Content-Type: multipart/mixed; boundary="-------------9905120939502" This is a multi-part message in MIME format. ---------------9905120939502 Content-Type: text/plain; name="99-168.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-168.keywords" Nodal surfaces, eigenfunctions, Dirichlet Laplacian ---------------9905120939502 Content-Type: application/postscript; name="nodal.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="nodal.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: nodal.dvi %%Pages: 14 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips nodal %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1999.05.12:1633 %%BeginProcSet: texc.pro /TeXDict 250 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 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 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 product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false} ifelse}{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 1000 300 300 (nodal.dvi) @start /Fa 13 118 df<127012F8A312700505788416>46 D<13F8EA03FC487EEA0F07 381C3B80EA38FF12793873C7C01383EAE701A73873838013C73879FF00EA38FEEA1C3838 0F03C0EA07FF6C1300EA00FC12197E9816>64 D97 D<133FA31307A4EA03C7EA0FF748B4FCEA3C1F487EEA700712E0A6EA700F12786C5A381F FFE0EA0FF7EA07C713197F9816>100 D<131E137F3801FF8013C7380383001380A2EA7F FFB5FCA2EA0380ACEA7FFC487E6C5A11197F9816>102 D<1203EA0780A2EA0300C7FCA4 EAFF80A31203ACEAFFFC13FE13FC0F1A7C9916>105 D<127E12FE127E120EA4EB7FE0A3 EB0F00131E5B5B5B120F7F13BC131EEA0E0E7F1480387F87F0EAFFCFEA7F871419809816 >107 D<38F9C38038FFEFC0EBFFE0EA3C78A2EA3870AA38FE7CF8A31512809116>109 DII<387F0FC038FF3FE0EA7F7F3807F040EBC0005BA290C7FCA8EA7F FC12FF127F13127F9116>114 DI117 D E /Fb 1 70 df69 D E /Fc 3 51 df<120FEA30C0EA6060A2EA4020EAC030A9EA4020EA6060A2EA30C0EA0F 000C137E9211>48 D<120C121C12EC120CAFEAFFC00A137D9211>I<121FEA60C01360EA F07013301260EA0070A2136013C012011380EA02005AEA08101210EA2020EA7FE012FF0C 137E9211>I E /Fd 2 81 df69 D80 D E /Fe 1 83 df82 D E /Ff 40 122 df<91390FE01FE091393838601891396079C01C9139C07B803C0101EB3300030713 38028014001303150EA3EB0700151E151C017FB612E0903907001C00130EEE01C01538A3 49EC0380A21570A2EE07005BA203E01308EE0E10A25B1720913801C006EE03C093C7FC01 E05B1403A201C090C8FCEA70C738F18F06EB0F0C38620618383C03E02E2D82A22B>14 D<1480EB010013025B5B5B13305B5BA2485A48C7FCA21206A2120E120C121C1218A21238 1230A21270A21260A212E0A35AAD12401260A21220123012107E113278A414>40 D<13087F130613021303A27F1480AD1303A31400A25BA21306A2130E130CA2131C131813 381330A25BA25B485AA248C7FC120612045A5A5A5A5A113280A414>I<120E121EA41202 A21204A21208A21210122012401280070F7D840F>44 DI< 127012F8A212F012E005057A840F>I<13011303A21306131E132EEA03CEEA001CA41338 A41370A413E0A4EA01C0A4EA0380A41207EAFFFC10217AA019>49 DI<14181438A21470A3 14E0A314C01301148013031400A21306A25BA25B1310EB3180EB61C0EB438013831201EA 03033802070012041208EA3FC7EA403E38800FF038000E00A25BA45BA31330152B7EA019 >52 DII<1207EA0F80A21300120EC7FCAB1270 12F8A25A5A09157A940F>58 D<90B512E090380F0038151C151E011E130E150FA349130E 151EA2153C4913781570EC01E0EC078090B5FC9038F001E0EC00F01578485A1538153CA2 48481378A315F0485AEC01E0EC03C0EC0700380F001EB512F020227DA122>66 D<9039FFF87FFC90390F000780A3011EEB0F00A449131EA4495BA490B512F89038F00078 A348485BA44848485AA44848485AA4000F130739FFF07FF826227DA124>72 D76 D<90B512E090380F0038151E150E01 1E1307A449130FA3151E5B153C157815E09038F003C09038FFFE0001F0C7FCA2485AA448 5AA4485AA4120FEAFFF020227DA121>80 D<001FB512F8391E03C0380018141812303820 0780A200401410A2EB0F001280A200001400131EA45BA45BA45BA4485AA41203B5FC1D22 77A123>84 D97 DI<137EEA01C138030180EA0703EA0E07121C00 3CC7FC12381278A35AA45B12701302EA300CEA1830EA0FC011157B9416>I<143CEB03F8 EB0038A31470A414E0A4EB01C013F9EA0185EA0705380E0380A2121C123C383807001278 A3EAF00EA31410EB1C201270133C38305C40138C380F078016237BA219>I<13F8EA0384 EA0E02121C123C1238EA7804EAF018EAFFE0EAF000A25AA41302A2EA6004EA7018EA3060 EA0F800F157A9416>I<143E144714CFEB018F1486EB0380A3EB0700A5130EEBFFF0EB0E 00A35BA55BA55BA55BA45B1201A2EA718012F100F3C7FC1262123C182D82A20F>II<13F0EA 0FE01200A3485AA4485AA448C7FC131FEB2180EBC0C0380F00E0A2120EA2381C01C0A438 380380A3EB070400701308130E1410130600E01320386003C016237DA219>I<13C0EA01 E013C0A2C7FCA8121E12231243A25AA3120EA25AA35AA21340EA7080A3EA71001232121C 0B217BA00F>I<13F0EA07E01200A3485AA4485AA448C7FCEB01E0EB0210EB0470380E08 F01310EB2060EB4000EA1D80001EC7FCEA1FC0EA1C70487EA27F142038703840A3EB1880 12E038600F0014237DA216>107 DI<391C0F80F8392610C10C3947606606 3987807807A2EB0070A2000EEBE00EA44848485AA3ED3820263803801340157016801530 3A7007003100D83003131E23157B9428>I<38380F80384C30C0384E4060388E8070EA8F 00128EA24813E0A4383801C0A3EB03840070138814081307EB031012E0386001E016157B 941B>I<137EEA01C338038180380701C0120E001C13E0123C12381278A338F003C0A214 80130700701300130E130CEA3018EA1870EA07C013157B9419>I<3801C1F03802621838 04741C3808780CEB700EA2141EEA00E0A43801C03CA3147838038070A2EBC0E0EBC1C038 072380EB1E0090C7FCA2120EA45AA3EAFFC0171F7F9419>I114 D<13FCEA018338020080EA0401EA0C03140090C7FC120F13F0EA07FC6C7EEA003E130F7F 1270EAF006A2EAE004EA4008EA2030EA1FC011157D9414>I<13C01201A4EA0380A4EA07 00EAFFF8EA0700A2120EA45AA45AA31310EA7020A213401380EA3100121E0D1F7C9E10> I<001E1360002313E0EA4380EB81C01283EA8701A238070380120EA3381C0700A31408EB 0E101218121CEB1E20EA0C263807C3C015157B941A>I<381C0180382603C0EA47071303 EA8701EA8E00A2000E13805AA338380100A31302A25B5B1218EA0C30EA07C012157B9416 >I<001EEB60E00023EBE0F0384380E1EB81C000831470D887011330A23907038020120E A3391C070040A31580A2EC0100130F380C0B02380613843803E0F81C157B9420>I<3803 C1E0380462103808347038103CF0EA203814601400C65AA45BA314203861C04012F11480 38E2C100EA4462EA383C14157D9416>I<001E133000231370EA438014E01283EA8700A2 380701C0120EA3381C0380A4EB0700A35BEA0C3EEA03CEEA000EA25B1260EAF038133048 5AEA80C0EA4380003EC7FC141F7B9418>I E /Fg 5 57 df<13C0A9B51280A23800C000 A911147E8F17>43 D<121FEA3180EA60C0EA4040EAC060A8EA4040EA60C0EA3180EA1F00 0B107F8F0F>48 D<1218127812981218AC12FF08107D8F0F>I<121FEA6180EA40C0EA80 6012C01200A213C0EA0180EA030012065AEA10201220EA7FC012FF0B107F8F0F>I<121F EA3180EA60C0A3EA3180EA3F00120EEA3380EA61C0EAC060A31340EA60C0EA1F000B107F 8F0F>56 D E /Fh 1 1 df0 D E /Fi 11 111 df15 D25 D28 D<124012E012601220A2 1240A2128003087D8209>59 D<380FFFC0380180601430EA0300A314E038060380EA07FF 380601C013005AA3EB018038180700EAFFFC14117E9017>66 D 68 D<380FFFC0EA030014401206A213081400485AEA0FF0EA0C10A2485A90C7FCA35A12 FE12117D9013>70 D<380F80FE3801C010A238026020A21330A238041840A2130CA23808 0680A21303A23818010012FE17117E9018>78 D<1320A21300A5EA0380EA04C01208A212 00EA0180A4EA0300A4124612CC12780B1780910D>106 D<3871F1F0389A1A18EA9C1CEA 9818121838303030A214321462386060641438170B7E8A1B>109 DI E /Fj 1 84 df 83 D E /Fk 45 122 df<136013E0A4EA03F8EA0FFE381FFF80383CE7C0EA78E13870E0 E012E013E1A2EBE0C0007013001278123F6C7EEA07FCC6B4FCEBEF80EBE3C013E1EBE0E0 12F0A312E03870E1C0EA78E3383CEF80381FFF006C5AEA03F0C65AA3136013277DA21A> 36 D<1338137813F0EA01E0EA03C0EA0780EA0F00120E5AA25AA25AA35AAA1270A37EA2 7EA27E120FEA0780EA03C0EA01E0EA00F8137813380D2878A21A>40 D<126012F012787E7E7EEA07801203EA01C0A2EA00E0A21370A31338AA1370A313E0A2EA 01C0A2EA03801207EA0F00121E5A5A5A12600D287CA21A>I<13E0A538F0E1E0EAFCE738 7EEFC0381FFF00EA07FCEA01F0EA07FCEA1FFF387EEFC038FCE7E0EAF0E13800E000A513 157D991A>I<13301378A8387FFFF0B512F8A26C13F038007800A8133015167E991A>I<12 18123E127E127F123F121F1207120EA2121C12FC12F81260080D77851A>I<1230127812 FCA212781230060676851A>46 D<14C0EB01E0A2130314C013071480130F1400A25B131E 133E133C137C1378A213F85B12015B12035BA212075B120F90C7FC5A121EA2123E123C12 7C127812F85AA2126013277DA21A>II<13C01201A212031207120F127F12FD12711201B2EA7FFF A3101E7B9D1A>III<383FFFC05AA20070C7FC A8EA71F8EA7FFEEBFF80387E07C0EA7801383000E0C7FC1470A3126012F014E0EAE00138 7003C0387C0F80383FFF00EA0FFEEA03F0141E7D9D1A>53 D<12E0B512F8A214F038E000 E0EB01C0EA0003EB0780EB0F00130E5BA25BA25BA25BA3485AA4485AA8151F7E9E1A>55 DII<1230127812FCA2127812301200A91230127812FCA212781230061576941A>I<1218 123C127EA2123C12181200A91218123C127EA2123E121E120EA2121C123C12F812F01260 071C77941A>I<387FFFF0B512F8A26C13F0C8FCA4387FFFF0B512F8A26C13F0150C7E94 1A>61 D<1338137CA2136C13EEA313C6A2EA01C7A3138300031380A4380701C0A213FFA2 4813E0EA0E00A3001E13F0001C1370387F01FC38FF83FE387F01FC171E7F9D1A>65 D68 D<387FFFFEB5FCA238E0380EA400001300B3A23803FF80A3171E7F9D1A>84 D91 D93 D97 D99 DIII<3801F87C3807FFFE5A381E078C381C0380 383801C0A5381C0380EA1E07381FFF005BEA39F80038C7FCA27E381FFF8014E04813F838 78007C0070131C48130EA40070131C0078133C003E13F8381FFFF0000713C00001130017 217F941A>I<127E12FE127E120EA6133EEBFF80000F13C013C1EB80E01300120EAC387F C3FC38FFE7FE387FC3FC171E7F9D1A>I<13C0487EA26C5A90C7FCA6EA7FE0A31200AF38 7FFF80B512C06C1380121F7C9E1A>I108 D<387CE0E038FFFBF8EA7FFF381F1F1CEA1E1EA2EA1C1CAC387F1F 1F39FFBFBF80397F1F1F00191580941A>IIII<387F81F838FF8FFC387F9FFE3803FE1EEBF80CEBE000A25B5BAAEA7FFF B5FC7E17157F941A>114 D<3807FB80EA1FFF127FEA7807EAE003A30078C7FCEA7FC0EA 1FFCEA07FE38003F801307386001C012E0A2EAF00338FC0780B51200EAEFFEEAE3F81215 7C941A>I<487E1203A6387FFFE0B5FCA238038000AA1470A43801C1E013FF6C1380EB3F 00141C7F9B1A>I<387E07E0EAFE0FEA7E07EA0E00AD1301EA0F033807FFFC6C13FE3800 FCFC17157F941A>I<387F83FC38FFC7FE387F83FC380E00E0A3380701C0A338038380A3 3801C700A3EA00EEA3137CA2133817157F941A>I<38FF83FEA338380038A26C1370A313 38137CA2380C6C60380EEEE0A413C6000613C0EA07C71383A217157F941A>I<387FC7F8 EBCFFCEBC7F8380703C038038380EBC700EA01EFEA00FE137C13781338137C13EE120113 C738038380000713C0EA0F01387FC7FC00FF13FE007F13FC17157F941A>I<387F83FC38 FFC7FE387F83FC380E00E0A27EEB01C0A2EA0381EB838013C31201EBC700EA00E7A213E6 1366136E133CA31338A35BA2EA30F0EA78E01271EA7FC06C5A001EC7FC17207F941A>I E /Fl 22 119 df<130813101320134013C0EA0180EA0300A21206A25AA2121C12181238 A212301270A4126012E0AF12601270A412301238A21218121C120CA27EA27EA2EA0180EA 00C013401320131013080D3B798117>0 D<7E12407E7E12187E7EA27EA2EA0180A213C0 120013E0A213601370A413301338AF13301370A4136013E0A213C012011380A2EA0300A2 1206A25A5A12105A5A5A0D3B7E8117>I II<14C0EB0700131C5B5B136013E0B2485A A2485A000FC7FC123812C01238120FEA03806C7EA26C7EB2136013707F7F1307EB00C012 3B7B811D>8 D<12C01238120FEA03806C7EA26C7EB2136013707F7F1307EB00C0EB0700 131C5B5B136013E0B2485AA2485A000FC7FC123812C0123B7B811D>I<12C0B3AE022079 8011>12 D18 D<12C012607E7E7E7E7E7F6C7E12007F13707FA27FA27F130F7F801303801301A2801300 80A214701478A21438143CA3141C141EA4140E140FA7801580B3A315005CA7140E141EA4 141C143CA314381478A2147014F0A25C13015CA213035C130791C7FC5B130E5BA25BA25B 5B5B1201485A90C8FC12065A5A5A5A5A19777F8125>I<15C0EC0180EC03005C140E140C 5C14385C146014E0495AA2495A130791C7FC5B130E131E131C133C13381378A25BA3485A A3485AA3485AA3120F90C8FCA35AA2121EA3123EA4123CA2127CA8127812F8B3AA127812 7CA8123CA2123EA4121EA3121FA27EA37F1207A36C7EA36C7EA36C7EA31378A21338133C 131C131E130E130F7F8013036D7EA26D7E1460147080141880140E8080EC0180EC00C01A 94758127>32 D<12C012607E12387E120C7E12076C7E12017F6C7EA2137013781338133C 131C131E130E130F7F80A26D7EA36D7EA36D7EA31478A3147C143CA3143EA2141EA3141F A480A21580A8140715C0B3AA1580140FA81500A25CA4141EA3143EA2143CA3147C1478A3 5CA3495AA3495AA3495AA291C7FC5B130E131E131C133C1338137813705BA2485A5B1203 48C8FC12065A121C5A12305A5A1A947F8127>III80 D<140FEC30C0EC60E014C1A2ECC0C001011300A25C1303A71307A691C7FC5BA9130E131E A6131CA713181338A2EA603012F05BEAE040EA6180001FC8FC1B377D7F18>82 D88 D<160FEE1880EE31C0EE63E016E3A2923801C1C0EEC000A24B5AA21507A293C7FCA25DA3 151EA4153E153CA3157CA3157815F8A54A5AA41403A25DA21407A45D140FA54A5AA54AC8 FCA5143E147EA4147CA214FCA25CA4495AA55C1303A35CA313075CA449C9FCA3130EA213 1E131CA35BA2EA7030EAF870136013E0EA70C0EA2180001ECAFC2B6F7D7F1C>90 D<1704170C1718A21730A21760A217C0A2EE0180A2EE0300A21606A35EA25EA25EA25EA2 5EA24B5AA24BC7FC1202000714065AD837805B126700875CEA03C05D6C7E5D6C7E5DA290 38780180A2133C4AC8FC131E1406A26D5AA2EB0798A2EB03F0A25C13015C13002E3C7B81 32>112 D<1704170CA21718A31730A31760A317C0A3EE0180A3EE0300A31606A45EA35E A35EA35EA35EA34B5AA34BC7FCA31202000714065A5AD837805B1267124700875CEA03C0 A25D6C7EA25DA26C7E5DA213784A5AA2133CA24AC8FCA2131E1406A2130F5CA3EB0798A3 EB03F0A35C1301A25C1300A22E5A7B8132>I<1503AB1203A25A7F120F121FEA13C01233 1223EA41E0A21281EA80F01200A21378A37FA37FA37FA3EB0780A3EB03C0A3EB01E0A3EB 00F0A31478A3143CA3141EA3140FA3EC0783A3EC03C3A3EC01E3A3EC00F3A3157BA3153F A3151FA3150FA31507A315031502205B7B8035>116 D<12C0B3AE02205D8035>I<387FFF E0B5FC00C0C7FCB3AB131F5D8135>I E /Fm 9 107 df0 D<12C012F01238120E6C7EEA01E0EA0078131C1307EB03C0EB00F0A2EB01C0EB0700131E 1378EA01E0EA0380000EC7FC123C127012C0C8FCA6387FFFE0B512F0141E7D951B>21 D<390F8007C03919E01C203920303010394018400839C00C8004EA80071400EB03801307 903804C00C394008600839203030103910E01E60390F8007C01E0E7E8D23>49 D<3801FF801207000EC7FC12185A5AA35AA2B51280A200C0C7FCA21260A37E7E120E3807 FF80120111167D9218>I<1303A21306A2130CA21318A21330A21360A213C0A2EA0180A2 EA0300A21206A25AA25AA25AA25AA25A1240101E7B9600>54 D<134013E0A2EA01B0A2EA 0318A2EA060CA2487EA2487EA238300180A2386000C0A2481360142013137E9218>94 D<137813C0EA0180EA0300AB12065A12F0120C7E7EABEA0180EA00C013780D217E9812> 102 D<12F0120C7E7EABEA0180EA00C0137813C0EA0180EA0300AB12065A12F00D217E98 12>I<12C0B3AF02217C980A>106 D E /Fn 30 113 df0 D<6C13026C13060060130C6C13186C13306C13606C13C03803018038018300EA00C6136C 1338A2136C13C6EA018338030180380600C048136048133048131848130C481306481302 1718789727>2 D14 D17 D<150C153C15F0EC03C0EC0F00143C14F0EB03C0010FC7FC133C13 F0EA03C0000FC8FC123C12F0A2123C120FEA03C0EA00F0133C130FEB03C0EB00F0143C14 0FEC03C0EC00F0153C150C1500A8007FB512F8B612FC1E287C9F27>20 D<12C012F0123C120FEA03C0EA00F0133C130FEB03C0EB00F0143C140FEC03C0EC00F015 3CA215F0EC03C0EC0F00143C14F0EB03C0010FC7FC133C13F0EA03C0000FC8FC123C1270 12C0C9FCA8007FB512F8B612FC1E287C9F27>I25 D<90380FFFFC137FD801F0C7FCEA03800006C8FC5A5A5AA25AA25AA81260A27EA27E 7E7E6C7EEA01E039007FFFFC131F1E1E7C9A27>I<38E001C0387000E0001C1338000F13 1E380380073900E001C090387800F0011C13380107130E903903C00780903900E001C091 38380070021E133C0207130E913901C00380913900F001E0A2913901C00380913907000E 00021E133C023813709138E001C0903903C00780902607000EC7FC011C1338017813F090 38E001C026038007C8FC380F001E001C1338007013E048485A2B207D9B32>29 D<4B7EA46F7EA2166082A2161C8282B812E0A2C9EA0700160E5E1630A25E5EA24B5AA42B 1A7D9832>33 D<1570033C136092380FFFE0030313C0ED00031607160E161C1638167016 E0ED01C0ED0380ED0700030E13605D4B13305D4B13104A4813004A5A4AC8FC140E5C5C5C 5C495A495A49C9FC130E5B5B5B5B485A485A48CAFC120E5A5A5A5A5A2C2C7DA332>37 D<12C07E12707E7E7E7E6C7E6C7E6C7E13707F7F7F7F6D7E6D7E6D7E1470808080806E7E 6E7E6E6C131003701330816F1360816F13C0ED0380ED01C0ED00E016701638161C160E16 071603ED03FF030F13E092383C0060037013002C2C7DA032>I50 D<1403A21406A2140CA21418A21430A21460A214C0A3EB01 80A2EB0300A21306A25BA25BA25BA25BA25BA2485AA248C7FCA31206A25AA25AA25AA25A A25A1240183079A300>54 D<0040141000C0143000601460A36C14C0A36CEB0180A26CEB 0300A33807FFFEA23803000CA36C6C5AA36C6C5AA2EB6060A36D5AA2EB1980A3010FC7FC A31306A21C2480A21D>56 DI<14801301EA01F938070F00EA0C03001813801307383006C0A238700CE0006013 60A338E01870A31330A31360A413C0A3EAE180A200611360EA6300007313E0A2003313C0 1236381E0180A2380C0300EA0F0EEA0DF80018C7FCA3142A7EA519>59 D<91387FFFFE49B6FC130790390C1E000C0130140801701400EBE01CA23801C03C1300C7 1238A21478A21470A291B5FCA24913FC02C0C7FCA213035CA249C8FCA2130EA25B1220EA 703812F86C5AEA7FC06C5A001EC9FC28247FA124>70 D<171E17FEEE01FC1603026014F8 D901E014C0EE06006E130401035CA3EB02F85E1478A2D9067C5B1304143CA2023E5B1308 141E141F5E497EA2158001200181C7FC140715C11340EC03E2A290388001F212010063C7 12FE007F5C48147C481478481430003891C8FC2F2980A629>78 D<0040144000C014C0B3 A40060EB0180A26CEB03006C1306000E131C380780783801FFE038007F801A1F7D9D21> 91 DI<130CA2131EA31333A2EB6180A2EBC0C0A338018060A248487EA3 00067FA2487FA3487FA2487FA248EB0180A348EB00C015401A1F7D9D21>94 D102 D<12F8120FEA03806C7E6C7EB113707F131EEB03C0EB1E00 13385B5BB1485A485A000FC7FC12F812317DA419>I<1320136013C0A3EA0180A3EA0300 A21206A35AA35AA25AA35AA35AA21260A37EA37EA27EA37EA37EA2EA0180A3EA00C0A313 6013200B327CA413>I<12C0A21260A37EA37EA27EA37EA37EA2EA0180A3EA00C0A31360 A213C0A3EA0180A3EA0300A21206A35AA35AA25AA35AA35AA20B327DA413>I<12C0B3B3 AD02317AA40E>II<12C0A21260A37EA37EA3 7EA37EA37EA36C7EA36C7EA31360A37FA37FA37FA37FA37FA3EB0180A3EB00C014401231 7DA419>110 D<160116031606A2160CA21618A21630A21660A216C0A2ED0180A2ED0300 A21506A25DA25DA25D1206001E5C122F004F5CEA87800007495AEA03C04AC7FCA23801E0 06A26C6C5AA2EB7818A26D5AA26D5AA26D5AA26D5AA26DC8FCA228327D812A>112 D E /Fo 25 86 df<1402EB7F043801C1C838070078000C131848132C0038136E003013 4600701387130100E0EB038013021304A213081310A21320D8604013003870C007383080 063839000E001B130C000E5B00065B380DC1C0D8087FC7FC48C8FC5A191D7E9A1E>31 D48 D<12035AB4FC1207B3A2EA7FF80D187D9713>III<1318A21338137813F813B8EA01381202A2 12041208121812101220124012C0B5FCEA0038A6EA03FF10187F9713>III<1240EA7FFF13FEA2EA4004EA80081310A2EA00201340A21380120113 005AA25A1206A2120EA5120410197E9813>III<130CA3131EA2132F1327A2EB4380A3EB81C0A200017F1300A248B47E 38020070A2487FA3487FA2003C131EB4EBFFC01A1A7F991D>65 D67 DIII<39FFE1FFC0390E001C00AB380FFFFC380E001CAC39FFE1 FFC01A1A7F991D>72 DI76 D<00FEEB7FC0000FEB0E001404EA 0B80EA09C0A2EA08E01370A21338131CA2130E1307EB0384A2EB01C4EB00E4A21474143C A2141C140C121C38FF80041A1A7F991D>78 D<137F3801C1C038070070000E7F487F003C 131E0038130E0078130F00707F00F01480A80078EB0F00A20038130E003C131E001C131C 6C5B6C5B3801C1C0D8007FC7FC191A7E991E>I82 DI< 007FB5FC38701C0700401301A200C0148000801300A300001400B13803FFE0191A7F991C >I<39FFE07FC0390E000E001404B200065B12076C5B6C6C5A3800E0C0013FC7FC1A1A7F 991D>I E /Fp 39 128 df<903801F03C9038071C47010C13C7EC19C690381C01801403 13181338A2EC0700A20003B512F03900700700A3140EA213E0A35CA2EA01C0A35CA2EA03 80A21430EB0070A248136038C630E038E638C038CC3180D8781EC7FC2025819C19>11 D45 D<1230127812F0126005047C830D>I<1418A21438A21478 A214B8EB0138A2EB023C141C1304130C13081310A21320A2EB7FFCEBC01C1380EA010014 1E0002130EA25A120C001C131EB4EBFFC01A1D7E9C1F>65 D<48B5FC39003C0380903838 01C0EC00E0A35B1401A2EC03C001E01380EC0F00141EEBFFFC3801C00E801580A2EA0380 A43907000F00140E141E5C000E13F0B512C01B1C7E9B1D>I<48B5FC39003C03C0903838 00E0A21570A24913781538A215785BA4484813F0A315E03803800115C014031580390700 0700140E5C5C000E13E0B512801D1C7E9B1F>68 D<48B512F038003C00013813301520A3 5BA214081500495AA21430EBFFF03801C020A439038040801400A2EC0100EA07005C1402 1406000E133CB512FC1C1C7E9B1C>I<3A01FFC3FF803A003C00780001381370A4495BA4 49485AA390B5FC3901C00380A4484848C7FCA43807000EA448131E39FFE1FFC0211C7E9B 1F>72 DI<3A01FFC07F803A003C001E000138131815205D5DD97002C7FC5C5C5CEB E04014C013E1EBE2E0EA01C4EBD07013E013C048487EA21418141CEA070080A348130F39 FFE07FC0211C7E9B20>75 D<3801FFC038003C001338A45BA45BA4485AA438038002A314 04EA0700140C14181438000E13F0B5FC171C7E9B1A>I79 D<3801FFFE39003C038090383801C0EC00E0A3EB7001A315C0EBE0031580EC0700141C38 01FFF001C0C7FCA3485AA448C8FCA45AEAFFE01B1C7E9B1C>I83 D<001FB512C0381C070138300E0000201480126012405B1280A2000014005BA45BA45BA4 485AA41203EA7FFE1A1C799B1E>I97 D<123F1207A2120EA45AA4EA39E0EA 3A18EA3C0C12381270130EA3EAE01CA31318133813301360EA60C0EA3180EA1E000F1D7C 9C13>I<13F8EA0304120EEA1C0EEA181CEA30001270A25AA51304EA60081310EA3060EA 0F800F127C9113>II<13F8EA0704120C EA1802EA38041230EA7008EA7FF0EAE000A5EA60041308EA30101360EA0F800F127C9113 >IIIII107 DI< 391C1E078039266318C0394683A0E0384703C0008E1380A2120EA2391C0701C0A3EC0380 D8380E1388A2EC0708151039701C032039300C01C01D127C9122>II<13F8EA030CEA0E06487E1218123000701380A238E00700A3130EA25BEA6018 5BEA30E0EA0F8011127C9115>I<380387803804C860EBD03013E0EA09C014381201A238 038070A31460380700E014C0EB0180EB8300EA0E86137890C7FCA25AA45AB4FC151A8091 15>IIII< 12035AA3120EA4EAFFE0EA1C00A35AA45AA4EAE080A2EAE100A2126612380B1A7C990E> I<381C0180EA2E03124EA2388E0700A2121CA2EA380EA438301C80A3EA383C38184D00EA 0F8611127C9116>II<381E0183382703871247148338870701A2120E A2381C0E02A31404EA180C131C1408EA1C1E380C26303807C3C018127C911C>I<381C01 80EA2E03124EA2388E0700A2121CA2EA380EA4EA301CA3EA383CEA1878EA0FB8EA003813 301370EAE0605BEA81800043C7FC123C111A7C9114>121 D127 D E /Fq 14 62 df<1318A2133CA2134EA21387A238010380000313C0EA02 01000613E0EA04004813F01470481378143848133C141C48131E387FFFFEB6FCA218177E 961D>1 D10 D<120112021204120C1218A21230A212701260A312E0AA1260A312701230A21218A2120C 12041202120108227D980E>40 D<12801240122012301218A2120CA2120E1206A31207AA 1206A3120E120CA21218A2123012201240128008227E980E>I<1330ABB512FCA2380030 00AB16187E931B>43 D48 D<1206120E12FE120EB1EAFFE00B157D9412> III<1330A2137013F012011370120212041208121812101220124012C0EAFF FEEA0070A5EA03FE0F157F9412>II56 DI61 D E /Fr 43 123 df14 D<137EEA0380EA0700120E5A5A12781270EAFFF0EAF000A25AA5 1270A2EA3806EA1C18EA07E00F157D9414>I<130FEB3180EB60C013E03801C0E0138012 03EA0700A25A120E121EA2EA1C01123CA3387FFFC0EA7803A33870078012F014005B130E A2485A12605BEA70305BEA30C0EA1180000FC7FC13237EA217>18 D21 D<000FB5FC5A5A3870418000401300EA8081A21200EA01831303A21203 A21206A2120EA2000C1380121CA2EA180118157E941C>25 DI<90387FFF8048B5FC5A390783C000EA0E0148 6C7E5AA25AA348485AA3495A91C7FCEA6007130EEA3018EA1870EA07C019157E941C>I< 380FFFF84813FC4813F8387020001240128013601200A25BA412015BA21203A348C7FC7E 16157E9415>I<000414704814F015F84814781538481418A2D840031310A21306A21520 EAC004010C134015C090381E018038E07607397FE7FF005C383F83FC381F01F01D158094 1E>33 D<127012F8A3127005057C840E>58 D<127012F812FCA212741204A41208A21210 A212201240060F7C840E>I<15181578EC01E0EC0780EC1E001478EB03E0EB0F80013CC7 FC13F0EA03C0000FC8FC123C12F0A2123C120FEA03C0EA00F0133CEB0F80EB03E0EB0078 141EEC0780EC01E0EC007815181D1C7C9926>I<14801301A2EB0300A31306A35BA35BA3 5BA35BA35BA3485AA448C7FCA31206A35AA35AA35AA35AA35AA311317DA418>I<12C012 F0123C120FEA03C0EA00F0133EEB0F80EB01E0EB0078141EEC0780EC01E0EC0078A2EC01 E0EC0780EC1E001478EB01E0EB0F80013EC7FC13F0EA03C0000FC8FC123C12F012C01D1C 7C9926>I64 D<90B512F090380F001E81ED0780011E1303A216C0A24914801507A2ED0F0049131E5D5D EC03E090B55A9038F000F0157881485A151C151EA248485BA35D485A5D4A5AEC0380000F 010FC7FCB512F822227DA125>66 D<027F138090390380810090380E0063013813274913 1F49130E485A485A48C7FC481404120E121E5A5D4891C7FCA35AA55A1520A25DA26C5C12 704AC7FC6C130200185B001C5B00061330380381C0D800FEC8FC21247DA223>I<90B512 F090380F003C150E81011EEB0380A2ED01C0A25B16E0A35BA449EB03C0A44848EB0780A2 16005D4848130E5D153C153848485B5D4A5A0207C7FC000F131CB512F023227DA128>I< 90B6128090380F00071501A2131EA21600A25BA2140192C7FCEB7802A21406140EEBFFFC EBF00CA33801E008A391C8FC485AA4485AA4120FEAFFFC21227DA120>70 D<9039FFF83FFE90390F0003C0A3011EEB0780A449EB0F00A449131EA490B512FC9038F0 003CA348485BA448485BA44848485AA4000F130339FFF83FFE27227DA128>72 D<9039FFF801FF010FC71278166016C0011EEB010015025D5D4913205D5D0202C7FC495A 5C141C147CEBF0BEEBF11E13F2EBF80FEA01F001E07F1407A248486C7EA36E7EEA078081 1400A2000F497E39FFF80FFF28227DA129>75 D77 D<9039FF8007FE010FEB00F016C0D90BC013 4001131480A2EB11E0A2903921F001001320A2147801401302147C143CA2496C5AA3140F D801005B15881407A20002EB03D0A215F01401485C1400A2120C001E1440EAFFC027227D A127>I<147F90380381C090380E0060013813385B49131C4848131E4848130E48C7FC48 140F120E5A123CA25AA348141EA3153CA3481478A215F06C14E01401EC03C00070EB0780 007814000038130E6C13386C5B380783C0D800FEC7FC20247DA225>I<90B512C090380F 0078151C81011E130F811680A249EB0F00A3151E495B5D15E0EC0380D9FFFCC7FCEBF007 6E7E8148486C7EA44848485AA44848485A1680A29138038100120F39FFF801C6C8127821 237DA125>82 D<903803F81090380E0420903818026090382001E0EB400001C013C05B12 01A200031480A21500A27FEA01F013FE3800FFE06D7EEB1FF8EB01FCEB003C141C80A300 20130CA3140800601318141000705B5C00C85BD8C603C7FCEA81FC1C247DA21E>I<3BFF F03FFC03FF3B1F0003E000786C4A137017601740A217800207EB0100A2020B130214135E 14236F5A02415BA202815B1381000701015B13825E018413E193C7FC018813E2A2019013 E413A015E801C013F86E5A495BA290C75A120600025C30237DA12E>87 D<90397FF803FF903907E000F84A13E0010314C09138E001800101EB03001506ECF00401 005B6E5AEC7820EC7C405D023DC7FC143E141E141FA2142FEC6F8014C790380187C01403 01027F1304EB080101107FEB2000497F49137848C7FC48147CD80F8013FC3AFFE003FFC0 28227FA128>I97 DI<133FEBE0 80380380C0EA0701EA0E03121C003CC7FCA25AA35AA400701340A23830018038380200EA 1C1CEA07E012157E9415>I<141E14FC141CA31438A41470A414E01378EA01C4EA030238 0601C0120E121C123C383803801278A338F00700A31408EB0E101270131E38302620EA18 C6380F03C017237EA219>I<137EEA038138070080120E5A5A38780100EA7006EAFFF800 F0C7FCA25AA41480A238700300EA3004EA1838EA0FC011157D9417>I<147014F0A21460 1400A9130FEB3180EB41C01381A2EA0101A238000380A4EB0700A4130EA45BA45BA3EA70 70EAF0605BEA6380003EC7FC142C81A114>106 D<13F0EA0FE01200A3485AA4485AA448 C7FC1478EB0184EB021C380E0C3C1310EB2018EB4000485A001FC7FC13E0EA1C38487EA2 7F140838701C10A3EB0C20EAE006386003C016237EA219>I<383C07C038461860384720 303887403813801300A2000E1370A44813E0A2EB01C014C1003813C2EB03821484130100 701388383000F018157E941D>110 D<133EEBC180380380C0380700E0120E4813F0123C A25AA338F001E0A214C0130300701380EB07001306EA381C6C5AEA07E014157E9417>I< 3803C0F03804631CEB740EEA0878EB7007A2140FEA00E0A43801C01EA3143C38038038A2 EBC07014E038072180EB1E0090C7FCA2120EA45AA3B47E181F819418>I<136013E0A4EA 01C0A4EA0380EAFFFCEA0380A2EA0700A4120EA45AA31308EA3810A21320EA184013C0EA 0F000E1F7F9E12>116 D<001E131800231338EA438014701283A2EA8700000713E0120E A3381C01C0A314C2EB0384A21307380C0988EA0E113803E0F017157E941C>I<3801E0F0 3806310C38081A1C0010133CEA201C14181400C65AA45BA314083860E01012F0142038E1 704038423080383C1F0016157E941C>120 D<001E131800231338EA438014701283EA87 00A2000713E0120EA3381C01C0A4EB0380A21307EA0C0B380E1700EA03E7EA0007A2130E 1260EAF01C1318485AEA8060EA41C0003FC7FC151F7E9418>II E /Fs 76 127 df<1460A214F0A2497E14 78EB027C143CEB043E141EEB081F8001107F140701207F140301407F140101807F140048 C77E15780002147C153C48143E151E48141F8148158015074815C01503007FB612E0A2B7 12F024237EA229>1 D10 D<90381FC1F090387037189038C03E3C3801807C0003137839 07003800A9B612C03907003800B2143C397FE1FFC01E2380A21C>II<90380FC07F90397031C0809039E00B00402601801E13E00003EB3E013807003C 91381C00C01600A7B712E03907001C011500B23A7FF1FFCFFE272380A229>14 D34 D<127012F812FCA212741204A41208A21210A212201240060F7CA20E>39 D<132013401380EA01005A12061204120CA25AA25AA312701260A312E0AE1260A3127012 30A37EA27EA2120412067E7EEA0080134013200B327CA413>I<7E12407E7E12187E1204 1206A27EA2EA0180A313C01200A313E0AE13C0A312011380A3EA0300A21206A21204120C 5A12105A5A5A0B327DA413>I<497EB0B612FEA23900018000B01F227D9C26>43 D<127012F812FCA212741204A41208A21210A212201240060F7C840E>II<127012F8A3127005057C840E>I48 D<13801203120F12F31203B3A9EA07C0EAFFFE0F217CA018>III<1303A25BA25B1317A21327136713 471387120113071202120612041208A212101220A2124012C0B512F838000700A7EB0F80 EB7FF015217FA018>I<00101380381E0700EA1FFF5B13F8EA17E00010C7FCA6EA11F8EA 120CEA1C07381803801210380001C0A214E0A4127012F0A200E013C01280EA4003148038 200700EA1006EA0C1CEA03F013227EA018>I<137EEA01C138030080380601C0EA0C0312 1C381801800038C7FCA212781270A2EAF0F8EAF30CEAF4067F00F81380EB01C012F014E0 A51270A3003813C0A238180380001C1300EA0C06EA070CEA01F013227EA018>I<124012 60387FFFE014C0A23840008038C0010012801302A2485A5BA25B5BA21360134013C0A212 01A25B1203A41207A76CC7FC13237DA118>III<127012F8A312701200AB127012F8A3127005157C940E>I61 D<497EA3497EA3EB05E0A2EB09F01308A2EB1078 A3497EA3497EA2EBC01F497EA248B51280EB0007A20002EB03C0A348EB01E0A348EB00F0 121C003EEB01F839FF800FFF20237EA225>65 DI<903807E0109038 381830EBE0063901C0017039038000F048C7FC000E1470121E001C1430123CA2007C1410 1278A200F81400A812781510127C123CA2001C1420121E000E14407E6C6C13803901C001 003800E002EB381CEB07E01C247DA223>IIII<903807F00890383C0C 18EBE0023901C001B839038000F848C71278481438121E15185AA2007C14081278A200F8 1400A7EC1FFF0078EB00F81578127C123CA27EA27E7E6C6C13B86C7E3900E0031890383C 0C08903807F00020247DA226>I<39FFFC3FFF390FC003F039078001E0AE90B5FCEB8001 AF390FC003F039FFFC3FFF20227EA125>II<3803FFE038001F007FB3A6127012F8A2130EEAF01EEA401C6C5AEA1870EA 07C013237EA119>IIII<39FF8007FF 3907C000F81570D805E01320EA04F0A21378137C133C7F131F7FEB0780A2EB03C0EB01E0 A2EB00F014F81478143C143E141E140FA2EC07A0EC03E0A21401A21400000E1460121FD8 FFE0132020227EA125>III82 D<3803F020380C0C60EA1802 383001E0EA70000060136012E0A21420A36C1300A21278127FEA3FF0EA1FFE6C7E000313 8038003FC0EB07E01301EB00F0A214707EA46C1360A26C13C07E38C8018038C60700EA81 FC14247DA21B>I<007FB512F839780780780060141800401408A300C0140C00801404A4 00001400B3A3497E3801FFFE1E227EA123>I<39FFFC07FF390FC000F86C4813701520B3 A5000314407FA2000114806C7E9038600100EB3006EB1C08EB03F020237EA125>I<3BFF F03FFC03FE3B1F8007E000F86C486C48137017206E7ED807801540A24A7E2603C0021480 A39039E004780100011600A2EC083CD800F01402A2EC101E01785CA2EC200F013C5CA202 60138890391E400790A216D090391F8003F0010F5CA2EC00016D5CA20106130001025C2F 237FA132>87 D89 D<12FEA212C0B3B3A912FEA207317BA40E>91 D<12FEA21206B3B3A912FEA207317FA40E>93 D97 D<120E12FE121E120EAB131FEB61C0EB8060380F0030000E1338143C141C141E A7141C143C1438000F1370380C8060EB41C038083F0017237FA21B>II<14E0130F13011300ABEA01F8EA0704EA0C02EA1C01EA38001278 127012F0A7127012781238EA1801EA0C0238070CF03801F0FE17237EA21B>II<133E13E33801C780EA0387130748C7FC A9EAFFF80007C7FCB27FEA7FF0112380A20F>I<14703803F198380E1E18EA1C0E383807 00A200781380A400381300A2EA1C0EEA1E1CEA33F00020C7FCA212301238EA3FFE381FFF C06C13E0383000F0481330481318A400601330A2003813E0380E03803803FE0015217F95 18>I<120E12FE121E120EABEB1F80EB60C0EB80E0380F0070A2120EAF38FFE7FF18237F A21B>I<121C123EA3121CC7FCA8120E127E121E120EB1EAFFC00A227FA10E>I<13E0EA01 F0A3EA00E01300A81370EA07F012001370B3A51260EAF0E013C0EA6180EA3F000C2C83A1 0F>I<120E12FE121E120EABEB03FCEB01F014C01480EB02005B5B5B133813F8EA0F1CEA 0E1E130E7F1480EB03C0130114E0EB00F014F838FFE3FE17237FA21A>I<120E12FE121E 120EB3ADEAFFE00B237FA20E>I<390E1FC07F3AFE60E183803A1E807201C03A0F003C00 E0A2000E1338AF3AFFE3FF8FFE27157F942A>I<380E1F8038FE60C0381E80E0380F0070 A2120EAF38FFE7FF18157F941B>III<3801F82038070460EA0E 02EA1C01003813E0EA7800A25AA71278A2EA3801121CEA0C02EA070CEA01F0C7FCA9EB0F FE171F7E941A>III<1202A41206A3120E121E 123EEAFFFCEA0E00AB1304A6EA07081203EA01F00E1F7F9E13>I<000E137038FE07F0EA 1E00000E1370AD14F0A238060170380382783800FC7F18157F941B>I<38FF80FE381E00 781430000E1320A26C1340A2EB80C000031380A23801C100A2EA00E2A31374A21338A313 1017157F941A>I<39FF8FF87F393E01E03C001CEBC01814E0000E1410EB026014700007 1420EB04301438D803841340EB8818141CD801C81380EBD00C140E3900F00F00497EA2EB 6006EB400220157F9423>I<38FF83FE381F00F0000E13C06C1380EB8100EA0383EA01C2 EA00E41378A21338133C134E138FEA0187EB0380380201C0000413E0EA0C00383E01F038 FF03FE17157F941A>I<38FF80FE381E00781430000E1320A26C1340A2EB80C000031380 A23801C100A2EA00E2A31374A21338A31310A25BA35B12F05B12F10043C7FC123C171F7F 941A>I<383FFFC038380380EA300700201300EA600EEA401C133C1338C65A5B12015B38 038040EA07005A000E13C04813805AEA7801EA7007B5FC12157F9416>I126 D E /Ft 25 122 df<144014E0A3497EA2497E EB0278A2497EA3497EA2497EA3496C7EA201407F1403A290B57EA239018001F090C7FCA2 00021478A34880A2001E143E3AFFC003FFE0A223237DA229>65 D<903803F80290381FFF 0690387E03863901F000CE4848133ED80780131E48C7FC48140E001E1406123E123C007C 1402A2127800F81400A81278007C1402A2123C123E001E1404121F6C14086C6C1318D803 E013306C6C136039007E03C090381FFF00EB03FC1F247CA227>67 D69 D73 D80 D82 D<1304130EA3131FA2EB2F801327A2EB43C0A2EBC3E01381A248 C67EA2487F13FF38020078487FA3487F1218003C131F00FEEB7FE01B1A7F991F>97 DIIIII<39FFC1FF80391E003C00AB381FFFFC381E003CAC39FFC1 FF80191A7E991F>104 DI108 D<00FEEB01FE001E14F0A2 00171302A238138004A33811C008A23810E010A3EB7020A3EB3840A2EB1C80A3EB0F00A2 1306123800FEEB07FE1F1A7E9925>I<00FEEB3F80001FEB0E00EB80041217EA13C0EA11 E013F012101378137C133C131E130F14841307EB03C4EB01E4A2EB00F4147CA2143C141C 140C123800FE1304191A7E991F>III114 DI< 007FB5FC38701E0700601301124000C0148000801300A300001400B0133F3803FFF0191A 7F991D>I<39FFC03F80391E000E001404B2000E5B120F6C5B6C6C5A3800E0C0013FC7FC 191A7E991F>I<39FFE07F80391F803E00000F1318000713106C6C5A13E06C6C5A00005B 13F9017DC7FC7FA27F7FEB1780EB37C01323EB41E0EB81F0EA0180EB00780002137C0006 7F000E131E001E133FB4EB7FE01B1A7F991F>120 D<39FFC007F0393F0003806C140038 0F800200071306EBC0046C6C5A12016D5A3800F830EB7820EB7C40133EEB1E80131F6DC7 FCAAEB7FE01C1A7F991F>I E /Fu 3 107 df0 D<12C012F0123C120FEA03C0EA00F01338130E6D7EEB01E0EB0078141EEC0780A2EC1E00 1478EB01E0EB0780010EC7FC133813F0EA03C0000FC8FC123C127012C0C9FCA8007FB5FC B6128019247D9920>21 D<12C0B3B3A502297B9E0C>106 D E /Fv 8 121 df<1338137FEB87803801030090C7FC7FA27F12007FA2137013F8EA03B8EA063C EA0C1C121812381270A212E0A413181338EA6030EA70606C5AEA0F80111E7F9D12>14 DI<380FFFE05A5A3860C0001240485A12001201A348C7FCA35AA3120E120613127E 9112>28 D<126012F0A2126004047C830C>58 D<48B5128039003C01E090383800701538 153C151C5B151EA35BA44848133CA3153848481378157015F015E039070001C0EC0380EC 0700141C000E1378B512C01F1C7E9B22>68 D79 D<3801FFFE39003C03C090383800E015F01570A24913F0A3EC01E001E013C0EC0780EC1E 00EBFFF03801C038140C140EA2EA0380A43807001E1508A2151048130FD8FFE01320C7EA 03C01D1D7E9B20>82 D<380787803808C8403810F0C03820F1E0EBE3C03840E1803800E0 00A2485AA43863808012F3EB810012E5EA84C6EA787813127E9118>120 D E /Fw 3 121 df15 D<3807FFE03800E0703801C018140CA2140EEA0380A43807001CA31438000E1330147014 E0EB01C0381C0700EAFFFC17147F931B>68 D120 D E /Fx 7 83 df48 D<1360EA01E0120F12FF12F312 03B3A2387FFF80A2111B7D9A18>II<1260387FFFE0A214C01480A238E00300EAC0065B5BC6 5AA25B13E0A212015B1203A41207A66C5A131C7D9B18>55 DI< EA03F8EA0FFEEA1E0F383C07801278EB03C012F8A214E0A4EA78071238EA3C0BEA0E1BEA 03E3EA000314C0A2EA3807007C13801400EA780FEA383CEA1FF8EA0FE0131B7E9A18>I< B512F014FE380FC03FEC0F806E7E81A55D4A5A023EC7FCEBFFF0EBC07C80143F6E7EA6ED 8180A2EC0FC13AFFFC07C300EC01FE211C7E9B24>82 D E /Fy 27 122 df<126012F0A212701210A41220A212401280040C7B830D>44 D48 D56 D<1303A3497EA2497E130BA2EB11E0A2EB 31F01320A2EB4078A3497EA23801003EEBFFFEEB001E00027FA348EB0780A2000C14C012 1E39FF803FFC1E1D7E9C22>65 D<90380FE02090387018603801C00439030003E0000613 01000E13004814605A15201278127000F01400A80070142012781238A26C14407E000614 806CEB01003801C00638007018EB0FE01B1E7D9C21>67 DI77 DI<13201370A313B8A3EA011CA2EA031EEA020EA2487EEA07FFEA040738080380 A2001813C01301123838FC07F815157F9419>97 DIIII< B51280EA1C031300A21440A213081400A21318EA1FF8EA1C181308A390C7FCA5EAFFC012 157F9416>II<38FF8FF8381C01C0 A9EA1FFFEA1C01A938FF8FF815157F9419>II<38 FF81F8381C01E01480140013025B5B5B1330137013B8EA1D1C121EEA1C0E7F14801303EB 01C014E014F038FF83FC16157F941A>107 D<00FEEB0FE0001E140000171317A3381380 27A23811C047A33810E087A2EB7107A3133AA2131CA2123839FE083FE01B157F941F> 109 D<38FC03F8381E00E014401217EA138013C01211EA10E01370A21338131CA2130E13 0714C0130313011300123800FE134015157F9419>III114 DI<387FFFF03860703000401310A200801308A300001300ADEA07FF 15157F9419>I<38FF83F8381C00E01440AE000C13C0000E138038060100EA0386EA00FC 15157F9419>I<38FF80FE381E0038000E1320000F13606C13403803808013C03801C100 13E212001374137C1338A848B4FC1715809419>121 D E /Fz 73 124 df11 D<5CEB3F83EBE0E23803803C3807001C000E7F4813 1748EB338014230078EB43C000701341148139F00101E0A213021304A213081310A2D870 2013C0134000781303D8388013801239391D0007006C130E6C5B6D5A3808E0E038183F80 0010C8FC1B207E9D20>31 D<1380EA0100120212065AA25AA25AA35AA412E0AC1260A47E A37EA27EA27E12027EEA0080092A7C9E10>40 D<7E12407E12307EA27EA27EA37EA41380 AC1300A41206A35AA25AA25A12205A5A092A7E9E10>I<1306ADB612E0A2D80006C7FCAD 1B1C7E9720>43 D<126012F0A212701210A41220A212401280040C7C830C>II<126012F0A2126004047C830C>I48 D<5A1207123F12C71207B3A5EAFFF80D1C7C9B15>III<130CA2131C133CA2135C13DC139CEA011C12 0312021204120C1208121012301220124012C0B512C038001C00A73801FFC0121C7F9B15 >II<13F0EA030CEA0404 EA0C0EEA181E1230130CEA7000A21260EAE3E0EAE430EAE818EAF00C130EEAE0061307A5 1260A2EA7006EA300E130CEA1818EA0C30EA03E0101D7E9B15>I<1240387FFF801400A2 EA4002485AA25B485AA25B1360134013C0A212015BA21203A41207A66CC7FC111D7E9B15 >III<126012F0A212601200AA126012F0A2126004127C910C>I61 D<1306A3130FA3EB1780A2EB37C01323A2EB43E01341A2EB80F0A3 38010078A2EBFFF83802003CA3487FA2000C131F80001E5BB4EBFFF01C1D7F9C1F>65 DI<90381F8080EBE0613801801938070007000E1303 5A14015A00781300A2127000F01400A8007014801278A212386CEB0100A26C13026C5B38 0180083800E030EB1FC0191E7E9C1E>IIII<90381F8080EBE061380180193807000700 0E13035A14015A00781300A2127000F01400A6ECFFF0EC0F80007013071278A212387EA2 7E6C130B380180113800E06090381F80001C1E7E9C21>I<39FFF0FFF0390F000F00AC90 B5FCEB000FAD39FFF0FFF01C1C7F9B1F>II<38 07FF8038007C00133CB3127012F8A21338EA7078EA4070EA30E0EA0F80111D7F9B15>I< 39FFF01FE0390F000780EC060014045C5C5C5C5C49C7FC13021306130FEB17801327EB43 C0EB81E013016D7E1478A280143E141E80158015C039FFF03FF01C1C7F9B20>IIIIII82 D<3807E080EA1C19EA30 051303EA600112E01300A36C13007E127CEA7FC0EA3FF8EA1FFEEA07FFC61380130FEB07 C0130313011280A300C01380A238E00300EAD002EACC0CEA83F8121E7E9C17>I<007FB5 12C038700F010060130000401440A200C014201280A300001400B1497E3803FFFC1B1C7F 9B1E>I<39FFF01FF0390F000380EC0100B3A26C1302138000035BEA01C03800E018EB70 60EB0F801C1D7F9B1F>I<39FFE00FF0391F0003C0EC01806C1400A238078002A213C000 035BA2EBE00C00011308A26C6C5AA213F8EB7820A26D5AA36D5AA2131F6DC7FCA21306A3 1C1D7F9B1F>I<3AFFE1FFC0FF3A1F003E003C001E013C13186C6D1310A32607801F1320 A33A03C0278040A33A01E043C080A33A00F081E100A39038F900F3017913F2A2017E137E 013E137CA2013C133C011C1338A20118131801081310281D7F9B2B>I<39FFF07FC0390F C01E003807800CEBC00800035B6C6C5A13F000005BEB7880137C013DC7FC133E7F7F80A2 EB13C0EB23E01321EB40F0497E14783801007C00027F141E0006131F001F148039FF807F F01C1C7F9B1F>I<39FFF003FC390F8001E00007EB00C06D13800003EB01006D5A000113 026C6C5A13F8EB7808EB7C18EB3C10EB3E20131F6D5A14C06D5AABEB7FF81E1C809B1F> I<387FFFF0EA7C01007013E0386003C0A238400780130F1400131E12005B137C13785BA2 485A1203EBC010EA0780A2EA0F00481330001E13205A14604813E0EAF803B5FC141C7E9B 19>I<12FEA212C0B3B312FEA207297C9E0C>I<12FEA21206B3B312FEA20729809E0C>93 D97 D<12FC121CAA137CEA1D87381E0180381C00C014E0 14601470A6146014E014C0381E018038190700EA10FC141D7F9C17>II< EB1F801303AAEA03F3EA0E0BEA1807EA30031270126012E0A6126012701230EA1807EA0E 1B3803E3F0141D7F9C17>II<13F8EA018CEA071E1206 EA0E0C1300A6EAFFE0EA0E00B0EA7FE00F1D809C0D>II<12FC121CAA 137C1387EA1D03001E1380121CAD38FF9FF0141D7F9C17>I<1218123CA21218C7FCA712 FC121CB0EAFF80091D7F9C0C>I<13C0EA01E0A2EA00C01300A7EA07E01200B3A21260EA F0C012F1EA6180EA3E000B25839C0D>I<12FC121CAAEB0FE0EB0780EB06005B13105B5B 13E0121DEA1E70EA1C781338133C131C7F130F148038FF9FE0131D7F9C16>I<12FC121C B3A9EAFF80091D7F9C0C>I<39FC7E07E0391C838838391D019018001EEBE01C001C13C0 AD3AFF8FF8FF8021127F9124>IIII<3803 E080EA0E19EA1805EA3807EA7003A212E0A61270A2EA38071218EA0E1BEA03E3EA0003A7 EB1FF0141A7F9116>III<1204A4120CA2121C123CEAFFE0EA1C00A91310 A5120CEA0E20EA03C00C1A7F9910>I<38FC1F80EA1C03AD1307120CEA0E1B3803E3F014 127F9117>I<38FF07E0383C0380381C0100A2EA0E02A2EA0F06EA0704A2EA0388A213C8 EA01D0A2EA00E0A3134013127F9116>I<39FF3FC7E0393C0703C0001CEB01801500130B 000E1382A21311000713C4A213203803A0E8A2EBC06800011370A2EB8030000013201B12 7F911E>I<38FF0FE0381E0700EA1C06EA0E046C5AEA039013B0EA01E012007F12011338 EA021C1204EA0C0E487E003C138038FE1FF014127F9116>I<38FF07E0383C0380381C01 00A2EA0E02A2EA0F06EA0704A2EA0388A213C8EA01D0A2EA00E0A31340A25BA212F000F1 C7FC12F312661238131A7F9116>I123 D E /FA 27 122 df14 DI<120E1203A2 EA0180A213C01200A21360A31330A213781398EA0118EA020C1204EA080EEA18061230EA E00312C010177E9615>21 D<380FFFC0123F382108001241485A1202A212061204A2EA0C 18A212181308120E7F8D14>25 D28 D<126012F0A212701210A21220A21240A2040A7D830A>59 D<143014F0EB03C0EB0700131C1378EA01E0EA0780000EC7FC123812F0A21238120E6C7E EA01E0EA0078131C1307EB03C0EB00F0143014167D921B>I<130813181330A31360A313 C0A3EA0180A3EA0300A21206A35AA35AA35AA35AA35AA20D217E9812>I<12C012F0123C 120EEA0380EA01E0EA0078131E1307EB01C0EB00F0A2EB01C0EB0700131E1378EA01E0EA 0380000EC7FC123C12F012C014167D921B>I64 D<3807FFF83800E00E14071403A2EA01C01407A2140E3803803C EBFFF0A2EB803C3807001C140EA3000E131CA214381470381C01E0B5120018177F961B> 66 D<3807FFF83800E00E140315801401D801C013C0A21400A238038001A43907000380 A215005C000E130E140C5C1470381C01C0B5C7FC1A177F961D>68 D<0007B512803800E003EC0100A3EA01C0A21440A248485A138113FF1381D80701C7FCA3 90C8FC120EA45AEAFFC019177F9616>70 D<3907FE1FF83900E00380A43901C00700A438 03800EA2EBFFFEEB800E48485AA4000E5BA4485B38FF83FE1D177F961D>72 D<3907F007F80000EB00C001B81380A2139C39011C0100131E130EA238020702A2EB0382 A2380401C4A2EB00E4A2481378A21438A20018131012FE1D177F961C>78 D<3807FFF03800E01C14061407A2EA01C0A3140E48485A1470EBFF80EB80E03807007080 A3000E5BA21580A248EB310038FF801E19177F961B>82 DI<3AFF83FC1FC03A1C00E00700150615 0401015B13025DEB04F0EC702001085BA201105BEA0E200271C7FC13401472EB8074120F EB0078120E1470000C1330142022177E9621>87 D<133E130CA41318A4EA0730EA18F0EA 30701260136012C0A3EA80C013C4A212C1EA46C8EA38700F177E9612>100 D<1318133813101300A6EA01C0EA0220EA0430A2EA08601200A313C0A4EA0180A4EA6300 12E312C612780D1D80960E>106 D<121F1206A45AA4EA181C1366138EEA190CEA320012 3C123FEA3180EA60C013C4A3EAC0C813700F177E9612>I<38383C1E3844C66338470281 38460301388E0703EA0C06A338180C061520140C154039301804C0EC07001B0E7F8D1F> 109 DI114 D<1203A21206A4EAFFC0EA0C00A35AA45A1380A2EA31001232121C0A147F930D>116 D120 DI E /FB 39 117 df<1238127C12FEA3127C123807077C8610>46 D<13181378EA01F812FFA21201B3A738 7FFFE0A213207C9F1C>49 DI<13FE3807FFC0380F07E038 1E03F0123FEB81F8A3EA1F0314F0120014E0EB07C0EB1F803801FE007F380007C0EB01F0 14F8EB00FCA2003C13FE127EB4FCA314FCEA7E01007813F8381E07F0380FFFC03801FE00 17207E9F1C>I<14E013011303A21307130F131FA21337137713E7EA01C71387EA030712 07120E120C12181238127012E0B6FCA2380007E0A790B5FCA218207E9F1C>I<00301320 383E01E0383FFFC0148014005B13F8EA33C00030C7FCA4EA31FCEA37FF383E0FC0383807 E0EA3003000013F0A214F8A21238127C12FEA200FC13F0A2387007E0003013C0383C1F80 380FFF00EA03F815207D9F1C>II<1470A2 14F8A3497EA2497EA3EB067FA2010C7F143FA2496C7EA201307F140F01707FEB6007A201 C07F90B5FC4880EB8001A2D803007F14004880000680A23AFFE007FFF8A225227EA12A> 65 DIIII IIII76 D78 DII82 D<3801FE023807FF86381F01FE383C007E007C131E0078130EA200F81306 A27E1400B4FC13E06CB4FC14C06C13F06C13F86C13FC000313FEEA003F1303EB007F143F A200C0131FA36C131EA26C133C12FCB413F838C7FFE00080138018227DA11F>I<007FB6 1280A2397E03F80F00781407007014030060140100E015C0A200C01400A400001500B3A2 48B512F0A222227EA127>II97 D99 DI<13FE3807FF 80380F87C0381E01E0003E13F0EA7C0014F812FCA2B5FCA200FCC7FCA3127CA2127E003E 13186C1330380FC0703803FFC0C6130015167E951A>I104 D<121C123E127FA3123E 121CC7FCA7B4FCA2121FB2EAFFE0A20B247EA310>I107 DI<3AFF07F007F090 391FFC1FFC3A1F303E303E01401340496C487EA201001300AE3BFFE0FFE0FFE0A22B167E 9530>I<38FF07E0EB1FF8381F307CEB403CEB803EA21300AE39FFE1FFC0A21A167E951F> I<13FE3807FFC0380F83E0381E00F0003E13F848137CA300FC137EA7007C137CA26C13F8 381F01F0380F83E03807FFC03800FE0017167E951C>I<38FF0FE0EB3FF8381FE07CEB80 3E497E1580A2EC0FC0A8EC1F80A29038803F00EBC03EEBE0FCEB3FF8EB0FC090C8FCA8EA FFE0A21A207E951F>I114 DI<487EA41203A21207A2120F123FB5FCA2EA0F80ABEB8180A5EB8300EA07C3EA 03FEEA00F811207F9F16>I E end %%EndProlog %%BeginSetup %%Feature: *Resolution 300dpi TeXDict begin %%PaperSize: A4 %%EndSetup %%Page: 1 1 1 0 bop -68 170 a FB(THE)23 b(NOD)n(AL)i(SURF)-6 b(A)n(CE)25 b(OF)g(THE)f(SECOND)g(EIGENFUNCTION)g(OF)h(THE)369 228 y(LAPLA)n(CIAN)h(IN)e(R)861 210 y FA(D)917 228 y FB(CAN)h(BE)f(CLOSED) 691 352 y Fz(S\037REN)18 b(F)o(OURNAIS)27 480 y Fy(Abstra)o(ct)p Fz(.)h(W)m(e)14 b(construct)i(a)e(set)h(in)f Fx(R)699 465 y Fw(D)744 480 y Fz(with)g(the)h(prop)q(ert)o(y)g(that)f(the)h(no)q (dal)f(surface)h(of)f(the)h(second)27 530 y(eigenfunction)k(of)f(the)i (Diric)o(hlet)e(Laplacian)g(is)h(closed,)i(i.e.)33 b(do)q(es)20 b(not)f(touc)o(h)g(the)h(b)q(oundary)f(of)g(the)27 580 y(domain.)d(The)e(construction)h(is)f(explicit)f(in)h(all)f(dimensions) f Fv(D)i Fu(\025)e Fz(2)h(and)h(w)o(e)g(obtain)g(explicit)f(con)o(trol) h(of)27 630 y(the)g(connectivit)o(y)g(of)f(the)i(domain.)762 867 y Ft(Contents)-73 955 y Fs(1.)49 b(In)o(tro)q(duction)1582 b(1)-73 1013 y(1.1.)49 b(Generalities)1557 b(1)-73 1071 y(1.2.)49 b(The)16 b(domain)1553 b(2)-73 1129 y(2.)49 b(Preliminary)13 b(Estimates)1371 b(4)-73 1187 y(3.)49 b(Pro)q(of)17 b(of)g(the)f(main)f(theorem)1279 b(7)-73 1245 y(4.)49 b(An)16 b(explicit)e(estimate)g(on)j(the)f(n)o(um)o(b)q (er)e(of)j(holes)f(for)h Fr(D)e Fs(=)f(2)665 b(11)-73 1303 y(App)q(endix)16 b(A.)47 b(A)16 b(n)o(umerical)e(calculation)1082 b(13)-73 1361 y(References)1686 b(14)680 1592 y(1.)28 b Ft(Intr)o(oduction)-73 1680 y Fs(It)22 b(is)g(a)h(famous)f (conjecture)g(b)o(y)g(P)o(a)o(yne)g([P)o(a)o(y67)o(])g(that)h(the)g(no) q(dal)g(surface)f(of)h(the)f(2nd)h(eigenfunc-)-123 1738 y(tion)18 b(of)f(the)g(Diric)o(hlet)f(Laplacian)h(on)h(a)g(b)q(ounded,) g(connected)f(domain)f(\012)i(in)f FB(R)1427 1720 y FA(D)1476 1738 y Fs(touc)o(hes)g(the)g(b)q(ound-)-123 1796 y(ary)k Fr(@)s Fs(\012.)34 b(F)l(or)20 b(con)o(v)o(ex)g(domains)g(in)g FB(R)635 1778 y Fq(2)675 1796 y Fs(this)g(conjecture)g(w)o(as)h(pro)o (v)o(ed)f(b)o(y)g(Melas)g([Mel92)o(].)34 b(Ho)o(w)o(ev)o(er,)-123 1854 y(M.)15 b(Ho\013mann-Ostenhof,)g(T.)h(Ho\013mann-Ostenhof)f(and)h (N.)f(Nadirash)o(vili)f([HOHON98)o(])h(constructed)g(a)-123 1912 y(non-simply)h(connected)g(coun)o(terexample)e(to)k(the)f(general) f(conjecture)g(in)h FB(R)1356 1894 y Fq(2)1376 1912 y Fs(.)23 b(The)17 b(2-dimensional)f(ex-)-123 1970 y(ample)d(in)i ([HOHON98)o(])f(relies)g(hea)o(vily)f(on)j(c)o(ho)q(osing)f(a)h(v)o (ery)d(symmetric)e(domain)j(and)i(using)f(symmetry)-123 2028 y(argumen)o(ts.)21 b(In)c(higher)f(dimensions)f(it)h(is)g(not)h(p) q(ossible)g(to)g(c)o(ho)q(ose)g(similar,)d(v)o(ery)h(symmetric)e (domains.)-123 2086 y(The)g(obstruction)g(b)q(eing)g(that)g(there)f (are)h(only)g(a)g(\014nite)f(n)o(um)o(b)q(er)f(of)i(regular)g(p)q (olyhedra)g(in)g(an)o(y)f(dimension)-123 2144 y(greater)j(than)g(or)g (equal)f(to)h(3)g(\(in)f(3)h(dimensions)e(these)h(are)h(the)f(platonic) g(solids\).)21 b(Belo)o(w)14 b(w)o(e)g(will)f(lo)q(ok)i(at)-123 2203 y(almost)i(the)g(same)f(domain)h(as)h(in)f(the)g(ab)q(o)o(v)o(e)g (men)o(tioned)f(pap)q(er.)25 b(Because)16 b(of)i(the)f(lac)o(k)g(of)g (symmetrie)o(s)-123 2261 y(the)g(argumen)o(t)f(from)g([HOHON98)o(])h (cannot)h(b)q(e)f(applied.)24 b(W)l(e)17 b(shall)g(use)g(an)h (alternativ)o(e,)e(and)i(in)e(a)i(w)o(a)o(y)-123 2319 y(more)d(direct,)g(argumen)o(t)g(to)i(reac)o(h)e(the)h(desired)g (conclusion.)-123 2434 y(1.1.)28 b FB(Generalities.)22 b Fs(F)l(or)e(a)g(b)q(ounded)h(connected)e(domain)g(\012)h(\(with)g (su\016cien)o(tly)e(regular)i(b)q(oundary\))-123 2492 y(w)o(e)c(will)f(lo)q(ok)i(at)g(the)f(Laplace)h(op)q(erator)g(with)f (Diric)o(hlet)f(conditions)h(at)h(the)f(b)q(oundary)l(.)23 b(This)16 b(de\014nes)g(a)p -123 2562 250 2 v -73 2609 a Fz(Do)q(cumen)o(t)d(v)o(ersion:)32 b(Ma)o(y)13 b(12,)g(1999.)-73 2659 y Fp(Key)i(wor)n(ds)f(and)i(phr)n(ases.)k Fz(No)q(dal)14 b(surfaces,)g(eigenfunctions,)g(Diric)o(hlet)f(Laplacian.)-73 2708 y(I)k(wish)g(to)g(thank)g(J.P)m(.)g(Solo)o(v)o(ej)f(for)h(useful)g (discussions)h(and)f(for)g(suggesting)h(this)f(problem)f(to)h(me.)27 b(I)17 b(am)f(also)g(v)o(ery)-123 2757 y(grateful)f(to)h(S.E.)f(Gra)o (v)o(ersen)h(for)g(teac)o(hing)f(me)g(all)f(I)i(kno)o(w)f(ab)q(out)h (sto)q(c)o(hastic)g(pro)q(cesses)j(and)c(for)h(help)f(and)h (discussions)-123 2807 y(on)e(questions)g(directly)g(related)h(to)f (this)g(w)o(ork.)-123 2857 y(P)o(artially)f(supp)q(orted)i(b)o(y)e(the) i(Europ)q(ean)g(Union,)d(gran)o(t)i(FMRX-960001.)873 3093 y Fo(1)p eop %%Page: 2 2 2 1 bop -5 -64 a Fo(2)809 b(S\037REN)17 b(F)o(OURNAIS)-5 40 y Fs(p)q(ositiv)o(e,)e(self)g(adjoin)o(t)g(op)q(erator)i Fn(\000)p Fs(\001)714 47 y Fq(\012)756 40 y Fs(with)f(domain)f Fr(W)1093 17 y Fq(1)p FA(;)p Fq(2)1086 53 y(0)1140 40 y Fs(\(\012\))g(\(see)g([GT83)q(])g(for)h(notation\))h(and)f(purely)-5 98 y(discrete)f(sp)q(ectrum.)20 b(W)l(e)c(denote)g(the)g(eigen)o(v)m (alues)g(\(eigenfunctions\))f(b)o(y)h Fn(f)p Fr(\025)1461 105 y FA(j)1480 98 y Fs(\(\012\))p Fn(g)1578 80 y Fm(1)1578 111 y FA(j)r Fq(=1)1657 98 y Fs(\()p Fn(f)p Fr(u)1729 105 y FA(j)1747 98 y Fs(\(\012\))p Fn(g)1845 80 y Fm(1)1845 111 y FA(j)r Fq(=1)1909 98 y Fs(\),)f(so)653 304 y Fn(\000)p Fs(\001)733 311 y Fq(\012)760 304 y Fr(u)788 311 y FA(j)806 304 y Fs(\(\012\))42 b(=)g Fr(\025)1029 311 y FA(j)1047 304 y Fs(\(\012\))p Fr(u)1148 311 y FA(j)1167 304 y Fs(\(\012\))16 b(in)g(\012)760 377 y Fr(u)788 384 y FA(j)806 377 y Fs(\(\012\))42 b Fn(\021)f Fs(0)16 b(on)h Fr(@)s Fs(\012)p Fr(;)-5 581 y Fs(and)22 b(the)f(eigen)o(v)m(alues)f(are)h(ordered)g(according)g(to) g(size:)30 b(0)23 b Fr(<)f(\025)1252 588 y Fq(1)1272 581 y Fs(\(\012\))g Fr(<)g(\025)1455 588 y Fq(2)1475 581 y Fs(\(\012\))g Fn(\024)g Fr(\025)1659 588 y Fq(3)1679 581 y Fs(\(\012\))g Fn(\024)g Fr(:::)d Fs(\(It)i(is)-5 639 y(a)g(general)e(result)h(that)g Fr(\025)488 646 y Fq(1)509 639 y Fs(\(\012\))g(is)f(simple)f(and)j(strictly)d(p)q(ositiv) o(e\).)32 b(W)l(e)20 b(ma)o(y)e(tak)o(e)i(the)f(eigenfunctions)-5 697 y(to)g(b)q(e)g(real)g(and)g(orthonormal:)27 b Fn(h)p Fr(u)672 704 y FA(j)690 697 y Fs(\(\012\))p Fr(;)8 b(u)813 704 y FA(k)834 697 y Fs(\(\012\))p Fn(i)19 b Fs(=)f Fr(\016)1023 704 y FA(j;k)1068 697 y Fs(,)h(where)g Fr(\016)h Fs(is)f(the)f(Kronec)o (k)o(er)f(delta.)29 b(Since)18 b(the)-5 755 y(eigenfunctions)c(are)g (real,)f(and)i(the)e(\014rst)i(can)f(b)q(e)g(c)o(hosen)f(p)q(ositiv)o (e,)h(the)f(second)h(eigenfunction)g Fr(u)1829 762 y Fq(2)1848 755 y Fs(\(\012\))g(has)-5 813 y(to)j(tak)o(e)e(b)q(oth)i(p)q (ositiv)o(e)e(and)h(negativ)o(e)f(v)m(alues.)21 b(According)16 b(to)g(Couran)o(t's)g(No)q(dal)h(Domains)e(Theorem)g(\012)-5 871 y(splits)h(in)o(to)g(exactly)f(t)o(w)o(o)i(connected)e(op)q(en)i (sets)g(\012)959 878 y Fq(+)989 871 y Fr(;)8 b Fs(\012)1046 878 y Fm(\000)1091 871 y Fs(suc)o(h)17 b(that)f Fr(u)1335 878 y Fq(2)1369 871 y Fr(>)e Fs(0)j(on)f(\012)1564 878 y Fq(+)1594 871 y Fs(,)g Fr(u)1652 878 y Fq(2)1686 871 y Fr(<)e Fs(0)i(on)h(\012)1881 878 y Fm(\000)1927 871 y Fs(and)p -5 890 36 2 v -5 930 a(\012)d(=)p 96 890 V 14 w(\012)131 937 y Fq(+)172 930 y Fn([)p 216 890 V 11 w Fs(\012)251 937 y Fm(\000)281 930 y Fs(.)-5 988 y(It)i(is)g(no)o(w)h (natural)f(to)h(study)f(the)g(geometry)f(of)i(the)f(no)q(dal)h(set)f Fn(N)7 b Fs(\()p Fr(u)1305 995 y Fq(2)1325 988 y Fs(\),)15 b(where)662 1198 y Fn(N)7 b Fs(\()p Fr(u)757 1205 y Fq(2)777 1198 y Fs(\))14 b(=)p 862 1154 464 2 v 14 w Fn(f)p Fr(x)f Fn(2)h Fs(\012)p Fn(j)p Fr(u)1052 1205 y Fq(2)1072 1198 y Fs(\(\012\)\()p Fr(x)p Fs(\))f(=)h(0)p Fn(g)p Fr(:)-5 1401 y Fs(Generically)e(\(see)h([Uhl72]\),)g(this)g(is)g(a)h(manifold)e (of)i(co)q(dimension)f(1,)h(and)g(one)f(ma)o(y)f(ask)i(w)o(ether)f(it)g (alw)o(a)o(ys)-5 1460 y(touc)o(hes)j(the)g(b)q(oundary)i(of)e(the)g (domain,)f(i.e.)20 b(w)o(ether)829 1663 y Fn(N)7 b Fs(\()p Fr(u)924 1670 y Fq(2)944 1663 y Fs(\))k Fn(\\)g Fr(@)s Fs(\012)i Fn(6)p Fs(=)h Fn(;)-5 1867 y Fs(alw)o(a)o(ys.)21 b(This)c(is)f(the)g(ab)q(o)o(v)o(e)g(men)o(tioned)e(conjecture)h(b)o(y) h(P)o(a)o(yne)g([P)o(a)o(y67].)-5 2204 y(1.2.)28 b FB(The)19 b(domain.)j Fs(W)l(e)16 b(c)o(ho)q(ose)h(0)d Fr(<)g(R)799 2211 y Fq(1)833 2204 y Fr(<)f(R)921 2211 y Fq(2)958 2204 y Fs(suc)o(h)j(that)507 2409 y Fr(\025)535 2416 y Fq(1)555 2409 y Fs(\()p Fr(B)s Fs(\()p Fr(R)670 2416 y Fq(1)689 2409 y Fs(\)\))e Fr(<)g(\025)821 2416 y Fq(1)841 2409 y Fs(\()p Fr(B)s Fs(\()p Fr(R)956 2416 y Fq(2)975 2409 y Fs(\))d Fn(n)p 1041 2366 135 2 v 11 w Fr(B)s Fs(\()p Fr(R)1137 2416 y Fq(1)1157 2409 y Fs(\)\))j Fr(<)f(\025)1288 2416 y Fq(2)1308 2409 y Fs(\()p Fr(B)s Fs(\()p Fr(R)1423 2416 y Fq(1)1443 2409 y Fs(\)\))p Fr(:)-5 2613 y Fs(F)l(urthermore)d(w) o(e)h(let)g Fr(N)19 b Fn(2)14 b FB(N)p Fs(,)e(and)h(let)d Fn(f)p Fr(x)790 2620 y Fq(1)810 2613 y Fr(;)e(:::;)g(x)924 2620 y FA(N)956 2613 y Fn(g)13 b(\032)h(f)p Fr(x)f Fn(2)h FB(R)1202 2595 y FA(D)1234 2571 y Fl(\014)1234 2601 y(\014)1251 2613 y Fn(j)p Fr(x)p Fn(j)f Fs(=)h Fr(R)1409 2620 y Fq(1)1429 2613 y Fn(g)p Fs(.)19 b(Then)12 b(w)o(e)f(let)g Fr(\017)j Fn(2)g FB(R)1866 2620 y Fq(+)1897 2613 y Fn([)r(f)p Fs(0)p Fn(g)-5 2672 y Fs(and)j(de\014ne)482 2855 y(\012)517 2862 y FA(\017)548 2855 y Fs(=)c(\()p Fr(B)s Fs(\()p Fr(R)714 2862 y Fq(2)734 2855 y Fs(\))e Fn(n)p 800 2812 V 11 w Fr(B)s Fs(\()p Fr(R)896 2862 y Fq(1)915 2855 y Fs(\)\))g Fn([)1008 2815 y Fl(\000)1031 2855 y Fn([)1064 2834 y FA(N)1064 2867 y(j)r Fq(=1)1128 2855 y Fr(B)s Fs(\()p Fr(x)1215 2862 y FA(j)1232 2855 y Fr(;)d(\017)p Fs(\))1293 2815 y Fl(\001)1327 2855 y Fn([)j Fr(B)s Fs(\()p Fr(R)1467 2862 y Fq(1)1486 2855 y Fs(\))p Fr(:)p eop %%Page: 3 3 3 2 bop 664 -64 a Fo(THE)17 b(NOD)o(AL)f(SURF)l(A)o(CE)767 b(3)268 37 y gsave currentpoint currentpoint translate -90 neg rotate neg exch neg exch translate 268 37 a 16 w @beginspecial 0 @llx 0 @lly 612 @urx 792 @ury 2988 @rwi 2988 @rhi @setspecial %%BeginDocument: /users/fournais/files/no_surface/nodald02.eps 20 dict begin gsave /m {moveto} def /l {lineto} def /C {setrgbcolor} def /Y /setcmykcolor where { %%ifelse Use built-in operator /setcmykcolor get }{ %%ifelse Emulate setcmykcolor with setrgbcolor { %%def 1 sub 3 { %%repeat 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor } bind } ifelse def /G {setgray} def /S {stroke} def /NP {newpath} def /P {gsave fill grestore reversepath stroke} def /stringbbox {gsave NP 0 0 m false charpath flattenpath pathbbox 4 2 roll pop pop 1.1 mul cvi exch 1.1 mul cvi exch grestore} def /thin 3 def /medium 7 def /thick 16 def /boundarythick 20 def %% thickness of bounding box 576 36 translate 90 rotate 0.108 0.108 scale 1 setlinejoin 1 setlinecap 0.0 setgray /inch {72 mul} def /fheight 0.35 inch neg def 0 G medium setlinewidth [] 0 setdash thick setlinewidth NP 3331 1538 m 3334 1538 l 3336 1538 l 3339 1538 l 3342 1538 l 3344 1538 l 3347 1538 l 3349 1538 l 3352 1538 l 3354 1538 l 3357 1538 l 3359 1538 l 3362 1538 l 3364 1538 l 3367 1538 l 3369 1538 l 3372 1538 l 3374 1538 l 3377 1538 l 3379 1538 l 3382 1538 l 3384 1538 l 3387 1538 l 3389 1539 l 3392 1539 l 3394 1539 l 3397 1539 l 3399 1539 l 3402 1539 l 3404 1539 l 3407 1539 l 3410 1539 l 3412 1539 l 3415 1540 l 3417 1540 l 3420 1540 l 3422 1540 l 3424 1540 l 3427 1540 l 3430 1540 l 3432 1541 l 3435 1541 l 3437 1541 l 3439 1541 l 3442 1541 l 3444 1541 l 3447 1541 l 3449 1542 l 3452 1542 l S NP 2295 1934 m 2296 1932 l 2298 1931 l 2299 1929 l 2301 1928 l 2302 1926 l 2304 1925 l 2305 1923 l 2307 1922 l 2308 1920 l 2310 1919 l 2311 1917 l 2313 1916 l 2314 1914 l 2316 1913 l 2317 1911 l 2319 1910 l 2320 1908 l 2322 1907 l 2324 1905 l 2325 1904 l 2327 1902 l 2328 1901 l 2330 1899 l 2331 1898 l 2333 1896 l 2334 1895 l 2336 1893 l 2338 1892 l 2339 1890 l 2341 1889 l 2343 1887 l 2344 1886 l 2346 1885 l 2347 1883 l 2349 1882 l 2351 1880 l 2352 1879 l 2354 1877 l 2355 1876 l 2357 1875 l 2359 1873 l 2360 1872 l 2362 1870 l 2364 1869 l 2365 1867 l 2367 1866 l 2369 1865 l 2370 1863 l S NP 2113 2202 m 2114 2200 l 2114 2198 l 2115 2196 l 2116 2195 l 2117 2193 l 2118 2191 l 2118 2189 l 2119 2187 l 2120 2186 l 2121 2184 l 2122 2182 l 2122 2180 l 2123 2178 l 2124 2177 l 2125 2175 l 2126 2173 l 2127 2171 l 2128 2170 l 2128 2168 l 2129 2166 l 2130 2164 l 2131 2162 l 2132 2161 l 2133 2159 l 2134 2157 l 2135 2156 l 2135 2154 l 2136 2152 l 2137 2150 l 2138 2148 l 2139 2147 l 2140 2145 l 2141 2143 l 2142 2141 l 2143 2140 l 2144 2138 l 2145 2136 l 2146 2134 l 2147 2133 l 2148 2131 l 2149 2129 l 2150 2127 l 2150 2126 l 2152 2124 l 2152 2122 l 2153 2120 l 2154 2119 l 2155 2117 l S NP 2578 1721 m 2580 1720 l 2582 1719 l 2584 1718 l 2586 1717 l 2589 1716 l 2591 1715 l 2593 1713 l 2595 1712 l 2597 1711 l 2599 1710 l 2601 1709 l 2603 1708 l 2605 1707 l 2607 1706 l 2609 1705 l 2611 1704 l 2613 1703 l 2615 1702 l 2617 1701 l 2620 1700 l 2622 1699 l 2624 1697 l 2626 1696 l 2628 1695 l 2630 1694 l 2632 1693 l 2634 1692 l 2636 1691 l 2638 1690 l 2641 1689 l 2643 1688 l 2645 1687 l 2647 1686 l 2649 1685 l 2651 1684 l 2653 1683 l 2655 1682 l 2658 1681 l 2660 1680 l 2662 1679 l 2664 1678 l 2666 1677 l 2668 1676 l 2671 1675 l 2673 1674 l 2675 1673 l 2677 1672 l 2679 1671 l S NP 2425 1819 m 2427 1818 l 2429 1816 l 2431 1815 l 2433 1814 l 2434 1812 l 2436 1811 l 2438 1810 l 2440 1809 l 2442 1807 l 2443 1806 l 2445 1805 l 2447 1803 l 2449 1802 l 2451 1801 l 2452 1799 l 2454 1798 l 2456 1797 l 2458 1795 l 2460 1794 l 2462 1793 l 2463 1792 l 2465 1790 l 2467 1789 l 2469 1788 l 2471 1787 l 2473 1785 l 2475 1784 l 2476 1783 l 2478 1782 l 2480 1780 l 2482 1779 l 2484 1778 l 2486 1776 l 2488 1775 l 2490 1774 l 2492 1773 l 2494 1772 l 2495 1770 l 2497 1769 l 2499 1768 l 2501 1767 l 2503 1765 l 2505 1764 l 2507 1763 l 2509 1762 l 2511 1761 l 2513 1759 l 2515 1758 l S NP 4613 2499 m 4613 2501 l 4613 2502 l 4613 2504 l 4613 2506 l 4613 2508 l 4613 2510 l 4613 2512 l 4613 2514 l 4613 2516 l 4613 2518 l 4613 2519 l 4613 2521 l 4612 2523 l 4612 2525 l 4612 2527 l 4612 2529 l 4612 2531 l 4612 2533 l 4612 2535 l 4612 2536 l 4612 2538 l 4612 2540 l 4612 2542 l 4611 2544 l 4611 2546 l 4611 2548 l 4611 2550 l 4611 2552 l 4611 2553 l 4611 2555 l 4611 2557 l 4610 2559 l 4610 2561 l 4610 2563 l 4610 2565 l 4610 2567 l 4610 2568 l 4609 2570 l 4609 2572 l 4609 2574 l 4609 2576 l 4609 2578 l 4608 2580 l 4608 2582 l 4608 2583 l 4608 2585 l 4607 2587 l 4607 2589 l S NP 2190 2062 m 2191 2061 l 2192 2059 l 2193 2057 l 2194 2056 l 2196 2054 l 2197 2052 l 2198 2051 l 2199 2049 l 2200 2047 l 2201 2046 l 2203 2044 l 2204 2042 l 2205 2041 l 2206 2039 l 2207 2037 l 2209 2036 l 2210 2034 l 2211 2032 l 2212 2031 l 2213 2029 l 2215 2027 l 2216 2026 l 2217 2024 l 2218 2023 l 2220 2021 l 2221 2019 l 2222 2018 l 2223 2016 l 2225 2014 l 2226 2013 l 2227 2011 l 2229 2009 l 2230 2008 l 2231 2006 l 2232 2005 l 2234 2003 l 2235 2001 l 2236 2000 l 2238 1998 l 2239 1997 l 2240 1995 l 2242 1993 l 2243 1992 l 2244 1990 l 2245 1989 l 2247 1987 l 2248 1985 l 2250 1984 l S NP 2066 2348 m 2066 2346 l 2067 2345 l 2067 2343 l 2067 2341 l 2068 2339 l 2068 2337 l 2069 2335 l 2069 2333 l 2070 2332 l 2070 2330 l 2070 2328 l 2071 2326 l 2071 2324 l 2072 2322 l 2072 2321 l 2073 2319 l 2073 2317 l 2074 2315 l 2074 2313 l 2075 2311 l 2075 2309 l 2076 2307 l 2076 2306 l 2077 2304 l 2077 2302 l 2078 2300 l 2078 2298 l 2079 2296 l 2079 2295 l 2080 2293 l 2080 2291 l 2081 2289 l 2081 2287 l 2082 2285 l 2083 2284 l 2083 2282 l 2084 2280 l 2084 2278 l 2085 2276 l 2085 2274 l 2086 2272 l 2087 2271 l 2087 2269 l 2088 2267 l 2088 2265 l 2089 2263 l 2090 2262 l 2090 2260 l S NP 2066 2649 m 2065 2647 l 2065 2645 l 2065 2644 l 2064 2642 l 2064 2640 l 2064 2638 l 2063 2636 l 2063 2634 l 2062 2632 l 2062 2630 l 2062 2629 l 2061 2627 l 2061 2625 l 2061 2623 l 2060 2621 l 2060 2619 l 2060 2617 l 2060 2615 l 2059 2614 l 2059 2612 l 2059 2610 l 2058 2608 l 2058 2606 l 2058 2604 l 2058 2602 l 2057 2601 l 2057 2599 l 2057 2597 l 2056 2595 l 2056 2593 l 2056 2591 l 2056 2589 l 2055 2587 l 2055 2585 l 2055 2584 l 2055 2582 l 2055 2580 l 2054 2578 l 2054 2576 l 2054 2574 l 2054 2572 l 2054 2570 l 2053 2569 l 2053 2567 l 2053 2565 l 2053 2563 l 2053 2561 l 2053 2559 l S NP 2113 2796 m 2112 2794 l 2111 2792 l 2110 2790 l 2110 2789 l 2109 2787 l 2108 2785 l 2107 2783 l 2107 2781 l 2106 2780 l 2105 2778 l 2105 2776 l 2104 2774 l 2103 2772 l 2102 2770 l 2102 2769 l 2101 2767 l 2100 2765 l 2100 2763 l 2099 2762 l 2098 2760 l 2097 2758 l 2097 2756 l 2096 2754 l 2095 2752 l 2095 2750 l 2094 2749 l 2094 2747 l 2093 2745 l 2092 2743 l 2092 2742 l 2091 2740 l 2090 2738 l 2090 2736 l 2089 2734 l 2088 2732 l 2088 2730 l 2087 2729 l 2087 2727 l 2086 2725 l 2085 2723 l 2085 2721 l 2084 2719 l 2084 2718 l 2083 2716 l 2083 2714 l 2082 2712 l 2081 2710 l 2081 2708 l S NP 2190 2935 m 2189 2933 l 2187 2932 l 2186 2930 l 2185 2928 l 2184 2927 l 2183 2925 l 2182 2923 l 2181 2922 l 2180 2920 l 2178 2918 l 2177 2917 l 2176 2915 l 2175 2913 l 2174 2911 l 2173 2910 l 2172 2908 l 2171 2906 l 2170 2905 l 2169 2903 l 2168 2901 l 2167 2899 l 2166 2898 l 2165 2896 l 2164 2894 l 2163 2892 l 2162 2891 l 2161 2889 l 2159 2887 l 2158 2886 l 2157 2884 l 2156 2882 l 2155 2880 l 2154 2879 l 2153 2877 l 2152 2875 l 2152 2874 l 2151 2872 l 2150 2870 l 2149 2868 l 2148 2867 l 2147 2865 l 2146 2863 l 2145 2861 l 2144 2860 l 2143 2858 l 2142 2856 l 2141 2854 l 2140 2853 l S NP 2295 3064 m 2293 3062 l 2292 3061 l 2290 3059 l 2289 3058 l 2287 3056 l 2286 3055 l 2285 3053 l 2283 3051 l 2282 3050 l 2280 3048 l 2279 3047 l 2277 3045 l 2276 3044 l 2274 3042 l 2273 3041 l 2272 3039 l 2270 3038 l 2269 3036 l 2267 3034 l 2266 3033 l 2265 3031 l 2263 3030 l 2262 3028 l 2260 3026 l 2259 3025 l 2258 3023 l 2256 3022 l 2255 3020 l 2254 3019 l 2252 3017 l 2251 3015 l 2250 3014 l 2248 3012 l 2247 3011 l 2245 3009 l 2244 3007 l 2243 3006 l 2242 3004 l 2240 3003 l 2239 3001 l 2238 2999 l 2236 2998 l 2235 2996 l 2234 2994 l 2232 2993 l 2231 2991 l 2230 2990 l 2228 2988 l S NP 2425 3178 m 2424 3177 l 2422 3176 l 2420 3174 l 2418 3173 l 2416 3172 l 2415 3170 l 2413 3169 l 2411 3168 l 2409 3166 l 2408 3165 l 2406 3164 l 2404 3162 l 2402 3161 l 2401 3159 l 2399 3158 l 2397 3157 l 2396 3155 l 2394 3154 l 2392 3153 l 2390 3151 l 2389 3150 l 2387 3148 l 2385 3147 l 2384 3146 l 2382 3144 l 2380 3143 l 2379 3141 l 2377 3140 l 2375 3139 l 2374 3137 l 2372 3136 l 2370 3134 l 2369 3133 l 2367 3132 l 2365 3130 l 2364 3129 l 2362 3127 l 2360 3126 l 2359 3124 l 2357 3123 l 2355 3121 l 2354 3120 l 2352 3119 l 2351 3117 l 2349 3116 l 2347 3114 l 2346 3113 l 2344 3111 l S NP 3131 3448 m 3128 3448 l 3126 3447 l 3124 3447 l 3121 3447 l 3119 3447 l 3116 3446 l 3114 3446 l 3111 3446 l 3109 3445 l 3106 3445 l 3104 3445 l 3101 3444 l 3099 3444 l 3096 3444 l 3094 3443 l 3091 3443 l 3089 3443 l 3086 3442 l 3084 3442 l 3081 3441 l 3079 3441 l 3077 3441 l 3074 3440 l 3072 3440 l 3069 3440 l 3067 3439 l 3064 3439 l 3062 3438 l 3059 3438 l 3057 3438 l 3054 3437 l 3052 3437 l 3049 3436 l 3047 3436 l 3045 3436 l 3042 3435 l 3040 3435 l 3037 3434 l 3035 3434 l 3032 3433 l 3030 3433 l 3027 3432 l 3025 3432 l 3023 3432 l 3020 3431 l 3018 3431 l 3015 3430 l 3013 3430 l S NP 2935 3413 m 2933 3412 l 2931 3412 l 2928 3411 l 2926 3410 l 2923 3410 l 2921 3409 l 2919 3409 l 2916 3408 l 2914 3407 l 2912 3407 l 2909 3406 l 2907 3406 l 2904 3405 l 2902 3404 l 2900 3404 l 2897 3403 l 2895 3402 l 2893 3402 l 2890 3401 l 2888 3400 l 2886 3400 l 2883 3399 l 2881 3399 l 2879 3398 l 2876 3397 l 2874 3397 l 2872 3396 l 2869 3395 l 2867 3394 l 2865 3394 l 2862 3393 l 2860 3392 l 2857 3392 l 2855 3391 l 2853 3390 l 2850 3390 l 2848 3389 l 2846 3388 l 2843 3387 l 2841 3387 l 2839 3386 l 2836 3385 l 2834 3385 l 2832 3384 l 2830 3383 l 2827 3382 l 2825 3382 l 2823 3381 l S NP 2750 3355 m 2747 3354 l 2745 3353 l 2743 3353 l 2741 3352 l 2738 3351 l 2736 3350 l 2734 3349 l 2732 3348 l 2730 3347 l 2727 3346 l 2725 3346 l 2723 3345 l 2721 3344 l 2718 3343 l 2716 3342 l 2714 3341 l 2712 3340 l 2710 3339 l 2708 3338 l 2705 3337 l 2703 3336 l 2701 3335 l 2699 3335 l 2697 3334 l 2694 3333 l 2692 3332 l 2690 3331 l 2688 3330 l 2686 3329 l 2684 3328 l 2681 3327 l 2679 3326 l 2677 3325 l 2675 3324 l 2673 3323 l 2671 3322 l 2668 3321 l 2666 3320 l 2664 3319 l 2662 3318 l 2660 3317 l 2658 3316 l 2656 3315 l 2653 3314 l 2651 3313 l 2649 3312 l 2647 3311 l 2645 3310 l S NP 769 2499 m 793 2236 l 852 2012 l 961 1768 l 1114 1535 l 1306 1321 l 1516 1142 l 1764 978 l 2047 835 l 2352 722 l 2684 639 l 2987 594 l 3333 576 l 3681 594 l 4010 645 l 4298 718 l 4624 839 l 4879 967 l 5148 1143 l 5357 1321 l 5551 1537 l 5697 1759 l 5808 2004 l 5871 2237 l 5894 2493 l 5871 2758 l 5811 2986 l 5706 3223 l 5555 3455 l 5368 3666 l 5154 3850 l 4882 4030 l 4611 4165 l 4298 4279 l 4000 4355 l 3662 4405 l 3339 4421 l 3001 4405 l 2677 4357 l 2348 4274 l 2049 4163 l 1766 4021 l 1512 3853 l 1307 3677 l 1108 3455 l 967 3241 l 857 3000 l 792 2760 l 769 2499 l S NP 3331 3460 m 3329 3460 l 3327 3460 l 3324 3460 l 3321 3460 l 3319 3460 l 3316 3460 l 3314 3460 l 3311 3460 l 3309 3460 l 3306 3460 l 3304 3460 l 3301 3460 l 3299 3460 l 3296 3460 l 3294 3459 l 3291 3459 l 3289 3459 l 3286 3459 l 3284 3459 l 3281 3459 l 3279 3459 l 3276 3459 l 3274 3459 l 3271 3459 l 3269 3459 l 3266 3459 l 3264 3459 l 3261 3458 l 3259 3458 l 3256 3458 l 3253 3458 l 3251 3458 l 3248 3458 l 3246 3458 l 3243 3458 l 3241 3458 l 3239 3457 l 3236 3457 l 3233 3457 l 3231 3457 l 3228 3457 l 3226 3457 l 3224 3456 l 3221 3456 l 3218 3456 l 3216 3456 l 3214 3456 l 3211 3456 l S NP 4597 2348 m 4598 2350 l 4598 2352 l 4598 2354 l 4599 2356 l 4599 2358 l 4599 2360 l 4600 2361 l 4600 2363 l 4600 2365 l 4601 2367 l 4601 2369 l 4602 2371 l 4602 2373 l 4602 2375 l 4602 2376 l 4603 2378 l 4603 2380 l 4603 2382 l 4604 2384 l 4604 2386 l 4604 2388 l 4605 2390 l 4605 2391 l 4605 2393 l 4605 2395 l 4606 2397 l 4606 2399 l 4606 2401 l 4606 2403 l 4607 2404 l 4607 2406 l 4607 2408 l 4607 2410 l 4608 2412 l 4608 2414 l 4608 2416 l 4608 2418 l 4609 2420 l 4609 2421 l 4609 2423 l 4609 2425 l 4609 2427 l 4610 2429 l 4610 2431 l 4610 2433 l 4610 2435 l 4610 2436 l 4610 2438 l S NP 3727 3413 m 3725 3413 l 3723 3414 l 3720 3415 l 3718 3415 l 3715 3416 l 3713 3416 l 3711 3417 l 3708 3417 l 3706 3418 l 3703 3419 l 3701 3419 l 3699 3420 l 3696 3420 l 3694 3421 l 3692 3421 l 3689 3422 l 3687 3422 l 3684 3423 l 3682 3423 l 3679 3424 l 3677 3424 l 3674 3425 l 3672 3425 l 3670 3426 l 3667 3426 l 3665 3427 l 3662 3427 l 3660 3428 l 3658 3428 l 3655 3429 l 3653 3429 l 3650 3430 l 3648 3430 l 3645 3431 l 3643 3431 l 3640 3432 l 3638 3432 l 3636 3432 l 3633 3433 l 3631 3433 l 3628 3434 l 3626 3434 l 3623 3435 l 3621 3435 l 3618 3435 l 3616 3436 l 3614 3436 l 3611 3437 l S NP 3532 3448 m 3529 3448 l 3527 3449 l 3525 3449 l 3522 3449 l 3519 3450 l 3517 3450 l 3515 3450 l 3512 3450 l 3510 3451 l 3507 3451 l 3505 3451 l 3502 3451 l 3499 3452 l 3497 3452 l 3495 3452 l 3492 3452 l 3490 3453 l 3487 3453 l 3485 3453 l 3482 3453 l 3480 3453 l 3477 3454 l 3475 3454 l 3472 3454 l 3470 3454 l 3467 3454 l 3465 3455 l 3462 3455 l 3460 3455 l 3457 3455 l 3454 3455 l 3452 3456 l 3449 3456 l 3447 3456 l 3445 3456 l 3442 3456 l 3440 3456 l 3437 3457 l 3435 3457 l 3432 3457 l 3429 3457 l 3427 3457 l 3425 3457 l 3422 3458 l 3420 3458 l 3417 3458 l 3415 3458 l 3412 3458 l S NP 4238 3178 m 4236 3180 l 4234 3181 l 4232 3182 l 4230 3184 l 4229 3185 l 4227 3186 l 4225 3188 l 4223 3189 l 4221 3190 l 4220 3192 l 4218 3193 l 4216 3194 l 4214 3196 l 4212 3197 l 4211 3198 l 4209 3199 l 4207 3201 l 4205 3202 l 4203 3203 l 4201 3205 l 4199 3206 l 4198 3207 l 4196 3208 l 4194 3210 l 4192 3211 l 4190 3212 l 4188 3213 l 4186 3215 l 4185 3216 l 4183 3217 l 4181 3219 l 4179 3220 l 4177 3221 l 4175 3222 l 4173 3223 l 4171 3225 l 4169 3226 l 4168 3227 l 4166 3228 l 4164 3230 l 4162 3231 l 4160 3232 l 4158 3233 l 4156 3235 l 4154 3236 l 4152 3237 l 4150 3238 l 4148 3239 l S NP 4085 3276 m 4083 3277 l 4081 3279 l 4079 3280 l 4077 3281 l 4074 3282 l 4072 3283 l 4070 3284 l 4068 3285 l 4066 3286 l 4064 3287 l 4062 3288 l 4060 3289 l 4058 3291 l 4056 3292 l 4054 3293 l 4052 3294 l 4050 3295 l 4048 3296 l 4046 3297 l 4043 3298 l 4041 3299 l 4039 3300 l 4037 3301 l 4035 3302 l 4033 3303 l 4031 3304 l 4029 3305 l 4027 3306 l 4025 3307 l 4022 3308 l 4020 3309 l 4018 3310 l 4016 3311 l 4014 3312 l 4012 3313 l 4010 3314 l 4007 3315 l 4005 3316 l 4003 3317 l 4001 3318 l 3999 3319 l 3997 3320 l 3995 3321 l 3992 3322 l 3990 3323 l 3988 3324 l 3986 3325 l 3984 3326 l S NP 3913 3355 m 3911 3356 l 3909 3357 l 3907 3358 l 3904 3359 l 3902 3359 l 3900 3360 l 3898 3361 l 3895 3362 l 3893 3363 l 3891 3364 l 3889 3364 l 3886 3365 l 3884 3366 l 3882 3367 l 3879 3368 l 3877 3368 l 3875 3369 l 3872 3370 l 3870 3371 l 3868 3372 l 3866 3372 l 3863 3373 l 3861 3374 l 3859 3375 l 3856 3376 l 3854 3376 l 3852 3377 l 3850 3378 l 3847 3379 l 3845 3379 l 3843 3380 l 3840 3381 l 3838 3382 l 3836 3382 l 3833 3383 l 3831 3384 l 3829 3385 l 3827 3385 l 3824 3386 l 3822 3387 l 3819 3387 l 3817 3388 l 3815 3389 l 3813 3390 l 3810 3390 l 3808 3391 l 3806 3392 l 3803 3392 l S NP 4368 3064 m 4367 3065 l 4365 3067 l 4364 3068 l 4362 3070 l 4361 3071 l 4359 3073 l 4358 3074 l 4356 3076 l 4355 3077 l 4353 3079 l 4352 3080 l 4350 3082 l 4349 3083 l 4347 3085 l 4346 3086 l 4344 3088 l 4343 3089 l 4341 3091 l 4339 3092 l 4338 3094 l 4336 3095 l 4335 3097 l 4333 3098 l 4332 3100 l 4330 3101 l 4328 3103 l 4327 3104 l 4325 3106 l 4324 3107 l 4322 3108 l 4320 3110 l 4319 3111 l 4317 3113 l 4316 3114 l 4314 3116 l 4312 3117 l 4311 3119 l 4309 3120 l 4308 3122 l 4306 3123 l 4304 3124 l 4303 3126 l 4301 3127 l 4299 3129 l 4298 3130 l 4296 3132 l 4294 3133 l 4293 3134 l S NP 4597 2649 m 4597 2651 l 4596 2653 l 4596 2655 l 4596 2657 l 4595 2658 l 4595 2660 l 4594 2662 l 4594 2664 l 4593 2666 l 4593 2668 l 4593 2670 l 4592 2671 l 4592 2673 l 4591 2675 l 4591 2677 l 4590 2679 l 4590 2681 l 4589 2683 l 4589 2684 l 4588 2686 l 4588 2688 l 4587 2690 l 4587 2692 l 4586 2694 l 4586 2696 l 4585 2697 l 4585 2699 l 4584 2701 l 4584 2703 l 4583 2705 l 4583 2707 l 4582 2708 l 4581 2710 l 4581 2712 l 4580 2714 l 4580 2716 l 4579 2718 l 4579 2719 l 4578 2721 l 4578 2723 l 4577 2725 l 4576 2727 l 4576 2729 l 4575 2730 l 4575 2732 l 4574 2734 l 4573 2736 l 4573 2738 l S NP 4550 2796 m 4549 2798 l 4549 2799 l 4548 2801 l 4547 2803 l 4546 2805 l 4545 2806 l 4545 2808 l 4544 2810 l 4543 2812 l 4542 2814 l 4541 2815 l 4541 2817 l 4540 2819 l 4539 2821 l 4538 2822 l 4537 2824 l 4536 2826 l 4535 2828 l 4535 2830 l 4534 2831 l 4533 2833 l 4532 2835 l 4531 2837 l 4530 2838 l 4529 2840 l 4528 2842 l 4528 2844 l 4527 2845 l 4526 2847 l 4525 2849 l 4524 2851 l 4523 2853 l 4522 2854 l 4521 2856 l 4520 2858 l 4519 2860 l 4518 2861 l 4517 2863 l 4516 2865 l 4515 2867 l 4514 2868 l 4513 2870 l 4513 2872 l 4511 2874 l 4511 2875 l 4510 2877 l 4509 2879 l 4508 2880 l S NP 4473 2935 m 4472 2937 l 4471 2938 l 4470 2940 l 4469 2942 l 4467 2944 l 4466 2945 l 4465 2947 l 4464 2948 l 4463 2950 l 4462 2952 l 4460 2953 l 4459 2955 l 4458 2957 l 4457 2959 l 4456 2960 l 4454 2962 l 4453 2963 l 4452 2965 l 4451 2967 l 4450 2968 l 4448 2970 l 4447 2972 l 4446 2973 l 4445 2975 l 4443 2977 l 4442 2978 l 4441 2980 l 4440 2981 l 4438 2983 l 4437 2985 l 4436 2986 l 4434 2988 l 4433 2990 l 4432 2991 l 4431 2993 l 4429 2994 l 4428 2996 l 4427 2998 l 4425 2999 l 4424 3001 l 4423 3003 l 4421 3004 l 4420 3006 l 4419 3007 l 4417 3009 l 4416 3011 l 4415 3012 l 4413 3014 l S NP 4473 2062 m 4474 2064 l 4475 2066 l 4477 2067 l 4478 2069 l 4479 2071 l 4480 2072 l 4481 2074 l 4482 2076 l 4483 2078 l 4485 2079 l 4486 2081 l 4487 2083 l 4488 2084 l 4489 2086 l 4490 2088 l 4491 2090 l 4492 2091 l 4493 2093 l 4494 2095 l 4495 2096 l 4496 2098 l 4497 2100 l 4498 2101 l 4499 2103 l 4500 2105 l 4501 2107 l 4502 2108 l 4503 2110 l 4504 2112 l 4505 2113 l 4507 2115 l 4508 2117 l 4509 2119 l 4509 2120 l 4511 2122 l 4511 2124 l 4512 2126 l 4513 2127 l 4514 2129 l 4515 2131 l 4516 2133 l 4517 2134 l 4518 2136 l 4519 2138 l 4520 2140 l 4521 2141 l 4522 2143 l 4523 2145 l S NP 4550 2202 m 4551 2204 l 4552 2205 l 4552 2207 l 4553 2209 l 4554 2211 l 4555 2212 l 4556 2214 l 4556 2216 l 4557 2218 l 4558 2220 l 4558 2221 l 4559 2223 l 4560 2225 l 4561 2227 l 4561 2229 l 4562 2231 l 4563 2232 l 4563 2234 l 4564 2236 l 4565 2238 l 4565 2240 l 4566 2241 l 4567 2243 l 4567 2245 l 4568 2247 l 4569 2249 l 4569 2250 l 4570 2252 l 4571 2254 l 4571 2256 l 4572 2258 l 4573 2260 l 4573 2262 l 4574 2263 l 4575 2265 l 4575 2267 l 4576 2269 l 4576 2271 l 4577 2273 l 4578 2274 l 4578 2276 l 4579 2278 l 4579 2280 l 4580 2282 l 4580 2283 l 4581 2285 l 4581 2287 l 4582 2289 l S NP 4368 1934 m 4370 1935 l 4371 1937 l 4373 1938 l 4374 1940 l 4376 1941 l 4377 1943 l 4378 1944 l 4380 1946 l 4381 1948 l 4383 1949 l 4384 1951 l 4386 1952 l 4387 1954 l 4389 1955 l 4390 1957 l 4391 1959 l 4393 1960 l 4394 1962 l 4396 1963 l 4397 1965 l 4398 1966 l 4400 1968 l 4401 1969 l 4402 1971 l 4404 1973 l 4405 1974 l 4407 1976 l 4408 1977 l 4409 1979 l 4411 1980 l 4412 1982 l 4413 1984 l 4415 1985 l 4416 1987 l 4417 1989 l 4419 1990 l 4420 1992 l 4421 1993 l 4423 1995 l 4424 1997 l 4425 1998 l 4427 2000 l 4428 2001 l 4429 2003 l 4431 2005 l 4432 2006 l 4433 2008 l 4434 2009 l S NP 2578 3276 m 2576 3275 l 2574 3274 l 2572 3273 l 2570 3272 l 2568 3271 l 2566 3270 l 2564 3269 l 2562 3267 l 2560 3266 l 2558 3265 l 2556 3264 l 2554 3263 l 2552 3262 l 2550 3260 l 2548 3259 l 2546 3258 l 2544 3257 l 2542 3256 l 2540 3255 l 2538 3254 l 2536 3252 l 2534 3251 l 2532 3250 l 2530 3249 l 2528 3248 l 2526 3247 l 2524 3245 l 2522 3244 l 2521 3243 l 2519 3242 l 2517 3240 l 2515 3239 l 2513 3238 l 2511 3237 l 2509 3236 l 2507 3235 l 2505 3233 l 2503 3232 l 2501 3231 l 2499 3230 l 2497 3228 l 2495 3227 l 2494 3226 l 2492 3225 l 2490 3223 l 2488 3222 l 2486 3221 l 2484 3220 l S NP 4238 1819 m 4239 1820 l 4241 1822 l 4243 1823 l 4245 1824 l 4247 1826 l 4248 1827 l 4250 1828 l 4252 1830 l 4253 1831 l 4255 1833 l 4257 1834 l 4259 1835 l 4260 1837 l 4262 1838 l 4264 1839 l 4266 1841 l 4267 1842 l 4269 1844 l 4271 1845 l 4272 1846 l 4274 1848 l 4276 1849 l 4278 1850 l 4279 1852 l 4281 1853 l 4283 1855 l 4284 1856 l 4286 1857 l 4288 1859 l 4289 1860 l 4291 1862 l 4293 1863 l 4294 1865 l 4296 1866 l 4298 1867 l 4299 1869 l 4301 1870 l 4303 1872 l 4304 1873 l 4306 1875 l 4308 1876 l 4309 1877 l 4311 1879 l 4312 1880 l 4314 1882 l 4316 1883 l 4317 1885 l 4319 1886 l S NP 4085 1721 m 4087 1722 l 4089 1723 l 4091 1724 l 4093 1726 l 4095 1727 l 4097 1728 l 4099 1729 l 4101 1730 l 4103 1731 l 4105 1732 l 4107 1733 l 4109 1735 l 4111 1736 l 4113 1737 l 4115 1738 l 4117 1739 l 4119 1740 l 4121 1742 l 4123 1743 l 4125 1744 l 4127 1745 l 4129 1746 l 4131 1747 l 4133 1749 l 4135 1750 l 4137 1751 l 4138 1752 l 4140 1753 l 4142 1755 l 4144 1756 l 4146 1757 l 4148 1758 l 4150 1759 l 4152 1761 l 4154 1762 l 4156 1763 l 4158 1764 l 4160 1765 l 4162 1767 l 4164 1768 l 4166 1769 l 4168 1770 l 4169 1771 l 4171 1773 l 4173 1774 l 4175 1775 l 4177 1776 l 4179 1778 l S NP 2050 2499 m 2050 2497 l 2050 2495 l 2050 2493 l 2050 2491 l 2050 2489 l 2050 2487 l 2050 2486 l 2050 2484 l 2050 2482 l 2050 2480 l 2050 2478 l 2050 2476 l 2050 2474 l 2051 2472 l 2051 2471 l 2051 2468 l 2051 2467 l 2051 2465 l 2051 2463 l 2051 2461 l 2051 2459 l 2051 2457 l 2051 2455 l 2051 2454 l 2052 2452 l 2052 2450 l 2052 2448 l 2052 2446 l 2052 2444 l 2052 2442 l 2052 2440 l 2053 2438 l 2053 2436 l 2053 2435 l 2053 2433 l 2053 2431 l 2053 2429 l 2054 2427 l 2054 2425 l 2054 2423 l 2054 2421 l 2054 2420 l 2055 2418 l 2055 2416 l 2055 2414 l 2055 2412 l 2055 2410 l 2056 2408 l S NP 3727 1585 m 3730 1585 l 3732 1586 l 3735 1586 l 3737 1587 l 3739 1588 l 3742 1588 l 3744 1589 l 3747 1589 l 3749 1590 l 3751 1591 l 3754 1591 l 3756 1592 l 3759 1593 l 3761 1593 l 3763 1594 l 3766 1594 l 3768 1595 l 3770 1596 l 3773 1596 l 3775 1597 l 3777 1598 l 3780 1598 l 3782 1599 l 3784 1600 l 3787 1600 l 3789 1601 l 3791 1602 l 3794 1602 l 3796 1603 l 3798 1604 l 3801 1604 l 3803 1605 l 3806 1606 l 3808 1606 l 3810 1607 l 3812 1608 l 3815 1609 l 3817 1609 l 3820 1610 l 3822 1611 l 3824 1611 l 3827 1612 l 3829 1613 l 3831 1614 l 3833 1614 l 3836 1615 l 3838 1616 l 3840 1617 l S NP 3913 1642 m 3916 1643 l 3918 1644 l 3920 1645 l 3922 1646 l 3925 1647 l 3927 1648 l 3929 1648 l 3931 1649 l 3933 1650 l 3936 1651 l 3938 1652 l 3940 1653 l 3942 1654 l 3945 1655 l 3947 1656 l 3949 1656 l 3951 1657 l 3953 1658 l 3955 1659 l 3958 1660 l 3960 1661 l 3962 1662 l 3964 1663 l 3966 1664 l 3969 1665 l 3971 1666 l 3973 1667 l 3975 1668 l 3977 1669 l 3979 1669 l 3982 1670 l 3984 1671 l 3986 1672 l 3988 1673 l 3990 1674 l 3992 1675 l 3995 1676 l 3997 1677 l 3999 1678 l 4001 1679 l 4003 1680 l 4005 1681 l 4007 1682 l 4010 1683 l 4012 1684 l 4014 1685 l 4016 1686 l 4018 1687 l S NP 3532 1549 m 3535 1550 l 3537 1550 l 3539 1550 l 3542 1551 l 3544 1551 l 3547 1551 l 3549 1552 l 3552 1552 l 3554 1552 l 3557 1553 l 3559 1553 l 3562 1553 l 3564 1554 l 3567 1554 l 3569 1554 l 3572 1555 l 3574 1555 l 3577 1555 l 3579 1556 l 3581 1556 l 3584 1556 l 3586 1557 l 3589 1557 l 3591 1558 l 3594 1558 l 3596 1558 l 3599 1559 l 3601 1559 l 3604 1559 l 3606 1560 l 3609 1560 l 3611 1561 l 3614 1561 l 3616 1562 l 3618 1562 l 3621 1562 l 3623 1563 l 3626 1563 l 3628 1564 l 3631 1564 l 3633 1565 l 3636 1565 l 3638 1565 l 3640 1566 l 3643 1566 l 3645 1567 l 3648 1567 l 3650 1568 l S NP 2750 1642 m 2752 1641 l 2754 1641 l 2756 1640 l 2759 1639 l 2761 1638 l 2763 1637 l 2765 1636 l 2768 1636 l 2770 1635 l 2772 1634 l 2774 1633 l 2777 1632 l 2779 1631 l 2781 1631 l 2783 1630 l 2786 1629 l 2788 1628 l 2790 1627 l 2793 1627 l 2795 1626 l 2797 1625 l 2800 1624 l 2802 1624 l 2804 1623 l 2807 1622 l 2809 1621 l 2811 1620 l 2813 1620 l 2816 1619 l 2818 1618 l 2820 1617 l 2823 1617 l 2825 1616 l 2827 1615 l 2830 1614 l 2832 1614 l 2834 1613 l 2836 1612 l 2839 1611 l 2841 1611 l 2843 1610 l 2846 1609 l 2848 1609 l 2850 1608 l 2853 1607 l 2855 1606 l 2857 1606 l 2860 1605 l S NP 2935 1585 m 2938 1584 l 2940 1583 l 2943 1583 l 2945 1582 l 2948 1582 l 2950 1581 l 2952 1581 l 2955 1580 l 2957 1579 l 2960 1579 l 2962 1578 l 2964 1578 l 2967 1577 l 2969 1577 l 2971 1576 l 2974 1576 l 2976 1575 l 2979 1575 l 2981 1574 l 2984 1574 l 2986 1573 l 2989 1573 l 2991 1572 l 2993 1572 l 2996 1571 l 2998 1571 l 3001 1570 l 3003 1570 l 3005 1569 l 3008 1569 l 3010 1568 l 3013 1568 l 3015 1567 l 3018 1567 l 3020 1566 l 3023 1566 l 3025 1565 l 3027 1565 l 3030 1565 l 3032 1564 l 3035 1564 l 3037 1563 l 3040 1563 l 3042 1562 l 3045 1562 l 3047 1562 l 3049 1561 l 3052 1561 l S NP 3131 1549 m 3134 1549 l 3136 1549 l 3138 1549 l 3141 1548 l 3144 1548 l 3146 1548 l 3148 1547 l 3151 1547 l 3153 1547 l 3156 1547 l 3158 1546 l 3161 1546 l 3163 1546 l 3166 1546 l 3168 1545 l 3171 1545 l 3173 1545 l 3176 1545 l 3178 1544 l 3181 1544 l 3183 1544 l 3186 1544 l 3188 1544 l 3191 1543 l 3193 1543 l 3196 1543 l 3198 1543 l 3201 1543 l 3203 1542 l 3206 1542 l 3208 1542 l 3211 1542 l 3214 1542 l 3216 1541 l 3218 1541 l 3221 1541 l 3223 1541 l 3226 1541 l 3228 1541 l 3231 1541 l 3234 1540 l 3236 1540 l 3238 1540 l 3241 1540 l 3243 1540 l 3246 1540 l 3248 1540 l 3251 1539 l S boundarythick setlinewidth %/bd boundarythick 2 idiv def %[] 0 setdash %NP bd bd m bd 5000 bd sub l %6666 bd sub 5000 bd sub l %6666 bd sub bd l %bd bd l S showpage grestore end %%EndDocument @endspecial 1546 37 a currentpoint grestore moveto 1546 37 a 677 1349 a Fk(A)26 b(typical)d(domain) -73 1518 y Fs(As)16 b(a)h(measure)e(of)h(the)g(distance)g(b)q(et)o(w)o (een)f(the)h(p)q(oin)o(ts)h(w)o(e)f(in)o(tro)q(duce:)330 1603 y Fr(\016)f Fs(=)f(inf)486 1562 y Fl(\010)515 1603 y Fr(\016)i(>)e Fs(0)629 1560 y Fl(\014)629 1590 y(\014)646 1603 y Fn(f)p Fr(x)f Fn(2)h FB(R)801 1582 y FA(D)833 1560 y Fl(\014)833 1590 y(\014)849 1603 y Fn(j)p Fr(x)p Fn(j)g Fs(=)f Fr(R)1007 1610 y Fq(1)1027 1603 y Fn(g)h(\032)g([)1152 1582 y FA(N)1152 1615 y(j)r Fq(=1)1215 1603 y Fr(B)s Fs(\()p Fr(x)1302 1610 y FA(j)1320 1603 y Fr(;)8 b(\016)r Fs(\))1385 1562 y Fl(\011)1421 1603 y Fr(:)-123 1687 y Fs(This)21 b(measures)f(the)h(maximal)c(distance)k(b)q(et)o(w)o(een)f (neigh)o(b)q(oring)h(p)q(oin)o(ts.)36 b(Let)21 b(us)g(also)h(in)o(tro)q (duce)e(the)-123 1745 y(simpler:)604 1809 y Fr(\032)14 b Fs(=)f(min)704 1841 y FA(j)r Fm(6)p Fq(=)p FA(k)776 1809 y Fn(f)p Fs(dist)879 1819 y Fj(S)901 1809 y Fi(N)s Fh(\000)p Fg(1)q Fs(\()p Fr(x)1015 1816 y FA(j)1033 1809 y Fr(;)8 b(x)1083 1816 y FA(k)1104 1809 y Fs(\))p Fn(g)p Fr(;)-123 1907 y Fs(where)17 b(dist)97 1916 y Fj(S)119 1907 y Fi(N)s Fh(\000)p Fg(1)h Fs(is)e(geo)q(desic)h(distance)g(on)g (the)g(sphere.)22 b(W)l(e)17 b(will)e(alw)o(a)o(ys)i(assume)f(that)h Fr(\032)e(>)g Fs(0)i(and)g(that)-123 1965 y Fr(\017)f(<)f(\032=)p Fs(2.)25 b(T)l(o)18 b(mak)o(e)e(sure)h(that)h(the)f(holes)g(are)h(ev)o (enly)d(distributed,)h(w)o(e)h(assume)g Fr(\016)r(=\032)f Fn(\024)f Fr(c)1604 1972 y Fq(0)1624 1965 y Fs(,)i(where)g Fr(c)1818 1972 y Fq(0)1855 1965 y Fs(is)-123 2023 y(some)e(constan)o(t) i(whic)o(h)f(will)f(b)q(e)h(assumed)g(\014xed)g(throughout)h(the)f(pap) q(er.)-73 2081 y(W)l(e)g(will)f(simplify)f(notation)j(b)o(y)e(writing)h Fr(\025)746 2088 y FA(j;\017)804 2081 y Fs(and)g Fr(u)926 2088 y FA(j;\017)983 2081 y Fs(instead)g(of)h(the)f(hea)o(vier)f Fr(\025)1487 2088 y FA(j)1505 2081 y Fs(\(\012)1559 2088 y FA(\017)1576 2081 y Fs(\))h(and)h Fr(u)1734 2088 y FA(j)1752 2081 y Fs(\(\012)1806 2088 y FA(\017)1822 2081 y Fs(\))-73 2147 y(Notice)g(that)j(\012)226 2154 y Fq(0)263 2147 y Fs(=)e(\()p Fr(B)s Fs(\()p Fr(R)434 2154 y Fq(2)454 2147 y Fs(\))12 b Fn(n)p 523 2103 135 2 v 13 w Fr(B)s Fs(\()p Fr(R)619 2154 y Fq(1)638 2147 y Fs(\)\))h Fn([)g Fr(B)s Fs(\()p Fr(R)831 2154 y Fq(1)850 2147 y Fs(\),)19 b(is)f(the)h(union)f(of)h(t)o(w)o(o)g(disjoin)o(t)f(domains)g(and)h (that)-123 2205 y(the)d(Diric)o(hlet)e(Laplacian)j(on)g(this)f(set)g (is)g(explicitly)e(solv)m(able)i(in)g(terms)e(of)j(Bessel)e(functions.) -73 2263 y(In)f([Sto95])g(it)f(w)o(as)i(pro)o(v)o(ed)e(that)h Fn(\000)p Fs(\001)623 2270 y FA(\017)653 2263 y Fs(con)o(v)o(erges)f (to)i Fn(\000)p Fs(\001)1007 2270 y Fq(0)1040 2263 y Fs(in)e(strong)i(resolv)o(en)o(t)e(sense)h(as)g Fr(\017)g Fn(&)g Fs(0.)20 b(Th)o(us)-123 2321 y(in)c(particular)766 2385 y Fr(\025)794 2392 y FA(j;\017)849 2385 y Fn(\045)e Fr(\025)941 2392 y FA(j;)p Fq(0)985 2385 y Fr(:)-123 2459 y Fs(F)l(rom)i(this)h(w)o(e)g(see)g(that)g(when)g Fr(\017)g Fs(is)g(su\016cien)o(tly)e(small,)h(then)h Fr(\025)1104 2466 y Fq(2)p FA(;\017)1165 2459 y Fs(is)g(a)h(simple)d (eigen)o(v)m(alue.)22 b(W)l(e)17 b(c)o(ho)q(ose)-123 2518 y Fr(u)-95 2525 y Fq(1)p FA(;)p Fq(0)-32 2518 y Fs(and)g Fr(u)91 2525 y Fq(2)p FA(;)p Fq(0)154 2518 y Fs(to)g(b)q(e)f(p)q(ositiv)o(e)g(functions.)-73 2588 y(De\014ne)g Fn(N)7 b Fs(\()p Fr(u)173 2595 y Fq(2)p FA(;\017)217 2588 y Fs(\))14 b(=)p 301 2540 434 2 v 13 w Fn(f)p Fr(x)g Fn(2)g Fs(\012)450 2595 y FA(\017)466 2546 y Fl(\014)466 2575 y(\014)483 2588 y Fr(u)511 2595 y Fq(2)p FA(;\017)555 2588 y Fs(\()p Fr(x)p Fs(\))f(=)h(0)p Fn(g)p Fs(,)i(then)g(the)g(result)g(of)g(this)g(pap)q(er)h(is:)-123 2679 y FB(Theorem)g(1.1.)23 b Ff(Ther)n(e)17 b(exists)i(a)e Fr(\016)564 2686 y Fq(0)601 2679 y Ff(such)h(that)f(if)g Fr(\016)f Fn(\024)d Fr(\016)970 2686 y Fq(0)1007 2679 y Ff(then)684 2763 y Fn(N)7 b Fs(\()p Fr(u)779 2770 y Fq(2)p FA(;\017)823 2763 y Fs(\))k Fn(\\)g Fr(@)s Fs(\012)961 2770 y FA(\017)991 2763 y Fs(=)j Fn(;)p Fr(;)-123 2847 y Ff(for)j(al)r(l)i Fr(\017)e Ff(su\016ciently)i(smal)r(l.)p eop %%Page: 4 4 4 3 bop -5 -64 a Fo(4)809 b(S\037REN)17 b(F)o(OURNAIS)-5 37 y FB(Remark)g(1.2.)23 b Fs(Th)o(us,)18 b(the)f(theorem)f(sa)o(ys,)i (that)g(if)f(w)o(e)g(cut)h(man)o(y)l(,)d(small)h(holes,)i(and)g(they)f (are)h(almost)-5 95 y(ev)o(enly)d(distributed)g(o)o(v)o(er)h(the)g (sphere,)f(then)h(the)g(no)q(dal)h(surface)g(will)e(b)q(e)h(closed.)45 173 y(In)f(section)f(4)h(w)o(e)g(get)g(an)g(explicit)e(upp)q(er)i(b)q (ound)h(on)f(the)g(minim)o(al)d(n)o(um)o(b)q(er)h(of)i(holes)g (necessary)g(whic)o(h)-5 232 y(is)h Fn(\031)e Fs(10)145 213 y Fq(9)165 232 y Fs(.)45 290 y(W)l(e)f(use)h([Bas95])f(and)h([PS78) q(])f(as)h(standard)h(references)d(for)i(results)f(on)h(sto)q(c)o (hastic)f(pro)q(cesses.)21 b(In)13 b(those)-5 348 y(b)q(o)q(oks)18 b(references)d(to)i(the)f(original)g(articles)f(can)h(b)q(e)h(found.) 681 447 y(2.)28 b Ft(Preliminar)m(y)16 b(Estima)m(tes)45 534 y Fs(The)g(most)g(imp)q(ortan)o(t)f(result)h(in)g(this)g(section)g (is)g(the)g(follo)o(wing)g(estimate:)-5 612 y FB(Lemma)g(2.1.)23 b Fn(9)p Fr(C)17 b(>)d Fs(0)k Ff(indep)n(endent)h(of)f Fr(\017;)8 b(\016)18 b Ff(such)g(that)539 735 y Fn(jh)p Fr(u)600 742 y FA(j;\017)641 735 y Fr(;)8 b(u)691 742 y Fq(2)p FA(;)p Fq(0)738 735 y Fn(ij)14 b(\024)f Fr(C)927 697 y(\025)955 671 y Fq(1+)p FA(D)q(=)p Fq(4)955 710 y FA(j;\017)p 881 724 233 2 v 881 770 a Fn(j)p Fr(\025)923 777 y FA(j;\017)974 770 y Fn(\000)e Fr(\025)1052 777 y Fq(2)p FA(;)p Fq(0)1100 770 y Fn(j)1119 735 y Fr(N)5 b(\017)1183 715 y FA(D)1226 735 y Fs(+)11 b Fr(O)q Fs(\()p Fr(\017)1352 715 y FA(D)q Fq(+1)1429 735 y Fs(\))p Fr(;)-5 845 y Ff(when)19 b Fr(\017)e Ff(is)g(su\016ciently)i(smal)r(l)g(\(dep.) j(on)c Fr(\016)r Ff(\).)k(Her)n(e)17 b Fr(C)k Ff(c)n(an)d(b)n(e)g (chosen)g(as:)674 933 y Fr(C)f Fs(=)d(\()p Fr(@)823 940 y FA(r)842 933 y Fr(u)870 940 y Fq(2)p FA(;)p Fq(0)917 933 y Fn(j)931 940 y FA(r)q Fq(=)p FA(R)1002 945 y Fg(1)1021 933 y Fs(\))p Fr(\033)1068 940 y FA(D)q Fm(\000)p Fq(1)1145 933 y Fr(e)1168 912 y Fq(1+)1227 898 y Fg(1)p 1217 904 35 2 v 1217 925 a(8)p Fi(\031)1259 933 y Fr(C)1294 940 y Fq(0)1313 933 y Fr(;)-5 1011 y Ff(wher)n(e)k Fr(C)168 1018 y Fq(0)205 1011 y Ff(is)f(the)h(c)n(onstant)g(given)h(in)f(L)n (emma)f(2.6)g(b)n(elow,)h(and)g Fr(\033)1241 1018 y FA(D)q Fm(\000)p Fq(1)1332 1011 y Fs(=)c(v)o(ol)1446 1020 y Fe(R)1476 1011 y Fi(D)q Fh(\000)p Fg(1)t Fs(\()p Fr(B)s Fs(\(1\)\))p Ff(.)-5 1091 y FB(Remark)j(2.2.)23 b Fr(\016)343 1073 y Fm(\000)p Fq(\()p FA(D)q Fm(\000)p Fq(1\))493 1091 y Fs(is)18 b(prop)q(ortional)h(to)f(the)g(n)o(um)o(b)q(er)f(of)h (holes,)g(so)h Fr(N)k Fs(in)18 b(the)g(ab)q(o)o(v)o(e)g(Lemma)e(can)-5 1149 y(b)q(e)h(c)o(hanged)f(to)h Fr(\016)334 1131 y Fm(\000)p Fq(\()p FA(D)q Fm(\000)p Fq(1\))465 1149 y Fs(,)f(up)g(to)h(a)g(c)o (hange)f(in)g(the)g(constan)o(t)g Fr(C)t Fs(.)45 1228 y(The)g(pro)q(of)i(of)e(Lemma)e(2.1)j(is)f(giv)o(en)f(in)h(the)g(rest)g (of)h(this)f(section)g(as)h(a)f(series)g(of)h(lemm)o(as.)-5 1306 y FB(Lemma)f(2.3.)556 1381 y Fn(h)p Fr(u)603 1388 y FA(j;\017)644 1381 y Fr(;)8 b(u)694 1388 y Fq(2)p FA(;)p Fq(0)741 1381 y Fn(i)14 b Fs(=)g Fn(\000)870 1348 y Fr(@)896 1355 y FA(r)914 1348 y Fr(u)942 1355 y Fq(2)p FA(;)p Fq(0)989 1348 y Fn(j)1003 1355 y FA(r)q Fq(=)p FA(R)1074 1360 y Fg(1)p 869 1370 224 2 v 879 1415 a Fr(\025)907 1422 y FA(j;\017)959 1415 y Fn(\000)c Fr(\025)1036 1422 y Fq(2)p FA(;)p Fq(0)1106 1314 y Fl(Z)1134 1426 y FA(S)1155 1430 y Fi(\017)1181 1381 y Fr(u)1209 1388 y FA(j;\017)1249 1381 y Fs(\()p Fr(y)r Fs(\))p Fr(d\033)r Fs(\()p Fr(y)r Fs(\))p Fr(;)-5 1487 y Ff(wher)n(e)18 b Fr(S)163 1494 y FA(\017)193 1487 y Fs(=)c Fn(f)p Fr(x)f Fn(2)h Fs(\012)393 1494 y FA(\017)410 1445 y Fl(\014)410 1475 y(\014)427 1487 y Fn(j)p Fr(x)p Fn(j)f Fs(=)h Fr(R)585 1494 y Fq(1)604 1487 y Fn(g)k Ff(and)f Fr(\033)i Ff(is)f(surfac)n(e)f(me)n(asur)n(e)g (on)g(the)h(spher)n(e)f Fn(fj)p Fr(x)p Fn(j)c Fs(=)h Fr(R)1679 1494 y Fq(1)1699 1487 y Fn(g)p Ff(.)-5 1566 y(Pr)n(o)n(of.)19 b Fs(This)e(is,)e(in)h(fact,)g(just)g(Green's)g(iden) o(tit)o(y:)373 1644 y Fr(\025)401 1651 y FA(j;\017)442 1644 y Fn(h)p Fr(u)489 1651 y FA(j;\017)529 1644 y Fr(;)8 b(u)579 1651 y Fq(2)p FA(;)p Fq(0)626 1644 y Fn(i)42 b Fs(=)g Fn(h\000)p Fs(\001)866 1651 y FA(\017)882 1644 y Fr(u)910 1651 y FA(j;\017)950 1644 y Fr(;)8 b(u)1000 1651 y Fq(2)p FA(;)p Fq(0)1047 1644 y Fn(i)687 1744 y Fs(=)42 b Fr(\025)795 1751 y Fq(2)p FA(;)p Fq(0)842 1744 y Fn(h)p Fr(u)889 1751 y FA(j;\017)930 1744 y Fr(;)8 b(u)980 1751 y Fq(2)p FA(;)p Fq(0)1026 1744 y Fn(i)k(\000)1107 1677 y Fl(Z)1134 1789 y FA(S)1155 1793 y Fi(\017)1181 1744 y Fr(u)1209 1751 y FA(j;\017)1249 1744 y Fs(\()p Fr(y)r Fs(\))p Fr(@)1339 1751 y FA(r)1358 1744 y Fr(u)1386 1751 y Fq(2)p FA(;)p Fq(0)1433 1744 y Fs(\()p Fr(y)r Fs(\))p Fr(d\033)r Fs(\()p Fr(y)r Fs(\))p Fr(:)-5 1853 y Fs(No)o(w)k(w)o(e)g(use)g(that)h Fr(u)396 1860 y Fq(2)p FA(;)p Fq(0)459 1853 y Fs(is)f(rotationally)h(symme)o(tric)c(to)k(reac) o(h)e(the)h(conclusion.)p 1971 1853 2 33 v 1973 1821 30 2 v 1973 1853 V 2001 1853 2 33 v 45 1936 a(Th)o(us)h(w)o(e)e(need)h (to)h(estimate)609 1896 y Fl(R)633 1953 y FA(S)654 1957 y Fi(\017)680 1936 y Fr(u)708 1943 y FA(j;\017)748 1936 y Fs(\()p Fr(y)r Fs(\))p Fr(d\033)r Fs(\()p Fr(y)r Fs(\).)45 1996 y(Notice)e(the)h(follo)o(wing)g(argumen)o(t:)-5 2075 y FB(Lemma)g(2.4.)23 b Fn(8)p Fr(y)14 b Fn(2)h Fs(\012)455 2082 y FA(\017)489 2075 y Ff(we)j(have:)698 2153 y Fn(j)p Fr(u)740 2160 y FA(j;\017)780 2153 y Fs(\()p Fr(y)r Fs(\))p Fn(j)13 b(\024)h(k)p Fr(u)977 2160 y FA(j;\017)1017 2153 y Fn(k)1042 2160 y Fm(1)1079 2153 y Fr(e\025)1130 2160 y FA(j;\017)1171 2153 y Fd(E)1201 2132 y FA(y)1225 2153 y Fs([)p Fr(\034)1260 2160 y FA(\017)1276 2153 y Fs(])p Fr(;)-5 2231 y Ff(wher)n(e)k Fr(\034)154 2238 y FA(\017)188 2231 y Ff(is)f(the)h(exit)g(time)g(of)g(Br)n(ownian)f(motion)h(fr)n(om) e Fs(\012)1120 2238 y FA(\017)1154 2231 y Ff(i.e.)740 2310 y Fr(\034)761 2317 y FA(\017)792 2310 y Fs(=)e(inf)s Fn(f)p Fr(t)f(>)h Fs(0)p Fn(j)p Fr(W)1095 2317 y FA(t)1129 2310 y Fr(=)-29 b Fn(2)14 b Fs(\012)1206 2317 y FA(\017)1222 2310 y Fn(g)p Fr(;)-5 2388 y Ff(wher)n(e)k Fr(W)179 2395 y FA(t)211 2388 y Ff(is)f Fr(D)q Ff(-dimensional)j(Br)n(ownian)e (motion)g(and)f Fd(E)25 b Ff(denotes)18 b(the)g(exp)n(e)n(ctation.)-5 2467 y(Pr)n(o)n(of.)623 2545 y Fr(u)651 2552 y FA(j;\017)691 2545 y Fs(\()p Fr(y)r Fs(\))42 b(=)g Fr(e)900 2525 y FA(t\025)934 2530 y Fi(j;\017)972 2545 y Fs(\()p Fr(e)1014 2525 y FA(t)p Fq(\001)1056 2529 y Fi(\017)1073 2545 y Fr(u)1101 2552 y FA(j;\017)1141 2545 y Fs(\)\()p Fr(y)r Fs(\))797 2622 y(=)g Fr(e)900 2601 y FA(t\025)934 2606 y Fi(j;\017)972 2622 y Fd(E)1002 2601 y FA(y)1026 2622 y Fs([)p Fr(u)1068 2629 y FA(j;\017)1107 2622 y Fs(\()p Fr(W)1172 2629 y FA(t)1187 2622 y Fs(\))p Fr(;)8 b(t)14 b(<)f(\034)1332 2629 y FA(\017)1349 2622 y Fs(])796 2698 y Fn(\024)42 b Fr(e)900 2677 y FA(t\025)934 2682 y Fi(j;\017)972 2698 y Fn(k)p Fr(u)1025 2705 y FA(j;\017)1065 2698 y Fn(k)1090 2705 y Fm(1)1127 2698 y Fd(E)1157 2677 y FA(y)1181 2698 y Fs([1)14 b Fr(<)g(\034)1306 2705 y FA(\017)1322 2698 y Fr(=t)p Fs(])796 2774 y Fn(\024)42 b Fr(e)900 2753 y FA(t\025)934 2758 y Fi(j;\017)972 2774 y Fn(k)p Fr(u)1025 2781 y FA(j;\017)1065 2774 y Fn(k)1090 2781 y Fm(1)1127 2774 y Fd(E)1157 2753 y FA(y)1181 2774 y Fs([)p Fr(\034)1216 2781 y FA(\017)1232 2774 y Fs(])p Fr(=t:)-5 2854 y Fs(No)o(w)16 b(w)o(e)g(put)h Fr(t)c Fs(=)h Fr(\025)379 2834 y Fm(\000)p Fq(1)379 2867 y FA(j;\017)443 2854 y Fs(to)i(get)h(the)f(lemm)o(a.)p 1971 2854 V 1973 2823 30 2 v 1973 2854 V 2001 2854 2 33 v eop %%Page: 5 5 5 4 bop 664 -64 a Fo(THE)17 b(NOD)o(AL)f(SURF)l(A)o(CE)767 b(5)-123 37 y FB(Lemma)16 b(2.5.)615 106 y Fn(k)p Fr(u)668 113 y FA(j;\017)708 106 y Fn(k)733 113 y Fm(1)784 106 y Fn(\024)d Fr(e)859 86 y Fq(1)p FA(=)p Fq(\(8)p FA(\031)q Fq(\))963 106 y Fs(\()p Fr(\025)1010 113 y FA(j;\017)1051 106 y Fs(\))1070 86 y FA(D)q(=)p Fq(4)1137 106 y Fr(:)-123 200 y Ff(Pr)n(o)n(of.)19 b Fs(This)e(w)o(as)g(pro)o(v)o(ed)e(in)h([Da)o (v89)q(][p.)21 b(63].)h(I)16 b(am)f(grateful)h(to)h(T.)f (Ho\013mann-Ostenhof)h(for)g(p)q(oin)o(ting)-123 258 y(m)o(y)d(atten)o(tion)i(to)h(this)f(reference.)p 1852 258 2 33 v 1854 227 30 2 v 1854 258 V 1883 258 2 33 v -73 355 a(Th)o(us)24 b(w)o(e)g(will)e(pro)o(v)o(e)h(a)i(b)q(ound)g Fd(E)613 337 y FA(y)637 355 y Fs([)p Fr(\034)672 362 y FA(\017)688 355 y Fs(])h(=)h Fr(O)q Fs(\()p Fr(\017)p Fs(\))d(when)g Fr(y)k Fn(2)g Fr(S)1191 362 y FA(\017)1207 355 y Fs(.)44 b(This)24 b(will)f(\014nish)h(the)f(pro)q(of)i(of)-123 414 y(Lemma)14 b(2.1.)-123 507 y FB(Lemma)i(2.6.)23 b Ff(L)n(et)17 b Fr(W)317 514 y FA(t)346 507 y Fs(=)d(\()p Fr(X)461 489 y Fq(1)457 520 y FA(t)481 507 y Fr(;)8 b(X)547 489 y Fq(2)543 520 y FA(t)567 507 y Fr(;)g(:::;)g(X)697 489 y FA(D)693 520 y(t)728 507 y Fs(\))17 b Ff(b)n(e)h Fr(D)q Ff(-dimensional)i(Br)n(ownian)d(motion.)23 b(L)n(et)622 597 y Fr(\034)643 604 y FA(\017)674 597 y Fs(=)13 b(inf)s Fn(f)p Fr(t)h(>)g Fs(0)p Fn(j)p Fr(W)977 604 y FA(t)1011 597 y Fr(=)-29 b Fn(2)14 b Fs(\012)1088 604 y FA(\017)1104 597 y Fn(g)p Fr(:)-123 687 y Ff(Then)k Fn(9)p Fr(C)67 694 y Fq(0)100 687 y Fr(>)c Fs(0)k Ff(such)g(that)f Fn(8)p Fr(\016)d(>)g Fs(0)627 777 y(sup)624 818 y FA(y)q Fm(2)p FA(S)687 822 y Fi(\017)711 777 y Fd(E)742 757 y FA(y)765 777 y Fs([)p Fr(\034)800 784 y FA(\017)816 777 y Fs(])g Fn(\024)f Fr(C)931 784 y Fq(0)951 777 y Fr(\017)e Fs(+)g Fr(O)q Fs(\()p Fr(\017)p Fs(\))p Fr(;)-123 903 y Ff(when)19 b Fr(\017)e Ff(is)g(su\016ciently)i(smal)r(l)g(\(dep)n(ending)f(on)g Fr(\016)r Ff(\).)k(Her)n(e)17 b Fr(C)1015 910 y Fq(0)1052 903 y Ff(c)n(an)h(b)n(e)g(chosen)g(as:)265 1035 y Fr(C)300 1042 y Fq(0)334 1035 y Fs(=)390 1001 y(\()p Fr(D)13 b Fn(\000)e Fs(2\))p Fr(D)p 390 1024 207 2 v 410 1034 a Fn(p)p 451 1034 128 2 v 451 1074 a Fr(D)i Fn(\000)e Fs(1)723 1001 y Fr(R)760 976 y Fm(\000)p Fq(\()p FA(D)q Fm(\000)p Fq(1\))760 1014 y(1)p 607 1024 400 2 v 607 1080 a Fr(R)644 1055 y Fm(\000)p Fq(\()p FA(D)q Fm(\000)p Fq(2\))644 1092 y(1)788 1080 y Fn(\000)g Fr(R)875 1055 y Fm(\000)p Fq(\()p FA(D)q Fm(\000)p Fq(2\))875 1092 y(2)1012 1035 y Fs(\()p Fr(R)1068 1015 y Fq(2)1068 1048 y(2)1100 1035 y Fn(\000)f Fr(R)1186 1015 y Fq(2)1186 1048 y(1)1207 1035 y Fs(\)\(1)h(+)g Fr(R)1366 1042 y Fq(2)1386 1035 y Fr(=R)1447 1042 y Fq(1)1467 1035 y Fs(\))p Fr(;)-123 1163 y Ff(for)17 b Fr(D)f Fn(\025)d Fs(3)18 b Ff(and)525 1259 y Fr(C)560 1266 y Fq(0)593 1259 y Fs(=)650 1226 y(1)p 650 1248 25 2 v 650 1293 a(2)679 1259 y(\()p Fr(R)735 1239 y Fq(2)735 1272 y(2)767 1259 y Fn(\000)11 b Fr(R)854 1239 y Fq(2)854 1272 y(1)874 1259 y Fs(\))p Fr(R)930 1239 y Fm(\000)p Fq(1)930 1272 y(1)991 1226 y Fs(1)g(+)g Fr(R)1112 1233 y Fq(2)1132 1226 y Fr(=R)1193 1233 y Fq(1)p 982 1248 240 2 v 982 1293 a Fs(log)q(\()p Fr(R)1101 1300 y Fq(2)1121 1293 y Fr(=R)1182 1300 y Fq(1)1202 1293 y Fs(\))1226 1259 y Fr(;)-123 1366 y Ff(for)17 b Fr(D)f Fs(=)d(2)p Ff(.)-123 1460 y(Pr)n(o)n(of.)19 b Fs(Let)g Fr(M)163 1467 y FA(t)197 1460 y Fs(=)f Fr(W)306 1442 y Fq(2)299 1472 y FA(t)338 1460 y Fn(\000)13 b Fr(D)q(t)p Fs(,)20 b(then)e Fr(M)643 1467 y FA(t)677 1460 y Fs(is)h(a)g (martingale)f(since)g(the)h(di\013eren)o(t)f(co)q(ordinates)i(of)f Fr(W)1789 1467 y FA(t)1823 1460 y Fs(are)-123 1518 y(indep)q(enden)o(t) e(1-dimensional)g(Bro)o(wnian)h(motions.)26 b(Let)19 b Fr(x)d Fn(2)h Fr(S)1111 1525 y FA(\017)1128 1518 y Fs(,)h(w)o(e)f(ma)o(y)g(c)o(ho)q(ose)h(the)g(co)q(ordinates)h(so)-123 1576 y(that)e Fr(x)c Fs(=)h(\()p Fr(R)132 1583 y Fq(1)152 1576 y Fr(;)8 b Fs(0)p Fr(;)g(::::;)g Fs(0\).)20 b(No)o(w,)15 b(b)o(y)h(the)g(martingale)f(prop)q(ert)o(y)l(,)635 1667 y Fd(E)665 1646 y FA(x)690 1667 y Fs([)p Fr(M)751 1674 y FA(t)765 1667 y Fs(])f(=)g Fd(E)875 1646 y FA(x)900 1667 y Fs([)p Fr(M)961 1674 y Fq(0)980 1667 y Fs(])g(=)f Fr(R)1096 1646 y Fq(2)1096 1679 y(1)1117 1667 y Fr(;)-123 1757 y Fs(so)286 1847 y Fr(D)q Fd(E)358 1826 y FA(x)383 1847 y Fs([)p Fr(\034)418 1854 y FA(\017)434 1847 y Fs(])41 b(=)h Fd(E)599 1826 y FA(x)624 1847 y Fs([)p Fr(W)691 1826 y Fq(2)684 1859 y FA(\034)700 1863 y Fi(\017)716 1847 y Fs(])11 b Fn(\000)g Fr(R)828 1826 y Fq(2)828 1859 y(1)489 1924 y Fs(=)42 b Fr(R)606 1903 y Fq(2)606 1936 y(1)626 1924 y Fd(P)660 1903 y FA(x)678 1924 y Fs([)p Fn(j)p Fr(W)752 1931 y FA(\034)768 1935 y Fi(\017)785 1924 y Fn(j)13 b Fs(=)h Fr(R)901 1931 y Fq(1)921 1924 y Fs(])d(+)g Fr(R)1032 1903 y Fq(2)1032 1936 y(2)1052 1924 y Fd(P)1085 1903 y FA(x)1104 1924 y Fs([)p Fn(j)p Fr(W)1178 1931 y FA(\034)1194 1935 y Fi(\017)1211 1924 y Fn(j)i Fs(=)h Fr(R)1327 1931 y Fq(2)1347 1924 y Fs(])d Fn(\000)f Fr(R)1458 1903 y Fq(2)1458 1936 y(1)489 1997 y Fs(=)42 b(\()p Fr(R)625 1976 y Fq(2)625 2009 y(2)656 1997 y Fn(\000)11 b Fr(R)743 1976 y Fq(2)743 2009 y(1)763 1997 y Fs(\))p Fd(P)816 1976 y FA(x)835 1997 y Fs([)p Fn(j)p Fr(W)909 2004 y FA(\034)925 2008 y Fi(\017)941 1997 y Fn(j)j Fs(=)f Fr(R)1057 2004 y Fq(2)1077 1997 y Fs(])p Fr(:)-123 2087 y Fs(W)l(e)j(will)f(no)o(w)i(pro)o(v)o(e)e (that)i(when)f Fr(\017)g Fs(is)g(su\016cien)o(tly)e(small,)h(then)305 2198 y Fd(P)339 2178 y FA(x)358 2198 y Fs([)p Fn(j)p Fr(W)432 2205 y FA(\034)448 2209 y Fi(\017)464 2198 y Fn(j)f Fs(=)g Fr(R)581 2205 y Fq(2)600 2198 y Fs(])g Fn(\024)f Fr(C)715 2205 y Fq(1)735 2198 y Fr(\017)e Fs(+)g(\()839 2165 y(1)p 839 2187 25 2 v 839 2233 a(2)879 2198 y(+)g Fr(O)q Fs(\()p Fr(\017)p Fs(\)\))g(sup)1051 2239 y FA(y)q Fm(2)p FA(S)1114 2243 y Fi(\017)1139 2198 y Fd(P)1172 2178 y FA(y)1190 2198 y Fs([)p Fn(j)p Fr(W)1264 2205 y FA(\034)1280 2209 y Fi(\017)1296 2198 y Fn(j)j Fs(=)g Fr(R)1413 2205 y Fq(2)1432 2198 y Fs(])p Fr(:)-123 2340 y Fs(Since)i Fr(x)d Fn(2)h Fr(S)123 2347 y FA(\017)156 2340 y Fs(w)o(as)i(arbitrary)l(,)g(this)g(pro)o(v)o(es)g(the)g(lemma,)d (with)j Fr(C)1113 2347 y Fq(0)1146 2340 y Fs(=)e(2)1227 2314 y FA(R)1254 2303 y Fg(2)1254 2325 y(2)1272 2314 y Fm(\000)p FA(R)1326 2303 y Fg(2)1326 2325 y(1)p 1227 2329 116 2 v 1270 2357 a FA(D)1348 2340 y Fr(C)1383 2347 y Fq(1)1403 2340 y Fr(:)-73 2398 y Fs(W)l(e)i(in)o(tro)q(duce)g(the)g (follo)o(wing)g(stopping)h(times:)-123 2456 y(Let)494 2540 y Fr(k)f Fs(=)685 2506 y Fr(\017)p 592 2528 208 2 v 592 2538 a Fl(p)p 641 2538 158 2 v 641 2581 a Fr(R)678 2564 y Fq(2)678 2594 y(1)710 2581 y Fn(\000)10 b Fr(\017)779 2567 y Fq(2)809 2506 y Fr(R)846 2513 y Fq(1)877 2506 y Fs(+)h Fr(R)963 2513 y Fq(2)p 809 2528 174 2 v 884 2574 a Fs(2)1001 2540 y Fn(\031)1059 2506 y Fr(R)1096 2513 y Fq(1)1127 2506 y Fs(+)g Fr(R)1213 2513 y Fq(2)p 1059 2528 V 1105 2574 a Fs(2)p Fr(R)1166 2581 y Fq(1)1238 2540 y Fr(\017;)-123 2656 y Fs(and)17 b(de\014ne)159 2799 y Fr(\034)186 2779 y FA(\017)180 2811 y Fq(1)216 2799 y Fs(=)d(inf)s Fn(f)p Fr(t)g(>)f Fs(0)p Fn(j)473 2696 y Fl(v)473 2724 y(u)473 2754 y(u)473 2784 y(t)p 526 2696 193 2 v 547 2737 a FA(D)526 2752 y Fl(X)531 2857 y FA(j)r Fq(=2)598 2799 y Fs(\()p Fr(X)661 2776 y FA(j)657 2810 y(t)680 2799 y Fs(\))699 2785 y Fq(2)732 2799 y Fs(=)h Fr(k)k Fs(or)f Fn(j)p Fr(X)945 2779 y Fq(1)941 2811 y FA(t)965 2799 y Fn(j)d Fs(=)1049 2765 y Fr(R)1086 2772 y Fq(1)1117 2765 y Fs(+)d Fr(R)1203 2772 y Fq(2)p 1049 2788 174 2 v 1124 2833 a Fs(2)1244 2799 y(or)17 b Fn(j)p Fr(X)1362 2779 y Fq(1)1358 2811 y FA(t)1382 2799 y Fn(j)d Fs(=)f Fr(R)1498 2806 y Fq(1)1518 2799 y Fr(=)p Fs(2)p Fn(g)p Fr(:)p eop %%Page: 6 6 6 5 bop -5 -64 a Fo(6)809 b(S\037REN)17 b(F)o(OURNAIS)170 37 y gsave currentpoint currentpoint translate -90 neg rotate neg exch neg exch translate 170 37 a 17 w @beginspecial 0 @llx 0 @lly 366 @urx 581 @ury 1992 @rwi 3985 @rhi @setspecial %%BeginDocument: stoptider.eps %Magnification: 1.00 /$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 -105.0 -70.0 translate 90 rotate 1 -1 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 /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def $F2psBegin 10 setmiterlimit n 0 595 m 0 0 l 842 0 l 842 595 l cp clip 0.06000 0.06000 sc 60.000 slw % Arc gs n 2400.0 4570.3 5412.7 -3.9 -30.8 arcn gs col-1 s gr gr % Arc gs n 1324.0 4687.6 9476.6 -17.7 19.2 arc gs col-1 s gr gr 15.000 slw % Arc gs n 3000.0 4800.0 1950.0 -22.6 22.6 arc gs col-1 s gr gr 60.000 slw % Arc gs n 1462.5 4912.5 6356.2 4.4 27.0 arc gs col-1 s gr gr 15.000 slw % Arc gs n 5587.5 4800.0 3787.5 -11.4 11.4 arc gs col-1 s gr gr % Polyline n 4800 5550 m 9300 5550 l gs col-1 s gr % Polyline [100.0] 0 sd n 1200 4800 m 9300 4050 l gs col-1 s gr [] 0 sd % Polyline [100.0] 0 sd n 1200 4800 m 9300 5550 l gs col-1 s gr [] 0 sd % Polyline n 4800 4050 m 9300 4050 l gs col-1 s gr $F2psEnd rs %%EndDocument @endspecial 1033 37 a currentpoint grestore moveto 1033 37 a 753 936 a Fk(The)25 b(stopping)d(time)15 b Fr(\034)1232 918 y FA(\017)1226 948 y Fq(1)45 1040 y Fs(De\014ne)h(furthermore)688 1121 y Fr(\034)709 1128 y Fq(1)770 1121 y Fs(=)42 b(inf)s Fn(f)p Fr(t)13 b(>)h Fs(0)p Fn(jj)p Fr(W)1115 1128 y FA(t)1130 1121 y Fn(j)g Fs(=)f Fr(R)1246 1128 y Fq(1)1266 1121 y Fn(g)p Fr(;)688 1194 y(\034)709 1201 y Fq(2)770 1194 y Fs(=)42 b(inf)s Fn(f)p Fr(t)13 b(>)h Fs(0)p Fn(jj)p Fr(W)1115 1201 y FA(t)1130 1194 y Fn(j)g Fs(=)f Fr(R)1246 1201 y Fq(2)1266 1194 y Fn(g)p Fr(:)-5 1275 y Fs(Then)528 1356 y Fd(P)561 1336 y FA(x)580 1356 y Fs([)p Fn(j)p Fr(W)654 1363 y FA(\034)670 1367 y Fi(\017)686 1356 y Fn(j)h Fs(=)g Fr(R)803 1363 y Fq(2)823 1356 y Fs(])41 b(=)h Fd(P)991 1336 y FA(x)1010 1356 y Fs([)p Fn(j)p Fr(W)1084 1363 y FA(\034)1100 1367 y Fi(\017)1117 1356 y Fn(j)13 b Fs(=)h Fr(R)1233 1363 y Fq(2)1264 1356 y Fn(^)d Fr(\034)1335 1336 y FA(\017)1329 1369 y Fq(1)1365 1356 y Fr(<)j(\034)1444 1336 y FA(\017)1460 1356 y Fs(])878 1429 y Fn(\024)41 b Fd(P)991 1409 y FA(x)1010 1429 y Fs([)p Fn(j)p Fr(W)1084 1436 y FA(\034)1100 1440 y Fi(\017)1117 1429 y Fn(j)10 b(\016)h Fr(\022)1200 1436 y FA(\034)1220 1425 y Fi(\017)1216 1447 y Fg(1)1251 1429 y Fs(=)i Fr(R)1339 1436 y Fq(2)1359 1429 y Fs(])878 1514 y(=)42 b Fd(E)988 1494 y FA(x)1013 1514 y Fs([)p Fd(P)1060 1489 y FA(W)1093 1496 y Fi(\034)1110 1488 y(\017)1107 1509 y Fg(1)1127 1514 y Fs([)p Fn(j)p Fr(W)1201 1521 y FA(\034)1217 1525 y Fi(\017)1233 1514 y Fn(j)14 b Fs(=)g Fr(R)1350 1521 y Fq(2)1369 1514 y Fs(]])p Fr(;)-5 1595 y Fs(where)i(w)o(e)g(used)g (the)g(strong)i(Mark)o(o)o(v)d(prop)q(ert)o(y)h(of)h(Bro)o(wnian)f (motion.)k(W)l(e)c(con)o(tin)o(ue)f(the)h(calculation:)103 1680 y Fd(E)133 1660 y FA(x)158 1680 y Fs([)p Fd(P)205 1655 y FA(W)238 1663 y Fi(\034)255 1654 y(\017)252 1675 y Fg(1)272 1680 y Fs([)p Fn(j)p Fr(W)346 1687 y FA(\034)362 1691 y Fi(\017)378 1680 y Fn(j)e Fs(=)f Fr(R)494 1687 y Fq(2)514 1680 y Fs(]])41 b(=)h Fd(E)693 1660 y FA(x)718 1680 y Fs([)p Fd(P)765 1655 y FA(W)798 1663 y Fi(\034)815 1654 y(\017)812 1675 y Fg(1)832 1680 y Fs([)p Fn(j)p Fr(W)906 1687 y FA(\034)922 1691 y Fi(\017)938 1680 y Fn(j)14 b Fs(=)g Fr(R)1055 1687 y Fq(2)1075 1680 y Fr(;)8 b(\034)1118 1687 y Fq(1)1151 1680 y Fr(<)14 b(\034)1224 1687 y Fq(2)1244 1680 y Fr(;)8 b(W)1312 1687 y FA(\034)1328 1692 y Fg(1)1361 1680 y Fn(2)14 b Fr(S)1438 1687 y FA(\017)1454 1680 y Fs(]])c(+)h Fd(E)1572 1660 y FA(x)1596 1680 y Fs([)p Fd(P)1643 1655 y FA(W)1676 1663 y Fi(\034)1693 1654 y(\017)1690 1675 y Fg(1)1710 1680 y Fs([)p Fr(\034)1745 1687 y Fq(1)1779 1680 y Fr(>)i(\034)1851 1687 y Fq(2)1871 1680 y Fs(]])583 1753 y Fn(\021)41 b Fr(a)11 b Fs(+)g Fr(b:)-5 1834 y Fs(Let)19 b(us)g(lo)q(ok)h(at)f(the)f(\014rst)h(term)e Fr(a)p Fs(.)29 b(Belo)o(w)17 b(w)o(e)i(will)e(once)i(again)h(use)e(the) h(strong)h(Mark)o(o)o(v)e(prop)q(ert)o(y)h(of)-5 1892 y(Bro)o(wnian)d(motion.)118 1973 y Fd(P)152 1948 y FA(W)185 1956 y Fi(\034)202 1947 y(\017)199 1968 y Fg(1)219 1973 y Fs([)p Fn(j)p Fr(W)293 1980 y FA(\034)309 1984 y Fi(\017)325 1973 y Fn(j)e Fs(=)g Fr(R)442 1980 y Fq(2)461 1973 y Fr(;)8 b(\034)504 1980 y Fq(1)538 1973 y Fr(<)14 b(\034)611 1980 y Fq(2)631 1973 y Fr(;)8 b(W)699 1980 y FA(\034)715 1985 y Fg(1)747 1973 y Fn(2)14 b Fr(S)824 1980 y FA(\017)841 1973 y Fs(])41 b(=)h Fd(P)1010 1948 y FA(W)1043 1956 y Fi(\034)1060 1947 y(\017)1057 1968 y Fg(1)1076 1973 y Fs([)p Fn(j)p Fr(W)1150 1980 y FA(\034)1166 1984 y Fi(\017)1183 1973 y Fn(j)11 b(\016)g Fr(\022)1267 1980 y FA(\034)1283 1985 y Fg(1)1315 1973 y Fs(=)j Fr(R)1404 1980 y Fq(2)1424 1973 y Fr(;)8 b(\034)1467 1980 y Fq(1)1500 1973 y Fr(<)14 b(\034)1573 1980 y Fq(2)1593 1973 y Fr(;)8 b(W)1661 1980 y FA(\034)1677 1985 y Fg(1)1710 1973 y Fn(2)14 b Fr(S)1787 1980 y FA(\017)1803 1973 y Fs(])896 2053 y(=)42 b Fd(E)1006 2027 y FA(W)1040 2035 y Fi(\034)1057 2027 y(\017)1054 2047 y Fg(1)1079 2053 y Fs([1)1117 2061 y Fm(f)p FA(\034)1151 2066 y Fg(1)1168 2061 y FA(<\034)1211 2066 y Fg(2)1228 2061 y Fm(g)1248 2053 y Fs(1)1272 2061 y Fm(f)p FA(W)1323 2065 y Fi(\034)1337 2072 y Fg(1)1357 2061 y Fm(2)p FA(S)1402 2065 y Fi(\017)1417 2061 y Fm(g)1437 2053 y Fd(P)1470 2032 y FA(W)1503 2036 y Fi(\034)1517 2043 y Fg(1)1537 2053 y Fs([)p Fn(j)p Fr(W)1611 2060 y FA(\034)1627 2064 y Fi(\017)1643 2053 y Fn(j)14 b Fs(=)f Fr(R)1759 2060 y Fq(2)1779 2053 y Fs(]])896 2139 y Fn(\024)41 b Fs(\()s(sup)995 2180 y FA(y)q Fm(2)p FA(S)1058 2184 y Fi(\017)1082 2139 y Fd(P)1116 2119 y FA(y)1133 2139 y Fs([)p Fn(j)p Fr(W)1207 2146 y FA(\034)1223 2150 y Fi(\017)1240 2139 y Fn(j)13 b Fs(=)h Fr(R)1356 2146 y Fq(2)1376 2139 y Fs(]\))p Fd(E)1439 2114 y FA(W)1472 2121 y Fi(\034)1489 2113 y(\017)1486 2134 y Fg(1)1511 2139 y Fs([1)1549 2147 y Fm(f)p FA(\034)1583 2152 y Fg(1)1600 2147 y FA(<\034)1643 2152 y Fg(2)1660 2147 y Fm(g)1680 2139 y Fs(1)1704 2147 y Fm(f)p FA(W)1755 2151 y Fi(\034)1769 2158 y Fg(1)1789 2147 y Fm(2)p FA(S)1834 2151 y Fi(\017)1849 2147 y Fm(g)1869 2139 y Fs(])896 2276 y Fn(\024)41 b Fs(\()1000 2242 y(1)p 1000 2264 25 2 v 1000 2310 a(2)1040 2276 y(+)11 b Fr(O)q Fs(\()p Fr(\017)p Fs(\)\)\()s(sup)1223 2316 y FA(y)q Fm(2)p FA(S)1286 2320 y Fi(\017)1311 2276 y Fd(P)1344 2255 y FA(y)1362 2276 y Fs([)p Fn(j)p Fr(W)1436 2283 y FA(\034)1452 2287 y Fi(\017)1468 2276 y Fn(j)j Fs(=)g Fr(R)1585 2283 y Fq(2)1604 2276 y Fs(]\))p Fr(;)-5 2392 y Fs(when)21 b Fr(\017)f Fs(is)g(su\016cien)o(tly)e(small)h (\(dep.)33 b(on)21 b Fr(\016)r Fs(\).)33 b(Here)19 b(w)o(e)h(used)g (that)h(due)f(to)h(symmetry)c(the)j(c)o(hance)g(of)-5 2451 y("falling)c(bac)o(k")f(in)o(to)h(the)f(hole)h(w)o(e)f(came)f (from,)g(is)i Fn(\024)e Fs(1)p Fr(=)p Fs(2,)i(and)g(the)g(probabilit)o (y)f(of)h(falling)f(in)o(to)g(another)-5 2509 y(hole)h(is)g Fr(O)q Fs(\()p Fr(\017)p Fs(\))h(as)g Fr(\017)c Fn(!)h Fs(0)j(\(this)f(follo)o(ws)g(from)f([PS78)q(][Thm.)k(3.1.)i(p.)h (102]\).)45 2567 y(Th)o(us)17 b(w)o(e)e(only)h(need)g(an)h(estimate)d (of)j(order)f Fr(\017)g Fs(of)h(the)f(term)e Fr(b)p Fs(.)21 b(This)c(is)f(easily)f(accomplished:)-5 2631 y(First)e(for)h Fr(D)i Fn(\025)d Fs(3.)21 b(Since)13 b Fn(j)p Fr(W)535 2638 y FA(t)549 2631 y Fn(j)563 2613 y Fm(\000)p Fq(\()p FA(D)q Fm(\000)p Fq(2\))709 2631 y Fs(is)g(a)h(martingale)e(and)i(lea)o (v)o(es)e([)p Fr(R)1314 2605 y Fm(\000)p Fq(\()p FA(D)q Fm(\000)p Fq(2\))1314 2643 y(2)1446 2631 y Fr(;)c(R)1505 2605 y Fm(\000)p Fq(\()p FA(D)q Fm(\000)p Fq(2\))1505 2643 y(1)1637 2631 y Fs(])13 b(with)h(probabilit)o(y)-5 2689 y(one,)i(w)o(e)g(get)g(\()h([Bas95][Cor.)k(4.10)c(p.33]\))472 2817 y Fd(P)505 2791 y FA(W)538 2799 y Fi(\034)555 2790 y(\017)552 2811 y Fg(1)572 2817 y Fs([)p Fr(\034)607 2824 y Fq(1)641 2817 y Fr(>)c(\034)713 2824 y Fq(2)733 2817 y Fs(])h(=)817 2779 y Fr(R)854 2754 y Fm(\000)p Fq(\()p FA(D)q Fm(\000)p Fq(2\))854 2791 y(1)998 2779 y Fn(\000)d(j)p Fr(W)1108 2786 y FA(\034)1128 2775 y Fi(\017)1124 2797 y Fg(1)1144 2779 y Fn(j)1158 2761 y Fm(\000)p Fq(\()p FA(D)q Fm(\000)p Fq(2\))p 817 2805 473 2 v 854 2861 a Fr(R)891 2836 y Fm(\000)p Fq(\()p FA(D)q Fm(\000)p Fq(2\))891 2874 y(1)1034 2861 y Fn(\000)g Fr(R)1121 2836 y Fm(\000)p Fq(\()p FA(D)q Fm(\000)p Fq(2\))1121 2874 y(2)1295 2817 y Fs(1)1319 2824 y Fm(fj)p FA(W)1380 2832 y Fi(\034)1397 2824 y(\017)1394 2845 y Fg(1)1415 2824 y Fm(j)p FA(>R)1479 2829 y Fg(1)1496 2824 y Fm(g)1516 2817 y Fr(:)p eop %%Page: 7 7 7 6 bop 664 -64 a Fo(THE)17 b(NOD)o(AL)f(SURF)l(A)o(CE)767 b(7)-123 37 y Fs(Therefore,)16 b(b)o(y)f(a)i(\014rst)f(order)h(T)l(a)o (ylor)f(expansion,)614 129 y Fr(b)41 b Fn(\024)g Fr(c)p Fd(E)807 108 y FA(x)832 129 y Fs([)p Fn(j)p Fr(X)904 108 y Fq(1)900 141 y FA(\034)920 130 y Fi(\017)916 152 y Fg(1)948 129 y Fn(\000)11 b Fr(R)1035 136 y Fq(1)1054 129 y Fn(j)p Fs(])676 229 y Fn(\024)41 b Fr(c)777 174 y Fl(q)p 827 174 325 2 v 55 x Fd(E)857 215 y FA(x)882 229 y Fs([)p Fn(j)p Fr(X)954 212 y Fq(1)950 242 y FA(\034)970 231 y Fi(\017)966 253 y Fg(1)998 229 y Fn(\000)10 b Fr(R)1084 236 y Fq(1)1104 229 y Fn(j)1118 215 y Fq(2)1138 229 y Fs(])676 327 y(=)42 b Fr(c)p Fd(E)807 306 y FA(x)832 327 y Fs([)846 285 y Fl(p)p 896 285 44 2 v 42 x Fr(\034)923 310 y FA(\017)917 339 y Fq(1)939 327 y Fs(])p Fr(;)-123 420 y Fs(b)o(y)15 b(the)g(Jensen)h(inequalit)o(y)l(,)d(since)i(\()p Fr(X)602 402 y Fq(1)598 432 y FA(t)622 420 y Fs(\))641 402 y Fq(2)670 420 y Fn(\000)9 b Fr(t)16 b Fs(is)f(a)h(martingale.)j (Here)c Fr(c)g Fs(can)h(b)q(e)g(c)o(hosen)f(\(up)h(to)g(errors)f(of) -123 478 y(higher)h(order)h(in)e Fr(\017)p Fs(\))h(as)627 592 y Fr(c)e Fs(=)752 558 y(\()p Fr(D)f Fn(\000)d Fs(2\))p Fr(R)953 533 y Fm(\000)p Fq(\()p FA(D)q Fm(\000)p Fq(1\))953 571 y(1)p 719 581 400 2 v 719 637 a Fr(R)756 612 y Fm(\000)p Fq(\()p FA(D)q Fm(\000)p Fq(2\))756 649 y(1)900 637 y Fn(\000)g Fr(R)986 612 y Fm(\000)p Fq(\()p FA(D)q Fm(\000)p Fq(2\))986 649 y(2)1124 592 y Fr(:)-123 739 y Fs(No)o(w)19 b Fr(\034)18 721 y FA(\017)12 751 y Fq(1)52 739 y Fn(\024)i Fs(~)-27 b Fr(\034)136 721 y FA(\017)130 751 y Fq(1)152 739 y Fs(,)19 b(where)i(~)-26 b Fr(\034)356 721 y FA(\017)350 751 y Fq(1)390 739 y Fs(=)18 b(inf)s Fn(f)p Fr(t)g(>)g Fs(0)p Fn(j)660 681 y Fl(q)p 710 681 237 2 v 21 x(P)762 715 y FA(D)762 754 y(j)r Fq(=2)826 739 y Fs(\()p Fr(X)889 715 y FA(j)885 750 y(t)907 739 y Fs(\))926 725 y Fq(2)964 739 y Fs(=)g Fr(k)r Fn(g)p Fs(.)29 b(Remem)n(b)q(er)16 b(that)j Fr(x)f Fs(=)g(\()p Fr(R)1628 746 y Fq(1)1648 739 y Fr(;)8 b Fs(0)p Fr(;)g(::;)g Fs(0\))18 b(By)-123 807 y(scaling)702 879 y Fd(E)733 859 y FA(x)757 879 y Fs([)s(~)-27 b Fr(\034)798 859 y FA(\017)792 891 y Fq(1)814 879 y Fs(])13 b(=)h Fr(k)920 859 y Fq(2)940 879 y Fd(E)970 859 y FA(x)995 879 y Fs([)p Fr(\034)6 b Fs(])p Fr(;)-123 981 y Fs(where)16 b Fr(\034)k Fs(=)13 b(inf)s Fn(f)p Fr(t)h(>)g Fs(0)p Fn(j)316 923 y Fl(q)p 366 923 V 21 x(P)418 957 y FA(D)418 996 y(j)r Fq(=2)482 981 y Fs(\()p Fr(X)545 958 y FA(j)541 993 y(t)563 981 y Fs(\))582 967 y Fq(2)616 981 y Fs(=)f(1)p Fn(g)p Fs(.)-73 1098 y(F)l(or)j Fr(D)g Fs(=)e(2)i(w)o(e)g(ha)o(v)o(e)g(to)g(use)h(log)9 b Fn(j)p Fr(W)622 1105 y FA(t)637 1098 y Fn(j)16 b Fs(instead)g(of)h Fn(j)p Fr(W)951 1105 y FA(t)965 1098 y Fn(j)979 1080 y Fm(\000)p Fq(\()p FA(D)q Fm(\000)p Fq(2\))1111 1098 y Fs(.)304 1223 y Fd(P)337 1197 y FA(W)370 1205 y Fi(\034)387 1197 y(\017)384 1218 y Fg(1)404 1223 y Fs([)p Fr(\034)439 1230 y Fq(1)472 1223 y Fr(>)d(\034)545 1230 y Fq(2)565 1223 y Fs(])42 b(=)705 1185 y(log)10 b Fn(j)p Fr(W)837 1192 y FA(\034)857 1181 y Fi(\017)853 1203 y Fg(1)873 1185 y Fn(j)h(\000)g Fs(log)e Fr(R)1056 1192 y Fq(1)p 705 1211 371 2 v 771 1257 a Fs(log)q(\()p Fr(R)890 1264 y Fq(2)910 1257 y Fr(=R)971 1264 y Fq(1)991 1257 y Fs(\))1081 1223 y(1)1105 1231 y Fm(fj)p FA(W)1166 1239 y Fi(\034)1183 1230 y(\017)1180 1251 y Fg(1)1201 1231 y Fm(j)p FA(>R)1265 1236 y Fg(1)1282 1231 y Fm(g)620 1368 y Fn(\024)41 b Fr(R)737 1347 y Fm(\000)p Fq(1)737 1380 y(1)790 1288 y Fl(\014)790 1318 y(\014)807 1331 y Fn(j)p Fr(W)867 1338 y FA(\034)887 1327 y Fi(\017)883 1349 y Fg(1)903 1331 y Fn(j)11 b(\000)g Fr(R)1015 1338 y Fq(1)1035 1288 y Fl(\014)1035 1318 y(\014)p 790 1357 262 2 v 801 1402 a Fs(log)q(\()p Fr(R)920 1409 y Fq(2)940 1402 y Fr(=R)1001 1409 y Fq(1)1021 1402 y Fs(\))1056 1368 y(1)1080 1376 y Fm(fj)p FA(W)1141 1384 y Fi(\034)1158 1375 y(\017)1155 1396 y Fg(1)1176 1376 y Fm(j)p FA(>R)1240 1381 y Fg(1)1257 1376 y Fm(g)620 1542 y Fn(\024)41 b Fr(R)737 1522 y Fm(\000)p Fq(1)737 1555 y(1)790 1433 y Fl(\014)790 1463 y(\014)790 1493 y(\014)807 1491 y Fn(j)p Fr(X)865 1473 y Fq(1)861 1503 y FA(\034)881 1492 y Fi(\017)877 1514 y Fg(1)898 1491 y Fn(j)11 b(\000)f Fr(R)1009 1498 y Fq(1)1029 1433 y Fl(\014)1029 1463 y(\014)1029 1493 y(\014)p 790 1531 256 2 v 799 1577 a Fs(log)q(\()p Fr(R)918 1584 y Fq(2)937 1577 y Fr(=R)998 1584 y Fq(1)1019 1577 y Fs(\))1051 1542 y(1)1075 1550 y Fm(fj)p FA(W)1136 1558 y Fi(\034)1153 1549 y(\017)1150 1570 y Fg(1)1171 1550 y Fm(j)p FA(>R)1235 1555 y Fg(1)1252 1550 y Fm(g)1282 1542 y Fs(+)h Fr(O)q Fs(\()p Fr(\017)1408 1522 y Fq(2)1429 1542 y Fs(\))p Fr(:)-73 1674 y Fs(No)o(w,)52 1637 y Fl(P)104 1650 y FA(D)104 1688 y(j)r Fq(=2)168 1674 y Fs(\()p Fr(X)231 1650 y FA(j)227 1685 y(t)249 1674 y Fs(\))268 1656 y Fq(2)299 1674 y Fn(\000)g Fs(\()p Fr(D)i Fn(\000)d Fs(1\))p Fr(t)17 b Fs(is)f(a)g(martingale,)f(th)o(us)481 1823 y Fd(E)511 1802 y FA(x)536 1823 y Fs([)p Fr(\034)6 b Fs(])13 b(=)712 1789 y(1)p 661 1811 128 2 v 661 1857 a Fr(D)g Fn(\000)e Fs(1)793 1823 y Fd(E)823 1802 y FA(x)848 1823 y Fs([)883 1761 y FA(D)862 1776 y Fl(X)867 1881 y FA(j)r Fq(=2)934 1823 y Fs(\()p Fr(X)997 1802 y FA(j)993 1835 y(\034)1015 1823 y Fs(\))1034 1802 y Fq(2)1054 1823 y Fs(])i(=)1189 1789 y(1)p 1138 1811 V 1138 1857 a Fr(D)g Fn(\000)e Fs(1)1270 1823 y Fr(;)-123 1965 y Fs(and)17 b(w)o(e)f(get)405 2076 y Fr(C)440 2083 y Fq(1)474 2076 y Fs(=)551 2042 y Fr(D)d Fn(\000)e Fs(2)p 530 2064 169 2 v 530 2074 a Fn(p)p 572 2074 128 2 v 41 x Fr(D)i Fn(\000)e Fs(1)824 2042 y Fr(R)861 2017 y Fm(\000)p Fq(\()p FA(D)q Fm(\000)p Fq(1\))861 2054 y(1)p 709 2064 400 2 v 709 2121 a Fr(R)746 2095 y Fm(\000)p Fq(\()p FA(D)q Fm(\000)p Fq(2\))746 2133 y(1)889 2121 y Fn(\000)g Fr(R)976 2095 y Fm(\000)p Fq(\()p FA(D)q Fm(\000)p Fq(2\))976 2133 y(2)1119 2042 y Fs(1)g(+)g Fr(R)1240 2049 y Fq(2)1260 2042 y Fr(=R)1321 2049 y Fq(1)p 1119 2064 223 2 v 1218 2110 a Fs(2)1346 2076 y Fr(;)-123 2195 y Fs(for)17 b Fr(D)e Fn(\025)f Fs(3,)i(and)575 2298 y Fr(C)610 2305 y Fq(1)643 2298 y Fs(=)777 2265 y Fr(R)814 2244 y Fm(\000)p Fq(1)814 2277 y(1)p 700 2287 240 2 v 700 2333 a Fs(log)q(\()p Fr(R)819 2340 y Fq(2)839 2333 y Fr(=R)900 2340 y Fq(1)920 2333 y Fs(\))949 2265 y(1)11 b(+)g Fr(R)1070 2272 y Fq(2)1090 2265 y Fr(=R)1151 2272 y Fq(1)p 949 2287 223 2 v 1048 2333 a Fs(2)1177 2298 y Fr(;)-123 2408 y Fs(for)17 b Fr(D)e Fs(=)f(2.)p 1852 2408 2 33 v 1854 2376 30 2 v 1854 2408 V 1883 2408 2 33 v 497 2545 a(3.)28 b Ft(Pr)o(oof)17 b(of)h(the)h(main)f(theorem)-73 2632 y Fs(Here)e(w)o(e)g(will)g(pro)o (v)o(e)g(the)h(follo)o(wing,)f(more)g(precise,)f(v)o(ersion)h(of)i(the) e(main)g(theorem.)21 b(Let)c(us)h(\014rst)f(\014x)-123 2690 y(the)d(relativ)o(e)e(magnitude)g(of)j(the)e(v)m(arious)i (parameters:)k(Since)12 b Fr(\032)i Fn(\031)g Fr(\016)r Fs(,)f Fr(N)19 b Fn(\031)14 b Fr(\016)1340 2672 y Fm(\000)p Fq(\()p FA(D)q Fm(\000)p Fq(1\))1485 2690 y Fs(and)h(w)o(e)e(can)h (express)-123 2748 y(ev)o(erything)h(in)h(terms)f(of)h Fr(\016)r Fs(.)-123 2843 y FB(Theorem)h(3.1.)23 b Fn(9)p Fr(\016)276 2850 y Fq(0)309 2843 y Fr(>)14 b Fs(0)k Fn(8)p Fr(\016)c(<)f(\016)541 2850 y Fq(0)578 2843 y Fn(9)p Fr(\017)g(>)h Fs(0)k Ff(such)g(that)f Fr(u)971 2850 y Fq(2)p FA(;\017)1015 2843 y Fs(\()p Fr(x)p Fs(\))c Fn(\025)h Fs(0)k Fn(8)p Fr(x)e Ff(with)i Fn(j)p Fr(x)p Fn(j)13 b Fs(=)h Fr(R)1526 2850 y Fq(1)1556 2843 y Fn(\000)d Fr(\016)r Ff(.)p eop %%Page: 8 8 8 7 bop -5 -64 a Fo(8)809 b(S\037REN)17 b(F)o(OURNAIS)-5 37 y FB(Remark)g(3.2.)23 b Fs(The)17 b(follo)o(wing)e(argumen)o(t)h (sho)o(ws)h(that)f(Theorem)f(3.1)i(implies)c(Theorem)i(1.1:)45 95 y(Let)h Fr(R)169 77 y FA(\017)169 107 y Fq(0)203 95 y Fr(<)e(R)292 102 y Fq(1)328 95 y Fs(b)q(e)i(c)o(hosen)g(suc)o(h)h (that)815 176 y Fr(\025)843 183 y Fq(1)863 176 y Fs(\()p Fr(B)s Fs(\()p Fr(R)978 156 y FA(\017)978 189 y Fq(0)997 176 y Fs(\)\))d(=)g Fr(\025)1129 183 y Fq(2)p FA(;\017)1173 176 y Fr(:)-5 258 y Fs(Then)k Fr(R)161 240 y FA(\017)161 270 y Fq(0)198 258 y Fn(&)f Fr(R)302 265 y Fq(0)322 258 y Fs(,)h(where)g Fr(\025)525 265 y Fq(1)545 258 y Fs(\()p Fr(B)s Fs(\()p Fr(R)660 265 y Fq(0)679 258 y Fs(\)\))f(=)g Fr(\025)817 265 y Fq(2)p FA(;)p Fq(0)865 258 y Fr(:)g Fs(Supp)q(ose)j(no)o(w)e(that)h Fr(u)1329 265 y Fq(2)p FA(;\017)1372 258 y Fs(\()p Fr(x)p Fs(\))e Fn(\025)f Fs(0)j Fn(8j)p Fr(x)p Fn(j)c(\024)i Fr(R)1746 265 y Fq(1)1778 258 y Fn(\000)12 b Fr(\016)r Fs(.)26 b(F)l(or)18 b Fr(\016)-5 316 y Fs(and)i Fr(\017)e Fs(small)f(enough)i(this)g(implies)d(that)j Fr(u)833 323 y Fq(2)p FA(;\017)877 316 y Fs(\()p Fr(x)p Fs(\))f(is)g(p)q(ositiv)o(e)g(on)i(an)f(op)q(en)g(set)g(con)o(taining)f Fr(B)s Fs(\()p Fr(R)1870 323 y Fq(0)1890 316 y Fs(\))g(and)-5 374 y(therefore,)c(b)o(y)f(Couran)o(t's)i(No)q(dal)g(Domains)f (Theorem,)e(that)j Fr(\025)1191 381 y Fq(2)p FA(;\017)1249 374 y Fr(>)f(\025)1329 381 y Fq(2)p FA(;)p Fq(0)1377 374 y Fs(.)20 b(This)14 b(is)h(a)f(con)o(tradiction,)g(th)o(us)-5 432 y Fr(u)23 439 y Fq(2)p FA(;\017)81 432 y Fs(tak)o(es)h(negativ)o(e) e(v)m(alues)i(inside)f(the)g(sphere)g(of)h(radius)g Fr(R)1144 439 y Fq(1)1172 432 y Fn(\000)8 b Fr(\016)15 b Fs(and)g(the)g(no)q(dal) g(surface)g(gets)g(trapp)q(ed.)45 521 y(The)h(strategy)h(of)f(the)g (pro)q(of)i(is)e(the)g(follo)o(wing:)-5 579 y(W)l(e)g(lo)q(ok)h(at)795 641 y(\()p Fn(\000)p Fs(\001)894 648 y FA(\017)910 641 y Fs(\))929 620 y Fm(\000)p Fq(\()p FA(n)p Fq(+1\))1052 641 y Fr(u)1080 648 y Fq(2)p FA(;)p Fq(0)1127 641 y Fs(\()p Fr(x)p Fs(\))p Fr(;)-5 712 y Fs(where)e Fn(j)p Fr(x)p Fn(j)f Fs(=)f Fr(R)293 719 y Fq(1)323 712 y Fn(\000)c Fr(\016)r Fs(.)20 b(On)c(one)f(hand,)h(w)o(e)f(can)h(express)f(this,)g (using)h(the)f(sp)q(ectral)h(theorem,)d(as)j(a)g(sum)f(of)-5 770 y(terms)g(of)i(the)f(form)813 809 y(1)p 777 832 98 2 v 777 880 a Fr(\025)805 859 y FA(n)p Fq(+1)805 892 y FA(j;\017)879 843 y Fn(h)p Fr(u)926 850 y FA(j;\017)966 843 y Fr(;)8 b(u)1016 850 y Fq(2)p FA(;)p Fq(0)1063 843 y Fn(i)p Fr(u)1110 850 y FA(j;\017)1150 843 y Fs(\()p Fr(x)p Fs(\))p Fr(;)-5 953 y Fs(where)16 b(w)o(e)g(exp)q(ect)f(the)h (term)f(with)h Fr(j)h Fs(=)d(2)i(to)h(b)q(e)f(the)g(most)g(imp)q(ortan) o(t)f(one.)22 b(If)15 b Fr(u)1548 960 y Fq(2)p FA(;\017)1592 953 y Fs(\()p Fr(x)p Fs(\))e Fn(\024)h Fs(0)j(w)o(e)e(get)479 1082 y(\()p Fn(\000)p Fs(\001)578 1089 y FA(\017)593 1082 y Fs(\))612 1062 y Fm(\000)p Fq(\()p FA(n)p Fq(+1\))736 1082 y Fr(u)764 1089 y Fq(2)p FA(;)p Fq(0)811 1082 y Fs(\()p Fr(x)p Fs(\))e Fn(\024)943 995 y Fl(\014)943 1025 y(\014)943 1055 y(\014)943 1085 y(\014)943 1115 y(\014)959 1035 y(X)965 1141 y FA(j)r Fm(6)p Fq(=2)1081 1048 y Fs(1)p 1045 1071 V 1045 1119 a Fr(\025)1073 1098 y FA(n)p Fq(+1)1073 1131 y FA(j;\017)1147 1082 y Fn(h)p Fr(u)1194 1089 y FA(j;\017)1234 1082 y Fr(;)8 b(u)1284 1089 y Fq(2)p FA(;)p Fq(0)1331 1082 y Fn(i)p Fr(u)1378 1089 y FA(j;\017)1418 1082 y Fs(\()p Fr(x)p Fs(\))1484 995 y Fl(\014)1484 1025 y(\014)1484 1055 y(\014)1484 1085 y(\014)1484 1115 y(\014)1509 1082 y Fr(;)-5 1221 y Fs(Whic)o(h)16 b(will)f(b)q(e)h(small)f(in)h(a)g(suitable)g(sense,)g (i.e.)k Fr(O)q Fs(\()p Fr(\017)1015 1203 y FA(D)1047 1221 y Fs(\).)-5 1279 y(On)d(the)f(other)g(hand,)g(w)o(e)g(can)h (estimate)795 1362 y(\()p Fn(\000)p Fs(\001)894 1369 y FA(\017)910 1362 y Fs(\))929 1341 y Fm(\000)p Fq(\()p FA(n)p Fq(+1\))1052 1362 y Fr(u)1080 1369 y Fq(2)p FA(;)p Fq(0)1127 1362 y Fs(\()p Fr(x)p Fs(\))p Fr(;)-5 1443 y Fs(from)h(b)q(elo)o(w,)h(rather)g(explicitely)l(,)d(using)j(Bro)o (wnian)g(motion)e(tec)o(hniques.)28 b(This)19 b(giv)o(es)f(a)h(lo)o(w)o (er)f(b)q(ound,)-5 1501 y(whic)o(h)h(is)g(also)h(of)f(order)h Fr(\017)500 1483 y FA(D)531 1501 y Fs(.)31 b(By)18 b(k)o(eeping)g(trac) o(k)h(of)h(ho)o(w)f(the)g(constan)o(ts)h(in)f(b)q(oth)h(b)q(ounds)h (dep)q(end)e(on)-5 1559 y Fr(\016)r Fs(,)g(w)o(e)f(get)g(a)h(con)o (tradiction,)g(for)g(small)e Fr(\016)r Fs(,)h(if)g Fr(u)904 1566 y Fq(2)p FA(;\017)948 1559 y Fs(\()p Fr(x)p Fs(\))f Fn(\024)h Fs(0.)28 b(Notice,)18 b(that)h(it)f(is)h(essen)o(tial)e(for)i (the)g(lo)o(w)o(er)-5 1617 y(b)q(ound)f(that)e Fr(u)279 1624 y Fq(2)p FA(;)p Fq(0)343 1617 y Fs(is)g(a)g(p)q(ositiv)o(e)g (function.)45 1675 y(No)o(w)g(w)o(e)g(will)f(giv)o(e)g(the)h(details:) -5 1764 y FB(Lemma)g(3.3.)23 b Ff(L)n(et)c Fr(y)f Fn(2)f Fr(S)513 1771 y FA(\017)545 1764 y Fn(\021)g(f)p Fr(x)f Fn(2)g Fs(\012)754 1771 y FA(\017)771 1722 y Fl(\014)771 1752 y(\014)788 1764 y Fn(j)p Fr(x)p Fn(j)f Fs(=)i Fr(R)951 1771 y Fq(1)971 1764 y Fn(g)p Ff(,)i(and)g(let)h Fr(d)d Fs(=)f(dist\()p Fr(y)r(;)8 b(@)s Fs(\012)1502 1771 y FA(\017)1517 1764 y Fs(\))17 b Fr(>)f Fs(0)p Ff(.)27 b(Then)20 b Fn(9)p Fr(c;)8 b(C)19 b(>)d Fs(0)-5 1823 y Ff(indep)n(endent)k(of)d Fr(\017;)8 b(\016)18 b Ff(such)g(that)551 1892 y Fs(1)p 515 1914 V 515 1962 a Fr(\025)543 1941 y FA(n)p Fq(+1)543 1974 y(2)p FA(;\017)617 1926 y Fn(h)p Fr(u)664 1933 y Fq(2)p FA(;\017)708 1926 y Fr(;)8 b(u)758 1933 y Fq(2)p FA(;)p Fq(0)805 1926 y Fn(i)p Fr(u)852 1933 y Fq(2)p FA(;\017)896 1926 y Fs(\()p Fr(y)r Fs(\))13 b Fn(\025)h Fr(cd)d Fn(\000)g Fr(C)t(\017)1192 1905 y Fq(2)1222 1926 y Fn(\000)g Fr(C)t(\017)1331 1905 y FA(D)q Fq(+1)1407 1926 y Fr(\016)1431 1905 y Fm(\000)p Fq(1)1477 1926 y Fr(:)-5 2049 y Ff(Her)n(e)18 b Fr(c)f Ff(c)n(an)h(b)n(e)g (chosen)g(as)f Fr(c)d Fs(=)644 2030 y Fq(1)p 613 2038 81 2 v 613 2071 a FA(\025)634 2056 y Fi(n)p Fg(+1)634 2081 y(2)p Fi(;)p Fg(0)703 2030 y Fq(1)p 703 2038 18 2 v 703 2067 a(8)726 2049 y Fr(@)752 2056 y FA(r)771 2049 y Fr(u)799 2056 y Fq(2)p FA(;)p Fq(0)846 2049 y Fn(j)860 2056 y FA(r)q Fq(=)p FA(R)931 2061 y Fg(1)-5 2156 y Ff(Pr)n(o)n(of.)19 b Fs(Let)e Fr(F)23 b Fs(b)q(e)17 b(a)h(b)q(o)o(x)f(around)h Fr(y)g Fs(with)f(sidelength)f Fr(d)h Fs(with)g(t)o(w)o(o)f(sides)h(at)g(righ)o(t)g(angles)g(to)g(the) g(v)o(ector)-5 2214 y(from)e Fr(y)i Fs(to)f(the)f(origin.)21 b(Let)16 b Fr(F)569 2221 y Fq(1)604 2214 y Fs(b)q(e)f(the)h(side)f(of)h Fr(F)22 b Fs(with)15 b(the)h(largest)f(distance)h(to)g(the)f(origin.)21 b(W)l(e)15 b(de\014ne)-5 2272 y(the)h(stopping)h(time)d Fr(\034)409 2279 y FA(F)455 2272 y Fs(as)j(the)f(exit)f(time)f(from)i Fr(F)22 b Fs(i.e.)669 2354 y Fr(\034)690 2361 y FA(F)719 2354 y Fs(\()p Fr(!)r Fs(\))14 b(=)g(inf)s Fn(f)p Fr(t)f Fn(\025)h Fs(0)1047 2311 y Fl(\014)1047 2341 y(\014)1064 2354 y Fr(W)1110 2361 y FA(t)1125 2354 y Fs(\()p Fr(!)r Fs(\))19 b Fr(=)-29 b Fn(2)14 b Fr(F)7 b Fn(g)p Fr(:)-5 2435 y Fs(Let)17 b(also)639 2517 y Fr(\034)660 2524 y Fq(0)680 2517 y Fs(\()p Fr(!)r Fs(\))42 b(=)f(inf)s Fn(f)p Fr(t)13 b Fn(\025)h Fs(0)1063 2474 y Fl(\014)1063 2504 y(\014)1080 2517 y Fr(W)1126 2524 y FA(t)1141 2517 y Fs(\()p Fr(!)r Fs(\))20 b Fr(=)-30 b Fn(2)14 b Fs(\012)1307 2524 y Fq(0)1327 2517 y Fn(g)643 2595 y Fr(\034)664 2602 y FA(\017)680 2595 y Fs(\()p Fr(!)r Fs(\))42 b(=)f(inf)s Fn(f)p Fr(t)13 b Fn(\025)h Fs(0)1063 2553 y Fl(\014)1063 2583 y(\014)1080 2595 y Fr(W)1126 2602 y FA(t)1141 2595 y Fs(\()p Fr(!)r Fs(\))20 b Fr(=)-30 b Fn(2)14 b Fs(\012)1307 2602 y FA(\017)1323 2595 y Fn(g)p Fr(:)-5 2683 y Fs(Notice)g(that)h Fr(\034)272 2690 y FA(F)315 2683 y Fn(\024)f Fr(\034)389 2690 y Fq(0)422 2683 y Fn(\024)g Fr(\034)496 2690 y FA(\017)513 2683 y Fs(.)20 b(W)l(e)15 b(lo)q(ok)g(at)g(the)f(iterated)g(resolv)o (en)o(t)f(\()p Fn(\000)p Fs(\001)1356 2690 y FA(\017)1372 2683 y Fs(\))1391 2665 y Fm(\000)p Fq(\()p FA(n)p Fq(+1\))1529 2683 y Fs(for)i(a)g(su\016cien)o(tly)d(big)j Fr(n)p Fs(:)484 2764 y Fl(\002)505 2805 y Fs(\()p Fn(\000)p Fs(\001)604 2812 y FA(\017)619 2805 y Fs(\))638 2784 y Fm(\000)p Fq(\()p FA(n)p Fq(+1\))762 2805 y Fr(u)790 2812 y Fq(2)p FA(;)p Fq(0)837 2764 y Fl(\003)866 2805 y Fs(\()p Fr(y)r Fs(\))e(=)1013 2742 y Fm(1)995 2757 y Fl(X)1000 2862 y FA(j)r Fq(=1)1116 2771 y Fs(1)p 1080 2793 98 2 v 1080 2841 a Fr(\025)1108 2821 y FA(n)p Fq(+1)1108 2854 y FA(j;\017)1182 2805 y Fn(h)p Fr(u)1229 2812 y Fq(2)p FA(;)p Fq(0)1276 2805 y Fr(;)8 b(u)1326 2812 y FA(j;\017)1366 2805 y Fn(i)p Fr(u)1413 2812 y FA(j;\017)1454 2805 y Fs(\()p Fr(y)r Fs(\))p eop %%Page: 9 9 9 8 bop 664 -64 a Fo(THE)17 b(NOD)o(AL)f(SURF)l(A)o(CE)767 b(9)-123 37 y Fs(F)l(rom)15 b(Section)h(2)h(w)o(e)e(get:)476 57 y Fl(\014)476 87 y(\014)476 117 y(\014)476 147 y(\014)476 177 y(\014)493 97 y(X)498 203 y FA(j)r Fm(6)p Fq(=2)614 111 y Fs(1)p 578 133 98 2 v 578 181 a Fr(\025)606 160 y FA(n)p Fq(+1)606 194 y FA(j;\017)680 145 y Fn(h)p Fr(u)727 152 y FA(j;\017)768 145 y Fr(;)8 b(u)818 152 y Fq(2)p FA(;)p Fq(0)864 145 y Fn(i)p Fr(u)911 152 y FA(j;\017)952 145 y Fs(\()p Fr(y)r Fs(\))1016 57 y Fl(\014)1016 87 y(\014)1016 117 y(\014)1016 147 y(\014)1016 177 y(\014)1046 145 y Fn(\024)13 b Fr(C)t(\017)1157 124 y FA(D)q Fq(+1)1234 145 y Fr(N)r(:)-123 270 y Fs(W)l(e)j(apply)g(the)g(follo)o(wing)g (elemen)o(tary)e(form)o(ula)g(to)j(the)f(resolv)o(en)o(t:)630 313 y Fl(Z)679 326 y Fm(1)657 425 y Fq(0)725 380 y Fr(t)743 360 y FA(n)766 380 y Fr(e)789 360 y Fm(\000)p FA(t\025)852 380 y Fr(dt)d Fs(=)975 347 y(1)p 965 369 45 2 v 965 415 a Fr(c)986 422 y FA(n)1056 347 y Fs(1)p 1020 369 98 2 v 1020 415 a Fr(\025)1048 400 y Fq(1+)p FA(n)1122 380 y Fr(;)-123 491 y Fs(where)i Fr(c)38 498 y FA(n)76 491 y Fs(is)f(a)h(normalisation.)20 b(Belo)o(w,)14 b(w)o(e)g(will)g(rep)q (eatedly)g(use)h(the)f(fact)h(that)g Fr(u)1434 498 y Fq(2)p FA(;)p Fq(0)1495 491 y Fn(\025)e Fs(0.)21 b(W)l(e)15 b(will)e(write)-123 549 y Fr(\034)22 b Fs(instead)16 b(of)h Fr(\034)165 556 y FA(\017)182 549 y Fs(.)199 617 y Fl(\002)220 657 y Fs(\()p Fn(\000)p Fs(\001)319 664 y FA(\017)334 657 y Fs(\))353 636 y Fm(\000)p Fq(\()p FA(n)p Fq(+1\))477 657 y Fr(u)505 664 y Fq(2)p FA(;)p Fq(0)552 617 y Fl(\003)581 657 y Fs(\()p Fr(y)r Fs(\))41 b(=)h Fr(c)787 664 y FA(n)819 589 y Fl(Z)869 602 y Fm(1)847 702 y Fq(0)914 657 y Fr(t)932 636 y FA(n)955 657 y Fs(\()p Fr(e)997 636 y FA(t)p Fq(\001)1039 640 y Fi(\017)1056 657 y Fr(u)1084 664 y Fq(2)p FA(;)p Fq(0)1131 657 y Fs(\)\()p Fr(y)r Fs(\))p Fr(dt)686 790 y Fs(=)g Fr(c)787 797 y FA(n)811 790 y Fd(E)841 769 y FA(y)864 790 y Fs([)878 722 y Fl(Z)928 735 y Fm(1)906 835 y Fq(0)974 790 y Fr(t)992 769 y FA(n)1015 790 y Fr(u)1043 797 y Fq(2)p FA(;)p Fq(0)1090 790 y Fs(\()p Fr(W)1155 797 y FA(t)p Fm(^)p FA(\034)1212 790 y Fs(\))p Fr(dt)p Fs(])686 922 y Fn(\025)f Fr(c)787 929 y FA(n)811 922 y Fd(E)841 902 y FA(y)864 922 y Fs([)878 854 y Fl(Z)928 868 y Fm(1)906 967 y FA(\034)922 973 y Fi(F)974 922 y Fr(t)992 902 y FA(n)1015 922 y Fr(u)1043 929 y Fq(2)p FA(;)p Fq(0)1090 922 y Fs(\()p Fr(W)1155 929 y FA(t)p Fm(^)p FA(\034)1212 922 y Fs(\))p Fr(dt)p Fs(])686 1060 y(=)h Fr(c)787 1067 y FA(n)811 1060 y Fd(E)841 1040 y FA(y)864 1060 y Fs([)878 993 y Fl(Z)928 1006 y Fm(1)906 1105 y Fq(0)965 1060 y Fs(\()p Fr(t)11 b Fs(+)g Fr(\034)1083 1067 y FA(F)1112 1060 y Fs(\))1131 1040 y FA(n)1155 1060 y Fr(u)1183 1067 y Fq(2)p FA(;)p Fq(0)1230 1060 y Fs(\()p Fr(W)1295 1068 y Fq(\()p FA(t)p Fq(+)p FA(\034)1365 1074 y Fi(F)1390 1068 y Fq(\))p Fm(^)p FA(\034)1449 1060 y Fs(\))p Fr(dt)p Fs(])686 1193 y Fn(\025)41 b Fr(c)787 1200 y FA(n)819 1125 y Fl(Z)869 1138 y Fm(1)847 1238 y Fq(0)914 1193 y Fr(t)932 1172 y FA(n)955 1193 y Fd(E)986 1172 y FA(y)1009 1193 y Fs([)p Fr(u)1051 1200 y Fq(2)p FA(;)p Fq(0)1098 1193 y Fs(\()p Fr(W)1163 1201 y Fq(\()p FA(t)p Fq(+)p FA(\034)1233 1207 y Fi(F)1258 1201 y Fq(\))p Fm(^)p FA(\034)1317 1193 y Fs(\)])p Fr(dt)686 1326 y Fs(=)h Fr(c)787 1333 y FA(n)819 1258 y Fl(Z)869 1271 y Fm(1)847 1371 y Fq(0)914 1326 y Fr(t)932 1305 y FA(n)955 1326 y Fd(E)986 1305 y FA(y)1009 1326 y Fs([)p Fd(E)1053 1305 y FA(y)1077 1326 y Fs([)p Fr(u)1119 1333 y Fq(2)p FA(;)p Fq(0)1165 1326 y Fs(\()p Fr(W)1230 1333 y Fq(\()p FA(t)p Fq(+)p FA(\034)1300 1339 y Fi(F)1325 1333 y Fq(\))p Fm(^)p FA(\034)1384 1326 y Fs(\))p Fn(jF)1453 1333 y FA(\034)1469 1339 y Fi(F)1496 1326 y Fs(]])p Fr(dt)686 1458 y Fs(=)g Fr(c)787 1465 y FA(n)819 1390 y Fl(Z)869 1404 y Fm(1)847 1503 y Fq(0)914 1458 y Fr(t)932 1438 y FA(n)955 1458 y Fd(E)986 1438 y FA(y)1009 1458 y Fs([)p Fd(E)1053 1438 y FA(W)1086 1442 y Fi(\034)1101 1450 y(F)1133 1458 y Fs([)p Fr(u)1175 1465 y Fq(2)p FA(;)p Fq(0)1222 1458 y Fs(\()p Fr(W)1287 1465 y FA(t)p Fm(^)p FA(\034)1345 1458 y Fs(\)]])p Fr(dt)686 1591 y Fn(\025)f Fr(c)787 1598 y FA(n)811 1591 y Fd(E)841 1570 y FA(y)864 1591 y Fs([)p Fd(E)908 1570 y FA(W)942 1574 y Fi(\034)956 1583 y(F)989 1591 y Fs([)1003 1523 y Fl(Z)1052 1536 y FA(\034)1068 1541 y Fg(0)1030 1636 y Fq(0)1095 1591 y Fr(t)1113 1570 y FA(n)1136 1591 y Fr(u)1164 1598 y Fq(2)p FA(;)p Fq(0)1211 1591 y Fs(\()p Fr(W)1276 1598 y FA(t)p Fm(^)p FA(\034)1334 1591 y Fs(\))p Fr(dt)p Fs(]])686 1700 y(=)h Fd(E)796 1679 y FA(y)820 1700 y Fs([\()p Fn(\000)p Fs(\001)933 1707 y Fq(0)952 1700 y Fs(\))971 1679 y Fm(\000)p Fq(\()p FA(n)p Fq(+1\))1094 1700 y Fr(u)1122 1707 y Fq(2)p FA(;)p Fq(0)1169 1700 y Fs(\()p Fr(W)1234 1707 y FA(\034)1250 1713 y Fi(F)1277 1700 y Fs(\)])686 1798 y(=)807 1765 y(1)p 771 1787 V 771 1835 a Fr(\025)799 1814 y FA(n)p Fq(+1)799 1847 y(2)p FA(;)p Fq(0)873 1798 y Fd(E)903 1778 y FA(y)927 1798 y Fs([)p Fr(u)969 1805 y Fq(2)p FA(;)p Fq(0)1015 1798 y Fs(\()p Fr(W)1080 1805 y FA(\034)1096 1811 y Fi(F)1124 1798 y Fs(\)])686 1938 y Fn(\025)807 1904 y Fs(1)p 771 1927 V 771 1975 a Fr(\025)799 1954 y FA(n)p Fq(+1)799 1987 y(2)p FA(;)p Fq(0)878 1904 y Fs(1)p 878 1927 25 2 v 878 1972 a(4)916 1938 y(min)n Fn(f)p Fr(u)1050 1945 y Fq(2)p FA(;)p Fq(0)1097 1938 y Fs(\()p Fr(z)r Fs(\))p Fn(j)p Fr(z)15 b Fn(2)f Fr(F)1291 1945 y Fq(1)1311 1938 y Fn(g)686 2080 y(\025)807 2047 y Fs(1)p 771 2069 98 2 v 771 2117 a Fr(\025)799 2096 y FA(n)p Fq(+1)799 2129 y(2)p FA(;)p Fq(0)878 2047 y Fs(1)p 878 2069 25 2 v 878 2114 a(4)912 2047 y Fr(d)p 912 2069 26 2 v 912 2114 a Fs(2)943 2080 y Fr(@)969 2087 y FA(r)987 2080 y Fr(u)1015 2087 y Fq(2)p FA(;)p Fq(0)1062 2080 y Fn(j)1076 2087 y FA(r)q Fq(=)p FA(R)1147 2092 y Fg(1)1178 2080 y Fs(+)d Fr(O)q Fs(\()p Fr(\017)1304 2060 y Fq(2)1324 2080 y Fs(\))p Fr(:)p 1852 2200 2 33 v 1854 2168 30 2 v 1854 2200 V 1883 2200 2 33 v -123 2291 a FB(Remark)17 b(3.4.)23 b Fs(Let)17 b(us)f(lo)q(ok)h(at)f(one)h (of)f(the)g(holes)596 2376 y Fr(H)i Fs(=)c Fn(fj)p Fr(y)r Fn(j)f Fs(=)h Fr(R)887 2383 y Fq(1)907 2376 y Fn(g)d(\\)g Fr(B)s Fs(\()p Fr(x)1074 2383 y FA(k)1095 2376 y Fr(;)d(\017)p Fs(\))p Fr(:)-123 2461 y Fs(Let)17 b Fr(d)p Fs(\()p Fr(y)r Fs(\))c(=)h(dist\()p Fr(y)r(;)8 b(@)s Fs(\012)329 2468 y FA(\017)344 2461 y Fs(\),)16 b(and)h(let)e Fr(d\033)j Fs(b)q(e)f(normalised)p 696 2469 231 2 v 15 w(surface)f(measure)f(on)i Fn(fj)p Fr(y)r Fn(j)c Fs(=)g Fr(R)1545 2468 y Fq(1)1565 2461 y Fn(g)p Fs(,)j(then)340 2505 y Fl(Z)368 2618 y FA(H)410 2573 y Fr(d)p Fs(\()p Fr(y)r Fs(\))g Fr(d\033)r Fs(\()p Fr(y)r Fs(\))41 b Fn(\025)755 2505 y Fl(Z)783 2618 y Fm(f)p FA(y)q Fm(2)p FA(H)s Fm(j)p FA(d)p Fq(\()p FA(y)q Fq(\))p Fm(\025)p FA(\017=)p Fq(2)p Fm(g)1053 2573 y Fr(\017=)p Fs(2)17 b Fr(d\033)r Fs(\()p Fr(y)r Fs(\))675 2700 y(=)42 b Fr(\017=)p Fs(2\()p Fr(\017=)p Fs(2\))929 2679 y FA(D)q Fm(\000)p Fq(1)1047 2666 y Fr(\033)1075 2673 y FA(D)q Fm(\000)p Fq(1)p 1012 2688 175 2 v 1012 2738 a Fr(R)1049 2717 y FA(D)q Fm(\000)p Fq(1)1049 2750 y(1)1129 2738 y Fs(~)-26 b Fr(\033)1155 2745 y FA(D)1203 2700 y Fs(+)11 b Fr(O)q Fs(\()p Fr(\017)1329 2679 y FA(D)q Fq(+1)1406 2700 y Fs(\))675 2824 y(=)42 b(2)779 2803 y Fm(\000)p FA(D)839 2824 y Fr(\017)859 2803 y FA(D)930 2790 y Fr(\033)958 2797 y FA(D)q Fm(\000)p Fq(1)p 896 2812 V 896 2861 a Fr(R)933 2841 y FA(D)q Fm(\000)p Fq(1)933 2874 y(1)1013 2861 y Fs(~)-27 b Fr(\033)1038 2868 y FA(D)1086 2824 y Fs(+)11 b Fr(O)q Fs(\()p Fr(\017)1212 2803 y FA(D)q Fq(+1)1289 2824 y Fs(\))p Fr(;)p eop %%Page: 10 10 10 9 bop -5 -64 a Fo(10)790 b(S\037REN)17 b(F)o(OURNAIS)-5 37 y Fs(where)f Fr(\033)164 44 y FA(m)211 37 y Fs(=)e(v)o(ol)325 44 y Fe(R)355 35 y Fi(m)387 37 y Fs(\()p Fr(B)s Fs(\(1\)\))i(and)j(~) -26 b Fr(\033)666 44 y FA(D)714 37 y Fs(is)16 b(the)g(surface)g (measure)f(of)i Fn(fj)p Fr(y)r Fn(j)c Fs(=)g(1)p Fn(g)k Fs(in)f FB(R)1565 19 y FA(D)-5 127 y FB(Lemma)g(3.5.)23 b Ff(L)n(et)18 b Fr(x)13 b Fn(2)h Fs(\012)513 134 y FA(\017)530 127 y Ff(,)j Fn(j)p Fr(x)p Fn(j)c Fs(=)h Fr(R)720 134 y Fq(1)751 127 y Fn(\000)d Fr(\016)r Ff(.)22 b(Then)c(ther)n(e)f (exists)h(a)g(p)n(ositive)f(c)n(onstant)i Fr(c)1677 134 y Fq(1)1714 127 y Ff(such)e(that)496 174 y Fl(\002)517 214 y Fs(\()p Fn(\000)p Fs(\001)616 221 y FA(\017)632 214 y Fs(\))651 194 y Fm(\000)p Fq(\()p FA(n)p Fq(+1\))774 214 y Fr(u)802 221 y Fq(2)p FA(;)p Fq(0)849 174 y Fl(\003)878 214 y Fs(\()p Fr(x)p Fs(\))d Fn(\025)f Fr(c)1031 221 y Fq(1)1051 214 y Fr(\017)1071 194 y FA(D)1102 214 y Fr(\016)1126 194 y Fm(\000)p Fq(\()p FA(D)q Fm(\000)p Fq(1\))1269 214 y Fs(+)e Fr(O)q Fs(\()p Fr(\017)1395 194 y FA(D)q Fq(+1)1472 214 y Fs(\))p Fr(:)-5 298 y Ff(Her)n(e)18 b Fr(c)133 305 y Fq(1)170 298 y Ff(c)n(an)g(b)n(e)f(chosen)i(as)468 401 y Fr(c)489 408 y Fq(1)522 401 y Fs(=)14 b(2)598 380 y Fm(\000)p Fq(\(2)p FA(D)q Fq(+3\))753 367 y Fr(\033)781 374 y FA(D)q Fm(\000)p Fq(1)p 753 389 105 2 v 778 435 a Fs(~)-26 b Fr(\033)804 442 y FA(D)863 401 y Fr(R)900 380 y Fm(\000)p Fq(1)900 413 y(1)948 401 y Fs(\(2)p Fr(R)1028 408 y Fq(1)1059 401 y Fn(\000)11 b Fr(\016)r Fs(\))1192 367 y(1)p 1157 389 98 2 v 1157 437 a Fr(\025)1185 417 y FA(n)p Fq(+1)1185 450 y(2)p FA(;)p Fq(0)1258 401 y Fs(\()p Fr(@)1303 408 y FA(r)1322 401 y Fr(u)1350 408 y Fq(2)p FA(;)p Fq(0)1397 401 y Fn(j)1411 408 y FA(r)q Fq(=)p FA(R)1482 413 y Fg(1)1501 401 y Fs(\))p Fr(:)-5 527 y Ff(Pr)n(o)n(of.)19 b Fs(The)12 b(same)e(calculus)h(as)h(ab)q(o)o (v)o(e,)f(again)h(using)g(the)f(strong)i(Mark)o(o)o(v)d(prop)q(ert)o(y) h(of)h(Bro)o(wnian)f(motion,)-5 585 y(pro)o(v)o(es)16 b(that)386 633 y Fl(\002)407 673 y Fs(\()p Fn(\000)p Fs(\001)506 680 y FA(\017)522 673 y Fs(\))541 652 y Fm(\000)p Fq(\()p FA(n)p Fq(+1\))664 673 y Fr(u)692 680 y Fq(2)p FA(;)p Fq(0)739 633 y Fl(\003)768 673 y Fs(\()p Fr(x)p Fs(\))e Fn(\025)f Fr(c)921 680 y FA(n)953 605 y Fl(Z)1003 618 y Fm(1)981 718 y Fq(0)1049 673 y Fr(t)1067 652 y FA(n)1090 673 y Fd(E)1120 652 y FA(x)1153 633 y Fl(\002)1174 673 y Fd(E)1204 652 y FA(W)1237 656 y Fi(\034)1252 665 y(B)1285 673 y Fs([)p Fr(u)1327 680 y Fq(2)p FA(;)p Fq(0)1374 673 y Fs(\()p Fr(W)1439 680 y FA(t)p Fm(^)p FA(\034)1497 673 y Fs(\)])1530 633 y Fl(\003)1558 673 y Fr(dt;)-5 772 y Fs(where)642 835 y Fr(\034)663 842 y FA(B)694 835 y Fs(\()p Fr(!)r Fs(\))h(=)f(inf)t Fn(f)p Fr(t)g Fn(\025)g Fs(0)1021 793 y Fl(\014)1021 823 y(\014)1038 835 y Fn(j)p Fr(W)1098 842 y FA(t)1113 835 y Fs(\()p Fr(!)r Fs(\))p Fn(j)h(\025)g Fr(R)1301 842 y Fq(1)1320 835 y Fn(g)p Fr(;)-5 909 y Fs(is)g(the)f(exit)f(time)g(from)g(the)h(ball)h(of)f (radius)h Fr(R)844 916 y Fq(1)864 909 y Fs(.)20 b(Using)14 b(the)f(in)o(tegral)g(form)o(ula)f(and)i(the)f(sp)q(ectral)h(theorem,) -5 967 y(the)i(righ)o(t)g(hand)h(side)f(is)g(equal)g(to)635 1093 y Fd(E)666 1073 y FA(x)699 1008 y Fl(")746 1031 y Fm(1)728 1046 y Fl(X)733 1151 y FA(j)r Fq(=1)849 1060 y Fs(1)p 813 1082 V 813 1130 a Fr(\025)841 1109 y FA(n)p Fq(+1)841 1143 y FA(j;\017)915 1093 y Fn(h)p Fr(u)962 1100 y Fq(2)p FA(;)p Fq(0)1009 1093 y Fr(;)8 b(u)1059 1100 y FA(j;\017)1099 1093 y Fn(i)p Fr(u)1146 1100 y FA(j;\017)1187 1093 y Fs(\()p Fr(W)1252 1100 y FA(\034)1268 1106 y Fi(B)1296 1093 y Fs(\))1315 1008 y Fl(#)1352 1093 y Fr(:)-5 1227 y Fs(No)o(w,)22 b(the)f(exit)g(distribution)g(of)g(the)g (ball)g(for)h(Bro)o(wnian)f(motion)g(started)g(at)h Fr(x)f Fs(is)g(explicitly)e(kno)o(wn)-5 1285 y(\()e([Bas95][p.92]\))e(so)i(w)o (e)f(get)188 1368 y Fl(\002)209 1408 y Fs(\()p Fn(\000)p Fs(\001)308 1415 y FA(\017)324 1408 y Fs(\))343 1388 y Fm(\000)p Fq(\()p FA(n)p Fq(+1\))466 1408 y Fr(u)494 1415 y Fq(2)p FA(;)p Fq(0)541 1368 y Fl(\003)570 1408 y Fs(\()p Fr(x)p Fs(\))41 b Fn(\025)776 1346 y Fm(1)757 1361 y Fl(X)763 1466 y FA(j)r Fq(=1)879 1375 y Fs(1)p 843 1397 V 843 1445 a Fr(\025)871 1424 y FA(n)p Fq(+1)871 1458 y FA(j;\017)945 1408 y Fn(h)p Fr(u)992 1415 y Fq(2)p FA(;)p Fq(0)1039 1408 y Fr(;)8 b(u)1089 1415 y FA(j;\017)1129 1408 y Fn(i)1156 1340 y Fl(Z)1184 1453 y FA(S)1205 1457 y Fi(\017)1231 1408 y Fr(K)t Fs(\()p Fr(x;)g(y)r Fs(\))p Fr(u)1418 1415 y FA(j;\017)1458 1408 y Fs(\()p Fr(y)r Fs(\))p Fr(d\033)r Fs(\()p Fr(y)r Fs(\))678 1564 y(=)799 1530 y(1)p 762 1552 V 762 1600 a Fr(\025)790 1580 y FA(n)p Fq(+1)790 1613 y(2)p FA(;\017)864 1564 y Fn(h)p Fr(u)911 1571 y Fq(2)p FA(;)p Fq(0)959 1564 y Fr(;)g(u)1009 1571 y Fq(2)p FA(;\017)1052 1564 y Fn(i)1079 1496 y Fl(Z)1107 1609 y FA(S)1128 1613 y Fi(\017)1154 1564 y Fr(K)t Fs(\()p Fr(x;)g(y)r Fs(\))p Fr(u)1341 1571 y Fq(2)p FA(;\017)1384 1564 y Fs(\()p Fr(y)r Fs(\))p Fr(d\033)r Fs(\()p Fr(y)r Fs(\))i(+)h Fr(O)q Fs(\()p Fr(\017)1703 1543 y FA(D)q Fq(+1)1780 1564 y Fs(\))p Fr(:)-5 1683 y Fs(where)16 b Fr(d\033)i Fs(is)e(normalised)p 256 1691 231 2 v 15 w(surface)g(measure)f(on)i Fr(S)955 1690 y FA(\017)988 1683 y Fs(and)723 1796 y Fr(K)t Fs(\()p Fr(x;)8 b(y)r Fs(\))13 b(=)h Fr(R)984 1775 y FA(D)q Fm(\000)p Fq(2)984 1808 y(1)1066 1762 y Fr(R)1103 1744 y Fq(2)1103 1775 y(1)1135 1762 y Fn(\000)c(j)p Fr(x)p Fn(j)1240 1744 y Fq(2)p 1066 1785 194 2 v 1076 1830 a Fn(j)p Fr(y)j Fn(\000)d Fr(x)p Fn(j)1218 1816 y FA(D)1265 1796 y Fr(:)-5 1907 y Fs(If)16 b(w)o(e)g(just)g(include)f(the)h(hole)g(nearest)h(to)f Fr(x)g Fs(in)g(the)g(in)o(tegral,)f(w)o(e)h(get:)312 1946 y Fl(Z)340 2059 y FA(S)361 2063 y Fi(\017)386 2014 y Fr(K)t Fs(\()p Fr(x;)8 b(y)r Fs(\))586 1980 y(1)p 550 2003 98 2 v 550 2051 a Fr(\025)578 2030 y FA(n)p Fq(+1)578 2063 y(2)p FA(;\017)652 2014 y Fn(h)p Fr(u)699 2021 y Fq(2)p FA(;)p Fq(0)746 2014 y Fr(;)g(u)796 2021 y Fq(2)p FA(;\017)839 2014 y Fn(i)p Fr(u)886 2021 y Fq(2)p FA(;\017)930 2014 y Fs(\()p Fr(y)r Fs(\))p Fr(d\033)r Fs(\()p Fr(y)r Fs(\))232 2175 y Fn(\025)312 2105 y Fl(\022)349 2175 y Fr(R)386 2155 y FA(D)q Fm(\000)p Fq(2)386 2187 y(1)474 2141 y Fs(2)p Fr(R)535 2148 y Fq(1)566 2141 y Fn(\000)j Fr(\016)p 468 2164 177 2 v 468 2209 a Fs(\()p Fr(\016)i Fs(+)e Fr(\016)r Fs(\))614 2195 y FA(D)650 2175 y Fr(\016)674 2105 y Fl(\023)d(\022)755 2175 y Fs(2)779 2155 y Fm(\000)p FA(D)839 2175 y Fr(\017)859 2155 y FA(D)930 2141 y Fr(\033)958 2148 y FA(D)q Fm(\000)p Fq(1)p 896 2164 175 2 v 896 2213 a Fr(R)933 2192 y FA(D)q Fm(\000)p Fq(1)933 2225 y(1)1013 2213 y Fs(~)-27 b Fr(\033)1038 2220 y FA(D)1075 2105 y Fl(\023)1120 2090 y( )1201 2141 y Fs(1)p 1164 2164 98 2 v 1164 2212 a Fr(\025)1192 2191 y FA(n)p Fq(+1)1192 2224 y(2)p FA(;)p Fq(0)1271 2141 y Fs(1)p 1271 2164 25 2 v 1271 2209 a(8)1301 2175 y Fr(@)1327 2182 y FA(r)1346 2175 y Fr(u)1374 2182 y Fq(2)p FA(;)p Fq(0)1420 2175 y Fn(j)1434 2182 y FA(r)q Fq(=)p FA(R)1505 2187 y Fg(1)1525 2090 y Fl(!)1575 2175 y Fs(+)11 b Fr(O)q Fs(\()p Fr(\017)1701 2155 y FA(D)q Fq(+1)1778 2175 y Fs(\))p Fr(:)-5 2302 y Fs(Th)o(us)17 b Fr(c)140 2309 y Fq(1)176 2302 y Fs(can)f(b)q(e)h(c)o (hosen)f(as)468 2404 y Fr(c)489 2411 y Fq(1)522 2404 y Fs(=)e(2)598 2384 y Fm(\000)p Fq(\(2)p FA(D)q Fq(+3\))753 2371 y Fr(\033)781 2378 y FA(D)q Fm(\000)p Fq(1)p 753 2393 105 2 v 778 2439 a Fs(~)-26 b Fr(\033)804 2446 y FA(D)863 2404 y Fr(R)900 2384 y Fm(\000)p Fq(1)900 2417 y(1)948 2404 y Fs(\(2)p Fr(R)1028 2411 y Fq(1)1059 2404 y Fn(\000)11 b Fr(\016)r Fs(\))1192 2371 y(1)p 1157 2393 98 2 v 1157 2441 a Fr(\025)1185 2420 y FA(n)p Fq(+1)1185 2453 y(2)p FA(;)p Fq(0)1258 2404 y Fs(\()p Fr(@)1303 2411 y FA(r)1322 2404 y Fr(u)1350 2411 y Fq(2)p FA(;)p Fq(0)1397 2404 y Fn(j)1411 2411 y FA(r)q Fq(=)p FA(R)1482 2416 y Fg(1)1501 2404 y Fs(\))p Fr(:)p 1971 2522 2 33 v 1973 2491 30 2 v 1973 2522 V 2001 2522 2 33 v 45 2612 a Fs(No)o(w,)16 b(w)o(e)g(can)g(pro)o(v)o(e)f(Theorem)g(3.1:)-5 2701 y Ff(Pr)n(o)n(of.)k Fs(If)d Fn(j)p Fr(x)p Fn(j)d Fs(=)h Fr(R)351 2708 y Fq(1)382 2701 y Fn(\000)d Fr(\016)18 b Fs(and)e Fr(\016)g Fn(\035)d Fr(\017)p Fs(,)j(then)676 2784 y Fn(j)p Fr(u)718 2791 y FA(j;\017)758 2784 y Fs(\()p Fr(x)p Fs(\))p Fn(j)41 b(\024)g(k)p Fr(u)1012 2791 y FA(j;\017)1052 2784 y Fn(k)1077 2791 y Fm(1)1114 2784 y Fr(e\025)1165 2791 y FA(j;\017)1206 2784 y Fd(E)1236 2764 y FA(x)1261 2784 y Fs([)p Fr(\034)1296 2791 y FA(\017)1312 2784 y Fs(])879 2857 y(=)h Fn(k)p Fr(u)1012 2864 y FA(j;\017)1052 2857 y Fn(k)1077 2864 y Fm(1)1114 2857 y Fr(e\025)1165 2864 y FA(j;\017)1206 2857 y Fr(o)1229 2864 y FA(\016)1248 2857 y Fs(\(1\))p Fr(;)p eop %%Page: 11 11 11 10 bop 664 -64 a Fo(THE)17 b(NOD)o(AL)f(SURF)l(A)o(CE)748 b(11)-123 37 y Fs(where)16 b Fr(o)41 44 y FA(\016)60 37 y Fs(\(1\))h(tends)f(to)h(zero)f(as)h Fr(\016)h Fs(gets)e(small)745 19 y Fq(1)763 37 y Fs(.)21 b(Therefore)408 82 y Fl(\014)408 112 y(\014)408 142 y(\014)408 172 y(\014)408 202 y(\014)424 122 y(X)430 228 y FA(j)r Fm(6)p Fq(=2)546 136 y Fs(1)p 509 158 98 2 v 509 206 a Fr(\025)537 185 y FA(n)p Fq(+1)537 219 y FA(j;\017)611 169 y Fn(h)p Fr(u)658 176 y FA(j;\017)699 169 y Fr(;)8 b(u)749 176 y Fq(2)p FA(;)p Fq(0)796 169 y Fn(i)p Fr(u)843 176 y FA(j;\017)883 169 y Fs(\()p Fr(x)p Fs(\))949 82 y Fl(\014)949 112 y(\014)949 142 y(\014)949 172 y(\014)949 202 y(\014)979 169 y Fn(\024)14 b Fr(\017)1052 149 y FA(D)1084 169 y Fr(\016)1108 149 y Fm(\000)p Fq(\()p FA(D)q Fm(\000)p Fq(1\))1239 169 y Fr(o)1262 176 y FA(\016)1281 169 y Fs(\(1\))p Fr(:)-123 307 y Fs(On)i(the)g(other)h(hand,)350 355 y Fl(\002)371 395 y Fs(\()p Fn(\000)p Fs(\001)470 402 y FA(\017)486 395 y Fs(\))505 375 y Fm(\000)p Fq(\()p FA(n)p Fq(+1\))628 395 y Fr(u)656 402 y Fq(2)p FA(;)p Fq(0)703 355 y Fl(\003)732 395 y Fs(\()p Fr(x)p Fs(\))41 b Fn(\025)h Fr(c)941 402 y Fq(1)960 395 y Fr(\017)980 375 y FA(D)1012 395 y Fr(\016)1036 375 y Fm(\000)p Fq(\()p FA(D)q Fm(\000)p Fq(1\))1179 395 y Fs(+)11 b Fr(O)q Fs(\()p Fr(\017)1305 375 y FA(D)q Fq(+1)1382 395 y Fs(\))p Fr(:)-123 482 y Fs(Cho)q(osing)18 b(no)o(w)f Fr(\017;)8 b(\016)17 b Fs(su\016cien)o(tly)d(small,)g(w)o(e)i(get)g(a)h(con)o(tradiction)f (if)g Fr(u)1222 489 y Fq(2)p FA(;\017)1279 482 y Fn(\024)d Fs(0.)p 1852 482 2 33 v 1854 451 30 2 v 1854 482 V 1883 482 2 33 v 145 605 a(4.)28 b Ft(An)18 b(explicit)g(estima)m(te)h(on)f (the)g(number)g(of)g(holes)g(f)o(or)g Fr(D)d Fs(=)f(2)-73 693 y(Let)h(us)h(redo)f(the)g(pro)q(of)h(of)f(Theorem)f(3.1)h(a)h (little)d(more)h(carefully)f(in)i(the)g(case)g(where)f Fr(D)i Fs(=)e(2)h(and)h(the)-123 751 y(holes)g(are)h(ev)o(enly)d (placed)i(i.e.)k(2)p Fr(\031)r(R)558 758 y Fq(1)591 751 y Fs(=)14 b Fr(N)5 b(\032)14 b Fs(=)g(2)p Fr(N)5 b(\016)-73 809 y Fs(First)16 b(w)o(e)g(estimate)593 821 y Fl(\014)593 851 y(\014)593 881 y(\014)593 911 y(\014)593 941 y(\014)609 861 y(X)615 967 y FA(j)r Fm(6)p Fq(=2)731 875 y Fs(1)p 694 897 98 2 v 694 945 a Fr(\025)722 925 y FA(n)p Fq(+1)722 958 y FA(j;\017)796 909 y Fn(h)p Fr(u)843 916 y FA(j;\017)884 909 y Fr(;)8 b(u)934 916 y Fq(2)p FA(;)p Fq(0)981 909 y Fn(i)p Fr(u)1028 916 y FA(j;\017)1068 909 y Fs(\()p Fr(x)p Fs(\))1134 821 y Fl(\014)1134 851 y(\014)1134 881 y(\014)1134 911 y(\014)1134 941 y(\014)1159 909 y Fr(:)-123 1039 y Fs(F)l(or)17 b Fn(j)p Fr(x)p Fn(j)c Fs(=)h Fr(R)123 1046 y Fq(1)153 1039 y Fn(\000)d Fr(\016)r Fs(,)16 b(where)f Fr(\016)h Fn(\035)d Fr(\017)p Fs(,)j(w)o(e)g(get)419 1126 y Fn(j)p Fr(u)461 1133 y FA(j;\017)501 1126 y Fs(\()p Fr(x)p Fs(\))p Fn(j)41 b(\024)g(k)p Fr(u)755 1133 y FA(j;\017)795 1126 y Fs(\()p Fr(x)p Fs(\))p Fn(k)886 1133 y Fm(1)923 1126 y Fr(e\025)974 1133 y FA(j;\017)1015 1126 y Fd(E)1045 1105 y FA(x)1070 1126 y Fs([)p Fr(\034)1105 1133 y FA(\017)1121 1126 y Fs(])623 1199 y(=)g Fn(k)p Fr(u)755 1206 y FA(j;\017)795 1199 y Fs(\()p Fr(x)p Fs(\))p Fn(k)886 1206 y Fm(1)923 1199 y Fr(e\025)974 1206 y FA(j;\017)1015 1199 y Fs(\()p Fd(E)1064 1178 y FA(x)1089 1199 y Fs([)p Fr(\034)1124 1206 y Fq(0)1143 1199 y Fs(])11 b(+)g Fr(O)q Fs(\()p Fr(\017)p Fs(\)\))p Fr(:)-123 1286 y Fs(No)o(w)640 1373 y Fn(j)p Fr(x)p Fn(j)696 1352 y Fq(2)756 1373 y Fs(=)42 b Fd(E)866 1352 y FA(x)891 1373 y Fs([)p Fr(W)951 1380 y FA(\034)967 1385 y Fg(0)997 1373 y Fn(\000)10 b Fs(2)p Fr(\034)1091 1380 y Fq(0)1112 1373 y Fs(])756 1446 y(=)42 b Fr(R)873 1425 y Fq(2)873 1458 y(1)904 1446 y Fn(\000)11 b Fs(2)p Fd(E)1008 1425 y FA(x)1033 1446 y Fs([)p Fr(\034)1068 1453 y Fq(0)1088 1446 y Fs(])p Fr(;)-123 1533 y Fs(so)680 1618 y Fd(E)710 1598 y FA(x)735 1618 y Fs([)p Fr(\034)770 1625 y Fq(0)789 1618 y Fs(])j(=)873 1585 y Fr(R)910 1567 y Fq(2)910 1597 y(1)942 1585 y Fn(\000)d(j)p Fr(x)p Fn(j)1048 1567 y Fq(2)p 873 1607 194 2 v 958 1653 a Fs(2)1072 1618 y Fr(;)-123 1710 y Fs(and)17 b(therefore)485 1811 y Fn(j)p Fr(u)527 1818 y FA(j;\017)567 1811 y Fs(\()p Fr(x)p Fs(\))p Fn(j)41 b(\024)g(k)p Fr(u)821 1818 y FA(j;\017)861 1811 y Fs(\()p Fr(x)p Fs(\))p Fn(k)952 1818 y Fm(1)989 1811 y Fr(e\025)1040 1818 y FA(j;\017)1086 1778 y Fs(2)p Fr(R)1147 1785 y Fq(1)1178 1778 y Fn(\000)11 b Fr(\016)p 1086 1800 166 2 v 1156 1845 a Fs(2)1256 1811 y Fr(\016)688 1932 y Fn(\024)41 b Fr(e)791 1912 y Fq(1+)851 1898 y Fg(1)p 841 1904 35 2 v 841 1925 a(8)p Fi(\031)887 1899 y Fs(2)p Fr(R)948 1906 y Fq(1)980 1899 y Fn(\000)10 b Fr(\016)p 887 1921 166 2 v 958 1967 a Fs(2)1058 1932 y Fr(\025)1086 1907 y Fq(3)p FA(=)p Fq(2)1086 1945 y FA(j;\017)1141 1932 y Fr(\016)o(;)-123 2034 y Fs(b)o(y)16 b(Lemma)e(2.5.)22 b(Therefore)64 2079 y Fl(\014)64 2109 y(\014)64 2139 y(\014)64 2168 y(\014)64 2198 y(\014)81 2119 y(X)86 2225 y FA(j)r Fm(6)p Fq(=2)202 2132 y Fs(1)p 166 2154 98 2 v 166 2202 a Fr(\025)194 2182 y FA(n)p Fq(+1)194 2215 y FA(j;\017)268 2166 y Fn(h)p Fr(u)315 2173 y FA(j;\017)355 2166 y Fr(;)8 b(u)405 2173 y Fq(2)p FA(;)p Fq(0)452 2166 y Fn(i)p Fr(u)499 2173 y FA(j;\017)540 2166 y Fs(\()p Fr(x)p Fs(\))606 2079 y Fl(\014)606 2109 y(\014)606 2139 y(\014)606 2168 y(\014)606 2198 y(\014)663 2166 y Fn(\024)42 b Fr(e)767 2145 y Fq(1+)826 2132 y Fg(1)p 816 2138 35 2 v 816 2158 a(8)p Fi(\031)863 2132 y Fs(2)p Fr(R)924 2139 y Fq(1)955 2132 y Fn(\000)11 b Fr(\016)p 863 2154 166 2 v 933 2200 a Fs(2)1033 2166 y Fr(C)1080 2119 y Fl(X)1085 2225 y FA(j)r Fm(6)p Fq(=2)1202 2132 y Fs(1)p 1165 2154 98 2 v 1165 2202 a Fr(\025)1193 2182 y FA(n)p Fq(+1)1193 2215 y FA(j;\017)1347 2127 y Fr(\025)1375 2102 y Fq(3)p FA(=)p Fq(2)1375 2140 y FA(j;\017)p 1272 2154 233 2 v 1272 2200 a Fn(j)p Fr(\025)1314 2207 y FA(j;\017)1366 2200 y Fn(\000)g Fr(\025)1444 2207 y Fq(2)p FA(;)p Fq(0)1491 2200 y Fn(j)1510 2166 y Fr(\025)1538 2140 y Fq(3)p FA(=)p Fq(2)1538 2179 y FA(j;\017)1594 2166 y Fr(N)5 b(\016)r(\017)1682 2145 y Fq(2)664 2335 y Fs(=)42 b Fr(e)767 2315 y Fq(1+)826 2301 y Fg(1)p 816 2307 35 2 v 816 2328 a(8)p Fi(\031)863 2302 y Fs(2)p Fr(R)924 2309 y Fq(1)955 2302 y Fn(\000)11 b Fr(\016)p 863 2324 166 2 v 933 2370 a Fs(2)1033 2335 y Fr(C)1080 2288 y Fl(X)1085 2394 y FA(j)r Fm(6)p Fq(=2)1233 2297 y Fr(\025)1261 2276 y Fq(2)p Fm(\000)p FA(n)1261 2310 y(j;\017)p 1165 2324 233 2 v 1165 2370 a Fn(j)p Fr(\025)1207 2377 y FA(j;\017)1259 2370 y Fn(\000)g Fr(\025)1337 2377 y Fq(2)p FA(;)p Fq(0)1384 2370 y Fn(j)1403 2335 y Fr(\031)r(R)1470 2342 y Fq(1)1489 2335 y Fr(\017)1509 2315 y Fq(2)1529 2335 y Fr(;)p -123 2454 250 2 v -73 2484 a Fc(1)-54 2499 y Fz(In)i(fact,)614 2574 y Fb(E)639 2557 y Fw(x)663 2574 y Fz([)p Fv(\034)693 2580 y Fw(\017)708 2574 y Fz(])41 b(=)h Fb(E)860 2557 y Fw(x)884 2574 y Fz([)p Fv(\034)914 2580 y Fc(0)932 2574 y Fz(])9 b(+)g Fv(O)q Fz(\()p Fv(\017)p Fz(\))761 2663 y(=)840 2635 y Fv(R)872 2620 y Fc(2)872 2645 y(1)900 2635 y Fu(\000)g(j)p Fv(x)p Fu(j)989 2620 y Fc(2)p 840 2653 167 2 v 906 2691 a Fv(D)1021 2663 y Fz(+)g Fv(O)q Fz(\()p Fv(\017)p Fz(\))761 2765 y(=)840 2737 y(2)p Fv(R)893 2743 y Fc(1)920 2737 y Fu(\000)h Fv(\016)p 840 2755 142 2 v 893 2793 a(D)987 2765 y(\016)h Fz(+)e Fv(O)q Fz(\()p Fv(\017)p Fz(\))p Fv(:)-123 2857 y Fz(This)14 b(is)g(pro)o(v)o(ed)g(in)f(section)i(4.)p eop %%Page: 12 12 12 11 bop -5 -64 a Fo(12)790 b(S\037REN)17 b(F)o(OURNAIS)-5 37 y Fs(where)f Fr(C)k Fs(is)c(the)g(constan)o(t)h(from)e(Lemma)f(2.1.) 21 b(Let)c(us)f(estimate)1333 17 y Fq(1)p 1258 25 168 2 v 1258 54 a Fm(j)p FA(\025)1289 59 y Fi(j;\017)1325 54 y Fm(\000)p FA(\025)1373 59 y Fg(2)p Fi(;)p Fg(0)1415 54 y Fm(j)1446 37 y Fs(b)o(y)229 161 y(max)o Fn(f)456 127 y Fs(1)p 350 149 237 2 v 350 195 a Fn(j)p Fr(\025)392 202 y Fq(1)p FA(;\017)447 195 y Fn(\000)11 b Fr(\025)525 202 y Fq(2)p FA(;)p Fq(0)573 195 y Fn(j)592 161 y Fr(;)724 127 y Fs(1)p 618 149 V 618 195 a Fn(j)p Fr(\025)660 202 y Fq(3)p FA(;\017)716 195 y Fn(\000)f Fr(\025)793 202 y Fq(2)p FA(;)p Fq(0)841 195 y Fn(j)860 161 y(g)j Fs(=)h(max)o Fn(f)1179 127 y Fs(1)p 1071 149 240 2 v 1071 195 a Fn(j)p Fr(\025)1113 202 y Fq(1)p FA(;)p Fq(0)1171 195 y Fn(\000)d Fr(\025)1249 202 y Fq(2)p FA(;)p Fq(0)1297 195 y Fn(j)1316 161 y Fr(;)1450 127 y Fs(1)p 1343 149 V 1343 195 a Fn(j)p Fr(\025)1385 202 y Fq(3)p FA(;)p Fq(0)1443 195 y Fn(\000)g Fr(\025)1521 202 y Fq(2)p FA(;)p Fq(0)1568 195 y Fn(j)1587 161 y(g)g Fs(+)g Fr(o)p Fs(\(1\))p Fr(;)-5 274 y Fs(as)17 b Fr(\017)d Fn(&)f Fs(0.)45 332 y(Let)j(us)h(\014x)f Fr(R)301 339 y Fq(2)321 332 y Fr(=R)382 339 y Fq(1)418 332 y Fs(in)g(suc)o(h)g(a)h(w)o(a)o(y)f(that)291 447 y Fr(\025)319 454 y Fq(2)p FA(;)p Fq(0)380 447 y Fs(=)e Fr(\025)460 454 y Fq(1)480 447 y Fs(\()p Fr(B)s Fs(\()p Fr(R)595 454 y Fq(2)614 447 y Fs(\))d Fn(n)p 680 403 135 2 v 11 w Fr(B)s Fs(\()p Fr(R)776 454 y Fq(1)796 447 y Fs(\)\))i(=)904 413 y Fr(\025)932 420 y Fq(1)p FA(;)p Fq(0)991 413 y Fs(+)e Fr(\025)1068 420 y Fq(3)p FA(;)p Fq(0)p 904 435 212 2 v 998 481 a Fs(2)1134 447 y(=)1191 413 y Fr(\025)1219 420 y Fq(1)1239 413 y Fs(\()p Fr(B)s Fs(\()p Fr(R)1354 420 y Fq(1)1374 413 y Fs(\)\))g(+)g Fr(\025)1500 420 y Fq(2)1520 413 y Fs(\()p Fr(B)s Fs(\()p Fr(R)1635 420 y Fq(1)1654 413 y Fs(\)\))p 1191 435 501 2 v 1429 481 a(2)1697 447 y Fr(:)-5 546 y Fs(F)l(urthermore,)j(w)o(e)i (ha)o(v)o(e)g(to)g(estimate)905 571 y Fl(X)910 677 y FA(j)r Fm(6)p Fq(=2)985 619 y Fr(\025)1013 598 y Fq(2)p Fm(\000)p FA(n)1013 631 y(j;\017)1082 619 y Fr(:)-5 744 y Fs(W)l(e)g(ma)o(y)f(tak)o(e)h Fr(n)e Fs(=)f(4,)k(and)f(get)658 790 y Fl(X)663 896 y FA(j)r Fm(6)p Fq(=2)738 837 y Fr(\025)766 816 y Fm(\000)p Fq(2)766 850 y FA(j;\017)855 837 y Fn(\024)936 790 y Fl(X)1016 837 y Fr(\025)1044 844 y FA(j)1063 837 y Fs(\()p Fr(B)s Fs(\()p Fr(R)1178 844 y Fq(2)1197 837 y Fs(\)\))1235 816 y Fm(\000)p Fq(2)855 1006 y Fn(\024)936 959 y Fl(X)964 1064 y FA(j)1016 936 y Fl(\022)1053 1006 y Fs(4)p Fr(\031)1112 973 y Fs(1)p 1112 995 25 2 v 1112 1040 a(2)1178 973 y Fr(j)p 1146 995 87 2 v 1146 1040 a(\031)r(R)1213 1023 y Fq(2)1213 1053 y(2)1237 936 y Fl(\023)1274 947 y Fm(\000)p Fq(2)1330 1006 y Fr(;)-5 1144 y Fs(b)o(y)g(an)h(inequalit)o(y)d(pro)o(v)o(ed)i(in)f([L)l(Y83)q (][Cor.)21 b(1].)g(Th)o(us)679 1213 y Fl(X)684 1319 y FA(j)r Fm(6)p Fq(=2)759 1260 y Fr(\025)787 1239 y Fm(\000)p Fq(2)787 1273 y FA(j;\017)848 1260 y Fn(\024)906 1226 y Fr(R)943 1208 y Fq(4)943 1239 y(2)p 906 1249 58 2 v 922 1294 a Fs(4)976 1213 y Fl(X)1004 1318 y FA(j)1057 1260 y Fr(j)1080 1239 y Fm(\000)p Fq(2)1141 1260 y Fs(=)1197 1226 y Fr(\031)1227 1208 y Fq(2)1247 1226 y Fr(R)1284 1208 y Fq(4)1284 1239 y(2)p 1197 1249 107 2 v 1226 1294 a Fs(24)1309 1260 y Fr(;)-5 1396 y Fs(and)c(w)o(e)f(\014nally)g(obtain) 586 1436 y Fl(\014)586 1466 y(\014)586 1496 y(\014)586 1526 y(\014)586 1556 y(\014)602 1476 y(X)608 1582 y FA(j)r Fm(6)p Fq(=2)710 1490 y Fs(1)p 688 1512 69 2 v 688 1558 a Fr(\025)716 1540 y Fq(5)716 1570 y FA(j;\017)761 1523 y Fn(h)p Fr(u)808 1530 y FA(j;\017)849 1523 y Fr(;)8 b(u)899 1530 y Fq(2)p FA(;)p Fq(0)946 1523 y Fn(i)p Fr(u)993 1530 y FA(j;\017)1033 1523 y Fs(\()p Fr(x)p Fs(\))1099 1436 y Fl(\014)1099 1466 y(\014)1099 1496 y(\014)1099 1526 y(\014)1099 1556 y(\014)505 1685 y Fn(\024)42 b Fr(e)609 1665 y Fq(1+)668 1651 y Fg(1)p 658 1657 35 2 v 658 1678 a(8)p Fi(\031)705 1652 y Fr(\031)735 1634 y Fq(3)p 705 1674 50 2 v 705 1720 a Fs(48)759 1685 y(\(2)p Fr(R)839 1692 y Fq(1)870 1685 y Fn(\000)11 b Fr(\016)r Fs(\))p Fr(C)t(R)1039 1692 y Fq(1)1058 1685 y Fr(R)1095 1665 y Fq(4)1095 1698 y(2)1214 1652 y Fs(2)p 1120 1674 213 2 v 1120 1720 a Fr(\025)1148 1727 y Fq(3)p FA(;)p Fq(0)1206 1720 y Fn(\000)g Fr(\025)1284 1727 y Fq(1)p FA(;)p Fq(0)1337 1685 y Fr(\017)1357 1665 y Fq(2)1387 1685 y Fs(+)g Fr(o)p Fs(\()p Fr(\017)1498 1665 y Fq(2)1518 1685 y Fs(\))-5 1799 y(On)17 b(the)f(other)g(hand)h(w)o(e)f(get)g(from) f(Lemma)f(3.5)j(that)329 1867 y Fl(\002)350 1908 y Fs(\()p Fn(\000)p Fs(\001)449 1915 y FA(\017)464 1908 y Fs(\))483 1887 y Fm(\000)p Fq(5)531 1908 y Fr(u)559 1915 y Fq(2)p FA(;)p Fq(0)605 1867 y Fl(\003)634 1908 y Fs(\()p Fr(x)p Fs(\))d Fn(\025)796 1874 y Fs(1)p 772 1896 74 2 v 772 1942 a(128)855 1874 y(2)p Fr(R)916 1881 y Fq(1)947 1874 y Fn(\000)d Fr(\016)p 855 1896 166 2 v 894 1942 a(\031)r(R)961 1949 y Fq(1)1056 1874 y Fs(1)p 1030 1896 76 2 v 1030 1942 a Fr(\025)1058 1925 y Fq(5)1058 1954 y(2)p FA(;)p Fq(0)1111 1908 y Fs(\()p Fr(@)1156 1915 y FA(r)1174 1908 y Fr(u)1202 1915 y Fq(2)p FA(;)p Fq(0)1249 1908 y Fn(j)1263 1915 y FA(r)q Fq(=)p FA(R)1334 1920 y Fg(1)1354 1908 y Fs(\))p Fr(\017)1393 1887 y Fq(2)1412 1908 y Fr(\016)1436 1887 y Fm(\000)p Fq(1)1494 1908 y Fs(+)g Fr(O)q Fs(\()p Fr(\017)1620 1887 y Fq(3)1640 1908 y Fs(\))p Fr(:)-5 2026 y Fs(Th)o(us)17 b(w)o(e)f(get)g(a)h(con)o(tradiction)e(when)362 2142 y Fr(\016)43 b(<)511 2109 y Fs(3)p 511 2131 25 2 v 511 2177 a(8)541 2142 y Fr(\031)571 2122 y Fm(\000)p Fq(4)617 2142 y Fs(\()p Fr(e)659 2122 y Fq(1+)718 2108 y Fg(1)p 709 2114 35 2 v 709 2135 a(8)p Fi(\031)750 2142 y Fs(\))769 2122 y Fm(\000)p Fq(2)816 2142 y Fr(\025)844 2122 y Fm(\000)p Fq(5)844 2155 y(2)p FA(;)p Fq(0)892 2142 y Fs(\()p Fr(\025)939 2149 y Fq(2)p FA(;)p Fq(0)998 2142 y Fn(\000)10 b Fr(\025)1075 2149 y Fq(1)p FA(;)p Fq(0)1123 2142 y Fs(\))1175 2109 y Fr(R)1212 2088 y Fm(\000)p Fq(1)1212 2121 y(1)1260 2109 y Fr(R)1297 2088 y Fm(\000)p Fq(4)1297 2121 y(2)1353 2109 y Fs(log)q(\()p Fr(R)1472 2116 y Fq(2)1492 2109 y Fr(=R)1553 2116 y Fq(1)1573 2109 y Fs(\))p 1147 2131 474 2 v 1147 2177 a(\()p Fr(R)1203 2159 y Fq(2)1203 2189 y(2)1234 2177 y Fn(\000)h Fr(R)1321 2159 y Fq(2)1321 2189 y(1)1341 2177 y Fs(\)\(1)h(+)f Fr(R)1501 2184 y Fq(2)1521 2177 y Fr(=R)1582 2184 y Fq(1)1602 2177 y Fs(\))1626 2142 y Fr(:)45 2254 y Fs(This)16 b(is)h(equiv)m(alen) o(t)d(to:)412 2373 y Fr(N)19 b(>)14 b Fs(\()p Fr(e)564 2352 y Fq(1+)623 2338 y Fg(1)p 613 2344 35 2 v 613 2365 a(8)p Fi(\031)655 2373 y Fs(\))674 2352 y Fq(2)698 2339 y Fs(8)p Fr(\031)752 2321 y Fq(5)p 698 2361 74 2 v 723 2407 a Fs(3)790 2339 y(1)d(+)h Fr(R)912 2346 y Fq(2)931 2339 y Fr(=R)992 2346 y Fq(1)p 782 2361 240 2 v 782 2407 a Fs(log)q(\()p Fr(R)901 2414 y Fq(2)921 2407 y Fr(=R)982 2414 y Fq(1)1002 2407 y Fs(\))1026 2373 y Fr(R)1063 2352 y Fq(2)1063 2385 y(1)1083 2373 y Fr(R)1120 2352 y Fq(4)1120 2385 y(2)1140 2373 y Fs(\()p Fr(R)1196 2352 y Fq(2)1196 2385 y(2)1228 2373 y Fn(\000)e Fr(R)1314 2352 y Fq(2)1314 2385 y(1)1335 2373 y Fs(\))1427 2334 y Fr(\025)1455 2316 y Fq(5)1455 2347 y(2)p FA(;)p Fq(0)p 1359 2361 213 2 v 1359 2407 a Fr(\025)1387 2414 y Fq(3)p FA(;)p Fq(0)1445 2407 y Fn(\000)h Fr(\025)1523 2414 y Fq(1)p FA(;)p Fq(0)1576 2373 y Fr(:)-5 2486 y Fs(If)16 b(w)o(e)g(tak)o(e)g Fr(R)259 2493 y Fq(1)292 2486 y Fs(=)e(1)j(then)f(w)o(e)g(get)g(from)f(a)i (table)f(of)g(Bessel)f(functions:)693 2571 y Fr(\025)721 2578 y Fq(1)p FA(;)p Fq(0)810 2571 y Fs(=)42 b(5)p Fr(:)p Fs(7831)15 b(=)f(\(2)p Fr(:)p Fs(4048\))1263 2550 y Fq(2)693 2646 y Fr(\025)721 2653 y Fq(3)p FA(;)p Fq(0)810 2646 y Fs(=)42 b(14)p Fr(:)p Fs(6819)15 b(=)f(\(3)p Fr(:)p Fs(8317\))1287 2625 y Fq(2)-5 2730 y Fs(Th)o(us,)h(w)o(e)f(get)h Fr(\025)309 2737 y Fq(2)p FA(;)p Fq(0)370 2730 y Fs(=)f(10)p Fr(:)p Fs(2325.)23 b(The)14 b(Maple)g(computation)g(in)h(the)f(app)q (endix)h(sho)o(ws)g(that)g Fr(R)1771 2737 y Fq(2)1805 2730 y Fs(=)f(1)p Fr(:)p Fs(9762,)-5 2788 y(and)j(w)o(e)f(get)806 2853 y Fr(N)j(>)14 b Fs(15)p Fr(:)p Fs(843)f Fn(\002)e Fs(10)1161 2832 y Fq(9)1181 2853 y Fr(:)p eop %%Page: 13 13 13 12 bop 664 -64 a Fo(THE)17 b(NOD)o(AL)f(SURF)l(A)o(CE)748 b(13)399 37 y Ft(Appendix)17 b Fs(A.)26 b Ft(A)19 b(numerical)e (calcula)m(tion)-73 124 y Fs(In)g(this)g(app)q(endix)h(w)o(e)f(ha)o(v)o (e)f(a)i(short)g(Maple)f(co)q(de)h(whic)o(h)f(giv)o(es)f(us)i(a)g (graph)g(where)f Fr(R)1600 131 y Fq(2)1637 124 y Fs(can)h(b)q(e)g(read) -123 182 y(o\013.)-50 341 y FA(>)29 345 y Fk(with\(plots)o(\):)-50 505 y FA(>)29 509 y Fk(e1:=diff\(u)o(\(x\))o(,x$)o(2\))o(+di)o(ff\()o (u\()o(x\),)o(x\)/)o(x+1)o(0.)o(232)o(5*u)o(\(x)k(\))j(=)h(0;)387 713 y Ff(e1)21 b Fs(:=)13 b(\()558 679 y Fr(@)587 661 y Fq(2)p 544 702 76 2 v 544 747 a Fr(@)s(x)601 733 y Fq(2)634 713 y Fs(u\()p Fr(x)p Fs(\)\))d(+)825 657 y FA(@)p 815 665 41 2 v 815 694 a(@)r(x)869 676 y Fs(u\()p Fr(x)p Fs(\))p 810 702 152 2 v 872 747 a Fr(x)978 713 y Fs(+)h(10)p Fr(:)p Fs(2325)d(u\()p Fr(x)p Fs(\))16 b(=)d(0)-50 1007 y FA(>)29 1011 y Fk(p:=dsolve\()o Fn(f)p Fk(e1,)o(u\()o(1\)=)o(0,D)o(\(u\))o(\(1)o(\)=1)o Fn(g)p Fk(,u)o(\(x\))o(,ty)o(pe=)o(n)22 b(umeric\);)588 1179 y Fr(p)14 b Fs(:=)g FB(pro)r(c)o Fs(\()p Ff(rkf45)p 937 1179 15 2 v 24 w(x)7 b Fs(\))16 b Fr(:)8 b(:)g(:)16 b FB(end)-50 1447 y FA(>)29 1451 y Fk(odeplot\(p,)o([x,)o(u\(x)o(\)])o (,1.)o(.2\))o(;)53 1455 y gsave currentpoint currentpoint translate 270 neg rotate neg exch neg exch translate 53 1455 a @beginspecial 0 @llx 0 @lly 612 @urx 792 @ury 3076 @rwi 3980 @rhi @setspecial %%BeginDocument: Bessel01.eps 20 dict begin gsave /m {moveto} def /l {lineto} def /C {setrgbcolor} def /Y /setcmykcolor where { %%ifelse Use built-in operator /setcmykcolor get }{ %%ifelse Emulate setcmykcolor with setrgbcolor { %%def 1 sub 3 { %%repeat 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor } bind } ifelse def /G {setgray} def /S {stroke} def /NP {newpath} def /P {gsave fill grestore reversepath stroke} def /stringbbox {gsave NP 0 0 m false charpath flattenpath pathbbox 4 2 roll pop pop 1.1 mul cvi exch 1.1 mul cvi exch grestore} def /thin 3 def /medium 7 def /thick 16 def /boundarythick 20 def %% thickness of bounding box 576 36 translate 90 rotate 0.108 0.108 scale 1 setlinejoin 1 setlinecap 0.0 setgray /inch {72 mul} def /fheight 0.35 inch neg def 0 G medium setlinewidth [] 0 setdash 0.3 G NP 769 810 m 873 1095 l 978 1374 l 1082 1644 l 1187 1906 l 1292 2158 l 1396 2400 l 1501 2631 l 1605 2850 l 1710 3057 l 1815 3251 l 1919 3431 l 2024 3597 l 2128 3749 l 2233 3886 l 2338 4007 l 2442 4113 l 2547 4204 l 2652 4279 l 2756 4338 l 2861 4381 l 2965 4409 l 3070 4421 l 3175 4418 l 3279 4399 l 3384 4365 l 3488 4317 l 3593 4255 l 3698 4179 l 3802 4089 l 3907 3987 l 4011 3873 l 4116 3747 l 4221 3610 l 4325 3463 l 4430 3307 l 4534 3142 l 4639 2968 l 4744 2788 l 4848 2602 l 4953 2410 l 5057 2213 l 5162 2012 l 5267 1809 l 5371 1604 l 5476 1397 l 5581 1190 l 5685 984 l 5790 779 l 5894 576 l S 0 G medium setlinewidth NP 666 500 m 666 4500 l S thin setlinewidth NP 645 528 m 687 528 l S NP 645 669 m 687 669 l S NP 623 810 m 709 810 l S NP 645 951 m 687 951 l S NP 645 1093 m 687 1093 l S NP 645 1234 m 687 1234 l S NP 645 1375 m 687 1375 l S NP 623 1516 m 709 1516 l S NP 645 1658 m 687 1658 l S NP 645 1799 m 687 1799 l S NP 645 1940 m 687 1940 l S NP 645 2081 m 687 2081 l S NP 623 2223 m 709 2223 l S NP 645 2364 m 687 2364 l S NP 645 2505 m 687 2505 l S NP 645 2646 m 687 2646 l S NP 645 2788 m 687 2788 l S NP 623 2929 m 709 2929 l S NP 645 3070 m 687 3070 l S NP 645 3211 m 687 3211 l S NP 645 3353 m 687 3353 l S NP 645 3494 m 687 3494 l S NP 623 3635 m 709 3635 l S NP 645 3776 m 687 3776 l S NP 645 3918 m 687 3918 l S NP 645 4059 m 687 4059 l S NP 645 4200 m 687 4200 l S NP 623 4341 m 709 4341 l S NP 645 4483 m 687 4483 l S medium setlinewidth NP 666 810 m 5999 810 l S thin setlinewidth NP 769 778 m 769 842 l S NP 974 794 m 974 826 l S NP 1179 794 m 1179 826 l S NP 1384 794 m 1384 826 l S NP 1589 794 m 1589 826 l S NP 1794 778 m 1794 842 l S NP 1999 794 m 1999 826 l S NP 2204 794 m 2204 826 l S NP 2409 794 m 2409 826 l S NP 2614 794 m 2614 826 l S NP 2819 778 m 2819 842 l S NP 3024 794 m 3024 826 l S NP 3229 794 m 3229 826 l S NP 3434 794 m 3434 826 l S NP 3639 794 m 3639 826 l S NP 3844 778 m 3844 842 l S NP 4049 794 m 4049 826 l S NP 4254 794 m 4254 826 l S NP 4459 794 m 4459 826 l S NP 4664 794 m 4664 826 l S NP 4869 778 m 4869 842 l S NP 5074 794 m 5074 826 l S NP 5279 794 m 5279 826 l S NP 5484 794 m 5484 826 l S NP 5689 794 m 5689 826 l S NP 5894 778 m 5894 842 l S medium setlinewidth /Courier findfont 115 scalefont setfont (0) dup stringbbox 810 exch sub exch 623 exch sub exch m show (0.05) dup stringbbox 2 idiv 1516 exch sub exch 623 exch sub exch m show (0.1) dup stringbbox 2 idiv 2223 exch sub exch 623 exch sub exch m show (0.15) dup stringbbox 2 idiv 2929 exch sub exch 623 exch sub exch m show (0.2) dup stringbbox 2 idiv 3635 exch sub exch 623 exch sub exch m show (0.25) dup stringbbox 2 idiv 4341 exch sub exch 623 exch sub exch m show (1) dup stringbbox 778 exch sub exch 2 idiv 769 exch sub exch m show (1.2) dup stringbbox 778 exch sub exch 2 idiv 1794 exch sub exch m show (1.4) dup stringbbox 778 exch sub exch 2 idiv 2819 exch sub exch m show (1.6) dup stringbbox 778 exch sub exch 2 idiv 3844 exch sub exch m show (1.8) dup stringbbox 778 exch sub exch 2 idiv 4869 exch sub exch m show (2) dup stringbbox 778 exch sub exch 2 idiv 5894 exch sub exch m show boundarythick setlinewidth /bd boundarythick 2 idiv def [] 0 setdash NP bd bd m bd 5000 bd sub l 6666 bd sub 5000 bd sub l 6666 bd sub bd l bd bd l S showpage grestore end %%EndDocument @endspecial 1335 1455 a currentpoint grestore moveto 1335 1455 a -50 2842 a FA(>)29 2846 y Fk(odeplot\(p,)o([x,)o(u\(x)o(\)])o(,1.)o(97.)o(.1)o(.98)o(\);)p eop %%Page: 14 14 14 13 bop -5 -64 a Fo(14)790 b(S\037REN)17 b(F)o(OURNAIS)171 37 y gsave currentpoint currentpoint translate 270 neg rotate neg exch neg exch translate 171 37 a @beginspecial 0 @llx 0 @lly 612 @urx 792 @ury 3076 @rwi 3980 @rhi @setspecial %%BeginDocument: Bessel02.eps 20 dict begin gsave /m {moveto} def /l {lineto} def /C {setrgbcolor} def /Y /setcmykcolor where { %%ifelse Use built-in operator /setcmykcolor get }{ %%ifelse Emulate setcmykcolor with setrgbcolor { %%def 1 sub 3 { %%repeat 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor } bind } ifelse def /G {setgray} def /S {stroke} def /NP {newpath} def /P {gsave fill grestore reversepath stroke} def /stringbbox {gsave NP 0 0 m false charpath flattenpath pathbbox 4 2 roll pop pop 1.1 mul cvi exch 1.1 mul cvi exch grestore} def /thin 3 def /medium 7 def /thick 16 def /boundarythick 20 def %% thickness of bounding box 576 36 translate 90 rotate 0.108 0.108 scale 1 setlinejoin 1 setlinecap 0.0 setgray /inch {72 mul} def /fheight 0.35 inch neg def 0 G medium setlinewidth [] 0 setdash 0.3 G NP 769 4421 m 873 4342 l 978 4264 l 1082 4185 l 1187 4107 l 1292 4028 l 1396 3949 l 1501 3871 l 1605 3792 l 1710 3714 l 1815 3635 l 1919 3556 l 2024 3478 l 2128 3399 l 2233 3321 l 2338 3242 l 2442 3164 l 2547 3085 l 2652 3007 l 2756 2928 l 2861 2850 l 2965 2771 l 3070 2693 l 3175 2614 l 3279 2536 l 3384 2457 l 3488 2379 l 3593 2300 l 3698 2222 l 3802 2143 l 3907 2065 l 4011 1987 l 4116 1908 l 4221 1830 l 4325 1751 l 4430 1673 l 4534 1595 l 4639 1516 l 4744 1438 l 4848 1359 l 4953 1281 l 5057 1203 l 5162 1124 l 5267 1046 l 5371 968 l 5476 890 l 5581 811 l 5685 733 l 5790 655 l 5894 576 l S 0 G medium setlinewidth NP 666 500 m 666 4500 l S thin setlinewidth NP 645 512 m 687 512 l S NP 645 620 m 687 620 l S NP 645 729 m 687 729 l S NP 623 837 m 709 837 l S NP 645 946 m 687 946 l S NP 645 1054 m 687 1054 l S NP 645 1162 m 687 1162 l S NP 645 1271 m 687 1271 l S NP 623 1379 m 709 1379 l S NP 645 1488 m 687 1488 l S NP 645 1596 m 687 1596 l S NP 645 1705 m 687 1705 l S NP 645 1813 m 687 1813 l S NP 623 1922 m 709 1922 l S NP 645 2030 m 687 2030 l S NP 645 2138 m 687 2138 l S NP 645 2247 m 687 2247 l S NP 645 2355 m 687 2355 l S NP 623 2464 m 709 2464 l S NP 645 2572 m 687 2572 l S NP 645 2681 m 687 2681 l S NP 645 2789 m 687 2789 l S NP 645 2898 m 687 2898 l S NP 623 3006 m 709 3006 l S NP 645 3115 m 687 3115 l S NP 645 3223 m 687 3223 l S NP 645 3331 m 687 3331 l S NP 645 3440 m 687 3440 l S NP 623 3548 m 709 3548 l S NP 645 3657 m 687 3657 l S NP 645 3765 m 687 3765 l S NP 645 3874 m 687 3874 l S NP 645 3982 m 687 3982 l S NP 623 4091 m 709 4091 l S NP 645 4199 m 687 4199 l S NP 645 4307 m 687 4307 l S NP 645 4416 m 687 4416 l S medium setlinewidth NP 666 1922 m 5999 1922 l S thin setlinewidth NP 769 1890 m 769 1954 l S NP 974 1906 m 974 1938 l S NP 1179 1906 m 1179 1938 l S NP 1384 1906 m 1384 1938 l S NP 1589 1906 m 1589 1938 l S NP 1794 1890 m 1794 1954 l S NP 1999 1906 m 1999 1938 l S NP 2204 1906 m 2204 1938 l S NP 2409 1906 m 2409 1938 l S NP 2614 1906 m 2614 1938 l S NP 2819 1890 m 2819 1954 l S NP 3024 1906 m 3024 1938 l S NP 3229 1906 m 3229 1938 l S NP 3434 1906 m 3434 1938 l S NP 3639 1906 m 3639 1938 l S NP 3844 1890 m 3844 1954 l S NP 4049 1906 m 4049 1938 l S NP 4254 1906 m 4254 1938 l S NP 4459 1906 m 4459 1938 l S NP 4664 1906 m 4664 1938 l S NP 4869 1890 m 4869 1954 l S NP 5074 1906 m 5074 1938 l S NP 5279 1906 m 5279 1938 l S NP 5484 1906 m 5484 1938 l S NP 5689 1906 m 5689 1938 l S NP 5894 1890 m 5894 1954 l S medium setlinewidth /Courier findfont 115 scalefont setfont (-0.002) dup stringbbox 2 idiv 837 exch sub exch 623 exch sub exch m show (-0.001) dup stringbbox 2 idiv 1379 exch sub exch 623 exch sub exch m show (0) dup stringbbox 1922 exch sub exch 623 exch sub exch m show (0.001) dup stringbbox 2 idiv 2464 exch sub exch 623 exch sub exch m show (0.002) dup stringbbox 2 idiv 3006 exch sub exch 623 exch sub exch m show (0.003) dup stringbbox 2 idiv 3548 exch sub exch 623 exch sub exch m show (0.004) dup stringbbox 2 idiv 4091 exch sub exch 623 exch sub exch m show (1.97) dup stringbbox 1890 exch sub exch 2 idiv 769 exch sub exch m show (1.972) dup stringbbox 1890 exch sub exch 2 idiv 1794 exch sub exch m show (1.974) dup stringbbox 1890 exch sub exch 2 idiv 2819 exch sub exch m show (1.976) dup stringbbox 1890 exch sub exch 2 idiv 3844 exch sub exch m show (1.978) dup stringbbox 1890 exch sub exch 2 idiv 4869 exch sub exch m show (1.98) dup stringbbox 1890 exch sub exch 2 idiv 5894 exch sub exch m show boundarythick setlinewidth /bd boundarythick 2 idiv def [] 0 setdash NP bd bd m bd 5000 bd sub l 6666 bd sub 5000 bd sub l 6666 bd sub bd l bd bd l S showpage grestore end %%EndDocument @endspecial 1453 37 a currentpoint grestore moveto 1453 37 a 855 1365 a Ft(References)-5 1444 y Fz([Bas95])112 b(Ric)o(hard)13 b(F.)g(Bass,)i Fp(Pr)n(ob)n(abilistic)e(Te)n(chniques)j(in)e(Analysis)p Fz(,)g(Spinger)g(V)m(erlag,)f(1995.)-5 1494 y([Da)o(v89])104 b(E.)13 b(B.)h(Da)o(vies,)f Fp(He)n(at)i(kernels)f(and)i(sp)n(e)n(ctr)n (al)e(the)n(ory)p Fz(,)f(Cam)o(bridge)f(Univ)o(ersit)o(y)i(Press,)h (1989.)-5 1543 y([GT83])115 b(D.)14 b(Gilbarg)g(and)i(N.S.)f(T)m (rudinger,)g Fp(El)r(liptic)g(Partial)h(Di\013er)n(ential)g(Equations)h (of)g(Se)n(c)n(ond)g(Or)n(der)p Fz(,)e(2nd)g(ed.,)239 1593 y(Grundlehren)f(der)h(mathematisc)o(hen)d(Wissensc)o(haften,)j (Springer)f(V)m(erlag,)f(1983.)-5 1643 y([HOHON98])21 b(M.)13 b(Ho\013mann-Ostenhof,)f(T.)h(Ho\013mann-Ostenhof,)g(and)g(N.)h (Nadirash)o(vili,)d Fp(The)k(no)n(dal)g(line)g(of)f(the)h(se)n(c)n(ond) 239 1693 y(eigenfunction)e(of)g(the)g(Laplacian)g(in)g Fx(R)881 1678 y Fc(2)912 1693 y Fp(c)n(an)g(b)n(e)g(close)n(d)p Fz(,)f(Duk)o(e)f(Math.)g(Journal)h Fx(90)f Fz(\(1998\),)g(no.)g(3,)g (631{640.)-5 1743 y([L)m(Y83])124 b(P)m(.)12 b(Li)h(and)g(S.)g(Y)m(au,) g Fp(On)h(the)h(Schr\177)-21 b(odinger)15 b(Equation)g(and)g(the)g (Eigenvalue)g(Pr)n(oblem)p Fz(,)d(Comm.)e(Math.)j(Ph)o(ys.)239 1792 y(\(1983\),)f(no.)h(88,)g(309{318.)-5 1842 y([Mel92])110 b(A.)9 b(D.)g(Melas,)h Fp(On)h(the)g(no)n(dal)g(line)g(of)g(the)g(se)n (c)n(ond)g(eigenfunction)h(of)f(the)g(Laplacian)g(in)g Fx(R)1655 1827 y Fc(2)1674 1842 y Fz(,)f(J.)f(Di\013.)g(Geometry)239 1892 y(\(1992\),)j(no.)h(35,)g(255{263.)-5 1942 y([P)o(a)o(y67])109 b(L.)13 b(E.)g(P)o(a)o(yne,)g Fp(Isop)n(erimetric)g(ine)n(qualities)h (and)i(their)d(applic)n(ations)p Fz(,)h(SIAM)f(Review)h Fx(9)g Fz(\(1967\),)e(no.)h(3,)g(453{)239 1992 y(488.)-5 2042 y([PS78])127 b(S.)11 b(C.)h(P)o(ort)g(and)g(C.)g(J.)g(Stone,)g Fp(Br)n(ownian)i(motion)f(and)h(classic)n(al)f(p)n(otential)g(the)n (ory)p Fz(,)f(Academic)g(Press,)h(1978.)-5 2091 y([Sto95])118 b(P)m(.)11 b(Stollmann,)e Fp(A)k(c)n(onver)n(genc)n(e)g(the)n(or)n(em)g (for)f(Dirichlet)g(forms)h(with)f(applic)n(ations)h(to)g(b)n(oundary)h (value)f(pr)n(ob-)239 2141 y(lems)h(with)g(varying)h(domains)p Fz(,)f(Math.)g(Z.)f Fx(219)h Fz(\(1995\),)e(no.)i(2,)f(275{287.)-5 2191 y([Uhl72])112 b(K.)13 b(Uhlen)o(b)q(ec)o(k,)i Fp(Eigenfunctions)g (of)g(Laplac)n(e)g(op)n(er)n(ators)p Fz(,)e(Bull.)g(AMS)h Fx(78)g Fz(\(1972\),)e(no.)i(6,)f(1073{1076.)45 2284 y Fp(E-mail)h(addr)n(ess)s Fz(:)19 b Fa(fournais@imf.au.d)o(k)45 2369 y Fy(Dep)m(ar)m(tment)e(of)f(Ma)m(thema)m(tics,)h(Ny)e(Munkegade,) j(8000)e(Aarhus)g(C,)g(Denmark)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9905120939502--