This is a multi-part message in MIME format. ---------------0202052011714 Content-Type: text/plain; name="02-53.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-53.keywords" Fermion point process, Glauber dynamics, ergodicity, log-Sobolev inequality ---------------0202052011714 Content-Type: application/postscript; name="glauber.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="glauber.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: glauber.dvi %%CreationDate: Wed Feb 06 11:06:11 2002 %%Pages: 18 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR10 CMTI10 CMBX10 CMR9 CMCSC10 CMSY9 CMMI9 CMSY6 %%+ CMSY10 CMMI10 CMMI8 CMSY8 CMR8 CMEX10 CMMI6 CMTI9 CMR6 MSAM10 MSBM10 %%+ CMBX9 CMTT9 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: C:\TEXMF\MIKTEX\BIN\DVIPS.EXE -P pdf glauber %DVIPSParameters: dpi=8000, compressed %DVIPSSource: TeX output 2002.02.06:1105 %%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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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX9 %!PS-AdobeFont-1.1: CMBX9 1.0 %%CreationDate: 1991 Aug 20 16:36:25 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /zero put dup 49 /one 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 readonly def /FontBBox{-58 -250 1195 750}readonly def /UniqueXX 5000767 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 69 /E 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 be30812eae30521c3c775adf1959595a43d8a9c4e12d9f9ece27a0c9b5f45e38 8e26a39cb0ad4e25ab8f21b25ad1b106ed22af7947ea8ae4dc17b9facbc5e22e 89a91b046ba4cddde69c1997fa7f4cd949ba3ac4b6e001475daf165b9432dc70 6e30aeead8cc342d94b20a4045a331175655d6aa279627efc0fb53a4ce66b68f a45b5765f0c1368c02c39418366880efada9c81e9dfcee2a9443eadb0b488090 c80d067f05e4109c21eeb21f68b041e3724b547601d146a5f1c69df08ef39e26 10622e79b515c550009d2374828748830726c63123e0a7bc1efa9af601e4abe2 2f7d65ee1801e045f9d5fa19458fbac1ae9186e6002efcc5e086e1047a707880 4bdd3d08e797da1b031e3ff27fbeddbf234a55ad85478c7d1a72f3f053ecc267 28227eda462ff302abae66e8f1ec2734f4b7b6fd71c8b32cd6032aa0a31c0229 07555e3edbec4fd6cb7ee430b1e7980d617a992dfcbd391db7827b796963ebd6 042d854f4f470d9341ca6cde05310bf5dbfc3749cdc597b3132b92b99e625baf 4886966c35191f332fa3025ab328c98cb16ca89e2c828a1d8db72bcd1a4843c9 f64eb45039643be12dae15ef8813c25e45ff067019f8819ee0af0ef89bf35ea6 835a40d690ca885afd4c94d354a394571bc86a3af905ce0293e23d97d2ea97d7 15cddad0cbb26ccd25625308b8550f9c24dd75af3e91e4aee8ad547c4e566119 da9bfcea6a1f5efe3b651f9ba57758468dedbf99a80da75de0bde2f827681c7e e02875230ff3811af7ee45d1f57ad8472c3e5efdfb752f4e9ce2f4b83ad30d4a 23f2d1e51fb251f043f1b2f8a2827ad8470943fd266afa75b060a04e037dbbff 730f 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 177 /slashbig put dup 179 /parenleftBig put dup 180 /parenrightBig put dup 181 /parenleftbigg put dup 182 /parenrightbigg put dup 183 /bracketleftbigg put dup 184 /bracketrightbigg put dup 195 /parenleftBigg put dup 33 /parenrightBigg put dup 40 /braceleftBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 56 /bracelefttp put dup 57 /bracerighttp put dup 58 /braceleftbt put dup 59 /bracerightbt put dup 60 /braceleftmid put dup 61 /bracerightmid put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 66 /parenleftex put dup 67 /parenrightex put dup 80 /summationtext put dup 81 /producttext put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 98 /hatwide put dup 101 /tildewide put dup 110 /braceleftBig put dup 111 /bracerightBig 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 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902ffda70f4594e3e6d20176e3f2f2bfe70828a97749020d480af0c3db5b5cb8 bebad5b02e303b4f0d6c39c213ca7ca9e95bac159bf57ea43d78f53c721b35e3 065c386368a60a73123bb64044f2224caa3a9f7b97d34cfc1c458d020cb16d9e 6aa628a7a2e9558332a49602c6b801394b41b05221a410db056356bbc256c284 b7edf71e60908ef03a2ffeaee89bf1c6ff521abc69c3bc3bf0768fe091b72ce9 29e95abf3cae82cb55b0d2453fdf043353341058c88e8797559cfb99f149233a 7188f49c8ec97fc2296d5c9144efeb709bb50236b151b9e8c42d925bca849b9f 437a777a49fef471c8456302e48885feaf007792475fdc6f8229ee1ce3f3ed50 408e958069a77f5f6da9fd13954619f97e6bfde6bbf951a8656042b064065874 b024449211e83343650d71a16da06ea89bf24ef15b321e01fe69a8e8cd95fdac 39f2fe4343544f92c1f5ff15fcf3371cf98a0b35bd936b2522a63bc3ea0a3a74 db0be44378aadbda1d6b3cb651fe4e0a12234452c9a554db1f3e8f5e7384b1ef 43f12c96de9a42a06a62d5efd92b56ba70422fa950b2fe50300b2d3712e116e5 0b40a3e2cb6ef0361b23d26bb2f7c98e34856772ce00e886ad25fc8fb1b53ff6 5e2fd39da0d85ef69d8290c639e626c8a184405f05afb48d66632b6292d85931 4259810d41baf7478010b1626ac5e8e54f2e04892018f9c04f31241d71988a86 ff1b5b0b1c1a69fb66c7c89ad3d81a59218bf6c411164e331aae4c5629022a5d e615c9e6f2853088bc8e34b3e0ffc82749ace2946177d1baa70affa21f57bf73 d080141ab6d82b1ae2c315e4d65bcf0fc0b4073556ea8fce8061da286c196e0d da1bb9bb37703e6a14739a67b8177ed254e0689e3fcfcee2de11b35e174a719b 0c42be8aab2839262f7e254e9d8b428fb1dc0650591c3a4f02379b58bbefefde 0bb4e9209d9bc73858d70190529b8f1e2e3006e812ff517049a7bbac421fef6f c7ecd0f0fc16746fbe9774a0e3dc40e1e80a4c8330f61ac142ab0d0c5daff15e 9f579ac0b168b7b7da00f67d5ec783dd51d3a5f257e48f5c09e0d2a6d978e845 812211c066dc2286e2701c705a8fc14516bbb5f1df65e718e67d4be77822e2c0 ef4df86db31b5884e8497dbe548c5df81f0661aa9030a6d7be9521acfa22a99e a2b97da54725d9e9cbb2b07e9dd13fe76d934f5a1fb13e5ab8bf9b5c7a4ee45d fe1517afd4b842420ed0651fd41ae503b7ad6383488c47da9d571692ffd75830 67642307761358d04f7328b79c3a69cd1209f0b8811a59eda9dcb086e742471d 7398c615d3d6669aacbc1f7007bb44ed47e52642b48e0d68485da35343e1b942 d60211cd8290c3176cce0eb60bcc4685320bbbc83be3b3bcdc7f930b7a755100 6821799c41645f681803284a1eaece56d615d9d8639aed00dc4d4747e0f6c347 30842263e2ca63f73a686ab19cc8750670bd4907b26cb6c69e7d704171dfb36b e87f65808af8cd0fa20faad8730c434e18d5ed2f0b6bd859faa84525c6bbdf88 236c8456c2e1da7c81064893acdffa433caeacba4192aacfde4cb3d6b1c26bb9 a821f89adde9c07b423a9e12c55d789a274cf0dc793b0eff3b2068b2913f07d0 086a7662a134f84adefed07d8a2943841cc6b1a7ba9a80cdc16e26a7fc385a0f 6ba5cbb430a53ddd9d20a423f91e9820c7c76b41181d2716bddca5ff04278184 5115b8857b21424111302b1ac11baa06e5340651714569c2bd16526412db1c7b 3d28254499be8a258e30b5ea706b375c6fdd5cb983729e9dfcc764d97fd6a0ed 654a942475b02c19efbd758df4bff9bd9893da3831927b3dde1acf9b430ad593 11b64db2d48512b510dc192eb1b5f3363f2d00b445f57bbb9cf3952f891445d9 9d3c073272cfeb914ad6fb4e61f3982182a00e9489eee3a609b114f198614567 3b9030a9 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 164 /square put dup 175 /squareminus put dup 177 /forcesbar put dup 179 /dblarrowheadright put dup 180 /dblarrowheadleft put dup 181 /dblarrowleft put dup 182 /dblarrowright put dup 183 /dblarrowup put dup 184 /dblarrowdwn put dup 195 /squiggleright 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 66a602a9f142a563fdae3c8d40de6a8aa65e60dc20706421c85ac0b17b190260 5bdbd60a5c254c7f3153a14d54b17bf35160ab618831432f8c9e0e8ff11dc197 8e22e9a1276aba0aebefc72b7faa87486e5316bf51ede92fd5f300b3d25e81ea a152b55ac82836b3a3be7e36fb0183aa7dda5c0b7e74ca75c84cbf42a0bed633 55ca68faccfa0d6f24d7918825724c2c56414de0653d0079c5714b32b6a4d391 4d3a44ad3829fb0b831a430b7f05828c613d940722b61eda9d0f7dc13d2c3438 11d2b0dd08bea58660256701a63664de4c315efb2f982f658d914f175be54bed ff5506345158f17143d427c065f0dfade9636d1fcc2a0aab330e2ef0021eb4c0 0f772a3a6d53bc18bb046abf646d51e4d6987d7a8fcb5ec56918cd6f42e9dc76 503dff3e002582979a0dafd5d9c5f79988bd6ef0e38c2225fd02285c0768d6c1 9637c882a5c3c21b1adb539f6e78b9eef1724859bb9ef8521bfdf87e8245fe23 748aeafd4523c7ce8799ab4517b6b3bea3f32dad76d700e70cfc9ae7ca67e1c7 53ba237768e3445717fa8b8f99d0511e52858c2428eac33915649d66559f5be5 1fde747e7eb87a1d8a4b87ec91b187c00836fd315748e914a63b777157c92738 9f29de59d007e0546b587f985e52f724355799d7c6183ad35353e7b6359c72dd deacc347a0d949c75111064e75de3f277496c091bca0b2fd9af20d8a50f6cec7 124cf952add26f3d904983e4e24f60f5f3ad2bda5baebdefb01f479f9fbb5d89 421bfb83399a9e1be9f7097cd6378d942822c2c33607e9c534356710d2784ffd 3dbd4e4aff22edd50701f8414e58f406ccf036985462633204484adb7c991ed8 2fe410be30b995532684eb07806a6be287fd1604db68628378272d7300279db0 6a87002c4b008d19dbdb707d1b5496dd5562c745c8fbe07a6f8b6d4eb3748a16 aa641174e6509951b59f522a2c797a624bb1305f7aed5f40fdaa4f68a4b8ad5c edad2b08aef3e240900c2ab3b8cc8dfa65fca3838cf2d7ad4d1ba6be1df16ba5 3ae0f248eade906c792f25787ae5a5a6ab49318dd640af593d205f06d2ddbcf3 f081f51fb5ff09d4ea07bc6828f4b940942327b902fef727d1eea6b2fd6b2271 6f9ee2131e9f98ddcdce58da2e6903ecc11ef2a92fb0f4ecdebcad001b287555 7216d0011761189aff6cb6085f8a305a0c7bc419276398705b341edebefa9d7f aaa6d7c4d62934a69421c01e2d7bc24211159e37233990d2a6bcbdd55f7a3174 af4733ac33f655b833ae0d9409f47af972351aa36f1946d7be2330bb0dfee186 fcdc3f16902403aa4d1d7ee4d1cd4dc3bda3902d4d5cd4fd9a59ae58782f5adb 2278358be54f97ff2827998c27910ac0143a7a5ae678bb57bc7df2a127429d1d 680f95b5558470b015d4ddb192a81f080ce51f5a7cb9e7ed92d2460b2a23a358 723706bca5950b9c650938d8f44db0e426fe09fc6fa77342383421eb4c174c43 d3072f9bdce12e324987c543c32b6b38be69ecd063d10abb1ee6d07751c323c3 d5528ecf7216f311239e9b63856d34fa6cace3777e457b5ccaef70377296845d 934224eec98a9b296aa0724052068892bd104d531d81157629eec652050f62cf eb545af43cd5d44775dde2b2b3a4f053711c56e157b47595985b28082730dc9b 0a51ce012862fbedd9211eab0a0ce2fa9b3ff8ec362f30403ace8f536649393e d8c3cf8ea8797a4d1fbc7a2c1911366c6a788de61ec03e49fabbbaf9ec92e188 c75c344b2c913d9d6bd40bc96252e89a33e0a675a18fb22a5f1c3bdb895987cb 222bb5bd440c5500b87554bca3e90e237d0819ad3871bab3cf51f3c53af1d7db 387446b38fe59777ce87b2d31fad0eed515e64008c83d834f05d9d036c2f20ad 505a417b125d8bd5719969eb02e3064bd81e649f587aca51688c4885eebf22f3 6c648d2a4100d429b7b7d8567364cc952de9c7a2540fb339e0598555cad6f4ac 26a06f9eae0890e29f6ca79f69f285977f4840382291ad0aa4d12622afb3f751 959ecf49012aad4a113f9f27749ec46c3c25882400179c85c30d03e4cf7ed3fe 3365408b71fd043b5bf809c4f03aeb10b1e60014c8a92c1a3df10f3f562489aa c7367a88f5897a364f21fde252ae30a56941af8af9473f6124fb6b9158307ee2 7283e5e20e075f14dc8c03883bb93df8173361ba166c3ab3377cdea1c17d3fd3 eddc551e2ad86ea50a218764b9195a3971ffee09b10e490c5cc5fd6114c9b650 de0e9573f5235cd2a55b36cbd71af47e6a1468a6fd0d045f72fe71c5 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 162 /Delta put dup 164 /Lambda put dup 175 /fi put dup 177 /ffi put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 195 /suppress put dup 40 /parenleft put dup 41 /parenright put dup 49 /one put dup 50 /two put dup 51 /three put dup 58 /colon put dup 61 /equal 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 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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 162 /Delta put dup 164 /Lambda put dup 175 /fi put dup 177 /ffi put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 195 /suppress put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 55 /seven put dup 57 /nine put dup 65 /A put dup 66 /B put dup 68 /D put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueXX 5000828 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3ffd20326947b1065649b533eb5e9e3 a88a87fe987918a687950b26f647d6e08bf14d983ed1d4a8ca1e2da8d985f944 2eb916f5b6ed0697ac7c33e1e36a3bf460d34ce45f1631871097cb04f18e3889 4cf4ac1538eb19481311d24fe3be7beaa4a3730e8b4831fe59d6d9ce2e46116b 629c7ba2f9ce3ecba2f43bc162a5a077ca1b2882a42afdcec3f4b75b5d63e0bc 8e5dc95257766d8ea467ad9cbaf47be60f797580cced6884b3a68f70c91f4fdf 80fd00ed9139e7f480dc3a76af72ad9b434187730bdfaefe4cbfe5c7edcaaf24 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI9 %!PS-AdobeFont-1.1: CMTI9 1.0 %%CreationDate: 1991 Aug 18 21:08:07 % 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 (CMTI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI9 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 162 /Delta put dup 164 /Lambda put dup 175 /fi put dup 177 /ffi put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 195 /suppress 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 52 /four put dup 55 /seven put dup 57 /nine put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 121 /y put readonly def /FontBBox{-35 -250 1148 750}readonly def /UniqueXX 5000827 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd800bcb325468c79ffacb9cd3036f2ed7bb65d cc883aff8d6350d6e8af1a9c1d2d7fcddc4c554c7e350f2420586f43e2124611 487cdb176d453854b51cb8f9100bf241899bbef87d57f1135e589813995c4d86 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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 162 /Delta put dup 164 /Lambda put dup 175 /beta put dup 177 /delta put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 187 /xi put dup 195 /psi put dup 59 /comma put dup 75 /K put dup 78 /N put dup 83 /S put dup 99 /c put dup 100 /d put dup 102 /f put dup 105 /i put dup 107 /k put dup 108 /l put dup 110 /n put dup 117 /u put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueXX 5087381 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 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7ced5cd346f4ceaaf645178638e68fba42c5dc73a87753a63a936de85c35febc 607b2d451200797ba5e60a3e2f8137605bff821145861dc8b95afc2b21694e21 d236172af2360613701291118829dd90770d0e9a9e9ff21f703ac6863cd77e5b d3ee6b63877c21ad4d32b7c357e184db00a73d2700334d41578483adc21115fe 5ac4ebbf0c3ef9a48b2fc01ef4b507ed18ca242f740a5afaec47bb885b96032e 51860b6f1c6f34238f5bc2e11390c7d579b3eb8dade909c690de9f53022af8de edd9bb7881d36a502abef20e65b5c66e23df17ad897fa923c286690624464f95 301aadc0ca31b50406c5fbcbd991b48e877aabea302c1bfeffe3ba324b758736 66ade209de304d9db63cf3b04b52a103256e659e014efed2d76eb5ce8b0b61ee 996e6ab2aa5930753241734f29ee0170fb4df845befb411d52542ebb97862a80 3e3ce651b78dad03a38bd029f24750cc9055b8907efbc483f92c19bbfb225f83 96706486f65065e36e3070069a62563abb4dbfd2e060a8381cef98cd2cd71dcf 885e1f28e660cf044e0a49a4914af4e070b62a7cacdf1ca8dab9022b8c0d604b 356056764880b66849b27517e6624e83c8995ebef6aeb962d1462041ea70938b da2a921da71606f987992cc305ba9d12543ae4c984b58b9349e5956dd57b208f 529ae276eaeb646af710cfd6dbd5ae67420ec0804f2318c5d9d7ab5e03044860 ec31c0b5a913ae08413ce95c4e2490ddd961e757c313617448e15571bed73fa9 df098aa3b14e4939d4f6d22621320e3c6791fe5ca3064dfcb1aa5f8d 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 162 /Delta put dup 164 /Lambda put dup 175 /fi put dup 177 /ffi put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 187 /cedilla put dup 195 /suppress put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 101 /e put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 114 /r put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueXX 5000791 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b094541f345ec367ae85ca9bd 24568a01d3b9f8095f7420e6c423c414b3dcce6da48dd1c89a56d078e0d0e2f2 62a13640a06d17e44ee3866c3471fb58fedf5a3b77294517651c16bdd7267d39 a54e7171752dbde63ac19bb4b3021ce95eb5fe67390b09ae4d9ed4d704a67443 f55dce17acd996c1f5e023c9e5a18cbeecc3097f23763acb86cdd7cd13381ae7 4e48495ec7fa520539d87f8a8dcb3c826275469b6800876a457e7d1e5be867c7 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42be73be7787011acdde9e46e14e0086110a2bdf91a69e6133fdcae5b4b1fcc5 e3863f6783fc942d6ff9653bc14f83d6d47bf8c38cbe89c25db09469fb705fc7 866613beed3ccceb01265f6715ffaa5ef526e59f9ceec921155054a139ed11f6 b305bbd8c08959fe67 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 162 /periodcentered put dup 164 /asteriskmath put dup 167 /plusminus put dup 175 /circledot put dup 177 /openbullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 187 /similar put dup 189 /propersubset put dup 195 /arrowleft put dup 33 /arrowright put dup 34 /arrowup put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 59 /emptyset put dup 70 /F put dup 82 /R put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 110 /backslash 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 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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 162 /Delta put dup 164 /Lambda put dup 167 /Sigma put dup 174 /alpha put dup 175 /beta put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 191 /tau put dup 195 /psi put dup 58 /period put dup 59 /comma put dup 61 /slash put dup 65 /A put dup 69 /E put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 81 /Q put dup 83 /S put dup 88 /X put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 102 /f put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 110 /n put dup 113 /q put dup 115 /s put dup 116 /t put dup 117 /u put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueXX 5087383 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a196cf14c95a4e02f9cd2da8c db2546c6df52e524745992e18d9ff87aa25e4e1800bbe4ebb357c6ef55ed6d03 6d3a00c1ee8073266c21d2f0ac85d656abf61d7e5a4fa87da8ec3b5329e434d0 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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 162 /Delta put dup 164 /Lambda put dup 167 /Sigma put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 195 /psi put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 71 /G put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 88 /X put dup 89 /Y put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 113 /q put dup 115 /s put dup 116 /t put dup 117 /u put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueXX 5087385 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930da4f8d58abefd45db119b10eb409dd89792 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /minus put dup 162 /periodcentered put dup 164 /asteriskmath put dup 167 /plusminus put dup 169 /circleplus put dup 173 /circlemultiply put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 178 /bullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 185 /precedesequal put dup 186 /followsequal put dup 187 /similar put dup 188 /approxequal put dup 189 /propersubset put dup 190 /propersuperset put dup 191 /lessmuch put dup 195 /arrowleft put dup 33 /arrowright put dup 34 /arrowup put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 54 /negationslash put dup 55 /mapsto put dup 59 /emptyset put dup 65 /A put dup 66 /B put dup 69 /E put dup 70 /F put dup 76 /L 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 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put dup 112 /radical put dup 114 /nabla put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueXX 5000820 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMCSC10 %!PS-AdobeFont-1.1: CMCSC10 1.0 %%CreationDate: 1991 Aug 18 17:46:49 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMCSC10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMCSC10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /Gamma put dup 162 /Delta put dup 164 /Lambda put dup 167 /Sigma put dup 169 /Phi put dup 173 /Omega put dup 174 /arrowup put dup 175 /arrowdown put dup 176 /quotesingle put dup 177 /exclamdown put dup 178 /questiondown put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 195 /suppress put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 52 /four put dup 53 /five put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 68 /D put dup 70 /F put dup 73 /I put dup 74 /J put dup 75 /K put dup 77 /M 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 89 /Y put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{14 -250 1077 750}readonly def /UniqueXX 5000772 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee 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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 164 /Lambda put dup 166 /Pi put dup 167 /Sigma put dup 169 /Phi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 195 /suppress put dup 33 /exclam 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 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 90 /Z 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 readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueXX 5000793 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306457c7e62c1d269f306bd3402e266de fc3b5e7d8a8d2f5bf0fe6ddd40d07391df4fad4a6018dce29a2b8f692b29f202 3a7c0e66de8ed85c14f1f8492167357f51a7e84cc5d92e0fee4d81cf7fbc8de5 2d2e7bb57142033993f9c08c315abade8dbc4a732e84e142d3bee51557910e12 cd8aa37c459a5e6b7f5269f59078aba3be4641a11ac48d0b625c8325b38ec08e 4c9e5e7fed976a5650d99d82114f449b9ca14c4ec957702295a39a93ef93f618 99b8ea06b092c3c1e503e6e436e0a9fa22576c8930ab3dc8c20f5d82b69cddf8 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fd27e3b636b9b66361a229e5fa548c043ecfcba59e00a67ab27fdb8dce2280a8 7a899ad713a2b40b7c0e59212c670a18d5ff468f23a33095315e14a1fcf08e6d 1300b519c9c3249b4cfca61572ac1531e90c5c4a2f296b0f47219d57d3643dcc 629341f79646303c1be336 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 161 /minus put dup 162 /periodcentered put dup 164 /asteriskmath put dup 166 /diamondmath put dup 167 /plusminus put dup 169 /circleplus put dup 173 /circlemultiply put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 178 /bullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 185 /precedesequal put dup 186 /followsequal put dup 187 /similar put dup 188 /approxequal put dup 189 /propersubset put dup 190 /propersuperset put dup 191 /lessmuch put dup 195 /arrowleft put dup 48 /prime put dup 50 /element put dup 54 /negationslash put dup 82 /R put dup 102 /braceleft put dup 103 /braceright put dup 110 /backslash 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 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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 161 /Gamma put dup 162 /Delta put dup 164 /Lambda put dup 166 /Pi put dup 167 /Sigma put dup 169 /Phi put dup 173 /Omega put dup 174 /alpha put dup 175 /beta put dup 176 /gamma put dup 177 /delta put dup 178 /epsilon1 put dup 179 /zeta put dup 180 /eta put dup 181 /theta put dup 182 /iota put dup 183 /kappa put dup 184 /lambda put dup 185 /mu put dup 186 /nu put dup 187 /xi put dup 188 /pi put dup 189 /rho put dup 190 /sigma put dup 191 /tau put dup 195 /psi put dup 59 /comma put dup 88 /X put dup 89 /Y put readonly def /FontBBox{-29 -250 1075 750}readonly def /UniqueXX 5087384 def currentdict end currentfile eexec 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f516a61bae40d482f5ca7e5ccdb0f1240e70138d44b731885f55a0b411e22b3e f6dcddec5f03cc3661f7d2459b6ee15dec2b0106f9adf0de807dc473f829dad7 a21472882706b9995bbeb669908c59b094d00380d5c5d53da7b13c548fae6c39 2126884d55dafa9eb098ae032f336c9bdab5f718d7b325252450cd1b6843f6a9 efdbf400bdf29032ed0bd2158939f79b4d519f8c55980b27883af6217780ebc7 805115042dbbc21049301773 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 161 /minus put dup 162 /periodcentered put dup 164 /asteriskmath put dup 166 /diamondmath put dup 167 /plusminus put dup 169 /circleplus put dup 173 /circlemultiply put dup 174 /circledivide put dup 175 /circledot put dup 176 /circlecopyrt put dup 177 /openbullet put dup 178 /bullet put dup 179 /equivasymptotic put dup 180 /equivalence put dup 181 /reflexsubset put dup 182 /reflexsuperset put dup 183 /lessequal put dup 184 /greaterequal put dup 185 /precedesequal put dup 186 /followsequal put dup 187 /similar put dup 188 /approxequal put dup 189 /propersubset put dup 190 /propersuperset put dup 191 /lessmuch put dup 195 /arrowleft put dup 82 /R put dup 102 /braceleft put dup 103 /braceright 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 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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 161 /Gamma put dup 162 /Delta put dup 164 /Lambda put dup 166 /Pi put dup 167 /Sigma put dup 169 /Phi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 195 /suppress put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 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 65 /A put dup 66 /B put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 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 90 /Z put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 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 196 /dieresis put readonly def /FontBBox{-39 -250 1036 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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 162 /Delta put dup 164 /Lambda put dup 166 /Pi put dup 167 /Sigma put dup 169 /Phi put dup 173 /Omega put dup 174 /ff put dup 175 /fi put dup 176 /fl put dup 177 /ffi put dup 178 /ffl put dup 179 /dotlessi put dup 180 /dotlessj put dup 181 /grave put dup 182 /acute put dup 183 /caron put dup 184 /breve put dup 185 /macron put dup 186 /ring put dup 187 /cedilla put dup 188 /germandbls put dup 189 /ae put dup 190 /oe put dup 191 /oslash put dup 195 /suppress put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 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 73 /I 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 89 /Y put dup 90 /Z put dup 97 /a put dup 101 /e put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 120 /x put dup 196 /dieresis put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueXX 5000768 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306462c563d51ece66fcc9c831843ecae 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont TeXDict begin 39158280 55380996 1000 8000 8000 (glauber.dvi) @start /Fa 134[523 3[523 523 523 523 1[523 523 523 523 2[523 523 523 2[523 1[523 1[523 32[523 17[523 46[{}17 996.264 /CMTT9 rf /Fb 198[589 589 589 589 589 589 589 1[589 589 48[{}9 996.264 /CMBX9 rf /Fc 186[808 69[{}1 1212.12 /MSBM10 rf /Fd 154[523 20[993 80[{}2 885.568 /CMEX10 rf /Fe 252[943 3[{}1 1212.12 /MSAM10 rf /Ff 194[621 2[234 6[406 406 406 7[320 320 36[552 1[664 1[{}9 664.176 /CMR6 rf /Fg 133[496 589 562 805 558 651 403 496 511 558 620 620 681 991 310 558 372 372 620 558 372 558 620 558 558 620 12[867 681 2[822 2[1087 760 2[467 901 938 791 1[915 1[853 901 7[620 1[620 1[620 620 620 620 620 2[372 434 372 2[496 496 372 24[1069 1[681 12[{}53 1212.12 /CMTI10 rf /Fh 134[497 1[680 470 549 340 418 431 1[523 523 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926 875 859 909 3[943 1[337 337 606 606 606 606 606 606 606 606 606 606 1[337 404 337 943 1[471 471 337 4[606 337 18[1010 673 673 707 5[909 1[842 1[1010 758{}83 1212.12 /CMR10 rf /Fr 145[424 6[424 424 19[705 27[0 3[553 1[243 47[640{}8 664.176 /CMSY6 rf /Fs 166[594 846 28[284 59[{}3 996.264 /CMMI9 rf /Ft 152[512 512 19[871 80[284 1[{}4 996.264 /CMSY9 rf /Fu 134[611 2[611 611 589 456 600 1[556 633 611 744 511 633 1[301 611 639 534 556 622 589 578 611 7[811 3[811 783 609 3[841 1[985 1[840 565 404 4[826 2[811 7[551 551 551 1[551 551 1[551 551 551 1[318 376 318 44[{}43 996.264 /CMCSC10 rf /Fv 128[512 3[512 455 540 540 740 540 569 398 404 401 540 569 512 569 853 284 540 1[284 569 512 313 455 569 455 569 512 3[284 1[284 626 768 1[1052 768 768 740 569 754 1[697 796 768 938 640 796 526 370 768 803 668 697 782 1[725 768 6[284 512 512 512 512 512 512 512 512 512 512 512 284 341 284 2[398 398 284 19[512 5[569 569 12[{}73 996.264 /CMR9 rf /Fw 135[736 3[542 550 574 1[774 697 774 1162 387 2[387 774 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Fo(2)337 b(f)p Fq(0)p Fn(;)202 b Fq(1)p Fo(g)p Fg(,)433 b Fn(\273)393 b Fo(2)337 b Fn(X)95 b Fg(.)1328 28857 y(\(b\))433 b(F)-93 b(or)433 b Fq(\244)337 b Fo(\275\275)f(R)434 b Fg(and)f(a)g(b)-62 b(ounde)g(d)432 b(me)-62 b(asur)g(able)431 b(function)h Fn(f)130 b Fg(,)434 b(de\257ne)0 31652 y Fq(\(3.4\))12297 b Fn(E)15683 31840 y Fk(\244)16391 31652 y Fn(f)130 b Fq(\()p Fn(\273)56 b Fq(\))338 b(:=)20596 30002 y Fj(Z)21270 32751 y Fm(X)22044 32900 y Ff(\244)22908 31652 y Fn(\260)23536 31840 y Fk(\244)24244 31652 y Fq(\()p Fn(\263)25245 31840 y Fk(\244)25953 31652 y Fq(;)202 b Fn(\273)27022 31840 y Fk(\244)27675 31587 y Fi(c)28157 31652 y Fq(\))p Fn(f)130 b Fq(\()p Fn(\263)30352 31840 y Fk(\244)31062 31652 y Fn(\273)31592 31840 y Fk(\244)32245 31587 y Fi(c)32727 31652 y Fq(\))202 b Fn(d\263)34561 31840 y Fk(\244)35269 31652 y Fn(;)0 34584 y Fg(wher)-62 b(e)612 b Fn(d\263)4763 34772 y Fk(\244)6083 34584 y Fg(is)g(the)f(c)-62 b(ounting)611 b(me)-62 b(asur)g(e.)1092 b(Then,)656 b Fn(E)25202 34772 y Fk(\244)25910 34584 y Fn(f)743 b Fg(is)612 b(a)g(version)f(of)g(the)h(c)-62 b(onditional)609 b(exp)-62 b(e)g(ctation)0 36023 y Fn(E)70 b Fq(\()p Fn(f)130 b Fo(jF)3367 36211 y Fk(\244)4020 35959 y Fi(c)4502 36023 y Fq(\))434 b Fg(and)g Fo(f)p Fn(E)9263 36211 y Fk(\244)9970 36023 y Fo(g)10576 36211 y Fk(\244)p Fl(\275\275R)13985 36023 y Fg(is)f(a)g(system)f(of)h(pr)-62 b(ob)g(ability)430 b(kernel)i(\(sp)-62 b(e)g(ci\257c)g(ation\))431 b Fq([6)o(])p Fg(.)1328 38312 y Fq(The)361 b(conditional)f(probabilit) -34 b(y)360 b(also)g(has)g(b)34 b(een)360 b(giv)-34 b(en)360 b(in)g(a)g(little)f(di\256eren)-34 b(t)360 b(form)g(in)g([22].)523 b(F)-101 b(or)360 b(the)g(pro)34 b(of)0 39751 y(of)404 b(the)h(prop)34 b(osition,)404 b(w)-34 b(e)405 b(prepare)f(t)-34 b(w)g(o)405 b(lemmas.)0 42040 y Fw(Lemma)488 b(3.2.)565 b Fg(L)-62 b(et)452 b Fn(K)540 b Fg(b)-62 b(e)452 b(a)h(symmetric)e(op) -62 b(er)g(ator)450 b(on)j Fn(`)26927 41600 y Fk(2)27453 42040 y Fq(\()p Fo(R)p Fq(\))h Fg(satisfying)c(the)i(c)-62 b(ondition)451 b(\(1.3\).)612 b(F)-93 b(or)453 b(any)0 43757 y Fn(\273)434 b Fo(2)378 b Fn(X)551 b Fg(and)456 b Fq(\242)378 b Fo(\275)g(R)p Fg(,)462 b Fn(K)11768 43118 y Fk(\()p Fm(\273)35 b Fk(\))11681 44138 y(\242)13460 43757 y Fg(is)455 b(invertible)f(in)j(the)e(pr)-62 b(oje)g(ction)454 b(sp)-62 b(ac)g(e)454 b Fn(P)33269 43945 y Fk(\242)34109 43757 y Fn(`)34614 43317 y Fk(2)35140 43757 y Fq(\()p Fo(R)p Fq(\))378 b Fo(\264)g Fn(`)39313 43317 y Fk(2)39839 43757 y Fq(\(\242\))457 b Fg(and)f(the)f(inverse)0 45552 y Fq(\()p Fn(K)1587 44913 y Fk(\()p Fm(\273)35 b Fk(\))1500 45934 y(\242)2823 45552 y Fq(\))3294 45112 y Fl(\241)p Fk(1)4986 45552 y Fg(satis\257es)431 b Fo(k)p Fq(\()p Fn(K)11732 44913 y Fk(\()p Fm(\273)35 b Fk(\))11645 45934 y(\242)12968 45552 y Fq(\))13439 45112 y Fl(\241)p Fk(1)14697 45552 y Fo(k)337 b(\267)f Fq(1)p Fn(=\270)p Fg(,)434 b(uniformly)e(in)i Fn(\273)393 b Fo(2)337 b Fn(X)529 b Fg(and)433 b Fq(\242)337 b Fo(\275)f(R)p Fg(.)0 47905 y(Pr)-62 b(o)g(of.)552 b Fq(W)-101 b(e)458 b(consider)f(the)h(case)g (\242)426 b(=)f Fo(R)p Fq(.)700 b(Giv)-34 b(en)458 b(a)f(v)-34 b(ector)458 b Fn(f)557 b Fo(2)426 b Fn(`)31503 47465 y Fk(2)32029 47905 y Fq(\()p Fo(R)p Fq(\),)472 b(w)-34 b(e)458 b(let)f Fn(f)39042 48093 y Fk(\244)40177 47905 y Fq(:=)425 b Fn(P)42660 48093 y Fk(\244)43368 47905 y Fn(f)589 b Fq(and)458 b Fn(f)47553 48093 y Fk(\244)48206 47840 y Fi(c)49114 47905 y Fq(:=)0 49344 y Fn(P)778 49532 y Fk(\244)1431 49279 y Fi(c)1913 49344 y Fn(f)130 b Fq(.)539 b(Denote)404 b(b)-34 b(y)405 b Fo(h\242)p Fn(;)202 b Fo(\242i)404 b Fq(the)g(usual)h(inner)f(pro)34 b(duct)405 b(in)f Fn(`)26741 48904 y Fk(2)27267 49344 y Fq(\()p Fo(R)p Fq(\).)539 b(W)-101 b(e)404 b(then)h(get)f(\(\244)337 b(:=)f(\244\()p Fn(\273)56 b Fq(\)\))8059 51654 y Fo(h)p Fn(f)63 b(;)202 b(K)10841 51153 y Fk(\()p Fm(\273)35 b Fk(\))12077 51654 y Fn(f)130 b Fo(i)1108 b Fq(=)f Fo(h)p Fn(f)17493 51842 y Fk(\244)18471 51654 y Fq(+)269 b Fn(f)20276 51842 y Fk(\244)20929 51589 y Fi(c)21411 51654 y Fn(;)202 b Fq(\()p Fn(P)23199 51842 y Fk(\244)23907 51654 y Fn(K)357 b Fq(+)269 b Fn(P)27283 51842 y Fk(\244)27936 51589 y Fi(c)28418 51654 y Fq(\()p Fn(I)364 b Fo(\241)269 b Fn(K)87 b Fq(\)\)\()p Fn(f)34120 51842 y Fk(\244)35099 51654 y Fq(+)269 b Fn(f)36904 51842 y Fk(\244)37557 51589 y Fi(c)38039 51654 y Fq(\))p Fo(i)14379 54083 y Fq(=)1107 b Fo(h)p Fn(f)17493 54271 y Fk(\244)18202 54083 y Fn(;)202 b(K)87 b(f)20450 54271 y Fk(\244)21158 54083 y Fo(i)269 b Fq(+)h Fo(h)p Fn(f)24175 54271 y Fk(\244)24828 54018 y Fi(c)25310 54083 y Fn(;)202 b Fq(\()p Fn(I)364 b Fo(\241)269 b Fn(K)87 b Fq(\))p Fn(f)30609 54271 y Fk(\244)31262 54018 y Fi(c)31745 54083 y Fo(i)270 b Fq(+)f(2)p Fn(i)202 b Fq(Im)o Fo(h)p Fn(f)37435 54271 y Fk(\244)38144 54083 y Fn(;)g(K)87 b(f)40392 54271 y Fk(\244)41045 54018 y Fi(c)41527 54083 y Fo(i)p Fn(:)-42335 b Fq(\(3.5\))0 56253 y(By)403 b(using)i(\(1.3\))g(w)-34 b(e)404 b(see)g(that)0 58564 y(\(3.6\))11693 b Fo(jh)p Fn(f)63 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b(The)396 b(case)f(of)h(general)f(\242)337 b Fo(\275)f(R)396 b Fq(also)f(follo)-34 b(ws)0 66666 y(from)404 b(the)h(observ)-67 b(ation)404 b(\(3.7\).)35229 b Fe(\244)0 69010 y Fw(Lemma)386 b(3.3.)499 b Fg(L)-62 b(et)371 b Fn(A)h Fg(b)-62 b(e)371 b(a)g(b)-62 b(ounde)g(d)370 b(op)-62 b(er)g(ator)369 b(on)j Fn(`)24699 68570 y Fk(2)25224 69010 y Fq(\()p Fo(R)p Fq(\))h Fg(with)e(b)-62 b(ounde)g(d)370 b(inverse.)535 b(Supp)-62 b(ose)370 b(that)g Fn(A)47211 68500 y Fl(\241)p Fk(1)47211 69391 y(\242)48840 69010 y Fg(ex-)0 70583 y(ists)343 b(in)i(the)f(pr)-62 b(oje)g(ction)342 b(sp)-62 b(ac)g(e)343 b Fn(`)14542 70143 y Fk(2)15067 70583 y Fq(\(\242\))j Fg(for)e(any)g(subset)f Fq(\242)i Fg(and)f(uniformly)f(b) -62 b(ounde)g(d,)360 b(i.e.,)h Fq(sup)43159 70874 y Fk(\242)p Fl(\275R)45735 70583 y Fo(k)p Fn(A)47250 70073 y Fl(\241)p Fk(1)47250 70964 y(\242)48508 70583 y Fo(k)337 b Fn(<)0 72257 y Fo(1)p Fg(.)597 b(Set)446 b(the)g(op)-62 b(er)g(ator)11365 71950 y Fj(e)11079 72257 y Fn(A)11988 72445 y Fk(\242)13188 72257 y Fq(:=)360 b Fn(A)15737 72445 y Fk(\242)16855 72257 y Fo(\251)279 b Fn(A)18986 72445 y Fk(\242)19770 72192 y Fi(c)20699 72257 y Fg(on)446 b Fn(`)22951 71817 y Fk(2)23477 72257 y Fq(\()p Fo(R)p Fq(\))362 b(=)e Fn(`)27616 71817 y Fk(2)28141 72257 y Fq(\(\242\))280 b Fo(\251)f Fn(`)32100 71817 y Fk(2)32626 72257 y Fq(\(\242)34107 71817 y Fm(c)34570 72257 y Fq(\))p Fg(.)597 b(Then,)449 b Fq(\()40315 71950 y Fj(e)40028 72257 y Fn(A)40937 72445 y Fk(\242)41777 72257 y Fq(\))42248 71817 y Fl(\241)p Fk(1)43953 72257 y Fg(c)-62 b(onver)g(ges)444 b(to)0 73696 y Fn(A)909 73256 y Fl(\241)p Fk(1)2600 73696 y Fg(in)434 b(the)f(str)-62 b(ong)432 b(op)-62 b(er)g(ator)432 b(top)-62 b(olo)g(gy)430 b(as)j Fq(\242)h Fg(tends)e(to)h Fo(R)p Fg(.)24962 75260 y Fk(5)p eop %%Page: 6 6 6 5 bop 0 393 a Fg(Pr)-62 b(o)g(of.)552 b Fq(Observ)-34 b(e)404 b(that)10810 2669 y(\()11568 2362 y Fj(e)11281 2669 y Fn(A)12190 2857 y Fk(\242)13030 2669 y Fq(\))13501 2168 y Fl(\241)p Fk(1)15029 2669 y 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49 v 355 w Fn(S)71 b Fq(.)926 b(Giv)-34 b(en)534 b(a)0 40425 y(function)354 b Fn(f)467 b Fo(2)337 b Fn(`)7406 39985 y Fl(1)8403 40425 y Fq(\()p Fn(S)70 b Fq(\),)363 b(w)-34 b(e)353 b(extend)f(it)h(to)g(a)f (function)24950 40105 y Fj(e)24723 40425 y Fn(f)468 b Fo(2)336 b Fn(`)27433 39985 y Fl(1)28430 40425 y Fq(\()p Fn(S)236 b Fo([)p 30854 39451 V 166 w Fn(S)71 b Fq(\))352 b(b)-34 b(y)34351 40105 y Fj(e)34123 40425 y Fn(f)131 b Fq(\()p Fn(x)p Fq(\))337 b(=)g Fn(f)130 b Fq(\()p Fn(x)p Fq(\))354 b(and)43343 40105 y Fj(e)43116 40425 y Fn(f)131 b Fq(\()p 44311 39758 693 49 v Fn(x)p Fq(\))337 b(=)f Fo(\241)p Fn(f)130 b Fq(\()p Fn(x)p Fq(\))0 41910 y(for)404 b(eac)-34 b(h)404 b Fn(x)337 b Fo(2)g Fn(S)70 b Fq(.)538 b(W)-101 b(e)404 b(consider)g(the)g(an)-34 b(ti-symmetric)405 b(subspace)g Fn(`)31274 41470 y Fl(1)31274 42209 y Fm(a)32270 41910 y Fq(\()p Fn(S)340 b Fo([)p 34901 40936 814 49 v 269 w Fn(S)70 b Fq(\))405 b(of)f Fn(`)38475 41470 y Fl(1)39472 41910 y Fq(\()p Fn(S)339 b Fo([)p 42103 40936 V 270 w Fn(S)70 b Fq(\))404 b(de\257ned)h(b)-34 b(y)0 44269 y(\(4.3\))6441 b Fn(`)9437 43769 y Fl(1)9437 44569 y Fm(a)10433 44269 y Fq(\()p Fn(S)340 b Fo([)p 13065 43296 V 270 w Fn(S)70 b Fq(\))337 b(=)f Fo(f)p Fn(g)381 b Fo(2)336 b Fn(`)19179 43769 y Fl(1)20176 44269 y Fq(\()p Fn(S)k Fo([)p 22807 43296 V 269 w Fn(S)70 b Fq(\))741 b(:)g Fn(g)43 b Fq(\()p 27002 43602 693 49 v Fn(x)p Fq(\))338 b(=)e Fo(\241)p Fn(g)43 b Fq(\()p Fn(x)p Fq(\))405 b(for)f(an)-34 b(y)405 b Fn(x)336 b Fo(2)h Fn(S)70 b Fo(g)p Fn(:)0 46629 y Fq(Note)523 b(that)h Fn(Q)e Fq(can)h(b)34 b(e)523 b(regarded)f(as)h (an)g(op)34 b(erator)522 b(on)h Fn(`)27214 46189 y Fl(1)27214 46929 y Fm(a)28211 46629 y Fq(\()p Fn(S)418 b Fo([)p 31000 45655 814 49 v 349 w Fn(S)70 b Fq(\).)894 b(W)-101 b(e)523 b(also)f(de\257ne)h(a)g(m)-34 b(ultiplication)0 48113 y(op)34 b(erator)404 b Fn(V)673 b Fq(on)405 b Fn(`)8458 47674 y Fl(1)9454 48113 y Fq(\()p Fn(S)340 b Fo([)p 12086 47140 V 270 w Fn(S)70 b Fq(\))404 b(with)h(a)f(function)0 50675 y(\(4.4\))9313 b Fn(V)269 b Fq(\()43 b Fj(e)-716 b Fn(x)p Fq(\))337 b(=)g Fn(A)p Fq(\()p Fn(x;)202 b(x)p Fq(\))269 b(+)22375 49524 y Fj(X)21289 52159 y Fm(y)32 b Fl(2)p Fm(S)50 b Fk(:)157 b Fm(y)32 b Fl(6)p Fk(=)p Fm(x)25414 50675 y Fo(j)p Fn(A)p Fq(\()p Fn(x;)202 b(y)43 b Fq(\))p Fo(j)p Fn(;)1414 b Fq(if)447 b Fj(e)-716 b Fn(x)337 b Fq(=)f Fn(x)404 b Fq(or)p 37561 50008 693 49 v 404 w Fn(x)o(;)0 54509 y Fq(whic)-34 b(h)405 b(lea)-34 b(v)g(es)404 b Fn(`)7448 54069 y Fl(1)7448 54809 y Fm(a)8444 54509 y Fq(\()p Fn(S)340 b Fo([)p 11075 53535 814 49 v 269 w Fn(S)71 b Fq(\))404 b(in)-34 b(v)-67 b(arian)-34 b(t.)539 b(W)-101 b(e)403 b(notice)h(that)i(b)-34 b(y)404 b(the)h(h)-34 b(yp)34 b(otheses)0 56869 y(\(4.5\))14380 b Fn(V)270 b Fq(\()43 b Fj(e)-716 b Fn(x)p Fq(\))337 b Fo(\267)f(\241)p Fn(\270;)1414 b Fq(for)404 b(all)447 b Fj(e)-716 b Fn(x)336 b Fo(2)h Fn(S)i Fo([)p 32373 55895 V 270 w Fn(S)70 b(:)1328 59183 y Fq(No)-34 b(w)405 b(w)-34 b(e)405 b(consider)f(the)g(follo)-34 b(wing)405 b(equation:)0 61497 y(\(4.6\))16593 b Fn(Af)130 b Fq(\()p Fn(x)p Fq(\))338 b(=)e Fn(h)p Fq(\()p Fn(x)p Fq(\))p Fn(;)1415 b(x)337 b Fo(2)f Fn(S)s(;)0 63812 y Fq(for)404 b Fn(h)337 b Fo(2)g Fn(`)4540 63372 y Fl(1)5536 63812 y Fq(\()p Fn(S)70 b Fq(\).)540 b(The)404 b(l.h.s.)538 b(of)405 b(\(4.6\))f(can)h(b)34 b(e)403 b(rewritten)i(as)0 67586 y(\(4.7\))3097 b Fn(Af)130 b Fq(\()p Fn(x)p Fq(\))338 b(=)11095 66434 y Fj(X)10512 69259 y Fd(e)-562 b Fm(y)32 b Fl(2)p Fm(S)50 b Fl([)p 12846 68534 622 40 v Fm(S)13670 67586 y Fn(Q)p Fq(\()p Fn(x;)251 b Fj(e)-722 b Fn(y)44 b Fq(\)\()18139 67266 y Fj(e)17911 67586 y Fn(f)131 b Fq(\()50 b Fj(e)-723 b Fn(y)44 b Fq(\))270 b Fo(\241)21925 67266 y Fj(e)21697 67586 y Fn(f)131 b Fq(\()p Fn(x)p Fq(\)\))271 b(+)26010 65149 y Fj(0)26010 67331 y(@)27070 67586 y Fn(A)p Fq(\()p Fn(x;)202 b(x)p Fq(\))270 b(+)33414 66434 y Fj(X)32328 69069 y Fm(y)32 b Fl(2)p Fm(S)50 b Fk(:)157 b Fm(y)32 b Fl(6)p Fk(=)p Fm(x)36452 67586 y Fo(j)p Fn(A)p Fq(\()p Fn(x;)202 b(y)43 b Fq(\))p Fo(j)40846 65149 y Fj(1)40846 67331 y(A)42337 67266 y(e)42109 67586 y Fn(f)131 b Fq(\()p Fn(x)p Fq(\))p Fn(:)0 71447 y Fq(Th)-34 b(us)406 b(\(4.6\))e(is)g (equiv)-67 b(alen)-34 b(t)404 b(to)0 73931 y(\(4.8\))13754 b(\()p Fn(Q)270 b Fq(+)f Fn(V)h Fq(\))20831 73612 y Fj(e)20604 73931 y Fn(f)131 b Fq(\()43 b Fj(e)-716 b Fn(x)p Fq(\))337 b(=)24559 73612 y Fj(e)24580 73931 y Fn(h)p Fq(\()43 b Fj(e)-716 b Fn(x)p Fq(\))p Fn(;)1458 b Fj(e)-717 b Fn(x)337 b Fo(2)g Fn(S)i Fo([)p 32999 72958 814 49 v 270 w Fn(S)70 b(:)24962 75260 y Fk(7)p eop %%Page: 8 8 8 7 bop 0 481 a Fq(for)1778 161 y Fj(e)1799 481 y Fn(h)337 b Fo(2)g Fn(`)4484 41 y Fl(1)4484 781 y Fm(a)5480 481 y Fq(\()p Fn(S)227 b Fo([)p 7886 -493 814 49 v 157 w Fn(S)70 b Fq(\).)520 b(Let)348 b Fo(f)p Fn(X)13753 663 y Fm(t)14148 481 y Fn(;)202 b(t)336 b Fo(\270)g Fq(0)p Fo(g)348 b Fq(b)34 b(e)348 b(the)g(Mark)-34 b(o)g(v)348 b(c)-34 b(hain)348 b(on)g Fn(S)227 b Fo([)p 32987 -493 V 157 w Fn(S)418 b Fq(whic)-34 b(h)348 b(is)g(generated)g(b)-34 b(y)348 b Fn(Q)p Fq(.)520 b(Then)0 1920 y(b)-34 b(y)405 b(the)f(F)-101 b(eynman-Kac)404 b(form)-34 b(ula,)405 b(w)-34 b(e)404 b(obtain)0 4622 y(\(4.9\))7883 4302 y Fj(e)7656 4622 y Fn(f)131 b Fq(\()43 b Fj(e)-716 b Fn(x)p Fq(\))337 b(=)f(\()p Fn(Q)270 b Fq(+)f Fn(V)h Fq(\))15990 4122 y Fl(\241)p Fk(1)17227 4302 y Fj(e)17248 4622 y Fn(h)p Fq(\()43 b Fj(e)-716 b Fn(x)p Fq(\))337 b(=)g Fo(\241)p Fc(E)22979 4844 y Fd(e)-553 b Fm(x)23736 2913 y Fj(\267)24375 2972 y(Z)25588 3323 y Fl(1)25049 5721 y Fk(0)26765 4302 y Fj(e)26786 4622 y Fn(h)p Fq(\()p Fn(X)28959 4804 y Fm(t)29355 4622 y Fq(\))202 b(exp)32082 2913 y Fj(\265)32975 2972 y(Z)34187 3323 y Fm(t)33648 5721 y Fk(0)34784 4622 y Fn(V)269 b Fq(\()p Fn(X)37235 4804 y Fm(s)37726 4622 y Fq(\))p Fn(ds)39396 2913 y Fj(\266)40491 4622 y Fn(dt)41560 2913 y Fj(\270)42401 4622 y Fn(:)0 7407 y Fq(In)463 b(order)h(to)f(get)h(the)g(matrix)f(comp)34 b(onen)-34 b(ts)465 b Fn(A)22217 6967 y Fl(\241)p Fk(1)23474 7407 y Fq(\()p Fn(x;)202 b(y)43 b Fq(\),)479 b(w)-34 b(e)464 b(tak)-34 b(e)464 b Fn(h)436 b Fq(=)f Fn(\261)34682 7589 y Fm(y)35234 7407 y Fq(,)478 b(the)464 b(delta)f(function)i(at)f (p)34 b(oin)-34 b(t)0 9067 y Fn(y)397 b Fo(2)353 b Fn(S)70 b Fq(.)569 b(Th)-34 b(us)6930 8747 y Fj(e)6951 9067 y Fn(h)354 b Fq(=)9322 8747 y Fj(e)9299 9067 y Fn(\261)9838 9249 y Fm(y)10743 9067 y Fq(=)f Fn(\261)12578 9249 y Fm(y)13406 9067 y Fo(\241)276 b Fn(\261)p 15164 8748 553 40 v 181 x Fm(y)15716 9067 y Fq(.)568 b(Let)414 b(us)g(de\257ne)h (the)f(pro)67 b(jection)415 b Fn(\274)397 b Fq(:)555 b Fn(S)346 b Fo([)p 35901 8093 814 49 v 276 w Fn(S)423 b Fo(!)354 b Fn(S)484 b Fq(b)-34 b(y)414 b Fn(\274)43 b Fq(\()p Fn(z)53 b Fq(\))355 b(=)e Fn(\274)43 b Fq(\()p 46702 8399 618 49 v Fn(z)53 b Fq(\))354 b(=)f Fn(z)53 b Fq(.)0 10506 y(Then)405 b(from)f(\(4.9\))h(and)g(b)-34 b(y)404 b(noting)h(that)h Fn(V)269 b Fq(\()p Fn(x)p Fq(\))337 b(=)g Fn(V)269 b Fq(\()p 25039 9838 693 49 v Fn(x)p Fq(\),)404 b Fn(x)337 b Fo(2)g Fn(S)70 b Fq(,)404 b(w)-34 b(e)404 b(ha)-34 b(v)g(e)405 b(the)f(b)34 b(ound:)0 13275 y(\(4.10\))6294 b Fo(j)p Fn(A)10637 12774 y Fl(\241)p Fk(1)11894 13275 y Fq(\()p Fn(x;)202 b(y)43 b Fq(\))p Fo(j)337 b(\267)g Fc(E)17467 13457 y Fm(x)18254 11566 y Fj(\267)18894 11625 y(Z)20106 11976 y Fl(1)19567 14374 y Fk(0)21304 13275 y Fn(\261)21843 13457 y Fm(y)22395 13275 y Fq(\()p Fn(\274)43 b Fq(\()p Fn(X)25075 13457 y Fm(t)25472 13275 y Fq(\)\))202 b(exp)28670 11566 y Fj(\265)29562 11625 y(Z)30775 11976 y Fm(t)30236 14374 y Fk(0)31372 13275 y Fn(V)269 b Fq(\()p Fn(\274)43 b Fq(\()p Fn(X)35028 13457 y Fm(s)35520 13275 y Fq(\)\))p Fn(ds)37661 11566 y Fj(\266)38756 13275 y Fn(dt)39825 11566 y Fj(\270)40666 13275 y Fn(:)0 15958 y Fq(W)-101 b(e)374 b(notice)h(that)g(the)g(probabilit)-34 b(y)376 b(la)-34 b(w)375 b(of)g(the)g(c)-34 b(hain)375 b Fo(f)p Fn(\274)43 b Fq(\()p Fn(X)27974 16140 y Fm(t)28370 15958 y Fq(\))p Fo(g)375 b Fq(on)g Fn(S)445 b Fq(is)374 b(the)h(same)g(as)g(that)h(of)f(the)g(Mark)-34 b(o)g(v)0 17553 y(c)g(hain)417 b(on)h Fn(S)487 b Fq(with)418 b(a)e Fn(Q)p Fq(-matrix)15496 17247 y Fj(b)15252 17553 y Fn(Q)h Fq(de\257ned)h(b)-34 b(y)22755 17247 y Fj(b)22512 17553 y Fn(Q)p Fq(\()p Fn(x;)202 b(y)43 b Fq(\))359 b(=)f Fo(j)p Fn(A)p Fq(\()p Fn(x;)202 b(y)43 b Fq(\))p Fo(j)p Fq(,)420 b Fn(x)358 b Fo(6)p Fq(=)g Fn(y)43 b Fq(,)420 b(and)39208 16644 y Fj(P)40487 17912 y Fm(y)32 b Fl(2)p Fm(S)42734 17247 y Fj(b)42491 17553 y Fn(Q)p Fq(\()p Fn(x;)202 b(y)43 b Fq(\))359 b(=)f(0)417 b(for)0 19037 y(all)404 b Fn(x)336 b Fo(2)h Fn(S)70 b Fq(.)538 b(By)404 b(using)g(the)h(strong)g(Mark)-34 b(o)g(v)404 b(prop)34 b(ert)-34 b(y)404 b(and)h(the)g(fact)f(that)h Fn(V)607 b Fo(\267)336 b(\241)p Fn(\270)404 b Fq(w)-34 b(e)404 b(get)0 21076 y(\(4.11\))9375 b Fo(j)p Fn(A)13718 20576 y Fl(\241)p Fk(1)14975 21076 y Fq(\()p Fn(x;)202 b(y)43 b Fq(\))p Fo(j)337 b(\267)g(k)p Fq(\()21061 20770 y Fj(b)20817 21076 y Fn(Q)270 b Fq(+)f Fn(V)h Fq(\))24705 20576 y Fl(\241)p Fk(1)25963 21076 y Fo(k)f(\242)g Fc(E)28252 21258 y Fm(x)28837 21076 y Fq([)p Fn(e)29738 20576 y Fl(\241)p Fm(\270\277)31437 20687 y Fi(y)32396 21076 y Fq(;)605 b Fn(\277)33868 21258 y Fm(y)34757 21076 y Fn(<)337 b Fo(1)p Fq(])p Fn(;)0 23235 y Fq(where)491 b Fn(\277)4122 23417 y Fm(y)5165 23235 y Fq(is)g(the)h(\257rst)g (hitting)g(time)f(at)h Fn(y)535 b Fq(of)491 b(the)h(c)-34 b(hain)491 b Fo(f)p Fn(\274)43 b Fq(\()p Fn(X)30583 23417 y Fm(t)30980 23235 y Fq(\))p Fo(g)491 b Fq(and)h Fo(k)p Fq(\()36313 22928 y Fj(b)36069 23235 y Fn(Q)328 b Fq(+)g Fn(V)269 b Fq(\))40073 22795 y Fl(\241)p Fk(1)41331 23235 y Fo(k)491 b Fq(is)g(the)h(op)34 b(erator)0 24881 y(norm)392 b(of)g(\()5239 24575 y Fj(b)4995 24881 y Fn(Q)245 b Fq(+)g Fn(V)269 b Fq(\))8833 24441 y Fl(\241)p Fk(1)10483 24881 y Fq(acting)392 b(on)g Fn(`)16283 24441 y Fk(2)16809 24881 y Fq(\()p Fn(S)70 b Fq(\),)394 b(whic)-34 b(h)393 b(is)f(b)34 b(ounded)393 b(b)-34 b(y)392 b(1)p Fn(=\270)p Fq(.)534 b(F)-101 b(or)392 b(simplicit)-34 b(y)391 b(w)-34 b(e)392 b(write)g Fn(u)46590 25063 y Fm(y)47142 24881 y Fq(\()p Fn(x)p Fq(\))337 b(:=)0 26401 y Fc(E)808 26583 y Fm(x)1393 26401 y Fq([)p Fn(e)2294 25961 y Fl(\241)p Fm(\270\277)3993 26072 y Fi(y)4952 26401 y Fq(;)605 b Fn(\277)6424 26583 y Fm(y)7313 26401 y Fn(<)337 b Fo(1)p Fq(].)538 b(Let)403 b(us)i(de\257ne)g(a)f(substo)34 b(c)-34 b(hastic)405 b(matrix)f(on)h Fn(S)474 b Fq(b)-34 b(y)0 29498 y(\(4.12\))11299 b(\246\()p Fn(x;)202 b(y)43 b Fq(\))338 b(:=)20070 27425 y Fj(\()21290 28371 y(b)21046 28677 y Fn(Q)q Fq(\()p Fn(x;)202 b(y)43 b Fq(\))p Fn(=)p Fo(j)26003 28371 y Fj(b)25759 28677 y Fn(Q)q Fq(\()p Fn(x;)202 b(x)p Fq(\))p Fo(j)p Fn(;)1211 b Fq(if)404 b Fn(x)337 b Fo(6)p Fq(=)f Fn(y)43 b(;)21046 30404 y Fq(0)p Fn(;)9481 b Fq(if)404 b Fn(x)337 b Fq(=)f Fn(y)43 b(;)0 32818 y Fq(if)429 b Fo(j)1716 32511 y Fj(b)1473 32818 y Fn(Q)p Fq(\()p Fn(x;)202 b(x)p Fq(\))p Fo(j)378 b(6)p Fq(=)h(0;)441 b(\246\()p Fn(x;)202 b(y)43 b Fq(\))380 b(=)e(0)429 b(for)g(ev)-34 b(ery)429 b Fn(y)422 b Fo(2)378 b Fn(S)500 b Fq(if)429 b Fo(j)25444 32511 y Fj(b)25201 32818 y Fn(Q)p Fq(\()p Fn(x;)202 b(x)p Fq(\))p Fo(j)378 b Fq(=)g(0.)614 b(Let)429 b Fn(\256)t Fq(\()p Fn(x)p Fq(\))h(b)34 b(e)429 b(a)g(random)h(v)-67 b(ariable)428 b(of)0 34413 y(exp)34 b(onen)-34 b(tial)363 b(distribution)i(with)f (parameter)f Fn(Q)p Fq(\()p Fn(x)p Fq(\))338 b(=)e Fo(j)26293 34107 y Fj(b)26050 34413 y Fn(Q)p Fq(\()p Fn(x;)202 b(x)p Fq(\))p Fo(j)363 b Fq(and)h(w)-34 b(e)364 b(use)f Fc(E)g Fq(for)h(the)f(exp)34 b(ectation.)525 b(W)-101 b(e)0 35852 y(then)405 b(ha)-34 b(v)g(e)405 b(the)f(follo)-34 b(wing)405 b(iden)-34 b(tit)g(y:)0 37983 y(\(4.13\))9757 b Fn(u)13548 38165 y Fm(y)14100 37983 y Fq(\()p Fn(x)p Fq(\))337 b(=)f Fc(E)p Fq([)p Fn(e)19060 37482 y Fl(\241)p Fm(\270\256)p Fk(\()p Fm(x)p Fk(\))22262 37983 y Fq(]\(\246\()p Fn(x;)202 b(y)43 b Fq(\))271 b(+)28273 36831 y Fj(X)28297 39467 y Fm(z)37 b Fl(6)p Fk(=)p Fm(y)30226 37983 y Fq(\246\()p Fn(x;)202 b(z)53 b Fq(\))p Fn(u)34620 38165 y Fm(y)35172 37983 y Fq(\()p Fn(z)g Fq(\)\))p Fn(:)0 41644 y Fq(By)408 b(the)i(h)-34 b(yp)34 b(otheses)410 b(w)-34 b(e)409 b(get)g Fc(E)p Fq([)p Fn(e)15703 41204 y Fl(\241)p Fm(\270\256)p Fk(\()p Fm(x)p Fk(\))18905 41644 y Fq(])344 b(=)21647 41047 y Fm(Q)p Fk(\()p Fm(x)p Fk(\))p 21007 41365 3283 49 v 21007 42078 a Fm(\270)p Fk(+)p Fm(Q)p Fk(\()p Fm(x)p Fk(\))24767 41644 y Fo(\267)26187 41166 y Fm(b)p Fl(\241)p Fm(\270)p 26187 41364 1684 49 v 26827 42062 a(b)28412 41644 y Fq(uniformly)409 b(in)g Fn(x)345 b Fo(2)f Fn(S)70 b Fq(.)554 b(W)-101 b(e)408 b(put)i(this)g(estimate)0 43319 y(in)-34 b(to)405 b(\(4.13\))g(and)g(mak)-34 b(e)403 b(an)i(in\257nite)g(iteration)f(to)g(get)0 45269 y(\(4.14\))17898 b Fn(u)21689 45451 y Fm(y)22241 45269 y Fq(\()p Fn(x)p Fq(\))337 b Fo(\267)g Fq(\241\()p Fn(x;)202 b(y)43 b Fq(\))p Fn(;)0 47338 y Fq(where)452 b(\241)417 b(:=)6425 46429 y Fj(P)7704 46779 y Fl(1)7704 47696 y Fm(n)p Fk(=1)9533 47338 y Fq(\(\()10608 46861 y Fm(b)p Fl(\241)p Fm(\270)p 10608 47059 V 11249 47756 a(b)12424 47338 y Fq(\)\246\))14275 46898 y Fm(n)14902 47338 y Fq(,)465 b(whic)-34 b(h)453 b(is)f(a)g(b)34 b(ounded)454 b(op)34 b(erator)452 b(on)h Fn(`)33717 46898 y Fl(1)34713 47338 y Fq(\()p Fn(S)70 b Fq(\))454 b(with)f(norm)g(less)f(than)48241 46861 y Fm(b)p Fl(\241)p Fm(\270)p 48241 47059 V 48808 47756 a(\270)50057 47338 y Fq(.)0 48777 y(By)403 b(using)i(\(4.11\))g(and)g (\(4.14\))f(w)-34 b(e)405 b(complete)f(the)g(pro)34 b(of)405 b(for)f(the)g(real)g(case.)1328 50216 y(W)-101 b(e)444 b(consider)f(the)i(case)e(where)h Fn(A)g Fq(is)f(a)h(complex)g(matrix.) 657 b(W)-101 b(e)443 b(write)h Fn(A)403 b Fq(=)g Fn(A)38314 50398 y Fk(1)39136 50216 y Fq(+)295 b Fn(iA)41701 50398 y Fk(2)42227 50216 y Fq(,)453 b(where)444 b Fn(A)47471 50398 y Fk(1)48441 50216 y Fq(and)0 51655 y Fn(A)909 51837 y Fk(2)1839 51655 y Fq(are)404 b(real)f(matrices.)538 b(Let)403 b Fn(S)14467 51837 y Fk(1)15398 51655 y Fq(and)h Fn(S)18497 51837 y Fk(2)19428 51655 y Fq(b)34 b(e)403 b(t)-34 b(w)g(o)406 b(copies)e(of)g Fn(S)70 b Fq(.)539 b(Then)405 b(w)-34 b(e)404 b(naturally)h(ha)-34 b(v)g(e)404 b(the)h(bijection)0 53606 y(\(4.15\))3725 b Fn(`)7327 53105 y Fk(2)7853 53606 y Fq(\()p Fn(S)70 b Fq(\))337 b Fo(3)g Fn(f)468 b Fq(=)336 b Fn(f)14023 53788 y Fk(1)14819 53606 y Fq(+)269 b Fn(if)17042 53788 y Fk(2)17904 53606 y Fo(7!)337 b Fn(f)20046 53788 y Fk(1)20842 53606 y Fo(\251)269 b Fn(f)22647 53788 y Fk(2)23510 53606 y Fo(2)337 b Fn(`)25160 53105 y Fk(2)25160 53914 y(real)26732 53606 y Fq(\()p Fn(S)27946 53788 y Fk(1)28472 53606 y Fq(\))270 b Fo(\251)f Fn(`)30930 53105 y Fk(2)30930 53914 y(real)32502 53606 y Fq(\()p Fn(S)33716 53788 y Fk(2)34243 53606 y Fq(\))35051 53272 y Fo(\273)35051 53661 y Fq(=)36330 53606 y Fn(`)36835 53105 y Fk(2)36835 53914 y(real)38407 53606 y Fq(\()p Fn(S)39621 53788 y Fk(1)40417 53606 y Fo([)h Fn(S)42238 53788 y Fk(2)42764 53606 y Fq(\))p Fn(;)0 55557 y Fq(where)362 b Fn(`)3968 55117 y Fk(2)3968 55926 y(real)5540 55557 y Fq(\()p Fo(\242)p Fq(\))h(means)f(the)h(real)f(Hilb)34 b(ert)361 b(space.)525 b(Under)362 b(this)h(map,)371 b Fn(A)362 b Fq(on)h Fn(`)35416 55117 y Fk(2)35941 55557 y Fq(\()p Fn(S)70 b Fq(\))364 b(is)d(equiv)-67 b(alen)-34 b(t)362 b(to)h(the)f(real)0 56996 y(matrix)0 59376 y(\(4.16\))17456 b Fo(A)336 b Fq(=)23137 57666 y Fj(\265)24029 58641 y Fn(A)24938 58823 y Fk(1)26571 58641 y Fo(\241)p Fn(A)28423 58823 y Fk(2)24029 60080 y Fn(A)24938 60262 y Fk(2)27042 60080 y Fn(A)27951 60262 y Fk(1)28949 57666 y Fj(\266)0 62160 y Fq(acting)473 b(on)f Fn(`)5961 61720 y Fk(2)5961 62529 y(real)7533 62160 y Fq(\()p Fn(S)8747 62342 y Fk(1)9589 62160 y Fo([)315 b Fn(S)11455 62342 y Fk(2)11981 62160 y Fq(\).)743 b(Let)472 b Fn(A)16681 61720 y Fl(\241)p Fk(1)18389 62160 y Fq(:=)450 b Fn(C)402 b Fq(+)314 b Fn(iD)34 b Fq(,)488 b(where)472 b Fn(C)560 b Fq(and)473 b Fn(D)505 b Fq(are)472 b(real)f(matrices.)743 b(It)472 b(is)g(easy)g(to)0 63599 y(c)-34 b(hec)g(k)404 b(that)0 65550 y(\(4.17\))12014 b Fn(C)424 b Fq(=)336 b Fn(P)18458 65738 y Fm(S)19030 65861 y Ff(1)19547 65550 y Fo(A)20515 65049 y Fl(\241)p Fk(1)21773 65550 y Fn(P)22551 65738 y Fm(S)23123 65861 y Ff(1)23640 65550 y Fn(;)1413 b(D)370 b Fq(=)337 b Fo(\241)p Fn(P)29765 65738 y Fm(S)30337 65861 y Ff(1)30854 65550 y Fo(A)31822 65049 y Fl(\241)p Fk(1)33079 65550 y Fn(P)33857 65738 y Fm(S)34429 65861 y Ff(2)34946 65550 y Fn(;)0 67500 y Fq(where)460 b Fn(P)4339 67688 y Fm(S)4911 67823 y Fi(i)5318 67500 y Fq(,)473 b Fn(i)429 b Fq(=)f(1)p Fn(;)202 b Fq(2,)473 b(are)459 b(the)h(pro)67 b(jections)461 b(of)f Fn(`)23365 67061 y Fk(2)23365 67869 y(real)24937 67500 y Fq(\()p Fn(S)26151 67682 y Fk(1)26984 67500 y Fo([)306 b Fn(S)28841 67682 y Fk(2)29367 67500 y Fq(\))460 b(to)g Fn(`)32340 67061 y Fk(2)32340 67869 y(real)33912 67500 y Fq(\()p Fn(S)35126 67682 y Fm(i)35503 67500 y Fq(\),)473 b Fn(i)429 b Fq(=)g(1)p Fn(;)202 b Fq(2,)473 b(resp)34 b(ectiv)-34 b(ely)-101 b(.)703 b(W)-101 b(e)0 68940 y(notice)552 b(that)i Fo(A)e Fq(satis\257es)i(the)e(h)-34 b(yp)34 b(otheses)554 b(in)f(the)f(lemma)h (with)g Fn(S)623 b Fq(b)34 b(eing)552 b(replaced)g(b)-34 b(y)553 b Fn(S)44235 69122 y Fk(1)45129 68940 y Fo([)369 b Fn(S)47049 69122 y Fk(2)47575 68940 y Fq(.)983 b(By)0 70379 y(applying)405 b(the)f(ab)34 b(o)-34 b(v)g(e)404 b(result)g(for)h(the)f(real)g(case)f(w)-34 b(e)405 b(pro)-34 b(v)g(e)404 b(the)h(lemma.)15753 b Fe(\244)1328 72492 y Fq(The)351 b(condition)f(\(4.18\))h(b)34 b(elo)-34 b(w)350 b(is)g(a)f(w)-34 b(ell)350 b(kno)-34 b(wn)351 b(su\261cien)-34 b(t)351 b(condition)g(for)f(the)g(construction)h(of)f (a)g(F)-101 b(eller)0 73931 y(Mark)-34 b(o)g(v)404 b(pro)34 b(cess)404 b(on)g Fn(C)87 b Fq(\()p Fn(X)95 b Fq(\))405 b(with)g(pre-generator)f Fo(L)g Fq(de\257ned)h(b)-34 b(y)405 b(\(1.7\))f([13].)24962 75260 y Fk(8)p eop %%Page: 9 9 9 8 bop 0 393 a Fw(Lemma)464 b(4.2.)554 b Fg(Supp)-62 b(ose)431 b(that)h(\(1.11\))f(holds.)556 b(Then)433 b(we)h(have)0 3219 y Fq(\(4.18\))12777 b(sup)15807 4262 y Fm(x)p Fl(2R)17996 2067 y Fj(X)17969 4702 y Fm(u)p Fl(6)p Fk(=)p Fm(x)20024 3219 y Fq(sup)19977 4271 y Fm(\273)35 b Fl(2)p Fm(X)22097 3219 y Fo(j)p Fn(c)p Fq(\()p Fn(x;)202 b(\273)56 b Fq(\))269 b Fo(\241)g Fn(c)p Fq(\()p Fn(x;)202 b(\273)30014 2718 y Fm(u)30614 3219 y Fq(\))p Fo(j)337 b Fn(<)f Fo(1)p Fn(:)0 7099 y Fg(Pr)-62 b(o)g(of.)552 b Fq(By)404 b(de\257nition)h(of)f (the)h(\260ip)g(rates)f(in)g(\(1.9\))g(and)h(\(3.3\),)g(the)f (condition)h(\(4.18\))g(is)f(equiv)-67 b(alen)-34 b(t)403 b(to)0 9925 y(\(4.19\))10070 b(sup)13099 10967 y Fm(x)p Fl(2R)15289 8773 y Fj(X)15261 11408 y Fm(u)p Fl(6)p Fk(=)p Fm(x)17316 9925 y Fq(sup)17270 10977 y Fm(\273)35 b Fl(2)p Fm(X)19390 9925 y Fo(j)p Fn(\260)20355 10107 y Fm(x)20939 9925 y Fq(\()p Fn(\273)21940 10107 y Fm(x)22526 9925 y Fq(;)202 b Fn(\273)23595 10171 y Fl(f)p Fm(x)p Fl(g)25065 9919 y Fi(c)25548 9925 y Fq(\))269 b Fo(\241)h Fn(\260)28129 10107 y Fm(x)28713 9925 y Fq(\()p Fn(\273)29714 10107 y Fm(x)30300 9925 y Fq(;)202 b Fn(\273)31425 9424 y Fm(u)31369 10282 y Fl(f)p Fm(x)p Fl(g)32839 10030 y Fi(c)33321 9925 y Fq(\))p Fo(j)337 b Fn(<)g Fo(1)p Fn(:)0 13861 y Fq(F)-101 b(rom)404 b(\(3.2\))h(w)-34 b(e)404 b(see)g(that)h(for)f(an)-34 b(y)405 b Fn(\263)89 b(;)404 b(\273)393 b Fo(2)337 b Fn(X)0 16717 y Fq(\(4.20\))4798 b Fn(\260)8523 16899 y Fm(x)9107 16717 y Fq(\()p Fn(\263)10108 16899 y Fm(x)10694 16717 y Fq(;)202 b Fn(\273)11763 16964 y Fl(f)p Fm(x)p Fl(g)13233 16712 y Fi(c)13716 16717 y Fq(\))337 b(=)f Fn(K)16919 16217 y Fk(\()p Fm(\263)61 b Fk(\))18181 16717 y Fq(\()p Fn(x;)202 b(x)p Fq(\))269 b Fo(\241)h Fn(K)23646 16079 y Fk(\()p Fm(\263)61 b Fk(\))23559 17158 y Fl(f)p Fm(x)p Fl(g)p Fm(;)p Fl(f)p Fm(x)p Fl(g)26760 16906 y Fi(c)27244 16717 y Fq(\()p Fn(x;)202 b Fo(\242)p Fq(\)\()p Fn(K)31342 16079 y Fk(\()p Fm(\273)35 b Fk(\))31255 17158 y Fl(f)p Fm(x)p Fl(g)32725 16906 y Fi(c)33208 16717 y Fq(\))33679 16217 y Fl(\241)p Fk(1)34937 16717 y Fn(K)36053 16079 y Fk(\()p Fm(\273)g Fk(\))35966 17158 y Fl(f)p Fm(x)p Fl(g)37436 16906 y Fi(c)37864 17158 y Fm(;)p Fl(f)p Fm(x)p Fl(g)39651 16717 y Fq(\()p Fo(\242)p Fn(;)202 b(x)p Fq(\))p Fn(;)0 19933 y Fq(where)313 b Fn(K)4530 19294 y Fk(\()p Fm(\263)61 b Fk(\))4443 20373 y Fl(f)p Fm(x)p Fl(g)p Fm(;)p Fl(f)p Fm(x)p Fl(g)7644 20121 y Fi(c)8128 19933 y Fq(\()p Fn(x;)202 b Fo(\242)p Fq(\))312 b(and)i Fn(K)14333 19294 y Fk(\()p Fm(\273)35 b Fk(\))14246 20373 y Fl(f)p Fm(x)p Fl(g)15716 20121 y Fi(c)16143 20373 y Fm(;)p Fl(f)p Fm(x)p Fl(g)17931 19933 y Fq(\()p Fo(\242)p Fn(;)202 b(x)p Fq(\))312 b(are)g(the)h Fn(x)p Fq(th)h(ro)-34 b(w)313 b(v)-34 b(ector)312 b(of)h Fn(K)35021 19493 y Fk(\()p Fm(\263)61 b Fk(\))36595 19933 y Fq(and)314 b(the)f Fn(x)p Fq(th)g(column)g(v)-34 b(ector)0 21810 y(of)449 b Fn(K)2541 21370 y Fk(\()p Fm(\273)35 b Fk(\))4224 21810 y Fq(remo)-34 b(ving)448 b(the)h Fn(x)p Fq(th)g(comp)34 b(onen)-34 b(t,)460 b(resp)34 b(ectiv)-34 b(ely)-101 b(.)669 b(By)447 b(inserting)i(the)f(form)-34 b(ula)449 b(\(4.20\))g(in)-34 b(to)449 b(\(4.19\))0 23249 y(w)-34 b(e)405 b(get)5679 25827 y Fn(\260)6307 26009 y Fm(x)6892 25827 y Fq(\()p Fn(\273)7893 26009 y Fm(x)8479 25827 y Fq(;)202 b Fn(\273)9548 26074 y Fl(f)p Fm(x)p Fl(g)11018 25822 y Fi(c)11500 25827 y Fq(\))270 b Fo(\241)f Fn(\260)14081 26009 y Fm(x)14666 25827 y Fq(\()p Fn(\273)15667 26009 y Fm(x)16252 25827 y Fq(;)202 b Fn(\273)17377 25327 y Fm(u)17321 26185 y Fl(f)p Fm(x)p Fl(g)18791 25933 y Fi(c)19274 25827 y Fq(\))6786 28271 y(=)1107 b Fn(K)9952 27632 y Fk(\()p Fm(\273)35 b Fk(\))9865 28711 y Fl(f)p Fm(x)p Fl(g)p Fm(;)p Fl(f)p Fm(x)p Fl(g)13066 28459 y Fi(c)13550 28271 y Fq(\()p Fn(x;)202 b Fo(\242)p Fq(\))16263 26925 y Fj(n)17071 28271 y Fq(\()p Fn(K)18658 27632 y Fk(\()p Fm(\273)19473 27319 y Fi(u)20018 27632 y Fk(\))18571 28711 y Fl(f)p Fm(x)p Fl(g)20041 28459 y Fi(c)20525 28271 y Fq(\))20996 27770 y Fl(\241)p Fk(1)22254 28271 y Fn(K)23370 27632 y Fk(\()p Fm(\273)24185 27319 y Fi(u)24729 27632 y Fk(\))23283 28711 y Fl(f)p Fm(x)p Fl(g)24753 28459 y Fi(c)25180 28711 y Fm(;)p Fl(f)p Fm(x)p Fl(g)26968 28271 y Fq(\()p Fo(\242)p 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Fn(:)24962 75260 y Fk(9)p eop %%Page: 10 10 10 9 bop 0 589 a Fq(W)-101 b(e)404 b(insert)g(\(4.25\))g(in)-34 b(to)405 b(\(4.21\))g(and)g(use)f(the)h(fact)f(that)i Fn(K)27598 -50 y Fk(\()p Fm(\273)28413 -363 y Fi(u)28957 -50 y Fk(\))27511 1029 y Fl(f)p Fm(x)p Fl(g)28981 777 y Fi(c)29408 1029 y Fm(;)p Fl(f)p Fm(x)p Fl(g)31196 589 y Fq(\()p Fo(\242)p Fn(;)202 b(x)p Fq(\))336 b(=)h Fn(Q)36281 149 y Fk(\()p Fm(u)p Fk(\))37613 589 y Fn(K)38729 -50 y Fk(\()p Fm(\273)35 b Fk(\))38642 1029 y Fl(f)p Fm(x)p Fl(g)40112 777 y Fi(c)40540 1029 y Fm(;)p Fl(f)p Fm(x)p Fl(g)42327 589 y Fq(\()p Fo(\242)p Fn(;)202 b(x)p Fq(\))404 b(to)g(get)4373 3008 y Fo(j)p Fn(\260)5338 3190 y Fm(x)5922 3008 y Fq(\()p Fn(\273)6923 3190 y Fm(x)7509 3008 y Fq(;)202 b Fn(\273)8578 3255 y Fl(f)p Fm(x)p Fl(g)10048 3003 y Fi(c)10530 3008 y Fq(\))270 b Fo(\241)f Fn(\260)13111 3190 y Fm(x)13696 3008 y Fq(\()p Fn(\273)14697 3190 y Fm(x)15282 3008 y Fq(;)202 b Fn(\273)16407 2508 y Fm(u)16351 3366 y Fl(f)p Fm(x)p Fl(g)17821 3114 y 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Fk(\()p Fm(\273)35 b Fk(\))19528 20199 y Fl(f)p Fm(x)p Fl(g)20998 19946 y Fi(c)21481 19758 y Fq(\))21952 19257 y Fl(\241)p Fk(1)23210 19758 y Fq(\()p Fn(y)43 b(;)202 b(z)53 b Fq(\))p Fo(j)338 b(\267)28083 18938 y Fq(1)p 28032 19479 V 28032 20589 a Fn(\270)28872 19758 y Fq(\241\()p Fn(y)43 b(;)202 b(z)53 b Fq(\))p Fn(;)0 22448 y Fq(where)404 b(\241)336 b(=)h(\(1)270 b Fo(\241)f Fn(\270)p Fq(\)\246\()p Fn(I)11529 22695 y Fl(f)p Fm(x)p Fl(g)12999 22443 y Fi(c)13751 22448 y Fo(\241)h Fq(\(1)f Fo(\241)g Fn(\270)p Fq(\)\246\))20080 22008 y Fl(\241)p Fk(1)21743 22448 y Fq(and)405 b(\246)f(is)g(a)g (substo)34 b(c)-34 b(hastic)406 b(matrix)d(on)i Fo(R)270 b(n)f(f)p Fn(x)p Fo(g)404 b Fq(giv)-34 b(en)404 b(b)-34 b(y)14524 26000 y(\246\()p Fn(y)43 b(;)202 b(z)53 b Fq(\))338 b(=)19785 23927 y Fj(\()24137 24429 y Fl(j)p Fm(K)63 b Fk(\()p Fm(y)32 b(;z)37 b Fk(\))p Fl(j)27480 24552 y Ff(1)p 20894 24747 10291 49 v 20894 24795 a Fd(P)21888 25736 y Fi(z)26 b Fr(2Rnf)p Fi(x)p Fr(g)p Ff(:)142 b Fi(z)26 b 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Fq(\)-Bernoulli)f(measure)g(on)h Fo(f)p Fq(0)p Fn(;)202 b Fq(1)p Fo(g)470 b Fq(with)i(a)e(Radon-)0 70697 y(Nik)-34 b(o)34 b(dym)465 b(deriv)-67 b(ativ)-34 b(e,)478 b(sa)-34 b(y)465 b Fn(\275)14033 70058 y Fk(\()p Fm(\273)35 b Fk(\))14033 71079 y(\244)15268 70697 y Fq(\()p Fn(\263)16269 70879 y Fm(x)16855 70697 y Fq(\).)721 b(By)464 b(the)i(same)e(argumen) -34 b(t)466 b(to)g(sho)-34 b(w)466 b(\(3.3\))f(w)-34 b(e)466 b(see)e(that)i Fn(\275)45821 70058 y Fk(\()p Fm(\273)35 b Fk(\))45821 71079 y(\244)47056 70697 y Fq(\()p Fn(\263)48057 70879 y Fm(x)48643 70697 y Fq(\))465 b(is)0 72492 y(uniformly)500 b(b)34 b(ounded)500 b(as)g Fn(\270)c Fo(\267)f Fn(\275)15517 71853 y Fk(\()p Fm(\273)35 b Fk(\))15517 72874 y(\244)16752 72492 y Fq(\()p Fn(\263)17753 72674 y Fm(x)18339 72492 y Fq(\))496 b Fo(\267)f Fq(1)333 b Fo(\241)g Fn(\270)499 b Fq(.)825 b(Th)-34 b(us)501 b(w)-34 b(e)500 b(ha)-34 b(v)g(e)500 b(a)f(uniform)h(log-Sob)34 b(olev)499 b(constan)-34 b(t.)0 73931 y(By)485 b(using)h(the)g(so)g (called)f(martingale)g(expansion)h(metho)34 b(d)487 b(dev)-34 b(elop)34 b(ed)485 b(in)h([14)o(],)506 b(w)-34 b(e)486 b(ha)-34 b(v)g(e)486 b(the)g(follo)-34 b(wing)24726 75260 y Fk(11)p eop %%Page: 12 12 12 11 bop 0 393 a Fq(b)34 b(ound:)10529 3938 y Fn(E)11424 4126 y Fm(N)12321 3938 y Fn(f)13044 3438 y Fk(2)13772 3938 y Fq(log)15540 2229 y Fj(\265)17461 3118 y Fn(f)18184 2678 y Fk(2)p 16565 3659 3041 49 v 16565 4770 a Fn(E)17460 4958 y Fm(N)18356 4770 y Fn(f)19079 4419 y Fk(2)19739 2229 y Fj(\266)21738 3938 y Fq(=)24254 2423 y Fm(N)23799 2787 y Fj(X)23788 5377 y Fm(n)p Fk(=1)25763 3938 y Fn(E)26658 4126 y Fm(N)27756 2229 y Fj(\265)28649 3938 y Fn(E)29544 4120 y Fm(n)30170 3938 y Fn(f)30893 3438 y Fk(2)30763 4238 y Fm(n)p Fl(\241)p Fk(1)32794 3938 y Fq(log)35455 3048 y Fn(f)36178 2608 y Fk(2)36048 3373 y Fm(n)p Fl(\241)p Fk(1)p 34694 3659 3944 49 v 34694 4793 a Fn(E)35589 4975 y Fm(n)36215 4793 y Fn(f)36938 4375 y Fk(2)36808 5140 y Fm(n)p Fl(\241)p Fk(1)38770 2229 y Fj(\266)21738 8211 y Fo(\267)1107 b Fn(c)24313 8393 y Fk(0)25506 6695 y Fm(N)25051 7059 y Fj(X)25040 9650 y Fm(n)p Fk(=1)27015 8211 y Fn(E)27910 8399 y Fm(N)28807 8211 y Fo(jr)30154 8393 y Fm(x)30684 8504 y Fi(n)31305 8211 y Fn(f)31898 8393 y Fm(n)p Fl(\241)p Fk(1)33727 8211 y Fo(j)34064 7710 y Fk(2)34590 8211 y Fn(:)-34927 b Fq(\(4.37\))0 11917 y(W)-101 b(e)404 b(use)g(Lemma)f(4.4,)h(the)h(Sc)-34 b(h)g(w)g(arz)405 b(inequalit)-34 b(y)-101 b(,)404 b(and)h(consistency) f(of)g(the)h(sp)34 b(eci\257cation,)403 b(to)i(get)0 15287 y(\(4.38\))10800 b Fn(E)14792 15475 y Fm(N)15689 15287 y Fn(f)16412 14787 y Fk(2)17140 15287 y Fq(log)17 b(\()20206 14467 y Fn(f)20929 14027 y Fk(2)p 19310 15008 3041 49 v 19310 16118 a Fn(E)20205 16306 y Fm(N)21102 16118 y Fn(f)21825 15768 y Fk(2)22484 15287 y Fq(\))337 b Fo(\267)g Fn(c)25097 15469 y Fk(0)25622 15287 y Fq(\()p Fn(\256)275 b Fq(+)269 b(1\))29432 14787 y Fk(2)29958 15287 y Fn(E)30853 15475 y Fm(N)31749 15287 y Fo(jr)33096 15475 y Fk(\244)33749 15624 y Fi(N)34573 15287 y Fn(f)130 b Fo(j)35633 14787 y Fk(2)36160 15287 y Fn(;)0 18669 y Fq(where)551 b Fo(jr)4999 18857 y Fk(\244)5652 19006 y Fi(N)6476 18669 y Fn(f)130 b Fo(j)7536 18229 y Fk(2)8645 18669 y Fq(=)10169 17760 y Fj(P)11449 19027 y Fm(x)p Fl(2)p Fk(\244)13259 19176 y Fi(N)14285 18669 y Fo(jr)15632 18851 y Fm(x)16217 18669 y Fn(f)g Fo(j)17277 18229 y Fk(2)17803 18669 y Fq(.)980 b(Th)-34 b(us)553 b(\(1.16\))f(is)f(pro)-34 b(v)g(en.)980 b(The)552 b(global)f(log-Sob)34 b(olev)551 b(inequalit)-34 b(y)0 20196 y(\(1.13\))405 b(follo)-34 b(ws)404 b(from)h(\(1.16\))f(b)-34 b(y)405 b(simple)f(appro)-34 b(ximation.)22125 b Fe(\244)21610 24612 y Fq(5.)606 b Fp(Appendix)1328 26771 y Fq(In)531 b(this)h(App)34 b(endix,)562 b(w)-34 b(e)531 b(giv)-34 b(e)531 b(the)g(pro)34 b(of)531 b(of)g(Lemma)g(4.4.)918 b(Under)531 b(the)g(condition)h(\(1.3\),)563 b(if)531 b Fn(q)43 b Fq(\()p Fn(K)87 b Fq(\))532 b(is)0 28210 y(su\261cien)-34 b(tly)404 b(small,)g(the)g(inequalit)-34 b(y)404 b(\(1.11\))h(holds)g(for)f(a)g(p)34 b(ossibly)404 b(di\256eren)-34 b(t)404 b Fn(\270)36976 28392 y Fk(1)37839 28210 y Fn(>)336 b Fq(0:)0 32302 y(\(5.1\))6906 b(inf)9154 33109 y Fm(x)p Fl(2R)11316 29817 y Fj(8)11316 30908 y(<)11316 33089 y(:)12394 32302 y Fq(min)p Fo(f)p Fn(K)87 b Fq(\()p Fn(x;)202 b(x)p Fq(\))p Fn(;)g Fq(1)269 b Fo(\241)g Fn(K)87 b Fq(\()p Fn(x;)202 b(x)p Fq(\))p Fo(g)270 b(\241)28798 31150 y Fj(X)27700 33785 y Fm(y)32 b Fl(2R)p Fk(:)p Fm(y)g Fl(6)p Fk(=)p Fm(x)31849 32302 y Fo(j)p Fn(K)87 b Fq(\()p Fn(x;)202 b(y)43 b Fq(\))p Fo(j)36450 32484 y Fk(1)36976 29817 y Fj(9)36976 30908 y(=)36976 33089 y(;)38391 32302 y Fo(\270)336 b Fn(\270)40377 32484 y Fk(1)40903 32302 y Fn(:)0 36393 y Fq(In)404 b(the)h(sequel,)e(the)h(constan)-34 b(ts)406 b(dep)34 b(end)405 b(only)f(on)h Fn(\270)f Fq(in)g(\(1.3\))g (and)h Fn(\270)32040 36575 y Fk(1)32970 36393 y Fq(in)f(\(5.1\).)1328 37832 y(Recall)g(that)h(Dobrushin's)g(indep)34 b(endence)404 b(matrix)g(is)g(giv)-34 b(en)404 b(b)-34 b(y)0 40487 y(\(5.2\))13087 b Fn(C)16444 40669 y Fm(xy)17863 40487 y Fq(:=)383 b(sup)19479 41540 y Fm(\273)35 b Fl(2)p Fm(X)22373 40487 y Fq(sup)21600 41530 y Fm(A)p Fl(2F)23604 41723 y Fr(f)p Fi(x)p Fr(g)25173 40487 y Fo(j)p Fn(E)26475 39987 y Fm(\273)26405 40787 y(x)26989 40487 y Fq(\()p Fn(A)p Fq(\))270 b Fo(\241)g Fn(E)31288 39987 y Fm(\273)31737 39674 y Fi(y)31218 40787 y Fm(x)32290 40487 y Fq(\()p Fn(A)p Fq(\))p Fo(j)p Fn(;)0 44519 y Fq(where)432 b Fn(E)4498 43929 y Fm(\273)4428 44677 y(x)5013 44519 y Fq(\()p Fn(A)p Fq(\))384 b(:=)f Fn(E)9806 44766 y Fl(f)p Fm(x)p Fl(g)11331 44519 y Fq(\(1)12408 44707 y Fm(A)13169 44519 y Fq(\)\()p Fn(\273)56 b Fq(\))433 b(\(see)f(\(3.4\)\))h(and)g(hereafter)f(w)-34 b(e)432 b(will)g(also)g(use)g Fn(E)38635 43929 y Fm(\273)38565 44901 y Fk(\244)39273 44519 y Fq(\()p Fo(\242)p Fq(\))g(for)h Fn(E)43763 44707 y Fk(\244)44470 44519 y Fq(\()p Fo(\242)p Fq(\)\()p Fn(\273)56 b Fq(\).)624 b(It)432 b(is)0 45958 y(easy)404 b(to)g(see)g(that)0 48590 y(\(5.3\))12325 b Fn(C)15682 48772 y Fm(xy)17101 48590 y Fq(=)383 b(sup)18381 49643 y Fm(\273)35 b Fl(2)p Fm(X)20501 48590 y Fo(j)p Fn(\260)21466 48772 y Fm(x)22050 48590 y Fq(\()p Fn(\273)23051 48772 y Fm(x)23637 48590 y Fq(;)202 b Fn(\273)24706 48837 y Fl(f)p Fm(x)p Fl(g)26176 48585 y Fi(c)26659 48590 y Fq(\))270 b Fo(\241)f Fn(\260)29240 48772 y Fm(x)29824 48590 y Fq(\()p Fn(\273)30825 48772 y Fm(x)31411 48590 y Fq(;)202 b Fn(\273)32536 48001 y Fm(y)32480 49031 y Fl(f)p Fm(x)p Fl(g)33950 48779 y Fi(c)34433 48590 y Fq(\))p Fo(j)p Fn(:)0 52070 y Fw(Lemma)464 b(5.1.)554 b Fg(Supp)-62 b(ose)431 b(that)h(\(5.1\))g(holds.)555 b(Then)0 54950 y Fq(\(5.4\))12728 b(sup)15151 55992 y Fm(x)p Fl(2R)17313 54950 y Fq(max\()20044 53798 y Fj(X)20040 56433 y Fm(y)32 b Fl(6)p Fk(=)p Fm(x)22001 54950 y Fn(C)22867 55132 y Fm(xy)23949 54950 y Fn(;)24694 53798 y Fj(X)24690 56433 y Fm(y)g Fl(6)p Fk(=)p Fm(x)26650 54950 y Fn(C)27516 55132 y Fm(y)g(x)28598 54950 y Fq(\))338 b Fo(\267)e Fn(c)31211 55132 y Fk(1)31736 54950 y Fn(q)43 b Fq(\()p Fn(K)87 b Fq(\))34378 54449 y Fk(2)34906 54950 y Fn(:)0 58860 y Fg(Pr)-62 b(o)g(of.)552 b Fq(By)404 b(\(5.3\))g(w)-34 b(e)405 b(follo)-34 b(w)405 b(the)f(argumen)-34 b(t)405 b(of)g(the)f(pro)34 b(of)405 b(of)f(Theorem)g(1.2)g(to)h(get)0 61740 y(\(5.5\))16652 b(sup)19075 62782 y Fm(x)p Fl(2R)21241 60588 y Fj(X)21237 63223 y Fm(y)32 b Fl(6)p Fk(=)p Fm(x)23198 61740 y Fn(C)24064 61922 y Fm(xy)25483 61740 y Fo(\267)336 b Fn(c)27287 61239 y Fl(0)27287 62039 y Fk(1)27812 61740 y Fn(q)43 b Fq(\()p Fn(K)87 b Fq(\))30454 61239 y Fk(2)30982 61740 y Fn(:)0 65662 y Fq(In)404 b(order)g(to)g(get)h(the)f(other)g(b) 34 b(ound)406 b(recall)d(that)0 68931 y(\(5.6\))15299 b Fn(\260)18418 69113 y Fm(x)19003 68931 y Fq(\()p Fn(\273)20004 69113 y Fm(x)20589 68931 y Fq(;)202 b Fn(\273)21658 69178 y Fl(f)p Fm(x)p Fl(g)23128 68926 y Fi(c)23611 68931 y Fq(\))337 b(=)524 b(lim)25699 69748 y Fk(\242)p Fl("R)28591 68111 y Fn(\271)p Fq(\()p Fn(\273)30322 68299 y Fk(\242)31163 68111 y Fq(\))p 28091 68652 4044 49 v 28091 69762 a Fn(\271)p Fq(\()p Fn(\273)29822 70009 y Fk(\242)p Fl(n)p Fm(x)31663 69762 y Fq(\))32267 68931 y Fn(:)0 72444 y Fq(It)374 b(is)g(v)-34 b(ery)373 b(con)-34 b(v)g(enien)g(t)375 b(to)f(write)g(the)g(r.h.s.)528 b(of)375 b(the)f(ab)34 b(o)-34 b(v)g(e)374 b(equation)h(just)g(b)-34 b(y)374 b Fn(\271)p Fq(\()p Fn(\273)56 b Fq(\))p Fn(=\271)p Fq(\()p Fn(\273)41159 72690 y Fl(f)p Fm(x)p Fl(g)42629 72438 y Fi(c)43114 72444 y Fq(\))374 b(ev)-34 b(en)374 b(though)0 73931 y(b)34 b(oth)378 b(n)-34 b(umerator)378 b(and)g(denominator)g (are)e(zero.)529 b(By)376 b(follo)-34 b(wing)378 b(this)g(con)-34 b(v)g(en)g(tion)378 b(and)g(b)-34 b(y)378 b(using)f(\(3.3\))h(and)24726 75260 y Fk(12)p eop %%Page: 13 13 13 12 bop 0 393 a Fq(the)405 b(fact)f(that)h(\(informal!\))540 b Fn(\271)p Fq(\()p Fn(\273)15068 639 y Fl(f)p Fm(x)p Fl(g)16538 387 y Fi(c)17021 393 y Fq(\))337 b(=)g Fn(\271)p Fq(\()p Fn(\273)56 b Fq(\))271 b(+)e Fn(\271)p Fq(\()p Fn(\273)24637 -47 y Fm(x)25222 393 y Fq(\))405 b(w)-34 b(e)404 b(see)g(that)5490 2763 y Fo(j)p Fn(\260)6455 2945 y Fm(x)7039 2763 y Fq(\()p Fn(\273)8040 2945 y Fm(x)8626 2763 y Fq(;)202 b Fn(\273)9695 3010 y Fl(f)p Fm(x)p Fl(g)11165 2758 y Fi(c)11647 2763 y Fq(\))270 b Fo(\241)f Fn(\260)14228 2945 y Fm(x)14813 2763 y Fq(\()p Fn(\273)15814 2945 y Fm(x)16400 2763 y Fq(;)202 b Fn(\273)17525 2173 y Fm(y)17469 3204 y Fl(f)p Fm(x)p Fl(g)18939 2951 y Fi(c)19421 2763 y Fq(\))p Fo(j)1107 b(\267)g Fq(\(1)270 b Fo(\241)f Fn(\270)p Fq(\))27123 2263 y Fk(2)27649 2763 y Fo(j)p Fn(\260)28614 2945 y Fm(x)29198 2763 y Fq(\()p Fn(\273)30199 2945 y Fm(x)30785 2763 y Fq(;)202 b Fn(\273)31854 3010 y Fl(f)p Fm(x)p Fl(g)33324 2758 y Fi(c)33807 2763 y Fq(\))34278 2263 y Fl(\241)p Fk(1)35805 2763 y Fo(\241)269 b Fn(\260)37645 2945 y Fm(x)38230 2763 y Fq(\()p Fn(\273)39231 2945 y Fm(x)39817 2763 y Fq(;)202 b Fn(\273)40942 2173 y Fm(y)40886 3204 y Fl(f)p Fm(x)p Fl(g)42356 2951 y Fi(c)42838 2763 y Fq(\))43309 2263 y Fl(\241)p Fk(1)44567 2763 y Fo(j)21336 5625 y Fq(=)1107 b(\(1)270 b Fo(\241)f Fn(\270)p Fq(\))27123 5124 y Fk(2)27649 5625 y Fo(j)28119 4805 y Fn(\271)p Fq(\()p Fn(\273)29906 4365 y Fm(x)30491 4805 y Fq(\))p 28119 5346 2845 49 v 28411 6456 a Fn(\271)p Fq(\()p Fn(\273)56 b Fq(\))31365 5625 y Fo(\241)32710 4805 y Fn(\271)p Fq(\()p Fn(\273)34497 4365 y Fm(xy)35580 4805 y Fq(\))p 32710 5346 3342 49 v 32975 6456 a Fn(\271)p Fq(\()p Fn(\273)34762 6106 y Fm(y)35315 6456 y Fq(\))36184 5625 y Fo(j)21336 8959 y Fq(=)1107 b(\(1)270 b Fo(\241)f Fn(\270)p Fq(\))27123 8458 y Fk(2)27649 8959 y Fo(j)28119 8139 y Fn(\271)p Fq(\()p Fn(\273)29906 7699 y Fm(y)30459 8139 y Fq(\))p 28119 8680 2812 49 v 28395 9790 a Fn(\271)p Fq(\()p Fn(\273)56 b Fq(\))31332 8959 y Fo(\241)32677 8139 y Fn(\271)p Fq(\()p Fn(\273)34464 7699 y Fm(xy)35547 8139 y Fq(\))p 32677 8680 3342 49 v 32926 9790 a Fn(\271)p Fq(\()p Fn(\273)34713 9440 y Fm(x)35298 9790 y Fq(\))36151 8959 y Fo(j)36621 8139 y Fn(\271)p Fq(\()p Fn(\273)38408 7699 y Fm(x)38993 8139 y Fq(\))p 36621 8680 2845 49 v 36637 9790 a Fn(\271)p Fq(\()p Fn(\273)38424 9440 y Fm(y)38977 9790 y Fq(\))21336 12541 y Fo(\267)23386 10832 y Fj(\265)24411 11721 y Fq(1)269 b Fo(\241)h Fn(\270)p 24411 12262 2795 49 v 25455 13373 a(\270)27338 10832 y Fj(\266)28231 11134 y Fk(4)28959 12541 y Fo(j)p Fn(\260)29924 12723 y Fm(y)30475 12541 y Fq(\()p Fn(\273)31476 12723 y Fm(y)32029 12541 y Fq(;)202 b Fn(\273)33098 12788 y Fl(f)p Fm(y)32 b Fl(g)34535 12536 y Fi(c)35018 12541 y Fq(\))270 b Fo(\241)f Fn(\260)37599 12723 y Fm(y)38150 12541 y Fq(\()p Fn(\273)39151 12723 y Fm(y)39704 12541 y Fq(;)202 b Fn(\273)40829 12041 y Fm(x)40773 12899 y Fl(f)p Fm(y)32 b Fl(g)42210 12647 y Fi(c)42693 12541 y Fq(\))p Fo(j)p Fn(;)-43838 b Fq(\(5.7\))0 15433 y(where)404 b Fn(\273)4091 14993 y Fm(xy)5510 15433 y Fq(=)336 b(\()p Fn(\273)7846 14993 y Fm(x)8432 15433 y Fq(\))8903 14993 y Fm(y)9859 15433 y Fq(and)405 b(w)-34 b(e)405 b(ha)-34 b(v)g(e)404 b(used)h(the)f(b)34 b(ound)0 18545 y(\(5.8\))20281 17725 y Fn(\271)p Fq(\()p Fn(\273)22068 17285 y Fm(x)22654 17725 y Fq(\))p 20281 18266 2845 49 v 20297 19376 a Fn(\271)p Fq(\()p Fn(\273)22084 19026 y Fm(y)22638 19376 y Fq(\))23595 18545 y Fo(\267)24875 16836 y Fj(\265)25900 17725 y Fq(1)269 b Fo(\241)g Fn(\270)p 25900 18266 2795 49 v 26943 19376 a(\270)28827 16836 y Fj(\266)29719 17138 y Fk(2)0 21789 y Fq(in)484 b(the)h(last)f (inequalit)-34 b(y)-101 b(.)779 b(In)484 b(fact,)505 b(w)-34 b(e)484 b(see)g(that)23438 21192 y Fm(\271)p Fk(\()p Fm(\273)24819 20879 y Fi(x)25337 21192 y Fk(\))p 23438 21510 2265 49 v 23448 22223 a Fm(\271)p Fk(\()p Fm(\273)24829 21971 y Fi(y)25328 22223 y Fk(\))26306 21789 y Fq(=)27852 21192 y Fm(\271)p Fk(\()p Fm(\273)29233 20879 y Fi(x)29751 21192 y Fk(\))p 27852 21510 V 27862 22223 a Fm(\271)p Fk(\()p Fm(\273)29243 21971 y Fi(y)29742 22223 y Fk(\))30383 21192 y Fm(\271)p Fk(\()p Fm(\273)31764 20879 y Fi(xy)32726 21192 y Fk(\))p 30383 21510 2710 49 v 30383 22223 a Fm(\271)p Fk(\()p Fm(\273)31764 21971 y Fi(xy)32726 22223 y Fk(\))33709 21789 y Fq(and)h Fn(\271)p Fq(\()p Fn(\273)37933 21349 y Fm(x)38519 21789 y Fq(\))p Fn(=\271)p Fq(\()p Fn(\273)41383 21349 y Fm(xy)42466 21789 y Fq(\),)504 b(for)485 b(example,)0 23639 y(is)552 b(equal)g(to)h Fn(\260)6972 23821 y Fm(y)7524 23639 y Fq(\()p Fn(\273)8525 23821 y Fm(y)9078 23639 y Fq(;)202 b Fn(\273)10203 23199 y Fm(x)10147 24057 y Fl(f)p Fm(y)32 b Fl(g)11584 23805 y Fi(c)12067 23639 y Fq(\))p Fn(=\260)13772 23821 y Fm(y)14324 23639 y Fq(\()p 14795 22652 587 49 v Fn(\273)15381 23930 y Fm(y)15933 23639 y Fq(;)202 b Fn(\273)17058 23199 y Fm(x)17002 24057 y Fl(f)p Fm(y)32 b Fl(g)18439 23805 y Fi(c)18922 23639 y Fq(\).)984 b(W)-101 b(e)552 b(then)h(use)g(\(3.3\).)984 b(No)-34 b(w)554 b(b)-34 b(y)552 b(using)i(\(5.7\))f(and)g(the)g(same)0 25298 y(argumen)-34 b(t)405 b(used)g(to)f(get)h(\(5.5\))f(w)-34 b(e)405 b(conclude)f(the)g(pro)34 b(of.)22906 b Fe(\244)1328 27687 y Fq(F)-101 b(rom)529 b(no)-34 b(w)530 b(on)f(w)-34 b(e)528 b(assume)h(that)h Fn(q)43 b Fq(\()p Fn(K)87 b Fq(\))530 b(is)e(su\261cien)-34 b(tly)529 b(small)f(so)h(that)g(the)g (Dobrushin)h(uniqueness)0 29126 y(condition)405 b(holds,)f(i.e.,)0 31466 y(\(5.9\))17286 b(sup)19709 32509 y Fm(x)p Fl(2R)23048 30314 y Fj(X)21871 32950 y Fm(y)32 b Fl(2R)p Fk(:)158 b Fm(y)32 b Fl(6)p Fk(=)p Fm(x)26177 31466 y Fn(C)27043 31648 y Fm(xy)28462 31466 y Fn(<)337 b Fq(1)p Fn(:)0 34916 y Fq(F)-101 b(or)404 b(an)-34 b(y)405 b(function)g Fn(f)468 b Fq(:)336 b Fn(X)432 b Fo(!)337 b Fw(R)404 b Fq(and)h(\242)337 b Fo(\275)g(R)p Fq(,)404 b(de\257ne)0 37075 y(\(5.10\))11848 b(V)-101 b(ar)16834 37263 y Fk(\242)17673 37075 y Fq(\()p Fn(f)130 b Fq(\))338 b(:=)2886 b(sup)21293 38128 y Fm(\273)35 b(;\263)61 b Fl(2)p Fm(X)69 b Fk(:)156 b Fm(\273)24779 38300 y Ff(\242)25443 38109 y Fi(c)25925 38128 y Fk(=)p Fm(\263)27071 38300 y Ff(\242)27735 38109 y Fi(c)28418 37075 y Fo(j)p Fn(f)130 b Fq(\()p Fn(\273)56 b Fq(\))271 b Fo(\241)e Fn(f)130 b Fq(\()p Fn(\263)89 b Fq(\))p Fo(j)p Fn(:)0 40162 y Fq(The)405 b(follo)-34 b(wing)404 b(result)h(is)e(giv)-34 b(en)404 b(in)h([9)o(]:)0 42444 y Fw(Theorem)g(5.2.)511 b Fg(Assume)384 b(the)h(Dobrushin)f (uniqueness)g(c)-62 b(ondition)384 b(\(5.9\))g(holds.)539 b(Then)386 b(for)f(any)h Fq(\244)336 b Fo(\275\275)g(R)0 43883 y Fg(and)433 b Fn(\273)393 b Fo(2)337 b Fn(X)529 b Fg(we)433 b(have)0 46223 y Fq(\(5.11\))12572 b Fo(j)p Fn(E)16971 45633 y Fm(\273)16901 46604 y Fk(\244)17608 46223 y Fq(\()p Fn(f)130 b Fq(;)202 b Fn(g)43 b Fq(\))p Fo(j)339 b(\267)22686 45071 y Fj(X)22388 47706 y Fm(k)24 b(;l)11 b Fl(2)p Fk(\244)24936 46223 y Fq(V)-101 b(ar)26825 46420 y Fm(k)27394 46223 y Fq(\()p Fn(f)130 b Fq(\))p Fn(D)30063 46420 y Fm(k)24 b(l)30925 46223 y Fq(V)-101 b(ar)32813 46420 y Fm(l)33160 46223 y Fq(\()p Fn(g)43 b Fq(\))0 50035 y Fg(with)433 b Fn(D)3637 50232 y Fm(k)24 b(l)4833 50035 y Fq(=)6113 49126 y Fj(P)7392 49477 y Fl(1)7392 50394 y Fm(n)p Fk(=0)9221 50035 y Fq(\()p Fn(C)10645 49596 y Fm(n)11272 50035 y Fq(\))11743 50232 y Fm(k)g(l)12603 50035 y Fg(,)434 b(wher)-62 b(e)432 b Fn(E)17796 49446 y Fm(\273)17726 50417 y Fk(\244)18434 50035 y Fq(\()p Fn(f)130 b Fq(;)202 b Fn(g)43 b Fq(\))339 b(=)d Fn(E)23842 49446 y Fm(\273)23772 50417 y Fk(\244)24480 50035 y Fq(\()p Fn(f)130 b(g)43 b Fq(\))271 b Fo(\241)f Fn(E)29215 49446 y Fm(\273)29145 50417 y Fk(\244)29852 50035 y Fq(\()p Fn(f)130 b Fq(\))p Fn(E)32482 49446 y Fm(\273)32412 50417 y Fk(\244)33122 50035 y Fq(\()p Fn(g)43 b Fq(\))p Fg(.)1328 52674 y Fq(F)-101 b(or)404 b Fn(x)472 b(=)-741 b Fo(2)337 b Fq(\244)f Fo(\275\275)g(R)405 b Fq(w)-34 b(e)404 b(in)-34 b(tro)34 b(duce)405 b(an)f(op)34 b(erator)404 b Fn(E)25222 52035 y Fk(\()p Fm(x)p Fk(\))25152 53056 y(\244)26943 52674 y Fq(b)-34 b(y)0 55190 y(\(5.12\))17508 b Fn(E)21570 54551 y Fk(\()p Fm(x)p Fk(\))21500 55571 y(\244)23223 55190 y Fq(:=)336 b Fn(R)25759 55372 y Fm(x)26344 55190 y Fn(E)27239 55378 y Fk(\244)27947 55190 y Fn(R)28867 55372 y Fm(x)29452 55190 y Fn(;)0 57349 y Fq(where)404 b Fn(R)4425 57531 y Fm(x)5414 57349 y Fq(is)g(the)h(in)-34 b(v)g(olution)405 b(de\257ned)g(b)-34 b(y)405 b Fn(R)21190 57531 y Fm(x)21775 57349 y Fn(f)130 b Fq(\()p Fn(\273)56 b Fq(\))338 b(=)f Fn(f)130 b Fq(\()p Fn(\273)27424 56909 y Fm(x)28010 57349 y Fq(\))405 b(for)f(all)f Fn(\273)393 b Fo(2)337 b Fn(X)95 b Fq(.)539 b(It)404 b(is)g(ob)-34 b(vious)405 b(that)0 60133 y(\(5.13\))11789 b Fn(E)15851 59494 y Fk(\()p Fm(x)p Fk(\))15781 60515 y(\244)17168 60133 y Fn(f)130 b Fq(\()p Fn(\273)56 b Fq(\))338 b(=)21037 58483 y Fj(Z)21710 61232 y Fm(X)22484 61381 y Ff(\244)23349 60133 y Fn(\260)23977 60321 y Fk(\244)24684 60133 y Fq(\()p Fn(\263)25685 60321 y Fk(\244)26394 60133 y Fq(;)202 b Fn(\273)27519 59633 y Fm(x)27463 60433 y Fk(\244)28116 60181 y Fi(c)28597 60133 y Fq(\))p Fn(f)130 b Fq(\()p Fn(\263)30792 60321 y Fk(\244)31502 60133 y Fn(\273)32032 60321 y Fk(\244)32685 60069 y Fi(c)33167 60133 y Fq(\))p Fn(d\263)34799 60321 y Fk(\244)0 63515 y Fq(and)420 b(hence)g Fn(E)6720 62877 y Fk(\()p Fm(x)p Fk(\))6650 63897 y(\244)8456 63515 y Fq(and)g Fn(E)11723 63703 y Fk(\244)12850 63515 y Fq(are)f(m)-34 b(utually)421 b(absolutely)f(con)-34 b(tin)g(uous)421 b(b)-34 b(y)420 b(\(3.2\))g(when)h(they)f(are)f (understo)34 b(o)g(d)0 64954 y(as)406 b(probabilit)-34 b(y)407 b(measures)f(on)g Fn(X)501 b Fq(as)406 b(usual.)545 b(W)-101 b(e)406 b(denote)g(b)-34 b(y)407 b Fn(u)29601 65142 y Fm(x;)p Fk(\244)31505 64954 y Fq(its)f(Radon-Nik)-34 b(o)34 b(dym)407 b(deriv)-67 b(ativ)-34 b(e,)405 b(that)0 66394 y(is,)0 69618 y(\(5.14\))12940 b Fn(u)16731 69806 y Fm(x;)p Fk(\244)18230 69618 y Fq(\()p Fn(\273)56 b Fq(\))337 b(:=)21844 68763 y Fn(dE)23440 68124 y Fk(\()p Fm(x)p Fk(\))23370 69145 y(\244)p 21844 69339 2913 49 v 22184 70449 a Fn(dE)23710 70637 y Fk(\244)24889 69618 y Fq(\()p Fn(\273)56 b Fq(\))338 b(=)28167 68786 y Fn(\260)28795 68974 y Fk(\244)29503 68786 y Fq(\()p Fn(\273)30504 68974 y Fk(\244)31212 68786 y Fq(;)202 b Fn(\273)32337 68346 y Fm(x)32281 69145 y Fk(\244)32934 68893 y Fi(c)33416 68786 y Fq(\))p 28167 69339 5720 49 v 28167 70449 a Fn(\260)28795 70637 y Fk(\244)29503 70449 y Fq(\()p Fn(\273)30504 70637 y Fk(\244)31212 70449 y Fq(;)g Fn(\273)32281 70637 y Fk(\244)32934 70385 y Fi(c)33416 70449 y Fq(\))34020 69618 y Fn(;)0 72492 y Fq(where)461 b(the)g(second)g(equalit)-34 b(y)460 b(follo)-34 b(ws)462 b(immediately)e(from)g(\(3.4\))i(and)f (\(5.13\).)709 b(It)461 b(should)g(b)34 b(e)461 b(noted)g(that)0 73931 y Fn(u)694 74119 y Fm(x;)p Fk(\244)2597 73931 y Fq(is)404 b(uniformly)g(b)34 b(ounded)405 b(in)g Fn(x)e Fq(and)i(\244.)24726 75260 y Fk(13)p eop %%Page: 14 14 14 13 bop 1328 393 a Fq(F)-101 b(or)404 b(an)-34 b(y)405 b Fn(x)336 b Fo(6)p Fq(=)h Fn(y)380 b Fo(2)337 b(R)p Fq(,)404 b(de\257ne)0 2523 y(\(5.15\))13456 b Fn(G)17506 2705 y Fm(xy)18924 2523 y Fq(:=)3017 b(sup)20541 3625 y Fk(\244)p Fl(\275\275R)p Fk(:)156 b Fm(y)32 b Fl(2)p Fk(\244)p Fm(;x)105 b(=)-575 b Fl(2)p Fk(\244)27928 2523 y Fq(V)-101 b(ar)29816 2705 y Fm(y)30369 2523 y Fq(\()p Fn(u)31534 2711 y Fm(x;)p Fk(\244)33033 2523 y Fq(\))p Fn(:)0 5765 y Fw(Lemma)464 b(5.3.)554 b Fg(Supp)-62 b(ose)431 b(that)h(\(5.1\))g(holds.)555 b(Then)0 8143 y Fq(\(5.16\))12136 b(sup)15166 9186 y Fm(x)p Fl(2R)17328 8143 y Fq(max\()20059 6992 y Fj(X)20055 9627 y Fm(y)32 b Fl(6)p Fk(=)p Fm(x)22015 8143 y Fn(G)22968 8325 y Fm(xy)24050 8143 y Fn(;)24593 6992 y Fj(X)24589 9627 y Fm(y)g Fl(6)p Fk(=)p Fm(x)26549 8143 y Fn(G)27502 8325 y Fm(y)g(x)28584 8143 y Fq(\))337 b Fo(\267)g Fn(c)31197 8325 y Fk(2)31722 8143 y Fn(q)43 b Fq(\()p Fn(K)87 b Fq(\))34364 7643 y Fk(2)34891 8143 y Fn(:)0 11767 y Fg(Pr)-62 b(o)g(of.)552 b Fq(F)-101 b(ollo)-34 b(wing)405 b(the)g(con)-34 b(v)g(en)g(tion)405 b(used)g(in)f(the)h(pro)34 b(of)404 b(of)g(Lemma)g(5.1)g(w)-34 b(e)404 b(see)g(from)g(\(5.14\))h(that)13364 13897 y(V)-101 b(ar)15253 14079 y Fm(y)15805 13897 y Fq(\()p Fn(u)16970 14085 y Fm(x;)p Fk(\244)18469 13897 y Fq(\))1108 b(=)1153 b(sup)22097 14950 y Fm(\273)35 b Fl(2)p Fm(X)24218 13897 y Fo(j)p Fn(u)25249 14085 y Fm(x;)p Fk(\244)26747 13897 y Fq(\()p Fn(\273)27804 13397 y Fm(y)28357 13897 y Fq(\))270 b Fo(\241)f Fn(u)31004 14085 y Fm(x;)p Fk(\244)32503 13897 y Fq(\()p Fn(\273)56 b Fq(\))p Fo(j)20048 17443 y Fq(=)1153 b(sup)22097 18496 y Fm(\273)35 b Fl(2)p Fm(X)24218 17443 y Fo(j)24688 16623 y Fn(\271)p Fq(\()p Fn(\273)26475 16183 y Fm(y)d(x)27557 16623 y Fq(\))p 24688 17164 3342 49 v 24952 18274 a Fn(\271)p Fq(\()p Fn(\273)26739 17924 y Fm(y)27292 18274 y Fq(\))28431 17443 y Fo(\241)29776 16623 y Fn(\271)p Fq(\()p Fn(\273)31563 16183 y Fm(x)32148 16623 y Fq(\))p 29776 17164 2845 49 v 30068 18274 a Fn(\271)p Fq(\()p Fn(\273)56 b Fq(\))32753 17443 y Fo(j)33223 16623 y Fn(\271)p Fq(\()p Fn(\273)34954 16811 y Fk(\244)35607 16558 y Fi(c)36089 16623 y Fq(\))p 33223 17164 3338 49 v 33223 18274 a Fn(\271)p Fq(\()p Fn(\273)35010 17857 y Fm(x)34954 18656 y Fk(\244)35607 18404 y Fi(c)36089 18274 y Fq(\))36693 17443 y Fn(:)-37030 b Fq(\(5.17\))0 20433 y(W)-101 b(e)404 b(notice)g(that)10555 22964 y Fo(j)11025 22144 y Fn(\271)p Fq(\()p Fn(\273)12812 21704 y Fm(y)32 b(x)13895 22144 y Fq(\))p 11025 22686 3342 49 v 11290 23796 a Fn(\271)p Fq(\()p Fn(\273)13077 23446 y Fm(y)13630 23796 y Fq(\))14768 22964 y Fo(\241)16113 22144 y Fn(\271)p Fq(\()p Fn(\273)17900 21704 y Fm(x)18486 22144 y Fq(\))p 16113 22686 2845 49 v 16406 23796 a Fn(\271)p Fq(\()p Fn(\273)56 b Fq(\))19090 22964 y Fo(j)1107 b Fq(=)g Fo(j)p Fn(\260)23549 23146 y Fm(x)24133 22964 y Fq(\()p Fn(\273)25134 23146 y Fm(x)25719 22964 y Fq(;)202 b Fn(\273)26844 22375 y Fm(y)26788 23405 y Fl(f)p Fm(x)p Fl(g)28258 23153 y Fi(c)28741 22964 y Fq(\))29212 22464 y Fl(\241)p Fk(1)30740 22964 y Fo(\241)269 b Fn(\260)32580 23146 y Fm(x)33164 22964 y Fq(\()p Fn(\273)34165 23146 y Fm(x)34751 22964 y Fq(;)202 b Fn(\273)35820 23211 y Fl(f)p Fm(x)p Fl(g)37290 22959 y Fi(c)37773 22964 y Fq(\))38244 22464 y Fl(\241)p Fk(1)39502 22964 y Fo(j)20534 25640 y(\267)1107 b Fn(\270)23291 25140 y Fl(\241)p Fk(2)24548 25640 y Fo(j)p Fn(\260)25513 25822 y Fm(x)26098 25640 y Fq(\()p Fn(\273)27099 25822 y Fm(x)27684 25640 y Fq(;)202 b Fn(\273)28809 25051 y Fm(y)28753 26081 y Fl(f)p Fm(x)p Fl(g)30223 25829 y Fi(c)30706 25640 y Fq(\))270 b Fo(\241)f Fn(\260)33287 25822 y Fm(x)33871 25640 y Fq(\()p Fn(\273)34872 25822 y Fm(x)35458 25640 y Fq(;)202 b Fn(\273)36527 25887 y Fl(f)p Fm(x)p Fl(g)37997 25635 y Fi(c)38480 25640 y Fq(\))p Fo(j)p Fn(:)-39625 b Fq(\(5.18\))0 27946 y(No)-34 b(w)301 b(b)-34 b(y)300 b(follo)-34 b(wing)300 b(exactly)f(the)h(same)g (w)-34 b(a)g(y)300 b(as)g(used)g(in)g(the)g(pro)34 b(of)300 b(of)g(Lemma)f(5.1)h(w)-34 b(e)300 b(\257nish)h(the)f(pro)34 b(of.)1212 b Fe(\244)1328 30297 y Fq(F)-101 b(or)404 b(\242)337 b Fo(\275)g Fq(\244,)403 b(de\257ne)0 32638 y(\(5.19\))15430 b Fn(\267)19225 31999 y Fk(\(\244\))19225 33020 y Fm(x;)p Fk(\242)21193 32638 y Fq(:=)336 b(V)-101 b(ar)24698 32826 y Fk(\242)25537 32638 y Fn(E)26432 32885 y Fk(\244)p Fl(n)p Fk(\242)28394 32638 y Fq(\()p Fn(u)29559 32826 y Fm(x;)p Fk(\244)31059 32638 y Fq(\))p Fn(;)0 34836 y Fq(and)405 b(for)f(an)-34 b(y)405 b Fn(x)336 b Fo(6)p Fq(=)h Fn(y)380 b Fo(2)337 b(R)p Fq(,)404 b(w)-34 b(e)404 b(put)0 37177 y(\(5.20\))14880 b Fn(\267)18675 37359 y Fm(xy)20094 37177 y Fq(:=)3016 b(sup)21710 38279 y Fk(\244)p Fl(\275\275R)p Fk(:)157 b Fm(y)32 b Fl(2)p Fk(\244)p Fm(;x)105 b(=)-575 b Fl(2)p Fk(\244)29097 37177 y Fn(\267)29795 36538 y Fk(\(\244\))29795 37617 y Fm(x;)p Fl(f)p Fm(y)32 b Fl(g)32080 37177 y Fn(:)0 40419 y Fw(Lemma)464 b(5.4.)554 b Fg(Supp)-62 b(ose)431 b(that)h(the)h(Dobrushin)e (uniqueness)h(c)-62 b(ondition)432 b(\(5.9\))f(holds.)556 b(Then)0 42797 y Fq(\(5.21\))12391 b(sup)15420 43840 y Fm(x)p Fl(2R)17582 42797 y Fq(max\()20313 41645 y Fj(X)20309 44280 y Fm(y)32 b Fl(6)p Fk(=)p Fm(x)22270 42797 y Fn(\267)22968 42979 y Fm(xy)24050 42797 y Fn(;)24593 41645 y Fj(X)24589 44280 y Fm(y)g Fl(6)p Fk(=)p Fm(x)26549 42797 y Fn(\267)27247 42979 y Fm(y)g(x)28330 42797 y Fq(\))337 b Fo(\267)f Fn(c)30942 42979 y Fk(3)31467 42797 y Fn(q)43 b Fq(\()p Fn(K)87 b Fq(\))34109 42296 y Fk(2)34637 42797 y Fn(:)0 46814 y Fg(Pr)-62 b(o)g(of.)552 b Fq(T)-101 b(o)405 b(estimate)f(the)h (quan)-34 b(tities)405 b Fn(\267)18767 46176 y Fk(\(\244\))18767 47255 y Fm(x;)p Fl(f)p Fm(y)32 b Fl(g)21455 46814 y Fq(w)-34 b(e)405 b(remark)e(that)6962 49582 y Fo(jr)8309 49764 y Fm(y)8861 49582 y Fn(E)9756 49828 y Fk(\244)p Fl(n)p Fm(y)11431 49582 y Fn(u)12125 49770 y Fm(x;)p Fk(\244)13624 49582 y Fo(j)1106 b(\267)h(j)p Fn(R)18374 49764 y Fm(y)18926 49582 y Fn(E)19821 49828 y Fk(\244)p Fl(n)p Fm(y)21496 49582 y Fn(u)22190 49770 y Fm(x;)p Fk(\244)23958 49582 y Fo(\241)270 b Fn(E)26136 48943 y Fk(\()p Fm(y)32 b Fk(\))26066 50022 y(\244)p Fl(n)p Fm(y)27740 49582 y Fn(u)28434 49770 y Fm(x;)p Fk(\244)29933 49582 y Fo(j)269 b Fq(+)g Fo(j)p Fn(E)33053 48943 y Fk(\()p Fm(y)32 b Fk(\))32983 50022 y(\244)p Fl(n)p Fm(y)34658 49582 y Fn(u)35352 49770 y Fm(x;)p Fk(\244)37120 49582 y Fo(\241)269 b Fn(E)39227 49828 y Fk(\244)p Fl(n)p Fm(y)40902 49582 y Fn(u)41596 49770 y Fm(x;)p Fk(\244)43095 49582 y Fo(j)15067 51623 y(\267)1107 b Fq(V)-101 b(ar)19006 51805 y Fm(y)19558 51623 y Fq(\()p Fn(u)20723 51811 y Fm(x;)p Fk(\244)22222 51623 y Fq(\))270 b(+)f Fo(j)p Fn(E)25407 51870 y Fk(\244)p Fl(n)p Fm(y)27082 51623 y Fq(\()p Fn(u)28247 51811 y Fm(x;)p Fk(\244)29746 51623 y Fq(;)202 b Fn(u)30979 51870 y Fm(y)32 b(;)p Fk(\244)p Fl(n)p Fm(y)33412 51623 y Fq(\))p Fo(j)p Fn(:)-34557 b Fq(\(5.22\))0 53803 y(Using)404 b(\(5.11\))h(w)-34 b(e)405 b(see)e(that)9957 56181 y Fo(j)p Fn(E)11189 56427 y Fk(\244)p Fl(n)p Fm(y)12864 56181 y Fq(\()p Fn(u)14029 56369 y Fm(x;)p Fk(\244)15528 56181 y Fq(;)202 b Fn(u)16761 56427 y Fm(y)32 b(;)p Fk(\244)p Fl(n)p Fm(y)19194 56181 y Fq(\))p Fo(j)1107 b(\267)23940 55029 y Fj(X)23159 57713 y Fm(k)24 b(;l)11 b Fl(2)p Fk(\244)p Fl(n)p Fm(y)26674 56181 y Fq(V)-101 b(ar)28563 56378 y Fm(k)29132 56181 y Fq(\()p Fn(u)30297 56369 y Fm(x;)p Fk(\244)31796 56181 y Fq(\))p Fn(D)33271 56378 y Fm(k)24 b(l)34131 56181 y Fq(V)-101 b(ar)36020 56378 y Fm(l)36367 56181 y Fq(\()p Fn(u)37532 56427 y Fm(y)32 b(;)p Fk(\244)p Fl(n)p Fm(y)39965 56181 y Fq(\))21109 59799 y Fo(\267)25325 58648 y Fj(X)23159 61332 y Fm(k)24 b Fl(2f)p Fm(x)p Fl(g)25770 61080 y Fi(c)26197 61332 y Fm(;l)11 b Fl(2f)p Fm(y)32 b Fl(g)28814 61080 y Fi(c)29443 59799 y Fn(G)30396 59996 y Fm(xk)31495 59799 y Fn(D)32499 59996 y Fm(k)24 b(l)33359 59799 y Fn(G)34312 59996 y Fm(y)32 b(l)35155 59799 y Fn(:)-35492 b Fq(\(5.23\))0 63319 y(Hence)404 b(w)-34 b(e)404 b(obtain)0 65449 y(\(5.24\))12477 b Fn(\267)16272 65631 y Fm(xy)17691 65449 y Fo(\267)336 b Fn(G)19923 65631 y Fm(xy)21275 65449 y Fq(+)24653 64298 y Fj(X)22487 66982 y Fm(k)24 b Fl(2f)p Fm(x)p Fl(g)25098 66730 y Fi(c)25525 66982 y Fm(;l)11 b Fl(2f)p Fm(y)32 b Fl(g)28142 66730 y Fi(c)28771 65449 y Fn(G)29724 65646 y Fm(xk)30823 65449 y Fn(D)31827 65646 y Fm(k)24 b(l)32687 65449 y Fn(G)33640 65646 y Fm(y)32 b(l)34483 65449 y Fn(:)0 68969 y Fq(No)-34 b(w)405 b(w)-34 b(e)404 b(use)h(Lemma)e(5.1)h(and)h(Lemma)f(5.3)g(to)g (\257nish)h(the)g(pro)34 b(of.)17986 b Fe(\244)1328 71409 y Fq(Recall)404 b(that)h(w)-34 b(e)404 b(ha)-34 b(v)g(e)405 b(de\257ned)g(a)f(nonlinear)g(exp)34 b(ectation)29474 71102 y Fj(e)29227 71409 y Fn(E)30122 71597 y Fk(\244)31234 71409 y Fq(b)-34 b(y)0 73628 y(\(5.25\))20673 73322 y Fj(e)20426 73628 y Fn(E)21321 73816 y Fk(\244)22029 73628 y Fn(f)467 b Fq(=)337 b(\()p Fn(E)25735 73816 y Fk(\244)26443 73628 y Fn(f)27166 73128 y Fk(2)27693 73628 y Fq(\))28164 73128 y Fk(1)p Fm(=)p Fk(2)29631 73628 y Fn(:)24726 75260 y Fk(14)p eop %%Page: 15 15 15 14 bop 0 468 a Fw(Lemma)464 b(5.5.)554 b Fg(The)433 b(op)-62 b(er)g(ator)15160 161 y Fj(e)14913 468 y Fn(E)15808 656 y Fk(\244)16949 468 y Fg(has)433 b(the)f(fol)62 b(lowing)432 b(pr)-62 b(op)g(erties.)0 2626 y(\(a\))432 b(If)i Fq(\244)4159 2186 y Fl(0)4806 2626 y Fo(\275)337 b Fq(\244)f Fo(\275\275)g(R)p Fg(,)11566 2319 y Fj(e)11319 2626 y Fn(E)12214 2814 y Fk(\244)13168 2319 y Fj(e)12922 2626 y Fn(E)13817 2829 y Fk(\244)14470 2577 y Fr(0)15159 2626 y Fq(=)16685 2319 y Fj(e)16439 2626 y Fn(E)17334 2814 y Fk(\244)18041 2626 y Fg(.)0 4851 y(\(b\))432 b Fn(R)2902 5033 y Fm(x)3825 4851 y Fo(\267)336 b Fn(\270)5811 4411 y Fl(\241)p Fk(1)p Fm(=)p Fk(2)8256 4545 y Fj(e)8010 4851 y Fn(E)8905 5033 y Fm(x)9923 4851 y Fg(for)433 b(al)62 b(l)433 b Fn(x)336 b Fo(2)h(R)p Fg(.)0 7077 y(\(c\))2229 6770 y Fj(e)1982 7077 y Fn(E)2877 7265 y Fk(\244)3832 6770 y Fj(e)3585 7077 y Fn(E)4480 7259 y Fm(x)5402 7077 y Fo(\267)f Fn(c)7206 7259 y Fk(4)7978 6770 y Fj(e)7731 7077 y Fn(E)8626 7259 y Fm(x)9458 6770 y Fj(e)9211 7077 y Fn(E)10106 7265 y Fk(\244)11247 7077 y Fg(for)434 b(al)62 b(l)432 b Fn(x)471 b(=)-740 b Fo(2)336 b Fq(\244)h Fo(\275\275)f(R)p Fg(.)0 9302 y(\(d\))432 b Fo(j)2628 8996 y Fj(e)2381 9302 y Fn(E)3276 9490 y Fk(\244)3984 9302 y Fq(\()p Fn(f)130 b Fq(;)202 b Fn(g)43 b Fq(\))p Fo(j)338 b(\267)8764 8299 y(p)p 9774 8299 607 49 v 1003 x Fq(2\()11098 8996 y Fj(e)10851 9302 y Fn(E)11746 9490 y Fk(\244)12454 9302 y Fn(g)43 b Fq(\))13546 8862 y Fl(\241)p Fk(1)14804 9302 y Fq(V)-101 b(ar)16693 9490 y Fk(\244)17401 9302 y Fq(\()p Fn(g)18493 8862 y Fk(2)19020 9302 y Fq(\))270 b Fo(\242)20613 8996 y Fj(e)20367 9302 y Fn(E)21262 9490 y Fk(\244)21969 9302 y Fq(\()p Fn(f)401 b Fo(\241)269 b Fn(E)25541 9490 y Fk(\244)26249 9302 y Fn(f)130 b Fq(\))p Fg(,)0 11528 y(wher)-62 b(e)3670 11221 y Fj(e)3423 11528 y Fn(E)4318 11716 y Fk(\244)5025 11528 y Fq(\()p Fn(f)130 b Fq(;)202 b Fn(g)43 b Fq(\))339 b(=)9715 11221 y Fj(e)9469 11528 y Fn(E)10364 11716 y Fk(\244)11071 11528 y Fn(f)130 b(g)314 b Fo(\241)14145 11221 y Fj(e)13898 11528 y Fn(E)14793 11716 y Fk(\244)15501 11528 y Fn(f)16471 11221 y Fj(e)16224 11528 y Fn(E)17119 11716 y Fk(\244)17827 11528 y Fn(g)478 b Fg(for)433 b(any)g(r)-62 b(e)g(al)433 b(me)-62 b(asur)g(able)430 b(function)j Fn(f)564 b Fg(and)434 b Fn(g)43 b Fg(.)0 13757 y(Pr)-62 b(o)g(of.)552 b Fq(\(a\))405 b(It)f(is)g(ob)-34 b(vious)405 b(from)f(the)h(consistency)f(of)h(sp)34 b(eci\257cation.) 538 b(\(b\))405 b(By)f(\(3.3\))g(w)-34 b(e)405 b(get)12651 16564 y Fj(e)12404 16871 y Fn(E)13299 17053 y Fm(x)13884 16871 y Fn(f)130 b Fq(\()p Fn(\273)56 b Fq(\))1108 b(=)19293 15162 y Fj(\265)20185 15221 y(Z)21600 16871 y Fn(d\263)22761 17053 y Fm(x)23346 16871 y Fn(\260)23974 17053 y Fm(x)24558 16871 y Fq(\()p Fn(\263)25559 17053 y Fm(x)26145 16871 y Fq(;)202 b Fn(\273)27214 17117 y Fl(f)p Fm(x)p Fl(g)28684 16865 y Fi(c)29167 16871 y Fq(\))p Fn(f)30361 16370 y Fk(2)30888 16871 y Fq(\()p Fn(\263)31889 17053 y Fm(x)32474 16871 y Fn(\273)33004 17117 y Fl(f)p Fm(x)p Fl(g)34474 16865 y Fi(c)34957 16871 y Fq(\))35428 15162 y Fj(\266)36321 15464 y Fk(1)p Fm(=)p Fk(2)17243 19657 y Fo(\270)1107 b Fn(\270)20000 19157 y Fk(1)p Fm(=)p Fk(2)21467 19657 y Fn(R)22387 19839 y Fm(x)22972 19657 y Fn(f)130 b Fq(\()p Fn(\273)56 b Fq(\))p Fn(:)-25560 b Fq(\(5.26\))0 21725 y(\(c\))396 b(It)g(is)g(easy)f(to)i(c)-34 b(hec)g(k)395 b(b)-34 b(y)397 b(using)f(\(3.3\))g(and)h(the)f(uniform)h(b)34 b(oundedness)398 b(of)e Fn(u)37653 21913 y Fm(x;)p Fk(\244)39152 21725 y Fq(.)535 b(\(d\))397 b(W)-101 b(e)396 b(\257rst)g(assume)247 23071 y Fj(e)0 23377 y Fn(E)895 23565 y Fk(\244)1603 23377 y Fn(g)380 b Fq(=)336 b(1)405 b(and)f(observ)-34 b(e)404 b(that)0 26182 y(\(5.27\))13424 b Fo(j)17104 25876 y Fj(e)16858 26182 y Fn(E)17753 26370 y Fk(\244)18460 26182 y Fn(f)130 b(g)314 b Fo(\241)21534 25876 y Fj(e)21287 26182 y Fn(E)22182 26370 y Fk(\244)22890 26182 y Fn(f)130 b Fo(j)337 b Fq(=)25700 25362 y Fo(j)p Fn(E)26932 25550 y Fk(\244)27639 25362 y Fn(f)28362 24923 y Fk(2)28889 25362 y Fq(\()p Fn(g)29981 24923 y Fk(2)30777 25362 y Fo(\241)269 b Fq(1\))p Fo(j)p 25700 25904 7704 49 v 26420 26925 a Fj(e)26173 27231 y Fn(E)27068 27419 y Fk(\244)27776 27231 y Fn(f)130 b(g)314 b Fq(+)30850 26925 y Fj(e)30603 27231 y Fn(E)31498 27419 y Fk(\244)32206 27231 y Fn(f)33536 26182 y(:)0 29388 y Fq(Denoting)405 b(b)-34 b(y)7166 29081 y Fj(b)6919 29388 y Fn(E)7814 29576 y Fk(\244)8926 29388 y Fq(a)404 b(cop)-34 b(y)404 b(of)h Fn(E)15040 29576 y Fk(\244)16151 29388 y Fq(w)-34 b(e)405 b(get)5628 31963 y Fo(j)p Fn(E)6860 32151 y Fk(\244)7567 31963 y Fn(f)8290 31463 y Fk(2)8817 31963 y Fq(\()p Fn(g)9909 31463 y Fk(2)10705 31963 y Fo(\241)269 b Fq(1\))p Fo(j)1107 b Fq(=)16621 31143 y(1)p 16621 31685 607 49 v 16621 32795 a(2)17360 31963 y Fo(j)p Fn(E)18592 32151 y Fk(\244)19569 31963 y Fo(\255)21027 31657 y Fj(b)20781 31963 y Fn(E)21676 32151 y Fk(\244)22383 31963 y Fq(\()p Fn(f)23577 31463 y Fk(2)24374 31963 y Fo(\241)25813 31644 y Fj(b)25586 31963 y Fn(f)26309 31463 y Fk(2)26836 31963 y Fq(\)\()p Fn(g)28399 31463 y Fk(2)29195 31963 y Fo(\241)277 b Fj(b)-681 b Fn(g)31028 31463 y Fk(2)31555 31963 y Fq(\))p Fo(j)14438 34867 y(\267)16621 34047 y Fq(1)p 16621 34588 V 16621 35698 a(2)17360 34867 y(V)-101 b(ar)19249 35055 y Fk(\244)19957 34867 y Fq(\()p Fn(g)21049 34366 y Fk(2)21575 34867 y Fq(\))p Fn(E)22941 35055 y Fk(\244)23919 34867 y Fo(\255)25378 34560 y Fj(b)25131 34867 y Fn(E)26026 35055 y Fk(\244)26734 34867 y Fq(\()p Fn(f)27928 34366 y Fk(2)28724 34867 y Fo(\241)30163 34547 y Fj(b)29936 34867 y Fn(f)30659 34366 y Fk(2)31186 34867 y Fq(\))14438 37770 y Fo(\267)16621 36950 y Fq(1)p 16621 37491 V 16621 38601 a(2)17360 37770 y(V)g(ar)19249 37958 y Fk(\244)19957 37770 y Fq(\()p Fn(g)21049 37269 y Fk(2)21575 37770 y Fq(\)\()p Fn(E)23412 37958 y Fk(\244)24390 37770 y Fo(\255)25849 37463 y Fj(b)25602 37770 y Fn(E)26497 37958 y Fk(\244)27205 37770 y Fq(\()p Fn(f)401 b Fq(+)30109 37450 y Fj(b)29882 37770 y Fn(f)131 b Fq(\))31077 37269 y Fk(2)31603 37770 y Fq(\))32074 37269 y Fk(1)p Fm(=)p Fk(2)33541 37770 y Fq(\()p Fn(E)34907 37958 y Fk(\244)35884 37770 y Fo(\255)37343 37463 y Fj(b)37096 37770 y Fn(E)37991 37958 y Fk(\244)38699 37770 y Fo(j)p Fn(f)400 b Fo(\241)41468 37450 y Fj(b)41241 37770 y Fn(f)131 b Fo(j)42302 37269 y Fk(2)42828 37770 y Fq(\))43299 37269 y Fk(1)p Fm(=)p Fk(2)14438 40253 y Fo(\267)16488 39191 y(p)p 17498 39191 V 1062 x Fq(2V)-101 b(ar)19993 40441 y Fk(\244)20701 40253 y Fq(\()p Fn(g)21793 39753 y Fk(2)22320 40253 y Fq(\))23038 39947 y Fj(e)22791 40253 y Fn(E)23686 40441 y Fk(\244)24394 40253 y Fn(f)400 b Fo(\242)26240 39947 y Fj(e)25993 40253 y Fn(E)26888 40441 y Fk(\244)27596 40253 y Fq(\()p Fn(f)h Fo(\241)269 b Fn(E)31168 40441 y Fk(\244)31875 40253 y Fn(f)130 b Fq(\))p Fn(:)-33406 b Fq(\(5.28\))0 42477 y(Hence)404 b(w)-34 b(e)404 b(obtain)h(\(d\))g (when)14590 42171 y Fj(e)14343 42477 y Fn(E)15238 42665 y Fk(\244)15946 42477 y Fn(g)380 b Fq(=)337 b(1.)538 b(The)405 b(general)e(case)h(follo)-34 b(ws)405 b(immediately)e(from)i (this.)4506 b Fe(\244)1328 44745 y Fq(In)523 b(order)f(to)h(pro)-34 b(v)g(e)523 b(the)g(sw)-34 b(eeping)523 b(our)g(relations,)551 b(w)-34 b(e)523 b(\257nally)f(prepare)h(the)g(follo)-34 b(wing)523 b(lemma.)893 b(W)-101 b(e)0 46184 y(remark)413 b(here)h(that)i(the)f(statemen)-34 b(ts)416 b(in)e(Lemma)g(5.6)g(and)i (Lemma)e(4.4)g(are)g(sligh)-34 b(tly)414 b(mo)34 b(di\257ed)415 b(from)f(that)0 47623 y(of)404 b([8],)g(but)h(the)f(pro)34 b(ofs)405 b(are)e(adapted)j(from)e([8].)0 49853 y Fw(Lemma)540 b(5.6.)589 b Fg(F)-93 b(or)495 b(any)f(r)-62 b(e)g(al)493 b(me)-62 b(asur)g(able)491 b(function)i Fn(f)578 b Fq(:)446 b Fn(X)542 b Fo(!)446 b Fw(R)p Fg(,)510 b Fq(\244)446 b Fo(\275\275)g(R)p Fg(,)510 b(and)494 b Fn(x)581 b(=)-741 b Fo(2)447 b Fq(\244)p Fg(,)508 b(we)495 b(have)d(a)0 51292 y(c)-62 b(onstant)432 b Fn(C)5698 51480 y Fk(\244)p Fm(;)p Fk(\244)p Fl(\\)p Fk(\244)8545 51636 y Fi(f)9564 51292 y Fg(such)g(that)0 53807 y Fq(\(5.29\))7325 b Fo(jr)11769 53989 y Fm(x)12601 53501 y Fj(e)12354 53807 y Fn(E)13249 53995 y Fk(\244)13957 53807 y Fn(f)130 b Fo(j)337 b(\267)16880 53501 y Fj(e)16634 53807 y Fn(E)17529 53995 y Fk(\244)18236 53807 y Fo(r)19246 53989 y Fm(x)19831 53807 y Fn(f)401 b Fq(+)269 b Fn(C)22903 53995 y Fk(\244)p Fm(;)p Fk(\244)p Fl(\\)p Fk(\244)25750 54151 y Fi(f)26335 53807 y Fn(\267)27033 53168 y Fk(\(\244\))27033 54189 y Fm(x;)p Fk(\244)p Fl(\\)p Fk(\244)29757 54345 y Fi(f)31462 52656 y Fj(X)30544 55281 y Fm(y)32 b Fl(2)p Fk(\244)p Fl(\\)p Fk(\244)33601 55437 y Fi(f)34580 53501 y Fj(e)34333 53807 y Fn(E)35228 53989 y Fm(x)36060 53501 y Fj(e)35813 53807 y Fn(E)36708 53995 y Fk(\244)37416 53807 y Fo(r)38426 53989 y Fm(y)38978 53807 y Fn(f)63 b(:)0 57268 y Fg(wher)-62 b(e)441 b Fq(\244)4273 57465 y Fm(f)5319 57268 y Fg(stands)f(for)h(the)f(supp)-62 b(ort)439 b(of)i Fn(f)130 b Fg(,)444 b(i.e.,)e Fq(\244)23640 57465 y Fm(f)24686 57268 y Fg(is)f(the)f(smal)62 b(lest)440 b(subset)f Fq(\242)351 b Fo(\275)f(R)442 b Fg(satisfying)d Fo(r)46834 57450 y Fm(x)47419 57268 y Fn(f)482 b Fq(=)351 b(0)0 58707 y Fg(for)433 b(any)g Fn(x)337 b Fo(2)f Fq(\242)7443 58268 y Fm(c)7906 58707 y Fg(.)0 60937 y(Pr)-62 b(o)g(of.)552 b Fq(In)405 b(almost)f(the)h(same)f(w)-34 b(a)g(y)404 b(as)h(sho)-34 b(wing)405 b(\(5.21\),)g(w)-34 b(e)404 b(see)g(that)7474 63161 y Fo(jr)8821 63343 y Fm(x)9652 62855 y Fj(e)9406 63161 y Fn(E)10301 63349 y Fk(\244)11008 63161 y Fn(f)130 b Fo(j)1108 b(\267)e(j)p Fn(R)16482 63343 y Fm(x)17314 62855 y Fj(e)17068 63161 y Fn(E)17963 63349 y Fk(\244)18670 63161 y Fn(f)400 b Fo(\241)21122 62855 y Fj(e)20876 63161 y Fn(E)21771 63349 y Fk(\244)22478 63161 y Fn(R)23398 63343 y Fm(x)23984 63161 y Fn(f)130 b Fo(j)269 b Fq(+)h Fo(j)27109 62855 y Fj(e)26863 63161 y Fn(E)27758 63349 y Fk(\244)28465 63161 y Fn(R)29385 63343 y Fm(x)29970 63161 y Fn(f)401 b Fo(\241)32422 62855 y Fj(e)32176 63161 y Fn(E)33071 63349 y Fk(\244)33778 63161 y Fn(f)130 b Fo(j)13176 65238 y Fq(=)1106 b Fo(j)15809 64931 y Fj(e)15562 65238 y Fn(E)16457 65426 y Fk(\244)17165 65238 y Fn(u)17859 64599 y Fk(1)p Fm(=)p Fk(2)17859 65619 y Fm(x;)p Fk(\244)19358 65238 y Fn(R)20278 65420 y Fm(x)20863 65238 y Fn(f)400 b Fo(\241)23315 64931 y Fj(e)23068 65238 y Fn(E)23963 65426 y Fk(\244)24671 65238 y Fn(R)25591 65420 y Fm(x)26176 65238 y Fn(f)130 b Fo(j)270 b Fq(+)f Fo(j)29302 64931 y Fj(e)29055 65238 y Fn(E)29950 65426 y Fk(\244)30658 65238 y Fn(R)31578 65420 y Fm(x)32163 65238 y Fn(f)400 b Fo(\241)34615 64931 y Fj(e)34368 65238 y Fn(E)35263 65426 y Fk(\244)35971 65238 y Fn(f)130 b Fo(j)337 b Fq(=:)g Fn(I)39518 65420 y Fk(1)40313 65238 y Fq(+)269 b Fn(I)42058 65420 y Fk(2)42583 65238 y Fn(:)-42920 b Fq(\(5.30\))0 67372 y(By)403 b(the)i(triangle)f(inequalit)-34 b(y)-101 b(,)403 b(w)-34 b(e)405 b(obtain)0 69529 y(\(5.31\))18667 b Fn(I)22297 69711 y Fk(2)23159 69529 y Fo(\267)24685 69223 y Fj(e)24439 69529 y Fn(E)25334 69717 y Fk(\244)26041 69529 y Fo(r)27051 69711 y Fm(x)27636 69529 y Fn(f)63 b(:)1328 71596 y Fq(F)-101 b(or)404 b Fn(I)4036 71778 y Fk(1)4966 71596 y Fq(w)-34 b(e)405 b(observ)-34 b(e)403 b(that)0 73820 y(\(5.32\))12657 b Fn(I)16287 74002 y Fk(1)17150 73820 y Fq(=)336 b Fo(j)19012 73514 y Fj(e)18766 73820 y Fn(E)19661 74008 y Fk(\244)20368 73820 y Fq(\()21086 73514 y Fj(e)20839 73820 y Fn(E)21734 74067 y Fk(\244)p Fl(n)p Fk(\244)23510 74223 y Fi(f)24096 73820 y Fn(u)24790 73181 y Fk(1)p Fm(=)p Fk(2)24790 74202 y Fm(x;)p Fk(\244)26289 73820 y Fq(\))p Fn(R)27680 74002 y Fm(x)28265 73820 y Fn(f)401 b Fo(\241)30717 73514 y Fj(e)30471 73820 y Fn(E)31366 74008 y Fk(\244)32073 73820 y Fn(R)32993 74002 y Fm(x)33579 73820 y Fn(f)130 b Fo(j)24726 75260 y Fk(15)p eop %%Page: 16 16 16 15 bop 0 589 a Fq(since)3216 282 y Fj(e)2970 589 y Fn(E)3865 835 y Fk(\244)p Fl(n)p Fk(\244)5641 991 y Fi(f)6226 589 y Fn(R)7146 771 y Fm(x)7731 589 y Fn(f)468 b Fq(=)336 b Fn(R)10991 771 y Fm(x)11576 589 y Fn(f)130 b Fq(.)540 b(W)-101 b(e)403 b(notice)h(that)21705 282 y Fj(e)21459 589 y Fn(E)22354 777 y Fk(\244)23061 589 y Fq(\()23779 282 y Fj(e)23532 589 y Fn(E)24427 835 y Fk(\244)p Fl(n)p Fk(\244)26203 991 y Fi(f)26789 589 y Fn(u)27483 -50 y Fk(1)p Fm(=)p Fk(2)27483 970 y Fm(x;)p Fk(\244)28981 589 y Fq(\))338 b(=)e(1)404 b(and)h(use)f(Lemma)g(5.5)g(\(d\))h(to)f (get)12480 3206 y Fn(I)13013 3388 y Fk(1)14646 3206 y Fo(\267)16695 2144 y(p)p 17705 2144 607 49 v 1062 x Fq(2)q(V)-101 b(ar)20200 3394 y Fk(\244)20908 3206 y Fq(\()21626 2899 y Fj(e)21379 3206 y Fn(E)22274 3452 y Fk(\244)p Fl(n)p Fk(\244)24050 3608 y Fi(f)24636 3206 y Fn(u)25330 2567 y Fk(1)p Fm(=)p Fk(2)25330 3587 y Fm(x;)p Fk(\244)26828 3206 y Fq(\))27299 2705 y Fk(2)28072 2899 y Fj(e)27826 3206 y Fn(E)28721 3394 y Fk(\244)29428 3206 y Fq(\()p Fn(R)30819 3388 y Fm(x)31405 3206 y Fn(f)400 b Fo(\241)269 b Fn(E)34505 3394 y Fk(\244)35213 3206 y Fn(R)36133 3388 y Fm(x)36718 3206 y Fn(f)130 b Fq(\))14646 5533 y Fo(\267)16695 4471 y(p)p 17705 4471 V 1062 x Fq(2)q Fn(\267)19010 4894 y Fk(\(\244\))19010 5915 y Fm(x;)p Fk(\244)p Fl(\\)p Fk(\244)21734 6071 y Fi(f)22566 5227 y Fj(e)22319 5533 y Fn(E)23214 5721 y Fk(\244)23922 5533 y Fq(\()p Fn(R)25313 5715 y Fm(x)25898 5533 y Fn(f)401 b Fo(\241)269 b Fn(E)28999 5721 y Fk(\244)29706 5533 y Fn(R)30626 5715 y Fm(x)31212 5533 y Fn(f)130 b Fq(\))p Fn(:)-32743 b Fq(\(5.33\))0 7869 y(F)-101 b(or)404 b(\242)337 b Fo(\275)f Fq(\244,)404 b(let)g Fn(m)9199 8057 y Fk(\244)p Fm(;)p Fk(\242)11356 7869 y Fq(b)34 b(e)404 b(the)h(sp)34 b(ectral)403 b(gap)i(constan)-34 b(t)406 b(of)e(the)h(measure)f Fn(E)35915 8057 y Fk(\244)36622 7869 y Fo(j)36959 8057 y Fl(F)37631 8250 y Ff(\(\244)p Fr(n)p Ff(\242\))39911 8059 y Fi(c)40448 7869 y Fq(,)g(i.e.,)0 10479 y(\(5.34\))17041 10173 y Fj(e)16794 10479 y Fn(E)17689 10667 y Fk(\244)18397 10479 y Fq(\()p Fn(f)c Fo(\241)269 b Fn(E)21968 10667 y Fk(\244)22676 10479 y Fn(f)130 b Fq(\))338 b Fo(\267)e Fn(m)26551 9840 y Fl(\241)p Fk(1)p Fm(=)p Fk(2)26551 10861 y(\244)p Fm(;)p Fk(\242)28997 10173 y Fj(e)28750 10479 y Fn(E)29645 10667 y Fk(\244)30353 10479 y Fo(jr)31700 10667 y Fk(\242)32539 10479 y Fn(f)130 b Fo(j)0 12897 y Fq(for)464 b(all)f Fn(f)594 b Fq(with)465 b(\244)309 b Fo(\\)g Fq(\244)10776 13094 y Fm(f)11817 12897 y Fo(\275)435 b Fq(\242,)479 b(where)463 b Fo(jr)19932 13085 y Fk(\242)20771 12897 y Fn(f)130 b Fo(j)437 b Fq(=)f(\()24118 11988 y Fj(P)25398 13255 y Fm(x)p Fl(2)p Fk(\242)27596 12897 y Fo(jr)28943 13079 y Fm(x)29528 12897 y Fn(f)130 b Fo(j)30588 12457 y Fk(2)31114 12897 y Fq(\))31585 12457 y Fk(1)p Fm(=)p Fk(2)33052 12897 y Fq(.)718 b(W)-101 b(e)463 b(notice)g(that)i Fn(m)43632 13085 y Fk(\244)p Fm(;)p Fk(\242)45849 12897 y Fq(do)34 b(es)464 b(not)0 14336 y(dep)34 b(end)405 b(on)f(the)h(b)34 b(oundary)405 b(conditions)g(and)g Fn(m)22657 14524 y Fk(\244)p Fm(;)p Fk(\242)24747 14336 y Fn(>)336 b Fq(0)404 b(b)-34 b(y)405 b(Lemma)f(5.7)f(b)34 b(elo)-34 b(w.)539 b(Therefore)12274 16759 y Fn(I)12807 16941 y Fk(1)14439 16759 y Fo(\267)16489 15697 y(p)p 17499 15697 V 1062 x Fq(2)p Fn(\267)18803 16120 y Fk(\(\244\))18803 17140 y Fm(x;)p Fk(\244)p Fl(\\)p Fk(\244)21527 17296 y Fi(f)22112 16759 y Fq(\()p Fn(m)23647 16947 y Fk(\244)p Fm(;)p Fk(\244)p Fl(\\)p Fk(\244)26494 17103 y Fi(f)27080 16759 y Fq(\))27551 16258 y Fl(\241)p Fk(1)p Fm(=)p Fk(2)29997 16452 y Fj(e)29750 16759 y Fn(E)30645 16947 y Fk(\244)31353 16759 y Fo(jr)32700 16947 y Fk(\244)p Fl(\\)p Fk(\244)34633 17103 y Fi(f)35217 16759 y Fn(R)36137 16941 y Fm(x)36723 16759 y Fn(f)130 b Fo(j)p Fn(:)-38120 b Fq(\(5.35\))0 19095 y(Using)404 b(Lemma)g(5.5)g(\(b\))h(and)g(\(c\),) f(w)-34 b(e)405 b(see)e(that)0 21340 y(\(5.36\))14468 21033 y Fj(e)14222 21340 y Fn(E)15117 21528 y Fk(\244)15824 21340 y Fn(R)16744 21522 y Fm(x)17330 21340 y Fo(jr)18677 21528 y Fk(\244)p Fl(\\)p Fk(\244)20610 21684 y Fi(f)21195 21340 y Fn(f)130 b Fo(j)337 b(\267)f Fn(c)24396 21522 y Fk(4)24922 21340 y Fn(\270)25629 20839 y Fl(\241)p Fk(1)p Fm(=)p Fk(2)28074 21033 y Fj(e)27827 21340 y Fn(E)28722 21522 y Fm(x)29554 21033 y Fj(e)29307 21340 y Fn(E)30202 21528 y Fk(\244)30910 21340 y Fo(jr)32257 21528 y Fk(\244)p Fl(\\)p Fk(\244)34190 21684 y Fi(f)34775 21340 y Fn(f)130 b Fo(j)p Fn(:)0 23482 y Fq(F)-101 b(rom)404 b(\(5.35\),)h(\(5.36\),)f (and)h(the)f(triangle)g(inequalit)-34 b(y)404 b(w)-34 b(e)405 b(get)0 25848 y(\(5.37\))11795 b Fn(I)15425 26030 y Fk(1)16287 25848 y Fo(\267)337 b Fn(C)18433 26036 y Fk(\244)p Fm(;)p Fk(\244)p Fl(\\)p Fk(\244)21280 26192 y Fi(f)21865 25848 y Fn(\267)22563 25209 y Fk(\(\244\))22563 26230 y Fm(x;)p Fk(\244)p Fl(\\)p Fk(\244)25287 26386 y Fi(f)26993 24697 y Fj(X)26075 27322 y Fm(y)32 b Fl(2)p Fk(\244)p Fl(\\)p Fk(\244)29132 27478 y Fi(f)30110 25542 y Fj(e)29864 25848 y Fn(E)30759 26030 y Fm(x)31590 25542 y Fj(e)31343 25848 y Fn(E)32238 26036 y Fk(\244)32946 25848 y Fo(r)33956 26030 y Fm(y)34508 25848 y Fn(f)63 b(;)0 29593 y Fq(where)404 b Fn(C)4371 29781 y Fk(\244)p Fm(;)p Fk(\244)p Fl(\\)p Fk(\244)7218 29937 y Fi(f)8140 29593 y Fq(=)9419 28590 y Fo(p)p 10430 28590 V 10430 29593 a Fq(2)p Fn(c)11561 29775 y Fk(4)12086 29593 y Fq(\()p Fn(\270m)14328 29781 y Fk(\244)p Fm(;)p Fk(\244)p Fl(\\)p Fk(\244)17175 29937 y Fi(f)17761 29593 y Fq(\))18232 29153 y Fl(\241)p Fk(1)p Fm(=)p Fk(2)20431 29593 y Fq(.)538 b(By)403 b(\(5.31\))i(and)g(\(5.37\))g(w)-34 b(e)404 b(complete)g(the)h(pro)34 b(of.)4791 b Fe(\244)0 31835 y Fw(Lemma)590 b(5.7.)613 b Fg(F)-93 b(or)535 b Fq(\242)519 b Fo(\275)g Fq(\244)g Fo(\275\275)g(R)p Fg(,)559 b(we)535 b(have)d(a)i(sp)-62 b(e)g(ctr)g(al)532 b(gap)h(c)-62 b(onstant)533 b Fn(m)37619 32023 y Fk(\244)p Fm(;)p Fk(\242)39892 31835 y Fn(>)519 b Fq(0)535 b Fg(of)e(the)g(me)-62 b(asur)g(e)0 33274 y Fn(E)895 33462 y Fk(\244)1603 33274 y Fo(j)1940 33462 y Fl(F)2612 33655 y Ff(\(\244)p Fr(n)p Ff(\242\))4892 33464 y Fi(c)5862 33274 y Fg(indep)g(endent)432 b(of)h(the)f(b)-62 b(oundary)432 b(c)-62 b(onditions.)555 b(In)435 b(p)-62 b(articular,)430 b(we)k(have)0 35490 y Fq(\(5.38\))19130 b Fn(C)23093 35736 y Fk(\244)p Fm(;)p Fl(f)p Fm(y)32 b Fl(g)25837 35490 y Fo(\267)336 b Fn(c)27641 35672 y Fk(5)0 37578 y Fg(uniformly)432 b(in)i Fn(y)381 b Fo(2)336 b Fq(\244)h Fo(\275\275)f(R)p Fg(.)0 39932 y(Pr)-62 b(o)g(of.)552 b Fq(F)-101 b(or)489 b(a)f(giv)-34 b(en)488 b(b)34 b(oundary)489 b(condition)g Fn(\273)533 b Fo(2)477 b Fn(X)95 b Fq(,)509 b(regard)488 b Fn(E)30414 39342 y Fm(\273)30344 40313 y Fk(\244)31052 39932 y Fq(\()p Fo(\242)p Fq(\))h(as)f(a)g(probabilit) -34 b(y)489 b(measure)f(on)h Fn(X)49349 40120 y Fk(\244)50057 39932 y Fq(.)0 41472 y(Let)466 b Fn(\272)2833 41660 y Fk(\244)3982 41472 y Fq(:=)5703 40563 y Fj(Q)6848 41830 y Fm(x)p Fl(2)p Fk(\244)8915 41472 y Fn(\272)9514 41654 y Fm(x)10565 41472 y Fq(b)34 b(e)467 b(the)g(pro)34 b(duct)468 b(of)f(\()21087 40995 y Fk(1)p 21087 41193 471 49 v 21087 41890 a(2)21691 41472 y Fn(;)22362 40995 y Fk(1)p 22362 41193 V 22362 41890 a(2)22966 41472 y Fq(\)-Bernoulli)f(measures)h(on)g Fo(f)p Fq(0)p Fn(;)202 b Fq(1)p Fo(g)p Fq(.)727 b(Since)466 b(the)i(Bernoulli)0 42911 y(measure)374 b(has)h(p)34 b(ositiv)-34 b(e)374 b(sp)34 b(ectral)374 b(gap,)380 b(it)375 b(is)f(enough)h(to)g(pro)-34 b(v)g(e)374 b(that)i(there)e (exist)g(p)34 b(ositiv)-34 b(e)374 b(constan)-34 b(ts)376 b Fn(c)p Fq(\(\242\))0 44350 y(and)405 b Fn(c)2882 43910 y Fl(0)3192 44350 y Fq(\(\242\))g(so)f(that)h(for)g(an)-34 b(y)404 b Fn(A)337 b Fo(\275)f Fn(X)17337 44538 y Fk(\242)18581 44350 y Fq(w)-34 b(e)405 b(ha)-34 b(v)g(e)0 46618 y(\(5.39\))7006 b Fn(c)p Fq(\(\242\))p Fn(\272)13179 46806 y Fk(\242)14019 46618 y Fq(\()p Fn(A)p Fq(\))337 b Fo(\267)g Fn(E)18452 46028 y Fm(\273)18382 46999 y Fk(\244)19089 46618 y Fq(\()p Fn(A)p Fq(\))h Fo(\267)e Fn(c)23082 46117 y Fl(0)23393 46618 y Fq(\(\242\))p Fn(\272)25944 46806 y Fk(\242)26784 46618 y Fq(\()p Fn(A)p Fq(\))1213 b(uniformly)404 b(in)g Fn(\273)393 b Fo(2)337 b Fn(X)95 b(:)0 49339 y Fq(W)-101 b(e)488 b(notice)h(that)8670 48655 y Fm(dE)9894 48221 y Fi(\273)9844 48963 y Ff(\244)p 8670 49060 1782 49 v 8784 49757 a Fm(d\272)9731 49906 y Ff(\244)10584 49339 y Fq(\()p Fn(\263)11585 49527 y Fk(\244)12294 49339 y Fq(\))478 b(=)g(2)15270 48899 y Fl(j)p Fk(\244)p Fl(j)16501 49339 y Fn(\260)17129 49527 y Fk(\244)17836 49339 y Fq(\()p Fn(\263)18837 49527 y Fk(\244)19546 49339 y Fq(;)202 b Fn(\273)20615 49527 y Fk(\244)21268 49274 y Fi(c)21749 49339 y Fq(\).)794 b(No)-34 b(w)489 b(b)-34 b(y)490 b(noticing)f(that) 35407 48362 y Fj(R)36417 49339 y Fn(d\263)37578 49586 y Fk(\244)p Fl(n)p Fk(\242)39541 49339 y Fn(\260)40169 49527 y Fk(\244)40876 49339 y Fq(\()p Fn(\263)41877 49527 y Fk(\244)42586 49339 y Fq(;)202 b Fn(\273)43655 49527 y Fk(\244)44308 49274 y Fi(c)44789 49339 y Fq(\))479 b(=)f Fn(\271)p Fq(\()p Fn(\263)48891 49527 y Fk(\242)50057 49339 y Fo(\242)0 50925 y Fn(\273)530 51113 y Fk(\244)1183 50861 y Fi(c)1665 50925 y Fq(\))p Fn(=\271)p Fq(\()p Fn(\273)4473 51113 y Fk(\244)5126 50861 y Fi(c)5609 50925 y Fq(\))361 b(\(see)f(the)h(con)-34 b(v)g(en)g(tion)361 b(in)g(Lemma)f(5.1\))g(and)h(using)g(the)g(same)f(argumen)-34 b(t)362 b(used)f(to)g(sho)-34 b(w)361 b(\(3.3\))0 52443 y(w)-34 b(e)405 b(easily)e(c)-34 b(hec)g(k)404 b(that)h(\(5.39\))g (holds)f(with)h Fn(c)p Fq(\(\242\))337 b(=)g(\(2)p Fn(\270)p Fq(\))26800 52003 y Fl(j)p Fk(\242)p Fl(j)28567 52443 y Fq(and)405 b Fn(c)31449 52003 y Fl(0)31759 52443 y Fq(\(\242\))337 b(=)g(\(2\(1)270 b Fo(\241)f Fn(\270)p Fq(\)\))40613 52003 y Fl(j)p Fk(\242)p Fl(j)41976 52443 y Fq(.)7138 b Fe(\244)1328 54739 y Fq(No)-34 b(w)405 b(w)-34 b(e)405 b(giv)-34 b(e)403 b(the)i(pro)34 b(of)404 b(for)h(the)f(sw)-34 b(eeping)405 b(out)g(relations.)0 57034 y Fg(Pr)-62 b(o)g(of)433 b(of)g(L)-62 b(emma)433 b(4.4.)552 b Fq(The)423 b(lemma)e(will)h(follo)-34 b(w)422 b(b)-34 b(y)422 b(using)h(Lemma)e(5.6)h(rep)34 b(eatedly)-101 b(.)591 b(Supp)34 b(ose)423 b Fo(j)p Fq(\244)p Fo(j)366 b Fq(=)g Fn(N)0 58473 y Fq(and)437 b(let)f Fo(f)p Fn(y)5372 58655 y Fk(1)5898 58473 y Fn(;)202 b Fo(\242)g(\242)g(\242)403 b Fn(;)202 b(y)9388 58661 y Fm(N)10284 58473 y Fo(g)436 b Fq(b)34 b(e)436 b(an)h(en)-34 b(umeration)437 b(of)f(the)h(p)34 b(oin)-34 b(ts)437 b(in)f(\244,)443 b(for)437 b(instance,)444 b(w)-34 b(e)436 b(let)g Fn(l)414 b Fo(7!)390 b Fn(d)p Fq(\()p Fn(x;)202 b(y)47894 58670 y Fm(l)48240 58473 y Fq(\))437 b(b)34 b(e)0 59912 y(decreasing,)442 b(where)435 b Fn(d)p Fq(\()p Fn(x;)202 b(y)43 b Fq(\))436 b(is)f(the)h(distance)f (from)g Fn(x)g Fq(to)h Fn(y)43 b Fq(.)632 b(Let)435 b(\244)31966 60109 y Fm(l)32700 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b(to)i(obtain)0 3583 y(\(5.42\))9728 b Fo(jr)14172 3765 y Fm(x)15004 3277 y Fj(e)14757 3583 y Fn(E)15652 3771 y Fk(\244)16360 3583 y Fn(f)130 b Fo(j)337 b(\267)19283 3277 y Fj(e)19036 3583 y Fn(E)19931 3771 y Fk(\244)20639 3583 y Fo(r)21649 3765 y Fm(x)22234 3583 y Fn(f)400 b Fq(+)270 b Fn(c)24965 3765 y Fk(6)26147 2068 y Fm(N)25692 2432 y Fj(X)25821 5067 y Fm(l)11 b Fk(=1)27645 3583 y Fn(\267)28343 3765 y Fm(xy)29338 3921 y Fi(l)29962 3277 y Fj(e)29715 3583 y Fn(E)30610 3765 y Fm(x)31442 3277 y Fj(e)31195 3583 y Fn(E)32090 3771 y Fk(\244)32798 3583 y Fo(r)33808 3765 y Fm(y)34273 3921 y Fi(l)34897 3277 y Fj(e)34650 3583 y Fn(E)35545 3771 y Fk(\244)36198 3927 y Fi(l)36575 3583 y Fn(f)63 b(:)0 6743 y Fq(W)-101 b(e)475 b(rep)34 b(eat)476 b(this)g(to)g(the)h(second)f(term)g(of)g(the)g(r.h.s.)754 b(of)476 b(\(5.42\).)754 b(F)-101 b(or)476 b(example,)493 b(giv)-34 b(en)476 b(1)456 b Fo(\267)g Fn(l)480 b Fo(\267)456 b Fn(N)132 b Fq(,)493 b(w)-34 b(e)0 8460 y(en)g(umerate)545 b(the)g(p)34 b(oin)-34 b(ts)546 b(in)f(\244)14445 8657 y Fm(l)15336 8460 y Fq(b)-34 b(y)545 b Fo(f)p Fn(y)18403 7821 y Fk(\(2\))18360 8807 y(1)19661 8460 y Fn(;)202 b(y)20837 7821 y Fk(\(2\))20794 8807 y(2)22095 8460 y Fn(;)g Fo(\242)g(\242)g(\242)403 b Fn(;)202 b(y)25628 7821 y Fk(\(2\))25585 8852 y Fm(N)94 b Fl(\241)p Fm(l)27505 8460 y Fo(g)545 b Fq(so)f(that)i Fn(j)641 b Fo(7!)571 b Fn(d)p Fq(\()p Fn(y)37670 8657 y Fm(l)38017 8460 y Fn(;)202 b(y)39193 7821 y Fk(\(2\))39150 8822 y Fm(j)40451 8460 y Fq(\))545 b(decreases.)959 b(After)0 10014 y(con)-34 b(tin)g(ual)406 b(rep)34 b(etitions,)403 b(w)-34 b(e)405 b(will)e(\257nally)h(get)h(the)f(b)34 b(ound)406 b(\(4.34\))e(with)0 13377 y(\(5.43\))9647 b Fn(\256)13523 12877 y Fk(\(\244\))13519 13677 y Fm(xy)15300 13377 y Fq(=)16867 11748 y Fl(j)p Fk(\244)p Fl(j)16580 12226 y Fj(X)16597 14861 y Fm(k)24 b Fk(=1)18532 13377 y Fn(A)19441 12877 y Fm(k)22835 12226 y Fj(X)20212 15215 y Fl(h)p Fm(y)21043 15338 y Ff(1)21505 15215 y Fm(;)p Fl(\242\242\242)158 b Fm(;y)23433 15371 y Fi(k)14 b Fr(\241)p Ff(1)24987 15215 y Fl(i)25353 14744 y Ff(\()p Fi(x;y)24 b Ff(\))25353 15523 y(\244)27410 13377 y Fn(\267)28108 13559 y Fm(xy)29103 13682 y Ff(1)29619 13377 y Fn(\267)30317 13559 y Fm(y)30782 13682 y Ff(1)31244 13559 y Fm(y)31709 13682 y Ff(2)32428 13377 y Fo(\242)202 b(\242)g(\242)f Fn(\267)34742 13559 y Fm(y)35207 13715 y Fi(k)14 b Fr(\241)p Ff(1)36761 13559 y Fm(y)37313 13377 y Fn(;)0 17594 y Fq(where)420 b Fn(A)h Fq(is)f(a)g(constan)-34 b(t)422 b(dep)34 b(ending)422 b(only)e(on)h Fn(\270)f Fq(\(and)i Fn(\270)26914 17776 y Fk(1)27440 17594 y Fq(\))e(and)i Fo(h)p Fn(y)31770 17776 y Fk(1)32296 17594 y Fn(;)202 b Fo(\242)g(\242)g(\242)403 b Fn(;)202 b(y)35786 17791 y Fm(k)24 b Fl(\241)p Fk(1)37557 17594 y Fo(i)38028 16955 y Fk(\()p Fm(x;y)32 b Fk(\))38028 17975 y(\244)40524 17594 y Fq(means)421 b(a)f(path)i(in)e(\244)0 19033 y(so)404 b(that)h(\()p Fn(y)5179 19215 y Fk(0)6043 19033 y Fq(:=)336 b Fn(x;)202 b(y)9485 19230 y Fm(k)10390 19033 y Fq(:=)336 b Fn(y)43 b Fq(\))0 21007 y(\(5.44\))10558 b Fn(d)p Fq(\()p Fn(y)15351 21189 y Fm(i)p Fl(\241)p Fk(1)16929 21007 y Fn(;)202 b(y)18062 21189 y Fm(i)18438 21007 y Fq(\))337 b(=)g(max)p Fo(f)p Fn(d)p Fq(\()p Fn(y)25084 21189 y Fm(i)p Fl(\241)p Fk(1)26662 21007 y Fn(;)202 b(y)27795 21189 y Fm(j)28282 21007 y Fq(\))337 b(:)539 b Fn(j)406 b Fq(=)336 b Fn(i;)202 b Fo(\242)g(\242)g(\242)403 b Fn(;)202 b(k)38 b Fo(g)0 22982 y Fq(for)404 b(all)g Fn(i)336 b Fq(=)g(1)p Fn(;)202 b(:)g(:)g(:)f(;)h(k)38 b Fq(.)539 b(Th)-34 b(us)405 b(w)-34 b(e)405 b(ha)-34 b(v)g(e)0 25881 y(\(5.45\))10755 b Fn(\256)14627 26063 y Fm(xy)16046 25881 y Fo(\267)17731 24366 y Fl(1)17326 24730 y Fj(X)17343 27365 y Fm(k)24 b Fk(=1)19279 25881 y Fn(A)20188 25381 y Fm(k)22653 24730 y Fj(X)20959 27414 y Fl(h)p Fm(y)21790 27537 y Ff(1)22251 27414 y Fm(;)p Fl(\242\242\242)158 b Fm(;y)24179 27570 y Fi(k)14 b Fr(\241)p Ff(1)25733 27414 y Fl(i)26301 25881 y Fn(\267)26999 26063 y Fm(xy)27994 26186 y Ff(1)28511 25881 y Fn(\267)29209 26063 y Fm(y)29674 26186 y Ff(1)30136 26063 y Fm(y)30601 26186 y Ff(2)31319 25881 y Fo(\242)202 b(\242)g(\242)f Fn(\267)33633 26063 y Fm(y)34098 26219 y Fi(k)14 b Fr(\241)p Ff(1)35652 26063 y Fm(y)36205 25881 y Fn(;)0 29339 y Fq(where)428 b Fo(h)p Fn(y)4594 29521 y Fk(1)5120 29339 y Fn(;)202 b Fo(\242)g(\242)g(\242) 403 b Fn(;)202 b(y)8610 29536 y Fm(k)24 b Fl(\241)p Fk(1)10381 29339 y Fo(i)428 b Fq(is)g(a)g(path)h(consisting)f(of)g Fn(k)323 b Fo(\241)285 b Fq(1)428 b(di\256eren)-34 b(t)428 b(p)34 b(oin)-34 b(ts)429 b(in)f Fo(R)g Fq(and)h(\(4.36\))f(follo)-34 b(ws)428 b(from)0 30778 y(Lemma)404 b(5.4)f(if)i(w)-34 b(e)404 b(tak)-34 b(e)404 b Fn(q)43 b Fq(\()p Fn(K)87 b Fq(\))406 b(small)e(enough.)27323 b Fe(\244)18739 33315 y Fq(6.)606 b Fp(A)-35 b(ckno)-27 b(wledgments)1328 35473 y Fq(The)447 b(authors)g(w)-34 b(ould)447 b(lik)-34 b(e)445 b(to)h(thank)h(Prof.)f(T.)g(Shiga)g(for)g(his)g(useful)h(suggestions,) 457 b(esp)34 b(ecially)444 b(for)i(the)0 36913 y(pro)34 b(of)588 b(of)g(Lemma)f(4.1.)1089 b(W)-101 b(e)587 b(also)g(thank)i (Prof.)e(B.)g(Zegarlinski)f(for)i(the)g(commen)-34 b(ts)588 b(on)g(log-Sob)34 b(olev)0 38352 y(inequalit)-34 b(y)474 b(and)h(giving)f(us)h(the)g(reference)e([8])h(and)h(Prof.)f(N.)g(Y)-101 b(oshida)475 b(and)h(Prof.)e(H.)g(Osada)g(for)h(useful)0 39791 y(discussions.)773 b(This)482 b(w)-34 b(ork)482 b(w)-34 b(as)483 b(supp)34 b(orted)483 b(in)f(part)g(b)-34 b(y)482 b(the)h(Ministry)e(of)i(Education,)502 b(Culture,)f(Sp)34 b(orts,)0 41230 y(Science)503 b(and)i(T)-101 b(ec)-34 b(hnology)505 b(of)f(Japan)h(under)f(the)h(Gran)-34 b(t-in-Aid)505 b(No.70302932)f(\(\257rst)h(author\),)530 b(and)505 b(b)-34 b(y)0 42669 y(Japan)405 b(So)34 b(ciet)-34 b(y)404 b(for)g(the)g (Promotion)h(of)f(Science,)f(Researc)-34 b(h)405 b(F)-101 b(ello)-34 b(wships)405 b(\(second)f(author\).)21663 45206 y Fp(References)512 47143 y Fv([1])513 b(Aizenman,)470 b(M.)444 b(and)g(Holley)-85 b(,)470 b(R.)444 b(\(1987\).)g(Rapid)g(con) -28 b(v)g(ergence)443 b(to)h(equilibrium)i(of)d(sto)28 b(c)-28 b(hastic)444 b(Ising)f(mo)28 b(dels)445 b(in)f(the)2105 48361 y(Dobrushin)386 b(Sholosman)i(regime,)400 b(in)387 b Fh(Per)-52 b(c)g(olation)408 b(the)-52 b(ory)407 b(and)h(er)-52 b(go)g(dic)408 b(the)-52 b(ory)408 b(of)f(in\257nite)h(p)-52 b(article)408 b(systems)p Fv(,)400 b(Kesten,)2105 49578 y(H.)341 b(\(ed.\).)i(IMS)e(V)-85 b(olume)342 b(in)g(Math.)g(and)f (Appl.)h Fb(8)p Fv(,)g(1-11,)g(Springer,)f(Berlin)p Ft(\242)p Fv(Heidelb)28 b(erg)p Ft(\242)p Fv(New)344 b(Y)-85 b(ork.)512 50796 y([2])513 b(Araki,)342 b(H)f(and)g(Matsui,)h(T.)f(\(1985\).)h (Groud)f(states)h(of)f(the)h Fs(X)78 b(Y)228 b Fv(-mo)28 b(del.)343 b Fh(Commun.)366 b(Math.)f(Phys.)340 b Fb(101)j Fv(213{245.)512 52014 y([3])513 b(Boro)28 b(din,)239 b(A,)g(Ok)-28 b(ounk)g(o)g(v,)238 b(A)h(and)f(Olshanski,)i(G.)f (\(2000\))g(Asymptotics)g(of)g(Planc)-28 b(herel)239 b(measures)f(for)g(symmetric)i(groups.)2105 53231 y Fh(J.)365 b(A)-26 b(mer.)367 b(Math.)e(So)-52 b(c.)340 b Fb(13)j Fv(no.)e(3,)h(481{515.)512 54449 y([4])513 b(Boro)28 b(din,)434 b(A)g(and)f(Olshanski,)j(G.)e(\(1998\).)h(P)-28 b(oin)g(t)433 b(pro)28 b(cesses)434 b(and)g(the)g(in\257nite)h (symmetric)g(group)e(part)g(I)28 b(I)g(I:)434 b(fermion)2105 55667 y(p)28 b(oin)-28 b(t)341 b(pro)28 b(cesses.)343 b(math.)f(R)-85 b(T/9804088.)512 56884 y([5])513 b(Deusc)-28 b(hel,)342 b(J.)g(D.)g(and)f(Stro)28 b(o)g(c)-28 b(k,)342 b(D.)g(W.)g(\(1989\).)g Fh(L)-52 b(ar)g(ge)366 b(deviations)p Fv(.)342 b(Academic)g(Press,)f(Boston.)512 58102 y([6])513 b(Georgii,)348 b(H-O.)f(\(1988\).)g Fh(Gibbs)372 b(Me)-52 b(asur)g(es)371 b(and)f(Phase)h(T)-78 b(r)-52 b(ansitions)p Fv(.)345 b(de)i(Gruyter)g(Studies)g(in)f(Mathematics)j Fb(9)p Fv(,)f(W)-85 b(alter)2105 59319 y(de)341 b(Gruyter,)h(Berlin)p Ft(\242)p Fv(New)h(Y)-85 b(ork.)512 60537 y([7])513 b(Gross,)341 b(L.\(1976\).)h(Logarithmic)g(Sob)28 b(olev)343 b(inequalities,)i Fh(A)-26 b(mer.)366 b(J.)g(Math.)340 b Fb(97)i Fv(1061{1083.)512 61755 y([8])513 b(Guionnet,)343 b(A.)e(and)g(Zegarlinski,)i(B.)f (\(2000\).)g(Lectures)f(on)h(logarithmic)g(Sob)28 b(olev)343 b(inequalities.)h(preprin)-28 b(t.)512 62972 y([9])513 b(F\304)-512 b(ollmer,)343 b(H.)e(\(1982\).)h(A)f(co)-28 b(v)-57 b(ariance)342 b(estimate)h(for)e(Gibbs)h(measures,)g Fh(J.)365 b(F)-78 b(unct.)365 b(A)-26 b(nal.)341 b Fb(46)h Fv(387{395.)0 64190 y([10])513 b(F)-85 b(ukushima,)279 b(M,)h(Oshima,)g(Y)e(and)h(T)-85 b(ak)-28 b(eda,)278 b(M.)i(\(1994\).)f Fh(Dirichlet)309 b(F)-78 b(orms)308 b(and)g(Symmetric)h(Markov)g(Pr)-52 b(o)g(c)g(esses)p Fv(.)280 b(W)-85 b(alter)2105 65408 y(de)341 b(Gruyter,)h(Berlin)h Ft(\242)e Fv(New)g(Y)-85 b(ork.)0 66625 y([11])513 b(Kato,)342 b(T.)f(\(1966\).)h Fh(Perturb)-52 b(ation)365 b(the)-52 b(ory)366 b(for)f(line)-52 b(ar)366 b(op)-52 b(er)g(ators)p Fv(.)341 b(Springer-V)-85 b(erlag,)342 b(Berlin)g Ft(\242)g Fv(Heidelb)28 b(erg)343 b Ft(\242)e Fv(New)g(Y)-85 b(ork.)0 67843 y([12])513 b(Laro)28 b(c)-28 b(he,)383 b(E.)375 b(\(1995\).)h(Hyp)28 b(ercon)-28 b(tractivit)g(\266)-484 b(e)376 b(p)28 b(our)375 b(des)g(systemes)h(de)f(spins)g(de)g(p)28 b(ort)-28 b(\266)-484 b(e)376 b(e)f(in\257nie,)385 b Fh(Pr)-52 b(ob.)396 b(Th.)g(R)-52 b(el.)397 b(Fields)2105 69061 y Fb(101)342 b Fv(89{132.)0 70278 y([13])513 b(Liggett,)343 b(T.)e(M.)h(\(1985\).)g Fh(Inter)-52 b(acting)366 b(Particle)h(Systems) p Fv(.)342 b(Springer)f(V)-85 b(erlag,)341 b(New)g(Y)-85 b(ork.)0 71496 y([14])513 b(Lu,)345 b(S.)g(L.)g(and)g(Y)-85 b(au,)345 b(H.)g(T.\(1993\).)g(Sp)28 b(ectral)346 b(gap)e(and)h (logarithmic)g(Sob)28 b(olev)346 b(inequalit)-28 b(y)347 b(for)d(Ka)-28 b(w)g(asaki)344 b(and)h(Glaub)28 b(er)2105 72714 y(dynamics,)342 b Fh(Commun.)366 b(Math.)f(Phys.)340 b Fb(156)j Fv(399{433.)0 73931 y([15])513 b(Macc)-28 b(hi,)342 b(O.)g(\(1975\).)g(The)f(coincidence)i(approac)-28 b(h)340 b(to)h(sto)28 b(c)-28 b(hastic)342 b(p)28 b(oin)-28 b(t)342 b(pro)28 b(cesses.)342 b Fh(A)-52 b(dv.)366 b(Appl.)f(Pr)-52 b(ob.)341 b Fb(7)h Fv(83{122.)24726 75260 y Fk(17)p eop %%Page: 18 18 18 17 bop 0 393 a Fv([16])513 b(Macc)-28 b(hi,)291 b(O.)f(\(1977\).)h (The)f(fermion)g(pro)28 b(cess)290 b({)g(a)g(mo)28 b(del)291 b(of)f(sto)28 b(c)-28 b(hastic)291 b(p)28 b(oin)-28 b(t)290 b(pro)28 b(cess)290 b(with)g(repulsiv)-28 b(e)290 b(p)28 b(oin)-28 b(ts.)291 b(In)e Fh(T)-78 b(r)-52 b(ans-)2105 1610 y(actions)424 b(of)h(the)g(Seventh)h(Pr)-52 b(ague)425 b(Confer)-52 b(enc)g(e)426 b(on)f(Information)f(The)-52 b(ory,)439 b(Statistic)-52 b(al)424 b(De)-52 b(cision)425 b(F)-78 b(unctions,)438 b(R)-52 b(andom)2105 2828 y(Pr)g(o)g(c)g(esses) 396 b(and)g(of)f(the)h(Eighth)f(Eur)-52 b(op)g(e)g(an)394 b(Me)-52 b(eting)397 b(of)e(Statisticians)f(\(T)-78 b(e)-52 b(ch.)396 b(Univ.)f(Pr)-52 b(ague,)404 b(Pr)-52 b(ague,)403 b(1974\))p Fv(.)373 b(V)-85 b(ol.)374 b(A,)2105 4046 y(391{398.)0 5263 y([17])513 b(Matsui,)246 b(T.)f(\(1986\).)h(Explicit) g(form)-28 b(ulas)245 b(for)f(correlation)h(functions)g(of)g(ground)f (states)i(of)f(the)g(1)g(dimensional)i Fs(X)78 b(Y)228 b Fv(-mo)28 b(del.)2105 6481 y Fh(A)-26 b(nn.)365 b(Inst.)h(Henri)g (Poinc)-52 b(ar)-26 b(\266)-497 b(e)342 b Fb(45)p Fv(\(1\))g(49{59.)0 7699 y([18])513 b(Martinelli,)344 b(F.)e(\(1997\).)g(Lectures)f(on)g (Glaub)28 b(er)343 b(dynamics)f(for)e(discrete)j(spin)e(mo)28 b(dels.)343 b(preprin)-28 b(t)340 b(mparc)h(97-587.)0 8916 y([19])513 b(Ma,)288 b(Z.)g(M.)g(and)g(R\304)-512 b(oc)-28 b(kner,)287 b(M.)i(\(1992\).)f(In)-28 b(tro)28 b(duction)287 b(to)h(the)g(Theory)f(of)g(\(non-symmetric\))j(Diric)-28 b(hlet)288 b(forms.)g(Springer-)2105 10134 y(V)-85 b(erlag,)341 b(Berlin-Heidelb)28 b(erg-New)344 b(Y)-85 b(ork.)0 11352 y([20])513 b(Osada,)273 b(H.)h(\(1996\))f(Diric)-28 b(hlet)274 b(form)f(approac)-28 b(h)271 b(to)j(in\257nite-dimensional)h(Wiener)f (pro)28 b(cesses)274 b(with)f(singular)g(in)-28 b(teractions.)2105 12569 y Fh(Commun.)366 b(Math.)e(Phys.)340 b Fb(176)j Fv(117{131.)0 13787 y([21])513 b(Shirai,)373 b(T)g(and)f(T)-85 b(ak)-57 b(ahashi,)373 b(Y.)g(Random)g(p)28 b(oin)-28 b(t)373 b(\257eld)h(asso)28 b(ciated)374 b(with)f(certain)g(F)-85 b(redholm)373 b(determinan)-28 b(t)374 b(I)f(:)g(fermion,)2105 15005 y(P)-28 b(oisson,)341 b(and)g(b)28 b(oson)341 b(p)28 b(oin)-28 b(t)342 b(pro)28 b(cesses.)342 b(preprin)-28 b(t.)0 16222 y([22])513 b(Shirai,)366 b(T)e(and)h(T)-85 b(ak)-57 b(ahashi,)366 b(Y.)g(Random)e(p)28 b(oin)-28 b(t)366 b(\257eld)g(asso)28 b(ciated)367 b(with)e(certain)h(F)-85 b(redholm)366 b(determinan)-28 b(t)366 b(I)28 b(I)365 b(:)g(fermion)2105 17440 y(shift)341 b(and)g(its)h(ergo)28 b(dic)342 b(and)f(Gibbs)h(prop)28 b(erties.)342 b(preprin)-28 b(t.)0 18658 y([23])513 b(Simon,)277 b(B.)260 b(\(1975\).)g(A)f(remark) h(on)g(Nelson's)g(b)28 b(est)260 b(h)-28 b(yp)28 b(ercon)-28 b(tractiv)g(e)260 b(estimates,)278 b Fh(Pr)-52 b(o)g(c.)291 b(A)-26 b(mer.)291 b(Math.)f(So)-52 b(c.)259 b Fb(55)i Fv(376{378.)0 19875 y([24])513 b(Soshnik)-28 b(o)g(v,)341 b(A.)h(\(2000\).)g(Determinan)-28 b(tal)342 b(random)f(p)28 b(oin)-28 b(t)342 b(\257elds.)g(h)-28 b(ttp://xxx.lanl.go)g (v/abs/math.PR/0002099.)0 21093 y([25])513 b(Sp)28 b(ohn,)322 b(H.)h(\(1987\).)g(In)-28 b(teracting)322 b(Bro)-28 b(wnian)321 b(P)-28 b(articles)323 b(:)446 b(A)322 b(Study)h(of)f(Dyson's)h(Mo)28 b(del.)324 b(In)e(,)653 b Fh(Hydr)-52 b(o)g(dynamic)347 b(Behavior)2105 22310 y(and)365 b(Inter)-52 b(acting)367 b(Particle)f(Systems,)g(e)-52 b(d.)366 b(G.)f(Pap)-52 b(anic)g(alaou)p Fv(.)0 23528 y([26])513 b(Stro)28 b(o)g(c)-28 b(k,)374 b(D.)368 b(W.)g(and)e(Zegarlinski,)376 b(B.)367 b(\(1992\).)h(The)f(logarithmic)h(Sob)28 b(olev)369 b(inequalit)-28 b(y)369 b(for)e(discrete)h(spin)f(systems)h(on)2105 24746 y(a)341 b(lattice,)j Fh(Commun.)365 b(Math.)g(Phys.)340 b Fb(149)j Fv(175{193.)0 25963 y([27])513 b(Stro)28 b(o)g(c)-28 b(k,)583 b(D.)534 b(W.)h(and)f(Zegarlinski,)583 b(B.)535 b(\(1992\).)g(The)e(equiv)-57 b(alence)537 b(of)d(the)h(logarithmic)g (Sob)28 b(olev)535 b(inequalit)-28 b(y)536 b(and)2105 27181 y(Dobrushin-Shlosman)341 b(mixing)h(condition,)h 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