Content-Type: multipart/mixed; boundary="-------------0311141042927" This is a multi-part message in MIME format. ---------------0311141042927 Content-Type: text/plain; name="03-501.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-501.keywords" Percolation, infinite graphs, Peierls argument ---------------0311141042927 Content-Type: application/postscript; name="ps7.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ps7.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: ps7.dvi %%CreationDate: Fri Nov 14 14:39:24 2003 %%Pages: 12 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMR12 CMR8 CMR10 CMBX10 CMTI10 CMSY10 CMMI10 %%+ MSBM10 CMMI8 CMSY8 CMSY6 CMMI6 MSAM10 MSBM7 CMR6 CMEX10 CMR9 CMMI9 %%+ CMSY9 CMBX12 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips.exe -P pdf ps7 %DVIPSParameters: dpi=8000, compressed %DVIPSSource: TeX output 2003.11.14:1439 %%BeginProcSet: tex.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: alt-rule.pro %! % Patch by TVZ % Makes dvips files draw rules with stroke rather than fill. % Makes narrow rules more predictable at low resolutions % after distilling to PDF. % May have unknown consequences for very thick rules. % Tested only with dvips 5.85(k). 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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 69 /E put dup 71 /G put dup 76 /L put dup 78 /N put dup 82 /R put dup 84 /T put dup 86 /V put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueXX 5031982 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d56491bdecdcafdc988206c5a457a19270b 37d0ab776e03eaa7eb568eeab6b5e79dec03b0dcbf923a2aa8e4f4deda2cb043 858d8b430efef0ed914a80ffc9818aba0fb30dae3694e5b31df3855892d59b89 7f82ff79fce8444c7926ffc4ea008e63bc518936c9098afc2d1c14030a2c15bc c9285c1b57e80570d2f6301009e63d9ddeba44e3251c75ca9616a2c5de9ade66 1a15db37b40b67c1bd71f77343912c62f47dd24be0e03dd833ea60d2eb020407 eaf042fed30f0c2f01956c83d46d982b50dd979c2026ed5f71979dd98c9c1eee 25054000a5eed5ce9498b2cb59a7ee55e0538ab47d86e69b2ac79229c8c6bfc9 3a24e822a2f1c9964fe630344579a25f66646113f73d1315c447f070c575de76 266c6773b01137e4a039df5e4c2f199e8e3396ef9fd001eebe090e04384c4089 ec9f13a07ad8733c6af775f432f154ef50ec4477e08760c03c006e4d1eb1c7cd acb8955ca4b0d62c7de046ff58d431bb9883d28586552c51809e3ced4dfc2198 b97ad9e52ea33c91b9e335a80d9b77d3430e0487930f992459664abca5dd9177 150b0fb502fbb04ccaa94ffe2697ed07906f95fd98e7b325bfbf10ea6b39fd76 c1e5295022cb39e9cd471b7b0b3f9d72da99deb0615d35cc22f42dbf5cfb70dc 6279fd520d99f32201abf25e621cfe1d37961d9c0ad36ef3d9264c547638152a be30812b1188a10f6327c9e8714449de2ae636420f56a44cebd774f0b131c629 818455cc8dd8d40939f3ba8aa5abc54590c7286b172193c23adb1b4f9931a9f7 73b960b1f5b760438028478e4442dc93a567b07e20017c18fca2a1daa2649dee 5a6ec51fd6c3f4b27367c31e95a56ba4bbeda0493bc02306037e2ba69add9b94 373364db316691e73d6811cd76fbc92b0d35764d92b73d7a8d21df2d89ae678f fa4fe75f55b0c6e63ee6d38aa7f7353f98aa433cb8ea0dd997522658fc64ea42 6f7f4332ea55cb036dc03f1e03afd67befd7c3705adceefc89cb8acb0aa84123 bafed35b1596315dc6e74355fe11e7bf176b49c5a1059da56e016b297f73e4a2 c4c9c612a98f6a63010fc775c38fe92f1f34d325e3d9451655795d72b48535d0 4b6e89fff86a08ae803f4c56b67fb81f4b065f34d7768d32abe37b77fa4a6673 68db980e32729d186d782188f57c41a0de7d1caaa5d40a612f42c3c9c72898cc 5a8a4bad26bff66fc4475c7287dfd55161a3161204c7fcfac40c333dfd101a82 d1cc1e5878dc5e20ea097b15d466791b65aa4e68f5493984848da7c2a062a874 83b2bc6b0b186d90c2dc26213abf3569ebc41a6836b010b395b601ed9f83293a f28ad87bdd8177b3bd447e89f24fa3bcb81ec035c6ad6c527cd781b599cdc3b9 8042d3909024fcbcd16c942c317bcdd62301dc3f05ff9f5e721b3b5c3428aa35 23c59e6c09c017ab322a8b36d35b5b487f40b1ba058efc919cf8e154e84dd0a8 c7f7a7f6ddd458051a728f8a7f70438916987ceabcbdebec17748ef2b0345ecc 9aaca27b6743f584d6ffe1108a77a05cdffe763fa7b63d95e4064cb32af3fe44 65601073000f09a6a23c44fe9133312d653ad4e45ed44bdcd6cd8eef4f9db166 04399fa4606d39471e76c5be497d76474b6528e8753b18a80b7204b4e2a927f9 915a72136ecd442a4b71abb8cc38750fbcdefbe8ff5c5088ea3f9e91df6efc10 cf2455d491b05a1806767204ee8fd45b6b5aa7422a7ccc86c9debe8592613807 416c4cc7e8cbd3ceef2a4e4aa5d0fc5f058ca6da3722381c4fa52e1c3567100e a17b02a14dba1f1666a1bb154f226c0dbb04a1347de882ee4d276c9d6f93ef40 1545fdffdeed99ee79c7e9a75308b4740fe819c5c87e57da7a4798352e2372ee 986ffd1fb314e34e1bde16eb881e9dba402a3a02a7a40e8e153346204e2b7e4c 5662a5f7a08065ec493da59285c60015172a8572a97ce6b80c61b7623387aeb5 93d81acacc2d0226eee3d2ef38eae338815568b116bf5b804613431b6cb1c16d 64d113fd5fe498f4ab4f12c1dc2fd9558073cc9c132b70eecc6ea375995f8d89 e79f4fe33bf4a7865653948bebc0abf9643f0e370600dafabf0731e227caee94 66645e0364c6410ecac922f504047b3354fb7867af491de2d37c6798215e6198 64cce09bec2da34264a5eab77a1b45763619b6b611981b719f690667db4260f2 ffbfdd3dc8483a1f9e644171c9ae9f802e040ab9e1f522375f2069a3bcf3cf47 8addbe617ee45408e945af5f3bca9ad27694cc80ef87c98b8be036ff33e16044 c2f323de6419d6e01178f55fbe138f57985703a6324e87e9128710e72b2a64f2 9bcb21fba23149b65d08fe5d7fc42af2b8e429e6a29ca5accb1862631881ad71 5194c4d2a2d6741096d2609049d3a525f9f77a0b58a135a73e0b0ab6539d42f2 9b40a8dc88d957b9bfabe9d3781c82bebfbfbdb2ea1aa11236056eb0a86fd6c3 28118be477e0306b7e1a3cd7bfe23dd26e0b433974cc1ed349e531dc3a20fe07 af78d1fe9c5a452a5748e7232065c0c4b32405cbfcc0e956dbd7619e76655633 f4fdf1e4ad9e0c4379e11c444e3fea813bc25d4a368ea8c8fc6eff06f78f87f5 3e1b789bdbed546197c18faeede527d46f73b164006da44e5f6adbb0dc4218a7 b07de0c8700e0ec9a848c7fd472a97cf0e780a2e1253a890cc1292d266899349 b986ebb94430dc27eb19f7dcc367db28f4608bc14fb6762019b6e787f3d0e837 5b9bff02c7f1c3e9fa05dddc125967a050ec175311c2f85a07e670d9b9e00f9f a84a170ecba41cfafb52d0bc9db94caf9ee86017a405e3efd18b27ce61a04860 7c2bd8d67c45b137cec49433908fcfb81d51a03f3f56e01d77d311642afa0333 7b752cf2edadf92fc797b5475612b1ef50ad899b97cb92460e4f74a59f7485fe 9056b833c442c099e2c250bce3dbeb407b1816353d2e5dc9663bce5b94e6ecc5 b65550bc54cb7fe71c3465b5e333f80b8d526a9cf9ef4558572a99ba424e9cdc 614ea93acec3eeb17bd78bea7b4e02f0cccad0a4607ac9e8f83a90a0199dbc88 baa585332a128df25a6c809825bcd6d68dc853485e27fc6106bbcaa680e66312 041d6b67706cdcb9888a3ad0313bfe5b342d43410c39385bd473ff862aa85bbd 5fd09015f5b2e2027c7cc36e487f27db5a2a112dfffe40ef8bf7942352e2b21e 97b3f863757ea917e41fdfae5ef4aa1fc06d9cbefba7291ebbc069f2f8ea45c7 2f13b4d7b0da420e9afbcf300777155ff712ca64a1a37428a6f05209dcd3682b fc332fe833f8ab7b7f361b1c9dd0101e1ecf47086e4e95d2be2e6781e071ecc0 5af9c711d7d861961a7a5f3d092fc96ee7a06e8e0f3bbff1551b030e1e913407 172c55d7b68da93a61a236ebb2f12a6db0f6899bb3622370ac3239dc3122209a 1b0887b0278a90fd0ab650cb25f5ba4fa46f764a98d739aafb165545d1627467 13995ef8d1c82ba35f9605e16f2337aa284a64914290362e21fc84c77b696ef0 33ac144e36531a914cc91856f774d389cffdbb0de379fbdd86c98cdc41cc9343 4d0ac6f58445bbfefc95f379cb2a33a91b3047061fc7c27b3bcc3bdbccb7fcf7 67514f8fcf473483d7933e0c8dd4a67110e36b4c09b726db724cf82f81a86ad5 daded5ef45ab7ae6ba16d90c21ca15d6830afaa2a4e8e599925a3c1f9df7c9da 91b535c3ecbd89bbf6901482dbb345ccdc28072c5433381095f134b6e1f85424 03709e4bb2a0fe5fe34db24bbe240aab15d6dfd9516901ac756cd59256ca33c1 afc7ef3acb7783b0499e114a42dfbe6fe203173e5d34e5c54d107a7ddca91213 98afe06d6e35d070e595dd1baadc600631aae246c04fe788ad41ad3ad0dc9cd7 3e7c7e672e7d68aa524232f4efb86f571a201613a5a27f0c48ff8257e6522453 c247067e3709f54d89702e0d41f9837a336221f997ccca1e00a1cac23a02dfad 8c5139a45925f3dc15f9d1265617518f68d6be8a5fe32c2cc1621e0fde25ddfd aaff8442cc988d8fe814182ca8767e937b54a28a3bc292653ebe80a5914137fa 7bb34a7d83f97d28a954b97935b2bd13b1ca39357e48ed8217d58fc6f3010cc1 517ca0f37da17f47ff62f3e48698fbb83e8f7c99f4d7f8bd922f3e5a9ce074fd 9e33b80308783e17221e760f500a185c89e3279101a0de477e268d236f 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY9 %!PS-AdobeFont-1.1: CMSY9 1.0 %%CreationDate: 1991 Aug 15 07:22:27 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 189 /propersubset put readonly def /FontBBox{-30 -958 1146 777}readonly def /UniqueXX 5000819 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8 3750d597b842d8d96c423ba1273ddc63eb43f34fa90ea73a234af35fd9b24eb7 4f19626e9124df7acb92629cc87b2cf81c4a89b27447806db34393ddf0402958 e2daee0b4bf315f9aa0a2650d2e11ee289ffaec940d0af160fe7f29411c1c458 69d1ac3bcfe2ae4da72422549095f1fc9fec606289316f106e719a4ae2fd45ee 868fddf07a80e9a626bb43cdfdaa9dcd208b88fd99af20b27f9d6b5cda6a3fa8 8d5acfce8b5eebb38854378cecc3bfdac526514f1be9fb28ae628918ba942a3e 05b78c3aff808cf416a5ed01be280231b75a7aee0c66b0549713bef6b3cbc188 e625ba81a0a7f34771e69cf51959595a11c3e703d4cc8f9ee73328e271bdd040 b6d6187379283791d3e1118ab776d5771a400da9e2c4dde9174abd1a4ae32164 a8f35390508c738d70ad7df7f5f294768bb50538585eacee06ba09533f9e88af 5d1c44f08f3f8f3c0a1ef28356d00560f860ef1c573c99db12d4bccf421f266d e4e7dca60220b836d81e004e9951785eba1e8f5aa779ad9c0db288ba0fa5fec7 c05d4b377df8a93a12b0caa0e3242615b60dfb120f73c1afc8671f0ad5a2fed0 fc4eff81752bf883bfba365964aec8b75112aa8702bfb67c71d7db2c9b1877f5 eeda089786277e6a2a50053209d15b6119189723d27370a8584cd6ef58b950db a939e8a8c869fa105434 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 189 /rho put dup 86 /V put dup 90 /Z put readonly def /FontBBox{-29 -250 1075 750}readonly def /UniqueXX 5087384 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930da4f8d58abefd45db119b10eb409dd89792 3c6e705479464a4b33ae3d31bfe98efe259f07f7950237bbaee4f7b64ffea83a 757fa717d50c3298392891bfd60e34a056b0d6021da3fd9b8b01bf78b0b23dc4 ea3b0605150d20b27ff9ea5f2524661019d982a2e47cd7c21ee5ca9ed9227821 f8d07119397de26838c11d7b35bcedc43e011626e300f8249ed846d7b5ccbd89 02550cda17ddf113df658fb13f8162681013766863784efac128e01ef997e1a4 30312afac8f3d948edecd7d5090ab5c864b91d7f6e80256f0e1fc99f1102b74c 61cc456e2e1bedf0e627d5f70f738f963a13666405c51ebf23ca077e97570a2f 1118536b92e585cf1dbd5fbf36a0a5d197538c05304e93a15f3c931168d91d42 483ef428b1a3f1b7e548fd1d23b80f5523c8b8bf450b02804fb689f915c86693 f7dfb1c43f681627c7f027021aa3a8179148f1819cd94b8890a9ac63dc1e8a68 f254b64f6f173c3d02e3ebc037da05074017c8b113e94f4e2d26b467ef9da07e afd7ea7c86e025af52c5ceeb835951e67e02a38daf1e9e698bd21996c1071e84 ff7fbe77f6ddf9d06fa92b16ecebcbe0370b5cd9d450d0f2906a105cadf93ed0 0abc7fcf2c996e7c15e452006e35390d99b9cfc59925f1444c20d9c3e1fc2910 32cf9fa9d580eac1bc956334da391b870a757c3bee6aa3e005b20988d8587876 3ced20c6edf2ac2b22890e37ff7aedeb0986d85ae05bc84339bf2116df48c5d1 cc888bb33d409b06dd70ffd0618b2080442aac439aeb39c20a87ad74ef10629c f1604db8611e8e4224c5b21ab0c7ebddc77fe9a283b3a8c6a2a9508f6836e3fc 3eae1366b4da4f44cb7a77fd81d278c17775d56cb28b70f18d894bfd1ec1827f fb1099ea73d550aae6858f31b1b18cd32c08de808831626d4c6672a6eddc541a 8e0f404bd0fd25aa6195ad04aedaf45403b4935b14860a6bc02b348199bbed87 08ebe3341c04a19a1319bf88f84e420edb7081519350ac9af1b82c3890db24b9 90f807f535a5d93c0891821db391ce2fb151e6cd958730a718427bc5129b6b49 38bb9bcd1ad9cdd295e8076f6db79fb911d6bc1a310c0ba3dd70db441f813c1c 0b7f58be9bf857e3f999230c4884510e5834386b3e7e93ac25e0ee45cb336a2f 3f5f00c349d73121c1bd3489d64a4f420136e4dbaf94a56845fff48aa354c85f 7201775ab7d8ce90d6bd16d433d1b94de49936b514f9aa97c60ba4d93ffb1161 537cd43fe8759a4597b7e39800972c511870d04e2a886d0d37edc07f5a7abbfc 274e6975f6682b0a650af9ec73d61c2084dc0687f2e5c5e5cd86a3deed583ac9 58d26b5f9f0e284b83a31e8885c6bf9adb6fea8d45fc5bece2e5bd54753885f0 a334c85d5d582e274060abe9321baf42777b78de5ee2047ae687abed9ff13e29 ae0b173df33c4c2a4ab9bebe8b0aba5f792b6a7f7553bf687aed0524d5668aac c9777192a28f00fd4bcf0f6e3665cfb11fde16859bb41eb718cb5a963ce50899 4c1a59d18789162323938469b9e50e9346f42a271605880d520a1e173fdeff4d 9bdac9e3549e985f3f5a59694d3c7e1dc7bb2b3c41edf3183aedff647ec39e34 d2c964d786f349762161d8bcdcf2a48825fe1939f9a81e7a79e0724af4b5fb5b 3baef5e0ce1db25857dec8c7fa972439001d32eb41b3d8926ad7d39fc69b0b26 aabc6cb485223b4b0dbbdb9276a29d55d15cfab4bf6528aec564b8f226167f5d e3fff6adf16c86a5ea3946a76f4fc3b5c13ce5977c555255b81d80d24a6df35c ed12d1e20dd90e264a948ceac9ec4ec15e2cad73ae5c5551 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 175 /fi put dup 189 /ae put dup 40 /parenleft put dup 41 /parenright put dup 65 /A put dup 80 /P put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put readonly def /FontBBox{-39 -250 1036 750}readonly def /UniqueXX 5000792 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b448685bdd20eb05d5e7c2126132e33a59a7170609 dcf4871a5d023c9ef57d3362d9f2d7a440bb69bf653364105f16f4d0f03582f9 aced3d05cc76489b16e3fa8a446094d30038b06ecceda269f2eab9d19a99c7f9 39f9548f206c5a457a19270b2b82c43b091dfc5573468eaa3e7a4a32f8042891 d85e4b180fcbcb3091d2800e54c87d84ce9cad6869b5aabbbe47f40c68799893 d22b765295e1e69e33aa048b7ed98ba480ceca91f3ebf8ef85fe9a3976909626 b95ac5940d53f9b02215d84a44837ba25ed15cce0d504f1d335065594f3bc824 5405407591cccb11cfd4645da60d960c0b93f187b0cf7b105543c0b70f89af5d 264b6c026e3ae646acf145950202ec73282111e3e601cf2bcde22ce3edf6db23 516481420f26552ff4472d749811f27768150450d0d0ebe3c79f999e99b5c0f2 2eaebb12d97782b1bd91b2a1f62a76412548ad53c0dd411d4a08c0f071c2c218 63d9adb75a4621803ecb84c2bb235b620b658984b2d8e0c4637e2811bc8f0d04 6c8935afc70141e1b2d9c23bdb251d304b3378faf8928bd09686aa0340fb0cb1 dc48c996ef91530ff078666fad227a3589f50b605267212d3a65ebc1019a8eec 9a0739a00279471a01e1505c17658c10030fae32f274fdc8c8774b0d5406c384 a1d17068dcb0c2575c562fdc5a2176609bad9a2c255e426a4325a4fc3053c7b3 2f35f7bbc9aa50135f30223360fa2fbf019294b3af98224c5c05d6038db08bc1 4103010094dc215ef9ccfb96b237851332c8ed81a9184cced4edbda26b381f0c 73a2bc4a92533b0e667e82ce49ddd8ac9e34d9d40fd4b5b9910eab3b1234bbf3 af6608d66eaa424c158eddb75fbe39462708da8f5c18d92fbdb2f534598a844a e076f96a741f8002af3b38d7e4bfa363f5f5d1193014f71b8117c85e49fc8ec8 1ad4b59f29c635688d1d20e0db4cbe33b9e0b767d2ed5417010be15d5740609d 2fad0e57d61b108d380c5b1493cc3b6a4f1b2ae824f777e5e029efb900bb970e e92bbd22f02332a8452896a84f5333cec02326fbf1b14d47d5e632dcb426aee0 343b0a1ae0454b4e13af8ffe864d05745afd313cdf214fcf880e51f4aa200c0c fb7aa82b9b1c39d9add18c1494c68861acbc4a026a5fe39c0675b6896f7c5c9e bdda10a2d8ca96a5696e1611ccfdee6c963fac39821baf55c2089db9bdeba4a5 95ecbde970f5be3356e13b8807593518f1c3333b03f8e7f2249e7a5131b9b500 db5d35ec3e30d07fb3a359faff3549513b6dd598bc8f58614176afeb24aa664e 528663f22c0ef6107f2bf22246f5f75af9fae271f8920c42e422f6133d4e7608 c24d68b684e036d837112c0ad7d35cdd6281dff6bc760bf45b644f84fbdcac9c 102d45567314bbfb8070974db73ce09fe2e30786a96d9c7822eaaaf051bc96f3 b1d3cdae606f160e09e92d7220097676d57edf498483fe8f737b9852cde371ea 50c713e27121818471cc8c7a0e46ecee921c83d3054069a96e7d9a3d349422d1 6d1a47db04fe08168c194ae43a37f0363fb2b93a6db82e2121174d490c8e9633 6c720d5a543e3cbf0ca8bd0e9c043d4c82a7701de89f32635a3abe5238877f5d 011542f17a9c38d8ced8a290f29b6d95959c802619967a2fba848ac7997e4dc6 feb092e43780c95920cc402c23543d8ebe3d3c39d57ccbafe9ae150b0550a8f7 c004d7f5d9b1681d18f347bb7a3e5ca83835e04c7539ec0a72ee4afb84075948 23e91ca04ce0c37ac8278c6243748d001d931ab2af57acc90f8e0596d2e9c8cc 06b2c7aef80aa5f151f9b2c9d0fedb1c1608128f03d3192f05b0479ab32fb82a 3bc5729dbd4becac171644ec9ce936e007313faee88426f9841cfbc119f563d4 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4be3db31b3d9f5279df227040d968a4f98958d4f3f9de14d63d5e940e5302c58 3ed8810c45cf19d8ac15dee919cbf649f1488a87bb7e1beacc99ccd1fd4a6f08 74d6dce99954b2d266e791cade045148fa4cc80d72158e1aa0558c37728e970c f96a66addf6821a7a151311e2f5bfaeb66e19ab0edc906651475614e5ba5a492 fb75b947b391134d02c4a1bb293c7f81c839c0158249ae5cb7cc90130b67c701 7b79493f6ed75ba6df1fb8519e61310e27a62f5f7b54b04c3eabda6f65e6fa4b a720bdff8635d7439216fe202915d70e85af2d790558ce347f0db4189d7f2cd6 38cb18527e11f42b8d6373d9c9865084a54afca80ff0f3e82978e50a74f09269 d7d05c6e5351c2043a83aaf77bfe58765ec9a0a00cc1bec77b2db598f3c23341 fd4c697ed0526f5c18ac64f4f383c24d946764e3518d5e651d57b4a2be142e3b c4c727018f2744c9f51555a2f5267bb91322301c78a5f7d566007f5a6064a48c ef968b4ca045a433b34f279e2f110a63e9580f76846aed82a8efb87df5c2a3fb 5d0f8230387a40e9235239fd8c7f8efef89be3eb90b9804155a7258d3b8d7e31 cf8b0507e93e4c960a11efe571f1f181ca665f08e8a46f62c81be6ed10cd398f a63ff124ba1f1ade181dc46463c8fc3b2ac1c7e11cb5dd186124052135951a72 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7f4ac200cc1db4a511741c4e63f9f9014443d0a6e815d4054042dcdea630ea57 ef2e024a7b34d01e78792c6c3c2b8a99927934d2044925635828551a3c0c25e1 2cb9dcb7d240cf097f2d0bf7748a95d0540b8924fc76c315a2fc19a36d6d9880 996834aa07ac537b0e7139114f3dbfba51af05443a0e137280045c6246821a30 ec2878bd9f1a400600a22cd8a4d4eafca09b6c5f8aa027168fe3ada413c43172 8015537a050cd680cbd051d4b27d56ec1299833ce21bd8e504cec0dc4afa5aa0 d78c025b9df8fb8b0495a8180589bf9c992ea222cd6b3656ea9bd05504778f55 e82a8c7f854d2ee92b298a548af6aa74556f52cb4e7d4d9b8fd52bf8e1af992b 0589f840b9c2baffb085d46b5db3a19f4e66b30fb52ce4d168e552c8e53b1fed d4774ea1521900d3b6516c4a69666213b1fdf4a5fb8d2a38aa2a900f4700555e 81292c5daa4a2fc5b2e5726363b9ad88597a65b6bf621af7815c0eb1e2fc22ab 90ee57f1bb1d04862d717ac92b366c516f8a736d6a0f13da12cd23850c8bb21d 5da7f9e4f70420e24cd1f9c0a34b5fc927c5aaa9fc016ea873497b6f0a195c2a 19c5f9b9af0fdf5f90ea5eb7def0c48c37579c91c6d9349abd64fc493da51ba2 c98b08aa363acb0464104012ac736aefe31a3612a73ea8a42534b545e29345af 87991f755b42dabfea949256c7f63f2712dc5b389f8130bdf49abba68b242fd7 6ed590d124691bbac28a56c0185843cb91d2cf6015e1ea9e8d33fa8e8118a525 757d7738b57593c6f16e7ba814acba099405705b22b7471c167719aea69fce4c 60f9b9de0800561795bab84e4a9ac3cebd74767843aa67e2adda24 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 175 /lessnotequal put dup 189 /lessnotdblequal put dup 69 /E put dup 71 /G put dup 84 /T put dup 86 /V put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueXX 5032014 def currentdict end currentfile eexec b5e2457af4a6caaa4536453c498a618ecdce7b5f6f2d5944265309f833bd3d50 3df8bd14c071f404d2177500681a801856cb8c9c9cf0112c9bd17ccba9ab8944 1759a320b2078df0fafdd3e357994bd81740d1a7522ddfafb75d58f5def046f7 084d10fa2f702e570b09a7073d70efe37b48c355a7089772238b7b4538c4912f cf6703a14fa416057ea2e1b94455913dd271b42846ee10d78270a8118e7e4bcb 8deb048bb280c240d59409d47046c890e5f79e5dd4b0be65a28a77aeb53dd33e ac506a6da87bb55e329f433627e06c3c94b078838b908538d93f888e504e25e0 3a8ccf7d2c02122ab3d485525564295390d1154e15fee35edff4eb016fadd32c 26d1b4c09de87943780954cebc4965aa890fe1f80b03eb07103e13d468245c7b c26455f229e8859585c6eeafd9474acdb27290fd62950dc52f893c99a2a614d4 2a7e3618266f3fb0c68a01a30373e048385ffe7dd5844e208a55d1a9263ab2db cb9b19038724f95af00349840157be45cb337871b55a304b87ade64b10ff0eba 21aa0eaeef74aa01f049303a8b25de0acc26bb3a33fcf267f479a2a73f7dece3 2aa23083687322b417fe86e8f5afc2136d045ad6b392364811a17630a94ff097 038eedeedb8da4ebf8fa5a96ce27a2ebf771d14f4a11fb1f9f48cdd207209934 885ef9118463552984b0fd3cf15d55de967fbb051549dd65f60dfe07f299690e 58252f032bca78935b7d2df852ed688bd13dd171b91cf7bd46d5433aef9f50f1 41a2517c4912b837793f84ddcf1116f70f7e9b1e6ec752dbc9183d4791d92602 397bafe29ac6f879ceb72c673c8a1aa0135d6f92d0db7791f9a2f1a5b55347e0 3e63723efd258754f5f4a9ad35873a98d54b6c54b17a39892dc1d38660597448 fdba9258493a0db0d560c01d49f2d889a99eaa8a5b5919cbd33f6b3d97bf27b3 e0b94b4dcc94213911ed25a52ec3a8ab35244b9d054a1b2bb41caac27c039234 04bd2b4100a350f1819dc2b4f825d7f5f006d12975fbe7a878e2303f70b34136 83bd51d67648f05ce83dc6a1affcff33ef690b66a70fbfeaaae99801ddc5b989 8b77fe83eb163fbaeaa41ed062b786bc3c6651b3055320f2df9adadbdb2ba7f7 587436e9387ae6db93858819131f36413647037c077ecd6d6c01ef6de4b51af6 d3fdfa15a81b56aead64c84912f607b3f064d15af2a1ab1be6c5cc2beb06448b 8849ec7647826016f6c9d87c46202987ad59ee4c9486a802b0d1c1e11509f49b ea056ba4805063939b1f88738472e51c6770124bcff6124e204705ff12398d1f deb21b7f7da431c9229e3aee30e28f7d88c3ac1114f0685da357379b580bc856 3cb04e8f2ef3a124fe26891cf6a993ca3fcc396d15f4a73049f62055c7b602fd a64ded5c20f9e8d51c241a0559397d35ce8cd25c6894e518fc6ee1e1fb2eb547 164b657b1b762437ec364c44c6bdedf6bd1f2b8d0d8050c3fb5bc0d136aebd0a f92375a9457fbb003f6ece517e319c0196720553324609170672b45137818d84 ea8513a943c003281d9e5c9b95244df86983f140b1fd488cb2e6230d6ae28c34 4e91b84408f8d57ab189b365f0d69f1779562a18cd1561c7b33cf0f78da051fd d348a937e6886744ce1bb88e123dcdbfdfb99bd9246288856bb1bfa788b28590 aba0f9e5ad52dfa631529bda8c215d91fea036a02ae6098b411587e0f3eaaff0 e183f080c8b1b5baa64f062df0ba23769963fbafa419c32f298c87f56d48d0a9 beac0ea7107d5967e298a60a69159693d9f7a671240abad030e6effa78cedf77 8f281b0b824600fc4bf1182984be3ece88fa71288f2b8bb1ed5ac18762dbb6fb d602525014e5435b4de1c0ad285660e34763a3857a1bcb93780287fb2397565c 0a6314798d27b0893c8e09b8631b68dcd72be029353ec6166d2b4ba60747c877 3cb26399598528b6c0867047bff625538be25657d5fdf47be97ccc85e1256df9 bfa1812b1369b8474c8b81c4c3534c74a8321f2309e8723c9b39963a52db9d5e 925c1f4e1c60c42ea4e46b00f678c2c66806f29c9ee5d67a8c75ef1cc014dc25 14b668b142a48ec636365a357f9c38a190b6a260a73a57878ff99851f12f4faf e214621d986749382d17609b351d52198eedbf3fb41cc41f4e90ee37241eef36 981c7b6e0e8455922b25924ded53353bcb8e1e70b9ab5eb61d5f52eb84f95dd4 755a1976ec845666260c902cd99a0b8b7416ea0149df4a9703319fe125b6c770 99dcb265e47e367d12d8dc0cffb2edb61fdda91dcdc839e09ed9c33879285336 9ccdc6826971e7ab94986362e697349a3c4708eb54fd14cd1ab6ce09ff02d830 4fac65f473b304ebd5cf15df95e42b8fb56110721088cfcfb6e39d031f9b94ca 198b5b34d1863b99e2bcf2056cd34660d53f87f335f3e2344514055ecbc156a9 7c187e01c5b7f622d66b6440a2654705d42ee405a51ea5013a27e0681a6cc469 d492e289c0b975c066d43733113a5c2c13ccee1ca61cc52e3892fbc2fbd07b93 d46828c7da58dbf4ece53bb994e75254bf814d5bbb0370b5ffb9d20bf0b08ec3 db4befa1ea6a967423867f491c5f7fbcfc0064326891c25a980973994bfb176c 8a4fc2d4e9cee913878a24e328150aacacaed71ebbc36199a7858c1097c17578 f8d33a5b2b361ede167f793fce7b2773055aee0f767ef1cbf972b345573ad8ec 463365d9c84217028b6b77cf9443f41f6590970b4ccb7ca6f50958631ff038d2 8f281ed831f564a8ae0f94c80e944a24696aff18c5a64a50b1a7c6fa4170c443 f7a1c4f41470894547b4ce62401f8e096177ac701441db99887592f870e58054 e91ef9192e5454225eb9a10005fb0ae9026de3efd2d57ae8636bcb725b539921 4965f80bea4a4c44f558ed37c59bcdc24aefa8a6b51ea6f71003aa1db7233e77 61b0637722184a6c2ff29af84e0511f4d962e4e207d8464c9a837cf99bc6d9ef 6ed2d9575493b6a7d6937041f15b380ca73f3b7cbf85d41a5280ee5e2a36c0d4 8fa89edf2900c8526038bf016ae461899f1e2060024c62656a5b4bf214da6972 308bf29a448ba146342b1a0d658616b4a58f474448b24082f175e84e80d69269 5f067fb3988d12125fdbeaa7ce8246592dc39273081a299e2d296c7056c04f66 5f61ad46a36b1407bc192a6d53f0ae6cb5c466720ca7c6fcb2c904f32f0a5ef7 1e3dd2c762df03b3ab2d345da3d55a0b928689f6b591ce511c19deda0cdcfd42 67dfe108451b71d45ba6d43f2127e57469a0c90d8944ffe0551301503c97aa49 cd00af1f61dae2a30b79fbb7a8c1b6d38d64728e0c7b06bfcbc0a72de9b6a738 8f5a85f5e6bdfe0bc7a1e00aa104e2b885f47b8e37a23b593a8a26593fed8762 7b0f6a30aa66e39417af4135e576ea836d9d8b961576b7b814c144ebcaf3cb6c b1b8b3166d551671367ae71b3ce6c74ca9a3f2de021e02335c9955f742433233 00cbfa51fc770b1b6b9635b97ed05f46a2e4455e4036ef8713fab36683dda3e6 4d729a42f20f13e5d1a571c244e29fa98411c62860e37f2618cb65005846503a 7bbcef9f2e68095b0c95c16977a8da0fe89882f0e4190dc8df3fd8fd292b923c 37a35234546556118d777eec5a57eb94f6ecdf16d1726e67dd4b346d459b94eb a0d997760c707e4fcf1e06609b6aa4cb96f59b31beb5bb15701c5d3da50f273e fba9103dc86e20bfb31687ac932ecffae32c4d5f7cfa7fbb2bc88dee1a1dc59f a0fc160bc61f8494f86ee1dd8c6cfe1aa1ee3dd8e49a01b7be982368dfa12497 5363bfb9a765835e2a25dd883d36a983906ad9dfea2dbbd849d9ac6e1b53a57f f18f29530eba6789aa888cb9494d51aa3e0eb8c51aa1db9655e1a0566adf79dc beec5d0ea53338651867 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 175 /vextendsingle put dup 183 /bracketleftbigg put dup 184 /bracketrightbigg put dup 189 /braceleftbigg put dup 80 /summationtext put dup 88 /summationdisplay put dup 112 /radicalbig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueXX 5000774 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d5649664203ab24acb938d58d246707ffe7 d62f04bec4b70c21ef75beb2b812622b3c74e969d72d3cd11bd7106294a99caf 0b1629bc7d4de6b96ca82930831d64575f23f4ad06a0e45e315b1d392411be8d 6d73c998789ff258a07a3c8c2057325784514c845500bfd1a971310cfc11d41c 1a167dbd5ff012c60add4e87325f6e5299032a839de65fb1473a166aae1876a4 414a434f22c1d241591fb36f857df6fa930608750ffc0c54f44994662b1f00f1 400bf752ea8d83ffc4cb77a290bc2d99981ae59a191748ba5c7ba1a9d2583fd2 1398452b6ff5d83a059f7eadcd2ef744e9dd22bdf9c79d049bf06835e878c32b 7765c69bdd8ef4deb4ea7cfff4cf9354a4ddffa689de961d16772491c7afbd7f ffde42400764c68e954ee5c455a5687959829bc3b319b2147deaab3628662c80 30c5e02fea09609abe4eaa12e217bc3af673f1bc36a7039eb13fcacb4218fe0f c5a3f9452d4edf46cc91db67b624d4f2d37502fb9f11af4da18ca40b61097f95 d44329375467ed13c5cb585ec53f62b83ef9502cc755af44bf32b87b8ae9f3f2 f8dbf72dab90acafbacd280db6aaffaefdff6d5eff26669bac56280a950560e3 d01714ba38659b0e4e28bf8cdc1f15ff7e3ea9d95a6afbbe3d8e6905aa6d531d 6b6dcab89e2036cc492c65f76ddcc3cfcff33fde3d9ffd10f7a90530fd5c7683 4329baa902e701ca42717476298c9ff511b1941244bb1dfd0e5cab92a43970cd 1bdb292eadd055c97972844fdcdb0aa06401e7b2632b9d134f96bce0064860f3 9859da7a9a82be0a840672a1331212752ae1b521b693b384a72dba8d37a86eaf a0341e7fc5f6ef7a8c566aacb91e8bf5005c18837894c4eacff41dba54e071ca 83f4d4be6c2568b5f01f2beefd6f5be299e4291b06f1d9155bc5270c267a3bed 746b9223b827f6afe6dfd727359ccab772edda6ee2e8af279673b578f4cc134a af8038c3a041f46896c863755143a7213105b94678445ddbd65519d1f5c7d63a be9cd4ae421bafca90d71048b8e3a092ce4fdfa3b145e1319ec2c37f00b493b2 41c041ddf19c0bc3ef63eb504d127efd1783945b706e599b446c3bfcb24ede6f cccd27ff555b646c2c34d9297eb445cf3ddcecea45003337d2a7e38be2b6be91 b0bde4f6e57d73f390945dacc0b0d0d439e700acd537201f198d3e33b41d0afc 434dab1dd23404df9a42a5671bffbdb9d8d0bdb398a6c5142000315d1945e04d 56b72847279ffd067db7c863ae6785f129939b978e2c2717cf1451c875c01ee3 6f8810082826f0a3fc689ac456ef69a86590382851e850c90c7dff6e360ef8e4 5450d18e8a47a201d371ffd78ef6bfb7ae5ce1b5c536a299ff4b51f774e3b2f1 794c7066f7e92562044e7bbe5182ce31c48c5a41ff5d5064d1881542e3d8336c 55a6b3955e567c29e66fa278dbedab1088675f5647fccd29811af6827599d84f 6528040f91dc273e391d11633d102a01bab2681c916fbe2bcd68d1d6562091fc 52f4ee95194ccea635121bde96fb25f5b786adc70569c4f3d26b8d096ad71a2e 5b6bb3e2ae01fc728e433e12761c1c5f30d135612e841ac6ea94f105548cecd9 be6752469a8adc8754226642d4515e45f3c60f23083c68d296c1d7aad60aa580 1ec0029a02e2089682995ea4f16fa6191e1a0cdc414eaa353ddf96a0c6fa2c92 8106cca81ef02228c48c59703c47350637b2ea5199bb57979652f9a1b4f6a0e1 b0063f520101aa19d56974cdcb5d8f617ba96eceda27819f48a7ea55e1f053d9 b7a6164af12f4be3fd74de092853bda99a1166f4e45977b468fe596d4eb03fa4 d132c43792ff111e16a4aadb3efd1a4bcfb96185494140dda7ae0da2d698c844 90d7197519797f98ccd55099b76cffc0f2efa8b5b4c31d5ebde81bb08d1d20d2 2b069ab0dd85b45d1d960ae4c51b5621d93cf0b0498745d757f2743a3b29579f f58609be031dd2d04ac279afd141c0ecf0e4dbe384f0a4f052c7f6bdf124d3de c5699749710c1c9479e8eafa17c080f026f92319833fe28d4b908389a374bfd6 ff7b92fb90df65e71cc7e988f48044425d2673cf80551403ab9f56eed6ac1438 5bc882e2dcf0384136846ca58c0b01376aeda20de9139ae6a1c688db2d666256 86c2cf0cf7505cbf4283a42cd31782a89e23b30572c3089b6fd1d2c350a1994a 28277255c11ae73f3801495893fce7782e938e5c5b569c808f19a79c750bd220 4efcda7d07207e8ad7f8d0fe3eaa94e4d8df687354fdfda446250b5abde44cec d1d0403c83bd59d635506d36988da63390af98ed2d83d49811d3cfc3e8aa42bb a39e226f1e1ce916969563373faed08e42b0c093111f1b2838b8d6c100b3632b 4513e9422badc343 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 175 /fi put dup 183 /caron put dup 184 /breve put dup 189 /ae put dup 40 /parenleft put dup 41 /parenright put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 58 /colon put dup 61 /equal put dup 99 /c put dup 100 /d put dup 101 /e put dup 110 /n put dup 111 /o put dup 116 /t put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueXX 5000789 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b031f3fc6c13d7da7277a94ba 018e9998b3dd888011a5d7c4204989f30f908b95533bda845746b673ab71ea57 65a0d14f4350707e47c8276305b28513cbe1bb0dbd269a53719bda46e536685d df78ca0146b6b93e760256b74d939d4e35b5e77238f04c92298dfdd188feea30 e053eefbcbb52f2011772b3aae39f5805597bbc1e8bb75a446ce014030f4f2f0 f49f9e962ee4a1024a746fa92a3628db5270732b54e43fe5ecfa524f127e5fcc 788e77e66098336ad67fe4cccaf0253272d5df79864bf4b734cb9a5859d557d8 bc11b8e00221ebc12e97de4b1f466ead83a4c894709363bca9040410a52d592e 34ee40cc7e5efa920546b981aa659513a24b1b85c221a1875b62d0b89e57a368 321b8043a5b094e0379760a443d632892b14ad6d19dacc8c78093243ad67e6a3 08e56e6b68412ee690b10dac6e17708754a00d51fc957b500eb80175716eef4b 2ca1ef867614659bee3f2b7319e97b6fdf1efc847bf3cee3156f72f21751da8e 5fb6898919e6799820d3de0642d756e09d6fae4ff08dd3deda3173bff4bb11f7 9109c97ddc05897af709ea199a90fcee8ce4c7a3c15b18170c41c04de2d3fba8 f34296a95b8e1e8de3739b17273f8f2c85e914615e8eac5e8bd2387ba3b1edf4 7968f06e2067d836d0f9f3e085cdfd2de06a62c81d786b304326f7002e83160a 36598589228b4dddddc43c85e1d126f8fe81b828028e26317af5894aaccf4f69 6301e1a9fc45935d8a414957f08febebbc3a72ada80f101e47447d019ade56e9 f4fab969bba2b44e47399fedf5caa1bcea216d7ba713d523dfb63f2cc8ee4b7d e1b136548c187765cf0ca0405b03fb2ba17ca922003907081814f9ba564c03d7 1f5b1283b8d2a0ad00641acef57168c4c872c3933355d19f71c4832aac70c507 209d58640bae9d77d7e383a655c3392940d8201f1a076161c53a331d1e7018da 7a5849c19333357415f69fd13eeda746b52a6a2a3cae53a83ce1a630e2183300 7875244334fe8e0e7d89082d1851c9ce22a710a148ed5d19fbf9fdfe0849ae82 0667c1d56c05f310d9f2805b3e5371d9d21661cebc56779aeef7c7e96e298048 4b356a247f2b7c7da1cc9f300b5aed7fb568647f668814d33a315feafb3468c6 727db3f27d152b8d9e46af6c2119bd30c6cbb32cfb2b380d7a81382fd8776c28 1ba0e742e4d485f61f6faa3557cb7f536899a344d8b76ad144a4f6545b91d71c e7cae57d1b77476e76250d2e66b950861d0a21f4089df886b664a76f39e91fc0 66e4ff230dc208e5a58e485b0de1de533111e6ec7487926e25ac33e07d937d79 233e6823debff9ecb17e4a65461c49ab041fe8dd9f3ca82b3bca798136cffd22 63260baacceee68b3f4e166bf71d34b87c961e611a2409602a4adc329dd0c622 68fe248a3376c30e3e2973add15a0751b25e19388e740c38f1ee2db4259382a3 a70839d1c604222bab1e243135f32a3a15a14c49df613b96ecfc641fb4c9ca15 53df299f1d2992c634b51ead95c246ae568e48720f1ac69d54e8022060d0533d 2165fcbb3f031b7974c92c12c909bad32d689de3663a7862e9e537c8155f973b 0abc21a38caf599d1490ae64214a5576026921de86dca2ca1981abdd1ca05f5c 2dafc4f42587c9750ae4b91fcf9c2543e5e6fa74d3cd4ee8c9f0cdeb9bec8471 5bab5edc5d46188f78821a9fab63d7c03f3fd181ecf4ae9271cedef8217ff165 4a7d5628a8c46eba12356ce98c7768bfe2afe1eb77920e0b62c683e4d1d9af26 d250664de046e9899f1610dc23337cc90453fbc20f7ea7224755bf649854ce87 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49c4bb81f76ea326f9e2f9fa6a66ad24dc1a2f157c3507fab11782d740b9ca0c c4f6d0893df23f12fd77418272b3a6bffaa4055fd61f5567fbed5d1bf17935e2 314ba23cf74889336e3df38bb4c31b16345299a59cc8fb152598fb6904d67eaf 0685c4fb98159deae3db8f141f09ed8c6f9e661e2240bdfdf946f712b38cbf3f c7d155562349f7276482f8a620cb27c9abaebce50b8ca75084dbb95c48103e46 fc1f69ac0d086b1f273b5e233df512abd2ce4df79454b9c9e692751329e9d4cd 99dfee3e620a30e41a4b01d5e1964498d7faa944233f18077cb1960e83ec291d 7d4d7f9da1e15f6bf9b638897eb9f0d56418821ae608e95939e2f00f4a4ba3c8 1cb8d2c0931d819d586e2e6c858281aa08000bd3aa73269e54efebeecd4ee454 4c7a114b61c30e0f963a1f4743a26d76186e5e9fa01c419dafa4f5a7ca274670 13d341ed7265c12b1d9d8dfb5ddf0eb5ac898f51f46c105247eeb090b1dcb5da 10b425bb9b72c3d7a026a1514ef5ce0b0071f06ad84cedc36badb9a911926ab6 54b6b6d65b4620d36e1f7510c4c63e8e35b11e5094c4e00420524e0ff293fd02 abecd941798f8b79b1a9ba44d12b58492124ed7f079db15e31d4bc012419bb2e 42bc537d9ad7accd344ad030f14ba46d768cd5a689264398b736a5d1bc057dde d9d554cc213fb21c3a34fc92dedf50e358c2fc6675cdefaabb3d933f5c058f35 579d4ba3913715dab7a8a16dda9d644d2c190cb307f900d071117d12f34fa84f 7ad241413ad31581a9b96e5015a288d0ae21bbdf866486534d99fab9c17098f3 587241fb8c2cb384e71ca1eaad 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 161 /squaredot put dup 164 /square put dup 175 /squareminus put dup 183 /dblarrowup put dup 184 /dblarrowdwn put dup 189 /arrowtailright put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueXX 5031981 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6f33457e3200f0935ea98f1a0cfaff49cfa2 d877ff805620cba84126dfd175a82e1b4c6ec58461fce97cdd749a012e2c4243 86b199a97c306b282350cbe5af90cf1d8e7371dcaedbd5fa3346dd3e558072c5 fd51a2edfee6890f62efd1b552699123f4f52c2817dadd5ece30b3bc178a3ef4 f45fd8ad0ee5bf30db5f9eee9ddefe249f3c349a886cf9130cba68ae559ad719 c439930ece1d52d38366116c390cbeabb8fdb36811cf3b8e2416825e8c880b24 6521ec6880ebe7f127c8741c756b29e1f89e3fd5d638a7bbade8c3a0fdbbf7ba ec37cd372afb2d9dd25d509726d20a05914e418f6b7e759c440beb61519df02a 06734ec1226890d17c51f25f775e6879fdc231bc4d7fcb4b33d04fa273bdb8e8 f124171714614f8a94e98a6c2d6bdc1254913999f9c181e42c0886556cdf48af 92a265c1eb06bb30fb2745840c17bd31c7b8bdfd37dfb9643a418a2eaae14094 6752e0a315207aef85c3c2630538907971ec1b05687208571a3504bb3369b5c8 9a43e5abe343d0599fd8d41a1bf01de49f5856bdd67d63a2b3ecea75d721a6ca 0c8751ab41e892e9cf674218dd44acfd40019bf59f4d69ea40d5cf9068a6ff96 9f8cc772b920baf17334833d732431a2969b9a30fe38bd44772cef653935a957 af61dd568f041847ccce3c587656cba7bbf46d8cefec936c6d34dbe51bc04929 0440354f3b29ed8e6b4f21ad53b5809779d67339f284563d54ed4d2a504b014d 974a31c0ceb57b364af519bad4e85c41d521a3ffc26dbac555b42dc191367152 1d60c30b95b5bbafb3c7d87a1c043ea5f0e6dab83a4de99a742e75f02837b923 c50eb478ac4131ee604e3d7b784bd8c93eb9430291d452c21be0495718546333 66a602a9f142a563fdae3c8d40de6a8aa0b92363e5935b01cf60f1515823a302 ecb87313b10f28dfbb78e1ffbe551a5038af8038c01ce1ef665a6c6f46b33b4f 8c5ca9628b7e91f40ce260235672cd59edde0e8ee701acc6b4921dd1b506b30e dd1192e251a5f68231cdbd2c08111d91447e1739a8bfcdb9b00c511170e80b73 39ddc843228561b464431136ae47c715e84ae422f7b07afcab7f212489c33ec9 73b658412566b3562eb8c6c3c46dbfefd744a46784befe60db4c0e0491d11193 e984247d95910e6404a5b0c0697569a270895ba9be7612770e47a56c88130a1f 491256d57f67340367f62ec022e052ed732a364ee8ad8462d21ac9c1546fd67b 76ef55cfad860ed8cb651a9a12d8738503c5e439baa4a3b1be2d30d0494786d3 8ca2892eb8fdadd6f1853d34e27610a7b2c22df9b6d1cfbce2608b5b62c5fc60 12c40e65caa075b8b84539ad248c2201c4d9f05a6d2339ada40e16edb9b61edf 5f3f2532e13ae5d98a1d1f9f65bdd70a25b359aad1051966df3c5960ea2f5794 823c7d1077c604017f973bb4fdbf3abc459c6f41ddfdd6e98c9b5c0dc7d61c38 34f1c6011791acbf32e8378d2bc42ab6d6943d532d0f354e8e448829dfc59f95 e96fb9e51a274ca24a61f3fa3248566b27221b4abff422967764f74b815fb784 899f79bb177164c73ba5af24c6d75cf8fe7a8ad7a30dcb8727621901c2106ef6 ef010e29694855867165d667d0289822eaee0b77c12431dcb9667ee0e1aa170e af2eb3d6a568f90fa1fa0228634c7e2cc5a6d5e0e2bf1fab40aa9042f2926237 f924b137f0869f3c4f21017f668fe7b00657fa7d24026a5f64a02e99cba14a13 73788df8e75321d8e805fe9664e40fbda77f2bab32b030eef52b52f795e03d98 0050605283e8d08d58741bc42bf0f960020be6f12b59a814d8d71555d7b06e4a 7892c7f9091b1cb704b91276e29345d422239db04eff500e01d07f80301ae9b5 c9d5f599610e1f1a66d7a7b79a90a590a86124c761ac83857fc53ecbc0b2cbe9 a8d774c58496d642d6dc55805bcef2e1506a3520316274eab1e4a3e20e5cc9d4 2feaf12b8c917491ed742896c37e4952a002a26bd1961d07333cd8bc9517ea1c ed19b8a39b6a9316813e1f0f26040ed86e56b86afbd22bd0819e4e55caf7748e a8fe7e5f2db44f85c3b40eeb207244a8c3010ecf25b16e8596f83fbd84ce648a 280bcc85ad80d336cffcddd5f2988d44325274bc7352c45ee8e198a68a4d23c3 6f8ebb2a3db9c3f7d7cb227a8287f9ba58d84300c59b038c652396f82ab34171 fe781a5de8fc871a0d9251b2b7017830dce13fe1374c 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 164 /Lambda put dup 175 /beta put dup 176 /gamma put dup 183 /kappa put dup 184 /lambda put dup 189 /rho put dup 59 /comma put dup 87 /W put dup 101 /e put dup 107 /k put dup 110 /n put dup 114 /r put dup 120 /x put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueXX 5087381 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 164 /Lambda put dup 175 /fi put dup 176 /fl put dup 183 /caron put dup 184 /breve put dup 189 /ae 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 65 /A put dup 67 /C put dup 80 /P put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 121 /y put readonly def /FontBBox{-301 -250 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /minus put dup 164 /asteriskmath put dup 175 /circledot put dup 176 /circlecopyrt put dup 183 /lessequal put dup 184 /greaterequal put dup 189 /propersubset put dup 48 /prime put dup 50 /element put dup 106 /bar put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueXX 5000816 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6a18979f258b8c6ed3 ccafbc21aca420c9c83eea371adc20e038b4d7b8ac303004b0aa205f04135140 76407216032fdd22e6219da8f16b28ca12524deb7bca073cc5eba65c102a5e85 fd48e6d062cd4283ee570a7774597e5bf0e3400b6be72db0115f3cb12db70ce0 83722870cddfadee715f10f1fcaf20e06f3c54afe5ca238539bfe2b596116e83 f5371ff18fa5003d8543226cfd4025f9940365b392a858d27f078d3abcffe4a1 54e78c7692d1a32bf935967c64f01b24788ff8325d61145e2d4a489fd986fb77 38e6b254522c77ca2797a504a9ce4676a77ebacb026eca94dde5922c936f8e90 c43e285195b345015bf9bb5e2a444de0213c34482117dd1113a9e09864dcd572 6a18621f8c5b560c80c7e96c593a69d95f1693c242a2f0a92ec904f4c95da15c 60c0be9ae75b3d02ae968bab991ac7824ede6d31b318317e6efa17bd6f563a8b 0cf3ae448b0e2e87e0073c535cc0b41d8d1ab59bcd630b7a3bc437511afc550b 96497e1de8bac2696572a64b20c358ea936457c99720949097303a40b3655ccb 65feea0643b75dc136cbeec7405b5a358b53499ac6075845f721b0433b4eaddb ce9b75024b150c92eeea86b2bb003bb4b643909fdd7f0d377124c0668eb0fc4d 2732d58b8186eb30ba5d7a3751252a4c30a7caf9ab2801982432016654cb9f2d c6c1c7f3e13a2886777dc3b887403cde7c008b7badab445fddf0bcea8b8d36dc c31cde22e792956737e0a766617cdf68abf442a88cb757f9fe94bd0b9d30a426 bb76c22d42ec31802d6dd6319bbca4172d5f7f46f87b5014a6fb38db5f22aa19 dc3ca58cb7857a0f4ec51356cb78ddd549eaabb1718ef833ed53e8c38f0243e3 734edd0b43c0920f067e182ecce03340e1addd19c6e14458cb89ee04a015ebc3 7d8a9b34cf0e66b457910cd685fe963e1e1ad8db48692f2cca7e606f42bb63fb 18cc65c9fee05a02f63e671dbd65ca3112c4ff8bfc94d8ceceb4afb780ba010d 6d210035684896abc566986ab6f24922f3022ea1db32c50e2882e2a670080939 30a01b20b16aa3f81d5e6af594c3ca63c729072dc268ad58889ad8fd55fd9316 6731e7c7a5660ec19d39c46cf77382b0c35a3da4636605d9336113d21a77cf18 2b2ca6429c74fab8b80ad464c82ddf56871a07389e425dd41179493d6d11a6ac 57bee69a10f8af65a3cba66feebd194932394d95cc11afc2bc860e6fc37831e1 5f9b5765d7ed1bf05b2c18886d9d971483ea5bb4db9d40dcd46299aa0830547d 4477b671c20e8d2a4c6f5f9e3723800fe589dfbb5d45a903ab565a1e1a462c85 61d2db2e502583cc6b320c3037c5e59136638484d932113aea727c667f6106ba e9f11e562ada67fbc5c811f28c85b0282b8c88d3095a3df93c9155f8e3fd3ddc 1fe7937d3af228fd6244c9c68ab289441679cf779c8ae4864f51083d116ce37e a871fc5365178c5b30acfb4795824f03a6bcb689bbb6336f33d2c728ea73fdac bec7ce274d87f19f6e5c4a5fd4e145a729a9b47d13ce2c5b5a27d0056e725fec aa98707c6d4847c538a800ef27854172eee94dbc362467f4dc15a68f415cb420 4f6f5f7fd9f81f84d261a684e5820732c1212caed664bd7f6ec41f358585f033 3c444cae70c92a5c75847781bd24d10632126c4656031ec234019949eb61f185 b34f63f8b561712f3528b074ed26032336d2a7ae68dd8f2b1ae45ccdc3cdec8a 39dbafe6bbfa98fd6c4fc1f5a2c51348b7496be34919154e33be6c0b72d5fb69 20d46b221a024ad713d5f0ab5e8eb3023740182de3724836fae64b03808f194f 0502f34095189c270d40482eb1d159f70821c81fe9840adf5af5f87fc332c34c 21e2aa6e9685f90fe8a9984e635b1ec0ca1fcf4be54e543e5c035c95eec02414 4d63707dcdbe62daac044ca020e33b7c31156c0ad5286a2af737d2f54fad4b0f f3137b87aa3d83673974755ac935f388028d02ff83be4addbadf6199728cfb05 37946d3f432a326b01e9f98b5a719cb72fefa903b035c6efc3aa202e0dfa7808 8f76822e2b73cb768c068902a89ead398ecb8ce1872ed97de26fa614a9657ec4 6d4bbf12d92452610089d73bedede8e07f2caeced062866170fabf2849c374f0 0ea445684f08c486ab37ce57632af04a37e90b8d0a26b02bb2ccf8ea82a41a5c d75c01ee2479ba1ad8e784c7802704124e1d573dc31d0244a043c052b97fe6c6 f23e28ef1aacc79a5a2acc1079afc3d58510de6e65c0f2119b09d103d42f942d f84a20c127548c34cf891e7243949f9ebe2e59548a810e73eac39824b77548fe 141804a8b8611337cbf45a4a65a7625217c9d9b916805870bdbc2cbc555c88d6 0bc65f70f2e8059a991c5677259c004090a651a0d5706abfb06cc0f4 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 161 /minus put dup 164 /asteriskmath put dup 175 /circledot put dup 176 /circlecopyrt put dup 183 /lessequal put dup 184 /greaterequal put dup 189 /propersubset put dup 48 /prime put dup 50 /element put dup 91 /union put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueXX 5000818 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b66566ca5ba42b1911a 5d7f1bf343015eece988b7a93bce0c7aa61344d48aed9c92c8698d4b7c9951c8 7d103f2414b39e1437f9d2e50c4ee5f218f2e6716926a79ea978f13b1f855345 191dd7d31d8f82c2e3343c7a5894d95bdc492c28226834efcb5c12fea36ac5cc 430e0aa604961e34888adf6c1f3954cbc2498e225d953cf5685852162346f474 5a2a7087d5d7ad486de16d2ca8e15cee26e012671ba3bdc7d95cc8c98bb774f5 08625e968aee27ff7d1a06e63bcfb5aa4876c3f8f13b30ccccee73c3caf4e70d 98e6ed2f422dbb4950bf789680e064150995941a9f4dd68a575949847a7d012b b910bf03a423f0988c943911f0aae5494ef9465ab04f591e6e844b8155df1f90 584866c5dd894c23a0ef626eddbedc47c01a3f1a55a925ae47a6ae5bc8f9e674 80a0a57393f694abdcbd2e00c402b8808b0cea7f6589418179cc310304b7f790 c38f8f98ce5338c2dbebf183f89d92f966247e82728f48c232c83975ab1e448b 41226acce21ed2da2237a830411b2b728e62b972cccd4f69c924a465584cb519 bac6956ed37b14ddd6b2e95caa26598aa8d3aafe2f589781ba86db76f7e7dd5b 58f85d81544bb5de7285e93d221925218e1e541bcc5d0190921ac17c3107171f 8d4b2fc5652a85e10e4ba32258620e0df73a7fd827609b4b6114bdd40cd538f7 2a4c05ed48cd2647c848ed23ae99f698bad96df092706b730dab25b6f42e3f2d cdcf1582cc040d15c9993b62b8d21f0ce241ba024102ca92488ebe8e05e100d1 4b2303a7c3a6c9ff5745a28670c3b3def43854fe261e68317c2b0b9def8927cd f9f94934b165c0b49dcb8b52cd34518a4348094c2367424296dc4fb91af49dbf ff0f0cb2f50673df20c811e2a7076af4ecbb83f0ef970c01c5d1878427394a82 729aa71af52272040b80ecdfa98f2b134943966e5bb0ac15948debc22e084a2a fe39245bfb23ecbd5de2a1de39d7ad63c9034461d5e036f26334ad427470f955 89118069b1ad5352a6be00a696936764ab9f4de0691c7e4e76d1135ef39cbfae 381ce3bcaa383a517c827c274338d774392c726d34291ea9c3cd45be38dffc0e e88f084b7d23e906b46796289c981a8ea0ed1b84dbbd80ba2743ecb038824d15 1a94dffe1c722a0d65100e5aa4499f664f93825e6a011ef2677d6e31a1dbfd94 f3c03d0873fe4ecd28214de806083751c5873bfef829a494624073183e6e8db3 90d9984016e4030ba922248186138ffe3b7823bf86cfbd6ece06107700980c31 f2825b121a851b7b21db2e024bddc132543049449780701c1ce3043fba97fa15 f413b301bf24fce0710aacd1a03f18cabfce47ef1beb2c64014391bc9a7e236f 06871a23dac580f9729a82cc49f86819ff54ab1987bf246dbe895383d46224cb c198bb96872eff39e45ec97d6266211e9a9e363e0c7f6b4145173fa2ee8fd930 cdef47ab9c4690fa8c74f93f81afd25105997b65de55bcf47fe49f485cedaee4 93ba85409f0b1ebeb68b43feed4ef119d1e18d43b8f8a21411719f8ec202f3ed 5b51fb92919874eea5d6ba2b3266b1abf2f6fdb203406fd07b46241a817061ed ddbdecffa1e796d0e5eaeaad269ec80ca0cbf6609e71301db09682cba6046606 26410dcd712a294d6e966aee7b3dc379c5339506ff88732d23f55b5b0c1ad15c a0ac5960e716cedaa823947475fe5909294b0f73f2ebb9ff2f8876a523a459ee a09bca92f381f0a531f8b90cfb0589377c1fe0aa6dfe59cdbc866876435c3867 cb275fb466eaeeb56041f9c52c73acec07b89ac5224a3c12e3607a2d042c5bf4 6b9960d05cb88bf4bb1f7b7e67205965a5cc661d0ae6ff4c7bed783e4c8a0e93 5f925ea8ffcf0337dc9c68852156619bb34e307cd6217392a11fa83b8b34ded4 79eb5baecb95347cadcd2c12c345ce0ad032c4e90d9bfc7c436b2291d94e384f 1e82f9ef8c10f922a998b5cf047be8a4ae4ee0de5568c7fab27dbae66246a1c6 0d7aa3bacc90e892d1919c50d1769839d5abf2699a32a957acb9508847527b91 97d5acdcbbd525760646ff34263d31d7007a1baeb01ac10547e4e35ecaba108d af8d1c371c03b25415fddd4dd1bacc09731b8f68c95297e2cc278e90e94baebb 0f54fcba4fdfff0569a0bff46cad7a84086bc3b26c6550c88c286ac1b60f9983 17ce30d2e534da8f8e660a560b1329bff80b234adc08329a2e34581efc389ad0 b50b5d6432c89cfe546f7b3a7579aaecc4800b319aa1385b2a7b2d39955b27ae 8d9757a708c4d1c99fac37b9dbbc4ad71a1b6de7d4e14663e5eb6fa5c34e3bda 8287bdba3375dab8f9040d55550c476c5435afeab6575b521edb912d37d07020 aa95038f4ffc283f966401be61d69550c0e94d8828cee1f8ecdc6d789193a85e d8d96f8be6e9ad47cb39accb08ee5fee790d75538470bc6c0f69f4689ab1356c d012f5e16246365fd6a322f66d8e6fe773ad89d3b47061e9286f9ddff995a637 37f5e6d11149e541d3707b607f4eaa0bb88babce00afeb4f69de4571c61ca05e 709f2004dbcbd4cbdde9dbe257a67c499306dfe2ea3455048fb6035b8f9678d6 32ea88d0f4e2ff5fab9b043d224899ad7e11b2f0237cb54613f5e01c9f00221d eef3885601d14736cc3c4ea6962a182252dad3ae7e1a8c201a1103d8cefa3d3b 5c0ab7d945ae629a5177deb689add62321e91bde2f5061d73972825f248f43ca a41dace25fb332082bc8e8fc52aab68e2fd9f1493c1eaebc6b51f66cd20dba7f 0487d5b6bff97b2e392bf71a0f7e4f58a7e2fd9af5e1d3838f25e99741ac8e51 6d3351649f03482cf929bf6fab3e1e1dec0a1c9aad6a246bd2f63b32c61e9a06 b63ede3e87ac85f321b17f3eb9f3f1c96f03143f309d76b3f707b660ec606e9f fc57176c718e60eecfcd25e7acbaa995f350eec38d13fa4701ed3612e607cfe4 92cf3f190ec41a6e3190bc5101e89b3eab8cfb43cd8714884d1996e91a694c5f 60492281ecf36255369cc1c185db2e6ffccb49ac0da7f3939b893a81102282a6 01cc626f696ceff91d6298e4efb7395397f2735cc619119bb8622418b622912a 7198650ba10c8cebfcb74772 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 164 /Lambda put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 183 /kappa put dup 184 /lambda put dup 189 /rho put dup 191 /tau put dup 59 /comma put dup 61 /slash put dup 71 /G put dup 73 /I put dup 82 /R put dup 86 /V put dup 87 /W put dup 99 /c put dup 100 /d put dup 101 /e put dup 105 /i put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 114 /r put dup 120 /x put dup 121 /y put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueXX 5087383 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a196cf14c95a4e02f9cd2da8c db2546c6df52e524745992e18d9ff87aa25e4e1800bbe4ebb357c6ef55ed6d03 6d3a00c1ee8073266c21d2f0ac85d656abf61d7e5a4fa87da8ec3b5329e434d0 d2adab706b42a2e5331be5295399d803ccac03f631f01f39a022fcdf63486b68 7d15ef284a77def7fde4898543e7b5f7ec267756103e477f547cfb8d2311c4b0 09deff56085f5d419697af1846c8b88c1bbbae149f0f19ca3c8dafe19cec48fe 6b38357246d8b5cef80b53bab4cbbbe8bc8318cc462a4158258f456e697f6058 86010493f252073ee4e4c9531f23485415ad6bc901bbaea659ef701f825f4b6d 5c9bed0dce7b63810f906b33ea4955531ca1ed1fc120cc56dc1e277a33222cc6 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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 161 /Gamma put dup 164 /Lambda put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 181 /theta put dup 183 /kappa put dup 184 /lambda put dup 188 /pi put dup 189 /rho put dup 191 /tau put dup 33 /omega put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 100 /d put dup 101 /e put dup 102 /f put dup 109 /m put dup 110 /n put dup 112 /p put dup 114 /r put dup 117 /u put dup 118 /v put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueXX 5087385 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930da4f8d58abefd45db119b10eb409dd89792 3c6e705479464a4b310b58348c4b42393988fef4925cf984423aaf65fea9f0e6 4629953bcf50b919d968d99bd185f83112b2759cc411764e9bde677f57c5ee5a c555448d5f81a16259ded1e11bf4119d53e8ab07a802df900d3a2d5ccc1c6876 d29c5e9effb7af3ef83400b0910c07873a8c56fa7b1eb1ba426043b00cc95dbe dc6e136cbbbcb3f67509964f9f281ebf81fe5b018122eaf66c4a838487e82e18 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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 161 /minus put dup 164 /asteriskmath put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 181 /reflexsubset put dup 183 /lessequal put dup 184 /greaterequal put dup 188 /approxequal put dup 189 /propersubset put dup 191 /lessmuch put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 55 /mapsto put dup 57 /existential put dup 59 /emptyset put dup 71 /G put dup 80 /P put dup 82 /R put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 112 /radical put dup 120 /section put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueXX 5000820 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8 3750d597b842d8d96c423ba1273ddd32f3a54a912a443fcd44f7c3a6fe3956b0 aa1e784aaec6fce08dae0c76da9d0a3eba57b98a6233d9e9f0c3f00fcc6b2c6a 9ba23af389e6dfff4efec3de05d6276c6be417703ce508377f25960ef4ed83b4 9b01b873f3a639ce00f356229b6477a081933fef3bb80e2b9dffa7f75567b1fa 4d739b772f8d674e567534c6c5bbf1cf615372be20b18472f7aa58be8c216dbd df81cc0a86b6d8318ca68fe22c8af13b54d7576fe4ca5a7af9005ea5cc4edb79 c0ab668e4fec4b7f5a9eb5f0e4c088cd818ecc4feb4b40ec8bd2981bf2336074 b64c43002591893a571bbe4dd05506a93c40f517d06cb123975ca3614a4e4dad 46370c34f8d1b9212235a51bf544f5601e8b068a3bc1c9d55b08d2115eae8b2f c460cfd0316bcb531869e2fa9a909134580c9c7a1cc20aea7cd103ce266e9e5e cb4d475e044235edd0f6e03ce474a98f2f7e556e1dbcce21453fcf427b5c5a87 de7f4cd1038644db050b91dba6a08f6333278ceeff84480b587a723e4f60af19 ed1ea2bb0aab1451c39e8e83b99c5eef929776a34da76b9a133bed1cf9be67ee 0a078374ac370da0758815b9f98711c7ef396986788021b5c7b3bdbffdc5fa24 e30ad38f76d398ca678b19126059b0a1375cbd25c9d4ddb17b171dfd4861dccc 2fb4ac73ffb638a569488d18f086f757c5d8a5eb745ea86472cc48cc0be08730 34eae5a86a1613f1f69ad107f420f2350d059e03bcf1e85823a8c24515352d28 a5bf50b5e7c145ef872654def59200ec48c511c689e7e1dcbe5caf7554315b97 776f4ace13efac2e46b2403e2c891b01312e4b030f3947262ff32527a2888b1a 98093d8adf221367dfdf1147f36ac76586c9c14c3cf9878d6e5270b88d7a93c4 a16a6bbf0f03543aee082f68321bbcb35256e626a63304d5158701589480a076 78627f685158ceb59376b23a90fb2a6ef2d2d1840419ded65edadcefa7307a1b 7e614d111afef1252188efda2f5434d426d913b057dc86f99cec3b1af2f9eaf3 914023ebae0d6e84fed85e46f9d3d26de4d69b4be98bfb98415946c1828ff415 7af4de746bd301b25828b7edf970e86d3ddc2f0e44677bf37bcfe62ea42f6177 b3c0ab7945c05be2f8033d85179503f477a97b5a14cb90f5e678e6572e47c338 20092dda27b1b5dc05b7f1a9945446b1a4dc31b52001d6511e40b2a2496910ef 754648040386ed483bedbd3d196ce645b068f6ab18863edf4bfe156ae0db0ac7 ec11b568779c06edcec87f26a4714121c24536ed20e50780d81a2003e850ff35 44f1e179c832e53edf0e695aa8df47a515754dfc748ee4c057aa4bb402cb60d5 b5cb0fb8d4ade40cbbefcf0524a05fdab9e02702fb2f7b291153e306eeda1263 23dd7d46db92408a3fdaee8c7091c2ceef22b734889e0c9788a9941187ae2717 74e00c95b11cccd4844916e9dac252773207a1dca8da651c089af0730d3761f8 70b9bf89bd0845cf3110e1c04293bf3651f9f23f064ec3c191eedd4771df52e5 fd42d200494baaca546b8239daff0e472a9ed7c03b7c14538908b689d7f73f24 822d653d8950ca089496fa573edad6430f567ad105b27bf73526249b7621ce84 47a3e9b9cc407b55f8900f241ab152092faad1c3f46fe06ab349a09a9f673016 30ede66bc4c4651e34896e09822992ad4d5cf4601d8939492a748d1494ab754c 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8454fb3641c8a5bf13f0ac82fbb85d58b354be47083a93ccbf80702711780a8c 40f5c87a4e51d6f231ef33c2510c88dfdbacd89b3056fa5c24e1d5fd00923378 8f60eeb3d8d515de4777100d8e90d8c1237fc0f59d18cbbf97f5ea4a745effef 9a083156c1c4bf553891d6abd064e7b08053e7344b96750e1d466fa1f5807cae 5a1324ffdd681191f1ce87adf3404f4b4d7ed6aadb48cec5f106395ad0db15fe 972ef0e630d8ebb90e33d1963da88ce85081cc054ed66de0b03d0ad382f5fe28 7b7534025cba071432af7e98cdff3eda4386e2e097d31f947825997bcbebb817 f666072fbe62083225bf50749fb422206a15fe4516dcbc12034ed81806d16515 7da2c6b238ad16761f09b75c1927c1a97e6d9907d6315625a2c02922a4c4a51d e87c6c010d2f17897810f6d76f899d941ce773bf153b6dbee71df49b05f8bd99 57b7fdfc4f11224b742275a29bd7f0968edef9453a3e45ce335c8e6c9832e609 f179dcb2fb5033479851e5ec94ed3f0220d5b2f1790bce00d4ef01d9de640190 a5644b47afc9097bed832705ba946aab4bb941d9605f0f31ae5bc51d9b390d02 7aed97f703a95704fec15ce3ae40c52626a6d59fe5cbb12888e1a5a716b9fa83 0010d3367ef2c0be29770ac805331c0e820f2982fe92cf911d995b5a2357435d 1214b516fc043d3dbadc342872b2d658f50d696e7db6dd022c76ce25777b2e38 6ce763e777f53f0673a509532e6fdf52576f88e76a1cbc569c801d56fdc8adeb e2c7b11a0a38862bce1b6a5f4fcb8d6b7cefd64d744f360b9f5593627834b13d f1e5ca4a66dad5788157bf037eabe912fee3d059419a1516af619ec96c2441a4 ab5e147f0af574287de87aa9ce98824b28025a1d024863c584b0335d180cc6f9 8e25aaa04c3bb9496943398637c1a2ef84ed8a914db6cd5be8ee70085ad4bb14 77624d6c6b9d11940e407c9f3e6067308ecd8851ec9a1a63ac75b767375f82a6 7d57d7ad52f5144ef87ad8d422e72854507f2937f2593c2a0d6541855db106de fd5db66f18b0e0972dab4987607ff59a42205cfd6e6825350d06672edb2cdfc9 dd9aa06eca9bdabc0b1a5a748e363affb0bf6a0a2c24243e57af14def19064e2 c02106e261173febd180f28eef312a05d3d2a38ff7e393a8232fc0d000c906df 6d847d06017224b292a8f654643c837cf753891f2e276ae4346a35ddb8de142f 31b424b0300332b665039b2346a46bd177b8873674a266eef063199ef363069c 46aff9c28f2692af89d2a3cb0834e7aa3295d593116c4b10a155cc5fceee4791 e21db036204be41eaee76191d47283347eab360f120c225c8040963f2e498f36 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509005a39ae675fefb718bc3a4d161995b3f00cb668d5e8eb1a49590aeb7c5ad e37ba9a4d22c7c5d714e32b1ece4c6d717cf57b9f2793c477da7b13acb330ec1 b9cbfa02c4ab3bf1e0ad35930210507e000665322cbf876c933b3a2f155e80e0 8eb8aed68f3e62825be275fcbb430cc2fb7dc9e356104094ad56fe56e7d51e84 0b8b4e74117bd060058a18040e649584efabcdf6a3044f29349546476951dac3 de1a3690971bc7197a2b147599acbf756009f96b36505e01e0ffbfa72714b0f6 020c18fe1e236df5a72fdf9db1243f508e27207165f10160de5fe6a8dd932815 83868fa9bc25c1c58cd57da362516c7d05ef6ef2b06e1cee0e08d69ea8e76e87 6f04d8bfc5d98081ab5395c6816475ce589e62b229ef88d7fc82e7ed0bafb3d5 e248bac2cdd60812acb4296fe868dd06b23e6f30c9629b7b60aca1dd5ef65cb3 dbc351fa86e93a0774720c0e3f746690971bca0709a5142b2176521e4e1a5b3c 3cdc6356f56586915320d4ff617fb932978000ac9dd85dba875aec3c243f0601 990b902449061bf9c01a522650549723654277f0f4cefcf6b0d327e46c3528e9 3d8d2917014d294c680e10a93c2741b2a8042d5fb254ba638bb7138f677d7e43 8d7915a12e0acab96ab4aeb39b9dc718e793ad46d2f6e6c14485d205e3dec14e 10865193b9dbe213718e3856e44a37f1c6958741d661cc645e9eebbec835b410 316f70a22dec572e01daf96773d297be2c8baa036bc1d03dcbfaf3fbb6e34b40 5d17028347183110f61a41b28d5d1c1487b52182a23d5b7b042cc57f3c5e529c ee8782dd61e7a590040aafc18bd40adae06db8ef286ffe326cc8b7ad2e786af6 03be92cfbbe45087fc641f29a3e989fdbb28e925a24da1354867c9a5c6dc5be1 a7ca54b90b351e41a3c070667fa846ab8cc173d6e66151991c6aec4087c254fa 681221cc129702efbd7ae12989127d1e1360fc137da68389e73bd53ade8feebf d16f67c922b84e6535d2d95584ef7740fc008c25d29bbf266c5192ea60db8d6d b2a12c607c154be81ea762b3f65ec27f01fa93cc587c09cfba732dc453e5cffa 3b18f5e7fde26f9a09922305b1d865f08bbf0b0c78bdef6fa2679bc58c873088 44676ea7166e903e8e059884e7ef92eac6c52181360e6a6423100118dc0141bc face21f5bf5ee5761ebefe6432990a14731393dff8cd8566e493731e00502e70 07347d58b7de02ebb91731341fa51ac53581139b71686a60857906721878100e 261dcebd0b34a4658bbbbfc3818deebce98b3c450e0c2f9d225d20b444715b6a 6536cec712641d37729cdd3999e8da4d9cfe44c07e649b66eb16399b82025f75 7ed1301a7968871eddfdb8b74f7f42ac4be38ab220ecc486ce64a045571fca87 589be9632c76f4bfeb482dbe7fbc7209ec41c0fdbb7a4bffa3d9cb31dd276cec bb49f1b3b3636392f893aff6e89fa93ed661885341b32e6cb96a51a2b7a40b72 1db81ea9c11d22bc46e9eb82a1c3d4b85e95287a282fa2606145770d86eae712 8bc6e4698aa0c19e42353bfc3c3587e938252722275c788fc838ed4f77a04416 8eebd65969300e09f56a69b49d2b75eacd7a68f1fd9937937c814614e8afb90c b41b58fb693e36ed947f2fbf80a4050c9d4d80e9c01cdde6f94ffd1d84ea738f e128a00261b36a7639d5d54457e9e8de36ac776d6c0785c7b542bd0c30f7155a 024124b938cd2c01698a747a0776f040f795708bc56dceb9fd466d5b57d29943 0973dbc151d1a397339e5ce51ac5679816a3b40dca3db046ab65158039dc6506 eec77a0e5409bd3db80e5606fc5efdb11555b96fbd362a55aae8de55ced3fbc6 19b3afd7112a0b2852228ad4249253ed886be147499b6783a5a2876f26314bc0 9c59c2f185f9b8abaf7137eb8fd662347d3db5f51f159c2c539c3f2ec091214e 4eaa2002ed650361ece6db9083c5fc1c4f06589b66023a8edd2dbcbb21f0f4e9 2db8401dd19f08e77b2d406ccebc2ec8a96030777ecf6f8744ad1a15823bc58c 8b05535414763bec89a2143ea362c0854b0ebe48e690c4531ef9e413877d3ad6 868cdfc9726c31af9b248eb4daa8040138a7cbe8208924bcd57daf7d0582ae32 3abcf56360770e819b97961427582381663fd701139315b50008c8b123e5d138 07b1061de911696d94524c8c7277466b77baaf29d61c1d92dcd0837239861638 271287a5de9f98b5f2884a68861b59a1a1f1c047688ed9714d4399d3b8928595 33383929c0675c45cd60be3316efb051e694717bd6ab211275e53fdedbd2b867 252fd78ee2646cdbf5d51767e42cfc9eca765d09df254fb9d42f106f60e1f5ad 1f4a6352fa 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 161 /Gamma put dup 164 /Lambda put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 181 /grave put dup 183 /caron put dup 184 /breve put dup 188 /germandbls put dup 189 /ae put dup 191 /oslash put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 52 /four put dup 58 /colon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 71 /G put dup 73 /I put dup 76 /L put dup 77 /M put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put 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2c2f6edb29f79d2f9d01abd19ea686f2646facf799bc 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 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 188 /germandbls put dup 189 /ae put dup 191 /oslash put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 126 /tilde put dup 196 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueXX 5000793 def currentdict 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Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (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 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 188 /germandbls put dup 189 /ae put dup 191 /oslash put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 58 /colon put dup 61 /equal put dup 84 /T put dup 101 /e put dup 105 /i put dup 110 /n put dup 116 /t put dup 120 /x put dup 196 /dieresis put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueXX 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493d1f9a0eec112ddbec72d6f02d1297177493d214eb7e7ff292076ec08b3bbc 3a530b9bc3ba56d2ba1efa106ec0334917a069652631cc9cf3c0bccc74eedcf1 d618cfab49b8d976692940e7ae9ba17ce09347e5258e7c13b62f92a62905f81f 348d9c405442a7878269 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 188 /germandbls put dup 189 /ae put dup 191 /oslash put dup 44 /comma put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 65 /A put dup 66 /B put dup 78 /N put dup 80 /P 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 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 116 /t put dup 118 /v put dup 196 /dieresis put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueXX 5000794 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204a520640fd7c4d05e35ea5486f a43d0e970787203855daccb1ed0b16325828b13b7c767a1a58c87902f0546ea5 404440fa6526975783c2d7a9572b8e5d5176bac598b8960609f49016aebf3763 89c1adfe6a1ffc0c53b3acd9d168ec3a278fd4f1d9b085dc2df701346c9d0887 3d230a161692b42a45bb5da343b4edd839dd60685a59aaa49fc65cd8e2ad2a9c 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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 161 /Gamma put dup 162 /Delta put dup 163 /Theta put dup 164 /Lambda put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 188 /germandbls put dup 189 /ae put dup 191 /oslash put dup 46 /period put dup 58 /colon put dup 73 /I put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 196 /dieresis put readonly def /FontBBox{-33 -250 945 749}readonly def /UniqueXX 5000795 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be356c29c3aec91ef17f5d31183b1e489f1fa 559a6693fbdd04d4f7e99886d8cb7cca13d4aa9ceb34a708af22e69ccef0e504 c3997075243585edf60b6447c8d01cffb04b6e3ddbaa7da5891286861322a795 e9d990fd56af2f5c2b35801a0b07874e4ac3170481d232e150443253c92a8b05 513d4a4154a99d91f629ccd30376500f5d16aaa9b203bad6350e4b19424c0e33 75f2949bc1927a6f6221252f8a96618e646aa1b0810753c98becc26b37837775 24bca72b818d314074c1c91f4e1f4148d3d703d12d179ee6970bd55d04aa9a9d 97f63f2c7dbd129ec2ed3ba5f8080270ea5854ea165dbcc061e2f47f1aaf0997 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Fs(\257nite)h Ft(if)g Fr(j)p Fq(V)269 b Fr(j)345 b Fq(<)h Fr(1)p Ft(,)411 b(and)f(in\257nite)h(otherwise.)555 b(Let)1575 50547 y Fq(G)381 b Ft(=)h(\()p Fq(V)67 b(;)202 b(E)70 b Ft(\))431 b(and)h Fq(G)11222 50107 y Fn(0)11914 50547 y Ft(=)381 b(\()p Fq(V)14686 50107 y Fn(0)14996 50547 y Fq(;)202 b(E)16500 50107 y Fn(0)16810 50547 y Ft(\))432 b(b)34 b(e)430 b(t)-34 b(w)g(o)433 b(graphs.)620 b(Then)432 b Fq(G)287 b Fr([)g Fq(G)32654 50107 y Fn(0)33346 50547 y Ft(=)382 b(\()p Fq(V)557 b Fr([)287 b Fq(V)38477 50107 y Fn(0)38788 50547 y Fq(;)202 b(E)357 b Fr([)287 b Fq(E)42639 50107 y Fn(0)42949 50547 y Ft(\).)620 b(If)431 b Fq(V)46592 50107 y Fn(0)47284 50547 y Fr(\265)381 b Fq(V)701 b Ft(and)1575 52052 y Fq(E)2540 51612 y Fn(0)3187 52052 y Fr(\265)336 b Fq(E)70 b Ft(,)404 b(then)g Fq(G)9885 51612 y Fn(0)10600 52052 y Ft(is)g(a)g Fs(sub)-62 b(gr)g(aph)401 b Ft(of)k Fq(G)p Ft(,)e(written)i(as)f Fq(G)27501 51612 y Fn(0)28149 52052 y Fr(\265)336 b Fq(G)p Ft(.)1575 53558 y(Tw)-34 b(o)352 b(v)-34 b(ertices)351 b Fq(x)g Ft(and)h Fq(y)396 b Ft(of)351 b Fq(G)h Ft(are)f Fs(adjac)-62 b(ent)349 b Ft(if)i Fr(f)p Fq(x;)202 b(y)43 b Fr(g)352 b Ft(is)f(an)h(edge)f(of)h Fq(G)p Ft(.)521 b(The)352 b Fs(de)-62 b(gr)g(e)g(e)349 b Fq(d)41975 53740 y Fo(x)42912 53558 y Ft(of)j(a)f(v)-34 b(ertex)351 b Fq(x)336 b Fr(2)h Fq(V)1575 55063 y Ft(in)409 b Fq(G)h Ft(is)g(the)g(n)-34 b(um)g(b)34 b(er)410 b(of)g(v)-34 b(ertices)409 b Fq(y)453 b Ft(adjacen)-34 b(t)412 b(to)e Fq(x)p Ft(.)554 b(A)410 b(graph)g Fq(G)346 b Ft(=)g(\()p Fq(V)67 b(;)202 b(E)70 b Ft(\))410 b(is)g Fs(lo)-62 b(c)g(al)62 b(ly)436 b(\257nite)409 b Ft(if)h Fq(d)47594 55245 y Fo(x)48525 55063 y Fq(<)346 b Ft(+)p Fr(1)1575 56569 y Ft(for)440 b(all)f Fq(x)396 b Fr(2)g Fq(V)269 b Ft(,)448 b(and)441 b(it)f(is)f Fs(b)-62 b(ounde)g(d)464 b(de)-62 b(gr)g(e)g(e,)473 b(with)465 b(maximum)h(de)-62 b(gr)g(e)g(e)465 b Ft(\242)p Fs(,)448 b Ft(if)440 b(max)39988 56757 y Fo(x)p Fn(2)p Fo(V)41957 56569 y Fr(f)p Fq(d)43194 56751 y Fo(x)43779 56569 y Fr(g)396 b(\267)g Ft(\242)g Fq(<)g Fr(1)p Ft(.)645 b(A)1575 58074 y(graph)469 b Fq(G)443 b Ft(=)g(\()p Fq(V)67 b(;)202 b(E)70 b Ft(\))469 b(is)f Fs(c)-62 b(onne)g(cte)g(d)466 b Ft(if)i(for)g(an)-34 b(y)469 b(pair)f Fq(B)61 b(;)202 b(C)555 b Ft(of)468 b(subsets)i(of)e Fq(V)738 b Ft(suc)-34 b(h)469 b(that)g Fq(B)373 b Fr([)313 b Fq(C)530 b Ft(=)443 b Fq(V)738 b Ft(and)1575 59580 y Fq(B)329 b Fr(\\)268 b Fq(C)424 b Ft(=)336 b Fr(;)p Ft(,)404 b(there)f(is)g(an)h(edge)f Fq(e)337 b Fr(2)g Fq(E)473 b Ft(suc)-34 b(h)404 b(that)h Fq(e)268 b Fr(\\)h Fq(B)397 b Fr(6)p Ft(=)337 b Fr(;)403 b Ft(and)i Fq(e)268 b Fr(\\)g Fq(C)424 b Fr(6)p Ft(=)336 b Fr(;)p Ft(.)539 b(Unless)403 b(otherwise)h(stated,)1575 61085 y(the)g(graphs)h(considered)f(hereafter)g(are)g(connected.)1575 62590 y(A)p Fs(p)-62 b(ath)402 b Ft(in)i(a)g(graph)h Fq(G)f Ft(is)g(a)g(sub-graph)i Fq(\277)474 b Ft(=)336 b(\()p Fq(V)23663 62772 y Fo(\277)24239 62590 y Fq(;)202 b(E)25673 62772 y Fo(\277)26247 62590 y Ft(\))405 b(of)f Fq(G)g Ft(suc)-34 b(h)405 b(that)9665 65013 y Fq(V)10372 65195 y Fo(\277)11284 65013 y Ft(=)337 b Fr(f)p Fq(x)13863 65195 y Fw(1)14388 65013 y Fq(;)202 b(x)15620 65195 y Fw(2)16146 65013 y Fq(;)g(:)g(:)g(:)e(;)i(x)19532 65195 y Fo(n)20158 65013 y Fr(g)1616 b Fq(E)23275 65195 y Fo(\277)24187 65013 y Ft(=)336 b Fr(f)q(f)p Fq(x)27372 65195 y Fw(1)27897 65013 y Fq(;)202 b(x)29129 65195 y Fw(2)29654 65013 y Fr(g)p Fq(;)g Fr(f)p Fq(x)32098 65195 y Fw(2)32624 65013 y Fq(;)g(x)33856 65195 y Fw(3)34381 65013 y Fr(g)p Fq(;)g(:)g(:)g(:)f (;)h Fr(f)p Fq(x)38980 65195 y Fo(n)p Fn(\241)p Fw(1)40808 65013 y Fq(;)g(x)42040 65195 y Fo(n)42666 65013 y Fr(gg)1575 67435 y Ft(where)438 b(all)g Fq(x)7525 67617 y Fo(i)8338 67435 y Ft(are)g(distinct.)641 b(The)439 b(v)-34 b(ertices)437 b Fq(x)22993 67617 y Fw(1)23957 67435 y Ft(and)i Fq(x)27041 67617 y Fo(n)28105 67435 y Ft(are)f(called)f(end-v)-34 b(ertices)438 b(of)h(the)f(path,)448 b(while)438 b(the)1575 68940 y(v)-34 b(ertices)421 b Fq(x)6673 69122 y Fw(2)7199 68940 y Fq(;)202 b(:)g(:)g(:)f(;)h(x)10586 69122 y Fo(n)p Fn(\241)p Fw(1)12836 68940 y Ft(are)422 b(called)g(the)h(inner)g(v)-34 b(ertices)421 b(of)i Fq(\277)560 b Ft(and)423 b(w)-34 b(e)423 b(sa)-34 b(y)423 b(that)g Fq(\277)560 b Ft(connects)423 b(\(or)f(links\))h Fq(x)51443 69122 y Fw(1)1575 70445 y Ft(to)347 b Fq(x)3692 70627 y Fo(n)4318 70445 y Ft(,)358 b(\(as)347 b(w)-34 b(ell)346 b(as)h Fq(\277)484 b Ft(is)346 b(a)h(path)h(from)f Fq(x)20146 70627 y Fw(1)21018 70445 y Ft(to)g Fq(x)23135 70627 y Fo(n)23761 70445 y Ft(\).)520 b(The)347 b(length)g Fr(j)p Fq(\277)137 b Fr(j)347 b Ft(of)g(a)f(path)i Fq(\277)474 b Ft(=)336 b(\()p Fq(V)41364 70627 y Fo(\277)41940 70445 y Fq(;)202 b(E)43374 70627 y Fo(\277)43949 70445 y Ft(\))347 b(is)f(the)h(n)-34 b(um)g(b)34 b(er)1575 71951 y(of)404 b(its)h(edges,)e(i.e.)538 b Fr(j)p Fq(\277)137 b Fr(j)336 b Ft(=)g Fr(j)p Fq(E)14497 72133 y Fo(\277)15072 71951 y Fr(j)p Ft(.)1575 73456 y(Giv)-34 b(en)541 b(a)g(graph)g Fq(G)565 b Ft(=)f(\()p Fq(V)67 b(;)202 b(E)70 b Ft(\))542 b(and)f(t)-34 b(w)g(o)543 b(distinct)e(v)-34 b(ertices)540 b Fq(x;)202 b(y)608 b Fr(2)565 b Fq(V)269 b Ft(,)575 b(w)-34 b(e)541 b(denote)h(b)-34 b(y)541 b Fr(P)45116 72867 y Fo(xy)45016 73838 y(G)46739 73456 y Ft(the)g(set)g(of)1575 74962 y(all)472 b(paths)i(in)f Fq(G)g Ft(connecting)g Fq(x)f Ft(to)i Fq(y)43 b Ft(.)745 b(The)473 b Fs(distanc)-62 b(e)471 b Fq(d)28083 75150 y Fo(G)28871 74962 y Ft(\()p Fq(x;)202 b(y)43 b Ft(\))474 b(in)f Fq(G)f Ft(b)34 b(et)-34 b(w)g(een)474 b Fq(x)f Ft(and)g Fq(y)517 b Ft(is)472 b(the)h(n)-34 b(um)g(b)34 b(er)26469 78283 y(2)p eop %%Page: 3 3 3 2 bop 1575 5313 a Fq(d)2206 5501 y Fo(G)2994 5313 y Ft(\()p Fq(x;)202 b(y)43 b Ft(\))338 b(=)e(min)q Fr(fj)p Fq(\277)137 b Fr(j)336 b Ft(:)g Fq(\277)474 b Fr(2)337 b(R)15575 4724 y Fo(xy)15575 5695 y(G)16657 5313 y Fr(g)p Ft(.)509 b(Note)315 b(that)h Fq(d)24117 5501 y Fo(G)24906 5313 y Ft(\()p Fq(x;)202 b(y)43 b Ft(\))338 b(=)e(1)315 b(if)g(and)h(only)f(if)g Fr(f)p Fq(x;)202 b(y)43 b Fr(g)338 b(2)e Fq(E)70 b Ft(.)509 b(Giv)-34 b(en)315 b(t)-34 b(w)g(o)316 b(edges)1575 6819 y Fq(e)430 b Ft(and)h Fq(e)5516 6379 y Fn(0)6256 6819 y Ft(of)f Fq(G)p Ft(,)436 b(w)-34 b(e)431 b(de\257ne)f Fq(d)15357 7007 y Fo(G)16145 6819 y Ft(\()p Fq(e;)202 b(e)18283 6379 y Fn(0)18595 6819 y Ft(\))380 b(=)f(min)p Fr(f)p Fq(d)24025 7007 y Fo(G)24814 6819 y Ft(\()p Fq(x;)202 b(y)43 b Ft(\))380 b(:)g Fq(x)f Fr(2)g Fq(e;)202 b(y)424 b Fr(2)380 b Fq(e)34854 6379 y Fn(0)35164 6819 y Fr(g)p Ft(.)616 b(If)430 b Fq(S)s(;)202 b(R)389 b Fr(\275)379 b Fq(V)699 b Ft(then)431 b Fq(d)46701 7007 y Fo(G)47489 6819 y Ft(\()p Fq(S)s(;)202 b(R)9 b Ft(\))381 b(=)1575 8324 y(min)p Fr(f)p Fq(d)4832 8512 y Fo(G)5620 8324 y Ft(\()p Fq(x;)202 b(y)43 b Ft(\))338 b(:)f Fq(x)f Fr(2)h Fq(S)s(;)202 b(y)380 b Fr(2)337 b Fq(R)9 b Fr(g)p Ft(.)539 b(If)404 b Fq(F)34 b(;)202 b(H)434 b Fr(\275)337 b Fq(E)474 b Ft(then)405 b Fq(d)27479 8512 y Fo(G)28267 8324 y Ft(\()p Fq(F)34 b(;)202 b(H)98 b Ft(\))337 b(=)g(min)p Fr(f)p Fq(d)36541 8512 y Fo(G)37330 8324 y Ft(\()p Fq(e;)202 b(e)39468 7884 y Fn(0)39779 8324 y Ft(\))337 b(:)f Fq(e)i Fr(2)e Fq(F)34 b(;)202 b(e)45222 7884 y Fn(0)45870 8324 y Fr(2)337 b Fq(H)98 b Fr(g)p Ft(.)1575 9830 y(Let)397 b Fq(G)337 b Ft(=)f(\()p Fq(V)67 b(;)202 b(E)70 b Ft(\))398 b(b)34 b(e)398 b(an)g Fs(in\257nite)g Ft(graph.)537 b(A)397 b Fs(r)-62 b(ay)397 b Fq(\275)336 b Ft(=)h(\()p Fq(V)28083 10012 y Fo(\275)28620 9830 y Fq(;)202 b(E)30054 10012 y Fo(\275)30591 9830 y Ft(\))398 b(in)g Fq(G)f Ft(is)h(an)g Fs(in\257nite)g Ft(sub-graph)h(of)f Fq(G)g Ft(suc)-34 b(h)1575 11335 y(that)7750 14346 y Fq(V)8457 14528 y Fo(\275)9331 14346 y Ft(=)337 b Fr(f)p Fq(x)11910 14528 y Fw(1)12435 14346 y Fq(;)202 b(x)13667 14528 y Fw(2)14193 14346 y Fq(;)g(:)g(:)g(:)f(;)h(x)17580 14528 y Fo(n)18205 14346 y Fq(;)g(:)g(:)g(:)o Fr(g)1616 b Fq(E)23275 14528 y Fo(\275)24149 14346 y Ft(=)337 b Fr(ff)p Fq(x)27334 14528 y Fw(1)27859 14346 y Fq(;)202 b(x)29091 14528 y Fw(2)29616 14346 y Fr(g)p Fq(;)g Fr(f)p Fq(x)32060 14528 y Fw(2)32586 14346 y Fq(;)g(x)33818 14528 y Fw(3)34343 14346 y Fr(g)p Fq(;)g(:)g(:)g(:)f(;)h Fr(f)p Fq(x)38942 14528 y Fo(n)p Fn(\241)p Fw(1)40770 14346 y Fq(;)g(x)42002 14528 y Fo(n)42628 14346 y Fr(g)p Fq(;)g(:)g(:)g(:)o Fr(g)1575 16571 y Ft(where)417 b(all)g Fq(x)7483 16753 y Fo(i)8276 16571 y Ft(are)g(distinct.)579 b(The)418 b(v)-34 b(ertex)417 b Fq(x)22093 16753 y Fw(1)23036 16571 y Ft(is)g(called)g(the)h Fs(starting)444 b(vertex)416 b Ft(of)i(the)g(ra)-34 b(y)417 b(and)i(w)-34 b(e)418 b(sa)-34 b(y)417 b(that)1575 18077 y Fq(\275)403 b Fs(starts)431 b(at)h Fq(x)8113 18259 y Fw(1)8639 18077 y Ft(.)538 b(W)-101 b(e)403 b(denote)h(b)-34 b(y)403 b Fr(R)18215 17637 y Fo(x)18215 18436 y(G)19408 18077 y Ft(the)g(set)h(of)g(all)e(ra)-34 b(ys)404 b(in)f Fq(G)g Ft(starting)h(at)g Fq(x)p Ft(.)538 b(A)403 b Fs(ge)-62 b(o)g(desic)430 b(r)-62 b(ay)403 b Ft(in)g Fq(G)g Ft(is)g(a)1575 19582 y(ra)-34 b(y)404 b Fq(\275)f Ft(suc)-34 b(h)405 b(that,)g(if)g Fq(V)12212 19764 y Fo(\275)13085 19582 y Ft(=)337 b Fr(f)p Fq(x)15664 19764 y Fw(0)16190 19582 y Fq(;)202 b(x)17422 19764 y Fw(1)17947 19582 y Fq(;)g(x)19179 19764 y Fw(2)19704 19582 y Fq(;)g(:)g(:)g(:)f(;)h(x)23091 19764 y Fo(n)23717 19582 y Fq(;)g(:)g(:)g(:)o Fr(g)p Ft(,)403 b(then)i Fq(d)30408 19770 y Fo(G)31197 19582 y Ft(\()p Fq(x)32361 19764 y Fw(0)32887 19582 y Fq(;)202 b(x)34119 19764 y Fo(n)34744 19582 y Ft(\))338 b(=)e Fq(n)404 b Ft(for)g(all)f Fq(n)337 b Fr(2)f Fp(N)p Ft(.)1575 21088 y(Let)350 b Fq(\275)f Ft(and)i Fq(\275)7599 20648 y Fn(0)8260 21088 y Ft(b)34 b(e)350 b(t)-34 b(w)g(o)351 b(geo)34 b(desic)350 b(ra)-34 b(ys)350 b(starting)h(at)f Fq(x)g Ft(with)h(v)-34 b(ertex)350 b(sets)g Fq(V)35595 21270 y Fo(\275)36469 21088 y Ft(=)337 b Fr(f)p Fq(x)f Ft(=)h Fq(x)41357 21270 y Fw(0)41882 21088 y Fq(;)202 b(x)43114 21270 y Fw(1)43640 21088 y Fq(;)g(x)44872 21270 y Fw(2)45397 21088 y Fq(;)g(:)g(:)g(:)f(;)h(x) 48784 21270 y Fo(n)49410 21088 y Fq(;)g(:)g(:)g(:)n Fr(g)1575 22593 y Ft(and)294 b Fq(V)4528 22797 y Fo(\275)5010 22545 y Fm(0)5701 22593 y Ft(=)336 b Fr(f)p Fq(x)h Ft(=)f Fq(y)10489 22775 y Fw(0)11015 22593 y Fq(;)202 b(y)12148 22775 y Fw(1)12674 22593 y Fq(;)g(y)13807 22775 y Fw(2)14333 22593 y Fq(;)g(:)g(:)g(:)f(;)h(y)17621 22775 y Fo(n)18247 22593 y Fq(;)g(:)g(:)g(:)o Fr(g)294 b Ft(resp)34 b(ectiv)-34 b(ely)-101 b(.)500 b(If)293 b Fq(V)29851 22775 y Fo(\275)30682 22593 y Ft(and)i Fq(V)33636 22797 y Fo(\275)34118 22545 y Fm(0)34765 22593 y Ft(are)f(suc)-34 b(h)294 b(that)h Fq(d)42449 22781 y Fo(G)43238 22593 y Ft(\()p Fq(x)44402 22775 y Fo(n)45028 22593 y Fq(;)202 b(y)46161 22775 y Fo(m)47049 22593 y Ft(\))337 b(=)f Fq(n)49 b Ft(+)g Fq(m)1575 24099 y Ft(for)515 b(an)-34 b(y)515 b Fr(f)p Fq(n;)202 b(m)p Fr(g)521 b(2)g Fp(N)p Ft(,)543 b(then)516 b(the)f(union)h Fq(\261)567 b Ft(=)521 b Fq(\275)343 b Fr([)g Fq(\275)26955 23659 y Fn(0)27780 24099 y Ft(is)515 b(called)f(a)h(a)f Fs(ge)-62 b(o)g(desic)533 b(diameter)512 b Ft(\(or)j Fs(bi-in\257nite)1575 25604 y(ge)-62 b(o)g(desic)p Ft(\))401 b(in)j Fq(G)p Ft(.)1575 27109 y(Giv)-34 b(en)422 b Fq(G)367 b Ft(=)f(\()p Fq(V)67 b(;)202 b(E)70 b Ft(\))423 b(connected)g(and)f Fq(R)377 b Fr(\275)367 b Fq(V)269 b Ft(,)426 b(let)c Fq(E)70 b Fr(j)26841 27297 y Fo(R)27977 27109 y Ft(=)366 b Fr(ff)p Fq(x;)202 b(y)43 b Fr(g)367 b(2)g Fq(E)437 b Ft(:)366 b Fq(x)h Fr(2)g Fq(R)9 b(;)202 b(y)411 b Fr(2)366 b Fq(R)9 b Fr(g)423 b Ft(and)g(de\257ne)g(the)1575 28615 y(graph)418 b Fq(G)p Fr(j)6316 28803 y Fo(R)7445 28615 y Ft(=)359 b(\()p Fq(R)9 b(;)202 b(E)70 b Fr(j)11988 28803 y Fo(R)12757 28615 y Ft(\).)580 b(Note)417 b(that)i Fq(G)p Fr(j)21017 28803 y Fo(R)22204 28615 y Ft(is)e(a)h(sub-graph)h (of)f Fq(G)p Ft(.)579 b(W)-101 b(e)417 b(call)g Fq(G)p Fr(j)39030 28803 y Fo(R)40217 28615 y Fs(the)445 b(r)-62 b(estriction)444 b(of)h Fq(G)h Fs(to)1575 30120 y Fq(R)9 b Ft(.)540 b(W)-101 b(e)404 b(sa)-34 b(y)405 b(that)g Fq(R)348 b Fr(\275)337 b Fq(V)674 b Ft(is)404 b Fs(c)-62 b(onne)g(cte)g(d)402 b Ft(if)j Fq(G)p Fr(j)23193 30308 y Fo(R)24367 30120 y Ft(is)f(connected.)540 b(F)-101 b(or)405 b(an)-34 b(y)404 b(non)i(empt)-34 b(y)405 b Fq(R)347 b Fr(\275)337 b Fq(V)270 b Ft(,)404 b(w)-34 b(e)405 b(further)1575 31626 y(denote)g(b)-34 b(y)404 b Fq(@)7808 31808 y Fo(e)8299 31626 y Fq(R)414 b Ft(the)405 b(\(edges\))g(b)34 b(oundary)404 b(of)h Fq(R)414 b Ft(de\257ned)405 b(b)-34 b(y)19446 34349 y Fq(@)20090 34531 y Fo(e)20582 34349 y Fq(R)346 b Ft(=)337 b Fr(f)p Fq(e)g Fr(2)g Fq(E)406 b Ft(:)336 b Fr(j)p Fq(e)270 b Fr(\\)f Fq(R)9 b Fr(j)337 b Ft(=)g(1)p Fr(g)15380 b Ft(\(2)p Fq(:)p Ft(1\))3457 37072 y(If)404 b Fq(R)346 b Fr(\275)337 b Fq(V)673 b Ft(w)-34 b(e)405 b(denote)19660 38578 y(diam\()p Fq(R)9 b Ft(\))339 b(=)1261 b(sup)25775 39679 y Fn(f)p Fo(x;y)32 b Fn(g\275)p Fo(R)29652 38578 y Fq(d)30283 38766 y Fo(G)31071 38578 y Ft(\()p Fq(x;)202 b(y)43 b Ft(\))1575 41761 y(If)533 b Fq(x)551 b Fr(2)h Fq(V)802 b Ft(w)-34 b(e)533 b(denote)h Fq(B)61 b Ft(\()p Fq(x;)202 b(r)16207 41943 y Fw(0)16733 41761 y Ft(\))533 b(the)g(ball)g(of)g(radius)h Fq(r)28271 41943 y Fw(0)29329 41761 y Ft(and)g(cen)-34 b(ter)533 b(at)g Fq(x)p Ft(,)565 b(namely)532 b Fq(B)61 b Ft(\()p Fq(x;)202 b(r)46322 41943 y Fw(0)46848 41761 y Ft(\))552 b(=)f Fr(f)p Fq(y)595 b Fr(2)1575 43267 y Fq(V)606 b Ft(:)740 b Fq(d)4596 43455 y Fo(G)5385 43267 y Ft(\()p Fq(x;)202 b(y)43 b Ft(\))337 b Fr(\267)g Fq(r)10360 43449 y Fw(0)10886 43267 y Fr(g)p Ft(.)1575 44772 y(A)404 b(graph)h(is)f Fs(planar)e Ft(if)j(it)f(can)g(b)34 b(e)404 b(dra)-34 b(wn)405 b(in)f(a)g(plane)h(without)h(edges)e(crossing.)1575 46278 y(It)535 b(is)h(p)34 b(ossible)535 b(for)h(a)f(planar)h(graph)h Fq(G)555 b Ft(=)h(\()p Fq(V)67 b(;)202 b(E)70 b Ft(\))536 b(to)g(de\257ne)g(the)g Fs(dual)e Ft(graph)i Fq(G)41149 45838 y Fn(\244)42231 46278 y Ft(=)556 b(\()p Fq(V)45177 45838 y Fn(\244)45703 46278 y Fq(;)202 b(E)47207 45838 y Fn(\244)47732 46278 y Ft(\))536 b(in)g(the)1575 47783 y(follo)-34 b(wing)306 b(w)-34 b(a)g(y:)489 b(let)305 b(a)g Fs(fac)-62 b(e)304 b Ft(of)h(a)g(planar)h(graph)f(b)34 b(e)305 b(eac)-34 b(h)305 b(simply)g(connected)h(region)e(of)i(the)f (plane)h(delimited)1575 49289 y(b)-34 b(y)347 b(the)h(edges)f(of)g Fq(G)p Ft(.)520 b(No)-34 b(w)348 b(tak)-34 b(e)347 b(as)g(set)g(of)h (the)f(v)-34 b(ertices)346 b Fq(V)28747 48849 y Fn(\244)29620 49289 y Ft(of)h Fq(G)31896 48849 y Fn(\244)32769 49289 y Ft(the)g(set)h(of)f(the)h(faces)f(of)g Fq(G)p Ft(;)366 b(t)-34 b(w)g(o)348 b(faces)f(in)1575 50794 y Fq(V)2551 50354 y Fn(\244)3371 50794 y Ft(forms)295 b(an)f(edge)g Fq(e)11393 50354 y Fn(\244)12256 50794 y Fr(2)337 b Fq(E)14366 50354 y Fn(\244)15185 50794 y Ft(if)295 b(they)f(ha)-34 b(v)g(e)295 b(an)f(edge)g Fq(e)337 b Fr(2)g Fq(E)364 b Ft(in)294 b(common.)502 b(With)295 b(this)f(de\257nitions,)317 b(eac)-34 b(h)295 b(edge)1575 52300 y Fq(e)2139 51860 y Fn(\244)3082 52300 y Fr(2)416 b Fq(E)5271 51860 y Fn(\244)6249 52300 y Ft(cuts)453 b(exactly)e(one)h(edges)g Fq(e)417 b Fr(2)g Fq(E)521 b Ft(and)453 b(vicev)-34 b(ersa,)463 b(i.e.)681 b(there)452 b(is)g(a)g(one-to-one)h(corresp)34 b(ondence)1575 53805 y(b)g(et)-34 b(w)g(een)412 b(edges)g(in)f Fq(E)481 b Ft(and)413 b(edges)e(in)h Fq(E)20304 53365 y Fn(\244)20829 53805 y Ft(.)561 b(Giv)-34 b(en)411 b Fq(\260)416 b Fr(\275)349 b Fq(E)481 b Ft(w)-34 b(e)412 b(denote)g(b)-34 b(y)412 b Fq(\260)37047 53365 y Fn(\244)37922 53805 y Fr(\275)348 b Fq(E)40178 53365 y Fn(\244)41115 53805 y Ft(the)412 b(unique)g(set)g(in)f Fq(E)51443 53365 y Fn(\244)1575 55311 y Ft(asso)34 b(ciated)475 b(to)g Fq(\260)542 b Ft(b)-34 b(y)475 b(suc)-34 b(h)476 b(one-to-one)f (corresp)34 b(ondence.)751 b(Note)475 b(also)f(that)i(the)g(p)34 b(ossibilit)-34 b(y)474 b Fq(e)46605 54871 y Fn(\244)47586 55311 y Ft(=)455 b Fr(f)p Fq(z)53 b(;)202 b(z)53 b Fr(g)1575 56816 y Ft(with)405 b Fq(z)389 b Fr(2)337 b Fq(V)7411 56376 y Fn(\244)8341 56816 y Ft(is)403 b(not)i(excluded;)f(it)g(o)34 b(ccurs)403 b(whenev)-34 b(er)405 b Fq(e)337 b Ft(=)f Fr(f)p Fq(x;)202 b(y)43 b Fr(g)405 b Ft(with)g Fq(d)36409 56998 y Fo(x)37330 56816 y Ft(=)337 b(1.)1575 59896 y Fr(x)p Ft(3)404 b Fs(Cut)433 b(sets,)f(minimal)h(cut)f(sets)h(and)g (Peierls)f(Contours)h(on)g(in\257nite)h(gr)-62 b(aphs)p Ft(.)1575 61402 y(In)330 b(order)g(to)h(apply)f(a)g(P)-34 b(eierls)329 b(t)-34 b(yp)34 b(e)331 b(argumen)-34 b(t)331 b(for)f(p)34 b(ercolation)330 b(on)h(general)e(graphs)i(w)-34 b(e)331 b(need)f(to)h(in)-34 b(tro)34 b(duce)1575 62907 y(the)422 b(concept)g(of)g(cut)g(sets)g(and)h(minimal)e(cut)h(sets)g (of)h(a)e(graph.)592 b(Suc)-34 b(h)423 b(cut)f(sets)g(ma)-34 b(y)422 b(b)34 b(e)422 b(regarded)f(as)h(the)1575 64413 y(generalization)403 b(of)i(the)f(concept)h(of)f(P)-34 b(eierls)403 b(con)-34 b(tours)405 b(used)g(in)f(the)h Fq(d)p Ft(-dimensional)f(Ising)g(mo)34 b(del.)1575 65918 y(Hereafter)403 b Fp(G)337 b Ft(=)f(\()p Fp(V)p Fq(;)202 b Fp(E)p Ft(\))405 b(will)f(denote)h(a)f(lo)34 b(cally)402 b(\257nite)j(in\257nite)g(connected)f(graph.)1575 67424 y(A)j(set)g Fq(\260)i Fr(\275)342 b Fp(E)407 b Ft(is)g(called)g(a)g Fs(cut)435 b(set)407 b Ft(if)g(the)g(graph)h Fp(G)25495 67606 y Fo(\260)26429 67424 y Ft(=)342 b(\()p Fp(V)p Fq(;)202 b Fp(E)272 b Fr(\241)f Fq(\260)67 b Ft(\))408 b(is)f(disconnected.)548 b(A)408 b(cut)f(set)h Fq(\260)h Fr(\275)341 b Fp(E)408 b Ft(is)1575 68929 y(called)433 b(a)h Fs(minimal)460 b(cut)g(set)433 b Ft(if)h(for)g(all)f Fq(e)387 b Fr(2)f Fq(\260)501 b Ft(the)434 b(set)g Fq(\260)356 b Fr(\241)289 b Fq(e)434 b Ft(is)g(not)h(a)f(cut)g(set.)628 b(A)434 b Fs(\257nite)f Ft(minimal)h(cut)g(set)1575 70435 y Fq(\260)403 b Fr(\275)337 b Fp(E)404 b Ft(in)g Fp(G)g Ft(ha)-34 b(v)g(e)405 b(the)f(follo)-34 b(wing)405 b(prop)34 b(ert)-34 b(y)-101 b(.)1575 72570 y Fl(Prop)39 b(osition)466 b(1)p Ft(:)540 b Fs(L)-62 b(et)434 b Fq(\260)501 b Fs(b)-62 b(e)434 b(a)g(\257nite)g(minimal)g(cut)f(set)g(in)i Fp(G)338 b Ft(=)f(\()p Fp(V)p Fq(;)202 b Fp(E)p Ft(\))p Fs(,)436 b(then)d(it)h(may)g(o)-62 b(c)g(cur)433 b(only)g(one)h(of)1575 74075 y(the)e(fol)62 b(lowing)432 b(two)i(p)-62 b(ossibilities:)26469 78283 y Ft(3)p eop %%Page: 4 4 4 3 bop 1575 5313 a Fs(i\))433 b Fp(G)3819 5495 y Fo(\260)4844 5313 y Fs(has)g(no)g(\257nite)h(c)-62 b(onne)g(cte)g(d)432 b(c)-62 b(omp)g(onents,)1575 7134 y(ii\))532 b Fp(G)4290 7316 y Fo(\260)5415 7134 y Fs(has)g(one)h(and)f(only)h(one)g(\257nite)g (c)-62 b(onne)g(cte)g(d)531 b(c)-62 b(omp)g(onent)532 b Fq(A)33512 7316 y Fo(\260)34621 7134 y Ft(=)517 b(\()p Fq(I)37085 7316 y Fo(\260)37677 7134 y Fq(;)202 b(E)39111 7316 y Fo(\260)39703 7134 y Ft(\))534 b Fs(with)e Fq(E)44335 7316 y Fo(\260)45444 7134 y Ft(=)517 b Fp(E)p Fr(j)48049 7322 y Fo(I)48457 7433 y Fk(\260)49043 7134 y Ft(\))534 b Fs(and)1575 8639 y Fq(\260)403 b Ft(=)337 b Fq(@)4530 8821 y Fo(e)5021 8639 y Fq(I)5554 8821 y Fo(\260)6146 8639 y Fs(.)1575 10775 y Fl(Pro)39 b(of)p Ft(.)521 b(Let)352 b(us)h(denote)g(b)-34 b(y)353 b Fp(G)15872 10335 y Fw(ext)15872 11074 y Fo(\260)17545 10775 y Ft(=)336 b(\()p Fp(V)20170 10335 y Fw(ext)20170 11074 y Fo(\260)21507 10775 y Fq(;)202 b Fp(E)22854 10335 y Fw(ext)22854 11074 y Fo(\260)24190 10775 y Ft(\))353 b(the)g(union)g(of)g(all)g(in\257nite)g(comp)34 b(onen)-34 b(ts)354 b(of)f Fp(G)46277 10957 y Fo(\260)46869 10775 y Ft(.)521 b(Let)352 b(no)-34 b(w)1575 12391 y Fp(G)2518 11951 y Fw(in)-26 b(t)2518 12691 y Fo(\260)4058 12391 y Ft(b)34 b(e)362 b(the)g(graph)g(with)h(v)-34 b(ertex)361 b(set)h Fq(I)19839 12573 y Fo(\260)20767 12391 y Ft(=)337 b Fp(V)184 b Fr(\241)g Fp(V)25108 11951 y Fw(ext)25108 12691 y Fo(\260)26807 12391 y Ft(and)362 b(with)h(edge)f(set)f Fq(E)37303 12573 y Fo(\260)38232 12391 y Ft(=)337 b Fp(E)184 b Fr(\241)g Ft(\()p Fq(\260)252 b Fr([)184 b Fp(E)44782 11951 y Fw(ext)44782 12691 y Fo(\260)46119 12391 y Ft(\).)525 b(If)361 b Fq(I)49154 12573 y Fo(\260)50083 12391 y Ft(=)336 b Fr(;)1575 13896 y Ft(\(and)405 b(consequen)-34 b(tly)405 b Fq(E)12477 14078 y Fo(\260)13406 13896 y Ft(=)336 b Fr(;)p Ft(\))405 b(then)g Fp(G)19871 14078 y Fo(\260)20867 13896 y Ft(has)f(no)h (\257nite)g(connected)f(comp)34 b(onen)-34 b(ts)406 b(and)f(w)-34 b(e)404 b(are)g(in)g(the)h(case)1575 15402 y Fs(i\))p Ft(.)565 b(Assume)413 b(that)i Fq(I)11014 15584 y Fo(\260)11958 15402 y Fr(6)p Ft(=)351 b Fr(;)p Ft(.)566 b(In)413 b(this)h(case)f Fp(G)22176 14962 y Fw(in)-26 b(t)22176 15702 y Fo(\260)23768 15402 y Ft(m)-34 b(ust)414 b(b)34 b(e)413 b(connected)h(since)f(b)-34 b(y)413 b(de\257nition)i(of)e(minimal)g(cut)1575 16907 y(set,)405 b(for)g(an)-34 b(y)405 b(edge)g Fq(e)338 b Fr(2)h Fq(\260)67 b Ft(,)404 b(the)i(graph)f(\()p Fp(V)21071 16468 y Fw(ext)21071 17207 y Fo(\260)22678 16907 y Fr([)270 b Fq(I)24289 17089 y Fo(\260)24881 16907 y Fq(;)202 b Fp(E)26228 16468 y Fw(ext)26228 17207 y Fo(\260)27834 16907 y Fr([)270 b Fq(E)29807 17089 y Fo(\260)30669 16907 y Fr([)g Fq(e)p Ft(\))339 b(=)e(\()p Fp(V)p Fq(;)202 b Fp(E)271 b Fr(\241)f Ft(\()p Fq(\260)337 b Fr(\241)270 b Fq(e)p Ft(\)\))406 b(is)e(connected)i(and)1575 18524 y(this)389 b(w)-34 b(ould)390 b(not)f(b)34 b(e)389 b(p)34 b(ossible)388 b(if)h Fp(G)17799 18084 y Fw(in)-26 b(t)17799 18823 y Fo(\260)19367 18524 y Ft(is)388 b(done)h(b)-34 b(y)389 b(more)g(than)g(one)g(connected)g(comp)34 b(onen)-34 b(ts.)535 b(Finally)-101 b(,)391 b(the)1575 20029 y(iden)-34 b(tit)g(y)405 b Fq(E)6948 20211 y Fo(\260)7877 20029 y Ft(=)336 b Fp(E)p Fr(j)10301 20217 y Fo(I)10709 20328 y Fk(\260)11700 20029 y Ft(follo)-34 b(ws)404 b(immediately)g(from)g (the)h(fact)f(that)i Fq(E)33447 20211 y Fo(\260)34308 20029 y Fr([)269 b Fp(E)36193 19589 y Fw(ext)36193 20329 y Fo(\260)37799 20029 y Fr([)g Fq(\260)404 b Ft(=)336 b Fp(E)p Ft(.)539 b Fj(\244)1575 22165 y Ft(If)470 b(case)f Fs(ii\))g Ft(o)34 b(ccurs)469 b(w)-34 b(e)470 b(sa)-34 b(y)470 b(that)h Fq(\260)537 b Ft(is)470 b Fs(a)493 b(Peierls)g(c)-62 b(ontour)p Ft(.)734 b(In)470 b(this)h(case)e(the)h(set)g Fq(I)41730 22347 y Fo(\260)42769 22165 y Fr(\275)446 b Fq(V)739 b Ft(is)470 b(called)f Fs(the)1575 23670 y(interior)401 b(of)g Fq(\260)437 b Ft(and)371 b(the)f(graph)g Fq(A)17150 23852 y Fo(\260)17743 23670 y Ft(,)376 b(uniquely)369 b(determined)i(b)-34 b(y)370 b Fq(\260)67 b Ft(,)376 b(is)370 b(called)f(the)h Fs(animal)401 b(asso)-62 b(ciate)g(d)398 b(to)k(the)1575 25175 y(Peierls)443 b(c)-62 b(ontour)444 b Fq(\260)67 b Ft(.)575 b(W)-101 b(e)416 b(sa)-34 b(y)417 b(that)g(a)g(P)-34 b(eierls)415 b(con)-34 b(tour)418 b Fq(\260)483 b Fs(surr)-62 b(ounds)414 b Fq(x)357 b Fr(2)h Fq(V)686 b Ft(if)416 b Fq(x)357 b Fr(2)g Fq(I)42064 25357 y Fo(\260)42656 25175 y Ft(.)576 b(P)-34 b(eierls)415 b(con)-34 b(tours)1575 26681 y(in)404 b Fp(G)g Ft(ha)-34 b(v)g(e)404 b(the)h(follo)-34 b(wing)405 b(prop)34 b(ert)-34 b(y)-101 b(.)1575 28816 y Fl(Prop)39 b(osition)494 b(2)p Ft(.)615 b Fs(L)-62 b(et)456 b Fq(\260)524 b Fs(b)-62 b(e)456 b(a)h(Peierls)e(c)-62 b(ontour)456 b(in)h Fp(G)g Fs(surr)-62 b(ounding)455 b Fq(x)p Fs(,)462 b(then)456 b(for)h(any)f(r)-62 b(ay)456 b Fq(\275)379 b Ft(=)g(\()p Fq(V)48989 28998 y Fo(\275)49526 28816 y Fq(;)202 b(E)50960 28998 y Fo(\275)51497 28816 y Ft(\))1575 30322 y Fs(in)434 b Fp(G)f Fs(starting)f(at)h Fq(x)g Fs(we)h(have)e(that)g Fq(E)19454 30504 y Fo(\275)20260 30322 y Fr(\\)270 b Fq(\260)403 b Fr(6)p Ft(=)337 b Fr(;)p Ft(.)1575 32142 y Fl(Pro)39 b(of.)553 b Ft(Supp)34 b(ose)410 b(b)-34 b(y)410 b(absurd)g(that)g Fq(E)19863 32324 y Fo(\275)20672 32142 y Fr(\\)273 b Fq(\260)412 b Ft(=)345 b Fr(;)p Ft(.)553 b(Then)410 b Fq(E)29642 32324 y Fo(\275)30524 32142 y Fr(\275)344 b Fq(E)32776 31702 y Fw(1)32706 32442 y Fo(\275)33574 32142 y Fr([)273 b Fq(E)35620 31702 y Fw(2)35550 32442 y Fo(\275)36555 32142 y Ft(with)409 b Fq(E)40285 31702 y Fw(1)40215 32442 y Fo(\275)41156 32142 y Fr(\275)344 b Fq(E)43338 32324 y Fo(\260)44339 32142 y Ft(and)410 b Fq(E)47666 31702 y Fw(2)47596 32442 y Fo(\275)48537 32142 y Fr(\275)49925 31829 y Ft(~)49824 32142 y Fp(E)50632 31702 y Fw(ext)50632 32442 y Fo(\260)1575 33847 y Ft(where)5470 33541 y(~)5196 33847 y Fq(G)6149 33408 y Fw(ext)6149 34147 y Fo(\260)8015 33847 y Ft(=)529 b(\()10093 33534 y(~)9958 33847 y Fp(V)10833 33408 y Fw(ext)10833 34147 y Fo(\260)12170 33847 y Fq(;)12810 33534 y Ft(~)12709 33847 y Fp(E)13517 33408 y Fw(ext)13517 34147 y Fo(\260)14853 33847 y Ft(\))520 b(is)g(some)g(\(in\257nite\))h(connected)g(comp)34 b(onen)-34 b(t)521 b(of)g Fp(G)40022 33408 y Fw(ext)40022 34147 y Fo(\260)41357 33847 y Ft(.)886 b(The)521 b(case)e Fq(E)48834 33408 y Fw(2)48764 34147 y Fo(\275)49890 33847 y Ft(=)529 b Fr(;)1575 35353 y Ft(w)-34 b(ould)474 b(imply)f(that)i Fq(E)12240 35535 y Fo(\275)13229 35353 y Fr(\275)452 b Fq(E)15519 35535 y Fo(\260)16584 35353 y Ft(whic)-34 b(h)475 b(is)e(imp)34 b(ossible)473 b(since)g Fq(E)31319 35535 y Fo(\275)32329 35353 y Ft(is)g(in\257nite)h(and)h Fq(E)41116 35535 y Fo(\260)42181 35353 y Ft(is)e(\257nite.)747 b(The)474 b(case)1575 36858 y Fq(E)2540 36419 y Fw(1)2470 37158 y Fo(\275)3449 36858 y Ft(=)384 b Fr(;)433 b Ft(is)f(imp)34 b(ossible)432 b(since)g(no)h(edge)f(in)h Fp(E)22746 36419 y Fw(ext)22746 37158 y Fo(\260)24514 36858 y Ft(has)g Fq(x)f Ft(as)h(one)g(of)f(its)h(end-p)34 b(oin)-34 b(ts.)625 b(Finally)432 b(the)h(last)g(case)1575 38564 y Fq(E)2540 38124 y Fw(1)2470 38863 y Fo(\275)3402 38564 y Fr(6)p Ft(=)336 b Fr(;)319 b Ft(and)h Fq(E)8843 38124 y Fw(2)8773 38863 y Fo(\275)9705 38564 y Fr(6)p Ft(=)337 b Fr(;)318 b Ft(is)h(imp)34 b(ossible)318 b(since)h(otherwise)g Fq(A)27969 38746 y Fo(\260)28660 38564 y Fr([)29735 38250 y Ft(~)29567 38564 y Fp(G)30510 38124 y Fw(ext)30510 38863 y Fo(\260)32165 38564 y Ft(w)-34 b(ould)320 b(b)34 b(e)318 b(connected)i(whic)-34 b(h)319 b(con)-34 b(tradicts)1575 40069 y(prop)34 b(osition)405 b(1.)538 b Fj(\244)1575 42204 y Ft(W)-101 b(e)459 b(denote)h(b)-34 b(y)460 b(\241)10176 42397 y Fi(G)11379 42204 y Ft(the)g(set)f(of)h(all)f(P)-34 b(eierls)458 b(con)-34 b(tours)461 b(in)f Fp(G)f Ft(and)h(b)-34 b(y)460 b(\241)35327 41765 y Fo(x)35327 42569 y Fi(G)36530 42204 y Ft(the)g(set)g(of)f(P)-34 b(eierls)459 b(con)-34 b(tours)460 b(in)1575 43710 y Fp(G)497 b Ft(surrounding)i Fq(x)p Ft(.)818 b(There)497 b(is)h(clearly)d(a)j(one-to-one)g(corresp) 34 b(ondence)497 b(b)34 b(et)-34 b(w)g(een)499 b(P)-34 b(eierls)496 b(con)-34 b(tours)499 b(and)1575 45215 y(connected)427 b(\257nite)h(sets)g(in)f Fp(G)p Ft(.)607 b(Namely)-101 b(,)432 b(if)427 b(\243)22879 45408 y Fi(G)24050 45215 y Ft(denotes)h(the)g(set)f(of)h(all)e(connected)i(and)g(\257nite)g (subsets)g(of)1575 46721 y Fp(V)p Ft(,)450 b(then)441 b(prop)34 b(osition)441 b(1)g(implies)f(immediately)g(that)h(the)g(map) h Fq(\260)464 b Fr(7!)398 b Fq(I)35333 46903 y Fo(\260)36365 46721 y Ft(is)440 b(bijection)h(from)g(\241)46266 46914 y Fi(G)47451 46721 y Ft(in)-34 b(to)441 b(\243)50888 46914 y Fi(G)51632 46721 y Ft(.)1575 48226 y(As)375 b(a)h(matter)f(of)h (fact,)381 b(if)375 b Fq(R)347 b Fr(\275)336 b Fp(V)376 b Ft(is)f(connected)h(and)g(\257nite,)381 b(then)376 b Fq(@)33134 48408 y Fo(e)33626 48226 y Fq(R)385 b Ft(is)375 b(a)g(P)-34 b(eierls)374 b(con)-34 b(tour)377 b(with)f(in)-34 b(terior)1575 49732 y Fq(R)9 b Ft(.)1575 52812 y Fr(x)p Ft(4)404 b Fs(Bernoul)62 b(li)432 b(Per)-62 b(c)g(olation)431 b(on)j Fp(G)f Fs(and)h(the)e(Peierls)g(ar)-62 b(gument)p Ft(.)1575 54317 y(A)491 b(p)34 b(ercolation)491 b(pro)34 b(cess)491 b(on)h Fp(G)f Ft(is)g(de\257ned)i(as)e(follo)-34 b(ws.)802 b(Supp)34 b(ose)492 b(that)h(eac)-34 b(h)491 b(edge)h(in)f Fp(G)g Ft(can)h(ha)-34 b(v)g(e)492 b(t)-34 b(w)g(o)1575 55823 y(p)34 b(ossible)519 b(states.)884 b(Namely)519 b(an)g(edge)g Fq(e)529 b Fr(2)g Fp(E)519 b Ft(can)g(b)34 b(e)519 b(either)g Fs(op)-62 b(en)518 b Ft(or)h Fs(close)-62 b(d)p Ft(.)881 b(A)519 b Fs(c)-62 b(on\257gur)g(ation)517 b Ft(of)j(the)1575 57328 y(system)412 b(is)g(a)h(function)g Fq(!)395 b Ft(:)350 b Fp(E)h Fr(!)f(f)p Ft(0)p Fq(;)202 b Ft(1)p Fr(g)412 b Ft(assigning)h(to)g(eac)-34 b(h)413 b(edge)f Fq(e)351 b Fr(2)f Fp(E)413 b Ft(either)f(the)g(v)-67 b(alue)412 b(1)g(\(op)34 b(en)413 b(edge\))1575 58834 y(or)404 b(0)g(\(closed)g(edge\).)539 b(Let)403 b(\255)i(b)34 b(e)404 b(the)g(set)g(of)h(all)e(con\257gurations)j(of)e(the)h(system.) 1575 60339 y(In)414 b(the)h(Bernoulli)f(b)34 b(ond)415 b(p)34 b(ercolation,)417 b(the)e(edges)f(of)h Fp(G)f Ft(are)g Fs(op)-62 b(en)414 b Ft(with)h(indep)34 b(enden)-34 b(t)416 b(probabilit)-34 b(y)415 b Fq(p)g Ft(and)1575 61845 y(closed)e(with)i(indep)34 b(enden)-34 b(t)415 b(probabilit)-34 b(y)414 b(1)276 b Fr(\241)g Fq(p)p Ft(.)567 b(The)414 b(pro)34 b(duct)414 b(measure)g(of)g(the)g(con\257gurations)h (of)f(edges)1575 63350 y(is)404 b(denoted)h(b)-34 b(y)404 b Fq(P)9834 63532 y Fo(p)10363 63350 y Ft(.)1575 64856 y(Let)451 b Fq(C)87 b Ft(\()p Fq(x)p Ft(\))453 b(b)34 b(e)451 b(the)h(op)34 b(en)452 b(cluster)g(of)g(the)g(v)-34 b(ertex)451 b Fq(x)p Ft(,)463 b(namely)452 b Fq(C)87 b Ft(\()p Fq(x)p Ft(\))452 b(is)g(the)g(connected)g(set)g(of)g(op)34 b(en)452 b(edges)1575 66361 y(\(p)34 b(ossibly)404 b(empt)-34 b(y\))405 b(con)-34 b(taining)405 b Fq(x)p Ft(.)539 b(Then)405 b(de\257ne)19393 69084 y Fq(\265)19962 69266 y Fo(x)20547 69084 y Ft(\()p Fq(p)p Ft(\))337 b(=)g Fq(P)24494 69266 y Fo(p)25023 69084 y Fr(f)p Fq(C)87 b Ft(\()p Fq(x)p Ft(\))405 b(is)f(in\257nite)p Fr(g)15327 b Ft(\(4)p Fq(:)p Ft(1\))1575 71807 y(and)405 b(the)f Fs(critic)-62 b(al)432 b(pr)-62 b(ob)g(ability)430 b(of)j(p)-62 b(er)g(c)g(olation)401 b Ft(on)j Fp(G)g Ft(as)20634 74531 y Fq(p)21244 74713 y Fo(c)22044 74531 y Ft(=)336 b(sup)q Fr(f)p Fq(p)h Ft(:)f Fq(\265)27943 74713 y Fo(x)28528 74531 y Ft(\()p Fq(p)p Ft(\))h(=)g(0)p Fr(g)16568 b Ft(\(4)p Fq(:)p Ft(2\))26469 78283 y(4)p eop %%Page: 5 5 5 4 bop 1575 5313 a Ft(W)-101 b(e)577 b(recall)g(that)j(b)-34 b(y)578 b(FK)-34 b(G)579 b(inequalit)-34 b(y)578 b Fq(p)21541 5495 y Fo(c)22582 5313 y Ft(is)g(indep)34 b(enden)-34 b(t)579 b(of)g(the)g(c)-34 b(hoice)578 b(of)g(the)h(v)-34 b(ertex)577 b Fq(x)h Ft(\(see)g(e.g.)1575 6819 y(Theorem)404 b(2.8)g(in)g([9]\).)1575 9269 y(Let)i Fq(\260)h Fr(2)340 b Ft(\241)6690 9462 y Fi(G)7840 9269 y Ft(b)34 b(e)406 b(a)g(P)-34 b(eierls)405 b(con)-34 b(tour)407 b(w)-34 b(e)407 b(denote)f(b)-34 b(y)407 b Fq(S)26944 9451 y Fo(\260)27943 9269 y Ft(the)f(subset)i(of)e Fp(V)h Ft(de\257ned)g(as)f Fq(S)42826 9451 y Fo(\260)43759 9269 y Ft(=)340 b Fr(f)p Fq(x)g Fr(2)g Fp(V)h Ft(:)745 b Fq(x)340 b Fr(2)1575 10775 y Fq(e)404 b Ft(for)g(some)g Fq(e)338 b Fr(2)e Fq(\260)67 b Fr(g)404 b Ft(and)h(call)e(it)i Fs(the)432 b(supp)-62 b(ort)402 b Ft(of)i Fq(\260)67 b Ft(.)1575 12280 y(Let)395 b Fq(W)505 b Fr(\275)336 b Fp(E)p Ft(,)397 b(and)g(de\257ne)f Fq(L)p Ft(\()p Fq(W)168 b Ft(\))338 b(=)e(sup)20573 12571 y Fo(W)21461 12694 y Fh(1)21922 12571 y Fn([)p Fo(W)23437 12694 y Fh(2)23898 12571 y Fw(=)p Fo(W)25906 12280 y Fq(d)26537 12473 y Fi(G)27281 12280 y Ft(\()p Fq(W)28897 12462 y Fw(1)29423 12280 y Fq(;)202 b(W)31107 12462 y Fw(2)31632 12280 y Ft(\).)536 b(A)396 b(set)g Fq(W)504 b Fr(2)337 b Fp(E)396 b Ft(is)f Fq(r)34 b Ft(-close)395 b(if)g Fq(L)p Ft(\()p Fq(W)168 b Ft(\))338 b Fr(\267)e Fq(r)34 b Ft(.)1575 13786 y(A)384 b(graph)g Fp(G)g Ft(has)g(the)g Fs(minimal)415 b(cut)f(set)g(pr)-62 b(op)g(erty)381 b Ft(if)j(it)g(exists)f Fq(r)31314 13968 y Fw(0)32177 13786 y Fq(<)336 b Fr(1)384 b Ft(suc)-34 b(h)385 b(that)g(an)-34 b(y)384 b(minimal)f(cut)i(set)f Fq(\260)1575 15291 y Ft(is)404 b Fq(r)3341 15473 y Fw(0)3866 15291 y Ft(-close.)1575 16797 y(When)424 b Fp(G)h Ft(is)f(planar,)429 b(if)c Fq(\260)437 b Fr(2)371 b Ft(\241)16007 16990 y Fi(G)17175 16797 y Ft(is)424 b(a)g(P)-34 b(eierls)423 b(con)-34 b(tour)426 b(then)f(its)g(dual)f Fq(\260)35698 16357 y Fn(\244)36595 16797 y Fr(\275)370 b Fp(E)38716 16357 y Fn(\244)39666 16797 y Ft(is)424 b(a)h Fs(simple)450 b(cycle)423 b Ft(in)h Fp(G)51106 16357 y Fn(\244)51632 16797 y Ft(,)1575 18302 y(i.e.)531 b(a)386 b(path)h(in)f Fp(G)9797 17862 y Fn(\244)10709 18302 y Ft(that)h(starts)g(in)f(a)g(v) -34 b(ertex)385 b(of)h Fp(V)24963 17862 y Fn(\244)25876 18302 y Ft(and)g(ends)h(in)f(the)g(same)g(v)-34 b(ertex)386 b(and)g(do)34 b(esn't)387 b(rep)34 b(eat)385 b(a)1575 19807 y(v)-34 b(ertex)403 b(except)h(for)g(the)h(\257rst)f(and)h(last.) 1575 21313 y(It)f(is)g(no)-34 b(w)405 b(easy)f(to)g(state)h(the)f(P)-34 b(eierls)403 b(argumen)-34 b(t)406 b(for)e(our)g(p)34 b(ercolation)404 b(pro)34 b(cess)403 b(on)i Fp(G)p Ft(.)538 b(Consider)19262 23596 y(1)269 b Fr(\241)g Fq(\265)21918 23778 y Fo(x)22503 23596 y Ft(\()p Fq(p)p Ft(\))338 b(=)e Fq(P)26450 23778 y Fo(p)26979 23596 y Fr(fj)p Fq(C)87 b Ft(\()p Fq(x)p Ft(\))p Fr(j)337 b Fq(<)f Fr(1g)1575 25880 y Ft(No)-34 b(w)555 b(observ)-34 b(e)555 b(that)h(for)f(an)-34 b(y)555 b(con\257guration)h Fq(!)598 b Ft(suc)-34 b(h)556 b(that)g Fr(j)p Fq(C)87 b Ft(\()p Fq(x)p Ft(\))p Fr(j)588 b Fq(<)f Fr(1)555 b Ft(it)g(exists)f(at)h(least)g(a)g(P)-34 b(eierls)1575 27385 y(con)g(tour)491 b Fq(\260)557 b Ft(of)490 b(closed)f(edges)h(surrounding)i Fq(x)p Ft(.)795 b(The)491 b(probabilit)-34 b(y)490 b(of)h(suc)-34 b(h)490 b(P)-34 b(eierls)489 b(con)-34 b(tour)491 b Fq(\260)557 b Ft(is)490 b(simply)1575 28891 y(\(1)270 b Fr(\241)f Fq(p)p Ft(\))5215 28451 y Fn(j)p Fo(\260)52 b Fn(j)6330 28891 y Ft(.)538 b(Hence)14505 31902 y(1)269 b Fr(\241)g Fq(\265)17161 32084 y Fo(x)17747 31902 y Ft(\()p Fq(p)p Ft(\))337 b Fr(\267)21202 30750 y Fg(X)20915 33376 y Fo(\260)52 b Fn(2)p Fw(\241)22667 33071 y Fk(x)22667 33688 y Ff(G)23239 31902 y Ft(\(1)270 b Fr(\241)f Fq(p)p Ft(\))26879 31401 y Fn(j)p Fo(\260)52 b Fn(j)28331 31902 y Ft(=)29621 30750 y Fg(X)29610 33363 y Fo(n)p Fn(\270)p Fw(1)31384 31902 y Ft(\(1)269 b Fr(\241)g Fq(p)p Ft(\))35023 31401 y Fo(n)36166 30750 y Fg(X)35990 33285 y Fk(\260)43 b Fm(2)p Fh(\241)37520 33056 y Fk(x)37520 33643 y Ff(G)35985 34261 y Fm(j)p Fk(\260)g Fm(j)p Fh(=)p Fk(n)38432 31902 y Ft(1)10439 b(\(4)p Fq(:)p Ft(3\))1575 36000 y(Therefore)379 b(in)g(order)g(to)h(pro)-34 b(v)g(e)380 b(that)g Fq(p)19494 36182 y Fo(c)20294 36000 y Fq(<)336 b Ft(1)380 b(one)f(has)h(simply)f (to)h(sho)-34 b(w)380 b(that)h(it)e(exists)h(a)f(constan)-34 b(t)381 b Fq(K)467 b Ft(suc)-34 b(h)1575 37506 y(that,)405 b(for)f(all)f Fq(n)337 b Fr(2)g Fp(N)23813 37973 y Fg(X)23637 40508 y Fk(\260)43 b Fm(2)p Fh(\241)25167 40279 y Fk(x)25167 40866 y Ff(G)23632 41484 y Fm(j)p Fk(\260)g Fm(j)p Fh(=)p Fk(n)26080 39125 y Ft(1)336 b Fr(\267)h Fq(K)29418 38624 y Fo(n)49477 39125 y Ft(\(4)p Fq(:)p Ft(4\))1575 43290 y(It)482 b(is)h(imp)34 b(ortan)-34 b(t)483 b(to)g(remark)f(that)h(the)g (tec)-34 b(hniques)483 b(dev)-34 b(elop)34 b(ed)483 b(in)f([7])g(and)i ([22)o(])f(could)f(b)34 b(e)483 b(used)g(also)f(to)1575 44796 y(further)401 b(pro)-34 b(v)g(e)402 b(\(easily\))e(that)j(the)e (function)h Fq(\265)23219 44978 y Fo(x)23749 45101 y Fh(0)24266 44796 y Ft(\()p Fq(p)p Ft(\))f(is)g(actually)f(analytic)h (in)g(the)g(neigh)-34 b(b)34 b(orho)g(o)g(d)402 b(of)f Fq(p)337 b Ft(=)f(1)1575 46301 y(and)405 b(so)f(are)f(the)i(\257nite)g (connectivit)-34 b(y)404 b(functions.)1575 49382 y Fr(x)p Ft(4)g Fs(R)-62 b(esults)p Ft(.)1575 50887 y(W)-101 b(e)379 b(will)h(pro)-34 b(v)g(e)380 b(in)h(this)f(section)h(three)f(theorems.) 530 b(The)381 b(\257rst)g(one)f(is)g(an)h(easy)e(consequence)h(of)h (the)f(results)1575 52392 y(in)388 b(section)g(4)f(of)i([2)o(])f(and)h (iden)-34 b(tify)388 b(a)g(class)f(of)i(b)34 b(ounded)389 b(degree)e(graphs)i(for)f(whic)-34 b(h)388 b Fq(p)41439 52574 y Fo(c)42239 52392 y Fq(<)336 b Ft(1.)533 b(Note)388 b(that)h(for)1575 53898 y(our)f(purp)34 b(oses)388 b(the)g(conditions)g (listed)g(in)g(prop)34 b(osition)388 b(8)g(in)f([2])g(ma)-34 b(y)388 b(b)34 b(e)387 b(sligh)-34 b(tly)388 b(w)-34 b(eak)g(ened.)534 b(The)388 b(second)1575 55403 y(theorem)395 b(iden)-34 b(ti\257es)396 b(a)f(di\256eren)-34 b(t)395 b(class)g(of)h(b)34 b(ounded)396 b(degree)e(graphs)i(for)f(whic)-34 b(h)396 b Fq(p)39873 55585 y Fo(c)40673 55403 y Fq(<)336 b Ft(1;)398 b(in)d(particular)g(the)1575 56909 y(request)411 b(of)h(existence)f(of)i(a)e(bi-in\257nite)i(geo)34 b(desic)411 b(is)g(replaced)h(b)-34 b(y)412 b(a)f(v)-34 b(ery)411 b(w)-34 b(eak)412 b(isop)34 b(erimetric)411 b(inequalit)-34 b(y)1575 58414 y(condition.)1575 59920 y(In)325 b(the)h(third)g (theorem)g(w)-34 b(e)325 b(sho)-34 b(w)327 b(that)g(for)e(planar)h (graphs)g(again)g Fq(p)32865 60102 y Fo(c)33664 59920 y Fq(<)337 b Ft(1)325 b(for)g(a)h(v)-34 b(ery)325 b(large)f(class)h(of) h(graphs,)1575 61425 y(including)404 b(graphs)h(lo)34 b(cally)403 b(\257nited)i(but)g(not)g(b)34 b(ounded)405 b(degree.)1575 64506 y Fl(Theorem)464 b(1)p Ft(.)538 b Fs(L)-62 b(et)434 b Fp(G)336 b Ft(=)g(\()p Fp(V)p Fq(;)202 b Fp(E)p Ft(\))435 b Fs(b)-62 b(e)433 b(b)-62 b(ounde)g(d)431 b(de)-62 b(gr)g(e)g(e,)432 b(with)h(maximum)g(de)-62 b(gr)g(e)g(e)432 b Ft(\242)p Fs(,)h(such)f(that:)1575 66011 y(i\))h Fp(G)g Fs(has)g(the)f(minimal)h(cut)g(set)f(pr)-62 b(op)g(erty.)1575 67516 y(ii\))433 b(Ther)-62 b(e)432 b(exists)h(a)g(bi-in\257nite)f(ge)-62 b(o)g(desic)431 b(in)j Fp(G)p Fs(.)1575 69022 y(Then)f(the)g(critic)-62 b(al)431 b(pr)-62 b(ob)g(ability)430 b Fq(p)17270 69204 y Fo(c)18167 69022 y Fs(on)j Fp(G)p Fs(,)g(de\257ne)-62 b(d)433 b(in)h(\(4.2\),)e(is)h(strictly)e(less)h(than)h(1.)1575 70527 y Fl(Pro)39 b(of)p Ft(.)538 b(T)-101 b(ak)-34 b(e)404 b Fq(x)g Ft(on)g(the)h(bi-in\257nite)g(geo)34 b(desic.)537 b(W)-101 b(e)404 b(w)-34 b(an)g(t)406 b(to)e(b)34 b(ound)25492 71773 y Fg(X)25317 74308 y Fk(\260)43 b Fm(2)p Fh(\241)26847 74079 y Fk(x)26847 74666 y Ff(G)25311 75283 y Fm(j)p Fk(\260)g Fm(j)p Fh(=)p Fk(n)27759 72925 y Ft(1)21112 b(\(4)p Fq(:)p Ft(1\))26469 78283 y(5)p eop %%Page: 6 6 6 5 bop 1575 5313 a Ft(First)606 b(note)g(that,)657 b(b)-34 b(y)606 b(ii\),)656 b(there)606 b(exists)f Fq(r)22264 5495 y Fw(0)23396 5313 y Ft(suc)-34 b(h)607 b(that)g(ev)-34 b(ery)604 b(P)-34 b(eierls)605 b(con)-34 b(tour)607 b Fq(\260)740 b Fr(2)673 b Ft(\241)44887 5506 y Fi(G)46236 5313 y Ft(is)606 b Fq(r)48204 5495 y Fw(0)48729 5313 y Ft(-close.)1575 6819 y(Therefore,)403 b(a)i(P)-34 b(eierls)403 b(con)-34 b(tour)405 b Fq(\260)471 b Ft(with)405 b(cardinalit)-34 b(y)404 b Fq(n)f Ft(has)i(ev)-34 b(ery)403 b(pair)h(of)g(edges)g(at)h (most)f Fq(r)45261 7001 y Fw(0)45787 6819 y Fq(n)g Ft(far)g(apart.)1575 8324 y(Consider)410 b(no)-34 b(w)411 b(the)g(geo)34 b(desic)409 b(diameter)g Fq(\261)392 b Ft(=)346 b Fq(\275)273 b Fr([)g Fq(\275)25913 7884 y Fn(0)26634 8324 y Ft(in)409 b Fp(G)h Ft(from)g Fq(x)p Ft(,)h(then,)h(b)-34 b(y)410 b(prop)34 b(osition)411 b(2,)g Fq(E)47141 8506 y Fo(\275)47951 8324 y Fr(\\)273 b Fq(\260)413 b Fr(6)p Ft(=)346 b Fr(;)1575 9830 y Ft(and)403 b Fq(E)4825 10033 y Fo(\275)5307 9781 y Fm(0)5926 9830 y Fr(\\)267 b Fq(\260)403 b Fr(6)p Ft(=)337 b Fr(;)p Ft(.)538 b(Let)402 b Fq(e)13527 10012 y Fo(x)14112 9830 y Ft(\()p Fq(\260)67 b Ft(\))403 b(\(let)g Fq(e)18937 9390 y Fn(0)18937 10129 y Fo(x)19522 9830 y Ft(\()p Fq(\260)67 b Ft(\)\))404 b(b)34 b(e)402 b(the)h(\257rst)g(edge,)f(in)h(the)g (natural)g(ra)-34 b(y)402 b(order,)g(of)h Fq(E)46977 10012 y Fo(\275)47917 9830 y Ft(\(of)g Fq(E)50662 10033 y Fo(\275)51144 9781 y Fm(0)51497 9830 y Ft(\))1575 11335 y(whic)-34 b(h)405 b(b)34 b(elongs)404 b(to)g Fq(\260)471 b Ft(and)405 b(de\257ne)10733 14346 y Fq(\261)11272 14528 y Fo(n)11898 14346 y Ft(\()p Fq(x)p Ft(\))337 b(=)g Fr(f)p Fq(e)g Fr(2)g Fq(E)18697 14528 y Fo(\275)19570 14346 y Ft(:)g Fr(9)p Fq(\260)404 b Fr(2)337 b Ft(\241)23852 13846 y Fo(x)23852 14650 y Fi(G)24999 14346 y Ft(suc)-34 b(h)405 b(that)810 b Fr(j)p Fq(\260)67 b Fr(j)336 b Ft(=)g Fq(n)404 b Ft(and)h Fq(e)337 b Ft(=)f Fq(e)39981 14528 y Fo(x)40566 14346 y Ft(\()p Fq(\260)67 b Ft(\))p Fr(g)1575 16571 y Ft(Th)-34 b(us)13406 16925 y Fg(X)13231 19460 y Fk(\260)43 b Fm(2)p Fh(\241)14761 19231 y Fk(x)14761 19818 y Ff(G)13225 20436 y Fm(j)p Fk(\260)g Fm(j)p Fh(=)p Fk(n)15673 18077 y Ft(1)337 b(=)18674 16925 y Fg(X)17895 19610 y Fo(e)p Fn(2)p Fo(\261)19374 19721 y Fk(n)19942 19610 y Fw(\()p Fo(x)p Fw(\))22958 16925 y Fg(X)21538 19460 y Fk(\260)43 b Fm(2)p Fh(\241)23068 19231 y Fk(x)23068 19818 y Ff(G)23640 19460 y Fh(:)141 b Fm(j)p Fk(\260)43 b Fm(j)p Fh(=)p Fk(n)22314 20436 y(e)22705 20547 y(x)23224 20436 y Fh(\()p Fk(\260)g Fh(\)=)p Fk(e)26463 18077 y Ft(1)337 b Fr(\267)f(j)p Fq(\261)29561 18259 y Fo(n)30187 18077 y Ft(\()p Fq(x)p Ft(\))p Fr(j)539 b Ft(sup)32783 19124 y Fo(e)p Fn(2)p Fi(E)36409 16925 y Fg(X)35059 19443 y Fk(\260)43 b Fm(2)p Fh(\241)36589 19672 y Ff(G)37397 19443 y Fm(j)p Fk(\260)g Fm(j)p Fh(=)p Fk(n)36574 20196 y(e)p Fm(2)p Fk(\260)39845 18077 y Ft(1)1575 22485 y(No)-34 b(w,)379 b(since)373 b Fq(\275)206 b Fr([)h Fq(\275)10061 22045 y Fn(0)10744 22485 y Ft(is)372 b(bi-in\257nite)i(geo)34 b(desic,)378 b Fr(j)p Fq(\261)23364 22667 y Fo(n)23989 22485 y Ft(\()p Fq(x)p Ft(\))p Fr(j)337 b(\267)g Fq(d)28209 22678 y Fi(G)28953 22485 y Ft(\()p Fq(e)29988 22667 y Fo(x)30574 22485 y Ft(\()p Fq(\260)67 b Ft(\))p Fq(;)202 b(e)33314 22045 y Fn(0)33314 22785 y Fo(x)33899 22485 y Ft(\()p Fq(\260)67 b Ft(\)\).)529 b(Since)373 b Fr(f)p Fq(e)41177 22667 y Fo(x)41762 22485 y Ft(\()p Fq(\260)67 b Ft(\))p Fq(;)202 b(e)44502 22045 y Fn(0)44502 22785 y Fo(x)45088 22485 y Ft(\()p Fq(\260)67 b Ft(\))p Fr(g)338 b(\275)e Fq(\260)440 b Ft(and)1575 23990 y Fq(\260)471 b Ft(is)404 b Fq(r)4440 24172 y Fw(0)5369 23990 y Ft(close,)f(this)i (means)f(that)i Fq(d)p Ft(\()p Fq(e)18975 24172 y Fo(x)19560 23990 y Ft(\()p Fq(\260)67 b Ft(\))p Fq(;)202 b(e)22300 23551 y Fn(0)22300 24290 y Fo(x)22886 23990 y Ft(\()p Fq(\260)67 b Ft(\))337 b Fr(\267)g Fq(r)26687 24172 y Fw(0)27213 23990 y Fq(n)403 b Ft(and)i(w)-34 b(e)405 b(obtain)g(the)f(b)34 b(ound)23327 27001 y Fr(j)p Fq(\261)24203 27183 y Fo(n)24828 27001 y Ft(\()p Fq(x)p Ft(\))p Fr(j)337 b(\267)g Fq(r)28964 27183 y Fw(0)29489 27001 y Fq(n)1575 29226 y Ft(T)-101 b(o)336 b(b)34 b(ound)337 b(the)f(n)-34 b(um)g(b)34 b(er)337 b(of)f(P)-34 b(eierls)335 b(con)-34 b(tour)337 b(of)f(cardinalit)-34 b(y)336 b Fq(n)f Ft(con)-34 b(taining)337 b(a)f(\257xed)g(edge)f Fq(e)p Ft(,)350 b(w)-34 b(e)336 b(pro)34 b(ceed)335 b(as)1575 30732 y(follo)-34 b(ws.)513 b(Let)326 b(no)-34 b(w)327 b Fp(G)11484 30914 y Fo(r)11909 31037 y Fh(0)12751 30732 y Ft(b)34 b(e)325 b(the)i(graph)f(with)h(v)-34 b(ertex)326 b(set)g Fp(V)g Ft(and)h(edge)f(set)g Fp(E)36569 30914 y Fo(r)36994 31037 y Fh(0)37847 30732 y Ft(=)336 b Fr(ff)p Fq(x;)202 b(y)43 b Fr(g)338 b Ft(:)662 b Fq(d)44781 30925 y Fi(G)45525 30732 y Ft(\()p Fq(x;)202 b(y)43 b Ft(\))338 b Fr(\267)e Fq(r)50500 30914 y Fw(0)51026 30732 y Fr(g)p Ft(.)1575 32237 y Fp(G)2518 32419 y Fo(r)2943 32542 y Fh(0)3980 32237 y Ft(is)520 b(a)h(connected)h(b)34 b(ounded)522 b(degree)e(graph)i(with)f(maxim)-34 b(um)522 b(degree)e(\242)38280 32419 y Fo(\275)38762 32542 y Fh(0)39810 32237 y Ft(=)531 b(max)43540 32430 y Fo(x)p Fn(2)p Fi(V)45594 32237 y Fr(j)p Fq(B)61 b Ft(\()p Fq(x;)202 b(r)49161 32419 y Fw(0)49686 32237 y Ft(\))p Fr(j)532 b(\267)1575 33743 y Ft(\242)2585 33303 y Fo(r)3010 33426 y Fh(0)3471 33303 y Fw(+1)4728 33743 y Ft(.)853 b(Consider)510 b(no)-34 b(w)510 b(the)g(supp)34 b(ort)510 b Fq(S)21243 33925 y Fo(\260)22344 33743 y Ft(of)f Fq(\260)67 b Ft(.)853 b(Since)509 b Fq(\260)576 b Ft(is)509 b Fq(r)32059 33925 y Fw(0)32585 33743 y Ft(-close,)534 b(then)510 b(the)f(restriction)g Fq(S)48070 33925 y Fo(\260)48662 33743 y Fr(j)48999 33936 y Fi(G)49688 34047 y Fk(r)50073 34246 y Fh(0)51154 33743 y Ft(is)1575 35248 y(connected)404 b(in)h Fp(G)9589 35430 y Fo(r)10014 35553 y Fh(0)10529 35248 y Ft(.)539 b(Moreo)-34 b(v)g(er)403 b(1)337 b Fr(\267)g(j)p Fq(S)20034 35430 y Fo(\260)20626 35248 y Fr(j)f(\267)h Ft(2)p Fq(n)p Ft(.)538 b(Therefore)12721 39071 y(sup)12806 40119 y Fo(e)p Fn(2)p Fi(E)16230 37920 y Fg(X)14880 40438 y Fk(\260)43 b Fm(2)p Fh(\241)16410 40667 y Ff(G)17218 40438 y Fm(j)p Fk(\260)g Fm(j)p Fh(=)p Fk(n)16395 41191 y(e)p Fm(2)p Fk(\260)19666 39071 y Ft(1)336 b Fr(\267)h Ft(sup)21902 40119 y Fo(x)p Fn(2)p Fi(V)24270 37556 y Fw(2)p Fo(n)23915 37920 y Fg(X)23932 40555 y Fo(k)24 b Fw(=1)29661 37920 y Fg(X)27996 40438 y Fk(W)108 b Fm(\275)p Ff(V)p Fh(:)403 b Fm(j)p Fk(W)108 b Fm(j)p Fh(=)p Fk(k)26001 41222 y(W)g Fm(j)27125 41451 y Ff(G)27642 41562 y Fk(r)28027 41761 y Fh(0)29002 41222 y(connected)o Fk(;)404 b(x)p Fm(2)p Fk(W)35541 39071 y Fr(\267)37175 37556 y Fw(2)p Fo(n)36820 37920 y Fg(X)36838 40555 y Fo(k)24 b Fw(=1)38773 39071 y Ft(\242)39783 38571 y Fw(2)p Fo(k)39783 39371 y(\275)40265 39494 y Fh(0)1575 43554 y Ft(i.e.)537 b(\257nally)14503 44956 y Fg(X)12774 47640 y Fo(\260)52 b Fn(2)p Fw(\241)14526 47335 y Fk(x)14526 47952 y Ff(G)15098 47640 y Fw(:)262 b Fn(j)p Fo(\260)52 b Fn(j)p Fw(=)p Fo(n)18185 46107 y Ft(1)337 b Fr(\267)f Ft(2)p Fq(R)9 b(n)23227 44592 y Fw(2)p Fo(n)22873 44956 y Fg(X)22890 47591 y Fo(k)24 b Fw(=1)24826 46107 y Ft(\242)25836 45607 y Fw(2)p Fo(k)25836 46407 y(\275)26318 46530 y Fh(0)27212 46107 y Fr(\267)336 b Ft(2)p Fq(R)9 b(n)31311 44592 y Fw(2)p Fo(n)30956 44956 y Fg(X)30974 47591 y Fo(k)24 b Fw(=1)32707 46107 y Ft([\242)34054 45607 y Fo(r)34479 45730 y Fh(0)34940 45607 y Fw(+1)36198 46107 y Ft(])36535 45607 y Fw(2)p Fo(k)37910 46107 y Fr(\267)337 b Fq(C)40143 45607 y Fo(n)1575 49745 y Ft(for)404 b(some)g(constan)-34 b(t)406 b Fq(C)491 b Ft(dep)34 b(ending)405 b(only)f(on)g(\242)h(and)f Fq(r)27199 49927 y Fw(0)27725 49745 y Ft(.)539 b Fj(\244)1575 52511 y Fl(Theorem)464 b(2)p Ft(.)538 b Fs(L)-62 b(et)434 b Fp(G)336 b Ft(=)g(\()p Fp(V)p Fq(;)202 b Fp(E)p Ft(\))435 b Fs(b)-62 b(e)433 b(b)-62 b(ounde)g(d)431 b(de)-62 b(gr)g(e)g(e,)432 b(with)h(maximum)g(de)-62 b(gr)g(e)g(e)432 b Ft(\242)p Fs(,)h(such)f(that:)1575 54016 y(i\))h Fp(G)g Fs(has)g(the)f(minimal)h (cut)g(set)f(pr)-62 b(op)g(erty)1575 55522 y(ii\))568 b(It)i(is)f(p)-62 b(ossible)566 b(to)j(\257nd)h(a)f(uniform)g(c)-62 b(onstant)568 b Fq(C)656 b Fs(and)569 b(a)g(vertex)g Fq(x)35475 55704 y Fw(0)36583 55522 y Fr(2)583 b Fp(V)570 b Fs(such)e(that,)601 b(for)569 b(al)62 b(l)569 b(\257nite)1575 57027 y(c)-62 b(onne)g(cte)g(d)432 b Fq(W)504 b Fr(\275)337 b Fp(V)434 b Fs(with)f Fq(x)14622 57209 y Fw(0)15484 57027 y Fr(2)337 b Fq(W)168 b Fs(,)433 b(the)g(ine)-62 b(quality)20211 60192 y Fr(j)p Fq(@)21192 60374 y Fo(e)21684 60192 y Fq(W)168 b Fr(j)336 b(\270)25256 59372 y Ft(1)p 25082 59913 954 49 v 25082 61024 a Fq(C)26804 60192 y Ft(ln)202 b(diam)434 b(\()p Fq(W)168 b Ft(\))16146 b(\(4)p Fq(:)p Ft(2\))1575 63194 y Fs(is)433 b(veri\257e)-62 b(d.)1575 64699 y(Then)433 b(the)g(critic)-62 b(al)431 b(pr)-62 b(ob)g(ability)430 b Fq(p)17270 64881 y Fo(c)18167 64699 y Fs(on)j Fp(G)p Fs(,)g(de\257ne)-62 b(d)433 b(in)h(\(4.2\),)e (is)h(strictly)e(less)h(than)h(1.)1575 66835 y Fl(Pro)39 b(of)p Ft(.)575 b(T)-101 b(o)417 b(b)34 b(ound)417 b(the)g(factor)g (\(4.1\))g(w)-34 b(e)417 b(no)-34 b(w)418 b(pro)34 b(ceed)415 b(as)i(follo)-34 b(ws.)576 b(Since)417 b Fp(G)f Ft(is)g(connected)h (and)g(lo)34 b(cally)1575 68340 y(\257nite,)541 b(for)513 b(an)-34 b(y)514 b Fq(x)519 b Fr(2)g Fp(V)513 b Ft(hence)h(in)f (particular)g(for)h Fq(x)26799 68522 y Fw(0)27324 68340 y Ft(,)541 b(there)513 b(exists)g(a)h(geo)34 b(desic)512 b(ra)-34 b(y)513 b Fq(\275)g Ft(starting)h(at)g Fq(x)51106 68522 y Fw(0)51632 68340 y Ft(.)1575 69845 y(Then,)378 b(since)370 b Fq(\260)403 b Fr(2)337 b Ft(\241)10920 69305 y Fo(x)11450 69428 y Fh(0)10920 70232 y Fi(G)11966 69845 y Ft(,)377 b(w)-34 b(e)371 b(ha)-34 b(v)g(e,)377 b(b)-34 b(y)371 b(prop)34 b(osition)372 b(2,)377 b(that)371 b Fq(E)30333 70027 y Fo(\275)31072 69845 y Fr(\\)203 b Fq(\260)403 b Fr(6)p Ft(=)337 b Fr(;)p Ft(.)527 b(Let)370 b Fq(e)38566 70027 y Fo(x)39096 70150 y Fh(0)39613 69845 y Ft(\()p Fq(\260)67 b Ft(\))371 b(b)34 b(e)370 b(the)h(\257rst)h(edge) e(\(in)1575 71351 y(the)404 b(natural)h(order)f(of)g(the)h(ra)-34 b(y\))404 b(in)g Fq(E)19419 71533 y Fo(\275)20360 71351 y Ft(whic)-34 b(h)405 b(b)34 b(elongs)405 b(to)f Fq(\260)471 b Ft(and)405 b(de\257ne)10084 74362 y Fq(r)10631 74544 y Fo(n)11258 74362 y Ft(\()p Fq(x)12422 74544 y Fw(0)12948 74362 y Ft(\))337 b(=)f Fr(f)p Fq(e)h Fr(2)g Fq(E)18582 74544 y Fo(\275)19456 74362 y Ft(:)f Fr(9)p Fq(\260)404 b Fr(2)337 b Ft(\241)23737 73821 y Fo(x)24267 73944 y Fh(0)23737 74748 y Fi(G)25187 74362 y Ft(suc)-34 b(h)405 b(that)809 b Fr(j)p Fq(\260)67 b Fr(j)336 b Ft(=)g Fq(n)404 b Ft(and)h Fq(e)337 b Ft(=)f Fq(e)40168 74544 y Fo(x)40698 74667 y Fh(0)41215 74362 y Ft(\()p Fq(\260)67 b Ft(\))p Fr(g)6019 b Ft(\(4)p Fq(:)p Ft(3\))26469 78283 y(6)p eop %%Page: 7 7 7 6 bop 1575 5313 a Ft(Hence)12684 5667 y Fg(X)12305 8402 y Fk(\260)43 b Fm(2)p Fh(\241)13835 7974 y Fk(x)14299 8173 y Fh(0)13835 8759 y Ff(G)12504 9377 y Fm(j)p Fk(\260)g Fm(j)p Fh(=)p Fk(n)15150 6819 y Ft(1)337 b(=)18385 5667 y Fg(X)17372 8352 y Fo(e)p Fn(2)p Fo(r)18860 8463 y Fk(n)19427 8352 y Fw(\()p Fo(x)20323 8475 y Fh(0)20784 8352 y Fw(\))23387 5667 y Fg(X)21485 8402 y Fk(\260)43 b Fm(2)p Fh(\241)23015 7974 y Fk(x)23479 8173 y Fh(0)23015 8759 y Ff(G)23995 8402 y Fk(x)p Fh(:)235 b Fm(j)p Fk(\260)43 b Fm(j)p Fh(=)p Fk(n)22743 9377 y(e)23134 9488 y(x)23654 9377 y Fh(\()p Fk(\260)g Fh(\)=)p Fk(e)27375 6819 y Ft(1)337 b Fr(\267)f(j)p Fq(r)30481 7001 y Fo(n)31107 6819 y Ft(\()p Fq(x)p Ft(\))p Fr(j)539 b Ft(sup)33704 7867 y Fo(e)p Fn(2)p Fi(E)37330 5667 y Fg(X)35980 8186 y Fk(\260)43 b Fm(2)p Fh(\241)37510 8414 y Ff(G)38318 8186 y Fm(j)p Fk(\260)g Fm(j)p Fh(=)p Fk(n)37495 8938 y(e)p Fm(2)p Fk(\260)40765 6819 y Ft(1)1575 11208 y(No)-34 b(w)405 b(observ)-34 b(e)404 b(that)17851 12713 y Fr(j)p Fq(r)18735 12895 y Fo(n)19361 12713 y Ft(\()p Fq(x)p Ft(\))p Fr(j)337 b(\267)2101 b Ft(sup)22949 13941 y Fo(\260)52 b Fn(2)p Fw(\241)24701 13437 y Fk(x)25165 13636 y Fh(0)24701 14253 y Ff(G)25681 13941 y Fw(:)p Fn(j)p Fo(\260)g Fn(j)p Fw(=)p Fo(n)28507 12713 y Ft(diam)404 b(\()p Fq(I)32541 12895 y Fo(\260)33134 12713 y Ft(\))269 b(+)g(1)1575 15985 y(and)405 b(b)-34 b(y)404 b(h)-34 b(yp)34 b(othesis)405 b(ii\))f(of)h(the)f(theorem,)g(i.e.)538 b(inequalit)-34 b(y)403 b(\(4.2\))i(w)-34 b(e)405 b(get)22349 18451 y Fr(j)p Fq(\275)23313 18633 y Fo(n)23939 18451 y Ft(\()p Fq(x)p Ft(\))p Fr(j)337 b(\267)f Fq(C)28480 17950 y Fo(n)29376 18451 y Ft(+)269 b(1)1575 20916 y(The)404 b(factor)h(sup)9363 21207 y Fo(e)p Fn(2)p Fi(E)11274 20007 y Fg(P)12687 20997 y Fk(\260)43 b Fm(2)p Fh(\241)14217 21226 y Ff(G)15024 20997 y Fm(j)p Fk(\260)g Fm(j)p Fh(=)p Fk(n)14201 21749 y(e)p Fm(2)p Fk(\260)17527 20916 y Ft(1)404 b(can)h(no)-34 b(w)405 b(b)34 b(e)404 b(b)34 b(ounded)405 b(exactly)e(as)h(in)h(theorem)f(1.)1346 b Fj(\244)1575 24158 y Fl(Theorem)586 b(3)p Ft(.)858 b Fs(L)-62 b(et)531 b Fp(G)514 b Ft(=)g(\()p Fp(V)p Fq(;)202 b Fp(E)p Ft(\))533 b Fs(b)-62 b(e)531 b(a)g(planar)f(and)i(lo)-62 b(c)g(al)62 b(ly)529 b(\257nite)i(c)-62 b(onne)g(cte)g(d)530 b(in\257nite)i(gr)-62 b(aph)530 b(with)h(the)1575 25664 y(fol)62 b(lowing)432 b(pr)-62 b(op)g(erties:)1575 27169 y(i\))433 b(Its)g(dual)f(gr)-62 b(aph)432 b Fp(G)11550 26729 y Fn(\244)12509 27169 y Fs(is)h(b)-62 b(ounde)g(d)432 b(de)-62 b(gr)g(e)g(e)431 b(with)i(maximum)h(de)-62 b(gr)g(e)g(e)432 b Ft(\242)34898 26729 y Fn(\244)35424 27169 y Fs(.)1575 28675 y(ii\))568 b(It)i(is)f(p)-62 b(ossible)566 b(to)j(\257nd)h(a)f(uniform)g(c)-62 b(onstant)568 b Fq(C)656 b Fs(and)569 b(a)g(vertex)g Fq(x)35475 28857 y Fw(0)36583 28675 y Fr(2)583 b Fp(V)570 b Fs(such)e(that,)601 b(for)569 b(al)62 b(l)569 b(\257nite)1575 30180 y(c)-62 b(onne)g(cte)g(d)432 b Fq(W)504 b Fr(\275)337 b Fp(V)434 b Fs(with)f Fq(x)14622 30362 y Fw(0)15484 30180 y Fr(2)337 b Fq(W)168 b Fs(,)433 b(the)g(ine)-62 b(quality)20428 33088 y Fr(j)p Fq(@)21409 33270 y Fo(e)21900 33088 y Fq(W)168 b Fr(j)337 b(\270)25473 32268 y Ft(1)p 25299 32809 954 49 v 25299 33919 a Fq(C)26587 33088 y Ft(ln)202 b(diam)434 b(\()p Fq(W)168 b Ft(\))16363 b(\(4)p Fq(:)p Ft(4\))1575 35832 y Fs(is)433 b(veri\257e)-62 b(d.)1575 37338 y(Then)433 b(the)g(critic)-62 b(al)431 b(pr)-62 b(ob)g(ability)430 b Fq(p)17270 37520 y Fo(c)18167 37338 y Fs(on)j Fp(G)p Fs(,)g(de\257ne)-62 b(d)433 b(in)h(\(4.2\),)e (is)h(strictly)e(less)h(than)h(1.)1575 39473 y Fl(Pro)39 b(of)p Ft(.)601 b(Pro)34 b(ceeding)424 b(as)i(in)f(theorem)g(2,)431 b(consider)425 b(the)g(geo)34 b(desic)425 b(ra)-34 b(y)425 b Fq(\275)f Ft(from)i Fq(x)39598 39655 y Fw(0)40123 39473 y Ft(.)602 b(Then,)431 b(since)425 b Fq(\260)439 b Fr(2)372 b Ft(\241)50586 38933 y Fo(x)51116 39056 y Fh(0)50586 39860 y Fi(G)51632 39473 y Ft(,)1575 40979 y(w)-34 b(e)466 b(ha)-34 b(v)g(e)467 b(that)g Fq(E)9861 41161 y Fo(\275)10708 40979 y Fr(\\)311 b Fq(\260)507 b Fr(6)p Ft(=)439 b Fr(;)p Ft(.)724 b(Let)466 b Fq(e)18809 41161 y Fo(x)19339 41284 y Fh(0)19856 40979 y Ft(\()p Fq(\260)67 b Ft(\))466 b(b)34 b(e)466 b(the)g(\257rst)h(edge)f(\(in)g(the)g(natural)h(order)f(of)g (the)g(ra)-34 b(y\))467 b(in)f Fq(E)51432 41161 y Fo(\275)1575 42484 y Ft(whic)-34 b(h)405 b(b)34 b(elongs)404 b(to)g Fq(\260)471 b Ft(and)405 b(de\257ne)g(the)g(edge)e(set)i Fq(r)25120 42666 y Fo(n)25746 42484 y Ft(\()p Fq(x)26910 42666 y Fw(0)27436 42484 y Ft(\))f(as)g(in)h(\(4.3\).)1575 43989 y(Then)12653 44343 y Fg(X)11759 47078 y Fk(\260)43 b Fm(2)p Fk(\260)g Fm(2)p Fh(\241)14318 46650 y Fk(x)14782 46849 y Fh(0)14318 47435 y Ff(G)12472 48053 y Fm(j)p Fk(\260)g Fm(j)p Fh(=)p Fk(n)15633 45495 y Ft(1)337 b(=)18869 44343 y Fg(X)17856 47028 y Fo(e)p Fn(2)p Fo(r)19344 47139 y Fk(n)19911 47028 y Fw(\()p Fo(x)20807 47151 y Fh(0)21267 47028 y Fw(\))23639 44343 y Fg(X)21968 47078 y Fk(\260)43 b Fm(2)p Fh(\241)23498 46650 y Fk(x)23962 46849 y Fh(0)23498 47435 y Ff(G)24478 47078 y Fh(:)236 b Fm(j)p Fk(\260)43 b Fm(j)p Fh(=)p Fk(n)22765 48053 y(e)23156 48164 y(x)23620 48363 y Fh(0)24136 48053 y(\()p Fk(\260)g Fh(\)=)p Fk(e)27395 45495 y Ft(1)337 b Fr(\267)g(j)p Fq(r)30502 45677 y Fo(n)31127 45495 y Ft(\()p Fq(x)32291 45677 y Fw(0)32817 45495 y Ft(\))p Fr(j)539 b Ft(sup)34250 46543 y Fo(e)p Fn(2)p Fi(E)37876 44343 y Fg(X)36526 46862 y Fk(\260)43 b Fm(2)p Fh(\241)38056 47091 y Ff(G)38864 46862 y Fm(j)p Fk(\260)g Fm(j)p Fh(=)p Fk(n)38041 47614 y(e)p Fm(2)p Fk(\260)41311 45495 y Ft(1)1575 50028 y(where,)404 b(as)g(b)34 b(efore,)22086 51533 y Fr(j)p Fq(\275)23050 51715 y Fo(n)23676 51533 y Ft(\()p Fq(x)24840 51715 y Fw(0)25366 51533 y Ft(\))p Fr(j)337 b(\267)f Fq(C)28743 51033 y Fo(n)29639 51533 y Ft(+)269 b(1)1575 53629 y(W)-101 b(e)435 b(no)-34 b(w)437 b(use)g(the)f(one-to-one)h(corresp)34 b(ondence)436 b(b)34 b(et)-34 b(w)g(een)437 b(P)-34 b(eierls)435 b(con)-34 b(tours)437 b(of)f(cardinalit)-34 b(y)436 b Fq(n)g Ft(in)g Fp(G)g Ft(and)1575 55135 y(simple)404 b(cycles)e(of)j(length)g Fq(n)e Ft(in)h Fp(G)17400 54695 y Fn(\244)17926 55135 y Ft(,)g(and)g(w)-34 b(e)405 b(write)17360 57714 y(sup)17445 58762 y Fo(e)p Fn(2)p Fi(E)21072 56563 y Fg(X)19722 59081 y Fk(\260)43 b Fm(2)p Fh(\241)21252 59310 y Ff(G)22060 59081 y Fm(j)p Fk(\260)g Fm(j)p Fh(=)p Fk(n)21237 59833 y(e)p Fm(2)p Fk(\260)24507 57714 y Ft(1)337 b(=)731 b(sup)26729 58762 y Fo(e)27165 58510 y Fm(\244)27645 58762 y Fn(2)p Fi(E)28862 58510 y Fm(\244)31686 56563 y Fg(X)29880 59121 y Fk(\260)30356 58930 y Fm(\244)30835 59121 y(2)p Fh(\241)31889 58892 y Fm(\244)31889 59479 y Ff(G)32650 59121 y Fm(j)p Fk(\260)33378 58930 y Fm(\244)33858 59121 y(j)p Fh(=)p Fk(n)31371 60098 y(e)31762 59907 y Fm(\244)32242 60098 y(2)p Fk(\260)33271 59907 y Fm(\244)35577 57714 y Ft(1)13294 b(\(4)p Fq(:)p Ft(5\))1575 62328 y(where)377 b(\241)5811 61888 y Fn(\244)5811 62692 y Fi(G)6932 62328 y Ft(is)f(the)i(set)f(of)g (all)g(simple)g(cycles)e(in)j Fp(G)24588 61888 y Fn(\244)25113 62328 y Ft(.)529 b(Using)378 b(no)-34 b(w)378 b(h)-34 b(yp)34 b(othesis)378 b Fs(i)p Ft(\))f(of)h(the)f(theorem,)382 b(it)377 b(is)g(no)-34 b(w)1575 63834 y(trivial)403 b(to)h(b)34 b(ound)406 b(the)e(l.)538 b(h.)h(s.)g(of)404 b(\(4.5\).)539 b(W)-101 b(e)404 b(get)20453 66481 y(sup)20059 67528 y Fo(e)20495 67276 y Fm(\244)20975 67528 y Fn(2)p Fi(E)22192 67276 y Fm(\244)24775 65329 y Fg(X)23209 67887 y Fk(\260)23685 67696 y Fm(\244)24165 67887 y(2)p Fh(\241)25219 67658 y Fm(\244)25219 68245 y Ff(G)25979 67887 y Fm(j)p Fk(\260)43 b Fm(j)p Fh(=)p Fk(n)24460 68864 y(e)24851 68673 y Fm(\244)25331 68864 y(2)p Fk(\260)26360 68673 y Fm(\244)28427 66481 y Ft(1)337 b Fr(\267)f Ft([\242)31996 65980 y Fn(\244)32522 66481 y Ft(])32859 65980 y Fo(n)49477 66481 y Ft(\(4)p Fq(:)p Ft(6\))1575 71027 y(Hence)403 b(w)-34 b(e)405 b(ha)-34 b(v)g(e)405 b(again)f(pro)-34 b(v)g(ed)405 b(that)19282 72455 y Fg(X)17349 75140 y Fo(\260)52 b Fn(2)p Fw(\241)19101 75251 y Fk(x)19565 75450 y Fh(0)20081 75140 y Fw(:)262 b Fn(j)p Fo(\260)52 b Fn(j)p Fw(=)p Fo(n)23168 73607 y Ft(1)337 b Fr(\267)f Ft(\()p Fq(C)26814 73106 y Fo(n)27710 73607 y Ft(+)270 b(1\)[\242)31347 73106 y Fn(\244)31873 73607 y Ft(])32210 73106 y Fo(n)33172 73607 y Fr(\267)337 b Fq(K)35568 73106 y Fo(n)26469 78283 y Ft(7)p eop %%Page: 8 8 8 7 bop 11023 19844 a @beginspecial 33 @llx 554 @lly 564 @urx 780 @ury 2834 @rwi 1417 @rhi @setspecial %%BeginDocument: fig1.eps %!PS-Adobe-3.0 EPSF-3.0 %%BoundingBox: 33 554 564 780 %%Creator: CorelDRAW %%Title: %%CreationDate: Mon Nov 03 17:06:13 2003 %%DocumentProcessColors: Cyan Yellow Black %%DocumentSuppliedResources: (atend) %%EndComments %%BeginProlog /AutoFlatness true def /AutoSteps 0 def /CMYKMarks false def /UseLevel2 false def %Color profile: _DEFAULT.CCM - Genérico CMYK %%BeginResource: procset wCorel6Dict 6.0 0 % Copyright (c)1992-96 Corel Corporation % All rights reserved. v6.0 r1.1 /wCorel6Dict 300 dict def wCorel6Dict begin/bd{bind def}bind def/ld{load def} bd/xd{exch def}bd/_ null def/rp{{pop}repeat}bd/@cp/closepath ld/@gs/gsave ld /@gr/grestore ld/@np/newpath ld/Tl/translate ld/$sv 0 def/@sv{/$sv save def}bd /@rs{$sv restore}bd/spg/showpage ld/showpage{}bd currentscreen/@dsp xd/$dsp /@dsp def/$dsa xd/$dsf xd/$sdf false def/$SDF false def/$Scra 0 def/SetScr /setscreen ld/setscreen{pop pop pop}bd/@ss{2 index 0 eq{$dsf 3 1 roll 4 -1 roll pop}if exch $Scra add exch load SetScr}bd/SepMode_5 where{pop}{/SepMode_5 0 def}ifelse/CurrentInkName_5 where{pop}{/CurrentInkName_5(Composite)def}ifelse /$ink_5 where{pop}{/$ink_5 -1 def}ifelse/$c 0 def/$m 0 def/$y 0 def/$k 0 def /$t 1 def/$n _ def/$o 0 def/$fil 0 def/$C 0 def/$M 0 def/$Y 0 def/$K 0 def/$T 1 def/$N _ def/$O 0 def/$PF false def/s1c 0 def/s1m 0 def/s1y 0 def/s1k 0 def /s1t 0 def/s1n _ def/$bkg false def/SK 0 def/SM 0 def/SY 0 def/SC 0 def/$op false def matrix currentmatrix/$ctm xd/$ptm matrix def/$ttm matrix def/$stm matrix def/$fst 128 def/$pad 0 def/$rox 0 def/$roy 0 def/$fmp 50 def/$ffpnt true def/CorelDrawReencodeVect[16#0/grave 16#5/breve 16#6/dotaccent 16#8/ring 16#A/hungarumlaut 16#B/ogonek 16#C/caron 16#D/dotlessi 16#27/quotesingle 16#60/grave 16#7C/bar 16#82/quotesinglbase/florin/quotedblbase/ellipsis/dagger/daggerdbl 16#88/circumflex/perthousand/Scaron/guilsinglleft/OE 16#91/quoteleft/quoteright/quotedblleft/quotedblright/bullet/endash/emdash 16#98/tilde/trademark/scaron/guilsinglright/oe 16#9F/Ydieresis 16#A1/exclamdown/cent/sterling/currency/yen/brokenbar/section 16#a8/dieresis/copyright/ordfeminine/guillemotleft/logicalnot/minus/registered/macron 16#b0/degree/plusminus/twosuperior/threesuperior/acute/mu/paragraph/periodcentered 16#b8/cedilla/onesuperior/ordmasculine/guillemotright/onequarter/onehalf/threequarters/questiondown 16#c0/Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/AE/Ccedilla 16#c8/Egrave/Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex/Idieresis 16#d0/Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis/multiply 16#d8/Oslash/Ugrave/Uacute/Ucircumflex/Udieresis/Yacute/Thorn/germandbls 16#e0/agrave/aacute/acircumflex/atilde/adieresis/aring/ae/ccedilla 16#e8/egrave/eacute/ecircumflex/edieresis/igrave/iacute/icircumflex/idieresis 16#f0/eth/ntilde/ograve/oacute/ocircumflex/otilde/odieresis/divide 16#f8/oslash/ugrave/uacute/ucircumflex/udieresis/yacute/thorn/ydieresis]def /L2?/languagelevel where{pop languagelevel 2 ge}{false}ifelse def /@BeginSysCorelDict{systemdict/Corel30Dict known{systemdict/Corel30Dict get exec}if}bd/@EndSysCorelDict{systemdict/Corel30Dict known{end}if}bd AutoFlatness {/@ifl{dup currentflat exch sub 10 gt{ ([Error: PathTooComplex; OffendingCommand: AnyPaintingOperator]\n)print flush @np exit}{currentflat 2 add setflat}ifelse}bd/@fill/fill ld/fill{currentflat{ {@fill}stopped{@ifl}{exit}ifelse}bind loop setflat}bd/@eofill/eofill ld/eofill {currentflat{{@eofill}stopped{@ifl}{exit}ifelse}bind loop setflat}bd/@clip /clip ld/clip{currentflat{{@clip}stopped{@ifl}{exit}ifelse}bind loop setflat} bd/@eoclip/eoclip ld/eoclip{currentflat{{@eoclip}stopped{@ifl}{exit}ifelse} bind loop setflat}bd/@stroke/stroke ld/stroke{currentflat{{@stroke}stopped {@ifl}{exit}ifelse}bind loop setflat}bd}if L2?{/@ssa{true setstrokeadjust}bd}{ /@ssa{}bd}ifelse/d/setdash ld/j/setlinejoin ld/J/setlinecap ld/M/setmiterlimit ld/w/setlinewidth ld/O{/$o xd}bd/R{/$O xd}bd/W/eoclip ld/c/curveto ld/C/c ld/l /lineto ld/L/l ld/rl/rlineto ld/m/moveto ld/n/newpath ld/N/newpath ld/P{11 rp} bd/u{}bd/U{}bd/A{pop}bd/q/@gs ld/Q/@gr ld/&{}bd/@j{@sv @np}bd/@J{@rs}bd/g{1 exch sub/$k xd/$c 0 def/$m 0 def/$y 0 def/$t 1 def/$n _ def/$fil 0 def}bd/G{1 sub neg/$K xd _ 1 0 0 0/$C xd/$M xd/$Y xd/$T xd/$N xd}bd/k{1 index type /stringtype eq{/$t xd/$n xd}{/$t 0 def/$n _ def}ifelse/$k xd/$y xd/$m xd/$c xd /$fil 0 def}bd/K{1 index type/stringtype eq{/$T xd/$N xd}{/$T 0 def/$N _ def} ifelse/$K xd/$Y xd/$M xd/$C xd}bd/x/k ld/X/K ld/sf{1 index type/stringtype eq{ /s1t xd/s1n xd}{/s1t 0 def/s1n _ def}ifelse/s1k xd/s1y xd/s1m xd/s1c xd}bd/i{ dup 0 ne{setflat}{pop}ifelse}bd/v{4 -2 roll 2 copy 6 -2 roll c}bd/V/v ld/y{2 copy c}bd/Y/y ld/@w{matrix rotate/$ptm xd matrix scale $ptm dup concatmatrix /$ptm xd 1 eq{$ptm exch dup concatmatrix/$ptm xd}if 1 w}bd/@g{1 eq dup/$sdf xd {/$scp xd/$sca xd/$scf xd}if}bd/@G{1 eq dup/$SDF xd{/$SCP xd/$SCA xd/$SCF xd} if}bd/@D{2 index 0 eq{$dsf 3 1 roll 4 -1 roll pop}if 3 copy exch $Scra add exch load SetScr/$dsp xd/$dsa xd/$dsf xd}bd/$ngx{$SDF{$SCF SepMode_5 0 eq{$SCA} {$dsa}ifelse $SCP @ss}if}bd/p{/$pm xd 7 rp/$pyf xd/$pxf xd/$pn xd/$fil 1 def} bd/@MN{2 copy le{pop}{exch pop}ifelse}bd/@MX{2 copy ge{pop}{exch pop}ifelse}bd /InRange{3 -1 roll @MN @MX}bd/wDstChck{2 1 roll dup 3 -1 roll eq{1 add}if}bd /@dot{dup mul exch dup mul add 1 exch sub}bd/@lin{exch pop abs 1 exch sub}bd /cmyk2rgb{3{dup 5 -1 roll add 1 exch sub dup 0 lt{pop 0}if exch}repeat pop}bd /rgb2cmyk{3{1 exch sub 3 1 roll}repeat 3 copy @MN @MN 3{dup 5 -1 roll sub neg exch}repeat}bd/rgb2g{2 index .299 mul 2 index .587 mul add 1 index .114 mul add 4 1 roll pop pop pop}bd/WaldoColor_5 where{pop}{/SetRgb/setrgbcolor ld/GetRgb /currentrgbcolor ld/SetGry/setgray ld/GetGry/currentgray ld/SetRgb2 systemdict /setrgbcolor get def/GetRgb2 systemdict/currentrgbcolor get def/SetHsb systemdict/sethsbcolor get def/GetHsb systemdict/currenthsbcolor get def /rgb2hsb{SetRgb2 GetHsb}bd/hsb2rgb{3 -1 roll dup floor sub 3 1 roll SetHsb GetRgb2}bd/setcmykcolor where{pop/SetCmyk_5/setcmykcolor ld}{/SetCmyk_5{ cmyk2rgb SetRgb}bd}ifelse/currentcmykcolor where{pop/GetCmyk/currentcmykcolor ld}{/GetCmyk{GetRgb rgb2cmyk}bd}ifelse/setoverprint where{pop}{/setoverprint{ /$op xd}bd}ifelse/currentoverprint where{pop}{/currentoverprint{$op}bd}ifelse /@tc_5{5 -1 roll dup 1 ge{pop}{4{dup 6 -1 roll mul exch}repeat pop}ifelse}bd /@trp{exch pop 5 1 roll @tc_5}bd/setprocesscolor_5{SepMode_5 0 eq{SetCmyk_5}{0 4 $ink_5 sub index exch pop 5 1 roll pop pop pop pop SepsColor true eq{$ink_5 3 gt{1 sub neg SetGry}{0 0 0 4 $ink_5 roll SetCmyk_5}ifelse}{1 sub neg SetGry} ifelse}ifelse}bd/findcmykcustomcolor where{pop}{/findcmykcustomcolor{5 array astore}bd}ifelse/setcustomcolor where{pop}{/setcustomcolor{exch aload pop SepMode_5 0 eq{pop @tc_5 setprocesscolor_5}{CurrentInkName_5 eq{4 index}{0} ifelse 6 1 roll 5 rp 1 sub neg SetGry}ifelse}bd}ifelse/@scc_5{dup type /booleantype eq{setoverprint}{1 eq setoverprint}ifelse dup _ eq{pop setprocesscolor_5 pop}{findcmykcustomcolor exch setcustomcolor}ifelse SepMode_5 0 eq{true}{GetGry 1 eq currentoverprint and not}ifelse}bd/colorimage where{pop /ColorImage{colorimage}def}{/ColorImage{/ncolors xd pop/dataaq xd{dataaq ncolors dup 3 eq{/$dat xd 0 1 $dat length 3 div 1 sub{dup 3 mul $dat 1 index get 255 div $dat 2 index 1 add get 255 div $dat 3 index 2 add get 255 div rgb2g 255 mul cvi exch pop $dat 3 1 roll put}for $dat 0 $dat length 3 idiv getinterval pop}{4 eq{/$dat xd 0 1 $dat length 4 div 1 sub{dup 4 mul $dat 1 index get 255 div $dat 2 index 1 add get 255 div $dat 3 index 2 add get 255 div $dat 4 index 3 add get 255 div cmyk2rgb rgb2g 255 mul cvi exch pop $dat 3 1 roll put}for $dat 0 $dat length ncolors idiv getinterval}if}ifelse}image}bd }ifelse/setcmykcolor{1 5 1 roll _ currentoverprint @scc_5/$ffpnt xd}bd /currentcmykcolor{0 0 0 0}bd/setrgbcolor{rgb2cmyk setcmykcolor}bd /currentrgbcolor{currentcmykcolor cmyk2rgb}bd/sethsbcolor{hsb2rgb setrgbcolor} bd/currenthsbcolor{currentrgbcolor rgb2hsb}bd/setgray{dup dup setrgbcolor}bd /currentgray{currentrgbcolor rgb2g}bd}ifelse/WaldoColor_5 true def/@sft{$tllx $pxf add dup $tllx gt{$pwid sub}if/$tx xd $tury $pyf sub dup $tury lt{$phei add}if/$ty xd}bd/@stb{pathbbox/$ury xd/$urx xd/$lly xd/$llx xd}bd/@ep{{cvx exec }forall}bd/@tp{@sv/$in true def 2 copy dup $lly le{/$in false def}if $phei sub $ury ge{/$in false def}if dup $urx ge{/$in false def}if $pwid add $llx le{/$in false def}if $in{@np 2 copy m $pwid 0 rl 0 $phei neg rl $pwid neg 0 rl 0 $phei rl clip @np $pn cvlit load aload pop 7 -1 roll 5 index sub 7 -1 roll 3 index sub Tl matrix currentmatrix/$ctm xd @ep pop pop pop pop}{pop pop}ifelse @rs}bd /@th{@sft 0 1 $tly 1 sub{dup $psx mul $tx add{dup $llx gt{$pwid sub}{exit} ifelse}loop exch $phei mul $ty exch sub 0 1 $tlx 1 sub{$pwid mul 3 copy 3 -1 roll add exch @tp pop}for pop pop}for}bd/@tv{@sft 0 1 $tlx 1 sub{dup $pwid mul $tx add exch $psy mul $ty exch sub{dup $ury lt{$phei add}{exit}ifelse}loop 0 1 $tly 1 sub{$phei mul 3 copy sub @tp pop}for pop pop}for}bd/@pf{@gs $ctm setmatrix $pm concat @stb eoclip Bburx Bbury $pm itransform/$tury xd/$turx xd Bbllx Bblly $pm itransform/$tlly xd/$tllx xd/$wid $turx $tllx sub def/$hei $tury $tlly sub def @gs $vectpat{1 0 0 0 0 _ $o @scc_5{eofill}if}{$t $c $m $y $k $n $o @scc_5{SepMode_5 0 eq $pfrg or{$tllx $tlly Tl $wid $hei scale <00> 8 1 false[8 0 0 1 0 0]{}imagemask}{/$bkg true def}ifelse}if}ifelse @gr $wid 0 gt $hei 0 gt and{$pn cvlit load aload pop/$pd xd 3 -1 roll sub/$phei xd exch sub /$pwid xd $wid $pwid div ceiling 1 add/$tlx xd $hei $phei div ceiling 1 add /$tly xd $psx 0 eq{@tv}{@th}ifelse}if @gr @np/$bkg false def}bd/@dlt_hsb{$fco nff ge{/$fco nff 1 sub def}if sub neg 2 div dup $fco div/kdb xd nff $fco sub div/kdb2 xd sub neg 2 div dup $fco div/kds xd nff $fco sub div/kds2 xd sub neg dup 0 eq{pop $frb 2 eq{.99}{-.99}ifelse}if dup $frb 2 eq exch 0 lt and{1 add} if dup $frb 1 eq exch 0 gt and{1 sub}if 2 div dup $fco div/kdh xd nff $fco sub div/kdh2 xd}bd/@dlt_cmyk{$fmp 50 eq{sub neg nff dup 1 gt{1 sub}if div/$dk xd sub neg nff dup 1 gt{1 sub}if div/$dy xd sub neg nff dup 1 gt{1 sub}if div/$dm xd sub neg nff dup 1 gt{1 sub}if div/$dc xd sub neg nff dup 1 gt{1 sub}if div /$dt xd}{$fco nff ge{/$fco nff 1 sub def}if sub neg 2 div dup $fco div/$dk xd nff $fco sub div/$dk2 xd sub neg 2 div dup $fco div/$dy xd nff $fco sub div /$dy2 xd sub neg 2 div dup $fco div/$dm xd nff $fco sub div/$dm2 xd sub neg 2 div dup $fco div/$dc xd nff $fco sub div/$dc2 xd sub neg 2 div dup $fco div /$dt xd nff $fco sub div/$dt2 xd}ifelse}bd/@dlt{$fse $fss sub dup/nff xd $fmp mul $fmp add 100 div round cvi dup 0 le{pop 1}if/$fco xd/$fi 0 def $frb dup 1 eq exch 2 eq or{$frt dup $frc $frm $fry $frk @tc_5 4 copy cmyk2rgb rgb2hsb 3 copy/myb xd/mys xd/myh xd $tot $toc $tom $toy $tok @tc_5 cmyk2rgb rgb2hsb 3 1 roll 4 1 roll 5 1 roll @dlt_hsb}{$frt dup $frc $frm $fry $frk @tc_5 5 copy $tot dup $toc $tom $toy $tok @tc_5 5 1 roll 6 1 roll 7 1 roll 8 1 roll 9 1 roll @dlt_cmyk}ifelse}bd/@ffnxt{/$fi $fi 1 add def $frb dup 1 eq exch 2 eq or{pop pop pop pop myh mys myb $fi $fco lt{kdb add 3 1 roll kds add 3 1 roll kdh add 3 1 roll}{kdb2 add 3 1 roll kds2 add 3 1 roll kdh2 add 3 1 roll}ifelse 3 copy /myb xd/mys xd/myh xd hsb2rgb rgb2cmyk}{$fi $fco lt $fmp 50 eq or{$dk add 5 1 roll $dy add 5 1 roll $dm add 5 1 roll $dc add 5 1 roll $dt add 5 1 roll}{$dk2 add 5 1 roll $dy2 add 5 1 roll $dm2 add 5 1 roll $dc2 add 5 1 roll $dt2 add 5 1 roll}ifelse}ifelse}bd/ffcol{5 copy $fsit 0 eq{setcmykcolor pop}{SepMode_5 0 ne {$frn findcmykcustomcolor exch setcustomcolor}{pop pop pop pop $frc $frm $fry $frk $frn findcmykcustomcolor exch setcustomcolor}ifelse}ifelse}bd/@ftl{1 index 4 index sub dup $pad mul dup/$pdw xd 2 mul sub $fst div/$wid xd 2 index sub /$hei xd pop Tl @dlt $fss 0 eq{ffcol n 0 0 m 0 $hei l $pdw $hei l $pdw 0 l @cp $ffpnt{fill}{@np}ifelse}if $fss $wid mul $pdw add 0 Tl nff{ffcol n 0 0 m 0 $hei l $wid $hei l $wid 0 l @cp $ffpnt{fill}{@np}ifelse $wid 0 Tl @ffnxt}repeat 5 rp $tot dup $toc $tom $toy $tok @tc_5 ffcol n 0 0 m 0 $hei l $pdw $hei l $pdw 0 l @cp $ffpnt{fill}{@np}ifelse 5 rp}bd/@ftrs{1 index 4 index sub dup $rox mul /$row xd 2 div 1 index 4 index sub dup $roy mul/$roh xd 2 div 2 copy dup mul exch dup mul add sqrt $row dup mul $roh dup mul add sqrt add dup/$hei xd $fst div/$wid xd 4 index add $roh add exch 5 index add $row add exch Tl $fan rotate pop pop pop pop @dlt $fss 0 eq{ffcol $fty 3 eq{$hei dup neg dup m 2 mul @sqr}{ 0 0 m 0 0 $hei 0 360 arc}ifelse $ffpnt{fill}{@np}ifelse}if 1.0 $pad 2 mul sub dup scale $hei $fss $wid mul sub/$hei xd nff{ffcol $fty 3 eq{n $hei dup neg dup m 2 mul @sqr}{n 0 0 m 0 0 $hei 0 360 arc}ifelse $ffpnt{fill}{@np}ifelse/$hei $hei $wid sub def @ffnxt}repeat 5 rp}bd/@ftc{1 index 4 index sub dup $rox mul /$row xd 2 div 1 index 4 index sub dup $roy mul/$roh xd 2 div 2 copy dup mul exch dup mul add sqrt $row dup mul $roh dup mul add sqrt add dup/$hei xd $fst div/$wid xd 4 index add $roh add exch 5 index add $row add exch Tl pop pop pop pop @dlt $fss 0 eq{ffcol $ffpnt{fill}{@np}ifelse}{n}ifelse/$dang 180 $fst 1 sub div def/$sang $dang -2 div 180 add def/$eang $dang 2 div 180 add def/$sang $sang $dang $fss mul add def/$eang $eang $dang $fss mul add def/$sang $eang $dang sub def nff{ffcol n 0 0 m 0 0 $hei $sang $fan add $eang $fan add arc $ffpnt{fill}{@np}ifelse 0 0 m 0 0 $hei $eang neg $fan add $sang neg $fan add arc $ffpnt{fill}{@np}ifelse/$sang $eang def/$eang $eang $dang add def @ffnxt} repeat 5 rp}bd/@fstAdjust{72 0 matrix defaultmatrix dtransform exch dup mul exch dup mul add sqrt currentscreen pop pop div dup mul 1 add cvi 2 256 InRange $fsc 1 sub dup 0 le{pop 1}if mul/opt$fst xd AutoSteps dup 1 eq exch 3 eq or{ opt$fst dup dup $fst dup 3 1 roll div exch 3 -1 roll gt{/$adj xd/$fst xd}{pop pop}ifelse}if AutoSteps dup 2 eq exch 3 eq or{opt$fst dup dup $fst dup 3 1 roll div exch 3 -1 roll lt{/$adj xd/$fst xd}{pop pop}ifelse}if}bd/@ff{/old$fst $fst def/$adj 1 def/$fss 0 def $o 1 eq setoverprint AutoSteps 0 gt{@fstAdjust}if 1 1 $fsc 1 sub{dup 1 sub $fsit 0 eq{$fsa exch 5 mul 5 getinterval aload pop pop /$frk xd/$fry xd/$frm xd/$frc xd/$frn _ def/$frt 1 def $fsa exch 5 mul 5 getinterval aload pop $adj mul cvi $fss add/$fse xd/$tok xd/$toy xd/$tom xd /$toc xd/$ton _ def/$tot 1 def}{$fsa exch 7 mul 7 getinterval aload pop pop /$frt xd/$frn xd/$frk xd/$fry xd/$frm xd/$frc xd $fsa exch 7 mul 7 getinterval aload pop $adj mul cvi $fss add/$fse xd/$tot xd/$ton xd/$tok xd/$toy xd/$tom xd /$toc xd}ifelse $fsit 0 eq SepMode_5 0 eq or dup not CurrentInkName_5 $frn eq and or{@sv $ctm setmatrix eoclip Bbllx Bblly Bburx Bbury $fty 2 eq{@ftc}{1 index 3 index m 2 copy l 3 index 1 index l 3 index 3 index l @cp $fty dup 1 eq exch 3 eq or{@ftrs}{pop pop pop pop $fan rotate pathbbox @ftl}ifelse}ifelse @rs /$fss $fse def}{1 0 0 0 0 _ $o @scc_5{fill}if}ifelse}for @np/$fst old$fst def} bd/@Pf{@sv SepMode_5 0 eq $Psc 0 ne or $ink_5 3 eq or{0 J 0 j[]0 d $t $c $m $y $k $n $o @scc_5 pop $ctm setmatrix 72 1000 div dup matrix scale dup concat dup Bburx exch Bbury exch itransform ceiling cvi/Bbury xd ceiling cvi/Bburx xd Bbllx exch Bblly exch itransform floor cvi/Bblly xd floor cvi/Bbllx xd $Prm aload pop $Psn load exec}{1 SetGry eofill}ifelse @rs @np}bd/F{matrix currentmatrix $sdf{$scf $sca $scp @ss}if $fil 1 eq{@pf}{$fil 2 eq{@ff}{$fil 3 eq{@Pf}{$t $c $m $y $k $n $o 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3 index m 3 index 1 index l 2 copy l 1 index 3 index l 3 index 3 index l clip pop pop pop pop}bd/tcc{@cc}def/@i{@sm @gs @ii 6 index 1 ne{/$frg true def pop pop}{1 eq{s1t s1c s1m s1y s1k s1n $O @scc_5 /$frg xd}{/$frg false def}ifelse 1 eq{@gs $ctm setmatrix F @gr}if}ifelse @np /$ury xd/$urx xd/$lly xd/$llx xd/$bts xd/$hei xd/$wid xd/$dat $wid $bts mul 8 div ceiling cvi string def $bkg $frg or{$SDF{$SCF $SCA $SCP @ss}if $llx $lly Tl $urx $llx sub $ury $lly sub scale $bkg{$t $c $m $y $k $n $o @scc_5 pop}if $wid $hei abs $bts 1 eq{$bkg}{$bts}ifelse[$wid 0 0 $hei neg 0 $hei 0 gt{$hei}{0}ifelse]/tcc load $bts 1 eq{imagemask}{image}ifelse $SDF{$dsf $dsa $dsp @ss}if}{$hei abs{tcc pop}repeat}ifelse @gr $ctm setmatrix}bd/@I{@sm @gs @ii @np/$ury xd/$urx xd/$lly xd/$llx xd/$ncl xd/$bts xd/$hei xd/$wid xd/$dat $wid $bts mul $ncl mul 8 div ceiling cvi string def $ngx $llx $lly Tl $urx $llx sub $ury $lly sub scale $wid $hei abs $bts[$wid 0 0 $hei neg 0 $hei 0 gt{$hei}{0}ifelse]/@cc load false $ncl ColorImage $SDF{$dsf $dsa $dsp @ss}if @gr $ctm setmatrix}bd L2?{/@I_2{@sm @gs @ii @np/$ury xd/$urx xd/$lly xd/$llx xd /$ncl xd/$bts xd/$hei xd/$wid xd/$dat $wid $bts mul $ncl mul 8 div ceiling cvi string def $ngx $ncl 1 eq{/DeviceGray}{$ncl 3 eq{/DeviceRGB}{/DeviceCMYK} ifelse}ifelse setcolorspace $llx $lly Tl $urx $llx sub $ury $lly sub scale 8 dict begin/ImageType 1 def/Width $wid def/Height $hei abs def/BitsPerComponent $bts def/Decode $ncl 1 eq{[0 1]}{$ncl 3 eq{[0 1 0 1 0 1]}{[0 1 0 1 0 1 0 1]} ifelse}ifelse def/ImageMatrix[$wid 0 0 $hei neg 0 $hei 0 gt{$hei}{0}ifelse]def /DataSource currentfile/ASCII85Decode filter def currentdict end image $SDF {$dsf $dsa $dsp @ss}if @gr $ctm setmatrix}bd}{/@I_2{}bd}ifelse/z{exch findfont exch scalefont setfont}bd/ZB{9 dict dup begin 4 1 roll/FontType 3 def /FontMatrix xd/FontBBox xd/Encoding 256 array def 0 1 255{Encoding exch /.notdef put}for/CharStrings 256 dict def CharStrings/.notdef{}put/Metrics 256 dict def Metrics/.notdef 3 -1 roll put/BuildChar{exch dup/$char exch/Encoding get 3 index get def dup/Metrics get $char get aload pop setcachedevice begin Encoding exch get CharStrings exch get end exec}def end definefont pop}bd /ZBAddChar{findfont begin dup 4 1 roll dup 6 1 roll Encoding 3 1 roll put CharStrings 3 1 roll put Metrics 3 1 roll put end}bd/Z{findfont dup maxlength 2 add dict exch dup{1 index/FID ne{3 index 3 1 roll put}{pop pop}ifelse}forall pop dup dup/Encoding get 256 array copy dup/$fe xd/Encoding exch put dup /Fontname 3 index put 3 -1 roll dup length 0 ne{0 exch{dup type 0 type eq{exch pop}{$fe exch 2 index exch put 1 add}ifelse}forall pop}if dup 256 dict dup /$met xd/Metrics exch put dup/FontMatrix get 0 get 1000 mul 1 exch div 3 index length 256 eq{0 1 255{dup $fe exch get dup/.notdef eq{pop pop}{5 index 3 -1 roll get 2 index mul $met 3 1 roll put}ifelse}for}if pop definefont pop pop}bd /@ftx{{currentpoint 3 -1 roll(0)dup 3 -1 roll 0 exch put dup @gs true charpath $ctm setmatrix @@txt @gr @np stringwidth pop 3 -1 roll add exch m}forall}bd /@ft{matrix currentmatrix exch $sdf{$scf $sca $scp @ss}if $fil 1 eq{/@@txt/@pf ld @ftx}{$fil 2 eq{/@@txt/@ff ld @ftx}{$fil 3 eq{/@@txt/@Pf ld @ftx}{$t $c $m $y $k $n $o @scc_5{show}{pop}ifelse}ifelse}ifelse}ifelse $sdf{$dsf $dsa $dsp @ss}if setmatrix}bd/@st{matrix currentmatrix exch $SDF{$SCF $SCA $SCP @ss}if $T $C $M $Y $K $N $O @scc_5{{currentpoint 3 -1 roll(0)dup 3 -1 roll 0 exch put dup @gs true charpath $ctm setmatrix $ptm concat stroke @gr @np stringwidth pop 3 -1 roll add exch m}forall}{pop}ifelse $SDF{$dsf $dsa $dsp @ss}if setmatrix}bd /@te{@ft}bd/@tr{@st}bd/@ta{dup @gs @ft @gr @st}bd/@t@a{dup @gs @st @gr @ft}bd /@tm{@sm concat}bd/e{/t{@te}def}bd/r{/t{@tr}def}bd/o{/t{pop}def}bd/a{/t{@ta} def}bd/@a{/t{@t@a}def}bd/t{@te}def/T{@np $ctm setmatrix/$ttm matrix def}bd/ddt {t}def/@t{/$stm $stm currentmatrix def 3 1 roll m $ttm concat ddt $stm setmatrix}bd/@n{/$ttm exch matrix rotate def}bd/@s{}bd/@l{}bd/@B{@gs S @gr F} bd/@b{@cp @B}bd/@sep{CurrentInkName_5(Composite)eq{/$ink_5 -1 def}{ CurrentInkName_5(Cyan)eq{/$ink_5 0 def}{CurrentInkName_5(Magenta)eq{/$ink_5 1 def}{CurrentInkName_5(Yellow)eq{/$ink_5 2 def}{CurrentInkName_5(Black)eq {/$ink_5 3 def}{/$ink_5 4 def}ifelse}ifelse}ifelse}ifelse}ifelse}bd/@whi{@gs -72000 dup m -72000 72000 l 72000 dup l 72000 -72000 l @cp 1 SetGry fill @gr} bd/@neg{[{1 exch sub}/exec cvx currenttransfer/exec cvx]cvx settransfer @whi} bd/currentscale{1 0 dtransform matrix defaultmatrix idtransform dup mul exch dup mul add sqrt 0 1 dtransform matrix defaultmatrix idtransform dup mul exch dup mul add sqrt}bd/@unscale{}bd/@smc{CMYKMarks{0 exch eq{1 1 1 1 1 _ 0 @scc_5 pop}{1 0 0 0 0 _ 0 @scc_5 pop}ifelse}{SetGry}ifelse}bd/@gmc{CMYKMarks{GetCmyk pop pop pop 1 eq{0}{1}ifelse}{GetGry}ifelse}bd/@sqr{dup 0 rl dup 0 exch rl neg 0 rl @cp}bd/corelsym{@gs @np Tl -90 rotate 7{45 rotate -.75 2 m 1.5 @sqr fill} repeat @gr}bd/@reg_cor{@gs @np Tl -6 -6 m 12 @sqr @gs 1 @gmc sub @smc fill @gr 4{90 rotate 0 4 m 0 4 rl}repeat stroke 0 0 corelsym @gr}bd/@reg_std{@gs @np Tl .3 w 0 0 5 0 360 arc @cp @gs 1 @gmc sub @smc fill @gr 4{90 rotate 0 0 m 0 8 rl }repeat stroke @gr}bd/@reg_inv{@gs @np Tl .3 w 0 0 5 0 360 arc @cp @gs 1 @gmc sub @smc fill @gr 4{90 rotate 0 0 m 0 8 rl}repeat stroke 0 0 m 0 0 5 90 180 arc @cp 0 0 5 270 360 arc @cp fill @gr}bd/@reg_sqr{@gs @np Tl .3 w -7 -7 m 14 @sqr fill @gs 1 @gmc sub @smc 0 0 5 0 360 arc @cp save fill restore 4{90 rotate 0 0 m 0 8 rl}repeat stroke @gr 4{90 rotate 0 0 m 0 5 rl}repeat stroke @gr}bd /@reg_lng{@gs @np 0 eq{/$Xlen 16 def/$Ylen 8 def}{/$Xlen 8 def/$Ylen 16 def} ifelse Tl .3 w 0 0 5 0 360 arc @cp @gs 1 @gmc sub @smc fill @gr 90 rotate 0 0 m 0 $Xlen rl 90 rotate 0 0 m 0 $Ylen rl 90 rotate 0 0 m 0 $Xlen rl 90 rotate 0 0 m 0 $Ylen rl stroke @gr}bd/@setdcol{0 $dcol eq{0 exch 1 exch sub 0 0 0 _ 0 @scc_5 pop}{1 $dcol eq{0 exch 0 exch 1 exch sub 0 0 _ 0 @scc_5 pop}{2 $dcol eq {0 exch 0 exch 0 exch 1 exch sub 0 _ 0 @scc_5 pop}{SetGry}ifelse}ifelse}ifelse }bd/$corelmeter[1 .95 .75 .50 .25 .05 0]def/@colormeter{@gs @np 0 @setdcol 0.3 w/Courier findfont 5 scalefont setfont/yy xd/xx xd 0 1 6{dup xx 20 sub yy m 20 @sqr @gs $corelmeter exch get dup @setdcol fill @gr stroke xx 18 sub yy 2 add m exch dup 3 ge{1 SetGry}{0 @setdcol}ifelse 3 eq{pop}{100 mul 100 exch sub cvi 20 string cvs show}ifelse/yy yy 20 add def}for @gr}bd/@calbar{@gs Tl @gs @np 0 0 m @gs 20 @sqr 1 1 0 0 0 _ 0 @scc_5 pop fill @gr 20 0 Tl 0 0 m @gs 20 @sqr 1 1 0 1 0 _ 0 @scc_5 pop fill @gr 20 0 Tl 0 0 m @gs 20 @sqr 1 0 0 1 0 _ 0 @scc_5 pop fill @gr 20 0 Tl 0 0 m @gs 20 @sqr 1 0 1 1 0 _ 0 @scc_5 pop fill @gr 20 0 Tl 0 0 m @gs 20 @sqr 1 0 1 0 0 _ 0 @scc_5 pop fill @gr 20 0 Tl 0 0 m @gs 20 @sqr 1 1 1 0 0 _ 0 @scc_5 pop fill @gr 20 0 Tl 0 0 m @gs 20 @sqr 1 1 1 1 0 _ 0 @scc_5 pop fill @gr @gr @np -84 0 Tl @gs 0 0 m 20 @sqr clip 1 1 0 0 0 _ 0 @scc_5 pop @gain @gr 20 0 Tl @gs 0 0 m 20 @sqr clip 1 0 1 0 0 _ 0 @scc_5 pop @gain @gr 20 0 Tl @gs 0 0 m 20 @sqr clip 1 0 0 1 0 _ 0 @scc_5 pop @gain @gr 20 0 Tl @gs 0 0 m 20 @sqr clip 1 0 0 0 1 _ 0 @scc_5 pop @gain @gr @gr}bd/@gain{10 10 Tl @np 0 0 m 0 10 360{0 0 15 4 -1 roll dup 5 add arc @cp}for fill}bd/@crop{@gs 10 div /$croplen xd .3 w 0 @smc Tl rotate 0 0 m 0 $croplen neg rl stroke @gr}bd /deflevel 0 def/@sax{/deflevel deflevel 1 add def}bd/@eax{/deflevel deflevel dup 0 gt{1 sub}if def deflevel 0 gt{/eax load}{eax}ifelse}bd/eax{{exec}forall} bd/@rax{deflevel 0 eq{@rs @sv}if}bd/@daq{dup type/arraytype eq{{}forall}if}bd /@BMP{/@cc xd 12 index 1 gt UseLevel2 true eq and{7 -2 roll pop pop @I_2}{11 index 1 eq{12 -1 roll pop @i}{7 -2 roll pop pop @I}ifelse}ifelse}bd systemdict /pdfmark known not{/pdfmark/cleartomark ld}if/@BeginEPS{$ngx 0 SetGry 0 J 0 j 1 w[]0 d 10 M false setoverprint @np/EPSDict 200 dict def EPSDict begin @sv SepMode_5 0 ne{/colorimage{@gs/sepdict 50 dict def sepdict begin[{1 exch sub} /exec cvx currenttransfer/exec cvx]cvx settransfer 3 eq/$rgb xd{$rgb{/daq0 xd /daq1 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$rgb{3}{4}ifelse add def /$out $out 1 add def}forall pop $data}bd}if}bd/@EndEPS{@rs end}bd end %%EndResource %%EndProlog %%BeginSetup wCorel6Dict begin @BeginSysCorelDict 2.6131 setmiterlimit 1.00 setflat /$fst 128 def %%EndSetup %%Page: 1 1 %LogicalPage: 1 %%BeginPageSetup @sv @sm @sv %%EndPageSetup @rax %Note: Object 281.09622 557.23578 281.09735 699.32665 @E 0 J 0 j [] 0 d 0 R 0 @G 0.00 0.00 0.00 0.20 K 0 11.33858 11.33858 0.00000 @w 281.09679 699.32665 m 281.09679 557.23578 L S @rax %Note: Object 175.62104 611.36561 227.23739 645.19682 @E 0 O 0 @g 0.00 0.00 0.00 0.20 k 0 J 0 j [] 0 d 0 R 0 @G 0.00 0.00 0.00 0.20 K 0 0.21543 0.21543 0.00000 @w 175.62104 645.19682 m 227.23739 645.19682 L 227.23739 611.36561 L 175.62104 611.36561 L 175.62104 645.19682 L @c B @rax %Note: Object 122.39065 618.38901 135.83339 641.05483 @E 0 O 0 @g 0.00 0.00 0.00 0.20 k 0 J 0 j [] 0 d 0 R 0 @G 0.00 0.00 0.00 0.20 K 0 0.21543 0.21543 0.00000 @w 122.39065 630.01672 m 128.93726 641.05483 L 135.83339 629.42712 L 129.28677 618.38901 L 122.39065 630.01672 L @c B @rax %Note: Object 188.56176 628.53874 233.40869 648.07002 @E 0 O 0 @g 0.00 0.00 0.00 0.50 k 0 J 0 j [] 0 d 0 R 0 @G 0.00 0.00 0.00 1.00 K 0 0.21600 0.21600 0.00000 @w 188.56176 628.53874 m 188.56176 628.53874 196.22296 646.37972 233.40869 648.07002 C S @rax %Note: Object 188.56176 609.38277 233.40869 628.91433 @E 0 O 0 @g 0.00 0.00 0.00 0.50 k 0 J 0 j [] 0 d 0 R 0 @G 0.00 0.00 0.00 1.00 K 0 0.21600 0.21600 0.00000 @w 188.56176 628.91433 m 188.56176 628.91433 196.22296 611.07307 233.40869 609.38277 C S @rax %Note: Object 181.47515 628.47921 240.49531 654.18293 @E 0 O 0 @g 0.00 0.00 0.00 0.50 k 0 J 0 j [] 0 d 0 R 0 @G 0.00 0.00 0.00 0.20 K 0 14.17323 14.17323 0.00000 @w 181.47515 628.47921 m 181.47515 628.47921 191.55770 651.95858 240.49531 654.18293 C S @rax %Note: Object 181.47515 603.27014 240.49531 628.97357 @E 0 O 0 @g 0.00 0.00 0.00 0.50 k 0 J 0 j [] 0 d 0 R 0 @G 0.00 0.00 0.00 0.20 K 0 14.17323 14.17323 0.00000 @w 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R 0 @G 0.00 0.00 0.00 0.20 K 0 14.17323 14.17323 0.00000 @w 153.12869 629.21112 m 153.12869 629.21112 172.89638 583.17902 268.84176 578.81820 C S @rax %Note: Object 160.21531 628.30120 261.75515 672.52167 @E 0 O 0 @g 0.00 0.00 0.00 0.50 k 0 J 0 j [] 0 d 0 R 0 @G 0.00 0.00 0.00 0.20 K 0 14.17323 14.17323 0.00000 @w 160.21531 628.30120 m 160.21531 628.30120 177.56164 668.69490 261.75515 672.52167 C S @rax %Note: Object 160.21531 584.93112 261.75515 629.15159 @E 0 O 0 @g 0.00 0.00 0.00 0.50 k 0 J 0 j [] 0 d 0 R 0 @G 0.00 0.00 0.00 0.20 K 0 14.17323 14.17323 0.00000 @w 160.21531 629.15159 m 160.21531 629.15159 177.56164 588.75789 261.75515 584.93112 C S @rax %Note: Object 198.48302 628.62180 223.48743 639.51080 @E 0 O 0 @g 0.00 0.00 0.00 0.50 k 0 J 0 j [] 0 d 0 R 0 @G 0.00 0.00 0.00 1.00 K 0 0.21600 0.21600 0.00000 @w 198.48302 628.62180 m 198.48302 628.62180 202.75455 638.56857 223.48743 639.51080 C S @rax %Note: Object 198.48302 617.94198 223.48743 628.83099 @E 0 O 0 @g 0.00 0.00 0.00 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O 0 @g 0.00 0.00 0.00 0.50 k 0 J 0 j [] 0 d 0 R 0 @G 0.00 0.00 0.00 0.20 K 0 19.84252 19.84252 0.00000 @w 144.15562 632.34624 m 144.15562 632.34624 164.37033 610.46589 262.48677 608.39291 C S @rax %Note: Object 207.85068 632.52794 263.64416 632.52907 @E 0 J 0 j [] 0 d 0 R 0 @G 0.00 0.00 0.00 0.20 K 0 14.17323 14.17323 0.00000 @w 207.85068 632.52850 m 263.64416 632.52850 L S @rax %Note: Object 271.64778 572.65427 271.64891 686.64784 @E 0 J 0 j [] 0 d 0 R 0 @G 0.00 0.00 0.00 0.20 K 0 19.84252 19.84252 0.00000 @w 271.64835 686.64784 m 271.64835 572.65427 L S @rax %Note: Object 121.45323 629.34860 288.18992 701.96315 @E 0 O 0 @g 0.00 0.00 0.00 0.50 k 0 J 0 j [] 0 d 0 R 0 @G 0.00 0.00 0.00 1.00 K 0 2.01600 2.01600 0.00000 @w 121.45323 629.34860 m 121.45323 629.34860 149.93745 695.67931 288.18992 701.96315 C S @rax %Note: Object 121.30696 556.51323 288.04365 629.12778 @E 0 O 0 @g 0.00 0.00 0.00 0.50 k 0 J 0 j [] 0 d 0 R 0 @G 0.00 0.00 0.00 1.00 K 0 2.01600 2.01600 0.00000 @w 121.30696 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Fp(R)14810 44053 y Fw(3)15336 44493 y Ft(.)539 b(Let)403 b(no)-34 b(w)406 b Fq(V)673 b Ft(b)34 b(e)404 b(the)h(subset)g(of)f Fp(Z)31932 44053 y Fw(3)32862 44493 y Ft(giv)-34 b(en)404 b(b)-34 b(y)9918 47216 y Fq(V)606 b Ft(=)337 b Fr(f)p Fq(v)380 b Ft(=)336 b(\()p Fq(m)16899 47398 y Fw(1)17425 47216 y Fq(;)202 b(m)19028 47398 y Fw(2)19554 47216 y Fq(;)g(m)21157 47398 y Fw(3)21683 47216 y Ft(\))337 b Fr(2)g Fp(Z)24444 46716 y Fw(3)25307 47216 y Ft(:)538 b Fq(m)27246 47398 y Fw(1)28109 47216 y Fr(\270)336 b Ft(0)p Fq(;)404 b(m)31799 46716 y Fw(2)31799 47516 y(2)32594 47216 y Ft(+)269 b Fq(m)34870 46716 y Fw(2)34870 47516 y(3)35733 47216 y Fr(\267)337 b Fq(\256)207 b Ft(ln\(1)270 b(+)f Fq(m)42628 47398 y Fw(1)43154 47216 y Ft(\))1575 49939 y(Namely)580 b Fq(V)849 b Ft(is)581 b(the)g(in\257nite)g(set)f(of)h(v)-34 b(ertices)580 b(in)g Fp(Z)26283 49500 y Fo(d)27403 49939 y Ft(\\inside")h(the)g(region)f(of) h Fp(R)40745 49500 y Fw(3)41851 49939 y Ft(delimitated)g(b)-34 b(y)581 b(the)1575 51445 y(rev)-34 b(olution)400 b(surface)g(of)g (equation)g Fq(r)18280 51005 y Fw(2)19142 51445 y Ft(=)337 b Fq(\256)206 b Ft(ln)q(\(1)261 b(+)f Fq(x)p Ft(\))400 b(where)g Fq(r)30601 51005 y Fw(2)31463 51445 y Ft(=)336 b Fq(y)33379 51005 y Fw(2)34167 51445 y Ft(+)260 b Fq(z)35987 51005 y Fw(2)36513 51445 y Ft(,)400 b Fq(\256)405 b Ft(is)400 b(a)f(p)34 b(ositiv)-34 b(e)400 b(constan)-34 b(t)402 b(and)1575 52950 y Fq(y)380 b Fr(\270)337 b Ft(0)404 b(\(see)g(\257gure)g(1\).)1575 54456 y(Consider)503 b(no)-34 b(w)504 b(the)f(graph)h Fq(G)d Ft(=)g Fp(L)18787 54016 y Fw(3)19313 54456 y Fr(j)19650 54644 y Fo(V)20964 54456 y Ft(whic)-34 b(h)504 b(is)f(the)g(restriction)f(of)i Fp(L)36226 54016 y Fw(3)37254 54456 y Ft(to)f Fq(V)270 b Ft(.)834 b(This)504 b(graph)f(do)34 b(es)503 b(not)1575 55961 y(admit)540 b(an)g(in\257nite)h(bi-geo)34 b(desic)539 b(and)h(is)g(not)h(planar,)573 b(so)540 b(theorem)g(1)g(and)h(theorem)e (3)h(are)f(not)i(useful.)1575 57467 y(But)407 b(it)g(is)g(easy)g(to)g (see)g(that)h Fq(G)f Ft(satis\257es)h(all)e(h)-34 b(yp)34 b(othesis)408 b(of)g(theorem)f(2)g(and)h(therefore)f(has)g(a)h(non)f (trivial)1575 58972 y(p)34 b(ercolation)403 b(threshold.)1575 60478 y(Indeed,)314 b(the)292 b(graph)g Fq(G)f Ft(is)g(b)34 b(ounded)293 b(degree)d(and)j(has)f(the)f(minimal)g(cut)h(set)g(prop)34 b(ert)-34 b(y)-101 b(.)501 b(It)291 b(is)g(also)h(immediate)1575 61983 y(to)398 b(c)-34 b(hec)g(k)398 b(that)i Fq(G)e Ft(satis\257es)h(the)f(isop)34 b(erimetric)397 b(condition)i(\(4.2\).) 537 b(The)399 b(w)-34 b(orst)399 b(cases)f(are)f(clearly)g(when)i(w)-34 b(e)1575 63489 y(c)g(ho)34 b(ose)404 b(connected)g(subsets)i(of)e Fq(V)674 b Ft(of)404 b(the)h(form)15108 66212 y Fq(W)16253 66394 y Fo(n)17215 66212 y Ft(=)337 b Fr(f)p Fq(v)380 b Ft(=)336 b(\()p Fq(m)22883 66394 y Fw(1)23409 66212 y Fq(;)202 b(m)25012 66394 y Fw(2)25538 66212 y Fq(;)g(m)27141 66394 y Fw(3)27667 66212 y Ft(\))337 b Fr(2)g Fq(V)606 b Ft(:)538 b Fq(m)32872 66394 y Fw(1)33735 66212 y Fr(\267)337 b Fq(n)269 b Ft(+)g(1)p Fr(g)1575 68935 y Ft(where)404 b Fq(n)336 b Fr(\270)h Ft(1)404 b(in)g(suc)-34 b(h)405 b(w)-34 b(a)g(y)405 b(that)g Fr(j)p Fq(W)19148 69117 y Fo(n)19774 68935 y Fr(j)336 b(\270)h Ft(2.)1575 70440 y(In)404 b(this)h(case)e(w)-34 b(e)405 b(ha)-34 b(v)g(e)404 b(that)i Fr(j)p Fq(@)16206 70622 y Fo(e)16697 70440 y Fq(W)17842 70622 y Fo(n)18468 70440 y Fr(j)336 b(\270)h Fq(\274)43 b(\256)207 b Ft(ln\(1)270 b(+)f Fq(n)p Ft(\))404 b(and)h(diam)p Fq(W)33437 70622 y Fo(n)34400 70440 y Fr(\267)336 b Ft(2\()p Fq(n)270 b Ft(+)f(1\).)539 b(Hence)18309 72981 y Fr(j)p Fq(@)19290 73163 y Fo(e)19781 72981 y Fq(W)20926 73163 y Fo(n)21552 72981 y Fr(j)p 16924 73522 6351 49 v 16924 74632 a Ft(ln\(diam)q Fq(W)22177 74814 y Fo(n)22803 74632 y Ft(\))23743 73801 y Fr(\270)25947 72981 y Fq(\274)43 b(\256)208 b Ft(ln\(1)270 b(+)f Fq(n)p Ft(\))p 25156 73522 8068 49 v 25156 74632 a(ln)202 b(2)269 b(+)g(ln)q(\(1)g(+)g Fq(n)p Ft(\))33693 73801 y Fr(\270)35105 72981 y Fq(\274)43 b(\256)p 35105 73522 1515 49 v 35559 74632 a Ft(2)26469 78283 y(8)p eop %%Page: 9 9 9 8 bop 1575 5313 a Ft(It)364 b(is)h(in)-34 b(teresting)365 b(to)g(notice)g(that)h(t)-34 b(w)g(o-dimensional)366 b(slices)e(of)h Fq(G)f Ft(are)h(not)g(exp)34 b(ected)364 b(to)h(ha)-34 b(v)g(e)365 b(a)g(non)g(trivial)1575 6819 y(p)34 b(ercolation)476 b(threshold,)496 b(i.e)476 b(theorem)h(11.55)g (of)g([9])f(is)h(not)g(applicable.)757 b(E.g.)g(tak)-34 b(e)477 b(the)g(bi-dimensional)1575 8324 y(\(in\257nite\))405 b(subgraph)h Fq(G)p Fr(j)13178 8512 y Fo(V)13726 8635 y Fh(0)14646 8324 y Ft(of)f Fq(G)f Ft(where)6423 11335 y Fq(V)7130 11517 y Fw(0)7993 11335 y Ft(=)336 b Fr(f)p Fq(v)380 b Ft(=)336 b(\()p Fq(m)13660 11517 y Fw(1)14187 11335 y Fq(;)202 b(m)15790 11517 y Fw(2)16316 11335 y Fq(;)g(m)17919 11517 y Fw(3)18444 11335 y Ft(\))337 b Fr(2)g Fp(Z)21205 10835 y Fw(3)22068 11335 y Ft(:)538 b Fq(m)24007 11517 y Fw(1)24870 11335 y Fr(\270)336 b Ft(0)p Fq(;)1010 b(m)29166 11517 y Fw(3)30029 11335 y Ft(=)336 b(0)p Fq(;)1010 b Ft(0)337 b Fr(\267)g Fq(m)36548 11517 y Fw(2)37410 11335 y Fr(\267)38690 10240 y Fg(p)p 39902 10240 6613 49 v 1095 x Fq(\256)207 b Ft(ln\(1)270 b(+)f Fq(m)45517 11517 y Fw(1)46043 11335 y Ft(\))p Fr(g)1575 13269 y Ft(The)383 b(graph)h Fq(G)p Fr(j)8752 13457 y Fo(V)9300 13580 y Fh(0)10199 13269 y Ft(can)f(b)34 b(e)383 b(view)-34 b(ed)383 b(as)g(a)g(subgraph)h(of)g Fp(Z)27846 12829 y Fw(2)28754 13269 y Ft(of)g(the)f(t)-34 b(yp)34 b(e)383 b Fq(G)p Ft(\()p Fq(f)130 b Ft(\))385 b(of)e(section)g(11.5)g (in)g([9])f(where)1575 14774 y Fq(f)130 b Ft(\()p Fq(u)p Ft(\))406 b(is)e(the)h(non)h(negativ)-34 b(e)405 b(function)h(on)f([0)p Fq(;)202 b Fr(1)p Ft(\))404 b(giv)-34 b(en)405 b(b)-34 b(y)405 b Fq(f)130 b Ft(\()p Fq(u)p Ft(\))339 b(=)33617 13738 y Fg(p)p 34829 13738 5717 49 v 1036 x Fq(\256)207 b Ft(ln\(1)269 b(+)h Fq(u)p Ft(\).)540 b(W)-101 b(e)404 b(immediately)g(see)1575 16280 y(that)473 b(this)g(graph)f(is)g(not)h (\\fat)g(enough")g(to)g(to)f(ensure)g(non)h(trivial)e(p)34 b(ercolation)472 b(threshold)h(b)-34 b(y)472 b(theorem)1575 17785 y(11.55)427 b(of)h([9],)k(since)427 b Fq(f)130 b Ft(\()p Fq(u)p Ft(\))p Fq(=)202 b Ft(ln)i Fq(u)376 b Fr(!)f Ft(0)428 b(as)f Fq(u)376 b Fr(!)f(1)p Ft(.)609 b(So)428 b(the)g(graph)g Fq(G)f Ft(could)h(pro)-34 b(vide)427 b(an)h(example)f(of)h(sub-)1575 19291 y(graph)409 b(of)h Fp(Z)7211 18851 y Fw(3)8145 19291 y Ft(whic)-34 b(h)410 b(p)34 b(ercolates)409 b(while)f(no)i(bi-dimensional)f(subsets)h(of)g (this)f(graph)h(p)34 b(ossibly)409 b(p)34 b(ercolates.)1575 20796 y(W)-101 b(e)403 b(also)h(remark)f(that)j(this)e(example)g(can)g (easily)f(generalized)g(to)i Fp(Z)34097 20356 y Fo(d)35040 20796 y Ft(and)g(other)g Fq(d)p Ft(-dimensional)f(lattice.)1575 23247 y(Concering)451 b(no)-34 b(w)453 b(planar)e(graphs)h(b)-34 b(y)452 b(this)f(theorem)h(3)f(one)g(can)h(establish)f(w)-34 b(eather)452 b(a)g(planar)f(graph)h(has)1575 24752 y(or)g(not)i(a)f (non)h(trivial)d(p)34 b(ercolation)453 b(threshold,)466 b(without)454 b(using)g(an)-34 b(y)453 b(particular)g(symmetry)f(prop) 34 b(ert)-34 b(y)453 b(of)1575 26258 y(the)404 b(graph)h(\(e.g.,)e (transitivit)-34 b(y\).)1575 27763 y(In)452 b(particular,)463 b(theorem)452 b(3)g(can)g(b)34 b(e)451 b(pro)-34 b(v)g(ed)453 b(easily)e(to)h(b)34 b(e)451 b(true)h(for)g(the)h(a)e(class)h(of)g (tilings)g(of)g(the)h(plane,)1575 29269 y(usually)431 b(referred)g(in)g(the)h(literature)e(as)i(the)f(class)g(of)h Fs(normal)458 b(tilings)430 b Ft(\(see)h(e.g.)620 b([11]\).)g(A)431 b(tiling)g Fp(T)g Ft(of)h(the)1575 30774 y(plane)416 b(is)g(a)h(coun)-34 b(table)417 b(family)f Fp(T)357 b Ft(=)g Fr(f)p Fq(T)20068 30956 y Fw(1)20594 30774 y Fq(;)202 b(T)21841 30956 y Fw(2)22367 30774 y Fq(;)g(:)g(:)g(:)f(;)h Fr(g)416 b Ft(of)h(closed)f(sets)g(\(called)g(tiles\))g(whic)-34 b(h)418 b(co)-34 b(v)g(er)415 b(the)i(plane)1575 32279 y(without)464 b(gaps)f(or)f(o)-34 b(v)g(erlaps.)714 b(A)463 b(tiling)f Fp(T)h Ft(is)f Fs(normal)g Ft(if)h(ev)-34 b(ery)461 b(tile)h Fq(T)603 b Fr(2)434 b Fp(T)463 b Ft(is)f(a)h(top)34 b(ological)462 b(disk)g(and)h(it)1575 33785 y(is)414 b(uniformly)g(b)34 b(ounded,)418 b(i.e.)567 b(there)414 b(exists)g(t)-34 b(w)g(o)415 b(p)34 b(ositiv)-34 b(e)414 b(n)-34 b(um)g(b)34 b(ers)415 b Fq(U)34747 33978 y Fi(T)35807 33785 y Ft(and)g Fq(u)38868 33978 y Fi(T)39927 33785 y Ft(suc)-34 b(h)415 b(that)h Fq(T)582 b Ft(con)-34 b(tains)415 b(a)1575 35290 y(disk)400 b(of)h(radius)g Fq(u)9817 35483 y Fi(T)10862 35290 y Ft(and)h(it)e(is)g(con)-34 b(tained)402 b(in)e(a)g(disk)h(of)f(radius)h Fq(U)31916 35478 y Fw(T)32651 35290 y Ft(.)537 b(F)-101 b(or)400 b(normal)h(tilings,)f(the)h(in)-34 b(tersection)1575 36796 y(of)501 b(an)-34 b(y)501 b(pair)g(of)g(tiles)f (is)h(empt)-34 b(y)501 b(or)g(ma)-34 b(y)501 b(consists)g(of)g(a)g(set) g(of)g(isolated)g(p)34 b(oin)-34 b(ts)502 b(or)e(arcs.)828 b(The)501 b(isolated)1575 38301 y(p)34 b(oin)-34 b(ts)405 b(of)f(a)g(tiling)g(are)g(called)f(v)-34 b(ertices)404 b(of)g(the)h(tiling)f(and)g(the)h(arcs)f(are)f(called)h(edges)g(of)g (the)h(tiling.)1575 39807 y(If)357 b Fq(V)3447 40000 y Fi(T)4450 39807 y Ft(denotes)h(the)g(set)f(of)h(all)e(v)-34 b(ertices)357 b(of)g(the)h(tiling)f(and)h Fq(E)29681 40000 y Fi(T)30684 39807 y Ft(the)g(set)f(of)h(all)e(edges)h(of)h(the)g (tiling,)366 b(then)358 b(the)1575 41312 y(graph)405 b Fq(G)5966 41505 y Fi(T)6948 41312 y Ft(=)336 b(\()p Fq(V)9405 41505 y Fi(T)10052 41312 y Fq(;)202 b(E)11486 41505 y Fi(T)12131 41312 y Ft(\))404 b(is)g(called)f(the)i Fs(1-skeleton)c Ft(of)j Fp(T)p Ft(.)539 b(Note)404 b(that)i Fq(G)34959 41505 y Fi(T)36008 41312 y Ft(is)e(lo)34 b(cally)403 b(\257nite)h(if)g Fp(T)g Ft(is)g(normal.)1575 42818 y(No)-34 b(w,)404 b(the)h(follo)-34 b(wing)405 b(corollary)d(is)i(an)h(easy)e (consequence)h(of)h(theorem)f(3.)1575 45268 y Fl(Corollary)563 b(1)p Ft(.)798 b Fs(L)-62 b(et)512 b Fp(T)h Fs(b)-62 b(e)513 b(a)f(normal)h(tiling.)795 b(Then)513 b(the)e(gr)-62 b(aph)512 b Fq(G)33670 45461 y Fi(T)34796 45268 y Ft(=)481 b(\()p Fq(V)37398 45461 y Fi(T)38044 45268 y Fq(;)202 b(E)39478 45461 y Fi(T)40123 45268 y Ft(\))513 b Fs(has)f(b)-62 b(ond)512 b(p)-62 b(er)g(c)g(olation)1575 46774 y(thr)g(eshold)430 b(less)j(than)f(1.)1575 48909 y Fl(Pro)39 b(of)p Ft(.)632 b(It)436 b(is)g(easy)f(to)h(see)f(that)i(if)f Fp(T)f Ft(is)h(a)g(normal)f(tiling)h(then)g Fq(G)33149 49102 y Fi(T)34231 48909 y Ft(is)f(b)34 b(ounded)437 b(degree)e(and)i Fq(G)47610 48469 y Fn(\244)47610 49273 y Fi(T)48691 48909 y Ft(is)f(also)1575 50414 y(b)34 b(ounded)423 b(degree.)589 b(T)-101 b(o)422 b(pro)-34 b(v)g(e)422 b(that)g(inequalit)-34 b(y)422 b(\(4.2\))g(is)f(satis\257ed)h(let)f(us)h(consider)f(a)h(set)f Fq(T)44765 50596 y Fo(n)45757 50414 y Fr(\275)366 b Fp(T)421 b Ft(formed)1575 51920 y(b)-34 b(y)405 b Fq(n)337 b Ft(=)h Fr(j)p Fq(T)6650 52102 y Fo(n)7276 51920 y Fr(j)404 b Ft(pairwise)h(connected)g(tiles)f(of)h Fp(T)g Ft(\(t)-34 b(w)g(o)407 b(tiles)d(are)g(pairwise)h(connected)g(if)f(they)h(ha)-34 b(v)g(e)405 b(an)h(edge)1575 53425 y(in)492 b(common\).)804 b(Denote)493 b(no)-34 b(w)493 b(b)-34 b(y)493 b Fq(W)18910 53607 y Fo(n)20020 53425 y Fr(\275)483 b Fq(V)22153 53618 y Fi(T)23292 53425 y Ft(the)492 b(set)h(of)g(v)-34 b(ertices)491 b(in)h Fq(T)35602 53607 y Fo(n)36721 53425 y Ft(and)h(b)-34 b(y)493 b Fq(P)41716 53607 y Fo(n)42835 53425 y Ft(the)g(subset)g(of)g (the)1575 54931 y(plane)516 b(whic)-34 b(h)516 b(is)g(the)g(union)g(of) g(the)g(tiles)f(in)h Fq(T)24108 55113 y Fo(n)24735 54931 y Ft(.)873 b(Without)516 b(loss)g(in)f(generalit)-34 b(y)-101 b(,)543 b(w)-34 b(e)516 b(can)g(supp)34 b(ose)516 b(that)1575 56436 y Fq(P)2353 56618 y Fo(n)3484 56436 y Ft(is)505 b(a)g(simply)g(connected)g(closed)g(set)g(of)h(the)f (plane.)842 b(Then)506 b(the)f(edge)g(b)34 b(oundary)506 b Fq(@)43518 56618 y Fo(e)44009 56436 y Fq(W)45154 56618 y Fo(n)46285 56436 y Ft(of)g Fq(W)48912 56618 y Fo(n)50043 56436 y Ft(is)e(a)1575 57942 y(P)-34 b(eierls)504 b(con)-34 b(tour)507 b(of)f Fq(G)12526 57502 y Fn(\244)12526 58306 y Fi(T)13171 57942 y Ft(.)843 b(By)505 b(h)-34 b(yp)34 b(othesis)507 b(the)f(dual)g(graph)g(is)f(b)34 b(ounded)507 b(degree,)530 b(i.e.)842 b(a)505 b(tile)g(can)h(ha)-34 b(v)g(e)1575 59447 y(at)541 b(most)h Fq(N)673 b Ft(v)-34 b(ertices,)574 b(so)541 b(that)h Fr(j)p Fq(W)18712 59629 y Fo(n)19338 59447 y Fr(j)564 b(\267)h Fq(N)132 b(n)p Ft(.)948 b(Let)541 b(no)-34 b(w)542 b Fq(A)p Ft(\()p Fq(P)31995 59629 y Fo(n)32622 59447 y Ft(\))f(b)34 b(e)541 b(the)g(area)g(of)g(the)g(set)h Fq(P)46961 59629 y Fo(n)48128 59447 y Ft(and)g(let)1575 60953 y Fq(L)p Ft(\()p Fq(P)3649 61135 y Fo(n)4276 60953 y Ft(\))480 b(b)34 b(e)480 b(the)g(length)h(of) g(the)f(b)34 b(oundary)481 b(of)g Fq(P)24285 61135 y Fo(n)24911 60953 y Ft(.)767 b(By)479 b(the)i(assumption)g(that)h(the)e (tiling)g(is)g(normal,)499 b(w)-34 b(e)1575 62458 y(ha)g(v)g(e)476 b(that)g Fq(A)p Ft(\()p Fq(P)9296 62640 y Fo(n)9923 62458 y Ft(\))456 b Fr(\270)f Fq(n)p Ft(\()p Fq(\274)43 b(u)14875 62018 y Fw(2)14875 62822 y Fi(T)15521 62458 y Ft(\))476 b(and)g(the)g(b)34 b(oundary)476 b(of)g Fq(P)28814 62640 y Fo(n)29915 62458 y Ft(con)-34 b(tains)477 b(at)f(least)f Fq(L)40025 62705 y Fw(\()40446 62458 y Fq(P)41224 62640 y Fo(n)41850 62458 y Ft(\))p Fq(=U)43755 62651 y Fi(T)44877 62458 y Ft(edges)g(so)g(that)1575 63964 y Fr(j)p Fq(@)2556 64146 y Fo(e)3047 63964 y Fq(W)4192 64146 y Fo(n)4818 63964 y Fr(j)367 b(\270)h Fq(L)p Ft(\()p Fq(P)8907 64146 y Fo(n)9534 63964 y Ft(\))p Fq(=U)11439 64157 y Fi(T)12084 63964 y Ft(.)595 b(Moreo)-34 b(v)g(er,)426 b(since)d Fq(P)22469 64146 y Fo(n)23518 63964 y Ft(is)f(some)h(closed)f(and)i (simply)e(connected)i(set)f(in)f(the)i(plane,)1575 65469 y(then)405 b(it)f(is)g(p)34 b(ossible)404 b(to)g(\257nd)h(a)f(constan) -34 b(t)406 b Fq(C)491 b Ft(suc)-34 b(h)405 b(that)h Fq(L)28673 65716 y Fw(\()29094 65469 y Fq(P)29872 65651 y Fo(n)30498 65469 y Ft(\))337 b Fr(\270)g Fq(C)87 b Ft([)p Fq(A)p Ft(\()p Fq(P)36034 65651 y Fo(n)36661 65469 y Ft(\)])37469 65029 y Fw(1)p Fo(=)p Fw(2)38935 65469 y Ft(.)539 b(Therefore)404 b(w)-34 b(e)404 b(get)6909 69156 y Fr(j)p Fq(@)67 b(W)9102 69338 y Fo(n)9728 69156 y Fr(j)336 b(\270)g Fq(L)p Ft(\()p Fq(P)13754 69338 y Fo(n)14381 69156 y Ft(\))p Fq(=U)16286 69349 y Fi(T)17269 69156 y Fr(\270)18941 68336 y Fq(C)p 18681 68878 1474 49 v 18681 69988 a(U)19509 70181 y Fi(T)20287 69156 y Ft([)p Fq(A)p Ft(\()p Fq(P)22782 69338 y Fo(n)23409 69156 y Ft(\)])24217 68656 y Fw(1)p Fo(=)p Fw(2)26020 69156 y Fr(\270)27693 68336 y Fq(C)p 27433 68878 V 27433 69988 a(U)28261 70181 y Fi(T)29039 69156 y Ft([)p Fq(n)p Ft(\()p Fq(\274)43 b(u)32003 68656 y Fw(2)32003 69460 y Fi(T)32648 69156 y Ft(\)])33456 68656 y Fw(1)p Fo(=)p Fw(2)35260 69156 y Fr(\270)336 b Fq(C)37492 68224 y Fr(p)p 38503 68224 735 49 v 38503 69156 a Fq(\274)39437 68336 y(u)40131 68529 y Fi(T)p 39370 68878 1474 49 v 39370 69988 a Fq(U)40198 70181 y Fi(T)41178 67447 y Fg(\267)41951 68336 y Fr(j)p Fq(W)43433 68518 y Fo(n)44058 68336 y Fr(j)p 41951 68878 2445 49 v 42620 69988 a Fq(N)44528 67447 y Fg(\270)45168 67749 y Fw(1)p Fo(=)p Fw(2)1575 71951 y Ft(Since)404 b(no)-34 b(w)405 b Fr(j)p Fq(W)8747 72133 y Fo(n)9373 71951 y Fr(j)9710 71511 y Fw(1)p Fo(=)p Fw(2)11513 71951 y Fr(\270)336 b Ft(ln[\(diam)q(\()p Fq(W)18853 72133 y Fo(n)19479 71951 y Ft(\)],)404 b(the)h(inequalit)-34 b(y)404 b(\(4.2\))g(follo)-34 b(ws.)539 b Fj(\244)1575 73456 y Ft(It)460 b(is)h(in)-34 b(teresting)461 b(to)g(observ)-34 b(e)461 b(that)g(the)g(class)g(of)g(tilings)f(satisfying)i(the)f(prop) 34 b(osition)461 b(3)g(includes)f(quasi-)1575 74962 y(p)34 b(erio)g(dic)403 b(tiling)h(of)g(the)h(plane)f(suc)-34 b(h)405 b(as)f(the)h(P)-34 b(enrose)404 b(tiling)g(of)g(\257gure)h(2.) 26469 78283 y(9)p eop %%Page: 10 10 10 9 bop 14173 29292 a @beginspecial 72 @llx 72 @lly 472 @urx 472 @ury 2267 @rwi 2267 @rhi @setspecial %%BeginDocument: fig2.eps %!PS-Adobe-2.0 %%Title: QuasiTiler %%Creator: W3Kit %%Pages: 1 1 %%BoundingBox: 72 72 472 472 %%EndComments %%BeginProcSet: Safety Net /Infinity { 0 } def /NaN { 0 } def /defineusername { pop pop } def %%EndProlog %%Page: 1 1 gsave 72 72 translate gsave 0 0 400 400 rectclip [ 50 0 0 50 200 200 ] concat 1 setgray -4 -4 8 8 rectfill 0 setlinewidth 0 setgray 0 /QuasiTilerTilePath defineusername /QuasiTilerTilePath { ucache -2 -2 2 2 setbbox 0 0 moveto 0.601501 -0.19544 rlineto 0.371748 0.511667 rlineto } cvlit def [ -5.096 5.35825 -3.52125 5.86992 -1.9465 5.35825 0 5.35825 1.57475 5.86992 3.1495 5.35825 4.72425 5.86992 -4.12275 4.01869 -2.548 3.50702 -0.973249 4.01869 0.973249 4.01869 2.548 3.50702 4.12275 4.01869 -4.12275 2.3629 -3.1495 1.65579 -0.973249 2.3629 0.601501 2.87456 1.57475 2.16746 4.12275 2.3629 -5.096 1.02333 -3.52125 0.511667 -1.57475 0.511667 -2.98023e-08 1.02333 1.57475 0.511667 3.1495 1.02333 -4.12275 -1.33956 -2.548 -0.827895 -3e-08 -0.632456 0.973249 -1.33956 2.548 -0.827895 4.12275 -1.33956 -5.096 -2.67912 ] aload pop 64 2 idiv { gsave translate QuasiTilerTilePath ustroke grestore } repeat /QuasiTilerTilePath { ucache -2 -2 2 2 setbbox 0 0 moveto 0.601501 -0.19544 rlineto 0.371748 0.511667 rlineto } cvlit def [ -2.548 -2.48369 -0.973249 -1.97202 0.601501 -2.48369 2.548 -2.48369 3.52125 -3.19079 -5.096 -4.33491 -3.52125 -3.82325 -1.57475 -3.82325 -5.96046e-08 -4.33491 1.57475 -3.82325 3.1495 -4.33491 -5.6975 -6.18614 -4.12275 -5.67448 -1.9465 -4.96737 -0.973249 -5.67448 1.57475 -5.47904 2.548 -6.18614 4.12275 -5.67448 ] aload pop 36 2 idiv { gsave translate QuasiTilerTilePath ustroke grestore } repeat /QuasiTilerTilePath { ucache -2 -2 2 2 setbbox 0 0 moveto 0.371748 -0.511667 rlineto 0.371748 0.511667 rlineto } cvlit def [ -6.06925 5.67448 -4.4945 5.16281 -2.91975 5.67448 -1.345 5.16281 -0.371748 5.86992 0.601501 5.16281 2.17625 5.67448 3.751 5.16281 5.32575 5.67448 -2.91975 4.01869 -0.371748 3.82325 2.17625 4.01869 5.32575 4.01869 -5.46775 2.16746 -4.4945 2.87456 -3.52125 2.16746 -1.9465 2.67912 -0.371748 2.16746 1.203 2.67912 2.77775 2.16746 3.751 2.87456 4.72425 2.16746 -4.4945 0.827895 -1.9465 1.02333 1.203 1.02333 3.751 0.827895 -4.4945 -0.827895 -2.91975 -0.316228 -1.345 -0.827895 -0.371748 -0.120788 0.601501 -0.827895 2.17625 -0.316228 ] aload pop 64 2 idiv { gsave translate QuasiTilerTilePath ustroke grestore } repeat /QuasiTilerTilePath { ucache -2 -2 2 2 setbbox 0 0 moveto 0.371748 -0.511667 rlineto 0.371748 0.511667 rlineto } cvlit def [ 3.751 -0.827895 5.32575 -0.316228 5.6975 -0.827895 -5.46775 -2.16746 -3.893 -2.67912 -2.91975 -1.97202 -1.9465 -2.67912 -0.371748 -2.16746 1.203 -2.67912 2.17625 -1.97202 3.1495 -2.67912 4.72425 -2.16746 -5.46775 -3.82325 -2.91975 -4.01869 -0.371748 -3.82325 2.17625 -4.01869 4.72425 -3.82325 -6.06925 -5.67448 -4.4945 -5.16281 -2.91975 -5.67448 -1.345 -5.16281 -0.371748 -5.86992 0.601501 -5.16281 2.17625 -5.67448 3.751 -5.16281 5.32575 -5.67448 6.299 -4.96737 ] aload pop 54 2 idiv { gsave translate QuasiTilerTilePath ustroke grestore } repeat /QuasiTilerTilePath { ucache -2 -2 2 2 setbbox 0 0 moveto 0.371748 -0.511667 rlineto 0.601501 0.19544 rlineto } cvlit def [ -6.67075 5.47904 -5.6975 6.18614 -4.12275 5.67448 -2.548 6.18614 -0.973249 5.67448 0.973249 5.67448 2.548 6.18614 4.12275 5.67448 -5.096 4.33491 -3.52125 3.82325 -1.9465 4.33491 1.19209e-07 4.33491 1.57475 3.82325 3.1495 4.33491 4.72425 3.82325 -6.06925 1.97202 -5.096 2.67912 -2.548 2.48369 -1.57475 3.19079 0 2.67912 2.17625 1.97202 3.1495 2.67912 5.096 2.67912 -5.6975 0.827895 -4.12275 1.33956 -2.548 0.827895 -0.973249 1.33956 0.601501 0.827895 2.548 0.827895 4.12275 1.33956 -5.096 -1.02333 -3.52125 -0.511667 ] aload pop 64 2 idiv { gsave translate QuasiTilerTilePath ustroke grestore } repeat /QuasiTilerTilePath { ucache -2 -2 2 2 setbbox 0 0 moveto 0.371748 -0.511667 rlineto 0.601501 0.19544 rlineto } cvlit def [ -1.9465 -1.02333 -0.973249 -0.316228 1.57475 -0.511667 3.1495 -1.02333 4.72425 -0.511667 -4.4945 -2.87456 -3.52125 -2.16746 -1.57475 -2.16746 0 -1.65579 1.57475 -2.16746 4.12275 -2.3629 5.096 -1.65579 -6.06925 -4.01869 -4.12275 -4.01869 -2.548 -3.50702 -0.973249 -4.01869 0.601501 -3.50702 2.548 -3.50702 4.12275 -4.01869 -5.096 -5.35825 -3.52125 -5.86992 -2.548 -5.16281 0 -5.35825 0.973249 -4.65114 3.1495 -5.35825 4.72425 -5.86992 5.6975 -5.16281 ] aload pop 54 2 idiv { gsave translate QuasiTilerTilePath ustroke grestore } repeat /QuasiTilerTilePath { ucache -2 -2 2 2 setbbox 0 0 moveto -3e-08 -0.632456 rlineto 0.601501 0.19544 rlineto } cvlit def [ -5.096 4.96737 -3.1495 4.96737 -1.9465 4.96737 1.19209e-07 4.96737 1.9465 4.96737 3.1495 4.96737 5.096 4.96737 -4.72425 3.82325 -3.1495 3.31158 -1.57475 3.82325 0 3.31158 1.9465 3.31158 3.52125 3.82325 5.096 3.31158 -5.6975 1.46035 -4.12275 1.97202 -2.17625 1.97202 -0.973249 1.97202 0.973249 1.97202 2.548 1.46035 4.12275 1.97202 -4.12275 0.316228 -2.17625 0.316228 -0.601501 0.827895 0.973249 0.316228 2.91975 0.316228 4.12275 0.316228 -4.72425 -1.535 -3.1495 -1.02333 -1.57475 -1.535 0 -1.02333 1.9465 -1.02333 ] aload pop 64 2 idiv { gsave translate QuasiTilerTilePath ustroke grestore } repeat /QuasiTilerTilePath { ucache -2 -2 2 2 setbbox 0 0 moveto -3e-08 -0.632456 rlineto 0.601501 0.19544 rlineto } cvlit def [ 3.52125 -1.535 5.096 -1.02333 -5.6975 -2.87456 -4.12275 -3.38623 -2.548 -2.87456 -0.601501 -2.87456 0.601501 -2.87456 2.548 -2.87456 4.4945 -2.87456 5.096 -2.67912 -5.6975 -4.53035 -3.751 -4.53035 -2.548 -4.53035 -0.601501 -4.53035 0.973249 -4.01869 2.548 -4.53035 4.4945 -4.53035 ] aload pop 34 2 idiv { gsave translate QuasiTilerTilePath ustroke grestore } repeat /QuasiTilerTilePath { ucache -2 -2 2 2 setbbox 0 0 moveto -3e-08 -0.632456 rlineto 0.601501 -0.19544 rlineto } cvlit def [ -3.751 5.16281 -2.548 5.16281 -0.601501 5.16281 1.345 5.16281 2.548 5.16281 4.4945 5.16281 -4.12275 4.01869 -2.548 3.50702 -0.601501 3.50702 0.973249 4.01869 2.548 3.50702 4.12275 4.01869 -4.72425 2.16746 -3.1495 1.65579 -1.57475 2.16746 0.371748 2.16746 1.57475 2.16746 3.52125 2.16746 -3.52125 0.511667 -1.57475 0.511667 -2.98023e-08 1.02333 1.57475 0.511667 3.52125 0.511667 -4.12275 -1.33956 -2.548 -0.827895 -0.601501 -0.827895 0.973249 -1.33956 2.548 -0.827895 4.12275 -1.33956 -5.096 -2.67912 -3.1495 -2.67912 -1.203 -2.67912 ] aload pop 64 2 idiv { gsave translate QuasiTilerTilePath ustroke grestore } repeat /QuasiTilerTilePath { ucache -2 -2 2 2 setbbox 0 0 moveto -3e-08 -0.632456 rlineto 0.601501 -0.19544 rlineto } cvlit def [ 0 -2.67912 1.9465 -2.67912 3.52125 -3.19079 -5.096 -4.33491 -3.1495 -4.33491 -1.57475 -3.82325 -5.96046e-08 -4.33491 1.9465 -4.33491 3.1495 -4.33491 ] aload pop 18 2 idiv { gsave translate QuasiTilerTilePath ustroke grestore } repeat /QuasiTilerTilePath { ucache -2 -2 2 2 setbbox 0 0 moveto 0.601501 0.19544 rlineto 0.371748 0.511667 rlineto } cvlit def [ -5.6975 5.16281 -2.548 5.16281 -0.973249 5.67448 2.548 5.16281 -3.52125 3.82325 0 3.31158 1.57475 3.82325 4.72425 3.82325 -6.06925 1.97202 -5.096 2.67912 -1.57475 2.16746 2.17625 1.97202 3.1495 2.67912 -4.12275 0.316228 -2.548 0.827895 0.601501 0.827895 4.12275 0.316228 -5.096 -1.02333 -1.9465 -1.02333 -0.973249 -0.316228 3.1495 -1.02333 -4.4945 -2.87456 -3.52125 -2.16746 0 -2.67912 1.57475 -2.16746 5.096 -2.67912 -6.06925 -4.01869 -2.548 -4.53035 -0.973249 -4.01869 2.548 -4.53035 4.12275 -4.01869 -3.52125 -5.86992 ] aload pop 64 2 idiv { gsave translate QuasiTilerTilePath ustroke grestore } repeat /QuasiTilerTilePath { ucache -2 -2 2 2 setbbox 0 0 moveto 0.601501 0.19544 rlineto 0.371748 0.511667 rlineto } cvlit def [ -1.19209e-07 -6.38158 0.973249 -5.67448 4.72425 -5.86992 5.6975 -5.16281 ] aload pop 8 2 idiv { gsave translate QuasiTilerTilePath ustroke grestore } repeat /QuasiTilerTilePath { ucache -2 -2 2 2 setbbox 0 0 moveto 0.601501 -0.19544 rlineto 0.601501 0.19544 rlineto } cvlit def [ -5.6975 5.16281 -3.751 5.16281 -2.548 5.16281 -0.601501 5.16281 1.345 5.16281 2.548 5.16281 4.4945 5.16281 -4.72425 2.16746 -3.1495 2.67912 -1.57475 2.16746 0.371748 2.16746 1.9465 2.67912 3.52125 2.16746 -4.12275 -0.316228 -2.17625 -0.316228 -0.601501 -0.827895 0.973249 -0.316228 2.91975 -0.316228 4.12275 -0.316228 -4.72425 -2.16746 -3.1495 -2.67912 -1.203 -2.67912 0 -2.67912 1.9465 -2.67912 3.52125 -2.16746 -5.6975 -5.16281 -3.751 -5.16281 -2.17625 -5.67448 -0.601501 -5.16281 0.973249 -5.67448 2.548 -5.16281 4.4945 -5.16281 ] aload pop 64 2 idiv { gsave translate QuasiTilerTilePath ustroke grestore } repeat /QuasiTilerTilePath { ucache -2 -2 2 2 setbbox 0 0 moveto 0.371748 -0.511667 rlineto 0.601501 -0.19544 rlineto } cvlit def [ -3.52125 5.86992 0 6.38158 1.57475 5.86992 4.72425 5.86992 -2.548 4.53035 -0.973249 4.01869 2.548 4.53035 -4.12275 3.38623 -3.1495 2.67912 0.601501 2.87456 4.12275 3.38623 -5.096 1.02333 -1.57475 1.535 1.57475 1.535 3.1495 1.02333 -4.12275 -0.316228 -5.96046e-08 0.390879 0.973249 -0.316228 4.12275 -0.316228 -2.548 -1.46035 -0.973249 -1.97202 2.548 -1.46035 3.52125 -2.16746 -5.096 -3.31158 -3.52125 -3.82325 0 -3.31158 1.57475 -3.82325 -5.6975 -5.16281 -1.9465 -4.96737 -0.973249 -5.67448 2.548 -5.16281 ] aload pop 62 2 idiv { gsave translate QuasiTilerTilePath ustroke grestore } repeat /QuasiTilerTilePath { ucache -2 -2 2 2 setbbox 0 0 moveto -3e-08 -0.632456 rlineto 0.371748 0.511667 rlineto } cvlit def [ -4.12275 4.65114 -0.973249 4.65114 0.973249 4.65114 4.12275 4.65114 5.6975 5.16281 -5.096 3.31158 -1.9465 3.31158 0.601501 3.50702 3.1495 3.31158 5.6975 3.50702 -5.096 1.65579 -2.548 1.46035 0 1.65579 3.1495 1.65579 5.096 1.65579 -2.91975 0.316228 -0.973249 0.316228 2.17625 0.316228 4.72425 0.511667 -3.52125 -1.535 -0.973249 -1.33956 1.57475 -1.535 4.72425 -1.535 -3.52125 -3.19079 -1.57475 -3.19079 1.57475 -3.19079 4.12275 -3.38623 -4.4945 -4.53035 -1.9465 -4.33491 0.601501 -4.53035 3.751 -4.53035 5.096 -4.33491 ] aload pop 64 2 idiv { gsave translate QuasiTilerTilePath ustroke grestore } repeat /QuasiTilerTilePath { ucache -2 -2 2 2 setbbox 0 0 moveto -3e-08 -0.632456 rlineto 0.371748 -0.511667 rlineto } cvlit def [ -4.4945 5.16281 -1.345 5.16281 0.601501 5.16281 3.751 5.16281 -3.52125 3.82325 -0.973249 4.01869 1.57475 3.82325 4.72425 3.82325 -6.06925 1.97202 -3.52125 2.16746 -0.371748 2.16746 2.17625 1.97202 4.72425 2.16746 -2.548 0.827895 0.601501 0.827895 2.548 0.827895 -5.096 -1.02333 -1.9465 -1.02333 0.601501 -0.827895 3.1495 -1.02333 5.6975 -0.827895 -4.4945 -2.87456 -1.9465 -2.67912 1.203 -2.67912 3.1495 -2.67912 -6.06925 -4.01869 -4.12275 -4.01869 -0.973249 -4.01869 1.57475 -3.82325 4.12275 -4.01869 ] aload pop 60 2 idiv { gsave translate QuasiTilerTilePath ustroke grestore } repeat grestore showpage grestore %%Trailer %%EndDocument %%EndDocument @endspecial 19658 30798 a Ft(Figure)404 b(2.)538 b Fe(The)341 b(P)-28 b(enrose)341 b(Tiling)p Ft(.)1575 35794 y(It)433 b(is)f(\257nally)h(imp)34 b(ortan)-34 b(t)434 b(to)f(remark)f(that,)441 b(as)433 b(a)f(p)34 b(ossible)433 b(further)g(dev)-34 b(elopmen)g(t)434 b(of)f(metho)34 b(ds)434 b(and)f(ideas)1575 37299 y(illustrated)346 b(in)h(this)g(pap)34 b(er,)357 b(one)346 b(could)h(try)f(to)h(mo)34 b(dify)346 b(theorem)g(3)h(b)-34 b(y)346 b(strengthening)i(the)f(condition)g(\(4.2\))1575 38804 y(\(whic)-34 b(h,)486 b(as)469 b(w)-34 b(e)470 b(ha)-34 b(v)g(e)469 b(tried)g(to)g(sho)-34 b(w,)487 b(is)468 b(a)h(a)g(v)-34 b(ery)469 b(mild)f(condition\))j(and)e(at)h (the)f(same)g(time)g(b)-34 b(y)469 b(trying)1575 40310 y(to)546 b(relax)e(the)i(condition)g(that)h(the)f(dual)g(b)34 b(e)545 b(b)34 b(ounded)547 b(degree)e(in)g(order)g(to)h(treat)g(pro) -34 b(ximit)g(y)546 b(random)1575 41815 y(graphs)505 b(\(whic)-34 b(h)505 b(in)f(general)g(ha)-34 b(v)g(e)505 b(a)f(dual)g(not)h(b)34 b(ounded)506 b(degree\))e(lik)-34 b(e)503 b(e.g.)838 b(the)505 b Fq(\257)64 b Ft(-sk)-34 b(eletons)504 b(on)h(p)34 b(oin)-34 b(t)1575 43321 y(P)g(oisson)341 b(pro)34 b(cesses)339 b(of)i(the)g(plane.)517 b(These)341 b(random)g(graphs)g(includes)g(the)f Fs(Gabriel)373 b(gr)-62 b(aph)339 b Ft(and)i(the)g Fs(r)-62 b(elative)1575 44826 y(neighb)g(orho)g(o)g(d)430 b(gr)-62 b(aph)402 b Ft(whic)-34 b(h)405 b(are)f(imp)34 b(ortan)-34 b(t)405 b(for)f(man)-34 b(y)405 b(applications.)1575 47907 y Fs(A)-62 b(cknow)62 b(le)-62 b(dgements)1575 49412 y Ft(This)301 b(w)-34 b(ork)302 b(w)-34 b(as)301 b(supp)34 b(orted)303 b(b)-34 b(y)301 b(MURST)h(\(Ministero)g(dell'Univ)-34 b(ersita')299 b(e)i(della)g(Ricerca)e(Scien)-34 b(ti\257ca)302 b(e)f(T)-101 b(ec-)1575 50918 y(nologica,)389 b(Italy\))d(and)h(b)-34 b(y)387 b(CNPq)g(\(Conselho)g(Nacional)f(de)g(Desen)-34 b(v)g(olvimen)g(to)387 b(Cien)-34 b(t)41868 50630 y(\266)42003 50918 y(i\257co)387 b(e)f(T)-101 b(ecnol\266)-606 b(ogico,)1575 52423 y(Brazil\).)777 b(W)-101 b(e)483 b(w)-34 b(an)g(t)486 b(to)f(thank)g(Itai)f(Benjamini)g(for)g(a)g(v)-67 b(aluable)483 b(suggestion)i(ab)34 b(out)485 b(the)g(form)-34 b(ulation)485 b(of)1575 53928 y(theorem)404 b(3)g(and)h(Eman)-34 b(uele)404 b(Caglioti)h(and)f(Lorenzo)g(Bertini)f(for)h(useful)h(discussions.)1575 57748 y Fa(References)2181 60453 y Ft([1])605 b(Alexander,)316 b(K.)295 b(S.:)484 b Fs(Per)-62 b(c)g(olation)330 b(and)j(minimal)g(sp) -62 b(anning)333 b(for)-62 b(ests)331 b(in)i(in\257nite)g(gr)-62 b(aphs)p Ft(.)293 b(Ann.)i(Probab.)4066 61959 y Fl(23)404 b Ft(\(1995\),)h(no.)f(1,)g(87{104.)2181 64461 y([2])605 b(Babson,)321 b(E.;)334 b(Benjamini,)320 b(I.:)486 b Fs(Cut)337 b(sets)f(and)h(norme)-62 b(d)337 b(c)-62 b(ohomolo)g(gy)333 b(with)k(applic)-62 b(ations)334 b(to)j(p)-62 b(er)g(c)g(olation)p Ft(.)4066 65966 y(Pro)34 b(c.)403 b(Amer.)h(Math.)g(So)34 b(c.)404 b Fl(127)g Ft(\(1999\),)h(no.)f(2,)g(589{597.)2181 68468 y([3])605 b(Benjamini,)368 b(I.;)373 b(Ly)-34 b(ons,)368 b(R.;)374 b(P)-34 b(eres,)367 b(Y.;)374 b(Sc)-34 b(hramm,)368 b(O.:)516 b Fs(Critic)-62 b(al)390 b(p)-62 b(er)g(c)g(olation)390 b(on)i(any)g(nonamenable)4066 69973 y(gr)-62 b(oup)432 b(has)h(no)g(in\257nite)h(clusters)p Ft(.)401 b(Ann.)k(Probab.)f Fl(27)g Ft(\(1999\),)h(no.)f(3,)g(1347{1356.)2181 72475 y([4])605 b(Benjamini,)448 b(I.;)456 b(Ly)-34 b(ons,)448 b(R.;)457 b(P)-34 b(eres,)447 b(Y.;)457 b(Sc)-34 b(hramm,)448 b(O.:)609 b Fs(Gr)-62 b(oup-invariant)463 b(p)-62 b(er)g(c)g(olation) 463 b(on)j(gr)-62 b(aphs)p Ft(.)4066 73980 y(Geom.)404 b(F)-101 b(unct.)405 b(Anal.,)f Fl(9)g Ft(\(1999\),)h(no.)f(1,)g(29{66) 26166 78283 y(10)p eop %%Page: 11 11 11 10 bop 2181 5313 a Ft([5])605 b(Benjamini,)346 b(I.)331 b(and)i(Sc)-34 b(hramm,)346 b(O.:)502 b Fs(Per)-62 b(c)g(olation)365 b(b)-62 b(eyond)365 b Fp(Z)32351 4874 y Fo(d)32891 5313 y Fs(,)380 b(many)366 b(questions)f(and)h(a)h(few)f(answers)p Ft(.)4066 6819 y(Electr.)403 b(Comm.)i(Probab.,)f Fl(1)g Ft(\(1996\),)h(71{82.)2181 9321 y([6])605 b(Benjamini,)724 b(I.)660 b(and)g(Sc)-34 b(hramm,)725 b(O.:)1050 b Fs(R)-62 b(e)g(c)g(ent)668 b(pr)-62 b(o)g(gr)g(ess)667 b(on)i(p)-62 b(er)g(c)g(olation)667 b(b)-62 b(eyond)667 b Fp(Z)45597 8881 y Fo(d)46137 9321 y Ft(,)723 b(Preprin)-34 b(t,)4066 10826 y(h)g(ttp://www.wisdom.w)g(eizmann.ac.il/math)g(users/sc)g (hramm/pap)34 b(ers/p)-34 b(y)g(ond-rep/.)2181 13328 y([7])605 b(Braga,G.)460 b(A.;)489 b(Pro)34 b(cacci,)473 b(A.;)488 b(Sanc)-34 b(his,)476 b(R.:)652 b Fs(A)-31 b(nalyticity)483 b(of)i(the)f(d-dimensional)f(b)-62 b(ond)485 b(p)-62 b(er)g(c)g(olation)4066 14833 y(pr)g(ob)g(ability)430 b(ar)-62 b(ound)433 b(p=1)p Ft(,)403 b(Journal)h(of)h(Statistical)f(Ph) -34 b(ysics,)404 b Fl(107)g Ft(\(2002\),)h(Nos)f(5/6,)g(1267{1282.)2181 17335 y([8])605 b(Grimmett,)384 b(G.)378 b(R.;)386 b(Stacey)-101 b(,)383 b(A.)378 b(M.:)525 b Fs(Critic)-62 b(al)408 b(pr)-62 b(ob)g(abilities)406 b(for)j(site)f(and)i(b)-62 b(ond)409 b(p)-62 b(er)g(c)g(olation)407 b(mo)-62 b(dels)p Ft(.)4066 18840 y(Ann.)405 b(Probab.)f Fl(26)g Ft(\(1998\),)h(no.)f(4,)g (1788{1812.)2181 21342 y([9])605 b(Grimmett,)404 b(G.)g(R.:)539 b Fs(Per)-62 b(c)g(olation)p Ft(,)401 b(I)34 b(I)403 b(edition,)i(Springer)f(V)-101 b(erlag,)403 b(New)h(Y)-101 b(ork,)403 b(1999.)1575 23844 y([10])605 b(Grimmett,)416 b(G.)d(R.;)k(Newman,)f(C.)e(M.:)556 b Fs(Per)-62 b(c)g(olation)440 b(in)i Fr(1)276 b Ft(+)f(1)442 b Fs(dimensions)p Ft(.)411 b(Disorder)i(in)g(ph)-34 b(ysical)4066 25349 y(systems,)404 b(167{190,)h(Oxford)f(Sci.)f(Publ.,)h(Oxford)g(Univ.)g(Press,)f(New)i (Y)-101 b(ork,)403 b(1990.)1575 27851 y([11])605 b(Gr)34 b(\304)-640 b(u)q(n)-34 b(baum,)456 b(B.;)465 b(Shephard,)456 b(G.)445 b(C.)g Fs(Tilings)470 b(and)h(p)-62 b(atterns.)470 b(A)-31 b(n)471 b(intr)-62 b(o)g(duction)p Ft(.)443 b(A)i(Series)f(of)h (Bo)34 b(oks)4066 29357 y(in)405 b(the)f(Mathematical)h(Sciences.)e(W.) g(H.)h(F)-101 b(reeman)404 b(and)h(Compan)-34 b(y)-101 b(,)405 b(New)g(Y)-101 b(ork)403 b(1989.)1575 31858 y([12])605 b(H\304)-606 b(a)q(ggstr\304)g(om,)557 b(O.:)784 b Fs(Markov)545 b(r)-62 b(andom)546 b(\257elds)g(and)g(p)-62 b(er)g(c)g(olation)544 b(on)j(gener)-62 b(al)546 b(gr)-62 b(aphs)p Ft(.)524 b(Adv.)j(in)g(Appl.)4066 33364 y(Probab.)405 b Fl(32)f Ft(\(2000\),)g(no.)h(1,)e(39{66.)1575 35866 y([13])605 b(H\304)-606 b(a)q(ggstr\304)g(om,)564 b(O.;)594 b(Sc)-34 b(honmann,)566 b(R.)532 b(H.;)595 b(Steif,)564 b(J.)532 b(E.:)793 b Fs(The)551 b(Ising)g(mo)-62 b(del)550 b(on)h(dilute)-62 b(d)549 b(gr)-62 b(aphs)549 b(and)4066 37371 y(str)-62 b(ong)433 b(amenability)p Ft(.)401 b(Ann.)k(Probab.)f Fl(28)g Ft(\(2000\),)h(no.)f(3,)g(1111{1137)1575 39873 y([14])605 b(H\304)-606 b(a)q(ggstr\304)g(om,)655 b(O.;)704 b(P)-34 b(eres)604 b(Y.;)704 b(and)606 b(Sc)-34 b(honmann,)657 b(R.)604 b(H.:)940 b Fs(Per)-62 b(c)g(olation)616 b(on)i(tr)-62 b(ansitive)616 b(gr)-62 b(aphs)616 b(as)4066 41378 y(a)559 b(c)-62 b(o)g(alesc)g(ent)556 b(pr)-62 b(o)g(c)g(ess:)806 b(r)-62 b(elentless)556 b(mer)-62 b(ging)559 b(fol)62 b(lowe)-62 b(d)556 b(by)j(simultane)-62 b(ous)555 b(uniqueness)p Ft(,)572 b(P)-34 b(erplexing)4066 42884 y(Probabilit)g(y)417 b(Problems:)562 b(F)-101 b(estsc)-34 b(hrift)418 b(in)e(Honor)h(of)f (Harry)g(Kesten,)j(M.)d(Bramson)g(and)h(R.)f(Durrett,)4066 44389 y(eds)405 b(\(1999\),)f(69{90,)g(Birkhuser,)f(Boston.)1575 46891 y([15])605 b(Jonasson,)451 b(J.:)p Fs(The)466 b(r)-62 b(andom)467 b(cluster)e(mo)-62 b(del)466 b(on)h(a)g(gener)-62 b(al)466 b(gr)-62 b(aph)465 b(and)i(a)g(phase)e(tr)-62 b(ansition)466 b(char)-62 b(ac-)4066 48396 y(terization)432 b(of)h(nonamenability)p Ft(.)401 b(Sto)34 b(c)-34 b(hastic)405 b(Pro)34 b(cess.)403 b(Appl.)h Fl(79)g Ft(\(1999\),)h(no.)f(2,)g (335{354.)1575 50898 y([16])605 b(Jonasson,)420 b(Johan;)j(Steif,)c (Je\256rey)c(E.:)562 b Fs(A)-31 b(menability)442 b(and)i(phase)f(tr)-62 b(ansition)443 b(in)i(the)e(Ising)i(mo)-62 b(del)p Ft(.)414 b(J.)4066 52404 y(Theoret.)405 b(Probab.)p Fl(12)f Ft(\(1999\),)h(no.)f (2,)g(549{559.)1575 54905 y([17])605 b(Ly)-34 b(ons,)541 b(R.:)756 b Fs(The)533 b(Ising)h(mo)-62 b(del)532 b(and)i(p)-62 b(er)g(c)g(olation)531 b(on)j(tr)-62 b(e)g(es)532 b(and)i(tr)-62 b(e)g(e-like)531 b(gr)-62 b(aphs)p Ft(.)510 b(Comm.)k(Math.)4066 56411 y(Ph)-34 b(ys.)405 b Fl(125)f Ft(\(1989\),)g(no.)h(2,)e(337{353.) 1575 58912 y([18])605 b(Ly)-34 b(ons,)384 b(R.:)525 b Fs(R)-62 b(andom)410 b(walks)f(and)g(the)g(gr)-62 b(owth)409 b(of)g(gr)-62 b(oups)p Ft(.)376 b(C.)i(R.)g(Acad.)h(Sci.)e(P)-34 b(aris)379 b(Sr.)f(I)g(Math.)h Fl(320)4066 60418 y Ft(\(1995\),)405 b(no.)f(11,)g(1361{1366.)1575 62920 y([19])605 b(Ly)-34 b(ons,)484 b(R.:)666 b Fs(Pr)-62 b(ob)g(abilistic)489 b(asp)-62 b(e)g(cts)490 b(of)i(in\257nite)g(tr)-62 b(e)g(es)491 b(and)h(some)g(applic)-62 b(ations)p Ft(.)464 b(\(V)-101 b(ersailles,)482 b(1995\),)4066 64425 y(81{94,)404 b(Progr.)g(Probab.,) g(40,)g(Birkhuser,)f(Basel,)f(1996.)1575 66927 y([20])605 b(Ly)-34 b(ons,)323 b(R.:)488 b Fs(Phase)339 b(tr)-62 b(ansitions)339 b(on)i(nonamenable)e(gr)-62 b(aphs.)338 b(Pr)-62 b(ob)g(abilistic)337 b(te)-62 b(chniques)338 b(in)i(e)-62 b(quilibrium)4066 68432 y(and)434 b(none)-62 b(quilibrium)431 b(statistic)-62 b(al)430 b(physics)p Ft(.)400 b(J.)k(Math.)h(Ph)-34 b(ys.)404 b Fl(41)g Ft(\(2000\),)h(no.)f (3,)g(1099{1126.)1575 70934 y([21])605 b(Muc)-34 b(hnik)445 b(R.;)464 b(P)-34 b(ak)444 b(I.:)618 b Fs(Per)-62 b(c)g(olation)469 b(on)h(Grigor)-62 b(chuk)468 b(gr)-62 b(oups)p Ft(.)442 b(Comm.)i(in)g(Algebra)g Fl(29)g Ft(\(2001\),)455 b(no.)4066 72439 y(2)404 b(661{671.)26166 78283 y(11)p eop %%Page: 12 12 12 11 bop 1575 5313 a Ft([22])605 b(Pro)34 b(cacci,)347 b(A.;)357 b(Scopp)34 b(ola,)349 b(B.;)357 b(Braga,)348 b(G.)335 b(A.;)357 b(Sanc)-34 b(his,)349 b(R.:)504 b Fs(Per)-62 b(c)g(olation)368 b(c)-62 b(onne)g(ctivity)367 b(in)j(the)f(highly)4066 6819 y(sup)-62 b(er)g(critic)g(al)430 b(r)-62 b(e)g(gime)p Ft(.)403 b(Preprin)-34 b(t.)1575 9321 y([23])605 b(Sc)-34 b(honmann,)333 b(R.)313 b(H.:)493 b Fs(Per)-62 b(c)g(olation)347 b(in)j Fr(1)87 b Ft(+)g(1)349 b Fs(dimensions)f(at)h(the)f(uniqueness)g(thr)-62 b(eshold)p Ft(.)309 b(P)-34 b(erplexing)4066 10826 y(Problems)505 b(in)h(Probabilit)-34 b(y)-101 b(.)505 b(F)-101 b(estsc)-34 b(hrift)506 b(in)f(honor)h(of)g(Harry)e(Kesten.)h(M.)g(Bramson,)530 b(R.)505 b(Durrett,)4066 12332 y(eds.)404 b(Birkhuser)g(\(1999\),)g (53{67.)1575 14833 y([24])605 b(Sc)-34 b(honmann,)628 b(R.)582 b(H.:)893 b Fs(Stability)594 b(of)j(in\257nite)g(clusters)d (in)j(sup)-62 b(er)g(critic)g(al)593 b(p)-62 b(er)g(c)g(olation)p Ft(.)578 b(Probabilit)-34 b(y)4066 16339 y(Theory)405 b(and)f(Related)h(Fields)f Fl(113)g Ft(\(1999\))h(287{300.)1575 18840 y([25])605 b(Sc)-34 b(honmann,)581 b(R.)544 b(H.:)819 b Fs(Multiplicity)558 b(of)k(phase)e(tr)-62 b(ansitions)561 b(and)h(me)-62 b(an-\257eld)561 b(critic)-62 b(ality)559 b(on)k(highly)4066 20346 y(non-amenable)432 b(gr)-62 b(aphs)p Ft(.)402 b(Comm.)j(Math.)f(Ph)-34 b(ys.)404 b Fl(219)g Ft(\(2001\),)h(no.)f(2,)g(271{322)1575 22848 y([26])605 b(Sc)-34 b(honmann,)515 b(R.)492 b(H.:)712 b Fs(Me)-62 b(an-\257eld)513 b(critic)-62 b(ality)510 b(for)k(p)-62 b(er)g(c)g(olation)511 b(on)j(planar)f(non-amenable)f(gr) -62 b(aphs)p Ft(.)4066 24353 y(Comm.)405 b(Math.)g(Ph)-34 b(ys.)404 b Fl(225)g Ft(\(2002\),)g(no.)h(3,)e(453{463.)26166 78283 y(12)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0311141042927--