Content-Type: multipart/mixed; boundary="-------------0309181038634" This is a multi-part message in MIME format. ---------------0309181038634 Content-Type: text/plain; name="03-431.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-431.keywords" Trotter-Kato Product Formula, Norm- and Trace-norm Convergence ---------------0309181038634 Content-Type: application/postscript; name="ICMP-ws-procs10x7.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ICMP-ws-procs10x7.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: ICMP-ws-procs10x7.dvi %%CreationDate: Thu Sep 18 17:26:45 2003 %%Pages: 8 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: C:\TEXMF\MIKTEX\BIN\DVIPS.EXE ICMP-ws-procs10x7 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.09.18:1726 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N /vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N /rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N /@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{ /hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B /@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{ /urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known {userdict/md get type/dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{ itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack} if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{ noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{ Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale }if 0 setgray}N/psfts{S 65781.76 div N}N/startTexFig{/psf$SavedState save N userdict maxlength dict begin/magscale true def normalscale currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts /psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR/showpage{}N/erasepage{}N/copypage{}N/p 3 def @MacSetUp}N/doclip{ psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N/endTexFig{end psf$SavedState restore}N/@beginspecial{SDict begin/SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count/ocount X/dcount countdictstack N}N/@setspecial{ CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR }{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if/showpage{}N/erasepage{}N/copypage{}N newpath}N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{end} repeat grestore SpecialSave restore end}N/@defspecial{SDict begin}N /@fedspecial{end}B/li{lineto}B/rl{rlineto}B/rc{rcurveto}B/np{/SaveX currentpoint/SaveY X N 1 setlinecap newpath}N/st{stroke SaveX SaveY moveto}N/fil{fill SaveX SaveY moveto}N/ellipse{/endangle X/startangle X /yrad X/xrad X/savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet TeXDict begin 39158280 55380996 1000 600 600 (ICMP-ws-procs10x7.dvi) @start %DVIPSBitmapFont: Fa cmbx9 9 9 /Fa 9 58 df48 D<147814F81303131FEA03FFB5FCA3EAFC1F1200B3B2007FB512 FEA41F317AB02C>III<151F5D5DA25D5C 5C5C5CA25C143D147D14F9EB01F114E1EB03C1EB0781130FEB1F01133E133C137813F012 01EA03E0EA07C01380EA0F00121E123E5A5AB712FEA4C700031300A80103B512FEA42731 7EB02C>I<000C140ED80FE013FE90B5FC5D5D5D5D5D92C7FC14FC14F091C8FC1380A6EB 87FE9038BFFFC090B512F09038FC0FF89038E003FE01C07F497E01001480000E6D13C0C8 FCA216E0A3121FEA7F807F487EA316C05B5CD87F801480D87C0014006C5B393F8007FE39 1FE01FFC0007B512F06C14C0C691C7FCEB1FF823327CB02C>II<123C123F90B612F8A44815F016 E016C0168016005D007CC7127E00785C4A5A00F8495A48495A4A5A4A5AC7FC4AC7FC147E 14FE5C13015C1303A2495AA2130FA2131FA25C133FA4137FA96D5AA2010FC8FC25337BB1 2C>I57 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb cmsy5 5 1 /Fb 1 1 df0 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc cmr5 5 4 /Fc 4 52 df<14E0B0B712C0A3C700E0C7FCB022237C9B2B>43 D<1360EA01E0120F12FF 12F11201B3A3387FFF80A2111C7B9B1C>49 DII E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd cmmi5 5 3 /Fd 3 116 df<0003B512F815FE3A003E001F80ED07C0ED03E0150149EB00F0A216F8A2 5BA4484814F01501A216E04848130316C0ED0780ED0F004848131E5D15F8EC07E0B61280 02FCC7FC251C7C9B2E>68 D78 D<137E3801FF80EA0381380703C0380E0780EB0300EA0F80EA07F86CB4FC6C1380EA000F EA3003127812F8EB0700EAF00EEA7FFCEA1FF012127C911C>115 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe msbm10 10 2 /Fe 2 83 df78 D<007FB612E0B712FE6C6F7E2703C01E0313 E0000190393C00F3F00238EB70F8EE783CEE381E83EE3C07161C18801703A617071800EE 3C0FEE380E173EEE78FCEEF7F892380FFFE0023FB5128004FCC7FC16B8913838F03CED70 1CED781EED380EED3C0FED1C07031E7FED0E03030F7FED0701EE81E0ED0380707E030113 701778EEE0380300133C707EEE700EEE780F9338380780EE3C03041C13C093381E01E000 03013C90380E00F0007FB539F00FFFFEB67F6C8137397DB836>82 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff cmex10 10 10 /Ff 10 90 df2 DI<00F01378B3B3B2153674812E> 13 D<151E153E157C15F8EC01F0EC03E01407EC0FC0EC1F8015005C147E5CA2495A495A A2495AA2495AA2495AA249C7FCA2137EA213FE5B12015BA212035BA21207A25B120FA35B 121FA45B123FA548C8FCA912FEB3A8127FA96C7EA5121F7FA4120F7FA312077FA21203A2 7F1201A27F12007F137EA27FA26D7EA26D7EA26D7EA26D7EA26D7E6D7EA2147E80801580 EC0FC0EC07E01403EC01F0EC00F8157C153E151E1F94718232>16 D<12F07E127C7E7E6C7E7F6C7E6C7E12017F6C7E137EA27F6D7EA26D7EA26D7EA26D7EA2 6D7EA26D7EA280147E147F80A21580141FA215C0A2140F15E0A3140715F0A4140315F8A5 EC01FCA9EC00FEB3A8EC01FCA9EC03F8A515F01407A415E0140FA315C0141FA21580A214 3F1500A25C147E14FE5CA2495AA2495AA2495AA2495AA2495AA249C7FC137EA25B485A5B 1203485A485A5B48C8FC123E5A5A5A1F947D8232>I32 D<12F07E127C7E123F7E6C7E6C7E6C7E7F12016C7E7F137E133E133F6D7E130F806D7EA2 6D7E80130180130080147E147F8081141F81140F81140781A2140381140181A2140081A2 157FA36F7EA382151FA282150FA3821507A382A21503A282A31501A282A31500A382A482 A21780A7163F17C0AC161F17E0B3B3A217C0163FAC1780167FA71700A25EA45EA31501A3 5EA21503A35EA21507A25EA3150F5EA3151F5EA2153F5EA34BC7FCA315FEA25D1401A25D 14035D1407A25D140F5D141F5D143F92C8FC5C147E14FE5C13015C13035C495AA2495A5C 131F49C9FC133E137E5B5B485A12035B485A485A48CAFC5A123E5A5A5A2BF87E8242>I< EE03C0160F163F167F923801FF004B5A4B5AED0FF04B5A4B5A4B5A4BC7FC5C5D4A5A1407 5D4A5AA2141F5DA24A5AA44A5AB3B3B3B214FF92C8FCA35B5CA2495AA25C13075C130F49 5A5C133F495A49C9FC485A485A485A485AEA1FC0485AB4CAFC12FCA2B4FCEA3F806C7EEA 0FF06C7E6C7E6C7E6C7E6D7E6D7E131F806D7E130780130380A26D7EA2807FA381147FB3 B3B3B26E7EA46E7EA281140FA26E7E8114036E7E81806F7E6F7E6F7E6F7EED07FC6F7E6F 7E9238007FC0163F160F16032AF8748243>40 D88 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg cmti10 10 47 /Fg 47 122 df12 DI<150C151C153815F0EC01E0EC03C0EC0780EC0F00 141E5C147C5C5C495A1303495A5C130F49C7FCA2133EA25BA25BA2485AA212035B12075B A2120F5BA2121FA290C8FCA25AA2123EA2127EA2127CA412FC5AAD1278A57EA3121C121E A2120E7EA26C7E6C7EA212001E5274BD22>40 D<140C140E80EC0380A2EC01C015E0A214 0015F0A21578A4157C153CAB157CA715FCA215F8A21401A215F0A21403A215E0A21407A2 15C0140F1580A2141F1500A2143EA25CA25CA2495AA2495A5C1307495A91C7FC5B133E13 3C5B5B485A12035B48C8FC120E5A12785A12C01E527FBD22>I44 D<387FFFF8A2B5FCA214F0150579941E>I<120EEA3F80127F12FFA31300 127E123C0909778819>I48 D<15181538157815F0140114031407EC0FE0141F147FEB03FF90383F EFC0148FEB1C1F13001580A2143FA21500A25CA2147EA214FEA25CA21301A25CA21303A2 5CA21307A25CA2130FA25CA2131FA25CA2133FA291C7FC497EB61280A31D3877B72A>I< EC03F8EC0FFE91383C0F809138F007C0903901E003E0D903C013F09038078001020013F8 130E131E90391C6000FCEB3870EC30011370A213F013E0EC70030001016013F813C014E0 ECC00701C114F0903881800F018314E09039C7001FC001FEEB3F80D80078140090C7127E 5D4A5A4A5AEC07C0EC1F80023EC7FC14FCEB01F0EB07C0495A011EC8FC137C4914C04848 13015B485A4848130348C71380000E1407001E140F48EC1F00D83FF85B397FFFC07E3978 3FFFFCEA700FD8F0075BD8E0015B6D13C0021FC7FC263A79B72A>II<16E0ED01F01503A3150716E0A3150F16C0A2151F16 80A2ED3F00A3157EA2157C15FC5D14015D14035D14075D140F5D141F92C7FC143EA25CEC F81C153E903801F07EEB03E014C090380780FE130F49485A133EEB7C01137801F05BEA01 E03803C003EA0FFE391FFFC3F04813FB267C01FF13403AF0003FFFE000601307C71400EC 0FE05DA3141F5DA3143F92C7FCA4143E141C24487DB72A>I<010314186E13F8903907F0 07F091B512E016C01600495B15F8010E13E0020CC7FC011EC8FC131CA3133C1338A31378 1370A2147F9038F3FFC09038EF83E09038FC01F0496C7E485A497F49137CC8FC157EA315 FEA41401000C5C123F5A1403485C5A4A5A12F800E05C140F4A5A5D6C49C7FC0070137E00 785B387C01F8383E07F0381FFFC06C90C8FCEA01F8253A77B72A>I<133C137E13FF5AA3 13FE13FCEA00701300B2120EEA3F80127F12FFA31300127E123C102477A319>58 DI65 D<0107B712FEA3903A000FF000074B130018 7C021F153CA25DA2143FA25D1838147FA292C8FCEE03804A130718004A91C7FCA201015C A24A131E163E010314FE91B5FC5EA2903807F800167C4A1378A2130FA24A1370A2011F14 F0A24A90C8FCA2133FA25CA2137FA291CAFCA25BA25B487EB6FCA337397BB836>70 DI<0103B5D8F80FB512E0A390260007F8C7381FE0004B5DA2020F153F615DA2021F 157F96C7FC5DA2023F5D605DA2027F14016092C7FCA24A1403605CA249B7FC60A202FCC7 12070103150F605CA20107151F605CA2010F153F605CA2011F157F95C8FC5CA2013F5D5F 5CA2017F14015F91C7FC491403007FD9FE01B512F8B55BA243397CB83E>I<0103B512F8 A390390007F8005DA2140FA25DA2141FA25DA2143FA25DA2147FA292C7FCA25CA25CA213 01A25CA21303A25CA21307A25CA2130FA25CA2131FA25CA2133FA25CA2137FA291C8FC49 7EB6FCA25C25397CB820>I<0103B500F890387FFFE0A21AC090260007F8C7380FFC004B 15E061020F4BC7FC183E4B5C18F0021F4A5A4D5A4BEB0F804DC8FC023F143C5F4B5B4C5A 027FEB07C04CC9FCED001E5E4A5BED01FCECFE0315070101497E151FECFC7C4B7E903903 FDE07FDAFFC07F1580ED003F49488014F84A131F83130F160F4A801607011F81A24A1303 83133F16014A80A2017F6E7EA291C8FC494A7F007F01FE011F13FCB55CA243397CB840> 75 D<0107B512FCA25E9026000FF8C7FC5D5D141FA25DA2143FA25DA2147FA292C8FCA2 5CA25CA21301A25CA21303A25CA21307A25CA2130F170C4A141CA2011F153C17384A1478 A2013F157017F04A14E01601017F140317C091C71207160F49EC1F80163F4914FF000102 071300B8FCA25E2E397BB834>I<0007B812E0A25AD9F800EB001F01C049EB07C0485AD9 00011403121E001C5C003C17801403123800785C00701607140700F01700485CA2140FC7 92C7FC5DA2141FA25DA2143FA25DA2147FA292C9FCA25CA25CA21301A25CA21303A25CA2 1307A25CA2130FA25CEB3FF0007FB512F8B6FCA2333971B83B>84 D<14F8EB07FE90381F871C90383E03FE137CEBF801120148486C5A485A120FEBC001001F 5CA2EA3F801403007F5C1300A21407485C5AA2140F5D48ECC1C0A2141F15831680143F15 87007C017F1300ECFF076C485B9038038F8E391F0F079E3907FE03FC3901F000F0222677 A42A>97 D<133FEA1FFFA3C67E137EA313FE5BA312015BA312035BA31207EBE0F8EBE7FE 9038EF0F80390FFC07C013F89038F003E013E0D81FC013F0A21380A2123F1300A214075A 127EA2140F12FE4814E0A2141F15C05AEC3F80A215005C147E5C387801F8007C5B383C03 E0383E07C0381E1F80D80FFEC7FCEA01F01C3B77B926>I<147F903803FFC090380FC1E0 90381F0070017E13784913383901F801F83803F003120713E0120FD81FC013F091C7FC48 5AA2127F90C8FCA35A5AA45AA3153015381578007C14F0007EEB01E0003EEB03C0EC0F80 6CEB3E00380F81F83803FFE0C690C7FC1D2677A426>II<147F903803FFC090380F C1E090383F00F0017E13785B485A485A485A120F4913F8001F14F0383F8001EC07E0EC1F 80397F81FF00EBFFF891C7FC90C8FC5A5AA55AA21530007C14381578007E14F0003EEB01 E0EC03C06CEB0F806CEB3E00380781F83803FFE0C690C7FC1D2677A426>IIIII<150E153F157FA3157E151C1500ABEC1F80EC7FC0ECF1F0EB01 C090380380F813071401130F130E131EEB1C03133C013813F0A2EB0007A215E0A2140FA2 15C0A2141FA21580A2143FA21500A25CA2147EA214FEA25CA21301A25CA213035C121C38 7E07E0A238FE0FC05C49C7FCEAF83EEA787CEA3FF0EA0FC0204883B619>IIIII<147F903803FFC090380FC1F090 381F00F8017E137C5B4848137E4848133E0007143F5B120F485AA2485A157F127F90C7FC A215FF5A4814FEA2140115FC5AEC03F8A2EC07F015E0140F007C14C0007EEB1F80003EEB 3F00147E6C13F8380F83F03803FFC0C648C7FC202677A42A>I<9039078007C090391FE0 3FF090393CF0787C903938F8E03E9038787FC00170497EECFF00D9F0FE148013E05CEA01 E113C15CA2D80003143FA25CA20107147FA24A1400A2010F5C5E5C4B5A131F5EEC80035E 013F495A6E485A5E6E48C7FC017F133EEC70FC90387E3FF0EC0F8001FEC9FCA25BA21201 A25BA21203A25B1207B512C0A3293580A42A>I<3903C003F0390FF01FFC391E783C0F38 1C7C703A3C3EE03F8038383FC0EB7F800078150000701300151CD8F07E90C7FCEAE0FE5B A2120012015BA312035BA312075BA3120F5BA3121F5BA3123F90C9FC120E212679A423> 114 D<14FE903807FF8090380F83C090383E00E04913F00178137001F813F00001130313 F0A215E00003EB01C06DC7FC7FEBFFC06C13F814FE6C7F6D13807F010F13C01300143F14 1F140F123E127E00FE1480A348EB1F0012E06C133E00705B6C5B381E03E06CB45AD801FE C7FC1C267AA422>II<13 F8D803FEEB01C0D8078FEB03E0390E0F8007121E121C0038140F131F007815C01270013F 131F00F0130000E015805BD8007E133FA201FE14005B5D120149137EA215FE120349EBFC 0EA20201131E161C15F813E0163CD9F003133814070001ECF07091381EF8F03A00F83C78 E090393FF03FC090390FC00F00272679A42D>I<01F0130ED803FC133FD8071EEB7F80EA 0E1F121C123C0038143F49131F0070140FA25BD8F07E140000E08013FEC6485B150E1201 5B151E0003141C5BA2153C000714385B5DA35DA24A5A140300035C6D48C7FC0001130E38 00F83CEB7FF8EB0FC0212679A426>I<01F01507D803FC903903801F80D8071E903907C0 3FC0D80E1F130F121C123C0038021F131F49EC800F00701607A249133FD8F07E168000E0 ED000313FEC64849130718000001147E5B03FE5B0003160E495BA2171E00070101141C01 E05B173C1738A217781770020314F05F0003010713016D486C485A000190391E7C078028 00FC3C3E0FC7FC90393FF81FFE90390FE003F0322679A437>I<903907E007C090391FF8 1FF89039787C383C9038F03E703A01E01EE0FE3803C01F018013C0D8070014FC48148000 0E1570023F1300001E91C7FC121CA2C75AA2147EA214FEA25CA21301A24A1370A2010314 F016E0001C5B007E1401010714C000FEEC0380010F1307010EEB0F0039781CF81E903838 7C3C393FF03FF03907C00FC027267CA427>I<13F0D803FCEB01C0D8071EEB03E0D80E1F 1307121C123C0038140F4914C01270A249131FD8F07E148012E013FEC648133F16001201 5B5D0003147E5BA215FE00075C5BA214015DA314035D14070003130FEBF01F3901F87FE0 38007FF7EB1FC7EB000F5DA2141F003F5C48133F92C7FC147E147C007E13FC387001F8EB 03E06C485A383C1F80D80FFEC8FCEA03F0233679A428>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fh cmr7 7 15 /Fh 15 127 df1 D<903803FF80011F13F090387E00FCD801F8133FD8 07E0EB0FC04848EB07E04848EB03F048C7EA01F8A2007EEC00FCA248157EA7007E15FCA3 6CEC01F8A26C6CEB03F0000F15E0A26C6CEB07C0000315806C6CEB0F00A26C6C131ED8C0 70EB1C060178133CD86038EB380C01181330011C13700070151CD87FFCEB7FFC003F15F8 A327297DA82F>10 D<1306130C13181330136013E0EA01C0EA0380A2EA07005A120E121E A2121C123CA35AA512F85AAB7E1278A57EA3121C121EA2120E120F7EEA0380A2EA01C0EA 00E0136013301318130C13060F3B7AAB1A>40 D<12C012607E7E7E120E7EEA0380A2EA01 C013E0120013F0A213701378A3133CA5133E131EAB133E133CA51378A3137013F0A213E0 120113C0EA0380A2EA0700120E120C5A5A5A5A0F3B7DAB1A>I<140EB3A2B812E0A3C700 0EC8FCB3A22B2B7DA333>43 D48 D<13381378EA01F8121F12FE12E01200B3 AB487EB512F8A215267BA521>I<13FF000313E0380E03F0381800F848137C48137E0078 7F12FC6CEB1F80A4127CC7FC15005C143E147E147C5C495A495A5C495A010EC7FC5B5B90 3870018013E0EA0180390300030012065A001FB5FC5A485BB5FCA219267DA521>I<13FF 000313E0380F01F8381C007C0030137E003C133E007E133FA4123CC7123E147E147C5C49 5AEB07E03801FF8091C7FC380001E06D7E147C80143F801580A21238127C12FEA2150048 5B0078133E00705B6C5B381F01F03807FFC0C690C7FC19277DA521>I<1438A2147814F8 1301A2130313071306130C131C131813301370136013C012011380EA03005A120E120C12 1C5A12305A12E0B612E0A2C7EAF800A7497E90383FFFE0A21B277EA621>I54 D61 D91 D111 D<380F8010381FF038383FFFF04813E038E07FC038400F80 15067BA621>126 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fi cmmi7 7 26 /Fi 26 122 df11 D20 D23 D<48B512F8000714FC4814F84814F0D8 3C07C7FC1270EAC006130E1200A3131E131CA2133CA35BA313F8A3485AA26C5A1E1A7D98 1F>28 D<1238127C12FEA3127C123807077A8614>58 D<1238127C12FE12FFA2127F123B 1203A31206A3120C121812381270122008127A8614>I61 D<12E012F812FEEA3F80EA0FE0EA03F8 EA00FEEB3F80EB0FE0EB03F8EB00FEEC3F80EC0FE0EC03F8EC00FEED3F80ED0FE0ED03F8 ED00FE163E16FEED03F8ED0FE0ED3F80EDFE00EC03F8EC0FE0EC3F8002FEC7FCEB03F8EB 0FE0EB3F8001FEC8FCEA03F8EA0FE0EA3F8000FEC9FC12F812E027277AA134>I64 D<4B7E1503150782150F151FA2153FA2156F15 CF82EC0187140315071406140E140C02187FA2EC30031460A214C013011480D903007F91 B5FC5B90380C0001A25B13380130805B01E013005B12011203000F4A7ED8FFF890381FFF E0A22B2A7DA932>I<013FB512F816FF903A01FC001FC04AEB07E0EE03F001031401A24A 14F8A2130717F04A130317E0010F1407EE0FC04AEB1F80EE7E00011F495A91B512F0A291 388001FC013FEB007E8291C7EA1F80160F4915C0A2137EA213FEEE1F805BEE3F00000115 3E16FE49EB01F84B5A0003EC1FC0B7C7FC15F82D287DA732>I<4AB41308020FEBE01891 397F80F038903A01F8001870D903E0EB0CF0D90F80130749C71203013E15E05B49140148 5A484815C0485A120F5B001F168090C8FC4892C7FCA2127EA4127C12FCA21606007C5DA3 5E007E5D123E5E6C5D6C6C495A00074AC7FCD803E0130E6C6C13383900FE01F090383FFF C0D907FCC8FC2D2A7DA830>I<013FB512FC16FF903A01FC001FC04AEB03E0EE01F80103 1400177C4A80A2010781A25CA2130F18805CA2011F1600A24A5CA2133F173E91C8127E17 7C4915FC5F017E14015F01FE4A5AA2494A5A4C5A00014BC7FC167C495CED03E00003EC1F C0B600FEC8FC15F031287DA736>I<903B3FFFF01FFFF8A2D901FCC7EAFE004A5CA20103 14015F5CA2010714035F5CA2010F14075F5CA2011F140F91B65AA2913880000F013F141F 5F91C7FCA249143F94C7FC137EA201FE5C167E5BA2000115FE5E5BA200031401B539C07F FFE0A235287DA736>72 D78 D<000FB712E05A9039800FE007D81E009038C001C05A0038011F130012300070 5C00601501023F148012E0481400A2C74890C7FCA2147EA214FEA25CA21301A25CA21303 A25CA21307A25CA2130FA25CA2131F001FB57EA22B287DA727>84 D102 D<1407EC0F80141FA21500140E91C7FC A7EB03E0EB07F8EB0C3C1318EB303E136013C0A248485AA2C7FCA25CA4495AA4495AA449 5AA4495AA21238D87C1FC7FC12FC133E485AEA70F8EA7FE0EA1F80193380A61B>106 D<133EEA07FEA2EA007CA213FCA25BA21201A25BA21203EC07809038E01FC0EC38600007 EB61E014C3EBC187EBC307D80FC613C09038CC038001B8C7FC13E0487E13FEEB3F80EB0F C0486C7E1303003E1460A2127EECC0C0127CECC18012FC903801E30038F800FE0070137C 1B297CA723>I<3B07801FC007E03B0FE07FF01FF83B18F0E0F8783C3B30F1807CE03E90 3AFB007D801ED860FEEB3F005B49133E00C14A133E5B1201A24848495BA35F4848485A18 30EE01F0A23C0F8003E003E060A218C0933801E180271F0007C013E3933800FF00000E6D 48137C341B7D993B>109 D<3907801FC0390FE07FF03918F0E0F83930F1807CEBFB00D8 60FE133C5B5B00C1147C5B1201A248485BA34A5AEA07C01660EC03E0A23A0F8007C0C0A2 EDC180913803C300D81F0013C7EC01FE000EEB00F8231B7D9929>I<9038F007C03901FC 1FF039031E78780006EBE03C90381FC01C000CEB801E14005B0018141F133E1200137E15 3E137CA213FC157C5B1578000114F0A2EC01E0EC03C03903FC07809038FE1F00EBE7FCEB E1F0D807E0C7FCA25BA2120FA25B121FEAFFF8A22025809922>112 D115 D<131C133EA25BA45BA4485AB512E0A23801F000 485AA4485AA4485AA448C7FC1460A214C0123EEB0180EB0300EA1E06EA1F1CEA0FF8EA03 E013267EA419>I<90387C03C03901FF0FF03907079C30390E03B078000CEBF0F8001813 E1123015F0396007C0E015001200A2495AA449C7FC15301238007C1460EAFC3E15C0EAF8 7E39F06F03803970C70700383F83FE381F01F81D1B7D9926>120 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fj cmsy7 7 12 /Fj 12 108 df0 D<1238127C12FEA3127C123807077A9114>I< 1338A50060130C00F8133E00FC137E00FE13FE383FBBF83807FFC000011300EA007C48B4 FC000713C0383FBBF838FE38FE00FC137E00F8133E0060130C00001300A517197B9A22> 3 D<160E163E16FEED03F8ED0FE0ED3F80EDFE00EC03F8EC0FE0EC3F8002FEC7FCEB03F8 EB0FE0EB3F8001FEC8FCEA03F8EA0FE0EA3F8000FEC9FC12F812FEEA3F80EA0FE0EA03F8 EA00FEEB3F80EB0FE0EB03F8EB00FEEC3F80EC0FE0EC03F8EC00FEED3F80ED0FE0ED03F8 ED00FE163E160E1600AB007FB612FCB712FEA227357AA734>20 D<12E012F812FEEA3F80 EA0FE0EA03F8EA00FEEB3F80EB0FE0EB03F8EB00FEEC3F80EC0FE0EC03F8EC00FEED3F80 ED0FE0ED03F8ED00FE163E16FEED03F8ED0FE0ED3F80EDFE00EC03F8EC0FE0EC3F8002FE C7FCEB03F8EB0FE0EB3F8001FEC8FCEA03F8EA0FE0EA3F80007EC9FC12F812E0CAFCAB00 7FB612FCB712FEA227357AA734>I<176017F01770A217781738173C171C171E83717E71 7E717EEF00F8BAFC19801900CB12F8EF01E04D5A4D5A4DC7FC171E171C173C1738177817 70A217F01760391F7C9D42>33 D<13E0EA01F0EA03F8A3EA07F0A313E0A2120F13C0A3EA 1F80A21300A25A123EA35AA3127812F8A25A12100D1E7D9F13>48 D<017F157F2601FFE0903803FFC0000701F890380FF1F0260F83FC90381F0038261E00FF 013C7F001890263F8078130C4890261FC0E07F007090260FE1C07F0060EB07E3913803F7 80486DB4C7EA01806E5A157E157F81824B7E0060DAF7E0EB0300913801E3F0DBC3F85B6C 90260381FC13066C90260F00FE5B001C011E90387F803C6C017C90381FE0F82607C7F86D B45A2601FFE0010313C06C6CC86CC7FC391B7C9942>I<49B5FC130F133F01FFC7FCEA01 F8EA03E0EA078048C8FC121E121C123C123812781270A212F05AA2B7FCA300E0C8FCA27E 1270A212781238123C121C121E7E6C7EEA03E0EA01F86CB4FC013FB5FC130F130120277A A12D>I<147EEB03FEEB0FE0EB1F00133E5BB35BA2485AEA07E0EAFF8000FCC7FCB47EEA 07E0EA01F06C7EA2137CB37F7FEB0FE0EB03FEEB007E173B7BAB22>102 D<12FCB47EEA0FE0EA01F06C7E137CB37FA27FEB0FC0EB03FEEB007EEB03FEEB0FC0EB1F 00133EA25BB35B485AEA0FE0EAFF8000FCC7FC173B7BAB22>I<38C00180EAE003B3B3B3 A3EAC001113B78AB22>107 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fk cmsy10 10 18 /Fk 18 113 df<007FB81280B912C0A26C17803204799641>0 D<121C127FEAFF80A5EA 7F00121C0909799917>I<020FB6128091B712C01303010F1680D91FF8C9FCEB7F8001FE CAFCEA01F8485A485A485A5B48CBFCA2123EA25AA2127812F8A25AA87EA21278127CA27E A27EA26C7E7F6C7E6C7E6C7EEA00FEEB7F80EB1FF86DB71280010316C01300020F158091 CAFCAE001FB812804817C0A26C1780324479B441>18 D20 D<126012F812FEEA7F80 EA3FE0EA0FF8EA03FEC66C7EEB3FE0EB0FF8EB03FE903800FF80EC3FE0EC0FF8EC03FE91 3800FF80ED3FE0ED0FF8ED03FE923800FF80EE3FE0EE0FF8EE03FE933800FF80EF3FC017 1FEF7F80933801FF00EE07FCEE1FF0EE7FC04B48C7FCED07FCED1FF0ED7FC04A48C8FCEC 07FCEC1FF0EC7FC04948C9FCEB07FCEB1FF0EB7FC04848CAFCEA07FCEA1FF0EA7FC048CB FC12FC1270CCFCAE007FB81280B912C0A26C1780324479B441>I<181EA4181F84A28518 0785727EA2727E727E85197E85F11F80F10FC0F107F0007FBA12FCBCFCA26C19FCCCEA07 F0F10FC0F11F80F13F00197E61614E5A4E5AA24E5A61180F96C7FCA260181EA4482C7BAA 53>33 D49 D<91381FFFFE91B6FC1303 010F14FED91FF0C7FCEB7F8001FEC8FCEA01F8485A485A485A5B48C9FCA2123EA25AA212 7812F8A25AA2B712FE16FFA216FE00F0C9FCA27EA21278127CA27EA27EA26C7E7F6C7E6C 7E6C7EEA00FEEB7F80EB1FF06DB512FE010314FF1300021F13FE283279AD37>I54 D<0238EB07FC02F890383FFF80010391B512C0010F01 0314E0011FEB0F81017B90391E003FF09026E3F078131F010349130FECF1E0902607F3C0 130714F7DAFF8014E092C7FC18C04A140F49481580EF1F004A141E5F4A5CEE01E0011F4A 5A4A010FC7FC163E9138C001F8ED0FFC013F90383FFF804AB57E028114F0DA83017F91C7 EA3FFC496E7E1607017E6E7E8201FE6E1380A249157FA2173F12015BA21800485AA2177E 4848157CA25F48484A5A01C75D019F4A5A261FBF80495A496C011EC7FC003F01F0137C91 38FC03F0D87E3FB512C0D87C1F91C8FCD8780713F8D8E00113C0343D7EBA37>66 DI72 D92 D102 D<12FCEAFFC0EA07F0EA 01FCEA007E7F80131F80130FB3A7801307806D7E6D7EEB007EEC1FF0EC07F8EC1FF0EC7E 00495A495A495A5C130F5CB3A7131F5C133F91C7FC137E485AEA07F0EAFFC000FCC8FC1D 537ABD2A>I<126012F0B3B3B3B3A91260045377BD17>106 D<0070131C00F0131EB3B3B3 B3A80070131C175277BD2A>I112 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fl cmmi10 10 45 /Fl 45 122 df11 D<1403EC3FF891387FFF80D901E313C014800103133F9138001F80ED07 0092C7FC80A280A2808013018080130080147F81143F8149B47E130790380F8FF0EB3E0F 496C7E13F83801F003D803E07F1207380FC0011380121FEA3F0014005A127EA212FE5D48 1301A35DA24813035D6C13075D127C4A5A6C91C7FC5C6C133E6C6C5A3807C0F03801FFE0 D8003FC8FC223D7DBB25>14 DI17 D<1338017E14F001FEEB07FC151F49133B 15F30001EB01C391380383F89039F80700E0020E90C7FC00035B1478EBF0E0EBF1C03807 F78001FEC9FCEBFFE014FE390FE1FFC09038E01FE09038C007F06E7E001F6D7EA2D98000 EB0180A2003F150302011400010013F8A2485D1606007E0100130E160C00FE151CED7C38 48EC1FF00038EC07C029267CA430>20 D<133F14C0EB07F06D7E801301A26D7EA3147FA3 6E7EA36E7EA36E7EA36E7EA36E7EA36E7EA26E7EA214014A7E5C4A7E91381E3F80143C14 784A6C7E1301EB03E049486C7EEB0F80EB1F00496D7E137E5B48486D7E485A485A000F6E 7E485A485A48C87E12FE167F4816800070151F293B7CB930>I<017E1438D83FFE147E16 FEA2D801FC14FC12000001140116F85BED03F0120315074914E0150F000715C0ED1F805B ED3F00000F147EA2495B4A5A001F495A5D49485A4A5A003F49C7FC143EEB00F8495A4848 5AEB0F80D87E3EC8FC13F8EAFFE0138000F8C9FC27257CA429>23 D<013FB512FE90B7FC5A5A4815FE260F801CC7FCEA1E005A00385B5A5A481378C7FC1470 14F0A4495AA31303A3495AA3130FA25C131FA3133FA291C8FC131E28257EA324>28 D34 D<121C127FEAFF80A5EA7F00121C0909798817> 58 D<121C127FEAFF80A213C0A3127F121C1200A412011380A2120313005A1206120E5A 5A5A12600A19798817>II<150C151E153EA2153C157CA2157815F8A215F01401A215E01403A215C01407A215 80140FA215005CA2141E143EA2143C147CA2147814F8A25C1301A25C1303A2495AA25C13 0FA291C7FC5BA2131E133EA2133C137CA2137813F8A25B1201A25B1203A25B1207A25B12 0FA290C8FC5AA2121E123EA2123C127CA2127812F8A25A12601F537BBD2A>I<126012FC B4FCEA7FC0EA1FF0EA07FCEA01FF38007FC0EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC 07FCEC01FF9138007FC0ED1FF0ED07FCED01FF9238007FC0EE1FF0EE07FCEE01FF933800 7F80EF1FC0A2EF7F80933801FF00EE07FCEE1FF0EE7FC04B48C7FCED07FCED1FF0ED7FC0 4A48C8FCEC07FCEC1FF0EC7FC04948C9FCEB07FCEB1FF0EB7FC04848CAFCEA07FCEA3FF0 EA7FC048CBFC12FC1270323279AD41>I64 D<1760177017F01601A216 03A21607160FA24C7EA216331673166316C3A2ED0183A2ED0303150683150C1601151815 30A21560A215C014011580DA03007FA202061300140E140C5C021FB5FC5CA20260C7FC5C 83495A8349C8FC1306A25BA25B13385B01F01680487E000716FFB56C013F13FF5EA2383C 7DBB3E>I<0103B77E4916F018FC903B0007F80003FE4BEB00FFF07F80020FED3FC0181F 4B15E0A2141FA25DA2143F19C04B143F1980027F157F190092C812FE4D5A4A4A5AEF0FF0 4AEC1FC005FFC7FC49B612FC5F02FCC7B4FCEF3FC00103ED0FE0717E5C717E1307844A14 01A2130F17035CA2131F4D5A5C4D5A133F4D5A4A4A5A4D5A017F4BC7FC4C5A91C7EA07FC 49EC3FF0B812C094C8FC16F83B397DB83F>I<9339FF8001C0030F13E0037F9038F80380 913A01FF807E07913A07F8000F0FDA1FE0EB079FDA3F80903803BF0002FFC76CB4FCD901 FC80495A4948157E495A495A4948153E017F163C49C9FC5B1201484816385B1207485A18 30121F4993C7FCA2485AA3127F5BA312FF90CCFCA41703A25F1706A26C160E170C171C5F 6C7E5F001F5E6D4A5A6C6C4A5A16076C6C020EC8FC6C6C143C6C6C5C6CB4495A90393FE0 0FC0010FB5C9FC010313FC9038007FC03A3D7CBA3B>I<0103B7FC4916E018F8903B0007 F80007FE4BEB00FFF03F80020FED1FC0180F4B15E0F007F0021F1503A24B15F81801143F 19FC5DA2147FA292C8FCA25C18035CA2130119F84A1507A2130319F04A150FA2010717E0 181F4A16C0A2010FEE3F80A24AED7F00187E011F16FE4D5A4A5D4D5A013F4B5A4D5A4A4A 5A057FC7FC017F15FEEE03FC91C7EA0FF049EC7FC0B8C8FC16FC16C03E397DB845>I<01 03B812F05BA290260007F8C7123F4B1407F003E0020F150118005DA2141FA25D19C0143F A24B1330A2027F1470190092C7126017E05C16014A495A160F49B6FCA25F9138FC000F01 031407A24A6DC8FCA201075C18034A130660010F160693C7FC4A150E180C011F161C1818 4A1538A2013F5E18F04A4A5AA2017F15074D5A91C8123F49913803FF80B9FCA295C7FC3C 397DB83D>I<0103B5D8F803B512F8495DA290260007F8C73807F8004B5DA2020F150F61 5DA2021F151F615DA2023F153F615DA2027F157F96C7FC92C8FCA24A5D605CA249B7FC60 A202FCC7120101031503605CA201071507605CA2010F150F605CA2011F151F605CA2013F 153F605CA2017F157F95C8FC91C8FC496C4A7EB690B6FCA345397DB845>72 D<0107B512FCA216F890390007F8005DA2140FA25DA2141FA25DA2143FA25DA2147FA292 C7FCA25CA25CA21301A25CA21303A25CA21307A25CA2130FA25CA2131FA25CA2133FA25C A2137FA291C8FC497EB6FCA326397DB824>I<0103B6FC5B5E90260007FCC8FC5D5D140F A25DA2141FA25DA2143FA25DA2147FA292C9FCA25CA25CA21301A25CA21303A25CA21307 18404A15C0A2010F150118804A1403A2011F16005F4A1406170E013F151E171C4A143C17 7C017F5D160391C7120F49EC7FF0B8FCA25F32397DB839>76 D<902603FFF891381FFFF8 496D5CA2D90007030113006FEC007C02061678DA0EFF157081020C6D1460A2DA1C3F15E0 705CEC181F82023815016F6C5C1430150702706D1303030392C7FC02607FA2DAE0015C70 1306ECC0008201016E130EEF800C5C163F0103EDC01C041F131891C713E0160F49EDF038 18300106140717F8010E02031370EFFC60130CEE01FE011C16E004005B011815FF177F13 38600130153FA20170151F95C8FC01F081EA07FCB512E01706A245397DB843>78 D<4BB4FC031F13F09238FE01FC913903F0007EDA07C0EB1F80DA1F80EB0FC0023EC7EA07 E002FCEC03F0495A4948EC01F8495A4948EC00FC495A49C912FE49167E13FE49167F1201 485AA2485AA2120F5B001F17FFA2485AA34848ED01FEA400FFEE03FC90C9FCA2EF07F8A2 EF0FF0A218E0171F18C0EF3F806C167F180017FE4C5A6C6C5D1603001F4B5A6D4A5A000F ED1F806C6C4AC7FC6D147E0003EC01F8D801FC495AD8007EEB0FC090263F807FC8FC9038 07FFF801001380383D7CBA3F>I<0003B812FEA25A903AF8003FC00101C0913880007E48 48163C90C7007F141C121E001C92C7FCA2485CA200305C007017180060130112E0485CA2 1403C716005DA21407A25DA2140FA25DA2141FA25DA2143FA25DA2147FA292C9FCA25CA2 5CA21301A25CA21303A25CEB0FFC003FB6FC5AA237397EB831>84 D<003FB56C48B51280485DA226007F80C7381FF00091C8EA07C0604993C7FCA2491506A2 0001160E170C5BA20003161C17185BA20007163817305BA2000F167017605BA2001F16E0 5F5BA2003F15015F5BA2007F150394C8FC90C8FCA25E4815065A160E160C161C16181638 5E127E5E4B5A6C4A5A4BC9FC6C6C131E6C6C5B6C6C13F83903F807E06CB55A6C6C48CAFC EB0FF0393B7BB839>I<147E903803FF8090390FC1C38090391F00EFC0017E137F49133F 485A4848EB1F8012075B000F143F48481400A2485A5D007F147E90C7FCA215FE485C5AA2 14015D48150CA21403EDF01C16181407007C1538007E010F1330003E131F027B13706C01 E113E03A0F83C0F9C03A03FF007F80D800FCEB1F0026267DA42C>97 D<133FEA1FFFA3C67E137EA313FE5BA312015BA312035BA31207EBE0FCEBE3FF9038E707 C0390FFE03E09038F801F001F013F8EBE000485A15FC5BA2123F90C7FCA214015A127EA2 140312FE4814F8A2140715F05AEC0FE0A215C0EC1F80143F00781400007C137E5C383C01 F86C485A380F07C06CB4C7FCEA01FC1E3B7CB924>I<163FED1FFFA3ED007F167EA216FE A216FCA21501A216F8A21503A216F0A21507A2027E13E0903803FF8790380FC1CF90381F 00EF017EEB7FC049133F485A4848131F000715805B000F143F485A1600485A5D127F90C7 127EA215FE5A485CA21401A248ECF80CA21403161CEDF0181407007C1538007E010F1330 003E131F027B13706C01E113E03A0F83C0F9C03A03FF007F80D800FCEB1F00283B7DB92B >100 DI<16F8ED03FEED0F8792381F0F80ED3E3F167F157CA215FC1700161C4A 48C7FCA414035DA414075DA20107B512F0A39026000FE0C7FC5DA4141F5DA4143F92C8FC A45C147EA514FE5CA413015CA4495AA45C1307A25C121E123F387F8F80A200FF90C9FC13 1E12FEEA7C3CEA7878EA1FF0EA07C0294C7CBA29>II107 DIIII<90390F8003F090391FE00FFC903939 F03C1F903A70F8700F80903AE0FDE007C09038C0FF80030013E00001491303018015F05C EA038113015CA2D800031407A25CA20107140FA24A14E0A2010F141F17C05CEE3F80131F EE7F004A137E16FE013F5C6E485A4B5A6E485A90397F700F80DA383FC7FC90387E1FFCEC 07E001FEC9FCA25BA21201A25BA21203A25B1207B512C0A32C3583A42A>I<3903E001F8 3907F807FE390E3C1E07391C3E381F3A183F703F800038EBE07F0030EBC0FF00705B0060 1500EC007E153CD8E07F90C7FCEAC07EA2120013FE5BA312015BA312035BA312075BA312 0F5BA3121F5B0007C9FC21267EA425>114 D<14FF010313C090380F80F090383E003801 78131C153C4913FC0001130113E0A33903F000F06D13007F3801FFE014FC14FF6C14806D 13C0011F13E013039038003FF014071403001E1301127FA24814E0A348EB03C012F800E0 EB07800070EB0F006C133E001E13F83807FFE0000190C7FC1E267CA427>II<13F8D803FE1438D8070F147C000E6D 13FC121C1218003814011230D8701F5C12601503EAE03F00C001005B5BD8007E1307A201 FE5C5B150F1201495CA2151F120349EC80C0A2153F1681EE0180A2ED7F0303FF13001201 4A5B3A00F8079F0E90397C0E0F1C90393FFC07F8903907F001F02A267EA430>I<903907 E001F090391FF807FC9039783E0E0F9039E01F1C1FD801C09038383F803A03800FF07F01 00EBE0FF5A000E4A1300000C157E021F133C001C4AC7FC1218A2C7123FA292C8FCA25CA2 147EA214FEA24A130CA20101141C001E1518003F5BD87F81143801835C00FF1560010714 E03AFE0E7C01C0D87C1C495A2778383E0FC7FC391FF00FFC3907C003F029267EA42F> 120 D<13F8D803FE1470D8070F14F8000EEB8001121C121800381403003015F0EA701F12 60013F130700E0010013E012C05BD8007E130F16C013FE5B151F000115805BA2153F0003 15005BA25D157EA315FE5D1401000113033800F80790387C1FF8EB3FF9EB0FE1EB00035D A2000E1307D83F805B007F495AA24A5A92C7FCEB003E007C5B00705B6C485A381E07C06C B4C8FCEA01FC25367EA429>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fm cmr10 10 74 /Fm 74 127 df<1506150FA24B7EA24B7EA24B7EA2EDDFF0A29138018FF8A291380307FC A291380603FEA291380E01FF140CDA1C007F141802386D7E143002706D7E146002E06D7E 5C01016E7E5C01036E7E91C7FC496E7E1306010E6E7E130C011C6E7F131801386F7E1330 01706F7E136001E06F7E5B170F484882170748C97F17030006831701488383481880001F B9FC4818C0A24818E0A2BA12F0A23C3C7CBB45>1 D10 DII<121C127FEA FF80A8EA7F00AB123EAB121CABC7FCA8121C127FEAFF80A5EA7F00121C093C79BB17>33 D<146014E0EB01C0EB0380EB0700130E131E5B5BA25B485AA2485AA212075B120F90C7FC A25A121EA2123EA35AA65AB2127CA67EA3121EA2121F7EA27F12077F1203A26C7EA26C7E 1378A27F7F130E7FEB0380EB01C0EB00E01460135278BD20>40 D<12C07E12707E7E7E12 0F6C7E6C7EA26C7E6C7EA21378A2137C133C133E131EA2131F7FA21480A3EB07C0A6EB03 E0B2EB07C0A6EB0F80A31400A25B131EA2133E133C137C1378A25BA2485A485AA2485A48 C7FC120E5A5A5A5A5A13527CBD20>I<15301578B3A6007FB812F8B912FCA26C17F8C800 78C8FCB3A6153036367BAF41>43 D<121C127FEAFF80A213C0A3127F121C1200A4120113 80A2120313005A1206120E5A5A5A12600A19798817>II<121C12 7FEAFF80A5EA7F00121C0909798817>I48 DIII<1538A2157815F8A2140114031407A2140F141F141B14331473 146314C313011483EB030313071306130C131C131813301370136013C01201EA03801300 5A120E120C5A123812305A12E0B712F8A3C73803F800AB4A7E0103B512F8A325397EB82A >I<0006140CD80780133C9038F003F890B5FC5D5D158092C7FC14FC38067FE090C9FCAB EB07F8EB3FFE9038780F803907E007E090388003F0496C7E12066E7EC87EA28181A21680 A4123E127F487EA490C71300485C12E000605C12700030495A00385C6C1303001E495A6C 6C485A3907E03F800001B5C7FC38007FFCEB1FE0213A7CB72A>II<12301238123E003FB612E0A316C05A168016000070C712060060140E5D15 1800E01438485C5D5DC712014A5A92C7FC5C140E140C141C5CA25CA214F0495AA21303A2 5C1307A2130FA3495AA3133FA5137FA96DC8FC131E233B7BB82A>III<121C127FEAFF80A5EA7F00121CC7FCB2121C127FEA FF80A5EA7F00121C092479A317>I<121C127FEAFF80A5EA7F00121CC7FCB2121C127F5A 1380A4127F121D1201A412031300A25A1206A2120E5A121812385A1260093479A317>I< 007FB812F8B912FCA26C17F8CCFCAE007FB812F8B912FCA26C17F836167B9F41>61 D<1538A3157CA315FEA34A7EA34A6C7EA202077FEC063FA2020E7FEC0C1FA2021C7FEC18 0FA202387FEC3007A202707FEC6003A202C07F1501A2D901807F81A249C77F167FA20106 810107B6FCA24981010CC7121FA2496E7EA3496E7EA3496E7EA213E0707E1201486C81D8 0FFC02071380B56C90B512FEA3373C7DBB3E>65 DI<913A01FF8001 80020FEBE003027F13F8903A01FF807E07903A03FC000F0FD90FF0EB039F4948EB01DFD9 3F80EB00FF49C8127F01FE153F12014848151F4848150FA248481507A2485A1703123F5B 007F1601A35B00FF93C7FCAD127F6DED0180A3123F7F001F160318006C7E5F6C7E17066C 6C150E6C6C5D00001618017F15386D6C5CD91FE05C6D6CEB03C0D903FCEB0F80902701FF 803FC7FC9039007FFFFC020F13F002011380313D7BBA3C>II70 DIII75 DII< B5913807FFFE8080C69238007FE06EEC1F80D9DFF0EC0F001706EBCFF8EBC7FCA2EBC3FE EBC1FFA201C07F6E7EA26E7E6E7E81140F6E7E8114036E7E168080ED7FC016E0153FED1F F0ED0FF8A2ED07FCED03FEA2ED01FF6F1386A2EE7FC6EE3FE6A2EE1FF6EE0FFEA2160716 03A216011600A2177E486C153E487ED80FFC151EB500C0140EA2170637397DB83E>I IIIII<003FB812E0A3D9C003EB001F273E0001FE130348EE01F0 0078160000701770A300601730A400E01738481718A4C71600B3B0913807FF80011FB612 E0A335397DB83C>II87 D<003FB7FCA39039FC0001FE01C0130349495A003E C7FC003C4A5A5E0038141F00784A5A12704B5A5E006014FF4A90C7FCA24A5A5DC712074A 5AA24A5A5D143F4A5AA24A5A92C8FC5B495AA2495A5C130F4948EB0180A2495A5C137F49 5A16034890C7FC5B1203485AEE0700485A495C001F5D48485C5E4848495A49130FB8FCA3 29397BB833>90 DI93 D97 DIIII<147E903803FF8090380FC1E0EB1F8790383F0FF0137EA213FCA23901F803C091C7FC ADB512FCA3D801F8C7FCB3AB487E387FFFF8A31C3B7FBA19>IIIIIII<2703F00FF0 EB1FE000FFD93FFCEB7FF8913AF03F01E07E903BF1C01F83803F3D0FF3800FC7001F8026 03F70013CE01FE14DC49D907F8EB0FC0A2495CA3495CB3A3486C496CEB1FE0B500C1B500 83B5FCA340257EA445>I<3903F00FF000FFEB3FFCECF03F9039F1C01F803A0FF3800FC0 3803F70013FE496D7EA25BA35BB3A3486C497EB500C1B51280A329257EA42E>II<3903F01FE000FFEB7FF89038F1E07E9039F3801F803A0FF7000FC0D803FEEB 07E049EB03F04914F849130116FC150016FEA3167FAA16FEA3ED01FCA26DEB03F816F06D 13076DEB0FE001F614C09039F7803F009038F1E07E9038F0FFF8EC1FC091C8FCAB487EB5 12C0A328357EA42E>I<3807E01F00FFEB7FC09038E1E3E09038E387F0380FE707EA03E6 13EE9038EC03E09038FC0080491300A45BB3A2487EB512F0A31C257EA421>114 DI<1318A5 1338A31378A313F8120112031207001FB5FCB6FCA2D801F8C7FCB215C0A93800FC011580 EB7C03017E13006D5AEB0FFEEB01F81A347FB220>IIIIII<003FB512FCA2EB8003D83E 0013F8003CEB07F00038EB0FE012300070EB1FC0EC3F800060137F150014FE495AA2C648 5A495AA2495A495A495AA290387F000613FEA2485A485A0007140E5B4848130C4848131C A24848133C48C7127C48EB03FC90B5FCA21F247EA325>II126 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fn cmbxti10 10 30 /Fn 30 122 df<383FFFFE5AA3B5FCA214FC7E1708789622>45 D<0103B712F04916FEF0 FF8019E0D9000790C713F0187FF03FF84A151F4B15FCA3141F5DA3023FED3FF85D19F018 7F027FEDFFE05D4D13C04D138002FF913807FE004BEB1FFCEFFFF092B612C0495E18F092 39C0001FF8717E496F7E4B6D7E1980A25B92C7FCA35B5C5FA2011F17004A5C60171F013F 5E4A4A5A177F4D5A017F02035B4A010F1380B9C7FC17FC17F094C8FC3E3979B844>66 DI<0103B812E04917F0A3D90007EB8000F03FE0180F5C93C71207A35C4B 15C0A3023F14F0EDFC01A21603027F168003F849C7FC1607A202FF130F9238F03FC092B5 FCA25B5FEDE07F163F49141F03C090C8FCA24C130F495EED803E60043C133E4991C7FC92 C8127E187C18FC495E4A1401170360013F15074A4A5A171F177F017F4A485A4A130FB95A A395C7FC3C3979B83F>69 D<0103B812C04917E0A3D90007EB80019438007FC0181F5C93 C7120FA35C4B1580A3143F4B13F017F81601027F160003F849C7FCA2160302FF13074B48 5A163F92B5FC5B5FA2EDE03F49141F03C05B160F161F5B038090C8FCA349141E92CAFCA3 5B5CA3133F5CA3137F5CB612FE815DA23B397AB83C>I<0103B500C049B512F0496E4914 F81BF082D9000792390007F80070EC03E019074A7F62EDBFFE190F91381F9FFF031F5E6F 7F191F4A6C7F023E94C7FC6F7F61DA7E017F027C163E6F7F197E02FC6D7E4A167C707E19 FC01016E7E4A5E701380188101036E13C14A5E7013E118E301076E13F34AEDFBE08218FF 010F814A5E83A2011F8191C86C5BA2835B013E6F90C8FCA283137ED801FE6F5AB512FC18 7EA2183C4D397AB84A>78 D<923803FF80033F13F84AB512FE0207903801FF80913A1FF8 007FE0DA3FE06D7EDAFF806D7E4990C76C7E49481407D90FFC81494814034A81133F495A 494816805A5C5A91C8FC5A5A5BA2121F495D123FA348484B1300A448484B5AA34D5AA260 49157F6017FF605E604C5B6C7E4C90C7FC4C5A003F4B5A6D4A5A001F4B5A6C6C4A5A6D49 13802707FF800790C8FC6C9038E03FFCC690B512F0013F14C0010F49C9FC010013E0393B 72B947>I<0103B712E04916FC18FF19C0D90007D9800313E0050013F04A157FF03FF893 C7FCA24A16FCA25DA2143FF07FF85DA2027F16F018FF4B15E0A202FF4A13C019804B4913 004D5A49ED1FFCEFFFF892B612E018804903FCC7FC03E0C9FC5DA25BA25DA25BA292CAFC A25BA25CA2133FA25CA2137FA25CB612F8A43E397AB841>I<0103B7FC4916F018FC18FF D90007D9800F1380050113C07113E04AED7FF093C7FCA219F85C5DA3023FEDFFF05DA219 E0027F5C4B15C04D1380190002FF4A5A4BEB1FFCEF3FF04CB45A4990B6C7FC17FC839238 E00FFF4902037F4B6C7F848249825DA25E495E1500A25E5B4A5DA3013F4A141C4AED803E A2197E017F6E147C4A16FCB6D8F801EBC0F870EBE3F094387FFFE0051F13C0CA00031300 3F3A79B847>82 D<48B9FC481880A21900489039807FF8039026FC00FF130001F0824848 4A7F5B5CD81F80163E01005C5A5C123E007E5D127C4A157E00FC177C00784B133CC793C7 FC5CA293C9FCA25CA25DA2143FA25DA2147FA25DA214FFA25DA25BA25DA25BA25DA25BA2 5D007FB612F8B7FCA25E393871B742>84 D97 D99 DII<167E923803FFC04B13E092381FE3F09238 3FC7F816CFED7F9FA215FF17F05CEE1FE0EE078093C7FCA25C5DA414075D011FB6128017 C01780A29026000FF8C7FCA5141F5DA4143F5DA4147FA25DA414FF5DA45B92C8FCA4495A A45C1307120F383F87F8EA7FC7EAFFCF5CA2EB8FE0EB9FC0EB1F80D8FC3FC9FCEA7FFEEA 1FF8EA07E02D4C81BA21>III<143C147F495A15805B1500A25C6D5AEB0070 91C7FCAB133FEBFFC000037F3807C7F0380F87F8EA1F07A2EA3E0FA2127C131F5C12FCEA F83F00005B137F5CA213FF5CA25A91C7FC5A5BEC0F801207EBFC1F1500120F495A143E5C 13F000075BEBF1F06CB45AC65B013EC7FC193C79BA1E>I108 DII< EC0FF891B5FC010314C090390FF81FF090393FE007F890397FC003FCD9FF8013FE48EB00 01484814FF485A120F5B121F5B123F5D127F5BA25D00FF15FE5BA2ED0FFCA25BED1FF816 F0153F16E0007FEC7FC016806C6CEBFF004A5A391FE007FC390FF81FF00003B512C0C691 C7FCEB1FF0282777A531>I<90390FC003FC903A3FF00FFF8090267FFC3F13E0903A7DFE FE0FF0903AF8FFF807F800019138F003FC4913C001F115FED803E113801500A2EA07E313 C35CEA000301071407A25CA2010F140F17FC5CA2011F141F17F85CEE3FF0133F17E04AEB 7FC0A2017FECFF806E4813004B5A6E485A9039FFFC0FF091B512C0029F90C7FCEC87F848 0180C8FCA291C9FCA25AA25BA21207A25B387FFFE0B57EA25C2F367EA531>I<01FCEB7F 803A03FF01FFF04801C713FC3A0F9FEFC0FE3A1F0FFF003F4A13FF003E495A131F4A5A00 7C13F0A216FED8FC3FEB01FC00F89038E000F000001500A2137F5CA313FF5CA35A91C8FC A35A5BA312075BA35B5BEA01E0282779A52A>114 D<14F0EB03F8130780495AA3131FA2 5CA2133FA25CA2137FA2B6128015C0A21580C6EB8000A25AA291C7FCA25AA25BA21207A2 5BA2120FA25BA2121FA29038F007C0A2003F130F1580EBE01F1500143E147EEBC07C6C6C 5A380FE3F0EBFFE000031380C648C7FC1A3778B520>116 D<133FD9FFC0130F000301F0 EB1FC02607E7F8133FD80F83147F1387381F07FC003E15FFA2D87E0F1580007C5B011F5B 00FC13F000F81600EA003F4A5AA2017F5C14C0150713FF02805BA2150F5A0200EBF83EA2 031F137E177C16F0A2033F13F8A26C91387FF1F0EC80FF903A7F83F7F3E0903A3FFFC3FF C0010F01811380903A01FC007E002F2779A534>I<017EEB01E03A01FF8003F8489038E0 07FC3907C7F00FD80F87EB1FFEEA1F0F14F8123E150F397C1FF0071503013F130100FC90 38E000FC12F8EA007F5C16F813FF5C15014815F01400A2ED03E05A5BED07C0A2ED0F80A2 ED1F00151E0001143E6D5B6C5C90387F81F06DB45A010F1380D901FEC7FC272779A52C> I<013F161ED9FFC0011EEB7F80000301F090393F80FFC02607E7F8137FD80F83DAFF8113 E01387381F07FC003E5C1700D87E0F167F007C01F8153F011F49141F00FC01F049EB0FC0 12F8EA003FECE0074C1480137F14C0030F141F01FF4A14001480A26048021F143E02005B 60A360A2033F495A6C5F9139807FF803017F01FF495A903B3FC1FBFC0F8090261FFFF1B5 C7FC01079038C07FFC01009038000FF03B2779A540>II<133FD9FFC0130F000301F0EB1FC02607E7F8133FD80F83147F1387381F07FC003E 15FF1780EA7E0F007C5B011F5B00FC01F0140012F8EA003F4A5A5E137F14C0150701FF5C 1480A2150F485D1400A2151F5EA3153F5E6C147FEC80FFEB7F836DB55A130F903801FC7F 90C712FF5EEA03C0260FF00190C7FC121F003F495AA24A5A4A5A01E05B49485A9038007F 80261FC0FFC8FC380FFFFC000313F0C613802A3779A52F>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fo cmti9 9 57 /Fo 57 123 df<130E131F133F137E13FCEA01F8EA03F0EA07E0EA0F80EA1F00123E5A5A 5A100E67B327>19 D<1560EC01E0EC03C0EC0700140E5C143C5C5C495A495A13075C49C7 FC5B131E5B137C137813F85B12015B12035B1207A25B120FA290C8FC5AA2121E123EA312 3C127CA31278A212F8A35AAF12701278A21238A2123C121CA27EA27E6C7E12011B4A75B7 1F>40 D<14301438A28080A2140F801580A2140315C0A4140115E0A81403A415C0A31407 A31580140FA315005CA3141E143EA2143C147CA25CA25C13015C13035C13075C130F91C7 FC131E133E133C5B5B485AA2485A485A48C8FC121E5A12705A5A1B4A7EB71F>I44 DI<121C127F12FFA412FE12380808778718> I48 DIII54 DIII<1370EA01FC1203A413F8EA00 E01300B0121C127F5AA45A12380E20779F18>I64 D<161C163CA2167C16FCA21501821503A2ED077E150F150E151CA21538A2 157015F015E0EC01C0A2913803807F82EC0700A2140E141E141C5CA25CA25C49B6FCA25B 913880003F49C7EA1F80A2130E131E131C133C13385B13F05B12011203D80FF0EC3FC0D8 FFFE903807FFFEA32F367BB539>I<0107B612C04915F017FC903A003F8000FE177FEF3F 8092C7121FA24A15C0A2147EA214FE18804A143FA20101ED7F00177E4A5C16010103EC03 F04C5A4AEB1FC091B6C7FC495C9139F0007F804AEB0FC0707E010F6E7E834A1301A2011F 81A25CA2133F5F91C71203A2494A5AA2017E4A5A4C5A01FE4A5A4CC7FC49EB01FE0001EC 07FC007FB612F0B712C04BC8FC32337BB236>II<0107B712F05B18E0903A003F80001F1707170392C7FC17015C18C0147EA214 FEA24A130EA20101EC1E03041C13804A91C7FC163C13035E9138F001F891B5FC5B5EECE0 011500130F5E5C1707011F01015BEEC00E0280141E92C7121C133F173C91C81238177849 5DA2017E14014C5A01FE14074C5A49141F00014AB45A007FB7FCB8FC94C7FC34337CB234 >69 D<0107B712E05B18C0903A003F80003F170F170792C7FC17035C1880147EA214FEA2 5C161C0101EC3C07043813004A91C7FCA20103147816704A13F0150349B5FCA25EECE003 130F6F5A14C0A2011F13035E1480A2013F90C9FCA291CAFCA25BA2137EA213FEA25B1201 387FFFFCB5FCA233337CB232>I<92391FE001809238FFF8030207EBFE07913A1FF01F0F 0091393F80079F9139FE0003DFD901F86DB4FCD907F05C49481300495A4948147E49C812 7C137E13FE485A48481578A2485AA248481570A2485A94C7FC123F5BA3127F90CBFCA400 FE91383FFFFCA25F9238003F8094C7FCA2007E5DA2167EA2007F15FE7E5E6C6C1301A26C 6C495A6D13076C6CEB0F786C6C133E3A00FF01FC3090387FFFF0011F01C0C8FCD903FEC9 FC313775B43B>I<010FB51280A216009038003FC05DA292C7FCA25CA2147EA214FEA25C A21301A25CA21303A25CA21307A25CA2130FA25CA2131FA25CA2133FA291C8FCA25BA213 7EA213FEA25B1201B512F8A25C21337BB21E>73 D<91381FFFFE5C16FC9138003F80A316 00A25D157EA315FE5DA314015DA314035DA314075DA3140F5DA3141F5DA3143F92C7FCA2 121C007E5B00FE137EA214FE485BEAF80100E05B495A387007E038780FC06C48C8FCEA1F FCEA07F0273579B228>I<0107B590380FFFF05B19E09026003FC0903803FE004B14F818 E092C7485A4DC7FC4A141E177C027E5CEE01E002FE495A4C5A4A011FC8FC163E01011478 5E4A485AED07C00103495A4B7EECF03F157F903907F1FFE0ECF3E79138E787F0ECEF0790 380FFE0302FC7FECF80114E0D91FC07F15004A7FA2013F147E167F91C77E835B161F017E 81160F13FE835B000182267FFFF090B57EB55B95C7FC3C337BB23B>I<0107B512C05BA2 9026003FC0C7FC5DA292C8FCA25CA2147EA214FEA25CA21301A25CA21303A25CA21307A2 5CA2130FA25C17E0011F140117C05C1603013F1580160791C7FCEE0F005B5E017E143EA2 01FE5CED01FC4913030001EC1FF8007FB6FCB7FC5E2B337CB230>I<902607FFC0ED7FFC 4917FF81D9003F4B1300611803023BED077CA2027BED0EFC610273151C1838DAF1F01439 F071F014E118E10101ED01C36102C1EC0383EF070301031607050E5BEC80F8171C0107ED 380F6102001470A249EDE01FDC01C090C7FC130EEE0380011E017C5C933807003E011C14 0EA2013C4A137E187C01385C5E017816FC6F485B1370ED3FC001F0EC80016000011500D8 07F81503277FFF803E90B512C0B5EB3C01151C46337BB245>I<902607FF8090383FFFC0 496D5BA2D9001F913803F8004A6C6D5A6060EC3BF0027B140360EC71F8A202F11407DAF0 FC91C7FC14E0A20101017E5B170E14C0810103151EEE801CEC801FA20107ECC03C030F13 38140016E049010713781770010E14F01503011E15F0705A011C1301A2013C14FD03005B 133816FF0178147F5F0170143FA213F070C8FC1201EA07F8267FFF807FB5140EA23A337B B239>II<0107B612C04915F883903A00 3F8001FEEE003FEF1F8092C713C0170F5C18E0147EA214FEEF1FC05CA201011680173F4A 1500177E010315FE5F4AEB03F8EE07E00107EC3FC091B6C7FC16F802E0C9FC130FA25CA2 131FA25CA2133FA291CAFCA25BA2137EA213FEA25B1201387FFFF0B5FCA233337CB234> I<0107B512FE49ECFFC017F0903A003F8007F8EE01FCEE007E92C7127F835C1880147EA2 14FEEF7F005CA2010115FE5F4A13015F01034A5AEE0FC04A495A04FEC7FC49B512F016C0 9138E003E0ED01F8010F6D7E167C4A137EA2131FA25CA2013F14FEA291C7FCA24913015E 137EEF01C001FE150318805B00011607277FFFF0001400B5ECFE0EEE7E1CC9EA1FF8EE07 E032357BB238>82 D<913901FC018091380FFF03023F13C791387E07EF903A01F801FF00 49487E4A7F495A4948133E131F91C7FC5B013E143CA3137E1638A293C7FC137FA26D7E14 E014FE90381FFFC06D13F86D7F01017F6D6C7E020F7F1400153F6F7E150FA4120EA2001E 5D121CA2151F003C92C7FCA2003E143E5D127E007F5C6D485A9038C007E039F3F80FC000 F0B5C8FC38E03FFC38C00FF029377AB42B>I<0003B812C05A1880903AF800FC003F260F C001141F0180150F01005B001EEE07001403121C003C4A5BA200380107140E127800705C A2020F141E00F0161CC74990C7FCA2141FA25DA2143FA292C9FCA25CA2147EA214FEA25C A21301A25CA21303A25CA21307A25C497E001FB512F05AA2323374B237>I<3B3FFFF801 FFFE485CA2D801FEC7EA1FC049EC0F80170049140EA2161E120349141CA2163C12074914 38A21678120F491470A216F0121F495CA21501123F90C75BA215035A007E5DA2150712FE 4892C7FCA25D150E48141E151C153C153815786C5C5D007C1301007E495A003EEB0F806C 011EC8FC380FC0FC6CB45A000113E06C6CC9FC2F3570B239>I97 D<137EEA0FFE121F5B1200A35BA21201A25BA21203A25BA21207A2EBC3E0EBCFF8380FDC 3EEBF81F497E01E01380EA1FC0138015C013005AA2123EA2007E131F1580127CA2143F00 FC14005AA2147EA25CA2387801F85C495A6C485A495A6C48C7FCEA0FFCEA03F01A3578B3 23>I<14FCEB07FF90381F078090383E03C0EBFC013801F8033803F0073807E00F13C012 0F391F80070091C7FC48C8FCA35A127EA312FE5AA4007C14C0EC01E0A2EC03C06CEB0F80 EC1F006C137C380F81F03803FFC0C648C7FC1B2278A023>III<151FED7FC0EDF0E0020113F0EC03E3A2EC07C316E0ED C1C091380FC0005DA4141F92C7FCA45C143E90381FFFFEA3D9007EC7FC147CA414FC5CA5 13015CA413035CA413075CA3130FA25CA3131F91C8FCA35B133E1238EA7E3CA2EAFE7812 FC485AEA78E0EA3FC0000FC9FC244582B418>I<143FECFF80903803E1E6903807C0FF90 380F807FEB1F00133E017E133F49133EA24848137EA24848137CA215FC12074913F8A214 01A2D80FC013F0A21403120715E01407140F141F3903E03FC00001137FEBF0FF38007FCF 90381F0F801300141FA21500A25C143E1238007E137E5C00FE5B48485A387803E0387C0F 80D81FFFC7FCEA07F820317CA023>III<133F EA07FF5A13FEEA007EA3137CA213FCA213F8A21201A213F0A21203A213E0A21207A213C0 A2120FA21380A2121FA21300A25AA2123EA2127EA2127C1318EAFC1C133CEAF838A21378 137012F013F0EAF8E01279EA3FC0EA0F00103579B314>108 D<2703C003F8137F3C0FF0 0FFE01FFC03C1E783C1F07C1E03C1C7CF00F8F01F03B3C3DE0079E0026383FC001FC7FD9 7F805B007001005B5E137ED8F0FC90380FC00100E05FD860F8148012000001021F130360 491400A200034A13076049013E130FF081800007027EEC83C0051F138049017C1403A200 0F02FC1407053E130049495CEF1E0E001F01015D183C010049EB0FF0000E6D48EB03E03A 227AA03F>I<3903C007F0390FF01FFC391E787C1E391C7CF01F393C3DE00F26383FC013 80EB7F8000781300EA707EA2D8F0FC131F00E01500EA60F8120000015C153E5BA2000314 7E157C4913FCEDF8180007153C0201133801C013F0A2000F1578EDE070018014F016E000 1FECE1C015E390C7EAFF00000E143E26227AA02B>I<14FCEB07FF90381F07C090383E03 E09038FC01F0EA01F83903F000F8485A5B120F484813FCA248C7FCA214014814F8127EA2 140300FE14F05AA2EC07E0A2007CEB0FC01580141FEC3F006C137E5C381F01F0380F83E0 3803FF80D800FCC7FC1E2278A027>I<011E137C90387F81FF9039F3C387C09039E3EF03 E03901E1FE01D9C1FC13F0EBC3F8000313F0018314F814E0EA07871307000313C0120001 0F130316F01480A2011F130716E01400A249EB0FC0A2013EEB1F80A2017EEB3F00017F13 3E5D5D9038FF81F09038FDC3E09038F8FF80027EC7FC000190C8FCA25BA21203A25BA212 07A25BB5FCA325307FA027>I<903803F01890380FF83890383E1C7890387C0EF89038F8 07F0EA01F0EA03E000071303D80FC013E0A2EA1F801407D83F0013C0A348130F007E1480 A300FE131F481400A35C143E147E127C14FE495AEA3C03EA3E07EA1F0E3807FCF8EA01F0 C7FC13015CA313035CA21307A25C48B5FCA25C1D3078A023>I<3903C00FC0390FF03FF0 391E78F078391C7DE03C393C3FC0FC00381380EB7F00007814F8D8707E13701500EAF0FC 12E0EA60F812001201A25BA21203A25BA21207A25BA2120FA25BA2121FA290C8FC120E1E 227AA020>II<1303EB0F80 A3131FA21400A25BA2133EA2137EA2137C387FFFF8A2B5FC3800F800A21201A25BA21203 A25BA21207A25BA2120FA25B1460001F13F014E01300130114C01303001E1380EB07005B EA0F1EEA07F8EA01E015307AAE19>II<01F01338D803FC13FCEA 0F1E120E121C123C0038147CEA783E0070143CA2137ED8F07C1338EA60FCC65A15780001 14705BA215F0000314E05BA2EC01C0A2EBC003158014071500EBE00EA26C6C5A3800F878 EB7FE0EB1F801E227AA023>I<011F137C90387FC1FF3A01E1E787803A03C0F703C09038 80FE0FEA07004813FC000E1580001E9038F80700001C91C7FC1301003C5B121812001303 5CA31307A25C1506010F130F150E14800038141ED87C1F131C00FC143C1538013F5B39F0 7FC0E03970F3C3C0393FE1FF80260F807EC7FC22227CA023>120 D<13F0D803FC1307D80F1E130F000E141F121C123C0038143FD8783E133E1270A2017E13 7ED8F07C137CEA60FCC65A15FC000114F85BA21401000314F013E0A2140315E0EA07C0A2 0003130715C0EBE00F141F0001133F9038F07F8038007FEFEB1F8FEB001F1500A25C003E 133E007E137E147C5C007C5BEA7001495A38380780D83C1FC7FCEA0FFCEA07F020317AA0 25>I<903807801C90381FE03C90383FF038017F13789038FFF8F03901F07CE0EBE01F39 03C003C09038800780EC0F00C7121E141C143C5C5C495AEB07C0495A011EC7FC5B5B4913 704913F0000114E0485A38078001390FC003C0381FF80790383E0F80393C1FFF00127838 700FFE38F007F838E001E01E227CA01F>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fp cmr9 9 62 /Fp 62 122 df<91393FE00FE0903A01FFF83FF8903A07E01EF83C903A1F800FF07E903A 3F001FE0FE017E133F4914C0485A1738484890381F8000ACB812C0A33B03F0001F8000B3 A7486C497EB50083B5FCA32F357FB42D>11 D<14C01301EB0380EB0F00130E5B133C5B5B A2485A485AA212075B120F90C7FC5AA2121E123EA3123C127CA55AB0127CA5123C123EA3 121E121FA27E7F12077F1203A26C7E6C7EA213787F131C7F130FEB0380EB01C01300124A 79B71E>40 D<12C07E1270123C121C7E120F6C7E6C7EA26C7E6C7EA27F1378137C133C13 3EA2131E131FA37F1480A5EB07C0B0EB0F80A514005BA3131E133EA2133C137C137813F8 5BA2485A485AA2485A48C7FC120E5A123C12705A5A124A7CB71E>I<123C127EB4FCA213 80A2127F123D1201A412031300A25A1206120E120C121C5A5A126009177A8715>44 DI<123C127E12FFA4127E123C08087A8715>I48 D<13075B5B137FEA07FFB5FC13BFEAF83F 1200B3B3A2497E007FB51280A319327AB126>IIII<000C14C0380FC00F90B5 128015005C5C14F014C0D80C18C7FC90C8FCA9EB0FC0EB7FF8EBF07C380FC03F9038001F 80EC0FC0120E000CEB07E0A2C713F01403A215F8A41218127E12FEA315F0140712F80060 14E01270EC0FC06C131F003C14806CEB7F00380F80FE3807FFF8000113E038003F801D34 7CB126>I<14FE903807FF80011F13E090383F00F0017C13703901F801F8EBF003EA03E0 1207EA0FC0EC01F04848C7FCA248C8FCA35A127EEB07F0EB1FFC38FE381F9038700F8090 38E007C039FFC003E0018013F0EC01F8130015FC1400A24814FEA5127EA4127F6C14FCA2 6C1301018013F8000F14F0EBC0030007EB07E03903E00FC03901F81F806CB51200EB3FFC EB0FE01F347DB126>I<1230123C003FB6FCA34814FEA215FC0070C71238006014301570 15E04814C01401EC0380C7EA07001406140E5C141814385CA25CA2495A1303A3495AA213 0FA3131F91C7FCA25BA55BA9131C20347CB126>III<123C127E12FFA4127E123C1200B012 3C127E12FFA4127E123C08207A9F15>I<15E0A34A7EA24A7EA34A7EA3EC0DFE140CA2EC 187FA34A6C7EA202707FEC601FA202E07FECC00FA2D901807F1507A249486C7EA301066D 7EA2010E80010FB5FCA249800118C77EA24981163FA2496E7EA3496E7EA2000182160748 7ED81FF04A7ED8FFFE49B512E0A333367DB53A>65 DIIIIIIII75 DIIIII82 D<90381FE00390387FFC0748B5FC3907F01FCF 390F8003FF48C7FC003E80814880A200788000F880A46C80A27E92C7FC127F13C0EA3FF0 13FF6C13F06C13FF6C14C06C14F0C680013F7F01037F9038003FFF140302001380157F15 3FED1FC0150F12C0A21507A37EA26CEC0F80A26C15006C5C6C143E6C147E01C05B39F1FC 03F800E0B512E0011F138026C003FEC7FC22377CB42B>I<007FB712FEA390398007F001 D87C00EC003E0078161E0070160EA20060160600E01607A3481603A6C71500B3AB4A7E01 1FB512FCA330337DB237>III89 D<003FB612FCA39039F80007F813C090C7EA 0FF0003EEC1FE0123C0038EC3FC00078EC7F801270EDFF004A5AA20060495AA24A5A4A5A C7FC4A5A4A5AA24A5A4AC7FCA2495A495AA2495A495AA24948130C495AA2495A49C7FCA2 4848141CA2485A485A1638485A4848147816F84848130148481307153FB7FCA326337CB2 2F>I97 DII<153FEC0FFFA3EC007F81AEEB07F0EB3FFCEBFC0F3901F003BF39 07E001FF48487E48487F8148C7FCA25A127E12FEAA127E127FA27E6C6C5BA26C6C5B6C6C 4813803A03F007BFFC3900F81E3FEB3FFCD90FE0130026357DB32B>III<151F90391FC07F809039FFF8E3C03901F07FC73907E03F033A0FC01F8380 9039800F8000001F80EB00074880A66C5CEB800F000F5CEBC01F6C6C48C7FCEBF07C380E FFF8380C1FC0001CC9FCA3121EA2121F380FFFFEECFFC06C14F06C14FC4880381F000100 3EEB007F4880ED1F8048140FA56C141F007C15006C143E6C5C390FC001F83903F007E0C6 B51280D91FFCC7FC22337EA126>IIIIII<2703F01FE013FF00FF90267FF80313C0903BF1E07C0F03E0 903BF3803E1C01F02807F7003F387FD803FE1470496D486C7EA2495CA2495CB3486C496C 487EB53BC7FFFE3FFFF0A33C217EA041>I<3903F01FC000FFEB7FF09038F1E0FC9038F3 807C3907F7007EEA03FE497FA25BA25BB3486CEB7F80B538C7FFFCA326217EA02B>II<3903F03F8000FFEBFFE09038 F3C0F89038F7007ED807FE7F6C48EB1F804914C049130F16E0ED07F0A3ED03F8A9150716 F0A216E0150F16C06D131F6DEB3F80160001FF13FC9038F381F89038F1FFE0D9F07FC7FC 91C8FCAA487EB512C0A325307EA02B>I<3803E07C38FFE1FF9038E38F809038E71FC0EA 07EEEA03ECA29038FC0F8049C7FCA35BB2487EB512E0A31A217FA01E>114 DI<1330A51370A313F0A21201 A212031207381FFFFEB5FCA23803F000AF1403A814073801F806A23800FC0EEB7E1CEB1F F8EB07E0182F7FAD1E>IIII<3A7FFF807FF8A33A07F8001FC00003EC0F80 0001EC070015066C6C5BA26D131C017E1318A26D5BA2EC8070011F1360ECC0E0010F5BA2 903807E180A214F3010390C7FC14FBEB01FEA26D5AA31478A21430A25CA214E05CA2495A 1278D8FC03C8FCA21306130EEA701CEA7838EA1FF0EA0FC025307F9F29>121 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fq cmbx10 10 48 /Fq 48 122 df<913803FFC0027F13F00103B512FC010FEB00FED93FF8133FD97FE0EBFF 8049485A5A1480484A13C04A6C1380A36F1300167E93C7FCA592383FFFC0B8FCA4000390 C7FCB3ABB5D8FC3F13FFA4303A7EB935>12 D45 DI<141E143E 14FE1307133FB5FCA313CFEA000FB3B3A6007FB61280A4213779B630>49 DIII58 D65 DIIIIII75 DIIIII82 D I<003FB91280A4D9F800EBF003D87FC09238007FC049161F007EC7150FA2007C1707A200 781703A400F818E0481701A4C892C7FCB3AE010FB7FCA43B387DB742>II97 D<13FFB5FCA412077EAF4AB47E020F13F0023F 13FC9138FE03FFDAF00013804AEB7FC00280EB3FE091C713F0EE1FF8A217FC160FA217FE AA17FCA3EE1FF8A217F06E133F6EEB7FE06E14C0903AFDF001FF80903AF8FC07FE009039 F03FFFF8D9E00F13E0D9C00390C7FC2F3A7EB935>I<903801FFC0010F13FC017F13FFD9 FF8013802603FE0013C048485AEA0FF8121F13F0123F6E13804848EB7F00151C92C7FC12 FFA9127FA27F123FED01E06C7E15036C6CEB07C06C6C14806C6C131FC69038C07E006DB4 5A010F13F00101138023257DA42A>II<903803FF80011F13F0017F13FC3901FF83FE3A03FE007F8048 48133F484814C0001FEC1FE05B003FEC0FF0A2485A16F8150712FFA290B6FCA301E0C8FC A4127FA36C7E1678121F6C6C14F86D14F000071403D801FFEB0FE06C9038C07FC06DB512 00010F13FC010113E025257DA42C>II<161FD907FEEBFFC090387FFFE348B6EAEFE02607FE07138F 260FF801131F48486C138F003F15CF4990387FC7C0EEC000007F81A6003F5DA26D13FF00 1F5D6C6C4890C7FC3907FE07FE48B512F86D13E0261E07FEC8FC90CAFCA2123E123F7F6C 7E90B512F8EDFF8016E06C15F86C816C815A001F81393FC0000F48C8138048157F5A163F A36C157F6C16006D5C6C6C495AD81FF0EB07FCD807FEEB3FF00001B612C06C6C91C7FC01 0713F02B377DA530>I<13FFB5FCA412077EAFED7FC0913803FFF8020F13FE91381F03FF DA3C01138014784A7E4A14C05CA25CA291C7FCB3A3B5D8FC3F13FFA4303A7DB935>II<141FEC7FC0ECFFE0A24913F0A56D13E0A2EC7FC0EC1F0091C7FCA9EC0F F0EB0FFFA4EB007F143FB3B0121FEA3F80EA7FC0EAFFE0EC7FE0A215C014FF6C48138090 3883FE006CB45A000F13F0000113801C4B86BA1D>I<13FFB5FCA412077EAF92380FFFE0 A4923803FC0016F0ED0FE0ED1F804BC7FC157E5DEC03F8EC07E04A5A141FEC7FE04A7E81 81A2ECCFFEEC0FFF496C7F806E7F6E7F82157F6F7E6F7E82150F82B5D8F83F13F8A42D3A 7EB932>I<13FFB5FCA412077EB3B3ACB512FCA4163A7DB91B>I<01FED97FE0EB0FFC00FF 902601FFFC90383FFF80020701FF90B512E0DA1F81903983F03FF0DA3C00903887801F00 0749DACF007F00034914DE6D48D97FFC6D7E4A5CA24A5CA291C75BB3A3B5D8FC1FB50083 B512F0A44C257DA451>I<01FEEB7FC000FF903803FFF8020F13FE91381F03FFDA3C0113 80000713780003497E6D4814C05CA25CA291C7FCB3A3B5D8FC3F13FFA430257DA435>I< 903801FFC0010F13F8017F13FFD9FF807F3A03FE003FE048486D7E48486D7E48486D7EA2 003F81491303007F81A300FF1680A9007F1600A3003F5D6D1307001F5DA26C6C495A6C6C 495A6C6C495A6C6C6CB45A6C6CB5C7FC011F13FC010113C029257DA430>I<9039FF01FF 80B5000F13F0023F13FC9138FE07FFDAF00113800007496C13C06C0180EB7FE091C713F0 EE3FF8A2EE1FFCA3EE0FFEAA17FC161FA217F8163F17F06E137F6E14E06EEBFFC0DAF003 13809139FC07FE0091383FFFF8020F13E0020390C7FC91C9FCACB512FCA42F357EA435> I<9038FE03F000FFEB0FFEEC3FFF91387C7F809138F8FFC000075B6C6C5A5CA29138807F 80ED3F00150C92C7FC91C8FCB3A2B512FEA422257EA427>114 D<90383FF0383903FFFE F8000F13FF381FC00F383F0003007E1301007C130012FC15787E7E6D130013FCEBFFE06C 13FCECFF806C14C06C14F06C14F81203C614FC131F9038007FFE140700F0130114007E15 7E7E157C6C14FC6C14F8EB80019038F007F090B512C000F8140038E01FF81F257DA426> I<130FA55BA45BA25B5BA25A1207001FEBFFE0B6FCA3000390C7FCB21578A815F86CEB80 F014816CEBC3E090383FFFC06D1380903803FE001D357EB425>I<01FFEC3FC0B5EB3FFF A4000714016C80B3A35DA25DA26C5C6E4813E06CD9C03E13FF90387FFFFC011F13F00103 138030257DA435>III121 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fr cmr8 8 50 /Fr 50 123 df<123C127EB4FCA21380A2127F123D1201A312031300A25A1206120E5A5A 5A126009157A8714>44 DI<123C127E12FFA4127E123C08087A 8714>I<15C0140114031580A214071500A25C140EA2141E141CA2143C143814781470A2 14F05CA213015CA213035C130791C7FCA25B130EA2131E131CA2133C1338A21378137013 F05BA212015BA212035BA2120790C8FC5A120EA2121E121CA2123C1238A212781270A212 F05AA21A437CB123>II<130C133C137CEA03FC12FFEAFC7C 1200B3B113FE387FFFFEA2172C7AAB23>III< 140EA2141E143EA2147E14FEA2EB01BE1303143E1306130E130C131813381330136013E0 13C0EA0180120313001206120E120C5A123812305A12E0B612FCA2C7EA3E00A9147F9038 1FFFFCA21E2D7EAC23>I<000CEB0180380FC01F90B512005C5C14F014C0D80C7EC7FC90 C8FCA8EB1FC0EB7FF8380DE07C380F801F01001380000E130F000CEB07C0C713E0A21403 15F0A4127812FCA448EB07E012E0006014C00070130F6C14806CEB1F006C133E380780F8 3801FFE038007F801C2D7DAB23>II<12 30123C003FB512F8A215F05A15E039700001C000601480140348EB0700140E140CC7121C 5C143014705C495AA2495AA249C7FCA25B130E131EA2133EA3133C137CA413FCA913781D 2E7CAC23>II<123C127E12FFA4127E123C1200AD123C127E12FFA4127E123C081D 7A9C14>58 D67 D69 D71 DII75 D77 D80 D82 D<90383F80303901FFF0703807C07C390F000EF0001E13074813034813011400127000F0 1470A315307EA26C1400127E127FEA3FE013FE381FFFE06C13FC6C13FF00011480D8003F 13E013039038003FF0EC07F81401140015FC157C12C0153CA37EA215787E6C14706C14F0 6CEB01E039F78003C039E3F00F0038E07FFE38C00FF01E2F7CAD27>I<007FB712F8A290 39000FC003007C150000701638A200601618A200E0161CA248160CA5C71500B3A94A7E01 1FB512E0A22E2D7EAC33>I87 D<13FF000713C0380F01F0381C00 F8003F137C80A2143F001E7FC7FCA4EB07FF137F3801FE1FEA07F0EA1FC0EA3F80EA7F00 127E00FE14065AA3143F7E007E137F007FEBEF8C391F83C7FC390FFF03F83901FC01E01F 207D9E23>97 DI I<15F8141FA214011400ACEB0FE0EB7FF83801F81E3803E0073807C003380F8001EA1F00 481300123E127EA25AA9127C127EA2003E13017EEB8003000F13073903E00EFC3A01F03C FFC038007FF090391FC0F800222F7EAD27>II< EB03F0EB0FFCEB3E1EEB7C3F13F8EA01F0A23803E00C1400AAB512E0A23803E000B3A648 7E387FFF80A2182F7FAE16>I<013F13F89038FFC3FE3903E1FF1E3807807C000F140C39 1F003E00A2003E7FA76C133EA26C6C5A00071378380FE1F0380CFFC0D81C3FC7FC90C8FC A3121E121F380FFFF814FF6C14C04814F0391E0007F848130048147C12F848143CA46C14 7C007C14F86CEB01F06CEB03E03907E01F803901FFFE0038003FF01F2D7E9D23>III<130FEB1F80EB3FC0A4 EB1F80EB0F0090C7FCA8EB07C013FFA2130F1307B3AD1230127838FC0F80A21400485AEA 783EEA3FF8EA07E0123C83AD16>I108 D<2607C07FEB07F03BFFC3FFC03FFC903AC783F0783F3C0FCE01F8E01F 803B07DC00F9C00F01F8D9FF8013C04990387F000749137EA249137CB2486C01FEEB0FE0 3CFFFE0FFFE0FFFEA2371E7E9D3C>I<3807C0FE39FFC3FF809038C703E0390FDE01F0EA 07F8496C7EA25BA25BB2486C487E3AFFFE1FFFC0A2221E7E9D27>II<3807C0FE39FFC7FF809038CF03E0390FDC01F03907F800FC49137E4913 3E49133FED1F80A3ED0FC0A8151F1680A2ED3F00A26D137E6D137C5D9038FC01F09038CE 07E09038C7FF80D9C1FCC7FC01C0C8FCA9487EEAFFFEA2222B7E9D27>I<380781F838FF 87FEEB8E3FEA0F9CEA07B813B0EBF01EEBE000A45BB0487EB5FCA2181E7E9D1C>114 D<3801FE183807FFB8381E01F8EA3C00481378481338A21418A27E7EB41300EA7FF06CB4 FC6C13C06C13F0000113F838001FFC130138C0007E143EA26C131EA27EA26C133CA26C13 7838FF01F038E3FFC000C0130017207E9E1C>I<1360A413E0A312011203A21207121FB5 12F0A23803E000AF1418A714383801F03014703800F860EB3FE0EB0F80152A7FA81B>I< D807C013F800FF131FA2000F130100071300B21401A314033803E007EC0EFC3A01F81CFF C038007FF890391FE0F800221F7E9D27>I<3AFFFC01FFC0A23A0FE0007E000007147C15 380003143015706C6C1360A26C6C5BA390387C0180A26D48C7FCA2EB3F07EB1F06A2EB0F 8CA214DCEB07D8A2EB03F0A36D5AA26D5A221E7F9C25>I<3BFFFC3FFE07FFA23B0FE003 F001F801C09038E000F00007010114E0812603E00314C0A2913807F8012701F006781380 A29039F80E7C030000D90C3C1300A290397C181E06A2151F6D486C5AA2168C90391F6007 98A216D890390FC003F0A36D486C5AA36DC75A301E7F9C33>I<3AFFFC07FF80A23A0FF0 03FC000003EB01F0000114C06D485A000091C7FCEB7C06EB3E0E6D5A14B8EB0FB0EB07E0 13036D7E497E1307EB067C497EEB1C1F01387FEB700F496C7E6E7ED803C07F00076D7E39 1FE003FC3AFFF007FFC0A2221D7F9C25>I<3AFFFC01FFC0A23A0FE0007E000007147C15 38000314306D137000011460A26C6C5BA2EBFC01017C5BEB7E03013E90C7FCA2EB1F06A2 148EEB0F8CA2EB07D8A2EB03F0A36D5AA26D5AA2495AA2130391C8FC1278EAFC06A25B13 1CEA7838EA7070EA3FE0EA0F80222B7F9C25>I<003FB51280A2EB003F003C1400003813 7E00305BEA700100605B495A495A130F00005B495A49C7FC5B137E9038FC0180EA01F812 0313F03807E003EA0FC0001F1400138048485A007E5B00FE133FB6FCA2191D7E9C1F>I E %EndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: A4 %%EndSetup %%Page: 1 1 1 0 bop -250 -22 a Fr(Septem)n(b)r(er)24 b(18,)f(2003)72 b(17:26)g(WSPC/T)-6 b(rim)22 b(Size:)31 b(10in)24 b(x)g(7in)g(for)f (Pro)r(ceedings)637 b(ICMP-ws-pro)r(cs10x7)p -300 -5 3 200 v -557 55 200 3 v 3957 55 V 3898 -5 3 200 v -300 6314 V -557 6055 200 3 v 3898 6314 3 200 v 3960 6055 200 3 v 289 1174 a Fq(TR)m(OTTER-KA)-8 b(TO)32 b(PR)m(ODUCT)h(F)m (ORMULA:)f(SOME)f(RECENT)g(RESUL)-8 b(TS)1469 1514 y Fp(V.)25 b(A.)h(ZA)n(GREBNO)n(V)1009 1631 y Fo(Universit)n(\023)-37 b(e)29 b(de)f(la)f(M)n(\023)-37 b(editerr)l(an)n(\023)g(ee)30 b(\(A)n(ix-Marseil)t(le)e(II\))1762 1722 y(and)796 1813 y(Centr)l(e)g(de)g(Physique)h(Th)n(\023)-37 b(eorique)29 b(-)e(CNRS)h(-)g(Luminy)f(-)h(Case)g(907,)1275 1905 y(13288)g(Marseil)t (le)g(Ce)l(dex)g(9,)f(F)-6 b(r)l(anc)l(e)1240 1996 y(E-mail:)35 b(zagr)l(ebnov@cpt.univ-mrs.fr)562 2246 y Fr(Recen)n(tly)23 b(new)e(results)g(ab)r(out)h(the)g(op)r(erator-norm)e(and)h(the)h (trace-norm)e(con)n(v)n(ergences)k(of)562 2329 y(the)30 b(T)-6 b(rotter-Kato)30 b(pro)r(duct)g(form)n(ulae)e(with)h(ultimate)g (optimal)f(error-b)r(ound)g(estimates)562 2412 y(ha)n(v)n(e)18 b(b)r(een)f(established.)30 b(They)17 b(concern)g(the)g(self-adjoin)n (t)f(strongly)g(con)n(tin)n(uous)i(and)f(Gibbs)562 2495 y(semigroups)25 b(on)i(Hilb)r(ert)f(spaces.)39 b(Explicit)27 b(dep)r(endence)h(of)e(the)i(rate)e(of)g(con)n(v)n(ergence)j(on)562 2578 y(the)c(generators)g(domains)d(is)h(found.)172 2896 y Fq(1.)47 b(Op)s(erator-Norm)31 b(Con)m(v)m(ergence)h(and)g(Optimal)e (Error)i(Bound)172 3045 y(1.1.)48 b Fn(Op)-5 b(er)g(ator-Norm)35 b(Conver)-5 b(genc)g(e)172 3195 y Fm(Consider)27 b(real-v)-5 b(alued,)26 b(Borel)h(measurable)f(functions)i Fl(f)36 b Fm(on)28 b([0)p Fl(;)14 b Fk(1)p Fm(\))27 b(satisfying)1110 3378 y(0)c Fk(\024)g Fl(f)9 b Fm(\()p Fl(s)p Fm(\))23 b Fk(\024)f Fm(1)p Fl(;)97 b(f)9 b Fm(\(0\))23 b(=)f(1)p Fl(;)97 b(f)2166 3344 y Fj(0)2189 3378 y Fm(\(0\))23 b(=)f Fk(\000)p Fm(1)p Fl(:)831 b Fm(\(1\))172 3561 y(Some)28 b(examples)f(of)g(functions)h(satisfying)f(\(1\))h(are)1028 3745 y Fl(f)9 b Fm(\()p Fl(s)p Fm(\))23 b(=)g Fl(e)1331 3710 y Fj(\000)p Fi(s)1418 3745 y Fl(;)97 b(f)9 b Fm(\()p Fl(s)p Fm(\))23 b(=)g(\(1)18 b(+)g Fl(k)2023 3710 y Fj(\000)p Fh(1)2112 3745 y Fl(s)p Fm(\))2183 3710 y Fj(\000)p Fi(k)2276 3745 y Fl(;)97 b(k)26 b(>)c Fm(0)p Fl(:)749 b Fm(\(2\))297 3928 y(W)-7 b(e)27 b(are)f(in)n(terested)g(in)h(those)f(functions)h Fl(f)36 b Fm(whic)n(h)26 b(satisfy)h(not)g(only)f(\(1\))h(but)g(also)f (that)h(for)f(ev)n(ery)172 4036 y(small)i Fl(")22 b(>)h Fm(0)k(there)h(exists)f(a)g(p)r(ositiv)n(e)g(constan)n(t)g Fl(\016)f Fm(=)d Fl(\016)s Fm(\()p Fl(")p Fm(\))g Fl(<)g Fm(1)k(suc)n(h)g(that)1382 4219 y Fl(f)9 b Fm(\()p Fl(s)p Fm(\))23 b Fk(\024)g Fm(1)17 b Fk(\000)h Fl(\016)s Fm(\()p Fl(")p Fm(\))p Fl(;)98 b(s)23 b Fk(\025)f Fl(";)1103 b Fm(\(3\))172 4402 y(and)28 b(that)g(for)f(some)g(\014xed)g(constan)n (t)g Fl(\024)h Fm(with)g(1)23 b Fl(<)f(\024)h Fk(\024)g Fm(2,)1257 4620 y([)p Fl(f)9 b Fm(])1353 4632 y Fi(\024)1419 4620 y Fm(:=)23 b(sup)1534 4688 y Fi(s>)p Fh(0)1678 4564 y Fk(j)p Fl(f)9 b Fm(\()p Fl(s)p Fm(\))19 b Fk(\000)f Fm(1)g(+)g Fl(s)p Fk(j)p 1678 4601 483 4 v 1879 4677 a Fl(s)1918 4653 y Fi(\024)2194 4620 y Fl(<)23 b Fk(1)p Fl(:)978 b Fm(\(4\))172 4846 y(A)42 b(function)f Fl(f)9 b Fm(\()p Fl(s)p Fm(\))42 b(satisfying)e(\(1\))h(has)f(prop)r(ert)n(y)g (\(3\),)45 b(if)c(it)h(is)f(non-increasing.)75 b(Of)41 b(course,)i(the)172 4954 y(functions)33 b(in)f(\(2\))g(ha)n(v)n(e)f (prop)r(erties)g(\(3\))i(and)e(\(4\).)51 b(Condition)32 b(\(3\))g(is)g(necessary)-7 b(.)49 b(F)-7 b(or)31 b(this)h(accoun)n(t) 172 5062 y(and)c(some)f(further)g(remarks)f(on)i(conditions)f(\(3\))g (and)h(\(4\))g(w)n(e)f(refer)g(to)g([1])h(and)f([2].)297 5170 y(The)h(follo)n(wing)e(theorem)h(is)h(pro)n(v)n(ed)e(b)n(y)h(Ic)n (hinose-T)-7 b(am)n(ura-T)g(am)n(ura-Zagrebno)m(v)21 b(in)28 b([2].)172 5364 y Fq(Theorem)e(1.1.)41 b Fg(L)l(et)25 b Fl(f)35 b Fg(and)26 b Fl(g)i Fg(b)l(e)e(functions)g(having)h(pr)l(op) l(erties)g(\(3\))f(and)h(\(4\))f(with)g Fl(\024)d Fm(=)g(2)i Fg(as)h(wel)t(l)h(as)172 5472 y(\(1\).)44 b(If)31 b Fl(A)h Fg(and)g Fl(B)j Fg(ar)l(e)d(non-ne)l(gative)f(self-adjoint)i(op)l(er)l (ators)f(in)f(a)h(Hilb)l(ert)f(sp)l(ac)l(e)h Fk(H)g Fg(with)g(domains) 172 5580 y Fl(dom)p Fm(\()p Fl(A)p Fm(\))40 b Fg(and)e Fl(dom)p Fm(\()p Fl(B)t Fm(\))i Fg(such)e(that)g(the)g(op)l(er)l(ator)h (sum)f Fl(C)44 b Fm(:=)38 b Fl(A)25 b Fm(+)f Fl(B)42 b Fg(is)d(self-adjoint)h(on)e(domain)1805 5781 y Fr(1)p eop %%Page: 2 2 2 1 bop -250 -22 a Fr(Septem)n(b)r(er)24 b(18,)f(2003)72 b(17:26)g(WSPC/T)-6 b(rim)22 b(Size:)31 b(10in)24 b(x)g(7in)g(for)f (Pro)r(ceedings)637 b(ICMP-ws-pro)r(cs10x7)p -300 -5 3 200 v -557 55 200 3 v 3957 55 V 3898 -5 3 200 v -300 6314 V -557 6055 200 3 v 3898 6314 3 200 v 3960 6055 200 3 v 172 636 a(2)172 856 y Fl(dom)p Fm(\()p Fl(C)6 b Fm(\))25 b(=)d Fl(dom)p Fm(\()p Fl(A)p Fm(\))d Fk(\\)f Fl(dom)p Fm(\()p Fl(B)t Fm(\))p Fg(,)31 b(then)e(the)h(T)-6 b(r)l(otter-Kato)29 b(pr)l(o)l(duct)g(formulae)i(c)l(onver)l(ge)f(in)f (op)l(er)l(ator)172 964 y(norm)h(with)g(the)g(err)l(or)h(b)l(ound)e Fl(O)r Fm(\()p Fl(n)1310 934 y Fj(\000)p Fh(1)1400 964 y Fm(\))h Fg(:)929 1134 y Fk(k)p Fm([)p Fl(g)s Fm(\()p Fl(tB)t(=)p Fm(2)p Fl(n)p Fm(\))p Fl(f)9 b Fm(\()p Fl(tA=n)p Fm(\))p Fl(g)s Fm(\()p Fl(tB)t(=)p Fm(2)p Fl(n)p Fm(\)])1991 1100 y Fi(n)2053 1134 y Fk(\000)18 b Fl(e)2175 1100 y Fj(\000)p Fi(tC)2307 1134 y Fk(k)23 b Fm(=)f Fl(O)r Fm(\()p Fl(n)2606 1100 y Fj(\000)p Fh(1)2696 1134 y Fm(\))p Fl(;)929 1267 y Fk(k)p Fm([)p Fl(f)9 b Fm(\()p Fl(tA=n)p Fm(\))p Fl(g)s Fm(\()p Fl(tB)t(=n)p Fm(\)])1611 1233 y Fi(n)1674 1267 y Fk(\000)18 b Fl(e)1796 1233 y Fj(\000)p Fi(tC)1928 1267 y Fk(k)23 b Fm(=)g Fl(O)r Fm(\()p Fl(n)2228 1233 y Fj(\000)p Fh(1)2318 1267 y Fm(\))p Fl(;)99 b(n)22 b Fk(!)i(1)p Fl(:)609 b Fm(\(5\))172 1437 y Fg(The)25 b(c)l(onver)l(genc) l(e)e(is)h(uniform)g(on)f(e)l(ach)h(c)l(omp)l(act)g Fl(t)p Fg(-interval)g(in)f(the)h(half-line)h Fm([0)p Fl(;)14 b Fk(1)p Fm(\))p Fg(.)36 b(If)24 b(the)f(op)l(er)l(ator)172 1545 y Fl(C)39 b Fg(is)32 b(strictly)g(p)l(ositive,)i(i.e.)46 b Fl(C)34 b Fk(\025)26 b Fl(\021)s(I)39 b Fg(for)33 b(some)f(c)l (onstant)f Fl(\021)g(>)26 b Fm(0)p Fg(,)32 b(the)h(c)l(onver)l(genc)l (e)f(is)g(uniform)g(on)172 1653 y(the)e(whole)h(half-line)h Fm([0)p Fl(;)14 b Fk(1)p Fm(\))p Fg(.)297 1837 y Fm(T)-7 b(aking)41 b(in)g(\(2\))h(the)g(exp)r(onen)n(tial)f(functions)h Fl(f)9 b Fm(\()p Fl(s)p Fm(\))46 b(=)g Fl(g)s Fm(\()p Fl(s)p Fm(\))g(=)g Fl(e)2486 1807 y Fj(\000)p Fi(s)2573 1837 y Fm(,)f(one)d(gets)f(the)h(follo)n(wing)172 1945 y(op)r(erator-norm)25 b(con)n(v)n(erging)g Fg(symmetric)33 b Fm(and)28 b Fg(nonsymmetric)k Fm(T)-7 b(rotter)27 b(form)n(ulae)g(:) 172 2129 y Fq(Corollary)j(1.1.)41 b Fg(F)-6 b(or)28 b(non-ne)l(gative)h (self-adjoint)h(op)l(er)l(ators)f Fl(A)f Fg(and)h Fl(B)j Fg(whose)d(op)l(er)l(ator)g(sum)e Fl(C)j Fm(:=)172 2237 y Fl(A)19 b Fm(+)f Fl(B)34 b Fg(is)c(self-adjoint)i(on)e Fl(dom)p Fm(\()p Fl(C)6 b Fm(\))24 b(=)f Fl(dom)p Fm(\()p Fl(A)p Fm(\))c Fk(\\)g Fl(dom)p Fm(\()p Fl(B)t Fm(\))31 b Fg(we)f(have)975 2407 y Fk(k)p Fm(\()p Fl(e)1088 2373 y Fj(\000)p Fi(tB)s(=)p Fh(2)p Fi(n)1330 2407 y Fl(e)1369 2373 y Fj(\000)p Fi(tA=n)1574 2407 y Fl(e)1613 2373 y Fj(\000)p Fi(tB)s(=)p Fh(2)p Fi(n)1855 2407 y Fm(\))1887 2373 y Fi(n)1951 2407 y Fk(\000)18 b Fl(e)2073 2373 y Fj(\000)p Fi(tC)2206 2407 y Fk(k)k Fm(=)h Fl(O)r Fm(\()p Fl(n)2505 2373 y Fj(\000)p Fh(1)2595 2407 y Fm(\))p Fl(;)975 2540 y Fk(k)p Fm(\()p Fl(e)1088 2506 y Fj(\000)p Fi(tA=n)1293 2540 y Fl(e)1332 2506 y Fj(\000)p Fi(tB)s(=n)1541 2540 y Fm(\))1573 2506 y Fi(n)1637 2540 y Fk(\000)18 b Fl(e)1759 2506 y Fj(\000)p Fi(tC)1892 2540 y Fk(k)k Fm(=)h Fl(O)r Fm(\()p Fl(n)2191 2506 y Fj(\000)p Fh(1)2281 2540 y Fm(\))p Fl(;)99 b(n)23 b Fk(!)g(1)p Fl(;)646 b Fm(\(6\))172 2710 y Fg(uniformly)37 b(on)e(e)l(ach)h(c)l(omp)l(act)f Fl(t)p Fg(-interval)h(in)f Fm([0)p Fl(;)14 b Fk(1)p Fm(\))p Fg(,)37 b(and)e(uniformly)h(on)g Fm([0)p Fl(;)14 b Fk(1)p Fm(\))35 b Fg(,)h(if)g Fl(C)42 b Fg(is)35 b(strictly)172 2818 y(p)l(ositive.)297 3002 y Fm(In)29 b([2])g(it)g(is)g(also)f(sho)n (wn)g(that)h(Theorem)f(1.1)g(allo)n(ws)g(a)g(generalization)f(to)i(the) h(case,)e(when)h Fl(C)j Fm(:=)172 3110 y Fl(A)234 3122 y Fh(1)290 3110 y Fm(+)18 b Fk(\001)c(\001)g(\001)19 b Fm(+)f Fl(A)634 3122 y Fi(m)725 3110 y Fm(is)27 b(self-adjoin)n(t)g (on)h Fl(dom)p Fm(\()p Fl(C)6 b Fm(\))24 b(=)f Fl(dom)p Fm(\()p Fl(A)1998 3122 y Fh(1)2036 3110 y Fm(\))18 b Fk(\\)33 b(\001)14 b(\001)g(\001)32 b(\\)19 b Fl(dom)p Fm(\()p Fl(A)2627 3122 y Fi(m)2691 3110 y Fm(\).)172 3377 y Fq(1.2.)48 b Fn(Optimality)33 b(of)i(the)g(Err)-5 b(or)33 b(Bound)172 3526 y Fm(Notice)28 b(that)g(the)g(results)f(of)h (the)f(pap)r(er)h([2)o(])g(are)f(w)n(onderfully)f(optimal)i(in)g(man)n (y)f(resp)r(ects.)172 3710 y Fq(Remark)36 b(1.1.)k Fm(One)31 b(can)g Fg(not)39 b Fm(relax)30 b(the)i(self-adjoin)n(tness)e(of)h(the) h(sum)g Fl(C)j Fm(:=)29 b Fl(A)21 b Fm(+)g Fl(B)36 b Fm(to)31 b Fg(essential)172 3818 y Fm(self-adjoin)n(tness,)23 b(since)e(in)i(this)f(case)f(there)h(is)g(an)f(example)h(of)g(a)f (couple)h(of)g(non-negativ)n(e)e(self-adjoin)n(t)172 3926 y(op)r(erators)26 b Fl(A)i Fm(and)f Fl(B)32 b Fm(suc)n(h)27 b(that)1072 4096 y(lim)14 b(inf)1100 4146 y Fi(n)p Fj(!1)1316 4096 y Fk(k)p Fm(\()p Fl(e)1429 4062 y Fj(\000)p Fi(tA=n)1634 4096 y Fl(e)1673 4062 y Fj(\000)p Fi(tB)s(=n)1882 4096 y Fm(\))1914 4062 y Fi(n)1978 4096 y Fk(\000)k Fl(e)2100 4062 y Fj(\000)p Fi(tC)2232 4096 y Fk(k)23 b(\025)g Fl(D)r Fm(\()p Fl(t)p Fm(\))p Fl(;)793 b Fm(\(7\))172 4287 y(where)28 b Fl(D)r Fm(\()p Fl(t)p Fm(\))g(is)f(p)r(ositiv)n(e)g(and)h(con)n(tin)n (uous)e(function)j(of)e Fl(t)c(>)g Fm(0.)36 b(This)28 b(example)f(is)h(due)f(to)h([3)o(].)172 4471 y(Notice)f(that)f(in)g (this)h(case)e(w)n(e)h(still)g(ha)n(v)n(e)f(the)i Fg(str)l(ong)33 b Fm(con)n(v)n(ergence)24 b(of)i(the)g(T)-7 b(rotter)25 b(pro)r(duct)h(form)n(ula)172 4579 y(\(see,)i(e.g.)36 b([4]\):)1263 4759 y(lim)1234 4809 y Fi(n)p Fj(!1)1408 4759 y Fm(\()p Fl(e)1479 4724 y Fj(\000)p Fi(tA=n)1685 4759 y Fl(e)1724 4724 y Fj(\000)p Fi(tB)s(=n)1933 4759 y Fm(\))1965 4724 y Fi(n)2010 4759 y Fl(u)23 b Fm(=)f Fl(e)2207 4724 y Fj(\000)p Fi(t)2299 4709 y Fh(~)2284 4724 y Fi(C)2340 4759 y Fl(u;)955 b Fm(\(8\))172 4964 y(for)36 b(an)n(y)e Fl(u)i Fk(2)h(H)q Fm(.)61 b(Here)1026 4943 y(~)1007 4964 y Fl(C)42 b Fm(is)36 b(the)g(closure)e(of)i(op)r (erator)e Fl(C)43 b Fm(:=)36 b Fl(A)24 b Fm(+)f Fl(B)40 b Fm(with)c(domain)f Fl(dom)p Fm(\()p Fl(C)6 b Fm(\))38 b(=)172 5072 y Fl(dom)p Fm(\()p Fl(A)p Fm(\))20 b Fk(\\)f Fl(dom)p Fm(\()p Fl(B)t Fm(\).)172 5256 y Fq(Remark)42 b(1.2.)e Fm(Let)c(op)r(erators)f Fl(A)i Fm(and)f Fl(B)41 b Fm(v)n(erify)36 b(the)h(conditions)f(of)g(Theorem)g(1.1.)63 b(If)37 b(they)g(do)172 5364 y(not)e(comm)n(ute,)i(then)e(it)h(is)f (easy)f(to)g(c)n(hec)n(k)g(that)i(in)f(the)g Fg(nonsymmetric)40 b Fm(case)34 b(\(see)h(\(6\)\))g(the)g(error)172 5472 y(b)r(ound)d(can)f(not)g(b)r(e)h(impro)n(v)n(ed.)47 b(Hence,)32 b(in)f(this)h(case)e(the)i(estimate)f Fl(O)r Fm(\()p Fl(n)2611 5441 y Fj(\000)p Fh(1)2701 5472 y Fm(\))h(is)f Fg(ultimate)i(optimal)p Fm(.)172 5580 y(It)c(is)e(m)n(uc)n(h)h(less)f (eviden)n(t)h(for)g(case)f(of)h(symmetric)f(T)-7 b(rotter)27 b(pro)r(duct)h(form)n(ula)f(\(see)h(\(6\)\).)38 b(One)27 b(of)h(the)p eop %%Page: 3 3 3 2 bop -250 -22 a Fr(Septem)n(b)r(er)24 b(18,)f(2003)72 b(17:26)g(WSPC/T)-6 b(rim)22 b(Size:)31 b(10in)24 b(x)g(7in)g(for)f (Pro)r(ceedings)637 b(ICMP-ws-pro)r(cs10x7)p -300 -5 3 200 v -557 55 200 3 v 3957 55 V 3898 -5 3 200 v -300 6314 V -557 6055 200 3 v 3898 6314 3 200 v 3960 6055 200 3 v 3437 636 a(3)172 856 y Fm(imp)r(ortan)n(t)29 b(result)h(of)f(the)h(pap)r(er)f([2)o(])h(is)f(the)h(construction)e(of) i Fg(a)h(c)l(ouple)37 b Fm(of)29 b Fg(unb)l(ounde)l(d)38 b Fm(op)r(erators)27 b Fl(A)172 964 y Fm(and)h Fl(B)k Fm(v)n(erifying)26 b(conditions)h(of)h(Corollary)d(1.1)i(suc)n(h)g (that)974 1134 y Fk(k)p Fm(\()p Fl(e)1087 1100 y Fj(\000)p Fi(tB)s(=)p Fh(2)p Fi(n)1329 1134 y Fl(e)1368 1100 y Fj(\000)p Fi(tA=n)1574 1134 y Fl(e)1613 1100 y Fj(\000)p Fi(tB)s(=)p Fh(2)p Fi(n)1855 1134 y Fm(\))1887 1100 y Fi(n)1951 1134 y Fk(\000)18 b Fl(e)2073 1100 y Fj(\000)p Fi(tC)2205 1134 y Fk(k)23 b(\025)f Fl(L)p Fm(\()p Fl(t)p Fm(\))p Fl(n)2558 1100 y Fj(\000)p Fh(1)2647 1134 y Fl(;)696 b Fm(\(9\))172 1305 y(where)27 b Fl(L)p Fm(\()p Fl(t)p Fm(\))g(is)g(p)r(ositiv)n(e)f(and)h(con)n(tin)n(uous)f(function)h(of)g Fl(t)c(>)g Fm(0.)36 b(This)27 b(pro)n(v)n(es)e(optimalit)n(y)h(of)h (the)g(error)172 1412 y(b)r(ound)h Fl(O)r Fm(\()p Fl(n)575 1382 y Fj(\000)p Fh(1)665 1412 y Fm(\))g(in)g(Theorem)f(1.1)g(\(and)g (Corollary)f(1.1\))h(in)g(the)h Fg(symmetric)33 b Fm(case.)172 1596 y Fq(Remark)k(1.3.)j Fm(W)-7 b(e)34 b(w)n(ould)e(lik)n(e)g(to)h(w) n(arn)f(the)h(reader)e(against)h(the)h(conclusion)f(that)h(for)g Fg(any)40 b Fm(t)n(w)n(o)172 1704 y(noncomm)n(utativ)n(e)18 b(op)r(erators)f Fl(A)i Fm(and)f Fl(B)23 b Fm(v)n(erifying)18 b(conditions)g(of)g(Theorem)g(1.1)g(the)h(error)e(b)r(ound)i(esti-)172 1812 y(mate)k Fl(O)r Fm(\()p Fl(n)522 1782 y Fj(\000)p Fh(1)612 1812 y Fm(\))g(is)g(the)g(b)r(est)g(p)r(ossible)g(for)f(the)h Fg(symmetric)28 b Fm(T)-7 b(rotter-Kato)20 b(pro)r(duct)j(form)n(ula.) 34 b(Whereas)172 1920 y(for)28 b Fg(nonsymmetric)k Fm(case)27 b(it)h(is)f(true.)172 2104 y Fq(Remark)i(1.4.)40 b Fm(The)26 b(pap)r(er)g([2)o(])g(co)n(v)n(ers)e(the)i(most)g(of)f(the)h(kno)n(wn)g (results)f(concerning)f(the)j(op)r(erator-)172 2212 y(norm)k(con)n(v)n (ergence)e(of)i(the)g(T)-7 b(rotter-Kato)29 b(pro)r(duct)i(form)n(ula)f (for)h(self-adjoin)n(t)g(semigroups.)46 b(More-)172 2320 y(o)n(v)n(er,)23 b(this)i(pap)r(er)e(pro)n(v)n(es)g(that)h Fl(O)r Fm(\()p Fl(n)1335 2290 y Fj(\000)p Fh(1)1425 2320 y Fm(\))g(is)g(optimal)g(error)e(b)r(ound)j(estimate)f(in)g(the)h(ab)r (o)n(v)n(e)d(sense.)36 b(See)172 2428 y(discussion)27 b(in)h([2].)172 2694 y Fq(2.)47 b(Op)s(erator-Norm)31 b(Con)m(v)m(ergence)h(and)g(F)-8 b(ractional)33 b(P)m(o)m(w)m(er)g(of)e (Self-Adjoin)m(t)294 2802 y(Generators)172 2952 y(2.1.)48 b Fn(F)-7 b(r)i(actional)34 b(Power)g(and)h(Conver)-5 b(genc)g(e)35 b(R)-5 b(ate)172 3101 y Fm(The)38 b(results)g(of)f([2])h (sho)n(w)f(ho)n(w)g(decisiv)n(e)g(is)h(the)g(relation)f(b)r(et)n(w)n (een)h(op)r(erators)e Fl(A)i Fm(and)g Fl(B)k Fm(for)37 b(the)172 3209 y(con)n(v)n(ergence)d(of)h(the)h(pro)r(duct)g(form)n (ula.)59 b(A)36 b(recen)n(t)f(Ic)n(hinose-Neidhardt-Zagrebno)n(v)d (theorem)j([5)o(])172 3317 y(extends)k([6)o(],)j(where)c(the)h(problem) f(of)g(relation)g(b)r(et)n(w)n(een)g(domains)g(of)g(self-adjoin)n(t)g (non-negativ)n(e)172 3425 y(op)r(erators)c Fl(A)j Fm(and)f Fl(B)k Fm(and)c(the)g(b)r(ound)g(on)g(the)g(con)n(v)n(ergence)e(rate)h (in)i(the)f(T)-7 b(rotter-Kato)34 b(pro)r(duct)172 3533 y(form)n(ula)27 b(w)n(as)g(considered)f(for)h(the)h(\014rst)g(time.)297 3641 y(Let)h Fl(A)g Fm(and)f Fl(B)33 b Fm(b)r(e)c(t)n(w)n(o)f (non-negativ)n(e)f(self-adjoin)n(t)h(op)r(erators)e(in)j(a)f(Hilb)r (ert)i(space)d Fk(H)j Fm(suc)n(h)e(that)172 3749 y Fl(dom)p Fm(\()p Fl(A)422 3719 y Fh(1)p Fi(=)p Fh(2)528 3749 y Fm(\))18 b Fk(\\)h Fl(dom)p Fm(\()p Fl(B)907 3719 y Fh(1)p Fi(=)p Fh(2)1012 3749 y Fm(\))28 b(is)g(dense)f(in)h Fk(H)q Fm(.)37 b(By)27 b Fl(H)35 b Fm(w)n(e)27 b(denote)g(the)h (form-sum)f(of)h Fl(A)g Fm(and)f Fl(B)t Fm(,)h(i.e.)1609 3941 y Fl(H)i Fm(=)23 b Fl(A)1903 3876 y Fj(\001)1881 3941 y Fm(+)g Fl(B)172 4111 y Fm(whic)n(h)38 b(is)g(in)g(turn)g(a)g (non-negativ)n(e)e(self-adjoin)n(t)h(op)r(erator)g(in)h Fk(H)q Fm(.)68 b(By)37 b(de\014nition)i Fl(dom)p Fm(\()p Fl(H)3231 4081 y Fh(1)p Fi(=)p Fh(2)3335 4111 y Fm(\))i(=)172 4219 y Fl(dom)p Fm(\()p Fl(A)422 4189 y Fh(1)p Fi(=)p Fh(2)528 4219 y Fm(\))18 b Fk(\\)h Fl(dom)p Fm(\()p Fl(B)907 4189 y Fh(1)p Fi(=)p Fh(2)1012 4219 y Fm(\).)38 b(In)27 b([5])h(w)n(e)f(assume)g(that)1165 4389 y Fl(dom)p Fm(\()p Fl(H)1429 4355 y Fi(\013)1477 4389 y Fm(\))c Fk(\022)g Fm(\()p Fl(dom)p Fm(\()p Fl(A)1902 4355 y Fi(\013)1950 4389 y Fm(\))c Fk(\\)g Fl(dom)p Fm(\()p Fl(B)2330 4355 y Fi(\013)2378 4389 y Fm(\)\))c Fl(;)845 b Fm(\(10\))172 4559 y(for)43 b(some)g Fl(\013)50 b Fk(2)g Fm(\(1)p Fl(=)p Fm(2)p Fl(;)14 b Fm(1\).)83 b(This)43 b(assumption)g(is)g Fg(str)l(onger)53 b Fm(than)43 b Fl(dom)p Fm(\()p Fl(H)2700 4529 y Fh(1)p Fi(=)p Fh(2)2805 4559 y Fm(\))50 b Fk(\022)f Fl(dom)p Fm(\()p Fl(A)3251 4529 y Fh(1)p Fi(=)p Fh(2)3356 4559 y Fm(\))29 b Fk(\\)172 4667 y Fl(dom)p Fm(\()p Fl(B)427 4637 y Fh(1)p Fi(=)p Fh(2)533 4667 y Fm(\),)43 b(but)e Fg(we)l(aker)51 b Fm(than)40 b(the)h(assumption)e Fl(dom)p Fm(\()p Fl(H)7 b Fm(\))45 b Fk(\022)f Fl(dom)p Fm(\()p Fl(A)p Fm(\))28 b Fk(\\)g Fl(dom)p Fm(\()p Fl(B)t Fm(\))41 b(used)f(in)h([1)o(])172 4775 y(and)c([2].)64 b(Notice)37 b(that)g(comparing)e(to)i(our)f(pap)r(er)g([6],)j(in)e([5])g(w)n(e)f (do)h(not)f(demand)h Fg(smal)t(lness)45 b Fm(of)172 4883 y(the)29 b(op)r(erator)d Fl(B)718 4853 y Fi(\013)794 4883 y Fm(with)i(resp)r(ect)g(to)g Fl(A)1432 4853 y Fi(\013)1480 4883 y Fm(.)38 b(W)-7 b(e)29 b(assume)e(that)h(the)h(Kato)e(functions)h (further)g(ful\014ll)h(the)172 4991 y(conditions)f(similar)e(to)i (those)f(of)h([2)o(]:)981 5201 y([)p Fl(f)9 b Fm(])1077 5213 y Fh(2)p Fi(\013)1180 5201 y Fm(:=)23 b(sup)1295 5268 y Fi(s>)p Fh(0)1440 5145 y Fk(j)p Fl(f)9 b Fm(\()p Fl(s)p Fm(\))18 b Fk(\000)g Fm(1)g(+)g Fl(s)p Fk(j)p 1440 5182 483 4 v 1621 5258 a Fl(s)1660 5234 y Fh(2)p Fi(\013)1955 5201 y Fl(<)23 b Fm(+)p Fk(1)p Fl(;)97 b(\013)23 b Fk(2)g Fm(\(0)p Fl(;)14 b Fm(1])p Fl(;)661 b Fm(\(11\))988 5533 y([)p Fl(g)s Fm(])1077 5545 y Fh(2)p Fi(\013)1180 5533 y Fm(:=)23 b(sup)1295 5600 y Fi(s>)p Fh(0)1440 5476 y Fk(j)p Fl(g)s Fm(\()p Fl(s)p Fm(\))18 b Fk(\000)g Fm(1)g(+)g Fl(s)p Fk(j)p 1440 5514 476 4 v 1618 5590 a Fl(s)1657 5566 y Fh(2)p Fi(\013)1948 5533 y Fl(<)23 b Fm(+)p Fk(1)p Fl(;)97 b(\013)23 b Fk(2)g Fm(\(0)p Fl(;)14 b Fm(1])p Fl(:)668 b Fm(\(12\))p eop %%Page: 4 4 4 3 bop -250 -22 a Fr(Septem)n(b)r(er)24 b(18,)f(2003)72 b(17:26)g(WSPC/T)-6 b(rim)22 b(Size:)31 b(10in)24 b(x)g(7in)g(for)f (Pro)r(ceedings)637 b(ICMP-ws-pro)r(cs10x7)p -300 -5 3 200 v -557 55 200 3 v 3957 55 V 3898 -5 3 200 v -300 6314 V -557 6055 200 3 v 3898 6314 3 200 v 3960 6055 200 3 v 172 636 a(4)172 856 y Fm(It)31 b(is)f(easily)f(seen)h(that)g (\(11\))g(and)f(\(12\))h(are)f(in)h(fact)h(conditions)e(at)h(the)g (neigh)n(b)r(orho)r(o)r(d)f(of)h(zero.)43 b(W)-7 b(e)172 964 y(set)1260 1129 y Fl(m)1333 1141 y Fi(f)1376 1129 y Fm(\()p Fl(x)p Fm(\))24 b(:=)85 b(sup)1622 1203 y Fi(y)r Fj(2)p Fh([)p Fi(x;)p Fj(1)p Fh(\))1884 1129 y Fl(f)9 b Fm(\()p Fl(y)s Fm(\))p Fl(;)97 b(x)24 b(>)e Fm(0)p Fl(:)172 1353 y Fm(Notice)34 b(that)h(the)f(examples)f(\(2\))h(satisfy) g(condition)g(\(11\))f(and)h Fl(m)2338 1365 y Fi(f)2381 1353 y Fm(\()p Fl(x)p Fm(\))h Fl(<)e Fm(1)h(for)f Fl(x)h(>)f Fm(0.)56 b(Then)34 b(the)172 1461 y(main)28 b(statemen)n(t)g(of)f(the)h ([5])f(is)h(the)g(follo)n(wing)172 1641 y Fq(Theorem)j(2.1.)41 b Fg(L)l(et)29 b Fl(A)h Fg(and)h Fl(B)j Fg(b)l(e)c(non-ne)l(gative)g (self-adjoint)h(op)l(er)l(ators)g(and)f(let)g Fl(H)g Fm(:=)23 b Fl(A)3238 1576 y Fi(:)3216 1641 y Fm(+)f Fl(B)34 b Fg(b)l(e)172 1749 y(their)c(form)h(sum.)38 b(Assume)28 b(that)i(for)g(some)g Fl(\013)23 b Fk(2)h Fm(\(1)p Fl(=)p Fm(2)p Fl(;)14 b Fm(1\))27 b Fg(the)j(c)l(ondition)h(\(10\))f(is)f (satis\014e)l(d.)39 b(F)-6 b(urther,)172 1857 y(let)32 b Fl(f)40 b Fg(and)33 b Fl(g)h Fg(b)l(e)e(Kato)g(functions)f(which)j (ob)l(ey)e(c)l(onditions)h(\(11\))f(and)h(\(12\).)45 b(If)32 b(in)g(addition)i(one)e(has)172 1965 y Fl(dom)p Fm(\()p Fl(A)422 1934 y Fh(1)p Fi(=)p Fh(2)528 1965 y Fm(\))26 b Fk(\022)f Fl(dom)p Fm(\()p Fl(B)931 1934 y Fh(1)p Fi(=)p Fh(2)1036 1965 y Fm(\))32 b Fg(and)g Fl(m)1336 1977 y Fi(f)1379 1965 y Fm(\()p Fl(x)p Fm(\))27 b Fl(<)e Fm(1)31 b Fg(for)h Fl(x)26 b(>)g Fm(0)p Fg(,)32 b(then)f(for)h(any)g (\014nite)e(interval)i Fm([0)p Fl(;)14 b(T)e Fm(])30 b Fg(ther)l(e)172 2072 y(is)h(a)f(c)l(onstant)f Fl(C)724 2084 y Fi(T)5 b(;)p Fh(2)p Fi(\013)p Fj(\000)p Fh(1)977 2072 y Fl(>)22 b Fm(0)30 b Fg(such)f(that)777 2201 y Ff(\015)777 2251 y(\015)823 2271 y Fm(\()p Fl(f)9 b Fm(\()p Fl(tA=)p Fm(2)p Fl(n)p Fm(\))p Fl(g)s Fm(\()p Fl(tB)t(=n)p Fm(\))p Fl(f)g Fm(\()p Fl(tA=)p Fm(2)p Fl(n)p Fm(\)\))1862 2230 y Fi(n)1925 2271 y Fk(\000)18 b Fl(e)2047 2237 y Fj(\000)p Fi(tH)2187 2201 y Ff(\015)2187 2251 y(\015)2256 2271 y Fk(\024)23 b Fl(C)2403 2283 y Fi(T)5 b(;)p Fh(2)p Fi(\013)p Fj(\000)p Fh(1)2730 2215 y Fm(1)p 2643 2252 216 4 v 2643 2328 a Fl(n)2693 2304 y Fh(2)p Fi(\013)p Fj(\000)p Fh(1)3325 2271 y Fm(\(13\))172 2456 y Fg(for)31 b Fl(t)23 b Fk(2)g Fm([0)p Fl(;)14 b(T)e Fm(])29 b Fg(and)h Fl(n)23 b Fm(=)g(1)p Fl(;)14 b Fm(2)p Fl(;)g(:)g(:)g(:)n Fg(.)172 2636 y Fm(The)32 b(estimate)f(\(13\))f(giv)n(es)g(the)i (ultimate)f(optimal)g(error)e(b)r(ound)j(for)e(the)i(con)n(v)n(ergence) c(rate,)k(whic)n(h)172 2744 y(can)g(b)r(e)g(seen)g(from)g(the)g (example)g(giv)n(en)f(in)h([3].)50 b(Theorem)31 b(2.1)h(impro)n(v)n(es) e(our)h(previous)g(result)h([6)o(],)172 2852 y(where)38 b(the)g(same)f Fg(optimal)48 b Fm(con)n(v)n(ergence)35 b(rate)i Fl(O)r Fm(\()p Fl(n)1907 2822 y Fj(\000)p Fh(\(2)p Fi(\013)p Fj(\000)p Fh(1\))2177 2852 y Fm(\))h(w)n(as)f(obtained)g(but) i(under)e(stronger)172 2960 y(conditions)29 b(on)g(the)g(op)r(erators)e Fl(A)i Fm(and)g Fl(B)k Fm(and)c(on)f(the)i(Kato)d(functions)j Fl(f)37 b Fm(and)29 b Fl(g)s Fm(.)40 b(Notice)29 b(also)f(that)172 3067 y(Theorem)33 b(2.1)f(treats)h(a)g(case)f(whic)n(h)h(is)g(not)g(co) n(v)n(ered)f(b)n(y)h([1)o(],)i([2].)54 b(Nev)n(ertheless,)33 b(setting)g(formally)172 3175 y Fl(\013)26 b Fm(=)f(1)k(in)g(Theorem)f (2.1)g(the)i(assumptions)e(and)h(results)f(of)h(that)g(theorem)g(turn)g (in)n(to)g(those)f(ones)h(of)172 3283 y([1],)35 b([2])e(except)h(that)f (assumption)g Fl(dom)p Fm(\()p Fl(A)1586 3253 y Fh(1)p Fi(=)p Fh(2)1691 3283 y Fm(\))g Fk(\022)g Fl(dom)p Fm(\()p Fl(B)2109 3253 y Fh(1)p Fi(=)p Fh(2)2214 3283 y Fm(\))h(b)r(ecomes)f Fg(sup)l(er\015uous)p Fm(.)53 b(In)34 b([7)o(])g(it)g(is)172 3391 y(conjectured)c(that)g(this)g(condition)g(is)g Fg(sup)l (er\015uous)36 b Fm(not)30 b(only)f(for)g Fl(\013)f Fm(=)e(1)k(but)g (also)f(for)g Fl(\013)f Fk(2)f Fm(\(1)p Fl(=)p Fm(2)p Fl(;)14 b Fm(1\).)172 3499 y(Ho)n(w)n(ev)n(er,)27 b(the)h(pro)r(of)g (of)g(this)g(conjecture)f(remains)h(still)g(an)g(op)r(en)g(problem.)37 b(Notice)28 b(that)h(condition)172 3607 y Fl(dom)p Fm(\()p Fl(A)422 3577 y Fh(1)p Fi(=)p Fh(2)528 3607 y Fm(\))d Fk(\022)g Fl(dom)p Fm(\()p Fl(B)932 3577 y Fh(1)p Fi(=)p Fh(2)1037 3607 y Fm(\))k(implies)g Fl(dom)p Fm(\()p Fl(H)1647 3577 y Fh(1)p Fi(=)p Fh(2)1751 3607 y Fm(\))d(=)f Fl(dom)p Fm(\()p Fl(A)2151 3577 y Fh(1)p Fi(=)p Fh(2)2256 3607 y Fm(\))k(whic)n(h)f(guaran)n(tees)e(the)j(densit)n(y)g(of)172 3715 y Fl(dom)p Fm(\()p Fl(H)436 3685 y Fh(1)p Fi(=)p Fh(2)541 3715 y Fm(\))f(in)g Fk(H)q Fm(.)39 b(The)29 b(pro)r(of)f(of)g(Theorem)g(2.1)f(relies)h(on)g(some)g(ideas)g(of)g (Cherno\013)h([8)o(],)g([9])f(and)h(of)172 3823 y(the)g (Neidhardt-Zagrebno)n(v)d(pap)r(er)i([10)o(].)39 b(In)29 b(man)n(y)f(asp)r(ects)g(it)h(follo)n(ws)e(the)i(line)g(of)f(reasoning) e(of)j([1)o(])172 3931 y(and)f([2)o(].)172 4111 y Fq(Remark)j(2.1.)315 4278 y Fm(\(i\))43 b(By)20 b(example)g(giv)n(en)f(in)i([3)o(])g(the)f (error)f(b)r(ound)h(estimate)h Fl(O)r Fm(\()p Fl(n)2346 4248 y Fj(\000)p Fh(\(2)p Fi(\013)p Fj(\000)p Fh(1\))2616 4278 y Fm(\))f(in)h(Theorem)e(2.1)h(cannot)445 4386 y(b)r(e)28 b(impro)n(v)n(ed,)e(i.e.)37 b(it)28 b(is)f(the)h Fg(ultimate)i(optimal) f Fm(one.)292 4494 y(\(ii\))43 b(This)25 b(optimal)h(error)e(b)r(ound)i (w)n(as)f(already)f(found)i(in)g([6)o(].)37 b(In)26 b([6)o(])g(it)g(w)n (as)f(assumed)g(that)h Fl(B)3343 4464 y Fi(\013)3417 4494 y Fm(is)445 4602 y Fg(smal)t(l)37 b Fm(with)28 b(resp)r(ect)f(to)h Fl(A)1309 4572 y Fi(\013)1356 4602 y Fm(,)g(i.e.)37 b Fl(dom)p Fm(\()p Fl(A)1800 4572 y Fi(\013)1848 4602 y Fm(\))24 b Fk(\022)e Fl(dom)p Fm(\()p Fl(B)2246 4572 y Fi(\013)2294 4602 y Fm(\))28 b(and)1194 4767 y Fk(k)p Fl(B)1303 4733 y Fi(\013)1350 4767 y Fl(u)p Fk(k)22 b(\024)g Fl(a)p Fk(k)p Fl(A)1697 4733 y Fi(\013)1744 4767 y Fl(u)p Fk(k)c Fm(+)g Fl(b)p Fk(k)p Fl(u)p Fk(k)p Fl(;)95 b(u)22 b Fk(2)i Fl(dom)p Fm(\()p Fl(A)2620 4733 y Fi(\013)2668 4767 y Fm(\))p Fl(;)602 b Fm(\(14\))445 4932 y(for)30 b(some)h Fl(\013)f Fk(2)f Fm(\(1)p Fl(=)p Fm(2)p Fl(;)14 b Fm(1\))30 b(and)h Fl(a)e Fk(2)h Fm(\(0)p Fl(;)14 b Fm(1\),)32 b Fl(b)d(>)f Fm(0.)48 b(This)31 b(condition)g(is)h(reluxed)e (in)i([5])f(to)g(the)445 5040 y(mild)19 b(sub)r(ordination)f(condition) h Fl(dom)p Fm(\()p Fl(A)1751 5010 y Fh(1)p Fi(=)p Fh(2)1856 5040 y Fm(\))24 b Fk(\022)e Fl(dom)p Fm(\()p Fl(B)2254 5010 y Fh(1)p Fi(=)p Fh(2)2360 5040 y Fm(\),)f(whic)n(h)e(is)g(ob)n (viously)e(implied)j(b)n(y)445 5148 y(\(14\).)36 b(The)26 b(y)n(et)g Fg(op)l(en)k(pr)l(oblem)k Fm(is)26 b(to)g(eliminate)h (completely)f(this)h(sub)r(ordination)f(condition.)269 5256 y(\(iii\))43 b(The)23 b(conditions)g(on)g(the)g(Kato)g(functions)g Fl(f)32 b Fm(and)23 b Fl(g)j Fm(in)e([5)o(])g(are)e(essen)n(tially)g (relaxed)h(compared)445 5364 y(to)k([6].)272 5472 y(\(iv\))42 b(Theorem)23 b(2.1)g(sho)n(ws)f(that)i(only)g(relations)f(b)r(et)n(w)n (een)g(certain)h Fg(domains)32 b Fm(related)23 b(to)h Fl(A)p Fm(,)h Fl(B)j Fm(and)445 5580 y Fl(H)34 b Fm(are)27 b(decisiv)n(e)f(for)i(the)g Fg(c)l(onver)l(genc)l(e)i(r)l(ate)d Fm(of)h(the)g(T)-7 b(rotter-Kato)25 b(pro)r(duct)j(form)n(ula.)p eop %%Page: 5 5 5 4 bop -250 -22 a Fr(Septem)n(b)r(er)24 b(18,)f(2003)72 b(17:26)g(WSPC/T)-6 b(rim)22 b(Size:)31 b(10in)24 b(x)g(7in)g(for)f (Pro)r(ceedings)637 b(ICMP-ws-pro)r(cs10x7)p -300 -5 3 200 v -557 55 200 3 v 3957 55 V 3898 -5 3 200 v -300 6314 V -557 6055 200 3 v 3898 6314 3 200 v 3960 6055 200 3 v 3437 636 a(5)295 856 y Fm(\(v\))42 b(Theorem)27 b(2.1)h(holds)g(for)f Fl(\013)e Fk(2)g Fm(\(1)p Fl(=)p Fm(2)p Fl(;)14 b Fm(1\).)37 b(The)29 b(metho)r(d)f(of)h(the)f(pro)r(of) g(do)r(es)g(not)h(allo)n(w)e(one)h(to)445 964 y(include)e(the)g(case)g Fl(\013)d Fm(=)g(1.)36 b(W)-7 b(e)26 b(considered)f(it)i(separately)d (in)j([2)o(].)37 b(It)26 b(is)g(remark)-5 b(able)25 b(that)h(one)445 1072 y(do)r(es)h(not)g(need)h(an)n(y)f Fg(sub)l(or)l(dination)35 b Fm(condition)27 b(in)h(this)g(case.)272 1180 y(\(vi\))42 b(F)-7 b(or)28 b Fl(\013)d Fm(=)f(1)p Fl(=)p Fm(2)j(w)n(e)i(cannot)f (exp)r(ect)h(op)r(erator-norm)d(con)n(v)n(ergence)g(in)j(general,)e (see)i([3)o(].)40 b(Ho)n(w-)445 1288 y(ev)n(er,)20 b(if)h(there)f(is)g (a)g Fg(sub)l(or)l(dination)28 b Fm(suc)n(h)20 b(as,)h(for)e(example,)j (op)r(erator)c Fl(B)25 b Fm(is)20 b(relativ)n(ely)f(compact)445 1396 y(with)28 b(resp)r(ect)f(to)g Fl(A)p Fm(,)h(then)h(op)r (erator-norm)24 b(con)n(v)n(ergence)i(holds,)h(see)g([10)o(].)172 1661 y Fq(2.2.)48 b Fn(Example)172 1811 y Fm(Let)35 b Fk(H)f Fm(:=)f Fl(L)610 1823 y Fh(2)647 1811 y Fm(\(\012\))h(where)g (\012)g(is)g(a)f(b)r(ounded)h(domain)g(in)g Fe(R)2115 1781 y Fh(1)2158 1811 y Fm(,)i Fl(l)f Fm(=)e(1)p Fl(;)14 b Fm(2)p Fl(;)g(:)g(:)g(:)o Fm(,)35 b(with)g(b)r(oundary)e Fl(@)5 b Fm(\012)34 b(of)172 1919 y Fl(C)237 1889 y Fj(1)308 1919 y Fm(-class.)k(By)28 b Fl(A)h Fm(w)n(e)f(denote)g(the)h(negativ)n (e)e(half)i(Laplacian)e(with)i(Diric)n(hlet)f(b)r(oundary)g(conditions) 172 2061 y(in)38 b Fl(L)336 2073 y Fh(2)372 2061 y Fm(\(\012\),)i(i.e)d Fl(A)i Fm(=)g Fk(\000)959 2028 y Fh(1)p 958 2042 34 4 v 958 2089 a(2)1001 2061 y Fm(\001)1070 2073 y Fi(D)1131 2061 y Fm(.)65 b(The)37 b(domain)f(is)h(giv)n(en)f(b)n(y)h Fl(dom)p Fm(\()p Fl(A)p Fm(\))j(:=)e Fl(H)2671 2031 y Fh(2)2664 2081 y(2)2708 2061 y Fm(\(\012\))25 b Fk(\\)2989 1977 y Fh(o)2960 2061 y Fl(H)3036 2031 y Fh(1)3029 2081 y(2)3073 2061 y Fm(\(\012\),)40 b(where)224 2131 y Fh(o)196 2215 y Fl(H)272 2185 y Fh(1)265 2236 y(2)332 2215 y Fm(\(\012\))28 b(is)f(the)h(closure)f(of)g Fl(C)1147 2185 y Fj(1)1141 2236 y Fh(0)1218 2215 y Fm(\(\012\))h(in)g(the)g(Leb)r(esgue)f(space)g Fl(H)2267 2185 y Fh(1)2260 2236 y(2)2304 2215 y Fm(\(\012\).)37 b(Using)28 b(the)g(space)f Fl(H)3163 2185 y Fh(2)3156 2238 y(2)p Fi(;B)3259 2246 y Fd(D)3314 2215 y Fm(\(\012\),)1080 2384 y Fl(H)1156 2350 y Fh(2)1149 2405 y(2)p Fi(;B)1252 2413 y Fd(D)1308 2384 y Fm(\(\012\))d(:=)e Fk(f)p Fl(u)g Fk(2)i Fl(H)1833 2350 y Fh(2)1826 2405 y(2)1870 2384 y Fm(\(\012\))f(:)h Fl(B)2127 2396 y Fi(D)2187 2384 y Fl(u)p Fk(j)2258 2396 y Fi(@)t Fh(\012)2371 2384 y Fm(=)f(0)p Fk(g)p Fl(;)172 2553 y Fm(where)28 b Fl(B)476 2565 y Fi(D)563 2553 y Fm(is)g(giv)n(en)f(b)n(y)g Fl(B)1042 2565 y Fi(D)1102 2553 y Fl(u)c Fm(:=)f Fl(u)p Fk(j)1354 2565 y Fi(@)t Fh(\012)1472 2553 y Fm(for)27 b Fl(u)c Fk(2)g Fl(H)1824 2523 y Fh(2)1817 2574 y(2)1861 2553 y Fm(\(\012\),)29 b(w)n(e)e(obtain)g Fl(dom)p Fm(\()p Fl(A)p Fm(\))d(:=)f Fl(H)2910 2523 y Fh(2)2903 2576 y(2)p Fi(;B)3006 2584 y Fd(D)3062 2553 y Fm(\(\012\).)297 2661 y(F)-7 b(urther,)30 b(let)f Fl(B)34 b Fm(b)r(e)c(the)f(negativ)n(e)f (half)i(Laplacian)e(with)i(Neumann)f(b)r(oundary)g(conditions,)g(i.e.,) 172 2769 y Fl(B)f Fm(:=)22 b Fk(\000)448 2736 y Fh(1)p 448 2750 V 448 2798 a(2)491 2769 y Fm(\001)560 2781 y Fi(N)623 2769 y Fm(.)37 b(One)27 b(has)g Fl(dom)p Fm(\()p Fl(B)t Fm(\))d(:=)f Fl(H)1504 2739 y Fh(2)1497 2792 y(2)p Fi(;B)1600 2800 y Fd(N)1658 2769 y Fm(\(\012\),)28 b(where)1078 2938 y Fl(H)1154 2904 y Fh(2)1147 2958 y(2)p Fi(;B)1250 2966 y Fd(N)1307 2938 y Fm(\(\012\))c(:=)f Fk(f)p Fl(u)f Fk(2)h Fl(H)1832 2904 y Fh(2)1825 2958 y(2)1869 2938 y Fm(\(\012\))h(:)f Fl(B)2126 2950 y Fi(N)2189 2938 y Fl(u)p Fk(j)2260 2950 y Fi(@)t Fh(\012)2373 2938 y Fm(=)g(0)p Fk(g)p Fl(:)172 3107 y Fm(The)32 b(b)r(oundary)f(op)r(erator)f Fl(B)1127 3119 y Fi(N)1222 3107 y Fm(is)i(giv)n(en)f(b)n(y)g(\()p Fl(B)1745 3119 y Fi(N)1809 3107 y Fl(u)p Fm(\)\()p Fl(x)p Fm(\))f(=)2198 3074 y Fi(@)p 2135 3088 166 4 v 2135 3135 a(@)t(\027)t Fh(\()p Fi(x)p Fh(\))2311 3107 y Fl(u)p Fm(\()p Fl(x)p Fm(\))p Fl(;)97 b(x)31 b Fk(2)f Fl(@)5 b Fm(\012,)33 b(for)e Fl(u)e Fk(2)i Fl(H)3288 3077 y Fh(2)3281 3127 y(2)3325 3107 y Fm(\(\012\),)172 3215 y(where)40 b Fl(\027)5 b Fm(\()p Fl(x)p Fm(\))41 b(is)f(the)g(outer)g (unit)g(normal)f(to)h(the)g(b)r(oundary)g Fl(@)5 b Fm(\012)39 b(at)h(the)g(p)r(oin)n(t)h Fl(x)j Fk(2)g Fl(@)5 b Fm(\012.)73 b(Since)172 3361 y Fl(dom)p Fm(\()p Fl(A)422 3331 y Fh(1)p Fi(=)p Fh(2)528 3361 y Fm(\))23 b(=)722 3277 y Fh(o)694 3361 y Fl(H)770 3331 y Fh(1)763 3382 y(2)807 3361 y Fm(\(\012\))28 b(and)f Fl(dom)p Fm(\()p Fl(B)1375 3331 y Fh(1)p Fi(=)p Fh(2)1481 3361 y Fm(\))c(=)f Fl(H)1699 3331 y Fh(1)1692 3382 y(2)1736 3361 y Fm(\(\012\),)29 b(one)e(gets)g Fl(dom)p Fm(\()p Fl(A)2485 3331 y Fh(1)p Fi(=)p Fh(2)2590 3361 y Fm(\))d Fk(\022)e Fl(dom)p Fm(\()p Fl(B)2988 3331 y Fh(1)p Fi(=)p Fh(2)3093 3361 y Fm(\).)38 b(Hence)933 3573 y Fl(dom)p Fm(\()p Fl(H)1197 3539 y Fh(1)p Fi(=)p Fh(2)1302 3573 y Fm(\))23 b(=)g Fl(dom)p Fm(\()p Fl(A)1695 3539 y Fh(1)p Fi(=)p Fh(2)1800 3573 y Fm(\))c Fk(\\)g Fl(dom)p Fm(\()p Fl(B)2180 3539 y Fh(1)p Fi(=)p Fh(2)2285 3573 y Fm(\))k(=)2479 3485 y Fh(o)2451 3573 y Fl(H)2527 3539 y Fh(1)2520 3594 y(2)2564 3573 y Fm(\(\012\))p Fl(;)172 3742 y Fm(and)28 b Fl(H)i Fm(:=)22 b Fl(A)651 3677 y Fi(:)629 3742 y Fm(+)g Fl(B)28 b Fm(=)22 b(2)p Fl(A)h Fm(=)g Fk(\000)p Fm(\001)1243 3754 y Fi(D)1303 3742 y Fm(.)37 b(Therefore)26 b Fl(dom)p Fm(\()p Fl(H)2003 3712 y Fi(\013)2051 3742 y Fm(\))d(=)g Fl(dom)p Fm(\()p Fl(A)2444 3712 y Fi(\013)2492 3742 y Fm(\))28 b(for)f(arbitrary)e Fl(\013)f Fk(2)f Fm([0)p Fl(;)14 b Fm(1].)297 3850 y(No)n(w)25 b(w)n(e)h(calculate)f(the)h(domains)f Fl(dom)p Fm(\()p Fl(A)1666 3820 y Fi(\013)1714 3850 y Fm(\))h(and)g Fl(dom)p Fm(\()p Fl(B)2187 3820 y Fi(\013)2235 3850 y Fm(\))g(for)f Fl(\013)f Fk(2)f Fm(\(1)p Fl(=)p Fm(2)p Fl(;)14 b Fm(1\).)35 b(By)26 b([11)o(])g(and)f([12)o(])172 3958 y(one)j(\014nds)f(:)37 b Fl(dom)p Fm(\()p Fl(A)833 3928 y Fi(\013)882 3958 y Fm(\))23 b(=)g Fl(H)1101 3928 y Fh(2)p Fi(\013)1094 3981 y Fh(2)p Fi(;B)1197 3989 y Fd(D)1252 3958 y Fm(\(\012\))p Fl(;)98 b(\013)23 b Fk(2)h Fm(\(1)p Fl(=)p Fm(2)p Fl(;)14 b Fm(1\),)26 b(and)1042 4215 y Fl(dom)p Fm(\()p Fl(B)1297 4180 y Fi(\013)1345 4215 y Fm(\))d(=)1488 4073 y Ff(\()1567 4137 y Fl(H)1643 4107 y Fh(2)p Fi(\013)1636 4158 y Fh(2)1723 4137 y Fm(\(\012\))p Fl(;)182 b(\013)24 b Fk(2)f Fm(\(1)p Fl(=)p Fm(2)p Fl(;)14 b Fm(3)p Fl(=)p Fm(4\))p Fl(;)1567 4292 y(H)1643 4262 y Fh(2)p Fi(\013)1636 4315 y Fh(2)p Fi(;B)1739 4323 y Fd(N)1797 4292 y Fm(\(\012\))p Fl(;)108 b(\013)24 b Fk(2)f Fm(\(3)p Fl(=)p Fm(4)p Fl(;)14 b Fm(1\))p Fl(:)172 4463 y Fm(Since)31 b Fl(H)468 4433 y Fh(2)p Fi(\013)461 4486 y Fh(2)p Fi(;B)564 4494 y Fd(D)620 4463 y Fm(\(\012\))d Fk(\022)f Fl(H)940 4433 y Fh(2)p Fi(\013)933 4484 y Fh(2)1020 4463 y Fm(\(\012\),)k(one)f(gets)g Fl(dom)p Fm(\()p Fl(H)1791 4433 y Fi(\013)1838 4463 y Fm(\))e(=)f Fl(dom)p Fm(\()p Fl(A)2240 4433 y Fi(\013)2288 4463 y Fm(\))h Fk(\022)f Fl(dom)p Fm(\()p Fl(B)2695 4433 y Fi(\013)2743 4463 y Fm(\))k(for)e Fl(\013)f Fk(2)g Fm(\(1)p Fl(=)p Fm(2)p Fl(;)14 b Fm(3)p Fl(=)p Fm(4\).)172 4571 y(Applying)28 b(no)n(w)f(Theorem)g(2.1)g(w)n(e)g(\014nd)h(that)857 4661 y Ff(\015)857 4711 y(\015)857 4761 y(\015)903 4664 y(\020)953 4757 y Fl(e)992 4722 y Fj(\000)p Fi(t)p Fh(\()p Fj(\000)p Fh(\001)1202 4730 y Fd(D)1252 4722 y Fh(\))p Fi(=)p Fh(2)p Fi(n)1391 4757 y Fl(e)1430 4722 y Fj(\000)p Fi(t)p Fh(\()p Fj(\000)p Fh(\001)1640 4730 y Fd(N)1692 4722 y Fh(\))p Fi(=)p Fh(2)p Fi(n)1830 4664 y Ff(\021)1880 4682 y Fi(n)1943 4757 y Fk(\000)18 b Fl(e)2065 4722 y Fj(\000)p Fi(t)p Fh(\()p Fj(\000)p Fh(\001)2275 4730 y Fd(D)2326 4722 y Fh(\))2356 4661 y Ff(\015)2356 4711 y(\015)2356 4761 y(\015)2425 4757 y Fm(=)23 b Fl(O)r Fm(\()p Fl(n)2660 4722 y Fj(\000)p Fi(\024)2756 4757 y Fm(\))172 4950 y(for)28 b(an)n(y)e Fl(\024)d Fm(:=)g(2)p Fl(\013)18 b Fk(\000)g Fm(1)23 b Fl(<)g(\024)1035 4962 y Fh(0)1095 4950 y Fm(:=)1215 4917 y Fh(1)p 1215 4931 34 4 v 1215 4979 a(2)1286 4950 y Fm(\()p Fl(\013)h Fm(=)1492 4917 y Fh(3)p 1492 4931 V 1492 4979 a(4)1535 4950 y Fm(\))k(uniformly)g (in)g Fl(t)23 b Fk(2)g Fm([0)p Fl(;)14 b(T)e Fm(])26 b(as)h Fl(n)c Fk(!)g(1)p Fm(.)297 5058 y(If)34 b Fl(\013)g Fm(=)f(3)p Fl(=)p Fm(4,)h(then)h(it)f(do)r(es)g(not)g(hold)f(that)h Fl(dom)p Fm(\()p Fl(A)2013 5028 y Fh(3)p Fi(=)p Fh(4)2119 5058 y Fm(\))g Fk(\022)f Fl(dom)p Fm(\()p Fl(B)2538 5028 y Fh(3)p Fi(=)p Fh(4)2643 5058 y Fm(\).)56 b(Hence)34 b Fl(dom)p Fm(\()p Fl(A)3257 5028 y Fh(3)p Fi(=)p Fh(4)3362 5058 y Fm(\))23 b Fk(\\)172 5166 y Fl(dom)p Fm(\()p Fl(B)427 5136 y Fh(3)p Fi(=)p Fh(4)533 5166 y Fm(\))g(is)g(a)g(prop)r(er)f (subset)i(of)f Fl(dom)p Fm(\()p Fl(H)1598 5136 y Fh(3)p Fi(=)p Fh(4)1703 5166 y Fm(\))g(whic)n(h)g Fg(do)l(es)k(not)k Fm(allo)n(w)22 b(one)h(to)g(apply)g(Theorem)f(2.1.)297 5274 y(If)28 b Fl(\013)23 b Fk(2)h Fm(\(3)p Fl(=)p Fm(4)p Fl(;)14 b Fm(1\),)26 b(then)518 5443 y Fl(dom)p Fm(\()p Fl(A)768 5409 y Fi(\013)817 5443 y Fm(\))18 b Fk(\\)h Fl(dom)p Fm(\()p Fl(B)1196 5409 y Fi(\013)1244 5443 y Fm(\))k(=)518 5576 y Fl(H)594 5541 y Fh(2)p Fi(\013)587 5598 y Fh(2)p Fi(;)p Fj(f)p Fi(B)724 5606 y Fd(D)776 5598 y Fi(;B)846 5606 y Fd(N)899 5598 y Fj(g)937 5576 y Fm(\(\012\))h(:=)f Fk(f)p Fl(u)f Fk(2)h Fl(H)1462 5541 y Fh(2)p Fi(\013)1455 5596 y Fh(2)1542 5576 y Fm(\(\012\))h(:)f Fl(B)1799 5588 y Fi(D)1859 5576 y Fl(u)p Fk(j)1930 5588 y Fi(@)t Fh(\012)2043 5576 y Fm(=)g(0)p Fl(;)14 b(B)2273 5588 y Fi(N)2335 5576 y Fl(u)p Fk(j)2406 5588 y Fi(@)t Fh(\012)2520 5576 y Fm(=)22 b(0)p Fk(g)g(\022)h Fl(H)2877 5541 y Fh(2)p Fi(\013)2870 5596 y Fh(2)p Fi(;B)2973 5604 y Fd(D)3029 5576 y Fm(\(\012\))p Fl(:)p eop %%Page: 6 6 6 5 bop -250 -22 a Fr(Septem)n(b)r(er)24 b(18,)f(2003)72 b(17:26)g(WSPC/T)-6 b(rim)22 b(Size:)31 b(10in)24 b(x)g(7in)g(for)f (Pro)r(ceedings)637 b(ICMP-ws-pro)r(cs10x7)p -300 -5 3 200 v -557 55 200 3 v 3957 55 V 3898 -5 3 200 v -300 6314 V -557 6055 200 3 v 3898 6314 3 200 v 3960 6055 200 3 v 172 636 a(6)172 856 y Fm(This)29 b(yields)f(that)h Fl(dom)p Fm(\()p Fl(A)1028 826 y Fi(\013)1076 856 y Fm(\))20 b Fk(\\)f Fl(dom)p Fm(\()p Fl(B)1457 826 y Fi(\013)1505 856 y Fm(\))29 b(is)f(a)h(prop)r(er)e(subset)i(of)f Fl(dom)p Fm(\()p Fl(H)2602 826 y Fi(\013)2650 856 y Fm(\))h(whic)n(h)f Fg(do)l(es)j(not)37 b Fm(allo)n(w)172 964 y(one)28 b(to)f(apply)g (Theorem)g(2.1,)g(either.)297 1072 y(If)h Fl(\013)c Fm(=)f(1,)28 b(then)g(one)g(gets)f(that)i Fl(dom)p Fm(\()p Fl(A)p Fm(\))19 b Fk(\\)g Fl(dom)p Fm(\()p Fl(B)t Fm(\))29 b(is)f(a)g(prop)r (er)f(subset)g(of)h Fl(dom)p Fm(\()p Fl(H)7 b Fm(\))29 b(to)r(o)f(whic)n(h)172 1180 y(yields)g Fl(H)i Fk(6)p Fm(=)22 b Fl(A)d Fm(+)f Fl(B)t Fm(.)37 b(Therefore)26 b(the)i(results)g(of)f([2])g(are)g(not)h(applicable.)297 1288 y(Notice)34 b(that)g(in)g(con)n(trast)f(to)h([6])g(and)f(to)h (examples)g(giv)n(en)f(there,)i(here)f(w)n(e)f(ha)n(v)n(e)g(an)h (example,)172 1396 y(when)28 b(the)g(op)r(erator)e Fl(B)934 1365 y Fi(\013)1009 1396 y Fm(is)i Fg(not)h(smal)t(l)37 b Fm(with)28 b(resp)r(ect)g(to)f Fl(A)2103 1365 y Fi(\013)2178 1396 y Fm(for)h Fl(\013)23 b Fk(2)g Fm(\(1)p Fl(=)p Fm(2)p Fl(;)14 b Fm(3)p Fl(=)p Fm(4\).)172 1654 y Fq(3.)47 b(T)-8 b(rotter)33 b(Pro)s(duct)g(F)-8 b(orm)m(ula)30 b(for)j(Gibbs)e (Semigroups)172 1803 y(3.1.)48 b Fn(T)-7 b(r)i(ac)g(e-Norm)35 b(Conver)-5 b(genc)g(e)35 b(R)-5 b(ate)172 1953 y Fm(Let)20 b Fk(H)h Fm(b)r(e)f(a)f(separable)f(Hilb)r(ert)j(space)e(with)h Fk(B)s Fm(\()p Fk(H)q Fm(\))f(the)h(algebra)e(of)i(all)f(b)r(ounded)h (op)r(erators)e(on)h Fk(H)q Fm(.)34 b(Let)172 2061 y Fk(C)216 2073 y Fj(1)286 2061 y Fm(\()p Fk(H)q Fm(\))22 b(denote)e(the)h(ideal)f(of)h(compact)f(op)r(erators)f(in)h Fk(B)s Fm(\()p Fk(H)q Fm(\).)35 b(Then)20 b(the)h(eigen)n(v)-5 b(alues)20 b Fk(f)p Fl(\025)3002 2073 y Fi(k)3043 2061 y Fm(\()p Fk(j)p Fl(C)6 b Fk(j)p Fm(\))p Fk(g)3260 2030 y Fj(1)3260 2084 y Fi(k)q Fh(=1)3406 2061 y Fm(of)172 2168 y(the)24 b(compact)e(self-adjoin)n(t)h(op)r(erator)e Fk(j)p Fl(C)6 b Fk(j)24 b Fm(=)1613 2098 y Fk(p)p 1682 2098 169 4 v 70 x Fl(C)1747 2145 y Fj(\003)1786 2168 y Fl(C)29 b Fm(are)22 b(kno)n(wn)h(as)f(the)h(singular)f(v)-5 b(alues)23 b Fk(f)p Fl(s)3136 2180 y Fi(k)3176 2168 y Fm(\()p Fl(C)6 b Fm(\))p Fk(g)3347 2138 y Fj(1)3347 2192 y Fi(k)q Fh(=1)172 2276 y Fm(of)32 b(the)g(op)r(erator)e Fl(C)6 b Fm(.)49 b(W)-7 b(e)32 b(denote)f(b)n(y)h Fk(C)1476 2288 y Fi(p)1514 2276 y Fm(\()p Fk(H)q Fm(\))g(\(1)d Fk(\024)h Fl(p)f(<)g Fk(1)p Fm(\))j(the)g(ideals)f(in)h Fk(B)s Fm(\()p Fk(H)q Fm(\))g(consisting)e(of)i(all)172 2384 y(compact)c(op)r(erators)d Fl(C)34 b Fm(suc)n(h)27 b(that)1550 2493 y Fj(1)1523 2518 y Ff(X)1523 2697 y Fi(k)q Fh(=1)1657 2597 y Fl(s)1696 2609 y Fi(k)1737 2597 y Fm(\()p Fl(C)6 b Fm(\))1866 2562 y Fi(p)1928 2597 y Fl(<)23 b Fk(1)p Fl(:)172 2826 y Fm(Then)28 b(eac)n(h)f(ideal)g Fk(C)818 2838 y Fi(p)856 2826 y Fm(\()p Fk(H)q Fm(\))i(is)e(a)g(Banac)n (h)g(space)f(with)j(the)f(norm:)1346 3078 y Fk(k)p Fl(C)6 b Fk(k)1494 3102 y Fi(p)1555 3078 y Fm(=)1643 2936 y Ff( )1736 2974 y Fj(1)1709 2999 y Ff(X)1709 3177 y Fi(k)q Fh(=1)1843 3078 y Fl(s)1882 3090 y Fi(k)1923 3078 y Fm(\()p Fl(C)g Fm(\))2052 3043 y Fi(p)2091 2936 y Ff(!)2157 2953 y Fh(1)p Fi(=p)2276 3078 y Fl(:)172 3307 y Fm(In)28 b(particular)f(the) g(ideal)h Fk(C)1045 3319 y Fh(1)1082 3307 y Fm(\()p Fk(H)q Fm(\))g(is)f(called)h(the)g Fg(tr)l(ac)l(e-class)p Fm(,)g(with)g(the)g (trace-norm)e Fk(k)p Fl(C)6 b Fk(k)3035 3332 y Fh(1)3095 3307 y Fm(=)23 b Fl(T)12 b(r)r Fk(j)p Fl(C)6 b Fk(j)p Fm(.)172 3482 y Fq(De\014nition)29 b(3.1.)40 b Fm(A)26 b(strongly)e(con)n(tin)n(uous)g(one-parameter)g(semigroup)g Fk(f)p Fl(U)2668 3494 y Fi(t)2696 3482 y Fk(g)2738 3494 y Fi(t)p Fj(\025)p Fh(0)2878 3482 y Fm(is)h(called)g(a)g(Gibbs)172 3590 y(semigroup)i(if)h Fl(U)703 3602 y Fi(t)755 3590 y Fk(2)23 b(C)877 3602 y Fh(1)914 3590 y Fm(\()p Fk(H)q Fm(\))28 b(for)f(an)n(y)g Fl(t)c(>)g Fm(0)k(\(see)h(e.g.)36 b([13)o(]\).)172 3765 y(There)28 b(are)e(sev)n(eral)g(results)h(on)g (the)h(trace-norm)e(con)n(v)n(ergence)f(of)j(the)g(T)-7 b(rotter)27 b(pro)r(duct)g(form)n(ula)g(for)172 3873 y(self-adjoin)n(t)d(Gibbs)g(semigroups,)f(see)h(e.g.)35 b([14)o(]-[17)o(].)h(A)24 b(recen)n(t)f(result)h(b)n(y)g(Zagrebno)n(v)d ([18)o(])j(is)g(related)172 3981 y(to)k(the)g(ab)r(o)n(v)n(e)e(optimal) i(op)r(erator-norm)c(con)n(v)n(ergence)i(of)h(the)h(T)-7 b(rotter-Kato)25 b(pro)r(duct)j(form)n(ula.)172 4156 y Fq(Theorem)j(3.1.)41 b Fg(L)l(et)29 b Fl(A)h Fg(and)g Fl(B)k Fg(b)l(e)c(non-ne)l(gative)g(self-adjoint)i(op)l(er)l(ators)e (such)g(that)g(:)172 4264 y(\(i\))g(the)g(op)l(er)l(ator)h(sum)e Fl(C)h Fm(:=)22 b Fl(A)d Fm(+)f Fl(B)34 b Fg(is)c(self-adjoint)i(on)d Fl(dom)p Fm(\()p Fl(C)6 b Fm(\))25 b(=)d Fl(dom)p Fm(\()p Fl(A)p Fm(\))e Fk(\\)f Fl(dom)p Fm(\()p Fl(B)t Fm(\))p Fg(;)172 4372 y(\(ii\))31 b(either)g Fl(e)597 4342 y Fj(\000)p Fi(\034)7 b(A)763 4372 y Fk(2)23 b(C)885 4384 y Fh(1)922 4372 y Fm(\()p Fk(H)q Fm(\))p Fg(,)31 b(or)f Fl(e)1259 4342 y Fj(\000)p Fi(\034)7 b(B)1428 4372 y Fk(2)23 b(C)1550 4384 y Fh(1)1587 4372 y Fm(\()p Fk(H)q Fm(\))p Fg(,)31 b(for)f Fl(\034)j(>)23 b Fm(0)p Fg(.)172 4480 y(Then)172 4588 y(\(a\))30 b(the)g(T)-6 b(r)l(otter)30 b(pr)l(o)l(duct)g(formula)g(c)l(onver)l(ge)h(in)e(the)h(tr)l(ac)l (e-norm)g(with)g(err)l(or)g(b)l(ound)g Fl(O)r Fm(\()p Fl(n)3116 4558 y Fj(\000)p Fh(1)3206 4588 y Fm(\))p Fg(:)931 4747 y Fk(k)p Fm(\()p Fl(e)1044 4712 y Fj(\000)p Fi(tA=n)1250 4747 y Fl(e)1289 4712 y Fj(\000)p Fi(tB)s(=n)1498 4747 y Fm(\))1530 4712 y Fi(n)1594 4747 y Fk(\000)18 b Fl(e)1716 4712 y Fj(\000)p Fi(tC)1848 4747 y Fk(k)1890 4759 y Fh(1)1950 4747 y Fm(=)23 b Fl(O)r Fm(\()p Fl(n)2185 4712 y Fj(\000)p Fh(1)2274 4747 y Fm(\))p Fl(;)100 b(n)22 b Fk(!)h(1)p Fl(;)612 b Fm(\(15\))172 4906 y Fg(uniformly)32 b(on)f(e)l(ach)h(c)l (omp)l(act)f Fl(t)p Fg(-interval)h(in)e Fm([0)p Fl(;)14 b Fk(1)p Fm(\))p Fg(.)42 b(F)-6 b(urther,)31 b(if)h Fl(C)37 b Fg(is)31 b(strictly)g(p)l(ositive,)i(then)e(\(15\))172 5013 y(holds)h(uniformly)e(on)g Fm([0)p Fl(;)14 b Fk(1)p Fm(\))p Fg(;)172 5121 y(\(b\))30 b(the)g(err)l(or)g(b)l(ound)g Fl(O)r Fm(\()p Fl(n)1044 5091 y Fj(\000)p Fh(1)1134 5121 y Fm(\))g Fg(is)g(ultimate)g(optimal.)172 5297 y Fm(The)e(pro)r(of)f (of)h(the)g(theorem)f(is)g(based)g(on)172 5472 y Fq(Lemma)44 b(3.1.)d Fg(L)l(et)f Fl(A)h Fg(b)l(e)g(a)g(self-adjoint)h(op)l(er)l (ator)g(such)f(that)g Fl(e)2368 5441 y Fj(\000)p Fi(tA)2541 5472 y Fk(2)j(C)2684 5484 y Fh(1)2721 5472 y Fm(\()p Fk(H)q Fm(\))d Fg(for)g Fl(t)i(>)g Fm(0)p Fg(.)71 b(L)l(et)172 5580 y Fl(B)42 b Fg(b)l(e)37 b(gener)l(ator)g(of)h(a)g(c)l(ontr)l (action)f(semigr)l(oup)h(and)g(let)f Fk(f)p Fl(e)2163 5549 y Fj(\000)p Fi(tH)2302 5580 y Fk(g)2344 5592 y Fi(t>)p Fh(0)2495 5580 y Fg(b)l(e)g(a)h(Gibbs)g(semigr)l(oup)g(with)p eop %%Page: 7 7 7 6 bop -250 -22 a Fr(Septem)n(b)r(er)24 b(18,)f(2003)72 b(17:26)g(WSPC/T)-6 b(rim)22 b(Size:)31 b(10in)24 b(x)g(7in)g(for)f (Pro)r(ceedings)637 b(ICMP-ws-pro)r(cs10x7)p -300 -5 3 200 v -557 55 200 3 v 3957 55 V 3898 -5 3 200 v -300 6314 V -557 6055 200 3 v 3898 6314 3 200 v 3960 6055 200 3 v 3437 636 a(7)172 864 y Fk(k)p Fl(e)253 833 y Fj(\000)p Fi(tH)393 864 y Fk(k)22 b(\024)h Fm(1)p Fg(.)38 b(L)l(et)876 1057 y Fl(\017)p Fm(\()p Fl(s;)14 b(t)p Fm(\))23 b(:=)204 b(sup)1259 1113 y Fc(2)p Fd(s)p 1224 1122 127 3 v 1224 1156 a Fc(2)p Fd(s)p Fc(+1)1361 1136 y Fi(t)p Fj(\024)p Fi(\034)7 b Fj(\024)1574 1113 y Fc(2)p Fd(s)p 1537 1122 130 3 v 1537 1156 a Fc(2)p Fd(s)p Fb(\000)p Fc(1)1676 1136 y Fi(t)1715 961 y Ff(\015)1715 1011 y(\015)1715 1061 y(\015)1761 964 y(\020)1811 1057 y Fl(e)1850 1022 y Fj(\000)p Fi(\034)g(A=s)2058 1057 y Fl(e)2097 1022 y Fj(\000)p Fi(\034)g(B)s(=s)2308 964 y Ff(\021)2358 982 y Fi(s)2412 1057 y Fk(\000)18 b Fl(e)2534 1022 y Fj(\000)p Fi(\034)7 b(H)2686 961 y Ff(\015)2686 1011 y(\015)2686 1061 y(\015)2746 1057 y Fl(;)556 b Fm(\(16\))172 1305 y Fg(for)31 b Fl(s)23 b Fk(2)g Fe(N)t Fg(,)37 b Fl(s)23 b(>)f Fm(1)29 b Fg(and)i Fl(t)23 b(>)f Fm(0)p Fg(.)38 b(Then)31 b(for)f(e)l(ach)h Fl(t)1754 1317 y Fh(0)1814 1305 y Fl(>)23 b Fm(0)29 b Fg(ther)l(e)h(exist)g Fl(l)r Fm(\()p Fl(t)2463 1317 y Fh(0)2499 1305 y Fm(\))g Fg(and)h Fl(l)2750 1275 y Fj(0)2772 1305 y Fm(\()p Fl(t)2834 1317 y Fh(0)2872 1305 y Fm(\))f Fg(such)g(that:)492 1403 y Ff(\015)492 1453 y(\015)492 1503 y(\015)538 1406 y(\020)588 1498 y Fl(e)627 1464 y Fj(\000)p Fi(tA=n)833 1498 y Fl(e)872 1464 y Fj(\000)p Fi(tB)s(=n)1081 1406 y Ff(\021)1130 1423 y Fi(n)1194 1498 y Fk(\000)18 b Fl(e)1316 1464 y Fj(\000)p Fi(tH)1455 1403 y Ff(\015)1455 1453 y(\015)1455 1503 y(\015)1502 1557 y Fh(1)1562 1498 y Fk(\024)23 b Fl(l)r Fm(\()p Fl(t)1739 1510 y Fh(0)1775 1498 y Fm(\))p Fl(\017)p Fm(\([)p Fl(n=)p Fm(2])p Fl(;)14 b(t=)p Fm(2\))j(+)h Fl(l)2363 1464 y Fj(0)2386 1498 y Fm(\()p Fl(t)2448 1510 y Fh(0)2486 1498 y Fm(\))p Fl(\017)p Fm(\([\()p Fl(n)g Fm(+)g(1\))p Fl(=)p Fm(2])p Fl(;)c(t=)p Fm(2\))171 b(\(17\))172 1686 y Fg(for)31 b(al)t(l)g Fl(n)23 b(>)f Fm(1)30 b Fg(and)g Fl(t)23 b Fk(\025)g Fl(t)988 1698 y Fh(0)1025 1686 y Fg(,)172 1871 y Fm(see)36 b([17)o(])f(and)h([19)o(].)61 b(No)n(w)35 b(conditions)g(of)h(the)g(Theorem)e(3.1)h(and)g(the)h (mini-max)g(principle)f(imply)172 1979 y(that)d(the)f(op)r(erator)e Fl(C)37 b Fm(is)31 b(generator)d(of)j(the)g(con)n(tracting)f(Gibbs)h (semigroup)e(with)2916 1909 y Ff(\015)2916 1959 y(\015)2962 1979 y Fl(e)3001 1949 y Fj(\000)p Fi(tC)3133 1909 y Ff(\015)3133 1959 y(\015)3208 1979 y Fk(\024)f Fm(1)i(for)172 2087 y Fl(t)j(>)f Fm(0.)53 b(On)32 b(the)i(other)f(hand,)h(b)n(y)f(Theorem)f (1.1)g(one)h(gets)g(that)g Fl(\017)p Fm(\()p Fl(s;)14 b(t)p Fm(\))33 b(=)e Fl(O)r Fm(\(1)p Fl(=s)p Fm(\))j(in)f(\(16\).)53 b(Then)172 2195 y(the)30 b(conditions)e(of)h(Lemma)f(3.1)g(are)g(v)n (eri\014ed)g(for)g Fl(C)j Fm(=)25 b Fl(H)36 b Fm(and)28 b Fl(\017)p Fm(\()p Fl(k)2359 2207 y Fi(n)2405 2195 y Fl(;)14 b(t=)p Fm(2\))24 b(=)g Fl(O)r Fm(\(1)p Fl(=n)p Fm(\),)30 b Fl(\017)p Fm(\()p Fl(m)3156 2207 y Fi(n)3201 2195 y Fl(;)14 b(t=)p Fm(2\))24 b(=)172 2303 y Fl(O)r Fm(\(1)p Fl(=n)p Fm(\),)41 b(where)d Fl(k)793 2315 y Fi(n)879 2303 y Fm(=)i([)p Fl(n=)p Fm(2])d(and)h Fl(m)1446 2315 y Fi(n)1532 2303 y Fm(=)i([\()p Fl(n)26 b Fm(+)f(1\))p Fl(=)p Fm(2].)67 b(This)38 b(pro)n(v)n(es)f(the)h(assertion)f(\(a\))h (of)g(the)172 2411 y(Theorem)27 b(3.1.)172 2679 y Fq(3.2.)48 b Fn(Optimality)33 b(of)i(the)g(Err)-5 b(or)33 b(Bound)172 2828 y Fm(Similar)27 b(to)g(Sec.)g(1.2)f(it)h(is)g(enough)g(to)f(pro)r (duce)h(t)n(w)n(o)f(op)r(erators)f Fl(A)i Fm(and)g Fl(B)k Fm(suc)n(h)c(that)g(one)g(can)g(\014nd)g(a)172 2936 y Fg(p)l(ositive)36 b Fl(L)533 2948 y Fh(1)570 2936 y Fm(\()p Fl(t)p Fm(\))28 b(ensuring)f(in)g(\(15\))h(for)f Fl(t)c(>)f Fm(0)28 b(the)g(lo)n(w)n(er)e(estimate)h(:)1094 3108 y Fk(k)p Fm(\()p Fl(e)1207 3074 y Fj(\000)p Fi(tA=n)1413 3108 y Fl(e)1452 3074 y Fj(\000)p Fi(tB)s(=n)1661 3108 y Fm(\))1693 3074 y Fi(n)1756 3108 y Fk(\000)18 b Fl(e)1878 3074 y Fj(\000)p Fi(tC)2011 3108 y Fk(k)2053 3120 y Fh(1)2113 3108 y Fk(\025)k Fl(L)2257 3120 y Fh(1)2294 3108 y Fm(\()p Fl(t)p Fm(\))p Fl(n)2438 3074 y Fj(\000)p Fh(1)2528 3108 y Fl(:)774 b Fm(\(18\))172 3280 y(T)-7 b(o)31 b(this)h(end)f(let)h Fk(H)g Fm(b)r(e)g Fg(\014nite-dimensional)p Fm(.)49 b(Then)31 b(b)n(y)g(the)g(Bak)n(er{Campb)r(ell{Hausdor\013)d(form)n(ula)172 3388 y(w)n(e)g(ha)n(v)n(e)e(for)h(small)g Fk(j)p Fl(t)p Fk(j)526 3560 y Fl(e)565 3526 y Fj(\000)p Fi(tA)696 3560 y Fl(e)735 3526 y Fj(\000)p Fi(tB)892 3560 y Fm(=)22 b(exp)1106 3493 y Ff(\002)1141 3560 y Fk(\000)p Fl(t)p Fm(\()p Fl(A)c Fm(+)g Fl(B)t Fm(\))h(+)1642 3527 y Fi(t)1667 3502 y Fc(2)p 1642 3541 58 4 v 1654 3588 a Fh(2)1709 3560 y Fm([)p Fl(A;)14 b(B)t Fm(])19 b Fk(\000)2033 3527 y Fi(t)2058 3502 y Fc(3)p 2033 3541 V 2046 3588 a Fh(6)2101 3560 y Fm([)p Fl(A)g Fk(\000)f Fl(B)t(;)2402 3527 y Fh(1)p 2402 3541 34 4 v 2402 3588 a(2)2445 3560 y Fm([)p Fl(A;)c(B)t Fm(]])19 b(+)f Fl(O)2845 3572 y Fi(p)2883 3560 y Fm(\()p Fk(j)p Fl(t)p Fk(j)2991 3526 y Fh(4)3029 3560 y Fm(\))3061 3493 y Ff(\003)3096 3560 y Fl(;)172 3732 y Fm(with)23 b([)p Fl(A;)14 b(B)t Fm(])24 b(=)e Fl(AB)12 b Fk(\000)c Fl(B)t(A)p Fm(,)23 b(where)f(and)g(b)r(elo)n(w)g Fl(O)1748 3744 y Fi(p)1786 3732 y Fm(\()p Fk(j)p Fl(t)p Fk(j)1894 3702 y Fi(k)1936 3732 y Fm(\),)h(for)f Fl(k)k(>)c Fm(0,)h(means)f(some) g(b)r(ounded)g(op)r(erator)172 3840 y(with)29 b(norm)e(of)g(order)f Fl(O)r Fm(\()p Fk(j)p Fl(t)p Fk(j)1063 3810 y Fi(k)1105 3840 y Fm(\).)37 b(Then)700 4011 y Fl(N)9 b Fm(\()p Fl(t)p Fm(\))23 b(:=)g(\()p Fl(e)1075 3977 y Fj(\000)p Fi(tA)1206 4011 y Fl(e)1245 3977 y Fj(\000)p Fi(tB)1379 4011 y Fm(\))1411 3977 y Fh(1)p Fi(=t)1530 4011 y Fm(=)g(exp)1745 3944 y Ff(\002)1779 4011 y Fk(\000)p Fm(\()p Fl(A)c Fm(+)f Fl(B)t Fm(\))h(+)2255 3979 y Fi(t)p 2251 3993 V 2251 4040 a Fh(2)2294 4011 y Fm([)p Fl(A;)14 b(B)t Fm(])19 b(+)f Fl(O)2671 4023 y Fi(p)2710 4011 y Fm(\()p Fk(j)p Fl(t)p Fk(j)2818 3977 y Fh(2)2855 4011 y Fm(\))2887 3944 y Ff(\003)2922 4011 y Fl(:)172 4183 y Fm(W)-7 b(e)28 b(understand)g Fl(N)9 b Fm(\(0\))23 b(=)f Fl(e)1081 4153 y Fj(\000)p Fh(\()p Fi(A)p Fh(+)p Fi(B)s Fh(\))1370 4183 y Fm(and)28 b(ha)n(v)n(e)473 4429 y Fl(E)534 4441 y Fh(1)572 4429 y Fm(\()p Fl(A)p Fm(;)14 b Fl(B)t Fm(\))24 b(:=)961 4373 y Fl(d)p 946 4410 74 4 v 946 4486 a(dt)1029 4429 y(N)9 b Fm(\()p Fl(t)p Fm(\))p Fk(j)1222 4441 y Fi(t)p Fh(=0)1359 4429 y Fm(=)1457 4373 y(1)p 1457 4410 42 4 v 1457 4486 a(2)1549 4325 y Fj(1)1522 4350 y Ff(X)1522 4529 y Fi(k)q Fh(=1)1667 4373 y Fm(\()p Fk(\000)p Fm(1\))1838 4343 y Fi(k)q Fj(\000)p Fh(1)p 1667 4410 297 4 v 1780 4486 a Fl(k)s Fm(!)2026 4325 y Fi(k)1991 4350 y Ff(Y)1987 4527 y Fi(j)s Fh(=1)2102 4429 y Fm(\()p Fl(A)19 b Fm(+)f Fl(B)t Fm(\))2397 4395 y Fi(j)s Fj(\000)p Fh(1)2517 4429 y Fm([)p Fl(A;)c(B)t Fm(]\()p Fl(A)20 b Fm(+)e Fl(B)t Fm(\))3025 4395 y Fi(k)q Fj(\000)p Fi(j)3149 4429 y Fl(;)153 b Fm(\(19\))172 4676 y(of)24 b(whic)n(h)g(the)h(righ)n(t-hand)d(side)i (is)g(norm-con)n(v)n(ergen)n(t)d(and)j(can)f(b)r(e)i(a)e(non-zero)f(op) r(erator)h(with)h(b)r(ound)172 4784 y(2)214 4754 y Fj(\000)p Fh(1)303 4784 y Fk(k)p Fm([)p Fl(A;)14 b(B)t Fm(])p Fk(k)p Fl(e)638 4754 y Fj(k)p Fi(A)p Fh(+)p Fi(B)s Fj(k)863 4784 y Fm(,)28 b(if)g Fl(A)g Fm(and)f Fl(B)32 b Fm(do)c(not)f(comm)n (ute)h(with)g(eac)n(h)f(other.)36 b(It)28 b(follo)n(ws)e(that)1122 4956 y Fl(N)9 b Fm(\()p Fl(t)p Fm(\))24 b(=)e Fl(e)1442 4922 y Fj(\000)p Fh(\()p Fi(A)p Fh(+)p Fi(B)s Fh(\))1722 4956 y Fm(+)c Fl(tE)1896 4968 y Fh(1)1934 4956 y Fm(\()p Fl(A)p Fm(;)c Fl(B)t Fm(\))19 b(+)f Fl(O)2329 4968 y Fi(p)2368 4956 y Fm(\()p Fl(t)2430 4922 y Fh(2)2467 4956 y Fm(\))p Fl(;)172 5128 y Fm(so)27 b(that)h(with)g Fl(t)23 b Fm(=)g(1)p Fl(=n)626 5300 y Fm(\()p Fl(e)697 5265 y Fj(\000)p Fi(A=n)878 5300 y Fl(e)917 5265 y Fj(\000)p Fi(B)s(=n)1101 5300 y Fm(\))1133 5265 y Fi(n)1202 5300 y Fm(=)f Fl(N)9 b Fm(\(1)p Fl(=n)p Fm(\))1563 5265 y Fi(n)1631 5300 y Fm(=)22 b Fl(e)1757 5265 y Fj(\000)p Fh(\()p Fi(A)p Fh(+)p Fi(B)s Fh(\))2037 5300 y Fm(+)c Fl(n)2170 5265 y Fj(\000)p Fh(1)2259 5300 y Fl(E)2320 5312 y Fh(1)2358 5300 y Fm(\()p Fl(A)p Fm(;)c Fl(B)t Fm(\))19 b(+)f Fl(O)2753 5312 y Fi(p)2792 5300 y Fm(\()p Fl(n)2874 5265 y Fj(\000)p Fh(2)2963 5300 y Fm(\))p Fl(:)307 b Fm(\(20\))172 5472 y(Since)28 b(for)f Fg(\014nite-dimensional)38 b Fm(Hilb)r(ert)28 b(space)f Fk(H)h Fm(one)f(has)g Fk(k\001k)2213 5497 y Fh(1)2273 5472 y Fm(=)22 b Fk(k\001k)p Fm(,)27 b(the)h(error)e(b)r(ound)i Fl(O)r Fm(\()p Fl(n)3267 5441 y Fj(\000)p Fh(1)3357 5472 y Fm(\))g(is)172 5580 y(optimal)g(for)f Fg(nonsymmetric)33 b Fm(T)-7 b(rotter)26 b(pro)r(duct)i(form)n(ula)e (for)i(Gibbs)f(semigroups.)p eop %%Page: 8 8 8 7 bop -250 -22 a Fr(Septem)n(b)r(er)24 b(18,)f(2003)72 b(17:26)g(WSPC/T)-6 b(rim)22 b(Size:)31 b(10in)24 b(x)g(7in)g(for)f (Pro)r(ceedings)637 b(ICMP-ws-pro)r(cs10x7)p -300 -5 3 200 v -557 55 200 3 v 3957 55 V 3898 -5 3 200 v -300 6314 V -557 6055 200 3 v 3898 6314 3 200 v 3960 6055 200 3 v 172 636 a(8)172 852 y Fq(Remark)46 b(3.1.)40 b Fm(The)g(pro)r(of)g(that)g(the)h Fg(symmetric)k Fm(T)-7 b(rotter)40 b(pro)r(duct)g(form)n(ula)f(con)n(v)n(erges)f(in)i(the)172 960 y(trace-norm)26 b(with)i(error)e(b)r(ound)i Fl(O)r Fm(\()p Fl(n)1393 930 y Fj(\000)p Fh(1)1483 960 y Fm(\):)775 1117 y Fk(k)p Fm(\()p Fl(e)888 1083 y Fj(\000)p Fi(tB)s(=)p Fh(2)p Fi(n)1130 1117 y Fl(e)1169 1083 y Fj(\000)p Fi(tA=n)1375 1117 y Fl(e)1414 1083 y Fj(\000)p Fi(tB)s(=)p Fh(2)p Fi(n)1656 1117 y Fm(\))1688 1083 y Fi(n)1752 1117 y Fk(\000)18 b Fl(e)1874 1083 y Fj(\000)p Fi(tC)2006 1117 y Fk(k)2048 1129 y Fh(1)2108 1117 y Fm(=)k Fl(O)r Fm(\()p Fl(n)2342 1083 y Fj(\000)p Fh(1)2432 1117 y Fm(\))p Fl(;)98 b(n)22 b Fk(!)i(1)p Fl(;)455 b Fm(\(21\))172 1275 y(follo)n(ws)24 b(along)f(the)h(same)g(line)h(of)f(reasoning)e(as)i(in)g(the)h(pro)r (of)f(of)g(assertion)e(\(a\))j(of)f(Theorem)g(3.1.)34 b(The)172 1383 y(pro)r(of)29 b(of)g(existence)f(of)h(the)h(couple)e(of) h(un)n(b)r(ounded)g(op)r(erators)e Fl(A)j Fm(and)e Fl(B)34 b Fm(suc)n(h)28 b(that)h(one)g(can)g(\014nd)g(a)172 1491 y Fg(p)l(ositive)36 b Fl(L)533 1503 y Fh(2)570 1491 y Fm(\()p Fl(t)p Fm(\))28 b(ensuring)f(in)g(\(21\))h(for)f Fl(t)c(>)f Fm(0)28 b(the)g(lo)n(w)n(er)e(estimate)h(:)746 1649 y Fk(k)p Fm(\()p Fl(e)859 1615 y Fj(\000)p Fi(tB)s(=)p Fh(2)p Fi(n)1101 1649 y Fl(e)1140 1615 y Fj(\000)p Fi(tA=n)1346 1649 y Fl(e)1385 1615 y Fj(\000)p Fi(tB)s(=)p Fh(2)p Fi(n)1627 1649 y Fm(\))1659 1615 y Fi(n)1723 1649 y Fk(\000)18 b Fl(e)1845 1615 y Fj(\000)p Fi(tC)1977 1649 y Fk(k)2019 1661 y Fh(1)2079 1649 y Fk(\025)k Fl(L)2223 1661 y Fh(2)2260 1649 y Fm(\()p Fl(t)p Fm(\))p Fl(n)2404 1615 y Fj(\000)p Fh(1)2494 1649 y Fl(;)97 b(n)23 b Fk(!)g(1)p Fl(;)426 b Fm(\(22\))172 1807 y(needs)28 b(a)f(more)g(subtle)h(reasoning)d (based)j(on)f(the)h(example)f([3].)172 1981 y(Notice)38 b(that)h(Remark)e(1.3)g(ab)r(out)h(optimalit)n(y)g(is)f(also)g(v)-5 b(alid)38 b(in)h(the)f(case)f(of)h Fg(symmetric)43 b Fm(T)-7 b(rotter)172 2089 y(pro)r(duct)28 b(form)n(ula)f(for)g(Gibbs)h (semigroups.)172 2346 y Fq(Ac)m(kno)m(wledgemen)m(ts)172 2496 y Fm(I)f(w)n(ould)g(lik)n(e)f(to)h(thank)g(Christian)f(Gerard)g (and)g(Ricardo)g(W)-7 b(eder)27 b(for)f(in)n(viting)h(me)g(to)f(giv)n (e)g(a)h(talk)f(in)172 2604 y(the)e(Session)e(on)h(Quan)n(tum)g(Mec)n (hanics)f(and)h(Sp)r(ectral)g(Theory)f(at)h(ICMP)g(2003.)34 b(The)23 b(presen)n(t)f(rep)r(ort)172 2712 y(is)28 b(an)f(extended)h(v) n(ersion)e(of)i(this)g(talk.)172 2969 y Fq(References)172 3102 y Fp(1.)51 b(T.)26 b(Ic)n(hinose)g(and)g(Hideo)f(T)-6 b(am)n(ura,)26 b Fo(Commun.)g(Math.)i(Phys.)e Fa(217)p Fp(,)h(489)f(\(2001\).)172 3193 y(2.)51 b(T.)26 b(Ic)n(hinose,)g(Hideo) f(T)-6 b(am)n(ura,)25 b(Hiroshi)h(T)-6 b(am)n(ura)25 b(and)f(V.)i(A.)f(Zagrebno)n(v,)g Fo(Commun.)i(Math.)g(Phys.)e Fa(221)p Fp(,)282 3284 y(499)i(\(2001\).)172 3376 y(3.)51 b(Hiroshi)26 b(T)-6 b(am)n(ura,)26 b Fo(Inte)l(gr.)i(Equ.)f(Op)l(er.)h (The)l(ory)f Fa(37)p Fp(,)f(350)h(\(2000\).)172 3467 y(4.)51 b(M.)26 b(Reed)g(and)f(B.)h(Simon,)f Fo(Metho)l(ds)k(of)f(Mo)l (dern)g(Mathematic)l(al)h(Physics)f(I.)f(F)-6 b(unctional)28 b(A)n(nalysis)p Fp(,)f(Aca-)282 3558 y(demic)e(Press,)i(New)f(Y)-6 b(ork,)25 b(1972.)172 3650 y(5.)51 b(T.)26 b(Ic)n(hinose,)h(H.)e (Neidhardt)g(and)g(V.)h(A.)f(Zagrebno)n(v,)h Fo(J.)i(F)-6 b(unct.)28 b(A)n(nal.)d Fa(204)p Fp(,)h(475)h(\(2003\))172 3741 y(6.)51 b(H.)25 b(Neidhardt)g(and)h(V.)f(A.)h(Zagrebno)n(v,)g Fo(Intger.)i(Equ.)f(Op)l(er.)h(The)l(ory.)e Fa(35)p Fp(,)h(209)f (\(1999\).)172 3832 y(7.)51 b(T.)20 b(Ic)n(hinose,)g(H.)f(Neidhardt,)h (and)f(V.)f(A.)h(Zagrebno)n(v,)i(Op)r(erator)e(norm)f(con)n(v)n (ergence)h(of)h(the)e(T)-6 b(rotter-Kato)282 3924 y(pro)r(duct)36 b(form)n(ula,)j Fo(Pr)l(o)l(c)l(e)l(e)l(dings)h(of)d(the)h (International)g(Confer)l(enc)l(e)h(on)f(F)-6 b(unctional)38 b(A)n(nalysis)f(\(Kiev,)282 4015 y(A)n(ugust)29 b(22)f(-)g(26,)f (2001\))p Fp(,)g(p.100)g(-)e(106,)i(Ukr.)e(Acad.)h(Press,)h(Kiev)e (2003.)172 4106 y(8.)51 b(P)-6 b(.)26 b(R.)f(Cherno\013,)h Fo(J.)i(F)-6 b(unct.)28 b(A)n(nal.)d Fa(2)p Fp(,)h(238)h(\(1968\).)172 4198 y(9.)51 b(P)-6 b(.)26 b(R.)f(Cherno\013,)h Fo(Bul)t(l.)h(A)n(mer.) g(Math.)h(So)l(c.)d Fa(76)p Fp(,)i(395)g(\(1970\).)172 4289 y(10.)51 b(H.)26 b(Neidhardt)f(and)g(V.)h(A.)f(Zagrebno)n(v,)h Fo(Commun.)h(Math.)h(Phys.)e Fa(205)p Fp(,)g(129)h(\(1999\).)172 4380 y(11.)51 b(D.)26 b(F)-6 b(ujiw)n(ara,)27 b Fo(Pr)l(o)l(c.)h(Jap)l (an)h(A)l(c)l(ad.)d Fa(43)p Fp(,)g(82)g(\(1967\).)172 4472 y(12.)51 b(P)-6 b(.)26 b(Grisv)l(ard,)g Fo(A)n(r)l(ch.)i(R)l (ational)g(Me)l(ch.)g(A)n(nal.)d Fa(25)p Fp(,)h(40)h(\(1967\)40.)172 4563 y(13.)51 b(V.)25 b(A.)g(Zagrebno)n(v,)h Fo(T)-6 b(opics)27 b(in)g(the)h(The)l(ory)g(of)e(Gibbs)i(Semigr)l(oups)p Fp(,)f(\(V)-6 b(ol.10,)26 b(Series)g(A)e(:)i(Mathematical)282 4654 y(Ph)n(ysics\))g(Leuv)n(en)e(Univ)n(ersit)n(y)h(Press,)i(Leuv)n (en,)e(2003.)172 4746 y(14.)51 b(V.)26 b(A.)f(Zagrebno)n(v,)h Fo(J.)i(Math.)f(Phys.)p Fp(,)g Fa(29)p Fp(,)f(888)h(\(1988\).)172 4837 y(15.)51 b(H.)26 b(Neidhardt,)f(V.)h(A.)f(Zagrebno)n(v,)h Fo(Commun.)h(Math.)h(Phys.)p Fp(,)e Fa(131)p Fp(,)g(333)h(\(1990\).)172 4928 y(16.)51 b(T.)27 b(Ic)n(hinose,)f(Hideo)g(T)-6 b(am)n(ura,)25 b Fo(Asymptotic)k(A)n(nalysis)p Fp(,)d Fa(17)p Fp(,)h(239)g(\(1998\).) 172 5020 y(17.)51 b(H.)26 b(Neidhardt,)f(V.)h(A.)f(Zagrebno)n(v,)h Fo(J.)h(F)-6 b(unct.)29 b(A)n(nal.)p Fp(,)c Fa(167)p Fp(,)i(113)f(\(1999\).)172 5111 y(18.)51 b(V.)26 b(A.)g(Zagrebno)n(v,)g (Ultimate)f(optimal)h(error)g(b)r(ound)f(for)i(the)f(T)-6 b(rotter)26 b(pro)r(duct)f(form)n(ula:)35 b(Gibbs)26 b(semi-)282 5202 y(groups,)h Fo(Pr)l(o)l(c)l(e)l(e)l(dings)j(of)e(the)h (International)g(Confer)l(enc)l(e)h(on)e(F)-6 b(unctional)29 b(A)n(nalysis)f(\(Kiev,)g(A)n(ugust)i(22)e(-)282 5294 y(26,)f(2001\))p Fp(,)g(p.94)g(-)e(99,)i(Ukr.)e(Acad.)h(Press,)h(Kiev)e (2003.)172 5385 y(19.)51 b(V.)26 b(Cac)n(hia,)h(V.)f(A.)f(Zagrebno)n (v,)h Fo(J.)h(L)l(ondon)i(Math.So)l(c.)p Fp(,)d Fa(64)p Fp(,)g(436)h(\(2001\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0309181038634--