%!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: diff1.dvi %%Pages: 28 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips diff1 -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.09.11:1023 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}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{dup dup 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 /IE 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 IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /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 dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{ 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup 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 /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 sub]/id ch-image N /rw ch-width 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 dup 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 dup gp add /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{ dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1 adv}B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 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}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{] }if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 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 dup 1 get dup mul exch 0 get dup 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 /IE 256 array N 2 string 0 1 255{IE S dup 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 /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V{}B /RV statusdict begin /product where{pop false[ (Display)(NeXT)(LaserWriter 16/600)]{dup length product length le{dup 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 rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail {dup /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 1095 600 600 (diff1.dvi) @start %DVIPSBitmapFont: Fa cmti8 8.76 16 /Fa 16 122 df<90261FFFC0EDFFFC6060D9007F17004E5ADA77E05D180E02F7151DA202 E7ED39F8A20101167318E302C75EDAC3F0EB01C3010316C7EF0387028391380707E0A201 07ED0E0F171C02035E1738496C6C141F1770010E03E05BA2011E913801C03FA2011CDA03 8090C7FCEE0700013C5E913800FC0E0138167E5E01784A13FEA201704A5BA201F0ECE001 EDFDC049017F5C16800001ED0003A20007027E5CD80FF8ED0FF8B5D97C01B512C0157815 3846327AB146>77 D<011FB512FEEEFFC017F0903A007F8007FCEE00FE92C7127F834A15 80171F5C18C01301EF3F805CA20103ED7F00A24A14FE5F01074A5A4C5A4AEB0FE0EE7F80 49B500FEC7FC16F8829138E003FE011FEB00FF707E5C163F133FA25CA2017F147FA291C7 90C7FCA2495CA2495C18E000011601A249ED03C0D807FEECFF07B500F090387F9F809338 3FFF00705AC9EA07F833337BB139>82 D97 D<13FF121F5BA21200A25BA21201A25BA21203 A25BA21207A25BA2380FE7F8EBFFFE809038FE3F80391FF81FC0EBF00F9038E007E013C0 EA3F80A21300A248130FA2127EA200FE131FA24814C0143F1580A2EC7F00007C5B5C387E 01FC383F07F8EB8FF0381FFFC06C5BD803FCC7FC1B3378B224>I<903801FF80010F13E0 013F13F0EB7FC33801FE033803FC07EA07F8EA0FF013E0D81FC013E0003F90C7FC5BA212 7F90C8FCA35A5AA3127E15381578007F14F86CEB03F09038800FE0391FE07FC06CB51200 000313FCC613E01D1F799E24>III<157F913801FFC05CEC07EFEC0FDFA2EC1F9FA2ED8F8091383F8000A292 C7FCA35CA2147EA314FE90387FFFFEA3D900FCC7FC1301A25CA41303A25CA31307A25CA3 130FA25CA3131FA25CA3133F91C8FCA4137E127CEAFE7CA213FC5BEAFDF0B45A6C5A6C5A 224181B218>I105 D108 D<3907F007F8391FF83FFE6D487E3A3EFEFC3F80393C7FF01F007813E0ECC00F00 70138039F0FF001F5B12E0C65A0001143FA24914005D1203157E4913FEA20007ECFC1E14 0101E0141CEDF83C120F167801C0EBF0F8EDFBF0EDFFE06E13C049EB3F00271F7A9E2C> 110 D<90383F80FFD9FFC313C002E713E03A01F7FFC7F001E3EB03F83903C3FE019138FC 00FCEB83F8380787F0A2EB07E01200010F1301A214C0A2011F1303A2028013F81507013F 14F0A29138000FE0151FD97F8013C09138C03F80EDFF00ECF1FE90B512F8495B9038FC7F 8091C8FC1201A25BA21203A25BA21207120FB57EA3262D7F9E28>112 D<3907F01FE0390FF87FF0391FFDFFF8393EFFF8FC383C7FE0007813C1148100701301EA F0FE15F8D8E0FC13F0000090C7FC1201A25BA21203A25BA21207A25BA2120FA25BA45B1E 1F7A9E21>114 DII121 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb msbm7 7.665 1 /Fb 1 83 df 82 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc cmbxsl10 10.95 12 /Fc 12 115 df46 D<913801FFE0021F13FE027FEBFF8049B612E0010715F090260FFE0713F890 261FF00113FC90383FC0006EEB7FFE496C133F01FF15FF6E7FA55CA25C6D5A011FC75A90 C813FE167F17FC16FF17F84B13F04B13E0A24B13804B13004B5A4B5A4B5AEDFFE04A5B4A 90C7FC4A5AEC0FF891391FF001F0EC7FC0ECFF804990380003E0EB03FC495AD90FE0EB07 C0495A4948130F90B7FC4816805A5A4816005A5A485DA2B7FC5EA2303C7BBB34>50 D<913803FFE0021F13FE027FEBFF8049B612C049018313E0902607FC0013F0D90FF0EB7F F8495AD93FF814FCA2497EA54AEBFFF8A24A5AD93FE014F06D485A90C714E04B13C04B13 804B13004B5A913801FFF891B512E0168016E016F89138003FFEED0FFF6F138017C06F13 E0A317F0A3EA07E0EA1FF8487E127F5D12FF17E0A24B13C0A249491380A249491300D87F C0495A6D48485A263FFC075B6CB65A6C15C0000392C7FCC614F8010F13802E3D7ABB34> I<137FEBFFC04813E05A5A14F05AA214E0A36C13C014806C1300EA00FC90C7FCAAEA07F0 487EEA3FFC7F127F12FFA66C5A5B6C5AEA1FC0142878A71D>58 D<011FB712FC49EEFFC0 19F019FC19FFD9001F90C76C13804B020F13C0847213E084023F17F0A25DA21AF8A2147F 4E13F05DA31AE002FF5D1AC05D4E13801A0060494C5AF0FFF84B01075B92B712C06106FC C7FC4916E003F0CAFC5DA45BA25DA45BA25DA45BA292CBFCA3007FB67EA4B7FC453E7DBD 47>80 D<0003BA12C0A35AA2DAC007EBC007D9FE001501D80FF8EE007F49EF3F80494915 1F5B48485DA290C7160F5A5D123E94C7FC127E127CA24BED1F005AC84991C7FCA4157FA2 5EA415FFA25EA45CA25EA45CA25EA45CA25EA348B8FC5FA25AA2423D76BC49>84 D101 DI104 D109 D111 D<903903FC0FF83A03FFF83FFE92B512 8002F914C0ECFBFD15F139001FFFC315831503A214FE4A6C1380013F1500ED007E4A90C7 FCA25CA2137FA25CA413FFA25CA45AA25CA3007F13FFB6FCA42A287DA72B>114 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd cmsy5 5.475 5 /Fd 5 113 df0 D<13E0A400E013E0EAF8E3EAFCE7B5FC001F13 00EA07FCA2EA1FFFB512E0EAFCE7EAF8E3EAE0E000001300A41314799421>3 D<1470B3ABB712F8A325207A9F31>63 D<12E0B3B3A9032D77A115>106 D<171C173CA2177817F817F0EE01E0A2EE03C016071780EE0F005E161E5EA25E16F85E4B 5A15035E00074A5AEA1F80003F4AC7FCB46C5B6D131E00075C6C6C137C15786C6C5B7F00 00495AEB7E035D6D485A148F011F90C8FCEB0FDEA2EB07FCA26D5A6D5AA26D5A2E2E7A82 32>112 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe cmbxsl10 11.9902 27 /Fe 27 122 df46 D<16F0ED03F81507ED3FF015FF140F017FB5FCA316E0EBFFE1EB0001A2 5CA216C0A45CA21680A45CA21600A45CA25DA4143FA25DA4147FA25DA414FFA25DA45BA2 5DA3007FB7FCA316FEB7FC284177C039>49 DII<171F177FA25F5E5E5E4C5A161F163F167F16 FF5D4B5B5DED0FBFED1F3F153E157C4B485AEC01F0EC03E0EC07C0EC0F80EC1F00023E49 5A5C14FC495A495A495A4948485BEB1F80EB3F00137E5B485A4848495B485A485A485A48 C7FC127EB912C0A5C8001FEB8000A294C7FCA45DA25EA349B7FC5FA25BA232407ABF39> I<0107ED0780D90FF0147F02FFEB0FFF92B61200A25F5F495D17E05F5F4CC7FC16F84914 E0168003F8C8FC91CAFCA35BA2137EA391380FFF80D9FE7F13F090B612FC16FFDAFC0F7F DAE0037F0280804848487E4981497F6C488190C8FC5D83A5EA07F0D81FF85B487E6D5D12 7FA25D00FF5EA25B4B5B6C485D5B01C0495B90C74890C7FC4B5AD83FC0495A261FE0035B D9FC0F5B6CB65A6C158000014AC8FC6C14F0011F90C9FC314278C039>I<923807FF8003 7F13F00203B57E020F804AEB83FE91397FFC007FDAFFF07F01039039C001FF8049495A49 495A49485B495AA2495A495A481700A248495C705A484990C8FCA25AA2EC803048903887 FFE0028F13FC02BF7F48497FDA7E037FDAF8017F4A6C7F4849804A137F835CA291C7487E A2485AA35BA25DA25F5BA34B5B127FA25F5D003F5EA24B5B6C6C4990C7FC6D495A6C6C49 5A6C9038C1FFF86C90B512E06C5D6C92C8FC013F13FC010313C0314277C039>III58 D<932603FFF814F093B6EA80010307EDE003033FEDF80F92B7EA FE1F02039139E00FFF7F020F9026FE000113FF023F01F06D6C13E04A01C0141F49B5C87E 4901FC814949814901E081494917C04949815B90B5CA127F5C485B48F13F805C5A5C5AA2 48491800A24896C7FC5CA25AA25CA3B5FCA291CDFCA71AF8A26C1801A2621903A26C6D4C 5AA26C180F6E4C5A6C616E163F6C6D4CC7FC6CEF01FE6E4B5A6C01FE4B5A6C6DED1FF001 3F01C0EC7FE06D01F8903803FF806D9026FF803F90C8FC010391B512FC010016F0023F15 C002034AC9FCDA001F13C04C4675C453>67 D<010FB712E0495EA4D9000701F8C8FC5EA4 5CA25EA45CA25EA45CA25EA45CA293C9FCA491B5FCA25DA449171F193E5DA2197CA25B19 FC4B15F8A21801A249EE03F018075DF00FE0181F183F49167F943801FFC04B5B170F94B5 FC007FB91280A31900BAFC40447CC345>76 D<010FB812C04917FEF1FFC01AF086D90007 9026F000077F4C01007F737E7313807313C05C854C16E0A45C615EA44A18C06193C81480 A24F1300A24A4C5A624B4A5B06075B061F5B95B5128091B848C7FC19F819E04EC8FC4BCB FCA25BA25DA45BA25DA45BA25DA45BA25DA3007FB612F8A4B7FC4B447CC34E>80 D<0003BB1280A35AA2DAE0019038FC001F02001603D80FFC8301F0711300494982498400 1F5E5BA290C7FC484A82123E5FA248611A3E5DA2485EC894C7FCA35DA25FA45DA25FA45D A294CAFCA492B5FCA25EA45CA25EA45CA25EA348B812F05AA260A2494375C250>84 D<91387FFFE00107B512FE011F6E7E4915E090267FF0037F6EC67F01FF6E7E163F707EA3 5CA26D5AEB3F8090C8FC163FED03FF020FB55A49B6FC130F017FEBF03F48B5EA007F4813 F8000F01E05C4813804890C7FC5B484814FF485A5F5B5DA25D5D6D495B6D5B6C6C90387F 7FF8273FFF83FEEBFFC06CD9FFFC14E06CECF03F0003DAC01F13C03A003FFE0007332C7B AB38>97 D<913803FFE0023F13FE49B61280010715C04901C113E0903A3FFE003FF0D97F F8EB1FF84948EB0FFC48491307484914FE485B48150391C7FC5AA2485AA348B7FCA4D8FF FCC9FCA25BA4127FA317F017F86C6C140116036C6CEC07F0000FED0FE06C6CEC3FC06C01 C0EBFF806CD9F00F13006C90B55A013F14F8010F14C0010001FCC7FC2F2C7AAB35>101 D104 DI108 D<9127FF800FFFEC1FFE48B5017F01E090B512C00381B5D8F80314F00383 6E488092268FF83F90381FF07F923D9FC01FFE3F803FFCD8000FD93F00EC7E006D017E5D 03F890260FFFF0131F4B5D49494949133FA24B5D4B92C7FCA292C75B49033F157FA24A4B 5DA4013F037F15FFA24A4B5DA4017F03FF5CA24A4B5DA401FF4A5DA24A4B5DA3007FD9FF E0B600C1B612804B03811500A2B600C11583A2592C7CAB5F>I<9139FF800FFF48B5017F 13E00381B512F803838092388FF83F92399FC01FFED8000FEB3F006D137E03F8130F5D49 49131FA25D5DA292C7FC49153FA24A5DA4013F157FA24A5DA4017F15FFA24A5DA401FF5C A24A5DA3007FD9FFE0B612C04B1580A2B612C1A23A2C7CAB40>I<913801FFE0021F13FE 91B612C0010381010F9038807FF8903A1FFC001FFCD93FF06D7E49486D7E4848487F4A15 8048814890C714C05A5B001F17E0A2123F495CA2127FA448484A13C0A418805E007F1700 A24C5AA2003F5E4C5A6C6C4A5A4C5A6C6C495B2707FF80075B6C9026E01FFEC7FCC690B5 12F8013F14E0010F91C8FC010013F0332C7AAB39>I<913A1FF007FF80D93FFF017F13F8 03F1B512FE03F7809226FFFC0F13C0DCE00313E00100DA800113F04BC7FC4B15F803F014 7F4917FC183F5D19FEA35BA25DA449167FA24B15FCA218FF19F85B5F92C714F05F4D13E0 19C0494B13806F5B6F4913006FEB7FFC9239F801FFF8DBFE0F5B4990B612C04A6C91C7FC 4A6C13F80307138092CAFCA2137FA25CA413FFA25CA4007FEBFFC0A2B6FCA33F3F80AB40 >I<9138FF01FF48B5000713E04B13F04B13F892383FBFFCED7E3F3A000FFEFC7F903807 FFF0A215E04913C017F81580EE3FF09238001FC093C7FC495AA35CA3133FA25CA4137FA2 5CA413FFA25CA3007FEBFFE0A4B6FC2E2C7DAB2F>114 D<91383FFE0F0103B6FC130F13 3F90387FF00F9038FF80014890C712FE49147E485AA312076D147C6D6C130014F0ECFF80 15F815FF6C15C016E06C15F06C15F87F6D14FC130F1301EB000F9138007FFE151F003E14 0FED07FC003F1403A3486CEB07F8A26DEB0FF06D131F6DEB7FE09039FE01FFC0B71280ED FE00486C13F8D8F80F1380282C7CAB2D>I<143EA4147EA2147C14FCA21301A21303495A 130FA2131F137F13FF00035B000F90B512C0B7FCA4000101F0C7FC5CA45AA25CA45AA25C A45AA291C8FCA2151FA25A153E5BA3157EA25DA24A5A380FFF03EC8FF06CEBFFE06C5CC6 91C7FCEB1FFC223F77BE2D>I<013FB5D8C001B5FC19FE5F491480A2010001F0C7EA3F80 19006F143E60027F15FC6F5C1701023F4A5A606F130760021F4A5A6F131F95C7FC6E5CEE 807E177C17FC6E5CEEC1F016C36E5C16E7EEEFC06E5C16FF94C8FC5E805EA26F5A5EA25E 6F5AA25E151F4BC9FC153E157E157CD807805B381FE001D83FF05B007F13034A5A00FF5C 4A5A141F027FCAFC38FDE0FE387FC7F8EBFFF06C13C0000F90CBFCEA03F8403F80AB3C> 121 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff cmmi6 6.57 2 /Ff 2 62 df15 D61 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg cmmi8 8.76 4 /Fg 4 121 df<903803FF80131F137F4848C7FCEA03FC485AEA0FE0485A123F5B127F90 C8FCB512FCA348C8FCA45A7EA3127E127F383F8007381FC01F380FF07F3807FFFE6C13F8 38007FE0191F7C9E1F>15 D<010FB712F8A3903A003FC0001F17034B13011700147F1870 92C8FCA25CA25CA20101020F13F0A24A010E1300161E1303163E4A5BED01FC49B5FCA25E ECF001010F130016784A1370A2011F14F0A24A5B93C8FC133FA25CA2137FA291CAFCA25B A25BEA03FFB512FEA335327DB131>70 D<010FB67E17F817FE903A003FC001FF9338003F C04BEB1FE0170F027FEC07F0A292C7FC18F84A15F0170F5CA20101ED1FE0A24AEC3FC018 800103ED7F0017FE4AEB03F8EE1FF049B612C04CC7FCA29139F000FF80010F147FEE1FC0 4A80160F131FA25CA2013F141FA25CA2017F143F5F91C7FC180E49161EA249EDE03CD803 FF167CB500FC90381FF1F893380FFFF07013E0CA138037337DB13A>82 D<90397F807F803A01FFE1FFC04801FB13E03A07E3FFE3F0D80F8013C348481387003EEB 7F07003C137E4813FE16E0007001FC13C0C790C7FC1301A25CA21303A25CA2D87C07EB01 E012FE9138E003C0130FED0780D8FC1F130F3AF83FF01F0039FC7FF87E39FFF9FFFCD87F F05B391FE03FC0241F7D9E2C>120 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fh cmsy7 7.665 20 /Fh 20 115 df0 D<127E12FFA6127E0808799316>I<131EA213 1F131EA2003013030078EB078000FCEB0FC000FF133F397F9E7F80391FEDFE003807FFF8 C613C0013FC7FCEBFFC0000713F8381FEDFE397F9E7F8039FF1E3FC000FC130F0078EB07 800030EB0300000090C7FCA2131F131EA21A1B7B9D25>3 D6 D<12F012FCB4FC13C0EA3FF0EA0FFCEA03FF C613C0EB3FF0EB0FFCEB03FE903800FF80EC3FE0EC0FF8EC07FE913801FF809138007FE0 ED1FF8ED07FE923801FF809238007FC0161F167F923801FF80923807FE00ED1FF8ED7FE0 913801FF80DA07FEC7FCEC0FF8EC3FE0ECFF80D903FEC8FCEB0FFCEB3FF0EBFFC0000390 C9FCEA0FFCEA3FF0EA7FC048CAFC12FC12F0CBFCAA007FB71280B812C0A32A3979AB39> 21 D35 D<13FCEA01FEA41203A213FCA2EA07F8A3EA0FF0A213E0A2121F13C0A21380123F1300A3 127EA2127CA212FC5AA25A12300F217DA215>48 DI<91B512C01307131F137FD9FFC0C7FC4848C8FCEA03F8 EA07E0485A5B48C9FC5A123E5AA2127812F8A25AA2B712C0A400F0C9FCA27EA21278127C A27E123F7E6C7E7F6C7EEA03F8EA01FE3800FFC06DB512C0131F13071300222C79A531> I54 D63 D 72 D83 D<0070151C00F0153CB3A96C157CA2007C15F8007E14016C 6CEB07F0D81FE0EB1FE03A0FFE01FFC06CB612800001ECFE006C6C13F80107138026277B A531>91 D<903807FF80017F13F848B512FE0007ECFF80260FFE0113C03A1FE0001FE0D8 3F80EB07F0007EC7EA01F8007C140048157CA248153CB3A90070151C26277BA531>I<12 F0B3B3B3AA044077AF16>106 D<38E001C0EAF003B3B3B3A8EAE001124077AF25>I<12F0 A27EA21278127CA2123C123EA2121E121FA26C7EA212077FA212037FA26C7EA212007FA2 1378137CA2133C133EA27FA27F80A2130780A2130380A26D7EA2130080A21478147CA280 A2141E141FA2801580A2140715C0A2140314011A407BAF25>110 D<1807180F181FA2183EA2187C18FC18F8EF01F0A2EF03E0A2EF07C0A2EF0F80A2EF1F00 A2173E177E177C5FA24C5AA24C5AA24C5AA24C5A01C0141F000393C7FC0007153EEA1FE0 003F5DB47E00F75DEAC3F800034A5A6C7E4B5A6C7E4B5A017F130F5E6D6C48C8FCA29038 1FC03EA26D6C5AA26D6C5AA2903803F9F0A26DB45AA26D5B5D147F92C9FC80143E141E38 417A833C>112 D114 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fi cmex10 10.95 27 /Fi 27 114 df<141E143E147E14FC14F8EB01F0EB03E01307EB0FC01480131FEB3F00A2 137EA25BA2485AA2485AA212075BA2120F5BA2121FA25BA2123FA448C7FCA812FEB3A412 7FA86C7EA4121FA27FA2120FA27F1207A27F1203A26C7EA26C7EA2137EA27FA2EB1F8013 0F14C0EB07E01303EB01F0EB00F814FC147E143E141E176C72832A>0 D<12F07E7E127E123E7E6C7E7F6C7E12037F6C7EA26C7EA2137EA27FA2EB1F80A214C013 0FA214E01307A214F0A21303A214F8A4EB01FCA8EB00FEB3A4EB01FCA8EB03F8A414F0A2 1307A214E0A2130F14C0A2131F1480A2EB3F00A2137EA25BA2485AA2485A5B1207485A5B 48C7FC123E127E5A5A5A176C7C832A>I<151F15FF1403EC0FFEEC3FF0EC7FC0ECFF8049 1300495A5C495AA25CB3AD130F5C131F133F495A495A000390C7FCEA07FCEA1FF8EAFFC0 90C8FCA213C0EA1FF8EA07FC6CB4FCC67F6D7E6D7E131F130F801307B3AD80A26D7E806D 7E6D1380EC7FC0EC3FF0EC0FFEEC03FF1400151F206C768335>8 D<12F8B4FC13E0EA3FF8EA0FFEEA03FFC67F6D7E6D7E131F6D7EA21307B3AD8013038080 6D7E6D1380EC7FC0EC3FF0EC0FFCEC03FF1400A21403EC0FFCEC3FF0EC7FC0ECFF804913 00495A5C5C13075CB3AD130FA2495A133F495A495A000390C7FCEA0FFEEA3FF8EAFFE090 C8FC12F8206C768335>I<00F0130FB3B3B3A5183B738133>13 D16 D<12F87E7E127F6C7E6C7E120F6C7E6C7E7F6C7E1200137E137F6D7EA26D7E6D7EA26D7E A26D7EA26D7EA26D7EA2147FA281143F81141FA281140FA281A2140781A21403A281A314 0181A46E7EA6ED7F80AAED3FC0B3A9ED7F80AAEDFF00A64A5AA45D1403A35DA21407A25D 140FA25DA2141F5DA2143F5D147F92C7FCA214FEA2495AA2495AA2495AA2495AA2495A49 5AA249C8FC137E5B1201485A5B485A485A121F485A48C9FC12FE5A5A22A37D8336>II<12F87E7E12 7F6C7E6C7E6C7E6C7E12036C7E6C7E7F137F6D7E131F806D7E1307806D7E6D7EA26D7EA2 147F6E7EA26E7EA26E7EA26E7EA26E7EA26E7EA28114008181A282153F82A2151F82A215 0F82A2150782A21503A282A2150182A4150082A4EE7F80A6EE3FC0A7EE1FE0AB17F0160F B3AC161F17E0ABEE3FC0A7EE7F80A6EEFF00A45E1501A45E1503A25EA21507A25E150FA2 5E151FA25E153FA25E157F93C7FCA25D5D14015DA24A5AA24A5AA24A5AA24A5AA24A5AA2 4AC8FC14FEA2495AA2495A495A5C130F495A5C133F49C9FC13FE5B485A485A1207485A48 5A485A48CAFC12FE5A5A2CDA7D8343>II I<17F81603160F163FEEFFF04B13C0030713004B5AED1FF84B5A4B5A4B5A4A5B4A90C7FC 5D4A5AA24A5AA24A5AA25D143FA25DB3B3B3A7147FA25DA214FFA292C8FC5B495AA2495A 5C130F495A495A495A495A4890C9FC485AEA0FFCEA1FF0EA7FE0EAFF8048CAFCA26C7EEA 7FE0EA1FF0EA0FFCEA03FE6C7E6C7F6D7E6D7E6D7E6D7E1307806D7EA26D7E7F81A2147F A281A2143FB3B3B3A781A2141F81A26E7EA26E7EA26E7E816E7F6E7F6F7E6F7E6F7EED0F FE6F7E030113C06F13F0EE3FF8160F160316002DDA758344>26 D[<173E177E17FEEE01 FCEE03F8EE07F0EE0FE0A2EE1FC0EE3F80EE7F005E5E4B5A15035E4B5A150F5E151F4B5A 5E157F93C7FC5D4A5AA24A5AA24A5AA24A5AA24A5AA24A5AA24A5AA34AC8FCA2495AA213 035CA213075CA2130F5CA2131F5CA2133F5CA2137FA25CA213FF91C9FCA25AA35BA21203 A25BA21207A45B120FA55B121FA65B123FA85B127FAD12FF5BB3B3A47F127FAD123F7FA8 121F7FA6120F7FA512077FA41203A27FA21201A27FA37EA280137FA280A2133FA280131F A280130FA2801307A2801303A2801301A26D7EA26E7EA36E7EA26E7EA26E7EA26E7EA26E 7EA26E7EA26E7E8182153F826F7E150F8215076F7E8215016F7E8282EE3F80EE1FC0EE0F E0A2EE07F0EE03F8EE01FCEE00FE177E173E>47 272 107 131 72 32 D[<12F87E7E127F6C7E6C7E6C7EA26C7E6C7E6C7E7F1200137F80133F6D7E80130F80 6D7E1303801301806D7EA26E7EA26E7EA26E7EA26E7EA26E7EA26E7EA36E7EA26E7EA282 157FA282153FA282151FA282150FA2821507A282A21503A2821501A282A381A21780A216 7FA217C0A4163F17E0A5161F17F0A6160F17F8A8160717FCAD17FE1603B3B3A4160717FC AD17F8160FA817F0161FA617E0163FA517C0167FA41780A216FFA21700A25DA35EA21503 5EA21507A25EA2150F5EA2151F5EA2153F5EA2157F5EA215FF93C7FCA24A5AA24A5AA34A 5AA24A5AA24A5AA24A5AA24A5AA24AC8FCA2495A5C13035C1307495A5C131F5C495A137F 91C9FC13FE12015B485A485A485AA2485A485A48CAFC12FE5A5A>47 272 125 131 72 I[27 272 104 131 53 I[27 272 127 131 53 I<0603B5FC067F14F80507B77E053F16F04CB812FE0407717E04 1F18E0047F18F84BB6D8000314FE030702E0D9001F6D7E4B49C8000180033F01F0DB003F 13F04B01C0040F7F92B5CA00037F020301FC050013FF4A01F0063F7F4A01C0060F7F4A49 727F4A48CC00017F4A48737F4A48747E4901E0F21FFE6F1A3F496D507E496D97B57E496D 4F806F61496E4E80D93FFB6D4E6D7E02F16D95381FFE3FD97FE06D4E486C7E49486C6C4E 486C7E6F6CF0FFF048496C6C4D496C7E4A6C6C4DEBC003486E6D4C01807F496D6D4C497E 00076E6D4C481580496D6D4C487F000F6F6C4C4815C0496E6C4C48147F706C4B5B48486E 6C4B49EC3FE0706D4A5B706D4A90C8FC48486E6D4A48ED1FF0706D4A5A716C4A5A496F6C 4A48150F007F706C494916F8716C495B716D485B496F6D4890C91207716D485A00FF706D 484817FC726C485A726C485A90CA6CB5481603725C725C7291CAFC725B725B735A4F7E4E 7F4E7F4E804E804E806DDC3FFC6D16074E486C7E4E486C7E007F4C496C6C17F84D496C7E 6D4B496C6D150F4D496C7F4D486D7F003F4C486D6D16F06D4B486E6C151F4D486E7E4C49 6E7E6C6C4A496E6CED3FE04C496E7F4C90C86C7F6C6C4A486F6DEC7FC04C486F7F6D4A48 706C14FF00074B48706C15806D4949706C5B00034A49706C15006D4949706D5A6C4A90CA 6C01C05B6E484871EBE0076C6D4848716D485A4B48F07FF86D6C4848726C485AD93FF149 726C485A02FB4995380FFF7F6DB54872495A6D91CC6C5C4B856D49735C6D497490C7FC6D 49745A4B1A1F6D01F8F27FFC6E6C505A6E6C4F5B6E6C6C06075B6E6D4E5B6E01F0063F5B 6E01FC95B5C8FC020001FF050313FC6F01C0040F5B6F01F0043F5B030F01FE4BB512C06F D9FFE0021F5C030102FF0103B548C9FC6F6C91B712F8041F18E00407188004014DCAFCDC 003F16F005071680DD007F02F8CBFC060391CCFC7E7F7B7F89>79 DIII88 D90 D<15C0EC03F0EC0FFC4A7E91387FFF8049B512E00107EB3FF890391FFC0FFE90393FE001 FF903AFF80007FC0D803FEC7EA1FF0D80FF8EC07FCD83FC0EC00FF48C9EA3F8000FCEE0F C000F01603321080C233>98 D104 DI<1CE01B011B 03A2F307C0A2F30F80A2F31F00A21B3EA263A263A2505AA2631A03A2505AA2505AA250C7 FCA21A3EA262A262A24F5AA24F5AA24F5AA24F5AA24FC8FCA2193EA261A261A24E5AA24E 5AA2611807131801384C5A137CD801FC4CC9FC487E0007173EEA1FFF003E5F5A26F07F80 5D1260C66C6C4A5AA26D6C4A5AA24D5A6D7E4D5A6D7E4DCAFC6D7E173EA26D6C5CA26D6C 5CA26E6C485AA24C5AEC3FC04C5AEC1FE05F91380FF00FA24CCBFCEC07F8163EEC03FC5E EC01FE5EA26E6C5AA26F5AA26F5AA25E151F93CCFC81536D76835B>112 D<1CE01B01A21B03A21CC01B07A21C801B0FA21C0063A21B1E1B3EA21B3C1B7CA21B781B F8A263A21A01A2631A03A2631A07A2631A0FA298C7FC62A21A1E1A3EA21A3C1A7CA21A78 1AF8A262A21901A2621903A2621907A262190FA297C8FC61A2191E193EA2193C197CA219 78A219F8A2611801A2611803A2611807130801185F0138160F133C017C94C9FC01FC5E12 01486C161E0007173E120FD80EFF163C121E48177C5A486C6C1578006017F812006D6C5D 1701A2606D6C1403A2606D6C1407A260170F6D7E95CAFC5F6D7E171EA2173E6D7E173C17 7C6D7E177817F8A26E6C5B1601A2DA3FC05B1603A25F91381FE007A25F91380FF00FA294 CBFC5EEC07F8161EA2913803FC3EA2163C167CEC01FE167816F8EC00FF5EA36F5AA36F5A A35E151FA293CCFC81A253A476835B>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fj cmmi5 5.475 25 /Fj 25 123 df14 DI19 D21 D26 D<127812FC12FEA3127E1206A2120E120C121CA21238127012 60070F788515>59 D<1438A21478A214F0A2EB01E0A214C01303A2EB0780A2EB0F00A213 0E131EA25BA25BA2137013F0A2485AA2485AA25B1207A248C7FCA2121EA2121C123CA25A A25AA25AA2152D7AA121>61 D<157015F0A214011403811406140CA214184A7EEC707C14 6014C01301EC807EEB030049133E13065B90381FFFFE815B903860001FA25B4848148000 03140F000F141FD8FFE0EBFFF85CA225207B9F2F>65 D<90B512FE48ECFFC016E03A000F 8007F8011FEB01FCED007C91C7123EA249143F161F133EA2137E163F137CA213FC167E5B 167C000115FC16F849EB01F015030003EC07E0ED0FC049EB3F000007EB01FE007FB512F8 B612E092C7FC281F7A9E32>68 D<48B512F815FF16C03A000F800FE0011FEB03F0150102 0013F8A25BED03F0133EED07E0017EEB0F80ED7F0090387FFFF85D9038FC03F8EC00FC49 137CA20001147EA25BA2000314FE16065B0007150ED8FFFE141EED7F3CED3FF8C8EA0FF0 27207A9E2F>82 D98 DIII<133C137CA313781300A6EA07E0EA1FF0EA3CF8127012 711261EAE1F0120313E01207A213C0120F1387EA1F86138E130E133CEA0FF8EA07E0101F 7C9E18>105 D<147014F8A214F0A21400A6133FEBFFC03801E7E0EA0383EA0703EA0607 120E000013C0A2130FA21480A2131FA21400A25BA2133EA2137EA2EA707CEAF8FCEAF9F8 485AB45A6CC7FC15287D9E1B>II<260F807F13FE3B3FE3FFC3FF 803B3BF787E78FC03A71FE03EE07D861FCEBFC03D863F8EBF807D8E3F013F0000314E013 E000070107130F178001C013C0161F000F010F140EEE3F0C01809038803E1CA217780100 9038001FF0EE0FC02F147C9338>109 D<3803E07E3907F9FF80390EFFC7C0391C7F03E0 38187E0101FC13F01238EA00F8A200011303A201F013E0A20003EB07C0A2EC0F80EC1F00 3807FC7EEBFFF8EBCFE001C0C7FC120FA25BA2121FEAFFF0A31C1D7E9322>112 D<380F81FC383FE7FF383BFF0F3971FC07803861F80FEA63F000E31400EA03E091C7FC12 07A25BA2120FA25BA390C8FCA219147C931F>114 DI<13F0A21201A412035B387FFF80B5FCA23807C0 00A3120F5BA3121F90C7FCA2EB0380003F1300EA3E07130EEA3F3CEA1FF8EA07E0111C7C 9B1A>I<3803F07E380FFCFF383C3FC739381F87800070130FEA603FD8E03E1300120091 C7FC137EA2137C12789038FC038000F81400495A00F1130E38F3FC3C387FBFF8383E0FE0 19147B9324>120 D<3907E003C0EA1FF0383CF8071270A20061130FD8E1F01380120313 E0141F0007140013C0A25C143EA214FEEA03E16CB45A6C137C130014FC000F5BEA1F0149 5A495A381E1FC0D81FFFC7FCEA07F81A1D7C9321>I<3801F8063803FE1E3807FFFC4813 F814F0C712E0EB01C0EB0780EB0E00133C5B13E0EA03C03807001C000E1338381FFFF848 13F04813E038F07FC038E01F0017147B931F>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fk cmr5 5.475 10 /Fk 10 127 df<1338137013E0EA01C01203EA078013005A121EA25AA35AA412F85AA97E 1278A47EA37EA27E7E1380EA03C01201EA00E0137013380D2D7AA119>40 D<12E012707E7E121E7E7E1380EA03C0A2EA01E0A3EA00F0A413F81378A913F813F0A4EA 01E0A3EA03C0A2EA078013005A121E121C5A5A5A0D2D7CA119>I<13FE3803FF80380F83 E0381E00F0001C1370003C1378007C137C0078133CA300F8133EAB0078133CA3007C137C 003C1378A26C13F0380F83E03803FF803800FE00171F7D9D1F>48 D<137013F0120712FFA212F91201B3A3B512C0A3121E7A9D1F>III<127812FCA412781200A81278 12FCA4127806147B9312>58 D<127812FCA412781200A8127812F812FCA3127C120CA212 1CA2121812381270A21260061D7B9312>I61 D<38078040381FE0E0383FFFC048138000E01300EA403C13067B9E1F>126 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fl cmr8 8.76 68 /Fl 68 128 df<91393FF80FF0903A03FFFE7FFC010F90B5FC903A3FF03FFCFED97F8013 F8D9FF0013E0485A4914C04848157C92381F80005BAAB812C0A33B03F0001F8000B3A648 6C497E267FFF83B5FCA32F337FB22D>11 DI<91383FFDC00103B5FC130F90383FF01F EB7FC0EBFF00485A5B4848130FA25BAAB7FCA33903F0000FB3A6486CEB1FE03A7FFF81FF FEA327337FB22B>I<137813FCA212011203EA07F8EA0FE0EA1FC0EA3F801300127C5A5A 5A0E0E70B227>19 D 40 D<12F07E127C127E7E7E6C7E6C7E7F12036C7EA26C7EA2137CA2137E133EA2133F7F A31480130FA414C0A41307AA130FA41480A4131F1400A35B133EA2137E137CA25BA2485A A2485A12075B485A48C7FC5A127E127C5A5A124A7CB61E>I<123E127FEAFF80A213C0A3 127F123F1201A21203A213801207A2120F13005A123E5A127812700A177A8815>44 DI<123E127FEAFF80A5EA7F00123E09097A8815>I48 D<13075B133F13FF120FB5FC133F12F012 00B3B2497EB612C0A31A3079AF27>II< EB3FF848B5FC3907F83FC0390FC01FE0391F0007F0001E14F8393F8003FCA213C01401A2 EB8003121FC7FC15F8140715F0140FEC1FE0EC3FC0903801FF8090383FFE0014F814FF90 38003FC0EC1FE0EC07F8EC03FCA2EC01FEA2EC00FFA3127EB4FCA4EC01FEA2481303007C 14FC007EEB07F8393F800FF0391FF03FE00007B512C00001140038003FF820317DAF27> II<001E14E0381FE00F90B5FC15C015801500 14FC14F0D81C7FC7FC90C8FCA9EB1FF0EBFFFE001D7F391FF87F809038C01FC09038800F E090380007F0001E1303C713F8A2140115FCA4127E12FEA4EC03F8A212F00078EB07F000 7CEB0FE06C131F391F803FC09038E0FF803907FFFE006C13F838007FC01E317CAF27>I< 903801FF80010F13E090381FC3F090387F00F801FC137C3901F801FCEA03F0EA07E0EA0F C0A2EA1F8091C7FC123F90C8FC5AA3007E13C038FE0FFE90383FFF804913E09038F80FF0 39FFF003F813E09038C001FC90388000FEA290C7127E157FA25AA3127EA2127FA27E157E 15FE6C7EEC01FCEA0FC09038E003F83907F007F03903FC1FE06CB512806C6C1300EB1FF8 20317DAF27>I<1238123E003FB61280A35A16005D0078C7123C0070147C5D5D48495A14 034A5AC7485A92C7FC141E143E143C5C14F85C13015C13035C1307A2130F5C131FA4133F 91C8FCA35BAA133E21327CB027>III<123E 127FEAFF80A5EA7F00123EC7FCAD123E127FEAFF80A5EA7F00123E091F7A9E15>I61 D64 D<15E04A7EA34A7EA24A7EA34A7E140EA2EC1C7FA2023C7FEC383FA202787F EC701FA24A6C7EA349486C7EA2010380EC8003A249486C7EA3010E6D7E010FB5FC4980A2 011CC77E496E7EA3496E7EA201F08149140FA200018200031507486C81D81FF84A7EB549 B512E0A333337DB23A>IIIIIII II76 DII80 D82 D<90383FF0073801FFFE48EBFF8F390FF03FFF381FC007EB8001 48C7FC007E80007C8000FC805A81A36C80A27E6C91C7FC6C7E13E013FC383FFFE06C13FE 6CEBFFC06C14F06C80C680011F7F01017FEB000F02011380EC007FED3FC0151FA200E014 0FA21507A27EA2150F6C15807E6CEC1F006C5C6D137E01E05B9038FE07F800F1B55AD8E0 7F13C0010790C7FC22347CB22B>I<007FB8FCA390398007F000D87E00153F007C820078 82007082A200F01780A2481603A6C792C7FCB3AA4A7E011FB512FCA331317DB038>IIII<003FB612FCA39038F8000701C0EB0FF849EB1FF090C7FC003EEC3FE0003C15C000 7C147F0078ECFF8016005C0070495A5D14074A5AC75B141F4A5A5D147F4A5A92C7FC5B49 5A5C1307495A5C011F141C495A5C137F495A91C7FC48153C485A5B1207484814785B001F 15F84848130115034848130F4848137F90B6FCA326327CB12F>90 D<3807FFE0001F13FC4813FF01807FEC3FC0141F6E7EEA1F00C71207A314FF133F48B5FC 3807FE07EA1FF0EA3FC0EA7F80130012FE16E05AA2140F7E6C131F387F807F3A3FE1FBFF C0381FFFF16C01E113000001EB00FC231F7D9E27>97 DI<90381FFF80017F13E048B512F03807FC07EA0FF0EA1F E013C0393F8003E0007F90C7FC90C8FCA25A5AA77E7EA26D13386C6C13786C7E6D13F038 0FF8013907FF0FE00001EBFFC06C6C1300EB1FFC1D1F7E9E22>I<153FEC07FFA3EC007F 81AFEB1FF890B5FC000314FF3807FC1F380FF0034848C6FC48487F497F48C7FCA312FEA9 127FA36C6C5B6D5B001F5B380FF0072607FC1F13806CB5EABFF8C6EBFE3FEB3FF025337D B22B>IIIIII107 DI<3C07C07FF001FFC028 FFC1FFFC0713F001C7D9FE1F7F903BCFC1FF3F07FC3B0FDF007F7C01D807DE6D486C7E01 FC5C496D48137E495CA2495CB1486C496C13FFB52703FFFC0F13F0A33C1F7D9E41>I<39 07C07FF039FFC1FFFC01C77F9038CFC1FF390FDF007FD807FEEB3F805B49131F5BA25BB1 486CEB3FC0B53803FFFCA3261F7D9E2B>II<3907E07FE039FFE3FFF801EF13FE9038FFC1FF3A0FFE007F806C48EB3FC001F0 EB1FE0A249EB0FF01507A216F81503A7150716F0150FA26DEB1FE06D133F6DEB7FC001FF EBFF8002C313009038EFFFFE01E313F801E013C091C8FCAA487EB5FCA3252D7E9E2B>I< 90381FF0079038FFFE0F0003EBFF8F3907FE1FCF390FF807FF381FF00148487E5B48487F 8190C7FC5A5AA77E7EA26D5B6C6C5B5C6C6C5A6C6C5A3907FC3FBF6CB5123FC613FCEB3F F090C7FCAAED7F80913807FFF8A3252D7D9E29>I<3807C1FC38FFC3FF01CF138014BF38 0FDE3FEA07DC13FC9038F81F0049C7FCA35BB0487EB512C0A3191F7E9E1E>I<3801FF8E 000F13FE123FEA7F81387E007E48133E48131E140E7E7E6C6CC7FC13FC387FFFC014F06C 13FC000F7F00037FEA003F01011380EB007F00E0133F141F6C130FA26C131F6C14006C5B 38FFC0FEEBFFFC00F75B00E013C0191F7E9E1E>I<1370A513F0A31201A212031207120F 121FB512FEA33803F000AF1407A6140F13F83801FC1EEBFE3E3800FFFCEB7FF8EB1FE018 2D7FAC1E>IIII<3A7FFF01FFF8A32607FE001380D801FCEBFE00000014F890387E01F0013F 5BEC83C090381FC780010F90C7FCEB07EFEB03FE5C1301130080497E497F9038079FC0EB 0F0F90381E07E06E7E013C7F9038F801FC00011300000314FF000F1580B46C4813FCA326 1F7F9E29>II<003FB512F0A3EB801FD83E0013E0003CEB3FC00038EB7F 800078EBFF005B00705B495A495A130FC6485A5C495A90387F807013FF4813005B485A48 4813F0120F4913E0381FE001EA3FC0387F800300FF131F90B5FCA31C1F7E9E22>I<007C 13F838FE01FCEAFF03A4EAFE01387C00F8160878B127>127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fm cmr6 6.57 7 /Fm 7 56 df<13381378EA01F812FFA212FE1200B3A9387FFFF8A315247AA321>49 D<48B4FC000F13E04813F0383F0FFCEA7C03387801FEEAFC006C137FA3143F147F1238C7 FCA214FEA2EB01FCEB03F8EB07F014E0EB0FC0EB1F80EB3E005B5B3801F007EA03E03807 800EEA0F00121E383FFFFE5AB512FCA318247CA321>II<1438147814F813011303A21307130E131CA21338137013E0EA 01C0A2EA0380EA0700120EA25A5A5A5AB612C0A3C7EAF800A790383FFFC0A31A247DA321 >I<003C131C383F80FC13FF14F814F014C01480EBFC00003CC7FCA5EBFF80003F13E0EB C3F0EB01F8383C00FC0038137EC7FC143FA4127C12FCA3147F147E00F013FE007813FCEA 7E01383F87F8381FFFE06C13C03803FE0018257CA321>II<1238123E003FB512C0A35A15803970 001F00141E5C48137C5CC6485A5C495A1307495A91C7FC5B131E133EA25BA313FCA25BA2 1201A86C5A1A267CA421>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fn cmti10 10.95 49 /Fn 49 122 df<933807FFC0043F13F093B512FC923903FE01FE923907F0003F4B5A4B48 13FF5E153F93C7FC5D037E14FE18FC03FE1400A25DA31401A25DA31403A20103B712F0A2 18E0903A0007F00007170F5D18C0A2020F141FA24B1480A2173FA2021F1500A24B5BA217 7E143F17FE92C7FC5FA24A1301181C027E14F8A20403133C14FEEFF0385C1878A2010116 F0A24A903801F9E0EE00FF18C00103ED3F804A91C7FCA313075CA2123F48485A12FF5CA2 49CAFC5BEAFC3E5BEA7FFCEA3FF0EA0FE0385383BF33>12 D 40 DI44 D<387FFFFEA3B5FCA21705799521>II48 D<1507150FA2151E153E157E15FEEC 01FC1407141F147F903803FFF8EB1FFB14E71407010013F0A2140FA215E0A2141FA215C0 A2143FA21580A2147FA21500A25CA25CA21301A25CA21303A25CA21307A25CA2130FA25C A2131FA2133FEBFFF0B612F0A215E0203C77BB2E>I<4AB4FC020713E0021F7F91383F87 F89138FE01FC903901F800FE4948137F5C4948EB3F80495A141CEB1E1EA2D93C0E14C0A2 1378A2D9F01EEB7F80A2EBE01C143C000115FF5C01C049130014F001E1495AEBE7E06CB4 48485A4A485AD97E005B90C7485A4B5A4B5A03FEC7FCEC03FCEC07F04A5AEC3F8002FFC8 FCEB01FCEB07F0EB0FC0495A013EC9FC49141C49143C485A48485C485A484814F890C748 5A001E1403D81FFF495A48EBF81F91B55AEA7C1FD878075C486C91C7FC01005B48EB3FFC EC0FF02A3D79BB2E>I<133FEB7F80EBFFC0A25AA21480A26C1300137E90C7FCB3EA1F80 487E127F12FFA56C5A007EC7FC122777A61C>58 D<1738173C177C17FCA21601A21603A2 1607160FA2161FA2EE3BFEA21671A216E1150116C1ED0381A2ED0701A2150E151E151C15 38831570825D14015D4A5AA24AC7FCA2140E141E021FB6FC5CA20270C77FA24A147F1301 5C13035C49C8FCA2130E131E5BA2137C01FC82000316FFD80FFF02037FB500E0017FEBFF 80A25C39407BBF44>65 D<49B712C018F818FE903B0007FE0001FF6E489038007F804BEC 3FC019E00207151FA24B15F0180F140FA25D181F141F19E04B143F19C0143FF07F804B14 FF4D1300027F4A5A4D5A92C7485AEF3FE04AECFF8092B548C7FC17F817FF4948C77FEF1F E04A6E7EA201036F7E844A1403A21307845CA2130F604A1407A2011F5E170F5C4D5A013F 5E173F4A4A5A4D5A017F4A90C7FC4C5A49C7EA0FF8486DEB3FF0B812C094C8FC16F83C3E 7BBD40>I<923A01FFC001C0031FEBF00392B538FC078002039038C0FE0F913A0FFE003F 1FDA1FF8EB0FBFDA7FE0903807FF00DAFF807F4990C7FC49488049485D49481400495A49 5A4948157C495AA24890C9FC48481678A2485A120F491670121F5B95C7FC123F5B127FA2 5BA212FFA25BA590CCFC170FA2170E171EA2171C173C5FA26C6C5D16015F6C6C4A5A1607 6C6C4A5A6C6C4AC8FC163ED807FC14FC6C6C495A3A01FF800FE03A007FE07FC06DB5C9FC 010F13FC010113E03A4072BE41>I<49B712C018F818FED9000790380003FF6E48903800 7F804BEC1FC0F00FE00207ED07F0A24BEC03F8F001FC140FA24BEC00FEA2141FA24B15FF A2143FA25DA2147F605DA202FF16FEA292C81203A25B19FC4A1507A2010317F8A24A150F 19F01307F01FE05CF03FC0130FF07F804A160060011F5E4D5A4A4A5A4D5A133F4D5A4AEC 3FC04DC7FC017F15FEEE03FC4948EB0FF0486DEB7FE0B8128004FCC8FC16E0403E7BBD45 >I<49B812F8A3D9000790C7123F6E48140F4B14031801020716F018005DA2140FA25D19 E0141FA24B1338A2023FEC7801A24B0170C7FC17F0147F16014B485A160702FF131F92B5 FC5FA249EB003F160F4A6D5AA21303A24A91C8FCF0038001074A1307A24A010EEB0F0093 C7FC010F160E181E5C60131F605C18F8013F4B5AA24A14034D5A017F150FEF3FC0494814 FF486D130FB95AA295C7FC3D3E7BBD3E>I<49B648B6FC495DA2D9000790268000071380 6E48C73803FE00A24B5D02071507A24B5DA2020F150FA24B5DA2021F151FA24B5DA2023F 153FA24B5DA2027F157FA24B5DA202FF15FF92B7FC96C7FCA24990C71201A24A5DA20103 1503A24A5DA201071507A24A5DA2010F150FA24A5DA2011F151FA24A5DA2013F153FA24A 5DA2017F157FA249484A5A4801E04913E0B66CB67EA3483E7BBD44>72 D<0103B6FCA3D900071380913803FE00A25D1407A25DA2140FA25DA2141FA25DA2143FA2 5DA2147FA25DA214FFA292C7FCA25BA25CA21303A25CA21307A25CA2130FA25CA2131FA2 5CA2133FA25CA2137FA2495A4813E0B67EA3283E7BBD23>I<49B6903807FFFE60A2D900 070180010113F06E48C81380F1FE004BEC01F802074B5A4E5A4BEC0F804EC7FC020F153E 18FC4BEB01F04D5A021F4A5A4D5A4B49C8FC177E023F14F84C5A4B485A4C5A027F495A04 3FC9FC4B5A4C7EECFF8103837F150F5D4990383E7FE0ED7C3FECFEF8DAFFF07F49EBE01F 158003007F4A130F495A834A1307A2010F8116035C83011F80A24A6D7FA2013F82177F5C 84017F153FA249484A7E486D4A7EB6D8800FEBFFC0A261473E7BBD46>75 D<49B612C0A3D900070180C7FC6E48C8FCA25D1407A25DA2140FA25DA2141FA25DA2143F A25DA2147FA25DA214FFA292C9FCA25BA25CA21303A25C18E001071501A24A15C0170313 0FEF07805CA2011FED0F00A24A5C173E133F177E4A5C1601017F4A5A160F4948133F486D 48B45AB8FCA25F333E7BBD39>I<49B5933807FFFC496062D90007F0FE006EEF3FF8EDBF 80F277F0020717EFA2033F923801CFE0A2020FEE039FF1071F020E606F6C140E021E173F 191C021C04385BA2023CEE707FA2023804E090C7FCF001C002785F6F6C495A02705FF007 0002F0ED0E01A202E0031C5BA20101EE380318704A6C6C5D18E001031707EF01C00280DA 03805BA20107923807000FA20200020E5C5F49D903F8141F5F010E4B5CA2011E4B133FA2 011CDAF9C05CEEFB80013C177F6FB4C7FC013895C8FC01785C4C5C13F8486C4A495AD80F FF4C6C7EB500F04948B512FE16E01500563E7BBD52>I<49B5020FB5FC495EA2D900036D 010113E09538007F80F13F0070143E4A163CEDBFE0033F1538A24A6C6C1478A2020E1670 6F7E021E16F01507021C6D5CA291263C03FE1301A202385E6F7E02781503A2DA70000180 5BA202F0017F130717C04A93C7FCEE3FE001015EA24A90381FF00EA20103020F131E17F8 4A151CEE07FC0107163CA291C73803FE38A2491678EE01FF010E167082011E16F0A2011C 6F5AA2133C173F01385E1378171F13F8486C6F5AEA0FFFB512F0170795C8FC483E7BBD44 >I<92391FF801C09238FFFE0302039038FF878091390FF81FCF91391FE007FF91383F80 034A486C130002FE7F494880495A4A147E0107153E5C130F4A143CA2131FA21738A28094 C7FC808080ECFF806D13F815FF6D14E016F86D806D806D6C7F141F020380EC003F03077F 1501ED007FA2163FA2161F12075AA2000E5EA2001E153F94C7FCA25E003F157E5E15016D 495A486C5C6DEB0FE06D495AD87DFC495A27F8FF81FFC8FC90383FFFFC486C13F0D8E001 13C032407ABE33>83 D<48B9FC5AA29139001FF803D807F84AC6FC494A133E01C0133F48 48161E5E90C7FC48147F121E4C131C5A15FF5A93C7FC183C485B1838485CC716001403A2 5DA21407A25DA2140FA25DA2141FA25DA2143FA25DA2147FA25DA214FFA292C9FCA25BA2 5CA21303A21307EB1FFF003FB67E5AA2383D71BC41>I<003FB5D8E001B512E0485DA2C6 01F0C7383FFC00D97FC0EC0FF0715A4A5D01FF5EA291C890C7FCA2485EA249150EA20003 161EA249151CA20007163CA2491538A2000F1678A2491570A2001F16F0A2495DA2003F15 01A2495DA2007F1503A2495DA200FF1507A290C890C8FCA25EA248151EA25EA25E16F85E 4B5A007F14034B5A6C6C495A6D013FC9FC001F147E390FF003F83907FC1FF06CB512C0C6 91CAFCEB1FF83B3F6EBD44>I<277FFFFE01B500FC90B512E0B5FCA200079026C0000F01 8090381FFE006C90C76C48C7EA07F86C484B6E5A04035E63631A0798C7FC0407150E1A1E 040F151C1A3C041F153862163B6D6016736C03F34A5A16E303014B5A04C31407DB038392 C8FC61DB0703140E715B150E04015C151C033C5D153803785D0370140103E05D1803DA01 C05D4EC9FCEC0380180EEC870060EB7F8E029E5D149C02B85D18F002F05D715A5C705B5C 95CAFC5C5F91C8FC017E5DA2017C5D5F13785F533F6EBD5B>87 D<14FF010313C0010FEB E7C090381FE3EF90383F81FF49C6FC01FE137F484814800003143F485AA248481400A248 485BA2003F147E5B007F14FEA2495BA200FF1301A201005BA21403163815F05A02071378 A291380FE070007E131F023F13F0007F137F6C9038FFE1E0EB81F73A1FC7E3F7C0390FFF C3FF6C018113803A01FE007F00252977A72E>97 DIIII<16FE923803FF805D92380FCFC0ED1F8F161FED3F3FA2157EA21780 9238FE1F0093C7FC5DA31401A25DA31403A25DA20103B512F8A390260007E0C7FCA3140F A25DA3141FA25DA3143FA292C8FCA35CA2147EA314FEA25CA31301A25CA31303A25CA313 075CA3495A123F127F00FF5B131F91C9FCA2133EEAFC7E5BEA7FF86C5AEA1FC02A5383BF 1C>III<14F8EB01FC1303A414F8EB01F090C7FCAD13FEEA03FF481380380FCFC0 130FEA1E07130F123CA21238EA781F1270EB3F8012F01400C65AA2137E13FEA25B1201A2 485AA25B00071338A213E0000F1378A2EBC07014F014E013811383EB87C0EBCF803807FF 005BEA01FC163E79BC1C>II108 DIII<903907F003FC90390FF80FFF90261FFC 3F1380903A3E7E7F1FC0013C9038FC0FE091383FF807017801E013F0A20170EBC0039038 F07F80030013F813E0027E1307000113FEA2C75AA20101140F17F05CA20103141FA24A14 E0A20107143F17C05CEE7F80130FEEFF005E1501496C485A5E6E485AED1FE090393FFE3F C0DABFFFC7FCEC1FFEEC07F049C9FCA2137EA213FEA25BA21201A25BA21203487EB512E0 A32D3A80A72E>I<02FF13E0010313C1010F13E390391FE3E7C090383F81FF49C6FC01FE 137F484814800003143F485AA248481400A248485BA2003F147E5B007F14FEA2495BA200 FF1301A201005BA21403A25D5A1407A24A5A007E131F143F007F137F6C495A1381EA1FC7 380FFFDF6CEB9F803801FE1FC7123FA292C7FCA25CA2147EA214FEA25CA21301497E90B5 12F05AA2233A77A72A>I<3903F801FE3A07FC07FF80260FFE1F13C090393F3F87E0001E EB7E0790381FF80F003CEBF01FA2003813E038783FC016C000709038800F8091C8FCEAF0 7FA2EA007EA213FEA25BA21201A25BA21203A25BA21207A25BA2120FA25BA2121FA25B12 0F232979A726>III<13FED803FF143E486D137E38 0FCFC0130FD81E0714FE130F003C5DA200381401EA781F00705DEB3F8000F014031400C6 485CA2017E130713FE5E5B0001140FA2495CA20003141FEEC1C0491481A2ED3F83A2EE03 80157FEDFF076D5A000149EB0F009038FC07DF3A00FE1F9F9E90397FFF0FFED93FFE5B90 390FF803F82A2979A731>I<01FEEB07E0D803FFEB0FF0481380390FCFC01F130FD81E07 14F8130F003C140FED07F000381403D8781F13011270EB3F8000F0140091C7FCC64814E0 A2017E130113FE16C05B00011403A249148015071203ED0F005B150E151EA25D5DA26C6C 5B14016D485A3900FF0FC06DB45A6D48C7FCEB07F8252979A72A>I<01FEED01F8D803FF 9138F803FC48EB8001260FCFC01407130FD81E07010314FE130F003CEDF003EF01FC0038 02071300D8781F167C00705DEB3F8000F0020F143C1400C6484A1338A2017E011F147813 FE4C13705B0001023F14F0A24991C712E0170100035C18C049017E1303A21880170715FE EF0F006D487E0001161E6D485C6D48EB807C3B00FF1F9FE1F8903A7FFF0FFFF06D486C5B 902607F8001380372979A73C>I<90390FF007F090393FFC1FFC90397FFE3FFE9039FC7E 7E3F3901F03F783A03E01FF07FD807C014FF01805BD80F005BA2001E15FEED807C001C15 00003C133FA2C790C7FCA25CA2147EA214FEA25CA2130116385C1678EA3E0300FF15704A 13F00107130116E090390FF803C0D8FE1F13073AFC3EFC0F8090397CFE3F00397FFC7FFE 9038F03FFC391FE00FF028297BA72A>I<13FED803FF147C486D13FC380FCFC0130FD81E 071301130F003C15F8A200381403EA781F007015F0EB3F8000F014071400C64814E0A201 7E130F13FE16C05B0001141FA2491480A20003143FA2491400A25DA2157E15FE1401A26C 6C485AEBFC0F3800FE3F90B5FC90383FFDF8EB0FF1EB0003A25D1407001F5C387F800F5D 141F5D00FF133F4948C7FC007E13FE387801FC387C03F8387F0FF0383FFFC06C5BD807FC C8FC263B79A72C>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fo cmmi7 7.665 40 /Fo 40 123 df11 D13 D<903803FF804913F05BEB1E1FEB1C07140391C7FCA2131E131FA2806D7E80A26D7E806D 7E497E133F90B5FCEA01FCEA03F03907E07F80EA0FC0EA1F80003F133F13005A007E131F 12FE143F481400A4147EA2007C5B127E383E01F8383F03F0381FC7E03807FFC0000190C7 FC1C2D7DAC21>II19 D<13FE6D7E6D7E133F6D7E130F80130780A2130380130180A21300808081A26E7EA2141F 81143F4A7E14FF1303903807F7F8EB0FE390381FC3FCEB3F83EB7F01D801FE7F3803FC00 48487F48487F48481480485A4848133F48C713C048141F4815E00078140F232C7CAB2B> 21 D26 D<48B6128012075A5A481500267F03C0C7FC12FC12F0EAE007A2 00005B130FA3131F91C8FCA35BA3137EA5137C211B7D9A22>28 D33 DI<127EB4FCA21380A412 7F1201A21203A213005A5A120E5A123C5A12700914798716>59 D II<1260 12F812FE6C7EEA7FE0EA1FF8EA07FE3801FF8038007FE0EB1FF8EB07FCEB01FF9038007F C0EC1FF0EC0FFCEC03FF020013C0ED3FF0ED0FFCED03FF030013C0163FA216FF03031300 ED0FFCED3FF0EDFFC0020390C7FCEC0FFCEC1FF0EC7FC04948C8FCEB07FCEB1FF8EB7FE0 3801FF80D807FEC9FCEA1FF8EA7FE0EAFF8048CAFC12F812602A2C79A539>I64 D<16F0A215011503A24B7EA2150F151FA2153B157BED73FC15E3EC01E115C1EC0381A2EC 0701140E82141CEC3C0014385C14F05C49487FA249B6FC5BA2010EC77E131E011C15805B 0178143F13705B12011207D81FF0EC7FC0D8FFFE90380FFFFEA25B2F2D7CAC37>I<013F B6FC17E017F8010190380003FC6D48EB00FE0101157E177F4A143FA21303A25C177F0107 157E17FE4AEB01FCEE03F8010FEC07F0EE1FC04AEB7F0091B512FC49809139E0003F804A EB0FC0EE07E0013FEC03F017F85C1601137FA291C71203A24915F0160749EC0FE0A20001 ED1FC0EE7F8049ECFF000003EC07FCB75A16E04BC7FC302C7CAB37>I<9238FFE007020F EBF80F023FEBFE1F9139FFE07F3F01039038001FFED90FF8130749481303D93FC0130149 4814FC49C8FC485A485A48481578485A5B121F491570123F5B007F1600A290CAFC5AA25A A4EE0380A21607A2EE0F00A2007E151E007F153E5E6C6C5C6C6C495A6D495A6C6CEB0FC0 D807FCEB3F802703FF81FEC7FCC6EBFFFC013F13E0010790C8FC302E7BAC34>I<013FB6 FC17E017F8010190380003FC6D48EB00FE0101153FEF1F804AEC0FC0A20103ED07E0A24A 140318F01307A25CA2130F17075CA2131F18E04A140FA2013F16C0171F5C1880017FED3F 00A291C8127E5F5B4C5A494A5A4C5A0001ED1FC04CC7FC4914FE0003EC0FF8B712E01680 03FCC8FC342C7CAB3B>I<013FB7FCA30101EB00016D48EB007F0101153E171E5C170E13 03A25CA20107ECF01EA24AEBE0001501130F15039138E00FC091B5FC5BA29138C01F8015 07133FA2028090C7FCA2137F92C8FC91C9FCA25BA25BA21201A25B487EB512FCA25C302C 7CAB2E>70 D78 DI<013FB512F816FF17C001019038000FF06D48EB 03F801016E7E16004A80A21303A25CA201075D16014A5C4C5A010F4A5A4C5A4A013FC7FC ED01FC49B512F0A29138C00FF8ED03FC013F6D7E15004A7F82017F5CA291C75AA2491301 A2495CA20001EE03C0A249ED0780487EB500F0EC0F00923800FF3EEE7FFCC9EA0FF0322D 7CAB37>82 D<90261FFFFC90B5FC5EA201009039F0007FF0DA3FC0EB3F80EF3E006E6C5B 17F06E6C485A4C5A6E6C485A4CC7FC913803FC1E161C6F5A02015B5E6E6C5A5E157FA26F 7EA24B7E15FF913801CFF0140391380787F8EC0F07141E4A6C7E14784A6C7EEB01E04948 6C7E495A49C76C7E133E496E7EEA01FCD807FEECFFF0B5D88003EBFF805D1800382C7DAB 3C>88 D97 DIII 102 DI<137FEA0FFFA25B1200A35BA21201A25BA21203A25BA200 07EBFF8001F713E09038EFC7F0EBFF03390FFC01F813F813F013E0001F130313C001805B A2003F1307A201005B140F5A91381FC380127E158700FE133F1600485C151EEC1FBEEC0F FC48EB07F0212C7BAB2B>III<137FEA0FFFA25B1200A35BA21201A25BA21203A25BA20007EB07F0 EC1FF89038E03E78EC78F8380FE1F1EBE3E3EBC783EBCF03EA1FDE9038FC01F049C7FC13 FF4813C0EB9FF0EB07F81303EA7F011538127EA200FE1478157000FC14F0A2903800F9E0 ECFFC048EB3F801D2C7BAB27>I<3907F00FF8390FFC3FFE391EFEFC7F393C7FF03F0038 9038E01F80007813C0007013801400D8F0FE133FA2C6481400A200015CA249137E15FE12 03913801FC1C13F0EDF83C00071303167801E013F016F0913801FBE0913800FFC049EB7F 00261B7E9A2D>110 D115 DI<9038FF03F80003EBC7FE 3907E7EF9F390F83FE1FD81F01133F001EEBFC7F003C13F81238EA7803153E0000EBF000 A21307A25CA2130F003E140E007F13C0151EEAFF1F153CD8FE3F137CD87C7F13F89038FF F3F0393FF3FFC0D81FC01300201B7D9A29>120 DI<90383F80789038FFC0F848EBF0F048EBFFE0A24814C09038800F809038001F 00C7123E5C495A495AEB07C0495A013FC7FC137E13F8484813E03803E0013807C003380F 800F48B512C05A486C1380D87C1F130038F00FFCEB03F81D1B7D9A23>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fp msbm10 10.95 5 /Fp 5 83 df67 D69 D80 DII E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fq cmmi10 10.95 55 /Fq 55 123 df11 D13 DII<133E137E13FEA31201A25BA21203A25BA21207A25B120F A25BA2121FA25B123FA25B007F130EA2EB001E5A143C5A147814F838FC01F0EB03E0EB0F C038FE3F80387FFF00EA3FFCEA0FF017297BA720>19 D21 D23 D<4A7E4A7EA25DA5EDFFF0020F13F8143F91B5FC1303903907FE 7FE0D90FF8C7FC495A495A495AA2495AA24890C8FCA25BA76C7EA290387FBFFCECFFFE7F 7F5B90387F7FF8D801FCC8FC485A5B485A485A485A90C9FC5A123E127E127C12FCA25AA2 7EA37E7E7F13E0EA7FF813FF6C13C06C13F86C13FE6CEBFFC0000114F06C6C13FC130F01 037FEB007F141F14031400157EA3131E90381F80FCECE1F86DB45A01035B9038007F8025 537EBF28>I<013FB7FC90B8128012035A481700A2281FE03803C0C7FC383F0078127E00 7C5D48EB70074813F05AEA000114E0150F010391C8FCA214C01307A2010F5B02807FA213 1FA2EB3F00A24980137E13FEA20001815B820003140F5BA26F5A6C485C31287DA634>I< ED3FE0913801FFF8020713FE91380FF0FF91393FC07F8091387F803F9139FE001FC04948 130F010315E05C495A130F4A14F0131F5C133F5C017F141F17E091C7FCA249143FA24915 C0167F120117804914FF170000034A5AA24B5A4B5A486C5C4B5A6DEB3FC06E485A270FF7 E1FFC7FC9038F3FFFC01E113F09038E07FC0001F90C9FCA25BA2123FA25BA2127FA290CA FCA25AA25AA3127C2C3B7EA72F>I<021FB6FC91B712801303130F491600137F9027FFF0 7FE0C7FC489038801FF03903FE000F496D7E48481303485A5B121F4848130115035B127F 90C7FCA2481407A2485D150FA25E151F5E153F5E4BC8FC007E5C007F495A6C495A6D485A 391FC01FE0390FF07FC06CB5C9FC000113FC38007FE031287DA634>I<013FB612C090B7 12E012035A4816C05AD9C00FC8FC383F001E127E127C48133E5A48133CC7127CA45CA313 01A25C1303A31307A25C130FA4495AA3133FA25C6DC9FC2B287DA628>I<1678A21670A2 16F0A25EA21501A25EA21503A25EA21507A293C7FCA25DA2150E913807FFF8023F13FF91 B61280010390381C3FE0D90FF8EB0FF0903A1FE03C07F8D93FC0EB01FC9038FF00384848 EC00FE48480178137E49157F48481370120F484801F0133FA248485B177F387F8001A201 005BA248010314FFA2484A13FE1601020714FC1603030013F8007E15074AEB0FF0007FED 1FE0020E14C0D83F80EC3F80021EEBFF00D81FC0495A3A0FE01C07FCD807F8495A3A03FE 3C7FE06CB612806C6C49C7FC010F13F0D90078C8FCA21470A214F0A25CA21301A25CA213 03A25CA21307A230527CBE36>30 D32 DI<123FEA7F80EAFFC0A6EA7F80EA3F000A0A79 8919>58 D<123FEA7F80EAFFC0A313E0A3127F123F1200A41201A213C01203A2EA0780A2 120F1300121E123E5A127812700B1C798919>I<1806181F187FEF01FFEF07FEEF3FF8EF FFE00403138093380FFE00EE3FF0EEFFC0030390C7FCED0FFCED7FF0913801FFC0020790 C8FCEC1FFCEC7FE0903801FF80D907FEC9FCEB1FF8EBFFE000031380D80FFECAFCEA3FF8 EAFFC090CBFCA213C0EA3FF8EA0FFE3803FF80C613E0EB1FF8EB07FE903801FF80903800 7FE0EC1FFCEC07FF020113C09138007FF0ED0FFCED03FF030013C0EE3FF0EE0FFE933803 FF80040013E0EF3FF8EF07FEEF01FFEF007F181F1806383679B147>II<126012F812FE 6C7EEA7FE0EA1FFCEA07FF000113C038007FF0EB0FFCEB03FF010013C0EC3FF0EC0FFE91 3803FF80020013E0ED3FF8ED07FE923801FF809238007FE0EE1FF8EE07FF040113C09338 007FF0EF1FFCEF03FF1700A21703EF1FFCEF7FF0933801FFC004071300EE1FF8EE7FE092 3801FF80DB07FEC7FCED3FF8EDFFE002031380DA0FFEC8FCEC3FF0ECFFC0010390C9FCEB 0FFCEB7FF03801FFC0000790CAFCEA1FFCEA7FE0EAFF8048CBFC12F81260383679B147> I<4AB47E020F13F091383F83FC9138FC007ED901F0EB3F804948EB1FC04A130F4948EB07 E049C7EA03F080D91FE0EB01F88017FC013F1400A25C17FE6D5A90C9FCA515E091383FFF 0191B51281010314C190390FF803E190391FE000F14948137B495AD801FEC7EA3BFC161F 485A485A4848EC0FF8A2485A17F0003F151F5B007F16E0A249143F17C012FFEE7F8090C8 FC17005E5E15015E6C4A5A15075E6C6C495A4B5A6C6CEB7F806C6C49C7FC9038F003FE39 07FC0FF80001B512E06C1480D91FFCC8FC2F417CBF30>64 D<17075F5FA24D7EA2177F17 FFA25EA25E5E177F160E84161C173F16381678167016E0A2ED01C015031680DB07007FA2 030E131F151E151C5DA25D15F05D4A4880A24A48130F140792C7FC4AB6FCA25C023CC712 0F14385C845C1707495A495AA249C8FC5B131EA2013E8213FE0001160F260FFF80EC3FFF B500F00107B512FCA219F83E407DBF44>I<49B712F8F0FF8019E0D9000390C7EA1FF06E 48EC07F84B6E7E727E020382844B1680A20207167FA24B15FFA2140F1A004B5C61021F15 03614B4A5A4E5A023F4B5A4E5A4BECFF80DD03FEC7FC027FEC0FFC92B612E095C8FC18E0 4AC7EA0FF8EF03FE4AEC00FF727E010183183F4A82181F1303A24A82A21307614A153FA2 010F5F187F4A5E18FF011F4B90C7FC604A14034D5A013FED1FF04D5A4948ECFFC0496C01 0790C8FCB812FC17F094C9FC413E7DBD45>II<49B712F818FF19C0D90003903980003FF06E90C7EA0FF84BEC03FCF000 FE0203167FF13F805DF11FC01407F10FE05D1AF0140F19075DA2021F17F8A25DA2023F16 0FA25DA2027F17F0A24B151FA214FF1AE092C9123FA24918C0197F5C1A80010317FF1A00 5C4E5A01075F18034A5E4E5A010F160F4E5A4A5E4E5A011F4CC7FC18FE4A4A5AEF07F801 3FED0FE0EF3FC0494849B4C8FC496CEB0FFCB812F017C004FCC9FC453E7DBD4B>I<49B9 FCA3D90003903880000F6E90C712014BEC007F193F0203163E191E5DA21407A25D191C14 0FA25D1707021F4A133CA24B010E1300171E143FA24B5B177C027F14FC160792B55AA291 B6FCED800792380001F016005BA24A5CA201031401A24A5CA2010791C9FCA25CA2130FA2 5CA2131FA25CA2133FA2137F497EB612F8A3403E7DBD3A>70 DI<49B6D8C03FB512F8A2058015F0D900 0301C0C7387FF8006E90C8EA3FE0A24B5E0203167FA24B5EA2020716FFA24B93C7FCA202 0F5DA24B5DA2021F1503A24B5DA2023F1507A24B5DA2027F150F92B7FC61A2DAFF80C712 1FA292C85BA249163FA24A5EA20103167FA24A5EA2010716FFA24A93C8FCA2010F5DA24A 5DA2011F1503A24A5DA2013F1507A249484A5AD9FFF0EC1FFEB6D8E01FB512FCA34D3E7D BD4C>I<49B600C090387FFFF896B5FC5FD9000301C0C7001F13806E90C8380FFC001AF0 4BED1FC002034CC7FC197E4B5DF001F00207ED07E04E5A4B4AC8FC183E020F15FC4D5A4B EB03E04D5A021FEC1F80053EC9FC4B5B4C5A023F495A4C5A4B487E163F027F497E16FF15 83923887EFFC9138FF8F8792389F07FE157E4B6C7E4913F04B7E03C0804B7E4948C77FA2 4A6E7EA20107153F844A141F84130F717E5C1707011F8217034A81A2013F6F7FA249484A 7FD9FFF04A13F0B600E0017F13FFA34D3E7DBD4D>75 D<49B612F0A3D9000301C0C7FC6E 90C8FCA25D1403A25DA21407A25DA2140FA25DA2141FA25DA2143FA25DA2147FA25DA214 FFA292C9FCA25BA25C180E0103161EA24A151C183C130718785CA2010F16F017015CEF03 E0011F1507A24AEC0FC0171F013FED7F8017FF49481307496C013F1300B9FCA25F373E7D BD3E>I79 D<49B77E18F818FFD90003D9800113C06E90C7EA3FE04BEC0FF0727E02036F7E727E5DA2 020782A25DA2140F4E5A5DA2021F4B5A614B140F61023F4B5A4E5A4B4AC7FCEF01FE027F EC07F8EF7FE092B6128005FCC8FC91B7FCDB80037F92C76C7E717E496F7E170F4A811707 1303A25CA20107150FA25C60010F151FA25CA2011F153FF103805C1907133F051FEB0F00 49485ED9FFF0EDF83EB600E090380FFC7E71B45A050113F0CB6C5A413F7DBD45>82 DI87 D<91B6D88007B512C0A302039026FC0001EBFC00020001F06D13C06F486EC7FC4C147E6F 6C5C19F06F6C495A4E5A6F6C495A4EC8FC180E6F6C5B606F6C5B6017016F6C485A4D5A6F 018FC9FC179E17BCEE7FF85F705AA3707EA283163F167FEEF7FCED01E7EEC3FEED0383ED 070392380E01FF151E4B6C7F5D5D4A486D7E4A5A4A486D7E92C7FC140E4A6E7E5C4A6E7E EB01F0495A49486E7E130F013F8201FF4B7E000701F0023F7FB500FC49B512FE5EA24A3E 7EBD4B>I97 DIIII<167F923801FFC04B13E09238 07F3F0ED0FC7ED1FCF169F153FA2ED7F1FA217E0EE0FC04BC7FC5DA41401A25DA31403A2 0107B512FEA390260007F8C7FCA25DA3140FA25DA4141FA25DA3143FA25DA3147FA292C8 FCA35CA25CA31301A25CA313035CA4495AEA1F87007F5B13C712FFEBCFC0138FEB9F8013 1F4848C9FCEA7C7EEA7FFC6C5AEA0FE02C537CBF2D>III<147E14FEEB01FFA25BA214FEEB01 FC14F890C7FCAC13FF4813804813C03807CFE0EA0F87381F07F0121E123C130F12380078 13E0131F127000F013C0133F1200EB7F80A214005BA25B1201A25B1203140EEA07F8A2EB F01E120F143C13E01478A214F013E13807E7E0EBFFC06C13803800FE00183E7EBC1F>I< ED07E0ED0FF0151FA516E0ED0FC092C7FCACEC0FF0EC3FFC4A7EECFC7FEB01F049487E02 C01380EB0780130FEC007F131E1600131C013C5BA290C75AA21401A25DA21403A25DA214 07A25DA2140FA25DA2141FA25DA2143FA25DA2147FA292C7FCA25CA2003F5BEA7F8100FF 5B1383EB87F85C138F495A38FE7F80D87FFFC8FCEA3FFCEA1FF0245081BC25>II110 DI115 D<14F813011303A41307A25CA2130FA25CA2131FA25CA2B612F0A215E039003F 8000137FA291C7FCA25BA25BA21201A25BA21203A25BA21207A25BA2120FEC01C013E014 03121FEC078013C0EC0F005C143E5C000F5BEBE3F06CB45A6C5BC690C7FC1C3A7EB821> I<137F2601FFC0EB0F80486DEB1FC0EA07E7380F87F00103143FEA1E07003C1680A20038 157FEA780F170000705BD8F01F5CA2C648485BA2EC8001137F5E1400491303A2495CA200 011407171C4914F0A2030F133CA2EEE038031F1378153F6C6C017F137003FF13F09039FF 01F7E1D97F87EBF3E0903A3FFFC3FFC06D1381903A03FE007F002E297EA734>I<017FEE 07C02601FFC0011FEB0FE0486D90393F801FF0EA07E7380F87F0D81F03147F121ED83C07 ED000F180700384B1303EA780F4C130100705BD8F01F01011400A226003FC04914E0A2DA 80031401137F4C14C014004901071403A2494A148018070001140F1900494A5B180E181E A260031F147C6C6C6E1378033F5C6D90387FF801903B7F80FBFC03E0903B3FE3F1FF0FC0 90261FFFE0EBFF806D496C90C7FC01019038000FF83C297EA741>119 DI<137F2601FFC0EB0F80486DEB1FC0EA07E7380F87F00103143FEA 1E07003C1680A20038157FEA780F170000705BD8F01F5CA2C648485BA2EC8001137F5E14 00491303A2495CA200011407A2495CA2150FA25E151F153F6C6C137F4B5A6D5AEB7F876D B5FC6DEBBF80903803FE3F90C7127FA293C7FC00035CD81FC05B383FE0015D14034A5A5D 49485A49485A393C003F80003E01FFC8FC381F83FEEBFFF8000713E0000190C9FC2A3B7E A72D>I I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fr cmsy10 10.95 32 /Fr 32 126 df0 D<123FEA7F80EAFFC0A6EA7F80EA3F000A0A79 9B19>I<0070167000F816F06C1501007E15036CED07E06C6CEC0FC06C6CEC1F806C6CEC 3F006C6C147E6C6C5C6C6C495A017E495A6D495A6D6C485A6D6C485A6D6C48C7FC903803 F07E6D6C5A903800FDF8EC7FF06E5A6E5AA24A7E4A7EECFDF8903801F8FC903803F07E49 487E49486C7E49486C7E49486C7E017E6D7E496D7E48486D7E4848147E4848804848EC1F 804848EC0FC048C8EA07E0007EED03F0481501481500007016702C2C73AC47>I10 D15 D17 D<0203B7FC143F49B8FC13074990C9FCEB3FF0EB7F8001FECA FC485AEA03F0485A485A5B48CBFCA2123EA25AA21278A212F8A25AA77EA21278A2127CA2 7EA27EA26C7E7F6C7E6C7EEA01FC6C7EEB7F80EB3FF0EB0FFF6D90B7FC1301EB003F1403 91CAFCAE003FB9FCA4384779B947>I<180F183F18FF1703EF1FFCEF7FF0933801FFC004 071300EE1FF8EE7FE0923801FF80DB07FEC7FCED3FF8EDFFE002031380DA0FFEC8FCEC3F F0ECFFC0010390C9FCEB0FFCEB7FF03801FFC0000790CAFCEA1FFCEA7FE0EAFF8048CBFC 6C7EEA7FE0EA1FFCEA07FF000113C038007FF0EB0FFCEB03FF010013C0EC3FF0EC0FFE91 3803FF80020013E0ED3FF8ED07FE923801FF809238007FE0EE1FF8EE07FF040113C09338 007FF0EF1FFCEF03FF1700183F180F1800AE007FB812FEBAFCA3384779B947>20 D<12F012FCB4FC13C0EA3FF8EA0FFE3803FF80C613E0EB1FF8EB07FE903801FF80903800 7FE0EC1FFCEC07FF020113C09138007FF0ED0FFCED03FF030013C0EE3FF0EE0FFE933803 FF80040013E0EF3FF8EF07FEEF01FFEF007FEF01FFEF07FEEF3FF8EFFFE0040313809338 0FFE00EE3FF0EEFFC0030390C7FCED0FFCED7FF0913801FFC0020790C8FCEC1FFCEC7FE0 903801FF80D907FEC9FCEB1FF8EBFFE000031380D80FFECAFCEA3FF8EA7FC048CBFC12FC 12F0CCFCAE007FB812FEBAFCA3384779B947>I<0207B7FC143F49B8FC13074948C9FCEB 3FF0EB7F8001FECAFC485AEA03F0485A485A5B48CBFCA2123EA25AA21278A212F8A25AA8 7EA21278A2127CA27EA27EA26C7E7F6C7E6C7EEA01FC6C7EEB7F80EB3FF0EB0FFF6D90B7 FC1301EB003F1403383679B147>26 D<141EB3B3B300E0ED01C000FC150FD8FF80147F01 C014FFD87FF001031380D80FF8903807FC00D803FCEB0FF0D800FFEB3FC0013F91C7FC90 381F9E7E90380FDEFC6DB45A6D5B6D5B6D5BA26E5AA26EC8FCA2141EA3140CA32A517FBE 2D>35 D49 D<0207B512FC143F49B6FC13074948 C8FCEB3FF0EB7F8001FEC9FC485AEA03F0485A485A5B48CAFCA2123EA25AA21278A212F8 A25AA2B812FCA400F0CAFCA27EA21278A2127CA27EA27EA26C7E7F6C7E6C7EEA01FC6C7E EB7F80EB3FF0EB0FFF6D90B512FC1301EB003F14032E3679B13D>I<007FB5FCB612F015 FE6C6E7EC700037F9138003FF0ED07F8ED01FC6F7E163FEE1F80EE0FC01607EE03E0A2EE 01F0A2EE00F8A21778A2177CA2173CA2B812FCA4CA123CA2177CA21778A217F8A2EE01F0 A2EE03E0A2EE07C0160FEE1F80EE3F0016FE4B5AED07F8ED3FF0913801FFC0007FB65AB6 48C7FC15F06C14802E3679B13D>I<173CA2177CA217F8A2EE01F0160317E0EE07C0A2EE 0F80A2EE1F00A2163EA25EA25E15015E4B5AA24B5AA24B5AA24BC7FCA2153EA25D15FC5D 4A5AA24A5AA24A5AA24A5AA24AC8FCA2143EA25C14FC5C495AA2495AA2495AA2495AA249 C9FCA2133E137E137C5BA2485AA2485AA2485AA2485AA248CAFC5A123E5AA25AA25A1270 2E5274BF00>54 D<0070EE01C000F016036C1607A200781780007C160FA2003C1700003E 5EA26C163EA26C163C6D157CA26C6C5DA200035E6D1401A200015E6D140390B7FC6C5EA2 6D5D017CC7120FA26D4AC7FCA2011E141E011F143EA26D143C6E137CA26D6C5BA201035C ECE001A26D6C485AA201005CECF807A202785BEC7C0FA26E48C8FCA2EC1E1EEC1F3EA2EC 0FFCA26E5AA36E5AA36E5AA2324080BE33>56 DI<151C153CA2157CA2903803FC7890380FFFF8133F9038FE 07F03901F801F8D803F07F13E0D807C07F000F497E1380001FECDF80010013CF14074815 C0158FA2007E90380F87E0A21507141FA2141E00FE15F0143EA2143C147CA21478A214F8 A214F01301A214E0A21303A214C01307A21480007E15E0130FA21400D87F1F130FA2D83F 1E14C0A2133E001F158001BC131FA2D80FFC14005D6C48133E00035C15FC0001495A3900 FE07F048B512C092C7FCEBE3FCD803E0C8FCA25BA3244B7CC42D>59 D72 D<0438F0018004F8180303031907A270F00F00650307 61656570601C011C03030F18071C0F704D5A150E1C3F031E187F7116FF047FEE01FB031C 943803F3F81B07033C6D16E3043FEE0FC70338EF1F87F33F0703786D037E5BDB701F16FC 1A0103F0EFF80F03E06DEC03F0040FED07E00201EF0FC003C0EE1F8071DA3F005B020301 075D038016FE714948131F02075FDB00034A5A4F5A4A6E495A020E01014A485C021E6E49 C7FC61021C04FE143F023C6DEB81FC0238ED83F8027891387FC7F0F0CFE04AEDFFC0715B 494893C8FC715A49485D0030705A267C07805DD87F0F6F4882B5C849EEE1C0714816F749 92CA13FF1E0049745A491BF86C48F20FE0D81FE097C8FCEA078062437DBE6D>77 D83 D<0070EE01C000F01603B3B317 077E1880007C160FA26CEE1F00003F5E6C6C157ED80FE04A5A6C6C4A5AD803FCEC0FF06C B46CEB7FE03B007FF807FF806DB6C7FC010F14FC010114E09026001FFEC8FC32377BB53D >91 DI 94 D<15FF1407143FEC7FF0ECFFC0491300495A5C495AA25CB3A8130FA2495AA2495AEB FF804890C7FCEA0FFEEAFFF813C013F8EA0FFEEA01FF6C7FEB3FC06D7EA26D7EA21307B3 A880A26D7E806D7E6D13C0EC7FF0EC3FFF14071400205B7AC32D>102 DI<12F0B3B3B3B3B3 045B76C319>106 D<00E0131C6C133CB3B3B3B3B048131C165A75C32D>I<12F0A27EA212 78127CA2123C123EA2121E121FA27E7FA212077FA212037FA212017FA212007FA2137813 7CA27FA2131E131FA27F80A2130780A2130380A2130180A2130080A21478147CA2143C14 3EA2141E141FA26E7EA2140781A2140381A2140181A2140081A21578157CA2153C153EA2 151E151FA2811680A2150716C0A215031501225B7BC32D>110 D<1A071A0F1A1FA21A3E A21A7CA21AF8A2F101F0A2F103E0A2F107C0A2F10F80A2F11F00A2193EA261A261A24E5A A24E5AA24E5AA24E5AA24EC7FCA2183EA260A260A24D5AA201184B5A137801F85ED803FC 15071207486C4B5A123F486C4BC8FC12FD00F86D143E12E0C66C6C5CA2013F5D804C5A6D 7E4C5A6D7E4C5A6D7E4C5A6D7E4CC9FC6D7E163E7F6F5AA26E6C5AA291383FE1F0A29138 1FF3E0A26EB45AA26E5BA26E90CAFCA25D14015D14005D1578485B7A834C>112 D114 D<150FA24B7EA24B7EA24B7EA2EDF9F002017F15F0913803E07CA24A 487E020F133F4B7E4A486C7E4A80023E13074A6D7E02FC8049486D7E4A13004948147C01 07157E49488049486E7E49C86C7E013E1507496F7E01FC8248486F7E48486F7E4848167E 4848824848EE1F8048CAEA0FC0007EEF07E048EF03F04817016C1703007EEF07E06CEF0F C06C6CEE1F806C6CEE3F006C6C167E6C6C5E6C6C4B5A6C6C4B5A017C5E6D4B5A013F150F 6D6C4A5A6D6C4AC7FC6D6C147E0103157C6D6C5C6E13016D6C495A027C5C6E495A023F13 0F6E5C6E6C48C8FC6F5A0207133E6E6C5AA26E6C5A15F902005BED7FE0A26F5AA26F5AA2 6FC9FCA23C4F7BC047>125 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fs cmbx10 10.95 48 /Fs 48 122 df39 DI<12F012FC7E127F6C7E6C7E6C7E6C7E7F6C7E12017F6C7EA2 6D7E80133F80131F80130F80A2130780A36D7EA380A27FA41580B21500A45BA25CA3495A A35C130FA25C131F5C133F5C137F5C49C7FCA2485A5B1203485A5B485A485A485A48C8FC 12FE5A12F0195A7AC329>I45 DI48 D<140F143F5C495A130F48B5FCB6FCA313F7EAFE071200B3B3A8B712F0A5243C 78BB34>I<903807FFC0013F13F848B6FC4881000F15E0261FFC1F7F263FE0037FD98001 7F486C6C6C7E13F000FF6E7E6D7FA2178081A46C5A6C5AEA0F80C8FC4B1300A34B5AA24B 5A5E4B5A4A5B5E4A5B4A90C7FC4A5AEC1FF84A5A4A5AECFF80499038000F80495A5CD907 F0EB1F00495A495A495A49C75A13FC48B7FC485D5A5A5A5A5AB7FC5EA4293C7BBB34>I< 903801FFF0011F13FE017FEBFFC090B67E4801817F3A03FC007FF8D807F06D7E486C806D 131F486C807FA76C485C6C48133F6C5AC8485A5E4B5A4A5B4A5B021F5B902607FFFEC7FC 15F815FF16C0D9000313F09138007FF8ED3FFE6F7EA26F138017C0A28117E0EA0FC0EA3F F0487EA2487EA44B13C0A36C484913804915006C485B6D495A6CB448485A6C90B55A0003 15E06C1580D8003F49C7FC010313E02B3D7CBB34>II<000F157801E0EB03F801FF133F91B5FC5E5EA25E5E4BC7FC5D15 F015C04AC8FC0180C9FCA9903881FFE0018F13FC01BF13FF90B612C002817FD9FC007F01 F0EB3FF84980496D7E5BC87F81A21780A3120FEA3FE0487E12FF7FA41700495BA26C485C 49133F90C75BD83FC0495A3A1FF001FFF0260FFC075B6CB65A6C5D6C4AC7FC6C6C13F001 0F1380293D7BBB34>II<121F 7F13F890B712F0A45A17E017C01780A217005E485D007EC7EA03F85E007C4A5A150F4B5A 4B5A484AC7FC15FE5DC7485A14034A5AA24A5A141F5D143F5D147F14FFA24990C8FCA35B A25BA25CA2130FA5131FAA6D5A6D5A6D5A2C3F7ABD34>I<903801FFF0010F13FF013F14 C04980D9FFC07F489038003FF8D803FCEB0FFC48486D7E120F496D7EA2001F80A27FA27F 7F6D6C5A02C05BECF00702FC5B6CEBFF0FED9FF86CECFFF05E6C15806C92C7FC6C15C06D 14F06D80498048B67E488148487E260FFC1F1480261FF80714C048487E49C6FC007F023F 13E049130F00FF801501497F167FA2163FA36D15C0167F6C7EEEFF806C7E6D01031300D8 1FFE495A3A0FFFC07FFC6C90B55A6C5DC615C0013F91C7FC010313F02B3D7CBB34>I<90 3801FFE0010F13FE013F6D7E90B67E4801E07F489038803FF048496C7E48486D7E484880 003F140782485A17808112FF17C0A517E0A55D127FA25D6C7EA2001F5C6C6C5B6C6C5B6C 6DB5FC6CEBFFFB6C14F3013F13E3010F018313C090380018031400A25DD803F01580487E 487E486C15005D5E151F5E4B5A49495A6C4813FFD9F0035BD9FC0F13806CB6C7FC6C5C6C 14F86C6C13E0D90FFEC8FC2B3D7CBB34>II<16FCA24B7EA24B7EA34B7FA24B7FA34B7FA24B7FA24B7F15 7D157C03FC7FEDF87F020180EDF03FA20203804B7E0207814B7E020F81158082021F814B 7E4A81143E82027E81027C7F02FC814A147F49B77EA34982A2D907E0C7001F7F5C83010F 834A80011F8391C87EA24983013E81017E83017C81B500FC91B612FCA5463E7CBD4F>65 DI<9226 0FFFC0131E4AB500FC133E020F02FF137E023FEDC0FE91B712F101039138803FFB010F90 39F80007FF4901E01301490180EB007F4948C8123F4948151F4849150F48491507484915 034A15015A48491500A291CA127E5A5B127F193EA348481700AE6C7E193EA3123F7F6C18 7E80197C6C6D16FC7E6EED01F86C6D15036C6DED07F06C6DED0FE06D6C151F6D6C6CEC3F C06D01E0ECFF806D01F80103130001039039FF803FFC010091B55A023F15E0020F158002 014AC7FCDA000F13E03F407ABE4C>III<92260FFFC0131E4AB500FC 133E020F02FF137E023FEDC0FE91B712F101039138803FFB010F9039F80007FF4901E013 01490180EB007F4948C8123F4948151F4849150F48491507484915034A15015A48491500 A291CA127E5A5B127F193EA3484894C7FCAC0407B612FCA26C7EA393C7383FFE006C7EA2 7E80A26C7F7E806C7F6C7F6C7F6D6C157F6D6C6C14FF6D01E05B6D01FC5B01039038FFC0 1F010091B512F3023F15E1020FED807E02019138FE001EDA000F01E090C7FC46407ABE52 >71 D78 D80 D82 D<003FB912FCA5903BFE003FFE 003FD87FF0EE0FFE01C0160349160190C71500197E127EA2007C183EA400FC183F48181F A5C81600B3AF010FB712F8A5403D7CBC49>84 D<90383FFFE048B512FE00076E7E4881D9 FC0313F0486CC67F153F82151F82150F6C5A6C5A6C5AC8FC157F91B5FC131F90B6FC0003 EBFE0F4813E0001F1380383FFE005B485A485AA25BA3151F7F153F6C6C137F903AFC01FB FF80273FFF07F313FE6CEBFFE3000714C1000114803A001FFC003F2F287DA733>97 D<13FFB5FCA512077EB0EDFFF0020F13FE023F6D7E91B612E003817F9139FC007FFC02F0 131F4A6D7E4A6D7E4A15808218C0A28218E0AB18C05EA21880A24C13006E5B6E5C6EEB3F FC02FC495A9139FF83FFF0D9FC7FB55A496C1480D9F00F01FCC7FCD9E00113E0333F7DBE 3A>IIIIII<13FFB5FCA512077EB0ED3FFC4AB5FC020714C04A80EC1FE14A C66C7E147E02F8804A133F5CA25CA35CB3A3B5D8FE0FB512E0A5333F7CBE3A>II<13FFB5FCA512077EB092380FFFFEA5DB01FEC7FCED07FC4B5A4B5AED3F C0EDFF804A90C8FC4A5A4A5AEC1FF0143F4A7E4A7E8181A202F77F02C37F02837F02017F 806F7E6F7E826F7E816F7F6F7F6F7F83B5D8FC07EBFFC0A5323F7DBE37>107 D<13FFB5FCA512077EB3B3AFB512FCA5163F7CBE1D>I<01FFD93FFC903801FFE0B548B5 D8800F13FC4A6E487F020FDAF07F6D7EDA1FE0ECFF0791293F807FF9FC037F0007903B7E 003FFBF0016C01F8DAFFC0804A6D497EA24A92C7FC4A5CA34A5CB3A3B5D8FE07B5D8F03F EBFF80A551287CA758>I<01FFEB3FFCB548B5FC020714C04A80EC1FE14AC66C7E000713 7E6C01F8804A133F5CA25CA35CB3A3B5D8FE0FB512E0A533287CA73A>II<01FFEBFFF0B5000F13FE023F6D7E91B612E003817F9139FC007FFC0003 01F0133F4A6D7E4A6D7E4A15808218C082A218E082A95E18C0A34C1380A24C1300806E49 5A6E495A02FCEBFFF8DAFF835B92B55A02BF1480028F01FCC7FC028113E00280C9FCADB5 12FEA5333A7DA73A>II<3901FE03FE00FF90380FFF 80023F13E04A13F0153F9138FC7FF83807FFF07E14E014C0A2ED3FF09138801FE0ED0FC0 92C7FC91C8FCB3A2B6FCA525287DA72B>I<90383FFE1E0003B512FE5A121F383FF00FEB 800148C7FC157E12FE153E7E7F01F090C7FC13FF14FCECFF806C14E015F86C806C800007 806C1580C6FC010F14C0EB003F140300F81300153F7E151F7EA26CEC3F806D137F01E0EB FF009038FC07FE90B55A5D00FC14E0D8F01F90C7FC22287DA729>III119 D121 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ft cmr10 10.95 92 /Ft 92 128 df0 D<16E04B7EA24B7EA24B7EA2 4B7EA24B7EA203397FA203707FA24B6C7EA24A486C7EA24A486C7EA24A486C7EA2020E6D 7EA24A6D7EA24A6D7FA24A6D7FA24A6E7EA249486E7EA249486E7EA249C86C7EA2010E6F 7EA2496F7EA2496F7FA2496F7FA249707EA24848707EA24848707EA248CA6C7EA2000E71 7E000FB9FC4884A2481980A24819C0A2BB12E043407CBF4C>I<153CA3157EA315FFA34A 7FA34A7FA34A7F157F153FA2020E7F151FA2021C7F150FA202387F1507A202707F1503A2 02F07F4A7EA201018114C0810103815C167F01078191C7FC163FA2010E81161FA2498116 0FA249811607A24981160313F0830001816D4A1380487ED81FFE020F13C0B5D8E001B6FC A338407DBF3F>3 D<011FB612E0A3D900070180C7FCDA01FEC8FCA8913807FFC0027F13 FC0103B67E903A0FFDFE7FE0D93FE1EB0FF8D9FF01EB01FED801FE6E7ED803FCED7F80D8 07F8ED3FC0D80FF0ED1FE0001F17F0D83FE0ED0FF8A2007F17FC01C0150700FF17FEA800 7F17FC01E0150F003F17F8A2D81FF0ED1FF0000F17E0D807F8ED3FC0D803FCED7F80D801 FEEDFF00D800FF4A5AD93FE1EB0FF8D90FFDEB7FE00103B61280D9007F01FCC7FC020713 C0DA01FEC8FCA8913807FF80011FB612E0A3373E7BBD42>8 D<913803FFE0023F13FE49 B612C00107018013F0903A1FFC001FFCD97FF0EB07FFD9FFC001017F48496D7F4890C86C 7E48486F7E48486F7EA248486F7EA248486F7EA3007F834981A96C6C4B5AA46C6C4B5AA2 000F5FA26C6C4B5AA200035F6D153F00015F00005F6D157F6D93C7FCA26D6C14FE011F5D A226E00FC0903901F80380A2010715F0267003E0903903E00700A2010115C0A200785F6C 6C6C903807801ED83FFFEDFFFEA46C5FA3393F7CBE42>10 D<913A01FFC00FF8021F9038 F07FFE91B500FDB5FC49903981FFFE7F902607FC009038F8FF804948485BD91FE05CD93F C05CEB7F805F0200ED7F00496D90C8FC4980ADB97EA3C648C76CC8FCB3AE486C4A7E007F D9FC3FEBFF80A339407FBF35>I<4AB47E021F13F091B57E49EB81FC903907FC007E495A D91FE013FF49485AEB7F80A214005B496D5A167C93C7FCAA167FB8FCA33900FE00018182 B3AC486CECFF80007FD9FC3F13FEA32F407FBF33>I<913803FFC0021F13FF91B6FC4913 80903807FC01EB0FF8EB1FE0EB3FC0EB7F80A291C7FC49805BADB8FCA3C648C77EB3AE48 6CECFF80007FD9FC3F13FEA32F407FBF33>I<912603FF80EBFFC0021FD9F00F13F891B5 D8F87F7F49903A83FCFFC0FE903C07FC007FFE003F4948495A90261FE00101F0EB7F80D9 3FC04A13FFD97F805CA202005C5B4992C7EA7F006F153E7091C7FCAAF13F80BBFCA3C648 C76CC7FC197F193FB3AC486C4A6CEB7FC0007FD9FC3FD9FE1FB5FCA348407FBF4C>I<13 3E133F137F13FFA2EA01FEEA03FCEA07F813F0EA0FE0EA1FC01380EA3E005A5A12701220 10116EBE2D>19 D22 D<003FEB1F80397F803FC039FFC07FE0A301 E013F0A3007F133F003F131F0000EB0070A4000114F0A24913E000031301A239078003C0 A2000F130701001380001EEB0F00003E5B48133E0078133C007013381C1C7DBE2D>34 D<123FEA7F80EAFFC0A313E0A3127F123F1200A41201A213C01203A2EA0780A2120F1300 121E123E5A127812700B1C79BE19>39 D<147814F8EB01F01303EB07E0EB0FC01480EB1F 005B137E137C13FC5B1201485AA2485AA25B120FA25B121FA348C7FCA4127EA612FE5AB2 7E127EA67EA46C7EA3120F7FA212077FA26C7EA26C7E12007F137C137E7F7FEB0F8014C0 EB07E0EB03F01301EB00F81478155A78C323>I<12F07E127C127E7E6C7E120F6C7E7F6C 7E12017F12007F137EA27FA27F1480A2130F14C0A3EB07E0A4EB03F0A614F81301B21303 14F0A6EB07E0A4EB0FC0A31480131FA214005BA2137EA25B5B12015B1203485A5B485A12 1F48C7FC127E127C5A5A155A7BC323>I<150FB3AABA12F0A4C8000FC9FCB3AA3C3C7BB4 47>43 D<123FEA7F80EAFFC0A313E0A3127F123F1200A41201A213C01203A2EA0780A212 0F1300121E123E5A127812700B1C798919>II<123FEA7F80EAFF C0A6EA7F80EA3F000A0A798919>I48 D<14E013011303130F137FEA07FFB5FC139FEAF81F1200B3B3ABEB7FF8B612FCA31E3C78 BB2D>II< EB0FFE90387FFFC03901FC1FF03903E007FC48486C7E48C66C7E001E6D1380EA1FC0486C EB7FC07FA66C5A6C5AC8EAFF80A34A13005D14034A5A5DEC1FF0EC7FE090381FFF804AC7 FCECFFC09038001FF0EC07FC6E7E6E7E6E138016C0157F16E0A2ED3FF0A3121FEA7FC0A2 487EA4ED7FE0A24914C06C4813FF007CC713806C5B003F491300391FC007FE3907FC1FF8 6CB55AC614C0D91FFEC7FC243D7CBB2D>I<151E153EA2157E15FEA21401A214031407A2 140E141E141C143C1478147014F0EB01E014C01303EB078014005B131E131C133C5B1370 13F05B485A12035B48C7FC5A120E121E5A123812785AB8FCA3C73801FE00AC4A7E0103B6 FCA3283D7EBC2D>I<0007140701C0133F9038FC01FF90B6FC5D5D5D15E05D92C7FCEB7F F890C9FCACEB03FF011F13C0017F13F09038FF0FF89038F803FEEBE001496C7E49EB7F80 90C713C0C8123F16E0A2151FA216F0A4127F487E7FA416E049133F90C7FC007015C00078 147F007C15806CECFF006C5B6D485A390FE007FC3907F83FF00001B55A6C1480D91FFCC7 FC243D7CBB2D>II<123812 3C123F90B612FCA44815F816F0A216E00078C7EA03C00070EC0780150F1600151E48143E 5D15785DC712015D4A5A14075D4AC7FC5C141E143E143C147CA25CA21301A2495AA31307 A3130F5CA2131FA5133FAB6D5A6D5A263F7BBD2D>III<123FEA7F80EAFFC0 A6EA7F80EA3F00C7FCB3123FEA7F80EAFFC0A6EA7F80EA3F000A2779A619>I<123FEA7F 80EAFFC0A6EA7F80EA3F00C7FCB3123FEA7F8012FF13C0A5127F123F1201A41203A21380 A21207A2EA0F00A2121EA25A127C127812700A3979A619>I61 D63 D<15074B7EA34B7EA34B7EA34B7EA34B7E15EFA2913801C7FCA391380383FEA39138 0701FFA2020F80EC0E00A2021E80021C137FA2023C800238133FA20278800270131FA202 F0804A130FA2494880160791B6FC4981A2913880000349C77F82A2010E6E7FA3496F7EA3 496F7EA201788201F8151F6D82D803FE153FD80FFFEDFFFCB500F0011FEBFFF8A33D407D BF44>65 DIIIIIIII<011FB512FEA3D90007 13806E130080B3B3A5121FEA7FC0EAFFE0A45C5D140701C05B387F800F397E001FF86C6C 485A391FE0FFE00007B512806C49C7FC38007FF0273F7DBD2F>IIIIIII82 DI<003FB91280A3D9F000EBE001018090397FC0003F 48C7ED1FC0007C1707A200781703A300701701A548EF00E0A5C81600B3B14B7E4A7F010F B612FEA33B3D7DBC42>IIII89 D<003FB712F8A391C7123F01F8EC7FF001E015E0018014FF90C813C0003E 5C4B1380003C1600007C5C00784A5AA24B5A5E0070143F4B5AA24B5A5EC75A4A5B93C7FC 5C5D140F4A5A5D143F4A5AA24A5A5D5B495B92C8FC5B4A141C130F495A5C133F495AA249 48143C5C5A485BA24890C812785B000F16F8485A491401003F150349140F007F153F4848 EB01FF90B7FCA32E3E7BBD38>II< D801C013E00003130100071303390F8007C001001380001EEB0F00003E5B003C131EA248 5BA20070133800F01378A2481370A439FF807FC001C013E001E013F0A3007F133FA3393F C01FE0391F800FC01C1C73BE2D>I I<1318133C137E13FF4813803803E7C03807C3E0380F81F0381F00F8003C133C48131E48 130F00601306180D76BD2D>I<123FEA7F80EAFFC0A6EA7F80EA3F000A0A79BD19>III< EA01FC12FFA3120712031201B1EC07FE91383FFFC04A13F09039FDFE0FFC9039FFF003FE 9138C000FF4AEB7F8091C7123F4915C0EE1FE05B160F17F0A3EE07F8ABEE0FF0A317E016 1F6D15C0163F6DEC7F806EEBFF006E485A9039F3F007FC9039E1FC1FF801E0B512E0D9C0 3F5BC7D807FEC7FC2D407EBE33>I<903803FF80011F13F8017F13FE9038FF80FF484848 7E4848EBFF80485A485A485AA2485AED7F004848133E92C7FCA348C9FCAA7F127FA36C6C EB01C015036C7EED07806C7E6C6CEB0F006D5B6CB4137EC6EBC1FC6DB45A011F13E00103 90C7FC222A7DA828>IIIIIII<14FCEB01FEEB03FFA6 EB01FEEB00FC1400AC147FEB7FFFA313017F147FB3B3A5127FEAFF8014FF14FEA2EB81FC A2EB03F8387F0FF0383FFFE06C13803803FE00185185BD1C>III<2701F803FF49B47E00FF010FD9E00713F0023FD9 F01F7F913B7E0FF83F07FC0007903BF807FC7C03FE2703F9E003EBF0013C01FBC001FDE0 000280D9FFC07F01FF6D8191C75B4992C7FCA3495CB3A5486C496CECFF80B5D8F87FD9FC 3F13FEA347287DA74C>I<3901F803FF00FF010F13E0023F7F91387E0FF800079038F807 FC3903F9E0033901FBC00102807F01FF130091C7FC5BA35BB3A5486C497EB5D8F87F13FC A32E287DA733>I<14FF010713E0011F13F890387F81FE9038FC003F4848EB1F804848EB 0FC04848EB07E04848EB03F0001F15F8491301003F15FCA248C812FEA44815FFA96C15FE A36C6CEB01FCA36C6CEB03F8000F15F06D13076C6CEB0FE06C6CEB1FC06C6CEB3F803A00 7F81FE006DB45A010F13F0010090C7FC282A7EA82D>I<3901FC07FE00FF90383FFFC04A 13F09039FDFE1FFC3A03FFF007FE6C9038C001FF4A6C138091C7127F4915C0EE3FE04914 1FA217F0160FA217F81607A9160F17F0A2161F17E0163F6D15C0167F6DECFF806E481300 6E485A9138F007FC9039FDFC1FF801FCB512E0023F5BDA07FEC7FC91C9FCAD487EB512F8 A32D3A7EA733>I<49B4131C010FEBE03C013F13F09038FFC1F8489038007C7C4848133E 4848EB1FFC4848130F48481307491303123F5B007F14015BA212FF90C7FCA97F127FA27F 123F15037F6C6C1307000F140F6C6C131F6D133F6CB413FDC6EBC3F990387FFFF1011F13 C1903803FE0190C7FCAD4B7E92B512F8A32D3A7DA730>I<3901F80FF000FFEB3FF8EC7F FCECFDFE3807F9F1000313E13801FBC114819038FF80FCEC007815005BA35BB3A4487EB5 12FEA31F287EA724>I<90387FE0E03803FFFD000F13FF381FE07F383F001F007E130700 7C130312FC481301A214007E7E7E01C0130013F8387FFFC014FC6C7F6CEBFF806C14C000 0314E0C6FC010713F0EB007FEC0FF800E0130714036C1301A214007EA26C130115F06C13 036C14E0EBC00F9038F03FC000FBB5128000F1EBFE0038E03FF01D2A7DA824>I<131CA5 133CA4137CA213FCA2120112031207121FB612C0A3D801FCC7FCB3A215E0A8140113FE00 0014C0EBFF03EB7F07ECCF8090383FFF006D5AEB03FC1B397EB723>IIIIII<001FB61280A3EBE000D98001130049485A001E495A121C003C 495A4A5A143F00385C4A5A4A5AA24990C7FCC6485A13075C495A495A013FEB038014E0EB 7FC0EBFF80A24813004848130712075B4848140048485BA248485B4848137F4848485A90 B6FCA321277EA628>II126 D<003E131F007FEB3F8039FF807FC0A5397F003F80003EEB1F001A0977BD2D>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fu cmr7 7.665 25 /Fu 25 127 df0 D<140FA34A7EA24A7EA34A7EA34A7E14E7A2903801 C7F814C3A290380383FC148101077F14011400497F010E7FA2011E80011C133FA2498015 1FA24980150FA2498015070001815B15030003811207D80FF080D8FFFC90387FFFF0A32C 2D7DAC32>3 D10 D<1307130F131E133E137C13F8EA01F0A2EA03E0EA07C0A2EA0F80A2EA1F00 A3123EA3127E127CA412FCA35AAA7EA3127CA4127E123EA37EA3EA0F80A2EA07C0A2EA03 E0EA01F0A2EA00F8137C133E131E130F1307104079AF1C>40 D<12E07E1278127C7E7EEA 0F80A2EA07C0EA03E0A2EA01F0A2EA00F8A3137CA3137E133EA4133FA3131FAA133FA313 3EA4137E137CA313F8A3EA01F0A2EA03E0A2EA07C0EA0F80A2EA1F00123E5A12785A5A10 407BAF1C>I43 D48 D<130E131E137EEA03FE12FFA2EAFC7E1200B3AD13FF007F13 FFA3182A7AA924>III<14075C5C5CA25C 5C5BA2EB03BFEB073F130EA2131C13381370A213E0EA01C0EA0380A2EA0700120E5AA25A 5A5AB612FCA3C7EA3F00A7EC7F8090381FFFFCA31E2A7DA924>I<001EEB0380381FC03F 90B5FC150014FC5C14E0001C90C7FC90C8FCA8EB7FE0381DFFFC381FE0FEEB803F391E00 1F80001C14C0C7EA0FE0A2140715F0A3127E12FEA4140F15E012F80078EB1FC0007CEB3F 80003F137F391FC1FF00380FFFFC000313F0C613C01C2B7CA924>II<1238123E003FB512F8A35A15F015E039700003C0EC0780140F48EB1F00141E5C C7127C5C495A5C13035C1307495AA2131F91C7FCA25BA3137EA413FEAA137C1D2C7BAA24 >I<127E12FFA6127E1200AB127E12FFA6127E081B799A15>58 D<127E12FFA6127E1200 AB127C12FE12FFA5127F1203A21207A21206120EA2121C123C12381270A20827799A15> I61 D91 D93 D<15F8143FA314031401ACEB1FF190B5FC3803FC3F3807E00F381FC007EB8003383F0001 127EA212FE5AA77E127EA27EEB8003381FC007390FE01FFC3A03F87FFFC03801FFFD3800 3FF1222C7DAB28>100 DI108 D<3903E0FF80D8FFE313E09038EFC7F0EBFF03D80FFC7F3807F80113F0A2 13E0B0B5381FFFC0A3221B7D9A28>110 D<13E0A51201A31203A21207120F381FFFF0B5 FCA23807E000AC1438A7EBF078000313F013FD3800FFE0EB3FC015277EA61C>116 D<3807C008380FE01C381FF878383FFFF038787FE038E01FC038400F80160779AA24> 126 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fv cmbx12 13.14 56 /Fv 56 122 df<92260FFFF0EB3FFC4AB5D8FE03B51280020FDAFF8F14C0023F92B612E0 9126FFFE009138FC3FF0010301F0013FEBE07F49018049903880FFF84990C7B55AD91FFC 5D133F4A5D5C137F4A4BEB7FF0057FEC3FE0053FEC0F80051F91C7FCABBB12E0A526007F F0C7D81FFCC8FCB3B3A3007FB5D8F01FB512FEA54D4C7ECB48>11 D19 D44 DII<15F014011407141F 147FEB03FF137FB6FCA313FC1380C7FCB3B3B2007FB712E0A52B4777C63D>49 D IIIII65 DI<93263FFF80EB03C00307B500F81307037F02FF130F 0203B7EAC01F020FEEF03F023F903AFE007FF87F91B500E090380FFEFF4991C70003B5FC 4901FC1400010F01F0814949151F490180814990C97E4948824A825A484982485B1A7F48 5B1A3F5A4A171F5AA24A170FA25AA297C7FC91CDFCB5FCAF7E80A2F207C07EA280A26C19 0F806C1A801A1F6C7F1A3F6C6D18006C6D5F6C197E6E5F6D6C16016D6D4B5A6D01E04B5A 6D6DED1FE0010301FCED7FC06D01FF4B5A6D02E0010790C7FC023F01FFEB7FFC020F91B5 5A020316E0DA007F1580030702FCC8FCDB003F13C04A4D79CB59>IIII<93263F FF80EB03C00307B500F81307037F02FF130F0203B7EAC01F020FEEF03F023F903AFE007F F87F91B500E090380FFEFF4991C70003B5FC4901FC1400010F01F0814949151F49018081 4990C97E4948824A825A484982485B1A7F485B1A3F5A4A171F5AA24A170FA25AA297C8FC 91CEFCB5FCAE050FB712C07E80A36C93C7001FEBC000A280A27E807EA26C7F807E6C7F6C 7F806D7F6D6D5D6D7F6D01F85D01036D92B5FC6DD9FF805B6D02F01307023F01FFEB7FFD 020F91B5EAF87F0203EEE03FDA007FED800F03079138FE0003DB003F01C090C8FC524D79 CB61>III75 DIIIII82 DI<003FBB12C0 A5DA80019038FC001FD9FC001601D87FF09438007FE001C0183F49181F90C7170FA2007E 1907A3007C1903A500FC1AF0481901A5C894C7FCB3B3A749B812FCA54C4A7CC955>I II<003FB600FC 013FB61280A5D8000701FCC8D83FF8C7FC6D6DED1FE0626D6D4B5A6D6E147F4FC8FC6E6D 5C6E6D495A7013036E5E6E6D495A6E6D130F4E5A6E6D5C6E6D495A187F6E6E48C9FC6F6D 5AEFE1FC6F13E36FEBF3F86FEBFFF060816F5C608181707FA2707F707FA2707F4C7FA24C 7F4C7F93B67E5E4B486C7F03036D7FED07F84C6C7F4B486C7F151F4C6C7F4B486C7F4B48 6C7F5D4B6D804A486E7F14034A486E7F4B6E7F4A5A021F6F7F4A486E7F4B6E7F4AC8FC4A 6F7F4A6F80495A0103717FD90FFE707FB600FE027FB612FCA5564B7DCA5D>88 DI<01 03B5FC013F14F090B612FC4815FF48D9801F13C048D9C0077F03017F6F7F167F83163F83 6C49131FA2C648C7FC90C8FCA5ED3FFF021FB5FC49B6FC131F017FEBF01F48B512004813 F8000F13E0485B4890C7FCA2485A5B12FF5BA3163FA26D147FA26C6C14F76D01037F6C90 26C00FE713C06C9026F03FC713FF000790B512876CECFE03C66CEBF80101079038C0003F 38317DB03C>97 DI<91380FFFF091B6FC 010715C0011F15E0017F9038007FF0D9FFF8EBFFF8485B485B485B485B91C7FC5AEE7FF0 485AEE1FC0007F92C7FCA25B12FFAD6C7EA3123F7F6C163EA26C6D147E6E14FE6C16FC6C 6DEB01F86C01F813076C01FEEB1FF0903A7FFF80FFE0011F90B5128001071500010014F8 020F13C02F317CB037>II<9138 1FFF8049B512F8010714FE011F6E7E017F01077F9026FFF8017F489039E0007FF048496D 7E48498048151F91C76C7E5A48488182A2127F4916808212FFA390B8FCA401FCCAFCA512 7FA27FA2123FA26C6CED0F80A26C6D141F6C6D143FEF7F006C01F014FE6C6D130126007F FEEB07FC903A3FFFC07FF0010F90B55A0103158001004AC7FC020713E031317DB038>I< EDFFF0020F13FE023F7F91B61280010301F013C049138149484813E0EB1FFC14F8133F14 F0137FA24A6C13C06F1380EE3E0093C7FCABB77EA526007FF0C8FCB3B3A3003FB512F8A5 2B4C7DCB26>II< EB7FC0B5FCA512037EB3A2923803FFC0031F13F84B13FE92B6FC02C1010F7F9126C3F803 7FDAC7E0809138CF8001ECDF0002DE8102FE7F5C5CA25CA35CB3A9B6D8C07FEBFFE0A53B 4C7BCB44>II107 DI<90287FC001FFE0EC7FF8B5 010F01FC0103B5FC033F01FF010F14C092B6D8803F8002C101079026C07FC17F9126C3F8 039026E0FE007F00039026C7E001D9E1F8137F6C9026CFC000D9F3F06D7E02DFC7EBF7C0 02DEDBFF808002FE6E151F4A93C7FC4A5DA24A5DA34A5DB3A9B6D8C03FB5D8F00FB512FC A55E317BB067>I<903A7FC003FFC0B5011F13F84B13FE92B6FC02C1010F7F9126C3F803 7F0003D9C7E0806C9038CF8001ECDF0002DE8102FE7F5C5CA25CA35CB3A9B6D8C07FEBFF E0A53B317BB044>I<913807FF80027F13F80103B6FC010F15C090263FFE0113F0903A7F F0003FF849486D7E48496D7E48496D7E4890C76C138048486E13C0A2001F17E0003F17F0 4980A2007F17F8A400FF17FCAA007F17F8A46C6C4A13F0A2001F17E0A26C6C4A13C06C17 806E5B6C6D4913006C6D495A26007FF8EB7FF8903A3FFE01FFF0010FB612C0010392C7FC D9007F13F80207138036317DB03D>I<90397FC00FFFB590B512F002C314FE02CF809126 DFFC1F13C09126FFC0077F0003DA00017F6C496D7F4A6E7E4A814A6E7E5C711380A28319 C0A319E083AB5F19C0A34D1380A219005F6E5D6E147F6E4A5A6E495B6E495BDBC00F5BDB F83F138002EFB6C7FC02E314FC02E014E0DB1FFEC8FC92CAFCB0B612C0A53B467CB044> II<9039FF803F E0B5EBFFFC02837F02877FDA8FF3138003C713C00003EB9F076C13BEA214FC14F8A24A6C 1380A2923800FE004A90C7FCA45CB3A7B612E0A52A317CB032>I<90390FFFC0F090B512 FB000314FF5A380FFE01391FF0003FD83FC0131F48481307A290C712035A15017F7F7F01 F890C7FC13FEEBFFF86CEBFFC015F86C14FEEDFF806C15C06C15E06C15F0000115F86C7E 010F14FCEB007F020313FEEC007F151F00F8140F15076C14031501A27EA26CEC03FC7F6D EB07F86D130F01F8EB1FF001FFEBFFE091B512C0481500D8FC3F13FCD8F00313E027317C B030>I<14F8A51301A41303A21307A2130FA2131F133F137F13FF1203000F90B512F0B7 FCA426007FF8C7FCB3A7167CA916FC80013F14F8ECFE01011FEB03F090380FFF8F6DEBFF E06D14C001001400EC1FFC26457EC430>III120 D<007FB500C090387FFFE0A5C601F0C73803F8006E5D017F5E6E1407013F5E80170F011F 5E6E141F6D93C7FC6F5B6D153E6F137E6D157C6F13FCA26D6D5B16016D5DEDF803027F5C EDFC07023F5CEDFE0F021F5C15FF161F6E91C8FC16BF6E13BE16FE6E5BA26E5BA36E5BA2 6F5AA26F5AA26F5AA393C9FC5D153E157E157CD81F8013FC486C5B387FE001D8FFF05B14 035D14074A5A49485A007F017FCAFC383FC1FEEBFFFC000F13F06C13C0C690CBFC3B467E B041>I E %EndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: a4 %%EndSetup %%Page: 1 1 1 0 bop 30 513 a Fv(MET)-10 b(AST)g(ABILITY)40 b(IN)h(REVERSIBLE)d (DIFFUSION)j(PR)m(OCESSES)f(I.)82 705 y(SHARP)g(ASYMPTOTICS)h(F)m(OR)f (CAP)-10 b(A)m(CITIES)39 b(AND)h(EXIT)h(TIMES)987 1252 y(An)m(ton)g(Bo)m(vier)1713 1219 y Fu(1)1754 1252 y Fv(,)g(Mic)m(hael)f (Ec)m(kho\013)2699 1219 y Fu(2)2739 1252 y Fv(,)914 1398 y(V)m(\023)-59 b(eronique)42 b(Ga)m(yrard)1953 1365 y Fu(3)1993 1398 y Fv(,)f(Markus)g(Klein)2804 1365 y Fu(4)1375 1673 y Ft(Septem)m(b)s(er)29 b(11,)j(2002;)g(10:23)50 2009 y Fs(Abstract:)54 b Ft(W)-8 b(e)39 b(dev)m(elop)e(a)h(p)s(oten)m (tial)e(theoretic)j(approac)m(h)e(to)h(the)g(problem)d(of)j (metastabilit)m(y)f(for)50 2154 y(rev)m(ersible)27 b(di\013usion)e(pro) s(cesses)k(with)e(generators)i(of)g(the)f(form)g Fr(\000)p Fq(\017)p Ft(\001)15 b(+)h Fr(r)p Fq(F)d Ft(\()p Fr(\001)p Ft(\))p Fr(r)29 b Ft(on)f Fp(R)3138 2121 y Fo(d)3215 2154 y Ft(or)g(subsets)g(of)50 2300 y Fp(R)109 2267 y Fo(d)158 2300 y Ft(,)35 b(where)e Fq(F)46 b Ft(is)33 b(a)h(smo)s(oth)g(function)e(with)h(\014nitely)e(man)m(y)j(lo)s(cal)f (minima.)49 b(In)33 b(analogy)h(to)g(previous)50 2445 y(w)m(ork)45 b(in)f(discrete)g(Mark)m(o)m(v)j(c)m(hains,)h(w)m(e)e(sho) m(w)f(that)g Fn(metastable)j(exit)e(times)f Ft(from)g(the)g(attractiv)m (e)50 2591 y(domains)32 b(of)h(the)h(minima)d(of)j Fq(F)46 b Ft(can)34 b(b)s(e)e(related,)j(up)d(to)i(m)m(ultiplicativ)m(e)d (errors)i(that)h(tend)f(to)h(one)f(as)50 2736 y Fq(\017)39 b Fr(#)g Ft(0,)i(to)e(the)g(capacities)g(of)f(suitably)f(constructed)i (sets.)65 b(W)-8 b(e)40 b(sho)m(w)e(that)h(this)f(capacities)h(can)f(b) s(e)50 2882 y(computed,)27 b(again)g(up)f(to)h(m)m(ultiplicativ)m(e)d (errors)j(that)g(tend)f(to)i(one,)g(in)d(terms)h(of)h(lo)s(cal)f(c)m (haracteristics)50 3027 y(of)i Fq(F)40 b Ft(at)28 b(the)g(starting)f (minim)m(um)e(and)i(the)h(relev)-5 b(an)m(t)28 b Fn(sadd)5 b(le)31 b(p)-5 b(oints)p Ft(.)41 b(As)28 b(a)g(result,)f(w)m(e)h(are)g (able)f(to)h(giv)m(e)50 3173 y(the)33 b(\014rst)f(rigorous)h(pro)s(of)f (of)h(the)g(classical)g Fn(Eyring-Kr)-5 b(amers)36 b(formula)e Ft(in)e(dimension)f(larger)h(than)h(1.)50 3318 y(The)j(estimates)i(on)f (capacities)g(mak)m(e)h(use)f(of)g(their)f(v)-5 b(ariational)36 b(represen)m(tation)h(and)g(monotonicit)m(y)50 3463 y(prop)s(erties)26 b(of)i(Diric)m(hlet)f(forms.)39 b(The)28 b(metho)s(ds)f(dev)m(elop)s (ed)g(here)h(are)g(extensions)f(of)h(our)g(earlier)f(w)m(ork)50 3609 y(on)j(discrete)g(Mark)m(o)m(v)i(c)m(hains)e(to)h(con)m(tin)m (uous)f(di\013usion)e(pro)s(cesses.)50 3827 y Fn(Keywor)-5 b(ds:)42 b Ft(Metastabilit)m(y)-8 b(,)31 b(di\013usion)d(pro)s(cesses,) j(p)s(oten)m(tial)e(theory)-8 b(,)32 b(capacit)m(y)-8 b(,)32 b(exit)e(times)50 4045 y Fn(AMS)i(Subje)-5 b(ct)32 b(Classi\014c)-5 b(ation:)43 b Ft(82C44,)32 b(60K35)p 50 4647 546 4 v 159 4709 a Fm(1)197 4735 y Fl(W)-6 b (eierstrass-Institut)28 b(f)r(\177)-41 b(ur)26 b(Angew)n(andte)f (Analysis)i(und)e(Sto)r(c)n(hastik,)i(Mohrenstrasse)g(39,)f(D-10117)d (Berlin,)50 4825 y(German)n(y)-6 b(.)33 b(e-mail:)i(b)r(o)n (vier@wias-b)r(erlin.de)159 4891 y Fm(2)197 4916 y Fl(Institut)f(f)r (\177)-41 b(ur)34 b(Angew)n(andte)f(Mathematik,)j(Univ)n(ersit\177)-39 b(at)35 b(Z)r(\177)-41 b(uric)n(h,)37 b(Win)n(terth)n(urer)e(Strasse)g (190,)h(CH-8057)31 b(Z)r(\177)-41 b(uric)n(h,)50 5007 y(Switzerland.)50 5098 y(e-mail:)35 b(mec)n(kho\013@amath.unizh.c)n(h) 159 5163 y Fm(3)197 5189 y Fl(FSB-IMB,)25 b(EPFL,)h(CH-1015)e (Lausanne,)i(Switzerland.)36 b(e-mail:)f(V)-6 b(eronique.Ga)n (yrard@ep\015.c)n(h,)50 5280 y(on)22 b(lea)n(v)n(e)h(from:)32 b(Cen)n(tre)23 b(de)g(Ph)n(ysique)f(Th)n(\023)-36 b(eorique,)23 b(CNRS,)g(Lumin)n(y)-6 b(,)23 b(Case)g(907,)f(F-13288)f(Marseille,)26 b(Cedex)d(9,)g(F)-6 b(rance.)159 5345 y Fm(4)197 5371 y Fl(Institut)26 b(f)r(\177)-41 b(ur)26 b(Mathematik,)f(Univ)n (ersit\177)-39 b(at)27 b(P)n(otsdam,)e(Am)g(Neuen)h(P)n(alais)g(10,)f (D-14469)f(P)n(otsdam,)h(German)n(y)-6 b(.)50 5462 y(e-mail:)35 b(mklein@felix.math.uni-p)r(otsdam.de)50 5571 y Fk(11)p Fj(=september)r(=)p Fk(2002;)29 b(10:23)1068 b Fl(1)p eop %%Page: 2 2 2 1 bop 50 295 a Fl(2)50 513 y Fv(1.)55 b(In)m(tro)s(duction.)159 731 y Ft(In)37 b(this)h(pap)s(er)f(and)g(a)i(follo)m(w-up)e(pap)s(er)g ([BGK])i(w)m(e)g(in)m(v)m(estigate)g(rev)m(ersible)e(di\013usion)f(pro) s(cesses)50 876 y Fq(X)125 890 y Fo(\017)160 876 y Ft(\()p Fq(t)p Ft(\),)31 b(giv)m(en)f(as)h(solutions)e(of)h(an)h(It^)-45 b(o)31 b(sto)s(c)m(hastic)g(di\013eren)m(tial)e(equation)1174 1096 y Fq(dX)1296 1110 y Fo(\017)1331 1096 y Ft(\()p Fq(t)p Ft(\))d(=)f Fr(r)p Fq(F)13 b Ft(\()p Fq(X)1813 1110 y Fo(\017)1848 1096 y Ft(\()p Fq(t)p Ft(\)\))p Fq(dt)22 b Ft(+)2178 1017 y Fr(p)p 2254 1017 83 4 v 79 x Ft(2)p Fq(\017)q(dW)13 b Ft(\()p Fq(t)p Ft(\))937 b(\(1)p Fq(:)p Ft(1\))50 1316 y(on)33 b(a)h(regular)f(domain)g(\012)d Fr(\022)g Fp(R)1154 1283 y Fo(d)1202 1316 y Ft(,)35 b(where)e(the)h (drift)e Fr(r)p Fq(F)46 b Ft(is)33 b(generated)h(b)m(y)g(a)g(p)s(oten)m (tial)f(function)f(that)50 1462 y(is)27 b(su\016cien)m(tly)g(regular.) 39 b(W)-8 b(e)30 b(are)e(in)m(terested)h(in)e(the)h(case)h(when)f(the)g (function)f Fq(F)13 b Ft(\()p Fq(x)p Ft(\))29 b(has)f(sev)m(eral)g(lo)s (cal)50 1607 y(minima.)38 b(W)-8 b(e)32 b(alw)m(a)m(ys)f(assume)f(that) h Fq(X)1447 1621 y Fo(\017)1513 1607 y Ft(is)e(killed)f(on)i(\012)2041 1574 y Fo(c)2108 1607 y Ft(if)f(it)h(exists.)159 1825 y(This)g(problem)g(is)h(a)h(sp)s(ecial)f(case)h(of)g(the)g(more)g (general)g(class)g(of)g Fn(smal)5 b(l)35 b(r)-5 b(andom)36 b(p)-5 b(erturb)g(ations)36 b(of)50 1971 y(dynamic)-5 b(al)32 b(systems)d Ft(studied)e(since)g(the)h(early)g(1970s)h(b)m(y)f (F)-8 b(reidlin)25 b(and)j(W)-8 b(en)m(tzell)29 b(\(see)g(their)e (standard)50 2116 y(text)41 b([FW]\))h(using)c(large)j(deviation)e (metho)s(ds.)70 b(Ho)m(w)m(ev)m(er,)45 b(in)m(v)m(estigations)40 b(in)m(to)g(this)f(problem)f(can)50 2262 y(b)s(e)f(traced)i(bac)m(k)f (m)m(uc)m(h)g(further)f(in)g(the)h(ph)m(ysical)e(and)i(c)m(hemical)g (literature)f([Ey)-8 b(,Kra].)64 b(One)37 b(of)i(the)50 2407 y(earliest)d(textb)s(o)s(ok)h(sources)g(is)f(the)h(b)s(o)s(ok)f(b) m(y)h(Eyring)f(et)h(al.)60 b([GLE].)38 b(T)m(ypical)d(questions)h (related)h(to)50 2553 y(this)29 b(problem)g(are)68 2771 y Fr(\017)46 b Ft(What)41 b(are)g(the)g(t)m(ypical)f(times)h(to)g(reac) m(h)g(the)g(neigh)m(b)s(orho)s(o)s(ds)d(of)j(minim)m(um)d Fq(a)j Ft(starting)f(from)g(a)159 2916 y(minim)m(um)27 b Fq(b)k Ft(of)f(the)h(function)e Fq(F)13 b Ft(?)40 b(\(a)m(v)m(erage,) 34 b(distribution\).)68 3135 y Fr(\017)46 b Ft(What)31 b(are)g(t)m(ypical)f(paths)g(for)g(suc)m(h)g(a)h(pro)s(cess?)68 3353 y Fr(\017)46 b Ft(What)32 b(is)f(the)h(nature)f(of)h(the)g(lo)m (w-lying)e(sp)s(ectrum)g(of)i(the)g(generator)h(of)f(this)e(pro)s (cess?)44 b(What)32 b(are)159 3498 y(the)e(eigenfunctions)f(asso)s (ciated)i(to)g(small)e(eigen)m(v)-5 b(alues?)159 3716 y(It)35 b(should)e(come)j(as)g(no)f(surprise)e(that)i(these)h (questions)e(are)i(on)f(a)g(qualitativ)m(e)g(lev)m(el)g(w)m(ell)f (under-)50 3862 y(sto)s(o)s(d.)40 b(Ho)m(w)m(ev)m(er,)32 b(there)e(is)e(at)j(presen)m(t)e(still)f(a)i(considerable)e(gap)i(b)s (et)m(w)m(een)g Fn(mathematic)-5 b(al)5 b(ly)35 b(rigor)-5 b(ous)50 4007 y Ft(and)32 b Fn(heuristic)68 b Ft(results.)46 b(Rigorous)33 b(results)e(are)i(mostly)g(based)f(on)h(the)g(theory)g (of)f(Large)i(Deviations)50 4153 y(dev)m(elop)s(ed)i(in)g(this)g(con)m (text)i(b)m(y)f(F)-8 b(reidlin)35 b(and)h(W)-8 b(en)m(tzell.)61 b(They)37 b(are)g(v)m(ery)h(\015exible)d(and)h(apply)g(in)f(a)50 4298 y(v)-5 b(ariet)m(y)41 b(of)g(situations)e(w)m(ell)h(b)s(ey)m(ond)g (the)h(setting)g(of)g(\(1.1\).)73 b(Ho)m(w)m(ev)m(er,)46 b(they)41 b(yield)e(generally)h(only)50 4444 y(rough)32 b(asymptotic)g(estimates)h(in)e(the)i(parameter)f Fq(\017)h Ft(\(`logarithmic)e(equiv)-5 b(alence'\))32 b(for)h(exp)s(onen)m (tially)50 4589 y(small)g(\(or)i(large\))g(quan)m(tities)f(suc)m(h)g (as)h(escap)s(e)g(times)f(or)h(small)e(eigen)m(v)-5 b(alues.)53 b(A)35 b(second,)h(v)m(ery)f(natu-)50 4735 y(ral)h(approac)m(h)i(that)g (w)m(as)g(initiated)e(v)m(ery)h(early)g(in)f(the)i(ph)m(ysical)e(and)h (c)m(hemical)g(literature)g(is)f(based)50 4880 y(on)f(what)g(is)g (called)f Fn(semi-classic)-5 b(al)39 b(analysis)e Ft(or)e Fn(WKB-the)-5 b(ory)37 b Ft(\(for)e(a)h(v)m(ery)g(recen)m(t)g(review)f (on)g(these)50 5025 y(metho)s(ds,)j(see)g(e.g.)62 b([Kolo]\).)f(This)35 b(metho)s(ds)i(pro)m(vide)f Fn(formal)j Ft(asymptotic)e(series)f (expansions)g(in)g Fq(\017)50 5171 y Ft(and)j(can)i(b)s(e)f(seen)g(as)g (an)g(in\014nite)e(dimensional)g(v)m(ersion)h(of)i(the)f(saddle)f(p)s (oin)m(t)g(metho)s(d.)70 b(In)40 b(man)m(y)50 5316 y(cases,)i(suc)m(h)d (expansions)e(can)i(to)s(da)m(y)h(b)s(e)e(justi\014ed)f(b)m(y)i(what)g (has)f(b)s(ecome)i(to)f(b)s(e)g(called)f Fn(micr)-5 b(olo)g(c)g(al)50 5462 y(analysis)p Ft(,)28 b(whic)m(h)c(w)m(as)i(mainly)d(dev)m(elop)s (ed)i(in)f(view)h(of)g(solving)f(quan)m(tum)h(mec)m(hanical)g (tunneling)e(prob-)p eop %%Page: 3 3 3 2 bop 3671 295 a Fl(3)50 513 y Ft(lems)24 b([HS1,HS2,HS3,HS4].)40 b(Unfortunately)-8 b(,)26 b(the)f(sto)s(c)m(hastic)g(tunneling)d (problem)h(b)s(et)m(w)m(een)i(p)s(oten)m(tial)50 658 y(w)m(ells)j(corresp)s(onds)h(to)h(a)g(particularly)d(in)m(tricate)j (quan)m(tum)f(mec)m(hanical)g(problem,)g(called)g(\\tunneling)50 804 y(through)h(non-resonan)m(t)i(w)m(ells".)42 b(In)31 b(this)f(situation,)g(classical)h(WKB)g(theory)h(breaks)f(do)m(wns,)g (since)f(it)50 949 y(is)j(not)h(p)s(ossible)e(to)j(\014nd)d(a)j(global) e(solution)g(based)g(on)h(a)h(single)d(p)s(o)m(w)m(er-series)i(ansatz.) 53 b(On)33 b(a)i(formal)50 1095 y(lev)m(el,)d(these)h(problems)d(can)i (b)s(e)g(solv)m(ed)g(using)e(matc)m(hed)j(series)f(expansions)e(where)i (di\013eren)m(t)f(ans\177)-45 b(atze)50 1240 y(in)27 b(di\013eren)m(t)g(domains)g(are)h(matc)m(hed)h(in)e(o)m(v)m(erlapping) g(regions)h(to)g(determine)f(co)s(e\016cien)m(ts)i(\(see)g(in)e(par-)50 1385 y(ticular)40 b([MatSc)m(h1,BuMa1,BuMa2,MS1]\).)78 b(Justifying)39 b(these)i(expansions)f(is,)j(ho)m(w)m(ev)m(er,)i(far)c (from)50 1531 y(trivial)f(and)i(constitutes,)k(as)d(Kolotsolk)m(o)m(v)i ([Kolo])e(p)s(oin)m(ts)e(out)i(\\one)g(of)g(the)g(main)e(and)h(still)e (op)s(en)50 1676 y(questions)34 b(of)h(the)g(theory",)i(except)f(in)d (the)j(case)g Fq(d)d Ft(=)f(1)k(where)e(considerable)f (simpli\014cations)f(o)s(ccur)50 1822 y([KoMak,BuMa1,BuMa2,KN].)k (Indeed,)c(while)d(it)j(app)s(ears)f(clear)g(that)i(the)f(metho)s(ds)f (in)m(tro)s(duced)f(in)50 1967 y(the)i(third)f(pap)s(er)g(on)i(quan)m (tum)f(mec)m(hanical)g(tunneling)e(b)m(y)i(Hel\013er)g(and)g(Sj\177)-45 b(ostrand)32 b([HS3])h(should)d(in)50 2113 y(principle)f(allo)m(w)k(to) h(solv)m(e)f(this)f(problem,)g(this)g(program)h(has)f(not)i(b)s(een)e (carried)g(out)h(in)f(this)g(con)m(text)50 2258 y(y)m(et.)159 2476 y(Here)41 b(w)m(e)g(tak)m(e)h(a)f(new)f(lo)s(ok)h(at)g(this)f(old) f(problem)g(using)g(neither)h(large)h(deviations,)h(nor)e(semi-)50 2622 y(classical)28 b(expansions,)h(but)g(some)h(rather)f(classical)g (ideas)g(from)g Fn(p)-5 b(otential)33 b(the)-5 b(ory)p Ft(.)43 b(The)29 b(deep)g(connec-)50 2767 y(tion)j(b)s(et)m(w)m(een)h (Mark)m(o)m(v)i(pro)s(cesses)d(and)g(p)s(oten)m(tial)g(theory)h(has)g (b)s(een)f(w)m(ell-kno)m(wn)f(since)h(at)i(least)f(the)50 2913 y(w)m(ork)26 b(of)g(Kakutani)g([Kaku])g(and)f(is)h(the)g(sub)5 b(ject)26 b(of)g(n)m(umerous)f(textb)s(o)s(oks)i(\(see)g(in)e (particular)f(the)j(fun-)50 3058 y(damen)m(tal)j(monograph)f(b)m(y)h (Do)s(ob)h([Do)s(o]\).)42 b(This)28 b(connection)i(has)g(found)e(n)m (umerous)h(and)h(widespread)50 3204 y(applications)e(\(see)k(e.g.)41 b([DS,Szni])30 b(and)g(references)h(wherein\).)159 3422 y(The)d(particular)e(approac)m(h)j(w)m(e)g(presen)m(t)f(here)g(is)g (distinguished)c(b)m(y)k(the)h(fact)g(that)g(it)f(largely)f(a)m(v)m (oids)50 3567 y(the)d(attempt)h(to)g(solv)m(e)f(the)h(b)s(oundary)d(v) -5 b(alue)23 b(problems)f(that)j(arise)f(in)e(this)h(connection)h(b)m (y)g(straigh)m(tfor-)50 3713 y(w)m(ard)k(PDE)i(metho)s(ds,)e(but)h (that)g(it)g(tries)f(to)i(reduce)f(most)g(problems)f(to)i(that)f(of)g (the)h(computation)f(of)50 3858 y Fn(Newtonian)g(c)-5 b(ap)g(acities)28 b Ft(whic)m(h)d(then)h(are)g(estimated)h(using)d Fn(variational)31 b(principles)c Ft(and)f Fn(monotonicity)50 4004 y(pr)-5 b(op)g(erties)p Ft(.)46 b(In)30 b(this)h(it)f(is)h(in)f (spirit)f(close)i(to)h(the)g(\\electric)f(net)m(w)m(ork")i(approac)m(h) f(used)e(extensiv)m(ely)h(in)50 4149 y(the)h(study)g(of)h(recurrence)f (and)g(transience)g(prop)s(erties)f(of)i(Mark)m(o)m(v)h(c)m(hains)e ([NS,DS].)h(This)d(approac)m(h)50 4295 y(to)f(the)h(metastabilit)m(y)e (problem)f(w)m(as)j(initiated)d(in)g(fact)j(in)e(t)m(w)m(o)i(preceding) e(pap)s(ers)g([BEGK1,BEGK2])50 4440 y(in)f(the)i(con)m(text)i(of)e Fn(discr)-5 b(ete)29 b Ft(Mark)m(o)m(v)i(c)m(hains,)d(including,)e(in)i (particular,)f(\(in)h([BEGK1]\))j(discrete)d(v)m(er-)50 4585 y(sions)i(of)i(\(1.1\).)47 b(In)31 b(fact,)i(the)f(discrete)g (setting)f(o\013ers)i(\(as)f(w)m(e)g(shall)e(p)s(oin)m(t)h(out)h(in)e (due)h(place\))h(sev)m(eral)50 4731 y(adv)-5 b(an)m(tages)37 b(for)e(this)f(approac)m(h)h(and)g(mak)m(es)g(it)g(app)s(ear)g (probabilistically)30 b(m)m(uc)m(h)35 b(more)h(transparen)m(t)50 4876 y(than)31 b(in)g(the)h(di\013usion)d(setting.)45 b(W)-8 b(e)33 b(susp)s(ect)e(that)h(this)f(ma)m(y)h(ha)m(v)m(e)h(b)s (een)e(the)h(reason)g(wh)m(y)f(the)h(ideas)50 5022 y(to)e(study)e(the)h (sp)s(ectral)g(problem)e(of)j(generators)g(of)f(Mark)m(o)m(v)i(c)m (hains)d(presen)m(ted)h(in)f(the)i(1973)h(pap)s(er)d(of)50 5167 y(W)-8 b(en)m(tzell)23 b([W)-8 b(en])24 b(and)e(that)i(are)f (somewhat)g(similar)d(to)j(our)f(approac)m(h)h(w)m(ere)g(apparen)m(tly) f(not)h(dev)m(elop)s(ed)50 5313 y(in)36 b(the)h(direction)f(w)m(e)i (are)f(going.)61 b(While)36 b(the)i(di\013usion)c(case)39 b(mak)m(es)f(probabilistic)33 b(in)m(terpretations)50 5458 y(more)h(complicated,)h(the)f(presen)m(t)g(pap)s(er)f(ma)m(y)i (clarify)d(our)i(approac)m(h)g(as)g(it)g(forces)g(us)f(to)i(dev)m(elop) f(in)p eop %%Page: 4 4 4 3 bop 50 295 a Fl(4)50 513 y Ft(m)m(uc)m(h)41 b(more)g(detail)f(the)h (fundamen)m(tal)f(p)s(oten)m(tial)g(theoretic)i(bac)m(kground)e(from)h (a)g(purely)e(analytic)50 658 y(p)s(oin)m(t)30 b(of)g(view.)42 b(Let)31 b(us)f(men)m(tion)g(that)i(in)d(our)h(view)g(the)h(approac)m (h)g(presen)m(ted)g(here)g(o\013ers)g(t)m(w)m(o)h(main)50 804 y(adv)-5 b(an)m(tages)41 b(o)m(v)m(er)f(the)f(micro-lo)s(cal)f (approac)m(h.)67 b(First,)41 b(it)d(is)g(tec)m(hnically)g(considerably) g(simpler,)g(as)50 949 y(w)m(e)f(hop)s(e)e(these)i(pap)s(ers)e(will)f (demonstrate,)39 b(and)d(second,)i(it)e(is)f(more)i(\015exible)e(and)g (can)i(b)s(e)f(applied)50 1095 y(in)e(a)i(broad)g(range)g(of)f (discrete)h(and)f(con)m(tin)m(uous)g(Mark)m(o)m(v)j(pro)s(cesses.)56 b(Its)36 b(dra)m(wbac)m(k,)i(on)e(the)f(other)50 1240 y(hand,)h(is)e(that)i(it)g(ma)m(y)g(not)f(readily)f(b)s(e)h(extended)h (to)g(yield)e(systematic)i(asymptotic)g(expansions)e(to)50 1385 y(all)h(orders)g(in)f Fq(\017)p Ft(.)57 b(Also,)38 b(w)m(e)e(mak)m(e)h(strongly)e(use)h(of)g(the)g(fact)h(that)f(w)m(e)g (in)m(v)m(estigate)h(a)g(sto)s(c)m(hastic)f(\(or)50 1531 y(sub-sto)s(c)m(hastic\))28 b(op)s(erator,)h(and)f(our)f(metho)s(d)g (cannot)i(b)s(e)e(extended)h(to)h(arbitrary)e(elliptic)e(op)s(erators.) 159 1749 y(W)-8 b(e)29 b(will)d(from)i(no)m(w)g(on)g(assume)g(that)h Fq(F)41 b Ft(is)28 b(at)h(least)f(three)h(times)f(con)m(tin)m(uously)f (di\013eren)m(tiable)f(and)50 1895 y(has)g(a)h(\014nite)e(set)i(of)g (lo)s(cal)f(minima)1257 1862 y Fu(5)1295 1895 y Ft(,)i(whic)m(h)d(w)m (e)i(denote)g(b)m(y)f Fr(M)g Ft(=)f Fr(f)p Fq(x)2473 1909 y Fu(1)2514 1895 y Fq(;)15 b(:)g(:)g(:)h(;)f(x)2767 1909 y Fo(n)2817 1895 y Fr(g)p Ft(.)40 b(W)-8 b(e)28 b(will)c(also)i(assume)50 2040 y(that)f Fq(F)38 b Ft(has)24 b(exp)s(onen)m(tially)f(tigh)m(t)i(lev)m(el)g(sets,)h(i.e.)39 b(that)2008 1967 y Fi(R)2051 2072 y Fo(y)r Fu(:)p Fo(F)10 b Fu(\()p Fo(y)r Fu(\))p Fh(\025)p Fo(a)2379 2040 y Ft(exp)o(\()p Fr(\000)p Fq(F)j Ft(\()p Fq(z)t Ft(\))p Fq(=\017)p Ft(\))p Fq(dz)32 b(<)25 b(C)7 b(e)3258 2007 y Fh(\000)p Fo(a=\017)3426 2040 y Ft(,)26 b(where)50 2185 y Fq(C)32 b Ft(=)25 b Fq(C)7 b Ft(\()p Fq(a)p Ft(\))25 b Fq(<)g Fr(1)30 b Ft(is)f(uniform)f (in)h Fq(\017)c Fr(\024)g Ft(1.)41 b(Our)29 b(main)g(in)m(terest)h(are) g(the)h(distribution)26 b(of)k(stopping)f(times)1366 2455 y Fq(\034)1406 2469 y Fo(A)1491 2455 y Fr(\021)c Ft(inf)20 b Fr(f)q Fq(t)25 b(>)g Ft(0)p Fr(j)p Fq(X)7 b Ft(\()p Fq(t)p Ft(\))27 b Fr(2)e Fq(A)p Fr(g)1130 b Ft(\(1)p Fq(:)p Ft(2\))50 2724 y(for)40 b(the)h(pro)s(cess)f(starting)g (in)f(one)i(minim)m(um,)g(sa)m(y)g Fq(x)h Fr(2)g(M)p Ft(,)h(of)e Fq(F)13 b Ft(,)43 b(when)c Fq(A)k Ft(=)f Fq(B)3114 2738 y Fo(\032)3156 2724 y Ft(\()p Fq(y)s Ft(\))f(is)e(a)i (small)50 2870 y(ball)36 b(around)h(another)h(minim)m(um,)f Fq(y)j Fr(2)d(M)p Ft(.)63 b(It)38 b(will)e(actually)h(b)s(ecome)h (apparen)m(t)g(that)g(the)g(precise)50 3015 y(c)m(hoice)24 b(of)g(the)g(hitting)f(set)h(is)f(often)h(not)g(imp)s(ortan)m(t,)g(and) f(that)i(the)f(problem)e(is)h(virtually)e(equiv)-5 b(alen)m(t)23 b(to)50 3161 y(considering)g(the)j(escap)s(e)g(from)e(a)i(suitably)d(c) m(hosen)j(neigh)m(b)s(orho)s(o)s(d)d(of)j Fq(x)p Ft(,)g(pro)m(vided)e (this)g(neigh)m(b)s(orho)s(o)s(d)50 3306 y(con)m(tains)30 b(the)h(relev)-5 b(an)m(t)31 b Fn(sadd)5 b(le)33 b(p)-5 b(oints)32 b Ft(connecting)f Fq(x)f Ft(and)g Fq(y)s Ft(.)159 3524 y(In)36 b(this)f(pap)s(er)g(w)m(e)i(will)d(study)h(the)i(mean)f(v) -5 b(alues)36 b(of)h(suc)m(h)f(stopping)f(times.)59 b(Our)35 b(approac)m(h)h(will)50 3670 y(consist)30 b(of)g(t)m(w)m(o)i(distinct)d (steps:)17 3888 y(\(i\))46 b(Using)27 b(v)-5 b(ariational)26 b(principles,)f(w)m(e)j(will)d(giv)m(e)j(v)m(ery)f(sharp)g(estimates)h (on)f(some)h(relev)-5 b(an)m(t)28 b Fn(c)-5 b(ap)g(acities)p Ft(.)-8 4106 y(\(ii\))45 b(W)-8 b(e)43 b(will)d(then)i(sho)m(w)g(that)h (exp)s(ected)f(times)g(of)h(in)m(terest)f(can)h(b)s(e)e(expressed)h(in) f(terms)h(of)g(these)159 4252 y(capacities)31 b(and)e Fn(e)-5 b(quilibrium)33 b(p)-5 b(otentials)p Ft(.)159 4470 y(In)20 b(the)h(follo)m(w-up)f(pap)s(er)g([BGK])h(w)m(e)h(will)c (consider)i(the)h(asso)s(ciated)g(sp)s(ectral)f(problems.)36 b(A)21 b(corollary)50 4615 y(will)28 b(then)i(sho)m(w)g(that)h (metastable)g(exit)f(ha)m(v)m(e)i(an)e(asymptotically)g(exp)s(onen)m (tial)f(distribution.)159 4833 y(T)-8 b(o)30 b(b)s(e)e(able)h(to)h (state)g(our)f(results,)f(w)m(e)i(need)f(to)h(recall)e(a)i(n)m(um)m(b)s (er)d(of)j(k)m(ey)g(concepts)g(from)e(p)s(oten)m(tial)50 4979 y(theory)i(whic)m(h)g(will)d(allo)m(w)j(us)g(to)h(establish)e (some)h(notation.)p 50 5102 546 4 v 159 5163 a Fm(5)197 5189 y Fl(F)-6 b(or)30 b(man)n(y)g(of)f(the)j(results)g(of)e(this)i (pap)r(er,)g(these)g(conditions)g(can)e(b)r(e)h(relaxed)h(greatly)-6 b(.)49 b(In)30 b(particular,)k(one)c(ma)n(y)50 5280 y(consider)i (functions)g Fg(F)42 b Fl(=)30 b Fg(F)925 5289 y Ff(\017)990 5280 y Fl(dep)r(ending)i(on)f Fg(\017)p Fl(,)i(and)e(one)g(ma)n(y)f (also)i(consider)g(cases)h(with)e(in\014nitely)i(man)n(y)d(minima.)50 5371 y(This)c(ma)n(y)-6 b(,)25 b(ho)n(w)n(ev)n(er,)g(lead)h(to)g (di\013eren)n(t)h(questions)f(and)g(di\013eren)n(t)h(results,)h(and)d (w)n(e)h(prefer)g(to)f(explain)h(our)g(metho)r(ds)f(in)50 5462 y(a)g(simple)i(and)e(w)n(ell-con\014ned)i(setting.)p eop %%Page: 5 5 5 4 bop 3671 295 a Fl(5)50 513 y Fs(Ac)m(kno)m(wledgemen)m(ts:)43 b Ft(A.)25 b(Bo)m(vier)g(thanks)f(the)h(EPFL)f(and)g(V.)h(Ga)m(yrard)g (the)f(WIAS)h(for)f(hospitalit)m(y)50 658 y(and)30 b(\014nancial)e (supp)s(ort)h(that)i(made)f(this)g(collab)s(oration)f(p)s(ossible.)50 911 y Fv(2.)55 b(Some)41 b(basic)f(bac)m(kground)h(on)h(p)s(oten)m (tial)f(theory)-10 b(.)159 1129 y Ft(In)32 b(this)h(section)g(w)m(e)h (collect)f(notations)h(and)e(form)m(ulas)h(from)f(p)s(oten)m(tial)h (theory)h(that)f(will)e(b)s(e)i(used)50 1274 y(throughout)k(the)h(pap)s (er.)63 b(All)36 b(of)i(these)h(results)d(are)j(standard)e(and)g(can)h (b)s(e)f(found)g(in)g(the)h(classical)50 1420 y(textb)s(o)s(oks)31 b(on)f(p)s(oten)m(tial)g(theory)-8 b(,)31 b(e.g.)42 b([BluGet,Do)s (o,Szni].)159 1638 y(The)30 b(generator)h(of)g(our)f(di\013usion)d(pro) s(cesses)k(are)f(linear)f(elliptic)f(op)s(erators)j Fq(L)2943 1652 y Fo(\017)3008 1638 y Ft(of)f(the)h(form)941 1887 y Fq(L)1003 1901 y Fo(\017)1063 1887 y Ft(=)25 b Fr(\000)p Fq(\017e)1309 1850 y Fo(F)10 b Fu(\()p Fh(\001)p Fu(\))p Fo(=\017)1516 1887 y Fr(r)p Fq(e)1634 1850 y Fh(\000)p Fo(F)g Fu(\()p Fh(\001)p Fu(\))p Fo(=\017)1897 1887 y Fr(r)25 b Ft(=)g Fr(\000)p Fq(\017)p Ft(\001)20 b(+)f(\()p Fr(r)p Fq(F)13 b Ft(\()p Fr(\001)p Ft(\))p Fq(;)i Fr(r)p Ft(\))707 b(\(2)p Fq(:)p Ft(1\))50 2137 y(de\014ned)29 b(\(a)i(priori\))d(on)j Fq(C)956 2104 y Fu(2)996 2137 y Ft(\(\012\),)g(where)f(\012)24 b Fr(\022)h Fp(R)1697 2104 y Fo(d)1745 2137 y Ft(,)31 b(and)f Fq(F)38 b Fr(2)25 b Fq(C)2232 2104 y Fu(2)2272 2137 y Ft(\(\012\).)41 b(\012,)30 b(and)g(in)f(fact)i(all)f(subsets)f(of)i Fp(R)3661 2104 y Fo(d)50 2283 y Ft(that)d(w)m(e)g(will)d(consider)h(in)h(this)f(pap)s (er)g(will)f(b)s(e)i(regular)g(\(A)h(set)g Fq(A)p Fr(\032)p Fp(R)2490 2250 y Fo(d)2565 2283 y Ft(is)f(called)g Fn(r)-5 b(e)g(gular)29 b Ft(if)d(and)h(only)g(if)50 2428 y(its)33 b(complemen)m(t)g(is)f(a)i(region)f(with)f(con)m(tin)m(uously)g (di\013eren)m(tiable)g(b)s(oundary\).)48 b(By)33 b(construction,)h Fq(L)3675 2442 y Fo(\017)50 2573 y Ft(is)29 b(symmetric)h(on)g Fq(L)772 2540 y Fu(2)813 2573 y Ft(\(\012)p Fq(;)15 b(e)996 2540 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(x)p Fu(\))p Fo(=\017)1279 2573 y Fq(dx)p Ft(\))31 b(with)e(Diric)m(hlet)h(b)s(oundary)e (conditions)h(on)h(\012)3063 2540 y Fo(c)3100 2573 y Ft(.)50 2792 y Fs(Green's)35 b(function.)41 b Ft(Consider)29 b(for)h Fq(\025)25 b Fr(2)g Fp(C)54 b Ft(the)31 b(Diric)m(hlet)e (problem)1300 3000 y(\()p Fq(L)1397 3014 y Fo(\017)1453 3000 y Fr(\000)19 b Fq(\025)p Ft(\))p Fq(f)10 b Ft(\()p Fq(x)p Ft(\))26 b(=)f Fq(g)s Ft(\()p Fq(x)p Ft(\))p Fq(;)108 b(x)25 b Fr(2)g Ft(\012)1632 3173 y Fq(f)10 b Ft(\()p Fq(x)p Ft(\))25 b(=)g(0)p Fq(;)107 b(x)25 b Fr(2)g Ft(\012)2336 3135 y Fo(c)3523 3086 y Ft(\(2)p Fq(:)p Ft(2\))50 3395 y(The)g(asso)s(ciated)h(Diric)m(hlet)f(Green's)h(function)f Fq(G)1780 3362 y Fo(\025)1780 3420 y Fu(\012)1836 3395 y Ft(\()p Fq(x;)15 b(y)s Ft(\))26 b(is)f(the)h(k)m(ernel)g(of)g(the)g (in)m(v)m(erse)f(of)h(the)g(op)s(erator)50 3540 y(\()p Fq(L)147 3554 y Fo(\017)202 3540 y Fr(\000)20 b Fq(\025)p Ft(\),)31 b(i.e.)40 b(for)31 b(an)m(y)f Fq(g)f Fr(2)c Fq(C)1127 3554 y Fu(0)1168 3540 y Ft(\(\012\),)1370 3800 y Fq(f)10 b Ft(\()p Fq(x)p Ft(\))25 b(=)1668 3676 y Fi(Z)1719 3883 y Fu(\012)1790 3800 y Fq(G)1862 3763 y Fo(\025)1862 3823 y Fu(\012)1918 3800 y Ft(\()p Fq(x;)15 b(y)s Ft(\))p Fq(g)s Ft(\()p Fq(y)s Ft(\))p Fq(dy)1139 b Ft(\(2)p Fq(:)p Ft(3\))50 4077 y(Note)32 b(that)e(the)h(Green's)g(function)e(is)g (symmetric)h(with)f(resp)s(ect)h(to)h(the)g(measure)f Fq(e)3025 4044 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(x)p Fu(\))p Fo(=\017)3308 4077 y Fq(dx)p Ft(,)31 b(i.e.)1185 4327 y Fq(G)1257 4289 y Fo(\025)1257 4349 y Fu(\012)1313 4327 y Ft(\()p Fq(x;)15 b(y)s Ft(\))26 b(=)f Fq(e)1687 4289 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(y)r Fu(\))p Fo(=\017)1968 4327 y Fq(G)2040 4289 y Fo(\025)2040 4349 y Fu(\012)2096 4327 y Ft(\()p Fq(y)s(;)15 b(x)p Ft(\))p Fq(e)2348 4289 y Fo(F)10 b Fu(\()p Fo(x)p Fu(\))p Fo(=\017)3523 4327 y Ft(\(2)p Fq(:)p Ft(4\))50 4576 y(Recall)35 b(that)h(the)g(sp)s (ectrum)e(of)i Fq(L)1266 4590 y Fo(\017)1336 4576 y Ft(\(more)g (precisely)e(the)i(Diric)m(hlet)e(sp)s(ectrum)h(of)g(the)h(restriction) f(of)50 4722 y Fq(L)112 4736 y Fo(\017)181 4722 y Ft(to)h(\012,)f(whic) m(h)f(w)m(e)h(will)d(sometimes)j(denote)g(b)m(y)g Fq(L)1937 4689 y Fu(\012)1937 4744 y Fo(\017)1993 4722 y Ft(\),)i(is)d(the)h (complemen)m(t)g(of)f(the)h(set)h(of)f(v)-5 b(alues)34 b Fq(\025)50 4867 y Ft(for)c(whic)m(h)f Fq(G)521 4834 y Fo(\025)521 4892 y Fu(\012)607 4867 y Ft(de\014nes)h(a)h(b)s(ounded)d (op)s(erator.)50 5085 y Fs(P)m(oisson)36 b(k)m(ernel.)41 b Ft(Consider)29 b(for)h Fq(\025)25 b Fr(2)g Fp(C)54 b Ft(the)30 b(b)s(oundary)f(v)-5 b(alue)30 b(problem)1278 5294 y(\()p Fq(L)1375 5308 y Fo(\017)1430 5294 y Fr(\000)20 b Fq(\025)p Ft(\))p Fq(f)10 b Ft(\()p Fq(x)p Ft(\))26 b(=)f(0)p Fq(;)106 b(x)26 b Fr(2)f Ft(\012)1610 5466 y Fq(f)10 b Ft(\()p Fq(x)p Ft(\))25 b(=)g Fq(\036)p Ft(\()p Fq(x)p Ft(\))p Fq(;)107 b(x)25 b Fr(2)g Ft(\012)2445 5429 y Fo(c)3523 5380 y Ft(\(2)p Fq(:)p Ft(5\))p eop %%Page: 6 6 6 5 bop 50 295 a Fl(6)50 513 y Ft(W)-8 b(e)31 b(denote)g(b)m(y)g Fq(H)709 480 y Fo(\025)702 538 y Fu(\012)788 513 y Ft(the)f(asso)s (ciated)h(solution)e(op)s(erator)i(whic)m(h)e(can)i(b)s(e)e(represen)m (ted)i(in)e(the)h(form)559 780 y Fq(f)10 b Ft(\()p Fq(x)p Ft(\))26 b(=)e(\()p Fq(H)975 742 y Fo(\025)968 802 y Fu(\012)1025 780 y Fq(\036)p Ft(\)\()p Fq(x)p Ft(\))i(=)f Fr(\000)p Fq(\017)1481 656 y Fi(Z)1531 863 y Fo(@)t Fu(\012)1646 780 y Fq(e)1688 742 y Fh(\000)p Fu([)p Fo(F)10 b Fu(\()p Fo(y)r Fu(\))p Fh(\000)p Fo(F)g Fu(\()p Fo(x)p Fu(\)])p Fo(=\017)2221 780 y Fq(\036)p Ft(\()p Fq(y)s Ft(\))p Fq(@)2441 796 y Fo(n)p Fu(\()p Fo(y)r Fu(\))2587 780 y Fq(G)2659 742 y Fo(\025)2659 802 y Fu(\012)2715 780 y Ft(\()p Fq(y)s(;)15 b(x)p Ft(\))p Fq(d\033)3024 794 y Fu(\012)3082 780 y Ft(\()p Fq(y)s Ft(\))323 b(\(2)p Fq(:)p Ft(6\))50 1044 y(where)27 b Fq(d\033)409 1058 y Fu(\012)494 1044 y Ft(denotes)h(the)g(Euclidean)e(surface)i(measure)f (on)h Fq(@)5 b Ft(\012,)29 b(and)e Fq(@)2566 1061 y Fo(n)p Fu(\()p Fo(y)r Fu(\))2739 1044 y Ft(denotes)i(the)f(deriv)-5 b(ativ)m(e)27 b(in)50 1190 y(the)35 b(direction)e(of)i(the)g Fn(exterior)h Ft(normal)e(v)m(ector)i(on)f Fq(@)5 b Ft(\012)34 b(at)i Fq(y)s Ft(,)g(acting)f(on)f(the)h(\014rst)f(argumen)m(t)i(of)f (the)50 1335 y(function)29 b Fq(G)478 1302 y Fo(\025)478 1360 y Fu(\012)534 1335 y Ft(\()p Fq(y)s(;)15 b(x)p Ft(\).)159 1553 y(The)34 b(relation)h(b)s(et)m(w)m(een)g(the)h(op)s(erator)f Fq(H)1660 1521 y Fo(\025)1653 1579 y Fu(\012)1744 1553 y Ft(and)f(the)i(Green)f(function)e(\(2.6\))k(is)d(a)i(consequence)g (of)50 1699 y(the)30 b(t)m(w)m(o)i(Green's)f(iden)m(tities)e(that)i (here)f(tak)m(e)i(the)e(form)177 1828 y Fi(Z)227 2034 y Fu(\012)299 1952 y Fq(dx)15 b(e)455 1914 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(x)p Fu(\))p Fo(=\017)738 1878 y Fi(\000)780 1952 y Fq(\017)p Fr(r)p Fq(\036)p Ft(\()p Fq(x)p Ft(\))21 b Fr(\001)f(r)p Fq( )s Ft(\()p Fq(x)p Ft(\))h Fr(\000)f Fq( )s Ft(\()p Fq(x)p Ft(\)\()p Fq(L)1788 1966 y Fo(\017)1825 1952 y Fq(\036)p Ft(\)\()p Fq(x)p Ft(\))2036 1878 y Fi(\001)2104 1952 y Ft(=)14 b Fq(\017)2241 1828 y Fi(Z)2292 2034 y Fo(@)t Fu(\012)2406 1952 y Fq(e)2448 1914 y Fh(\000)p Fo(F)c Fu(\()p Fo(x)p Fu(\))p Fo(=\017)2732 1952 y Fq( )s Ft(\()p Fq(x)p Ft(\))p Fq(@)2964 1968 y Fo(n)p Fu(\()p Fo(x)p Fu(\))3112 1952 y Fq(\036)p Ft(\()p Fq(x)p Ft(\))15 b Fq(d\033)3402 1966 y Fu(\012)3460 1952 y Ft(\()p Fq(x)p Ft(\))3523 2112 y(\(2)p Fq(:)p Ft(7\))50 2257 y(\(\014rst)30 b(Green's)g(iden)m(tit)m(y\))h(and)464 2398 y Fi(Z)515 2605 y Fu(\012)586 2522 y Fq(e)628 2484 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(x)p Fu(\))p Fo(=\017)912 2522 y Fq(dx)1026 2448 y Fi(\000)1068 2522 y Fq(\036)p Ft(\()p Fq(x)p Ft(\)\()p Fq(L)1341 2536 y Fo(\017)1397 2522 y Fr(\000)20 b Fq(\025)p Ft(\))p Fq( )s Ft(\()p Fq(x)p Ft(\))h Fr(\000)f Fq( )s Ft(\()p Fq(x)p Ft(\)\()p Fq(L)2153 2536 y Fo(\017)2210 2522 y Fr(\000)g Fq(\025)p Ft(\))p Fq(\036)p Ft(\()p Fq(x)p Ft(\))2565 2448 y Fi(\001)1051 2765 y Ft(=)40 b Fq(\017)1214 2641 y Fi(Z)1265 2848 y Fo(@)t Fu(\012)1379 2765 y Fq(e)1421 2727 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(x)p Fu(\))p Fo(=\017)1704 2765 y Ft(\()p Fq( )s Ft(\()p Fq(x)p Ft(\))p Fq(@)1971 2781 y Fo(n)p Fu(\()p Fo(x)p Fu(\))2121 2765 y Fq(\036)p Ft(\()p Fq(x)p Ft(\))21 b Fr(\000)f Fq(\036)p Ft(\()p Fq(x)p Ft(\))p Fq(@)2633 2781 y Fo(n)p Fu(\()p Fo(x)p Fu(\))2781 2765 y Fq( )s Ft(\()p Fq(x)p Ft(\)\))15 b Fq(d\033)3114 2779 y Fu(\012)3173 2765 y Ft(\()p Fq(x)p Ft(\))3523 2646 y(\(2)p Fq(:)p Ft(8\))50 3040 y(\(second)31 b(Green's)f(iden)m (tit)m(y\),)h(where)f Fq(\036;)15 b( )29 b Fr(2)c Fq(C)1708 3007 y Fu(2)1748 3040 y Ft(\(\012\).)50 3258 y Fs(Equilibrium)37 b(p)s(oten)m(tial)g(and)h(equilibrium)g(measure.)47 b Ft(Let)33 b Fq(A;)15 b(D)33 b Fr(\032)c Fp(R)2837 3225 y Fo(d)2918 3258 y Ft(b)s(e)k(regular)f(and)g(suc)m(h)50 3404 y(that)f(\()p Fq(A)20 b Fr([)g Fq(D)s Ft(\))564 3371 y Fo(c)626 3404 y Fr(\032)25 b Ft(dom\()p Fq(F)13 b Ft(\).)41 b(Then)30 b(the)g(equilibrium)c(p)s(oten)m(tial)j(\(of)i (the)g(capacitor)g(\()p Fq(A;)15 b(D)s Ft(\)\),)32 b Fq(h)3451 3371 y Fo(\025)3451 3429 y(A;D)3593 3404 y Ft(,)f(is)50 3549 y(de\014ned)e(as)i(the)f(solution)f(of)i(the)f(Diric) m(hlet)f(problem)1148 3767 y(\()p Fq(L)1245 3781 y Fo(\017)1300 3767 y Fr(\000)20 b Fq(\025)p Ft(\))p Fq(h)1531 3730 y Fo(\025)1531 3790 y(A;D)1674 3767 y Ft(\()p Fq(x)p Ft(\))26 b(=)f(0)p Fq(;)106 b(x)26 b Fr(2)e Ft(\()p Fq(A)d Fr([)f Fq(D)s Ft(\))2575 3730 y Fo(c)1479 3940 y Fq(h)1531 3903 y Fo(\025)1531 3963 y(A;D)1674 3940 y Ft(\()p Fq(x)p Ft(\))26 b(=)f(1)p Fq(;)106 b(x)26 b Fr(2)e Fq(A)1479 4113 y(h)1531 4075 y Fo(\025)1531 4135 y(A;D)1674 4113 y Ft(\()p Fq(x)p Ft(\))i(=)f(0)p Fq(;)106 b(x)26 b Fr(2)e Fq(D)3523 3939 y Ft(\(2)p Fq(:)p Ft(9\))50 4361 y(Note)32 b(that)e(\(2.9\))j(has)d(a)h(unique)d(solution)h(pro)m(vided)g Fq(\025)h Ft(is)g(not)g(in)f(the)i(sp)s(ectrum)e(of)i Fq(L)3097 4314 y Fu(\()p Fo(A)p Fh([)p Fo(B)s Fu(\))3315 4286 y Fj(c)3097 4372 y Fo(\017)3354 4361 y Ft(.)159 4579 y(The)f(equilibrium)25 b(measure,)31 b Fq(e)1250 4546 y Fo(\025)1250 4604 y(A;D)1392 4579 y Ft(,)g(is)e(de\014ned)h(as)g (the)h(unique)d(measure)i(on)h Fq(@)5 b(A)p Ft(,)31 b(suc)m(h)f(that) 1210 4859 y Fq(h)1262 4821 y Fo(\025)1262 4881 y(A;D)1405 4859 y Ft(\()p Fq(x)p Ft(\))c(=)1649 4735 y Fi(Z)1699 4942 y Fo(@)t(A)1816 4859 y Fq(G)1888 4821 y Fo(\025)1888 4881 y(D)1949 4863 y Fj(c)1988 4859 y Ft(\()p Fq(x;)15 b(y)s Ft(\))p Fq(e)2240 4821 y Fo(\025)2240 4881 y(A;D)2383 4859 y Ft(\()p Fq(dy)s Ft(\))929 b(\(2)p Fq(:)p Ft(10\))159 5207 y(If)30 b(w)m(e)h(consider)e Fq(L)800 5221 y Fo(\017)865 5207 y Ft(as)i(a)f(map)g(from)g Fq(H)1552 5174 y Fo(n)1602 5207 y Ft(\(\012\))g(to)h Fq(H)1962 5174 y Fo(n)p Fh(\000)p Fu(2)2105 5207 y Ft(\(\012\),)g(\(2.10\))i(ma)m(y)d(also)h(b)s(e)f (written)f(as)1322 5462 y Fq(e)1364 5424 y Fo(\025)1364 5484 y(A;D)1506 5462 y Ft(\()p Fq(dy)s Ft(\))d(=)f(\()p Fq(L)1890 5476 y Fo(\017)1946 5462 y Fr(\000)19 b Fq(\025)p Ft(\))p Fq(h)2176 5424 y Fo(\025)2176 5484 y(A;D)2319 5462 y Ft(\()p Fq(y)s Ft(\))1040 b(\(2)p Fq(:)p Ft(11\))p eop %%Page: 7 7 7 6 bop 3671 295 a Fl(7)50 513 y Ft(where)22 b(of)h(course)g(b)s(oth)f (sides)f(are)i(to)h(b)s(e)e(in)m(terpreted)g(as)h(measures)f(equipp)s (ed)f(with)g(the)i(w)m(eak)h(top)s(ology)-8 b(.)50 658 y(A)22 b(simple)e(computation)i(using)f(the)h(second)g(Green's)h(iden)m (tit)m(y)e(and)h(the)g(P)m(oisson)g(k)m(ernel)g(represen)m(tation)50 804 y(\(2.6\))32 b(allo)m(ws)e(to)h(compute)f(the)h(righ)m(t)f(hand)f (side)h(of)g(\(2.11\))j(as)835 1038 y(\()p Fq(L)932 1052 y Fo(\017)987 1038 y Fr(\000)20 b Fq(\025)p Ft(\))p Fq(h)1218 1000 y Fo(\025)1218 1060 y(A;D)1361 1038 y Ft(\()p Fq(x)p Ft(\))26 b(=)f Fq(\017@)1690 1054 y Fo(n)p Fu(\()p Fo(x)p Fu(\))1838 1038 y Fq(h)1890 1000 y Fo(\025)1890 1060 y(A;D)2032 1038 y Ft(\()p Fq(x)p Ft(\))p Fq(d\033)2253 1052 y Fo(A)p Fh([)p Fo(D)2424 1038 y Ft(\()p Fq(x)p Ft(\))c Fr(\000)f Fq(\025)p Ft(1)-23 b(I)2766 1052 y Fo(A)2825 1038 y Fq(dx)553 b Ft(\(2)p Fq(:)p Ft(12\))50 1345 y Fs(Capacit)m(y)-9 b(.)46 b Ft(Giv)m(en)32 b(a)g(capacitor,)i(\() p Fq(A;)15 b(D)s Ft(\),)34 b(and)e Fq(\025)c Fr(2)g Fp(R)s Ft(,)39 b(the)32 b Fq(\025)p Ft(-capacit)m(y)i(of)e(the)h(capacitor)g (is)e(de\014ned)50 1490 y(as)1227 1636 y(cap)1363 1598 y Fo(\025)1363 1658 y(A)1422 1636 y Ft(\()p Fq(D)s Ft(\))26 b Fr(\021)1692 1512 y Fi(Z)1742 1718 y Fo(@)t(A)1860 1636 y Fq(e)1902 1598 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(y)r Fu(\))p Fo(=\017)2183 1636 y Fq(e)2225 1598 y Fo(\025)2225 1658 y(A;D)2367 1636 y Ft(\()p Fq(dy)s Ft(\))945 b(\(2)p Fq(:)p Ft(13\))50 1850 y(Using)29 b(\(2.12\))k(and)d(the)g(Green's)h (second)f(iden)m(tit)m(y)-8 b(,)31 b(one)g(obtains)e(from)h(\(2.13\))j (that)108 2103 y(cap)244 2065 y Fo(\025)244 2125 y(A)303 2103 y Ft(\()p Fq(D)s Ft(\))26 b(=)f Fq(\017)625 1979 y Fi(Z)676 2185 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))897 2167 y Fj(c)951 2103 y Fq(dx)15 b(e)1107 2065 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(x)p Fu(\))p Fo(=\017)1406 1974 y Fi(\024)1454 2025 y(\015)1454 2080 y(\015)1504 2103 y Fr(r)p Fq(h)1632 2065 y Fo(\025)1632 2125 y(A;D)1775 2103 y Ft(\()p Fq(x)p Ft(\))1897 2025 y Fi(\015)1897 2080 y(\015)1948 2043 y Fu(2)1948 2142 y(2)2009 2103 y Fr(\000)2111 2041 y Fq(\025)p 2111 2082 54 4 v 2119 2165 a(\017)2190 2029 y Fi(\000)2231 2103 y Fq(h)2283 2065 y Fo(\025)2283 2125 y(A;D)2426 2103 y Ft(\()p Fq(x)p Ft(\))2548 2029 y Fi(\001)2590 2043 y Fu(2)2631 1974 y Fi(\025)2704 2103 y Fr(\021)25 b Ft(\010)2866 2065 y Fo(\025)2866 2127 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))3087 2109 y Fj(c)3142 2029 y Fi(\000)3183 2103 y Fq(h)3235 2065 y Fo(\025)3235 2125 y(A;D)3378 2029 y Fi(\001)3477 2103 y Ft(\(2)p Fq(:)p Ft(14\))50 2376 y(\010)116 2343 y Fo(\025)116 2401 y Fu(\012)202 2376 y Ft(is)30 b(called)f(the)i (Diric)m(hlet)e(form)h(\(or)h(energy\))g(for)f(the)g(op)s(erator)h Fq(L)2495 2390 y Fo(\017)2550 2376 y Fr(\000)20 b Fq(\025)30 b Ft(on)g(\012.)159 2595 y(A)37 b(fundamen)m(tal)f(consequence)i(of)f (\(2.14\))j(is)c(the)h(v)-5 b(ariational)36 b(represen)m(tation)i(of)f (the)g(capacit)m(y)i(if)50 2740 y Fp(R)34 b Fr(3)25 b Fq(\025)g Fr(\024)g Ft(0,)31 b(namely)1276 2885 y(cap)1412 2848 y Fo(\025)1412 2908 y(A)1471 2885 y Ft(\()p Fq(D)s Ft(\))26 b(=)109 b(inf)1741 2945 y Fo(h)p Fh(2H)1895 2954 y Fj(A;D)2035 2885 y Ft(\010)2101 2848 y Fo(\025)2101 2910 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))2322 2892 y Fj(c)2361 2885 y Ft(\()p Fq(h)p Ft(\))994 b(\(2)p Fq(:)p Ft(15\))50 3099 y(where)30 b Fr(H)390 3113 y Fo(A;D)562 3099 y Ft(denotes)h(the)f(set)h(of)g(function)732 3333 y Fr(H)809 3347 y Fo(A;D)976 3333 y Fr(\021)1072 3259 y Fi(\010)1125 3333 y Fq(h)26 b Fr(2)f Fq(W)1388 3295 y Fu(1)p Fo(;)p Fu(2)1486 3333 y Ft(\(\012\))g(:)h Fq(h)p Ft(\()p Fq(x)p Ft(\))g(=)f(0)p Fq(;)15 b(x)26 b Fr(2)f Fq(D)s(;)15 b(h)p Ft(\()p Fq(x)p Ft(\))27 b(=)e(1)p Fq(;)15 b(x)26 b Fr(2)e Fq(A)2974 3259 y Fi(\011)3477 3333 y Ft(\(2)p Fq(:)p Ft(16\))50 3567 y(where)g Fq(W)406 3534 y Fo(k)r(;n)517 3567 y Ft(\(\012\))h(denotes)g(the)f(space)i(of)e Fq(k)s Ft(-times)h(w)m(eakly)g(di\013eren)m(tiable)e(functions)g(whose) h(deriv)-5 b(ativ)m(es)50 3712 y(of)30 b(order)g Fr(\024)25 b Fq(k)34 b Ft(are)c(in)g Fq(L)888 3679 y Fo(n)937 3712 y Ft(\(\012\).)50 3931 y Fs(Probabilistic)j(in)m(terpretation:)45 b(equilibrium)32 b(p)s(oten)m(tial.)39 b Ft(Note)29 b(that)g Fq(L)2876 3945 y Fo(\017)2938 3931 y Ft(generates)h(a)e(Mark)m(o)m(v)50 4076 y(di\013usion)d(pro)s(cess)j Fq(X)804 4090 y Fo(\017)839 4076 y Ft(\()p Fq(t)p Ft(\))h(on)f(\012)f(\(killed)g(on)h Fq(@)5 b Ft(\012\).)40 b(If)27 b Fq(\025)f Ft(=)f(0,)k(the)f (equilibrium)c(p)s(oten)m(tial)j(has)h(a)h(natural)50 4222 y(probabilistic)e(in)m(terpretation)j(in)f(terms)h(of)g(hitting)f (probabilities)e(of)k(this)e(pro)s(cess,)h(namely)-8 b(,)1203 4456 y Fq(h)1255 4470 y Fo(A;D)1397 4456 y Ft(\()p Fq(x)p Ft(\))26 b Fr(\021)f Fq(h)1693 4418 y Fu(0)1693 4478 y Fo(A;D)1836 4456 y Ft(\()p Fq(x)p Ft(\))g(=)g Fp(P)2134 4470 y Fo(x)2181 4456 y Ft([)p Fq(\034)2246 4470 y Fo(A)2330 4456 y Fq(<)g(\034)2466 4470 y Fo(D)2532 4456 y Ft(])920 b(\(2)p Fq(:)p Ft(17\))50 4690 y(The)30 b(equilibrium)25 b(measure)31 b(also)f(has)g(an)g(in)m(terpretation,)g (namely)1102 4924 y Fq(e)1144 4938 y Fo(A;D)1286 4924 y Ft(\()p Fq(dy)s Ft(\))d(=)e(lim)1586 4983 y Fo(t)p Fh(#)p Fu(0)1715 4924 y Fq(t)1748 4887 y Fh(\000)p Fu(1)1845 4924 y Fp(E)1900 4938 y Fo(y)1950 4924 y Fp(P)2005 4940 y Fo(X)2065 4949 y Fj(\017)2097 4940 y Fu(\()p Fo(t)p Fu(\))2186 4924 y Ft([)p Fq(\034)2251 4938 y Fo(D)2342 4924 y Fq(<)g(\034)2478 4938 y Fo(A)2537 4924 y Ft(])p Fq(dy)823 b Ft(\(2)p Fq(:)p Ft(18\))50 5171 y(\(see)35 b(e.g.)55 b([Szni],)35 b(Section)f(2.3.\).)56 b(While)33 b(this)h(giv)m(es)h(in)e(principle)e(a)k(probabilistic)d(in)m (terpretation)i(of)50 5316 y(the)c(capacit)m(y)i(as)e(w)m(ell,)g(this)f (is)g(m)m(uc)m(h)h(less)g(useful)e(than)i(in)f(the)i(discrete)f(space,) h(discrete)f(time)g(setting)50 5462 y(\(see)h([BEGK2]\).)p eop %%Page: 8 8 8 7 bop 50 295 a Fl(8)159 513 y Ft(If)35 b Fq(\025)g(<)g Ft(0,)j(the)e(equilibrium)31 b(p)s(oten)m(tial)k(still)f(has)i(a)h (probabilistic)32 b(in)m(terpretation)k(in)e(term)i(of)h(the)50 658 y(sub-sto)s(c)m(hastic)25 b(pro)s(cess)h Fq(X)1027 625 y Fo(\025)1020 681 y(\017)1075 658 y Ft(\()p Fq(t)p Ft(\))g(obtained)f(b)m(y)g(killing)e(the)i(pro)s(cess)g Fq(X)2503 672 y Fo(\017)2539 658 y Ft(\()p Fq(t)p Ft(\))h(with)e(rate)i Fr(\000)p Fq(\025)f Ft(\(and)h(on)f Fq(@)5 b Ft(\012\).)50 804 y(If)30 b Fq(\034)40 b Ft(denotes)31 b(the)f(time)g(when)g Fq(X)1233 771 y Fo(\025)1226 826 y(\017)1311 804 y Ft(is)g(killed,)e(w) m(e)j(ha)m(v)m(e)g(that)1345 1064 y Fq(h)1397 1027 y Fo(\025)1397 1087 y(A;D)1540 1064 y Ft(\()p Fq(x)p Ft(\))26 b(=)f Fp(P)1839 1027 y Fo(\025)1839 1087 y(x)1887 1064 y Ft([)p Fq(\034)1952 1078 y Fo(A)2036 1064 y Fq(<)g(\034)2172 1078 y Fo(D)2258 1064 y Fr(^)20 b Fq(\034)10 b Ft(])1063 b(\(2)p Fq(:)p Ft(19\))50 1325 y(More)31 b(imp)s(ortan)m(tly)-8 b(,)29 b(w)m(e)i(ha)m(v)m(e)h(for)e(general)g Fq(\025)p Ft(,)h(that)1374 1586 y Fq(h)1426 1548 y Fo(\025)1426 1608 y(A;D)1568 1586 y Ft(\()p Fq(x)p Ft(\))26 b(=)f Fp(E)1867 1600 y Fo(x)1919 1586 y Fq(e)1961 1548 y Fo(\025\034)2038 1557 y Fj(A)2094 1586 y Ft(1)-23 b(I)2149 1600 y Fo(\034)2183 1609 y Fj(A)2234 1600 y Fo(<\034)2325 1609 y Fj(D)3477 1586 y Ft(\(2)p Fq(:)p Ft(20\))50 1847 y(for)29 b Fq(x)c Fr(2)g Ft(\()p Fq(A)18 b Fr([)f Fq(D)s Ft(\))663 1814 y Fo(c)700 1847 y Ft(,)30 b(whenev)m(er)f(the)g(righ)m(t-hand)f(side)g (exists.)40 b(so)29 b(that)h Fq(h)2579 1814 y Fo(\025)2656 1847 y Ft(can)f(b)s(e)g(seen)g(as)g(the)h(Laplace)50 1992 y(transform)f(of)i(the)g(hitting)e(time)h Fq(\034)1273 2006 y Fo(A)1362 1992 y Ft(of)g(the)h(pro)s(cess)f(starting)g(in)f Fq(x)h Ft(and)g(killed)e(in)h Fq(D)s Ft(.)159 2210 y(Note)j(that)f (\(2.20\))h(implies)c(that)1388 2420 y Fq(d)p 1362 2461 101 4 v 1362 2544 a(d\025)1473 2482 y(h)1525 2444 y Fo(\025)p Fu(=0)1525 2504 y Fo(A;D)1667 2482 y Ft(\()p Fq(x)p Ft(\))e(=)f Fp(E)1966 2496 y Fo(x)2018 2482 y Fq(\034)2058 2496 y Fo(A)2117 2482 y Ft(1)-23 b(I)2172 2496 y Fo(\034)2206 2505 y Fj(A)2257 2496 y Fo(<\034)2348 2505 y Fj(D)3477 2482 y Ft(\(2)p Fq(:)p Ft(21\))50 2746 y(Di\013eren)m(tiating)30 b(the)g(de\014ning)f(equation)h(of)h Fq(h)1682 2713 y Fo(\025)1682 2771 y(A;D)1855 2746 y Ft(then)f(implies)d(that)k(the)g (function)1026 3044 y Fq(w)1091 3058 y Fo(A;D)1234 3044 y Ft(\()p Fq(x)p Ft(\))25 b(=)1477 2916 y Fi(\032)1576 2976 y Fp(E)1630 2990 y Fo(x)1682 2976 y Fq(\034)1722 2990 y Fo(A)1781 2976 y Ft(1)-23 b(I)1837 2990 y Fo(\034)1871 2999 y Fj(A)1922 2990 y Fo(<\034)2013 2999 y Fj(D)2073 2976 y Fq(;)92 b(x)25 b Fr(2)g Ft(\()p Fq(A)20 b Fr([)g Fq(D)s Ft(\))2670 2943 y Fo(c)1576 3112 y Ft(0)p Fq(;)544 b(x)25 b Fr(2)g Fq(A)20 b Fr([)g Fq(D)3477 3044 y Ft(\(2)p Fq(:)p Ft(22\))50 3328 y(solv)m(es)j(the)h(inhomogeneous)e(Diric)m (hlet)g(problem)g(\(to)i(simplify)c(notation,)25 b(w)m(e)f(set)f(from)g (no)m(w)g(on)g Fq(h)3471 3342 y Fo(A;D)3639 3328 y Fr(\021)50 3473 y Fq(h)102 3440 y Fu(0)102 3498 y Fo(A)p Fu(;)p Fo(D)243 3473 y Ft(,)31 b(etc.\))1123 3603 y Fq(L)1185 3617 y Fo(\017)1219 3603 y Fq(w)1284 3617 y Fo(A;D)1427 3603 y Ft(\()p Fq(x)p Ft(\))26 b(=)f Fq(h)1723 3617 y Fo(A;D)1865 3603 y Ft(\()p Fq(x)p Ft(\))p Fq(;)107 b(x)25 b Fr(2)g Ft(\()p Fq(A)c Fr([)f Fq(D)s Ft(\))2600 3565 y Fo(c)1219 3776 y Fq(w)1284 3790 y Fo(A;D)1427 3776 y Ft(\()p Fq(x)p Ft(\))26 b(=)f(0)p Fq(;)106 b(x)26 b Fr(2)e Fq(A)d Fr([)f Fq(D)3477 3689 y Ft(\(2)p Fq(:)p Ft(23\))50 3950 y(Therefore,)29 b(the)g(mean)g(hitting)f(time)g(in)g Fq(A)h Ft(of)g(the)g(pro)s(cess)f(killed)f(in)g Fq(D)32 b Ft(can)d(b)s(e)f(represen)m(ted)h(in)f(terms)50 4095 y(of)i(the)h(Green's)g(function)e(as)932 4356 y Fp(E)986 4370 y Fo(x)1038 4356 y Fq(\034)1078 4370 y Fo(A)1137 4356 y Ft(1)-23 b(I)1193 4370 y Fo(\034)1227 4379 y Fj(A)1278 4370 y Fo(<\034)1369 4379 y Fj(D)1455 4356 y Ft(=)1551 4232 y Fi(Z)1601 4439 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))1822 4421 y Fj(c)1877 4356 y Fq(dy)s(G)2044 4372 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))2265 4354 y Fj(c)2304 4356 y Ft(\()p Fq(x;)15 b(y)s Ft(\))p Fq(h)2566 4370 y Fo(A;D)2709 4356 y Ft(\()p Fq(y)s Ft(\))650 b(\(2)p Fq(:)p Ft(24\))50 4647 y(Note)32 b(that)e(in)g(the)g(particular)f(case) i(when)f Fq(D)e Ft(=)d Fr(;)p Ft(,)31 b(w)m(e)f(get)i(the)f(familiar)d (Diric)m(hlet)h(problem)1425 4867 y Fq(L)1487 4881 y Fo(\017)1522 4867 y Fq(w)1587 4881 y Fo(A)1646 4867 y Ft(\()p Fq(x)p Ft(\))d(=)f(1)p Fq(;)106 b(x)26 b Fr(2)f Fq(A)2298 4829 y Fo(c)1522 5039 y Fq(w)1587 5053 y Fo(A)1646 5039 y Ft(\()p Fq(x)p Ft(\))h(=)f(0)p Fq(;)106 b(x)26 b Fr(2)f Fq(A)3477 4953 y Ft(\(2)p Fq(:)p Ft(25\))50 5259 y(and)30 b(the)g(represen)m(tation)1402 5415 y Fp(E)1456 5429 y Fo(x)1508 5415 y Fq(\034)1548 5429 y Fo(A)1632 5415 y Ft(=)1728 5292 y Fi(Z)1779 5498 y Fo(A)1834 5480 y Fj(c)1887 5415 y Fq(dy)s(G)2054 5429 y Fo(A)2109 5411 y Fj(c)2147 5415 y Ft(\()p Fq(x;)15 b(y)s Ft(\))1120 b(\(2)p Fq(:)p Ft(26\))p eop %%Page: 9 9 9 8 bop 3671 295 a Fl(9)159 513 y Ft(The)39 b(full)e(b)s(eaut)m(y)j(of) g(all)e(this)h(comes)h(out)g(when)f(com)m(bining)f(\(2.10\))k(with)c (\(2.24\),)44 b(resp.)68 b(\(2.26\).)50 658 y(Namely)-8 b(,)31 b(let)f Fq(B)601 672 y Fo(\032)643 658 y Ft(\()p Fq(x)p Ft(\))h(b)s(e)f(the)h(ball)d(of)j(radius)d Fq(\032)j Ft(cen)m(tered)g(at)g Fq(x)p Ft(.)41 b(Then,)29 b(using)g(F)-8 b(ubini's)29 b(theorem,)179 791 y Fi(Z)230 998 y Fo(@)t(B)328 1007 y Fj(\032)365 998 y Fu(\()p Fo(x)p Fu(\))483 915 y Fq(e)525 878 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(z)s Fu(\))p Fo(=\017)804 915 y Fp(E)858 929 y Fo(z)906 915 y Fq(\034)946 929 y Fo(A)1005 915 y Fq(e)1047 932 y Fo(B)1102 941 y Fj(\032)1140 932 y Fu(\()p Fo(x)p Fu(\))p Fo(;A)1319 915 y Ft(\()p Fq(dz)t Ft(\))27 b(=)1604 791 y Fi(Z)1655 998 y Fo(A)1710 980 y Fj(c)1763 915 y Fq(dy)s(e)1900 878 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(y)r Fu(\))p Fo(=\017)2197 791 y Fi(Z)2247 998 y Fo(@)t(B)2345 1007 y Fj(\032)2383 998 y Fu(\()p Fo(x)p Fu(\))2500 915 y Fq(G)2572 929 y Fo(A)2627 911 y Fj(c)2665 915 y Ft(\()p Fq(y)s(;)15 b(z)t Ft(\))p Fq(e)2911 932 y Fo(B)2966 941 y Fj(\032)3005 932 y Fu(\()p Fo(x)p Fu(\))p Fo(;A)3184 915 y Ft(\()p Fq(dz)t Ft(\))1509 1176 y(=)1604 1052 y Fi(Z)1655 1259 y Fo(A)1710 1241 y Fj(c)1763 1176 y Fq(dy)s(e)1900 1139 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(y)r Fu(\))p Fo(=\017)2182 1176 y Fq(h)2234 1193 y Fo(B)2289 1202 y Fj(\032)2327 1193 y Fu(\()p Fo(x)p Fu(\))p Fo(;A)2505 1176 y Ft(\()p Fq(y)s Ft(\))3477 1048 y(\(2)p Fq(:)p Ft(27\))50 1429 y(and)161 1549 y Fi(Z)211 1756 y Fo(@)t(B)309 1765 y Fj(\032)346 1756 y Fu(\()p Fo(x)p Fu(\))464 1673 y Fq(e)506 1635 y Fh(\000)p Fo(F)g Fu(\()p Fo(z)s Fu(\))p Fo(=\017)786 1673 y Fp(E)840 1687 y Fo(z)888 1673 y Fq(\034)928 1687 y Fo(A)987 1673 y Ft(1)-23 b(I)1043 1687 y Fo(\034)1077 1696 y Fj(A)1128 1687 y Fo(<\034)1219 1696 y Fj(D)1279 1673 y Fq(e)1321 1689 y Fo(B)1376 1698 y Fj(\032)1414 1689 y Fu(\()p Fo(x)p Fu(\))p Fo(;A)p Fh([)p Fo(D)1703 1673 y Ft(\()p Fq(dz)t Ft(\))27 b(=)1989 1549 y Fi(Z)2039 1756 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))2260 1738 y Fj(c)2315 1673 y Fq(dy)s(e)2452 1635 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(y)r Fu(\))p Fo(=\017)2733 1673 y Fq(h)2785 1689 y Fo(B)2840 1698 y Fj(\032)2878 1689 y Fu(\()p Fo(x)p Fu(\))p Fo(;A)p Fh([)p Fo(D)3167 1673 y Ft(\()p Fq(y)s Ft(\))p Fq(h)3337 1687 y Fo(A;D)3480 1673 y Ft(\()p Fq(y)s Ft(\))3477 1851 y(\(2)p Fq(:)p Ft(28\))50 1996 y(Notice)36 b(that)g(in)d(the)j(case)g(of)f(discrete)g(Mark)m(o)m(v)i(pro)s (cesses,)f(w)m(e)f(can)h(replace)f(the)g(ball)f Fq(B)3253 2010 y Fo(\032)3295 1996 y Ft(\()p Fq(x)p Ft(\))i(b)m(y)f(the)50 2142 y(single)c(p)s(oin)m(t)g Fq(x)p Ft(.)46 b(In)31 b(that)i(case)g(\(2.27\))h(and)e(\(2.28\))i(yield)d(directly)g(form)m (ulae)g(for)h(mean)g(hitting)f(times)50 2287 y(in)d(terms)h(of)h (capacities)f(and)g(equilibrium)c(p)s(oten)m(tials.)39 b(In)29 b(this)f(con)m(text)k(they)d(pro)m(vided)f(the)i(basis)e(for)50 2433 y(connecting)35 b(in)e(a)i(precise)f(w)m(a)m(y)i(capacities)f(and) f(mean)h(exit)f(times,)i(and,)f(ultimately)-8 b(,)35 b(eigen)m(v)-5 b(alues)35 b(of)50 2578 y Fq(L)112 2592 y Fo(\017)182 2578 y Ft([BEGK2].)57 b(In)34 b(the)i(di\013usion)d (case,)38 b(the)d(usefulness)f(of)h(these)h(equations)f(will)e(b)s (ecome)j(apparen)m(t)50 2724 y(only)29 b(when)h(w)m(e)g(ha)m(v)m(e)i (some)f(a)f(priori)e(regularit)m(y)i(estimates)h(for)f(the)g(mean)h (times)f(as)g(functions)f(of)i(the)50 2869 y(starting)f(p)s(oin)m(t.)50 3119 y Fv(3.)55 b(Results.)159 3337 y Ft(W)-8 b(e)32 b(are)f(no)m(w)g(ready)f(to)i(state)g(the)f(main)e(results)h(of)h(this) e(pap)s(er.)41 b(One)30 b(k)m(ey)i(notion)e(w)m(e)h(will)d(need)j(is)50 3483 y(that)g(of)g(the)f(saddle)g(b)s(et)m(w)m(een)h(t)m(w)m(o)h(sets)f Fq(A;)15 b(B)30 b Fr(\032)c Fp(R)1850 3450 y Fo(d)1899 3483 y Ft(.)41 b(W)-8 b(e)31 b(sa)m(y)h(that)f Fq(z)2522 3450 y Fh(\003)2589 3483 y Ft(=)26 b Fq(z)2732 3450 y Fh(\003)2774 3483 y Ft(\()p Fq(A;)15 b(B)5 b Ft(\))31 b(is)e(a)i(saddle)f(p)s(oin)m(t)50 3628 y(b)s(et)m(w)m(een)h Fq(A)f Ft(and)g Fq(B)5 b Ft(,)30 b(if)808 3849 y Fi(b)790 3872 y Fq(F)13 b Ft(\()p Fq(A;)i(B)5 b Ft(\))26 b Fr(\021)f Fq(F)13 b Ft(\()p Fq(z)1387 3834 y Fh(\003)1430 3872 y Ft(\()p Fq(A;)i(B)5 b Ft(\)\))26 b(=)261 b(inf)1839 3935 y Fo(!)r Fu(:)p Fo(!)r Fu(\(0\))p Fh(2)p Fo(A;!)r Fu(\(1\))p Fh(2)p Fo(B)2475 3872 y Ft(sup)2437 3952 y Fo(t)p Fh(2)p Fu([0)p Fo(;)p Fu(1])2664 3872 y Fq(F)13 b Ft(\()p Fq(!)s Ft(\()p Fq(t)p Ft(\)\))555 b(\(3)p Fq(:)p Ft(1\))50 4153 y(where)30 b(the)h(in\014m)m(um)e(is)h(o)m(v)m(er)i(all) e(con)m(tin)m(uous)g(paths)g(going)h(from)g Fq(A)g Ft(to)g Fq(B)5 b Ft(.)42 b(Note)32 b(that)f Fq(z)3217 4120 y Fh(\003)3260 4153 y Ft(\()p Fq(A;)15 b(B)5 b Ft(\))31 b(ma)m(y)50 4298 y(not)f(b)s(e)g(uniquely)e(de\014ned;)h(w)m(e)i(call)f Fr(S)7 b Ft(\()p Fq(A;)15 b(B)5 b Ft(\))31 b(the)f(set)h(of)g(all)e(p)s (ossible)f(solutions.)159 4516 y(T)-8 b(o)43 b(a)m(v)m(oid)f (complications)f(that)i(are)f(not)h(our)f(main)f(concern)h(here,)j(w)m (e)e(will)c(mak)m(e)k(the)g(general)50 4662 y(assumption)29 b(that)i(all)e(saddle)g(p)s(oin)m(ts)g(w)m(e)i(will)d(deal)i(with)f (are)i(non-degenerate)g(in)e(the)i(sense)f(that)50 4880 y Fs(Assumption)35 b(\(ND\))p Ft(:)17 5098 y(\(i\))46 b(The)40 b(Hessian)h(matrix)f(of)h Fq(F)54 b Ft(at)41 b(all)f(saddle)g(p)s(oin)m(ts)g Fq(z)2108 5065 y Fh(\003)2190 5098 y Ft(is)g(non-degenerate)i(\(i.e.)73 b(has)40 b(non)h(zero-)159 5244 y(eigen)m(v)-5 b(alues\).)-8 5462 y(\(ii\))45 b(Along)30 b(an)m(y)h(optimal)e(path)h Fq(!)j Ft(con)m(tributing)c(in)g(\(3.1\),)k Fq(F)13 b Ft(\()p Fq(!)s Ft(\()p Fq(t)p Ft(\)\))31 b(has)f(as)h(unique) d(maxim)m(um.)p eop %%Page: 10 10 10 9 bop 50 295 a Fl(10)159 513 y Ft(The)28 b(basis)f(for)h(the)g (success)g(of)h(our)e(approac)m(h)i(is)e(the)i(fact)g(that)f (capacities)h(can)f(b)s(e)g(estimated)g(v)m(ery)50 658 y(sharply)-8 b(.)50 876 y Fe(Theorem)37 b(3.1:)91 b Fn(Assume)33 b(that)h Fq(A;)15 b(B)30 b Fr(\032)25 b Fp(R)1693 843 y Fo(d)1774 876 y Fn(ar)-5 b(e)34 b(close)-5 b(d)34 b(and)10 1095 y(\(i\))62 b Ft(dist)14 b(\()q Fr(S)7 b Ft(\()p Fq(A;)15 b(B)5 b Ft(\))p Fq(;)15 b(A)21 b Fr([)f Fq(B)5 b Ft(\))25 b Fr(\025)g Fq(\016)k(>)c Ft(0)33 b Fn(for)g(some)g Fq(\016)k Fn(indep)-5 b(endent)34 b(of)f Fq(\017)-18 1313 y Fn(\(ii\))47 b(b)-5 b(oth)34 b Fq(A)e Fn(and)i Fq(B)j Fn(c)-5 b(ontain)34 b(a)f(close)-5 b(d)34 b(b)-5 b(al)5 b(l)33 b(of)g(r)-5 b(adius)34 b(at)f(le)-5 b(ast)34 b Fq(\017)p Fn(.)159 1531 y(Then,)44 b(if)d Fr(S)7 b Ft(\()p Fq(A;)15 b(B)5 b Ft(\))43 b(=)e Fr(f)p Fq(z)1092 1498 y Fh(\003)1088 1553 y Fu(1)1135 1531 y Fq(;)15 b(:)g(:)g(:)h(;)f (z)1382 1498 y Fh(\003)1378 1553 y Fo(n)1428 1531 y Fr(g)p Fn(,)45 b(and)d(Assumption)h Fs(\(ND\))e Fn(holds)i(for)f(these)g(sadd) 5 b(le)43 b(p)-5 b(oints,)50 1676 y(then)424 1852 y Ft(cap)560 1875 y Fo(A)619 1852 y Ft(\()p Fq(B)5 b Ft(\))26 b(=)f Fq(e)927 1815 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(z)1106 1787 y Fd(\003)1144 1815 y Fu(\()p Fo(x;y)r Fu(\)\))p Fo(=\017)1414 1791 y Ft(\(2)p Fq(\031)s(\017)p Ft(\))1621 1758 y Fo(d=)p Fu(2)p 1414 1831 325 4 v 1526 1915 a Ft(2)p Fq(\031)1809 1739 y Fo(k)1764 1766 y Fi(X)1771 1960 y Fo(i)p Fu(=1)2119 1791 y Fr(j)p Fq(\025)2197 1758 y Fh(\003)2197 1813 y Fu(1)2239 1791 y Ft(\()p Fq(z)2320 1758 y Fh(\003)2316 1815 y Fo(i)2363 1791 y Ft(\))p Fr(j)p 1921 1831 700 4 v 1921 1850 a Fi(p)p 2012 1850 610 4 v 76 x Fr(j)15 b Ft(det)q(\()p Fr(r)2290 1900 y Fu(2)2331 1926 y Fq(F)e Ft(\()p Fq(z)2483 1895 y Fh(\003)2479 1951 y Fo(i)2525 1926 y Ft(\)\))p Fr(j)2647 1779 y Fi(\000)2689 1852 y Ft(1)21 b(+)e Fq(O)s Ft(\()2952 1782 y Fr(p)p 3028 1782 37 4 v 70 x Fq(\017)p Fr(j)c Ft(ln)g Fq(\017)p Fr(j)p Ft(\))3294 1779 y Fi(\001)3523 1852 y Ft(\(3)p Fq(:)p Ft(2\))50 2104 y Fn(wher)-5 b(e)34 b Fq(\025)360 2071 y Fh(\003)360 2127 y Fu(1)401 2104 y Ft(\()p Fq(z)482 2071 y Fh(\003)478 2128 y Fo(i)525 2104 y Ft(\))f Fn(denotes)g(the)g (ne)-5 b(gative)33 b(eigenvalue)f(of)h(the)g(Hessian)g(at)g Fq(z)2616 2071 y Fh(\003)2612 2128 y Fo(i)2658 2104 y Fn(.)50 2323 y Fs(Remark:)47 b Ft(In)30 b(cases)i(when)e(some)h(saddle) f(p)s(oin)m(t)g(are)h(degenerate,)i(one)e(can)h(also)e(obtain)h (precise,)f(but)50 2468 y(somewhat)h(less)e(explicit)g(expressions,)g (as)i(will)d(b)s(e)h(clear)i(from)f(the)g(pro)s(of.)159 2686 y(Our)f(next)i(result)e(concerns)h(the)h(mean)f(metastable)h(exit) g(times)f(from)g(a)g(minim)m(um)e Fq(x)3204 2700 y Fo(i)3234 2686 y Ft(.)50 2904 y Fe(Theorem)33 b(3.2:)89 b Fn(L)-5 b(et)30 b Fq(x)1005 2918 y Fo(i)1065 2904 y Fn(b)-5 b(e)30 b(a)h(minimum)f(of)h Fq(F)43 b Fn(and)31 b(let)f Fq(D)j Fn(b)-5 b(e)30 b(any)h(close)-5 b(d)31 b(subset)f(of)g Fp(R)3246 2871 y Fo(d)3324 2904 y Fn(such)h(that:)10 3123 y(\(i\))47 b(If)36 b Fr(M)367 3137 y Fo(i)430 3123 y Fr(\021)d(f)p Fq(y)624 3137 y Fu(1)665 3123 y Fq(;)15 b(:)g(:)g(:)h(;)f(y)911 3137 y Fo(k)956 3123 y Fr(g)33 b(\032)f(M)37 b Fn(enumer)-5 b(ates)38 b(al)5 b(l)37 b(those)h(minima)f(of)g Fq(F)50 b Fn(such)36 b(that)i Fq(F)13 b Ft(\()p Fq(y)3248 3137 y Fo(j)3287 3123 y Ft(\))33 b Fr(\024)f Fq(F)13 b Ft(\()p Fq(x)3616 3137 y Fo(i)3647 3123 y Ft(\))p Fn(,)159 3268 y(then)33 b Fr([)422 3235 y Fo(k)422 3292 y(j)t Fu(=1)552 3268 y Fq(B)621 3282 y Fo(\017)656 3268 y Ft(\()p Fq(y)736 3282 y Fo(j)774 3268 y Ft(\))26 b Fr(\032)f Fq(D)s Fn(,)32 b(and)-18 3486 y(\(ii\))62 b Ft(dist)14 b(\()q Fr(S)7 b Ft(\()p Fq(x)521 3500 y Fo(i)551 3486 y Fq(;)15 b Fr(M)700 3500 y Fo(i)731 3486 y Ft(\))p Fq(;)g(D)s Ft(\))26 b Fr(\025)f Fq(\016)k(>)c Ft(0)33 b Fn(for)g(some)g Fq(\016)k Fn(indep)-5 b(endent)34 b(of)f Fq(\017)p Fn(.)159 3704 y(If)43 b(Assumption)h Fs(\(ND\))90 b Fn(holds)44 b(for)g(the)f(sadd)5 b(les)45 b Fq(S)5 b Ft(\()p Fq(x)2143 3718 y Fo(i)2173 3704 y Fq(;)15 b(D)s Ft(\))44 b Fn(and)g(if)e(the)i(Hessian)f(of)g Fq(F)56 b Fn(at)44 b Fq(x)3572 3718 y Fo(i)3645 3704 y Fn(is)50 3850 y(non-de)-5 b(gener)g(ate,)34 b(then)596 4117 y Fp(E)651 4131 y Fo(x)692 4140 y Fj(i)731 4117 y Fq(\034)771 4131 y Fo(D)862 4117 y Ft(=)1365 4055 y(2)p Fq(\031)s(e)1507 4022 y Fu([)p Fo(F)10 b Fu(\()p Fo(z)1650 3995 y Fd(\003)1688 4022 y Fu(\))p Fh(\000)p Fo(F)g Fu(\()p Fo(x)1898 4031 y Fj(j)1932 4022 y Fu(\)])p Fo(=\017)p 969 4096 1481 4 v 969 4137 a Fi(p)p 1060 4137 538 4 v 78 x Ft(det\()p Fr(r)1297 4188 y Fu(2)1338 4215 y Fq(F)j Ft(\()p Fq(x)1496 4229 y Fo(i)1527 4215 y Ft(\)\))1612 4146 y Fi(P)1708 4169 y Fo(k)1708 4242 y(j)t Fu(=1)2023 4155 y Fh(j)p Fo(\025)2088 4133 y Fd(\003)2088 4177 y Fk(1)2126 4155 y Fu(\()p Fo(z)2192 4133 y Fd(\003)2189 4178 y Fj(j)2230 4155 y Fu(\))p Fh(j)p 1865 4194 574 4 v 1865 4205 a Fr(p)p 1941 4205 498 4 v 53 x Fh(j)f Fu(det\()p Fh(r)2164 4240 y Fk(2)2200 4258 y Fo(F)e Fu(\()p Fo(z)2322 4237 y Fd(\003)2319 4281 y Fj(j)2360 4258 y Fu(\)\))p Fh(j)2475 4043 y Fi(\000)2517 4117 y Ft(1)20 b(+)g Fq(O)s Ft(\()2780 4047 y Fr(p)p 2856 4047 37 4 v 70 x Fq(\017)p Fr(j)15 b Ft(ln)g Fq(\017)p Fr(j)p Ft(\))3122 4043 y Fi(\001)3523 4117 y Ft(\(3)p Fq(:)p Ft(3\))50 4616 y Fs(Remark:)62 b Ft(In)36 b(the)i(case)g(when)f(there)g(is)f(a)i (single)e(saddle)g(p)s(oin)m(t)h Fq(z)2488 4583 y Fh(\003)2530 4616 y Ft(,)i(this)d(reduces)h(to)h(the)g(classical)50 4762 y(Eyring)29 b(form)m(ula)g([GLE,MS1])498 5039 y Fp(E)552 5053 y Fo(x)593 5062 y Fj(i)633 5039 y Fq(\034)673 5053 y Fo(D)764 5039 y Ft(=)972 4978 y(2)p Fq(\031)p 870 5018 304 4 v 870 5101 a Fr(j)p Fq(\025)948 5115 y Fu(1)990 5101 y Ft(\()p Fq(z)1071 5070 y Fh(\003)1067 5127 y Fo(i)1113 5101 y Ft(\))p Fr(j)1195 4898 y Fi(p)p 1286 4898 610 4 v 78 x Fr(j)15 b Ft(det)q(\()p Fr(r)1564 4950 y Fu(2)1605 4976 y Fq(F)e Ft(\()p Fq(z)1757 4950 y Fh(\003)1799 4976 y Ft(\)\))p Fr(j)p 1195 5018 700 4 v 1231 5036 a Fi(p)p 1322 5036 538 4 v 78 x Ft(det\()p Fr(r)1559 5088 y Fu(2)1600 5114 y Fq(F)g Ft(\()p Fq(x)1758 5128 y Fo(i)1789 5114 y Ft(\)\))1906 5039 y Fq(e)1948 5002 y Fu([)p Fo(F)d Fu(\()p Fo(z)2091 4974 y Fd(\003)2129 5002 y Fu(\))p Fh(\000)p Fo(F)g Fu(\()p Fo(x)2339 5011 y Fj(j)2374 5002 y Fu(\)])p Fo(=\017)2510 4938 y Fi(\020)2564 5039 y Ft(1)21 b(+)f Fq(O)s Ft(\()p Fq(\017)2865 5002 y Fu(1)p Fo(=)p Fu(2)2979 5039 y Fr(j)15 b Ft(ln)f Fq(\017)p Fr(j)p Ft(\))3207 4938 y Fi(\021)3523 5039 y Ft(\(3)p Fq(:)p Ft(4\))50 5316 y(Note)33 b(that)g(the)f(co)s(e\016cien)m(t)h(2)p Fq(\031)i Ft(di\013ers)c(from)g(the)h Fq(\031)j Ft(that)e(is)e(found)f (in)h([MS1])i(b)m(y)e(a)i(factor)g(2)f(since)f(w)m(e)50 5462 y(consider)e(the)i(transition)e(through,)h(and)f(not)i(just)f(the) g(arriv)-5 b(al)29 b(at)i(the)g(saddle)e(p)s(oin)m(t.)p eop %%Page: 11 11 11 10 bop 3633 295 a Fl(11)50 513 y Fv(4.)55 b(Some)41 b(useful)g(to)s(ols)h(and)f(a-priori)g(estimates)159 731 y Ft(This)i(section)i(collects)g(a)g(n)m(um)m(b)s(er)f(of)h(to)s (ols)g(and)f(a-priori)f(estimates)i(that)h(extend)f(the)g(simple)50 876 y(probabilistic)27 b(instrumen)m(ts)i(used)g(in)g(the)i(discrete)f (con)m(text)i(of)38 b([BEGK2])31 b(to)g(the)g(di\013usion)d(setting.)81 1095 y Fs(Regularit)m(y)36 b(estimates.)41 b Ft(T)-8 b(o)31 b(b)s(e)g(able)f(to)i(pass)f(from)f(the)h(discrete)g(setting)g (of)38 b([BEGK1,)33 b(BEGK2])50 1240 y(to)40 b(the)g(setting)g(of)f (di\013usion)e(pro)s(cesses,)42 b(w)m(e)e(will)d(need)i(some)h(a)g (priory)e(con)m(trol)i(on)g(the)f(regularit)m(y)50 1385 y(prop)s(erties)30 b(of)i(solutions)e(of)h(the)h(Diric)m(hlet)f (problems)f(in)m(tro)s(duced)g(b)s(efore.)44 b(F)-8 b(ortunately)g(,)33 b(this)e(theory)50 1531 y(is)i(w)m(ell)g(dev)m(elop)s(ed)g(in)f(the)i (general)g(setting)g(of)g(second)g(order)g(linear)e(elliptic)f (di\013eren)m(tial)i(equations,)50 1676 y(and)d(w)m(e)g(can)h(dra)m(w)f (on)g(standard)g(results.)159 1895 y(The)h(follo)m(wing)g(t)m(w)m(o)i (k)m(ey)g(lemmata)f(are)g(tak)m(en)h(from)f([GT],)g(more)g(precisely)f (Corollaries)f(9.24)j(and)50 2040 y(9.25.)58 b(They)35 b(concern)h(second)g(order)f(elliptic)e(op)s(erators)j Fq(L)e Ft(=)g Fq(a)2374 2054 y Fo(ij)2438 2040 y Ft(\()p Fq(x)p Ft(\))p Fq(D)2635 2054 y Fo(ij)2724 2040 y Ft(+)23 b Fq(b)2857 2054 y Fo(i)2887 2040 y Ft(\()p Fq(x)p Ft(\))p Fq(D)3084 2054 y Fo(i)3139 2040 y Ft(+)h Fq(c)p Ft(\()p Fq(x)p Ft(\).)57 b(where)50 2185 y Fq(a)98 2199 y Fo(ij)187 2185 y Fr(2)25 b Fq(C)345 2152 y Fu(0)385 2185 y Ft(\(\012\))p Fq(;)15 b(b)600 2199 y Fo(i)631 2185 y Fq(;)g(c)26 b Fr(2)f Fq(L)884 2152 y Fh(1)961 2185 y Ft(\(\012\).)41 b(Assume)30 b(that)1031 2436 y(\003\()p Fq(\030)t(;)15 b(\030)t Ft(\))26 b Fr(\025)f Ft(\()p Fq(\030)t(;)15 b(a)p Ft(\()p Fq(x)p Ft(\))p Fq(\030)t Ft(\))27 b Fr(\025)e Fq(\025)p Ft(\()p Fq(\030)t(;)15 b(\030)t Ft(\))27 b Fq(>)e Ft(0)91 b Fr(8)p Fq(\030)29 b Fr(2)c Fp(R)2680 2399 y Fo(d)3523 2434 y Ft(\(4)p Fq(:)p Ft(1\))50 2704 y(let)33 b(moreo)m(v)m(er)i Fq(\015)g Ft(=)773 2668 y Fu(\003)p 773 2683 50 4 v 776 2735 a Fo(\025)833 2704 y Ft(,)f(and)f(c)m(ho)s(ose)h Fq(\027)39 b Ft(suc)m(h)33 b(that)1855 2603 y Fi(\020)1920 2660 y Fh(k)p Fo(b)p Fh(k)p 1920 2683 107 4 v 1952 2735 a Fo(\025)2038 2603 y Fi(\021)2092 2622 y Fu(2)2163 2704 y Fr(\024)c Fq(\027)6 b Ft(,)34 b(and)2564 2660 y Fh(j)p Fo(c)p Fh(j)p 2564 2683 76 4 v 2580 2735 a Fo(\025)2680 2704 y Fr(\024)c Fq(\027)6 b Ft(.)49 b(Let)34 b Fq(W)3171 2671 y Fu(2)p Fo(;n)3278 2704 y Ft(\(\012\))f(denote)50 2849 y(the)k(Banac)m(h)i (space)f(of)f(t)m(w)m(o-times)i(\(w)m(eakly\))f(di\013eren)m(tiable)e (functions)g(whose)h(deriv)-5 b(ativ)m(es)37 b(of)g(order)50 2995 y Fr(\024)25 b Ft(2)31 b(are)f(in)f Fq(L)541 2962 y Fo(n)591 2995 y Ft(\(\012\).)50 3213 y Fe(Lemma)38 b(4.1:)53 b Ft(\()p Fs(Corollary)36 b(9.25)c Ft(in)d([GT]\))p Fn(If)34 b Fq(u)26 b Fr(2)g Fq(W)2094 3180 y Fu(2)p Fo(;n)2201 3213 y Ft(\(\012\))33 b Fn(is)g(p)-5 b(ositive)34 b(and)g(satis\014es)g Fq(Lu)26 b Ft(=)g(0)34 b Fn(in)50 3358 y Ft(\012)p Fn(,)e(then)h(for)g (any)g(b)-5 b(al)5 b(l)34 b Fq(B)934 3372 y Fu(2)p Fo(R)1030 3358 y Ft(\()p Fq(y)s Ft(\))25 b Fr(\032)g Ft(\012)p Fn(,)1379 3607 y Ft(sup)1303 3688 y Fo(z)s Fh(2)p Fo(B)1445 3697 y Fj(R)1496 3688 y Fu(\()p Fo(y)r Fu(\))1607 3607 y Fq(u)p Ft(\()p Fq(z)t Ft(\))h Fr(\024)f Fq(C)111 b Ft(inf)1984 3670 y Fo(z)s Fh(2)p Fo(B)2126 3679 y Fj(R)2176 3670 y Fu(\()p Fo(y)r Fu(\))2288 3607 y Fq(u)p Ft(\()p Fq(z)t Ft(\))1067 b(\(4)p Fq(:)p Ft(2\))50 3899 y Fn(wher)-5 b(e)34 b(the)f(c)-5 b(onstant)34 b Fq(C)e Ft(=)25 b Fq(C)7 b Ft(\()p Fq(n;)15 b(\015)5 b(;)15 b(\027)6 b(R)1428 3866 y Fu(2)1468 3899 y Ft(\))26 b Fq(<)f Fr(1)32 b Fn(dep)-5 b(ends)34 b(only)g(on)f Fq(\015)38 b Fn(and)33 b Fq(\027)6 b(R)2797 3866 y Fu(2)50 4117 y Fe(Lemma)38 b(4.2:)51 b Ft(\()p Fs(Corollary)36 b(9.24)31 b Ft(in)e([GT]\))p Fn(If)34 b Fq(u)25 b Fr(2)g Fq(W)2089 4084 y Fu(2)p Fo(;n)2196 4117 y Ft(\(\012\))33 b Fn(is)f(p)-5 b(ositive)34 b(and)g(satis\014es)f Fq(Lu)26 b Ft(=)f Fq(f)42 b Fn(in)50 4263 y(a)33 b(b)-5 b(al)5 b(l)33 b Fq(B)365 4277 y Fo(R)420 4286 y Fk(0)460 4263 y Ft(\()p Fq(x)p Ft(\))p Fn(,)g(then)g(for)g(any)g(b)-5 b(al)5 b(l)34 b Fq(B)1401 4277 y Fo(R)1460 4263 y Ft(\()p Fq(x)p Ft(\))p Fn(,)f Fq(R)26 b Fr(\024)f Fq(R)1903 4277 y Fu(0)1944 4263 y Fn(,)718 4534 y(osc)843 4551 y Fo(B)898 4560 y Fj(R)950 4551 y Fu(\()p Fo(x)p Fu(\))1052 4534 y Fq(u)h Fr(\024)f Fq(C)1312 4406 y Fi(\022)1410 4473 y Fq(R)p 1390 4514 110 4 v 1390 4597 a(R)1459 4611 y Fu(0)1511 4406 y Fi(\023)1578 4425 y Fo(\013)1645 4434 y Fi(\020)1699 4534 y Fn(osc)1824 4548 y Fo(B)1879 4560 y Fj(R)1926 4574 y Fk(0)1962 4560 y(\()p Fj(x)p Fk(\))2056 4534 y Fq(u)20 b Ft(+)g Fq(R)2288 4548 y Fu(0)2329 4534 y Fr(k)p Fq(f)30 b Fr(\000)20 b Fq(cu)p Fr(k)2676 4551 y Fo(n;B)2798 4560 y Fj(R)2845 4574 y Fk(0)2885 4551 y Fu(\()p Fo(x)p Fu(\))2987 4434 y Fi(\021)3523 4534 y Ft(\(4)p Fq(:)p Ft(3\))50 4807 y Fn(wher)-5 b(e)34 b(osc)432 4821 y Fo(A)491 4807 y Fq(u)25 b Fr(\021)g Ft(sup)801 4829 y Fo(A)876 4807 y Fq(u)20 b Fr(\000)g Ft(inf)1149 4821 y Fo(A)1224 4807 y Fq(u)32 b Fn(and)i(the)f(c)-5 b(onstants)35 b Fq(\013)25 b Ft(=)g Fq(\013)p Ft(\()p Fq(n;)15 b(\015)5 b(;)15 b(\027)6 b(R)2616 4774 y Fu(2)2615 4830 y(0)2658 4807 y Ft(\))25 b Fq(>)g Ft(0)33 b Fn(and)50 4953 y Fq(C)f Ft(=)25 b Fq(C)7 b Ft(\()p Fq(n;)15 b(\015)5 b(;)15 b(\027)6 b(R)658 4920 y Fu(2)657 4975 y(0)698 4953 y Ft(\))26 b Fq(<)f Fr(1)32 b Fn(dep)-5 b(end)34 b(only)g(on)f Fq(\015)k Fn(and)d Fq(\027)6 b(R)1990 4920 y Fu(2)1989 4975 y(0)2030 4953 y Fn(.)159 5171 y Ft(The)29 b(w)m(a)m(y)i(w)m(e)g(will)c(use)i(these)i(lemmata)f(is)f(to)i (consider)d(domains)h(dep)s(ending)e(on)j Fq(\017)g Ft(c)m(hosen)g(in)f (suc)m(h)50 5316 y(a)h(w)m(a)m(y)h(that)f(the)g(n)m(umerical)e(constan) m(ts)j Fq(C)36 b Ft(and)30 b Fq(\013)g Ft(are)g(indep)s(enden)m(t)d(of) j Fq(\017)p Ft(.)41 b(Since)29 b(for)g(the)h(op)s(erator)g Fq(L)3675 5330 y Fo(\017)50 5462 y Ft(\003)25 b(=)g Fq(\025)g Ft(=)g Fq(\017)p Ft(,)31 b Fq(\015)f Ft(=)25 b(1,)31 b(and)f(w)m(e)h(can)f(c)m(ho)s(ose)i Fq(\027)f Ft(=)25 b Fq(\017)1750 5429 y Fh(\000)p Fu(2)1862 5462 y Ft(sup)1999 5484 y Fo(y)r Fh(2)p Fu(\012)2159 5462 y Fr(kr)p Fq(F)13 b Ft(\()p Fq(y)s Ft(\))p Fr(k)2515 5416 y Fu(2)2515 5489 y Fh(1)2592 5462 y Ft(.)p eop %%Page: 12 12 12 11 bop 50 295 a Fl(12)50 513 y Fs(An)40 b(analytic)g(renew)m(al)g (estimate.)53 b Ft(In)34 b(this)g(section)i(w)m(e)f(consider)f(only)g (the)h(case)i Fq(\025)c Fr(\021)f Ft(0)k(and)e(w)m(e)50 658 y(omit)41 b(the)g(sup)s(erscript)d(0.)74 b(One)40 b(of)h(the)h(most)f(useful)e(form)m(ulas)i(used)f(in)g(our)g(analysis)g (of)h(discrete)50 804 y(Mark)m(o)m(v)32 b(c)m(hains)e(w)m(as)g(the)h Fn(r)-5 b(enewal)34 b(e)-5 b(quation)1286 1047 y Fp(P)1341 1061 y Fo(x)1387 1047 y Ft([)p Fq(\034)1452 1061 y Fo(A)1536 1047 y Fq(<)25 b(\034)1672 1061 y Fo(D)1738 1047 y Ft(])g(=)1895 985 y Fp(P)1950 999 y Fo(x)1997 985 y Ft([)p Fq(\034)2062 999 y Fo(A)2146 985 y Fq(<)g(\034)2282 999 y Fo(D)r Fh([)p Fo(x)2438 985 y Ft(])p 1895 1026 568 4 v 1895 1109 a Fp(P)1950 1123 y Fo(x)1997 1109 y Ft([)p Fq(\034)2062 1123 y Fo(A)p Fh([)p Fo(D)2256 1109 y Fq(<)g(\034)2392 1123 y Fo(x)2438 1109 y Ft(])3523 1047 y(\(4)p Fq(:)p Ft(4\))50 1296 y(obtained)33 b(from)h(decomp)s(osing)f(the)i(ev)m(en)m (t)h Fr(f)p Fq(\034)1678 1310 y Fo(A)1769 1296 y Fq(<)31 b(\034)1911 1310 y Fo(D)1977 1296 y Fr(g)j Ft(according)g(to)h(whether) f(the)g(pro)s(cess)g(visits)f Fq(x)50 1441 y Ft(b)s(efore)38 b(going)h(to)h Fq(A)f Ft(or)g(not)g(and)f(using)g(the)h(Mark)m(o)m(v)h (prop)s(ert)m(y)-8 b(.)67 b(While)37 b(this)h(form)m(ula)g(is)g(still)f (true)50 1587 y(in)43 b(the)h(di\013usion)d(case)k(\(if)f Fq(d)k(>)g Ft(1\),)g(it)c(is)f(useless,)k(since)c(the)h(denominator)f (equals)h(one)g(and)g(the)50 1732 y(n)m(umerator)29 b(equals)f(the)i (left-hand)e(side.)39 b(A)29 b(natural)f(idea)h(in)f(this)g(situation)f (w)m(ould)h(b)s(e)h(to)h(decomp)s(ose)50 1878 y(not)40 b(according)f(to)i(whether)e(the)h(starting)f(p)s(oin)m(t)g Fq(x)g Ft(is)g(revisited,)i(but)d(whether)h(a)h(suitably)e(c)m(hosen)50 2023 y(small)26 b(neigh)m(b)s(orho)s(o)s(d)f(of)i Fq(x)g Ft(is)g(revisited)f Fn(after)58 b Ft(a)27 b(suitably)f(c)m(hosen)i (short)f(time,)h(or)f(not)h(\(in)e(analogy)i(to)50 2169 y(the)i(probabilistic)c(represen)m(tation)k(of)g(capacit)m(y\).)42 b(Ho)m(w)m(ev)m(er,)33 b(an)m(y)d(suc)m(h)f(pro)s(cedure)g(runs)f(quic) m(kly)h(in)m(to)50 2314 y(problems,)g(as)h(it)g(is)g(imp)s(ossible)d (to)k(obtain)f(an)g(exact)i(renew)m(al)e(argumen)m(t.)159 2532 y(F)-8 b(ortunately)g(,)34 b(it)f(is)e(rather)i(easy)g(to)g (obtain)f(a)h(useful)e(analogue)i(of)g(\(4.4\))i(b)m(y)d(purely)f (analytic)h(con-)50 2678 y(siderations.)39 b(In)30 b(fact)h(w)m(e)g (will)c(pro)m(v)m(e)32 b(the)e(follo)m(wing)f(prop)s(osition:)50 2896 y Fe(Prop)s(osition)47 b(4.3:)55 b Fn(L)-5 b(et)40 b Fq(A;)15 b(D)43 b Fn(b)-5 b(e)39 b(disjoint)h(close)-5 b(d)41 b(sets)f(whose)g(c)-5 b(omplement)41 b(is)f(r)-5 b(e)g(gular,)42 b(and)f(let)50 3041 y Fq(x)28 b Fr(2)g Ft(\()p Fq(A)22 b Fr([)f Fq(D)s Ft(\))539 3008 y Fo(c)576 3041 y Fn(,)34 b(such)g(that)51 b Ft(dist)o(\()p Fq(x;)15 b(A)22 b Fr([)f Fq(D)s Ft(\))29 b Fq(>)e(c\017)p Fn(.)47 b(L)-5 b(et)35 b Fq(B)2114 3055 y Fo(\032)2156 3041 y Ft(\()p Fq(x)p Ft(\))g Fn(denote)g(the)f(b)-5 b(al)5 b(l)35 b(of)g(r)-5 b(adius)35 b Fq(\032)f Fn(c)-5 b(enter)g(e)g(d)50 3187 y(at)34 b Fq(x)p Fn(.)44 b(Then)34 b(for)g(any)g Fq(\032)27 b Fr(\024)g Fq(c\017)p Fn(,)33 b Fq(c)28 b(<)e Fr(1)p Fn(,)34 b(ther)-5 b(e)34 b(exists)g(a)g(\014nite)f(p)-5 b(ositive)35 b(c)-5 b(onstant)35 b(\(dep)-5 b(ending)35 b(only)f(on)50 3332 y Fq(c)f Fn(and)g(on)g(the)g(value)g(of)g Fr(kr)p Fq(F)13 b Ft(\()p Fq(x)p Ft(\))p Fr(k)1280 3346 y Fh(1)1358 3332 y Fn(\),)33 b(such)g(that)1366 3589 y Fq(h)1418 3603 y Fo(A;D)1561 3589 y Ft(\()p Fq(x)p Ft(\))26 b Fr(\024)f Fq(C)1892 3519 y Ft(cap)2044 3536 y Fo(B)2099 3545 y Fj(\032)2136 3536 y Fu(\()p Fo(x)p Fu(\))2239 3519 y Ft(\()p Fq(A)p Ft(\))p 1887 3568 496 4 v 1887 3651 a(cap)2039 3668 y Fo(B)2094 3677 y Fj(\032)2131 3668 y Fu(\()p Fo(x)p Fu(\))2234 3651 y Ft(\()p Fq(D)s Ft(\))3523 3589 y(\(4)p Fq(:)p Ft(5\))50 4039 y Fc(Pro)s(of:)47 b Ft(W)-8 b(e)32 b(b)s(egin)d(b)m(y)h(pro)m(ving)g(the)g(follo)m(wing)f (lemma.)50 4258 y Fe(Lemma)38 b(4.4:)91 b Fn(With)33 b(the)g(notation)h(of)f(the)g(pr)-5 b(op)g(osition,)469 4495 y Fq(h)521 4509 y Fo(A;D)664 4495 y Ft(\()p Fq(x)p Ft(\))25 b Fr(\024)116 b Ft(sup)907 4575 y Fo(z)s Fh(2)p Fo(@)t(B)1092 4584 y Fj(r)1128 4575 y Fu(\()p Fo(x)p Fu(\))1241 4495 y Fq(G)1313 4511 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))1534 4493 y Fj(c)1573 4495 y Ft(\()p Fq(z)t(;)15 b(x)p Ft(\))p Fq(e)1823 4457 y Fo(F)10 b Fu(\()p Fo(x)p Fu(\))p Fo(=")2070 4371 y Fi(Z)2121 4578 y Fo(@)t(B)2219 4587 y Fj(\032)2256 4578 y Fu(\()p Fo(x)p Fu(\))2374 4495 y Fq(e)2416 4457 y Fh(\000)p Fo(F)g Fu(\()p Fo(y)r Fu(\))p Fo(=")2701 4495 y Fq(e)2743 4511 y Fo(D)r Fh([)p Fo(B)2908 4520 y Fj(\032)2946 4511 y Fu(\()p Fo(x)p Fu(\))p Fo(;A)3125 4495 y Ft(\()p Fq(dy)s Ft(\))469 4761 y Fq(h)521 4775 y Fo(A;D)664 4761 y Ft(\()p Fq(x)p Ft(\))25 b Fr(\025)129 b Ft(inf)907 4823 y Fo(z)s Fh(2)p Fo(@)t(B)1092 4832 y Fj(r)1128 4823 y Fu(\()p Fo(x)p Fu(\))1241 4761 y Fq(G)1313 4777 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))1534 4759 y Fj(c)1573 4761 y Ft(\()p Fq(z)t(;)15 b(x)p Ft(\))p Fq(e)1823 4723 y Fo(F)10 b Fu(\()p Fo(x)p Fu(\))p Fo(=")2070 4637 y Fi(Z)2121 4843 y Fo(@)t(B)2219 4852 y Fj(\032)2256 4843 y Fu(\()p Fo(x)p Fu(\))2374 4761 y Fq(e)2416 4723 y Fh(\000)p Fo(F)g Fu(\()p Fo(y)r Fu(\))p Fo(=")2701 4761 y Fq(e)2743 4777 y Fo(D)r Fh([)p Fo(B)2908 4786 y Fj(\032)2946 4777 y Fu(\()p Fo(x)p Fu(\))p Fo(;A)3125 4761 y Ft(\()p Fq(dy)s Ft(\))3523 4639 y(\(4)p Fq(:)p Ft(6\))159 5098 y Fn(wher)-5 b(e)34 b Fq(e)458 5115 y Fo(A)p Fh([)p Fo(D)r(;B)700 5124 y Fj(r)736 5115 y Fu(\()p Fo(x)p Fu(\))871 5098 y Fn(is)f(the)g(e)-5 b(quilibrium)32 b(me)-5 b(asur)g(e)34 b(de\014ne)-5 b(d)34 b(in)e(\(2.10\).)50 5316 y Fc(Pro)s(of:)52 b Ft(Let)33 b(\012)f(b)s(e)g(a)h(regular)f (domain,)g(and)g(let)g Fq(f)42 b Ft(b)s(e)32 b(a)h(function)e (de\014ned)g(on)h Fq(@)5 b Ft(\012.)47 b(Recall)32 b(that)i(the)50 5462 y(op)s(erator)i Fq(H)498 5476 y Fu(\012)589 5462 y Fr(\021)e Fq(H)777 5429 y Fo(\025)p Fu(=0)770 5487 y(\012)953 5462 y Ft(de\014ned)g(in)h(\(2.6\))j(can)e(b)s(e)f(seen)h (as)h(mapping)d(a)i(function)f Fq(f)45 b Ft(de\014ned)35 b(on)h Fq(@)5 b Ft(\012)p eop %%Page: 13 13 13 12 bop 3633 295 a Fl(13)50 513 y Ft(to)33 b(a)g(harmonic)f(function) f(\(with)h(resp)s(ect)g(to)i(the)e(op)s(erator)h Fq(L)2262 527 y Fo(\017)2297 513 y Ft(\))g(on)f(\012.)48 b(W)-8 b(e)33 b(call)f Fq(H)3035 527 y Fu(\012)3091 513 y Fq(f)42 b Ft(the)33 b Fn(harmonic)50 658 y(extension)e Ft(of)f Fq(f)10 b Ft(.)159 876 y(Cho)s(osing)22 b(\012)j Fr(\021)f Ft(\()p Fq(A)6 b Fr([)g Fq(D)s Ft(\))1024 843 y Fo(c)1061 876 y Ft(,)25 b(w)m(e)f(see)f(that)h(the)f(equilibrium)c(p)s(oten)m (tial)j Fq(h)2620 890 y Fo(A;D)2786 876 y Ft(satis\014es)h(the)g (mean-v)-5 b(alue)50 1022 y(prop)s(ert)m(y)1321 1167 y Fq(h)1373 1181 y Fo(A;D)1516 1167 y Ft(\()p Fq(x)p Ft(\))26 b(=)f Fq(H)1836 1184 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))2057 1165 y Fj(c)2096 1167 y Fq(h)2148 1181 y Fo(A;D)2290 1167 y Ft(\()p Fq(x)p Ft(\))p Fq(;)1086 b Ft(\(4)p Fq(:)p Ft(7\))50 1347 y(No)m(w)32 b(let)f Fq(C)7 b Fr(\032)o Ft(\()p Fq(A)21 b Fr([)g Fq(D)s Ft(\))851 1314 y Fo(c)919 1347 y Ft(b)s(e)30 b(a)i(regular)e(neigh)m(b)s(orho)s (o)s(d)f(of)j Fq(x)p Ft(.)43 b(No)m(w)32 b(since)e Fq(h)2709 1361 y Fo(A;D)r Fh([)p Fo(C)2988 1347 y Ft(and)h Fq(h)3218 1361 y Fo(A;D)3392 1347 y Ft(coincide)50 1493 y(on)f Fq(@)5 b Ft(\()p Fq(A)21 b Fr([)f Fq(D)s Ft(\),)31 b(it)f(is)f(ob)m (vious)h(that)1404 1682 y Fq(h)1456 1696 y Fo(A;D)1624 1682 y Ft(=)25 b Fq(H)1796 1698 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))2017 1680 y Fj(c)2056 1682 y Fq(h)2108 1696 y Fo(A;D)r Fh([)p Fo(C)3523 1682 y Ft(\(4)p Fq(:)p Ft(8\))50 1871 y(holds)36 b(on)h(\()p Fq(A)25 b Fr([)f Fq(D)k Fr([)c Fq(C)7 b Ft(\))936 1838 y Fo(c)973 1871 y Ft(.)61 b(Using)37 b(the)g(\014rst)f(Green's)i(iden)m(tit)m(y)f(\(2.7\))i(for)e(\012)f(=)g (\000)h Fr(\021)f Ft(\()p Fq(A)25 b Fr([)g Fq(D)i Fr([)e Fq(C)7 b Ft(\))3648 1838 y Fo(c)3685 1871 y Ft(,)50 2016 y Fq(\036)25 b Fr(\021)g Fq(G)297 2033 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))518 2015 y Fj(c)557 2016 y Ft(\()p Fq(x;)15 b Fr(\001)p Ft(\))32 b(and)e Fq( )e Fr(\021)d Fq(h)1188 2030 y Fo(A;D)r Fh([)p Fo(C)1436 2016 y Ft(,)31 b(w)m(e)g(get)163 2225 y Fq(H)239 2242 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))460 2224 y Fj(c)499 2225 y Fq(h)551 2239 y Fo(A;D)r Fh([)p Fo(C)799 2225 y Ft(\()p Fq(x)p Ft(\))26 b(=)20 b Fr(\000)g Fq(\017)1181 2102 y Fi(Z)1232 2308 y Fo(@)t Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))1516 2225 y Fq(e)1558 2188 y Fu(\()p Fo(F)10 b Fu(\()p Fo(x)p Fu(\))p Fh(\000)p Fo(F)g Fu(\()p Fo(y)r Fu(\)\))p Fo(=")2054 2225 y Fq(h)2106 2239 y Fo(A;D)r Fh([)p Fo(C)2354 2225 y Ft(\()p Fq(y)s Ft(\))p Fq(@)2520 2242 y Fo(n)p Fu(\()p Fo(y)r Fu(\))2666 2225 y Fq(G)2738 2242 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))2959 2224 y Fj(c)2998 2225 y Ft(\()p Fq(y)s(;)15 b(x)p Ft(\))p Fq(d\033)3307 2239 y Fo(A)p Fh([)p Fo(D)3478 2225 y Ft(\()p Fq(y)s Ft(\))947 2484 y(=)20 b Fr(\000)g Fq(\017)1181 2360 y Fi(Z)1232 2567 y Fo(@)t(C)1351 2484 y Fq(e)1393 2447 y Fu(\()p Fo(F)10 b Fu(\()p Fo(x)p Fu(\))p Fh(\000)p Fo(F)g Fu(\()p Fo(y)r Fu(\)\))p Fo(=")1889 2484 y Fq(G)1961 2501 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))2182 2482 y Fj(c)2221 2484 y Ft(\()p Fq(y)s(;)15 b(x)p Ft(\))p Fq(@)2479 2501 y Fo(n)p Fu(\()p Fo(y)r Fu(\))2625 2484 y Fq(h)2677 2498 y Fo(A;D)r Fh([)p Fo(C)2925 2484 y Ft(\()p Fq(y)s Ft(\))p Fq(d\033)3142 2498 y Fo(C)3204 2484 y Ft(\()p Fq(y)s Ft(\))1038 2727 y Fr(\000)1129 2603 y Fi(Z)1180 2810 y Fo(@)t(C)1299 2727 y Fq(e)1341 2690 y Fu(\()p Fo(F)10 b Fu(\()p Fo(x)p Fu(\))p Fh(\000)p Fo(F)g Fu(\()p Fo(y)r Fu(\)\))p Fo(=")1836 2727 y Fq(G)1908 2744 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))2129 2725 y Fj(c)2168 2727 y Ft(\()p Fq(y)s(;)15 b(x)p Ft(\))p Fq(e)2420 2741 y Fo(A;D)r Fh([)p Fo(C)2669 2727 y Ft(\()p Fq(dy)s Ft(\))3523 2887 y(\(4)p Fq(:)p Ft(9\))50 3033 y(where)33 b Fq(n)p Ft(\()p Fq(y)s Ft(\))h(is)e(the)i(inner)e(unit)g(normal)g(at)j Fq(y)e Fr(2)d Fq(@)5 b Ft(\()p Fq(A)24 b Fr([)d Fq(D)26 b Fr([)c Fq(C)7 b Ft(\).)50 b(Here)34 b(w)m(e)g(ha)m(v)m(e)h(used)d (that)j Fq(h)3462 3047 y Fo(A;D)r Fh([)p Fo(C)50 3178 y Ft(v)-5 b(anishes)25 b(on)h Fq(@)5 b(C)34 b Ft(and)25 b(that)j(the)e(Green's)h(function)e(v)-5 b(anishes)25 b(when)h Fq(x)f Fr(2)g Fq(@)5 b Ft(\()p Fq(A)12 b Fr([)g Fq(D)s Ft(\).)41 b(The)26 b(last)g(equalit)m(y)50 3324 y(follo)m(ws)j(from)h(\(2.12\))j(together)f(with)d(\(2.11\).)159 3542 y(W)-8 b(e)41 b(no)m(w)f(c)m(ho)s(ose)h Fq(C)48 b Fr(\021)41 b Fq(B)1116 3556 y Fo(\032)1158 3542 y Ft(\()p Fq(x)p Ft(\).)70 b(If)40 b(w)m(e)g(could)f(replace)h Fq(G)2261 3558 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))2482 3540 y Fj(c)2521 3542 y Ft(\()p Fq(y)s(;)15 b(x)p Ft(\))41 b(b)m(y)f(a)g(constan)m(t)h(v)-5 b(alue)40 b(on)50 3687 y Fq(@)5 b(B)172 3701 y Fo(\032)214 3687 y Ft(\()p Fq(x)p Ft(\),)33 b(w)m(e)e(could)g(extract)h(this)e(v)-5 b(alue)31 b(from)g(the)g(in)m(tegral;)h(the)f(remaining)f(in)m(tegral)g(then)h(w) m(ould)f(b)s(e)50 3833 y(some)g(partial)f(capacit)m(y)-8 b(.)43 b(In)29 b(fact,)i(in)e(the)h(discrete)g(case)h(w)m(e)g(could)e (c)m(ho)s(ose)i(instead)f(of)g(the)g(ball)f Fq(B)3545 3847 y Fo(\032)3587 3833 y Ft(\()p Fq(x)p Ft(\))50 3978 y(just)k(the)g(p)s(oin)m(t)f Fq(x)p Ft(,)j(and)d(then)h(this)g(problem) e(w)m(as)j(absen)m(t,)h(and)e(w)m(e)g(w)m(ould)f(readily)g(get)j (\(4.4\).)51 b(In)33 b(the)50 4124 y(presen)m(t)d(situation)g(w)m(e)g (still)f(get)i(t)m(w)m(o)h(b)s(ounds,)c(namely)309 4318 y Fq(h)361 4332 y Fo(A;D)504 4318 y Ft(\()p Fq(x)p Ft(\))e Fr(\025)f(\000)106 b Ft(sup)834 4398 y Fo(z)s Fh(2)p Fo(@)t(B)1019 4407 y Fj(\032)1055 4398 y Fu(\()p Fo(x)p Fu(\))1168 4318 y Fq(G)1240 4334 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))1461 4316 y Fj(c)1500 4318 y Ft(\()p Fq(z)t(;)15 b(x)p Ft(\))p Fq(e)1750 4280 y Fo(F)10 b Fu(\()p Fo(x)p Fu(\))p Fo(=")1997 4194 y Fi(Z)2048 4401 y Fo(@)t(B)2146 4410 y Fj(\032)2183 4401 y Fu(\()p Fo(x)p Fu(\))2301 4318 y Fq(e)2343 4280 y Fh(\000)p Fo(F)g Fu(\()p Fo(y)r Fu(\))p Fo(=")2628 4318 y Fq(e)2670 4334 y Fo(A;D)r Fh([)p Fo(B)2912 4343 y Fj(\032)2949 4334 y Fu(\()p Fo(x)p Fu(\))3052 4318 y Ft(\()p Fq(dy)s Ft(\))309 4586 y Fq(h)361 4600 y Fo(A;D)504 4586 y Ft(\()p Fq(x)p Ft(\))26 b Fr(\024)f(\000)119 b Ft(inf)834 4648 y Fo(z)s Fh(2)p Fo(@)t(B)1019 4657 y Fj(\032)1055 4648 y Fu(\()p Fo(x)p Fu(\))1168 4586 y Fq(G)1240 4602 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))1461 4584 y Fj(c)1500 4586 y Ft(\()p Fq(z)t(;)15 b(x)p Ft(\))p Fq(e)1750 4548 y Fo(F)10 b Fu(\()p Fo(x)p Fu(\))p Fo(=")1997 4462 y Fi(Z)2048 4668 y Fo(@)t(B)2146 4677 y Fj(\032)2183 4668 y Fu(\()p Fo(x)p Fu(\))2301 4586 y Fq(e)2343 4548 y Fh(\000)p Fo(F)g Fu(\()p Fo(y)r Fu(\))p Fo(=")2628 4586 y Fq(e)2670 4602 y Fo(A;D)r Fh([)p Fo(B)2912 4611 y Fj(\032)2949 4602 y Fu(\()p Fo(x)p Fu(\))3052 4586 y Ft(\()p Fq(dy)s Ft(\))3477 4463 y(\(4)p Fq(:)p Ft(10\))50 4807 y(But,trivially)-8 b(,)34 b Fq(h)644 4821 y Fo(A)p Fh([)p Fo(B)s(;C)921 4807 y Ft(=)e(1)24 b Fr(\000)f Fq(h)1239 4821 y Fo(C)q(;A)p Fh([)p Fo(B)1480 4807 y Ft(,)36 b(and)e(hence,)i(b)m (y)e(\(2.11\))j(with)d Fq(\025)e Ft(=)g(0,)37 b Fr(\000)p Fq(e)3022 4821 y Fo(A)p Fh([)p Fo(B)s(;C)3299 4807 y Ft(=)32 b Fq(e)3444 4821 y Fo(C)q(;A)p Fh([)p Fo(B)3685 4807 y Ft(,)50 4953 y(whic)m(h)d(implies)f(\(4.6\).)42 b Fr(})159 5171 y Ft(A)m(t)33 b(this)e(p)s(oin)m(t)g(it)h(is)f(clear)h (that)h(w)m(e)g(will)c(need)j(to)h(b)s(e)f(able)g(to)g(con)m(trol)h (the)f(Green's)h(function)e(near)50 5316 y(the)j(diagonal.)50 b(Before)35 b(turning)d(to)j(these)f(estimates,)i(w)m(e)e(bring)e (\(4.10\))k(in)d(a)h(sligh)m(tly)e(more)i(suitable)50 5462 y(form.)40 b(Namely)30 b(w)m(e)h(will)d(sho)m(w)i(that)p eop %%Page: 14 14 14 13 bop 50 295 a Fl(14)50 513 y Fe(Lemma)38 b(4.5:)91 b Fn(Within)33 b(the)g(situation)h(of)f(the)g(pr)-5 b(evious)33 b(lemma)846 771 y Fq(h)898 785 y Fo(A;D)1040 771 y Ft(\()p Fq(x)p Ft(\))26 b Fr(\024)116 b Ft(sup)1284 851 y Fo(z)s Fh(2)p Fo(@)t(B)1469 860 y Fj(\032)1506 851 y Fu(\()p Fo(x)p Fu(\))1619 771 y Fq(G)1691 787 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))1912 769 y Fj(c)1951 771 y Ft(\()p Fq(z)t(;)15 b(x)p Ft(\))p Fq(e)2201 733 y Fo(F)10 b Fu(\()p Fo(x)p Fu(\))p Fo(=\017)2429 771 y Ft(cap)2580 787 y Fo(B)2635 796 y Fj(\032)2673 787 y Fu(\()p Fo(x)p Fu(\))2775 771 y Ft(\()p Fq(A)p Ft(\))564 b(\(4)p Fq(:)p Ft(11\))80 1068 y Fc(Pro)s(of:)48 b Ft(By)30 b(\(2.18\),)j(it)d(is)f(ob)m(vious)h (that)h Fq(e)1590 1084 y Fo(D)r Fh([)p Fo(B)1755 1093 y Fj(\032)1793 1084 y Fu(\()p Fo(x)p Fu(\))p Fo(;A)1972 1068 y Ft(\()p Fq(dy)s Ft(\))26 b Fr(\024)f Fq(e)2301 1084 y Fo(B)2356 1093 y Fj(\032)2394 1084 y Fu(\()p Fo(x)p Fu(\))p Fo(;A)2573 1068 y Ft(\()p Fq(dy)s Ft(\).)41 b(But)31 b(then)284 1224 y Fi(Z)335 1431 y Fo(@)t(B)433 1440 y Fj(\032)470 1431 y Fu(\()p Fo(x)p Fu(\))588 1348 y Fq(e)630 1310 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(y)r Fu(\))p Fo(=")915 1348 y Fq(e)957 1364 y Fo(D)r Fh([)p Fo(B)1122 1373 y Fj(\032)1160 1364 y Fu(\()p Fo(x)p Fu(\))p Fo(;A)1339 1348 y Ft(\()p Fq(dy)s Ft(\))26 b Fr(\024)1626 1224 y Fi(Z)1677 1431 y Fo(@)t(B)1775 1440 y Fj(\032)1812 1431 y Fu(\()p Fo(x)p Fu(\))1930 1348 y Fq(e)1972 1310 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(x)p Fu(\))p Fo(=\017)2255 1348 y Fq(e)2297 1362 y Fo(B)2352 1371 y Fj(\032)2390 1362 y Fo(;A)2470 1348 y Ft(\()p Fq(dy)s Ft(\))26 b(=)f(cap)2909 1364 y Fo(B)2964 1373 y Fj(\032)3001 1364 y Fu(\()p Fo(x)p Fu(\))3104 1348 y Ft(\()p Fq(A)p Ft(\))3477 1359 y(\(4)p Fq(:)p Ft(12\))50 1638 y(Th)m(us)k(the)i(upp)s(er)d(b)s(ound)g(in)h (\(4.6\))k(implies)27 b(\(4.11\).)p Fr(})159 1857 y Ft(A)m(t)e(this)e (p)s(oin)m(t)h(w)m(e)g(w)m(an)m(t)h(to)g(express)f(the)h(Green's)f (function)f(in)g(the)i(b)s(ounds)d(of)i(Lemma)h(4.4)g(in)e(terms)50 2002 y(of)30 b(capacities,)h(to)s(o.)42 b(W)-8 b(e)31 b(pro)s(ceed)f(as)h(in)e(\(2.27\),)k(to)e(get)h(this)d(time)84 2270 y Fq(e)126 2233 y Fo(F)10 b Fu(\()p Fo(x)p Fu(\))p Fo(=\017)367 2146 y Fi(Z)418 2353 y Fo(@)t(B)516 2362 y Fj(\032)553 2353 y Fu(\()p Fo(x)p Fu(\))671 2270 y Fq(e)713 2233 y Fh(\000)p Fo(F)g Fu(\()p Fo(z)s Fu(\))p Fo(=\017)992 2270 y Fq(G)1064 2287 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))1285 2268 y Fj(c)1324 2270 y Ft(\()p Fq(x;)15 b(z)t Ft(\))p Fq(e)1574 2287 y Fo(B)1629 2296 y Fj(\032)1668 2287 y Fu(\()p Fo(x)p Fu(\))p Fo(;A)p Fh([)p Fo(D)1957 2270 y Ft(\()p Fq(dz)t Ft(\))27 b(=)2243 2146 y Fi(Z)2293 2353 y Fo(@)t(B)2391 2362 y Fj(\032)2429 2353 y Fu(\()p Fo(x)p Fu(\))2546 2270 y Fq(G)2618 2287 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))2839 2268 y Fj(c)2878 2270 y Ft(\()p Fq(z)t(;)15 b(x)p Ft(\))p Fq(e)3128 2287 y Fo(B)3183 2296 y Fj(\032)3222 2287 y Fu(\()p Fo(x)p Fu(\))p Fo(;A)p Fh([)p Fo(D)3512 2270 y Ft(\()p Fq(dz)t Ft(\))1302 2485 y(=)25 b Fq(h)1450 2501 y Fo(B)1505 2510 y Fj(\032)1543 2501 y Fu(\()p Fo(x)p Fu(\))p Fo(;A)p Fh([)p Fo(D)1832 2485 y Ft(\()p Fq(x)p Ft(\))h(=)f(1)3477 2596 y(\(4)p Fq(:)p Ft(13\))50 2742 y(This)j(implies)g(that)599 3005 y(1)d Fr(\025)g Fq(e)807 2967 y Fo(F)10 b Fu(\()p Fo(x)p Fu(\))p Fo(=\017)1132 3005 y Ft(inf)1049 3068 y Fo(z)s Fh(2)p Fo(B)1191 3077 y Fj(\032)1227 3068 y Fu(\()p Fo(x)p Fu(\))1341 3005 y Fq(G)1413 3021 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))1634 3003 y Fj(c)1673 3005 y Ft(\()p Fq(x;)15 b(z)t Ft(\))1896 2881 y Fi(Z)1948 3087 y Fo(B)2003 3096 y Fj(\032)2040 3087 y Fu(\()p Fo(x)p Fu(\))2158 3005 y Fq(dz)t(e)2293 2967 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(z)s Fu(\))p Fo(=\017)2573 3005 y Fq(e)2615 3021 y Fo(B)2670 3030 y Fj(\032)2708 3021 y Fu(\()p Fo(x)p Fu(\))p Fo(;A)p Fh([)p Fo(D)2997 3005 y Ft(\()p Fq(dz)t Ft(\))669 3227 y(=)25 b Fq(e)807 3190 y Fo(F)10 b Fu(\()p Fo(x)p Fu(\))p Fo(=\017)1132 3227 y Ft(inf)1049 3290 y Fo(z)s Fh(2)p Fo(B)1191 3299 y Fj(\032)1227 3290 y Fu(\()p Fo(x)p Fu(\))1341 3227 y Fq(G)1413 3243 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))1634 3225 y Fj(c)1673 3227 y Ft(\()p Fq(x;)15 b(z)t Ft(\)cap)2034 3243 y Fo(B)2089 3252 y Fj(\032)2126 3243 y Fu(\()p Fo(x)p Fu(\))2229 3227 y Ft(\()p Fq(A)20 b Fr([)g Fq(D)s Ft(\))3477 3122 y(\(4)p Fq(:)p Ft(14\))50 3499 y(i.e.)918 3648 y Fq(e)960 3611 y Fo(F)10 b Fu(\()p Fo(x)p Fu(\))p Fo(=\017)1284 3648 y Ft(inf)1201 3711 y Fo(z)s Fh(2)p Fo(B)1343 3720 y Fj(\032)1380 3711 y Fu(\()p Fo(x)p Fu(\))1493 3648 y Fq(G)1565 3665 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))1786 3646 y Fj(c)1825 3648 y Ft(\()p Fq(x;)15 b(z)t Ft(\))27 b Fr(\024)2476 3587 y Ft(1)p 2167 3627 665 4 v 2167 3711 a(cap)2318 3727 y Fo(B)2373 3736 y Fj(\032)2411 3727 y Fu(\()p Fo(x)p Fu(\))2513 3711 y Ft(\()p Fq(A)21 b Fr([)f Fq(D)s Ft(\))3477 3648 y(\(4)p Fq(:)p Ft(15\))50 3884 y(It)29 b(is)g(clear)h(at)g(this)e (p)s(oin)m(t)h(that)h(w)m(e)g(cannot)g(con)m(tin)m(ue)g(unless)e(w)m(e) i(can)g(compare)g(the)f(in\014m)m(um)f(and)h(the)50 4030 y(suprem)m(um)k(of)i Fq(G)677 4046 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))898 4028 y Fj(c)937 4030 y Ft(\()p Fq(z)t(;)15 b(x)p Ft(\))36 b(with)e Fq(z)j Fr(2)c Fq(B)1635 4044 y Fo(\032)1677 4030 y Ft(\()p Fq(x)p Ft(\).)55 b(But)35 b(suc)m(h)g(a)g(result)f(is)g(pro)m(vided)g(b)m(y)h(the)g(Harnac)m(k)50 4175 y(inequalities.)50 4393 y Fe(Lemma)42 b(4.6:)107 b Fn(If)36 b Fq(\032)31 b Ft(=)g Fq(c\017)p Fn(,)37 b(for)f(some)h Fq(c)31 b(<)g Fr(1)p Fn(,)36 b(then)h(ther)-5 b(e)36 b(exists)h(a)f(c)-5 b(onstant)38 b Fq(C)k Fn(dep)-5 b(ending)37 b(only)50 4539 y(on)c Fq(c)p Fn(,)g(such)f(that)1013 4688 y Ft(sup)943 4768 y Fo(z)s Fh(2)p Fo(B)1085 4777 y Fj(\032)1121 4768 y Fu(\()p Fo(x)p Fu(\))1235 4688 y Fq(G)1307 4704 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))1528 4686 y Fj(c)1567 4688 y Ft(\()p Fq(z)t(;)15 b(x)p Ft(\))26 b Fr(\024)f Fq(C)105 b Ft(inf)1984 4751 y Fo(z)s Fh(2)p Fo(B)2126 4760 y Fj(\032)2163 4751 y Fu(\()p Fo(x)p Fu(\))2276 4688 y Fq(G)2348 4704 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))2569 4686 y Fj(c)2608 4688 y Ft(\()p Fq(z)t(;)15 b(x)p Ft(\))661 b(\(4)p Fq(:)p Ft(16\))50 5118 y Fc(Pro)s(of:)46 b Ft(W)-8 b(e)29 b(will)c(apply)h(Lemma)i(4.1.)41 b(If)27 b(w)m(e)h(c)m(ho)s(ose)h Fq(R)d Fr(\024)f Fq(\017)p Ft(,)j(w)m(e)h(can) f(use)f(\(4.2\))j(with)c(a)i(constan)m(t)h(that)50 5264 y(do)s(es)h(not)g(dep)s(end)f(on)h Fq(\017)895 5231 y Fu(6)936 5264 y Ft(.)p 50 5372 546 4 v 159 5436 a Fm(6)197 5462 y Fl(If)25 b Fg(x)g Fl(is)i(a)e(\(quadratic\))h(critical)i(p)r (oin)n(t)d(of)h Fg(F)11 b Fl(,)25 b(then)h(w)n(e)g(can)g(ev)n(en)g(c)n (ho)r(ose)g Fg(R)c Fl(=)g Fg(\017)2591 5436 y Fm(1)p Ff(=)p Fm(2)2697 5462 y Fl(.)p eop %%Page: 15 15 15 14 bop 3633 295 a Fl(15)159 513 y Ft(Note)39 b(that)g Fq(u)p Ft(\()p Fq(z)t Ft(\))g Fr(\021)e Fq(G)978 529 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))1199 511 y Fj(c)1238 513 y Ft(\()p Fq(z)t(;)15 b(x)p Ft(\))40 b(is)d(harmonic)g(in)f(\()p Fq(A)26 b Fr([)f Fq(D)s Ft(\))2433 480 y Fo(c)2470 513 y Fr(n)p Fq(x)p Ft(.)64 b(Th)m(us)37 b(if)g Fq(\032)h(>)g Ft(2)p Fq(R)q Ft(,)i Fq(u)e Ft(is)f(har-)50 658 y(monic)d(in)g Fq(B)502 672 y Fu(2)p Fo(R)598 658 y Ft(\()p Fq(z)t Ft(\),)k(for)d(an)m (y)h Fq(z)h Fr(2)c Fq(@)5 b(B)1393 672 y Fo(\032)1436 658 y Ft(\()p Fq(x)p Ft(\).)56 b(No)m(w)36 b(let)f Fq(a;)15 b(b)34 b Fr(2)f Fq(@)5 b(B)2364 672 y Fo(\032)2407 658 y Ft(\()p Fq(x)p Ft(\).)56 b(Assume)35 b(that)h Fq(a)f Ft(is)f(suc)m(h)h(that)50 804 y(sup)187 826 y Fo(z)s Fh(2)p Fo(@)t(B)372 835 y Fj(\032)408 826 y Fu(\()p Fo(x)p Fu(\))526 804 y Fq(u)p Ft(\()p Fq(z)t Ft(\))j(=)e Fq(u)p Ft(\()p Fq(a)p Ft(\),)41 b(and)36 b(inf)1369 820 y Fo(z)s Fh(2)p Fo(@)t(B)1554 829 y Fj(\032)1590 820 y Fu(\()p Fo(x)p Fu(\))1708 804 y Fq(u)p Ft(\()p Fq(z)t Ft(\))i(=)f Fq(u)p Ft(\()p Fq(b)p Ft(\).)62 b(Then)36 b(w)m(e)i(can)g(\014nd)e Fq(k)k Fr(\024)c Fq(\031)s(\032=R)j Ft(p)s(oin)m(ts)50 949 y Fq(x)102 963 y Fu(1)143 949 y Fq(;)15 b(:)g(:)g(:)h(;)f(x)396 963 y Fo(k)467 949 y Fr(2)24 b Fq(@)5 b(B)674 963 y Fo(\032)717 949 y Ft(\()p Fq(x)p Ft(\))31 b(suc)m(h)f(that)h Fq(x)1324 963 y Fu(1)1390 949 y Ft(=)25 b Fq(a)p Ft(,)30 b Fq(b)c Fr(2)f Fq(B)1809 963 y Fo(R)1868 949 y Ft(\()p Fq(x)1955 963 y Fo(k)2000 949 y Ft(\),)31 b(and)f Fq(B)2337 963 y Fo(R)2396 949 y Ft(\()p Fq(x)2483 963 y Fo(i)2514 949 y Ft(\))21 b Fr(\\)e Fq(B)2719 963 y Fo(R)2779 949 y Ft(\()p Fq(x)2866 963 y Fo(i)p Fu(+1)2989 949 y Ft(\))25 b Fr(6)p Ft(=)g Fr(;)p Ft(.)1205 2170 y @beginspecial 0 @llx 0 @lly 148 @urx 130 @ury 1480 @rwi @setspecial %%BeginDocument: diff1.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: diff1.eps %%Creator: fig2dev Version 3.2 Patchlevel 0-beta2 %%CreationDate: Mon Feb 11 14:42:24 2002 %%For: bovier@hilbert.wias-berlin.de (Anton Bovier,403,547,) %%Orientation: Portrait %%BoundingBox: 0 0 148 130 %%Pages: 0 %%BeginSetup %%IncludeFeature: *PageSize Letter %%EndSetup %%Magnification: 0.60 %%EndComments /MyAppDict 100 dict dup begin def /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save -37.0 174.0 translate 1 -1 scale .9 .9 scale % to make patterns same scale as in xfig % This junk string is used by the show operators /PATsstr 1 string def /PATawidthshow { % cx cy cchar rx ry string % Loop over each character in the string { % cx cy cchar rx ry char % Show the character dup % cx cy cchar rx ry char char PATsstr dup 0 4 -1 roll put % cx cy cchar rx ry char (char) false charpath % cx cy cchar rx ry char /clip load PATdraw % Move past the character (charpath modified the % current point) currentpoint % cx cy cchar rx ry char x y newpath moveto % cx cy cchar rx ry char % Reposition by cx,cy if the character in the string is cchar 3 index eq { % cx cy cchar rx ry 4 index 4 index rmoveto } if % Reposition all characters by rx ry 2 copy rmoveto % cx cy cchar rx ry } forall pop pop pop pop pop % - currentpoint newpath moveto } bind def /PATcg { 7 dict dup begin /lw currentlinewidth def /lc currentlinecap def /lj currentlinejoin def /ml currentmiterlimit def /ds [ currentdash ] def /cc [ currentrgbcolor ] def /cm matrix currentmatrix def end } bind def % PATdraw - calculates the boundaries of the object and % fills it with the current pattern /PATdraw { % proc save exch PATpcalc % proc nw nh px py 5 -1 roll exec % nw nh px py newpath PATfill % - restore } bind def % PATfill - performs the tiling for the shape /PATfill { % nw nh px py PATfill - PATDict /CurrentPattern get dup begin setfont % Set the coordinate system to Pattern Space PatternGState PATsg % Set the color for uncolored pattezns PaintType 2 eq { PATDict /PColor get PATsc } if % Create the string for showing 3 index string % nw nh px py str % Loop for each of the pattern sources 0 1 Multi 1 sub { % nw nh px py str source % Move to the starting location 3 index 3 index % nw nh px py str source px py moveto % nw nh px py str source % For multiple sources, set the appropriate color Multi 1 ne { dup PC exch get PATsc } if % Set the appropriate string for the source 0 1 7 index 1 sub { 2 index exch 2 index put } for pop % Loop over the number of vertical cells 3 index % nw nh px py str nh { % nw nh px py str currentpoint % nw nh px py str cx cy 2 index show % nw nh px py str cx cy YStep add moveto % nw nh px py str } repeat % nw nh px py str } for 5 { pop } repeat end } bind def % PATkshow - kshow with the current pattezn /PATkshow { % proc string exch bind % string proc 1 index 0 get % string proc char % Loop over all but the last character in the string 0 1 4 index length 2 sub { % string proc char idx % Find the n+1th character in the string 3 index exch 1 add get % string proe char char+1 exch 2 copy % strinq proc char+1 char char+1 char % Now show the nth character PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr) false charpath % string proc char+1 char char+1 /clip load PATdraw % Move past the character (charpath modified the current point) currentpoint newpath moveto % Execute the user proc (should consume char and char+1) mark 3 1 roll % string proc char+1 mark char char+1 4 index exec % string proc char+1 mark... cleartomark % string proc char+1 } for % Now display the last character PATsstr dup 0 4 -1 roll put % string proc (char+1) false charpath % string proc /clip load PATdraw neewath pop pop % - } bind def % PATmp - the makepattern equivalent /PATmp { % patdict patmtx PATmp patinstance exch dup length 7 add % We will add 6 new entries plus 1 FID dict copy % Create a new dictionary begin % Matrix to install when painting the pattern TilingType PATtcalc /PatternGState PATcg def PatternGState /cm 3 -1 roll put % Check for multi pattern sources (Level 1 fast color patterns) currentdict /Multi known not { /Multi 1 def } if % Font dictionary definitions /FontType 3 def % Create a dummy encoding vector /Encoding 256 array def 3 string 0 1 255 { Encoding exch dup 3 index cvs cvn put } for pop /FontMatrix matrix def /FontBBox BBox def /BuildChar { mark 3 1 roll % mark dict char exch begin Multi 1 ne {PaintData exch get}{pop} ifelse % mark [paintdata] PaintType 2 eq Multi 1 ne or { XStep 0 FontBBox aload pop setcachedevice } { XStep 0 setcharwidth } ifelse currentdict % mark [paintdata] dict /PaintProc load % mark [paintdata] dict paintproc end gsave false PATredef exec true PATredef grestore cleartomark % - } bind def currentdict end % newdict /foo exch % /foo newlict definefont % newfont } bind def % PATpcalc - calculates the starting point and width/height % of the tile fill for the shape /PATpcalc { % - PATpcalc nw nh px py PATDict /CurrentPattern get begin gsave % Set up the coordinate system to Pattern Space % and lock down pattern PatternGState /cm get setmatrix BBox aload pop pop pop translate % Determine the bounding box of the shape pathbbox % llx lly urx ury grestore % Determine (nw, nh) the # of cells to paint width and height PatHeight div ceiling % llx lly urx qh 4 1 roll % qh llx lly urx PatWidth div ceiling % qh llx lly qw 4 1 roll % qw qh llx lly PatHeight div floor % qw qh llx ph 4 1 roll % ph qw qh llx PatWidth div floor % ph qw qh pw 4 1 roll % pw ph qw qh 2 index sub cvi abs % pw ph qs qh-ph exch 3 index sub cvi abs exch % pw ph nw=qw-pw nh=qh-ph % Determine the starting point of the pattern fill %(px, py) 4 2 roll % nw nh pw ph PatHeight mul % nw nh pw py exch % nw nh py pw PatWidth mul exch % nw nh px py end } bind def % Save the original routines so that we can use them later on /oldfill /fill load def /oldeofill /eofill load def /oldstroke /stroke load def /oldshow /show load def /oldashow /ashow load def /oldwidthshow /widthshow load def /oldawidthshow /awidthshow load def /oldkshow /kshow load def % These defs are necessary so that subsequent procs don't bind in % the originals /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def /PATredef { MyAppDict begin { /fill { /clip load PATdraw newpath } bind def /eofill { /eoclip load PATdraw newpath } bind def /stroke { PATstroke } bind def /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def /ashow { 0 0 null 6 3 roll PATawidthshow } bind def /widthshow { 0 0 3 -1 roll PATawidthshow } bind def /awidthshow { PATawidthshow } bind def /kshow { PATkshow } bind def } { /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def } ifelse end } bind def false PATredef % Conditionally define setcmykcolor if not available /setcmykcolor where { pop } { /setcmykcolor { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor - pop } bind def } ifelse /PATsc { % colorarray aload length % c1 ... cn length dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /PATsg { % dict begin lw setlinewidth lc setlinecap lj setlinejoin ml setmiterlimit ds aload pop setdash cc aload pop setrgbcolor cm setmatrix end } bind def /PATDict 3 dict def /PATsp { true PATredef PATDict begin /CurrentPattern exch def % If it's an uncolored pattern, save the color CurrentPattern /PaintType get 2 eq { /PColor exch def } if /CColor [ currentrgbcolor ] def end } bind def % PATstroke - stroke with the current pattern /PATstroke { countdictstack save mark { currentpoint strokepath moveto PATpcalc % proc nw nh px py clip newpath PATfill } stopped { (*** PATstroke Warning: Path is too complex, stroking with gray) = cleartomark restore countdictstack exch sub dup 0 gt { { end } repeat } { pop } ifelse gsave 0.5 setgray oldstroke grestore } { pop restore pop } ifelse newpath } bind def /PATtcalc { % modmtx tilingtype PATtcalc tilematrix % Note: tiling types 2 and 3 are not supported gsave exch concat % tilingtype matrix currentmatrix exch % cmtx tilingtype % Tiling type 1 and 3: constant spacing 2 ne { % Distort the pattern so that it occupies % an integral number of device pixels dup 4 get exch dup 5 get exch % tx ty cmtx XStep 0 dtransform round exch round exch % tx ty cmtx dx.x dx.y XStep div exch XStep div exch % tx ty cmtx a b 0 YStep dtransform round exch round exch % tx ty cmtx a b dy.x dy.y YStep div exch YStep div exch % tx ty cmtx a b c d 7 -3 roll astore % { a b c d tx ty } } if grestore } bind def /PATusp { false PATredef PATDict begin CColor PATsc end } bind def % this is the pattern fill program from the Second edition Reference Manual % with changes to call the above pattern fill % left30 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 32 16 true [ 32 0 0 -16 0 16 ] {} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P1 exch def 1.1111 1.1111 scale %restore scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n 0 4847 m 0 0 l 5153 0 l 5153 4847 l cp clip 0.03600 0.03600 sc 7.500 slw % Ellipse n 3450 1800 546 546 0 360 DrawEllipse gs col0 s gr % Ellipse n 1800 2550 525 525 0 360 DrawEllipse gs col0 s gr % Ellipse n 1575 3375 525 525 0 360 DrawEllipse gs col0 s gr % Ellipse n 1575 3300 75 75 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 100.00 215.00] PATmp PATsp ef gr PATusp gs col0 s gr % Ellipse n 4575 3375 75 75 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P1 [16 0 0 -8 300.00 220.00] PATmp PATsp ef gr PATusp gs col0 s gr % Ellipse n 3075 3300 900 900 0 360 DrawEllipse gs col0 s gr % Ellipse n 4275 2400 530 530 0 360 DrawEllipse gs col0 s gr % Ellipse n 2475 1950 474 474 0 360 DrawEllipse gs col0 s gr % Ellipse n 3075 3300 1500 1500 0 360 DrawEllipse gs col0 s gr % Polyline gs clippath 2287 4325 m 2191 4403 l 2239 4289 l 2142 4419 l 2190 4455 l cp clip n 3075 3225 m 2175 4425 l gs col0 s gr gr % arrowhead n 2287 4325 m 2191 4403 l 2239 4289 l 2263 4307 l 2287 4325 l cp gs 0.00 setgray ef gr col0 s % Ellipse n 4575 3300 530 530 0 360 DrawEllipse gs col0 s gr % Polyline gs clippath 3540 3837 m 3584 3952 l 3491 3872 l 3584 4004 l 3633 3970 l cp clip n 3075 3225 m 3600 3975 l gs col0 s gr gr % arrowhead n 3540 3837 m 3584 3952 l 3491 3872 l 3516 3855 l 3540 3837 l cp gs 0.00 setgray ef gr col0 s /Times-Roman ff 210.00 scf sf 3450 1875 m gs 1 -1 sc ( 3) col0 sh gr % Polyline gs clippath 4509 2150 m 4626 2113 l 4539 2201 l 4678 2118 l 4647 2067 l cp clip n 4275 2325 m 4650 2100 l gs col0 s gr gr % arrowhead n 4509 2150 m 4626 2113 l 4539 2201 l 4524 2176 l 4509 2150 l cp gs 0.00 setgray ef gr col0 s /Times-Bold ff 240.00 scf sf 4650 3600 m gs 1 -1 sc (a) col0 sh gr /Times-Bold ff 240.00 scf sf 1350 3600 m gs 1 -1 sc (b) col0 sh gr /Times-Bold ff 240.00 scf sf 4125 2475 m gs 1 -1 sc (x) col0 sh gr /Times-Bold ff 240.00 scf sf 3375 1800 m gs 1 -1 sc (x) col0 sh gr /Times-Bold ff 240.00 scf sf 4425 2175 m gs 1 -1 sc (R) col0 sh gr /Times-Bold ff 240.00 scf sf 2700 3675 m gs 1 -1 sc (rho) col0 sh gr /Times-Bold ff 240.00 scf sf 3075 3150 m gs 1 -1 sc (x) col0 sh gr /Times-Roman ff 210.00 scf sf 4275 2550 m gs 1 -1 sc (2) col0 sh gr $F2psEnd rs end %%EndDocument @endspecial 159 2389 a(Clearly)k(then)209 2673 y Fq(u)p Ft(\()p Fq(a)p Ft(\))d Fr(\024)f Fq(C)111 b Ft(inf)587 2736 y Fo(z)s Fh(2)p Fo(B)729 2745 y Fj(R)780 2736 y Fu(\()p Fo(a)p Fu(\))902 2673 y Fr(\024)25 b Fq(C)255 b Ft(inf)1084 2736 y Fo(z)s Fh(2)p Fo(B)1226 2745 y Fj(R)1277 2736 y Fu(\()p Fo(a)p Fu(\))p Fh(\\)p Fo(B)1476 2745 y Fj(R)1529 2736 y Fu(\()p Fo(x)1598 2745 y Fk(2)1634 2736 y Fu(\))1677 2673 y Fq(u)p Ft(\()p Fq(z)t Ft(\))27 b Fr(\024)d Fq(C)117 b Ft(sup)2054 2754 y Fo(z)s Fh(2)p Fo(B)2196 2763 y Fj(R)2247 2754 y Fu(\()p Fo(x)2316 2763 y Fk(2)2352 2754 y Fu(\))2395 2673 y Fq(u)p Ft(\()p Fq(z)t Ft(\))27 b Fr(\024)e Fq(C)2758 2636 y Fu(2)2921 2673 y Ft(inf)2813 2736 y Fo(z)s Fh(2)p Fo(B)2955 2745 y Fj(R)3006 2736 y Fu(\()p Fo(x)3075 2745 y Fk(2)3111 2736 y Fu(\))3154 2673 y Fq(u)p Ft(\()p Fq(z)t Ft(\))i Fr(\024)e Fq(:)15 b(:)g(:)395 2897 y Fr(\001)g(\001)g(\001)26 b(\024)f Fq(C)694 2860 y Fo(k)r Fh(\000)p Fu(1)943 2897 y Ft(sup)846 2978 y Fo(z)s Fh(2)p Fo(B)988 2987 y Fj(R)1039 2978 y Fu(\()p Fo(x)1108 2988 y Fj(k)1149 2978 y Fu(\))1192 2897 y Fq(u)p Ft(\()p Fq(z)t Ft(\))h Fr(\024)f Fq(C)1554 2860 y Fo(k)1724 2897 y Ft(inf)1614 2960 y Fo(z)s Fh(2)p Fo(B)1756 2969 y Fj(R)1806 2960 y Fu(\()p Fo(x)1875 2970 y Fj(k)1916 2960 y Fu(\))1959 2897 y Fq(u)p Ft(\()p Fq(z)t Ft(\))i(=)e Fq(u)p Ft(\()p Fq(b)p Ft(\))3477 3080 y(\(4)p Fq(:)p Ft(17\))50 3225 y(Th)m(us)33 b Fq(u)p Ft(\()p Fq(a)p Ft(\))f Fr(\024)e Fq(C)659 3192 y Fo(\032=R)793 3225 y Fq(u)p Ft(\()p Fq(b)p Ft(\))35 b(Therefore,)g(if)e Fq(\032)e Ft(=)f Fq(c\017)p Ft(,)36 b(for)d(some)i(\014nite)d(constan)m (t)k Fq(c)p Ft(,)f(and)e Fq(R)f Ft(=)f Fq(\017)p Ft(,)k(sup)d(and)50 3371 y(inf)d(are)h(related)h(b)m(y)f(at)h(most)g(a)g(\014nite)e Fq(\017)p Ft(-indep)s(enden)m(t)f(constan)m(t.)42 b(This)29 b(pro)m(v)m(es)i(the)f(lemma.)p Fr(})159 3589 y Ft(Com)m(bining)21 b(no)m(w)j(Lemma)f(4.6)i(with)d(Lemma)i(4.5)g(and)f(equation)h (\(4.14\),)j(w)m(e)d(arriv)m(e)f(at)h(the)g(assertion)50 3735 y(of)30 b(the)h(prop)s(osition.)p Fr(})50 3953 y Fs(A-priori)43 b(b)s(ounds)h(on)g(capacities.)62 b Ft(T)-8 b(o)38 b(mak)m(e)h(use)e(of)h(the)f(renew)m(al)h(estimate)g(\(4.4\))h (w)m(e)f(need)g(of)50 4098 y(course)h(some)h(b)s(ounds)c(on)j(the)h (copacities.)67 b(The)38 b(next)i(prop)s(osition)c(pro)m(vides)j(a)g (\014rst)f(set)i(of)f(rough)50 4244 y(b)s(ounds,)32 b(that)h(pro)m (vide)f(the)i(necessary)f(estimates)h(in)d(the)j(equilibrium)28 b(p)s(oten)m(tial)k(that)i(will)c(later)j(b)s(e)50 4389 y(used)c(to)j(get)f(sharp)e(b)s(ounds)f(on)j(capacities.)50 4607 y Fe(Prop)s(osition)45 b(4.7:)51 b Fn(L)-5 b(et)38 b Fq(D)j Fn(b)-5 b(e)37 b(a)i(close)-5 b(d)38 b(set,)h(and)g Fq(x)c Fr(2)f Fq(D)2271 4574 y Fo(c)2308 4607 y Fn(.)57 b(Denote)37 b(by)h Fq(z)2880 4574 y Fh(\003)2956 4607 y Ft(=)d Fq(z)3108 4574 y Fh(\003)3150 4607 y Ft(\()p Fq(x;)15 b(D)s Ft(\))38 b Fn(a)g(p)-5 b(oint)50 4753 y(such)33 b(that)1028 4991 y Fq(F)13 b Ft(\()p Fq(z)1180 4954 y Fh(\003)1223 4991 y Ft(\))25 b(=)251 b(inf)1379 5054 y Fo(\015)t Fu(:)p Fo(\015)t Fu(\(0\)=)p Fo(x;\015)t Fu(\(1\))p Fh(2)p Fo(D)1956 4836 y Fi( )2066 4991 y Ft(sup)2028 5072 y Fo(t)p Fh(2)p Fu([0)p Fo(;)p Fu(1])2255 4991 y Ft([)p Fq(F)2338 5005 y Fo(N)2408 4991 y Ft(\()p Fq(\015)5 b Ft(\()p Fq(t)p Ft(\)\)])2659 4836 y Fi(!)3477 4991 y Ft(\(4)p Fq(:)p Ft(18\))50 5316 y Fn(wher)-5 b(e)31 b(the)f(in\014mum)g(is)f(over)h(al)5 b(l)30 b(c)-5 b(ontinuous)31 b(p)-5 b(aths)32 b(le)-5 b(ading)30 b(fr)-5 b(om)31 b Fq(x)f Fn(to)g Fq(D)s Fn(.)41 b(L)-5 b(et)29 b Fq(\032)d Fr(\024)f Fq(c\017=)p Fr(kr)p Fq(F)13 b Ft(\()p Fq(z)3481 5283 y Fh(\003)3524 5316 y Ft(\))p Fr(k)3604 5330 y Fh(1)3682 5316 y Fn(.)50 5462 y(Then)33 b(ther)-5 b(e)33 b(is)g(a)g(c)-5 b(onstant)34 b Fq(C)e(>)25 b Ft(0)33 b Fn(such)g(that)p eop %%Page: 16 16 16 15 bop 50 295 a Fl(16)10 513 y Fn(\(i\))936 672 y Ft(cap)1087 689 y Fo(B)1142 698 y Fj(\032)1180 689 y Fu(\()p Fo(x)p Fu(\))1282 672 y Ft(\()p Fq(D)s Ft(\))26 b Fr(\025)f Fq(C)7 b Ft(\()p Fr(kr)p Fq(F)13 b Ft(\()p Fq(z)1932 635 y Fh(\003)1974 672 y Ft(\))p Fr(k)2054 686 y Fh(1)2152 672 y Ft(+)2243 602 y Fr(p)p 2319 602 37 4 v 70 x Fq(\017)p Ft(\))p Fq(\032)2438 635 y Fo(d)p Fh(\000)p Fu(1)2574 672 y Fq(e)2616 635 y Fh(\000)p Fo(F)d Fu(\()p Fo(z)2795 607 y Fd(\003)2833 635 y Fu(\))p Fo(=\017)3477 672 y Ft(\(4)p Fq(:)p Ft(19\))-18 968 y Fn(\(ii\))1301 1128 y Ft(cap)1453 1144 y Fo(B)1508 1153 y Fj(\032)1545 1144 y Fu(\()p Fo(x)p Fu(\))1648 1128 y Ft(\()p Fq(D)s Ft(\))25 b Fr(\024)g Fq(\017C)7 b(\032)2073 1090 y Fo(d)p Fh(\000)p Fu(2)2208 1128 y Fq(e)2250 1090 y Fh(\000)p Fo(F)j Fu(\()p Fo(z)2429 1063 y Fd(\003)2467 1090 y Fu(\))p Fo(=\017)3477 1128 y Ft(\(4)p Fq(:)p Ft(20\))50 1569 y Fc(Pro)s(of:)52 b Ft(T)-8 b(o)34 b(pro)m(v)m(e)f(the)g(lo)m(w)m(er)g (b)s(ound)d(w)m(e)j(use)g(the)g(v)-5 b(ariational)31 b(represen)m(tation)i(of)g(capacities)f(\(2.15\))50 1714 y(and)37 b(some)g(ob)m(vious)g(monotonicit)m(y)h(prop)s(erties.)60 b(W)-8 b(e)38 b(b)s(egin)e(b)m(y)h(c)m(ho)s(osing)h(a)f(smo)s(oth)g (path)h Fq(!)i Ft(going)50 1860 y(from)g Fq(x)h Ft(to)h Fq(D)h Ft(in)d(suc)m(h)g(a)i(w)m(a)m(y)f(that)h(it)e(remains)g(in)g (the)h(lev)m(el)f(set)i Fq(F)13 b Ft(\()p Fq(z)t Ft(\))44 b Fr(\024)e Fq(F)13 b Ft(\()p Fq(z)3030 1827 y Fh(\003)3073 1860 y Ft(\),)44 b(with)39 b(equalit)m(y)50 2005 y(holding)34 b(only)h(when)f(passing)h Fq(z)1197 1972 y Fh(\003)1239 2005 y Ft(.)57 b(In)35 b(fact,)k(the)d(canonical)f(path)h(can)g(b)s(e)f (constructed)i(using)d(pieces)50 2151 y(of)c(the)h(deterministic)d(tra) 5 b(jectory)31 b(of)g(the)f(unp)s(erturb)s(ed)d(equation)j Fq(dX)2559 2165 y Fo(\017)2595 2151 y Ft(\()p Fq(t)p Ft(\))25 b(=)g Fr(r)p Fq(F)13 b Ft(\()p Fq(X)3076 2165 y Fo(\017)3112 2151 y Ft(\()p Fq(t)p Ft(\)\))p Fq(dt)31 b Ft(in)e(rather)50 2296 y(ob)m(vious)i(manner,)f(but)h(this)f(is)g (not)i(imp)s(ortan)m(t)e(at)i(the)g(momen)m(t.)44 b(Giv)m(en)31 b(this)f(path,)i(w)m(e)g(parametrize)50 2442 y(it)e(b)m(y)g (arc-length,)h(so)f(that)h Fr(k)18 b Ft(_)-43 b Fq(!)t Ft(\()p Fq(t)p Ft(\))p Fr(k)1284 2456 y Fu(2)1351 2442 y Ft(=)25 b(1)30 b(for)h(all)e(time.)159 2660 y(Giv)m(en)h Fq(!)s Ft(\()p Fq(t)p Ft(\),)h(w)m(e)g(construct)g(the)f(tub)s(e)g(of)g (width)f Fq(\032)h Ft(around)g Fq(!)s Ft(\()p Fq(t)p Ft(\),)1106 2928 y Fq(!)1166 2891 y Fo(\032)1234 2928 y Fr(\021)24 b(f)p Fq(z)30 b Fr(2)25 b Fp(R)1592 2891 y Fo(d)1640 2928 y Fr(j9)1716 2945 y Fo(t)p Fh(2)p Fu([0)p Fo(;)p Fh(j)p Fo(!)r Fh(j)p Fu(])1987 2928 y Fr(k)p Fq(!)s Ft(\()p Fq(t)p Ft(\))c Fr(\000)f Fq(z)t Fr(k)2398 2942 y Fu(2)2465 2928 y Fr(\024)25 b Fq(\032)p Fr(g)824 b Ft(\(4)p Fq(:)p Ft(21\))50 3197 y(Let)42 b(us)e(denote)i(the)f Fq(d)28 b Fr(\000)f Ft(1-dimensional)39 b(disk)h(of)i(radius)d Fq(\032)i Ft(cen)m(tered)h(at)g(the)g(origin)d(b)m(y)i Fq(D)3412 3211 y Fo(\032)3455 3197 y Ft(.)73 b(The)50 3342 y(imp)s(ortan)m(t)29 b(p)s(oin)m(t)h(to)h(notice)g(is)e(that)1095 3643 y Fr(kr)p Fq(h)p Ft(\()p Fq(!)s Ft(\()p Fq(t)p Ft(\))21 b(+)f Fq(z)1620 3657 y Fh(?)1682 3643 y Ft(\))p Fr(k)1762 3605 y Fu(2)1762 3665 y(2)1829 3643 y Fr(\025)1925 3515 y Fi(\024)2000 3581 y Fq(d)p 1984 3622 81 4 v 1984 3705 a(dt)2075 3643 y(h)p Ft(\()p Fq(!)s Ft(\()p Fq(t)p Ft(\))h(+)f Fq(z)2479 3657 y Fh(?)2541 3643 y Ft(\))2576 3515 y Fi(\025)2624 3533 y Fu(2)3477 3643 y Ft(\(4)p Fq(:)p Ft(22\))50 3924 y(Therefore)30 b(w)m(e)h(ma)m(y)g(b)s(ound)d(the)j(Diric)m(hlet)e(form) 733 4225 y(\010)799 4239 y Fu(\012)856 4225 y Ft(\()p Fq(h)p Ft(\))d Fr(\025)f Fq(\017)1152 4101 y Fi(Z)1202 4308 y Fo(D)1261 4317 y Fj(\032)1319 4225 y Fq(dz)1408 4239 y Fh(?)1485 4101 y Fi(Z)1576 4123 y Fh(j)p Fo(!)r Fh(j)1536 4308 y Fu(0)1687 4225 y Fq(dte)1809 4187 y Fo(F)10 b Fu(\()p Fo(!)r Fu(\()p Fo(t)p Fu(\)+)p Fo(z)2115 4197 y Fd(?)2171 4187 y Fu(\))p Fo(=\017)2286 4097 y Fi(\024)2362 4163 y Fq(d)p 2345 4204 V 2345 4287 a(dt)2436 4225 y(h)p Ft(\()p Fq(!)s Ft(\()p Fq(t)p Ft(\))21 b(+)f Fq(z)2840 4239 y Fh(?)2902 4225 y Ft(\))2937 4097 y Fi(\025)2985 4115 y Fu(2)3477 4225 y Ft(\(4)p Fq(:)p Ft(23\))50 4521 y(The)37 b(minimization)e(problem)h(is)h(no)m(w)h(trivial,)g(i.e.)62 b(it)38 b(decomp)s(oses)f(for)h(eac)m(h)h(\014xed)e Fq(z)3169 4535 y Fh(?)3268 4521 y Ft(in)m(to)h(a)g(one-)50 4666 y(dimensional)i(problem)i(whose)h(solution)f(is)h(w)m(ell)f(kno)m(wn.) 79 b(In)43 b(fact,)48 b(the)43 b(minimizer)e Fq(h)3239 4680 y Fo(z)3274 4690 y Fd(?)3332 4666 y Ft(\()p Fq(t)p Ft(\))j(is)f(the)50 4812 y(solution)29 b(of)h(the)h(1-dimensional)d (Diric)m(hlet)h(problem)1102 4965 y Fi(\024)1150 5094 y Fr(\000)p Fq(\017)1285 5032 y(d)p 1269 5073 V 1269 5156 a(dt)1379 5094 y Ft(+)1498 5032 y Fq(d)p 1481 5073 V 1481 5156 a(dt)1572 5094 y(F)13 b Ft(\()p Fq(!)s Ft(\()p Fq(t)p Ft(\))21 b(+)f Fq(z)1995 5108 y Fh(?)2057 5094 y Ft(\))2092 4965 y Fi(\025)2183 5032 y Fq(d)p 2166 5073 V 2166 5156 a(dt)2257 5094 y(h)2309 5108 y Fo(z)2344 5118 y Fd(?)2403 5094 y Ft(\()p Fq(t)p Ft(\))26 b(=)f(0)1087 5294 y Fq(h)1139 5308 y Fo(z)1174 5318 y Fd(?)1232 5294 y Ft(\(0\))h(=)f(1)1087 5466 y Fq(h)1139 5480 y Fo(z)1174 5490 y Fd(?)1232 5466 y Ft(\()p Fr(j)p Fq(!)s Fr(j)p Ft(\))h(=)f(0)3477 5253 y(\(4)p Fq(:)p Ft(24\))p eop %%Page: 17 17 17 16 bop 3633 295 a Fl(17)50 513 y Ft(whose)30 b(solution)f(is)g (readily)g(found)g(to)i(b)s(e)1298 757 y Fq(h)1350 771 y Fo(z)1385 781 y Fd(?)1443 757 y Ft(\()p Fq(t)p Ft(\))26 b(=)1679 616 y Fi(R)1739 639 y Fh(j)p Fo(!)r Fh(j)1722 722 y Fo(t)1850 689 y Fq(dse)1982 656 y Fo(F)10 b Fu(\()p Fo(!)r Fu(\()p Fo(s)p Fu(\)+)p Fo(z)2295 666 y Fd(?)2350 656 y Fu(\))p Fo(=\017)p 1679 736 773 4 v 1679 772 a Fi(R)1739 795 y Fh(j)p Fo(!)r Fh(j)1722 878 y Fu(0)1850 846 y Fq(dse)1982 819 y Fo(F)g Fu(\()p Fo(!)r Fu(\()p Fo(s)p Fu(\)+)p Fo(z)2295 829 y Fd(?)2350 819 y Fu(\))p Fo(=\017)3477 757 y Ft(\(4)p Fq(:)p Ft(25\))50 1007 y(Inserting)29 b(this)g(solution)g(in)m(to)h(the)h(lo)m(w)m(er)g(b)s(ound)d(\(4.23\)) 33 b(yields)884 1267 y(cap)1036 1284 y Fo(B)1091 1293 y Fj(\032)1128 1284 y Fu(\()p Fo(x)p Fu(\))1231 1267 y Ft(\()p Fq(D)s Ft(\))26 b Fr(\025)f Fq(\017)1553 1144 y Fi(Z)1603 1350 y Fo(D)1662 1359 y Fj(\032)1720 1267 y Fq(dz)1809 1281 y Fh(?)1886 1112 y Fi(")1939 1144 y(Z)2030 1166 y Fh(j)p Fo(!)r Fh(j)1989 1350 y Fu(0)2141 1267 y Fq(dte)2263 1230 y Fo(F)10 b Fu(\()p Fo(!)r Fu(\()p Fo(t)p Fu(\)+)p Fo(z)2569 1240 y Fd(?)2624 1230 y Fu(\))p Fo(=\017)2725 1112 y Fi(#)2778 1131 y Fh(\000)p Fu(1)3477 1267 y Ft(\(4)p Fq(:)p Ft(26\))50 1505 y(F)-8 b(rom)29 b(here)f(the)h(stated)g(lo)m(w)m(er)f(b)s(ounds)f(results)g(follo)m(w)g (from)h(simple)f(saddle)g(p)s(oin)m(t)g(ev)-5 b(aluations)28 b(of)h(the)50 1650 y(in)m(tegral)h(in)f(the)i(denominator.)159 1868 y(T)-8 b(o)31 b(pro)m(v)m(e)h(the)f(upp)s(er)e(b)s(ound,)h(just)g (note)i(that)f(in)f(the)h(case)h(when)e Fq(z)2585 1835 y Fh(\003)2653 1868 y Ft(=)c Fq(x)p Ft(,)32 b(w)m(e)f(can)g(alw)m(a)m (ys)h(c)m(ho)s(ose)50 2014 y(a)g(function)f Fq(h)i Ft(that)g(is)e (equal)h(to)h(one)f(on)h Fq(B)1583 2028 y Fo(\032)1625 2014 y Ft(\()p Fq(x)p Ft(\))g(and)e(that)i(deca)m(ys)g(to)g(zero)g(o)m (v)m(er)h(a)f(distance)e Fq(\032)p Ft(.)47 b(Then)50 2159 y Fr(kr)p Fq(h)p Fr(k)268 2173 y Fu(2)335 2159 y Fr(\024)25 b Ft(1)p Fq(=\032)f Ft(on)g(a)g(set)g(of)g(v)m(olume)f Fq(C)7 b(\032)1438 2126 y Fo(d)1480 2159 y Ft(,)25 b(and)e(zero)i (elsewhere.)38 b(The)23 b(upp)s(er)e(b)s(ound)h(\(4.20\))k(follo)m(ws)c (im-)50 2305 y(mediately)-8 b(.)39 b(If)26 b Fq(z)626 2272 y Fh(\003)694 2305 y Fr(6)p Ft(=)f Fq(x)p Ft(,)i(w)m(e)h(c)m(ho)s (ose)f(a)h(trial)d(function)h(that)h(c)m(hanges)h(from)e(0)h(to)h(1)f (in)f(a)h Fq(\032)p Ft(-neigh)m(b)s(orho)s(o)s(d)50 2450 y(of)32 b(the)g(saddle)f Fq(z)639 2417 y Fh(\003)681 2450 y Ft(;)i(a)m(w)m(a)m(y)h(from)d Fq(z)1230 2417 y Fh(\003)1304 2450 y Ft(the)i(c)m(hange)g(tak)m(es)g(place)f(in)f(a)h (set)h(where)e Fq(F)13 b Ft(\()p Fq(y)s Ft(\))29 b Fq(>)e(F)13 b Ft(\()p Fq(z)3294 2417 y Fh(\003)3337 2450 y Ft(\),)33 b(so)f(that)50 2595 y(the)f(resulting)d(additional)h(con)m(tribution)g (to)i(the)g(Diric)m(hlet)f(form)g(is)f(exp)s(onen)m(tially)g (suppressed.)40 b(This)50 2741 y(also)30 b(yields)f(\(4.20\).)43 b Fr(})50 2959 y Fs(Remark:)50 b Ft(This)31 b(estimate)i(is)e(in)g (general)h(quite)g(p)s(o)s(or,)g(in)e(particular)h(when)g Fq(z)2870 2926 y Fh(\003)2941 2959 y Fr(6)p Ft(=)d Fq(x)p Ft(.)46 b(W)-8 b(e)33 b(will)d(pro)m(v)m(e)50 3105 y(sharp)e(results)h (in)f(that)j(case)g(in)d(Section)i(6.)41 b(The)29 b(crude)g(b)s(ounds)f (serv)m(e)i(t)m(w)m(o)h(purp)s(oses:)39 b(1\))30 b(to)h(yield)d(an)50 3250 y(a)35 b(priori)d(b)s(ound)h(on)h(the)h(equilibrium)30 b(p)s(oten)m(tial)k(\(in)g(conjunction)g(with)f(Prop)s(osition)g(4.3\)) j(that)f(will)50 3395 y(then)e(b)s(e)f(used)g(to)i(pro)m(v)m(e)g(a)g (sharp)e(estimate)i(on)f(capacities,)h(and)f(2\))h(to)f(get)i(an)e(a)g (priori)e(estimate)j(on)50 3541 y(the)c(sp)s(ectrum)g(of)g(certain)g (Diric)m(hlet)g(op)s(erators.)50 3759 y Fs(Bounds)h(on)g(the)f (equilibrium)g(p)s(oten)m(tial.)38 b Ft(Com)m(bining)25 b(the)h(renew)m(al)g(b)s(ound)f(on)h(the)h(equilibrium)50 3905 y(obtained)37 b(in)f(Prop)s(osition)g(4.5)j(with)d(the)i(b)s(ound) e(on)h(capacities)h(from)f(prop)s(osition)f(4.7)i(yields)e(v)m(ery)50 4050 y(sharp)26 b(estimates)j(on)f(the)g(equilibrium)23 b(p)s(oten)m(tial)k(in)f(the)i(lev)m(el)g(set)g(of)g(the)g(saddle)f(b)s (et)m(w)m(een)h(the)g(sets)h Fq(A)50 4195 y Ft(and)h Fq(D)s Ft(.)50 4414 y Fe(Corollary)j(4.8:)49 b Fn(L)-5 b(et)29 b Fq(A)g Fn(and)h Fq(D)i Fn(b)-5 b(e)29 b(close)-5 b(d)30 b(sets)g(and)g(assume)g(that)g Fq(z)2572 4381 y Fh(\003)2614 4414 y Ft(\()p Fq(A;)15 b(D)s Ft(\))26 b Fr(62)f Fq(A)13 b Fr([)g Fq(D)s Fn(.)39 b(Then)30 b(ther)-5 b(e)50 4559 y(is)32 b(a)h(\014nite)g(p)-5 b(ositive)34 b(c)-5 b(onstant)34 b Fq(C)39 b Fn(such)33 b(that,)h(for)f Fq(x)25 b Fr(62)g Fq(A)20 b Fr([)g Fq(D)s Fn(,)32 b(and)i Fq(z)2514 4526 y Fh(\003)2556 4559 y Ft(\()p Fq(x;)15 b(D)s Ft(\))26 b Fr(6)p Ft(=)f Fq(x)p Fn(,)1021 4756 y Fq(h)1073 4770 y Fo(A;D)1215 4756 y Ft(\()p Fq(x)p Ft(\))h Fr(\024)f Fq(C)7 b(\017)1568 4718 y Fh(\000)p Fu(1)p Fo(=)p Fu(2)1738 4756 y Fq(e)1780 4718 y Fh(\000)p Fu([)p Fo(F)j Fu(\()p Fo(z)1980 4691 y Fd(\003)2018 4718 y Fu(\()p Fo(x;A)p Fu(\)\))p Fh(\000)p Fo(F)g Fu(\()p Fo(z)2399 4691 y Fd(\003)2437 4718 y Fu(\()p Fo(x;D)r Fu(\)\)])p Fo(=\017)3477 4756 y Ft(\(4)p Fq(:)p Ft(27\))50 5171 y Fs(Remark:)56 b Ft(This)32 b(b)s(ound)h(is)h(useful)e(only)i (when)g Fq(F)13 b Ft(\()p Fq(z)1952 5138 y Fh(\003)1994 5171 y Ft(\()p Fq(x;)i(D)s Ft(\)\))34 b Fq(>)e(F)13 b Ft(\()p Fq(x)p Ft(\).)54 b(If)34 b(this)f(is)h(not)h(the)g(case)g(one) 50 5316 y(ma)m(y)d(use)f(that)h Fq(h)656 5330 y Fo(A;D)799 5316 y Ft(\()p Fq(y)s Ft(\))27 b(=)g(1)21 b Fr(\000)g Fq(h)1252 5330 y Fo(D)r(;A)1394 5316 y Ft(\()p Fq(y)s Ft(\))32 b(and)f(apply)f(\(4.27\))k(on)d Fq(h)2416 5330 y Fo(D)r(;A)2559 5316 y Ft(\()p Fq(y)s Ft(\).)44 b(This)29 b(yields)h(go)s(o)s(d)h(con)m(trol)50 5462 y(whenev)m(er)f Fq(x)g Ft(is)g(b)s(elo)m(w)f(the)i(lev)m(el)f(set)h(of)f(the)h(saddle)e Fq(z)1970 5429 y Fh(\003)2012 5462 y Ft(\()p Fq(A;)15 b(D)s Ft(\).)p eop %%Page: 18 18 18 17 bop 50 295 a Fl(18)50 513 y Fc(Pro)s(of:)46 b Ft(The)28 b(pro)s(of)f(is)g(straigh)m(tforw)m(ard.)40 b(W)-8 b(e)29 b(just)f(insert)f(the)h(b)s(ounds)e(on)i(capacities)g(of)h(Prop)s (osition)50 658 y(4.7)g(in)m(to)g(the)g(renew)m(al)f(b)s(ound)e(on)j (the)f(equilibrium)c(p)s(oten)m(tial)k(of)h(Prop)s(osition)d(4.5,)31 b(c)m(ho)s(osing)d Fq(\032)d Ft(=)g Fq(C)7 b(\017)p Ft(.)50 804 y Fr(})50 1041 y Fv(5.)55 b(Sharp)41 b(estimates)f(on)i (capacities.)159 1260 y Ft(In)g(this)f(section)i(w)m(e)h(sho)m(w)e(ho)m (w)h(to)g(get)h(coinciding)d(upp)s(er)f(and)j(lo)m(w)m(er)f(b)s(ound)f (on)i(the)g(relativ)m(e)50 1405 y(Newtonian)28 b(capacit)m(y)i(of)f(t)m (w)m(o)h(balls)d(of)i(radius)e Fq(\032)i Ft(cen)m(tered)g(at)h(the)f (lo)s(cal)f(minima)e Fq(x;)15 b(y)32 b Ft(of)d(the)g(function)50 1551 y Fq(F)13 b Ft(.)51 b(W)-8 b(e)35 b(assume)e(that)i Fq(\032)f Ft(is)e(so)j(small)d(that)i Fq(z)1653 1518 y Fh(\003)1695 1551 y Ft(\()p Fq(x;)15 b(y)s Ft(\))35 b(is)e(not)h(con)m(tained)g(in)e(these)j(balls,)e(and)g(that)i(the)50 1696 y(radii)28 b(are)j(at)g(least)g Fq(\017)775 1663 y Fu(7)815 1696 y Ft(.)41 b(Let)31 b(us)f(denote)h(these)f(sets)h(b)m (y)f Fq(B)2059 1710 y Fo(x)2135 1696 y Ft(and)g Fq(B)2381 1710 y Fo(y)2425 1696 y Ft(,)g(resp)s(ectiv)m(ely)-8 b(.)159 1914 y(W)g(e)29 b(denote)g(b)m(y)f Fr(S)783 1928 y Fo(x;y)917 1914 y Ft(the)g(set)h(of)f(p)s(oin)m(ts)f(that)h(realize)g (the)g(minimax)e(in)h(the)h(de\014nition)3264 1891 y Fi(b)3246 1914 y Fq(F)13 b Ft(\()p Fq(x;)i(y)s Ft(\))29 b(\(c.f.)50 2060 y(\(3.1\)\).)43 b(W)-8 b(e)31 b(will)d(assume)i(that)h Fr(S)1232 2074 y Fo(x;y)1369 2060 y Ft(is)e(a)i(\(\014nite\))f(union)f (of)h(p)s(oin)m(ts.)50 2278 y Fe(Theorem)35 b(5.1:)50 b Fn(L)-5 b(et)32 b Fq(s)961 2245 y Fh(\003)961 2300 y Fu(1)1002 2278 y Fq(;)15 b(:)g(:)g(:)i(;)e(s)1247 2245 y Fh(\003)1247 2304 y Fo(k)1323 2278 y Fn(denote)32 b(the)g(sadd)5 b(le)32 b(p)-5 b(oints)33 b(c)-5 b(onne)g(cting)32 b Fq(x)f Fn(to)h Fq(y)s Fn(,)f(and)i(supp)-5 b(ose)32 b(that)50 2423 y(Assumption)j Fs(\(ND\))e Fn(holds)i(for)f Fq(S)5 b Ft(\()p Fq(xy)s Ft(\))p Fn(.)46 b(L)-5 b(et)34 b Fq(\025)1723 2390 y Fh(\003)1723 2446 y Fu(1)1765 2423 y Ft(\()p Fq(s)1843 2390 y Fh(\003)1843 2447 y Fo(i)1885 2423 y Ft(\))g Fn(denote)g(the)g (unique)f(ne)-5 b(gative)34 b(eigenvalue)f(of)h(the)50 2569 y(Hessian)f(of)f Fq(F)46 b Fn(at)33 b Fq(s)754 2536 y Fh(\003)754 2592 y Fo(i)796 2569 y Fn(.)41 b(Then,)33 b(under)g(the)g(ab)-5 b(ove)33 b(hyp)-5 b(othesis)36 b(on)d(the)g(function)f Fq(F)13 b Fn(,)444 2840 y Ft(cap)596 2854 y Fo(B)651 2863 y Fj(x)696 2840 y Ft(\()p Fq(B)800 2854 y Fo(y)844 2840 y Ft(\))25 b(=)g Fq(e)1042 2802 y Fh(\000)1108 2798 y Fi(b)1099 2802 y Fo(F)10 b Fu(\()p Fo(x;y)r Fu(\))p Fo(=\017)1397 2778 y Ft(\(2)p Fq(\031)s(\017)p Ft(\))1604 2745 y Fo(d=)p Fu(2)p 1397 2819 325 4 v 1509 2902 a Ft(4)p Fq(\031)1793 2726 y Fo(k)1747 2754 y Fi(X)1754 2947 y Fo(i)p Fu(=1)2102 2778 y Fr(j)p Fq(\025)2180 2745 y Fh(\003)2180 2801 y Fu(1)2223 2778 y Ft(\()p Fq(s)2301 2745 y Fh(\003)2301 2802 y Fo(i)2342 2778 y Ft(\))p Fr(j)p 1905 2819 697 4 v 1905 2837 a Fi(p)p 1995 2837 606 4 v 1995 2914 a Fr(j)15 b Ft(det)q(\()p Fr(r)2273 2887 y Fu(2)2314 2914 y Fq(F)e Ft(\()p Fq(s)2463 2882 y Fh(\003)2463 2939 y Fo(i)2505 2914 y Ft(\)\))p Fr(j)2627 2766 y Fi(\000)2669 2840 y Ft(1)20 b(+)g Fq(O)s Ft(\()2932 2770 y Fr(p)p 3008 2770 37 4 v 70 x Fq(\017)p Fr(j)15 b Ft(ln)f Fq(\017)p Fr(j)p Ft(\))3273 2766 y Fi(\001)3523 2840 y Ft(\(5)p Fq(:)p Ft(1\))50 3275 y Fc(Pro)s(of:)47 b Ft(The)30 b(capacit)m(y)i (cap)1076 3289 y Fo(B)1131 3298 y Fj(x)1176 3275 y Ft(\()p Fq(B)1280 3289 y Fo(y)1324 3275 y Ft(\))f(v)m(eri\014es)f(the)g(Diric)m (hlet)f(principle)e(\(2.15\))1403 3492 y(cap)1554 3506 y Fo(B)1609 3515 y Fj(x)1654 3492 y Ft(\()p Fq(B)1758 3506 y Fo(y)1802 3492 y Ft(\))f(=)67 b(inf)1959 3552 y Fo(h)p Fh(2H)2113 3533 y Fj(x)2113 3570 y(y)2168 3492 y Ft(\010\()p Fq(h)p Ft(\))1167 b(\(5)p Fq(:)p Ft(2\))50 3736 y(\(where)26 b(w)m(e)g(abbreviate)g(for)f(simplicit)m(y)f(\010)g Fr(\021)h Ft(\010)1713 3753 y Fu(\()p Fo(B)1796 3762 y Fj(x)1837 3753 y Fh([)p Fo(B)1941 3762 y Fj(y)1980 3753 y Fu(\))2008 3735 y Fj(c)2047 3736 y Ft(\))h(where)g(the)g(space)g Fr(H)2835 3703 y Fo(x)2834 3759 y(y)2907 3736 y Ft(is)f(the)h(function) e(space)645 3953 y Fr(H)723 3916 y Fo(x)722 3976 y(y)793 3953 y Fr(\021)889 3880 y Fi(\010)942 3953 y Fq(h)i Fr(2)f Fq(W)1205 3916 y Fu(1)p Fo(;)p Fu(2)1303 3953 y Ft(\()p Fp(R)1398 3916 y Fo(d)1446 3953 y Fq(;)15 b Fp(Q)10 b Ft(\()p Fq(dx)p Ft(\)\))32 b(:)26 b Fq(h)p Ft(\()p Fq(z)t Ft(\))g Fr(2)f Ft([0)p Fq(;)15 b Ft(1])p Fq(;)g(h)2391 3970 y Fh(j)p Fo(B)2468 3979 y Fj(x)2541 3953 y Fr(\021)24 b Ft(1)p Fq(;)15 b(h)2773 3970 y Fh(j)p Fo(B)2850 3979 y Fj(y)2920 3953 y Ft(=)25 b(0)3061 3880 y Fi(\011)3523 3953 y Ft(\(5)p Fq(:)p Ft(3\))159 4243 y(F)-8 b(or)33 b(simplicit)m(y)c(w)m(e)k(consider)e(the)h(case)h(of)f(a)h(single)e (saddle)g(p)s(oin)m(t,)h Fq(s)2610 4210 y Fh(\003)2651 4243 y Ft(,)h(\014rst.)45 b(Without)32 b(restriction)50 4389 y(of)e(generalit)m(y)h(w)m(e)g(can)f(c)m(ho)s(ose)i(co-ordinates)f (suc)m(h)f(that)h Fq(s)2118 4356 y Fh(\003)2184 4389 y Ft(=)25 b(0)31 b(and)f(that)1002 4654 y Fq(F)13 b Ft(\()p Fq(z)t Ft(\))27 b(=)e Fq(F)13 b Ft(\(0\))21 b Fr(\000)1621 4592 y(j)p Fq(\025)1699 4559 y Fh(\003)1699 4615 y Fu(1)1741 4592 y Fr(j)p 1621 4633 146 4 v 1671 4716 a Ft(2)1777 4654 y Fq(z)1823 4616 y Fu(2)1819 4676 y(1)1884 4654 y Ft(+)2022 4540 y Fo(d)1975 4568 y Fi(X)1982 4761 y Fo(i)p Fu(=2)2133 4592 y Fq(\025)2186 4559 y Fh(\003)2186 4616 y Fo(i)p 2133 4633 95 4 v 2157 4716 a Ft(2)2238 4654 y Fq(z)2284 4616 y Fu(2)2280 4676 y Fo(i)2346 4654 y Ft(+)f Fq(O)s Ft(\()p Fr(k)p Fq(z)t Fr(k)2680 4616 y Fu(3)2680 4676 y(2)2722 4654 y Ft(\))766 b(\(5)p Fq(:)p Ft(4\))50 4919 y(for)30 b(small)f Fr(k)p Fq(z)t Fr(k)563 4933 y Fu(2)605 4919 y Ft(.)40 b(De\014ne)31 b(a)g(neigh)m(b)s(orho)s (o)s(d)d(of)i(zero)835 5189 y Fq(C)900 5203 y Fo(\016)965 5189 y Fr(\021)1061 5061 y Fi(\024)1109 5189 y Fr(\000)p Fq(\016)s(=)1268 5081 y Fi(q)p 1360 5081 146 4 v 1360 5189 a Fr(j)p Fq(\025)1438 5158 y Fh(\003)1438 5214 y Fu(1)1480 5189 y Fr(j)p Fq(;)15 b(\016)s(=)1633 5081 y Fi(q)p 1726 5081 V 1726 5189 a Fr(j)p Fq(\025)1804 5158 y Fh(\003)1804 5214 y Fu(1)1846 5189 y Fr(j)1871 5061 y Fi(\025)1984 5076 y Fo(d)1934 5103 y Fi(O)1944 5297 y Fo(i)p Fu(=2)2087 5088 y Fi(h)2130 5189 y Fr(\000)p Ft(2)p Fq(\016)s(=)2334 5111 y Fi(p)p 2426 5111 95 4 v 2426 5189 a Fq(\025)2479 5158 y Fh(\003)2479 5215 y Fo(i)2521 5189 y Fq(;)g Ft(2)p Fq(\016)s(=)2694 5111 y Fi(p)p 2787 5111 V 2787 5189 a Fq(\025)2840 5158 y Fh(\003)2840 5215 y Fo(i)2882 5088 y Fi(i)3523 5189 y Ft(\(5)p Fq(:)p Ft(5\))p 50 5375 546 4 v 159 5436 a Fm(7)197 5462 y Fl(It)25 b(will)i(b)r(ecome)f(clear)h(from)d(the)j(pro)r(of)d (that)i(the)g(precise)i(form)c(of)h(these)i(sets)g(is)g(irrelev)l(an)n (t)f(for)g(the)g(result.)p eop %%Page: 19 19 19 18 bop 3633 295 a Fl(19)50 513 y Ft(Since)44 b(w)m(e)h(ha)m(v)m(e)h (assumed)e(that)i(there)f(is)f(a)h(single)e(saddle)h(p)s(oin)m(t)g(at)i (the)f(comm)m(unication)f(heigh)m(t)50 658 y(b)s(et)m(w)m(een)31 b Fq(x)g Ft(and)f Fq(y)s Ft(,)h(it)f(is)g(p)s(ossible)e(to)j(c)m(ho)s (ose)h Fq(\016)e(>)25 b Ft(0)31 b(so)g(small)e(that)j(there)e(exists)h (a)g(strip)e Fr(S)3308 672 y Fo(\016)3379 658 y Ft(of)i(width)50 804 y(2)p Fq(\016)s(=)183 727 y Fi(p)p 275 727 146 4 v 275 804 a Fr(j)p Fq(\025)353 772 y Fh(\003)353 828 y Fu(1)396 804 y Fr(j)j Ft(con)m(taining)f(0)h(and)g(separating)f Fq(x)h Ft(and)g Fq(y)i Ft(in)d(the)h(sense)g(that)g(an)m(y)h(path)e (connecting)h(these)50 949 y(p)s(oin)m(ts)23 b(m)m(ust)i(cross)g Fr(S)810 963 y Fo(\016)850 949 y Ft(,)h(and)e(that)h(for)g(all)f Fq(z)29 b Fr(2)c(S)1730 963 y Fo(\016)1770 949 y Fr(n)p Fq(C)1880 963 y Fo(\016)1920 949 y Ft(,)h Fq(F)13 b Ft(\()p Fq(z)t Ft(\))27 b Fr(\025)e Fq(\016)2324 916 y Fu(2)2365 949 y Ft(.)39 b(Let)25 b Fq(D)2661 963 y Fo(x)2732 949 y Ft(and)f Fq(D)2978 963 y Fo(y)3047 949 y Ft(b)s(e)g(the)h(connected) 50 1095 y(comp)s(onen)m(ts)30 b(of)h Fp(R)714 1062 y Fo(d)762 1095 y Fr(nS)862 1109 y Fo(\016)933 1095 y Ft(con)m(taining)f Fq(x)g Ft(and)g Fq(y)s Ft(,)g(resp)s(ectiv)m(ely)-8 b(.)50 1313 y Fs(The)31 b(upp)s(er)g(b)s(ound.)40 b Ft(T)-8 b(o)28 b(pro)m(v)m(e)g(an)g(upp)s(er)d(b)s(ound)g(on)i(the)h(capacit)m (y)g(w)m(e)g(just)f(c)m(ho)s(ose)h(a)g(function)e Fq(h)3649 1280 y Fu(+)50 1458 y Ft(to)31 b(our)f(con)m(v)m(enience.)42 b(W)-8 b(e)31 b(will)d(mak)m(e)j(the)g(c)m(hoice)882 1653 y Fq(h)934 1615 y Fu(+)995 1653 y Ft(\()p Fq(z)t Ft(\))26 b(=)f(1)p Fq(;)15 b(z)30 b Fr(2)25 b Fq(D)1551 1667 y Fo(x)1597 1653 y Fq(;)106 b(h)1780 1615 y Fu(+)1841 1653 y Ft(\()p Fq(z)t Ft(\))26 b(=)f(0)p Fq(;)15 b(z)31 b Fr(2)25 b Fq(D)2398 1667 y Fo(y)882 1837 y Fq(h)934 1800 y Fu(+)1010 1837 y Ft(on)30 b Fr(S)1191 1851 y Fo(\016)1231 1837 y Fr(n)p Fq(C)1341 1851 y Fo(\016)1396 1837 y Ft(arbitrary)-8 b(,)30 b(except)16 b Fr(kr)p Fq(h)2244 1800 y Fu(+)2305 1837 y Fr(k)2350 1851 y Fu(2)2417 1837 y Fr(\024)25 b Fq(c)2552 1728 y Fi(q)p 2643 1728 V 109 x Fr(j)p Fq(\025)2721 1806 y Fh(\003)2721 1861 y Fu(1)2763 1837 y Fr(j)p Fq(=\016)882 2010 y Ft(for)45 b Fq(z)30 b Fr(2)25 b Fq(C)1259 2024 y Fo(\016)1298 2010 y Fq(;)15 b(h)1390 1972 y Fu(+)1452 2010 y Ft(\()p Fq(z)t Ft(\))26 b(=)f Fq(f)10 b Ft(\()p Fq(z)1822 2024 y Fu(1)1862 2010 y Ft(\))3523 1831 y(\(5)p Fq(:)p Ft(6\))50 2222 y(where)30 b Fq(f)39 b Ft(is)30 b(the)g(solution)f(of)i(the)f(one-dimensional)e(Diric)m(hlet)i(problem) 1009 2361 y Fi(\022)1076 2489 y Fr(\000)p Fq(\017)1236 2428 y(d)p 1195 2468 131 4 v 1195 2552 a(dz)1284 2566 y Fu(1)1356 2489 y Ft(+)1499 2428 y Fq(d)p 1458 2468 V 1458 2552 a(dz)1547 2566 y Fu(1)1599 2489 y Fq(F)13 b Ft(\()p Fq(z)1747 2503 y Fu(1)1788 2489 y Ft(;)i(0)p Fq(;)g(:)g(:)g(:)j(;)d Ft(0\))2156 2361 y Fi(\023)2292 2428 y Fq(d)p 2250 2468 V 2250 2552 a(dz)2339 2566 y Fu(1)2391 2489 y Fq(f)10 b Ft(\()p Fq(z)2523 2503 y Fu(1)2564 2489 y Ft(\))26 b(=)f(0)1175 2728 y Fq(f)10 b Ft(\()p Fr(\000)p Fq(\016)s(=)1424 2619 y Fi(q)p 1516 2619 146 4 v 1516 2728 a Fr(j)p Fq(\025)1594 2697 y Fh(\003)1594 2752 y Fu(1)1636 2728 y Fr(j)p Ft(\))26 b(=)f(1)1175 2932 y Fq(f)10 b Ft(\(+)p Fq(\016)s(=)1424 2823 y Fi(q)p 1516 2823 V 1516 2932 a Fr(j)p Fq(\025)1594 2900 y Fh(\003)1594 2956 y Fu(1)1636 2932 y Fr(j)p Ft(\))26 b(=)f(0)3523 2693 y(\(5)p Fq(:)p Ft(7\))50 3164 y(The)30 b(solution)f(of)h(this)f (problem)g(is)h(ob)m(viously)1312 3493 y Fq(f)10 b Ft(\()p Fq(z)1444 3507 y Fu(1)1485 3493 y Ft(\))25 b(=)1672 3343 y Fi(R)1732 3365 y Fo(\016)r(=)1804 3306 y Fr(p)p 1881 3306 125 4 v 1881 3365 a Fh(j)p Fo(\025)1946 3343 y Fd(\003)1946 3387 y Fk(1)1983 3365 y Fh(j)1715 3448 y Fo(z)1750 3457 y Fk(1)2025 3416 y Fq(e)2067 3383 y Fo(F)10 b Fu(\()p Fo(t;)p Fu(0\))p Fo(=\017)2337 3416 y Fq(dt)p 1652 3472 785 4 v 1652 3535 a Fi(R)1713 3549 y Fo(\016)r(=)1785 3490 y Fr(p)p 1861 3490 125 4 v 59 x Fh(j)p Fo(\025)1926 3527 y Fd(\003)1926 3571 y Fk(1)1964 3549 y Fh(j)1695 3658 y(\000)p Fo(\016)r(=)1824 3599 y Fr(p)p 1900 3599 V 59 x Fh(j)p Fo(\025)1965 3636 y Fd(\003)1965 3680 y Fk(1)2003 3658 y Fh(j)2044 3609 y Fq(e)2086 3582 y Fo(F)g Fu(\()p Fo(t;)p Fu(0\))p Fo(=\017)2357 3609 y Fq(dt)3523 3493 y Ft(\(5)p Fq(:)p Ft(8\))50 3872 y(Inserting)29 b(this)g(function)g(in)m(to)i(\(5.2\),)h(w)m(e)f(see)g(that)202 4180 y(cap)354 4194 y Fo(B)409 4203 y Fj(x)454 4180 y Ft(\()p Fq(B)558 4194 y Fo(y)602 4180 y Ft(\))25 b Fr(\024)g Fq(\017)810 4056 y Fi(Z)901 4078 y Fu(2)p Fo(\016)r(=)1009 4021 y Fr(p)p 1086 4021 82 4 v 1086 4078 a Fo(\025)1129 4056 y Fd(\003)1129 4100 y Fk(2)861 4262 y Fh(\000)p Fu(2)p Fo(\016)r(=)1026 4205 y Fr(p)p 1102 4205 V 57 x Fo(\025)1145 4241 y Fd(\003)1145 4284 y Fk(2)1203 4180 y Fq(dz)1292 4194 y Fu(2)1334 4180 y Fr(\001)15 b(\001)g(\001)1455 4056 y Fi(Z)1546 4078 y Fu(2)p Fo(\016)r(=)1654 4022 y Fr(p)p 1730 4022 V 56 x Fo(\025)1773 4056 y Fd(\003)1773 4103 y Fj(d)1505 4262 y Fh(\000)p Fu(2)p Fo(\016)r(=)1670 4207 y Fr(p)p 1747 4207 V 1747 4262 a Fo(\025)1790 4241 y Fd(\003)1790 4287 y Fj(d)1848 4180 y Fq(dz)1937 4194 y Fo(d)1995 4024 y Fi( )2067 4056 y(Z)2158 4078 y Fo(\016)r(=)2230 4020 y Fr(p)p 2306 4020 125 4 v 58 x Fh(j)p Fo(\025)2371 4056 y Fd(\003)2371 4100 y Fk(1)2409 4078 y Fh(j)2117 4262 y(\000)p Fo(\016)r(=)2246 4204 y Fr(p)p 2322 4204 V 58 x Fh(j)p Fo(\025)2387 4241 y Fd(\003)2387 4284 y Fk(1)2425 4262 y Fh(j)2466 4180 y Fq(dz)2555 4194 y Fu(1)2597 4180 y Fq(e)2639 4142 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(z)s Fu(\))p Fo(=\017)2933 4180 y Fr(k)q Fq(f)3034 4142 y Fh(0)3059 4180 y Ft(\()p Fq(z)3136 4194 y Fu(1)3177 4180 y Ft(\))p Fr(k)3258 4129 y Fu(2)3299 4024 y Fi(!)657 4454 y Ft(+)20 b Fq(\017c\016)867 4416 y Fh(\000)p Fu(2)981 4330 y Fi(Z)1032 4537 y Fh(S)1077 4547 y Fj(\016)1112 4537 y Fh(n)p Fo(C)1201 4547 y Fj(\016)1257 4454 y Fq(dz)t(e)1392 4416 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(z)s Fu(\))p Fo(=\017)3523 4305 y Ft(\(5)p Fq(:)p Ft(9\))50 4760 y(The)30 b(second)g(term)h(is)e (b)s(ounded)f(b)m(y)i Fq(\017c\016)1455 4727 y Fh(\000)p Fu(2)1554 4760 y Fq(e)1596 4727 y Fh(\000)p Fo(\016)1688 4699 y Fk(2)1724 4727 y Fo(=\017)1796 4760 y Fq(const:)g Ft(b)m(y)g(assumption)f(on)h Fq(F)13 b Ft(.)159 4978 y(The)30 b(\014rst)f(term)i(is)e(giv)m(en)i(b)m(y)1139 5271 y(\010)1205 5285 y Fo(C)1257 5295 y Fj(\016)1298 5271 y Ft(\()p Fq(h)1385 5234 y Fu(+)1446 5271 y Ft(\))25 b(=)g Fq(\017)1651 5121 y Fi(R)1694 5226 y Fo(C)1746 5236 y Fj(\016)1802 5194 y Fq(dz)t(e)1937 5161 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(z)s Fu(\))p Fo(=\017)2217 5194 y Fq(e)2259 5161 y Fu(2)p Fo(F)g Fu(\()p Fo(z)2414 5170 y Fk(1)2450 5161 y Fo(;)p Fu(0\))p Fo(=\017)p 1650 5251 960 4 v 1650 5284 a Fi(\022)1717 5339 y(R)1778 5352 y Fo(\016)r(=)1850 5293 y Fr(p)p 1926 5293 125 4 v 59 x Fh(j)p Fo(\025)1991 5330 y Fd(\003)1991 5374 y Fk(1)2029 5352 y Fh(j)1760 5461 y(\000)p Fo(\016)r(=)1889 5402 y Fr(p)p 1965 5402 V 59 x Fh(j)p Fo(\025)2030 5439 y Fd(\003)2030 5483 y Fk(1)2068 5461 y Fh(j)2109 5412 y Fq(e)2151 5386 y Fo(F)g Fu(\()p Fo(t;)p Fu(0\))p Fo(=\017)2421 5412 y Fq(dt)2501 5284 y Fi(\023)2568 5302 y Fu(2)3477 5271 y Ft(\(5)p Fq(:)p Ft(10\))p eop %%Page: 20 20 20 19 bop 50 295 a Fl(20)50 513 y Ft(No)m(w)31 b(on)f Fq(C)448 527 y Fo(\016)518 513 y Ft(w)m(e)h(ha)m(v)m(e)h(that)844 751 y Fq(F)13 b Ft(\()p Fq(z)t Ft(\))26 b(=)f Fq(F)13 b Ft(\(0\))22 b(+)1462 690 y Fr(\000j)p Fq(\025)1611 657 y Fh(\003)1611 712 y Fu(1)1653 690 y Fr(j)p Fq(z)1724 657 y Fu(2)1720 712 y(1)1786 690 y Ft(+)e Fq(\025)1930 657 y Fh(\003)1930 712 y Fu(2)1971 690 y Fq(z)2017 657 y Fu(2)2013 712 y(2)2074 690 y Fr(\001)15 b(\001)g(\001)21 b Ft(+)f Fq(\025)2344 657 y Fh(\003)2344 715 y Fo(d)2386 690 y Fq(z)2432 657 y Fu(2)2428 715 y Fo(d)p 1462 730 1011 4 v 1945 813 a Ft(2)2504 751 y(+)g Fq(O)s Ft(\()p Fr(k)p Fq(z)t Fr(k)2838 714 y Fu(3)2838 774 y(2)2880 751 y Ft(\))562 b(\(5)p Fq(:)p Ft(11\))50 977 y(and)30 b(th)m(us)610 1122 y Fq(F)13 b Ft(\()p Fq(z)t Ft(\))21 b Fr(\000)f Ft(2)p Fq(F)13 b Ft(\()p Fq(z)1102 1136 y Fu(1)1144 1122 y Fq(;)i Ft(0\))27 b(=)e Fr(\000)p Fq(F)13 b Ft(\(0\))21 b(+)1767 1061 y Fr(j)p Fq(\025)1845 1028 y Fh(\003)1845 1083 y Fu(1)1887 1061 y Fr(j)p Fq(z)1958 1028 y Fu(2)1954 1083 y(1)2019 1061 y Ft(+)f Fq(\025)2163 1028 y Fh(\003)2163 1083 y Fu(2)2205 1061 y Fq(z)2251 1028 y Fu(2)2247 1083 y(2)2307 1061 y Fr(\001)15 b(\001)g(\001)21 b Ft(+)f Fq(\025)2577 1028 y Fh(\003)2577 1086 y Fo(d)2620 1061 y Fq(z)2666 1028 y Fu(2)2662 1086 y Fo(d)p 1767 1101 941 4 v 2214 1185 a Ft(2)2738 1122 y(+)g Fq(O)s Ft(\()p Fr(k)p Fq(z)t Fr(k)3072 1085 y Fu(3)3072 1145 y(2)3114 1122 y Ft(\))328 b(\(5)p Fq(:)p Ft(12\))50 1329 y(But)30 b(on)g Fq(C)421 1343 y Fo(\016)461 1329 y Ft(,)h Fr(k)p Fq(z)t Fr(k)653 1343 y Fu(2)720 1329 y Fr(\024)25 b Fq(C)888 1296 y Fh(0)913 1329 y Fq(\016)34 b Ft(and)c(if)f(w)m(e)h(c) m(ho)s(ose)i Fq(\016)d Ft(=)c Fq(K)1918 1252 y Fi(p)p 2008 1252 231 4 v 2008 1329 a Fq(\017)p Fr(j)15 b Ft(ln)g Fq(\017)p Fr(j)30 b Ft(for)g(some)h(constan)m(t)g Fq(K)7 b Ft(,)30 b(the)h(n)m(umerator)50 1475 y(in)e(\(5.10\))k(satis\014es)c (the)i(b)s(ound)129 1599 y Fi(Z)180 1805 y Fo(C)256 1722 y Fq(dz)t(e)391 1685 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(z)s Fu(\))p Fo(=\017)671 1722 y Fq(e)713 1685 y Fu(2)p Fo(F)g Fu(\()p Fo(z)868 1694 y Fk(1)904 1685 y Fo(;)p Fu(0\))p Fo(=\017)1088 1722 y Fr(\024)25 b Fq(e)1226 1685 y Fh(\000)p Fo(F)10 b Fu(\(0\))p Fo(=\017)1504 1722 y Fq(e)1546 1685 y Fo(C)t(\017)1632 1657 y Fk(1)p Fj(=)p Fk(2)1733 1685 y Fh(j)i Fu(ln)h Fo(\017)p Fh(j)1892 1657 y Fk(3)p Fj(=)p Fk(2)2011 1599 y Fi(Z)2062 1805 y Fb(R)2112 1787 y Fj(d)2166 1722 y Ft(exp)2320 1594 y Fi(\022)2387 1722 y Fr(\000)2469 1661 y(j)p Fq(\025)2547 1628 y Fh(\003)2547 1684 y Fu(1)2588 1661 y Fr(j)p Fq(z)2659 1628 y Fu(2)2655 1684 y(1)2721 1661 y Ft(+)20 b Fr(\001)15 b(\001)g(\001)21 b Ft(+)f Fq(\025)3082 1628 y Fh(\003)3082 1687 y Fo(d)3124 1661 y Fq(z)3170 1628 y Fu(2)3166 1687 y Fo(d)p 2468 1702 744 4 v 2799 1785 a Ft(2)p Fq(\017)3222 1594 y Fi(\023)3304 1722 y Fq(dz)1088 1987 y Ft(=)25 b Fq(e)1226 1950 y Fh(\000)p Fo(F)10 b Fu(\(0\))p Fo(=\017)1582 1926 y Ft(\(2)p Fq(\031)s(\017)p Ft(\))1789 1893 y Fo(d=)p Fu(2)p 1515 1967 460 4 v 1515 1999 a Fi(Q)1600 2022 y Fo(d)1600 2095 y(i)p Fu(=1)1738 1991 y Fi(p)p 1829 1991 146 4 v 76 x Fr(j)p Fq(\025)1907 2036 y Fh(\003)1907 2093 y Fo(i)1949 2067 y Fr(j)2000 1887 y Fi(\020)2055 1987 y Ft(1)20 b(+)g Fq(O)2298 1887 y Fi(\020)2352 1987 y Fq(\017)2389 1950 y Fu(1)p Fo(=)p Fu(2)2504 1987 y Fr(j)15 b Ft(ln)f Fq(\017)p Fr(j)2697 1950 y Fu(3)p Fo(=)p Fu(2)2811 1887 y Fi(\021)q(\021)3477 1864 y Ft(\(5)p Fq(:)p Ft(13\))50 2249 y(Similarly)-8 b(,)27 b(the)k(in)m(tegral)f(in)f(the)h(denominator)g(is)f(b)s(ounded)g (from)g(b)s(elo)m(w)h(b)m(y)137 2399 y Fi(Z)228 2422 y Fo(\016)r(=)300 2363 y Fr(p)p 376 2363 125 4 v 59 x Fh(j)p Fo(\025)441 2400 y Fd(\003)441 2444 y Fk(1)479 2422 y Fh(j)187 2606 y(\000)p Fo(\016)r(=)316 2547 y Fr(p)p 392 2547 V 59 x Fh(j)p Fo(\025)457 2584 y Fd(\003)457 2628 y Fk(1)495 2606 y Fh(j)536 2523 y Fq(e)578 2486 y Fo(F)10 b Fu(\()p Fo(t;)p Fu(0\))p Fo(=\017)849 2523 y Fq(dt)25 b Fr(\025)g Fq(e)1092 2486 y Fh(\000)p Fo(C)t(\017)1235 2458 y Fk(1)p Fj(=)p Fk(2)1336 2486 y Fh(j)12 b Fu(ln)h Fo(\017)p Fh(j)1495 2458 y Fk(3)p Fj(=)p Fk(2)1600 2523 y Fq(e)1642 2486 y Fu(+)p Fo(F)d Fu(\(0\))p Fo(=\017)1934 2368 y Fi( )2017 2462 y Ft(\(2)p Fq(\031)s(\017)p Ft(\))2224 2429 y Fu(1)p Fo(=)p Fu(2)p 2017 2502 323 4 v 2060 2520 a Fi(p)p 2151 2520 146 4 v 77 x Fr(j)p Fq(\025)2229 2566 y Fh(\003)2229 2622 y Fu(1)2271 2597 y Fr(j)2370 2523 y(\000)20 b Ft(2)2521 2399 y Fi(Z)2613 2422 y Fh(1)2572 2606 y Fo(\016)r(=)2644 2547 y Fr(p)p 2721 2547 125 4 v 2721 2606 a Fh(j)p Fo(\025)2786 2584 y Fd(\003)2786 2628 y Fk(1)2823 2606 y Fh(j)2865 2523 y Fq(dt)15 b(e)3002 2486 y Fh(\000j)p Fo(\025)3124 2458 y Fd(\003)3124 2504 y Fk(1)3162 2486 y Fh(j)p Fo(t)3211 2458 y Fk(2)3246 2486 y Fo(=\017)3319 2368 y Fi(!)858 2832 y Fr(\025)p Fq(e)971 2795 y Fh(\000)p Fo(C)t(\017)1114 2767 y Fk(1)p Fj(=)p Fk(2)1215 2795 y Fh(j)d Fu(ln)h Fo(\017)p Fh(j)1374 2767 y Fk(3)p Fj(=)p Fk(2)1478 2832 y Fq(e)1520 2795 y Fu(+)p Fo(F)d Fu(\(0\))p Fo(=\017)1813 2677 y Fi( )1896 2771 y Ft(\(2)p Fq(\031)s(\017)p Ft(\))2103 2738 y Fu(1)p Fo(=)p Fu(2)p 1896 2811 323 4 v 1939 2830 a Fi(p)p 2030 2830 146 4 v 76 x Fr(j)p Fq(\025)2108 2875 y Fh(\003)2108 2931 y Fu(1)2150 2906 y Fr(j)2249 2832 y(\000)2356 2771 y Fq(e)2398 2738 y Fh(\000)p Fo(\016)2490 2710 y Fk(2)2526 2738 y Fo(=\017)p 2351 2811 252 4 v 2351 2896 a Fq(\016)s(\017)2431 2870 y Fh(\000)p Fu(1)p Fo(=)p Fu(2)2614 2677 y Fi(!)954 3126 y Ft(=)25 b Fq(e)1092 3089 y Fu(+)p Fo(F)10 b Fu(\(0\))p Fo(=\017)1392 2990 y Fr(p)p 1468 2990 138 4 v 75 x Ft(2)p Fq(\031)s(\017)p 1380 3105 237 4 v 1380 3124 a Fi(p)p 1471 3124 146 4 v 77 x Fr(j)p Fq(\025)1549 3169 y Fh(\003)1549 3225 y Fu(1)1591 3201 y Fr(j)1643 3025 y Fi(\020)1697 3126 y Ft(1)21 b(+)e Fq(O)1941 3025 y Fi(\020)1995 3126 y Fq(\017)2032 3089 y Fu(1)p Fo(=)p Fu(2)2146 3126 y Fr(j)c Ft(ln)f Fq(\017)p Fr(j)2339 3089 y Fu(3)p Fo(=)p Fu(2)2454 3025 y Fi(\021\021)3477 2815 y Ft(\(5)p Fq(:)p Ft(14\))50 3386 y(Com)m(bining)28 b(the)j(estimates)g(\(5.13\),)h (\(5.14\),)h(and)d(\(5.9\),)i(w)m(e)f(arriv)m(e)f(at)h(the)g(upp)s(er)d (b)s(ound)353 3628 y(\010)419 3642 y Fo(C)471 3652 y Fj(\016)511 3628 y Ft(\()p Fq(h)598 3591 y Fu(+)659 3628 y Ft(\))e Fr(\024)f Fq(e)858 3591 y Fh(\000)p Fo(F)10 b Fu(\(0\))p Fo(=\017)1136 3628 y Ft(\(2)p Fq(\031)s(\017)p Ft(\))1343 3591 y Fo(d=)p Fu(2)1777 3567 y Fr(j)p Fq(\025)1855 3534 y Fh(\003)1855 3589 y Fu(1)1897 3567 y Fr(j)p 1471 3607 758 4 v 1471 3703 a Ft(2)p Fq(\031)1571 3625 y Fi(p)p 1662 3625 567 4 v 78 x Fr(j)15 b Ft(det)q(\()p Fr(r)1940 3677 y Fu(2)1981 3703 y Fq(F)e Ft(\(0\)\))p Fr(j)2255 3527 y Fi(\020)2309 3628 y Ft(1)21 b(+)f Fq(O)2553 3527 y Fi(\020)2607 3628 y Fq(\017)2644 3591 y Fu(1)p Fo(=)p Fu(2)2758 3628 y Fr(j)15 b Ft(ln)f Fq(\017)p Fr(j)2951 3591 y Fu(3)p Fo(=)p Fu(2)3066 3527 y Fi(\021\021)3477 3628 y Ft(\(5)p Fq(:)p Ft(15\))50 3883 y(Since)30 b(this)g(results)g (coincides)h(with)f(the)h(heuristic)f(results,)g(w)m(e)i(ma)m(y)g(exp)s (ect)g(to)g(get)g(a)g(corresp)s(onding)50 4029 y(lo)m(w)m(er)e(b)s (ound.)50 4247 y Fs(The)k(lo)m(w)m(er)h(b)s(ound.)41 b Ft(F)-8 b(or)31 b(the)g(lo)m(w)m(er)f(b)s(ound)f(w)m(e)i(will)c (consider)j(a)g(di\013eren)m(t)g(domain)573 4480 y Fi(b)555 4503 y Fq(C)620 4517 y Fo(\016)685 4503 y Fr(\021)781 4375 y Fi(\024)829 4503 y Fr(\000)p Ft(2)p Fq(\016)s(=)1033 4394 y Fi(q)p 1126 4394 146 4 v 1126 4503 a Fr(j)p Fq(\025)1204 4472 y Fh(\003)1204 4527 y Fu(1)1246 4503 y Fr(j)p Fq(;)15 b Ft(2)p Fq(\016)s(=)1444 4394 y Fi(q)p 1537 4394 V 1537 4503 a Fr(j)p Fq(\025)1615 4472 y Fh(\003)1615 4527 y Fu(1)1657 4503 y Fr(j)1682 4375 y Fi(\025)1795 4390 y Fo(d)1746 4417 y Fi(O)1755 4610 y Fo(i)p Fu(=2)1898 4375 y Fi(\024)1946 4503 y Fr(\000)p Fq(\016)s(=)2105 4395 y Fi(q)p 2197 4395 370 4 v 2197 4503 a Ft(\()p Fq(d)21 b Fr(\000)f Ft(1\))p Fq(\025)2524 4472 y Fh(\003)2524 4529 y Fo(i)2566 4503 y Fq(;)15 b(\016)s(=)2694 4395 y Fi(q)p 2787 4395 V 2787 4503 a Ft(\()p Fq(d)21 b Fr(\000)f Ft(1\))p Fq(\025)3114 4472 y Fh(\003)3114 4529 y Fo(i)3156 4375 y Fi(\025)685 4788 y Fr(\021)781 4660 y Fi(\024)829 4788 y Fr(\000)p Ft(2)p Fq(\016)s(=)1033 4679 y Fi(q)p 1126 4679 146 4 v 1126 4788 a Fr(j)p Fq(\025)1204 4756 y Fh(\003)1204 4812 y Fu(1)1246 4788 y Fr(j)p Fq(;)15 b Ft(2)p Fq(\016)s(=)1444 4679 y Fi(q)p 1537 4679 V 1537 4788 a Fr(j)p Fq(\025)1615 4756 y Fh(\003)1615 4812 y Fu(1)1657 4788 y Fr(j)1682 4660 y Fi(\025)1751 4788 y Fr(\012)1860 4765 y Fi(b)1841 4788 y Fq(C)1913 4750 y Fh(?)1906 4810 y Fo(\016)3477 4628 y Ft(\(5)p Fq(:)p Ft(16\))50 5032 y(Let)33 b Fq(h)267 4999 y Fh(\003)343 5032 y Ft(denote)g(the)g(minimizer)e(of)i(the)g(v)-5 b(ariational)32 b(problem)f(\(5.2\),)36 b(i.e.)49 b(the)33 b(equilibrium)28 b(p)s(oten)m(tial)50 5177 y(of)i(the)h(capacitor)g(\() p Fq(B)808 5191 y Fo(x)854 5177 y Fq(;)15 b(B)963 5191 y Fo(y)1007 5177 y Ft(\).)41 b(Then)1325 5403 y(inf)1283 5462 y Fo(h)p Fh(2H)1437 5444 y Fj(x)1437 5480 y(y)1492 5403 y Ft(\010\()p Fq(h)p Ft(\))26 b(=)f(\010\()p Fq(h)1955 5365 y Fh(\003)1997 5403 y Ft(\))h Fr(\025)f Ft(\010)2229 5437 y Fi(b)2220 5441 y Fo(C)2272 5451 y Fj(\016)2312 5403 y Ft(\()p Fq(h)2399 5365 y Fh(\003)2442 5403 y Ft(\))1000 b(\(5)p Fq(:)p Ft(17\))p eop %%Page: 21 21 21 20 bop 3633 295 a Fl(21)50 513 y Ft(Ob)m(viously)-8 b(,)128 779 y(\010)203 813 y Fi(b)194 817 y Fo(C)246 827 y Fj(\016)286 779 y Ft(\()p Fq(h)p Ft(\))27 b Fr(\025)541 756 y Ft(\026)531 779 y(\010)606 813 y Fi(b)597 817 y Fo(C)649 827 y Fj(\016)689 779 y Ft(\()p Fq(h)p Ft(\))f Fr(\021)f Fq(\017)985 655 y Fi(Z)1045 857 y(b)1036 861 y Fo(C)1088 871 y Fj(\016)1143 779 y Fq(dz)t(e)1278 741 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(z)s Fu(\))p Fo(=\017)1573 651 y Fi(\022)1651 717 y Fq(@)5 b(h)p Ft(\()p Fq(z)t Ft(\))p 1651 758 223 4 v 1694 841 a Fq(@)g(z)1789 855 y Fu(1)1885 651 y Fi(\023)1952 670 y Fu(2)435 1075 y Ft(=)25 b Fq(\017)583 951 y Fi(Z)642 1154 y(b)633 1158 y Fo(C)689 1136 y Fd(?)685 1182 y Fj(\016)763 1075 y Fq(dz)852 1089 y Fh(?)929 920 y Fi( )1001 951 y(Z)1092 974 y Fu(2)p Fo(\016)r(=)1200 915 y Fr(p)p 1277 915 125 4 v 1277 974 a Fh(j)p Fo(\025)1342 952 y Fd(\003)1342 996 y Fk(1)1380 974 y Fh(j)1052 1158 y(\000)p Fu(2)p Fo(\016)r(=)1217 1099 y Fr(p)p 1293 1099 V 59 x Fh(j)p Fo(\025)1358 1136 y Fd(\003)1358 1180 y Fk(1)1396 1158 y Fh(j)1437 1075 y Fq(dz)1526 1089 y Fu(1)1568 1075 y Fq(e)1610 1038 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(z)s Fu(\))p Fo(=\017)1904 943 y Fi(\015)1904 998 y(\015)1904 1052 y(\015)1904 1107 y(\015)1966 1014 y Fq(@)5 b(h)p Ft(\()p Fq(z)2148 1028 y Fu(1)2190 1014 y Fq(;)15 b(z)2272 1028 y Fh(?)2334 1014 y Ft(\))p 1966 1054 404 4 v 2099 1138 a Fq(@)5 b(z)2194 1152 y Fu(1)2380 943 y Fi(\015)2380 998 y(\015)2380 1052 y(\015)2380 1107 y(\015)2430 966 y Fu(2)2471 920 y Fi(!)435 1393 y Fr(\025)25 b Fq(\017)583 1269 y Fi(Z)642 1471 y(b)633 1475 y Fo(C)689 1453 y Fd(?)685 1500 y Fj(\016)763 1393 y Fq(dz)852 1407 y Fh(?)929 1237 y Fi( )1467 1393 y Ft(inf)1001 1474 y Fo(f)7 b Fu(:)p Fo(f)1106 1481 y Ft(\()1142 1474 y Fh(\006)p Fo(\016)r(=)1271 1415 y Fr(p)p 1348 1415 125 4 v 1348 1474 a Fh(j)p Fo(\025)1413 1452 y Fd(\003)1413 1496 y Fk(1)1450 1474 y Fh(j)1472 1481 y Ft(\))1507 1474 y Fu(=)p Fo(h)1606 1456 y Fd(\003)1644 1481 y Ft(\()1679 1474 y Fh(\006)p Fo(\016)r(=)1808 1415 y Fr(p)p 1884 1415 V 59 x Fh(j)p Fo(\025)1949 1452 y Fd(\003)1949 1496 y Fk(1)1987 1474 y Fh(j)2009 1481 y Ft(\))2059 1269 y Fi(Z)2150 1291 y Fu(2)p Fo(\016)r(=)2258 1232 y Fr(p)p 2335 1232 V 2335 1291 a Fh(j)p Fo(\025)2400 1269 y Fd(\003)2400 1313 y Fk(1)2437 1291 y Fh(j)2110 1475 y(\000)p Fu(2)p Fo(\016)r(=)2275 1417 y Fr(p)p 2351 1417 V 58 x Fh(j)p Fo(\025)2416 1453 y Fd(\003)2416 1497 y Fk(1)2454 1475 y Fh(j)2495 1393 y Fq(dz)2584 1407 y Fu(1)2625 1393 y Fq(e)2667 1355 y Fh(\000)p Fo(F)j Fu(\()p Fo(z)s Fu(\))p Fo(=\017)2962 1393 y Fr(k)p Fq(f)3062 1355 y Fh(0)3087 1393 y Ft(\()p Fq(z)3164 1407 y Fu(1)3205 1393 y Ft(\))p Fr(k)3286 1342 y Fu(2)3327 1237 y Fi(!)3477 1093 y Ft(\(5)p Fq(:)p Ft(18\))50 1674 y(The)38 b(minimization)e (problem)h(for)h(\014xed)g(v)-5 b(alues)38 b(of)g Fq(z)1973 1688 y Fh(?)2073 1674 y Ft(is)g(of)h(course)g(the)f(solution)f(of)i (the)g(Diric)m(hlet)50 1819 y(problem)704 1850 y Fi(\022)771 1978 y Fr(\000)p Fq(\017)931 1916 y(d)p 890 1957 131 4 v 890 2040 a(dz)979 2054 y Fu(1)1051 1978 y Ft(+)1194 1916 y Fq(d)p 1153 1957 V 1153 2040 a(dz)1242 2054 y Fu(1)1294 1978 y Fq(F)13 b Ft(\()p Fq(z)1442 1992 y Fu(1)1484 1978 y Ft(;)i Fq(z)1566 1992 y Fh(?)1628 1978 y Ft(\))1663 1850 y Fi(\023)1798 1916 y Fq(d)p 1756 1957 V 1756 2040 a(dz)1845 2054 y Fu(1)1897 1978 y Fq(f)10 b Ft(\()p Fq(z)2029 1992 y Fu(1)2070 1978 y Ft(\))26 b(=)f(0)1460 2232 y Fq(f)1530 2104 y Fi(\022)1597 2232 y Fr(\000)p Ft(2)p Fq(\016)s(=)1801 2124 y Fi(q)p 1893 2124 146 4 v 1893 2232 a Fr(j)p Fq(\025)1971 2201 y Fh(\003)1971 2257 y Fu(1)2013 2232 y Fr(j)2038 2104 y Fi(\023)2131 2232 y Ft(=)g Fq(h)2279 2195 y Fh(\003)2336 2104 y Fi(\022)2403 2232 y Fr(\000)p Ft(2)p Fq(\016)s(=)2607 2124 y Fi(q)p 2699 2124 V 2699 2232 a Fr(j)p Fq(\025)2777 2201 y Fh(\003)2777 2257 y Fu(1)2819 2232 y Fr(j)q Fq(;)15 b(z)2927 2246 y Fh(?)2989 2104 y Fi(\023)1460 2487 y Fq(f)1530 2359 y Fi(\022)1597 2487 y Ft(+2)p Fq(\016)s(=)1801 2378 y Fi(q)p 1893 2378 V 1893 2487 a Fr(j)p Fq(\025)1971 2456 y Fh(\003)1971 2511 y Fu(1)2013 2487 y Fr(j)2038 2359 y Fi(\023)2131 2487 y Ft(=)25 b Fq(h)2279 2449 y Fh(\003)2336 2359 y Fi(\022)2403 2487 y Ft(2)p Fq(\016)s(=)2536 2378 y Fi(q)p 2628 2378 V 2628 2487 a Fr(j)p Fq(\025)2706 2456 y Fh(\003)2706 2511 y Fu(1)2749 2487 y Fr(j)p Fq(;)15 b(z)2856 2501 y Fh(?)2918 2359 y Fi(\023)3477 2232 y Ft(\(5)p Fq(:)p Ft(19\))50 2721 y(The)30 b(solution)g(of)h(this)e (Diric)m(hlet)h(problem)g(is)f(readily)h(obtained:)41 b(let)31 b(us)f(set)h Fq(a)26 b Ft(=)g Fq(h)3011 2688 y Fh(\003)3053 2721 y Ft(\()p Fr(\000)p Ft(2)p Fq(\016)s(=)3292 2644 y Fi(p)p 3385 2644 V 3385 2721 a Fr(j)p Fq(\025)3463 2690 y Fh(\003)3463 2746 y Fu(1)3505 2721 y Fr(j)p Fq(;)15 b(z)3612 2735 y Fh(?)3674 2721 y Ft(\))50 2867 y(and)37 b Fq(b)g Ft(=)f Fq(h)469 2834 y Fh(\003)511 2867 y Ft(\(2)p Fq(\016)s(=)679 2790 y Fi(p)p 772 2790 V 772 2867 a Fr(j)p Fq(\025)850 2836 y Fh(\003)850 2891 y Fu(1)893 2867 y Fr(j)p Fq(;)15 b(z)1000 2881 y Fh(?)1062 2867 y Ft(\),)39 b(and)e Fq(g)s Ft(\()p Fq(z)1468 2881 y Fu(1)1510 2867 y Ft(\))h(=)e Fq(F)13 b Ft(\()p Fq(z)1838 2881 y Fu(1)1880 2867 y Ft(;)i Fq(z)1962 2881 y Fh(?)2024 2867 y Ft(\).)62 b(The)37 b(general)g(solution)f(of)i(the)f(di\013eren)m(tial)50 3012 y(equation)30 b(in)f(\(5.19\))k(is)1457 3173 y Fq(f)10 b Ft(\()p Fq(z)1589 3187 y Fu(1)1630 3173 y Ft(\))25 b(=)g Fq(c)1840 3049 y Fi(Z)1932 3072 y Fo(s)1891 3256 y(z)1926 3265 y Fk(1)1986 3173 y Fq(e)2028 3135 y Fo(g)r Fu(\()p Fo(t)p Fu(\))p Fo(=\017)2222 3173 y Fq(dt)1175 b Ft(\(5)p Fq(:)p Ft(20\))50 3392 y(where)30 b(the)g(constan)m(ts)i Fq(c)e Ft(and)g Fq(s)g Ft(are)h(determined)e(b)m(y)h(the)h(b)s(oundary) d(conditions,)h(i.e.)1392 3636 y Fq(c)1446 3513 y Fi(Z)1537 3535 y Fo(s)1497 3719 y Fh(\000)p Fu(2)p Fo(\016)r(=)1662 3660 y Fr(p)p 1738 3660 125 4 v 59 x Fh(j)p Fo(\025)1803 3697 y Fd(\003)1803 3741 y Fk(1)1841 3719 y Fh(j)1882 3636 y Fq(e)1924 3599 y Fo(g)r Fu(\()p Fo(t)p Fu(\))p Fo(=\017)2119 3636 y Fq(dt)c Ft(=)g Fq(a)1392 3913 y(c)1446 3789 y Fi(Z)1537 3811 y Fo(s)1497 3995 y Fu(2)p Fo(\016)r(=)1605 3937 y Fr(p)p 1681 3937 V 58 x Fh(j)p Fo(\025)1746 3974 y Fd(\003)1746 4017 y Fk(1)1784 3995 y Fh(j)1825 3913 y Fq(e)1867 3875 y Fo(g)r Fu(\()p Fo(t)p Fu(\))p Fo(=\017)2062 3913 y Fq(dt)g Ft(=)g Fq(b)3477 3788 y Ft(\(5)p Fq(:)p Ft(21\))50 4185 y(from)30 b(whic)m(h)f(w)m(e)i(get)g(that)1416 4331 y Fq(c)26 b Ft(=)1936 4269 y Fq(a)p 1588 4310 745 4 v 1588 4326 a Fi(R)1648 4348 y Fo(s)1631 4431 y Fh(\000)p Fu(2)p Fo(\016)r(=)1796 4372 y Fr(p)p 1872 4372 125 4 v 59 x Fh(j)p Fo(\025)1937 4409 y Fd(\003)1937 4453 y Fk(1)1975 4431 y Fh(j)2016 4399 y Fq(e)2058 4373 y Fo(g)r Fu(\()p Fo(t)p Fu(\))p Fo(=\017)2252 4399 y Fq(dt)3477 4331 y Ft(\(5)p Fq(:)p Ft(22\))50 4587 y(while)i Fq(s)i Ft(is)g(determined)f(through)g(the)i(equation)1440 4710 y Fi(R)1501 4732 y Fo(s)1483 4816 y Fu(2)p Fo(\016)r(=)1591 4757 y Fr(p)p 1668 4757 V 1668 4816 a Fh(j)p Fo(\025)1733 4794 y Fd(\003)1733 4837 y Fk(1)1770 4816 y Fh(j)1812 4783 y Fq(e)1854 4750 y Fo(g)r Fu(\()p Fo(t)p Fu(\))p Fo(=\017)2048 4783 y Fq(dt)p 1012 4859 1545 4 v 1012 4922 a Fi(R)1073 4936 y Fu(2)p Fo(\016)r(=)1181 4877 y Fr(p)p 1257 4877 125 4 v 59 x Fh(j)p Fo(\025)1322 4914 y Fd(\003)1322 4957 y Fk(1)1360 4936 y Fh(j)1055 5045 y(\000)p Fu(2)p Fo(\016)r(=)1220 4986 y Fr(p)p 1297 4986 V 1297 5045 a Fh(j)p Fo(\025)1362 5023 y Fd(\003)1362 5067 y Fk(1)1399 5045 y Fh(j)1441 4995 y Fq(e)1483 4969 y Fo(g)r Fu(\()p Fo(t)p Fu(\))p Fo(=\017)1677 4995 y Fq(dt)20 b Ft(+)1868 4922 y Fi(R)1929 4945 y Fo(s)1911 5028 y Fu(2)p Fo(\016)r(=)2019 4969 y Fr(p)p 2096 4969 V 2096 5028 a Fh(j)p Fo(\025)2161 5006 y Fd(\003)2161 5050 y Fk(1)2198 5028 y Fh(j)2240 4995 y Fq(e)2282 4969 y Fo(g)r Fu(\()p Fo(t)p Fu(\))p Fo(=\017)2476 4995 y Fq(dt)2593 4879 y Ft(=)2704 4818 y Fq(b)p 2699 4859 49 4 v 2699 4942 a(a)3477 4879 y Ft(\(5)p Fq(:)p Ft(23\))50 5210 y(or)978 5263 y Fi(Z)1068 5286 y Fo(s)1028 5470 y Fu(2)p Fo(\016)r(=)1136 5411 y Fr(p)p 1213 5411 125 4 v 1213 5470 a Fh(j)p Fo(\025)1278 5448 y Fd(\003)1278 5492 y Fk(1)1315 5470 y Fh(j)1357 5387 y Fq(e)1399 5349 y Fo(g)r Fu(\()p Fo(t)p Fu(\))p Fo(=\017)1593 5387 y Fq(dt)25 b Ft(=)1885 5325 y Fq(b)p 1805 5366 199 4 v 1805 5449 a(a)c Fr(\000)f Fq(b)2030 5263 y Fi(Z)2120 5286 y Fu(2)p Fo(\016)r(=)2228 5227 y Fr(p)p 2305 5227 125 4 v 2305 5286 a Fh(j)p Fo(\025)2370 5264 y Fd(\003)2370 5308 y Fk(1)2408 5286 y Fh(j)2080 5470 y(\000)p Fu(2)p Fo(\016)r(=)2245 5411 y Fr(p)p 2321 5411 V 59 x Fh(j)p Fo(\025)2386 5448 y Fd(\003)2386 5492 y Fk(1)2424 5470 y Fh(j)2466 5387 y Fq(e)2508 5349 y Fo(g)r Fu(\()p Fo(t)p Fu(\))p Fo(=\017)2702 5387 y Fq(dt)695 b Ft(\(5)p Fq(:)p Ft(24\))p eop %%Page: 22 22 22 21 bop 50 295 a Fl(22)50 513 y Ft(and)30 b(th)m(us)949 566 y Fi(Z)1040 589 y Fo(s)1000 773 y Fh(\000)p Fu(2)p Fo(\016)r(=)1165 714 y Fr(p)p 1241 714 125 4 v 59 x Fh(j)p Fo(\025)1306 751 y Fd(\003)1306 795 y Fk(1)1344 773 y Fh(j)1385 690 y Fq(e)1427 652 y Fo(g)r Fu(\()p Fo(t)p Fu(\))p Fo(=\017)1621 690 y Fq(dt)c Ft(=)1909 628 y Fq(a)p 1834 669 199 4 v 1834 752 a(a)20 b Fr(\000)g Fq(b)2058 566 y Fi(Z)2149 589 y Fu(2)p Fo(\016)r(=)2257 530 y Fr(p)p 2333 530 125 4 v 59 x Fh(j)p Fo(\025)2398 567 y Fd(\003)2398 610 y Fk(1)2436 589 y Fh(j)2108 773 y(\000)p Fu(2)p Fo(\016)r(=)2273 714 y Fr(p)p 2350 714 V 2350 773 a Fh(j)p Fo(\025)2415 751 y Fd(\003)2415 795 y Fk(1)2453 773 y Fh(j)2494 690 y Fq(e)2536 652 y Fo(g)r Fu(\()p Fo(t)p Fu(\))p Fo(=\017)2730 690 y Fq(dt)667 b Ft(\(5)p Fq(:)p Ft(25\))50 943 y(Inserting)29 b(this)g(solution)g(in)m(to)h(\(5.18\))j(yields)129 1302 y(\010)204 1336 y Fi(b)195 1340 y Fo(C)247 1350 y Fj(\016)287 1302 y Ft(\()p Fq(h)374 1264 y Fh(\003)417 1302 y Ft(\))25 b Fr(\025)g Fq(\017)625 1178 y Fi(Z)685 1380 y(b)676 1384 y Fo(C)732 1363 y Fd(?)728 1409 y Fj(\016)806 1302 y Fq(dz)895 1316 y Fh(?)972 1178 y Fi(Z)1063 1201 y Fu(2)p Fo(\016)r(=)1171 1142 y Fr(p)p 1247 1142 V 59 x Fh(j)p Fo(\025)1312 1179 y Fd(\003)1312 1223 y Fk(1)1350 1201 y Fh(j)1022 1385 y(\000)p Fu(2)p Fo(\016)r(=)1187 1326 y Fr(p)p 1264 1326 V 1264 1385 a Fh(j)p Fo(\025)1329 1363 y Fd(\003)1329 1407 y Fk(1)1367 1385 y Fh(j)1408 1302 y Fq(dz)1497 1316 y Fu(1)1549 1207 y Fq(e)1591 1174 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(z)1767 1183 y Fk(1)1803 1174 y Fo(;z)1860 1184 y Fd(?)1913 1174 y Fu(\))p Fo(=\017)2029 1106 y Fi(\020)2083 1207 y Fq(h)2135 1174 y Fh(\003)2177 1207 y Ft(\()p Fr(\000)p Ft(2)p Fq(\016)s(=)2416 1130 y Fi(p)p 2509 1130 146 4 v 2509 1207 a Fr(j)p Fq(\025)2587 1176 y Fh(\003)2587 1232 y Fu(1)2629 1207 y Fr(j)p Fq(;)15 b(z)2736 1221 y Fh(?)2798 1207 y Ft(\))2833 1106 y Fi(\021)2888 1125 y Fu(2)2944 1207 y Fq(e)2986 1174 y Fu(2)p Fo(F)10 b Fu(\()p Fo(z)3141 1183 y Fk(1)3177 1174 y Fo(;z)3234 1184 y Fd(?)3287 1174 y Fu(\))p Fo(=\017)p 1549 1281 1839 4 v 1944 1314 a Fi(\022)2011 1369 y(R)2072 1392 y Fo(s)p Fu(\()p Fo(z)2169 1402 y Fd(?)2223 1392 y Fu(\))2054 1488 y Fh(\000)p Fu(2)p Fo(\016)r(=)2219 1429 y Fr(p)p 2296 1429 125 4 v 2296 1488 a Fh(j)p Fo(\025)2361 1466 y Fd(\003)2361 1510 y Fk(1)2398 1488 y Fh(j)2440 1442 y Fq(e)2482 1416 y Fo(F)g Fu(\()p Fo(t;z)2650 1426 y Fd(?)2704 1416 y Fu(\))p Fo(=\017)2804 1442 y Fq(dt)2884 1314 y Fi(\023)2951 1333 y Fu(2)477 1786 y Ft(=)25 b Fq(\017)621 1615 y Fi(R)673 1722 y(b)664 1726 y Fo(C)720 1705 y Fd(?)716 1751 y Fj(\016)794 1688 y Fq(dz)883 1702 y Fh(?)960 1587 y Fi(\020)1014 1688 y Fq(h)1066 1655 y Fh(\003)1124 1587 y Fi(\020)1178 1688 y Fr(\000)p Ft(2)p Fq(\016)t(=)1383 1611 y Fi(p)p 1474 1611 146 4 v 77 x Fr(j)p Fq(\025)1552 1657 y Fh(\003)1552 1712 y Fu(1)1594 1688 y Fr(j)q Fq(;)15 b(z)1702 1702 y Fh(?)1764 1587 y Fi(\021)1838 1688 y Fr(\000)20 b Fq(h)1981 1655 y Fh(\003)2038 1587 y Fi(\020)2093 1688 y Ft(2)p Fq(\016)t(=)2227 1611 y Fi(p)p 2318 1611 V 77 x Fr(j)p Fq(\025)2396 1657 y Fh(\003)2396 1712 y Fu(1)2439 1688 y Fr(j)p Fq(;)15 b(z)2546 1702 y Fh(?)2608 1587 y Fi(\021\021)2716 1606 y Fu(2)p 621 1766 2137 4 v 1252 1829 a Fi(R)1313 1843 y Fu(2)p Fo(\016)r(=)1421 1784 y Fr(p)p 1498 1784 125 4 v 1498 1843 a Fh(j)p Fo(\025)1563 1821 y Fd(\003)1563 1864 y Fk(1)1600 1843 y Fh(j)1295 1952 y(\000)p Fu(2)p Fo(\016)r(=)1460 1893 y Fr(p)p 1537 1893 V 1537 1952 a Fh(j)p Fo(\025)1602 1930 y Fd(\003)1602 1974 y Fk(1)1640 1952 y Fh(j)1681 1902 y Fq(e)1723 1876 y Fo(F)10 b Fu(\()p Fo(t;z)1891 1886 y Fd(?)1945 1876 y Fu(\))p Fo(=\017)2046 1902 y Fq(dt)3477 1555 y Ft(\(5)p Fq(:)p Ft(26\))50 2167 y(But)30 b(using)f(again)i(\(5.4\),)h(w)m(e)f(see)g(that)327 2352 y Fi(Z)418 2375 y Fu(2)p Fo(\016)r(=)526 2316 y Fr(p)p 602 2316 V 59 x Fh(j)p Fo(\025)667 2353 y Fd(\003)667 2397 y Fk(1)705 2375 y Fh(j)377 2559 y(\000)p Fu(2)p Fo(\016)r(=)542 2500 y Fr(p)p 619 2500 V 619 2559 a Fh(j)p Fo(\025)684 2537 y Fd(\003)684 2581 y Fk(1)722 2559 y Fh(j)763 2476 y Fq(e)805 2439 y Fo(F)10 b Fu(\()p Fo(t;z)973 2449 y Fd(?)1027 2439 y Fu(\))p Fo(=\017)1128 2476 y Fq(dt)25 b Ft(=)1329 2348 y Fi(\022)1396 2476 y Fq(e)1438 2432 y Fu(+)1506 2371 y Fi(P)1602 2393 y Fj(d)1602 2466 y(i)p Fk(=2)1732 2384 y Fj(\025)1770 2366 y Fd(\003)1770 2411 y Fj(i)1808 2384 y(z)1841 2366 y Fk(2)1839 2411 y Fj(i)p 1732 2418 145 4 v 1775 2454 a Fk(2)p Fj(\017)1912 2476 y Ft(+)20 b Fq(O)s Ft(\()p Fq(\016)2153 2439 y Fu(3)2195 2476 y Fq(=\017)p Ft(\))2312 2348 y Fi(\023)2395 2352 y(Z)2486 2375 y Fu(2)p Fo(\016)r(=)2594 2316 y Fr(p)p 2670 2316 125 4 v 59 x Fh(j)p Fo(\025)2735 2353 y Fd(\003)2735 2397 y Fk(1)2773 2375 y Fh(j)2445 2559 y(\000)p Fu(2)p Fo(\016)r(=)2610 2500 y Fr(p)p 2687 2500 V 2687 2559 a Fh(j)p Fo(\025)2752 2537 y Fd(\003)2752 2581 y Fk(1)2790 2559 y Fh(j)2831 2476 y Fq(dte)2953 2439 y Fh(\000)3021 2392 y Fd(j)p Fj(\025)3080 2373 y Fd(\003)3080 2417 y Fk(1)3118 2392 y Fd(j)p Fj(t)3165 2373 y Fk(2)p 3021 2424 179 4 v 3081 2460 a(2)p Fj(\017)337 2768 y Fr(\024)455 2631 y(p)p 531 2631 138 4 v 75 x Ft(2)p Fq(\031)s(\017)p 444 2747 237 4 v 444 2765 a Fi(p)p 535 2765 146 4 v 77 x Fr(j)p Fq(\025)613 2811 y Fh(\003)613 2866 y Fu(1)655 2842 y Fr(j)691 2768 y Fq(e)733 2724 y Fu(+)802 2663 y Fi(P)898 2685 y Fj(d)898 2758 y(i)p Fk(=2)1027 2676 y Fj(\025)1065 2658 y Fd(\003)1065 2703 y Fj(i)1104 2676 y(z)1137 2658 y Fk(2)1135 2703 y Fj(i)p 1027 2710 145 4 v 1070 2746 a Fk(2)p Fj(\017)1208 2768 y Ft(+)g Fq(O)s Ft(\()p Fq(\016)1449 2730 y Fu(3)1490 2768 y Fq(=\017)p Ft(\))3477 2612 y(\(5)p Fq(:)p Ft(27\))50 3052 y(and)30 b(so)553 3229 y(\010)628 3264 y Fi(b)619 3268 y Fo(C)671 3278 y Fj(\016)711 3229 y Ft(\()p Fq(h)798 3192 y Fh(\003)841 3229 y Ft(\))25 b Fr(\025)1008 3088 y Fi(p)p 1099 3088 183 4 v 77 x Fq(\017)p Fr(j)p Fq(\025)1214 3134 y Fh(\003)1214 3189 y Fu(1)1256 3165 y Fr(j)p 1008 3208 274 4 v 1057 3226 a(p)p 1132 3226 101 4 v 1132 3302 a Ft(2)p Fq(\031)1307 3229 y Ft(exp)1461 3074 y Fi( )1533 3229 y Fr(\000)1666 3116 y Fo(d)1619 3143 y Fi(X)1626 3336 y Fo(i)p Fu(=2)1777 3168 y Fq(\025)1830 3135 y Fh(\003)1830 3191 y Fo(i)1871 3168 y Fq(z)1917 3135 y Fu(2)1913 3191 y Fo(i)p 1777 3208 182 4 v 1826 3292 a Ft(2)p Fq(\017)1990 3229 y Ft(+)19 b Fq(O)s Ft(\()p Fq(\016)2230 3192 y Fu(3)2272 3229 y Fq(=\017)p Ft(\))2389 3074 y Fi(!)896 3532 y Fr(\002)987 3409 y Fi(Z)1047 3611 y(b)1038 3615 y Fo(C)1094 3593 y Fd(?)1090 3639 y Fj(\016)1168 3532 y Fq(dz)1257 3546 y Fh(?)1334 3404 y Fi(\022)1401 3532 y Fq(h)1453 3495 y Fh(\003)1510 3404 y Fi(\022)1577 3532 y Fr(\000)p Ft(2)p Fq(\016)t(=)1782 3424 y Fi(q)p 1873 3424 146 4 v 108 x Fr(j)p Fq(\025)1951 3501 y Fh(\003)1951 3557 y Fu(1)1993 3532 y Fr(j)q Fq(;)c(z)2101 3546 y Fh(?)2163 3404 y Fi(\023)2250 3532 y Fr(\000)20 b Fq(h)2393 3495 y Fh(\003)2450 3404 y Fi(\022)2517 3532 y Ft(2)p Fq(\016)t(=)2651 3424 y Fi(q)p 2743 3424 V 2743 3532 a Fr(j)p Fq(\025)2821 3501 y Fh(\003)2821 3557 y Fu(1)2863 3532 y Fr(j)p Fq(;)15 b(z)2970 3546 y Fh(?)3032 3404 y Fi(\023\023)3166 3423 y Fu(2)3477 3374 y Ft(\(5)p Fq(:)p Ft(28\))50 3781 y(No)m(w)31 b(w)m(e)g(use)f(the)g(fact)i(that)f Fq(h)1133 3748 y Fh(\003)1175 3781 y Ft(\()p Fq(z)t Ft(\))26 b(=)f Fp(P)1468 3795 y Fo(z)1510 3781 y Ft([)p Fq(\034)1575 3795 y Fo(B)1630 3804 y Fj(x)1701 3781 y Fq(<)g(\034)1837 3795 y Fo(B)1892 3804 y Fj(y)1935 3781 y Ft(])h(=)f Fq(h)2134 3795 y Fo(B)2189 3804 y Fj(x)2230 3795 y Fo(;B)2307 3804 y Fj(y)2350 3781 y Ft(\()p Fq(z)t Ft(\).)42 b(Then)29 b(Corollary)g(4.8)i(implies)50 3999 y Fe(Lemma)38 b(5.2:)91 b Fn(Uniformly)34 b(in)e Fq(z)1312 4013 y Fh(?)1399 3999 y Fr(2)1503 3976 y Fi(b)1485 3999 y Fq(C)1557 3966 y Fh(?)1550 4025 y Fo(\016)1618 3999 y Fn(,)1019 4276 y Ft(1)21 b Fr(\000)f Fq(h)1228 4238 y Fh(\003)1285 4147 y Fi(\022)1352 4276 y Fr(\000)p Ft(2)p Fq(\016)t(=)1557 4167 y Fi(q)p 1648 4167 V 109 x Fr(j)p Fq(\025)1726 4244 y Fh(\003)1726 4300 y Fu(1)1768 4276 y Fr(j)q Fq(;)15 b(z)1876 4290 y Fh(?)1938 4147 y Fi(\023)2030 4276 y Fr(\024)25 b Fq(C)7 b(\017)2235 4238 y Fh(\000)p Fu(1)p Fo(=)p Fu(2)2405 4276 y Fq(e)2447 4238 y Fh(\000)p Fo(\016)2539 4211 y Fk(2)2575 4238 y Fo(=)p Fu(\(4)p Fo(\017)p Fu(\))1246 4530 y Fq(h)1298 4493 y Fh(\003)1356 4402 y Fi(\022)1423 4530 y Ft(2)p Fq(\016)t(=)1557 4421 y Fi(q)p 1648 4421 V 109 x Fr(j)p Fq(\025)1726 4499 y Fh(\003)1726 4554 y Fu(1)1768 4530 y Fr(j)q Fq(;)15 b(z)1876 4544 y Fh(?)1938 4402 y Fi(\023)2030 4530 y Fr(\024)25 b Fq(C)7 b(\017)2235 4493 y Fh(\000)p Fu(1)p Fo(=)p Fu(2)2405 4530 y Fq(e)2447 4493 y Fh(\000)p Fo(\016)2539 4465 y Fk(2)2575 4493 y Fo(=)p Fu(\(4)p Fo(\017)p Fu(\))3477 4403 y Ft(\(5)p Fq(:)p Ft(29\))159 4954 y(As)30 b(an)h(immediate)e(consequence,)i(w)m(e)g(see)g(that)50 5271 y(\010)125 5305 y Fi(b)116 5309 y Fo(C)168 5319 y Fj(\016)208 5271 y Ft(\()p Fq(h)295 5233 y Fh(\003)337 5271 y Ft(\))26 b Fr(\025)494 5170 y Fi(\020)548 5271 y Ft(1)21 b Fr(\000)f Fq(C)7 b(\017)814 5233 y Fh(\000)p Fu(1)p Fo(=)p Fu(2)984 5271 y Fq(e)1026 5233 y Fh(\000)p Fo(\016)1118 5206 y Fk(2)1154 5233 y Fo(=)p Fu(\(4)p Fo(\017)p Fu(\))1319 5170 y Fi(\021)1374 5189 y Fu(2)1430 5271 y Fq(e)1472 5233 y Fh(\000)p Fo(O)r Fu(\()p Fo(\016)1649 5206 y Fk(3)1685 5233 y Fo(=\017)p Fu(\))1797 5209 y Ft(\(2)p Fq(\031)s(\017)p Ft(\))2004 5176 y Fo(d=)p Fu(2)p 1797 5250 325 4 v 1908 5333 a Ft(2)p Fq(\031)2236 5130 y Fi(p)p 2326 5130 146 4 v 2326 5207 a Fr(j)p Fq(\025)2404 5176 y Fh(\003)2404 5231 y Fu(1)2447 5207 y Fr(j)p 2143 5250 423 4 v 2143 5283 a Fi(Q)2228 5305 y Fo(d)p Fh(\000)p Fu(1)2228 5378 y Fo(i)p Fu(=2)2379 5277 y Fi(p)p 2470 5277 95 4 v 74 x Fq(\025)2523 5319 y Fh(\003)2523 5376 y Fo(i)2591 5115 y Fi( )2663 5271 y Ft(1)20 b Fr(\000)2830 5130 y Fi(p)p 2921 5130 312 4 v 78 x Fq(\017)p Ft(\()p Fq(d)h Fr(\000)f Ft(1\))p 2830 5250 403 4 v 3010 5333 a Fq(\016)3244 5271 y(e)3286 5226 y Fh(\000)3431 5202 y Fj(\016)3462 5183 y Fk(2)p 3354 5212 222 4 v 3354 5250 a(2\()p Fj(d)p Fd(\000)p Fk(1\))p Fj(\017)3591 5115 y Fi(!)3663 5134 y Fo(d)p Fh(\000)p Fu(1)3477 5462 y Ft(\(5)p Fq(:)p Ft(30\))p eop %%Page: 23 23 23 22 bop 3633 295 a Fl(23)50 513 y Ft(Cho)s(osing)24 b(as)i(b)s(efore)f Fq(\016)860 480 y Fu(2)927 513 y Ft(=)g Fq(C)7 b(\017)p Fr(j)15 b Ft(ln)f Fq(\017)p Fr(j)p Ft(,)27 b(w)m(e)f(see)g(that)g(to)h(leading)d(order)h(\(5.30\))j(coincides)d (with)f(the)i(upp)s(er)50 658 y(b)s(ound)i(\(5.15\),)33 b(whic)m(h)c(pro)m(v)m(es)i(the)g(theorem)g(in)e(the)h(case)i Fq(k)c Ft(=)d(1.)159 876 y(The)39 b(generalization)f(of)i(this)e (estimate)i(to)f(the)h(case)g(when)e(sev)m(eral)i(saddle)e(p)s(oin)m (ts)g(exist)h(on)g(the)50 1022 y(comm)m(unication)25 b(heigh)m(t)g(is)g(completely)g(straigh)m(tforw)m(ard)g(and)g(will)e(b) s(e)i(left)g(to)h(the)g(reader.)39 b(The)25 b(result)50 1167 y(is)k(the)i(form)m(ula)e(stated)j(in)d(the)h(theorem.)41 b Fr(})50 1423 y Fv(6.)55 b(Metastable)41 b(exit)f(times)g(and)h (capacities.)159 1641 y Ft(In)d(this)f(section)h(w)m(e)h(compute)g(the) f(mean)h(v)-5 b(alue)38 b(of)g(certain)g Fn(metastable)k(exit)e(times)e Ft(in)f(terms)i(of)50 1786 y(capacities.)54 b(This)33 b(will)f(b)s(e)i(largely)g(analogous)h(to)g(the)g(results)f(on)g(mean)h (transition)e(times)h(obtained)50 1932 y(in)d([BEGK1,BEGK2].)48 b(The)32 b(only)f(new)h(ingredien)m(t)f(needed)h(is)f(the)h(follo)m (wing)f(sharp)s(ening)f(of)i(\(2.27\),)50 2077 y(resp.)40 b(\(2.28\))32 b(when)e(a)h(pro)s(cess)f(starts)g(in)f(a)i(lo)s(cal)f (minim)m(um)d(of)k Fq(F)13 b Ft(.)50 2295 y Fe(Prop)s(osition)36 b(6.1:)p Fn(L)-5 b(et)32 b Fq(x)g Fn(b)-5 b(e)32 b(a)g(\(non-de)-5 b(gener)g(ate)33 b(quadr)-5 b(atic\))33 b(critic)-5 b(al)33 b(p)-5 b(oint)33 b(of)f Fq(F)44 b Fn(and)33 b(let)f Fq(A;)15 b(D)35 b Fn(b)-5 b(e)50 2441 y(close)g(d)34 b(sets.)42 b(Then)33 b(ther)-5 b(e)33 b(exists)g Fq(\013)26 b(>)f Ft(0)33 b Fn(such)g(that)900 2711 y Fp(E)954 2725 y Fo(x)1006 2711 y Fq(\034)1046 2725 y Fo(D)1137 2711 y Ft(=)1244 2570 y Fi(R)1287 2676 y Fo(D)1348 2658 y Fj(c)1402 2644 y Fq(dy)s(e)1539 2611 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(y)r Fu(\))p Fo(=\017)1820 2644 y Fq(h)1872 2660 y Fo(B)1927 2669 y Fj(\017)1960 2660 y Fu(\()p Fo(x)p Fu(\))p Fo(;D)2145 2644 y Ft(\()p Fq(y)s Ft(\))p 1244 2691 1021 4 v 1509 2774 a(cap)1660 2790 y Fo(B)1715 2799 y Fj(\017)1748 2790 y Fu(\()p Fo(x)p Fu(\))1850 2774 y Ft(\()p Fq(D)s Ft(\))2290 2611 y Fi(\020)2344 2711 y Ft(1)21 b(+)f Fq(O)s Ft(\()p Fq(\017)2645 2674 y Fo(\013=)p Fu(2)2770 2711 y Ft(\))2805 2611 y Fi(\021)3523 2711 y Ft(\(6)p Fq(:)p Ft(1\))50 2988 y Fn(and)465 3162 y Fp(E)520 3176 y Fo(x)572 3162 y Fq(\034)612 3176 y Fo(D)677 3162 y Ft(1)-23 b(I)732 3176 y Fo(\034)766 3185 y Fj(D)822 3176 y Fo(<\034)913 3185 y Fj(A)994 3162 y Ft(=)1101 3005 y Fi(R)1144 3111 y Fu(\()p Fo(A)p Fh([)p Fo(D)r Fu(\))1365 3093 y Fj(c)1420 3078 y Fq(dy)s(e)1557 3045 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(y)r Fu(\))p Fo(=\017)1838 3078 y Fq(h)1890 3095 y Fo(B)1945 3104 y Fj(\017)1978 3095 y Fu(\()p Fo(x)p Fu(\))p Fo(;D)r Fh([)p Fo(A)2267 3078 y Ft(\()p Fq(y)s Ft(\))p Fq(h)2437 3092 y Fo(D)r(;A)2580 3078 y Ft(\()p Fq(y)s Ft(\))p 1101 3141 1598 4 v 1570 3224 a(cap)1722 3241 y Fo(B)1777 3250 y Fj(\017)1809 3241 y Fu(\()p Fo(x)p Fu(\))1912 3224 y Ft(\()p Fq(A)20 b Fr([)g Fq(D)s Ft(\))2725 3061 y Fi(\020)2779 3162 y Ft(1)g(+)g Fq(O)s Ft(\()p Fq(\017)3079 3125 y Fo(\013=)p Fu(2)3205 3162 y Ft(\))3240 3061 y Fi(\021)3523 3162 y Ft(\(6)p Fq(:)p Ft(2\))50 3590 y Fc(Pro)s(of:)45 b Ft(The)25 b(pro)s(ofs)h(of)g(\(6.1\))i(and)e (\(6.2\))i(are)e(completely)g(analogous,)i(and)d(w)m(e)i(will)c(only)i (consider)h(the)50 3735 y(former.)51 b(Let)34 b(us)g(write)f Fq(w)975 3749 y Fo(D)1041 3735 y Ft(\()p Fq(y)s Ft(\))e Fr(\021)g Fp(E)1347 3749 y Fo(y)1397 3735 y Fq(\034)1437 3749 y Fo(D)1502 3735 y Ft(,)k Fq(y)f Fr(2)d Fq(D)1811 3702 y Fo(c)1848 3735 y Ft(.)51 b(Recall)34 b(that)h Fq(w)2468 3749 y Fo(D)2534 3735 y Ft(\()p Fq(y)s Ft(\))f(solv)m(es)g (the)g(inhomogeneous)50 3881 y(Diric)m(hlet)23 b(problem)g(\(2.25\))j (\(with)e Fq(A)h Ft(=)g Fr(;)p Ft(\).)40 b(W)-8 b(e)25 b(will)d(consider)h(this)g(function)g(on)i(a)f(ball)f Fq(B)3209 3895 y Fo(R)3264 3904 y Fk(0)3304 3881 y Ft(\()p Fq(x)p Ft(\),)j(where)50 4026 y Fq(x)32 b Ft(is)f(a)h(critical)f(p)s (oin)m(t)g(of)h Fq(F)13 b Ft(.)46 b(This)30 b(implies)g(that)i(for)g (some)h(constan)m(t)g Fq(K)7 b Ft(,)32 b(sup)2830 4048 y Fo(y)r Fh(2)p Fo(B)2973 4057 y Fj(R)3020 4071 y Fk(0)3060 4048 y Fu(\()p Fo(x)p Fu(\))3178 4026 y Fr(kr)p Fq(F)13 b Ft(\()p Fq(y)s Ft(\))p Fr(k)3533 4040 y Fh(1)3639 4026 y Fr(\024)50 4172 y Fq(K)7 b(R)203 4186 y Fu(0)274 4172 y Ft(\(if)30 b Fq(R)462 4186 y Fu(0)533 4172 y Ft(is)f(small\).)40 b(Th)m(us)30 b(the)g(H\177)-45 b(older)30 b(and)g(Harnac)m(k)i (inequalities)27 b(Lemmata)32 b(\(4.23\))h(and)c(\(4.22\))50 4317 y(ha)m(v)m(e)i(uniform)e(constan)m(ts)i(if)e Fq(R)1155 4331 y Fu(0)1221 4317 y Fr(\024)1317 4252 y(p)p 1393 4252 37 4 v 65 x Fq(\017)p Ft(.)159 4535 y(No)m(w)35 b(note)h(\014rst)e(that)h(due)f(to)i(\(2.26\),)i Fq(w)1620 4549 y Fo(D)1686 4535 y Ft(\()p Fq(y)s Ft(\))e(inherits)c(from)i(Lemma) h(4.6)h(the)f(uniform)e(Harnac)m(k)50 4681 y(b)s(ound)1281 4827 y(sup)1188 4907 y Fo(y)r Fh(2)p Fo(B)1331 4891 y Fd(p)p 1381 4891 28 3 v 33 x Fj(\017)1413 4907 y Fu(\()p Fo(x)p Fu(\))1526 4827 y Fq(w)1591 4841 y Fo(D)1657 4827 y Ft(\()p Fq(y)s Ft(\))26 b Fr(\024)f Fq(C)128 b Ft(inf)1984 4890 y Fo(y)r Fh(2)p Fo(B)2127 4873 y Fd(p)p 2177 4873 V 33 x Fj(\017)2209 4890 y Fu(\()p Fo(x)p Fu(\))2322 4827 y Fq(w)2387 4841 y Fo(D)2453 4827 y Ft(\()p Fq(y)s Ft(\))952 b(\(6)p Fq(:)p Ft(3\))50 5084 y(No)m(w)31 b(use)f(Lemma)g (\(4.22\))j(with)c Fq(R)d Ft(=)f Fq(\017)p Ft(,)31 b(since)e Fq(w)1779 5098 y Fo(D)1876 5084 y Ft(solv)m(es)h Fq(L)2196 5098 y Fo(\017)2231 5084 y Fq(w)2296 5098 y Fo(D)2387 5084 y Ft(=)25 b(1.)41 b(This)29 b(yields)f(that)995 5385 y(osc)1116 5401 y Fo(B)1171 5410 y Fj(\017)1204 5401 y Fu(\()p Fo(x)p Fu(\))1306 5385 y Fq(w)1371 5399 y Fo(D)1463 5385 y Fr(\024)d Fq(C)7 b(\017)1668 5347 y Fo(\013=)p Fu(2)1807 5229 y Fi( )1975 5385 y Ft(sup)1879 5465 y Fo(y)r Fh(2)p Fo(B)2022 5474 y Fj(R)2069 5488 y Fk(0)2109 5465 y Fu(\()p Fo(x)p Fu(\))2222 5385 y Fq(w)2287 5399 y Fo(D)2354 5385 y Ft(\()p Fq(y)s Ft(\))20 b(+)g Fq(R)2652 5399 y Fu(0)2693 5229 y Fi(!)3523 5385 y Ft(\(6)p Fq(:)p Ft(4\))p eop %%Page: 24 24 24 23 bop 50 295 a Fl(24)50 513 y Ft(This)28 b(implies)g(immediately)h (that)922 734 y(sup)853 814 y Fo(y)r Fh(2)p Fo(B)996 823 y Fj(\017)1029 814 y Fu(\()p Fo(x)p Fu(\))1142 734 y Fq(w)1207 748 y Fo(D)1273 734 y Ft(\()p Fq(y)s Ft(\))d Fr(\024)f Fq(w)1578 748 y Fo(D)1644 734 y Ft(\()p Fq(x)p Ft(\))c(+)f Fq(C)1950 696 y Fu(2)1990 734 y Fq(e)2032 696 y Fo(\013=)p Fu(2)2157 734 y Fq(w)2222 748 y Fo(D)2288 734 y Ft(\()p Fq(x)p Ft(\))h(+)f Fq(C)7 b(\017)2631 696 y Fu(1)p Fo(=)p Fu(2+)p Fo(\013=)p Fu(2)935 961 y Ft(inf)853 1024 y Fo(y)r Fh(2)p Fo(B)996 1033 y Fj(\017)1029 1024 y Fu(\()p Fo(x)p Fu(\))1142 961 y Fq(w)1207 975 y Fo(D)1273 961 y Ft(\()p Fq(y)s Ft(\))26 b Fr(\025)f Fq(w)1578 975 y Fo(D)1644 961 y Ft(\()p Fq(x)p Ft(\))c Fr(\000)f Fq(C)1950 923 y Fu(2)1990 961 y Fq(e)2032 923 y Fo(\013=)p Fu(2)2157 961 y Fq(w)2222 975 y Fo(D)2288 961 y Ft(\()p Fq(x)p Ft(\))h Fr(\000)f Fq(C)7 b(\017)2631 923 y Fu(1)p Fo(=)p Fu(2+)p Fo(\013=)p Fu(2)3523 871 y Ft(\(6)p Fq(:)p Ft(5\))50 1235 y(Using)29 b(these)i(estimates)g(in)e(\(2.27\))k(with)c Fq(\032)c Ft(=)g Fq(\017)30 b Ft(resp.)40 b(\(2.28\))33 b(pro)m(v)m(es)e(the)g(prop)s(osition.)p Fr(})159 1453 y Ft(By)i(the)g(preceding)f(prop)s(osition,)g(all)g(w)m(e)h(need)g(to)h (kno)m(w)f(in)e(order)i(to)h(compute)f(the)g(mean)g(arriv)-5 b(al)50 1599 y(times)30 b(are)h(the)f(capacities)h(and)f(the)g (equilibrium)c(p)s(oten)m(tial.)40 b(The)30 b(latter)h(is)f(quite)f(w)m (ell)h(con)m(trolled)g(b)m(y)50 1744 y(Prop)s(osition)d(4.3)j(and)f (the)g(rough)g(estimates)h(on)f(capacities)g(Prop)s(osition)f(4.7,)i (and)f(this)f(will)f(allo)m(w)h(us)50 1890 y(to)j(get)g(already)f (quite)g(remark)-5 b(able)30 b(form)m(ulae.)50 2108 y Fe(Theorem)45 b(6.2:)68 b Fn(L)-5 b(et)40 b Fq(x)1006 2122 y Fo(j)1044 2108 y Fn(,)g Fq(j)i Ft(=)36 b(1)p Fq(;)15 b(:)g(:)g(:)j(;)d(n)38 b Fn(b)-5 b(e)39 b(the)g(lo)-5 b(c)g(al)41 b(minima)f(of)f Fq(F)13 b Fn(.)60 b(L)-5 b(et)39 b Fr(S)2956 2122 y Fo(k)3038 2108 y Ft(=)d Fr([)3206 2075 y Fo(k)3206 2132 y(i)p Fu(=1)3328 2108 y Fq(B)3397 2122 y Fo(\032)3439 2108 y Ft(\()p Fq(x)3526 2122 y Fo(i)3557 2108 y Ft(\))j Fn(b)-5 b(e)50 2253 y(a)36 b(c)-5 b(ol)5 b(le)-5 b(ction)37 b(of)f(b)-5 b(al)5 b(ls)36 b Fq(B)923 2267 y Fo(\032)966 2253 y Ft(\()p Fq(x)1053 2267 y Fo(i)1083 2253 y Ft(\))g Fn(wher)-5 b(e)37 b Fq(\032)31 b Fr(\025)g Fq(\017)k Fn(and)i(no)f(b)-5 b(al)5 b(l)36 b(c)-5 b(ontains)38 b(any)e(other)h(minimum)f(or)h(sadd)5 b(le)50 2399 y(p)-5 b(oint)34 b(of)f Fq(F)13 b Fn(.)41 b(Assume)33 b(mor)-5 b(e)g(over)34 b(that)g(for)g(a)f(given)e Fq(j)5 b Fn(,)34 b(al)5 b(l)33 b Fq(i)25 b(>)g(k)s Fn(,)33 b Fq(i)25 b Fr(6)p Ft(=)g Fq(j)5 b Fn(,)33 b(either)962 2653 y Fq(F)13 b Ft(\()p Fq(z)1114 2616 y Fh(\003)1156 2653 y Ft(\()p Fq(x)1243 2667 y Fo(i)1274 2653 y Fq(;)i(x)1366 2667 y Fo(j)1405 2653 y Ft(\)\))21 b Fr(\000)e Fq(F)13 b Ft(\()p Fq(x)1744 2667 y Fo(j)1783 2653 y Ft(\))26 b Fq(>)f(F)13 b Ft(\()p Fq(z)2092 2616 y Fh(\003)2134 2653 y Ft(\()p Fq(x)2221 2667 y Fo(i)2252 2653 y Fq(;)i Fr(S)2347 2667 y Fo(k)2392 2653 y Ft(\)\))21 b Fr(\000)f Fq(F)13 b Ft(\()p Fq(x)2732 2667 y Fo(i)2762 2653 y Ft(\))726 b(\(6)p Fq(:)p Ft(6\))50 2908 y Fn(or)1301 3053 y Fq(F)13 b Ft(\()p Fq(z)1453 3016 y Fh(\003)1495 3053 y Ft(\()p Fq(x)1582 3067 y Fo(i)1613 3053 y Fq(;)i(x)1705 3067 y Fo(j)1744 3053 y Ft(\)\))26 b Fq(<)f(F)13 b Ft(\()p Fq(z)2088 3016 y Fh(\003)2130 3053 y Ft(\()p Fq(x)2217 3067 y Fo(i)2248 3053 y Fq(;)i Fr(S)2343 3067 y Fo(k)2388 3053 y Ft(\)\))1065 b(\(6)p Fq(:)p Ft(7\))50 3262 y Fn(Then,)33 b(for)g Fq(j)e(>)25 b(k)s Fn(,)92 3475 y Fp(E)146 3489 y Fo(x)188 3498 y Fj(j)232 3475 y Fq(\034)272 3489 y Fh(S)317 3499 y Fj(k)386 3475 y Ft(=)117 3686 y(=)474 3624 y(1)p 224 3665 546 4 v 224 3748 a(cap)375 3764 y Fo(B)430 3773 y Fj(\017)463 3764 y Fu(\()p Fo(x)532 3773 y Fj(j)566 3764 y Fu(\))599 3748 y Ft(\()p Fr(S)689 3762 y Fo(k)734 3748 y Ft(\))1198 3599 y Fi(X)796 3799 y Fo(i)p Fu(:)p Fo(F)10 b Fu(\()p Fo(z)965 3780 y Fd(\003)1002 3799 y Fu(\()p Fo(x)1071 3808 y Fj(i)1100 3799 y Fo(;)p Fh(S)1167 3809 y Fj(k)1206 3799 y Fu(\)\))p Fo(>F)g Fu(\()p Fo(z)1441 3780 y Fd(\003)1479 3799 y Fu(\()p Fo(x)1548 3808 y Fj(i)1578 3799 y Fo(;x)1641 3808 y Fj(j)1675 3799 y Fu(\)\))1910 3624 y Ft(\(2)p Fq(\031)s(\017)p Ft(\))2117 3591 y Fo(d=)p Fu(2)p 1758 3665 629 4 v 1758 3683 a Fi(p)p 1848 3683 538 4 v 1848 3761 a Ft(det)q(\()p Fr(r)2086 3734 y Fu(2)2127 3761 y Fq(F)j Ft(\()p Fq(x)2285 3775 y Fo(i)2315 3761 y Ft(\)\))2397 3686 y Fq(e)2439 3648 y Fh(\000)p Fo(F)d Fu(\()p Fo(x)2621 3657 y Fj(i)2650 3648 y Fu(\))p Fo(=\017)2751 3686 y Ft(\(1)21 b(+)f Fq(O)s Ft(\()p Fq(\017)3087 3648 y Fu(1)p Fo(=)p Fu(2)3201 3686 y Fr(j)15 b Ft(ln)f Fq(\017)p Fr(j)p Fq(;)h(\017)3471 3648 y Fo(\013=)p Fu(2)3597 3686 y Ft(\)\))3523 3903 y(\(6)p Fq(:)p Ft(8\))50 4048 y Fn(Note)37 b(that)i(the)f(sum)f (always)j(includes)d(the)h(term)g Fq(i)d Ft(=)e Fq(j)5 b Fn(.)57 b(In)38 b(p)-5 b(articular,)40 b(if)d(for)h(al)5 b(l)38 b Fq(i)d(>)e(k)s Fn(,)39 b Fq(F)13 b Ft(\()p Fq(x)3539 4062 y Fo(i)3570 4048 y Ft(\))34 b Fq(>)50 4194 y(F)13 b Ft(\()p Fq(x)208 4208 y Fo(j)247 4194 y Ft(\))p Fn(,)32 b(then)438 4480 y Fp(E)492 4494 y Fo(x)533 4503 y Fj(j)578 4480 y Fq(\034)618 4494 y Fh(S)663 4504 y Fj(k)732 4480 y Ft(=)1089 4419 y(1)p 839 4459 546 4 v 839 4542 a(cap)990 4559 y Fo(B)1045 4568 y Fj(\017)1078 4559 y Fu(\()p Fo(x)1147 4568 y Fj(j)1181 4559 y Fu(\))1214 4542 y Ft(\()p Fr(S)1304 4556 y Fo(k)1349 4542 y Ft(\))1559 4419 y(\(2)p Fq(\031)s(\017)p Ft(\))1766 4386 y Fo(d=)p Fu(2)p 1406 4459 629 4 v 1406 4477 a Fi(p)p 1497 4477 538 4 v 78 x Ft(det)q(\()p Fr(r)1735 4529 y Fu(2)1776 4555 y Fq(F)13 b Ft(\()p Fq(x)1934 4569 y Fo(i)1964 4555 y Ft(\)\))2046 4480 y Fq(e)2088 4443 y Fh(\000)p Fo(F)d Fu(\()p Fo(x)2270 4452 y Fj(j)2304 4443 y Fu(\))p Fo(=\017)2405 4480 y Ft(\(1)21 b(+)f Fq(O)s Ft(\()p Fq(\017)2741 4443 y Fu(1)p Fo(=)p Fu(2)2855 4480 y Fr(j)15 b Ft(ln)g Fq(\017)p Fr(j)p Fq(;)g(\017)3126 4443 y Fo(\013=)p Fu(2)3251 4480 y Ft(\)\))202 b(\(6)p Fq(:)p Ft(9\))50 4953 y Fs(Remark:)49 b Ft(A)32 b(transition)e(to)i(a)g (set)h Fq(D)h Ft(for)d(whic)m(h)g(\(6.9\))i(holds)d(will)f(b)s(e)i (called)g(a)h Fn(metastable)k(exit)31 b Ft(and)50 5098 y(the)f(form)m(ula)g(\(6.9\))i(is)e(the)g(mean)h(metastable)g(exit)f (time)g(from)g(the)g(minim)m(um)e Fq(j)5 b Ft(.)50 5316 y Fc(Pro)s(of:)62 b Ft(Let)37 b(us)f(consider)g(the)h(set)h(\000)1416 5330 y Fo(j)1490 5316 y Fr(\021)d(f)p Fq(y)40 b Ft(:)c Fq(F)13 b Ft(\()p Fq(y)s Ft(\))36 b Fq(>)g(F)13 b Ft(\()p Fq(z)2271 5283 y Fh(\003)2314 5316 y Ft(\()p Fq(x)2401 5330 y Fo(j)2439 5316 y Fq(;)i Fr(M)p Ft(\)\))26 b(+)e Fq(\016)s Fr(g)38 b Ft(for)f(some)g(su\016cien)m(tly)50 5462 y(small)c Fq(\016)k(>)c Ft(0.)55 b(Let)36 b(\000)823 5476 y Fo(j)861 5462 y Ft(\()p Fq(i)p Ft(\))g(denote)g(the)f(connected) h(comp)s(onen)m(t)f(of)h(\000)2521 5476 y Fo(j)2594 5462 y Ft(that)f(con)m(tains)h Fq(x)3209 5476 y Fo(i)3239 5462 y Ft(.)55 b(Note)36 b(that)p eop %%Page: 25 25 25 24 bop 3633 295 a Fl(25)50 513 y Ft(some)34 b(of)g(these)g(sets)h (ma)m(y)f(b)s(e)f(empt)m(y)-8 b(,)36 b(and)e(some)g(ma)m(y)g(coincide.) 51 b(Let)34 b Fr(f)p Ft(\000)2708 527 y Fo(j)2747 513 y Ft(\()n(~)-43 b Fq(\023)p Ft(\))p Fr(g)2894 543 y Fu(~)-35 b Fo(\023)2960 513 y Ft(b)s(e)33 b(an)h(en)m(umeration)50 658 y(of)c(the)h(distinct)e(non-empt)m(y)h(mem)m(b)s(ers)g(of)g(this)g (collection.)40 b(Let)31 b(us)f(write)572 802 y Fi(Z)623 1009 y Fh(S)673 987 y Fj(c)668 1033 y(k)727 926 y Fq(dy)s(e)864 888 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(y)r Fu(\))p Fo(=\017)1145 926 y Fq(h)1197 942 y Fo(B)1252 951 y Fj(\017)1285 942 y Fu(\()p Fo(x)1354 951 y Fj(j)1389 942 y Fu(\))p Fo(;)p Fh(S)1484 952 y Fj(k)1527 926 y Ft(\()p Fq(y)s Ft(\))26 b(=)1767 802 y Fi(Z)1818 1009 y Fu(\000)1863 987 y Fj(c)1863 1031 y(j)1916 926 y Fq(dy)s(e)2053 888 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(y)r Fu(\))p Fo(=\017)2335 926 y Fq(h)2387 942 y Fo(B)2442 951 y Fj(\017)2474 942 y Fu(\()p Fo(x)2543 951 y Fj(j)2578 942 y Fu(\))p Fo(;)p Fh(S)2673 952 y Fj(k)2717 926 y Ft(\()p Fq(y)s Ft(\))1671 1200 y(=)1767 1114 y Fi(X)1819 1308 y Fu(~)-36 b Fo(\023)1914 1077 y Fi(Z)1964 1283 y Fu(\000)2009 1292 y Fj(j)2043 1283 y Fu(\()o(~)h Fo(\023)p Fu(\))p Fh(nS)2208 1293 y Fj(k)2268 1200 y Fq(dy)s(e)2405 1163 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(y)r Fu(\))p Fo(=\017)2686 1200 y Fq(h)2738 1217 y Fo(B)2793 1226 y Fj(\017)2826 1217 y Fu(\()p Fo(x)2895 1226 y Fj(j)2930 1217 y Fu(\))p Fo(;)p Fh(S)3025 1227 y Fj(k)3069 1200 y Ft(\()p Fq(y)s Ft(\))3477 1082 y(\(6)p Fq(:)p Ft(10\))50 1512 y(The)35 b(\014rst)f(in)m(tegral)h(is)g(b)s (ounded)e(b)m(y)i Fq(C)22 b Ft(exp\()p Fr(\000)p Ft([)p Fq(F)13 b Ft(\()p Fq(z)1883 1479 y Fh(\003)1926 1512 y Ft(\()p Fq(x)2013 1526 y Fo(j)2051 1512 y Fq(;)i Fr(M)p Ft(\)\))25 b(+)e Fq(\016)s Ft(])p Fq(=\017)p Ft(\))38 b(and)c(will)f(b)s(e)i(negligible.)53 b(The)50 1657 y(remaining)30 b(con)m(tributions)h(will)f(b)s(e)h(split)g(in)m(to)h(those)h(for)f (whic)m(h)f Fq(F)13 b Ft(\()p Fq(z)2517 1624 y Fh(\003)2559 1657 y Ft(\()p Fq(x)2645 1671 y Fu(~)-35 b Fo(\023)2678 1657 y Fq(;)15 b Fr(S)2773 1671 y Fo(k)2818 1657 y Ft(\)\))29 b Fq(>)f(F)13 b Ft(\()p Fq(z)3168 1624 y Fh(\003)3211 1657 y Ft(\()p Fq(x)3297 1671 y Fu(~)-35 b Fo(\023)3330 1657 y Fq(;)15 b(x)3422 1671 y Fo(j)3460 1657 y Ft(\)\))33 b(and)50 1803 y(those)i(for)f(whic)m(h)f(the)i(con)m(trary)g(is)f (true.)53 b(The)34 b(p)s(oin)m(t)f(is)h(that)h(for)f(the)h(former)f Fq(h)2944 1819 y Fo(B)2999 1828 y Fj(\017)3031 1819 y Fu(\()p Fo(x)3100 1828 y Fj(j)3135 1819 y Fu(\))p Fo(;)p Fh(S)3230 1829 y Fj(k)3274 1803 y Ft(\()p Fq(y)s Ft(\))h(is)e(close)50 1948 y(to)39 b(one,)j(while)37 b(for)i(the)g(latter,)i(it)e(is)f(t)m (ypically)g(v)m(ery)h(small.)65 b(Here)39 b(w)m(e)h(mak)m(e)g(use)e(of) h(the)g(fact)h(that)50 2094 y(if)33 b Fq(y)i Fr(2)c Ft(\000)366 2108 y Fo(j)404 2094 y Ft(\()n(~)-43 b Fq(\023)p Ft(\),)36 b(and)e Fq(F)13 b Ft(\()p Fq(z)900 2061 y Fh(\003)942 2094 y Ft(\()p Fq(x)1028 2108 y Fu(~)-35 b Fo(\023)1061 2094 y Fq(;)15 b Fr(S)1156 2108 y Fo(k)1201 2094 y Ft(\)\))33 b Fq(>)e(F)13 b Ft(\()p Fq(z)1558 2061 y Fh(\003)1601 2094 y Ft(\()p Fq(x)1687 2108 y Fu(~)-35 b Fo(\023)1719 2094 y Fq(;)15 b(x)1811 2108 y Fo(j)1850 2094 y Ft(\)\),)36 b(then)e Fq(z)2238 2061 y Fh(\003)2280 2094 y Ft(\()p Fq(y)s(;)15 b Fr(S)2458 2108 y Fo(k)2504 2094 y Ft(\))32 b(=)f Fq(z)2719 2061 y Fh(\003)2761 2094 y Ft(\()p Fq(x)2847 2108 y Fu(~)-35 b Fo(\023)2880 2094 y Fq(;)15 b Fr(S)2975 2108 y Fo(k)3020 2094 y Ft(\))35 b(and)e Fq(z)3316 2061 y Fh(\003)3358 2094 y Ft(\()p Fq(y)s(;)15 b(x)3533 2108 y Fo(j)3572 2094 y Ft(\))32 b(=)50 2239 y Fq(z)96 2206 y Fh(\003)138 2239 y Ft(\()p Fq(x)224 2253 y Fu(~)-35 b Fo(\023)257 2239 y Fq(;)15 b(x)349 2253 y Fo(j)387 2239 y Ft(\).)41 b(Then)997 2429 y Fi(X)594 2628 y Fo(i)p Fu(:)p Fo(F)10 b Fu(\()p Fo(z)763 2610 y Fd(\003)800 2628 y Fu(\()p Fo(x)869 2637 y Fk(~)-31 b Fj(\023)899 2628 y Fo(;)p Fh(S)966 2638 y Fj(k)1005 2628 y Fu(\)\))p Fo(>F)10 b Fu(\()p Fo(z)1240 2610 y Fd(\003)1278 2628 y Fu(\()p Fo(x)1347 2637 y Fk(~)-31 b Fj(\023)1377 2628 y Fo(;x)1440 2637 y Fj(j)1474 2628 y Fu(\)\))1546 2391 y Fi(Z)1597 2598 y Fu(\000)1642 2607 y Fj(j)1676 2598 y Fu(\()o(~)c Fo(\023)p Fu(\))p Fh(nS)1841 2608 y Fj(k)1901 2515 y Fq(dy)s(e)2038 2477 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(y)r Fu(\))p Fo(=\017)2319 2515 y Fq(h)2371 2531 y Fo(B)2426 2540 y Fj(\017)2459 2531 y Fu(\()p Fo(x)2528 2540 y Fj(j)2562 2531 y Fu(\))p Fo(;)p Fh(S)2657 2541 y Fj(k)2701 2515 y Ft(\()p Fq(y)s Ft(\))604 2815 y(=)1103 2729 y Fi(X)700 2928 y Fo(i)p Fu(:)p Fo(F)g Fu(\()p Fo(z)869 2910 y Fd(\003)906 2928 y Fu(\()p Fo(x)975 2937 y Fk(~)-31 b Fj(\023)1005 2928 y Fo(;)p Fh(S)1072 2938 y Fj(k)1111 2928 y Fu(\)\))p Fo(>F)10 b Fu(\()p Fo(z)1346 2910 y Fd(\003)1384 2928 y Fu(\()p Fo(x)1453 2937 y Fk(~)-31 b Fj(\023)1484 2928 y Fo(;x)1547 2937 y Fj(j)1580 2928 y Fu(\)\))1652 2691 y Fi(Z)1703 2898 y Fu(\000)1748 2907 y Fj(j)1782 2898 y Fu(\()o(~)c Fo(\023)p Fu(\))p Fh(nS)1947 2908 y Fj(k)2007 2815 y Fq(dy)s(e)2144 2777 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(y)r Fu(\))p Fo(=\017)2440 2741 y Fi(\000)2482 2815 y Ft(1)20 b Fr(\000)g Fq(h)2690 2831 y Fh(S)2735 2841 y Fj(k)2775 2831 y Fo(;B)2852 2840 y Fj(\017)2884 2831 y Fu(\()p Fo(x)2953 2840 y Fj(j)2988 2831 y Fu(\))3021 2815 y Ft(\()p Fq(y)s Ft(\))3139 2741 y Fi(\001)3477 2696 y Ft(\(6)p Fq(:)p Ft(11\))50 3144 y(No)m(w)31 b(b)m(y)f(Corollary)f(4.8,)794 3407 y(0)d Fr(\024)f Fq(h)1013 3423 y Fh(S)1058 3433 y Fj(k)1098 3423 y Fo(;B)1175 3432 y Fj(\017)1206 3423 y Fu(\()p Fo(x)1275 3432 y Fj(j)1310 3423 y Fu(\))1343 3407 y Ft(\()p Fq(y)s Ft(\))h Fr(\024)f Fq(C)7 b(\017)1692 3369 y Fh(\000)p Fu(1)p Fo(=)p Fu(2)1862 3407 y Fq(e)1904 3369 y Fh(\000)p Fu([)p Fo(F)j Fu(\()p Fo(z)2104 3342 y Fd(\003)2142 3369 y Fu(\()p Fo(x)2211 3378 y Fk(~)-31 b Fj(\023)2241 3369 y Fo(;)p Fh(S)2308 3379 y Fj(k)2347 3369 y Fu(\)\))p Fh(\000)p Fo(F)10 b Fu(\()p Fo(z)2582 3342 y Fd(\003)2620 3369 y Fu(\()p Fo(x)2689 3378 y Fk(~)-31 b Fj(\023)2719 3369 y Fo(;x)2782 3378 y Fj(j)2816 3369 y Fu(\)\)])p Fo(=\017)3477 3407 y Ft(\(6)p Fq(:)p Ft(12\))50 3669 y(whic)m(h)21 b(b)m(y)h(assumption)f(is)h(exp)s(onen)m(tially)f(small.) 36 b(On)22 b(the)h(other)f(hand,)i(if)d Fq(x)2695 3683 y Fu(~)-35 b Fo(\023)2750 3669 y Ft(is)22 b(the)g(absolute)h(minim)m (um)50 3815 y(of)30 b Fq(F)44 b Ft(within)28 b(\000)595 3829 y Fo(j)633 3815 y Ft(\()n(~)-43 b Fq(\023)p Ft(\),)31 b(and)f(if)f(the)i(Hessian,)f Fr(r)1641 3782 y Fu(2)1681 3815 y Fq(F)13 b Ft(\()p Fq(x)1838 3829 y Fu(~)-35 b Fo(\023)1871 3815 y Ft(\),)31 b(at)g(this)f(minim)m(um)d(is)j (non-degenerate,)573 3985 y Fi(Z)623 4192 y Fu(\000)668 4201 y Fj(j)702 4192 y Fu(\()o(~)-35 b Fo(\023)p Fu(\))p Fh(nS)867 4202 y Fj(k)927 4109 y Fq(dy)s(e)1064 4072 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(y)r Fu(\))p Fo(=\017)1371 4109 y Ft(=)1630 4048 y(\(2)p Fq(\031)s(\017)p Ft(\))1837 4015 y Fo(d=)p Fu(2)p 1477 4088 630 4 v 1477 4106 a Fi(p)p 1568 4106 539 4 v 78 x Ft(det)q(\()p Fr(r)1806 4158 y Fu(2)1846 4184 y Fq(F)j Ft(\()p Fq(x)2003 4198 y Fu(~)-35 b Fo(\023)2036 4184 y Ft(\)\))2118 4109 y Fq(e)2160 4072 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(x)2342 4081 y Fk(~)-31 b Fj(\023)2372 4072 y Fu(\))p Fo(=\017)2473 4109 y Ft(\(1)21 b(+)f Fq(O)s Ft(\()p Fq(\017)2809 4072 y Fu(1)p Fo(=)p Fu(2)2923 4109 y Fr(j)15 b Ft(ln)f Fq(\017)p Fr(j)p Ft(\)\))291 b(\(6)p Fq(:)p Ft(13\))50 4401 y(b)m(y)30 b(standard)g(Laplace)g (asymptotics.)41 b(Th)m(us)785 4582 y Fi(X)380 4782 y Fu(~)-35 b Fo(\023)p Fu(:)p Fo(F)10 b Fu(\()p Fo(z)551 4763 y Fd(\003)588 4782 y Fu(\()p Fo(x)657 4791 y Fk(~)-31 b Fj(\023)687 4782 y Fo(;)p Fh(S)754 4792 y Fj(k)793 4782 y Fu(\)\))p Fo(>F)10 b Fu(\()p Fo(z)1028 4763 y Fd(\003)1067 4782 y Fu(\()p Fo(x)1136 4791 y Fk(~)-31 b Fj(\023)1166 4782 y Fo(;x)1229 4791 y Fj(j)1263 4782 y Fu(\)\))1335 4545 y Fi(Z)1385 4751 y Fu(\000)1430 4760 y Fj(j)1464 4751 y Fu(\()o(~)c Fo(\023)p Fu(\))p Fh(nS)1629 4761 y Fj(k)1689 4669 y Fq(dy)s(e)1826 4631 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(y)r Fu(\))p Fo(=\017)2107 4669 y Fq(h)2159 4685 y Fo(B)2214 4694 y Fj(\017)2247 4685 y Fu(\()p Fo(x)2316 4694 y Fj(j)2351 4685 y Fu(\))p Fo(;)p Fh(S)2446 4695 y Fj(k)2490 4669 y Ft(\()p Fq(y)s Ft(\))391 4987 y(=)891 4901 y Fi(X)486 5100 y Fu(~)-35 b Fo(\023)p Fu(:)p Fo(F)10 b Fu(\()p Fo(z)657 5082 y Fd(\003)694 5100 y Fu(\()p Fo(x)763 5109 y Fk(~)-31 b Fj(\023)793 5100 y Fo(;)p Fh(S)860 5110 y Fj(k)899 5100 y Fu(\)\))p Fo(>F)10 b Fu(\()p Fo(z)1134 5082 y Fd(\003)1173 5100 y Fu(\()p Fo(x)1242 5109 y Fk(~)-31 b Fj(\023)1272 5100 y Fo(;x)1335 5109 y Fj(j)1369 5100 y Fu(\)\))1604 4926 y Ft(\(2)p Fq(\031)s(\017)p Ft(\))1811 4893 y Fo(d=)p Fu(2)p 1452 4966 630 4 v 1452 4984 a Fi(p)p 1543 4984 539 4 v 78 x Ft(det\()p Fr(r)1780 5036 y Fu(2)1821 5062 y Fq(F)13 b Ft(\()p Fq(x)1978 5076 y Fu(~)-35 b Fo(\023)2011 5062 y Ft(\)\))2092 4987 y Fq(e)2134 4950 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(x)2316 4959 y Fk(~)-31 b Fj(\023)2346 4950 y Fu(\))p Fo(=\017)2447 4987 y Ft(\(1)21 b(+)f Fq(O)s Ft(\()p Fq(\017)2783 4950 y Fu(1)p Fo(=)p Fu(2)2897 4987 y Fr(j)15 b Ft(ln)f Fq(\017)p Fr(j)p Ft(\)\))3477 4859 y(\(6)p Fq(:)p Ft(14\))50 5316 y(The)30 b(remaining)f(terms)h(cannot)h (b)s(e)f(computed)h(as)g(precisely;)e(ho)m(w)m(ev)m(er,)j(often)f(the)g (upp)s(er)e(b)s(ound)f(will)50 5462 y(sho)m(w)g(that)h(they)g(are)g (totally)g(negligible)d(\(but)i(this)g(is)f(not)i(alw)m(a)m(ys)g(the)g (case\).)42 b(Using)27 b(again)i(Corollary)p eop %%Page: 26 26 26 25 bop 50 295 a Fl(26)3225 b Fa(R)l(efer)l(enc)l(es)50 513 y Ft(4.8,)32 b(when)d Fq(F)13 b Ft(\()p Fq(z)611 480 y Fh(\003)653 513 y Ft(\()p Fq(x)739 527 y Fu(~)-35 b Fo(\023)772 513 y Fq(;)15 b Fr(S)867 527 y Fo(k)912 513 y Ft(\)\))26 b Fr(\024)f Fq(F)13 b Ft(\()p Fq(z)1256 480 y Fh(\003)1299 513 y Ft(\()p Fq(x)1385 527 y Fu(~)-35 b Fo(\023)1418 513 y Fq(;)15 b(x)1510 527 y Fo(j)1548 513 y Ft(\)\),)80 678 y Fi(Z)131 884 y Fu(\000)176 893 y Fj(j)210 884 y Fu(\()o(~)-35 b Fo(\023)p Fu(\))p Fh(nS)375 894 y Fj(k)435 802 y Fq(dy)s(e)572 764 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(y)r Fu(\))p Fo(=\017)853 802 y Fq(h)905 818 y Fo(B)960 827 y Fj(\017)993 818 y Fu(\()p Fo(x)1062 827 y Fj(j)1096 818 y Fu(\))p Fo(;)p Fh(S)1191 828 y Fj(k)1235 802 y Ft(\()p Fq(y)s Ft(\))26 b Fr(\024)f Fq(C)7 b(\017)1584 764 y Fh(\000)p Fu(1)p Fo(=)p Fu(2)1770 678 y Fi(Z)1820 884 y Fu(\000)1865 893 y Fj(j)1899 884 y Fu(\()o(~)-35 b Fo(\023)p Fu(\))p Fh(nS)2064 894 y Fj(k)2124 802 y Fq(dy)s(e)2261 764 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(y)r Fu(\))p Fo(=\017)2542 802 y Fq(e)2584 764 y Fh(\000)p Fu([)p Fo(F)g Fu(\()p Fo(z)2784 737 y Fd(\003)2822 764 y Fu(\()p Fo(y)r(;x)2952 773 y Fj(j)2986 764 y Fu(\)\))p Fh(\000)p Fo(F)g Fu(\()p Fo(z)3221 737 y Fd(\003)3259 764 y Fu(\()p Fo(y)r(;)p Fh(S)3393 774 y Fj(k)3433 764 y Fu(\)\)])p Fo(=\017)90 1063 y Ft(=)25 b Fq(C)7 b(\017)295 1025 y Fh(\000)p Fu(1)p Fo(=)p Fu(2)481 939 y Fi(Z)531 1145 y Fu(\000)576 1154 y Fj(j)610 1145 y Fu(\()o(~)-35 b Fo(\023)q Fu(\))p Fh(nS)776 1155 y Fj(k)816 1145 y Fu(:)p Fo(z)875 1127 y Fd(\003)912 1145 y Fu(\()p Fo(y)r(;)p Fh(S)1046 1155 y Fj(k)1085 1145 y Fu(\)=)p Fo(y)1228 1063 y Fq(dy)s(e)1365 1025 y Fh(\000)p Fo(F)10 b Fu(\()p Fo(z)1544 998 y Fd(\003)1582 1025 y Fu(\()p Fo(x)1651 1034 y Fk(~)-31 b Fj(\023)1681 1025 y Fo(;x)1744 1034 y Fj(j)1778 1025 y Fu(\)\))p Fo(=\017)85 1323 y Ft(+)20 b Fq(C)7 b(\017)285 1286 y Fh(\000)p Fu(1)p Fo(=)p Fu(2)456 1323 y Fq(e)498 1286 y Fh(\000)p Fo(F)j Fu(\()p Fo(x)680 1295 y Fj(j)714 1286 y Fu(\))p Fo(=\017)830 1200 y Fi(Z)880 1406 y Fu(\000)925 1415 y Fj(j)959 1406 y Fu(\()o(~)-35 b Fo(\023)q Fu(\))p Fh(nS)1125 1416 y Fj(k)1165 1406 y Fu(:)p Fo(z)1224 1388 y Fd(\003)1261 1406 y Fu(\()p Fo(y)r(;)p Fh(S)1395 1416 y Fj(k)1434 1406 y Fu(\))p Fh(6)p Fu(=)p Fo(y)1577 1323 y Fq(dy)s(e)1714 1286 y Fh(\000)p Fu([)p Fo(F)10 b Fu(\()p Fo(y)r Fu(\))p Fh(\000)p Fo(F)g Fu(\()p Fo(x)2125 1295 y Fk(~)-31 b Fj(\023)2156 1286 y Fu(\)])p Fo(=\017)2277 1323 y Fq(e)2319 1286 y Fh(\000)p Fu([)p Fo(F)10 b Fu(\()p Fo(z)2519 1258 y Fd(\003)2556 1286 y Fu(\()p Fo(x)2625 1295 y Fk(~)-31 b Fj(\023)2655 1286 y Fo(;x)2718 1295 y Fj(j)2752 1286 y Fu(\)\))p Fh(\000)p Fo(F)10 b Fu(\()p Fo(x)2990 1295 y Fj(j)3025 1286 y Fu(\))p Fh(\000)p Fo(F)g Fu(\()p Fo(z)3232 1258 y Fd(\003)3270 1286 y Fu(\()p Fo(x)3339 1295 y Fk(~)-31 b Fj(\023)3369 1286 y Fo(;)p Fh(S)3436 1296 y Fj(k)3475 1286 y Fu(\)\)+)p Fo(F)10 b Fu(\()p Fo(x)3712 1295 y Fk(~)-31 b Fj(\023)3743 1286 y Fu(\)])p Fo(=\017)90 1547 y Ft(=)25 b Fq(C)7 b(\017)295 1509 y Fh(\000)p Fu(1)p Fo(=)p Fu(2)466 1547 y Fq(e)508 1509 y Fh(\000)p Fo(F)j Fu(\()p Fo(z)687 1482 y Fd(\003)724 1509 y Fu(\()p Fo(x)793 1518 y Fk(~)-31 b Fj(\023)824 1509 y Fo(;x)887 1518 y Fj(j)920 1509 y Fu(\)\))p Fo(=\017)1064 1547 y Fr(j)q(f)p Ft(\000)1192 1561 y Fo(j)1230 1547 y Ft(\()n(~)-43 b Fq(\023)q Ft(\))p Fr(nS)1433 1561 y Fo(k)1504 1547 y Ft(:)25 b Fq(z)1600 1509 y Fh(\003)1642 1547 y Ft(\()p Fq(y)s(;)15 b Fr(S)1820 1561 y Fo(k)1866 1547 y Ft(\))25 b(=)g Fq(y)s Fr(gj)85 1757 y Ft(+)20 b Fq(C)7 b(\017)285 1720 y Fh(\000)p Fu(1)p Fo(=)p Fu(2)456 1757 y Fq(e)498 1720 y Fh(\000)p Fo(F)j Fu(\()p Fo(x)680 1729 y Fj(j)714 1720 y Fu(\))p Fo(=\017)815 1757 y Fq(e)857 1720 y Fh(\000)p Fu([)p Fo(F)g Fu(\()p Fo(z)1057 1692 y Fd(\003)1094 1720 y Fu(\()p Fo(x)1163 1729 y Fk(~)-31 b Fj(\023)1193 1720 y Fo(;x)1256 1729 y Fj(j)1290 1720 y Fu(\)\))p Fh(\000)p Fo(F)10 b Fu(\()p Fo(x)1528 1729 y Fj(j)1563 1720 y Fu(\))p Fh(\000)p Fo(F)g Fu(\()p Fo(z)1770 1692 y Fd(\003)1808 1720 y Fu(\()p Fo(x)1877 1729 y Fk(~)-31 b Fj(\023)1907 1720 y Fo(;)p Fh(S)1974 1730 y Fj(k)2013 1720 y Fu(\)\)+)p Fo(F)10 b Fu(\()p Fo(x)2250 1729 y Fk(~)-31 b Fj(\023)2281 1720 y Fu(\)])p Fo(=\017)2566 1696 y Ft(\(2)p Fq(\031)s(\017)p Ft(\))2773 1663 y Fo(d=)p Fu(2)p 2413 1736 630 4 v 2413 1754 a Fi(p)p 2504 1754 539 4 v 78 x Ft(det\()p Fr(r)2741 1806 y Fu(2)2782 1832 y Fq(F)13 b Ft(\()p Fq(x)2939 1846 y Fu(~)-35 b Fo(\023)2972 1832 y Ft(\)\))3053 1757 y(\(1)21 b(+)f Fq(O)s Ft(\()p Fq(\017)3389 1720 y Fu(1)p Fo(=)p Fu(2)3504 1757 y Fr(j)15 b Ft(ln)f Fq(\017)p Fr(j)p Ft(\)\))3477 1937 y(\(6)p Fq(:)p Ft(15\))50 2083 y(The)29 b(\014rst)h(summand)e(is)h(alw)m(a)m (ys)i(exp)s(onen)m(tially)d(negligible)g(compared)i(to)h(the)f (principle)d(terms,)j(since)50 2228 y(of)k(course)h Fq(F)13 b Ft(\()p Fq(z)592 2195 y Fh(\003)635 2228 y Ft(\()p Fq(x)721 2242 y Fu(~)-35 b Fo(\023)753 2228 y Fq(;)15 b(x)845 2242 y Fo(j)884 2228 y Ft(\)\))33 b Fq(>)f(F)13 b Ft(\()p Fq(x)1248 2242 y Fo(j)1286 2228 y Ft(\).)54 b(The)34 b(second)g(summand)f(is)g(negligible)g(only)g(when)h(\(6.6\))i (holds,)50 2374 y(whic)m(h)f(will)e(b)s(e)i(the)h(case)i(in)c(the)i (main)f(applications.)56 b(This)34 b(implies)g(\(6.8\).)59 b(\(6.9\))38 b(is)d(an)h(immediate)50 2519 y(consequence.)41 b Fr(})50 2737 y Fc(Pro)s(of)36 b(of)f(Theorem)f(3.2:)89 b Ft(The)30 b(pro)s(of)g(of)g(Theorem)h(3.2)g(is)f(immediate)g(b)m(y)g (inserting)f(the)i(form)m(ula)50 2883 y(for)36 b(the)h(Capacit)m(y)g (of)f(Theorem)g(3.1)i(in)m(to)e(\(6.8\),)k(except)e(for)e(the)h(error)f (terms)g(of)h(order)e Fq(\017)3318 2850 y Fo(\013=)p Fu(2)3480 2883 y Ft(whic)m(h)50 3028 y(w)m(e)d(will)c(no)m(w)j(sho)m(w) h(can)f(b)s(e)g(remo)m(v)m(ed)h(easily)-8 b(.)43 b(Namely)-8 b(,)32 b(note)g(that)g(nothing)e(c)m(hanges)j(in)d(the)h(pro)s(of)g(of) 50 3174 y(Theorem)g(6.2)h(if)e(w)m(e)i(replace)f(the)g(starting)g(p)s (oin)m(t)f Fq(x)1897 3188 y Fo(j)1967 3174 y Ft(b)m(y)h(some)g(p)s(oin) m(t)g Fq(x)26 b Fr(2)g Fq(B)2795 3148 y Fh(p)p 2855 3148 31 4 v 47 x Fo(\017)2890 3174 y Ft(\()p Fq(y)s Ft(\).)43 b(Also,)32 b(insp)s(ecting)50 3319 y(the)g(pro)s(of)f(of)h(Theorem)g (5.1)h(one)f(sees)g(that)h(the)f(di\013erence)f(b)s(et)m(w)m(een)h(cap) 2707 3333 y Fo(B)2762 3342 y Fj(\017)2794 3333 y Fo(x)2835 3342 y Fj(j)2874 3319 y Ft(\()p Fr(S)2964 3333 y Fo(k)3009 3319 y Ft(\))g(and)g(cap)3406 3333 y Fo(B)3461 3342 y Fj(\017)3494 3333 y Fo(x)3539 3319 y Ft(\()p Fr(S)3629 3333 y Fo(k)3674 3319 y Ft(\))50 3464 y(for)e Fq(x)25 b Fr(2)g Fq(B)421 3439 y Fh(p)p 481 3439 V 47 x Fo(\017)516 3464 y Ft(\()p Fq(y)s Ft(\))30 b(is)g(in)f(fact)i(m)m(uc)m(h)g(smaller) d(than)j(the)f(error)g(terms.)41 b(Th)m(us)29 b(w)m(e)i(get)g(that)g (in)e(fact)944 3740 y(osc)1066 3756 y Fo(x)p Fh(2)p Fo(B)1211 3740 y Fd(p)p 1261 3740 28 3 v 33 x Fj(\017)1293 3756 y Fu(\()p Fo(x)1362 3765 y Fj(j)1397 3756 y Fu(\))1430 3740 y Fp(E)1485 3754 y Fo(x)1536 3740 y Fq(\034)1576 3754 y Fh(S)1621 3764 y Fj(k)1690 3740 y Fr(\024)c Fq(C)7 b Ft(\()p Fq(\017)1930 3702 y Fo(\013=)p Fu(2)2075 3740 y Ft(+)20 b Fq(\017)2203 3702 y Fu(1)p Fo(=)p Fu(2)2318 3740 y Fr(j)15 b Ft(ln)f Fq(\017)p Fr(j)p Ft(\))p Fp(E)2601 3754 y Fo(x)2642 3763 y Fj(j)2687 3740 y Fq(\034)2727 3754 y Fh(S)2772 3764 y Fj(k)3477 3740 y Ft(\(6)p Fq(:)p Ft(16\))50 4016 y(whic)m(h)33 b(impro)m(v)m(es)i(the)g(input)d(in)i (the)h(H\177)-45 b(older)34 b(estimate)h(b)m(y)g(a)g(factor)h Fq(\017)2561 3983 y Fo(\013=)p Fu(2)2686 4016 y Ft(,)g(whic)m(h)e(in)f (turm)h(allo)m(ws)g(to)50 4161 y(impro)m(v)m(e)h(the)g(error)g (estimates)h(in)e(Theorem)h(6.2)h(from)f Fq(\017)2103 4128 y Fo(\013=)p Fu(2)2264 4161 y Ft(to)h Fq(\017)2417 4128 y Fo(\013)2468 4161 y Ft(.)56 b(Iterating)35 b(this)f(pro)s (ceedure,)i(w)m(e)50 4306 y(can)f(reduce)g(this)f(errors)g(un)m(til)f (they)i(are)h(of)f(the)g(same)g(order)g(as)g(the)g Fq(\017)2602 4273 y Fu(1)p Fo(=)p Fu(2)2716 4306 y Fr(j)15 b Ft(ln)g Fq(\017)p Fr(j)35 b Ft(terms.)54 b(This)33 b(pro)m(v)m(es)50 4452 y(Theorem)d(3.2.)42 b Fr(})50 4735 y Fv(References)-251 4953 y Ft([BEGK1])47 b(A.)33 b(Bo)m(vier,)h(M.)f(Ec)m(kho\013,)h(V.)f (Ga)m(yrard,)h(and)e(M.)h(Klein,)f(\\Metastabilit)m(y)h(in)f(sto)s(c)m (hastic)h(dynamics)159 5098 y(of)d(disordered)f(mean-\014eld)g(mo)s (dels",)h(Probab.)40 b(Theor.)g(Rel.)h(Fields)28 b Fs(119)p Ft(,)k(99{161\(2001\).)-251 5316 y([BEGK2])47 b(A.)28 b(Bo)m(vier,)h(M.)g(Ec)m(kho\013,)g(V.)f(Ga)m(yrard,)h(and)f(M.)g (Klein,)f(\\Metastabilit)m(y)h(and)g(lo)m(w-lying)e(sp)s(ectra)i(in)159 5462 y(rev)m(ersible)h(Mark)m(o)m(v)j(c)m(hains",)e(Comm)m(un.)40 b(Math.)i(Ph)m(ys.)e Fs(228)p Ft(,)31 b(219-255)j(\(2002\).)p eop %%Page: 27 27 27 26 bop 50 295 a Fa(Metastability)27 b(and)g(sp)l(e)l(ctr)l(a)2736 b Fl(27)-143 513 y Ft([BGK])46 b(A.)35 b(Bo)m(vier,)h(V.)f(Ga)m(yrard,) i(and)d(M.)h(Klein,)f(\\Metastabilit)m(y)h(in)f(rev)m(ersible)f (di\013usion)f(pro)s(cesses)i(I)s(I.)159 658 y(Precise)c(asymptotics)h (for)f(small)f(eigen)m(v)-5 b(alues,)30 b(preprin)m(t)f(\(2002\).)-224 876 y([BluGet])46 b(R.M.)34 b(Blumen)m(thal)e(and)g(R.K.)h(Geto)s(or,)i (\\Mark)m(o)m(v)h(pro)s(cesses)d(and)f(p)s(oten)m(tial)h(theory",)h (Academic)159 1022 y(Press,)c(New)h(Y)-8 b(ork,)31 b(London,)f(1968.) -226 1240 y([BuMa1])47 b(V.A.)32 b(Buslo)m(v)f(and)g(K.A.)h(Mak)-5 b(aro)m(v,)33 b(\\A)f(time-scale)g(hierarc)m(h)m(y)f(with)f(small)g (di\013usion")f(\(Russian\),)159 1385 y(T)-8 b(eoret.)45 b(Mat.)h(Fiz.)e Fs(76)p Ft(,)32 b(219{230)j(\(1988\);)g(translation)c (in)f(Theoret.)44 b(Math.)h(Ph)m(ys.)f Fs(76)p Ft(,)33 b(818{826)159 1531 y(\(1989\).)-226 1749 y([BuMa2])47 b(V.A.)31 b(Buslo)m(v)f(and)g(K.A.)h(Mak)-5 b(aro)m(v,)32 b(\\Life)e(spans)f(and)h(least)g(eigen)m(v)-5 b(alues)30 b(of)h(an)f(op)s(erator)h(of)f(small)159 1895 y(di\013usion")36 b(\(Russian\),)j(Mat.)64 b(Zametki)38 b Fs(51)g Ft(,)i(20{31)g (\(1992\);)k(translation)37 b(in)g(Math.)64 b(Notes)39 b Fs(51)p Ft(,)159 2040 y(14{21)32 b(\(1992\).)-57 2258 y([DS])46 b(P)-8 b(.G.)31 b(Do)m(yle)g(and)e(J.L.)h(Snell,)f(\\Random)h (w)m(alks)f(and)h(electrical)f(net)m(w)m(orks",)j(Carus)d(Mathematical) 159 2404 y(Monographs,)i(22,)g(Mathematical)g(Asso)s(ciation)f(of)h (America,)f(W)-8 b(ashington,)31 b(DC,)g(1984.)-100 2622 y([Do)s(o])47 b(J.L.)26 b(Do)s(ob,)i(\\Classical)d(p)s(oten)m(tial)h (theory)h(and)f(its)f(probabilistic)e(coun)m(terpart",)29 b(Grundlehren)23 b(der)159 2767 y(mathematisc)m(hen)31 b(Wissensc)m(haften)f(262,)i(Springer)c(V)-8 b(erlag,)31 b(Berlin,)e(1984.)-47 2985 y([Ey])46 b(H.)35 b(Eyring,)g(\\The)f(activ) -5 b(ated)36 b(complex)e(in)g(c)m(hemical)g(reactions",)j(J.)e(Chem.)52 b(Ph)m(ys.)i Fs(3)35 b Ft(,)h(107{115)159 3131 y(\(1935\).)-90 3349 y([FW])47 b(M.I.)25 b(F)-8 b(reidlin)21 b(and)j(A.D.)h(W)-8 b(en)m(tzell,)27 b(\\Random)d(p)s(erturbations)e(of)i(dynamical)f (systems",)j(Springer,)159 3495 y(Berlin-Heidelb)s(erg-New)i(Y)-8 b(ork,)31 b(1984.)-127 3713 y([GLE])46 b(S.)33 b(Glasstone,)h(K.J.)f (Laidler,)g(and)f(H.)i(Eyring,)f(\\The)g(theory)g(of)h(rate)g(pro)s (cesses",)g(McGra)m(w-Hill,)159 3858 y(New)c(Y)-8 b(ork,)32 b(\(1941\).)-74 4076 y([GT])46 b(D.)c(Gilbarg)e(and)h(N.S.)g(T)-8 b(rudinger,)42 b(\\Elliptic)d(partial)h(di\013eren)m(tial)g(equations)h (of)h(second)f(order.)159 4222 y(Second)33 b(edition.)47 b(Grundlehren)30 b(der)i(Mathematisc)m(hen)i(Wissensc)m(haften,)g(224.) 50 b(Springer-V)-8 b(erlag,)159 4367 y(Berlin,)29 b(1983.)-101 4585 y([HS1])46 b(B.)26 b(Hel\013er,)g(and)f(J.)g(Sj\177)-45 b(ostrand,)25 b(\\J.)h(Multiple)d(w)m(ells)h(in)g(the)h(semiclassical)f (limit.)36 b(I.)26 b(Comm.)38 b(P)m(artial)159 4731 y(Di\013eren)m (tial)30 b(Equations)f Fs(9)p Ft(,)i(337{408)i(\(1984\).)-101 4949 y([HS2])46 b(B.)d(Hel\013er,)j(and)c(J.)h(Sj\177)-45 b(ostrand,)45 b(\\Puits)d(m)m(ultiples)e(en)j(limite)e(semi-classique.) 76 b(I)s(I.)42 b(In)m(teraction)159 5095 y(mol)m(\023)-43 b(eculaire.)77 b(Sym)m(\023)-43 b(etries.)77 b(P)m(erturbation",)45 b(Ann.)77 b(Inst.)f(H.)43 b(P)m(oincar)m(\023)-43 b(e)44 b(\(Ph)m(ys.)77 b(Th)m(\023)-43 b(eor.\))78 b Fs(42)p Ft(,)159 5240 y(127{212)33 b(\(1985\).)-101 5458 y([HS3])46 b(B.)40 b(Hel\013er,)i(and)d(J.)g(Sj\177)-45 b(ostrand,)42 b(\\Multiple)37 b(w)m(ells)i(in)f(the)i(semiclassical)e(limit.)66 b(I)s(I)s(I.)38 b(In)m(teraction)p eop %%Page: 28 28 28 27 bop 50 295 a Fl(28)3225 b Fa(R)l(efer)l(enc)l(es)159 513 y Ft(through)30 b(nonresonan)m(t)g(w)m(ells",)g(Math.)41 b(Nac)m(hr.)g Fs(124)p Ft(,)32 b(263{313)h(\(1985\).)-101 731 y([HS4])46 b(B.)32 b(Hel\013er,)f(and)f(J.)h(Sj\177)-45 b(ostrand,)31 b(\\Puits)f(m)m(ultiples)e(en)j(m)m(\023)-43 b(ecanique)32 b(semi-classique.)41 b(IV.)3347 708 y(\023)3339 731 y(Etude)31 b(du)159 876 y(complexe)f(de)h(Witten",)g(Comm.)40 b(P)m(artial)30 b(Di\013eren)m(tial)g(Equations)f Fs(10)p Ft(,)j(245{340)h(\(1985\).)-152 1095 y([Kaku])46 b(S.)32 b(Kakutani,)h(\\Mark)m(o)m(v)i(pro)s(cesses)d(and)g(the)h(Diric)m(hlet) f(problem",)g(Pro)s(c.)48 b(Jap.)f(Acad.)h Fs(21)p Ft(,)34 b(227{)159 1240 y(233)e(\(1941\).)-89 1458 y([Kra])46 b(H.A.)35 b(Kramers,)g(\\Bro)m(wnian)f(motion)g(in)f(a)i(\014eld)e(of)i (force)g(and)f(the)g(di\013usion)e(mo)s(del)h(of)i(c)m(hemical)159 1604 y(reactions",)c(Ph)m(ysica)f Fs(7)p Ft(,)h(284{304)j(\(1940\).) -124 1822 y([Kolo])46 b(V.N.)22 b(Kolok)m(oltso)m(v,)j(\\Semiclassical) 19 b(analysis)h(for)i(di\013usions)c(and)j(sto)s(c)m(hastic)h(pro)s (cesses",)i(Springer,)159 1967 y(Berlin,)29 b(2000.)-260 2185 y([KoMak])77 b(V.N.)37 b(Kolok)m(oltso)m(v)h(and)e(K.A.)h(Mak)-5 b(aro)m(v,)40 b(\\Asymptotic)d(sp)s(ectral)f(analysis)f(of)h(a)h(small) e(di\013usion)159 2331 y(op)s(erator)24 b(and)f(the)h(life)e(times)h (of)h(the)g(corresp)s(onding)d(di\013usion)g(pro)s(cess",)k(Russian)d (J.)i(Math.)39 b(Ph)m(ys.)159 2476 y Fs(4)p Ft(,)31 b(341{360)i (\(1996\).)-76 2695 y([KN])46 b(C.)29 b(Kipnis)e(and)i(C.M.)g(Newman,)h (\\The)g(metastable)g(b)s(eha)m(vior)f(of)g(infrequen)m(tly)f(observ)m (ed,)i(w)m(eakly)159 2840 y(random,)g(one-dimensional)e(di\013usion)g (pro)s(cesses",)i(SIAM)h(J.)f(Appl.)39 b(Math.)i Fs(45)p Ft(,)31 b(972{982)j(\(1985\).)-286 3058 y([MatSc)m(h1])48 b(B.J.)38 b(Matk)m(o)m(wsky)i(and)d(Z.)h(Sc)m(h)m(uss,)h(\\The)f(exit)f (problem:)54 b(a)38 b(new)g(approac)m(h)g(to)g(di\013usion)d(across)159 3204 y(p)s(oten)m(tial)30 b(barriers",)f(SIAM)h(J.)g(Appl.)40 b(Math.)h Fs(36)p Ft(,)31 b(604{623)i(\(1979\).)-116 3422 y([MS1])46 b(R.S.)c(Maier)g(and)g(D.L.)h(Stein,)i(\\Limiting)40 b(exit)j(lo)s(cation)e(distributions)e(in)i(the)i(sto)s(c)m(hastic)g (exit)159 3567 y(problem",)29 b(SIAM)i(J.)f(Appl.)39 b(Math.,)31 b Fs(57)p Ft(,)g(752{790)j(\(1997\).)-56 3785 y([NS])46 b(C.St.J.A.)31 b(Nash-Williams,)e(\\Random)i(w)m(alk)g (and)f(electric)h(curren)m(ts)f(in)g(net)m(w)m(orks")i(Pro)s(c.)42 b(Cam)m(b.)159 3931 y(Phil.)c(So)s(c.)j Fs(55)p Ft(,)31 b(181-194)i(\(1959\).)12 4149 y([S])46 b(Ch.)27 b(Sc)m(h)s(\177)-48 b(utte,)29 b(\\Conformational)e(dynamics:)38 b(mo)s(delling,)25 b(theory)-8 b(,)30 b(algorithm,)d(and)g(application)f(to)159 4295 y(biomolecules",)j(preprin)m(t)g(SC)g(99-18,)k(ZIB-Berlin)c (\(1999\).)-185 4513 y([SFHD])47 b(Ch.)27 b(Sc)m(h)s(\177)-48 b(utte,)29 b(A.)g(Fisc)m(her,)f(W.)h(Huisinga,)e(and)h(P)-8 b(.)28 b(Deu\015hard,)g(\\A)g(direct)g(approac)m(h)g(to)h(conforma-)159 4658 y(tional)g(dynamics)h(based)g(on)g(h)m(ybrid)e(Mon)m(te)k(Carlo",) e(J.)g(Comput.)40 b(Ph)m(ys.)h Fs(151)p Ft(,)31 b(146{168)j(\(1999\).) -104 4876 y([Szni])45 b(A.-S.)32 b(Sznitman,)g(\\Bro)m(wnian)g(motion,) g(obstacles)h(and)e(random)g(media",)i(Springer)d(Monographs)159 5022 y(in)f(Mathematics.)42 b(Springer,)28 b(Berlin,)h(1998.)-114 5240 y([W)-8 b(en])47 b(A.D.)29 b(W)-8 b(en)m(tzell,)29 b(\\F)-8 b(orm)m(ulas)28 b(for)f(eigenfunctions)f(and)h(eigenmeasures)h (that)g(are)g(connected)g(with)e(a)159 5385 y(Mark)m(o)m(v)32 b(pro)s(cess",)f(T)-8 b(eor.)41 b(V)-8 b(ero)5 b(jatnost.)42 b(i)30 b(Primenen.)39 b Fs(18)p Ft(,)31 b(329)h(\(1973\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF