Content-Type: multipart/mixed; boundary="-------------0305220519147" This is a multi-part message in MIME format. ---------------0305220519147 Content-Type: text/plain; name="03-231.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-231.keywords" Non-Equilibrium Steady States, Stability, Positive Commutator, Lattice Fermion System ---------------0305220519147 Content-Type: application/postscript; name="editsta.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="editsta.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.78 p1.4 Copyright 1996-98 ASCII Corp.(www-ptex@ascii.co.jp) %%based on dvipsk 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: editsta.dvi %%Pages: 44 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX12 CMR12 CMTI10 CMR10 CMMI10 CMTT9 CMSY10 CMTI12 %%+ CMMI12 CMSY8 CMSY7 CMMI7 CMBX10 CMR7 CMEX10 CMMI5 MSBM10 EUFM10 CMR5 %%+ CMSY5 MSBM7 MSAM10 EUFM7 CMR8 Ryumin-Light-H %%EndComments %DVIPSCommandLine: dvipsk -D600 -t a4paper -P dl editsta.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.05.21:1905 %%BeginProcSet: texc.pro %! 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 82 /R put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 def currentdict end currentfile eexec b5e2457af4a6caaa4536453c498a618ecdce7b5f6f2d5944265309f833bd3d50 3df8bd14c071f404d2177500681a801856cb8c9c9cf0112c9bd17ccba9ab8944 1759a320b2078df0fafdd3e357994bd81740d1a7522ddfafb75d58f5def046f7 084d10fa2f702e570b09a7073d70efe37b48c355a7089772238b7b4538c4912f cf6703a14fa416057ea2e1b94455913dd271b42846ee10d78270a8118e7e4bcb 8deb048bb280c240d59409d47046c890e5f79e5dd4b0be65a28a77aeb53dd33e ac506a6da87bb55e329f433627e06c3c94b078838b908538d93f888e504e25e0 3a8ccf7d2c02122ab3d485525564295390d1154e15fee35edff4eb016fadd32c 26d1b4c09de87943780954cebc4965aa890fe1f80b03eb07103e13d468245c7b c26455f229e8859585c6eeafd9474acdb27290fd62950dc52f893c99a2a614d4 2a7e3618266f3fb0c68a01a30373e048385ffe7dd5844e208a55d1a9263ab2db cb9b19038724f95af00349840157be45cb337871b55a304b87ade64b10ff0eba 21aa0eaeef74aa01f049303a8b25de0acc26bb3a33fcf267f479a2a73f7dece3 2aa23083687322b417fe86e8f5afc2136d045ad6b392364811a17630a94ff097 038eedeedb8da4ebf8fa5a96ce27a2ebf771d14f4a11fb1f9f48cdd207209934 885ef9118463552984b0fd3cf15d55de967fbb051549dd65f60dfe07f299690e 58252f032bca78935b7d2df852ed688bd13dd171b91cf7bd46d5433aef9f50f1 41a2517c4912b837793f84ddcf1116f70f7e9b1e6ec752dbc9183d4791d92602 397bafe29ac6f879ceb72c673c8a1aa0135d6f92d0db7791f9a2f1a5b55347e0 3e63723efd258754f5f4a9ad35873a98d54b6c54b17a39892dc1d38660597448 fdba9258493a0db0d560c01d49f2d889a99eaa8a5b5919cbd33f6b3d97bf27b3 e0b94b48d5d21d6c29aee4faca71bd775bb9defa37ff627f1e4926f10d59313e 77d81a511027ef9a8ed5b7fa9b5bd67758cd13bab6ac18411c974a65d0c90085 67bf5860d0930d7f43e557ee45e76ecb2806d21f4732bb282ac8b591f2b8775d 4b3ce2a37ebb371cb74a3cc513bdd70a0115de4760c21f034f5080d769f7ff83 554af524184702a02e534d1eb98dd3f7c5d51900409354dea37de4ac74a12c3a 69320913de6a728bd887c57f4ee5dcb8772249587ce3190bb662ac853abcff77 d4bd77985a4a805f44f8cc8597416c488866ff8429f65cd0edeec1fe2f0015a3 2bb098b0b3d8c913007f5846ca593c491e1efabc8f420a5fa338c47fd2427813 3279d27f12022195e15827797fe71d898f17b8af955e5bd5ce1da5fae7b3ece8 87aa52db81daa819242c6acb7fbed098d59a27bbcb12dca67e16915a4709fa5e 16311a55ddda46359ab51be5964874f1d3612ec10d2444576c508ae6a285a57f b978458b6f5f93c00882f2227d43d42c648437715a27cefd8d4ffb490e3e9980 1d1b4f1cacbc8b9e59c04c677a04047cc4b655fd8985d8e619a99e71cad67c94 2744577e185ac8b225e5352ae93ef0d804b73b943353266c74677b899d45b286 869cbe8ce9ad231007381de74bd3c6c84958f07ffce0620e42902a01fd15c98a 421ff110e5de48cd65568060afaa1e11e9893e720f22b754c47c334a1f5d0933 9b1b1b91b036b981a6be8a3276634fd4e39af829e2f2ac2b2dde35379abd1bf5 3a1071f07537db5fff44d145fce9713d8a9e2660900c41b83f948e16731d3048 2c8fa2aef8de49e3cf649e8ff62a2748cd9e299c7b 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 48 /prime put readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueID 5000815 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6784decd2fb36dec8f 7478da853762e00bc9c52ba86ef50d222ff142abc18e4fb95b86b3e57d9390ca 86e8de6347bf7f48f93322a372c5d7830521b14e8fa0771961b027c81365426d d95d16b950e216a90f28ce806244ff28dd969cb512a23e05045e7b2bcaf7d6f0 ca8b51494f2ec0681bdfde60420628d9bce98a0ca1ecca9140f452a88db19a30 ed46ae881b098c843a51d06f9a153489357fd24de6a0c36cb5a9eb9b3b64dd75 18f83018ae7cefa1c12fd205df6880a705e28457af7b869a6dd28f6b37d82cc3 e51cba348dcdb34efb119b8ce5a996f15492b0f5d34538850a2c1b6a509fe5fa 015eceaf28b6f9385526e98445876548d0f2a09cfea14b84be3721b51afa2185 90c7b9ca643c7f4fe67cd8985f27855c40ae7ea592fadfbba29301071039d18a 8a6880cc0bff22fc8af50fd2802aff5a96381fb4bef240dd26b34a5461188ca0 c85e4b3f803f94dce14e8ddb2284e22abfe148560aff784967ed795332cc0995 80b762f864b2c15bc640a4cd7f6927c7dd95aef777fd226254b8e6e1b8ec6483 2750d9929627c4cecf9f5c8feb57a2c7ee5d1315593f3fa16c7c7292a594ce14 2c396d9b787665569dbd0f710822ae3155a3aa53d6f055cb19e897b235b65bce 4d507a4d41edc6a09dff1f737f2711478e6e998218b7f3e7c1aaf3938f68e9cf 9ea46e4a2904c459281cc156f5ce80d6356e357011cd0bc383318129bc8ce38a 758c05d1205192a457f4b1b31df97baec7611905cac0 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /Lambda put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 56 /eight put dup 57 /nine put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueID 5000788 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 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0f50c112d0354b7a2c9a456446cc15358395fa2dcd9c243223aeb8c3b8279567 665fc5596ac8cc7d99e4c48903f6610ee5d6be351c18b3c6aa2a591c1cfcf821 18431454407ba878a62f11293c3dcf4d8c9bf0f38212a5d954ff9ec5686a23af 698a84f8e69f808d9605abe2f3bd874cb84419e4d6ac6eb96389df5c76587369 02327bff0911cb733f15e7753ded2ce52d8d9700e3193d201b9a2954d17bacd1 d45654f6c1ca90a11f768cd51f88b6019b82a7a50d10a9e526531e7264944e52 535ac3eaee7d909b405a2ca87112b56c2c3a1642e9b239b73eb7743fbd4d21ce 91ce17d49e815a036d7c342de592647c7f23b55239a3838936d19c27df9fb443 c6365d2e179e55bcdb097595ddccea441c0103aadda283bd5b5a4f13af19615c 41481d3c4098aaa0310eb6806c4d84f1c14cb035b909b5a68ca312b7d59a037d c820f95aae977eac1b9416f6be1ac28d5e2b24b647c216661756ff4a944ef6ad c2928d722b8e8994869771699e90e0c549a8ccae4c301ca08db88a1ca7fc89c0 1a43e659700a90eaa5046266d868235656f071d2c4c82fa07cd0e53dc1240b30 6501b8ed634bdd8fda657c5fd9ff3f3f3d5f895c774d0d92fc238681b7ed1624 3e0f8ef4cdd1b53afb9e629134d46fa8a8e300ede04f067f1c82a274ce8ede87 483c73f8653526475824903a9f684ce10990d32b082200b7254f7d5db1ce6cd6 b1e8f9c9417efb9779dcd137616871a342e2196fcc92bdee305d110b1158905c ff8c085c0f6902f07a 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: EUFM10 %!PS-AdobeFont-1.1: EUFM10 2.1 %%CreationDate: 1992 Nov 20 17:36:20 % Euler fonts were designed by Hermann Zapf. % 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 (EUFM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 65 /A put dup 72 /H put dup 104 /h put dup 106 /j put readonly def /FontBBox{-26 -224 1055 741}readonly def /UniqueID 5031986 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6f33457e320fe93acc64a4016412366d1ad2 8776cfdce37d3a89b95e5bb9373a99730cba9c11192589e22438b44f4ecc6a8d 1f72a7ea6a698d9580bcafcc2366c5964e5d3094b67d86eec82e16761e4b6f4d fd245d3c0b4188a0543ee88fa787fb10793268e415e389ff071cdd46bf6b260e 1e9c9ac037edba327c6379836655308ddf35e4a4488ad32d3ab301c67a2fa71e 8ae00daeead4271df276f5ce395a57e414f85237cf8f1be070309ed5c3078506 c9d38c2adeeac618d04c0be2ff891dcfc9ada7588efb91f739188a933e1d37bf 632cf042106df1bae0cdfb79c578e565014cead123bf856cc0c14983f5548a20 c0d45446e0dd2b017f528e46d394a88a2d5f86fc669ac7d489f1851c100b29a0 ee3ec82faf87eba944a57dc14ebc94f2445a8a21ee68eef1848d534bdd192ea1 181220351ff93c51e53fcd5ee6ec92bd644d121eac52bc163acb6948a7a4c256 7243c7113b1b5650aeeedb14ffd9b6ce33605907595230a06db01bfb510a460e 76598fb2bb67498f79f0f5b5c5fdb5515cf2257aad324ef1a877a296e2851d38 f8995123e527c4956247fc8ab910206a9ce8736563ff0d4cd5d00579d60bca51 650af348ac4d6422c4cd74d924b532d4a28ba95eff24aca4b3cd1d28e207c7f8 906f8dd99aa04c135fe46c1dc8ee89bd768df17240a70f1a96382defcc180b9a f1b6fa57a9b0a2440bae0797790f91733970b16e742e6341d95faa40c583dac5 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 63 /emptyset put dup 67 /C put dup 82 /R put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d56491bdecdcafdc988206c5a457a19270b 37d0ab776e03eaa7eb568eeab6b5e79dec03b0dcbf923a2aa8e4f4deda2cb043 858d8b430efef0ed914a80ffc9818aba0fb30dae3694e5b31df3855892d59b89 7f82ff79fce8444c7926ffc4ea008e63bc518936c9098afc2d1c14030a2c15bc c9285c1b57e80570d2f6301009e63d9ddeba44e3251c75ca9616a2c5de9ade66 1a15db37b40b67c1bd71f77343912c62f47dd24be0e03dd833ea60d2eb020407 eaf042fed30f0c2f01956c83d46d982b50dd979c2026ed5f71979dd98c9c1eee 25054000a5eed5ce9498b2cb59a7ee55e0538ab47d86e69b2ac79229c8c6bfc9 3a24e822a2f1c9964fe630344579a25f66646113f73d1315c447f070c575de76 266c6773b01137e4a039df5e4c2f199e8e3396ef9fd001eebe090e04384c4089 ec9f13a07ad8733c6af775f432f154ef50ec4477e08760c03c006e4d1eb1c7cd acb8955ca4b0d62c7de046ff58d431bb9883d28586552c51809e3ced4dfc2198 b97ad9e52ea33c91b9e335a80d9b77d3430e0487930f992459664abca5dd9177 150b0fb502fbb04ccaa94ffe2697ed07906f95fd98e7b325bfbf10ea6b39fd76 c1e5295022cb39e9cd471b7b0b3f9d72da99deb0615d35cc22f42dbf5cfb70dc 6279fd520d99f32201abf25e621cfe1d37961d9c0ad36ef3d9264c547638152a be30812eae30521c3c775adf1959595a43d8a9c4e12d9f9ece27a0c9b5f45e38 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0b4f0941233f397908e4acce5f2c4508487753e0c461fe5d3bb63f33a32f9766 20c526b222d1c8da4aa9f21fb6e6f733016e0507952d30769a46761d0d57eb18 4a7a48c6130ac708b1a7369faebddc0a61b416deec7b693b3c618401b6f0b5d4 f865a31ebb35087d6fbc7dbd108a3537ed416eaea1a83098e84df3be8af2eac0 42bf525fa422d0485dde44cffeee2725096a44e7ca4a4c71b2c7e08f8a8ac1f5 ee772375d90161ce61a8de439ae5b57fe78958e8e218736447eed8cb4802bd6c 7b580f6f550c977bb550d9151e768e1700dbd8a6f60303f8ea16c514f5ae34ef 347c0e3dea950e7f5e986eb04f94c135b804ed27bd1730e177ce2d3687082e74 0ef8e0d22ab9e5cb0488a2320652b2afdd5cbcccf7769daccb0cb45221e993d4 1baa7815a9a9fa4b7b3b3ccab718f354efb631b40dceeefcd859c154ae0ad39b de78649c8d769e8f90b7d03127a9c18b6a5e690f31f2563473031a305ed3614d 92fe7f2cc8996267a9e89e7c4bd3021fa50133bb999e2f03249b4714a4f13196 eb4936bb70e6d4743e43a0013b1862c37c023e4e81e8e933ba86600cef81a77d 734eab7e682189e3d27a2d17d16d3338ff978102db1bdac93c717d6eac7aa758 36e4a82e8766036387c24837b3132c49692ed4089cb4593ffcd685333cfc4182 4ae3ab8290642a2e9f482daf66c4e43f00f4b478383695dafc3aeaa4ff9d3a4c 7c477e1154d930fd7ae47c18b740d23ceee6571c06b2a40890215a53e925b172 34f262805597958001b67052a5e6c6490318f242f2bb676b8683e938d5c5bbfe 3754c3e19c8521b434785417124614f4b490e99f92d071dffee35a734fba27ac 97e4c5b24b983c7d0ab24d09c9967cbdd8428258eb93339df1f66de61b384ad9 b5eb01f62788fe84c2430951f79db93a5bd3866095c496001b25781f57e8fcd7 9b08e49ca859205353b624da14a9004aea198e9cbf5401b99340b7b82d3bca93 176089b9de07034901b6cff7d1a19f1bde96be45994aed57ae068ca4c9a40658 778ae00645f9d7868e5b82 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 18 /theta put dup 23 /nu put dup 59 /comma put dup 78 /N put dup 83 /S put dup 86 /V put dup 99 /c put dup 101 /e put dup 102 /f put dup 105 /i put dup 106 /j put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 114 /r put dup 118 /v put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueID 5087380 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311e252234d51741e18db3a68e8ad10242 29e5817a10e796a78d2c7f7c1f50961b9a57aa604c9f821dbf5a9295197bc666 31678d7d2c7e1f8f2151ce0c29efce46270570f4301c5dad1b38884732e53dad 05dfc36b020e726ca575f71f04abf2b49e1f1d6d2a08a477658ca9a48f1c8cce 14382f0a42201de56e2821ee2b0e91a818b1b753d810ec2428dae5153b66af66 9442192024becf743a8f6264a58a0b3aa823510a4957d1bab3e1b83135362888 3e392f49926471a8021fdb48637baf0abcf31204b1b214f48c99f44af7782323 69d772ae4f49be53a1bf584747bf29b94f530d59a5be5370b8dd0691e32a21ba 9d0d592cf3bb50a99a3ecd18d6c4114cb4cfecc49705da90d7edb10e3a28f9eb a49347a09d31eec896f2a28e9607d99d42914aea16545c16e136bdd0b75fe486 61e24c7a9e9b233a76754c760822d0a88c575e08ae3e3f081d7bfb80ec4c03ac a08b65fb6474f71fc95c7c2462f2eb3754e647c29498288f2ced0bd3a2bca5c3 2f80a61fa667802d376cacd34de1f9572d1b3ab09d6f352750db4a768f869638 a87457889aa502850427b40e1294c5ba33e6b17532b2d13b5f50e71582f93daa 231e86597872a2a968127afb0f36257a76ce1eb24e6c4f29fce2aaf6532fb5cf df84da03f03a45b915209f67d398c7c4f7a3b0623750259087a478dca684b29e a8b7f52060bae9394369ed1cecc645d113cca3772a8dc36c6d916b5c5e65454c 0a188a33395fd3da67ad0a31ddab6a8434dd80f0bd0ea1a2a62eecc331d88b3e 369552daa1c575a98fe84383548626b65be7c24ae7eb61159f149872cfd87cf7 6c1a7c155c7186c7600e58cb61b8cb0c5dea98268a39e928bf6fc53985eaa763 2b28c1eb3d82e4d3fe225223f4266d2b6147e344aad2d4884a9308aadc7801af a606838925e3e50ee7aaf73f090baefcfd4e34b795cd92d83c944d91a4940f2f c955d727ea5bd76f0c8057b12621eceb0947b0a5dd10dbcbc52957dd978d7db0 c794660afdd4e946b3d7fc1ba60907fb3f756aab0406c0e7bce5a2238fb3e12e 07e2c0a7f465e90057642d704434ae0b57bd46c8ea91ab2a386e24c6deaf9cb0 8cdaa8fc4ec3fadc06bf28659cb878453db16f4b03925b2127b83be8fc240f33 23ea950b51d70c8fe861e30c346efa0a0125137598caca5478991e35f5bbbc19 4c787488ba339830bca15f8333431902e82b66a514a41c0bff92719a6a0d9587 4cff62c622fc61a0308a69f9fb7ef51bb274ce9e059c04b64e817abdbba3ec52 edea66add61b1350782f905b2e6a5d1c2acc85be7a12d2f3a628eba6aae29099 ba5b4562237108747997f779ec8da904b00855a0e0b3a535cc6ee97df9c57554 95efdb32151f0aa3ca119986db9d833297c00f5bf1b48218725fa9708acb358f 391f5b8056bf75f06b9f14a4301890083e032ae62f958c41d8f0817078c76448 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61d7b2e41c5e9b8644832041e4d1c5ab1b1eb489e97473157a9db2c61b9a02e4 0d9ca16bb76923801f2a7f45bb874493422839164fad8635101460ec4c7b083d 035eb7721ceaadeaaaecc60655957a9676771ceed9e5b2ed4c7163cfff930693 12d63af3a030fff962f62efe7991683effe04c5b8d43364aef4f1e9149a9e467 a49565841441865a95e80828973a1e0729af43f09333bcadbb587be481f65deb 425edfb3bab76ea339dba1be12c01234781a24839b3afbba61631a030cd0ca71 826a4acd5ce50af33407d9cc8d02e8a3f5031d06cd3b1040e460b107464d12f3 888faabb3c05fe405efdb1730fe2b3fef7b41b56d373c380d2be02024cfff592 68effa5b9fea4efe5957228a5db0354969152b42b29fcff6e19e2a274ddfc1da 2f4f109f6fc1f989a0b2503c71ebb8032d1921ab878c706c0efe269733673877 607bce71c4cad04ac77474798d6b0eb60ccb479b1b0bd70f122f87c449485c0d 05905b7c3221d82d2e991c3d0bcfcc55692c66c0b8fef1721304675c68513a9c 5b2dc4467ed1cafcd89bf9f3735ff7f3d8a6fd032e5334a2b2da754a300b0480 3525f3d507e7bc2b65054276b68572c966285f046d9a7f31862cd631e1dd8009 6f6967a7f3cfd181a6b1fd9334f2ce68c74719d61e6a3e0069622b4838b8e07c 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533019de426be91988dbb6e104870ddba616ccba0d617dc5b7685105f21f36aa 15ac725b286d0c451710d41eeffddd8c55e24d21881126812d18b8eb4202cd2d 5039f62d96b8849b9488c0cf23bd082e09d614ab523dc42c3e175c7d30a35c8c f909987721a4f63551e2a868418542121c3699ec2bef00eaf74d35ed30e0ac2a 4183a7914e0231b33acd7be8ef672314930fbee810f24097338ebb0bfc24df74 761ad04259e38c1c566b25631bafa929fe95ff9c4ab8b37fa8d9162a0f5c6daf 1a2d23a3b16fcf913436e0fec1a4e45ab675c001b939a2a46f33d92389d8de65 8f887822fe2c94dca026037790770e201145194f91d2936caca8d905d5a829b4 3891faeb9a21a2766ec4af1b249d572d66804b36f1a7d7542b07db343c23fc4c ded5bde99edd5eccadb25f296ad9a12af15e02215c7e151a7bf541a5ca4ae648 b91a0bcca891c5bf5c05b69737ec8b176642df9c356bdf534cc73bce3eeaf7ca 1ac3a191e00baa472736c6e86de644f2cbcf07d42ccc5a354cff3cb671b4b761 d65b79512d4910542b39500f1a01be27786acb1e6a670dbf2c1df3847a2d4992 ab0e3c00b307801bf81d0fe58d0d2ffc98f2b4f8ca6377776ed61d0abe02e3b1 8cd341b17c14de84dddac1a6b05da413b6dbc706a87a850399fad0b3a0aad995 3e55f92ad4144769683400b79c1824540a06d0e538df1680db1c7e6c6762e2a0 2824ac6bec3275fec389f4acefa1d4e8410b190064967b6cb8866f95d6f58904 8b95a0cd1610e32de17f7a3ba5a14c23feb3de811723d4075268d68b638c4ab8 242842ade3c33e651d9f55e4849e20d75a29a0305530faefd1c1b92a562ccf4e 9e248f64af242a55fd4ef88d21c734248c97f7064eda2f1a7579607762a47ca5 b3b0ccc54ce8d2098173636d7ef3c397861707f4e0af2af7e718d5e4a725982b d2da883d181f1b5d2ea58c140e89355df0c4154f7d1f55789d4d3eec59d050a9 49e9f055c7e30492e4324c9738bd2c91042fa5f9252794f09fd9599aed7c5aae 0a7a87adeaad416d06e53fddf809bcfa6121e6f6ec1e67af2a576eac8e209551 113b01790f15aa91202161e5c1232a5b6bc6bc0f3035f356bc8da2796cfcb336 24d216f5fa7862f35605eb3b9f4e0dd7e50500851e39014a96b5d1c74176dffc 3fd6d9471cd70211c680b9bce2137c9b704facdd8514ba64e1222d21fc914402 851e04795ad6fc60c6244979771a6c3d612f42c291d30e4237ecd5b85073aac4 d2715f222d97b80702ccebf806740bd183e6d2e5935ef3ee1807c537fa28594a a6889a5510b3c0f8cab3e0f7e8c6c29578189c1d4df7b4c9f1288d315ba97387 7ef6eb6b513bee8b3263c8eff4c5922ba9c81a58ec9a23863445aea45159e616 6702eb40bff990c45271be88da4ca9c4951a0164f3f083e8ae0680153e799936 58f6a25475e1d98d415dc2e95f66b64b7de525e0d03b34de2b7d0ef8f1000712 b58a54c1853eec5008ab33428b78be0bbe43840f9333844e8684eae19cd6456a 4b4182bd718b5678aac45606abd781f94944e47dcbdb441e5a5ab2f696f5aa7f 2a7b12128e42afcc4d78162329f955d9cfb56c42f09b8e551ee2caddb9a63242 f696fbad96853f5cf7c56dd13eb91af31c8e6612c3929f50b7dc1a4754aa6a25 d1706b05169f6fc026c4fe7e2e5e59f724d4c8b423f92eeae816d1f26cbb0a2a 5c2c02a7421d46c8410e45c757ce44b4e27d4f67667ef1d8b827bd8dc3740397 5e4981b8bff9d5982306dd1e89fe666c4a1e41db4feb3c090ecfeecc58 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 10 /angbracketleftbig put dup 11 /angbracketrightbig put dup 12 /vextendsingle put dup 13 /vextenddouble put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 28 /angbracketleftbigg put dup 29 /angbracketrightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 40 /braceleftBigg put dup 41 /bracerightBigg put dup 42 /angbracketleftBigg put dup 43 /angbracketrightBigg put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 68 /angbracketleftBig put dup 69 /angbracketrightBig put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 99 /hatwider put dup 100 /hatwidest put dup 104 /bracketleftBig put dup 105 /bracketrightBig put dup 112 /radicalbig put dup 113 /radicalBig put dup 114 /radicalbigg put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueID 5000774 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d5649664203ab24acb938d58d246707ffe7 d62f04bec4b70c21ef75beb2b812622b3c74e969d72d3cd11bd7106294a99caf 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0375d286f813a0a89bb2a58fdd1289c7bf 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /Delta put dup 3 /Lambda put dup 10 /Omega put dup 22 /macron put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 57 /nine put dup 61 /equal put dup 99 /c put dup 111 /o put dup 115 /s put dup 126 /tilde put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueID 5000790 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b448685940729e6e22f4be0e08c6505cba868f7950 93f556b64b808dd710eb936d3ac83e5783b5f7e56d363a59a026a5619a357d21 c4e77bea121eb24b09027d1aad93158edf912940e91cd2f17922b35885894644 7773730bb19af9e70453e1b78d5edc123b273a979891d6d5e624f4a388f3897e 13b045e34b4ee2f5e4fc5e996d3549640010c671de14d18b0b3cd300e57dac03 22e59bfbf7b29422230870f5897fcfaad4b50c7c1c58edcd119218163d6f6e56 25ccb882db7b6d09a1586508e8cb642a11c29a46915e4a96e282079cb21922c1 c2e360b487a45302fd22ec8c5fab63e54b5e844d4b17ca2fff37c69c366dd72a d02922c14c0932f65ed03e4219c117962edbad2dcdeaa9c10ce8af38a4ae52e2 b377245b0be19a77d6c936e7530cc4d0b78d0cc4a92698fa2870fa54f2d8503e 2d17b3d52fb2febb09f2b2af0c2a1892039ebe19a690098799a858e3d39631bd 6925a154d161df3918074ada6bd52baddd0adc3f07e2d9f15e27cbf7fe8b98c4 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8355bacae8f008bc9c1e3bb0bf77e39b1f2a759f8865dba8b82a0ae3e4a68fe8 4781e3192c860e0f6176167f4f3497de61334cf3827fef42c18955a07a5374ce d57568d0953da14f4de5e27f8301d48eb14f2a72a86fb32fe3d0ef55e573511e 845b0ae845205929a5da359ed1aef23868707001c1f50e9170a3cebf23bfd1f9 f191462f09c86488edde3ae41a9f68531fb29997d81a2f6effbbdfdf9d8b7385 136d2233788907c6969587084eee65fc6fd2112061e917746a627768643ecdce 3d5873a2fc4fc013a4e51d1fa8ef25746601536d8f0a6c91a3c36d62a4d1590c ffb179d44bd438a9c7da51a8cdda30c446efdf53780042cd075c5f23e768dafd 5e5a1479133d11dfc5d19da7e2f875f6b8cb46698a700fd01c0973cd5a7ad866 cef3d1a4fbcd48956d7235c9400a3b8ab29f0236cf48b52b3961f50a4a190509 d113acd8dac15ee8b42696d79abff7f37f9f6aa4f763745fa9ac73bbbf59b695 2c7815ce210c8031d3f27bbc1d0202ca10538e326a1af7b65b548931229ab755 42d0aec0ada5d818ce7147c06152c5231670816ed46113d92d153edb2a522f0b ef485525429e5e2b7f531edac9a477c34da3160a41fdd50a1e447c7f7f853d0b 43d074ddfbb508713299a40c2323174cbfff41c1fde9e5061a036e7775eccf47 41db37e3d05bc6da3112b4de4ec8e45ec2980679044098f024213393d01a6f5f f212f72848f5f457e6c153513840514ded3e030763962b269c10fabb07c46bca b0d7fcea9d5b7f16b832c719f4423383b423523229fe7b1a587d74239f574f23 0365de8f987d4cce359879544360f948a600ef2e0300bfa447d628d64b54bbbb 9a26b4e262bdf82628f565f2aa10e08452f37de8067d081ef7779af2b1dfcf2f 2a361bd63ec7107f8d8cb405637252df82b72f18c57182443c6163d6609328c3 c42c8e6d2ebb39cfb612c2e275dbcd3a5ba03906abdf2a946fcf6178deac4e95 0c8077617249651a8fbd17356ee6305638c52cd1969233a188a6fd05d055b6ab e8a36d6c8f81d1176ca133466fff7daec90ce036cfc6c5e1c534da14fe43ade1 9bf16acf93b56945431bcd609dd70a9ba075ef66bd73263752084dd1843fe0d8 3194db008da78b5753da2279f42795037bbd14bbbcee84dd0f411afda65ed439 3b74f5868e710fcde30cb6b16d8279cded96694e34c12ac46732732f422806d6 f2a1acc7829b4dc7130b93fb9955e829642f4cb130fe0d64bd3c225db95525f0 e3c92ff76e9b0c566695c0fde30b5b05cb8ec68f224a5439bba674cb1ea3b977 eb5eeb6cb9cac0ef0317dc27234972c2226f6525a3b029e9d3abf3bca1f6f16d f9bfe278465ed1520937277125b0ddf86fa0235020581da752a259f02f90a38d a6749e64f1f2a4db9b13b5823f484c96372620d8f468e4bd86b55b1ad757d676 c2cd27689eb37cd86175a294db57cd15a3124fa909eac7b4187f4ab330f0f239 18ac2a4eeb04098675227356258ab60c70c8e6e544db3672f9857e23b58673f1 71f48cc3064fabd3e709348de617ff5500158b30336eabef20ad5fd16a5c373b 30a3cdb574caf0ccee4aad70cc30be67e67fdedc7e4ff565e872d05d0b430e4b 78a9da5977508bb083c588ecb92e4f419dd6988c54c42dc2357b8ab102ed5598 2076a395870c9c85267cd4c86b883ebba43632061c96c81bd6df3dd2fb813057 3d771de24b8aaf493a0e6afdcf9a44f3336dbaf37b9f2b5630e0502677913566 a7e0ebc49550fd87b4a4581686ff3c6fc708100af0c6d654b0477078f981468e c5b6e15f3751e7fa0c7125a4fe2c313acae9b62ec60e111692f5f16264229ed5 f3d4bab1bb9782f5a46f85c77c4ff2a992b1baceb74ee43408275d958554ab11 242c25d725e1151d2ed5d1e38833025d754649c09c03fa04d83259075d9b01b3 0ea837aa1946f0775108486816d7e9bfa9a7cd7f70c7b2ff937c0401912f95b4 a0b019c6f09cbd1fd8e95e3028303abf6528cd6460e26678d1b0352bc390d394 30f97b8492f3b1a9895df6114cd5947a88758976c131c1d8e0bafcc76a80991b bae6faacd8dd17f3760ed5861195061d82923e9ac21aa4adbe617d875f5b6730 f68aef8cea611a6bb710059b7439f684c51caf79790f822e4ac3f6bc1f778c4f c77065cd47d942 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 12 /fi put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 76 /L put dup 80 /P put dup 82 /R put dup 84 /T put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 121 /y put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueID 5000768 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306462c563d51ece66fcc9c831843ecae 1fefc1a232e2724f7baee428ae03aadb95c3035345c15e9922fe49e1f2cfc980 237316572dbc57064edac9b0db8913a5e2d45e97e19a91435ccf8adfc835b585 48e74b291446d689c7f2f8c4325e8356e974ae30c3e2977477baeaa33d141fb7 80f59351e84bfc88c87b3a4a1d25e168b9d134554f6a581378c7d2d6eca8ac09 045cc3a5e0ed86f147133f094029e3483bebc81cfde69942cf645ea20d0a2b64 5140ccc2f9cb4a3d969f4b0a79cd3329b440dfcff5285494406f3dd08d804c86 df191e57232f6f1f83df6430f214e03fa5412b1e13d83c60c52a698bc9a38159 19a64162af7b83a9468fae9507c6cb55ea80df8c696dfbc3296c663902d77e89 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 15 /epsilon1 put dup 17 /eta put dup 18 /theta put dup 21 /lambda put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 28 /tau put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 59 /comma put dup 61 /slash put dup 65 /A put dup 67 /C put dup 69 /E 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 86 /V put dup 93 /sharp put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 118 /v put dup 120 /x put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueID 5087382 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930f01bdaae649990604a33aa9eaffbe5b5489 e5c7c9ff9d9be01b08220832c41caac64816b53bbc087ae4b621d2b44b21ae5e 3f7fe4ddb05c675abfe30510eee4a7c874bb57b2ffe521a6875eddfdfd18c781 25bfca5a097aad361dd94df46f14026c25ea988194899b95001209cb7009cead 4b4ea03473ea120e7a7e341d4e3926796347fdb3a124c29660ac09f4648b313c b58e824884382f24ce08d2edc686923acace1201c16a07a42803288cd0b946f7 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 8 /circleplus put dup 21 /greaterequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 63 /perpendicular put dup 70 /F put dup 72 /H put dup 107 /bardbl put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueID 5000817 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /asteriskmath put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b66566ca5ba42b1911a 5d7f1bf343015eece988b7a93bce0c7aa61344d48aed9c92c8698d4b7c9951c8 7d103f2414b39e1437f9d2e50c4ee5f218f2e6716926a79ea978f13b1f855345 191dd7d31d8f82c2e3343c7a5894d95bdc492c28226834efcb5c12fea36ac5cc 430e0aa604961e34888adf6c1f3954cbc2498e225d953cf5685852162346f474 5a2a7087d5d7ad486de16d2ca8e15cee26e012671ba3bdc7d95cc8c98bb774f5 08625e968aee27ff7d1a06e63bcfb5aa4876c3f8f13b30ccccee73c3caf4e70d 98e6ed2f422dbb4950bf789680e064150995941a9f4dd68a575949847a7d012b b910bf03a7a227d51386469ec9ef415f3bb849d30411890d902a21cacd3199c8 a0b3222b3f90fbd0b0d7bfde7e95c959ab3f1d43337fafcf6b73f775178403f3 cd511cc4c14a5016ec884f124e74a90daa9eaf02e90f140ef78ee3a2d0f926f7 81c57fb30e0301ba29ab222b8009ea925c001dfcd07b450530ce85bbb36d9a7b a6e900c2e645e97f09a0399e51f6c092cebda8e9f62e8045c311fddddacb7402 62073411423cfd611fc5544b2628b33986c71ba4e7c0bbd30709b5e33297e26e 4c7a4ad8b5cf33981f783559f091b6e7a695c85af31eb31376a8079bc423fa38 2261fe61e935c1e32b3c957cde0cdc1ee93180d6bdc58962524884b3706d24bd 9d317dfe7877767ce952aac0d2e2a33be72fe117a577bef1e7837685c89585be 47f83f5ed42ffba53be3b6ade5f83c66b3ec2a23735a888a277d712f8b07104d 5518ce6942ce7a39ad52cdb3124d367009c67b1532d4c6160af4db2040 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 76 /L put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueID 5087386 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7bcb09b7064e91fa0ded228e42 09aae407a7aaca60b1076299ac4abd23ef02f108765f0e3d91f92f3afbfded37 2fcf6e4b1416901517da8f2fb3c9fe7a87bcbe6fd36cd5b5823fdb74229036a6 3c3346a1093e6b1036902c1bf42fc317c80abf04020a47b344c36de42f05c490 a0ff44ab6d5249e9f552a8707bb7661e242644814001c8430ebd5e5f0b944ceb 666ee64359d663e355b2f17093a964139d17287f6ca6a024767eba4fe4873855 babe2f07b91560f68300b06dfe27264c163195d446980c35bca0b48f7806626a e72636593a05ba403ce1c0f8b2cea3ecd586e90ac17d034ba4af708304f23131 3459fbbbfb97d4834d0395754ab3f22d6495d2144087d448616fa1ce27bc50d3 46543287e3860d99b433624119bb9920a2113604c0e260fd275ba55e0fd19c83 e19addc3baa1f32f6b7284038845ccee71a3311ddb17b84975f7a984bed7c6ec 2a06e5b335a763d081c6273f86a46632fd9141a27902074fc860df3a2eb59b89 774c767022dbb577e30da128bd7706a43af886d0c256b50fa968ef06776aac0b a5387e9011eb2334c1f42c090f06a1125c207ea6324e87f46414050d88070346 7b9a6e4d2045feac295d15ca55b900da594770ed9843365bbc81bff2190b731b 8790385f7c313b4f64683aededa87b8c04da713754a05b7ad08b814b94b7be0a 5f20abfa50f767a51aa084b330955da5231cf6660a76c929e56afc63797a999f 757018dd00ca061ace2eae761c8e56838535e4f0dde8afaf899c418727ddfd13 eaf227fd767d803230aa511e59084b0bec6f25be7f971add8493041b2f4da439 f1984cf9c80d6dc1750d3381148af5b4c59da1353c90dfec4f982316f33843d0 060f63213e18fbb710f13620ecc25090699b11725c6c0f4a9f4804c6de55baa4 e5254e2e2dc54d2d53ad76327e8675797a162bc50ae239fbd4cd16a686de0e1e 25ea39892ae86a3c1a8ab6d6bae04333949a12a1762e80c15a2d986d91c8d147 d750c4b0d1f416f84a20df354d821cceca1b5e104311fb73bbedc9043105f871 b9b3369b81149286d613e56f8adcfba643dc5cc6d388d6245e63cf6e00bfde49 737084739801e523801edbe0c6591df1188efb345bc6c54010bcc4433c32a0c0 c1d31c95ea8376a57920bf59a23a168e708d86e1a1fcdd93716f069bb117bb22 20a55cded5a70dec9fefbfb0e6e8ea25eb2b10ac7d2e87eb8ddc4e4700d9ff2b e1d5c9c4b1de278f653e83fe22b7390ef657a85c1215722dee4b3a01f0ea0b33 4c38e38f544ad8de0e72e843843c86d6661131b2bf0181c5b5611e74bd2331b8 f623f516bad0ea8b6e7048d8c138529483fa00b636f2ba89f8f33cbb0b10a851 03b309620f8fd1e24e37c3a7370197c1b465c4c6e8683a18280e568922e646e1 e51250f2244d80ee5d62c73fa758b7571adf219d60e2535db2431f0f0fd78f07 6377474320f48c035495c4dcf51df2efa799c57bf33eb6970cb6df674206a372 4b 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 45 /hyphen put dup 69 /E put dup 77 /M put dup 78 /N put dup 80 /P put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 121 /y put dup 122 /z put readonly def /FontBBox{-36 -251 1103 750}readonly def /UniqueID 5000829 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 66 /B put dup 67 /C put dup 69 /E put dup 71 /G put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 121 /y put dup 122 /z put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueID 5000769 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5bfae5c1f476ee6f05eb1d2caeb 269958b194521197b312fcced4867f3c8fbd030bd715d8ffda1dcd454b174e7a 1a97b59fe770e67702519d9d9b23d61ac08424d555242a8ca08c49aef300945d 99b999a79ce74804ae6bfde623f4463371442f6523a5f6ce19c839a708c02513 2e22c696c8ccade45680e5197189d0f98e7f0d5f955e353970b392cf530a68cc 56b0035ddfbf206c3074beeb0739dcbca272a6e629fb7aea2c5ba7bae50c7b4c 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% Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 6 /plusminus put dup 8 /circleplus put dup 10 /circlemultiply put dup 14 /openbullet put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 33 /arrowright put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 56 /universal put dup 70 /F put dup 72 /H put dup 77 /M put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 85 /U put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 112 /radical put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueID 5000820 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8 3750d597b842d8d96c423ba1273ddd32f3a54a912a443fcd44f7c3a6fe3956b0 aa1e784aaec6fce08dae0c76da9d0a3eba57b98a6233d9e9f0c3f00fcc6b2c6a 9ba23af389e6dfff4efec3de05d6276c6be417703ce508377f25960ef4ed83b4 9b01b873f3a639ce00f356229b6477a081933fef3bb80e2b9dffa7f75567b1fa 4d739b772f8d674e567534c6c5bbf1cf615372be20b18472f7aa58be8c216dbd df81cc0a86b6d8318ca68fe22c8af13b54d7576fe4ca5a7af9005ea5cc4edb79 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9a51f3298d466ec454dd1fb7d9a66a9da1cb6dc92f968872c55c3c5eafba03c6 62f8e8deec64164b01939854f2dfd2bb3231ac9e52276d84a29f163784144ee8 a67e53d0f6386c00d2d27c077732c655b6d7 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTT9 %!PS-AdobeFont-1.1: CMTT9 1.0 %%CreationDate: 1991 Aug 20 16:46:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTT9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch true def end readonly def /FontName /CMTT9 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 45 /hyphen put dup 46 /period put dup 58 /colon put dup 64 /at put dup 97 /a put dup 99 /c put dup 101 /e put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 115 /s put dup 116 /t put dup 117 /u put dup 121 /y put readonly def /FontBBox{-6 -233 542 698}readonly def /UniqueID 5000831 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be465959b1188f76df7824c135a7fb382c4e1 1b0ddfe856b6f34552ca48c24b57dd8448fac257c4d93fb122e1218fd36b99b2 d79dadb8a66613ef11039be77816a7a2d6781fe0d69cecc36232ae87a172afe0 95532a8c7ff40bdd4a5fcfb6ccfd8f73c572231734e0609c6743ee11010f6579 820b367c49069ac1d45760c0ce8dc1e67e2beb72be6d287910547e1b4bb52465 b75947d224998223275a2c340a1c14c04328cc2935c53ee2723f1de6357a291e 14863f6c1411477b3ea074052b30a4c1ef03ab324d678307e1cb02c7aaffa90b 73975378b9debcd3f6aab626a361be8068c60edd0b2be4a7bbf4efa21cc23607 bf7e2556c17b5923d7b0fc036d5fea743c55f2bbf72e8a61995bd63d7104a085 b8dd5a83d9077376a4ae8f0c048606bb9e84c640c52d0a95b35d48cfafd6ebe3 d6eb0ed88c1212818ca93e40614740a77c446b792d02945642014a5d55ba970e f97dcdd5b839f621a052d1085a1e668161d3ac0227c7867324ede883589b39e5 d152793009cb5a08eb6719a44dd01b6147e3436bdbb28215ce3d29a3b1dfc72b d942f8cb3376b4e12cff53a793d935346ceb2afc3428cd9313c671ec1911252c 19de341556999aaf7417aa5085daa35ee71353fdc2c7d83ec2dedaabb1b89b99 e4cb5282159cbfb45a286f8d6bf3abdedccbdcebaf6994b17f1be07d10f00cba 47f2fbf66c0d83921a5a0a4e2054a8a2dcf492cc40de855b476ae77bfe12ce22 d1195e3f9d51272541e21267b2924e293e7a1090b4eefdb707d2c29ad62a240f ad89834a3b49e3715ed086a4bcfb0b5118744fda024989692cdf4e88876c2dd4 e1f3dbf5792893854a5c4495c99a9e9e2410ad797d5c79d940621c6e94fd03a3 3430a835efb9f316b5b63cb95d20565feeb96595259df476e095051144f7735a 893c135adf531b8f4cab1b78d6bcf1d9776b5360954efa386bae4f2e30a33074 59e73070a5a4d46ee76b59fc0eb0835ce221476365352bc9c88aa4b887566160 941ff7a1d888cf3ef7b3726d4da0bfe63c454b139814f278641da1a3959e3089 8373bc9b16aa0dfb02602224295d2bfe8ad98885cea92c362d1c595f08b14d6d 04776dba03b249f8480e8aeec58c1728a8880cf5c1335f4ec20ce008a61e9417 1171a2d9ee425be5822df255b7a7625104e3bdc0fc1693dcd042bbe431a8c030 1c4e131ca7ec0ee38ed4f55160b0b7d7fba81f91a8bea19d607aa6aff563d82a 34d5dbdd66cef7b1512aecc3753433b6082d2a545343bd6e6a418e56e42c65d2 d75ba557953d9173276c47df8a3ef843a7480ff0af3589e4aea1de780f27af42 9d369530cc8f9aaff5770eac045fd893a096cf23d1e23e7b80d1952571925e7e f9b20c6b63a165dc5e2985490ff62905c5457735daaba35966bb04953cf448f0 32ed41f2387a16389eedab3e73df8ba976287431c38a0a27ba401e699a0eb124 afabe8469a17b839df0e2c65e24f8fce4e5aa26ad4940e88c158e8ffa3ab94c5 e5158072ccf3512a2f7c56f08b1d863f898b5f326081bbcb6332098fe2175966 c591cde59dda6cb775b69ac6bd546cb7c45a61044a45ceacca146343a80d9119 452347b884a0da099161a2569266633edb1c9e34e891729f53c5009db268acd2 bb17f9cf8d66b0ecb37301fba60050318f316f0dbad1c30263437c60b2ae00c5 7a8450aecf9febc5a77681b5a828a2a091d33a275c9f01005f60b41c911bfde3 6e71b7938481ada5cb68149e29c15032bc96f9fbd13301a9850d75e25741044d a7966b594c705b488e0bbd5b53ae20c1f273565031529881242dff8706c5ba48 ea5c8c38343c552b270005becd5cc555a57932abba57104a17de40396e69be92 fc10bbbef9baf80a47f4ed84fcf92d659aabad6846e11371fdfadd25fd6666b7 d84b8addc03a88d67fe6c0f1cf4c03a45a90863a431df2cdc92ecccc180db98c 8996f84964a8a11ac3979e4c8e05bc78907486a234aa443cb1413cd27888cf4a b5bac15395271492fb2609104c5006ebb6b468dccf81089e66459352eab383ec 263a5d25e897add0abcbae21ba125294ea1c9f1534da69b15626a58bc18c2a2a a2058280a8dc42bc125c58ce9f51637eb736ec79cdbcaf31b1ac9d317dd36933 c1a4f13c324c896d6bf72f580444370bca6bd6835ee96f6c1e344ca14ee2bacd 8d81153245e5dae549b3e2d8e35dafe90a9395bc9f9a449bc9534d0436ce6cae a38fa777cfcf988bdf50c0e071e2b630654a02f510fc5e0827a883f1beffec29 cb3f89ccc64d0a1245f1696adf5e0df87ae4c885b70e0668eb40506c55d0fa8a 83e177f319de38095f696f5f99bcff80dce9d09aed98dc2603386b7e887740d5 9a23c48b4312c63ceb79b88a902f764cd4757aa9fa9a994ffeff082917104de4 be61bd874efd805c5f3c111d7f4ef4e5f8e368d87f702e37a18f683f6b9c8131 7c10d15740ec8a87e06cff860b6d6e1778119e05956d3d2a0cc54caeef28c4fd 4d0015e2212ad8c737fb6ee6f9a238e611ae9aec5903a9c809304920956cd1e7 54de52aa3f3bedea1ba07f08e65f03d37381e9e2d9b72be60a318f930da1c42e 1822e24dc86004b4ea0f90e4ad78331286cd246f61b81351d1b09dd94f6430d2 5749eb02c882d6a3a69c1f99486523ace6016a8ba6ebb1a1461bdc45413c954e 15488dcbc24729063f0f9111733172e29e19aafc98053719d4a6b90f331b144e 54bf0b11ffe13e7587d2c7130583c04c5c75106da547f76514bcf60ef7773d13 825137395b2a4bbffdc8acada6715e46e373c2b6cc4203892ee95cc0e2f6b208 9ee37d20f0a6c68c7b6abf6956d311179ac36fa9ca2e843507540be1c04ad3a3 b22ce8f3d392eda65cf9b96a0f7dbf0bb62018f3a6c0b6a11fd99265e4cdbc07 74a774585e2e70bd23f64c6fa534ff60bb20dad5e25fbdea2e1b641108cc85d3 e9602515683eb54cb40e296fcbe628333188101d2f8b935af2effdf26463c60e 75f43674064d1c215d2df14498bc12a34f85e14d276c5be718dddb2560d3b480 100cb3700bcefcebcd24055c051dcbb6b57150fe6522f6c99b9b88d9ae36450e 42bc10351135bdeab4af0d0145ff7843774c5491263a4a2e29d147accb4d121c a8e28959e92ad3571b48f8aaace21934d815e9d48bffb7611c880edff958cb42 6d8c503cab7e024668c5b1fe45262da75661f9cf94e5626bfed93cfd2d9973b5 0e671495ac347ff2d3a560d41bf4dbe72ef310e4677451fedf9f6de043fa3ace d8d7499c4a10292129967a096cc1d5fa874372775364846eeaa6177922ce1ced 13f556726d1d31fa31df2c6e80aa2055f666d39e41fe0774b3e50cd12e448bd0 c974e68653ccf7dca0b9c367e56cc6dcbe4aa067fb1df5cfb03066991270358e e027244119d7b59793ec920f469763843c1d00310a5bf1d73a2a52b77025ee6c 0a614a6f0f30540815b61db56c9ff96e693a821c8fb0099ac4b4dec7f204ed92 ca3b566c91b5f939178219eecb34331c8256982e1c7ef599c88d6c9fdd3bff6c 90b6d5baa5ffbfb1b2b32dcbbed545d18acb0ad15834b2e8063fb40f54c554bd 72e65764356c6f63e7fa671d3d373adf06fa067828b38803ad3cdc8dd508679a fbb220c4763e903d768d1fc72a61b54467f372d15557a19255605d7bddb63049 569d56d5667115d1b9dfe403d19ab2b696d6451c816003081433fdb010da2fb4 da15b1770260dd9ed7095acec477835e964dd64cc656be53a4ead016797f4c42 b926790169cd56aba377ea7e53f3d3c1d848f2e7994b760c800ca6c5006b0fc7 a1ed2877311c1c248ee41b7a4c944bebbe4e975e57e6a4559084d72bd7f3eb12 1d1c958993ee7e2b46bc42005cca0fe6cd384a65ffb3af18e26e82bd6527ea13 348d425f1649fcc1282d84a3374f5d1c0abbeea05ea2e4f3876e811f76d16a2f 66cc5265690e6fd4b238f10381231b9a4b4918514f687b4024e8e912f7e6e010 71326f2928cb334813ea69f3cd092def7a9fb46e9c741b999d8a5ed4593f402e ca7f9953a0fc195fdac3ba91e38c191074d3ad1af03eb6faffebc6d1d016babd a997e40d4496dfbaacb3141cbb2293706edfa8e495ef49a95ffe5f9930f69bdb ac6a750d2a010fd81bbd1f11320fbc80ad2bc5d480af4a3d3b2303521153fdf5 128385444dbd9689544d8b2e2c64fd4801bc8b57a9fad43c6693d668ec63c82b f3d33bd49b9c7d9f4c71a15724507856afb35ee0a08d396c554d2f4b15fd3796 f78a376a421d4516dda356f9b7787330c2d39cd1d322d4cb27abce9cf955ca95 39d6d93ea6a2147b36dbca6c1b80db348fb0dd8850d335967c645ff167b3bb31 fdded0ac2837c2ddf260e8eeac20ad57b82b9f37da5148fabc73551f37ff46b4 72afa5ece992d262320beb1c611618a86199bcc40b2a352c64bddb742022022f 518a03f9bd858ba4159c8e4a17b5e4bbd2d9b607ac0bdedb2727ef5d73a0354c 3dc3ea40ab2de8dce7ad6e2e4ea83d4087e037fdad5e3c08e8e726b58e5a0313 f343cecebf32a1e9154b5b7628c3be87df9f4ceb969fec272d6c880e4bc46347 6d7db8c37e5eaea9937d270610a74a872d570fb7ad92a8705a6bff0bb73cda02 34682e276c847739ed4267c8ac17dd5acdb8b689a0bc9e7e9367ac21a1b11fdb 640445dd62e167e3a53afd1b5fa4b018feb7556803c8211f3f375a2b18269516 b9e43459b4469e99c02d129aa0e33add15095b5843b794cd088aad376fa0e4ba 5a616decc9d146b213f86cc3f2c7fbf568af5990fb275349bbb243098c9e96e9 7466d443ecb0cd8cd34a7a17d84ed1ca69b71a029a9c6fed1245970e551f0091 9e8d0534c995a503dbf8fb9ec47f9e5968d27ab37851144de21cdab6123541da fba9edb5d39c428d1ca4776ce43db6fcf6faf29702954a42e3d4aaea0674f741 79c0edea29086a4f3d1b51bf91cd0e125181abbedc5cf590d3d97f82c12a8cc1 a62cddd7c21bea7d5b13b4a238b003c5680f289cca67c8e1f0af8a4a071d4950 d8839ff868c787317ba851f812edc39fb203616188c020a017535544710131cd 37da9b53761b94c90969ae78ba06e94337135a07dcc168b94bc74ade83543346 d494badaeab491832ce622e1bacc21870db00c365e9111c8b58d885edf29a651 5ceea357618ff4a12507fb4715ab256d04c6bbfce4ce3fb0210bf0912da33f3a 134d76e96c16238a97af897f057fc6642e2561b7e58a03dfd7bf31c8ff4ce540 c27202bb8ddca026 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 17 /eta put dup 18 /theta put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 /theta1 put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 78 /N put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 122 /z put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueID 5087385 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 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c63c79fbaf30951c4fb3d8105335 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 3 /Lambda put dup 6 /Sigma put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 19 /acute put dup 20 /caron put dup 22 /macron put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 42 /asterisk put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 63 /question put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 91 /bracketleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueID 5000793 def 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 55 /seven put dup 56 /eight put dup 58 /colon put dup 59 /semicolon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 88 /X put dup 89 /Y put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueID 5000828 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3ffd20326947b1065649b533eb5e9e3 a88a87fe987918a687950b26f647d6e08bf14d983ed1d4a8ca1e2da8d985f944 2eb916f5b6ed0697ac7c33e1e36a3bf460d34ce45f1631871097cb04f18e3889 4cf4ac1538eb19481311d24fe3be7beaa4a3730e8b4831fe59d6d9ce2e46116b 629c7ba2f9ce3ecba2f43bc162a5a077ca1b2882a42afdcec3f4b75b5d63e0bc 8e5dc95257766d8ea467ad9cbaf47be60f797580cced6884b3a68f70c91f4fdf 80fd00ed9139e7f480dc3a76af72ad9b434187730bdfaefe4cbfe5c7edcaaf24 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/Fe 173[ 42 82[{ } 1 58.1154 /MSBM7 rf /Ff 207[ 18 47[ 45{ } 2 41.511 /CMSY5 rf /Fg 198[ 28 28 1[ 28 28 28 28 28 28 28 4[ 43 1[ 22 22 36[ 38 3[{ } 13 41.511 /CMR5 rf /Fh 149[ 23 1[ 43 31[ 60 6[ 60 65[{ } 4 83.022 /EUFM10 rf /Fi 165[ 55 7[ 60 14[ 60 3[ 65 63[{ } 4 83.022 /MSBM10 rf /Fj 137[ 30 3[ 28 2[ 29 37 51 20 1[ 24 22 2[ 28 28 1[ 27 12[ 34 2[ 36 4[ 45 18[ 19 35[ 30 4[ 28 5[ 33 38 11[{ } 19 41.511 /CMMI5 rf /Fk 141[ 83 83 83 6[ 39 39 3[ 120 83 7[ 92 46 1[ 120 18[ 51 51 5[ 74 1[ 74 1[ 74 1[ 74 12[ 67 67 67 67 6[ 66 66 2[ 62 62 62 62 4[ 44 44 61 61 50 50 2[ 46 28 39 39 6[ 35 35 38 38{ } 40 83.022 /CMEX10 rf /Fl 129[ 33 10[ 26 3[ 33 11[ 30 37[ 51 3[ 33 1[ 33 33 33 33 33 33 33 33 4[ 51 1[ 26 26 17[ 33 11[ 47 6[ 45 1[ 55 1[{ } 21 58.1154 /CMR7 rf /Fm 134[ 50 1[ 69 1[ 53 37 38 39 1[ 53 48 53 80 27 50 1[ 27 53 48 1[ 44 53 42 1[ 46 12[ 66 1[ 72 1[ 65 3[ 57 7[ 73 69 68 72 7[ 48 48 48 48 48 48 48 48 48 48 1[ 27 33[ 53 12[{ } 39 83.022 /CMBX10 rf /Fn 135[ 38 1[ 33 1[ 25 31 31 1[ 34 33 41 59 21 35 27 23 39 1[ 32 31 35 30 29 4[ 27 6[ 39 1[ 39 41 1[ 52 42 1[ 53 1[ 46 2[ 29 54 2[ 49 1[ 48 1[ 50 3[ 34 1[ 20 25[ 42 44 41 2[ 31 1[ 34 39 30 33 1[ 39 2[ 32 34 1[ 28 1[ 35 38 43 11[{ } 49 58.1154 /CMMI7 rf /Fo 148[ 34 34[ 57 1[ 48 6[ 52 8[ 0 3[ 45 66 19 14[ 66 11[ 52 12[ 52 4[ 34 2[ 52{ } 13 58.1154 /CMSY7 rf /Fp 252[ 35 3[{ } 1 66.4176 /CMSY8 rf /Fq 179[ 66 8[ 70 67[{ } 2 99.6264 /CMMI12 rf /Fr 133[ 40 47 1[ 65 45 52 32 40 41 1[ 50 50 55 80 25 2[ 30 50 45 30 45 50 45 45 50 12[ 70 55 2[ 66 1[ 72 87 7[ 66 23[ 35 45[{ } 29 99.6264 /CMTI12 rf /Fs 133[ 50 59 1[ 81 59 62 44 44 46 59 62 56 62 93 31 2[ 31 62 1[ 34 51 1[ 50 62 54 13[ 62 84 1[ 77 1[ 88 4[ 42 3[ 74 1[ 81 9[ 56 1[ 56 2[ 56 56 56 56 2[ 31 37 45[{ } 36 99.6264 /CMBX12 rf /Ft 143[ 69 4[ 42 23 32 32 42 42 9[ 55 55 5[ 52 1[ 50 70 1[ 58 66 1[ 100 4[ 70 1[ 60 13[ 46 1[ 0 3[ 55 83 7[ 83 83 6[ 83 6[ 65 4[ 65 65 2[ 65 2[ 42 3[ 65 1[ 65 1[ 65 2[ 42 65 23 65{ } 36 83.022 /CMSY10 rf /Fu 133[ 60 71 2[ 71 75 52 53 55 71 75 67 75 112 37 2[ 37 75 67 41 61 75 60 75 65 10[ 102 1[ 94 75 100 101 92 101 105 128 81 2[ 50 1[ 106 1[ 88 1[ 97 96 102 7[ 67 67 67 67 67 67 67 67 67 37[ 78 11[{ } 48 119.552 /CMBX12 rf /Fv 134[ 39 3[ 39 39 39 2[ 39 39 39 39 39 39 39 39 39 39 1[ 39 1[ 39 1[ 39 32[ 39 5[ 39 11[ 39 39 45[{ } 21 74.7198 /CMTT9 rf /Fw 133[ 39 1[ 47 59 40 48 30 39 1[ 37 42 40 50 73 25 43 34 29 48 40 41 39 43 36 36 44 6[ 57 48 69 78 48 57 49 51 63 66 53 1[ 67 1[ 57 71 46 36 69 1[ 53 61 69 59 63 62 2[ 65 42 65 23 23 18[ 54 3[ 49 39 52 54 52 49 1[ 36 47 43 47 36 41 50 48 2[ 39 41 1[ 34 37 43 47 53 11[{ } 74 83.022 /CMMI10 rf /Fx 128[ 42 42 2[ 42 37 44 44 60 44 46 32 33 33 44 46 42 46 69 23 44 25 23 46 42 25 37 46 37 46 42 1[ 23 42 23 1[ 23 1[ 62 62 85 62 62 60 46 61 65 57 65 62 76 52 65 43 30 62 65 54 57 63 60 59 62 1[ 39 1[ 65 1[ 23 23 42 42 42 42 42 42 42 42 42 42 42 23 28 23 65 42 32 32 23 4[ 42 11[ 42 1[ 42 42 4[ 69 1[ 46 48 60 65 60 1[ 60 2[ 58 1[ 69 52{ } 95 83.022 /CMR10 rf /Fy 133[ 34 40 39 55 38 45 28 34 35 38 42 42 47 68 21 38 25 25 42 38 25 38 42 38 38 42 7[ 62 62 83 1[ 62 59 47 61 64 56 64 62 74 52 64 44 32 62 64 54 56 63 59 58 62 5[ 25 25 1[ 42 42 2[ 42 42 42 42 42 1[ 25 30 25 2[ 34 34 25 24[ 73 1[ 47 51 11[{ } 68 83.022 /CMTI10 rf /Fz 139[ 46 46 3[ 59 3[ 62 1[ 33 65 59 5[ 59 7[ 88 9[ 91 79[{ } 10 119.552 /CMR12 rf /FA 134[ 123 3[ 129 90 92 95 123 1[ 116 129 194 65 2[ 65 129 1[ 71 106 129 1[ 129 113 12[ 162 129 4[ 182 8[ 153 23[ 77 45[{ } 22 206.559 /CMBX12 rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: a4 %%EndSetup %%Page: 1 1 1 0 bop FA 1147 583 a(The) p 1621 583 a(Stabilit) p 2361 583 a(y) p 2560 583 a(of) 889 1096 y(the) p 1292 1096 a(Non-Equilibrium) 1253 1487 y(Steady) p 2020 1487 a(States) p Fz 1573 2217 a(Y) p 1651 2217 a(oshik) m(o) p 2009 2217 a(Ogata) p Fy 1015 2416 a(Dep) l(artment) p 1469 2416 a(of) p 1567 2416 a(Physics,) p 1893 2416 a(Gr) l(aduate) p 2255 2416 a(Scho) l(ol) p 2514 2416 a(of) p 2611 2416 a(Scienc) l(e) 1504 2516 y(The) p 1674 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a(=) p Ft 892 4128 a(f) p Fw(!) p Ft 989 4128 a(g) p Fx(.) p 1099 4128 a(T) p 1152 4128 a(o) p 1225 4128 a(in) n(v) n (estigate) p 1642 4128 a(the) p 1788 4128 a(stabilit) n(y) p 2115 4128 a(of) p Fw 2213 4128 a(!) p Fx 2268 4128 a(,) p 2322 4128 a(w) n(e) p 2448 4128 a(prepare) p 2752 4128 a(an) p 2870 4128 a(external) p 3195 4128 a(\014nite) 515 4227 y(system) p 792 4227 a(describ) r(ed) p 1162 4227 a(in) p 1261 4227 a(a) p 1332 4227 a(\014nite) p 1547 4227 a(dimensional) p 2006 4227 a(Hilb) r(ert) p 2294 4227 a(space) p Fi 2518 4227 a(C) p Fn 2578 4197 a(d) p Fx 2617 4227 a(.) p 2684 4227 a(W) p 2762 4227 a(e) p 2829 4227 a(connect) p 3135 4227 a(it) p 3221 4227 a(to) p Fw 3324 4227 a(!) p Fx 515 4327 a(through) p 837 4327 a(a) p 914 4327 a(b) r(ounded) p 1261 4327 a(in) n(teraction) p Fw 1686 4327 a(V) p Fx 1789 4327 a(and) p 1958 4327 a(in) n(v) n (estigate) p 2380 4327 a(if) p 2464 4327 a(\006) p Fn 2524 4339 a(\034) p Fj 2555 4347 a(V) p Fx 2609 4327 a(\() p Fw(!) p Fx 2696 4327 a(\)) p 2763 4327 a(has) p 2919 4327 a(an) p Fw 3043 4327 a(!) p Fx 3098 4327 a(-normal) 515 4427 y(state) p 729 4427 a(or) p 842 4427 a(not.) p 1055 4427 a(If) p 1150 4427 a(there) p 1373 4427 a(is) p 1468 4427 a(no) p Fw 1594 4427 a(!) p Fx 1649 4427 a(-normal) p 1969 4427 a(state) p 2184 4427 a(in) p 2292 4427 a(\006) p Fn 2352 4439 a(\034) p Fj 2383 4447 a(V) p Fx 2437 4427 a(\() p Fw(!) p Fx 2524 4427 a(\),) p Fw 2621 4427 a(!) p Fx 2714 4427 a(is) p 2809 4427 a(macroscopically) 515 4526 y(unstable.) p Fs 515 4753 a(2.2) p Fr 770 4753 a(The) p 969 4753 a(mo) p 1094 4753 a(del) p Fx 515 4907 a(In) p 614 4907 a(this) p 772 4907 a(pap) r(er) p 1001 4907 a(w) n(e) p 1119 4907 a(consider) p 1438 4907 a(the) p 1577 4907 a(free) p 1732 4907 a(lattice) p 1981 4907 a(F) p 2028 4907 a(ermion) p 2301 4907 a(system) p 2572 4907 a(in) p 2665 4907 a(one) p 2813 4907 a(dimension.) p 3236 4907 a(The) 515 5006 y(explicit) p 808 5006 a(form) p 1004 5006 a(of) p 1098 5006 a(the) p 1241 5006 a(NESS) p 1480 5006 a(is) p 1564 5006 a(kno) n(wn) p 1826 5006 a(for) p 1954 5006 a(this) p 2115 5006 a(system) p 2391 5006 a([19) o(].) 1926 5255 y(3) p eop %%Page: 4 4 4 3 bop Fx 639 523 a(Let) p Fh 791 523 a(h) p Ft 863 523 a(\021) p Fw 955 523 a(l) p Fl 982 493 a(2) p Fx 1019 523 a(\() p Fi(Z) p Fx(\)) p 1170 523 a(b) r(e) p 1286 523 a(a) p 1358 523 a(Hilb) r(ert) p 1648 523 a(space) p 1873 523 a(of) p 1970 523 a(a) p 2043 523 a(single) p 2277 523 a(F) p 2324 523 a(ermion.) p 2643 523 a(By) p 2776 523 a(F) p 2823 523 a(ourier) p 3067 523 a(transfor-) 515 623 y(mation,) p 822 623 a(it) p 908 623 a(is) p 994 623 a(unitary) p 1290 623 a(equiv) p 1479 623 a(alen) n(t) p 1687 623 a(to) p Fw 1791 623 a(L) p Fl 1848 593 a(2) p Fx 1885 623 a(\([) p Ft(\000) p Fw(\031) s(;) p 2092 623 a(\031) p Fx 2142 623 a(\)\).) p 2275 623 a(The) p 2448 623 a(Hamiltonian) p Fw 2928 623 a(h) p Fx 3007 623 a(of) p 3104 623 a(a) p 3176 623 a(single) 515 722 y(F) p 562 722 a(ermion) p 839 722 a(is) p 922 722 a(giv) n(en) p 1139 722 a(b) n(y) 1123 939 y(\() p Fw(hf) p Fx 1253 939 a(\)) p 1299 939 a(\() p Fw(n) p Fx(\)) p 1436 939 a(=) 1534 883 y(1) p 1534 920 42 4 v 1534 996 a(2) 1599 939 y(\() p Fw 1632 939 a(f) p Fx 1695 939 a(\() p Fw(n) p Ft 1796 939 a(\000) p Fx 1879 939 a(1\)) p 1971 939 a(+) p Fw 2054 939 a(f) p Fx 2117 939 a(\() p Fw 2150 939 a(n) p Fx 2218 939 a(+) p 2301 939 a(1\)\)) p Ft 2425 939 a(\000) p Fw 2508 939 a(\015) p 2556 939 a(f) p Fx 2619 939 a(\() p Fw 2652 939 a(n) p Fx(\)) p Fw 2748 939 a(;) p Fx 3287 939 a(\(1\)) 515 1145 y(whic) n(h) p 752 1145 a(is) p 836 1145 a(describ) r(ed) p 1203 1145 a(in) p 1300 1145 a(F) p 1347 1145 a(ourier) p 1588 1145 a(represen) n(tation) p 2130 1145 a(as) p Fk 1579 1315 a(c) p Fw 1572 1337 a(hf) p Fx 1683 1337 a(\() p Fw 1716 1337 a(k) p Fx 1762 1337 a(\)) p 1817 1337 a(=) p Fw 1904 1337 a(!) p Fx 1973 1337 a(\() p Fw(k) p Fx 2051 1337 a(\)) 2115 1315 y(^) p Fw 2097 1337 a(f) p Fx 2161 1337 a(\() p Fw(k) p Fx 2239 1337 a(\)) p Fw 2285 1337 a(;) 1615 1462 y(!) p Fx 1683 1462 a(\() p Fw 1716 1462 a(k) p Fx 1762 1462 a(\)) p 1817 1462 a(=) p 1904 1462 a(cos) p 2029 1462 a(\() p Fw 2062 1462 a(k) p Fx 2108 1462 a(\)) p Ft 2158 1462 a(\000) p Fw 2241 1462 a(\015) p 2289 1462 a(:) p Fx 515 1644 a(The) p Fw 690 1644 a(\015) p Fx 738 1644 a(-term) p 969 1644 a(represen) n(ts) p 1365 1644 a(the) p 1513 1644 a(in) n(teraction) p 1936 1644 a(with) p 2130 1644 a(the) p 2278 1644 a(external) p 2603 1644 a(\014eld.) p 2830 1644 a(The) p 3006 1644 a(free) p 3170 1644 a(F) p 3217 1644 a(ermi) 515 1744 y(gas) p 658 1744 a(is) p 742 1744 a(describ) r(ed) p 1109 1744 a(as) p 1211 1744 a(the) p 1354 1744 a(CAR-algebra) p Ft 1856 1744 a(O) p Fn 1922 1756 a(f) p Fx 1993 1744 a(o) n(v) n(er) p Fh 2170 1744 a(h) p Fx(.) p 2274 1744 a(And) p 2456 1744 a(its) p 2572 1744 a(dynamics) p 2939 1744 a(is) p 3022 1744 a(giv) n(en) p 3239 1744 a(b) n(y) p Fw 1544 1939 a(\013) p Fn 1597 1899 a(f) 1597 1960 y(t) p Fx 1654 1939 a(\() p Fw(a) p Fx 1744 1939 a(\() p Fw(f) p Fx 1826 1939 a(\)\)) p 1914 1939 a(=) p Fw 2001 1939 a(a) p Fk 2059 1872 a(\000) p Fx 2097 1939 a(e) p Fn 2134 1905 a(ith) p Fw 2225 1939 a(f) p Fk 2275 1872 a(\001) p Fw 2327 1939 a(:) p Fx 3273 1939 a(\(2\)) 639 2122 y(In) p 736 2122 a(the) p 873 2122 a(initial) p 1106 2122 a(state) p Fw 1303 2122 a(!) p Fl 1355 2134 a(0) p Fx 1392 2122 a(,) p 1437 2122 a(the) p 1574 2122 a(lattice) p 1821 2122 a(is) p 1898 2122 a(separated) p 2265 2122 a(in) n(to) p 2427 2122 a(the) p 2564 2122 a(left) p 2702 2122 a(and) p 2857 2122 a(the) p 2994 2122 a(righ) n(t.) p 3224 2122 a(And) 515 2221 y(they) p 704 2221 a(are) p 844 2221 a(k) n(ept) p 1031 2221 a(at) p 1134 2221 a(di\013eren) n(t) p 1464 2221 a(in) n(v) n(erse) p 1741 2221 a(temp) r(erature) p Fw 2216 2221 a(\014) p Fo 2263 2233 a(\000) p Fw 2319 2221 a(;) p 2356 2221 a(\014) p Fl 2403 2233 a(+) p Fx 2458 2221 a(,) p 2511 2221 a(resp) r(ectiv) n(ely) p 2930 2221 a(.) p 2995 2221 a(The) p 3168 2221 a(NESS) p Fw 515 2321 a(!) p Fn 567 2333 a(\032) p Fx 638 2321 a(asso) r(ciated) p 1039 2321 a(with) p Fw 1233 2321 a(!) p Fl 1285 2333 a(0) p Fx 1355 2321 a(is) p 1444 2321 a(realized) p 1754 2321 a(as) p 1861 2321 a(the) p 2009 2321 a(asymptotic) p 2442 2321 a(state) p 2650 2321 a(under) p 2891 2321 a(the) p 3040 2321 a(dynamics) 515 2421 y(\(2\).) p 680 2421 a(The) p 848 2421 a(explicit) p 1138 2421 a(form) p 1332 2421 a(of) p Fw 1424 2421 a(!) p Fn 1476 2433 a(\032) p Fx 1540 2421 a(w) n(as) p 1697 2421 a(obtained) p 2035 2421 a(b) n(y) p 2148 2421 a(Ho) p 2277 2421 a(and) p 2436 2421 a(Araki:) p Fw 2698 2421 a(!) p Fn 2750 2433 a(\032) p Fx 2813 2421 a(is) p 2894 2421 a(a) p 2961 2421 a(state) p 3162 2421 a(whose) p Fw 515 2520 a(n) p Fx(-p) r(oin) n(t) p 809 2520 a(functions) p 1167 2520 a(ha) n(v) n(e) p 1358 2520 a(a) p 1428 2520 a(structure) p Fw 933 2703 a(!) p Fn 985 2715 a(\032) p Fx 1038 2703 a(\() p Fw(a) p Fx(\() p Fw(f) p Fn 1187 2715 a(n) p Fx 1232 2703 a(\)) p Fo 1264 2669 a(\003) p Ft 1316 2703 a(\001) p 1353 2703 a(\001) p 1390 2703 a(\001) p Fw 1427 2703 a(a) p Fx(\() p Fw(f) p Fl 1544 2715 a(1) p Fx 1581 2703 a(\)) p Fo 1613 2669 a(\003) p Fw 1651 2703 a(a) p Fx(\() p Fw(g) p Fl 1767 2715 a(1) p Fx 1804 2703 a(\)) p Ft 1850 2703 a(\001) p 1887 2703 a(\001) p 1924 2703 a(\001) p Fw 1961 2703 a(a) p Fx(\() p Fw(g) p Fn 2077 2715 a(m) p Fx 2140 2703 a(\)\)) p 2228 2703 a(=) p Fw 2315 2703 a(\016) p Fn 2352 2715 a(nm) p Fx 2470 2703 a(det\() p Ft(h) p Fw(f) p Fn 2690 2715 a(i) p Fw 2718 2703 a(;) p 2755 2703 a(\032g) p Fn 2838 2715 a(j) p Ft 2873 2703 a(i) p Fx(\)) p Fw(;) p Fx 3287 2703 a(\(3\)) 515 2886 y(where) p Fw 755 2886 a(\032) p Fx 825 2886 a(is) p 909 2886 a(represen) n(ted) p 1350 2886 a(as) p 1452 2886 a(a) p 1521 2886 a(m) n(ultiplication) p 2054 2886 a(op) r(erator,) p Fw 1180 3197 a(\032) p Fx(\() p Fw(k) p Fx 1301 3197 a(\)) p 1356 3197 a(=) p Fk 1444 3002 a(8) 1444 3076 y(>) 1444 3101 y(<) 1444 3251 y(>) 1444 3276 y(:) 1565 3013 y(\020) p Fx 1614 3105 a(1) p 1674 3105 a(+) p Fw 1757 3105 a(e) p Fo 1796 3071 a(\000) p Fn(\014) p Fg 1886 3079 a(+) p Fn 1932 3071 a(\017) p Fl(\() p Fn(k) p Fl 2022 3071 a(\)) p Fk 2053 3013 a(\021) p Fo 2102 3031 a(\000) p Fl(1) p Fw 2288 3105 a(k) p Ft 2357 3105 a(2) p Fx 2436 3105 a([0) p Fw(;) p 2538 3105 a(\031) p Fx 2588 3105 a(\)) p Fk 1531 3213 a(\020) p Fx 1581 3305 a(1) p 1641 3305 a(+) p Fw 1724 3305 a(e) p Fo 1763 3271 a(\000) p Fn(\014) p Ff 1853 3279 a(\000) p Fn 1901 3271 a(\017) p Fl(\() p Fn(k) p Fl 1991 3271 a(\)) p Fk 2022 3213 a(\021) p Fo 2071 3230 a(\000) p Fl(1) p Fw 2257 3305 a(k) p Ft 2326 3305 a(2) p Fx 2405 3305 a([) p Ft(\000) p Fw(\031) s(;) p Fx 2580 3305 a(0\)) p Fw 2653 3197 a(;) p Fx 3273 3197 a(\(4\)) p Fw 1189 3462 a(\017) p Fx(\() p Fw(k) p Fx 1301 3462 a(\)) p 1356 3462 a(=) p 1444 3462 a(cos) p 1569 3462 a(\() p Fw(k) p Fx 1647 3462 a(\)) p Ft 1698 3462 a(\000) p Fw 1781 3462 a(\015) p 1829 3462 a(;) p Fx 515 3645 a(in) p 610 3645 a(the) p 751 3645 a(F) p 798 3645 a(ourier) 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Fn 2336 3955 a(S) p Fx 2422 3943 a(=) p Fi 2524 3943 a(C) p Fn 2584 3913 a(d) p Fx 2623 3943 a(.) p 2710 3943 a(W) p 2788 3943 a(e) p 2861 3943 a(denote) p 3138 3943 a(b) n(y) p Fw 3262 3943 a(H) p Fn 3331 3955 a(S) p Fx 515 4043 a(the) p 658 4043 a(free) p 817 4043 a(Hamiltonian) p 1294 4043 a(of) p 1389 4043 a(the) p 1532 4043 a(system.) p 1840 4043 a(The) p 2010 4043 a(free) p 2170 4043 a(dynamics) p Fw 2537 4043 a(\013) p Fn 2590 4013 a(S) 2590 4064 y(t) p Fx 2666 4043 a(is) p 2749 4043 a(giv) n(en) p 2966 4043 a(b) n(y) p Fw 1516 4226 a(\013) p Fn 1569 4191 a(S) 1569 4246 y(t) p Fx 1631 4226 a(\() p Fw 1664 4226 a(A) p Fx(\)) p 1781 4226 a(=) p 1869 4226 a(e) p Fn 1906 4191 a(itH) p Fj 2008 4199 a(S) p Fw 2055 4226 a(A) p Fx(e) p Fo 2154 4191 a(\000) p Fn(itH) p Fj 2308 4199 a(S) p Fw 2355 4226 a(:) p Fx 515 4408 a(The) p 685 4408 a(com) n(bined) p 1057 4408 a(system) p Ft 1332 4408 a(S) p Fx 1407 4408 a(+) p Ft 1490 4408 a(R) p Fx 1588 4408 a(is) p 1672 4408 a(describ) r(ed) p 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1926 5255 y(4) p eop %%Page: 5 5 5 4 bop Fx 639 523 a(Let) p 784 523 a(us) p 887 523 a(consider) p 1207 523 a(the) p 1346 523 a(dynamics) p 1710 523 a(including) p 2066 523 a(the) p 2205 523 a(in) n(teraction) p 2619 523 a(b) r(et) n(w) n(een) p Ft 2936 523 a(S) p Fx 3016 523 a(and) p Ft 3174 523 a(R) p Fx(.) p 3303 523 a(In) 515 623 y(this) p 677 623 a(pap) r(er,) p 933 623 a(w) n(e) p 1055 623 a(de\014ne) p 1295 623 a(the) p 1438 623 a(in) n(teraction) p 1856 623 a(term) p Fw 2054 623 a(V) p Fx 2149 623 a(b) n(y) p Fw 1427 805 a(V) p Fx 1517 805 a(=) p Fw 1605 805 a(\025) p Ft 1672 805 a(\001) p Fw 1713 805 a(Y) p Ft 1798 805 a(\012) p Fx 1881 805 a(\() p Fw 1914 805 a(a) p Fx 1972 805 a(\() p Fw(f) p Fx 2054 805 a(\)) p 2104 805 a(+) p Fw 2187 805 a(a) p Fo 2231 771 a(\003) p Fx 2283 805 a(\() p Fw(f) p Fx 2365 805 a(\)\)) p Fw 2443 805 a(:) p Fx 515 988 a(Here) p Fw 708 988 a(\025) p Fx 781 988 a(is) p 861 988 a(a) p 927 988 a(coupling) p 1256 988 a(constan) n(t) p 1588 988 a(and) 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a(is) p 1453 1187 a(expanded) p 1825 1187 a(as) p 1927 1187 a(follo) n(ws;) p Fw 515 1428 a(\013) p Fn 568 1440 a(t) p Fx 611 1428 a(\() p Fw(A) p Fx(\)) p Ft 761 1428 a(\021) p Fw 848 1428 a(\013) p Fl 901 1393 a(0) p Fn 901 1448 a(t) p Fx 953 1428 a(\() p Fw(A) p Fx(\)) p 1098 1428 a(+) p Fk 1184 1349 a(X) p Fn 1181 1527 a(n) p Fo(\025) p Fl(1) p Fw 1321 1428 a(i) p Fn 1350 1393 a(n) p Fk 1409 1315 a(Z) p Fn 1492 1335 a(t) p Fl 1455 1503 a(0) p Fw 1535 1428 a(dt) p Fl 1608 1440 a(1) p Fk 1659 1315 a(Z) p Fn 1742 1335 a(t) p Fg 1767 1343 a(1) p Fl 1705 1503 a(0) p Fw 1818 1428 a(dt) p Fl 1891 1440 a(2) p Ft 1942 1428 a(\001) p 1979 1428 a(\001) p 2016 1428 a(\001) p Fk 2053 1315 a(Z) p Fn 2136 1335 a(t) p Fj 2161 1343 a(n) p Ff(\000) p Fg(1) p Fl 2099 1503 a(0) p Fw 2293 1428 a(dt) p Fn 2366 1440 a(n) p Fk 2425 1360 a(\002) p Fw 2459 1428 a(\013) p Fl 2512 1393 a(0) p Fn 2512 1448 a(t) p Fj 2537 1456 a(n) p Fx 2596 1428 a(\() p Fw 2629 1428 a(V) p Fx 2695 1428 a(\)) p Fw 2742 1428 a(;) p Fk 2779 1360 a(\002) p Ft 2813 1428 a(\001) p 2850 1428 a(\001) p 2887 1428 a(\001) p Fw 2938 1428 a(;) p Fk 2975 1360 a(\002) p Fw 3009 1428 a(\013) p Fl 3062 1393 a(0) p Fn 3062 1448 a(t) p Fg 3087 1456 a(1) p Fx 3138 1428 a(\() p Fw(V) p Fx 3237 1428 a(\)) p Fw 3283 1428 a(;) p 3320 1428 a(\013) p Fl 3373 1393 a(0) p Fn 3373 1448 a(t) p Fx 3424 1428 a(\() p Fw 3457 1428 a(A) p Fx(\)) p Fk 3551 1360 a(\003\003) o(\003) p Fw 3669 1428 a(:) p Fx 3273 1624 a(\(5\)) 515 1803 y(The) p 682 1803 a(righ) n(t) p 880 1803 a(hand) p 1085 1803 a(side) p 1249 1803 a(con) n(v) n(erges) p 1618 1803 a(in) p 1711 1803 a(norm) p 1926 1803 a(top) r(ology) p 2264 1803 a(in) p Ft 2358 1803 a(O) p Fx 2426 1803 a(.) p Fw 2485 1803 a(\013) p Fn 2538 1815 a(t) p Fx 2592 1803 a(is) p 2673 1803 a(strongly) p 2991 1803 a(con) n(tin) n(uous) 515 1902 y(one) p 667 1902 a(parameter) p 1064 1902 a(group) p 1300 1902 a(of) p 1394 1902 a(automorphisms.) p Fs 515 2135 a(2.3) p Fr 770 2135 a(Positive) p 1142 2135 a(c) p 1182 2135 a(ommutator) p 1684 2135 a(metho) p 1936 2135 a(d) p Fx 515 2288 a(By) p 652 2288 a(the) p 802 2288 a(standard) p 1155 2288 a(theory) p 1423 2288 a(\(see) p 1597 2288 a(Section) p 1894 2288 a(3\),) p 2027 2288 a(the) p 2177 2288 a(problem) p 2507 2288 a(of) p 2609 2288 a(the) p 2759 2288 a(existence) p 3119 2288 a(of) p 3221 2288 a(a) p Fw 3297 2288 a(!) p Fx 3352 2288 a(-) 515 2388 y(normal) p 802 2388 a(state) p 1010 2388 a(in) p 1112 2388 a(\006) p Fn 1172 2400 a(\034) p Fx 1214 2388 a(\() p Fw(!) p Fx 1301 2388 a(\)) p 1366 2388 a(is) p 1454 2388 a(translated) p 1852 2388 a(to) p 1959 2388 a(the) p 2107 2388 a(problem) p 2435 2388 a(of) p 2535 2388 a(the) p 2683 2388 a(non) n(trivialit) n(y) p 3164 2388 a(of) p 3264 2388 a(the) 515 2487 y(Liouville) p 856 2487 a(op) r(erator's) p 1248 2487 a(k) n(ernel.) p 1530 2487 a(T) p 1583 2487 a(o) p 1653 2487 a(in) n(v) n(estigate) p 2068 2487 a(the) p 2213 2487 a(k) n(ernel,) p 2482 2487 a(w) n(e) p 2606 2487 a(use) p 2751 2487 a(the) p 2895 2487 a(p) r(ositiv) n(e) p 3204 2487 a(com-) 515 2587 y(m) n(utator) p 842 2587 a(metho) r(d.) p 1195 2587 a(In) p 1305 2587 a(order) p 1528 2587 a(to) p 1636 2587 a(construct) p 2007 2587 a(the) p 2156 2587 a(p) r(ositiv) n(e) p 2470 2587 a(comm) n(utator) p 2944 2587 a(for) p 3078 2587 a(the) p 3227 2587 a(\014eld) 515 2687 y(with) p 702 2687 a(disp) r(ersion) p 1090 2687 a(relation) p Fw 1393 2687 a(!) p Fx 1448 2687 a(\() p Fw(k) p Fx 1526 2687 a(\)) p 1581 2687 a(=) p 1668 2687 a(cos) p Fw 1793 2687 a(k) p Ft 1853 2687 a(\000) p Fw 1932 2687 a(\015) p Fx 1980 2687 a(,) p 2028 2687 a(w) n(e) p 2148 2687 a(carry) p 2361 2687 a(out) p 2506 2687 a(the) p 2647 2687 a(v) p 2686 2687 a(ariable) p 2956 2687 a(transforma-) 515 2786 y(tion) p 684 2786 a(from) p Fw 880 2786 a(k) p Ft 949 2786 a(2) p Fx 1027 2786 a([) p Ft(\000) p Fw(\031) s(;) p 1202 2786 a(\031) p Fx 1252 2786 a(\)) p 1311 2786 a(to) p Fw 1412 2786 a(t) p Ft 1465 2786 a(2) p Fi 1543 2786 a(R) p Fx(,) p 1653 2786 a(b) n(y) p Fw 1768 2786 a(t) p Fx 1821 2786 a(=) p 1908 2786 a(tan) p Fn 2052 2754 a(k) p 2052 2768 37 4 v Fl 2054 2815 a(2) p Fx 2099 2786 a(.) p 2159 2786 a(Under) p 2409 2786 a(this) p 2570 2786 a(ariable) p 2842 2786 a(translation,) p 3284 2786 a(w) n(e) 515 2886 y(iden) n(tify) p Fh 822 2886 a(h) p Fx 896 2886 a(=) p Fw 992 2886 a(L) p Fl 1049 2856 a(2) p Fx 1085 2886 a(\([) p Ft(\000) p Fw(\031) s(;) p 1292 2886 a(\031) p Fx 1342 2886 a(\)) p Fw(;) p 1411 2886 a(dk) p Fx 1500 2886 a(\)) p 1566 2886 a(with) p Fw 1760 2886 a(L) p Fl 1817 2856 a(2) p Fx 1853 2886 a(\() p Fi(R) p Fw(;) p Fl 2047 2853 a(2) p 1992 2867 142 4 v Fn 1992 2915 a(t) p Fg 2017 2898 a(2) p Fl 2050 2915 a(+1) p Fw 2144 2886 a(dt) p Fx(\).) p 2324 2886 a(Then) p 2545 2886 a(w) n(e) p 2672 2886 a(de\014ne) p 2917 2886 a(the) p 3065 2886 a(rescaling) 515 2986 y(group) p 757 2986 a(with) p 953 2986 a(resp) r(ect) p 1244 2986 a(to) p Fw 1353 2986 a(t) p Fx 1417 2986 a(\(Section) p 1747 2986 a(5\).) p 1901 2986 a(The) p 2079 2986 a(p) r(ositiv) n(e) p 2393 2986 a(comm) n(utator) p 2868 2986 a(is) p 2959 2986 a(constructed) 515 3085 y(with) p 707 3085 a(the) p 853 3085 a(aid) p 994 3085 a(of) p 1091 3085 a(the) p 1237 3085 a(generator) p 1609 3085 a(of) p 1707 3085 a(the) p 1852 3085 a(unitary) p 2148 3085 a(represen) n(tation) p 2694 3085 a(of) p 2792 3085 a(this) p 2956 3085 a(group.) p 3233 3085 a(Due) 515 3185 y(to) p 616 3185 a(the) p 758 3185 a(prop) r(ert) n(y) p 1097 3185 a(of) p 1190 3185 a(the) p 1333 3185 a(disp) r(ersion) p 1723 3185 a(relation) p Fw 2027 3185 a(!) p Fx 2082 3185 a(\() p Fw(k) p Fx 2160 3185 a(\)) p 2215 3185 a(=) p 2303 3185 a(cos) p Fw 2428 3185 a(k) p Ft 2490 3185 a(\000) p Fw 2572 3185 a(\015) p Fx 2620 3185 a(,) p 2670 3185 a(w) n(e) p 2791 3185 a(need) p 2984 3185 a(a) p 3053 3185 a(cut) p 3195 3185 a(o\013) p 3312 3185 a(of) 515 3284 y(the) p 658 3284 a(in) n(teraction) p 1075 3284 a(term.) p 1306 3284 a(\(See) p 1486 3284 a(Section) p 1777 3284 a(6.\)) p Fs 515 3517 a(2.4) p Fr 770 3517 a(Main) p 1027 3517 a(The) p 1187 3517 a(or) p 1273 3517 a(em) p Fx 515 3670 a(F) p 562 3670 a(or) p 676 3670 a(the) p 831 3670 a(analysis,) p 1181 3670 a(w) n(e) p 1316 3670 a(need) p 1522 3670 a(sev) n(eral) p 1807 3670 a(assumptions) p 2291 3670 a(on) p 2418 3670 a(the) p 2573 3670 a(small) p 2803 3670 a(system) p 3090 3670 a(and) p 3264 3670 a(the) 515 3770 y(in) n(teraction) p Fw 932 3770 a(V) p Fx 999 3770 a(.) p Fm 515 3936 a(Assumption) p 1044 3936 a(2.1) p Fy 1208 3936 a(Under) p 1453 3936 a(the) p 1582 3936 a(identi\014c) l(ation) p Fh 2075 3936 a(h) p Fx 2141 3936 a(=) p Fw 2228 3936 a(L) p Fl 2285 3906 a(2) p Fx 2322 3936 a(\([) p Ft(\000) p Fw(\031) s(;) p 2529 3936 a(\031) p Fx 2579 3936 a(\)) p Fw(;) p 2648 3936 a(dk) p Fx 2737 3936 a(\)) p 2793 3936 a(=) p Fw 2881 3936 a(L) p Fl 2938 3906 a(2) p Fx 2975 3936 a(\() p Fi(R) p Fw(;) p Fl 3168 3903 a(2) p 3114 3917 V Fn 3114 3964 a(t) p Fg 3139 3948 a(2) p Fl 3171 3964 a(+1) p Fw 3265 3936 a(dt) p Fx(\)) p Fy(,) 515 4035 y(let) p Fw 631 4035 a(u) p Fx(\() p Fw(\022) p Fx 752 4035 a(\)) p Fy 815 4035 a(b) l(e) p 916 4035 a(a) p 988 4035 a(str) l(ongly) p 1299 4035 a(c) l(ontinuous) p 1717 4035 a(one) p 1874 4035 a(p) l(ar) l(ameter) p 2265 4035 a(unitary) p 2556 4035 a(gruop) p 2789 4035 a(on) p Fh 2908 4035 a(h) p Fy 2981 4035 a(de\014ne) l(d) p 3261 4035 a(by) p Fx 598 4285 a(\() p Fw(u) p Fx(\() p Fw(\022) p Fx 751 4285 a(\)) p Fw(f) p Fx 833 4285 a(\)) p 880 4285 a(\() p Fw(t) p Fx(\)) p Ft 997 4285 a(\021) p Fx 1085 4285 a(e) p Fj 1132 4228 a(\022) p 1132 4237 30 3 v Fg 1133 4271 a(2) p Fk 1175 4154 a(r) p 1258 4154 338 4 v Fw 1322 4229 a(t) p Fl 1352 4205 a(2) p Fx 1408 4229 a(+) p 1491 4229 a(1) p 1268 4266 318 4 v 1268 4342 a(e) p Fl 1305 4318 a(2) p Fn(\022) p Fw 1376 4342 a(t) p Fl 1406 4318 a(2) p Fx 1462 4342 a(+) p 1545 4342 a(1) p Fw 1596 4285 a(f) p Fk 1659 4218 a(\000) p Fx 1697 4285 a(e) p Fn 1734 4251 a(\022) p Fw 1772 4285 a(t) p Fk 1802 4218 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p 1839 3888 a(p) p Fn 1881 3921 a(N) p Fg 1940 3891 a(\() p Fj(n) p Fg(\)) p Fj 1934 3930 a(r) p Fk 2030 3793 a(\014) 2030 3843 y(\014) 2030 3892 y(\014) p Fn 2057 3946 a(p) p Fj 2091 3982 a(N) p Fg 2141 3957 a(\() p Fj(n) p Fg(\)) p Fj 2136 3994 a(r) p Fo 2231 3946 a(H) p Fj 2288 3954 a(S) p Fk 2333 3746 a(!) p Fx 2418 3888 a(+) p 2526 3888 a(min) p Fn 2501 3942 a(m) p Fo(2) p Fn(N) p Fj 2658 3950 a(r) p Fx 2704 3888 a(inf) p Fw 2819 3888 a(\033) p Fk 2883 3746 a( ) p Fw 2959 3888 a(p) p Fn 3001 3921 a(N) p Fg 3060 3891 a(\() p Fj(m) p Fg(\)) p Fj 3054 3941 a(l) p Fw 3164 3888 a(Y) p 3230 3888 a(p) p Fn 3272 3900 a(N) p Fj 3331 3881 a(c) 3325 3921 y(l) p Fw 3366 3888 a(Y) p 3433 3888 a(p) p Fn 3475 3921 a(N) p Fg 3534 3891 a(\() p Fj(m) p Fg(\)) p Fj 3528 3941 a(l) p Fk 3638 3793 a(\014) 3638 3843 y(\014) 3638 3892 y(\014) p Fn 3665 3946 a(p) p Fj 3699 3982 a(N) p Fg 3749 3957 a(\() p Fj(m) p Fg(\)) p Fj 3744 4005 a(l) p Fo 3853 3946 a(H) p Fj 3910 3954 a(S) p Fk 3956 3746 a(!) p Fw 4045 3888 a(>) p Fx 4132 3888 a(0) p Fw(:) p Fy 613 4182 a(2.) p 722 4182 a(L) l(et) p Fw 866 4182 a(Y) p Fn 914 4194 a(mn) p Ft 1041 4182 a(\021) p 1129 4182 a(h) p Fw(') p Fn 1215 4194 a(m) p Ft 1278 4182 a(j) p Fw(Y) p Ft 1368 4182 a(j) p Fw(') p Fn 1445 4194 a(n) p Ft 1491 4182 a(i) p Fy(.) p 1586 4182 a(We) p 1737 4182 a(assume) p Ft 2028 4182 a(j) p Fw(Y) p Fn 2099 4194 a(mn) p Ft 2203 4182 a(j) p Fw 2249 4182 a(>) p Fx 2337 4182 a(0) p Fy 2408 4182 a(for) p 2541 4182 a(al) t(l) p Fw 2660 4182 a(m) p Ft 2756 4182 a(6) p Fx(=) p Fw 2843 4182 a(n) p Fx 515 4357 a(W) p 593 4357 a(e) p 658 4357 a(obtain) p 916 4357 a(the) p 1059 4357 a(follo) n(wing) p 1410 4357 a(results.) p Fm 515 4516 a(Theorem) p 920 4516 a(2.2) p Fy 1084 4516 a(The) p 1253 4516 a(NESS) p 1495 4516 a(which) p 1728 4516 a(is) p 1817 4516 a(far) p 1950 4516 a(fr) l(om) p 2146 4516 a(e) l(quilibrium) p 2576 4516 a(is) p 2665 4516 a(macr) l(osc) l(opic) l(al) t(ly) p 3258 4516 a(un-) 515 4616 y(stable.) p Fm 515 4791 a(Theorem) p 920 4791 a(2.3) p Fy 1084 4791 a(Supp) l(ose) p 1400 4791 a(that) p 1569 4791 a(Assumption) p 2026 4791 a(2.1) p 2166 4791 a(to) p 2265 4791 a(2.4) p 2405 4791 a(ar) l(e) p 2546 4791 a(satis\014e) l(d.) p 2896 4791 a(Then) p 3113 4791 a(for) p 3245 4791 a(suf-) 515 4890 y(\014ciently) p 830 4890 a(smal) t(l) p Fw 1051 4890 a(\025) p Fy(,) p 1156 4890 a(the) p 1295 4890 a(system) p 1567 4890 a(shows) p 1806 4890 a(r) l(eturn) p 2059 4890 a(to) p 2160 4890 a(e) l(quilibrium,i.e.,any) p 2885 4890 a(state) p 3085 4890 a(which) p 3320 4890 a(is) 515 4990 y(normal) p 800 4990 a(to) p 900 4990 a(the) p 1038 4990 a(e) l(quilibrium) p 1467 4990 a(state) p 1667 4990 a(c) l(onver) l(ges) p 2037 4990 a(to) p 2137 4990 a(the) p 2275 4990 a(e) l(quilibrium) p 2704 4990 a(state.) p Fx 1926 5255 a(6) p eop %%Page: 7 7 7 6 bop Fu 515 523 a(3) p 716 523 a(The) p 991 523 a(Standard) p 1573 523 a(Theory) p 2040 523 a(and) p 2300 523 a(NESS) p Fx 515 705 a(In) p 612 705 a(this) p 767 705 a(section,) p 1062 705 a(w) n(e) p 1177 705 a(recapitulate) p 1627 705 a(the) p 1764 705 a(standard) p 2104 705 a(theory) p 2358 705 a(\(see) p 2517 705 a([14) o(]\)) p 2700 705 a(and) p 2854 705 a(explain) p 3136 705 a(its) p 3245 705 a(role) 515 805 y(in) p 606 805 a(the) p 743 805 a(in) n(v) n(estigation) p 1224 805 a(of) p 1313 805 a(the) p 1450 805 a(NESS.) p 1706 805 a(Let) p Ft 1849 805 a(M) p Fx 1971 805 a(b) r(e) p 2078 805 a(a) p 2141 805 a(v) n(on) p 2292 805 a(Neumann) p 2663 805 a(algebra) p 2948 805 a(on) p 3057 805 a(a) p 3120 805 a(Hilb) r(ert) 515 904 y(space) p Fh 741 904 a(H) p Fx 833 904 a(with) p 1026 904 a(a) p 1099 904 a(cyclic) p 1332 904 a(and) p 1497 904 a(separating) p 1904 904 a(v) n(ector) p 2157 904 a(\012.) p 2290 904 a(A) p 2384 904 a(p) r(ositiv) n(e) p 2695 904 a(linear) p 2930 904 a(functional) p Fw 3324 904 a(!) p Fx 515 1004 a(on) p Ft 630 1004 a(M) p Fx 757 1004 a(is) p 841 1004 a(said) p 1012 1004 a(to) p 1113 1004 a(b) r(e) p Fy 1226 1004 a(normal) p Fx 1510 1004 a(if) p 1586 1004 a(there) p 1798 1004 a(exists) p 2028 1004 a(a) p 2097 1004 a(densit) n(y) p 2383 1004 a(matrix) p Fw 2653 1004 a(\032) p Fx 2724 1004 a(suc) n(h) p 2911 1004 a(that) p Fw 1648 1186 a(!) p Fx 1716 1186 a(\() p Ft 1749 1186 a(\001) p Fx(\)) p 1827 1186 a(=) p 1915 1186 a(T) p 1968 1186 a(r) p 2014 1186 a(\() p Fw(\032) p Fx(\() p Ft(\001) p Fx(\)\)) p Fw 2223 1186 a(:) p Fx 515 1369 a(W) p 593 1369 a(e) p 666 1369 a(denote) p 943 1369 a(b) n(y) p Ft 1067 1369 a(M) p Fo 1167 1381 a(\003) p Fl(+) p Fx 1292 1369 a(the) p 1443 1369 a(set) p 1582 1369 a(of) p 1685 1369 a(all) p 1809 1369 a(normal) p 2099 1369 a(p) r(ositiv) n(e) p 2415 1369 a(linear) p 2655 1369 a(functional.) p 3103 1369 a(W) p 3181 1369 a(e) p 3254 1369 a(can) 515 1469 y(de\014ne) p 755 1469 a(an) p 870 1469 a(op) r(erator) p Fw 1205 1469 a(S) p Fl 1256 1481 a(0) p Fx 1321 1469 a(on) p 1436 1469 a(a) p 1505 1469 a(dense) p 1732 1469 a(set) p Ft 1861 1469 a(M) p Fx(\012) p 2048 1469 a(b) n(y) p Fw 1709 1651 a(S) p Fl 1760 1663 a(0) p Fw 1798 1651 a(x) p Fx(\012) p 1928 1651 a(=) p Fw 2016 1651 a(x) p Fo 2063 1617 a(\003) p Fx 2101 1651 a(\012) p Fw(:) 515 1834 y(S) p Fl 566 1846 a(0) p Fx 631 1834 a(is) p 714 1834 a(closable) p 1024 1834 a(and) p 1185 1834 a(w) n(e) p 1307 1834 a(represen) n(t) p 1666 1834 a(the) p 1809 1834 a(p) r(olar) p 2023 1834 a(decomp) r(osition) p 2570 1834 a(of) p 2665 1834 a(the) p 2808 1834 a(closure) 3099 1813 y(\026) p Fw 3085 1834 a(S) p Fl 3136 1846 a(0) p Fx 3201 1834 a(as) 1762 1996 y(\026) p Fw 1748 2017 a(S) p Fl 1799 2029 a(0) p Fx 1859 2017 a(=) p Fw 1947 2017 a(J) p Fx 2001 2017 a(\001) p Fg 2080 1960 a(1) p 2080 1969 29 3 v 2080 2002 a(2) p Fw 2123 2017 a(:) 515 2199 y(J) p Fx 600 2199 a(is) p 687 2199 a(called) p 926 2199 a(the) p Fy 1073 2199 a(mo) l(dular) p 1398 2199 a(c) l(onjugation) p Fx 1845 2199 a(and) p 2010 2199 a(\001) p 2111 2199 a(is) p 2198 2199 a(called) p 2437 2199 a(the) p Fy 2583 2199 a(mo) l(dular) p 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Fx 921 2697 a(b) n(y) p Fw 1463 2880 a(!) p Fn 1515 2892 a(\030) p Fx 1565 2880 a(\() p Fw(x) p Fx(\)) p 1700 2880 a(=) p Ft 1787 2880 a(h) p Fw 1820 2880 a(\030) t(;) p 1897 2880 a(x\030) p Ft 1984 2880 a(i) p 2113 2880 a(8) p Fw(x) p Ft 2230 2880 a(2) p 2308 2880 a(M) p Fw(:) p Fx 515 3063 a(W) p 593 3063 a(e) p 658 3063 a(ha) n(v) n(e) p 849 3063 a(the) p 992 3063 a(follo) n(wing) p 1343 3063 a(Theorems;) p Fm 515 3229 a(Theorem) p 920 3229 a(3.1) p Fy 1084 3229 a(F) p 1132 3229 a(or) p 1246 3229 a(any) p Fw 1413 3229 a(!) p Ft 1504 3229 a(2) p 1596 3229 a(M) p Fo 1696 3241 a(\003) p Fl(+) p Fy 1785 3229 a(,) p 1849 3229 a(ther) l(e) p 2064 3229 a(exists) p 2298 3229 a(a) p 2378 3229 a(unique) p Fw 2653 3229 a(\030) p Fx 2693 3229 a(\() p Fw(!) p Fx 2780 3229 a(\)) p Ft 2849 3229 a(2) p 2941 3229 a(P) p Fy 3043 3229 a(such) p 3239 3229 a(that) p Fw 515 3328 a(!) p Fx 592 3328 a(=) p Fw 680 3328 a(!) p Fn 732 3343 a(\030) p Fl 764 3343 a(\() p Fn(!) p Fl 834 3343 a(\)) p Fy 864 3328 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a(of) p Ft 2191 3894 a(M) p Fy(,) p 2346 3894 a(ther) l(e) p 2552 3894 a(exists) p 2779 3894 a(a) p 2850 3894 a(unique) p 3117 3894 a(unitary) 515 3993 y(op) l(er) l(ator) p Fw 841 3993 a(U) p Fx 907 3993 a(\() p Fw(\013) p Fx(\)) p Fy 1055 3993 a(satisfying) p 1424 3993 a(the) p 1562 3993 a(fol) t(lowing) p 1914 3993 a(pr) l(op) l(erty:) 613 4159 y(1.) p Fw 722 4159 a(U) p Fx 802 4159 a(\() p Fw(\013) p Fx(\)) p Fw 934 4159 a(xU) p Fx 1061 4159 a(\() p Fw(\013) p Fx(\)) p Fo 1179 4118 a(\003) p Fx 1240 4159 a(=) p Fw 1328 4159 a(\013) p Fx 1395 4159 a(\() p Fw(x) p Fx(\)) p Fw 1521 4159 a(;) p 1642 4159 a(x) p Ft 1713 4159 a(2) p 1791 4159 a(M) p Fx(;) p Fy 613 4325 a(2.) p Fw 722 4325 a(U) p Fx 802 4325 a(\() p Fw(\013) p Fx(\)) p Ft 934 4325 a(P) p 1021 4325 a(\032) p 1109 4325 a(P) p Fy 1227 4325 a(and) p Fw 1388 4325 a(U) p Fx 1468 4325 a(\() p Fw(\013) p Fx(\)) p Fw 1599 4325 a(\030) p Fx 1653 4325 a(\() p Fw 1686 4325 a(!) p Fx 1741 4325 a(\)) p 1796 4325 a(=) p Fw 1883 4325 a(\030) p Fk 1937 4258 a(\000) p Fw 1975 4325 a(\013) p Fo 2028 4295 a(\000) p Fl(1) p Fo(\003) p Fx 2166 4325 a(\() p Fw(!) p Fx 2253 4325 a(\)) p Fk 2285 4258 a(\001) p Fw 2360 4325 a(!) p Ft 2438 4325 a(2) p 2516 4325 a(M) p Fo 2616 4337 a(\003) p Fl(+) p Fy 2705 4325 a(,) 722 4425 y(wher) l(e) p Fx 957 4425 a(\() p Fw(\013) p Fo 1042 4395 a(\003) p Fw 1081 4425 a(!) p Fx 1136 4425 a(\)\() p Fw(A) p Fx(\)) p Ft 1317 4425 a(\021) p Fw 1405 4425 a(!) p Fx 1460 4425 a(\() p Fw(\013) p Fx(\() p Fw(A) p Fx(\)\)) p Fy(,) 613 4591 y(3.) p Fx 722 4591 a([) p Fw(U) p Fx 825 4591 a(\() p Fw(\013) p Fx(\)) p Fw 957 4591 a(;) p 994 4591 a(J) p Fx 1048 4591 a(]) p 1094 4591 a(=) p 1181 4591 a(0) 515 4840 y(The) p 685 4840 a(follo) n(wing) p 1036 4840 a(corollary) p 1380 4840 a(can) p 1532 4840 a(b) r(e) p 1645 4840 a(easily) p 1874 4840 a(c) n(hec) n(k) n(ed.) 1926 5255 y(7) p eop %%Page: 8 8 8 7 bop Fm 515 523 a(Corollary) p 939 523 a(3.3) p Fy 1103 523 a(If) p 1190 523 a(unitary) p 1482 523 a(op) l(er) l(ator) p Fw 1809 523 a(U) p Fy 1904 523 a(satis\014es) p Fw 1076 697 a(U) p 1142 697 a(xU) p Fo 1255 663 a(\003) p Fx 1317 697 a(=) p Fw 1404 697 a(\013) p Fx 1471 697 a(\() p Fw 1504 697 a(x) p Fx(\)) p Fw 1597 697 a(;) p 1719 697 a(U) p 1785 697 a(j) p Fx 1824 697 a(\() p Fw(x) p Fx(\)) p Fw(U) p Fo 2001 663 a(\003) p Fx 2063 697 a(=) p Fw 2150 697 a(j) p Fx 2203 697 a(\() p Fw(\013) p Fx 2303 697 a(\() p Fw(x) p Fx(\)) q(\)) p Fw 2546 697 a(x) p Ft 2616 697 a(2) p 2695 697 a(M) p Fw(;) 1832 946 y(U) p Fx 1898 946 a(\012) p Ft 1981 946 a(2) p 2059 946 a(P) p Fw 2124 946 a(;) p Fy 515 1120 a(then) p Fw 698 1120 a(U) p Fy 791 1120 a(is) p 879 1120 a(the) p 1015 1120 a(unique) p 1280 1120 a(unitary) p 1570 1120 a(op) l(er) l(ator) p 1895 1120 a(which) p 2127 1120 a(satis\014es) p 2437 1120 a(the) p 2573 1120 a(c) l(onditions) p 2968 1120 a(of) p 3064 1120 a(The) l(or) l(em) 515 1220 y(3.2.) p Fx 515 1379 a(Let) p Fw 677 1379 a(\013) p Fn 730 1391 a(t) p Fx 800 1379 a(b) r(e) p 926 1379 a(a) p 1009 1379 a(one) p 1174 1379 a(parameter) p 1585 1379 a(group) p 1833 1379 a(of) p 1941 1379 a(automorphisms) p 2537 1379 a(and) p 2711 1379 a(let) p Fw 2845 1379 a(U) p Fx 2911 1379 a(\() p Fw(t) p Fx(\)) p 3046 1379 a(b) r(e) p 3172 1379 a(a) p 3254 1379 a(one) 515 1479 y(parameter) p 910 1479 a(group) p 1144 1479 a(of) p 1237 1479 a(unitaries) p 1577 1479 a(asso) r(ciated) p 1970 1479 a(with) p Fw 2158 1479 a(\013) p Fn 2211 1491 a(t) p Fx 2240 1479 a(.) p 2299 1479 a(If) p Fw 2381 1479 a(U) p Fx 2447 1479 a(\() p Fw(t) p Fx(\)) p 2567 1479 a(is) p 2649 1479 a(strongly) p 2968 1479 a(con) n(tin) n(uous,) 515 1579 y(it) p 595 1579 a(is) p 677 1579 a(written) p 965 1579 a(as) p Fw 1065 1579 a(U) p Fx 1131 1579 a(\() p Fw(t) p Fx(\)) p 1248 1579 a(=) p 1336 1579 a(e) p Fn 1373 1548 a(itL) p Fx 1496 1579 a(with) p 1683 1579 a(self) p 1826 1579 a(adjoin) n(t) p 2105 1579 a(op) r(erator) p Fw 2438 1579 a(L) p Fx(.) p 2554 1579 a(W) p 2632 1579 a(e) p 2694 1579 a(call) p Fw 2844 1579 a(L) p Fx 2926 1579 a(the) p 3067 1579 a(Liouville) 515 1678 y(op) r(erator) p 850 1678 a(of) p Fw 944 1678 a(\013) p Fn 997 1690 a(t) p Fx 1026 1678 a(.) 639 1778 y(No) n(w) p 835 1778 a(w) n(e) p 963 1778 a(mo) n(v) n(e) p 1184 1778 a(from) p Fw 1387 1778 a(W) p Fo 1477 1748 a(\003) p Fx 1515 1778 a(-dynamical) p 1948 1778 a(systems) p 2262 1778 a(to) p Fw 2370 1778 a(C) p Fo 2435 1748 a(\003) p Fx 2473 1778 a(-dynamical) p 2906 1778 a(systems,) p 3245 1778 a(and) 515 1877 y(explain) p 809 1877 a(the) p 958 1877 a(role) p 1126 1877 a(of) p 1227 1877 a(the) p 1376 1877 a(standard) p 1728 1877 a(theory) p 1995 1877 a(in) p 2098 1877 a(the) p 2247 1877 a(in) n(v) n(estigation) p 2740 1877 a(of) p 2841 1877 a(the) p 2990 1877 a(NESS.) p 3258 1877 a(Let) p Fw 515 1977 a(\013) p Fn 568 1989 a(t) p Fx 628 1977 a(b) r(e) p 744 1977 a(a) p 817 1977 a(one) p 972 1977 a(parameter) p 1373 1977 a(group) p 1612 1977 a(of) p 1709 1977 a(automorphisms) p 2295 1977 a(whic) n(h) p 2536 1977 a(describ) r(es) p 2893 1977 a(the) p 3040 1977 a(dynamics) 515 2077 y(of) p 615 2077 a(a) p 689 2077 a(unital) p Fw 935 2077 a(C) p Fo 1000 2047 a(\003) p Fx 1038 2077 a(-algebra) p Ft 1362 2077 a(O) p Fx 1430 2077 a(,) p 1488 2077 a(and) p 1655 2077 a(let) p Fw 1781 2077 a(!) p Fx 1868 2077 a(b) r(e) p 1987 2077 a(a) p 2061 2077 a(state) p 2270 2077 a(of) p Ft 2370 2077 a(O) p Fx 2438 2077 a(.) p 2515 2077 a(What) p 2754 2077 a(w) n(e) p 2881 2077 a(are) p 3025 2077 a(in) n(terested) 515 2176 y(in) p 616 2176 a(is) p 703 2176 a(roughly) p 1009 2176 a(sp) r(eaking,) p 1379 2176 a(the) p 1526 2176 a(asymptotic) p 1957 2176 a(state) p Fw 2164 2176 a(w) p Ft 2247 2176 a(\000) p Fx 2333 2176 a(lim) p Fn 2448 2188 a(T) p Fo 2496 2188 a(!1) p Fw 2647 2176 a(!) p Ft 2722 2176 a(\016) p Fw 2785 2176 a(\013) p Fn 2838 2188 a(T) p Fx 2890 2176 a(.) p 2962 2176 a(In) p 3070 2176 a(practice,) 515 2276 y(the) p 657 2276 a(existence) p 1010 2276 a(of) p 1104 2276 a(the) p 1247 2276 a(limit) p 1445 2276 a(p) r(oin) n(t) p 1661 2276 a(is) p 1744 2276 a(not) p 1891 2276 a(trivial.) p 2170 2276 a(Ho) n(w) n(ev) n(er,) p 2528 2276 a(as) p 2629 2276 a(the) p 2772 2276 a(space) p 2993 2276 a(of) p 3087 2276 a(states) p 3323 2276 a(is) 515 2376 y(w) n(eak-) p Ft(\003) p Fx 788 2376 a(compact,) p 1140 2376 a(the) p 1280 2376 a(set) p Ft 1406 2376 a(f) p Fw(!) p Ft 1513 2376 a(\016) p Fw 1566 2376 a(\013) p Fn 1619 2388 a(t) p Fw 1694 2376 a(t) p Ft 1747 2376 a(2) p Fi 1826 2376 a(R) p Ft(g) p Fx 1951 2376 a(has) p 2096 2376 a(accum) n(ulation) p 2602 2376 a(p) r(oin) n(ts.) p 2882 2376 a(But) p 3044 2376 a(they) p 3227 2376 a(ma) n(y) 515 2475 y(not) p 662 2475 a(b) r(e) p 775 2475 a(in) n(v) p 881 2475 a(arian) n(t) p 1124 2475 a(under) p Fw 1359 2475 a(\013) p Fn 1412 2487 a(t) p Fx 1442 2475 a(.) p 1502 2475 a(T) p 1555 2475 a(o) p 1624 2475 a(mak) n(e) p 1841 2475 a(in) n(v) p 1947 2475 a(arian) n(t) p 2189 2475 a(states,) p 2448 2475 a(w) n(e) p 2570 2475 a(in) n(tegrate) p 2919 2475 a(the) p 3062 2475 a(states;) p Fw 1561 2706 a(!) p Fn 1616 2672 a(T) p Fx 1691 2706 a(=) 1798 2650 y(1) p 1788 2687 61 4 v Fw 1788 2763 a(T) p Fk 1872 2593 a(Z) p Fn 1955 2614 a(T) p Fl 1918 2782 a(0) p Fw 2021 2706 a(!) p Ft 2094 2706 a(\016) p Fw 2154 2706 a(\013) p Fn 2207 2718 a(t) p Fw 2237 2706 a(dt:) p Fx 515 2935 a(Again) p Fw 764 2935 a(!) p Fn 819 2905 a(T) p Fx 905 2935 a(has) p 1060 2935 a(w) n(eak-) p Ft(\003) p Fx 1343 2935 a(accum) n(ulation) p 1860 2935 a(p) r(oin) n(ts,) p 2141 2935 a(whic) n(h) p 2385 2935 a(are) p 2531 2935 a(in) n(v) p 2637 2935 a(arian) n(t) p 2886 2935 a(under) p Fw 3128 2935 a(\013) p Fn 3181 2947 a(t) p Fx 3245 2935 a(this) 515 3034 y(time.) p 747 3034 a(W) p 825 3034 a(e) p 893 3034 a(call) p 1049 3034 a(them) p 1265 3034 a(the) p 1411 3034 a(non-equilibrium) p 2017 3034 a(steady) p 2281 3034 a(states) p 2521 3034 a(\(NESS\)) p 2828 3034 a(and) p 2993 3034 a(denote) p 3264 3034 a(the) 515 3134 y(set) p 644 3134 a(of) p 739 3134 a(them) p 951 3134 a(b) n(y) p 1066 3134 a(\006) p Fl 1126 3104 a(+) p Fx 1181 3134 a(\() p Fw(!) p Fx 1268 3134 a(\).) 639 3234 y(Let) p 801 3234 a(\() p Ft(H) p Fn 903 3246 a(!) p Fw 951 3234 a(;) p 988 3234 a(\031) p Fn 1035 3246 a(!) p Fw 1084 3234 a(;) p Fx 1121 3234 a(\012) p Fn 1181 3246 a(!) p Fx 1229 3234 a(\)) p 1301 3234 a(b) r(e) p 1427 3234 a(the) p 1583 3234 a(GNS) p 1797 3234 a(triple) p 2032 3234 a(of) p Fw 2139 3234 a(!) p Fx 2194 3234 a(.) p 2293 3234 a(Supp) r(ose) p 2631 3234 a(that) p 2824 3234 a(there) p 3049 3234 a(exists) p 3291 3234 a(an) 515 3333 y(extension) p 890 3333 a(~) p Fw 881 3333 a(\013) p Fn 934 3345 a(t) p Fx 991 3333 a(of) p Fw 1085 3333 a(\013) p Fn 1138 3345 a(t) p Fx 1194 3333 a(to) p 1295 3333 a(the) p 1437 3333 a(v) n(on) p 1593 3333 a(Neumann) p 1969 3333 a(algebra) p Fw 2259 3333 a(\031) p Fn 2306 3345 a(!) p Fx 2354 3333 a(\() p Ft(O) p Fx 2454 3333 a(\)) p Ff 2486 3278 a(0) r(0) p Fx 2532 3333 a(.) p 2592 3333 a(Then) p 2808 3333 a(b) n(y) p 2923 3333 a(Theorem) p 3273 3333 a(3.2) 515 3433 y(there) p 727 3433 a(is) p 811 3433 a(one) p 963 3433 a(parameter) p 1360 3433 a(group) p 1595 3433 a(of) p 1690 3433 a(unitary) p 1983 3433 a(op) r(erators) p Fw 2351 3433 a(U) p Fn 2408 3445 a(t) p Fx 2464 3433 a(suc) n(h) p 2652 3433 a(that) 1424 3607 y(~) p Fw 1416 3607 a(\013) p Fn 1469 3619 a(t) p Fx 1512 3607 a(\() p Fw 1545 3607 a(x) p Fx(\)) p 1647 3607 a(=) p Fw 1735 3607 a(U) p Fn 1792 3619 a(t) p Fw 1821 3607 a(xU) p Fo 1934 3573 a(\003) p Fn 1925 3628 a(t) p Fw 2055 3607 a(x) p Ft 2126 3607 a(2) p Fw 2204 3607 a(\031) p Fn 2251 3619 a(!) p Fx 2300 3607 a(\() p Ft(O) p Fx 2400 3607 a(\)) p Ff 2432 3548 a(0) q(0) p Fw 1416 3732 a(U) p Fo 1482 3697 a(\003) p Fn 1473 3752 a(t) p Ft 1520 3732 a(P) p 1608 3732 a(\032) p 1695 3732 a(P) p Fw 1760 3732 a(:) p Fx 3273 3732 a(\(6\)) 515 3906 y(In) p 622 3906 a(addition,) p 978 3906 a(supp) r(ose) p 1294 3906 a(that) p Fw 1477 3906 a(U) p Fn 1534 3918 a(t) p Fx 1595 3906 a(is) p 1682 3906 a(strongly) p 2007 3906 a(con) n(tin) n(uous) p 2426 3906 a(and) p 2591 3906 a(let) p Fw 2715 3906 a(L) p Fx 2803 3906 a(b) r(e) p 2920 3906 a(the) p 3067 3906 a(Liouville) 515 4006 y(op) r(erator.) p 882 4006 a(If) p Fw 965 4006 a(!) p Fo 1017 4018 a(1) p Ft 1110 4006 a(2) p Fx 1188 4006 a(\006) p Fl 1248 3975 a(+) p Fx 1303 4006 a(\() p Fw(!) p Fx 1390 4006 a(\)) p 1450 4006 a(is) p Fw 1534 4006 a(!) p Fx 1589 4006 a(-normal,) p 1921 4006 a(there) p 2133 4006 a(exists) p 2362 4006 a(a) p 2432 4006 a(v) n(ector) p Fw 2681 4006 a(\030) p Fo 2717 4018 a(1) p Fx 2815 4006 a(in) p Ft 2912 4006 a(P) p Fx 3004 4006 a(s.t.) p Fw 1283 4180 a(!) p Fo 1335 4192 a(1) p Fx 1419 4180 a(\() p Fw(A) p Fx(\)) p 1569 4180 a(=) p Ft 1657 4180 a(h) p Fw(\030) p Fo 1725 4192 a(1) p Fw 1796 4180 a(;) p 1833 4180 a(\031) p Fn 1880 4192 a(!) p Fx 1942 4180 a(\() p Fw(A) p Fx(\)) p Fw 2082 4180 a(\030) p Fo 2118 4192 a(1) p Ft 2189 4180 a(i) p Fw 2235 4180 a(;) p 2355 4180 a(A) p Ft 2441 4180 a(2) p 2519 4180 a(O) p Fw 2587 4180 a(;) p Fx 515 4354 a(b) n(y) p 635 4354 a(Theorem) p 991 4354 a(3.1.) p 1171 4354 a(As) p Fw 1299 4354 a(!) p Fo 1351 4366 a(1) p Ft 1452 4354 a(2) p Fx 1538 4354 a(\006) p Fl 1598 4324 a(+) p Fx 1653 4354 a(\() p Fw(!) p Fx 1740 4354 a(\),) p Fw 1829 4354 a(!) p Fo 1881 4366 a(1) p Fx 1984 4354 a(is) p 2072 4354 a(in) n(v) p 2178 4354 a(arian) n(t) p 2426 4354 a(under) p Fw 2666 4354 a(\013) p Fn 2719 4366 a(t) p Fx 2748 4354 a(:) p Fw 2818 4354 a(!) p Fo 2870 4366 a(1) p Ft 2962 4354 a(\016) p Fw 3025 4354 a(\013) p Fn 3078 4366 a(t) p Fx 3138 4354 a(=) p Fw 3234 4354 a(!) p Fo 3286 4366 a(1) p Fx 3356 4354 a(.) 515 4454 y(Hence) p 761 4454 a(w) n(e) p 884 4454 a(ha) n(v) n(e) p Ft 556 4628 a(h) p Fw 589 4628 a(U) p Fo 655 4594 a(\003) p Fn 646 4648 a(t) p Fw 692 4628 a(\030) p Fo 728 4640 a(1) p Fw 799 4628 a(;) p 836 4628 a(\031) p Fn 883 4640 a(!) p Fx 945 4628 a(\() p Fw(A) p Fx(\)) p Fw 1086 4628 a(U) p Fo 1152 4594 a(\003) p Fn 1143 4648 a(t) p Fw 1190 4628 a(\030) p Fo 1226 4640 a(1) p Ft 1296 4628 a(i) p Fw 1343 4628 a(:) p Fx 1389 4628 a(=) p Fw 1476 4628 a(!) p Fo 1528 4640 a(1) p Ft 1617 4628 a(\016) p Fw 1677 4628 a(\013) p Fn 1730 4640 a(t) p Fx 1773 4628 a(\() p Fw(A) p Fx(\)) p 1923 4628 a(=) p Fw 2010 4628 a(!) p Fo 2062 4640 a(1) p Fx 2146 4628 a(\() p Fw 2179 4628 a(A) p Fx(\)) p 2296 4628 a(=) p Ft 2384 4628 a(h) p Fw(\030) p Fo 2452 4640 a(1) p Fw 2523 4628 a(;) p 2560 4628 a(\031) p Fn 2607 4640 a(!) p Fx 2669 4628 a(\() p Fw(A) p Fx(\)) p Fw 2810 4628 a(\030) p Fo 2846 4640 a(1) p Ft 2916 4628 a(i) p Fw 2962 4628 a(;) p 3082 4628 a(A) p Ft 3168 4628 a(2) p 3246 4628 a(O) p Fw 3314 4628 a(:) p Fx 515 4802 a(As) p Fw 637 4802 a(\030) p Fo 673 4814 a(1) p Fw 744 4802 a(;) p 781 4802 a(U) p Fn 838 4814 a(t) p Fw 867 4802 a(\030) p Fo 903 4814 a(1) p Ft 997 4802 a(2) p 1075 4802 a(P) p Fx 1140 4802 a(,) p 1190 4802 a(w) n(e) p 1313 4802 a(get) p Fw 1721 4976 a(U) p Fo 1787 4942 a(\003) p Fn 1778 4997 a(t) p Fw 1825 4976 a(\030) p Fo 1861 4988 a(1) p Fx 1955 4976 a(=) p Fw 2043 4976 a(\030) p Fo 2079 4988 a(1) p Fw 2149 4976 a(;) p Fx 1926 5255 a(8) p eop %%Page: 9 9 9 8 bop Fx 515 523 a(b) n(y) p 630 523 a(Theorem) p 981 523 a(3.1.) p 1147 523 a(This) p 1336 523 a(means) p Fw 1736 706 a(\030) p Fo 1772 718 a(1) p Ft 1866 706 a(2) p Fx 1944 706 a(Ker) p Fw 2078 706 a(L:) p Fx 515 888 a(In) p 619 888 a(other) p 836 888 a(w) n(ords,) p 1097 888 a(w) n(e) p 1219 888 a(ha) n(v) n(e) p 1410 888 a(the) p 1553 888 a(follo) n(wing) p 1904 888 a(prop) r(osition:) p Fm 515 1054 a(Prop) s(osition) p 1031 1054 a(3.4) p Fy 1194 1054 a(If) p Fx 1282 1054 a(Ker) p Fw 1416 1054 a(L) p Fx 1495 1054 a(=) p Ft 1583 1054 a(f) p Fx(0) p Ft(g) p Fy(,) p 1762 1054 a(any) p 1922 1054 a(state) p 2121 1054 a(in) p Fx 2223 1054 a(\006) p Fl 2283 1024 a(+) p Fx 2338 1054 a(\() p Fw(!) p Fx 2425 1054 a(\)) p Fy 2487 1054 a(is) p Fw 2576 1054 a(!) p Fy 2631 1054 a(-singular.) p Fu 515 1329 a(4) p 716 1329 a(The) 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3310 a(c) l(onjugation) p Fw 1925 3310 a(J) p Fn 1971 3322 a(S) p Fy 2049 3310 a(is) p Fw 1586 3493 a(J) p Fn 1632 3505 a(S) p Fx 1694 3493 a(\() p Fw(\036) p Ft 1794 3493 a(\012) p Fw 1877 3493 a( ) p Fx 1934 3493 a(\)) p 1989 3493 a(=) 2094 3471 y(\026) p Fw 2077 3493 a( ) p Ft 2152 3493 a(\012) p Fx 2246 3471 a(\026) p Fw 2235 3493 a(\036) q(;) p Fy 515 3694 a(the) p 653 3694 a(dual) p 833 3694 a(r) l(epr) l(esentation) p Fw 1374 3694 a(\031) p Fn 1424 3654 a(]) 1421 3718 y(S) p Fx 1492 3694 a(=) p Fw 1580 3694 a(J) p Fn 1626 3706 a(S) p Fw 1674 3694 a(\031) p Fn 1721 3706 a(S) p Fw 1770 3694 a(J) p Fn 1816 3706 a(S) p Fy 1894 3694 a(is) p Fw 1659 3890 a(\031) p Fn 1709 3850 a(]) 1706 3914 y(S) p Fx 1769 3890 a(\() p Fw(A) p Fx(\)) p 1918 3890 a(=) p 2006 3890 a(1) p Ft 2066 3890 a(\012) p Fx 2171 3869 a(\026) p Fw 2149 3890 a(A;) p Fy 515 4073 a(and) p 676 4073 a(the) p 814 4073 a(natur) l(al) p 1099 4073 a(p) l(ositive) p 1398 4073 a(c) l(one) p 1589 4073 a(is) p Ft 1384 4325 a(P) p Fn 1442 4337 a(S) p Fx 1513 4325 a(=) p Fk 1601 4183 a(\() 1668 4247 y(X) p Fn 1716 4423 a(i) p Fw 1802 4325 a(') p Fn 1856 4337 a(i) p Ft 1902 4325 a(\012) p Fx 1998 4325 a(\026) p Fw 1985 4325 a(') p Fn 2039 4337 a(i) p Fx 2067 4325 a(;) p Fw 2104 4325 a(') p Fn 2158 4337 a(i) p Ft 2209 4325 a(2) p 2287 4325 a(H) p Fn 2357 4337 a(S) p Fk 2406 4183 a(\)) p Fw 2486 4325 a(:) p Fx 515 4585 a(As) p 635 4585 a(the) p 776 4585 a(dimension) p 1167 4585 a(of) p Fh 1259 4585 a(H) p Fn 1319 4597 a(S) p Fx 1393 4585 a(is) p 1474 4585 a(\014nite,) p Fw 1708 4585 a(\031) p Fn 1755 4597 a(S) p Fx 1803 4585 a(\() p Ft(O) p Fn 1901 4597 a(S) p Fx 1950 4585 a(\)) p 2007 4585 a(is) p 2089 4585 a(itself) p 2288 4585 a(a) p 2355 4585 a(v) n(on) p 2510 4585 a(Neumann) p 2884 4585 a(algebra.) p 3206 4585 a(Note) 515 4685 y(that) p Ft 688 4685 a(O) p Fn 754 4697 a(S) p Fx 824 4685 a(has) p 965 4685 a(no) p 1074 4685 a(non-trivial) p 1476 4685 a(t) n(w) n(o-sided) p 1839 4685 a(ideal) p 2031 4685 a(and) p 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515 523 a(Let) p 663 523 a(an) p 779 523 a(op) r(erator) p Fw 1114 523 a(L) p Fn 1171 535 a(S) p Fx 1246 523 a(on) p Ft 1361 523 a(H) p Fn 1431 535 a(S) p Fx 1507 523 a(b) r(e) p Fw 1517 681 a(L) p Fn 1574 693 a(S) p Fx 1645 681 a(=) p Fw 1732 681 a(H) p Fn 1801 693 a(S) p Ft 1868 681 a(\012) p Fx 1951 681 a(1) p Ft 2011 681 a(\000) p Fx 2094 681 a(1) p Ft 2154 681 a(\012) p Fw 2237 681 a(H) p Fn 2306 693 a(S) p Fw 2354 681 a(:) p Fx 515 839 a(Then,) p 755 839 a(w) n(e) p 877 839 a(ha) n(v) n(e) 1203 997 y(e) p Fn 1240 963 a(itL) p Fj 1334 971 a(S) p Fw 1380 997 a(\031) p Fn 1427 1009 a(S) p Fx 1489 997 a(\() p Fw(A) p Fx(\)) p 1630 997 a(e) p Fo 1667 963 a(\000) p Fn(itL) p Fj 1813 971 a(S) p Fx 1882 997 a(=) p Fw 1969 997 a(\031) p Fn 2016 1009 a(S) p Fk 2078 930 a(\000) p Fx 2116 997 a(e) p Fn 2153 963 a(itH) p Fj 2255 971 a(S) p Fw 2302 997 a(A) p Fx(e) p Fo 2401 963 a(\000) p Fn(itH) p Fj 2555 971 a(S) p Fk 2602 930 a(\001) p Fw 2654 997 a(;) p Fx 1203 1140 a(e) p Fn 1240 1106 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r(erator) p 1901 2242 a(de\014ned) p 2194 2242 a(b) n(y) p Fw 2316 2242 a(L) p Fn 2373 2254 a(f) p Ft 2450 2242 a(\021) p Fw 2549 2242 a(d) p Fx(\000\() p Fw(h) p Fx(\)) p Ft 2780 2242 a(\012) p Fx 2867 2242 a(1) p Ft 2932 2242 a(\000) p Fx 3019 2242 a(1) p Ft 3084 2242 a(\012) p Fw 3171 2242 a(d) p Fx(\000\() p Fw(h) p Fx(\)) 515 2342 y(where) p Fw 755 2342 a(d) p Fx(\000\() p Fw(h) p Fx(\)) p 990 2342 a(is) p 1073 2342 a(the) p 1216 2342 a(second) p 1484 2342 a(quan) n(tization) p 1964 2342 a(of) p Fw 2059 2342 a(h) p Fx 2134 2342 a(in) p 2231 2342 a(\(1\).) p Fw 2397 2342 a(L) p Fn 2454 2354 a(f) p Fx 2524 2342 a(satis\014es) 1232 2507 y(e) p Fn 1269 2473 a(itL) p Fj 1363 2482 a(f) p Fx 1405 2507 a(\012) p 1488 2507 a(=) p 1576 2507 a(\012) p Fw(;) p Fx 1232 2650 a(e) p Fn 1269 2616 a(itL) p Fj 1363 2625 a(f) p Fw 1405 2650 a(x) p Fx(e) p Fo 1489 2616 a(\000) p Fn(itL) p Fj 1635 2625 a(f) p Fx 1700 2650 a(=) p 1796 2650 a(~) p Fw 1788 2650 a(\013) p Fn 1841 2610 a(f) 1841 2671 y(t) 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4919 a(f) p Fx 2463 4907 a(.) 639 5006 y(Then) p 856 5006 a(w) n(e) p 978 5006 a(ha) n(v) n(e) p 1170 5006 a(the) p 1313 5006 a(follo) n(wing) p 1663 5006 a(prop) r(osition.) 1905 5255 y(11) p eop %%Page: 12 12 12 11 bop Fm 515 523 a(Prop) s(osition) p 1031 523 a(4.4) p Fy 1194 523 a(Supp) l(ose) p 1513 523 a(that) p Fw 1684 523 a(!) p Fn 1736 535 a(S) p Fy 1816 523 a(is) p 1907 523 a(faithful,) p 2219 523 a(and) p Fx 2382 523 a(0) p Fw 2450 523 a(<) p 2541 523 a(\032) p 2611 523 a(<) p Fx 2702 523 a(1) p Fy(.) p 2813 523 a(The) p 2985 523 a(GNS) p 3189 523 a(triple) p Fx 515 623 a(\() p Ft(H) p Fn 617 635 a(!) p Fw 665 623 a(;) p 702 623 a(\031) p Fn 749 635 a(!) p Fw 797 623 a(;) p Fx 834 623 a(\012) p Fn 894 635 a(!) p Fx 942 623 a(\)) p Fy 1004 623 a(asso) l(ciate) l(d) p 1393 623 a(with) p 1573 623 a(the) p 1711 623 a(state) p Fw 1910 623 a(!) p Fx 1988 623 a(=) p Fw 2076 623 a(!) p Fn 2128 635 a(S) p Ft 2194 623 a(\012) p Fw 2277 623 a(!) p Fn 2329 635 a(\032) p Fy 2397 623 a(is) 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p Ft 1700 4976 a(\000) p Fw 1783 4976 a(J) p 1837 4976 a(I) p 1880 4976 a(J) o(;) p Ft 1962 4976 a(\001) p Fx(]) p Fw 2022 4976 a(;) p 2225 4976 a(I) p Fx 2291 4976 a(=) p Fw 2379 4976 a(\031) p Fn 2426 4988 a(\032) p Fx 2478 4976 a(\() p Fw 2511 4976 a(V) p Fx 2578 4976 a(\)) p Fw 2624 4976 a(;) p Fx 1905 5255 a(12) p eop %%Page: 13 13 13 12 bop Fx 515 523 a(with) p Fw 704 523 a(D) p Fx 775 523 a(\() 811 501 y(~) p Fw 807 523 a(\016) p Fx 847 523 a(\)) p 903 523 a(=) p Fw 990 523 a(D) p Fx 1061 523 a(\() 1097 501 y(~) p Fw 1093 523 a(\016) p Fl 1130 535 a(0) p Fx 1168 523 a(\).) p 1260 523 a(Note) p 1461 523 a(that) p Fw 1136 706 a(I) p Fx 1202 706 a(=) p Fw 1289 706 a(\031) p Fn 1336 718 a(\032) p Fx 1389 706 a(\() p Fw(V) p Fx 1488 706 a(\)) p 1544 706 a(=) p 1631 706 a(\() p Fw(Y) p Ft 1749 706 a(\012) p Fx 1832 706 a(1\)) p Ft 1924 706 a(\012) p Fx 2007 706 a(\() p Fw(a) p Fx(\() p Fw(g) p Fl 2155 718 a(1) p Fx 2192 706 a(\)) p 2243 706 a(+) p Fw 2326 706 a(a) p Fo 2370 671 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1517 a(:) p Fx 3231 1517 a(\(10\)) 515 1699 y(The) p 685 1699 a(restriction) p 1083 1699 a(of) p 1186 1699 a(~) p Fw 1178 1699 a(\013) p Fn 1231 1711 a(t) p Fx 1288 1699 a(to) p Ft 1389 1699 a(O) p Fx 1485 1699 a(is) p Fw 1569 1699 a(\013) p Fn 1622 1711 a(t) p Fx 1651 1699 a(.) 639 1799 y(On) p 778 1799 a(the) p 921 1799 a(other) p 1138 1799 a(hands,) p 1401 1799 a(the) p 1544 1799 a(p) r(erturb) r(ed) p 1932 1799 a(Liouville) p 2272 1799 a(op) r(erator) p Fw 2607 1799 a(L) p Fx 2691 1799 a(is) p 2775 1799 a(de\014ned) p 3061 1799 a(b) n(y) p Fw 1608 1981 a(L) p Ft 1688 1981 a(\021) p Fw 1775 1981 a(L) p Fn 1832 1993 a(f) p Fx 1893 1981 a(+) p Fw 1976 1981 a(I) p Ft 2038 1981 a(\000) p Fw 2121 1981 a(J) p 2175 1981 a(I) p 2218 1981 a(J) o(:) 515 2172 y(L) p Fx 604 2172 a(is) p 692 2172 a(a) p 766 2172 a(self-adjoin) n(t) p 1198 2172 a(op) r(erator,) p 1562 2172 a(and) p 1728 2172 a(e) p Fn 1765 2142 a(itL) p Fx 1896 2172 a(is) p 1985 2172 a(an) p 2105 2172 a(unitary) p 2403 2172 a(op) 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2820 a(;) p Fx 3245 2820 a(\(12\)) 515 3011 y(and) p 676 3011 a([e) p Fn 736 2981 a(itL) p Fg 830 2989 a(0) p Fw 867 3011 a(;) p 904 3011 a(J) p Fx 958 3011 a(]) p 1004 3011 a(=) p 1091 3011 a(0,) p 1184 3011 a(w) n(e) p 1306 3011 a(ha) n(v) n(e) p Fk 1712 3127 a(\002) p Fx 1746 3194 a(e) p Fn 1783 3159 a(itL) p Fw 1881 3194 a(;) p 1918 3194 a(J) p Fk 1972 3127 a(\003) p Fx 2030 3194 a(=) p 2118 3194 a(0) p Fw(:) p Fx 3231 3194 a(\(13\)) 515 3376 y(So) p 630 3376 a(w) n(e) p 752 3376 a(get) 1526 3559 y(e) p Fn 1563 3525 a(itL) p Ft 1661 3559 a(P) p 1749 3559 a(\032) p 1836 3559 a(P) p Fw 1901 3559 a(;) p Fx 1526 3696 a(e) p Fn 1563 3662 a(itL) p Fw 1661 3696 a(\030) p Fx 1715 3696 a(\() p Fw(!) p Fx 1802 3696 a(\)) p 1857 3696 a(=) p Fw 1945 3696 a(\030) p Fx 1999 3696 a(\() p 2040 3696 a(~) p Fw 2031 3696 a(\013) p Fo 2084 3708 a(\000) p Fn(t) p Fx 2179 3696 a(\() p Fw 2212 3696 a(!) p Fx 2267 3696 a(\)\)) p Fw 2345 3696 a(:) p Fx 3231 3696 a(\(14\)) 515 3879 y(Unitary) p 829 3879 a(op) r(erator) p 1169 3879 a(satisfying) p 1544 3879 a(\(11\),\(13\),\(14\)) p 2065 3879 a(is) p 2154 3879 a(unique) p 2429 3879 a(b) n(y) p 2549 3879 a(Theorem) p 2905 3879 a(3.2,) p 3068 3879 a(and) p Fw 3234 3879 a(L) p Fx 3323 3879 a(is) 515 3978 y(the) p 658 3978 a(Liouville) p 998 3978 a(op) r(erator) p 1333 3978 a(of) p 1436 3978 a(~) p Fw 1427 3978 a(\013) p Fn 1480 3990 a(t) p Fx 1510 3978 a(.) 639 4078 y(Hence) p 885 4078 a(b) n(y) p 999 4078 a(Section) p 1289 4078 a(3,) p 1380 4078 a(the) p 1522 4078 a(problem) p 1844 4078 a(of) p 1937 4078 a(in) n(v) n(estigating) p 2423 4078 a(the) p 2565 4078 a(stabilit) n(y) p 2888 4078 a(of) p Fw 2981 4078 a(!) p Fx 3062 4078 a(under) p Fw 3297 4078 a(\013) p Fn 3350 4090 a(t) p Fx 515 4178 a(is) p 598 4178 a(translated) p 991 4178 a(to) p 1092 4178 a(the) p 1235 4178 a(problem) p 1559 4178 a(of) p 1653 4178 a(in) n(v) n (estigating) p 2140 4178 a(the) p 2283 4178 a(sp) r(ectral) p 2595 4178 a(prop) r(ert) n(y) p 2935 4178 a(of) p Fw 3029 4178 a(L) p Fx(;) p Fw 602 4381 a(L) p Fx 681 4381 a(=) p 769 4381 a(\() p Fw(H) p Fn 870 4393 a(S) p Ft 937 4381 a(\012) p Fx 1020 4381 a(1\)) p Ft 1112 4381 a(\012) p Fx 1195 4381 a(1) p Ft 1255 4381 a(\000) p Fx 1338 4381 a(\(1) p Ft 1430 4381 a(\012) p Fw 1513 4381 a(H) p Fn 1582 4393 a(S) p Fx 1630 4381 a(\)) p Ft 1681 4381 a(\012) p Fx 1764 4381 a(1) p 1824 4381 a(+) p 1907 4381 a(1) p Ft 1967 4381 a(\012) p Fw 2050 4381 a(d) p Fx(\000) p Fk 2159 4288 a(\020) p Fx 2209 4359 a(~) p Fw 2209 4381 a(h) p Fk 2256 4288 a(\021) p Fx 620 4549 a(+) p Fw 703 4549 a(\025) p Fx 765 4549 a(\() p Fw 798 4549 a(Y) p Ft 883 4549 a(\012) p Fx 966 4549 a(1\)) p Ft 1058 4549 a(\012) p Fx 1141 4549 a(\() p Fw(a) p Fx 1231 4549 a(\() p Fw(g) p Fl 1303 4561 a(1) p Fx 1340 4549 a(\)) p 1391 4549 a(+) p Fw 1474 4549 a(a) p Fo 1518 4515 a(\003) p Fx 1570 4549 a(\() p Fw(g) p Fl 1642 4561 a(1) p Fx 1679 4549 a(\)\)) p Ft 1762 4549 a(\000) p Fw 1845 4549 a(\025) p Fk 1907 4482 a(\000) p Fx 1945 4549 a(1) p Ft 2005 4549 a(\012) p Fx 2101 4528 a(\026) p Fw 2088 4549 a(Y) p Fk 2155 4482 a(\001) p Ft 2211 4549 a(\012) p Fx 2295 4549 a(\() p Ft(\000) p Fx(1\)) p Fn 2465 4508 a(N) p Fx 2542 4549 a(\() p Fw(a) p Fx 2632 4549 a(\() p Fw(g) p Fl 2704 4561 a(2) p Fx 2741 4549 a(\)) p Ft 2792 4549 a(\000) p Fw 2875 4549 a(a) p Fo 2919 4515 a(\003) p Fx 2971 4549 a(\() p Fw(g) p Fl 3043 4561 a(2) p Fx 3080 4549 a(\)\)) p 3245 4549 a(\(15\)) 515 4732 y(on) p Ft 630 4732 a(H) p Fn 700 4744 a(!) p Fx 748 4732 a(,) p 799 4732 a(where) 1726 4893 y(~) p Fw 1725 4915 a(h) p Fx 1796 4915 a(=) p Fw 1884 4915 a(h) p Ft 1950 4915 a(\010) p 2033 4915 a(\000) p Fw(h:) p Fx 3231 4915 a(\(16\)) 1905 5255 y(13) p eop %%Page: 14 14 14 13 bop Fu 515 523 a(5) p 716 523 a(Rescaling) p 1318 523 a(Group) p Fx 515 705 a(In) p 620 705 a(order) p 838 705 a(to) p 940 705 a(construct) p 1307 705 a(the) p 1451 705 a(p) r(ositiv) n(e) p 1759 705 a(comm) n(utator) p 2228 705 a(for) p 2356 705 a(the) p 2500 705 a(lattice) p 2755 705 a(system,) p 3055 705 a(w) n(e) p 3178 705 a(in) n(tro-) 515 805 y(duce) p 703 805 a(a) p 767 805 a(strongly) p 1082 805 a(con) n(tin) n(uous) p 1492 805 a(one) p 1639 805 a(parameter) p 2031 805 a(group) p 2261 805 a(of) p 2350 805 a(unitaries.) p 2722 805 a(W) p 2800 805 a(e) p 2859 805 a(\014rst) p 3025 805 a(transform) 515 904 y(the) p 658 904 a(v) p 697 904 a(ariable) p Fw 969 904 a(k) p Ft 1038 904 a(2) p Fx 1117 904 a([) p Ft(\000) p Fw(\031) s(;) p 1292 904 a(\031) p Fx 1342 904 a(\)) p 1402 904 a(to) p Fw 1503 904 a(t) p Ft 1556 904 a(2) p Fi 1635 904 a(R) p Fx 1722 904 a(b) n(y) p Fw 1765 1130 a(t) p Fx 1818 1130 a(=) p 1906 1130 a(tan) p Fw 2050 1074 a(k) p 2050 1111 46 4 v Fx 2052 1187 a(2) p Fw 2105 1130 a(:) p Fx 515 1336 a(By) p 645 1336 a(this) p 807 1336 a(v) p 846 1336 a(ariable) p 1118 1336 a(transformation,) p 1707 1336 a(w) n(e) p 1830 1336 a(ha) n(v) n(e) p Fh 1189 1519 a(h) p Fx 1255 1519 a(=) p Fw 1343 1519 a(L) p Fl 1400 1484 a(2) p Fx 1451 1519 a(\([) p Ft(\000) p Fw(\031) s(;) p 1658 1519 a(\031) p Fx 1708 1519 a(\)) p Fw 1754 1519 a(;) p Fx 1791 1519 a(d) p Fw(k) p Fx 1883 1519 a(\)) p 1938 1519 a(=) p Fw 2026 1519 a(L) p Fl 2083 1484 a(2) p Fx 2134 1519 a(\([) p Ft(\0001) p Fw(;) p Ft 2374 1519 a(1) p Fx(\)) p Fw 2503 1519 a(;) p Fx 2540 1519 a(d) p Fw(\026) p Fx(\)) p Fw 2682 1519 a(;) p Fx 515 1701 a(where) p Fw 755 1701 a(\026) p Fx 833 1701 a(is) p 916 1701 a(the) p 1059 1701 a(p) r(ositiv) n(e) p 1366 1701 a(measure) p 1690 1701 a(on) p Fi 1805 1701 a(R) p Fx 1893 1701 a(giv) n(en) p 2109 1701 a(b) n(y) 1625 1918 y(d) p Fw(\026) p Fx 1735 1918 a(\() p Fw(t) p Fx(\)) p 1852 1918 a(=) 2034 1862 y(2) p 1950 1899 211 4 v Fw 1950 1975 a(t) p Fl 1980 1951 a(2) p Fx 2035 1975 a(+) p 2118 1975 a(1) 2170 1918 y(d) p Fw(t:) p Fx 515 2136 a(The) p 690 2136 a(m) n(ultiplication) p 1227 2136 a(op) r(erator) p Fw 1566 2136 a(m) p Fx(\() p Fw(k) p Fx 1717 2136 a(\)) p 1782 2136 a(is) p 1870 2136 a(transformed) p 2338 2136 a(as) p Fw 2445 2136 a(m) p Fx(\() p Fw(k) p Fx 2596 2136 a(\)) p Ft 2658 2136 a(!) p Fw 2772 2136 a(m) p Fx(\() p Fw(k) p Fx 2923 2136 a(\() p Fw(t) p Fx(\)\).) p 3123 2136 a(In) p 3231 2136 a(par-) 515 2235 y(ticular,) p 801 2235 a(the) p 944 2235 a(single) p 1175 2235 a(particle) p 1475 2235 a(energy) p Fw 1740 2235 a(h) p Fx 1816 2235 a(is) p 1899 2235 a(transformed) p 2364 2235 a(to) p Fw 1030 2453 a(h) p Fx 1092 2453 a(\() p Fw(k) p Fx 1170 2453 a(\)) p 1225 2453 a(=) p 1313 2453 a(cos) p Fw 1438 2453 a(k) p Ft 1502 2453 a(\000) p Fw 1585 2453 a(\015) p Ft 1738 2453 a(!) p Fw 1928 2453 a(h) p Fx 1990 2453 a(\() p Fw(k) p Fx 2081 2453 a(\() p Fw 2114 2453 a(t) p Fx(\)\)) p 2231 2453 a(=) 2413 2396 y(2) p 2329 2434 V Fw 2329 2510 a(t) p Fl 2359 2486 a(2) p Fx 2414 2510 a(+) p 2497 2510 a(1) p Ft 2567 2453 a(\000) p Fx 2650 2453 a(1) p Ft 2710 2453 a(\000) p Fw 2793 2453 a(\015) p 2841 2453 a(:) p Fx 515 2670 a(No) n(w) p 704 2670 a(w) n(e) p 826 2670 a(in) n(tro) r(duce) p 1195 2670 a(a) p 1264 2670 a(unitary) p 1558 2670 a(op) r(erator) p Fw 1892 2670 a(u) p Fx(\() p Fw(\022) p Fx 2013 2670 a(\)) p 2073 2670 a(on) p Fh 2189 2670 a(h) p Fx 2260 2670 a(de\014ned) p 2546 2670 a(b) n(y) 705 2920 y(\() p Fw(u) p Fx(\() p Fw(\022) p Fx 858 2920 a(\)) p Fw(f) p Fx 940 2920 a(\)) p 986 2920 a(\() p Fw 1019 2920 a(t) p Fx(\)) p Ft 1104 2920 a(\021) p Fx 1192 2920 a(e) p Fj 1239 2863 a(\022) p 1239 2872 30 3 v Fg 1240 2905 a(2) p Fk 1282 2789 a(r) p 1365 2789 338 4 v Fw 1429 2864 a(t) p Fl 1459 2840 a(2) p Fx 1515 2864 a(+) p 1598 2864 a(1) p 1375 2901 318 4 v 1375 2977 a(e) p Fl 1412 2953 a(2) p Fn(\022) p Fw 1483 2977 a(t) p Fl 1513 2953 a(2) p Fx 1568 2977 a(+) p 1651 2977 a(1) p Fw 1703 2920 a(f) p Fk 1766 2853 a(\000) p Fx 1804 2920 a(e) p Fn 1841 2885 a(\022) p Fw 1879 2920 a(t) p Fk 1909 2853 a(\001) p Fw 1961 2920 a(;) p 2081 2920 a(\022) p Ft 2145 2920 a(2) p Fi 2223 2920 a(R) p Fw(;) p 2403 2920 a(f) p Ft 2476 2920 a(2) p Fw 2554 2920 a(L) p Fl 2611 2885 a(2) p Fx 2648 2920 a(\([) p Ft(\0001) p Fw(;) p Ft 2888 2920 a(1) p Fx(\)) p Fw(;) p 3040 2920 a(d\026) p Fx(\)) p Fw(:) p Fx 515 3134 a(By) p 645 3134 a(an) p 760 3134 a(elemen) n(tary) p 1185 3134 a(calculation,) p 1628 3134 a(w) n(e) p 1750 3134 a(ha) n(v) n(e) p 1941 3134 a(the) p 2084 3134 a(follo) n(wing) p 2435 3134 a(lemma;) p Fm 515 3300 a(Lemma) p 853 3300 a(5.1) p Fw 1017 3300 a(u) p Fx(\() p Fw(\022) p Fx 1138 3300 a(\)) p Fy 1200 3300 a(is) p 1289 3300 a(a) p 1361 3300 a(str) l(ongly) p 1672 3300 a(c) l(ontinuous) p 2090 3300 a(one) p 2247 3300 a(p) l(ar) l(ameter) p 2637 3300 a(gr) l(oup) p 2865 3300 a(of) p 2963 3300 a(unitaries.) p Fx 515 3466 a(F) p 562 3466 a(rom) p 736 3466 a(this) p 902 3466 a(lemma,) p Fw 1197 3466 a(u) p Fx(\() p Fw(\022) p Fx 1318 3466 a(\)) p 1381 3466 a(is) p 1468 3466 a(written) p 1763 3466 a(as) p Fw 1868 3466 a(u) p Fx(\() p Fw(\022) p Fx 1989 3466 a(\)) p 2051 3466 a(=) p 2144 3466 a(e) p Fn 2181 3436 a(i\022) r(p) p Fx 2307 3466 a(with) p 2500 3466 a(a) p 2573 3466 a(selfadjoin) n(t) p 2976 3466 a(op) r(erator) p Fw 3314 3466 a(p) p Fx(.) 515 3565 y(The) p 685 3565 a(action) p 934 3565 a(of) p Fw 1029 3565 a(p) p Fx 1098 3565 a(on) p Fw 1214 3565 a(C) p Fo 1279 3535 a(1) p Fl 1273 3586 a(0) p Fx 1363 3565 a(\() p Fi 1396 3565 a(R) p Fx(\)) p 1515 3565 a(is) 969 3801 y(\() p Fw(pf) p Fx 1093 3801 a(\)) p 1139 3801 a(\() p Fw(t) p Fx(\)) p 1256 3801 a(=) 1354 3745 y(1) p 1354 3782 42 4 v 1354 3858 a(2) p Fw 1406 3801 a(f) p Fx 1469 3801 a(\() p Fw(t) p Fx(\)) p Ft 1582 3801 a(\000) p Fw 1746 3745 a(t) p Fl 1776 3715 a(2) p 1675 3782 211 4 v Fw 1675 3858 a(t) p Fl 1705 3834 a(2) p Fx 1761 3858 a(+) p 1844 3858 a(1) p Fw 1895 3801 a(f) p Fx 1959 3801 a(\() p Fw(t) p Fx(\)) p 2072 3801 a(+) p Fw 2155 3801 a(tf) p Fo 2235 3767 a(0) p Fx 2271 3801 a(\() p Fw 2304 3801 a(t) p Fx(\)) p Fw 2463 3801 a(f) p Ft 2535 3801 a(2) p Fw 2614 3801 a(C) p Fo 2679 3767 a(1) p Fl 2673 3822 a(0) p Fx 2763 3801 a(\() p Fi(R) p Fx(\)) p Fw 2902 3801 a(:) p Fx 515 4019 a(Let) p 669 4019 a(~) p Fw 663 4019 a(u) p Fx(\() p Fw(\022) p Fx 784 4019 a(\)) p 845 4019 a(b) r(e) p 958 4019 a(a) p 1027 4019 a(unitary) p 1320 4019 a(op) r(erator) p 1655 4019 a(on) p Fh 1770 4019 a(h) p Ft 1832 4019 a(\010) p Fh 1915 4019 a(h) p Fx 1986 4019 a(de\014ned) p 2272 4019 a(b) n(y) 1572 4201 y(~) p Fw 1567 4201 a(u) p Fx 1614 4201 a(\() p Fw(\022) p Fx 1687 4201 a(\)) p Ft 1743 4201 a(\021) p Fw 1831 4201 a(u) p Fx(\() p Fw(\022) p Fx 1952 4201 a(\)) p Ft 2003 4201 a(\010) p Fw 2086 4201 a(u) p Fx(\() p Ft(\000) p Fw(\022) p Fx 2272 4201 a(\)) p Fw(:) p Fx 515 4384 a(The) p 693 4384 a(generator) p 1069 4384 a(of) p 1177 4384 a(~) p Fw 1171 4384 a(u) p Fx(\() p Fw(\022) p Fx 1292 4384 a(\)) p 1360 4384 a(is) p 1457 4384 a(~) p Fw 1450 4384 a(p) p Ft 1527 4384 a(\021) p Fw 1627 4384 a(p) p Ft 1692 4384 a(\010) p 1780 4384 a(\000) p Fw(p) p Fx(.) p 1968 4384 a(W) p 2046 4384 a(e) p 2119 4384 a(de\014ne) p 2366 4384 a(a) p 2442 4384 a(unitary) p 2743 4384 a(op) r(erator) p Fw 3085 4384 a(U) p Fx 3151 4384 a(\() p Fw(\022) p Fx 3224 4384 a(\)) p 3291 4384 a(on) p Ft 515 4484 a(F) p Fx 583 4484 a(\() p Fh(h) p Ft 677 4484 a(\010) p Fh 760 4484 a(h) p Fx(\)) p 863 4484 a(b) n(y) p 978 4484 a(the) p 1121 4484 a(second) p 1389 4484 a(quan) n(tization) p 1869 4484 a(of) p 1969 4484 a(~) p Fw 1963 4484 a(u) p Fx(\() p Fw(\022) p Fx 2084 4484 a(\):) p Fw 1632 4666 a(U) p Fx 1711 4666 a(\() p Fw(\022) p Fx 1784 4666 a(\)) p Ft 1840 4666 a(\021) p Fx 1928 4666 a(\000) p 1994 4666 a(\() p 2031 4666 a(~) p Fw 2026 4666 a(u) p Fx 2087 4666 a(\() p Fw(\022) p Fx 2160 4666 a(\)) q(\)) p Fw 2239 4666 a(:) p Fx 515 4849 a(Since) p Fw 734 4849 a(U) p Fx 800 4849 a(\() p Fw(\022) p Fx 873 4849 a(\)) p 935 4849 a(is) p 1021 4849 a(a) p 1093 4849 a(strongly) p 1416 4849 a(con) n(tin) n(uous) p 1834 4849 a(one) p 1989 4849 a(parameter) p 2388 4849 a(unitary) p 2683 4849 a(group,) p 2945 4849 a(it) p 3030 4849 a(is) p 3116 4849 a(written) 515 4949 y(as) p Fw 617 4949 a(U) p Fx 683 4949 a(\() p Fw(\022) p Fx 756 4949 a(\)) p 811 4949 a(=) p 899 4949 a(e) p Fn 936 4918 a(\022) r(A) p Fg 1020 4926 a(0) p Fx 1056 4949 a(,) p 1106 4949 a(where) p Fw 1347 4949 a(A) p Fl 1409 4961 a(0) p Fx 1474 4949 a(is) p 1557 4949 a(an) p 1673 4949 a(an) n(ti-selfadjoin) n(t) p 2240 4949 a(op) r(erator.) 1905 5255 y(14) p eop %%Page: 15 15 15 14 bop Fx 639 523 a(No) n(w) p 826 523 a(w) n(e) p 946 523 a(in) n(tro) r(duce) p 1313 523 a(the) p 1454 523 a(comm) n(utator) p 1920 523 a(of) p Fw 2012 523 a(L) p Fn 2069 535 a(f) p Fx 2137 523 a(and) p Fw 2296 523 a(A) p Fl 2358 535 a(0) p Fx 2421 523 a(as) p 2521 523 a(a) p 2587 523 a(quadratic) p 2957 523 a(form) p 3151 523 a(on) p 3264 523 a(the) 515 623 y(dense) p 738 623 a(set) p 863 623 a(\() p Fw(D) p Fx 966 623 a(\() p Fw(N) p Fx 1074 623 a(\)) p Ft 1117 623 a(\\) p Fw 1183 623 a(D) p Fx 1254 623 a(\() p Fw(A) p Fl 1348 635 a(0) p Fx 1388 623 a(\)\)) p Ft 1486 623 a(\002) p Fx 1585 623 a(\() p Fw(D) p Fx 1688 623 a(\() p Fw(N) p Fx 1796 623 a(\)) p Ft 1839 623 a(\\) p Fw 1905 623 a(D) p Fx 1976 623 a(\() p Fw(A) p Fl 2070 635 a(0) p Fx 2109 623 a(\)\).) p 2232 623 a(By) p 2359 623 a(straigh) n(t) p 2663 623 a(forw) n(ard) p 2964 623 a(calculation,) 515 722 y(the) p 658 722 a(action) p 907 722 a(of) p 1001 722 a(Ad) p 1115 722 a(~) p Fw 1110 722 a(u) p Fx 1157 722 a(\() p Fw(\022) p Fx 1230 722 a(\)) p 1291 722 a(on) p 1406 722 a(a) p 1475 722 a(m) n(ultiplication) p 2008 722 a(op) r(erator) p Fw 2343 722 a(m) p Fl 2416 734 a(1) p Ft 2471 722 a(\010) p Fw 2554 722 a(m) p Fl 2627 734 a(2) p Fx 2692 722 a(is) 999 905 y(\() p 1036 905 a(~) p Fw 1031 905 a(u) p Fx 1093 905 a(\() p Ft(\000) p Fw(\022) p Fx 1231 905 a(\)) p 1277 905 a(\() p Fw(m) p Fl 1382 917 a(1) p Ft 1438 905 a(\010) p Fw 1521 905 a(m) p Fl 1594 917 a(2) p Fx 1631 905 a(\)) p 1682 905 a(~) p Fw 1677 905 a(u) p Fx 1738 905 a(\() p Fw 1771 905 a(\022) p Fx 1812 905 a(\)\)) p 1890 905 a(\() p Fw 1923 905 a(t) p Fx(\)) p 2008 905 a(=) 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Fw(m) p Fl 1592 1382 a(1) p Ft 1648 1370 a(\010) p Fw 1731 1370 a(m) p Fl 1804 1382 a(2) p Fx 1841 1370 a(\)) p Fw 1887 1370 a(U) p Fx 1967 1370 a(\() p Fw(\022) p Fx 2040 1370 a(\)) p 2096 1370 a(=) p 2183 1370 a(d\000) p Fk 2295 1303 a(\000) p Fw 2333 1370 a(m) p Fo 2406 1334 a(\000) p Fn(\022) p Fl 2406 1392 a(1) p Ft 2514 1370 a(\010) p Fw 2597 1370 a(m) p Fn 2670 1336 a(\022) p Fl 2670 1390 a(2) p Fk 2707 1303 a(\001) p Fw 2759 1370 a(;) p Fx 3245 1370 a(\(17\)) 515 1553 y(where) 1487 1735 y(\() p Fw(m) p Fn 1592 1701 a(\022) 1592 1756 y(i) p Fw 1630 1735 a(f) p Fx 1680 1735 a(\)) p 1726 1735 a(\() p Fw(t) p Fx(\)) p 1843 1735 a(=) p Fw 1931 1735 a(m) p Fn 2004 1747 a(i) p Fk 2045 1668 a(\000) p Fx 2083 1735 a(e) p Fn 2120 1701 a(\022) p Fw 2158 1735 a(t) p Fk 2188 1668 a(\001) p Fw 2239 1735 a(f) p Fx 2289 1735 a(\() p Fw(t) p Fx(\)) p Fw(:) p Fx 515 1918 a(W) p 593 1918 a(e) p 658 1918 a(ha) n(v) n(e) p 849 1918 a(the) p 992 1918 a(follo) n(wing) p 1343 1918 a(prop) r(osition:) p Fm 515 2084 a(Prop) s(osition) p 1031 2084 a(5.2) p Fy 1194 2084 a(Supp) l(ose) p 1508 2084 a(that) p Fw 1675 2084 a(m) p Fl 1748 2096 a(1) p Fw 1785 2084 a(;) p 1822 2084 a(m) p Fl 1895 2096 a(2) p Fy 1959 2084 a(ar) l(e) p 2097 2084 a(b) l(ounde) l(d) p 2411 2084 a(multiplic) l(ation) p 2932 2084 a(op) l(er) l(ators) p 3290 2084 a(on) p Fw 515 2183 a(L) p Fl 572 2153 a(2) p Fx 608 2183 a(\() p Fi(R) p Fw(;) p 737 2183 a(d\026) p Fx(\)) p Fy 893 2183 a(which) p 1126 2183 a(ar) l(e) p 1267 2183 a(di\013er) l(entiable) p 1763 2183 a(and) p 1924 2183 a(satisfy) p Fx 1570 2366 a(sup) p Fn 1695 2386 a(t) p Fo(2) p Fe(R) p Ft 1825 2366 a(j) p Fw(q) s(m) p Fn 1961 2378 a(i) p Fx 1989 2366 a(\() p Fw(t) p Fx(\)) p Ft(j) p Fw 2130 2366 a(<) p Ft 2217 2366 a(1) p Fw(;) p Fy 515 2549 a(wher) l(e) p Fw 1597 2731 a(q) s(m) p Fn 1710 2743 a(i) p Fx 1751 2731 a(\() p Fw 1784 2731 a(t) p Fx(\)) p Ft 1869 2731 a(\021) p 1957 2731 a(\000) p Fw(tm) p Fo 2125 2697 a(0) p Fn 2125 2752 a(i) p Fx 2165 2731 a(\() p Fw 2198 2731 a(t) p Fx(\)) p Fw 2274 2731 a(:) p Fy 515 2914 a(Then) p 731 2914 a(we) p 854 2914 a(have) p Ft 563 3097 a(h) p Fx 596 3097 a(d\000) p 708 3097 a(\() p Fw(m) p Fl 813 3109 a(1) p Ft 868 3097 a(\010) p Fw 951 3097 a(m) p Fl 1024 3109 a(2) p Fx 1061 3097 a(\)) p Fw 1108 3097 a(';) p 1199 3097 a(A) p Fl 1261 3109 a(0) p Fw 1298 3097 a( ) p Ft 1355 3097 a(i) p Fx 1406 3097 a(+) p Ft 1489 3097 a(h) p Fw(A) p Fl 1583 3109 a(0) p Fw 1621 3097 a(';) p Fx 1712 3097 a(d\000) p 1824 3097 a(\() p Fw(m) p Fl 1929 3109 a(1) p Ft 1985 3097 a(\010) p Fw 2068 3097 a(m) p Fl 2141 3109 a(2) p Fx 2178 3097 a(\)) p Fw 2224 3097 a( ) p Ft 2281 3097 a(i) p Fx 2336 3097 a(=) p Ft 2424 3097 a(h) p Fw(';) p Fx 2547 3097 a(d\000) p 2659 3097 a(\() p Fw 2692 3097 a(q) s(m) p Fl 2805 3109 a(1) p Ft 2860 3097 a(\010) p 2943 3097 a(\000) p Fw(q) s(m) p Fl 3121 3109 a(2) p Fx 3158 3097 a(\)) p Fw 3204 3097 a( ) p Ft 3261 3097 a(i) p Fw 3307 3097 a(;) p Fx 3245 3196 a(\(18\)) p Fy 515 3380 a(for) p 647 3380 a(al) t(l) p Fw 766 3380 a(';) p 857 3380 a( ) p Ft 937 3380 a(2) p Fw 1016 3380 a(D) p Fx 1087 3380 a(\() p Fw(N) p Fx 1195 3380 a(\)) p Ft 1246 3380 a(\\) p Fw 1320 3380 a(D) p Fx 1391 3380 a(\() p Fw(A) p Fl 1485 3392 a(0) p Fx 1522 3380 a(\)) p Fy(.) 515 3546 y(Pr) l(o) l(of) p Fx 515 3645 a(As) p Fw 637 3645 a(U) p Fx 703 3645 a(\() p Fw(\022) p Fx 776 3645 a(\)) p 836 3645 a(preserv) n(es) p 1192 3645 a(n) n(um) n(b) r(er,) p 1517 3645 a(w) n(e) p 1639 3645 a(ha) n(v) n(e) p Fw 1830 3645 a(U) p Fx 1896 3645 a(\() p Fw(\022) p Fx 1969 3645 a(\)) p Fw(D) p Fx 2072 3645 a(\() p Fw(N) p Fx 2180 3645 a(\)) p Ft 2236 3645 a(\032) p Fw 2323 3645 a(D) p Fx 2394 3645 a(\() p Fw(N) p Fx 2502 3645 a(\)) p 2562 3645 a(and) p 2723 3645 a(ha) n(v) n(e) p 2914 3645 a(the) p 3056 3645 a(follo) n(wing) 515 3745 y(relation:) 539 3982 y(lim) p Fn 530 4036 a(\022) p Fo 564 4036 a(!) p Fl(0) p Fk 677 3840 a(*) p Fw 744 3982 a(') p Fx 808 3920 a(d\000) p Fk 920 3853 a(\000) p Fw 958 3920 a(m) p Fo 1031 3885 a(\000) p Fn(\022) p Fl 1031 3942 a(1) p Ft 1139 3920 a(\010) p Fw 1222 3920 a(m) p Fn 1295 3890 a(\022) p Fl 1295 3941 a(2) p Fk 1332 3853 a(\001) p Ft 1388 3920 a(\000) p Fx 1472 3920 a(d\000) p 1584 3920 a(\() p Fw(m) p Fl 1689 3932 a(1) p Ft 1744 3920 a(\010) p Fw 1827 3920 a(m) p Fl 1900 3932 a(2) p Fx 1937 3920 a(\)) p 808 3963 1162 4 v Fw 1368 4039 a(\022) 1980 3982 y( ) p Fk 2037 3840 a(+) p Fx 539 4239 a(=) p 636 4239 a(lim) p Fn 627 4293 a(\022) p Fo 661 4293 a(!) p Fl(0) p Fk 774 4122 a(\034) p Fx 836 4239 a(d\000) p 948 4239 a(\() p Fw(m) p Fl 1053 4251 a(1) p Ft 1109 4239 a(\010) p Fw 1192 4239 a(m) p Fl 1265 4251 a(2) p Fx 1302 4239 a(\)) p Fw 1348 4239 a(U) p Fx 1427 4239 a(\() p Fw 1460 4239 a(\022) p Fx 1501 4239 a(\)) p Fw 1547 4239 a(';) 1648 4183 y(U) p Fx 1728 4183 a(\() p Fw(\022) p Fx 1801 4183 a(\)) p Ft 1852 4183 a(\000) p Fx 1935 4183 a(1) p 1648 4220 329 4 v Fw 1792 4296 a(\022) 1987 4239 y( ) p Fk 2044 4122 a(\035) p Fx 2124 4239 a(+) p Fk 2207 4122 a(\034) p Fw 2280 4183 a(U) p Fx 2359 4183 a(\() p Fw(\022) p Fx 2432 4183 a(\)) p Ft 2484 4183 a(\000) p Fx 2567 4183 a(1) p 2280 4220 V Fw 2423 4296 a(\022) 2618 4239 y(';) p Fx 2709 4239 a(d\000) p 2821 4239 a(\() p Fw(m) p Fl 2926 4251 a(1) p Ft 2982 4239 a(\010) p Fw 3065 4239 a(m) p Fl 3138 4251 a(2) p Fx 3175 4239 a(\)) p Fw 3221 4239 a( ) p Fk 3278 4122 a(\035) p Fw 3354 4239 a(:) p Fx 3231 4421 a(\(19\)) 515 4600 y(Using) p 748 4600 a(dominated) p 1159 4600 a(con) n(v) n(ergence) p 1618 4600 a(theorem,) p 1964 4600 a(w) n(e) p 2086 4600 a(ha) n(v) n(e) p 2278 4600 a(the) p 2421 4600 a(follo) n(wing) p 2771 4600 a(statemen) n(ts:) 616 4766 y(1.) p 722 4766 a(F) p 769 4766 a(or) p 871 4766 a(eac) n(h) p Fw 1058 4766 a( ) p Ft 1138 4766 a(2) p Fw 1217 4766 a(D) p Fx 1288 4766 a(\() p Fw(N) p Fx 1396 4766 a(\)) p 1456 4766 a(and) p 1617 4766 a(b) r(ounded) p 1956 4766 a(op) r(erator) p Fw 2291 4766 a(m) p Fx 2392 4766 a(on) p Fh 2507 4766 a(h) p Ft 2569 4766 a(\010) p Fh 2652 4766 a(h) p Fx(,) 1389 4948 y(d\000) p 1501 4948 a(\() p Fw(m) p Fx(\)) p Fw 1652 4948 a(U) p Fx 1731 4948 a(\() p Fw 1764 4948 a(\022) p Fx 1805 4948 a(\)) p Fw 1851 4948 a( ) p Ft 1931 4948 a(!) p Fx 2037 4948 a(d\000) p 2149 4948 a(\() p Fw 2182 4948 a(m) p Fx(\)) p Fw 2301 4948 a( ) s(;) p 2478 4948 a(\022) p Ft 2542 4948 a(!) p Fx 2648 4948 a(0) p Fw(;) p Fx 1905 5255 a(15) p eop %%Page: 16 16 16 15 bop Fx 616 523 a(2.) p 722 523 a(F) p 769 523 a(or) p 871 523 a(eac) n(h) p Fw 1058 523 a( ) p Ft 1138 523 a(2) p Fw 1217 523 a(D) p Fx 1288 523 a(\() p Fw(A) p Fl 1382 535 a(0) p Fx 1419 523 a(\),) 1638 745 y(lim) p Fn 1630 799 a(\022) p Fo 1664 799 a(!) p Fl(0) p Fw 1786 689 a(U) p Fx 1866 689 a(\() p Fw(\022) p Fx 1939 689 a(\)) p Ft 1990 689 a(\000) p Fx 2073 689 a(1) p 1786 726 329 4 v Fw 1930 802 a(\022) 2124 745 y( ) p Fx 2205 745 a(=) p Fw 2292 745 a(A) p Fl 2354 757 a(0) p Fw 2392 745 a( ) s(;) p Fx 616 984 a(3.) p 722 984 a(F) p 769 984 a(or) p 871 984 a(eac) n(h) p Fw 1058 984 a( ) p Ft 1138 984 a(2) p Fw 1217 984 a(D) p Fx 1288 984 a(\() p Fw(N) p Fx 1396 984 a(\),) 788 1233 y(lim) p Fn 779 1287 a(\022) p Fo 813 1287 a(!) p Fl(0) p Fk 926 1088 a(\015) 926 1138 y(\015) 926 1188 y(\015) 926 1237 y(\015) 926 1287 y(\015) 972 1091 y( ) p Fx 1048 1172 a(d\000) p Fk 1160 1105 a(\000) p Fw 1198 1172 a(m) p Fo 1271 1136 a(\000) p Fn(\022) p Fl 1271 1194 a(1) p Ft 1378 1172 a(\010) p Fw 1461 1172 a(m) p Fn 1534 1142 a(\022) p Fl 1534 1192 a(2) p Fk 1572 1105 a(\001) p Ft 1628 1172 a(\000) p Fx 1711 1172 a(d\000) p 1823 1172 a(\() p Fw 1856 1172 a(m) p Fl 1929 1184 a(1) p Ft 1984 1172 a(\010) p Fw 2067 1172 a(m) p Fl 2140 1184 a(2) p Fx 2177 1172 a(\)) p 1048 1214 1162 4 v Fw 1608 1290 a(\022) p Ft 2238 1233 a(\000) p Fx 2321 1233 a(d\000) p 2433 1233 a(\() p Fw(q) s(m) p Fl 2578 1245 a(1) p Ft 2634 1233 a(\010) p 2717 1233 a(\000) p Fw(q) s(m) p Fl 2895 1245 a(2) p Fx 2932 1233 a(\)) p Fk 2964 1091 a(!) p Fw 3043 1233 a( ) p Fk 3100 1088 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p Ft 2153 1788 a(i) p Fx 2208 1788 a(=) p Ft 2296 1788 a(h) p Fw(';) p 2419 1788 a(S) p Fl 2470 1800 a(1) p Fw 2508 1788 a( ) p Ft 2565 1788 a(i) p Fw 2611 1788 a(:) p Fx 515 1962 a(Here) p Fw 720 1962 a(S) p Fl 771 1974 a(1) p Fx 844 1962 a(is) p 936 1962 a(the) p 1088 1962 a(second) p 1364 1962 a(quan) n(tization) p Fw 1852 1962 a(d) p Fx(\000) p 1961 1962 a(\() p Fw 1994 1962 a(s) p Fl 2033 1974 a(1) p Ft 2088 1962 a(\010) p Fw 2171 1962 a(s) p Fl 2210 1974 a(1) p Fx 2247 1962 a(\)) p 2316 1962 a(of) p Fw 2419 1962 a(s) p Fl 2458 1974 a(1) p Ft 2519 1962 a(\010) p Fw 2608 1962 a(s) p Fl 2647 1974 a(1) p Fx 2720 1962 a(on) p Fh 2844 1962 a(h) p Ft 2912 1962 a(\010) p Fh 3000 1962 a(h) p Fx(,) p 3105 1962 a(with) p Fw 3303 1962 a(s) p Fl 3342 1974 a(1) p Fx 515 2061 a(de\014ned) p 801 2061 a(b) n(y) 1207 2284 y(\() p Fw(s) p Fl 1278 2296 a(1) p Fw 1315 2284 a(f) p Fx 1365 2284 a(\)) p 1411 2284 a(\() p Fw(t) p Fx(\)) p 1528 2284 a(=) p Fw 1616 2284 a(s) p Fl 1655 2296 a(1) p Fx 1692 2284 a(\() p Fw(t) p Fx(\)) p Fw(f) p Fx 1836 2284 a(\() p Fw(t) p Fx(\)) p Fw(;) p 2051 2284 a(s) p Fl 2090 2296 a(1) p Fx 2127 2284 a(\() p Fw(t) p Fx(\)) p 2245 2284 a(=) 2444 2228 y(4) p Fw(t) p Fl 2516 2198 a(2) p 2342 2265 313 4 v Fx 2342 2341 a(\(1) p 2434 2341 a(+) p Fw 2517 2341 a(t) p Fl 2547 2317 a(2) p Fx 2585 2341 a(\)) p Fl 2617 2317 a(2) p Fw 2664 2284 a(:) p Fx 515 2502 a(Note) p 715 2502 a(that) p Fw 895 2502 a(S) p Fl 946 2514 a(1) p Fx 1011 2502 a(is) p 1095 2502 a(p) r(ositiv) n(e.) p 1434 2502 a(This) p 1624 2502 a(is) p 1707 2502 a(the) p 1850 2502 a(main) p 2058 2502 a(p) r(oin) n(t) p 2274 2502 a(of) p 2369 2502 a(this) p 2531 2502 a(comm) n(utator.) 639 2602 y(Similarly) p 958 2602 a(,) p 1008 2602 a(w) n(e) p 1131 2602 a(ha) n(v) n(e) p Ft 1264 2776 a(h) p Fw 1297 2776 a(S) p Fl 1348 2788 a(1) p Fw 1385 2776 a(';) p 1476 2776 a(A) p Fl 1538 2788 a(0) p Fw 1576 2776 a( ) p Ft 1633 2776 a(i) p Fx 1683 2776 a(+) p Ft 1766 2776 a(h) p Fw 1799 2776 a(A) p Fl 1861 2788 a(0) p Fw 1898 2776 a(';) p 1989 2776 a(S) p Fl 2040 2788 a(1) p Fw 2078 2776 a( ) p Ft 2135 2776 a(i) p Fx 2190 2776 a(=) p Ft 2278 2776 a(h) p Fw(';) p 2401 2776 a(S) p Fl 2452 2788 a(2) p Fw 2489 2776 a( ) p Ft 2546 2776 a(i) p Fw 2592 2776 a(;) p Ft 1264 2901 a(h) p Fw 1297 2901 a(S) p Fl 1348 2913 a(2) p Fw 1385 2901 a(';) p 1476 2901 a(A) p Fl 1538 2913 a(0) p Fw 1576 2901 a( ) p Ft 1633 2901 a(i) p Fx 1683 2901 a(+) p Ft 1766 2901 a(h) p Fw 1799 2901 a(A) p Fl 1861 2913 a(0) p Fw 1898 2901 a(';) p 1989 2901 a(S) p Fl 2040 2913 a(2) p Fw 2078 2901 a( ) p Ft 2135 2901 a(i) p Fx 2190 2901 a(=) p Ft 2278 2901 a(h) p Fw(';) p 2401 2901 a(S) p Fl 2452 2913 a(3) p Fw 2489 2901 a( ) p Ft 2546 2901 a(i) p Fw 2592 2901 a(;) p Fx 515 3075 a(where) p Fw 1046 3249 a(S) p Fl 1097 3261 a(2) p Fx 1157 3249 a(=) p 1245 3249 a(d\000) p 1357 3249 a(\() p Fw(s) p Fl 1428 3261 a(2) p Ft 1484 3249 a(\010) p 1567 3249 a(\000) p Fw(s) p Fl 1671 3261 a(2) p Fx 1707 3249 a(\)) p Fw 1754 3249 a(;) p Fx 1060 3444 a(\() p Fw(s) p Fl 1131 3456 a(2) p Fw 1168 3444 a(f) p Fx 1218 3444 a(\)) p 1264 3444 a(\() p Fw(t) p Fx(\)) p 1382 3444 a(=) p Fw 1469 3444 a(s) p Fl 1508 3456 a(2) p Fx 1546 3444 a(\() p Fw(t) p Fx(\)) p Fw(f) p Fx 1690 3444 a(\() p Fw(t) p Fx(\)) p Fw(;) p 1904 3444 a(s) p Fl 1943 3456 a(2) p Fx 1980 3444 a(\() p Fw(t) p Fx(\)) p 2098 3444 a(=) 2195 3383 y(8) p Fk 2251 3316 a(\000) p Fx 2289 3383 a(1) p Ft 2349 3383 a(\000) p Fw 2432 3383 a(t) p Fl 2462 3353 a(2) p Fk 2499 3316 a(\001) p Fw 2551 3383 a(t) p Fl 2581 3353 a(2) p 2195 3425 423 4 v Fx 2251 3514 a(\(1) p 2343 3514 a(+) p Fw 2426 3514 a(t) p Fl 2456 3490 a(2) p Fx 2493 3514 a(\)) p Fl 2526 3473 a(3) p Fw 2628 3444 a(;) 1046 3638 y(S) p Fl 1097 3650 a(3) p Fx 1157 3638 a(=) p 1245 3638 a(d\000) p 1357 3638 a(\() p Fw(s) p Fl 1428 3650 a(3) p Ft 1484 3638 a(\010) p Fw 1567 3638 a(s) p Fl 1606 3650 a(3) p Fx 1643 3638 a(\)) p Fw 1689 3638 a(;) p Fx 1060 3833 a(\() p Fw(s) p Fl 1131 3845 a(3) p Fw 1168 3833 a(f) p Fx 1218 3833 a(\)) p 1264 3833 a(\() p Fw(t) p Fx(\)) p 1382 3833 a(=) p Fw 1469 3833 a(s) p Fl 1508 3845 a(3) p Fx 1546 3833 a(\() p Fw(t) p Fx(\)) p Fw(f) p Fx 1690 3833 a(\() p Fw(t) p Fx(\)) p Fw(;) p 1904 3833 a(s) p Fl 1943 3845 a(3) p Fx 1980 3833 a(\() p Fw(t) p Fx(\)) p 2098 3833 a(=) 2195 3772 y(16) p Fk 2293 3704 a(\000) p Fw 2330 3772 a(t) p Fl 2360 3742 a(2) p Ft 2416 3772 a(\000) p Fx 2499 3772 a(4) p Fw(t) p Fl 2571 3742 a(4) p Fx 2626 3772 a(+) p Fw 2709 3772 a(t) p Fl 2739 3742 a(6) p Fk 2777 3704 a(\001) p 2195 3814 620 4 v Fx 2349 3903 a(\(1) p 2441 3903 a(+) p Fw 2524 3903 a(t) p Fl 2554 3879 a(2) p Fx 2592 3903 a(\)) p Fl 2624 3861 a(4) p Fw 2825 3833 a(:) p Fx 515 4064 a(Note) p 715 4064 a(that) p Fw 895 4064 a(S) p Fl 946 4076 a(1) p Fw 983 4064 a(;) p 1020 4064 a(S) p Fl 1071 4076 a(2) p Fw 1108 4064 a(;) p 1145 4064 a(S) p Fl 1196 4076 a(3) p Fx 1261 4064 a(are) p 1400 4064 a(all) p Fw 1515 4064 a(N) p Fx 1591 4064 a(-b) r(ounded.) 639 4164 y(The) p 819 4164 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4463 a(\() p Fw(a) p Fx 1328 4463 a(\() p Fw(g) p Fl 1400 4475 a(1) p Fx 1437 4463 a(\)) p 1488 4463 a(+) p Fw 1571 4463 a(a) p Fo 1615 4429 a(\003) p Fx 1667 4463 a(\() p Fw(g) p Fl 1739 4475 a(1) p Fx 1776 4463 a(\)\)) p Ft 1859 4463 a(\000) p Fk 1942 4396 a(\000) p Fx 1980 4463 a(1) p Ft 2040 4463 a(\012) p Fx 2135 4442 a(\026) p Fw 2123 4463 a(Y) p Fk 2190 4396 a(\001) p Ft 2246 4463 a(\012) p Fx 2329 4463 a(\() p Ft(\000) p Fx(1\)) p Fn 2500 4421 a(N) p Fx 2577 4463 a(\() p Fw(a) p Fx 2667 4463 a(\() p Fw(g) p Fl 2739 4475 a(2) p Fx 2776 4463 a(\)) p Ft 2826 4463 a(\000) p Fw 2909 4463 a(a) p Fo 2953 4429 a(\003) p Fx 3005 4463 a(\() p Fw 3038 4463 a(g) p Fl 3078 4475 a(2) p Fx 3114 4463 a(\)) q(\)) p Fk 3179 4371 a(\021) p Fw 3242 4463 a(:) p Fx 3231 4632 a(\(20\)) 515 4802 y(W) p 593 4802 a(e) p 658 4802 a(ha) n(v) n(e) p Ft 1293 4976 a(h) p Fw(';) p 1416 4976 a(I) p Fl 1452 4988 a(0) p Fw 1490 4976 a(A) p Fl 1552 4988 a(0) p Fw 1590 4976 a( ) p Ft 1647 4976 a(i) p Fx 1697 4976 a(+) p Ft 1781 4976 a(h) p Fw(A) p Fl 1875 4988 a(0) p Fw 1912 4976 a(';) p 2003 4976 a(I) p Fl 2039 4988 a(0) p Fw 2077 4976 a( ) p Ft 2134 4976 a(i) p Fx 2190 4976 a(=) p Ft 2277 4976 a(h) p Fw 2310 4976 a(';) p 2401 4976 a(I) p Fl 2437 4988 a(1) p Fw 2475 4976 a( ) p Ft 2532 4976 a(i) p Fw 2578 4976 a(;) p Fx 1905 5255 a(16) p eop %%Page: 17 17 17 16 bop Fx 515 523 a(on) p Fw 630 523 a(D) p Fx 701 523 a(\() p Fw(A) p Fl 795 535 a(0) p Fx 833 523 a(\)) p Ft 884 523 a(\002) p Fw 967 523 a(D) p Fx 1038 523 a(\() p Fw(A) p Fx(\).) p 1224 523 a(Here) p Fw 1421 523 a(I) p Fl 1457 535 a(1) p Fx 1522 523 a(is) p 1606 523 a(de\014ned) p 1892 523 a(b) n(y) p Fw 515 717 a(I) p Fl 551 729 a(1) p Ft 612 717 a(\021) p Fx 699 717 a(\() p Fw(Y) p Ft 817 717 a(\012) p Fx 900 717 a(1\)) p Ft 992 717 a(\012) p Fx 1075 717 a(\() p Fw(a) p Fx 1165 717 a(\() p Ft(\000) p Fw(i) p Fx 1298 717 a(~) p Fw 1291 717 a(p) o(g) p Fl 1372 729 a(1) p Fx 1409 717 a(\)) p 1460 717 a(+) p Fw 1543 717 a(a) p Fo 1587 682 a(\003) p Fx 1639 717 a(\() p Ft(\000) p Fw(i) p Fx 1772 717 a(~) p Fw 1765 717 a(p) o(g) p Fl 1846 729 a(1) p Fx 1883 717 a(\)\)) p Ft 1966 717 a(\000) p Fk 2049 649 a(\000) p Fx 2087 717 a(1) p Ft 2147 717 a(\012) p Fx 2242 696 a(\026) p Fw 2230 717 a(Y) p Fk 2296 649 a(\001) p Ft 2353 717 a(\012) p Fx 2436 717 a(\() p Ft(\000) p Fx(1\)) p Fn 2607 675 a(N) p Fx 2683 717 a(\() p Fw 2716 717 a(a) p Fx 2774 717 a(\() p Ft(\000) p Fw(i) p Fx 2907 717 a(~) p Fw 2900 717 a(p) o(g) p Fl 2981 729 a(2) p Fx 3017 717 a(\)) p Ft 3068 717 a(\000) p Fw 3151 717 a(a) p Fo 3195 682 a(\003) p Fx 3247 717 a(\() p Ft(\000) p Fw(i) p Fx 3380 717 a(~) p Fw 3373 717 a(p) o(g) p Fl 3454 729 a(2) p Fx 3491 717 a(\)\)) p Fw 3570 717 a(:) p Fx 515 899 a(Similarly) p 834 899 a(,) p 884 899 a(w) n(e) p 1006 899 a(ha) n(v) n(e) p Ft 591 1082 a(h) p Fw(';) p 714 1082 a(I) p Fl 750 1094 a(1) p Fw 788 1082 a(A) p Fl 850 1094 a(0) p Fw 887 1082 a( ) p Ft 944 1082 a(i) p Fx 995 1082 a(+) p Ft 1078 1082 a(h) p 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a(\000) p Fx 1281 1447 a(~) p Fw 1274 1447 a(p) p Fl 1316 1413 a(2) p Fw 1352 1447 a(g) p Fl 1392 1459 a(1) p Fk 1429 1380 a(\001) p Fx 1486 1447 a(+) p Fw 1569 1447 a(a) p Fo 1613 1413 a(\003) p Fk 1665 1380 a(\000) p Ft 1703 1447 a(\000) p Fx 1775 1447 a(~) p Fw 1768 1447 a(p) p Fl 1810 1413 a(2) p Fw 1846 1447 a(g) p Fl 1886 1459 a(1) p Fk 1923 1380 a(\001\001) p Ft 2018 1447 a(\000) p Fk 2101 1380 a(\000) p Fx 2139 1447 a(1) p Ft 2199 1447 a(\012) p Fx 2294 1426 a(\026) p Fw 2282 1447 a(Y) p Fk 2348 1380 a(\001) p Ft 2405 1447 a(\012) p Fx 2488 1447 a(\() p Ft(\000) p Fx(1\)) p Fn 2658 1405 a(N) p Fk 2735 1380 a(\000) p Fw 2773 1447 a(a) p Fk 2831 1380 a(\000) p Ft 2869 1447 a(\000) p Fx 2941 1447 a(~) p Fw 2934 1447 a(p) p Fl 2976 1413 a(2) p Fw 3013 1447 a(g) p Fl 3053 1459 a(2) p Fk 3089 1380 a(\001) p Ft 3146 1447 a(\000) p Fw 3229 1447 a(a) p Fo 3273 1413 a(\003) p Fk 3325 1380 a(\000) p Ft 3363 1447 a(\000) p Fx 3435 1447 a(~) p Fw 3428 1447 a(p) p Fl 3470 1413 a(2) p Fw 3507 1447 a(g) p Fl 3547 1459 a(2) p Fk 3583 1380 a(\001\001) p Fw 3673 1447 a(;) 515 1592 y(I) p Fl 551 1604 a(3) p Ft 612 1592 a(\021) p Fx 699 1592 a(\() p Fw(Y) p Ft 817 1592 a(\012) p Fx 900 1592 a(1\)) p Ft 992 1592 a(\012) p Fk 1075 1524 a(\000) p Fw 1113 1592 a(a) p Fk 1171 1524 a(\000) p Fw 1209 1592 a(i) p Fx 1245 1592 a(~) p Fw 1238 1592 a(p) p Fl 1280 1557 a(3) p Fw 1316 1592 a(g) p Fl 1356 1604 a(1) p Fk 1393 1524 a(\001) p Fx 1450 1592 a(+) p Fw 1533 1592 a(a) p Fo 1577 1557 a(\003) p Fk 1629 1524 a(\000) p Fw 1667 1592 a(i) p Fx 1703 1592 a(~) p Fw 1696 1592 a(p) p Fl 1738 1557 a(3) p Fw 1774 1592 a(g) p Fl 1814 1604 a(1) p Fk 1851 1524 a(\001\001) p Ft 1946 1592 a(\000) p Fk 2029 1524 a(\000) p Fx 2067 1592 a(1) p Ft 2127 1592 a(\012) p Fx 2222 1571 a(\026) p Fw 2210 1592 a(Y) p Fk 2276 1524 a(\001) p Ft 2333 1592 a(\012) p Fx 2416 1592 a(\() p Ft(\000) p Fx(1\)) p Fn 2586 1550 a(N) p Fk 2663 1524 a(\000) p Fw 2701 1592 a(a) p Fk 2759 1524 a(\000) p Fw 2797 1592 a(i) p Fx 2833 1592 a(~) p Fw 2826 1592 a(p) p Fl 2868 1557 a(3) p Fw 2905 1592 a(g) p Fl 2945 1604 a(2) p Fk 2982 1524 a(\001) p Ft 3038 1592 a(\000) p Fw 3121 1592 a(a) p Fo 3165 1557 a(\003) p Fk 3217 1524 a(\000) p Fw 3255 1592 a(i) p Fx 3291 1592 a(~) p Fw 3284 1592 a(p) p Fl 3326 1557 a(3) p Fw 3363 1592 a(g) p Fl 3403 1604 a(2) p Fk 3440 1524 a(\001\001) p Fw 3529 1592 a(:) p Fu 515 1866 a(6) p 716 1866 a(Cut) p 985 1866 a(O\013) p 1210 1866 a(of) p 1363 1866 a(the) p 1596 1866 a(In) l(teraction) p 2287 1866 a(and) p 2547 1866 a(Eigen) l(v) l(ector) p Fx 515 2048 a(In) p 629 2048 a(case) p 816 2048 a(the) p 969 2048 a(disp) r(ersion) p 1371 2048 a(relation) p 1686 2048 a(of) p 1792 2048 a(the) p 1945 2048 a(\014eld) p Fw 2136 2048 a(L) p Fn 2193 2060 a(f) p Fx 2274 2048 a(is) p Fw 2368 2048 a(!) p Fx 2423 2048 a(\() p Fw(k) p Fx 2501 2048 a(\)) p 2574 2048 a(=) p Ft 2680 2048 a(j) p Fw(k) p Ft 2749 2048 a(j) p Fx(,) p Fw 2836 2048 a(A) p Fl 2898 2060 a(0) p Fx 2974 2048 a(is) p 3068 2048 a(tak) n(en) p 3305 2048 a(as) 515 2148 y(the) p 664 2148 a(generator) p 1039 2148 a(of) p 1140 2148 a(the) p 1289 2148 a(the) p 1439 2148 a(shift) p 1632 2148 a(op) r(erator.) p 2018 2148 a(And) p 2206 2148 a(the) p 2355 2148 a(comm) n(utator) p 2830 2148 a(is) p 2919 2148 a(giv) n(en) p 3142 2148 a(b) n(y) p 3264 2148 a(the) 515 2247 y(n) n(um) n(b) r(er) p 825 2247 a(op) r(erator) p Fw 1169 2247 a(N) p Fx 1281 2247 a(:) p 1357 2247 a([) p Fw(L) p Fn 1437 2259 a(f) p Fw 1480 2247 a(;) p 1517 2247 a(A) p Fl 1579 2259 a(0) p Fx 1616 2247 a(]) p 1677 2247 a(=) p Fw 1778 2247 a(N) p Fx 1854 2247 a(.) p 1939 2247 a(Note) p 2148 2247 a(that) p 2337 2247 a(the) p 2488 2247 a(disp) r(ersion) p 2887 2247 a(relation) p 3200 2247 a(of) p Fw 3303 2247 a(N) p Fx 515 2347 a(has) p 666 2347 a(a) p 738 2347 a(strictly) p 1025 2347 a(p) r(ositiv) n(e) p 1335 2347 a(sp) r(ectral) p 1649 2347 a(gap) p 1809 2347 a(1) p Fw 1878 2347 a(>) p Fx 1970 2347 a(0.) p 2080 2347 a(Ho) n(w) n(ev) n(er) p 2418 2347 a(in) p 2517 2347 a(our) p 2668 2347 a(case,) p 2870 2347 a(the) p 3016 2347 a(disp) r(ersion) 515 2447 y(relation) p 821 2447 a(is) p Fw 906 2447 a(!) p Fx 961 2447 a(\() p Fw(k) p Fx 1039 2447 a(\)) p 1097 2447 a(=) p 1187 2447 a(cos) o(\() p Fw(k) p Fx 1376 2447 a(\)) p Ft 1428 2447 a(\000) p Fw 1512 2447 a(\015) p Fx 1560 2447 a(,) p 1612 2447 a(and) p 1775 2447 a(the) p 1919 2447 a(comm) n(utator) p 2389 2447 a(is) p 2474 2447 a([) p Fw(L) p Fn 2554 2459 a(f) p Fw 2597 2447 a(;) p 2634 2447 a(A) p Fl 2696 2459 a(0) p Fx 2733 2447 a(]) p 2782 2447 a(=) p Fw 2872 2447 a(S) p Fl 2923 2459 a(1) p Fx 2960 2447 a(.) p 3025 2447 a(Note) p 3227 2447 a(that) 515 2546 y(the) p 659 2546 a(disp) r(ersion) p 1050 2546 a(relation) p 1356 2546 a(of) p Fw 1451 2546 a(S) p Fl 1502 2558 a(1) p Fx 1568 2546 a(is) p 1652 2546 a(p) r(ositiv) n(e,) p 1984 2546 a(but) p 2137 2546 a(it) p 2221 2546 a(do) r(es) p 2409 2546 a(not) p 2557 2546 a(ha) n(v) n(e) p 2750 2546 a(a) p 2820 2546 a(sp) r(ectral) p 3133 2546 a(gap:) p 3324 2546 a(it) 515 2646 y(attains) p 790 2646 a(zero) p 963 2646 a(at) p Fw 1063 2646 a(t) p Fx 1116 2646 a(=) p 1203 2646 a(0) p 1270 2646 a(and) p Fw 1430 2646 a(t) p Fx 1483 2646 a(=) p Ft 1570 2646 a(\0061) p Fx(.) p 1777 2646 a(The) p 1946 2646 a(existence) p 2297 2646 a(of) p 2389 2646 a(the) p 2530 2646 a(sp) r(ectral) p 2840 2646 a(gap) p 2995 2646 a(is) p 3076 2646 a(essen) n(tial) 515 2745 y(in) p 611 2745 a(application) p 1039 2745 a(of) p 1133 2745 a(p) r(ositiv) n(e) p 1439 2745 a(comm) n(utator) p 1907 2745 a(metho) r(d) p 2208 2745 a(as) p 2309 2745 a(seen) p 2489 2745 a(in) p 2585 2745 a(the) p 2727 2745 a(argumen) n(ts) p 3131 2745 a(in) p 3227 2745 a([10) o(].) 515 2845 y(So) p 630 2845 a(this) p 793 2845 a(situation) p 1139 2845 a(causes) p 1394 2845 a(a) p 1464 2845 a(problem.) p 1821 2845 a(T) p 1874 2845 a(o) p 1943 2845 a(o) n(v) n(ercome) p 2306 2845 a(this) p 2468 2845 a(di\016culit) n(y) p 2802 2845 a(,) p 2854 2845 a(w) n(e) p 2976 2845 a(assume) p 3264 2845 a(the) 515 2945 y(Assumption) p 975 2945 a(2.1) p 1108 2945 a(on) p 1224 2945 a(the) p 1367 2945 a(in) n(teraction.) p Fm 515 3111 a(Assumption2.1) p Fy 1208 3111 a(We) p 1353 3111 a(assume) p 1638 3111 a(that) p Fw 1802 3111 a(g) p Fn 1845 3071 a(j) 1842 3134 y(i) p Fx 1903 3111 a(\() p Fw(i;) p 2001 3111 a(j) p Fx 2063 3111 a(=) p 2150 3111 a(1) p Fw(;) p Fx 2229 3111 a(2\)) p Fy 2327 3111 a(ar) l(e) p 2462 3111 a(in) p 2558 3111 a(the) p 2691 3111 a(domain) p Fw 2982 3111 a(D) p Fx 3053 3111 a(\() p Fw(p) p Fl 3127 3081 a(3) p Fx 3164 3111 a(\)) p Fy(,) p 3247 3111 a(and) 515 3221 y(the) p 653 3221 a(supp) l(ort) p 946 3221 a(of) p Fw 1044 3221 a(g) p Fn 1087 3181 a(j) 1084 3244 y(i) p Fy 1151 3221 a(satisfy) p Fx 1707 3413 a(supp) p Fw(g) p Fn 1921 3374 a(j) 1918 3437 y(i) p Ft 1979 3413 a(\032) p Fx 2066 3413 a(\003) p Fn 2124 3425 a(v) p Fw 2164 3413 a(;) p Fy 515 3596 a(for) p 647 3596 a(some) p Fx 860 3596 a(0) p Fw 924 3596 a(<) p 1012 3596 a(v) p 1078 3596 a(<) p Fx 1166 3596 a(1) p Fy(.) p Fx 515 3762 a(No) n(w) p 704 3762 a(w) n(e) p 826 3762 a(decomp) r(ose) 1245 3740 y(~) p Fh 1244 3762 a(h) p Fx 1310 3762 a(=) p Fh 1398 3762 a(h) p Ft 1460 3762 a(\010) p Fh 1543 3762 a(h) p Fx 1614 3762 a(with) p 1803 3762 a(resp) r(ect) p 2087 3762 a(to) p 2189 3762 a(\003) p Fn 2247 3774 a(v) p Fx 2286 3762 a(;) 743 3923 y(~) p Fh 742 3945 a(h) p Fx 808 3945 a(=) 897 3923 y(~) p Fh 896 3945 a(h) p Fl 939 3957 a(\003) p Fj 984 3965 a(v) p Ft 1043 3945 a(\010) p Fx 1127 3923 a(~) p Fh 1126 3945 a(h) p Fl 1169 3957 a(\003) p Fj 1214 3941 a(c) 1214 3974 y(v) p Fw 1254 3945 a(;) p Fx 786 4063 a(~) p Fh 785 4085 a(h) p Fl 828 4097 a(\003) p Fj 873 4105 a(v) p Fx 937 4085 a(=) p Fw 1024 4085 a(L) p Fl 1081 4051 a(2) p Fx 1118 4085 a(\(\003) p Fn 1208 4097 a(v) p Fw 1248 4085 a(;) p 1285 4085 a(d\026) p Fx(\)) p Ft 1428 4085 a(\010) p Fw 1511 4085 a(L) p Fl 1568 4051 a(2) p Fx 1605 4085 a(\(\003) p Fn 1695 4097 a(v) p Fw 1735 4085 a(;) p 1772 4085 a(d\026) p Fx(\)) p Fw(;) p Fx 2018 4063 a(~) p Fh 2017 4085 a(h) p Fl 2060 4097 a(\003) p Fj 2105 4081 a(c) 2105 4114 y(v) p Fx 2168 4085 a(=) p Fw 2256 4085 a(L) p Fl 2313 4051 a(2) p Fx 2350 4085 a(\(\003) p Fn 2440 4051 a(c) 2440 4105 y(v) p Fw 2479 4085 a(;) p 2516 4085 a(d\026) p Fx(\)) p Ft 2660 4085 a(\010) p Fw 2743 4085 a(L) p Fl 2800 4051 a(2) p Fx 2837 4085 a(\(\003) p Fn 2927 4051 a(c) 2927 4105 y(v) p Fw 2966 4085 a(;) p 3003 4085 a(d\026) p Fx(\)) p Fw(;) p Fx 515 4284 a(where) p 758 4284 a(\003) p Fn 816 4253 a(c) 816 4304 y(v) p Fx 886 4284 a(is) p 973 4284 a(the) p 1119 4284 a(complemen) n(t) p 1585 4284 a(of) p 1683 4284 a(\003) p Fn 1741 4296 a(v) p Fx 1811 4284 a(in) p Fi 1911 4284 a(R) p Fx(.) p 2041 4284 a(Note) p 2244 4284 a(that) 2428 4262 y(~) p Fw 2427 4284 a(h) p Fx 2506 4284 a(\(16\)) p 2684 4284 a(is) p 2771 4284 a(decomp) r(osed) p 3238 4284 a(in) n(to) 516 4361 y(~) p Fw 515 4383 a(h) p Fx 596 4383 a(=) 695 4361 y(~) p Fw 694 4383 a(h) p Fl 742 4395 a(\003) p Fj 787 4403 a(v) p Ft 850 4383 a(\010) p Fx 938 4361 a(~) p Fw 937 4383 a(h) p Fl 985 4395 a(\003) p Fj 1030 4379 a(c) 1030 4412 y(v) p Fw 1070 4383 a(;) p Fx 1127 4383 a(with) p 1323 4383 a(resp) r(ect) p 1613 4383 a(to) p 1721 4383 a(this) p 1889 4383 a(decomp) r(osition) p 2442 4383 a(of) p 2543 4383 a(the) p 2692 4383 a(Hilb) r(ert) p 2985 4383 a(space.) p 3258 4383 a(Let) p Fw 515 4483 a(N) p Fl 582 4495 a(\003) p Fj 627 4503 a(v) p Fx 694 4483 a(b) r(e) p Fw 1632 4665 a(N) p Fl 1699 4677 a(\003) p Fj 1744 4685 a(v) p Ft 1807 4665 a(\021) p Fw 1895 4665 a(d) p Fx(\000\(1) p Ft 2082 4665 a(\010) p Fx 2165 4665 a(0\)) p Fw(;) p Fx 515 4848 a(with) p 705 4848 a(resp) r(ect) p 991 4848 a(to) p 1094 4848 a(this) p 1257 4848 a(decomp) r(osition.) p 1840 4848 a(Belo) n(w) p 2087 4848 a(w) n(e) p 2211 4848 a(denote) p 2479 4848 a(b) n(y) p Ft 2596 4848 a(U) p Fx 2656 4848 a(\() p Fh(h) p Fo 2731 4818 a(0) p Fx 2755 4848 a(\)) p 2816 4848 a(a) p 2887 4848 a(set) p 3018 4848 a(of) 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638 y(F) p Fx 583 638 a(\() 616 616 y(~) p Fh 615 638 a(h) p Fl 658 650 a(\003) p Fj 703 633 a(c) 703 667 y(v) p Fx 743 638 a(\)) p Fy 805 638 a(which) p 1039 638 a(satis\014es) p 1351 638 a(the) p 1489 638 a(fol) t(lowing) p 1841 638 a(c) l(onditions;) 613 804 y(1.) p 722 804 a(F) p 770 804 a(or) p 877 804 a(any) p Fw 1036 804 a(u) p Fl 1084 816 a(\003) p Fj 1129 824 a(v) p Ft 1192 804 a(2) p 1271 804 a(U) p Fx 1331 804 a(\() 1364 782 y(~) p Fh 1363 804 a(h) p Fl 1406 816 a(\003) p Fj 1451 824 a(v) p Fx 1492 804 a(\)) p Fy 1554 804 a(and) p Fw 1715 804 a(u) p Fl 1763 816 a(\003) p Fj 1808 800 a(c) 1808 833 y(v) p Ft 1871 804 a(2) p 1949 804 a(U) p Fx 2009 804 a(\() 2042 782 y(~) p Fh 2041 804 a(h) p Fl 2084 816 a(\003) p Fj 2129 800 a(c) 2129 833 y(v) p Fx 2170 804 a(\)) p Fy(,) p Fw 1357 968 a(U) p Fx 1423 968 a(\000\() p Fw(u) p Fl 1555 980 a(\003) p Fj 1600 988 a(v) p Ft 1658 968 a(\010) p Fw 1741 968 a(u) p Fl 1789 980 a(\003) p Fj 1834 964 a(c) 1834 997 y(v) p Fx 1874 968 a(\)) p Fw(U) p Fo 1972 934 a(\003) p Fx 2033 968 a(=) p 2121 968 a(\000\() p Fw(u) p Fl 2253 980 a(\003) p Fj 2298 988 a(v) p Fx 2338 968 a(\)) p Ft 2389 968 a(\012) p Fx 2472 968 a(\000\() p Fw(u) p Fl 2604 980 a(\003) p Fj 2649 964 a(c) 2649 997 y(v) p Fx 2689 968 a(\)) p Fw(;) p Fy 613 1177 a(2.) p 722 1177 a(F) p 770 1177 a(or) p 877 1177 a(any) p Fw 1036 1177 a(f) p Ft 1109 1177 a(2) p Fx 1188 1156 a(~) p Fh 1187 1177 a(h) p Fl 1230 1189 a(\003) p Fj 1275 1197 a(v) p Fy 1346 1177 a(and) p Fw 1507 1177 a(g) p Ft 1573 1177 a(2) p Fx 1652 1156 a(~) p Fh 1651 1177 a(h) p Fl 1694 1189 a(\003) p Fj 1739 1173 a(c) 1739 1206 y(v) p Fw 1268 1354 a(U) p 1334 1354 a(a) p Fx(\() p Fw(f) p Ft 1478 1354 a(\010) p Fw 1561 1354 a(g) p Fx 1604 1354 a(\)) p Fw(U) p Fo 1702 1319 a(\003) p Fx 1763 1354 a(=) p Fw 1850 1354 a(a) p Fx(\() p Fw(f) p Fx 1976 1354 a(\)) p Ft 2027 1354 a(\012) p Fx 2110 1354 a(1) p 2170 1354 a(+) p 2253 1354 a(\() p Ft(\000) p Fx(1\)) p Fn 2424 1319 a(N) p Fg 2477 1327 a(\003) p Fj 2515 1335 a(v) p Ft 2576 1354 a(\012) p Fw 2659 1354 a(a) p Fx(\() p Fw(g) p Fx 2778 1354 a(\)) p Fw(:) p Fx 515 1600 a(By) p 645 1600 a(this) p 807 1600 a(prop) r(osition,) p 1267 1600 a(w) n(e) p 1389 1600 a(ha) n(v) n(e) p Fw 1107 1779 a(U) p Fx 1173 1779 a(\000\(e) p Fn 1294 1745 a(it) p Fl 1343 1729 a(~) p Fn 1342 1745 a(h) p Fg 1381 1753 a(\003) p Fj 1419 1761 a(v) p Ft 1481 1779 a(\010) p Fx 1564 1779 a(e) p Fn 1601 1743 a(it) p Fl 1650 1728 a(~) p Fn 1649 1743 a(h) p Fg 1688 1753 a(\003) p Fj 1726 1741 a(c) 1726 1770 y(v) p Fx 1770 1779 a(\)) p Fw(U) p Fo 1868 1745 a(\003) p Fx 1930 1779 a(=) p 2017 1779 a(\000\(e) p Fn 2138 1745 a(it) p Fl 2187 1729 a(~) p Fn 2186 1745 a(h) p Fg 2225 1753 a(\003) p Fj 2263 1761 a(v) p Fx 2308 1779 a(\)) p Ft 2358 1779 a(\012) p Fx 2441 1779 a(\000\(e) p Fn 2562 1743 a(it) p Fl 2611 1728 a(~) p Fn 2610 1743 a(h) p Fg 2649 1753 a(\003) p Fj 2687 1741 a(c) 2687 1770 y(v) p Fx 2732 1779 a(\)) p Fw(:) p Fx 515 1959 a(Recall) p 766 1959 a(that) p 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3157 a(2) p Ft 3373 3145 a(\012) p Fx 3456 3145 a(\() p Ft(\000) p Fx(1\)) p Fn 3627 3109 a(N) p Fg 3680 3119 a(\003) p Fj 3718 3107 a(c) 3718 3136 y(v) p Fw 3761 3145 a(;) p Fx 515 3330 a(and) p 676 3330 a(1) p Ft 736 3330 a(\012) p Fw 819 3330 a(N) p Fn 895 3299 a(c) p Fl 886 3352 a(\003) p Fx 963 3330 a(strongly) p 1285 3330 a(comm) n(ute,) p 1665 3330 a(where) p Fw 1906 3330 a(V) p Fl 1954 3342 a(1) p Fw 1991 3330 a(;) p 2028 3330 a(V) p Fl 2076 3342 a(2) p Fx 2142 3330 a(are) p 2280 3330 a(b) r(ounded) p 2619 3330 a(op) r(erators.) p 3020 3330 a(First) p 3222 3330 a(note) 515 3429 y(that) 1094 3594 y(e) p Fn 1131 3559 a(itL) p Fg 1225 3567 a(0) p Fx 1284 3594 a(=) p Fk 1372 3501 a(\020) p Fx 1421 3594 a(e) p Fn 1458 3559 a(itL) p Fj 1552 3567 a(S) p Ft 1617 3594 a(\012) p Fx 1700 3594 a(\000) p Fl 1752 3606 a(\003) p Fj 1797 3614 a(v) p Fk 1851 3501 a(\020) p Fx 1900 3594 a(e) p Fn 1937 3559 a(it) p Fl 1986 3544 a(~) p Fn 1985 3559 a(h) p Fg 2024 3567 a(\003) p Fj 2062 3575 a(v) p 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a(means) p Fw 961 4436 a(L) p Fl 1018 4448 a(0) p Fx 1083 4436 a(and) p Fw 1246 4436 a(N) p Fl 1313 4448 a(\003) p Fj 1358 4432 a(c) 1358 4466 y(v) p Fx 1426 4436 a(strongly) p 1749 4436 a(comm) n(ute) p 2107 4436 a([16) o(].) p 2300 4436 a(Next) p 2504 4436 a(w) n(e) p 2627 4436 a(sho) n(w) p 2834 4436 a(that) p Fw 3015 4436 a(L) p Fx 3100 4436 a(and) p Fw 3263 4436 a(N) p Fn 3339 4406 a(c) p Fl 3330 4459 a(\003) p Fx 515 4536 a(strongly) p 836 4536 a(comm) n(ute.) p 1226 4536 a(By) p 1356 4536 a(the) p 1499 4536 a(p) r(erturbation) p 1986 4536 a(theory) p 2213 4536 a(,) p 2263 4536 a(w) n(e) p 2385 4536 a(ha) n(v) n(e) 752 4763 y(e) p Fn 789 4728 a(iLt) p Fx 910 4763 a(=) p 998 4763 a(e) p Fn 1035 4728 a(iL) p Fg 1104 4736 a(0) p Fn 1136 4728 a(t) p Fx 1184 4763 a(+) p Fo 1296 4659 a(1) p Fk 1269 4684 a(X) p Fn 1267 4859 a(n) p Fl(=1) p Fk 1406 4650 a(Z) p Fn 1489 4670 a(t) p Fl 1452 4838 a(0) p Fw 1532 4763 a(dt) p Fl 1605 4775 a(1) p Fk 1656 4650 a(Z) p Fn 1739 4670 a(t) p Fg 1764 4678 a(1) p Fl 1702 4838 a(0) p Fw 1814 4763 a(dt) p Fl 1887 4775 a(2) p Ft 1939 4763 a(\001) p 1976 4763 a(\001) p 2013 4763 a(\001) p Fk 2049 4650 a(Z) p Fn 2132 4670 a(t) p Fj 2157 4678 a(n) p Ff(\000) p Fg(1) p Fl 2095 4838 a(0) p Fw 2289 4763 a(dt) p Fn 2362 4775 a(n) p Fw 2408 4763 a(V) p Fx 2475 4763 a(\() p Fw(t) p Fn 2537 4775 a(n) p Fx 2582 4763 a(\)) p Ft 2628 4763 a(\001) p 2665 4763 a(\001) p 2702 4763 a(\001) p Fw 2739 4763 a(V) p Fx 2806 4763 a(\() p Fw(t) p Fl 2868 4775 a(1) p Fx 2905 4763 a(\)e) p Fn 2974 4728 a(itL) p Fg 3068 4736 a(0) p Fw 3105 4763 a(;) 1520 4976 y(V) p Fx 1587 4976 a(\() p Fw(t) p Fx(\)) p 1704 4976 a(=) p 1792 4976 a(e) p Fn 1829 4942 a(itL) p Fg 1923 4950 a(0) p Fk 1973 4909 a(\000) p Fw 2012 4976 a(V) p Fl 2060 4988 a(1) p Ft 2116 4976 a(\012) p Fx 2199 4976 a(1) p 2259 4976 a(+) p Fw 2342 4976 a(V) p Fl 2390 4988 a(2) p Ft 2446 4976 a(\012) p Fx 2529 4976 a(\() p Ft(\000) p Fx(1\)) p Fn 2700 4941 a(N) p Fg 2753 4951 a(\003) p Fj 2791 4939 a(c) 2791 4967 y(v) p Fk 2834 4909 a(\001) p Fx 2886 4976 a(e) p Fo 2923 4942 a(\000) p Fn(itL) p Fg 3069 4950 a(0) p Fw 3105 4976 a(;) p Fx 1905 5255 a(18) p eop %%Page: 19 19 19 18 bop Fx 515 523 a(where) p 755 523 a(the) p 898 523 a(righ) n(t) p 1100 523 a(hand) p 1307 523 a(side) p 1474 523 a(con) n(v) n(erges) p 1846 523 a(in) p 1944 523 a(norm) p 2161 523 a(b) r(ecause) p 2468 523 a(of) p 2563 523 a(the) p 2706 523 a(b) r(oundedness) p 3194 523 a(of) p Fw 3289 523 a(V) p Fx 3356 523 a(.) 515 623 y(Using) p 748 623 a(\(21\),) p 946 623 a(w) n(e) p 1069 623 a(ha) n(v) n(e) p Fk 630 732 a(\002) p Fw 664 799 a(V) p Fx 731 799 a(\() p Fw(t) p Fx(\)) p Fw(;) p Fx 862 799 a(1) p Ft 922 799 a(\012) p Fx 1005 799 a(e) p Fn 1042 763 a(isN) p Fg 1149 773 a(\003) p Fj 1187 761 a(c) 1187 790 y(v) p Fk 1232 732 a(\003) p Fx 1289 799 a(=) p 1377 799 a(e) p Fn 1414 765 a(itL) p Fg 1508 773 a(0) p Fk 1558 732 a(\002) p Fw 1593 799 a(V) p Fl 1641 811 a(1) p Ft 1697 799 a(\012) p Fx 1780 799 a(1) p 1840 799 a(+) p Fw 1923 799 a(V) p Fl 1971 811 a(2) p Ft 2027 799 a(\012) p Fx 2110 799 a(\() p Ft(\000) p Fx(1\)) p Fn 2281 763 a(N) p Fg 2334 773 a(\003) p Fj 2372 761 a(c) 2372 790 y(v) p Fw 2415 799 a(;) p Fx 2452 799 a(1) p Ft 2512 799 a(\012) p Fx 2595 799 a(e) p Fn 2632 763 a(isN) p Fg 2739 773 a(\003) p Fj 2777 761 a(c) 2777 790 y(v) p Fk 2821 732 a(\003) p Fx 2869 799 a(e) p Fo 2906 765 a(\000) p Fn(itL) p Fg 3052 773 a(0) p Fx 3112 799 a(=) p 3200 799 a(0) p Fw(:) p Fx 515 975 a(Substituting) p 990 975 a(this) p 1152 975 a(to) p 1254 975 a(the) p 1397 975 a(p) r(erturbation) p 1884 975 a(expansion,) p 2292 975 a(w) n(e) p 2415 975 a(obtain) p Fk 1554 1084 a(\002) p Fx 1589 1151 a(e) p Fn 1626 1117 a(iLt) p Fw 1724 1151 a(;) p Fx 1761 1151 a(1) p Ft 1821 1151 a(\012) p Fx 1904 1151 a(e) p Fn 1941 1116 a(isN) p Fg 2048 1126 a(\003) p Fj 2086 1114 a(c) 2086 1142 y(v) p Fk 2130 1084 a(\003) p Fx 2187 1151 a(=) p 2275 1151 a(0) p Fw(:) p Fx 515 1327 a(This) p 704 1327 a(sho) n(ws) p 943 1327 a(that) p Fw 1123 1327 a(L) p Fx 1207 1327 a(and) p Fw 1368 1327 a(N) p Fl 1435 1339 a(\003) p Fj 1480 1323 a(c) 1480 1357 y(v) p Fx 1548 1327 a(strongly) p 1869 1327 a(comm) n(ute.) p Fd 2425 1327 a(\003) p Fx 515 1427 a(Similarly) p 834 1427 a(,) p Fw 891 1427 a(N) p Fl 958 1439 a(\003) p Fj 1003 1423 a(c) 1003 1456 y(v) p Fx 1077 1427 a(strongly) p 1404 1427 a(comm) n(utes) p 1800 1427 a(with) p Fw 1995 1427 a(L) p Fn 2052 1439 a(S) p Fx 2099 1427 a(,) p Fw 2157 1427 a(L) p Fn 2214 1439 a(f) p Fx 2257 1427 a(,) p Fw 2315 1427 a(L) p Fl 2372 1439 a(0) p Fx 2409 1427 a(,) p Fw 2467 1427 a(\025I) p Fx 2558 1427 a(,) p Fw 2616 1427 a(\025J) p 2718 1427 a(I) p 2761 1427 a(J) p Fx 2815 1427 a(,) p Fw 2874 1427 a(N) p Fx 2950 1427 a(,) p Fw 3008 1427 a(S) p Fl 3059 1439 a(1) p Fx 3096 1427 a(.) p 3173 1427 a(Using) 515 1527 y(this) p 677 1527 a(fact,) p 863 1527 a(w) n(e) p 986 1527 a(can) p 1138 1527 a(deriv) n(e) p 1383 1527 a(the) p 1526 1527 a(form) p 1722 1527 a(of) p Fw 1816 1527 a(L) p Fx('s) p 1956 1527 a(eign) n(v) n(ector;) p Fm 515 1688 a(Prop) s(osition) p 1031 1688 a(6.3) p Fy 1194 1688 a(Supp) l(ose) p Fw 1511 1688 a( ) p Fy 1597 1688 a(is) p 1686 1688 a(an) p 1805 1688 a(eigenve) l(ctor) p 2237 1688 a(of) p Fw 2335 1688 a(L) p Fy(.) p 2455 1688 a(Then) p Fw 2671 1688 a( ) p Fy 2758 1688 a(satis\014es) p Fw 1587 1864 a(P) p Fk 1665 1797 a(\000) p Fw 1703 1864 a(N) p Fl 1770 1876 a(\003) p Fj 1815 1860 a(c) 1815 1893 y(v) p Fx 1878 1864 a(=) p 1966 1864 a(0) p Fk 2008 1797 a(\001) p Fw 2059 1864 a( ) p Fx 2139 1864 a(=) p Fw 2227 1864 a( ) s(;) p Fy 515 2048 a(wher) l(e) p Fw 749 2048 a(P) p Fk 828 1981 a(\000) p Fw 866 2048 a(N) p Fl 933 2060 a(\003) p Fj 978 2044 a(c) 978 2077 y(v) p Fx 1041 2048 a(=) p 1129 2048 a(0) p Fk 1171 1981 a(\001) p Fy 1238 2048 a(is) p 1327 2048 a(the) p 1465 2048 a(sp) l(e) l(ctr) l (al) p 1761 2048 a(pr) l(oje) l(ction) p 2146 2048 a(of) p Fw 2243 2048 a(N) p Fl 2310 2060 a(\003) p Fj 2355 2044 a(c) 2355 2077 y(v) p Fy 2425 2048 a(c) l(orr) l(esp) l(onding) p 2949 2048 a(to) p Fw 3049 2048 a(N) p Fl 3116 2060 a(\003) p Fj 3161 2044 a(c) 3161 2077 y(v) p Fx 3224 2048 a(=) p 3311 2048 a(0) p Fy(.) 515 2209 y(Pr) l(o) l(of) p Fx 515 2309 a(Let) p Fw 657 2309 a(P) p Fn 710 2321 a(e) p Fx 767 2309 a(\(resp.) p Fw(P) p Fn 1024 2321 a(o) p Fx 1061 2309 a(\)) p 1114 2309 a(b) r(e) p 1220 2309 a(the) p 1357 2309 a(sp) r(ectral) p 1662 2309 a(pro) p 1788 2309 a(jection) p 2050 2309 a(of) p Fw 2138 2309 a(N) p Fl 2205 2321 a(\003) p Fj 2250 2305 a(c) 2250 2338 y(v) p Fx 2311 2309 a(on) n(to) p Fw 2491 2309 a(N) p Fl 2558 2321 a(\003) p Fj 2603 2305 a(c) 2603 2338 y(v) p Fx 2666 2309 a(=) p 2753 2309 a(ev) n(en) p 2936 2309 a(\(resp.) p Fw(N) p Fl 3207 2321 a(\003) p Fj 3252 2305 a(c) 3252 2338 y(v) p Fx 3314 2309 a(=) 515 2408 y(o) r(dd\),) p 736 2408 a(and) p 899 2408 a(let) p Fw 1021 2408 a(E) p Fn 1082 2420 a(n) p Fx 1157 2408 a(b) r(e) p 1272 2408 a(the) p 1417 2408 a(sp) r(ectral) p 1731 2408 a(pro) p 1857 2408 a(jection) p 2127 2408 a(on) n(to) p 2316 2408 a(the) p 2461 2408 a(in) n(terv) p 2669 2408 a(al) p 2763 2408 a([) p Ft(\000) p Fw(n;) p 2938 2408 a(n) p Fx(]) p 3039 2408 a(of) p Fw 3136 2408 a(L) p Fx(.) p 3258 2408 a(Let) 515 2508 y(us) p 621 2508 a(decomp) r(ose) p 1039 2508 a(the) p 1182 2508 a(Hilb) r(ert) p 1468 2508 a(space) p Ft 1690 2508 a(H) p Fx 1789 2508 a(as) p Ft 1633 2684 a(H) p Fx 1727 2684 a(=) p Fw 1815 2684 a(P) p Fn 1868 2696 a(e) p Ft 1904 2684 a(H) p 1993 2684 a(\010) p Fw 2076 2684 a(P) p Fn 2129 2696 a(o) p Ft 2166 2684 a(H) p Fw 2237 2684 a(:) p Fx 515 2861 a(Then) p Fw 732 2861 a(L) p Fx 816 2861 a(is) p 899 2861 a(decomp) r(osed) p 1363 2861 a(in) n(to) p Fw 1655 3037 a(L) p Fx 1734 3037 a(=) p Fw 1822 3037 a(P) p Fn 1875 3049 a(e) p Fw 1911 3037 a(L) p Ft 1986 3037 a(\010) p Fw 2069 3037 a(P) p Fn 2122 3049 a(o) p Fw 2159 3037 a(L;) p Fx 3245 3037 a(\(22\)) 515 3213 y(with) p 707 3213 a(resp) r(ect) p 995 3213 a(to) p 1100 3213 a(the) p 1246 3213 a(subspaces) p Fw 1629 3213 a(P) p Fn 1682 3225 a(e) p Ft 1718 3213 a(H) p Fx 1820 3213 a(and) p Fw 1985 3213 a(P) p Fn 2038 3225 a(o) p Ft 2076 3213 a(H) p Fx 2147 3213 a(.) p 2217 3213 a(This) p 2410 3213 a(is) p 2497 3213 a(deriv) n(ed) p 2791 3213 a(b) n(y) p 2910 3213 a(the) p 3056 3213 a(follo) n(wing) 515 3313 y(consideration:) p 1055 3313 a(for) p 1182 3313 a(all) p Fw 1297 3313 a(') p Ft 1375 3313 a(2) p Fw 1453 3313 a(D) p Fx 1524 3313 a(\() p Fw(L) p Fx(\),) p 1696 3313 a(w) n(e) p 1818 3313 a(ha) n(v) n(e) 1394 3489 y(lim) p Fn 1365 3539 a(n) p Fo(!1) p Fw 1552 3489 a(LE) p Fn 1670 3501 a(n) p Fw 1715 3489 a(') p Fx 1792 3489 a(=) p 1909 3489 a(lim) p Fn 1880 3539 a(n) p Fo(!1) p Fw 2067 3489 a(E) p Fn 2128 3501 a(n) p Fw 2174 3489 a(L') p Fx 2308 3489 a(=) p Fw 2395 3489 a(L':) p Fx 515 3689 a(By) p 645 3689 a(the) p 788 3689 a(strong) p 1042 3689 a(comm) n(utativit) n(y) p 1600 3689 a(of) p Fw 1695 3689 a(L) p Fx 1779 3689 a(and) p Fw 1941 3689 a(N) p Fl 2008 3701 a(\003) p Fj 2053 3685 a(c) 2053 3718 y(v) p Fx 2093 3689 a(,) p Fw 922 3866 a(P) p Fn 975 3878 a(e) p Fw 1011 3866 a(L') p Fx 1145 3866 a(=) p Fw 1233 3866 a(P) p Fn 1286 3878 a(e) p Fx 1365 3866 a(lim) p Fn 1336 3915 a(n) p Fo(!1) p Fw 1523 3866 a(LE) p Fn 1641 3878 a(n) p Fw 1686 3866 a(') p Fx 1763 3866 a(=) p 1880 3866 a(lim) p Fn 1851 3915 a(n) p Fo(!1) p Fw 2038 3866 a(P) p Fn 2091 3878 a(e) p Fw 2127 3866 a(LE) p Fn 2245 3878 a(n) p Fw 2290 3866 a(') p Fx 2367 3866 a(=) p 2484 3866 a(lim) p Fn 2455 3915 a(n) p Fo(!1) p Fw 2642 3866 a(LE) p Fn 2760 3878 a(n) p Fw 2805 3866 a(P) p Fn 2858 3878 a(e) p Fw 2894 3866 a(':) p Fx 515 4066 a(On) p 653 4066 a(the) p 796 4066 a(other) p 1013 4066 a(hands,) p 1276 4066 a(w) n(e) p 1399 4066 a(ha) n(v) n(e) 1619 4242 y(lim) p Fn 1590 4292 a(n) p Fo(!1) p Fw 1777 4242 a(E) p Fn 1838 4254 a(n) p Fw 1884 4242 a(P) p Fn 1937 4254 a(e) p Fw 1973 4242 a(') p Fx 2050 4242 a(=) p Fw 2138 4242 a(P) p Fn 2191 4254 a(e) p Fw 2226 4242 a(':) p Fx 515 4448 a(Hence,) p 785 4448 a(as) p Fw 886 4448 a(L) p Fx 971 4448 a(is) p 1054 4448 a(closed,) p 1322 4448 a(w) n(e) p 1444 4448 a(ha) n(v) n(e) p Fw 1636 4448 a(P) p Fn 1689 4460 a(e) p Fw 1725 4448 a(') p Ft 1802 4448 a(2) p Fw 1880 4448 a(D) p Fx 1951 4448 a(\() p Fw(L) p Fx(\)) p 2100 4448 a(and) p Fw 1680 4624 a(LP) p Fn 1790 4636 a(e) p Fw 1826 4624 a(') p Fx 1903 4624 a(=) p Fw 1991 4624 a(P) p Fn 2044 4636 a(e) p Fw 2080 4624 a(L':) p Fx 515 4800 a(Similarly) p 834 4800 a(,) p 884 4800 a(w) n(e) p 1006 4800 a(ha) n(v) n(e) p Fw 1198 4800 a(P) p Fn 1251 4812 a(o) p Fw 1288 4800 a(') p Ft 1365 4800 a(2) p Fw 1444 4800 a(D) p Fx 1515 4800 a(\() p Fw(L) p Fx(\)) p 1663 4800 a(and) p Fw 1679 4976 a(LP) p Fn 1789 4988 a(o) p Fw 1826 4976 a(') p Fx 1903 4976 a(=) p Fw 1991 4976 a(P) p Fn 2044 4988 a(o) p Fw 2081 4976 a(L':) p Fx 1905 5255 a(19) p eop %%Page: 20 20 20 19 bop Fx 515 523 a(Hence) p 772 523 a(w) n(e) p 904 523 a(obtain) p 1173 523 a(\(22\).) p 1412 523 a(In) p 1526 523 a(particular,) p 1945 523 a(if) p Fw 2032 523 a( ) p Fx 2127 523 a(is) p 2221 523 a(an) p 2346 523 a(eigen) n(v) n(ector) p 2788 523 a(of) p Fw 2893 523 a(L) p Fx(,) p Fw 3013 523 a(P) p Fn 3066 535 a(e) p Fw 3102 523 a( ) p Fx 3197 523 a(is) p 3291 523 a(an) 515 623 y(eigen) n(v) n(ector) p 948 623 a(of) p Fw 1043 623 a(P) p Fn 1096 635 a(e) p Fw 1132 623 a(L) p Fx 1218 623 a(and) p Fw 1381 623 a(P) p Fn 1434 635 a(o) p Fw 1471 623 a( ) p Fx 1557 623 a(is) p 1642 623 a(an) p 1758 623 a(eigen) n(v) n(ector) p 2191 623 a(of) p Fw 2287 623 a(P) p Fn 2340 635 a(o) p Fw 2377 623 a(L) p Fx(.) p 2498 623 a(In) p 2603 623 a(the) p 2747 623 a(represen) n (tation) p 3291 623 a(on) p Ft 515 722 a(H) p Fn 585 734 a(S) p Ft 651 722 a(\012) p 734 722 a(F) p Fx 802 722 a(\() 835 701 y(~) p Fh 834 722 a(h) p Fl 877 734 a(\003) p Fj 922 742 a(v) p Fx 963 722 a(\)) p Ft 1014 722 a(\012) p 1097 722 a(F) p Fx 1165 722 a(\() 1198 701 y(~) p Fh 1197 722 a(h) p Fl 1240 734 a(\003) p Fj 1285 718 a(c) 1285 751 y(v) p Fx 1326 722 a(\),) p Fw 1409 722 a(P) p Fn 1462 734 a(e) p Fw 1498 722 a(L) p Fx 1582 722 a(on) p Fw 1697 722 a(P) p Fn 1750 734 a(e) p Ft 1786 722 a(H) p Fx 1885 722 a(is) p Fw 525 922 a(P) p Fn 578 934 a(e) p Fw 614 922 a(L) p Ft(j) p Fn 693 947 a(P) p Fj 735 955 a(e) p Fo 768 947 a(H) p Fx 852 922 a(=) p Fk 939 830 a(\020) p Fx 989 922 a(\() p Fw(L) p Fn 1078 934 a(S) p Ft 1144 922 a(\012) p Fx 1227 922 a(1\)) p 1320 922 a(+) p 1403 922 a(1) p Ft 1463 922 a(\012) p Fx 1546 922 a(d\000) p Fl 1644 934 a(\003) p Fj 1689 942 a(v) p Fk 1743 830 a(\020) p Fx 1793 900 a(~) p Fw 1792 922 a(h) p Fl 1840 934 a(\003) p Fj 1885 942 a(v) p Fk 1925 830 a(\021) p Fx 1993 922 a(+) p Fw 2076 922 a(V) p Fl 2124 934 a(1) p Fx 2180 922 a(+) p Fw 2264 922 a(V) p Fl 2312 934 a(2) p Fk 2349 830 a(\021) p Ft 2417 922 a(\012) p Fx 2500 922 a(1) p 2560 922 a(+) p 2643 922 a(\(1) p Ft 2735 922 a(\012) p Fx 2818 922 a(1\)) p Ft 2911 922 a(\012) p Fx 2994 922 a(d\000) p Fl 3092 934 a(\003) p Fj 3137 918 a(c) 3137 951 y(v) p Fk 3191 830 a(\020) p Fx 3241 900 a(~) p Fw 3240 922 a(h) p Fl 3288 934 a(\003) p Fj 3333 918 a(c) 3333 951 y(v) p Fk 3373 830 a(\021) p Ft 3437 922 a(j) p Fo 3460 934 a(H) p Fj 3517 942 a(e) p Fx 515 1109 a(b) r(ecause) p Fw 1483 1275 a(P) p Fn 1536 1287 a(e) p Fx 1595 1275 a(=) p 1682 1275 a(1) p Ft 1742 1275 a(\012) p Fw 1825 1275 a(P) p Fx 1890 1275 a(\() p Fw(N) p Fl 1989 1287 a(\003) p Fj 2034 1271 a(c) 2034 1304 y(v) p Fx 2098 1275 a(=) p Fw 2185 1275 a(ev) s(en) p Fx(\)) p Fw(:) p Fx 515 1442 a(On) p 653 1442 a(the) p 796 1442 a(other) p 1013 1442 a(hand,) p Fw 1244 1442 a(P) p Fn 1297 1454 a(o) p Fw 1334 1442 a(L) p Fx 1418 1442 a(on) p Ft 1533 1442 a(H) p Fn 1603 1454 a(o) p Fx 1668 1442 a(is) p Fw 525 1629 a(P) p Fn 578 1641 a(o) p Fw 615 1629 a(L) p Ft(j) p Fo 694 1654 a(H) p Fj 751 1662 a(o) p Fx 812 1629 a(=) p Fk 900 1537 a(\020) p Fw 949 1629 a(L) p Fn 1006 1641 a(S) p Ft 1072 1629 a(\012) p Fx 1155 1629 a(1) p 1215 1629 a(+) p 1298 1629 a(1) p Ft 1358 1629 a(\012) p Fx 1441 1629 a(d\000) p Fl 1539 1641 a(\003) p Fj 1584 1649 a(v) p Fk 1638 1537 a(\020) p Fx 1689 1607 a(~) p Fw 1688 1629 a(h) p Fl 1736 1641 a(\003) p Fj 1781 1649 a(v) p Fk 1821 1537 a(\021) p Fx 1889 1629 a(+) p Fw 1972 1629 a(V) p Fl 2020 1641 a(1) p Ft 2076 1629 a(\000) p Fw 2159 1629 a(V) p Fl 2207 1641 a(2) p Fk 2245 1537 a(\021) p Ft 2313 1629 a(\012) p Fx 2396 1629 a(1) p 2456 1629 a(+) p 2539 1629 a(\(1) p Ft 2631 1629 a(\012) p Fx 2714 1629 a(1\)) p Ft 2806 1629 a(\012) p Fx 2889 1629 a(d\000) p Fl 2987 1641 a(\003) p Fj 3032 1625 a(c) 3032 1658 y(v) p Fx 3072 1629 a(\() 3105 1607 y(~) p Fw 3104 1629 a(h) p Fl 3152 1641 a(\003) p Fj 3197 1625 a(c) 3197 1658 y(v) p Fx 3238 1629 a(\)) p Ft(j) p Fo 3293 1641 a(H) p Fj 3350 1649 a(o) p Fx 515 1816 a(b) r(ecause) p Fw 1504 1982 a(P) p Fn 1557 1994 a(o) p Fx 1617 1982 a(=) p 1705 1982 a(1) p Ft 1765 1982 a(\012) p Fw 1848 1982 a(P) p Fx 1913 1982 a(\() p Fw(N) p Fl 2012 1994 a(\003) p Fj 2057 1978 a(c) 2057 2011 y(v) p Fx 2120 1982 a(=) p Fw 2208 1982 a(odd) p Fx(\)) p Fw(:) p Fx 515 2164 a(Note) p 717 2164 a(that) p 899 2164 a(d\000) p Fl 997 2176 a(\003) p Fj 1042 2160 a(c) 1042 2194 y(v) p Fx 1082 2164 a(\() 1115 2143 y(~) p Fw 1114 2164 a(h) p Fl 1162 2176 a(\003) p Fj 1207 2160 a(c) 1207 2194 y(v) p Fx 1247 2164 a(\)) p 1309 2164 a(and) p Fw 1473 2164 a(N) p Fl 1540 2176 a(\003) p Fj 1585 2160 a(c) 1585 2194 y(v) p Fx 1654 2164 a(strongly) p 1977 2164 a(comm) n(ute.So) p 2447 2164 a(d\000) p Fl 2545 2176 a(\003) p Fj 2590 2160 a(c) 2590 2194 y(v) p Fx 2630 2164 a(\() 2663 2143 y(~) p Fw 2662 2164 a(h) p Fl 2710 2176 a(\003) p Fj 2755 2160 a(c) 2755 2194 y(v) p Fx 2796 2164 a(\)) p 2857 2164 a(is) p 2943 2164 a(decomp) r(osed) 515 2279 y(in) n(to) p Fw 679 2279 a(P) p Fn 732 2291 a(e) p Fx 768 2279 a(d\000) p Fl 866 2291 a(\003) p Fj 911 2275 a(c) 911 2308 y(v) p Fx 951 2279 a(\() 984 2257 y(~) p Fw 983 2279 a(h) p Fl 1031 2291 a(\003) p Fj 1076 2275 a(c) 1076 2308 y(v) p Fx 1116 2279 a(\)) p Ft 1158 2279 a(\010) p Fw 1233 2279 a(P) p Fn 1286 2291 a(o) p Fx 1324 2279 a(d\000) p Fl 1422 2291 a(\003) p Fj 1467 2275 a(c) 1467 2308 y(v) p Fx 1507 2279 a(\() 1540 2257 y(~) p Fw 1539 2279 a(h) p Fl 1587 2291 a(\003) p Fj 1632 2275 a(c) 1632 2308 y(v) p Fx 1673 2279 a(\).) p 1763 2279 a(On) p 1898 2279 a(the) p 2037 2279 a(other) p 2250 2279 a(hand,) p 2477 2279 a(d\000) p Fl 2575 2291 a(\003) p Fj 2620 2275 a(c) 2620 2308 y(v) p Fx 2660 2279 a(\() 2693 2257 y(~) p Fw 2692 2279 a(h) p Fl 2740 2291 a(\003) p Fj 2785 2275 a(c) 2785 2308 y(v) p Fx 2825 2279 a(\)) p 2881 2279 a(in) p Ft 2974 2279 a(F) p Fx 3042 2279 a(\() 3075 2258 y(~) p Fh 3074 2279 a(h) p Fl 3117 2291 a(\003) p Fj 3162 2275 a(c) 3162 2308 y(v) p Fx 3203 2279 a(\)) p 3259 2279 a(has) 515 2394 y(unique) p 777 2394 a(eigen) n(v) n(ector) p 1201 2394 a(\012) p Fl 1261 2406 a(\003) p Fj 1306 2390 a(c) 1306 2423 y(v) p Fx 1366 2394 a(whic) n(h) p 1596 2394 a(is) p 1672 2394 a(the) p 1807 2394 a(v) p 1846 2394 a(acuum) p 2106 2394 a(v) n(ector) p 2348 2394 a(of) p Ft 2434 2394 a(F) p Fx 2502 2394 a(\() 2535 2372 y(~) p Fh 2534 2394 a(h) p Fl 2577 2406 a(\003) p Fj 2622 2390 a(c) 2622 2423 y(v) p Fx 2663 2394 a(\).) p 2753 2394 a(So) p 2870 2394 a(d\000) p Fl 2968 2406 a(\003) p Fj 3013 2390 a(c) 3013 2423 y(v) p Fk 3054 2323 a(\014) 3054 2373 y(\014) p Fn 3081 2427 a(P) p Fj 3123 2435 a(e) p Fo 3156 2427 a(F) p Fl 3210 2427 a(\() 3238 2412 y(~) p Fc 3236 2427 a(h) p Fg 3272 2437 a(\003) p Fj 3310 2425 a(c) 3310 2454 y(v) p Fl 3349 2427 a(\)) p Fx 515 2532 a(has) p 659 2532 a(the) p 798 2532 a(unique) p 1064 2532 a(eigen) n(v) n(ector) p 1492 2532 a(\012) p Fl 1552 2544 a(\003) p Fj 1597 2528 a(c) 1597 2561 y(v) p Fx 1637 2532 a(,) p 1685 2532 a(while) p 1908 2532 a(d\000) p Fl 2006 2544 a(\003) p Fj 2051 2528 a(c) 2051 2561 y(v) p Fk 2091 2462 a(\014) 2091 2512 y(\014) p Fn 2119 2566 a(P) p Fj 2161 2574 a(o) p Fo 2194 2566 a(F) p Fl 2248 2566 a(\() 2276 2550 y(~) p Fc 2274 2566 a(h) p Fg 2310 2576 a(\003) p Fj 2348 2564 a(c) 2348 2592 y(v) p Fl 2388 2566 a(\)) p Fx 2442 2532 a(has) p 2586 2532 a(no) p 2697 2532 a(eigen) n(v) n(ector.) p 3160 2532 a(Hence) 515 2658 y(b) n(y) p 630 2658 a(Theorem) p 981 2658 a(A.1,) p 1158 2658 a(if) p 1234 2658 a(there) p 1447 2658 a(exists) p 1676 2658 a(an) p 1791 2658 a(eigen) n(v) n (ector) p Fw 2223 2658 a( ) p Fn 2277 2670 a(e) p Fx 2340 2658 a(of) p Fw 2435 2658 a(P) p Fn 2488 2670 a(e) p Fw 2524 2658 a(L) p Fx(,) p 2631 2658 a(it) p 2714 2658 a(is) p 2798 2658 a(of) p 2892 2658 a(the) p 3035 2658 a(form) p Fw 1685 2825 a( ) p Fn 1739 2837 a(e) p Fx 1797 2825 a(=) p Fw 1885 2825 a(') p Ft 1958 2825 a(\012) p Fx 2041 2825 a(\012) p Fl 2101 2837 a(\003) p Fj 2146 2821 a(c) 2146 2854 y(v) p Fw 2186 2825 a(;) p Fx 515 2991 a(and) p Fw 676 2991 a(P) p Fn 729 3003 a(o) p Fw 766 2991 a(L) p Fx 851 2991 a(has) p 999 2991 a(no) p 1114 2991 a(eigen) n(v) n(ector.) p 1578 2991 a(Hence) p 1824 2991 a(if) p Fw 1901 2991 a( ) p Fx 1985 2991 a(is) p 2069 2991 a(the) p 2212 2991 a(eigen) p 2424 2991 a(v) n(ector) p 2673 2991 a(of) p Fw 2768 2991 a(L) p Fx(,) p 2875 2991 a(w) n(e) p 2997 2991 a(ha) n(v) n(e) p Fw 1454 3158 a(P) p Fn 1507 3170 a(e) p Fw 1543 3158 a( ) p Fx 1623 3158 a(=) p Fw 1710 3158 a(') p Ft 1783 3158 a(\012) p Fx 1866 3158 a(\012) p Fl 1926 3170 a(\003) p Fj 1971 3154 a(c) 1971 3187 y(v) p Fw 2011 3158 a(;) p 2117 3158 a(P) p Fn 2170 3170 a(o) p Fw 2208 3158 a( ) p Fx 2288 3158 a(=) p 2375 3158 a(0) p Fw(:) p Fx 515 3324 a(That) p 722 3324 a(is,) p Fw 1701 3491 a( ) p Fx 1781 3491 a(=) p Fw 1869 3491 a(') p Ft 1941 3491 a(\012) p Fx 2025 3491 a(\012) p Fl 2085 3503 a(\003) p Fj 2130 3487 a(c) 2130 3520 y(v) p Fw 2170 3491 a(:) p Fx 515 3657 a(This) p 704 3657 a(means) p Fw 696 3824 a(P) p Fx 761 3824 a(\() p Fw(N) p Fl 860 3836 a(\003) p Fj 905 3820 a(c) 905 3853 y(v) p Fx 969 3824 a(=) p 1056 3824 a(0\)) p Fw( ) p Fx 1210 3824 a(=) p Fw 1298 3824 a(') p Ft 1370 3824 a(\012) p Fw 1453 3824 a(P) p Fx 1518 3824 a(\() p Fw(N) p Fl 1617 3836 a(\003) p Fj 1662 3820 a(c) 1662 3853 y(v) p Fx 1726 3824 a(=) p 1813 3824 a(0\)\012) p Fl 1947 3836 a(\003) p Fj 1992 3820 a(c) 1992 3853 y(v) p Fx 2055 3824 a(=) p Fw 2143 3824 a(') p Ft 2216 3824 a(\012) p Fx 2299 3824 a(\012) p Fl 2359 3836 a(\003) p Fj 2404 3820 a(c) 2404 3853 y(v) p Fx 2467 3824 a(=) p Fw 2554 3824 a( ) s(:) p Fd 3133 3824 a(\003) p Fu 515 4096 a(7) p 716 4096 a(P) l(ositiv) l(e) p 1225 4096 a(Comm) l(utator) p Fx 515 4278 a(In) p 621 4278 a(this) p 786 4278 a(section,) p 1090 4278 a(w) n(e) p 1215 4278 a(construct) p 1583 4278 a(strictly) p 1871 4278 a(p) r(ositiv) n(e) p 2181 4278 a(comm) n(utator.) p 2690 4278 a(First) p 2896 4278 a(w) n(e) p 3021 4278 a(de\014ne) p 3264 4278 a(the) 515 4377 y(op) r(erator) p 850 4377 a([) p Fw(L;) p 967 4377 a(A) p Fl 1029 4389 a(0) p Fx 1066 4377 a(]) p 1116 4377 a(on) p Fw 1232 4377 a(D) p Fx 1303 4377 a(\([) p Fw(L;) p 1452 4377 a(A) p Fl 1514 4389 a(0) p Fx 1551 4377 a(]\)) p Ft 1629 4377 a(\021) p Fw 1717 4377 a(D) p Fx 1788 4377 a(\() p Fw(N) p Fx 1896 4377 a(\)) p 1956 4377 a(b) n(y) 1605 4544 y([) p Fw(L;) p 1722 4544 a(A) p Fl 1784 4556 a(0) p Fx 1821 4544 a(]) p Ft 1867 4544 a(\021) p Fw 1954 4544 a(S) p Fl 2005 4556 a(1) p Fx 2061 4544 a(+) p Fw 2144 4544 a(\025I) p Fl 2228 4556 a(1) p Fw 2266 4544 a(:) p Fx 515 4710 a(Note) p 722 4710 a(that) p 909 4710 a(this) p 1078 4710 a(is) p 1168 4710 a(de\014ned) p 1461 4710 a(as) p 1570 4710 a(an) p 1692 4710 a(op) r(erator) p 2034 4710 a(of) p 2135 4710 a(the) p 2285 4710 a(righ) n(t) p 2493 4710 a(hand) p 2708 4710 a(side,) p 2906 4710 a(and) p 3074 4710 a(not) p 3229 4710 a(as) p 3337 4710 a(a) 515 4810 y(comm) n(utator) p Fw 983 4810 a(LA) p Fl 1102 4822 a(0) p Ft 1158 4810 a(\000) p Fw 1241 4810 a(A) p Fl 1303 4822 a(0) p Fw 1340 4810 a(L) p Fx(.) p 1457 4810 a(Ho) n(w) n(ev) n(er) p 1791 4810 a(the) p 1934 4810 a(argumen) n(ts) p 2338 4810 a(in) p 2435 4810 a(Section) p 2726 4810 a(5) p 2795 4810 a(guaran) n(tees) p Ft 903 4976 a(h) p Fw(L';) p 1083 4976 a(A) p Fl 1145 4988 a(0) p Fw 1182 4976 a( ) p Ft 1239 4976 a(i) p Fx 1290 4976 a(+) p Ft 1373 4976 a(h) p Fw 1406 4976 a(A) p Fl 1468 4988 a(0) p Fw 1505 4976 a(';) p 1596 4976 a(L ) p Ft 1710 4976 a(i) p Fx 1765 4976 a(=) p Ft 1853 4976 a(h) p Fw(';) p 1976 4976 a(S) p Fl 2027 4988 a(1) p Fx 2083 4976 a(+) p Fw 2166 4976 a(\025I) p Fl 2250 4988 a(1) p Fw 2288 4976 a( ) p Ft 2345 4976 a(i) p Fx 2401 4976 a(=) p Ft 2488 4976 a(h) p Fw(';) p Fx 2611 4976 a([) p Fw 2635 4976 a(L;) p 2729 4976 a(A) p Fl 2791 4988 a(0) p Fx 2828 4976 a(]) p Fw 2865 4976 a( ) p Ft 2922 4976 a(i) p Fw 2968 4976 a(;) p Fx 1905 5255 a(20) p eop %%Page: 21 21 21 20 bop Fx 515 523 a(for) p 648 523 a(all) p Fw 769 523 a( ) s(;) p 863 523 a(') p Ft 951 523 a(2) p Fw 1039 523 a(D) p Fx 1110 523 a(\() p Fw(A) p Fl 1204 535 a(0) p Fx 1242 523 a(\)) p Ft 1297 523 a(\\) p Fw 1375 523 a(D) p Fx 1446 523 a(\() p Fw(N) p Fx 1554 523 a(\).) p 1664 523 a(Next) p 1873 523 a(w) n(e) p 2001 523 a(in) n(tro) r(duce) p 2376 523 a(the) p 2526 523 a(mo) r(di\014cation) p 3009 523 a(term.) p 3258 523 a(Let) p Fw 515 623 a(b) p Fx(\() p Fw(e) p Fx(\)) p 681 623 a(b) r(e) p 794 623 a(a) p 863 623 a(b) r(ounded) p 1202 623 a(op) r(erator) p 1537 623 a(de\014ned) p 1823 623 a(b) n(y) p Fw 1297 780 a(b) p Fx 1355 780 a(=) p Fw 1443 780 a(b) p Fx(\() p Fw(e) p Fx(\)) p 1605 780 a(=) p Fw 1693 780 a(\022) r(\025) p Fx(\() 1834 759 y(\026) p Fw 1814 780 a(QR) p Fl 1944 746 a(2) p Fn 1943 801 a(\017) p Fw 1981 780 a(I) p Fl 2017 792 a(0) p Fw 2055 780 a(Q) p Ft 2139 780 a(\000) p Fw 2222 780 a(QI) p Fl 2324 792 a(0) p Fw 2361 780 a(R) p Fl 2425 746 a(2) p Fn 2424 801 a(\017) p Fx 2481 759 a(\026) p Fw 2462 780 a(Q) p Fx(\)) p Fw(;) 1629 967 y(R) p Fn 1692 979 a(\017) p Fx 1747 967 a(=) p Fk 1835 875 a(\020) p Fx 1884 967 a(\() p Fw 1917 967 a(L) p Fl 1974 979 a(0) p Ft 2029 967 a(\000) p Fw 2112 967 a(e) p Fx(\)) p Fl 2183 926 a(2) p Fx 2239 967 a(+) p Fw 2322 967 a(\017) p Fl 2356 933 a(2) p Fk 2393 875 a(\021) p Fo 2442 892 a(\000) p Fg 2504 870 a(1) p 2504 879 29 3 v 2504 912 a(2) p Fw 2560 967 a(;) p Fx 515 1156 a(where) p Fw 765 1156 a(Q) p Ft 872 1156 a(\021) p Fw 977 1156 a(P) p Fx 1042 1156 a(\() p Fw(L) p Fn 1131 1168 a(S) p Fx 1220 1156 a(=) p Fw 1325 1156 a(e) p Fx(\)) p Ft 1422 1156 a(\012) p Fw 1512 1156 a(P) p Fl 1565 1168 a(\012) p Fw 1617 1156 a(;) p Fx 1756 1135 a(\026) p Fw 1737 1156 a(Q) p Ft 1843 1156 a(\021) p Fx 1948 1156 a(1) p Ft 2015 1156 a(\000) p Fw 2106 1156 a(Q) p Fx 2210 1156 a(and) p Fw 2382 1156 a(I) p Fl 2418 1168 a(0) p Fx 2494 1156 a(is) p 2588 1156 a(de\014ned) p 2884 1156 a(at) p 2997 1156 a(\(20\).) p 3236 1156 a(The) 515 1255 y(parameters) p Fw 951 1255 a(\022) p Fx 1026 1255 a(and) p Fw 1194 1255 a(\017) p Fx 1262 1255 a(will) p 1425 1255 a(b) r(e) p 1544 1255 a(determined) p 1984 1255 a(later.) p 2230 1255 a(W) p 2308 1255 a(e) p 2379 1255 a(de\014ne) p 2625 1255 a([) p Fw(L;) p 2742 1255 a(b) p Fx(\() p Fw(e) p Fx(\)]) p 2938 1255 a(on) p Fw 3059 1255 a(D) p Fx 3130 1255 a(\() p Fw(N) p Fx 3238 1255 a(\)) p 3305 1255 a(as) 515 1355 y([) p Fw(L;) p 632 1355 a(b) p Fx(\() p Fw(e) p Fx(\)]) p Ft 816 1355 a(\021) p Fw 904 1355 a(Lb) p Fx(\() p Fw(e) p Fx(\)) p Ft 1118 1355 a(\000) p Fw 1202 1355 a(b) p Fx(\() p Fw(e) p Fx(\)) p Fw(L) p Fx(.) p 1457 1355 a(Note) p 1658 1355 a(that) p Fw 1838 1355 a(L) p Fn 1895 1367 a(S) p Fw 1943 1355 a(b) p Fx(\() p Fw(e) p Fx(\)) p Ft 2100 1355 a(\000) p Fw 2183 1355 a(b) p Fx(\() p Fw(e) p Fx(\)) p Fw(L) p Fn 2379 1367 a(S) p Fx 2455 1355 a(and) p Fw 2616 1355 a(\025I) p Fl 2700 1367 a(0) p Fw 2739 1355 a(b) p Fx(\() p Fw(e) p Fx(\)) p Ft 2896 1355 a(\000) p Fw 2979 1355 a(b) p Fx(\() p Fw(e) p Fx(\)) p Fw(\025I) p Fl 3202 1367 a(0) p Fx 3268 1355 a(are) 515 1455 y(b) r(ounded.) p 886 1455 a(F) p 933 1455 a(or) p 1035 1455 a(all) p Fw 1150 1455 a( ) p Ft 1230 1455 a(2) p Fw 1309 1455 a(D) p Fx 1380 1455 a(\() p Fw(N) p Fx 1488 1455 a(\),) 515 1637 y([) p Fw(L) p Fn 595 1649 a(f) p Fw 637 1637 a(;) p 674 1637 a(b) p Fx(\() p Fw(e) p Fx(\)]) p Fw( ) p Fx 916 1637 a(=) p Fw 1004 1637 a(\022) r(\025) p Fk 1107 1570 a(\000) p Fx 1164 1616 a(\026) p Fw 1146 1637 a(Q) o(R) p Fl 1275 1603 a(2) p Fn 1274 1658 a(\017) p Fx 1326 1637 a([) p Fw(L) p Fn 1406 1649 a(f) p Fw 1449 1637 a(;) p 1486 1637 a(I) p Fl 1522 1649 a(0) p Fx 1559 1637 a(]) p Fw 1596 1637 a(Q) p Ft 1680 1637 a(\000) p Fw 1763 1637 a(Q) p Fx 1843 1637 a([) p Fw(L) p Fn 1923 1649 a(f) p Fw 1965 1637 a(;) p 2002 1637 a(I) p Fl 2038 1649 a(0) p Fx 2076 1637 a(]) p Fw 2113 1637 a(R) p Fl 2177 1603 a(2) p Fn 2176 1658 a(\017) p Fx 2233 1616 a(\026) p Fw 2214 1637 a(Q) p Fk 2279 1570 a(\001) p Fw 2331 1637 a( ) p Fx 2411 1637 a(=) p Fw 2499 1637 a(\022) r(\025) p Fk 2602 1545 a(\020) p Fx 2671 1616 a(\026) p Fw 2652 1637 a(QR) p Fl 2782 1603 a(2) p Fn 2781 1658 a(\017) p Fx 2829 1616 a(~) p Fw 2819 1637 a(I) p Fl 2855 1649 a(1) p Fw 2892 1637 a(Q) p Ft 2976 1637 a(\000) p Fw 3059 1637 a(Q) p 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a(<) p 2361 2918 a(t) p Fl 2391 2930 a(4) p Fw 2429 2918 a(;) p 2550 2918 a(\022) p Fo 2591 2884 a(\000) p Fl(1) p Fw 2681 2918 a(\017) p Fx 2733 2918 a(+) p Fw 2816 2918 a(\022) r(\025) p Fl 2905 2884 a(2) p Fw 2943 2918 a(\017) p Fo 2977 2884 a(\000) p Fl(3) p Fw 3089 2918 a(<) p 3176 2918 a(t) p Fl 3206 2930 a(5) p Fw 3244 2918 a(;) p Fy 515 3092 a(then) p Fw 1261 3298 a(P) p 1326 3298 a(E) p Fl 1392 3264 a(0) 1387 3319 y(\001) p Fw 1446 3298 a(B) t(E) p Fl 1579 3264 a(0) 1574 3319 y(\001) p Fw 1633 3298 a(P) p Ft 1721 3298 a(\025) p Fw 1818 3242 a(\022) r(\025) p Fl 1907 3212 a(2) p 1818 3279 127 4 v Fw 1865 3355 a(\017) 1955 3298 y(\015) p Fn 1998 3310 a(e) p Fk 2048 3206 a(\020) p Fx 2097 3298 a(1) p Ft 2157 3298 a(\000) p Fx 2240 3298 a(7) 2300 3277 y(~) p Fw 2282 3298 a(P) p Fn 2335 3310 a(e) p Fk 2371 3206 a(\021) p Fw 2434 3298 a(E) p Fl 2500 3264 a(0) 2495 3319 y(\001) p Fw 2554 3298 a(P) r(;) p Fy 544 3514 a(wher) l(e) p Fx 797 3493 a(~) p Fw 779 3514 a(P) p Fn 832 3526 a(e) p Fy 898 3514 a(is) p Fx 1385 3667 a(~) p Fw 1366 3688 a(P) p Fn 1419 3700 a(e) p Fx 1478 3688 a(=) p Fw 1566 3688 a(P) p Fx 1631 3688 a(\() p Fw(L) p Fn 1720 3700 a(S) p Fx 1791 3688 a(=) p Fw 1878 3688 a(e) p Fx(\)) p Ft 1968 3688 a(\001) p Fx 2009 3688 a(Ker) o(\000\() p Fw(e) p Fx(\)) p Ft 2317 3688 a(\012) p Fw 2400 3688 a(P) p Fl 2453 3700 a(\012) p Fw 2505 3688 a(;) p Fy 515 3862 a(with) p Fw 695 3862 a(P) p Fl 748 3874 a(\012) p Fy 800 3862 a(,) p 855 3862 a(the) p 993 3862 a(pr) l(oje) l(ction) p 1378 3862 a(onto) p Fx 1567 3862 a(\012) p Fy(.) p Fm 515 4035 a(Remark) p 878 4035 a(7.3) p Fy 1041 4035 a(By) p 1174 4035 a(the) p 1317 4035 a(de\014nition,) p Fx 1720 4035 a(\000\() p Fw(e) p Fx(\)) p Fy 1910 4035 a(is) p 2003 4035 a(p) l(ositive.) p 2351 4035 a(As) p Fw 2481 4035 a(\015) p Fn 2524 4047 a(e) p Fy 2594 4035 a(is) p 2688 4035 a(the) p 2831 4035 a(minimal) p 3163 4035 a(eigen-) 515 4135 y(value) p 729 4135 a(ab) l(ove) p 954 4135 a(0,) p 1052 4135 a(it) p 1134 4135 a(is) p 1223 4135 a(strictly) p 1502 4135 a(p) l(ositive) p 1802 4135 a(if) p Fx 1882 4135 a(\000\() p Fw(e) p Fx(\)) p Ft 2060 4135 a(6) p Fx(=) p 2148 4135 a(0) p Fy(.) p Fx 515 4309 a(The) p 691 4309 a(pro) r(of) p 913 4309 a(go) r(es) p 1101 4309 a(parallel) p 1402 4309 a(to) p 1508 4309 a([10) o(].) p 1713 4309 a(The) p 1889 4309 a(di\013erence) p 2266 4309 a(emerges) p 2586 4309 a(from) p 2787 4309 a(the) p 2935 4309 a(di\013erence) p 3312 4309 a(of) 515 4408 y(the) p 657 4408 a(comm) n(utator) p 1124 4408 a([) p Fw(L) p Fl 1204 4420 a(0) p Fw 1241 4408 a(;) p 1278 4408 a(A) p Fl 1340 4420 a(0) p Fx 1377 4408 a(]:) p 1460 4408 a(in) p 1556 4408 a([10) o(],) p 1735 4408 a(it) p 1817 4408 a(w) n(as) p 1975 4408 a([) p Fw(L) p Fl 2055 4420 a(0) p Fw 2092 4408 a(;) p 2129 4408 a(A) p Fl 2191 4420 a(0) p Fx 2228 4408 a(]) p 2274 4408 a(=) p Fw 2362 4408 a(N) p Fx 2438 4408 a(,) p 2488 4408 a(while) p 2704 4408 a(w) n(e) p 2825 4408 a(ha) n(v) n(e) p 3015 4408 a([) p Fw(L) p Fl 3095 4420 a(0) p Fw 3132 4408 a(;) p 3169 4408 a(A) p Fl 3231 4420 a(0) p Fx 3268 4408 a(]) p 3314 4408 a(=) p Fw 515 4508 a(S) p Fl 566 4520 a(1) p Fx 603 4508 a(.) p 695 4508 a(Whereas) p Fw 1046 4508 a(N) p Fx 1160 4508 a(has) p 1319 4508 a(a) p 1399 4508 a(sp) r(ectral) p 1722 4508 a(gap,) p Fw 1916 4508 a(S) p Fl 1967 4520 a(1) p Fx 2042 4508 a(has) p 2201 4508 a(no) p 2327 4508 a(sp) r(ectral) p 2650 4508 a(gap.) p 2872 4508 a(Since) p 3099 4508 a(p) r(ositiv) n(e) 515 4608 y(comm) n(utator) p 989 4608 a(metho) r(d) p 1297 4608 a(mak) n(es) p 1553 4608 a(use) p 1702 4608 a(of) p 1803 4608 a(the) p 1951 4608 a(\014nite) p 2170 4608 a(sp) r(ectral) p 2488 4608 a(gap,) p 2675 4608 a(the) p 2824 4608 a(lac) n(k) p 3000 4608 a(of) p 3101 4608 a(the) p 3250 4608 a(gap) 515 4707 y(causes) p 770 4707 a(a) p 840 4707 a(di\016cult) n(y) p 1151 4707 a(.) p 1212 4707 a(Ho) n(w) n(ev) n(er,) p 1571 4707 a(if) p 1647 4707 a(w) n(e) p 1770 4707 a(restrict) p 2058 4707 a(ourselv) n(es) p 2410 4707 a(to) p 2512 4707 a(the) p 2655 4707 a(subspace) p Fw 3003 4707 a(P) p Ft 3068 4707 a(H) p Fx 3167 4707 a(where) p Fw 515 4807 a(P) p Fx 604 4807 a(=) p Fw 693 4807 a(P) p Fx 758 4807 a(\() p Fw(N) p Fn 866 4777 a(c) 857 4827 y(v) p Fx 924 4807 a(=) p 1012 4807 a(0\),) p Fw 1138 4807 a(S) p Fl 1189 4819 a(1) p Ft 1226 4807 a(j) p Fn 1249 4819 a(P) p Fo 1300 4819 a(H) p Fx 1390 4807 a(has) p 1539 4807 a(a) p 1608 4807 a(sp) r(ectral) p 1921 4807 a(gap.) p 2112 4807 a(That) p 2321 4807 a(is) p 2405 4807 a(wh) n(y) p 2581 4807 a(w) n(e) p 2704 4807 a(in) n(tro) r(duced) p 3120 4807 a(the) p 3264 4807 a(cut) 515 4907 y(o\013) p 634 4907 a(of) p 731 4907 a(the) p 875 4907 a(in) n(teraction) p 1295 4907 a(in) p 1393 4907 a(Section) p 1686 4907 a(6.) p 1793 4907 a(T) p 1846 4907 a(o) p 1917 4907 a(pro) n(v) n(e) p 2143 4907 a(the) p 2287 4907 a(theorem,) p 2636 4907 a(w) n(e) p 2760 4907 a(use) p 2905 4907 a(the) p 3050 4907 a(F) p 3097 4907 a(esh) n(bac) n(h) 515 5006 y(map) p 699 5006 a(theorem) p 1022 5006 a([13) o(];) 1905 5255 y(22) p eop %%Page: 23 23 23 22 bop Fm 515 523 a(Theorem) p 920 523 a(7.4) p Fy 1084 523 a(L) l(et) p Fw 1223 523 a(H) p Fy 1325 523 a(b) l(e) p 1423 523 a(a) p 1492 523 a(close) l(d) p 1730 523 a(op) l(er) l(ator) p 2053 523 a(densely) p 2340 523 a(de\014ne) l(d) p 2617 523 a(on) p 2732 523 a(a) p 2801 523 a(Hilb) l(ert) p 3070 523 a(sp) l(ac) l(e) p Ft 3283 523 a(H) p Fy 3354 523 a(,) 515 623 y(and) p 673 623 a(let) p Fw 787 623 a(P) p Fy 879 623 a(b) l(e) p 979 623 a(a) p 1048 623 a(pr) l(oje) l(ction) p 1431 623 a(op) l(er) l(ator) p 1755 623 a(such) p 1941 623 a(that) p Fx 2108 623 a(Ran) p Fw(P) p Ft 2345 623 a(\032) p Fw 2432 623 a(D) p Fx 2503 623 a(\() p Fw(H) p Fx 2611 623 a(\)) p Fy(.) p 2707 623 a(We) p 2855 623 a(set) p Fx 3000 602 a(\026) p Fw 2982 623 a(P) p Fx 3069 623 a(=) p 3157 623 a(1) p Ft 3212 623 a(\000) p Fw 3290 623 a(P) p Fy 3355 623 a(.) 515 722 y(Then) p Fw 730 722 a(H) p Fl 813 727 a(\026) p Fn 799 742 a(P) p Ft 877 722 a(\021) p Fx 984 701 a(\026) p Fw 965 722 a(P) p 1030 722 a(H) p Fx 1124 701 a(\026) p Fw 1106 722 a(P) p Fy 1199 722 a(is) p 1287 722 a(a) p 1358 722 a(densely) p 1648 722 a(de\014ne) l(d) p 1927 722 a(op) l(er) l(ator) p 2252 722 a(on) p Fx 2388 701 a(\026) p Fw 2370 722 a(P) p Ft 2435 722 a(H) p Fy 2506 722 a(.) p 2569 722 a(L) l(et) p Fw 2711 722 a(z) p Fy 2782 722 a(b) l(e) p 2883 722 a(an) p 3000 722 a(element) p 3307 722 a(in) 515 822 y(the) p 653 822 a(r) l(esolvent) p 1000 822 a(set) p Fw 1129 822 a(\032) p Fx(\() p Fw(H) p Fl 1288 827 a(\026) p Fn 1273 842 a(P) p Fx 1329 822 a(\)) p Fy 1391 822 a(of) p Fw 1488 822 a(H) p Fl 1572 827 a(\026) p Fn 1557 842 a(P) p Fy 1642 822 a(on) p Fx 1780 801 a(\026) p Fw 1761 822 a(P) p Ft 1826 822 a(H) p Fy 1897 822 a(.) p 1960 822 a(Assume) p Fk 869 924 a(\015) 869 974 y(\015) 869 1024 y(\015) p Fw 915 1020 a(P) p 980 1020 a(H) p Fx 1074 999 a(\026) p Fw 1056 1020 a(P) p Fx 1134 1020 a(\() p Fw(H) p Fl 1250 1025 a(\026) p Fn 1235 1040 a(P) p Ft 1309 1020 a(\000) p Fw 1392 1020 a(z) p Fx 1435 1020 a(\)) p Fo 1467 978 a(\000) p Fl(1) p Fx 1588 999 a(\026) p Fw 1570 1020 a(P) p Fk 1635 924 a(\015) 1635 974 y(\015) 1635 1024 y(\015) p Fw 1704 1020 a(<) p Ft 1791 1020 a(1) p Fw(;) p Fk 1996 924 a(\015) 1996 974 y(\015) 1996 1024 y(\015) p Fx 2061 999 a(\026) p Fw 2042 1020 a(P) p Fx 2121 1020 a(\() p Fw(H) p Fl 2237 1025 a(\026) p Fn 2222 1040 a(P) p Ft 2296 1020 a(\000) p Fw 2379 1020 a(z) p Fx 2422 1020 a(\)) p Fo 2454 978 a(\000) p Fl(1) p Fx 2575 999 a(\026) p Fw 2557 1020 a(P) p 2621 1020 a(H) p 2697 1020 a(P) p Fk 2762 924 a(\015) 2762 974 y(\015) 2762 1024 y(\015) p Fw 2831 1020 a(<) p Ft 2919 1020 a(1) p Fw(:) p Fy 515 1213 a(Then) p 731 1213 a(the) p 869 1213 a(F) p 917 1213 a(eshb) l(ach) p 1218 1213 a(map) p Fw 1028 1386 a(f) p Fn 1069 1398 a(P) p Fx 1124 1386 a(\() p Fw(H) p Ft 1250 1386 a(\000) p Fw 1333 1386 a(z) p Fx 1376 1386 a(\)) p Ft 1431 1386 a(\021) p Fw 1518 1386 a(P) p Fx 1583 1386 a(\() p 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748 1832 a(\() p Fw(H) p Fx 856 1832 a(\)) p Ft 935 1832 a(\() p 1004 1832 a(\)) p Fx 1133 1832 a(0) p Ft 1197 1832 a(2) p Fw 1276 1832 a(\033) p Fx 1326 1832 a(\() p Fw(f) p Fn 1399 1844 a(P) p Fx 1455 1832 a(\() p Fw(H) p Ft 1581 1832 a(\000) p Fw 1664 1832 a(z) p Fx 1707 1832 a(\)\)) p Fw(;) p 1893 1832 a(z) p Ft 1958 1832 a(2) p Fw 2036 1832 a(\033) p Fn 2083 1844 a(P) p 2134 1844 a(P) p Fx 2190 1832 a(\() p Fw(H) p Fx 2298 1832 a(\)) p Ft 2377 1832 a(\() p 2446 1832 a(\)) p Fx 2575 1832 a(0) p Ft 2640 1832 a(2) p Fw 2718 1832 a(\033) p Fn 2765 1844 a(P) p 2816 1844 a(P) p Fx 2872 1832 a(\() p Fw(f) p Fn 2945 1844 a(P) p Fx 3000 1832 a(\() p Fw(H) p Ft 3127 1832 a(\000) p Fw 3210 1832 a(z) p Fx 3253 1832 a(\)\)) p Fw(:) p Fy 515 2005 a(Her) l(e) p Fw 713 2005 a(\033) p Fy 793 2005 a(and) p Fw 955 2005 a(\033) p Fn 1002 2017 a(P) p 1053 2017 a(P) p Fy 1138 2005 a(r) l(epr) l(esent) p 1495 2005 a(sp) l(e) l(ctrum) p 1844 2005 a(and) p 2005 2005 a(pur) l(e) p 2191 2005 a(p) l(oint) p 2401 2005 a(sp) l(e) l(ctrum) p 2750 2005 a(r) l(esp) l(e) l(ctively.) p Fx 515 2163 a(T) p 568 2163 a(o) p 637 2163 a(apply) p 865 2163 a(the) p 1008 2163 a(theorem,) p 1354 2163 a(w) n(e) p 1477 2163 a(in) n(tro) r(duce) p Fw 1652 2337 a(\037) p Fn 1704 2349 a(\027) p Ft 1769 2337 a(\021) p Fw 1856 2337 a(P) p Fx 1921 2337 a(\() p Fw(N) p Ft 2052 2337 a(\024) p Fw 2140 2337 a(\027) p Fx 2186 2337 a(\)) p Fw(:) p Fx 515 2525 a(Pro) p 652 2525 a(jections) 971 2504 y(~) p Fw 952 2525 a(P) p Fn 1005 2537 a(e) p Fw 1041 2525 a(;) p 1078 2525 a(\037) p Fn 1130 2537 a(\027) p Fw 1172 2525 a(;) p 1209 2525 a(P) r(;) p 1301 2525 a(Q;) p 1404 2525 a(E) p Fl 1470 2494 a(0) 1465 2547 y(\001) p Fx 1551 2525 a(are) p 1690 2525 a(in) p 1787 2525 a(relation) p 2091 2525 a(that) 1043 2677 y(~) p Fw 1025 2698 a(P) p Fn 1078 2710 a(e) p Ft 1137 2698 a(\024) p Fw 1224 2698 a(\037) p Fn 1276 2710 a(\027) p Fw 1318 2698 a(;) p Fx 1456 2677 a(~) p Fw 1437 2698 a(P) p Fn 1490 2710 a(e) p Ft 1549 2698 a(\024) p Fw 1637 2698 a(P) r(;) p Fx 1831 2677 a(~) p Fw 1813 2698 a(P) p Fn 1866 2710 a(e) p Ft 1925 2698 a(\024) p Fw 2012 2698 a(Q;) p 2198 2698 a(Q) p Ft 2287 2698 a(\024) p Fw 2374 2698 a(E) p Fl 2440 2663 a(0) 2435 2718 y(\001) p Fw 2494 2698 a(;) p 2614 2698 a(Q) p Ft 2703 2698 a(\024) p Fw 2790 2698 a(P) r(:) p Fx 515 2871 a(Hence) p 761 2871 a(w) n(e) p 884 2871 a(ha) n(v) n(e) p 1075 2871 a(the) p 1218 2871 a(follo) n(wings;) 1216 3023 y(\026) p Fw 1197 3044 a(Q) p Fx 1281 3023 a(~) p Fw 1263 3044 a(P) p Fn 1316 3056 a(e) p Fx 1375 3044 a(=) p 1463 3044 a(0) p Fw(;) p Fx 1643 3023 a(\026) p Fw 1624 3044 a(Q) p Fx 1708 3023 a(~) p Fw 1690 3044 a(P) p Fo 1755 3009 a(?) p Fn 1743 3064 a(e) p Fx 1834 3044 a(=) 1940 3023 y(\026) p Fw 1921 3044 a(Q;) p 2107 3044 a(E) p Fl 2173 3009 a(0) 2168 3064 y(\001) p Fw 2227 3044 a(P) p 2292 3044 a(\037) p Fn 2344 3056 a(\027) p Fx 2403 3023 a(~) p Fw 2385 3044 a(P) p Fn 2438 3056 a(e) p Fx 2497 3044 a(=) 2603 3023 y(~) p Fw 2584 3044 a(P) p Fn 2637 3056 a(e) p Fw 2673 3044 a(:) p Fx 515 3217 a(As) p Fw 638 3217 a(L) p Fl 695 3229 a(0) p Fx 732 3217 a(,) p Fw 783 3217 a(N) p Fl 850 3229 a(\003) p Fj 895 3213 a(c) 895 3246 y(v) p Fx 935 3217 a(,) p Fw 987 3217 a(N) p Fx 1063 3217 a(,) p Fw 1114 3217 a(L) p Fn 1171 3229 a(S) p Fx 1247 3217 a(strongly) p 1569 3217 a(comm) n(ute,) p 1951 3217 a(the) p 2094 3217 a(pro) p 2220 3217 a(jections) p Fw 2522 3217 a(E) p Fl 2588 3187 a(0) 2583 3240 y(\001) p Fx 2642 3217 a(,) p Fw 2694 3217 a(Q) p Fx(,) p Fw 2811 3217 a(P) p Fx 2876 3217 a(,) p Fw 2928 3217 a(\037) p Fn 2980 3229 a(\027) p Fx 3049 3217 a(comm) n(ute) 515 3316 y(eac) n(h) p 702 3316 a(other.) p 951 3316 a(So) p 1066 3316 a(w) n(e) p 1188 3316 a(can) p 1341 3316 a(de\014ne) p 1580 3316 a(the) p 1723 3316 a(follo) n(wing) p 2074 3316 a(pro) p 2200 3316 a(jections) p Fw 1318 3489 a(Q) p Fl 1384 3501 a(1) p Ft 1444 3489 a(\021) p Fw 1532 3489 a(QE) p Fl 1664 3455 a(0) 1659 3510 y(\001) p Fw 1717 3489 a(P) p 1782 3489 a(\037) p Fn 1834 3501 a(\027) p Fw 1875 3489 a(;) p 1995 3489 a(Q) p Fl 2061 3501 a(2) p Ft 2121 3489 a(\021) p Fx 2228 3468 a(\026) p Fw 2209 3489 a(QE) p Fl 2341 3455 a(0) 2336 3510 y(\001) p Fw 2394 3489 a(P) p 2459 3489 a(\037) p Fn 2511 3501 a(\027) p Fw 2553 3489 a(;) p Fx 3245 3489 a(\(24\)) 515 3663 y(whic) n(h) p 752 3663 a(satisfy) p Fw 1229 3836 a(Q) p Fl 1295 3848 a(1) p Fx 1350 3815 a(~) p Fw 1332 3836 a(P) p Fn 1385 3848 a(e) p Fx 1444 3836 a(=) 1550 3815 y(~) p Fw 1532 3836 a(P) p Fn 1585 3848 a(e) p Fw 1620 3836 a(;) p 1740 3836 a(Q) p Fl 1806 3848 a(2) p Fx 1862 3815 a(~) p Fw 1843 3836 a(P) p Fn 1896 3848 a(e) p Fx 1955 3836 a(=) p 2043 3836 a(0) p Fw(;) p 2204 3836 a(Q) p Fl 2270 3848 a(2) p Fx 2326 3815 a(~) p Fw 2307 3836 a(P) p Fo 2372 3801 a(?) p Fn 2360 3856 a(e) p Fx 2451 3836 a(=) p Fw 2539 3836 a(Q) p Fl 2605 3848 a(2) p Fw 2642 3836 a(:) p Fx 3231 3836 a(\(25\)) 515 4009 y(The) p 685 4009 a(relation) p Fw 990 4009 a(N) p 1066 4009 a(P) p Fx 1154 4009 a(=) p Fw 1241 4009 a(N) p Fl 1308 4021 a(\003) p Fj 1353 4029 a(v) p Fw 1393 4009 a(P) p Fx 1486 4009 a(implies) p Fw 1095 4182 a(N) p Fl 1162 4194 a(\003) p Fj 1207 4202 a(v) p Fw 1247 4182 a(Q) p Fl 1313 4194 a(2) p Fx 1373 4182 a(=) p Fw 1461 4182 a(N) p Fl 1528 4194 a(\003) p Fj 1573 4202 a(v) p Fw 1612 4182 a(P) p Fx 1696 4161 a(\026) p Fw 1677 4182 a(QE) p Fl 1809 4147 a(0) 1804 4202 y(\001) p Fw 1863 4182 a(\037) p Fn 1915 4194 a(\027) p Fx 1979 4182 a(=) p Fw 2067 4182 a(N) p 2143 4182 a(P) p Fx 2226 4161 a(\026) p Fw 2208 4182 a(Q) o(E) p Fl 2339 4147 a(0) 2334 4202 y(\001) p Fw 2393 4182 a(\037) p Fn 2445 4194 a(\027) p Fx 2509 4182 a(=) p Fw 2597 4182 a(N) p 2673 4182 a(Q) p Fl 2739 4194 a(2) p Fw 2776 4182 a(:) p Fx 515 4355 a(As) p Fw 637 4355 a(P) p Fx 702 4355 a(\() p Fw(N) p Fx 833 4355 a(=) p 921 4355 a(0\)) p 1018 4355 a(=) p 1105 4355 a(1) p Ft 1165 4355 a(\012) p Fw 1248 4355 a(P) p Fl 1301 4367 a(\012) p Fx 1353 4355 a(,) p 1404 4355 a(and) p 1565 4355 a(\001) p Ft 1653 4355 a(\\) p Fw 1727 4355 a(\033) p Fx 1777 4355 a(\() p Fw(L) p Fl 1866 4367 a(0) p Fx 1903 4355 a(\)) p 1959 4355 a(=) p Ft 2046 4355 a(f) p Fw(e) p Ft(g) p Fx(,) p 2219 4355 a(w) n(e) p 2341 4355 a(ha) n(v) n(e) p Fw 573 4544 a(P) p Fx 638 4544 a(\() p Fw(N) p Fx 769 4544 a(=) p 856 4544 a(0\)) 949 4523 y(\026) p Fw 930 4544 a(Q) p Fx 1019 4544 a(=) p 1107 4544 a(\(1) p Ft 1199 4544 a(\000) p Fw 1282 4544 a(P) p Fx 1347 4544 a(\() p Fw(L) p Fn 1436 4556 a(S) p Fx 1507 4544 a(=) p Fw 1594 4544 a(e) p Fx(\)\)) p Ft 1716 4544 a(\012) p Fw 1799 4544 a(P) p Fl 1852 4556 a(\012) p Fx 1927 4544 a(=) p Fk 2015 4466 a(X) p Fn 2014 4644 a(f) p Fo 2053 4644 a(6) p Fl(=) p Fn(e) p Fw 2150 4544 a(P) p Fx 2215 4544 a(\() p Fw(L) p Fn 2304 4556 a(S) p Fx 2374 4544 a(=) p Fw 2462 4544 a(f) p Fx 2512 4544 a(\)) p Ft 2562 4544 a(\012) p Fw 2645 4544 a(P) p Fl 2698 4556 a(\012) p Ft 2773 4544 a(\024) p Fw 2861 4544 a(P) p Fx 2926 4544 a(\() p Fw(L) p Fl 3015 4556 a(0) p Ft 3075 4544 a(\032) p Fx 3163 4544 a(\001) p Fn 3232 4510 a(c) p Fx 3266 4544 a(\)) p Fw(;) p Fx 515 4803 a(whic) n(h) p 752 4803 a(implies) p Fw 1223 4976 a(P) p Fx 1288 4976 a(\() p Fw(N) p Fx 1419 4976 a(=) p 1507 4976 a(0\)) p Fw(Q) p Fl 1647 4988 a(2) p Fx 1707 4976 a(=) p Fw 1794 4976 a(P) p Fx 1859 4976 a(\() p Fw(N) p Fx 1990 4976 a(=) p 2078 4976 a(0\)) 2171 4955 y(\026) p Fw 2152 4976 a(Q) o(E) p Fl 2283 4942 a(0) 2278 4997 y(\001) p Fw 2337 4976 a(P) p 2402 4976 a(\037) p Fn 2454 4988 a(\027) p Fx 2518 4976 a(=) p 2606 4976 a(0) p Fw(:) p Fx 1905 5255 a(23) p eop %%Page: 24 24 24 23 bop Fx 515 523 a(It) p 605 523 a(follo) n(ws) p 877 523 a(that) p Fw 1556 706 a(N) p 1632 706 a(Q) p Fl 1698 718 a(2) p Fx 1758 706 a(=) p Fw 1846 706 a(N) p Fl 1913 718 a(\003) p Fj 1958 726 a(v) p Fw 1998 706 a(Q) p Fl 2064 718 a(2) p Ft 2124 706 a(\025) p Fw 2211 706 a(Q) p Fl 2277 718 a(2) p Fw 2314 706 a(:) p Fx 3231 706 a(\(26\)) p Fy 515 888 a(Pr) l(o) l(of) p 738 888 a(of) p 835 888 a(The) l(or) l(em) p 1180 888 a(7.2) p Fx 515 988 a(W) p 593 988 a(e) p 658 988 a(apply) p 886 988 a(Theorem) p 1237 988 a(7.4) p 1371 988 a(to) p 1472 988 a(the) p 1615 988 a(op) r(erator) p Fw 1950 988 a(B) p Fo 2017 958 a(0) p Fx 2068 988 a(de\014ned) p 2354 988 a(b) n(y) p Fw 1624 1171 a(B) p Fo 1691 1136 a(0) p Ft 1737 1171 a(\021) p Fw 1825 1171 a(B) p Ft 1910 1171 a(\000) p Fw 1994 1171 a(\016) p Fn 2031 1183 a(e) p Ft 2085 1171 a(\001) p Fx 2145 1150 a(~) p Fw 2126 1171 a(P) p Fo 2191 1136 a(?) p Fn 2179 1191 a(e) p Fw 2247 1171 a(:) 515 1353 y(\016) p Fn 552 1365 a(e) p Fx 615 1353 a(will) p 772 1353 a(b) r(e) p 885 1353 a(de\014ned) p 1171 1353 a(later.) p 1397 1353 a(W) p 1475 1353 a(e) p 1540 1353 a(de\014ne) p 1780 1353 a(op) r(erators) p Fw 2147 1353 a(B) p Fo 2214 1323 a(0) p Fn 2210 1375 a(ij) p Fx 2297 1353 a(b) n(y) p Fw 1458 1547 a(B) p Fo 1525 1513 a(0) p Fn 1521 1568 a(i;j) p Ft 1622 1547 a(\021) p Fw 1710 1547 a(Q) p Fn 1776 1559 a(i) p Fw 1803 1547 a(B) p Fo 1870 1513 a(0) p Fw 1894 1547 a(Q) p Fn 1960 1559 a(j) p Fw 2077 1547 a(i;) p 2143 1547 a(j) p Fx 2205 1547 a(=) p 2293 1547 a(1) p Fw(;) p Fx 2372 1547 a(2) p Fw(:) p Fx 515 1730 a(If) p Fw 598 1730 a(#) p Ft 670 1730 a(2) p Fw 748 1730 a(\032) p Fx(\() p Fw(B) p Fo 890 1700 a(0) p Fl 886 1750 a(22) p Fx 957 1730 a(\),) p 1040 1730 a(then) p Fw 1327 1913 a(") p Fx 1380 1913 a(\() p Fw(#) p Fx(\)) p Ft 1517 1913 a(\021) p Fw 1604 1913 a(B) p Fo 1671 1878 a(0) p Fl 1667 1933 a(11) p Ft 1756 1913 a(\000) p Fw 1839 1913 a(B) p Fo 1906 1878 a(0) p Fl 1902 1933 a(12) p Fx 1972 1913 a(\() p Fw(B) p Fo 2071 1878 a(0) p Fl 2067 1933 a(22) p Ft 2156 1913 a(\000) p Fw 2239 1913 a(#) p Fx(\)) p Fo 2320 1878 a(\000) p Fl(1) p Fw 2410 1913 a(B) p Fo 2477 1878 a(0) p Fl 2473 1933 a(21) p Fw 2543 1913 a(;) p Fx 515 2095 a(is) p 598 2095 a(w) n(ell-de\014ned) p 1053 2095 a(and) p Fw 1446 2278 a(") p Fx 1499 2278 a(\() p Fw(#) p Fx(\)) p 1635 2278 a(=) p Fw 1723 2278 a(f) p Fn 1764 2290 a(Q) p Fg 1816 2298 a(1) p Fx 1866 2278 a(\() p Fw(B) p Fo 1965 2244 a(0) p Ft 2007 2278 a(\000) p Fw 2090 2278 a(#) p Fx(\)) p 2190 2278 a(+) p Fw 2273 2278 a(#Q) p Fl 2388 2290 a(1) p Fw 2425 2278 a(:) p Fx 515 2460 a(First) p 718 2460 a(w) n(e) p 842 2460 a(in) n(v) n(estigate) p 1256 2460 a(the) p 1400 2460 a(lo) n(w) n(er) p 1619 2460 a(b) r(ound) p 1876 2460 a(of) p Fw 1972 2460 a(B) p Fo 2039 2430 a(0) p Fl 2035 2481 a(22) p Fx 2134 2460 a(in) p 2232 2460 a(order) p 2450 2460 a(to) p 2553 2460 a(study) p 2783 2460 a(the) p 2927 2460 a(resolv) n(en) n(t) p 3277 2460 a(set) p Fw 515 2560 a(\032) p Fx(\() p Fw(B) p Fo 657 2530 a(0) p Fl 653 2581 a(22) p Fx 723 2560 a(\).) p Fm 515 2726 a(Prop) s(osition) p 1031 2726 a(7.5) p Fy 1194 2726 a(We) p 1345 2726 a(have) p Fw 1167 2954 a(B) p Fo 1234 2920 a(0) p Fl 1230 2975 a(22) p Ft 1323 2954 a(\025) p Fk 1411 2837 a(\022) p Fx 1503 2898 a(9) p 1482 2935 84 4 v 1482 3011 a(10) p Fw 1575 2954 a(v) p Ft 1637 2954 a(\000) p Fw 1720 2954 a(\016) p Fn 1757 2966 a(e) p Ft 1811 2954 a(\000) p Fw 1894 2954 a(C) p Fl 1953 2966 a(1) p Fk 2004 2887 a(\000) p Fw 2042 2954 a(\025) p Fx 2109 2954 a(+) p Fw 2192 2954 a(\022) r(\025) p Fl 2281 2920 a(2) p Fw 2319 2954 a(\017) p Fo 2353 2920 a(\000) p Fl(2) p Fk 2442 2887 a(\001) 2480 2837 y(\023) p Ft 2559 2954 a(\001) p Fw 2601 2954 a(Q) p Fl 2667 2966 a(2) p Fw 2704 2954 a(;) p Fy 515 3182 a(wher) l(e) p Fw 1555 3364 a(C) p Fl 1614 3376 a(1) p Fx 1674 3364 a(=) p Ft 1762 3364 a(k) p Fw(I) p Fl 1840 3376 a(1) p Ft 1877 3364 a(k) p Fx 1937 3364 a(+) p 2020 3364 a(2) p Ft 2080 3364 a(\001) p 2122 3364 a(k) p Fw(I) p Fl 2200 3376 a(0) p Ft 2237 3364 a(k) p Fl 2279 3330 a(2) p Fw 2316 3364 a(:) p Fy 515 3547 a(Pr) l(o) l(of) p Fx 515 3647 a(W) p 593 3647 a(e) p 658 3647 a(estimate) p 990 3647 a(the) p 1133 3647 a(eac) n(h) p 1320 3647 a(term) p 1519 3647 a(of) p Fw 1155 3848 a(B) p Fo 1222 3813 a(0) p Fl 1218 3868 a(22) p Fx 1311 3848 a(=) p Fw 1399 3848 a(Q) p Fl 1465 3860 a(2) p Fx 1502 3848 a(\() 1565 3792 y(9) p 1544 3829 V 1544 3905 a(10) p Fw 1637 3848 a(S) p Fl 1688 3860 a(1) p Fx 1744 3848 a(+) p Fw 1827 3848 a(\025I) p Fl 1911 3860 a(1) p Fx 1967 3848 a(+) p 2050 3848 a([) p Fw 2074 3848 a(L;) p 2168 3848 a(b) p Fx(]) p Ft 2244 3848 a(\000) p Fw 2327 3848 a(\016) p Fn 2364 3860 a(e) p Ft 2418 3848 a(\001) p Fx 2478 3827 a(~) p Fw 2460 3848 a(P) p Fo 2525 3813 a(?) p Fn 2513 3868 a(e) p Fx 2580 3848 a(\)) p Fw(Q) p Fl 2678 3860 a(2) p Fw 2716 3848 a(:) p Fx 515 4058 a(F) p 562 4058 a(rom) p 744 4058 a(the) p 898 4058 a(inequalit) n(y) p Fw 1295 4058 a(S) p Fl 1346 4070 a(1) p Ft 1425 4058 a(\025) p Fw 1531 4058 a(v) p Ft 1601 4058 a(\001) p Fw 1650 4058 a(N) p Fl 1717 4070 a(\003) p Fj 1762 4078 a(v) p Fx 1841 4058 a(and) p 2013 4058 a(the) p 2168 4058 a(inequalit) n(y) p 2564 4058 a(\(26\),) p 2776 4058 a(the) p 2931 4058 a(\014rst) p 3113 4058 a(term) p 3323 4058 a(is) 515 4158 y(b) r(ounded) p 859 4158 a(b) r(elo) n(w) p 1100 4158 a(as) p Fl 1234 4125 a(9) p 1218 4139 67 4 v 1218 4187 a(10) p Fw 1294 4158 a(Q) p Fl 1360 4170 a(2) p Fw 1397 4158 a(S) p Fl 1448 4170 a(1) p Fw 1485 4158 a(Q) p Fl 1551 4170 a(2) p Ft 1620 4158 a(\025) p Fl 1743 4125 a(9) p 1727 4139 V 1727 4187 a(10) p Fw 1803 4158 a(v) s(Q) p Fl 1912 4170 a(2) p Fx 1949 4158 a(.) p 2026 4158 a(The) p 2202 4158 a(lo) n(w) n(er) p 2424 4158 a(b) r(ound) p 2686 4158 a(of) p 2786 4158 a(the) p 2935 4158 a(second) p 3208 4158 a(term) 515 4258 y(is) p 606 4258 a(giv) n(en) p 831 4258 a(b) n(y) p Fw 955 4258 a(\025I) p Fl 1039 4270 a(1) p Ft 1114 4258 a(\025) p 1215 4258 a(\000) p Fw(\025) p Ft 1352 4258 a(\001) p 1399 4258 a(k) p Fw(I) p Fl 1477 4270 a(1) p Ft 1514 4258 a(k) p Fx(,) p 1616 4258 a(as) p Fw 1726 4258 a(I) p Fl 1762 4270 a(1) p Fx 1836 4258 a(is) p 1928 4258 a(b) r(ounded.) p 2323 4258 a(Let) p 2480 4258 a(us) p 2595 4258 a(estimate) p 2936 4258 a(the) p 3087 4258 a(norm) p 3312 4258 a(of) 515 4357 y(the) p 666 4357 a(third) p 883 4357 a(term.) p 1139 4357 a(Substituting) p Fw 1623 4357 a(L) p Fx 1717 4357 a(=) p Fw 1819 4357 a(L) p Fl 1876 4369 a(0) p Fx 1937 4357 a(+) p Fw 2026 4357 a(\025I) p Fl 2110 4369 a(0) p Fx 2148 4357 a(,) p Fw 2209 4357 a(L) p Fl 2266 4369 a(0) p Fw 2303 4357 a(Q) p Fx 2406 4357 a(=) p Fw 2508 4357 a(QL) p Fl 2631 4369 a(0) p Fx 2704 4357 a(and) p Fw 2874 4357 a(Q) p Fl 2940 4369 a(2) p Fw 2977 4357 a(Q) p Fx 3080 4357 a(=) p 3181 4357 a(0,) p 3284 4357 a(w) n(e) 515 4457 y(obtain) p Fw 771 4457 a(Q) p Fl 837 4469 a(2) p Fx 887 4457 a([) p Fw(L;) p 1004 4457 a(b) p Fx(]) p Fw 1076 4457 a(Q) p Fl 1142 4469 a(2) p Fx 1202 4457 a(=) p Ft 1290 4457 a(\000) p Fw(\022) r(\025) p Fl 1444 4427 a(2) p Fw 1482 4457 a(Q) p Fl 1548 4469 a(2) p Fx 1584 4457 a(\() p Fw(I) p Fl 1652 4469 a(0) p Fw 1690 4457 a(QI) p Fl 1792 4469 a(0) p Fw 1830 4457 a(R) p Fl 1894 4427 a(2) p Fn 1893 4477 a(\017) p Fx 1950 4436 a(\026) p Fw 1931 4457 a(Q) p Fx 2010 4457 a(+) 2107 4436 y(\026) p Fw 2089 4457 a(Q) o(R) p Fl 2218 4427 a(2) p Fn 2217 4477 a(\017) p Fw 2255 4457 a(I) p Fl 2291 4469 a(0) p Fw 2329 4457 a(QI) p Fl 2431 4469 a(0) p Fx 2468 4457 a(\)) p Fw(Q) p Fl 2566 4469 a(2) p Fw 2603 4457 a(:) p Fx 2652 4457 a(Then) p 2866 4457 a(it) p 2947 4457 a(is) p 3028 4457 a(estimated) 515 4556 y(as) p Ft 625 4556 a(k) p Fw(Q) p Fl 733 4568 a(2) p Fx 783 4556 a([) p Fw(L;) p 900 4556 a(b) p Fx(]) p Fw 972 4556 a(Q) p Fl 1038 4568 a(2) p Ft 1075 4556 a(k) p 1153 4556 a(\024) p Fx 1254 4556 a(2) p Ft 1319 4556 a(\001) p 1366 4556 a(k) p Fw(I) p Fl 1444 4568 a(0) p Ft 1481 4556 a(k) p Fl 1523 4526 a(2) p Fw 1560 4556 a(\022) r(\025) p Fl 1649 4526 a(2) p Fw 1687 4556 a(\017) p Fo 1721 4526 a(\000) p Fl(2) p Fx 1810 4556 a(,) p 1870 4556 a(scince) p 2119 4556 a(w) n(e) p 2249 4556 a(ha) n(v) n(e) p Ft 2448 4556 a(k) p Fw(R) p Fl 2554 4526 a(2) p Fn 2553 4577 a(\017) p Ft 2591 4556 a(k) p 2669 4556 a(\024) p Fw 2770 4556 a(\017) p Fo 2804 4526 a(\000) p Fl(2) p Fx 2892 4556 a(.) p 2976 4556 a(The) p 3155 4556 a(fourth) 515 4656 y(term) p 713 4656 a(is) p Fw 797 4656 a(Q) p Fl 863 4668 a(2) p Fx 918 4635 a(~) p Fw 900 4656 a(P) p Fo 965 4626 a(?) p Fn 953 4677 a(e) p Fw 1021 4656 a(Q) p Fl 1087 4668 a(2) p Fx 1146 4656 a(=) p Fw 1234 4656 a(Q) p Fl 1300 4668 a(2) p Fx 1365 4656 a(b) n(y) p 1480 4656 a(\(25\).) p 1688 4656 a(Hence) p 1934 4656 a(w) n(e) p 2057 4656 a(obtain) p Fw 1201 4889 a(B) p Fo 1268 4854 a(0) p Fl 1264 4909 a(22) p Ft 1358 4889 a(\025) p Fk 1445 4771 a(\022) p Fx 1537 4832 a(9) p 1517 4869 84 4 v 1517 4945 a(10) p Fw 1610 4889 a(v) p Ft 1671 4889 a(\000) p Fw 1754 4889 a(C) p Fl 1813 4901 a(1) p Fk 1865 4821 a(\000) p Fw 1903 4889 a(\025) p Fx 1970 4889 a(+) p Fw 2053 4889 a(\022) r(\025) p Fl 2142 4854 a(2) p Fw 2180 4889 a(\017) p Fo 2214 4854 a(\000) p Fl(2) p Fk 2302 4821 a(\001) p Ft 2359 4889 a(\000) p Fw 2442 4889 a(\016) p Fn 2479 4901 a(e) p Fk 2514 4771 a(\023) p Fw 2589 4889 a(Q) p Fl 2655 4901 a(2) p Fx 1905 5255 a(24) p eop %%Page: 25 25 25 24 bop Fx 515 523 a(where) p Fw 755 523 a(C) p Fl 814 535 a(1) p Fx 874 523 a(=) p Ft 962 523 a(k) p Fw(I) p Fl 1040 535 a(1) p Ft 1077 523 a(k) p Fx 1137 523 a(+) p 1220 523 a(2) p Ft 1280 523 a(\001) p 1322 523 a(k) p Fw(I) p Fl 1400 535 a(0) p Ft 1437 523 a(k) p Fl 1479 493 a(2) p Fw 1516 523 a(:) p Fd 1705 523 a(\003) p Fx 515 623 a(Supp) r(ose) p Fw 840 623 a(\025;) p 925 623 a(\022) r(;) p 1003 623 a(\017) p Fx 1065 623 a(satis\014es) p Fw 1647 805 a(\025) p Fx 1713 805 a(+) p Fw 1796 805 a(\022) r(\025) p Fl 1885 771 a(2) p Fw 1923 805 a(\017) p Fo 1957 771 a(\000) p Fl(2) p Fw 2069 805 a(<) p 2157 805 a(t) p Fl 2187 817 a(1) p Fw 2224 805 a(;) p Fx 515 997 a(where) p Fw 754 997 a(t) p Fl 784 1009 a(1) p Fx 844 997 a(=) p Fn 980 964 a(v) p 942 978 114 4 v Fl 942 1025 a(5) p Fn(C) p Fg 1023 1033 a(1) p Fw 1065 997 a(:) p Fx 1114 997 a(Then) p 1330 997 a(if) p Fw 1405 997 a(#) p Fx 1480 997 a(satis\014es) p Fw 1785 997 a(#) p Ft 1857 997 a(\024) p Fl 1955 964 a(1) p 1955 978 34 4 v 1955 1025 a(2) p Fw 1998 997 a(v) p Ft 2057 997 a(\000) p Fw 2138 997 a(\016) p Fn 2175 1009 a(e) p Fx 2210 997 a(,) p 2260 997 a(it) p 2342 997 a(is) p 2424 997 a(in) p 2520 997 a(the) p 2662 997 a(resolv) n(en) n(t) p 3010 997 a(set) p Fw 3138 997 a(\032) p Fx(\() p Fw(B) p Fo 3280 967 a(0) p Fl 3276 1017 a(22) p Fx 3347 997 a(\)) 515 1096 y(of) p Fw 609 1096 a(B) p Fo 676 1066 a(0) p Fl 672 1117 a(22) p Fx 770 1096 a(b) n(y) p 886 1096 a(this) p 1047 1096 a(prop) r(osition.) 639 1196 y(Next) p 842 1196 a(w) n(e) p 964 1196 a(estimate) p 1297 1196 a(the) p 1440 1196 a(lo) n(w) n(er) p 1657 1196 a(b) r(ound) p 1913 1196 a(of) p Fw 2008 1196 a(") p Fx 2061 1196 a(\() p Fw 2092 1196 a(#) p Fx(\)) q(.) p Fm 515 1362 a(Prop) s(osition) p 1031 1362 a(7.6) p Fy 1194 1362 a(Supp) l(ose) p Fw 1511 1362 a(\025;) p 1596 1362 a(\022) r(;) p 1674 1362 a(\017) p Fy 1737 1362 a(satis\014es) p Fw 1647 1545 a(\025) p Fx 1713 1545 a(+) p Fw 1796 1545 a(\022) r(\025) p Fl 1885 1510 a(2) p Fw 1923 1545 a(\017) p Fo 1957 1510 a(\000) p Fl(2) p Fw 2069 1545 a(<) p 2157 1545 a(t) p Fl 2187 1557 a(1) p Fw 2224 1545 a(;) p Fy 515 1736 a(wher) l(e) p Fw 747 1736 a(t) p Fl 777 1748 a(1) p Fx 837 1736 a(=) p Fn 974 1703 a(v) p 935 1717 114 4 v Fl 935 1765 a(5) p Fn(C) p Fg 1016 1773 a(1) p Fw 1058 1736 a(:) p Fy 1109 1736 a(Then) p 1323 1736 a(if) p Fw 1402 1736 a(#) p Fy 1478 1736 a(satis\014es) p Fw 1788 1736 a(#) p Ft 1860 1736 a(\024) p Fl 1958 1703 a(1) p 1958 1717 34 4 v 1958 1765 a(2) p Fw 2001 1736 a(v) p Ft 2058 1736 a(\000) p Fw 2137 1736 a(\016) p Fn 2174 1748 a(e) p Fy 2209 1736 a(,) p 2263 1736 a(it) p 2343 1736 a(is) p 2430 1736 a(in) p 2530 1736 a(the) p 2666 1736 a(r) l(esolvent) p 3011 1736 a(set) p Fw 3138 1736 a(\032) p Fx(\() p Fw(B) p Fo 3280 1706 a(0) p Fl 3276 1757 a(22) p Fx 3347 1736 a(\)) p Fy 515 1836 a(of) p Fw 612 1836 a(B) p Fo 679 1805 a(0) p Fl 675 1856 a(22) p Fy 775 1836 a(and) p 937 1836 a(the) p 1075 1836 a(fol) t(lowing) p 1427 1836 a(ine) l(quality) p 1802 1836 a(holds;) p Fw 598 2093 a(") p Fx 651 2093 a(\() p Fw(#) p Fx(\)) p Ft 787 2093 a(\025) p Fx 875 2093 a(2) p Fw(\022) r(\025) p Fl 1006 2059 a(2) p Fw 1043 2093 a(Q) p Fl 1109 2105 a(1) p Fk 1160 1976 a(\024) p Fw 1204 2093 a(I) p Fl 1240 2105 a(0) p Fx 1295 2072 a(\026) p Fw 1277 2093 a(R) p Fl 1341 2059 a(2) p Fn 1340 2113 a(\017) p Fw 1378 2093 a(I) p Fl 1414 2105 a(0) p Ft 1471 2093 a(\000) p Fx 1627 2037 a(1) p 1564 2074 169 4 v 1564 2150 a(2) p Fw(\022) r(\025) p Fl 1695 2126 a(2) p Fw 1742 2093 a(\016) p Fn 1779 2105 a(e) p Ft 1833 2093 a(\001) p Fx 1893 2072 a(~) p Fw 1874 2093 a(P) p Fo 1939 2059 a(?) p Fn 1927 2113 a(e) p Fk 1995 1976 a(\025) p Fw 2053 2093 a(Q) p Fl 2119 2105 a(1) p Ft 2156 2093 a(\000) p Fw(C) p Fl 2280 2105 a(2) p Fk 2331 2026 a(\000) p Fw 2369 2093 a(\022) p Fl 2410 2059 a(2) p Fw 2447 2093 a(\025) p Fl 2495 2059 a(2) p Fw 2533 2093 a(Q) p Fl 2599 2105 a(1) p Fw 2636 2093 a(I) p Fl 2672 2105 a(0) p Fx 2728 2072 a(\026) p Fw 2710 2093 a(R) p Fl 2774 2059 a(2) p Fn 2773 2113 a(\017) p Fw 2811 2093 a(I) p Fl 2847 2105 a(0) p Fw 2884 2093 a(Q) p Fl 2950 2105 a(1) p Fx 3006 2093 a(+) p 3089 2093 a(\() p Fw(\025) p Fl 3169 2059 a(2) p Fx 3225 2093 a(+) p Fw 3308 2093 a(\022) p Fl 3349 2059 a(2) p Fw 3387 2093 a(\025) p Fl 3435 2059 a(4) p Fw 3472 2093 a(\017) p Fo 3506 2059 a(\000) p Fl(4) p Fx 3595 2093 a(\)) p Fw(Q) p Fl 3693 2105 a(1) p Fk 3730 2026 a(\001) p Fy 515 2331 a(wher) l(e) p Fw 749 2331 a(C) p Fl 808 2343 a(2) p Fx 869 2331 a(=) p 956 2331 a(10) p Fw(v) p Fo 1083 2300 a(\000) p Fl(1) p Ft 1172 2331 a(k) p Fw(I) p Fl 1250 2343 a(1) p Ft 1287 2331 a(k) p Fl 1329 2300 a(2) p Fx 1384 2331 a(+) p 1467 2331 a(20) p Fw(v) p Fo 1594 2300 a(\000) p Fl(1) p Fx 1701 2331 a(+) p 1784 2331 a(80) p Fw(v) p Fo 1911 2300 a(\000) p Fl(1) p Ft 2000 2331 a(k) p Fw(I) p Fl 2078 2343 a(0) p Ft 2115 2331 a(k) p Fl 2157 2300 a(4) p Fw 2194 2331 a(:) p Fy 515 2497 a(Pr) l(o) l(of) p Fx 515 2596 a(First) p Fw 717 2596 a(B) p Fo 784 2566 a(0) p Fl 780 2617 a(11) p Fx 878 2596 a(is) p 962 2596 a(giv) n(en) p 1178 2596 a(b) n(y) p Fw 1112 2829 a(B) p Fo 1179 2794 a(0) p Fl 1175 2849 a(11) p Fx 1269 2829 a(=) p Fw 1356 2829 a(Q) p Fl 1422 2841 a(1) p Fk 1473 2712 a(\022) p Fx 1565 2772 a(9) p 1544 2809 84 4 v 1544 2886 a(10) p Fw 1637 2829 a(S) p Fl 1688 2841 a(1) p Fx 1744 2829 a(+) p Fw 1827 2829 a(\025I) p Fl 1911 2841 a(1) p Fx 1967 2829 a(+) p 2050 2829 a([) p Fw 2074 2829 a(L;) p 2168 2829 a(b) p Fx(]) p Ft 2244 2829 a(\000) p Fw 2327 2829 a(\016) p Fn 2364 2841 a(e) p Ft 2418 2829 a(\001) p Fx 2478 2808 a(~) p Fw 2460 2829 a(P) p Fo 2525 2794 a(?) p Fn 2513 2849 a(e) p Fk 2580 2712 a(\023) p Fw 2655 2829 a(Q) p Fl 2721 2841 a(1) p Fw 2758 2829 a(:) p Fx 515 3056 a(As) p Fw 642 3056 a(S) p Fl 693 3068 a(1) p Fw 730 3056 a(Q) p Fx 825 3056 a(=) p 920 3056 a(0) p 993 3056 a(and) p Fw 1159 3056 a(aQ) p Fx 1298 3056 a(=) p Fw 1393 3056 a(Qa) p Fo 1503 3026 a(\003) p Fx 1570 3056 a(=) p 1665 3056 a(0,) p 1762 3056 a(the) p 1909 3056 a(\014rst) p 2085 3056 a(and) p 2250 3056 a(the) p 2397 3056 a(second) p 2669 3056 a(term) p 2872 3056 a(v) p 2911 3056 a(anish.) p 3173 3056 a(Using) 534 3135 y(\026) p Fw 515 3156 a(Q) o(Q) p Fl 646 3168 a(1) p Fx 706 3156 a(=) p 794 3156 a(0,) p Fw 886 3156 a(L) p Fl 943 3168 a(0) p Fw 980 3156 a(Q) p Fx 1069 3156 a(=) p Fw 1156 3156 a(QL) p Fl 1279 3168 a(0) p Fx 1316 3156 a(,) p 1366 3156 a(w) n(e) p 1489 3156 a(obtain) p Fw 1173 3384 a(B) p Fo 1240 3350 a(0) p Fl 1236 3404 a(11) p Fx 1329 3384 a(=) p 1417 3384 a(2) p Fw(\022) r(\025) p Fl 1548 3350 a(2) p Fw 1585 3384 a(Q) p Fl 1651 3396 a(1) p Fk 1702 3267 a(\024) p Fw 1746 3384 a(I) p Fl 1782 3396 a(0) p Fx 1837 3363 a(\026) p Fw 1819 3384 a(R) p Fl 1883 3350 a(2) p Fn 1882 3404 a(\017) p Fw 1920 3384 a(I) p Fl 1956 3396 a(0) p Ft 2013 3384 a(\000) p Fx 2169 3328 a(1) p 2106 3365 169 4 v 2106 3441 a(2) p Fw(\022) r(\025) p Fl 2237 3417 a(2) p Fw 2284 3384 a(\016) p Fn 2321 3396 a(e) p Ft 2375 3384 a(\001) p Fx 2435 3363 a(~) p Fw 2417 3384 a(P) p Fo 2482 3350 a(?) p Fn 2470 3404 a(e) p Fk 2537 3267 a(\025) p Fw 2595 3384 a(Q) p Fl 2661 3396 a(1) p Fw 2698 3384 a(:) p Fx 3231 3384 a(\(27\)) 515 3633 y(Second,) p Fw 819 3633 a(Q) p Fl 885 3645 a(2) p Fx 936 3633 a(\() p Fw(B) p Fo 1035 3603 a(0) p Fl 1031 3654 a(22) p Ft 1120 3633 a(\000) p Fw 1203 3633 a(#) p Fx(\)) p Fo 1284 3592 a(\000) p Fl(1) p Fw 1387 3633 a(Q) p Fl 1453 3645 a(2) p Fx 1518 3633 a(is) p 1601 3633 a(ev) p 1677 3633 a(aluated) p 1973 3633 a(as) p 2075 3633 a(follo) n(ws;) p Fm 515 3799 a(Lemma) p 853 3799 a(7.7) p Fy 1017 3799 a(Supp) l(ose) p Fw 1337 3799 a(\025) p Fx 1406 3799 a(+) p Fw 1492 3799 a(\022) r(\025) p Fl 1581 3769 a(2) p Fw 1619 3799 a(\017) p Fo 1653 3769 a(\000) p Fl(2) p Fw 1771 3799 a(<) p 1866 3799 a(t) p Fl 1896 3811 a(1) p Fy 1966 3799 a(wher) l(e) p Fw 2205 3799 a(t) p Fl 2235 3811 a(1) p Fx 2302 3799 a(=) p Fn 2619 3767 a(v) p 2406 3781 462 4 v Fl 2406 3828 a(5\() p Fo(k) p Fn(I) p Fg 2528 3836 a(1) p Fo 2561 3828 a(k) p Fl(+2) p Fo(k) p Fn(I) p Fg 2742 3836 a(0) p Fo 2775 3828 a(k) p Fg 2809 3812 a(2) p Fl 2841 3828 a(\)) p Fw 2877 3799 a(:) p Fy 2933 3799 a(Then) p 3154 3799 a(for) p 3290 3799 a(al) t(l) p Fw 515 3921 a(#) p Ft 587 3921 a(\024) p Fl 684 3888 a(1) p 684 3902 34 4 v 684 3950 a(2) p Fw 728 3921 a(v) p Ft 789 3921 a(\000) p Fw 872 3921 a(\016) p Fn 909 3933 a(e) p Fy 945 3921 a(,) p Fw 1000 3921 a(#) p Fy 1079 3921 a(is) p 1168 3921 a(in) p 1270 3921 a(the) p 1408 3921 a(r) l(esolvent) p 1755 3921 a(set) p Fw 1884 3921 a(\032) p Fx(\() p Fw(B) p Fo 2026 3891 a(0) p Fl 2022 3942 a(22) p Fx 2092 3921 a(\)) p Fy 2154 3921 a(of) p Fw 2252 3921 a(B) p Fo 2319 3891 a(0) p Fl 2315 3942 a(22) p Fy 2415 3921 a(and) p Fx 903 4139 a(0) p Ft 967 4139 a(\024) p Fk 1055 4047 a(D) p Fw 1106 4139 a( ) s(;) p 1200 4139 a(Q) p Fl 1266 4151 a(2) p Fx 1316 4139 a(\() p Fw 1349 4139 a(B) p Fo 1416 4104 a(0) p Fl 1412 4159 a(22) p Ft 1500 4139 a(\000) p Fw 1583 4139 a(#) p Fx(\)) p Fo 1665 4093 a(\000) p Fl(1) p Fw 1768 4139 a(Q) p Fl 1834 4151 a(2) p Fw 1871 4139 a( ) p Fk 1928 4047 a(E) p Ft 2001 4139 a(\024) p Fx 2089 4139 a(5) p Ft(k) p Fw(S) p Fo 2229 4090 a(\000) p Fg 2290 4068 a(1) p 2290 4077 29 3 v 2290 4110 a(2) p Fl 2224 4161 a(1) p Fw 2332 4139 a(Q) p Fl 2398 4151 a(2) p Fw 2435 4139 a( ) p Ft 2492 4139 a(k) p Fl 2534 4104 a(2) p Fw 2571 4139 a(;) p Ft 2692 4139 a(8) p Fw( ) p Ft 2819 4139 a(2) p 2897 4139 a(H) p Fw 2968 4139 a(:) p Fy 515 4487 a(Pr) l(o) l(of) p Fx 515 4587 a(As) p Fw 643 4587 a(S) p Fl 694 4599 a(1) p Fw 731 4587 a(Q) p Fl 797 4599 a(2) p Ft 865 4587 a(\025) p Fw 961 4587 a(v) s(N) p Fl 1071 4599 a(\003) p Fj 1116 4607 a(v) p Fw 1156 4587 a(Q) p Fl 1222 4599 a(2) p Ft 1291 4587 a(\025) p Fw 1387 4587 a(v) s(Q) p Fl 1496 4599 a(2) p Fx 1533 4587 a(,) p Fw 1590 4587 a(S) p Fl 1641 4599 a(1) p Fx 1711 4587 a(is) p 1799 4587 a(in) n(v) n(ertible) p 2171 4587 a(on) p Fw 2292 4587 a(Q) p Fl 2358 4599 a(2) p Ft 2395 4587 a(H) p Fx 2466 4587 a(.) p 2541 4587 a(The) p 2717 4587 a(pro) r(of) p 2939 4587 a(is) p 3027 4587 a(similar) p 3305 4587 a(to) 515 4687 y(that) p 700 4687 a(of) p 800 4687 a([10) o(].) p 1005 4687 a(In) p 1114 4687 a(facts,) p 1341 4687 a(it) p 1429 4687 a(is) p 1518 4687 a(easier) p 1755 4687 a(b) r(ecause) p 2067 4687 a(w) n(e) p 2195 4687 a(are) p 2339 4687 a(considering) p 2779 4687 a(F) p 2826 4687 a(ermion) p 3108 4687 a(system,) 515 4786 y(whose) p 760 4786 a(in) n(teraction) p 1177 4786 a(terms) p 1409 4786 a(are) p 1547 4786 a(b) r(ounded.) p 1919 4786 a(W) p 1997 4786 a(e) p 2061 4786 a(omit) p 2255 4786 a(the) p 2398 4786 a(details.) p Fd(\003) p Fx 515 4886 a(Third,) p Fk 773 4790 a(\015) 773 4840 y(\015) 773 4890 y(\015) p Fw 819 4886 a(S) p Fo 875 4837 a(\000) p Fg 937 4815 a(1) p 937 4824 V 937 4857 a(2) p Fl 870 4908 a(1) p Fw 979 4886 a(B) p Fo 1046 4856 a(0) p Fl 1042 4907 a(21) p Fw 1113 4886 a( ) p Fk 1170 4790 a(\015) 1170 4840 y(\015) 1170 4890 y(\015) p Fx 1243 4886 a(is) p 1327 4886 a(estimated) p 1706 4886 a(as) p 1808 4886 a(follo) n(ws;) 1905 5255 y(25) p eop %%Page: 26 26 26 25 bop Fm 515 523 a(Lemma) p 853 523 a(7.8) p Fk 787 603 a(\015) 787 653 y(\015) 787 702 y(\015) p Fw 833 698 a(S) p Fo 889 650 a(\000) p Fg 950 627 a(1) p 950 636 29 3 v 950 670 a(2) p Fl 884 720 a(1) p Fw 993 698 a(B) p Fo 1060 664 a(0) p Fl 1056 719 a(21) p Fw 1126 698 a( ) p Fk 1183 603 a(\015) 1183 653 y(\015) 1183 702 y(\015) p Fl 1229 621 a(2) p Ft 1290 698 a(\024) p Fw 1377 698 a(C) p Fl 1436 710 a(3) p Fk 1488 631 a(\000) p Fw 1526 698 a(\022) p Fl 1567 664 a(2) p Fw 1604 698 a(\025) p Fl 1652 664 a(2) p Ft 1690 698 a(k) p Fx 1749 677 a(\026) p Fw 1732 698 a(R) p Fn 1795 710 a(\017) p Fw 1826 698 a(I) p Fl 1862 710 a(0) p Fw 1900 698 a(Q) p Fl 1966 710 a(1) p Fw 2003 698 a( ) p Ft 2060 698 a(k) p Fl 2102 664 a(2) p Fx 2157 698 a(+) p Fw 2240 698 a(\022) p Fl 2281 664 a(2) p Fw 2319 698 a(\025) p Fl 2367 664 a(4) p Fw 2404 698 a(\017) p Fo 2438 664 a(\000) p Fl(4) p Ft 2527 698 a(k) p Fw( ) p Ft 2626 698 a(k) p Fl 2668 664 a(2) p Fx 2723 698 a(+) p Fw 2806 698 a(\025) p Fl 2854 664 a(2) p Ft 2892 698 a(k) p Fw( ) p Ft 2991 698 a(k) p Fl 3033 664 a(2) p Fk 3069 631 a(\001) p Fy 515 880 a(wher) l(e) p Fw 749 880 a(C) p Fl 808 892 a(3) p Fx 869 880 a(=) p 956 880 a(2) p Fw(v) p Fo 1041 850 a(\000) p Fl(1) p Ft 1130 880 a(k) p Fw(I) p Fl 1208 892 a(1) p Ft 1246 880 a(k) p Fl 1288 850 a(2) p Fx 1343 880 a(+) p 1426 880 a(4) p Fw(v) p Fo 1511 850 a(\000) p Fl(1) p Fx 1618 880 a(+) p 1701 880 a(16) p Fw(v) p Fo 1828 850 a(\000) p Fl(1) p Ft 1917 880 a(k) p Fw(I) p Fl 1995 892 a(0) p Ft 2032 880 a(k) p Fl 2074 850 a(4) p Fw 2111 880 a(:) p Fy 515 1022 a(Pr) l(o) l(of) p Fx 515 1122 a(The) p 685 1122 a(pro) r(of) p 902 1122 a(is) p 986 1122 a(the) p 1129 1122 a(same) p 1337 1122 a(as) p 1439 1122 a(that) p 1619 1122 a(of) p 1713 1122 a([10) o(].) p Fd(\003) p Fx 639 1221 a(No) n(w) p 835 1221 a(let) p 961 1221 a(us) p 1074 1221 a(complete) p 1430 1221 a(the) p 1580 1221 a(pro) r(of) p 1803 1221 a(of) p 1904 1221 a(Prop) r(osition) p 2357 1221 a(7.6.) p 2542 1221 a(Com) n(bining) p 2970 1221 a(Lemma) p 3273 1221 a(7.7) 515 1321 y(and) p 678 1321 a(7.8,) p 838 1321 a(w) n(e) p 962 1321 a(obtain) p 1222 1321 a(the) p 1367 1321 a(follo) n(wing) p 1720 1321 a(ev) p 1796 1321 a(aluation:) p 2156 1321 a(if) p Fw 2234 1321 a(\025;) p 2319 1321 a(\022) r(;) p 2397 1321 a(\017) p Fx 2461 1321 a(satis\014es) p Fw 2769 1321 a(\025) p Fx 2837 1321 a(+) p Fw 2921 1321 a(\022) r(\025) p Fl 3010 1291 a(2) p Fw 3048 1321 a(\017) p Fo 3082 1291 a(\000) p Fl(2) p Fw 3198 1321 a(<) p 3289 1321 a(t) p Fl 3319 1333 a(1) p Fx 3356 1321 a(,) 515 1420 y(then) p 704 1420 a(for) p 831 1420 a(all) p Fw 946 1420 a(#) p Ft 1018 1420 a(\024) p Fl 1116 1388 a(1) p 1116 1402 34 4 v 1116 1449 a(2) p Fw 1159 1420 a(v) p Ft 1221 1420 a(\000) p Fw 1304 1420 a(\016) p Fn 1341 1432 a(e) p Fx 1376 1420 a(,) p 1427 1420 a(w) n(e) p 1549 1420 a(ha) n(v) n(e) p Fw 1741 1420 a(#) p Ft 1813 1420 a(2) p Fw 1891 1420 a(\032) p Fx(\() p Fw(B) p Fo 2033 1390 a(0) p Fl 2029 1441 a(22) p 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Fw 2958 1605 a(\025) p Fl 3006 1571 a(4) p Fw 3043 1605 a(\017) p Fo 3077 1571 a(\000) p Fl(4) p Fx 3166 1605 a(\)) p Ft(k) p Fw( ) p Ft 3297 1605 a(k) p Fl 3339 1571 a(2) p Fk 3376 1513 a(\021) p Fx 515 1787 a(where) p Fw 759 1787 a(C) p Fl 818 1799 a(2) p Fx 885 1787 a(=) p 978 1787 a(10) p Fw(v) p Fo 1105 1757 a(\000) p Fl(1) p Ft 1194 1787 a(k) p Fw(I) p Fl 1272 1799 a(1) p Ft 1309 1787 a(k) p Fl 1351 1757 a(2) p Fx 1409 1787 a(+) p 1494 1787 a(20) p Fw(v) p Fo 1621 1757 a(\000) p Fl(1) p Fx 1731 1787 a(+) p 1816 1787 a(80) p Fw(v) p Fo 1943 1757 a(\000) p Fl(1) p Ft 2032 1787 a(k) p Fw(I) p Fl 2110 1799 a(0) p Ft 2147 1787 a(k) p Fl 2189 1757 a(4) p Fx 2226 1787 a(.) p 2297 1787 a(Com) n(bining) p 2723 1787 a(this) p 2888 1787 a(with) p 3081 1787 a(\(27\),) p 3284 1787 a(w) n(e) 515 1887 y(obtain) p 773 1887 a(the) p 916 1887 a(required) p 1242 1887 a(estimation.) p Fd 1680 1887 a(\003) p Fx 639 1987 a(No) n(w) p 821 1987 a(let) p 934 1987 a(us) p 1033 1987 a(complete) p 1376 1987 a(the) p 1512 1987 a(pro) r(of) p 1722 1987 a(of) p 1809 1987 a(Theorem) p 2153 1987 a(7.2.) p 2316 1987 a(Supp) r(ose) p 2635 1987 a(that) p Fw 2807 1987 a(\025) p Fx 2859 1987 a(+) p Fw 2928 1987 a(\022) r(\025) p Fl 3017 1957 a(2) p Fw 3055 1987 a(\017) p Fo 3089 1957 a(\000) p Fl(2) p Fw 3201 1987 a(<) p 3289 1987 a(t) p Fl 3319 1999 a(1) p Fx 3356 1987 a(.) 515 2086 y(Then) p 732 2086 a(b) n(y) p 847 2086 a(Prop) r(osition) p 1294 2086 a(7.6,) p 1451 2086 a(for) p 1578 2086 a(all) p Fw 1693 2086 a(#) p Ft 1765 2086 a(\024) p Fl 1863 2054 a(1) p 1863 2068 V 1863 2115 a(2) p Fw 1906 2086 a(v) p Ft 1967 2086 a(\000) p Fw 2050 2086 a(\016) p Fn 2087 2098 a(e) p Fx 2123 2086 a(,) p 2174 2086 a(w) n(e) p 2296 2086 a(ha) n(v) n(e) p Fw 2487 2086 a(#) p Ft 2560 2086 a(2) p Fw 2638 2086 a(\032) p Fx(\() p Fw(B) p Fo 2780 2056 a(0) p Fl 2776 2107 a(22) p Fx 2846 2086 a(\)) p 2906 2086 a(and) p Fw 1761 2238 a(") p Fx 1814 2238 a(\() p Fw(#) p Fx(\)) p Ft 1951 2238 a(\025) p Fw 2038 2238 a(D) r(;) p Fx 515 2390 a(where) p Fw 515 2571 a(D) p Fx 609 2571 a(=) p 696 2571 a(2) p Fw(\022) r(\025) p Fl 827 2537 a(2) p Fw 865 2571 a(Q) p Fl 931 2583 a(1) p Fk 982 2454 a(\024) p Fw 1025 2571 a(I) p Fl 1061 2583 a(0) p Fx 1117 2550 a(\026) p Fw 1099 2571 a(R) p Fl 1163 2537 a(2) p Fn 1162 2592 a(\017) p Fw 1200 2571 a(I) p Fl 1236 2583 a(0) p Ft 1292 2571 a(\000) p Fx 1449 2515 a(1) p 1385 2552 169 4 v 1385 2628 a(2) p Fw(\022) r(\025) p Fl 1516 2604 a(2) p Fw 1564 2571 a(\016) p Fn 1601 2583 a(e) p Ft 1655 2571 a(\001) p Fx 1715 2550 a(~) p Fw 1696 2571 a(P) p Fo 1761 2537 a(?) p Fn 1749 2592 a(e) p Fk 1817 2454 a(\025) p Fw 1875 2571 a(Q) p Fl 1941 2583 a(1) p Ft 1996 2571 a(\000) p Fw 2079 2571 a(C) p Fl 2138 2583 a(2) p Fk 2190 2504 a(\000) p Fw 2228 2571 a(\022) p Fl 2269 2537 a(2) p Fw 2306 2571 a(\025) p Fl 2354 2537 a(2) p Fw 2392 2571 a(Q) p Fl 2458 2583 a(1) p Fw 2495 2571 a(I) p Fl 2531 2583 a(0) p Fx 2586 2550 a(\026) p Fw 2568 2571 a(R) p Fl 2632 2537 a(2) p Fn 2631 2592 a(\017) p Fw 2669 2571 a(I) p Fl 2705 2583 a(0) p Fw 2743 2571 a(Q) p Fl 2809 2583 a(1) p Fx 2864 2571 a(+) p 2947 2571 a(\() p Fw(\025) p Fl 3027 2537 a(2) p Fx 3084 2571 a(+) p Fw 3167 2571 a(\022) p Fl 3208 2537 a(2) p Fw 3245 2571 a(\025) p Fl 3293 2537 a(4) p Fw 3331 2571 a(\017) p Fo 3365 2537 a(\000) p Fl(4) p Fx 3454 2571 a(\)) p Fw(Q) p Fl 3552 2583 a(1) p Fk 3589 2504 a(\001) p Fw 3641 2571 a(:) p Fx 515 2773 a(Note) p 715 2773 a(that) p Fw 895 2773 a(D) p Fx 994 2773 a(is) p Fw 1077 2773 a(#) p Fx(-indep) r(enden) n(t.) p 1657 2773 a(Let) p Fw 1806 2773 a(#) p Fl 1855 2785 a(0) p Ft 1915 2773 a(\021) p Fx 2003 2773 a(inf) p Fw 2117 2773 a(\033) p Fx 2167 2773 a(\() p Fw(B) p Fo 2266 2743 a(0) p Ft 2291 2773 a(j) p Fl 2314 2788 a(\() p Fn(Q) p Fg 2392 2796 a(1) p Fo 2424 2788 a(\010) p Fn(Q) p Fg 2528 2796 a(2) p Fl 2561 2788 a(\)) p Fo(H) p Fx 2648 2773 a(\).) p 2740 2773 a(Here) p 2936 2773 a(w) n(e) p 3058 2773 a(ha) n(v) n(e) p 3250 2773 a(t) n(w) n(o) 515 2873 y(cases,) 616 3014 y(1.) p Fw 722 3014 a(#) p Fl 771 3026 a(0) p Ft 832 3014 a(\025) p Fl 929 2982 a(1) p 929 2996 34 4 v 929 3043 a(2) p Fw 972 3014 a(v) p Ft 1034 3014 a(\000) p Fw 1117 3014 a(\016) p Fn 1154 3026 a(e) p Fx 1190 3014 a(.) 616 3168 y(2.) p Fl 732 3135 a(1) p 732 3149 V 732 3197 a(2) p Fw 775 3168 a(v) p Ft 839 3168 a(\000) p Fw 924 3168 a(\016) p Fn 961 3180 a(e) p Fw 1024 3168 a(>) p 1116 3168 a(#) p Fl 1165 3180 a(0) p Fx 1202 3168 a(.) p 1270 3168 a(In) p 1377 3168 a(this) p 1541 3168 a(case,) p Fw 1743 3168 a(#) p Fl 1792 3180 a(0) p Ft 1857 3168 a(2) p Fw 1940 3168 a(\032) p Fx(\() p Fw(B) p Fo 2082 3138 a(0) p Fl 2078 3189 a(22) p Fx 2149 3168 a(\)) p 2211 3168 a(and) p Fw 2375 3168 a(") p Fx 2428 3168 a(\() p Fw(#) p Fl 2509 3180 a(0) p Fx 2546 3168 a(\)) p 2609 3168 a(is) p 2695 3168 a(w) n(ell-de\014ned.) p 3190 3168 a(Then) p Fw 722 3268 a(f) p Fn 763 3280 a(Q) p Fg 815 3288 a(1) p Fx 852 3268 a(\() p Fw(B) p Fo 951 3238 a(0) p Ft 993 3268 a(\000) p Fw 1076 3268 a(#) p Fl 1125 3280 a(0) p Fx 1162 3268 a(\)) p 1222 3268 a(satis\014es) p Fw 1057 3420 a(f) p Fn 1098 3432 a(Q) p Fg 1150 3440 a(1) p Fx 1187 3420 a(\() p Fw(B) p Fo 1286 3385 a(0) p Ft 1328 3420 a(\000) p Fw 1411 3420 a(#) p Fl 1460 3432 a(0) p Fx 1497 3420 a(\)) p 1553 3420 a(=) p Fw 1640 3420 a(") p Fx 1693 3420 a(\() p Fw(#) p Fl 1774 3432 a(0) p Fx 1811 3420 a(\)) p Ft 1862 3420 a(\000) p Fw 1945 3420 a(#) p Fl 1994 3432 a(0) p Ft 2050 3420 a(\001) p Fw 2091 3420 a(Q) p Fl 2157 3432 a(1) p Ft 2217 3420 a(\025) p Fx 2305 3420 a(\(inf) p Fw 2452 3420 a(\033) p Fx 2502 3420 a(\() p Fw(D) p Fx 2605 3420 a(\)) p Ft 2657 3420 a(\000) p Fw 2740 3420 a(#) p Fl 2789 3432 a(0) p Fx 2826 3420 a(\)) p Ft 2877 3420 a(\001) p Fw 2918 3420 a(Q) p Fl 2984 3432 a(1) p Fw 3021 3420 a(:) p Fx 722 3572 a(On) p 854 3572 a(the) p 990 3572 a(other) p 1200 3572 a(hand,) p 1425 3572 a(note) p 1602 3572 a(that) p Fw 1775 3572 a(#) p Fl 1824 3584 a(0) p Ft 1884 3572 a(2) p Fw 1963 3572 a(\033) p Fx 2013 3572 a(\() p Fw(B) p Fo 2112 3542 a(0) p Ft 2136 3572 a(j) p Fl 2159 3587 a(\() p Fn(Q) p Fg 2237 3595 a(1) p Fo 2269 3587 a(\010) p Fn(Q) p Fg 2373 3595 a(2) p Fl 2406 3587 a(\)) p Fo(H) p Fx 2493 3572 a(\).) p 2583 3572 a(By) p 2706 3572 a(F) p 2753 3572 a(esh) n(bac) n(h) p 3056 3572 a(Theorem) 722 3671 y(7.4,) p 875 3671 a(this) p 1032 3671 a(implies) p 1309 3671 a(0) p Ft 1374 3671 a(2) p Fw 1452 3671 a(\033) p Fx 1502 3671 a(\() p Fw(f) p Fn 1575 3683 a(Q) p Fg 1627 3691 a(1) p Fx 1664 3671 a(\() p Fw(B) p Fo 1763 3641 a(0) p Ft 1796 3671 a(\000) p Fw 1870 3671 a(#) p Fl 1919 3683 a(0) p Fx 1955 3671 a(\)\).) p 2078 3671 a(Hence) p Fw 2320 3671 a(f) p Fn 2361 3683 a(Q) p Fg 2413 3691 a(1) p Fx 2449 3671 a(\() p Fw(B) p Fo 2548 3641 a(0) p Ft 2581 3671 a(\000) p Fw 2655 3671 a(#) p Fl 2704 3683 a(0) p Fx 2740 3671 a(\)) p 2795 3671 a(is) p 2874 3671 a(not) p 3017 3671 a(in) n(v) n(ertible,) 722 3771 y(and) p 884 3771 a(w) n(e) p 1006 3771 a(get) p Fw 1790 3923 a(#) p Fl 1839 3935 a(0) p Ft 1900 3923 a(\025) p Fx 1987 3923 a(inf) p Fw 2102 3923 a(\033) p Fx 2152 3923 a(\() p Fw(D) p Fx 2255 3923 a(\)) p Fw 2288 3923 a(:) p Fx 515 4102 a(So) p 630 4102 a(w) n(e) p 752 4102 a(ha) n(v) n(e) p Fw 1418 4272 a(#) p Fl 1467 4284 a(0) p Ft 1527 4272 a(\025) p Fx 1615 4272 a(min) p Ft(f) p Fx 1805 4216 a(1) p 1805 4253 42 4 v 1805 4329 a(2) p Fw 1856 4272 a(v) p Ft 1918 4272 a(\000) p Fw 2001 4272 a(\016) p Fn 2038 4284 a(e) p Fw 2073 4272 a(;) p Fx 2110 4272 a(inf) p Fw 2225 4272 a(\033) p Fx 2275 4272 a(\() p Fw(D) p Fx 2378 4272 a(\)) p Ft 2411 4272 a(g) p Fw(;) p Fx 515 4445 a(i.e.,) p Fw 1237 4632 a(B) p Fo 1304 4597 a(0) p Ft 1328 4632 a(j) p Fl 1351 4647 a(\() p Fn(Q) p Fg 1429 4655 a(1) p Fo 1461 4647 a(\010) p Fn(Q) p Fg 1565 4655 a(2) p Fl 1598 4647 a(\)) p Fo(H) p Ft 1708 4632 a(\025) p Fx 1796 4632 a(min) p Ft(f) p Fx 1986 4576 a(1) p 1986 4613 V 1986 4689 a(2) p Fw 2037 4632 a(v) p Ft 2099 4632 a(\000) p Fw 2182 4632 a(\016) p Fn 2219 4644 a(e) p Fw 2254 4632 a(;) p Fx 2291 4632 a(inf) p Fw 2406 4632 a(\033) p Fx 2456 4632 a(\() p Fw(D) p Fx 2559 4632 a(\)) p Ft 2592 4632 a(g) p Fw(:) p Fx 3231 4632 a(\(28\)) 515 4807 y(The) p 695 4807 a(next) p 891 4807 a(problem) p 1224 4807 a(is) p 1317 4807 a(to) p 1428 4807 a(in) n(v) n(estigate) p 1851 4807 a(the) p 2004 4807 a(lo) n(w) n(er) p 2231 4807 a(b) r(ound) p 2496 4807 a(of) p Fw 2600 4807 a(D) p Fx 2671 4807 a(.) p 2760 4807 a(F) p 2807 4807 a(or) p 2919 4807 a(the) p 3072 4807 a(purp) r(ose,) 515 4907 y(w) n(e) p 646 4907 a(estimate) p Fw 987 4907 a(Q) p Fl 1053 4919 a(1) p Fw 1090 4907 a(I) p Fl 1126 4919 a(0) p Fx 1182 4886 a(\026) p Fw 1164 4907 a(R) p Fl 1228 4876 a(2) p Fn 1227 4927 a(\017) p Fw 1265 4907 a(I) p 1308 4907 a(Q) p Fl 1374 4919 a(1) p Fx 1411 4907 a(.) p 1497 4907 a(Under) p 1758 4907 a(the) p 1910 4907 a(Assumption) p 2379 4907 a(2.2) p 2521 4907 a(and) p 2692 4907 a(using) p 2918 4907 a(pull) p 3093 4907 a(through) 515 5006 y(form) n(ula,) p 842 5006 a(w) n(e) p 965 5006 a(obtain) p 1223 5006 a(the) p 1366 5006 a(follo) n(wing) p 1716 5006 a(b) r(ound;) 1905 5255 y(26) p eop %%Page: 27 27 27 26 bop Fm 515 523 a(Prop) s(osition) p 1031 523 a(7.9) p Fy 1194 523 a(Under) p 1448 523 a(Assumption) p 1904 523 a(2.2) p 2045 523 a(we) p 2168 523 a(have) p Fw 1222 719 a(Q) p Fl 1288 731 a(1) p Fw 1324 719 a(I) p Fl 1360 731 a(0) p Fx 1416 698 a(\026) p Fw 1398 719 a(R) p Fl 1462 685 a(2) p Fn 1461 740 a(\017) p Fw 1499 719 a(I) p Fl 1535 731 a(0) p Fw 1573 719 a(Q) p Fl 1639 731 a(1) p Ft 1699 719 a(\025) p Fw 1796 663 a(\031) p 1796 700 51 4 v 1804 776 a(\017) 1857 719 y(Q) p Fl 1923 731 a(1) p Fk 1973 627 a(h) p Fx 2013 719 a(\000\() p Fw(e) p Fx(\)) p Ft 2186 719 a(\000) p Fw 2269 719 a(C) p Fl 2328 731 a(3) p Fx 2366 719 a(\() p Fw(\017) p Fg 2442 663 a(1) p 2442 672 29 3 v 2442 705 a(4) p Fx 2484 719 a(\)) p Fk 2516 627 a(i) p Fw 2569 719 a(Q) p Fl 2635 731 a(1) p Fy 515 918 a(wher) l(e) p Fw 749 918 a(C) p Fl 808 930 a(3) p Fy 875 918 a(is) p 965 918 a(a) p 1037 918 a(p) l(ositive) p 1336 918 a(c) l(onstant.) p Fx 515 1079 a(Assume) p 822 1079 a(that) p Fw 1002 1079 a(\022) r(;) p 1080 1079 a(\025;) p 1165 1079 a(\017) p Fx 1227 1079 a(satis\014es) p 1533 1079 a(the) p 1676 1079 a(follo) n(wings.) p Fm 515 1239 a(Assumption) p 1044 1239 a(7.1) p Fw 627 1415 a(\025) p Fx 694 1415 a(+) p Fw 777 1415 a(\022) r(\025) p Fl 866 1381 a(2) p Fw 904 1415 a(\017) p Fo 938 1381 a(\000) p Fl(2) p Fw 1050 1415 a(<) p 1137 1415 a(t) p Fl 1167 1427 a(1) p Fw 1205 1415 a(;) p 1326 1415 a(\022) p 1391 1415 a(<) p 1478 1415 a(t) p Fl 1508 1427 a(2) p Fw 1546 1415 a(;) p 1667 1415 a(\022) r(\025) p Fl 1756 1381 a(2) p Fw 1794 1415 a(\017) p Fo 1828 1381 a(\000) p Fl(1) p Fw 1940 1415 a(<) p 2028 1415 a(t) p Fl 2058 1427 a(3) p Fw 2095 1415 a(;) p 2217 1415 a(\017) p 2274 1415 a(<) p 2361 1415 a(t) p Fl 2391 1427 a(4) p Fw 2429 1415 a(;) p 2550 1415 a(\022) p Fo 2591 1381 a(\000) p Fl(1) p Fw 2681 1415 a(\017) p Fx 2733 1415 a(+) p Fw 2816 1415 a(\022) r(\025) p Fl 2905 1381 a(2) p Fw 2943 1415 a(\017) p Fo 2977 1381 a(\000) p Fl(3) p Fw 3089 1415 a(<) p 3176 1415 a(t) p Fl 3206 1427 a(5) p Fw 3244 1415 a(:) p Fy 515 1591 a(wher) l(e) p Fw 1034 1785 a(t) p Fl 1064 1797 a(2) p Ft 1125 1785 a(\021) p Fx 1250 1729 a(1) p 1222 1766 97 4 v Fw 1222 1842 a(C) p Fl 1281 1854 a(2) p Fw 1329 1785 a(;) p 1451 1785 a(t) p Fl 1481 1797 a(3) p Ft 1541 1785 a(\021) p Fw 1702 1729 a(v) p 1639 1766 171 4 v Fx 1639 1842 a(8) p Fw(\031) s(\015) p Fn 1774 1854 a(e) p Fw 1819 1785 a(;) p 1941 1785 a(t) p Fl 1971 1797 a(4) p Ft 2031 1785 a(\021) p Fx 2119 1785 a(\() p Fw 2191 1729 a(\015) p Fn 2234 1741 a(e) p 2161 1766 139 4 v Fx 2161 1842 a(4) p Fw(C) p Fl 2262 1854 a(3) p Fx 2309 1785 a(\)) p Fl 2341 1751 a(4) p Fw 2378 1785 a(;) p 2500 1785 a(t) p Fl 2530 1797 a(5) p Ft 2591 1785 a(\021) p Fw 2693 1729 a(\031) s(\015) p Fn 2786 1741 a(e) p 2688 1766 V Fx 2688 1842 a(4) p Fw(C) p Fl 2789 1854 a(2) p Fw 2836 1785 a(:) p Fx 515 2122 a(The) p 685 2122 a(second) p 953 2122 a(assumption) p 1392 2122 a(implies) p 1673 2122 a(2) p Fw(\022) r(\025) p Fl 1804 2091 a(2) p Ft 1860 2122 a(\000) p Fw 1942 2122 a(C) p Fl 2001 2134 a(2) p Fw 2039 2122 a(\022) p Fl 2080 2091 a(2) p Fw 2117 2122 a(\025) p Fl 2165 2091 a(2) p Fw 2226 2122 a(>) p 2314 2122 a(\022) r(\025) p Fl 2403 2091 a(2) p Fw 2441 2122 a(:) p Fx 2491 2122 a(Using) p 2724 2122 a(this) p 2886 2122 a(and) p 3047 2122 a(the) p 3190 2122 a(lo) n(w) n(er) 515 2221 y(b) r(ound) p 771 2221 a(of) p Fw 865 2221 a(I) p Fl 901 2233 a(0) p Fx 957 2200 a(\026) p Fw 939 2221 a(R) p Fl 1003 2191 a(2) p Fn 1002 2242 a(\017) p Fw 1040 2221 a(I) p Fl 1076 2233 a(0) p Fx 1141 2221 a(of) p 1236 2221 a(Prop) r(osition) p 1683 2221 a(7.9,) p 1840 2221 a(w) n(e) p 1962 2221 a(obtain) p Fw 598 2447 a(D) p Ft 701 2447 a(\025) p Fw 798 2447 a(\022) r(\025) p Fl 887 2412 a(2) p Fw 935 2390 a(\031) p 935 2428 51 4 v 943 2504 a(\017) 995 2447 y(Q) p Fl 1061 2459 a(1) p Fk 1112 2354 a(h) p Fx 1151 2447 a(\000\() p Fw(e) p Fx(\)) p Ft 1325 2447 a(\000) p Fw 1408 2447 a(C) p Fl 1467 2459 a(3) p Fx 1504 2447 a(\() p Fw(\017) p Fg 1580 2390 a(1) p 1580 2399 29 3 v 1580 2432 a(4) p Fx 1623 2447 a(\)) p Fk 1655 2354 a(i) p Fw 1708 2447 a(Q) p Fl 1774 2459 a(1) p Ft 1833 2447 a(\000) p Fw 1920 2447 a(\016) p Fn 1957 2459 a(e) p Ft 2015 2447 a(\001) p Fw 2060 2447 a(Q) p Fl 2126 2459 a(1) p Fx 2181 2426 a(~) p Fw 2163 2447 a(P) p Fo 2228 2412 a(?) p Fn 2216 2467 a(e) p Fw 2284 2447 a(Q) p Fl 2350 2459 a(1) p Ft 2409 2447 a(\000) p Fw 2495 2447 a(C) p Fl 2554 2459 a(2) p Fk 2606 2379 a(\000) p Fw 2644 2447 a(\025) p Fl 2692 2412 a(2) p Fx 2748 2447 a(+) p Fw 2831 2447 a(\022) p Fl 2872 2412 a(2) p Fw 2910 2447 a(\025) p Fl 2958 2412 a(4) p Fw 2995 2447 a(\017) p Fo 3029 2412 a(\000) p Fl(4) p Fk 3118 2379 a(\001) p Fw 3170 2447 a(Q) p Fl 3236 2459 a(1) p Fw 3273 2447 a(:) p Fx 3231 2582 a(\(29\)) 515 2766 y(Recall) p 765 2766 a(that) p 946 2766 a(\000\() p Fw(e) p Fx(\)) p 1129 2766 a(is) p 1213 2766 a(b) r(ounded) p 1553 2766 a(b) r(elo) n(w) p 1788 2766 a(as) p 1891 2766 a(\000\() p Fw(e) p Fx(\)) p Ft 2070 2766 a(\025) p Fw 2158 2766 a(\015) p Fn 2201 2778 a(e) p Ft 2256 2766 a(\001) p Fx 2298 2766 a(\(Ker) o(\000\() p Fw(e) p Fx(\)\)) p Fo 2651 2736 a(?) p Fx 2707 2766 a(.) p 2769 2766 a(Substituting) p 3245 2766 a(this) 515 2866 y(to) p 616 2866 a(\(29\),) p 815 2866 a(w) n(e) p 937 2866 a(get) p Fw 598 3117 a(D) p Ft 692 3117 a(\025) p Fw 779 3117 a(\022) r(\025) p Fl 868 3082 a(2) p Fw 916 3060 a(\031) p 916 3098 51 4 v 924 3174 a(\017) 977 3117 y(Q) p Fl 1043 3129 a(1) p Fk 1093 3000 a(\024) 1137 3024 y(\020) p Fw 1187 3117 a(\015) p Fn 1230 3129 a(e) p Ft 1284 3117 a(\000) p Fw 1367 3117 a(\016) p Fn 1404 3129 a(e) p Fw 1521 3060 a(\017) p 1449 3098 178 4 v 1449 3174 a(\031) s(\022) r(\025) p Fl 1588 3150 a(2) p Fk 1637 3024 a(\021) p Ft 1705 3117 a(\001) p Fx 1765 3096 a(~) p Fw 1746 3117 a(P) p Fo 1811 3082 a(?) p Fn 1799 3137 a(e) p Ft 1885 3117 a(\000) p Fw 1969 3117 a(C) p Fl 2028 3129 a(3) p Fx 2065 3117 a(\() p Fw(\017) p Fg 2141 3060 a(1) p 2141 3069 29 3 v 2141 3102 a(4) p Fx 2183 3117 a(\)) p Ft 2234 3117 a(\000) p Fw 2327 3060 a(C) p Fl 2386 3072 a(2) p 2327 3098 97 4 v Fw 2350 3174 a(\031) p Fk 2448 3049 a(\000) p Fw 2486 3117 a(\022) p Fo 2527 3082 a(\000) p Fl(1) p Fw 2616 3117 a(\017) p Fx 2668 3117 a(+) p Fw 2751 3117 a(\022) r(\025) p Fl 2840 3082 a(2) p Fw 2878 3117 a(\017) p Fo 2912 3082 a(\000) p Fl(3) p Fk 3001 3049 a(\001) 3039 3000 y(\025) p Fw 3097 3117 a(Q) p Fl 3163 3129 a(1) p Fw 3200 3117 a(:) p Fx 3306 3117 a(\(30\)) 515 3338 y(No) n(w) p 704 3338 a(w) n(e) p 826 3338 a(determine) p Fw 1214 3338 a(\016) p Fn 1251 3350 a(e) p Fx 1314 3338 a(as) p Fw 1231 3558 a(\016) p Fn 1268 3570 a(e) p Fx 1326 3558 a(=) p Fw 1424 3502 a(\031) s(\022) r(\025) p Fl 1563 3472 a(2) p 1424 3539 178 4 v Fw 1496 3615 a(\017) p Fx 1625 3558 a(\() p Fw(\015) p Fn 1700 3570 a(e) p Fx 1754 3558 a(+) p Fw 1837 3558 a(a) p 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2837 3974 a(1) p Fw 2874 3962 a(:) p Fx 515 4188 a(Hence) p 761 4188 a(inf) p Fw 876 4188 a(\033) p Fx 926 4188 a(\() p Fw(D) p Fx 1029 4188 a(\)) p 1090 4188 a(is) p 1174 4188 a(b) r(ounded) p 1513 4188 a(b) r(elo) n(w) p 1748 4188 a(b) n(y) 882 4414 y(inf) p Fw 997 4414 a(\033) p Fx 1047 4414 a(\() p Fw(D) p Fx 1150 4414 a(\)) p Ft 1206 4414 a(\025) p Fw 1293 4414 a(\022) r(\025) p Fl 1382 4379 a(2) p Fw 1430 4357 a(\031) p 1430 4394 51 4 v 1438 4471 a(\017) p Fk 1504 4297 a(\024) p Ft 1548 4414 a(\000) p Fw(a) p Ft 1675 4414 a(\000) p Fw 1758 4414 a(C) p Fl 1817 4426 a(3) p Fx 1855 4414 a(\() p Fw(\017) p Fg 1931 4357 a(1) p 1931 4366 29 3 v 1931 4399 a(4) p Fx 1973 4414 a(\)) p Ft 2024 4414 a(\000) p Fw 2117 4357 a(C) p Fl 2176 4369 a(2) p 2117 4394 97 4 v Fw 2140 4471 a(\031) p Fk 2237 4346 a(\000) p Fw 2275 4414 a(\022) p Fo 2316 4379 a(\000) p Fl(1) p Fw 2405 4414 a(\017) p Fx 2458 4414 a(+) p Fw 2541 4414 a(\022) r(\025) p Fl 2630 4379 a(2) p Fw 2668 4414 a(\017) p Fo 2702 4379 a(\000) p Fl(3) p Fk 2790 4346 a(\001) 2828 4297 y(\025) p Fw 2886 4414 a(Q) p Fl 2952 4426 a(1) p Fw 2989 4414 a(:) p Fx 515 4639 a(Recall) p 770 4639 a(the) p 919 4639 a(lo) n(w) n(er) p 1141 4639 a(b) r(ound) p 1402 4639 a(\(28\)) p 1583 4639 a(of) p Fw 1683 4639 a(B) p Fo 1750 4609 a(0) p Fx 1773 4639 a(.) p 1849 4639 a(As) p 1977 4639 a(the) p 2125 4639 a(third) p 2338 4639 a(condition) p 2708 4639 a(of) p 2808 4639 a(Assumption) p 3273 4639 a(7.1) 515 4739 y(implies) 1622 4893 y(1) p 1622 4930 42 4 v 1622 5006 a(2) p Fw 1674 4949 a(v) p Ft 1735 4949 a(\000) p Fw 1818 4949 a(\016) p Fn 1855 4961 a(e) p Ft 1914 4949 a(\025) p Fx 2012 4893 a(1) p 2012 4930 V 2012 5006 a(4) p Fw 2063 4949 a(v) p 2129 4949 a(>) p Fx 2217 4949 a(0) p Fw(;) p Fx 1905 5255 a(27) p eop %%Page: 28 28 28 27 bop Fx 515 523 a(and) 1063 735 y(0) p Ft 1127 735 a(\025) p Fw 1215 735 a(\022) r(\025) p Fl 1304 700 a(2) p Fw 1352 679 a(\031) p 1352 716 51 4 v 1360 792 a(\017) p Fk 1426 618 a(\024) p Ft 1470 735 a(\000) p Fw(a) p Ft 1597 735 a(\000) p Fw 1680 735 a(C) p Fl 1739 747 a(3) p Fx 1776 735 a(\() p Fw(\017) p Fg 1852 678 a(1) p 1852 687 29 3 v 1852 720 a(4) p Fx 1895 735 a(\)) p Ft 1945 735 a(\000) p Fw 2038 679 a(C) p Fl 2097 691 a(2) p 2038 716 97 4 v Fw 2062 792 a(\031) p Fk 2159 667 a(\000) p Fw 2197 735 a(\022) p Fo 2238 700 a(\000) p Fl(1) p Fw 2327 735 a(\017) p Fx 2379 735 a(+) p Fw 2462 735 a(\022) r(\025) p Fl 2551 700 a(2) p Fw 2589 735 a(\017) p Fo 2623 700 a(\000) p Fl(3) p Fk 2712 667 a(\001) 2750 618 y(\025) p Fw 2808 735 a(;) p Fx 515 963 a(the) p 658 963 a(lo) n(w) n(er) p 875 963 a(b) r(ound) p 1131 963 a(is) p Fw 860 1174 a(B) p Fo 927 1140 a(0) p Ft 950 1174 a(j) p Fl 973 1189 a(\() p Fn(Q) p Fg 1051 1197 a(1) p Fo 1084 1189 a(\010) p Fn(Q) p Fg 1188 1197 a(2) p Fl 1220 1189 a(\)) p Fo(H) p Ft 1330 1174 a(\025) p Fw 1418 1174 a(\022) r(\025) p Fl 1507 1140 a(2) p Fw 1555 1118 a(\031) p 1555 1155 51 4 v 1563 1231 a(\017) p Fk 1629 1057 a(\024) p Ft 1673 1174 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a(Assumption) p 3130 1421 a(7.1,) p 3284 1421 a(w) n(e) 515 1521 y(get) p Fw 598 1773 a(P) p 663 1773 a(E) p Fl 729 1739 a(0) 724 1794 y(\001) p Fw 783 1773 a(B) t(E) p Fl 916 1739 a(0) 911 1794 y(\001) p Fw 970 1773 a(P) p Ft 1058 1773 a(\025) p Fw 1145 1773 a(\022) r(\025) p Fl 1234 1739 a(2) p Fw 1282 1717 a(\031) p 1282 1754 51 4 v 1290 1830 a(\017) 1342 1773 y(P) p 1407 1773 a(E) p Fl 1473 1739 a(0) 1468 1794 y(\001) p Fk 1541 1656 a(\024) p Ft 1585 1773 a(\000) p Fw(a) p Ft 1712 1773 a(\000) p Fw 1795 1773 a(C) p Fl 1854 1785 a(3) p Fx 1891 1773 a(\() p Fw(\017) p Fg 1967 1717 a(1) p 1967 1726 29 3 v 1967 1759 a(4) p Fx 2010 1773 a(\)) p Ft 2060 1773 a(\000) p Fw 2153 1717 a(C) p Fl 2212 1729 a(2) p 2153 1754 97 4 v Fw 2177 1830 a(\031) p Fk 2274 1706 a(\000) p Fw 2312 1773 a(\022) p Fo 2353 1739 a(\000) p Fl(1) p Fw 2442 1773 a(\017) p Fx 2494 1773 a(+) p Fw 2577 1773 a(\022) r(\025) p Fl 2666 1739 a(2) p Fw 2704 1773 a(\017) p Fo 2738 1739 a(\000) p Fl(3) p Fk 2827 1706 a(\001) p Fx 2884 1773 a(+) p 2967 1773 a(\() p Fw(\015) p Fn 3042 1785 a(e) p Fx 3096 1773 a(+) p Fw 3179 1773 a(a) p Fx(\)) p Ft 3274 1773 a(\001) p Fx 3334 1752 a(~) p Fw 3315 1773 a(P) p Fo 3380 1739 a(?) p Fn 3368 1794 a(e) p Fk 3436 1656 a(\025) p Fw 3494 1773 a(E) p Fl 3560 1739 a(0) 3555 1794 y(\001) p Fw 3614 1773 a(P) p Fx 679 2006 a(=) p Fw 766 2006 a(\022) r(\025) p Fl 855 1971 a(2) p Fw 903 1950 a(\031) p 903 1987 51 4 v 911 2063 a(\017) 964 2006 y(P) p 1029 2006 a(E) p Fl 1095 1971 a(0) 1090 2026 y(\001) p Fk 1162 1889 a(\024) p Fw 1206 2006 a(\015) p Fn 1249 2018 a(e) p Ft 1303 2006 a(\000) p Fw 1386 2006 a(C) p Fl 1445 2018 a(3) p Fx 1483 2006 a(\() p Fw(\017) p Fg 1559 1949 a(1) p 1559 1958 29 3 v 1559 1992 a(4) p Fx 1601 2006 a(\)) p Ft 1652 2006 a(\000) p Fw 1745 1950 a(C) p Fl 1804 1962 a(2) p 1745 1987 97 4 v Fw 1768 2063 a(\031) p Fk 1865 1939 a(\000) p Fw 1903 2006 a(\022) p Fo 1944 1971 a(\000) p Fl(1) p Fw 2034 2006 a(\017) p Fx 2086 2006 a(+) p Fw 2169 2006 a(\022) 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a(7.2.) p Fd(\003) p Fs 515 2649 a(7.2) p 770 2649 a(P) m(ositiv) m(e) p 1194 2649 a(Comm) m(utator) p 1858 2649 a(with) p 2113 2649 a(resp) s(ect) p 2501 2649 a(to) p 2638 2649 a(the) p 2832 2649 a(sp) s(ectral) p 3254 2649 a(lo-) 770 2765 y(calization) p 1271 2765 a(in) p Fq 1402 2765 a(L) p Fx 515 2918 a(In) p 619 2918 a(this) p 780 2918 a(subsection) p 1183 2918 a(w) n(e) p 1305 2918 a(complete) p 1655 2918 a(the) p 1798 2918 a(pro) r(of) p 2015 2918 a(of) p 2110 2918 a(Theorem) p 2461 2918 a(7.1.) 515 3018 y(Let) p 670 3018 a(\001) p Fo 739 2988 a(0) p Fx 797 3018 a(b) r(e) p 916 3018 a(an) p 1038 3018 a(in) n(terv) p 1246 3018 a(al) p 1345 3018 a(s.t.) p 1512 3018 a(\001) p Ft 1615 3018 a(\032) p Fx 1714 3018 a(\001) p Fo 1783 2988 a(0) p Fx 1806 3018 a(,) p 1865 3018 a(\001) p Fo 1934 2988 a(0) p Ft 1981 3018 a(\\) p Fw 2059 3018 a(\033) p Fx 2109 3018 a(\() p Fw(L) p Fn 2198 3030 a(S) p Fx 2246 3018 a(\)) p 2312 3018 a(=) p Ft 2411 3018 a(f) p Fw(e) p Ft(g) p Fx 2567 3018 a(and) p 2735 3018 a(supp) p Fw(h) p Ft 2987 3018 a(\032) p Fx 3086 3018 a(\001) p Fo 3155 2988 a(0) p Fx 3178 3018 a(.) p 3258 3018 a(Let) p Fw 515 3117 a(F) p Fl 568 3129 a(\001) p Ff 623 3113 a(0) p Ft 682 3117 a(2) p Fw 770 3117 a(C) p Fo 835 3087 a(1) p Fl 829 3138 a(0) p Fx 905 3117 a(\() p Fi(R) p Fx(\)) p 1063 3117 a(b) r(e) p 1182 3117 a(a) p 1257 3117 a(smo) r(oth) p 1556 3117 a(function) p 1887 3117 a(0) p Ft 1961 3117 a(\024) p Fw 2058 3117 a(F) p Fl 2111 3129 a(\001) p Ff 2166 3113 a(0) p Ft 2225 3117 a(\024) p Fx 2322 3117 a(1) p 2397 3117 a(whic) n(h) p 2641 3117 a(satis\014es) p Fw 2952 3117 a(F) p Fl 3005 3129 a(\001) p Ff 3060 3113 a(0) p Fx 3119 3117 a(=) p 3216 3117 a(1) p 3291 3117 a(on) 515 3217 y(supp) p Fw 716 3217 a(h) p Fx 796 3217 a(and) p 962 3217 a(supp) p Fw 1164 3217 a(F) p Fl 1217 3229 a(\001) p Ff 1272 3213 a(0) p Ft 1329 3217 a(\032) p Fx 1424 3217 a(\001) p Fo 1493 3187 a(0) p Fx 1517 3217 a(.) p 1590 3217 a(W) p 1668 3217 a(e) p 1737 3217 a(set) p Fw 1871 3217 a(F) p Fl 1936 3187 a(0) 1924 3240 y(\001) p Ff 1979 3223 a(0) p Ft 2036 3217 a(\021) p Fw 2132 3217 a(F) p Fl 2185 3229 a(\001) p Ff 2240 3213 a(0) p Fx 2266 3217 a(\() p Fw(L) p Fl 2355 3229 a(0) p Fx 2392 3217 a(\),) p 2481 3217 a(and) 2665 3196 y(\026) p Fw 2647 3217 a(F) p Fl 2712 3187 a(0) 2700 3240 y(\001) p Ff 2755 3223 a(0) p Ft 2812 3217 a(\021) p Fx 2907 3217 a(1) p Ft 2970 3217 a(\000) p Fw 3056 3217 a(F) p Fl 3121 3187 a(0) 3109 3240 y(\001) p Ff 3164 3223 a(0) p Fx 3190 3217 a(.) p 3264 3217 a(W) p 3342 3217 a(e) 515 3317 y(ev) p 591 3317 a(aluate) p 840 3317 a(eac) n(h) p 1027 3317 a(term) p 1225 3317 a(of) p 1320 3317 a(the) p 1463 3317 a(follo) n(wing) p 1813 3317 a(equation;) p Fw 1013 3534 a(P) p 1078 3534 a(h) p Fx 1140 3534 a([) p Fw(L;) p 1257 3534 a(A) p Fx(]) p Fw 1355 3534 a(hP) p Fx 1491 3534 a(=) p Fw 1579 3534 a(P) p 1644 3534 a(hB) t(hP) p Fx 1890 3534 a(+) 2003 3478 y(1) p 1983 3515 84 4 v 1983 3591 a(10) p Fw 2076 3534 a(P) p 2141 3534 a(hS) p Fl 2240 3546 a(1) p Fw 2277 3534 a(hP) p Fx 1491 3696 a(=) p Fw 1579 3696 a(P) p 1644 3696 a(hF) p Fl 1757 3661 a(0) 1745 3716 y(\001) p Ff 1800 3700 a(0) p Fw 1826 3696 a(B) t(F) p Fl 1958 3661 a(0) 1946 3716 y(\001) p Ff 2001 3700 a(0) p Fw 2028 3696 a(hP) p Fx 3231 3696 a(\(31\)) 1487 3868 y(+) p Fw 1570 3868 a(P) p 1635 3868 a(h) p Fx 1701 3847 a(\026) p Fw 1683 3868 a(F) p Fl 1748 3834 a(0) 1736 3889 y(\001) p Ff 1791 3872 a(0) p Fw 1817 3868 a(B) t(F) p Fl 1949 3834 a(0) 1937 3889 y(\001) p Ff 1992 3872 a(0) p Fw 2018 3868 a(hP) p Fx 2150 3868 a(+) 2263 3812 y(1) p 2243 3849 V 2243 3925 a(40) p Fw 2336 3868 a(P) p 2401 3868 a(hS) p Fl 2500 3880 a(1) p Fw 2536 3868 a(hP) p Fx 2667 3868 a(+) p Fw 2751 3868 a(h:c:) p Fx 3231 3868 a(\(32\)) 1487 4068 y(+) p Fw 1570 4068 a(P) p 1635 4068 a(h) p Fx 1701 4047 a(\026) p Fw 1683 4068 a(F) p Fl 1748 4034 a(0) 1736 4089 y(\001) p Ff 1791 4072 a(0) p Fw 1817 4068 a(B) p Fx 1903 4047 a(\026) p Fw 1884 4068 a(F) p Fl 1949 4034 a(0) 1937 4089 y(\001) p Ff 1992 4072 a(0) p Fw 2018 4068 a(hP) p Fx 2150 4068 a(+) 2263 4012 y(1) p 2243 4049 V 2243 4125 a(20) p Fw 2336 4068 a(P) p 2401 4068 a(hS) p Fl 2500 4080 a(1) p Fw 2536 4068 a(hP) p Fx 3231 4068 a(\(33\)) 515 4279 y(F) p 562 4279 a(or) p 664 4279 a(the) p 807 4279 a(\014rst) p 978 4279 a(part) p 1158 4279 a(\(31\),) p 1356 4279 a(w) n(e) p 1479 4279 a(ha) n(v) n(e) p 1670 4279 a(the) p 1813 4279 a(follo) n(wing) p 2164 4279 a(lemma.) p Fm 515 4445 a(Lemma) p 853 4445 a(7.10) p Fy 1065 4445 a(Supp) l(ose) p 1381 4445 a(that) p Fw 1550 4445 a(\025;) p 1635 4445 a(\017;) p 1706 4445 a(\022) p Fy 1777 4445 a(satisfy) p 2036 4445 a(Assumption) p 2492 4445 a(7.1.) p 2666 4445 a(Then) p 2883 4445 a(we) p 3006 4445 a(have) p Fx 969 4676 a(\(31\)) p 1139 4676 a(=) p Fw 1227 4676 a(P) p 1292 4676 a(hF) p Fl 1405 4642 a(0) 1393 4697 y(\001) p Ff 1448 4680 a(0) p Fw 1474 4676 a(B) t(F) p Fl 1606 4642 a(0) 1594 4697 y(\001) p Ff 1649 4680 a(0) p Fw 1676 4676 a(hP) p Ft 1812 4676 a(\025) p Fw 1909 4620 a(\022) r(\025) p Fl 1998 4590 a(2) p 1909 4657 127 4 v Fw 1956 4733 a(\017) 2046 4676 y(\015) p Fn 2089 4688 a(e) p Fw 2125 4676 a(P) p 2190 4676 a(h) p Fx(\(1) p Ft 2330 4676 a(\000) p Fw 2413 4676 a(\025c) p Fl 2497 4688 a(1) p Ft 2553 4676 a(\000) p Fx 2636 4676 a(7) 2696 4655 y(~) p Fw 2678 4676 a(P) p Fn 2731 4688 a(e) p Fx 2766 4676 a(\)) p Fw(hP) r(;) p Fy 515 4887 a(wher) l(e) p Fw 749 4887 a(c) p Fl 785 4899 a(1) p Fy 852 4887 a(is) p 941 4887 a(a) p 1013 4887 a(c) l(onstant) p 1344 4887 a(which) p 1578 4887 a(dep) l(ends) p 1887 4887 a(only) p 2068 4887 a(on) p Fw 2187 4887 a(F) p Fl 2240 4899 a(\001) p Ff 2295 4882 a(0) p Fy 2351 4887 a(and) p Ft 2512 4887 a(k) p Fw(I) p Fl 2590 4899 a(0) p Ft 2627 4887 a(k) p Fy(.) p Fx 1905 5255 a(28) p eop %%Page: 29 29 29 28 bop Fy 515 523 a(Pr) l(o) l(of) p Fx 515 623 a(The) p 685 623 a(pro) r(of) p 902 623 a(is) p 986 623 a(the) p 1129 623 a(same) p 1337 623 a(as) p 1439 623 a(that) p 1619 623 a(of) p 1713 623 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2718 1055 a(+) p Fw 2801 1055 a(\017) p Fx 2853 1055 a(+) p Fw 2936 1055 a(\025\017) p Fo 3018 1021 a(\000) p Fl(1) p Fx 3108 1055 a(\)) p Fw(P) p 3205 1055 a(h) p Fl 3253 1021 a(2) p Fw 3290 1055 a(P) p Fy 515 1234 a(wher) l(e) p Fw 749 1234 a(c) p Fl 785 1246 a(2) p Fy 852 1234 a(is) p 941 1234 a(a) p 1013 1234 a(c) l(onstant) p 1344 1234 a(which) p 1578 1234 a(dep) l(ends) p 1887 1234 a(only) p 2068 1234 a(on) p Fw 2187 1234 a(F) p Fl 2240 1246 a(\001) p Ff 2295 1230 a(0) p Fy 2351 1234 a(and) p Fw 2512 1234 a(I) p Fl 2548 1246 a(0) p Fy 2586 1234 a(.) 515 1379 y(Pr) l(o) l(of) p Fx 515 1479 a(W) p 593 1479 a(e) p 658 1479 a(divide) p 904 1479 a(the) p 1047 1479 a(second) p 1315 1479 a(part) p 1496 1479 a(\(32\)) p 1671 1479 a(in) n(to) p 1839 1479 a(three) p 2052 1479 a(parts;) 1139 1673 y(\(32\)) p 1310 1673 a(=) 1428 1617 y(9) p 1408 1654 84 4 v 1408 1730 a(10) p Fw 1501 1673 a(P) p 1566 1673 a(h) p Fx 1632 1653 a(\026) p Fw 1614 1673 a(F) p Fl 1679 1639 a(0) 1667 1694 y(\001) p 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a(0) p Fw 1945 2035 a(hP) r(:) p Fx 2090 2035 a(+) p Fw 2173 2035 a(h:c:) p Fx 3231 2035 a(\(36\)) 515 2191 y(Using) p 753 2191 a(the) p 902 2191 a(fact) p 1071 2191 a(that) p 1256 2191 a(0) p Ft 1329 2191 a(\024) p Fw 1425 2191 a(F) p Fl 1490 2161 a(0) 1478 2214 y(\001) p Ff 1533 2197 a(0) p Ft 1592 2191 a(\024) p Fx 1688 2191 a(1,) p 1787 2191 a(w) n(e) p 1914 2191 a(get) p 2058 2191 a(\(34\)) p Ft 2237 2191 a(\025) p Fx 2334 2191 a(0) p 2408 2191 a(as) p 2515 2191 a(in) p 2617 2191 a(the) p 2766 2191 a(case) p 2946 2191 a(\(57\)) p 3127 2191 a(of) p 3227 2191 a([10) o(].) 515 2291 y(T) p 568 2291 a(o) p 642 2291 a(ev) p 718 2291 a(aluate) p 971 2291 a(\(35\),) p 1175 2291 a(w) n(e) p 1302 2291 a(apply) p 1535 2291 a(Lemma) p 1836 2291 a(B.1) p 1992 2291 a(with) p Fw 2185 2291 a(') p Fl 2239 2303 a(1) p Fx 2308 2291 a(=) p Fw 2403 2291 a(F) p Fl 2468 2261 a(0) 2456 2314 y(\001) p Ff 2511 2297 a(0) p Fw 2538 2291 a(P) p 2603 2291 a( ) p Fx 2660 2291 a(,) p 2716 2291 a(and) p Fw 2882 2291 a(') p Fl 2936 2303 a(2) p Fx 3004 2291 a(=) 3118 2270 y(\026) p Fw 3100 2291 a(F) p Fl 3165 2261 a(0) 3153 2314 y(\001) p Ff 3208 2297 a(0) p Fw 3234 2291 a(P) p 3299 2291 a( ) p Fx 3356 2291 a(.) 515 2390 y(As) p Fw 642 2390 a(L) p Fl 699 2402 a(0) p Fx 768 2390 a(and) p Fw 935 2390 a(N) p Fl 1002 2402 a(\003) p Fj 1047 2386 a(c) 1047 2419 y(v) p Fx 1119 2390 a(strongly) p 1445 2390 a(comm) n(ute) p 1807 2390 a(with) p Fw 2001 2390 a(N) p Fx 2109 2390 a(resp) r(ectiv) n(ely) p 2528 2390 a(,) p Fw 2584 2390 a(F) p Fl 2649 2360 a(0) 2637 2413 y(\001) p Ff 2692 2397 a(0) p Fw 2719 2390 a(P) p 2784 2390 a( ) s(;) p Fx 2896 2369 a(\026) p Fw 2878 2390 a(F) p Fl 2943 2360 a(0) 2931 2413 y(\001) p Ff 2986 2397 a(0) p Fw 3012 2390 a(P) p 3077 2390 a( ) p Fx 3166 2390 a(are) p 3310 2390 a(in) p Fw 515 2490 a(D) p Fx 586 2490 a(\() p Fw(N) p Fx 694 2490 a(\).) p 786 2490 a(By) p 916 2490 a(Lemma) p 1213 2490 a(B.1,) p 1387 2490 a(w) n(e) p 1509 2490 a(ha) n(v) n(e) p Fk 612 2600 a(\014) 612 2650 y(\014) 639 2603 y(\012) p Fw 678 2671 a( ) p Fk 749 2600 a(\014) 749 2650 y(\014) p Fw 777 2671 a(\025P) p Fx 909 2650 a(\026) p Fw 890 2671 a(F) p Fl 955 2636 a(0) 943 2691 y(\001) p Ff 998 2675 a(0) p Fw 1025 2671 a(I) p Fl 1061 2683 a(1) p Fw 1099 2671 a(F) p Fl 1164 2636 a(0) 1152 2691 y(\001) p Ff 1207 2675 a(0) p Fw 1233 2671 a(P) p Fk 1298 2600 a(\014) 1298 2650 y(\014) p Fw 1339 2671 a( ) p Fk 1396 2603 a(\013) 1436 2600 y(\014) 1436 2650 y(\014) p Ft 1486 2671 a(\024) 538 2869 y(\024) p Fx 625 2869 a(3) p Fw(c) p Ft(k) p Fw(Y) p Ft 811 2869 a(k) p Fl 853 2835 a(2) p Fk 904 2752 a(\022) 965 2774 y(\015) 965 2823 y(\015) 965 2873 y(\015) p Fw 1011 2869 a(N) p Fg 1097 2798 a(1) p 1097 2807 29 3 v 1097 2841 a(2) p Fl 1078 2894 a(\003) p Fj 1123 2902 a(v) p Fw 1163 2869 a(F) p Fl 1228 2835 a(0) 1216 2890 y(\001) p Ff 1271 2873 a(0) p Fw 1297 2869 a(P) p 1362 2869 a( ) p Fk 1419 2774 a(\015) 1419 2823 y(\015) 1419 2873 y(\015) p Fl 1465 2792 a(2) p Fx 1521 2869 a(+) p Fk 1604 2774 a(\015) 1604 2823 y(\015) 1604 2873 y(\015) p Fw 1650 2869 a(N) p Fg 1736 2798 a(1) p 1736 2807 V 1736 2841 a(2) p Fl 1717 2894 a(\003) p Fj 1762 2902 a(v) p Fx 1821 2848 a(\026) p Fw 1802 2869 a(F) p Fl 1867 2835 a(0) 1855 2890 y(\001) p Ff 1910 2873 a(0) p Fw 1937 2869 a(P) p 2002 2869 a( ) p Fk 2059 2774 a(\015) 2059 2823 y(\015) 2059 2873 y(\015) p Fl 2105 2792 a(2) p Fk 2142 2752 a(\023) p Fx 2203 2869 a(+3) p Fw(\025) p Fl 2358 2835 a(2) p Ft 2405 2813 a(k) p Fx 2453 2813 a(~) p Fw 2446 2813 a(pg) p Fl 2528 2825 a(1) p Ft 2565 2813 a(k) p Fl 2606 2771 a(2) p 2405 2850 239 4 v Fw 2506 2926 a(c) p Fk 2667 2777 a(\020) 2717 2799 y(\015) 2717 2848 y(\015) p Fw 2763 2869 a(F) p Fl 2828 2835 a(0) 2816 2890 y(\001) p Ff 2871 2873 a(0) p Fw 2898 2869 a(P) p 2963 2869 a( ) p Fk 3020 2799 a(\015) 3020 2848 y(\015) p Fl 3066 2818 a(2) p Fx 3121 2869 a(+) p Fk 3204 2799 a(\015) 3204 2848 y(\015) p Fx 3269 2848 a(\026) p Fw 3250 2869 a(F) p Fl 3315 2835 a(0) 3303 2890 y(\001) p Ff 3358 2873 a(0) p Fw 3385 2869 a(P) p 3450 2869 a( ) p Fk 3507 2799 a(\015) 3507 2848 y(\015) p Fl 3553 2818 a(2) p Fk 3590 2777 a(\021) p Fx 515 3117 a(+3) p Fw(c) p Ft(k) p Fx 713 3096 a(\026) p Fw 700 3117 a(Y) p Ft 765 3117 a(k) p Fl 807 3082 a(2) p Fk 858 3000 a(\022) 919 3021 y(\015) 919 3071 y(\015) 919 3121 y(\015) p Fw 965 3117 a(N) p Fg 1050 3046 a(1) p 1050 3055 29 3 v 1050 3088 a(2) p Fl 1032 3141 a(\003) p Fj 1077 3149 a(v) p Fw 1117 3117 a(F) p Fl 1182 3082 a(0) 1170 3137 y(\001) p Ff 1225 3121 a(0) p Fw 1251 3117 a(P) p 1316 3117 a( ) p Fk 1373 3021 a(\015) 1373 3071 y(\015) 1373 3121 y(\015) p Fl 1419 3039 a(2) p Fx 1475 3117 a(+) p Fk 1558 3021 a(\015) 1558 3071 y(\015) 1558 3121 y(\015) p Fw 1604 3117 a(N) p Fg 1690 3046 a(1) p 1690 3055 V 1690 3088 a(2) p Fl 1671 3141 a(\003) p Fj 1716 3149 a(v) p Fx 1775 3096 a(\026) p Fw 1756 3117 a(F) p Fl 1821 3082 a(0) 1809 3137 y(\001) p Ff 1864 3121 a(0) p Fw 1890 3117 a(P) p 1955 3117 a( ) p Fk 2012 3021 a(\015) 2012 3071 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p Fw 1448 3343 a(Y) p Ft 1514 3343 a(k) p Fl 1556 3309 a(2) p Fk 1593 3276 a(\001) 1645 3248 y(\015) 1645 3298 y(\015) 1645 3347 y(\015) p Fw 1691 3343 a(N) p Fg 1777 3272 a(1) p 1777 3281 29 3 v 1777 3315 a(2) p Fl 1758 3368 a(\003) p Fj 1803 3376 a(v) p Fw 1843 3343 a(P) p 1908 3343 a( ) p Fk 1965 3248 a(\015) 1965 3298 y(\015) 1965 3347 y(\015) p Fl 2011 3266 a(2) p Fx 2067 3343 a(+) p Fw 2150 3343 a(\025) p Fl 2198 3309 a(2) p Fx 2246 3287 a(6) p 2246 3324 42 4 v Fw 2249 3400 a(c) p Ft 2311 3343 a(k) p Fw 2352 3343 a(P) p 2417 3343 a( ) p Ft 2474 3343 a(k) p Fl 2516 3302 a(2) p Fk 2567 3251 a(\020) p Ft 2616 3343 a(k) p Fx 2665 3343 a(~) p Fw 2658 3343 a(pg) p Fl 2740 3355 a(1) p Ft 2777 3343 a(k) p Fl 2818 3302 a(2) p Fx 2874 3343 a(+) p Ft 2957 3343 a(k) p Fx 3005 3343 a(~) p Fw 2998 3343 a(pg) p Fl 3080 3355 a(2) p Ft 3117 3343 a(k) p Fl 3158 3302 a(2) p Fk 3196 3251 a(\021) p Fw 3259 3343 a(;) p Fx 515 3522 a(for) p 639 3522 a(arbitrary) p Fw 990 3522 a(c) p 1048 3522 a(>) p Fx 1136 3522 a(0.) p 1237 3522 a(W) p 1315 3522 a(e) p 1377 3522 a(used) p 1563 3522 a(the) p 1703 3522 a(strong) p 1955 3522 a(comm) n(utativit) n(y) p 2510 3522 a(of) p Fw 2602 3522 a(N) p Fl 2669 3534 a(\003) p Fj 2714 3542 a(v) p Fx 2778 3522 a(and) p Fw 2937 3522 a(L) p Fl 2994 3534 a(0) p Fx 3031 3522 a(,) p 3079 3522 a(together) 515 3622 y(with) p 700 3622 a(the) p 839 3622 a(fact) p 999 3622 a(that) p 1175 3622 a(0) p Ft 1239 3622 a(\024) p Fw 1327 3622 a(F) p Fl 1392 3592 a(0) 1380 3645 y(\001) p Ff 1435 3628 a(0) p Ft 1484 3622 a(\024) p Fx 1572 3622 a(1.) p 1672 3622 a(T) p 1725 3622 a(aking) p Fw 1945 3622 a(c) p Fx 2005 3622 a(as) p Fw 2103 3622 a(c) p Fx 2162 3622 a(=) p Fw 2249 3622 a(v) s(=) p Fx(240\() p Ft(k) p Fw(Y) p Ft 2599 3622 a(k) p Fl 2641 3592 a(2) p Fx 2688 3622 a(+) p Ft 2764 3622 a(k) p Fx 2819 3601 a(\026) p Fw 2806 3622 a(Y) p Ft 2871 3622 a(k) p Fl 2913 3592 a(2) p Fx 2950 3622 a(\),) p 3030 3622 a(w) n(e) p 3148 3622 a(obtain) p Fk 583 3753 a(\014) 583 3803 y(\014) 611 3757 y(\012) p Fw 650 3824 a( ) p Fk 721 3753 a(\014) 721 3803 y(\014) p Fw 748 3824 a(\025P) p Fx 880 3803 a(\026) p Fw 861 3824 a(F) p Fl 926 3790 a(0) 914 3845 y(\001) p Ff 969 3828 a(0) p Fw 996 3824 a(I) p Fl 1032 3836 a(1) p Fw 1070 3824 a(F) p Fl 1135 3790 a(0) 1123 3845 y(\001) p Ff 1178 3828 a(0) p Fw 1204 3824 a(P) p Fk 1269 3753 a(\014) 1269 3803 y(\014) p Fw 1311 3824 a( ) p Fk 1368 3757 a(\013) 1407 3753 y(\014) 1407 3803 y(\014) p Ft 1458 3824 a(\024) p Fw 1575 3768 a(v) p 1555 3805 84 4 v Fx 1555 3881 a(40) p Fk 1662 3729 a(\015) 1662 3778 y(\015) 1662 3828 y(\015) p Fw 1708 3824 a(N) p Fg 1794 3753 a(1) p 1794 3762 29 3 v 1794 3795 a(2) p Fl 1775 3848 a(\003) p Fj 1820 3856 a(v) p Fw 1860 3824 a(P) p 1925 3824 a( ) p Fk 1982 3729 a(\015) 1982 3778 y(\015) 1982 3828 y(\015) p Fl 2028 3747 a(2) p Fx 2084 3824 a(+) p Fw 2167 3824 a(c) p Fo 2203 3790 a(0) p Fl 2203 3845 a(1) p Fw 2240 3824 a(\025) p Fl 2288 3790 a(2) p Ft 2340 3824 a(k) p Fw 2381 3824 a(P) p 2446 3824 a( ) p Ft 2503 3824 a(k) p Fl 2544 3782 a(2) p Fk 2596 3732 a(\020) p Ft 2645 3824 a(k) p Fx 2694 3824 a(~) p Fw 2687 3824 a(p) o(g) p Fl 2768 3836 a(1) p Ft 2805 3824 a(k) p Fl 2847 3782 a(2) p Fx 2902 3824 a(+) p Ft 2986 3824 a(k) p Fx 3034 3824 a(~) p Fw 3027 3824 a(pg) p Fl 3109 3836 a(2) p Ft 3146 3824 a(k) p Fl 3187 3782 a(2) p Fk 3224 3732 a(\021) p Fw 3288 3824 a(;) p Fx 515 4016 a(where) p Fw 755 4016 a(c) p Fo 791 3986 a(0) p Fl 791 4036 a(1) p Fx 851 4016 a(=) p 939 4016 a(1440) p Fl 1118 3983 a(1) p 1115 3997 36 4 v Fn 1115 4044 a(v) p Fx 1160 4016 a(\() p Ft(k) p Fw(Y) p Ft 1300 4016 a(k) p Fl 1342 3986 a(2) p Fx 1398 4016 a(+) p Ft 1481 4016 a(k) p Fx 1536 3995 a(\026) p Fw 1523 4016 a(Y) p Ft 1589 4016 a(k) p Fl 1631 3986 a(2) p Fx 1668 4016 a(\).) p 1760 4016 a(Hence) p 2007 4016 a(w) n(e) p 2129 4016 a(ha) n(v) n(e) p Fw 515 4218 a(\025P) p Fx 647 4197 a(\026) p Fw 628 4218 a(F) p Fl 693 4184 a(0) 681 4239 y(\001) p Ff 736 4222 a(0) p Fw 763 4218 a(I) p Fl 799 4230 a(1) p Fw 836 4218 a(F) p Fl 901 4184 a(0) 889 4239 y(\001) p Ff 944 4222 a(0) p Fw 971 4218 a(P) p Fx 1054 4218 a(+) 1168 4162 y(1) p 1147 4199 84 4 v 1147 4275 a(40) p Fw 1240 4218 a(P) p 1305 4218 a(S) p Fl 1356 4230 a(1) p Fw 1393 4218 a(P) p Fx 1476 4218 a(+) p Fw 1559 4218 a(h:c) p Ft 538 4418 a(\025) p Fx 656 4362 a(1) p 635 4399 V 635 4475 a(20) p Fw 728 4418 a(P) p 793 4418 a(S) p Fl 844 4430 a(1) p Fw 881 4418 a(P) p Ft 965 4418 a(\000) p Fw 1078 4362 a(v) p 1058 4399 V Fx 1058 4475 a(20) p Fw 1151 4418 a(P) p 1216 4418 a(N) p Fl 1283 4430 a(\003) p Fj 1328 4438 a(v) p Fw 1367 4418 a(P) p Ft 1451 4418 a(\000) p Fx 1534 4418 a(2) p Fw(c) p Fo 1612 4384 a(0) p Fl 1612 4439 a(1) p Fw 1648 4418 a(\025) p Fl 1696 4384 a(2) p Fw 1734 4418 a(P) p Fk 1813 4351 a(\000) p Ft 1851 4418 a(k) p Fx 1900 4418 a(~) p Fw 1893 4418 a(p) o(g) p Fl 1974 4430 a(1) p Ft 2011 4418 a(k) p Fl 2053 4384 a(2) p Fx 2108 4418 a(+) p Ft 2191 4418 a(k) p Fx 2240 4418 a(~) p Fw 2233 4418 a(p) o(g) p Fl 2314 4430 a(2) p Ft 2351 4418 a(k) p Fl 2393 4384 a(2) p Fk 2430 4351 a(\001) p Ft 2491 4418 a(\025) p 2579 4418 a(\000) p Fx(2) p Fw(c) p Fo 2722 4384 a(0) p Fl 2722 4439 a(1) p Fw 2758 4418 a(\025) p Fl 2806 4384 a(2) p Fw 2844 4418 a(P) p Fk 2922 4326 a(\020) p Ft 2972 4418 a(k) p Fx 3020 4418 a(~) p Fw 3013 4418 a(pg) p Fl 3095 4430 a(1) p Ft 3132 4418 a(k) p Fl 3174 4377 a(2) p Fx 3229 4418 a(+) p Ft 3312 4418 a(k) p Fx 3361 4418 a(~) p Fw 3354 4418 a(pg) p Fl 3436 4430 a(2) p Ft 3472 4418 a(k) p Fl 3514 4377 a(2) p Fk 3551 4326 a(\021) p Fw 3615 4418 a(:) p Fx 515 4597 a(Sandwic) n(hing) p 1001 4597 a(this) p 1167 4597 a(inequalit) n(y) p 1556 4597 a(with) p Fw 1749 4597 a(h) p Fx(,) p 1852 4597 a(and) p 2018 4597 a(using) p 2239 4597 a(the) p 2386 4597 a(comm) n(utativit) n(y) p 2948 4597 a(of) p Fw 3046 4597 a(h) p Fx 3126 4597 a(and) p Fw 3291 4597 a(P) p Fx 3356 4597 a(,) 515 4697 y(w) n(e) p 637 4697 a(obtain) 515 4871 y(\(35\)) p 685 4871 a(=) p Fw 773 4871 a(\025P) p 886 4871 a(h) p Fx 953 4850 a(\026) p Fw 934 4871 a(F) p Fl 999 4837 a(0) 987 4892 y(\001) p Ff 1042 4875 a(0) p Fw 1069 4871 a(I) p Fl 1105 4883 a(1) p Fw 1142 4871 a(F) p Fl 1207 4837 a(0) 1195 4892 y(\001) p Ff 1250 4875 a(0) p Fw 1277 4871 a(hP) p Fx 1408 4871 a(+) 1522 4815 y(1) p 1501 4852 V 1501 4928 a(40) p Fw 1594 4871 a(P) p 1659 4871 a(hS) p Fl 1758 4883 a(1) p Fw 1795 4871 a(hP) p Fx 1926 4871 a(+) p Fw 2009 4871 a(h:c:) p Ft 2162 4871 a(\025) p 2250 4871 a(\000) p Fx(2) p Fw(c) p Fo 2393 4837 a(0) p Fl 2393 4892 a(1) p Fw 2429 4871 a(\025) p Fl 2477 4837 a(2) p Fw 2514 4871 a(h) p Fl 2562 4837 a(2) p Fw 2600 4871 a(P) p Fk 2678 4779 a(\020) p Ft 2728 4871 a(k) p Fx 2776 4871 a(~) p Fw 2769 4871 a(pg) p Fl 2851 4883 a(1) p Ft 2888 4871 a(k) p Fl 2929 4829 a(2) p Fx 2985 4871 a(+) p Ft 3068 4871 a(k) p Fx 3117 4871 a(~) p Fw 3110 4871 a(p) o(g) p Fl 3191 4883 a(2) p Ft 3228 4871 a(k) p Fl 3270 4829 a(2) p Fk 3307 4779 a(\021) p Fw 3370 4871 a(:) p Fx 3231 5006 a(\(37\)) 1905 5255 y(29) p eop %%Page: 30 30 30 29 bop Fx 515 523 a(The) p 685 523 a(last) p 843 523 a(part) p 1023 523 a(\(36\)) p 1198 523 a(can) p 1350 523 a(b) r(e) p 1463 523 a(estimated) p 1842 523 a(in) p 1939 523 a(a) p 2008 523 a(manner) p 2308 523 a(of) p 2403 523 a(Prop) r(osition) p 2850 523 a(5.2) p 2983 523 a(in) p 3080 523 a([10) o(]:) 598 780 y(\(36\)) p 768 780 a(=) p Fw 856 780 a(P) p 921 780 a(h) p Fx 987 759 a(\026) p Fw 969 780 a(F) p Fl 1034 746 a(0) 1022 801 y(\001) p Ff 1077 784 a(0) p Fx 1103 780 a([) p Fw(L;) p 1220 780 a(b) p Fx(]) p Fw(F) p Fl 1344 746 a(0) 1332 801 y(\001) p Ff 1387 784 a(0) p Fw 1413 780 a(hP) p Fx 1526 780 a(+) p Fw(h:c:) p Fx 1743 780 a(=) p Fw 1831 780 a(P) p 1896 780 a(h) p Fk 1958 663 a(\022) 2018 678 y(q) p 2101 678 293 4 v Fx 2120 759 a(\026) p Fw 2101 780 a(F) p Fl 2154 792 a(\001) p Ff 2209 776 a(0) p Fx 2236 780 a(\() p Fw(L) p Fl 2325 792 a(0) p Fx 2362 780 a(\)) p Ft 2413 780 a(\000) p Fk 2496 678 a(q) p 2579 678 256 4 v Fx 2597 759 a(\026) p Fw 2579 780 a(F) p Fl 2632 792 a(\001) p Ff 2687 776 a(0) p Fx 2713 780 a(\() p Fw(L) p Fx(\)) p Fk 2834 663 a(\023) 2909 679 y(q) p 2992 679 135 4 v Fx 3011 759 a(\026) p Fw 2992 780 a(F) p Fl 3057 752 a(0) 3045 805 y(\001) p Ff 3100 788 a(0) p Fx 3127 780 a([) p Fw(L;) p 3244 780 a(b) p Fx(]) p Fw(F) p Fl 3368 746 a(0) 3356 801 y(\001) p Ff 3411 784 a(0) p Fw 3437 780 a(hP) p Fx 3550 780 a(+) p Fw(h:c:) p Ft 1354 1016 a(\025) p 1442 1016 a(\000) p Fw(c) p Ff 1543 957 a(00) p Fl 1543 1037 a(2) p Fk 1601 949 a(\000) p Fw 1639 1016 a(\022) r(\025) p Fl 1728 982 a(2) p Fx 1784 1016 a(+) p Fw 1867 1016 a(\022) r(\025) p Fl 1956 982 a(3) p Fw 1994 1016 a(\017) p Fo 2028 982 a(\000) p Fl(2) p Fk 2117 949 a(\001) p Fw 2169 1016 a(h) p Fl 2217 982 a(2) p Fw 2254 1016 a(P) p Fx 2342 1016 a(=) p Ft 2430 1016 a(\000) p Fw(c) p Ff 2531 957 a(00) p Fl 2531 1037 a(2) p Fw 2585 960 a(\022) r(\025) p Fl 2674 930 a(2) p 2585 997 127 4 v Fw 2632 1073 a(\017) p Fk 2736 949 a(\000) p Fw 2774 1016 a(\017) p Fx 2826 1016 a(+) p Fw 2909 1016 a(\025\017) p Fo 2991 982 a(\000) p Fl(1) p Fk 3080 949 a(\001) p Fw 3132 1016 a(h) p Fl 3180 982 a(2) p Fw 3217 1016 a(P) r(:) p Fx 515 1222 a(Hence) p 761 1222 a(w) n(e) p 884 1222 a(ha) n(v) n(e) 515 1437 y(\(32\)) p 685 1437 a(=) p 773 1437 a(\(34\)) p 939 1437 a(+) p 1022 1437 a(\(35\)) p 1188 1437 a(+) p 1271 1437 a(\(36\)) p Ft 1442 1437 a(\025) p 1529 1437 a(\000) p Fx(2) p Fw(c) p Fo 1672 1403 a(0) p Fl 1672 1458 a(1) p Fw 1709 1437 a(\025) p Fl 1757 1403 a(2) p Fw 1794 1437 a(h) p Fl 1842 1403 a(2) p Fw 1879 1437 a(P) p Fk 1958 1345 a(\020) p Ft 2008 1437 a(k) p Fx 2056 1437 a(~) p Fw 2049 1437 a(pg) p Fl 2131 1449 a(1) p Ft 2168 1437 a(k) p Fl 2209 1395 a(2) p Fx 2265 1437 a(+) p Ft 2348 1437 a(k) p Fx 2397 1437 a(~) p Fw 2390 1437 a(p) o(g) p Fl 2471 1449 a(2) p Ft 2508 1437 a(k) p Fl 2550 1395 a(2) p Fk 2587 1345 a(\021) p Ft 2655 1437 a(\000) p Fw 2738 1437 a(c) p Ff 2774 1378 a(00) p Fl 2774 1458 a(2) p Fw 2829 1381 a(\022) r(\025) p Fl 2918 1351 a(2) p 2829 1418 V Fw 2875 1494 a(\017) p Fk 2980 1370 a(\000) p Fw 3018 1437 a(\017) p Fx 3070 1437 a(+) p Fw 3153 1437 a(\025\017) p Fo 3235 1403 a(\000) p Fl(1) p Fk 3324 1370 a(\001) p Fw 3376 1437 a(h) p Fl 3424 1403 a(2) p Fw 3461 1437 a(P) p Ft 2300 1651 a(\025) p 2388 1651 a(\000) p Fw(c) p Fl 2489 1663 a(2) p Fw 2536 1595 a(\022) r(\025) p Fl 2625 1565 a(2) p 2536 1632 V Fw 2582 1708 a(\017) p Fk 2686 1584 a(\000) p Fw 2724 1651 a(\022) p Fo 2765 1617 a(\000) p Fl(1) p Fw 2855 1651 a(\017) p Fx 2907 1651 a(+) p Fw 2990 1651 a(\017) p Fx 3042 1651 a(+) p Fw 3125 1651 a(\025\017) p Fo 3207 1617 a(\000) p Fl(1) p Fk 3296 1584 a(\001) p Fw 3348 1651 a(h) p Fl 3396 1617 a(2) p Fw 3433 1651 a(P) r(;) p Fx 515 1857 a(where) p Fw 755 1857 a(c) p Fl 791 1869 a(2) p Fx 856 1857 a(dep) r(ends) p 1177 1857 a(only) p 1359 1857 a(on) p Fw 1474 1857 a(I) p Fl 1510 1869 a(0) p Fx 1576 1857 a(and) p Fw 1737 1857 a(F) p Fl 1790 1869 a(\001) p Ff 1845 1853 a(0) p Fx 1872 1857 a(.) p Fd(\003) p Fx 639 1957 a(Finally) p 887 1957 a(,) p 938 1957 a(the) p 1081 1957 a(third) p 1289 1957 a(part) p 1469 1957 a(\(33\)) p 1644 1957 a(can) p 1796 1957 a(b) r(e) p 1909 1957 a(estimated) p 2288 1957 a(as) p 2390 1957 a(follo) n(ws.) p Fm 515 2123 a(Lemma) p 853 2123 a(7.12) p Fw 877 2338 a(P) p 942 2338 a(h) p Fx 1008 2317 a(\026) p Fw 990 2338 a(F) p Fl 1055 2304 a(0) 1043 2358 y(\001) p Ff 1098 2342 a(0) p Fw 1124 2338 a(B) p Fx 1210 2317 a(\026) p Fw 1191 2338 a(F) p Fl 1256 2304 a(0) 1244 2358 y(\001) p Ff 1299 2342 a(0) p Fw 1326 2338 a(hP) p Fx 1457 2338 a(+) 1571 2282 y(1) p 1550 2319 84 4 v 1550 2395 a(20) p Fw 1643 2338 a(P) p 1708 2338 a(hS) p Fl 1807 2350 a(1) p Fw 1844 2338 a(hP) p Ft 1979 2338 a(\025) p 2067 2338 a(\000) p Fw(c) p Fl 2168 2350 a(3) p Fw 2215 2282 a(\022) r(\025) p Fl 2304 2252 a(2) p 2215 2319 127 4 v Fw 2261 2395 a(\017) p Fx 2352 2338 a(\() p Fw(\022) p Fo 2425 2304 a(\000) p Fl(1) p Fw 2514 2338 a(\017) p Fx 2567 2338 a(+) p Fw 2650 2338 a(\025\017) p Fo 2732 2304 a(\000) p Fl(1) p Fx 2821 2338 a(\)) p Fw(h) p Fl 2901 2304 a(2) p Fw 2938 2338 a(P) r(:) p Fy 515 2544 a(wher) l(e) p Fw 749 2544 a(c) p Fl 785 2556 a(3) p Fy 852 2544 a(dep) l(ends) p 1162 2544 a(only) p 1342 2544 a(on) p Fw 1461 2544 a(I) p Fl 1497 2556 a(0) p Fy 1564 2544 a(and) p Fw 1726 2544 a(F) p Fl 1779 2556 a(\001) p Ff 1834 2540 a(0) p Fy 1860 2544 a(.) 515 2710 y(Pr) l(o) l(of) p Fx 515 2809 a(The) p 685 2809 a(same) p 893 2809 a(as) p 995 2809 a(that) p 1175 2809 a(of) p 1270 2809 a(Prop) r(osition) p 1717 2809 a(5.2) p 1850 2809 a(in) p 1947 2809 a([10) o(].) p Fd 2136 2809 a(\003) p Fx 515 2909 a(No) n(w) p 706 2909 a(let) p 827 2909 a(us) p 936 2909 a(complete) p 1288 2909 a(the) p 1433 2909 a(pro) r(of) p 1651 2909 a(of) p 1748 2909 a(Theorem) p 2100 2909 a(7.1.) p 2271 2909 a(By) p 2403 2909 a(Lemma) p 2701 2909 a(7.10) p 2878 2909 a(to) p 2982 2909 a(7.12,) p 3182 2909 a(if) p Fw 3260 2909 a(\025;) p 3345 2909 a(\017) p Fx 515 3009 a(and) p Fw 676 3009 a(\022) p Fx 745 3009 a(satisfy) p 1004 3009 a(Assumption) p 1464 3009 a(7.1,) p 1621 3009 a(w) n(e) p 1743 3009 a(obtain) p Fw 515 3240 a(P) p 580 3240 a(h) p Fx 642 3240 a([) p Fw 664 3240 a(L;) p 758 3240 a(A) p Fx(]) p Fw 857 3240 a(hP) p Fx 993 3240 a(=) p 1080 3240 a(\(31\)) p 1246 3240 a(+) p 1329 3240 a(\(32\)) p 1495 3240 a(+) p 1578 3240 a(\(33\)) p Ft 1749 3240 a(\025) p Fw 1847 3184 a(\022) r(\025) p Fl 1936 3154 a(2) p 1847 3221 V Fw 1893 3297 a(\017) 1984 3240 y(\015) p Fn 2027 3252 a(e) p Fw 2062 3240 a(P) p 2127 3240 a(h) p Fk 2189 3148 a(\020) p Fx 2238 3240 a(1) p Ft 2298 3240 a(\000) p Fx 2381 3240 a(7) 2441 3219 y(~) p Fw 2423 3240 a(P) p Fn 2476 3252 a(e) p Ft 2530 3240 a(\000) p Fw 2613 3240 a(C) p Fl 2672 3252 a(1) p Fx 2710 3240 a(\() p Fw(\025) p Fx 2809 3240 a(+) p Fw 2892 3240 a(\017\022) p Fo 2967 3206 a(\000) p Fl(1) p Fx 3074 3240 a(+) p Fw 3158 3240 a(\017) p Fx 3210 3240 a(+) p Fw 3293 3240 a(\025\017) p Fo 3375 3206 a(\000) p Fl(1) p Fx 3464 3240 a(\)) p Fk 3496 3148 a(\021) p Fw 3560 3240 a(hP) r(:) p Fx 515 3446 a(Let) p Fw 663 3446 a(\017) p Fx(,) p Fw 748 3446 a(\022) p Fx 817 3446 a(b) r(e) p Fw 1570 3628 a(\017) p Fx 1627 3628 a(=) p Fw 1714 3628 a(\025) p Fg 1787 3572 a(44) p 1773 3581 85 3 v 1773 3614 a(100) p Fw 1872 3628 a(;) p 1992 3628 a(\022) p Fx 2056 3628 a(=) p Fw 2143 3628 a(\025) p Fg 2216 3572 a(26) p 2202 3581 V 2202 3614 a(100) p Fw 2301 3628 a(:) p Fx 515 3811 a(Then) p Fw 515 3994 a(\017\022) p Fo 590 3959 a(\000) p Fl(1) p Fx 702 3994 a(=) p Fw 790 3994 a(\025) p Fg 862 3937 a(18) p 848 3946 V 848 3979 a(100) p Fw 947 3994 a(;) p 1067 3994 a(\025\017) p Fo 1149 3959 a(\000) p Fl(1) p Fx 1261 3994 a(=) p Fw 1349 3994 a(\025) p Fg 1421 3937 a(56) p 1407 3946 V 1407 3979 a(100) p Fw 1506 3994 a(;) p 1626 3994 a(\022) r(\025) p Fl 1715 3959 a(2) p Fw 1753 3994 a(\017) p Fo 1787 3959 a(\000) p Fl(1) p Fx 1899 3994 a(=) p Fw 1986 3994 a(\025) p Fg 2044 3937 a(182) p 2045 3946 V 2045 3979 a(100) p Fw 2144 3994 a(;) p 2264 3994 a(\022) r(\025) p Fl 2353 3959 a(2) p Fw 2390 3994 a(\017) p Fo 2424 3959 a(\000) p Fl(2) p Fx 2536 3994 a(=) p Fw 2624 3994 a(\025) p Fg 2682 3937 a(138) p 2682 3946 V 2682 3979 a(100) p Fw 2781 3994 a(;) p 2901 3994 a(\022) r(\025) p Fl 2990 3959 a(2) p Fw 3028 3994 a(\017) p Fo 3062 3959 a(\000) p Fl(3) p Fx 3174 3994 a(=) p Fw 3262 3994 a(\025) p Fg 3334 3937 a(94) p 3320 3946 V 3320 3979 a(100) p Fx 515 4176 a(and) p 681 4176 a(if) p Fw 762 4176 a(\025) p Fx 843 4176 a(is) p 931 4176 a(su\016cien) n(tly) p 1352 4176 a(small,) p 1598 4176 a(Assumption) p 2063 4176 a(7.1) p 2201 4176 a(is) p 2290 4176 a(satis\014ed.) p 2656 4176 a(F) p 2703 4176 a(urther) p 2962 4176 a(more,) p 3199 4176 a(for) p Fw 3331 4176 a(\025) p Fx 515 4276 a(small) p 732 4276 a(enough,) p Fw 1408 4493 a(C) p Fl 1467 4505 a(1) p Fx 1504 4493 a(\() p Fw(\025) p Fx 1603 4493 a(+) p Fw 1687 4493 a(\017\022) p Fo 1762 4459 a(\000) p Fl(1) p Fx 1869 4493 a(+) p Fw 1952 4493 a(\017) p Fx 2004 4493 a(+) p Fw 2087 4493 a(\025\017) p Fo 2169 4459 a(\000) p Fl(1) p Fx 2259 4493 a(\)) p Fw 2314 4493 a(<) p Fx 2411 4437 a(1) p 2411 4474 42 4 v 2411 4550 a(2) p Fw 2463 4493 a(:) p Fx 515 4699 a(is) p 598 4699 a(satis\014ed.) p 950 4699 a(Hence) p 1196 4699 a(w) n(e) p 1319 4699 a(\014nally) p 1570 4699 a(obtain) p Fw 610 4930 a(P) p 675 4930 a(h) p Fx 737 4930 a([) p Fw(L;) p 854 4930 a(A) p Fx(]) p Fw 952 4930 a(hP) p Ft 1088 4930 a(\025) p Fw 1186 4874 a(\022) r(\025) p Fl 1275 4844 a(2) p 1186 4911 127 4 v Fw 1232 4987 a(\017) 1333 4874 y(\015) p Fn 1376 4886 a(e) p 1333 4911 79 4 v Fx 1351 4987 a(2) p Fw 1421 4930 a(P) p 1486 4930 a(h) p Fk 1548 4838 a(\020) p Fx 1597 4930 a(1) p Ft 1657 4930 a(\000) p Fx 1740 4930 a(14) 1842 4909 y(~) p Fw 1824 4930 a(P) p Fn 1877 4942 a(e) p Fk 1912 4838 a(\021) p Fw 1976 4930 a(hP) p Fx 2111 4930 a(=) p Fw 2199 4930 a(\025) p Fg 2257 4874 a(182) p 2257 4883 85 3 v 2257 4916 a(100) p Fw 2366 4874 a(\015) p Fn 2409 4886 a(e) p 2366 4911 79 4 v Fx 2385 4987 a(2) p Fw 2455 4930 a(P) p 2520 4930 a(h) p Fk 2582 4838 a(\020) p Fx 2631 4930 a(1) p Ft 2691 4930 a(\000) p Fx 2774 4930 a(14) 2876 4909 y(~) p Fw 2858 4930 a(P) p Fn 2911 4942 a(e) p Fk 2946 4838 a(\021) p Fw 3009 4930 a(hP) r(:) p Fd 3219 4930 a(\003) p Fx 1905 5255 a(30) p eop %%Page: 31 31 31 30 bop Fu 515 523 a(8) p 716 523 a(Virial) p 1095 523 a(Theorem) p Fx 515 705 a(In) p 623 705 a(this) p 789 705 a(section,) p 1095 705 a(w) n(e) p 1222 705 a(apply) p 1455 705 a(the) p 1602 705 a(new) p 1778 705 a(metho) r(d) p 2084 705 a(in) n(tro) r(duced) p 2504 705 a(b) n(y) p 2624 705 a(M.Merkli) p 2991 705 a([10) o(],) p 3176 705 a(to) p 3282 705 a(in-) 515 805 y(v) n(estigate) p 860 805 a(the) p 1003 805 a(eigen) n(v) n(ectors) p 1466 805 a(of) p Fw 1560 805 a(L) p Fx(.) p 1676 805 a(Our) p 1846 805 a(situation) p 2192 805 a(here) p 2372 805 a(is) p 2454 805 a(di\013eren) n(t) p 2781 805 a(from) p 2977 805 a(that) p 3156 805 a(of) p 3250 805 a([10) o(]) 515 904 y(in) p 615 904 a(the) p 761 904 a(follo) n(wing) p 1114 904 a(sense:) p 1365 904 a(in) p 1465 904 a([10) o(],) p 1649 904 a(the) p 1795 904 a(double) p 2065 904 a(comm) n(utator) p 2536 904 a(of) p Fw 2634 904 a(L) p Fl 2691 916 a(0) p Fx 2758 904 a(with) p Fw 2950 904 a(A) p Fl 3012 916 a(0) p Fx 3081 904 a(v) p 3120 904 a(anishes) 515 1004 y(i.e.,) p 678 1004 a([[) p Fw(L) p Fl 781 1016 a(0) p Fw 817 1004 a(;) p 854 1004 a(A) p Fl 916 1016 a(0) p Fx 954 1004 a(]) p Fw(;) p 1014 1004 a(A) p Fl 1076 1016 a(0) p Fx 1113 1004 a(]) p 1168 1004 a(=) p 1263 1004 a(0,) p 1362 1004 a(while) p 1583 1004 a(in) p 1685 1004 a(our) p 1838 1004 a(case) p 2018 1004 a(it) p 2106 1004 a(do) r(es) p 2298 1004 a(not) p 2451 1004 a([[) p Fw(L) p Fl 2554 1016 a(0) p Fw 2591 1004 a(;) p 2628 1004 a(A) p Fl 2690 1016 a(0) p Fx 2727 1004 a(]) p Fw(;) p 2787 1004 a(A) p Fl 2849 1016 a(0) p Fx 2886 1004 a(]) p 2941 1004 a(=) p Fw 3036 1004 a(S) p Fl 3087 1016 a(2) p Fx 3125 1004 a(.) p 3199 1004 a(That) 515 1103 y(is) p 605 1103 a(the) p 755 1103 a(reason) p 1020 1103 a(wh) n(y) p 1203 1103 a(w) n(e) p 1332 1103 a(still) p 1500 1103 a(need) p 1701 1103 a(the) p 1850 1103 a(scaling) p 2130 1103 a(factor) p Fw 2374 1103 a(n) p Fx 2459 1103 a(\(see) p 2632 1103 a(b) r(elo) n(w\)) p 2906 1103 a(although) p 3259 1103 a(our) 515 1203 y(in) n(teraction) p 932 1203 a(is) p 1016 1203 a(b) r(ounded.) 639 1303 y(Assume) p 959 1303 a(that) p Fw 1152 1303 a( ) p Fx 1249 1303 a(is) p 1345 1303 a(a) p 1427 1303 a(normalized) p 1864 1303 a(eigen) n(v) n(ector) p 2308 1303 a(of) p Fw 2415 1303 a(L) p Fx 2511 1303 a(with) p 2713 1303 a(eigen) n(v) p 2935 1303 a(alue) p Fw 3122 1303 a(e) p Fx(.) p 3258 1303 a(Let) p Fw 515 1402 a(g) p Ft 590 1402 a(2) p Fw 677 1402 a(C) p Fo 742 1372 a(1) p Fl 736 1423 a(0) p Fx 813 1402 a(\() p Fi(R) p Fx(\)) p 971 1402 a(b) r(e) p 1089 1402 a(a) p 1164 1402 a(smo) r(oth) p 1463 1402 a(function) p 1794 1402 a(suc) n(h) p 1987 1402 a(that) p 2172 1402 a(supp) p Fw(g) p Ft 2418 1402 a(\032) p Fx 2515 1402 a([) p Ft(\000) p Fx(2) p Fw(;) p Fx 2682 1402 a(2],) p 2803 1402 a(0) p Ft 2877 1402 a(\024) p Fw 2974 1402 a(g) p Ft 3049 1402 a(\024) p Fx 3146 1402 a(1,) p 3245 1402 a(and) p Fw 515 1502 a(g) p Fx 596 1502 a(=) p 700 1502 a(1) p 779 1502 a(on) p 904 1502 a([) p Ft(\000) p Fx(1) p Fw(;) p Fx 1071 1502 a(1].) p 1223 1502 a(Let) p Fw 1381 1502 a(f) p Ft 1470 1502 a(2) p Fw 1565 1502 a(C) p Fo 1630 1472 a(1) p Fn 1624 1525 a(b) p Fx 1700 1502 a(\() p Fi(R) p Fx(\)) p 1862 1502 a(b) r(e) p 1985 1502 a(a) p 2063 1502 a(smo) r(oth) p 2366 1502 a(function) p 2701 1502 a(whic) n(h) p 2948 1502 a(satis\014es) p 3264 1502 a(the) 515 1602 y(follo) n(wing) p 865 1602 a(conditions;) p Fw 515 1784 a(f) p Fo 565 1750 a(0) p Ft 611 1784 a(\025) p Fx 698 1784 a(0) p Fw(;) p 860 1784 a(f) p Fo 910 1750 a(0) p Fx 933 1784 a(\(0\)) p 1062 1784 a(=) p 1149 1784 a(1) p Fw(;) p Ft 1311 1784 a(k) p Fw(f) p Fl 1403 1750 a(\() p Fn(k) p Fl 1465 1750 a(\)) p Ft 1495 1784 a(k) p Fn 1537 1796 a(k) p Fo 1573 1796 a(\000) p Fl(4) p Fw 1685 1784 a(<) p Ft 1773 1784 a(1) p Fw(;) p 1976 1784 a(k) p Fx 2045 1784 a(=) p 2132 1784 a(0) p Fw(;) p Ft 2211 1784 a(\001) p 2248 1784 a(\001) p 2285 1784 a(\001) p Fw 2335 1784 a(;) p Fx 2372 1784 a(5) p Fw(;) p Ft 2533 1784 a(k) p Fw(f) p Fl 2625 1750 a(\() p Fn(l) p Fl(\)) p Ft 2702 1784 a(k) p Fn 2744 1796 a(l) p Fo(\000) p Fl(3) p Fw 2877 1784 a(<) p Ft 2965 1784 a(1) p Fw(;) p 3168 1784 a(l) p Fx 3217 1784 a(=) p 3305 1784 a(0) p Fw(;) p Ft 3384 1784 a(\001) p 3421 1784 a(\001) p 3458 1784 a(\001) p Fw 3508 1784 a(;) p Fx 3545 1784 a(4) p Fw(:) p Fx 515 1967 a(where) p 755 1967 a(w) n(e) p 877 1967 a(used) p 1067 1967 a(the) p 1210 1967 a(notation) p Ft 1153 2171 a(k) p Fw(f) p Ft 1245 2171 a(k) p Fn 1287 2183 a(m) p Ft 1372 2171 a(\021) p Fk 1460 2058 a(Z) p Fw 1557 2171 a(dx) p Ft(h) p Fw(x) p Ft(i) p Fn 1758 2137 a(m) p Ft 1822 2171 a(j) p Fw(f) p Fx 1895 2171 a(\() p Fw(x) p Fx(\)) p Ft(j) p Fw(;) p Ft 2150 2171 a(h) p Fw(x) p Ft(i) p Fx 2285 2171 a(=) p 2373 2171 a(\(1) p 2465 2171 a(+) p Fw 2548 2171 a(x) p Fl 2595 2137 a(2) p Fx 2633 2171 a(\)) p Fg 2675 2115 a(1) p 2675 2124 29 3 v 2675 2157 a(2) p Fw 2717 2171 a(:) p Fx 515 2403 a(F) p 562 2403 a(urthermore,) p 1019 2403 a(w) n(e) p 1142 2403 a(assume) p 1428 2403 a(that) p Fw 1608 2403 a(h) p Fx 1679 2403 a(=) p Ft 1767 2341 a(p) p 1836 2341 73 4 v Fw 1836 2403 a(f) p Fo 1886 2379 a(0) p Fx 1937 2403 a(satis\014es) p 2242 2403 a(that) p Fw 852 2586 a(h) p Fl 900 2552 a(\() p Fn(k) p Fl 962 2552 a(\)) p Ft 1016 2586 a(2) p Fw 1094 2586 a(C) p Fo 1159 2552 a(1) p Fn 1153 2606 a(b) p Fx 1230 2586 a(\() p Fi(R) p Fx(\)) p Fw(;) p 1474 2586 a(k) p Fx 1543 2586 a(=) p 1631 2586 a(0) p Fw(;) p Fx 1710 2586 a(1) p Fw(;) p Ft 1871 2586 a(k) p Fw(h) p Fl 1961 2552 a(\() p Fn(l) p Fl(\)) p Ft 2037 2586 a(k) p Fn 2079 2598 a(l) p Fo(\000) p Fl(2) p Fw 2213 2586 a(<) p Ft 2300 2586 a(1) p Fw(;) p 2503 2586 a(l) p Fx 2553 2586 a(=) p 2640 2586 a(0) p Fw(;) p Ft 2719 2586 a(\001) p 2756 2586 a(\001) p 2793 2586 a(\001) p Fw 2843 2586 a(;) p Fx 2880 2586 a(3) p Fw(;) p Ft 1034 2727 a(k) p Fw(h) p Fl 1124 2693 a(\() p Fn(m) p Fl(\)) p Ft 1238 2727 a(k) p Fn 1280 2739 a(m) p Fo(\000) p Fl(3) p Fw 1451 2727 a(<) p Ft 1539 2727 a(1) p Fw(;) p 1741 2727 a(m) p Fx 1837 2727 a(=) p 1925 2727 a(0) p Fw(;) p Ft 2004 2727 a(\001) p 2041 2727 a(\001) p 2078 2727 a(\001) p Fw 2128 2727 a(;) p Fx 2165 2727 a(4) p Fw(;) p Fx 2326 2727 a(sup) p Fn 2327 2797 a(x) p Fo(2) p Fe(R) p Ft 2465 2727 a(j) p Fw(x) p Ft(j) p 2577 2727 a(\001) p 2619 2727 a(j) p Fw(h) p Fx(\() p Fw(x) p Fx(\)) p Ft(j) p Fw 2848 2727 a(<) p Ft 2935 2727 a(1) p Fw(:) p Fx 515 2959 a(No) n(w) p 704 2959 a(let) p Fw 824 2959 a(\013;) p 914 2959 a(\027) p Fx 988 2959 a(b) r(e) p 1101 2959 a(p) r(ositiv) n(e) p 1408 2959 a(parameters.) p 1870 2959 a(W) p 1948 2959 a(e) p 2013 2959 a(de\014ne) p 2253 2959 a(the) p 2396 2959 a(follo) n(wing) p 2747 2959 a(op) r(erators;) p Fw 518 3141 a(f) p Fn 559 3153 a(\013) p Ft 629 3141 a(\021) p Fw 716 3141 a(f) p Fx 766 3141 a(\() p Fw(i\013A) p Fl 942 3153 a(0) p Fx 980 3141 a(\)) p Fw(;) p 1132 3141 a(f) p Fo 1182 3107 a(0) p Fn 1173 3162 a(\013) p Ft 1243 3141 a(\021) p Fw 1331 3141 a(f) p Fo 1381 3107 a(0) p Fx 1404 3141 a(\() p Fw(i\013A) p Fl 1580 3153 a(0) p Fx 1617 3141 a(\)) p Fw(;) p 1770 3141 a(f) p Ff 1820 3082 a(0) o(0) p Fn 1811 3162 a(\013) p Ft 1887 3141 a(\021) p Fw 1975 3141 a(f) p Ff 2025 3082 a(00) p Fx 2069 3141 a(\() p Fw(i\013A) p Fl 2245 3153 a(0) p Fx 2283 3141 a(\)) p Fw(;) p 2435 3141 a(h) p Fn 2483 3153 a(\013) p Ft 2554 3141 a(\021) p Fk 2641 3068 a(p) p 2724 3068 88 4 v Fw 2724 3141 a(f) p Fo 2774 3117 a(0) p Fn 2765 3162 a(\013) p Fw 2812 3141 a(;) p 2932 3141 a(g) p Fn 2972 3153 a(\027) p Ft 3036 3141 a(\021) p Fw 3124 3141 a(g) p Fx 3167 3141 a(\() p Fw(\027) p 3245 3141 a(N) p Fx 3321 3141 a(\)) p Fw(:) p Fx 515 3324 a(By) p 645 3324 a(the) p 788 3324 a(condition) p 1152 3324 a(that) p 1332 3324 a(sup) p Fn 1457 3344 a(x) p Fo(2) p Fe(R) p Ft 1600 3324 a(j) p Fw(x) p Ft(j) p 1712 3324 a(\001) p 1753 3324 a(j) p Fw(h) p Fx(\() p Fw(x) p Fx(\)) p Ft(j) p Fw 1982 3324 a(<) p Ft 2070 3324 a(1) p Fx(,) p 2204 3324 a(w) n(e) p 2326 3324 a(ha) n(v) n(e) p Fw 2517 3324 a(h) p Fn 2565 3336 a(\013) p Ft 2612 3324 a(H) p 2706 3324 a(\032) p Fw 2794 3324 a(D) p Fx 2865 3324 a(\() p Fw(A) p Fl 2959 3336 a(0) p Fx 2997 3324 a(\).) 639 3424 y(F) p 686 3424 a(or) p 792 3424 a(simplicit) n(y) p 1136 3424 a(,) p 1191 3424 a(b) r(elo) n(w,) p 1454 3424 a(w) n(e) p 1580 3424 a(use) p 1727 3424 a(the) p 1874 3424 a(notation) p Fw 2210 3424 a(K) p Ft 2316 3424 a(\021) p Fx 2410 3424 a([) p Fw(L;) p 2527 3424 a(A) p Fl 2589 3436 a(0) p Fx 2625 3424 a(]) p 2678 3424 a(=) p Fw 2772 3424 a(S) p Fl 2823 3436 a(1) p Fx 2881 3424 a(+) p Fw 2966 3424 a(I) p Fn 3009 3393 a(\025) p Fl 3002 3444 a(1) p Fx 3053 3424 a(,) p Ft 3108 3424 a(h) p Fw(A) p Ft(i) p Fn 3234 3436 a( ) p Ft 3314 3424 a(\021) 515 3523 y(h) p Fw( ) s(;) p 641 3523 a(A ) p Ft 760 3523 a(i) p Fx(,) p 844 3523 a(and) p Fw 1006 3523 a( ) p Fn 1060 3535 a(\013;\027) p Ft 1189 3523 a(\021) p Fw 1277 3523 a(h) p Fn 1325 3535 a(\013) p Fw 1373 3523 a(g) p Fn 1413 3535 a(\027) p Fw 1454 3523 a( ) p Fx 1511 3523 a(.) p 1573 3523 a(W) p 1651 3523 a(e) p 1716 3523 a(also) p 1883 3523 a(use) p 2028 3523 a(the) p 2171 3523 a(notation) p Fw 2504 3523 a(I) p Fn 2547 3493 a(\025) 2540 3545 y(j) p Ft 2615 3523 a(\021) p Fw 2703 3523 a(\025I) p Fn 2787 3535 a(j) p Fx 2823 3523 a(.) p 2885 3523 a(W) p 2963 3523 a(e) p 3029 3523 a(\014rst) p 3201 3523 a(sho) n(w) 515 3623 y(the) p 658 3623 a(follo) n(wing) p 1008 3623 a(prop) r(osition:) p Fm 515 3789 a(Prop) s(osition) p 1031 3789 a(8.1) p Fy 1194 3789 a(L) l(et) p Fw 1338 3789 a(\027) p Fx 1407 3789 a(=) p Fw 1495 3789 a(\013) p Fg 1558 3736 a(3) p 1558 3745 29 3 v 1558 3779 a(2) p Fw 1600 3789 a(:) p Fy 1653 3789 a(If) p Fw 1740 3789 a( ) p Fy 1827 3789 a(is) p 1916 3789 a(an) p 2035 3789 a(eigenve) l(ctor) p 2467 3789 a(of) p Fw 2564 3789 a(L) p Fy(,) p 2676 3789 a(then) p Fx 1094 3971 a(lim) p Fn 1080 4023 a(\013) p Fo(!) p Fl(0) p Ft 1236 3971 a(h) p Fw 1269 3971 a(K) p Ft 1346 3971 a(i) p Fn 1377 3996 a( ) p Fj 1421 4004 a(\013\027) p Fx 1523 3971 a(=) p 1624 3971 a(lim) p Fn 1611 4023 a(\013) p Fo(!) p Fl(0) p Ft 1767 3971 a(h) p Fw( ) p Ft 1870 3971 a(j) p Fw(g) p Fn 1933 3983 a(\027) p Fw 1974 3971 a(h) p Fn 2022 3983 a(\013) p Fx 2083 3971 a([) p Fw(L;) p 2200 3971 a(A) p Fl 2262 3983 a(0) p Fx 2299 3971 a(]) p Fw 2336 3971 a(h) p Fn 2384 3983 a(\013) p Fw 2431 3971 a(g) p Fn 2471 3983 a(\027) p Ft 2512 3971 a(j) p Fw 2549 3971 a( ) p Ft 2606 3971 a(i) p Fx 2661 3971 a(=) p 2749 3971 a(0) p Fw(:) p Fy 515 4180 a(Pr) l(o) l(of) p Fx 515 4280 a(Let) p Fw 675 4280 a( ) p Fx 771 4280 a(b) r(e) p 895 4280 a(a) p 975 4280 a(unit) p 1161 4280 a(eigen) n(v) n(ector) p 1604 4280 a(of) p Fw 1710 4280 a(L) p Fx 1805 4280 a(with) p 2005 4280 a(eigen) n(v) p 2227 4280 a(alue) p Fw 2413 4280 a(e) p Fx(.) p 2545 4280 a(W) p 2623 4280 a(e) p 2699 4280 a(ha) n(v) n(e) p 2902 4280 a(the) p 3056 4280 a(follo) n(wing) 515 4379 y(lemma.) p Fm 515 4545 a(Lemma) p 853 4545 a(8.2) p Ft 1376 4728 a(jh) p Fw( ) p Ft 1502 4728 a(j) p Fw(g) p Fn 1565 4740 a(\027) p Fw 1606 4728 a(f) p Fn 1647 4740 a(\013) p Fx 1694 4728 a(\() p Fw(L) p Ft 1801 4728 a(\000) p Fw 1885 4728 a(e) p Fx(\)) p Fw(g) p Fn 1996 4740 a(\027) p Ft 2036 4728 a(j) p Fw 2073 4728 a( ) p Ft 2130 4728 a(i) q(j) p 2209 4728 a(\024) p Fw 2296 4728 a(C) p Fl 2355 4740 a(1) p Ft 2411 4728 a(\001) p Fw 2453 4728 a(\027) q(:) p Fy 515 4911 a(wher) l(e) p Fw 749 4911 a(C) p Fl 808 4923 a(1) p Fy 875 4911 a(is) p 965 4911 a(a) p 1037 4911 a(p) l(ositive) p 1336 4911 a(c) l(onstant) p 1667 4911 a(which) p 1900 4911 a(is) p 1989 4911 a(indep) l(endent) p 2450 4911 a(of) p Fw 2547 4911 a(\027) p Fy 2623 4911 a(and) p Fw 2785 4911 a(\013) p Fy(.) p Fx 1905 5255 a(31) p eop %%Page: 32 32 32 31 bop Fy 515 523 a(Pr) l(o) l(of) p Fx 515 623 a(As) p Fw 637 623 a(L) p Fl 694 635 a(0) p Fx 759 623 a(and) p Fw 920 623 a(N) p Fx 1024 623 a(strongly) p 1345 623 a(comm) n(ute) p 1702 623 a(and) p Fw 1864 623 a( ) p Ft 1944 623 a(2) p Fw 2022 623 a(D) p Fx 2093 623 a(\() p Fw(L) p Fx(\)) p 2238 623 a(=) p Fw 2325 623 a(D) p Fx 2396 623 a(\() p Fw(L) p Fl 2485 635 a(0) p Fx 2522 623 a(\),) p 2605 623 a(w) n(e) p 2728 623 a(ha) n(v) n(e) 1444 793 y(\() p Fw(L) p Fl 1533 805 a(0) p Ft 1588 793 a(\000) p Fw 1671 793 a(e) p Fx(\)) p Fw(g) p Fn 1782 805 a(\027) p Fw 1823 793 a( ) p Fx 1903 793 a(=) p Fw 1991 793 a(g) p Fn 2031 805 a(\027) p Fx 2072 793 a(\() p Fw(L) p Fl 2161 805 a(0) p Ft 2216 793 a(\000) p Fw 2299 793 a(e) p Fx(\)) p Fw( ) s(:) p Fx 515 964 a(As) p Fw 637 964 a(g) p Ft 703 964 a(2) p Fw 781 964 a(C) p Fo 846 934 a(1) p Fl 840 985 a(0) p Fx 917 964 a(\() p Fi(R) p Fx(\),) p 1092 964 a(using) p 1310 964 a(op) r(erator) p 1644 964 a(calculous,) p 2023 964 a(w) n(e) p 2145 964 a(obtain) p 2404 964 a(the) p 2547 964 a(b) r(ound) p Ft 1614 1135 a(k) p Fx([) p Fw(I) p Fn 1722 1101 a(\025) p Fl 1715 1155 a(0) p Fw 1765 1135 a(;) p 1802 1135 a(g) p Fn 1842 1147 a(\027) p Fx 1883 1135 a(]) p Ft(k) p Fw 1971 1135 a(<) p 2058 1135 a(C) p Fl 2117 1147 a(1) p Ft 2173 1135 a(\001) p Fw 2215 1135 a(\027) q(;) p Fx 515 1306 a(where) p Fw 755 1306 a(C) p Fl 814 1318 a(1) p Fx 879 1306 a(is) p 963 1306 a(a) p 1032 1306 a(constan) n(t) p 1367 1306 a(whic) n(h) p 1604 1306 a(is) p 1688 1306 a(indep) r(enden) n(t) p 2158 1306 a(of) p Fw 2253 1306 a(\027) p Fx 2327 1306 a(and) p Fw 2488 1306 a(\013) p Fx(.) p 2601 1306 a(Hence,) p 2871 1306 a(w) n(e) p 2993 1306 a(ha) n(v) n(e) p Ft 872 1476 a(jh) p Fw( ) p Ft 998 1476 a(j) p Fw(g) p Fn 1061 1488 a(\027) p Fw 1102 1476 a(f) p Fn 1143 1488 a(\013) p Fx 1190 1476 a(\() p Fw(L) p Ft 1297 1476 a(\000) p Fw 1380 1476 a(e) p Fx(\)) p Fw(g) p Fn 1491 1488 a(\027) p Ft 1532 1476 a(j) p Fw 1569 1476 a( ) p Ft 1626 1476 a(ij) p Fx 1704 1476 a(=) p Fk 1792 1406 a(\014) 1792 1456 y(\014) 1820 1409 y(\012) p Fw 1859 1476 a( ) p Fk 1930 1406 a(\014) 1930 1456 y(\014) p Fw 1958 1476 a(g) p Fn 1998 1488 a(\027) p Fw 2038 1476 a(f) p Fn 2079 1488 a(\013) p Fx 2126 1476 a([) p Fw(I) p Fn 2192 1442 a(\025) p Fl 2185 1497 a(0) p Fw 2236 1476 a(;) p 2273 1476 a(g) p Fn 2313 1488 a(\027) p Fx 2354 1476 a(]) p Fk 2377 1406 a(\014) 2377 1456 y(\014) p Fw 2418 1476 a( ) p Fk 2475 1409 a(\013) 2515 1406 y(\014) 2515 1456 y(\014) p Ft 2565 1476 a(\024) p Fw 2653 1476 a(C) p Fl 2712 1488 a(1) p Ft 2768 1476 a(\001) p Fw 2810 1476 a(\027) q(:) p Fd 2957 1476 a(\003) p Fx 515 1647 a(On) p 653 1647 a(the) p 796 1647 a(other) p 1013 1647 a(hands,) p 1276 1647 a(w) n(e) p 1399 1647 a(ha) n(v) n(e) p 1590 1647 a(the) p 1733 1647 a(follo) n(wing) p 2084 1647 a(lemma.) p Fm 515 1804 a(Lemma) p 853 1804 a(8.3) p Fk 612 1904 a(\014) 612 1954 y(\014) 612 2003 y(\014) p Ft 639 1999 a(h) p Fw 672 1999 a(K) p Ft 749 1999 a(i) p Fn 780 2024 a( 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a(j) p Fw 2412 2164 a( ) p Ft 2469 2164 a(i) q(j) p Fx 2543 2164 a(+) p Fw 2626 2164 a(\013) p Fl 2679 2130 a(2) p Fw 2717 2164 a(\027) p Fo 2763 2130 a(\000) p Fl(1) p Ft 2871 2164 a(\001) p Fw 2912 2164 a(b) p Fl 2948 2176 a(0) p Fx 3004 2164 a(+) p Fw 3087 2164 a(\013) p Ft 3159 2164 a(\001) p Fx 3197 2142 a(~) p Fw 3200 2164 a(b) p Fl 3236 2176 a(0) p Fw 3273 2164 a(:) p Fy 515 2350 a(wher) l(e) p Fw 749 2350 a(b) p Fl 785 2362 a(0) p Fy 852 2350 a(and) p Fx 1010 2329 a(~) p Fw 1013 2350 a(b) p Fl 1049 2362 a(0) p Fy 1116 2350 a(ar) l(e) p 1257 2350 a(p) l(ositive) p 1556 2350 a(c) l(onstants) p 1921 2350 a(which) p 2154 2350 a(dep) l(end) p 2430 2350 a(only) p 2610 2350 a(on) p Fw 2729 2350 a(f) p Fy 2808 2350 a(and) p Fw 2970 2350 a(I) p Fl 3006 2362 a(0) p Fy 3043 2350 a(.) 515 2507 y(Pr) l(o) l(of) p Fx 515 2607 a(First) p 717 2607 a(w) n(e) p 839 2607 a(consider) p 1163 2607 a(the) p 1306 2607 a(\014eld) p 1486 2607 a(term:) p 1717 2607 a(Im) p Ft 1830 2607 a(h) p Fw( ) p Ft 1933 2607 a(j) p Fw(g) p Fn 1996 2619 a(\027) p Fw 2037 2607 a(f) p Fn 2078 2619 a(\013) p Fx 2125 2607 a(\() p Fw(L) p Fn 2214 2619 a(f) p Ft 2275 2607 a(\000) p Fw 2358 2607 a(e) p Fx(\)) p Fw(g) p Fn 2469 2619 a(\027) p Ft 2510 2607 a(j) p Fw 2547 2607 a( ) p Ft 2604 2607 a(i) p Fx(.) p 2696 2607 a(Observ) n(e) p 3015 2607 a(that) 598 2802 y(2Im) p Ft 752 2802 a(h) p Fw 785 2802 a( ) p Ft 855 2802 a(j) p Fw 879 2802 a(g) p Fn 919 2814 a(\027) p Fw 959 2802 a(f) p Fn 1000 2814 a(\013) p Fx 1047 2802 a(\() p Fw(L) p Fn 1136 2814 a(f) p Ft 1198 2802 a(\000) p Fw 1281 2802 a(e) p Fx(\)) p Fw(g) p Fn 1392 2814 a(\027) p Ft 1433 2802 a(j) p Fw 1469 2802 a( ) p Ft 1526 2802 a(i) p Fx 1582 2802 a(=) p Ft 1669 2802 a(\000) p Fw(i) p Ft 1777 2802 a(h) p Fw( ) p Ft 1866 2802 a(j) p Fw(g) p Fn 1929 2814 a(\027) p Fw 1970 2802 a(f) p Fn 2011 2814 a(\013) p Fx 2058 2802 a(\() p Fw(L) p Fn 2147 2814 a(f) p Ft 2208 2802 a(\000) p Fw 2291 2802 a(e) p Fx(\)) p Fw(g) p Fn 2402 2814 a(\027) p Fw 2443 2802 a( ) p Ft 2500 2802 a(i) p Fx(+) p Fw(i) p Ft 2640 2802 a(h) p Fx(\() p Fw(L) p Fn 2761 2814 a(f) p Ft 2822 2802 a(\000) p Fw 2905 2802 a(e) p Fx(\)) p Fw(g) p Fn 3016 2814 a(\027) p Fw 3057 2802 a( ) p Ft 3114 2802 a(j) p Fw(f) p Fn 3178 2814 a(\013) p Fw 3225 2802 a(g) p Fn 3265 2814 a(\027) p Fw 3306 2802 a( ) p Ft 3363 2802 a(i) p Fx 1448 2927 a(=) p Ft 1536 2927 a(h) p Fw(g) p Fn 1608 2939 a(\027) p Fw 1649 2927 a( ) p Ft 1706 2927 a(j) p Fx 1743 2927 a([) p Fw(L) p Fn 1823 2939 a(f) p Fw 1866 2927 a(;) p 1903 2927 a(if) p Fn 1973 2939 a(\013) p Fx 2019 2927 a(]) p Fw 2056 2927 a(g) p Fn 2096 2939 a(\027) p Fw 2137 2927 a( ) p Ft 2194 2927 a(i) p Fx 2249 2927 a(=) p 2337 2927 a(Re) p Ft 2449 2927 a(h) p Fw( ) p Ft 2538 2927 a(j) p Fw(g) p Fn 2601 2939 a(\027) p Fx 2642 2927 a([) p Fw(L) p Fn 2722 2939 a(f) p Fw 2765 2927 a(;) p 2802 2927 a(if) p Fn 2872 2939 a(\013) p Fx 2918 2927 a(]) p Fw(g) p Fn 2981 2939 a(\027) p Fw 3022 2927 a( ) p Ft 3079 2927 a(i) p Fw 3125 2927 a(:) 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3329 a(3) p Fw 3119 3317 a(g) p Fn 3159 3329 a(\027) p Fw 3200 3317 a( ) s(;) 1145 3520 y(R) p Fl 1208 3532 a(3) p Fx 1268 3520 a(=) p Fk 1356 3407 a(Z) p Fw 1453 3520 a(d) p Fx 1514 3498 a(~) p Fw 1496 3520 a(f) p Fx 1545 3520 a(\() p Fw(z) p Fx 1620 3520 a(\)\() p Fw(z) p Ft 1745 3520 a(\000) p Fw 1828 3520 a(i\013A) p Fl 1972 3532 a(0) p Fx 2010 3520 a(\)) p Fo 2042 3486 a(\000) p Fl(3) p Fw 2131 3520 a(S) p Fl 2182 3532 a(3) p Fx 2219 3520 a(\() p Fw(z) p Ft 2312 3520 a(\000) p Fw 2395 3520 a(i\013A) p Fl 2539 3532 a(0) p Fx 2577 3520 a(\)) p Fo 2609 3486 a(\000) p Fl(1) p Fw 2698 3520 a(:) p Fx 515 3729 a(Belo) n(w,) p 783 3729 a(w) n(e) p 906 3729 a(estimate) p 1238 3729 a(eac) n(h) p 1425 3729 a(term) p 1624 3729 a(of) 932 3899 y(Re) p Ft 1044 3899 a(h) p Fw( ) p Ft 1133 3899 a(j) p Fw(g) p Fn 1196 3911 a(\027) p Fx 1237 3899 a([) p Fw(L) p Fn 1317 3911 a(f) p Fw 1359 3899 a(;) p 1396 3899 a(if) p Fn 1466 3911 a(\013) p Fx 1513 3899 a(]) p Fw(g) p Fn 1576 3911 a(\027) p Fw 1617 3899 a( ) p Ft 1674 3899 a(i) p Fx 1729 3899 a(=) p Ft 1812 3899 a(\000) p Fw 1895 3899 a(\013) p Fx(Re) p Ft 2061 3899 a(h) p Fw( ) p Ft 2150 3899 a(j) p Fw(g) p Fn 2213 3911 a(\027) p Fw 2254 3899 a(f) p Fo 2304 3865 a(0) p Fx 2327 3899 a(\() p Fw(i\013A) p Fl 2503 3911 a(0) p Fx 2541 3899 a(\)) p Fw(S) p Fl 2624 3911 a(1) p Fw 2661 3899 a(g) p Fn 2701 3911 a(\027) p Fw 2742 3899 a( ) p Ft 2799 3899 a(i) p Fx 3231 3899 a(\(39\)) 1812 4086 y(+) p Fw 1905 4030 a(\013) p Fl 1958 4000 a(2) p 1905 4067 V Fx 1930 4143 a(2) 2006 4086 y(Im) p Fk 2119 3994 a(D) p Fw 2170 4086 a( ) p Ft 2227 4086 a(j) p Fw(g) p Fn 2290 4098 a(\027) p Fw 2331 4086 a(f) p Ff 2381 4027 a(00) p Fx 2425 4086 a(\() p Fw(i\013A) p Fl 2601 4098 a(0) p Fx 2639 4086 a(\)) p Fw(S) p Fl 2722 4098 a(2) p Fw 2759 4086 a(g) p Fn 2799 4098 a(\027) p Fw 2840 4086 a( ) p Fk 2897 3994 a(E) p Fx 3231 4086 a(\(40\)) 1812 4248 y(+) p Fw 1895 4248 a(\013) p Fl 1948 4214 a(3) p Fx 1986 4248 a(Re) p Ft 2098 4248 a(h) p Fw( ) p Ft 2187 4248 a(j) p Fw(g) p Fn 2250 4260 a(\027) p Fw 2291 4248 a(R) p Fl 2354 4260 a(3) p Fw 2391 4248 a(g) p Fn 2431 4260 a(\027) p Fw 2472 4248 a( ) p Ft 2529 4248 a(i) p Fw 2576 4248 a(:) p Fx 3231 4248 a(\(41\)) 515 4419 y(First) p 717 4419 a(w) n(e) p 839 4419 a(estimate) p 1172 4419 a(\(39\).) p 1380 4419 a(Note) p 1580 4419 a(that) 568 4589 y(Re) p Ft 680 4589 a(h) p Fw( ) p Ft 769 4589 a(j) p Fw(g) p Fn 832 4601 a(\027) p Fw 873 4589 a(f) p Fo 923 4555 a(0) p Fx 946 4589 a(\() p Fw(i\013A) p Fl 1122 4601 a(0) p Fx 1160 4589 a(\)) p Fw(S) p Fl 1243 4601 a(1) p Fw 1280 4589 a(g) p Fn 1320 4601 a(\027) p Fw 1361 4589 a( ) p Ft 1418 4589 a(i) p Fx 1473 4589 a(=) p 1561 4589 a(Re) p Ft 1673 4589 a(h) p Fw( ) p Ft 1762 4589 a(j) p Fw(g) p Fn 1825 4601 a(\027) p Fw 1866 4589 a(h) p Fn 1914 4601 a(\013) p Fx 1961 4589 a([) p Fw(h) p Fn 2032 4601 a(\013) p Fw 2080 4589 a(;) p 2117 4589 a(S) p Fl 2168 4601 a(1) p Fx 2205 4589 a(]) p Fw(g) p Fn 2268 4601 a(\027) p Fw 2309 4589 a( ) p Ft 2366 4589 a(i) p Fx 2416 4589 a(+) p Ft 2499 4589 a(h) p Fw 2532 4589 a( ) p Ft 2589 4589 a(j) p Fw(g) p Fn 2652 4601 a(\027) p Fw 2693 4589 a(h) p Fn 2741 4601 a(\013) p Fw 2788 4589 a(S) p Fl 2839 4601 a(1) p Fw 2876 4589 a(h) p Fn 2924 4601 a(\013) p Fw 2971 4589 a(g) p Fn 3011 4601 a(\027) p Fw 3052 4589 a( ) p Ft 3109 4589 a(i) p Fw 3155 4589 a(:) p Fx 3231 4589 a(\(42\)) 515 4760 y(The) p 685 4760 a(\014rst) p 857 4760 a(term) 1018 4949 y(Re) p Ft 1129 4949 a(h) p Fw 1162 4949 a( ) p Ft 1219 4949 a(j) p Fw(g) p Fn 1282 4961 a(\027) p Fw 1323 4949 a(h) p Fn 1371 4961 a(\013) p Fx 1418 4949 a([) p Fw(h) p Fn 1489 4961 a(\013) p Fw 1536 4949 a(;) p 1573 4949 a(S) p Fl 1624 4961 a(1) p Fx 1661 4949 a(]) p Fw(g) p Fn 1724 4961 a(\027) p Fw 1765 4949 a( ) p Ft 1822 4949 a(i) p Fx 1878 4949 a(=) 1975 4893 y(1) p 1975 4930 42 4 v 1975 5006 a(2) p Ft 2040 4949 a(h) p Fw 2073 4949 a( ) p Ft 2130 4949 a(j) p Fw(g) p Fn 2193 4961 a(\027) p Fx 2248 4949 a([) p Fw(h) p Fn 2319 4961 a(\013) p Fw 2366 4949 a(;) p Fx 2403 4949 a([) p Fw(h) p Fn 2474 4961 a(\013) p Fw 2521 4949 a(;) p 2558 4949 a(S) p Fl 2609 4961 a(1) p Fx 2646 4949 a(]]) p Fw 2706 4949 a(g) p Fn 2746 4961 a(\027) p Fw 2787 4949 a( ) p Ft 2844 4949 a(i) p Fx 3245 4949 a(\(43\)) 1905 5255 y(32) p eop %%Page: 33 33 33 32 bop Fx 515 523 a(is) p 598 523 a(estimated) p 977 523 a(as) p 1079 523 a(follo) n(ws.) p 1384 523 a(By) p 1514 523 a(op) r(erator) p 1849 523 a(calculous,) p 2228 523 a(w) n(e) p 2350 523 a(ha) n(v) n(e) 928 697 y([) p Fw(h) p Fn 999 709 a(\013) p Fw 1046 697 a(;) p 1083 697 a(S) p Fl 1134 709 a(1) p Fx 1171 697 a(]) p Fw(') p Fx 1272 697 a(=) p 1359 697 a([) p Fw(h) p Fx(\() p Fw(i\013A) p Fl 1606 709 a(0) p Fx 1644 697 a(\)) p Fw(;) p 1713 697 a(S) p Fl 1764 709 a(1) p Fx 1801 697 a(]) p Fw(') p Fx 1902 697 a(=) p Ft 1990 697 a(\000) p Fw(i\013h) p Fo 2185 663 a(0) p Fx 2207 697 a(\() p Fw(i\013A) p Fl 2383 709 a(0) p Fx 2421 697 a(\)) p Fw(S) p Fl 2504 709 a(2) p Fw 2542 697 a(') p Fx 2614 697 a(+) p Fw 2697 697 a(\013) p Fl 2750 663 a(2) p Fw 2788 697 a(R) p Fl 2851 709 a(2) p Fw 2888 697 a(';) p Fx 3245 697 a(\(44\)) 515 871 y(for) p 648 871 a(all) p Fw 769 871 a(') p Ft 856 871 a(2) p Fw 945 871 a(D) p Fx 1016 871 a(\() p Fw(N) p Fx 1124 871 a(\).) p 1234 871 a(Here) p Fw 1436 871 a(R) p Fl 1499 883 a(2) p Fx 1570 871 a(is) p 1660 871 a(a) p Fw 1735 871 a(N) p Fx 1811 871 a(-b) r(ounded) p 2183 871 a(op) r(erator) p 2524 871 a(and) p 2692 871 a(w) n(e) p 2820 871 a(ha) n(v) n(e) p Ft 3017 871 a(k) p Fw(R) p Fl 3122 883 a(2) p Fw 3159 871 a(g) p Fn 3199 883 a(\027) p Ft 3240 871 a(k) p 3314 871 a(\024) p Fw 515 1007 a(\027) p Fo 561 977 a(\000) p Fl(1) p Ft 668 1007 a(\001) p Fn 719 965 a(b) p Ff 748 923 a(00) p Fg 748 982 a(1) p 719 988 75 4 v Fl 739 1036 a(2) p Fx 803 1007 a(.) p 863 1007 a(On) p 1001 1007 a(the) p 1143 1007 a(other) p 1360 1007 a(hand,) p 1590 1007 a(w) n(e) p 1712 1007 a(ha) n(v) n(e) p Ft 1903 1007 a(k) p Fw(\013) p Fx 2012 1007 a([) p Fw(h) p Fn 2083 1019 a(\013) p Fw 2130 1007 a(;) p 2167 1007 a(h) p Fo 2215 977 a(0) p Fn 2215 1028 a(\013) p Fw 2262 1007 a(S) p Fl 2313 1019 a(2) p Fx 2351 1007 a(]) p Fw 2387 1007 a(g) p Fn 2427 1019 a(\027) p Ft 2468 1007 a(k) p 2533 1007 a(\024) p Fw 2621 1007 a(\013) p Fl 2674 977 a(2) p Fw 2711 1007 a(\027) p Fo 2757 977 a(\000) p Fl(1) p Ft 2864 1007 a(\001) p Fw 2905 1007 a(b) p Fo 2941 977 a(0) p Fl 2941 1028 a(2) p Fx 2978 1007 a(.) p 3038 1007 a(Hence) p 3284 1007 a(w) n(e) 515 1107 y(obtain) p Ft 573 1299 a(j) p Fx(\(43\)) p Ft(j) p Fx 790 1299 a(=) p Ft 877 1299 a(j) p Fx(Re) p Ft 1012 1299 a(h) p Fw 1045 1299 a( ) p Ft 1102 1299 a(j) p Fw(g) p Fn 1165 1311 a(\027) p Fw 1206 1299 a(h) p Fn 1254 1311 a(\013) p Fx 1301 1299 a([) p Fw(h) p Fn 1372 1311 a(\013) p Fw 1419 1299 a(;) p 1456 1299 a(S) p Fl 1507 1311 a(1) p Fx 1544 1299 a(]) p Fw(g) p Fn 1607 1311 a(\027) p Fw 1648 1299 a( ) p Ft 1705 1299 a(ij) p Fx 1783 1299 a(=) 1881 1243 y(1) p 1881 1280 42 4 v 1881 1356 a(2) p Ft 1946 1299 a(jh) p Fw 2002 1299 a( ) p Ft 2059 1299 a(j) p Fw(g) p Fn 2122 1311 a(\027) p Fx 2177 1299 a([) p Fw(h) p Fn 2248 1311 a(\013) p Fw 2295 1299 a(;) p Fx 2332 1299 a([) p Fw(h) p Fn 2403 1311 a(\013) p Fw 2450 1299 a(;) p 2487 1299 a(S) p Fl 2538 1311 a(1) p Fx 2575 1299 a(]]) p Fw 2635 1299 a(g) p Fn 2675 1311 a(\027) p Fw 2716 1299 a( ) p Ft 2773 1299 a(ij) p 2851 1299 a(\024) p Fw 2939 1299 a(\013) p Fl 2992 1265 a(2) p Fw 3030 1299 a(\027) p Fo 3076 1265 a(\000) p Fl(1) p Ft 3183 1299 a(\001) p Fw 3225 1299 a(b) p Fl 3261 1311 a(1) p Fw 3298 1299 a(:) p Fx 3231 1434 a(\(45\)) 515 1605 y(where) p Fw 755 1605 a(b) p Fl 791 1617 a(1) p Fx 855 1605 a(is) p 939 1605 a(a) p 1008 1605 a(constan) n(t) p 1343 1605 a(whic) n(h) p 1580 1605 a(dep) r(ends) p 1901 1605 a(only) p 2084 1605 a(on) p 2199 1605 a(the) p 2342 1605 a(function) p Fw 2667 1605 a(h) p Fx(.) 515 1704 y(Next) p 718 1704 a(let) p 838 1704 a(us) p 944 1704 a(estimate) p 1277 1704 a(\(40\).) p 1484 1704 a(The) p 1655 1704 a(imaginary) p 2045 1704 a(part) p 2225 1704 a(is) p 2308 1704 a(deriv) n(ed) p 2599 1704 a(b) n(y) p 2715 1704 a(the) p 2858 1704 a(comm) n(utator,) 1092 1903 y(2Im) p Fk 1246 1811 a(D) p Fw 1297 1903 a( ) p Fk 1368 1808 a(\014) 1368 1857 y(\014) 1368 1907 y(\014) p Fw 1395 1903 a(g) p Fn 1435 1915 a(\027) p Fw 1476 1903 a(f) p Ff 1526 1844 a(00) p Fn 1517 1924 a(\013) p Fw 1571 1903 a(S) p Fl 1622 1915 a(2) p Fw 1659 1903 a(g) p Fn 1699 1915 a(\027) p Fw 1740 1903 a( ) p Fk 1797 1811 a(E) p Fx 1881 1903 a(=) p Fw 1968 1903 a(i) p Fk 2011 1811 a(D) p Fw 2061 1903 a( ) p Fk 2132 1808 a(\014) 2132 1857 y(\014) 2132 1907 y(\014) p Fw 2160 1903 a(g) p Fn 2200 1915 a(\027) p Fk 2255 1811 a(h) p Fw 2294 1903 a(S) p Fl 2345 1915 a(2) p Fw 2382 1903 a(g) p Fn 2422 1915 a(\027) p Fw 2463 1903 a(;) p 2500 1903 a(f) p Ff 2550 1844 a(00) p Fn 2541 1924 a(\013) p Fk 2594 1811 a(i) p Fw 2648 1903 a( ) p Fk 2705 1811 a(E) p Fw 2779 1903 a(:) p Fx 515 2135 a(whic) n(h) p 751 2135 a(is) p 833 2135 a(b) r(ounded) p 1171 2135 a(using) p 1387 2135 a(op) r(erator) p 1721 2135 a(calculous) p 2076 2135 a(b) n(y) p Fk 2190 2040 a(\015) 2190 2090 y(\015) 2190 2139 y(\015) 2236 2043 y(h) p Fw 2275 2135 a(S) p Fl 2326 2147 a(2) p Fw 2363 2135 a(g) p Fn 2403 2147 a(\027) p Fw 2444 2135 a(;) p 2481 2135 a(f) p Ff 2531 2080 a(00) p Fn 2522 2156 a(\013) p Fk 2576 2043 a(i) 2615 2040 y(\015) 2615 2090 y(\015) 2615 2139 y(\015) p Ft 2684 2135 a(\024) p Fx 2772 2135 a(4) p Fw(\013\027) p Fo 2913 2105 a(\000) p Fl(1) p Ft 3018 2135 a(\001) p Fw 3057 2135 a(b) p Fl 3093 2147 a(2) p Fx 3130 2135 a(.) p 3190 2135 a(Then) 515 2249 y(w) n(e) p 637 2249 a(get) p Ft 1604 2423 a(j) p Fx(\(40\)) p Ft(j) p 1820 2423 a(\024) p Fw 1908 2423 a(\013) p Fl 1961 2388 a(3) p Fw 1999 2423 a(\027) p Fo 2045 2388 a(\000) p Fl(1) p Ft 2153 2423 a(\001) p Fw 2194 2423 a(b) p Fl 2230 2435 a(2) p Fw 2267 2423 a(:) p Fx 3231 2423 a(\(46\)) 515 2597 y(Finally) p 763 2597 a(,) p 813 2597 a(the) p 956 2597 a(last) p 1114 2597 a(term) p 1312 2597 a(\(41\)) p 1487 2597 a(is) p 1571 2597 a(estimated) p 1950 2597 a(as) p Ft 1187 2771 a(j) p Fx(\(41\)) p Ft(j) p Fx 1404 2771 a(=) p Fk 1492 2700 a(\014) 1492 2750 y(\014) p Fw 1520 2771 a(\013) p Fl 1573 2736 a(3) p Fx 1610 2771 a(Re) p Ft 1722 2771 a(h) p Fw( ) p Ft 1811 2771 a(j) p Fw(g) p Fn 1874 2783 a(\027) p Fw 1915 2771 a(R) p Fl 1978 2783 a(3) p Fw 2016 2771 a(g) p Fn 2056 2783 a(\027) p Fw 2097 2771 a( ) p Ft 2154 2771 a(i) p Fk 2186 2700 a(\014) 2186 2750 y(\014) p Ft 2237 2771 a(\024) p Fw 2324 2771 a(\013) p Fl 2377 2736 a(3) p Fw 2415 2771 a(\027) p Fo 2461 2736 a(\000) p Fl(1) p Ft 2569 2771 a(\001) p Fw 2610 2771 a(b) p Fl 2646 2783 a(3) p Fw 2683 2771 a(:) p Fx 3231 2771 a(\(47\)) 515 2945 y(Note) p 715 2945 a(that) p 895 2945 a(from) p 1092 2945 a(\(38\)) p 1267 2945 a(to) p 1368 2945 a(\(42\),) p Ft 515 3119 a(h) p Fw( ) p Ft 604 3119 a(j) p Fw(g) p Fn 667 3131 a(\027) p Fw 708 3119 a(h) p Fn 756 3131 a(\013) p Fw 803 3119 a(S) p Fl 854 3131 a(1) p Fw 891 3119 a(h) p Fn 939 3131 a(\013) p Fw 986 3119 a(g) p Fn 1026 3131 a(\027) p Fw 1067 3119 a( ) p Ft 1124 3119 a(i) p Fx 1175 3119 a(+) p 1258 3119 a(2) p Fw(\013) p Fo 1353 3084 a(\000) p Fl(1) p Fx 1442 3119 a(Im) p Ft 1555 3119 a(h) p Fw 1588 3119 a( ) p Ft 1658 3119 a(j) p Fw 1682 3119 a(g) p Fn 1722 3131 a(\027) p Fw 1762 3119 a(f) p Fn 1803 3131 a(\013) p Fx 1850 3119 a(\() p Fw(L) p Fn 1939 3131 a(f) p Ft 2001 3119 a(\000) p Fw 2084 3119 a(e) p Fx(\)) p Fw(g) p Fn 2195 3131 a(\027) p Ft 2236 3119 a(j) p Fw 2272 3119 a( ) p Ft 2329 3119 a(i) p Fx 2385 3119 a(=) p Fw 2472 3119 a(\013) p Fo 2525 3084 a(\000) p Fl(1) p Fx 2615 3119 a(\(40\)) p 2781 3119 a(+) p Fw 2864 3119 a(\013) p Fo 2917 3084 a(\000) p Fl(1) p Fx 3007 3119 a(\(41\)) p Ft 3173 3119 a(\000) p Fx 3256 3119 a(\(43\)) p Fw(:) p Fx 515 3293 a(Hence) p 761 3293 a(b) n(y) p 877 3293 a(ev) p 953 3293 a(aluations) p 1309 3293 a(\(45\),\(46\),) p 1678 3293 a(and) p 1839 3293 a(\(47\),) p 2037 3293 a(w) n(e) p 2159 3293 a(ha) n(v) n(e) p 2351 3293 a(the) p 2494 3293 a(estimate) p Fk 749 3396 a(\014) 749 3446 y(\014) p Ft 777 3467 a(h) p Fw( ) p Ft 866 3467 a(j) p Fw(g) p Fn 929 3479 a(\027) p Fw 970 3467 a(h) p Fn 1018 3479 a(\013) p Fw 1065 3467 a(S) p Fl 1116 3479 a(1) p Fw 1154 3467 a(h) p Fn 1202 3479 a(\013) p Fw 1249 3467 a(g) p Fn 1289 3479 a(\027) p Fw 1330 3467 a( ) p Ft 1387 3467 a(i) p Fx 1437 3467 a(+) p 1520 3467 a(2) p Fw(\013) p Fo 1615 3432 a(\000) p Fl(1) p Fx 1705 3467 a(Im) p Ft 1818 3467 a(h) p Fw( ) p Ft 1921 3467 a(j) p Fw(g) p Fn 1984 3479 a(\027) p Fw 2025 3467 a(f) p Fn 2066 3479 a(\013) p Fx 2113 3467 a(\() p Fw(L) p Fn 2202 3479 a(f) p Ft 2263 3467 a(\000) p Fw 2346 3467 a(e) p Fx(\)) p Fw(g) p Fn 2457 3479 a(\027) p Ft 2498 3467 a(j) p Fw 2535 3467 a( ) p Ft 2592 3467 a(i) p Fk 2624 3396 a(\014) 2624 3446 y(\014) p Ft 2675 3467 a(\024) p Fw 2762 3467 a(\013) p Fl 2815 3432 a(2) p Fw 2853 3467 a(\027) p Fo 2899 3432 a(\000) p Fl(1) p Ft 3007 3467 a(\001) p Fw 3048 3467 a(b) p Fl 3084 3479 a(0) p Fw 3121 3467 a(:) p Fx 515 3641 a(Similarly) p 834 3641 a(,) p 884 3641 a(w) n(e) p 1006 3641 a(obtain) p 1264 3641 a(the) p 1407 3641 a(b) r(ound) p Fk 825 3744 a(\014) 825 3794 y(\014) 852 3748 y(\012) p Fw 891 3815 a( ) p Ft 948 3815 a(j) p Fw(g) p Fn 1011 3827 a(\027) p Fw 1052 3815 a(h) p Fn 1100 3827 a(\013) p Fw 1148 3815 a(I) p Fn 1191 3781 a(\025) p Fl 1184 3835 a(1) p Fw 1234 3815 a(h) p Fn 1282 3827 a(\013) p Fw 1329 3815 a(g) p Fn 1369 3827 a(\027) p Fw 1410 3815 a( ) p Fk 1467 3748 a(\013) p Fx 1525 3815 a(+) p 1608 3815 a(2) p Fw(\013) p Fo 1703 3781 a(\000) p Fl(1) p Fx 1792 3815 a(Im) p Fk 1905 3748 a(\012) p Fw 1944 3815 a( ) p Fk 2015 3744 a(\014) 2015 3794 y(\014) p Fw 2043 3815 a(g) p Fn 2083 3827 a(\027) p Fw 2124 3815 a(f) p Fn 2165 3827 a(\013) p Fx 2212 3815 a(\() p Fw(I) p Fn 2287 3781 a(\025) p Fl 2280 3835 a(0) p Ft 2349 3815 a(\000) p Fw 2432 3815 a(e) p Fx(\)) p Fw(g) p Fn 2543 3827 a(\027) p Fk 2584 3744 a(\014) 2584 3794 y(\014) p Fw 2625 3815 a( ) p Fk 2682 3748 a(\013) 2722 3744 y(\014) 2722 3794 y(\014) p Ft 2772 3815 a(\024) p Fw 2860 3815 a(\013) p Ft 2932 3815 a(\001) p Fx 2970 3793 a(~) p Fw 2973 3815 a(b) p Fl 3009 3827 a(0) p Fw 3046 3815 a(:) p Fx 515 3989 a(Hence) p 761 3989 a(w) n(e) p 884 3989 a(\014nally) p 1135 3989 a(obtain) p Fk 515 4092 a(\014) 515 4142 y(\014) 542 4096 y(\012) p Fw 582 4163 a( ) p Ft 639 4163 a(j) p Fw(g) p Fn 702 4175 a(\027) p Fw 743 4163 a(h) p Fn 791 4175 a(\013) p Fk 852 4096 a(\000) p Fw 890 4163 a(S) p Fl 941 4175 a(1) p Fx 996 4163 a(+) p Fw 1079 4163 a(I) p Fn 1122 4129 a(\025) p Fl 1115 4183 a(1) p Fk 1166 4096 a(\001) p Fw 1218 4163 a(h) p Fn 1266 4175 a(\013) p Fw 1313 4163 a(g) p Fn 1353 4175 a(\027) p Fw 1394 4163 a( ) p Fk 1451 4096 a(\013) 1490 4092 y(\014) 1490 4142 y(\014) p Ft 1541 4163 a(\024) p Fx 1628 4163 a(2) p Fw(\013) p Fo 1723 4129 a(\000) p Fl(1) p Ft 1826 4163 a(j) p Fx(Im) p Ft 1962 4163 a(h) p Fw 1995 4163 a( ) p Ft 2066 4163 a(j) p Fw(g) p Fn 2129 4175 a(\027) p Fw 2170 4163 a(f) p Fn 2211 4175 a(\013) p Fx 2257 4163 a(\() p Fw(L) p Ft 2365 4163 a(\000) p Fw 2448 4163 a(e) p Fx(\)) p Fw(g) p Fn 2559 4175 a(\027) p Ft 2600 4163 a(j) p Fw 2636 4163 a( ) p Ft 2693 4163 a(i) q(j) p Fx 2767 4163 a(+) p Fw 2850 4163 a(\013) p Fl 2903 4129 a(2) p Fw 2941 4163 a(\027) p Fo 2987 4129 a(\000) p Fl(1) p Ft 3095 4163 a(\001) p Fw 3136 4163 a(b) p Fl 3172 4175 a(0) p Fx 3228 4163 a(+) p Fw 3311 4163 a(\013) p Ft 3383 4163 a(\001) p Fx 3421 4141 a(~) p Fw 3424 4163 a(b) p Fl 3460 4175 a(0) p Fw 3497 4163 a(:) p Fd 3686 4163 a(\003) p Fx 515 4337 a(No) n(w) p 707 4337 a(let) p 829 4337 a(us) p 938 4337 a(complete) p 1292 4337 a(the) p 1437 4337 a(pro) r(of) p 1657 4337 a(of) p 1754 4337 a(Prop) r(osition) p 2204 4337 a(8.1.) p 2378 4337 a(By) p 2511 4337 a(Lemma) p 2810 4337 a(8.2) p 2946 4337 a(and) p 3110 4337 a(Lemma) 515 4437 y(8.3,) p 672 4437 a(w) n(e) p 794 4437 a(obtain) p Fk 811 4540 a(\014) 811 4590 y(\014) 839 4543 y(\012) p Fw 878 4611 a( ) p Ft 935 4611 a(j) p Fw(g) p Fn 998 4623 a(\027) p Fw 1039 4611 a(h) p Fn 1087 4623 a(\013) p Fk 1148 4543 a(\000) p Fw 1186 4611 a(S) p Fl 1237 4623 a(1) p Fx 1293 4611 a(+) p Fw 1376 4611 a(I) p Fn 1419 4576 a(\025) p Fl 1412 4631 a(1) p Fk 1462 4543 a(\001) p Fw 1514 4611 a(h) p Fn 1562 4623 a(\013) p Fw 1609 4611 a(g) p Fn 1649 4623 a(\027) p Fw 1690 4611 a( ) p Fk 1747 4543 a(\013) 1786 4540 y(\014) 1786 4590 y(\014) p Ft 1837 4611 a(\024) p Fx 1925 4611 a(2) p Fw(C) p Fl 2026 4623 a(1) p Ft 2081 4611 a(\001) p Fw 2123 4611 a(\013) p Fo 2176 4576 a(\000) p Fl(1) p Fw 2265 4611 a(\027) p Fx 2330 4611 a(+) p Fw 2413 4611 a(\013) p Fl 2466 4576 a(2) p Fw 2504 4611 a(\027) p Fo 2550 4576 a(\000) p Fl(1) p Ft 2658 4611 a(\001) p Fw 2699 4611 a(b) p Fl 2735 4623 a(0) p Fx 2790 4611 a(+) p Fw 2873 4611 a(\013) p Ft 2945 4611 a(\001) p Fx 2984 4589 a(~) p Fw 2987 4611 a(b) p Fl 3023 4623 a(0) p Fw 3060 4611 a(:) p Fx 515 4802 a(No) n(w) p 704 4802 a(let) p Fw 824 4802 a(\027) p Fx 898 4802 a(b) r(e) p Fw 1011 4802 a(\027) p Fx 1080 4802 a(=) p Fw 1168 4802 a(\013) p Fg 1231 4750 a(3) p 1231 4759 29 3 v 1231 4792 a(2) p Fx 1273 4802 a(.) p 1333 4802 a(Then) p 1550 4802 a(w) n(e) p 1672 4802 a(ha) n(v) n(e) p Fk 913 4906 a(\014) 913 4956 y(\014) 940 4909 y(\012) p Fw 980 4976 a( ) p Ft 1037 4976 a(j) p Fw(g) p Fn 1100 4988 a(\027) p Fw 1141 4976 a(h) p Fn 1189 4988 a(\013) p Fk 1250 4909 a(\000) p Fw 1288 4976 a(S) p Fl 1339 4988 a(1) p Fx 1394 4976 a(+) p Fw 1477 4976 a(I) p Fn 1520 4942 a(\025) p Fl 1513 4997 a(1) p Fk 1564 4909 a(\001) p Fw 1616 4976 a(h) p Fn 1664 4988 a(\013) p Fw 1711 4976 a(g) p Fn 1751 4988 a(\027) p Fw 1792 4976 a( ) p Fk 1849 4909 a(\013) 1888 4906 y(\014) 1888 4956 y(\014) p Ft 1939 4976 a(\024) p Fx 2026 4976 a(2) p Fw(C) p Fl 2127 4988 a(1) p Ft 2183 4976 a(\001) p Fw 2224 4976 a(\013) p Fg 2287 4920 a(1) p 2288 4929 V 2288 4962 a(2) p Fx 2349 4976 a(+) p Fw 2432 4976 a(\013) p Fg 2495 4920 a(1) p 2495 4929 V 2495 4962 a(2) p Ft 2556 4976 a(\001) p Fw 2597 4976 a(b) p Fl 2633 4988 a(0) p Fx 2689 4976 a(+) p Fw 2772 4976 a(\013) p Ft 2844 4976 a(\001) p Fx 2882 4954 a(~) p Fw 2885 4976 a(b) p Fl 2921 4988 a(0) p Fw 2958 4976 a(:) p Fx 1905 5255 a(33) p eop %%Page: 34 34 34 33 bop Fx 515 523 a(T) p 568 523 a(aking) p Fw 791 523 a(\013) p Ft 868 523 a(!) p Fx 974 523 a(0,) p 1066 523 a(w) n(e) p 1189 523 a(obtain) 889 692 y(lim) p Fn 876 744 a(\013) p Fo(!) p Fl(0) p Ft 1032 692 a(h) p Fw(K) p Ft 1141 692 a(i) p Fn 1173 717 a( ) p Fj 1217 725 a(\013;\027) p Fx 1338 692 a(=) p 1439 692 a(lim) p Fn 1425 744 a(\013) p Fo(!) p Fl(0) p Fk 1582 625 a(\012) p Fw 1621 692 a( ) p Ft 1678 692 a(j) p Fw(g) p Fn 1741 704 a(\027) p Fw 1782 692 a(h) p Fn 1830 704 a(\013) p Fk 1891 625 a(\000) p Fw 1929 692 a(S) p Fl 1980 704 a(1) p Fx 2035 692 a(+) p Fw 2118 692 a(I) p Fn 2161 658 a(\025) p Fl 2154 713 a(1) p Fk 2205 625 a(\001) p Fw 2257 692 a(h) p Fn 2305 704 a(\013) p Fw 2352 692 a(g) p Fn 2392 704 a(\027) p Fw 2433 692 a( ) p Fk 2490 625 a(\013) p Fx 2552 692 a(=) p 2640 692 a(0) p Fw(:) p Fd 2954 692 a(\003) p Fx 515 892 a(Next) p 715 892 a(w) n(e) p 834 892 a(calculate) p 1177 892 a(the) p 1317 892 a(lo) n(w) n(er) p 1531 892 a(b) r(ound) p 1784 892 a(of) p Ft 1875 892 a(h) p Fw(K) p Ft 1984 892 a(i) p Fn 2016 917 a( ) p Fj 2060 925 a(\013;\027) p Fx 2158 892 a(.) p 2217 892 a(W) p 2295 892 a(e) p 2357 892 a(use) p 2497 892 a(Theorem) p 2845 892 a(7.1.) p 3010 892 a(F) p 3057 892 a(or) p Fw 3156 892 a(e) p Ft 3218 892 a(2) p Fi 3296 892 a(R) p Fx(,) 515 992 y(let) p 639 992 a(~) p Fw 636 992 a(e) p Ft 699 992 a(\021) p Fw 789 992 a(e) p Fx(,) p 880 992 a(if) p Fw 957 992 a(e) p Fx 1025 992 a(is) p 1109 992 a(an) p 1226 992 a(eigen) n(v) p 1448 992 a(alue) p 1623 992 a(of) p Fw 1719 992 a(L) p Fn 1776 1004 a(S) p Fx 1824 992 a(,) p 1876 992 a(and) p 2038 992 a(let) p 2163 992 a(~) p Fw 2160 992 a(e) p Fx 2227 992 a(b) r(e) p 2341 992 a(an) p 2458 992 a(eigen) n(v) p 2680 992 a(alue) p 2855 992 a(of) p Fw 2951 992 a(L) p Fn 3008 1004 a(S) p Fx 3085 992 a(whic) n(h) p 3323 992 a(is) 515 1091 y(next) p 702 1091 a(to) p Fw 803 1091 a(e) p Fx(,) p 892 1091 a(in) p 989 1091 a(case) p 1165 1091 a(that) p Fw 1345 1091 a(e) p Fx 1411 1091 a(is) p 1495 1091 a(not) p 1642 1091 a(an) p 1758 1091 a(eigen) n(v) p 1980 1091 a(alue) p 2154 1091 a(of) p Fw 2249 1091 a(L) p Fn 2306 1103 a(S) p Fx 2353 1091 a(.) p Fm 515 1247 a(Prop) s(osition) p 1031 1247 a(8.4) p Fy 1194 1247 a(L) l(et) p Fw 1349 1247 a(\027) p Ft 1438 1247 a(\021) p Fw 1545 1247 a(\013) p Fg 1608 1194 a(3) p 1608 1203 29 3 v 1608 1237 a(2) p Fy 1691 1247 a(and) p 1863 1247 a(supp) l(ose) p 2178 1247 a(the) p 2326 1247 a(Assumption) p 2794 1247 a(2.1) p 2945 1247 a(and) p 3117 1247 a(2.2) p 3268 1247 a(ar) l(e) 515 1346 y(satis\014e) l(d.) p 865 1346 a(Then) p 1081 1346 a(for) p 1214 1346 a(su\016ciently) p 1633 1346 a(smal) t(l) p Fw 1854 1346 a(\025) p Fy(,) p 1957 1346 a(we) p 2080 1346 a(have) p Fx 1261 1536 a(lim) p Fn 1247 1587 a(\013) p Fo(!) p Fl(0) p Ft 1404 1536 a(h) p Fw(K) p Ft 1513 1536 a(i) p Fn 1545 1561 a( ) p Fj 1589 1569 a(\013;\027) p Ft 1709 1536 a(\025) p Fw 1797 1536 a(b) p Fl 1836 1548 a(~) p Fn 1833 1548 a(e) p Fk 1882 1444 a(\020) p Ft 1932 1536 a(k) p Fw( ) p Ft 2031 1536 a(k) p Fl 2073 1502 a(2) p Ft 2128 1536 a(\000) p Fx 2211 1536 a(14) p Ft(k) p Fx 2354 1515 a(~) p Fw 2337 1536 a(P) p Fl 2393 1548 a(~) p Fn 2390 1548 a(e) p Fw 2424 1536 a( ) p Ft 2481 1536 a(k) p Fl 2523 1502 a(2) p Fk 2560 1444 a(\021) p Fw 2623 1536 a(;) p Fy 515 1753 a(wher) l(e) p Fw 749 1753 a(b) p Fl 788 1765 a(~) p Fn 785 1765 a(e) p Fx 844 1753 a(=) p Fl 941 1720 a(1) p 941 1734 34 4 v 941 1781 a(2) p Ft 1003 1753 a(\001) p Fw 1044 1753 a(\025) p Fg 1102 1700 a(182) p 1103 1709 85 3 v 1103 1743 a(100) p Ft 1220 1753 a(\001) p Fw 1261 1753 a(\015) p Fl 1307 1765 a(~) p Fn 1304 1765 a(e) p Fy 1340 1753 a(.) 515 1908 y(Pr) l(o) l(of) p Fx 515 2008 a(Let) p Fw 663 2008 a(\037) p Ft 738 2008 a(2) p Fw 817 2008 a(C) p Fo 882 1978 a(1) p Fl 876 2028 a(0) p Fx 953 2008 a(\() p Fi(R) p Fx(\)) p 1105 2008 a(b) r(e) p 1218 2008 a(a) p 1287 2008 a(smo) r(oth) p 1580 2008 a(function) p 1905 2008 a(suc) n(h) p 2093 2008 a(that) 1145 2277 y(0) p Ft 1210 2277 a(\024) p Fw 1297 2277 a(\037) p Ft 1372 2277 a(\024) p Fx 1460 2277 a(1) p 1667 2277 a(and) p Fk 1981 2107 a(8) 1981 2181 y(<) 1981 2331 y(:) p Fw 2070 2216 a(\037) p Fx(\() p Fw(t) p Fx(\)) p 2240 2216 a(=) p 2327 2216 a(1) p Ft 2452 2216 a(j) p Fw(t) p Ft(j) p 2551 2216 a(\024) p Fx 2649 2160 a(1) p 2649 2197 42 4 v 2649 2273 a(2) p Fw 2069 2376 a(\037) p Fx(\() p Fw(t) p Fx(\)) p 2238 2376 a(=) p 2326 2376 a(0) p 2450 2376 a(1) p Ft 2515 2376 a(\024) p 2603 2376 a(j) p Fw(t) p Ft(j) p Fw(:) 2726 2277 y(;) p Fx 515 2541 a(Supp) r(ose) p 840 2541 a(that) p 1020 2541 a(the) p 1163 2541 a(function) p 1488 2541 a(0) p Ft 1553 2541 a(\024) p Fx 1650 2541 a(\026) p Fw 1641 2541 a(\037) p Ft 1716 2541 a(\024) p Fx 1803 2541 a(1) p 1872 2541 a(de\014ned) p 2158 2541 a(b) n(y) p Fw 1719 2711 a(\037) p Fl 1771 2676 a(2) p Fx 1827 2711 a(+) p 1920 2711 a(\026) p Fw 1910 2711 a(\037) p Fl 1962 2676 a(2) p Fx 2022 2711 a(=) p 2110 2711 a(1) p Fw(;) p Fx 515 2891 a(b) r(elongs) p 823 2891 a(to) p Fw 936 2891 a(C) p Fo 1001 2861 a(1) p Fn 995 2914 a(b) p Fx 1071 2891 a(\() p Fi(R) p Fx(\).) p 1288 2891 a(Let) p Fw 1448 2891 a(\037) p Fn 1500 2903 a(n) p Ft 1586 2891 a(\021) p Fw 1692 2891 a(\037) p Fx(\() p Fn 1786 2858 a(N) p 1786 2872 59 4 v 1795 2920 a(n) p Fx 1855 2891 a(\)) p 1925 2891 a(and) p 2107 2891 a(\026) p Fw 2098 2891 a(\037) p Fn 2150 2903 a(n) p Ft 2236 2891 a(\021) p Fx 2351 2891 a(\026) p Fw 2341 2891 a(\037) p Fx(\() p Fn 2435 2858 a(N) p 2436 2872 V 2445 2920 a(n) p Fx 2504 2891 a(\).) p 2629 2891 a(In) p 2744 2891 a(the) p 2897 2891 a(follo) n(wing,) p 3284 2891 a(w) n(e) 515 2991 y(assume) p 805 2991 a(that) p Fw 989 2991 a(n) p 1067 2991 a(<) p 1161 2991 a(\027) p Fo 1207 2960 a(\000) p Fl(1) p Fx 1296 2991 a(.) p 1367 2991 a(Recall) p 1620 2991 a(that) p Fw 1804 2991 a(g) p Ft 1875 2991 a(2) p Fw 1959 2991 a(C) p Fo 2024 2960 a(1) p Fl 2018 3011 a(0) p Fx 2095 2991 a(\() p Fi(R) p Fx(\)) p 2251 2991 a(is) p 2338 2991 a(a) p 2410 2991 a(smo) r(oth) p 2707 2991 a(function) p 3036 2991 a(suc) n(h) p 3227 2991 a(that) 515 3090 y(supp) p Fw 715 3090 a(g) p Ft 787 3090 a(\032) p Fx 880 3090 a([) p Ft(\000) p Fx(2) p Fw(;) p Fx 1047 3090 a(2],) p 1166 3090 a(0) p Ft 1237 3090 a(\024) p Fw 1331 3090 a(g) p Ft 1402 3090 a(\024) p Fx 1496 3090 a(1,) p 1593 3090 a(and) p Fw 1758 3090 a(g) p Fx 1830 3090 a(=) p 1923 3090 a(1) p 1996 3090 a(on) p 2115 3090 a([) p Ft(\000) p Fx(1) p Fw(;) p Fx 2282 3090 a(1].) p 2417 3090 a(Then) p 2637 3090 a(w) n(e) p 2763 3090 a(ha) n(v) n(e) p Fw 2958 3090 a(g) p Fn 2998 3102 a(\027) p Fw 3039 3090 a(\037) p Fn 3091 3102 a(n) p Fx 3165 3090 a(=) p Fw 3259 3090 a(\037) p Fn 3311 3102 a(n) p Fw 3356 3090 a(:) p Fx 515 3190 a(First) p 717 3190 a(w) n(e) p 839 3190 a(pro) n(v) n(e) p 1063 3190 a(the) p 1206 3190 a(follo) n(wing) p 1557 3190 a(Lemma.) p Fm 515 3345 a(Lemma) p 853 3345 a(8.5) p Ft 1017 3345 a(h) p Fw(K) p Ft 1126 3345 a(i) p Fn 1158 3370 a( ) p Fj 1202 3378 a(\013;\027) p Fy 1329 3345 a(is) p 1418 3345 a(b) l(ounde) l(d) p 1734 3345 a(b) l(elow) p 1955 3345 a(by) p Ft 653 3539 a(h) p Fw(K) p Ft 762 3539 a(i) p Fn 794 3564 a( ) p Fj 838 3572 a(\013;\027) p Ft 959 3539 a(\025) p 1047 3539 a(h) p Fw(K) p Ft 1156 3539 a(i) p Fn 1188 3564 a(\037) p Fj 1229 3572 a(n) p Fn 1270 3564 a(P) p 1321 3564 a( ) p Fj 1365 3572 a(\013;\027) p Ft 1481 3539 a(\000) p 1564 3539 a(k) p Fw(I) p Fn 1649 3505 a(\025) p Fl 1642 3560 a(1) p Ft 1692 3539 a(k) p 1752 3539 a(\001) p 1794 3539 a(k) p Fx 1854 3518 a(\026) p Fw 1836 3539 a(P) p 1900 3539 a( ) p Fn 1954 3551 a(\013;\027) p Ft 2058 3539 a(k) p Fl 2100 3505 a(2) p Ft 2155 3539 a(\000) p 2238 3539 a(k) p Fw(I) p Fn 2323 3505 a(\025) p Fl 2316 3560 a(1) p Ft 2366 3539 a(k) p 2426 3539 a(\001) p 2468 3539 a(k) p Fx 2520 3539 a(\026) p Fw 2510 3539 a(\037) p Fn 2562 3551 a(n) p Fw 2607 3539 a( ) p Fn 2661 3551 a(\013;\027) p Ft 2765 3539 a(k) p Fl 2807 3505 a(2) p Ft 2862 3539 a(\000) p Fw 2945 3539 a(n) p Fo 2995 3505 a(\000) p Fl(2) p Ft 3103 3539 a(\001) p Fw 3144 3539 a(C) p Fl 3203 3551 a(2) p Fy 515 3729 a(wher) l(e) p Fw 749 3729 a(C) p Fl 808 3741 a(2) p Fy 875 3729 a(is) p 965 3729 a(a) p 1037 3729 a(p) l(ositive) p 1336 3729 a(c) l(onstant) p 1667 3729 a(which) p 1900 3729 a(dep) l(ends) p 2210 3729 a(only) p 2390 3729 a(on) p Fw 2509 3729 a(I) p Fn 2552 3699 a(\025) p Fl 2545 3749 a(0) p Fy 2625 3729 a(and) p Fw 2787 3729 a(\037) p Fy(.) 515 3884 y(Pr) l(o) l(of) p Fx 515 3984 a(As) p Fw 637 3984 a(\037) p Fn 689 3996 a(n) p Fw 735 3984 a(D) p Fx 806 3984 a(\() p Fw(N) p Fx 914 3984 a(\)) p Ft 969 3984 a(\032) p Fw 1057 3984 a(D) p Fx 1128 3984 a(\() p Fw(N) p Fx 1236 3984 a(\),) p 1329 3984 a(\026) p Fw 1319 3984 a(\037) p Fn 1371 3996 a(n) p Fw 1416 3984 a(D) p Fx 1487 3984 a(\() p Fw(N) p Fx 1595 3984 a(\)) p Ft 1650 3984 a(\032) p Fw 1738 3984 a(D) p Fx 1809 3984 a(\() p Fw(N) p Fx 1917 3984 a(\)) p 1977 3984 a(and) p Fw 2138 3984 a(K) p Fx 2243 3984 a(is) p Fw 2326 3984 a(N) p Fx 2402 3984 a(-b) r(ounded) p 2768 3984 a(w) n(e) p 2891 3984 a(ha) n(v) n(e) p Ft 604 4203 a(h) p Fw(K) p Ft 713 4203 a(i) p Fn 745 4228 a( ) p Fj 789 4236 a(\013;\027) p Fx 909 4203 a(=) p Fk 997 4086 a(\034) p Fw 1059 4203 a(\037) p Fn 1111 4215 a(n) p Fw 1156 4203 a(K) p 1233 4203 a(\037) p Fn 1285 4215 a(n) p Fx 1349 4203 a(+) p 1441 4203 a(\026) p Fw 1432 4203 a(\037) p Fn 1484 4215 a(n) p Fw 1529 4203 a(K) p Fx 1615 4203 a(\026) p Fw 1606 4203 a(\037) p Fn 1658 4215 a(n) p Fx 1721 4203 a(+) 1814 4147 y(1) p 1814 4184 42 4 v 1814 4260 a(2) 1865 4203 y([) p Fw(\037) p Fn 1940 4215 a(n) p Fw 1985 4203 a(;) p Fx 2022 4203 a([) p Fw(\037) p Fn 2097 4215 a(n) p Fw 2142 4203 a(;) p 2179 4203 a(K) p Fx 2256 4203 a(]]) p 2320 4203 a(+) 2413 4147 y(1) p 2413 4184 V 2413 4260 a(2) 2465 4203 y([) p 2498 4203 a(\026) p Fw 2488 4203 a(\037) p Fn 2540 4215 a(n) p Fw 2585 4203 a(;) p Fx 2622 4203 a([) p 2655 4203 a(\026) p Fw 2645 4203 a(\037) p Fn 2697 4215 a(n) p Fw 2742 4203 a(;) p 2779 4203 a(K) p Fx 2856 4203 a(]]) p Fk 2902 4086 a(\035) p Fn 2964 4286 a( ) p Fj 3008 4294 a(\013;\027) p Fw 3119 4203 a(:) p Fx 3231 4203 a(\(48\)) 515 4438 y(By) p 640 4438 a(the) p 777 4438 a(same) p 979 4438 a(argumen) n(ts) p 1378 4438 a(as) p 1475 4438 a(w) n(e) p 1591 4438 a(had) p 1747 4438 a(in) p 1839 4438 a(Lemma) p 2129 4438 a(6.2,) p Fw 2282 4438 a(K) p Fx 2380 4438 a(and) p Fw 2536 4438 a(N) p Fl 2603 4450 a(\003) p Fj 2648 4434 a(c) 2648 4467 y(v) p Fx 2710 4438 a(strongly) p 3026 4438 a(comm) n(ute.) 515 4537 y(No) n(w) p 704 4537 a(w) n(e) p 826 4537 a(estimate) p 1159 4537 a(eac) n(h) p 1345 4537 a(term) p 1544 4537 a(of) p 1639 4537 a(\(48\).) 639 4637 y(Let) p 788 4637 a(us) p 895 4637 a(decomp) r(ose) p 1312 4637 a(the) p 1455 4637 a(\014rst) p 1627 4637 a(term) p 1825 4637 a(of) p 1920 4637 a(\(48\)) p 2095 4637 a(as) p Ft 612 4831 a(h) p Fw(\037) p Fn 696 4843 a(n) p Fw 741 4831 a(K) p 818 4831 a(\037) p Fn 870 4843 a(n) p Ft 915 4831 a(i) p Fn 947 4856 a( ) p Fj 991 4864 a(\013;\027) p Fx 1112 4831 a(=) p Ft 1199 4831 a(h) p Fw 1232 4831 a( ) p Fn 1286 4843 a(\013;\027) p Fw 1390 4831 a(;) p 1427 4831 a(P) p 1492 4831 a(\037) p Fn 1544 4843 a(n) p Fw 1589 4831 a(K) p 1666 4831 a(\037) p Fn 1718 4843 a(n) p Fw 1762 4831 a(P) p 1827 4831 a( ) p Fn 1881 4843 a(\013;\027) p Ft 1985 4831 a(i) p Fx 2036 4831 a(+) p Fk 2119 4764 a(\012) p Fw 2158 4831 a( ) p Fn 2212 4843 a(\013;\027) p Fw 2317 4831 a(;) p Fx 2372 4810 a(\026) p Fw 2354 4831 a(P) p 2418 4831 a(\037) p Fn 2470 4843 a(n) p Fw 2515 4831 a(K) p 2592 4831 a(\037) p Fn 2644 4843 a(n) p Fw 2689 4831 a(P) p 2754 4831 a( ) p Fn 2808 4843 a(\013;\027) p Fk 2912 4764 a(\013) p Fx 1410 4976 a(+) p Fk 1493 4909 a(\012) p Fw 1532 4976 a( ) p Fn 1586 4988 a(\013;\027) p Fw 1691 4976 a(;) p 1728 4976 a(P) p 1793 4976 a(\037) p Fn 1845 4988 a(n) p Fw 1889 4976 a(K) p 1966 4976 a(\037) p Fn 2018 4988 a(n) p Fx 2082 4955 a(\026) p Fw 2063 4976 a(P) p 2128 4976 a( ) p Fn 2182 4988 a(\013;\027) p Fk 2286 4909 a(\013) p Fx 2344 4976 a(+) p Fk 2427 4909 a(\012) p Fw 2466 4976 a( ) p Fn 2520 4988 a(\013;\027) p Fw 2624 4976 a(;) p Fx 2680 4955 a(\026) p Fw 2661 4976 a(P) p 2726 4976 a(\037) p Fn 2778 4988 a(n) p Fw 2823 4976 a(K) p 2900 4976 a(\037) p Fn 2952 4988 a(n) p Fx 3015 4955 a(\026) p Fw 2997 4976 a(P) p 3062 4976 a( ) p Fn 3116 4988 a(\013;\027) p Fk 3220 4909 a(\013) p Fw 3273 4976 a(:) p Fx 1905 5255 a(34) p eop %%Page: 35 35 35 34 bop Fx 515 523 a(By) p 645 523 a(the) p 788 523 a(strong) p 1042 523 a(comm) n(utativit) n(y) p 1600 523 a(of) p Fw 1695 523 a(N) p Fl 1762 535 a(\003) p Fj 1807 519 a(c) 1807 552 y(v) p Fx 1875 523 a(with) p 2064 523 a(and) p Fw 2225 523 a(K) p Fx 2329 523 a(and) p Fw 2491 523 a(N) p Fx 2567 523 a(,) p 2617 523 a(w) n(e) p 2739 523 a(ha) n(v) n(e) p Fk 972 624 a(\012) p Fw 1011 691 a( ) p Fn 1065 703 a(\013;\027) p Fw 1169 691 a(;) p Fx 1225 670 a(\026) p Fw 1206 691 a(P) p 1271 691 a(\037) p Fn 1323 703 a(n) p Fw 1368 691 a(K) p 1445 691 a(\037) p Fn 1497 703 a(n) p Fw 1542 691 a(P) p 1607 691 a( ) p Fn 1661 703 a(\013;\027) p Fk 1765 624 a(\013) p Fx 1827 691 a(=) p Fk 1915 624 a(\012) p Fw 1954 691 a( ) p Fn 2008 703 a(\013;\027) p Fw 2112 691 a(;) p Fx 2168 670 a(\026) p Fw 2149 691 a(P) p 2214 691 a(P) p 2279 691 a(\037) p Fn 2331 703 a(n) p Fw 2376 691 a(K) p 2453 691 a(\037) p Fn 2505 703 a(n) p Fw 2549 691 a( ) p Fn 2603 703 a(\013;\027) p Fk 2708 624 a(\013) p Fx 2770 691 a(=) p 2858 691 a(0) p Fw(:) p Fx 515 860 a(Similarly) p Fk 868 792 a(\012) p Fw 907 860 a( ) p Fn 961 872 a(\013;\027) p Fw 1065 860 a(;) p 1102 860 a(P) p 1167 860 a(\037) p Fn 1219 872 a(n) p Fw 1264 860 a(K) p 1341 860 a(\037) p Fn 1393 872 a(n) p Fx 1456 839 a(\026) p Fw 1438 860 a(P) p 1503 860 a( ) p 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a( ) p Fn 2816 1044 a(\013;\027) p Fk 2920 964 a(\013) p Ft 1295 1169 a(\025) p Fk 1383 1102 a(\012) p Fw 1422 1169 a( ) p Fn 1476 1181 a(\013;\027) p Fw 1581 1169 a(;) p Fx 1636 1148 a(\026) p Fw 1618 1169 a(P) p 1682 1169 a(\037) p Fn 1734 1181 a(n) p Fw 1779 1169 a(I) p Fn 1822 1135 a(\025) p Fl 1815 1190 a(1) p Fw 1866 1169 a(\037) p Fn 1918 1181 a(n) p Fx 1982 1148 a(\026) p Fw 1963 1169 a(P) p 2028 1169 a( ) p Fn 2082 1181 a(\013;\027) p Fk 2186 1102 a(\013) p Ft 2248 1169 a(\025) p 2336 1169 a(\000k) p Fw(I) p Fn 2486 1135 a(\025) p Fl 2479 1190 a(1) p Ft 2529 1169 a(k) p 2589 1169 a(\001) p 2630 1169 a(k) p Fx 2690 1148 a(\026) p Fw 2672 1169 a(P) p 2736 1169 a( ) p Fn 2790 1181 a(\013;\027) p Ft 2895 1169 a(k) p Fl 2937 1135 a(2) p Fx 515 1329 a(Hence) p 761 1329 a(w) n(e) p 884 1329 a(obtain) p Ft 1096 1489 a(h) p Fw 1129 1489 a(\037) p Fn 1181 1501 a(n) p Fw 1226 1489 a(K) p 1303 1489 a(\037) p Fn 1355 1501 a(n) p Ft 1399 1489 a(i) p Fn 1432 1513 a( ) p Fj 1476 1521 a(\013;\027) p Ft 1596 1489 a(\025) p 1684 1489 a(h) p Fw(K) p Ft 1793 1489 a(i) p Fn 1825 1513 a(\037) p Fj 1866 1521 a(n) p Fn 1907 1513 a(P) p 1958 1513 a( ) p Fj 2002 1521 a(\013;\027) p Ft 2118 1489 a(\000) p 2201 1489 a(k) p Fw(I) p Fn 2286 1454 a(\025) p Fl 2279 1509 a(1) p Ft 2330 1489 a(k) p 2390 1489 a(\001) p 2431 1489 a(k) p Fx 2491 1468 a(\026) p Fw 2473 1489 a(P) p 2537 1489 a( ) p Fn 2591 1501 a(\013;\027) p Ft 2696 1489 a(k) p Fl 2738 1454 a(2) p Fw 2774 1489 a(:) p Fx 515 1660 a(The) p 685 1660 a(estimation) p 1092 1660 a(of) p 1186 1660 a(the) p 1329 1660 a(second) p 1597 1660 a(term) p 1796 1660 a(of) p 1890 1660 a(\(48\)) p 2066 1660 a(is) p 2149 1660 a(the) p 2292 1660 a(same) p 2500 1660 a(as) p 2602 1660 a(ab) r(o) n(v) n(e;) p Ft 568 1820 a(h) p Fx 610 1820 a(\026) p Fw 600 1820 a(\037) p Fn 652 1832 a(n) p Fw 697 1820 a(K) p Fx 784 1820 a(\026) p Fw 774 1820 a(\037) p Fn 826 1832 a(n) p Ft 871 1820 a(i) p Fn 903 1845 a( ) p Fj 947 1853 a(\013;\027) p Fx 1068 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a(2) p Fw 3303 1820 a(:) p Fx 515 2000 a(Using) p 748 2000 a(op) r(erator) p 1083 2000 a(calculous,) p 1462 2000 a(the) p 1605 2000 a(third) p 1812 2000 a(and) p 1974 2000 a(the) p 2117 2000 a(forth) p 2322 2000 a(term) p 2521 2000 a(of) p 2615 2000 a(\(48\)) p 2791 2000 a(is) p 2874 2000 a(b) r(ounded) p 3213 2000 a(as) p Ft 1119 2181 a(k) p Fx([) p Fw(\037) p Fn 1236 2193 a(n) p Fw 1280 2181 a(;) p Fx 1317 2181 a([) p Fw(\037) p Fn 1392 2193 a(n) p Fw 1438 2181 a(;) p 1475 2181 a(K) p Fx 1552 2181 a(]]) p Ft(k) p 1662 2181 a(\024) p Fw 1759 2125 a(c) p Fl 1795 2137 a(11) p 1759 2162 107 4 v Fw 1769 2238 a(n) p Fl 1819 2214 a(2) p Fw 1875 2181 a(;) p Ft 1995 2181 a(k) p Fx([) p 2070 2181 a(\026) p Fw 2060 2181 a(\037) p Fn 2112 2193 a(n) p Fw 2157 2181 a(;) p Fx 2194 2181 a([) p 2227 2181 a(\026) p Fw 2217 2181 a(\037) p Fn 2269 2193 a(n) p Fw 2314 2181 a(;) p 2351 2181 a(K) p Fx 2428 2181 a(]]) p Ft(k) p 2538 2181 a(\024) p Fw 2636 2125 a(c) p Fl 2672 2137 a(12) p 2636 2162 V Fw 2646 2238 a(n) p Fl 2696 2214 a(2) p Fw 2752 2181 a(:) p Fx 515 2364 a(Hence) p 761 2364 a(w) n(e) p 884 2364 a(obtain) p Ft 612 2548 a(h) p Fw(K) p Ft 721 2548 a(i) p Fn 753 2573 a( ) p Fj 797 2581 a(\013;\027) p Ft 926 2548 a(\025) p 1023 2548 a(h) p Fw(K) p Ft 1132 2548 a(i) p Fn 1164 2573 a(\037) p Fj 1205 2581 a(n) p Fn 1246 2573 a(P) p 1297 2573 a( ) p Fj 1341 2581 a(\013;\027) p Ft 1461 2548 a(\000) p 1547 2548 a(k) p Fw(I) p Fn 1632 2514 a(\025) p Fl 1625 2569 a(1) p Ft 1675 2548 a(k) p 1739 2548 a(\001) p 1784 2548 a(k) p Fx 1844 2527 a(\026) p Fw 1826 2548 a(P) p 1890 2548 a( ) p Fn 1944 2560 a(\013;\027) p Ft 2048 2548 a(k) p Fl 2090 2514 a(2) p Ft 2149 2548 a(\000) p 2236 2548 a(k) p Fw(I) p Fn 2321 2514 a(\025) p Fl 2314 2569 a(1) p Ft 2364 2548 a(k) p 2427 2548 a(\001) p 2472 2548 a(k) p Fx 2524 2548 a(\026) p Fw 2514 2548 a(\037) p Fn 2566 2560 a(n) p Fw 2611 2548 a( ) p Fn 2665 2560 a(\013;\027) p Ft 2769 2548 a(k) p Fl 2811 2514 a(2) p Ft 2870 2548 a(\000) p Fw 2957 2548 a(n) p Fo 3007 2514 a(\000) p Fl(2) p Ft 3117 2548 a(\001) p Fw 3162 2548 a(C) p Fl 3221 2560 a(2) p Fw 3259 2548 a(;) 515 2737 y(C) p Fl 574 2749 a(2) p Fx 639 2737 a(is) p 722 2737 a(a) p 792 2737 a(constan) n(t) p 1127 2737 a(whic) n(h) p 1364 2737 a(dep) r(ends) p 1685 2737 a(only) p 1867 2737 a(on) 1993 2716 y(^) p Fw 1983 2737 a(I) p Fn 2026 2706 a(\025) p Fl 2019 2757 a(1) p Fx 2097 2737 a(and) p Fw 2258 2737 a(\037) p Fx(.) p Fd 2370 2737 a(\003) p Fx 639 2836 a(F) p 686 2836 a(urthermore) p 1134 2836 a(w) n(e) p 1269 2836 a(estimate) p 1615 2836 a(the) p 1771 2836 a(lo) n(w) n(er) p 2001 2836 a(b) r(ound) p 2270 2836 a(of) p Ft 2378 2836 a(h) p Fw(K) p Ft 2487 2836 a(i) p Fn 2519 2861 a(\037) p Fj 2560 2869 a(n) p Fn 2601 2861 a(P) p 2652 2861 a( ) p Fj 2696 2869 a(\013;\027) p Fx 2794 2836 a(.) p 2893 2836 a(Let) p 3055 2836 a(\001) p 3165 2836 a(b) r(e) p 3291 2836 a(an) 515 2936 y(in) n(terv) p 723 2936 a(al) p 814 2936 a(whic) n(h) p 1050 2936 a(con) n(tains) p 1374 2936 a(the) p 1516 2936 a(eigen) n(v) p 1738 2936 a(alue) p Fw 1912 2936 a(e) p Fx 1977 2936 a(of) p Fw 2070 2936 a(L) p Fx 2153 2936 a(corresp) r(onding) p 2685 2936 a(to) p Fw 2786 2936 a( ) p Fx 2843 2936 a(.) p 2902 2936 a(Supp) r(ose) p 3227 2936 a(that) 515 3035 y(\001) p 611 3035 a(con) n(tains) p 937 3035 a(exactly) p 1222 3035 a(one) p 1374 3035 a(eigen) n(v) p 1596 3035 a(alue) p 1774 3035 a(~) p Fw 1770 3035 a(e) p Fx 1836 3035 a(of) p Fw 1931 3035 a(L) p Fn 1988 3047 a(S) p Fx 2063 3035 a(i.e.,) p 2219 3035 a(\001) p Ft 2306 3035 a(\\) p Fw 2379 3035 a(\033) p Fx 2429 3035 a(\() p Fw(L) p Fn 2518 3047 a(S) p Fx 2566 3035 a(\)) p 2622 3035 a(=) p Ft 2709 3035 a(f) p Fx 2754 3035 a(~) p Fw 2751 3035 a(e) p Ft(g) p Fx(,) p 2882 3035 a(and) p 3046 3035 a(~) p Fw 3043 3035 a(e) p Fx 3109 3035 a(b) r(elongs) 515 3135 y(to) p 616 3135 a(the) p 759 3135 a(in) n(terior) p 1053 3135 a(of) p 1147 3135 a(\001.) p 1276 3135 a(Let) p Fw 1425 3135 a(\037) p Fl 1477 3147 a(\001) p Ft 1559 3135 a(2) p Fw 1637 3135 a(C) p Fo 1702 3105 a(1) p Fl 1696 3156 a(0) p Fx 1773 3135 a(\() p Fi(R) p Fx(\)) p 1925 3135 a(b) r(e) p 2038 3135 a(a) p 2107 3135 a(smo) r(oth) p 2400 3135 a(function) p 2726 3135 a(suc) n(h) p 2913 3135 a(that) 700 3295 y(0) p Ft 765 3295 a(\024) p Fw 852 3295 a(\037) p Fl 904 3307 a(\001) p Ft 986 3295 a(\024) p Fx 1074 3295 a(1) p Fw(;) p 1318 3295 a(\037) p Fl 1370 3307 a(\001) p Fx 1429 3295 a(\() p Fw(t) p Fx(\)) p 1546 3295 a(=) p 1634 3295 a(1) p 1759 3295 a(for) p Fw 1941 3295 a(t) p Ft 1994 3295 a(2) p Fx 2072 3295 a(\001) p Fw(;) p Fx 2345 3295 a(supp) p Fw(\037) p Fl 2568 3307 a(\001) p Ft 2645 3295 a(\\) p Fw 2719 3295 a(\033) p Fx 2769 3295 a(\() p Fw(L) p Fn 2858 3307 a(S) p Fx 2906 3295 a(\)) p 2961 3295 a(=) p Ft 3049 3295 a(f) p Fx 3094 3295 a(~) p Fw 3091 3295 a(e) p Ft 3129 3295 a(g) p Fw(;) p Fx 515 3454 a(and) p 676 3454 a(supp) r(ose) p 988 3454 a(that) p 1168 3454 a(the) p 1311 3454 a(function) p 1636 3454 a(0) p Ft 1701 3454 a(\024) p Fx 1798 3454 a(\026) p Fw 1789 3454 a(\037) p Fl 1841 3466 a(\001) p Ft 1922 3454 a(\024) p Fx 2010 3454 a(1) p 2079 3454 a(de\014ned) p 2365 3454 a(b) n(y) p Fw 1709 3614 a(\037) p Fl 1761 3580 a(2) 1761 3634 y(\001) p Fx 1838 3614 a(+) p 1931 3614 a(\026) p Fw 1921 3614 a(\037) p Fl 1973 3580 a(2) 1973 3634 y(\001) p Fx 2055 3614 a(=) p 2143 3614 a(1) 515 3773 y(b) r(elongs) p 816 3773 a(to) p Fw 920 3773 a(C) p Fo 985 3743 a(1) p Fn 979 3797 a(b) p Fx 1056 3773 a(\() p Fi(R) p Fx(\).) p 1249 3773 a(W) p 1327 3773 a(e) p 1395 3773 a(set) p Fw 1528 3773 a(\037) p Fl 1580 3785 a(\001) p Ft 1666 3773 a(\021) p Fw 1759 3773 a(\037) p Fl 1811 3785 a(\001) p Fx 1870 3773 a(\() p Fw(L) p Fx(\)) p 2021 3773 a(and) p 2196 3773 a(\026) p Fw 2186 3773 a(\037) p Fl 2238 3785 a(\001) p Ft 2325 3773 a(\021) p Fx 2427 3773 a(\026) p Fw 2417 3773 a(\037) p Fl 2469 3785 a(\001) p Fx 2528 3773 a(\() p Fw(L) p Fx(\).) p 2718 3773 a(Then) p 2938 3773 a(b) n(y) p 3056 3773 a(Theorem) 515 3873 y(7.1,) p 672 3873 a(if) p 748 3873 a(w) n(e) p 870 3873 a(tak) n(e) p Fw 1050 3873 a(\017) p Fx(,) p Fw 1134 3873 a(\022) p Fx 1203 3873 a(and) p Fw 1365 3873 a(\025) p Fx 1441 3873 a(suitably) p 1723 3873 a(,) p 1773 3873 a(w) n(e) p 1895 3873 a(ha) n(v) n(e) p Fw 920 4053 a(P) p 985 4053 a(\037) p Fl 1037 4065 a(\001) p Fx 1096 4053 a(\() p Fw(L) p Fx(\)) p 1231 4053 a([) p Fw(L;) p 1348 4053 a(A) p Fx(]) p Fw 1446 4053 a(\037) p Fl 1498 4065 a(\001) p Fx 1557 4053 a(\() p Fw(L) p Fx(\)) p Fw(P) p Ft 1766 4053 a(\025) p Fw 1853 4053 a(b) p Fl 1892 4065 a(~) p Fn 1889 4065 a(e) p Fw 1925 4053 a(P) p 1990 4053 a(\037) p Fl 2042 4065 a(\001) p Fx 2100 4053 a(\() p Fw(L) p Fx(\)) p Fk 2235 3961 a(\020) p Fx 2285 4053 a(1) p Ft 2345 4053 a(\000) p Fx 2428 4053 a(14) 2529 4032 y(~) p Fw 2512 4053 a(P) p Fl 2568 4065 a(~) p Fn 2565 4065 a(e) p Fk 2600 3961 a(\021) p Fw 2663 4053 a(\037) p Fl 2715 4065 a(\001) p Fx 2774 4053 a(\() p Fw(L) p Fx(\)) p Fw(P) r(:) p Fx 3231 4053 a(\(49\)) 515 4233 y(for) p 642 4233 a(some) p Fw 850 4233 a(b) p Fl 889 4245 a(~) p Fn 886 4245 a(e) p Fw 944 4233 a(>) p Fx 1032 4233 a(0.) p 1133 4233 a(W) p 1211 4233 a(e) p 1276 4233 a(ha) n(v) n(e) p 1468 4233 a(the) p 1611 4233 a(follo) n(wing) p 1961 4233 a(lemma.) p Fm 515 4380 a(Lemma) p 853 4380 a(8.6) p Fy 1017 4380 a(Supp) l(ose) p 1332 4380 a(the) p 1468 4380 a(Assumption) p 1924 4380 a(2.1) p 2062 4380 a(and) p 2223 4380 a(2.2) p 2361 4380 a(ar) l(e) p 2501 4380 a(satis\014e) l(d.) p 2851 4380 a(Then) p 3066 4380 a(for) p 3198 4380 a(su\016-) 515 4480 y(ciently) p 782 4480 a(smal) t(l) p Fw 1003 4480 a(\025) p Fy 1081 4480 a(we) p 1204 4480 a(have) p Ft 612 4685 a(h) p Fw(K) p Ft 721 4685 a(i) p Fn 753 4710 a(\037) p Fj 794 4718 a(n) p Fn 835 4710 a(P) p 886 4710 a( ) p Fj 930 4718 a(\013;\027) p Ft 1051 4685 a(\025) p Fw 1138 4685 a(b) p Fl 1177 4697 a(~) p Fn 1174 4697 a(e) p Fk 1223 4593 a(\020) p Ft 1273 4685 a(k) p Fw(\037) p Fl 1367 4697 a(\001) p Fw 1425 4685 a(P) p 1490 4685 a(\037) p Fn 1542 4697 a(n) p Fw 1587 4685 a( ) p Fn 1641 4697 a(\013\027) p Ft 1726 4685 a(k) p Fl 1768 4651 a(2) p Ft 1823 4685 a(\000) p Fx 1906 4685 a(14) p Ft(k) p Fx 2049 4664 a(~) p Fw 2032 4685 a(P) p Fl 2088 4697 a(~) p Fn 2085 4697 a(e) p Fw 2119 4685 a(\037) p Fl 2171 4697 a(\001) p Fw 2230 4685 a(P) p 2295 4685 a(\037) p Fn 2347 4697 a(n) p Fw 2392 4685 a( ) p Fn 2446 4697 a(\013\027) p Ft 2531 4685 a(k) p Fl 2573 4651 a(2) p Fk 2609 4593 a(\021) p Ft 1520 4847 a(\000) p 1603 4847 a(k) p Fw(I) p Fn 1688 4812 a(\025) p Fl 1681 4867 a(1) p Ft 1731 4847 a(k) p 1791 4847 a(\001) p 1833 4847 a(k) p Fw(P) p Fx 1949 4847 a(\026) p Fw 1940 4847 a(\037) p Fl 1992 4859 a(\001) p Fw 2050 4847 a(\037) p Fn 2102 4859 a(n) p Fw 2147 4847 a( ) p Fn 2201 4859 a(\013\027) p Ft 2285 4847 a(k) p Fl 2327 4812 a(2) p Ft 2382 4847 a(\000) p Fw 2465 4847 a(C) p Fl 2524 4859 a(3) p Fx 2562 4847 a(\() p Fw(n) p Fo 2644 4812 a(\000) p Fl(1) p Fx 2752 4847 a(+) p Fw 2835 4847 a(\013) p Fx 2907 4847 a(+) p Fw 2990 4847 a(\013n) p Fx 3111 4847 a(+) p Fw 3194 4847 a(\027) p Fx 3240 4847 a(\)) p Fw(:) p Fx 1905 5255 a(35) p eop %%Page: 36 36 36 35 bop Fy 515 523 a(Pr) l(o) l(of) p Fx 515 623 a(W) p 593 623 a(e) p 658 623 a(ha) n(v) n(e) p Fw 849 623 a(\037) p Fl 901 635 a(\001) p Fw 960 623 a(D) p Fx 1031 623 a(\() p Fw(N) p Fx 1139 623 a(\)) p Ft 1194 623 a(\032) p Fw 1282 623 a(D) p Fx 1353 623 a(\() p Fw(N) p Fx 1461 623 a(\)) p 1521 623 a(and) p 1682 623 a(the) p 1825 623 a(follo) n(wing) p 2176 623 a(decomp) r(osition) p 2723 623 a(is) p 2806 623 a(w) n(ell-de\014ned:) p Ft 905 802 a(h) p Fw 938 802 a(K) p Ft 1015 802 a(i) p Fn 1046 827 a(\037) p Fj 1087 835 a(n) p Fn 1129 827 a(P) p 1180 827 a( ) p Fj 1224 835 a(\013;\027) p Fx 1345 802 a(=) p Ft 1432 802 a(h) p Fw(\037) p Fl 1516 814 a(\001) p Fx 1589 802 a(\() p Fw(K) p Fx 1716 802 a(+) p 1799 802 a([) p Fw(L;) p 1916 802 a(b) p Fx(]\)) p Fw 2020 802 a(\037) p Fl 2072 814 a(\001) p Ft 2131 802 a(i) p Fn 2163 827 a(\037) p Fj 2204 835 a(n) p 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2638 1024 a(\035) p Fn 2699 1224 a(\037) p Fj 2740 1232 a(n) p Fn 2782 1224 a(P) p 2833 1224 a( ) p Fj 2877 1232 a(\013;\027) p Fx 3231 1141 a(\(52\)) p Ft 1340 1343 a(\000) p 1423 1343 a(h) p Fw(\037) p Fl 1507 1355 a(\001) p Fx 1566 1343 a([) p Fw(L;) p 1683 1343 a(b) p Fx(]) p Fw(\037) p Fl 1794 1355 a(\001) p Ft 1852 1343 a(i) p Fn 1884 1368 a(\037) p Fj 1925 1376 a(n) p Fn 1966 1368 a(P) p 2017 1368 a( ) p Fj 2061 1376 a(\013;\027) p Fx 3231 1343 a(\(53\)) 515 1534 y(The) p 685 1534 a(\014rst) p 857 1534 a(part) p 1037 1534 a(\(50\)) p 1212 1534 a(is) p 1296 1534 a(estimated) p 1674 1534 a(b) n(y) p 1790 1534 a(\(49\)) p 1965 1534 a(as) 598 1738 y(\(50\)) p 768 1738 a(=) p Ft 856 1738 a(h) p Fw(\037) p Fl 940 1750 a(\001) p Fx 1013 1738 a(\() p Fw(K) p Fx 1140 1738 a(+) p 1223 1738 a([) p Fw(L;) p 1340 1738 a(b) p Fx(]\)) p Fw 1444 1738 a(\037) p Fl 1496 1750 a(\001) p Ft 1555 1738 a(i) p Fn 1587 1763 a(\037) p Fj 1628 1771 a(n) p Fn 1669 1763 a(P) p 1720 1763 a( ) p Fj 1764 1771 a(\013;\027) p Fx 1885 1738 a(=) p Ft 1973 1738 a(h) p Fw(\037) p Fn 2057 1750 a(n) p Fw 2102 1738 a( ) p Fn 2156 1750 a(\013;\027) p Ft 2275 1738 a(j) p Fw(P) p 2363 1738 a(\037) p Fl 2415 1750 a(\001) p Fx 2473 1738 a([) p Fw(L;) p 2590 1738 a(A) p Fx(]) p Fw(\037) p Fl 2727 1750 a(\001) p Fw 2785 1738 a(P) p Ft 2850 1738 a(j) p Fw 2887 1738 a(\037) p Fn 2939 1750 a(n) p Fw 2984 1738 a( ) p Fn 3038 1750 a(\013;\027) p Ft 3143 1738 a(i) 1651 1908 y(\025) p Fw 1738 1908 a(b) p Fl 1777 1920 a(~) p Fn 1774 1920 a(e) p Fk 1824 1816 a(\020) p Ft 1873 1908 a(k) p Fw(\037) p Fl 1967 1920 a(\001) p Fw 2025 1908 a(P) p 2090 1908 a(\037) p Fn 2142 1920 a(n) p Fw 2187 1908 a( ) p Fn 2241 1920 a(\013\027) p Ft 2326 1908 a(k) p Fl 2368 1874 a(2) p Ft 2423 1908 a(\000) p Fx 2506 1908 a(14) p Ft(k) p Fx 2649 1887 a(~) p Fw 2632 1908 a(P) p Fl 2688 1920 a(~) p Fn 2685 1920 a(e) p Fw 2720 1908 a(\037) p Fl 2772 1920 a(\001) p Fw 2830 1908 a(P) p 2895 1908 a(\037) p Fn 2947 1920 a(n) p Fw 2992 1908 a( ) p Fn 3046 1920 a(\013\027) p Ft 3131 1908 a(k) p Fl 3173 1874 a(2) p Fk 3209 1816 a(\021) p Fw 3273 1908 a(:) p Fx 515 2112 a(The) p 685 2112 a(estimation) p 1092 2112 a(of) p 1186 2112 a(the) p 1329 2112 a(second) p 1597 2112 a(part) p 1777 2112 a(\(51\)) p 1953 2112 a(is) p 2036 2112 a(due) p 2193 2112 a(to) p Fw 2294 2112 a(S) p Fl 2345 2124 a(1) p Ft 2406 2112 a(\025) p Fx 2493 2112 a(0;) 526 2291 y(\(51\)) p 697 2291 a(=) p Fk 784 2224 a(\012) p Fx 833 2291 a(\026) p Fw 823 2291 a(\037) p Fl 875 2303 a(\001) p Fx 934 2291 a(\() p Fw(S) p Fl 1017 2303 a(1) p Fx 1073 2291 a(+) p Fw 1156 2291 a(I) p Fn 1199 2257 a(\025) p Fl 1192 2311 a(1) p Fx 1243 2291 a(\)) p 1285 2291 a(\026) p Fw 1275 2291 a(\037) p Fl 1327 2303 a(\001) p Fk 1386 2224 a(\013) p Fn 1425 2324 a(\037) p Fj 1466 2332 a(n) p Fn 1507 2324 a(P) p 1558 2324 a( ) p Fj 1602 2332 a(\013;\027) p Ft 1723 2291 a(\025) p Fk 1810 2224 a(\012) p Fx 1860 2291 a(\026) p Fw 1850 2291 a(\037) p Fl 1902 2303 a(\001) p Fw 1960 2291 a(I) p Fn 2003 2257 a(\025) p Fl 1996 2311 a(1) p Fx 2057 2291 a(\026) p Fw 2047 2291 a(\037) p Fl 2099 2303 a(\001) p Fk 2157 2224 a(\013) p Fn 2197 2324 a(\037) p Fj 2238 2332 a(n) p Fn 2279 2324 a(P) p 2330 2324 a( ) p Fj 2374 2332 a(\013;\027) p Ft 2495 2291 a(\025) p 2582 2291 a(\000k) p Fw(I) p Fn 2732 2257 a(\025) p Fl 2725 2311 a(1) p Ft 2775 2291 a(kk) p Fw(P) p Fx 2933 2291 a(\026) p Fw 2924 2291 a(\037) p Fl 2976 2303 a(\001) p Fw 3034 2291 a(\037) p Fn 3086 2303 a(n) p Fw 3131 2291 a( ) p Fn 3185 2303 a(\013;\027) p Ft 3289 2291 a(k) p Fl 3331 2257 a(2) p Fx 515 2490 a(No) n(w) p 704 2490 a(w) n(e) p 826 2490 a(estimate) p 1159 2490 a(the) p 1302 2490 a(third) p 1509 2490 a(part) p 1689 2490 a(\(52\).) p 1897 2490 a(Note) p 2098 2490 a(that) p Fk 515 2602 a(\034) p Fx 587 2663 a(1) p 587 2700 V 587 2776 a(2) 638 2719 y([) p 671 2719 a(\026) p Fw 661 2719 a(\037) p Fl 713 2731 a(\001) p Fw 772 2719 a(;) p Fx 809 2719 a([) p 842 2719 a(\026) p Fw 832 2719 a(\037) p Fl 884 2731 a(\001) p Fw 943 2719 a(;) p 980 2719 a(K) p Fx 1057 2719 a(]]) p Fk 1103 2602 a(\035) p Fn 1164 2802 a(\037) p Fj 1205 2810 a(n) p Fn 1247 2802 a(P) p 1298 2802 a( ) p Fj 1342 2810 a(\013;\027) p Fx 1463 2719 a(=) 1560 2663 y(1) p 1560 2700 V 1560 2776 a(2) p Ft 1626 2719 a(h) p Fx 1668 2719 a(\026) p Fw 1658 2719 a(\037) p Fl 1710 2731 a(\001) p Fx 1768 2719 a([) p 1801 2719 a(\026) p Fw 1791 2719 a(\037) p Fl 1843 2731 a(\001) p Fw 1902 2719 a(;) p 1939 2719 a(K) p Fx 2016 2719 a(]) p Ft 2057 2719 a(\000) p Fx 2140 2719 a([) p 2173 2719 a(\026) p Fw 2163 2719 a(\037) p Fl 2215 2731 a(\001) p Fw 2274 2719 a(;) p 2311 2719 a(K) p Fx 2388 2719 a(]) p 2421 2719 a(\026) p Fw 2411 2719 a(\037) p Fl 2463 2731 a(\001) p Ft 2521 2719 a(i) p Fn 2553 2744 a(\037) p Fj 2594 2752 a(n) p Fn 2635 2744 a(P) p 2686 2744 a( ) p Fj 2730 2752 a(\013;\027) p Fx 2851 2719 a(=) p 2939 2719 a(Re) p Ft 3051 2719 a(h) p Fx 3093 2719 a(\026) p Fw 3083 2719 a(\037) p Fl 3135 2731 a(\001) p Fx 3194 2719 a([) p 3227 2719 a(\026) p Fw 3217 2719 a(\037) p Fl 3269 2731 a(\001) p Fw 3327 2719 a(;) p 3364 2719 a(K) p Fx 3441 2719 a(]) p Ft(i) p Fn 3496 2744 a(\037) p Fj 3537 2752 a(n) p Fn 3578 2744 a(P) p 3629 2744 a( ) p Fj 3673 2752 a(\013;\027) p Fx 3245 2897 a(\(54\)) 515 3072 y(As) p Fw 637 3072 a(N) p Fl 704 3084 a(\003) p Fj 749 3068 a(c) 749 3101 y(v) p Fx 817 3072 a(strongly) p 1138 3072 a(comm) n(utes) p 1528 3072 a(with) p Fw 1718 3072 a(N) p Fx 1794 3072 a(,) p Fw 1844 3072 a(L) p Fx(,) p 1951 3072 a(and) p Fw 2113 3072 a(K) p Fx 2190 3072 a(,) p 2240 3072 a(w) n(e) p 2362 3072 a(ha) n(v) n(e) p Ft 515 3251 a(h) p Fx 557 3251 a(\026) p Fw 547 3251 a(\037) p Fl 599 3263 a(\001) p Fx 658 3251 a([) p 691 3251 a(\026) p Fw 681 3251 a(\037) p Fl 733 3263 a(\001) p Fw 791 3251 a(;) p 828 3251 a(K) p Fx 905 3251 a(]) p Ft(i) p Fn 960 3276 a(\037) p Fj 1001 3284 a(n) p Fn 1042 3276 a(P) p 1093 3276 a( ) p Fj 1137 3284 a(\013;\027) p Fx 1258 3251 a(=) p Ft 1346 3251 a(h) p Fw(\037) p Fn 1430 3263 a(n) p Fw 1475 3251 a(P) p 1540 3251 a( ) p Fn 1594 3263 a(\013;\027) p Ft 1712 3251 a(j) p Fx 1745 3251 a(\026) p Fw 1735 3251 a(\037) p Fl 1787 3263 a(\001) p Fx 1846 3251 a([) p 1879 3251 a(\026) p Fw 1869 3251 a(\037) p Fl 1921 3263 a(\001) p Fw 1980 3251 a(;) p 2017 3251 a(K) p Fx 2094 3251 a(]) p Ft(j) p Fw 2153 3251 a(\037) p Fn 2205 3263 a(n) p Fw 2250 3251 a(P) p 2315 3251 a( ) p Fn 2369 3263 a(\013;\027) p Ft 2473 3251 a(i) p Fx 2528 3251 a(=) p Ft 2616 3251 a(h) p Fw(\037) p Fn 2700 3263 a(n) p Fw 2746 3251 a( ) p Fn 2800 3263 a(\013;\027) p Ft 2918 3251 a(j) p Fx 2951 3251 a(\026) p Fw 2941 3251 a(\037) p Fl 2993 3263 a(\001) p Fw 3051 3251 a(P) p Fx 3116 3251 a([) p 3149 3251 a(\026) p Fw 3139 3251 a(\037) p Fl 3191 3263 a(\001) p Fw 3250 3251 a(;) p 3287 3251 a(K) p Fx 3364 3251 a(]) p Ft(j) p Fw 3423 3251 a(\037) p Fn 3475 3263 a(n) p Fw 3520 3251 a( ) p Fn 3574 3263 a(\013;\027) p Ft 3679 3251 a(i) p Fw 3725 3251 a(;) p Fx 515 3430 a(and) p 676 3430 a(get) p 815 3430 a(the) p 958 3430 a(b) r(ound) p Fk 926 3533 a(\014) 926 3583 y(\014) 926 3633 y(\014) p Ft 954 3629 a(h) p Fx 996 3629 a(\026) p Fw 986 3629 a(\037) p Fl 1038 3641 a(\001) p Fx 1097 3629 a([) p 1130 3629 a(\026) p Fw 1120 3629 a(\037) p Fl 1172 3641 a(\001) p Fw 1231 3629 a(;) p 1268 3629 a(K) p Fx 1345 3629 a(]) p Ft(i) p Fn 1399 3654 a(\037) p Fj 1440 3662 a(n) p Fn 1482 3654 a(P) p 1533 3654 a( ) p Fj 1577 3662 a(\013;\027) p Fk 1674 3533 a(\014) 1674 3583 y(\014) 1674 3633 y(\014) p Ft 1725 3629 a(\024) p 1813 3629 a(k) p Fx 1864 3629 a(\026) p Fw 1854 3629 a(\037) p Fl 1906 3641 a(\001) p Fw 1965 3629 a(\037) p Fn 2017 3641 a(n) p Fw 2062 3629 a( ) p Fn 2116 3641 a(\013;\027) p Ft 2220 3629 a(k) p 2280 3629 a(\001) p 2322 3629 a(k) p Fx 2363 3629 a([) p 2396 3629 a(\026) p Fw 2386 3629 a(\037) p Fl 2438 3641 a(\001) p Fw 2497 3629 a(;) p 2534 3629 a(K) p Fx 2611 3629 a(]) p Fw(\037) p Fn 2686 3641 a(n) p Fw 2731 3629 a( ) p Fn 2785 3641 a(\013;\027) p Ft 2889 3629 a(k) p Fw 2944 3629 a(:) p Fx 515 3833 a(Let) p 666 3833 a(us) p 776 3833 a(estimate) p Ft 1111 3833 a(k) p Fx 1163 3833 a(\026) p Fw 1153 3833 a(\037) p Fl 1205 3845 a(\001) p Fw 1263 3833 a(\037) p Fn 1315 3845 a(n) p Fw 1360 3833 a( ) p Fn 1414 3845 a(\013;\027) p Ft 1519 3833 a(k) p Fx(.) p 1628 3833 a(As) p Fw 1754 3833 a(L ) p Fx 1895 3833 a(=) p Fw 1987 3833 a(e ) p Fx 2114 3833 a(and) p 2288 3833 a(\026) p Fw 2278 3833 a(\037) p Fl 2330 3845 a(\001) p Fx 2388 3833 a(\() p Fw(e) p Fx(\)) p 2519 3833 a(=) p 2612 3833 a(0,) p 2707 3833 a(w) n(e) p 2833 3833 a(ha) n(v) n(e) p 3037 3833 a(\026) p Fw 3027 3833 a(\037) p Fl 3079 3845 a(\001) p Fw 3137 3833 a( ) p Fx 3222 3833 a(=) p 3314 3833 a(0.) 515 3932 y(So) p 630 3932 a(w) n(e) p 752 3932 a(get) 819 4111 y(\026) p Fw 810 4111 a(\037) p Fl 862 4123 a(\001) p Fw 920 4111 a(\037) p Fn 972 4123 a(n) p Fw 1017 4111 a( ) p Fn 1071 4123 a(\013;\027) p Fx 1199 4111 a(=) p 1286 4111 a([) p 1319 4111 a(\026) p Fw 1309 4111 a(\037) p Fl 1361 4123 a(\001) p Fw 1420 4111 a(;) p 1457 4111 a(\037) p Fn 1509 4123 a(n) p Fx 1554 4111 a(]) p Fw(h) p Fn 1625 4123 a(\013) p Fw 1672 4111 a(g) p Fn 1712 4123 a(\027) p Fw 1753 4111 a( ) p Fx 1829 4111 a(+) p Fw 1912 4111 a(\037) p Fn 1964 4123 a(n) p Fx 2009 4111 a([) p 2042 4111 a(\026) p Fw 2032 4111 a(\037) p Fl 2084 4123 a(\001) p Fw 2143 4111 a(;) p 2180 4111 a(h) p Fn 2228 4123 a(\013) p Fx 2275 4111 a(]) p Fw(g) p Fn 2338 4123 a(\027) p Fw 2379 4111 a( ) p Fx 2454 4111 a(+) p Fw 2537 4111 a(\037) p Fn 2589 4123 a(n) p Fw 2634 4111 a(h) p Fn 2682 4123 a(\013) p Fx 2729 4111 a([) p 2762 4111 a(\026) p Fw 2752 4111 a(\037) p Fl 2804 4123 a(\001) p Fw 2863 4111 a(;) p 2900 4111 a(g) p Fn 2940 4123 a(\027) p Fx 2981 4111 a(]) p Fw( ) s(:) p Fx 515 4290 a(By) p 645 4290 a(op) r(erator) p 980 4290 a(calculous,) p 1359 4290 a(w) n(e) p 1481 4290 a(ha) n(v) n(e) p Ft 565 4469 a(k) p Fx 606 4469 a([) p 639 4469 a(\026) p Fw 629 4469 a(\037) p Fl 681 4481 a(\001) p Fw 740 4469 a(;) p 777 4469 a(\037) p Fn 829 4481 a(n) p Fx 874 4469 a(]) p Ft(k) p 961 4469 a(\024) p Fw 1049 4469 a(n) p Fo 1099 4435 a(\000) p Fl(1) p Ft 1206 4469 a(\001) p Fw 1248 4469 a(c) p Fo 1284 4435 a(0) p Fl 1284 4490 a(2) p Fw 1321 4469 a(;) p Ft 1404 4469 a(k) p Fw(\037) p Fn 1498 4481 a(n) p Fx 1543 4469 a([) p 1576 4469 a(\026) p Fw 1566 4469 a(\037) p Fl 1618 4481 a(\001) p Fw 1677 4469 a(;) p 1714 4469 a(h) p Fn 1762 4481 a(\013) p Fx 1809 4469 a(]) p Ft(k) p 1896 4469 a(\024) p Fw 1984 4469 a(c) p Fo 2020 4435 a(0) p Fl 2020 4490 a(3) p Ft 2075 4469 a(\001) p Fw 2117 4469 a(\013) p Fx(\(1) p 2263 4469 a(+) p Fw 2346 4469 a(n) p Fx(\)) p Fw(;) p Ft 2511 4469 a(k) p Fx 2552 4469 a([) p 2585 4469 a(\026) p Fw 2575 4469 a(\037) p Fl 2627 4481 a(\001) p Fw 2686 4469 a(;) p 2723 4469 a(g) p Fn 2763 4481 a(\027) p Fx 2804 4469 a(]) p Ft(k) p 2891 4469 a(\024) p Fw 2979 4469 a(\027) p Ft 3044 4469 a(\001) p Fw 3085 4469 a(c) p Fo 3121 4435 a(0) p Fl 3121 4490 a(4) p Fw 3159 4469 a(;) p Fx 3245 4469 a(\(55\)) 515 4648 y(and) p 676 4648 a(hence) p 907 4648 a(w) n(e) p 1029 4648 a(obtain) p Ft 1263 4827 a(k) p Fx 1315 4827 a(\026) p Fw 1305 4827 a(\037) p Fl 1357 4839 a(\001) p Fw 1416 4827 a(\037) p Fn 1468 4839 a(n) p Fw 1513 4827 a( ) p Fn 1567 4839 a(\013;\027) p Ft 1671 4827 a(k) p 1736 4827 a(\024) p Fw 1823 4827 a(c) p Fo 1859 4793 a(0) p Fl 1859 4848 a(5) p Fx 1896 4827 a(\() p Fw(n) p Fo 1978 4793 a(\000) p Fl(1) p Fx 2086 4827 a(+) p Fw 2169 4827 a(\013) p Fx 2241 4827 a(+) p Fw 2324 4827 a(\013n) p Fx 2446 4827 a(+) p Fw 2529 4827 a(\027) p Fx 2575 4827 a(\)) p Fw(;) p Fx 515 5006 a(where) p Fw 755 5006 a(c) p Fo 791 4976 a(0) p Fl 791 5027 a(5) p Fx 856 5006 a(is) p 939 5006 a(a) p 1008 5006 a(p) r(ositiv) n(e) p 1316 5006 a(constan) n(t.) 1905 5255 y(36) p eop %%Page: 37 37 37 36 bop Fx 639 523 a(On) p 778 523 a(the) p 921 523 a(other) p 1138 523 a(hand,) p Ft 1368 523 a(k) p Fx([) p 1443 523 a(\026) p Fw 1433 523 a(\037) p Fl 1485 535 a(\001) p Fw 1543 523 a(;) p 1580 523 a(K) p Fx 1657 523 a(]) p Fw(\037) p Fn 1732 535 a(n) p Fw 1777 523 a( ) p Fn 1831 535 a(\013;\027) p Ft 1935 523 a(k) p Fx 2004 523 a(is) p 2088 523 a(estimated) p 2467 523 a(as) p Ft 1539 693 a(k) p Fx([) p 1614 693 a(\026) p Fw 1604 693 a(\037) p Fl 1656 705 a(\001) p Fw 1714 693 a(;) p 1751 693 a(K) p Fx 1828 693 a(]) p Fw(\037) p Fn 1903 705 a(n) p Fw 1948 693 a( ) p Fn 2002 705 a(\013;\027) p Ft 2106 693 a(k) p 2171 693 a(\024) p Fw 2258 693 a(c) p Fo 2294 659 a(0) p Fl 2294 714 a(6) p Fw 2332 693 a(:) p Fx 515 864 a(Hence) p 761 864 a(w) n(e) p 884 864 a(get) p Fk 515 962 a(\034) p Fx 587 1023 a(1) p 587 1060 42 4 v 587 1136 a(2) 638 1079 y([) p 671 1079 a(\026) p Fw 661 1079 a(\037) p Fl 713 1091 a(\001) p Fw 772 1079 a(;) p Fx 809 1079 a([) p 842 1079 a(\026) p Fw 832 1079 a(\037) p Fl 884 1091 a(\001) p Fw 943 1079 a(;) p 980 1079 a(K) p Fx 1057 1079 a(]]) p Fk 1103 962 a(\035) p Fn 1164 1162 a(\037) p Fj 1205 1170 a(n) p Fn 1247 1162 a(P) p 1298 1162 a( ) p Fj 1342 1170 a(\013;\027) p Fx 1463 1079 a(=) p 1550 1079 a(Re) p Ft 1662 1079 a(h) p Fx 1704 1079 a(\026) p Fw 1694 1079 a(\037) p Fl 1746 1091 a(\001) p Fx 1805 1079 a([) p 1838 1079 a(\026) p Fw 1828 1079 a(\037) p Fl 1880 1091 a(\001) p Fw 1939 1079 a(;) p 1976 1079 a(K) p Fx 2053 1079 a(]) p Ft(i) p Fn 2107 1104 a(\037) p Fj 2148 1112 a(n) p Fn 2190 1104 a(P) p 2241 1104 a( ) p Fj 2285 1112 a(\013;\027) p Ft 2406 1079 a(\025) p 2493 1079 a(\000) p 2572 1079 a(k) p Fx 2623 1079 a(\026) p Fw 2613 1079 a(\037) p Fl 2665 1091 a(\001) p Fw 2724 1079 a(\037) p Fn 2776 1091 a(n) p Fw 2821 1079 a( ) p Fn 2875 1091 a(\013;\027) p Ft 2979 1079 a(k) p 3039 1079 a(\001) p 3081 1079 a(k) p Fx 3122 1079 a([) p 3155 1079 a(\026) p Fw 3145 1079 a(\037) p Fl 3197 1091 a(\001) p Fw 3256 1079 a(;) p 3293 1079 a(K) p Fx 3370 1079 a(]) p Fw(\037) p Fn 3445 1091 a(n) p Fw 3489 1079 a( ) p Fn 3543 1091 a(\013;\027) p Ft 3648 1079 a(k) 1779 1307 y(\025) p 1866 1307 a(\000) p Fk 1945 1212 a(\014) 1945 1261 y(\014) 1945 1311 y(\014) p Ft 1972 1307 a(h) p Fx 2015 1307 a(\026) p Fw 2005 1307 a(\037) p Fl 2057 1319 a(\001) p Fx 2115 1307 a([) p 2148 1307 a(\026) p Fw 2138 1307 a(\037) p Fl 2190 1319 a(\001) p Fw 2249 1307 a(;) p 2286 1307 a(K) p Fx 2363 1307 a(]) p Ft(i) p Fn 2418 1332 a(\037) p Fj 2459 1340 a(n) p Fn 2500 1332 a(P) p 2551 1332 a( ) p Fj 2595 1340 a(\013;\027) p Fk 2693 1212 a(\014) 2693 1261 y(\014) 2693 1311 y(\014) p Ft 2744 1307 a(\025) p 2831 1307 a(\000) p Fw(c) p Fo 2932 1273 a(0) p Fl 2932 1328 a(7) p Fx 2969 1307 a(\() p Fw(n) p Fo 3051 1273 a(\000) p Fl(1) p Fx 3159 1307 a(+) p Fw 3242 1307 a(\013) p Fx 3314 1307 a(+) p Fw 3397 1307 a(\013n) p Fx 3518 1307 a(+) p Fw 3601 1307 a(\027) p Fx 3647 1307 a(\)) p Fw(:) p Fx 515 1510 a(The) p 688 1510 a(same) p 900 1510 a(argumen) n(t) p 1274 1510 a(is) p 1361 1510 a(adaptable) p 1747 1510 a(for) p Fk 1877 1443 a(\012) p Fl 1926 1478 a(1) p 1926 1492 34 4 v 1926 1539 a(2) p Fx 1969 1510 a([) p Fw(\037) p Fl 2044 1522 a(\001) p Fw 2103 1510 a(;) p Fx 2140 1510 a([) p Fw(\037) p Fl 2215 1522 a(\001) p Fw 2273 1510 a(;) p 2310 1510 a(K) p Fx 2387 1510 a(]]) p Fk 2433 1443 a(\013) p Fx 2472 1510 a(,) p 2527 1510 a(and) p 2691 1510 a(the) p 2837 1510 a(third) p 3048 1510 a(part) p 3231 1510 a(\(52\)) 515 1610 y(is) p 598 1610 a(b) r(ounded) p 937 1610 a(b) n(y) 515 1826 y(\(52\)) p 685 1826 a(=) p Fk 773 1708 a(\034) p Fx 845 1769 a(1) p 845 1806 42 4 v 845 1882 a(2) 897 1826 y([) p Fw(\037) p Fl 972 1838 a(\001) p Fw 1030 1826 a(;) p Fx 1067 1826 a([) p Fw(\037) p Fl 1142 1838 a(\001) p Fw 1201 1826 a(;) p 1238 1826 a(K) p Fx 1315 1826 a(]]) p 1379 1826 a(+) 1472 1769 y(1) p 1472 1806 V 1472 1882 a(2) 1523 1826 y([) p 1556 1826 a(\026) p Fw 1546 1826 a(\037) p Fl 1598 1838 a(\001) p Fw 1657 1826 a(;) p Fx 1694 1826 a([) p 1727 1826 a(\026) p Fw 1717 1826 a(\037) p Fl 1769 1838 a(\001) p Fw 1828 1826 a(;) p 1865 1826 a(K) p Fx 1942 1826 a(]]) p Fk 1988 1708 a(\035) p Fn 2049 1909 a(\037) p Fj 2090 1917 a(n) p Fn 2132 1909 a(P) p 2183 1909 a( ) p Fj 2227 1917 a(\013;\027) p Ft 2348 1826 a(\025) p 2435 1826 a(\000) p Fw(C) p Fo 2565 1791 a(0) p Fl 2559 1846 a(3) p Ft 2615 1826 a(\001) p Fx 2656 1826 a(\() p Fw(n) p Fo 2738 1791 a(\000) p Fl(1) p Fx 2846 1826 a(+) p Fw 2929 1826 a(\013) p Fx 3001 1826 a(+) p Fw 3084 1826 a(\013n) p Fx 3206 1826 a(+) p Fw 3289 1826 a(\027) p Fx 3335 1826 a(\)) p Fw(:) p Fx 515 2066 a(The) p 685 2066 a(fourth) p 937 2066 a(term) p 1136 2066 a(\(53\)is) p 1367 2066 a(b) r(ounded) p 1706 2066 a(b) n(y) p Fk 612 2191 a(\014) 612 2240 y(\014) 612 2290 y(\014) p Ft 639 2286 a(h) p Fw 672 2286 a(\037) p Fl 724 2298 a(\001) p Fx 782 2286 a([) p Fw(L;) p 899 2286 a(b) p Fx(]) p Fw(\037) p Fl 1010 2298 a(\001) p Ft 1068 2286 a(i) p Fn 1100 2311 a(\037) p Fj 1141 2319 a(n) p Fn 1182 2311 a(P) p 1233 2311 a( ) p Fj 1277 2319 a(\013;\027) p Fk 1375 2191 a(\014) 1375 2240 y(\014) 1375 2290 y(\014) p Fx 1426 2286 a(=) p Fk 1514 2191 a(\014) 1514 2240 y(\014) 1514 2290 y(\014) p Ft 1541 2286 a(h) p Fw 1574 2286 a(\037) p Fl 1626 2298 a(\001) p Fx 1684 2286 a([) p Fw(L) p Ft 1782 2286 a(\000) p Fw 1865 2286 a(e;) p 1941 2286 a(b) p Fx(]) p Fw(\037) p Fl 2052 2298 a(\001) p Ft 2110 2286 a(i) p Fn 2143 2311 a(\037) p Fj 2184 2319 a(n) p Fn 2225 2311 a(P) p 2276 2311 a( ) p Fj 2320 2319 a(\013;\027) p Fk 2418 2191 a(\014) 2418 2240 y(\014) 2418 2290 y(\014) p Ft 1847 2443 a(\024) p Fx 1935 2443 a(2) p Ft(k) p Fx(\() p Fw(L) p Ft 2125 2443 a(\000) p Fw 2208 2443 a(e) p Fx(\)) p Fw(\037) p Fl 2331 2455 a(\001) p Fw 2390 2443 a(\037) p Fn 2442 2455 a(n) p Fw 2487 2443 a( ) p Fn 2541 2455 a(\013;\027) p Ft 2645 2443 a(k) p 2705 2443 a(\001) p 2746 2443 a(k) p Fw(b\037) p Fl 2876 2455 a(\001) p Fw 2934 2443 a(\037) p Fn 2986 2455 a(n) p Fw 3031 2443 a(P) p 3096 2443 a( ) p Fn 3150 2455 a(\013;\027) p Ft 3254 2443 a(k) p Fx 515 2613 a(W) p 593 2613 a(e) p 658 2613 a(estimate) p Ft 990 2613 a(k) p Fx(\() p Fw(L) p Ft 1139 2613 a(\000) p Fw 1222 2613 a(e) p Fx(\)) p Fw(\037) p Fl 1345 2625 a(\001) p Fw 1404 2613 a(\037) p Fn 1456 2625 a(n) p Fw 1501 2613 a( ) p Fn 1555 2625 a(\013;\027) p Ft 1659 2613 a(k) p Fx(.) p 1760 2613 a(W) p 1838 2613 a(e) p 1903 2613 a(ha) n(v) n(e) 632 2784 y(\() p Fw(L) p Ft 739 2784 a(\000) p Fw 822 2784 a(e) p Fx(\)) p Fw(\037) p Fl 945 2796 a(\001) p Fw 1004 2784 a(\037) p Fn 1056 2796 a(n) p Fw 1101 2784 a( ) p Fn 1155 2796 a(\013;\027) p Fx 1282 2784 a(=) p Fw 1370 2784 a(\037) p Fl 1422 2796 a(\001) p Fx 1480 2784 a([) p Fw(L;) p 1597 2784 a(\037) p Fn 1649 2796 a(n) p Fx 1694 2784 a(]) p Fw( ) p Fn 1771 2796 a(\013;\027) p Fx 1894 2784 a(+) p Fw 1977 2784 a(\037) p Fl 2029 2796 a(\001) p Fw 2087 2784 a(\037) p Fn 2139 2796 a(n) p Fx 2184 2784 a([) p Fw(L;) p 2301 2784 a(h) p Fn 2349 2796 a(\013) p Fx 2396 2784 a(]) p Fw(g) p Fn 2459 2796 a(\027) p Fw 2500 2784 a( ) p Fx 2576 2784 a(+) p Fw 2659 2784 a(\037) p Fl 2711 2796 a(\001) p Fw 2769 2784 a(\037) p Fn 2821 2796 a(n) p Fw 2866 2784 a(h) p Fn 2914 2796 a(\013) p Fx 2961 2784 a([) p Fw(L;) p 3078 2784 a(g) p Fn 3118 2796 a(\027) p Fx 3159 2784 a(]) p Fw( ) s(:) p Fx 515 2954 a(By) p 645 2954 a(op) r(erator) p 980 2954 a(calculous) p 1336 2954 a(w) n(e) p 1458 2954 a(get) p Ft 762 3124 a(k) p Fx([) p Fw(L;) p 921 3124 a(\037) p Fn 973 3136 a(n) p Fx 1017 3124 a(]) p Ft(k) p 1105 3124 a(\024) p Fw 1192 3124 a(n) p Fo 1242 3090 a(\000) p Fl(1) p Fw 1331 3124 a(c) p Ff 1367 3065 a(0) q(0) p Fl 1367 3145 a(1) p Fw 1412 3124 a(;) p Ft 1495 3124 a(k) p Fx([) p Fw(L;) p 1654 3124 a(g) p Fn 1694 3136 a(\027) p Fx 1734 3124 a(]) p Ft(k) p 1822 3124 a(\024) p Fw 1909 3124 a(\027) p 1955 3124 a(c) p Ff 1991 3065 a(0) q(0) p Fl 1991 3145 a(2) p Fw 2037 3124 a(;) p Ft 2120 3124 a(k) p Fw(\037) p Fn 2214 3136 a(n) p Fx 2258 3124 a([) p Fw(L;) p 2375 3124 a(h) p Fn 2423 3136 a(\013) p Fx 2470 3124 a(]) p Ft(k) p 2557 3124 a(\024) p Fw 2645 3124 a(c) p Ff 2681 3065 a(00) p Fl 2681 3145 a(3) p Fx 2726 3124 a(\() p Fw(\013) p Ft 2830 3124 a(\001) p Fw 2872 3124 a(n) p Fx 2940 3124 a(+) p Fw 3023 3124 a(\013) p Fx(\)) p Fw(;) p Fx 515 3294 a(and) p 676 3294 a(obtain) p Ft 1067 3464 a(k) p Fx(\() p Fw(L) p Ft 1215 3464 a(\000) p Fw 1298 3464 a(e) p Fx(\)) p Fw(\037) p Fl 1421 3476 a(\001) p Fw 1480 3464 a(\037) p Fn 1532 3476 a(n) p Fw 1577 3464 a(P) p 1642 3464 a( ) p Fn 1696 3476 a(\013;\027) p Ft 1800 3464 a(k) p 1865 3464 a(\024) p Fw 1952 3464 a(c) p Ff 1988 3405 a(0) q(0) p Fl 1988 3485 a(4) p Fx 2033 3464 a(\() p Fw(n) p Fo 2115 3430 a(\000) p Fl(1) p Fx 2223 3464 a(+) p Fw 2306 3464 a(\027) p Fx 2371 3464 a(+) p Fw 2454 3464 a(\013) p Ft 2526 3464 a(\001) p Fw 2567 3464 a(n) p Fx 2635 3464 a(+) p Fw 2718 3464 a(\013) p Fx(\)) p Fw(:) p Fx 515 3651 a(On) p 650 3651 a(the) p 790 3651 a(other) p 1004 3651 a(hand,) p 1232 3651 a(w) n(e) p 1351 3651 a(ha) n(v) n(e) p Ft 1540 3651 a(k) p Fw(b\037) p Fl 1670 3663 a(\001) p Fw 1728 3651 a(\037) p Fn 1780 3663 a(n) p Fw 1825 3651 a(P) p 1890 3651 a( ) p Fn 1944 3663 a(\013;\027) p Ft 2048 3651 a(k) p 2112 3651 a(\024) p Fw 2200 3651 a(c) p Ff 2236 3596 a(00) p Fl 2236 3672 a(5) p Fx 2281 3651 a(,) p 2329 3651 a(since) p Fw 2530 3651 a(b) p Fx 2590 3651 a(is) p 2670 3651 a(b) r(ounded.) p 3041 3651 a(Hence) p 3284 3651 a(w) n(e) 515 3751 y(obtain) p Fk 1044 3826 a(\014) 1044 3876 y(\014) 1044 3925 y(\014) p Ft 1071 3921 a(h) p Fw 1104 3921 a(\037) p Fl 1156 3933 a(\001) p Fx 1214 3921 a([) p Fw(L;) p 1331 3921 a(b) p Fx(]) p Fw(\037) p Fl 1442 3933 a(\001) p Ft 1500 3921 a(i) p Fn 1532 3946 a(\037) p Fj 1573 3954 a(n) p Fn 1615 3946 a(P) p 1666 3946 a( ) p Fj 1710 3954 a(\013;\027) p Fk 1807 3826 a(\014) 1807 3876 y(\014) 1807 3925 y(\014) p Ft 1858 3921 a(\024) p Fw 1946 3921 a(C) p Ff 2011 3862 a(00) p Fl 2005 3942 a(3) p Fx 2056 3921 a(\() p Fw(n) p Fo 2138 3887 a(\000) p Fl(1) p Fx 2246 3921 a(+) p Fw 2329 3921 a(\027) p Fx 2394 3921 a(+) p Fw 2477 3921 a(\013) p Ft 2548 3921 a(\001) p Fw 2590 3921 a(n) p Fx 2658 3921 a(+) p Fw 2741 3921 a(\013) p Fx(\)) p Fw(;) p Fx 515 4112 a(and) p 676 4112 a(get) 882 4282 y(\(53\)) p 1053 4282 a(=) p Ft 1141 4282 a(\000) p 1220 4282 a(h) p Fw 1251 4282 a(\037) p Fl 1303 4294 a(\001) p Fx 1362 4282 a([) p Fw(L;) p 1479 4282 a(b) p Fx(]) p Fw(\037) p Fl 1590 4294 a(\001) p Ft 1648 4282 a(i) p Fn 1680 4307 a(\037) p Fj 1721 4315 a(n) p Fn 1762 4307 a(P) p 1813 4307 a( ) p Fj 1857 4315 a(\013;\027) p Ft 1978 4282 a(\025) p 2066 4282 a(\000) p Fw(C) p Ff 2196 4223 a(00) p Fl 2190 4303 a(3) p Fx 2241 4282 a(\() p Fw(n) p Fo 2323 4248 a(\000) p Fl(1) p Fx 2430 4282 a(+) p Fw 2513 4282 a(\027) p Fx 2578 4282 a(+) p Fw 2661 4282 a(\013) p Ft 2733 4282 a(\001) p Fw 2774 4282 a(n) p Fx 2843 4282 a(+) p Fw 2926 4282 a(\013) p Fx(\)) 515 4452 y(Gathering) p 908 4452 a(the) p 1051 4452 a(estimations,) p 1513 4452 a(w) n(e) p 1635 4452 a(obtain) p Ft 612 4647 a(h) p Fw(K) p Ft 721 4647 a(i) p Fn 753 4672 a(\037) p Fj 794 4680 a(n) p Fn 835 4672 a(P) p 886 4672 a( ) p Fj 930 4680 a(\013;\027) p Fx 1051 4647 a(=) p 1138 4647 a(\(50\)) p 1304 4647 a(+) p 1387 4647 a(\(51\)) p 1553 4647 a(+) p 1637 4647 a(\(52\)) p 1803 4647 a(+) p 1886 4647 a(\(53\)) p Ft 743 4817 a(\025) p Fw 831 4817 a(b) p Fl 870 4829 a(~) p Fn 867 4829 a(e) p Fk 916 4725 a(\020) p Ft 966 4817 a(k) p Fw(\037) p Fl 1060 4829 a(\001) p Fw 1118 4817 a(P) p 1183 4817 a(\037) p Fn 1235 4829 a(n) p Fw 1280 4817 a( ) p Fn 1334 4829 a(\013\027) p Ft 1418 4817 a(k) p Fl 1460 4783 a(2) p Ft 1515 4817 a(\000) p Fx 1598 4817 a(14) p Ft(k) p Fx 1742 4796 a(~) p Fw 1724 4817 a(P) p Fl 1780 4829 a(~) p Fn 1777 4829 a(e) p Fw 1812 4817 a(\037) p Fl 1864 4829 a(\001) p Fw 1923 4817 a(P) p 1988 4817 a(\037) p Fn 2040 4829 a(n) p Fw 2085 4817 a( ) p Fn 2139 4829 a(\013\027) p Ft 2223 4817 a(k) p Fl 2265 4783 a(2) p Fk 2302 4725 a(\021) p Ft 2370 4817 a(\000) p 2453 4817 a(k) p Fw(I) p Fn 2538 4783 a(\025) p Fl 2531 4838 a(1) p Ft 2581 4817 a(kk) p Fw(P) p Fx 2739 4817 a(\026) p Fw 2730 4817 a(\037) p Fl 2782 4829 a(\001) p Fw 2839 4817 a(\037) p Fn 2891 4829 a(n) p Fw 2937 4817 a( ) p Fn 2991 4829 a(\013;\027) p Ft 3095 4817 a(k) p Fl 3137 4783 a(2) p Ft 2092 4976 a(\000) p Fw 2175 4976 a(C) p Fl 2234 4988 a(3) p Fx 2271 4976 a(\() p Fw(n) p Fo 2353 4942 a(\000) p Fl(1) p Fx 2461 4976 a(+) p Fw 2544 4976 a(\027) p Fx 2609 4976 a(+) p Fw 2692 4976 a(\013) p Ft 2764 4976 a(\001) p Fw 2805 4976 a(n) p Fx 2874 4976 a(+) p Fw 2957 4976 a(\013) p Fx(\)) p Fw(:) p Fd 3231 4976 a(\003) p Fx 1905 5255 a(37) p eop %%Page: 38 38 38 37 bop Fx 515 523 a(No) n(w) p 705 523 a(let) p 825 523 a(us) p 933 523 a(complete) p 1284 523 a(the) p 1428 523 a(pro) r(of) p 1646 523 a(of) p 1741 523 a(Prop) r(osition) p 2189 523 a(8.4.) p 2357 523 a(By) p 2488 523 a(substituting) p Fw 2951 523 a(\027) p Fx 3022 523 a(=) p Fw 3111 523 a(\013) p Fg 3174 471 a(3) p 3174 480 29 3 v 3174 513 a(2) p Fx 3245 523 a(and) p Fw 515 623 a(n) p Fx 588 623 a(=) p Fw 675 623 a(\013) p Fo 728 593 a(\000) p Fg 791 570 a(1) p 791 579 V 791 613 a(2) p Fx 861 623 a(in) n(to) p 1029 623 a(Lemma) p 1325 623 a(8.5) p 1459 623 a(and) p 1620 623 a(8.6,) p 1777 623 a(w) n(e) p 1900 623 a(ha) n(v) n(e) p Ft 612 851 a(h) p Fw(K) p Ft 721 851 a(i) p Fn 753 875 a( ) p Fj 797 883 a(\013;\027) p Ft 917 851 a(\025) p Fw 1005 851 a(b) p Fl 1044 863 a(~) p Fn 1041 863 a(e) p Fk 1090 758 a(\020) p Ft 1140 851 a(k) p Fw(\037) p Fl 1234 863 a(\001) p Fw 1292 851 a(P) p 1357 851 a(\037) p Fn 1409 863 a(n) p Fw 1454 851 a( ) p Fn 1508 863 a(\013\027) p Ft 1592 851 a(k) p Fl 1634 816 a(2) p Ft 1690 851 a(\000) p Fx 1773 851 a(14) p Ft(k) p Fx 1916 830 a(~) p Fw 1899 851 a(P) p Fl 1955 863 a(~) p Fn 1952 863 a(e) p Fw 1986 851 a(\037) p Fl 2038 863 a(\001) p Fw 2097 851 a(P) p 2162 851 a(\037) p Fn 2214 863 a(n) p Fw 2259 851 a( ) p Fn 2313 863 a(\013\027) p Ft 2397 851 a(k) p Fl 2439 816 a(2) p Fk 2476 758 a(\021) p Ft 2542 851 a(\000) p 2623 851 a(k) p Fw(I) p Fn 2708 816 a(\025) p Fl 2701 871 a(1) p Ft 2751 851 a(k) p 2808 851 a(\001) p 2848 851 a(k) p Fw(P) p Fx 2964 851 a(\026) p Fw 2955 851 a(\037) p Fl 3007 863 a(\001) p Fw 3065 851 a(\037) p Fn 3117 863 a(n) p Fw 3162 851 a( ) p Fn 3216 863 a(\013\027) p Ft 3300 851 a(k) p Fl 3342 816 a(2) p Ft 661 1021 a(\000) p Fw 744 1021 a(C) p Fl 803 1033 a(3) p Fx 841 1021 a(\() p Fw(\013) p Fg 936 964 a(1) p 936 973 V 936 1007 a(2) p Fx 997 1021 a(+) p Fw 1080 1021 a(\013) p Fx 1152 1021 a(+) p Fw 1235 1021 a(\013) p Fg 1298 964 a(1) p 1299 973 V 1299 1007 a(2) p Fx 1359 1021 a(+) p Fw 1442 1021 a(\013) p Fg 1505 964 a(3) p 1506 973 V 1506 1007 a(2) p Fx 1548 1021 a(\)) p Ft 1599 1021 a(\000) p 1682 1021 a(k) p Fw(I) p Fn 1767 987 a(\025) p Fl 1760 1042 a(1) p Ft 1810 1021 a(k) p 1870 1021 a(\001) p 1911 1021 a(k) p Fx 1972 1000 a(\026) p Fw 1953 1021 a(P) p 2018 1021 a( ) p Fn 2072 1033 a(\013;\027) p Ft 2176 1021 a(k) p Fl 2218 987 a(2) p Ft 2273 1021 a(\000) p 2356 1021 a(k) p Fw(I) p Fn 2441 987 a(\025) p Fl 2434 1042 a(1) p Ft 2484 1021 a(k) p 2544 1021 a(\001) p 2586 1021 a(k) p Fx 2638 1021 a(\026) p Fw 2628 1021 a(\037) p Fn 2680 1033 a(n) p Fw 2724 1021 a( ) p Fn 2778 1033 a(\013;\027) p Ft 2883 1021 a(k) p Fl 2925 987 a(2) p Ft 2980 1021 a(\000) p Fw 3063 1021 a(\013) p Ft 3135 1021 a(\001) p Fw 3176 1021 a(C) p Fl 3235 1033 a(2) p Fw 3273 1021 a(:) p Fx 515 1204 a(As) 793 1386 y(lim) p Fn 605 1438 a(\013) p Fo(!) p Fl(0) p Fn(;\027) p Fo 804 1438 a(!) p Fl(0) p Fn(;n) p Fo(!1) p Fw 1110 1386 a(\037) p Fn 1162 1398 a(n) p Fw 1208 1386 a( ) p Fn 1262 1398 a(\013\027) p Fx 1369 1386 a(=) p Fw 1457 1386 a( ) s(;) p Fx 1822 1386 a(lim) p Fn 1634 1438 a(\013) p Fo(!) p Fl(0) p Fn(;\027) p Fo 1833 1438 a(!) p Fl(0) p Fn(;n) p Fo(!1) p Fx 2149 1386 a(\026) p Fw 2139 1386 a(\037) p Fn 2191 1398 a(n) p Fw 2236 1386 a( ) p Fn 2290 1398 a(\013\027) p Fx 2398 1386 a(=) p 2486 1386 a(0) p Fw(;) p Fx 2739 1386 a(lim) p Fn 2647 1438 a(\013) p Fo(!) p Fl(0) p Fn(;\027) p Fo 2846 1438 a(!) p Fl(0) p Fw 2960 1386 a( ) p Fn 3014 1398 a(\013\027) p Fx 3121 1386 a(=) p Fw 3209 1386 a( ) s(;) p Fx 515 1606 a(w) n(e) p 637 1606 a(obtain) 528 1789 y(lim) p Fn 515 1840 a(\013) p Fo(!) p Fl(0) p Ft 671 1789 a(h) p Fw(K) p Ft 780 1789 a(i) p Fn 812 1814 a( ) p Fj 856 1822 a(\013;\027) p Ft 977 1789 a(\025) p Fw 1064 1789 a(b) p Fl 1103 1801 a(~) p Fn 1100 1801 a(e) p Fk 1150 1697 a(\020) p Ft 1199 1789 a(k) p Fw(\037) p Fl 1293 1801 a(\001) p Fw 1351 1789 a(P) p 1416 1789 a( ) p Ft 1473 1789 a(k) p Fl 1515 1755 a(2) p Ft 1570 1789 a(\000) p Fx 1653 1789 a(14) p Ft(k) p Fx 1796 1768 a(~) p Fw 1779 1789 a(P) p Fl 1835 1801 a(~) p Fn 1832 1801 a(e) p Fw 1867 1789 a(\037) p Fl 1919 1801 a(\001) p Fw 1977 1789 a(P) p 2042 1789 a( ) p Ft 2099 1789 a(k) p Fl 2141 1755 a(2) p Fk 2178 1697 a(\021) p Ft 2246 1789 a(\000) p 2329 1789 a(k) p Fw(I) p Fn 2414 1755 a(\025) p Fl 2407 1809 a(1) p Ft 2457 1789 a(k) p 2517 1789 a(\001) p 2559 1789 a(k) p Fw(P) p Fx 2675 1789 a(\026) p Fw 2666 1789 a(\037) p Fl 2718 1801 a(\001) p Fw 2776 1789 a( ) p Ft 2833 1789 a(k) p Fl 2875 1755 a(2) p Ft 2930 1789 a(\000) p 3013 1789 a(k) p Fw(I) p Fn 3098 1755 a(\025) p Fl 3091 1809 a(1) p Ft 3141 1789 a(k) p 3201 1789 a(\001) p 3242 1789 a(k) p Fx 3303 1768 a(\026) p Fw 3284 1789 a(P) p 3349 1789 a( ) p Ft 3406 1789 a(k) p Fl 3448 1755 a(2) p Fw 3485 1789 a(:) p Fx 515 1997 a(Substituting) p Fw 990 1997 a(\037) p Fl 1042 2009 a(\001) p Fw 1101 1997 a( ) p Fx 1181 1997 a(=) p Fw 1269 1997 a( ) p Fx 1326 1997 a(,) p 1386 1997 a(\026) p Fw 1376 1997 a(\037) p Fl 1428 2009 a(\001) p Fw 1487 1997 a( ) p Fx 1567 1997 a(=) p 1655 1997 a(0) p 1724 1997 a(and) p Fw 1885 1997 a(P) p 1950 1997 a( ) p Fx 2030 1997 a(=) p Fw 2118 1997 a( ) p Fx 2175 1997 a(,) p 2226 1997 a(w) n(e) p 2348 1997 a(obtain) p 2606 1997 a(the) p 2749 1997 a(required) p 3075 1997 a(result) 1146 2200 y(lim) p Fn 1132 2252 a(\013) p Fo(!) p Fl(0) p Ft 1288 2200 a(h) p Fw 1321 2200 a(K) p Ft 1398 2200 a(i) p Fn 1429 2225 a( ) p Fj 1473 2233 a(\013;\027) p Ft 1594 2200 a(\025) p Fw 1682 2200 a(b) p Fl 1721 2212 a(~) p Fn 1718 2212 a(e) p Fk 1767 2108 a(\020) p Ft 1816 2200 a(k) p Fw( ) p Ft 1915 2200 a(k) p Fl 1957 2166 a(2) p Ft 2012 2200 a(\000) p Fx 2095 2200 a(14) p Ft(k) p Fx 2238 2179 a(~) p Fw 2221 2200 a(P) p Fl 2277 2212 a(~) p Fn 2274 2212 a(e) p Fw 2309 2200 a( ) p Ft 2366 2200 a(k) p Fl 2408 2166 a(2) p Fk 2445 2108 a(\021) p Fw 2508 2200 a(:) p Fd 2697 2200 a(\003) p Fx 515 2409 a(By) p 645 2409 a(Prop) r(osition) p 1092 2409 a(8.1) p 1226 2409 a(and) p 1387 2409 a(8.4,) p 1544 2409 a(w) n(e) p 1666 2409 a(ha) n(v) n(e) 1171 2612 y(0) p 1236 2612 a(=) p 1337 2612 a(lim) p Fn 1324 2663 a(\013) p Fo(!) p Fl(0) p Ft 1480 2612 a(h) p Fw(K) p Ft 1589 2612 a(i) p Fn 1621 2637 a( ) p Fj 1665 2645 a(\013;\027) p Ft 1786 2612 a(\025) p Fw 1873 2612 a(b) p Fl 1912 2624 a(~) p Fn 1909 2624 a(e) p Fk 1958 2520 a(\020) p Ft 2008 2612 a(k) p Fw( ) p Ft 2107 2612 a(k) p Fl 2149 2578 a(2) p Ft 2204 2612 a(\000) p Fx 2287 2612 a(14) p Ft(k) p Fx 2430 2591 a(~) p Fw 2413 2612 a(P) p Fl 2469 2624 a(~) p Fn 2466 2624 a(e) p Fw 2500 2612 a( ) p Ft 2557 2612 a(k) p Fl 2599 2578 a(2) p Fk 2636 2520 a(\021) p Fw 2699 2612 a(:) p Fx 515 2839 a(If) 616 2818 y(~) p Fw 598 2839 a(P) p Fl 654 2851 a(~) p Fn 651 2851 a(e) p Fw 687 2839 a( ) p Fx 767 2839 a(=) p 854 2839 a(0,) p 947 2839 a(this) p 1109 2839 a(is) p 1192 2839 a(a) p 1261 2839 a(con) n(tradiction.) p 1799 2839 a(Then) p 2016 2839 a(w) n(e) p 2138 2839 a(ha) n(v) n(e) p 2329 2839 a(the) p 2472 2839 a(follo) n(wing) p 2823 2839 a(Theorem:) p Fm 515 3005 a(Theorem) p 920 3005 a(8.7) p Fy 1084 3005 a(F) p 1132 3005 a(or) p Fw 1248 3005 a(e) p Ft 1326 3005 a(2) p Fi 1421 3005 a(R) p Fy(,) p 1548 3005 a(let) p Fx 1677 3005 a(~) p Fw 1674 3005 a(e) p Ft 1752 3005 a(\021) p Fw 1857 3005 a(e) p Fy(,) p 1962 3005 a(if) p Fw 2052 3005 a(e) p Fy 2129 3005 a(is) p 2228 3005 a(an) p 2356 3005 a(eigenvalue) p 2766 3005 a(of) p Fw 2873 3005 a(L) p Fn 2930 3017 a(S) p Fy 2977 3005 a(,) p 3044 3005 a(and) p 3214 3005 a(let) p Fx 3344 3005 a(~) p Fw 3340 3005 a(e) p Fy 515 3105 a(b) l(e) p 621 3105 a(an) p 744 3105 a(eigenvalue) p 1149 3105 a(of) p Fw 1250 3105 a(L) p Fn 1307 3117 a(S) p Fy 1389 3105 a(which) p 1626 3105 a(is) p 1719 3105 a(next) p 1904 3105 a(to) p Fw 2008 3105 a(e) p Fy(,) p 2107 3105 a(in) p 2213 3105 a(c) l(ase) p 2395 3105 a(that) p Fw 2569 3105 a(e) p Fy 2641 3105 a(is) p 2734 3105 a(not) p 2885 3105 a(an) p 3008 3105 a(eigenvalue) 515 3205 y(of) p Fw 616 3205 a(L) p Fn 673 3217 a(S) p Fy 721 3205 a(.) p 796 3205 a(Supp) l(ose) p 1116 3205 a(that) p 1289 3205 a(Assumption) p 1749 3205 a(2.1) p 1893 3205 a(and) p 2058 3205 a(2.2) p 2202 3205 a(ar) l(e) p 2347 3205 a(satis\014e) l(d.) p 2709 3205 a(Then) p 2929 3205 a(if) p Fw 3013 3205 a(\025) p Fy 3095 3205 a(is) p 3188 3205 a(smal) t(l) 515 3304 y(enough,) p 821 3304 a(ther) l(e) p 1025 3304 a(is) p 1113 3304 a(no) p 1229 3304 a(eigenve) l(ctor) p 1659 3304 a(with) p 1837 3304 a(eigenvalue) p Fw 2236 3304 a(e) p Fy(,) p 2328 3304 a(which) p 2560 3304 a(is) p 2647 3304 a(ortho) l(gonal) p 3051 3304 a(to) p Fx 3167 3283 a(~) p Fw 3148 3304 a(P) p Fl 3204 3316 a(~) p Fn 3201 3316 a(e) p Fy 3237 3304 a(.) p 3300 3304 a(In) 515 3404 y(p) l(articular,) p 920 3404 a(if) p Fx 1019 3383 a(~) p Fw 1001 3404 a(P) p Fl 1057 3416 a(~) p Fn 1054 3416 a(e) p Fx 1113 3404 a(=) p 1200 3404 a(0) p Fy(,) p Fw 1297 3404 a(e) p Fy 1365 3404 a(is) p 1454 3404 a(not) p 1601 3404 a(an) p 1720 3404 a(eigenvalue) p 2121 3404 a(of) p Fw 2218 3404 a(L) p Fy(.) p Fu 515 3678 a(9) p 716 3678 a(The) p 991 3678 a(Stabilit) l(y) p 1535 3678 a(of) p 1688 3678 a(the) p 1921 3678 a(NESS) p Fx 515 3860 a(In) p 622 3860 a(this) p 788 3860 a(section) p 1069 3860 a(w) n(e) p 1196 3860 a(in) n(v) n(estigate) p 1613 3860 a(our) p 1764 3860 a(ph) n(ysical) p 2087 3860 a(in) n(terest:) p 2425 3860 a(the) p 2572 3860 a(stabilit) n(y) p 2900 3860 a(of) p 2998 3860 a(the) p 3145 3860 a(NESS.) 515 3960 y(Recall) p 759 3960 a(that) p 932 3960 a(the) p 1069 3960 a(NESS) p 1301 3960 a(of) p 1389 3960 a(the) p 1526 3960 a(free) p 1679 3960 a(F) p 1726 3960 a(ermion) p 1997 3960 a(mo) r(del) p 2237 3960 a(is) p 2314 3960 a(giv) n(en) p 2525 3960 a(b) n(y) p Fw 2633 3960 a(n) p Fx(-p) r(oin) n(t) p 2921 3960 a(functions) p 3273 3960 a(\(3\)) 515 4059 y(with) p 705 4059 a(a) p 775 4059 a(distribution) p 1228 4059 a(function) p 1554 4059 a(\(4\).) p 1723 4059 a(W) p 1801 4059 a(e) p 1867 4059 a(sho) n(w) p 2073 4059 a(if) p 2150 4059 a(the) p 2294 4059 a(NESS) p 2534 4059 a(is) p 2618 4059 a(far) p 2746 4059 a(from) p 2943 4059 a(equilibrium,) 515 4159 y(i.e.,) p 674 4159 a(the) p 818 4159 a(in) n(v) n(erse) p 1095 4159 a(temp) r(erature) p Fw 1570 4159 a(\014) p Fo 1617 4171 a(\000) p Fx 1702 4159 a(and) p Fw 1865 4159 a(\014) p Fl 1912 4171 a(+) p Fx 1997 4159 a(are) p 2137 4159 a(di\013eren) n(t,) p 2490 4159 a(it) p 2574 4159 a(is) p 2659 4159 a(macroscopically) p 3259 4159 a(un-) 515 4259 y(stable) p 750 4259 a(\(Theorem) p 1128 4259 a(2.2\).) p 1325 4259 a(Strictly) p 1618 4259 a(sp) r(eaking,) p 1978 4259 a(w) n(e) p 2096 4259 a(sho) n(w) p 2296 4259 a(the) p 2434 4259 a(follo) n(wing:) p 2815 4259 a(w) n(e) p 2932 4259 a(can) p 3079 4259 a(c) n(ho) r(ose) p 3337 4259 a(a) 515 4358 y(suitable) p 820 4358 a(\014nite) p 1028 4358 a(system) p 1299 4358 a(suc) n(h) p 1482 4358 a(that) p 1658 4358 a(the) p 1797 4358 a(NESS) p 2031 4358 a(is) p 2110 4358 a(destro) n(y) n(ed) p 2478 4358 a(macroscopically) p 3072 4358 a(when) p 3284 4358 a(w) n(e) 515 4458 y(add) p 678 4458 a(it) p 763 4458 a(with) p 954 4458 a(suitable) p 1266 4458 a(in) n(teraction.) p 1721 4458 a(This) p 1913 4458 a(results) p 2179 4458 a(from) p 2377 4458 a(the) p 2522 4458 a(lac) n(k) p 2695 4458 a(of) p 2791 4458 a(detailed) p 3107 4458 a(balance) 515 4558 y(of) p 607 4558 a(the) p 747 4558 a(NESS) p 983 4558 a(far) p 1107 4558 a(from) p 1301 4558 a(equilibrium.) p 1773 4558 a(On) p 1908 4558 a(the) p 2049 4558 a(other) p 2263 4558 a(hand,) p 2491 4558 a(for) p 2616 4558 a(a) p 2682 4558 a(class) p 2874 4558 a(of) p 2966 4558 a(in) n(teraction,) 515 4657 y(w) n(e) p 637 4657 a(sho) n(w) p 843 4657 a("return) p 1138 4657 a(to) p 1240 4657 a(equilibrium".) 1905 5255 y(38) p eop %%Page: 39 39 39 38 bop Fs 515 523 a(9.1) p 770 523 a(Instabilit) m(y) p 1309 523 a(of) p 1437 523 a(the) p 1631 523 a(NESS) p Fx 515 676 a(First) p 720 676 a(let) p 842 676 a(us) p 952 676 a(in) n(v) n(estigate) p 1368 676 a(the) p 1513 676 a(instabilit) n(y) p 1908 676 a(of) p 2006 676 a(NESS) p 2247 676 a(under) p 2485 676 a(in) n(teraction) p 2905 676 a(with) p 3097 676 a(suitable) 515 776 y(small) p 739 776 a(system.) p 1067 776 a(By) p 1205 776 a(Theorem) p 1562 776 a(8.7) p 1703 776 a(and) p 1871 776 a(Prop) r(osition) p 2325 776 a(3.4,) p 2491 776 a(it) p 2581 776 a(su\016ces) p 2870 776 a(to) p 2978 776 a(sho) n(w) 3209 755 y(~) p Fw 3191 776 a(P) p Fn 3244 788 a(e) p Fx 3314 776 a(=) 515 876 y(Ker) o(\000\() p Fw(e) p Fx(\)) p Ft 827 876 a(\012) p Fw 915 876 a(P) p Fl 968 888 a(\012) p Fx 1054 876 a(=) p 1153 876 a(0) p 1229 876 a(for) p 1363 876 a(all) p 1486 876 a(eigen) n(v) p 1708 876 a(alue) p Fw 1889 876 a(e) p Fx 1962 876 a(of) p Fw 2064 876 a(L) p Fn 2121 888 a(S) p Fx 2168 876 a(.) p 2249 876 a(F) p 2296 876 a(or) p 2405 876 a(the) p 2555 876 a(purp) r(ose,) p 2899 876 a(w) n(e) p 3028 876 a(study) p 3264 876 a(the) 515 991 y(k) n(ernel) p 760 991 a(of) p 856 991 a(\000\() p Fw(e) p Fx(\).) p 1074 991 a(Recall) p 1325 991 a(the) p 1469 991 a(de\014nition) p 1838 991 a(of) p Fw 1934 991 a(') p Fn 1988 1003 a(n) p Fw 2033 991 a(;) p 2070 991 a(E) p Fn 2131 1003 a(n) p Fw 2177 991 a(;) p 2214 991 a(E) p Fn 2275 1003 a(nm) p Fw 2379 991 a(;) p 2416 991 a(N) p Fl 2492 948 a(\() p Fn(j) p Fl 2548 948 a(\)) p Fn 2483 1016 a(l) p Fw 2579 991 a(;) p 2616 991 a(N) p Fl 2692 948 a(\() p Fn(i) p Fl(\)) p Fn 2683 1001 a(r) p Fw 2771 991 a(;) p 2808 991 a(N) p Fn 2875 1003 a(l) p Fw 2900 991 a(;) p 2937 991 a(N) p Fn 3004 1003 a(r) p Fx 3040 991 a(,) p 3092 991 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Fw(k) p Fn 1342 3205 a(nm) p Fx 1447 3193 a(\)) p Ft 1497 3193 a(\000) p Fw 1580 3193 a(c) p Fn 1616 3205 a(n) p Ft 1680 3193 a(\001) p Fw 1722 3193 a(\032) p Fg 1775 3137 a(1) p 1774 3146 V 1774 3179 a(2) p Fx 1817 3193 a(\() p Fw(k) p Fn 1892 3205 a(mn) p Fx 1996 3193 a(\)) p Fk 2028 3098 a(\014) 2028 3148 y(\014) 2028 3197 y(\014) p Fl 2056 3118 a(2) p Fx 2117 3193 a(=) p Fk 2204 3098 a(\014) 2204 3148 y(\014) 2204 3197 y(\014) p Fw 2232 3193 a(c) p Fn 2268 3205 a(m) p Ft 2349 3193 a(\001) p Fx 2391 3193 a(\(1) p Ft 2483 3193 a(\000) p Fw 2566 3193 a(\032) p Fx(\)) p Fg 2651 3129 a(1) p 2651 3138 V 2651 3171 a(2) p Fx 2708 3193 a(\() p Ft(\000) p Fw(k) p Fn 2848 3205 a(mn) p Fx 2952 3193 a(\)) p Ft 3002 3193 a(\000) p Fw 3085 3193 a(c) p Fn 3121 3205 a(n) p Ft 3185 3193 a(\001) p Fw 3227 3193 a(\032) p Fg 3280 3137 a(1) p 3279 3146 V 3279 3179 a(2) p Fx 3322 3193 a(\() p Ft(\000) p Fw(k) p Fn 3462 3205 a(mn) p Fx 3566 3193 a(\)) p Fk 3598 3098 a(\014) 3598 3148 y(\014) 3598 3197 y(\014) p Fl 3626 3118 a(2) p Fx 515 3393 a(=) p Fk 602 3297 a(\014) 602 3347 y(\014) 602 3397 y(\014) p Fw 630 3393 a(c) p Fn 666 3405 a(m) p Ft 747 3393 a(\001) p Fw 789 3393 a(\032) p Fg 842 3336 a(1) p 842 3345 V 842 3378 a(2) p Fx 884 3393 a(\() p Fw(k) p Fn 959 3405 a(nm) p Fx 1064 3393 a(\)) p Ft 1115 3393 a(\000) p Fw 1198 3393 a(c) p Fn 1234 3405 a(n) p Ft 1297 3393 a(\001) p Fx 1339 3393 a(\(1) p Ft 1431 3393 a(\000) p Fw 1514 3393 a(\032) p Fx(\)) p Fg 1599 3329 a(1) p 1599 3338 V 1599 3371 a(2) p Fx 1655 3393 a(\() p Fw(k) p Fn 1730 3405 a(nm) p Fx 1835 3393 a(\)) p Fk 1867 3297 a(\014) 1867 3347 y(\014) 1867 3397 y(\014) p Fl 1895 3318 a(2) p Fx 1955 3393 a(=) p Fk 2043 3297 a(\014) 2043 3347 y(\014) 2043 3397 y(\014) p Fw 2071 3393 a(c) p Fn 2107 3405 a(m) p Ft 2188 3393 a(\001) p Fw 2229 3393 a(\032) p Fg 2282 3336 a(1) p 2282 3345 V 2282 3378 a(2) p Fx 2325 3393 a(\() p Ft(\000) p Fw(k) p Fn 2465 3405 a(nm) p Fx 2569 3393 a(\)) p Ft 2620 3393 a(\000) p Fw 2703 3393 a(c) p Fn 2739 3405 a(n) p Ft 2802 3393 a(\001) p Fx 2844 3393 a(\(1) p Ft 2936 3393 a(\000) p Fw 3019 3393 a(\032) p Fx(\)) p Fg 3104 3329 a(1) p 3104 3338 V 3104 3371 a(2) p Fx 3160 3393 a(\() p Ft(\000) p Fw(k) p Fn 3300 3405 a(nm) p Fx 3405 3393 a(\)) p Fk 3437 3297 a(\014) 3437 3347 y(\014) 3437 3397 y(\014) p Fl 3465 3318 a(2) p Fx 3525 3393 a(=) p 3612 3393 a(0) p Fw(:) p Fx 515 3589 a(This) p 704 3589 a(implies) p 986 3589 a(the) p 1129 3589 a(follo) n(wing:) p Fw 534 3780 a(c) p Fn 570 3792 a(n) p 525 3817 99 4 v Fw 525 3893 a(c) p Fn 561 3905 a(m) p Fx 657 3836 a(=) 754 3780 y(\() q(1) p Ft 847 3780 a(\000) p Fw 930 3780 a(\032) p Fx(\)) p Fg 1015 3716 a(1) p 1015 3725 29 3 v 1015 3758 a(2) p Fx 1071 3780 a(\() p Fw(k) p Fn 1146 3792 a(mn) p Fx 1251 3780 a(\)) p 754 3817 529 4 v Fw 865 3905 a(\032) p Fg 918 3853 a(1) p 918 3862 29 3 v 918 3896 a(2) p Fx 960 3905 a(\() p Fw(k) p Fn 1035 3917 a(mn) p Fx 1140 3905 a(\)) 1316 3836 y(=) 1413 3780 y(\() q(1) p Ft 1506 3780 a(\000) p Fw 1589 3780 a(\032) p Fx(\)) p Fg 1674 3716 a(1) p 1674 3725 V 1674 3758 a(2) p Fx 1730 3780 a(\() p Ft(\000) p Fw(k) p Fn 1870 3792 a(mn) p Fx 1974 3780 a(\)) p 1413 3817 594 4 v Fw 1524 3905 a(\032) p Fg 1577 3853 a(1) p 1577 3862 29 3 v 1577 3896 a(2) p Fx 1619 3905 a(\() p Ft(\000) p Fw(k) p Fn 1759 3917 a(mn) p Fx 1864 3905 a(\)) 2040 3836 y(=) p Fw 2248 3780 a(\032) p Fg 2301 3727 a(1) p 2301 3736 V 2301 3770 a(2) p Fx 2343 3780 a(\() p Fw(k) p Fn 2418 3792 a(nm) p Fx 2523 3780 a(\)) p 2137 3817 529 4 v 2137 3918 a(\(1) p Ft 2229 3918 a(\000) p Fw 2312 3918 a(\032) p Fx(\)) p Fg 2398 3853 a(1) p 2398 3862 29 3 v 2398 3896 a(2) p Fx 2454 3918 a(\() p Fw(k) p Fn 2529 3930 a(nm) p Fx 2633 3918 a(\)) 2699 3836 y(=) p Fw 2907 3780 a(\032) p Fg 2960 3727 a(1) p 2960 3736 V 2960 3770 a(2) p Fx 3002 3780 a(\() p Ft(\000) p Fw(k) p Fn 3142 3792 a(nm) p Fx 3247 3780 a(\)) p 2796 3817 594 4 v 2796 3918 a(\() q(1) p Ft 2889 3918 a(\000) p Fw 2972 3918 a(\032) p Fx(\)) p Fg 3057 3853 a(1) p 3057 3862 29 3 v 3057 3896 a(2) p Fx 3113 3918 a(\() p Ft(\000) p Fw(k) p Fn 3253 3930 a(nm) p Fx 3357 3918 a(\)) 3231 4016 y(\(58\)) 515 4188 y(for) p Fw 642 4188 a(n) p Ft 715 4188 a(6) p Fx(=) p Fw 802 4188 a(m) p Fx(.) p 935 4188 a(Recall) p 1185 4188 a(that) p 1365 4188 a(the) p 1508 4188 a(F) p 1555 4188 a(ermion) p 1833 4188 a(distribution) p Fw 2285 4188 a(\032) p Fx 2356 4188 a(in) p 2453 4188 a(the) p 2596 4188 a(NESS) p 2834 4188 a(is) p 2918 4188 a(giv) n(en) p 3135 4188 a(b) n(y) p Fw 1075 4492 a(\032) p Fx(\() p Fw(k) p Fx 1196 4492 a(\)) p Ft 1251 4492 a(\021) p Fk 1339 4297 a(8) 1339 4372 y(>) 1339 4397 y(<) 1339 4546 y(>) 1339 4571 y(:) 1469 4309 y(\020) p Fx 1518 4401 a(1) p 1578 4401 a(+) p Fw 1661 4401 a(e) p Fo 1700 4367 a(\000) p Fn(\014) p Ff 1790 4375 a(\000) p Fl 1839 4367 a(\(cos\() p Fn(k) p Fl 2016 4367 a(\)) p Fo(\000) p Fn(\015) p Fl 2133 4367 a(\)) p Fk 2163 4309 a(\021) p Fo 2212 4326 a(\000) p Fl(1) p Fw 2315 4401 a(;) p 2435 4401 a(k) p Ft 2504 4401 a(2) p Fx 2582 4401 a([0) p Fw(;) p 2684 4401 a(\031) p Fx 2734 4401 a(\)) p Fk 1426 4508 a(\020) p Fx 1476 4600 a(1) p 1536 4600 a(+) p Fw 1619 4600 a(e) p Fo 1658 4566 a(\000) p Fn(\014) p Fg 1748 4574 a(+) p Fl 1794 4566 a(\(cos\() p Fn(k) p Fl 1971 4566 a(\)) p Fo(\000) p Fn(\015) p Fl 2088 4566 a(\)) p Fk 2118 4508 a(\021) p Fo 2167 4525 a(\000) p Fl(1) p Fw 2270 4600 a(;) p 2390 4600 a(k) p Ft 2459 4600 a(2) p Fx 2537 4600 a([) p Ft(\000) p Fw(\031) s(;) p Fx 2712 4600 a(0\)) p Fw(:) p Fx 515 4801 a(The) p 685 4801 a(condition) p 1050 4801 a(\(58\)) p 1225 4801 a(requires) p Fw 1559 4976 a(e) p Fo 1598 4942 a(\000) p Fn(\014) p Ff 1688 4950 a(\000) p Fn 1737 4942 a(E) p Fj 1786 4950 a(nm) p Fx 1904 4976 a(=) p Fw 1992 4976 a(e) p Fo 2031 4942 a(\000) p Fn(\014) p Fg 2121 4950 a(+) p Fn 2167 4942 a(E) p Fj 2216 4950 a(nm) p Fw 2311 4976 a(;) p Fx 1905 5255 a(40) p eop %%Page: 41 41 41 40 bop Fx 515 523 a(i.e.,) p Fw(\014) p Fo 691 535 a(\000) p Fx 792 523 a(=) p Fw 902 523 a(\014) p Fl 949 535 a(+) p Fx 1004 523 a(.) p 1104 523 a(That) p 1325 523 a(is,) p 1448 523 a(\000\(0\)) p 1647 523 a(has) p 1808 523 a(non) p 1983 523 a(trivial) p 2243 523 a(k) n(ernel) p 2501 523 a(only) p 2697 523 a(if) p 2786 523 a(the) p 2943 523 a(NESS) p 3195 523 a(is) p 3291 523 a(an) 515 623 y(equilibrium) p 957 623 a(state) p 1163 623 a(indeed.) p 1463 623 a(If) p Fw 1548 623 a(\014) p Fo 1595 635 a(\000) p Ft 1677 623 a(6) p Fx(=) p Fw 1768 623 a(\014) p Fl 1815 635 a(+) p Fx 1870 623 a(,) p 1923 623 a(then,) p 2138 623 a(w) n(e) p 2262 623 a(ha) n(v) n(e) 2474 602 y(~) p Fw 2455 623 a(P) p Fn 2508 635 a(e) p Fx 2570 623 a(=) p 2661 623 a(0) p 2732 623 a(for) p 2861 623 a(all) p 2978 623 a(eigen) p 3192 623 a(v) p 3231 623 a(alue) p Fw 515 722 a(e) p Fx 585 722 a(of) p Fw 684 722 a(L) p Fn 741 734 a(S) p Fx 789 722 a(.) p 862 722 a(Hence) p 1113 722 a(the) p 1261 722 a(Liouville) p 1605 722 a(op) r(erator) p Fw 1944 722 a(L) p Fx 2033 722 a(corresp) r(onding) p 2571 722 a(to) p 2677 722 a(the) p 2824 722 a(NESS) p 3067 722 a(do) r(es) p 3259 722 a(not) 515 822 y(ha) n(v) n(e) p 712 822 a(an) p 833 822 a(eigen) n(v) n(ector,) p 1294 822 a(b) n(y) p 1415 822 a(Theorem) p 1772 822 a(8.7.) p 1955 822 a(Then,) p 2201 822 a(b) n(y) p 2322 822 a(Prop) r(osition) p 2775 822 a(3.4,) p 2939 822 a(w) n(e) p 3067 822 a(ha) n(v) n(e) p 3264 822 a(the) 515 922 y(follo) n(wing) p 865 922 a(Theorem:) p Fm 515 1086 a(Theorem) p 920 1086 a(2.2) p Fy 1115 1086 a(The) p 1285 1086 a(NESS) p 1526 1086 a(far) p 1659 1086 a(fr) l(om) p 1855 1086 a(e) l(quilibrium) p 2285 1086 a(is) p 2374 1086 a(macr) l(osc) l(opic) l(al) t(ly) p 2967 1086 a(unstable.) p Fs 515 1318 a(9.2) p 770 1318 a(Return) p 1156 1318 a(to) p 1293 1318 a(equilibrium) p Fx 515 1471 a(In) p 618 1471 a(this) p 779 1471 a(section) p 1056 1471 a(w) n(e) p 1178 1471 a(in) n(v) n(estigate) p 1590 1471 a(the) p 1733 1471 a(equilibrium) p 2173 1471 a(case,) p 2371 1471 a(i.e.,) p Fw 2528 1471 a(\014) p Fl 2575 1483 a(+) p Fx 2653 1471 a(=) p Fw 2740 1471 a(\014) p Fo 2787 1483 a(\000) p Fx 2843 1471 a(.) p 2903 1471 a(By) p 3033 1471 a(the) p 3175 1471 a(result) 515 1571 y(of) p 615 1571 a(the) p 764 1571 a(previous) p 1101 1571 a(subsection,) p 1534 1571 a(w) n(e) p 1662 1571 a(can) p 1820 1571 a(sho) n(w) p Fy 2032 1571 a(r) l(eturn) p 2290 1571 a(to) p 2396 1571 a(e) l(quilibrium) p Fx 2829 1571 a(for) p 2962 1571 a(the) p 3111 1571 a(class) p 3312 1571 a(of) 515 1670 y(in) n(teraction) p 936 1670 a(w) n(e) p 1062 1670 a(in) n(tro) r(duced.) p 1521 1670 a(First,) p 1752 1670 a(w) n(e) p 1878 1670 a(in) n(tro) r(duce) p 2251 1670 a(the) p 2398 1670 a(follo) n(wing) p 2752 1670 a(w) n(ell-kno) n(wn) p 3187 1670 a(facts.) 515 1770 y(The) p 685 1770 a(follo) n(wing) p 1036 1770 a(Theorem) p 1387 1770 a(w) n(as) p 1546 1770 a(sho) n(wn) p 1798 1770 a(b) n(y) p 1913 1770 a(H.Araki) p 2230 1770 a([17) o(],[18) o(]:) p Fm 515 1934 a(Theorem) p 920 1934 a(9.1) p Fy 1084 1934 a(L) l(et) p Fx 1227 1934 a(\() p Fh(A) p Fw(;) p 1356 1934 a(\034) p Fx 1401 1934 a(\)) p Fy 1462 1934 a(b) l(e) p 1564 1934 a(a) p Fw 1636 1934 a(C) p Fo 1701 1904 a(\003) p Fy 1739 1934 a(-dynamic) l(al) p 2161 1934 a(system) p 2432 1934 a(and) p 2593 1934 a(let) p Fw 2709 1934 a(!) p Fy 2793 1934 a(b) l(e) p 2894 1934 a(a) p Fx 2966 1934 a(\() p Fw(\014) t(;) p 3086 1934 a(\034) p Fx 3131 1934 a(\)) p Fy(-KMS) 515 2034 y(state) p 721 2034 a(with) p 908 2034 a(GNS-r) l(epr) l(esentation) p Fx 1657 2034 a(\() p Ft(H) p Fw 1760 2034 a(;) p 1797 2034 a(\031) s(;) p Fx 1884 2034 a(\012\)) p Fy(.) p 2060 2034 a(L) l(et) p Fw 2210 2034 a(L) p Fy 2303 2034 a(b) l(e) p 2411 2034 a(the) p 2555 2034 a(Liouvil) t(le) p 2905 2034 a(op) l(er) l(ator) p 3238 2034 a(c) l(or-) 515 2133 y(r) l(esp) l(onding) p 936 2133 a(to) p Fw 1044 2133 a(\034) p Fy 1089 2133 a(.) p 1180 2133 a(If) p Fw 1276 2133 a(P) p Fx 1379 2133 a(=) p Fw 1483 2133 a(P) p Fo 1548 2103 a(\003) p Ft 1624 2133 a(2) p Fh 1718 2133 a(A) p Fy 1816 2133 a(then) p Fx 2010 2133 a(\012) p Ft 2108 2133 a(2) p Fw 2202 2133 a(D) p Fx 2273 2133 a(\(e) p Fn 2342 2103 a(\014) p Fl 2383 2103 a(\() p Fn(L) p Fl(+) p Fn(\031) p Fl 2547 2103 a(\() p Fn(P) p Fl 2624 2103 a(\)\)) p Fn(=) p Fl(2) p Fx 2747 2133 a(\)) p Fy(.) p 2869 2133 a(L) l(et) p Fw 3021 2133 a(\034) p Fn 3066 2103 a(P) p Fy 3160 2133 a(b) l(e) p 3271 2133 a(the) 515 2233 y(p) l(erturb) l(e) l(d) p 881 2233 a(automorphism) p 1433 2233 a(gr) l(oup) p 1668 2233 a(by) p Fw 1784 2233 a(P) p Fy 1849 2233 a(,) p 1913 2233 a(and) p 2082 2233 a(let) p Fx 2206 2233 a(\012) p Fn 2266 2203 a(P) p Ft 2358 2233 a(\021) p Fx 2459 2233 a(e) p Fn 2496 2203 a(\014) p Fl 2537 2203 a(\() p Fn(L) p Fl(+) p Fn(\031) p Fl 2701 2203 a(\() p Fn(P) p Fl 2778 2203 a(\)\)) p Fn(=) p Fl(2) p Fx 2901 2233 a(\012) p Fy(.) p 3047 2233 a(Then) p 3271 2233 a(the) 515 2332 y(state) p Fw 714 2332 a(!) p Fn 769 2302 a(P) p Fy 854 2332 a(de\014ne) l(d) p 1134 2332 a(by) p Fw 1566 2564 a(!) p Fn 1621 2529 a(P) p Fx 1676 2564 a(\() p Fw(A) p Fx(\)) p 1826 2564 a(=) 1923 2507 y(\(\012) p Fn 2015 2477 a(P) p Fw 2071 2507 a(;) p 2108 2507 a(A) p Fx(\012) p Fn 2230 2477 a(P) p Fx 2286 2507 a(\)) p 1923 2544 395 4 v 1955 2621 a(\(\012) p Fn 2047 2597 a(P) p Fw 2102 2621 a(;) p Fx 2139 2621 a(\012) p Fn 2199 2597 a(P) p Fx 2254 2621 a(\)) p Fy 515 2800 a(is) p 604 2800 a(a) p Fx 676 2800 a(\() p Fw(\014) t(;) p 796 2800 a(\034) p Fn 841 2770 a(P) p Fx 898 2800 a(\)) p Fy(-KMS) p 1174 2800 a(state.) p Fx 515 2964 a(On) p 659 2964 a(the) p 808 2964 a(other) p 1031 2964 a(hand,) p 1270 2964 a(the) p 1419 2964 a(follo) n(wing) p 1775 2964 a(theorem) p 2105 2964 a(concerning) p Fy 2526 2964 a(r) l(eturn) p 2784 2964 a(to) p 2889 2964 a(e) l(quilibrium) p Fx 3323 2964 a(is) 515 3064 y(kno) n(wn) p 778 3064 a([9) o(]:) p Fm 515 3228 a(Prop) s(osition) p 1031 3228 a(9.2) p Fy 1194 3228 a(L) l(et) p Fx 1344 3228 a(\() p Fh(A) p Fw(;) p 1473 3228 a(\034) p Fx 1518 3228 a(\)) p Fy 1587 3228 a(b) l(e) p 1695 3228 a(a) p Fw 1773 3228 a(C) p Fo 1838 3198 a(\003) p Fy 1877 3228 a(-dynamic) l(al) p 2305 3228 a(system) p 2583 3228 a(and) p 2751 3228 a(let) p Fw 2874 3228 a(!) p Fy 2964 3228 a(b) l(e) p 3072 3228 a(a) p Fx 3151 3228 a(\() p Fw(\014) t(;) p 3271 3228 a(\034) p Fx 3316 3228 a(\)) p Fy(-) 515 3328 y(KMS) p 736 3328 a(state) p 942 3328 a(with) p 1130 3328 a(the) p 1274 3328 a(GNS-r) l(epr) l(esentation) p Fx 2024 3328 a(\() p Ft(H) p Fw 2127 3328 a(;) p 2164 3328 a(\031) s(;) p Fx 2251 3328 a(\012\)) p Fy(.) p 2428 3328 a(Assume) p 2745 3328 a(that) p 2921 3328 a(the) p 3066 3328 a(Liouvil) t(le) 515 3427 y(op) l(er) l(ator) p Fw 845 3427 a(L) p Fy 935 3427 a(of) p Fw 1036 3427 a(\034) p Fy 1115 3427 a(has) p 1268 3427 a(a) p 1343 3427 a(simple) p 1606 3427 a(eigenvalue) p Fx 2011 3427 a(0) p Fy 2085 3427 a(c) l(orr) l(esp) l(onding) p 2613 3427 a(to) p 2717 3427 a(the) p 2858 3427 a(eigenve) l(ctor) p Fx 3294 3427 a(\012) p Fy(,) 515 3527 y(and) p 681 3527 a(that) p 855 3527 a(the) p 998 3527 a(r) l(est) p 1163 3527 a(of) p 1266 3527 a(the) p 1408 3527 a(sp) l(e) l(ctrum) p 1762 3527 a(of) p Fw 1865 3527 a(L) p Fy 1956 3527 a(is) p 2050 3527 a(c) l(ontinuous.) p 2516 3527 a(Then) p 2738 3527 a(for) p 2875 3527 a(any) p Fw 3039 3527 a(!) p Fy 3094 3527 a(-normal) 515 3626 y(state) p Fw 714 3626 a(\032) p Fy(,) p 812 3626 a(we) p 935 3626 a(have) p 1126 3626 a(the) p 1264 3626 a(r) l(eturn) p 1517 3626 a(to) p 1616 3626 a(e) l(quilibrium) p 2046 3626 a(in) p 2148 3626 a(an) p 2267 3626 a(er) l(go) l(dic) p 2548 3626 a(me) l(an) p 2768 3626 a(sense:) p Fx 1394 3864 a(lim) p Fn 1362 3917 a(T) p Fo 1410 3917 a(!1) p Fx 1575 3808 a(1) p 1566 3845 61 4 v Fw 1566 3921 a(T) p Fk 1650 3751 a(Z) p Fn 1733 3771 a(T) p Fl 1696 3939 a(0) p Fw 1799 3864 a(\032) p Fx 1856 3864 a(\() p Fw(\013) p Fn 1941 3876 a(t) p Fx 1971 3864 a(\() p Fw(A) p Fx(\)\)) p Fw 2143 3864 a(dt) p Fx 2240 3864 a(=) p Fw 2327 3864 a(!) p Fx 2382 3864 a(\() p Fw(A) p Fx(\)) p Fw(:) p Fx 639 4109 a(No) n(w) p 826 4109 a(let) p 943 4109 a(us) p 1047 4109 a(return) p 1299 4109 a(to) p 1398 4109 a(our) p 1543 4109 a(system.) p 1850 4109 a(Note) p 2048 4109 a(that) p Fw 2225 4109 a(!) p Fn 2277 4121 a(\032) p Fx 2341 4109 a(is) p 2422 4109 a(a) p 2488 4109 a(\() p Fw(\014) t(;) p 2608 4109 a(\013) p Fn 2661 4069 a(f) 2661 4129 y(t) p Fx 2705 4109 a(\)-KMS) p 2977 4109 a(state) p 3178 4109 a(of) p Ft 3270 4109 a(O) p Fn 3336 4121 a(f) p Fx 515 4208 a(if) p Fw 593 4208 a(\014) p Fo 640 4220 a(\000) p Fx 723 4208 a(=) p Fw 815 4208 a(\014) p Fl 862 4220 a(+) p Fx 944 4208 a(=) p Fw 1035 4208 a(\014) p Fx 1086 4208 a(.) p 1154 4208 a(W) p 1232 4208 a(e) p 1299 4208 a(denote) p 1569 4208 a(b) n(y) p Fw 1687 4208 a(!) p Fn 1742 4178 a(S) 1739 4232 y(\014) p Fx 1819 4208 a(the) p 1965 4208 a(\() p Fw(\014) t(;) p 2085 4208 a(\013) p Fn 2138 4220 a(S) p Fx 2186 4208 a(\)-KMS) p 2463 4208 a(state) p 2669 4208 a(o) n(v) n(er) p Ft 2849 4208 a(O) p Fn 2915 4220 a(S) p Fx 2963 4208 a(.) p 3030 4208 a(The) p 3203 4208 a(state) p Fw 515 4321 a(!) p Fn 570 4291 a(S) 567 4345 y(\014) p Ft 641 4321 a(\012) p Fw 728 4321 a(!) p Fn 780 4333 a(\032) p Fx 853 4321 a(is) p 943 4321 a(a) p 1019 4321 a(\() p Fw(\014) t(;) p 1139 4321 a(\013) p Fl 1192 4333 a(0) p Fx 1230 4321 a(\)-KMS) p 1512 4321 a(state) p 1722 4321 a(o) n(v) n(er) p Ft 1907 4321 a(O) p Fn 1973 4333 a(S) p Ft 2044 4321 a(\012) p 2132 4321 a(O) p Fn 2198 4333 a(f) p Fx 2241 4321 a(.) p 2321 4321 a(As) p 2451 4321 a(our) p 2606 4321 a(p) r(erturbation) p Fw 3100 4321 a(V) p Fx 3201 4321 a(is) p 3291 4321 a(an) 515 4421 y(elemen) n(t) p 818 4421 a(of) p Ft 908 4421 a(O) p Fx 976 4421 a(,) p 1025 4421 a(the) p 1164 4421 a(non) n(triviarit) n(y) p 1645 4421 a(of) p 1736 4421 a(Ker) p Fw 1870 4421 a(L) p Fx 1950 4421 a(is) p 2030 4421 a(guaran) n(teed) p 2453 4421 a(b) n(y) p 2564 4421 a(Theorem) p 2911 4421 a(9.1.) p 3076 4421 a(The) p 3243 4421 a(fact) 515 4520 y(that) p 691 4520 a(0) p 756 4520 a(is) p 835 4520 a(the) p 974 4520 a(simple) p 1229 4520 a(eigen) n(v) p 1451 4520 a(alue) p 1622 4520 a(of) p Fw 1712 4520 a(L) p Fx 1793 4520 a(can) p 1941 4520 a(b) r(e) p 2050 4520 a(deriv) n(ed) p 2337 4520 a(as) p 2435 4520 a(follo) n(ws) p 2703 4520 a(:) p 2761 4520 a(\014rst) p 2929 4520 a(the) p 3068 4520 a(equation) 515 4620 y(\(57\)) p 690 4620 a(implies) p 972 4620 a(that) p 1152 4620 a(Ker) o(\000\(0\)) p 1472 4620 a(is) p 1556 4620 a(one-dimensional) p 2165 4620 a(and) p 2327 4620 a(is) p 2411 4620 a(spanned) p 2734 4620 a(b) n(y) p 2850 4620 a(a) p 2919 4620 a(v) n(ector) p 3169 4620 a(of) p 3264 4620 a(the) 515 4720 y(form) p Fw 1621 4900 a(') p Fx 1699 4900 a(=) p Fk 1786 4821 a(X) p Fn 1835 4998 a(i) p Fw 1920 4900 a(c) p Fn 1956 4912 a(i) p Fw 1984 4900 a(') p Fn 2038 4912 a(i) p Ft 2084 4900 a(\012) p Fw 2167 4900 a(') p Fn 2221 4912 a(i) p Fw 2249 4900 a(;) p Fx 1905 5255 a(41) p eop %%Page: 42 42 42 41 bop Fx 515 523 a(with) p Fw 1318 706 a(c) p Fn 1354 718 a(n) p Fx 1422 706 a(=) p Fw 1510 706 a(Z) p Fn 1567 718 a(\014) p Fo 1612 669 a(\000) p Fg 1673 647 a(1) p 1673 656 29 3 v 1673 689 a(2) p Fx 1716 706 a(e) p Fj 1763 645 a(\014) p 1763 659 35 3 v Fg 1766 691 a(2) p Fn 1808 671 a(E) p Fj 1857 679 a(n) p Fw 1901 706 a(;) p 2021 706 a(Z) p Fn 2078 718 a(\014) p Fx 2146 706 a(=) p Fk 2233 627 a(X) p Fn 2273 801 a(n) p Fx 2367 706 a(e) p Fj 2414 645 a(\014) p 2414 659 V Fg 2417 691 a(2) p Fn 2459 671 a(E) p Fj 2508 679 a(n) p Fw 2552 706 a(:) p Fx 515 958 a(Let) p Fw 670 958 a( ) p Fl 724 970 a(1) p Fw 761 958 a(;) p 798 958 a( ) p Fl 852 970 a(2) p Fx 924 958 a(b) r(e) p 1044 958 a(eigen) n(v) n(ectors) p 1514 958 a(of) p Fw 1616 958 a(L) p Fx 1706 958 a(with) p 1902 958 a(eigen) n(v) p 2124 958 a(alue) p 2306 958 a(0.) p 2427 958 a(By) p 2564 958 a(Theorem) p 2921 958 a(8.7,) p 3086 958 a(w) n(e) p 3215 958 a(ha) n(v) n(e) 539 1037 y(~) p Fw 515 1058 a(P) p Fl 568 1070 a(0) p Fw 605 1058 a( ) p Fn 659 1070 a(i) p Fx 710 1058 a(=) p Fw 798 1058 a(c) p Fn 834 1070 a(i) p Fw 861 1058 a(') p Ft 920 1058 a(\012) p Fx 990 1058 a(\012,) p 1096 1058 a(with) p Fw 1279 1058 a(c) p Fn 1315 1070 a(i) p Ft 1365 1058 a(6) p Fx(=) p 1453 1058 a(0.) p 1552 1058 a(Then) p Fn 1772 1021 a( ) p Fg 1816 1029 a(1) p 1772 1039 77 4 v Fn 1780 1087 a(c) p Fg 1810 1095 a(1) p Ft 1864 1058 a(\000) p Fn 1944 1021 a( ) p Fg 1988 1029 a(2) p 1944 1039 V Fn 1951 1087 a(c) p Fg 1981 1095 a(2) p Fx 2051 1058 a(is) p 2128 1058 a(an) p 2237 1058 a(eigen) n(v) n(ector) p 2662 1058 a(of) p Fw 2750 1058 a(L) p Fx 2827 1058 a(with) p 3010 1058 a(eigen) n(v) p 3232 1058 a(alue) 515 1180 y(0,) p 602 1180 a(whic) n(h) p 833 1180 a(is) p 909 1180 a(orthogonal) p 1318 1180 a(to) 1438 1159 y(~) p Fw 1413 1180 a(P) p Fl 1466 1192 a(0) p Fx 1504 1180 a(.) p 1561 1180 a(By) p 1685 1180 a(Theorem) p 2029 1180 a(8.7,) p 2181 1180 a(this) p 2336 1180 a(en) n(tails) p Fw 2590 1180 a( ) p Fl 2644 1192 a(1) p Fx 2705 1180 a(=) p Fn 2802 1146 a(c) p Fg 2832 1154 a(1) p 2802 1161 63 4 v Fn 2802 1208 a(c) p Fg 2832 1216 a(2) p Fw 2874 1180 a( ) p Fl 2928 1192 a(2) p Fx 2966 1180 a(.) p 3024 1180 a(Hence) p 3264 1180 a(the) 515 1279 y(Liouville) p 860 1279 a(op) r(erator) p Fw 1201 1279 a(L) p Fx 1290 1279 a(has) p 1444 1279 a(a) p 1519 1279 a(simple) p 1783 1279 a(eigen) n(v) p 2005 1279 a(alue) p 2185 1279 a(0.) p 2303 1279 a(W) p 2381 1279 a(e) p 2452 1279 a(denote) p 2725 1279 a(the) p 2873 1279 a(corresp) r(onding) 515 1379 y(eigen) n(v) n(ector) p 957 1379 a(b) n(y) p 1082 1379 a(\012) p Fn 1142 1391 a(V) p Fx 1238 1379 a(and) p 1410 1379 a(the) p 1563 1379 a(state) p 1777 1379 a(corresp) r(onding) p 2321 1379 a(to) p 2433 1379 a(\012) p Fn 2493 1391 a(V) p Fx 2589 1379 a(b) n(y) p Fw 2715 1379 a(!) p Fn 2767 1391 a(V) p Fx 2824 1379 a(.) p 2915 1379 a(By) p 3056 1379 a(Theorem) 515 1479 y(9.1,) p Fw 676 1479 a(!) p Fn 728 1491 a(V) p Fx 817 1479 a(is) p 904 1479 a(a) p 977 1479 a(\() p Fw(\014) t(;) p 1097 1479 a(\013) p Fx(\)-KMS) p 1429 1479 a(state.) p 1676 1479 a(On) p 1818 1479 a(the) p 1964 1479 a(other) p 2185 1479 a(hand,) p 2420 1479 a(as) p 2526 1479 a(\000\() p Fw(e) p Fx(\)) p 2713 1479 a(do) r(es) p 2903 1479 a(not) p 3055 1479 a(ha) n(v) n(e) p 3250 1479 a(an) n(y) 515 1578 y(eigen) n(v) n(ector) p 939 1578 a(for) p Fw 1059 1578 a(e) p Ft 1121 1578 a(6) p Fx(=) p 1209 1578 a(0,) p 1295 1578 a(and) 1468 1557 y(~) p Fw 1450 1578 a(P) p Fn 1503 1590 a(e) p Fx 1562 1578 a(=) p 1649 1578 a(0.) p 1748 1578 a(So) p 1857 1578 a(the) p 1993 1578 a(rest) p 2148 1578 a(of) p 2235 1578 a(the) p 2371 1578 a(sp) r(ectrum) p 2727 1578 a(of) p Fw 2814 1578 a(L) p Fx 2892 1578 a(is) p 2968 1578 a(con) n(tin) n (uous.) 515 1678 y(Accordingly) p 943 1678 a(,) p 992 1678 a(from) p 1189 1678 a(the) p 1332 1678 a(Prop) r(osition) p 1779 1678 a(9.2,) p 1935 1678 a(w) n(e) p 2058 1678 a(obtain) p 2316 1678 a(the) p Fy 2459 1678 a(r) l(eturn) p 2711 1678 a(to) p 2811 1678 a(e) l(quilibrium) p Fx(:) p Fm 515 1844 a(Theorem) p 920 1844 a(2.3) p Fy 1115 1844 a(Supp) l(ose) p 1438 1844 a(that) p 1614 1844 a(Assumption) p 2077 1844 a(2.1) p 2223 1844 a(and) p 2391 1844 a(2.4) p 2537 1844 a(ar) l(e) p 2684 1844 a(satis\014e) l(d.) p 3053 1844 a(Then) p 3276 1844 a(for) 515 1944 y(su\016ciently) p 928 1944 a(smal) t(l) p Fw 1144 1944 a(\025) p Fy(,) p 1243 1944 a(the) p 1376 1944 a(system) p 1642 1944 a(exhibits) p 1936 1944 a(r) l(eturn) p 2183 1944 a(to) p 2278 1944 a(e) l(quilibrium) p 2702 1944 a(in) p 2799 1944 a(an) p 2912 1944 a(er) l(go) l(dic) p 3188 1944 a(me) l(an) 515 2043 y(sense.) p Fm 515 2209 a(Ac) m(kno) m(wledgemen) m(t.) p Fx 515 2309 a(The) p 691 2309 a(author) p 965 2309 a(thanks) p 1241 2309 a(for) p 1373 2309 a(A.Arai,) p 1676 2309 a(F.Hiroshima,) p 2183 2309 a(and) p 2350 2309 a(T.Matsui) p 2718 2309 a(for) p 2851 2309 a(useful) p 3095 2309 a(advices.) 515 2408 y(Man) n(y) p 748 2408 a(thanks) p 1018 2408 a(also) p 1184 2408 a(go) p 1295 2408 a(to) p 1397 2408 a(M.Merkli) p 1759 2408 a(for) p 1886 2408 a(helpful) p 2160 2408 a(argumen) n(ts) p 2565 2408 a(and) p 2726 2408 a(commen) n(ts.) p Fu 515 2683 a(A) p 751 2683 a(T) p 834 2683 a(ensor) p 1189 2683 a(Pro) t(duct) p 1713 2683 a(of) p 1867 2683 a(Linear) p 2286 2683 a(Op) t(erators) p Fx 515 2865 a(Ab) r(out) p 779 2865 a(tensor) p 1034 2865 a(pro) r(duct) p 1352 2865 a(of) p 1452 2865 a(linear) p 1688 2865 a(op) r(erators,) p 2086 2865 a(w) n(e) p 2214 2865 a(ha) n(v) n(e) p 2411 2865 a(the) p 2560 2865 a(follo) n(wing) p 2916 2865 a(Theorem.) p 3317 2865 a(It) 515 2964 y(can) p 667 2964 a(b) r(e) p 780 2964 a(pro) n(v) n(ed) p 1050 2964 a(using) p 1267 2964 a(sp) r(ectral) p 1579 2964 a(theorem) p 1902 2964 a(giv) n(en) p 2119 2964 a(in) p 2216 2964 a([16) o(].) p Fm 515 3131 a(Theorem) p 920 3131 a(A.1) p Fy 1108 3131 a(L) l(et) p Ft 1254 3131 a(H) p Fn 1324 3143 a(i) p Fx 1351 3131 a(\() p Fw(i) p Fx 1439 3131 a(=) p 1531 3131 a(1) p Fw(;) p Fx 1610 3131 a(2\)) p Fy 1715 3131 a(b) l(e) p 1819 3131 a(sep) l(ar) l(able) p 2174 3131 a(Hilb) l(ert) p 2450 3131 a(sp) l(ac) l(es,) p 2728 3131 a(and) p 2892 3131 a(let) p Fw 3010 3131 a(L) p Fn 3067 3143 a(i) p Fy 3126 3131 a(b) l(e) p 3230 3131 a(self-) 515 3230 y(adjoint) p 795 3230 a(op) l(er) l(ators) p 1154 3230 a(on) p Ft 1271 3230 a(H) p Fn 1341 3242 a(i) p Fy 1369 3230 a(.) p 1432 3230 a(L) l(et) p Fw 1574 3230 a(L) p Fy 1658 3230 a(b) l(e) p 1759 3230 a(the) p 1895 3230 a(self-adjoint) p 2324 3230 a(op) l(er) l(ator) p 2649 3230 a(on) p Ft 2766 3230 a(H) p Fl 2836 3242 a(1) p Ft 2888 3230 a(\012) p 2967 3230 a(H) p Fl 3037 3242 a(2) p Fy 3075 3230 a(,) p 3129 3230 a(de\014ne) l(d) 515 3330 y(by) p Fw 1564 3512 a(L) p Ft 1644 3512 a(\021) p Fw 1731 3512 a(L) p Fl 1788 3524 a(1) p Ft 1844 3512 a(\012) p Fx 1927 3512 a(1) p 1987 3512 a(+) p 2070 3512 a(1) p Ft 2130 3512 a(\012) p Fw 2213 3512 a(L) p Fl 2270 3524 a(2) p Fw 2306 3512 a(:) p Fy 515 3695 a(Supp) l(ose) p 831 3695 a(that) p Fw 1001 3695 a(L) p Fl 1058 3707 a(2) p Fy 1125 3695 a(has) p 1273 3695 a(a) p 1346 3695 a(unique) p 1613 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a(B.1) p Fy 1037 523 a(F) p 1085 523 a(or) p 1192 523 a(al) t(l) p Fw 1311 523 a(') p Fl 1365 535 a(1) p Fw 1402 523 a(;) p 1439 523 a(') p Fl 1493 535 a(2) p Ft 1554 523 a(2) p Fw 1632 523 a(D) p Fx 1703 523 a(\() p Fw(N) p Fx 1811 523 a(\)) p Fy(,) p Fw 1899 523 a(g) p Ft 1964 523 a(2) p Fh 2043 523 a(h) p Fl 2086 535 a(\003) p Fj 2131 543 a(v) p Fy 2171 523 a(,) p 2226 523 a(and) p Fw 2388 523 a(c) p 2447 523 a(>) p Fx 2534 523 a(0) p Fy 2606 523 a(we) p 2729 523 a(have) p Ft 515 770 a(jh) p Fw(') p Fl 624 782 a(1) p Ft 662 770 a(j) p Fw(X) p Ft 778 770 a(\012) p Fx 861 770 a(\() p Fw 894 770 a(a) p Fx(\() p Fw(g) p Fx 1013 770 a(\)) p 1063 770 a(+) p Fw 1146 770 a(a) p Fo 1190 736 a(\003) p Fx 1228 770 a(\() p Fw(g) p Fx 1303 770 a(\)\)) p Fw 1382 770 a(') p Fl 1436 782 a(2) p Ft 1473 770 a(ij) p 1552 770 a(\024) p Fx 1639 770 a(3) p Fw(c) p Ft(k) p Fw(X) p Ft 1835 770 a(k) p Fl 1877 736 a(2) p Fk 1926 653 a(\022) 1987 675 y(\015) 1987 725 y(\015) 1987 775 y(\015) p Fw 2033 770 a(N) p Fg 2119 700 a(1) p 2119 709 29 3 v 2119 742 a(2) p Fl 2100 795 a(\003) p Fj 2145 803 a(v) p Fw 2185 770 a(') p Fl 2239 782 a(1) p Fk 2277 675 a(\015) 2277 725 y(\015) 2277 775 y(\015) p Fl 2323 693 a(2) p Fx 2379 770 a(+) p Fk 2462 675 a(\015) 2462 725 y(\015) 2462 775 y(\015) p Fw 2508 770 a(N) p Fg 2593 700 a(1) p 2593 709 V 2593 742 a(2) p Fl 2575 795 a(\003) p Fj 2620 803 a(v) p Fw 2660 770 a(') p Fl 2714 782 a(2) p Fk 2751 675 a(\015) 2751 725 y(\015) 2751 775 y(\015) p Fl 2797 693 a(2) p Fk 2835 653 a(\023) p Fx 2914 770 a(+) p 2997 770 a(3) p Ft 3049 714 a(k) p Fw 3090 714 a(g) p Ft 3133 714 a(k) p Fl 3174 673 a(2) p 3049 751 163 4 v Fw 3112 827 a(c) p Fk 3235 678 a(\020) p Ft 3285 770 a(k) p Fw 3326 770 a(') p Fl 3380 782 a(1) p Ft 3418 770 a(k) p Fl 3459 729 a(2) p Fx 3515 770 a(+) p Ft 3598 770 a(k) p Fw(') p Fl 3694 782 a(2) p Ft 3731 770 a(k) p Fl 3773 729 a(2) p Fk 3810 678 a(\021) 515 1088 y(\014) 515 1138 y(\014) 542 1092 y(\012) p Fw 582 1159 a(') p Fl 636 1171 a(1) p Ft 673 1159 a(j) p Fw(X) p Ft 790 1159 a(\012) p Fx 873 1159 a(\() p Ft(\000) p Fx(1\)) p Fn 1044 1125 a(N) p Fx 1107 1159 a(\() p Fw(a) p Fx(\() p Fw(g) p Fx 1258 1159 a(\)) p Ft 1308 1159 a(\000) p Fw 1391 1159 a(a) p Fo 1435 1125 a(\003) p Fx 1473 1159 a(\() p Fw(g) p Fx 1548 1159 a(\)\)) p Fw(') p Fl 1666 1171 a(2) p Fk 1704 1092 a(\013) 1744 1088 y(\014) 1744 1138 y(\014) p Ft 1794 1159 a(\024) p Fx 1882 1159 a(3) p Fw(c) p Ft(k) p Fw(X) p Ft 2078 1159 a(k) p Fl 2120 1125 a(2) p Fk 2169 1042 a(\022) 2230 1064 y(\015) 2230 1113 y(\015) 2230 1163 y(\015) p Fw 2276 1159 a(N) p Fg 2362 1088 a(1) p 2362 1097 29 3 v 2362 1130 a(2) p Fl 2343 1183 a(\003) p Fj 2388 1191 a(v) p Fw 2428 1159 a(') p Fl 2482 1171 a(1) p Fk 2520 1064 a(\015) 2520 1113 y(\015) 2520 1163 y(\015) p Fl 2566 1082 a(2) p Fx 2621 1159 a(+) p Fk 2704 1064 a(\015) 2704 1113 y(\015) 2704 1163 y(\015) p Fw 2751 1159 a(N) p Fg 2836 1088 a(1) p 2836 1097 V 2836 1130 a(2) p Fl 2818 1183 a(\003) p Fj 2863 1191 a(v) p Fw 2902 1159 a(') p Fl 2956 1171 a(2) p Fk 2994 1064 a(\015) 2994 1113 y(\015) 2994 1163 y(\015) p Fl 3040 1082 a(2) p Fk 3077 1042 a(\023) p Fx 3157 1159 a(+) p 3240 1159 a(3) p Ft 3292 1103 a(k) p Fw 3333 1103 a(g) p Ft 3376 1103 a(k) p Fl 3417 1061 a(2) p 3291 1140 163 4 v Fw 3355 1216 a(c) p Fk 3478 1067 a(\020) p Ft 3528 1159 a(k) p Fw 3569 1159 a(') p Fl 3623 1171 a(1) p Ft 3661 1159 a(k) p Fl 3702 1117 a(2) p Fx 3758 1159 a(+) p Ft 3841 1159 a(k) p Fw 3882 1159 a(') p Fl 3936 1171 a(2) p Ft 3974 1159 a(k) p Fl 4016 1117 a(2) p Fk 4053 1067 a(\021) p Fu 515 1616 a(References) p Fx 708 1798 a([1]) p 838 1798 a(D.Ruelle,) p Fy(A) p 1262 1798 a(variational) p 1677 1798 a(formulation) p 2123 1798 a(of) p 2217 1798 a(e) l(quilibrium) p 2643 1798 a(statistic) l(al) p 3010 1798 a(me) l(chanics) 838 1898 y(and) p 999 1898 a(the) p 1137 1898 a(Gibbs) p 1366 1898 a(phase) p 1596 1898 a(rule.) p Fx 1788 1898 a(Comm) n(un.) p 2168 1898 a(Math.) p 2415 1898 a(Ph) n(ys.) p Fm 2643 1898 a(5) p Fx(,324{329,\(1967\)) 708 2064 y([2]) p 838 2064 a(W.Pusz,) p 1168 2064 a(and) p 1329 2064 a(S.L.W) p 1551 2064 a(orono) n(wicz,) p Fy 1959 2064 a(Passive) p 2256 2064 a(states) p 2488 2064 a(and) p 2649 2064 a(KMS) p 2862 2064 a(states) p 3095 2064 a(for) p 3226 2064 a(gen-) 838 2163 y(er) l(al) p 1000 2163 a(quantum) p 1341 2163 a(systems) p Fx 1645 2163 a(Comm) n(un.Math.Ph) n(ys.) p Fm(58) p Fx(,273-290,\(1978\)) 708 2329 y([3]) p 838 2329 a(O.Lanford) p 1242 2329 a(and) p 1407 2329 a(D.W.Robinson,) p Fy 1995 2329 a(Statistic) l(al) p 2382 2329 a(me) l(chanics) p 2785 2329 a(of) p 2886 2329 a(quantum) p 3230 2329 a(spin) 838 2429 y(systems.) p 1169 2429 a(III) p Fx 1292 2429 a(Comm) n(un.Math.Ph) n(ys.) p Fm(9) p Fx(,327-338,\(1968\)) 708 2595 y([4]) p 838 2595 a(H.Araki,) p Fy(On) p 1311 2595 a(the) p 1471 2595 a(e) l(quivalenc) l (e) p 1923 2595 a(of) p 2042 2595 a(the) p 2202 2595 a(KMS) p 2439 2595 a(c) l(ondition) p 2823 2595 a(and) p 3006 2595 a(the) p 3166 2595 a(varia-) 838 2695 y(tional) p 1069 2695 a(princip) l(al) p 1411 2695 a(for) p 1540 2695 a(quantum) p 1877 2695 a(lattic) l(e) p 2120 2695 a(systems) p Fx 2419 2695 a(Comm) n(un.Math.Ph) n(ys.) p Fm(38) p Fx(,1-) 838 2794 y(10,\(1974\)) 708 2960 y([5]) p 838 2960 a(R.Haag,) p 1163 2960 a(D.Kastler,) p 1567 2960 a(and) p 1732 2960 a(B.T) p 1867 2960 a(ryc) n(h-P) n(ohlmey) n (er,) p Fy 2488 2960 a(Stability) p 2815 2960 a(and) p 2979 2960 a(e) l(quilibrium) 838 3060 y(states.) p Fx 1094 3060 a(Comm) n(un.) p 1475 3060 a(Math.) p 1722 3060 a(Ph) n(ys.) p Fm 1949 3060 a(38) p Fx 2072 3060 a(\(1974\),) p 2354 3060 a(173{193.) 708 3226 y([6]) p 838 3226 a(H.) p 1035 3226 a(Araki,) p Fy(On) p 1479 3226 a(the) p Fw 1695 3226 a(X) p 1771 3226 a(Y) p Fy 1837 3226 a(-mo) l(del) p 2182 3226 a(on) p 2379 3226 a(two-side) l(d) p 2820 3226 a(in\014nite) p 3184 3226 a(chain) p Fx 838 3326 a(Publ.RIMS,Ky) n(oto) p 1518 3326 a(Univ,) p Fm 1744 3326 a(20) p Fx 1867 3326 a(277-296) p 2171 3326 a(\(1984\).) 708 3492 y([7]) p 838 3492 a(V.) p 967 3492 a(Jak) l(\024) p 1097 3492 a(si) n(\023) p 1154 3492 a(c,) p 1258 3492 a(and) p 1435 3492 a(C.) p 1562 3492 a(A.) p 1691 3492 a(Pillet,) p Fy 1956 3492 a(On) p 2111 3492 a(a) p 2198 3492 a(mo) l(del) p 2450 3492 a(for) p 2597 3492 a(quantum) p 2953 3492 a(friction.) p 3290 3492 a(II:) 838 3591 y(F) p 886 3591 a(ermi's) p 1148 3591 a(golden) p 1413 3591 a(rule) p 1588 3591 a(and) p 1756 3591 a(dynamics) p 2130 3591 a(at) p 2236 3591 a(p) l(ositive) p 2542 3591 a(temp) l(er) l(atur) l(e) p Fx(,) p 3026 3591 a(Comm) n(un.) 838 3691 y(Math.) p 1084 3691 a(Ph) n(ys.) p 1312 3691 a(176,) p 1487 3691 a(619-644) p 1792 3691 a(\(1996\).) 708 3857 y([8]) p 838 3857 a(V.) p 964 3857 a(Jak) l(\024) p 1094 3857 a(si) n(\023) p 1151 3857 a(c,) p 1252 3857 a(and) p 1427 3857 a(C.) p 1550 3857 a(A.) p 1677 3857 a(Pillet,) p Fy 1939 3857 a(On) p 2091 3857 a(a) p 2175 3857 a(mo) l(del) p 2425 3857 a(for) p 2570 3857 a(quantum) p 2924 3857 a(friction.) p 3258 3857 a(III:) 838 3957 y(Er) l(go) l(dic) p 1131 3957 a(pr) l(op) l(erties) p 1507 3957 a(of) p 1599 3957 a(the) p 1731 3957 a(spin-b) l(oson) p 2132 3957 a(system) p Fx 2396 3957 a(Comm) n(un.) p 2770 3957 a(Math.) p 3010 3957 a(Ph) n(ys.) p 3231 3957 a(178,) 838 4056 y(627-651) p 1142 4056 a(\(1996\).) 708 4222 y([9]) p 838 4222 a(V.Bac) n(h,) p 1164 4222 a(J.F) p 1277 4222 a(r\177) p 1310 4222 a(ohlic) n(h,) p 1584 4222 a(I.M.Sigal,) p Fy 1972 4222 a(R) l(eturn) p 2257 4222 a(to) p 2363 4222 a(e) l(quilibrium.) p Fx 2824 4222 a(J.) p 2925 4222 a(Math.) p 3179 4222 a(Ph) n(ys.) p Fm 838 4322 a(41) p Fx 961 4322 a(no.) p 1099 4322 a(6,) p 1191 4322 a(3985{4060) p 1593 4322 a(\(2000\).) 667 4488 y([10]) p 838 4488 a(M.Merkli,) p Fy(Positive) p 1511 4488 a(c) l(ommutators) p 2004 4488 a(in) p 2104 4488 a(non-e) l(quilibrium) p 2698 4488 a(quantum) p 3037 4488 a(statistic) l(al) 838 4588 y(me) l(chanics) p Fx 1234 4588 a(Comm) n(un.) p 1615 4588 a(Math.Ph) n(ys.) p Fm(223) p Fx 2205 4588 a(327-362,) p 2532 4588 a(\(2001\)) 667 4754 y([11]) p 838 4754 a(V.Jak) l(\024) p 1053 4754 a(si) n(\023) p 1110 4754 a(c) p 1174 4754 a(and) p 1339 4754 a(C.A.Pillet,) p Fy 1757 4754 a(Non-e) l(quilibrium) p 2370 4754 a(ste) l(ady) p 2623 4754 a(states) p 2860 4754 a(of) p 2960 4754 a(\014nite) p 3177 4754 a(quan-) 838 4853 y(tum) p 1004 4853 a(systems) p 1306 4853 a(c) l(ouple) l(d) p 1594 4853 a(to) p 1690 4853 a(thermal) p 1991 4853 a(r) l(eservoirs.) p Fx 2392 4853 a(Comm) n(un.) p 2769 4853 a(Math.) p 3012 4853 a(Ph) n(ys.) p Fm 3236 4853 a(226) p Fa 878 4953 a<2121>p Fx 937 4953 a(no.1,) p 1140 4953 a(131-162,\(2002\)) 1905 5255 y(43) p eop %%Page: 44 44 44 43 bop Fx 667 523 a([12]) p 838 523 a(V.Jak) l(\024) p 1053 523 a(si) n(\023) p 1110 523 a(c) p 1167 523 a(and) p 1324 523 a(C.A.Pillet,) p Fy 1734 523 a(Mathematic) l(al) p 2246 523 a(the) l(ory) p 2493 523 a(of) p 2586 523 a(non-e) l (quilibrium) p 3177 523 a(quan-) 838 623 y(tum) p 1002 623 a(statistic) l(al) p 1367 623 a(me) l(chanics.) p Fx 1783 623 a(J.) p 1871 623 a(Statist.) p 2156 623 a(Ph) n(ys.) p 2377 623 a(108) p 2523 623 a(\(2002\),) p 2800 623 a(no.) p 2932 623 a(5-6,) p 3088 623 a(787{829) 667 789 y([13]) p 838 789 a(V.Bac) n(h,) p 1166 789 a(J.F) p 1279 789 a(r\177) p 1312 789 a(ohlic) n(h) p 1563 789 a(and) p 1734 789 a(I.M.Sigal,) p 2124 789 a(Quan) n(tum) p 2504 789 a(electro) r(dynamics) p 3123 789 a(of) p 3227 789 a(con-) 838 888 y(\014ned) p 1041 888 a(nonrelativistic) p 1583 888 a(particles.) p 1939 888 a(Adv.) p 2142 888 a(Math.) p Fm 2389 888 a(137) p Fx 2560 888 a(no.) p 2698 888 a(2,) p 2790 888 a(299{395) p 3108 888 a(\(1998\)) 667 1054 y([14]) p 838 1054 a(O.) p 949 1054 a(Bratteli) p 1253 1054 a(and) p 1411 1054 a(D.) p 1521 1054 a(W.) p 1653 1054 a(Robinson,) p Fy 2039 1054 a(Op) l(er) l(ator) p 2383 1054 a(A) n(lgebr) l(as) p 2712 1054 a(and) p 2869 1054 a(Quantum) p 3233 1054 a(Sta-) 838 1154 y(tistic) l(al) p 1105 1154 a(Me) l(chanics) p 1510 1154 a(1) p Fx(,) p 1604 1154 a(Springer-V) p 1993 1154 a(erlag,) p 2218 1154 a(1986.) 667 1320 y([15]) p 838 1320 a(O.) p 949 1320 a(Bratteli) p 1253 1320 a(and) p 1411 1320 a(D.) p 1521 1320 a(W.) p 1653 1320 a(Robinson,) p Fy 2039 1320 a(Op) l(er) l(ator) p 2383 1320 a(A) n(lgebr) l(as) p 2712 1320 a(and) p 2869 1320 a(Quantum) p 3233 1320 a(Sta-) 838 1420 y(tistic) l(al) p 1105 1420 a(Me) l(chanics) p 1510 1420 a(2) p Fx(,) p 1604 1420 a(Springer-V) p 1993 1420 a(erlag,) p 2218 1420 a(1996.) 667 1586 y([16]) p 838 1586 a(M.Reed) p 1160 1586 a(and) p 1336 1586 a(B.Simon,) p Fy 1712 1586 a(Metho) l(ds) p 2052 1586 a(of) p 2163 1586 a(mo) l(dern) p 2474 1586 a(mathematic) l(al) p 2996 1586 a(physics.) p 3322 1586 a(I.) 838 1685 y(F) p 886 1685 a(unctional) p 1255 1685 a(analysis.) p Fx 1599 1685 a(Second) p 1885 1685 a(edition.) p 2190 1685 a(Academic) p 2575 1685 a(Press,) p 2823 1685 a(Inc.) p 2991 1685 a(New) p 3183 1685 a(Y) p 3238 1685 a(ork,) 838 1785 y(1980) 667 1951 y([17]) p 838 1951 a(H.) p 964 1951 a(Araki:) p Fy 1253 1951 a(R) l(elative) p 1582 1951 a(Hamiltonian) p 2074 1951 a(for) p 2219 1951 a(faithful) p 2515 1951 a(normal) p 2812 1951 a(states) p 3058 1951 a(of) p 3168 1951 a(a) p 3252 1951 a(von) 838 2051 y(Neumann) p 1215 2051 a(algebr) l(a) p Fx(,) p 1518 2051 a(Publ.) p 1740 2051 a(R.I.M.S.,) p 2096 2051 a(Ky) n(oto) p 2346 2051 a(Univ.) p Fm 2572 2051 a(9) p Fx 2647 2051 a(165-209) p 2951 2051 a(\(1973\).) 667 2217 y([18]) p 838 2217 a(H.) p 946 2217 a(Araki:) p Fy 1207 2217 a(Positive) p 1520 2217 a(c) l(one,) p 1733 2217 a(R) l(adon-Niko) l(dym) p 2332 2217 a(the) l(or) l(ems,) p 2701 2217 a(r) l(elative) p 2988 2217 a(Hamiltoni-) 838 2316 y(ans) p 993 2316 a(and) p 1156 2316 a(the) p 1296 2316 a(Gibbs) p 1528 2316 a(c) l(ondition) p 1893 2316 a(in) p 1997 2316 a(statistic) l(al) p 2370 2316 a(me) l(chanics) p Fx 2769 2316 a(C*-Algebras) p 3245 2316 a(and) 838 2416 y(their) p 1029 2416 a(Applications) p 1505 2416 a(to) p 1599 2416 a(Statistical) p 1985 2416 a(Mec) n(hanics) p 2379 2416 a(and) p 2534 2416 a(Quan) n(tum) p 2898 2416 a(Field) p 3102 2416 a(Theory) p 3357 2416 a(,) 838 2516 y(edited) p 1087 2516 a(b) n(y) p 1202 2516 a(D.) p 1316 2516 a(Kastler) p 1607 2516 a(\(Editrice) p 1955 2516 a(Comp) r(ositori,) p 2451 2516 a(Bologna,) p 2796 2516 a(1975\).) 667 2682 y([19]) p 838 2682 a(T.G.Ho,andH.Araki,) p Fy(Asymptotic) p 2032 2682 a(time) p 2234 2682 a(evolution) p 2604 2682 a(of) p 2715 2682 a(a) p 2800 2682 a(p) l(artitione) l(d) p 3230 2682 a(in\014-) 838 2781 y(nite) p 1003 2781 a(two-side) l(d) p 1364 2781 a(isotr) l(opic) p Fw 1700 2781 a(X) p 1776 2781 a(Y) p Fy 1842 2781 a(-chain) p Fx 2092 2781 a(Pro) r(c.Steklo) n (vInst.Math,) p Fm(228) p Fx 3102 2781 a(191-204) 838 2881 y(\(2000\).) 667 3047 y([20]) p 838 3047 a(W.H.) p 1066 3047 a(Asc) n(h) n(bac) n(her) p 1512 3047 a(C.A.) p 1715 3047 a(Pillet) p 1945 3047 a(,) p Fy 2005 3047 a(Non-Equilibrium) p 2644 3047 a(Ste) l(ady) p 2914 3047 a(States) p 3167 3047 a(of) p 3271 3047 a(the) 838 3147 y(XY) p 991 3147 a(Chain) p Fx 1235 3147 a(mp-arc) p 1516 3147 a(/02-459) p 1821 3147 a(,\(2002\).) 1905 5255 y(44) p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0305220519147--