Content-Type: multipart/mixed; boundary="-------------0302110648569" This is a multi-part message in MIME format. ---------------0302110648569 Content-Type: text/plain; name="03-48.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-48.keywords" Quantum Shannon-McMillan Theorem, von Neumann mean entropy, entropy-typical subspaces ---------------0302110648569 Content-Type: application/postscript; name="I_Bjelakovic_030204.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="I_Bjelakovic_030204.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: uainventneu.dvi %%Pages: 20 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Times-Bold CMR8 CMMI8 CMR6 Times-Roman CMMI6 CMSY9 %%+ Courier MSBM10 CMMI10 CMR10 CMSY10 CMSY8 CMEX10 CMBX10 CMBX8 MSBM7 %%+ CMSY6 MSAM10 Times-Italic CMBX6 CMMI9 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips uainventneu.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.02.03:1053 %%BeginProcSet: texc.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/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/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]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/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: 8r.enc % @@psencodingfile@{ % author = "S. 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All Rights Reserved) readonly def /FullName (CMBX6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX6 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 110 /n put dup 120 /x put readonly def /FontBBox{-49 -250 1367 753}readonly def /UniqueXX 5000764 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be1f91da738298311c1360c8158d61cb73427 3c21bce59b0f2e1d3c853547e19715d3d570cef600ee263180d3e7a38d50b375 8dd0cefd9101435d4abab6d6fc24a6e3b161ca0bc1bf52f71a10b608905028a2 2381239adcbc00060d6c1e7ef5829015a5afa1d1ab3473c01363d08e71452ff9 fc63cff9ce95e7afaf4e8a1722bd03fe5dd4048821cda268cc9ff1bb832da53f 174d9e4af1ba9d2b589b8b429ce7368ac70433b0ff5cc83a19f4048fe72e7708 d7a8660449ecec43435f71d329eca654dfd2d005e7e97113ef3fdea187a5b834 be9e2c464878232f477bedc0c5ab6629d49e7fcb218c5353d120c77a5988d359 3721bf0112b242344df6bb8204624de09d31e54700f4ae64872a0d8c6794f65d 5825f1dd4c2a95c1bd09358efed666b41467c451dacf553a7219a9b6f15c9e05 53d54ee3fc6fb3ab2600ff64cde6edc96e198f0655a6354613d9ee365ff117ea fbca446ea8a72885313152306723dfabb26561a0887f00184c5639975db79b5f cfb46295842fd49f43ca585f36c6322f3b062b09630dba1e9942d3bd7f998163 731692475cedfafc8c07dc96855dcadcaef131b0c7545c37f5b5e82f34fad7ae 9da5183f3bf5fc3a9534d0803e680198feac773a9b4cb8dd21574f71a5132e84 2bb805b079d9ef80aba443cc74ce606e48e3ebd33a3aed020ea919845dafc115 bcbc62c1bb2867d722037939b720ea1c2d87233d0c3463b13e48e2b08a45b825 4496aea4d0caf8afe2be8c52ea28d3ee8a9cd6410cfdfe1f7b2f2ee631bd20b0 64e2312cf06de8aa4e46e1286864fb0dad8b85ccc2020107ef45acd5a8e7fa08 1f1ca294726dc0d287f3e57b9104ca9106bcb78475e4ba3818be3d84d558ffcb 5ce3bba032ee1e079e6a53a6dc18df00cf15138e79b3ca2d53ea8602e7b5c345 39d5848179585aab57bc814b1b1dcba6bcba20cacd3a0c87b95b2b99ac11b19b b53fa0ef48bfb23895b8b03a40fd771287bcc77ab0d7c39233b4cf5e831c52ef fcfa25d3c20c65ee9ff215555a9696f25c015d8efaa90d7dc8d5d54204af1d8f fb9352ad34e6e301066d5d5a0be73eed2d1946017a4371750bfd496a314dbdac 0cae43247b1ba024c5bac7f7b6fed09312da9d8b5be034e201d1cbb3bc3a93ed dc74d2f142b14d778b465ec9ee8430021012dd4f931f2cd0e6a18d2c12438693 5548edc654c8700cf77bac6332d5678297c8d3328e9457388ddf63e17814b623 c8e92ca634edf6e6930d670c8fa3e2363ebac6998c98b4e2f959e660a97820c4 1f6554eaa60796a3807df8bd258d90e1c1aaa642065af2fc766133c97ef7f521 0353c8f624d3a70bfd5e6dbbc56794ba790cf2a1e9899a4adbf6c3bfe004a249 7913545bbb8773019f6dab11ac1daab0f6f0361a819f80f7e3b1842fd2c84fe1 3614c8f0841270c1444a064c5377fbd93e8abe05a61144754ad98e959e218708 0b3dc0654d1d1d6c7955be292c4911a72fac347a3e29c03ed0c0732571763242 660c9c20037a9085c2a84bfa4ac2930a76b95ca32dc9d94fa46521feae3b53f6 c4ed451e2954cd899b9d67b1d5ab9e8d822c6a22673090723f1f8e9e8ed1a232 ff4faaae 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 22 /harpoonupright 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 66a602a9f142a563fdae3c8d40de6a8aa581ff3172367ca325e6d23102c5c6a2 02cc995819e3cf8ba1deca8d0b8e75bb167d18fcc8fe48b040b39a2fdea4444b b3074f4c31e57cef6b463d8e7deaf58bb3d4162c9de18792de834b87b8720323 c01a537d6ac479819cece5bfad6e498a402aaff52a5e322125084bbacffb7a7e 34b0dae164452a73bb9e6d7aa32ec4fb5b4306adc025d7f2a391a7622d374889 f5a721791b49e84dee4bc5884e8d9fec9b236dffc09fc6029cf15f25028fdf6e dcd6594738dcdf795b921a633a8ba8eb5c4dbd19fff5b63b65b0594e9e24d6df 1c79f275b9f94561ae97ed3d7af26b084888a86173b7777983281765e31d20f3 c2 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 3 /asteriskmath put dup 48 /prime put dup 50 /element put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueXX 5000816 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6a18979f258b8c6ed3 ccafbc21aca420c9c83eea371adc20e038b4d7b8ac303004b0aa205f04135140 76407216032fdd22e6219da8f16b28ca12524deb7bca073cc5eba65c102a5e85 fd48e6d062cd4283ee570a7774597e5bf0e3400b6be72db0115f3cb12db70ce0 83722870cddfadee715f10f1fcaf20e06f3c54afe5ca238539bfe2b596116e83 f5371ff18fa5003d8543226cfd4025f9940365b392a858d27f078d3abcffe4a1 54e78c7692d1a32bf935967c64f01b24788ff8325d61145e2d4a489fd986fb77 38e6b254522c77ca2797a504a9ce4676a77ebacb026eca94dde5922c936f8e90 c43e2851905abd26842613d8fb644512275d43f312eea72f726eaf53115dde29 c74a50749f3eca9b14bb1eb49c785918a8404fee10cbf4310ade381339155d53 7de8dc61263eaeb8ee74c8a4f3e126daffafe7f610bcb804a5bfe0eb4cf7c629 0ea9712301057b1c807557a63a51e1b335888033a0ccdef61e791476a5d9ac39 bb0ecd0742d33f21838886b56d9081c5e789a123efde396b6043160f0e00a08d c5b116c6207c7bbfc8959cee8174bf39e0134a988d76e93f422d57100fcd4a03 8e3417bef12ae183f68ce93552db1efcd570d4330a27d4c6405a28bce0376d8f af7eb70940da22fe7863874791891817af0e01274d9aba9c32df213c376663c7 17532eea92bb055474d3d1cc3cb21177f7ac86fd0e2c5fdd1469d5b2aa14428e 493742681de8490fbc7173a2104987946e1f5d1d659e0160b94c38cabbe565e1 65970d60c53294e50709be44c97d1dcf3c205888ae8354d556fa6ebb117bce2e 4d372702391c0dd0651739f4d3b0581a8548d32bfb0cddc8ea589f98850c410a f14c42e6ca7ad78b023e7553dfc942cbd4bb6f3d53f60c6f940c7a446d28222d 76431919661bc669e98c4b72ed7edaf90bfde06af8695dc555ab6b65c314dfc2 79ba5c39a136c746605d6a0f0ad7b1b1ef9b2937913dcbac35ee36319acfe8af 302258cebc468bdce5a4175e6d9a155f6edc288e0e8d8262e9bc48320776e579 584d634e96f32f5ff415d40c8db317d57654bda04d18cba39d1506340dda260a 9c7a4d1f06b5ef4025bd4ab5a90e9bbe781d65aa2114e040829982768301b598 2655d8595904185faf09b85a3836039ce1494c42a71b 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 78 /N put dup 90 /Z put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueXX 5032014 def currentdict end currentfile eexec b5e2457af4a6caaa4536453c498a618ecdce7b5f6f2d5944265309f833bd3d50 3df8bd14c071f404d2177500681a801856cb8c9c9cf0112c9bd17ccba9ab8944 1759a320b2078df0fafdd3e357994bd81740d1a7522ddfafb75d58f5def046f7 084d10fa2f702e570b09a7073d70efe37b48c355a7089772238b7b4538c4912f cf6703a14fa416057ea2e1b94455913dd271b42846ee10d78270a8118e7e4bcb 8deb048bb280c240d59409d47046c890e5f79e5dd4b0be65a28a77aeb53dd33e ac506a6da87bb55e329f433627e06c3c94b078838b908538d93f888e504e25e0 3a8ccf7d2c02122ab3d485525564295390d1154e15fee35edff4eb016fadd32c 26d1b4c09de87943780954cebc4965aa890fe1f80b03eb07103e13d468245c7b c26455f229e8859585c6eeafd9474acdb27290fd62950dc52f893c99a2a614d4 2a7e3618266f3fb0c68a01a30373e048385ffe7dd5844e208a55d1a9263ab2db cb9b19038724f95af00349840157be45cb337871b55a304b87ade64b10ff0eba 21aa0eaeef74aa01f049303a8b25de0acc26bb3a33fcf267f479a2a73f7dece3 2aa23083687322b417fe86e8f5afc2136d045ad6b392364811a17630a94ff097 038eedeedb8da4ebf8fa5a96ce27a2ebf771d14f4a11fb1f9f48cdd207209934 885ef9118463552984b0fd3cf15d55de967fbb051549dd65f60dfe07f299690e 58252f032bca78935b7d2df852ed688bd13dd171b91cf7bd46d5433aef9f50f1 41a2517c4912b837793f84ddcf1116f70f7e9b1e6ec752dbc9183d4791d92602 397bafe29ac6f879ceb72c673c8a1aa0135d6f92d0db7791f9a2f1a5b55347e0 3e63723efd258754f5f4a9ad35873a98d54b6c54b17a39892dc1d38660597448 fdba9258493a0db0d560c01d49f2d889a99eaa8a5b5919cbd33f6b3d97bf27b3 e0b94b4dcf3fe48b07398eec882c325567e28b9eb4d699111343b1910da9117d 74d6e9de75247ad908f8ce2cfc49e0faa22dbf4d1abe2c035b8f8eee2d1967e0 7cb9b5d8de5e0b1c3a86c7a08a580e51be75a8afdaba2cda08f87d352652a3c7 9bda6bcd32fd59e8079fee3fe7fa161ff626461e85ff8b9b60e2e6bc72884c6f dbaa5e8ad5912206cdd26b204cb34a5403c4fe9cc6d0a2e6805a98624c56baf0 3e673abaa926e941739ba23126dd78ca82499f376d15bfc7cc70f1586f5f10a1 fe99eed1a1f44ace1fc2bb8017599cf6ff3def1dcd7c683d21b5652c6c93b8fd f9fc70c3d5ade5f3509e03de2b93dc0c472061d17beec5a88b851b793206ce75 b32359f056a3c5e76af7a9627a6b2b293cd072e4904b8ea1ee2d0f2cee2175ca 61159ca39298a4f6af8a0ebb1b58b27e885e5708431c42838394cd45ffac06b8 154636f0b299cfec94493715c60025f42809234a89f13b3e1fbceadecd031fab 6a6283d5240402df432ecdc150513a5f917cc45341849958fe523cfb4f12c69b 2e88beef9104f76fd48daf72a990c050b4e5c2d83a06ea0d371d7c1af50e2d67 40cb2c9abc6d77dd70772fc9278e6d49da30c0212c2f94b8982c5712e4371957 2ba955ff6c60656c47616dc710c883796ac55106bf68c080194622842b1ad038 52341556036b02a3f7eb6f38b2309d6b6e99045f5bd4174518ba63d83666766c 93132f0f05f1f0fe8c7474a85a586aa3cd51ef9224707a0708a9966f537d0894 dcf9c38a780a51ba59a43c90bd24f1d6a2050e126c5fc34584ad719d39cbe7e1 143ef5e04ce291fabd56ec617cef4fdae6985b1b9578a354a47004f11f22070f 73ea58d77c881753362b0eba27d7ba8844f5f61ebe2db5bbeb5393a0ed2fa354 6983c159dd941027ea44d54767355c6ed7b16c94627eb33e2096be435cc31619 8a18f365e4150c0163b423490f452a5d417fa9ed181b136c268f5815afb45eb9 4e058d13a0627c4987680c1bc8330dbb569320979a83eed5cad9f45d88d50516 de0e929e0da1466cfcd7538a8d2ba843bafb1409a1b5cb2f705701e887c39ece 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX8 %!PS-AdobeFont-1.1: CMBX8 1.0 %%CreationDate: 1991 Aug 20 16:36:07 % 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 (CMBX8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 107 /k put dup 109 /m put dup 110 /n put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-59 -250 1235 750}readonly def /UniqueXX 5000766 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be1f91da738298311c1360c8158d61cb73427 3c21bce59b0f2e1d38c2fbc0984ce6a04f5d3bc84e2601397499a981b200b307 771141805f1f5cd7b12befe020f7121ec103dc2baf17f0aca1d0740c92a8dd61 12fa9c0d30cdae4f2af757b9f713c127d6ca02ef315205b81eff629104b21462 17d2b0f823133bccc80c9e400d6840db5fdf828f845313290fedee040d34f309 8f1b5a6b8941c81d1e607dcb310b14169cb433329994b4d1ac1b491e7422d518 2a8faf7ed3cbc2e23c58a2e310b56da54f51360556d27d58783694c51aaf62a3 a2db91ac8df80303c35cfd503341005020a5cf501fc0b64ff4f497951df6cd50 f00fafb04725d23bdd36616c7c06cd9052305682f6b6e521882922377b4979a2 e6f031b91b2a86d8631e81de17f90158ca05045c3b823da8b2d2aefd37a32728 45bace7594ceaafaecbf6140212789b3937a24cdd8dee327c934a9b0e328fd83 a1b535d9bbea6f0204ce5ee74cc30769734d4fa3dd0cac3986510c99ba062868 9285146cbe64caa89cc2f18a61d924e7f312fa189e0ae4deb913ed1cdfeae61b 4e0fe25f2775a8b2edabadebab4eb2ecf47258c1a77101c810cccd3b06633907 f765fc80bb1a2527f8a59c52760fffabbbac1d65a11708a3bea4557f24536773 d17dfd0eeee4e22f406e04765cf582d6ca4414f2441b6ff463c56018e44aa2fd c9af3f0480da99856aaa97e56b4bc28d9a74e6d74e5ee199932ba9548c9f5a24 4ad6d8509e006d51a96e77cf6decfe21ff0391441c8b95ace9dcb2d93694eb75 e76e0ef6fcdfb8a73c860e09dc49dcfea499450162b7373cccfe5e771eb05086 9552a2ff03a1c1c6a08405ec3ad93fa9c951b4aa746b4c4bf3f98a916ba7134e 93d777c339423c4fad91e1968b981744bda6f085de1cc197203fe456b7725df5 7be00d9fa5da5c3ada4cde11fb7defacfe12e8bf0c3ef6d7691cabc03af5fae5 b4811327905939cd130e7b9814a9b8f31acbbd1a301c2c8eb0fb72761a872e86 e737d65b4d729a99a4bd4618d42d76638700afa849695d320ca010a750a8d506 37b29d0d09b818d3d42e175a33b3195b311a1ef5db7da0ed66e69d808be010ac 7285112c83f00f287105cb05c7767c0de21c9d466b07cfe2e1ec5bcfc5500941 787c9db97e4bafd4b54f28a6e14a640cd461c641e816d1b7ffb4a7197e0ff5be eb1b7bc33f36a585a4bdac07585cb85987e41c21ec69c24612babd8852c0dc68 60dbcceb1d4786a47d63334d825662878d6f9c9c7690f5ffaf12cb099fad2364 76b6fccfa6357ed179c02491fd40c00e450310ece846af4cf7a065feda7e4042 c30d99f6d0a9efa8f31f8b442311ecdc2e21066996921a558e9b91dcc838749e 5712dc84ffe0304a42ce3b4bbafdea721851df7ad7e3f83c03cbacbefed8ce16 8a24a72a0709141fde8c6e64f51a9f474c93d4d02927894af833ca683778b687 063e2685d0c4e44cf8ea9ed4ff296038e5cd2fc3e85b5073778690e53af039b0 3a6f6b4eb565dbab0b4349bb78c90aa6d6162b675b5111664b500ca08d9e255f a85c59a07974748f6f3a42fa0d282cd02b21bd80dbe98bc1d32f8750d6cd12a4 9e848bfd9639e19f4847e261e0e0d6556b8357d88f66991aaeffc1e292fb4ea8 6ef2d4c33c2cf3b785b394e03570eb47773fbd3fce150498048d334cfc5fbfa5 59fb90b97fb481c41339d539d03844acfafba7b738c4fa2357447c9dbf1e1a2c 0a5dbff3aca01e1c122ec88230b6395ba3e7f3ebaf3d162e71d28166c70b2109 dafc286f73f7de00063c3f880304c0bf66696ffd909ad3de119b2f3cb23fce80 4a230a3cbdf41448b6960167778afbdc9c3975f41277e93b6cf414e483edd9a4 f9f4ad50e15b58c66957f3764184d79582f9d3923355ec5140133a75f68aad6a 41b51eea672f4b70534a5a8d04a9321bf9cf9441e32e273ac5d6e075a909145d ce62213911564042b42b581525e98526e207711190f42f4f395289a110709068 8f559aeb44779840f1e0e44c91282cc9b7a68f4d38005f675f4543cb4683beb5 41fa2caced24b5b15eb0ca7d1780a713a1f3ef75b72bf5a88f45f05f399b7f75 0ca573c315fa0f04b9f14f0b552f0ecc671dd3a37f83b388fae473266fa1971c f7a664843b9205c13a1cecec42304acedda8dd687753b9abdb99dcc3cbdde016 ff0e94ad4a2d9a1f54f9c0c2e4d4f601a6d42297ee0f997f2c2cf2873e98f2d1 74ead8f5ce6cd60cf7e754f8966f6681204440d42345a361623b63b67bf39834 a910795f1154e14a995b3ec1be84fe69308f9a755d39df5c652d708e754d73bf e281b7f1492ba362be4f960978d451745ff5f67c37f78a7ccb8e4eb0bd9242d6 5113e227b3046cc25cafa24d535b8ccaecded47a0f9758736a465f989e9f833c 9716003f4a9d41a03e8159131fec97cd46322093498d44d864cb1435dbb31272 6e835e298af865ed47d4ce77a815c939cec3495bdd79ff3636214c11c5cf7f74 128234c2a0805587659255c9ccdcba10ef 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 49 /one put dup 78 /N put dup 107 /k put dup 109 /m put dup 110 /n put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueXX 5000768 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306462c563d51ece66fcc9c831843ecae 1fefc1a232e2724f7baee428ae03aadb95c3035345c15e9922fe49e1f2cfc980 237316572dbc57064edac9b0db8913a5e2d45e97e19a91435ccf8adfc835b585 48e74b291446d689c7f2f8c4325e8356e974ae30c3e2977477baeaa33d141fb7 80f59351e84bfc88c87b3a4a1d25e168b9d134554f6a581378c7d2d6eca8ac09 045cc3a5e0ed86f147133f094029e3483bebc81cfde69942cf645ea20d0a2b64 5140ccc2f9cb4a3d969f4b0a79cd3329b440dfcff5285494406f3dd08d804c86 df191e57232f6f1f83df6430f214e03fa5412b1e13d83c60c52a698bc9a38159 19a64162af7b83a9468fae9507c6cb55ea80df8c696dfbc3296c663902d77e89 a181502627281181ac8624f65be9de5fb32f056cc7dd7cbc7366a91717b5892a 34ef6b441871c6993da3a81afc9bf70ff3842a21a29c0e49da1ccba37dd6b920 2702db510b8b1109acf4c27ee8d676bdbd16abe189f34a9b61532eb777457698 cee0561838eaab2d3939c5093afbb215bac433d1918c739b62566906bb2187f3 378521fb52e70c18886e1aa15f53a1f8a169528cf2ddd54c203613ff88e619f2 6c9c50c2c3b8b8c868a88afab1dc8472e7b6f3f992a161723a3c0904865a7c42 3281601eee91f35222c4a71f343739f102eb43145a47e4fac3bd666e4c114a90 5950618ecb1bdc29f90f6ebed1d8325fe6b52e14a8870e9b4bb088e585c71b91 493c12845a101937e3473ac5e9ae24f7c358f060dd64c64d18d21aefa5b5d6bc cf502464ead10c6f4f24a3933338c6ceb6c4b669f86dd725818d916a863d3315 04f6895b442011d2ea93eb4554d6d45f65bb6b9ea45fbe3ba440cbae227b717d 1e08823f69d5d226928d0ac33ce73d829b4b0163d57c1caa5ccce1476e9ae82f aec78eeb28686f8ac3d8c7556d9ded998dbf0c8e92bc3505c8f9590aa877a5d8 1d46e1000b691d86a13245271ba232b16d59ad8072cd228ad336bdcee06af13a 1576a75569bbbbd14e58b7e83fe1e3ac0a664ad6b54413fb0997938f75dd4ca7 25bdddec45987db041d8cd5e7c9b687ea977b02a30bac56d562adc89cf49a7bc f7a45f2692c7853746a8b6fc9a92594d14dd1b98c3693ca7b8a95347f606461d fb36404d7878fd231ee4ec246f703b5d6cab26c0997fee2c87de65f40b63ffb9 bdb8b74d40e0aa1c5e31c5a6d41596e1fed06d82368be51ae5706e155116cf47 83ccd5a374303d9ed776b61913ce9ee838b9bc1f6353213a46c3323ad8b0457a 1a3a89607be0bc7b7d18a22d76dd096a79ee1393ca7bf624681d185bb9e717a9 b5d40a1a60f530c1faa0218c529f90d5ac337c4aabdc5ae8052004a2bebcaa99 4f069361f494ceaaee336a7d2468bae0de542d8ebfa862c4a115112e1674a32b 878910a600d296be16856c2c93b98c44c2f1c163e2a256273e766e54dc8d03d8 0b5fe36256bebc673a5fd557decccc0b3aaadfa0736e394ea49d5aa064d57283 516cf913102782fe87a9ff09f599d05eea874e64324a44d83380210cca47e327 253053e1485d2e9bd5371cfbe651f11cc96c78773cf36c0225d4b334b6ed860c 84390459e3dbefc3dae899eb6e98ebdf61301023949b01555bb68ccf68fdfd26 939801c8de028d60d3fe0174a0bf5516279f24ab23cbae7af1aa51e091e636a1 8ec46f27846101d36c403d1a0ff1e47ead20c23e00541f1ccb95fa659931cbb8 2ecb6772d4bfebc83bee1e53153de95e2797db3c30bc7dca8615da72fe9f47d7 13cfc016ad63fe619ceab2fb1662ffa7bf8455e9aa9ef19d9520d21e46616966 1afceffb120364d30d7dfd35cacb836a0c201c246c530ee0203c01ce6a597ce4 b2248929874fcf3a3b6b90769a836a7e068666ab7d8607ddc2308187757d4531 91d11527cb6ce435b5f585ac14bf613cad8e1995ce4cba990189856ad7c8f392 603676394e1c80276226e242aaa927790a3fe6599613b32efa8d944c6dffea12 0e20e7d2f0e9234293267e816d5d479ea4719b84436d13a34fc76a00335e2c04 454a849b46f8f5f0cf0f3af47479ad728774b96639d100623ce486ed33a3f6dc 86227c92c9a8681ceb5a8650b3e102e021355bcf8b97eb771d041ed0f5976b81 18358571dc9d2f6926fb75c7e44042aea60b0e148d9128262ccbcb028ca6d179 a99f919c50b8f5190bfdb6c9eef317b39294232d53fcaefab6c564a9d354afb7 1019a1e5d958a4ac06531f66c07bcc04c928bf10800db6c0fae98dd18876db04 8c8e9c91c9bfeefb143ef59d9fe9e900645bb25ffb1d5b31387c810c21181424 2e3050210c022f18e4089745159cfb0290ea7a4c92ef7414ea361679ef189b69 69763cc125df6446ef226f0c2147deefa1bf69e03cef811d02edb2e06b283119 43a53ad502b89fe79a31d28a6a964ef329c45dfe1ecd87c24c4bf8ed81c4eacb 7f60c7ab5be5cdc9d546d5e8e6ae3bee50cce885363bada372c4842538b717c8 a539690ce0917231fa9a3a94e06b2428e98a772fb690570e886c04386e99ea00 cc54b85b9f24a18fe7c8203f00e1ace9056339248345ee6f33b8629c2c0ab0fc be746bc31f63e7a4c357c3cfe42c9ef50b5977adeb261c856d41726633771b5a 37e68103db9e8568557bffe8d0152c48c231ce55c080454e2f023648a2c76842 b1611e61f2eac35bd9d04e122fc0432218faeaa0e078c9fddc8fc1b0b362e346 04f971a36dab180d195f327640fcc910cacc70157cd27eb44543c115dc25d50a ac23d1c074b01c98bd745b28b69bec5341102783d9ed18439fabaa4da6424818 e673e743669905269ef073ea4938fce6d42cd7c897cfd50743d0e107d2a3c8b8 dacbd0627be9eee1eaed7d74780a7a4775954e9d8229a1024e596c226b080452 1f9e314133e93ac80f19ac5c43b5326ef4d9becb4a6c8eb18be8168f69d3881a f56b30e0984f4dc09359630a21519b59be0e9c46f4a8b7af7668f9d1353c47a3 3b213c2782a4fe1af854018f9270c39da873a0feecdf2a694448828045c40f37 81ab70357b6fb944ac40db5fa0d5a8e7fa4ce499415f528ba576246099e976ff 80f141cd607cf2b8c2f975c6a2d88b97a38ca2181b513986e65ca7c76725e0bd 0309fdf6c954c2a5dd4fec6eabca492b0bcde887908928f661fa3d5d40c8edc8 520b8e09bd07712f66b25f2b201bf35f0e03a2327846d039b989f7e5ec069e33 0fbd9e59941edfc72c1d2ed33a98 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 76 /circleplustext put dup 77 /circleplusdisplay put dup 78 /circlemultiplytext put dup 79 /circlemultiplydisplay put dup 80 /summationtext put dup 81 /producttext put dup 83 /uniontext put dup 87 /logicalortext put dup 88 /summationdisplay put dup 95 /logicalordisplay 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 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 3 /asteriskmath put dup 20 /lessequal put dup 26 /propersubset put dup 33 /arrowright put dup 37 /arrownortheast put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 63 /perpendicular 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 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8a50e98a1e29ee510aae9e5d1e543b88ef4cf9de59a6a8e4cfc3166e9a87e09a 04d5ddd2342e3aeff19ec49c31ab88dbb9f5b837b856dc03f04d784344718ac7 0be9c06df01779a2631789a0d0416dd3bde99e4208e68bee5c1c99e16b7cc198 95e4a258bb8d0f67ad5ff5e0441f6b1c9fc3523bd2b083f266cb1ba3d4b0d7f5 91f7bfcd5ae8cab3b9f1866144dd0f472764420ac2be5b3410ac0260c84f7f5f 110ac30b75f15e893dbd536fcb016f5db23cf2401591f1b4cea9e9a7f6d1679a 74455ff851c7aa9284baca1e15d94026f6b171d87ec26e4eec0b5aad45263798 cd22d55281221f4873f5b95fbf9d158e1daffc9ec56aa935606b5cb1c66b9fdc 28908bd66556bb2bcf8ce4b062ce10c2371690bd76dca4307eb0fcd11650307c 286842f35ca3921cacd6a48842dc1082c8cc384f06bc24938c28c2869da7d91f 8595b4bb79a37b1955d7887455928f3994e4c81c1a6a4e40a9d52d2da96a4bc7 a0527d7affafec45fd1c91a441cab8ec5c7f458d618bc43eae9c693bc9b2ca36 23419fd7390f77d5e0d2d36da561fc0c5584f095446ffaf3c327916164bc2d66 09df844b2cdccf86cbda59b1aecc644a47d73d261be3d0254160d1ccd17c6386 0a0e303f0b442ea0e874f41d2c9ea13e882af69a904dcdceda8a8a2abfeea0bd 7275e2901d8b4cc4856e2d8a7ed2206860723085a1bccf939b5fc644d3c634f0 5e8a0298754bc60dea3391c69586fef8d3ea86d521928a264a7a321df53b376e 7f85ec8a510ec1ee790ee8677130499266152cc0494d02afe3a65fffcd9a1b34 62251a8b30ed8833ff798753a3e023a6665769e9d332fb6b2f9d79b300a0c13a 3c7ae27b7cf07a6b9528af6b8be39acd5237f9d62d65ded096f7b7e6572baaca e873246eda33f8ddecc91c8355b30585fbd61838840a67f83664a101c1cb4947 6563ca96c512c11f9ad52e9b6f791dc83df777ff7d0fb2f557d038aaff20bc33 909682c1c102f079e76d4b201c62f8403e23b5c9b5601b265c04b286abb8888f c379e3defe30e0253334d86a39e53419cc32eaa0c9a58cc269f8d3226c61e7ee 455fb68a964469a70ea2512be6979a2ad42d701f2a80b7bd65523c29819dae88 846c325aa0e89f85f1183b4bcea68061c615322f1f1eb0c612302faa9d4883fb 8156fa4e8beba9babaf527b492508ef86e57b47fa4eef51b3cb0c75b7730aa8f 73338ba36cc8c713eec4ba8c13d9fda1a5239270d00f1b54716a815719a96705 304e7e1b8a2c8649e7abd413901d3ca98ede786bc2a1b8b688641ba744916df2 5f623c10569b126d4dc62443b0880d3879457ef86c26e41628ea1b9a69b0ac1b 764dd986642c6158d3cf4573f49d798d365b6be35ca5a1edfa7b9feb4e659380 08fa7a3b9ae1dc57e8e6d1deefc97d7990379050dbe3c054919713aaabdfe16e c2f1b03f008f12ed98e5cf82018d0e3870fdc5b5081065e577bd167eaff5ab66 350a5d65bb7fb9ab82fd8cf438c59667b79e30b83480d8d8651bea1953d1c67b 0792e264a9df2b7f58e910e20377799263223c79b2247a7917f8a4827f8ef6aa 5262cf3c4e5b6e273bdf4ab7c6a99778e9d7ad30925f9c8133bc96e5da048b85 1d29be79d7fea301896a320df6289dc90a9e49c2475cb048a54dca5e98f73165 ce662b2069f5e2e40f6f5ad91b74b53c01c38f26ee41a09c9e3f5d47806424b0 d9a66a55ffaff4e14d36f279e75b32854d08eac607a3c03ca235d57bd5f4f016 16ca86767170f1c84868f26316bfc6355a9fc7b42eb3c5b40a065159c2c9d323 631521e4603c05246cd011137ac5094c5deba8da3da091b6fd6320d7836fa55f 512b472fcd068849f57b0b56ccb5e9c729e19e80b23681d214f2695548aaf15b ac36fd2a31cd6352ba5598504589b4acfb49ea72f9e225992093e880e44efecf 61cb724c57496df4145b8b17c3d581e8d19dca067eb724db520bb546f1bea734 ec16fe986a86dfa953de4baa3fdaef426670b7f880a49eab0aaceff52fa5f760 ec3e62f7d673efc37cc3314f820bfc9772db67feb1e66c8b58b0da463a887d31 ab8e855ab0fd0a109dd2e48fbd3e021be0b6009181f97f753bc41d8b0749f422 103ec44c413f045b5a1abc10d131b084c3d57091ea6649f7b01f17d5e28066fd e3c122b567fcf0e71f7b68f926c44704f4e4c728234fe2719f1e8a0ccae9c1e9 bfd1721f3977c0a7250485601af8269d613518939f07 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 3 /asteriskmath put dup 14 /openbullet put dup 18 /reflexsubset put dup 19 /reflexsuperset put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 26 /propersubset put dup 27 /propersuperset put dup 33 /arrowright put dup 37 /arrownortheast put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 59 /emptyset put dup 65 /A put dup 66 /B put dup 72 /H put dup 78 /N put dup 80 /P put dup 83 /S put dup 84 /T 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 dup 116 /unionsq put dup 117 /intersectionsq put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueXX 5000820 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8 3750d597b842d8d96c423ba1273ddd32f3a54a912a443fcd44f7c3a6fe3956b0 aa1e784aaec6fce08dae0c76da9d0a3eba57b98a6233d9e9f0c3f00fcc6b2c6a 9ba23af389e6dfff4efec3de05d6276c6be417703ce508377f25960ef4ed83b4 9b01b873f3a639ce00f356229b6477a081933fef3bb80e2b9dffa7f75567b1fa 4d739b772f8d674e567534c6c5bbf1cf615372be20b18472f7aa58be8c216dbd 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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 22 /macron put dup 35 /numbersign 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 58 /colon put dup 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g 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 115 /s put dup 117 /u put dup 120 /x put dup 126 /tilde put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueXX 5000793 def currentdict end currentfile eexec 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160e3a0626e7abb01f2308440acde942345667d8db69112ad425be4f82eee415 b42d110610de8de4ea08643a00d3b113b0c466adf8973624375f53b573746314 2bcbecc4989eaeb7fafe3836ba7593ce5493ccf2550bce1d4261917623ff85bb 883f71a856787e82d9daf8d3a7201ca30f70bee9e7ce2ebb6cd7efde808aa06f f064f10c1ad94c45a2cc3c96d23a13ecf26ce74bdf731db5795a982063ab946d 3b05d913f35a99f961ed89bf981a471774ce469f2f947bafe3c57ab734fc237f 12812d7551a9ac81df891d9b5f3e65dec15280d38715fea9542f7cb2be5399a6 1bcb5c56d29ec34a650fe13015ac086c537e17846731cd4cff9c446dd5dfa970 5cf17316fa9fb59ce786d684136bada0da4992b44e24a4d4c8d07f93d25d3a5a 51018c437ed07d3cf3d547ad1903941c2a734728cfbf93cff7ce209565d4c3b2 146ee35d4f9f28a28a3390578d8fc5d4d9e7925d31fa1969f3680797c1927e88 11ed2644e4546669afb4b7d6482baccebc36181d697bc9bf49edee3d77501683 744cd84eae2f94adc596d073b53853337894418dd77143aaafbcbb775e9b419b 99bbdcf65da7f85a7ef58a2dfcde8072c209489454f39260006cbdeb51ffbccc b1cc96e8308f0f9ba8995abf1121c2c22834d411464ad64ae63cf0b1c1ac4400 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /Lambda put dup 9 /Psi put dup 10 /Omega put dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 17 /eta put dup 21 /lambda put dup 23 /nu put dup 25 /pi put dup 28 /tau put dup 33 /omega put dup 34 /epsilon put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 71 /G put dup 72 /H put dup 73 /I put dup 76 /L put dup 78 /N put dup 80 /P put dup 81 /Q put dup 83 /S put dup 84 /T put dup 85 /U put dup 92 /natural put dup 97 /a put dup 100 /d put dup 101 /e 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 115 /s put dup 118 /v put dup 120 /x put dup 121 /y put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueXX 5087385 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930da4f8d58abefd45db119b10eb409dd89792 3c6e705479464a4b310b58348c4b42393988fef4925cf984423aaf65fea9f0e6 4629953bcf50b919d968d99bd185f83112b2759cc411764e9bde677f57c5ee5a c555448d5f81a16259ded1e11bf4119d53e8ab07a802df900d3a2d5ccc1c6876 d29c5e9effb7af3ef83400b0910c07873a8c56fa7b1eb1ba426043b00cc95dbe 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 78 /N put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueXX 5031982 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d56491bdecdcafdc988206c5a457a19270b 37d0ab776e03eaa7eb568eeab6b5e79dec03b0dcbf923a2aa8e4f4deda2cb043 858d8b430efef0ed914a80ffc9818aba0fb30dae3694e5b31df3855892d59b89 7f82ff79fce8444c7926ffc4ea008e63bc518936c9098afc2d1c14030a2c15bc c9285c1b57e80570d2f6301009e63d9ddeba44e3251c75ca9616a2c5de9ade66 1a15db37b40b67c1bd71f77343912c62f47dd24be0e03dd833ea60d2eb020407 eaf042fed30f0c2f01956c83d46d982b50dd979c2026ed5f71979dd98c9c1eee 25054000a5eed5ce9498b2cb59a7ee55e0538ab47d86e69b2ac79229c8c6bfc9 3a24e822a2f1c9964fe630344579a25f66646113f73d1315c447f070c575de76 266c6773b01137e4a039df5e4c2f199e8e3396ef9fd001eebe090e04384c4089 ec9f13a07ad8733c6af775f432f154ef50ec4477e08760c03c006e4d1eb1c7cd acb8955ca4b0d62c7de046ff58d431bb9883d28586552c51809e3ced4dfc2198 b97ad9e52ea33c91b9e335a80d9b77d3430e0487930f992459664abca5dd9177 150b0fb502fbb04ccaa94ffe2697ed07906f95fd98e7b325bfbf10ea6b39fd76 c1e5295022cb39e9cd471b7b0b3f9d72da99deb0615d35cc22f42dbf5cfb70dc 6279fd520d99f32201abf25e621cfe1d37961d9c0ad36ef3d9264c547638152a be30812b1188a10f6327c9e8714449de2ae636420f56a44cebd774f0b131c629 818455cc8dd8d40939f3ba8aa5abc54590c7286b172193c23adb1b4f9931a9f7 73b960b1f5b760438028478e4442dc93a567b07e20017c18fca2a1daa2649dee 5a6ec51fd6c3f4b27367c31e95a56ba4bbeda0493bc02306037e2ba69add9b94 373364db316691e73d6811cd76fbc92b0c67ca18dcdf05807f17d1b309456ea9 d113ea0fccd6eb73bd8dfdbd5905a5e38bbf61cfe58fb01ec95a4bdd9565433d 4a7abb8dca6f76c45e50c1c9ba538078a5b7b67307730c84cfc5f62677924a48 28a9bb5be0b65603fd59e525ee6d0fdbef578ed48c15b1a344bd3e748631dc30 08992697b3543551e2a8740b14bdcc1df5780e9b12aed9da00183a46d0f2c727 ba64ce49e2dc7ae49ddec532db0ff9043ad3307040d1c84f4f7904c6250e4190 3f3cc5b97aa240986d464d116eb09e28c4a357bfc705238f21d179c1e8efa07a e338e737da2d04db865b77e9a539e9499aef03ebf58dce169a522fe6d2b43aff 0d806a114cf5af8368256694185c39f8eb6f61376624e3fa5572e82b68906652 e42d01549282e0bed67ed3692a53d6089395fa4e372a719e5b5e0224d57006cd f9736a43baf17b1991ae1d3e369c421fe0303d1c99fc9a131f48f07b54ed6e03 677f59d7599c753759e1b2e70a1c8781495a47d1d1ecc959d1fffd02d90ef97e d64051366bd579214b3a765d141bf635bb8f5e37e8ac4ca526cabcc4952f6d96 00f48de74285669150e7573b96739e8070aa8345003a001eb68af57ac4a9fdbf e39c159f52aea9d039c6649a8658950d75bacbfc078edf61ce692ea376694e7b 3b6c69f74cc753e0ee2c984c2495fa75c58028513718d676376add5a49c8f69b 2fdf21d07a0fb882bb9187709918da7840e1b40411532b9a5a2358b5cdf1de91 01c6b8ae55b866b2c6bd35cf9c3b48ec213929100b9ce305126f7795f8493a17 7ae5e1d525c518141e6464360ee250ae8c35f41820ffaee87bb999a091a86b96 96816c80ef31d1072b1fa4e55369c97fe3bdc3424bde0fa8591ed9433868a9f3 ab2cb902e6ce 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 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 e625ba8091f481ffd2b7a47220a8ad34d58daa96e5249e4f0706f1d535a25c6f 06cf9e2ff58fac111fadba4bd9095019791eada3119a041682c621b0e1831c3a f432fbe9d742963d816a1692e1010b567e2ae4d40d61f8a54b0401377c4b13fb 4fe1cf1aa15a3083562f28c7372e6479154306f1edee2664c2e9ae3a06f2c7f2 0dbba0d0d18e6c51c5090d4e888bd6ff50f9886b327dfe843b14c432b1c7ecfe f5a88cf8aabbc7cfcd35557663beee65e1d4ee72b329014e72b21350d36304c8 d9440a569c79eaeac097fcaf616f64f4ad6761a5cc902258bed6e689d5f82f1f 176fd5faff805c7590a821283fc148b5a020f24070d304a6867ae385ed6f6198 726d78540234ae574ae384a45bba776c084b94decf657985f4bac7c6a5b0cf8c 6ef5103d9c8f116ba6b100224a223b497ed5a1e8cbdfb23133e0e4a5f54e8b2c 1e21d829fb064945bf3c21290d4e3fd41494f15b1187b99f042af42e01429edd 83c9ef1aeae04d89cdf86aedd9c7cc9cded0d2dceff5612ba22c36d4ed921d7e e7a0e0520c92da27df2665bac00c1e9a6a00801182727134a52b5d54c4d882ce b948a97f4ff72598df13e7f0685ffd7e5b0d130673d686357ae147424314475a a5e927759ccac76ffa01ab6d16d0a325aace0d9ce0cc13eabb2b9eefbf3dc196 dc45e847dd7717e417f6767320127aeae1394e3392476d0761bd587c60a78883 531c073d45135fb89e1263a5f6027994ae37ec3ae8d553ecc1867060542d90f1 dd273f59 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /Lambda put dup 17 /eta put dup 23 /nu put dup 34 /epsilon put dup 59 /comma put dup 63 /star put dup 99 /c put dup 106 /j put dup 108 /l put dup 120 /x put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueXX 5087381 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 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f1c41b78f91ceda2cbd7e51f75f40dba296368bec3aa3d2f153091f5e09531b5 db1851dbf3d575d90650de63991f1a3c5ad405b0d62dd54dcfd770ec36c4ae96 0ec434cfaa1cfb72ddbd76e09e6b0ce15a7da2870f25383eab33235959ec1da9 47e6b210fab8d52dfad90d8aee2e42ef437ac7b988bd87c4406d75fc0afee052 94ac3984901632bb71b6835f9d881acf019cfb1e459822f7c0f1314cacfb6495 5c30d0269041682e2362862d9aa4d2f578e187fb05e50721bd1980751bb4345e 98c87df42fa15ea6dc83aade65016e09f4055b55c2104b1b3e57b46f41f0d9b1 c6cd65f59a 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 40 /parenleft put dup 41 /parenright put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueXX 5000789 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b031f3fc6c13d7da7277a94ba 018e9998b3dd888011a5d7c4204989f30f908b95533bda845746b673ab71ea57 65a0d14f4350707e47c8276305b28513cbe1bb0dbd269a53719bda46e536685d df78ca0146b6b93e760256b74d939d4e35b5e77238f04c92298dfdd188feea30 e053eefbcbb52f2011772b3aae39f5805597bbc1e8bb75a446ce014030f4f2f0 f49f9e962ee4a1024a746fa92a3628db5270732b54e43fe5ecfa524f127e5fcc 788e77e66098336ad67fe4cccaf0253272d5df79864bf4b734cb9a5859d557d8 bc11b8e00221ebc12e97de4b1f466ead83a4c894709363bca9040410a52d592e 34ee40cc7e5efa920546b981aa659513a24b1b85c221a1875b62d0b89e57a368 321b8043a5b094e0379760a443d632892b14ad6d19dacc8c78093243ad67e6a3 08e56e6b68412ee690b10dac6e17708754a00d51fc957b500eb80175716eef4b 2ca1ef867614659bee3f2b7319e97b6fdf1efc847bf3cee3156f72f21751da8e 5fb6898919e6799820d3de0642d756e09d6fae4ff08dd3deda3173bff4bb11f7 9109c97ddc05897af709ea199a90fcee8ce4c7a3c15b18170c41c04de2d3fba8 f34296a95b8e1e8de3739b17273f8f2c85e914615e8eac5e8bd2387ba3b1edf4 7968f06e2067d836d0f9f3e085cdfd2de06a62c81d786b304326f7002e83160a 36598589228b4dddddc43c85e1d126f8fe81b828028e26317af5894aaccf4f69 6301e1a9fc45935d8a414957f08febebbc3a72ada80f101e47447d019ade56e9 f4fab969bba2b44e47399fedf5caa1bcea216d7ba713d523da5d7776ec0bb452 fc23761a3e881b097d783e661787355609fe4e3270aa73c02b7c89ec92921bd1 9b8aae71ec457cbe8a993052d4b21509e7fe162582425768464215c3a2cf4ef8 1a3586a9c8a201c5a7e6fb48a5383051d293f18a5e3352b1aa21cbf91cbd0426 36816c6bacada687aba9c3f14117b0285d2e0e5c63c438593394ec3de48dec62 7f482d2a752814752fad8095c2bc2996863a5227c94d6bd6dada75f6c13b01ab 0b1db80cc7307e76a2eade5144476f790f32fe3ebcaceb8ef5fb5e26e563efef 0d98769714ef968bea05dbb0bcb1aad92d8d5a049f5f1e5df559092e76b50af5 2d9e83917f0215ed15c7d6b51a68d5970c24e46bcd7f71d6327dd362ecbb9c09 c7e1f7dbcd1ca88a67ef3872e9dc70488541348892881e735c14738d08863072 31db7482cc49f8425fe94afbae4bb3b6497e338a17762b4ad496dd34fc104fed 2cb8e390ae090d21083b097d4a09c4aec4e7d4fcfc530799a06dbeea9384b649 cc0ce2927d6342f9bc2c5cb3a3d425f046242aea900607c4eee559fdc8ca481b bcad022b495c241b9550b92fb6724db22b3a17f4d6dfdcb942ef25dab62404a5 109ba9b80e5c71f0571aa39b0ec7321f822348b8df3d01870b7c6127e04d19b0 67d53bf2bfe7f5f10fed71a37cccf58a7b0b206506d348258c5a9e864198b142 70a62da2e7ea5f675eabb0c6f122e7e22ab7f38fa4295ba4dc39989f2036585d e102bd8f329707d76d31601db52feb05134a7f2b3881fd5800b47a4e37ec03b4 012e7a716843bfcbf86528329e7e52455c7219810952a849abd2f7e2aaaca09b 12974b185e486c07bb61b8977df7c8e4f2ace12b770e494d1a0e1923a6caba9d 9973fd1661841eed71a437247317c16354c3b6adcb72592e603d10b825bec5ac b5b5e9a668118b277ee0827f1cf012deb9fbe738232a732440a2ca51fed12166 c466140e53361096464fb43c43b533e0ad5b80ab717bfc8e99f0cadf17fcceba 45eee0a7f73d452327ed141a83ae9e29eb4f880ad1ae27e6dd8c813afdf5dd5d 310b210af856080fd8686dd7616acd363e73f6afa82a44a09cb6f3e4422b4282 ce539962787ddb022cc6cf10a4af4395c7f924ed1c6a6eb9c27754dff108b28c 8e0c8bf56c8bd37a8759175d424dba9caec3650a2f8b3ac22f8ccadacdabe3fe c1091405052388c49c2321d641b2e6a82d98d7fae9e6b3f9b4be5a76b558a9b4 428155d6052c526263f0ae407b6949e6f24d566c7f4f6801d4724ac07cd42328 530b534325690780b171c4632c3304d2917c95508947667b0c03982e5ab5095f a4cc0e62bb79d4c50b78772a5a5b2990b498b57cc746d22e033adac3efa74f40 98c31b22e9f4255e5aa6ca3dc6376a058401 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 3 /Lambda put dup 9 /Psi put dup 14 /delta put dup 17 /eta put dup 23 /nu put dup 34 /epsilon put dup 59 /comma put dup 65 /A put dup 67 /C put dup 71 /G put dup 73 /I put dup 77 /M put dup 81 /Q put dup 92 /natural put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 115 /s put dup 120 /x put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueXX 5087383 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 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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. 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Fi(x)p Fo(\))p Fp(QU)1442 3902 y Fl(\003)1432 3962 y Fx(\011)1488 3940 y Fo(\()p Fi(x)p Fo(\))p Fp(\031)1665 3954 y Fx(\011)1720 3940 y Fo(\()p Fp(a)p Fo(\))92 b(\()p Fr(by)24 b Fo(\(3)p Fp(:)p Fo(4\)\))0 4106 y Fr(holds)46 b(for)f(e)n(v)o(ery)g Fp(a)65 b Fn(2)f(A)944 4073 y Fl(1)1062 4106 y Fr(and)45 b Fo([)p Fp(U)1324 4120 y Fx(\011)1380 4106 y Fo(\()p Fi(y)q Fo(\))p Fp(;)15 b(U)1608 4120 y Fx(\011)1665 4106 y Fo(\()p Fi(x)p Fo(\))p Fp(QU)1934 4073 y Fl(\003)1924 4132 y Fx(\011)1979 4106 y Fo(\()p Fi(x)p Fo(\)])66 b(=)e(0)45 b Fr(is)f(ob)o(vi-)0 4205 y(ous)50 b(by)g Fo([)p Fp(U)402 4219 y Fx(\011)458 4205 y Fo(\()p Fi(y)q Fo(\))p Fp(;)15 b(U)686 4219 y Fx(\011)743 4205 y Fo(\()p Fi(x)p Fo(\)])75 b(=)e(0)50 b Fr(for)g(all)g Fi(y)75 b Fn(2)f Fp(G)1838 4220 y Fx(l)1913 4205 y Fr(and)51 b Fi(x)74 b Fn(2)f Fq(Z)2418 4172 y Fx(\027)2461 4205 y Fr(.)49 b(Thus)0 4305 y Fn(f)p Fp(U)107 4319 y Fx(\011)163 4305 y Fo(\()p Fi(x)p Fo(\))p Fp(QU)432 4272 y Fl(\003)422 4332 y Fx(\011)478 4305 y Fo(\()p Fi(x)p Fo(\))p Fn(g)648 4323 y Fh(x)p Fl(2)p Fx(\003)p Fy(\()p Fx(l)q Fy(\))890 4305 y Fr(is)21 b(a)h(f)o(amily)g(of)g(mutually)h(commuting)g (projectors.)h(The)0 4416 y(Gelf)o(and)32 b(isomorphism)i(represents)g (the)d(projectors)j Fp(U)1824 4430 y Fx(\011)1880 4416 y Fo(\()p Fi(x)p Fo(\))p Fp(QU)2149 4383 y Fl(\003)2139 4443 y Fx(\011)2195 4416 y Fo(\()p Fi(x)p Fo(\))d Fr(as)g(contin-)0 4516 y(uous)e(characteristic)j(functions)e Fo(1)1121 4530 y Fx(Q)1177 4538 y Fu(x)1248 4516 y Fr(on)e(some)g(compact)h (\(totally)h(disconnected\))0 4626 y(space.)e(De\002ne)540 4603 y Fo(\026)519 4626 y Fp(Q)j Fo(:=)750 4558 y Fj(W)825 4653 y Fh(x)p Fl(2)p Fx(\003)p Fy(\()p Fx(l)q Fy(\))1060 4626 y Fp(U)1122 4640 y Fx(\011)1178 4626 y Fo(\()p Fi(x)p Fo(\))p Fp(QU)1447 4593 y Fl(\003)1437 4653 y Fx(\011)1493 4626 y Fo(\()p Fi(x)p Fo(\))p Fr(,)26 b(which)h(has)h(the)f (representation)0 4752 y(as)100 4684 y Fj(W)176 4779 y Fh(x)p Fl(2)p Fx(\003)p Fy(\()p Fx(l)q Fy(\))411 4752 y Fo(1)456 4766 y Fx(Q)512 4774 y Fc(x)585 4752 y Fo(=)h(1)729 4721 y Fk(S)788 4792 y Fc(x)p Ff(2)p Fu(\003)p Fw(\()p Fu(l)p Fw(\))993 4771 y Fx(Q)1049 4779 y Fc(x)1093 4752 y Fr(.)c(Note)i(that)1523 4729 y Fo(\026)1502 4752 y Fp(Q)f Fr(is)g(in)l(v)n(ariant)i(under)g(the)e(action)i(of)p eop %%Page: 9 9 9 8 bop 0 17 a Fv(The)19 b(Shannon-McMillan)i(Theorem)e(for)g(Er)o (godic)g(Quantum)h(Lattice)e(Systems)518 b(9)0 248 y Fp(U)62 262 y Fx(\011)117 248 y Fo(\()p Fq(Z)213 215 y Fx(\027)256 248 y Fo(\))p Fr(.)25 b(From)g(the)i Fq(Z)759 215 y Fx(\027)801 248 y Fr(-er)n(godicity)j(of)25 b Fp(\011)35 b Fr(we)25 b(deduce)i(that)2011 225 y Fo(\026)1990 248 y Fp(Q)i Fo(=)g Fi(1)p Fr(.)c(If)h(we)f(trans-)0 348 y(late)f(back)h(the)f(\002nite)f(subadditi)n(vity)28 b(of)23 b(probability)k(measures)f(to)d(the)h(e)o(xpectation)0 448 y(v)n(alues)h(of)e(the)h(projectors)i Fp(U)933 462 y Fx(\011)988 448 y Fo(\()p Fi(x)p Fo(\))p Fp(QU)1257 415 y Fl(\003)1247 475 y Fx(\011)1303 448 y Fo(\()p Fi(x)p Fo(\))e Fr(we)f(obtain:)429 645 y Fo(1)j(=)f Fn(h)p Fp(\012)701 659 y Fx(\011)757 645 y Fp(;)818 622 y Fo(\026)797 645 y Fp(Q\012)939 659 y Fx(\011)994 645 y Fn(i)g(\024)1192 558 y Fj(X)1150 760 y Fh(x)p Fl(2)p Fx(\003)p Fy(\()p Fx(l)q Fy(\))1366 645 y Fn(h)p Fp(\012)1471 659 y Fx(\011)1526 645 y Fp(;)15 b(U)1628 659 y Fx(\011)1684 645 y Fo(\()p Fi(x)p Fo(\))p Fp(QU)1953 607 y Fl(\003)1943 667 y Fx(\011)1999 645 y Fo(\()p Fi(x)p Fo(\))p Fp(\012)2194 659 y Fx(\011)2250 645 y Fn(i)1054 887 y Fo(=)25 b Fp(l)1179 849 y Fx(\027)1242 887 y Fn(\001)c(h)p Fp(\012)1393 901 y Fx(\011)1449 887 y Fp(;)15 b(Q\012)1631 901 y Fx(\011)1686 887 y Fn(i)91 b Fo(\()p Fr(by)24 b Fo(\(3)p Fp(:)p Fo(3\)\))p Fp(:)125 1053 y Fr(Thus)f(\(3.8\))h(is)f(clear)-5 b(.)0 1152 y(Combining)34 b(the)f(results)h(\(3.7\))g(and)f(\(3.8\))g(we)f(get)h(the)g(e)o (xistence)i(of)d(an)h(orthog-)0 1266 y(onal)45 b Fi(1)p Fr(-decomposition)j Fn(f)p Fp(P)964 1280 y Fx(i)993 1266 y Fn(g)1038 1223 y Fx(n)1081 1235 y Fu(l)1106 1223 y Fl(\000)p Fy(1)1038 1293 y Fx(i)p Fy(=0)1243 1266 y Fr(in)c Fp(n)1412 1281 y Fx(l)1501 1266 y Fn(\024)63 b Fp(l)1664 1233 y Fx(\027)1751 1266 y Fr(minimal)44 b(projectors)j Fp(P)2562 1280 y Fx(i)2654 1266 y Fn(2)0 1366 y(P)63 1380 y Fx(\011)t(;G)185 1392 y Fu(l)213 1366 y Fn(nf)p Fo(0)p Fp(;)15 b Fi(1)p Fn(g)p Fr(.)25 b(Here)e(we)g(use)h(the)g (standard)i(de\002nition)f(of)e(minimality:)84 1542 y Fp(P)35 b Fr(minimal)24 b(projector)i(in)e Fn(N)1036 1505 y Fl(0)1023 1565 y Fx(\011)t(;G)1145 1577 y Fu(l)1256 1542 y Fo(:)p Fn(\()-15 b(\))61 b Fo(0)25 b Fn(6)p Fo(=)g Fp(P)39 b Fn(2)25 b(P)1921 1556 y Fx(\011)t(;G)2043 1568 y Fu(l)2093 1542 y Fr(and)1220 1679 y Fp(Q)g Fn(\024)g Fp(P)60 b Fn(\))26 b Fp(Q)f Fo(=)g Fp(P)104 b Fn(8)p Fp(Q)24 b Fn(2)h(P)2299 1693 y Fx(\011)t(;G)2421 1705 y Fu(l)2449 1679 y Fn(nf)p Fo(0)p Fn(g)0 1845 y Fr(The)41 b(abelianness)k(of)d Fn(N)847 1812 y Fl(0)834 1872 y Fx(\011)t(;G)956 1884 y Fu(l)1024 1845 y Fr(implies)g(the)g(uniqueness) j(of)d(the)g(orthogonal)i Fi(1)p Fr(-)0 1979 y(decomposition)34 b Fn(f)p Fp(P)668 1993 y Fx(i)697 1979 y Fn(g)742 1936 y Fx(n)785 1948 y Fu(l)810 1936 y Fl(\000)p Fy(1)742 2006 y Fx(i)p Fy(=0)904 1979 y Fr(.)c(Further)h(it)g(follo)n(ws)g(that) h Fn(f)p Fp(P)1905 1993 y Fx(i)1934 1979 y Fn(g)1979 1936 y Fx(n)2022 1948 y Fu(l)2046 1936 y Fl(\000)p Fy(1)1979 2006 y Fx(i)p Fy(=0)2170 1979 y Fr(is)f(a)f(generating)0 2079 y(subset)25 b(for)f Fn(P)442 2093 y Fx(\011)t(;G)564 2105 y Fu(l)615 2079 y Fr(in)f(the)h(follo)n(wing)h(sense:)152 2332 y Fp(Q)h Fn(2)e(P)398 2346 y Fx(\011)t(;G)520 2358 y Fu(l)597 2332 y Fo(=)-15 b Fn(\))47 b(9f)p Fp(P)945 2346 y Fx(i)969 2356 y Fu(j)1006 2332 y Fn(g)1051 2289 y Fx(s)p Fl(\024)p Fx(n)1182 2301 y Fu(l)1206 2289 y Fl(\000)p Fy(1)1051 2359 y Fx(j)t Fy(=0)1326 2332 y Fn(\032)25 b(P)1485 2346 y Fx(\011)t(;G)1607 2358 y Fu(l)1680 2332 y Fr(such)g(that)f Fp(Q)h Fo(=)2271 2218 y Fx(s)2222 2245 y Fj(X)2226 2441 y Fx(j)t Fy(=0)2368 2332 y Fp(P)2426 2346 y Fx(i)2450 2356 y Fu(j)2487 2332 y Fp(:)k Fr(\(3.9\))0 2606 y(De\002ne)23 b Fp(p)311 2620 y Fx(i)364 2606 y Fo(:=)i Fn(h)p Fp(\012)590 2620 y Fx(\011)646 2606 y Fp(;)15 b(P)744 2620 y Fx(i)773 2606 y Fp(\012)843 2620 y Fx(\011)898 2606 y Fn(i)24 b Fr(and)g(order)g(the)g(minimal)g (projectors)i Fp(P)2229 2620 y Fx(i)2280 2606 y Fr(such)f(that)684 2772 y Fp(p)730 2786 y Fy(0)795 2772 y Fn(\024)g Fp(p)937 2786 y Fx(i)965 2772 y Fp(;)197 b Fn(8)p Fp(i)25 b Fn(2)g(f)p Fo(1)p Fp(;)15 b(:)g(:)g(:)i(;)e(n)1727 2787 y Fx(l)1774 2772 y Fn(\000)k Fo(1)p Fn(g)p Fp(:)516 b Fr(\(3.10\))0 2938 y(Let)571 3037 y Fp(G)p Fo(\()p Fp(P)735 3051 y Fy(0)776 3037 y Fo(\))25 b(:=)h Fn(f)p Fi(x)f Fn(2)g Fq(Z)1230 3000 y Fx(\027)1273 3037 y Fn(j)e Fp(U)10 b Fo(\()p Fi(x)p Fo(\))p Fp(P)1576 3051 y Fy(0)1617 3037 y Fp(U)1689 3000 y Fl(\003)1728 3037 y Fo(\()p Fi(x)p Fo(\))26 b(=)f Fp(P)2033 3051 y Fy(0)2073 3037 y Fn(g)p Fp(:)0 3180 y Fr(Note)k(that)g Fp(G)p Fo(\()p Fp(P)533 3194 y Fy(0)574 3180 y Fo(\))f Fr(is)h(a)f(subgroup)k(of)d Fq(Z)1326 3147 y Fx(\027)1397 3180 y Fr(and)g(contains)i Fp(G)1958 3195 y Fx(l)1984 3180 y Fr(,)d(since)i Fp(P)2308 3194 y Fy(0)2383 3180 y Fn(2)35 b(P)2542 3194 y Fx(\011)t(;G)2664 3206 y Fu(l)2692 3180 y Fr(.)0 3279 y(This)23 b(leads)i(to)e(the)h (representation)240 3525 y Fp(G)p Fo(\()p Fp(P)404 3539 y Fy(0)444 3525 y Fo(\))i(=)650 3412 y Fx(\027)601 3439 y Fj(M)608 3634 y Fx(j)t Fy(=1)754 3525 y Fp(k)801 3539 y Fx(j)837 3525 y Fo(\()p Fp(l)r Fo(\))p Fq(Z)p Fp(;)107 b Fr(with)23 b Fp(k)1360 3539 y Fx(j)1397 3525 y Fo(\()p Fp(l)r Fo(\))p Fn(j)p Fp(l)117 b Fr(for)23 b(all)h Fp(j)31 b Fn(2)25 b(f)p Fo(1)p Fp(;)15 b(:)g(:)g(:)i(;)e(\027)6 b Fn(g)p Fp(;)0 3799 y Fr(where)24 b(the)g(inte)o(gers)h Fp(k)735 3813 y Fx(j)772 3799 y Fo(\()p Fp(l)r Fo(\))e Fr(are)h(gi)n(v)o(en)g(by)70 3965 y Fp(k)117 3979 y Fx(j)154 3965 y Fo(\()p Fp(l)r Fo(\))i(:=)f(min)o Fn(f)p Fp(x)648 3979 y Fx(j)685 3965 y Fn(j)e Fp(x)785 3979 y Fx(j)845 3965 y Fr(is)g(the)h(j-th)g(component)h(of)f Fi(x)h Fn(2)g Fp(G)p Fo(\()p Fp(P)2066 3979 y Fy(0)2107 3965 y Fo(\))e Fr(and)h Fp(x)2371 3979 y Fx(j)2433 3965 y Fp(>)h Fo(0)p Fn(g)p Fp(:)0 4131 y Fr(F)o(or)43 b Fp(P)226 4145 y Fy(0)266 4131 y Fr(,)g(as)i(an)f(element)h(of)g Fn(P)1095 4145 y Fx(\011)t(;G)1217 4157 y Fu(l)1245 4131 y Fr(,)e Fn(f)p Fp(U)1418 4145 y Fx(\011)1474 4131 y Fo(\()p Fi(x)p Fo(\))p Fp(P)1657 4145 y Fy(0)1697 4131 y Fp(U)1769 4098 y Fl(\003)1759 4158 y Fx(\011)1814 4131 y Fo(\()p Fi(x)p Fo(\))p Fn(g)1984 4150 y Fh(x)p Fl(2)p Fx(\003)p Fy(\()p Fh(k)p Fy(\()p Fx(l)q Fy(\)\))2367 4131 y Fn(\022)64 b(P)2565 4145 y Fx(\011)t(;G)2687 4157 y Fu(l)0 4243 y Fr(for)35 b Fi(k)p Fo(\()p Fp(l)r Fo(\))47 b(=)e(\()p Fp(k)539 4257 y Fy(1)580 4243 y Fo(\()p Fp(l)r Fo(\))p Fp(;)15 b(:)g(:)g(:)i(;)e(k)928 4257 y Fx(\027)972 4243 y Fo(\()p Fp(l)r Fo(\)\))p Fr(.)34 b(Thus)h(by)g(\(3.9\))g(each)g Fp(U)1974 4257 y Fx(\011)2030 4243 y Fo(\()p Fi(x)p Fo(\))p Fp(P)2213 4257 y Fy(0)2253 4243 y Fp(U)2325 4210 y Fl(\003)2315 4270 y Fx(\011)2371 4243 y Fo(\()p Fi(x)p Fo(\))p Fr(,)f Fi(x)46 b Fn(2)0 4343 y Fp(\003)p Fo(\()p Fi(k)p Fo(\()p Fp(l)r Fo(\)\))p Fr(,)31 b(can)h(be)f(represented)j(as)c(a)h(sum)g(of)f(minimal)i (projectors.)h(But)d(then)i(by)0 4442 y(linearity)k(of)d(the)h(e)o (xpectation)j(v)n(alues)d(and)g(the)g(assumed)h(ordering)h(\(3.10\))e (each)0 4542 y Fp(U)62 4556 y Fx(\011)117 4542 y Fo(\()p Fi(x)p Fo(\))p Fp(P)300 4556 y Fy(0)341 4542 y Fp(U)413 4509 y Fl(\003)403 4569 y Fx(\011)458 4542 y Fo(\()p Fi(x)p Fo(\))40 b Fr(must)h(be)f(a)g(minimal)g(projector)j(for)d Fi(x)57 b Fn(2)f Fp(\003)p Fo(\()p Fi(k)p Fo(\()p Fp(l)r Fo(\)\))p Fr(.)40 b(Other)n(-)0 4641 y(wise)35 b(there)i(w)o(ould)f(be) g(a)f(contradiction)40 b(to)35 b Fn(h)p Fp(\012)1600 4655 y Fx(\011)1656 4641 y Fp(;)15 b(U)1758 4655 y Fx(\011)1813 4641 y Fo(\()p Fi(x)p Fo(\))p Fp(P)1996 4655 y Fy(0)2037 4641 y Fp(U)2109 4608 y Fl(\003)2099 4668 y Fx(\011)2154 4641 y Fo(\()p Fi(x)p Fo(\))p Fp(\012)2349 4655 y Fx(\011)2406 4641 y Fn(i)48 b Fo(=)f Fp(p)2653 4655 y Fy(0)2692 4641 y Fr(.)0 4762 y(Consequently)d Fn(f)p Fp(U)647 4776 y Fx(\011)703 4762 y Fo(\()p Fi(x)p Fo(\))p Fp(P)886 4776 y Fy(0)927 4762 y Fp(U)999 4729 y Fl(\003)989 4789 y Fx(\011)1044 4762 y Fo(\()p Fi(x)p Fo(\))p Fn(g)1214 4781 y Fh(x)p Fl(2)p Fx(\003)p Fy(\()p Fh(k)p Fy(\()p Fx(l)q Fy(\)\))1590 4762 y Fn(\022)58 b(f)p Fp(P)1822 4776 y Fx(i)1851 4762 y Fn(g)1896 4719 y Fx(n)1939 4731 y Fu(l)1963 4719 y Fl(\000)p Fy(1)1896 4790 y Fx(i)p Fy(=0)2057 4762 y Fr(.)40 b(Consider)2509 4740 y Fo(\026)2489 4762 y Fp(P)2547 4776 y Fy(0)2644 4762 y Fo(=)p eop %%Page: 10 10 10 9 bop 0 17 a Fv(10)2093 b(I.)18 b(Bjelak)o(o)o(vi)t(\264)-29 b(c)20 b(et)e(al.)0 180 y Fj(P)96 275 y Fh(x)p Fl(2)p Fx(\003)p Fy(\()p Fh(k)p Fy(\()p Fx(l)q Fy(\)\))429 248 y Fp(U)491 262 y Fx(\011)546 248 y Fo(\()p Fi(x)p Fo(\))p Fp(P)729 262 y Fy(0)770 248 y Fp(U)842 215 y Fl(\003)832 275 y Fx(\011)887 248 y Fo(\()p Fi(x)p Fo(\))p Fr(.)27 b(Ob)o(viously)1486 225 y Fo(\026)1465 248 y Fp(P)1523 262 y Fy(0)1590 248 y Fr(is)f(in)l(v)n(ariant)k(under)e(the)g(action)g (of)0 368 y Fp(U)62 382 y Fx(\011)117 368 y Fo(\()p Fq(Z)213 335 y Fx(\027)256 368 y Fo(\))23 b Fr(and)h(because)i(of)d(the)h Fq(Z)1072 335 y Fx(\027)1115 368 y Fr(-er)n(godicity)j(of)c Fp(\011)1229 520 y Fo(\026)1209 543 y Fp(P)1267 557 y Fy(0)1332 543 y Fo(=)i Fi(1)p Fp(:)0 718 y Fr(It)e(follo)n(ws)h(by)g (the)g(uniqueness)j(of)c(the)h(orthogonal)j Fi(1)p Fr(-decomposition) 614 907 y Fn(f)p Fp(U)721 921 y Fx(\011)777 907 y Fo(\()p Fi(x)p Fo(\))p Fp(P)960 921 y Fy(0)1001 907 y Fp(U)1073 869 y Fl(\003)1063 929 y Fx(\011)1118 907 y Fo(\()p Fi(x)p Fo(\))p Fn(g)1288 925 y Fh(x)p Fl(2)p Fx(\003)p Fy(\()p Fh(k)p Fy(\()p Fx(l)q Fy(\)\))1632 907 y Fo(=)e Fn(f)p Fp(P)1831 921 y Fx(j)1868 907 y Fn(g)1913 864 y Fx(n)1956 876 y Fu(l)1981 864 y Fl(\000)p Fy(1)1913 934 y Fx(j)t Fy(=0)2075 907 y Fp(:)0 1098 y Fr(Ob)o(viously)i Fp(n)457 1113 y Fx(l)510 1098 y Fo(=)g(#\()p Fp(\003)p Fo(\()p Fi(k)p Fo(\()p Fp(l)r Fo(\)\)\))f Fr(and)g(for)f(each)g Fp(P)1601 1112 y Fx(i)1630 1098 y Fr(,)f Fp(i)k Fn(2)f(f)p Fo(0)p Fp(;)15 b(:)g(:)g(:)j(;)d(n)2172 1113 y Fx(l)2219 1098 y Fn(\000)21 b Fo(1)p Fn(g)p Fr(,)j(there)i(is)0 1198 y(only)e(one)g Fi(x)i Fn(2)f Fp(\003)p Fo(\()p Fi(k)p Fo(\()p Fp(l)r Fo(\))q(\))e Fr(such)h(that)798 1373 y Fp(P)856 1387 y Fx(i)910 1373 y Fo(=)h Fp(U)1068 1387 y Fx(\011)1123 1373 y Fo(\()p Fi(x)p Fo(\))p Fp(P)1306 1387 y Fy(0)1347 1373 y Fp(U)1419 1335 y Fl(\003)1409 1395 y Fx(\011)1464 1373 y Fo(\()p Fi(x)p Fo(\))h(=:)f Fp(P)1794 1387 y Fh(x)1842 1373 y Fp(:)628 b Fr(\(3.11\))0 1547 y(It)23 b(follo)n(ws)h Fp(p)417 1561 y Fx(i)471 1547 y Fo(=)h Fp(p)613 1561 y Fy(0)674 1547 y Fr(for)f(all)g Fp(i)h Fn(2)g(f)p Fo(0)p Fp(;)15 b(:)g(:)g(:)j(;)d(n)1407 1562 y Fx(l)1453 1547 y Fn(\000)20 b Fo(1)p Fn(g)k Fr(and)g(hence)414 1774 y Fp(p)460 1788 y Fx(i)513 1774 y Fo(=)637 1712 y(1)p 619 1753 81 4 v 619 1836 a Fp(n)674 1851 y Fx(l)735 1774 y Fo(=)1035 1712 y(1)p 841 1753 436 4 v 841 1836 a(#\()p Fp(\003)p Fo(\()p Fi(k)p Fo(\()p Fp(l)r Fo(\)\)\))1286 1774 y Fp(;)197 b(i)26 b Fn(2)e(f)p Fo(0)p Fp(;)15 b(:)g(:)g(:)j(;)d(n) 1998 1789 y Fx(l)2044 1774 y Fn(\000)20 b Fo(1)p Fn(g)p Fp(:)0 2010 y Fr(Finally)-6 b(,)24 b(set)g(for)g(e)n(v)o(ery)g Fi(x)h Fn(2)g Fp(\003)p Fo(\()p Fi(k)p Fo(\()p Fp(l)r Fo(\)\))333 2185 y Fp(\011)389 2199 y Fh(x)436 2185 y Fo(\()p Fp(a)p Fo(\))g(:=)g(#\()p Fp(\003)p Fo(\()p Fi(k)p Fo(\()p Fp(l)r Fo(\)\)\))p Fn(h)p Fp(\012)1238 2199 y Fx(\011)1296 2185 y Fp(;)15 b(P)1394 2199 y Fh(x)1442 2185 y Fp(\031)1494 2199 y Fx(\011)1549 2185 y Fo(\()p Fp(a)p Fo(\))p Fp(\012)1737 2199 y Fx(\011)1793 2185 y Fn(i)p Fp(;)197 b(a)26 b Fn(2)e(A)2282 2147 y Fl(1)2357 2185 y Fp(:)0 2360 y Fr(From)f(\(3.11\),)h(\(3.3\))g(and)g(\(3.4\))g (we)e(get)495 2534 y Fp(\011)551 2548 y Fh(x)598 2534 y Fo(\()p Fp(a)p Fo(\))j(=)g(#\()p Fp(\003)p Fo(\()p Fi(k)p Fo(\()p Fp(l)r Fo(\)\)\))p Fn(h)p Fp(\012)1375 2548 y Fx(\011)1433 2534 y Fp(;)15 b(P)1531 2548 y Fh(x)1579 2534 y Fp(\031)1631 2548 y Fx(\011)1686 2534 y Fo(\()p Fp(a)p Fo(\))p Fp(\012)1874 2548 y Fx(\011)1930 2534 y Fn(i)741 2659 y Fo(=)25 b(#\()p Fp(\003)p Fo(\()p Fi(k)p Fo(\()p Fp(l)r Fo(\)\)\))p Fn(h)p Fp(\012)1375 2673 y Fx(\011)1433 2659 y Fp(;)15 b(P)1531 2673 y Fy(0)1571 2659 y Fp(\031)1623 2673 y Fx(\011)1678 2659 y Fo(\()p Fp(T)e Fo(\()p Fn(\000)p Fi(x)p Fo(\))p Fp(a)p Fo(\))p Fp(\012)2128 2673 y Fx(\011)2184 2659 y Fn(i)741 2784 y Fo(=)25 b Fp(\011)893 2798 y Fy(0)932 2784 y Fo(\()p Fp(T)13 b Fo(\()p Fn(\000)p Fi(x)p Fo(\))p Fp(a)p Fo(\))p Fp(;)198 b(a)26 b Fn(2)f(A)1768 2746 y Fl(1)1842 2784 y Fp(;)0 2958 y Fr(hence)217 3101 y Fo(1)p 23 3142 V 23 3225 a(#\()p Fp(\003)p Fo(\()p Fi(k)p Fo(\()p Fp(l)r Fo(\)\)\))574 3076 y Fj(X)483 3278 y Fh(x)p Fl(2)p Fx(\003)p Fy(\()p Fh(k)p Fy(\()p Fx(l)q Fy(\)\))796 3163 y Fn(h)p Fp(\012)901 3177 y Fx(\011)957 3163 y Fp(;)15 b(P)1055 3177 y Fh(x)1103 3163 y Fp(\031)1155 3177 y Fx(\011)1210 3163 y Fo(\()p Fp(a)p Fo(\))p Fp(\012)1398 3177 y Fx(\011)1454 3163 y Fn(i)25 b Fo(=)g Fn(h)p Fp(\012)1715 3177 y Fx(\011)1771 3163 y Fp(;)15 b Fo(\()1937 3076 y Fj(X)1846 3278 y Fh(x)p Fl(2)p Fx(\003)p Fy(\()p Fh(k)p Fy(\()p Fx(l)q Fy(\)\))2175 3163 y Fp(P)2233 3177 y Fh(x)2280 3163 y Fo(\))p Fp(\031)2367 3177 y Fx(\011)2423 3163 y Fo(\()p Fp(a)p Fo(\))p Fp(\012)2611 3177 y Fx(\011)2667 3163 y Fn(i)1514 3405 y Fo(=)25 b Fp(\011)10 b Fo(\()p Fp(a)p Fo(\))p Fp(:)0 3580 y Fr(Thus)24 b(we)e(arri)n(v)o(e)i(at)g(the)g(con)l(v)o(e)o(x)g(decomposition)k(of) 23 b Fp(\011)10 b Fr(:)605 3803 y Fp(\011)35 b Fo(=)997 3741 y(1)p 802 3782 V 802 3865 a(#\()p Fp(\003)p Fo(\()p Fi(k)p Fo(\()p Fp(l)r Fo(\)\)\))1354 3716 y Fj(X)1262 3918 y Fh(x)p Fl(2)p Fx(\003)p Fy(\()p Fh(k)p Fy(\()p Fx(l)q Fy(\)\))1591 3803 y Fp(\011)1647 3817 y Fy(0)1706 3803 y Fn(\016)21 b Fp(T)13 b Fo(\()p Fn(\000)p Fi(x)p Fo(\))p Fp(:)0 4089 y Fr(By)18 b(construction)k(this)c(is)h(a)f Fp(G)947 4104 y Fx(l)973 4089 y Fr(-er)n(godic)j(decomposition)h(of)c Fp(\011)10 b Fr(.)17 b(It)h(remains)h(to)g(pro)o(v)o(e)0 4189 y(the)27 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Fp(\003)p Fo(\()p Fi(k)p Fo(\()p Fp(l)r Fo(\)\))p Fp(:)p eop %%Page: 11 11 11 10 bop 0 17 a Fv(The)19 b(Shannon-McMillan)i(Theorem)e(for)g(Er)o (godic)g(Quantum)h(Lattice)e(Systems)481 b(11)0 248 y Fr(By)23 b(the)h(de\002nition)h(of)e(the)h(mean)g(entrop)o(y)h(this)f (is)g(equi)n(v)n(alent)i(to)d(the)h(statement)458 469 y Fn(j)p Fp(S)5 b Fo(\()p Fp(\011)645 421 y Fy(\()p Fx(l)q Fh(n)p Fy(\))635 482 y Fh(x)771 469 y Fo(\))21 b Fn(\000)f Fp(S)5 b Fo(\()p Fp(\011)1080 421 y Fy(\()p Fx(l)q Fh(n)p Fy(\))1070 495 y(0)1206 469 y Fo(\))p Fn(j)26 b Fo(=)f Fp(o)p Fo(\()p Fn(j)p Fi(n)p Fn(j)p Fo(\))183 b Fr(as)23 b Fi(n)j Fn(!)f(1)p Fp(:)288 b Fr(\(3.12\))0 697 y(This)33 b(can)g(be)g(seen)g(as)g(follo)n(ws:)h(In)f(vie)n(w)f(of)h(the)g (de\002nition)i(of)65 b Fp(\011)2251 649 y Fy(\()p Fx(l)q Fh(n)p Fy(\))2241 709 y Fh(x)2408 697 y Fr(we)32 b(ha)n(v)o(e)0 815 y Fp(S)5 b Fo(\()p Fp(\011)162 767 y Fy(\()p Fx(l)q Fh(n)p Fy(\))152 828 y Fh(x)288 815 y Fo(\))34 b(=)g Fp(S)5 b Fo(\()p Fp(\011)624 767 y Fy(\()p Fx(\003)p Fy(\()p Fx(l)q Fh(n)p Fy(\)\))614 828 y Fh(x)854 815 y Fo(\))34 b(=)g Fp(S)5 b Fo(\()p Fp(\011)1190 767 y Fy(\()p Fx(\003)p Fy(\()p Fx(l)q Fh(n)p Fy(\))p Fl(\000)p Fh(x)p Fy(\))1180 841 y(0)1517 815 y Fo(\))p Fr(.)28 b(W)-7 b(e)27 b(introduce)k(the)e(box)2446 792 y Fo(~)2422 815 y Fp(\003)f Fr(being)0 915 y(concentric)g(with)d Fp(\003)p Fo(\()p Fp(l)r Fi(n)p Fo(\))p Fp(;)h Fr(with)f(all)g(edges)h (enlar)n(ged)i(by)d Fp(l)i Fr(on)e(both)h(directions,)i(i.e.)0 1044 y(an)i Fp(l)r Fr(-neighborhood)35 b(of)30 b Fp(\003)p Fo(\()p Fp(l)r Fi(n)p Fo(\))p Fp(:)h Fr(The)f(tw)o(o)f(e)o(xpressions) 34 b Fp(S)5 b Fo(\()p Fp(\011)2040 996 y Fy(\()p Fx(l)q Fh(n)p Fy(\))2030 1057 y Fh(x)2166 1044 y Fo(\))30 b Fr(and)g Fp(S)5 b Fo(\()p Fp(\011)2553 996 y Fy(\()p Fx(l)q Fh(n)p Fy(\))2543 1070 y(0)2679 1044 y Fo(\))0 1191 y Fr(are)31 b(v)n(on)g(Neumann)g(entropies)i(of)e(the)f (restrictions)k(of)d Fp(\011)1903 1143 y Fy(\()1946 1126 y Fk(e)1930 1143 y Fx(\003)p Fy(\))1893 1217 y(0)2040 1191 y Fr(to)f(the)h(smaller)g(sets)0 1290 y Fp(\003)p Fo(\()p Fp(l)r Fi(n)p Fo(\))24 b Fr(and)g Fp(\003)p Fo(\()p Fp(l)r Fi(n)p Fo(\))d Fn(\000)f Fi(x)p Fr(,)j(respecti)n(v)o(ely)-6 b(.)27 b(On)c(the)h(other)h(hand)f(we)f(consider)j(the)e(box)21 1387 y Fj(b)0 1410 y Fp(\003)h Fr(being)i(concentric)i(with)c Fp(\003)p Fo(\()p Fp(l)r Fi(n)p Fo(\))h Fr(with)f(all)h(edges)h (shortened)i(by)c Fp(l)i Fr(at)f(both)g(sides.)0 1554 y Fp(S)5 b Fo(\()p Fp(\011)162 1506 y Fy(\()206 1489 y Fk(b)189 1506 y Fx(\003)p Fy(\))152 1580 y(0)270 1554 y Fo(\))31 b Fr(is)h(the)g(v)n(on)h(Neumann)f(entrop)o(y)i(of)e Fp(\011)1601 1506 y Fy(\()p Fx(\003)p Fy(\()p Fx(l)q Fh(n)p Fy(\)\))1591 1580 y(0)1862 1554 y Fr(and)g Fp(\011)2090 1506 y Fy(\()p Fx(\003)p Fy(\()p Fx(l)q Fh(n)p Fy(\)\))2080 1567 y Fh(x)2351 1554 y Fr(after)g(their)0 1686 y(restriction)j(to)c (the)h(set)797 1663 y Fj(b)775 1686 y Fp(\003)p Fr(.)f Fp(S)5 b Fo(\()p Fp(\011)1054 1638 y Fy(\()p Fx(\003)p Fy(\()p Fx(l)q Fh(n)p Fy(\)\))1044 1699 y Fh(x)1284 1686 y Fo(\))31 b Fr(and)h Fp(S)5 b Fo(\()p Fp(\011)1674 1638 y Fy(\()p Fx(\003)p Fy(\()p Fx(l)q Fh(n)p Fy(\)\))1664 1712 y(0)1904 1686 y Fo(\))32 b Fr(can)g(be)g(estimated)h(si-)0 1785 y(multaneously)27 b(using)d(the)g(subadditi)n(vity)j(of)d(the)g(v) n(on)g(Neumann)g(entrop)o(y)121 2021 y Fp(S)5 b Fo(\()p Fp(\011)283 1973 y Fy(\()327 1957 y Fk(e)310 1973 y Fx(\003)p Fy(\))273 2047 y(0)391 2021 y Fo(\))20 b Fn(\000)g Fo(log)e Fr(tr)744 2034 y Fy(~)725 2051 y Fx(\003)p Fl(n)p Fx(\003)p Fy(\()p Fx(l)q Fh(n)p Fy(\))985 2021 y Fi(1)25 b Fn(\024)g Fp(S)5 b Fo(\()p Fp(\011)1320 1973 y Fy(\()p Fx(l)q Fh(n)p Fy(\))1310 2051 y Fx(\\)1446 2021 y Fo(\))26 b Fn(\024)f Fp(S)5 b Fo(\()p Fp(\011)1765 1973 y Fy(\()1809 1957 y Fk(b)1792 1973 y Fx(\003)p Fy(\))1755 2047 y(0)1873 2021 y Fo(\))20 b(+)g(log)e Fr(tr)2207 2055 y Fx(\003)p Fy(\()p Fx(l)q Fh(n)p Fy(\))p Fl(n)2430 2038 y Fk(b)2412 2055 y Fx(\003)2467 2021 y Fi(1)p Fp(;)0 2243 y Fr(where)24 b Fp(\\)h Fn(2)g(f)p Fi(x)p Fp(;)15 b Fo(0)p 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b(box)g(of)f(side)i(length)g Fp(l)f Fr(v)n(anishes)i(asymptotically)h(in)c Fp(l)r Fr(.)p eop %%Page: 12 12 12 11 bop 0 17 a Fv(12)2093 b(I.)18 b(Bjelak)o(o)o(vi)t(\264)-29 b(c)20 b(et)e(al.)0 248 y Fz(Lemma)23 b(3.1.)i Fd(If)e Fp(\011)33 b Fd(is)23 b(a)g Fq(Z)861 215 y Fx(\027)904 248 y Fd(-er)m(godic)i(state)g(on)e Fn(A)1612 215 y Fl(1)1687 248 y Fd(,)f(then)1011 474 y Fo(lim)993 535 y Fx(l)q Fl(!1)1249 411 y Fo(#)p Fp(A)1393 426 y Fx(l)q(;\021)p 1181 453 365 4 v 1181 536 a Fo(#)p Fp(\003)p Fo(\()p Fi(k)p Fo(\()p Fp(l)r Fo(\)\))1581 474 y(=)j(0)0 701 y Fd(holds)g(for)f(e)o(very)g Fp(\021)k(>)d Fo(0)p Fd(.)0 860 y Fz(Pr)n(oof)20 b(of)g(Lemma)e(3.1:)i Fr(W)-7 b(e)18 b(suppose)k(on)e(the)f(contrary)j(that)e(there)h(is)e(some)h Fp(\021)2509 874 y Fy(0)2573 860 y Fp(>)25 b Fo(0)0 989 y Fr(such)i(that)g Fo(lim)15 b(sup)632 1010 y Fx(l)714 934 y Fy(#)p Fx(A)826 946 y Fu(l;\021)897 961 y Fw(0)p 683 968 283 4 v 683 1021 a Fy(#)p Fx(\003)p Fy(\()p Fh(k)p Fy(\()p Fx(l)q Fy(\)\))1006 989 y Fo(=)30 b Fp(a)g(>)g Fo(0)p Fr(.)25 b(Then)i(there)g(e)o(xists)g(a)f(subsequence)j Fo(\()p Fp(l)2642 1003 y Fx(j)2679 989 y Fo(\))0 1108 y Fr(with)23 b(the)h(property)980 1270 y Fo(lim)956 1329 y Fx(j)t Fl(!1)1209 1200 y Fo(#)p Fp(A)1353 1215 y Fx(l)1374 1225 y Fu(j)1406 1215 y Fx(;\021)1461 1224 y Fw(0)p 1155 1249 400 4 v 1155 1333 a Fo(#)p Fp(\003)p Fo(\()p Fi(k)p Fo(\()p Fp(l)1446 1347 y Fx(j)1483 1333 y Fo(\)\))1589 1270 y(=)h Fp(a:)0 1478 y Fr(By)e(the)h(conca)n(vity)i(of)d(the)h(v)n (on)h(Neumann)f(entrop)o(y)h(we)e(obtain)180 1678 y Fo(#)p Fp(\003)p Fo(\()p Fi(k)p Fo(\()p Fp(l)471 1692 y Fx(j)508 1678 y Fo(\)\))e Fn(\001)g Fp(s)688 1641 y Fy(\()p Fx(l)736 1651 y Fu(j)768 1641 y Fy(\))825 1678 y Fn(\025)1027 1592 y Fj(X)920 1793 y Fh(x)p Fl(2)p Fx(\003)p Fy(\()p Fh(k)p Fy(\()p Fx(l)1177 1803 y Fu(j)1211 1793 y Fy(\)\))1281 1678 y Fp(s)1324 1627 y Fy(\()p Fx(l)1372 1637 y Fu(j)1404 1627 y Fy(\))1324 1691 y Fh(x)825 1956 y Fo(=)997 1870 y Fj(X)920 2067 y Fh(x)p Fl(2)p Fx(A)1063 2079 y Fu(l)1083 2095 y(j)1116 2079 y(;\021)1167 2094 y Fw(0)1221 1956 y Fp(s)1264 1905 y Fy(\()p Fx(l)1312 1915 y Fu(j)1344 1905 y Fy(\))1264 1969 y Fh(x)1396 1956 y Fo(+)1564 1870 y Fj(X)1487 2067 y Fh(x)p Fl(2)p Fx(A)1630 2044 y Fu(c)1630 2091 y(l)1650 2107 y(j)1682 2091 y(;\021)1733 2106 y Fw(0)1787 1956 y Fp(s)1830 1905 y Fy(\()p Fx(l)1878 1915 y Fu(j)1911 1905 y Fy(\))1830 1969 y Fh(x)825 2259 y Fn(\025)j Fo(#)p Fp(A)1064 2274 y Fx(l)1085 2284 y Fu(j)1118 2274 y Fx(;\021)1173 2283 y Fw(0)1232 2259 y Fn(\001)c Fo(\()p Fp(s)g Fo(+)g Fp(\021)1511 2273 y Fy(0)1551 2259 y Fo(\))g(+)g(#)p Fp(A)1841 2222 y Fx(c)1841 2283 y(l)1862 2293 y Fu(j)1895 2283 y Fx(;\021)1950 2292 y Fw(0)2009 2259 y Fn(\001)87 b Fo(min)2054 2320 y Fh(x)p Fl(2)p Fx(A)2197 2297 y Fu(c)2197 2344 y(l)2217 2360 y(j)2250 2344 y(;\021)2301 2359 y Fw(0)2355 2259 y Fp(s)2398 2208 y Fy(\()p Fx(l)2446 2218 y Fu(j)2478 2208 y Fy(\))2398 2272 y Fh(x)2510 2259 y Fp(:)0 2521 y Fr(Here)21 b(we)f(made)h(use)g (of)g(\(3.14\))h(at)f(the)g(last)g(step.)h(Using)f(that)h(for)f(the)g (mean)g(entrop)o(y)0 2620 y(holds)131 2822 y Fp(s)p Fo(\()p Fp(\011)265 2836 y Fh(x)312 2822 y Fp(;)15 b(G)423 2837 y Fx(l)450 2822 y Fo(\))25 b(=)127 b(lim)606 2887 y Fx(\003)p Fy(\()p Fh(m)p Fy(\))p Fl(\045)p Fg(N)896 2868 y Fu(\027)1086 2760 y Fo(1)p 960 2801 297 4 v 960 2884 a(#)p Fp(\003)p Fo(\()p Fi(m)p Fo(\))1267 2822 y Fp(S)5 b Fo(\()p Fp(\011)1429 2774 y Fy(\()p Fx(l)q Fh(m)p Fy(\))1419 2834 y Fh(x)1578 2822 y Fo(\))25 b(=)56 b(inf)1734 2887 y Fx(\003)p Fy(\()p Fh(m)p Fy(\))2056 2760 y Fo(1)p 1931 2801 V 1931 2884 a(#)p Fp(\003)p Fo(\()p Fi(m)p Fo(\))2238 2822 y Fp(S)5 b Fo(\()p Fp(\011)2400 2774 y Fy(\()p Fx(l)q Fh(m)p Fy(\))2390 2834 y Fh(x)2548 2822 y Fo(\))0 3055 y Fr(we)23 b(obtain)i(a)e(further) i(estimation)g(for)f(the)g(second)h(term)f(on)f(the)h(right)g(hand)h (side:)323 3275 y Fo(#)p Fp(A)467 3237 y Fx(c)467 3298 y(l)488 3308 y Fu(j)520 3298 y Fx(;\021)575 3307 y Fw(0)634 3275 y Fn(\001)87 b Fo(min)679 3335 y Fh(x)p Fl(2)p Fx(A)822 3312 y Fu(c)822 3359 y(l)842 3375 y(j)875 3359 y(;\021)926 3374 y Fw(0)980 3275 y Fp(s)1023 3224 y Fy(\()p Fx(l)1071 3234 y Fu(j)1103 3224 y Fy(\))1023 3287 y Fh(x)1160 3275 y Fn(\025)24 b Fo(#)p Fp(A)1399 3237 y Fx(c)1399 3298 y(l)1420 3308 y Fu(j)1453 3298 y Fx(;\021)1508 3307 y Fw(0)1567 3275 y Fn(\001)87 b Fo(min)1612 3335 y Fh(x)p Fl(2)p Fx(A)1755 3312 y Fu(c)1755 3359 y(l)1775 3375 y(j)1808 3359 y(;\021)1859 3374 y Fw(0)1936 3213 y Fo(1)p 1923 3254 73 4 v 1923 3337 a Fp(l)1952 3306 y Fx(\027)1950 3364 y(j)2005 3275 y Fp(s)p Fo(\()p Fp(\011)2139 3289 y Fh(x)2185 3275 y Fp(;)15 b(G)2296 3290 y Fx(l)2317 3300 y Fu(j)2355 3275 y Fo(\))1160 3495 y(=)24 b(#)p Fp(A)1399 3457 y Fx(c)1399 3518 y(l)1420 3528 y Fu(j)1453 3518 y Fx(;\021)1508 3527 y Fw(0)1567 3495 y Fn(\001)c Fp(s)p Fo(\()p Fp(\011)10 b Fo(\))205 b Fr(\(by\(3.13\)\))s Fp(:)0 3701 y Fr(After)33 b(di)n(viding)i Fo(#)p Fp(\003)p Fo(\()p Fi(k)p Fo(\()p Fp(l)848 3715 y Fx(j)885 3701 y 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Fd(be)24 b(an)f(er)m(godic)i(state)g(on) f Fn(A)1610 4398 y Fl(1)1684 4431 y Fd(.)e(Then)i(for)g(e)o(very)g Fp(")h Fn(2)g Fo(\(0)p Fp(;)15 b Fo(1\))760 4654 y(lim)g(sup)746 4737 y Fx(\003)p Fy(\()p Fh(n)p Fy(\))p Fl(\045)p Fg(N)1013 4718 y Fu(\027)1188 4593 y Fo(1)p 1077 4633 268 4 v 1077 4717 a(#)p Fp(\003)p Fo(\()p Fi(n)p Fo(\))1354 4654 y Fp(\014)1405 4668 y Fx(";)p Fh(n)1507 4654 y Fo(\()p Fp(\011)10 b Fo(\))26 b Fn(\024)f Fp(s)p Fo(\()p Fp(\011)10 b Fo(\))p Fp(:)p eop %%Page: 13 13 13 12 bop 0 17 a Fv(The)19 b(Shannon-McMillan)i(Theorem)e(for)g(Er)o (godic)g(Quantum)h(Lattice)e(Systems)481 b(13)0 248 y Fz(Pr)n(oof)20 b(of)g(Lemma)f(3.2:)h Fr(W)-7 b(e)19 b(\002x)g Fp(")26 b(>)f Fo(0)19 b Fr(and)i(choose)g(arbitrary)h Fp(\021)s Fr(,)d Fp(\016)29 b(>)c Fo(0)p Fr(.)20 b(Consider)0 348 y(the)k Fp(G)205 363 y Fx(l)231 348 y Fr(-er)n(godic)i (decomposition)850 566 y Fp(\011)35 b Fo(=)1206 505 y(1)p 1047 546 365 4 v 1047 629 a(#)p Fp(\003)p Fo(\()p Fi(k)p Fo(\()p Fp(l)r Fo(\)\))1528 480 y Fj(X)1436 681 y Fh(x)p Fl(2)p Fx(\003)p Fy(\()p Fh(k)p Fy(\()p Fx(l)q Fy(\)\))1765 566 y Fp(\011)1821 580 y Fx(x)0 849 y Fr(of)25 b Fp(\011)34 b Fr(for)26 b(inte)o(gers)h Fp(l)j Fn(\025)e Fo(1)p Fr(.)d(By)f(Lemma)g (3.1)h(there)h(is)f(an)h(inte)o(ger)g Fp(L)i Fn(\025)h Fo(1)24 b Fr(such)i(that)0 949 y(for)e(an)o(y)f Fp(l)28 b Fn(\025)c Fp(L)886 1035 y(")p 885 1076 46 4 v 885 1159 a Fo(2)966 1097 y Fn(\025)1231 1035 y Fo(1)p 1072 1076 365 4 v 1072 1159 a(#)p Fp(\003)p Fo(\()p Fi(k)p Fo(\()p Fp(l)r Fo(\)\))1446 1097 y(#)p Fp(A)1590 1112 y Fx(l)q(;\021)1698 1097 y Fn(\025)h Fo(0)0 1299 y Fr(holds,)f(where)g Fp(A)555 1314 y Fx(l)q(;\021)662 1299 y Fr(is)f(de\002ned)h(by)g(\(3.14\).)g (This)f(inequality)k(implies)869 1464 y Fo(1)p 710 1505 V 710 1588 a(#)p Fp(\003)p Fo(\()p Fi(k)p Fo(\()p Fp(l)r Fo(\)\))1084 1526 y(#)p Fp(A)1228 1488 y Fx(c)1228 1549 y(l)q(;\021)1332 1526 y Fn(\001)20 b Fo(\(1)h Fn(\000)1580 1464 y Fp(")p 1579 1505 46 4 v 1579 1588 a Fo(2)1634 1526 y(\))26 b Fn(\025)f 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Fr(and)f Fp(m)1494 3127 y Fy(\()p Fh(n)p Fy(\))1494 3187 y Fh(x)1623 3175 y Fo(:=)h Fp(\011)1800 3189 y Fh(x)1857 3175 y Fe(\026)g Fn(B)1983 3127 y Fy(\()p Fh(n)p Fy(\))1980 3187 y Fh(x)0 3345 y Fr(for)h Fi(x)31 b Fn(2)f Fp(A)376 3312 y Fx(c)376 3372 y(l)q(;\021)485 3345 y Fr(and)c Fi(n)31 b Fn(2)f Fq(N)887 3312 y Fx(\027)929 3345 y Fr(.)c(The)f(states)j Fp(m)1452 3359 y Fh(x)1524 3345 y Fr(are)e Fp(G)1731 3360 y Fx(l)1758 3345 y Fr(-er)n(godic)j (since)e(the)o(y)f(are)h(re-)0 3456 y(strictions)i(of)c Fp(G)530 3471 y Fx(l)557 3456 y Fr(-er)n(godic)j(states)f Fp(\011)1166 3470 y Fh(x)1238 3456 y Fr(on)f(a)g(quasi-local)j (algebra.)e(This)f(easily)i(fol-)0 3555 y(lo)n(ws)d(from)h(Theorem)g (4.3.17.)f(in)h([4)q(].)e(Moreo)o(v)o(er)l(,)j(by)f(the)f(Gelf)o(and)i (isomorphism)0 3655 y(and)22 b(Riesz)h(representation)j(theorem,)c(we)g (can)g(identify)i(the)e(states)h Fp(m)2267 3669 y Fh(x)2335 3655 y Fr(with)f(prob-)0 3755 y(ability)i(measures)g(on)f 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4283 y Fr(\(3.17\))0 4528 y Fp(m)80 4542 y Fh(x)127 4528 y Fr(-almost)31 b(surely)g(and)f(in)g Fp(L)1002 4495 y Fy(1)1042 4528 y Fo(\()p Fp(m)1157 4542 y Fh(x)1204 4528 y Fo(\))f Fr(for)i(all)f Fi(x)37 b Fn(2)g Fp(A)1782 4495 y Fx(c)1782 4556 y(l)q(;\021)1865 4528 y Fr(,)29 b(where)h Fp(h)2220 4542 y Fh(x)2297 4528 y Fr(denotes)i(the)0 4663 y(K)m(olmogoro)o(v-Sinai)g(entrop)o(y)g(of)e Fp(m)1197 4677 y Fh(x)1244 4663 y Fr(,)f(and)h Fp(!)1513 4677 y Fh(n)1599 4663 y Fn(2)37 b Fp(B)1766 4677 y Fh(x)1813 4627 y Fx(\003)p Fy(\()p Fh(n)p Fy(\))1995 4663 y Fr(are)30 b(the)g(components)0 4762 y(of)36 b Fp(!)51 b Fn(2)d Fp(B)402 4729 y Fg(Z)446 4706 y Fu(\027)397 4785 y Fh(x)523 4762 y Fr(corresponding)41 b(to)36 b(the)g(box)g Fp(\003)p Fo(\()p Fi(n)p Fo(\))p Fr(.)g(Actually)-6 b(,)37 b(as)f(we)f(shall)i (see,)p eop %%Page: 14 14 14 13 bop 0 17 a Fv(14)2093 b(I.)18 b(Bjelak)o(o)o(vi)t(\264)-29 b(c)20 b(et)e(al.)0 248 y Fr(we)27 b(need)i(the)f(theorem)i(cited)f (abo)o(v)o(e)f(only)h(in)f(its)g(weak)o(er)h(form)f(\(con)l(v)o(er)n (gence)k(in)0 348 y(probability\))37 b(kno)n(wn)d(as)g (Shannon-McMillan)i(theorem.)f(F)o(or)d(each)j Fp(n)d Fr(and)i Fi(x)44 b Fn(2)0 448 y Fp(A)68 415 y Fx(c)68 475 y(l)q(;\021)174 448 y Fr(let)62 682 y Fp(C)134 634 y Fy(\()p Fh(n)p Fy(\))127 695 y Fh(x)263 682 y Fo(:=)25 b Fn(f)p Fp(!)486 696 y Fh(n)560 682 y Fn(2)g Fp(B)720 634 y Fy(\()p Fh(n)p Fy(\))715 695 y Fh(x)824 682 y Fn(j)e(j)d(\000) 1129 621 y Fo(1)p 1018 661 268 4 v 1018 744 a(#)p Fp(\003)p Fo(\()p Fi(n)p Fo(\))1311 682 y(log)e Fp(m)1524 634 y Fy(\()p Fh(n)p Fy(\))1524 695 y Fh(x)1628 682 y Fo(\()p Fp(!)1720 696 y Fh(n)1769 682 y Fo(\))i Fn(\000)g Fp(h)1967 696 y Fh(x)2015 682 y Fn(j)25 b Fp(<)g(\016)s Fn(g)275 898 y Fo(=)38 b Fn(f)p Fp(!)486 912 y Fh(n)560 898 y Fn(2)25 b Fp(B)720 850 y Fy(\()p Fh(n)p Fy(\))715 911 y Fh(x)824 898 y Fn(j)e Fp(e)914 861 y Fl(\000)p Fy(#)p Fx(\003)p Fy(\()p Fh(n)p Fy(\))p Fl(\001)p Fy(\()p Fx(h)1264 869 y Fc(x)1305 861 y 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Fu(\027)969 2023 y Fo(1)p 858 2063 V 858 2146 a(#)p Fp(\003)p Fo(\()p Fi(n)p Fo(\))1135 2084 y Fp(H)7 b Fo(\()p Fp(m)1333 2036 y Fy(\()p Fh(n)p Fy(\))1333 2097 y Fh(x)1438 2084 y Fo(\))25 b(=)44 b(inf)1594 2149 y Fx(\003)p Fy(\()p Fh(n)p Fy(\))1880 2023 y Fo(1)p 1768 2063 V 1768 2146 a(#)p Fp(\003)p Fo(\()p Fi(n)p Fo(\))2046 2084 y Fp(H)7 b Fo(\()p Fp(m)2244 2036 y Fy(\()p Fh(n)p Fy(\))2244 2097 y Fh(x)2348 2084 y Fo(\))p Fp(;)0 2311 y Fr(\(cf.)46 b([21)q(]\),)23 b(and)h(by)g(\(3.16\).)g(Here)f Fp(H)30 b Fr(denotes)25 b(the)f(Shannon)h(entrop)o(y)-6 b(.)0 2410 y(From)28 b(\(3.17\))j(it)d(follo)n(ws)i(that)g(there)g(is)f (an)h Fp(N)45 b Fn(2)36 b Fq(N)28 b Fr(\(depending)k(on)e Fp(l)r Fr(\))e(such)i(that)0 2510 y(for)24 b(all)f Fi(n)j Fn(2)f Fq(N)478 2477 y Fx(\027)543 2510 y Fr(with)e Fp(\003)p Fo(\()p Fi(n)p Fo(\))k Fn(\033)e Fp(\003)p Fo(\()p Fp(N)10 b Fo(\))651 2707 y Fp(m)731 2659 y Fy(\()p Fh(n)p Fy(\))731 2720 y Fh(x)835 2707 y Fo(\()p Fp(C)942 2659 y Fy(\()p Fh(n)p Fy(\))935 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Fh(y)1621 4231 y Fo(\()p Fp(\011)10 b Fo(\))33 b Fr(is)g(satis\002ed.)h(Moreo)o(v)o(er)g(by)0 4331 y(\(3.18\))262 4496 y Fp(\014)313 4510 y Fx(";)p Fh(y)414 4496 y Fo(\()p Fp(\011)10 b Fo(\))25 b Fn(\024)g Fo(log)17 b Fr(tr)859 4510 y Fh(y)907 4496 y Fo(\()p Fp(q)983 4510 y Fh(y)1031 4496 y Fo(\))575 4651 y Fn(\024)25 b Fo(log)847 4565 y Fj(X)803 4762 y Fh(x)p Fl(2)p Fx(A)946 4739 y Fu(c)946 4785 y(l;\021)1038 4651 y Fp(e)1080 4613 y Fy(#)p Fx(\003)p Fy(\()p Fh(n)p Fy(\))p Fl(\001)p Fy(\()p Fx(h)1375 4621 y Fc(x)1417 4613 y Fy(+)p Fx(\016)r Fy(\))1557 4651 y Fo(+)20 b Fr(tr)1703 4670 y Fx(\003)p Fy(\()p Fh(y)q Fy(\))p Fl(n)p Fx(\003)p Fy(\()p Fx(l)q Fh(n)p Fy(\))2062 4651 y Fi(1)p eop %%Page: 15 15 15 14 bop 0 17 a Fv(The)19 b(Shannon-McMillan)i(Theorem)e(for)g(Er)o (godic)g(Quantum)h(Lattice)e(Systems)481 b(15)575 253 y Fn(\024)25 b Fo(log)17 b(#)p Fp(A)947 215 y Fx(c)947 276 y(l)q(;\021)1051 253 y Fn(\001)j Fp(e)1138 215 y Fy(#)p Fx(\003)p Fy(\()p Fh(n)p Fy(\)\()p Fx(l)1394 192 y Fu(\027)1434 215 y Fy(\()p Fx(s)p Fy(\()p Fx(\011)7 b Fy(\)+)p Fx(\021)r Fy(\)+)p Fx(\016)r Fy(\))1840 253 y Fo(\))21 b(+)f Fr(tr)2042 271 y Fx(\003)p Fy(\()p Fh(y)q Fy(\))p Fl(n)p Fx(\003)p Fy(\()p Fx(l)q Fh(n)p Fy(\))2401 253 y Fi(1)575 390 y Fn(\024)25 b Fo(log)17 b(#)p Fp(A)947 353 y Fx(c)947 413 y(l)q(;\021)1051 390 y Fo(+)j(#)p Fp(\003)p Fo(\()p Fi(n)p Fo(\)\()p Fp(l)1473 353 y Fx(\027)1517 390 y Fo(\()p Fp(s)p Fo(\()p Fp(\011)10 b Fo(\))20 b(+)g Fp(\021)s Fo(\))h(+)f Fp(\016)s Fo(\))671 515 y(+)p Fr(tr)797 533 y Fx(\003)p Fy(\()p Fh(y)q Fy(\))p Fl(n)p Fx(\003)p Fy(\()p Fx(l)q Fh(n)p Fy(\))1155 515 y Fi(1)575 651 y Fn(\024)25 b Fo(log)17 b(#)p Fp(A)947 614 y Fx(c)947 674 y(l)q(;\021)1051 651 y Fo(+)j(#\()p Fp(\003)p Fo(\()p Fi(y)q Fo(\)\)\()p Fp(s)p Fo(\()p Fp(\011)10 b Fo(\))22 b(+)e Fp(\021)k Fo(+)c Fp(\016)s Fo(\))671 776 y(+)p Fr(tr)797 794 y Fx(\003)p Fy(\()p Fh(y)q Fy(\))p Fl(n)p Fx(\003)p Fy(\()p Fx(l)q Fh(n)p Fy(\))1155 776 y Fi(1)p Fp(:)0 946 y Fr(W)-7 b(e)23 b(can)h(conclude)h(from)f(this)g(that)571 1147 y Fo(lim)15 b(sup)557 1230 y Fx(\003)p Fy(\()p Fh(y)q Fy(\))p Fl(\045)p Fg(N)823 1211 y Fu(\027)1033 1086 y Fo(1)p 887 1126 337 4 v 887 1209 a(#\()p Fp(\003)p Fo(\()p Fi(y)q Fo(\)\))1234 1147 y Fp(\014)1285 1161 y Fx(";)p Fh(y)1386 1147 y Fo(\()p Fp(\011)10 b Fo(\))26 b Fn(\024)f Fp(s)p Fo(\()p Fp(\011)10 b Fo(\))20 b(+)g Fp(\021)j Fo(+)d Fp(\016)n(;)0 1403 y Fr(because)42 b Fo(#)p Fp(A)471 1370 y Fx(c)471 1431 y(l)q(;\021)593 1403 y Fr(does)f(not)f(depend)i (on)e Fi(n)f Fr(and)h Fp(\003)p Fo(\()p Fi(y)q Fo(\))57 b Fn(\045)f Fq(N)2112 1370 y Fx(\027)2194 1403 y Fr(if)39 b(and)i(only)f(if)0 1520 y Fp(\003)p Fo(\()p Fi(n)p Fo(\))26 b Fn(\045)f Fq(N)399 1487 y Fx(\027)442 1520 y Fr(.)e(This)g(leads)i (to)761 1742 y Fo(lim)15 b(sup)747 1825 y Fx(\003)p Fy(\()p Fh(y)q Fy(\))p Fl(\045)p Fg(N)1013 1806 y Fu(\027)1188 1681 y Fo(1)p 1078 1721 267 4 v 1078 1805 a(#)p Fp(\003)p Fo(\()p Fi(y)q Fo(\))1354 1742 y Fp(\014)1405 1756 y Fx(";)p Fh(y)1506 1742 y Fo(\()p Fp(\011)10 b Fo(\))25 b Fn(\024)g Fp(s)p Fo(\()p Fp(\011)10 b Fo(\))p Fp(;)0 1993 y Fr(since)25 b Fp(\021)s Fr(,)d Fp(\016)29 b(>)c Fo(0)f Fr(were)f(chosen)i(arbitrarily)-6 b(.)208 b Fn(u)-61 b(t)125 2193 y Fr(Let)32 b Fp(\027)50 b Fn(2)43 b Fq(N)p Fr(.)33 b(F)o(or)f Fi(n)44 b Fo(=)g(\()p Fp(n)1063 2207 y Fy(1)1102 2193 y Fp(;)15 b(:)g(:)g(:)i(;)e(n)1359 2207 y Fx(\027)1402 2193 y Fo(\))44 b Fn(2)f Fq(N)1651 2160 y Fx(\027)1727 2193 y Fr(we)33 b(de\002ne)h Fn(j)p Fi(n)p Fn(j)44 b Fo(:=)2413 2124 y Fj(Q)2499 2151 y Fx(\027)2499 2219 y(i)p Fy(=1)2632 2193 y Fp(n)2687 2207 y Fx(i)0 2292 y Fr(and)27 b(write)g Fi(n)k Fn(!)g(1)26 b Fr(alternati)n(v)o(ely) k(for)d Fp(\003)p Fo(\()p Fi(n)p Fo(\))32 b Fn(\045)f Fq(N)1718 2259 y Fx(\027)1760 2292 y Fr(.)26 b(Further)i(we)d (introduce)30 b(the)0 2392 y(notation)530 2563 y Fi(n)25 b Fn(\025)g Fi(m)48 b Fo(:)p Fn(\()-15 b(\))48 b Fp(n)1139 2577 y Fx(i)1192 2563 y Fn(\025)25 b Fp(m)1368 2577 y Fx(i)1396 2563 y Fp(;)107 b Fn(8)p 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Fh(n)p Fl(!1)491 4087 y Fy(1)p 466 4102 V 466 4155 a Fl(j)p Fh(n)p Fl(j)561 4123 y Fp(H)7 b Fo(\()p Fp(P)750 4090 y Fy(\()p Fh(n)p Fy(\))854 4123 y Fo(\))26 b(=)f Fp(h)h(<)e Fn(1)31 4263 y Fd(2.)i Fo(lim)15 b(sup)403 4285 y Fh(n)p Fl(!1)643 4227 y Fy(1)p 618 4242 V 618 4295 a Fl(j)p Fh(n)p Fl(j)713 4263 y Fp(\013)771 4277 y Fx(";)p Fh(n)872 4263 y Fo(\()p Fp(P)978 4230 y Fy(\()p Fh(n)p Fy(\))1083 4263 y Fo(\))26 b Fn(\024)f Fp(h;)197 b Fn(8)p Fp(")25 b Fn(2)g Fo(\(0)p Fp(;)15 b Fo(1\))0 4455 y Fd(then)24 b(for)g(e)o(very)g Fp(")i Fn(2)f Fo(\(0)p Fp(;)15 b Fo(1\))895 4677 y(lim)865 4732 y Fh(n)p Fl(!1)1108 4616 y Fo(1)p 1077 4656 109 4 v 1077 4740 a Fn(j)p Fi(n)p Fn(j)1195 4677 y Fp(\013)1253 4691 y Fx(";)p Fh(n)1355 4677 y Fo(\()p Fp(P)1461 4640 y Fy(\()p Fh(n)p Fy(\))1566 4677 y Fo(\))25 b(=)g Fp(h:)696 b Fr(\(3.21\))p eop %%Page: 16 16 16 15 bop 0 17 a Fv(16)2093 b(I.)18 b(Bjelak)o(o)o(vi)t(\264)-29 b(c)20 b(et)e(al.)0 248 y Fr(Note)23 b(that)h(we)f(do)h(not)g(e)o (xpect)g(either)h Fn(f)p Fp(A)1341 215 y Fy(\()p Fh(n)p Fy(\))1446 248 y Fn(g)1491 262 y Fh(n)p Fl(2)p Fg(N)1631 244 y Fu(\027)1697 248 y Fr(or)e Fn(f)p Fp(P)1911 215 y Fy(\()p Fh(n)p Fy(\))2016 248 y Fn(g)2061 262 y Fh(n)p Fl(2)p Fg(N)2201 244 y Fu(\027)2267 248 y Fr(to)g(ful\002ll)h(an)o(y)0 348 y(consistenc)o(y)j(conditions.)g(W)-7 b(e)24 b(will)g(see)g(later)h (on)g(that)f(this)h(is)f(the)h(important)h(point)0 448 y(for)e(why)f(Lemma)f(3.3)h(will)h(be)f(useful)i(in)e(the)h (non-commutati)n(v)o(e)j(setting.)0 647 y Fz(Pr)n(oof)d(of)g(Lemma)e (3.3:)i Fr(Let)f Fp(\016)29 b(>)c Fo(0)e Fr(and)h(de\002ne)203 839 y Fp(A)271 791 y Fy(\()q Fh(n)p Fy(\))271 865 y(1)376 839 y Fo(\()p Fp(\016)s Fo(\))i(:=)636 738 y Fj(n)697 839 y Fp(a)f Fn(2)g Fp(A)924 802 y Fy(\()p Fh(n)p Fy(\))1053 839 y Fn(j)h Fp(P)1175 802 y Fy(\()p Fh(n)p Fy(\))1294 839 y Fo(\()q Fp(a)p Fo(\))f Fp(>)g(e)1576 802 y Fl(\000j)p Fh(n)p Fl(j)p Fy(\()p Fx(h)p Fl(\000)p Fx(\016)r Fy(\))1904 738 y Fj(o)1980 839 y Fp(;)203 1036 y(A)271 988 y 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3538 y Fy(\()p Fh(n)p Fy(\))1410 3612 y(3)1524 3498 y Fp(P)1595 3461 y Fy(\()p Fh(n)p Fy(\))1699 3498 y Fo(\()p Fp(A)1802 3450 y Fy(\()p Fh(n)p Fy(\))1802 3524 y(3)1907 3498 y Fo(\))44 3820 y(=)55 b(lim)140 3874 y Fh(n)p Fl(!1)383 3758 y Fo(1)p 351 3799 109 4 v 351 3882 a Fn(j)p Fi(n)p Fn(j)485 3719 y Fj(\020)539 3820 y Fp(P)610 3782 y Fy(\()p Fh(n)p Fy(\))715 3820 y Fo(\()p Fp(A)818 3772 y Fy(\()p Fh(n)p Fy(\))818 3846 y(3)922 3820 y Fo(\))15 b(log)j(#)p Fp(A)1249 3772 y Fy(\()p Fh(n)p Fy(\))1249 3846 y(3)1374 3820 y Fn(\000)i Fp(P)1536 3782 y Fy(\()p Fh(n)p Fy(\))1640 3820 y Fo(\()p Fp(A)1743 3772 y Fy(\()p Fh(n)p Fy(\))1743 3846 y(3)1848 3820 y Fo(\))15 b(log)j Fp(P)2102 3782 y Fy(\()p Fh(n)p Fy(\))2206 3820 y Fo(\()p Fp(A)2309 3772 y Fy(\()p Fh(n)p Fy(\))2309 3846 y(3)2414 3820 y Fo(\))2449 3719 y Fj(\021)2529 3820 y Fo(=)25 b(0)p Fp(:)0 4091 y Fr(Here)33 b(we)g(used)h(the)f(f)o(act)h(that)1037 4050 y Fy(log)13 b(#)p Fx(A)1252 4026 y Fw(\()p Fc(n)p Fw(\))p 1037 4070 305 4 v 1147 4123 a Fl(j)p Fh(n)p Fl(j)1385 4091 y Fr(stays)34 b(bounded)h(and)f Fn(\000)2194 4023 y Fj(P)2305 4091 y Fp(p)2351 4105 y Fx(i)2394 4091 y Fo(log)18 b Fp(p)2573 4105 y Fx(i)2644 4091 y Fn(\024)0 4217 y(\000)86 4148 y Fj(P)197 4217 y Fp(p)243 4231 y Fx(i)286 4217 y Fo(log)g Fp(q)460 4231 y Fx(i)544 4217 y Fr(for)59 b(\002nite)f(v)o(ectors)h Fo(\()p Fp(p)1351 4231 y Fx(i)1379 4217 y Fo(\))16 b Fp(;)f Fo(\()p Fp(q)1546 4231 y Fx(i)1574 4217 y Fo(\))58 b Fr(with)1885 4148 y Fj(P)1981 4243 y Fx(i)2025 4217 y Fp(p)2071 4231 y Fx(i)2188 4217 y Fo(=)2347 4148 y Fj(P)2443 4243 y Fx(i)2486 4217 y Fp(q)2527 4231 y Fx(i)2644 4217 y Fn(\024)0 4349 y Fo(1)e Fr(and)i Fp(p)335 4363 y Fx(i)363 4349 y Fp(;)15 b(q)444 4363 y Fx(i)559 4349 y Fn(\025)86 b Fo(0)p Fr(.)56 b(Since)h Fp(A)1166 4301 y Fy(\()q Fh(n)p Fy(\))1166 4375 y(3)1327 4349 y Fr(does)g(not)g(contrib)n(ute)j(to)d(the)g(entrop) o(y)0 4488 y(one)50 b(easily)h(concludes)i(that)80 b Fo(lim)1039 4543 y Fh(n)p Fl(!1)1241 4488 y Fp(P)1312 4455 y Fy(\()p Fh(n)p Fy(\))1431 4387 y Fj(\020)1485 4488 y Fp(A)1553 4440 y Fy(\()q Fh(n)p Fy(\))1553 4514 y(1)1658 4387 y Fj(\021)1786 4488 y Fo(=)73 b(0)50 b Fr(because)h(otherwise)0 4643 y Fo(lim)15 b(inf)252 4657 y Fh(n)p Fl(!1)492 4607 y Fy(1)p 468 4622 85 4 v 468 4676 a Fl(j)o Fh(n)p Fl(j)562 4643 y Fp(H)660 4569 y Fj(\000)702 4643 y Fp(P)773 4610 y Fy(\()p Fh(n)p Fy(\))877 4569 y Fj(\001)966 4643 y Fp(<)47 b(h)35 b Fr(w)o(ould)h(hold.)g (Recall)g(that)g Fp(\016)51 b(>)c Fo(0)35 b Fr(w)o(as)g(cho-)0 4762 y(sen)24 b(arbitrarily)-6 b(.)p eop %%Page: 17 17 17 16 bop 0 17 a Fv(The)19 b(Shannon-McMillan)i(Theorem)e(for)g(Er)o (godic)g(Quantum)h(Lattice)e(Systems)481 b(17)0 248 y Fr(Thus)631 423 y Fo(lim)601 478 y Fh(n)p Fl(!1)803 423 y Fp(P)874 386 y Fy(\()p Fh(n)p Fy(\))993 322 y Fj(\020)1047 423 y Fp(A)1115 375 y Fy(\()q Fh(n)p Fy(\))1115 449 y(2)1220 423 y Fo(\()p Fp(\016)s Fo(\))1333 322 y Fj(\021)1414 423 y Fo(=)25 b(1)p Fp(;)198 b Fn(8)p Fp(\016)28 b(>)d Fo(0)p Fp(:)432 b Fr(\(3.22\))0 639 y(Consequently)57 b(the)d(lemma)g(follo)n(ws)g(since)h Fp(P)1652 606 y Fy(\()p Fh(n)p Fy(\))1771 639 y Fo(\()q Fp(\012)5 b Fo(\))81 b Fn(\025)g Fo(1)43 b Fn(\000)g Fp(")53 b Fr(implies)0 774 y Fp(P)71 741 y Fy(\()p Fh(n)p Fy(\))190 673 y Fj(\020)245 774 y Fp(\012)25 b Fn(\\)19 b Fp(A)488 726 y Fy(\()q Fh(n)p Fy(\))488 800 y(2)593 774 y Fo(\()p Fp(\016)s Fo(\))706 673 y Fj(\021)787 774 y Fn(\025)25 b Fo(\(1)17 b Fn(\000)f Fp(")p Fo(\))1144 741 y Fy(2)1205 774 y Fr(for)23 b Fn(j)p Fi(n)p Fn(j)f Fr(suf)n(\002ciently)j(lar)n(ge)e(and)g(one)g (needs)45 b(at)0 939 y(least)27 b Fo(\(1)21 b Fn(\000)f Fp(")p Fo(\))461 897 y Fy(2)523 939 y Fn(\001)i Fp(e)612 906 y Fl(j)p Fh(n)p Fl(j)p Fy(\()p Fx(h)p Fl(\000)p Fx(\016)r Fy(\))910 939 y Fr(elements)27 b(from)f Fp(A)1528 891 y Fy(\()q Fh(n)p Fy(\))1528 965 y(2)1633 939 y Fo(\()p Fp(\016)s Fo(\))g Fr(to)g(co)o(v)o(er)g Fp(\012)h Fn(\\)21 b Fp(A)2339 891 y Fy(\()q Fh(n)p Fy(\))2339 965 y(2)2444 939 y Fo(\()p Fp(\016)s Fo(\))27 b Fr(and)0 1039 y Fp(\016)g Fr(can)c(be)h(chosen)h(arbitrarily)h(small.)206 b Fn(u)-61 b(t)0 1238 y Fz(Pr)n(oof)45 b(of)g(Pr)n(oposition)h(2.1:)e Fn(A)1131 1205 y Fy(\()p Fh(n)p Fy(\))1279 1238 y Fr(as)g(a)h(\002nite) f(dimensional)k Fp(C)2274 1205 y Fl(\003)2313 1238 y Fr(-algebra)e(is)0 1364 y(isomorphic)39 b(to)e(a)f(\002nite)h(direct)i (sum)1282 1296 y Fj(L)1383 1322 y Fx(M)1383 1391 y(j)t Fy(=1)1525 1364 y Fn(B)s Fo(\()p Fn(H)1701 1316 y Fy(\()p Fh(n)p Fy(\))1700 1391 y Fx(j)1805 1364 y Fo(\))p Fr(,)d(where)h(each)h Fn(H)2438 1316 y Fy(\()p Fh(n)p Fy(\))2437 1391 y Fx(j)2578 1364 y Fr(is)e(a)0 1510 y(Hilbert)c(space)g(with)f Fo(dim)14 b Fn(H)956 1462 y Fy(\()p Fh(n)p Fy(\))955 1537 y Fx(j)1099 1510 y Fo(=)39 b Fp(d)1256 1462 y Fy(\()p Fh(n)p Fy(\))1256 1537 y Fx(j)1400 1510 y Fp(<)g Fn(1)30 b Fr(and)h(an)o(y)g(minimal)h (projector)h(in)0 1656 y Fn(A)73 1623 y Fy(\()p Fh(n)p Fy(\))195 1656 y Fr(is)20 b(represented)i(by)e(a)f(one-dimensional)k (projector)f(on)e Fn(H)2040 1623 y Fy(\()p Fh(n)p Fy(\))2169 1656 y Fo(:=)2290 1588 y Fj(L)2391 1614 y Fx(M)2391 1683 y(j)t Fy(=1)2533 1656 y Fn(H)2611 1608 y Fy(\()p Fh(n)p Fy(\))2610 1683 y Fx(j)0 1802 y Fr(with)k Fo(dim)15 b Fn(H)430 1769 y Fy(\()p Fh(n)p Fy(\))560 1802 y Fo(=)656 1734 y Fj(P)752 1760 y Fx(M)752 1829 y(j)t Fy(=1)894 1802 y Fp(d)941 1754 y Fy(\()p Fh(n)p Fy(\))941 1829 y Fx(j)1072 1802 y Fo(=:)26 b Fp(d)1241 1816 y Fh(n)1291 1802 y Fr(.)d(Note)h(that)1696 1734 y Fj(L)1797 1760 y Fx(M)1797 1829 y(j)t Fy(=1)1939 1802 y Fn(B)s Fo(\()p Fn(H)2115 1754 y Fy(\()p Fh(n)p Fy(\))2114 1829 y Fx(j)2219 1802 y Fo(\))i Fn(\032)g(B)s Fo(\()p Fn(H)2553 1769 y Fy(\()p Fh(n)p Fy(\))2657 1802 y Fo(\))p Fr(.)0 1933 y(Consider)i(the)f(spectral)i(representation)i(of)26 b(the)h(density)i(operator)f Fp(D)2271 1947 y Fh(n)2347 1933 y Fr(of)e Fp(\011)2514 1900 y Fy(\()p Fh(n)p Fy(\))2644 1933 y Fr(in)0 2044 y Fn(B)s Fo(\()p Fn(H)176 2011 y Fy(\()p Fh(n)p Fy(\))280 2044 y Fo(\))p Fr(:)218 2305 y Fp(D)293 2319 y Fh(n)368 2305 y Fo(=)490 2191 y Fx(d)526 2199 y Fc(n)464 2219 y Fj(X)472 2414 y Fx(i)p Fy(=1)610 2305 y Fp(\025)663 2257 y Fy(\()p Fh(n)p Fy(\))663 2332 y Fx(i)767 2305 y Fn(j)p Fp(q)836 2257 y Fy(\()p Fh(n)p Fy(\))833 2332 y Fx(i)941 2305 y Fn(ih)p Fp(q)1055 2257 y Fy(\()p Fh(n)p Fy(\))1052 2332 y Fx(i)1159 2305 y Fn(j)p Fp(;)198 b(\025)1460 2257 y Fy(\()p Fh(n)p Fy(\))1460 2332 y Fx(i)1589 2305 y Fn(2)25 b Fo([0)p Fp(;)15 b Fo(1])p Fp(;)40 b Fn(j)p Fp(q)1989 2257 y Fy(\()p Fh(n)p Fy(\))1986 2332 y Fx(i)2094 2305 y Fn(i)25 b(2)g(H)2318 2268 y Fy(\()p Fh(n)p Fy(\))2422 2305 y Fp(:)0 2575 y Fr(F)o(or)34 b Fi(n)46 b Fo(=)g(\()p Fp(n)470 2589 y Fy(1)509 2575 y Fp(;)15 b(:)g(:)g(:)i(;)e(n)766 2589 y Fx(\027)809 2575 y Fo(\))46 b Fn(2)g Fq(N)1063 2542 y Fx(\027)1140 2575 y Fr(let)34 b Fp(A)1332 2542 y Fy(\()p Fh(n)p Fy(\))1471 2575 y Fr(be)h(the)g(\002nite)f(set)h(consisting)j(of)d(the)0 2704 y(eigen-projectors)28 b Fp(q)657 2656 y Fy(\()p Fh(n)p Fy(\))654 2731 y Fx(i)787 2704 y Fo(:=)d Fn(j)p Fp(q)977 2656 y Fy(\()p Fh(n)p Fy(\))974 2731 y Fx(i)1081 2704 y Fn(ih)p Fp(q)1195 2656 y Fy(\()p Fh(n)p Fy(\))1192 2731 y Fx(i)1300 2704 y Fn(j)e Fr(of)h Fp(\011)1513 2671 y Fy(\()p Fh(n)p Fy(\))1616 2704 y Fr(,)f(i.e.)982 2898 y Fp(A)1050 2860 y Fy(\()p Fh(n)p Fy(\))1179 2898 y Fo(:=)j Fn(f)p Fp(q)1390 2850 y Fy(\()p Fh(n)p Fy(\))1387 2925 y Fx(i)1494 2898 y Fn(g)1539 2858 y Fx(d)1575 2866 y Fc(n)1539 2925 y Fx(i)p Fy(=1)1658 2898 y Fp(:)812 b Fr(\(3.23\))0 3077 y(Let)23 b Fp(P)215 3044 y Fy(\()p Fh(n)p Fy(\))342 3077 y Fr(be)g(the)h(probability)j(distrib)n(ution)g (on)d Fp(A)1628 3044 y Fy(\()p Fh(n)p Fy(\))1755 3077 y Fr(gi)n(v)o(en)g(by:)716 3264 y Fp(P)787 3226 y Fy(\()p Fh(n)p Fy(\))891 3264 y Fo(\()p Fp(q)970 3216 y Fy(\()p Fh(n)p Fy(\))967 3291 y Fx(i)1075 3264 y Fo(\))h(:=)h Fp(\011)1323 3226 y Fy(\()p Fh(n)p Fy(\))1426 3264 y Fo(\()p Fp(q)1505 3216 y Fy(\()p Fh(n)p Fy(\))1502 3291 y Fx(i)1610 3264 y Fo(\))f(=)g Fp(\025)1819 3216 y Fy(\()p Fh(n)p Fy(\))1819 3291 y Fx(i)1924 3264 y Fp(:)546 b Fr(\(3.24\))0 3443 y(Recall)24 b(that)h Fn(j)p Fi(n)p Fn(j)i Fo(=)647 3374 y Fj(Q)733 3401 y Fx(\027)733 3470 y(i)p Fy(=1)866 3443 y Fp(n)921 3457 y Fx(i)949 3443 y Fr(.)c(Let)g Fp(D)29 b Fo(:=)d(log)s(\(dim)15 b Fn(H)1763 3410 y Fy(\(0\))1857 3443 y Fo(\))p Fr(,)23 b(then)2152 3407 y Fy(1)p 2128 3422 85 4 v 2128 3475 a Fl(j)p Fh(n)p Fl(j)2237 3443 y Fo(log)18 b(#)p Fp(A)2514 3410 y Fy(\()p Fh(n)p Fy(\))2644 3443 y Fn(\024)0 3583 y Fp(D)k Fr(for)f(all)f Fi(n)26 b Fn(2)f Fq(N)569 3550 y Fx(\027)611 3583 y Fr(.)20 b(W)-7 b(e)19 b(sho)n(w)h(that)h(the)g(f)o(amily)g Fn(f)p Fo(\()p Fp(A)1692 3550 y Fy(\()p Fh(n)p Fy(\))1797 3583 y Fp(;)15 b(P)1908 3550 y Fy(\()p Fh(n)p Fy(\))2013 3583 y Fo(\))p Fn(g)2093 3597 y Fh(n)p Fl(2)p Fg(N)2233 3578 y Fu(\027)2296 3583 y Fr(ful\002lls)21 b(both)0 3683 y(conditions)27 b(in)c(Lemma)f(3.3)i(and)g(consequently)278 3891 y Fo(lim)248 3946 y Fh(n)p Fl(!1)491 3830 y Fo(1)p 459 3870 109 4 v 459 3954 a Fn(j)p Fi(n)p Fn(j)578 3891 y Fp(\013)636 3905 y Fx(";)p Fh(n)738 3891 y Fo(\()p Fp(P)844 3854 y Fy(\()p Fh(n)p Fy(\))948 3891 y Fo(\))i(=)55 b(lim)1105 3946 y Fh(n)p Fl(!1)1348 3830 y Fo(1)p 1316 3870 V 1316 3954 a Fn(j)p Fi(n)p Fn(j)1435 3891 y Fp(H)7 b Fo(\()p Fp(P)1624 3854 y Fy(\()p Fh(n)p Fy(\))1728 3891 y Fo(\))p Fp(;)198 b Fn(8)p Fp(")25 b Fn(2)g Fo(\(0)p Fp(;)15 b Fo(1\))p Fp(:)80 b Fr(\(3.25\))0 4143 y(It)23 b(is)h(clear)g(that)g Fp(H)7 b Fo(\()p Fp(P)709 4110 y Fy(\()p Fh(n)p Fy(\))814 4143 y Fo(\))25 b(=)g Fn(\000)1056 4075 y Fj(P)1152 4101 y Fx(d)1188 4109 y Fc(n)1152 4170 y Fx(i)p Fy(=1)1286 4143 y Fp(\025)1339 4095 y Fy(\()p Fh(n)p Fy(\))1339 4170 y Fx(i)1458 4143 y Fo(log)18 b Fp(\025)1644 4095 y Fy(\()p Fh(n)p Fy(\))1644 4170 y Fx(i)1773 4143 y Fo(=)25 b Fp(S)5 b Fo(\()p Fp(\011)2031 4110 y Fy(\()p Fh(n)p Fy(\))2135 4143 y Fo(\))p Fr(.)23 b(Thus)741 4360 y Fp(h)j Fo(:=)55 b(lim)940 4414 y Fh(n)p Fl(!1)1183 4298 y Fo(1)p 1151 4339 V 1151 4422 a Fn(j)p Fi(n)p Fn(j)1270 4360 y Fp(H)7 b Fo(\()p Fp(P)1459 4322 y Fy(\()p Fh(n)p Fy(\))1563 4360 y Fo(\))26 b(=)f Fp(s)p Fo(\()p Fp(\011)10 b Fo(\))p Fp(:)571 b Fr(\(3.26\))0 4576 y(Ne)o(xt)23 b(assume)h(the)g(follo)n(wing)h(ordering:)896 4762 y Fp(i)g(<)g(j)54 b Fo(=)-15 b Fn(\))47 b Fp(\025)1386 4715 y Fy(\()p Fh(n)p Fy(\))1386 4790 y Fx(i)1516 4762 y Fn(\025)25 b Fp(\025)1665 4715 y Fy(\()p Fh(n)p Fy(\))1665 4790 y Fx(j)p eop %%Page: 18 18 18 17 bop 0 17 a Fv(18)2093 b(I.)18 b(Bjelak)o(o)o(vi)t(\264)-29 b(c)20 b(et)e(al.)0 248 y Fr(and)24 b(de\002ne)g(for)g Fp(")h Fn(2)g Fo(\(0)p Fp(;)15 b Fo(1\))406 528 y Fp(n)461 542 y Fx(";)p Fh(n)587 528 y Fo(:=)25 b(min)p Fn(f)p Fp(k)k Fn(2)c(f)p Fo(1)p Fp(;)15 b(:)g(:)g(:)i(;)e(d)1406 542 y Fh(n)1456 528 y Fn(gj)1611 415 y Fx(k)1565 442 y Fj(X)1569 637 y Fx(j)t Fy(=1)1711 528 y Fp(\025)1764 480 y Fy(\()p Fh(n)p Fy(\))1764 555 y Fx(j)1894 528 y Fn(\025)25 b Fo(1)20 b Fn(\000)g Fp(")p Fn(g)p Fp(:)0 845 y Fr(Thus)k Fp(\013)263 859 y Fx(";)p Fh(n)364 845 y Fo(\()p Fp(P)470 812 y Fy(\()p Fh(n)p Fy(\))575 845 y Fo(\))h(=)g(log)18 b(#\()p Fn(f)p Fp(q)1064 797 y Fy(\()p Fh(n)p Fy(\))1061 872 y Fx(i)1169 845 y Fn(g)1214 796 y Fx(n)1257 804 y Fu(";)p Fc(n)1214 872 y Fx(i)p Fy(=1)1352 845 y Fo(\))25 b(=)g(log)18 b Fp(n)1696 859 y Fx(";)p Fh(n)1797 845 y Fr(.)k(W)-7 b(e)23 b(claim)h(:)546 1048 y Fp(\013)604 1062 y Fx(";)p Fh(n)705 1048 y Fo(\()p Fp(P)811 1010 y Fy(\()p Fh(n)p Fy(\))916 1048 y Fo(\))h(=)g Fp(\014)1123 1062 y Fx(";)p Fh(n)1225 1048 y Fo(\()p Fp(\011)1326 1010 y Fy(\()p Fh(n)p Fy(\))1431 1048 y Fo(\))p Fp(;)197 b Fn(8)p Fp(")25 b Fn(2)g Fo(\(0)p Fp(;)15 b Fo(1\))p Fp(:)378 b Fr(\(3.27\))0 1260 y(From)29 b Fp(\011)292 1227 y Fy(\()p Fh(n)p Fy(\))396 1260 y Fo(\()431 1192 y Fj(P)528 1211 y Fx(n)571 1219 y Fu(";)p Fc(n)528 1288 y Fx(i)p Fy(=1)680 1260 y Fp(q)724 1213 y Fy(\()p Fh(n)p Fy(\))721 1288 y Fx(i)828 1260 y Fo(\))39 b Fn(\025)e Fo(1)26 b Fn(\000)f Fp(")30 b Fr(and)g(tr)1465 1274 y Fh(n)1529 1192 y Fj(P)1625 1211 y Fx(n)1668 1219 y Fu(";)p Fc(n)1625 1288 y Fx(i)p Fy(=1)1778 1260 y Fp(q)1822 1213 y Fy(\()p Fh(n)p Fy(\))1819 1288 y Fx(i)1964 1260 y Fo(=)37 b Fp(n)2127 1274 y Fx(";)p Fh(n)2258 1260 y Fr(it)30 b(is)g(ob)o(vious)0 1379 y(that)24 b Fp(\014)210 1393 y Fx(";)p Fh(n)312 1379 y Fo(\()p Fp(\011)413 1346 y Fy(\()p Fh(n)p Fy(\))517 1379 y Fo(\))i Fn(\024)f Fp(\013)732 1393 y Fx(";)p Fh(n)834 1379 y Fo(\()p Fp(P)940 1346 y Fy(\()p Fh(n)p Fy(\))1044 1379 y Fo(\))p Fr(.)0 1496 y(Assume)i Fp(\014)370 1510 y Fx(";)p Fh(n)472 1496 y Fo(\()p Fp(\011)573 1463 y Fy(\()p Fh(n)p Fy(\))677 1496 y Fo(\))k Fp(<)g(\013)903 1510 y Fx(";)p Fh(n)1005 1496 y Fo(\()p Fp(P)1111 1463 y Fy(\()p Fh(n)p Fy(\))1215 1496 y Fo(\))p Fr(.)26 b(Then)h(there)h(e)o(xists)f(a)f(projector)j Fp(q)34 b Fn(2)d(A)2611 1463 y Fy(\()p Fh(n)p Fy(\))0 1614 y Fr(with)i Fp(\011)260 1581 y Fy(\()p Fh(n)p Fy(\))363 1614 y Fo(\()p Fp(q)s Fo(\))43 b Fn(\025)f Fo(1)28 b Fn(\000)e Fp(")32 b Fr(such)i(that)f Fp(m)42 b Fo(:=)g Fr(tr)1560 1628 y Fh(n)1641 1614 y Fp(q)j(<)d(n)1895 1628 y Fx(";)p Fh(n)1996 1614 y Fr(.)32 b(Let)2204 1546 y Fj(P)2300 1572 y Fx(m)2300 1641 y(i)p Fy(=1)2433 1614 y Fn(j)p Fp(q)2499 1628 y Fx(i)2527 1614 y Fn(ih)p Fp(q)2638 1628 y Fx(i)2667 1614 y Fn(j)p Fr(,)0 1732 y(where)d Fn(j)p Fp(q)316 1746 y Fx(i)344 1732 y Fn(i)35 b(2)f(H)587 1699 y Fy(\()p Fh(n)p Fy(\))690 1732 y Fr(,)28 b(be)g(the)h(spectral)i (representation)h(of)d Fp(q)s Fr(.)e(F)o(or)g Fp(D)2271 1746 y Fh(n)2349 1732 y Fr(as)h(density)0 1846 y(matrix)j(on)g Fn(H)466 1813 y Fy(\()p Fh(n)p Fy(\))600 1846 y Fr(we)f(use)h(K)n(y)f (F)o(an')-5 b(s)31 b(maximum)g(principle,)h([2)q(],)e(and)h(obtain)i (the)0 1945 y(contradiction)135 2186 y Fo(1)20 b Fn(\000)g Fp(")26 b Fn(\024)f Fp(\011)521 2149 y Fy(\()p Fh(n)p Fy(\))625 2186 y Fo(\()p Fp(q)s Fo(\))h(=)f Fr(tr)916 2200 y Fh(n)966 2186 y Fp(D)1041 2200 y Fh(n)1090 2186 y Fp(q)j Fo(=)1290 2072 y Fx(m)1255 2100 y Fj(X)1264 2295 y Fx(i)p Fy(=1)1387 2186 y Fn(h)p Fp(q)1463 2200 y Fx(i)1491 2186 y Fp(;)15 b(D)1606 2200 y Fh(n)1656 2186 y Fp(q)1697 2200 y Fx(i)1724 2186 y Fn(i)26 b(\024)1915 2072 y Fx(m)1881 2100 y Fj(X)1890 2295 y Fx(i)p Fy(=1)2027 2186 y Fp(\025)2080 2138 y Fy(\()p Fh(n)p Fy(\))2080 2213 y Fx(i)2210 2186 y Fp(<)f Fo(1)c Fn(\000)f Fp(":)0 2455 y(\011)32 b Fr(is)24 b(er)n(godic.)h(Thus)f(we)e(can)i(apply)h (Lemma)e(3.2:)333 2683 y Fo(lim)15 b(sup)319 2765 y Fx(\003)p Fy(\()p Fh(n)p Fy(\))p Fl(\045)p Fg(N)586 2746 y Fu(\027)797 2621 y Fo(1)p 650 2662 339 4 v 650 2745 a(#\()p Fp(\003)p Fo(\()p Fi(n)p Fo(\)\))999 2683 y Fp(\014)1050 2697 y Fx(";)p Fh(n)1152 2683 y Fo(\()p Fp(\011)1253 2645 y Fy(\()p Fh(n)p Fy(\))1357 2683 y Fo(\))25 b Fn(\024)g Fp(s)p Fo(\()p Fp(\011)10 b Fo(\))p Fp(;)198 b Fn(8)p Fp(")25 b Fn(2)g Fo(\(0)p Fp(;)15 b Fo(1\))p Fp(:)151 b Fr(\(3.28\))0 2947 y(Setting)23 b(\(3.27\))f(and)g(\(3.26\))h(in)f (\(3.28\))g(and)g(using)h(that)g Fo(#\()p Fp(\003)p Fo(\()p Fi(n)p Fo(\)\))j(=)f Fn(j)p Fi(n)p Fn(j)d Fr(we)f(obtain)505 3178 y Fo(lim)15 b(sup)551 3251 y Fh(n)p Fl(!1)840 3117 y Fo(1)p 809 3157 109 4 v 809 3241 a Fn(j)p Fi(n)p Fn(j)927 3178 y Fp(\013)985 3192 y Fx(";)p Fh(n)1087 3178 y Fo(\()p Fp(P)1193 3141 y Fy(\()p Fh(n)p Fy(\))1297 3178 y Fo(\))26 b Fn(\024)f Fp(h;)198 b Fn(8)p Fp(")25 b Fn(2)g Fo(\(0)p Fp(;)15 b Fo(1\))p Fp(:)337 b Fr(\(3.29\))0 3414 y(W)l(ith)30 b(\(3.26\))g(and)g(\(3.29\))h(both)f(conditions)j(in)c(Lemma)g(3.3)g (are)h(satis\002ed.)g(It)g(fol-)0 3513 y(lo)n(ws)23 b(\(3.25\).)h(No)n (w)e(we)h(set)h(back)g(\(3.27\))h(and)f(\(3.26\))g(in)f(\(3.25\))i(and) f(arri)n(v)o(e)g(at)365 3738 y Fo(lim)275 3803 y Fx(\003)p Fy(\()p Fh(n)p Fy(\))p Fl(\045)p Fg(N)542 3784 y Fu(\027)753 3676 y Fo(1)p 606 3717 339 4 v 606 3800 a(#\()p Fp(\003)p Fo(\()p Fi(n)p Fo(\)\))954 3738 y Fp(\014)1005 3752 y Fx(";)p Fh(n)1107 3738 y Fo(\()p Fp(\011)10 b Fo(\))26 b(=)f Fp(s)p Fo(\()p Fp(\011)10 b Fo(\))p Fp(;)197 b Fn(8)p Fp(")25 b Fn(2)g Fo(\(0)p Fp(;)15 b Fo(1\))p Fp(:)184 b Fn(u)-61 b(t)0 3979 y Fz(Pr)n(oof)24 b(of)g(the)f(Quantum)e (Shannon-McMillan)i(Theor)n(em:)0 4079 y Fr(Fix)36 b Fp(\016)54 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