1176 a(,) p 2148 1176 a(w) n(e) p 2270 1176 a(use) p 2413 1176 a(an) p 2529 1176 a(\(irregular\)) p 2931 1176 a(notation) p Fo 1350 1347 a(~) p 1351 1347 a(x) p Fy 1399 1347 a(\() p Fo 1426 1331 a(~) 1431 1347 y(t) p Fy(\)) p 1516 1347 a(=) p 1604 1347 a(\() p Fo(x) p Fp 1683 1359 a(I) p Fy 1736 1347 a(exp) o(\() p Fo(\025) p Fn 1942 1312 a(\000) p Fp(n) p Fo 2040 1347 a(t) p Fp 2070 1359 a(I) p Fy 2108 1347 a(\)) p Fo(;) p 2205 1347 a(I) p Fk 2271 1347 a(2) p 2350 1347 a(I) p Fp 2395 1359 a(d) p Fy 2433 1347 a(\)) p Fo(:) p Fy 3692 1347 a(\(39\)) p Fz 0 1527 a(Prop) s(osition) p 516 1527 a(20) p Fi 653 1527 a(If) p Fo 738 1527 a(~) p 739 1527 a(x) p Fk 809 1527 a(2) p 888 1527 a(D) p Fi 954 1527 a(om) p Fy 1071 1527 a(\() p Fo 1102 1527 a(~) p 1103 1527 a(x) p Fp 1150 1539 a(c) p Fy 1184 1527 a(\)) p Fi 1245 1527 a(then) p Fo 1446 1506 a(~) 1428 1527 y(X) p Fp 1497 1539 a(n) p Fy 1542 1527 a(\() p Fo 1573 1527 a(~) p 1574 1527 a(x) p Fy 1622 1527 a(\() p Fo 1649 1512 a(~) 1654 1527 y(t) p Fy(\)\)) p Fi 1777 1527 a(c) l(onver) l(ges) p 2146 1527 a(uniformly) p 2523 1527 a(in) p Fo 2618 1512 a(~) 2623 1527 y(t) p Fk 2676 1527 a(2) p Fq 2755 1527 a(C) p Fn 2815 1493 a(I) p Fm 2853 1502 a(d) p Fi 2920 1527 a(for) p 3051 1527 a(any) p 3208 1527 a(c) l(omp) l(act) p 3527 1527 a(subset) p 3772 1527 a(of) p Fq 0 1627 a(C) p Fn 60 1593 a(I) p Fm 98 1602 a(d) p Fi 137 1627 a(.) p 200 1627 a(F) p 248 1627 a(urthermor) l(e,) 73 1784 y(\(i\)) p 208 1784 a(If) p Fo 295 1784 a(I) p Fk 361 1784 a(2) p 439 1784 a(I) p Fp 484 1796 a(d) p Fi 553 1784 a(satis\014es) p Fo 865 1784 a(x) p Fp 912 1796 a(c;I) p Fy 1023 1784 a(=) p 1110 1784 a(0) p Fi 1182 1784 a(\(i.e.,) p 1385 1784 a(if) p Fo 1466 1784 a(I) p Fk 1532 1784 a(62) p 1610 1784 a(K) p Fp 1672 1797 a(~) p 1673 1797 a(x) p Fm 1711 1805 a(c) p Fi 1747 1784 a(\),) p 1836 1784 a(then) p Fy 1502 1955 a(lim) p Fp 1473 2005 a(n) p Fn(!1) p Fo 1661 1955 a(X) p Fp 1730 1967 a(n;I) p Fy 1828 1955 a(\() p Fo 1859 1955 a(~) p 1860 1955 a(x) p Fy 1908 1955 a(\() p Fo 1935 1939 a(~) 1940 1955 y(t) p Fy(\)\)) p 2058 1955 a(=) p 2145 1955 a(0) p Fo(;) 2278 1939 y(~) 2283 1955 y(t) p Fk 2336 1955 a(2) p Fq 2415 1955 a(C) p Fn 2475 1921 a(I) p Fm 2513 1930 a(d) p Fo 2551 1955 a(:) p Fi 47 2186 a(\(ii\)) p 208 2186 a(De\014ne) p Fo 477 2186 a(~) p 470 2186 a(') p Fn 524 2156 a(\003) p Fy 585 2186 a(=) p Fo 680 2186 a(~) p 673 2186 a(') p Fn 727 2156 a(\003) p Fp 726 2211 a(~) p 727 2211 a(x) p Fy 792 2186 a(=) p 880 2186 a(\() p Fo(') p Fn 966 2156 a(\003) p Fp 965 2211 a(~) p 966 2211 a(x;I) p Fo 1076 2186 a(;) p 1142 2186 a(I) p Fk 1208 2186 a(2) p 1287 2186 a(I) p Fp 1332 2198 a(d) p Fy 1371 2186 a(\)) p Fi 1433 2186 a(by) p Fo 952 2447 a(') p Fn 1006 2413 a(\003) p Fp 1005 2468 a(~) p 1006 2468 a(x;I) p Fy 1102 2447 a(\() p Fo 1129 2432 a(~) 1134 2447 y(t) p Fy(\)) p 1220 2447 a(=) p Fg 1307 2305 a(\() p Fj 1476 2352 a(1) p 1426 2366 135 4 v Fo 1426 2413 a(x) p Fp 1473 2425 a(c;I) p Fy 1613 2385 a(lim) p Fp 1584 2434 a(n) p Fn(!1) p Fo 1771 2385 a(X) p Fp 1840 2397 a(n;I) p Fy 1939 2385 a(\() p Fo 1970 2385 a(~) p 1971 2385 a(x) p Fy 2019 2385 a(\() p Fo 2046 2369 a(~) 2051 2385 y(t) p Fy(\)\)) p Fo(;) 2237 2369 y(~) 2242 2385 y(t) p Fk 2295 2385 a(2) p Fq 2373 2385 a(C) p Fn 2433 2350 a(I) p Fm 2471 2359 a(d) p Fo 2510 2385 a(;) p Fi 2646 2385 a(if) p Fo 2726 2385 a(I) p Fk 2792 2385 a(2) p 2871 2385 a(K) p Fp 2933 2398 a(~) p 2934 2398 a(x) p Fm 2972 2406 a(c) p Fo 3021 2385 a(;) p Fy 1416 2512 a(0) p Fo(;) p Fi 2646 2512 a(if) p Fo 2726 2512 a(I) p Fk 2792 2512 a(62) p 2871 2512 a(K) p Fp 2933 2525 a(~) p 2934 2525 a(x) p Fm 2972 2533 a(c) p Fo 3021 2512 a(:) p Fi 208 2692 a(Then) p Fo 431 2692 a(~) p 424 2692 a(') p Fn 478 2657 a(\003) p Fi 546 2692 a(is) p 635 2692 a(entir) l(e) p 872 2692 a(and) p 1033 2692 a(satis\014es) p Fo 1220 2863 a(x) p Fp 1267 2875 a(c;I) p Fo 1355 2863 a(') p Fn 1409 2828 a(\003) p Fp 1409 2883 a(I) p Fy 1447 2863 a(\() p Fo(\025) 1522 2847 y(~) 1527 2863 y(t) p Fy 1558 2863 a(\)) p 1613 2863 a(=) p 1701 2863 a(\010) p Fp 1761 2875 a(I) p Fy 1799 2863 a(\() p Fo 1830 2863 a(~) p 1831 2863 a(x) p Fp 1878 2875 a(c) p Fo 1919 2863 a(~) p 1912 2863 a(') p Fn 1966 2828 a(\003) p Fy 2005 2863 a(\() p Fo 2032 2847 a(~) 2037 2863 y(t) p Fy(\)\)) p Fo(;) 2223 2847 y(~) 2228 2863 y(t) p Fk 2281 2863 a(2) p Fq 2359 2863 a(C) p Fn 2419 2828 a(I) p Fm 2457 2837 a(d) p Fo 2496 2863 a(;) p 2563 2863 a(I) p Fk 2629 2863 a(2) p 2707 2863 a(I) p Fp 2752 2875 a(d) p Fo 2805 2863 a(:) p Fi 208 3034 a(wher) l(e,) p 467 3034 a(the) p 605 3034 a(notation) p 936 3034 a(of) p 1034 3034 a(the) p 1172 3034 a(variable) p 1483 3034 a(for) p Fy 1615 3034 a(\010) p Fp 1675 3046 a(I) p Fi 1743 3034 a(is) p 1832 3034 a(as) p 1938 3034 a(in) p 2040 3034 a(The) l(or) l(em) p 2385 3034 a(9.) p 2491 3034 a(It) p 2580 3034 a(also) p 2750 3034 a(holds) p 2962 3034 a(that) p Fo 1355 3243 a(x) p Fp 1402 3255 a(c;I) p Fo 1514 3187 a(@) p 1563 3187 a(') p Fn 1617 3157 a(\003) p Fp 1617 3210 a(I) p 1514 3224 142 4 v Fo 1522 3300 a(@) p 1571 3300 a(t) p Fp 1601 3312 a(J) p Fy 1665 3243 a(\() p Fo 1691 3226 a(~) p Fy 1697 3243 a(0\)) p 1794 3243 a(=) p Fo 1882 3243 a(x) p Fp 1929 3255 a(J) p Fy 1990 3243 a(\003) p Fp 2048 3255 a(I) p 2082 3255 a(J) p Fy 2128 3243 a(\() p Fo 2159 3243 a(~) p 2160 3243 a(x) p Fy(\)) p Fo(;) p 2336 3243 a(I) p 2379 3243 a(;) p 2416 3243 a(J) p Fk 2493 3243 a(2) p 2571 3243 a(I) p Fp 2616 3255 a(d) p Fo 2669 3243 a(;) p Fi 208 3455 a(wher) l(e) p Fy 442 3455 a(\003) p Fi 529 3455 a(is) p 618 3455 a(de\014ne) l(d) p 899 3455 a(in) p 1000 3455 a(\(24\).) -42 3626 y(Pr) l(o) l(of.) p Fy 219 3626 a(F) p 266 3626 a(or) p Fo 366 3626 a(~) p 368 3626 a(z) p Fk 433 3626 a(2) p Fq 511 3626 a(C) p Fn 571 3591 a(I) p Fm 609 3600 a(d) p Fy 648 3626 a(,) p 699 3626 a(de\014ne) p Fk 1567 3737 a(j) p Fo 1589 3737 a(~) p 1590 3737 a(z) p Fk 1633 3737 a(j) p Fn 1656 3749 a(\003) p Fy 1717 3737 a(=) p Fg 1820 3658 a(X) p Fp 1804 3837 a(I) p Fn 1838 3837 a(2I) p Fm 1921 3846 a(d) p Fo 1969 3737 a(v) p Fp 2009 3749 a(L;I) p Fk 2127 3737 a(j) p Fo(z) p Fp 2189 3749 a(I) p Fk 2227 3737 a(j) p Fo(;) p Fy 0 3967 a(where) p Fo 243 3967 a(v) p Fp 283 3979 a(L;I) p Fy 416 3967 a(is) p 502 3967 a(as) p 607 3967 a(in) p 706 3967 a(\(FP2\).) p 990 3967 a(It) p 1082 3967 a(is) p 1168 3967 a(easy) p 1353 3967 a(to) p 1457 3967 a(see) p 1594 3967 a(that) p 1776 3967 a(\(FP2\)) p 2023 3967 a(implies) p 2307 3967 a(that) p Fk 2490 3967 a(j) p 2533 3967 a(\001) p 2576 3967 a(j) p Fn 2599 3979 a(\003) p Fy 2667 3967 a(is) p 2753 3967 a(a) p 2825 3967 a(norm.) p 3081 3967 a(\(Note) p 3317 3967 a(that) p Fo 3499 3967 a(v) p Fp 3539 3979 a(L;I) p Fo 3670 3967 a(>) p Fy 3762 3967 a(0) p 3817 3967 a(,) p Fo 0 4067 a(I) p Fk 66 4067 a(2) p 144 4067 a(I) p Fp 189 4079 a(d) p Fy 242 4067 a(,) p 293 4067 a(implies) p 575 4067 a(uniqueness,) p 1016 4067 a(i.e.,) p 1173 4067 a(that) p Fk 1353 4067 a(j) p Fo(a) p Fk(j) p Fn 1443 4079 a(\003) p Fy 1504 4067 a(=) p 1592 4067 a(0) p 1661 4067 a(implies) p Fo 1943 4067 a(a) p Fy 2010 4067 a(=) p 2097 4067 a(0) p 2153 4067 a(.\)) 125 4166 y(Using) p 358 4166 a(non-negativit) n(y) p 909 4166 a(of) p 1004 4166 a(elemen) n(ts) p 1343 4166 a(of) p Fk 1438 4166 a(J) p Fy 1509 4166 a(,) p 1560 4166 a(w) n(e) p 1682 4166 a(ha) n(v) n(e) p Fk 1153 4349 a(j) p Fy(\() p Fk(J) p Fy 1280 4349 a(\() p Fo 1311 4349 a(~) p 1312 4349 a(x) p Fp 1359 4361 a(c) p Fy 1393 4349 a(\)) p Fo 1437 4349 a(~) p 1439 4349 a(z) p Fy 1482 4349 a(\)) p Fp 1514 4361 a(I) p Fk 1552 4349 a(j) p Fl 1598 4349 a(5) p Fg 1706 4270 a(X) p Fp 1686 4449 a(J) p Fn 1728 4449 a(2I) p Fm 1811 4458 a(d) p Fk 1859 4349 a(J) p Fy 1931 4349 a(\() p Fo 1962 4349 a(~) p 1963 4349 a(x) p Fp 2010 4361 a(c) p Fy 2044 4349 a(\)) p Fp 2076 4361 a(I) p 2110 4361 a(J) p Fk 2171 4349 a(j) p Fo(z) p Fp 2233 4361 a(J) p Fk 2279 4349 a(j) p Fo(;) p 2422 4349 a(I) p Fk 2488 4349 a(2) p 2566 4349 a(I) p Fp 2611 4361 a(d) p Fo 2664 4349 a(:) p Fy 0 4602 a(Hence) p Fk 1394 4702 a(jJ) p Fy 1488 4702 a(\() p Fo 1519 4702 a(~) p 1520 4702 a(x) p Fp 1567 4714 a(c) p Fy 1602 4702 a(\)) p Fo 1646 4702 a(~) p 1648 4702 a(z) p Fk 1690 4702 a(j) p Fn 1713 4714 a(\003) p Fl 1774 4702 a(5) p Fo 1862 4702 a(\025) p Fk(j) p Fo 1932 4702 a(~) p 1933 4702 a(z) p Fk 1976 4702 a(j) p Fn 1999 4714 a(\003) p Fo 2051 4702 a(;) p 2142 4702 a(~) p 2143 4702 a(z) p Fk 2208 4702 a(2) p Fq 2287 4702 a(C) p Fn 2347 4667 a(I) p Fm 2385 4676 a(d) p Fo 2423 4702 a(:) p Fy 3692 4702 a(\(40\)) 125 4844 y(W) p 203 4844 a(e) p 271 4844 a(no) n(w) p 447 4844 a(see) p 585 4844 a(that) p 768 4844 a(Prop) r(osition) p 1218 4844 a(20) p 1333 4844 a(is) p 1419 4844 a(a) p 1492 4844 a(consequence) p 1964 4844 a(of) p 2062 4844 a(the) p 2208 4844 a(follo) n(wing,) p 2586 4844 a(whic) n(h) p 2827 4844 a(is) p 2914 4844 a(pro) n(v) n(ed) p 3187 4844 a(in) p 3288 4844 a(the) p 3434 4844 a(pro) r(of) p 3654 4844 a(of) p 3752 4844 a([6,) 0 4944 y(Prop) r(osition) p 447 4944 a(4.4].) p Fz 0 5100 a(Lemma) p 339 5100 a(21) p Fi 476 5100 a(L) l(et) p Fo 628 5100 a(p) p Fk 709 5100 a(2) p Fq 804 5100 a(N) p Fi(,) p 930 5100 a(and) p 1101 5100 a(let) p Fo 1226 5100 a(H) p Fj 1302 5070 a(\() p Fp(n) p Fj(\)) p Fy 1439 5100 a(:) p Fq 1540 5100 a(C) p Fp 1600 5070 a(p) p Fk 1677 5100 a(!) p Fq 1800 5100 a(C) p Fp 1860 5070 a(p) p Fi 1898 5100 a(,) p Fo 1965 5100 a(n) p Fk 2054 5100 a(2) p Fq 2149 5100 a(N) p Fi(,) p 2275 5100 a(b) l(e) p 2386 5100 a(a) p 2467 5100 a(se) l(quenc) l(e) p 2814 5100 a(of) p 2920 5100 a(p) l(olynomials) p 3381 5100 a(with) p 3571 5100 a(p) l(ositive) 0 5200 y(c) l(o) l(e\016cients,) p 449 5200 a(and) p 610 5200 a(e) l(ach) p 797 5200 a(term) p 995 5200 a(with) p 1175 5200 a(de) l(gr) l(e) l(e) p Fy 1423 5200 a(2) p Fi 1494 5200 a(or) p 1601 5200 a(lar) l(ger,) p 1862 5200 a(which) p 2095 5200 a(satis\014es) p 2407 5200 a(a) p 2479 5200 a(r) l(e) l(cursion) p 2840 5200 a(r) l(elation) p Fo 1363 5371 a(H) p Fj 1439 5337 a(\() p Fp(n) p Fj(+1\)) p Fy 1643 5371 a(=) p Fo 1730 5371 a(H) p Fj 1806 5337 a(\(1\)) p Fk 1914 5371 a(\016) p Fo 1974 5371 a(H) p Fj 2050 5337 a(\() p Fp(n) p Fj(\)) p Fo 2147 5371 a(;) p 2243 5371 a(n) p Fk 2316 5371 a(2) p Fq 2394 5371 a(N) p Fo(:) p Fy 3692 5371 a(\(41\)) p Fi 125 5542 a(L) l(et) p Fo 268 5542 a(@) p 317 5542 a(H) p Fj 393 5512 a(\(1\)) p Fi 511 5542 a(b) l(e) p 613 5542 a(the) p 751 5542 a(Jac) l(obi) p 1007 5542 a(matrix) p 1273 5542 a(of) p Fo 1371 5542 a(H) p Fj 1447 5512 a(\(1\)) p Fi 1536 5542 a(:) p Fo 1083 5781 a(@) p 1132 5781 a(H) p Fj 1208 5738 a(\(1\)) p Fp 1201 5804 a(ij) p Fy 1297 5781 a(\() p Fo(z) p Fy 1372 5781 a(\)) p 1427 5781 a(=) p Fo 1524 5725 a(@) p 1573 5725 a(H) p Fj 1649 5682 a(\(1\)) p Fp 1642 5748 a(i) p 1524 5762 214 4 v Fo 1570 5838 a(@) p 1619 5838 a(z) p Fp 1658 5850 a(j) p Fy 1748 5781 a(\() p Fo(z) p Fy 1823 5781 a(\)) p Fo(;) p 1951 5781 a(i;) p 2017 5781 a(j) p Fk 2079 5781 a(2) p Fy 2157 5781 a(1) p Fo(;) p Fk 2236 5781 a(\001) p 2273 5781 a(\001) p 2310 5781 a(\001) p Fo 2346 5781 a(;) p 2383 5781 a(p;) p 2491 5781 a(z) p Fk 2557 5781 a(2) p Fq 2635 5781 a(C) p Fp 2695 5747 a(p) p Fo 2734 5781 a(:) p Fy 1878 6120 a(17) p 90 rotate dyy eop %%Page: 18 18 18 17 bop Fi 0 83 a(Assume) p 310 83 a(that) p Fo 479 83 a(a) p Fk 546 83 a(2) p Fq 624 83 a(R) p Fp 684 43 a(p) p Fj 684 104 a(+) p Fi 769 83 a(is) p 857 83 a(a) p 929 83 a(\014xe) l(d) p 1120 83 a(p) l(oint) p 1330 83 a(of) p Fo 1427 83 a(H) p Fj 1503 53 a(\(1\)) p Fi 1592 83 a(,) p 1646 83 a(and) p 1807 83 a(that) p 1977 83 a(ther) l(e) p 2183 83 a(exists) p 2410 83 a(a) p 2482 83 a(norm) p Fk 2703 83 a(j) p 2744 83 a(\001) p 2784 83 a(j) p Fn 2807 95 a(\003) p Fi 2875 83 a(on) p Fq 2993 83 a(C) p Fp 3053 53 a(p) p Fi 3121 83 a(and) p Fo 3282 83 a(\025) p 3353 83 a(>) p Fy 3441 83 a(1) p Fi 3512 83 a(such) p 3700 83 a(that) p Fk 1365 280 a(j) p Fo(@) p 1437 280 a(H) p Fj 1513 246 a(\(1\)) p Fy 1602 280 a(\() p Fo(a) p Fy(\)) p Fo(z) p Fk 1753 280 a(j) p Fn 1776 292 a(\003) p Fl 1837 280 a(5) p Fo 1925 280 a(\025) p Fk(j) p Fo(z) p Fk 2039 280 a(j) p Fn 2062 292 a(\003) p Fo 2113 280 a(;) p 2210 280 a(z) p Fk 2275 280 a(2) p Fq 2353 280 a(C) p Fp 2413 246 a(p) p Fo 2452 280 a(;) p Fy 3692 280 a(\(42\)) p Fi 0 462 a(and) p 161 462 a(that) p Fo 331 462 a(x) p Fk 401 462 a(2) p Fq 480 462 a(R) p Fp 540 423 a(p) p Fj 540 483 a(+) p Fi 625 462 a(is) p 714 462 a(a) p 786 462 a(p) l(oint) p 996 462 a(for) p 1129 462 a(which) p 1362 462 a(ther) l(e) p 1569 462 a(exist) p 1762 462 a(p) l(ositive) p 2061 462 a(c) l(onstants) p Fo 2426 462 a(\032) p Fi 2499 462 a(and) p Fo 2660 462 a(C) p Fi 2725 462 a(,) p 2781 462 a(satisfying) p Fo 3150 462 a(\032) p 3216 462 a(<) p Fy 3303 462 a(1) p Fi(,) p 3400 462 a(such) p 3589 462 a(that) p Fk 1109 668 a(j) p Fo(H) p Fj 1208 625 a(\() p Fp(n) p Fj(\)) p Fp 1201 691 a(i) p Fy 1304 668 a(\() p Fo(x) p Fy(\)) p Fk 1435 668 a(\000) p Fo 1518 668 a(a) p Fp 1562 680 a(i) p Fk 1589 668 a(j) p Fl 1636 668 a(5) p Fo 1723 668 a(C) p 1788 668 a(\032) p Fp 1831 634 a(n) p Fo 1877 668 a(;) p 1973 668 a(i) p Fy 2025 668 a(=) p 2112 668 a(1) p Fo(;) p Fk 2191 668 a(\001) p 2228 668 a(\001) p 2265 668 a(\001) p Fo 2301 668 a(;) p 2338 668 a(p;) p 2447 668 a(n) p Fk 2519 668 a(2) p Fq 2598 668 a(Z) p Fj 2653 680 a(+) p Fo 2708 668 a(;) p Fy 3692 668 a(\(43\)) p Fi 0 851 a(and) p 161 851 a(for) p 294 851 a(which) p 527 851 a(the) p 665 851 a(limit) p Fy 1447 950 a(\003\() p Fo(x) p Fy(\)) p 1639 950 a(=) p 1756 950 a(lim) p Fp 1727 1000 a(n) p Fn(!1) p Fo 1914 950 a(\025) p Fn 1962 916 a(\000) p Fp(n) p Fo 2060 950 a(@) p 2109 950 a(H) p Fj 2185 916 a(\() p Fp(n) p Fj(\)) p Fy 2281 950 a(\() p Fo(x) p Fy(\)) p 3692 950 a(\(44\)) p Fi 0 1121 a(exists.) 125 1221 y(Then) p 341 1221 a(ther) l(e) p 548 1221 a(exists) p 775 1221 a(an) p 894 1221 a(entir) l(e) p 1131 1221 a(function) p Fo 1457 1221 a(H) p Fn 1533 1186 a(\003) p Fy 1594 1221 a(=) p 1682 1221 a(\() p Fo(H) p Fn 1790 1186 a(\003) p Fj 1783 1241 a(1) p Fo 1828 1221 a(;) p Fk 1865 1221 a(\001) p 1902 1221 a(\001) p 1939 1221 a(\001) p Fo 1976 1221 a(;) p 2013 1221 a(H) p Fn 2089 1186 a(\003) p Fp 2082 1241 a(p) p Fy 2127 1221 a(\)) p 2182 1221 a(:) p Fq 2258 1221 a(C) p Fp 2318 1186 a(p) p Fk 2379 1221 a(!) p Fq 2485 1221 a(C) p Fp 2545 1186 a(p) p Fi 2613 1221 a(such) p 2802 1221 a(that) p Fy 970 1422 a(lim) p Fp 940 1471 a(n) p Fn(!1) p Fo 1128 1422 a(H) p Fj 1204 1387 a(\() p Fp(n) p Fj(\)) p Fy 1301 1422 a(\() p Fo(x) p Fj 1380 1434 a(1) p Fy 1431 1422 a(exp\() p Fo(\025) p Fn 1638 1387 a(\000) p Fp(n) p Fo 1736 1422 a(t) p Fj 1766 1434 a(1) p Fy 1803 1422 a(\)) p Fo(;) p Fk 1872 1422 a(\001) p 1909 1422 a(\001) p 1946 1422 a(\001) p Fo 1983 1422 a(;) p 2020 1422 a(x) p Fp 2067 1434 a(p) p Fy 2120 1422 a(exp\() p Fo(\025) p Fn 2327 1387 a(\000) p Fp(n) p Fo 2424 1422 a(t) p Fp 2454 1434 a(p) p Fy 2493 1422 a(\)\)) p 2580 1422 a(=) p Fo 2668 1422 a(H) p Fn 2744 1387 a(\003) p Fy 2782 1422 a(\() p Fo(t) p Fy(\)) p Fo(;) p Fi 0 1633 a(uniformly) p 379 1633 a(in) p Fo 481 1633 a(t) p Fy 534 1633 a(=) p 621 1633 a(\() p Fo(t) p Fj 683 1645 a(1) p Fo 721 1633 a(;) p Fk 758 1633 a(\001) p 795 1633 a(\001) p 832 1633 a(\001) p Fo 868 1633 a(;) p 905 1633 a(t) p Fp 935 1645 a(p) p Fy 974 1633 a(\)) p Fi 1036 1633 a(on) p 1154 1633 a(any) p 1314 1633 a(c) l(omp) l(act) p 1634 1633 a(set) p 1763 1633 a(of) p Fq 1861 1633 a(C) p Fp 1921 1603 a(p) p Fi 1959 1633 a(.) p Fo 125 1733 a(H) p Fn 201 1703 a(\003) p Fi 268 1733 a(satis\014es) p Fo 1343 1833 a(H) p Fn 1419 1798 a(\003) p Fy 1457 1833 a(\() p Fo(\025t) p Fy(\)) p 1623 1833 a(=) p Fo 1710 1833 a(H) p Fj 1786 1798 a(\(1\)) p Fy 1875 1833 a(\() p Fo(H) p Fn 1983 1798 a(\003) p Fy 2021 1833 a(\() p Fo(t) p Fy(\)\)) p Fo(;) p 2244 1833 a(t) p Fk 2298 1833 a(2) p Fq 2376 1833 a(C) p Fp 2436 1798 a(p) p Fo 2474 1833 a(;) p Fi 0 1982 a(and) p Fo 1198 2048 a(@) p 1247 2048 a(H) p Fn 1323 2018 a(\003) p Fp 1316 2070 a(i) p 1198 2085 163 4 v Fo 1222 2161 a(@) p 1271 2161 a(t) p Fp 1301 2173 a(j) p Fy 1370 2104 a(\(0\)) p 1499 2104 a(=) p Fo 1587 2104 a(x) p Fp 1634 2116 a(j) p Fy 1670 2104 a(\003) p Fp 1728 2116 a(ij) p Fy 1786 2104 a(\() p Fo(x) p Fy(\)) p Fo(;) p 1994 2104 a(i;) p 2060 2104 a(j) p Fk 2122 2104 a(2) p 2200 2104 a(f) p Fy(1) p Fo(;) p Fy 2321 2104 a(2) p Fo(;) p Fk 2400 2104 a(\001) p 2437 2104 a(\001) p 2474 2104 a(\001) p Fo 2509 2104 a(;) p 2546 2104 a(p) p Fk(g) p Fo(:) p Fi 0 2305 a(F) p 48 2305 a(urthermor) l(e,) p 505 2305 a(if) p Fo 586 2305 a(a) p Fp 630 2317 a(i) p Fy 680 2305 a(=) p 768 2305 a(0) p Fi 839 2305 a(for) p 972 2305 a(some) p Fo 1184 2305 a(i) p Fk 1236 2305 a(2) p 1314 2305 a(f) p Fy(1) p Fo(;) p Fk 1435 2305 a(\001) p 1472 2305 a(\001) p 1509 2305 a(\001) p Fo 1545 2305 a(;) p 1582 2305 a(p) p Fk(g) p Fi(,) p 1720 2305 a(then) p Fo 1904 2305 a(H) p Fn 1980 2275 a(\003) p Fp 1973 2327 a(i) p Fy 2018 2305 a(\() p Fo(t) p Fy(\)) p 2136 2305 a(=) p 2224 2305 a(0) p Fi 2279 2305 a(.) -42 2488 y(R) l(emark.) p Fy 303 2488 a(The) p 469 2488 a(pro) r(of) p 681 2488 a(of) p 770 2488 a([6,) p 881 2488 a(Prop) r(osition) p 1323 2488 a(4.4]) p 1474 2488 a(is) p 1552 2488 a(for) p 1674 2488 a(3SG,) p 1873 2488 a(but) p 2020 2488 a(the) p 2157 2488 a(pro) r(of) p 2369 2488 a(is) p 2447 2488 a(v) p 2486 2488 a(alid) p 2643 2488 a(for) p 2764 2488 a(the) p 2902 2488 a(general) p 3183 2488 a(case) p 3354 2488 a(of) p 3443 2488 a(Lemma) p 3734 2488 a(21.) 0 2587 y(In) p 104 2587 a(the) p 247 2587 a(pro) r(of) p 465 2587 a(of) p 560 2587 a([6) o(,) p 676 2587 a(Prop) r(osition) p 1123 2587 a(4.4],) p 1303 2587 a([6,) p 1419 2587 a(Theorem) p 1770 2587 a(3.5]) p 1927 2587 a(is) p 2011 2587 a(referred) p 2319 2587 a(to,) p 2444 2587 a(whic) n(h) p 2682 2587 a(should) p 2945 2587 a(b) r(e) p 3059 2587 a(replaced,) p 3410 2587 a(in) p 3507 2587 a(the) p 3651 2587 a(pro) r(of) 0 2687 y(of) p 94 2687 a(Lemma) p 390 2687 a(21,) p 523 2687 a(b) n(y) p 666 2687 a(lim) p Fp 637 2737 a(n) p Fn(!1) p Fo 825 2687 a(H) p Fj 901 2653 a(\() p Fp(n) p Fj(\)) p Fy 997 2687 a(\() p Fo(x) p Fy(\)) p 1132 2687 a(=) p Fo 1220 2687 a(a) p Fy(,) p 1314 2687 a(whic) n(h) p 1551 2687 a(is) p 1634 2687 a(a) p 1702 2687 a(consequence) p 2170 2687 a(of) p 2264 2687 a(the) p 2406 2687 a(assumption) p 2845 2687 a(\(43\).) p 3052 2687 a(Similarly) p 3371 2687 a(,) p 3421 2687 a(Prop) r(osition) 0 2816 y(3.8,) p 157 2816 a(Prop) r(osition) p 604 2816 a(4.3,) p 761 2816 a(and) p 922 2816 a(\(4.4\)) p 1120 2816 a(in) p 1217 2816 a([6]) p 1333 2816 a(are) p 1471 2816 a(replaced) p 1799 2816 a(b) n(y) p 1914 2816 a(\(43\),) p 2112 2816 a(\(44\),) p 2311 2816 a(and) p 2472 2816 a(\(41\),) p 2671 2816 a(resp) r(ectiv) n(ely) p 3090 2816 a(.) p Fd 3774 2915 a(3) p Fy 125 3198 a(T) p 178 3198 a(o) p 255 3198 a(apply) p 491 3198 a(Lemma) p 795 3198 a(21) p 914 3198 a(to) p 1023 3198 a(the) p 1174 3198 a(pro) r(of) p 1399 3198 a(of) p 1502 3198 a(Prop) r(osition) p 1957 3198 a(20,) p 2100 3198 a(w) n(e) p 2231 3198 a(only) p 2421 3198 a(need) p 2622 3198 a(to) p 2732 3198 a(note) p 2924 3198 a(that) p 3112 3198 a(Prop) r(osition) p 3567 3198 a(4,) p 3669 3198 a(\(40\),) 0 3297 y(Lemma) p 296 3297 a(19,) p 430 3297 a(and) p 592 3297 a(Prop) r(osition) p 1038 3297 a(17,) p 1172 3297 a(corresp) r(ond) p 1595 3297 a(to) p 1697 3297 a(\(41\),) p 1895 3297 a(\(42\),) p 2093 3297 a(\(43\),) p 2292 3297 a(and) p 2453 3297 a(\(44\),) p 2651 3297 a(resp) r(ectiv) n(ely) p 3070 3297 a(.) p Fd 3778 3397 a(2) p Fi -42 3762 a(Pr) l(o) l(of) p 181 3762 a(of) p 279 3762 a(The) l(or) l(em) p 624 3762 a(9.) p Fy 733 3762 a(Assume) p 1041 3762 a(that) p Fo 1221 3762 a(x) p Fp 1268 3774 a(c;I) p Fk 1379 3762 a(6) p Fy(=) p 1466 3762 a(0) p 1535 3762 a(and) p Fo 1696 3762 a(~) p 1697 3762 a(x) p Fk 1767 3762 a(2) p 1846 3762 a(D) p Fi 1912 3762 a(om) p Fy 2029 3762 a(\() p Fo 2060 3762 a(~) p 2061 3762 a(x) p Fp 2108 3774 a(c) p Fy 2142 3762 a(\)) p 2189 3762 a(.) 125 3862 y(Denote) p 417 3862 a(b) n(y) p Fo 539 3862 a(p) p Fp 580 3875 a(~) p 581 3875 a(x;n;I) p Fy 737 3862 a(,) p 797 3862 a(the) p 947 3862 a(join) n(t) p 1148 3862 a(distribution) p 1608 3862 a(of) p 1710 3862 a(\() p Fo(\025) p Fn 1790 3828 a(\000) p Fp(n) p Fo 1888 3862 a(s) p Fp 1927 3874 a(J) p Fy 1973 3862 a(,) p Fo 2033 3862 a(J) p Fk 2122 3862 a(2) p 2213 3862 a(I) p Fp 2258 3874 a(d) p Fy 2297 3862 a(\)) p 2364 3862 a(under) p Fo 2607 3862 a(\026) p Fp 2656 3875 a(~) p 2657 3875 a(x) o(;n;I) p Fy 2827 3862 a(.) p 2908 3862 a(Its) p 3038 3862 a(generating) p 3452 3862 a(function) p 3784 3862 a(is) 0 3962 y(expressed,) p 396 3962 a(with) p 585 3962 a(the) p 728 3962 a(de\014nitions) p 1129 3962 a(\(9\)) p 1263 3962 a(and) p 1425 3962 a(\(23\),) p 1623 3962 a(as) p Fg 1162 4091 a(Z) p Fn 1245 4112 a(1) p Fj 1208 4280 a(0) p Fo 1329 4204 a(e) p Fp 1364 4159 a(~) 1368 4170 y(t) p Fn(\001) p Fp 1415 4155 a(~) 1413 4170 y(\030) p Fo 1449 4204 a(p) p Fp 1490 4217 a(~) p 1491 4217 a(x) o(;n;I) p Fy 1647 4204 a([) p Fo(d) 1715 4182 y(~) 1713 4204 y(\030) p Fy 1753 4204 a(]) p 1799 4204 a(=) p Fo 1897 4148 a(X) p Fp 1966 4160 a(n;I) p Fy 2065 4148 a(\() p Fo 2096 4148 a(~) p 2097 4148 a(x) p Fy(\() p Fo 2171 4133 a(~) 2176 4148 y(t) p Fy 2207 4148 a(\)\)) p 1897 4185 375 4 v Fo 1944 4261 a(X) p Fp 2013 4273 a(n;I) p Fy 2112 4261 a(\() p Fo 2143 4261 a(~) p 2144 4261 a(x) p Fy 2192 4261 a(\)) p Fo 2295 4204 a(;) 2382 4189 y(~) 2387 4204 y(t) p Fk 2440 4204 a(2) p Fq 2519 4204 a(C) p Fn 2579 4170 a(I) p Fm 2617 4179 a(d) p Fo 2655 4204 a(;) p Fy 3692 4204 a(\(45\)) 0 4451 y(where) p 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4761 a(20) p 921 4761 a(and) p 1085 4761 a(\(D) n(A1\)) p 1344 4761 a(in) p 1443 4761 a(\(45\),) p 1643 4761 a(w) n(e) p 1767 4761 a(see) p 1904 4761 a(that) p 2085 4761 a(the) p 2230 4761 a(righ) n(t) p 2433 4761 a(hand) p 2643 4761 a(side) p 2811 4761 a(of) p 2908 4761 a(\(45\)) p 3085 4761 a(con) n(v) n(erges) p 3459 4761 a(to) p Fo 3562 4761 a(') p Fn 3616 4726 a(\003) p Fp 3616 4781 a(I) p Fy 3655 4761 a(\() p Fo 3682 4745 a(~) 3687 4761 y(t) p Fy(\)) p 3775 4761 a(=) p Fo 0 4860 a(') p Fn 54 4826 a(\003) p Fp 53 4881 a(~) p 54 4881 a(x;I) p Fy 150 4860 a(\() p Fo 177 4845 a(~) 182 4860 y(t) p Fy(\)) p 274 4860 a(of) p 370 4860 a(Prop) r(osition) p 819 4860 a(20,) p 955 4860 a(uniformly) p 1335 4860 a(in) p Fo 1428 4845 a(~) 1433 4860 y(t) p Fy 1493 4860 a(on) p 1610 4860 a(compact) p 1943 4860 a(sets.) p 2143 4860 a(Hence) p Fo 2392 4860 a(p) p Fp 2433 4873 a(~) p 2434 4873 a(x) o(;n;I) p Fy 2619 4860 a(con) n(v) n(erges) p 2993 4860 a(w) n(eakly) p 3269 4860 a(to) p 3372 4860 a(a) p 3443 4860 a(probabilit) n(y) 0 4975 y(measure) p Fo 324 4975 a(p) p Fn 366 4941 a(\003) p Fp 365 4996 a(~) p 366 4996 a(x) o(;I) p Fy 488 4975 a(whose) p 733 4975 a(generating) p 1139 4975 a(function) p 1465 4975 a(is) p Fo 1548 4975 a(') p Fn 1602 4941 a(\003) p Fp 1602 4996 a(I) p Fy 1641 4975 a(\() p Fo 1668 4960 a(~) 1673 4975 y(t) p Fy(\).) 125 5075 y(W) p 203 5075 a(e) p 276 5075 a(therefore) p 633 5075 a(see) p 775 5075 a(that) p 963 5075 a(the) p 1115 5075 a(claims) p 1377 5075 a(in) p 1482 5075 a(Theorem) p 1841 5075 a(9,) p 1944 5075 a(except) p 2213 5075 a(those) p 2438 5075 a(on) p 2562 5075 a(the) p 2713 5075 a(existence) p 3074 5075 a(and) p 3244 5075 a(p) r(ositivit) n(y) p 3622 5075 a(of) p 3725 5075 a(the) 0 5174 y(densit) n(y) p 289 5174 a(of) p 393 5174 a(\026) p Fo 386 5174 a(p) p Fn 428 5140 a(\003) p Fp 427 5197 a(~) p 428 5197 a(x;) p Fj(\(1\)) p Fy 575 5174 a(,) p 629 5174 a(are) p 770 5174 a(direct) p 1008 5174 a(consequences) p 1512 5174 a(of) p 1609 5174 a(Prop) r(osition) p 2059 5174 a(17) p 2172 5174 a(and) p 2336 5174 a(Prop) r(osition) p 2786 5174 a(20.) p 2936 5174 a(That) p 3147 5174 a(\(25\)) p 3324 5174 a(determines) p Fo 3754 5174 a(~) p 3748 5174 a(') p Fn 3802 5140 a(\003) p Fy 0 5274 a(follo) n(ws) p 279 5274 a(from) p 481 5274 a(the) p 630 5274 a(fact) p 800 5274 a(that) p 986 5274 a(its) p 1108 5274 a(T) p 1161 5274 a(a) n(ylor) p 1375 5274 a(co) r(e\016cien) n(ts) p 1801 5274 a(are) p 1945 5274 a(determined) p 2385 5274 a(recursiv) n(ely) p 2805 5274 a(b) n(y) p 2926 5274 a(\(25\).) p 3152 5274 a(\(Note) p 3391 5274 a(that) p 3577 5274 a(\010) p Fp 3637 5286 a(I) p Fy 3709 5274 a(is) p 3798 5274 a(a) 0 5374 y(p) r(olynomial) p 426 5374 a(with) p 613 5374 a(eac) n(h) p 797 5374 a(term) p 993 5374 a(ha) n(ving) p 1258 5374 a(degree) p 1514 5374 a(2) p 1581 5374 a(or) p 1680 5374 a(larger.) p 1947 5374 a(Hence) p 2191 5374 a(when) p 2406 5374 a(comparing) p 2809 5374 a(the) p 2950 5374 a(co) r(e\016cien) n(ts) p 3367 5374 a(of) p Fo 3459 5374 a(t) p Fp 3489 5344 a(N) p Fy 3577 5374 a(in) p 3672 5374 a(b) r(oth) 0 5473 y(hand) p 209 5473 a(sides) p 410 5473 a(of) p 506 5473 a(\(25\),) p 707 5473 a(the) p 851 5473 a(righ) n(t) p 1054 5473 a(hand) p 1263 5473 a(side) p 1431 5473 a(con) n(tains) p 1759 5473 a(co) r(e\016cien) n(ts) p 2180 5473 a(for) p 2309 5473 a(lo) n(w) n(er) p 2528 5473 a(order) p 2747 5473 a(than) p Fo 2942 5473 a(N) p Fy 3018 5473 a(,) p 3071 5473 a(hence) p 3303 5473 a(the) p 3447 5473 a(co) r(e\016cien) n(ts) 0 5573 y(of) p Fo 101 5573 a(~) p 95 5573 a(') p Fn 149 5539 a(\003) p Fy 215 5573 a(is) p 298 5573 a(determined) p 732 5573 a(inductiv) n(ely) p 1159 5573 a(in) p Fo 1255 5573 a(N) p Fy 1331 5573 a(.\)) 125 5698 y(T) p 178 5698 a(o) p 247 5698 a(pro) n(v) n(e) p 471 5698 a(that) p 658 5698 a(\026) p Fo 651 5698 a(p) p Fn 693 5663 a(\003) p Fp 692 5718 a(~) p 693 5718 a(x) o(;I) p Fy 815 5698 a(has) p 964 5698 a(a) p Fo 1033 5698 a(C) p Fn 1098 5667 a(1) p Fy 1196 5698 a(densit) n(y) p 1448 5698 a(,) p 1499 5698 a(w) n(e) p 1621 5698 a(prepare) p 1921 5698 a(the) p 2064 5698 a(follo) n(wing.) 1878 6120 y(18) p 90 rotate dyy eop %%Page: 19 19 19 18 bop Fy 125 83 a(Denote) p 409 83 a(b) n(y) p 532 83 a(\026) p Fo 525 83 a(p) p Fp 566 96 a(~) p 567 96 a(x) o(;n;I) p Fy 723 83 a(,) p 773 83 a(the) p 916 83 a(distribution) p 1369 83 a(of) p Fo 1464 83 a(\025) p Fn 1512 53 a(\000) p Fp(n) p Fo 1609 83 a(L) p Fy 1693 83 a(under) p Fo 1929 83 a(\026) p Fp 1978 96 a(~) p 1979 96 a(x) o(;n;I) p Fy 2149 83 a(.) p 2209 83 a(Its) p 2331 83 a(generating) p 2738 83 a(function) p 3063 83 a(is,) p 3169 83 a(from) p 3366 83 a(\(7\)) p 3499 83 a(and) p 3661 83 a(\(45\),) p Fg 1191 202 a(Z) p Fn 1274 222 a(1) p Fj 1237 390 a(0) p Fo 1358 315 a(e) p Fp 1397 281 a(t\030) p Fy 1465 315 a(\026) p Fo 1458 315 a(p) p Fp 1499 328 a(~) p 1500 328 a(x) o(;n;I) p Fy 1656 315 a([) p Fo(d\030) p Fy 1762 315 a(]) p 1808 315 a(=) p Fo 1906 259 a(X) p Fp 1975 271 a(n;I) p Fy 2074 259 a(\() p Fo 2105 259 a(~) p 2106 259 a(x) p Fy(\() p Fo(t) m(~) p 2215 259 a(e) p Fy(\)\)) p 1906 296 413 4 v Fo 1973 372 a(X) p Fp 2042 384 a(n;I) p Fy 2140 372 a(\() p Fo 2171 372 a(~) p 2172 372 a(x) p Fy 2220 372 a(\)) p Fo 2343 315 a(;) p 2435 315 a(t) p Fk 2488 315 a(2) p Fq 2566 315 a(C) p Fo(;) p Fy 3692 315 a(\(46\)) 0 547 y(where) p Fo 237 547 a(~) p 240 547 a(e) p Fy 302 547 a(=) p 389 547 a(\() p Fo(e) p Fp 460 559 a(J) p Fo 507 547 a(;) p 567 547 a(J) p Fk 644 547 a(2) p 722 547 a(I) p Fp 767 559 a(d) p Fy 806 547 a(\)) p 866 547 a(is) p 950 547 a(giv) n(en) p 1166 547 a(b) n(y) p Fo 1761 646 a(e) p Fp 1800 658 a(J) p Fy 1869 646 a(=) p Fk 1956 646 a(j) p Fo(J) p Fk 2033 646 a(j) p Fo(:) p Fy 3692 646 a(\(47\)) 125 794 y(According) p 519 794 a(to) p 621 794 a(what) p 828 794 a(has) p 976 794 a(b) r(een) p 1172 794 a(pro) n(v) n(ed) p 1442 794 a(b) r(elo) n(w) p 1678 794 a(\(45\),) p 1876 794 a(\(46\)) p 2051 794 a(con) n(v) n(erges) p 2423 794 a(to) p 2525 794 a(an) p 2640 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1254 a(g) p Fp 855 1266 a(I) p Fy 906 1254 a(.) 125 1354 y(Prop) r(osition) p 571 1354 a(20) p 682 1354 a(also) p 849 1354 a(implies) p 1130 1354 a(that) p Fo 800 1576 a(x) p Fp 847 1588 a(c;I) p Fo 958 1520 a(dg) p Fp 1041 1532 a(I) p 958 1557 121 4 v Fo 982 1633 a(dt) p Fy 1089 1576 a(\(0\)) p 1218 1576 a(=) p Fo 1306 1576 a(x) p Fp 1353 1588 a(c;I) p Fg 1468 1463 a(Z) p Fn 1552 1484 a(1) p Fj 1515 1652 a(0) p Fo 1636 1576 a(\030) p Fy 1683 1576 a(\026) p Fo 1676 1576 a(p) p Fn 1718 1542 a(\003) p Fp 1717 1597 a(~) p 1718 1597 a(x) o(;I) p Fy 1813 1576 a([) p Fo(d\030) p Fy 1919 1576 a(]) p 1966 1576 a(=) p Fg 2073 1498 a(X) p Fp 2053 1676 a(J) p Fn 2095 1676 a(2I) p Fm 2178 1685 a(d) p Fo 2226 1576 a(e) p Fp 2265 1588 a(J) p Fo 2325 1576 a(x) p Fp 2372 1588 a(J) p Fy 2433 1576 a(\003) p Fp 2491 1588 a(I) p 2525 1588 a(J) p Fy 2571 1576 a(\() p Fo 2602 1576 a(~) p 2603 1576 a(x) p Fy 2651 1576 a(\)) p Fo(;) p 2775 1576 a(I) p Fk 2841 1576 a(2) p 2920 1576 a(I) p Fp 2965 1588 a(d) p 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p Fy 1646 2267 a(=) p 1733 2267 a(0) p Fo(;) p Fy 1895 2267 a(if) p Fo 1971 2267 a(x) p Fp 2018 2279 a(c;J) p Fy 2137 2267 a(=) p 2225 2267 a(0) p Fo 2281 2267 a(:) p Fz 0 2497 a(Lemma) p 339 2497 a(22) p Fi 548 2497 a(\(i\)) p Fo 683 2497 a(M) p Fp 764 2509 a(I) p Fy 825 2497 a(=) p Fo 912 2497 a(M) p Fp 993 2509 a(I) p Fy 1031 2497 a(\() p Fo 1062 2497 a(~) p 1063 2497 a(x) p Fy 1111 2497 a(\)) p 1166 2497 a(:=) p Fg 1277 2384 a(Z) p Fn 1360 2405 a(1) p Fj 1323 2573 a(0) p Fo 1444 2497 a(\030) p Fy 1491 2497 a(\026) p Fo 1484 2497 a(p) p Fn 1526 2463 a(\003) p Fp 1525 2518 a(~) p 1526 2518 a(x;I) p Fy 1621 2497 a([) p Fo(d\030) p Fy 1727 2497 a(]) p Fo 1774 2497 a(>) p Fy 1862 2497 a(0) p Fo(;) p 2029 2497 a(I) p Fk 2095 2497 a(2) p 2174 2497 a(K) p Fp 2236 2510 a(~) p 2237 2510 a(x) p Fm 2275 2518 a(c) p Fo 2324 2497 a(;) p 2389 2497 a(~) p 2390 2497 a(x) p Fk 2461 2497 a(2) p Fy 2539 2497 a(\004) p Fp 2594 2509 a(d) p Fk 2652 2497 a(\\) p 2725 2497 a(D) p Fi 2791 2497 a(om) p Fy 2908 2497 a(\() p Fo 2939 2497 a(~) p 2940 2497 a(x) p Fp 2987 2509 a(c) p Fy 3022 2497 a(\)) p Fo(:) p Fi 47 2709 a(\(ii\)) p 208 2709 a(If) p Fo 294 2709 a(~) p 296 2709 a(x) p Fk 367 2709 a(2) p Fy 447 2709 a(\004) p Fp 502 2721 a(d) p Fk 560 2709 a(\\) p 635 2709 a(D) p Fi 701 2709 a(om) p Fy 818 2709 a(\() p Fo 849 2709 a(~) p 850 2709 a(x) p Fp 897 2721 a(c) p Fy 931 2709 a(\)) p Fi 994 2709 a(and) p Fo 1156 2709 a(I) p Fk 1224 2709 a(2) p 1304 2709 a(K) p Fp 1366 2722 a(~) p 1367 2722 a(x) p Fm 1405 2730 a(c) p Fi 1470 2709 a(then) p Fy 1663 2709 a(\026) p Fo 1656 2709 a(p) p Fn 1698 2675 a(\003) p Fp 1697 2730 a(~) p 1698 2730 a(x) o(;I) p Fi 1823 2709 a(is) p 1913 2709 a(not) p 2060 2709 a(c) l(onc) l(entr) l(ate) l(d) p 2538 2709 a(on) p 2657 2709 a(a) p 2730 2709 a(single) p 2964 2709 a(p) l(oint) p 3175 2709 a(\(i.e.,) p 3380 2709 a(has) p 3529 2709 a(non-zer) l(o) 208 2809 y(varianc) l(e\).) p 607 2809 a(Namely,) p Fo 649 3036 a(V) p Fp 697 3048 a(I) p Fy 758 3036 a(=) p 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4033 a(0) p Fo(;) p Fy 208 4233 a(where) p 448 4233 a(w) n(e) p 570 4233 a(used) p 759 4233 a(\(47\)) p 935 4233 a(for) p Fo 1062 4233 a(e) p Fj 1101 4248 a(\(1\)) p Fo 1213 4233 a(>) p Fy 1300 4233 a(0,) p Fo 1391 4233 a(~) p 1392 4233 a(x) p Fk 1463 4233 a(2) p Fy 1541 4233 a(\004) p Fp 1596 4245 a(d) p Fk 1654 4233 a(\\) p 1728 4233 a(D) p Fi 1794 4233 a(om) p Fy 1911 4233 a(\() p Fo 1942 4233 a(~) p 1943 4233 a(x) p Fp 1990 4245 a(c) p Fy 2024 4233 a(\)) p 2084 4233 a(for) p Fo 2211 4233 a(x) p Fj 2258 4248 a(\(1\)) p Fo 2371 4233 a(>) p Fy 2459 4233 a(0,) p 2551 4233 a(and) p 2712 4233 a(\(36\)) p 2888 4233 a(for) p 3015 4233 a(\003) p Fp 3073 4248 a(I) p Fj 3118 4248 a(\(1\)) p Fy 3207 4233 a(\() p Fo 3238 4233 a(~) p 3239 4233 a(x) p Fy(\)) p Fo 3342 4233 a(>) p Fy 3429 4233 a(0) p 3485 4233 a(.) 55 4398 y(\(ii\)) p 208 4398 a(By) p 338 4398 a(de\014nition) p 707 4398 a(\(Prop) r(osition) p 1186 4398 a(4) p 1255 4398 a(and) p 1417 4398 a(\(9\)\),) p 1606 4398 a(w) n(e) p 1728 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4998 a(s) p Fy(\() p Fo(w) p Fy 1957 4998 a(\)) p 2013 4998 a(=) p Fo 2101 4998 a(m) p Fk(g) p Fo(;) p 2335 4998 a(I) p Fk 2401 4998 a(2) p 2479 4998 a(I) p Fp 2524 5010 a(d) p Fo 2577 4998 a(;) p 2642 4998 a(m) p Fk 2738 4998 a(2) p Fq 2816 4998 a(Z) p Fn 2871 4960 a(I) p Fm 2909 4969 a(d) p Fj 2871 5018 a(+) p Fo 2948 4998 a(:) p Fy 208 5146 a(Then) p 424 5146 a(\(50\)) p 600 5146 a(implies) p Fo 926 5342 a(x) p Fp 973 5354 a(c;I) p Fo 1061 5342 a(g) p Fp 1101 5354 a(I) p Fy 1138 5342 a(\() p Fo(\025t) p Fy(\)) p 1304 5342 a(=) p 1392 5342 a(\010) p Fp 1452 5354 a(I) p Fy 1490 5342 a(\() p Fo 1521 5342 a(~) p 1522 5342 a(x) p Fp 1569 5354 a(c) p Fo 1614 5342 a(~) p 1617 5342 a(g) p Fy 1660 5342 a(\() p Fo(t) p Fy(\)\)) p 1810 5342 a(=) p Fg 1945 5264 a(X) p Fp 1898 5465 a(m) p Fn(2) p Fc(Z) p Fe 2041 5430 a(I) p Fm 2075 5445 a(d) p Ff 2041 5484 a(+) p Fo 2141 5342 a(C) p Fp 2200 5354 a(I) p 2234 5354 a(;m) p Fg 2357 5264 a(Y) p Fp 2331 5442 a(J) p Fn 2373 5442 a(2I) p Fm 2456 5451 a(d) p Fo 2504 5342 a(g) p Fp 2544 5354 a(J) p Fy 2590 5342 a(\() p Fo(t) p Fy(\)) p Fp 2684 5308 a(m) p Fm 2743 5316 a(J) p Fo 2787 5342 a(;) p 2907 5342 a(t) p Fk 2960 5342 a(2) p Fq 3039 5342 a(C) p Fo(;) p Fy 3692 5342 a(\(51\)) 208 5646 y(where) p 448 5646 a(w) n(e) p 570 5646 a(wrote) p Fo 1610 5745 a(C) p Fp 1669 5757 a(I) p 1703 5757 a(;m) p Fy 1809 5745 a(=) p Fo 1897 5745 a(b) p Fp 1933 5757 a(I) p 1967 5757 a(;m) p Fg 2103 5666 a(Y) p Fp 2077 5844 a(J) p Fn 2119 5844 a(2I) p Fm 2202 5853 a(d) p Fo 2250 5745 a(x) p Fp 2297 5708 a(m) p Fm 2356 5716 a(J) p Fp 2297 5770 a(c;J) p Fo 2415 5745 a(:) p Fy 1878 6120 a(19) p 90 rotate dyy eop %%Page: 20 20 20 19 bop Fy 208 83 a(No) n(w) p 407 83 a(assume) p 704 83 a(that) p 901 83 a(\026) p Fo 894 83 a(p) p Fp 935 96 a(~) p 936 96 a(x) o(;I) p Fy 1068 83 a(is) p 1162 83 a(concen) n(trated) p 1663 83 a(on) p 1789 83 a(a) p 1868 83 a(single) p 2109 83 a(p) r(oin) n(t) p 2336 83 a(for) p 2473 83 a(some) p Fo 2691 83 a(I) p Fk 2774 83 a(2) p 2869 83 a(K) p Fp 2931 96 a(~) p 2932 96 a(x) p Fm 2970 104 a(c) p Fy 3019 83 a(.) p 3109 83 a(Then) p 3336 83 a(its) p 3462 83 a(generating) 208 183 y(function) p 533 183 a(is) p Fo 1788 282 a(g) p Fp 1828 294 a(I) p Fy 1866 282 a(\() p Fo(t) p Fy(\)) p 1984 282 a(=) p Fo 2071 282 a(e) p Fp 2110 248 a(M) p Fm 2173 256 a(I) p Fp 2207 248 a(t) p Fo 2236 282 a(:) p Fy 3692 282 a(\(52\)) 208 432 y(Di\013eren) n(tiating) p 762 432 a(\(51\)) p 939 432 a(b) n(y) p Fo 1056 432 a(t) p Fy 1115 432 a(up) p 1237 432 a(to) p 1340 432 a(t) n(wice,) p 1579 432 a(putting) p Fo 1876 432 a(t) p Fy 1932 432 a(=) p 2023 432 a(0,) p 2117 432 a(and) p 2280 432 a(using) p 2499 432 a(\(52\)) p 2676 432 a(on) p 2793 432 a(the) p 2938 432 a(left) p 3085 432 a(hand) p 3294 432 a(side,) p 3485 432 a(w) n(e) p 3609 432 a(obtain) 208 531 y(equations,) p 602 531 a(whic) n(h) p 840 531 a(ev) n(en) n(tually) p 1237 531 a(lead) p 1412 531 a(to) 914 726 y(\(0) p Fl 1010 726 a(5) p Fy(\)) p Fg 1176 647 a(X) p Fp 1207 821 a(m) p Fo 1310 726 a(C) p Fp 1369 738 a(I) p 1403 738 a(;m) p Fy 1486 726 a(\() p Fo(\025M) p Fp 1647 738 a(I) p Fk 1704 726 a(\000) p Fg 1787 647 a(X) p Fp 1826 825 a(J) p Fo 1921 726 a(m) p Fp 1994 738 a(J) p Fo 2040 726 a(M) p Fp 2121 738 a(J) p Fy 2167 726 a(\)) p Fj 2199 692 a(2) p Fy 2260 726 a(=) p Fk 2347 726 a(\000) p Fg 2426 647 a(X) p Fp 2456 821 a(m) p Fo 2559 726 a(C) p Fp 2618 738 a(I) p 2652 738 a(;m) p Fg 2749 647 a(X) p Fp 2788 825 a(J) p Fo 2883 726 a(m) p Fp 2956 738 a(J) p Fo 3002 726 a(V) p Fp 3050 738 a(J) p Fo 3111 726 a(:) p Fy 208 982 a(Since) p Fo 423 982 a(C) p Fp 482 994 a(I) p 516 994 a(;m) p Fy 599 982 a(,) p Fo 649 982 a(m) p Fp 722 994 a(J) p Fy 769 982 a(,) p 819 982 a(and) p Fo 979 982 a(V) p Fp 1027 994 a(J) p Fy 1100 982 a(are) p 1238 982 a(non-negativ) n(e,) p 1749 982 a(it) p 1831 982 a(follo) n(ws) p 2103 982 a(that) p Fo 2282 982 a(V) p Fp 2330 994 a(J) p Fy 2400 982 a(=) p 2487 982 a(0) p 2555 982 a(for) p 2681 982 a(all) p Fo 2796 982 a(J) p Fk 2873 982 a(2) p 2951 982 a(I) p Fp 2996 994 a(d) p Fy 3062 982 a(suc) n(h) p 3248 982 a(that) p 3427 982 a(there) p 3638 982 a(exists) p Fo 208 1092 a(m) p Fk 304 1092 a(2) p Fq 382 1092 a(Z) p Fn 437 1054 a(I) p Fm 475 1063 a(d) p Fj 437 1112 a(+) p Fy 540 1092 a(satisfying) p Fo 909 1092 a(m) p Fp 982 1104 a(J) p Fo 1051 1092 a(>) p Fy 1139 1092 a(0) p 1207 1092 a(and) p Fo 1367 1092 a(m) p Fp 1440 1104 a(K) p Fy 1527 1092 a(=) p 1614 1092 a(0) p 1682 1092 a(if) p Fo 1757 1092 a(K) p Fk 1857 1092 a(62) p 1935 1092 a(K) p Fp 1997 1105 a(~) p 1998 1105 a(x) p Fm 2036 1113 a(c) p Fy 2085 1092 a(.) p 2145 1092 a(In) p 2247 1092 a(other) p 2463 1092 a(w) n(ords,) p Fo 2723 1092 a(V) p Fp 2771 1104 a(J) p Fy 2841 1092 a(=) p 2928 1092 a(0) p 2996 1092 a(for) p 3122 1092 a(all) p Fo 3236 1092 a(J) p Fk 3313 1092 a(2) p 3392 1092 a(I) p Fp 3437 1104 a(d) p Fy 3502 1092 a(suc) n(h) p 3688 1092 a(that) 208 1191 y(\010) p Fp 268 1203 a(I) p Fy 306 1191 a(\() p Fo 337 1191 a(~) p 338 1191 a(x) p Fy(\)) p 448 1191 a(has) p 598 1191 a(a) p 670 1191 a(term) p 870 1191 a(whic) n(h) p 1110 1191 a(con) n(tains) p 1438 1191 a(a) p 1510 1191 a(p) r(ositiv) n(e) p 1819 1191 a(p) r(o) n(w) n(er) p 2064 1191 a(of) p Fo 2161 1191 a(x) p Fp 2208 1203 a(J) p Fy 2285 1191 a(and) p 2449 1191 a(comp) r(osed) p 2832 1191 a(of) p 2929 1191 a(those) p Fo 3148 1191 a(x) p Fp 3195 1203 a(K) p Fy 3290 1191 a(with) p Fo 3481 1191 a(K) p Fk 3585 1191 a(2) p 3667 1191 a(K) p Fp 3729 1204 a(~) p 3730 1204 a(x) p Fm 3768 1212 a(c) p Fy 3817 1191 a(.) 208 1291 y(With) p 428 1291 a(\(FP3\)) p 678 1291 a(w) n(e) p 806 1291 a(see) p 946 1291 a(that) p Fo 1131 1291 a(J) p Fy 1218 1291 a(=) p 1315 1291 a(\(1\)) p 1454 1291 a(has) p 1608 1291 a(suc) n(h) p 1801 1291 a(prop) r(ert) n(y) p 2146 1291 a(for) p 2279 1291 a(all) p Fo 2400 1291 a(I) p Fk 2475 1291 a(2) p 2563 1291 a(K) p Fp 2625 1304 a(~) p 2626 1304 a(x) p Fm 2664 1312 a(c) p Fy 2699 1291 a(.) p 2776 1291 a(Hence) p Fo 3029 1291 a(V) p Fj 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p 1434 2237 a(if) p Fo 1601 2432 a(\025) p 1663 2432 a(M) p Fj 1744 2447 a(\(1\)) p Fy 1856 2432 a(=) p Fg 1987 2353 a(X) p Fp 1944 2531 a(J) p Fn 1986 2531 a(2K) p Fm 2081 2541 a(~) p 2082 2541 a(x) 2115 2549 y(c) p Fo 2163 2432 a(M) p Fp 2244 2444 a(J) p Fo 2304 2432 a(m) p Fp 2377 2444 a(J) p Fo 2423 2432 a(;) p Fy 208 2706 a(for) p 335 2706 a(all) p Fo 450 2706 a(m) p Fy 551 2706 a(suc) n(h) p 738 2706 a(that) p Fo 918 2706 a(C) p Fj 977 2721 a(\(1\)) p Fp(;m) p Fo 1168 2706 a(>) p Fy 1255 2706 a(0) p 1311 2706 a(.) p 1371 2706 a(Prop) r(osition) p 1818 2706 a(4) p 1887 2706 a(implies) p 2169 2706 a(that) p Fo 2349 2706 a(C) p Fj 2408 2721 a(\(1\)) p Fp(;m) p Fo 2599 2706 a(>) p Fy 2686 2706 a(0) p 2756 2706 a(for) p Fo 2883 2706 a(m) p Fy 2983 2706 a(suc) n(h) p 3170 2706 a(that;) 243 2872 y(\(a\)) p Fo 390 2872 a(m) p Fj 463 2887 a(\(1\)) p Fy 575 2872 a(=) p 663 2872 a(2) p 732 2872 a(and) p Fo 894 2872 a(m) p Fp 967 2884 a(J) p Fy 1036 2872 a(=) p 1123 2872 a(0,) p Fo 1216 2872 a(J) p Fk 1293 2872 a(6) p Fy(=) p 1380 2872 a(\(1\),) 238 3005 y(\(b\)) p Fo 390 3005 a(m) p Fj 463 3020 a(\(1\)) p Fy 575 3005 a(=) p Fo 663 3005 a(d) p Fy 725 3005 a(+) p 808 3005 a(1) p 877 3005 a(and) p Fo 1038 3005 a(m) p Fp 1111 3017 a(J) p Fy 1181 3005 a(=) p 1268 3005 a(0,) p Fo 1360 3005 a(J) p Fk 1438 3005 a(6) p Fy(=) p 1525 3005 a(\(1\).) 208 3171 y(Therefore) p 597 3171 a(2) p Fo(M) p Fj 720 3186 a(\(1\)) p Fy 854 3171 a(=) p 964 3171 a(\() p Fo(d) p Fy 1067 3171 a(+) p 1159 3171 a(1\)) p Fo(M) p Fj 1314 3186 a(\(1\)) p Fy 1416 3171 a(,) p 1484 3171 a(hence) p Fo 1728 3171 a(M) p Fj 1809 3186 a(\(1\)) p Fy 1943 3171 a(=) p 2053 3171 a(0) p 2136 3171 a(for) p Fo 2277 3171 a(d) p Fl 2365 3171 a(=) p Fy 2476 3171 a(2,) p 2585 3171 a(whic) n(h) p 2836 3171 a(is) p 2933 3171 a(a) p 3015 3171 a(con) n(tradiction,) p 3560 3171 a(b) r(ecause) 208 3271 y(Prop) r(osition) p 654 3271 a(14) p 765 3271 a(implies) p 1047 3271 a(\(1\)) p Fk 1176 3271 a(2) p 1255 3271 a(K) p Fp 1317 3284 a(~) p 1318 3284 a(x) p Fm 1356 3292 a(c) p Fy 1391 3271 a(,) p 1441 3271 a(whic) n(h,) p 1702 3271 a(with) p 1891 3271 a(the) p 2034 3271 a(preceding) p 2408 3271 a(result,) p 2662 3271 a(implies) p Fo 2944 3271 a(M) p Fj 3025 3286 a(\(1\)) p Fo 3137 3271 a(>) p Fy 3225 3271 a(0) p 3280 3271 a(.) 208 3404 y(This) p 397 3404 a(pro) n(v) n(es) p 654 3404 a(that) p 834 3404 a(if) p Fo 910 3404 a(I) p Fk 976 3404 a(2) p 1054 3404 a(K) p Fp 1116 3417 a(~) p 1117 3417 a(x) p Fm 1155 3425 a(c) p Fy 1218 3404 a(and) p Fo 1378 3404 a(~) p 1380 3404 a(x) p Fk 1450 3404 a(2) p 1528 3404 a(D) p Fi 1594 3404 a(om) p Fy 1712 3404 a(\() p Fo 1743 3404 a(~) p 1744 3404 a(x) p Fp 1791 3416 a(c) p Fy 1825 3404 a(\)) p Fk 1876 3404 a(\\) p Fy 1950 3404 a(\004) p Fp 2005 3416 a(d) p Fy 2044 3404 a(,) p 2094 3404 a(then) p 2291 3404 a(\026) p Fo 2283 3404 a(p) p Fp 2324 3417 a(~) p 2325 3417 a(x;I) p Fy 2448 3404 a(is) p 2532 3404 a(not) p 2679 3404 a(concen) n(trated) p 3171 3404 a(on) p 3286 3404 a(a) p 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a(functions) p 2781 4538 a(are) p 2920 4538 a(no) p 3035 4538 a(greater) p 3317 4538 a(than) p 3511 4538 a(1\),) 1189 4721 y(max) p Fp 1167 4774 a(I) p Fn 1201 4774 a(2K) p Fm 1296 4784 a(~) p 1297 4784 a(x) 1330 4792 y(c) p Fk 1379 4721 a(j) p Fo(g) p Fp 1442 4733 a(I) p Fy 1479 4721 a(\() p Fk 1511 4651 a(p) p 1581 4651 107 4 v 1581 4721 a(\000) p Fy(1) p Fo 1687 4721 a(\025t) p Fy(\)) p Fk(j) p Fl 1844 4721 a(5) p Fy 1953 4721 a(max) p Fp 1931 4774 a(I) p Fn 1965 4774 a(2K) p Fm 2060 4784 a(~) p 2061 4784 a(x) 2094 4792 y(c) p Fk 2143 4721 a(j) p Fo(g) p Fp 2206 4733 a(I) p Fy 2243 4721 a(\() p Fk 2275 4651 a(p) p 2345 4651 V 2345 4721 a(\000) p Fy(1) p Fo 2451 4721 a(t) p Fy(\)) p Fk(j) p Fj 2536 4686 a(2) p Fo 2574 4721 a(;) p 2666 4721 a(t) p Fk 2719 4721 a(2) p Fq 2797 4721 a(R) p Fo(:) p Fy 208 4954 a(This) p 397 4954 a(and) p 559 4954 a(\(53\)) p 734 4954 a(imply) p Fk 1093 5137 a(j) p Fo(g) p Fp 1156 5149 a(I) p Fy 1193 5137 a(\() p Fk 1225 5068 a(p) p 1295 5068 V 1295 5137 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a(\003) p Fp 2692 256 a(~) p 2693 256 a(x) o(;) p Fj(\(1\)) p Fy 2853 230 a(.) 125 463 y(T) p 178 463 a(o) p 247 463 a(pro) n(v) n(e) p 471 463 a(the) p 614 463 a(p) r(ositivit) n (y) p 983 463 a(of) p 1085 463 a(\026) p Fo 1078 463 a(\032) p Fn 1121 433 a(\003) p Fp 1120 490 a(~) p 1121 490 a(x) o(;) p Fj(\(1\)) p Fy 1281 463 a(,) p 1332 463 a(substitute) p Fo 1720 463 a(g) p Fj 1760 478 a(\(1\)) p Fy 1849 463 a(\() p Fo(t) p Fy(\)) p 1966 463 a(=) p Fg 2054 350 a(Z) p Fn 2137 370 a(1) p Fj 2100 538 a(0) p Fo 2221 463 a(e) p Fp 2260 429 a(t\030) p Fy 2329 463 a(\026) p Fo 2321 463 a(\032) p Fn 2364 429 a(\003) p Fp 2363 485 a(~) p 2364 485 a(x;) p Fj(\(1\)) p Fy 2511 463 a(\() p Fo(\030) p Fy 2583 463 a(\)) p Fo 2629 463 a(d\030) p Fy 2740 463 a(in) p 2837 463 a(\(50\),) p 3035 463 a(to) p 3137 463 a(\014nd) p Fo 0 704 a(\025) p Fn 48 670 a(\000) p Fj(1) p Fy 145 704 a(\026) p Fo 138 704 a(\032) p Fn 181 670 a(\003) p Fp 180 727 a(~) p 181 727 a(x) o(;) p Fj(\(1\)) p Fy 327 704 a(\() 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Fj(\(1\)) p Fy 2589 704 a(\)\() p Fo(\030) p Fy 2693 704 a(\)) r(+) p 2823 704 a(non-negativ) n(e) p 3312 704 a(terms) p Fo 3515 704 a(;) p 3611 704 a(\030) p Fl 3674 704 a(=) p Fy 3762 704 a(0) p Fo 3818 704 a(;) p Fy 0 887 a(where) p 242 887 a(the) p 386 887 a(op) r(eration) p Fk 759 887 a(\003) p Fy 830 887 a(denotes) p 1132 887 a(con) n(v) n(olution,) p 1602 887 a(and) p 1765 887 a(w) n(e) p 1889 887 a(also) p 2057 887 a(used) p 2249 887 a(Prop) r(osition) p 2697 887 a(4.) p 2804 887 a(Note) p 3006 887 a(\014rst) p 3179 887 a(that) p 3360 887 a(since) p 3573 887 a(\026) p Fo 3565 887 a(\032) p Fn 3608 857 a(\003) p Fp 3607 914 a(~) p 3608 914 a(x;) p Fj(\(1\)) p Fy 3784 887 a(is) 0 999 y(con) n(tin) n(uous) p 417 999 a(and) p Fo 579 999 a(M) p Fj 660 1014 a(\(1\)) p Fo 773 999 a(>) p Fy 863 999 a(0,) p 956 999 a(there) p 1170 999 a(is) p 1254 999 a(a) p Fo 1324 999 a(\030) p Fj 1360 1011 a(0) p Fo 1422 999 a(>) p Fy 1512 999 a(0) p 1582 999 a(for) p 1710 999 a(whic) n(h) p 1956 999 a(\026) p Fo 1948 999 a(\032) p Fn 1991 969 a(\003) p Fp 1990 1026 a(~) p 1991 1026 a(x;) p Fj(\(1\)) p Fy 2138 999 a(\() p Fo(\030) p Fy 2210 999 a(\)) p Fo 2267 999 a(>) p Fy 2357 999 a(0) p 2427 999 a(in) p 2525 999 a(a) p 2595 999 a(neigh) n(b) r(orho) r(o) r(d) p 3117 999 a(of) p Fo 3213 999 a(\030) p Fj 3249 1011 a(0) p Fy 3300 999 a(.) p 3363 999 a(F) p 3410 999 a(urthermore,) 0 1111 y(the) p 151 1111 a(ab) r(o) n(v) n(e) p 394 1111 a(relation) p 707 1111 a(implies) p 997 1111 a(that) p 1185 1111 a(if) p 1277 1111 a(\026) p Fo 1269 1111 a(\032) p Fn 1312 1081 a(\003) p Fp 1311 1138 a(~) p 1312 1138 a(x;) p Fj(\(1\)) p Fy 1494 1111 a(is) p 1586 1111 a(p) r(ositiv) n(e) p 1901 1111 a(in) p 2006 1111 a(a) p 2083 1111 a(neigh) n(b) r(orho) r(o) r(d) p 2613 1111 a(of) p Fo 2715 1111 a(\030) p Fj 2751 1123 a(1) p Fo 2789 1111 a(;) p 2826 1111 a(\030) p Fj 2862 1123 a(2) p Fl 2936 1111 a(=) p Fy 3037 1111 a(0,) p 3139 1111 a(then) p 3336 1111 a(it) p 3427 1111 a(also) p 3602 1111 a(is) p 3694 1111 a(in) p 3798 1111 a(a) 0 1228 y(neigh) n(b) r(orho) r(o) r(d) p 529 1228 a(of) p Fo 632 1228 a(\025) p Fn 680 1198 a(\000) p Fj(1) p Fy 769 1228 a(\() p Fo(\030) p Fj 837 1240 a(1) p Fy 899 1228 a(+) p Fo 987 1228 a(\030) p Fj 1023 1240 a(2) p Fy 1061 1228 a(\).) p 1177 1228 a(With) p 1399 1228 a(Prop) r(osition) p 1854 1228 a(12) p 1972 1228 a(and) p 2142 1228 a(the) p 2293 1228 a(con) n(tin) n(uit) n(y) p 2693 1228 a(of) p 2803 1228 a(\026) p Fo 2795 1228 a(\032) p Fn 2838 1198 a(\003) p Fp 2837 1255 a(~) p 2838 1255 a(x;) p Fj(\(1\)) p Fy 2998 1228 a(,) p 3059 1228 a(w) n(e) p 3189 1228 a(therefore) p 3546 1228 a(see) p 3688 1228 a(that) 8 1328 y(\026) p Fo 0 1328 a(\032) p Fn 43 1298 a(\003) p Fp 42 1355 a(~) p 43 1355 a(x;) p Fj(\(1\)) p Fy 225 1328 a(is) p 317 1328 a(p) r(ositiv) n(e) p 633 1328 a(in) p 738 1328 a(a) p 815 1328 a(neigh) n(b) r(orho) r(o) r(d) p 1345 1328 a(of) p 1448 1328 a(0) p 1503 1328 a(.) p 1588 1328 a(The) p 1767 1328 a(ab) r(o) n(v) n(e) p 2010 1328 a(relation) p 2323 1328 a(also) p 2498 1328 a(implies) p 2788 1328 a(that) p 2976 1328 a(if) p 3068 1328 a(\026) p Fo 3061 1328 a(\032) p Fn 3104 1298 a(\003) p Fp 3103 1355 a(~) p 3104 1355 a(x) o(;) p Fj(\(1\)) p Fy 3286 1328 a(is) p 3378 1328 a(p) r(ositiv) n(e) p 3693 1328 a(in) p 3798 1328 a(a) 0 1445 y(neigh) n(b) r(orho) r(o) r(d) p 523 1445 a(of) p Fo 619 1445 a(\030) p Fj 655 1457 a(1) p Fo 693 1445 a(;) p 730 1445 a(\030) p Fj 766 1457 a(2) p Fo 803 1445 a(;) p Fk 840 1445 a(\001) p 877 1445 a(\001) p 914 1445 a(\001) p Fo 951 1445 a(;) p 988 1445 a(\030) p Fp 1024 1457 a(d) p Fj(+1) p Fl 1172 1445 a(=) p Fy 1262 1445 a(0,) p 1357 1445 a(then) p 1547 1445 a(it) p 1632 1445 a(also) p 1800 1445 a(is) p 1885 1445 a(in) p 1983 1445 a(a) p 2054 1445 a(neigh) n(b) r(orho) r(o) r(d) p 2577 1445 a(of) p Fo 2673 1445 a(\025) p Fn 2721 1415 a(\000) p Fj(1) p Fy 2810 1445 a(\() p Fo(\030) p Fj 2878 1457 a(1) p Fy 2936 1445 a(+) p Fo 3020 1445 a(\030) p Fj 3056 1457 a(2) p Fy 3113 1445 a(+) p Fk 3197 1445 a(\001) p 3234 1445 a(\001) p 3271 1445 a(\001) p Fy 3313 1445 a(+) p Fo 3397 1445 a(\030) p Fp 3433 1457 a(d) p Fj(+1) p Fy 3556 1445 a(\).) p 3653 1445 a(With) 0 1545 y(Prop) r(osition) p 447 1545 a(12,) p 581 1545 a(w) n(e) p 703 1545 a(inductiv) n(ely) p 1130 1545 a(see) p 1264 1545 a(that) p 1451 1545 a(\026) p Fo 1444 1545 a(\032) p Fn 1487 1515 a(\003) p Fp 1486 1572 a(~) p 1487 1572 a(x) o(;) p Fj(\(1\)) p Fy 1661 1545 a(is) p 1744 1545 a(p) r(ositiv) n(e) p 2051 1545 a(on) p 2167 1545 a(\(0) p Fo(;) p Fk 2278 1545 a(1) p Fy(\).) 125 1644 y(This) p 314 1644 a(completes) p 697 1644 a(a) p 767 1644 a(pro) r(of) p 984 1644 a(of) p 1078 1644 a(Theorem) p 1429 1644 a(9.) p Fd 3778 1744 a(2) p Fr 0 2159 a(4.3) p 255 2159 a(Exp) s(onen) m(t) p 763 2159 a(for) p 936 2159 a(mean) p 1235 2159 a(square) p 1589 2159 a(displacemen) m(t.) p Fy 0 2312 a(Here) p 207 2312 a(w) n(e) p 341 2312 a(pro) n(v) n(e) p 576 2312 a(Theorem) p 937 2312 a(10) p 1059 2312 a(and) p 1232 2312 a(Theorem) p 1593 2312 a(11.) p 1770 2312 a(Since) p 1997 2312 a(the) p 2152 2312 a(pro) r(ofs) p 2412 2312 a(are) p 2562 2312 a(similar) p 2846 2312 a(for) p 2984 2312 a(the) p 3138 2312 a(full) p 3294 2312 a(mo) r(del) p 3552 2312 a(and) p 3725 2312 a(the) 0 2412 y(restricted) p 370 2412 a(mo) r(del,) p 640 2412 a(w) n(e) p 762 2412 a(will) p 919 2412 a(concen) n(trate) p 1364 2412 a(on) p 1479 2412 a(the) p 1622 2412 a(full) p 1768 2412 a(mo) r(del.) 125 2512 y(W) p 203 2512 a(e) p 268 2512 a(in) n(tro) r(duce) p 637 2512 a(the) p 780 2512 a(follo) n(wing) p 1130 2512 a(notation) p 1462 2512 a(whic) n(h) p 1700 2512 a(w) n(e) p 1822 2512 a(use) p 1966 2512 a(throughout) p 2399 2512 a(this) p 2561 2512 a(section.) 125 2611 y(De\014ne) p Fo 382 2611 a(\027) p Fy 451 2611 a(:) p Fo 525 2611 a(W) p Fj 615 2581 a(\(0\)) p Fk 727 2611 a(!) p Fq 833 2611 a(Z) p Fj 888 2623 a(+) p Fk 962 2611 a([) p 1036 2611 a(f1g) p Fy 1229 2611 a(b) n(y) p Fo 552 2794 a(\027) p Fy 598 2794 a(\() p Fo(w) p Fy 691 2794 a(\)) p 748 2794 a(=) p 835 2794 a(min) p Fk 974 2794 a(f) p Fo(n) p Fk 1088 2794 a(2) p Fq 1166 2794 a(Z) p Fj 1221 2806 a(+) p Fk 1295 2794 a([) p 1369 2794 a(f1g) p 1558 2794 a(j) p Fo 1604 2794 a(w) p Fy 1665 2794 a(\() p Fo(k) p Fy 1743 2794 a(\)) p Fk 1800 2794 a(2) p Fo 1878 2794 a(G) p Fp 1943 2806 a(n) p Fo 2002 2794 a(;) p 2067 2794 a(k) p Fy 2136 2794 a(=) p 2223 2794 a(0) p Fo(;) p Fy 2302 2794 a(1) p Fo(;) p Fy 2381 2794 a(2) p Fo(;) p Fk 2460 2794 a(\001) p 2497 2794 a(\001) p 2534 2794 a(\001) p Fo 2569 2794 a(;) p 2606 2794 a(L) p Fy(\() p Fo(w) p Fy 2756 2794 a(\)) p Fk(g) p Fo(;) p 2923 2794 a(w) p Fk 3007 2794 a(2) p Fo 3086 2794 a(W) p Fj 3176 2760 a(\(0\)) p Fo 3265 2794 a(:) p Fy 3692 2794 a(\(54\)) 0 2977 y(Note) p 201 2977 a(the) p 344 2977 a(ob) n(vious) p 644 2977 a(relation) 1122 3076 y(2) p Fp 1164 3042 a(\027) p Fj 1201 3042 a(\() p Fp(w) p Fj 1277 3042 a(\)) p Fn(\000) p Fj(1) p Fo 1415 3076 a(<) p 1502 3076 a(L) p Fy(\() p Fo(w) p Fy 1652 3076 a(\)) p Fl 1708 3076 a(5) p Fo 1796 3076 a(d) p Fy(\() p Fo(d) p Fy 1933 3076 a(+) p 2016 3076 a(1\)) p Fp 2090 3042 a(\027) p Fj 2127 3042 a(\() p Fp(w) p Fj 2203 3042 a(\)) p Fo 2232 3076 a(;) p 2352 3076 a(w) p Fk 2437 3076 a(2) p Fo 2516 3076 a(W) p Fj 2606 3042 a(\(0\)) p Fo 2695 3076 a(:) p Fy 3692 3076 a(\(55\)) 0 3226 y(\(The) p 204 3226 a(second) p 473 3226 a(inequalit) n(y) p 859 3226 a(is) p 944 3226 a(b) r(ecause) p 1252 3226 a(there) p 1466 3226 a(are) p 1605 3226 a(\() p Fo(d) p Fy 1700 3226 a(+) p 1784 3226 a(1\)) p Fp 1858 3195 a(n) p Fy 1931 3226 a(unit) p 2108 3226 a(simplices) p 2460 3226 a(in) p Fo 2558 3226 a(F) p Fp 2611 3238 a(n) p Fy 2657 3226 a(,) p 2709 3226 a(and) p 2871 3226 a(within) p 3131 3226 a(eac) n(h) p 3319 3226 a(unit) p 3495 3226 a(simplex) p 3798 3226 a(a) 0 3325 y(self-a) n(v) n(oiding) p 476 3325 a(w) n(alk) p 669 3325 a(can) p 821 3325 a(sp) r(end) p 1059 3325 a(at) p 1161 3325 a(most) p Fo 1364 3325 a(d) p Fy 1435 3325 a(steps.\)) 125 3425 y(F) p 172 3425 a(or) p Fo 274 3425 a(w) p Fk 359 3425 a(2) p Fo 438 3425 a(W) p Fj 528 3395 a(\(0\)) p Fo 617 3425 a(;) p Fy 668 3425 a(let) p Fk 789 3425 a(k) p Fo 830 3425 a(w) p Fk 891 3425 a(k) p Fy 957 3425 a(=) p 1045 3425 a(max) p Fk(fj) p Fo(w) p Fy 1326 3425 a(\() p Fo(k) p Fy 1404 3425 a(\)) p Fk(j) p 1483 3425 a(j) p Fo 1530 3425 a(k) p Fy 1599 3425 a(=) p 1688 3425 a(0) p Fo(;) p Fy 1767 3425 a(1) p Fo(;) p Fy 1846 3425 a(2) p Fo(;) p Fk 1925 3425 a(\001) p 1962 3425 a(\001) p 1999 3425 a(\001) p Fo 2034 3425 a(;) p 2071 3425 a(L) p Fy(\() p Fo(w) p Fy 2221 3425 a(\)) p Fk(g) p Fy(,) p 2346 3425 a(where) p Fk 2586 3425 a(j) p 2628 3425 a(\001) p 2670 3425 a(j) p Fy 2721 3425 a(denotes) p 3022 3425 a(the) p 3165 3425 a(\(Euclidean\)) p 3614 3425 a(length) 0 3524 y(in) p Fq 97 3524 a(R) p Fp 157 3494 a(d) p Fk 219 3524 a(\033) p Fo 306 3524 a(F) p Fy 371 3524 a(.) p 431 3524 a(By) p 561 3524 a(de\014nition,) 1499 3624 y(2) p Fp 1541 3590 a(\027) p Fj 1578 3590 a(\() p Fp(w) p Fj 1654 3590 a(\)) p Fn(\000) p Fj(1) p Fo 1791 3624 a(<) p Fk 1879 3624 a(k) p Fo 1920 3624 a(w) p Fk 1981 3624 a(k) p Fl 2046 3624 a(5) p Fy 2134 3624 a(2) p Fp 2176 3590 a(\027) p Fj 2213 3590 a(\() p Fp(w) p Fj 2289 3590 a(\)) p Fo 2318 3624 a(:) p Fy 3692 3624 a(\(56\)) 0 3774 y(\(T) p 85 3774 a(o) p 155 3774 a(see) p 291 3774 a(this,) p 477 3774 a(note) p 662 3774 a(that) p 843 3774 a(b) n(y) p 959 3774 a(de\014nition) p 1329 3774 a(of) p Fo 1425 3774 a(\027) p Fy 1471 3774 a(\() p Fo(w) p Fy 1564 3774 a(\),) p Fo 1649 3774 a(w) p Fy 1740 3774 a(is) p 1824 3774 a(con) n(tained) p 2201 3774 a(in) p 2299 3774 a(the) p 2443 3774 a(ball) p 2605 3774 a(of) p 2701 3774 a(radius) p 2952 3774 a(2) p Fp 2994 3743 a(\027) p Fj 3031 3743 a(\() p Fp(w) p Fj 3107 3743 a(\)) p Fy 3164 3774 a(cen) n(tered) p 3495 3774 a(at) p Fo 3598 3774 a(O) p Fy 3663 3774 a(,) p 3715 3774 a(but) 0 3873 y(not) p 148 3873 a(in) p 244 3873 a(the) p 387 3873 a(ball) p 549 3873 a(of) p 643 3873 a(radius) p 893 3873 a(2) p Fp 935 3843 a(\027) p Fj 972 3843 a(\() p Fp(w) p Fj 1048 3843 a(\)) p Fn(\000) p Fj(1) p Fy 1162 3873 a(.\)) p Fz 0 4089 a(4.3.1) p 292 4089 a(T) p 350 4089 a(aub) s(erian) p 746 4089 a(t) m(yp) s(e) p 962 4089 a(estimates) p 1384 4089 a(and) p 1568 4089 a(n) m(um) m(b) s(er) p 1919 4089 a(of) p 2028 4089 a(paths.) p Fy 0 4242 a(Here) p 196 4242 a(w) n(e) p 318 4242 a(pro) n(v) n(e) p 542 4242 a(Theorem) p 893 4242 a(10) p 1004 4242 a(for) p 1131 4242 a(the) p 1274 4242 a(full) p 1419 4242 a(mo) r(del.) 125 4342 y(Let) p Fo 273 4342 a(\025) p Fy 349 4342 a(b) r(e) p 462 4342 a(as) p 564 4342 a(b) r(efore,) p Fo 837 4342 a(\014) p Fp 884 4354 a(c) p Fy 945 4342 a(b) r(e) p 1058 4342 a(a) p 1127 4342 a(critical) p 1404 4342 a(p) r(oin) n(t) p 1621 4342 a(of) p 1716 4342 a(the) p 1859 4342 a(full) p 2004 4342 a(mo) r(del,) p 2274 4342 a(and) p Fo 2434 4342 a(~) p 2435 4342 a(x) p Fp 2482 4354 a(can) p Fy 2593 4342 a(\() p Fo(\014) p Fp 2672 4354 a(c) p Fy 2707 4342 a(\)) p 2762 4342 a(=) p Fo 2848 4342 a(~) p 2850 4342 a(x) p Fp 2897 4354 a(can;) p Fn(I) p Fm 3062 4363 a(d) p Fy 3100 4342 a(\() p Fo(\014) p Fp 3179 4354 a(c) p Fy 3213 4342 a(\)) p 3273 4342 a(b) r(e) p 3386 4342 a(as) p 3488 4342 a(in) p 3585 4342 a(\(17\).) p Fz 0 4550 a(Prop) s(osition) p 516 4550 a(23) p Fi 653 4550 a(De\014ne,) p 945 4550 a(for) p Fo 1081 4550 a(b) p 1146 4550 a(>) p Fy 1239 4550 a(0) p Fi(,) p Fo 1340 4550 a(n) p Fk 1419 4550 a(2) p Fq 1504 4550 a(Z) p Fj 1559 4562 a(+) p Fi 1614 4550 a(,) p Fo 1673 4550 a(\030) p Fk 1743 4550 a(2) p Fq 1827 4550 a(R) p Fi(,) p Fo 1947 4550 a(c) p Fp 1983 4562 a(n) p Fy 2057 4550 a(=) p Fo 2150 4550 a(b\025) p Fn 2234 4516 a(\000) p Fp(n) p Fk 2332 4486 a(p) p 2401 4486 50 4 v Fo 2401 4550 a(n) p Fi 2484 4550 a(and) p Fo 2648 4550 a(h) p Fp 2696 4562 a(n) p Fy 2741 4550 a(\() p Fo(\030) p Fy 2813 4550 a(\)) p 2875 4550 a(=) 3079 4494 y(1) p 2979 4531 243 4 v Fk 2979 4548 a(p) p 3048 4548 92 4 v Fy 3048 4616 a(2) p Fo(\031) s(c) p Fp 3176 4628 a(n) p Fo 3231 4550 a(e) p Fn 3270 4516 a(\000) p Fp(\030) p Ff 3354 4491 a(2) p Fp 3386 4516 a(=) p Fj(\(2) p Fp(c) p Ff 3509 4491 a(2) p Fm 3509 4532 a(n) p Fj 3549 4516 a(\)) p Fi 3580 4550 a(.) p 3653 4550 a(Then) 0 4695 y(for) p 133 4695 a(su\016ciently) p 552 4695 a(lar) l(ge) p Fo 752 4695 a(b) p Fi 817 4695 a(it) p 900 4695 a(holds) p 1112 4695 a(that) p Fy 996 4877 a(lim) p Fp 967 4927 a(n) p Fn(!1) p Fy 1140 4877 a(\() p 1179 4877 a(\026) p Fo 1172 4877 a(p) p Fp 1213 4892 a(~) p 1214 4892 a(x) p Fm 1252 4900 a(can) p Fj 1352 4892 a(\() p Fp(\014) p Fm 1416 4900 a(c) p Fj 1446 4892 a(\)) p Fp(;n;I) p Fk 1609 4877 a(\003) p Fo 1669 4877 a(h) p Fp 1717 4889 a(n) p Fy 1762 4877 a(\)\() p Fo(\030) p Fy 1866 4877 a(\)) p 1922 4877 a(=) p 2017 4877 a(\026) p Fo 2010 4877 a(\032) p Fn 2053 4843 a(\003) p Fp 2052 4900 a(~) p 2053 4900 a(x) p Fm 2091 4908 a(can) p Fj 2190 4900 a(\() p Fp(\014) p Fm 2254 4908 a(c) p Fj 2285 4900 a(\)) p Fp(;I) p Fy 2369 4877 a(\() p Fo(\030) p Fy 2441 4877 a(\)) p Fo(;) p 2570 4877 a(I) p Fk 2636 4877 a(2) p 2714 4877 a(K) p Fp 2776 4890 a(~) p 2777 4890 a(x) p Fm 2815 4898 a(c) p Fo 2850 4877 a(;) p Fi 0 5084 a(uniformly) p 379 5084 a(in) p Fo 481 5084 a(\030) p Fk 544 5084 a(2) p Fq 622 5084 a(R) p Fi(,) p 737 5084 a(wher) l(e) p Fy 1189 5222 a(\() p 1228 5222 a(\026) p Fo 1221 5222 a(p) p Fp 1262 5235 a(~) p 1263 5235 a(x) o(;n;I) p Fk 1437 5222 a(\003) p Fo 1497 5222 a(h) p Fp 1545 5234 a(n) p Fy 1590 5222 a(\)\() p Fo(\030) p Fy 1694 5222 a(\)) p 1750 5222 a(=) p Fg 1838 5109 a(Z) p Fc 1884 5297 a(R) p Fo 1944 5222 a(h) p Fp 1992 5234 a(n) p Fy 2037 5222 a(\() p Fo(\030) p Fk 2128 5222 a(\000) p Fo 2211 5222 a(\021) p Fy 2255 5222 a(\)) p 2308 5222 a(\026) p Fo 2301 5222 a(p) p Fp 2342 5235 a(~) p 2343 5235 a(x;n;I) p Fy 2499 5222 a(\() p Fo(d\021) p Fy 2618 5222 a(\)) p Fi 0 5413 a(is) p 89 5413 a(a) p 161 5413 a(c) l(onvolution.) p Fy 1878 6120 a(21) p 90 rotate dyy eop %%Page: 22 22 22 21 bop Fi -42 83 a(Pr) l(o) l(of.) p Fy 219 83 a(F) p 266 83 a(or) p Fo 366 83 a(n) p Fk 439 83 a(2) p Fq 517 83 a(Z) p Fj 572 95 a(+) p Fy 654 83 a(and) p Fo 814 83 a(I) p Fk 880 83 a(2) p 958 83 a(I) p Fp 1003 95 a(d) p Fy 1056 83 a(,) p 1106 83 a(denote) p 1372 83 a(the) p 1513 83 a(c) n(haracteristic) p 2022 83 a(function) p 2346 83 a(of) p 2439 83 a(\() p 2478 83 a(\026) p Fo 2471 83 a(p) p Fp 2512 98 a(~) p 2513 98 a(x) p Fm 2551 106 a(can) p Fj 2651 98 a(\() p Fp(\014) p Fm 2715 106 a(c) p Fj 2745 98 a(\)) p Fp(;n;I) p Fk 2905 83 a(\003) p Fo 2962 83 a(h) p Fp 3010 95 a(n) p Fy 3055 83 a(\)\() p Fo(\030) p Fy 3159 83 a(\)) p Fo 3205 83 a(d\030) p Fy 3315 83 a(b) n(y) p Fo 3429 83 a(\036) p Fp 3478 95 a(n;I) p Fy 3591 83 a(.) p 3651 83 a(Then) p Fo 672 317 a(\036) p Fp 721 329 a(n;I) p Fy 821 317 a(\() p Fo(t) p Fy(\)) p 938 317 a(=) p Fg 1026 204 a(Z) p Fo 1123 317 a(e) p Fn 1162 238 a(p) p 1216 238 85 3 v 1216 283 a(\000) p Fj(1) p Fp(\030) r(t) p Fy 1362 317 a(\() p 1401 317 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2652 a(x) p Fn 1442 2618 a(00) p Fp 1442 2673 a(I) p Fy 1508 2652 a(=) p Fg 1596 2535 a(\032) p Fo 1700 2602 a(x) p Fp 1747 2614 a(c;I) p Fk 1853 2602 a(\000) p Fo 1936 2602 a(\017;) p 2076 2602 a(I) p Fk 2142 2602 a(2) p 2220 2602 a(K) p Fp 2282 2615 a(~) p 2283 2615 a(x) p Fm 2321 2623 a(c) p Fo 2370 2602 a(;) 1700 2701 y(\016) p 2076 2701 a(I) p Fk 2142 2701 a(62) p 2220 2701 a(K) p Fp 2282 2714 a(~) p 2283 2714 a(x) p Fm 2321 2722 a(c) p Fo 2370 2701 a(;) p Fy 0 2852 a(then) p Fo 188 2852 a(~) p 189 2852 a(x) p Fn 236 2822 a(00) p Fk 302 2852 a(2) p Fo 380 2852 a(D) p Fp 451 2822 a(o) p Fy 488 2852 a(.) p 548 2852 a(Also,) p 759 2852 a(b) n(y) p 874 2852 a(the) p 1017 2852 a(de\014nition) p 1386 2852 a(of) p Fk 1481 2852 a(K) p Fp 1543 2865 a(~) p 1544 2865 a(x) p Fm 1582 2873 a(c) p Fy 1644 2852 a(and) p 1806 2852 a(\(CS1\),) p 2069 2852 a(there) p 2281 2852 a(exists) p Fo 2510 2852 a(n) p Fj 2560 2864 a(0) p Fl 2620 2852 a(=) p Fo 2708 2852 a(n) p Fj 2758 2864 a(1) p Fy 2823 2852 a(suc) n(h) p 3010 2852 a(that) p Fk 531 3035 a(j) p Fo(Z) p Fp 611 3047 a(n;I) p Fy 710 3035 a(\() p Fo(\014) p Fp 789 3047 a(c) p Fk 842 3035 a(\000) 925 2965 y(p) p 994 2965 V 994 3035 a(\000) p Fy(1) p Fo 1100 3035 a(\025) p Fn 1148 3000 a(\000) p Fp(n) p Fo 1245 3035 a(t) p Fy(\)) p Fk(j) p Fl 1354 3035 a(5) p Fo 1441 3035 a(Z) p Fp 1498 3047 a(n;I) p Fy 1597 3035 a(\() p Fo(\014) p Fp 1676 3047 a(c) p Fy 1710 3035 a(\)) p Fo 1765 3035 a(<) p 1853 3035 a(\016) o(;) p 2008 3035 a(I) p Fk 2074 3035 a(62) p 2153 3035 a(K) p Fp 2215 3048 a(~) p 2216 3048 a(x) p Fm 2254 3056 a(c) p Fo 2303 3035 a(;) p 2367 3035 a(n) p Fy 2440 3035 a(=) p Fo 2528 3035 a(n) p Fj 2578 3047 a(0) p Fo 2615 3035 a(;) p 2652 3035 a(n) p Fj 2702 3047 a(0) p Fy 2757 3035 a(+) p 2840 3035 a(1) p Fo(;) p Fk 2919 3035 a(\001) p 2956 3035 a(\001) p 2993 3035 a(\001) p Fo 3030 3035 a(;) p 3094 3035 a(t) p Fk 3147 3035 a(2) p Fq 3226 3035 a(R) p Fo(:) p Fy 3692 3035 a(\(60\)) 0 3217 y(Prop) r(osition) p 450 3217 a(16,) p 588 3217 a(\(59\),) p 791 3217 a(and) p 956 3217 a(\(60\)) p 1134 3217 a(then) p 1327 3217 a(imply) p 1525 3217 a(,) p 1580 3217 a(with) p Fo 1771 3217 a(~) p 1772 3217 a(x) p Fn 1819 3187 a(00) p Fk 1891 3217 a(2) p Fo 1975 3217 a(D) p Fp 2046 3187 a(o) p Fy 2083 3217 a(,) p 2138 3217 a(that) p 2321 3217 a(there) p 2537 3217 a(exists) p 2769 3217 a(p) r(ositiv) n(e) p 3080 3217 a(constan) n(ts) p Fo 3451 3217 a(C) p Fj 3510 3229 a(1) p Fy 3579 3217 a(and) p Fo 3743 3217 a(C) p Fj 3802 3229 a(2) p Fy 0 3317 a(suc) n(h) p 187 3317 a(that) p Fk 456 3488 a(j) p Fo(Z) p Fp 536 3500 a(m) p Fj(+) p Fp(n;I) p Fy 745 3488 a(\() p Fo(\014) p Fp 824 3500 a(c) p Fk 876 3488 a(\000) 959 3419 y(p) p 1029 3419 V 1029 3488 a(\000) p Fy(1) p Fo 1135 3488 a(\025) p Fn 1183 3454 a(\000) p Fp(n) p Fo 1280 3488 a(t) p Fy(\)) p Fk(j) p Fy 1389 3488 a(=) p Fk 1476 3488 a(j) p Fo(X) p Fp 1568 3500 a(m;I) p Fy 1685 3488 a(\() p Fo 1728 3467 a(~) 1717 3488 y(Z) p Fp 1774 3500 a(n) p Fy 1819 3488 a(\() p Fo(\014) p Fp 1898 3500 a(c) p Fk 1950 3488 a(\000) 2034 3419 y(p) p 2103 3419 V 2103 3488 a(\000) p Fy(1) p Fo 2209 3488 a(\025) p Fn 2257 3454 a(\000) p Fp(n) p Fo 2354 3488 a(t) p Fy(\)) p Fk(j) p Fl 2463 3488 a(5) p Fo 2550 3488 a(X) p Fp 2619 3500 a(m;I) p Fy 2736 3488 a(\() p Fo 2767 3488 a(~) p 2768 3488 a(x) p Fn 2815 3454 a(00) p Fy 2858 3488 a(\)) p Fl 2913 3488 a(5) p Fo 3001 3488 a(C) p Fj 3060 3500 a(1) p Fo 3098 3488 a(e) p Fn 3137 3454 a(\000) p Fp(C) p Ff 3237 3462 a(2) p Fj 3268 3454 a(2) p Fm 3301 3429 a(m) p Fo 3360 3488 a(;) 539 3588 y(m) p Fk 635 3588 a(2) p Fq 714 3588 a(Z) p Fj 769 3600 a(+) p Fo 824 3588 a(;) p 889 3588 a(n) p Fy 962 3588 a(=) p Fo 1049 3588 a(n) p Fj 1099 3600 a(0) p Fo 1136 3588 a(;) p 1173 3588 a(n) p Fj 1223 3600 a(0) p Fy 1279 3588 a(+) p 1362 3588 a(1) p Fo(;) p Fk 1441 3588 a(\001) p 1478 3588 a(\001) p 1515 3588 a(\001) p Fo 1551 3588 a(;) p 1616 3588 a(I) p Fk 1682 3588 a(2) p 1760 3588 a(I) p Fp 1805 3600 a(d) p Fo 1844 3588 a(;) p 1909 3588 a(t) p Fk 1962 3588 a(2) p Fo 2040 3588 a(A:) p Fy 3692 3534 a(\(61\)) 125 3830 y(Let) p Fo 267 3830 a(n) p Fy 339 3830 a(b) r(e) p 446 3830 a(an) p 555 3830 a(in) n(teger) p 824 3830 a(satisfying) p Fo 1188 3830 a(n) p 1261 3830 a(>) p 1349 3830 a(n) p Fj 1399 3842 a(0) p Fy 1449 3830 a(,) p 1495 3830 a(and) p Fo 1651 3830 a(t) p Fy 1703 3830 a(b) r(e) p 1810 3830 a(a) p 1873 3830 a(real) p 2029 3830 a(satisfying) p Fk 2393 3830 a(j) p Fo(t) p Fk(j) p 2492 3830 a(2) p Fy 2570 3830 a([1) p Fo(;) p 2672 3830 a(\025) p Fp 2720 3800 a(n) p Fn(\000) p Fp(n) p Ff 2854 3808 a(0) p Fn 2886 3800 a(\000) p Fj(1) p Fy 2976 3830 a(\),) p 3054 3830 a(and) p Fo 3209 3830 a(m) p Fy 3305 3830 a(=) p Fg 3393 3713 a(\024) p Fy 3447 3774 a(log) p Fk 3568 3774 a(j) p Fo(t) p Fk(j) p 3447 3811 198 4 v Fy 3461 3887 a(log) p Fo 3582 3887 a(\025) p Fg 3654 3713 a(\025) p Fy 3704 3830 a(+) p 3776 3830 a(1,) 0 3983 y(where) p 243 3983 a([) p Fo(x) p Fy(]) p 368 3983 a(is) p 454 3983 a(the) p 600 3983 a(largest) p 872 3983 a(in) n(teger) p 1150 3983 a(not) p 1300 3983 a(exceeding) p Fo 1679 3983 a(x) p Fy(.) p 1796 3983 a(Then) p Fo 2016 3983 a(n) p Fk 2087 3983 a(\000) p Fo 2172 3983 a(m) p Fl 2273 3983 a(=) p Fo 2366 3983 a(n) p Fj 2416 3995 a(0) p Fy 2484 3983 a(and) p Fo 2649 3983 a(\025) p Fn 2697 3953 a(\000) p Fp(m) p Fo 2812 3983 a(t) p Fk 2870 3983 a(2) p Fo 2954 3983 a(A) p Fy 3030 3983 a(.) p 3099 3983 a(Hence) p 3349 3983 a(\(61\)) p 3528 3983 a(and) p 3692 3983 a(\(57\)) 0 4083 y(imply) p Fk 227 4320 a(j) p Fo(\036) p Fp 299 4332 a(n;I) p Fy 398 4320 a(\() p Fo(t) p Fy(\)) p Fk(j) p Fl 539 4320 a(5) p Fk 636 4264 a(j) p Fo(Z) p Fp 716 4276 a(m) p Fj(+) p Fp(n) p Fn(\000) p Fp(m;I) p Fy 1035 4264 a(\() p Fo(\014) p Fp 1114 4276 a(c) p Fk 1167 4264 a(\000) 1250 4199 y(p) p 1319 4199 107 4 v 1319 4264 a(\000) p Fy(1) p Fo 1425 4264 a(\025) p Fn 1473 4234 a(\000) p Fj(\() p Fp(n) p Fn(\000) p Fp(m) p Fj(\)) p Fy 1733 4264 a(\() p Fo(\025) p Fn 1813 4234 a(\000) p Fp(m) p Fo 1929 4264 a(t) p Fy(\)\)) p Fk(j) p 636 4301 1411 4 v Fo 1191 4377 a(Z) p Fp 1248 4389 a(n;I) p Fy 1347 4377 a(\() p Fo(\014) p Fp 1426 4389 a(c) p Fy 1460 4377 a(\)) p Fl 2080 4320 a(5) p Fo 2280 4264 a(C) p Fj 2339 4276 a(1) p 2177 4301 301 4 v Fo 2177 4377 a(Z) p Fp 2234 4389 a(n;I) p Fy 2333 4377 a(\() p Fo(\014) p Fp 2412 4389 a(c) p Fy 2446 4377 a(\)) p Fo 2502 4320 a(e) p Fn 2541 4286 a(\000) p Fp(C) p Ff 2641 4294 a(2) p Fj 2673 4286 a(2) p Fm 2706 4261 a(m) p Fl 2788 4320 a(5) p Fo 2988 4264 a(C) p Fj 3047 4276 a(1) p 2886 4301 V Fo 2886 4377 a(Z) p Fp 2943 4389 a(n;I) p Fy 3041 4377 a(\() p Fo(\014) p Fp 3120 4389 a(c) p Fy 3154 4377 a(\)) p Fo 3210 4320 a(e) p Fn 3249 4286 a(\000) p Fp(C) p Ff 3349 4294 a(2) p Fn 3381 4286 a(j) p Fp(t) p Fn(j) p Ff 3446 4261 a(1) p Fm(=d) 3536 4269 y(w) p Fo 3590 4320 a(;) 282 4472 y(n) p 355 4472 a(>) p 442 4472 a(n) p Fj 492 4484 a(0) p Fo 530 4472 a(;) p 594 4472 a(I) p Fk 660 4472 a(2) p 739 4472 a(I) p Fp 784 4484 a(d) p Fo 836 4472 a(;) p Fk 901 4472 a(j) p Fo(t) p Fk(j) p 1000 4472 a(2) p Fy 1078 4472 a([1) p Fo(;) p 1180 4472 a(\025) p Fp 1228 4438 a(n) p Fn(\000) p Fp(n) p Ff 1362 4446 a(0) p Fn 1395 4438 a(\000) p Fj(1) p Fy 1484 4472 a(\)) p Fo(:) p Fy 0 4656 a(On) p 139 4656 a(the) p 282 4656 a(other) p 500 4656 a(hand,) p 731 4656 a(w) n(e) p 854 4656 a(ha) n(v) n(e) p 1075 4656 a(lim) p Fp 1046 4706 a(n) p Fn(!1) p Fo 1233 4656 a(Z) p Fp 1290 4668 a(n;I) p Fy 1388 4656 a(\() p Fo(\014) p Fp 1467 4668 a(c) p Fy 1501 4656 a(\)) p 1558 4656 a(=) p Fo 1646 4656 a(x) p Fp 1693 4668 a(c;I) p Fy 1795 4656 a(,) p 1846 4656 a(and) p 2037 4656 a(lim) p Fp 2008 4706 a(n) p Fn(!1) p Fo 2195 4656 a(\036) p Fp 2244 4668 a(n;I) p Fy 2344 4656 a(\() p Fo(t) p Fy(\)) p 2462 4656 a(=) p Fo 2550 4656 a(g) p Fp 2589 4671 a(~) p 2590 4671 a(x) p Fm 2628 4679 a(can) p Fj 2727 4671 a(\() p Fp(\014) p Fm 2791 4679 a(c) p Fj 2822 4671 a(\)) p Fp(;I) p Fy 2906 4656 a(\() p Fo(t) p Fy(\),) p Fo 3052 4656 a(t) p Fk 3106 4656 a(2) p Fq 3185 4656 a(R) p Fy(,) p 3296 4656 a(b) n(y) p 3412 4656 a(\(58\).) p 3621 4656 a(Hence) 0 4780 y(the) p 143 4780 a(dominated) p 553 4780 a(con) n(v) n(ergence) p 1013 4780 a(theorem) p 1336 4780 a(implies) 688 5000 y(lim) p Fp 659 5050 a(n) p Fn(!1) p Fg 846 4887 a(Z) p Fc 892 5076 a(R) p Fk 952 5000 a(j) p Fo(\037) p Fj 1027 5018 a([) p Fn(\000) p Fp(\025) p Fm 1137 4999 a(n) p Fe(\000) p Fm(n) p Ff 1256 5011 a(0) p Fe 1288 4999 a(\000) p Ff(1) p Fp 1365 5018 a(;\025) p Fm 1424 4999 a(n) p Fe(\000) p Fm(n) p Ff 1543 5011 a(0) p Fe 1575 4999 a(\000) p Ff(1) p Fj 1652 5018 a(]) p Fy 1675 5000 a(\() p Fo(t) p Fy(\)) p Fo(\036) p Fp 1818 5012 a(n;I) p Fy 1918 5000 a(\() p Fo(t) p Fy(\)) p Fk 2031 5000 a(\000) p Fo 2114 5000 a(g) p Fp 2153 5015 a(~) p 2154 5015 a(x) p Fm 2192 5023 a(can) p Fj 2291 5015 a(\() p Fp(\014) p Fm 2355 5023 a(c) p Fj 2386 5015 a(\)) p Fp(;I) p Fy 2470 5000 a(\() p Fo(t) p Fy(\)) p Fk(j) p Fo 2601 5000 a(dt) p Fy 2697 5000 a(=) p 2785 5000 a(0) p Fo(;) p 2864 5000 a(I) p Fk 2929 5000 a(2) p 3008 5000 a(K) p Fp 3070 5013 a(~) p 3071 5013 a(x) p Fm 3109 5021 a(c) p Fo 3158 5000 a(;) p Fy 0 5225 a(where) p Fo 240 5225 a(\037) p Fp 292 5237 a(A) p Fy 374 5225 a(denotes) p 674 5225 a(the) p 817 5225 a(indicator) p 1168 5225 a(function) p 1493 5225 a(of) p 1588 5225 a(a) p 1657 5225 a(set) p Fo 1786 5225 a(A) p Fy(.) 125 5325 y(On) p 263 5325 a(the) p 406 5325 a(other) p 623 5325 a(hand,) p Fg 647 5437 a(Z) p Fc 693 5625 a(R) p Fk 753 5550 a(j) p Fo(\037) p Fj 828 5567 a([) p Fn(\000) p Fp(\025) p Fm 938 5549 a(n) p Fe(\000) p Fm(n) p Ff 1057 5561 a(0) p Fe 1089 5549 a(\000) p Ff(1) p Fp 1166 5567 a(;\025) p Fm 1225 5549 a(n) p Fe(\000) p Fm(n) p Ff 1344 5561 a(0) p Fe 1376 5549 a(\000) p Ff(1) p Fj 1453 5567 a(]) p Fy 1476 5550 a(\() p Fo(t) p Fy(\)) p Fo(\036) p Fp 1619 5562 a(n;I) p Fy 1719 5550 a(\() p Fo(t) p Fy(\)) p Fk 1832 5550 a(\000) p Fo 1915 5550 a(\036) p Fp 1964 5562 a(n;I) p Fy 2063 5550 a(\() p Fo(t) p Fy(\)) p Fk(j) p Fo 2194 5550 a(dt) p Fl 2291 5550 a(5) p Fy 2379 5550 a(2) p Fg 2435 5437 a(Z) p Fn 2517 5457 a(1) p Fp 2480 5625 a(\025) p Fm 2519 5607 a(n) p Fe(\000) p Fm(n) p Ff 2638 5619 a(0) p Fe 2670 5607 a(\000) p Ff(1) p Fo 2765 5550 a(e) p Fn 2804 5515 a(\000) p Fp(c) p Ff 2886 5490 a(2) p Fm 2886 5532 a(n) p Fp 2926 5515 a(t) p Ff 2951 5490 a(2) p Fp 2984 5515 a(=) p Fj(2) p Fo 3069 5550 a(dt) p Fl 647 5743 a(5) p Fy 735 5743 a(2) p Fg 791 5630 a(Z) p Fn 873 5650 a(1) p Fp 836 5818 a(\025) p Fm 875 5800 a(n) p Fe(\000) p Fm(n) p Ff 994 5812 a(0) p Fe 1026 5800 a(\000) p Ff(1) p Fo 1121 5743 a(te) p Fn 1190 5708 a(\000) p Fp(c) p Ff 1272 5683 a(2) p Fm 1272 5725 a(n) p Fp 1312 5708 a(t) p Ff 1337 5683 a(2) p Fp 1370 5708 a(=) p Fj(2) p Fo 1455 5743 a(dt) p Fy 1551 5743 a(=) 1689 5687 y(2) p 1649 5724 123 4 v Fo 1649 5800 a(b) p Fj 1685 5776 a(2) p Fo 1722 5800 a(n) 1781 5743 y(e) p Fn 1820 5708 a(\000) p Fp(n) p Fj(\() p Fp(b) p Ff 1968 5683 a(2) p Fp 2001 5708 a(\025) p Fe 2040 5683 a(\000) p Ff(2) p Fm(n) p Ff 2150 5695 a(0) p Fe 2182 5683 a(\000) p Ff(2) p Fp 2259 5708 a(=) p Fj(2) p Fn(\000) p Fj(2) p 2423 5708 a(log) p Fp 2520 5708 a(\025) p Fj(\)) p Fk 2640 5743 a(!) p Fy 2774 5743 a(0) p Fo(;) p 2908 5743 a(n) p Fk 2981 5743 a(!) p 3087 5743 a(1) p Fo(;) p Fy 1878 6120 a(22) p 90 rotate dyy eop %%Page: 23 23 23 22 bop Fy 0 83 a(for) p 127 83 a(su\016cien) n(tly) p 543 83 a(large) p Fo 746 83 a(b) p Fy(.) p 841 83 a(Therefore) 1244 303 y(lim) p Fp 1215 353 a(n) p Fn(!1) p Fg 1402 190 a(Z) p Fc 1448 379 a(R) p Fk 1508 303 a(j) p Fo(\036) p Fp 1580 315 a(n;I) p Fy 1680 303 a(\() p Fo(t) p Fy(\)) p Fk 1793 303 a(\000) p Fo 1876 303 a(g) p Fp 1915 318 a(~) p 1916 318 a(x) p Fm 1954 326 a(can) p Fj 2053 318 a(\() p Fp(\014) p Fm 2117 326 a(c) p Fj 2147 318 a(\)) p Fp(;I) p Fy 2231 303 a(\() p Fo(t) p Fy(\)) p Fk(j) p Fo 2362 303 a(dt) p Fy 2459 303 a(=) p 2547 303 a(0) p Fo 2603 303 a(:) p Fy 0 528 a(Using) p 233 528 a(the) p 376 528 a(in) n(v) n(ersion) p 725 528 a(form) n(ula) p 1030 528 a(w) n(e) p 1152 528 a(therefore) p 1501 528 a(ha) n(v) n(e) 138 749 y(sup) p Fp 142 819 a(\030) p Fn 174 819 a(2) p Fc(R) p Fk 277 749 a(j) p Fy(\() p 339 749 a(\026) p Fo 332 749 a(p) p Fp 373 764 a(~) p 374 764 a(x) p Fm 412 772 a(can) p Fj 512 764 a(\() p Fp(\014) p Fm 576 772 a(c) p Fj 607 764 a(\)) p Fp(;n;I) p Fk 770 749 a(\003) p Fo 830 749 a(h) p Fp 878 761 a(n) p Fy 923 749 a(\)\() p Fo(\030) p Fy 1027 749 a(\)) p Fk 1078 749 a(\000) p Fy 1169 749 a(\026) p Fo 1161 749 a(\032) p Fn 1204 714 a(\003) p Fp 1203 771 a(~) p 1204 771 a(x) p Fm 1242 779 a(can) p Fj 1341 771 a(\() p Fp(\014) p Fm 1405 779 a(c) p Fj 1436 771 a(\)) p Fp(;I) p Fy 1520 749 a(\() p Fo(\030) p Fy 1592 749 a(\)) p Fk(j) p Fl 1671 749 a(5) p Fg 1758 636 a(Z) p Fc 1804 824 a(R) p Fk 1864 749 a(j) p Fo(e) p Fn 1926 714 a(\000) 1978 670 y(p) p 2033 670 85 3 v 2033 714 a(\000) p Fj(1) p Fp(\030) r(t) p Fo 2179 749 a(\036) p Fp 2228 761 a(n;I) p Fy 2327 749 a(\() p Fo(t) p Fy(\)) p Fk 2440 749 a(\000) p Fy 2537 749 a(\026) p Fo 2523 749 a(') p Fp 2576 764 a(~) p 2577 764 a(x) p Fm 2615 772 a(can) p Fj 2715 764 a(\() p Fp(\014) p Fm 2779 772 a(c) p Fj 2810 764 a(\)) p Fp(;I) p Fy 2894 749 a(\() p Fo(t) p Fy(\)) p Fk(j) p Fo 3025 749 a(dt) p Fk 3149 749 a(!) p Fy 3283 749 a(0) p Fo(;) p 3416 749 a(n) p Fk 3489 749 a(!) p 3595 749 a(1) p Fo(:) p Fd 3778 967 a(2) p Fy 125 1249 a(W) p 203 1249 a(e) p 268 1249 a(will) p 424 1249 a(use) p 568 1249 a(the) p 711 1249 a(follo) n(wing) p 1061 1249 a(in) p 1158 1249 a(Section) p 1449 1249 a(4.3.2.) p Fz 0 1415 a(Corollary) p 425 1415 a(24) p Fi 634 1415 a(\(i\)) p 769 1415 a(Ther) l(e) p 1008 1415 a(exists) p 1235 1415 a(p) l(ositive) p 1534 1415 a(c) l(onstants) p Fo 1899 1415 a(C) p Fj 1958 1427 a(1) p Fi 2026 1415 a(and) p Fo 2187 1415 a(C) p Fj 2246 1427 a(2) p Fi 2313 1415 a(such) p 2502 1415 a(that) p Fo 1045 1598 a(Z) p Fp 1102 1610 a(m) p Fj(+) p Fp(n;I) p Fy 1310 1598 a(\() p Fo(\014) p Fp 1389 1610 a(c) p Fy 1442 1598 a(+) p Fo 1525 1598 a(\025) p Fn 1573 1563 a(\000) p Fp(n) p Fy 1671 1598 a(\)) p Fl 1726 1598 a(5) p Fo 1813 1598 a(C) p Fj 1872 1610 a(1) p Fo 1910 1598 a(e) p Fn 1949 1563 a(\000) p Fp(C) p Ff 2049 1571 a(2) p Fj 2081 1563 a(2) p Fm 2114 1538 a(m) p Fo 2173 1598 a(;) p 2299 1598 a(m;) p 2409 1598 a(n) p Fk 2482 1598 a(2) p Fq 2560 1598 a(Z) p Fj 2615 1610 a(+) p Fo 2671 1598 a(;) p 2737 1598 a(I) p Fk 2803 1598 a(2) p 2882 1598 a(I) p Fp 2927 1610 a(d) p Fo 2979 1598 a(:) p Fi 47 1814 a(\(ii\)) p 208 1814 a(Ther) l(e) p 446 1814 a(exists) p 674 1814 a(a) p 746 1814 a(p) l(ositive) p 1045 1814 a(c) l(onstant) p Fo 1376 1814 a(C) p Fi 1471 1814 a(such) p 1660 1814 a(that) p Fo 1304 1996 a(Z) p Fp 1361 2008 a(n;I) p Fy 1460 1996 a(\() p Fo(\014) p Fp 1539 2008 a(c) p Fk 1591 1996 a(\000) p Fo 1674 1996 a(\025) p Fn 1722 1962 a(\000) p Fp(n) p Fy 1820 1996 a(\)) p Fl 1875 1996 a(5) p Fo 1963 1996 a(C) q(;) p 2150 1996 a(n) p Fk 2222 1996 a(2) p Fq 2301 1996 a(Z) p Fj 2356 2008 a(+) p Fo 2411 1996 a(;) p 2478 1996 a(I) p Fk 2544 1996 a(2) p 2622 1996 a(I) p Fp 2667 2008 a(d) p Fo 2720 1996 a(:) p Fi -42 2179 a(Pr) l(o) l(of.) p Fy 297 2179 a(\(i\)) p 426 2179 a(Since) p Fo 644 2179 a(~) p 645 2179 a(x) p Fp 692 2191 a(can) p Fy 803 2179 a(\() p Fo(\014) p Fp 882 2191 a(c) p Fy 936 2179 a(+) p Fo 1021 2179 a(\025) p Fn 1069 2149 a(\000) p Fp(n) p Fo 1166 2179 a(t) p Fy(\)) p Fk 1255 2179 a(2) p Fo 1337 2179 a(D) p Fp 1408 2149 a(o) p Fk 1465 2179 a(\\) p Fy 1541 2179 a(\004) p Fp 1596 2191 a(d) p Fy 1664 2179 a(for) p Fo 1794 2179 a(t) p 1850 2179 a(>) p Fy 1942 2179 a(0,) p 2037 2179 a(w) n(e) p 2161 2179 a(ma) n(y) p 2343 2179 a(use) p 2489 2179 a(Prop) r(osition) p 2938 2179 a(16) p 3051 2179 a(in) p 3150 2179 a(a) p 3221 2179 a(similar) p 3496 2179 a(w) n(a) n(y) p 3667 2179 a(as) p 3771 2179 a(in) 208 2278 y(the) p 351 2278 a(pro) r(of) p 568 2278 a(of) p 662 2278 a(\(61\)) p 837 2278 a(to) p 939 2278 a(pro) n(v) n(e) p 1163 2278 a(the) p 1306 2278 a(\014rst) p 1477 2278 a(claim.) 55 2445 y(\(ii\)) p 208 2445 a(The) p 375 2445 a(second) p 640 2445 a(claim) p 858 2445 a(is) p 939 2445 a(a) p 1005 2445 a(direct) p 1237 2445 a(consequence) p 1703 2445 a(of) p 1794 2445 a(Prop) r(osition) p 2238 2445 a(20) p 2345 2445 a(for) p Fo 2464 2429 a(~) 2469 2445 y(t) p Fy 2522 2445 a(=) p Fo 2607 2445 a(~) p 2610 2445 a(e) p Fy 2673 2445 a(\(de\014ned) p 2988 2445 a(in) p 3082 2445 a(\(47\)\),) p 3310 2445 a(com) n(bined) p 3679 2445 a(with) 208 2544 y(\(19\).) p Fd 3778 2644 a(2) p Fi -42 3009 a(Pr) l(o) l(of) p 181 3009 a(of) p 279 3009 a(The) l(or) l(em) p 624 3009 a(10.) p Fy 776 3009 a(T) p 829 3009 a(o) p 898 3009 a(pro) n(v) n(e) p 1122 3009 a(the) p 1265 3009 a(upp) r(er) p 1502 3009 a(b) r(ound,) p 1782 3009 a(de\014ne) p Fo 1217 3204 a(\020) p Fp 1253 3216 a(n) p Fy 1322 3204 a(=) p Fg 1605 3125 a(X) p Fp 1409 3311 a(w) p Fn 1459 3311 a(2) p Fp(W) p Ff 1575 3295 a(\(0\)) p Fj 1652 3311 a(;) p Fp 1690 3311 a(\027) p Fj 1727 3311 a(\() p Fp(w) p Fj 1803 3311 a(\)) p Fb(5) p Fp(n) p Fo 1935 3204 a(e) p Fn 1974 3169 a(\000) p Fp(\014) p Fm 2064 3177 a(c) p Fp 2095 3169 a(L) p Fj(\() p Fp(w) p Fj 2217 3169 a(\)) p Fo 2246 3204 a(;) p 2338 3204 a(n) p Fk 2411 3204 a(2) p Fq 2489 3204 a(Z) p Fj 2544 3216 a(+) p Fo 2600 3204 a(:) p Fy 3692 3204 a(\(62\)) 0 3487 y(Then) p 221 3487 a(a) p 295 3487 a(consideration) p 807 3487 a(similar) p 1085 3487 a(to) p 1191 3487 a(that) p 1375 3487 a(in) p 1476 3487 a(the) p 1624 3487 a(pro) r(of) p 1845 3487 a(of) p 1944 3487 a(Prop) r(osition) p 2395 3487 a(4) p 2469 3487 a(pro) n(v) n(es) p 2730 3487 a(that) p 2915 3487 a(there) p 3131 3487 a(exists) p 3365 3487 a(a) p 3439 3487 a(p) r(olynomial) p Fo 0 3587 a(f) p Fj 41 3599 a(1) p Fy 101 3587 a(:) p Fq 175 3587 a(R) p Fn 235 3557 a(I) p Fm 273 3566 a(d) p Fk 334 3587 a(!) p Fq 440 3587 a(R) p Fy 528 3587 a(with) p 717 3587 a(p) r(ositiv) n(e) p 1024 3587 a(co) r(e\016cien) n(ts,) p 1468 3587 a(of) p 1562 3587 a(maximal) p 1901 3587 a(degree) p Fo 2160 3587 a(d) p Fy 2221 3587 a(+) p 2304 3587 a(1,) p 2397 3587 a(satisfying) p Fo 2767 3587 a(f) p Fj 2808 3599 a(1) p Fy 2844 3587 a(\() p Fo 2870 3569 a(~) p Fy 2876 3587 a(0\)) p 2974 3587 a(=) p 3061 3587 a(1,) p 3153 3587 a(suc) n(h) p 3341 3587 a(that) p Fo 1358 3779 a(\020) p Fp 1394 3791 a(n) p Fj(+1) p Fl 1546 3779 a(5) p Fo 1634 3779 a(f) p Fj 1675 3791 a(1) p Fy 1712 3779 a(\() p Fo 1755 3758 a(~) 1744 3779 y(Z) p Fp 1801 3791 a(n) p Fy 1846 3779 a(\() p Fo(\014) p Fp 1925 3791 a(c) p Fy 1959 3779 a(\)\)) p Fo(\020) p Fp 2059 3791 a(n) p Fo 2105 3779 a(;) p 2197 3779 a(n) p Fk 2270 3779 a(2) p Fq 2349 3779 a(Z) p Fj 2404 3791 a(+) p Fo 2459 3779 a(:) p Fy 3692 3779 a(\(63\)) 0 3972 y(The) p 168 3972 a(assumption) p 603 3972 a(\(CS1\)) p 840 3972 a(implies) p 1119 3972 a(that) p Fo 1307 3951 a(~) 1296 3972 y(Z) p Fp 1353 3984 a(n) p Fy 1397 3972 a(\() p Fo(\014) p Fp 1476 3984 a(c) p Fy 1511 3972 a(\)) p 1567 3972 a(is) p 1648 3972 a(b) r(ounded,) p 2007 3972 a(hence) p 2235 3972 a(there) p 2444 3972 a(exist) p 2637 3972 a(p) r(ositiv) n(e) p 2941 3972 a(constan) n(ts) p Fo 3306 3972 a(A) p Fj 3368 3984 a(1) p Fy 3430 3972 a(and) p Fo 3588 3972 a(A) p Fj 3650 3984 a(2) p Fl 3711 3972 a(=) p Fy 3798 3972 a(1) 0 4072 y(suc) n(h) p 187 4072 a(that) p Fo 1525 4171 a(\020) p Fp 1561 4183 a(n) p Fl 1630 4171 a(5) p Fo 1717 4171 a(A) p Fj 1779 4183 a(1) p Fo 1817 4171 a(A) p Fp 1879 4137 a(n) p Fj 1879 4192 a(2) p Fo 1938 4171 a(;) p 2030 4171 a(n) p Fk 2103 4171 a(2) p Fq 2182 4171 a(Z) p Fj 2237 4183 a(+) p Fo 2292 4171 a(:) p Fy 3692 4171 a(\(64\)) p Fo 0 4321 a(L) p 80 4321 a(>) p Fy 167 4321 a(2) p Fp 209 4290 a(\027) p Fn 246 4290 a(\000) p Fj(1) p Fy 363 4321 a(in) p 460 4321 a(\(55\)) p 635 4321 a(therefore) p 983 4321 a(implies) p Fo 727 4520 a(e) p Fn 766 4486 a(\000) p Fp(\014) p Fm 856 4494 a(c) p Fp 887 4486 a(k) p Fo 927 4520 a(N) p Fy 1003 4520 a(\() p Fo(k) p Fy 1081 4520 a(\)) p Fl 1137 4520 a(5) p Fg 1423 4441 a(X) p Fp 1224 4627 a(w) p Fn 1274 4627 a(2) p Fp(W) p Ff 1390 4611 a(\(0\)) p Fj 1467 4627 a(;) p Fp 1505 4627 a(L) p Fj(\() p Fp(w) p Fj 1627 4627 a(\)) p Fb(5) p Fp(k) p Fo 1755 4520 a(e) p Fn 1794 4486 a(\000) p Fp(\014) p Fm 1884 4494 a(c) p Fp 1914 4486 a(L) p Fj(\() p Fp(w) p Fj 2036 4486 a(\)) p Fl 2088 4520 a(5) p Fo 2176 4520 a(\020) p Fj 2212 4535 a([log) p Ff 2317 4550 a(2) p Fp 2360 4535 a(k) p Fj 2396 4535 a(]+1) p Fl 2527 4520 a(5) p Fo 2615 4520 a(A) p Fj 2677 4532 a(1) p Fo 2714 4520 a(A) p Fj 2776 4532 a(2) p Fo 2828 4520 a(k) p Fj 2874 4486 a(log) p Ff 2959 4501 a(2) p Fp 3003 4486 a(A) p Ff 3053 4494 a(2) p Fo 3090 4520 a(;) p Fy 0 4803 a(whic) n(h) p 238 4803 a(pro) n(v) n(es) p 494 4803 a(the) p 637 4803 a(upp) r(er) p 875 4803 a(b) r(ound.) 125 4903 y(T) p 178 4903 a(o) p 245 4903 a(pro) n(v) n(e) p 467 4903 a(the) p 608 4903 a(lo) n(w) n(er) p 823 4903 a(b) r(ound,) p 1100 4903 a(let) p Fo 1218 4903 a(b) p 1277 4903 a(>) p Fy 1365 4903 a(0) p 1432 4903 a(b) r(e) p 1543 4903 a(a) p 1610 4903 a(constan) n(t) p 1943 4903 a(suc) n(h) p 2128 4903 a(that) p 2306 4903 a(the) p 2447 4903 a(b) r(ound) p 2701 4903 a(in) p 2796 4903 a(Prop) r(osition) p 3241 4903 a(23) p 3350 4903 a(holds,) p 3588 4903 a(and) p 3748 4903 a(let) p Fo 0 5002 a(d) p Fp 43 5014 a(n) p Fy 111 5002 a(=) p Fg 199 4927 a(p) p 282 4927 275 4 v Fy 282 5002 a(2) p Fo(n) p Fy 388 5002 a(log) p Fo 508 5002 a(\025) q(c) p Fp 593 5014 a(n) p Fy 661 5002 a(=) p Fg 749 4927 a(p) p 832 4927 225 4 v Fy 832 5002 a(2) p 888 5002 a(log) p Fo 1008 5002 a(\025) p 1070 5002 a(b) p 1120 5002 a(n) p 1184 5002 a(\025) p Fn 1232 4968 a(\000) p Fp(n) p Fy 1329 5002 a(,) p Fo 1380 5002 a(n) p Fk 1453 5002 a(2) p Fq 1531 5002 a(Z) p Fj 1586 5014 a(+) p Fy 1641 5002 a(.) p 1701 5002 a(Then) p 1918 5002 a(Prop) r(osition) p 2365 5002 a(23) p 2476 5002 a(implies) 739 5228 y(lim) p Fp 710 5278 a(n) p Fn(!1) p Fy 897 5228 a(sup) p Fp 900 5298 a(\030) p Fn 932 5298 a(2) p Fc(R) p Fk 1036 5228 a(j) p Fg 1073 5115 a(Z) p Fj 1119 5304 a([) p Fp(\030) p Fn 1170 5304 a(\000) p Fp(d) p Fm 1257 5312 a(n) p Fp 1297 5304 a(;\030) p Fj 1349 5304 a(+) p Fp(d) p Fm 1435 5312 a(n) p Fj 1475 5304 a(]) p Fo 1512 5228 a(h) p Fp 1560 5240 a(n) p Fy 1605 5228 a(\() p Fo(\030) p Fk 1696 5228 a(\000) p Fo 1779 5228 a(\021) p Fy 1823 5228 a(\)) p 1876 5228 a(\026) p Fo 1869 5228 a(p) p Fp 1910 5243 a(~) p 1911 5243 a(x) p Fm 1949 5251 a(can) p Fj 2049 5243 a(\() p Fp(\014) p Fm 2113 5251 a(c) p Fj 2143 5243 a(\)) p Fp(;n;) p Fj(\(1\)) p Fy 2339 5228 a(\() p Fo(d\021) p Fy 2458 5228 a(\)) p Fk 2509 5228 a(\000) p Fy 2600 5228 a(\026) p Fo 2592 5228 a(\032) p Fn 2635 5194 a(\003) p Fp 2634 5251 a(~) p 2635 5251 a(x) p Fm 2673 5259 a(can) p Fj 2773 5251 a(\() p Fp(\014) p Fm 2837 5259 a(c) p Fj 2867 5251 a(\)) p Fp(;) p Fj(\(1\)) p Fy 3002 5228 a(\() p Fo(\030) p Fy 3074 5228 a(\)) p Fk(j) p Fl 710 5431 a(5) p Fy 827 5431 a(lim) p Fp 798 5481 a(n) p Fn(!1) p Fy 985 5431 a(sup) p Fp 988 5501 a(\030) p Fn 1020 5501 a(2) p Fc(R) p Fg 1124 5318 a(Z) p Fc 1170 5507 a(R) p Fn(n) p Fj([) p Fp(\030) p Fn 1297 5507 a(\000) p Fp(d) p Fm 1384 5515 a(n) p Fp 1424 5507 a(;\030) p Fj 1476 5507 a(+) p Fp(d) p Fm 1562 5515 a(n) p Fj 1602 5507 a(]) p Fo 1639 5431 a(h) p Fp 1687 5443 a(n) p Fy 1732 5431 a(\() p Fo(\030) p Fk 1823 5431 a(\000) p Fo 1906 5431 a(\021) p Fy 1950 5431 a(\)) p 2003 5431 a(\026) p Fo 1996 5431 a(p) p Fp 2037 5446 a(~) p 2038 5446 a(x) p Fm 2076 5454 a(can) p Fj 2175 5446 a(\() p Fp(\014) p Fm 2239 5454 a(c) p Fj 2270 5446 a(\)) p Fp(;n;) p Fj(\(1\)) p Fy 2466 5431 a(\() p Fo(d\021) p Fy 2585 5431 a(\)) p Fl 710 5631 a(5) p Fy 827 5631 a(lim) p Fp 798 5681 a(n) p Fn(!1) p Fo 985 5631 a(h) p Fp 1033 5643 a(n) p Fy 1078 5631 a(\() p Fo(d) p Fp 1153 5643 a(n) p Fy 1199 5631 a(\)) p 1254 5631 a(=) p 1371 5631 a(lim) p Fp 1342 5681 a(n) p Fn(!1) p Fy 1660 5575 a(1) p 1539 5612 284 4 v Fk 1539 5628 a(p) p 1608 5628 215 4 v Fy 1608 5701 a(2) p Fo(\031) s(b) p Fj 1736 5677 a(2) p Fo 1773 5701 a(n) p Fy 1855 5631 a(=) p 1943 5631 a(0) p Fo 1999 5631 a(:) p Fy 1878 6120 a(23) p 90 rotate dyy eop %%Page: 24 24 24 23 bop Fy 0 83 a(With) p 214 83 a(the) p 357 83 a(p) r(ositivit) n (y) p 726 83 a(of) p 827 83 a(\026) p Fo 820 83 a(\032) p Fn 863 53 a(\003) p Fp 862 110 a(~) p 863 110 a(x) p Fm 901 118 a(can) p Fj 1000 110 a(\() p Fp(\014) p Fm 1064 118 a(c) p Fj 1095 110 a(\)) p Fp(;) p Fj(\(1\)) p Fy 1257 83 a(\(Theorem) p 1640 83 a(9\),) p 1764 83 a(this) p 1925 83 a(implies) p 2207 83 a(that) p 2386 83 a(there) p 2598 83 a(exists) p 2827 83 a(an) p 2942 83 a(in) n(teger) p Fo 3216 83 a(n) p Fj 3266 95 a(2) p Fy 3331 83 a(and) p 3492 83 a(a) p 3560 83 a(p) r(ositiv) n(e) 0 183 y(constan) n(t) p Fo 335 183 a(\017) p Fy 396 183 a(suc) n(h) p 583 183 a(that) 850 364 y(\026) p Fo 843 364 a(p) p Fp 884 379 a(~) p 885 379 a(x) p Fm 923 387 a(can) p Fj 1022 379 a(\() p Fp(\014) p Fm 1086 387 a(c) p Fj 1117 379 a(\)) p Fp(;n;) p Fj(\(1\)) p Fy 1312 364 a(\([) p Fo(\030) p Fk 1426 364 a(\000) p Fo 1509 364 a(d) p Fp 1552 376 a(n) p Fo 1598 364 a(;) p 1635 364 a(\030) p Fy 1693 364 a(+) p Fo 1776 364 a(d) p Fp 1819 376 a(n) p Fy 1864 364 a(]\)) p Fl 1943 364 a(=) p Fo 2031 364 a(c) p Fp 2067 376 a(n) p Fo 2112 364 a(\017;) p 2238 364 a(n) p Fl 2310 364 a(=) p Fo 2398 364 a(n) p Fj 2448 376 a(2) p Fo 2499 364 a(;) p 2564 364 a(\030) p Fk 2627 364 a(2) p Fy 2705 364 a([) p Fo(\025) p Fn 2776 329 a(\000) p Fj(1) p Fo 2866 364 a(;) p 2903 364 a(\025) p Fy(]) p Fo(:) p Fy 125 545 a(Let) p Fo 276 545 a(k) p Fy 352 545 a(b) r(e) p 468 545 a(a) p 539 545 a(p) r(ositiv) n(e) p 849 545 a(in) n(teger,) p 1150 545 a(and) p 1314 545 a(let) p Fo 1437 545 a(n) p Fy 1517 545 a(b) r(e) p 1632 545 a(a) p 1704 545 a(p) r(ositiv) n(e) p 2014 545 a(in) n(teger) p 2291 545 a(satisfying) p Fo 2664 545 a(\025) p Fn 2712 515 a(\000) p Fp(n) p Fo 2809 545 a(k) p Fk 2883 545 a(2) p Fy 2965 545 a([1) p Fo(;) p 3067 545 a(\025) p Fy(\).) p 3215 545 a(Assume) p 3526 545 a(that) p Fo 3708 545 a(k) p Fy 3784 545 a(is) 0 644 y(su\016cien) n(tly) p 416 644 a(large) p 619 644 a(so) p 721 644 a(that) p Fo 901 644 a(n) p Fl 973 644 a(=) p Fo 1061 644 a(n) p Fj 1111 656 a(2) p Fy 1176 644 a(and) p Fo 1337 644 a(d) p Fp 1380 656 a(n) p Fl 1449 644 a(5) p Fy 1536 644 a(1) p Fk 1596 644 a(\000) p Fo 1679 644 a(\025) p Fn 1727 614 a(\000) p Fj(1) p Fy 1817 644 a(.) p 1877 644 a(Then) 1187 826 y(\026) p Fo 1180 826 a(p) p Fp 1221 841 a(~) p 1222 841 a(x) p Fm 1260 849 a(can) p Fj 1359 841 a(\() p Fp(\014) p Fm 1423 849 a(c) p Fj 1454 841 a(\)) p Fp(;n;) p Fj(\(1\)) p Fy 1650 826 a(\([) p Fo(\025) p Fn 1753 791 a(\000) p Fp(n) p Fo 1850 826 a(k) p Fk 1915 826 a(\000) p Fy 1998 826 a(2) p Fo(d) p Fp 2083 838 a(n) p Fo 2128 826 a(;) p 2165 826 a(\025) p Fn 2213 791 a(\000) p Fp(n) p Fo 2310 826 a(k) p Fy 2356 826 a(]\)) p Fl 2434 826 a(=) p Fo 2522 826 a(c) p Fp 2558 838 a(n) p Fo 2603 826 a(\017:) p Fy 0 1032 a(Note) p 203 1032 a(that) p 385 1032 a(w) n(e) p 509 1032 a(can) p 663 1032 a(construct) p 1030 1032 a(an) p 1147 1032 a(injection) p Fk 1488 1032 a(f) p Fo(w) p Fk 1618 1032 a(2) p Fo 1700 1032 a(W) p Fj 1790 988 a(\() p Fp(n) p Fj(\)) 1778 1060 y(\(1\)) p Fk 1913 1032 a(j) p Fo 1963 1032 a(L) p Fy(\() p Fo(w) p Fy 2113 1032 a(\)) p Fl 2172 1032 a(5) p Fo 2263 1032 a(k) p Fk 2309 1032 a(g) p 2377 1032 a(!) p 2486 1032 a(f) p Fo(w) p Fk 2616 1032 a(2) p Fo 2698 1032 a(W) p Fj 2788 997 a(\(0\)) p Fk 2903 1032 a(j) p Fo 2953 1032 a(L) p Fy(\() p Fo(w) p Fy 3103 1032 a(\)) p 3162 1032 a(=) p Fo 3253 1032 a(k) p Fk 3299 1032 a(g) p Fy 3370 1032 a(b) n(y) p 3487 1032 a(extending) 0 1140 y(the) p 143 1140 a(path.) p 369 1140 a(Hence) p Fo 632 1333 a(Z) p Fp 689 1348 a(n;) p Fj(\(1\)) p Fy 839 1333 a(\() p Fo(\014) p Fp 918 1345 a(c) p Fy 952 1333 a(\)) p Fo(c) p Fp 1020 1345 a(n) p Fo 1065 1333 a(\017) p Fl 1122 1333 a(5) p Fg 1577 1255 a(X) p Fp 1210 1454 a(w) p Fn 1260 1454 a(2) p Fp(W) p Ff 1376 1424 a(\() p Fm(n) p Ff(\)) 1367 1477 y(\(1\)) p Fj 1461 1454 a(;) p Fp 1499 1454 a(k) p Fn 1535 1454 a(\000) p Fj(2) p Fp(d) p Fm 1655 1462 a(n) p Fp 1696 1454 a(\025) p Fm 1735 1438 a(n) p Fb 1776 1454 a(5) p Fp(L) p Fj(\() p Fp(w) p Fj 1950 1454 a(\)) p Fb(5) p Fp(k) p Fo 2077 1333 a(e) p Fn 2116 1299 a(\000) p Fp(\014) p Fm 2206 1307 a(c) p Fp 2236 1299 a(L) p Fj(\() p Fp(w) p Fj 2358 1299 a(\)) p Fl 2411 1333 a(5) p Fo 2498 1333 a(e) p Fj 2537 1299 a(2) p Fp(\014) p Fm 2608 1307 a(c) p Fp 2639 1299 a(d) p Fm 2674 1307 a(n) p Fp 2714 1299 a(\025) p Fm 2753 1274 a(n) p Fo 2798 1333 a(e) p Fn 2837 1299 a(\000) p Fp(\014) p Fm 2927 1307 a(c) p Fp 2958 1299 a(k) p Fo 2999 1333 a(N) p Fy 3075 1333 a(\() p Fo(k) p Fy 3153 1333 a(\)) p Fo(:) p Fy 0 1699 a(Note) p 201 1699 a(that) p Fo 381 1699 a(\025) p Fn 429 1669 a(\000) p Fp(n) p Fo 526 1699 a(k) p Fl 595 1699 a(=) p Fy 683 1699 a(1) p 752 1699 a(implies) p Fo 1034 1699 a(n) p Fl 1106 1699 a(5) p Fy 1205 1643 a(log) p Fo 1326 1643 a(k) p 1204 1680 170 4 v Fy 1204 1756 a(log) p Fo 1325 1756 a(\025) p Fy 1384 1699 a(,) p 1434 1699 a(whic) n(h) p 1672 1699 a(further) p 1951 1699 a(implies) p Fo 1479 1944 a(d) p Fp 1522 1956 a(n) p Fo 1568 1944 a(\025) p Fp 1616 1909 a(n) p Fl 1684 1944 a(5) p Fg 1772 1869 a(p) p 1855 1869 267 4 v Fy 1855 1944 a(2) p Fo(=) p Fy 1953 1944 a(log) p Fo 2073 1944 a(\025b) p Fy 2171 1944 a(log) p Fo 2292 1944 a(k) s(:) p Fy 0 2125 a(Since) p Fo 217 2125 a(n) p Fl 290 2125 a(=) p Fy 377 2125 a(1) p 446 2125 a(for) p 574 2125 a(su\016cien) n(tly) p 989 2125 a(large) p Fo 1192 2125 a(k) p Fy 1238 2125 a(,) p Fo 1289 2125 a(\025) p Fn 1337 2095 a(\000) p Fp(n) p Fo 1434 2125 a(k) p Fl 1503 2125 a(=) p Fy 1591 2125 a(1) p 1660 2125 a(also) p 1827 2125 a(implies) p Fo 2108 2125 a(c) p Fp 2144 2137 a(n) p Fl 2213 2125 a(=) p Fo 2300 2125 a(bk) p Fn 2382 2095 a(\000) p Fj(1) p Fy 2471 2125 a(.) p 2531 2125 a(Hence) p 2778 2125 a(\(for) p 2937 2125 a(su\016cien) n(tly) p 3352 2125 a(large) p Fo 3556 2125 a(k) p Fy 3602 2125 a(\),) p Fo 1174 2333 a(N) p Fy 1250 2333 a(\() p Fo(k) p Fy 1328 2333 a(\)) p Fl 1383 2333 a(=) p Fo 1471 2333 a(e) p Fp 1510 2299 a(\014) p Fm 1548 2307 a(c) p Fp 1578 2299 a(k) p Fo 1619 2333 a(k) p Fn 1665 2299 a(\000) p Fj(1) p Fn(\000) p Fj(2) p Fp(b\014) p Fm 1902 2307 a(c) p Fk 1933 2242 a(p) p 2002 2242 216 4 v Fj 2002 2299 a(2) p Fp(=) p Fj 2081 2299 a(log) p Fp 2178 2299 a(\025) p Fo 2221 2333 a(Z) p Fp 2278 2348 a(n;) p Fj(\(1\)) p Fy 2428 2333 a(\() p Fo(\014) p Fp 2507 2345 a(c) p Fy 2541 2333 a(\)) p Fo(b\017;) p Fy 0 2514 a(whic) n(h) p 238 2514 a(implies) p 519 2514 a(the) p 662 2514 a(lo) n(w) n(er) p 879 2514 a(b) r(ound.) p Fd 3778 2614 a(2) p Fz 0 3009 a(4.3.2) p 292 3009 a(Large) p 558 3009 a(deviation) p 975 3009 a(t) m(yp) s(e) p 1191 3009 a(estimates) p 1612 3009 a(on) p 1745 3009 a(long) p 1952 3009 a(paths) p 2211 3009 a(and) p 2395 3009 a(short) p 2642 3009 a(paths.) p Fy 0 3162 a(De\014ne,) p 280 3162 a(for) p Fo 407 3162 a(n;) p 494 3162 a(m) p Fk 590 3162 a(2) p Fq 668 3162 a(Z) p Fj 723 3174 a(+) p Fy 779 3162 a(,) p Fo 516 3358 a(U) p Fp 573 3370 a(n;m) p Fy 719 3358 a(=) p Fg 1005 3280 a(X) p Fp 934 3474 a(w) p Fn 984 3474 a(2) p Fp(W) p Ff 1100 3449 a(\(0\)) p Fj 1177 3474 a(;) p Fp 939 3542 a(\027) p Fj 976 3542 a(\() p Fp(w) p Fj 1052 3542 a(\)) p Fb(5) p Fp(n;) 827 3623 y(L) p Fj(\() p Fp(w) p Fj 949 3623 a(\)) p Fb(=) p Fp(\025) p Fm 1066 3598 a(n) p Ff(+\() p Fm(m=) p Ff(2\)) p Fo 1337 3358 a(e) p Fn 1376 3324 a(\000) p Fp(\014) p Fm 1466 3332 a(c) p Fp 1496 3324 a(L) p Fj(\() p Fp(w) p Fj 1618 3324 a(\)) p Fy 1896 3358 a(and) p Fo 2279 3358 a(V) p Fp 2327 3370 a(n;m) p Fy 2475 3358 a(=) p Fg 2710 3280 a(X) p Fp 2639 3474 a(w) p Fn 2689 3474 a(2) p Fp(W) p Ff 2805 3449 a(\(0\)) p Fj 2882 3474 a(;) p Fp 2602 3538 a(\027) p Fj 2639 3538 a(\() p Fp(w) p Fj 2715 3538 a(\)=) p Fp(n) p Fj(+1) p Fp(;) 2582 3611 y(L) p Fj(\() p Fp(w) p Fj 2704 3611 a(\)) p Fb(5) p Fp(\025) p Fm 2821 3586 a(n) p Fe(\000) p Fm(m) p Fo 2991 3358 a(e) p Fn 3030 3324 a(\000) p Fp(\014) p Fm 3120 3332 a(c) p Fp 3150 3324 a(L) p Fj(\() p Fp(w) p Fj 3272 3324 a(\)) p Fo 3301 3358 a(:) p Fz 0 3802 a(Prop) s(osition) p 516 3802 a(25) p Fi 653 3802 a(Ther) l(e) p 892 3802 a(exist) p 1085 3802 a(p) l(ositive) p 1384 3802 a(c) l(onstants) p Fo 1749 3802 a(C) p Fn 1814 3772 a(0) p Fi 1868 3802 a(and) p Fo 2029 3802 a(C) p Fn 2094 3772 a(00) p Fi 2166 3802 a(such) p 2355 3802 a(that) p Fo 740 3998 a(U) p Fp 797 4010 a(n;m) p Fl 943 3998 a(5) p Fo 1031 3998 a(C) p Fn 1096 3964 a(0) p Fo 1120 3998 a(A) p Fp 1182 3964 a(n) p Fj 1182 4019 a(2) p Fo 1227 3998 a(e) p Fn 1266 3964 a(\000) p Fp(C) p Fe 1370 3939 a(00) p Fp 1410 3964 a(\025) p Fm 1449 3939 a(m=) p Ff(2) p Fo 1568 3998 a(;) p Fi 1664 3998 a(and) p Fo 1855 3998 a(V) p Fp 1903 4010 a(n;m) p Fl 2050 3998 a(5) p Fo 2138 3998 a(C) p Fn 2203 3964 a(0) p Fo 2226 3998 a(A) p Fp 2288 3964 a(n) p Fj 2288 4019 a(2) p Fo 2334 3998 a(e) p Fn 2373 3964 a(\000) p Fp(C) p Fe 2477 3939 a(00) p Fj 2517 3964 a(2) p Fm 2550 3939 a(m) p Fo 2609 3998 a(;) p 2705 3998 a(n;) p 2792 3998 a(m) p Fk 2888 3998 a(2) p Fq 2967 3998 a(Z) p Fj 3022 4010 a(+) p Fo 3077 3998 a(;) p Fi 0 4179 a(wher) l(e) p Fo 234 4179 a(A) p Fj 296 4191 a(2) p Fi 364 4179 a(is) p 453 4179 a(as) p 559 4179 a(in) p 661 4179 a(\(64\).) -42 4423 y(Pr) l(o) l(of.) p Fy 219 4423 a(Let) p Fo 367 4423 a(C) p Fn 432 4393 a(00) p Fy 503 4423 a(b) r(e) p 616 4423 a(a) p 685 4423 a(p) r(ositiv) n(e) p 992 4423 a(constan) n(t) p 1327 4423 a(satisfying) p Fo 1697 4423 a(C) p Fn 1762 4389 a(00) p Fl 1828 4423 a(5) p Fo 1925 4367 a(\025) p Fk 1992 4367 a(\000) 2075 4296 y(p) p 2144 4296 49 4 v Fo 2144 4367 a(\025) p 1925 4404 268 4 v 1966 4480 a(d) p Fy 2028 4480 a(+) p 2111 4480 a(1) 2217 4423 y(.) 125 4553 y(T) p 178 4553 a(o) p 247 4553 a(pro) n(v) n(e) p 471 4553 a(the) p 614 4553 a(b) r(ound) p 870 4553 a(for) p Fo 997 4553 a(U) p Fp 1054 4565 a(n;m) p Fy 1191 4553 a(,) p 1242 4553 a(de\014ne) p Fo 748 4754 a(S) p Fp 799 4766 a(n;m;I) p Fy 1000 4754 a(=) p Fg 1457 4675 a(X) p Fp 1087 4875 a(w) p Fn 1137 4875 a(2) p Fp(W) p Ff 1253 4844 a(\() p Fm(n) p Ff(\)) p Fm 1244 4894 a(I) p Fp 1339 4875 a(;) p 1377 4875 a(L) p Fj(\() p Fp(w) p Fj 1499 4875 a(\)) p Fb(=) p Fp(C) p Fe 1629 4858 a(00) p Fp 1669 4875 a(\025) p Fm 1708 4858 a(n) p Ff(+\() p Fm(m=) p Ff(2\)) p Fo 1960 4754 a(e) p Fn 1999 4720 a(\000) p Fp(\014) p Fm 2089 4728 a(c) p Fp 2119 4720 a(L) p Fj(\() p Fp(w) p Fj 2241 4720 a(\)) p Fo 2270 4754 a(;) p 2390 4754 a(n;) p 2477 4754 a(m) p Fk 2573 4754 a(2) p Fq 2651 4754 a(Z) p Fj 2706 4766 a(+) p Fo 2762 4754 a(;) p 2827 4754 a(I) p Fk 2893 4754 a(2) p 2971 4754 a(I) p Fp 3016 4766 a(d) p Fo 3069 4754 a(:) p Fy 0 5055 a(Then,) p 240 5055 a(in) p 337 5055 a(a) p 406 5055 a(similar) p 679 5055 a(w) n(a) n(y) p 847 5055 a(as) p 949 5055 a(deriving) p 1270 5055 a(\(63\),) p 1468 5055 a(w) n(e) p 1590 5055 a(\014nd) p Fo 606 5279 a(U) p Fp 663 5291 a(n) p Fj(+1) p Fp(;m) p Fl 894 5279 a(5) p Fo 981 5279 a(f) p Fj 1022 5291 a(1) p Fy 1059 5279 a(\() p Fo 1102 5258 a(~) 1091 5279 y(Z) p Fp 1148 5291 a(n) p Fy 1193 5279 a(\() p Fo(\014) p Fp 1272 5291 a(c) p Fy 1306 5279 a(\)\)) p Fo(U) p Fp 1427 5291 a(n;m) p Fj(+1) p Fy 1654 5279 a(+) p Fg 1753 5200 a(X) p Fp 1737 5378 a(I) p Fn 1771 5378 a(2I) p Fm 1854 5387 a(d) p Fo 1902 5279 a(S) p Fp 1953 5291 a(n;m;I) p Fo 2158 5223 a(@) p 2207 5223 a(f) p Fj 2248 5235 a(1) p 2154 5260 135 4 v Fo 2154 5336 a(@) p 2203 5336 a(x) p Fp 2250 5348 a(I) p Fy 2298 5279 a(\() p Fo 2341 5258 a(~) 2330 5279 y(Z) p Fp 2387 5291 a(n) p Fy 2433 5279 a(\() p Fo(\014) p Fp 2512 5291 a(c) p Fy 2546 5279 a(\)\)) p Fo 2624 5279 a(\020) p Fp 2660 5291 a(n) p Fo 2719 5279 a(;) p 2839 5279 a(n;) p 2926 5279 a(m) p Fk 3022 5279 a(2) p Fq 3100 5279 a(Z) p Fj 3155 5291 a(+) p Fo 3211 5279 a(:) p Fy 0 5547 a(This) p 190 5547 a(with) p 379 5547 a(\(64\)) p 554 5547 a(implies) p 836 5547 a(that) p 1016 5547 a(there) p 1228 5547 a(exists) p Fo 1457 5547 a(C) p Fj 1516 5559 a(1) p Fo 1577 5547 a(>) p Fy 1665 5547 a(0) p 1734 5547 a(suc) n(h) p 1921 5547 a(that) p Fo 831 5745 a(A) p Fn 893 5710 a(\000) p Fp(n) p Fn(\000) p Fj(1) 893 5767 y(2) p Fo 1076 5745 a(U) p Fp 1133 5757 a(n) p Fj(+1) p Fp(;m) p Fl 1363 5745 a(5) p Fo 1451 5745 a(A) p Fn 1513 5710 a(\000) p Fp(n) p Fj 1513 5767 a(2) p Fo 1610 5745 a(U) p Fp 1667 5757 a(n;m) p Fj(+1) p Fy 1893 5745 a(+) p Fo 1976 5745 a(C) p Fj 2035 5757 a(1) p Fg 2102 5666 a(X) p Fp 2087 5844 a(I) p Fn 2121 5844 a(2I) p Fm 2204 5853 a(d) p Fo 2252 5745 a(S) p Fp 2303 5757 a(n;m;I) p Fo 2494 5745 a(;) p 2614 5745 a(n;) p 2701 5745 a(m) p Fk 2797 5745 a(2) p Fq 2875 5745 a(Z) p Fj 2930 5757 a(+) p Fo 2985 5745 a(:) p Fy 1878 6120 a(24) p 90 rotate dyy eop %%Page: 25 25 25 24 bop Fy 125 83 a(Note) p 325 83 a(that) p Fo 505 83 a(L) p Fy(\() p Fo(w) p Fy 655 83 a(\)) p Fl 711 83 a(5) p Fo 798 83 a(d) p Fy 855 83 a(\() p Fo(d) p Fy 949 83 a(+) p 1033 83 a(1\)) p Fp 1107 49 a(\027) p Fj 1144 49 a(\() p Fp(w) p Fj 1220 49 a(\)) p Fy 1277 83 a(in) p 1374 83 a(\(55\)) p 1549 83 a(implies) p Fo 1303 262 a(U) p Fp 1360 274 a(n;m) p Fy 1506 262 a(=) p 1594 262 a(0) p Fo(;) p 1728 262 a(\025) p Fp 1776 228 a(n) p Fj(+\() p Fp(m=) p Fj(2\)) p Fo 2073 262 a(>) p 2161 262 a(d) p Fy 2218 262 a(\() p Fo(d) p Fy 2312 262 a(+) p 2395 262 a(1\)) p Fp 2469 228 a(n) p Fo 2514 262 a(;) p Fy 0 441 a(whic) n(h) p 238 441 a(holds) p 455 441 a(if) p Fo 1574 563 a(m) p 1670 563 a(>) p Fy 1767 507 a(2) p 1823 507 a(log) p Fo 1944 507 a(d) p 1767 544 220 4 v Fy 1792 620 a(log) p Fo 1913 620 a(\025) p Fy 2015 563 a(+) p 2098 563 a(2) p Fo(\013n;) p Fy 0 810 a(where) p 240 810 a(w) n(e) p 362 810 a(wrote) p Fo 593 810 a(\013) p Fy 670 810 a(=) 767 754 y(log\() p Fo(d) p Fy 968 754 a(+) p 1051 754 a(1\)) p 767 791 359 4 v 861 867 a(log) p Fo 983 867 a(\025) p Fk 1154 810 a(\000) p Fy 1237 810 a(1) p 1292 810 a(.) p 1352 810 a(\(Note) p 1585 810 a(that) p 1765 810 a(Prop) r(osition) p 2212 810 a(12) p 2322 810 a(implies) p Fo 2604 810 a(\013) p 2681 810 a(>) p Fy 2768 810 a(0.\)) p 2902 810 a(Therefore,) p 3302 810 a(if) p 3378 810 a(w) n(e) p 3500 810 a(write) p Fo 1177 1091 a(k) p Fj 1220 1103 a(0) p Fy 1257 1091 a(\() p Fo(n;) p 1376 1091 a(m) p Fy(\)) p 1505 1091 a(=) p Fg 1592 974 a(\024) p Fy 1646 1035 a(2) p Fo(\013n) p Fk 1809 1035 a(\000) p Fo 1892 1035 a(m) p 1646 1072 320 4 v Fy 1687 1148 a(2) p Fo(\013) p Fy 1800 1148 a(+) p 1883 1148 a(1) 1994 1091 y(+) 2220 1035 y(2) p 2276 1035 a(log) p Fo 2396 1035 a(d) p 2087 1072 486 4 v Fy 2087 1148 a(\(2) p Fo(\013) p Fy 2232 1148 a(+) p 2315 1148 a(1\)) p 2403 1148 a(log) p Fo 2524 1148 a(\025) p Fg 2582 974 a(\025) p Fo 2640 1091 a(;) p Fy 0 1315 a(then) p Fo 961 1486 a(U) p Fp 1018 1498 a(n;m) p Fl 1165 1486 a(5) p Fo 1252 1486 a(C) p Fj 1311 1498 a(1) p Fo 1349 1486 a(A) p Fp 1411 1452 a(n) p Fj 1411 1507 a(2) p Fp 1470 1375 a(k) p Ff 1505 1383 a(0) p Fj 1538 1375 a(\() p Fp(n;m) p Fj(\)) p Fg 1530 1408 a(X) p Fp 1530 1586 a(k) p Fj 1566 1586 a(=0) p Fg 1739 1408 a(X) p Fp 1723 1586 a(I) p Fn 1757 1586 a(2I) p Fm 1840 1595 a(d) p Fo 1888 1486 a(S) p Fp 1939 1498 a(n) p Fn(\000) p Fp(k) p Fn 2068 1498 a(\000) p Fj(1) p Fp(;m) p Fj(+) p Fp(k) q(;I) p Fo 2392 1486 a(;) p 2484 1486 a(n;) p 2571 1486 a(m) p Fk 2667 1486 a(2) p Fq 2745 1486 a(Z) p Fj 2800 1498 a(+) p Fo 2856 1486 a(:) p Fy 3692 1486 a(\(65\)) 125 1716 y(On) p 263 1716 a(the) p 406 1716 a(other) p 623 1716 a(hand,) p Fo 375 1910 a(S) p Fp 426 1922 a(n;m;I) p Fl 626 1910 a(5) p Fg 1083 1832 a(X) p Fp 714 2031 a(w) p Fn 764 2031 a(2) p Fp(W) p Ff 880 2001 a(\() p Fm(n) p Ff(\)) p Fm 871 2051 a(I) p Fp 965 2031 a(;) p 1004 2031 a(L) p Fj(\() p Fp(w) p Fj 1126 2031 a(\)) p Fb(=) p Fp(C) p Fe 1256 2015 a(00) p Fp 1295 2031 a(\025) p Fm 1334 2015 a(n) p Ff(+\() p Fm(m=) p Ff(2\)) p Fo 1587 1910 a(e) p Fn 1626 1876 a(\000) p Fj(\() p Fp(\014) p Fm 1742 1884 a(c) p Fn 1772 1876 a(\000) p Fp(\025) p Fe 1863 1851 a(\000) p Fm(n) p Fj 1949 1876 a(\)) p Fp(L) p Fj(\() p Fp(w) p Fj 2097 1876 a(\)) p Fo 2126 1910 a(e) p Fn 2165 1876 a(\000) p Fp(C) p Fe 2269 1851 a(00) p Fp 2309 1876 a(\025) p Fm 2348 1851 a(m=) p Ff(2) p Fl 2489 1910 a(5) p Fo 2577 1910 a(Z) p Fp 2634 1922 a(n;I) p Fy 2733 1910 a(\() p Fo(\014) p Fp 2812 1922 a(c) p Fk 2864 1910 a(\000) p Fo 2947 1910 a(\025) p Fn 2995 1876 a(\000) p Fp(n) p Fy 3093 1910 a(\)) p Fo(e) p Fn 3164 1876 a(\000) p Fp(C) p Fe 3268 1851 a(00) p Fp 3308 1876 a(\025) p Fm 3347 1851 a(m=) p Ff(2) p Fl 375 2150 a(5) p Fo 462 2150 a(C) p 527 2150 a(e) p Fn 566 2116 a(\000) p Fp(C) p Fe 670 2090 a(00) p Fp 711 2116 a(\025) p Fm 750 2090 a(m=) p Ff(2) p Fo 868 2150 a(;) p 988 2150 a(n;) p 1075 2150 a(m) p Fk 1171 2150 a(2) p Fq 1249 2150 a(Z) p Fj 1304 2162 a(+) p Fo 1360 2150 a(;) p 1424 2150 a(I) p Fk 1490 2150 a(2) p 1569 2150 a(I) p Fp 1614 2162 a(d) p Fo 1666 2150 a(;) p Fy 0 2330 a(for) p 127 2330 a(a) p 196 2330 a(p) r(ositiv) n(e) p 503 2330 a(constan) n(t) p Fo 838 2330 a(C) p Fy 903 2330 a(,) p 954 2330 a(where) p 1194 2330 a(w) n(e) p 1317 2330 a(used) p 1506 2330 a(Corollary) p 1873 2330 a(24.) p 2016 2330 a(This) p 2206 2330 a(and) p 2367 2330 a(\(65\)) p 2543 2330 a(imply) p Fo 1259 2575 a(U) p Fp 1316 2587 a(n;m) p Fl 1462 2575 a(5) p Fo 1550 2575 a(C) p 1615 2575 a(C) p Fj 1674 2587 a(1) p Fo 1712 2575 a(]) p Fk(I) p Fp 1789 2587 a(d) p Fo 1842 2575 a(A) p Fp 1904 2541 a(n) p Fj 1904 2595 a(2) p Fn 1991 2471 a(1) p Fg 1964 2496 a(X) p Fp 1963 2675 a(k) p Fj 1999 2675 a(=0) p Fo 2098 2575 a(e) p Fn 2137 2541 a(\000) p Fp(C) p Fe 2241 2516 a(00) p Fp 2281 2541 a(\025) p Ff 2320 2516 a(\() p Fm(m) p Ff(+) p Fm(k) p Ff 2468 2516 a(\)) p Fm(=) p Ff(2) p Fo 2558 2575 a(:) p Fy 125 2828 a(Note) p 325 2828 a(that) p 505 2828 a(there) p 717 2828 a(exists) p 947 2828 a(a) p 1016 2828 a(constan) n(t) p Fo 1351 2828 a(k) p Fj 1394 2840 a(0) p Fy 1459 2828 a(suc) n(h) p 1646 2828 a(that) p Fo 1074 3007 a(\025) p Fj 1122 2973 a(\() p Fp(m) p Fj(+) p Fp(k) p Fj 1294 2973 a(\)) p Fp(=) p Fj(2) p Fl 1416 3007 a(=) p Fo 1503 3007 a(\025) p Fj 1551 2973 a(\() p Fp(m) p Fn(\000) p Fp(k) p Ff 1723 2981 a(0) p Fj 1756 2973 a(\)) p Fp(=) p Fj(2) p Fy 1872 3007 a(+) p Fo 1955 3007 a(\025) p Fj 2003 2973 a(\() p Fp(k) p Fn 2065 2973 a(\000) p Fp(k) p Ff 2152 2981 a(0) p Fj 2185 2973 a(\)) p Fp(=) p Fj(2) p Fo 2283 3007 a(;) p 2375 3007 a(m;) p 2485 3007 a(k) p Fk 2554 3007 a(2) p Fq 2632 3007 a(Z) p Fj 2687 3019 a(+) p Fo 2743 3007 a(:) p Fy 0 3186 a(\(This) p 222 3186 a(is) p 305 3186 a(b) r(ecause) p Fo 575 3365 a(\025) p Fj 623 3331 a(\() p Fp(m) p Fj(+) p Fp(k) p Fj 795 3331 a(\)) p Fp(=) p Fj(2) p Fk 912 3365 a(\000) p Fy 995 3365 a(\() p Fo(\025) p Fp 1075 3331 a(m=) p Fj(2) p Fn(\000) p Fp(k) p Ff 1288 3339 a(0) p Fy 1344 3365 a(+) p Fo 1427 3365 a(\025) p Fp 1475 3331 a(k) q(=) p Fj(2) p Fn(\000) p Fp(k) p Ff 1665 3339 a(0) p Fy 1703 3365 a(\)) p 1758 3365 a(=) p 1846 3365 a(\() p Fo(\025) p Fp 1926 3331 a(m=) p Fj(2) p Fk 2075 3365 a(\000) p Fo 2158 3365 a(\025) p Fn 2206 3331 a(\000) p Fp(k) p Ff 2293 3339 a(0) p Fp 2326 3331 a(=) p Fj(2) p Fy 2397 3365 a(\)\() p Fo(\025) p Fp 2509 3331 a(k) q(=) p Fj(2) p Fk 2637 3365 a(\000) p Fo 2720 3365 a(\025) p Fn 2768 3331 a(\000) p Fp(k) p Ff 2855 3339 a(0) p Fp 2888 3331 a(=) p Fj(2) p Fy 2959 3365 a(\)) p Fk 3010 3365 a(\000) p Fo 3093 3365 a(\025) p Fn 3141 3331 a(\000) p Fp(k) p Ff 3228 3339 a(0) p Fy 0 3544 a(is) p 84 3544 a(increasing) p 473 3544 a(in) p Fo 570 3544 a(m) p Fy 671 3544 a(and) p Fo 833 3544 a(k) p Fy 879 3544 a(,) p 930 3544 a(hence) p 1161 3544 a(it) p 1245 3544 a(is) p 1329 3544 a(su\016cien) n(t) p 1678 3544 a(to) p 1780 3544 a(c) n(ho) r(ose) p Fo 2044 3544 a(k) p Fj 2087 3556 a(0) p Fy 2152 3544 a(suc) n(h) p 2340 3544 a(that) p 2520 3544 a(1) p Fl 2586 3544 a(=) p Fy 2674 3544 a(2) p Fo(\025) p Fn 2764 3514 a(\000) p Fp(k) p Ff 2851 3522 a(0) p Fp 2884 3514 a(=) p Fj(2) p Fy 2955 3544 a(.\)) p 3049 3544 a(Therefore) p 3426 3544 a(there) p 3638 3544 a(exists) 0 3643 y(a) p 69 3643 a(p) r(ositiv) n(e) p 376 3643 a(constan) n(t) p Fo 711 3643 a(C) p Fn 776 3613 a(0) p Fy 827 3643 a(suc) n(h) p 1015 3643 a(that) p Fo 449 3884 a(U) p Fp 506 3896 a(n;m) p Fl 652 3884 a(5) p Fo 740 3884 a(C) p 805 3884 a(C) p Fj 864 3896 a(1) p Fo 902 3884 a(]) p Fk(I) p Fp 979 3896 a(d) p Fn 1059 3780 a(1) p Fg 1032 3805 a(X) p Fp 1032 3984 a(k) p Fj 1068 3984 a(=0) p Fo 1166 3884 a(e) p Fn 1205 3850 a(\000) p Fp(C) p Fe 1309 3825 a(00) p Fp 1349 3850 a(\025) p Ff 1388 3825 a(\() p Fm(k) p Fe 1442 3825 a(\000) p Fm(k) p Ff 1518 3837 a(0) 1552 3825 y(\)) p Fm(=) p Ff(2) p Fo 1656 3884 a(A) p Fp 1718 3850 a(n) p Fj 1718 3905 a(2) p Fo 1763 3884 a(e) p Fn 1802 3850 a(\000) p Fp(C) p Fe 1906 3825 a(00) p Fp 1946 3850 a(\025) p Ff 1985 3825 a(\() p Fm(m) p Fe(\000) p Fm(k) p Ff 2134 3837 a(0) 2168 3825 y(\)) p Fm(=) p Ff(2) p Fl 2280 3884 a(5) p Fo 2368 3884 a(C) p Fn 2433 3850 a(0) p Fo 2457 3884 a(A) p Fp 2519 3850 a(n) p Fj 2519 3905 a(2) p Fo 2564 3884 a(e) p Fn 2603 3850 a(\000) p Fp(C) p Fe 2707 3825 a(00) p Fp 2747 3850 a(\025) p Fm 2786 3825 a(m=) p Ff(2) p Fo 2905 3884 a(;) p 2997 3884 a(n;) p 3084 3884 a(m) p Fk 3179 3884 a(2) p Fq 3258 3884 a(Z) p Fj 3313 3896 a(+) p Fo 3368 3884 a(:) p Fy 125 4162 a(T) p 178 4162 a(o) p 247 4162 a(pro) n(v) n(e) p 471 4162 a(the) p 614 4162 a(b) r(ound) p 870 4162 a(for) p Fo 997 4162 a(V) p Fp 1045 4174 a(n;m) p Fy 1183 4162 a(,) p 1234 4162 a(de\014ne) p Fo 697 4360 a(T) p Fp 746 4372 a(n;m;I) p Fy 946 4360 a(=) p Fg 1307 4281 a(X) p Fp 1034 4481 a(w) p Fn 1084 4481 a(2) p Fp(W) p Ff 1200 4451 a(\() p Fm(n) p Ff(\)) p Fm 1191 4501 a(I) p Fp 1285 4481 a(;) p 1324 4481 a(L) p Fj(\() p Fp(w) p Fj 1446 4481 a(\)) p Fb(5) p Fp(\025) p Fm 1563 4464 a(n) p Fe(\000) p Fm(m) p Fo 1713 4360 a(e) p Fn 1752 4326 a(\000) p Fp(\014) p Fm 1842 4334 a(c) p Fp 1872 4326 a(L) p Fj(\() p Fp(w) p Fj 1994 4326 a(\)) p Fo 2023 4360 a(;) p 2143 4360 a(n;) p 2230 4360 a(m) p Fk 2326 4360 a(2) p Fq 2404 4360 a(Z) p Fj 2459 4372 a(+) p Fy 2515 4360 a(;) p Fo 2579 4360 a(m) p Fl 2675 4360 a(5) p Fo 2763 4360 a(n;) p 2877 4360 a(I) p Fk 2943 4360 a(2) p 3022 4360 a(I) p Fp 3067 4372 a(d) p Fo 3120 4360 a(;) p Fy 0 4677 a(and) p 161 4677 a(write) p Fo 384 4656 a(~) 374 4677 y(T) p Fp 423 4689 a(n;m) p Fy 569 4677 a(=) p 657 4677 a(\() p Fo(T) p Fp 738 4689 a(n;m;I) p Fo 929 4677 a(;) p 989 4677 a(I) p Fk 1055 4677 a(2) p 1133 4677 a(I) p Fp 1178 4689 a(d) p Fy 1217 4677 a(\).) p 1310 4677 a(Then,) p 1549 4677 a(again) p 1771 4677 a(in) p 1868 4677 a(a) p 1937 4677 a(similar) p 2210 4677 a(w) n(a) n(y) p 2378 4677 a(as) p 2480 4677 a(deriving) p 2801 4677 a(\(63\),) p 2999 4677 a(w) n(e) p 3121 4677 a(\014nd) p Fo 1030 4868 a(V) p Fp 1078 4880 a(n;m) p Fl 1225 4868 a(5) p Fy 1313 4868 a(\() p Fo(f) p Fj 1386 4880 a(1) p Fy 1423 4868 a(\() p Fo 1465 4847 a(~) 1455 4868 y(T) p Fp 1504 4880 a(n;m) p Fy 1627 4868 a(\)) p Fk 1678 4868 a(\000) p Fy 1761 4868 a(1\)) p Fo(\020) p Fp 1871 4880 a(n) p Fy 1940 4868 a(=) p 2027 4868 a(\() p Fo(f) p Fj 2100 4880 a(1) p Fy 2137 4868 a(\() p Fo 2179 4847 a(~) 2169 4868 y(T) p Fp 2218 4880 a(n;m) p Fy 2342 4868 a(\)) p Fk 2393 4868 a(\000) p Fo 2476 4868 a(f) p Fj 2517 4880 a(1) p Fy 2554 4868 a(\() p Fo 2580 4851 a(~) p Fy 2586 4868 a(0) o(\)\)) p Fo(\020) p Fp 2727 4880 a(n) p Fo 2787 4868 a(:) p Fy 0 5047 a(On) p 138 5047 a(the) p 281 5047 a(other) p 498 5047 a(hand,) p 729 5047 a(the) p 872 5047 a(condition) p Fo 1236 5047 a(L) p Fy(\() p Fo(w) p Fy 1386 5047 a(\)) p Fl 1442 5047 a(5) p Fo 1530 5047 a(\025) p Fp 1578 5013 a(n) p Fn(\000) p Fp(m) p Fy 1762 5047 a(in) p 1859 5047 a(the) p 2002 5047 a(de\014nition) p 2371 5047 a(of) p Fo 2465 5047 a(T) p Fp 2514 5059 a(n;m;I) p Fy 2719 5047 a(implies) p Fo 893 5246 a(T) p Fp 942 5258 a(n;m;I) p Fl 1142 5246 a(5) p Fg 1295 5167 a(X) p Fp 1229 5367 a(w) p Fn 1279 5367 a(2) p Fp(W) p Ff 1395 5336 a(\() p Fm(n) p Ff(\)) p Fm 1386 5386 a(I) p Fo 1495 5246 a(e) p Fn 1534 5211 a(\000) p Fj(\() p Fp(\014) p Fm 1650 5219 a(c) p Fj 1680 5211 a(+) p Fp(\025) p Fm 1770 5186 a(m) p Fe(\000) p Fm(n) p Fj 1907 5211 a(\)) p Fp(L) p Fj(\() p Fp(w) p Fj 2055 5211 a(\)+1) p Fy 2191 5246 a(=) p Fo 2279 5246 a(eZ) p Fp 2375 5258 a(n;I) p Fy 2473 5246 a(\() p Fo(\014) p Fp 2552 5258 a(c) p Fy 2604 5246 a(+) p Fo 2687 5246 a(\025) p Fp 2735 5211 a(m) p Fn(\000) p Fp(n) p Fy 2892 5246 a(\)) p Fo(:) p Fy 0 5545 a(With) p 214 5545 a(Corollary) p 582 5545 a(24) p 692 5545 a(and) p 854 5545 a(\(63\)) p 1029 5545 a(w) n(e) p 1151 5545 a(ha) n(v) n(e) p 1343 5545 a(the) p 1486 5545 a(statemen) n(t.) p Fd 3778 5644 a(2) p Fy 1878 6120 a(25) p 90 rotate dyy eop %%Page: 26 26 26 25 bop Fz 0 83 a(Prop) s(osition) p 516 83 a(26) p Fi 653 83 a(F) p 701 83 a(or) p 808 83 a(su\016ciently) p 1227 83 a(lar) l(ge) p 1428 83 a(p) l(ositive) p 1727 83 a(c) l(onstant) p Fo 2058 83 a(\013) p Fi(,) p 2166 83 a(it) p 2249 83 a(holds) p 2461 83 a(that) p Fy 474 263 a(lim) p 474 276 116 4 v Fp 447 334 a(k) p Fn 483 334 a(!1) p Fy 616 263 a(\(log) p Fo 770 263 a(k) p Fy 816 263 a(\)) p Fp 848 229 a(s\013) p Fo 927 263 a(k) p Fn 973 229 a(\000) p Fp(s) p Fo 1060 263 a(E) p Fp 1121 275 a(k) p Fy 1162 263 a([) p Fk(k) p Fo 1226 263 a(w) p Fk 1287 263 a(k) p Fp 1330 221 a(sd) p Fm 1396 229 a(w) p Fy 1447 263 a(]) p Fo 1493 263 a(>) p Fy 1581 263 a(0) p Fo(;) p Fi 1808 263 a(and) p 2128 195 V Fy 2128 263 a(lim) p Fp 2101 317 a(k) p Fn 2137 317 a(!1) p Fy 2270 263 a(\(log) p Fo 2424 263 a(k) p Fy 2470 263 a(\)) p Fn 2502 229 a(\000) p Fp(s\013) p Fo 2633 263 a(k) p Fn 2679 229 a(\000) p Fp(s) p Fo 2766 263 a(E) p Fp 2827 275 a(k) p Fy 2868 263 a([) p Fk(k) p Fo 2932 263 a(w) p Fk 2993 263 a(k) p Fp 3036 221 a(sd) p Fm 3102 229 a(w) p Fy 3153 263 a(]) p Fo 3199 263 a(<) p Fk 3287 263 a(1) p Fo(;) p Fi 0 493 a(for) p 133 493 a(al) t(l) p Fo 251 493 a(s) p Fl 313 493 a(=) p Fy 401 493 a(0) p Fi(,) p 498 493 a(wher) l(e) p Fo 732 493 a(E) p Fp 793 505 a(k) p Fy 834 493 a([) p Fk(\001) p Fy(]) p Fi 933 493 a(is) p 1022 493 a(the) p 1160 493 a(exp) l(e) l(ctation) p 1589 493 a(de\014ne) l(d) p 1869 493 a(in) p 1971 493 a(The) l(or) l(em) p 2316 493 a(11.) -42 724 y(Pr) l(o) l(of.) p Fy 219 724 a(F) p 266 724 a(or) p Fo 368 724 a(k) p Fk 436 724 a(2) p Fq 515 724 a(N) p Fy(,) p 626 724 a(de\014ne) p 870 724 a(~) p Fo 865 724 a(n) p Fy(\() p Fo(k) p Fy 993 724 a(\)) p 1049 724 a(=) p Fg 1136 606 a(\024) p Fy 1191 667 a(log) p Fo 1313 667 a(k) p 1190 704 170 4 v Fy 1190 780 a(log) p Fo 1311 780 a(\025) p Fg 1370 606 a(\025) p Fy 1413 724 a(.) p 1473 724 a(Ob) n(viously) p 1830 724 a(,) p Fo 1880 724 a(\025) p Fj 1932 689 a(~) p Fp 1928 689 a(n) p Fj(\() p Fp(k) p Fj 2031 689 a(\)) p Fl 2085 724 a(5) p Fo 2173 724 a(k) p 2242 724 a(<) p 2329 724 a(\025) p Fj 2381 689 a(~) p Fp 2377 689 a(n) p Fj 2419 689 a(\() p Fp(k) p Fj 2481 689 a(\)+1) p Fy 2595 724 a(.) 125 873 y(F) p 172 873 a(or) p 274 873 a(p) r(ositiv) n(e) p 581 873 a(in) n(tegers) p Fo 888 873 a(m) p Fy 989 873 a(and) p Fo 1150 873 a(k) p Fy 1224 873 a(satisfying) p Fo 1594 873 a(m) p Fl 1689 873 a(5) p Fy 1781 873 a(~) p Fo 1777 873 a(n) p Fy(\() p Fo(k) p Fy 1905 873 a(\),) p 1988 873 a(Prop) r(osition) p 2435 873 a(25) p 2546 873 a(implies) p Fo 810 1061 a(]) p Fk(f) p Fo(w) p Fk 968 1061 a(2) p Fo 1047 1061 a(W) p Fj 1137 1027 a(\(0\)) p Fk 1249 1061 a(j) p Fo 1295 1061 a(L) p Fy(\() p Fo(w) p Fy 1445 1061 a(\)) p 1501 1061 a(=) p Fo 1588 1061 a(k) s(;) p 1699 1061 a(\027) p Fy 1745 1061 a(\() p Fo(w) p Fy 1838 1061 a(\)) p Fl 1894 1061 a(5) p Fy 1986 1061 a(~) p Fo 1982 1061 a(n) p Fy(\() p Fo(k) p Fy 2110 1061 a(\)) p Fk 2161 1061 a(\000) p Fo 2244 1061 a(m) p Fk(g) p Fl 2381 1061 a(5) p Fo 2469 1061 a(e) p Fp 2508 1027 a(\014) p Fm 2546 1035 a(c) p Fp 2576 1027 a(k) p Fo 2617 1061 a(U) p Fj 2678 1076 a(~) p Fp 2674 1076 a(n) p Fj(\() p Fp(k) p Fj 2777 1076 a(\)) p Fn(\000) p Fp(m;) p Fj(2) p Fp(m) p Fl 810 1161 a(5) p Fo 897 1161 a(C) p Fn 962 1126 a(0) p Fy 1000 1161 a(exp\() p Fo(\014) p Fp 1206 1173 a(c) p Fo 1240 1161 a(k) p Fy 1304 1161 a(+) p 1387 1161 a(\() t(~) p Fo 1419 1161 a(n) p Fy(\() p Fo(k) p Fy 1547 1161 a(\)) p Fk 1598 1161 a(\000) p Fo 1681 1161 a(m) p Fy(\)) p 1800 1161 a(log) p Fo 1921 1161 a(A) p Fj 1983 1173 a(2) p Fk 2039 1161 a(\000) p Fo 2122 1161 a(C) p Fn 2187 1126 a(00) p Fo 2230 1161 a(\025) p Fp 2278 1126 a(m) p Fy 2341 1161 a(\)) p Fl 810 1260 a(5) p Fo 897 1260 a(C) p Fn 962 1226 a(0) p Fy 1000 1260 a(exp\() p Fo(\014) p Fp 1206 1272 a(c) p Fo 1240 1260 a(k) p Fy 1304 1260 a(+) p 1387 1260 a(log) p Fj 1494 1281 a(log) p Fp 1592 1281 a(\025) p Fo 1649 1260 a(A) p Fj 1711 1272 a(2) p Fy 1762 1260 a(log) p Fo 1883 1260 a(k) p Fk 1948 1260 a(\000) p Fo 2031 1260 a(C) p Fn 2096 1226 a(00) p Fo 2138 1260 a(\025) p Fp 2186 1226 a(m) p Fy 2250 1260 a(\)) p Fo(:) p Fy 0 1438 a(This) p 190 1438 a(and) p 351 1438 a(Theorem) p 702 1438 a(10) p 812 1438 a(imply) p 1045 1438 a(that) p 1225 1438 a(for) p 1352 1438 a(su\016cien) n(tly) p 1768 1438 a(large) p Fo 1971 1438 a(\013) p Fy 2052 1438 a(and) p 2214 1438 a(for) p 2341 1438 a(an) n(y) p 2497 1438 a(real) p Fo 2659 1438 a(\017) p Fy(,) p 2744 1438 a(there) p 2956 1438 a(exists) p Fo 3185 1438 a(C) p 3274 1438 a(>) p Fy 3361 1438 a(0) p 3430 1438 a(suc) n(h) p 3618 1438 a(that) 285 1598 y(~) p Fo 266 1619 a(P) p Fp 319 1631 a(k) p Fy 361 1619 a([) p Fo(\027) p Fy 430 1619 a(\() p Fo(w) p Fy 523 1619 a(\)) p Fl 579 1619 a(5) p Fy 671 1619 a(~) p Fo 667 1619 a(n) p Fy(\() p Fo(k) p Fy 795 1619 a(\)) p Fk 845 1619 a(\000) p Fo 929 1619 a(\013) p Fy 996 1619 a(log) p 1117 1619 a(log) p Fo 1238 1619 a(k) p Fy 1302 1619 a(+) p Fo 1385 1619 a(\017) p Fy(]) p Fl 1465 1619 a(5) p Fo 1553 1619 a(C) p Fn 1618 1585 a(0) p Fo 1641 1619 a(C) p Fn 1706 1584 a(\000) p Fj(1) 1700 1642 y(1) p Fy 1809 1619 a(exp\(\(log) p Fj 2108 1640 a(log) p Fp 2205 1640 a(\025) p Fo 2263 1619 a(A) p Fj 2325 1631 a(2) p Fk 2381 1619 a(\000) p Fo 2464 1619 a(C) p Fj 2523 1631 a(3) p Fy 2560 1619 a(\)) p 2606 1619 a(log) p Fo 2727 1619 a(k) p Fk 2792 1619 a(\000) p Fo 2875 1619 a(C) p Fn 2940 1585 a(00) p Fo 2982 1619 a(\025) p Fn 3030 1585 a(\000) p 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786 2127 a(\(log) p Fo 940 2127 a(k) p Fy 986 2127 a(\)) p Fn 1018 2093 a(\000) p Fp(\013=d) p Fm 1182 2101 a(w) p Fo 1233 2127 a(k) p Fj 1279 2093 a(1) p Fp(=d) p Fm 1381 2101 a(w) p Fy 1432 2127 a(]) p Fo 1469 2127 a(e) p Fj 1508 2093 a(\(log) p Fp 1630 2093 a(k) p Fj 1666 2093 a(\)) p Ff 1692 2068 a(2) p Fl 1753 2127 a(5) p Fy 1867 2127 a(lim) p Fp 1840 2182 a(k) p Fn 1876 2182 a(!1) p Fy 2042 2106 a(~) p Fo 2023 2127 a(P) p Fp 2076 2139 a(k) p Fy 2117 2127 a([2) p Fp 2182 2093 a(\027) p Fj 2219 2093 a(\() p Fp(w) p Fj 2295 2093 a(\)) p Fo 2348 2127 a(<) p Fy 2435 2127 a(2\(log) p Fo 2630 2127 a(k) p Fy 2676 2127 a(\)) p Fn 2708 2093 a(\000) p Fp(\013=d) p Fm 2872 2101 a(w) p Fo 2923 2127 a(k) p Fj 2969 2093 a(1) p Fp(=d) p Fm 3071 2101 a(w) p Fy 3122 2127 a(]) p Fo 3159 2127 a(e) p Fj 3198 2093 a(\(log) p Fp 3321 2093 a(k) p Fj 3357 2093 a(\)) p Ff 3383 2068 a(2) p Fy 3443 2127 a(=) p 3531 2127 a(0) p Fo 3587 2127 a(:) p Fy 0 2341 a(Chebishev's) p 455 2341 a(inequalit) n(y) p 840 2341 a(implies,) p 1145 2341 a(for) p Fo 1272 2341 a(s) p Fl 1334 2341 a(=) p Fy 1422 2341 a(0,) p Fo 786 2533 a(E) p Fp 847 2545 a(k) p Fy 888 2533 a([) p Fk(k) p Fo 952 2533 a(w) p Fk 1013 2533 a(k) p Fp 1055 2491 a(sd) p Fm 1121 2499 a(w) p Fy 1173 2533 a(]) p Fl 1219 2533 a(=) p Fy 1306 2533 a(\(\(log) p Fo 1492 2533 a(k) p Fy 1538 2533 a(\)) p Fn 1570 2499 a(\000) p Fp(\013) p Fo 1669 2533 a(k) p Fy 1715 2533 a(\)) p Fp 1747 2499 a(s) p Fy 1783 2533 a(\(1) p Fk 1875 2533 a(\000) p Fy 1977 2512 a(~) p Fo 1958 2533 a(P) p Fp 2011 2545 a(k) p Fy 2052 2533 a([) p Fk(k) p Fo(w) p Fk 2178 2533 a(k) p Fl 2243 2533 a(5) p Fy 2331 2533 a(\(log) p Fo 2484 2533 a(k) p Fy 2530 2533 a(\)) p Fn 2562 2499 a(\000) p Fp(\013=d) p Fm 2726 2507 a(w) p Fo 2777 2533 a(k) p Fj 2823 2499 a(1) p Fp(=d) p Fm 2925 2507 a(w) p Fy 2976 2533 a(]\)) p Fo(:) p Fy 0 2713 a(Therefore,) 1199 2813 y(lim) p 1199 2826 116 4 v Fp 1172 2883 a(k) p Fn 1208 2883 a(!1) p Fy 1342 2813 a(\(log) p Fo 1495 2813 a(k) p Fy 1541 2813 a(\)) p Fp 1573 2778 a(s\013) p Fo 1652 2813 a(k) p Fn 1698 2778 a(\000) p Fp(s) p Fo 1785 2813 a(E) p Fp 1846 2825 a(k) p Fy 1887 2813 a([) p Fk(k) p Fo(w) p Fk 2013 2813 a(k) p Fp 2055 2771 a(sd) p Fm 2121 2779 a(w) p Fy 2172 2813 a(]) p Fo 2218 2813 a(>) p Fy 2306 2813 a(0) p Fo(;) p 2439 2813 a(s) p Fl 2501 2813 a(=) p Fy 2589 2813 a(0) p Fo 2645 2813 a(:) p Fy 125 3019 a(Next,) p 362 3019 a(for) p 497 3019 a(p) r(ositiv) n(e) p 813 3019 a(in) n(tegers) p Fo 1130 3019 a(m) p Fy(,) p Fo 1264 3019 a(k) p Fy 1310 3019 a(,) p 1372 3019 a(and) p Fo 1542 3019 a(`) p Fy(,) p 1638 3019 a(Prop) r(osition) p 2094 3019 a(25) p 2214 3019 a(implies,) p 2529 3019 a(with) p 2727 3019 a(an) p 2851 3019 a(ob) n(vious) p 3160 3019 a(inequalit) n(y) p 3554 3019 a(2) p Fp 3596 2989 a(m) p Fj(+) p Fp(`) p Fl 3775 3019 a(=) p Fy 0 3118 a(2) p Fp 42 3088 a(m) p Fn(\000) p Fj(1) p Fy 208 3118 a(+) p 291 3118 a(2) p Fp 333 3088 a(`) p Fn(\000) p Fj(1) p Fy 449 3118 a(,) p Fo 500 3118 a(m;) p 610 3118 a(`) p Fk 668 3118 a(2) p Fq 746 3118 a(Z) p Fj 801 3130 a(+) p Fy 857 3118 a(,) p Fo 571 3304 a(]) p Fk(f) p Fo(w) p Fk 729 3304 a(2) p Fo 808 3304 a(W) p Fj 898 3270 a(\(0\)) p Fk 1010 3304 a(j) p Fo 1056 3304 a(L) p Fy(\() p Fo(w) p Fy 1206 3304 a(\)) p 1262 3304 a(=) p Fo 1349 3304 a(k) s(;) p 1460 3304 a(\027) p Fy 1506 3304 a(\() p Fo(w) p Fy 1599 3304 a(\)) p 1655 3304 a(=) p 1747 3304 a(~) p Fo 1743 3304 a(n) p Fy(\() p Fo(k) p Fy 1871 3304 a(\)) p 1922 3304 a(+) p Fo 2005 3304 a(m) p Fy 2096 3304 a(+) p Fo 2179 3304 a(`) p Fy 2232 3304 a(+) p 2315 3304 a(2) p Fk(g) p Fl 2421 3304 a(5) p Fo 2509 3304 a(e) p Fp 2548 3270 a(\014) p Fm 2586 3278 a(c) p Fp 2616 3270 a(k) p Fo 2657 3304 a(V) p Fj 2709 3319 a(~) p Fp 2705 3319 a(n) p Fj 2747 3319 a(\() p Fp(k) p Fj 2809 3319 a(\)+) p Fp(m) p Fj(+) p Fp(`) p Fj(+1) p Fp(;m) p Fj(+) p Fp(`) p Fl 571 3409 a(5) p Fo 658 3409 a(C) p Fn 723 3374 a(0) p Fy 761 3409 a(exp\() p Fo(\014) p Fp 967 3421 a(c) p Fo 1001 3409 a(k) p Fy 1065 3409 a(+) p 1148 3409 a(\() t(~) p Fo 1180 3409 a(n) p Fy(\() p Fo(k) p Fy 1308 3409 a(\)) p 1359 3409 a(+) p Fo 1442 3409 a(m) p Fy 1533 3409 a(+) p Fo 1616 3409 a(`) p Fy 1669 3409 a(+) p 1752 3409 a(1\)) p 1840 3409 a(log) p Fo 1961 3409 a(A) p Fj 2023 3421 a(2) p Fk 2079 3409 a(\000) p Fo 2162 3409 a(C) p Fn 2227 3374 a(00) p Fy 2270 3409 a(2) p Fp 2312 3374 a(m) p Fj(+) p Fp(`) p Fy 2453 3409 a(\)) p Fl 571 3514 a(5) p Fo 658 3514 a(C) p Fn 723 3480 a(0) p Fy 761 3514 a(exp\() p Fo(\014) p Fp 967 3526 a(c) p Fo 1001 3514 a(k) p Fy 1065 3514 a(+) p 1148 3514 a(log) p Fj 1255 3535 a(log) p Fp 1352 3535 a(\025) p Fo 1410 3514 a(A) p Fj 1472 3526 a(2) p Fy 1523 3514 a(log) p Fo 1644 3514 a(k) p Fy 1708 3514 a(+) p Fo 1791 3514 a(m) p Fy 1878 3514 a(log) p Fo 1999 3514 a(A) p Fj 2061 3526 a(2) p Fk 2117 3514 a(\000) p Fo 2200 3514 a(C) p Fn 2265 3480 a(00) p Fy 2308 3514 a(2) p Fp 2350 3480 a(m) p Fn(\000) p Fj(1) p Fy 2498 3514 a(\)) p Fo(A) p Fp 2592 3479 a(`) p Fj(+1) 2592 3537 y(2) p Fy 2722 3514 a(exp\() p Fk(\000) p Fo(C) p Fn 3011 3480 a(00) p Fy 3053 3514 a(2) p Fp 3095 3480 a(`) p Fn(\000) p Fj(1) p Fy 3212 3514 a(\)) p Fo(:) p Fy 0 3692 a(This) p 190 3692 a(and) p 351 3692 a(Theorem) p 702 3692 a(10) p 812 3692 a(imply) p 1045 3692 a(that) p 1225 3692 a(there) p 1438 3692 a(exists) p 1667 3692 a(a) p 1736 3692 a(p) r(ositiv) n(e) p 2043 3692 a(constan) n(t) p Fo 2378 3692 a(C) p Fy 2471 3692 a(suc) n(h) p 2658 3692 a(that) 379 3913 y(~) p Fo 360 3934 a(P) p Fp 413 3946 a(k) p Fy 454 3934 a([) p Fo(\027) p Fy 523 3934 a(\() p Fo(w) p Fy 616 3934 a(\)) p Fl 673 3934 a(=) p Fy 765 3934 a(~) p Fo 761 3934 a(n) p Fy(\() p Fo(k) p Fy 889 3934 a(\)) p 939 3934 a(+) p Fo 1087 3878 a(\013) p 1032 3915 163 4 v Fy 1032 3991 a(log) p 1153 3991 a(2) 1219 3934 y(log) p 1340 3934 a(log) p Fo 1461 3934 a(k) p Fy 1507 3934 a(]) p 1553 3934 a(=) p Fn 1667 3830 a(1) p Fg 1641 3855 a(X) p Fp 1645 4034 a(`) p Fj(=0) p Fy 1793 3913 a(~) p Fo 1774 3934 a(P) p Fp 1827 3946 a(k) p Fy 1868 3934 a([) p Fo(\027) p Fy 1937 3934 a(\() p Fo(w) p Fy 2030 3934 a(\)) p 2087 3934 a(=) p 2179 3934 a(~) p Fo 2175 3934 a(n) p Fy(\() p Fo(k) p Fy 2303 3934 a(\)) p 2353 3934 a(+) p Fo 2436 3934 a(`) p Fy 2490 3934 a(+) p Fo 2637 3878 a(\013) p 2582 3915 V Fy 2582 3991 a(log) p 2704 3991 a(2) 2769 3934 y(log) p 2890 3934 a(log) p Fo 3011 3934 a(k) p Fy 3057 3934 a(]) p Fl 360 4179 a(5) p Fo 512 4123 a(C) p Fn 577 4093 a(0) p 458 4160 197 4 v Fo 458 4236 a(C) p Fj 517 4248 a(1) p Fo 554 4236 a(A) p Fj 616 4248 a(2) p Fn 705 4076 a(1) p Fg 678 4100 a(X) p Fp 682 4279 a(`) p Fj(=0) p Fo 812 4179 a(A) p Fp 874 4145 a(`) p Fj 874 4200 a(2) p Fo 911 4179 a(e) p Fn 950 4145 a(\000) p Fp(C) p Fe 1054 4120 a(00) p Fj 1094 4145 a(2) p Fm 1127 4120 a(`) p Fe(\000) p Ff(1) p Fy 1247 4179 a(exp\(\(log) p Fj 1546 4200 a(log) p Fp 1643 4200 a(\025) p Fo 1701 4179 a(A) p Fj 1763 4191 a(2) p Fk 1818 4179 a(\000) p Fo 1902 4179 a(C) p Fj 1961 4191 a(3) p Fy 1998 4179 a(\)) p 2044 4179 a(log) p Fo 2165 4179 a(k) p Fy 2230 4179 a(+) p Fo 2313 4179 a(\013) p Fy 2380 4179 a(log) p Fj 2487 4200 a(2) p Fo 2538 4179 a(A) p Fj 2600 4191 a(2) p Fy 2652 4179 a(log) p 2773 4179 a(log) p Fo 2894 4179 a(k) p Fk 2958 4179 a(\000) p Fo 3051 4123 a(C) p Fn 3116 4093 a(00) p 3051 4160 108 4 v Fy 3084 4236 a(8) 3169 4179 y(\(log) p Fo 3322 4179 a(k) p Fy 3368 4179 a(\)) p Fp 3400 4145 a(\013) p Fy 3447 4179 a(\)) p Fl 360 4406 a(5) p Fo 448 4406 a(C) p Fy 527 4406 a(exp\(\(log) p Fj 826 4426 a(log) p Fp 923 4426 a(\025) p Fo 980 4406 a(A) p Fj 1042 4418 a(2) p Fk 1098 4406 a(\000) p Fo 1181 4406 a(C) p Fj 1240 4418 a(3) p Fy 1278 4406 a(\)) p 1324 4406 a(log) p Fo 1445 4406 a(k) p Fy 1509 4406 a(+) p Fo 1592 4406 a(\013) p Fy 1660 4406 a(log) p Fj 1767 4426 a(2) p Fo 1818 4406 a(A) p Fj 1880 4418 a(2) p Fy 1931 4406 a(log) p 2052 4406 a(log) p Fo 2173 4406 a(k) p Fk 2238 4406 a(\000) p Fo 2331 4350 a(C) p Fn 2396 4320 a(00) p 2331 4387 V Fy 2364 4463 a(8) 2448 4406 y(\(log) p Fo 2602 4406 a(k) p Fy 2648 4406 a(\)) p Fp 2680 4372 a(\013) p Fy 2727 4406 a(\)) p Fo(;) p 2852 4406 a(k) p Fk 2921 4406 a(2) p Fq 2999 4406 a(N) p Fo(:) p Fy 0 4614 a(This,) p 213 4614 a(with) p 402 4614 a(\(56\),) p 600 4614 a(implies) p 882 4614 a(for) p 1009 4614 a(su\016cien) n(tly) p 1425 4614 a(large) p Fo 1628 4614 a(\013) p Fy(,) 330 4809 y(lim) p Fp 304 4863 a(k) p Fn 340 4863 a(!1) p Fy 505 4788 a(~) p Fo 486 4809 a(P) p Fp 539 4821 a(k) p Fy 581 4809 a([) p Fk(k) p Fo 645 4809 a(w) p Fk 706 4809 a(k) p Fl 771 4809 a(=) p Fy 859 4809 a(\(log) p Fo 1012 4809 a(k) p Fy 1058 4809 a(\)) p Fp 1090 4775 a(\013=d) p Fm 1202 4783 a(w) p Fo 1254 4809 a(k) p Fj 1300 4775 a(1) p Fp(=d) p Fm 1402 4783 a(w) p Fy 1452 4809 a(]) p Fo 1489 4809 a(e) p Fj 1528 4775 a(\(log) p Fp 1651 4775 a(k) p Fj 1687 4775 a(\)) p Ff 1713 4750 a(2) p Fl 1773 4809 a(5) p Fy 1888 4809 a(lim) p Fp 1861 4863 a(k) p Fn 1897 4863 a(!1) p Fy 2062 4788 a(~) p Fo 2044 4809 a(P) p Fp 2097 4821 a(k) p Fy 2138 4809 a([2) p Fp 2203 4775 a(\027) p Fj 2240 4775 a(\() p Fp(w) p Fj 2316 4775 a(\)) p Fl 2368 4809 a(=) p Fy 2456 4809 a(\(log) p Fo 2609 4809 a(k) p Fy 2655 4809 a(\)) p Fp 2687 4775 a(\013=d) p Fm 2799 4783 a(w) p Fo 2851 4809 a(k) p Fj 2897 4775 a(1) p Fp(=d) p Fm 2999 4783 a(w) p Fy 3049 4809 a(]) p Fo 3086 4809 a(e) p Fj 3125 4775 a(\(log) p Fp 3248 4775 a(k) p Fj 3284 4775 a(\)) p Ff 3310 4750 a(2) p Fy 3370 4809 a(=) p 3458 4809 a(0) p Fo 3514 4809 a(:) p Fy 125 5023 a(Let) p Fo 273 5023 a(s) p Fl 335 5023 a(=) p Fy 423 5023 a(0.) p 524 5023 a(Note) p 725 5023 a(that) p Fk 905 5023 a(k) p Fo 946 5023 a(w) p Fk 1007 5023 a(k) p Fl 1073 5023 a(5) p Fo 1160 5023 a(L) p Fy(\() p Fo(w) p Fy 1310 5023 a(\),) p 1394 5023 a(whic) n(h) p 1631 5023 a(implies) p Fo 900 5215 a(E) p Fp 961 5227 a(k) p Fy 1003 5215 a([) p Fk(k) p Fo 1067 5215 a(w) p Fk 1128 5215 a(k) p Fp 1170 5173 a(sd) p Fm 1236 5181 a(w) p Fy 1287 5215 a(]) p Fl 1334 5215 a(5) p Fy 1421 5215 a(\(\(log) p Fo 1607 5215 a(k) p Fy 1653 5215 a(\)) p Fp 1685 5181 a(\013) p Fo 1732 5215 a(k) p Fy 1778 5215 a(\)) p Fp 1810 5181 a(s) p Fy 1864 5215 a(+) p Fo 1947 5215 a(k) p Fp 1993 5181 a(sd) p Fm 2059 5189 a(w) p Fy 2129 5194 a(~) p Fo 2110 5215 a(P) p Fp 2163 5227 a(k) p Fy 2205 5215 a([) p Fk(k) p Fo 2269 5215 a(w) p Fk 2330 5215 a(k) p Fp 2372 5173 a(d) p Fm 2407 5181 a(w) p Fl 2481 5215 a(=) p Fy 2569 5215 a(\(log) p Fo 2722 5215 a(k) p Fy 2768 5215 a(\)) p Fp 2800 5181 a(\013) p Fo 2848 5215 a(k) p Fy 2894 5215 a(]) p Fo(:) p Fy 0 5395 a(Therefore,) p 1301 5427 116 4 v 1301 5495 a(lim) p Fp 1274 5549 a(k) p Fn 1310 5549 a(!1) p Fy 1443 5495 a(\(log) p Fo 1597 5495 a(k) p Fy 1643 5495 a(\)) p Fn 1675 5460 a(\000) p Fp(s\013) p Fo 1805 5495 a(k) p Fn 1851 5460 a(\000) p Fp(s) p Fo 1939 5495 a(E) p Fp 2000 5507 a(k) p Fy 2041 5495 a([) p Fk(k) p Fo 2105 5495 a(w) p Fk 2166 5495 a(k) p Fp 2209 5453 a(sd) p Fm 2275 5461 a(w) p Fy 2326 5495 a(]) p Fo 2372 5495 a(<) p Fk 2460 5495 a(1) p Fo(;) p Fd 3778 5642 a(2) p Fy 1878 6120 a(26) p 90 rotate dyy eop %%Page: 27 27 27 26 bop Fz 0 83 a(4.3.3) p 292 83 a(Re\015ection) p 743 83 a(principle.) p Fy 0 236 a(The) p 172 236 a(de\014nition) p 542 236 a(of) p 638 236 a(pre-) p Fo(d) p Fy(SG) p 965 236 a(in) p 1064 236 a(\(1\)) p 1199 236 a(and) p 1362 236 a(\(2\)) p 1497 236 a(induces) p 1794 236 a(a) p 1864 236 a(natural) p 2157 236 a(co) r(ordinate) p 2566 236 a(system) p 2843 236 a(on) p Fo 2960 236 a(G) p Fy 3054 236 a(whic) n(h) p 3293 236 a(is) p 3378 236 a(an) p 3495 236 a(on) n(to) p 3683 236 a(map) p Fo 0 336 a(\031) p Fy 87 336 a(:) p Fk 183 336 a(f) p Fy(0) p Fo(;) p Fy 304 336 a(1) p Fo(;) p Fy 383 336 a(2) p Fo(;) p Fk 462 336 a(\001) p 499 336 a(\001) p 536 336 a(\001) p Fo 570 336 a(;) p 607 336 a(d) p Fk(g) p Fc 692 306 a(Z) p Ff 731 314 a(+) p Fk 819 336 a(!) p Fo 939 336 a(G) p Fy 1040 336 a(de\014ned) p 1334 336 a(as) p 1444 336 a(follo) n(ws.) p 1774 336 a(F) p 1821 336 a(or) p 1931 336 a(eac) n(h) p Fo 2126 336 a(v) p Fj 2166 348 a(0) p Fp(;i) p Fy 2284 336 a(=) p Fo 2385 336 a(v) p Fp 2425 348 a(i) p Fk 2490 336 a(2) p Fo 2582 336 a(G) p Fj 2647 348 a(0) p Fy 2721 336 a(\() p Fo(i) p Fy 2819 336 a(=) p 2920 336 a(0) p Fo(;) p Fy 2999 336 a(1) p Fo(;) p Fy 3078 336 a(2) p Fo(;) p Fk 3157 336 a(\001) p 3194 336 a(\001) p 3231 336 a(\001) p Fo 3266 336 a(;) p 3303 336 a(d) p Fy(\)) p 3414 336 a(w) n(e) p 3545 336 a(assign) p 3798 336 a(a) 0 436 y(co) r(ordinate) p 408 436 a(\() p Fo(i;) p Fy 506 436 a(0) p Fo(;) p Fy 585 436 a(0) p Fo(;) p Fy 664 436 a(0) p Fo(;) p Fk 743 436 a(\001) p 780 436 a(\001) p 817 436 a(\001) p Fy 838 436 a(\);) p Fo 1182 535 a(\031) p Fy 1232 535 a(\() p Fo(i;) p Fy 1330 535 a(0) p Fo(;) p Fy 1409 535 a(0) p Fo(;) p Fy 1488 535 a(0) p Fo(;) p Fk 1567 535 a(\001) p 1604 535 a(\001) p 1641 535 a(\001) p Fy 1663 535 a(\)) p 1718 535 a(=) p Fo 1806 535 a(v) p Fj 1846 547 a(0) p Fp(;i) p Fo 1926 535 a(;) p 1991 535 a(i) p Fy 2042 535 a(=) p 2130 535 a(0) p Fo(;) p Fy 2209 535 a(1) p Fo(;) p Fy 2288 535 a(2) p Fo(;) p Fy 2367 535 a(3) p Fo(;) p Fk 2446 535 a(\001) p 2483 535 a(\001) p 2520 535 a(\001) p Fo 2554 535 a(;) p 2591 535 a(d:) p Fy 125 685 a(F) p 172 685 a(or) p Fo 285 685 a(n) p Fy 377 685 a(=) p 483 685 a(0) p Fo(;) p Fy 562 685 a(1) p Fo(;) p Fy 641 685 a(2) p Fo(;) p Fk 720 685 a(\001) p 757 685 a(\001) p 794 685 a(\001) p Fy 854 685 a(and) p Fo 1027 685 a(i) p Fy 1097 685 a(=) p 1204 685 a(0) p Fo(;) p Fy 1283 685 a(1) p Fo(;) p Fy 1362 685 a(2) p Fo(;) p Fy 1441 685 a(3) p Fo(;) p Fk 1520 685 a(\001) p 1557 685 a(\001) p 1594 685 a(\001) p Fo 1628 685 a(;) p 1665 685 a(d) p Fy(,) p 1773 685 a(put) p Fo 1937 685 a(G) p Fp 2002 697 a(n;i) p Fy 2132 685 a(=) p Fo 2239 685 a(G) p Fp 2304 697 a(n) p Fy 2375 685 a(+) p 2466 685 a(2) p Fp 2508 654 a(n) p Fo 2552 685 a(v) p Fj 2592 697 a(0) p Fp(;i) p Fy 2687 685 a(,) p 2752 685 a(and) p 2925 685 a(de\014ne) p 3176 685 a(a) p 3256 685 a(1) p 3340 685 a(:) p 3405 685 a(1) p 3485 685 a(on) n(to) p 3683 685 a(map) p Fo 0 784 a(\023) p Fp 29 796 a(n;i) p Fy 141 784 a(:) p Fo 212 784 a(G) p Fp 277 796 a(n;i) p Fk 389 784 a(!) p Fo 495 784 a(G) p Fp 560 796 a(n) p Fy 630 784 a(b) n(y) p Fo 743 784 a(\023) p Fp 772 796 a(n;i) p Fy 861 784 a(\() p Fo(v) p Fy 936 784 a(\)) p 992 784 a(=) p Fo 1079 784 a(v) p Fk 1136 784 a(\000) p Fy 1214 784 a(2) p Fp 1256 754 a(n) p Fo 1301 784 a(v) p Fj 1341 796 a(0) p Fp(;i) p Fy 1435 784 a(.) p Fo 1494 784 a(\023) p Fp 1523 796 a(n;i) p Fy 1637 784 a(naturally) p 1992 784 a(induces) p 2285 784 a(a) p 2352 784 a(1) p 2416 784 a(:) p 2463 784 a(1) p 2529 784 a(on) n(to) p 2713 784 a(map) p Fo 2895 784 a(B) p Fp 2958 796 a(n;i) p Fk 3070 784 a(!) p Fo 3176 784 a(B) p Fp 3239 796 a(n) p Fy 3298 784 a(,) p 3347 784 a(whic) n(h) p 3582 784 a(w) n(e) p 3701 784 a(also) 0 884 y(denote) p 268 884 a(b) n(y) p Fo 383 884 a(\023) p Fp 412 896 a(n;i) p Fy 501 884 a(.) 125 983 y(W) p 203 983 a(e) p 271 983 a(pro) r(ceed) p 582 983 a(with) p 775 983 a(b) n(y) p 894 983 a(induction) p 1267 983 a(in) p Fo 1368 983 a(n) p Fy 1449 983 a(and) p 1614 983 a(assume) p 1905 983 a(that) p 2089 983 a(a) p 2162 983 a(co) r(ordinate) p 2574 983 a(system) p Fo 2853 983 a(\031) p Fy 2935 983 a(on) p Fo 3054 983 a(G) p Fp 3119 995 a(n) p Fn(\000) p Fj(1) p Fy 3281 983 a(\(=) p Fo 3408 983 a(G) p Fp 3473 995 a(n) p Fn(\000) p Fj(1) p Fp(;) p Fj(0) p Fy 3656 983 a(\)) p 3720 983 a(has) 0 1083 y(b) r(een) p 194 1083 a(de\014ned) p 478 1083 a(for) p 604 1083 a(an) p Fo 717 1083 a(n) p Fl 790 1083 a(=) p Fy 878 1083 a(1,) p 969 1083 a(in) p 1064 1083 a(suc) n(h) p 1249 1083 a(a) p 1317 1083 a(w) n(a) n(y) p 1483 1083 a(that) p Fo 1661 1083 a(\031) p Fy 1711 1083 a(\() p Fo(v) p Fy 1786 1083 a(\)) p Fk 1842 1083 a(2) p Fo 1921 1083 a(G) p Fp 1986 1095 a(n) p Fn(\000) p Fj(1) p Fy 2142 1083 a(holds) p 2358 1083 a(for) p 2483 1083 a(an) n(y) p Fo 2638 1083 a(v) p Fy 2704 1083 a(=) p 2792 1083 a(\() p Fo(i) p Fj 2853 1095 a(0) p Fo 2890 1083 a(;) p 2927 1083 a(i) p Fj 2956 1095 a(1) p Fo 2993 1083 a(;) p 3030 1083 a(i) p Fj 3059 1095 a(2) p Fo 3095 1083 a(;) p Fk 3132 1083 a(\001) p 3169 1083 a(\001) p 3206 1083 a(\001) p Fo 3243 1083 a(;) p 3280 1083 a(i) p Fp 3309 1095 a(n) p Fn(\000) p Fj(1) p Fo 3439 1083 a(;) p Fy 3476 1083 a(0) p Fo(;) p Fy 3555 1083 a(0) p Fo(;) p Fy 3634 1083 a(0) p Fo(;) p Fk 3713 1083 a(\001) p 3750 1083 a(\001) p 3787 1083 a(\001) p Fy 3808 1083 a(\)) 0 1183 y(with) p Fo 189 1183 a(i) p Fp 218 1195 a(k) p Fk 282 1183 a(2) p 360 1183 a(f) p Fy(0) p Fo(;) p Fy 481 1183 a(1) p Fo(;) p Fy 560 1183 a(2) p Fo(;) p Fk 639 1183 a(\001) p 676 1183 a(\001) p 713 1183 a(\001) p Fo 747 1183 a(;) p 784 1183 a(d) p Fk(g) p Fy(,) p Fo 920 1183 a(k) p Fy 989 1183 a(=) p 1076 1183 a(0) p Fo(;) p Fy 1155 1183 a(1) p Fo(;) p Fy 1234 1183 a(2) p Fo(;) p Fk 1313 1183 a(\001) p 1350 1183 a(\001) p 1387 1183 a(\001) p Fo 1422 1183 a(;) p 1459 1183 a(n) p Fk 1527 1183 a(\000) p Fy 1610 1183 a(1.) p 1712 1183 a(F) p 1759 1183 a(or) p Fo 1861 1183 a(v) p Fk 1927 1183 a(2) p Fo 2006 1183 a(G) p Fp 2071 1195 a(n) p Fn(\000) p Fj(1) p Fp(;j) p Fy 2252 1183 a(,) p 2302 1183 a(with) p Fo 2491 1183 a(j) p Fk 2553 1183 a(2) p 2632 1183 a(f) p Fy(0) p Fo(;) p Fy 2753 1183 a(1) p Fo(;) p Fy 2832 1183 a(2) p Fo(;) p Fk 2911 1183 a(\001) p 2948 1183 a(\001) p 2985 1183 a(\001) p Fo 3019 1183 a(;) p 3056 1183 a(d) p Fk(g) p Fy(,) p 3192 1183 a(de\014ne) p Fo 424 1365 a(\031) p Fy 474 1365 a(\() p Fo(i) p Fj 535 1377 a(0) p Fo 573 1365 a(;) p 610 1365 a(i) p Fj 639 1377 a(1) p Fo 676 1365 a(;) p 713 1365 a(i) p Fj 742 1377 a(2) p Fo 778 1365 a(;) p Fk 815 1365 a(\001) p 852 1365 a(\001) p 889 1365 a(\001) p Fo 926 1365 a(;) p 963 1365 a(i) p Fp 992 1377 a(n) p Fn(\000) p Fj(1) p Fo 1122 1365 a(;) p 1159 1365 a(j;) p Fy 1230 1365 a(0) p Fo(;) p Fy 1309 1365 a(0) p Fo(;) p Fk 1388 1365 a(\001) p 1425 1365 a(\001) p 1462 1365 a(\001) p Fy 1483 1365 a(\)) p 1539 1365 a(=) p Fo 1626 1365 a(v) s(;) p Fy 1762 1365 a(if) p Fo 1866 1365 a(\031) p Fy 1916 1365 a(\() p Fo(i) p Fj 1977 1377 a(0) p Fo 2014 1365 a(;) p 2051 1365 a(i) p Fj 2080 1377 a(1) p Fo 2117 1365 a(;) p 2154 1365 a(i) p Fj 2183 1377 a(2) p Fo 2219 1365 a(;) p Fk 2256 1365 a(\001) p 2293 1365 a(\001) p 2330 1365 a(\001) p Fo 2367 1365 a(;) p 2404 1365 a(i) p Fp 2433 1377 a(n) p Fn(\000) p Fj(1) p Fo 2563 1365 a(;) p Fy 2600 1365 a(0) p Fo(;) p Fy 2679 1365 a(0) p Fo(;) p Fy 2758 1365 a(0) p Fo(;) p Fk 2837 1365 a(\001) p 2874 1365 a(\001) p 2911 1365 a(\001) p Fy 2932 1365 a(\)) p 2987 1365 a(=) p Fo 3075 1365 a(\023) p Fp 3104 1377 a(n) p Fn(\000) p Fj(1) p Fp(;j) p Fy 3285 1365 a(\() p Fo(v) p Fy 3360 1365 a(\)) p Fo(:) p Fy 0 1548 a(Note) p 201 1548 a(that) p 381 1548 a(this) p 542 1548 a(de\014nition) p 911 1548 a(is) p 995 1548 a(compatible) p 1419 1548 a(with) p Fo 1608 1548 a(G) p Fp 1673 1560 a(n) p Fn(\000) p Fj(1) p Fp(;) p Fj(0) p Fy 1880 1548 a(=) p Fo 1967 1548 a(G) p Fp 2032 1560 a(n) p Fn(\000) p Fj(1) p Fk 2186 1548 a(\032) p Fo 2273 1548 a(G) p Fp 2338 1560 a(n) p Fy 2398 1548 a(.) 125 1648 y(Eac) n(h) p 331 1648 a(p) r(oin) n(t) p 548 1648 a(in) p Fo 645 1648 a(G) p Fk 728 1648 a(n) p 788 1648 a(f) p Fo(O) p Fk 895 1648 a(g) p Fy 965 1648 a(has) p 1113 1648 a(exactly) p 1399 1648 a(t) n(w) n(o) p 1555 1648 a(co) r(ordinate) p 1964 1648 a(represen) n(tations,) p 2563 1648 a(b) r(ecause) p Fo 400 1830 a(\031) p Fy 450 1830 a(\() p Fo(j;) p 553 1830 a(j;) p Fk 624 1830 a(\001) p 661 1830 a(\001) p 698 1830 a(\001) p Fo 736 1830 a(;) p 773 1830 a(j;) p 844 1830 a(i;) p Fy 910 1830 a(0) p Fo(;) p Fk 989 1830 a(\001) p 1026 1830 a(\001) p 1063 1830 a(\001) p Fy 1085 1830 a(\)) p 1140 1830 a(=) p Fo 1228 1830 a(\031) p Fy 1278 1830 a(\() p Fo(i;) p 1376 1830 a(i;) p Fk 1442 1830 a(\001) p 1479 1830 a(\001) p 1516 1830 a(\001) p Fo 1552 1830 a(;) p 1589 1830 a(i;) p 1655 1830 a(j;) p Fy 1726 1830 a(0) p Fo(;) p Fk 1805 1830 a(\001) p 1842 1830 a(\001) p 1879 1830 a(\001) p Fy 1901 1830 a(\)) p Fk 1956 1830 a(2) p Fo 2034 1830 a(G) p Fp 2099 1842 a(m;i) p Fk 2224 1830 a(\\) p Fo 2298 1830 a(G) p Fp 2363 1842 a(m;j) p Fo 2491 1830 a(;) p Fy 2583 1830 a(0) p Fl 2648 1830 a(5) p Fo 2735 1830 a(i) p 2787 1830 a(<) p 2875 1830 a(j) p Fl 2937 1830 a(5) p Fo 3024 1830 a(d;) p 3132 1830 a(m) p Fk 3228 1830 a(2) p Fq 3307 1830 a(Z) p Fj 3362 1842 a(+) p Fo 3417 1830 a(:) p Fy 0 2013 a(Note) p 201 2013 a(also) p 367 2013 a(that) p 547 2013 a(if) p Fo 623 2013 a(\031) p Fy 673 2013 a(\() p Fo(i) p Fj 734 2025 a(0) p Fo 772 2013 a(;) p 809 2013 a(i) p Fj 838 2025 a(1) p Fo 874 2013 a(;) p 911 2013 a(i) p Fj 940 2025 a(2) p Fo 977 2013 a(;) p Fk 1014 2013 a(\001) p 1051 2013 a(\001) p 1088 2013 a(\001) p Fy(\)) p Fk 1166 2013 a(2) p Fo 1245 2013 a(G) p Fp 1310 2025 a(n) p Fy 1383 2013 a(then) p Fo 1572 2013 a(i) p Fp 1601 2025 a(k) p Fy 1664 2013 a(=) p 1752 2013 a(0,) p Fo 1844 2013 a(k) p Fy 1913 2013 a(=) p Fo 2001 2013 a(n) p Fy 2069 2013 a(+) p 2152 2013 a(1) p Fo(;) p 2231 2013 a(n) p Fy 2299 2013 a(+) p 2382 2013 a(2) p Fo(;) p Fk 2461 2013 a(\001) p 2498 2013 a(\001) p 2535 2013 a(\001) p Fy 2557 2013 a(.) 28 5692 y @beginspecial 72 @llx 72 @lly 526 @urx 500 @ury 4540 @rwi @setspecial %%BeginDocument: figRP.ps %!PS-Adobe-2.0 %%BoundingBox: 72 72 526 500 %%Pages: 1 %%For: Tetsuya Hattori %%CreationDate: 20020510--12; % based on sympFIG.ps ver20011207; %%Title: Hattori-Tsuda for JSP %%EndComments % original %BoundingBox: 72 72 526 781 /font14 {/Times-Roman findfont 14 scalefont setfont} def /font10 {/Times-Roman findfont 10 scalefont setfont} def /font9 {/Times-Roman findfont 9 scalefont setfont} def /font8 {/Times-Roman findfont 8 scalefont setfont} def /font6 {/Times-Roman findfont 6 scalefont setfont} def /fontI10 {/Times-Italic findfont 10 scalefont setfont} def /fontI6 {/Times-Italic findfont 6 scalefont setfont} def /greek10 {/Symbol findfont 10 scalefont setfont} def /greek9 {/Symbol findfont 9 scalefont setfont} def /greek8 {/Symbol findfont 8 scalefont setfont} def /greek6 {/Symbol findfont 6 scalefont setfont} def /sin60 { 3 sqrt mul 2 div } def % multiplication op by sin 60deg /rmove60 { dup 2 div exch sin60 rmoveto } def % 60 deg rmove op /rmove120 { dup -2 div exch sin60 rmoveto } def % 120 deg rmove op /rline60 { dup 2 div exch sin60 rlineto } def % length rline60 draws line of length in 60 deg direction /rline120 { dup -2 div exch sin60 rlineto } def % length rline60 draws line of length in 120 deg direction /triangle { dup 0 rlineto rline120 closepath } def % triangle with length param assumed /F1 { dup triangle dup 0 rmoveto dup triangle dup rmove120 dup triangle -1 mul rmove60 } def /F2 { dup F1 dup 2 mul 0 rmoveto dup F1 dup 2 mul rmove120 dup F1 -2 mul rmove60 } def /blob1 {1.5 0 360 arc fill} def % blob op /square {dup mul} def /xyajirushi {2 copy 2 copy rlineto -1 mul exch -1 mul exch rmoveto -1 mul rlineto 0.2 setlinewidth stroke} def % x-yajirushi op /yyajirushi {2 copy rlineto -1 mul exch -1 mul exch 2 copy rmoveto -1 mul rlineto 0.2 setlinewidth stroke} def % y-yajirushi op /genyajirushi { % gen-yajirushi op vx vy x y genyajirushi % (vx,vy): direction vector of main axis % (x,y): size and relative direction of yajirushi % normalize vx vy: 4 2 roll 2 copy square exch square add sqrt dup 3 1 roll div 3 1 roll div exch 4 2 roll % generate stack seq x vx, x vy, y vx, y vy: 4 2 roll 2 copy 6 5 roll dup 3 1 roll mul 3 1 roll mul exch 5 2 roll 3 2 roll dup 3 1 roll mul 3 1 roll mul exch % generate stack seq -x vx -y vy, -x vy +y vx, -x vx +y vy, -x vy -y vx: 2 copy -1 mul exch -1 mul exch 6 4 roll -1 mul exch -1 mul exch 2 copy 8 7 roll add 7 1 roll 6 5 roll add 5 1 roll 4 3 roll add 3 1 roll add 4 1 roll % draw yajirushi: 2 copy rmoveto -1 mul exch -1 mul exch rlineto 2 copy rlineto -1 mul exch -1 mul exch rlineto } def /tan7h { 6 sqrt 2 sub 3 sqrt sub 2 sqrt add } def % constant tan7.5deg /sin7h { 2 sqrt 2 mul 3 sqrt sub 1 sub 4 div 2 sqrt div sqrt } def % constant sin7.5deg /tan15 { 2 3 sqrt sub } def % constant tan15deg /sin15 { 3 sqrt 1 sub 2 div 2 sqrt div } def % constant sin15deg % u v a urangepointer := upper cone indicating the range (u,v)-(u+a,v) % sets final point at (u,v) /urangepointer { dup 4 1 roll % copy a for yajirushi % arc 2 div dup dup 5 4 roll add 4 1 roll 3 2 roll exch tan15 div sub exch sin15 div 90 15 sub 90 15 add arc % yajirushi -1 -1 tan15 mul 3 1 genyajirushi dup 0 rmoveto 1 -1 tan15 mul 3 1 genyajirushi -1 mul 0 rmoveto } def % u v a drangepointer := lower cone indicating the range (u,v)-(u+a,v) % sets final point at (u,v) /drangepointer { dup 4 1 roll % copy a for yajirushi %arc 2 div dup dup 5 4 roll add 4 1 roll 3 2 roll exch tan15 div add exch sin15 div 270 15 sub 270 15 add arc %yajirushi 1 tan15 3 1 genyajirushi -1 mul 0 rmoveto -1 tan15 3 1 genyajirushi } def % u v r upointpointer := upper circle of radius r indicating the point (u,v) /upointpointer { 3 1 roll 2 copy 5 1 roll 5 1 roll 3 2 roll % copy (u,v) dup 4 1 roll add 3 2 roll 0 360 arc %yajirushi moveto 1 0 2.8 1.3 genyajirushi -1 0 2.8 1.3 genyajirushi } def % -------------------- % 1 2 3 3 1 roll -> 3 1 2 % check for roll operator /checkroll { 0 50 moveto 1 2 3 { 10 0 rlineto -10 10 rmoveto } repeat pop pop 1 setlinewidth stroke 50 50 moveto 1 2 3 3 1 roll { 10 0 rlineto -10 10 rmoveto } repeat pop pop 2 setlinewidth stroke 100 50 moveto 1 2 3 3 2 roll { 10 0 rlineto -10 10 rmoveto } repeat pop pop 3 setlinewidth stroke } def % -------------------- /coordl {greek6 (p) show font6 <28> show} def /coordr {show font6 <29> show} def %%Page: 1 1 % Definition of reflection in the proof of reflection principle newpath 72 25.4 div dup scale % let 1 stand for 1 mm: 25.4 25.4 translate % now we have 160mm*250mm sheet to write 0 50 translate /usdlen { 50 } def % F2; thinlines 0 0 moveto usdlen 2 div F2 usdlen 2 mul 0 rmoveto usdlen 2 div F1 0.05 setlinewidth stroke % yajirushi usdlen 0 usdlen urangepointer 0.5 setlinewidth stroke usdlen 1.25 mul usdlen 2 div sin60 usdlen 2 div urangepointer 0.5 setlinewidth stroke usdlen 2 div -3 usdlen 2 mul drangepointer 0.5 setlinewidth stroke usdlen 1.5 mul usdlen sin60 usdlen 7 div upointpointer 0.5 setlinewidth stroke % sample path 0 0 moveto usdlen 1.5 mul dup sin60 exch 2 div exch rlineto usdlen 2 div 0 rlineto usdlen dup sin60 -1 mul exch 2 div exch rlineto usdlen 2 div dup sin60 -1 mul exch -2 div exch rlineto usdlen 2 div -1 mul 0 rlineto usdlen 2 div dup sin60 exch -2 div exch rlineto 1 setlinewidth stroke % caption % R usdlen dup 1.62 mul exch 1.3 mul sin60 moveto font8 (R) show 0 -1 rmoveto font6 (2,i) show usdlen dup 1.38 mul exch 0.6 mul sin60 moveto font8 (R) show 0 -1 rmoveto font6 (2,i) show usdlen 1.38 mul -18 moveto font8 (R) show 0 -1 rmoveto font6 (2,i) show usdlen 1.38 mul -13.5 moveto greek8 (~) show % G usdlen dup 0.3 mul exch 0.2 mul moveto font9 (G) show 0 -2 rmoveto font6 (1) show usdlen dup 1.3 mul exch 0.2 mul moveto font9 (G) show 0 -2 rmoveto font6 (1,i) show usdlen dup 2.3 mul exch 0.2 mul moveto greek9 (i) show gsave 0 3 rmoveto greek6 (-) show font6 (1) show grestore 0 -2 rmoveto font6 (2,i) show 0 2 rmoveto font9 <28> show (G) show 0 -1 rmoveto font6 (1) show 0 1 rmoveto font9 <29> show % sample path newpath usdlen 0.75 mul usdlen 2 div sin60 blob1 usdlen 0.75 mul 5 sub usdlen 2 div sin60 3 add moveto font6 (w) show <28> show (L) show <29> show newpath usdlen 0 blob1 usdlen 1 sub 5 moveto font6 (T) show <28> show (w) show <29> show % coordinates usdlen 2 div 10 sub -12 moveto coordl (i) coordr (=) show coordl (0i) coordr usdlen 9 sub -6 moveto coordl (ii) coordr (=) show coordl (00i) coordr usdlen 2 mul 21 sub -6 moveto coordl (000i) coordr (=) show coordl (iii) coordr usdlen 2.5 mul 14 sub -12 moveto coordl (0i0i) coordr (=) show coordl (i00i) coordr usdlen 1.25 mul 15 sub usdlen 2 div sin60 moveto coordl (j0i) coordr usdlen 1.75 mul 2 add usdlen 2 div sin60 moveto coordl (jii) coordr usdlen 1.5 mul 8 add usdlen sin60 moveto coordl (jji) coordr usdlen 4 div -40 moveto font10 (Reflection in the i-j hyperplane) show 0 -50 translate showpage clear %%PageTrailor %%Trailor %%EOF %%EndDocument @endspecial 1878 6120 a(27) p 90 rotate dyy eop %%Page: 28 28 28 27 bop Fy 125 83 a(W) p 203 83 a(e) p 279 83 a(no) n(w) p 463 83 a(de\014ne) p 714 83 a(a) p 794 83 a(re\015ection) p 1166 83 a(map) p 1361 83 a(\(see) p 1539 83 a(the) p 1693 83 a(Figure\)) p 1999 83 a(with) p 2199 83 a(whic) n(h) p 2448 83 a(w) n(e) p 2581 83 a(form) n(ulate) p 2966 83 a(a) p 3047 83 a(re\015ection) p 3418 83 a(principle) p 3771 83 a(in) 0 183 y(Theorem) p 351 183 a(28.) p 494 183 a(F) p 541 183 a(or) p 643 183 a(eac) n(h) p Fo 830 183 a(i) p Fy 881 183 a(=) p 969 183 a(1) p Fo(;) p Fy 1048 183 a(2) p Fo(;) p Fy 1127 183 a(3) p Fo(;) p Fk 1206 183 a(\001) p 1243 183 a(\001) p 1280 183 a(\001) p Fo 1315 183 a(;) p 1352 183 a(d) p Fy 1423 183 a(de\014ne) p Fo 1662 183 a(R) p Fj 1725 195 a(0) p Fp(;i) p Fy 1829 183 a(:) p Fk 1903 183 a(f) p Fy(0) p Fo(;) p Fy 2024 183 a(1) p Fo(;) p Fy 2103 183 a(2) p Fo(;) p Fk 2182 183 a(\001) p 2219 183 a(\001) p 2256 183 a(\001) p Fo 2290 183 a(;) p 2327 183 a(d) p Fk(g) p 2435 183 a(!) p 2541 183 a(f) p Fy(0) p Fo(;) p Fy 2662 183 a(1) p Fo(;) p Fy 2741 183 a(2) p Fo(;) p Fk 2820 183 a(\001) p 2857 183 a(\001) p 2894 183 a(\001) p Fo 2928 183 a(;) p 2965 183 a(d) p Fk(g) p Fy 3078 183 a(b) n(y) p Fg 1468 298 a(8) 1468 372 y(<) 1468 522 y(:) p Fo 1584 367 a(R) p Fj 1647 379 a(0) p Fp(;i) p Fy 1727 367 a(\(0\)) p 1856 367 a(=) p Fo 1944 367 a(i;) 1584 467 y(R) p Fj 1647 479 a(0) p Fp(;i) p Fy 1727 467 a(\() p Fo(i) p Fy(\)) p 1843 467 a(=) p 1931 467 a(0) p Fo(;) 1584 567 y(R) p Fj 1647 579 a(0) p Fp(;i) p Fy 1727 567 a(\() p Fo(j) p Fy 1798 567 a(\)) p 1854 567 a(=) p Fo 1941 567 a(j;) p 2040 567 a(j) p Fk 2102 567 a(6) p Fy(=) p 2190 567 a(0) p Fo(;) p 2269 567 a(i:) p Fy 125 764 a(F) p 172 764 a(or) p Fo 284 764 a(n) p Fy 375 764 a(=) p 481 764 a(1) p Fo(;) p Fy 560 764 a(2) p Fo(;) p Fy 639 764 a(3) p Fo(;) p Fk 718 764 a(\001) p 755 764 a(\001) p 792 764 a(\001) p Fy 851 764 a(and) p Fo 1023 764 a(i) p Fy 1093 764 a(=) p 1199 764 a(1) p Fo(;) p Fy 1278 764 a(2) p Fo(;) p Fk 1357 764 a(\001) p 1394 764 a(\001) p 1431 764 a(\001) p Fo 1466 764 a(;) p 1503 764 a(d) p Fy(,) p 1610 764 a(w) n(e) p 1743 764 a(de\014ne) p 1994 764 a(1) p 2076 764 a(:) p 2140 764 a(1) p 2220 764 a(maps) p Fo 2448 764 a(R) p Fp 2511 776 a(n;i) p Fy 2641 764 a(:) p Fo 2743 764 a(G) p Fk 2850 764 a(!) p Fo 2974 764 a(G) p Fy 3077 764 a(and) 3268 743 y(~) p Fo 3250 764 a(R) p Fp 3313 776 a(n;i) p Fy 3442 764 a(:) p Fo 3544 764 a(G) p Fk 3651 764 a(!) p Fo 3775 764 a(G) p Fy 0 864 a(\(`partial) p 322 864 a(re\015ections') p 736 864 a(with) p 923 864 a(resp) r(ect) p 1206 864 a(to) p 1306 864 a(a) p 1374 864 a(h) n(yp) r(erplane) p 1799 864 a(con) n(taining) p Fo 2201 864 a(\031) p Fy 2251 864 a(\() p Fo(j;) p 2354 864 a(j;) p Fk 2425 864 a(\001) p 2462 864 a(\001) p 2499 864 a(\001) p Fo 2537 864 a(;) p 2574 864 a(j;) p 2645 864 a(i;) p Fy 2711 864 a(0) p Fo(;) p Fy 2790 864 a(0) p Fo(;) p Fk 2869 864 a(\001) p 2906 864 a(\001) p 2943 864 a(\001) p Fy 2964 864 a(\)) p Fk 3020 864 a(2) p Fo 3098 864 a(G) p Fp 3163 876 a(n) p Fn(\000) p Fj(1) p Fp(;i) p Fy 3351 864 a(,) p Fo 3400 864 a(j) p Fk 3462 864 a(6) p Fy(=) p 3550 864 a(0) p Fo(;) p 3629 864 a(i) p Fy(,) p 3706 864 a(and) 0 964 y(`p) r(erp) r(endicular) p 549 964 a(to) p Fo 651 964 a(i) p Fy(-th) p 813 964 a(axis'\),) p 1060 964 a(b) n(y:) p Fo 595 1145 a(R) p Fp 658 1157 a(n;i) p Fy 746 1145 a(\() p Fo(\031) p Fy 828 1145 a(\() p Fo(x) p Fj 907 1157 a(0) p Fo 946 1145 a(;) p 983 1145 a(x) p Fj 1030 1157 a(1) p Fo 1067 1145 a(;) p 1104 1145 a(x) p Fj 1151 1157 a(2) p Fo 1189 1145 a(;) p Fk 1226 1145 a(\001) p 1263 1145 a(\001) p 1300 1145 a(\001) p Fy 1322 1145 a(\)\)) 595 1346 y(=) p Fg 682 1175 a(8) 682 1250 y(<) 682 1399 y(:) p Fo 797 1245 a(\031) p Fy 847 1245 a(\() p Fo(x) p Fj 926 1257 a(0) p Fo 965 1245 a(;) p 1002 1245 a(x) p Fj 1049 1257 a(1) p Fo 1086 1245 a(;) p 1123 1245 a(x) p Fj 1170 1257 a(2) p Fo 1208 1245 a(;) p Fk 1245 1245 a(\001) p 1282 1245 a(\001) p 1319 1245 a(\001) p Fy(\)) p Fo(;) p Fy 1439 1245 a(if) p Fo 1515 1245 a(\031) p Fy 1565 1245 a(\() p Fo(x) p Fj 1644 1257 a(0) p Fo 1682 1245 a(;) p 1719 1245 a(x) p Fj 1766 1257 a(1) p Fo 1804 1245 a(;) p 1841 1245 a(x) p Fj 1888 1257 a(2) p Fo 1925 1245 a(;) p Fk 1962 1245 a(\001) p 1999 1245 a(\001) p 2036 1245 a(\001) p Fo 2073 1245 a(;) p 2110 1245 a(x) p Fp 2157 1257 a(n) p Fo 2202 1245 a(;) p Fk 2239 1245 a(\001) p 2276 1245 a(\001) p 2313 1245 a(\001) p Fy(\)) p Fk 2391 1245 a(62) p Fo 2470 1245 a(G) p Fp 2535 1257 a(n) p Fn(\000) p Fj(1) p Fp(;i) p Fo 2708 1245 a(;) 797 1345 y(\031) p Fy 847 1345 a(\() p Fo(R) p Fj 942 1357 a(0) p Fp(;i) p Fy 1024 1345 a(\() p Fo(x) p Fj 1103 1357 a(0) p Fy 1141 1345 a(\)) p Fo(;) p 1210 1345 a(R) p Fj 1273 1357 a(0) p Fp(;i) p Fy 1353 1345 a(\() p Fo(x) p Fj 1432 1357 a(1) p Fy 1470 1345 a(\)) p Fo(;) p Fk 1539 1345 a(\001) p 1576 1345 a(\001) p 1613 1345 a(\001) p Fo 1650 1345 a(;) p 1687 1345 a(R) p Fj 1750 1357 a(0) p Fp(;i) p Fy 1830 1345 a(\() p Fo(x) p Fp 1909 1357 a(n) p Fn(\000) p Fj(1) p Fy 2040 1345 a(\)) p Fo(;) p 2109 1345 a(x) p Fp 2156 1357 a(n) p Fo 2202 1345 a(;) p Fy 2239 1345 a(0) p Fo(;) p Fy 2318 1345 a(0) p Fo(;) p Fy 2397 1345 a(0) p Fo(;) p Fk 2476 1345 a(\001) p 2513 1345 a(\001) p 2550 1345 a(\001) p Fy 2571 1345 a(\)) p Fo(;) p Fy 1425 1444 a(if) p Fo 1501 1444 a(\031) p Fy 1551 1444 a(\() p Fo(x) p Fj 1630 1456 a(0) p Fo 1668 1444 a(;) p 1705 1444 a(x) p Fj 1752 1456 a(1) p Fo 1790 1444 a(;) p 1827 1444 a(x) p Fj 1874 1456 a(2) p Fo 1911 1444 a(;) p Fk 1948 1444 a(\001) p 1985 1444 a(\001) p 2022 1444 a(\001) p Fo 2059 1444 a(;) p 2096 1444 a(x) p Fp 2143 1456 a(n) p Fo 2188 1444 a(;) p Fk 2225 1444 a(\001) p 2262 1444 a(\001) p 2299 1444 a(\001) p Fy(\)) p Fk 2378 1444 a(2) p Fo 2456 1444 a(G) p Fp 2521 1456 a(n) p Fn(\000) p Fj(1) p Fp(;i) p Fy 2750 1444 a(and) p Fo 2939 1444 a(x) p Fp 2986 1456 a(n) p Fy 3055 1444 a(=) p Fo 3142 1444 a(i;) p Fy 3692 1296 a(\(66\)) 0 1623 y(and) 385 1687 y(~) p Fo 367 1708 a(R) p Fp 430 1720 a(n;i) p Fy 519 1708 a(\() p Fo(\031) p Fy 601 1708 a(\() p Fo(x) p Fj 680 1720 a(0) p Fo 718 1708 a(;) p 755 1708 a(x) p Fj 802 1720 a(1) p Fo 840 1708 a(;) p 877 1708 a(x) p Fj 924 1720 a(2) p Fo 962 1708 a(;) p Fk 999 1708 a(\001) p 1036 1708 a(\001) p 1073 1708 a(\001) p Fy 1095 1708 a(\)\)) 367 1916 y(=) p Fg 455 1746 a(8) 455 1820 y(<) 455 1970 y(:) p Fo 570 1811 a(\031) p Fy 620 1811 a(\() p Fo(x) p Fj 699 1823 a(0) p Fo 738 1811 a(;) p 775 1811 a(x) p Fj 822 1823 a(1) p Fo 859 1811 a(;) p 896 1811 a(x) p Fj 943 1823 a(2) p Fo 981 1811 a(;) p Fk 1018 1811 a(\001) p 1055 1811 a(\001) p 1092 1811 a(\001) p Fy(\)) p Fo(;) p Fy 1211 1811 a(if) p Fo 1287 1811 a(\031) p Fy 1337 1811 a(\() p Fo(x) p Fj 1416 1823 a(0) p Fo 1455 1811 a(;) p 1492 1811 a(x) p Fj 1539 1823 a(1) p Fo 1576 1811 a(;) p 1613 1811 a(x) p Fj 1660 1823 a(2) p Fo 1698 1811 a(;) p Fk 1735 1811 a(\001) p 1772 1811 a(\001) p 1809 1811 a(\001) p Fo 1846 1811 a(;) p 1883 1811 a(x) p Fp 1930 1823 a(n) p Fo 1975 1811 a(;) p Fk 2012 1811 a(\001) p 2049 1811 a(\001) p 2086 1811 a(\001) p Fy(\)) p Fk 2164 1811 a(62) p Fo 2243 1811 a(G) p Fp 2308 1823 a(n) p Fn(\000) p Fj(1) p Fk 2456 1811 a([) p Fo 2530 1811 a(\023) p Fn 2559 1775 a(\000) p Fj(1) p Fp 2559 1834 a(n;i) p Fy 2649 1811 a(\() p Fo(G) p Fp 2746 1823 a(n) p Fn(\000) p Fj(1) p Fy 2877 1811 a(\)) p Fo(;) 570 1915 y(\031) p Fy 620 1915 a(\() p Fo(R) p Fj 715 1927 a(0) p Fp(;i) p Fy 796 1915 a(\() p Fo(x) p Fj 875 1927 a(0) p Fy 913 1915 a(\)) p Fo(;) p 982 1915 a(R) p Fj 1045 1927 a(0) p Fp(;i) p Fy 1126 1915 a(\() p Fo(x) p Fj 1205 1927 a(1) p Fy 1243 1915 a(\)) p Fo(;) p Fk 1312 1915 a(\001) p 1349 1915 a(\001) p 1386 1915 a(\001) p Fo 1423 1915 a(;) p 1460 1915 a(R) p Fj 1523 1927 a(0) p Fp(;i) p Fy 1603 1915 a(\() p Fo(x) p Fp 1682 1927 a(n) p Fn(\000) p Fj(1) p Fy 1813 1915 a(\)) p Fo(;) p 1882 1915 a(x) p Fp 1929 1927 a(n) p Fo 1975 1915 a(;) p 2012 1915 a(R) p Fj 2075 1927 a(0) p Fp(;i) p Fy 2155 1915 a(\() p Fo(x) p Fp 2234 1927 a(n) p Fj(+1) p Fy 2364 1915 a(\)) p Fo(;) p Fy 2433 1915 a(0) p Fo(;) p Fy 2512 1915 a(0) p Fo(;) p Fk 2591 1915 a(\001) p 2628 1915 a(\001) p 2665 1915 a(\001) p Fy 2687 1915 a(\)) p Fo(;) p Fy 1225 2018 a(if) p Fo 1301 2018 a(\031) p Fy 1351 2018 a(\() p Fo(x) p Fj 1430 2030 a(0) p Fo 1469 2018 a(;) p 1506 2018 a(x) p Fj 1553 2030 a(1) p Fo 1590 2018 a(;) p 1627 2018 a(x) p Fj 1674 2030 a(2) p Fo 1712 2018 a(;) p Fk 1749 2018 a(\001) p 1786 2018 a(\001) p 1823 2018 a(\001) p Fo 1859 2018 a(;) p 1896 2018 a(x) p Fp 1943 2030 a(n) p Fo 1989 2018 a(;) p Fk 2026 2018 a(\001) p 2063 2018 a(\001) p 2100 2018 a(\001) p Fy(\)) p Fk 2178 2018 a(2) p Fo 2256 2018 a(G) p Fp 2321 2030 a(n) p Fn(\000) p Fj(1) p Fk 2470 2018 a([) p Fo 2544 2018 a(\023) p Fn 2573 1982 a(\000) p Fj(1) p Fp 2573 2041 a(n;i) p Fy 2663 2018 a(\() p Fo(G) p Fp 2760 2030 a(n) p Fn(\000) p Fj(1) p Fy 2890 2018 a(\)) p 2978 2018 a(and) p Fo 3139 2018 a(x) p Fp 3186 2030 a(n) p Fy 3255 2018 a(=) p 3343 2018 a(0) p Fo 3399 2018 a(:) p Fy 3692 1863 a(\(67\)) 0 2183 y(Note) p 197 2183 a(that) p Fo 374 2183 a(G) p Fp 439 2195 a(n) p Fn(\000) p Fj(1) p Fy 583 2183 a(,) p Fo 631 2183 a(G) p Fp 696 2195 a(n) p Fn(\000) p Fj(1) p Fp(;i) p Fy 884 2183 a(,) p 932 2183 a(and) p Fo 1090 2183 a(\023) p Fn 1119 2148 a(\000) p Fj(1) p Fp 1119 2206 a(n;i) p Fy 1209 2183 a(\() p Fo(G) p Fp 1306 2195 a(n) p Fn(\000) p Fj(1) p Fy 1436 2183 a(\)) p 1493 2183 a(are) p 1628 2183 a(three) p 1837 2183 a(copies) p 2079 2183 a(of) p Fo 2170 2183 a(G) p Fp 2235 2195 a(n) p Fn(\000) p Fj(1) p Fy 2390 2183 a(aligned) p 2673 2183 a(in) p 2766 2183 a(`) p Fo(i) p Fy(-th) p 2949 2183 a(axis') p 3137 2183 a(direction,) p 3504 2183 a(suc) n(h) p 3688 2183 a(that) p Fo 464 2458 a(G) p Fp 529 2470 a(n) p Fn(\000) p Fj(1) p Fk 677 2458 a(\\) p Fo 751 2458 a(G) p Fp 816 2470 a(n) p Fn(\000) p Fj(1) p Fp(;i) p Fy 1013 2458 a(=) p Fk 1101 2458 a(f) p Fo(\031) p Fy 1193 2458 a(\() 1225 2364 y(0) p Fo(;) p Fy 1304 2364 a(1) p Fo(;) p Fy 1383 2364 a(2) 1225 2458 y(0) p Fo(;) p Fy 1304 2458 a(0) p Fo(;) p Fy 1383 2458 a(0) p Fo 1423 2458 a(;) p Fk 1460 2458 a(\001) p 1497 2458 a(\001) p 1534 2458 a(\001) p Fo 1571 2458 a(;) p Fy 1608 2458 a(0) p Fo(;) 1687 2379 y(n) 1697 2458 y(i) p 1736 2458 a(;) p Fy 1773 2458 a(0) p Fo(;) p Fy 1852 2458 a(0) p Fo(;) p Fk 1931 2458 a(\001) p 1968 2458 a(\001) p 2005 2458 a(\001) p Fy 2026 2458 a(\)) p Fk(g) p Fy 2123 2458 a(=) p Fk 2211 2458 a(f) p Fo(\031) p Fy 2303 2458 a(\() 2335 2363 y(0) p Fo(;) p Fy 2414 2363 a(1) p Fo(;) p Fy 2493 2363 a(2) p Fo 2354 2458 a(i;) p 2420 2458 a(i;) p 2486 2458 a(i) p 2533 2458 a(;) p Fk 2570 2458 a(\001) p 2607 2458 a(\001) p 2644 2458 a(\001) p Fo 2681 2458 a(;) 2718 2372 y(n) p Fk 2786 2372 a(\000) p Fy 2869 2372 a(1) p Fo 2800 2458 a(i) p 2911 2458 a(;) p Fy 2948 2458 a(0) p Fo(;) p Fy 3027 2458 a(0) p Fo(;) p Fy 3106 2458 a(0) p Fo(;) p Fk 3185 2458 a(\001) p 3222 2458 a(\001) p 3259 2458 a(\001) p Fy 3280 2458 a(\)) p Fk(g) p Fo(;) p Fy 3692 2458 a(\(68\)) 0 2641 y(and) p Fo 378 2799 a(G) p Fp 443 2811 a(n) p Fn(\000) p Fj(1) p Fp(;i) p Fk 636 2799 a(\\) p Fo 709 2799 a(\023) p Fn 738 2763 a(\000) p Fj(1) p Fp 738 2822 a(n;i) p Fy 828 2799 a(\() p Fo(G) p Fp 925 2811 a(n) p Fn(\000) p Fj(1) p Fy 1056 2799 a(\)) p 1111 2799 a(=) p Fk 1199 2799 a(f) p Fo(\031) p Fy 1291 2799 a(\() 1323 2704 y(0) p Fo(;) p Fy 1402 2704 a(1) p Fo(;) p Fy 1481 2704 a(2) 1323 2799 y(0) p Fo(;) p Fy 1402 2799 a(0) p Fo(;) p Fy 1481 2799 a(0) p Fo 1521 2799 a(;) p Fk 1558 2799 a(\001) p 1595 2799 a(\001) p 1632 2799 a(\001) p Fo 1669 2799 a(;) p Fy 1706 2799 a(0) p Fo(;) 1785 2712 y(n) p Fy 1852 2712 a(+) p 1935 2712 a(1) p Fo 1866 2799 a(i) p 1977 2799 a(;) p Fy 2014 2799 a(0) p Fo(;) p Fy 2093 2799 a(0) p Fo(;) p Fk 2172 2799 a(\001) p 2209 2799 a(\001) p 2246 2799 a(\001) p Fy 2267 2799 a(\)) p Fk(g) p Fy 2364 2799 a(=) p Fk 2452 2799 a(f) p Fo(\031) p Fy 2544 2799 a(\() 2576 2703 y(0) p Fo(;) p Fy 2655 2703 a(1) p Fo(;) p Fy 2734 2703 a(2) p Fo 2595 2799 a(i;) p 2661 2799 a(i;) p 2727 2799 a(i) p 2774 2799 a(;) p Fk 2811 2799 a(\001) p 2848 2799 a(\001) p 2885 2799 a(\001) p Fo 2922 2799 a(;) p 2959 2799 a(i;) 3025 2719 y(n) 3035 2799 y(i) p 3074 2799 a(;) p Fy 3111 2799 a(0) p Fo(;) p Fy 3190 2799 a(0) p Fo(;) p Fk 3269 2799 a(\001) p 3306 2799 a(\001) p 3343 2799 a(\001) p Fy 3365 2799 a(\)) p Fk(g) p Fo(:) p Fy 3692 2799 a(\(69\)) 0 2948 y(Note) p 192 2948 a(also) p 350 2948 a(that,) p 546 2948 a(b) n(y) p 653 2948 a(construction) p 1120 2948 a(all) p 1227 2948 a(the) p 1361 2948 a(p) r(oin) n(ts) p 1602 2948 a(in) p Fo 1690 2948 a(G) p Fp 1755 2960 a(n) p Fn(\000) p Fj(1) p Fp(;i) p Fy 1948 2948 a(can) p 2091 2948 a(b) r(e) p 2196 2948 a(written) p 2478 2948 a(as) p Fo 2571 2948 a(\031) p Fy 2621 2948 a(\() p Fo(x) p Fj 2700 2960 a(0) p Fo 2739 2948 a(;) p 2776 2948 a(x) p Fj 2823 2960 a(1) p Fo 2860 2948 a(;) p Fk 2897 2948 a(\001) p 2934 2948 a(\001) p 2971 2948 a(\001) p Fo 3008 2948 a(;) p 3045 2948 a(x) p Fp 3092 2960 a(n) p Fo 3137 2948 a(;) p Fy 3174 2948 a(0) p Fo(;) p Fy 3253 2948 a(0) p Fo(;) p Fk 3332 2948 a(\001) p 3369 2948 a(\001) p 3406 2948 a(\001) p Fy 3428 2948 a(\)) p 3479 2948 a(with) p Fo 3660 2948 a(x) p Fp 3707 2960 a(n) p Fy 3775 2948 a(=) p Fo 0 3048 a(i) p Fy(,) p 77 3048 a(those) p 290 3048 a(in) p Fo 384 3048 a(G) p Fp 449 3060 a(n) p Fn(\000) p Fj(1) p Fy 603 3048 a(as) p Fo 702 3048 a(\031) p Fy 752 3048 a(\() p Fo(x) p Fj 831 3060 a(0) p Fo 869 3048 a(;) p Fk 906 3048 a(\001) p 943 3048 a(\001) p 980 3048 a(\001) p Fo 1017 3048 a(;) p 1054 3048 a(x) p Fp 1101 3060 a(n) p Fo 1146 3048 a(;) p Fy 1183 3048 a(0) p Fo(;) p Fy 1262 3048 a(0) p Fo(;) p Fk 1341 3048 a(\001) p 1378 3048 a(\001) p 1415 3048 a(\001) p Fy 1437 3048 a(\)) p 1493 3048 a(with) p Fo 1679 3048 a(x) p Fp 1726 3060 a(n) p Fy 1794 3048 a(=) p 1882 3048 a(0,) p 1971 3048 a(and) p 2129 3048 a(those) p 2343 3048 a(in) p Fo 2436 3048 a(\023) p Fn 2465 3012 a(\000) p Fj(1) p Fp 2465 3071 a(n;i) p Fy 2555 3048 a(\() p Fo(G) p Fp 2652 3060 a(n) p Fn(\000) p Fj(1) p Fy 2783 3048 a(\)) p 2839 3048 a(as) p Fo 2938 3048 a(\031) p Fy 2988 3048 a(\() p Fo(x) p Fj 3067 3060 a(0) p Fo 3105 3048 a(;) p Fk 3142 3048 a(\001) p 3179 3048 a(\001) p 3216 3048 a(\001) p Fo 3252 3048 a(;) p 3289 3048 a(x) p Fp 3336 3060 a(n) p Fo 3382 3048 a(;) p 3419 3048 a(x) p Fp 3466 3060 a(n) p Fj(+1) p Fo 3596 3048 a(;) p Fy 3633 3048 a(0) p Fo(;) p Fk 3712 3048 a(\001) p 3749 3048 a(\001) p 3786 3048 a(\001) p Fy 3808 3048 a(\)) 0 3147 y(with) p Fo 189 3147 a(x) p Fp 236 3159 a(n) p Fy 305 3147 a(=) p 392 3147 a(0) p 462 3147 a(and) p Fo 623 3147 a(x) p Fp 670 3159 a(n) p Fj(+1) p Fy 823 3147 a(=) p Fo 910 3147 a(i) p Fy(.) p Fz 0 3330 a(Prop) s(osition) p 516 3330 a(27) p Fo 653 3330 a(R) p Fp 716 3342 a(n;i) p Fi 834 3330 a(and) p Fy 1014 3309 a(~) p Fo 996 3330 a(R) p Fp 1059 3342 a(n;i) p Fi 1177 3330 a(ar) l(e) p Fy 1318 3330 a(1) p 1382 3330 a(:) p 1428 3330 a(1) p Fi 1500 3330 a(maps.) p 1750 3330 a(Mor) l(e) l(over,) p 2141 3330 a(the) p 2279 3330 a(fol) t(lowing) p 2631 3330 a(hold.) 73 3505 y(\(i\)) p 208 3505 a(If) p Fo 295 3505 a(x) p Fk 365 3505 a(2) p Fo 444 3505 a(G) p Fp 509 3517 a(n) p Fn(\000) p Fj(1) p Fi 669 3505 a(then) p Fy 871 3484 a(~) p Fo 853 3505 a(R) p Fp 916 3517 a(n;i) p Fy 1005 3505 a(\() p Fo(x) p Fy(\)) p Fk 1140 3505 a(2) p Fo 1218 3505 a(\023) p Fn 1247 3469 a(\000) p Fj(1) p Fp 1247 3528 a(n;i) p Fy 1337 3505 a(\() p Fo(G) p Fp 1434 3517 a(n) p Fn(\000) p Fj(1) p Fy 1565 3505 a(\)) p Fi(.) 47 3691 y(\(ii\)) p 208 3691 a(If) p Fy 295 3691 a(\() p Fo(x;) p 411 3691 a(y) p Fy 455 3691 a(\)) p Fk 510 3691 a(2) p Fo 589 3691 a(B) p Fp 652 3703 a(n) p Fn(\000) p Fj(1) p Fi 782 3691 a(,) p 837 3691 a(then) p Fy 1022 3691 a(\() 1072 3670 y(~) p Fo 1054 3691 a(R) p Fp 1117 3703 a(n;i) p Fy 1205 3691 a(\() p Fo(x) p Fy(\)) p Fo(;) p Fy 1372 3670 a(~) p Fo 1353 3691 a(R) p Fp 1416 3703 a(n;i) p Fy 1506 3691 a(\() p Fo(y) p Fy 1582 3691 a(\)\)) p Fk 1669 3691 a(2) p Fo 1748 3691 a(B) p Fi 1815 3691 a(.) 22 3865 y(\(iii\)) p 208 3865 a(If) p Fo 295 3865 a(x) p Fk 365 3865 a(2) p Fo 444 3865 a(G) p Fp 509 3877 a(n) p Fn(\000) p Fj(1) p Fk 658 3865 a(\\) p Fo 731 3865 a(G) p Fp 796 3877 a(n) p Fn(\000) p Fj(1) p Fp(;i) p Fi 1000 3865 a(then) p Fo 1184 3865 a(R) p Fp 1247 3877 a(n;i) p Fy 1336 3865 a(\() p Fo(x) p Fy(\)) p 1471 3865 a(=) 1576 3844 y(~) p Fo 1558 3865 a(R) p Fp 1621 3877 a(n;i) p Fy 1710 3865 a(\() p Fo(x) p Fy(\)) p Fi(.) 35 4031 y(\(iv\)) p 208 4031 a(If) p Fo 295 4031 a(x) p Fk 365 4031 a(2) p Fo 444 4031 a(G) p Fp 509 4043 a(n) p Fi 584 4031 a(then) p Fo 768 4031 a(R) p Fp 831 4043 a(n;i) p Fy 920 4031 a(\() p Fo(x) p Fy(\)) p Fk 1055 4031 a(2) p Fo 1133 4031 a(G) p Fp 1198 4043 a(n) p Fi 1258 4031 a(.) 60 4217 y(\(v\)) p 208 4217 a(If) p Fo 295 4217 a(x) p Fk 365 4217 a(2) p Fo 444 4217 a(G) p Fp 509 4229 a(n) p Fn(\000) p Fj(1) p Fp(;i) p Fk 701 4217 a(\\) p Fg 807 4138 a([) p Fp 775 4317 a(j) p Fn 805 4317 a(6) p Fj(=0) p Fp(;i) p Fo 946 4217 a(G) p Fp 1011 4229 a(n) p Fn(\000) p Fj(1) p Fp(;j) p Fi 1222 4217 a(then) p Fo 1407 4217 a(R) p Fp 1470 4229 a(n;i) p Fy 1558 4217 a(\() p Fo(x) p Fy(\)) p 1693 4217 a(=) p Fo 1781 4217 a(x) p Fi(.) 35 4474 y(\(vi\)) p 208 4474 a(If) p Fy 295 4474 a(\() p Fo(x;) p 411 4474 a(y) p Fy 455 4474 a(\)) p Fk 510 4474 a(2) p Fo 589 4474 a(B) p Fp 652 4486 a(n) p Fn(\000) p Fj(1) p Fp(;i) p Fi 855 4474 a(then) p Fy 1039 4474 a(\() p Fo(R) p Fp 1134 4486 a(n;i) p Fy 1223 4474 a(\() p Fo(x) p Fy(\)) p Fo(;) p 1371 4474 a(R) p Fp 1434 4486 a(n;i) p Fy 1524 4474 a(\() p Fo(y) p Fy 1600 4474 a(\)\)) p Fk 1687 4474 a(2) p Fo 1766 4474 a(B) p Fi 1833 4474 a(.) -42 4665 y(Pr) l(o) l(of.) p Fy 219 4665 a(By) p 349 4665 a(de\014nition,) p Fo 741 4665 a(R) p Fj 805 4635 a(2) p Fp 804 4687 a(n;i) p Fy 892 4665 a(\() p Fo(x) p Fy(\)) p 1027 4665 a(=) 1133 4644 y(~) p Fo 1115 4665 a(R) p Fj 1179 4635 a(2) p Fp 1178 4687 a(n;i) p Fy 1267 4665 a(\() p Fo(x) p Fy(\)) p 1402 4665 a(=) p Fo 1489 4665 a(x) p Fy(,) p Fo 1587 4665 a(x) p Fk 1658 4665 a(2) p Fo 1736 4665 a(G) p Fy(,) p 1852 4665 a(hence,) p 2106 4665 a(in) p 2203 4665 a(particular,) p Fo 2609 4665 a(R) p Fp 2672 4677 a(n;i) p Fy 2788 4665 a(and) 2968 4644 y(~) p Fo 2950 4665 a(R) p Fp 3013 4677 a(n;i) p Fy 3129 4665 a(are) p 3268 4665 a(1) p 3332 4665 a(:) p 3378 4665 a(1) p 3447 4665 a(maps.) 78 4831 y(\(i\)) p 208 4831 a(If) p Fo 288 4831 a(x) p Fk 359 4831 a(2) p Fo 437 4831 a(G) p Fp 502 4843 a(n) p Fn(\000) p Fj(1) p Fy 658 4831 a(then) p Fo 845 4831 a(x) p Fy 918 4831 a(has) p 1064 4831 a(a) p 1131 4831 a(co) r(ordinate) p 1537 4831 a(of) p 1629 4831 a(the) p 1770 4831 a(form) p Fo 1964 4831 a(x) p Fy 2035 4831 a(=) p Fo 2122 4831 a(\031) p Fy 2172 4831 a(\() p Fo(x) p Fj 2251 4843 a(0) p Fo 2290 4831 a(;) p 2327 4831 a(x) p Fj 2374 4843 a(1) p Fo 2411 4831 a(;) p Fk 2448 4831 a(\001) p 2485 4831 a(\001) p 2522 4831 a(\001) p Fo 2559 4831 a(;) p 2596 4831 a(x) p Fp 2643 4843 a(n) p Fn(\000) p Fj(1) p Fo 2773 4831 a(;) p Fy 2810 4831 a(0) p Fo(;) p Fy 2889 4831 a(0) p Fo(;) p Fy 2968 4831 a(0) p Fo(;) p Fk 3047 4831 a(\001) p 3084 4831 a(\001) p 3121 4831 a(\001) p Fy 3142 4831 a(\),) p 3224 4831 a(whic) n(h,) p 3482 4831 a(with) p 3669 4831 a(\(67\),) 208 4931 y(implies) 507 4910 y(~) p Fo 489 4931 a(R) p Fp 552 4943 a(n;i) p Fy 641 4931 a(\() p Fo(x) p Fy(\)) p 776 4931 a(=) p Fo 864 4931 a(\031) p Fy 914 4931 a(\() p Fo(x) p Fj 993 4943 a(0) p Fo 1031 4931 a(;) p 1068 4931 a(x) p Fj 1115 4943 a(1) p Fo 1152 4931 a(;) p Fk 1189 4931 a(\001) p 1226 4931 a(\001) p 1263 4931 a(\001) p Fo 1300 4931 a(;) p 1337 4931 a(x) p Fp 1384 4943 a(n) p Fn(\000) p Fj(1) p Fo 1515 4931 a(;) p Fy 1552 4931 a(0) p Fo(;) p 1631 4931 a(i;) p Fy 1697 4931 a(0) p Fo(;) p Fk 1776 4931 a(\001) p 1813 4931 a(\001) p 1850 4931 a(\001) p Fy 1871 4931 a(\),) p 1954 4931 a(whic) n(h) p 2191 4931 a(is) p 2275 4931 a(in) p Fo 2371 4931 a(\023) p Fn 2400 4895 a(\000) p Fj(1) p Fp 2400 4954 a(n;i) p Fy 2490 4931 a(\() p Fo(G) p Fp 2587 4943 a(n) p Fn(\000) p Fj(1) p Fy 2718 4931 a(\)) p 2764 4931 a(.) 55 5097 y(\(ii\)) p 208 5097 a(let) p 327 5097 a(\() p Fo(x;) p 443 5097 a(y) p Fy 487 5097 a(\)) p Fk 543 5097 a(2) p Fo 622 5097 a(B) p Fp 685 5109 a(n) p Fn(\000) p Fj(1) p Fy 829 5097 a(.) p 888 5097 a(Then) p Fo 1105 5097 a(x;) p 1189 5097 a(y) p Fk 1256 5097 a(2) p Fo 1335 5097 a(G) p Fp 1400 5109 a(n) p Fn(\000) p Fj(1) p Fy 1530 5097 a(,) p 1581 5097 a(hence) p 1812 5097 a(their) p 2010 5097 a(co) r(ordinates) p 2451 5097 a(can) p 2604 5097 a(b) r(e) p 2717 5097 a(written) p 3007 5097 a(as) p Fo 768 5279 a(x) p Fy 838 5279 a(=) p Fo 926 5279 a(\031) p Fy 976 5279 a(\() p Fo(x) p Fj 1055 5291 a(0) p Fo 1093 5279 a(;) p 1130 5279 a(x) p Fj 1177 5291 a(1) p Fo 1215 5279 a(;) p Fk 1252 5279 a(\001) p 1289 5279 a(\001) p 1326 5279 a(\001) p Fo 1362 5279 a(;) p 1399 5279 a(x) p Fp 1446 5291 a(n) p Fn(\000) p Fj(1) p Fo 1577 5279 a(;) p Fy 1614 5279 a(0) p Fo(;) p Fy 1693 5279 a(0) p Fo(;) p Fy 1772 5279 a(0) p Fo(;) p Fk 1851 5279 a(\001) p 1888 5279 a(\001) p 1925 5279 a(\001) p Fy 1946 5279 a(\)) p Fo(;) p 2070 5279 a(y) p Fy 2137 5279 a(=) p Fo 2225 5279 a(\031) p Fy 2275 5279 a(\() p Fo(y) p Fj 2348 5291 a(0) p Fo 2385 5279 a(;) p 2422 5279 a(y) p Fj 2463 5291 a(1) p Fo 2500 5279 a(;) p Fk 2537 5279 a(\001) p 2574 5279 a(\001) p 2611 5279 a(\001) p Fo 2648 5279 a(;) p 2685 5279 a(y) p Fp 2726 5291 a(n) p Fn(\000) p Fj(1) p Fo 2856 5279 a(;) p Fy 2893 5279 a(0) p Fo(;) p Fy 2972 5279 a(0) p Fo(;) p Fy 3051 5279 a(0) p Fo(;) p Fk 3130 5279 a(\001) p 3167 5279 a(\001) p 3204 5279 a(\001) p Fy 3224 5279 a(\)) p Fo(:) p Fy 208 5462 a(With) p 422 5462 a(\(67\),) p 620 5462 a(w) n(e) p 743 5462 a(ha) n(v) n(e) 732 5624 y(~) p Fo 714 5645 a(R) p Fp 777 5657 a(n;i) p Fy 866 5645 a(\() p Fo(x) p Fy(\)) p 1001 5645 a(=) p Fo 1088 5645 a(\031) p Fy 1138 5645 a(\() p Fo(R) p Fj 1233 5657 a(0) p Fp(;i) p Fy 1314 5645 a(\() p Fo(x) p Fj 1393 5657 a(0) p Fy 1431 5645 a(\)) p Fo(;) p 1500 5645 a(R) p Fj 1563 5657 a(0) p Fp(;i) p Fy 1644 5645 a(\() p Fo(x) p Fj 1723 5657 a(1) p Fy 1761 5645 a(\)) p Fo(;) p Fk 1830 5645 a(\001) p 1867 5645 a(\001) p 1904 5645 a(\001) p Fo 1941 5645 a(;) p 1978 5645 a(R) p Fj 2041 5657 a(0) p Fp(;i) p Fy 2121 5645 a(\() p Fo(x) p Fp 2200 5657 a(n) p Fn(\000) p Fj(1) p Fy 2331 5645 a(\)) p Fo(;) p Fy 2400 5645 a(0) p Fo(;) p 2479 5645 a(i;) p Fy 2545 5645 a(0) p Fo(;) p Fy 2624 5645 a(0) p Fo(;) p Fk 2703 5645 a(\001) p 2740 5645 a(\001) p 2777 5645 a(\001) p Fy 2798 5645 a(\)) p Fk 2853 5645 a(2) p Fo 2932 5645 a(\023) p Fn 2961 5609 a(\000) p Fj(1) p Fp 2961 5668 a(n;i) p Fy 3050 5645 a(\() p Fo(G) p Fp 3147 5657 a(n) p Fn(\000) p Fj(1) p Fy 3278 5645 a(\)) p Fo(;) p Fy 1878 6120 a(28) p 90 rotate dyy eop %%Page: 29 29 29 28 bop Fy 208 83 a(and) p 374 83 a(a) p 449 83 a(similar) p 727 83 a(expression) p 1132 83 a(holds) p 1354 83 a(for) p Fo 1487 83 a(y) p Fy 1531 83 a(.) p 1606 83 a(Noting) p 1885 83 a(that) p Fo 2070 83 a(\023) p Fn 2099 48 a(\000) p Fj(1) p Fp 2099 106 a(n;i) p Fy 2189 83 a(\() p Fo(G) p Fp 2286 95 a(n) p Fn(\000) p Fj(1) p Fy 2417 83 a(\)) p 2482 83 a(is) p 2571 83 a(a) p 2645 83 a(cop) n(y) p 2844 83 a(of) p Fo 2944 83 a(G) p Fp 3009 95 a(n) p Fn(\000) p Fj(1) p Fy 3139 83 a(,) p 3196 83 a(it) p 3284 83 a(then) p 3479 83 a(su\016ces) p 3766 83 a(to) 208 183 y(pro) n(v) n(e) 546 365 y(\() p Fo(\031) p Fy 628 365 a(\() p Fo(R) p Fj 723 377 a(0) p Fp(;i) p Fy 804 365 a(\() p Fo(x) p Fj 883 377 a(0) p Fy 921 365 a(\)) p Fo(;) p Fk 990 365 a(\001) p 1027 365 a(\001) p 1064 365 a(\001) p Fo 1101 365 a(;) p 1138 365 a(R) p Fj 1201 377 a(0) p Fp(;i) p Fy 1281 365 a(\() p Fo(x) p Fp 1360 377 a(n) p Fn(\000) p Fj(1) p Fy 1491 365 a(\)) p Fo(;) p Fy 1560 365 a(0) p Fo(;) p Fy 1639 365 a(0) p Fo(;) p Fk 1718 365 a(\001) p 1755 365 a(\001) p 1792 365 a(\001) p Fy 1814 365 a(\)) p Fo(;) p 1883 365 a(\031) p Fy 1933 365 a(\() p Fo(R) p Fj 2028 377 a(0) p Fp(;i) p Fy 2109 365 a(\() p Fo(y) p Fj 2182 377 a(0) p Fy 2220 365 a(\)) p Fo(;) p Fk 2289 365 a(\001) p 2326 365 a(\001) p 2363 365 a(\001) p Fo 2399 365 a(;) p 2436 365 a(R) p Fj 2499 377 a(0) p Fp(;i) p Fy 2580 365 a(\() p Fo(y) p Fp 2653 377 a(n) p Fn(\000) p Fj(1) p Fy 2783 365 a(\)) p Fo(;) p Fy 2852 365 a(0) p Fo(;) p Fy 2931 365 a(0) p Fo(;) p Fk 3010 365 a(\001) p 3047 365 a(\001) p 3084 365 a(\001) p Fy 3106 365 a(\)\)) p Fk 3193 365 a(2) p Fo 3272 365 a(B) p Fp 3335 377 a(n) p Fn(\000) p Fj(1) p Fo 3479 365 a(:) p Fy 208 547 a(Noting) p 482 547 a(the) p 625 547 a(de\014nition) p 994 547 a(of) p Fo 1089 547 a(B) p Fp 1152 559 a(n) p Fn(\000) p Fj(1) p Fy 1282 547 a(,) p 1332 547 a(w) n(e) p 1455 547 a(ma) n(y) p 1635 547 a(assume,) p 1944 547 a(from) p 2141 547 a(\() p Fo(x;) p 2257 547 a(y) p Fy 2301 547 a(\)) p Fk 2356 547 a(2) p Fo 2435 547 a(B) p Fp 2498 559 a(n) p Fn(\000) p Fj(1) p Fy 2628 547 a(,) p 2679 547 a(that) p Fo 2859 547 a(x) p Fp 2906 559 a(n) p Fn(\000) p Fj(1) p Fy 3059 547 a(=) p Fo 3147 547 a(y) p Fp 3188 559 a(n) p Fn(\000) p Fj(1) p Fy 3345 547 a(and) 765 729 y(\() p Fo(\031) p Fy 847 729 a(\() p Fo(x) p Fj 926 741 a(0) p Fo 965 729 a(;) p 1002 729 a(x) p Fj 1049 741 a(1) p Fo 1086 729 a(;) p Fk 1123 729 a(\001) p 1160 729 a(\001) p 1197 729 a(\001) p Fo 1234 729 a(;) p 1271 729 a(x) p Fp 1318 741 a(n) p Fn(\000) p Fj(2) p Fo 1448 729 a(;) p Fy 1485 729 a(0) p Fo(;) p Fy 1564 729 a(0) p Fo(;) p Fy 1643 729 a(0) p Fo(;) p Fk 1722 729 a(\001) p 1759 729 a(\001) p 1796 729 a(\001) p Fy 1817 729 a(\)) p Fo(;) p 1886 729 a(\031) p Fy 1936 729 a(\() p Fo(y) p Fj 2009 741 a(0) p Fo 2047 729 a(;) p 2084 729 a(y) p Fj 2125 741 a(1) p Fo 2162 729 a(;) p Fk 2199 729 a(\001) p 2236 729 a(\001) p 2273 729 a(\001) p Fo 2310 729 a(;) p 2347 729 a(y) p Fp 2388 741 a(n) p Fn(\000) p Fj(2) p Fo 2517 729 a(;) p Fy 2554 729 a(0) p Fo(;) p Fy 2633 729 a(0) p Fo(;) p Fy 2712 729 a(0) p Fo(;) p Fk 2791 729 a(\001) p 2828 729 a(\001) p 2865 729 a(\001) p Fy 2886 729 a(\)\)) p Fk 2974 729 a(2) p Fo 3052 729 a(B) p Fp 3115 741 a(n) p Fn(\000) p Fj(2) p Fo 3259 729 a(:) p Fy 208 911 a(Inductiv) n(ely) p 607 911 a(,) p 657 911 a(it) p 740 911 a(ev) n(en) n(tually) p 1137 911 a(turns) p 1354 911 a(out) p 1502 911 a(that) p 1682 911 a(it) p 1765 911 a(su\016ces) p 2047 911 a(to) p 2149 911 a(pro) n(v) n(e) 499 1093 y(\() p Fo(\031) p Fy 581 1093 a(\() p Fo(R) p Fj 676 1105 a(0) p Fp(;i) p Fy 757 1093 a(\() p Fo(x) p Fy(\)) p Fo(;) p Fy 905 1093 a(0) p Fo(;) p Fy 984 1093 a(0) p Fo(;) p Fk 1063 1093 a(\001) p 1100 1093 a(\001) p 1137 1093 a(\001) p Fy(\)) p Fo(;) p 1229 1093 a(\031) p Fy 1279 1093 a(\() p Fo(R) p Fj 1374 1105 a(0) p Fp(;i) p Fy 1455 1093 a(\() p Fo(y) p Fy 1531 1093 a(\)) p Fo(;) p Fy 1600 1093 a(0) p Fo(;) p Fy 1679 1093 a(0) p Fo(;) p Fk 1758 1093 a(\001) p 1795 1093 a(\001) p 1832 1093 a(\001) p Fy 1854 1093 a(\)\)) p Fk 1941 1093 a(2) p Fo 2020 1093 a(B) p Fj 2083 1105 a(0) p Fy 2203 1093 a(if) p 2307 1093 a(\() p Fo(\031) p Fy 2389 1093 a(\() p Fo(x;) p Fy 2505 1093 a(0) p Fo(;) p Fy 2584 1093 a(0) p Fo(;) p Fk 2663 1093 a(\001) p 2700 1093 a(\001) p 2737 1093 a(\001) p Fy(\)) p Fo(;) p 2829 1093 a(\031) p Fy 2879 1093 a(\() p Fo(y) s(;) p Fy 2992 1093 a(0) p Fo(;) p Fy 3071 1093 a(0) p Fo(;) p Fk 3150 1093 a(\001) p 3187 1093 a(\001) p 3224 1093 a(\001) p Fy 3246 1093 a(\)\)) p Fk 3333 1093 a(2) p Fo 3412 1093 a(B) p Fj 3475 1105 a(0) p Fo 3526 1093 a(;) p Fy 208 1275 a(whic) n(h) p 445 1275 a(is) p 529 1275 a(ob) n(vious) p 829 1275 a(from) p 1025 1275 a(the) p 1168 1275 a(de\014nition) p 1537 1275 a(of) p Fo 1632 1275 a(F) p Fj 1685 1287 a(0) p Fy 1745 1275 a(=) p 1833 1275 a(\() p Fo(G) p Fj 1930 1287 a(0) p Fo 1968 1275 a(;) p 2005 1275 a(B) p Fj 2068 1287 a(0) p Fy 2105 1275 a(\).) 32 1441 y(\(iii\)) p 208 1441 a(Applying) p 567 1441 a(\(68\)) p 743 1441 a(in) p 839 1441 a(\(66\)) p 1015 1441 a(and) p 1176 1441 a(\(67\),) p 1374 1441 a(w) n(e) p 1497 1441 a(ha) n(v) n(e) p Fo 752 1702 a(R) p Fp 815 1714 a(n;i) p Fy 903 1702 a(\() p Fo(x) p Fy(\)) p 1038 1702 a(=) p Fo 1126 1702 a(R) p Fp 1189 1714 a(n;i) p Fy 1277 1702 a(\() p Fo(\031) p Fy 1359 1702 a(\() 1391 1607 y(0) p Fo(;) p Fy 1470 1607 a(1) p Fo(;) p Fy 1549 1607 a(2) 1391 1702 y(0) p Fo(;) p Fy 1470 1702 a(0) p Fo(;) p Fy 1549 1702 a(0) p Fo(;) p Fk 1628 1702 a(\001) p 1665 1702 a(\001) p 1702 1702 a(\001) p Fo 1738 1702 a(;) p Fy 1775 1702 a(0) p Fo(;) 1854 1622 y(n) 1864 1702 y(i) p 1903 1702 a(;) p Fy 1940 1702 a(0) p Fo(;) p Fy 2019 1702 a(0) p Fo(;) p Fk 2098 1702 a(\001) p 2135 1702 a(\001) p 2172 1702 a(\001) p Fy 2194 1702 a(\)\)) p 2281 1702 a(=) p Fo 2369 1702 a(\031) p Fy 2419 1702 a(\() 2451 1606 y(0) p Fo(;) p Fy 2530 1606 a(1) p Fo(;) p Fy 2609 1606 a(2) p Fo 2471 1702 a(i;) p 2537 1702 a(i;) p 2603 1702 a(i) p 2650 1702 a(;) p Fk 2687 1702 a(\001) p 2724 1702 a(\001) p 2761 1702 a(\001) p Fo 2798 1702 a(;) p 2835 1702 a(i;) 2901 1622 y(n) 2911 1702 y(i) p 2950 1702 a(;) p Fy 2987 1702 a(0) p Fo(;) p Fy 3066 1702 a(0) p Fo(;) p Fk 3145 1702 a(\001) p 3182 1702 a(\001) p 3219 1702 a(\001) p Fy 3240 1702 a(\)) p Fo(;) p Fy 208 1884 a(and) 632 2021 y(~) p Fo 614 2042 a(R) p Fp 677 2054 a(n;i) p Fy 765 2042 a(\() p Fo(x) p Fy(\)) p 900 2042 a(=) 1006 2021 y(~) p Fo 988 2042 a(R) p Fp 1051 2054 a(n;i) p Fy 1139 2042 a(\() p Fo(\031) p Fy 1221 2042 a(\() 1253 1946 y(0) p Fo(;) p Fy 1332 1946 a(1) p Fo(;) p Fy 1411 1946 a(2) p Fo 1274 2042 a(i;) p 1340 2042 a(i;) p 1406 2042 a(i) p 1453 2042 a(;) p Fk 1490 2042 a(\001) p 1527 2042 a(\001) p 1564 2042 a(\001) p Fo 1600 2042 a(;) 1637 1955 y(n) p Fk 1705 1955 a(\000) p Fy 1788 1955 a(1) p Fo 1719 2042 a(i) p 1830 2042 a(;) p Fy 1867 2042 a(0) p Fo(;) p Fy 1946 2042 a(0) p Fo(;) p Fy 2025 2042 a(0) p Fo(;) p Fk 2104 2042 a(\001) p 2141 2042 a(\001) p 2178 2042 a(\001) p Fy 2199 2042 a(\)\)) p 2287 2042 a(=) p Fo 2374 2042 a(\031) p Fy 2424 2042 a(\() 2456 1947 y(0) p Fo(;) p Fy 2535 1947 a(1) p Fo(;) p Fy 2614 1947 a(2) 2456 2042 y(0) p Fo(;) p Fy 2535 2042 a(0) p Fo(;) p Fy 2614 2042 a(0) p Fo 2655 2042 a(;) p Fk 2692 2042 a(\001) p 2729 2042 a(\001) p 2766 2042 a(\001) p Fo 2803 2042 a(;) p Fy 2840 2042 a(0) p Fo(;) p Fy 2919 2042 a(0) p Fo(;) 2998 1955 y(n) p Fy 3065 1955 a(+) p 3148 1955 a(1) p Fo 3079 2042 a(i) p 3189 2042 a(;) p Fy 3226 2042 a(0) p Fo(;) p Fk 3305 2042 a(\001) p 3342 2042 a(\001) p 3379 2042 a(\001) p Fy 3401 2042 a(\)) 614 2221 y(=) p Fo 701 2221 a(\031) p Fy 751 2221 a(\() 783 2125 y(0) p Fo(;) p Fy 862 2125 a(1) p Fo(;) p Fy 941 2125 a(2) p Fo 803 2221 a(i;) p 869 2221 a(i;) p 935 2221 a(i) p 982 2221 a(;) p Fk 1019 2221 a(\001) p 1056 2221 a(\001) p 1093 2221 a(\001) p Fo 1130 2221 a(;) p 1167 2221 a(i;) 1233 2141 y(n) 1243 2221 y(i) p 1282 2221 a(;) p Fy 1319 2221 a(0) p Fo(;) p Fy 1398 2221 a(0) p Fo(;) p Fk 1477 2221 a(\001) p 1514 2221 a(\001) p 1551 2221 a(\001) p Fy 1573 2221 a(\)) p Fo(:) p Fy 208 2385 a(Hence) p Fo 454 2385 a(R) p Fp 517 2397 a(n;i) p Fy 606 2385 a(\() p Fo(x) p Fy(\)) p 741 2385 a(=) 846 2364 y(~) p Fo 828 2385 a(R) p Fp 891 2397 a(n;i) p Fy 980 2385 a(\() p Fo(x) p Fy(\).) 35 2551 y(\(iv\)) p 208 2551 a(If) p Fo 291 2551 a(x) p Fk 361 2551 a(2) p Fo 440 2551 a(G) p Fp 505 2563 a(n) p Fy 578 2551 a(then) p Fo 767 2551 a(x) p Fy 842 2551 a(has) p 990 2551 a(a) p 1059 2551 a(co) r(ordinate) p Fo 1468 2551 a(x) p Fy 1538 2551 a(=) p Fo 1626 2551 a(\031) p Fy 1676 2551 a(\() p Fo(x) p Fj 1755 2563 a(0) p Fo 1793 2551 a(;) p 1830 2551 a(x) p Fj 1877 2563 a(1) p Fo 1915 2551 a(;) p Fk 1952 2551 a(\001) p 1989 2551 a(\001) p 2026 2551 a(\001) p Fo 2062 2551 a(;) p 2099 2551 a(x) p Fp 2146 2563 a(n) p Fo 2192 2551 a(;) p Fy 2229 2551 a(0) p Fo(;) p Fy 2308 2551 a(0) p Fo(;) p Fk 2387 2551 a(\001) p 2424 2551 a(\001) p 2461 2551 a(\001) p Fy 2482 2551 a(\).) p 2575 2551 a(Then) p 2791 2551 a(\(66\)) p 2967 2551 a(implies) p 3249 2551 a(that) p Fo 3428 2551 a(R) p Fp 3491 2563 a(n;i) p Fy 3580 2551 a(\() p Fo(x) p Fy(\)) p 3720 2551 a(has) 208 2650 y(a) p 277 2650 a(similar) p 550 2650 a(co) r(ordinate,) p 981 2650 a(hence) p Fo 1212 2650 a(R) p Fp 1275 2662 a(n;i) p Fy 1363 2650 a(\() p Fo(x) p Fy(\)) p Fk 1498 2650 a(2) p Fo 1577 2650 a(G) p Fp 1642 2662 a(n) p Fy 1701 2650 a(.) 58 2864 y(\(v\)) p 208 2864 a(Noting) p 480 2864 a(\(68\)) p 654 2864 a(and) p 814 2864 a(\(69\),) p 1011 2864 a(w) n(e) p 1132 2864 a(see) p 1264 2864 a(that) p Fo 1443 2864 a(x) p Fy 1516 2864 a(has) p 1663 2864 a(a) p 1730 2864 a(co) r(ordinate) p Fo 2137 2864 a(x) p Fy 2207 2864 a(=) p Fo 2295 2864 a(\031) p Fy 2345 2864 a(\() p Fo(j;) p 2448 2864 a(j;) p Fk 2519 2864 a(\001) p 2556 2864 a(\001) p 2593 2864 a(\001) p Fo 2631 2864 a(;) p 2668 2864 a(j;) 2739 2784 y(n) 2750 2864 y(i) p 2789 2864 a(;) p Fy 2826 2864 a(0) p Fo(;) p Fy 2905 2864 a(0) p Fo(;) p Fk 2984 2864 a(\001) p 3021 2864 a(\001) p 3058 2864 a(\001) p Fy 3079 2864 a(\),) p Fo 3161 2864 a(j) p Fk 3223 2864 a(6) p Fy(=) p 3311 2864 a(0) p Fo(;) p 3390 2864 a(i) p Fy(.) p 3477 2864 a(Then) p 3692 2864 a(\(66\)) 208 2964 y(implies) p Fo 489 2964 a(R) p Fp 552 2976 a(n;i) p Fy 641 2964 a(\() p Fo(x) p Fy(\)) p 776 2964 a(=) p Fo 864 2964 a(x) p Fy(.) 35 3130 y(\(vi\)) p 208 3130 a(This) p 397 3130 a(can) p 549 3130 a(b) r(e) p 662 3130 a(pro) n(v) n(ed) p 932 3130 a(in) p 1029 3130 a(a) p 1098 3130 a(similar) p 1371 3130 a(w) n(a) n(y) p 1540 3130 a(as) p 1642 3130 a(\(ii\).) p Fd 3778 3295 a(2) p Fy 125 3576 a(F) p 172 3576 a(or) p 273 3576 a(eac) n(h) p 460 3576 a(self-a) n(v) n(oiding) p 935 3576 a(path) p Fo 1128 3576 a(w) p Fk 1213 3576 a(2) p Fo 1291 3576 a(W) p Fj 1381 3546 a(\(0\)) p Fy 1498 3576 a(satisfying) p Fk 1867 3576 a(j) p Fo(w) p Fy 1951 3576 a(\() p Fo(L) p Fy(\() p Fo(w) p Fy 2133 3576 a(\)\)) p Fk(j) p Fo 2245 3576 a(<) p Fy 2333 3576 a(2) p Fp 2375 3542 a(\027) p Fj 2412 3542 a(\() p Fp(w) p Fj 2488 3542 a(\)) p Fn(\000) p Fj(1) p Fy 2602 3576 a(,) p 2653 3576 a(w) n(e) p 2775 3576 a(w) n(an) n(t) p 2977 3576 a(to) p 3078 3576 a(assign) p 3323 3576 a(a) p 3392 3576 a(self-a) n(v) n (oiding) 0 3676 y(path) p Fo 196 3676 a(R) p Fy 260 3676 a(\() p Fo(w) p Fy 353 3676 a(\)) p Fk 413 3676 a(2) p Fo 495 3676 a(W) p Fj 585 3646 a(\(0\)) p Fy 704 3676 a(\(`re\015ected) p 1094 3676 a(path'\)) p 1346 3676 a(suc) n(h) p 1535 3676 a(that) p Fk 1717 3676 a(j) p Fo(R) p Fy 1804 3676 a(\() p Fo(w) p Fy 1897 3676 a(\)\() p Fo(L) p Fy(\() p Fo(R) p Fy 2114 3676 a(\() p Fo(w) p Fy 2207 3676 a(\)\)\)) p Fk(j) p Fo 2356 3676 a(>) p Fy 2447 3676 a(2) p Fp 2489 3642 a(\027) p Fj 2526 3642 a(\() p Fp(R) p Fj(\() p Fp(w) p Fj 2678 3642 a(\)\)) p Fn(\000) p Fj(1) p Fy 2849 3676 a(\(the) p 3027 3676 a(righ) n(t) p 3230 3676 a(hand) p 3440 3676 a(side) p 3608 3676 a(stands) 0 3776 y(for) p 127 3776 a(the) p 270 3776 a(Euclidean) p 654 3776 a(distance) p 977 3776 a(of) p 1072 3776 a(the) p 1215 3776 a(endp) r(oin) n(ts) p 1594 3776 a(of) p Fo 1688 3776 a(R) p Fy 1752 3776 a(\() p Fo(w) p Fy 1845 3776 a(\)\).) p 1970 3776 a(This) p 2160 3776 a(is) p 2243 3776 a(p) r(ossible) p 2556 3776 a(using) p 2773 3776 a(\(66\)) p 2948 3776 a(and) p 3110 3776 a(\(67\),) p 3308 3776 a(as) p 3410 3776 a(follo) n(ws.) 125 3875 y(Giv) n(en) p Fo 365 3875 a(w) p Fy 450 3875 a(=) p 537 3875 a(\() p Fo(w) p Fy 630 3875 a(\(0\)) p Fo(;) p 773 3875 a(w) p Fy 834 3875 a(\(1\)) p Fo(;) p Fk 977 3875 a(\001) p 1014 3875 a(\001) p 1051 3875 a(\001) p Fo 1090 3875 a(;) p 1127 3875 a(w) p Fy 1188 3875 a(\() p Fo(L) p Fy(\() p Fo(w) p Fy 1370 3875 a(\)\)\)) p Fk 1491 3875 a(2) p Fo 1569 3875 a(W) p Fj 1659 3845 a(\(0\)) p Fy 1776 3875 a(with) p 1965 3875 a(the) p 2108 3875 a(prop) r(ert) n(y) p Fo 1033 4139 a(w) p Fy 1094 4139 a(\() p Fo(L) p Fy(\() p Fo(w) p Fy 1276 4139 a(\)\)) p Fk 1365 4139 a(2) p Fo 1443 4139 a(G) p Fp 1508 4151 a(n) p Fn(\000) p Fj(1) p Fk 1657 4139 a(n) p Fp 1754 4036 a(d) p Fg 1725 4060 a([) p Fp 1717 4237 a(i) p Fj(=1) p Fo 1839 4139 a(G) p Fp 1904 4151 a(n) p Fn(\000) p Fj(1) p Fp(;i) p Fo 2091 4139 a(;) p Fy 2183 4139 a(where) p Fo 2451 4139 a(n) p Fy 2524 4139 a(=) p Fo 2612 4139 a(\027) p Fy 2658 4139 a(\() p Fo(w) p Fy 2751 4139 a(\)) p Fo(;) p Fy 3692 4139 a(\(70\)) 0 4399 y(w) n(e) p 122 4399 a(shall) p 316 4399 a(de\014ne) p 556 4399 a(a) p 625 4399 a(path) p Fo 819 4399 a(R) p Fy 883 4399 a(\() p Fo(w) p Fy 976 4399 a(\)) p 1037 4399 a(as) p 1139 4399 a(follo) n(ws.) 125 4498 y(Let) p Fo 273 4498 a(T) p Fy 334 4498 a(\() p Fo(w) p Fy 427 4498 a(\)) p 487 4498 a(b) r(e) p 600 4498 a(a) p 669 4498 a(p) r(ositiv) n(e) p 977 4498 a(in) n(teger) p 1251 4498 a(satisfying) p Fo 800 4680 a(w) p Fy 861 4680 a(\() p Fo(k) p Fy 939 4680 a(\)) p Fk 995 4680 a(2) p Fo 1073 4680 a(G) p Fp 1138 4692 a(n) p Fn(\000) p Fj(1) p Fo 1269 4680 a(;) p 1333 4680 a(T) p Fy 1394 4680 a(\() p Fo(w) p Fy 1487 4680 a(\)) p Fl 1543 4680 a(5) p Fo 1630 4680 a(k) p Fl 1699 4680 a(5) p Fo 1787 4680 a(L) p Fy(\() p Fo(w) p Fy 1937 4680 a(\)) p Fo(;) p Fy 2062 4680 a(and) p Fo 2251 4680 a(w) p Fy 2312 4680 a(\() p Fo(T) p Fy 2405 4680 a(\() p Fo(w) p Fy 2498 4680 a(\)) p Fk 2550 4680 a(\000) p Fy 2633 4680 a(1\)) p Fk 2730 4680 a(62) p Fo 2808 4680 a(G) p Fp 2873 4692 a(n) p Fn(\000) p Fj(1) p Fo 3017 4680 a(:) p Fy 0 4862 a(The) p 171 4862 a(condition) p 537 4862 a(\(70\)) p 713 4862 a(implies) p 995 4862 a(that) p 1176 4862 a(suc) n(h) p 1364 4862 a(an) p 1480 4862 a(in) n(teger) p 1756 4862 a(\(uniquely\)) p 2158 4862 a(exists.) p 2422 4862 a(Since) p Fo 2640 4862 a(w) p Fy 2701 4862 a(\() p Fo(T) p Fy 2794 4862 a(\() p Fo(w) p Fy 2887 4862 a(\)\)) p Fk 2977 4862 a(2) p Fo 3056 4862 a(G) p Fp 3121 4874 a(n) p Fn(\000) p Fj(1) p Fy 3252 4862 a(,) p Fo 3304 4862 a(w) p Fy 3365 4862 a(\() p Fo(T) p Fy 3458 4862 a(\() p Fo(w) p Fy 3551 4862 a(\)) p Fk 3603 4862 a(\000) p Fy 3686 4862 a(1\)) p Fk 3785 4862 a(2) p Fp 36 4940 a(d) p Fg 8 4965 a([) p Fp 0 5142 a(i) p Fj(=1) p Fo 121 5044 a(G) p Fp 186 5056 a(n) p Fn(\000) p Fj(1) p Fp(;i) p Fy 374 5044 a(.) p 434 5044 a(Let) p Fo 581 5044 a(i) p Fy(\() p Fo(w) p Fy 703 5044 a(\)) p Fk 759 5044 a(2) p 838 5044 a(f) p Fy(1) p Fo(;) p Fy 959 5044 a(2) p Fo(;) p Fk 1038 5044 a(\001) p 1075 5044 a(\001) p 1112 5044 a(\001) p Fo 1147 5044 a(;) p 1184 5044 a(d) p Fk(g) p Fy 1295 5044 a(b) r(e) p 1407 5044 a(suc) n(h) p 1594 5044 a(that) p Fo 1773 5044 a(w) p Fy 1834 5044 a(\() p Fo(T) p Fy 1927 5044 a(\() p Fo(w) p Fy 2020 5044 a(\)) p Fk 2070 5044 a(\000) p Fy 2151 5044 a(1\)) p Fk 2248 5044 a(2) p Fo 2326 5044 a(G) p Fp 2391 5059 a(n) p Fn(\000) p Fj(1) p Fp(;i) p Fj(\() p Fp(w) p Fj 2636 5059 a(\)) p Fy 2680 5044 a(.) p 2740 5044 a(Clearly) p 2995 5044 a(,) p 3044 5044 a(suc) n(h) p 3230 5044 a(an) p 3345 5044 a(in) n(teger) p 3619 5044 a(is) p 3701 5044 a(also) 0 5221 y(unique.) p 302 5221 a(Also) p 489 5221 a(the) p 632 5221 a(de\014nitions) p 1034 5221 a(of) p Fo 1129 5221 a(T) p Fy 1190 5221 a(\() p Fo(w) p Fy 1283 5221 a(\)) p 1343 5221 a(and) p Fo 1504 5221 a(i) p Fy(\() p Fo(w) p Fy 1626 5221 a(\)) p 1687 5221 a(imply) p Fo 1381 5403 a(w) p Fy 1442 5403 a(\() p Fo(T) p Fy 1535 5403 a(\() p Fo(w) p Fy 1628 5403 a(\)\)) p Fk 1716 5403 a(2) p Fo 1795 5403 a(G) p Fp 1860 5415 a(n) p Fn(\000) p Fj(1) p Fk 2009 5403 a(\\) p Fo 2082 5403 a(G) p Fp 2147 5418 a(n) p Fn(\000) p Fj(1) p Fp(;i) p Fj(\() p Fp(w) p Fj 2392 5418 a(\)) p Fo 2436 5403 a(:) p Fy 3692 5403 a(\(71\)) 0 5585 y(De\014ne) p Fo 257 5585 a(R) p Fy 321 5585 a(\() p Fo(w) p Fy 414 5585 a(\)) p 475 5585 a(b) n(y) p Fo 1009 5738 a(R) p Fy 1073 5738 a(\() p Fo(w) p Fy 1166 5738 a(\)\() p Fo(k) p Fy 1276 5738 a(\)) p 1333 5738 a(=) p Fg 1420 5621 a(\032) p Fo 1524 5684 a(R) p Fp 1587 5699 a(n;i) p Fj(\() p Fp(w) p Fj 1747 5699 a(\)) p Fy 1777 5684 a(\() p Fo(w) p Fy 1870 5684 a(\() p Fo(k) p Fy 1948 5684 a(\)\)) p Fo(;) p Fy 2120 5684 a(0) p Fl 2184 5684 a(5) p Fo 2272 5684 a(k) p 2341 5684 a(<) p 2429 5684 a(T) p Fy 2490 5684 a(\() p Fo(w) p Fy 2583 5684 a(\)) p Fo(;) p Fy 1542 5769 a(~) p Fo 1524 5790 a(R) p Fp 1587 5805 a(n;i) p Fj(\() p Fp(w) p Fj 1747 5805 a(\)) p Fy 1777 5790 a(\() p Fo(w) p Fy 1870 5790 a(\() p Fo(k) p Fy 1948 5790 a(\)\)) p Fo(;) p 2120 5790 a(T) p Fy 2181 5790 a(\() p Fo(w) p Fy 2274 5790 a(\)) p Fl 2329 5790 a(5) p Fo 2417 5790 a(k) p Fl 2486 5790 a(5) p Fo 2573 5790 a(L) p Fy(\() p Fo(w) p Fy 2723 5790 a(\)) p Fo(:) p Fy 1878 6120 a(29) p 90 rotate dyy eop %%Page: 30 30 30 29 bop Fz 0 83 a(Theorem) p 405 83 a(28) p Fi 542 83 a(F) p 590 83 a(or) p 703 83 a(e) l(ach) p Fo 895 83 a(w) p Fk 989 83 a(2) p Fo 1078 83 a(W) p Fj 1168 53 a(\(0\)) p Fi 1292 83 a(satisfying) p Fk 1666 83 a(j) p Fo(w) p Fy 1750 83 a(\() p Fo(L) p Fy(\() p Fo(w) p Fy 1932 83 a(\)\)) p Fk(j) p Fo 2054 83 a(<) p Fy 2152 83 a(2) p Fp 2194 49 a(\027) p Fj 2231 49 a(\() p Fp(w) p Fj 2307 49 a(\)) p Fn(\000) p Fj(1) p Fi 2421 83 a(,) p Fo 2483 83 a(R) p Fy 2547 83 a(\() p Fo(w) p Fy 2640 83 a(\)) p Fk 2705 83 a(2) p Fo 2794 83 a(W) p Fj 2884 53 a(\(0\)) p Fi 3008 83 a(\(i.e.,) p 3218 83 a(is) p 3313 83 a(a) p 3390 83 a(self-avoiding) 0 183 y(p) l(ath) p 180 183 a(starting) p 487 183 a(fr) l(om) p Fo 683 183 a(O) p Fi 748 183 a(\)) p 812 183 a(which) p 1046 183 a(satis\014es) p Fo 1358 183 a(L) p Fy(\() p Fo(R) p Fy 1511 183 a(\() p Fo(w) p Fy 1604 183 a(\)\)) p 1692 183 a(=) p Fo 1779 183 a(L) p Fy(\() p Fo(w) p Fy 1929 183 a(\)) p Fi(,) p Fo 2017 183 a(\027) p Fy 2063 183 a(\() p Fo(R) p Fy 2159 183 a(\() p Fo(w) p Fy 2252 183 a(\)\)) p 2341 183 a(=) p Fo 2429 183 a(\027) p Fy 2475 183 a(\() p Fo(w) p Fy 2568 183 a(\)) p 2620 183 a(+) p 2703 183 a(1) p Fi(,) p 2799 183 a(and) p Fy 2961 183 a(2) p Fp 3003 153 a(\027) p Fj 3040 153 a(\() p Fp(w) p Fj 3116 153 a(\)) p Fo 3168 183 a(<) p Fk 3256 183 a(j) p Fo(R) p Fy 3343 183 a(\() p Fo(w) p Fy 3436 183 a(\)\() p Fo(L) p Fy(\() p Fo(w) p Fy 3650 183 a(\)\)) p Fk(j) p Fi 3753 183 a(.) -42 365 y(Pr) l(o) l(of.) p Fy 219 365 a(The) p 393 365 a(non-trivial) p 805 365 a(part) p 989 365 a(of) p 1087 365 a(the) p 1234 365 a(claim) p 1459 365 a(is) p 1546 365 a(that) p Fo 1730 365 a(R) p Fy 1794 365 a(\() p Fo(w) p Fy 1887 365 a(\)) p 1951 365 a(is) p 2038 365 a(a) p 2111 365 a(self-a) n(v) n(oiding) p 2590 365 a(path.) p 2828 365 a(All) p 2967 365 a(the) p 3114 365 a(other) p 3335 365 a(prop) r(erties) p 3729 365 a(are) 0 465 y(ob) n(vious) p 300 465 a(from) p 497 465 a(the) p 640 465 a(de\014nition) p 1008 465 a(of) p 1103 465 a(the) p 1246 465 a(re\015ections.) 125 565 y(W) p 203 565 a(e) p 270 565 a(\014rst) p 445 565 a(pro) n(v) n(e) p 672 565 a(that) p Fo 854 565 a(R) p Fy 918 565 a(\() p Fo(w) p Fy 1011 565 a(\)) p 1075 565 a(is) p 1161 565 a(self-a) n(v) n(oiding,) p 1664 565 a(when) p Fo 1883 565 a(w) p Fy 1975 565 a(is) p 2062 565 a(self-a) n(v) n(oiding.) p 2578 565 a(Let) p Fo 2730 565 a(n) p Fy 2808 565 a(=) p Fo 2900 565 a(\027) p Fy 2946 565 a(\() p Fo(w) p Fy 3039 565 a(\).) p 3142 565 a(Note) p 3345 565 a(that) p 3528 565 a(\(71\)) p 3706 565 a(and) 0 664 y(Prop) r(osition) p 447 664 a(27) p 558 664 a(\(iii\)) p 719 664 a(imply) p Fo 1223 764 a(R) p Fp 1286 779 a(n;i) p Fj(\() p Fp(w) p Fj 1446 779 a(\)) p Fy 1476 764 a(\() p Fo(w) p Fy 1569 764 a(\() p Fo(T) p Fy 1662 764 a(\() p Fo(w) p Fy 1755 764 a(\)\)\)) p 1876 764 a(=) 1982 743 y(~) p Fo 1964 764 a(R) p Fp 2027 779 a(n;i) p Fj(\() p Fp(w) p Fj 2187 779 a(\)) p Fy 2217 764 a(\() p Fo(w) p Fy 2310 764 a(\() p Fo(T) p Fy 2403 764 a(\() p Fo(w) p Fy 2496 764 a(\)\)\)) p Fo(:) p Fy 3692 764 a(\(72\)) 0 928 y(Since) p Fo 220 928 a(R) p Fp 283 940 a(n;i) p Fy 402 928 a(and) 585 907 y(~) p Fo 567 928 a(R) p Fp 630 940 a(n;i) p Fy 749 928 a(are) p 890 928 a(1) p 960 928 a(:) p 1011 928 a(1) p 1084 928 a(maps) p 1304 928 a(\(Prop) r(osition) p 1786 928 a(27\),) p 1956 928 a(the) p 2102 928 a(path) p 2299 928 a(segmen) n(ts) p Fo 2656 928 a(w) p Fj 2715 940 a(1) p Fy 2781 928 a(=) p Fk 2873 928 a(f) p Fo(R) p Fy 2979 928 a(\() p Fo(w) p Fy 3072 928 a(\)\() p Fo(k) p Fy 3182 928 a(\)) p Fo(;) p Fy 3283 928 a(0) p Fl 3353 928 a(5) p Fo 3445 928 a(k) p Fl 3519 928 a(5) p Fo 3612 928 a(T) p Fy 3673 928 a(\() p Fo(w) p Fy 3766 928 a(\)) p Fk(g) p Fy 0 1027 a(and) p Fo 161 1027 a(w) p Fj 220 1039 a(2) p Fy 281 1027 a(=) p Fk 369 1027 a(f) p Fo(R) p Fy 475 1027 a(\() p Fo(w) p Fy 568 1027 a(\)\() p Fo(k) p Fy 678 1027 a(\)) p Fo(;) p 775 1027 a(T) p Fy 836 1027 a(\() p Fo(w) p Fy 929 1027 a(\)) p Fl 985 1027 a(5) p Fo 1072 1027 a(k) p Fl 1141 1027 a(5) p Fo 1229 1027 a(L) p Fy(\() p Fo(w) p Fy 1379 1027 a(\)) p Fk(g) p Fy 1481 1027 a(are) p 1619 1027 a(self-a) n(v) n(oiding.) 125 1127 y(The) p 307 1127 a(de\014nition) p 688 1127 a(of) p Fo 795 1127 a(T) p Fy 856 1127 a(\() p Fo(w) p Fy 949 1127 a(\)) p 1021 1127 a(implies) p Fo 1314 1127 a(w) p Fy 1375 1127 a(\() p Fo(k) p Fy 1453 1127 a(\)) p Fk 1530 1127 a(2) p Fo 1628 1127 a(G) p Fp 1693 1139 a(n) p Fn(\000) p Fj(1) p Fy 1823 1127 a(,) p Fo 1889 1127 a(T) p Fy 1950 1127 a(\() p Fo(w) p Fy 2043 1127 a(\)) p Fl 2118 1127 a(5) p Fo 2226 1127 a(k) p Fl 2315 1127 a(5) p Fo 2423 1127 a(L) p Fy(\() p Fo(w) p Fy 2573 1127 a(\),) p 2671 1127 a(whic) n(h,) p 2947 1127 a(with) p 3148 1127 a(Prop) r(osition) p 3607 1127 a(27) p 3729 1127 a(\(i\),) 0 1227 y(implies) 305 1206 y(~) p Fo 287 1227 a(R) p Fp 350 1239 a(n;i) p Fy 438 1227 a(\() p Fo(w) p Fy 531 1227 a(\() p Fo(k) p Fy 609 1227 a(\)\)) p Fk 706 1227 a(2) p Fo 793 1227 a(\023) p Fn 822 1191 a(\000) p Fj(1) p Fp 822 1250 a(n;i) p Fy 912 1227 a(\() p Fo(G) p Fp 1009 1239 a(n) p Fn(\000) p Fj(1) p Fy 1139 1227 a(\),) p 1229 1227 a(hence,) p 1489 1227 a(in) p 1591 1227 a(particular,) p Fo 2003 1227 a(w) p Fj 2062 1239 a(2) p Fk 2122 1227 a(\\) p Fo 2199 1227 a(G) p Fp 2264 1239 a(n) p Fy 2341 1227 a(=) p Fk 2437 1227 a(f) p Fo(R) p Fp 2542 1239 a(n;i) p Fy 2630 1227 a(\() p Fo(w) p Fy 2723 1227 a(\() p Fo(T) p Fy 2816 1227 a(\() p Fo(w) p Fy 2909 1227 a(\)\)\)) p Fk(g) p Fy(.) p 3124 1227 a(On) p 3267 1227 a(the) p 3415 1227 a(other) p 3637 1227 a(hand,) 0 1326 y(Prop) r(osition) p 447 1326 a(27) p 557 1326 a(\(iv\)) p 716 1326 a(implies) p Fo 997 1326 a(w) p Fj 1056 1338 a(1) p Fk 1117 1326 a(\032) p Fo 1205 1326 a(G) p Fp 1270 1338 a(n) p Fy 1329 1326 a(.) p 1389 1326 a(Therefore) p Fo 1765 1326 a(w) p Fj 1824 1338 a(1) p Fy 1889 1326 a(and) p Fo 2050 1326 a(w) p Fj 2109 1338 a(2) p Fy 2174 1326 a(are) p 2313 1326 a(m) n(utually) p 2663 1326 a(a) n(v) n(oiding.) p 3025 1326 a(This) p 3214 1326 a(pro) n(v) n(es) p 3470 1326 a(that) p Fo 3650 1326 a(R) p Fy 3714 1326 a(\() p Fo(w) p Fy 3807 1326 a(\)) 0 1426 y(is) p 83 1426 a(self-a) n(v) n(oiding.) 125 1526 y(W) p 203 1526 a(e) p 268 1526 a(are) p 406 1526 a(left) p 551 1526 a(with) p 741 1526 a(pro) n(ving) p 1041 1526 a(that) p Fo 1220 1526 a(R) p Fy 1284 1526 a(\() p Fo(w) p Fy 1377 1526 a(\)) p 1438 1526 a(is) p 1522 1526 a(a) p 1591 1526 a(path,) p 1807 1526 a(i.e.,) 930 1708 y(\() p Fo(R) p Fy 1026 1708 a(\() p Fo(w) p Fy 1119 1708 a(\)\() p Fo(k) p Fy 1229 1708 a(\)) p Fo(;) p 1298 1708 a(R) p Fy 1362 1708 a(\() p Fo(w) p Fy 1455 1708 a(\)\() p Fo(k) p Fy 1586 1708 a(+) p 1669 1708 a(1\)\)) p Fk 1798 1708 a(2) p Fo 1876 1708 a(B) t(;) p 2008 1708 a(k) p Fy 2077 1708 a(=) p 2165 1708 a(0) p Fo(;) p Fy 2244 1708 a(1) p Fo(;) p Fy 2323 1708 a(2) p Fo(;) p Fk 2402 1708 a(\001) p 2439 1708 a(\001) p 2476 1708 a(\001) p Fo 2511 1708 a(;) p 2548 1708 a(L) p Fy(\() p Fo(w) p Fy 2698 1708 a(\)) p Fk 2749 1708 a(\000) p Fy 2832 1708 a(1) p Fo 2888 1708 a(:) p Fy 3692 1708 a(\(73\)) 0 1891 y(De\014nition) p 382 1891 a(of) p Fo 472 1891 a(T) p Fy 533 1891 a(\() p Fo(w) p Fy 626 1891 a(\)) p 681 1891 a(and) p 838 1891 a(Prop) r(osition) p 1280 1891 a(27) p 1387 1891 a(\(ii\)) p 1520 1891 a(imply) p 1749 1891 a(\(73\)) p 1920 1891 a(for) p Fo 2042 1891 a(T) p Fy 2103 1891 a(\() p Fo(w) p Fy 2196 1891 a(\)) p Fl 2251 1891 a(5) p Fo 2339 1891 a(k) p 2408 1891 a(<) p 2496 1891 a(L) p Fy(\() p Fo(w) p Fy 2646 1891 a(\).) p 2737 1891 a(Hence) p 2979 1891 a(w) n(e) p 3097 1891 a(ma) n(y) p 3272 1891 a(assume) p 3554 1891 a(0) p Fl 3619 1891 a(5) p Fo 3707 1891 a(k) p 3775 1891 a(<) 0 1991 y(T) p Fy 61 1991 a(\() p Fo(w) p Fy 154 1991 a(\).) p 256 1991 a(De\014nition) p 645 1991 a(of) p Fo 743 1991 a(T) p Fy 804 1991 a(\() p Fo(w) p Fy 897 1991 a(\)) p 960 1991 a(and) p 1125 1991 a(\(72\)) p 1304 1991 a(imply) p Fo 1540 1991 a(R) p Fy 1604 1991 a(\() p Fo(w) p Fy 1697 1991 a(\)\() p Fo(k) p Fy 1807 1991 a(\)) p 1869 1991 a(=) p Fo 1962 1991 a(R) p Fp 2025 2006 a(n;i) p Fj(\() p Fp(w) p Fj 2185 2006 a(\)) p Fy 2215 1991 a(\() p Fo(w) p Fy 2308 1991 a(\() p Fo(k) p Fy 2386 1991 a(\)\),) p 2506 1991 a(0) p Fl 2576 1991 a(5) p Fo 2669 1991 a(k) p Fl 2743 1991 a(5) p Fo 2836 1991 a(T) p Fy 2897 1991 a(\() p Fo(w) p Fy 2990 1991 a(\).) p 3092 1991 a(If) p Fo 3179 1991 a(w) p Fy 3240 1991 a(\() p Fo(k) p Fy 3318 1991 a(\)) p Fk 3379 1991 a(62) p Fo 3463 1991 a(G) p Fp 3528 2006 a(n) p Fn(\000) p Fj(1) p Fp(;i) p Fj(\() p Fp(w) p Fj 3773 2006 a(\)) p Fy 3817 1991 a(,) 0 2090 y(then) p 181 2090 a(\() p Fo(w) p Fy 274 2090 a(\() p Fo(k) p Fy 352 2090 a(\)) p Fo(;) p 421 2090 a(w) p Fy 482 2090 a(\() p Fo(k) p Fy 561 2090 a(+) q(1\)\)) p Fk 758 2090 a(2) p Fo 837 2090 a(B) p Fy 923 2090 a(and) p Fo 1076 2090 a(n) p Fy 1149 2090 a(=) p Fo 1236 2090 a(\027) p Fy 1282 2090 a(\() p Fo(w) p Fy 1375 2090 a(\)) p 1428 2090 a(and) p 1581 2090 a(\(71\)) p 1748 2090 a(with) p 1928 2090 a(self-a) n(v) n(oiding) p 2395 2090 a(prop) r(ert) n(y) p 2726 2090 a(of) p Fo 2812 2090 a(w) p Fy 2893 2090 a(imply) p Fo 3117 2090 a(w) p Fy 3178 2090 a(\() p Fo(k) p Fy 3257 2090 a(+) q(1\)) p Fk 3421 2090 a(62) p Fo 3500 2090 a(G) p Fp 3565 2105 a(n) p Fn(\000) p Fj(1) p Fp(;i) p Fj(\() p Fp(w) p Fj 3810 2105 a(\)) p Fy 0 2215 a(or) p Fo 105 2215 a(w) p Fy 166 2215 a(\() p Fo(k) p Fy 266 2215 a(+) p 352 2215 a(1\)) p Fk 455 2215 a(2) p Fo 540 2215 a(G) p Fp 605 2230 a(n) p Fn(\000) p Fj(1) p Fp(;i) p Fj(\() p Fp(w) p Fj 850 2230 a(\)) p Fk 901 2215 a(\\) p Fg 1061 2136 a([) p Fp 977 2318 a(j) p Fn 1007 2318 a(6) p Fj(=0) p Fp(;i) p Fj(\() p Fp(w) p Fj 1210 2318 a(\)) p Fo 1251 2215 a(G) p Fp 1316 2227 a(n) p Fn(\000) p Fj(1) p Fp(;j) p Fy 1510 2215 a(.) p 1582 2215 a(The) p 1756 2215 a(de\014nition) p 2129 2215 a(of) p Fo 2227 2215 a(R) p Fp 2290 2230 a(n;i) p Fj(\() p Fp(w) p Fj 2450 2230 a(\)) p Fy 2512 2215 a(and) p 2677 2215 a(Prop) r(osition) p 3128 2215 a(27\(v\)) p 3350 2215 a(imply) p 3587 2215 a(that) p 3771 2215 a(in) 0 2412 y(either) p 238 2412 a(case) p Fo 415 2412 a(R) p Fp 478 2427 a(n;i) p Fj(\() p Fp(w) p Fj 638 2427 a(\)) p Fy 668 2412 a(\() p Fo(w) p Fy 761 2412 a(\() p Fo(k) p Fy 860 2412 a(+) p 945 2412 a(1\)\)) p 1077 2412 a(=) p Fo 1168 2412 a(w) p Fy 1229 2412 a(\() p Fo(k) p Fy 1328 2412 a(+) p 1413 2412 a(1\).) p 1553 2412 a(Also) p Fo 1742 2412 a(w) p Fy 1803 2412 a(\() p Fo(k) p Fy 1881 2412 a(\)) p Fk 1941 2412 a(62) p Fo 2023 2412 a(G) p Fp 2088 2427 a(n) p Fn(\000) p Fj(1) p Fp(;i) p Fj(\() p Fp(w) p Fj 2333 2427 a(\)) p Fy 2392 2412 a(implies) p Fo 2676 2412 a(R) p Fp 2739 2427 a(n;i) p Fj(\() p Fp(w) p Fj 2899 2427 a(\)) p Fy 2929 2412 a(\() p Fo(w) p Fy 3022 2412 a(\() p Fo(k) p Fy 3100 2412 a(\)\)) p 3192 2412 a(=) p Fo 3284 2412 a(w) p Fy 3345 2412 a(\() p Fo(k) p Fy 3423 2412 a(\).) p 3522 2412 a(Hence) p 3771 2412 a(in) 0 2511 y(this) p 162 2511 a(case) p 338 2511 a(\(73\)) p 513 2511 a(holds.) 125 2611 y(The) p 306 2611 a(remaining) p 704 2611 a(case) p 890 2611 a(is) p 983 2611 a(when) p 1211 2611 a(0) p Fl 1292 2611 a(5) p Fo 1397 2611 a(k) p 1483 2611 a(<) p 1588 2611 a(T) p Fy 1649 2611 a(\() p Fo(w) p Fy 1742 2611 a(\)) p 1813 2611 a(and) p Fo 1984 2611 a(w) p Fy 2045 2611 a(\() p Fo(k) p Fy 2123 2611 a(\)) p Fk 2197 2611 a(2) p Fo 2293 2611 a(G) p Fp 2358 2626 a(n) p Fn(\000) p Fj(1) p Fp(;i) p Fj(\() p Fp(w) p Fj 2603 2626 a(\)) p Fy 2671 2611 a(hold.) p 2919 2611 a(If) p Fo 3012 2611 a(w) p Fy 3073 2611 a(\() p Fo(k) p Fy 3177 2611 a(+) p 3267 2611 a(1\)) p Fk 3381 2611 a(62) p Fo 3477 2611 a(G) p Fp 3542 2626 a(n) p Fn(\000) p Fj(1) p Fp(;i) p Fj(\() p Fp(w) p Fj 3787 2626 a(\)) p Fy 3817 2611 a(,) 0 2711 y(then) p 194 2711 a(a) p 269 2711 a(similar) p 546 2711 a(reasoning) p 921 2711 a(as) p 1028 2711 a(ab) r(o) n(v) n(e) p 1269 2711 a(applies.) p 1594 2711 a(Therefore) p 1975 2711 a(w) n(e) p 2102 2711 a(ma) n(y) p 2287 2711 a(also) p 2459 2711 a(assume) p Fo 2751 2711 a(w) p Fy 2812 2711 a(\() p Fo(k) p Fy 2913 2711 a(+) p 2999 2711 a(1\)) p Fk 3105 2711 a(2) p Fo 3192 2711 a(G) p Fp 3257 2726 a(n) p Fn(\000) p Fj(1) p Fp(;i) p Fj(\() p Fp(w) p Fj 3502 2726 a(\)) p Fy 3546 2711 a(.) p 3621 2711 a(Hence) 0 2810 y(\() p Fo(w) p Fy 93 2810 a(\() p Fo(k) p Fy 171 2810 a(\)) p Fo(;) p 240 2810 a(w) p Fy 301 2810 a(\() p Fo(k) p Fy 400 2810 a(+) p 483 2810 a(1\)\)) p Fk 612 2810 a(2) p Fo 690 2810 a(B) p Fp 753 2825 a(n) p Fn(\000) p Fj(1) p Fp(;i) p Fj(\() p Fp(w) p Fj 998 2825 a(\)) p Fy 1028 2810 a(,) p 1079 2810 a(in) p 1176 2810 a(whic) n(h) p 1413 2810 a(case) p 1589 2810 a(Prop) r(osition) p 2036 2810 a(27) p 2146 2810 a(\(vi\)) p 2306 2810 a(applies) p 2583 2810 a(and) p 2744 2810 a(w) n(e) p 2866 2810 a(ha) n(v) n(e) p 3058 2810 a(\(73\).) p Fd 3778 2910 a(2) p Fz 0 3275 a(Corollary) p 425 3275 a(29) p Fi 562 3275 a(If) p Fo 649 3275 a(s) p Fl 711 3275 a(=) p Fy 799 3275 a(0) p Fi 870 3275 a(and) p Fo 1031 3275 a(k) p Fk 1100 3275 a(2) p Fq 1178 3275 a(N) p Fi(,) p Fo 688 3458 a(E) p Fp 749 3470 a(k) p Fy 790 3458 a([2) p Fp 855 3423 a(s) p Fj(\() p Fp(\027) p Fj 949 3423 a(\() p Fp(w) p Fj 1025 3423 a(\)) p Fn(\000) p Fj(1\)) p Fy 1165 3458 a(;) p Fk 1232 3458 a(j) p Fo(w) p Fy 1316 3458 a(\() p Fo(k) p Fy 1394 3458 a(\)) p Fk(j) p Fl 1473 3458 a(5) p Fy 1561 3458 a(2) p Fp 1603 3423 a(\027) p Fj 1640 3423 a(\() p Fp(w) p Fj 1716 3423 a(\)) p Fn(\000) p Fj(1) p Fy 1830 3458 a(]) p Fl 1876 3458 a(5) p Fo 1964 3458 a(E) p Fp 2025 3470 a(k) p Fy 2066 3458 a([2) p Fp 2131 3423 a(s) p Fj(\() p Fp(\027) p Fj 2225 3423 a(\() p Fp(w) p Fj 2301 3423 a(\)) p Fn(\000) p Fj(1\)) p Fy 2442 3458 a(;) p Fk 2508 3458 a(j) p Fo(w) p Fy 2592 3458 a(\() p Fo(k) p Fy 2670 3458 a(\)) p Fk(j) p Fl 2749 3458 a(=) p Fy 2837 3458 a(2) p Fp 2879 3423 a(\027) p Fj 2916 3423 a(\() p Fp(w) p Fj 2992 3423 a(\)) p Fn(\000) p Fj(1) p Fy 3106 3458 a(]) p Fo(;) p Fi 0 3640 a(wher) l(e) p Fo 234 3640 a(E) p Fp 295 3652 a(k) p Fy 337 3640 a([) p Fk(\001) p Fy(]) p Fi 435 3640 a(is) p 525 3640 a(the) p 662 3640 a(exp) l(e) l(ctation) p 1091 3640 a(de\014ne) l(d) p 1371 3640 a(in) p 1473 3640 a(The) l(or) l(em) p 1818 3640 a(11.) -42 3823 y(Pr) l(o) l(of.) p Fy 219 3823 a(Recalling) p 580 3823 a(the) p 723 3823 a(de\014nition) p 1092 3823 a(of) p 1186 3823 a(the) p 1329 3823 a(measure) 1671 3802 y(~) p Fo 1653 3823 a(P) p Fp 1706 3835 a(k) p Fy 1761 3823 a(,) p 1811 3823 a(w) n(e) p 1934 3823 a(\014nd) p 2100 3823 a(from) p 2296 3823 a(Theorem) p 2647 3823 a(28,) p Fo 429 4068 a(E) p Fp 490 4080 a(k) p Fy 532 4068 a([2) p Fp 597 4033 a(s) p Fj(\() p Fp(\027) p Fj 691 4033 a(\() p Fp(w) p Fj 767 4033 a(\)) p Fn(\000) p Fj(1\)) p Fy 907 4068 a(;) p Fk 972 4068 a(j) p 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1074 4310 a(\() p Fo(w) p Fy 1167 4310 a(\)) p 1224 4310 a(=) p Fo 1311 4310 a(n) p Fy 1380 4310 a(+) p 1463 4310 a(1) p Fo(;) p Fk 1569 4310 a(j) p Fo(w) p Fy 1653 4310 a(\() p Fo(k) p Fy 1731 4310 a(\)) p Fk(j) p Fo 1810 4310 a(>) p Fy 1898 4310 a(2) p Fp 1940 4276 a(n) p Fy 1984 4310 a(]) p 2030 4310 a(=) p 2118 4310 a(2) p Fn 2160 4276 a(\000) p Fp(s) p Fo 2247 4310 a(E) p Fp 2308 4322 a(k) p Fy 2349 4310 a([2) p Fp 2414 4276 a(s) p Fj(\() p Fp(\027) p Fj 2508 4276 a(\() p Fp(w) p Fj 2584 4276 a(\)) p Fn(\000) p Fj(1\)) p Fy 2725 4310 a(;) p Fk 2789 4310 a(j) p Fo(w) p Fy 2873 4310 a(\() p Fo(k) p Fy 2951 4310 a(\)) p Fk(j) p Fo 3031 4310 a(>) p Fy 3118 4310 a(2) p Fp 3160 4276 a(\027) p Fj 3197 4276 a(\() p Fp(w) p Fj 3273 4276 a(\)) p Fn(\000) p Fj(1) p Fy 3387 4310 a(]) p Fl 429 4493 a(5) p Fo 517 4493 a(E) p Fp 578 4505 a(k) p Fy 619 4493 a([2) p Fp 684 4459 a(s) p Fj(\() p Fp(\027) p Fj 778 4459 a(\() p Fp(w) p Fj 854 4459 a(\)) p Fn(\000) p Fj(1\)) p Fy 995 4493 a(;) p Fk 1059 4493 a(j) p Fo(w) p Fy 1143 4493 a(\() p Fo(k) p Fy 1221 4493 a(\)) p Fk(j) p Fo 1301 4493 a(>) p Fy 1388 4493 a(2) p Fp 1430 4459 a(\027) p Fj 1467 4459 a(\() p Fp(w) p Fj 1543 4459 a(\)) p Fn(\000) p Fj(1) p Fy 1658 4493 a(]) p Fo(:) p Fy 0 4671 a(Hence) p Fo 69 4840 a(E) p Fp 130 4852 a(k) p Fy 171 4840 a([2) p Fp 236 4806 a(s) p Fj(\() p Fp(\027) p Fj 330 4806 a(\() p Fp(w) p Fj 406 4806 a(\)) p Fn(\000) p Fj(1\)) p Fy 546 4840 a(;) p Fk 611 4840 a(j) p Fo(w) p Fy 695 4840 a(\() p Fo(k) p Fy 773 4840 a(\)) p Fk(j) p Fl 852 4840 a(5) p Fy 940 4840 a(2) p Fp 982 4806 a(\027) p Fj 1019 4806 a(\() p Fp(w) p Fj 1095 4806 a(\)) p Fn(\000) p Fj(1) p Fy 1209 4840 a(]) p 1255 4840 a(=) p Fo 1343 4840 a(E) p Fp 1404 4852 a(k) p Fy 1445 4840 a([2) p Fp 1510 4806 a(s) p Fj(\() p Fp(\027) p Fj 1604 4806 a(\() p Fp(w) p Fj 1680 4806 a(\)) p Fn(\000) p Fj(1\)) p Fy 1821 4840 a(;) p Fk 1885 4840 a(j) p Fo(w) p Fy 1969 4840 a(\() p Fo(k) p Fy 2047 4840 a(\)) p Fk(j) p Fy 2126 4840 a(=) p 2214 4840 a(2) p Fp 2256 4806 a(\027) p Fj 2293 4806 a(\() p Fp(w) p Fj 2369 4806 a(\)) p Fn(\000) p Fj(1) p Fy 2483 4840 a(]) p 2525 4840 a(+) p Fo 2608 4840 a(E) p Fp 2669 4852 a(k) p Fy 2710 4840 a([2) p Fp 2775 4806 a(s) p Fj(\() p Fp(\027) p Fj 2869 4806 a(\() p Fp(w) p Fj 2945 4806 a(\)) p Fn(\000) p Fj(1\)) p Fy 3086 4840 a(;) p Fk 3150 4840 a(j) p Fo(w) p Fy 3234 4840 a(\() p Fo(k) p Fy 3312 4840 a(\)) p Fk(j) p Fo 3391 4840 a(<) p Fy 3479 4840 a(2) p Fp 3521 4806 a(\027) p Fj 3558 4806 a(\() p Fp(w) p Fj 3634 4806 a(\)) p Fn(\000) p Fj(1) p Fy 3748 4840 a(]) p Fl 69 4948 a(5) p Fo 156 4948 a(E) p Fp 217 4960 a(k) p Fy 258 4948 a([2) p Fp 323 4914 a(s) p Fj(\() p Fp(\027) p Fj 417 4914 a(\() p Fp(w) p Fj 493 4914 a(\)) p Fn(\000) p Fj(1\)) p Fy 634 4948 a(;) p Fk 699 4948 a(j) p Fo(w) p Fy 783 4948 a(\() p Fo(k) p Fy 861 4948 a(\)) p Fk(j) p Fl 940 4948 a(=) p Fy 1027 4948 a(2) p Fp 1069 4914 a(\027) p Fj 1106 4914 a(\() p Fp(w) p Fj 1182 4914 a(\)) p Fn(\000) p Fj(1) p Fy 1297 4948 a(]) p Fo(:) p Fd 3778 5106 a(2) p Fi -42 5471 a(Pr) l(o) l(of) p 181 5471 a(of) p 279 5471 a(The) l(or) l(em) p 624 5471 a(11.) p Fy 776 5471 a(Note) p 976 5471 a(an) p 1092 5471 a(ob) n(vious) p 1392 5471 a(b) r(ound) p Fk 1238 5654 a(j) p Fo(w) p Fy 1322 5654 a(\() p Fo(L) p Fy(\() p Fo(w) p Fy 1504 5654 a(\)\)) p Fk(j) p Fl 1617 5654 a(5) p Fk 1704 5654 a(k) p Fo(w) p Fk 1807 5654 a(k) p Fl 1872 5654 a(5) p Fy 1960 5654 a(2) p Fp 2002 5620 a(\027) p Fj 2039 5620 a(\() p Fp(w) p Fj 2115 5620 a(\)) p Fo 2144 5654 a(;) p 2236 5654 a(w) p Fk 2321 5654 a(2) p Fo 2399 5654 a(W) p Fj 2489 5620 a(\(0\)) p Fo 2578 5654 a(:) p Fy 1878 6120 a(30) p 90 rotate dyy eop %%Page: 31 31 31 30 bop Fy 0 83 a(This) p 190 83 a(and) p 351 83 a(Corollary) p 718 83 a(29) p 829 83 a(imply) 255 262 y(2) p Fn 297 228 a(\000) p Fp(s) p Fn(\000) p Fj(1) p Fo 469 262 a(E) p Fp 530 274 a(k) p Fy 571 262 a([) p Fk(k) p Fo 635 262 a(w) p Fk 696 262 a(k) p Fp 738 220 a(s) p Fy 774 262 a(]) p Fl 255 402 a(5) p Fy 342 402 a(2) p Fn 384 367 a(\000) p Fp(s) p Fn(\000) p Fj(1) p Fo 556 402 a(E) p Fp 617 414 a(k) p Fy 658 402 a([2) p Fp 723 367 a(s\027) p Fj 791 367 a(\() p Fp(w) p Fj 867 367 a(\)) p Fy 897 402 a(]) p 943 402 a(=) 1041 346 y(1) p 1041 383 42 4 v 1041 459 a(2) 1092 402 y(\() p Fo(E) p Fp 1185 414 a(k) p Fy 1227 402 a([2) p Fp 1292 367 a(s) p Fj(\() p Fp(\027) p Fj 1386 367 a(\() p Fp(w) p Fj 1462 367 a(\)) p Fn(\000) p Fj(1\)) p Fy 1602 402 a(;) p Fk 1667 402 a(j) p Fo(w) p Fy 1751 402 a(\() p Fo(k) p Fy 1829 402 a(\)) p Fk(j) p Fl 1908 402 a(5) p Fy 1996 402 a(2) p Fp 2038 367 a(\027) p Fj 2075 367 a(\() p Fp(w) p Fj 2151 367 a(\)) p Fn(\000) p Fj(1) p Fy 2265 402 a(]) p 2306 402 a(+) p Fo 2389 402 a(E) p Fp 2450 414 a(k) p Fy 2492 402 a([2) p Fp 2557 367 a(s) p Fj(\() p Fp(\027) p Fj 2651 367 a(\() p Fp(w) p Fj 2727 367 a(\)) p Fn(\000) p Fj(1\)) p Fy 2867 402 a(;) p Fk 2932 402 a(j) p Fo(w) p Fy 3016 402 a(\() p Fo(k) p Fy 3094 402 a(\)) p Fk(j) p Fo 3173 402 a(>) p Fy 3261 402 a(2) p 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4175 a(I) p 1198 4175 a(;i) p Fi 1260 4163 a(,) p Fo 1320 4163 a(I) p Fy 1394 4163 a(=) p 1490 4163 a(\(1\)) p Fo(;) p Fy 1633 4163 a(\(11\)) p Fi(,) p Fo 1841 4163 a(i) p Fy 1900 4163 a(=) p 1996 4163 a(0) p Fo(;) p Fy 2075 4163 a(1) p Fo(;) p Fy 2154 4163 a(2) p Fo(;) p Fy 2233 4163 a(3) p Fo(;) p Fy 2312 4163 a(4) p Fi(,) p 2411 4163 a(of) p 2513 4163 a(p) l(ositive) p 2817 4163 a(c) l(o) l(e\016cients,) p 3271 4163 a(such) p 3464 4163 a(that) p Fy 3638 4163 a(\010) p Fj 3698 4178 a(\(1\)) p Fp(;) p Fj(0) p Fi 208 4262 a(c) l(ontains) p 536 4262 a(a) p 609 4262 a(term) p Fo 807 4262 a(x) p Fj 854 4232 a(2) 854 4289 y(\(1\)) p Fi 944 4262 a(,) p Fy 999 4262 a(\010) p Fj 1059 4277 a(\(11\)) p Fp(;) p Fj(0) p Fi 1263 4262 a(c) l(ontains) p 1592 4262 a(a) p 1664 4262 a(term) p Fo 1863 4262 a(x) p Fj 1910 4232 a(4) 1910 4289 y(\(1\)) p Fi 1999 4262 a(,) p 2054 4262 a(and) p Fy 342 4465 a(\010) p Fj 402 4480 a(\(1\)) p Fy 491 4465 a(\() p Fo 522 4465 a(~) p 523 4465 a(x) p Fy 571 4465 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p 208 83 a(The) p 375 83 a(explicit) p 665 83 a(forms) p 891 83 a(\(74\)) p 1064 83 a(of) p 1155 83 a(\010) p Fp 1215 95 a(I) p Fy 1253 83 a(\() p Fo(x;) p 1369 83 a(y) s(;) p Fy 1450 83 a(0) p Fo(;) p Fy 1529 83 a(0) p Fo(;) p Fy 1608 83 a(0) p Fo(;) p Fy 1687 83 a(0\),) p Fo 1808 83 a(I) p Fy 1874 83 a(=) p 1961 83 a(\(1\)) p Fo(;) p Fy 2104 83 a(\(1) p Fo(;) p Fy 2215 83 a(1\),) p 2337 83 a(are) p 2473 83 a(obtained) p 2811 83 a(b) n(y) p 2924 83 a(explicit) p 3214 83 a(coun) n(ting) p 3550 83 a(of) p 3641 83 a(paths) 208 192 y(in) p 313 192 a(the) p 465 192 a(righ) n(t) p 675 192 a(hand) p 891 192 a(side) p 1066 192 a(of) p 1170 192 a(the) p 1322 192 a(de\014nition) p 1699 192 a(\(9\)) p 1842 192 a(for) p Fo 1981 171 a(~) p Fy 1978 192 a(\010) p 2075 192 a(=) p Fo 2195 171 a(~) 2178 192 y(X) p Fj 2247 204 a(1) p Fy 2298 192 a(,) p 2359 192 a(with) p 2557 192 a(aid) p 2704 192 a(of) p 2808 192 a(computer) p 3186 192 a(calculations.) p 3697 192 a(The) 208 292 y(explicit) p 500 292 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1893 a(to) p 1408 1893 a(\001) p Fj 1477 1908 a(\(3\)) p Fy 1581 1893 a(,) p 1629 1893 a(then) p Fo 1816 1893 a(w) p Fy 1903 1893 a(has) p 2049 1893 a(a) p 2116 1893 a(form) p 2310 1847 359 4 v Fo 2310 1893 a(v) p Fp 2350 1905 a(a) p Fo 2390 1893 a(v) p Fj 2430 1905 a(1) p Fp(;) p Fj(1) p Fo 2520 1893 a(v) p Fp 2560 1905 a(c) p Fo 2594 1893 a(v) p Fp 2634 1905 a(b) p Fy 2693 1893 a(or) p 2793 1847 V Fo 2793 1893 a(v) p Fp 2833 1905 a(a) p Fo 2873 1893 a(v) p Fp 2913 1905 a(c) p Fo 2947 1893 a(v) p Fj 2987 1905 a(1) p Fp(;) p Fj(1) p Fo 3077 1893 a(v) p Fp 3117 1905 a(b) p Fy 3176 1893 a(in) p 3271 1893 a(the) p 3411 1893 a(simplex.) p 3745 1893 a(T) p 3798 1893 a(o) 208 2017 y(this) p Fo 368 2017 a(w) p Fy 455 2017 a(assign) p 699 2017 a(a) p 766 2017 a(path) p Fo 958 2017 a(w) p Fn 1019 1987 a(0) p Fy 1069 2017 a(obtained,) p 1432 2017 a(as) p 1532 2017 a(b) r(efore,) p 1803 2017 a(b) n(y) p 1916 2017 a(shortcutting) p Fo 2386 2017 a(v) p Fj 2426 2029 a(1) p Fp(;) p Fj(1) p Fy 2530 2017 a(,) p 2579 2017 a(to) p 2679 2017 a(obtain) p 2935 1971 229 4 v Fo 2935 2017 a(v) p Fp 2975 2029 a(a) p Fo 3016 2017 a(v) p Fp 3056 2029 a(c) p Fo 3090 2017 a(v) p Fp 3130 2029 a(b) p Fy 3177 2017 a(.) p 3236 2017 a(Then) p Fo 3451 2017 a(w) p Fn 3512 1987 a(0) p Fk 3559 2017 a(2) p Fo 3638 2017 a(W) p Fj 3728 1974 a(\(1\)) 3716 2045 y(\(1\)) p Fy 3817 2017 a(,) 208 2126 y(and) p 365 2126 a(con) n(tributes) p 792 2126 a(a) p 857 2126 a(term) p 1052 2126 a(to) p 1149 2126 a(\010) p Fj 1209 2141 a(\(1\)) p Fy 1321 2126 a(similar) p 1590 2126 a(to) p 1687 2126 a(the) p 1826 2126 a(con) n(tribution) p 2294 2126 a(of) p Fo 2385 2126 a(w) p Fy 2470 2126 a(to) p 2567 2126 a(\010) p Fj 2627 2141 a(\(2\)) p Fy 2739 2126 a(but) p 2887 2126 a(with) p Fo 3072 2126 a(x) p Fj 3119 2141 a(\(3\)) p Fy 3232 2126 a(replaced) p 3555 2126 a(b) n(y) p Fo 3667 2126 a(x) p Fj 3714 2141 a(\(2\)) p Fy 3817 2126 a(.) 208 2273 y(This) p 397 2273 a(corresp) r(ondence) p 977 2273 a(is) p 1060 2273 a(2) p 1130 2273 a(to) p 1231 2273 a(1,) p 1323 2273 a(whic) n(h) p 1561 2273 a(explains) p 1882 2273 a(the) p 2025 2273 a(factor) 2273 2217 y(1) p 2273 2254 42 4 v 2273 2330 a(2) 2352 2273 y(in) p 2449 2273 a(the) p 2592 2273 a(expression) p 2992 2273 a(of) p 3086 2273 a(\010) p Fj 3146 2288 a(\(1\)) p Fy 3263 2273 a(in) p 3360 2273 a(\(75\).) 208 2433 y(The) p 378 2433 a(other) p 595 2433 a(p) r(ossibilities) p 1042 2433 a(\(4\)) p 1175 2433 a(and) p 1337 2433 a(\(12\)) p 1512 2433 a(are) p 1651 2433 a(similar.) 208 2565 y(That) p Fo 419 2565 a(x) p Fj 466 2535 a(2) 466 2592 y(\(1\)) p Fy 587 2565 a(is) p 674 2565 a(con) n(tained) p 1053 2565 a(in) p 1154 2565 a(\010) p Fj 1214 2580 a(\(1\)) p Fy 1303 2565 a(\() p Fo 1334 2565 a(~) p 1335 2565 a(x) p Fy(\)) p 1446 2565 a(is) p 1533 2565 a(pro) n(v) n(ed) p 1807 2565 a(in) p 1907 2565 a(Prop) r(osition) p 2358 2565 a(4.) p 2470 2565 a(This) p 2663 2565 a(term) p 2865 2565 a(can) p 3021 2565 a(not) p 3172 2565 a(b) r(e) p 3289 2565 a(obtained) p 3633 2565 a(b) n(y) p 3752 2565 a(an) 208 2677 y(ab) r(o) n(v) n(e) p 443 2677 a(men) n(tioned) p 846 2677 a(`shortcutting') p 1364 2677 a(pro) r(cedure) p 1749 2677 a(on) p 1865 2677 a(paths,) p 2114 2677 a(so) p 2216 2677 a(it) p 2299 2677 a(m) n(ust) p 2505 2677 a(b) r(e) p 2618 2677 a(con) n(tained) p 2994 2677 a(in) p 3090 2677 a(\010) p Fj 3150 2692 a(\(1\)) p Fp(;) p Fj(0) p Fy 3292 2677 a(\() p Fo 3323 2677 a(~) p 3324 2677 a(x) p Fy 3372 2677 a(\).) 208 2809 y(The) p 378 2809 a(expression) p 778 2809 a(for) p 905 2809 a(\010) p Fj 965 2824 a(\(11\)) p Fy 1115 2809 a(follo) n(ws) p 1388 2809 a(in) p 1485 2809 a(a) p 1554 2809 a(similar) p 1827 2809 a(w) n(a) n(y) p 1962 2809 a(.) 35 2998 y(\(iv\)) p Fo 208 2998 a(W) p Fj 298 2955 a(\(1\)) 286 3027 y(\(1\)) p Fy 406 2998 a(con) n(tains) p 723 2998 a(paths) p 941 2926 595 4 v Fo 941 2998 a(O) r(v) p Fj 1046 3010 a(1) p Fy 1084 2998 a(\() p Fo(v) p Fj 1156 3010 a(1) p Fy 1212 2998 a(+) p Fo 1295 2998 a(v) p Fj 1335 3010 a(2) p Fy 1373 2998 a(\)) p Fo(v) p Fj 1445 3010 a(1) p Fp(;) p Fj(1) p Fy 1549 2998 a(,) p 1593 2926 916 4 v Fo 1593 2998 a(O) r(v) p Fj 1698 3010 a(1) p Fy 1736 2998 a(\() p Fo(v) p Fj 1808 3010 a(1) p Fy 1865 2998 a(+) p Fo 1948 2998 a(v) p Fj 1988 3010 a(2) p Fy 2025 2998 a(\)\() p Fo(v) p Fj 2129 3010 a(1) p Fy 2186 2998 a(+) p Fo 2269 2998 a(v) p Fj 2309 3010 a(3) p Fy 2346 2998 a(\)) p Fo(v) p Fj 2418 3010 a(1) p Fp(;) p Fj(1) p Fy 2523 2998 a(,) p 2567 2926 1237 4 v Fo 2567 2998 a(O) r(v) p Fj 2672 3010 a(1) p Fy 2710 2998 a(\() p Fo(v) p Fj 2782 3010 a(1) p Fy 2838 2998 a(+) p Fo 2921 2998 a(v) p Fj 2961 3010 a(2) p Fy 2998 2998 a(\)\() p Fo(v) p Fj 3102 3010 a(1) p Fy 3159 2998 a(+) p Fo 3242 2998 a(v) p Fj 3282 3010 a(3) p Fy 3319 2998 a(\)\() p Fo(v) p Fj 3423 3010 a(1) p Fy 3480 2998 a(+) p Fo 3563 2998 a(v) p Fj 3603 3010 a(4) p Fy 3641 2998 a(\)) p Fo(v) p Fj 3713 3010 a(1) p Fp(;) p Fj(1) p Fy 3817 2998 a(,) 208 3128 y(and) p 367 3056 827 4 v Fo 367 3128 a(O) r(v) p Fj 472 3140 a(2) p Fy 510 3128 a(\() p Fo(v) p Fj 582 3140 a(2) p Fy 638 3128 a(+) p Fo 721 3128 a(v) p Fj 761 3140 a(3) p Fy 798 3128 a(\)) p Fo(v) p Fj 870 3140 a(3) p Fo 908 3128 a(v) p Fj 948 3140 a(4) p Fo 986 3128 a(v) p Fj 1026 3140 a(1) p Fo 1063 3128 a(v) p Fj 1103 3140 a(1) p Fp(;) p Fj(1) p Fy 1207 3128 a(,) p 1256 3128 a(whic) n(h) p 1491 3128 a(resp) r(ectiv) n(ely) p 1942 3128 a(con) n(tribute) p 2339 3128 a(terms) p Fo 2567 3128 a(x) p Fj 2614 3143 a(\(1\)) p Fo 2704 3128 a(x) p Fj 2751 3143 a(\() p Fp(i) p Fj(\)) p Fy 2845 3128 a(,) p Fo 2894 3128 a(i) p Fy 2945 3128 a(=) p 3033 3128 a(2) p Fo(;) p Fy 3112 3128 a(3) p Fo(;) p Fy 3191 3128 a(4,) p 3280 3128 a(and) p Fo 3439 3128 a(x) p Fj 3486 3098 a(3) 3486 3155 y(\(1\)) p Fo 3576 3128 a(x) p Fj 3623 3143 a(\(12\)) p Fy 3771 3128 a(in) 208 3228 y(\010) p Fj 268 3243 a(\(1\)) p Fy 370 3228 a(.) 208 3360 y(Similarly) p 527 3360 a(,) p 577 3360 a(paths) p 803 3360 a(sets) 1167 3463 y(\() p 1199 3397 274 4 v Fo(O) r(v) p Fj 1304 3475 a(1) p Fo 1343 3463 a(v) p Fj 1383 3475 a(1) p Fp(;) p Fj(1) p Fo 1487 3463 a(;) p 1524 3391 903 4 v 1524 3463 a(v) p Fj 1564 3475 a(1) p Fp(;) p Fj(2) p Fy 1654 3463 a(\() p Fo(v) p Fj 1726 3475 a(3) p Fy 1782 3463 a(+) p Fo 1865 3463 a(v) p Fj 1905 3475 a(2) p Fy 1943 3463 a(\)\() p Fo(v) p Fj 2047 3475 a(3) p Fy 2103 3463 a(+) p Fo 2186 3463 a(v) p Fj 2226 3475 a(1) p Fy 2264 3463 a(\)) p Fo(v) p Fj 2336 3475 a(1) p Fp(;) p Fj(3) p Fy 2426 3463 a(\)) p Fo(;) p Fy 1167 3572 a(\() p 1199 3505 274 4 v Fo(O) r(v) p Fj 1304 3584 a(1) p Fo 1343 3572 a(v) p Fj 1383 3584 a(1) p Fp(;) p Fj(1) p Fo 1487 3572 a(;) p 1524 3500 1224 4 v 1524 3572 a(v) p Fj 1564 3584 a(1) p Fp(;) p Fj(2) p Fy 1654 3572 a(\() p Fo(v) p Fj 1726 3584 a(3) p Fy 1782 3572 a(+) p Fo 1865 3572 a(v) p Fj 1905 3584 a(2) p Fy 1943 3572 a(\)\() p Fo(v) p Fj 2047 3584 a(3) p Fy 2103 3572 a(+) p Fo 2186 3572 a(v) p Fj 2226 3584 a(1) p Fy 2264 3572 a(\)\() p Fo(v) p Fj 2368 3584 a(3) p Fy 2424 3572 a(+) p Fo 2507 3572 a(v) p Fj 2547 3584 a(4) p Fy 2585 3572 a(\)) p Fo(v) p Fj 2657 3584 a(1) p Fp(;) p Fj(3) p Fy 2747 3572 a(\)) p Fo(;) p Fy 1167 3681 a(\() p 1199 3614 274 4 v Fo(O) r(v) p Fj 1304 3693 a(1) p Fo 1343 3681 a(v) p Fj 1383 3693 a(1) p Fp(;) p Fj(1) p Fo 1487 3681 a(;) p 1524 3609 1302 4 v 1524 3681 a(v) p Fj 1564 3693 a(1) p Fp(;) p Fj(2) p Fy 1654 3681 a(\() p Fo(v) p Fj 1726 3693 a(3) p Fy 1782 3681 a(+) p Fo 1865 3681 a(v) p Fj 1905 3693 a(2) p Fy 1943 3681 a(\)\() p Fo(v) p Fj 2047 3693 a(3) p Fy 2103 3681 a(+) p Fo 2186 3681 a(v) p Fj 2226 3693 a(1) p Fy 2264 3681 a(\)\() p Fo(v) p Fj 2368 3693 a(3) p Fy 2424 3681 a(+) p Fo 2507 3681 a(v) p Fj 2547 3693 a(4) p Fy 2585 3681 a(\)) p Fo(v) p Fj 2657 3693 a(3) p Fo 2695 3681 a(v) p Fj 2735 3693 a(1) p Fp(;) p Fj(3) p Fy 2825 3681 a(\)) p Fo(;) p Fy 1167 3784 a(\() p 1199 3717 274 4 v Fo(O) r(v) p Fj 1304 3796 a(1) p Fo 1343 3784 a(v) p Fj 1383 3796 a(1) p Fp(;) p Fj(1) p Fo 1487 3784 a(;) p 1524 3738 494 4 v 1524 3784 a(v) p Fj 1564 3796 a(1) p Fp(;) p Fj(2) p Fo 1654 3784 a(v) p Fj 1694 3796 a(2) p Fo 1732 3784 a(v) p Fj 1772 3796 a(4) p Fo 1809 3784 a(v) p Fj 1849 3796 a(3) p Fo 1887 3784 a(v) p Fj 1927 3796 a(1) p Fp(;) p Fj(3) p Fy 2017 3784 a(\)) p Fo(;) p Fy 208 3937 a(resp) r(ectiv) n(ely) p 660 3937 a(con) n(tribute) p 1059 3937 a(terms) p Fo 1291 3937 a(x) p Fj 1338 3907 a(3) 1338 3964 y(\(1\)) p Fo 1427 3937 a(x) p Fj 1474 3952 a(\() p Fp(i) p Fj(\)) p Fy 1568 3937 a(,) p Fo 1619 3937 a(i) p Fy 1671 3937 a(=) p 1758 3937 a(2) p Fo(;) p Fy 1837 3937 a(3) p Fo(;) p Fy 1916 3937 a(4,) p 2007 3937 a(and) p Fo 2169 3937 a(x) p Fj 2216 3907 a(3) 2216 3964 y(\(1\)) p Fo 2305 3937 a(x) p Fj 2352 3952 a(\(12\)) p Fy 2503 3937 a(in) p 2599 3937 a(\010) p Fj 2659 3952 a(\(11\)) p Fy 2796 3937 a(.) 58 4114 y(\(v\)) p 208 4114 a(The) p 378 4114 a(\014rst) p 548 4114 a(part) p 728 4114 a(follo) n(ws) p 999 4114 a(from) p 1195 4114 a(\(75\)) p 1370 4114 a(and) p 1530 4114 a(similar) p 1802 4114 a(expressions) p 2235 4114 a(for) p 2361 4114 a(\010) p Fj 2421 4129 a(\(3\)) p Fy 2510 4114 a(,) p 2560 4114 a(\010) p Fj 2620 4129 a(\(4\)) p Fy 2723 4114 a(.) p 2783 4114 a(The) p 2953 4114 a(second) p 3220 4114 a(part) p 3400 4114 a(follo) n(ws) p 3671 4114 a(from) 208 4213 y(an) p 327 4213 a(observ) p 557 4213 a(ation) p 773 4213 a(similar) p 1050 4213 a(to) p 1156 4213 a(that) p 1340 4213 a(used) p 1534 4213 a(to) p 1639 4213 a(pro) n(v) n(e) p 1868 4213 a(\(75\).) p 2088 4213 a(That) p 2300 4213 a(\010) p Fj 2360 4228 a(\(2\)) p Fy 2481 4213 a(con) n(tains) p 2811 4213 a(a) p 2884 4213 a(term) p Fo 3087 4213 a(x) p Fj 3134 4183 a(3) 3134 4240 y(\(1\)) p Fo 3224 4213 a(x) p Fj 3271 4228 a(\(11\)) p Fo 3393 4213 a(x) p Fj 3440 4228 a(\(12\)) p Fy 3595 4213 a(follo) n(ws) 208 4313 y(from) p 404 4313 a(an) p 519 4313 a(existence) p 872 4313 a(of) p 967 4313 a(a) p 1036 4313 a(path) p 1244 4419 1522 4 v Fo 1244 4491 a(O) r(v) p Fj 1349 4503 a(3) p Fy 1388 4491 a(\() p Fo(v) p Fj 1460 4503 a(1) p Fy 1516 4491 a(+) p Fo 1599 4491 a(v) p Fj 1639 4503 a(3) p Fy 1676 4491 a(\)\() p Fo(v) p Fj 1780 4503 a(1) p Fy 1837 4491 a(+) p Fo 1920 4491 a(v) p Fj 1960 4503 a(4) p Fy 1997 4491 a(\)) p Fo(v) p Fj 2069 4503 a(4) p Fo 2107 4491 a(v) p Fj 2147 4503 a(1) p Fo 2185 4491 a(v) p Fj 2225 4503 a(1) p Fp(;) p Fj(1) p Fy 2315 4491 a(\() p Fo(v) p Fj 2387 4503 a(1) p Fy 2443 4491 a(+) p Fo 2526 4491 a(v) p Fj 2566 4503 a(2) p Fy 2604 4491 a(\)) p Fo(v) p Fj 2676 4503 a(1) p Fp(;) p Fj(2) p Fo 2780 4491 a(;) p Fy 208 4670 a(and) p 369 4670 a(\010) p Fj 429 4685 a(\(12\)) p Fy 579 4670 a(con) n(tains) p 904 4670 a(a) p 974 4670 a(term) p Fo 1172 4670 a(x) p Fj 1219 4640 a(4) 1219 4696 y(\(1\)) p Fo 1309 4670 a(x) p Fj 1356 4685 a(\(2\)) p Fy 1473 4670 a(b) r(ecause) p 1780 4670 a(of) p 1875 4670 a(a) p 1944 4670 a(path) 1301 4877 y(\() p 1333 4810 274 4 v Fo(O) r(v) p Fj 1438 4889 a(1) p Fo 1477 4877 a(v) p Fj 1517 4889 a(1) p Fp(;) p Fj(1) p Fo 1621 4877 a(;) p 1658 4805 1033 4 v 1658 4877 a(v) p Fj 1698 4889 a(1) p Fp(;) p Fj(2) p Fy 1788 4877 a(\() p Fo(v) p Fj 1860 4889 a(2) p Fy 1916 4877 a(+) p Fo 1999 4877 a(v) p Fj 2039 4889 a(3) p Fy 2077 4877 a(\)) p Fo(v) p Fj 2149 4889 a(1) p Fp(;) p Fj(3) p Fy 2239 4877 a(\() p Fo(v) p Fj 2311 4889 a(3) p Fy 2368 4877 a(+) p Fo 2451 4877 a(v) p Fj 2491 4889 a(4) p Fy 2528 4877 a(\)) p Fo(v) p Fj 2600 4889 a(1) p Fp(;) p Fj(4) p Fy 2691 4877 a(\)) p Fo(:) p Fd 3778 5055 a(2) p Fz 0 5409 a(Theorem) p 405 5409 a(31) p Fi 615 5409 a(\(i\)) p Fy 750 5409 a(\010) p Fj 810 5424 a(\(1\)) p Fy 899 5409 a(\() p Fo(x;) p 1015 5409 a(y) s(;) p Fy 1096 5409 a(0) p Fo(;) p Fy 1175 5409 a(0) p Fo(;) p Fy 1254 5409 a(0) p Fo(;) p Fy 1333 5409 a(0\)) p 1452 5409 a(=) p Fo 1564 5409 a(x) p Fi(,) p 1683 5409 a(and) p Fy 1858 5409 a(\010) p Fj 1918 5424 a(\(11\)) p Fy 2040 5409 a(\() p Fo(x;) p 2156 5409 a(y) s(;) p Fy 2237 5409 a(0) p Fo(;) p Fy 2316 5409 a(0) p Fo(;) p Fy 2395 5409 a(0) p Fo(;) p Fy 2474 5409 a(0\)) p 2593 5409 a(=) p Fo 2705 5409 a(y) p Fi 2792 5409 a(has) p 2954 5409 a(a) p 3039 5409 a(unique) p 3320 5409 a(solution) p Fo 3646 5409 a(~) p 3647 5409 a(x) p Fp 3694 5421 a(c) p Fy 3775 5409 a(=) 208 5524 y(\() p Fo(x) p Fp 287 5536 a(c) p Fo 321 5524 a(;) p 358 5524 a(y) p Fp 399 5536 a(c) p Fo 433 5524 a(;) p Fy 470 5524 a(0) p Fo(;) p Fy 549 5524 a(0) p Fo(;) p Fy 628 5524 a(0) p Fo(;) p Fy 707 5524 a(0\)) p Fi 808 5524 a(in) p Fk 910 5524 a(f) p Fy(\() p Fo(x;) p 1068 5524 a(y) s(;) p Fy 1149 5524 a(0) p Fo(;) p Fy 1228 5524 a(0) p Fo(;) p Fy 1307 5524 a(0) p Fo(;) p Fy 1386 5524 a(0\)) p Fk 1481 5524 a(2) p Fq 1559 5524 a(R) p Fn 1619 5487 a(K) p Fm 1670 5495 a(r) q(es) p Fj 1619 5544 a(+) p Fk 1786 5524 a(j) p Fy 1833 5524 a(\() p Fo(x;) p 1949 5524 a(y) s(;) p Fy 2030 5524 a(0) p Fo(;) p Fy 2109 5524 a(0) p Fo(;) p Fy 2188 5524 a(0) p Fo(;) p Fy 2267 5524 a(0\)) p Fk 2362 5524 a(2) p Fy 2440 5524 a(\004) p Fj 2495 5536 a(4) p Fk 2551 5524 a(n) p 2611 5524 a(f) p Fo 2647 5506 a(~) p Fy 2653 5524 a(0) p Fk 2694 5524 a(gg) p Fi(.) p Fo 208 5656 a(x) p Fp 255 5668 a(c) p Fy 312 5656 a(=) p 400 5656 a(0) p Fo(:) p Fy(326490898) p Fk 857 5656 a(\001) p 894 5656 a(\001) p 931 5656 a(\001) p Fi 978 5656 a(and) p Fo 1139 5656 a(y) p Fp 1180 5668 a(c) p Fy 1237 5656 a(=) p 1325 5656 a(0) p Fo(:) p Fy(027929572) p Fk 1782 5656 a(\001) p 1819 5656 a(\001) p 1856 5656 a(\001) p Fi 1903 5656 a(ar) l(e) p 2044 5656 a(p) l(ositive.) p 2377 5656 a(\(In) p 2520 5656 a(p) l(articular,) p Fk 2925 5656 a(K) p Fp 2988 5668 a(r) r(es) p Fy 3111 5656 a(=) p Fk 3198 5656 a(K) p Fp 3260 5669 a(~) p 3261 5669 a(x) p Fm 3299 5677 a(c) p Fi 3349 5656 a(.\)) 47 5820 y(\(ii\)) p Fo 206 5820 a(~) p 208 5820 a(x) p Fp 255 5832 a(c) p Fi 319 5820 a(is) p 408 5820 a(a) p 480 5820 a(self-avoiding) p 959 5820 a(\014xe) l(d) p 1151 5820 a(p) l(oint,) p 1386 5820 a(i.e.,) p 1556 5820 a(satis\014es) p 1868 5820 a(\(FP1\)) p 2118 5820 a({) p 2190 5820 a(\(FP4\).) p Fy 1878 6120 a(32) p 90 rotate dyy eop %%Page: 33 33 33 32 bop Fi 22 83 a(\(iii\)) p 208 83 a(Ther) l(e) p 446 83 a(exists) p 674 83 a(a) p 746 83 a(critic) l(al) p 1025 83 a(p) l(oint) p 1235 83 a(of) p 1332 83 a(the) p 1470 83 a(r) l(estricte) l(d) p 1833 83 a(mo) l(del) p Fo 2071 83 a(\014) p Fp 2118 95 a(c;r) r(es) p Fi 2281 83 a(.) -42 300 y(Pr) l(o) l(of.) p Fy 297 300 a(\(i\)) p 426 300 a(Since) p 643 300 a(\(74\)) p 818 300 a(implies) p Fo 1100 300 a(x) p 1171 300 a(>) p 1258 300 a(x) p Fj 1305 270 a(2) p Fy 1361 300 a(+) p 1444 300 a(3) p Fo(x) p Fj 1533 270 a(3) p Fy 1589 300 a(+) p 1672 300 a(6) p Fo(x) p Fj 1761 270 a(4) p Fy 1817 300 a(+) p 1900 300 a(6) p Fo(x) p Fj 1989 270 a(5) p Fy 2026 300 a(,) p 2077 300 a(it) p 2160 300 a(follo) n(ws) p 2432 300 a(that) p 2612 300 a(0) p Fo 2677 300 a(<) p 2764 300 a(x) p 2835 300 a(<) p Fy 2932 244 a(3) p 2932 281 42 4 v 2932 357 a(8) 2998 300 y(.) 208 459 y(Eliminating) p Fo 661 459 a(y) p Fy 733 459 a(from) p 930 459 a(\010) p Fj 990 474 a(\(1\)) p Fy 1079 459 a(\() p Fo(x;) p 1195 459 a(y) s(;) p Fy 1276 459 a(0) p Fo(;) p Fy 1355 459 a(0) p Fo(;) p Fy 1434 459 a(0) p Fo(;) p Fy 1513 459 a(0\)) p 1609 459 a(=) p Fo 1697 459 a(x) p Fy 1773 459 a(and) p 1935 459 a(\010) p Fj 1995 474 a(\(11\)) p Fy 2117 459 a(\() p Fo(x;) p 2233 459 a(y) s(;) p Fy 2314 459 a(0) p Fo(;) p Fy 2393 459 a(0) p Fo(;) p Fy 2472 459 a(0) p Fo(;) p Fy 2551 459 a(0\)) p 2647 459 a(=) p Fo 2736 459 a(y) p Fy 2808 459 a(\(in) p 2937 459 a(a) p 3007 459 a(similar) p 3280 459 a(spirit) p 3498 459 a(with) p 3688 459 a(that) 208 559 y(of) p 304 559 a(the) p 450 559 a(pro) r(of) p 669 559 a(in) p 768 559 a([6,) p 886 559 a(Prop) r(osition) p 1336 559 a(3.1],) p 1519 559 a(but) p 1673 559 a(with) p 1865 559 a(length) n(y) p 2162 559 a(calculations\),) p 2673 559 a(w) n(e) p 2797 559 a(obtain) p 3058 559 a(an) p 3176 559 a(algebraic) p 3529 559 a(equation) p Fo 208 659 a(g) p Fy 251 659 a(\() p Fo(x) p Fy(\)) p 385 659 a(=) p 473 659 a(0,) p 565 659 a(where) p Fo 251 833 a(g) p Fy 294 833 a(\() p Fo(x) p Fy(\)) p 489 833 a(=) p Fk 636 833 a(\000) p Fy(3162456) p 1010 833 a(+) p 1093 833 a(3162456) p Fo(x) p Fy 1449 833 a(+) p 1532 833 a(27935028) p Fo(x) p Fj 1915 803 a(2) p Fy 1968 833 a(+) p 2051 833 a(82351390) p Fo(x) p Fj 2434 803 a(3) p Fy 2486 833 a(+) p 2569 833 a(534340195) p Fo(x) p Fj 2994 803 a(4) p Fk 3046 833 a(\000) p Fy 3129 833 a(22712313853) p Fo(x) p Fj 3638 803 a(5) p Fk 655 935 a(\000) p Fy 738 935 a(22749190609) p Fo(x) p Fj 1247 900 a(6) p Fy 1298 935 a(+) p 1381 935 a(173488539516) p Fo(x) p Fj 1932 900 a(7) p Fy 1982 935 a(+) p 2065 935 a(520491536505) p Fo(x) p Fj 2616 900 a(8) p Fy 2666 935 a(+) p 2749 935 a(159919155293) p Fo(x) p Fj 3300 900 a(9) p Fk 655 1036 a(\000) p Fy 738 1036 a(1067593750255) p Fo 1283 1036 a(x) p Fj 1330 1002 a(10) p Fk 1414 1036 a(\000) p Fy 1497 1036 a(3355567112768) p Fo 2042 1036 a(x) p Fj 2089 1002 a(11) p Fk 2173 1036 a(\000) p Fy 2256 1036 a(7117707818273) p Fo 2801 1036 a(x) p Fj 2848 1002 a(12) p Fk 2931 1036 a(\000) p Fy 3014 1036 a(8049744033921) p Fo(x) p Fj 3607 1002 a(13) p Fy 655 1138 a(+) p 738 1138 a(3218074725393) p Fo 1283 1138 a(x) p Fj 1330 1104 a(14) p Fy 1414 1138 a(+) p 1497 1138 a(2913259733292) o(0) p Fo(x) p Fj 2131 1104 a(15) p Fy 2214 1138 a(+) p 2297 1138 a(58986824992938) p Fo 2884 1138 a(x) p Fj 2931 1104 a(16) p Fy 3014 1138 a(+) p 3098 1138 a(7477844713214) o(4) p Fo(x) p Fj 3732 1104 a(17) p Fy 655 1240 a(+) p 738 1240 a(7089721441855) o(2) p Fo(x) p Fj 1372 1205 a(18) p Fy 1455 1240 a(+) p 1538 1240 a(55063893147408) p Fo 2125 1240 a(x) p Fj 2172 1205 a(19) p Fy 2256 1240 a(+) p 2339 1240 a(3609614096514) o(0) p Fo(x) p Fj 2973 1205 a(20) p Fy 655 1341 a(+) p 738 1341 a(1966948232569) o(2) p Fo(x) p Fj 1372 1307 a(21) p Fy 1455 1341 a(+) p 1538 1341 a(7841354208804) p Fo(x) p Fj 2131 1307 a(22) p Fy 2214 1341 a(+) p 2297 1341 a(1771680351168) p Fo(x) p Fj 2890 1307 a(23) p Fy 2973 1341 a(+) p 3056 1341 a(149567809608) p Fo(x) p Fj 3607 1307 a(24) p Fo 3672 1341 a(;) p Fy 208 1517 a(and) p Fo 369 1517 a(y) p Fy 440 1517 a(is) p 524 1517 a(expressed) p 896 1517 a(b) n(y) p 1012 1517 a(a) p 1081 1517 a(rational) p 1390 1517 a(function) p 1715 1517 a(of) p Fo 1810 1517 a(x) p Fy(,) p 1908 1517 a(sa) n(y) p Fo 2051 1517 a(h) p Fy(\() p Fo(x) p Fy(\).) 208 1648 y(The) p 378 1648 a(signs) p 581 1648 a(of) p 676 1648 a(the) p 818 1648 a(co) r(e\016cien) n(ts) p 1238 1648 a(imply) p 1471 1648 a(\(b) n(y) p 1618 1648 a(an) p 1733 1648 a(argumen) n(t) p 2104 1648 a(from) p 2300 1648 a(elemen) n(tary) p 2724 1648 a(calculus) p 3038 1648 a(similar) p 3310 1648 a(to) p 3411 1648 a(those) p 3628 1648 a(in) p 3725 1648 a(the) 208 1748 y(pro) r(of) p 426 1748 a(of) p 523 1748 a([6) o(,) p 640 1748 a(Prop) r(osition) p 1089 1748 a(3.1]\)) p 1280 1748 a(that) p Fo 1462 1748 a(g) p Fn 1505 1718 a(0) p Fy 1528 1748 a(\() p Fo(x) p Fy(\)) p 1669 1748 a(c) n(hanges) p 1978 1748 a(sign) p 2151 1748 a(no) p 2268 1748 a(more) p 2478 1748 a(than) p 2674 1748 a(4) p 2745 1748 a(times) p 2968 1748 a(on) p Fo 3085 1748 a(x) p Fl 3159 1748 a(=) p Fy 3250 1748 a(0) p 3321 1748 a(\(i.e.,) p Fo 3512 1748 a(g) p Fn 3555 1718 a(0) p Fy 3578 1748 a(\() p Fo(x) p Fy(\)) p 3720 1748 a(has) 208 1848 y(no) p 327 1848 a(more) p 539 1848 a(than) p 738 1848 a(4) p 811 1848 a(p) r(ositiv) n(e) p 1123 1848 a(ro) r(ots\).) p 1411 1848 a(Since,) p 1656 1848 a(b) n(y) p 1776 1848 a(explicit) p 2074 1848 a(calculation,) p Fo 2522 1848 a(g) p Fn 2565 1818 a(0) p Fy 2588 1848 a(\(0\)) p Fo 2724 1848 a(>) p Fy 2819 1848 a(0,) p Fo 2917 1848 a(g) p Fn 2960 1818 a(0) p Fy 2983 1848 a(\(1) p Fo(=) p Fy(5\)) p Fo 3202 1848 a(<) p Fy 3297 1848 a(0,) p Fo 3395 1848 a(g) p Fn 3438 1818 a(0) p Fy 3461 1848 a(\(3) p Fo(=) p Fy(8\)) p Fo 3680 1848 a(>) p Fy 3775 1848 a(0,) p Fo 208 1947 a(g) p Fn 251 1917 a(0) p Fy 273 1947 a(\(3) p Fo(=) p Fy(7\)) p Fo 494 1947 a(<) p Fy 591 1947 a(0,) p 690 1947 a(and) p Fo 856 1947 a(g) p Fn 899 1917 a(0) p Fy 922 1947 a(\(+) p Fk(1) p Fy(\)) p Fo 1166 1947 a(>) p Fy 1263 1947 a(0,) p Fo 1362 1947 a(g) p Fn 1405 1917 a(0) p Fy 1427 1947 a(\() p Fo(x) p Fy(\)) p 1572 1947 a(c) n(hanges) p 1885 1947 a(sign) p 2061 1947 a(4) p 2136 1947 a(times) p 2363 1947 a(on) p Fo 2483 1947 a(x) p Fl 2563 1947 a(=) p Fy 2659 1947 a(0,) p 2758 1947 a(and) p 2925 1947 a(2) p 2999 1947 a(times) p 3226 1947 a(on) p 3347 1947 a(0) p Fl 3420 1947 a(5) p Fo 3517 1947 a(x) p Fl 3596 1947 a(5) p Fy 3692 1947 a(3) p Fo(=) p Fy(8.) 208 2047 y(Also) p 392 2047 a(w) n(e) p 512 2047 a(ha) n(v) n(e) p Fo 701 2047 a(g) p Fy 744 2047 a(\(0\)) p Fo 873 2047 a(<) p Fy 961 2047 a(0,) p Fo 1051 2047 a(g) p Fy 1094 2047 a(\(1) p Fo(=) p Fy(5\)) p Fo 1306 2047 a(<) p Fy 1394 2047 a(0,) p 1484 2047 a(and) p Fo 1643 2047 a(g) p Fy 1686 2047 a(\(3) p Fo(=) p Fy(8\)) p Fo 1898 2047 a(>) p Fy 1985 2047 a(0) p 2041 2047 a(.) p 2100 2047 a(In) p 2201 2047 a(particular,) p 2606 2047 a(w) n(e) p 2726 2047 a(ha) n(v) n(e) p 2915 2047 a(one) p 3065 2047 a(and) p 3224 2047 a(only) p 3404 2047 a(one) p 3554 2047 a(solution) p Fo 208 2147 a(x) p Fy 278 2147 a(=) p Fo 366 2147 a(x) p Fp 413 2159 a(c) p Fy 475 2147 a(of) p Fo 569 2147 a(g) p Fy 612 2147 a(\() p Fo(x) p Fy(\)) p 747 2147 a(=) p 834 2147 a(0) p 904 2147 a(in) p Fo 1001 2147 a(x) p Fk 1071 2147 a(2) p Fy 1149 2147 a(\(1) p Fo(=) p Fy(5) p Fo(;) p Fy 1344 2147 a(3) p Fo(=) p Fy(8\).) 208 2278 y(Again) p 450 2278 a(b) n(y) p 565 2278 a(explicit) p 858 2278 a(calculation,) p 1301 2278 a(w) n(e) p 1423 2278 a(ha) n(v) n(e) p Fo 1615 2278 a(g) p Fn 1658 2248 a(0) p Fy 1681 2278 a(\(21) p Fo(=) p Fy(200\)) p Fo 2018 2278 a(>) p Fy 2105 2278 a(0) p 2175 2278 a(and) p Fo 2336 2278 a(g) p Fn 2379 2248 a(0) p Fy 2402 2278 a(\(11) p Fo(=) p Fy(100\)) p Fo 2739 2278 a(<) p Fy 2827 2278 a(0,) p 2919 2278 a(therefore,) p 3291 2278 a(the) p 3434 2278 a(other) p 3651 2278 a(p) r(oin) n(t) 208 2378 y(that) p Fo 393 2378 a(g) p Fn 436 2347 a(0) p Fy 458 2378 a(\() p Fo(x) p Fy(\)) p 603 2378 a(c) n(hanges) p 916 2378 a(sign) p 1092 2378 a(in) p 1194 2378 a(0) p Fl 1267 2378 a(5) p Fo 1363 2378 a(x) p Fl 1442 2378 a(5) p Fy 1538 2378 a(3) p Fo(=) p Fy(8) p 1696 2378 a(is) p 1784 2378 a(in) p 1886 2378 a(the) p 2034 2378 a(in) n(terv) p 2242 2378 a(al) p 2340 2378 a(\(21) p Fo(=) p Fy(200) p Fo(;) p Fy 2661 2378 a(11) p Fo(=) p Fy(100\).) p 3015 2378 a(By) p 3150 2378 a(explicit) p 3448 2378 a(calculation) p Fo 208 2477 a(g) p Fn 251 2447 a(0) p Fy 273 2477 a(\() p Fo(x) p Fy(\)) p Fl 429 2477 a(5) p Fo 538 2477 a(g) p Fn 581 2447 a(0) p Fy 603 2477 a(\(21) p Fo(=) p Fy(200\)) p Fl 961 2477 a(5) p Fy 1069 2477 a(9) p Fk 1137 2477 a(\002) p Fy 1229 2477 a(10) p Fj 1313 2447 a(5) p Fy 1349 2477 a(,) p Fo 1415 2477 a(x) p Fk 1506 2477 a(2) p Fy 1605 2477 a([21) p Fo(=) p Fy(200) p Fo(;) p Fy 1917 2477 a(11) p Fo(=) p Fy(100],) p 2253 2477 a(and) p Fo 2427 2477 a(g) p Fy 2470 2477 a(\(21) p Fo(=) p Fy(200\)) p Fl 2827 2477 a(5) p Fk 2935 2477 a(\000) p Fy(2) p Fk 3068 2477 a(\002) p Fy 3159 2477 a(10) p Fj 3243 2447 a(6) p Fy 3279 2477 a(.) p 3377 2477 a(Therefore,) p 3792 2477 a(if) p Fo 208 2577 a(x) p Fk 278 2577 a(2) p Fy 356 2577 a([21) p Fo(=) p Fy(200) p Fo(;) p Fy 668 2577 a(11) p Fo(=) p Fy(100],) p 984 2577 a(then) p Fo 1168 2577 a(g) p Fy 1211 2577 a(\() p Fo(x) p Fy(\)) p Fl 1346 2577 a(5) p Fk 1434 2577 a(\000) p Fy(2) p Fk 1549 2577 a(\002) p Fy 1622 2577 a(10) p Fj 1706 2547 a(6) p Fy 1750 2577 a(+) p 1823 2577 a(4) p Fo(:) p Fy(5) p Fk 1938 2577 a(\002) p Fy 2011 2577 a(10) p Fj 2095 2547 a(3) p Fo 2153 2577 a(<) p Fy 2241 2577 a(0.) p 2341 2577 a(This) p 2525 2577 a(implies) p Fo 2802 2577 a(g) p Fy 2845 2577 a(\() p Fo(x) p Fy(\)) p Fo 2980 2577 a(<) p Fy 3067 2577 a(0,) p 3155 2577 a(0) p Fl 3220 2577 a(5) p Fo 3308 2577 a(x) p Fl 3378 2577 a(5) p Fy 3466 2577 a(1) p Fo(=) p Fy(5,) p 3637 2577 a(hence) p Fo 208 2676 a(x) p Fp 255 2688 a(c) p Fy 315 2676 a(obtained) p 655 2676 a(ab) r(o) n(v) n(e) p 888 2676 a(is) p 970 2676 a(the) p 1112 2676 a(unique) p 1380 2676 a(solution) p 1693 2676 a(to) p Fo 1793 2676 a(g) p Fy 1836 2676 a(\() p Fo(x) p Fy(\)) p 1970 2676 a(=) p 2058 2676 a(0) p 2126 2676 a(on) p Fo 2239 2676 a(x) p Fl 2310 2676 a(=) p Fy 2397 2676 a(0) p 2453 2676 a(.) p 2512 2676 a(With) p Fo 2725 2676 a(y) p Fp 2766 2688 a(c) p Fy 2823 2676 a(=) p Fo 2910 2676 a(h) p Fy(\() p Fo(x) p Fp 3037 2688 a(c) p Fy 3072 2676 a(\),) p 3154 2676 a(\() p Fo(x) p Fp 3233 2688 a(c) p Fo 3267 2676 a(;) p 3304 2676 a(y) p Fp 3345 2688 a(c) p Fy 3379 2676 a(\)) p 3437 2676 a(is) p 3519 2676 a(therefore) 208 2776 y(the) p 353 2776 a(unique) p 625 2776 a(solution) p 942 2776 a(to) p 1046 2776 a(\010) p Fj 1106 2791 a(\(1\)) p Fy 1195 2776 a(\() p Fo(x;) p 1311 2776 a(y) s(;) p Fy 1392 2776 a(0) p Fo(;) p Fy 1471 2776 a(0) p Fo(;) p Fy 1550 2776 a(0) p Fo(;) p Fy 1629 2776 a(0\)) p 1728 2776 a(=) p Fo 1820 2776 a(x) p Fy(,) p 1921 2776 a(and) p 2085 2776 a(\010) p Fj 2145 2791 a(\(11\)) p Fy 2267 2776 a(\() p Fo(x;) p 2383 2776 a(y) s(;) p Fy 2464 2776 a(0) p Fo(;) p Fy 2543 2776 a(0) p Fo(;) p Fy 2622 2776 a(0) p Fo(;) p Fy 2701 2776 a(0\)) p 2801 2776 a(=) p Fo 2892 2776 a(y) p Fy 2966 2776 a(on) p Fq 3084 2776 a(R) p Fj 3144 2746 a(2) 3144 2797 y(+) p Fy 3199 2776 a(.) p 3266 2776 a(It) p 3359 2776 a(is) p 3444 2776 a(easy) p 3630 2776 a(to) p 3733 2776 a(see) 208 2891 y(that) p Fo 387 2891 a(x) p Fp 434 2903 a(c) p Fy 492 2891 a(=) p 579 2891 a(0) p Fo(:) p Fy(326490898) p Fk 1036 2891 a(\001) p 1073 2891 a(\001) p 1110 2891 a(\001) p Fy 1156 2891 a(and) p Fo 1317 2891 a(y) p Fp 1358 2903 a(c) p Fy 1415 2891 a(=) p 1503 2891 a(0) p Fo(:) p Fy(027929572) p Fk 1960 2891 a(\001) p 1997 2891 a(\001) p 2034 2891 a(\001) p Fy 2051 2891 a(,) p 2102 2891 a(whic) n(h) p 2340 2891 a(pro) n(v) n(e) p 2564 2891 a(that) p 2744 2891 a(\() p Fo(x) p Fp 2823 2903 a(c) p Fo 2857 2891 a(;) p 2894 2891 a(y) p Fp 2935 2903 a(c) p Fo 2969 2891 a(;) p Fy 3006 2891 a(0) p Fo(;) p Fy 3085 2891 a(0) p Fo(;) p Fy 3164 2891 a(0) p Fo(;) p Fy 3243 2891 a(0\)) p Fk 3338 2891 a(2) p Fy 3416 2891 a(\004) p Fj 3471 2903 a(4) p Fk 3527 2891 a(n) p 3587 2891 a(f) p Fo 3623 2873 a(~) p Fy 3629 2891 a(0) p Fk 3670 2891 a(g) p Fy 3725 2891 a(.) 55 3054 y(\(ii\)) p 208 3054 a(W) p 286 3054 a(e) p 351 3054 a(pro) n(v) n(e) p 574 3054 a(eac) n(h) p 761 3054 a(of) p 856 3054 a(\(FP1\)) p 1100 3054 a({) p 1169 3054 a(\(FP4\)) p 1414 3054 a(in) p 1511 3054 a(turn.) 243 3227 y(\(a\)) p 390 3227 a(Prop) r(osition) p 831 3227 a(30) p 936 3227 a(\(v\)) p 1066 3227 a(implies) p 1341 3227 a(that) p 1515 3227 a(if) p Fo 1585 3227 a(I) p Fk 1651 3227 a(62) p 1729 3227 a(K) p Fp 1792 3239 a(r) r(es) p Fy 1915 3227 a(=) p Fk 2003 3227 a(f) p Fy(\(1\)) p Fo(;) p Fy 2188 3227 a(\(11\)) p Fk(g) p Fy 2398 3227 a(then) p 2581 3227 a(\010) p Fp 2641 3239 a(I) p Fy 2679 3227 a(\() p Fo 2710 3227 a(~) p 2711 3227 a(x) p Fp 2758 3239 a(c) p Fy 2793 3227 a(\)) p 2848 3227 a(=) p 2936 3227 a(0.) p 3035 3227 a(Therefore) p Fo 3408 3206 a(~) p Fy 3405 3227 a(\010\() p Fo 3496 3227 a(~) p 3497 3227 a(x) p Fp 3544 3239 a(c) p Fy 3579 3227 a(\)) p 3634 3227 a(=) p Fo 3721 3227 a(~) p 3722 3227 a(x) p Fp 3769 3239 a(c) p Fy 3817 3227 a(.) 238 3357 y(\(b\)) p 390 3357 a(Prop) r(osition) p 837 3357 a(30) p 948 3357 a(\(v\)) p 1084 3357 a(implies) p Fk 1452 3532 a(J) p Fj 1508 3547 a(\(3\)) p Fp(J) p Fy 1640 3532 a(\() p Fo 1671 3532 a(~) p 1672 3532 a(x) p Fp 1719 3544 a(c) p Fy 1753 3532 a(\)) p 1809 3532 a(=) p Fk 1896 3532 a(J) p Fj 1952 3547 a(\(4\)) p Fp(J) p Fy 2084 3532 a(\() p Fo 2115 3532 a(~) p 2116 3532 a(x) p Fp 2163 3544 a(c) p Fy 2197 3532 a(\)) p 2253 3532 a(=) p 2340 3532 a(0) p Fo(;) p 2502 3532 a(J) p Fk 2579 3532 a(2) p 2657 3532 a(I) p Fp 2702 3544 a(d) p Fo 2755 3532 a(;) p Fy 390 3707 a(and) p Fk 1410 3807 a(J) p Fp 1466 3822 a(I) p Fj 1500 3822 a(\(1\)) p Fy 1589 3807 a(\() p Fo 1620 3807 a(~) p 1621 3807 a(x) p Fp 1668 3819 a(c) p Fy 1703 3807 a(\)) p 1758 3807 a(=) p Fk 1846 3807 a(J) p Fp 1902 3822 a(I) p Fj 1936 3822 a(\(11\)) p Fy 2058 3807 a(\() p Fo 2089 3807 a(~) p 2090 3807 a(x) p Fp 2137 3819 a(c) p Fy 2172 3807 a(\)) p 2227 3807 a(=) p 2315 3807 a(0) p Fo(;) p 2476 3807 a(I) p Fk 2542 3807 a(62) p 2621 3807 a(K) p Fp 2684 3819 a(r) r(es) p Fo 2797 3807 a(:) p Fy 390 3951 a(Therefore,) p Fo 790 3951 a(B) p Fy 880 3951 a(=) p Fk 967 3951 a(J) p Fy 1039 3951 a(\() p Fo 1070 3951 a(~) p 1071 3951 a(x) p Fp 1118 3963 a(c) p Fy 1153 3951 a(\)) p 1213 3951 a(has) p 1361 3951 a(a) p 1430 3951 a(form) p Fo 1351 4376 a(B) p Fy 1441 4376 a(=) p Fg 1529 4059 a(0) 1529 4205 y(B) 1529 4255 y(B) 1529 4305 y(B) 1529 4355 y(B) 1529 4405 y(B) 1529 4455 y(B) 1529 4508 y(@) p Fo 1643 4126 a(p) p Fy 1768 4126 a(3) p Fo(q) p 1932 4126 a(B) p Fj 1995 4138 a(13) p Fo 2149 4126 a(B) p Fj 2212 4138 a(14) p Fo 2365 4126 a(B) p Fj 2428 4138 a(15) p Fo 2581 4126 a(B) p Fj 2644 4138 a(16) p Fo 1644 4225 a(q) p 1788 4225 a(r) p 1932 4225 a(B) p Fj 1995 4237 a(23) p Fo 2149 4225 a(B) p Fj 2212 4237 a(24) p Fo 2365 4225 a(B) p Fj 2428 4237 a(25) p Fo 2581 4225 a(B) p Fj 2644 4237 a(26) p Fy 1643 4325 a(0) p 1788 4325 a(0) p Fo 1932 4325 a(B) p Fj 1995 4337 a(33) p Fo 2149 4325 a(B) p Fj 2212 4337 a(34) p Fo 2365 4325 a(B) p Fj 2428 4337 a(35) p Fo 2581 4325 a(B) p Fj 2644 4337 a(36) p Fy 1643 4425 a(0) p 1788 4425 a(0) p 1978 4425 a(0) p 2194 4425 a(0) p 2411 4425 a(0) p 2627 4425 a(0) 1643 4524 y(0) p 1788 4524 a(0) p 1978 4524 a(0) p 2194 4524 a(0) p 2411 4524 a(0) p 2627 4524 a(0) 1643 4624 y(0) p 1788 4624 a(0) p Fo 1932 4624 a(B) p Fj 1995 4636 a(63) p Fo 2149 4624 a(B) p Fj 2212 4636 a(64) p Fo 2365 4624 a(B) p Fj 2428 4636 a(65) p Fo 2581 4624 a(B) p Fj 2644 4636 a(66) p Fg 2756 4059 a(1) 2756 4205 y(C) 2756 4255 y(C) 2756 4305 y(C) 2756 4355 y(C) 2756 4405 y(C) 2756 4455 y(C) 2756 4508 y(A) p Fo 2856 4376 a(;) p Fy 390 4800 a(where,) p 653 4800 a(b) n(y) p 769 4800 a(\(74\),) p Fo 495 4976 a(p) p Fy 560 4976 a(=) p 647 4976 a(2) p Fo(x) p Fp 736 4988 a(c) p Fy 789 4976 a(+) p 872 4976 a(9) p Fo(x) p Fj 961 4942 a(2) p Fp 961 4997 a(c) p Fy 1016 4976 a(+) p 1099 4976 a(24) p Fo(x) p Fj 1230 4942 a(3) p Fp 1230 4997 a(c) p Fy 1285 4976 a(+) p 1368 4976 a(30) p Fo(x) p Fj 1499 4942 a(4) p Fp 1499 4997 a(c) p Fy 1555 4976 a(+) p 1638 4976 a(36) p Fo(x) p Fj 1769 4942 a(2) p Fp 1769 4997 a(c) p Fo 1805 4976 a(y) p Fp 1846 4988 a(c) p Fy 1898 4976 a(+) p 1981 4976 a(120) p Fo(x) p Fj 2154 4942 a(3) p Fp 2154 4997 a(c) p Fo 2191 4976 a(y) p Fp 2232 4988 a(c) p Fy 2283 4976 a(+) p 2367 4976 a(36) p Fo(x) p Fp 2498 4988 a(c) p Fo 2531 4976 a(y) p Fj 2575 4942 a(2) p Fp 2572 4997 a(c) p Fy 2630 4976 a(+) p 2713 4976 a(234) p Fo(x) p Fj 2886 4942 a(2) p Fp 2886 4997 a(c) p Fo 2922 4976 a(y) p Fj 2966 4942 a(2) p Fp 2963 4997 a(c) p Fy 3022 4976 a(+) p 3105 4976 a(192) p Fo(x) p Fp 3278 4988 a(c) p Fo 3311 4976 a(y) p Fj 3355 4942 a(3) p Fp 3352 4997 a(c) p Fy 3410 4976 a(+) p 3493 4976 a(132) p Fo(y) p Fj 3663 4942 a(4) p Fp 3660 4997 a(c) p Fo 3712 4976 a(;) 495 5078 y(q) p Fy 558 5078 a(=) p 646 5078 a(4) p Fo(x) p Fj 735 5044 a(3) p Fp 735 5098 a(c) p Fy 790 5078 a(+) p 873 5078 a(10) p Fo(x) p Fj 1004 5044 a(4) p Fp 1004 5098 a(c) p Fy 1059 5078 a(+) p 1142 5078 a(12) p Fo(x) p Fj 1273 5044 a(2) p Fp 1273 5098 a(c) p Fo 1310 5078 a(y) p Fp 1351 5090 a(c) p Fy 1403 5078 a(+) p 1486 5078 a(52) p Fo(x) p Fj 1617 5044 a(3) p Fp 1617 5098 a(c) p Fo 1654 5078 a(y) p Fp 1695 5090 a(c) p Fy 1747 5078 a(+) p 1830 5078 a(96) p Fo(x) p Fj 1961 5044 a(2) p Fp 1961 5098 a(c) p Fo 1998 5078 a(y) p Fj 2042 5044 a(2) p Fp 2039 5098 a(c) p Fy 2097 5078 a(+) p 2180 5078 a(176) p Fo(x) p Fp 2353 5090 a(c) p Fo 2386 5078 a(y) p Fj 2430 5044 a(3) p Fp 2427 5098 a(c) p Fy 2485 5078 a(+) p 2568 5078 a(220) p Fo(y) p Fj 2738 5044 a(4) p Fp 2735 5098 a(c) p Fo 2787 5078 a(;) 495 5179 y(r) p Fy 558 5179 a(=) p 645 5179 a(4) p Fo(x) p Fj 734 5145 a(3) p Fp 734 5200 a(c) p Fy 790 5179 a(+) p 873 5179 a(13) p Fo(x) p Fj 1004 5145 a(4) p Fp 1004 5200 a(c) p Fy 1059 5179 a(+) p 1142 5179 a(64) p Fo(x) p Fj 1273 5145 a(3) p Fp 1273 5200 a(c) p Fo 1310 5179 a(y) p Fp 1351 5191 a(c) p Fy 1403 5179 a(+) p 1486 5179 a(264) p Fo(x) p Fj 1659 5145 a(2) p Fp 1659 5200 a(c) p Fo 1695 5179 a(y) p Fj 1739 5145 a(2) p Fp 1736 5200 a(c) p Fy 1794 5179 a(+) p 1877 5179 a(88) p Fo(y) p Fj 2005 5145 a(3) p Fp 2002 5200 a(c) p Fy 2060 5179 a(+) p 2143 5179 a(880) p Fo(x) p Fp 2316 5191 a(c) p Fo 2349 5179 a(y) p Fj 2393 5145 a(3) p Fp 2390 5200 a(c) p Fy 2448 5179 a(+) p 2531 5179 a(930) p Fo(y) p Fj 2701 5145 a(4) p Fp 2698 5200 a(c) p Fo 2750 5179 a(;) p Fy 390 5355 a(and) p Fo 552 5355 a(B) p Fp 615 5367 a(ij) p Fy 673 5355 a('s) p 757 5355 a(are) p 895 5355 a(non-negativ) n(e,) p 1407 5355 a(and) p 1569 5355 a(\(75\)) p 1744 5355 a(implies) p Fo 1515 5533 a(B) p Fj 1578 5545 a(33) p Fy 1671 5533 a(=) p Fk 1759 5533 a(J) p Fj 1815 5548 a(\(2\)\(2\)) p Fy 1989 5533 a(\() p Fo 2020 5533 a(~) p 2021 5533 a(x) p Fp 2068 5545 a(c) p Fy 2103 5533 a(\)) p 2158 5533 a(=) p 2246 5533 a(\010) p Fj 2306 5548 a(\(1\)) p Fp(;) p Fj(1) p Fy 2447 5533 a(\() p Fo 2478 5533 a(~) p 2479 5533 a(x) p Fp 2526 5545 a(c) p Fy 2561 5533 a(\)) p Fo(;) 1515 5632 y(B) p Fj 1578 5644 a(36) p Fy 1671 5632 a(=) p Fk 1759 5632 a(J) p Fj 1815 5647 a(\(2\)\(12\)) p Fy 2022 5632 a(\() p Fo 2053 5632 a(~) p 2054 5632 a(x) p Fp 2101 5644 a(c) p Fy 2136 5632 a(\)) p 2191 5632 a(=) p 2279 5632 a(\010) p Fj 2339 5647 a(\(1\)) p Fp(;) p Fj(4) p Fy 2481 5632 a(\() p Fo 2512 5632 a(~) p 2513 5632 a(x) p Fp 2560 5644 a(c) p Fy 2594 5632 a(\)) p Fo(;) 1515 5732 y(B) p Fj 1578 5744 a(63) p Fy 1671 5732 a(=) p Fk 1759 5732 a(J) p Fj 1815 5747 a(\(12\)\(2\)) p Fy 2022 5732 a(\() p Fo 2053 5732 a(~) p 2054 5732 a(x) p Fp 2101 5744 a(c) p Fy 2136 5732 a(\)) p 2191 5732 a(=) p 2279 5732 a(\010) p Fj 2339 5747 a(\(11\)) p Fp(;) p Fj(1) p Fy 2514 5732 a(\() p Fo 2545 5732 a(~) p 2546 5732 a(x) p Fp 2593 5744 a(c) p Fy 2627 5732 a(\)) p Fo(;) 1515 5832 y(B) p Fj 1578 5844 a(66) p Fy 1671 5832 a(=) p Fk 1759 5832 a(J) p Fj 1815 5847 a(\(12\)\(12\)) p Fy 2055 5832 a(\() p Fo 2086 5832 a(~) p 2087 5832 a(x) p Fp 2134 5844 a(c) p Fy 2169 5832 a(\)) p 2224 5832 a(=) p 2312 5832 a(\010) p Fj 2372 5847 a(\(11\)) p Fp(;) p Fj(4) p Fy 2547 5832 a(\() p Fo 2578 5832 a(~) p 2579 5832 a(x) p Fp 2626 5844 a(c) p Fy 2660 5832 a(\)) p Fo(:) p Fy 1878 6120 a(33) p 90 rotate dyy eop %%Page: 34 34 34 33 bop Fy 390 120 a(A) p 477 120 a(2) p Fk 531 120 a(\002) p Fy 608 120 a(2) p 672 120 a(matrix) p Fg 939 3 a(\022) p Fo 1042 70 a(p) p Fy 1167 70 a(3) p Fo(q) 1043 169 y(q) p 1188 169 a(r) p Fg 1290 3 a(\023) p Fy 1375 120 a(has) p 1520 120 a(eigen) n(v) p 1742 120 a(alues) p Fo 1946 120 a(\025) p Fy 2017 120 a(=) p 2105 120 a(3) p Fo(:) p Fy(17282866849) p Fk 2646 120 a(\001) p 2683 120 a(\001) p 2720 120 a(\001) p Fo 2760 120 a(>) p Fy 2848 120 a(1) p 2914 120 a(and) p Fo 3072 120 a(\025) p Fj 3120 132 a(2) p Fy 3180 120 a(=) p 3268 120 a(0) p Fo(:) p Fy(249393708) p Fk 3725 120 a(\001) p 3762 120 a(\001) p 3799 120 a(\001) p Fy 3817 120 a(.) 390 279 y(In) p 494 279 a(fact,) p 681 279 a(they) p 868 279 a(are) p 1006 279 a(the) p 1149 279 a(solutions) p 1496 279 a(of) p Fo 1591 279 a(x) p Fj 1638 245 a(2) p Fk 1694 279 a(\000) p Fy 1777 279 a(\() p Fo(p) p Fy 1869 279 a(+) p Fo 1952 279 a(r) p Fy 1991 279 a(\)) p Fo(x) p Fy 2090 279 a(+) p Fo 2173 279 a(pr) p Fk 2273 279 a(\000) p Fy 2356 279 a(3) p Fo(q) p Fj 2438 245 a(2) p Fy 2498 279 a(=) p 2586 279 a(0) p 2641 279 a(.) 390 445 y(Consider) p 733 445 a(a) p 799 445 a(2) p Fk 853 445 a(\002) p Fy 930 445 a(2) p 995 445 a(matrix) p Fo 1261 445 a(B) p Fn 1328 411 a(0) p Fy 1375 445 a(=) p Fg 1462 328 a(\022) p Fo 1565 394 a(B) p Fj 1628 406 a(33) p Fo 1781 394 a(B) p Fj 1844 406 a(36) p Fo 1565 494 a(B) p Fj 1628 506 a(63) p Fo 1781 494 a(B) p Fj 1844 506 a(66) p Fg 1956 328 a(\023) p Fy 2017 445 a(.) p 2076 445 a(Its) p 2196 445 a(c) n(haracteristic) p 2703 445 a(p) r(olynomial) p Fo 3128 445 a(f) p Fy 3178 445 a(\() p Fo(x) p Fy(\)) p 3313 445 a(=) p 3401 445 a(det\() p Fo(x) p 3609 445 a(I) p Fk 3664 445 a(\000) p Fo 3741 445 a(B) p Fy 3808 445 a(\)) 390 587 y(is) p Fo 390 786 a(f) p Fy 440 786 a(\() p Fo(x) p Fy(\)) p 575 786 a(=) p Fo 663 786 a(x) p Fj 710 752 a(2) p Fk 755 786 a(\000) p Fy 828 786 a(\() p Fo(B) p Fj 923 798 a(33) p Fy 1002 786 a(+) p Fo 1075 786 a(B) p Fj 1138 798 a(66) p Fy 1208 786 a(\)) p Fo(x) p Fy 1295 786 a(+) p 1368 786 a(\() p Fo(B) p Fj 1463 798 a(33) p Fo 1535 786 a(B) p Fj 1598 798 a(66) p Fk 1676 786 a(\000) p Fo 1749 786 a(B) p Fj 1812 798 a(36) p Fo 1883 786 a(B) p Fj 1946 798 a(63) p Fy 2016 786 a(\)) p 2071 786 a(=) p 2159 786 a(\() p Fo(x) p Fk 2246 786 a(\000) p Fy 2330 730 a(1) p 2330 767 42 4 v 2330 843 a(2) 2381 786 y(\() p Fo(B) p Fj 2476 798 a(33) p Fy 2555 786 a(+) p Fo 2628 786 a(B) p Fj 2691 798 a(66) p Fy 2761 786 a(\)\)) p Fj 2825 752 a(2) p Fk 2871 786 a(\000) p Fy 2954 730 a(1) p 2954 767 V 2954 843 a(4) 3006 786 y(\() p Fo(B) p Fj 3101 798 a(33) p Fk 3180 786 a(\000) p Fo 3253 786 a(B) p Fj 3316 798 a(66) p Fy 3386 786 a(\)) p Fj 3418 752 a(2) p Fk 3464 786 a(\000) p Fo 3537 786 a(B) p Fj 3600 798 a(36) p Fo 3670 786 a(B) p Fj 3733 798 a(63) p Fo 3817 786 a(:) p Fy 3692 914 a(\(76\)) 390 1029 y(Note) p 591 1029 a(that) p 771 1029 a(\(75\)) p 946 1029 a(implies) p Fo 850 1211 a(x) p Fp 897 1223 a(c) p Fy 955 1211 a(=) p 1042 1211 a(\010) p Fj 1102 1226 a(\(1\)) p Fy 1192 1211 a(\() p Fo 1223 1211 a(~) p 1224 1211 a(x) p Fp 1271 1223 a(c) p Fy 1305 1211 a(\)) p 1360 1211 a(=) p Fo 1448 1211 a(B) p Fj 1511 1223 a(33) p Fo 1595 1211 a(x) p Fp 1642 1223 a(c) p Fy 1695 1211 a(+) p Fo 1778 1211 a(B) p Fj 1841 1223 a(36) p Fo 1925 1211 a(y) p Fp 1966 1223 a(c) p Fy 2018 1211 a(+) p 2101 1211 a(\010) p Fj 2161 1226 a(\(1\)) p Fp(;) p Fj(0) p Fy 2303 1211 a(\() p Fo 2334 1211 a(~) p 2335 1211 a(x) p Fp 2382 1223 a(c) p Fy 2417 1211 a(\)) p Fl 2472 1211 a(=) p Fo 2560 1211 a(B) p Fj 2623 1223 a(33) p Fo 2707 1211 a(x) p Fp 2754 1223 a(c) p Fy 2806 1211 a(+) p Fo 2889 1211 a(B) p Fj 2952 1223 a(36) p Fo 3037 1211 a(y) p Fp 3078 1223 a(c) p Fy 3130 1211 a(+) p Fo 3213 1211 a(x) p Fj 3260 1177 a(2) p Fp 3260 1231 a(c) p Fo 3297 1211 a(;) 850 1310 y(y) p Fp 891 1322 a(c) p Fy 948 1310 a(=) p 1036 1310 a(\010) p Fj 1096 1325 a(\(11\)) p Fy 1218 1310 a(\() p Fo 1249 1310 a(~) p 1250 1310 a(x) p Fp 1297 1322 a(c) p Fy 1331 1310 a(\)) p 1387 1310 a(=) p Fo 1474 1310 a(B) p Fj 1537 1322 a(63) p Fo 1622 1310 a(x) p Fp 1669 1322 a(c) p Fy 1721 1310 a(+) p Fo 1804 1310 a(B) p Fj 1867 1322 a(66) p Fo 1951 1310 a(y) p Fp 1992 1322 a(c) p Fy 2044 1310 a(+) p 2127 1310 a(\010) p Fj 2187 1325 a(\(11\)) p Fp(;) p Fj(0) p Fy 2362 1310 a(\() p Fo 2393 1310 a(~) p 2394 1310 a(x) p Fp 2441 1322 a(c) p Fy 2476 1310 a(\)) p Fl 2531 1310 a(=) p Fo 2619 1310 a(B) p Fj 2682 1322 a(63) p Fo 2766 1310 a(x) p Fp 2813 1322 a(c) p Fy 2866 1310 a(+) p Fo 2949 1310 a(B) p Fj 3012 1322 a(66) p Fo 3096 1310 a(y) p Fp 3137 1322 a(c) p Fy 3189 1310 a(+) p Fo 3272 1310 a(x) p Fj 3319 1280 a(4) p Fp 3319 1331 a(c) p Fo 3357 1310 a(;) p Fy 390 1487 a(whic) n(h) p 628 1487 a(further) p 907 1487 a(imply) 523 1716 y(0) p Fl 588 1716 a(5) p Fo 676 1716 a(x) p Fp 723 1728 a(a) p Fy 786 1716 a(:=) 907 1660 y(1) p 907 1697 V 907 1773 a(2) 958 1716 y(\() p Fo(B) p Fj 1053 1728 a(33) p Fy 1142 1716 a(+) p Fo 1225 1716 a(B) p Fj 1288 1728 a(66) p Fy 1359 1716 a(\)) p Fl 1414 1716 a(5) p Fy 1512 1660 a(1) p 1512 1697 V 1512 1773 a(2) 1563 1716 y(\(\(1) p Fk 1688 1716 a(\000) p Fo 1771 1716 a(x) p Fp 1818 1728 a(c) p Fy 1852 1716 a(\)) p 1903 1716 a(+) p 1986 1716 a(\(1) p Fk 2078 1716 a(\000) p Fo 2171 1660 a(x) p Fj 2218 1629 a(4) p Fp 2218 1680 a(c) p 2171 1697 85 4 v Fo 2176 1773 a(y) p Fp 2217 1785 a(c) p Fy 2266 1716 a(\)\)) p Fo 2353 1716 a(<) p Fy 2441 1716 a(1) p Fo(;) p Fy 523 1913 a(1) p Fk 583 1913 a(\000) p Fo 666 1913 a(B) p Fj 729 1925 a(33) p Fk 818 1913 a(\000) p Fo 901 1913 a(B) p Fj 964 1925 a(66) p Fy 1053 1913 a(+) p Fo 1136 1913 a(B) p Fj 1199 1925 a(33) p Fo 1269 1913 a(B) p Fj 1332 1925 a(66) p Fy 1426 1913 a(=) p 1513 1913 a(\(1) p Fk 1605 1913 a(\000) p Fo 1688 1913 a(B) p Fj 1751 1925 a(33) p Fy 1822 1913 a(\)\(1) p Fk 1946 1913 a(\000) p Fo 2029 1913 a(B) p Fj 2092 1925 a(66) p Fy 2163 1913 a(\)) p Fl 2218 1913 a(=) p Fy 2306 1913 a(\() p Fo 2351 1856 a(y) p Fp 2392 1868 a(c) p 2348 1893 82 4 v Fo 2348 1970 a(x) p Fp 2395 1982 a(c) p Fo 2439 1913 a(B) p Fj 2502 1925 a(36) p Fy 2591 1913 a(+) p Fo 2674 1913 a(x) p Fp 2721 1925 a(c) p Fy 2755 1913 a(\)\() p Fo 2829 1856 a(x) p Fp 2876 1868 a(c) p 2830 1893 V Fo 2833 1970 a(y) p Fp 2874 1982 a(c) p Fo 2921 1913 a(B) p Fj 2984 1925 a(63) p Fy 3073 1913 a(+) p Fo 3166 1856 a(x) p Fj 3213 1826 a(4) p Fp 3213 1877 a(c) p 3166 1893 85 4 v Fo 3171 1970 a(y) p Fp 3212 1982 a(c) p Fy 3260 1913 a(\)) p Fo 3316 1913 a(>) p 3403 1913 a(B) p Fj 3466 1925 a(36) p Fo 3537 1913 a(B) p Fj 3600 1925 a(63) p Fo 3684 1913 a(:) p Fy 390 2137 a(Note) p 591 2137 a(also) p 757 2137 a(that) p 937 2137 a(Prop) r(osition) p 1384 2137 a(30) p 1495 2137 a(\(v\)) p 1631 2137 a(implies) p 1913 2137 a(that) p Fo 1588 2317 a(B) p Fj 1651 2329 a(36) p Fl 1744 2317 a(=) p Fo 1832 2317 a(x) p Fj 1879 2283 a(3) p Fp 1879 2338 a(c) p Fo 1916 2317 a(y) p Fp 1957 2329 a(c) p Fy 2074 2317 a(and) p Fo 2291 2317 a(B) p Fj 2354 2329 a(63) p Fl 2447 2317 a(=) p Fo 2535 2317 a(x) p Fj 2582 2283 a(4) p Fp 2582 2338 a(c) p Fo 2619 2317 a(:) p Fy 390 2514 a(Using) p 622 2514 a(these) p 833 2514 a(estimates) p 1197 2514 a(in) p 1292 2514 a(\(76\)) p 1465 2514 a(w) n(e) p 1586 2514 a(\014nd) p 1750 2514 a(that) p Fo 1929 2514 a(f) p Fy 1979 2514 a(\() p Fo(x) p Fy(\)) p 2116 2514 a(is) p 2198 2514 a(increasing) p 2584 2514 a(on) p Fo 2698 2514 a(x) p Fl 2769 2514 a(=) p Fo 2856 2514 a(x) p Fp 2903 2526 a(a) p Fy 2944 2514 a(,) p 2993 2514 a(symmetric) p 3396 2514 a(with) p 3583 2514 a(resp) r(ect) 390 2613 y(to) p Fo 493 2613 a(x) p Fy 565 2613 a(=) p Fo 654 2613 a(x) p Fp 701 2625 a(a) p Fy 770 2613 a(\(hence) p 1034 2613 a(in) p 1131 2613 a(particular,) p Fo 1539 2613 a(f) p Fy 1589 2613 a(\() p Fk(\000) p Fy(1\)) p Fo 1783 2613 a(>) p 1873 2613 a(f) p Fy 1923 2613 a(\() p Fk(\000) p Fy(1) p 2079 2613 a(+) p 2163 2613 a(2) p Fo(x) p Fp 2252 2625 a(a) p Fy 2292 2613 a(\)) p 2349 2613 a(=) p Fo 2438 2613 a(f) p Fy 2488 2613 a(\(1\)\),) p Fo 2678 2613 a(f) p Fy 2728 2613 a(\() p Fo(x) p Fp 2807 2625 a(a) p Fy 2847 2613 a(\)) p Fl 2904 2613 a(5) p Fk 2993 2613 a(\000) p Fo(B) p Fj 3121 2625 a(36) p Fo 3191 2613 a(B) p Fj 3254 2625 a(63) p Fl 3349 2613 a(5) p Fk 3438 2613 a(\000) p Fo(x) p Fj 3550 2579 a(7) p Fp 3550 2634 a(c) p Fo 3587 2613 a(y) p Fp 3628 2625 a(c) p Fo 3686 2613 a(<) p Fy 3775 2613 a(0,) 390 2713 y(and) p Fo 545 2713 a(f) p Fy 595 2713 a(\(1\)) p Fo 724 2713 a(>) p Fy 811 2713 a(0) p 867 2713 a(.) p 924 2713 a(Therefore) p 1294 2713 a(the) p 1430 2713 a(t) n(w) n(o) p 1581 2713 a(eigen) n(v) p 1803 2713 a(alues) p Fo 2003 2713 a(\025) p Fj 2051 2725 a(3) p Fy 2110 2713 a(and) p Fo 2265 2713 a(\025) p Fj 2313 2725 a(4) p Fy 2372 2713 a(of) p Fo 2459 2713 a(B) p Fn 2526 2683 a(0) p Fy 2571 2713 a(are) p 2703 2713 a(real,) p 2882 2713 a(distinct,) p 3200 2713 a(and) p 3355 2713 a(ha) n(v) n(e) p 3540 2713 a(absolute) 390 2812 y(v) p 429 2812 a(alues) p 637 2812 a(less) p 791 2812 a(than) p 984 2812 a(1) p 1040 2812 a(.) 390 2928 y(The) p 553 2928 a(other) p 763 2928 a(t) n(w) n(o) p 912 2928 a(eigen) n(v) p 1134 2928 a(alues) p 1334 2928 a(are) p 1465 2928 a(0,) p 1552 2928 a(for) p 1671 2928 a(whic) n(h) p 1901 2928 a(w) n(e) p 2016 2928 a(ha) n(v) n(e) p 2200 2928 a(2) p 2262 2928 a(indep) r(enden) n(t) p 2725 2928 a(left) p 2863 2928 a(eigen) p 3067 2928 a(v) n(ectors) p 3342 2928 a(\(0) p Fo(;) p Fy 3453 2928 a(0) p Fo(;) p Fy 3532 2928 a(0) p Fo(;) p Fy 3611 2928 a(1) p Fo(;) p Fy 3690 2928 a(0) p Fo(;) p Fy 3769 2928 a(0\)) 390 3028 y(and) p 552 3028 a(\(0) p Fo(;) p Fy 663 3028 a(0) p Fo(;) p Fy 742 3028 a(0) p Fo(;) p Fy 821 3028 a(0) p Fo(;) p Fy 900 3028 a(1) p Fo(;) p Fy 979 3028 a(0\).) 390 3144 y(Therefore,) p Fo 787 3144 a(B) p Fy 877 3144 a(=) p Fk 965 3144 a(J) p Fy 1036 3144 a(\() p Fo 1067 3144 a(~) p 1068 3144 a(x) p Fp 1115 3156 a(c) p Fy 1150 3144 a(\)) p 1206 3144 a(is) p 1286 3144 a(diagonalizable) p 1822 3144 a(b) n(y) p 1934 3144 a(an) p 2046 3144 a(in) n(v) n(ertible) p 2409 3144 a(matrix,) p 2699 3144 a(whose) p 2941 3144 a(eigen) n(v) p 3163 3144 a(alues) p 3366 3144 a(are) p Fo 3501 3144 a(\025) p Fy(,) p Fo 3598 3144 a(\025) p Fj 3646 3156 a(2) p Fy 3683 3144 a(,) p Fo 3731 3144 a(\025) p Fj 3779 3156 a(3) p Fy 3817 3144 a(,) p Fo 390 3243 a(\025) p Fj 438 3255 a(4) p Fy 476 3243 a(,) p 526 3243 a(0,) p 617 3243 a(0,) p 708 3243 a(and) p 869 3243 a(the) p 1010 3243 a(eigen) n(v) p 1232 3243 a(alue) p Fo 1406 3243 a(\025) p Fy 1481 3243 a(whic) n(h) p 1717 3243 a(is) p 1800 3243 a(largest) p 2067 3243 a(in) p 2162 3243 a(absolute) p 2489 3243 a(v) p 2528 3243 a(alue) p 2703 3243 a(satis\014es) p Fo 3007 3243 a(\025) p 3079 3243 a(>) p Fy 3166 3243 a(1) p 3234 3243 a(and) p 3395 3243 a(all) p 3509 3243 a(the) p 3651 3243 a(other) 390 3343 y(eigen) n(v) p 612 3343 a(alues) p 820 3343 a(ha) n(v) n(e) p 1011 3343 a(absolute) p 1339 3343 a(v) p 1378 3343 a(alues) p 1586 3343 a(strictly) p 1871 3343 a(less) p 2024 3343 a(than) p 2217 3343 a(1) p 2273 3343 a(.) 390 3483 y(Denote) p 677 3483 a(the) p 822 3483 a(left) p 970 3483 a(eigen) n(v) n(ector) p 1403 3483 a(of) p Fo 1500 3483 a(B) p Fy 1597 3483 a(for) p Fo 1726 3483 a(\025) p Fy 1804 3483 a(b) n(y) p Fo 1918 3483 a(~) p 1921 3483 a(v) p Fp 1961 3495 a(L) p Fy 2038 3483 a(=) p 2129 3483 a(\() p Fo(v) p Fp 2201 3498 a(L;) p Fj(\(1\)) p Fo 2356 3483 a(;) p 2393 3483 a(v) p Fp 2433 3498 a(L;) p Fj(\(11\)) p Fo 2621 3483 a(;) p 2658 3483 a(v) p Fp 2698 3498 a(L;) p Fj(\(2\)) p Fo 2852 3483 a(;) p 2889 3483 a(v) p Fp 2929 3498 a(L;) p Fj(\(3\)) p Fo 3084 3483 a(;) p 3121 3483 a(v) p Fp 3161 3498 a(L;) p Fj(\(4\)) p Fo 3315 3483 a(;) p 3352 3483 a(v) p Fp 3392 3498 a(L;) p Fj(\(12\)) p Fy 3580 3483 a(\)) p 3626 3483 a(.) p 3692 3483 a(It) p 3784 3483 a(is) 390 3583 y(de\014ned) p 676 3583 a(b) n(y) p Fo 1053 3659 a(v) p Fp 1093 3674 a(L;) p Fj(\(11\)) p 1053 3702 228 4 v Fo 1070 3778 a(v) p Fp 1110 3793 a(L;) p Fj(\(1\)) p Fy 1314 3721 a(=) p Fo 1412 3665 a(\025) p Fk 1478 3665 a(\000) p Fo 1561 3665 a(p) p 1412 3702 192 4 v 1487 3778 a(q) p Fy 1636 3721 a(=) 1788 3665 y(3) p Fo(q) p 1734 3702 190 4 v 1734 3778 a(\025) p Fk 1801 3778 a(\000) p Fo 1884 3778 a(r) 1947 3721 y(;) 1043 3917 y(v) p Fp 1083 3932 a(L;) p Fj(\(2\)) p Fy 1261 3917 a(=) 1479 3861 y(1) p 1359 3898 284 4 v Fo 1359 3974 a(\025) p Fk 1425 3974 a(\000) p Fo 1508 3974 a(B) p Fj 1571 3986 a(33) p Fy 1652 3917 a(\() p Fo(B) p Fj 1747 3929 a(13) p Fo 1817 3917 a(v) p Fp 1857 3932 a(L;) p Fj(\(1\)) p Fy 2030 3917 a(+) p Fo 2113 3917 a(B) p Fj 2176 3929 a(23) p Fo 2247 3917 a(v) p Fp 2287 3932 a(L;) p Fj(\(11\)) p Fy 2493 3917 a(+) p Fo 2576 3917 a(B) p Fj 2639 3929 a(63) p Fo 2709 3917 a(v) p Fp 2749 3932 a(L;) p Fj(\(12\)) p Fy 2937 3917 a(\)) p Fo 1043 4096 a(v) p Fp 1083 4111 a(L;) p Fj(\(3\)) p Fy 1261 4096 a(=) 1362 4040 y(1) p 1359 4077 49 4 v Fo 1359 4153 a(\025) p Fy 1417 4096 a(\() p Fo(B) p Fj 1512 4108 a(14) p Fo 1583 4096 a(v) p Fp 1623 4111 a(L;) p Fj(\(1\)) p Fy 1796 4096 a(+) p Fo 1879 4096 a(B) p Fj 1942 4108 a(24) p Fo 2012 4096 a(v) p Fp 2052 4111 a(L;) p Fj(\(11\)) p Fy 2258 4096 a(+) p Fo 2341 4096 a(B) p Fj 2404 4108 a(34) p Fo 2474 4096 a(v) p Fp 2514 4111 a(L;) p Fj(\(2\)) p Fy 2688 4096 a(+) p Fo 2771 4096 a(B) p Fj 2834 4108 a(64) p Fo 2904 4096 a(v) p Fp 2944 4111 a(L;) p Fj(\(12\)) p Fy 3132 4096 a(\)) p Fo(;) 1043 4263 y(v) p Fp 1083 4278 a(L;) p Fj(\(4\)) p Fy 1261 4263 a(=) 1362 4207 y(1) p 1359 4244 V Fo 1359 4320 a(\025) p Fy 1417 4263 a(\() p Fo(B) p Fj 1512 4275 a(15) p Fo 1583 4263 a(v) p Fp 1623 4278 a(L;) p Fj(\(1\)) p Fy 1796 4263 a(+) p Fo 1879 4263 a(B) p Fj 1942 4275 a(25) p Fo 2012 4263 a(v) p Fp 2052 4278 a(L;) p Fj(\(11\)) p Fy 2258 4263 a(+) p Fo 2341 4263 a(B) p Fj 2404 4275 a(35) p Fo 2474 4263 a(v) p Fp 2514 4278 a(L;) p Fj(\(2\)) p Fy 2688 4263 a(+) p Fo 2771 4263 a(B) p Fj 2834 4275 a(65) p Fo 2904 4263 a(v) p Fp 2944 4278 a(L;) p Fj(\(12\)) p Fy 3132 4263 a(\)) p Fo(;) 1043 4429 y(v) p Fp 1083 4444 a(L;) p Fj(\(12\)) p Fy 1294 4429 a(=) 1512 4373 y(1) p 1392 4410 284 4 v Fo 1392 4486 a(\025) p Fk 1459 4486 a(\000) p Fo 1542 4486 a(B) p Fj 1605 4498 a(66) p Fy 1685 4429 a(\() p Fo(B) p Fj 1780 4441 a(16) p Fo 1850 4429 a(v) p Fp 1890 4444 a(L;) p Fj(\(1\)) p Fy 2063 4429 a(+) p Fo 2147 4429 a(B) p Fj 2210 4441 a(26) p Fo 2280 4429 a(v) p Fp 2320 4444 a(L;) p Fj(\(11\)) p Fy 2526 4429 a(+) p Fo 2609 4429 a(B) p Fj 2672 4441 a(36) p Fo 2742 4429 a(v) p Fp 2782 4444 a(L;) p Fj(\(2\)) p Fy 2937 4429 a(\)) p Fo(:) p Fy 390 4643 a(Ob) n(viously) p 747 4643 a(,) p 795 4643 a(w) n(e) p 914 4643 a(can) p 1064 4643 a(tak) n(e) p Fo 1241 4643 a(v) p Fp 1281 4658 a(L;) p Fj(\(1\)) p Fy 1459 4643 a(=) p 1546 4643 a(1) p 1602 4643 a(.) p Fo 1661 4643 a(\025) p Fy 1735 4643 a(is) p 1815 4643 a(the) p 1956 4643 a(larger) p 2189 4643 a(solution) p 2501 4643 a(of) p Fo 2593 4643 a(x) p Fj 2640 4609 a(2) p Fk 2691 4643 a(\000) p Fy 2769 4643 a(\() p Fo(p) p Fy 2856 4643 a(+) p Fo 2934 4643 a(r) p Fy 2973 4643 a(\)) p Fo(x) p Fy 3065 4643 a(+) p Fo 3143 4643 a(pr) p Fk 3239 4643 a(\000) p Fy 3317 4643 a(3) p Fo(q) p Fj 3399 4609 a(2) p Fy 3459 4643 a(=) p 3547 4643 a(0,) p 3637 4643 a(hence) p Fo 390 4743 a(\025) p 462 4743 a(>) p 549 4743 a(p) p Fy(.) p 651 4743 a(Therefore,) p Fo 1050 4743 a(v) p Fp 1090 4758 a(L;) p Fj(\(11\)) p Fo 1301 4743 a(>) p Fy 1389 4743 a(0) p 1444 4743 a(.) p 1504 4743 a(Note) p 1705 4743 a(that) p 1885 4743 a(Prop) r(osition) p 2332 4743 a(30) p 2442 4743 a(\(iv\)) p 2602 4743 a(implies) p Fo 1529 4923 a(B) p Fp 1592 4935 a(i;j) p Fo 1693 4923 a(>) p Fy 1781 4923 a(0) p Fo(;) p 1914 4923 a(i) p Fy 1966 4923 a(=) p 2054 4923 a(1) p Fo(;) p Fy 2133 4923 a(2) p Fo(;) p 2238 4923 a(j) p Fy 2300 4923 a(=) p 2388 4923 a(3) p Fo(;) p Fy 2467 4923 a(4) p Fo(;) p Fy 2546 4923 a(5) p Fo(;) p Fy 2625 4923 a(6) p Fo 2681 4923 a(:) p Fy 390 5103 a(Note) p 591 5103 a(also) p 757 5103 a(that) p Fo 937 5103 a(\025) p 1009 5103 a(>) p Fy 1096 5103 a(1) p Fo 1161 5103 a(>) p Fy 1249 5103 a(max) p Fk 1403 5103 a(f) p Fo(B) p Fj 1508 5115 a(33) p Fo 1578 5103 a(;) p 1615 5103 a(B) p Fj 1678 5115 a(66) p Fk 1748 5103 a(g) p Fy 1803 5103 a(.) p 1863 5103 a(Therefore) p Fo 2240 5103 a(v) p Fp 2280 5115 a(L;I) p Fo 2406 5103 a(>) p Fy 2494 5103 a(0,) p Fo 2586 5103 a(I) p Fy 2652 5103 a(=) p 2740 5103 a(\(2\)) p Fo(;) p Fy 2883 5103 a(\(3\)) p Fo(;) p Fy 3026 5103 a(\(4\)) p Fo(;) p Fy 3169 5103 a(\(12\).) 390 5471 y(Finally) p 638 5471 a(,) p 693 5471 a(note) p 880 5471 a(that) p 1063 5471 a(the) p 1209 5471 a(righ) n(t) p 1413 5471 a(eigen) n(v) n(ector) p 1847 5471 a(of) p Fo 1945 5471 a(B) p Fy 2042 5471 a(for) p Fo 2172 5471 a(\025) p Fy 2251 5471 a(can) p 2407 5471 a(b) r(e) p 2523 5471 a(c) n(hosen) p 2791 5471 a(to) p 2896 5471 a(b) r(e) p Fg 3012 5155 a(0) 3012 5301 y(B) 3012 5351 y(B) 3012 5401 y(B) 3012 5451 y(B) 3012 5500 y(B) 3012 5550 y(B) 3012 5603 y(@) p Fy 3181 5221 a(3) p Fo(q) 3126 5321 y(\025) p Fk 3193 5321 a(\000) p Fo 3276 5321 a(p) p Fy 3201 5421 a(0) 3201 5520 y(0) 3201 5620 y(0) 3201 5720 y(0) p Fg 3359 5155 a(1) 3359 5301 y(C) 3359 5351 y(C) 3359 5401 y(C) 3359 5451 y(C) 3359 5500 y(C) 3359 5550 y(C) 3359 5603 y(A) p Fy 3432 5471 a(,) p 3486 5471 a(hence) p 3720 5471 a(has) 390 5820 y(p) r(ositiv) n(e) p 697 5820 a(\(non-zero\)) p 1099 5820 a(\(1\)-comp) r(onen) n(t.) 1878 6120 y(34) p 90 rotate dyy eop %%Page: 35 35 35 34 bop Fy 247 83 a(\(c\)) p 390 83 a(The) p 561 83 a(explicit) p 854 83 a(form) p 1050 83 a(\(74\)) p 1225 83 a(sho) n(ws) p 1464 83 a(that) p 1644 83 a(\010) p Fj 1704 98 a(\(1\)) p Fy 1820 83 a(and) p 1982 83 a(\010) p Fj 2042 98 a(\(11\)) p Fy 2192 83 a(con) n(tain) p 2485 83 a(terms) p Fo 2716 83 a(x) p Fj 2763 53 a(2) 2763 110 y(\(1\)) p Fy 2880 83 a(and) p Fo 3042 83 a(x) p Fj 3089 53 a(4) 3089 110 y(\(1\)) p Fy 3178 83 a(,) p 3229 83 a(resp) r(ectiv) n(ely) p 3648 83 a(.) 238 228 y(\(d\)) p 390 228 a(W) p 468 228 a(e) p 533 228 a(ha) n(v) n(e) p 725 228 a(already) p 1018 228 a(sho) n(wn) p Fo 1268 228 a(~) p 1270 228 a(x) p Fp 1317 240 a(c) p Fk 1374 228 a(2) p Fy 1452 228 a(\004) p Fj 1507 240 a(4) p Fk 1563 228 a(\\) p Fq 1637 228 a(R) p Fn 1697 191 a(K) p Fm 1748 199 a(r) q(es) p Fj 1697 248 a(+) p Fy 1869 228 a(and) p 2030 228 a(since) p Fo 2234 228 a(x) p Fp 2281 240 a(c) p Fo 2338 228 a(>) p Fy 2426 228 a(0,) p Fo 2517 228 a(~) p 2518 228 a(x) p Fp 2565 240 a(c) p Fk 2622 228 a(6) p Fy(=) p Fo 2704 210 a(~) p Fy 2710 228 a(0) o(.) 32 379 y(\(iii\)) p 208 379 a(Theorem) p 564 379 a(15) p 681 379 a(implies) p 969 379 a(that) p 1155 379 a(there) p 1373 379 a(exists) p 1609 379 a(one) p 1767 379 a(and) p 1934 379 a(only) p 2122 379 a(one) p Fo 2281 379 a(\014) p Fj 2328 391 a(0) p Fk 2398 379 a(2) p Fq 2486 379 a(R) p Fy 2580 379 a(suc) n(h) p 2773 379 a(that) p Fo 2958 379 a(~) p 2959 379 a(x) p Fp 3006 391 a(can;) p Fn(K) p Fm 3184 399 a(r) q(es) p Fy 3277 379 a(\() p Fo(\014) p Fj 3356 391 a(0) p Fy 3394 379 a(\)) p Fk 3459 379 a(2) p Fo 3548 379 a(@) p 3597 379 a(D) p Fy 3668 379 a(.) p 3745 379 a(T) p 3798 379 a(o) 208 478 y(pro) n(v) n(e) p 436 478 a(that) p Fo 621 478 a(\014) p Fj 668 490 a(0) p Fy 737 478 a(is) p 825 478 a(the) p 973 478 a(critical) p 1254 478 a(p) r(oin) n(t) p 1476 478 a(of) p 1575 478 a(the) p 1723 478 a(restricted) p 2097 478 a(mo) r(del,) p 2373 478 a(it) p 2461 478 a(is) p 2549 478 a(su\016cien) n(t) p 2902 478 a(to) p 3008 478 a(pro) n(v) n(e,) p 3261 478 a(from) p 3462 478 a(\(CS2\)) p 3706 478 a(and) 208 578 y(\(D) n(A1\)) p 465 578 a({) p 534 578 a(\(D) n(A2\),) p 814 578 a(that) 1522 687 y(lim) p Fp 1493 737 a(n) p Fn(!1) p Fo 1698 666 a(~) 1680 687 y(X) p Fp 1749 699 a(n) p Fy 1794 687 a(\() p Fo 1825 687 a(~) p 1826 687 a(x) p Fp 1873 699 a(can;) p Fn(K) p Fm 2051 707 a(r) q(es) p Fy 2145 687 a(\() p Fo(\014) p Fj 2224 699 a(0) p Fy 2261 687 a(\)\)) p 2349 687 a(=) p Fo 2435 687 a(~) p 2436 687 a(x) p Fp 2483 699 a(c) p Fo 2531 687 a(;) p Fy 3692 687 a(\(77\)) 208 838 y(and) p Fo 1483 938 a(x) p Fp 1530 950 a(can;) p Fn(K) p Fm 1708 958 a(r) q(es) p Fp 1797 950 a(;I) p Fy 1855 938 a(\() p Fo(\014) p Fj 1934 950 a(0) p Fy 1971 938 a(\)) p Fk 2027 938 a(6) p Fy(=) p 2114 938 a(0) p Fo(;) p 2220 938 a(I) p Fk 2286 938 a(2) p 2365 938 a(K) p Fp 2428 950 a(r) r(es) p Fo 2541 938 a(:) p Fy 208 1065 a(The) p 378 1065 a(latter) p 604 1065 a(ob) n(viously) p 972 1065 a(holds,) p 1212 1065 a(b) r(ecause) p Fo 1334 1209 a(~) p 1335 1209 a(x) p Fp 1382 1221 a(can;) p Fn(K) p Fm 1560 1229 a(r) q(es) p Fy 1653 1209 a(\() p Fo(\014) p Fj 1732 1221 a(0) p Fy 1769 1209 a(\)) p 1825 1209 a(=) p 1912 1209 a(\() p Fo(e) p Fn 1983 1175 a(\000) p Fp(\014) p Ff 2073 1183 a(0) p Fo 2109 1209 a(;) p 2146 1209 a(e) p Fn 2185 1175 a(\000) p Fj(2) p Fp(\014) p Ff 2308 1183 a(0) p Fo 2344 1209 a(;) p Fy 2381 1209 a(0) p Fo(;) p Fy 2460 1209 a(0) p Fo(;) p Fy 2539 1209 a(0) p Fo(;) p Fy 2618 1209 a(0\)) p Fo(:) p Fy 208 1354 a(Hence) p 454 1354 a(it) p 537 1354 a(su\016ces) p 820 1354 a(to) p 921 1354 a(pro) n(v) n(e) p 1145 1354 a(\(77\).) 208 1479 y(Since) p Fq 429 1479 a(R) p Fn 489 1442 a(K) p Fm 540 1450 a(r) q(es) p Fj 489 1500 a(+) p Fk 663 1479 a(\032) p Fq 757 1479 a(R) p Fn 817 1442 a(I) p Ff 855 1450 a(4) p Fj 817 1500 a(+) p Fy 923 1479 a(is) p 1011 1479 a(an) p 1130 1479 a(in) n(v) p 1236 1479 a(arian) n(t) p 1483 1479 a(subset) p 1742 1479 a(\(Prop) r(osition) p 2225 1479 a(6) p 2299 1479 a(or) p 2404 1479 a(Prop) r(osition) p 2855 1479 a(8\),) p 2985 1479 a(w) n(e) p 3112 1479 a(ma) n(y) p 3296 1479 a(restrict) p 3587 1479 a(\(77\)) p 3766 1479 a(to) p Fq 208 1592 a(R) p Fn 268 1555 a(K) p Fm 319 1563 a(r) q(es) p Fj 268 1612 a(+) p Fy 426 1592 a(.) p 486 1592 a(De\014ne) p Fo 627 1742 a(x) p Fp 674 1754 a(n) p Fy 719 1742 a(\() p Fo(x;) p 835 1742 a(y) p Fy 879 1742 a(\)) p 935 1742 a(=) p Fo 1023 1742 a(X) p Fp 1092 1757 a(n;) p Fj(\(1\)) p Fy 1241 1742 a(\() p Fo(x;) p 1357 1742 a(y) s(;) p Fy 1438 1742 a(0) p Fo(;) p Fy 1517 1742 a(0) p Fo(;) p Fy 1596 1742 a(0) p Fo(;) p Fy 1675 1742 a(0\)) p Fo(;) p 1868 1742 a(y) p Fp 1909 1754 a(n) p Fy 1953 1742 a(\() p Fo(x;) p 2069 1742 a(y) p Fy 2113 1742 a(\)) p 2169 1742 a(=) p Fo 2257 1742 a(X) p Fp 2326 1757 a(n;) p Fj(\(11\)) p Fy 2508 1742 a(\() p Fo(x;) p 2624 1742 a(y) s(;) p Fy 2705 1742 a(0) p Fo(;) p Fy 2784 1742 a(0) p Fo(;) p Fy 2863 1742 a(0) p Fo(;) p Fy 2942 1742 a(0\)) p Fo(;) 631 1831 y(~) 627 1853 y(\036) p Fy 699 1853 a(=) p 787 1853 a(\() p Fo(\036) p Fj 868 1865 a(1) p Fo 906 1853 a(;) p 943 1853 a(\036) p Fj 992 1865 a(2) p Fy 1030 1853 a(\);) p Fo 1182 1853 a(\036) p Fj 1231 1865 a(1) p Fy 1268 1853 a(\() p Fo(x;) p 1384 1853 a(y) p Fy 1428 1853 a(\)) p 1484 1853 a(=) p 1572 1853 a(\010) p Fj 1632 1868 a(\(1\)) p Fy 1721 1853 a(\() p Fo(x;) p 1837 1853 a(y) s(;) p Fy 1918 1853 a(0) p Fo(;) p Fy 1997 1853 a(0) p Fo(;) p Fy 2076 1853 a(0) p Fo(;) p Fy 2155 1853 a(0\)) p Fo(;) p 2319 1853 a(\036) p Fj 2368 1865 a(2) p Fy 2406 1853 a(\() p Fo(x;) p 2522 1853 a(y) p Fy 2566 1853 a(\)) p 2622 1853 a(=) p 2709 1853 a(\010) p Fj 2769 1868 a(\(11\)) p Fy 2892 1853 a(\() p Fo(x;) p 3008 1853 a(y) s(;) p Fy 3089 1853 a(0) p Fo(;) p Fy 3168 1853 a(0) p Fo(;) p Fy 3247 1853 a(0) p Fo(;) p Fy 3326 1853 a(0\)) p Fo(:) p Fy 208 2018 a(Then) p 424 2018 a(\() p Fo(x) p Fj 503 2030 a(0) p Fy 541 2018 a(\() p Fo(x;) p 657 2018 a(y) p Fy 701 2018 a(\)) p Fo(;) p 770 2018 a(y) p Fj 811 2030 a(0) p Fy 849 2018 a(\() p Fo(x;) p 965 2018 a(y) p Fy 1009 2018 a(\)\)) p 1097 2018 a(=) p 1184 2018 a(\() p Fo(x;) p 1300 2018 a(y) p Fy 1344 2018 a(\)) p 1405 2018 a(and) p 1566 2018 a(\() p Fo(x) p Fp 1645 2030 a(n) p Fj(+1) p Fo 1775 2018 a(;) p 1812 2018 a(y) p Fp 1853 2030 a(n) p Fj(+1) p Fy 1982 2018 a(\)) p 2037 2018 a(=) p Fo 2129 1996 a(~) 2125 2018 y(\036) p Fk 2193 2018 a(\016) p Fy 2253 2018 a(\() p Fo(x) p Fp 2332 2030 a(n) p Fo 2378 2018 a(;) p 2415 2018 a(y) p Fp 2456 2030 a(n) p Fy 2500 2018 a(\),) p Fo 2584 2018 a(n) p Fk 2656 2018 a(2) p Fq 2735 2018 a(Z) p Fj 2790 2030 a(+) p Fy 2845 2018 a(.) 208 2143 y(Next) p 410 2143 a(de\014ne) 798 2243 y(\004) p Fj 853 2200 a(\(2\)) 853 2265 y(4) p Fy 966 2243 a(=) p Fk 1053 2243 a(f) p Fy(\() p Fo(x;) p 1211 2243 a(y) p Fy 1255 2243 a(\)) p Fk 1311 2243 a(2) p Fq 1389 2243 a(R) p Fj 1449 2208 a(2) 1449 2263 y(+) p Fk 1527 2243 a(j) p Fy 1573 2243 a(\() p Fo(x;) p 1689 2243 a(y) s(;) p Fy 1770 2243 a(0) p Fo(;) p Fy 1849 2243 a(0) p Fo(;) p Fy 1928 2243 a(0) p Fo(;) p Fy 2007 2243 a(0\)) p Fk 2103 2243 a(2) p Fy 2181 2243 a(\004) p Fj 2236 2255 a(4) p Fk 2274 2243 a(g) p Fy 2338 2243 a(=) p Fk 2426 2243 a(f) p Fy(\() p Fo(x;) p 2584 2243 a(y) p Fy 2628 2243 a(\)) p Fk 2683 2243 a(2) p Fq 2761 2243 a(R) p Fj 2821 2208 a(2) 2821 2263 y(+) p Fk 2900 2243 a(j) p Fo 2946 2243 a(x) p Fj 2993 2208 a(2) p Fl 3053 2243 a(=) p Fo 3141 2243 a(y) p Fk 3185 2243 a(g) p Fo(;) p Fy 208 2369 a(and) p Fo 1242 2469 a(D) p Fj 1313 2435 a(\(2\)) p Fy 1425 2469 a(=) p Fk 1513 2469 a(f) p Fy(\() p Fo(x;) p 1671 2469 a(y) p Fy 1715 2469 a(\)) p Fk 1770 2469 a(2) p Fy 1849 2469 a(\004) p Fj 1904 2426 a(\(2\)) 1904 2491 y(4) p Fk 2016 2469 a(j) p Fy 2062 2469 a(\() p Fo(x;) p 2178 2469 a(y) s(;) p Fy 2259 2469 a(0) p Fo(;) p Fy 2338 2469 a(0) p Fo(;) p Fy 2417 2469 a(0) p Fo(;) p Fy 2496 2469 a(0\)) p Fk 2591 2469 a(2) p Fo 2670 2469 a(D) p Fk 2741 2469 a(g) p Fo(;) p Fy 208 2596 a(where) p Fo 450 2596 a(D) p Fk 547 2596 a(\032) p Fq 637 2596 a(R) p Fj 697 2566 a(6) 697 2616 y(+) p Fy 782 2596 a(is) p 867 2596 a(as) p 971 2596 a(in) p 1070 2596 a(\(31\).) p 1283 2596 a(Denote) p 1570 2596 a(b) n(y) p Fo 1687 2596 a(D) p Fj 1758 2566 a(\(2\)) p Fp(o) p Fy 1880 2596 a(,) p Fo 1933 2596 a(D) p Fj 2004 2566 a(\(2\)) p Fp(c) p Fy 2123 2596 a(,) p Fo 2176 2596 a(@) p 2225 2596 a(D) p Fj 2296 2566 a(\(2\)) p Fy 2385 2596 a(,) p 2438 2596 a(the) p 2583 2596 a(in) n(terior,) p 2902 2596 a(exterior,) p 3234 2596 a(and) p 3397 2596 a(b) r(oundary) p 3773 2596 a(of) p Fo 208 2718 a(D) p Fj 279 2688 a(\(2\)) p Fy 395 2718 a(in) p 492 2718 a(\004) p Fj 547 2675 a(\(2\)) 547 2740 y(4) p Fy 637 2718 a(.) 208 2843 y(Note) p 408 2843 a(that) 1625 2943 y(\() p Fo(e) p Fn 1696 2909 a(\000) p Fp(\014) p Ff 1786 2917 a(0) p Fo 1822 2943 a(;) p 1859 2943 a(e) p Fn 1898 2909 a(\000) p Fj(2) p Fp(\014) p Ff 2021 2917 a(0) p Fy 2057 2943 a(\)) p Fk 2112 2943 a(2) p Fo 2191 2943 a(@) p 2240 2943 a(D) p Fj 2311 2909 a(\(2\)) p Fo 2399 2943 a(:) p Fy 208 3070 a(T) p 261 3070 a(o) p 330 3070 a(pro) n(v) n(e) p 554 3070 a(\(77\),) p 752 3070 a(it) p 835 3070 a(then) p 1024 3070 a(su\016ces) p 1306 3070 a(to) p 1408 3070 a(pro) n(v) n(e) 1140 3214 y(lim) p Fp 1111 3264 a(n) p Fn(!1) p Fy 1285 3214 a(\() p Fo(x) p Fp 1364 3226 a(n) p Fy 1410 3214 a(\() p Fo(x;) p 1526 3214 a(y) p Fy 1570 3214 a(\)) p Fo(;) p 1639 3214 a(y) p Fp 1680 3226 a(n) p Fy 1725 3214 a(\() p Fo(x;) p 1841 3214 a(y) p Fy 1885 3214 a(\)\)) p 1973 3214 a(=) p 2061 3214 a(\() p Fo(x) p Fp 2140 3226 a(c) p Fo 2174 3214 a(;) p 2211 3214 a(y) p Fp 2252 3226 a(c) p Fy 2286 3214 a(\)) p Fo(;) p Fy 2410 3214 a(\() p Fo(x;) p 2526 3214 a(y) p Fy 2570 3214 a(\)) p Fk 2626 3214 a(2) p Fo 2704 3214 a(@) p 2753 3214 a(D) p Fj 2824 3180 a(\(2\)) p Fo 2913 3214 a(:) p Fy 3692 3214 a(\(78\)) 208 3414 y(The) p 378 3414 a(de\014nition) p 747 3414 a(and) p 909 3414 a(Theorem) p 1259 3414 a(15) p 1370 3414 a(implies) p 1652 3414 a(that) p 1832 3414 a(these) p 2044 3414 a(sets) p 2207 3414 a(are) p 2345 3414 a(in) n(v) p 2451 3414 a(arian) n(t) p 2694 3414 a(sets) p 2856 3414 a(of) p 2951 3414 a(\() p Fo(\036) p Fj 3032 3426 a(1) p Fo 3070 3414 a(;) p 3107 3414 a(\036) p Fj 3156 3426 a(2) p Fy 3193 3414 a(\)) p 3253 3414 a(and) p 3415 3414 a(that) p Fo 984 3574 a(D) p Fj 1055 3540 a(\(2\)) p Fy 1168 3574 a(=) p Fk 1255 3574 a(f) p Fy(\() p Fo(x;) p 1413 3574 a(y) p Fy 1457 3574 a(\)) p Fk 1512 3574 a(2) p Fy 1591 3574 a(\004) p Fj 1646 3531 a(\(2\)) 1646 3597 y(4) p Fk 1758 3574 a(j) p Fy 1828 3574 a(sup) p Fp 1805 3644 a(n) p Fn(2) p Fc(Z) p Ff 1930 3652 a(+) p Fy 1990 3574 a(max) p Fk 2144 3574 a(f) p Fo(x) p Fp 2233 3586 a(n) p Fy 2278 3574 a(\() p Fo(x;) p 2394 3574 a(y) p Fy 2438 3574 a(\)) p Fo(;) p 2507 3574 a(y) p Fp 2548 3586 a(n) p Fy 2594 3574 a(\() p Fo(x;) p 2710 3574 a(y) p Fy 2754 3574 a(\)) p Fk(g) p Fo 2851 3574 a(<) p Fk 2939 3574 a(1g) p Fy 984 3755 a(=) p Fk 1072 3755 a(f) p Fy(\() p Fo(x;) p 1230 3755 a(y) p Fy 1274 3755 a(\)) p Fk 1329 3755 a(2) p Fy 1408 3755 a(\004) p Fj 1463 3712 a(\(2\)) 1463 3777 y(4) p Fk 1575 3755 a(j) p Fy 1644 3755 a(sup) p Fp 1621 3825 a(n) p Fn(2) p Fc(Z) p Ff 1746 3833 a(+) p Fy 1806 3755 a(max) p Fk(f) p Fo(x) p Fp 2050 3767 a(n) p Fy 2095 3755 a(\() p Fo(x;) p 2211 3755 a(y) p Fy 2255 3755 a(\)) p 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p Fj 1326 5048 a(1) p Fo 1364 5036 a(;) p 1401 5036 a(\036) p Fj 1450 5048 a(2) p Fy 1487 5036 a(\)) p 1196 5073 324 4 v Fo 1237 5149 a(@) p Fy 1286 5149 a(\() p Fo(x;) p 1402 5149 a(y) p Fy 1446 5149 a(\)) 1530 5092 y(\() p Fo(x;) p 1646 5092 a(y) p Fy 1690 5092 a(\)) p Fg 1722 4975 a(\023) p Fy 848 5242 a(=) p 996 5242 a(8) p Fo(x) p Fj 1085 5208 a(4) p Fy 1140 5242 a(+) p 1223 5242 a(62) p Fo(x) p Fj 1354 5208 a(5) p Fy 1410 5242 a(+) p 1493 5242 a(165) p Fo(x) p Fj 1666 5208 a(6) p Fy 1720 5242 a(+) p 1803 5242 a(192) p Fo(x) p Fj 1976 5208 a(7) p Fy 2031 5242 a(+) p 2114 5242 a(90) p Fo(x) p Fj 2245 5208 a(8) p Fy 2300 5242 a(+) p 2383 5242 a(128) p Fo(x) p Fj 2556 5208 a(4) p Fo 2592 5242 a(y) p Fy 2655 5242 a(+) p 2738 5242 a(432) p Fo(x) p Fj 2911 5208 a(5) p Fo 2947 5242 a(y) p Fy 3009 5242 a(+) p 3092 5242 a(516) p Fo(x) p Fj 3265 5208 a(6) p Fo 3301 5242 a(y) p Fy 3363 5242 a(+) p 3446 5242 a(360) p Fo(x) p Fj 3619 5208 a(7) p Fo 3656 5242 a(y) p Fy 1014 5344 a(+) p 1097 5344 a(528) p Fo(x) p Fj 1270 5310 a(3) p Fo 1306 5344 a(y) p Fj 1350 5310 a(2) p Fy 1406 5344 a(+) p 1489 5344 a(2088) p Fo(x) p Fj 1704 5310 a(4) p Fo 1740 5344 a(y) p Fj 1784 5310 a(2) p Fy 1839 5344 a(+) p 1922 5344 a(3996) p Fo(x) p Fj 2137 5310 a(5) p Fo 2173 5344 a(y) p Fj 2217 5310 a(2) p Fy 2272 5344 a(+) p 2355 5344 a(4770) p Fo(x) p Fj 2570 5310 a(6) p Fo 2606 5344 a(y) p Fj 2650 5310 a(2) p Fy 2705 5344 a(+) p 2788 5344 a(176) p Fo(xy) p Fj 3005 5310 a(3) p Fy 3060 5344 a(+) p 3143 5344 a(2552) p Fo(x) p Fj 3358 5310 a(2) p Fo 3393 5344 a(y) p Fj 3437 5310 a(3) p Fy 1014 5446 a(+) p 1097 5446 a(10032) p Fo(x) p Fj 1354 5411 a(3) p Fo 1389 5446 a(y) p Fj 1433 5411 a(3) p Fy 1489 5446 a(+) p 1572 5446 a(25200) p Fo(x) p Fj 1829 5411 a(4) p Fo 1864 5446 a(y) p Fj 1908 5411 a(3) p Fy 1964 5446 a(+) p 2047 5446 a(35040) p Fo(x) p Fj 2304 5411 a(5) p Fo 2339 5446 a(y) p Fj 2383 5411 a(3) p Fy 2438 5446 a(+) p 2521 5446 a(1860) p Fo(xy) p Fj 2780 5411 a(4) p Fy 2834 5446 a(+) p 2917 5446 a(11538) p Fo(x) p Fj 3174 5411 a(2) p Fo 3209 5446 a(y) p Fj 3253 5411 a(4) p Fy 3309 5446 a(+) p 3392 5446 a(56640) p Fo(x) p Fj 3649 5411 a(3) p Fo 3684 5446 a(y) p Fj 3728 5411 a(4) p Fy 1014 5547 a(+) p 1097 5547 a(113520) p Fo(x) p Fj 1396 5513 a(4) p Fo 1431 5547 a(y) p Fj 1475 5513 a(4) p Fy 1530 5547 a(+) p 1613 5547 a(3168) p Fo(xy) p Fj 1872 5513 a(5) p Fy 1926 5547 a(+) p 2009 5547 a(69912) p Fo(x) p Fj 2266 5513 a(2) p Fo 2302 5547 a(y) p Fj 2346 5513 a(5) p Fy 2401 5547 a(+) p 2484 5547 a(206640) p Fo(x) p Fj 2783 5513 a(3) p Fo 2818 5547 a(y) p Fj 2862 5513 a(5) p Fy 2917 5547 a(+) p 3000 5547 a(50376) p Fo(xy) p Fj 3301 5513 a(6) p Fy 1014 5649 a(+) p 1097 5649 a(201780) p Fo(x) p Fj 1396 5615 a(2) p Fo 1431 5649 a(y) p Fj 1475 5615 a(6) p Fy 1530 5649 a(+) p 1613 5649 a(11616) p Fo(y) p Fj 1867 5615 a(7) p Fy 1920 5649 a(+) p 2003 5649 a(62400) p Fo(xy) p Fj 2304 5615 a(7) p Fk 2358 5649 a(\000) p Fy 2441 5649 a(22440) p Fo(y) p Fj 2695 5615 a(8) p Fo 2729 5649 a(:) p Fy 208 5820 a(W) p 286 5820 a(e) p 351 5820 a(note) p 535 5820 a(the) p 678 5820 a(follo) n(wing) p 1029 5820 a(fact) p 1192 5820 a(pro) n(v) n(ed) p 1462 5820 a(b) n(y) p 1578 5820 a(an) p 1693 5820 a(elemen) n(tary) p 2118 5820 a(argumen) n(t) p 2489 5820 a(in) p 2586 5820 a([6) o(,) p 2701 5820 a(Prop) r(osition) p 3148 5820 a(3.3].) 1878 6120 y(35) p 90 rotate dyy eop %%Page: 36 36 36 35 bop Fz 208 83 a(Lemma) p 546 83 a(32) p Fi 683 83 a(If) p 770 83 a(ther) l(e) p 977 83 a(exists) p Fo 1204 83 a(\017) p 1261 83 a(>) p Fy 1349 83 a(0) p Fi 1420 83 a(such) p 1609 83 a(that) p Fk 1479 266 a(J) p Fj 1550 231 a(\(2\)) p Fy 1639 266 a(\() p Fo(x;) p 1755 266 a(y) p Fy 1799 266 a(\)) p Fo 1855 266 a(>) p 1943 266 a(\017;) p Fy 2043 266 a(\() p Fo(x;) p 2159 266 a(y) p Fy 2203 266 a(\)) p Fk 2259 266 a(2) p Fo 2337 266 a(@) p 2386 266 a(D) p Fj 2457 231 a(\(2\)) p Fo 2546 266 a(;) p Fy 3692 266 a(\(80\)) p Fi 208 448 a(then,) p Fo 418 448 a(\036) p Fj 467 460 a(1) p Fy 504 448 a(\() p Fo(x) p Fj 583 460 a(1) p Fo 621 448 a(;) p 658 448 a(y) p Fj 699 460 a(1) p Fy 736 448 a(\)) p Fo 791 448 a(<) p 879 448 a(\036) p Fj 928 460 a(1) p Fy 966 448 a(\() p Fo(x) p Fj 1045 460 a(2) p Fo 1083 448 a(;) p 1120 448 a(y) p Fj 1161 460 a(2) p Fy 1198 448 a(\)) p Fi 1260 448 a(holds) p 1472 448 a(for) p 1604 448 a(any) p Fy 1764 448 a(\() p Fo(x) p Fj 1843 460 a(1) p Fo 1881 448 a(;) p 1918 448 a(y) p Fj 1959 460 a(1) p Fy 1995 448 a(\)) p Fk 2051 448 a(2) p Fo 2129 448 a(@) p 2178 448 a(D) p Fj 2249 418 a(\(2\)) p Fi 2368 448 a(and) p Fy 2529 448 a(\() p Fo(x) p Fj 2608 460 a(2) p Fo 2646 448 a(;) p 2683 448 a(y) p Fj 2724 460 a(2) p Fy 2761 448 a(\)) p Fk 2816 448 a(2) p Fo 2895 448 a(@) p 2944 448 a(D) p Fj 3015 418 a(\(2\)) p Fi 3133 448 a(satisfying) p Fo 3502 448 a(x) p Fj 3549 460 a(1) p Fo 3610 448 a(<) p 3698 448 a(x) p Fj 3745 460 a(2) p Fi 3796 448 a(.) 166 664 y(R) l(emark.) p Fy 511 664 a(The) p 678 664 a(pro) r(of) p 891 664 a(of) p 981 664 a([6,) p 1093 664 a(Prop) r(osition) p 1536 664 a(3.3]) p 1689 664 a(applies) p 1962 664 a(to) p 2059 664 a(the) p 2198 664 a(presen) n(t) p 2483 664 a(case.) p 2690 664 a(The) p 2856 664 a(pro) r(of) p 3069 664 a(there) p 3277 664 a(\(after) p 3502 664 a([6,) p 3614 664 a(\(3.7\)]\)) 208 764 y(uses) p 384 764 a(only) p 566 764 a(\(79\)) p 741 764 a(and) p 903 764 a(the) p 1046 764 a(fact) p 1209 764 a(that) p Fo 1389 764 a(\036) p Fj 1438 776 a(1) p Fy 1504 764 a(and) p Fo 1665 764 a(\036) p Fj 1714 776 a(2) p Fy 1779 764 a(are) p 1918 764 a(p) r(olynomials) p 2380 764 a(of) p 2474 764 a(p) r(ositiv) n(e) p 2781 764 a(co) r(e\016cien) n(ts.) p Fd 3774 764 a(3) p Fy 208 1079 a(T) p 261 1079 a(o) p 335 1079 a(pro) n(v) n(e) p 563 1079 a(\(80\),) p 768 1079 a(note) p 957 1079 a(that) p 1142 1079 a(the) p 1289 1079 a(only) p 1476 1079 a(negativ) n(e) p 1809 1079 a(term) p 2012 1079 a(in) p Fk 2114 1079 a(J) p Fj 2185 1049 a(\(2\)) p Fy 2274 1079 a(\() p Fo(x;) p 2390 1079 a(y) p Fy 2434 1079 a(\)) p 2499 1079 a(is) p Fk 2588 1079 a(\000) p Fy(22440) p Fo(y) p Fj 2907 1049 a(8) p Fy 2941 1079 a(.) p 3015 1079 a(If) p 3103 1079 a(\() p Fo(x;) p 3219 1079 a(y) p Fy 3263 1079 a(\)) p Fk 3327 1079 a(2) p Fo 3413 1079 a(@) p 3462 1079 a(D) p Fj 3533 1049 a(\(2\)) p Fy 3622 1079 a(,) p 3679 1079 a(then) p Fo 208 1179 a(y) p Fy 274 1179 a(=) p Fo 362 1179 a(p) p Fy(\() p Fo(x) p Fy(\)) p Fo 539 1179 a(<) p 626 1179 a(p) p Fy(\(0\)) p Fo 797 1179 a(<) p Fy 885 1179 a(1) p Fo(=) p Fy(2,) p 1056 1179 a(hence) p 1281 1179 a(22440) p Fo(y) p Fj 1535 1149 a(8) p Fo 1592 1179 a(<) p Fy 1680 1179 a(11220) p Fo(y) p Fj 1934 1149 a(7) p Fo 1991 1179 a(<) p Fy 2079 1179 a(11616) p Fo(y) p Fj 2333 1149 a(7) p Fy 2367 1179 a(.) p 2425 1179 a(With) p 2634 1179 a(other) p 2845 1179 a(p) r(ositiv) n(e) p 3147 1179 a(terms) p 3372 1179 a(in) p Fk 3464 1179 a(J) p Fj 3535 1149 a(\(2\)) p Fy 3624 1179 a(\() p Fo(x;) p 3740 1179 a(y) p Fy 3784 1179 a(\),) 208 1279 y(\(80\)) p 383 1279 a(follo) n(ws) p 655 1279 a(easily) p 850 1279 a(.) 208 1436 y(W) p 286 1436 a(e) p 358 1436 a(are) p 505 1436 a(ready) p 741 1436 a(to) p 850 1436 a(pro) n(v) n(e) p 1082 1436 a(\(78\).) p 1312 1436 a(Let) p 1469 1436 a(\() p Fo(x;) p 1585 1436 a(y) p Fy 1629 1436 a(\)) p Fk 1697 1436 a(2) p Fo 1789 1436 a(@) p 1838 1436 a(D) p Fj 1909 1406 a(\(2\)) p Fy 1998 1436 a(,) p 2058 1436 a(and) p 2227 1436 a(assume) p 2522 1436 a(that) p Fk 2709 1436 a(f) p Fy(\() p Fo(x) p Fp 2830 1448 a(n) p Fy 2876 1436 a(\() p Fo(x;) p 2992 1436 a(y) p Fy 3036 1436 a(\)) p Fo(;) p 3105 1436 a(y) p Fp 3146 1448 a(n) p Fy 3191 1436 a(\() p Fo(x;) p 3307 1436 a(y) p Fy 3351 1436 a(\)\)) p Fk 3452 1436 a(j) p Fo 3511 1436 a(n) p Fk 3597 1436 a(2) p Fq 3688 1436 a(Z) p Fj 3743 1448 a(+) p Fk 3798 1436 a(g) p Fy 208 1547 a(accum) n(ulates) p 675 1547 a(at) p 775 1547 a(\() p Fo(x) p Fp 854 1559 a(a) p Fo 895 1547 a(;) p 932 1547 a(y) p Fp 973 1559 a(a) p Fy 1013 1547 a(\)) p Fk 1068 1547 a(6) p Fy(=) p 1156 1547 a(\() p Fo(x) p Fp 1235 1559 a(c) p Fo 1269 1547 a(;) p 1306 1547 a(y) p Fp 1347 1559 a(c) p Fy 1381 1547 a(\).) p 1473 1547 a(Let) p 1620 1547 a(\() p Fo(x) p Fp 1699 1559 a(b) p Fo 1733 1547 a(;) p 1770 1547 a(y) p Fp 1811 1559 a(b) p Fy 1844 1547 a(\)) p 1900 1547 a(=) p Fo 1992 1525 a(~) 1987 1547 y(\036) p Fy 2037 1547 a(\() p Fo(x) p Fp 2116 1559 a(a) p Fo 2157 1547 a(;) p 2194 1547 a(y) p Fp 2235 1559 a(a) p Fy 2274 1547 a(\).) p 2366 1547 a(Since) p 2582 1547 a(Theorem) p 2932 1547 a(31) p 3041 1547 a(implies) p 3322 1547 a(that) p 3500 1547 a(\() p Fo(x) p Fp 3579 1559 a(c) p Fo 3614 1547 a(;) p 3651 1547 a(y) p Fp 3692 1559 a(c) p Fy 3725 1547 a(\)) p 3784 1547 a(is) 208 1662 y(the) p 351 1662 a(only) p 533 1662 a(\014xed) p 733 1662 a(p) r(oin) n(t) p 950 1662 a(in) p 1047 1662 a(\004) p Fj 1102 1619 a(\(2\)) 1102 1684 y(4) p Fk 1210 1662 a(n) p 1270 1662 a(f) p Fo 1306 1645 a(~) p Fy 1312 1662 a(0) p Fk 1353 1662 a(g) p Fy(,) p 1445 1662 a(it) p 1528 1662 a(follo) n(ws) p 1801 1662 a(that) p Fo 1981 1662 a(x) p Fp 2028 1674 a(b) p Fk 2085 1662 a(6) p Fy(=) p Fo 2172 1662 a(x) p Fp 2219 1674 a(a) p Fy 2274 1662 a(.) 208 1807 y(Assume) p 508 1807 a(that) p Fo 680 1807 a(x) p Fp 727 1819 a(b) p Fo 784 1807 a(>) p 872 1807 a(x) p Fp 919 1819 a(a) p Fy 959 1807 a(.) p 1017 1807 a(F) p 1064 1807 a(or) p Fo 1158 1807 a(\017) p 1215 1807 a(>) p Fy 1303 1807 a(0) p 1364 1807 a(denote) p 1624 1807 a(the) p Fo 1760 1807 a(\017) p Fy 1814 1807 a(neigh) n(b) r(orho) r(o) r(d) p 2328 1807 a(of) p 2415 1807 a(\() p Fo(x;) p 2531 1807 a(y) p Fy 2575 1807 a(\)) p 2628 1807 a(b) n(y) p Fo 2735 1807 a(U) p Fp 2792 1819 a(\017) p Fy 2824 1807 a(\() p Fo(x;) p 2940 1807 a(y) p Fy 2984 1807 a(\).) p 3074 1807 a(Since) p Fo 3288 1785 a(~) 3283 1807 y(\036) p Fy 3353 1807 a(is) p 3429 1807 a(con) n(tin) n(uous,) 208 1917 y(for) p 335 1917 a(an) n(y) p Fo 493 1917 a(\017) p 551 1917 a(>) p Fy 640 1917 a(0) p 709 1917 a(there) p 923 1917 a(exists) p Fo 1152 1917 a(\016) p 1217 1917 a(>) p Fy 1306 1917 a(0) p 1375 1917 a(suc) n(h) p 1563 1917 a(that) p Fo 1749 1896 a(~) 1744 1917 y(\036) p Fy(\() p Fo(U) p Fp 1882 1929 a(\016) p Fy 1919 1917 a(\() p Fo(x) p Fp 1998 1929 a(a) p Fo 2039 1917 a(;) p 2076 1917 a(y) p Fp 2117 1929 a(a) p Fy 2157 1917 a(\)\)) p Fk 2245 1917 a(\032) p Fo 2334 1917 a(U) p Fp 2391 1929 a(\017) p Fy 2423 1917 a(\() p Fo(x) p Fp 2502 1929 a(b) p Fo 2536 1917 a(;) p 2573 1917 a(y) p Fp 2614 1929 a(b) p Fy 2647 1917 a(\).) p 2741 1917 a(T) p 2794 1917 a(aking) p Fo 3018 1917 a(\017) p Fy 3081 1917 a(small) p 3298 1917 a(enough) p 3585 1917 a(so) p 3688 1917 a(that) 208 2017 y(\() p Fo(x) p Fn 287 1987 a(0) p Fo 311 2017 a(;) p 348 2017 a(y) p Fn 392 1987 a(0) p Fy 414 2017 a(\)) p Fk 470 2017 a(2) p Fo 548 2017 a(U) p Fp 605 2029 a(\017) p Fy 637 2017 a(\() p Fo(x) p Fp 716 2029 a(b) p Fo 750 2017 a(;) p 787 2017 a(y) p Fp 828 2029 a(b) p Fy 861 2017 a(\)) p 918 2017 a(implies) p Fo 1197 2017 a(x) p Fn 1244 1987 a(0) p Fo 1291 2017 a(>) p 1378 2017 a(x) p Fp 1425 2029 a(m) p Fy 1512 2017 a(:=) p 1622 2017 a(\() p Fo(x) p Fp 1701 2029 a(a) p Fy 1755 2017 a(+) p Fo 1833 2017 a(x) p Fp 1880 2029 a(b) p Fy 1913 2017 a(\)) p Fo(=) p Fy(2,) p 2077 2017 a(w) n(e) p 2196 2017 a(therefore) p 2542 2017 a(see) p 2673 2017 a(that) p 2850 2017 a(there) p 3060 2017 a(exists) p Fo 3286 2017 a(\016) p 3349 2017 a(>) p Fy 3437 2017 a(0) p 3503 2017 a(suc) n(h) p 3688 2017 a(that) 208 2117 y(\() p Fo(x;) p 324 2117 a(y) p Fy 368 2117 a(\)) p Fk 424 2117 a(2) p Fo 504 2117 a(U) p Fp 561 2129 a(\016) p Fy 597 2117 a(\() p Fo(x) p Fp 676 2129 a(a) p Fo 717 2117 a(;) p 754 2117 a(y) p Fp 795 2129 a(a) p Fy 834 2117 a(\)) p Fk 885 2117 a(\\) p Fo 960 2117 a(@) p 1009 2117 a(D) p Fj 1080 2087 a(\(2\)) p Fy 1197 2117 a(implies) p Fo 1479 2117 a(\036) p Fj 1528 2129 a(1) p Fy 1566 2117 a(\() p Fo(x;) p 1682 2117 a(y) p Fy 1726 2117 a(\)) p Fo 1783 2117 a(>) p 1871 2117 a(x) p Fp 1918 2129 a(m) p Fy 1982 2117 a(.) p 2043 2117 a(Therefore) p 2420 2117 a(if) p Fo 2497 2117 a(\036) p Fj 2546 2129 a(1) p Fy 2584 2117 a(\() p Fo(x;) p 2700 2117 a(y) p Fy 2744 2117 a(\)) p Fo 2800 2117 a(>) p 2889 2117 a(x) p Fy(,) p 2988 2117 a(then) p 3178 2117 a(Lemma) p 3475 2117 a(32) p 3586 2117 a(implies) p Fo 208 2227 a(\036) p Fj 257 2239 a(1) p Fy 294 2227 a(\() p Fo 331 2205 a(~) 326 2227 y(\036) p Fy 376 2227 a(\() p Fo(x;) p 492 2227 a(y) p Fy 536 2227 a(\)\)) p Fo 624 2227 a(>) p 712 2227 a(\036) p Fj 761 2239 a(1) p Fy 798 2227 a(\() p Fo(x;) p 914 2227 a(y) p Fy 958 2227 a(\)) p 1019 2227 a(\() p Fo(>) p 1139 2227 a(x) p Fp 1186 2239 a(m) p Fy 1249 2227 a(\),) p 1332 2227 a(hence) p 1563 2227 a(b) n(y) p 1678 2227 a(induction,) p Fo 1015 2420 a(\036) p Fj 1064 2432 a(1) p Fy 1101 2420 a(\() p Fo 1138 2398 a(~) 1133 2420 y(\036) p Fp 1182 2386 a(n) p Fy 1228 2420 a(\() p Fo(x;) p 1344 2420 a(y) p Fy 1388 2420 a(\)\)) p Fo 1476 2420 a(>) p 1564 2420 a(x) p Fp 1611 2432 a(m) p Fo 1688 2420 a(;) p 1780 2420 a(n) p Fk 1853 2420 a(2) p Fq 1932 2420 a(N) p Fo(;) p Fy 2056 2420 a(\() p Fo(x;) p 2172 2420 a(y) p Fy 2216 2420 a(\)) p Fk 2272 2420 a(2) p Fo 2350 2420 a(U) p Fp 2407 2432 a(\017) p Fy 2439 2420 a(\() p Fo(x) p Fp 2518 2432 a(a) p Fo 2559 2420 a(;) p 2596 2420 a(y) p Fp 2637 2432 a(a) p Fy 2677 2420 a(\)) p Fk 2727 2420 a(\\) p Fo 2801 2420 a(@) p 2850 2420 a(D) p Fj 2921 2386 a(\(2\)) p Fo 3010 2420 a(:) p Fy 208 2636 a(By) p 343 2636 a(assumption) p 788 2636 a(that) p Fk 974 2636 a(f) p Fy(\() p Fo(x) p Fp 1095 2648 a(n) p Fy 1140 2636 a(\() p Fo(x;) p 1256 2636 a(y) p Fy 1300 2636 a(\)) p Fo(;) p 1369 2636 a(y) p Fp 1410 2648 a(n) p Fy 1456 2636 a(\() p Fo(x;) p 1572 2636 a(y) p Fy 1616 2636 a(\)\)) p Fk 1713 2636 a(j) p Fo 1768 2636 a(n) p Fk 1851 2636 a(2) p Fq 1938 2636 a(Z) p Fj 1993 2648 a(+) p Fk 2049 2636 a(g) p Fy 2124 2636 a(accum) n(ulates) p 2598 2636 a(at) p 2705 2636 a(\() p Fo(x) p Fp 2784 2648 a(a) p Fo 2825 2636 a(;) p 2862 2636 a(y) p Fp 2903 2648 a(a) p Fy 2943 2636 a(\),) p 3033 2636 a(there) p 3251 2636 a(exists) p 3486 2636 a(a) p 3560 2636 a(p) r(ositiv) n(e) 208 2736 y(in) n(teger) p Fo 487 2736 a(N) p Fy 595 2736 a(suc) n(h) p 787 2736 a(that) p 971 2736 a(\() p Fo(x) p Fp 1050 2748 a(N) p Fy 1114 2736 a(\() p Fo(x;) p 1230 2736 a(y) p Fy 1274 2736 a(\)) p Fo(;) p 1343 2736 a(y) p Fp 1384 2748 a(N) p Fy 1447 2736 a(\() p Fo(x;) p 1563 2736 a(y) p Fy 1607 2736 a(\)\)) p Fk 1703 2736 a(2) p Fo 1789 2736 a(U) p Fp 1846 2748 a(\017) p Fy 1878 2736 a(\() p Fo(x) p Fp 1957 2748 a(a) p Fo 1998 2736 a(;) p 2035 2736 a(y) p Fp 2076 2748 a(a) p Fy 2115 2736 a(\)) p Fk 2169 2736 a(\\) p Fo 2246 2736 a(@) p 2295 2736 a(D) p Fj 2366 2706 a(\(2\)) p Fy 2455 2736 a(.) p 2529 2736 a(Therefore) p Fo 2910 2736 a(x) p Fp 2957 2748 a(n) p Fy 3002 2736 a(\() p Fo(x;) p 3118 2736 a(y) p Fy 3162 2736 a(\)) p Fo 3226 2736 a(>) p 3321 2736 a(x) p Fp 3368 2748 a(m) p Fy 3446 2736 a(,) p Fo 3502 2736 a(n) p Fy 3583 2736 a(=) p Fo 3678 2736 a(N) p Fy 3775 2736 a(+) 208 2835 y(1) p Fo(;) p 287 2835 a(N) p Fy 380 2835 a(+) p 463 2835 a(2) p Fo(;) p Fk 542 2835 a(\001) p 579 2835 a(\001) p 616 2835 a(\001) p Fy 638 2835 a(,) p 689 2835 a(whic) n(h) p 927 2835 a(con) n(tradicts) p 1354 2835 a(the) p 1497 2835 a(assumption) p 1936 2835 a(that) p Fk 2116 2835 a(f) p Fo(x) p Fp 2205 2847 a(n) p Fy 2250 2835 a(\() p Fo(x;) p 2366 2835 a(y) p Fy 2410 2835 a(\)) p Fk 2466 2835 a(j) p Fo 2512 2835 a(n) p Fk 2585 2835 a(2) p Fq 2663 2835 a(Z) p Fj 2718 2847 a(+) p Fk 2774 2835 a(g) p Fy 2843 2835 a(accum) n(ulates) p 3312 2835 a(at) p Fo 3413 2835 a(x) p Fp 3460 2847 a(a) p Fo 3524 2835 a(<) p 3611 2835 a(x) p Fp 3658 2847 a(m) p Fy 3736 2835 a(.) 208 2968 y(Similar) p 504 2968 a(argumen) n(t) p 885 2968 a(also) p 1061 2968 a(holds) p 1288 2968 a(for) p 1425 2968 a(if) p 1511 2968 a(w) n(e) p 1644 2968 a(assume) p Fo 1940 2968 a(x) p Fp 1987 2980 a(b) p Fo 2061 2968 a(<) p 2165 2968 a(x) p Fp 2212 2980 a(a) p Fy 2252 2968 a(.) p 2342 2968 a(Therefore) p 2728 2968 a(\() p Fo(x) p Fp 2807 2980 a(c) p Fo 2842 2968 a(;) p 2879 2968 a(y) p Fp 2920 2980 a(c) p Fy 2953 2968 a(\)) p 3023 2968 a(is) p 3116 2968 a(the) p 3269 2968 a(only) p 3461 2968 a(p) r(oin) n(t) p 3688 2968 a(that) p Fk 208 3068 a(f) p Fy(\() p Fo(x) p Fp 329 3080 a(n) p Fy 374 3068 a(\() p Fo(x;) p 490 3068 a(y) p Fy 534 3068 a(\)) p Fo(;) p 603 3068 a(y) p Fp 644 3080 a(n) p Fy 689 3068 a(\() p Fo(x;) p 805 3068 a(y) p Fy 849 3068 a(\)\)) p Fk 937 3068 a(j) p Fo 983 3068 a(n) p Fk 1056 3068 a(2) p Fq 1135 3068 a(Z) p Fj 1190 3080 a(+) p Fk 1245 3068 a(g) p Fy 1314 3068 a(accum) n(ulates,) p 1806 3068 a(whic) n(h) p 2044 3068 a(pro) n(v) n(es) p 2300 3068 a(\(78\).) p Fd 3778 3168 a(2) p Fy 125 3549 a(Theorem) p 478 3549 a(31) p 592 3549 a(and) p 757 3549 a(the) p 903 3549 a(results) p 1170 3549 a(in) p 1270 3549 a(Section) p 1564 3549 a(3.2) p 1700 3549 a(imply) p 1936 3549 a(the) p 2083 3549 a(follo) n(wing) p 2436 3549 a(results) p 2703 3549 a(on) p 2822 3549 a(the) p 2968 3549 a(asymptotic) p 3398 3549 a(b) r(eha) n(viors) p 3773 3549 a(of) 0 3649 y(restricted) p 370 3649 a(self-a) n(v) n (oiding) p 845 3649 a(paths) p 1072 3649 a(on) p 1187 3649 a(the) p 1330 3649 a(4) p 1399 3649 a(dimensional) p 1856 3649 a(pre-Sierpi) r(\023) p 2208 3649 a(nski) p 2381 3649 a(gask) n(et.) p Fz 0 3815 a(Theorem) p 405 3815 a(33) p Fi 542 3815 a(L) l(et) p Fo 687 3815 a(~) p 688 3815 a(x) p Fp 735 3827 a(c) p Fy 796 3815 a(=) p 888 3815 a(\() p Fo(x) p Fp 967 3827 a(c) p Fo 1001 3815 a(;) p 1038 3815 a(y) p Fp 1079 3827 a(c) p Fo 1113 3815 a(;) p Fy 1150 3815 a(0) p Fo(;) p Fy 1229 3815 a(0) p Fo(;) p Fy 1308 3815 a(0) p Fo(;) p Fy 1387 3815 a(0\)) p Fk 1486 3815 a(2) p Fq 1568 3815 a(R) p Fj 1628 3785 a(6) 1628 3836 y(+) p Fi 1715 3815 a(and) p Fo 1878 3815 a(\014) p Fp 1925 3827 a(c;r) r(es) p Fy 2101 3815 a(=) p Fo 2193 3815 a(\014) p Fp 2240 3827 a(c;) p Fn(K) p Fm 2341 3835 a(r) q(es) p Fk 2460 3815 a(2) p Fq 2543 3815 a(R) p Fi 2635 3815 a(b) l(e) p 2739 3815 a(the) p 2879 3815 a(c) l(onstants) p 3246 3815 a(de\014ne) l(d) p 3528 3815 a(in) p 3632 3815 a(The) l(o-) 0 3915 y(r) l(em) p 166 3915 a(31.) p 314 3915 a(Then) p 530 3915 a(the) p 667 3915 a(fol) t(lowing) p 1019 3915 a(holds) p 1230 3915 a(for) p 1362 3915 a(the) p 1499 3915 a(r) l(estricte) l(d) p 1861 3915 a(self-avoiding) p 2340 3915 a(p) l(aths) p 2554 3915 a(on) p 2672 3915 a(the) p Fy 2809 3915 a(4) p Fi 2880 3915 a(dimensional) p 3341 3915 a(pr) l(e-Sierpi) r(\023) p 3693 3915 a(nski) 0 4014 y(gasket.) 73 4197 y(\(i\)) p 208 4197 a(If) p Fo 294 4197 a(~) p 295 4197 a(x) p Fk 365 4197 a(2) p 444 4197 a(D) p Fi 510 4197 a(om) p Fy 627 4197 a(\() p Fo 658 4197 a(~) p 659 4197 a(x) p Fp 706 4209 a(c) p Fy 740 4197 a(\)) p Fi(,) p 828 4197 a(then) p 1012 4197 a(the) p 1150 4197 a(fol) t(lowing) p 1502 4197 a(hold.) 208 4330 y(F) p 256 4330 a(or) p Fo 358 4330 a(I) p Fk 424 4330 a(2) p 502 4330 a(K) p Fp 565 4342 a(r) r(es) p Fy 688 4330 a(=) p Fk 776 4330 a(f) p Fy(\(1\)) p Fo(;) p Fy 961 4330 a(\(11\)) p Fk(g) p Fi(,) p 1200 4330 a(the) p 1333 4330 a(joint) p 1526 4330 a(distribution) p 1965 4330 a(of) p 2058 4330 a(sc) l(ale) l(d) p 2291 4330 a(gener) l(alize) l(d) p 2707 4330 a(p) l(ath) p 2882 4330 a(length) p Fy 3121 4330 a(\() p Fo(\025) p Fn 3201 4300 a(\000) p Fp(n) p Fo 3299 4330 a(s) p Fj 3338 4345 a(\(1\)) p Fo 3441 4330 a(;) p 3501 4330 a(\025) p Fn 3549 4300 a(\000) p Fp(n) p Fo 3647 4330 a(s) p Fj 3686 4345 a(\(11\)) p Fy 3808 4330 a(\)) p Fi 208 4439 a(under) p Fo 444 4439 a(\026) p Fp 493 4452 a(~) p 494 4452 a(x;n;I) p Fi 680 4439 a(c) l(onver) l(ges) p 1050 4439 a(we) l(akly) p 1311 4439 a(to) p 1411 4439 a(a) p 1483 4439 a(Bor) l(el) p 1704 4439 a(pr) l(ob) l(ability) p 2104 4439 a(me) l(asur) l(e) p Fo 2425 4439 a(p) p Fn 2467 4405 a(\003) p Fp 2466 4460 a(~) p 2467 4460 a(x;I) p Fi 2592 4439 a(on) p Fq 2711 4439 a(R) p Fj 2771 4405 a(6) p Fi 2838 4439 a(as) p Fo 2944 4439 a(n) p Fk 3017 4439 a(!) p 3123 4439 a(1) p Fi(.) 208 4572 y(The) p 377 4572 a(gener) l(ating) p 779 4572 a(function) p Fo 1106 4572 a(') p Fn 1160 4538 a(\003) p Fp 1159 4593 a(~) p 1160 4593 a(x;I) p Fi 1256 4572 a(,) p 1311 4572 a(de\014ne) l(d) p 1591 4572 a(by) p Fo 1374 4813 a(') p Fn 1428 4779 a(\003) p Fp 1427 4834 a(~) p 1428 4834 a(x;I) p Fy 1523 4813 a(\() p Fo 1550 4798 a(~) 1555 4813 y(t) p Fy 1586 4813 a(\)) p 1641 4813 a(=) p Fg 1729 4700 a(Z) p Fn 1812 4721 a(1) p Fj 1775 4889 a(0) p Fo 1896 4813 a(e) p Fp 1931 4768 a(~) 1935 4779 y(t) p Fn(\001) p Fp 1982 4763 a(~) 1980 4779 y(\030) p Fo 2016 4813 a(p) p Fn 2058 4779 a(\003) p Fp 2057 4834 a(~) p 2058 4834 a(x) o(;I) p Fy 2153 4813 a([) p Fo(d) 2221 4791 y(~) 2219 4813 y(\030) p Fy 2259 4813 a(]) p Fo(;) 2403 4798 y(~) 2408 4813 y(t) p Fk 2461 4813 a(2) p Fq 2540 4813 a(C) p Fj 2600 4779 a(6) p Fo 2651 4813 a(;) p Fi 208 5055 a(is) p 298 5055 a(an) p 418 5055 a(entir) l(e) p 656 5055 a(function) p 985 5055 a(in) p Fo 1083 5039 a(~) 1088 5055 y(t) p Fi(,) p 1175 5055 a(and) p 1338 5055 a(the) p 1477 5055 a(set) p 1608 5055 a(of) p 1707 5055 a(functions) p Fy 2069 5055 a(\() p Fo(') p Fn 2155 5025 a(\003) p Fp 2154 5081 a(~) p 2155 5081 a(x;) p Fj(\(1\)) p Fo 2316 5055 a(;) p 2378 5055 a(') p Fn 2432 5025 a(\003) p Fp 2431 5081 a(~) p 2432 5081 a(x;) p Fj(\(11\)) p Fy 2612 5055 a(\)) p Fi 2676 5055 a(ar) l(e) p 2818 5055 a(uniquely) p 3149 5055 a(determine) l(d) p 3578 5055 a(by) p 3687 5055 a(\(25\)) 208 5154 y(for) p Fo 340 5154 a(d) p Fy 406 5154 a(=) p 494 5154 a(4) p Fi 550 5154 a(.) 208 5299 y(If) p Fo 292 5299 a(~) p 293 5299 a(x) p Fk 364 5299 a(2) p 442 5299 a(D) p Fi 508 5299 a(om) p Fy 625 5299 a(\() p Fo 656 5299 a(~) p 657 5299 a(x) p Fp 704 5311 a(c) p Fy 739 5299 a(\)) p Fk 786 5299 a(\\) p Fy 857 5299 a(\004) p Fj 912 5311 a(4) p Fi 977 5299 a(and) p Fo 1137 5299 a(I) p Fk 1203 5299 a(2) p 1282 5299 a(K) p Fp 1345 5311 a(r) r(es) p Fi 1459 5299 a(,) p 1513 5299 a(then) p 1696 5299 a(the) p 1832 5299 a(distribution) p 2275 5299 a(of) p Fo 2371 5299 a(\025) p Fn 2419 5264 a(\000) p Fp(n) p Fo 2517 5299 a(L) p Fy(\() p Fo(w) p Fy 2667 5299 a(\)) p Fi(,) p 2753 5299 a(the) p 2890 5299 a(sc) l(ale) l(d) p 3126 5299 a(length) p 3368 5299 a(of) p Fo 3464 5299 a(w) p Fk 3549 5299 a(2) p Fo 3628 5299 a(W) p Fj 3718 5256 a(\() p Fp(n) p Fj(\)) p Fp 3706 5323 a(I) p Fi 3815 5299 a(,) 208 5398 y(under) p Fo 444 5398 a(\026) p Fp 493 5411 a(~) p 494 5411 a(x;n;I) p Fi 680 5398 a(c) l(onver) l(ges) p 1050 5398 a(we) l(akly) p 1311 5398 a(to) p 1411 5398 a(a) p 1483 5398 a(Bor) l(el) p 1704 5398 a(pr) l(ob) l(ability) p 2104 5398 a(me) l(asur) l(e) p Fy 2432 5398 a(\026) p Fo 2425 5398 a(p) p Fn 2467 5364 a(\003) p Fp 2466 5419 a(~) p 2467 5419 a(x;I) p Fi 2577 5398 a(,) p 2632 5398 a(which) p 2865 5398 a(has) p 3014 5398 a(a) p Fo 3086 5398 a(C) p Fn 3151 5368 a(1) p Fi 3251 5398 a(density) p Fy 3543 5398 a(\026) p Fo 3535 5398 a(\032) p Fn 3578 5364 a(\003) p Fp 3577 5419 a(~) p 3578 5419 a(x;I) p Fi 3688 5398 a(.) 208 5531 y(In) p 316 5531 a(p) l(articular,) p Fy 729 5531 a(\026) p Fo 721 5531 a(\032) p Fn 764 5497 a(\003) p Fp 763 5554 a(~) p 764 5554 a(x;) p Fj(\(1\)) p Fy 911 5531 a(\() p Fo(\030) p Fy 983 5531 a(\)) p Fo 1038 5531 a(>) p Fy 1126 5531 a(0) p Fi(,) p Fo 1223 5531 a(\030) p 1286 5531 a(>) p Fy 1374 5531 a(0) p Fi 1429 5531 a(.) p Fy 1878 6120 a(36) p 90 rotate dyy eop %%Page: 37 37 37 36 bop Fi 47 83 a(\(ii\)) p 208 83 a(F) p 256 83 a(or) p Fo 363 83 a(I) p Fk 429 83 a(2) p 507 83 a(I) p Fj 552 95 a(4) p Fy 613 83 a(=) p Fk 700 83 a(f) p Fy(\(1\)) p Fo(;) p Fy 885 83 a(\(11\)) p Fo(;) p Fy 1070 83 a(\(2\)) p Fo(;) p Fy 1213 83 a(\(3\)) p Fo(;) p Fy 1356 83 a(\(4\)) p Fo(;) p Fy 1499 83 a(\(12\)) p Fk(g) p Fi 1701 83 a(,) p Fy 1286 315 a(lim) p Fp 1257 365 a(n) p Fn(!1) p Fo 1444 315 a(Z) p Fn 1501 327 a(K) p Fm 1552 335 a(r) q(es) p Fp 1641 327 a(;n;I) p Fy 1759 315 a(\() p Fo(\014) p Fy 1842 315 a(\)) p 1898 315 a(=) p Fg 1986 198 a(\032) p Fy 2090 265 a(0) p Fo(;) p 2344 265 a(\014) p 2419 265 a(>) p 2507 265 a(\014) p Fp 2554 277 a(c;r) r(es) p Fo 2716 265 a(;) 2090 364 y(x) p Fp 2137 376 a(c;I) p Fo 2238 364 a(;) p 2344 364 a(\014) p Fy 2419 364 a(=) p Fo 2507 364 a(\014) p Fp 2554 376 a(c;r) r(es) p Fo 2716 364 a(;) p Fi 208 543 a(and) p Fy 1299 701 a(lim) p Fp 1270 751 a(n) p Fn(!1) p Fj 1500 597 a(4) p Fg 1457 622 a(X) p Fp 1463 799 a(i) p Fj(=1) p Fo 1591 701 a(Z) p Fn 1648 716 a(K) p Fm 1699 724 a(r) q(es) p Fp 1787 716 a(;n;) p Fj(\() p Fp(i) p Fj(\)) p Fy 1947 701 a(\() p Fo(\014) p Fy 2030 701 a(\)) p 2086 701 a(=) p Fk 2174 701 a(1) p Fo(;) p 2383 701 a(\014) p 2457 701 a(<) p 2545 701 a(\014) p Fp 2592 713 a(c;r) r(es) p Fo 2755 701 a(:) p Fi 22 961 a(\(iii\)) p 208 961 a(The) p 377 961 a(numb) l(er) p Fo 673 961 a(N) p Fp 740 973 a(r) r(es) p Fy 840 961 a(\() p Fo(k) p Fy 918 961 a(\)) p 973 961 a(=) p Fo 1061 961 a(N) p Fn 1128 973 a(K) p Fm 1179 981 a(r) q(es) p Fy 1272 961 a(\() p Fo(k) p Fy 1350 961 a(\)) p Fi 1412 961 a(of) p 1509 961 a(r) l(estricte) l(d) p 1872 961 a(self-avoiding) p 2352 961 a(p) l(aths) p 2566 961 a(of) p 2663 961 a(length) p Fo 2907 961 a(k) p Fi 2983 961 a(starting) p 3290 961 a(fr) l(om) p Fy 3486 961 a(0) p Fi 3557 961 a(satis\014es) p Fy 1542 1183 a(lim) p Fp 1515 1237 a(k) p Fn 1551 1237 a(!1) p Fy 1710 1127 a(1) p 1708 1164 46 4 v Fo 1708 1240 a(k) p Fy 1791 1183 a(log) p Fo 1912 1183 a(N) p Fp 1979 1195 a(r) r(es) p Fy 2079 1183 a(\() p Fo(k) p Fy 2157 1183 a(\)) p 2212 1183 a(=) p Fo 2300 1183 a(\014) p Fp 2347 1195 a(c;r) r(es) p Fo 2510 1183 a(:) p Fi 35 1484 a(\(iv\)) p 208 1484 a(The) p 380 1484 a(exp) l(onent) p 730 1484 a(for) p 865 1484 a(me) l(an) p 1089 1484 a(squar) l(e) p 1350 1484 a(displac) l(ement) p 1843 1484 a(for) p 1979 1484 a(the) p 2120 1484 a(r) l(estricte) l(d) p 2486 1484 a(mo) l(del) p 2726 1484 a(is) p Fo 2819 1484 a(d) p Fp 2862 1496 a(w) p Fy 2944 1484 a(=) 3047 1427 y(log) p Fo 3168 1427 a(\025) p 3047 1465 170 4 v Fy 3050 1541 a(log) p 3172 1541 a(2) 3255 1484 y(=) p 3349 1484 a(1) p Fo(:) p Fy(6657696) p Fk 3722 1484 a(\001) p 3759 1484 a(\001) p 3796 1484 a(\001) p Fi 3815 1484 a(,) 208 1623 y(in) p 309 1623 a(the) p 447 1623 a(sense) p 668 1623 a(that) p Fy 1263 1741 a(lim) p Fp 1236 1795 a(k) p Fn 1272 1795 a(!1) p Fy 1492 1684 a(1) p 1429 1722 167 4 v 1429 1798 a(log) p Fo 1550 1798 a(k) p Fy 1620 1741 a(log) p Fo 1741 1741 a(E) p Fp 1802 1753 a(r) r(es;k) p Fy 1958 1741 a([) p Fk(j) p Fo(w) p Fy 2065 1741 a(\() p Fo(k) p Fy 2143 1741 a(\)) p Fk(j) p Fp 2198 1706 a(s) p 2242 1706 a(d) p Fm 2277 1714 a(w) p Fy 2328 1741 a(]) p 2374 1741 a(=) p Fo 2462 1741 a(s;) p 2597 1741 a(s) p Fl 2659 1741 a(=) p Fy 2747 1741 a(0) p Fo(;) p Fi 208 1929 a(wher) l(e) p Fo 444 1929 a(E) p Fp 505 1941 a(r) r(es;k) p Fi 693 1929 a(is) p 784 1929 a(the) p 923 1929 a(exp) l(e) l(ctation) p 1354 1929 a(with) p 1536 1929 a(r) l(esp) l(e) l(ct) p 1809 1929 a(to) p 1910 1929 a(the) p 2050 1929 a(pr) l(ob) l(ability) p 2452 1929 a(me) l(asur) l(e) p 2775 1929 a(with) p 2957 1929 a(e) l(qual) p 3169 1929 a(weight) p 3428 1929 a(on) p 3548 1929 a(length) p Fo 3794 1929 a(k) p Fi 208 2029 a(r) l(estricte) l(d) p 570 2029 a(self-avoiding) p 1050 2029 a(p) l(aths) p 1264 2029 a(starting) p 1571 2029 a(at) p Fo 1670 2029 a(O) p Fi 1735 2029 a(.) -42 2212 y(R) l(emark.) p Fy 303 2212 a(The) p 479 2212 a(con) n(v) n(ergence) p 943 2212 a(of) p Fo 1042 2212 a(\026) p Fp 1091 2225 a(~) p 1092 2225 a(x;n;I) p Fy 1281 2212 a(and) p 1447 2212 a(the) p 1595 2212 a(prop) r(erties) p 1990 2212 a(of) p 2089 2212 a(the) p 2237 2212 a(limit) p 2440 2212 a(measure) p 2769 2212 a(holds) p 2991 2212 a(b) r(oth) p 3191 2212 a(for) p 3323 2212 a(the) p 3471 2212 a(full) p 3621 2212 a(mo) r(del) 0 2311 y(and) p 161 2311 a(the) p 304 2311 a(restricted) p 674 2311 a(mo) r(del) p 921 2311 a(\(if) p Fo 1028 2311 a(~) p 1029 2311 a(x) p Fk 1100 2311 a(2) p 1178 2311 a(D) p Fi 1244 2311 a(om) p Fy 1361 2311 a(\() p Fo 1392 2311 a(~) p 1393 2311 a(x) p Fp 1440 2323 a(c) p Fy 1475 2311 a(\)\),) p 1590 2311 a(b) r(ecause) p 1897 2311 a(they) p 2084 2311 a(hold) p 2269 2311 a(indep) r(enden) n(tly) p 2806 2311 a(of) p 2901 2311 a(\(CS2\).) p Fd 3774 2411 a(3) p Fs 0 2868 a(References) p Fy 42 3050 a([1]) p 171 3050 a(D.) p 290 3050 a(Dhar,) p Fi 530 3050 a(Self-avoiding) p 1027 3050 a(R) l(andom) p 1359 3050 a(Walks:) p 1651 3050 a(Some) p 1880 3050 a(Exactly) p 2179 3050 a(Soluble) p 2465 3050 a(Cases,) p Fy 2733 3050 a(Journ.) p 2997 3050 a(Math.) p 3249 3050 a(Ph) n(ys.) p Fz 3481 3050 a(19) p Fy 3609 3050 a(\(1978\)) 171 3150 y(5{11.) 42 3316 y([2]) p 171 3316 a(T.) p 282 3316 a(Hara,) p 511 3316 a(G.) p 628 3316 a(Slade,) p Fi 873 3316 a(Self-avoiding) p 1366 3316 a(walk) p 1553 3316 a(in) p 1655 3316 a(\014ve) p 1809 3316 a(or) p 1916 3316 a(mor) l(e) p 2126 3316 a(dimensions,) p 2585 3316 a(I,) p 2673 3316 a(the) p 2811 3316 a(critic) l(al) p 3091 3316 a(b) l(ehaviour,) p Fy 3487 3316 a(Comm) n(un.) 171 3415 y(Math.) p 417 3415 a(Ph) n(ys.) p Fz 645 3415 a(147) p Fy 816 3415 a(\(1992\)) p 1074 3415 a(101{136.) 42 3581 y([3]) p 171 3581 a(T.) p 284 3581 a(Hara,) p 515 3581 a(G.) p 633 3581 a(Slade,) p Fi 880 3581 a(The) p 1052 3581 a(self-avoiding-walk) p 1720 3581 a(and) p 1884 3581 a(p) l(er) l(c) l(olation) p 2305 3581 a(critic) l(al) p 2586 3581 a(p) l(oints) p 2832 3581 a(in) p 2936 3581 a(high) p 3116 3581 a(dimensions,) p Fy 3575 3581 a(Com) n(bi-) 171 3681 y(natrics,) p 467 3681 a(Probabilit) n(y) p 901 3681 a(and) p 1063 3681 a(Computing) p Fz 1498 3681 a(4) p Fy 1573 3681 a(\(1995\)) p 1831 3681 a(197{215.) 42 3847 y([4]) p 171 3847 a(K.) p 283 3847 a(Hattori,) p 596 3847 a(T.) p 703 3847 a(Hattori,) p Fi 1017 3847 a(Self-avoiding) p 1506 3847 a(pr) l(o) l(c) l(ess) p 1784 3847 a(on) p 1900 3847 a(the) p 2034 3847 a(Sierpi) r(\023) p 2245 3847 a(nski) p 2418 3847 a(gasket,) p Fy 2687 3847 a(Probabilit) n(y) p 3119 3847 a(Theory) p 3404 3847 a(and) p 3562 3847 a(Related) 171 3947 y(Fields) p Fz 414 3947 a(88) p Fy 538 3947 a(\(1991\)) p 796 3947 a(405{428.) 42 4113 y([5]) p 171 4113 a(K.) p 277 4113 a(Hattori,) p 586 4113 a(T.) p 687 4113 a(Hattori,) p 996 4113 a(S.) p 1084 4113 a(Kusuok) p 1355 4113 a(a,) p Fi 1439 4113 a(Self-avoiding) p 1923 4113 a(p) l(aths) p 2129 4113 a(on) p 2239 4113 a(the) p 2369 4113 a(pr) l(e-Sierpi) r (\023) p 2721 4113 a(nski) p 2889 4113 a(gasket,) p Fy 3153 4113 a(Probabilit) n(y) p 3579 4113 a(Theory) 171 4212 y(and) p 332 4212 a(Related) p 638 4212 a(Fields) p Fz 881 4212 a(84) p Fy 1005 4212 a(\(1990\)) p 1263 4212 a(1{26.) 42 4378 y([6]) p 171 4378 a(K.) p 293 4378 a(Hattori,) p 618 4378 a(T.) p 736 4378 a(Hattori,) p 1060 4378 a(S.) p 1164 4378 a(Kusuok) p 1435 4378 a(a,) p Fi 1535 4378 a(Self-avoiding) p 2034 4378 a(p) l(aths) p 2255 4378 a(on) p 2380 4378 a(the) p 2524 4378 a(thr) l(e) l(e) p 2733 4378 a(dimensional) p 3202 4378 a(Sierpi) r(\023) p 3413 4378 a(nski) p 3596 4378 a(gasket,) p Fy 171 4478 a(Publications) p 647 4478 a(of) p 742 4478 a(RIMS) p Fz 983 4478 a(29) p Fy 1106 4478 a(\(1993\)) p 1364 4478 a(455{509.) 42 4644 y([7]) p 171 4644 a(T.) p 282 4644 a(Hattori,) p 599 4644 a(S.) p 697 4644 a(Kusuok) p 968 4644 a(a,) p Fi 1060 4644 a(The) p 1231 4644 a(exp) l(onent) p 1578 4644 a(for) p 1711 4644 a(me) l(an) p 1933 4644 a(squar) l(e) p 2191 4644 a(displac) l(ement) p 2682 4644 a(of) p 2780 4644 a(self-avoiding) p 3260 4644 a(r) l(andom) p 3563 4644 a(walk) p 3751 4644 a(on) 171 4744 y(Sierpi) r(\023) p 382 4744 a(nski) p 558 4744 a(gasket,) p Fy 830 4744 a(Probabilit) n(y) p 1264 4744 a(Theory) p 1553 4744 a(and) p 1714 4744 a(Related) p 2020 4744 a(Fields) p Fz 2264 4744 a(93) p Fy 2387 4744 a(\(1992\)) p 2645 4744 a(273{284.) 42 4910 y([8]) p 171 4910 a(T.) p 290 4910 a(Hattori,) p 616 4910 a(H.) p 738 4910 a(Nak) p 881 4910 a(a) p 928 4910 a(jima,) p Fi 1147 4910 a(T) p 1200 4910 a(r) l(ansition) p 1559 4910 a(density) p 1851 4910 a(of) p 1956 4910 a(di\013usion) p 2305 4910 a(on) p 2432 4910 a(the) p 2577 4910 a(Sierpi) r(\023) p 2788 4910 a(nski) p 2972 4910 a(gasket) p 3228 4910 a(and) p 3397 4910 a(extension) p 3772 4910 a(of) 171 5009 y(Flory's) p 453 5009 a(formula,) p Fy 785 5009 a(Ph) n(ysical) p 1114 5009 a(Review) p 1404 5009 a(E) p Fz 1488 5009 a(52) p Fy 1611 5009 a(\(1995\)) p 1869 5009 a(1202{1205.) 42 5175 y([9]) p 171 5175 a(S.) p 273 5175 a(Kumar,) p 584 5175 a(Y.) p 702 5175 a(Singh,) p 962 5175 a(Y.) p 1080 5175 a(P) p 1130 5175 a(.) p 1185 5175 a(Joshi,) p 1428 5175 a(Critical) p 1734 5175 a(exp) r(onen) n(ts) p 2129 5175 a(of) p 2229 5175 a(self-a) n(v) n(oiding) p 2709 5175 a(w) n(alks) p 2941 5175 a(on) p 3061 5175 a(a) p 3135 5175 a(family) p 3394 5175 a(of) p 3494 5175 a(trancated) p Fo 171 5275 a(n) p Fy(-simplex) p 551 5275 a(lattices,) p 860 5275 a(Journal) p 1161 5275 a(of) p 1256 5275 a(Ph) n(ysics) p Fz 1553 5275 a(A) p 1657 5275 a(23) p Fy 1780 5275 a(\(1990\)) p 2039 5275 a(2987{3002.) 0 5441 y([10]) p 171 5441 a(M.) p 297 5441 a(C.) p 408 5441 a(Irwin,) p Fi 651 5441 a(Smo) l(oth) p 945 5441 a(Dynamic) l(al) p 1358 5441 a(Systems) p Fy(,) p 1698 5441 a(Academic) p 2078 5441 a(Press,) p 2320 5441 a(London,) p 2644 5441 a(1980.) 0 5607 y([11]) p 171 5607 a(N.) p 284 5607 a(Madras,) p 605 5607 a(G.) p 721 5607 a(Slade,) p Fi 965 5607 a(The) p 1135 5607 a(self-avoiding) p 1614 5607 a(walk,) p Fy 1825 5607 a(Birkh\177) p 2030 5607 a(auser,) p 2269 5607 a(1993.) 1878 6120 y(37) p 90 rotate dyy eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0205130553254--