Content-Type: multipart/mixed; boundary="-------------0909070243369" This is a multi-part message in MIME format. ---------------0909070243369 Content-Type: application/postscript; name="TrHeatKernel090906.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="TrHeatKernel090906.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.96.1 Copyright 2007 Radical Eye Software %%Title: heat090906.dvi %%CreationDate: Mon Sep 7 09:22:09 2009 %%Pages: 29 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX10 CMR8 CMCSC10 CMMI8 TeX-cmex8 CMMI6 CMSY6 MSBM7 %%+ CMMI5 CMSY8 CMR6 CMR10 MSBM10 CMMI7 CMMI10 CMSY10 CMR7 CMEX10 CMR5 %%+ CMSY7 CMTI8 CMSY5 CMTI10 MSAM10 CMBX8 CMTT8 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o heat090906.ps heat090906.dvi %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2009.09.07:0922 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro 0 0 %! 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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746B8A594DBCBB948EF8F3E2B69DBA77A5CBA4B09A7BBECC19AD70D87094011E 76 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /square put readonly def /FontBBox{8 -463 1331 1003}readonly def currentdict end currentfile eexec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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 12 /fi put dup 44 /comma put dup 46 /period put dup 49 /one put dup 50 /two put dup 53 /five put dup 70 /F put dup 76 /L put dup 80 /P put dup 82 /R put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-163 -250 1146 969}readonly def currentdict end currentfile eexec 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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 78 /N put dup 82 /R put readonly def /FontBBox{0 -504 2615 1004}readonly def currentdict end currentfile eexec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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 3 /asteriskmath put dup 48 /prime put dup 106 /bar put dup 112 /radical put readonly def /FontBBox{21 -944 1448 791}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052F09F9C8ADE9D907C058B87E9B6964 7D53359E51216774A4EAA1E2B58EC3176BD1184A633B951372B4198D4E8C5EF4 A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85 E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A 221A37D9A807DD01161779DDE7D5FC1B2109839E5B52DFBAF552B11EFFB6A16C F03FB920C15AE724EFDF0CCBF00A838D34440FF9FED532F44036AD22561184C5 283722DDFA7285E62754372D716D704AC0E00B2F6AB67154241C7449AA047833 94CEDB08E8C92907FE72A0B05AE36A7B9226ACD6E7890A0B528FDDE84A950FC6 801DE75CF2E739E9121149CCB8B1C87A106822648D84A3D3FBF295EE6C4BF403 BBE9A1C1F6DAEDD1E642ACC486E609703D7612BFFD10C324F5DC710811F7F614 3691B400E3773987424C0D2B0D8A736873C6371DDB2442F05E018A2B5CA9A4AA 17AABB95D09E5890CFFFED5AC01495D89A53D3CCD9EC861CF221D6FB2632C961 8F5D58EA8E1970765BFDEF2373AC6C82383B7847D37831E5F85F7A0E9F862B00 6440BFC2189D8D40FE5780243289B9906FAEE6C3FE5B1C4F7ABF8CFFA041F84E 5C729FE70876A9D6F8CD950A1919C7FBB7DF94552FF997CC04397E7FEC0715C9 477695F7AF229AE6897933896E307D9BFEBEDCE2D34DFB7D0D18F2C2A1109654 5229E2364AE66563E9C3974D26014C93FF82A9F7BA8A200900A69C94E1B0DB95 62286963E271705CC9C4C691DE7C4F0DEE9A2B9AB445147B7FA743E96E67C2D5 959CE81973705187C75FC661F5FA9D10E130006C87899709A0BED8DCD18DCCF7 95A63FBFE12A1A2140D680F196508F13585E1328813DDFBCFF295CA1672DE35C 027A1EAAB40FA1BCC97AE3660E9477425371A597F404B59C69E1A8A339FA0324 4C14959968872CBA860DEA5595F6E08883B163C80BC140D51ABB9D5E5B894601 776E4586D222C582CF0E227693BE592018C555302139C1EDA208817D15B82866 CFA5B98E17BE2CBC765F1526BCB7A8482E46129B018D4C57F863DFC50C6022A8 7178C350B216B4EDA5A4E7F6FEB0AA002636CFC65B8E99B6A667EA110CE7ECEE 3CD97CFFC6837526FAA8E7720137BE3F58F308E6FA2F583B23C46CA2F0810B2E 5828BC252D6D8E8C7BB0435B2B315F228B82F1709AFEB9B9ADD8A1BBCF14AB0D CF1783C3E80A20E25BCEF6EFCDB8FC4A9D94199CF6DF10DBF076C33122495902 F5915DADDEA9EA1891FC52952161C92AB13E6F39A96E6695CE6CCC7904D72540 00C571E804A7048B3A3111998911FA3F1E9FA9F7AFB6F0EE6A14FF090566A16B 85B4DC63C8F58EB723E568D06B8F41040380340A597D0F0C04F0F66B641E0FAC 58796A57CEA61D56112E9D4FD702765A9665AEE2F28F9FC644E8A89285CA5F17 13510D46CB5C9E952348273665332F0068F69E5FBC5D794351FF517F42CAB2E8 8275EA476AB7B3065C7CA463010C00C3CDC75B73FD02B168B0B8FDE717F19C4D 2C8F4656426F285515F495F11F647A543882A0F54A93DF6181F48C9654 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI8 %!PS-AdobeFont-1.1: CMTI8 1.0 %%CreationDate: 1991 Aug 18 21:07:42 % 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 (CMTI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI8 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 13 /fl put dup 44 /comma put dup 45 /hyphen put dup 58 /colon 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 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 89 /Y put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 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 127 /dieresis put readonly def /FontBBox{-35 -250 1190 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 525003F3DAD7933EB57E7DB1462E9D906F6D8F5BF740206C1EC5F36E00AAFF68 F3EF6F3A2540E5F9564D1C215BC1E7E69C7D04DA5DB1CF195613C9CBF4BAA360 84AEF3E10E24877FBE36AD731DC97305BDE6DB1F934909FAF60B8E28561FBC57 0F5B6225425BFDE8F0C71CD4507B82FF803E9A301397975E38A259DE1E1B4FC0 06BB1DC2D45B987A2268A77CE8DC025CB0D1B39788BBEE149103950650171C94 5FED1063050A90BD38605BD9365D1C2AE42A7DA3DDB9A263FE10BD487F63D908 D4F02758BD9D7BE53E6353A25BEFA29E42B50C1D078A8B3A746EEDB381CCE36C 93FC4BD1A8D1D6ACD0D355E948CFE397B74D243EB51597D251D0BAE6884D70E8 FEE119462E1939A9783414DF59EEA5FF1529F13869D1FF0A44935C5198CC5DAA E1FD2A17AEEBF4974052D06734A409E26C457C06700C55896C90BB33E044A737 46590D9FA242819B9527FD59818D8829D4EB2D26A34CC05A2CC063E66F2DF193 8E4F9670014BC243180E45B51DDB05AD1E6A7F619EE1CE09CFC1A4F02CB7270E 4482FEDF673EDED38EF3173C475C34BFB3F6623C5E942A7797FEDDD0EF1D54E9 1D90D7076C0A9687E334907C22F2E7C603388D8D626B0E5A62B7543DCBB575D7 9B3AF95BEA0DC284A53FB40DA68D22E0D0DDA4D06897AA6D0C822190D6AC890C BE4D5A361119B81C6191E2BC6A5BCB6EDB448EFF624A39B084282AA774EBD006 978CBDECE14A9709A6474CFDFA08AD4C76ABC85B16EA7CEB75BD87EC9BA26352 397C0CBA4112C39DFE9A7B2952D41647E886D869E8740C7822CC5C5753BBA2B1 22D756EC9562EACD19CA523D47DBC02C6830F26E441BEC7CA9E4469EE899070B 59620BC5F575C81D5287762E8C9EFBD81127ECD4D72302A413BE4BE3A8AC7BE3 244B800EC5E90558E7286A9E28B0D5D93ED40E29B877E6999D89D2703B1562BD 31C856CDA3046A9AD431D05D72C09D1D5C7C0E0207FC59EFD80F98DF54F0B5D9 5D8643F32AA06DD2E8FC2EF1A57CC150EEB3BDCE86FBDABBAFAC0690CC8AB961 BC9EDC23AE50D61AA18700BBAF79E80BF0C6B395D667A755D4BBB2FA44789495 AF69DCEEC71DC71A351938E7D6591F96CBB39FA51ABBCA12F51057CC4B8668D4 DA1964CEABA75457A2DAD63189AFBED3C5F42506C91B5979EEA16EBBE2D3CAE9 B75CE98CF4FC73690A3B855EA8A4ADBA868E335D4E3C6CB04939075DFD4181CB B88807BBF7E44B47EFA6EE6FD915D14A98DBD00399B95A8669D65998103270C5 8D94B6F41644D10067F1D9AFBB7BFCE423C1F0A38017F29D0A319FD4205CB438 E4B1F3C21C26C5D48571B56553192252E94CBC32FF5B2D455D147E7C9D0418FF 65A3BFCE4231FD3C7C52143FD7FFFFBA0C916BB018BAFCAE37A99131B4F945E0 A8D4A8E98C1140FE729F4122DE483FDE2CB5F321E842F07D4B930DE68931E3D1 0986E05BCDA906F827AABFB1FCB3115B90DF4E48821398B85448F55EB3237F3B E682EAA1DFB3B84068FB6726DC93C43F09A687ED90CAB0711D4DDE6BF6826D63 CA6A0AAC60FF869C0B735B2ADF260339E29474766C8A571829799BB5CF7859E0 76221FCDF4ACB969915C38F1565E3B4B2A9328C548019E230862F0C80D2CCE20 42A460512ED5064BB1DDAFA87E4F4EB79EFD7E45632FBE8573730B3DFA2B778C 4299018473FC2C83AAB105F6CB3A8E24991A550DE24932FF29446FAE8B4E8844 C7F0CBBED0AEBF4CDDC42B93387FA130B2FB9608642D171C78F5D5B761A5779E 308325606D1B09CEA6C0A0CA9BD5E86C68B5EB87D41149A60D700A6881656263 B58D17EB1367A5E561915A38CBFC5C72A2896F6EC955672A9604CB8E2D687083 53715B1436386F8F75C657CFA4C0DE168E8BF5EAB461F0F9289C78F5FB60BE13 50147DAA1DFE080BAD6A0B81BAF5BF497CAA461EA698C0EB34DB50FE819CBC20 B69FEC5ABB601E134529A0565CF3A5D865A001EE677C33D1EFC136F2FE4A9B4E A27B882FA7B033D30BC85FB54432E0508F6AE87C556317E056C5956611452D45 B79E0FA02AD184B2F8F0E91E184BA606DC56A70274CD0E3B13ACF522E1F3416F 011D6FD58D7527DC92801A37CDDDD1063E63FEDF54A8C2EAA19417C5773BEB3F 1BCAC5AB8BA2D64784C37B9E83C5E8BBC4491BEEA392A8E4D32A236299C56372 CF5F8C4516CB9E5B38C21F4DF40B15A2920DB25DB434526CA8DAAC691E67169D 33E683938C6B541236520DF657A771F8EFDBA4B19E30FB37697DC5BB41C4363F 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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 10 /Omega put dup 43 /plus put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 55 /seven put readonly def /FontBBox{-341 -250 1304 965}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF7158F1163BC1F3352E22A1452E73FECA8A4 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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 8 /braceleftbig put dup 9 /bracerightbig put dup 12 /vextendsingle 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 32 /parenleftBigg put dup 33 /parenrightBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 80 /summationtext put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 113 /radicalBig put readonly def /FontBBox{-24 -2960 1454 772}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CAC6A7BEB5D02276E511FFAF2AE11910 DE076F24311D94D07CACC323F360887F1EA11BDDA7927FF3325986FDB0ABDFC8 8E4B40E7988921D551EC0867EBCA44C05657F0DC913E7B3004A5F3E1337B6987 FEBC45F989C8DC6DC0AD577E903F05D0D54208A0AE7F28C734F130C133B48422 BED48639A2B74E4C08F2E710E24A99F347E0F4394CE64EACB549576E89044E52 EABE595BC964156D9D8C2BAB0F49664E951D7C1A3D1789C47F03C7051A63D5E8 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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 38 /ampersand put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 91 /bracketleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 109 /m put dup 120 /x put readonly def /FontBBox{-27 -250 1122 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 8 /circleplus put dup 10 /circlemultiply put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 27 /propersuperset put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 76 /L put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put dup 112 /radical put dup 114 /nabla put readonly def /FontBBox{-29 -960 1116 775}readonly def currentdict end 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cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 16 /zeta put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 24 /xi put dup 25 /pi put dup 27 /sigma put dup 28 /tau put dup 31 /chi put dup 33 /omega put dup 34 /epsilon put dup 44 /arrowhookleft put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 72 /H put dup 73 /I put dup 74 /J put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 82 /R put dup 83 /S put dup 87 /W 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 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec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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 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 25 /pi put dup 27 /sigma put dup 28 /tau put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 65 /A put dup 74 /J put dup 76 /L put dup 78 /N put dup 82 /R put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 112 /p put dup 115 /s put dup 116 /t put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{0 -250 1171 750}readonly def currentdict end currentfile eexec 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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 73 /I put dup 78 /N put dup 82 /R put readonly def /FontBBox{-55 -420 2343 920}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMCSC10 %!PS-AdobeFont-1.1: CMCSC10 1.0 %%CreationDate: 1991 Aug 18 17:46:49 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMCSC10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMCSC10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 46 /period put dup 49 /one put dup 50 /two put dup 52 /four put dup 65 /A put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R 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 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 120 /x put dup 121 /y put readonly def /FontBBox{14 -250 1077 750}readonly def currentdict end currentfile eexec 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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 8 /Phi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex 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 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /Delta put dup 43 /plus put dup 49 /one put readonly def /FontBBox{-20 -250 1193 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 21 /greaterequal put dup 26 /propersubset put dup 106 /bar put readonly def /FontBBox{-30 -955 1185 779}readonly def currentdict end currentfile eexec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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 21 /lambda put dup 22 /mu put dup 25 /pi put dup 65 /A put dup 82 /R put dup 100 /d put dup 101 /e put dup 106 /j put dup 107 /k put dup 115 /s put dup 116 /t put readonly def /FontBBox{37 -250 1349 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA06DA87FC7163A5A2A756A598FAB07633 89DE8BB201D5DB4627484A80A431B6AFDBBBF23D4157D4AFE17E6B1C853DD417 25F84CD55402AB88AB7EEFDEDBF2C2C731BD25567C53B474CCF739188A930039 098A197F9C4BE7594D79442B2C8A67447DE44698321145D7689B91EF235EA80E B600AA8E238064F154284096C4C2554EFE8DDF13AFF8D3CE30E0999375C0FEE6 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 50 /element put readonly def /FontBBox{-4 -948 1329 786}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 21 /lambda put dup 27 /sigma put dup 100 /d put dup 107 /k put dup 110 /n put dup 116 /t put readonly def /FontBBox{11 -250 1241 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 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CB8FF0D5A6401779FD528980C795EE3D50904F308F627A23C986A2AEBF5B8D2C AB98632C9E1AE7E48D4984A2049D8B17B9BBD6797F7D783197B3FBC0074412A4 F1C7EEE750AF35E34A34A98B8FD5FF9B282AFE9945AED07C12831BF3B1B99AC2 9561FD34C3465F1F79CF27F9E48AC9BF529AAE117BF6049D89A58C43E32428BA 1813EBFCEC2ACB4F66319A99612A73879446E3991CA389BCB4A543BBC997BA1E A69F5A3FA6DD34B934EE1ACB6ED9681F1C698697C3918DB165CE183037D39D9B 14C26662C9C53F4154FD86E451A1F21C89BA2B8C6B3E4BFC0EEBDB2B0964A9C7 DBD2869A6A385DE255561E6270453CD809AF86B4D9FAE4E5A5343508020340A1 47992EC4CFA1D045957DA5A38A674BFD8220D8A38C616016A19390542B9B8D0B EC784B59940F58881382A9BF8265736F13948B096C2DDDBBF5159D45C34FBC33 F7CBA110561B71CE6E86A19B57FBB33D3793D4271813BCBA180C0909B3B7A2A1 6F6D0FBEF0B70CB73FE399F9E09943B468F47A310AF35C6B43A2E14B77FC32C2 BF2D5DA466BD07D82BB87E85121A239E5FFB70C13191D22E256946C13C1F8A3D F217B1202302B1568FFF7C548020DE165F007D93504A4F9BD73CCEB485D98DB7 39E9326BBB59BC79F950E60123C9249D072B5C073535C6C840BC4CF620BFDDE1 54C57032DF4286CB539099 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: TeX-cmex8 %!PS-AdobeFont-1.0: TeX-cmex8 001.001 % Filtered by type1fix.pl 0.05 %%EndComments 13 dict dup begin /FontInfo 16 dict dup begin /Copyright (see\040copyright\040of\040original\040TeX\040font) def /FamilyName (TeX\040cmex8) def /FullName (TeX\040cmex8\040Regular) def /ItalicAngle 0 def /Notice (converted\040after\040April\0402001) def /UnderlinePosition -100 def /UnderlineThickness 50 def /Weight (Regular) def /isFixedPitch false def /version (001.001) def end readonly def /FontName /TeX-cmex8 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 80 /P put readonly def /FontBBox {-29 -2957 1554 772} readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 21 /lambda put dup 27 /sigma put dup 61 /slash put dup 90 /Z put dup 100 /d put dup 101 /e put dup 114 /r put dup 116 /t put readonly def /FontBBox{-24 -250 1110 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /Lambda put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 16 /dotlessi put dup 19 /acute put dup 20 /caron put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 127 /dieresis put readonly def /FontBBox{-36 -250 1070 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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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 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 69 /E put dup 70 /F put dup 72 /H put dup 73 /I put dup 76 /L put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 97 /a put dup 101 /e put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 121 /y put readonly def 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cleartomark %%EndFont TeXDict begin 39158280 55380996 1000 600 600 (heat090906.dvi) @start /Fa 128[35 9[35 35 35 35 3[35 35 35 35 1[35 35 35 1[35 35 2[35 7[35 1[35 35 35 35 35 35 1[35 1[35 35 35 35 1[35 35 35 35 35 35 1[35 35 35 5[35 35 1[35 35 35 4[35 1[35 35 35 44[{}45 66.4176 /CMTT8 rf /Fb 198[41 41 41 41 41 41 41 41 41 41 48[{}10 66.4176 /CMBX8 rf /Fc 252[65 3[{}1 83.022 /MSAM10 rf /Fd 134[40 39 55 38 45 28 34 35 1[42 42 47 68 21 38 1[25 42 38 25 38 42 38 38 42 12[59 1[61 1[56 3[52 5[54 16[42 2[42 42 2[25 1[25 31[47 12[{}34 83.022 /CMTI10 rf /Fe 173[42 82[{}1 58.1154 /MSBM7 rf /Ff 143[46 5[19 57[18 44[31 2[45{}5 41.511 /CMSY5 rf /Fg 128[36 4[29 35 33 47 33 38 24 29 30 33 36 36 40 58 18 33 1[22 36 33 22 33 36 33 33 36 7[53 3[53 51 40 52 2[55 53 64 45 55 1[27 53 55 46 48 54 51 50 53 1[36 4[22 12[26 22 30[42 40 44 11[{}52 66.4176 /CMTI8 rf /Fh 143[55 1[34 3[20 55[45 66 19 14[66 11[52 52 16[34 2[52{}11 58.1154 /CMSY7 rf /Fi 200[28 1[28 28 28 28 28 5[43 32[40 6[38 3[{}9 41.511 /CMR5 rf /Fj 142[83 1[55 55 18[92 46 1[120 5[39 1[88 14[73 73 3[74 1[74 1[74 6[73 73 14[66 66 4[62 62 4[44 44 61 61 50 50 3[28 2[48 48 6[38 38{}30 83.022 /CMEX10 rf /Fk 135[35 10[55 11[33 2[33 19 1[19 3[67 49 49 47 37 48 1[45 51 49 60 41 2[24 49 51 43 45 50 47 46 49 3[51 1[19 19 33 33 33 33 33 33 33 33 33 33 4[51 1[26 26 1[51 27[47 6[45 1[55 1[{}46 58.1154 /CMR7 rf /Fl 141[69 1[69 1[42 2[42 23 32 32 42 42 9[55 55 14[57 21[0 3[55 83 15[83 5[65 65 4[65 65 9[65 1[65 5[65 23 65{}25 83.022 /CMSY10 rf /Fm 133[39 41 47 2[48 30 39 37 1[42 40 50 73 25 43 34 29 2[41 39 43 36 36 44 6[57 2[78 3[51 63 2[63 67 81 57 1[46 36 69 1[53 1[69 59 63 62 44 1[65 42 65 23 23 13[23 9[39 52 1[52 2[36 47 1[47 36 1[50 48 48 3[36 1[37 43 47 53 11[{}59 83.022 /CMMI10 rf /Fn 133[32 34 38 3[25 31 2[34 3[21 35 27 23 2[32 31 35 30 1[36 14[50 3[53 1[46 1[37 8[50 2[52 34 52 20 30[31 38 1[39 2[40 39 39 6[35 38 43 11[{}33 58.1154 /CMMI7 rf /Fo 173[60 3[60 4[32 73[{}3 83.022 /MSBM10 rf /Fp 134[51 51 2[51 49 38 50 1[46 53 51 62 43 53 1[25 51 1[44 46 52 49 48 51 12[65 1[66 1[62 1[68 82 57 2[34 68 6[68 12[46 1[46 46 49[{}32 83.022 /CMCSC10 rf /Fq 128[42 42 3[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 2[42 23 42 23 51 62 1[85 62 62 60 46 61 1[57 65 62 76 52 65 43 30 62 65 54 57 63 60 59 62 3[65 1[23 23 42 42 42 42 42 42 42 42 42 42 1[23 28 23 65 1[32 32 23 4[42 19[69 46 46 48 60 1[60 4[58 1[69 52{}86 83.022 /CMR10 rf /Fr 206[30 5[47 41[50 1[{}3 49.8132 /CMR6 rf /Fs 173[48 82[{}1 66.4176 /MSBM7 rf /Ft 149[20 79[55 4[55 19[20 55{}5 66.4176 /CMSY8 rf /Fu 139[23 28 7[31 24 4[28 31 17[43 16[43 39[35 2[35 35 21[{}11 41.511 /CMMI5 rf /Fv 177[36 78[{}1 49.8132 /MSBM7 rf /Fw 205[42 49[48{}2 49.8132 /CMSY6 rf /Fx 139[24 5[38 2[33 6[32 72[36 5[37 21[{}6 49.8132 /CMMI6 rf /Fy 175[74 80[{}1 66.4176 /TeX-cmex8 rf /Fz 139[25 1[32 12[33 36 9[48 28[35 33[40 5[41 21[{}8 66.4176 /CMMI8 rf /FA 139[39 30 40 14[39 39 41 31[54 18[21 46[{}8 66.4176 /CMCSC10 rf /FB 128[35 3[35 31 37 37 51 37 39 27 28 28 37 39 35 39 59 20 37 1[20 39 35 22 31 39 31 39 35 3[20 1[20 43 53 53 72 53 53 51 39 52 1[48 55 53 65 44 55 36 25 53 55 46 48 54 51 50 53 3[55 2[20 35 35 35 35 35 35 35 35 35 35 1[20 24 20 2[27 27 20 55 17[35 35 2[20 2[39 39 41 51 6[49 3[{}81 66.4176 /CMR8 rf /FC 134[50 4[37 38 39 1[53 48 53 80 27 2[27 53 2[44 3[46 10[72 73 66 53 72 1[65 72 75 1[57 2[36 75 1[60 63 73 69 68 72 7[48 48 48 48 48 48 48 48 48 48 1[27 46[{}41 83.022 /CMBX10 rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%BeginPaperSize: a4 /setpagedevice where { pop << /PageSize [595 842] >> setpagedevice } { /a4 where { pop a4 } if } ifelse %%EndPaperSize end %%EndSetup %%Page: 1 1 TeXDict begin 1 0 bop 594 716 a FC(UNIVERSAL)42 b(BOUNDS)f(F)m(OR)g (TRA)m(CES)h(OF)f(THE)h(DIRICHLET)1430 832 y(LAPLA)m(CE)h(OPERA)-8 b(TOR)1256 1081 y FB(LEANDER)23 b(GEISINGER)48 b(&)f(TIMO)24 b(WEIDL)754 1311 y FA(Abstra)o(ct.)42 b FB(W)-6 b(e)29 b(deriv)n(e)f(upp)r(er)g(b)r(ounds)g(for)f(the)i(trace)f(of)g(the)g (heat)h(k)n(ernel)f Fz(Z)5 b FB(\()p Fz(t)p FB(\))28 b(=)754 1345 y Fy(P)829 1415 y Fx(k)q Fw(2)p Fv(N)956 1394 y Fz(e)989 1371 y Fw(\000)p Fx(\025)1074 1383 y Fu(k)1110 1371 y Fx(t)1138 1394 y FB(,)h(where)g(\()p Fz(\025)1464 1406 y Fx(k)1502 1394 y FB(\))1529 1406 y Fx(k)q Fw(2)p Fv(N)1674 1394 y FB(denote)h(the)f(eigen)n(v)l(alues)h (of)e(the)h(Diric)n(hlet)f(Laplace)754 1486 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Fm(\025)1258 3838 y Fk(1)1296 3826 y Fq(,)h(where)f Fm(j)1619 3838 y Fn(k)q(;)p Fk(1)1740 3826 y Fq(denotes)g(the)h(\014rst)f(zero)g (of)g(the)h(Bessel-function)f Fm(J)3380 3838 y Fn(k)3421 3826 y Fq(.)555 3926 y(F)-7 b(or)27 b Fm(r)f Fl(2)d Fo(R)28 b Fq(put)g(\()p Fm(r)r Fq(\))1188 3938 y Fk(+)1268 3926 y Fq(=)22 b(max)p Fl(f)p Fm(r)n(;)14 b Fq(0)p Fl(g)26 b Fq(and)i(for)f Fm(d)c Fl(2)g Fo(N)28 b Fq(let)456 4160 y(\(12\))858 b Fm(\033)1509 4172 y Fn(d)1585 4160 y Fq(=)1686 3990 y Fj(8)1686 4065 y(<)1686 4214 y(:)1802 4060 y Fq(5)p Fm(=)p Fq(2)81 b(if)j Fm(d)23 b Fq(=)g(2)1843 4159 y(2)124 b(if)84 b Fm(d)23 b Fq(=)g(3)1802 4259 y(3)p Fm(=)p Fq(2)81 b(if)j Fm(d)23 b Fl(\025)g Fq(4)2415 4160 y Fm(:)456 4403 y Fq(Finally)-7 b(,)27 b(let)h(\012)23 b Fl(\032)g Fo(R)1105 4373 y Fn(d)1171 4403 y Fq(b)r(e)28 b(an)f(arbitrary)f(op)r (en)i(set)f(with)h(\014nite)h(v)n(olume)e Fl(j)p Fq(\012)p Fl(j)p Fq(.)456 4559 y FC(Theorem)k(1.)41 b Fd(L)l(et)29 b Fm(\025)24 b Fl(2)f Fq([)1296 4537 y(~)1293 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a(2)2417 1381 y Fl(\000)g Fm(r)r Fq(\()p Fm(t)p Fq(\))456 1606 y(with)35 b(an)f(explicit)h(remainder)e(term)i Fm(r)r Fq(\()p Fm(t)p Fq(\))h(=)e Fm(O)r Fq(\()p Fm(t)2074 1576 y Fh(\000)2137 1550 y Fu(d)p Ff(\000)p Fi(3)p 2137 1563 V 2175 1596 a(2)2255 1606 y Fq(\))h(as)f Fm(t)h Fl(!)g Fq(0+.)57 b(W)-7 b(e)35 b(note)f(that)h(the)456 1706 y(b)r(ound)24 b(captures)f(the)i(main)f(asymptotic)f(b)r(eha)n (viour)g(of)h Fm(Z)6 b Fq(\()p Fm(t)p Fq(\))25 b(as)e Fm(t)h Fq(tends)h(to)f(zero:)34 b(The)24 b(\014rst)456 1805 y(term)i(equals)g(the)h(leading)f(term)h(of)g(the)g(short)f(time)h (asymptotics)f(of)h Fm(Z)6 b Fq(\()p Fm(t)p Fq(\))26 b(and)h(the)g(second)456 1905 y(term)c(sho)n(ws)g(the)h(correct)e (order)h(in)h Fm(t)f Fq(compared)g(with)h(the)g(second)f(term)h(of)g (the)g(asymptotic)456 2005 y(expansion.)555 2104 y(Moreo)n(v)n(er,)i (note)j(that)g(in)g(view)f(of)h(\(9\))g(the)g(b)r(ound)g(from)f (Theorem)g(1)g(deca)n(ys)g(exp)r(onen-)456 2204 y(tially)33 b(as)g Fm(t)h Fq(tends)g(to)g(in\014nit)n(y)-7 b(.)56 b(More)33 b(precisely)-7 b(,)35 b(it)f(follo)n(ws)f(that)h(the)g(b)r (ound)g(is)g(of)g(order)456 2306 y Fm(O)r Fq(\()p Fm(t)583 2276 y Fn(\033)621 2285 y Fu(d)657 2276 y Fk(+1)759 2306 y Fq(exp\()p Fl(\000)986 2284 y Fq(~)983 2306 y Fm(\025)14 b(t)p Fq(\)\))28 b(as)f Fm(t)c Fl(!)g(1)p Fq(.)456 2463 y Fd(R)l(emark.)48 b Fq(If)38 b(w)n(e)f(c)n(ho)r(ose)g Fm(\025)j Fq(=)1503 2441 y(~)1499 2463 y Fm(\025)e Fq(in)n(tro)r(duced) g(in)g(\(11\))f(w)n(e)h(arriv)n(e)e(at)h(a)h(univ)n(ersal)e(upp)r(er) 456 2563 y(b)r(ound)d(on)f Fm(Z)6 b Fq(\()p Fm(t)p Fq(\))33 b(dep)r(ending)g(only)g(on)f Fl(j)p Fq(\012)p Fl(j)h Fq(and)g(not)f(including)h(an)n(y)f(explicit)h(information)456 2663 y(on)c Fm(\025)621 2675 y Fk(1)659 2663 y Fq(.)43 b(F)-7 b(or)29 b(the)h(explicit)g(statemen)n(t)f(see)g(Corollary)f(9)h (in)h(section)f(4.)42 b(This)30 b(result)f(implies)456 2762 y(the)f(conjectured)f(inequalit)n(y)h(\(6\))g(for)f(dimensions)g Fm(d)d Fl(\024)e Fq(633.)555 2913 y(As)34 b(stated)g(ab)r(o)n(v)n(e,)g (our)f(pro)r(of)g(of)h(Theorem)f(1)g(relies)g(on)h(impro)n(v)n(ed)e(b)r (ounds)i(for)g(Riesz)456 3013 y(means)27 b(of)g(eigen)n(v)-5 b(alues.)36 b(Let)28 b(us)f(state)h(the)g(corresp)r(onding)d(result.) 456 3173 y FC(Theorem)31 b(2.)41 b Fd(L)l(et)29 b Fm(\025)24 b Fl(2)f Fq([)1296 3151 y(~)1293 3173 y Fm(\025;)14 b(\025)1426 3185 y Fk(1)1464 3173 y Fq(])30 b Fd(and)g Fm(\033)d(>)22 b(\033)1886 3185 y Fn(d)1955 3173 y Fd(and)30 b(put)g Fm(\034)2297 3185 y Fk(\012)2371 3173 y Fq(=)2470 3140 y Fn(\031)2511 3115 y Fi(2)2543 3140 y Fn(d)2578 3115 y Fi(2)p 2469 3154 142 4 v 2469 3228 a Fh(j)p Fk(\012)p Fh(j)2567 3184 y Fi(2)p 2566 3191 31 3 v 2566 3222 a Fu(d)2621 3173 y Fd(.)38 b(Then)31 b(the)f(estimate)758 3409 y Fm(R)821 3421 y Fn(\033)866 3409 y Fq(\(\003\))37 b Fl(\024)f Fm(L)1183 3374 y Fn(cl)1183 3429 y(\033)o(;d)1292 3409 y Fl(j)p Fq(\012)p Fl(j)1432 3388 y Fq(^)1412 3409 y Fm(B)1493 3292 y Fj(\022)1569 3353 y Fm(\025)p 1564 3390 58 4 v 1564 3466 a Fq(\003)1632 3409 y Fm(;)14 b(\033)1716 3421 y Fn(d)1773 3409 y Fq(+)1866 3353 y Fm(d)p 1866 3390 44 4 v 1867 3466 a Fq(2)1938 3409 y(+)k(1)p Fm(;)c(\033)21 b Fl(\000)d Fm(\033)2298 3421 y Fn(d)2337 3292 y Fj(\023)2426 3409 y Fq(\003)2484 3374 y Fn(\033)r Fk(+)2585 3352 y Fu(d)p 2585 3361 31 3 v 2586 3394 a Fi(2)2662 3409 y Fl(\000)32 b Fq(\()p Fm(S)5 b Fq(\(\003)p Fm(;)14 b(\025)p Fq(\)\))3086 3421 y Fk(+)456 3591 y Fd(holds)31 b(true)e(for)h(al)t(l)h Fq(\003)23 b Fl(\025)g Fm(\025)p Fd(,)30 b(wher)l(e)456 3787 y Fq(\(14\))41 b Fm(S)5 b Fq(\(\003)p Fm(;)14 b(\025)p Fq(\))37 b(=)g Fm(L)1104 3753 y Fn(cl)1104 3808 y(\033)o(;d)p Fh(\000)p Fk(1)1298 3787 y Fl(j)p Fq(\012)p Fl(j)1414 3727 y Fu(d)p Ff(\000)p Fi(1)p 1414 3740 104 3 v 1450 3773 a Fu(d)1546 3787 y Fq(\003)1604 3753 y Fn(\033)r Fk(+)1705 3727 y Fu(d)p Ff(\000)p Fi(1)p 1705 3740 V 1743 3773 a(2)1833 3726 y Fm(B)1914 3658 y Fj(\000)1962 3693 y Fk(1)p 1962 3707 34 4 v 1962 3754 a(2)2005 3726 y Fm(;)14 b(\033)2089 3738 y Fn(d)2146 3726 y Fq(+)2239 3693 y Fn(d)p Fk(+1)p 2239 3707 119 4 v 2282 3754 a(2)2368 3658 y Fj(\001)p 1833 3768 574 4 v 2099 3844 a Fq(2)2436 3766 y(^)2416 3787 y Fm(B)2497 3670 y Fj(\022)2573 3731 y Fm(\025)p 2568 3768 58 4 v 2568 3844 a Fq(\003)2636 3787 y Fm(;)g(\033)2720 3799 y Fn(d)2777 3787 y Fq(+)2870 3731 y Fm(d)19 b Fq(+)f(1)p 2870 3768 187 4 v 2942 3844 a(2)3066 3787 y Fm(;)c(\033)22 b Fl(\000)c Fm(\033)3302 3799 y Fn(d)3341 3670 y Fj(\023)456 3969 y Fd(if)30 b Fm(\025)24 b Fl(\025)e Fm(\034)731 3981 y Fk(\012)813 3969 y Fd(,)456 4151 y Fq(\(15\))362 b Fm(S)5 b Fq(\(\003)p Fm(;)14 b(\025)p Fq(\))37 b(=)f Fm(L)1424 4117 y Fn(cl)1424 4172 y(\033)o(;d)1533 4151 y Fl(j)p Fq(\012)p Fl(j)14 b Fq(\003)1711 4117 y Fn(\033)r Fk(+)1812 4095 y Fu(d)p 1812 4104 31 3 v 1813 4137 a Fi(2)1882 4095 y Fq(1)p 1881 4132 44 4 v 1881 4208 a Fm(d)1968 4130 y Fq(^)1948 4151 y Fm(B)2029 4034 y Fj(\022)2105 4095 y Fm(\025)p 2100 4132 58 4 v 2100 4208 a Fq(\003)2168 4151 y Fm(;)g(\033)2252 4163 y Fn(d)2309 4151 y Fq(+)2402 4095 y Fm(d)p 2402 4132 44 4 v 2403 4208 a Fq(2)2474 4151 y(+)k(1)p Fm(;)c(\033)21 b Fl(\000)d Fm(\033)2834 4163 y Fn(d)2873 4034 y Fj(\023)456 4333 y Fd(if)30 b Fm(\025)24 b(<)e(\034)731 4345 y Fk(\012)813 4333 y Fd(and)30 b Fq(\003)23 b Fm(<)f(\034)1178 4345 y Fk(\012)1230 4333 y Fd(,)30 b(or)582 4530 y Fm(S)5 b Fq(\(\003)p Fm(;)14 b(\025)p Fq(\))23 b(=)14 b Fm(L)1004 4495 y Fn(cl)1004 4550 y(\033)o(;d)p Fh(\000)p Fk(1)1197 4530 y Fl(j)p Fq(\012)p Fl(j)1313 4470 y Fu(d)p Ff(\000)p Fi(1)p 1313 4483 104 3 v 1350 4515 a Fu(d)1431 4530 y Fq(\003)1489 4495 y Fn(\033)r Fk(+)1591 4470 y Fu(d)p Ff(\000)p Fi(1)p 1591 4483 V 1628 4515 a(2)1719 4468 y Fm(B)1799 4401 y Fj(\000)1847 4436 y Fk(1)p 1847 4450 34 4 v 1847 4497 a(2)1891 4468 y Fm(;)g(\033)1975 4480 y Fn(d)2032 4468 y Fq(+)2125 4436 y Fn(d)p Fk(+1)p 2125 4450 119 4 v 2168 4497 a(2)2254 4401 y Fj(\001)p 1719 4511 574 4 v 1984 4587 a Fq(2)2321 4509 y(^)2302 4530 y Fm(B)2383 4413 y Fj(\022)2454 4474 y 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b(<)e(\034)731 4956 y Fk(\012)813 4944 y Fd(and)30 b Fq(\003)23 b Fl(\025)f Fm(\034)1178 4956 y Fk(\012)1230 4944 y Fd(.)456 5095 y(R)l(emark.)48 b Fq(Again)37 b(w)n(e)g(can)g(c)n (ho)r(ose)f Fm(\025)i Fq(as)f(in)g(\(11\))h(and)f(w)n(e)g(arriv)n(e)f (at)h(a)g(univ)n(ersal)f(b)r(ound)456 5195 y(dep)r(ending)28 b(only)f(on)g Fl(j)p Fq(\012)p Fl(j)p Fq(.)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 456 251 a Fk(6)814 b(LEANDER)22 b(GEISINGER)44 b(&)h(TIMO)23 b(WEIDL)456 450 y Fd(R)l(emark.)42 b Fq(In)27 b(view)h(of)f(\(10\))h(Theorem)e(2)i(can)f(b)r(e)h(read)f (as)594 634 y Fm(R)657 646 y Fn(\033)702 634 y Fq(\(\003\))37 b Fl(\024)g Fm(L)1020 600 y Fn(cl)1020 655 y(\033)o(;d)1129 634 y Fl(j)p Fq(\012)p Fl(j)14 b Fq(\003)1307 600 y Fn(\033)r Fk(+)1408 578 y Fu(d)p 1408 587 31 3 v 1409 620 a Fi(2)1471 634 y Fl(\000)1564 578 y Fq(1)p 1564 615 42 4 v 1564 691 a(2)1629 634 y Fm(B)1710 517 y Fj(\022)1781 578 y Fq(1)p 1781 615 V 1781 691 a(2)1833 634 y Fm(;)g(\033)1917 646 y Fn(d)1974 634 y 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Fq(for)456 1245 y Fm(x)23 b Fl(2)h Fo(R)665 1215 y Fn(d)703 1245 y Fq(.)37 b(F)-7 b(or)27 b(a)g(giv)n(en)g(\003)c Fm(>)f Fq(0)28 b(de\014ne)1726 1388 y Fm(l)1751 1400 y Fk(\003)1837 1388 y Fq(=)37 b Fm(\031)s Fq(\003)2047 1354 y Fh(\000)2109 1332 y Fi(1)p 2109 1341 29 3 v 2109 1374 a(2)2151 1388 y Fm(:)555 1523 y Fq(No)n(w)23 b(consider)e(an)i(op) r(en)g(set)f(\012)h Fl(\032)g Fo(R)1720 1492 y Fn(d)1759 1523 y Fq(.)35 b(Eac)n(h)22 b(section)g(\012\()p Fm(x)2430 1492 y Fh(0)2454 1523 y Fq(\))h(=)g Fl(f)p Fm(x)2686 1535 y Fn(d)2748 1523 y Fl(2)g Fo(R)37 b Fq(:)g(\()p Fm(x)3062 1492 y Fh(0)3086 1523 y Fm(;)14 b(x)3170 1535 y Fn(d)3209 1523 y Fq(\))24 b Fl(2)f Fq(\012)p Fl(g)456 1622 y Fq(is)g(a)h(one-dimensional)f(op)r(en)h(set)g(and)f(consists)g (of)h(at)g(most)g(coun)n(tably)f(man)n(y)g(op)r(en)h(disjoin)n(t)456 1722 y(in)n(terv)-5 b(als)26 b Fm(J)834 1734 y Fn(k)875 1722 y Fq(\()p Fm(x)954 1692 y Fh(0)978 1722 y Fq(\),)i Fm(k)e Fq(=)d(1)p Fm(;)14 b(:)g(:)g(:)f(;)h(N)9 b Fq(\()p Fm(x)1599 1692 y Fh(0)1623 1722 y Fq(\))23 b Fl(\024)g(1)p Fq(.)37 b(Let)28 b Fm(\024)p Fq(\()p Fm(x)2185 1692 y Fh(0)2209 1722 y Fm(;)14 b Fq(\003\))23 b Fl(\032)f Fo(N)28 b Fq(b)r(e)g(the)g(subset)g(of)f(all)h(those)456 1821 y(indices)22 b Fm(k)s Fq(,)i(for)e(whic)n(h)h(the)g(corresp)r(onding)d (in)n(terv)-5 b(al)23 b Fm(J)2179 1833 y Fn(k)2219 1821 y Fq(\()p Fm(x)2298 1791 y Fh(0)2323 1821 y Fq(\))g(is)f(strictly)h (longer)e(than)i Fm(l)3194 1833 y Fk(\003)3243 1821 y Fq(.)35 b(The)456 1921 y(n)n(um)n(b)r(er)27 b(these)h(indices)f(is)h (denoted)f(b)n(y)h Fm(\037)p Fq(\()p Fm(x)1887 1891 y Fh(0)1911 1921 y Fm(;)14 b Fq(\003\).)36 b(Put)850 2072 y(\012)910 2084 y Fk(\003)959 2072 y Fq(\()p Fm(x)1038 2038 y Fh(0)1062 2072 y Fq(\))23 b(=)1308 1993 y Fj([)1205 2175 y Fn(k)q Fh(2)p Fn(\024)p Fk(\()p Fn(x)1389 2159 y Ff(0)1411 2175 y Fn(;)p Fk(\003\))1516 2072 y Fm(J)1562 2084 y Fn(k)1603 2072 y Fq(\()p Fm(x)1682 2038 y Fh(0)1706 2072 y Fq(\))84 b(and)e(\012)2098 2084 y Fk(\003)2171 2072 y Fq(=)2340 1993 y Fj([)2259 2177 y Fn(x)2297 2161 y Ff(0)2319 2177 y Fh(2)p Fe(R)2406 2161 y Fu(d)p Ff(\000)p Fi(1)2513 2072 y Fl(f)p Fm(x)2602 2038 y Fh(0)2626 2072 y Fl(g)18 b(\002)g Fq(\012)2829 2084 y Fk(\003)2878 2072 y Fq(\()p Fm(x)2957 2038 y Fh(0)2981 2072 y Fq(\))c Fm(:)456 2299 y Fq(Ob)n(viously)24 b(the)j(set)f(\012)1174 2311 y Fk(\003)1249 2299 y Fq(is)g(the)g(subset)g(of)g(\012,)h(where)e(\012) h(is)g("wide)g(enough")f(in)h Fm(x)3036 2311 y Fn(d)3075 2299 y Fq(-direction.)456 2399 y(The)h(quan)n(tit)n(y)1467 2536 y Fm(d)1510 2548 y Fk(\003)1560 2536 y Fq(\(\012\))d(=)1795 2423 y Fj(Z)1841 2612 y Fe(R)1883 2595 y Fu(d)p Ff(\000)p Fi(1)2009 2536 y Fm(\037)14 b Fq(\()p Fm(x)2154 2502 y Fh(0)2178 2536 y Fm(;)g Fq(\003\))g Fm(dx)2409 2502 y Fh(0)456 2695 y Fq(is)29 b(an)g(e\013ectiv)n(e)h(area)e(of)h(the)h (pro)5 b(jection)29 b(of)g(\012)1957 2707 y Fk(\003)2036 2695 y Fq(on)n(to)g(the)h Fm(d)20 b Fl(\000)f Fq(1-dimensional)28 b(h)n(yp)r(erplane)456 2795 y(\()p Fm(x)535 2765 y Fh(0)559 2795 y Fm(;)14 b Fq(0\))27 b(coun)n(ting)g(also)g(the)g(m)n (ultiplicities)i(of)e(the)h(su\016cien)n(tly)g(long)f(in)n(terv)-5 b(als)27 b Fm(J)3054 2807 y Fn(k)3094 2795 y Fq(\()p Fm(x)3173 2765 y Fh(0)3197 2795 y Fq(\).)555 2894 y(Moreo)n(v)n(er,)e (for)i Fm(\026)c Fl(\025)g Fq(2)k(put)456 3124 y(\(17\))335 b Fm(")p Fq(\()p Fm(\026)p Fq(\))24 b(=)39 b(inf)1203 3178 y Fn(A)p Fh(\025)p Fk(1)1352 2957 y Fj(0)1352 3106 y(@)1425 3011 y(Z)1508 3031 y Fn(A)1471 3199 y Fk(0)1576 3032 y Fj(\020)1625 3124 y Fq(1)18 b Fl(\000)1791 3091 y Fn(t)1816 3066 y Fi(2)p 1778 3105 83 4 v 1778 3152 a Fn(A)1828 3136 y Fi(2)1871 3032 y Fj(\021)1920 3049 y Fn(\026)1920 3182 y Fk(+)1989 3124 y Fm(dt)h Fl(\000)2165 3045 y Fj(X)2164 3224 y Fn(k)q Fh(\025)p Fk(1)2299 3032 y Fj(\020)2349 3124 y Fq(1)f Fl(\000)2508 3091 y Fn(k)2544 3066 y Fi(2)p 2502 3105 V 2502 3152 a Fn(A)2552 3136 y Fi(2)2594 3032 y Fj(\021)2644 3049 y Fn(\026)2644 3182 y Fk(+)2699 2957 y Fj(1)2699 3106 y(A)2795 3124 y Fm(>)k Fq(0)14 b Fm(:)456 3353 y Fq(W)-7 b(e)42 b(are)f(no)n(w)g(in)h(the)g(p) r(osition)g(to)g(state)f(the)i(impro)n(v)n(ed)d(Berezin-Li-Y)-7 b(au)41 b(b)r(ound)h(from)456 3453 y([W)-7 b(ei08]:)456 3585 y FC(Prop)s(osition)30 b(3.)41 b Fd(F)-6 b(or)30 b(any)g(op)l(en)g(domain)h Fq(\012)23 b Fl(\032)g Fo(R)2125 3555 y Fn(d)2163 3585 y Fd(,)30 b Fm(\033)d Fl(\025)22 b Fq(3)p Fm(=)p Fq(2)29 b Fd(and)h(al)t(l)h Fq(\003)22 b Fm(>)h Fq(0)29 b Fd(the)h(b)l(ound)456 3738 y Fq(\(18\))217 b Fm(R)884 3750 y Fn(\033)929 3738 y Fq(\(\003\))37 b Fl(\024)g Fm(L)1247 3704 y Fn(cl)1247 3758 y(\033)o(;d)1369 3738 y Fl(j)p Fq(\012)1452 3750 y Fk(\003)1502 3738 y Fl(j)28 b Fq(\003)1611 3704 y Fn(\033)r Fk(+)1712 3681 y Fu(d)p 1712 3690 31 3 v 1713 3724 a Fi(2)1789 3738 y Fl(\000)k Fm(")1939 3671 y Fj(\000)1976 3738 y Fm(\033)22 b Fq(+)2138 3705 y Fn(d)p Fh(\000)p Fk(1)p 2138 3719 120 4 v 2181 3766 a(2)2268 3671 y Fj(\001)2333 3738 y Fm(L)2390 3704 y Fn(cl)2390 3758 y(\033)o(;d)p Fh(\000)p Fk(1)2570 3738 y Fm(d)2613 3750 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b(in)n(terc)n(hange)e(no)n(w)h (the)g(roles)g(of)g Fm(x)1798 915 y Fn(d)1866 903 y Fq(and)g Fm(x)2075 915 y Fn(i)2131 903 y Fq(for)g Fm(i)c Fq(=)g(1)p Fm(;)14 b(:)g(:)g(:)f(;)h(d)19 b Fl(\000)g Fq(1)28 b(and)g(in)n(tro)r (duce)g(in)456 1003 y(the)d(same)g(w)n(a)n(y)f(the)i(distribution)g (functions)f Fm(m)1988 1015 y Fn(i)2016 1003 y Fq(\()p Fl(\001)p Fq(;)14 b(\012\))26 b(for)f(\012)g(measured)g(along)f(the)i Fm(x)3211 1015 y Fn(i)3239 1003 y Fq(-axes.)456 1102 y(Finally)-7 b(,)27 b(put)1102 1244 y Fm(M)1183 1256 y Fn(i)1210 1244 y Fq(\()p Fm(y)s Fq(;)14 b(\012\))37 b(=)1554 1131 y Fj(Z)1637 1152 y Fn(y)1600 1320 y Fk(0)1691 1244 y Fm(m)1764 1256 y Fn(i)1791 1244 y Fq(\()p Fm(\034)9 b Fq(;)14 b(\012\))g Fm(d\034)94 b Fq(for)82 b Fm(i)23 b Fq(=)g(1)p Fm(;)14 b(:)g(:)g(:)f(;)h(d:)456 1408 y Fq(With)28 b(this)g(notation)f(w)n(e)g(can)g(form)n(ulate)g(a)h(result) f(similar)g(to)g(\(18\):)456 1537 y FC(Prop)s(osition)j(5.)41 b Fd(F)-6 b(or)30 b(any)g(op)l(en)g(domain)h Fq(\012)23 b Fl(\032)g Fo(R)2125 1507 y Fn(d)2163 1537 y Fd(,)30 b Fm(\033)d Fl(\025)22 b Fq(3)p Fm(=)p Fq(2)29 b Fd(and)h(al)t(l)h Fq(\003)22 b Fm(>)h Fq(0)754 1722 y Fm(R)817 1734 y Fn(\033)862 1722 y Fq(\(\003\))37 b Fl(\024)f Fm(L)1179 1687 y Fn(cl)1179 1742 y(\033)o(;d)1302 1609 y Fj(Z)1385 1629 y Fh(1)1382 1775 y Fu(\031)p 1358 1784 84 3 v 1358 1792 a Ff(p)p 1404 1792 38 3 v 35 x Fi(\003)1469 1722 y Fm(m)1542 1734 y Fn(i)1570 1722 y Fq(\()p Fm(\034)9 b Fq(;)14 b(\012\))g Fm(d\034)25 b Fq(\003)1952 1687 y Fn(\033)r Fk(+)2053 1665 y Fu(d)p 2053 1674 31 3 v 2054 1707 a Fi(2)2116 1722 y Fq(+)18 b Fm(\016)2236 1734 y Fn(\033)o(;d)2361 1722 y Fm(m)2434 1734 y Fn(i)2475 1605 y Fj(\022)2585 1666 y Fm(\031)p 2546 1703 127 4 v 2546 1719 a Fl(p)p 2616 1719 58 4 v 2616 1789 a Fq(\003)2683 1722 y(;)c(\012)2780 1605 y Fj(\023)2869 1722 y Fq(\003)2927 1687 y Fn(\033)r Fk(+)3028 1662 y Fu(d)p Ff(\000)p Fi(1)p 3028 1675 104 3 v 3066 1707 a(2)456 1947 y Fd(holds)31 b(true)e(for)h Fm(i)23 b Fq(=)g(1)p Fm(;)14 b(:)g(:)g(:)f(;)h(d)30 b Fd(with)g Fm(\016)1631 1959 y Fn(\033)o(;d)1749 1947 y Fq(=)23 b Fm(\031)s(L)1944 1917 y Fn(cl)1944 1971 y(\033)o(;d)2058 1947 y Fl(\000)18 b Fm(")2194 1880 y Fj(\000)2231 1947 y Fm(\033)k Fq(+)2393 1914 y Fn(d)p Fh(\000)p Fk(1)p 2393 1928 120 4 v 2436 1976 a(2)2523 1880 y Fj(\001)2575 1947 y Fm(L)2632 1917 y Fn(cl)2632 1971 y(\033)o(;d)p Fh(\000)p Fk(1)2811 1947 y Fd(.)456 2093 y(R)l(emark.)39 b Fq(Note)24 b(that)g(in)h(the)g(case)e(of)h Fm(")1721 2026 y Fj(\000)1759 2093 y Fm(\033)e Fq(+)1921 2060 y Fn(d)p Fh(\000)p Fk(1)p 1921 2074 V 1964 2122 a(2)2050 2026 y Fj(\001)2112 2093 y Fq(=)2211 2060 y Fk(1)p 2211 2074 34 4 v 2211 2122 a(2)2268 2093 y Fm(B)2349 2026 y Fj(\000)2387 2093 y Fm(\033)g Fq(+)2549 2060 y Fn(d)p Fk(+1)p 2549 2074 119 4 v 2592 2122 a(2)2678 2093 y Fm(;)2724 2060 y Fk(1)p 2724 2074 34 4 v 2724 2122 a(2)2768 2026 y Fj(\001)2830 2093 y Fq(w)n(e)i(ha)n(v)n(e)f Fm(\016)3174 2105 y Fn(\033)o(;d)3292 2093 y Fq(=)g(0.)456 2193 y(In)h(view)g(of)g (Lemma)g(4)g(this)h(o)r(ccurs,)f(in)h(particular,)e(if)i Fm(\033)i Fq(=)22 b Fm(\033)2408 2205 y Fn(d)2472 2193 y Fq(with)j Fm(\033)2705 2205 y Fn(d)2768 2193 y Fq(in)n(tro)r(duced)f (in)h(\(12\))o(.)456 2322 y Fd(R)l(emark.)42 b Fq(F)-7 b(or)27 b(domians)g(\012)g(with)h(\014nite)g(v)n(olume)f(\(19\))h (yields)1240 2394 y Fj(Z)1323 2414 y Fh(1)1319 2560 y Fu(\031)p 1296 2569 84 3 v 1296 2577 a Ff(p)p 1341 2577 38 3 v 1341 2612 a Fi(\003)1407 2507 y Fm(m)1480 2519 y Fn(i)1508 2507 y Fq(\()p Fm(\034)9 b Fq(;)14 b(\012\))p Fm(d\034)34 b Fq(=)22 b Fl(j)p Fq(\012)p Fl(j)d(\000)f Fm(M)2203 2519 y Fn(i)2244 2390 y Fj(\022)2353 2450 y Fm(\031)p 2315 2487 127 4 v 2315 2504 a Fl(p)p 2384 2504 58 4 v 70 x Fq(\003)2452 2507 y(;)c(\012)2549 2390 y Fj(\023)2637 2507 y Fm(:)456 2717 y Fq(Th)n(us)27 b(w)n(e)g(arriv)n(e)f (at)649 2880 y Fm(R)712 2892 y Fn(\033)757 2880 y Fq(\(\003\))37 b Fl(\024)f Fm(L)1074 2846 y Fn(cl)1074 2901 y(\033)o(;d)1183 2763 y Fj(\022)1244 2880 y Fl(j)p Fq(\012)p Fl(j)19 b(\000)f Fm(M)1533 2892 y Fn(i)1574 2763 y Fj(\022)1683 2824 y Fm(\031)p 1645 2861 127 4 v 1645 2878 a Fl(p)p 1714 2878 58 4 v 70 x Fq(\003)1782 2880 y(;)c(\012)1879 2763 y Fj(\023\023)2014 2880 y Fq(\003)2072 2846 y Fn(\033)r Fk(+)2174 2824 y Fu(d)p 2174 2833 31 3 v 2175 2866 a Fi(2)2237 2880 y Fq(+)k Fm(\016)2357 2892 y Fn(\033)o(;d)2466 2880 y Fm(m)2539 2892 y Fn(i)2580 2763 y Fj(\022)2690 2824 y Fm(\031)p 2651 2861 127 4 v 2651 2878 a Fl(p)p 2721 2878 58 4 v 2721 2948 a Fq(\003)2788 2880 y(;)c(\012)2885 2763 y Fj(\023)2974 2880 y Fq(\003)3032 2846 y Fn(\033)r Fk(+)3133 2820 y Fu(d)p Ff(\000)p Fi(1)p 3133 2833 104 3 v 3171 2866 a(2)456 3060 y Fq(for)27 b Fm(i)22 b Fq(=)h(1)p Fm(;)14 b(:)g(:)g(:)f(;)h(d)p Fq(.)37 b(Av)n(eraging)26 b(o)n(v)n(er)g(all)h(directions)g(one)g(claims)456 3240 y(\(20\))130 b Fm(R)797 3252 y Fn(\033)842 3240 y Fq(\(\003\))37 b Fl(\024)g Fm(L)1160 3205 y Fn(cl)1160 3260 y(\033)o(;d)1268 3122 y Fj(\022)1329 3240 y Fl(j)p Fq(\012)p Fl(j)19 b(\000)f Fm(M)1640 3122 y Fj(\022)1750 3183 y Fm(\031)p 1712 3220 127 4 v 1712 3237 a Fl(p)p 1781 3237 58 4 v 70 x Fq(\003)1848 3240 y(;)c(\012)1945 3122 y Fj(\023\023)2081 3240 y Fq(\003)2139 3205 y Fn(\033)r Fk(+)2240 3183 y Fu(d)p 2240 3192 31 3 v 2241 3225 a Fi(2)2304 3240 y Fq(+)k Fm(\016)2424 3252 y Fn(\033)o(;d)2533 3240 y Fm(m)2620 3122 y Fj(\022)2729 3183 y Fm(\031)p 2691 3220 127 4 v 2691 3237 a Fl(p)p 2760 3237 58 4 v 70 x Fq(\003)2827 3240 y(;)c(\012)2924 3122 y Fj(\023)3013 3240 y Fq(\003)3071 3205 y Fn(\033)r Fk(+)3172 3180 y Fu(d)p Ff(\000)p Fi(1)p 3172 3193 104 3 v 3210 3225 a(2)3290 3240 y Fm(;)456 3419 y Fq(where)863 3572 y Fm(m)p Fq(\()p Fm(t)p Fq(;)g(\012\))83 b(=)1368 3516 y(1)p 1367 3553 44 4 v 1367 3629 a Fm(d)1434 3572 y Fq(\()q Fm(m)1540 3584 y Fk(1)1577 3572 y Fq(\()p Fm(t)p Fq(;)14 b(\012\))19 b(+)f Fl(\001)c(\001)g(\001)k Fq(+)g Fm(m)2141 3584 y Fn(d)2180 3572 y Fq(\()p Fm(t)p Fq(;)c(\012\)\))28 b Fm(;)832 3780 y(M)9 b Fq(\()p Fm(y)s Fq(;)14 b(\012\))83 b(=)1368 3724 y(1)p 1367 3761 V 1367 3837 a Fm(d)1434 3780 y Fq(\()q Fm(M)1548 3792 y Fk(1)1584 3780 y Fq(\()p Fm(y)s Fq(;)14 b(\012\))19 b(+)f Fl(\001)c(\001)g(\001)k Fq(+)g Fm(M)2170 3792 y Fn(d)2209 3780 y Fq(\()p Fm(y)s Fq(;)c(\012\)\))23 b(=)2557 3667 y Fj(Z)2640 3687 y Fn(y)2603 3855 y Fk(0)2694 3780 y Fm(m)p Fq(\()p Fm(t)p Fq(;)14 b(\012\))p Fm(dt)g(:)555 3956 y Fq(Although)33 b(Prop)r(osition)f(5)g (is,)i(in)g(general,)f(not)g(as)f(sharp)g(as)g(\(18\),)i(w)n(e)f (cannot)f(deduce)456 4055 y(it)c(directly)f(quoting)g(Prop)r(osition)f (3,)i(but)g(w)n(e)f(ha)n(v)n(e)g(to)h(mo)r(dify)g(the)g(resp)r(ectiv)n (e)f(pro)r(of)g(from)456 4155 y([W)-7 b(ei08],)27 b(whic)n(h)g(relies)g (on)h(op)r(erator-v)-5 b(alued)25 b(Lieb-Thirring)h(inequalities)i (from)f([L)-9 b(W00].)456 4305 y Fd(Pr)l(o)l(of)30 b(of)h(Pr)l(op)l (osition)g(5.)43 b Fq(Consider)26 b(the)i(quadratic)f(form)520 4477 y Fl(kr)p Fm(u)p Fl(k)720 4435 y Fk(2)720 4502 y Fn(L)766 4485 y Fi(2)798 4502 y Fk(\(\012\))920 4477 y Fl(\000)18 b Fq(\003)c Fl(k)o Fm(u)p Fl(k)1205 4435 y Fk(2)1205 4502 y Fn(L)1251 4485 y Fi(2)1283 4502 y Fk(\(\012\))1423 4477 y Fq(=)37 b Fl(k)o(r)1635 4443 y Fh(0)1659 4477 y Fm(u)p Fl(k)1748 4431 y Fk(2)1748 4502 y Fn(L)1794 4485 y Fi(2)1825 4502 y Fk(\(\012\))1947 4477 y Fq(+)2030 4364 y Fj(Z)2077 4552 y Fe(R)2119 4536 y Fu(d)p Ff(\000)p Fi(1)2245 4477 y Fm(dx)2335 4443 y Fh(0)2372 4364 y Fj(Z)2418 4552 y Fk(\012\()p Fn(x)2529 4536 y Ff(0)2552 4552 y Fk(\))2596 4385 y Fj(\020)2645 4477 y Fl(j)p Fm(@)2712 4489 y Fn(x)2750 4498 y Fu(d)2789 4477 y Fm(u)p Fl(j)2860 4435 y Fk(2)2915 4477 y Fl(\000)19 b Fq(\003)p Fl(j)p Fm(u)p Fl(j)3151 4443 y Fk(2)3187 4385 y Fj(\021)3250 4477 y Fm(dx)3340 4489 y Fn(d)456 4673 y Fq(on)33 b(functions)i Fm(u)e Fq(from)h(the)g(form)g(core)f Fm(C)1825 4642 y Fh(1)1819 4693 y Fk(0)1896 4673 y Fq(\(\012\).)57 b(Here)33 b Fl(r)2371 4642 y Fh(0)2429 4673 y Fq(and)h(\001)2666 4642 y Fh(0)2723 4673 y Fq(denote)g(the)h(gradien)n(t)456 4772 y(and)d(the)h(Laplace)e(op)r(erator)g(in)i(the)f(\014rst)h Fm(d)22 b Fl(\000)f Fq(1)32 b(directions.)51 b(The)32 b(functions)h Fm(u)p Fq(\()p Fm(x)3162 4742 y Fh(0)3185 4772 y Fm(;)14 b Fl(\001)p Fq(\))33 b(sat-)456 4872 y(isfy)26 b(Diric)n(hlet)h(b)r(oundary)f(conditions)g(at)g(the)h(endp)r(oin)n(ts) g(of)f(eac)n(h)g(in)n(terv)-5 b(al)26 b Fm(J)2962 4884 y Fn(k)3003 4872 y Fq(\()p Fm(x)3082 4842 y Fh(0)3106 4872 y Fq(\))h(forming)456 4971 y(\012\()p Fm(x)595 4941 y Fh(0)619 4971 y Fq(\).)35 b(Let)23 b(the)g(b)r(ounded,)h(non-negativ) n(e)d(op)r(erators)g Fm(W)2274 4983 y Fn(k)2315 4971 y Fq(\()p Fm(x)2394 4941 y Fh(0)2418 4971 y Fm(;)14 b Fq(\003\))23 b(b)r(e)g(the)g(negativ)n(e)e(parts)3322 4939 y Fk(4)3378 4971 y Fq(of)p 456 5050 499 4 v 555 5124 a Fk(4)588 5149 y FB(The)k(negativ)n(e)h(part)f(of)f(a)g(real)g(n) n(um)n(b)r(er)h Fz(r)h FB(is)e(giv)n(en)h(b)n(y)f Fz(r)2148 5158 y Fw(\000)2221 5149 y FB(=)d(\()p Ft(j)p Fz(r)r Ft(j)16 b(\000)g Fz(r)r FB(\))p Fz(=)p FB(2)21 b Ft(\025)g FB(0.)33 b(F)-6 b(or)24 b(op)r(erators)h(w)n(e)g(use)456 5232 y(the)f(same)g(con)n(v)n(en)n(tion)i(in)d(the)i(sp)r(ectral)f (sense.)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 456 251 a Fk(8)814 b(LEANDER)22 b(GEISINGER)44 b(&)h(TIMO)23 b(WEIDL)456 450 y Fq(the)29 b(Sturm-Liouville)f(Op)r(erators)f Fl(\000)p Fm(@)1701 420 y Fk(2)1696 477 y Fn(x)1734 486 y Fu(d)1768 477 y Fn(;J)1825 486 y Fu(k)1861 477 y Fk(\()p Fn(x)1925 460 y Ff(0)1947 477 y Fk(\))1996 450 y Fl(\000)19 b Fq(\003)28 b(with)h(Diric)n(hlet)g(b)r(oundary)f(conditions)456 562 y(on)f Fm(J)617 574 y Fn(k)658 562 y Fq(\()p Fm(x)737 532 y Fh(0)761 562 y Fq(\).)37 b(Then)1440 698 y Fm(W)12 b Fq(\()p Fm(x)1609 664 y Fh(0)1633 698 y Fm(;)i Fq(\003\))23 b(=)g Fl(\010)1936 655 y Fn(N)6 b Fk(\()p Fn(x)2059 630 y Ff(0)2080 655 y Fk(\))1936 723 y Fn(k)q Fk(=1)2110 698 y Fm(W)2188 710 y Fn(k)2230 698 y Fq(\()p Fm(x)2309 664 y Fh(0)2333 698 y Fm(;)14 b Fq(\003\))456 831 y(is)27 b(the)h(negativ)n(e)f(part)g(of)1182 1001 y Fl(\000)p Fm(@)1296 967 y Fk(2)1291 1024 y Fn(x)1329 1033 y Fu(d)1363 1024 y Fn(;)p Fk(\012\()p Fn(x)1494 1007 y Ff(0)1516 1024 y Fk(\))1565 1001 y Fl(\000)18 b Fq(\003)23 b(=)f Fl(\010)1881 958 y Fn(N)6 b Fk(\()p Fn(x)2004 933 y Ff(0)2026 958 y Fk(\))1881 1026 y Fn(k)q Fk(=1)2070 909 y Fj(\020)2119 1001 y Fl(\000)p Fm(@)2233 967 y Fk(2)2228 1024 y Fn(x)2266 1033 y Fu(d)2300 1024 y Fn(;J)2357 1033 y Fu(k)2393 1024 y Fk(\()p Fn(x)2457 1007 y Ff(0)2479 1024 y Fk(\))2528 1001 y Fl(\000)18 b Fq(\003)2669 909 y Fj(\021)456 1176 y Fq(sub)5 b(ject)30 b(to)h(Diric)n(hlet)f(b)r(oundary)g(conditions)g (on)g(the)h(endp)r(oin)n(ts)g(of)f(the)h(in)n(terv)-5 b(als)30 b Fm(J)3245 1188 y Fn(k)3286 1176 y Fq(\()p Fm(x)3365 1146 y Fh(0)3389 1176 y Fq(\),)456 1276 y Fm(k)c Fq(=)c(1)p Fm(;)14 b(:)g(:)g(:)f(;)h(N)9 b Fq(\()p Fm(x)993 1246 y Fh(0)1017 1276 y Fq(\),)28 b(that)g(is)f(on)h Fm(@)5 b Fq(\012\()p Fm(x)1667 1246 y Fh(0)1690 1276 y Fq(\).)38 b(Then)848 1356 y Fj(Z)894 1544 y Fk(\012\()p Fn(x)1005 1528 y Ff(0)1028 1544 y Fk(\))1072 1377 y Fj(\020)1121 1469 y Fl(j)p Fm(@)1188 1481 y Fn(x)1226 1490 y Fu(d)1265 1469 y Fm(u)p Fl(j)1336 1427 y Fk(2)1391 1469 y Fl(\000)18 b Fq(\003)p Fl(j)p Fm(u)p Fl(j)1626 1434 y Fk(2)1663 1377 y Fj(\021)1726 1469 y Fm(dx)1816 1481 y Fn(d)1892 1469 y Fl(\025)37 b(\000h)p Fm(W)12 b(u)p Fq(\()p Fm(x)2308 1434 y Fh(0)2331 1469 y Fm(;)i Fl(\001)p Fq(\))p Fm(;)g(u)p Fq(\()p Fm(x)2587 1434 y Fh(0)2611 1469 y Fm(;)g Fl(\001)p Fq(\))p Fl(i)2735 1484 y Fn(L)2781 1467 y Fi(2)2813 1484 y Fk(\(\012\()p Fn(x)2950 1467 y Ff(0)2973 1484 y Fk(\)\))3029 1469 y Fm(:)456 1675 y Fq(and)27 b(consequen)n(tly)456 1863 y(\(21\))41 b Fl(k)o(r)p Fm(u)p Fl(k)844 1822 y Fk(2)844 1888 y Fn(L)890 1872 y Fi(2)922 1888 y Fk(\(\012\))1029 1863 y Fl(\000)s Fq(\003)14 b Fl(k)o Fm(u)p Fl(k)1299 1822 y Fk(2)1299 1888 y Fn(L)1345 1872 y Fi(2)1377 1888 y Fk(\(\012\))1518 1863 y Fl(\025)22 b(kr)1716 1829 y Fh(0)1739 1863 y Fm(u)p Fl(k)1828 1817 y Fk(2)1828 1888 y Fn(L)1874 1872 y Fi(2)1906 1888 y Fk(\(\012\))2013 1863 y Fl(\000)2081 1750 y Fj(Z)2127 1939 y Fe(R)2169 1922 y Fu(d)p Ff(\000)p Fi(1)2295 1863 y Fm(dx)2385 1829 y Fh(0)2409 1863 y Fl(h)p Fm(W)12 b(u)p Fq(\()p Fm(x)2658 1829 y Fh(0)2682 1863 y Fm(;)i Fl(\001)p Fq(\))p Fm(;)g(u)p Fq(\()p Fm(x)2938 1829 y Fh(0)2962 1863 y Fm(;)g Fl(\001)p Fq(\))p Fl(i)3086 1878 y Fn(L)3132 1862 y Fi(2)3164 1878 y Fk(\(\012\()p Fn(x)3301 1862 y Ff(0)3324 1878 y Fk(\)\))3380 1863 y Fm(:)456 2068 y Fq(No)n(w)28 b(w)n(e)g(can)g(extend)h(this)f (quadratic)g(form)g(b)n(y)g(zero)f(to)i Fm(C)2385 2038 y Fh(1)2379 2089 y Fk(0)2469 2001 y Fj(\000)2507 2068 y Fo(R)2567 2038 y Fn(d)2624 2068 y Fl(n)18 b Fm(@)5 b Fq(\012)2793 2001 y Fj(\001)2831 2068 y Fq(,)29 b(whic)n(h)f(is)h(a)f (form)456 2176 y(core)g(for)762 2109 y Fj(\000)800 2176 y Fl(\000)p Fq(\001)934 2193 y Fe(R)976 2177 y Fu(d)1011 2193 y Fh(n)p Fk(\012)1096 2109 y Fj(\001)1154 2176 y Fl(\010)20 b Fq(\()p Fl(\000)p Fq(\001)1405 2188 y Fk(\012)1475 2176 y Fl(\000)e Fq(\003\).)43 b(This)29 b(op)r(erator)f(corresp)r (onds)g(to)i(the)g(left)g(hand)g(side)456 2278 y(of)c(\(21\),)g(while)h (the)g(semi-b)r(ounded)f(form)h(on)f(the)h(righ)n(t)e(hand)i(side)f(is) h(closed)e(on)i(the)f(larger)456 2378 y(domain)h Fm(H)827 2348 y Fk(1)878 2311 y Fj(\000)916 2378 y Fo(R)976 2348 y Fn(d)p Fh(\000)p Fk(1)1099 2378 y Fm(;)14 b(L)1193 2348 y Fk(2)1230 2378 y Fq(\()p Fo(R)p Fq(\))1354 2311 y Fj(\001)1393 2378 y Fq(,)27 b(where)g(it)h(corresp)r(onds)e(to)i(the) g(op)r(erator)456 2540 y(\(22\))526 b Fl(\000)p Fq(\001)1264 2506 y Fh(0)1306 2540 y Fl(\012)18 b Fo(I)h Fl(\000)f Fm(W)12 b Fq(\()p Fm(x)1692 2506 y Fh(0)1716 2540 y Fm(;)i Fq(\003\))83 b(on)f Fm(L)2153 2506 y Fk(2)2204 2473 y Fj(\000)2242 2540 y Fo(R)2302 2506 y Fn(d)p Fh(\000)p Fk(1)2426 2540 y Fm(;)14 b(L)2520 2506 y Fk(2)2556 2540 y Fq(\()p Fo(R)p Fq(\))2680 2473 y Fj(\001)2746 2540 y Fm(:)456 2691 y Fq(Due)23 b(to)f(the)i(p)r(ositivit)n(y)e(of)h Fl(\000)p Fq(\001)1448 2708 y Fe(R)1490 2691 y Fu(d)1524 2708 y Fh(n)p Fk(\012)1633 2691 y Fq(the)g(v)-5 b(ariational)21 b(principle)i(implies)g(that)g(for)f(an)n(y)g Fm(\033)k Fl(\025)d Fq(0)988 2856 y(T)-7 b(r)14 b(\()p Fl(\000)p Fq(\001)1254 2868 y Fk(\012)1324 2856 y Fl(\000)k Fq(\003\))1497 2814 y Fn(\033)1497 2881 y Fh(\000)1636 2856 y Fq(=)82 b(T)-7 b(r)1883 2789 y Fj(\000\000)1959 2856 y Fl(\000)p Fq(\001)2093 2873 y Fe(R)2135 2856 y Fu(d)2170 2873 y Fh(n)p Fk(\012)2255 2789 y Fj(\001)2312 2856 y Fl(\010)18 b Fq(\()p Fl(\000)p Fq(\001)2561 2868 y Fk(\012)2631 2856 y Fl(\000)g Fq(\003\))2804 2789 y Fj(\001)2842 2806 y Fn(\033)2842 2891 y Fh(\000)1636 3003 y Fl(\024)82 b Fq(T)-7 b(r)14 b(\()p Fl(\000)p Fq(\001)2049 2968 y Fh(0)2091 3003 y Fl(\012)k Fo(I)g Fl(\000)g Fm(W)12 b Fq(\()p Fm(x)2476 2968 y Fh(0)2500 3003 y Fm(;)i Fq(\003\)\))2660 2957 y Fn(\033)2660 3028 y Fh(\000)2729 3003 y Fm(:)456 3153 y Fq(W)-7 b(e)35 b(can)h(no)n(w)e(apply)i(a)f(sharp)f (Lieb-Thirring)g(inequalit)n(y)h(to)h(the)g(Sc)n(hr\177)-42 b(odinger)33 b(op)r(erator)456 3252 y(\(22\))27 b(with)h(the)g(op)r (erator-v)-5 b(alued)25 b(p)r(oten)n(tial)j Fl(\000)p Fm(W)12 b Fq(\()p Fm(x)2143 3222 y Fh(0)2166 3252 y Fm(;)i Fq(\003\),)28 b(see)f([L)-9 b(W00],)27 b(and)h(claim)f(that)609 3445 y(T)-7 b(r)14 b(\()p Fl(\000)p Fq(\001)875 3411 y Fh(0)917 3445 y Fl(\012)k Fo(I)g Fl(\000)g Fm(W)12 b Fq(\()p Fm(x)1302 3411 y Fh(0)1326 3445 y Fm(;)i Fq(\003\)\))1485 3399 y Fn(\033)1485 3470 y Fh(\000)1578 3445 y Fl(\024)37 b Fm(L)1737 3411 y Fn(cl)1737 3466 y(\033)o(;d)p Fh(\000)p Fk(1)1931 3332 y Fj(Z)1977 3521 y Fe(R)2019 3504 y Fu(d)p Ff(\000)p Fi(1)2145 3445 y Fq(T)-7 b(r)13 b Fm(W)2334 3411 y Fn(\033)r Fk(+)2436 3385 y Fu(d)p Ff(\000)p Fi(1)p 2436 3398 104 3 v 2473 3431 a(2)2554 3445 y Fq(\()p Fm(x)2633 3411 y Fh(0)2657 3445 y Fm(;)h Fq(\003\))g Fm(dx)2888 3411 y Fh(0)2911 3445 y Fm(;)97 b(\033)27 b Fl(\025)3202 3389 y Fq(3)p 3202 3426 42 4 v 3202 3502 a(2)3268 3445 y Fm(:)555 3642 y Fq(No)n(w)k(let)g Fm(p)913 3654 y Fn(d)952 3642 y Fq(\()p Fm(x)1031 3612 y Fh(0)1055 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Ff(0)1770 3985 y Fk(\))1812 3958 y Fl(\000)11 b Fq(\003)23 b(on)h(the)g(in)n(terv)-5 b(al)23 b Fm(L)p Fq(\()p Fm(x)2652 3928 y Fh(0)2676 3958 y Fq(\))g(=)g([0)p Fm(;)14 b(p)2963 3970 y Fn(d)3001 3958 y Fq(\()p Fm(x)3080 3928 y Fh(0)3104 3958 y Fq(\)])24 b(sub)5 b(ject)456 4066 y(to)27 b(Diric)n(hlet)h(conditions)f(at)g(the)h(endp)r(oin)n(t)g (of)g(this)g(one)f(in)n(terv)-5 b(al)27 b(only)-7 b(.)37 b(Th)n(us,)1244 4229 y(T)-7 b(r)14 b Fm(W)1434 4195 y Fn(\033)r Fk(+)1535 4169 y Fu(d)p Ff(\000)p Fi(1)p 1535 4182 104 3 v 1573 4215 a(2)1653 4229 y Fq(\()p Fm(x)1732 4195 y Fh(0)1756 4229 y Fm(;)g Fq(\003\))23 b Fl(\024)g Fq(T)-7 b(r)2117 4208 y(~)2093 4229 y Fm(W)2183 4195 y Fn(\033)r Fk(+)2285 4169 y Fu(d)p Ff(\000)p Fi(1)p 2285 4182 V 2322 4215 a(2)2403 4229 y Fq(\()p Fm(x)2482 4195 y Fh(0)2506 4229 y Fm(;)14 b Fq(\003\))p Fm(;)456 4385 y Fq(where)721 4364 y(~)697 4385 y Fm(W)e Fq(\()p Fm(x)866 4355 y Fh(0)890 4385 y Fm(;)i Fq(\003\))28 b(is)h(the)g (negativ)n(e)e(part)h(of)h Fl(\000)p 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Fj(\023)2812 5027 y Fn(\033)r Fk(+)2913 5001 y Fu(d)p Ff(\000)p Fi(1)p 2913 5014 104 3 v 2951 5047 a(2)2812 5216 y Fk(+)3045 5130 y Fm(dx)3135 5096 y Fh(0)3159 5130 y Fm(:)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 640 251 a Fk(UNIVERSAL)28 b(BOUNDS)h(F)n(OR)h (TRA)n(CES)f(OF)g(THE)g(DIRICHLET)h(LAPLA)n(CE)f(OPERA)-5 b(TOR)150 b(9)456 450 y Fq(Note)24 b(that)h(the)h(righ)n(t)e(hand)g (side)h(of)g(this)g(b)r(ound)g(v)-5 b(anishes)24 b(if)i Fm(p)2480 462 y Fn(d)2518 450 y Fq(\()p Fm(x)2597 420 y Fh(0)2621 450 y Fq(\))d Fl(\024)g Fm(l)2789 462 y Fk(\003)2838 450 y Fq(.)36 b(F)-7 b(or)25 b Fm(p)3086 462 y Fn(d)3124 450 y Fq(\()p Fm(x)3203 420 y Fh(0)3227 450 y Fq(\))e Fm(>)g(l)3395 462 y Fk(\003)456 550 y Fq(w)n(e)k(ha)n(v)n(e)f(in)i (view)g(of)34 b(\(17\))682 699 y Fj(X)684 877 y Fn(j)s Fh(\025)p Fk(1)816 661 y Fj(\022)877 778 y Fq(1)18 b Fl(\000)1062 722 y Fm(l)1089 692 y Fk(2)1087 745 y(\003)1137 722 y Fm(j)1176 692 y Fk(2)p 1030 759 216 4 v 1030 835 a Fm(p)1072 806 y Fk(2)1072 860 y 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Fn(j)2399 5191 y Fq(\()p Fm(x)2478 5203 y Fn(d)2517 5191 y Fm(;)2563 5170 y Fq(^)2554 5191 y(\012\))23 b Fm(>)g(s)2796 5099 y Fj(o)2851 5095 y(\014)2851 5145 y(\014)2851 5195 y(\014)2879 5249 y Fk(1)2939 5191 y Fq(=:)39 b(^)-58 b Fm(m)3123 5203 y Fn(j)3158 5191 y Fq(\()p Fm(s)p Fq(;)3275 5170 y(^)3266 5191 y(\012\))14 b Fm(:)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 640 251 a Fk(UNIVERSAL)28 b(BOUNDS)h(F)n(OR)h (TRA)n(CES)f(OF)g(THE)g(DIRICHLET)h(LAPLA)n(CE)f(OPERA)-5 b(TOR)117 b(11)456 450 y Fq(The)27 b(corresp)r(onding)e(notions)i(w)n (e)f(use)h(also)f(with)i(resp)r(ect)f(to)g(the)h(domains)e(\012)2984 420 y Fh(\003)3049 450 y Fq(and)3220 429 y(^)3210 450 y(\012)3270 420 y Fh(\003)3308 450 y Fq(.)37 b(In)456 553 y(con)n(trast)31 b(to)h(the)g(preserv)-5 b(ation)31 b(of)i(length)f(in)h(the)f Fm(d)p Fq(th)h(direction)f(the)h(v)-5 b(alues)32 b(of)39 b(^)-49 b Fm(p)3162 565 y Fn(j)3197 553 y Fq(\()p Fm(x)3276 565 y Fn(d)3315 553 y Fq(;)3361 532 y(^)3352 553 y(\012\))456 664 y(and)34 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5249 y Fk(0)1203 5173 y Fq(^)-58 b Fm(m)1260 5185 y Fn(j)1295 5173 y Fq(\()p Fm(s)p Fq(;)1412 5152 y(^)1403 5173 y(\012\))14 b Fm(ds)23 b Fl(\025)1702 5078 y Fj(\014)1702 5127 y(\014)1702 5177 y(\014)1739 5152 y Fq(^)1730 5173 y(\012)1790 5185 y Fn(y)1829 5078 y Fj(\014)1829 5127 y(\014)1829 5177 y(\014)1880 5173 y Fq(=)1968 5078 y Fj(\014)1968 5127 y(\014)1968 5177 y(\014)2005 5152 y Fq(^)1995 5173 y(\012)2055 5139 y Fh(\003)2055 5194 y Fn(y)2095 5078 y Fj(\014)2095 5127 y(\014)2095 5177 y(\014)2146 5173 y Fq(=)2234 5060 y Fj(Z)2317 5081 y Fn(y)2280 5249 y Fk(0)2386 5173 y Fq(^)-58 b Fm(m)2443 5185 y Fn(j)2478 5173 y Fq(\()p Fm(s)p Fq(;)2596 5152 y(^)2586 5173 y(\012)2646 5139 y Fh(\003)2685 5173 y Fq(\))14 b Fm(ds)g(:)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 456 251 a Fk(12)781 b(LEANDER)22 b(GEISINGER)44 b(&)h(TIMO)23 b(WEIDL)456 450 y Fq(This)k(sho)n(ws)g (that)h(\(26\))f(holds)g(true)h(and)f(the)h(pro)r(of)f(is)g(complete.) 798 b Fc(\003)555 627 y Fq(No)n(w)27 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2744 y Fq(;)14 b(\012)2421 2710 y Fh(\003)2459 2744 y Fq(\()p Fm(\034)9 b Fq(\)\))24 b Fm(>)f(s)p Fl(gj)2816 2756 y Fn(d)p Fh(\000)p Fk(2)2953 2744 y Fm(d\034)1287 2970 y Fq(=)1435 2857 y Fj(Z)1518 2878 y Fh(1)1481 3046 y Fk(0)1618 2970 y Fq(~)-58 b Fm(m)1675 2982 y Fn(j)1710 2970 y Fq(\()p Fm(s)p Fq(;)14 b(\012)1878 2936 y Fh(\003)1916 2970 y Fq(\()p Fm(\034)9 b Fq(\)\))14 b Fm(d\034)25 b(;)180 b(j)28 b Fq(=)23 b(1)p Fm(;)14 b(:)g(:)g(:)f(;)h(d)19 b Fl(\000)f Fq(1)c Fm(:)-2522 b Fq(\(32\))456 3190 y(Put)43 b(~)-58 b Fm(m)p Fq(\()p Fm(s)p Fq(;)808 3169 y(~)799 3190 y(\012\))23 b(=)g(\()p Fm(d)c Fl(\000)f Fq(1\))1253 3160 y Fh(\000)p Fk(1)1356 3128 y Fj(P)1443 3149 y Fn(d)p Fh(\000)p Fk(1)1443 3215 y Fn(j)s Fk(=1)1597 3190 y Fq(~)-58 b Fm(m)1654 3202 y Fn(j)1689 3190 y Fq(\()p Fm(s)p Fq(;)1806 3169 y(~)1797 3190 y(\012\).)37 b(By)28 b(induction)g(assumption)f(w)n (e)g(ha)n(v)n(e)456 3434 y(\(33\))849 3413 y(~)818 3434 y Fm(M)8 b Fq(\()p Fm(y)s Fq(;)1030 3413 y(~)1020 3434 y(\012\))24 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Ff(\000)p Fi(1)2208 1606 y Fn(d)2350 1581 y Fq(\()p Fm(\034)9 b Fq(;)14 b(\012)2524 1546 y Fh(\003)2563 1581 y Fq(\))g Fm(d\034)24 b(:)-2279 b Fq(\(34\))456 1760 y(In)27 b(particular,)g(in)h(the)g(case)e(of)i Fm(\034)1521 1730 y Fh(\003)1583 1760 y Fl(\024)22 b Fm(y)31 b Fq(w)n(e)c(see)g(from)g(the)h(previous)f(b)r(ound)h(that)1621 1897 y Fm(M)9 b Fq(\()p Fm(y)s Fq(;)14 b(\012\))23 b Fl(\025)g Fm(d)2070 1863 y Fh(\000)p Fk(1)2173 1897 y Fl(j)p Fq(\012)p Fl(j)456 2035 y Fq(and)32 b(the)h(assertion)e(is)h (pro)n(v)n(en.)50 b(Hence,)34 b(let)f(us)f(consider)f(the)i(remaining)f (case)f Fm(\034)3133 2004 y Fh(\003)3203 2035 y Fm(>)g(y)k Fq(in)456 2134 y(more)26 b(detail.)37 b(F)-7 b(or)27 b Fm(\034)1120 2104 y Fh(\003)1182 2134 y Fm(>)c(y)30 b Fq(w)n(e)d(ha)n(v)n(e)1428 2272 y Fm(m)1501 2284 y Fn(d)1539 2272 y Fq(\()p Fm(y)s Fq(;)14 b(\012)1712 2237 y Fh(\003)1750 2272 y Fq(\))24 b Fl(\025)e Fq(\()p Fm(d)d Fl(\000)f Fq(1\))2144 2237 y Fn(d)p Fh(\000)p Fk(1)2268 2272 y Fm(y)2312 2237 y Fn(d)p Fh(\000)p Fk(1)2449 2272 y Fm(:)456 2409 y Fq(Because)26 b(of)i(the)g(monotonicit)n(y)e(of)i Fm(m)1681 2421 y Fn(d)1747 2409 y Fq(w)n(e)g(conclude)f(that)533 2476 y Fj(Z)616 2496 y Fn(y)579 2664 y Fk(0)670 2589 y Fm(m)743 2601 y Fn(d)781 2589 y Fq(\()p Fm(\034)9 b Fq(;)14 b(\012)955 2554 y Fh(\003)994 2589 y Fq(\))1036 2527 y Fu(d)p Ff(\000)p Fi(2)p 1036 2540 V 1036 2573 a Fu(d)p Ff(\000)p Fi(1)1154 2589 y Fm(d\034)47 b Fl(\025)37 b Fm(y)16 b(m)1522 2507 y Fu(d)p Ff(\000)p Fi(2)p 1522 2520 V 1522 2553 a Fu(d)p Ff(\000)p Fi(1)1512 2614 y Fn(d)1640 2589 y Fq(\()p Fm(y)s Fq(;)e(\012)1813 2554 y Fh(\003)1851 2589 y Fq(\))83 b(and)2197 2476 y Fj(Z)2280 2496 y Fn(y)2243 2664 y Fk(0)2334 2589 y Fm(m)2407 2601 y Fn(d)2445 2589 y Fq(\()p Fm(\034)9 b Fq(;)14 b(\012)2619 2554 y Fh(\003)2658 2589 y Fq(\))p Fm(d\034)47 b Fl(\025)37 b Fm(y)16 b(m)3048 2601 y Fn(d)3087 2589 y Fq(\()p Fm(y)s Fq(;)e(\012)3260 2554 y Fh(\003)3298 2589 y Fq(\))g Fm(:)456 2773 y Fq(Let)27 b(us)h(rewrite)f(inequalit)n(y)h(\(34\))f(as)g(follo)n (ws)659 2958 y Fm(M)9 b Fq(\()p Fm(y)s Fq(;)14 b(\012\))23 b Fl(\025)1074 2902 y(j)p Fq(\012)p Fl(j)p 1074 2939 107 4 v 1106 3015 a Fm(d)1274 2958 y Fq(+)1431 2902 y Fm(d)c Fl(\000)f Fq(1)p 1431 2939 187 4 v 1503 3015 a Fm(d)1641 2958 y(y)1699 2845 y Fj(Z)1782 2865 y Fn(y)1745 3033 y Fk(0)1835 2958 y Fm(m)1918 2877 y Fu(d)p Ff(\000)p Fi(2)p 1918 2890 104 3 v 1918 2922 a Fu(d)p Ff(\000)p Fi(1)1908 2983 y Fn(d)2036 2958 y Fq(\()p Fm(\034)9 b Fq(;)14 b(\012)2210 2924 y Fh(\003)2249 2958 y Fq(\))p Fm(d\034)1274 3203 y Fq(+)1431 3147 y Fm(d)19 b Fl(\000)f Fq(1)p 1431 3184 187 4 v 1503 3260 a Fm(d)1641 3203 y(y)1699 3090 y Fj(Z)1782 3111 y Fn(\034)1820 3086 y Ff(\003)1745 3279 y Fn(y)1872 3203 y Fm(m)1955 3122 y Fu(d)p Ff(\000)p Fi(2)p 1954 3135 104 3 v 1954 3168 a Fu(d)p Ff(\000)p Fi(1)1945 3229 y Fn(d)2073 3203 y Fq(\()p Fm(\034)9 b Fq(;)14 b(\012)2247 3169 y Fh(\003)2286 3203 y Fq(\))p Fm(d\034)28 b Fl(\000)2519 3147 y Fq(1)p 2518 3184 44 4 v 2518 3260 a Fm(d)2585 3090 y Fj(Z)2668 3111 y Fn(\034)2706 3086 y Ff(\003)2631 3279 y Fn(y)2758 3203 y Fm(m)2831 3215 y Fn(d)2870 3203 y Fq(\()p Fm(\034)9 b Fq(;)14 b(\012)3044 3169 y Fh(\003)3083 3203 y Fq(\))p Fm(d\034)24 b(:)456 3427 y Fq(Put)j Fm(A)d Fq(=)791 3360 y Fj(R)846 3381 y Fn(\034)884 3356 y Ff(\003)830 3457 y Fn(y)937 3427 y Fm(m)1010 3439 y Fn(d)1062 3427 y Fq(\()p Fm(\034)9 b Fq(;)14 b(\012)1236 3397 y Fh(\003)1275 3427 y Fq(\))g Fm(d\034)9 b Fq(.)38 b(Then)456 3651 y(\(35\))194 b(0)23 b Fm(<)g(A)g Fq(=)1123 3538 y Fj(Z)1207 3558 y Fn(\034)1245 3533 y Ff(\003)1170 3727 y Fk(0)1297 3651 y Fm(m)1370 3663 y Fn(d)1422 3651 y Fq(\()p Fm(\034)9 b Fq(;)14 b(\012)1596 3617 y Fh(\003)1635 3651 y Fq(\))g Fm(d\034)28 b Fl(\000)1871 3538 y Fj(Z)1955 3558 y Fn(y)1918 3727 y Fk(0)2008 3651 y Fm(m)2081 3663 y Fn(d)2134 3651 y Fq(\()p Fm(\034)9 b Fq(;)14 b(\012)2308 3617 y Fh(\003)2347 3651 y Fq(\))g Fm(d\034)33 b Fl(\024)22 b(j)p Fq(\012)p Fl(j)d(\000)f Fm(y)e(m)2930 3663 y Fn(d)2969 3651 y Fq(\()p Fm(y)s Fq(;)e(\012)3142 3617 y Fh(\003)3180 3651 y Fq(\))g Fm(:)456 3829 y Fq(Moreo)n(v)n(er,)600 4028 y Fm(M)9 b Fq(\()p Fm(y)s Fq(;)14 b(\012\))23 b Fl(\025)1016 3972 y(j)p Fq(\012)p Fl(j)p 1016 4009 107 4 v 1047 4085 a Fm(d)1150 4028 y Fq(+)1243 3972 y Fm(d)c Fl(\000)f Fq(1)p 1243 4009 187 4 v 1315 4085 a Fm(d)1453 4028 y(y)1497 3994 y Fk(2)1548 4028 y Fm(m)1631 3947 y Fu(d)p Ff(\000)p Fi(2)p 1631 3960 104 3 v 1631 3993 a Fu(d)p Ff(\000)p Fi(1)1621 4053 y Fn(d)1749 4028 y Fq(\()p Fm(y)s Fq(;)c(\012)1922 3994 y Fh(\003)1960 4028 y Fq(\))19 b(+)2104 3972 y Fm(d)f Fl(\000)g Fq(1)p 2104 4009 187 4 v 2175 4085 a Fm(d)2313 4028 y(y)2371 3915 y Fj(Z)2454 3936 y Fn(\034)2492 3911 y Ff(\003)2417 4104 y Fn(y)2544 4028 y Fm(m)2627 3947 y Fu(d)p Ff(\000)p Fi(2)p 2627 3960 104 3 v 2627 3993 a Fu(d)p Ff(\000)p Fi(1)2617 4053 y Fn(d)2745 4028 y Fq(\()p Fm(\034)9 b Fq(;)14 b(\012)2919 3994 y Fh(\003)2958 4028 y Fq(\))p Fm(d\034)29 b Fl(\000)3191 3972 y Fm(A)p 3191 4009 63 4 v 3201 4085 a(d)3277 4028 y(:)456 4224 y Fq(Due)i(to)f(the)h(monotonicit)n(y)f(of)g Fm(m)1559 4236 y Fn(d)1629 4224 y Fq(w)n(e)g(ha)n(v)n(e,)g(in)h(particular,)f Fm(m)2555 4236 y Fn(d)2594 4224 y Fq(\()p Fm(\034)9 b Fq(;)14 b(\012)2768 4193 y Fh(\003)2807 4224 y Fq(\))28 b Fl(\024)f Fm(m)3032 4236 y Fn(d)3071 4224 y Fq(\()p Fm(y)s Fq(;)14 b(\012)3244 4193 y Fh(\003)3282 4224 y Fq(\))31 b(for)456 4323 y Fm(y)25 b Fl(\024)e Fm(\034)37 b Fq(and)591 4427 y Fj(Z)674 4447 y Fn(\034)712 4422 y Ff(\003)637 4615 y Fn(y)764 4540 y Fm(m)847 4459 y Fu(d)p Ff(\000)p Fi(2)p 847 4472 104 3 v 847 4504 a Fu(d)p Ff(\000)p Fi(1)837 4565 y Fn(d)965 4540 y Fq(\()p Fm(\034)9 b Fq(;)14 b(\012)1139 4505 y Fh(\003)1178 4540 y Fq(\))p Fm(d\034)94 b Fq(=)82 b Fm(m)1613 4459 y Fu(d)p Ff(\000)p Fi(2)p 1613 4472 V 1613 4504 a Fu(d)p Ff(\000)p Fi(1)1603 4565 y Fn(d)1731 4540 y Fq(\()p Fm(y)s Fq(;)14 b(\012)1904 4505 y Fh(\003)1942 4540 y Fq(\))1988 4427 y Fj(Z)2071 4447 y Fn(\034)2109 4422 y Ff(\003)2034 4615 y Fn(y)2161 4423 y Fj(\022)2232 4484 y Fm(m)2305 4496 y Fn(d)2344 4484 y Fq(\()p Fm(\034)9 b Fq(;)14 b(\012)2518 4453 y Fh(\003)2557 4484 y Fq(\))p 2232 4521 357 4 v 2233 4597 a Fm(m)2306 4609 y Fn(d)2345 4597 y Fq(\()p Fm(y)s Fq(;)g(\012)2518 4573 y Fh(\003)2556 4597 y Fq(\))2599 4423 y Fj(\023)2670 4414 y Fu(d)p Ff(\000)p Fi(2)p 2670 4427 104 3 v 2670 4460 a Fu(d)p Ff(\000)p Fi(1)2802 4540 y Fm(d\034)1383 4797 y Fl(\025)82 b Fm(m)1613 4716 y Fu(d)p Ff(\000)p Fi(2)p 1613 4729 V 1613 4761 a Fu(d)p Ff(\000)p Fi(1)1603 4822 y Fn(d)1731 4797 y Fq(\()p Fm(y)s Fq(;)14 b(\012)1904 4762 y Fh(\003)1942 4797 y Fq(\))1988 4684 y Fj(Z)2071 4704 y Fn(\034)2109 4679 y Ff(\003)2034 4872 y Fn(y)2171 4741 y Fm(m)2244 4753 y Fn(d)2283 4741 y Fq(\()p Fm(\034)9 b Fq(;)14 b(\012)2457 4710 y Fh(\003)2496 4741 y Fq(\))p 2171 4778 357 4 v 2172 4854 a Fm(m)2245 4866 y Fn(d)2284 4854 y Fq(\()p Fm(y)s Fq(;)g(\012)2457 4830 y Fh(\003)2495 4854 y Fq(\))2552 4797 y Fm(d\034)33 b Fq(=)22 b Fm(m)2850 4716 y Ff(\000)p 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Fq(Therefore)26 b(w)n(e)h(claim)557 3532 y Fj(Z)640 3552 y Fh(1)603 3721 y Fn(\025t)724 3645 y Fm(e)763 3611 y Fh(\000)p Fn(s)850 3645 y Fm(s)889 3611 y Fn(\033)927 3620 y Fu(d)962 3611 y Fk(+)1023 3588 y Fu(d)p 1023 3597 31 3 v 1024 3631 a Fi(2)1088 3624 y Fq(~)1068 3645 y Fm(B)1149 3528 y Fj(\022)1252 3589 y Fm(\031)1302 3559 y Fk(2)1353 3589 y Fm(t)p 1220 3626 196 4 v 1220 3702 a Fq(4)p Fm(R)1326 3678 y Fk(2)1376 3702 y Fm(s)1425 3645 y(;)1472 3589 y Fq(1)p 1472 3626 42 4 v 1472 3702 a(2)1524 3645 y Fm(;)1570 3589 y(d)19 b Fq(+)f(1)p 1570 3626 187 4 v 1643 3702 a(2)1767 3528 y Fj(\023)1842 3645 y Fm(ds)580 3902 y Fl(\025)677 3841 y Fq(\000)743 3774 y Fj(\000)791 3808 y Fn(d)p 791 3822 35 4 v 792 3870 a Fk(2)854 3841 y Fq(+)g(1)979 3774 y Fj(\001)p 677 3883 340 4 v 707 3966 a Fq(\000)773 3899 y Fj(\000)821 3934 y Fn(d)p Fk(+1)p 821 3948 119 4 v 863 3995 a(2)949 3899 y Fj(\001)1041 3760 y( )1116 3779 y Fl(p)p 1185 3779 81 4 v 67 x Fm(\031)s(t)p 1116 3883 150 4 v 1159 3959 a(R)1276 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Fi(2)1706 4417 y Fq(^)1700 4438 y(\000)1766 4321 y Fj(\022)1827 4438 y Fm(\033)1874 4450 y Fn(d)1932 4438 y Fq(+)2025 4382 y Fm(d)p 2025 4419 44 4 v 2026 4495 a Fq(2)2096 4438 y(+)18 b(1)p Fm(;)c(\025t)2336 4321 y Fj(\023)2416 4438 y Fl(\000)k Fm(R)q Fq(\()p Fm(t;)c(\025)p Fq(\))456 4649 y(with)28 b Fm(R)q Fq(\()p Fm(t;)14 b(\025)p Fq(\))23 b(=)g Fm(r)1036 4661 y Fk(1)1074 4649 y Fq(\()p Fm(t;)14 b(\025)p Fq(\))19 b Fl(\000)f Fm(r)1392 4661 y Fk(2)1430 4649 y Fq(\()p Fm(t;)c(\025)p Fq(\))28 b(and)909 4861 y Fm(r)946 4873 y Fk(1)984 4861 y Fq(\()p Fm(t;)14 b(\025)p Fq(\))24 b(=)1352 4805 y Fl(j)p Fq(\012)p Fl(j)p 1284 4842 242 4 v 1284 4932 a Fq(\(4)p Fm(\031)s(t)p Fq(\))1480 4881 y Fu(d)p 1480 4890 31 3 v 1481 4923 a Fi(2)1545 4800 y Fq(\000)1611 4732 y Fj(\000)1659 4767 y Fn(d)p 1659 4781 35 4 v 1660 4828 a Fk(2)1722 4800 y Fq(+)18 b(1)1847 4732 y Fj(\001)p 1545 4842 340 4 v 1575 4925 a Fq(\000)1641 4858 y Fj(\000)1688 4892 y Fn(d)p Fk(+1)p 1688 4906 119 4 v 1731 4954 a(2)1817 4858 y Fj(\001)1904 4737 y Fl(p)p 1974 4737 81 4 v 1974 4805 a Fm(\031)s(t)p 1904 4842 150 4 v 1947 4918 a(R)2074 4800 y Fq(\000)2140 4732 y Fj(\000)2178 4800 y Fm(\033)2225 4812 y Fn(d)2282 4800 y Fq(+)2375 4767 y Fn(d)p Fk(+1)p 2375 4781 119 4 v 2418 4828 a(2)2504 4800 y Fm(;)c(\025t)2619 4732 y Fj(\001)p 2074 4842 584 4 v 2102 4925 a Fq(\000)2168 4858 y Fj(\000)2206 4925 y Fm(\033)2253 4937 y Fn(d)2310 4925 y Fq(+)2403 4892 y Fn(d)p 2403 4906 35 4 v 2404 4954 a Fk(2)2466 4925 y Fq(+)k(1)2591 4858 y Fj(\001)909 5123 y Fm(r)946 5135 y Fk(2)984 5123 y Fq(\()p Fm(t;)c(\025)p Fq(\))24 b(=)1352 5066 y Fl(j)p Fq(\012)p Fl(j)p 1284 5103 242 4 v 1284 5194 a Fq(\(4)p Fm(\031)s(t)p Fq(\))1480 5142 y Fu(d)p 1480 5151 31 3 v 1481 5185 a Fi(2)1545 5061 y Fq(\000)1611 4994 y Fj(\000)1659 5028 y Fn(d)p 1659 5042 35 4 v 1660 5090 a Fk(2)1722 5061 y Fq(+)18 b(1)1847 4994 y Fj(\001)p 1545 5103 340 4 v 1575 5187 a Fq(\000)1641 5119 y Fj(\000)1688 5154 y Fn(d)p Fk(+1)p 1688 5168 119 4 v 1731 5215 a(2)1817 5119 y Fj(\001)1904 5066 y Fq(\()p Fm(d)h Fl(\000)f Fq(1\))p Fm(\031)2215 5014 y Fi(5)p 2215 5023 29 3 v 2215 5056 a(2)2258 5066 y Fm(t)2298 5014 y Fi(3)p 2298 5023 V 2298 5056 a(2)p 1904 5103 436 4 v 2030 5180 a Fq(24)p Fm(R)2178 5156 y Fk(3)2360 5061 y Fq(\000)2426 4994 y Fj(\000)2464 5061 y Fm(\033)2511 5073 y Fn(d)2568 5061 y Fq(+)2661 5028 y Fn(d)p Fh(\000)p Fk(1)p 2661 5042 120 4 v 2705 5090 a(2)2791 5061 y Fm(;)c(\025t)2906 4994 y Fj(\001)p 2360 5103 585 4 v 2389 5187 a Fq(\000)2455 5119 y Fj(\000)2492 5187 y Fm(\033)2539 5199 y Fn(d)2597 5187 y Fq(+)2690 5154 y Fn(d)p 2690 5168 35 4 v 2691 5215 a Fk(2)2753 5187 y Fq(+)k(1)2878 5119 y Fj(\001)2968 5123 y Fm(:)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 640 251 a Fk(UNIVERSAL)28 b(BOUNDS)h(F)n(OR)h (TRA)n(CES)f(OF)g(THE)g(DIRICHLET)h(LAPLA)n(CE)f(OPERA)-5 b(TOR)117 b(17)456 450 y Fq(F)-7 b(rom)27 b Fl(j)p Fm(B)760 462 y Fn(R)814 450 y Fl(j)c Fq(=)g Fl(j)p Fq(\012)p Fl(j)28 b Fq(w)n(e)f(deduce)456 677 y(\(41\))919 b Fm(R)23 b 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Fn(d)p 1933 1032 35 4 v 1934 1079 a Fk(2)1996 1050 y Fq(+)k(1)2121 983 y Fj(\001)2169 972 y Fu(d)p Ff(\000)p Fi(1)p 2169 985 104 3 v 2206 1018 a Fu(d)p 1820 1093 468 4 v 1913 1176 a Fq(\000)1979 1108 y Fj(\000)2027 1143 y Fn(d)p Fk(+1)p 2027 1157 119 4 v 2070 1204 a(2)2155 1108 y Fj(\001)2302 1091 y Fq(^)2297 1112 y(\000)2363 995 y Fj(\022)2424 1112 y Fm(\033)2471 1124 y Fn(d)2528 1112 y Fq(+)2621 1056 y Fm(d)h Fq(+)f(1)p 2621 1093 187 4 v 2694 1169 a(2)2817 1112 y Fm(;)c(\025t)2932 995 y Fj(\023)517 1408 y Fm(r)554 1420 y Fk(2)592 1408 y Fq(\()p Fm(t;)g(\025)p Fq(\))24 b(=)932 1352 y Fl(j)p Fq(\012)p Fl(j)1048 1296 y Fu(d)p Ff(\000)p Fi(3)p 1048 1309 104 3 v 1085 1342 a Fu(d)p 892 1389 315 4 v 892 1483 a Fq(\(4)p Fm(\031)s(t)p Fq(\))1088 1428 y Fu(d)p Ff(\000)p Fi(3)p 1089 1441 104 3 v 1126 1474 a(2)1227 1347 y Fm(\031)1277 1317 y Fk(2)1314 1347 y Fq(\()p Fm(d)19 b Fl(\000)f Fq(1\))p Fm(B)1646 1280 y Fj(\000)1694 1314 y Fk(1)p 1694 1328 34 4 v 1694 1376 a(2)1737 1347 y Fm(;)c(\033)1821 1359 y Fn(d)1878 1347 y Fq(+)1971 1314 y Fn(d)p Fk(+1)p 1971 1328 119 4 v 2014 1376 a(2)2100 1280 y Fj(\001)p 1227 1389 912 4 v 1401 1465 a Fq(96\(2)p Fm(\033)1606 1477 y Fn(d)1662 1465 y Fq(+)k Fm(d)h Fl(\000)f Fq(1\))2158 1347 y(\000)2224 1280 y Fj(\000)2272 1314 y Fn(d)p 2272 1328 35 4 v 2273 1376 a Fk(2)2335 1347 y Fq(+)g(1)2460 1280 y Fj(\001)2507 1269 y Fu(d)p Ff(\000)p Fi(3)p 2507 1282 104 3 v 2544 1314 a Fu(d)p 2158 1389 468 4 v 2251 1472 a Fq(\000)2317 1405 y Fj(\000)2365 1440 y Fn(d)p Fk(+1)p 2365 1454 119 4 v 2408 1501 a(2)2494 1405 y Fj(\001)2640 1387 y Fq(^)2635 1408 y(\000)2701 1291 y Fj(\022)2762 1408 y Fm(\033)2809 1420 y Fn(d)2867 1408 y Fq(+)2960 1352 y Fm(d)g Fl(\000)g Fq(1)p 2960 1389 187 4 v 3032 1465 a(2)3156 1408 y Fm(;)c(\025t)3271 1291 y Fj(\023)3360 1408 y Fm(:)456 1623 y Fq(T)-7 b(o)20 b(complete)h(the)h(pro)r(of)e(it)h(remains)f(to)h(note)g(that)h(in)f (view)g(of)g(\(38\))f(w)n(e)h(can)f(alw)n(a)n(ys)g(estimate)456 1722 y(the)28 b(remainder)e(term)i Fm(R)q Fq(\()p Fm(t;)14 b(\025)p Fq(\))28 b(from)f(ab)r(o)n(v)n(e)f(b)n(y)i(zero.)1200 b Fc(\003)456 1891 y Fd(R)l(emark.)42 b Fq(According)26 b(to)i(\(11\))f(w)n(e)g(can)g(c)n(ho)r(ose)1529 2101 y(~)1525 2123 y Fm(\025)d Fq(=)1884 2037 y Fm(\031)17 b(j)1987 2007 y Fk(2)1992 2049 y Fu(d)p 1992 2058 31 3 v 1993 2091 a Fi(2)2033 2071 y Fh(\000)p Fk(1)p Fn(;)p Fk(1)p 1694 2104 671 4 v 1694 2210 a Fq(\000)1760 2143 y Fj(\000)1808 2178 y Fn(d)p 1808 2192 35 4 v 1809 2239 a Fk(2)1871 2210 y Fq(+)h(1)1996 2143 y Fj(\001)2033 2158 y Fk(2)p Fn(=d)2153 2210 y Fl(j)p Fq(\012)p Fl(j)2259 2186 y Fk(2)p Fn(=d)456 2354 y Fq(as)27 b(a)g(suitable)g(lo)n(w)n(er)f (b)r(ound)i(on)g Fm(\025)1573 2366 y Fk(1)1610 2354 y Fq(.)37 b(With)29 b(this)f(sp)r(ecial)f(c)n(hoice)g(of)g(parameter)f(w) n(e)h(\014nd)456 2511 y FC(Corollary)33 b(9.)40 b Fd(F)-6 b(or)30 b(any)h(op)l(en)f(set)f Fq(\012)23 b Fl(\032)g Fo(R)1867 2481 y Fn(d)1935 2511 y Fd(with)31 b(\014nite)e(volume)h(and) g(al)t(l)h Fm(t)23 b(>)g Fq(0)456 2705 y(\(42\))495 b Fm(Z)6 b Fq(\()p Fm(t)p Fq(\))37 b Fl(\024)1472 2648 y(j)p Fq(\012)p Fl(j)p 1404 2686 242 4 v 1404 2776 a Fq(\(4)p Fm(\031)s(t)p Fq(\))1600 2724 y Fu(d)p 1601 2733 31 3 v 1602 2767 a Fi(2)1675 2684 y Fq(^)1669 2705 y(\000)1735 2588 y Fj(\022)1796 2705 y Fm(\033)1843 2717 y Fn(d)1901 2705 y Fq(+)1994 2648 y Fm(d)p 1994 2686 44 4 v 1995 2762 a Fq(2)2065 2705 y(+)18 b(1)p Fm(;)2230 2683 y Fq(~)2227 2705 y Fm(\025)c(t)2319 2588 y Fj(\023)2426 2705 y Fl(\000)32 b Fq(\()p Fm(R)q Fq(\()p Fm(t)p Fq(\)\))2745 2717 y Fk(+)456 2911 y Fd(holds)f(true)e(with)606 3109 y Fm(R)q Fq(\()p Fm(t)p Fq(\))23 b(=)g Fm(c)911 3121 y Fk(1)p Fn(;d)1053 3052 y Fl(j)p Fq(\012)p Fl(j)1169 2997 y Fu(d)p Ff(\000)p Fi(1)p 1169 3010 104 3 v 1205 3042 a Fu(d)p 1012 3090 315 4 v 1012 3183 a Fq(\(4)p Fm(\031)s(t)p Fq(\))1208 3128 y Fu(d)p Ff(\000)p Fi(1)p 1209 3141 104 3 v 1247 3174 a(2)1342 3088 y Fq(^)1337 3109 y(\000)1402 2992 y Fj(\022)1464 3109 y Fm(\033)1511 3121 y Fn(d)1568 3109 y Fq(+)1661 3052 y Fm(d)c Fq(+)f(1)p 1661 3090 187 4 v 1733 3166 a(2)1857 3109 y Fm(;)1898 3087 y Fq(~)1894 3109 y Fm(\025)q(t)1973 2992 y Fj(\023)2052 3109 y Fl(\000)g Fm(c)2171 3121 y Fk(2)p Fn(;d)2313 3052 y Fl(j)p Fq(\012)p Fl(j)2429 2997 y Fu(d)p Ff(\000)p Fi(3)p 2429 3010 104 3 v 2465 3042 a Fu(d)p 2273 3090 315 4 v 2273 3183 a Fq(\(4)p Fm(\031)s(t)p Fq(\))2469 3128 y Fu(d)p Ff(\000)p Fi(3)p 2469 3141 104 3 v 2507 3174 a(2)2602 3088 y Fq(^)2597 3109 y(\000)2663 2992 y Fj(\022)2724 3109 y Fm(\033)2771 3121 y Fn(d)2828 3109 y Fq(+)2921 3052 y Fm(d)h Fl(\000)f Fq(1)p 2921 3090 187 4 v 2994 3166 a(2)3118 3109 y Fm(;)3158 3087 y Fq(~)3155 3109 y Fm(\025t)3233 2992 y Fj(\023)456 3318 y Fd(and)30 b(c)l(onstants)f Fm(c)1018 3330 y Fk(1)p Fn(;d)1109 3318 y Fm(;)f(c)1196 3330 y Fk(2)p Fn(;d)1317 3318 y Fd(given)j(explicitly)g (in)f(The)l(or)l(em)g(1.)555 3476 y Fq(Finally)-7 b(,)26 b(w)n(e)f(can)g(apply)h(\(7\))f(to)h(kno)n(wn)e(estimates)i(on)f Fm(Z)6 b Fq(\()p Fm(t)p Fq(\))26 b(and)f(compare)f(the)i(resulting)456 3575 y(univ)n(ersal)g(b)r(ounds)i(with)g(the)g(result)f(from)g (Corollary)e(9.)555 3675 y(T)-7 b(o)26 b(analyse)e(the)i(asymptotics)f (of)h Fm(Z)6 b Fq(\()p Fm(t)p Fq(\))25 b(for)h Fm(t)d Fl(!)g Fq(0+)i(on)g(con)n(v)n(ex)f(domains)h(v)-5 b(an)26 b(den)g(Berg)456 3774 y(pro)n(v)n(ed)g([vdB84b])g(that)i(for)f(all)h (con)n(v)n(ex)e(domains)h Fm(D)e Fl(\032)d Fo(R)2341 3744 y Fn(d)2408 3774 y Fq(and)27 b(all)g Fm(t)d(>)e Fq(0)1138 3971 y Fm(Z)6 b Fq(\()p Fm(t)p Fq(\))23 b Fl(\024)1477 3914 y(j)p Fm(D)r Fl(j)p 1415 3951 242 4 v 1415 4042 a Fq(\(4)p Fm(\031)s(t)p Fq(\))1611 3990 y Fu(d)p 1612 3999 31 3 v 1613 4033 a Fi(2)1685 3971 y Fl(\000)1873 3914 y(j)p Fm(@)5 b(D)r Fl(j)p 1778 3951 356 4 v 1778 4045 a Fq(4\(4)p Fm(\031)s(t)p Fq(\))2016 3990 y Fu(d)p Ff(\000)p Fi(1)p 2016 4003 104 3 v 2054 4036 a(2)2162 3971 y Fq(+)2255 3914 y(\()p Fm(d)19 b Fl(\000)f Fq(1\))c Fl(j)p Fm(@)5 b(D)r Fl(j)14 b Fm(t)p 2255 3951 475 4 v 2319 4042 a Fq(\(4)p Fm(\031)s(t)p Fq(\))2515 3990 y Fu(d)p 2515 3999 31 3 v 2516 4033 a Fi(2)2560 4042 y Fq(2)p Fm(R)2739 3971 y(;)456 4180 y Fq(where)34 b Fm(@)5 b(D)37 b Fq(denotes)e(the)g(b)r(oundary)f(of)i Fm(D)h Fq(and)e(at)g(eac)n(h)f(p)r(oin)n(t)h(of)g Fm(@)5 b(D)37 b Fq(the)e(curv)-5 b(ature)35 b(is)456 4280 y(b)r(ounded)c(b)n (y)934 4247 y Fk(1)p 926 4261 51 4 v 926 4308 a Fn(R)986 4280 y Fq(.)46 b(T)-7 b(o)30 b(pro)n(v)n(e)f(b)r(ounds)i(for)f(general) f(domains)h(\012)h(w)n(e)f(can)g(apply)g(this)h(b)r(ound)456 4379 y(to)c(the)h(ball.)37 b(Note)27 b(that)1496 4528 y Fl(j)p Fm(@)5 b(B)1631 4540 y Fn(R)1685 4528 y Fl(j)37 b Fq(=)f Fm(d)14 b(\031)1963 4472 y Fu(d)p 1963 4481 31 3 v 1964 4514 a Fi(2)2094 4472 y Fm(R)2158 4442 y Fn(d)p Fh(\000)p Fk(1)p 2018 4509 340 4 v 2018 4592 a Fq(\000)2084 4525 y Fj(\000)2132 4559 y Fn(d)p 2132 4573 35 4 v 2133 4621 a Fk(2)2195 4592 y Fq(+)k(1)2320 4525 y Fj(\001)2381 4528 y Fm(:)456 4723 y Fq(In)27 b(view)h(of)f(\(7\))h (and)g(\(41\))f(w)n(e)g(\014nd)456 4880 y FC(Corollary)33 b(10.)40 b Fd(F)-6 b(or)30 b(any)g(op)l(en)g(domain)h Fq(\012)24 b Fl(\032)e Fo(R)2082 4850 y Fn(d)2151 4880 y Fd(and)30 b(any)g Fm(t)23 b(>)g Fq(0)783 5094 y Fm(Z)6 b Fq(\()p Fm(t)p Fq(\))23 b Fl(\024)1128 5038 y(j)p Fq(\012)p Fl(j)p 1061 5075 242 4 v 1061 5165 a Fq(\(4)p Fm(\031)s(t)p Fq(\))1257 5114 y Fu(d)p 1257 5123 31 3 v 1258 5156 a Fi(2)1330 5094 y Fl(\000)1539 5038 y Fm(d)1582 4978 y Fl(p)p 1651 4978 51 4 v 60 x Fm(\031)p 1423 5075 395 4 v 1423 5187 a Fq(\000)1489 5119 y Fj(\000)1537 5154 y Fn(d)p 1537 5168 35 4 v 1538 5215 a Fk(2)1600 5187 y Fq(+)18 b(1)1725 5119 y Fj(\001)1774 5112 y Fi(1)p 1772 5121 31 3 v 1772 5154 a Fu(d)1898 5038 y Fl(j)p Fq(\012)p Fl(j)2014 4982 y Fu(d)p Ff(\000)p Fi(1)p 2014 4995 104 3 v 2051 5028 a Fu(d)p 1837 5075 356 4 v 1837 5168 a Fq(4\(4)p Fm(\031)s(t)p Fq(\))2075 5114 y Fu(d)p Ff(\000)p Fi(1)p 2075 5127 104 3 v 2113 5159 a(2)2221 5094 y Fq(+)2364 5038 y Fm(d)p Fq(\()p Fm(d)i Fl(\000)e Fq(1\))p 2314 5075 395 4 v 2314 5187 a(\000)2380 5119 y Fj(\000)2428 5154 y Fn(d)p 2428 5168 35 4 v 2429 5215 a Fk(2)2491 5187 y Fq(+)g(1)2616 5119 y Fj(\001)2665 5112 y Fi(2)p 2664 5121 31 3 v 2664 5154 a Fu(d)2789 5038 y Fl(j)p Fq(\012)p Fl(j)2905 4982 y Fu(d)p Ff(\000)p Fi(2)p 2905 4995 104 3 v 2942 5028 a Fu(d)p 2728 5075 356 4 v 2728 5168 a Fq(8\(4)p Fm(\031)s(t)p Fq(\))2966 5114 y Fu(d)p Ff(\000)p Fi(2)p 2966 5127 104 3 v 3004 5159 a(2)3094 5094 y Fm(:)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 456 251 a Fk(18)781 b(LEANDER)22 b(GEISINGER)44 b(&)h(TIMO)23 b(WEIDL)456 450 y Fd(R)l(emark.)46 b Fq(The)36 b(b)r(ounds)g(from)f(Corollary)f(9)h(and)g(Corollary)e(10)i (b)r(oth)h(capture)f(the)i(main)456 550 y(asymptotic)e(b)r(eha)n(viour) g(of)h Fm(Z)6 b Fq(\()p Fm(t)p Fq(\))37 b(as)e Fm(t)h Fq(tends)h(to)f(zero.)61 b(Moreo)n(v)n(er,)35 b(they)i(con)n(tain)e (order-)456 649 y(sharp)27 b(remainder)h(terms.)40 b(Actually)-7 b(,)29 b(in)g(the)g(regime)f Fm(t)d Fl(!)g Fq(0+)i(the)j(b)r(ound)f (form)f(Corollary)456 749 y(10)d(is)h(stronger)f(than)i(\(42\))o(.)37 b(On)26 b(the)h(other)f(hand)g(the)h(b)r(ound)g(from)f(Corollary)e(10)h (do)r(es)h(not)456 849 y(sho)n(w)g(an)i(exp)r(onen)n(tial)f(deca)n(y)f (as)h Fm(t)h Fq(tends)g(to)f(in\014nit)n(y)-7 b(.)555 1003 y(Moreo)n(v)n(er,)20 b(one)g(can)h(use)f(the)i(ideas)e(of)h ([Mel03])f(and)h([HH07])f(to)h(deriv)n(e)f(un)n(viersal)f(b)r(ounds)456 1102 y(on)24 b Fm(Z)6 b Fq(\()p Fm(t)p Fq(\).)37 b(W)-7 b(e)25 b(can)f(emplo)n(y)h(inequalit)n(y)f(\(5\))h(and)g(the)g(result)g (of)g(Luttinger)g(\(7\).)36 b(F)-7 b(or)24 b(the)i(ball)456 1202 y Fm(B)519 1214 y Fn(R)599 1202 y Fl(\032)g Fo(R)750 1172 y Fn(d)819 1202 y Fq(with)k Fl(j)p Fm(B)1096 1214 y Fn(R)1150 1202 y Fl(j)d Fq(=)f Fl(j)p Fq(\012)p Fl(j)k Fq(the)g(second)f(momen)n(t)g Fm(I)21 b Fq(\()p Fm(B)2316 1214 y Fn(R)2371 1202 y Fq(\))30 b(can)f(b)r(e)h(calculated)f (explicitly)-7 b(.)456 1301 y(If)28 b(w)n(e)f(insert)g(the)h(result)f (in)n(to)h(\(5\))g(w)n(e)f(\014nd)456 1494 y(\(43\))715 b Fm(Z)6 b Fq(\()p 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Fk(1)1932 3691 y Fl(\002)18 b Fo(R)23 b Fq(:)g Fl(j)p Fm(x)2214 3703 y Fn(d)2253 3691 y Fl(j)g Fm(<)2397 3635 y Fq(1)p 2397 3672 42 4 v 2397 3748 a(2)2448 3691 y Fm(f)g Fq(\()p Fl(j)p Fm(x)2614 3657 y Fh(0)2638 3691 y Fl(j)p Fq(\))2693 3574 y Fj(\033)2783 3691 y Fm(:)456 3896 y Fq(Then)682 3875 y(~)672 3896 y(\012)732 3908 y Fn(f)803 3896 y Fq(represen)n(ts)j(an)h(example)h(of) f(a)g(domain)h(with)g(the)g(distribution)f(function)1590 4046 y Fm(m)1663 4058 y Fn(d)1702 4046 y Fq(\()p Fm(\034)9 b Fq(;)1826 4025 y(~)1816 4046 y(\012)1876 4058 y Fn(f)1919 4046 y Fq(\))38 b(=)e Fm(m)p Fq(\()p Fm(\034)9 b Fq(\))14 b Fm(:)456 4218 y Fq(In)27 b(this)h(case,)f(to)h(study)f(explicit)h (examples)f(w)n(e)g(c)n(ho)r(ose)g Fm(f)2328 4230 y Fn(\026)2372 4218 y Fq(\()p Fm(s)p Fq(\))c(=)2586 4151 y Fj(\000)2624 4218 y Fm(!)2676 4230 y Fn(d)p Fh(\000)p Fk(1)2799 4218 y Fm(s)2838 4188 y Fn(d)p Fh(\000)p Fk(1)2962 4151 y Fj(\001)3000 4168 y Fh(\000)3066 4146 y Fi(1)p 3062 4155 36 3 v 3062 4188 a Fu(\026)3111 4218 y Fq(.)456 4372 y FC(Theorem)31 b(12.)41 b Fd(F)-6 b(or)30 b(any)g Fm(\026)23 b(>)f Fq(1)30 b Fd(and)g(al)t(l)h Fm(t)23 b(>)f Fq(0)1029 4619 y Fm(Z)6 b Fq(\()p Fm(t)p Fq(;)1200 4598 y(~)1191 4619 y(\012)1251 4631 y Fn(f)1283 4639 y Fu(\026)1327 4619 y Fq(\))23 b Fl(\024)1565 4563 y Fq(1)p 1480 4600 212 4 v 1480 4690 a(\(4)p Fm(\031)s Fq(\))1646 4639 y Fu(d)p 1646 4648 31 3 v 1647 4681 a Fi(2)1718 4563 y Fm(\031)1768 4533 y Fk(1)p Fh(\000)p Fn(\026)p 1711 4600 194 4 v 1711 4676 a Fm(\026)18 b Fl(\000)h Fq(1)1924 4533 y(\000)1990 4441 y Fj(\020)2039 4533 y Fm(\033)2086 4545 y Fn(d)2144 4533 y Fq(+)2237 4496 y Fn(d)p Fk(+)p Fn(\026)p Fk(+1)p 2237 4514 210 4 v 2325 4561 a(2)2457 4441 y Fj(\021)p 1924 4600 583 4 v 1952 4683 a Fq(\000)2018 4616 y Fj(\000)2055 4683 y Fm(\033)2102 4695 y Fn(d)2160 4683 y Fq(+)2253 4650 y Fn(d)p 2253 4664 35 4 v 2254 4712 a Fk(2)2316 4683 y Fq(+)f(1)2441 4616 y Fj(\001)2546 4619 y Fm(t)2576 4585 y Fh(\000)2638 4558 y Fu(d)p Ff(\000)p Fi(1+)p Fu(\026)p 2638 4572 182 3 v 2715 4605 a Fi(2)2848 4619 y Fm(:)456 4850 y Fd(Pr)l(o)l(of.)43 b Fq(The)22 b(de\014nition)h(of)1343 4829 y(~)1334 4850 y(\012)1394 4862 y Fn(f)1426 4870 y Fu(\026)1492 4850 y Fq(and)g(the)f(c)n(hoice)g (of)g Fm(f)2158 4862 y Fn(\026)2225 4850 y Fq(implies)g Fm(m)2574 4862 y Fn(d)2613 4850 y Fq(\()p Fm(\034)5 b(;)14 b Fq(\012)2783 4862 y Fn(f)2815 4870 y Fu(\026)2859 4850 y Fq(\))23 b(=)g Fm(\034)3047 4820 y Fh(\000)p Fn(\026)3144 4850 y Fq(.)35 b(Hence,)456 4950 y(w)n(e)27 b(can)g(emplo)n(y)g(Prop)r (osition)f(5)h(with)h Fm(\033)f Fq(=)22 b Fm(\033)1929 4962 y Fn(d)1996 4950 y Fq(and)28 b(\014nd)835 5140 y Fm(R)898 5152 y Fn(\033)936 5161 y Fu(d)975 5140 y Fq(\(\003\))37 b Fl(\024)g Fm(L)1293 5105 y Fn(cl)1293 5160 y(\033)1331 5169 y Fu(d)1366 5160 y Fn(;d)1438 5027 y Fj(Z)1521 5047 y Fh(1)1518 5193 y Fu(\031)p 1494 5202 84 3 v 1494 5210 a Ff(p)p 1540 5210 38 3 v 35 x Fi(\003)1605 5140 y Fm(\034)1650 5105 y Fh(\000)p Fn(\026)1761 5140 y Fm(d\034)24 b Fq(\003)1922 5105 y Fn(\033)1960 5114 y Fu(d)1995 5105 y Fk(+)2056 5083 y Fu(d)p 2056 5092 31 3 v 2057 5125 a Fi(2)2138 5140 y Fq(=)36 b Fm(L)2296 5105 y Fn(cl)2296 5160 y(\033)2334 5169 y Fu(d)2369 5160 y Fn(;d)2444 5083 y Fm(\031)2494 5053 y Fk(1)p Fh(\000)p Fn(\026)p 2437 5120 194 4 v 2437 5197 a Fm(\026)19 b Fl(\000)f Fq(1)2640 5140 y(\003)2698 5105 y Fn(\033)2736 5114 y Fu(d)2771 5105 y Fk(+)2832 5078 y Fu(d)p Fi(+)p Fu(\026)p Ff(\000)p Fi(1)p 2832 5092 182 3 v 2909 5125 a(2)3042 5140 y Fm(:)p eop end %%Page: 21 21 TeXDict begin 21 20 bop 640 251 a Fk(UNIVERSAL)28 b(BOUNDS)h(F)n(OR)h (TRA)n(CES)f(OF)g(THE)g(DIRICHLET)h(LAPLA)n(CE)f(OPERA)-5 b(TOR)117 b(21)456 450 y Fq(T)-7 b(o)23 b(this)h(inequalit)n(y)f(w)n(e) g(can)g(apply)h(the)g(Laplace)e(transformation)g(and)i(simplify)g(the)g (result-)456 550 y(ing)j(b)r(ound)h(on)f Fm(Z)6 b Fq(\()p Fm(t)p Fq(\))28 b(to)g(arriv)n(e)d(at)j(the)g(claimed)f(result.)1117 b Fc(\003)456 733 y Fd(R)l(emark.)44 b Fq(In)31 b(dimension)g Fm(d)e Fq(=)f(2,)k(according)d(to)i([Sim83,)13 b(vdB87,)g(ST90],)30 b(the)h(asymptotics)456 839 y(\(45\))i(and)g(\(46\))g(are)g(v)-5 b(alid)33 b(for)1479 818 y(~)1470 839 y(\012)1530 851 y Fn(f)1606 839 y Fq(as)g(w)n(ell.)54 b(In)34 b(this)g(case)f(the)h(b)r (ound)f(from)h(Theorem)e(12)456 938 y(reads)26 b(as)1212 1103 y Fm(Z)6 b Fq(\()p Fm(t)p Fq(;)1383 1082 y(~)1374 1103 y(\012)1434 1115 y Fn(f)1466 1123 y Fu(\026)1510 1103 y Fq(\))37 b Fl(\024)1836 1046 y Fq(4)p 1690 1083 333 4 v 1690 1171 a(105)14 b Fm(\031)1880 1143 y Fn(\026)p Fk(+)1980 1120 y Fi(1)p 1980 1129 29 3 v 1980 1163 a(2)2056 1041 y Fq(\000)2122 974 y Fj(\000)2160 1041 y Fq(4)k(+)2313 1004 y Fn(\026)p 2313 1022 41 4 v 2317 1070 a Fk(2)2363 974 y Fj(\001)p 2056 1083 346 4 v 2132 1160 a Fm(\026)h Fl(\000)f Fq(1)2439 1103 y Fm(t)2469 1068 y Fh(\000)2531 1041 y Fu(\026)p Fi(+1)p 2531 1055 107 3 v 2570 1088 a(2)2665 1103 y Fm(:)456 1288 y Fq(In)25 b(view)g(of)g(\(45\))g(this)h (b)r(ound)g(sho)n(ws)e(again)g(the)h(correct)f(order)g(in)i Fm(t)f Fq(as)g Fm(t)e Fl(!)g Fq(0+.)35 b(Moreo)n(v)n(er,)456 1387 y(if)28 b(w)n(e)f(compare)f(the)i(constan)n(ts)852 1602 y Fm(b)888 1614 y Fk(1)925 1602 y Fq(\()p Fm(\026)p Fq(\))c(:=)1329 1546 y(4)p 1183 1583 333 4 v 1183 1671 a(105)14 b Fm(\031)1373 1642 y Fn(\026)p Fk(+)1473 1620 y Fi(1)p 1473 1629 29 3 v 1473 1662 a(2)1536 1541 y Fq(\000)1602 1474 y Fj(\000)1639 1541 y Fq(4)k(+)1792 1504 y Fn(\026)p 1792 1522 41 4 v 1796 1570 a Fk(2)1843 1474 y Fj(\001)p 1536 1583 346 4 v 1612 1659 a Fm(\026)g Fl(\000)g Fq(1)1974 1602 y(and)83 b Fm(b)2227 1568 y Fh(0)2227 1623 y Fk(1)2263 1602 y Fq(\()p Fm(\026)p Fq(\))24 b(:=)2522 1541 y(\000)2588 1474 y Fj(\000)2626 1541 y Fq(1)18 b(+)2779 1504 y Fn(\026)p 2779 1522 41 4 v 2783 1570 a Fk(2)2829 1474 y Fj(\001)2881 1541 y Fm(\020)6 b Fq(\()p Fm(\026)p Fq(\))p 2522 1583 516 4 v 2655 1671 a(2)14 b Fm(\031)2761 1642 y Fn(\026)p Fk(+)2862 1620 y Fi(1)p 2862 1629 29 3 v 2862 1662 a(2)456 1805 y Fq(from)27 b(the)h(b)r(ound)g(ab)r(o)n(v)n(e)e(and)h(the)h (asymptotics)f(\(45\))h(w)n(e)f(\014nd)1011 2013 y(lim)973 2064 y Fn(\026)p Fh(!)p Fk(1+)1178 1896 y Fj(\022)1249 1957 y Fm(b)1285 1969 y Fk(1)1322 1957 y Fq(\()p Fm(\026)p Fq(\))p 1249 1994 188 4 v 1249 2070 a Fm(b)1285 2041 y Fh(0)1285 2092 y Fk(1)1322 2070 y Fq(\()p Fm(\026)p Fq(\))1446 1896 y Fj(\023)1531 2013 y Fq(=)60 b(lim)1618 2064 y Fn(\026)p Fh(!)p Fk(1+)1823 1896 y Fj(\022)1894 1957 y Fq(\()p Fm(\026)18 b Fq(+)g(6\)\()p Fm(\026)h Fq(+)f(4\)\()p Fm(\026)h Fq(+)f(2\))p 1894 1994 773 4 v 2073 2070 a(\()p Fm(\026)h Fl(\000)f Fq(1\))p Fm(\020)6 b Fq(\()p Fm(\026)p Fq(\))2676 1896 y Fj(\023)2761 2013 y Fq(=)22 b(1)14 b Fm(:)555 2221 y Fq(In)27 b(order)f(to)h(state)g(an)f (example)h(for)f(un)n(b)r(ounded)i(domains)e(with)i(\014nite)f(v)n (olume,)g(c)n(ho)r(ose)456 2321 y Fm(f)497 2333 y Fn(e)532 2321 y Fq(\()p Fm(s)p Fq(\))c(=)g(exp)887 2254 y Fj(\000)925 2321 y Fl(\000)p Fm(!)1042 2333 y Fn(d)p Fh(\000)p Fk(1)1165 2321 y Fm(s)1204 2291 y Fn(d)p Fh(\000)p Fk(1)1327 2254 y Fj(\001)1365 2321 y Fq(.)37 b(In)28 b(the)g(same)f(w)n(a)n(y)g(as)f (ab)r(o)n(v)n(e)h(one)g(can)g(sho)n(w)456 2484 y FC(Theorem)k(13.)41 b Fd(F)-6 b(or)30 b(al)t(l)g Fm(t)23 b(>)g Fq(0)29 b Fd(the)h(estimate)675 2689 y Fm(Z)6 b Fq(\()p Fm(t;)846 2668 y Fq(~)837 2689 y(\012)897 2701 y Fn(f)929 2709 y Fu(e)965 2689 y Fq(\))24 b Fl(\024)1218 2633 y Fq(1)p 1118 2670 242 4 v 1118 2760 a(\(4)p Fm(\031)s(t)p Fq(\))1314 2709 y Fu(d)p 1315 2718 31 3 v 1316 2751 a Fi(2)1375 2668 y Fq(^)1369 2689 y(\000)1435 2572 y Fj(\022)1496 2689 y Fm(\033)1543 2701 y Fn(d)1601 2689 y Fq(+)1694 2633 y Fm(d)p 1694 2670 44 4 v 1695 2746 a Fq(2)1765 2689 y(+)18 b(1)p Fm(;)c(\031)1977 2655 y Fk(2)2014 2689 y Fm(t)2044 2572 y Fj(\023)2124 2689 y Fq(+)2265 2573 y Fl(p)p 2334 2573 51 4 v 60 x Fm(\031)31 b Fq(ln)14 b Fm(t)p 2217 2670 356 4 v 2217 2763 a Fq(4\(4)p Fm(\031)s(t)p Fq(\))2455 2709 y Fu(d)p Ff(\000)p Fi(1)p 2455 2722 104 3 v 2493 2754 a(2)2593 2628 y Fq(\000)2659 2560 y Fj(\000)2697 2628 y Fm(\033)2744 2640 y Fn(d)2801 2628 y Fq(+)2894 2595 y Fn(d)p Fk(+1)p 2894 2609 119 4 v 2937 2656 a(2)3023 2628 y Fm(;)g(\031)3110 2597 y Fk(2)3147 2628 y Fm(t)3177 2560 y Fj(\001)p 2593 2670 623 4 v 2641 2753 a Fq(\000)2707 2686 y Fj(\000)2744 2753 y Fm(\033)2791 2765 y Fn(d)2849 2753 y Fq(+)2942 2720 y Fn(d)p 2942 2734 35 4 v 2943 2781 a Fk(2)3005 2753 y Fq(+)k(1)3130 2686 y Fj(\001)1101 2952 y Fq(+)1194 2836 y Fl(p)p 1263 2836 51 4 v 60 x Fm(\031)f Fq(\()q(ln)d Fm(\031)21 b Fl(\000)d Fq(1\))p 1194 2933 475 4 v 1253 3026 a(2\(4)p Fm(\031)s(t)p Fq(\))1491 2971 y Fu(d)p Ff(\000)p Fi(1)p 1491 2984 104 3 v 1529 3017 a(2)1688 2890 y Fq(\000)1754 2823 y Fj(\000)1792 2890 y Fm(\033)1839 2902 y Fn(d)1896 2890 y Fq(+)1989 2858 y Fn(d)p Fk(+1)p 1989 2872 119 4 v 2032 2919 a(2)2118 2890 y Fm(;)c(\031)2205 2860 y Fk(2)2242 2890 y Fm(t)2272 2823 y Fj(\001)p 1688 2933 623 4 v 1736 3016 a Fq(\000)1802 2948 y Fj(\000)1840 3016 y Fm(\033)1887 3028 y Fn(d)1944 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3685 y(since)c Fl(j)p Fq(\012)741 3697 y Fn(f)773 3705 y Fu(e)810 3685 y Fl(j)d Fq(=)f(1.)37 b(In)26 b(dimension)g Fm(d)e Fq(=)e(2)k(w)n(e)g(can)h(use)f(\(8\))g (and)h(\(46\))f(to)g(p)r(oin)n(t)h(out)f(that)h(ev)n(en)456 3785 y(the)h(second)f(term)g(of)h(the)g(b)r(ound)g(captures)e(the)i (righ)n(t)f(order)g(in)g Fm(t)h Fq(as)f Fm(t)h Fq(tends)f(to)h(zero.) 1462 4022 y(6.)41 b Fp(Pr)n(oof)31 b(of)g(Theorem)h(2)555 4172 y Fq(Here)23 b(w)n(e)g(use)g(the)g(results)g(from)g(section)f(3)h (to)g(deriv)n(e)f(univ)n(ersal)g(b)r(ounds)i(with)f(correction)456 4271 y(terms)34 b(on)g(the)g(Riesz)h(means)f Fm(R)1515 4283 y Fn(\033)1559 4271 y Fq(\(\003\).)58 b(First)34 b(w)n(e)g(note)g(that)h(Prop)r(osition)e(5)h(and)g(Lemma)456 4380 y(7)d(immediately)h(imply)g(the)g(follo)n(wing)f(estimate.)50 b(Recall)31 b(that)h Fm(\034)2597 4392 y Fk(\012)2679 4380 y Fq(=)e Fm(d)2817 4349 y Fk(2)2854 4380 y Fm(\031)2904 4349 y Fk(2)2942 4380 y Fl(j)p Fq(\012)p Fl(j)3048 4349 y Fh(\000)3111 4327 y Fi(2)p 3110 4336 V 3110 4369 a Fu(d)3186 4380 y Fq(and)i(let)456 4482 y Fm(\033)g Fl(\025)638 4449 y Fk(3)p 638 4463 34 4 v 638 4510 a(2)712 4482 y Fq(satisfy)e Fm(")1027 4414 y Fj(\000)1065 4482 y Fm(\033)22 b Fq(+)1227 4449 y Fn(d)p Fh(\000)p Fk(1)p 1227 4463 120 4 v 1270 4510 a(2)1356 4414 y Fj(\001)1423 4482 y Fq(=)1526 4449 y Fk(1)p 1526 4463 34 4 v 1526 4510 a(2)1569 4482 y Fm(B)1650 4414 y Fj(\000)1698 4449 y Fk(1)p 1698 4463 V 1698 4510 a(2)1741 4482 y Fm(;)14 b(\033)22 b Fq(+)1940 4449 y Fn(d)p Fk(+1)p 1940 4463 119 4 v 1983 4510 a(2)2068 4414 y Fj(\001)2107 4482 y Fq(,)31 b(hence)g Fm(\016)2432 4494 y Fn(\033)o(;d)2556 4482 y Fq(=)d(0.)47 b(Then)31 b(for)f(an)n(y)g(op)r(en)456 4590 y(domain)d(\012)c Fl(\032)f Fo(R)981 4559 y Fn(d)1048 4590 y Fq(and)27 b(all)h(\003)22 b Fm(>)h Fq(0)k(w)n(e)g(\014nd)693 4774 y Fm(R)756 4786 y Fn(\033)800 4774 y Fq(\(\003\))51 b Fl(\024)g Fm(L)1146 4739 y Fn(cl)1146 4794 y(\033)o(;d)1264 4717 y Fm(d)19 b Fl(\000)f Fq(1)p 1264 4755 187 4 v 1336 4831 a Fm(d)1474 4774 y Fl(j)p Fq(\012)p Fl(j)c Fq(\003)1652 4739 y Fn(\033)r Fk(+)1754 4717 y Fu(d)p 1754 4726 31 3 v 1755 4759 a Fi(2)2658 4774 y Fq(if)28 b(\003)23 b Fm(<)g(\034)2939 4786 y Fk(\012)3073 4774 y Fq(and)693 4953 y Fm(R)756 4965 y Fn(\033)800 4953 y Fq(\(\003\))51 b Fl(\024)g Fm(L)1146 4918 y Fn(cl)1146 4973 y(\033)o(;d)1255 4953 y Fl(j)p Fq(\012)p Fl(j)14 b Fq(\003)1433 4918 y Fn(\033)r Fk(+)1534 4896 y Fu(d)p 1534 4905 V 1535 4938 a Fi(2)1597 4953 y Fl(\000)k Fm(\031)f(L)1801 4918 y Fn(cl)1801 4973 y(\033)o(;d)1910 4953 y Fl(j)p Fq(\012)p Fl(j)2026 4893 y Fu(d)p Ff(\000)p Fi(1)p 2026 4906 104 3 v 2062 4938 a Fu(d)2144 4953 y Fq(\003)2202 4918 y Fn(\033)r Fk(+)2303 4893 y Fu(d)p Ff(\000)p Fi(1)p 2303 4906 V 2341 4938 a(2)2658 4953 y Fq(if)28 b(\003)23 b Fl(\025)g Fm(\034)2939 4965 y Fk(\012)2990 4953 y Fm(:)555 5116 y Fq(Next)30 b(w)n(e)f(discuss,)h(ho)n(w)f(a)g(tric)n(k)g(b)n(y)g (Aizenmann)h(and)f(Lieb)h([AL78])f(can)g(b)r(e)h(applied)g(to)456 5216 y(inequalities)d(for)g(eigen)n(v)-5 b(alue)27 b(means)g Fm(R)1733 5228 y Fn(\015)1775 5216 y Fq(\(\003\))h(with)g(remainder)f (terms.)p eop end %%Page: 22 22 TeXDict begin 22 21 bop 456 251 a Fk(22)781 b(LEANDER)22 b(GEISINGER)44 b(&)h(TIMO)23 b(WEIDL)456 450 y FC(Lemma)31 b(14.)41 b Fd(L)l(et)29 b Fm(\015)f(>)22 b(\033)27 b Fl(\025)1430 417 y Fk(3)p 1430 431 34 4 v 1430 479 a(2)1473 450 y Fd(,)k Fm(\025)1577 462 y Fk(1)1637 450 y Fl(\025)23 b Fm(\025)g Fl(\025)g Fq(0)29 b Fd(and)h Fq(\003)23 b Fl(\025)g Fm(\025)p Fd(.)39 b(Then)513 655 y Fm(R)576 667 y Fn(\015)619 655 y Fq(\(\003\))83 b Fl(\024)g Fm(L)1029 621 y Fn(cl)1029 676 y(\015)t(;d)1139 655 y Fl(j)p Fq(\012)p Fl(j)14 b Fq(\003)1317 621 y Fn(\015)t Fk(+)1416 598 y Fu(d)p 1416 607 31 3 v 1417 641 a Fi(2)1481 634 y Fq(^)1461 655 y Fm(B)1542 538 y Fj(\022)1618 599 y Fm(\025)p 1613 636 58 4 v 1613 712 a Fq(\003)1681 655 y Fm(;)g(\033)21 b Fq(+)1879 599 y Fm(d)p 1879 636 44 4 v 1880 712 a Fq(2)1951 655 y(+)d(1)p Fm(;)c(\015)22 b Fl(\000)c Fm(\033)2311 538 y Fj(\023)972 905 y Fl(\000)1272 838 y Fm(L)1329 807 y Fn(cl)1329 861 y(\033)o(;d)p 1060 886 576 4 v 1060 962 a Fm(B)g Fq(\()q Fm(\033)j Fq(+)d(1)p Fm(;)c(\015)23 b Fl(\000)18 b Fm(\033)s Fq(\))1659 792 y Fj(Z)1742 812 y Fk(\003)p Fh(\000)p Fn(\025)1706 980 y Fk(0)1897 905 y Fm(\034)1942 870 y Fn(\015)t Fh(\000)p Fn(\033)r Fh(\000)p Fk(1)2163 905 y Fm(M)2266 788 y Fj(\022)2449 849 y Fm(\031)p 2337 886 275 4 v 2337 902 a Fl(p)p 2407 902 205 4 v 2407 969 a Fq(\003)g Fl(\000)g Fm(\034)2621 905 y Fq(;)c(\012)2718 788 y Fj(\023)2793 905 y Fq(\(\003)19 b Fl(\000)f Fm(\034)9 b Fq(\))3063 863 y Fn(\033)r Fk(+)3164 841 y Fu(d)p 3164 850 31 3 v 3165 883 a Fi(2)3223 905 y Fm(d\034)972 1150 y Fq(+)1282 1093 y Fm(\016)1319 1105 y Fn(\033)o(;d)p 1060 1131 576 4 v 1060 1207 a Fm(B)18 b Fq(\()q Fm(\033)j Fq(+)d(1)p Fm(;)c(\015)23 b Fl(\000)18 b Fm(\033)s Fq(\))1659 1037 y Fj(Z)1742 1057 y Fk(\003)p Fh(\000)p Fn(\025)1706 1225 y Fk(0)1897 1150 y Fm(\034)1942 1115 y Fn(\015)t Fh(\000)p Fn(\033)r Fh(\000)p Fk(1)2163 1150 y Fm(m)2250 1033 y Fj(\022)2433 1093 y Fm(\031)p 2321 1131 275 4 v 2321 1147 a Fl(p)p 2390 1147 205 4 v 67 x Fq(\003)g Fl(\000)g Fm(\034)2605 1150 y Fq(;)c(\012)2702 1033 y Fj(\023)2776 1150 y Fq(\()q(\003)k Fl(\000)g Fm(\034)9 b Fq(\))3046 1108 y Fn(\033)r Fk(+)3147 1082 y Fu(d)p Ff(\000)p Fi(1)p 3147 1095 104 3 v 3185 1128 a(2)3279 1150 y Fm(d\034)c(:)-2930 b Fq(\(48\))456 1356 y Fd(Pr)l(o)l(of.)43 b Fq(W)-7 b(e)28 b(start)f(from)g(the)h(w)n(ell-kno)n(wn)e(iden)n(tit)n (y)i([AL78])908 1550 y Fm(R)971 1562 y Fn(\015)1014 1550 y Fq(\(\003\))83 b(=)1644 1494 y(1)p 1377 1531 576 4 v 1377 1607 a Fm(B)18 b Fq(\()p Fm(\033)k Fq(+)c(1)p Fm(;)c(\015)22 b Fl(\000)c Fm(\033)s Fq(\))1976 1437 y Fj(Z)2059 1458 y Fh(1)2022 1626 y Fk(0)2143 1550 y Fm(\034)2188 1516 y Fn(\015)t Fh(\000)p Fn(\033)r Fh(\000)p Fk(1)2423 1550 y Fm(R)2486 1562 y Fn(\033)2530 1550 y Fq(\(\003)h Fl(\000)f Fm(\034)9 b Fq(\))14 b Fm(d\034)1219 1794 y Fq(=)1644 1738 y(1)p 1377 1775 V 1377 1851 a Fm(B)k Fq(\()p Fm(\033)k Fq(+)c(1)p Fm(;)c(\015)22 b Fl(\000)c Fm(\033)s Fq(\))1976 1681 y Fj(Z)2059 1702 y Fk(\003)p Fh(\000)p Fn(\025)2022 1870 y Fk(0)2214 1794 y Fm(\034)2259 1760 y Fn(\015)t Fh(\000)p Fn(\033)r Fh(\000)p Fk(1)2493 1794 y Fm(R)2556 1806 y Fn(\033)2601 1794 y Fq(\(\003)g Fl(\000)g Fm(\034)9 b Fq(\))14 b Fm(d\034)5 b(:)-2534 b Fq(\(49\))456 2004 y(Here)33 b(w)n(e)h(ha)n(v)n(e)f(tak)n(en)g(in)n (to)h(accoun)n(t)f(that)h Fm(R)1954 2016 y Fn(\033)1999 2004 y Fq(\()2039 1983 y(~)2031 2004 y(\003)o(\))g(=)g(0)f(for)2470 1983 y(~)2462 2004 y(\003)g Fl(\024)g Fm(\025)h Fl(\024)f Fm(\025)2879 2016 y Fk(1)2917 2004 y Fq(.)56 b(No)n(w)33 b(w)n(e)h(can)456 2104 y(apply)27 b(Prop)r(osition)f(5)h(and)g(\014nd) 513 2313 y Fm(R)576 2325 y Fn(\015)619 2313 y Fq(\(\003\))83 b Fl(\024)1134 2246 y Fm(L)1191 2216 y Fn(cl)1191 2270 y(\033)o(;d)1299 2246 y Fl(j)p Fq(\012)p Fl(j)p 982 2294 V 982 2370 a Fm(B)18 b Fq(\()p Fm(\033)k Fq(+)c(1)p Fm(;)c(\015)22 b Fl(\000)d Fm(\033)s Fq(\))1595 2200 y Fj(Z)1678 2221 y Fk(\003)p Fh(\000)p Fn(\025)1641 2389 y Fk(0)1833 2313 y Fm(\034)1878 2279 y Fn(\015)t Fh(\000)p Fn(\033)r Fh(\000)p Fk(1)2126 2313 y Fq(\(\003)f Fl(\000)g Fm(\034)9 b Fq(\))2395 2272 y Fn(\033)r Fk(+)2497 2249 y Fu(d)p 2497 2258 31 3 v 2498 2292 a Fi(2)2569 2313 y Fm(d\034)972 2562 y Fl(\000)1272 2495 y Fm(L)1329 2465 y Fn(cl)1329 2518 y(\033)o(;d)p 1060 2543 576 4 v 1060 2619 a Fm(B)18 b Fq(\()q Fm(\033)j Fq(+)d(1)p Fm(;)c(\015)23 b Fl(\000)18 b Fm(\033)s Fq(\))1659 2449 y Fj(Z)1742 2469 y Fk(\003)p Fh(\000)p Fn(\025)1706 2638 y Fk(0)1897 2562 y Fm(\034)1942 2528 y Fn(\015)t Fh(\000)p Fn(\033)r Fh(\000)p Fk(1)2163 2562 y Fm(M)2266 2445 y Fj(\022)2449 2506 y Fm(\031)p 2337 2543 275 4 v 2337 2559 a Fl(p)p 2407 2559 205 4 v 2407 2626 a Fq(\003)g Fl(\000)g Fm(\034)2621 2562 y Fq(;)c(\012)2718 2445 y Fj(\023)2793 2562 y Fq(\(\003)19 b Fl(\000)f Fm(\034)9 b Fq(\))3063 2520 y Fn(\033)r Fk(+)3164 2498 y Fu(d)p 3164 2507 31 3 v 3165 2540 a Fi(2)3223 2562 y Fm(d\034)972 2807 y Fq(+)1282 2751 y Fm(\016)1319 2763 y Fn(\033)o(;d)p 1060 2788 576 4 v 1060 2864 a Fm(B)18 b Fq(\()q Fm(\033)j Fq(+)d(1)p Fm(;)c(\015)23 b Fl(\000)18 b Fm(\033)s Fq(\))1659 2694 y Fj(Z)1742 2714 y Fk(\003)p Fh(\000)p Fn(\025)1706 2883 y Fk(0)1897 2807 y Fm(\034)1942 2773 y Fn(\015)t Fh(\000)p Fn(\033)r Fh(\000)p Fk(1)2163 2807 y Fm(m)2250 2690 y Fj(\022)2433 2751 y Fm(\031)p 2321 2788 275 4 v 2321 2804 a Fl(p)p 2390 2804 205 4 v 67 x Fq(\003)g Fl(\000)g Fm(\034)2605 2807 y Fq(;)c(\012)2702 2690 y Fj(\023)2776 2807 y Fq(\()q(\003)k Fl(\000)g Fm(\034)9 b Fq(\))3046 2765 y Fn(\033)r Fk(+)3147 2739 y Fu(d)p Ff(\000)p Fi(1)p 3147 2752 104 3 v 3185 2785 a(2)3279 2807 y Fm(d\034)c(:)456 2999 y Fq(Finally)30 b(let)i(us)f(ev)-5 b(aluate)30 b(the)i(\014rst)e(term)h(on)g(the)g(righ)n(t)g(hand)g(side) f(of)h(this)h(expression.)45 b(A)456 3099 y(substitution)28 b(of)f(the)h(in)n(tegration)f(v)-5 b(ariable)26 b Fm(s)d Fq(=)2053 3066 y Fn(\034)p 2049 3080 46 4 v 2049 3128 a Fk(\003)2132 3099 y Fq(giv)n(es)1051 3260 y Fm(L)1108 3230 y Fn(cl)1108 3283 y(\033)o(;d)1203 3260 y Fl(j)p Fq(\012)p Fl(j)p 892 3308 576 4 v 892 3384 a Fm(B)18 b Fq(\()p Fm(\033)k Fq(+)c(1)p Fm(;)c(\015)23 b Fl(\000)18 b Fm(\033)s Fq(\))1491 3327 y(\003)1549 3293 y Fn(\015)t Fk(+)1648 3270 y Fu(d)p 1648 3279 31 3 v 1649 3313 a Fi(2)1721 3214 y Fj(Z)1804 3235 y Fk(1)p Fh(\000)1900 3212 y Fu(\025)p 1899 3221 38 3 v 1899 3255 a Fi(\003)1767 3403 y Fk(0)1964 3327 y Fm(s)2003 3293 y Fn(\015)t Fh(\000)p Fn(\033)r Fh(\000)p Fk(1)2237 3327 y Fq(\(1)g Fl(\000)g Fm(s)p Fq(\))2483 3293 y Fn(\033)r Fk(+)2585 3270 y Fu(d)p 2585 3279 31 3 v 2586 3313 a Fi(2)2644 3327 y Fm(ds)734 3608 y Fq(=)83 b Fl(j)p Fq(\012)p Fl(j)14 b Fq(\003)1060 3574 y Fn(\015)t Fk(+)1159 3552 y Fu(d)p 1159 3561 V 1160 3594 a Fi(2)1204 3608 y Fm(L)1261 3574 y Fn(cl)1261 3629 y(\033)o(;d)1366 3547 y Fm(B)t Fq(\()p Fm(\015)23 b Fl(\000)18 b Fm(\033)n(;)c(\033)23 b Fq(+)1859 3514 y Fn(d)p 1859 3528 35 4 v 1860 3576 a Fk(2)1922 3547 y Fq(+)18 b(1\))p 1366 3589 713 4 v 1441 3665 a Fm(B)t Fq(\()p Fm(\033)23 b Fq(+)18 b(1)p Fm(;)c(\015)22 b Fl(\000)c Fm(\033)s Fq(\))2102 3441 y Fj(0)2102 3591 y(@)2175 3608 y Fq(1)g Fl(\000)2328 3479 y Fj(R)2395 3478 y Fu(\025)p 2393 3487 38 3 v 2393 3520 a Fi(\003)2367 3576 y Fk(0)2445 3546 y Fq(\(1)g Fl(\000)g Fm(t)p Fq(\))2682 3516 y Fn(\015)t Fh(\000)p Fn(\033)r Fh(\000)p Fk(1)2903 3546 y Fm(t)2933 3516 y Fn(\033)r Fk(+)3034 3494 y Fu(d)p 3034 3503 31 3 v 3035 3536 a Fi(2)3079 3546 y Fm(dt)p 2328 3589 825 4 v 2384 3672 a(B)t Fq(\()p Fm(\015)23 b Fl(\000)18 b Fm(\033)n(;)c(\033)23 b Fq(+)2877 3639 y Fn(d)p 2877 3653 35 4 v 2878 3701 a Fk(2)2940 3672 y Fq(+)18 b(1\))3162 3441 y Fj(1)3162 3591 y(A)734 3866 y Fq(=)83 b Fl(j)p Fq(\012)p Fl(j)14 b Fq(\003)1060 3831 y Fn(\015)t Fk(+)1159 3809 y Fu(d)p 1159 3818 31 3 v 1160 3851 a Fi(2)1204 3866 y Fm(L)1261 3831 y Fn(cl)1261 3886 y(\015)t(;d)1385 3773 y Fj(\020)1434 3866 y Fq(1)k Fl(\000)1597 3845 y Fq(~)1577 3866 y Fm(B)1658 3798 y Fj(\000)1709 3833 y Fn(\025)p 1706 3847 46 4 v 1706 3894 a Fk(\003)1762 3866 y Fm(;)c(\033)22 b Fq(+)1961 3833 y Fn(d)p 1961 3847 35 4 v 1962 3894 a Fk(2)2024 3866 y Fq(+)c(1)p Fm(;)c(\015)22 b Fl(\000)c Fm(\033)2384 3798 y Fj(\001)2422 3773 y(\021)2486 3866 y Fm(:)3380 4031 y Fc(\003)555 4199 y Fq(If)29 b(w)n(e)e(apply)h (this)g(Lemma)g(with)g Fm(\033)f Fq(=)d Fm(\033)1848 4211 y Fn(d)1915 4199 y Fq(then)k(b)r(ecause)g(of)g Fm(\016)2544 4211 y Fn(\033)2582 4220 y Fu(d)2617 4211 y Fn(;d)2699 4199 y Fq(=)23 b(0)28 b(the)g(last)g(term)g(on)456 4299 y(the)c(righ)n(t)f(hand)h(side)g(of)30 b(\(48\))24 b(v)-5 b(anishes.)35 b(This)24 b(enables)g(us)g(to)f(\014nish)i(the)f(pro)r (of)f(of)h(Theorem)456 4398 y(2.)456 4566 y Fd(Pr)l(o)l(of)30 b(of)h(The)l(or)l(em)g(2.)42 b Fq(Inequalit)n(y)f(\(48\))e(with)i Fm(\015)48 b(>)43 b(\033)48 b Fq(=)43 b Fm(\033)2474 4578 y Fn(d)2553 4566 y Fq(and)d(with)g(a)g(substitution)456 4665 y Fm(y)25 b Fq(=)694 4633 y Fn(\031)p 620 4647 190 4 v 620 4655 a Fh(p)p 674 4655 135 3 v 674 4701 a Fk(\003)p Fh(\000)p Fn(\034)847 4665 y Fq(giv)n(es)519 4887 y Fm(R)582 4899 y Fn(\015)625 4887 y Fq(\(\003\))83 b Fl(\024)g Fm(L)1035 4853 y Fn(cl)1035 4907 y(\015)t(;d)1145 4887 y Fl(j)p Fq(\012)p Fl(j)14 b Fq(\003)1323 4853 y Fn(\015)t Fk(+)1422 4830 y Fu(d)p 1422 4839 31 3 v 1423 4873 a Fi(2)1487 4866 y Fq(^)1467 4887 y Fm(B)1548 4770 y Fj(\022)1624 4831 y Fm(\025)p 1619 4868 58 4 v 1619 4944 a Fq(\003)1687 4887 y Fm(;)g(\033)1771 4899 y Fn(d)1828 4887 y Fq(+)1921 4831 y Fm(d)p 1921 4868 44 4 v 1922 4944 a Fq(2)1993 4887 y(+)k(1)p Fm(;)c(\015)22 b Fl(\000)c Fm(\033)2350 4899 y Fn(d)2389 4770 y Fj(\023)978 5140 y Fl(\000)1096 5072 y Fq(2)p Fm(\031)1188 5042 y Fk(2)p Fn(\033)1259 5051 y Fu(d)1294 5042 y Fk(+)p Fn(d)p Fk(+2)1468 5072 y Fm(L)1525 5042 y Fn(cl)1525 5096 y(\033)1563 5105 y Fu(d)1598 5096 y Fn(;d)p 1052 5120 647 4 v 1052 5197 a Fm(B)g Fq(\()q Fm(\033)1213 5209 y Fn(d)1270 5197 y Fq(+)g(1)p Fm(;)c(\015)23 b Fl(\000)18 b Fm(\033)1628 5209 y Fn(d)1667 5197 y Fq(\))1737 5027 y Fj(Z)1852 5025 y Fu(\031)p 1830 5034 81 3 v 1830 5041 a Ff(p)p 1875 5041 35 3 v 1875 5077 a Fu(\025)1817 5193 y(\031)p 1793 5202 84 3 v 1793 5210 a Ff(p)p 1839 5210 38 3 v 35 x Fi(\003)1938 5023 y Fj(\022)1999 5140 y Fq(\003)g Fl(\000)2168 5083 y Fm(\031)2218 5053 y Fk(2)p 2168 5120 88 4 v 2171 5197 a Fm(y)2215 5173 y Fk(2)2266 5023 y Fj(\023)2327 5036 y Fn(\015)t Fh(\000)p Fn(\033)2456 5045 y Fu(d)2491 5036 y Fh(\000)p Fk(1)2594 5140 y Fm(M)9 b Fq(\()p Fm(y)s Fq(;)14 b(\012\))g Fm(y)2947 5105 y Fh(\000)p Fk(2)p Fn(\033)3070 5114 y Fu(d)3104 5105 y Fh(\000)p Fn(d)p Fh(\000)p Fk(3)3294 5140 y Fm(dy)p eop end %%Page: 23 23 TeXDict begin 23 22 bop 640 251 a Fk(UNIVERSAL)28 b(BOUNDS)h(F)n(OR)h (TRA)n(CES)f(OF)g(THE)g(DIRICHLET)h(LAPLA)n(CE)f(OPERA)-5 b(TOR)117 b(23)456 450 y Fq(for)40 b(all)h(\003)k Fl(\025)g Fm(\025)p Fq(.)77 b(First)41 b(w)n(e)f(assume)h Fm(\025)k Fl(\025)g Fm(\034)1977 462 y Fk(\012)2029 450 y Fq(,)f(i.e.)2316 417 y Fn(\031)p 2289 431 94 4 v 2289 440 a Fh(p)p 2344 440 40 3 v 50 x Fn(\025)2439 450 y Fl(\024)2559 417 y Fk(1)p 2559 431 35 4 v 2559 479 a Fn(d)2603 450 y Fl(j)p Fq(\012)p Fl(j)2709 420 y Fk(1)p Fn(=d)2815 450 y Fq(.)77 b(Then)41 b(w)n(e)g(ha)n(v)n(e)456 588 y Fm(y)16 b Fl(j)p Fq(\012)p Fl(j)629 532 y Fu(d)p Ff(\000)p Fi(1)p 629 545 104 3 v 666 578 a Fu(d)770 588 y Fl(\024)869 556 y Fk(1)p 868 570 35 4 v 868 617 a Fn(d)912 588 y Fl(j)p Fq(\012)p Fl(j)28 b Fq(for)f(all)1328 556 y Fn(\031)p 1298 570 100 4 v 1298 578 a Fh(p)p 1353 578 46 3 v 49 x Fk(\003)1431 588 y Fl(\024)c Fm(y)j Fl(\024)1710 556 y Fn(\031)p 1683 570 94 4 v 1683 578 a Fh(p)p 1738 578 40 3 v 50 x Fn(\025)1815 588 y Fq(and)h(in)h(view)g(of)f(Lemma)h(7)f(w) n(e)g(get)524 827 y Fm(R)587 839 y Fn(\015)630 827 y Fq(\(\003\))83 b Fl(\024)g Fm(L)1040 793 y Fn(cl)1040 848 y(\015)t(;d)1150 827 y Fl(j)p Fq(\012)p Fl(j)14 b Fq(\003)1328 793 y Fn(\015)t Fk(+)1427 771 y Fu(d)p 1427 780 31 3 v 1428 813 a Fi(2)1492 806 y Fq(^)1472 827 y Fm(B)1553 710 y Fj(\022)1629 771 y Fm(\025)p 1624 808 58 4 v 1624 884 a Fq(\003)1692 827 y Fm(;)g(\033)1776 839 y Fn(d)1833 827 y Fq(+)1926 771 y Fm(d)p 1926 808 44 4 v 1927 884 a Fq(2)1998 827 y(+)k(1)p Fm(;)c(\015)22 b Fl(\000)c Fm(\033)2355 839 y Fn(d)2394 710 y Fj(\023)983 1080 y Fl(\000)1115 1013 y Fq(2)p Fm(\031)1207 983 y Fk(2)p Fn(\033)1278 992 y Fu(d)1313 983 y Fk(+)p Fn(d)p Fk(+2)1486 1013 y Fm(L)1543 983 y Fn(cl)1543 1036 y(\033)1581 1045 y Fu(d)1616 1036 y Fn(;d)p 1071 1061 647 4 v 1071 1137 a Fm(B)g Fq(\()p Fm(\033)1231 1149 y Fn(d)1289 1137 y Fq(+)g(1)p Fm(;)c(\015)23 b Fl(\000)18 b Fm(\033)1647 1149 y Fn(d)1686 1137 y Fq(\))1742 1080 y Fl(j)p Fq(\012)p Fl(j)1858 1020 y Fu(d)p Ff(\000)p Fi(1)p 1858 1033 104 3 v 1894 1066 a Fu(d)1990 967 y Fj(Z)2105 965 y Fu(\031)p 2083 974 81 3 v 2083 982 a Ff(p)p 2128 982 35 3 v 2128 1017 a Fu(\025)2070 1133 y(\031)p 2046 1142 84 3 v 2046 1150 a Ff(p)p 2091 1150 38 3 v 2091 1185 a Fi(\003)2191 963 y Fj(\022)2252 1080 y Fq(\003)g Fl(\000)2421 1024 y Fm(\031)2471 994 y Fk(2)p 2421 1061 88 4 v 2424 1137 a Fm(y)2468 1113 y Fk(2)2519 963 y Fj(\023)2580 977 y Fn(\015)t Fh(\000)p Fn(\033)2709 986 y Fu(d)2744 977 y Fh(\000)p Fk(1)2860 1080 y Fm(y)2904 1046 y Fh(\000)p Fk(2)p Fn(\033)3027 1055 y Fu(d)3062 1046 y Fh(\000)p Fn(d)p Fh(\000)p Fk(2)3252 1080 y Fm(dy)f(:)456 1347 y Fq(If)37 b(w)n(e)f(substitute)i Fm(s)h Fq(=)1288 1315 y Fn(\031)1329 1289 y Fi(2)p 1268 1329 114 4 v 1268 1376 a Fn(y)1304 1359 y Fi(2)1336 1376 y Fk(\003)1428 1347 y Fq(and)e(simplify)g(the)h(expression)d(of)i(the)g(remainder)f(term)h (w)n(e)456 1451 y(arriv)n(e)25 b(at)848 1645 y Fm(R)911 1657 y Fn(\015)954 1645 y Fq(\(\003\))37 b Fl(\024)f Fm(L)1271 1611 y Fn(cl)1271 1665 y(\015)t(;d)1382 1645 y Fl(j)p Fq(\012)p Fl(j)14 b Fq(\003)1560 1611 y Fn(\015)t Fk(+)1659 1588 y Fu(d)p 1659 1597 31 3 v 1660 1631 a Fi(2)1723 1624 y Fq(^)1703 1645 y Fm(B)1784 1528 y Fj(\022)1860 1589 y Fm(\025)p 1856 1626 58 4 v 1856 1702 a Fq(\003)1923 1645 y Fm(;)g(\033)2007 1657 y Fn(d)2065 1645 y Fq(+)2158 1589 y Fm(d)p 2158 1626 44 4 v 2159 1702 a Fq(2)2229 1645 y(+)k(1)p Fm(;)c(\015)23 b Fl(\000)18 b Fm(\033)2587 1657 y Fn(d)2626 1528 y Fj(\023)2705 1645 y Fl(\000)g Fm(S)5 b Fq(\(\003)p Fm(;)14 b(\025)p Fq(\))456 1859 y(with)28 b Fm(S)5 b Fq(\(\003)p Fm(;)14 b(\025)p Fq(\))28 b(as)f(stated)g(in)h(\(14\).)555 1959 y(Next)i(w)n(e)f(assume)f Fm(\025)e(<)f(\034)1372 1971 y Fk(\012)1454 1959 y Fq(and)k(pro)r(ceed) f(in)h(t)n(w)n(o)g(steps.)41 b(If)30 b(at)f(the)h(same)e(time)i(\003)25 b Fm(<)g(\034)3369 1971 y Fk(\012)3421 1959 y Fq(,)456 2068 y(that)37 b(means)948 2035 y Fn(\031)p 919 2049 100 4 v 919 2058 a Fh(p)p 973 2058 46 3 v 973 2107 a Fk(\003)1068 2068 y Fm(>)1182 2035 y Fk(1)p 1182 2049 35 4 v 1182 2097 a Fn(d)1226 2068 y Fl(j)p Fq(\012)p Fl(j)1332 2038 y Fk(1)p Fn(=d)1438 2068 y Fq(,)j(w)n(e)d(ha)n(v)n(e)f Fm(y)16 b Fl(j)p Fq(\012)p Fl(j)2007 2012 y Fu(d)p Ff(\000)p Fi(1)p 2007 2025 104 3 v 2044 2058 a Fu(d)2165 2068 y Fm(>)2279 2035 y Fk(1)p 2278 2049 35 4 v 2278 2097 a Fn(d)2323 2068 y Fl(j)p Fq(\012)p Fl(j)37 b Fq(for)g(all)2768 2035 y Fn(\031)p 2738 2049 100 4 v 2738 2058 a Fh(p)p 2793 2058 46 3 v 49 x Fk(\003)2887 2068 y Fm(<)i(y)3071 2035 y Fn(\031)p 3044 2049 94 4 v 3044 2058 a Fh(p)p 3099 2058 40 3 v 50 x Fn(\025)3186 2068 y Fq(and)e(b)n(y)456 2179 y(similar)26 b(calculations)h(as)g(ab)r(o)n(v)n(e)f(w)n(e)h(arriv) n(e)f(at)597 2373 y Fm(R)660 2385 y Fn(\015)703 2373 y Fq(\(\003\))83 b Fl(\024)f Fm(L)1112 2339 y Fn(cl)1112 2394 y(\015)t(;d)1223 2373 y Fl(j)p Fq(\012)p Fl(j)14 b Fq(\003)1401 2339 y Fn(\015)t Fk(+)1500 2317 y Fu(d)p 1500 2326 31 3 v 1501 2359 a Fi(2)1564 2352 y Fq(^)1544 2373 y Fm(B)1625 2256 y Fj(\022)1701 2317 y Fm(\025)p 1696 2354 58 4 v 1696 2430 a Fq(\003)1764 2373 y Fm(;)g(\033)1848 2385 y Fn(d)1906 2373 y Fq(+)1998 2317 y Fm(d)p 1998 2354 44 4 v 1999 2430 a Fq(2)2070 2373 y(+)k(1)p Fm(;)c(\015)23 b Fl(\000)18 b Fm(\033)2428 2385 y Fn(d)2467 2256 y Fj(\023)1055 2626 y Fl(\000)1187 2559 y Fq(2)p Fm(\031)1279 2529 y Fk(2)p Fn(\033)1350 2538 y Fu(d)1385 2529 y Fk(+)p Fn(d)p Fk(+2)1559 2559 y Fm(L)1616 2529 y Fn(cl)1616 2582 y(\033)1654 2591 y Fu(d)1689 2582 y Fn(;d)p 1144 2607 647 4 v 1144 2683 a Fm(B)g Fq(\()p Fm(\033)1304 2695 y Fn(d)1362 2683 y Fq(+)g(1)p Fm(;)c(\015)22 b Fl(\000)c Fm(\033)1719 2695 y Fn(d)1758 2683 y Fq(\))1824 2570 y Fl(j)p Fq(\012)p Fl(j)p 1824 2607 107 4 v 1856 2683 a Fm(d)1954 2513 y Fj(Z)2069 2511 y Fu(\031)p 2047 2520 81 3 v 2047 2528 a Ff(p)p 2093 2528 35 3 v 35 x Fu(\025)2034 2679 y(\031)p 2010 2688 84 3 v 2010 2696 a Ff(p)p 2056 2696 38 3 v 35 x Fi(\003)2156 2509 y Fj(\022)2217 2626 y Fq(\003)g Fl(\000)2386 2570 y Fm(\031)2436 2540 y Fk(2)p 2386 2607 88 4 v 2389 2683 a Fm(y)2433 2659 y Fk(2)2483 2509 y Fj(\023)2544 2523 y Fn(\015)t Fh(\000)p Fn(\033)2673 2532 y Fu(d)2708 2523 y Fh(\000)p Fk(1)2825 2626 y Fm(y)2869 2592 y Fh(\000)p Fk(2)p Fn(\033)2992 2601 y Fu(d)3027 2592 y Fh(\000)p Fn(d)p Fh(\000)p Fk(3)3216 2626 y Fm(dy)456 2871 y Fq(and)37 b(w)n(e)f(obtain)h(the)h(claimed)e(inequalit)n(y)h (with)h Fm(S)5 b Fq(\(\003)p Fm(;)14 b(\025)p Fq(\))37 b(giv)n(en)f(in)i(\(15\))o(.)66 b(On)37 b(the)g(other)456 2971 y(hand,)27 b(if)h Fm(\025)c(<)e(\034)957 2983 y Fk(\012)1032 2971 y Fl(\024)h Fq(\003)k(w)n(e)h(get)517 3181 y Fm(R)580 3193 y Fn(\015)622 3181 y Fq(\(\003\))84 b Fl(\024)e Fm(L)1032 3146 y Fn(cl)1032 3201 y(\015)t(;d)1142 3181 y Fl(j)p Fq(\012)p Fl(j)14 b Fq(\003)1320 3146 y Fn(\015)t Fk(+)1419 3124 y Fu(d)p 1419 3133 31 3 v 1420 3166 a Fi(2)1484 3160 y Fq(^)1464 3181 y Fm(B)1545 3063 y Fj(\022)1621 3124 y Fm(\025)p 1616 3161 58 4 v 1616 3237 a Fq(\003)1684 3181 y Fm(;)g(\033)1768 3193 y Fn(d)1825 3181 y Fq(+)1918 3124 y Fm(d)p 1918 3161 44 4 v 1919 3237 a Fq(2)1990 3181 y(+)k(1)p Fm(;)c(\015)22 b Fl(\000)c Fm(\033)2347 3193 y Fn(d)2387 3063 y Fj(\023)975 3484 y Fl(\000)1107 3416 y Fq(2)p Fm(\031)1199 3386 y Fk(2)p Fn(\033)1270 3395 y Fu(d)1305 3386 y Fk(+)p Fn(d)p Fk(+2)1479 3416 y Fm(L)1536 3386 y Fn(cl)1536 3440 y(\033)1574 3449 y Fu(d)1609 3440 y Fn(;d)p 1064 3465 647 4 v 1064 3541 a Fm(B)g Fq(\()p Fm(\033)1224 3553 y Fn(d)1281 3541 y Fq(+)g(1)p Fm(;)c(\015)23 b Fl(\000)18 b Fm(\033)1639 3553 y Fn(d)1678 3541 y Fq(\))1734 3484 y Fl(j)p Fq(\012)p Fl(j)1850 3424 y Fu(d)p Ff(\000)p Fi(1)p 1850 3437 104 3 v 1887 3469 a Fu(d)1982 3371 y Fj(Z)2075 3362 y Ff(j)p Fi(\012)p Ff(j)2164 3319 y Fi(1)p 2163 3326 31 3 v 2163 3358 a Fu(d)p 2075 3378 133 3 v 2126 3411 a(d)2062 3537 y(\031)p 2038 3546 84 3 v 2038 3554 a Ff(p)p 2084 3554 38 3 v 35 x Fi(\003)2236 3367 y Fj(\022)2297 3484 y Fq(\003)g Fl(\000)2466 3427 y Fm(\031)2516 3397 y Fk(2)p 2466 3465 88 4 v 2469 3541 a Fm(y)2513 3517 y Fk(2)2563 3367 y Fj(\023)2624 3380 y Fn(\015)t Fh(\000)p Fn(\033)2753 3389 y Fu(d)2788 3380 y Fh(\000)p Fk(1)2905 3484 y Fm(y)2949 3449 y Fh(\000)p Fk(2)p Fn(\033)3072 3458 y Fu(d)3107 3449 y Fh(\000)p Fn(d)p Fh(\000)p Fk(2)3296 3484 y Fm(dy)975 3767 y Fl(\000)1107 3700 y Fq(2)p Fm(\031)1199 3669 y Fk(2)p Fn(\033)1270 3678 y Fu(d)1305 3669 y Fk(+)p Fn(d)p Fk(+2)1479 3700 y Fm(L)1536 3669 y Fn(cl)1536 3723 y(\033)1574 3732 y Fu(d)1609 3723 y Fn(;d)p 1064 3748 647 4 v 1064 3824 a Fm(B)g Fq(\()p Fm(\033)1224 3836 y Fn(d)1281 3824 y Fq(+)g(1)p Fm(;)c(\015)23 b Fl(\000)18 b Fm(\033)1639 3836 y Fn(d)1678 3824 y Fq(\))1744 3711 y Fl(j)p Fq(\012)p Fl(j)p 1744 3748 107 4 v 1776 3824 a Fm(d)1874 3654 y Fj(Z)1989 3652 y Fu(\031)p 1967 3661 81 3 v 1967 3669 a Ff(p)p 2013 3669 35 3 v 35 x Fu(\025)1930 3813 y Ff(j)p Fi(\012)p Ff(j)2019 3771 y Fi(1)p 2018 3778 31 3 v 2018 3809 a Fu(d)p 1930 3829 133 3 v 1981 3862 a(d)2091 3650 y Fj(\022)2152 3767 y Fq(\003)g Fl(\000)2321 3711 y Fm(\031)2371 3680 y Fk(2)p 2321 3748 88 4 v 2324 3824 a Fm(y)2368 3800 y Fk(2)2418 3650 y Fj(\023)2480 3663 y Fn(\015)t Fh(\000)p Fn(\033)2609 3672 y Fu(d)2643 3663 y Fh(\000)p Fk(1)2760 3767 y Fm(y)2804 3732 y Fh(\000)p Fk(2)p Fn(\033)2927 3741 y Fu(d)2962 3732 y Fh(\000)p Fn(d)p Fh(\000)p Fk(3)3151 3767 y Fm(dy)s(:)456 3998 y Fq(In)28 b(this)h(case)f(after)g(a)g (simpli\014cation)h(w)n(e)f(arriv)n(e)e(at)j Fm(S)5 b Fq(\(\003)p Fm(;)14 b(\025)p Fq(\))29 b(as)e(stated)i(in)g(\(16\))o(.) 40 b(Finally)-7 b(,)29 b(if)456 4098 y(w)n(e)e(apply)h(\(49\))g (directly)f(to)g(\(3\))h(w)n(e)f(claim)984 4312 y Fm(R)1047 4324 y Fn(\015)1090 4312 y Fq(\(\003\))51 b Fl(\024)f Fm(L)1435 4278 y Fn(cl)1435 4333 y(\015)t(;d)1546 4312 y Fl(j)p Fq(\012)p Fl(j)14 b Fq(\003)1724 4278 y Fn(\015)t Fk(+)1823 4256 y Fu(d)p 1823 4265 31 3 v 1824 4298 a Fi(2)1901 4291 y Fq(^)1881 4312 y Fm(B)1962 4195 y Fj(\022)2038 4256 y Fm(\025)p 2033 4293 58 4 v 2033 4369 a Fq(\003)2101 4312 y Fm(;)g(\033)2185 4324 y Fn(d)2242 4312 y Fq(+)2335 4256 y Fm(d)p 2335 4293 44 4 v 2336 4369 a Fq(2)2407 4312 y(+)k(1)p Fm(;)c(\015)22 b Fl(\000)d Fm(\033)2765 4324 y Fn(d)2804 4195 y Fj(\023)2892 4312 y Fm(:)456 4527 y Fq(Hence,)h(in)f(the)g(\014nal)f(b)r(ound)h Fm(S)5 b Fq(\(\003)p Fm(;)14 b(\025)p Fq(\))19 b(can)f(b)r(e)g(replaced)g(b)n (y)g(its)h(p)r(ositiv)n(e)f(part)g(\()p Fm(S)5 b Fq(\(\003)p Fm(;)14 b(\025)p Fq(\)\))3218 4539 y Fk(+)3274 4527 y Fq(.)83 b Fc(\003)555 4720 y Fq(Again)27 b(in)h(view)g(of)f(\(11\))g(w) n(e)h(can)f(c)n(ho)r(ose)1529 4946 y(~)1525 4968 y Fm(\025)d Fq(=)1891 4882 y Fm(\031)s(j)1980 4852 y Fk(2)1985 4894 y Fu(d)p 1985 4903 31 3 v 1986 4936 a Fi(2)2026 4916 y Fh(\000)p Fk(1)p Fn(;)p Fk(1)p 1694 4949 671 4 v 1694 5055 a Fq(\000)1760 4988 y Fj(\000)1808 5023 y Fn(d)p 1808 5037 35 4 v 1809 5084 a Fk(2)1871 5055 y Fq(+)18 b(1)1996 4988 y Fj(\001)2033 5003 y Fk(2)p Fn(=d)2153 5055 y Fl(j)p Fq(\012)p Fl(j)2259 5031 y Fk(2)p Fn(=d)456 5216 y Fq(as)27 b(a)g(lo)n(w)n(er)f(b)r(ound)i(on)f Fm(\025)1263 5228 y Fk(1)1301 5216 y Fq(.)37 b(Th)n(us)27 b(w)n(e)g(\014nd)p eop end %%Page: 24 24 TeXDict begin 24 23 bop 456 251 a Fk(24)781 b(LEANDER)22 b(GEISINGER)44 b(&)h(TIMO)23 b(WEIDL)456 450 y FC(Corollary)30 b(15.)38 b Fd(L)l(et)27 b Fq(\012)c Fl(\032)g Fo(R)1412 420 y Fn(d)1478 450 y Fd(b)l(e)28 b(an)f(op)l(en)h(set)f(with)h (\014nite)f(volume.)39 b(Then)28 b(for)g Fm(\015)g(>)23 b(\033)3270 462 y Fn(d)3336 450 y Fd(the)456 550 y(estimate)806 785 y Fm(R)869 797 y Fn(\015)911 785 y Fq(\(\003\))37 b Fl(\024)g Fm(L)1229 751 y Fn(cl)1229 805 y(\015)t(;d)1339 785 y Fl(j)p Fq(\012)p Fl(j)1479 764 y Fq(^)1459 785 y Fm(B)1540 643 y Fj( )1624 707 y Fq(~)1620 729 y Fm(\025)p 1616 766 58 4 v 1616 842 a Fq(\003)1683 785 y Fm(;)14 b(\033)1767 797 y Fn(d)1825 785 y Fq(+)1918 729 y Fm(d)p 1918 766 44 4 v 1919 842 a Fq(2)1989 785 y(+)k(1)p Fm(;)c(\015)23 b Fl(\000)18 b Fm(\033)2347 797 y Fn(d)2386 643 y Fj(!)2466 785 y Fq(\003)2524 751 y Fn(\015)t Fk(+)2623 728 y Fu(d)p 2623 737 31 3 v 2624 771 a Fi(2)2700 785 y Fl(\000)32 b Fq(\()p Fm(S)5 b Fq(\(\003\)\))3039 797 y Fk(+)456 1056 y Fd(holds)31 b(for)f(al)t(l)h Fq(\003)23 b Fl(\025)1091 1034 y Fq(~)1088 1056 y Fm(\025)p Fd(,)30 b(wher)l(e)1012 1307 y Fm(S)5 b Fq(\(\003\))37 b(=)g Fm(L)1386 1273 y Fn(cl)1386 1328 y(\015)t(;d)1496 1307 y Fl(j)p 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Fm(;)g(\033)1933 1750 y Fn(d)1990 1738 y Fq(+)2083 1705 y Fn(d)p Fk(+1)p 2083 1719 119 4 v 2126 1767 a(2)2212 1671 y Fj(\001)p 1677 1780 574 4 v 1943 1856 a Fq(2)2294 1778 y(^)2274 1799 y Fm(B)2355 1682 y Fj(\022)2426 1743 y Fm(\034)2462 1755 y Fk(\012)p 2426 1780 88 4 v 2441 1856 a Fq(\003)2524 1799 y Fm(;)g(\033)2608 1811 y Fn(d)2665 1799 y Fq(+)2758 1743 y Fm(d)19 b Fq(+)f(1)p 2758 1780 187 4 v 2831 1856 a(2)2954 1799 y Fm(;)c(\015)23 b Fl(\000)18 b Fm(\033)3187 1811 y Fn(d)3227 1682 y Fj(\023)938 2057 y Fq(+)g Fm(L)1078 2023 y Fn(cl)1078 2077 y(\015)t(;d)1188 2057 y Fl(j)p Fq(\012)p Fl(j)c Fq(\003)1366 2023 y Fn(\015)t Fk(+)1465 2000 y Fu(d)p 1465 2009 31 3 v 1466 2043 a Fi(2)1535 2001 y Fq(1)p 1534 2038 44 4 v 1534 2114 a Fm(d)1620 2036 y Fq(~)1601 2057 y Fm(B)1682 1915 y Fj( )1765 1979 y Fq(~)1762 2001 y Fm(\025)p 1757 2038 58 4 v 1757 2114 a Fq(\003)1825 2057 y Fm(;)1872 2001 y(\034)1908 2013 y Fk(\012)p 1872 2038 88 4 v 1887 2114 a Fq(\003)1970 2057 y Fm(;)g(\033)2054 2069 y Fn(d)2111 2057 y Fq(+)2204 2001 y Fm(d)p 2204 2038 44 4 v 2205 2114 a Fq(2)2276 2057 y(+)k(1)p Fm(;)c(\015)22 b Fl(\000)c Fm(\033)2633 2069 y Fn(d)2672 1915 y Fj(!)456 2308 y Fd(if)30 b Fq(\003)23 b Fl(\025)g Fm(\034)741 2320 y Fk(\012)792 2308 y Fd(.)456 2484 y(R)l(emark.)44 b Fq(W)-7 b(e)32 b(can)f(no)n(w)g(compare)f(this)i (result)f(with)h(estimate)g(\(18\))f(from)g(Prop)r(osition)f(3.)456 2584 y(In)k(b)r(oth)g(b)r(ounds)g(the)h(high)f(energy)f(asymptotics)g (\003)g Fl(!)h(1)g Fq(is)g(dominated)g(b)n(y)g(the)g(sharp)456 2684 y(\014rst)e(term.)50 b(In)33 b(view)f(of)39 b(\(10\))31 b(also)h(the)g(remainder)f(terms)h(sho)n(w)g(the)g(correct)f(order)g (as)h(\003)456 2783 y(tends)g(to)f(in\014nit)n(y)-7 b(.)51 b(In)32 b(this)g(limit)h(the)f(b)r(ound)g(from)g(Corollary)d(15)i(is)h (stronger)e(than)i(\(18\))456 2883 y(whenev)n(er)1005 3058 y Fm(")1058 2941 y Fj(\022)1119 3058 y Fm(\015)23 b Fq(+)1278 3002 y Fm(d)18 b Fl(\000)g Fq(1)p 1278 3039 187 4 v 1350 3115 a(2)1474 2941 y Fj(\023)1549 3058 y Fm(d)1592 3070 y Fk(\003)1641 3058 y Fq(\(\012\))24 b Fm(<)1887 3002 y Fq(1)p 1887 3039 42 4 v 1887 3115 a(2)1952 3058 y Fm(B)2033 2941 y Fj(\022)2104 3002 y Fq(1)p 2104 3039 V 2104 3115 a(2)2155 3058 y Fm(;)14 b(\033)2239 3070 y Fn(d)2297 3058 y Fq(+)2390 3002 y Fm(d)19 b Fq(+)f(1)p 2390 3039 187 4 v 2462 3115 a(2)2586 2941 y Fj(\023)2661 3058 y Fl(j)p Fq(\012)p Fl(j)2777 2998 y Fu(d)p Ff(\000)p Fi(1)p 2777 3011 104 3 v 2814 3044 a Fu(d)456 3283 y Fq(holds)g(true.)34 b(W)-7 b(e)19 b(remark)e(that)i(the)g(righ)n(t)f (hand)h(side)f(is)h(indep)r(enden)n(t)g(of)g Fm(\015)k Fq(while)c Fm(")3080 3215 y Fj(\000)3118 3283 y Fm(\015)k Fq(+)3277 3250 y Fn(d)p Fh(\000)p Fk(1)p 3277 3264 120 4 v 3320 3311 a(2)3406 3215 y Fj(\001)456 3382 y Fq(tends)c(to)g(zero)f (as)h Fm(\015)24 b Fq(tends)19 b(to)g(in\014nit)n(y)h(and)f Fm(d)1870 3394 y Fk(\003)1920 3382 y Fq(\(\012\))h(is)f(b)r(ounded)h (from)e(ab)r(o)n(v)n(e)g(b)n(y)h(the)h(diameter)456 3482 y(of)27 b(\012.)37 b(Hence)28 b(the)g(condition)f(ab)r(o)n(v)n(e)f (will)i(b)r(e)g(satis\014ed)f(for)g(large)g(enough)g Fm(\015)5 b Fq(.)555 3582 y(Moreo)n(v)n(er,)25 b(the)j(b)r(ound)g(from) f(Corollary)e(15)i(con)n(tains)g(the)h(factor)1466 3812 y(^)1447 3833 y Fm(B)1528 3691 y Fj( )1611 3755 y Fq(~)1608 3777 y Fm(\025)p 1603 3814 58 4 v 1603 3890 a Fq(\003)1671 3833 y Fm(;)g(\033)1769 3845 y Fn(d)1826 3833 y Fq(+)1919 3777 y Fm(d)p 1919 3814 44 4 v 1920 3890 a Fq(2)1991 3833 y(+)18 b(1)p Fm(;)c(\015)22 b Fl(\000)c Fm(\033)2348 3845 y Fn(d)2388 3691 y Fj(!)456 4104 y Fq(whic)n(h)h(deca)n(ys)e(exp)r (onen)n(tially)i(if)g(\003)k Fl(!)1701 4082 y Fq(~)1697 4104 y Fm(\025)q Fq(+)18 b(and)h(whic)n(h)g(impro)n(v)n(es)f(the)h(b)r (ound)h(from)e(Theorem)456 4206 y(2)27 b(in)h(comparison)e(to)h(\(18\)) g(for)g(v)-5 b(alues)28 b(of)f(\003)g(close)g(to)2195 4184 y(~)2192 4206 y Fm(\025)p Fq(.)1291 4509 y Fp(Appendix)k Fq(A.)42 b Fp(Pr)n(oof)31 b(of)h(Lemma)f(4)555 4658 y Fq(Here)c(w)n(e)h(pro)n(v)n(e)e(Lemma)h(4:)36 b(F)-7 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a(2)1857 2048 y(\()p Fm(f)9 b Fq(\()p Fm(k)21 b Fl(\000)d Fq(1\))h(+)f Fm(f)9 b Fq(\()p Fm(k)s Fq(\)\))2486 1906 y Fj(!)2570 2048 y Fq(+)2663 1992 y(1)p 2663 2029 V 2663 2105 a(2)2715 2048 y Fm(f)g Fq(\(0\))18 b Fl(\000)2982 1992 y Fq(1)p 2982 2029 V 2982 2105 a(2)3033 2048 y Fm(f)k Fq(\()q([)p Fm(A)p Fq(]\))14 b Fm(:)456 2297 y Fq(The)27 b(monotonicit)n(y)g(of)h Fm(f)36 b Fq(implies)28 b Fm(f)22 b Fq(\([)p Fm(A)p Fq(]\))i Fl(\024)f Fm(f)9 b Fq(\()p Fm(A)18 b Fl(\000)g Fq(1\))23 b(=)2360 2230 y Fj(\000)2408 2264 y Fk(2)p Fn(A)p Fh(\000)p Fk(1)p 2408 2278 169 4 v 2451 2325 a Fn(A)2501 2309 y Fi(2)2586 2230 y Fj(\001)2624 2245 y Fn(\026)2696 2297 y Fq(and)k(th)n(us)h(w)n(e)f(get)456 2553 y(\(51\))128 b Fm(F)785 2565 y Fn(\026)830 2553 y Fq(\()p Fm(A)p Fq(\))24 b Fl(\025)1077 2497 y Fq(1)p 1077 2534 42 4 v 1077 2610 a(2)1147 2553 y Fl(\000)1240 2497 y Fq(1)p 1240 2534 V 1240 2610 a(2)1306 2436 y Fj(\022)1377 2497 y Fq(2)p Fm(A)18 b Fl(\000)g Fq(1)p 1377 2534 247 4 v 1450 2610 a Fm(A)1512 2586 y Fk(2)1633 2436 y Fj(\023)1695 2453 y Fn(\026)1758 2553 y Fq(+)1857 2442 y Fk([)p Fn(A)p Fk(])1841 2474 y Fj(X)1841 2653 y Fn(k)q Fk(=1)1975 2411 y Fj( )2041 2440 y(Z)2124 2460 y Fn(k)2087 2628 y(k)q Fh(\000)p Fk(1)2227 2553 y Fm(f)9 b Fq(\()p Fm(t)p Fq(\))p Fm(dt)18 b Fl(\000)2555 2497 y Fq(1)p 2555 2534 42 4 v 2555 2610 a(2)2621 2553 y(\()p Fm(f)9 b Fq(\()p Fm(k)21 b Fl(\000)d Fq(1\))g(+)h Fm(f)9 b Fq(\()p Fm(k)s Fq(\)\))3250 2411 y Fj(!)456 2783 y Fq(for)28 b(all)i(0)25 b Fl(\024)h Fm(t)g Fl(\024)f Fm(A)p Fq(.)43 b(Lets)29 b(\014x)h Fm(\026)c Fl(\025)f Fq(3.)42 b(Then)30 b(the)f(function)h Fm(f)38 b Fq(can)29 b(b)r(e)h(di\013eren)n(tiated)g(t)n(wice)456 2892 y(and)c(w)n(e)h(claim)f(that)i(for)e(0)d Fl(\024)f Fm(t)h Fl(\024)g Fm(A)k Fq(the)g(function)h Fm(f)2163 2861 y Fh(00)2205 2892 y Fq(\()p Fm(t)p Fq(\))23 b(=)2433 2859 y Fn(d)2468 2834 y Fi(2)p 2420 2873 93 4 v 2420 2920 a Fn(dt)2480 2904 y Fi(2)2522 2892 y Fm(f)9 b Fq(\()p Fm(t)p Fq(\))27 b(attains)g(its)g(maxim)n(um)456 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y(\014nd)f(that)g(for)f(all)g Fm(k)f Fl(\025)c Fq(1)456 3796 y(\(52\))1069 3683 y Fj(Z)1152 3703 y Fn(k)1115 3871 y(k)q Fh(\000)p Fk(1)1255 3796 y Fm(f)9 b Fq(\()p Fm(t)p Fq(\))p Fm(dt)18 b Fl(\000)1583 3740 y Fq(1)p 1583 3777 42 4 v 1583 3853 a(2)1649 3796 y(\()p Fm(f)9 b Fq(\()p Fm(k)21 b Fl(\000)d Fq(1\))g(+)g Fm(f)9 b Fq(\()p Fm(k)s Fq(\)\))38 b(=)e Fl(\000)2511 3740 y Fq(1)p 2491 3777 84 4 v 2491 3853 a(12)2584 3796 y Fm(f)2634 3761 y Fh(00)2689 3796 y Fq(\()q Fm(\030)2758 3808 y Fn(k)2799 3796 y Fq(\))456 4005 y(with)28 b Fm(\030)681 4017 y Fn(k)745 4005 y Fl(2)23 b Fq([)p Fm(k)f Fl(\000)c Fq(1)p Fm(;)c(k)s Fq(].)36 b(Inserting)27 b(this)h(estimate)f(in)n(to)h (\(51\))f(yields)563 4252 y Fm(F)616 4264 y Fn(\026)661 4252 y Fq(\()p Fm(A)p Fq(\))d Fl(\025)909 4196 y Fq(1)p 909 4233 42 4 v 909 4309 a(2)979 4252 y Fl(\000)1072 4196 y Fq(1)p 1072 4233 V 1072 4309 a(2)1137 4135 y Fj(\022)1208 4196 y Fq(2)p Fm(A)18 b Fl(\000)g Fq(1)p 1208 4233 247 4 v 1282 4309 a Fm(A)1344 4285 y Fk(2)1465 4135 y Fj(\023)1526 4152 y Fn(\026)1589 4252 y Fl(\000)1703 4196 y Fq(1)p 1682 4233 84 4 v 1682 4309 a(12)1805 4141 y Fk([)p Fn(A)p Fk(])1789 4173 y Fj(X)1789 4352 y Fn(k)q Fk(=1)1923 4252 y Fm(f)1973 4218 y Fh(00)2015 4252 y Fq(\()p Fm(\030)2083 4264 y Fn(k)2125 4252 y Fq(\))23 b Fl(\025)2278 4196 y Fq(1)p 2278 4233 42 4 v 2278 4309 a(2)2348 4252 y Fl(\000)2441 4196 y Fq(1)p 2441 4233 V 2441 4309 a(2)2506 4135 y Fj(\022)2577 4196 y Fq(2)p Fm(A)18 b Fl(\000)g Fq(1)p 2577 4233 247 4 v 2651 4309 a Fm(A)2713 4285 y Fk(2)2834 4135 y Fj(\023)2895 4152 y Fn(\026)2958 4252 y Fl(\000)3051 4196 y Fq([)p Fm(A)p Fq(])p 3051 4233 109 4 v 3064 4309 a(12)3169 4252 y Fm(f)3219 4218 y Fh(00)3210 4273 y Fk(max)811 4531 y Fl(\025)909 4475 y Fq(1)p 909 4512 42 4 v 909 4588 a(2)979 4531 y Fl(\000)1072 4475 y Fq(1)p 1072 4512 V 1072 4588 a(2)1137 4414 y Fj(\022)1208 4475 y Fq(2)p Fm(A)g Fl(\000)g Fq(1)p 1208 4512 247 4 v 1282 4588 a Fm(A)1344 4564 y Fk(2)1465 4414 y Fj(\023)1526 4431 y Fn(\026)1589 4531 y Fl(\000)1709 4475 y Fm(\026)p 1682 4512 104 4 v 1682 4588 a Fq(3)p Fm(A)1810 4414 y Fj(\022)1881 4475 y Fq(2)p Fm(\026)g Fl(\000)g Fq(4)p 1881 4512 235 4 v 1881 4588 a(2)p Fm(\026)g Fl(\000)g Fq(1)2125 4414 y Fj(\023)2186 4431 y Fn(\026)p Fh(\000)p Fk(2)2330 4531 y Fm(:)456 4748 y Fq(In)31 b(a)g(\014rst)g(step)h(w)n(e)f(assume)f Fm(A)g Fl(\025)f Fm(\026)p Fq(.)48 b(Note)32 b(that)f(for)g Fm(\026)f Fl(\025)f Fq(3)i(the)g(factor)2878 4711 y Fk(2)p Fn(\026)p Fh(\000)p Fk(4)p 2878 4729 159 4 v 2878 4777 a(2)p Fn(\026)p Fh(\000)p Fk(1)3078 4748 y Fq(is)g(p)r(ositiv)n(e)456 4869 y(and)23 b(for)g Fm(A)g Fl(\025)g Fm(\026)g Fq(the)h(functions) 1485 4836 y Fk(2)p Fn(A)p Fh(\000)p Fk(1)p 1485 4850 169 4 v 1528 4898 a Fn(A)1578 4881 y Fi(2)1686 4869 y Fq(and)1875 4832 y Fn(\026)p 1854 4850 84 4 v 1854 4898 a Fk(3)p Fn(A)1970 4869 y Fq(are)f(non-increasing)e(and)i(w)n(e)g(can)g (put)h Fm(A)g Fq(=)e Fm(\026)456 4969 y Fq(in)27 b(the)h(b)r(ound)g(ab) r(o)n(v)n(e.)36 b(W)-7 b(e)28 b(arriv)n(e)e(at)1097 5170 y Fm(F)1150 5182 y Fn(\026)1195 5170 y Fq(\()p Fm(A)p Fq(\))38 b Fl(\025)1470 5114 y Fq(1)p 1470 5151 42 4 v 1470 5227 a(2)1540 5170 y Fl(\000)1633 5114 y Fq(1)p 1633 5151 V 1633 5227 a(2)1699 5053 y Fj(\022)1770 5114 y Fq(2)p Fm(\026)18 b Fl(\000)g Fq(1)p 1770 5151 235 4 v 1843 5227 a Fm(\026)1893 5203 y Fk(2)2014 5053 y Fj(\023)2075 5070 y Fn(\026)2138 5170 y Fl(\000)2231 5114 y Fq(1)p 2231 5151 42 4 v 2231 5227 a(3)2297 5053 y Fj(\022)2368 5114 y Fq(2)p Fm(\026)g Fl(\000)g Fq(4)p 2368 5151 235 4 v 2368 5227 a(2)p Fm(\026)g Fl(\000)g Fq(1)2612 5053 y Fj(\023)2673 5070 y Fn(\026)p Fh(\000)p Fk(2)p eop end %%Page: 26 26 TeXDict begin 26 25 bop 456 251 a Fk(26)781 b(LEANDER)22 b(GEISINGER)44 b(&)h(TIMO)23 b(WEIDL)456 450 y Fq(for)k(all)g Fm(A)c Fl(\025)g Fm(\026)28 b Fq(and)f(put)1099 677 y(\010)1159 689 y Fk(1)1196 677 y Fq(\()p Fm(\026)p Fq(\))d(=)1432 621 y(1)p 1432 658 42 4 v 1432 734 a(2)1502 677 y Fl(\000)1595 621 y Fq(1)p 1595 658 V 1595 734 a(2)1660 560 y Fj(\022)1731 621 y Fq(2)p Fm(\026)18 b Fl(\000)g Fq(1)p 1731 658 235 4 v 1805 734 a Fm(\026)1855 710 y Fk(2)1975 560 y Fj(\023)2037 577 y Fn(\026)2099 677 y Fl(\000)2192 621 y Fq(1)p 2192 658 42 4 v 2192 734 a(3)2258 560 y Fj(\022)2329 621 y Fq(2)p Fm(\026)g Fl(\000)g Fq(4)p 2329 658 235 4 v 2329 734 a(2)p Fm(\026)g Fl(\000)g Fq(1)2573 560 y Fj(\023)2634 577 y Fn(\026)p Fh(\000)p Fk(2)2778 677 y Fm(:)456 928 y Fq(In)33 b(a)g(second)g(step)h(w)n(e)f(assume)1517 867 y Fl(p)p 1586 867 V 61 x Fq(2)p Fm(\026)18 b Fl(\000)g Fq(1)33 b Fl(\024)f Fm(A)i(<)e(\026)p Fq(.)55 b(Put)34 b Fm(t)2471 897 y Fh(\003)2542 928 y Fq(=)2639 836 y Fj(q)p 2722 836 179 4 v 2787 895 a Fn(A)p 2732 909 159 4 v 2732 956 a Fk(2)p Fn(\026)p Fh(\000)p Fk(1)2934 928 y Fq(and)g(note)f(that)456 1053 y Fm(f)506 1023 y Fh(00)548 1053 y Fq(\()p Fm(t)p Fq(\))27 b Fl(\024)g Fq(0)j(holds)g(for)f(all)h (0)d Fl(\024)g Fm(t)g Fl(\024)g Fm(t)1640 1023 y Fh(\003)1708 1053 y Fq(and)j Fm(f)1922 1023 y Fh(00)1964 1053 y Fq(\()p Fm(t)p Fq(\))e Fl(\025)f Fq(0)j(holds)g(for)f(all)h Fm(t)2747 1023 y Fh(\003)2813 1053 y Fl(\024)c Fm(t)i Fl(\024)e Fm(A)p Fq(.)45 b(Cho)r(ose)456 1153 y Fm(k)502 1122 y Fh(\003)563 1153 y Fq(=)22 b([)p Fm(t)703 1122 y Fh(\003)742 1153 y Fq(])27 b(so)g(that)h Fm(f)1124 1122 y Fh(00)1166 1153 y Fq(\()p Fm(t)p Fq(\))c Fl(\024)e Fq(0)28 b(for)f(all)g(0)22 b Fl(\024)h Fm(t)g Fl(\024)g Fm(k)2022 1122 y Fh(\003)2060 1153 y Fq(.)37 b(F)-7 b(rom)27 b(\(52\))g(w)n(e)g(deduce)1269 1267 y Fj(Z)1352 1287 y Fn(k)1316 1455 y(k)q Fh(\000)p Fk(1)1455 1380 y Fm(f)9 b Fq(\()p Fm(t)p Fq(\))p Fm(dt)19 b Fl(\000)1784 1324 y Fq(1)p 1784 1361 42 4 v 1784 1437 a(2)1849 1380 y(\()q Fm(f)9 b Fq(\()p Fm(k)21 b Fl(\000)d Fq(1\))g(+)g Fm(f)9 b Fq(\()p Fm(k)s Fq(\)\))23 b Fl(\025)g Fq(0)456 1598 y(for)k(all)g(1)c Fl(\024)f Fm(k)k Fl(\024)d Fm(k)1053 1568 y Fh(\003)1091 1598 y Fq(.)37 b(Using)27 b(\(52\))g(once)g(more)g(for)g Fm(k)f Fq(=)d Fm(k)2286 1568 y Fh(\003)2342 1598 y Fq(+)18 b(1)27 b(w)n(e)h(get)761 1712 y Fj(Z)844 1733 y Fn(k)807 1901 y(k)q Fh(\000)p Fk(1)947 1825 y Fm(f)9 b Fq(\()p Fm(t)p Fq(\))p Fm(dt)19 b Fl(\000)1276 1769 y Fq(1)p 1276 1806 V 1276 1882 a(2)1341 1825 y(\()p Fm(f)9 b Fq(\()p Fm(k)22 b Fl(\000)c Fq(1\))g(+)g Fm(f)9 b Fq(\()p Fm(k)s Fq(\)\))37 b(=)g Fl(\000)2204 1769 y Fq(1)p 2184 1806 84 4 v 2184 1882 a(12)2276 1825 y Fm(f)2326 1791 y Fh(00)2382 1825 y Fq(\()p Fm(\030)2450 1837 y Fn(k)2486 1821 y Ff(\003)2522 1837 y Fk(+1)2610 1825 y Fq(\))g Fl(\025)g(\000)2876 1769 y Fq(1)p 2856 1806 V 2856 1882 a(12)2948 1825 y Fm(f)2998 1791 y Fh(00)2989 1846 y Fk(max)3116 1825 y Fm(:)456 2044 y Fq(Finally)26 b(for)g Fm(k)g Fl(\025)d Fm(k)1066 2013 y Fh(\003)1120 2044 y Fq(+)16 b(2)26 b(and)h Fm(t)c Fl(2)g Fq([)p Fm(k)d Fl(\000)c Fq(1)p Fm(;)e(k)s Fq(])26 b(w)n(e)g(use)g(the)h(monotonicit)n (y)f(of)h Fm(f)35 b Fq(and)27 b(estimate)456 2271 y(\(53\))867 2158 y Fj(Z)950 2178 y Fn(k)913 2346 y(k)q Fh(\000)p Fk(1)1052 2271 y Fm(f)9 b Fq(\()p Fm(t)p Fq(\))p Fm(dt)19 b Fl(\000)1381 2215 y Fq(1)p 1381 2252 42 4 v 1381 2328 a(2)1446 2271 y(\()q Fm(f)9 b Fq(\()p Fm(k)21 b Fl(\000)d Fq(1\))g(+)g Fm(f)9 b Fq(\()p Fm(k)s Fq(\)\))37 b Fl(\025)g(\000)2289 2215 y Fq(1)p 2289 2252 V 2289 2328 a(2)2354 2271 y(\()p Fm(f)9 b Fq(\()p Fm(k)21 b Fl(\000)d Fq(1\))g Fl(\000)g Fm(f)9 b Fq(\()p Fm(k)s Fq(\)\))28 b Fm(:)456 2484 y Fq(Hence)741 2609 y Fk([)p Fn(A)p Fk(])725 2641 y Fj(X)724 2820 y Fn(k)q Fk(=1)859 2578 y Fj( )925 2607 y(Z)1008 2627 y Fn(k)971 2796 y(k)q Fh(\000)p Fk(1)1110 2720 y Fm(f)9 b Fq(\()p Fm(t)p Fq(\))p Fm(dt)19 b Fl(\000)1439 2664 y Fq(1)p 1439 2701 V 1439 2777 a(2)1505 2720 y(\()p Fm(f)9 b Fq(\()p Fm(k)21 b Fl(\000)d Fq(1\))g(+)g Fm(f)9 b Fq(\()p Fm(k)s Fq(\)\))2134 2578 y Fj(!)747 3024 y Fl(\025)23 b(\000)930 2968 y Fq(1)p 910 3005 84 4 v 910 3081 a(12)1003 3024 y Fm(f)1053 2990 y Fh(00)1044 3045 y Fk(max)1188 3024 y Fl(\000)1281 2968 y Fq(1)p 1281 3005 42 4 v 1281 3081 a(2)1424 2913 y Fk([)p Fn(A)p Fk(])1408 2945 y Fj(X)1347 3124 y Fn(k)q Fk(=)p Fn(k)1470 3107 y Ff(\003)1506 3124 y Fk(+2)1604 3024 y Fq(\()p Fm(f)9 b Fq(\()p Fm(k)21 b Fl(\000)d Fq(1\))g Fl(\000)g Fm(f)9 b Fq(\()p Fm(k)s Fq(\)\))747 3282 y(=)23 b Fl(\000)930 3226 y Fq(1)p 910 3263 84 4 v 910 3339 a(12)1003 3282 y Fm(f)1053 3248 y Fh(00)1044 3303 y Fk(max)1188 3282 y Fl(\000)1281 3226 y Fq(1)p 1281 3263 42 4 v 1281 3339 a(2)1333 3282 y Fm(f)f Fq(\()q Fm(k)1475 3248 y Fh(\003)1531 3282 y Fq(+)c(1\))g(+)1799 3226 y(1)p 1799 3263 V 1799 3339 a(2)1851 3282 y Fm(f)k Fq(\([)p Fm(A)p Fq(]\))i Fl(\025)f(\000)2293 3226 y Fq(1)p 2273 3263 84 4 v 2273 3339 a(12)2365 3282 y Fm(f)2415 3248 y Fh(00)2406 3303 y Fk(max)2551 3282 y Fl(\000)2644 3226 y Fq(1)p 2644 3263 42 4 v 2644 3339 a(2)2696 3282 y Fm(f)f Fq(\()p Fm(t)2821 3248 y Fh(\003)2859 3282 y Fq(\))d(+)3003 3226 y(1)p 3003 3263 V 3003 3339 a(2)3055 3282 y Fm(f)j Fq(\([)p Fm(A)p Fq(]\))29 b Fm(;)456 3475 y Fq(since)e Fm(k)705 3445 y Fh(\003)761 3475 y Fq(+)18 b(1)23 b Fm(>)g(t)1027 3445 y Fh(\003)1065 3475 y Fq(.)37 b(Inserting)27 b(this)h(inequalit)n (y)f(in)n(to)g(\(51\))g(w)n(e)h(arriv)n(e)d(at)902 3679 y Fm(F)955 3691 y Fn(\026)1000 3679 y Fq(\()p Fm(A)p Fq(\))83 b Fl(\025)1367 3623 y Fq(1)p 1367 3660 V 1367 3736 a(2)1437 3679 y Fl(\000)1551 3623 y Fq(1)p 1530 3660 84 4 v 1530 3736 a(12)1623 3679 y Fm(f)1673 3644 y Fh(00)1664 3699 y Fk(max)1809 3679 y Fl(\000)1902 3623 y Fq(1)p 1902 3660 42 4 v 1902 3736 a(2)1953 3679 y Fm(f)9 b Fq(\()p Fm(t)2065 3644 y Fh(\003)2103 3679 y Fq(\))1209 3906 y(=)1367 3850 y(1)p 1367 3887 V 1367 3963 a(2)1437 3906 y Fl(\000)1575 3850 y Fm(\026)p 1530 3887 142 4 v 1530 3963 a Fq(3)p Fm(A)1634 3939 y Fk(2)1695 3789 y Fj(\022)1766 3850 y Fq(2)p Fm(\026)18 b Fl(\000)g Fq(4)p 1766 3887 235 4 v 1766 3963 a(2)p Fm(\026)g Fl(\000)g Fq(1)2010 3789 y Fj(\023)2071 3806 y Fn(\026)p Fh(\000)p Fk(2)2219 3906 y Fl(\000)2312 3850 y Fq(1)p 2312 3887 42 4 v 2312 3963 a(2)2378 3789 y Fj(\022)2449 3850 y Fq(2)p Fm(\026)g Fl(\000)g Fq(2)p 2449 3887 235 4 v 2449 3963 a(2)p Fm(\026)g Fl(\000)g Fq(1)2693 3789 y Fj(\023)2754 3806 y Fn(\026)1209 4156 y Fl(\025)1367 4099 y Fq(1)p 1367 4136 42 4 v 1367 4213 a(2)1437 4156 y Fl(\000)1675 4099 y Fm(\026)p 1530 4136 341 4 v 1530 4213 a Fq(3\(2)p Fm(\026)g Fl(\000)g Fq(1\))1894 4038 y Fj(\022)1965 4099 y Fq(2)p Fm(\026)g Fl(\000)g Fq(4)p 1965 4136 235 4 v 1965 4213 a(2)p Fm(\026)g Fl(\000)g Fq(1)2210 4038 y Fj(\023)2271 4056 y Fn(\026)p Fh(\000)p Fk(2)2419 4156 y Fl(\000)2512 4099 y Fq(1)p 2512 4136 42 4 v 2512 4213 a(2)2577 4038 y Fj(\022)2648 4099 y Fq(2)p Fm(\026)g Fl(\000)g Fq(2)p 2648 4136 235 4 v 2648 4213 a(2)p Fm(\026)g Fl(\000)g Fq(1)2893 4038 y Fj(\023)2954 4056 y Fn(\026)456 4375 y Fq(for)27 b(all)698 4315 y Fl(p)p 767 4315 V 60 x Fq(2)p Fm(\026)18 b Fl(\000)g Fq(1)23 b Fl(\024)f Fm(A)i(<)e(\026)p Fq(.)37 b(W)-7 b(e)28 b(put)950 4605 y(\010)1010 4617 y Fk(2)1047 4605 y Fq(\()p Fm(\026)p Fq(\))23 b(=)1282 4549 y(1)p 1282 4586 42 4 v 1282 4662 a(2)1352 4605 y Fl(\000)1590 4549 y Fm(\026)p 1445 4586 341 4 v 1445 4662 a Fq(3\(2)p Fm(\026)18 b Fl(\000)g Fq(1\))1809 4488 y Fj(\022)1880 4549 y Fq(2)p Fm(\026)g Fl(\000)g Fq(4)p 1880 4586 235 4 v 1880 4662 a(2)p Fm(\026)g Fl(\000)g Fq(1)2125 4488 y Fj(\023)2186 4505 y Fn(\026)p Fh(\000)p Fk(2)2334 4605 y Fl(\000)2427 4549 y Fq(1)p 2427 4586 42 4 v 2427 4662 a(2)2492 4488 y Fj(\022)2563 4549 y Fq(2)p Fm(\026)g Fl(\000)g Fq(2)p 2563 4586 235 4 v 2563 4662 a(2)p Fm(\026)g Fl(\000)g Fq(1)2808 4488 y Fj(\023)2869 4505 y Fn(\026)2927 4605 y Fm(:)555 4830 y Fq(In)30 b(a)e(third)i(step) f(w)n(e)g(assume)1530 4778 y Fl(p)p 1599 4778 51 4 v 52 x Fm(\026)d Fl(\024)f Fm(A)h(<)1943 4769 y Fl(p)p 2013 4769 235 4 v 2013 4830 a Fq(2)p Fm(\026)18 b Fl(\000)g Fq(1)o(.)42 b(Then)29 b(w)n(e)g(ha)n(v)n(e)f Fm(t)2877 4800 y Fh(\003)2941 4830 y Fq(=)3122 4797 y Fn(A)p 3041 4811 213 4 v 3041 4820 a Fh(p)p 3096 4820 159 3 v 42 x Fk(2)p Fn(\026)p Fh(\000)p Fk(1)3290 4830 y Fm(<)d Fq(1,)456 4946 y(hence)i Fm(k)732 4916 y Fh(\003)793 4946 y Fq(=)c([)p Fm(t)934 4916 y Fh(\003)972 4946 y Fq(])g(=)g(0.)36 b(W)-7 b(e)28 b(claim)g(that)588 5057 y Fj(Z)671 5077 y Fk(1)634 5246 y(0)722 5170 y Fm(f)9 b Fq(\()p Fm(t)p Fq(\))p Fm(dt)19 b Fl(\000)1051 5114 y Fq(1)p 1051 5151 42 4 v 1051 5227 a(2)1116 5170 y(\()p Fm(f)9 b Fq(\(0\))18 b(+)g Fm(f)9 b Fq(\(1\)\))23 b(=)1704 5057 y Fj(Z)1787 5077 y Fk(1)1750 5246 y(0)1838 5053 y Fj(\022)1899 5170 y Fq(1)18 b Fl(\000)2068 5114 y Fm(t)2098 5084 y Fk(2)p 2052 5151 100 4 v 2052 5227 a Fm(A)2114 5203 y Fk(2)2162 5053 y Fj(\023)2223 5067 y Fn(\026)2281 5170 y Fm(dt)h Fl(\000)2466 5114 y Fq(1)p 2466 5151 42 4 v 2466 5227 a(2)2536 5170 y Fl(\000)2629 5114 y Fq(1)p 2629 5151 V 2629 5227 a(2)2694 5053 y Fj(\022)2755 5170 y Fq(1)f Fl(\000)2937 5114 y Fq(1)p 2908 5151 100 4 v 2908 5227 a Fm(A)2970 5203 y Fk(2)3017 5053 y Fj(\023)3079 5070 y Fn(\026)3146 5170 y Fm(>)23 b Fq(0)14 b Fm(:)p eop end %%Page: 27 27 TeXDict begin 27 26 bop 640 251 a Fk(UNIVERSAL)28 b(BOUNDS)h(F)n(OR)h (TRA)n(CES)f(OF)g(THE)g(DIRICHLET)h(LAPLA)n(CE)f(OPERA)-5 b(TOR)117 b(27)456 467 y Fq(This)27 b(can)h(b)r(e)g(seen)g(b)n(y)f (expanding)1607 375 y Fj(\020)1656 467 y Fq(1)18 b Fl(\000)1822 434 y Fn(t)1847 409 y Fi(2)p 1809 448 83 4 v 1809 496 a Fn(A)1859 479 y Fi(2)1902 375 y Fj(\021)1951 392 y Fn(\026)2024 467 y Fq(and)2185 400 y Fj(\000)2223 467 y Fq(1)g Fl(\000)2401 434 y Fk(1)p 2376 448 V 2376 496 a Fn(A)2426 479 y Fi(2)2469 400 y Fj(\001)2507 417 y Fn(\026)2579 467 y Fq(as)27 b(series)g(and)h(comparing)456 592 y(co)r(e\016cien)n(ts.)36 b(Here,)27 b(using)h(\(53\))f(for)g Fm(k)f Fl(\025)d Fm(k)1850 561 y Fh(\003)1906 592 y Fq(+)18 b(2)23 b(=)f(2)27 b(yields)833 736 y Fk([)p Fn(A)p Fk(])817 768 y Fj(X)817 947 y Fn(k)q Fk(=1)951 705 y Fj( )1017 734 y(Z)1100 755 y Fn(k)1063 923 y(k)q Fh(\000)p Fk(1)1203 847 y Fm(f)9 b Fq(\()p Fm(t)p Fq(\))p Fm(dt)19 b Fl(\000)1532 791 y Fq(1)p 1532 828 42 4 v 1532 904 a(2)1597 847 y(\()p Fm(f)9 b Fq(\()p Fm(k)21 b Fl(\000)d Fq(1\))h(+)f Fm(f)9 b Fq(\()p Fm(k)s Fq(\)\))2226 705 y Fj(!)2315 847 y Fl(\025)22 b(\000)2477 791 y Fq(1)p 2477 828 V 2477 904 a(2)2528 847 y Fm(f)9 b Fq(\(1\))18 b(+)2795 791 y(1)p 2795 828 V 2795 904 a(2)2847 847 y Fm(f)k Fq(\()q([)p Fm(A)p Fq(]\))456 1089 y(and)27 b(inserting)g(this)h(estimate)f(in)n(to)h(\(51\))f(w)n(e) g(get)891 1307 y Fm(F)944 1319 y Fn(\026)989 1307 y Fq(\()p Fm(A)p Fq(\))d Fl(\025)1236 1251 y Fq(1)p 1236 1288 V 1236 1364 a(2)1306 1307 y Fl(\000)1399 1251 y Fq(1)p 1399 1288 V 1399 1364 a(2)1464 1190 y Fj(\022)1526 1307 y Fq(1)17 b Fl(\000)1707 1251 y Fq(1)p 1678 1288 100 4 v 1678 1364 a Fm(A)1740 1340 y Fk(2)1788 1190 y Fj(\023)1849 1207 y Fn(\026)1917 1307 y Fl(\025)2014 1251 y Fq(1)p 2014 1288 42 4 v 2014 1364 a(2)2084 1307 y Fl(\000)2177 1251 y Fq(1)p 2177 1288 V 2177 1364 a(2)2242 1190 y Fj(\022)2314 1251 y Fq(2)p Fm(\026)h Fl(\000)g Fq(2)p 2314 1288 235 4 v 2314 1364 a(2)p Fm(\026)g Fl(\000)g Fq(1)2558 1190 y Fj(\023)2619 1207 y Fn(\026)2687 1307 y Fl(\025)k Fq(\010)2834 1319 y Fk(2)2872 1307 y Fq(\()p Fm(\026)p Fq(\))p Fm(:)555 1520 y Fq(Finally)-7 b(,)33 b(w)n(e)d(deal)h(with)h(the)g(remaining)e (case)h(1)e Fl(\024)f Fm(A)i Fl(\024)2424 1468 y(p)p 2493 1468 51 4 v 52 x Fm(\026)p Fq(.)48 b(In)32 b(view)f(of)g(\(50\))g (w)n(e)g(can)456 1620 y(write)877 1855 y Fm(F)930 1867 y Fn(\026)975 1855 y Fq(\()p Fm(A)p Fq(\))83 b Fl(\025)1349 1744 y Fk([)p Fn(A)p Fk(])1332 1776 y Fj(X)1332 1954 y Fn(k)q Fk(=1)1467 1713 y Fj( )1532 1742 y(Z)1615 1762 y Fn(k)1578 1930 y(k)q Fh(\000)p Fk(1)1718 1855 y Fm(f)9 b Fq(\()p Fm(t)p Fq(\))14 b Fm(dt)19 b Fl(\000)f Fm(f)9 b Fq(\()p Fm(k)s Fq(\))2211 1713 y Fj(!)1184 2159 y Fq(=)1332 2046 y Fj(Z)1415 2066 y Fk(1)1378 2234 y(0)1466 2159 y Fm(f)g Fq(\()p Fm(t)p Fq(\))14 b Fm(dt)19 b Fl(\000)f Fm(f)9 b Fq(\(1\))18 b(+)2072 2048 y Fk([)p Fn(A)p Fk(])2056 2080 y Fj(X)2056 2259 y Fn(k)q Fk(=2)2190 2017 y Fj( )2256 2046 y(Z)2339 2066 y Fn(k)2302 2234 y(k)q Fh(\000)p Fk(1)2442 2159 y Fm(f)9 b Fq(\()p Fm(t)p Fq(\))p Fm(dt)19 b Fl(\000)f Fm(f)9 b Fq(\()p Fm(k)s Fq(\))2921 2017 y Fj(!)3000 2159 y 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Fq(\(1)9 b Fl(\000)g Fm(s)p Fq(\))2755 2696 y Fn(\026)2813 2730 y Fm(ds)g Fl(\000)2978 2613 y Fj(\022)3039 2730 y Fq(1)18 b Fl(\000)3221 2674 y Fq(1)p 3192 2711 100 4 v 3192 2787 a Fm(A)3254 2763 y Fk(2)3302 2613 y Fj(\023)3363 2630 y Fn(\026)3421 2730 y Fm(:)456 2943 y Fq(Lets)25 b(denote)h(the)f(righ)n(t)g(hand)h (side)f(with)h Fm(S)1848 2955 y Fn(\026)1893 2943 y Fq(\()p Fm(A)p Fq(\))g(and)g(analyse)e(it)i(in)g(more)e(detail.)37 b(W)-7 b(e)26 b(ha)n(v)n(e)662 3124 y Fm(d)p 631 3162 106 4 v 631 3238 a(dA)746 3181 y(S)797 3193 y Fn(\026)841 3181 y Fq(\()p Fm(A)p Fq(\))e(=)1089 3124 y(1)p 1089 3162 42 4 v 1089 3238 a(2)1154 3068 y Fj(Z)1271 3066 y Fi(1)p 1247 3075 76 3 v 1247 3117 a Fu(A)1290 3105 y Fi(2)1200 3256 y Fk(0)1351 3181 y Fm(s)1390 3146 y Fh(\000)1451 3124 y Fi(1)p 1451 3133 29 3 v 1451 3166 a(2)1494 3181 y Fq(\(1)18 b Fl(\000)g Fm(s)p Fq(\))1740 3146 y Fn(\026)1785 3181 y Fm(ds)g Fl(\000)1989 3124 y Fq(1)p 1978 3162 63 4 v 1978 3238 a Fm(A)2064 3064 y Fj(\022)2126 3181 y 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Fm(\026)2598 1046 y Fj(\023)2659 1063 y Fn(\026)2717 1163 y Fm(:)555 1346 y Fq(Lets)28 b(sum)f(up)h(the)g(results.)37 b(W)-7 b(e)28 b(ha)n(v)n(e)e(sho)n(wn)h(that)h(for)f Fm(\026)c Fl(\025)g Fq(3)k(and)g(all)h Fm(A)23 b Fl(\025)f Fq(1)967 1529 y Fm(F)1020 1541 y Fn(\026)1065 1529 y Fq(\()p Fm(A)p Fq(\))i Fl(\025)f Fq(min)1455 1412 y Fj(\032)1527 1473 y Fq(1)p 1527 1510 42 4 v 1527 1586 a(2)1579 1529 y Fm(B)1660 1412 y Fj(\022)1731 1473 y Fq(1)p 1731 1510 V 1731 1586 a(2)1782 1529 y Fm(;)14 b(\026)19 b Fq(+)f(1)2013 1412 y Fj(\023)2087 1529 y Fm(;)c Fq(\010)2184 1541 y Fk(1)2221 1529 y Fq(\()p Fm(\026)p Fq(\))p Fm(;)g Fq(\010)2432 1541 y Fk(2)2470 1529 y Fq(\()p Fm(\026)p Fq(\))p Fm(;)g Fq(\010)2681 1541 y Fk(3)2719 1529 y Fq(\()p Fm(\026)p Fq(\))2833 1412 y Fj(\033)2909 1529 y Fm(:)456 1712 y Fq(Note)27 b(that)1219 1879 y(\010)1279 1891 y Fk(1)1316 1879 y Fq(\()p Fm(\026)p Fq(\))d Fl(\025)1566 1822 y Fq(1)p 1566 1859 V 1566 1936 a(2)1617 1879 y Fm(B)1698 1761 y Fj(\022)1769 1822 y Fq(1)p 1769 1859 V 1769 1936 a(2)1821 1879 y Fm(;)14 b(\026)k Fq(+)g(1)2051 1761 y Fj(\023)2208 1879 y Fq(for)28 b(all)13 b Fm(\026)23 b Fl(\025)g Fq(5)1219 2111 y(\010)1279 2123 y Fk(2)1316 2111 y Fq(\()p Fm(\026)p Fq(\))h Fl(\025)1566 2055 y Fq(1)p 1566 2092 V 1566 2168 a(2)1617 2111 y Fm(B)1698 1994 y Fj(\022)1769 2055 y Fq(1)p 1769 2092 V 1769 2168 a(2)1821 2111 y Fm(;)14 b(\026)k Fq(+)g(1)2051 1994 y Fj(\023)2208 2111 y Fq(for)28 b(all)13 b Fm(\026)23 b Fl(\025)g Fq(30)1219 2343 y(\010)1279 2355 y Fk(3)1316 2343 y Fq(\()p Fm(\026)p Fq(\))h Fl(\025)1566 2287 y Fq(1)p 1566 2324 V 1566 2400 a(2)1617 2343 y Fm(B)1698 2226 y Fj(\022)1769 2287 y Fq(1)p 1769 2324 V 1769 2400 a(2)1821 2343 y Fm(;)14 b(\026)k Fq(+)g(1)2051 2226 y Fj(\023)2208 2343 y Fq(for)28 b(all)13 b Fm(\026)23 b Fl(\025)g Fq(4)p Fm(:)456 2526 y Fq(Therefore)j(the)i(claim)f(of)h (Lemma)f(4)g(holds)g(true)g(for)g(all)h Fm(\026)23 b Fl(2)g Fo(R)p Fq(,)28 b Fm(\026)23 b Fl(\025)g Fq(30.)35 b(T)-7 b(o)28 b(complete)f(the)456 2626 y(pro)r(of)i(w)n(e)g(are)g (left)h(with)h(the)f(\014nite)g(n)n(um)n(b)r(er)g(of)f(case)g Fm(\026)e Fq(=)f Fm(n=)p Fq(2)j(with)h Fm(n)d Fl(2)g Fo(N)p Fq(,)j(6)c Fl(\024)h Fm(n)f Fl(\024)g Fq(59.)456 2726 y(In)33 b(these)h(cases)e(the)i(claim)f(follo)n(ws)g(from)g(n)n (umerical)f(considerations)g(that)i(can)f(b)r(e)h(made)456 2825 y(rigorous)25 b(b)n(y)i(elemen)n(tary)g(analytic)g(metho)r(ds.) 1708 3004 y Fp(References)456 3137 y FB([AL78])118 b(M.)19 b(Aizenmann)j(and)e(E.H.)g(Lieb,)h Fg(On)h(semi-classic)l(al)h(b)l (ounds)h(for)f(eigenvalues)g(of)g(Schr\177)-36 b(odinger)781 3220 y(op)l(er)l(ators)p FB(,)25 b(Ph)n(ys.)f(Lett.)g Fb(66)f FB(\(1978\),)i(427{429.)456 3303 y([AS64])123 b(M.)23 b(Abramo)n(witz)g(and)i(I.A.)e(Stegun,)h Fg(Handb)l(o)l(ok)k (of)e(mathematic)l(al)h(functions)p FB(,)c(1964.)456 3386 y([BC86])114 b(J.)21 b(Brossard)g(and)h(R.)f(Carmona,)h Fg(Can)i(one)f(he)l(ar)i(the)e(dimension)i(of)f(a)g(fr)l(actal?)p FB(,)e(Comm.)f(Math.)781 3469 y(Ph)n(ys)j Fb(104)e FB(\(1986\),)j(no.)f (1,)f(103{122.)456 3552 y([Bro93])102 b(R.)24 b(M.)h(Bro)n(wn,)g Fg(The)i(tr)l(ac)l(e)h(of)f(the)g(he)l(at)h(kernel)e(in)h(Lipschitz)g (domains)p FB(,)g(T)-6 b(rans.)25 b(Amer.)f(Math.)781 3635 y(So)r(c.)g Fb(339)e FB(\(1993\),)j(no.)f(2,)f(889{900.)456 3718 y([Da)n(v85])91 b(E.)18 b(B.)g(Da)n(vies,)i Fg(T)-5 b(r)l(ac)l(e)21 b(pr)l(op)l(erties)i(of)e(the)g(Dirichlet)g(laplacian)p FB(,)g(Math.)e(Z.)g Fb(188)e FB(\(1985\),)22 b(245{251.)456 3801 y([Da)n(v89])p 781 3801 212 4 v 314 w(,)h Fg(He)l(at)j(kernels)f (and)i(sp)l(e)l(ctr)l(al)g(the)l(ory)p FB(,)d(Cam)n(bridge)f(Univ.)g (Press,)g(1989.)456 3884 y([F)-6 b(ab23])101 b(G.)22 b(F)-6 b(ab)r(er,)22 b Fg(Beweis,)i(dass)h(unter)f(al)t(len)h(homo)l (genen)g(Membr)l(anen)p FB(,)e(Sitzungsb)r(er.)f(Ba)n(y)n(er.)g(Ak)l (ad.)781 3967 y(Wiss.)h(M)r(\177)-37 b(unc)n(hen,)24 b(Math.-Ph)n(ys.)f(Kl.,)f(1923,)j(pp.)e(169{172.)456 4050 y([FLU02])72 b(J.)22 b(K.)h(F)-6 b(reeric)n(ks,)22 b(E.)g(H.)h(Lieb,)f(and)i(D.)e(Ueltsc)n(hi,)h Fg(Se)l(gr)l(e)l(gation)j (in)e(the)h(Falic)l(ov-Kimb)l(al)t(l)h(mo)l(del)p FB(,)781 4133 y(Comm.)d(Math.)g(Ph)n(ys.)h Fb(227)e FB(\(2002\),)k(no.)d(2,)h (243{279.)456 4216 y([FL)-8 b(V95])80 b(J.)32 b(Flec)n(kinger,)j(M.)d (Levitin,)j(and)e(D.)f(V)-6 b(assiliev,)34 b Fg(He)l(at)g(e)l(quation)g (on)h(the)e(triadic)h(von)h(Ko)l(ch)781 4299 y(snow\015ake:)43 b(Asymptotic)31 b(and)g(aumeric)l(al)h(analysis)p FB(,)f(Pro)r(c.)d (London)i(Math.)f(So)r(c.)h Fb(71)e FB(\(1995\),)781 4382 y(no.)23 b(3,)h(372{396.)456 4465 y([HH07])109 b(E.)19 b(M.)h(Harrell)e(and)j(L.)f(Hermi,)f Fg(On)j(Riesz)g(me)l(ans)h(of)g (eigenvalues)p FB(,)d(arXiv:0712.4088v1)i(\(2007\).)456 4548 y([Ivr98])125 b(V.)23 b(Ivrii,)h Fg(Micr)l(olo)l(c)l(al)k (analysis)f(and)g(pr)l(e)l(cise)g(sp)l(e)l(ctr)l(al)g(asymptotics)p FB(,)e(Springer)f(Monographs)h(in)781 4631 y(Mathematics,)f(Springer-V) -6 b(erlag,)23 b(Berlin,)f(1998.)456 4714 y([Kac51])94 b(M.)24 b(Kac,)i Fg(On)h(some)h(c)l(onne)l(ctions)g(b)l(etwe)l(en)f(pr) l(ob)l(ability)h(the)l(ory)g(and)g(di\013er)l(ential)g(and)g(inte)l(gr) l(al)781 4797 y(e)l(quations)p FB(,)c(Pro)r(ceedings)h(of)e(the)i (Second)h(Berk)n(eley)e(Symp)r(osium)g(on)g(Mathematical)h(Statistics) 781 4880 y(and)g(Probabilit)n(y)-6 b(,)24 b(1950,)h(Univ)n(ersit)n(y)g (of)f(California)f(Press,)h(Berk)n(eley)h(and)g(Los)f(Angeles,)h(1951,) 781 4963 y(pp.)e(189{215.)456 5046 y([Kac66])p 781 5046 V 317 w(,)g Fg(Can)j(one)g(he)l(ar)h(the)e(shap)l(e)i(of)f(a)g(drum?)p FB(,)f(Amer.)d(Math.)i(Mon)n(thly)h Fb(73)e FB(\(1966\),)i(1{23.)456 5133 y([Kra25])97 b(E.)24 b(Krahn,)1125 5116 y Fg(\177)1113 5133 y(Ub)l(er)i(eine)g(von)g(Rayleigh)h(formulierte)f (Minimaleigenschaft)h(des)f(Kr)l(eises)p FB(,)e(Math.)781 5216 y(Ann.)f Fb(94)g FB(\(1925\),)i(97{100.)p eop end %%Page: 29 29 TeXDict begin 29 28 bop 640 251 a Fk(UNIVERSAL)28 b(BOUNDS)h(F)n(OR)h (TRA)n(CES)f(OF)g(THE)g(DIRICHLET)h(LAPLA)n(CE)f(OPERA)-5 b(TOR)117 b(29)456 450 y FB([KVW08])35 b(H.)26 b(Ko)n(v)l(a)l(\024)-31 b(r)-8 b(\023)-27 b(\020k,)28 b(S.)f(V)-6 b(ugalter,)28 b(and)g(T.)e(W)-6 b(eidl,)28 b Fg(Two)i(dimensional)g(Ber)l (ezin-Li-Yau)e(ine)l(qualities)781 533 y(with)e(a)g(c)l(orr)l(e)l (ction)g(term)p FB(,)d(arXiv:)31 b(0802.2792v1)26 b(\(2008\).)456 616 y([Lut73])105 b(J.)18 b(M.)f(Luttinger,)j Fg(Gener)l(alize)l(d)i (isop)l(erimetric)f(ine)l(qualities)p FB(,)e(J.)f(Math.)h(Ph)n(ys.)f Fb(14)f FB(\(1973\),)k(no.)e(5,)781 699 y(586{593.)456 782 y([L)-8 b(W00])107 b(A.)30 b(Laptev)i(and)f(T.)f(W)-6 b(eidl,)32 b Fg(Sharp)i(Lieb-Thirring)d(ine)l(qualities)h(in)g(high)g (dimensions)p FB(,)h(Acta)781 865 y(Math.)23 b Fb(184)g FB(\(2000\),)i(no.)f(1,)f(87{111.)456 948 y([Mel03])99 b(A.D.)22 b(Mel\023)-35 b(as,)22 b Fg(A)j(lower)h(b)l(ound)g(for)g (sums)f(of)h(eigenvalues)f(of)g(the)g(laplacian)p FB(,)f(Amer.)e(Math.) i(So)r(c)781 1031 y Fb(131)e FB(\(2003\),)k(631{636.)456 1114 y([Min54])91 b(S.)21 b(Minakshisundaram,)g Fg(Eigenfunctions)i(on) h(Riemannian)g(manifolds)p FB(,)g(J.)d(Indian)h(Math.)f(So)r(c.)781 1197 y(\(N.S.\))i Fb(17)g FB(\(1954\),)j(159{165.)456 1280 y([MS67])111 b(H.)31 b(P)-6 b(.)31 b(McKean)h(and)g(I.)f(M.)f (Singer,)j Fg(Curvatur)l(e)h(and)g(the)e(eigenvalues)h(of)g(the)g (laplacian)p FB(,)i(J.)781 1363 y(Di\013eren)n(tial)23 b(Geometry)j Fb(1)d FB(\(1967\),)j(no.)d(1,)h(43{69.)456 1446 y([Sim83])97 b(B.)22 b(Simon,)h Fg(Non-classic)l(al)j(eigenvalue)f (asymptotics)p FB(,)e(J.)g(F)-6 b(unctional)24 b(Anal.)e Fb(53)h FB(\(1983\),)h(84{98.)456 1529 y([Smi81])97 b(L.)24 b(Smith,)h Fg(The)h(asymptotics)h(of)g(the)f(he)l(at)h(e)l(quation)h (for)e(a)h(b)l(oundary)h(value)f(pr)l(oblem)p FB(,)f(In)n(v)n(en)n(t.) 781 1612 y(Math.)d Fb(63)g FB(\(1981\),)j(no.)d(3,)h(467{493.)456 1695 y([ST90])125 b(F.)24 b(Steiner)i(and)g(P)-6 b(.)24 b(T)-6 b(rillen)n(b)r(erg,)24 b Fg(R)l(e\014ne)l(d)k(asymptotic)f(exp)l (ansion)i(for)e(the)g(p)l(artition)g(function)781 1778 y(of)f(unb)l(ounde)l(d)h(quantum)g(bil)t(liar)l(ds)p FB(,)d(J.)f(Math.)h(Ph)n(ys.)f Fb(31)g FB(\(1990\),)i(no.)f(7,)f (1670{1676.)456 1861 y([vdB84a])54 b(M.)26 b(v)l(an)h(den)h(Berg,)f Fg(On)h(the)g(sp)l(e)l(ctrum)i(of)f(the)f(Dirichlet)g(laplacian)i(for)f (horn-shap)l(e)l(d)j(r)l(e)l(gions)781 1944 y(in)25 b Fs(R)916 1921 y Fx(n)984 1944 y Fg(with)h(in\014nite)f(volume)p FB(,)f(J.)f(F)-6 b(unct.)25 b(Anal.)e Fb(58)g FB(\(1984\),)i(150{156.) 456 2028 y([vdB84b])p 781 2028 212 4 v 273 w(,)d Fg(A)i(uniform)h(b)l (ound)h(on)f(tr)l(ac)l(e)g FB(\()p Fz(e)1933 2004 y Fx(t)p Fr(\001)2010 2028 y FB(\))g Fg(for)g(c)l(onvex)f(r)l(e)l(gions)h(in)f Fs(R)2777 2004 y Fx(n)2844 2028 y Fg(with)h(smo)l(oth)h(b)l(ound-)781 2111 y(aries)p FB(,)d(Comm.)g(Math.)h(Ph)n(ys)g Fb(92)f FB(\(1984\),)i(no.)e(4,)h(525{530.)456 2194 y([vdB87])p 781 2194 V 312 w(,)f Fg(On)h(the)h(asymptotics)g(of)h(the)e(he)l(at)i (e)l(quation)f(and)h(b)l(ounds)h(on)e(tr)l(ac)l(es)g(asso)l(ciate)l(d)i (with)781 2277 y(the)e(Dirichlet)g(laplacian)p FB(,)g(J.)f(F)-6 b(unct.)24 b(Anal.)f Fb(71)g FB(\(1987\),)i(no.)f(2,)f(279{293.)456 2360 y([W)-6 b(ei08])98 b(T.)25 b(W)-6 b(eidl,)26 b Fg(Impr)l(ove)l(d)k (Ber)l(ezin-Li-Yau)c(ine)l(qualities)i(with)g(a)g(r)l(emainder)g(term)p FB(,)e(Amer.)e(Math.)781 2443 y(So)r(c.)g(T)-6 b(ransl.)22 b Fb(225)h FB(\(2008\),)i(no.)f(2,)f(253{263.)503 3030 y Fa(UNIVERSIT)818 3018 y(\177)818 3030 y(AT)39 b(STUTTGART,)f(FB)e (MATHEMATIK,)i(PFAFFENWALDRING)i(57,)c(70569)h(STUTTGART,)h(GERMANY) 1015 3113 y(E-MAIL:)f(leander.geisinger@mathe)q(mat)q(ik.u)q(ni-)q (stut)q(tga)q(rt.d)q(e)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0909070243369 Content-Type: application/octet-stream; name="09-155.keywords" Content-Transfer-Encoding: base64 Content-Disposition: attachment; filename="09-155.keywords" c3BlY3RyYWwgaW5lcXVhbGl0aWVzLCB0cmFjZSBlc3RpbWF0ZXMsIGhlYXQga2VybmVsLCBS aWVzeiBtZWFucw== ---------------0909070243369--