Content-Type: multipart/mixed; boundary="-------------9904211331363" This is a multi-part message in MIME format. ---------------9904211331363 Content-Type: text/plain; name="99-123.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-123.comments" Jarvenpaa=J\"arvenp\"a\"a ---------------9904211331363 Content-Type: text/plain; name="99-123.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-123.keywords" fractals, porosity, measures ---------------9904211331363 Content-Type: application/postscript; name="texas.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="texas.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.47 Copyright 1986-91 Radical Eye Software %%Title: preprint.dvi %%Pages: 11 1 %%BoundingBox: 0 0 596 843 %%DocumentFonts: Times-Bold Times-Roman Times-Italic %%EndComments %%BeginProcSet: texc.pro /TeXDict 200 dict def TeXDict begin /N /def load def /B{bind def}N /S /exch load def /X{S N}B /TR /translate load N /isls false N /vsize 10 N /@rigin{ isls{[0 1 -1 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale Resolution VResolution vsize neg mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@letter{/vsize 10 N}B /@landscape{/isls true N /vsize -1 N}B /@a4{/vsize 10.6929133858 N}B /@a3{ /vsize 15.5531 N}B /@ledger{/vsize 16 N}B /@legal{/vsize 13 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 add]/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}bind{adv 1 chg nd}bind{1 add chg}bind{1 add chg nd}bind{adv lsh}bind{ adv lsh nd}bind{adv rsh}bind{adv rsh nd}bind{1 add adv}bind{/rc X nd}bind{1 add set}bind{1 add clr}bind{adv 2 chg}bind{adv 2 chg nd}bind{pop nd}bind]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}N /eop{clear SI restore showpage userdict /eop-hook known{eop-hook}if}N /@start{userdict /start-hook known{start-hook}if /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}N /p /show load 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 statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval (Display)eq}{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 /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{clear SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro TeXDict begin /rf{655360 div mul Resolution mul 7227 div /PixPerEm X findfont dup length 1 add dict /nn X{1 index /FID ne{nn 3 1 roll put}{pop pop}ifelse} forall 256 dict begin nn /Encoding get 0 1 255{2 copy get 3 index 2 index get 1000 mul PixPerEm div def pop}for pop pop nn /Metrics currentdict put end /fontname X /nn dup nn definefont[PixPerEm 0 0 PixPerEm neg 0 0]makefont N fontname{/foo setfont}2 array copy cvx N fontname load 0 nn put}N /ObliqueSlant{dup sin S cos div neg}B /SlantFont{/foo X[1 0 foo 1 0 0] TransFont}N /ExtendFont{/foo X 3 2 roll[S{foo div}forall]3 1 roll[foo 0 0 1 0 0]TransFont}N /TransFont{S findfont S makefont dup length dict /nn X{1 index /FID ne{nn 3 1 roll put}{pop pop}ifelse}forall dup nn definefont pop}N 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 false N /BBcalc false N /p 3 def}B /@scaleunit 100 N /@hscale{@scaleunit div /hsc X}B /@vscale{ @scaleunit div /vsc X}B /@hsize{/hs X /CLIP true N}B /@vsize{/vs X /CLIP true N}B /@hoffset{/ho X}B /@voffset{/vo X}B /@angle{/ang X}B /@rwi{10 div /rwi X} B /@llx{/llx X}B /@lly{/lly X}B /@urx{/urx X}B /@ury{/ury X /BBcalc true N}B /magscale true def end /@MacSetUp{userdict /md known{userdict /md get type /dicttype eq{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}N /psfts{S 65536 div N}N /startTexFig{/psf$SavedState save N userdict maxlength dict begin /magscale false def normalscale currentpoint TR /psf$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}N /@setspecial{CLIP{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 BBcalc{rwi urx llx sub div dup scale llx neg lly neg TR}if /showpage{}N /erasepage{}N /copypage{}N newpath}N /@endspecial{grestore clear 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 1000 600 600 @start /Fa 4 51 df<130C1338137013E0EA01C0EA03801300 5A120EA25AA25AA312781270A312F0AB1270A312781238A37EA27EA27E7E1380EA01C0EA00E013 701338130C0E317AA418>40 D<12C012707E7E7E7E7E1380EA01C0A2EA00E0A21370A313781338 A3133CAB1338A313781370A313E0A2EA01C0A2EA038013005A120E5A5A5A12C00E317CA418>I< 13E01201120712FF12F91201B3A7487EB512C0A212217AA01E>49 D I E /Fb 2 92 df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c [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 23 29 34 34 57 53 23 23 23 34 46 17 23 17 19 34 34 34 34 34 34 34 34 34 34 23 23 46 46 46 34 63 42 42 45 49 42 42 49 49 23 30 45 38 57 45 49 42 49 42 34 38 49 42 57 42 38 38 26 19 26 29 34 23 34 34 30 34 30 19 34 34 19 19 30 19 49 34 34 34 34 26 26 19 34 30 45 30 30 26 27 19 27 37 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 26 34 34 11 34 34 34 34 15 38 34 23 23 34 34 0 34 34 34 17 0 36 24 23 38 38 34 60 68 0 34 0 23 23 23 23 23 23 23 23 0 23 23 0 23 23 23 60 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 60 0 19 0 0 0 0 38 49 64 21 0 0 0 0 0 45 0 0 0 19 0 0 19 34 45 34 0 0 0 0 ] /Times-Italic 1000 536560 rf /Fd 3 111 df<131FEBFF8C3801E0DE3803807E38 07007C48133C121E123E003C5B127CA3485BA215401560903801E0C012781303393807E180391C 1CF300380FF87F3807E03C1B177E9522>97 D<1338137CA2137813701300A7EA0780EA1FC0EA38 E01230EA60F0EAC1E0A3EA03C0A3EA0780A2EA0F0013041306EA1E0CA21318121CEA1E70EA0FE0 EA07800F237DA116>105 D<000F13FC381FC3FF3931C707803861EC0301F813C0EAC1F0A213E0 3903C00780A3EC0F00EA0780A2EC1E041506D80F00130C143C15181538001EEB1C70EC1FE0000C EB07801F177D9526>110 D E /Fe 27 121 df<1618163CA2167EA216FFA24B7FA24B6C7EA292 38063FE0A24B6C7EA24B6C7EA292383807FC153092387003FE15609238E001FF15C002016D7F5D 02036E7E92C7FC4A6E7E1406020E6E7E140C021C6E7E141802386E7E143002706E7E146002E06E 7E5C01016F7F5C0103707E91C9FC183F010683181F4983180F49831807498318034983A249707E A24848701380A248CBEA7FC0A20006F03FE0A248F01FF0A2001FBA12F8A24819FCA24819FEA2BC FC48477CC651>1 D<140C141C1438147014E0EB01C01303EB0780EB0F00A2131E5BA25B13F85B 12015B1203A2485AA3485AA348C7FCA35AA2123EA2127EA4127CA312FCB3A2127CA3127EA4123E A2123FA27EA36C7EA36C7EA36C7EA212017F12007F13787FA27F7FA2EB0780EB03C01301EB00E0 14701438141C140C166476CA26>40 D<12C07E12707E7E7E120F6C7E6C7EA26C7E6C7EA2137813 7C133C133E131E131FA2EB0F80A3EB07C0A3EB03E0A314F0A21301A214F8A41300A314FCB3A214 F8A31301A414F0A21303A214E0A3EB07C0A3EB0F80A3EB1F00A2131E133E133C137C13785BA248 5A485AA2485A48C7FC120E5A5A5A5A5A16647BCA26>I<16C04B7EB3AB007FBAFCBB1280A26C19 00C8D801E0C9FCB3AB6F5A41407BB84C>43 D<14FF010713E090381F81F890383E007C01FC133F 4848EB1F8049130F4848EB07C04848EB03E0A2000F15F0491301001F15F8A2003F15FCA390C8FC 4815FEA54815FFB3A46C15FEA56D1301003F15FCA3001F15F8A26C6CEB03F0A36C6CEB07E00003 15C06D130F6C6CEB1F806C6CEB3F00013E137C90381F81F8903807FFE0010090C7FC28447CC131 >48 D<143014F013011303131F13FFB5FC13E713071200B3B3B0497E497E007FB6FCA3204278C1 31>II<49B4FC010F13E0013F13FC9038FE01FE3A01F0007F80D803C0EB3F C048C7EA1FE0120EED0FF0EA0FE0486C14F8A215077F5BA26C48130FEA03C0C813F0A3ED1FE0A2 ED3FC01680ED7F0015FE4A5AEC03F0EC1FC0D90FFFC7FC15F090380001FCEC007FED3F80ED1FC0 ED0FE016F0ED07F816FC150316FEA2150116FFA3121EEA7F80487EA416FE491303A2007EC713FC 00701407003015F80038140F6C15F06CEC1FE06C6CEB3FC0D803E0EB7F803A01FE01FE0039007F FFF8010F13E0010190C7FC28447CC131>I<121EEA7F80A2EAFFC0A4EA7F80A2EA1E00C7FCB3A5 121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A2B78AA1B>58 D<007FBAFCBB1280A26C1900CEFCB0 007FBAFCBB1280A26C190041187BA44C>61 D91 D93 D97 D<167FED3FFFA315018182B3EC7F80903803FFF090380FC07C 90383F000E017E1307496D5AD803F87F48487F5B000F81485AA2485AA2127FA290C8FC5AAB7E7F A2123FA26C7EA2000F5D7F6C6C5B00035C6C6C9038077F806C6C010E13C0013F011C13FE90380F C0F8903803FFE09026007F0013002F467DC436>100 D102 DI105 D108 DI<3901FC01FE00FF903807 FFC091381E07F091383801F8000701707F0003EBE0002601FDC07F5C01FF147F91C7FCA25BA35B B3A8486CECFF80B5D8F83F13FEA32F2C7DAB36>II<3901FC03FC00FF90 380FFF8091383C07E091387001F83A07FDE000FE00010180137F01FFEC3F8091C7EA1FC04915E0 49140F17F0160717F8160317FCA3EE01FEABEE03FCA3EE07F8A217F0160F6D15E0EE1FC06D143F 17806EEB7E00D9FDC05B9039FCF003F891383C0FE091381FFF80DA03FCC7FC91C9FCAE487EB512 F8A32F3F7DAB36>I<3903F803F000FFEB1FFCEC3C3EEC707F0007EBE0FF3803F9C000015B13FB EC007E153C01FF13005BA45BB3A748B4FCB512FEA3202C7DAB26>114 D<90383FE0183901FFFC 383907E01F78390F0003F8001E1301481300007C1478127800F81438A21518A27EA27E6C6C1300 6C7E13FC383FFFE06C13FC6C13FF6C14C06C14E0C614F0011F13F81300EC0FFC140300C0EB01FE 1400157E7E153EA27EA36C143C6C147C15786C14F86CEB01F039F38003E039F1F00F8039E07FFE 0038C00FF01F2E7DAC26>I<1306A5130EA4131EA3133E137EA213FE12011207001FB512F0B6FC A2C648C7FCB3A4150CAA017E131C017F1318A26D133890381F8030ECC070903807E0E0903801FF C09038007F001E3E7EBC26>II120 D E /Ff 1 83 df82 D E /Fg 20 121 df<13031307130C131C1338137013E0A2EA01C0EA0380A2EA0700A25A120E12 1EA2121C123CA35AA512F8A25AAB7EA21278A57EA3121C121EA2120E120F7EA2EA0380A2EA01C0 EA00E0A213701338131C130C1307130310437AB11B>40 D<7E7E126012707E7E7EA27EEA0380A2 EA01C0A213E0120013F0A213701378A3133CA5133EA2131EAB133EA2133CA51378A3137013F0A2 13E0120113C0A2EA0380A2EA0700120EA25A5A5A12605A5A0F437BB11B>I48 D<130C133C137CEA03FC12FFEAFC7C1200B3B113FE387FFFFEA2172C7AAB23>III<140EA2141E 143E147EA214FE130114BEEB033E13071306130C131C131813301370136013C012011380EA0300 5A120E120C121C5A12305A12E0B612FCA2C7EA3E00A9147F90381FFFFCA21E2C7EAB23>I<000C EB0180380FC01F90B512005C5C14F014C0D80C7EC7FC90C8FCA8EB1FC0EB7FF0380DE07C380F80 1E380E000F000C14800008EB07C0C7FC15E01403A215F0A31238127C12FCA215E05A0060130715 C06CEB0F8015006C131E000E5B380780F83801FFE038007F801C2D7DAB23>II57 D61 D72 D97 D<15F890383F03FC3901FFE70E3903C0FC1E38078078390F003C0C 48EB3E00001E131E003E131FA6001E131E001F133E6C133C6C6C5A6D5A380DFFE0D81C3FC7FC90 C8FCA47E380FFFFC6CEBFF806C14C0000F14E0391E0007F00038EB00F80078147848143CA50078 14786C14F0001FEB03E03907E01F803901FFFE0038003FF01F2C7E9D23>103 D108 D<2607C07FEB07F03BFF C3FFC03FFC903AC781E0781E3B0FCC00F0C00FD807D8D9F980138001F090397F0007C049137EA2 49137CB2486C01FEEB0FE03CFFFE0FFFE0FFFEA2371D7E9C3C>I 111 D<3807C0FE39FFC7FF809038CF03E0390FD800F0D807F0137849137C497F811680A2150F16 C0A71680151F16005D153E6D5B6D5B9038D801F09038CE07E09038C7FF80D9C1FCC7FC01C0C8FC AA487EEAFFFEA2222A7E9C27>I<380781F838FF83FCEB8E1E380F9C3FEA079813B0141EEBE000 A35BB0487EB5FCA2181D7E9C1C>114 D<3AFFFC07FF80A23A07F003FC003903E001E0D801F05B 01F85B000049C7FCEB7C06EB3E0E6D5A14B8EB0FB0EB07E013036D7E497E1307EB067C497EEB1C 1F01387FEB700F90386007C0496C7E00016D7E000780001F497E3AFFF007FFC0A2221D7F9C25> 120 D E /Fh 17 123 df26 D34 D<123C127E12FFA4127E123C08087A8714>58 D<123C127EB4FCA21380A2127F123D1201A312031300A25A1206120E5A5A5A126009157A8714> I<1670A216F01501A24B7EA21507150DA2151915391531ED61FC156015C0EC0180A2EC03005C14 064A7F167E5C5CA25C14E05C4948137F91B6FC5B0106C7123FA25B131C1318491580161F5B5B12 0112031207000FED3FC0D8FFF8903807FFFEA22F2F7DAE35>65 D<013FB6FC17C0903A00FE0007 F0EE01F84AEB00FC177E1301177F5CA21303177E4A14FEA20107EC01FC17F84AEB03F0EE07E001 0FEC1FC0EE7F009138C003FC91B55A4914FE9139C0003F804AEB0FC017E0013F140717F091C7FC 16035BA2017E1407A201FE15E0160F4915C0161F0001ED3F80EE7F004914FEED03F80003EC0FF0 B712C003FCC7FC302D7CAC35>I97 D<1307EB0F80EB1FC0A2EB0F80EB070090C7FCA9EA01E0EA07F8EA0E3CEA1C3E123812301270EA 607EEAE07C12C013FC485A120012015B12035BA21207EBC04014C0120F13801381381F01801303 EB0700EA0F06131EEA07F8EA01F0122E7EAC18>105 D<15E0EC01F01403A3EC01C091C7FCA914 7CEB03FE9038078F80EB0E07131C013813C01330EB700F0160138013E013C0EB801F13001500A2 5CA2143EA2147EA2147CA214FCA25CA21301A25CA21303A25CA2130700385BEAFC0F5C49C7FCEA F83EEAF0F8EA7FF0EA1F801C3B81AC1D>I<131FEA03FFA2EA003FA2133EA2137EA2137CA213FC A25BA2120115F89038F003FCEC0F0E0003EB1C1EEC387EEBE07014E03807E1C09038E3803849C7 FC13CEEA0FDC13F8A2EBFF80381F9FE0EB83F0EB01F81300481404150C123EA2007E141C151800 7CEBF038ECF83000FC1470EC78E048EB3FC00070EB0F801F2F7DAD25>I<3907C007E0391FE03F F83918F8783E393879E01E39307B801F38707F00126013FEEAE0FC12C05B00815C0001143E5BA2 0003147E157C5B15FC0007ECF8081618EBC00115F0000F1538913803E0300180147016E0001F01 0113C015E390C7EAFF00000E143E251F7E9D2B>110 D<90387C01F89038FE07FE3901CF8E0F3A 03879C0780D907B813C0000713F000069038E003E0EB0FC0000E1380120CA2D8081F1307120014 00A249130F16C0133EA2017EEB1F80A2017C14005D01FC133E5D15FC6D485A3901FF03E09038FB 87C0D9F1FFC7FCEBF0FC000390C8FCA25BA21207A25BA2120FA2EAFFFCA2232B829D24>112 D<903807E03090381FF87090387C1CF0EBF80D3801F00F3903E007E0EA07C0000F1303381F8007 15C0EA3F00A248130F007E1480A300FE131F481400A35C143E5A147E007C13FE5C1301EA3E07EA 1F0E380FFCF8EA03F0C7FC13015CA313035CA21307A2EBFFFEA21C2B7D9D20>I<3807C01F390F F07FC0391CF8E0E0383879C138307B8738707F07EA607E13FC00E0EB03804848C7FCA212811201 5BA21203A25BA21207A25BA2120FA25BA2121FA290C8FC120E1B1F7E9D20>II<013F137C9038FFC1FF3A01C1E383803A0380F703C0390700F60F000E13 FE4813FC12180038EC0700003049C7FCA2EA200100005BA313035CA301075B5D14C000385CD87C 0F130600FC140E011F130C011B131C39F03BE038D8707113F0393FE0FFC0260F803FC7FC221F7E 9D28>120 D<011E1330EB3F809038FFC07048EBE0E0ECF1C03803C0FF9038803F809038000700 48130EC75A5C5C5C495A495A49C7FC131E13385B491340484813C0485A38070001000EEB038038 0FE007391FF81F0038387FFF486C5A38601FFC38E00FF038C003C01C1F7D9D21>122 D E /Fi 5 50 df0 D<170EA3170F8384170384170184717E1878187C 84180FF007C0BA12F819FC19F8CBEA07C0F00F00183E601878604D5A60170360170795C7FC5F17 0EA33E237CA147>33 D35 D<137813FE1201A3120313FCA3EA07F8A313F0A2EA0FE0A313C0121F1380A3EA3F00A3123E127E 127CA35AA35A0F227EA413>48 DI E /Fj 31 123 df11 DI<1578913807FFE0021F13FC91383C7FFEEC7007EC60 03ECE0004A13381600A280A380A280147CA2147E143E143F816E7EA26E7E81140781EC3FFC14FF 903803E1FEEB07C190381F00FF133E49EB7F805B0001143F485A484814C049131F120F485AA248 C7FC150F5A127EA300FEEC1F805AA316005A5DA2153E157E157CA26C5C127C4A5A6C495AA26C49 5A6C6C485A6C6C48C7FC3803E07C3800FFF0EB1FC027487CC62B>14 D<147002F8140E0101153F A301035DA24A147EA2010715FEA24A5CA2010F1401A24A5CA2011F1403A24A5CA2013F1407A291 C75BA249140FA2017E5DA201FE021F1318183849ED8030A20001033F13701860EE7F005E486C16 E0DB01DF13C09238039F016DD9071F1380489039801E0F83903BF7C078078700903AE1FFE003FE 903AE07F8000F8000F90CAFCA25BA2121FA25BA2123FA290CBFCA25AA2127EA212FEA25A123835 417DAB3B>22 DI26 D34 D<121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A0A 78891B>58 D<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413E013C0A312011380120313 005A1206120E5A5A5A12600B1D78891B>II<1618163C167CA2167816F8A216F01501A216E01503 A216C01507A21680150FA2ED1F00A2151E153EA2153C157CA2157815F8A25D1401A24A5AA25D14 07A25D140FA292C7FC5CA2141E143EA2143C147CA25CA25C1301A25C1303A25C1307A25C130FA2 91C8FC5BA2133EA2133C137CA2137813F8A25B1201A25B1203A2485AA25B120FA290C9FC5AA212 1E123EA2123C127CA2127812F8A25A126026647BCA31>I<127012FCB4FCEA7FC0EA1FF0EA07FC EA01FF38007FC0EB1FF0EB07FE903801FF809038007FE0EC1FF8EC03FE913800FF80ED3FE0ED0F F8ED03FF030013C0EE3FF0EE0FFCEE01FF9338007FC0EF1FF0EF07FCEF01FF9438007FC0F01FE0 A2F07FC0943801FF00EF07FCEF1FF0EF7FC04C48C7FCEE0FFCEE3FF0EEFFC0030390C8FCED0FF8 ED3FE0EDFF80DA03FEC9FCEC1FF8EC7FE0903801FF80D907FECAFCEB1FF0EB7FC04848CBFCEA07 FCEA1FF0EA7FC048CCFC12FC12703B3878B44C>I<1830187018F0A217011703A24D7EA2170F17 1FA21737A2176717E717C793380187FCA2EE0307EE07031606160CA216181638163004607FA216 C0030113011680ED0300A21506150E150C5D845D03707F15605DA24A5A4AB7FCA25C0206C87F5C 021C157F14185CA25C14E05C495A8549C9FC49163F1306130E5B133C137C01FE4C7ED807FFED01 FF007F01F0027FEBFFC0B5FC5C42477DC649>65 D<91B87E19F019FC02009039C00003FF6F4801 00138003FFED3FC01AE093C8121FF10FF0A24A17F84B1507A314035D190FA2020717F04B151F1A E0193F020F17C04BED7F80F1FF004E5A021F4B5A4B4A5AF01FF0F03FC0023F4AB4C7FC4BEB1FFC 92B612F018FEDA7FC0C7EA7F804BEC1FC0F00FF0727E02FF6F7E92C8FC727EA249835CA313035C A301075F4A1503A24E5A130F4A4B5A4E5AA2011F4C5A4A4B5A4D485A013F4B48C7FCEF0FFC4AEC 3FF801FF913801FFE0B9128005FCC8FC17C045447CC34A>I<4CB46C1318043F01F013384BB512 FC0307D9007E1378DB1FF090380F80F0DB7F80EB03C1DA01FEC7EA01C34A48EC00E7DA0FF0ED7F E04A48153F4A5A02FFC9121F494817C04948160F495A130F4A178049481607495A137F49481700 91CAFC5A485A1906485AA2485A96C7FC121F5BA2123F5BA3127F5BA4485AA419C0A2180161127F 180396C7FC6018066C6C160E601818001F17386D5E000F5F6D4B5A6C6C4B5A00034CC8FC6C6C15 0E6C6C153C017F5DD93FC0EB01E0D91FF0EB0FC0D907FE017FC9FC0101B512FCD9003F13E00207 90CAFC45487CC546>I<91B87E19F019FC02009039C00007FF6F489038007FC003FFED1FE0737E 93C86C7E737E19014A707E5D1A7FA20203EF3F805DA21BC014075DA3140F4B17E0A3141F4B17C0 A3143F4B167FA3027F18804B16FFA302FF180092C95A62A24917034A5F19076201034D5A5C4F5A 620107173F4A5F4FC7FC19FE010F4C5A4A15034E5AF00FE0011F4C5A4A4B5A06FFC8FC013FED01 FCEF0FF84AEC3FE001FF913803FF80B848C9FC17F094CAFC4B447CC351>I97 DIII<141E143F5C5CA3147E14 3891C7FCAE133EEBFF803801C3C0380781E0380601F0120E121CEA180312381230A2EA70070060 5BA2EAE00F00C05BEA001F5CA2133F91C7FCA25B137E13FE5BA212015BEC03800003140013F012 07495A1406140E140CEBC01C141814385C00035BEBE1C0C6B45A013EC7FC19437DC121>105 D<163C16FEA21501A316FCED00701600AE15FCEC03FF91380F0780021C13C091383803E0147014 E014C01301EC8007130314005B0106130F130E010C14C090C7FC151FA21680A2153FA21600A25D A2157EA215FEA25DA21401A25DA21403A25DA21407A25DA2140FA25DA2141F5DA2143F001C91C7 FC127F48137E5CA248485AEB03E038F807C038781F80D83FFEC8FCEA07F0275681C128>I<14FE 137FA3EB01FC13001301A25CA21303A25CA21307A25CA2130FA25CA2131FA25C163F013FECFFC0 923803C0E09138000703ED1E0F491338ED701F017E13E0EC01C001FE018013C00203EB07004948 C8FC140E00015B5C495A5C3803FBC001FFC9FC8014F83807F1FE9038F03F809038E00FE06E7E00 0F130381EBC001A2001FED01C017801380A2003F15031700010013F05E481506160E007E150C16 1C00FE01005BED787048EC3FE00038EC0F802B467BC433>I<01F8EB03FCD803FEEB1FFFD8071F 90387C0FC03B0E0F80E007E03A0C07C3C003001CD9C7007F001801CE1301003801DC80003013D8 EB0FF800705B00605BA200E0491303D8C01F5D5C12001607013F5D91C7FCA2160F495D137E161F 5F13FE49143F94C7FC187000014B136049147E16FE4C13E0000317C049150104F8138017030007 1700495D170EEE781C000FED7C3849EC1FF0D80380EC07C0342D7DAB3A>110 D112 D<91380FC00391383FF0 079138F83C0F903903E00E1E90390FC0063E90381F800790393F00037E4914FC01FE1301485AA2 484814F812075B000F140316F0485AA2003F14074914E0A3007F140F4914C0A3151F90C713805A A2153F6C1500A2127E5D007F14FE6C1301A214036C6C485A000F131E3807C0383803E0F13901FF C1F838003F01130014035DA314075DA3140F5DA2141FA2143F011FB51280A21600283F7DAB2B> I<01F8EB0FC0D803FEEB7FF0D8070FEBF038000E903883C07C3A0C07C701FC001C13CE0018EBDC 03003813D8003013F8D90FF013F800709038E000E0006015005C12E0EAC01F5C1200A2133F91C8 FCA35B137EA313FE5BA312015BA312035BA312075BA3120F5BEA0380262D7DAB2C>II<02FCEB07E0903A03FF801FFC903A0F07C0781E903A1C03E0E01F903A38 01F1C07FD9700013804901FB13FF4848EBFF00495B000316FE90C71438484A130012061401000E 5C120CC7FC14035DA314075DA3140F5DA3021F143817305D1770023F1460121E003F16E0267F80 7FEB01C0026F148000FF01EF1303D901CFEB070000FE903887C00E267C03835B3A3C0F01E0783A 1FFC00FFE0D803F0EB3F80302D7EAB37>120 D<133ED9FF8014E02603C3C0EB03F0380703E038 0601F0000E1507001C16E0EA180312380030150F007016C0EA60075C161FD8E00F158000C05BEA 001F4A133F1700133F91C7FC5E49147E137EA216FE01FE5C5BA215015E485AA215035EA2000014 07150F6D5C017C131F153F6D13FF90391F03CFC0903807FF8F903801FC0F90C7121F5EA2153F93 C7FCD807C05BD81FE0137E5DA24848485A4A5A01805B39380007C00018495A001C49C8FC6C137C 380781F83803FFE0C66CC9FC2C407DAB30>I<027CEB018049B413034901801300010F6D5A49EB E00E6F5A90393F03F838903978007EF80170EB1FF00160EB01E001E05C49495A90C748C7FC150E 5D5D5D5D4A5A4A5A4AC8FC140E5C5C5C5CEB03C049C9FC130E49141C4914185B49143848481430 491470D8039014F048B4495A3A0FEFC007C0391E03F01FD81C01B55A486C91C7FC485C00606D5A 00E0EB3FF048EB0FC0292D7CAB2D>I E /Fk 15 111 df<007FB912E0BA12F0A26C18E03C0478 9A4D>0 D<121FEA3F80EA7FC0EAFFE0A5EA7FC0EA3F80EA1F000B0B789E1C>I<49B4FC010F13E0 013F13F8497F48B6FC4815804815C04815E04815F0A24815F84815FCA3B712FEAA6C15FCA36C15 F86C15F0A26C15E06C15C06C15806C15006C6C13FC6D5B010F13E0010190C7FC27267BAB32>15 D<007FBA1280BB12C0A26C1980CEFCB0007FBA1280BB12C0A26C1980CEFCB0007FBA1280BB12C0 A26C1980422C7BAE4D>17 D<19E0F003F0180FF03FE0F0FF80943803FE00EF0FF8EF3FE0EFFF80 DC03FEC7FCEE0FF8EE3FE0EEFF80DB03FEC8FCED1FF8ED7FE0913801FF80DA07FEC9FCEC1FF0EC 7FC04948CAFCEB07FCEB1FF0EB7FC04848CBFCEA07FCEA1FF0EA7FC048CCFCA2EA7FC0EA1FF0EA 07FCEA01FF38007FC0EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC07FC913801FF809138007FE0 ED1FF8ED07FE923800FF80EE3FE0EE0FF8EE03FE933800FF80EF3FE0EF0FF8EF03FE943800FF80 F03FE0F00FF01803F000E01900B0007FB912E0BA12F0A26C18E03C4E78BE4D>20 D<127012FCB4FCEA7FC0EA1FF0EA07FCEA01FF38007FC0EB1FF0EB07FCEB01FF9038007FC0EC1F F0EC07FC913801FF809138007FE0ED1FF8ED07FE923800FF80EE3FE0EE0FF8EE03FE933800FF80 EF3FE0EF0FF8EF03FE943800FF80F03FE0F00FF0A2F03FE0F0FF80943803FE00EF0FF8EF3FE0EF FF80DC03FEC7FCEE0FF8EE3FE0EEFF80DB03FEC8FCED1FF8ED7FE0913801FF80DA07FEC9FCEC1F F0EC7FC04948CAFCEB07FCEB1FF0EB7FC04848CBFCEA07FCEA1FF0EA7FC048CCFC12FC1270CDFC B0007FB912E0BA12F0A26C18E03C4E78BE4D>I<037FB612E00207B712F0143F91B812E0010301 C0C9FCD907FCCAFCEB0FE0EB3F8049CBFC13FC485A485A485A5B485A121F90CCFC123EA2123C12 7CA2127812F8A25AA87EA21278127CA2123C123EA27E7F120F6C7E7F6C7E6C7E6C7E137E6D7EEB 1FE0EB07FC6DB47E010090B712E0023F16F01407020016E03C3A78B54D>26 D<49B4EF3FC0010F01E0923803FFF8013F01FC030F13FE4901FF92383FE01F48B66C91397E0007 C02603F80301E0D901F8EB01E02807E0007FF049486D7E01806D6CD907C0147048C76C6C494880 001EDA07FE49C87E001C6E6C013E150C486E6D48150E71481506486E01E0160793387FF1F00060 92263FF3E08193381FFBC000E004FF1780486F4915017090C9FC82707F8482717E844D7E6C4B6D 1503006004EF1700933803E7FE0070922607C7FF5DDC0F837F003004816D140E00384BC6FC0018 033E6D6C5C001C4B6D6C143C6C4BD91FFC5C6C4A486D6C5C6DD907E06D6C13036C6C49486D9038 E00FE0D801F0013FC890B55A27007C03FE6F91C7FC90263FFFF8031F5B010F01E0030313F8D901 FECAEA7FC0592D7BAB64>49 D<92B6FC02071580143F91B7120001030180C8FCD907FCC9FCEB1F E0EB3F80017ECAFC5B485A485A485A5B485A121F90CBFC123EA2123C127CA2127812F8A25AA2B9 FC1880A2180000F0CBFCA27EA21278127CA2123C123EA27E7F120F6C7E7F6C7E6C7E6C7E137E6D 7EEB1FE0EB07FC6DB47E010090B6FC023F1580140702001500313A78B542>I<1706170F171FA2 173EA2177CA217F8A2EE01F0A2EE03E0A2EE07C0A2EE0F80A2EE1F00A2163EA25EA25EA24B5AA2 4B5AA24B5AA24B5AA24BC7FCA2153EA25DA25DA24A5AA24A5AA24A5AA24A5AA24AC8FCA2143EA2 5CA25CA2495AA2495AA2495AA2495AA249C9FCA2133EA25BA25BA2485AA2485AA2485AA2485AA2 48CAFCA2123EA25AA25AA25A1260305C72C600>54 D76 D102 D<12FEEAFFE0EA07F8EA00FEEB7F806D7E6D7E130F6D7EA26D7EB3AD6D 7EA26D7E806E7E6E7EEC0FE0EC03FC913800FFE0A2913803FC00EC0FE0EC3FC04A5A4AC7FC5C49 5AA2495AB3AD495AA2495A131F495A495A01FEC8FCEA07F8EAFFE048C9FC236479CA32>I<1260 12F0B3B3B3B3B3A81260046474CA1C>106 D<126012F07EA21278127CA2123C123EA2121E121F A26C7EA212077FA212037FA212017FA26C7EA21378137CA2133C133EA2131E131FA26D7EA21307 80A2130380A2130180A26D7EA21478147CA2143C143EA280A28081A2140781A2140381A26E7EA2 140081A21578157CA2153C153EA281A2811680A2150716C0A2150316E0A2ED01F0A2150016F8A2 1678167CA2163C163EA2161E160C27647BCA32>110 D E /Fl [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 25 34 43 51 51 85 79 34 34 34 51 69 25 34 25 28 51 51 51 51 51 51 51 51 51 51 34 34 69 69 69 51 94 62 62 68 74 62 62 74 74 34 45 68 57 85 68 74 62 74 62 51 57 74 62 85 62 57 57 40 28 40 43 51 34 51 51 45 51 45 28 51 51 28 28 45 28 74 51 51 51 51 40 40 28 51 45 68 45 45 40 41 28 41 55 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 40 51 51 17 51 51 51 51 22 57 51 34 34 51 51 0 51 51 51 25 0 53 36 34 57 57 51 91 102 0 51 0 34 34 34 34 34 34 34 34 0 34 34 0 34 34 34 91 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 91 0 28 0 0 0 0 57 74 96 32 0 0 0 0 0 68 0 0 0 28 0 0 28 51 68 51 0 0 0 0 ] /Times-Italic 1000 804840 rf /Fm [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 25 34 42 51 51 85 79 34 34 34 51 58 25 34 25 28 51 51 51 51 51 51 51 51 51 51 28 28 58 58 58 45 94 74 68 68 74 62 57 74 74 34 40 74 62 91 74 74 57 74 68 57 62 74 74 96 74 74 62 34 28 34 48 51 34 45 51 45 51 45 34 51 51 28 28 51 28 79 51 51 51 51 34 40 28 51 51 74 51 51 45 49 20 49 55 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 34 51 51 17 51 51 51 51 18 45 51 34 34 57 57 0 51 51 51 25 0 46 36 34 45 45 51 102 102 0 45 0 34 34 34 34 34 34 34 34 0 34 34 0 34 34 34 102 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 91 0 28 0 0 0 0 62 74 91 32 0 0 0 0 0 68 0 0 0 28 0 0 28 51 74 51 0 0 0 0 ] /Times-Roman 1000 804840 rf /Fn [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 23 28 34 34 57 53 23 23 23 34 38 17 23 17 19 34 34 34 34 34 34 34 34 34 34 19 19 38 38 38 30 63 49 45 45 49 42 38 49 49 23 26 49 42 60 49 49 38 49 45 38 42 49 49 64 49 49 42 23 19 23 32 34 23 30 34 30 34 30 23 34 34 19 19 34 19 53 34 34 34 34 23 26 19 34 34 49 34 34 30 33 14 33 37 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 23 34 34 11 34 34 34 34 12 30 34 23 23 38 38 0 34 34 34 17 0 31 24 23 30 30 34 68 68 0 30 0 23 23 23 23 23 23 23 23 0 23 23 0 23 23 23 68 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 60 0 19 0 0 0 0 42 49 60 21 0 0 0 0 0 45 0 0 0 19 0 0 19 34 49 34 0 0 0 0 ] /Times-Roman 1000 536560 rf /Fo [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 19 25 32 38 38 64 59 25 25 25 38 52 19 25 19 21 38 38 38 38 38 38 38 38 38 38 25 25 52 52 52 38 70 47 47 51 55 47 47 55 55 25 34 51 43 64 51 55 47 55 47 38 43 55 47 64 47 43 43 30 21 30 32 38 25 38 38 34 38 34 21 38 38 21 21 34 21 55 38 38 38 38 30 30 21 38 34 51 34 34 30 31 21 31 41 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 30 38 38 13 38 38 38 38 16 43 38 25 25 38 38 0 38 38 38 19 0 40 27 25 43 43 38 68 76 0 38 0 25 25 25 25 25 25 25 25 0 25 25 0 25 25 25 68 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 68 0 21 0 0 0 0 43 55 72 24 0 0 0 0 0 51 0 0 0 21 0 0 21 38 51 38 0 0 0 0 ] /Times-Italic 1000 603630 rf /Fp [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 19 25 31 38 38 64 59 25 25 25 38 43 19 25 19 21 38 38 38 38 38 38 38 38 38 38 21 21 43 43 43 34 70 55 51 51 55 47 43 55 55 25 30 55 47 68 55 55 43 55 51 43 47 55 55 72 55 55 47 25 21 25 36 38 25 34 38 34 38 34 25 38 38 21 21 38 21 59 38 38 38 38 25 30 21 38 38 55 38 38 34 37 15 37 41 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 25 38 38 13 38 38 38 38 14 34 38 25 25 43 43 0 38 38 38 19 0 35 27 25 34 34 38 76 76 0 34 0 25 25 25 25 25 25 25 25 0 25 25 0 25 25 25 76 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 68 0 21 0 0 0 0 47 55 68 24 0 0 0 0 0 51 0 0 0 21 0 0 21 38 55 38 0 0 0 0 ] /Times-Roman 1000 603630 rf /Fq [ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 25 34 57 51 51 102 85 34 34 34 51 58 25 34 25 28 51 51 51 51 51 51 51 51 51 51 34 34 58 58 58 51 95 74 68 74 74 68 62 79 79 40 51 79 68 96 74 79 62 79 74 57 68 74 74 102 74 74 68 34 28 34 59 51 34 51 57 45 57 45 34 51 57 28 34 57 28 85 57 51 57 57 45 40 34 57 51 74 51 51 45 40 22 40 53 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 34 51 51 17 51 51 51 51 28 51 51 34 34 57 57 0 51 51 51 25 0 55 36 34 51 51 51 102 102 0 51 0 34 34 34 34 34 34 34 34 0 34 34 0 34 34 34 102 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 102 0 31 0 0 0 0 68 79 102 34 0 0 0 0 0 74 0 0 0 28 0 0 28 51 74 57 0 0 0 0 ] /Times-Bold 1000 804840 rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 600 TeXDict begin @a4 %%EndSetup %%Page: 1 1 bop 171 592 a Fq(POR)m(OSITIES)24 b(AND)h(DIMENSIONS)e(OF)k(MEASURES)c(SA)-10 b(TISFYING)24 b(THE)i(DOUBLING)1755 708 y(CONDITION)1614 1045 y Fp(JEAN-PIERRE)19 b(ECKMANN)1065 1194 y Fo(Univer)o(sity)h(of)g(Gene)o(va,) f(Departments)i(of)e(Physics)g(and)h(Mathematics,)1625 1285 y(1211)g(Gene)o(va)f(4,)h(Switzerland)1307 1377 y(E-mail:)30 b(J)n(ean-Pierr)m(e)o(.Ec)n(kmann@physics.unig)o(e)o(.c)o(h)1271 1592 y Fp(ESA)19 b(JA)1480 1576 y(\310)1520 1592 y(R)-6 b(VENPA)1780 1576 y(\310)1820 1592 y(A)1835 1576 y(\310)1894 1592 y(AND)19 b(MAARIT)h(JA)2444 1576 y(\310)2484 1592 y(R)-6 b(VENPA)2744 1576 y(\310)2784 1592 y(A)2799 1576 y(\310)1241 1742 y Fo(Univer)o(sity)20 b(of)g(Jyva)-32 b(\310)7 b(sk)o(yla)-32 b(\310)7 b(,)20 b(Department)g(of)g (Mathematics,)1388 1833 y(P)-10 b(.O.)19 b(Box)g(35,)h(FIN-40351)h(Jyva)-32 b(\310)7 b(sk)o(yla)-32 b(\310)7 b(,)19 b(F)m(inland)1336 1925 y(E-mail:)29 b(esaj@math.jyu.\256)22 b(and)d(amj@math.jyu.\256)795 2152 y Fn(This)e(paper)g(e)o(xplains)h(the)f(implications)h(of)e(a)i (mathematical)f(theory)g(\(by)g(the)g(same)h(authors\))f(of)g(holes)795 2231 y(in)23 b(fractals)f(and)h(their)f(relation)g(to)h(dimension)f(for)g (measurements.)42 b(The)22 b(no)o(v)o(elty)h(of)f(our)g(approach)795 2310 y(is)e(to)g(consider)g(the)g(fractal)g(measure)g(on)g(a)g(set)g(rather)g (than)g(just)g(the)g(support)f(of)h(that)g(measure.)33 b(This)795 2389 y(should)23 b(tak)o(e)g(into)f(account)h(in)f(a)h(more)f(precise)h(w)o (ay)g(the)f(distrib)o(ution)g(of)g(data)h(points)f(in)h(measured)795 2468 y(sets,)18 b(such)g(as)f(the)g(distrib)o(ution)f(of)h(galaxies.)165 2839 y Fq(1)102 b(Intr)n(oduction)165 3103 y Fm(The)35 b(aim)h(of)e(this)h (talk)g(is)g(to)f(describe)i(a)e(mathematics)j(paper)e([2])f(written)i(by)e (the)h(same)g(authors)g(to)165 3219 y(a)41 b(community)h(of)f(scientists)g (not)g(necessarily)g(specialized)i(in)e(the)g(theory)g(of)f(fractals.)84 b(W)-8 b(e)41 b(hope)165 3336 y(that)28 b(the)g(concepts)g(de)m(v)n(eloped)h (here)e(can)h(help)f(to)h(understand)f(some)g(issues)g(which)g(ha)n(v)n(e)h (occurred)g(in)165 3452 y(connection)f(with)f(the)g(debate)h(about)f(the)g (dimension)g(of)g(the)g(g)o(alaxy)g(distrib)n(ution.)315 3610 y(W)-8 b(e)30 b(w)o(ant)g(to)h(e)n(xplain)g(the)f(intuiti)m(v)n(e)j(moti)m(v) m(ations)f(of)e(our)f(de\256nitions)i(and)f(results)g(in)g(the)g(conte)n(xt) 165 3726 y(of)37 b(actual)h(ph)o(ysical)f(problems.)71 b(These)37 b(moti)m(v)m(ations)i(will)e(be)g(underlined)g(by)g(simple)g(b)n(ut)g(rele)m (v)m(ant)165 3842 y(e)n(xamples.)315 4000 y(The)31 b(observ)m(ation)g(and)g (study)f(of)g(fractal)i(sets,)f Fl(i.e)n(.)p Fm(,)j(sets)c(with)h(possibly)f (non-inte)n(ger)i(dimension,)165 4116 y(is)26 b(of)f(course)h(widespread.)37 b(The)27 b(present)e(w)o(ork)g(deals)h(with)g(tw)o(o)g(ne)m(w)f(aspects)h(of) g(fractal)h(sets:)360 4358 y Fk(\017)f Fm(Can)f(the)i(presence)f(of)f(v)n (oids)g(be)h(used)g(to)f(say)h(something)g(about)g(the)g(dimension)h(of)e(a)h (set?)360 4542 y Fk(\017)g Fm(Ho)m(w)f(do)g(v)n(oids)h(which)f(are)h(almost)h (empty)f(account)h(for)e(the)h(dimension)h(of)e(a)h(set?)165 4784 y(W)-8 b(e)34 b(will)g(gi)m(v)n(e)h(some)f(answers)f(in)h(the)g(case)g (of)g(sets)f(which)h(are)g(not)g(too)g(wild,)i(namely)f(satisfying)f(a)165 4900 y(certain)27 b(doubling)f(condition.)315 5058 y(Our)k(interest)h(in)g (the)f(question)h(of)f(v)n(oids)g(in)g(fractals)i(\(porosity\))e(w)o(as)f (raised)i(by)f(questions)g(about)165 5174 y(the)36 b(fractal)g(dimension)g (of)e(g)o(alaxy)i(distrib)n(utions.)66 b(In)35 b(vie)m(w)g(of)g(the)h(heated) f(debate)h(in)f(this)h(subject,)3894 5554 y Fq(1)p eop %%Page: 2 2 bop 165 533 a Fm(see)29 b Fl(e)n(.g)n(.)h Fm([1,10],)f(it)g(seems)f(adequate) i(to)e(pro)n(vide)h(as)f(man)n(y)h(analytical)i(tools)d(as)g(possible)g(with) h(which)165 649 y(the)38 b(fractal)g(dimension)f(can)g(be)g(estimated.)73 b(This)37 b(paper)g(pro)n(vides)g(a)g(ne)m(w)g(such)g(tool,)j(namely)e(the) 165 765 y Fl(por)-5 b(osity)32 b(of)f(the)h(measur)l(e)e(on)h(the)g(set)p Fm(.)54 b(The)32 b(general)g(idea)f(is)g(that)h(sets)f(with)g(lar)n(ge)h(v)n (oids)f(must)g(ha)n(v)n(e)165 882 y(small)j(dimension.)60 b(These)33 b(v)n(oids,)i Fl(e)n(.g)n(.)g Fm(in)e(g)o(alaxy)h(distrib)n(utions,)i(are)d (tak)o(en)h(as)e(indicators)i(of)f(small)165 998 y(dimension)40 b(in)f(a)g(sense)g(we)f(mak)o(e)i(precise)g(belo)m(w)-7 b(,)43 b(and)c(our)g(theory)g(allo)m(ws)h(a)f(systematic)h(w)o(ay)f(to)165 1114 y(disre)n(g)o(ard)h(occasional)f(points)f(\(g)o(alaxies\))h(inside)g(a)f (\(lar)n(ge\))g(v)n(oid.)75 b(W)-8 b(e)38 b(present)g(here)h(a)f(formalism) 165 1230 y(which)24 b(is)f(tailored)i(for)e(this)h(situation,)i(by)d (presenting)h(algorithms)h(for)e(measures)h(rather)g(than)g(for)f(their)165 1347 y(supports.)165 1665 y Fq(2)102 b(Measur)n(es)26 b(and)f(sets)165 1928 y Fm(The)17 b(main)g(idea)g([6,11])g(describing)g(the)g(relation)g (between)g(the)g(porosity)g(and)g(the)g(dimension)g(goes)g(about)165 2044 y(as)35 b(follo)m(ws,)i(and)e(we)f(present)h(some)g(intuiti)m(v)n(e)i(e) n(xamples)f(which)f(can)g(guide)g(the)g(reader)g(unf)o(amiliar)165 2161 y(with)c(this)g(problem.)52 b(T)-8 b(ak)o(e)31 b(the)g(well-kno)m(wn)g (middle)g(third)g(Cantor)g(set)g Fj(C)7 b Fm(.)51 b(Clearly)-7 b(,)34 b(if)c(we)g(consider)165 2277 y(the)d(v)n(oids)f(in)h(this)f(set,)h(e) m(v)n(ery)h(point)e(in)h Fj(C)33 b Fm(is)26 b(close)h(to)f(a)h(relati)m(v)n (ely)i(lar)n(ge)e(v)n(oid)f(\(namely)h(to)g(the)g(middle)165 2393 y(third)h(which)f(has)g(been)g(tak)o(en)h(out\).)42 b(W)-8 b(e)28 b(will)f(gi)m(v)n(e)i(a)e(more)h(precise)f(de\256nition)h(belo)m(w)-7 b(.)43 b(Another)27 b(w)o(ay)165 2509 y(to)c(vie)m(w)h(porosity)-7 b(,)24 b(which)f(is)g(closer)g(to)g(the)g(actual)h(de\256nition)g(is)f(as)f (follo)m(ws:)37 b(Suppose)22 b(we)g(ha)n(v)n(e)i(found)165 2625 y(a)i(v)n(oid)f(of)g(diameter)i Fj(\032)e Fm(inside)h(some)f(minimal)i (ball)f(of)f(radius)g Fj(r)2490 2589 y Fi(0)2539 2625 y Fm(around)g(a)g(gi)m (v)n(en)i(point)e(of)g(the)h(fractal.)165 2742 y(Then)35 b(the)f(question)g (is:)54 b(Ho)m(w)33 b(big)h(a)g(radius)g(we)g(ha)n(v)n(e)h(to)f(tak)o(e)g(in) g(order)g(to)g(see)g(for)g(the)g(\256rst)g(time)h(a)165 2858 y(bigger)26 b(v)n(oid)g(than)g(the)g(one)g(we)f(ha)n(v)n(e)i(already)f(seen?) 38 b(See)25 b(also)h(Fig.)g(3)f(belo)m(w)-7 b(.)315 3016 y(It)22 b(is)f(clear)i(that)f(if)g(we)f(tak)o(e)i(out)e(more)h(of)g(the)g(middle,)h (we)f(mak)o(e)g(the)g(dimension)g(of)g(the)g(set)f(smaller)l(,)165 3132 y(since)g(the)f(dimension)g(depends)g(in)g(a)g(well-kno)m(wn)g(f)o (ashion)g(on)g(the)g(length)h(ratio)f(of)g(v)n(oids)g(to)g(non-v)n(oids,)165 3248 y(namely)-7 b(,)35 b(if)d(we)f(remo)n(v)n(e)i(an)f(interv)m(al)h(of)e (length)1975 3209 y Fh(k)r Fi(\000)p Fg(2)p 1975 3225 129 4 v 2021 3283 a Fh(k)2145 3248 y Fm(from)h(the)g(middle,)i Fl(i.e)n(.)p Fm(,)i(lea)n(v)n(e)d(tw)o(o)e(interv)m(als)i(of)165 3389 y(length)461 3350 y Fg(1)p 459 3366 39 4 v 459 3424 a Fh(k)508 3389 y Fm(,)d(then)g(the)g (dimension)g(is)f(\(in)h Ff(R)o Fm(\))f Fe(log)17 b(2)p Fj(=)g Fe(log)f Fj(k)s Fm(.)48 b(F)n(or)29 b(e)n(xample)i(for)e(the)h(middle)h (third)e(Cantor)165 3506 y(set)37 b(the)h(dimension)f(is)g Fe(log)17 b(2)p Fj(=)g Fe(log)f(3)p Fm(.)71 b(Thus,)40 b Fl(lar)l(g)o(e)d (voids)g(imply)g(small)g(dimension)p Fm(.)71 b(Ho)m(we)m(v)n(er)l(,)41 b(the)165 3622 y(contrary)31 b(is)f Fl(not)g Fm(true,)i(as)e(we)g(shall)g(e)n (xplain)i(in)e(Example)i(3)d(belo)m(w)-7 b(.)52 b(One)30 b(can)g(construct)h (a)f(sequence)165 3738 y(of)f(re)n(gular)h(fractals,)h(all)f(of)f(the)g(same) h(dimension,)g(b)n(ut)g(with)f(porosity)g(decreasing)g(to)h(0.)47 b(Thus,)30 b Fl(a)f(set)165 3854 y(can)f(have)f(small)h(dimension)f(without)h (any)g(por)-5 b(osity)p Fm(.)43 b(It)27 b(is)g(this)h(aspect)g(which)f(is)g (connected)i(with)e(the)165 3971 y(contro)n(v)n(ersy)g(about)f(the)g (dimension)h(of)e(the)h(g)o(alaxy)h(distrib)n(ution.)315 4128 y(The)f(requirement)h(of)e(obtaining)i(information)f(about)g(e)n (xperimentally)j(measurable)d(objects)g(leads)165 4245 y(us)36 b(to)g(consider)g(measures,)j(or)c(mass)h(distrib)n(utions,)k(rather)c(than)g (sets.)68 b(This)36 b(issue)g(w)o(as)f(addressed)165 4361 y(earlier)j([3])e (in)g(the)h(conte)n(xt)h(of)e(dimension)h(measurements.)71 b(F)n(or)36 b(e)n(xample,)41 b(one)36 b(can)h(compute)g(the)165 4477 y(dimension)31 b(of)g(the)f Fl(support)g Fm(of)g(a)h(set,)h Fl(i.e)n(.)p Fm(,)i(of)c(the)h(complement)h(of)e(the)h(open)g(sets)f(of)g (zero)h(measure.)165 4593 y(But,)k(as)d(e)n(xplained)i(in)e([3],)j(the)d (so-called)h(correlation)h(dimension)f(which)f(is)h(based)f(on)g(the)h Fl(mass)e Fm(in)165 4709 y(balls)e(seems)f(to)h(be)f(the)h(natural)g (quantity)h(for)e(questions)g(of)g(e)n(xperimental)j(nature)e(and)g(is)f (commonly)165 4826 y(used)e(in)f(the)h(Grassber)n(ger)n(-Procaccia)g(method)h ([5].)37 b(Therefore,)26 b(we)g(shall)f(study)h(here)g(the)f(porosity)h(of) 165 4942 y(a)31 b Fl(measur)l(e)p Fm(,)g(and)g(not)f(only)h(the)g(porosity)f (of)g(the)h(support)f(of)g(the)h(measure)61 b([3].)52 b(W)-8 b(e)30 b(then)h(sho)m(w)f(that)165 5058 y(lar)n(ge)d(porosity)e(implies)h(a)g (non-tri)m(vial)h(upper)e(bound)g(on)h(the)f(dimension)h(\(in)g(f)o(act)g(on) f(all)h(multi-fractal)165 5174 y(dimensions)39 b Fj(D)749 5189 y Fh(q)787 5174 y Fm(,)i Fj(q)49 b(>)44 b Fe(1)p Fm(\).)75 b(Finally)-7 b(,)43 b(we)38 b(e)n(xplain)h(ho)m(w)f(porosity)g(is)g (estimated)i(for)e(a)g(gi)m(v)n(en)i(set)e(of)3894 5554 y Fq(2)p eop %%Page: 3 3 bop 165 533 a Fm(e)n(xperimental)29 b(points.)165 849 y Fq(3)102 b(P)n(or)n(osities)27 b(of)e(measur)n(es)165 1112 y Fm(Let)i Fj(\026)e Fm(be)h(a)f(probability)i(measure)g(on)e Ff(R)1653 1076 y Fh(n)1700 1112 y Fm(.)37 b(W)-8 b(e)26 b(de\256ne)f(for)h Fj(x)i Fk(2)g Ff(R)2601 1076 y Fh(n)2673 1112 y Fm(and)e Fj(r)m(;)17 b(")27 b(>)h Fe(0)p Fm(:)873 1274 y Fe(p)s(or)36 b(\()p Fj(\026;)17 b(x;)g(r)m(;)g(")p Fe(\))43 b(=)h(sup)q Fk(f)p Fj(p)27 b Fk(\025)h Fe(0)53 b(:)28 b Fm(there)e(is)g Fj(z)32 b Fk(2)c Ff(R)2738 1238 y Fh(n)2811 1274 y Fm(such)d(that)1041 1415 y Fj(B)5 b Fe(\()p Fj(z)t(;)17 b(pr)s Fe(\))28 b Fk(\032)g Fj(B)5 b Fe(\()p Fj(x;)17 b(r)s Fe(\))26 b Fm(and)f Fj(\026)p Fe(\()p Fj(B)5 b Fe(\()p Fj(z)t(;)17 b(pr)s Fe(\)\))28 b Fk(\024)g Fj("\026)p Fe(\()p Fj(B)5 b Fe(\()p Fj(x;)17 b(r)s Fe(\)\))p Fk(g)24 b Fj(:)165 1577 y Fm(In)29 b(other)h(w)o(ords,)f(we)g(consider)h(the)f(ball)h Fj(B)5 b Fe(\()p Fj(x;)17 b(r)s Fe(\))29 b Fm(of)g(radius)g Fj(r)i Fm(centered)g(at)e Fj(x)k Fk(2)g Ff(R)3224 1541 y Fh(n)3300 1577 y Fm(and)c(observ)n(e)g(the)165 1693 y(mass)20 b Fj(\026)p Fe(\()p Fj(B)5 b Fe(\()p Fj(x;)17 b(r)s Fe(\)\))i Fm(contained)i(in)f(it.)36 b(W)-8 b(e)20 b(no)m(w)f(look)h(for)g(the)g(lar)n(gest)h(ball)f(of)g(radius)g Fj(p)9 b Fk(\001)g Fj(r)22 b Fm(\(fully)e(contained)165 1809 y(in)28 b Fj(B)5 b Fe(\()p Fj(x;)17 b(r)s Fe(\))p Fm(\))28 b(around)f(a)h(point)h Fj(z)j Fm(such)27 b(that)i(the)f(mass)g(of)f(that)i (ball)f(does)g(not)g(e)n(xceed)h Fj(")e Fm(times)i(the)f(mass)165 1926 y Fj(\026)p Fe(\()p Fj(B)5 b Fe(\()p Fj(x;)17 b(r)s Fe(\)\))25 b Fm(of)h Fj(B)5 b Fe(\()p Fj(x;)17 b(r)s Fe(\))p Fm(.)1063 1889 y Fh(a)1141 1926 y Fm(See)26 b(Fig.)g(1.)37 b(One)26 b(then)g(de\256nes) 1254 2087 y Fe(p)s(or\()p Fj(\026;)17 b(x)p Fe(\))44 b(=)g(lim)1812 2147 y Fh(")p Fi(#)p Fg(0)1965 2087 y Fe(lim)17 b(inf)2048 2147 y Fh(r)r Fi(#)p Fg(0)2252 2087 y Fe(p)s(or)o(\()p Fj(\026;)g(x;)g(r)m(;) g(")p Fe(\))25 b Fj(;)970 b Fm(\(1\))165 2307 y(and)26 b(\256nally)833 2469 y Fe(p)s(or\()p Fj(\026)p Fe(\))44 b(=)g(inf)6 b Fk(f)p Fj(s)53 b Fe(:)g(p)s(or\()p Fj(\026;)17 b(x)p Fe(\))27 b Fk(\024)h Fj(s)e Fm(for)f Fj(\026)p Fm(-almost)h(all)h Fj(x)h Fk(2)g Ff(R)3128 2428 y Fh(n)3174 2469 y Fk(g)e Fj(:)549 b Fm(\(2\))165 2631 y(Note)33 b(that)f(when)g Fj(x)h Fm(is)f(not)g(in)g(the)h(support)f(of)g (the)g(measure,)j(the)e(de\256nition)g(is)e(not)i(v)n(ery)f(interesting)165 2747 y(since)21 b(for)g(small)h(enough)e Fj(r)k Fm(the)d(measure)g(of)g Fj(\026)p Fe(\()p Fj(B)5 b Fe(\()p Fj(x;)17 b(r)s Fe(\)\))j Fm(is)g(zero)h(and)g(then)g(an)n(y)h(ball)f(in)g(it)h(is)e(also)h(empty)-7 b(.)165 2905 y Fq(De\256nition)25 b(1)49 b Fl(The)26 b(quantity)g Fe(p)s(or\()p Fj(\026)p Fe(\))f Fl(is)g(called)i(the)f(por)-5 b(osity)26 b(of)g(the)g(measur)l(e)f Fj(\026)p Fl(.)315 3063 y Fm(It)i(is)g(not)g(dif)m(\256cult)h(to)f(see)f(that)i(the)f(de\256nition)h (of)e(the)h Fl(por)-5 b(osity)28 b(of)e(a)h(set)g Fm(\(see)g(De\256nition)g (5)f(belo)m(w\))165 3179 y(amounts)g(to)g(using)g(Eq.\(1\))g(with)g(the)g (limits)g(tak)o(en)h(in)f(the)g(opposite)g(order)l(,)g(that)g(is,)1151 3341 y Fe(p)s(or\(spt\()p Fj(\026)p Fe(\))p Fj(;)17 b(x)p Fe(\))44 b(=)g(lim)17 b(inf)1982 3401 y Fh(r)r Fi(#)p Fg(0)2203 3341 y Fe(lim)2219 3401 y Fh(")p Fi(#)p Fg(0)2355 3341 y Fe(p)s(or)o(\()p Fj(\026;)g(x;)g(r)m(;)g(")p Fe(\))25 b Fj(:)867 b Fm(\(3\))165 3567 y(Because)35 b(the)g(central)h(point)e(in)h Fj(B)5 b Fe(\()p Fj(x;)17 b(r)s Fe(\))34 b Fm(is)g(in)g(the)h(support)f Fe(spt)q(\()p Fj(\026)p Fe(\))f Fm(and)i(hence)g(occupied)g(when)f(we)165 3683 y(compute)g Fe(p)s(or)o(\(spt)q(\()p Fj(\026)p Fe(\))p Fj(;)17 b(x)p Fe(\))p Fm(,)34 b(one)e(\256nds)g(that)h Fe(p)s(or)o(\(spt)q (\()p Fj(\026)p Fe(\)\))j Fk(\024)2432 3644 y Fg(1)p 2432 3660 36 4 v 2432 3717 a(2)2477 3683 y Fm(.)58 b(One)32 b(can)g(also)h(sho)m(w)-7 b(,)34 b(using)e(density)165 3824 y(ar)n(guments,)e(that)e Fe(p)s(or\()p Fj(\026)p Fe(\))h Fk(\024)1248 3785 y Fg(1)p 1248 3801 V 1248 3859 a(2)1293 3824 y Fm(.)43 b(W)-8 b(e)28 b(also)g(note)g(that)g Fe(p)s(or)f Fm(is)g(determined)i Fl(\256r)o(st)e Fm(with)h(\252dust\272)g(of)f(relati)m(v)n(e)165 3941 y(weight)f Fj(")g Fm(and)f(only)h(then)g Fj(")f Fm(is)h(tak)o(en)g(to)g(0.)315 4098 y(The)j(tw)o(o)e(porosities)h(we)g(consider)g(satisfy)g(clearly)h Fe(p)s(or\(spt\()p Fj(\026)p Fe(\)\))h Fk(\024)i Fe(p)s(or)o(\()p Fj(\026)p Fe(\))p Fm(.)44 b(In)27 b(other)h(w)o(ords)f(the)165 4215 y(porosity)h(of)f(a)h(measure)g(is)f(lar)n(ger)h(than)g(that)g(of)g(the) g(support)f(of)g(the)h(measure,)h(precisely)f(because)g(the)165 4331 y(former)e(ne)n(glects)h(occasional)g(dust.)165 4534 y Fq(Example)40 b(1)80 b Fm(Let)40 b Fj(\016)939 4549 y Fg(0)1019 4534 y Fm(be)g(the)g(Dirac)g(measure)g(at)h(the)f(origin,)k(that)d(is,)j Fj(\016)2927 4549 y Fg(0)2966 4534 y Fe(\()p Fj(A)p Fe(\))j(=)g(1)40 b Fm(if)g Fe(0)46 b Fk(2)i Fj(A)40 b Fm(and)165 4650 y Fj(\016)208 4665 y Fg(0)248 4650 y Fe(\()p Fj(A)p Fe(\))51 b(=)h(0)42 b Fm(if)i Fe(0)63 b Fj(=)-61 b Fk(2)52 b Fj(A)p Fm(.)90 b(Let)44 b Fj(\026)e Fm(be)i(the)f(sum)g(of)g Fj(\016)2150 4665 y Fg(0)2233 4650 y Fm(and)g(the)g(Lebesgue)h(measure)g Fk(L)3476 4614 y Fh(n)3565 4650 y Fm(restricted)165 4767 y(to)37 b Fj(B)5 b Fe(\(0)p Fj(;)17 b Fe(1\))p Fm(,)40 b(that)e(is,)i Fj(\026)i Fe(=)h Fj(C)7 b Fe(\()p Fj(\016)1344 4782 y Fg(0)1412 4767 y Fe(+)28 b Fk(L)1585 4730 y Fh(n)1632 4767 y Fk(j)1660 4782 y Fh(B)s Fg(\(0)p Fh(;)p Fg(1\))1865 4767 y Fe(\))37 b Fm(where)g Fj(C)43 b Fm(is)37 b(the)h(normalization)h(constant.)72 b(Clearly)165 4883 y Fe(p)s(or\()p Fj(\026;)17 b Fe(0\))37 b(=)697 4844 y Fg(1)p 697 4860 V 697 4917 a(2)775 4883 y Fm(and)d Fe(p)s(or)o(\()p Fj(\026;)17 b(x)p Fe(\))38 b(=)f(0)c Fm(for)g(all)g Fj(x)39 b Fk(6)p Fe(=)e(0)c Fm(with)g Fk(j)p Fj(x)p Fk(j)38 b Fj(<)f Fe(1)p Fm(.)59 b(Thus)33 b Fe(p)s(or\()p Fj(\026)p Fe(\))k(=)3427 4844 y Fg(1)p 3427 4860 V 3427 4917 a(2)3472 4883 y Fm(.)59 b(Ho)m(we)m(v)n(er)l(,)165 4999 y Fe(p)s(or\(spt)q(\()p Fj(\026)p Fe(\)\))27 b(=)g(p)s(or\()p Fj(B)5 b Fe(\(0)p Fj(;)17 b Fe(1\)\))27 b(=)g(0)p Fm(.)p 165 5098 1512 4 v 165 5151 a Fd(a)203 5174 y Fn(The)17 b(con)m(v)o(entional)g(porosity)g(asks)h(for)e(the)h(lar)o(gest)g (ball)g(in)g Fh(B)s Fg(\()p Fh(x;)12 b(r)r Fg(\))17 b Fn(which)h(does)f(not)g (contain)g Fc(any)g Fn(point)g(of)g(the)g(set)g(in)g(question;)g(see)h(also)g (belo)n(w)l(.)3894 5554 y Fq(3)p eop %%Page: 4 4 bop 402 2783 a @beginspecial -3 @llx -3 @lly 277 @urx 277 @ury 2800 @rwi @setspecial %%BeginDocument: fig1p.ps /$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 -33.0 364.0 translate 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 /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def $F2psBegin 10 setmiterlimit n -1000 6763 m -1000 -1000 l 5863 -1000 l 5863 6763 l cp clip 0.06299 0.06299 sc 7.500 slw n 1530 3555 45 45 0 360 DrawEllipse gs col0 s gr n 2700 3600 2155 2155 0 360 DrawEllipse gs col0 s gr n 1530 3555 826 826 0 360 DrawEllipse gs col0 s gr n 2700 3600 64 64 0 360 DrawEllipse gs col0 s gr gs clippath 4297 2458 m 4432 2400 l 4331 2507 l 4484 2401 l 4450 2352 l cp clip n 2700 3600 m 4455 2385 l gs col0 s gr gr n 4297 2458 m 4432 2400 l 4331 2507 l 4334 2469 l 4297 2458 l cp gs 0.00 setgray ef gr col0 s gs clippath 1793 2934 m 1879 2814 l 1848 2958 l 1924 2788 l 1869 2764 l cp clip n 1530 3555 m 1890 2790 l gs col0 s gr gr n 1793 2934 m 1879 2814 l 1848 2958 l 1830 2924 l 1793 2934 l cp gs 0.00 setgray ef gr col0 s $F2psEnd rs %%EndDocument @endspecial 924 1740 a Fh(z)854 1504 y(p)11 b(r)1560 1740 y(x)1870 1292 y(r)1456 3294 y Fn(Figure)16 b(1.)34 b(The)17 b(ball)g Fh(B)s Fg(\()p Fh(z)s(;)12 b(pr)r Fg(\))17 b Fn(inside)g Fh(B)s Fg(\()p Fh(x;)12 b(r)r Fg(\))p Fn(.)165 3614 y Fq(Example)36 b(2)f Fm(W)-8 b(e)36 b(ne)n(xt)h(w)o(ant)e(to)h(ar)n(gue)g(that)h(v)n(oids)e (are)h(accounted)h(for)e(in)h(a)g(more)g(reasonable)g(w)o(ay)165 3730 y(in)h(the)f(measure)h(theoretic)h(de\256nition)e(of)g(porosity)-7 b(.)70 b(T)-8 b(o)36 b(illustrate)h(this)g(with)f(a)g(concrete)h(e)n(xample,) 165 3846 y(consider)25 b(the)f(celebrated)i(middle)f(third)f(Cantor)g(set)g (which)g(is)g(obtained)h(by)f(starting)h(with)f(the)g(interv)m(al)165 3962 y Fe([0)p Fj(;)17 b Fe(1])36 b Fm(and)h(taking)g(out)f(the)h(open)g (interv)m(al)h Fe(\()1827 3923 y Fg(1)p 1827 3939 36 4 v 1827 3997 a(3)1872 3962 y Fj(;)1926 3923 y Fg(2)p 1926 3939 V 1926 3997 a(3)1971 3962 y Fe(\))p Fm(.)69 b(Then)37 b(each)g(of)f(the)h(remaining) h(tw)o(o)e(interv)m(als)i(is)165 4104 y(di)m(vided)28 b(into)e(three)h (pieces)g(and)g(the)f(middle)i(one)e(\(of)g(length)2433 4064 y Fg(1)p 2433 4080 V 2433 4138 a(9)2478 4104 y Fm(\))g(is)g(discarded.)40 b(Going)26 b(on)g(recursi)m(v)n(ely)165 4220 y(\(and)46 b(inde\256nitely\))h (in)f(this)g(f)o(ashion,)51 b(we)45 b(get)h(the)g(middle)h(third)f(Cantor)g (set.)98 b(Its)46 b(dimension)g(is)165 4336 y Fj(d)51 b Fk(\021)406 4293 y Fg(log)13 b(2)p 406 4313 139 4 v 406 4370 a(log)g(3)554 4336 y Fm(.)90 b(Clearly)-7 b(,)49 b(this)43 b(set)g Fj(C)50 b Fm(has)43 b(v)n(oids)g(and)g(its)g(porosity)g(equals)h(in)f(f)o(act)3279 4297 y Fg(1)p 3279 4313 36 4 v 3279 4370 a(4)3324 4336 y Fm(.)89 b(W)-8 b(e)44 b(ne)n(xt)g(set)165 4483 y Fj(C)235 4498 y Fh(x)313 4483 y Fe(=)34 b Fk(f)p Fj(y)67 b Fe(:)d Fj(y)37 b Fe(=)d Fj(x)25 b Fe(+)g Fj(z)t(;)17 b(z)39 b Fk(2)34 b Fj(C)7 b Fk(g)p Fm(,)32 b(in)e(other)h(w)o(ords,)g Fj(C)2240 4498 y Fh(x)2314 4483 y Fm(is)f(the)g(translate)i(of)e Fj(C)37 b Fm(by)30 b Fj(x)p Fm(.)52 b(Clearly)31 b(each)165 4599 y Fj(C)235 4614 y Fh(x)302 4599 y Fm(has)23 b(ag)o(ain)i(dimension)f Fj(d)p Fm(.)36 b(From)23 b(the)h(general)g(theory)g(of)f(fractals)h([4,7])g(we)f(get)g(that)i(an)n(y)e (countable)165 4715 y(union)g(of)g(such)f(sets)h(has)f(still)i(dimension)f Fj(d)p Fm(.)36 b(In)22 b(particular)l(,)k(we)c(can)h(enumerate)h(the)f (rationals)h(in)f Fe([0)p Fj(;)17 b Fe(1])p Fm(,)165 4832 y(for)26 b(e)n(xample)h(calling)g(them)g Fj(x)1267 4847 y Fh(j)1303 4832 y Fm(,)f Fj(j)34 b Fe(=)28 b(1)p Fj(;)17 b Fe(2)p Fj(;)g(:)g(:)g(:)24 b Fm(and)h(construct)i(then)f(the)g(set)1765 5087 y Fj(D)47 b Fe(=)2039 4979 y Fi(1)2028 5004 y Fb([)2013 5186 y Fh(j)t Fg(=1)2152 5087 y Fj(C)2222 5102 y Fh(x)2262 5112 y Fd(i)2318 5087 y Fj(:)3894 5554 y Fq(4)p eop %%Page: 5 5 bop 165 533 a Fm(From)22 b(what)g(we)f(said)h(before,)h(and)f(from)g(the)g(w) o(ay)f(we)h(constructed)g(it,)i(the)e(set)g Fj(D)i Fm(has)e(dimension)g Fj(d)28 b(<)f Fe(1)165 649 y Fm(and)i Fl(no)g(voids)p Fm(.)48 b(Thus,)30 b(we)f(see)g(that)h(a)f Fl(set)g Fm(can)g(ha)n(v)n(e)h(small)g (dimension)f(and)g(no)g(v)n(oids.)48 b(The)29 b(e)n(xample)165 765 y(we)j(ha)n(v)n(e)g(just)g(gi)m(v)n(en)h(does)e Fl(not)h(work)g Fm(in)f(the)i(case)f(of)f(porosity)h(of)f(measures)h(and)g(this)g(is)f(one)h (of)f(the)165 882 y(reasons)24 b(wh)o(y)f(the)h(porosity)g(of)f(measures)h (is)f(a)h(more)g(useful)f(concept)i(than)f(that)g(of)f(sets.)37 b(Ho)m(we)m(v)n(er)l(,)25 b(we)165 998 y(will)32 b(construct)g(re)n(gular)g (fractals)g(in)f(Example)i(3)e(where)g(the)h(porosity)f(of)g(the)g(measure)h (is)f(arbitrarily)165 1114 y(small)c(b)n(ut)f(the)g(dimension)g(is)f(al)o(w)o (ays)1576 1075 y Fg(1)p 1576 1091 36 4 v 1576 1149 a(2)1621 1114 y Fm(.)315 1297 y(In)f(f)o(act,)h(one)f(sho)m(ws)f(easily)-7 b(,)25 b(see)f([2],)h(that)f(the)g(follo)m(wing)h(measure)f(has)g(porosity) 3307 1258 y Fg(1)p 3307 1274 V 3307 1331 a(2)3353 1297 y Fm(.)36 b(W)-8 b(e)24 b(construct)165 1413 y(a)29 b(measure)g(on)g(the)g(set)g Fj(D)i Fm(by)e(gi)m(ving)g(successi)m(v)n(ely)h(lo)m(wer)f(weight)h(to)e(the) h(translates)h(of)f Fj(C)7 b Fm(.)46 b(Let)29 b Fj(\026)g Fm(be)165 1529 y(the)21 b(usual)g(measure)h(associated)f(with)g(the)g(Cantor)g(set)g Fj(C)7 b Fm(,)22 b Fl(i.e)n(.)p Fm(,)i(the)d(measure)h(which)e(gi)m(v)n(es)i (equal)f(weight)165 1646 y(to)26 b(all)h(the)f(pieces)g(in)g(the)g(recursi)m (v)n(e)h(construction.)38 b(Then)26 b(we)g(de\256ne)1721 1874 y Fj(\027)50 b Fe(=)1964 1766 y Fi(1)1940 1791 y Fb(X)1943 1974 y Fh(i)p Fg(=1)2076 1874 y Fe(2)2125 1833 y Fi(\000)p Fh(i)2208 1874 y Fj(\026)2267 1889 y Fh(x)2307 1899 y Fd(i)2362 1874 y Fj(;)165 2112 y Fm(where)34 b Fj(\026)507 2127 y Fh(x)584 2112 y Fm(is)g(the)g(translate)h(of)f(the)g(measure)h Fj(\026)e Fm(by)h Fj(x)p Fm(.)62 b(Clearly)-7 b(,)38 b(the)c(support)g(of)f(this)h (measure)h(is)e(at)165 2229 y(least)27 b(all)g(of)f(the)h(interv)m(al)h Fe([0)p Fj(;)17 b Fe(1])25 b Fm(\(with)h(some)h(o)n(v)n(erhangs)g(from)f(the) g(translation\))i(and)e(has)g(therefore)h(no)165 2345 y(v)n(oids.)51 b(But)31 b(the)g(porosity)f Fe(p)s(or\()p Fj(\027)6 b Fe(\))p Fm(,)32 b(as)e(de\256ned)g(in)h(Eq.\(2\),)h(is)e(strictly)i(positi)m(v)n(e.) 53 b(The)30 b(reason)h(for)f(this)165 2461 y(is)f(that)h(if)f(we)f(consider)i (a)f(v)n(oid)g(of)f(the)i(original)g(set)f Fj(C)35 b Fm(and)29 b(look)g(for)g(a)g(point)g(in)g(one)g(of)g(the)g Fj(C)3666 2476 y Fh(x)3706 2486 y Fd(i)3765 2461 y Fm(v)n(ery)165 2577 y(close)f(to)g(the)g(boundary)f(of)h(this)f(v)n(oid,)i(then)f(we)f(must)h (tak)o(e)g(in)g(general)g(a)g(high)g(inde)n(x)g Fj(i)g Fm(to)f(\256nd)g(such) h(a)165 2693 y(point.)38 b(But)26 b(then)g(the)g(associated)h(weight)f(is)f (smaller)i(than)f Fe(2)2378 2657 y Fi(\000)p Fh(i)2461 2693 y Fm(,)g(and)f(if)h Fj(i)g Fm(is)f(lar)n(ge)i(enough,)f(then)g(this)g(is)165 2810 y(smaller)31 b(than)f(an)n(y)f Fj(")g Fm(which)h(w)o(as)f(gi)m(v)n(en)h (in)g(the)g(de\256nition)g(of)f(Eq.\(1\).)49 b(Therefore,)32 b(the)e(set)f Fj(C)3618 2825 y Fh(x)3658 2835 y Fd(i)3717 2810 y Fm(is)h(not)165 2926 y(counted)d(in)e(this)h(consideration,)i(and)d(the)h (measure)h(will)f(ha)n(v)n(e)g(the)g(same)g(porosity)g(as)f Fj(C)33 b Fm(itself.)315 3084 y(Ha)n(ving)42 b(described)h(the)f (de\256nitions,)47 b(we)42 b(can)g(no)m(w)g(ask)f(more)i(precisely)f(our)g (question)g(about)165 3200 y(the)34 b(relation)h(between)f(the)g(porosity)f (of)g(the)h(measure)g Fj(\026)f Fm(and)g(the)h(packing)g(dimension)g(of)g (the)f(same)165 3316 y(measure.)38 b(Our)25 b(general)i(aim)f(is)g(to)g(sho)m (w)f(the)h(follo)m(wing)165 3474 y Fq(Conjectur)n(e)g(2)49 b Fl(If)32 b(the)g(por)-5 b(osity)33 b(of)f Fj(\026)f Fl(is)h(lar)l(g)o(e)o (,)i(then)f(the)f(pac)n(king)h(dimension)f(of)g Fj(\026)f Fl(is)h(smaller)g (than)165 3590 y(the)26 b(dimension)g Fj(n)g Fl(of)g(the)g(ambient)g(space)n (.)315 3748 y Fm(Our)h(results)g(will)h(f)o(all)g(some)m(what)g(short)f(of)g (this)g(conjecture.)44 b(In)27 b(order)g(to)h(be)f(able)h(to)f(formulate)i(a) 165 3864 y(positi)m(v)n(e)f(result,)e(we)g(need)g(the)g(follo)m(wing)g (concept:)165 4022 y Fq(De\256nition)f(3)49 b Fl(The)26 b(pr)-5 b(obability)27 b(measur)l(e)e Fj(\026)g Fl(on)g Ff(R)2036 3986 y Fh(n)2109 4022 y Fl(satis\256es)g(the)h(local)g(doubling)g(condition)h(at)f Fj(x)g Fl(if)1487 4247 y Fe(lim)17 b(sup)1584 4326 y Fh(r)r Fi(#)p Fg(0)1813 4179 y Fj(\026)p Fe(\()p Fj(B)5 b Fe(\()p Fj(x;)17 b Fe(2)p Fj(r)s Fe(\)\))p 1813 4223 485 4 v 1837 4315 a Fj(\026)p Fe(\()p Fj(B)5 b Fe(\()p Fj(x;)17 b(r)s Fe(\)\))2351 4247 y Fj(<)44 b Fk(1)25 b Fj(:)1203 b Fm(\(4\))165 4478 y Fl(It)26 b(satis\256es)g(the)g(\(global\))g(doubling)g(condition)g(if)h (Eq.\(4\))f(holds)f(for)h Fj(\026)p Fl(-almost)f(e)n(very)h Fj(x)p Fl(.)315 4636 y Fm(The)20 b(bound)g(need)f(not)h(be)g(uniform)g(in)f (those)h Fj(x)p Fm(.)36 b(Note)19 b(that)i(the)f(doubling)g(condition)g(is)g (also)f(implied)165 4752 y(by)26 b(the)g(stronger)f(condition)1277 4905 y Fe(0)44 b Fj(<)g(ar)1588 4864 y Fh(s)1669 4905 y Fk(\024)h Fj(\026)p Fe(\()p Fj(B)5 b Fe(\()p Fj(x;)17 b(r)s Fe(\)\))43 b Fk(\024)i Fj(br)2480 4864 y Fh(s)2561 4905 y Fj(<)f Fk(1)25 b Fj(:)993 b Fm(\(5\))165 5058 y(In)32 b(ph)o(ysics,)j(it)e(is)f(generally)i (assumed)e(that)h(the)g(stronger)f(condition)i(\(5\))e(holds.)57 b(See)32 b(belo)m(w)h(for)f(the)165 5174 y(rele)m(v)m(ance)c(of)e(these)g (conditions)g(in)g(Nature.)38 b(Our)25 b(main)h(result)g(is)g(the)g(follo)m (wing)3894 5554 y Fq(5)p eop %%Page: 6 6 bop 165 533 a Fq(Theor)n(em)26 b(4)49 b Fl(Ther)l(e)23 b(is)f(a)g(function)i Fe(\001)1543 548 y Fh(n)1612 533 y Fl(de\256ned)f(for)g Fj(p)k Fk(2)h Fe(\(0)p Fj(;)17 b Fe(1)p Fj(=)p Fe(2\))k Fl(with)i(values)g(in)g Fe([0)p Fj(;)17 b Fe(1])22 b Fl(and)g(satisfying)1720 697 y Fe(lim)1709 770 y Fh(p)p Fi(!)1826 743 y Fa(1)p 1825 755 31 4 v 1825 796 a(2)1882 697 y Fe(\001)1963 712 y Fh(n)2010 697 y Fe(\()p Fj(p)p Fe(\))45 b(=)f(1)25 b Fj(;)165 946 y Fl(suc)n(h)h(that)g(if) h(a)e(Bor)l(el)h(pr)-5 b(obability)27 b(measur)l(e)f Fj(\026)f Fl(on)g Ff(R)2104 909 y Fh(n)2176 946 y Fl(satis\256es)h(the)g(global)g (doubling)g(condition)h(then)1184 1109 y Fe(dim)1348 1124 y Fg(H)1404 1109 y Fe(\()p Fj(\026)p Fe(\))44 b Fk(\024)g Fe(dim)1868 1124 y Fg(p)1910 1109 y Fe(\()p Fj(\026)p Fe(\))g Fk(\024)h Fj(n)22 b Fk(\000)h Fe(\001)2472 1124 y Fh(n)2519 1109 y Fe(\(p)s(or\()p Fj(\026)p Fe(\)\))i Fj(:)900 b Fm(\(6\))165 1314 y(Here,)27 b Fe(dim)579 1329 y Fg(H)634 1314 y Fe(\()p Fj(\026)p Fe(\))f Fm(is)f(the)i(Hausdorf)m(f)e(dimension)i(of)f(the)g(measure)g(and)g Fe(dim)2889 1329 y Fg(p)2931 1314 y Fe(\()p Fj(\026)p Fe(\))g Fm(is)f(its)h(packing)h(dimen-)165 1431 y(sion.)38 b(The)26 b(inequality)h(among)f(those)g(tw)o(o)f(is)h(ob)n(vious)f(from)h(their)h (de\256nition:)801 1667 y Fj(d)p 801 1680 51 4 v Fe(\()p Fj(\026;)17 b(x)p Fe(\))41 b(=)g(lim)17 b(inf)1327 1727 y Fh(r)r Fi(#)p Fg(0)1541 1600 y Fe(log)f Fj(\026)p Fe(\()p Fj(B)5 b Fe(\()p Fj(x;)17 b(r)s Fe(\)\))p 1541 1644 578 4 v 1735 1736 a(log)g Fj(r)2154 1667 y(;)p 801 1868 51 4 v 801 1947 a(d)p Fe(\()p Fj(\026;)g(x)p Fe(\))49 b(=)h(lim)17 b(sup)1358 2026 y Fh(r)r Fi(#)p Fg(0)1586 1880 y Fe(log)g Fj(\026)p Fe(\()p Fj(B)5 b Fe(\()p Fj(x;)17 b(r)s Fe(\)\))p 1586 1924 578 4 v 1781 2015 a(log)f Fj(r)2200 1947 y(;)731 2172 y Fe(dim)895 2187 y Fg(H)951 2172 y Fe(\()p Fj(\026)p Fe(\))41 b(=)g(sup)q Fk(f)p Fj(s)27 b Fk(\025)i Fe(0)53 b(:)g Fj(d)p 1802 2185 51 4 v Fe(\()p Fj(\026;)17 b(x)p Fe(\))27 b Fk(\025)h Fj(s)d Fm(for)h Fj(\026)p Fm(-almost)g(all)g Fj(x)i Fk(2)h Ff(R)3214 2131 y Fh(n)3260 2172 y Fk(g)c Fj(;)745 2313 y Fe(dim)909 2328 y Fg(p)951 2313 y Fe(\()p Fj(\026)p Fe(\))41 b(=)g(sup)q Fk(f)p Fj(s)27 b Fk(\025)i Fe(0)53 b(:)p 1802 2234 V 53 w Fj(d)p Fe(\()p Fj(\026;)17 b(x)p Fe(\))27 b Fk(\025)h Fj(s)d Fm(for)h Fj(\026)p Fm(-almost)g(all)g Fj(x)i Fk(2)h Ff(R)3214 2272 y Fh(n)3260 2313 y Fk(g)c Fj(:)315 2519 y Fm(There)i(is)e(an)h(e)n(xplicit)i(lo)m(wer)e(bound)f(for)h(the)g(function) g Fe(\001)2400 2534 y Fh(n)2473 2519 y Fm(in)f([11]:)1346 2723 y Fe(\001)1427 2738 y Fh(n)1474 2723 y Fe(\()p Fj(p)p Fe(\))44 b Fk(\025)h Fe(max)o Fk(f)p Fe(1)22 b Fk(\000)2324 2656 y Fj(c)2366 2671 y Fh(n)p 2177 2700 383 4 v 2177 2792 a Fe(log\()2414 2753 y Fg(1)p 2351 2769 161 4 v 2351 2826 a(1)p Fi(\000)p Fg(2)p Fh(p)2522 2792 y Fe(\))2570 2723 y Fj(;)17 b Fe(0)p Fk(g)24 b Fj(;)165 2980 y Fm(where)h Fj(c)481 2995 y Fh(n)556 2980 y Fj(>)j Fe(0)c Fm(is)h(a)h(constant)f(depending)h(only)g(on)e Fj(n)p Fm(.)38 b(According)25 b(to)h(Theorem)g(4)f(if)g(the)g(porosity)h(of)f (a)165 3097 y(measure)g Fj(\026)g Fm(which)f(satis\256es)g(the)h(doubling)g (condition)h(is)e(close)h(to)2619 3057 y Fg(1)p 2619 3073 36 4 v 2619 3131 a(2)2664 3097 y Fm(,)g(then)g(the)g(packing)h(dimension)f(of) 165 3213 y Fj(\026)h Fm(is)f(not)h(much)g(bigger)g(than)g Fj(n)c Fk(\000)h Fe(1)p Fm(.)165 3418 y Fq(Remark)i Fm(F)n(or)g(sets,)g(a)h (relation)g(between)f(porosity)h(and)f(dimension)g(has)g(been)h(established)f (by)g(Mattila)165 3534 y([6])h(and)f(Salli)i([11])e(using)h(the)g(follo)m (wing)g(de\256nition)h(of)e(porosity:)165 3692 y Fq(De\256nition)g(5)49 b Fl(The)26 b(por)-5 b(osity)26 b(of)g(a)f(set)h Fj(A)i Fk(\032)g Ff(R)1865 3656 y Fh(n)1937 3692 y Fl(at)e(a)f(point)h Fj(x)i Fk(2)g Ff(R)2614 3656 y Fh(n)2686 3692 y Fl(is)e(de\256ned)g(by)1366 3856 y Fe(p)s(or\()p Fj(A;)17 b(x)p Fe(\))44 b(=)g(lim)17 b(inf)2005 3916 y Fh(r)r Fi(#)p Fg(0)2209 3856 y Fe(p)s(or\()p Fj(A;)g(x;)g(r)s Fe(\))25 b Fj(;)165 4078 y Fl(wher)l(e)438 4242 y Fe(p)s(or\()p Fj(A;)17 b(x;)g(r)s Fe(\))27 b(=)g(sup)q Fk(f)p Fj(p)h Fk(\025)g Fe(0)f(:)70 b Fl(ther)l(e)26 b(is)g Fj(z)32 b Fk(2)c Ff(R)2183 4201 y Fh(n)2255 4242 y Fl(suc)n(h)e(that)g Fj(B)5 b Fe(\()p Fj(z)t(;)17 b(pr)s Fe(\))28 b Fk(\032)g Fj(B)5 b Fe(\()p Fj(x;)17 b(r)s Fe(\))22 b Fk(n)g Fj(A)p Fk(g)p Fj(:)165 4406 y Fl(Her)l(e)o(,)i Fj(B)5 b Fe(\()p Fj(z)t(;)17 b(\013)q Fe(\))24 b Fl(is)f(the)h(closed)f(ball) h(with)g(r)n(adius)f Fj(\013)h Fl(and)f(with)h(center)g(at)g Fj(z)t Fl(.)37 b(The)23 b(por)-5 b(osity)24 b(of)f Fj(A)28 b Fk(\032)g Ff(R)3807 4369 y Fh(n)3877 4406 y Fl(is)1346 4569 y Fe(p)s(or)o(\()p Fj(A)p Fe(\))g(=)g(inf)6 b Fk(f)p Fe(p)s(or)o(\()p Fj(A;)17 b(x)p Fe(\))28 b(:)g Fj(x)g Fk(2)g Fj(A)p Fk(g)d Fj(:)165 4775 y Fm(W)-8 b(e)26 b(do)g(not)f(kno)m(w)h(an)n(y)g(e)n(xample)i(where)360 4994 y Fk(\017)e Fm(The)g(porosity)f(of)h(the)g(measure)g(is)1709 4955 y Fg(1)p 1709 4971 V 1709 5028 a(2)1754 4994 y Fm(,)360 5174 y Fk(\017)g Fm(and)f(the)i(dimension)f(of)f(the)h(measure)h(is)e Fj(n)h Fm(\(in)g(the)g(ambient)h(space)f Ff(R)2991 5138 y Fh(n)3037 5174 y Fm(\).)3894 5554 y Fq(6)p eop %%Page: 7 7 bop 165 2915 a @beginspecial 34 @llx 46 @lly 531 @urx 501 @ury 3230 @rwi @setspecial %%BeginDocument: doublingp.ps /m {moveto} bind def /l {lineto} bind def /RJ { stringwidth neg exch neg exch rmoveto } bind def /CS { stringwidth 2 div neg exch 2 div neg exch rmoveto } bind def 0.25 0.25 scale 1 setlinecap mark /ISOLatin1Encoding 8#000 1 8#054 {StandardEncoding exch get} for /minus 8#056 1 8#217 {StandardEncoding exch get} for /dotlessi 8#301 1 8#317 {StandardEncoding exch get} for /space /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen /registered /macron /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedilla /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis /ISOLatin1Encoding where not {256 array astore def} if cleartomark /makeISOEncoded { findfont /curfont exch def /newfont curfont maxlength dict def /ISOLatin1 (-ISOLatin1) def /curfontname curfont /FontName get dup length string cvs def /newfontname curfontname length ISOLatin1 length add string dup 0 curfontname putinterval dup curfontname length ISOLatin1 putinterval def curfont { exch dup /FID ne { dup /Encoding eq { exch pop ISOLatin1Encoding exch } if dup /FontName eq { exch pop newfontname exch } if exch newfont 3 1 roll put } { pop pop } ifelse } forall newfontname newfont definefont } def /Times-Roman makeISOEncoded pop /Times-Bold makeISOEncoded pop /Times-Italic makeISOEncoded pop /Times-BoldItalic makeISOEncoded pop /Helvetica makeISOEncoded pop /Helvetica-Bold makeISOEncoded pop /Helvetica-Oblique makeISOEncoded pop /Helvetica-BoldOblique makeISOEncoded pop 0.000000 0.000000 0.000000 setrgbcolor 1 setlinewidth /Times-Italic-ISOLatin1 findfont 60 scalefont setfont [] 0 setdash 410 349 m 410 1748 l 2090 1748 l 2090 349 l 410 349 l stroke [] 0 setdash 410 349 m 410 359 l 620 349 m 620 359 l 829 349 m 829 359 l 1040 349 m 1040 359 l 1250 349 m 1250 359 l 1460 349 m 1460 359 l 1669 349 m 1669 359 l 1880 349 m 1880 359 l 2090 349 m 2090 359 l 410 1748 m 410 1738 l 620 1748 m 620 1738 l 829 1748 m 829 1738 l 1040 1748 m 1040 1738 l 1250 1748 m 1250 1738 l 1460 1748 m 1460 1738 l 1669 1748 m 1669 1738 l 1880 1748 m 1880 1738 l 2090 1748 m 2090 1738 l stroke 410 349 m 410 369 l 829 349 m 829 369 l 1250 349 m 1250 369 l 1669 349 m 1669 369 l 2090 349 m 2090 369 l 410 1748 m 410 1728 l 829 1748 m 829 1728 l 1250 1748 m 1250 1728 l 1669 1748 m 1669 1728 l 2090 1748 m 2090 1728 l /Times-Italic-ISOLatin1 findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont stroke 410 299 m gsave 410 299 translate 0 rotate 0 -20 m (0) CS (0) show grestore newpath 829 299 m gsave 829 299 translate 0 rotate 0 -20 m (4) CS (4) show grestore newpath 1250 299 m gsave 1250 299 translate 0 rotate 0 -20 m (8) CS (8) show grestore newpath 1669 299 m gsave 1669 299 translate 0 rotate 0 -20 m (12) CS (12) show grestore newpath 2090 299 m gsave 2090 299 translate 0 rotate 0 -20 m (16) CS (16) show grestore newpath /Helvetica-ISOLatin1 findfont 60 scalefont setfont /Helvetica-Bold-ISOLatin1 findfont 60 scalefont setfont 1250 199 m gsave 1250 199 translate 0 rotate 0 0 m /Times-Roman-ISOLatin1 findfont 60 scalefont setfont (r [Mpc/h]) CS /Times-Italic-ISOLatin1 findfont 60 scalefont setfont (r) show /Times-Roman-ISOLatin1 findfont 60 scalefont setfont ( [Mpc/h]) show grestore newpath 410 349 m 420 349 l 410 489 m 420 489 l 410 629 m 420 629 l 410 769 m 420 769 l 410 909 m 420 909 l 410 1049 m 420 1049 l 410 1188 m 420 1188 l 410 1328 m 420 1328 l 410 1468 m 420 1468 l 410 1608 m 420 1608 l 410 1748 m 420 1748 l 2090 349 m 2080 349 l 2090 489 m 2080 489 l 2090 629 m 2080 629 l 2090 769 m 2080 769 l 2090 909 m 2080 909 l 2090 1049 m 2080 1049 l 2090 1188 m 2080 1188 l 2090 1328 m 2080 1328 l 2090 1468 m 2080 1468 l 2090 1608 m 2080 1608 l 2090 1748 m 2080 1748 l stroke 410 349 m 430 349 l 410 629 m 430 629 l 410 909 m 430 909 l 410 1188 m 430 1188 l 410 1468 m 430 1468 l 410 1748 m 430 1748 l 2090 349 m 2070 349 l 2090 629 m 2070 629 l 2090 909 m 2070 909 l 2090 1188 m 2070 1188 l 2090 1468 m 2070 1468 l 2090 1748 m 2070 1748 l /Helvetica-Bold-ISOLatin1 findfont 60 scalefont setfont /Helvetica-ISOLatin1 findfont 60 scalefont setfont stroke 372 349 m gsave 372 349 translate 0 rotate 0 -20 m (0.0) RJ (0.0) show grestore newpath 372 629 m gsave 372 629 translate 0 rotate 0 -20 m (2.0) RJ (2.0) show grestore newpath 372 909 m gsave 372 909 translate 0 rotate 0 -20 m (4.0) RJ (4.0) show grestore newpath 372 1188 m gsave 372 1188 translate 0 rotate 0 -20 m (6.0) RJ (6.0) show grestore newpath 372 1468 m gsave 372 1468 translate 0 rotate 0 -20 m (8.0) RJ (8.0) show grestore newpath 372 1748 m gsave 372 1748 translate 0 rotate 0 -20 m (10.0) RJ (10.0) show grestore newpath /Helvetica-ISOLatin1 findfont 60 scalefont setfont /Helvetica-Bold-ISOLatin1 findfont 60 scalefont setfont 199 1048 m gsave 199 1048 translate 90 rotate 0 0 m /T{/Times-Italic-ISOLatin1 findfont 60 scalefont setfont}def /S{/Symbol findfont 60 scalefont setfont}def (mB\(\(2r\)\)/m\(BG\(r\)\)) CS S (m\() show T (B) show S (\() show T (2r) show S (\)\) / m\() show T (B) show S (\()show T (r) show S (\)\)) show grestore newpath /Times-Roman-ISOLatin1 findfont 90 scalefont setfont /Helvetica-ISOLatin1 findfont 90 scalefont setfont 1250 1916 m gsave 1250 1916 translate 0 rotate 0 0 m (CfA2South-SSRS2 VL19) CS (CfA2South-SSRS2 VL19) show grestore newpath 441 511 m 445 511 l 448 511 l 452 511 l 456 511 l 460 511 l 465 511 l 470 511 l 476 511 l 483 511 l 490 511 l 497 673 l 506 673 l 515 673 l 525 673 l 537 673 l 549 673 l 562 673 l 577 511 l 593 673 l 611 673 l 630 673 l 652 834 l 675 996 l 701 1320 l 729 1400 l 760 1077 l 793 1239 l 830 1400 l stroke 1.000000 0.000000 0.000000 setrgbcolor 436 511 m 439 511 l 441 511 l 445 511 l 448 511 l 452 673 l 456 673 l 460 673 l 465 673 l 470 673 l 476 673 l 483 673 l 490 592 l 497 592 l 506 592 l 515 592 l 525 592 l 537 592 l 549 592 l 562 565 l 577 565 l 593 619 l 611 619 l 630 619 l 652 619 l 675 673 l 701 713 l 729 754 l 760 705 l 793 705 l 830 770 l 871 867 l 916 861 l 964 727 l 1018 851 l 1077 1084 l 1141 1320 l stroke 0.000000 0.000000 0.000000 setrgbcolor 0.000000 1.000000 0.000000 setrgbcolor 510 511 m 520 511 l 531 673 l 543 673 l 555 996 l 570 1320 l 585 1643 l 602 1643 l 620 1643 l 641 996 l 663 1239 l 688 834 l 714 673 l 744 632 l 776 713 l 812 774 l 850 834 l 893 761 l 940 821 l 991 1037 l 1047 1192 l 1108 1131 l 1176 1235 l 1250 1360 l 1331 1337 l 1419 1259 l 1517 1101 l 1624 1014 l stroke 0.000000 0.000000 0.000000 setrgbcolor 0.000000 0.000000 1.000000 setrgbcolor 460 673 m 465 673 l 470 834 l 476 834 l 483 834 l 490 834 l 497 592 l 506 592 l 515 673 l 525 619 l 537 619 l 549 619 l 562 619 l 577 619 l 593 780 l 611 754 l 630 834 l 652 931 l 675 964 l 701 996 l 729 1061 l 760 895 l 793 851 l 830 737 l 871 789 l 916 826 l 964 899 l 1018 930 l 1077 912 l 1141 934 l 1212 938 l 1289 884 l 1374 889 l stroke 0.000000 0.000000 0.000000 setrgbcolor 1.000000 1.000000 0.000000 setrgbcolor 479 834 m 486 834 l 494 996 l 502 996 l 510 754 l 520 673 l 531 673 l 543 673 l 555 727 l 570 673 l 585 754 l 602 834 l 620 780 l 641 888 l 663 996 l 688 1019 l 714 976 l 744 915 l 776 780 l 812 834 l 850 778 l 893 740 l 940 734 l 991 756 l 1047 774 l 1108 810 l 1176 794 l 1250 828 l stroke 0.000000 0.000000 0.000000 setrgbcolor 0.737255 0.560784 0.560784 setrgbcolor 445 673 m 448 673 l 452 673 l 456 673 l 460 673 l 465 673 l 470 511 l 476 592 l 483 592 l 490 592 l 497 592 l 506 592 l 515 592 l 525 754 l 537 619 l 549 619 l 562 673 l 577 673 l 593 673 l 611 673 l 630 543 l 652 543 l 675 576 l 701 565 l 729 565 l 760 700 l 793 861 l 830 1131 l 871 1320 l stroke 0.000000 0.000000 0.000000 setrgbcolor 0.580392 0.000000 0.827451 setrgbcolor 443 511 m 446 511 l 450 511 l 454 511 l 458 511 l 463 511 l 468 511 l 473 511 l 479 511 l 486 511 l 494 673 l 502 834 l 510 834 l 520 834 l 531 834 l 543 834 l 555 834 l 570 592 l 585 511 l 602 511 l 620 511 l 641 511 l 663 511 l 688 511 l 714 511 l 744 565 l stroke 0.000000 0.000000 0.000000 setrgbcolor showpage %%EndDocument @endspecial 510 x Fn(Figure)14 b(2.)34 b(V)-8 b(eri\256cation)14 b(of)g(the)g(doubling)g(condition)f(for)h(se)n(v)o(eral)g(central)g(points)h (of)e(tw)o(o)i(galaxy)f(catalogs.)24 b(W)-5 b(e)14 b(thank)g(M.)g(Montuori)g (and)g(F)-5 b(.)14 b(Sylos-Labini)165 3504 y(for)20 b(these)i(calculations.) 36 b(The)20 b(\257uctuations)h(seem)g(quite)g(bounded,)h(e)o(xcept)f(at)f (short)h(distances,)i(where)e(there)f(are)h(problems)f(because)i(one)f(runs)g (out)f(of)165 3583 y(data.)165 3929 y Fm(Of)25 b(course,)h(this)g(w)o(ould)g (ha)n(v)n(e)g(to)g(be)g(a)f(measure)i(which)e(violates)i(the)f(doubling)g (condition.)315 4087 y(The)i(doubling)g(condition)g(is)f(a)h(bound)f(on)g (the)g(amplitudes)i(of)e(the)h(\257uctuations)g(of)f(the)h(inte)n(grated)165 4203 y(density)40 b(of)e(the)i(measure.)77 b(The)40 b(role)f(of)f(this)h (condition)h(in)f(e)n(xperiments)i(in)e(g)o(alaxy)h(distrib)n(utions)165 4319 y(is)d(some)m(what)h(obscure.)71 b(But,)40 b(for)c(e)n(xample,)42 b(in)37 b([9])g(the)g(authors)g(measured)g(2)g(periods)g(of)f(density)165 4436 y(\257uctuations)22 b(\(of)e(about)h(the)g(same)g(amplitude\))h(and)e (so)g(in)h(a)g(v)n(ery)g(weak)f(sense,)i(the)f(doubling)g(condition)165 4552 y(seems)42 b(e)n(xperimentally)i(satis\256ed.)84 b(Another)41 b(e)n(xample)i(is)e(gi)m(v)n(en)h(in)g(Fig.)f(2)g(where)g(the)h(results)f(of) 165 4668 y(measuring)27 b(the)f(doubling)g(condition)g(for)g(certain)h (catalogs)f(of)g(g)o(alaxies)h(are)f(sho)m(wn.)315 4826 y(F)n(or)41 b(re)n(gular)g(recursi)m(v)n(ely)h(constructed)f(fractals)h(the)e(doubling)h (condition)h(is)e(al)o(w)o(ays)g(satis\256ed.)165 4942 y(Finally)-7 b(,)34 b(we)d(belie)m(v)n(e)i(that)e(fractals)h(formed)f(in)g(Nature)g(by)g (a)f(ph)o(ysical)i(la)n(w)g(ha)n(v)n(e)f(not)g(only)g(the)g(same)165 5058 y(dimension)c(e)m(v)n(erywhere,)g(b)n(ut)f(also)g(satisfy)g(the)g (doubling)g(condition.)39 b(The)27 b(reason)e(for)h(this)g(is)f(that)i(for) 165 5174 y(e)n(xample)35 b(an)e(attractor)i(looks)e(e)m(v)n(erywhere)h (similar)g(because)g(it)f(is)g(created)i(by)d(a)i(\(smooth\))f(ph)o(ysical) 3894 5554 y Fq(7)p eop %%Page: 8 8 bop 165 533 a Fm(la)n(w)-7 b(,)41 b(which)36 b(transports)h(the)g(structure)g (of)f(the)h(fractal)h(around)e(in)h(space)g(\(making)g(at)g(most)g(smooth)165 649 y(coordinate)j(changes)e(locally\).)77 b(This)38 b(point)h(of)f(vie)m(w)h (has)f(been)h(adv)n(ocated)g(in)f([3],)k(and)c(has)g(been)165 765 y(rigorously)26 b(v)n(eri\256ed)g(for)g(a)g(fe)m(w)f(non-tri)m(vial)j(e)n (xamples)f(of)e(dynamical)i(systems.)315 923 y(Another)41 b(class)g(of)f (measurements)i(tak)o(es)f(the)g(density)g(\257uctuations)h(of)e(the)h(mass)g (itself)g(as)g(an)165 1039 y(indicator)35 b(of)f(the)g(dimension)g(of)g(the)g (measure.)62 b(This)34 b(seems)g(a)f(mathematically)38 b(inaccessible)d (\(and)165 1156 y(probably)46 b(wrong\))g(criterion)h(for)e(dimension)i (measurements.)99 b(It)46 b(might)g(be)g(that)h(this)e(idea)i(is)f(a)165 1272 y(consequence)26 b(of)f(the)g(re)n(gular)h(oscillations)h(one)e(gets)g (for)g(re)n(gular)h(Cantor)f(sets.)37 b(The)25 b(only)g(case)h(where)165 1388 y(a)g(rigorous)f(result)h(is)g(kno)m(wn)f(is)h(that)g(of)g(inte)n(ger)h (dimension:)165 1565 y Fq(Theor)n(em)f(6)49 b Fl(\(Mar)o(str)n(and\))17 b(Let)g Fj(s)g Fl(be)g(a)g(positive)g(inte)l(g)o(er)-11 b(.)33 b(Suppose)17 b(that)g(ther)l(e)g(e)n(xists)g(a)g(Radon)g(measur)l(e)165 1681 y Fj(\026)26 b Fl(on)f Ff(R)461 1645 y Fh(n)533 1681 y Fl(suc)n(h)h(that)g(the)g(density)1751 1927 y Fe(lim)1767 1987 y Fh(r)r Fi(#)p Fg(0)1913 1859 y Fj(\026)p Fe(\()p Fj(B)5 b Fe(\()p Fj(x;)17 b(r)s Fe(\)\))p 1913 1903 436 4 v 2089 1995 a Fj(r)2136 1966 y Fh(s)165 2160 y Fl(e)n(xists)26 b(and)g(is)f(positive)i (and)e(\256nite)i(in)f(a)f(set)h(of)g(positive)g Fj(\026)p Fl(-measur)l(e)n(.)37 b(Then)26 b Fj(s)f Fl(is)h(an)f(inte)l(g)o(er)-11 b(.)315 2336 y Fm(\(F)n(or)26 b(the)g(proof)f(see)h([7])f(Theorem)i(14.10.\)) 315 2494 y(It)37 b(is)g(well-kno)m(wn)h(that)f(one)h(cannot)f(e)n(xpect)i(an) e(inequality)h(in)g(the)f(sense)g(opposite)g(to)g(the)h(one)165 2610 y(stated)25 b(in)g(the)g(theorem.)38 b(That)25 b(is,)g(big)g(v)n(oids)f (imply)h(small)g(dimension,)h(b)n(ut)f(small)g(v)n(oids)f(do)g(not)h(imply) 165 2726 y(big)h(dimension.)38 b(This)26 b(is)g(illustrated)h(by)e(the)h (follo)m(wing)h(e)n(xample)g(in)f Ff(R)p Fm(,)g Fl(i.e)n(.)p Fm(,)i(in)e(one)f(dimension.)165 2938 y Fq(Example)j(3)g Fm(W)-8 b(e)28 b(will)h(construct)f(a)g(sequence)h(of)e(measures)h Fj(\026)2459 2902 y Fg(\()p Fh(n)p Fg(\))2561 2938 y Fm(,)h(all)g(of)e (dimension)i Fe(dim)3474 2953 y Fg(p)3516 2938 y Fe(\()p Fj(\026)3613 2902 y Fg(\()p Fh(n)p Fg(\))3714 2938 y Fe(\))i(=)3900 2899 y Fg(1)p 3900 2915 36 4 v 3900 2972 a(2)165 3083 y Fm(in)22 b Ff(R)f Fm(with)g(porosity)h Fe(p)s(or)o(\()p Fj(\026)1169 3047 y Fg(\()p Fh(n)p Fg(\))1271 3083 y Fe(\))27 b Fk(\024)1455 3044 y Fg(1)p 1451 3060 43 4 v 1451 3117 a Fh(n)1504 3083 y Fm(.)36 b(The)22 b(set)f Fj(A)1952 3047 y Fg(\()p Fh(n)p Fg(\))2076 3083 y Fm(is)g(a)g(Cantor)h(set)g(obtained)g(recursi)m(v)n(ely)h(as)e(follo)m (ws:)165 3199 y(Di)m(vide)27 b(the)f(interv)m(al)i Fe([0)p Fj(;)17 b Fe(1])25 b Fm(into)i Fj(n)1415 3163 y Fg(2)1480 3199 y Fm(equal)g(subinterv)m(als)g(and)e(select)i Fj(n)f Fm(of)g(these)g (subinterv)m(als,)i(namely)165 3345 y(the)d Fe(1)363 3309 y Fm(st)435 3345 y(,)f Fj(n)19 b Fe(+)g(1)705 3309 y Fm(st)777 3345 y(,)25 b(and)f(so)f(on.)37 b(The)24 b(measure)h(at)g(this)f(le)m(v)n(el) i(of)e(the)g(construction)h(is)f(obtained)h(by)f(gi)m(ving)165 3462 y(the)j(same)g(weight)870 3422 y Fg(1)p 866 3438 V 866 3496 a Fh(n)945 3462 y Fm(to)g(each)g(subinterv)m(al)h(of)f Fj(A)1926 3426 y Fg(\()p Fh(n)p Fg(\))2028 3462 y Fm(.)40 b(In)26 b(other)h(w)o(ords,)f(the)h(unit)g(measure)h(is)e(uniformly)165 3578 y(distrib)n(uted)h(on)e(the)i Fj(n)e Fm(interv)m(als)i(constructed)g(so) e(f)o(ar)-6 b(.)315 3736 y(No)m(w)25 b(repeat)i(inducti)m(v)n(ely)h(the)e (procedure)f(for)h(each)g(of)f(the)h Fj(n)f Fm(interv)m(als,)j(di)m(viding)e (it)g(into)g Fj(n)3659 3699 y Fg(2)3724 3736 y Fm(equal)165 3888 y(pieces)h(and)e(selecting)i(each)g Fj(n)1277 3851 y Fm(th)1386 3888 y(among)f(them.)38 b(Gi)m(v)n(e)27 b(each)f(of)f(these)h(interv)m(als)h (weight)3440 3848 y Fg(1)p 3419 3864 78 4 v 3419 3922 a Fh(n)3462 3903 y Fa(2)3507 3888 y Fm(.)315 4074 y(Continuing)33 b(inde\256nitely)g(in)f (this)f(f)o(ashion,)j(one)e(obtains)g(the)g(Cantor)g(set)g Fj(A)3137 4038 y Fg(\()p Fh(n)p Fg(\))3270 4074 y Fm(and)g(the)g(measure)165 4191 y Fj(\026)224 4154 y Fg(\()p Fh(n)p Fg(\))354 4191 y Fm(on)c(it.)46 b(The)29 b(dimension)g(of)f(this)h(measure)g(is)f Fe(dim)2168 4206 y Fg(p)2210 4191 y Fe(\()p Fj(\026)2307 4154 y Fg(\()p Fh(n)p Fg(\))2408 4191 y Fe(\))k(=)f(log\()p Fj(n)p Fe(\))p Fj(=)17 b Fe(log)o(\()p Fj(n)3132 4154 y Fg(2)3172 4191 y Fe(\))31 b(=)3358 4151 y Fg(1)p 3358 4167 36 4 v 3358 4225 a(2)3404 4191 y Fm(,)e(and)f(it)h(is)g(not)165 4332 y(dif)m(\256cult)d(to)e(check)g (that)h(the)g(porosity)f(is)g(less)g(than)2040 4293 y Fg(1)p 2036 4309 43 4 v 2036 4366 a Fh(n)2089 4332 y Fm(.)36 b(\(Since)25 b(the)f(g)o(aps)h(become)g(smaller)g(for)e(lar)n(ger)i Fj(n)p Fm(,)165 4477 y(see)g([2].\))36 b(In)24 b(this)h(case,)g(it)g(is)f(easy)h(to) f(compute)h(numerically)h(the)f(porosity)g(of)f(the)g(sets)g Fj(A)3423 4441 y Fg(\()p Fh(n)p Fg(\))3550 4477 y Fm(when)g Fj(n)g Fm(is)165 4593 y(not)29 b(too)f(lar)n(ge.)46 b(In)27 b(Fig.)i(3)f(we)g(sho)m(w)f(the)i(quotient)g Fe(p)s(or\()p Fj(\026;)17 b(x;)g(r)m(;)g(")30 b Fe(=)h(0\))c Fm(as)h(a)g(function)h(of)f Fj(r)j Fm(when)c Fj(x)i Fm(is)165 4709 y(the)23 b(leftmost)h(point)f(of)f Fj(A)1081 4673 y Fg(\()p Fh(n)p Fg(\))1183 4709 y Fm(.)37 b(Similar)l(,)24 b(b)n(ut)f(more)g(irre)n(gular)h(pictures)f(are)g(obtained)g(when)f(one)h (chooses)165 4826 y(another)j(point)f Fj(x)p Fm(.)38 b(One)24 b(can)i(understand)f(the)g(origin)g(of)g(the)g(oscillations)i(by)d(looking)i (at)f(Fig.)g(4,)h(where)165 4942 y(we)35 b(plot)h(the)f(radius)g Fj(\032)h Fm(of)f(the)g(lar)n(gest)h(empty)g(interv)m(al)h(as)e(a)g(function) h(of)e Fj(r)s Fm(.)66 b(W)-8 b(e)35 b(see)g(that)h Fj(\032)f Fm(gro)m(ws)165 5058 y(linearly)-7 b(,)32 b(until)d(a)g(point)g(of)g(the)g (Cantor)g(set)f(is)h(hit,)h(and)f(then)g(it)g(stays)g(constant)g(until)g(a)g (bigger)g(v)n(oid)g(is)165 5174 y(found.)38 b(Then)26 b(Fig.)g(3)f(is)g (obtained)i(by)f(di)m(viding)g(the)h(v)m(alues)f(obtained)h(in)e(Fig.)h(4)g (by)f Fj(r)s Fm(.)3894 5554 y Fq(8)p eop %%Page: 9 9 bop 402 2862 a @beginspecial 5 @llx 1 @lly 534 @urx 537 @ury 2856 @rwi @setspecial %%BeginDocument: lacundivp.ps /X-CENTER-PLOT 0 def /Y-CENTER-PLOT 0 def clippath pathbbox /page-height exch def /page-width exch def pop pop 538.583 page-width gt { % Rotation required Y-CENTER-PLOT 1 eq { % Center in y page-height 538.583 sub 2 div } { % Don't center in y 0 } ifelse X-CENTER-PLOT 1 eq { % Center in x page-width 538.583 sub 2 div } { % Don't center in x 0 } ifelse translate } { % No rotation - just handle centering X-CENTER-PLOT 1 eq { % Center in x page-width 538.583 sub 2 div } { % Don't center in x 0 } ifelse Y-CENTER-PLOT 1 eq { % Center in y page-height 538.583 sub 2 div } { % Don't center in y 0 } ifelse translate } ifelse 0.06 0.06 scale /choose-font % stack: fontsize fontname => --- { findfont exch scalefont setfont newpath 0 0 moveto (0) true charpath flattenpath pathbbox /top exch def pop /bottom exch def pop bottom top bottom top add 2 div /center-font-val exch def /upper-font-val exch def /lower-font-val exch def } def /center-x-just % stack: (string) x y => (string) newx y { exch 2 index stringwidth pop 2 div sub exch } def /left-x-just % stack: (string) x y => (string) newx y { } def /right-x-just % stack: (string) x y => (string) newx y { exch 2 index stringwidth pop sub exch } def /center-y-just % stack: (string) x y => (string) x newy { center-font-val sub } def /lower-y-just % stack: (string) x y => (string) x newy { lower-font-val sub } def /upper-y-just % stack: (string) x y => (string) x newy { upper-font-val sub } def /just-string % stack: (string) x y just => --- { dup 0 eq { pop center-x-just center-y-just } if dup 1 eq { pop left-x-just center-y-just } if dup 2 eq { pop left-x-just upper-y-just } if dup 3 eq { pop center-x-just upper-y-just } if dup 4 eq { pop right-x-just upper-y-just } if dup 5 eq { pop right-x-just center-y-just } if dup 6 eq { pop right-x-just lower-y-just } if dup 7 eq { pop center-x-just lower-y-just } if dup 8 eq { pop left-x-just lower-y-just } if moveto show } def /mark0 {/size exch def /y exch def /x exch def newpath x size sub y size sub moveto size size add 0 rlineto 0 size size add rlineto 0 size size add sub 0 rlineto closepath fill} def /mark1 {/size exch def /y exch def /x exch def newpath x size sub y size sub moveto size size add 0 rlineto 0 size size add rlineto 0 size size add sub 0 rlineto closepath stroke} def /mark2 {/size exch def /y exch def /x exch def newpath x y moveto x y size 0 360 arc stroke} def /mark3 {/size exch def /y exch def /x exch def newpath x size sub y size sub moveto x size add y size add lineto x size sub y size add moveto x size add y size sub lineto stroke} def /mark4 {/size exch def /y exch def /x exch def newpath x size sub y moveto x y size add lineto x size add y lineto x y size sub lineto closepath stroke} def /mark5 {/size exch def /y exch def /x exch def x y size mark1 newpath x size sub y moveto size size add 0 rlineto stroke} def /mark6 {/size exch def /y exch def /x exch def newpath x y moveto x y size 0 360 arc fill} def /mark7 {/size exch def /y exch def /x exch def newpath x y moveto x size sub y size sub lineto x size add y size sub lineto closepath fill newpath x y moveto x size add y size add lineto x size sub y size add lineto closepath fill} def 0 setgray 300 /Times-Bold choose-font () 4488 8856 3 just-string 300 /Times-Roman choose-font /sss{setrgbcolor}def /l{lineto}def 14 setlinewidth [] 0 setdash 1 0 1 sss newpath 8061 7986 moveto 8886 7986 lineto stroke 14 setlinewidth 1 0 0 sss [] 0 setdash newpath 8061 7594 moveto 8886 7594 lineto stroke 14 setlinewidth 0 1 0 sss [] 0 setdash newpath 8061 7202 moveto 8886 7202 lineto stroke 14 setlinewidth 0 0 1 sss [] 0 setdash newpath 8061 6810 moveto 8886 6810 lineto stroke 0 0 0 sss (0.00) 1245 820 5 just-string (0.10) 1245 1499 5 just-string (0.20) 1245 2179 5 just-string (0.30) 1245 2859 5 just-string (0.40) 1245 3538 5 just-string (0.50) 1245 4218 5 just-string (0.60) 1245 4898 5 just-string (0.70) 1245 5577 5 just-string (0.80) 1245 6257 5 just-string (0.90) 1245 6937 5 just-string (1.00) 1245 7616 5 just-string (1e-06) 2501 90 7 just-string (1e-04) 4224 90 7 just-string (1e-02) 5947 90 7 just-string (1e+00) 7669 90 7 just-string [] 0 setdash 8 setlinewidth newpath 1335 820 moveto 7971 820 l stroke [] 0 setdash 2 setlinewidth newpath 1335 1499 moveto 7971 1499 l stroke [] 0 setdash 2 setlinewidth newpath 1335 2179 moveto 7971 2179 l stroke [] 0 setdash 2 setlinewidth newpath 1335 2859 moveto 7971 2859 l stroke [] 0 setdash 2 setlinewidth newpath 1335 3538 moveto 7971 3538 l stroke [] 0 setdash 2 setlinewidth newpath 1335 4218 moveto 7971 4218 l stroke [] 0 setdash 2 setlinewidth newpath 1335 4898 moveto 7971 4898 l stroke [] 0 setdash 2 setlinewidth newpath 1335 5577 moveto 7971 5577 l stroke [] 0 setdash 2 setlinewidth newpath 1335 6257 moveto 7971 6257 l stroke [] 0 setdash 2 setlinewidth newpath 1335 6937 moveto 7971 6937 l stroke [] 0 setdash 2 setlinewidth newpath 1335 7616 moveto 7971 7616 l stroke [] 0 setdash 2 setlinewidth newpath 2501 7956 moveto 2501 480 l stroke [] 0 setdash 2 setlinewidth newpath 4224 7956 moveto 4224 480 l stroke [] 0 setdash 2 setlinewidth newpath 5947 7956 moveto 5947 480 l stroke [] 0 setdash 2 setlinewidth newpath 7669 7956 moveto 7669 480 l stroke 14 setlinewidth [] 0 setdash 1 0 1 sss newpath 1896 4218 moveto 1905 4135 l 1914 4056 l 1923 3981 l 1932 3909 l 1940 3840 l 1948 3775 l 1956 3712 l 1964 3652 l 1972 3594 l 1979 3538 l 1987 3485 l 1994 3434 l 2001 3384 l 2008 3337 l 2015 3291 l 2022 3247 l 2028 3205 l 2035 3163 l 2041 3124 l 2048 3085 l 2155 2519 l 2239 2179 l 2307 1953 l 2393 3115 l 2463 3882 l 2522 4425 l 2572 4831 l 2617 5145 l 2657 5395 l 2693 5599 l 2726 5769 l 2756 5913 l 2784 6036 l 2810 6142 l 2835 6235 l 2858 6317 l 2879 6389 l 2900 6454 l 2919 6513 l 2937 6566 l 2955 6614 l 2971 6658 l 2977 6672 l 2987 6514 l 2997 6364 l 3007 6222 l 3017 6087 l 3026 5959 l 3237 3746 l 3242 3706 l 3247 3667 l 3252 3629 l 3257 3592 l 3262 3556 l 3388 2771 l 3392 2753 l 3395 2735 l 3398 2718 l 3402 2701 l 3405 2684 l 3496 2283 l 3498 2273 l 3501 2263 l 3504 2253 l 3506 2243 l 3509 2234 l 3579 1990 l 3581 1984 l 3583 1977 l 3586 1971 l 3588 1965 l 3590 1959 l 3648 1795 l 3733 2986 l 3803 3772 l 3861 4330 l 3912 4747 l 3957 5070 l 3997 5327 l 4033 5537 l 4066 5712 l 4096 5860 l 4124 5986 l 4150 6096 l 4174 6192 l 4197 6276 l 4218 6351 l 4239 6418 l 4258 6478 l 4276 6533 l 4294 6582 l 4311 6627 l 4318 6646 l 4318 6642 l 4318 6637 l 4319 6633 l 4319 6628 l 4319 6624 l 4328 6489 l 4328 6484 l 4329 6480 l 4329 6476 l 4329 6472 l 4330 6467 l 4338 6339 l 4338 6335 l 4339 6331 l 4339 6327 l 4339 6323 l 4339 6319 l 4348 6198 l 4348 6194 l 4348 6190 l 4349 6186 l 4349 6183 l 4349 6179 l 4357 6063 l 4358 6060 l 4358 6056 l 4358 6053 l 4358 6049 l 4359 6045 l 4367 5936 l 4367 5932 l 4367 5929 l 4367 5925 l 4368 5922 l 4368 5918 l 4382 5734 l 4395 5562 l 4408 5403 l 4420 5253 l 4432 5113 l 4444 4982 l 4455 4858 l 4466 4741 l 4477 4632 l 4487 4528 l 4497 4429 l 4507 4336 l 4516 4247 l 4526 4163 l 4535 4083 l 4544 4006 l 4552 3933 l 4561 3863 l 4569 3797 l 4577 3733 l 4577 3732 l 4577 3731 l 4578 3730 l 4578 3728 l 4578 3727 l 4582 3693 l 4582 3692 l 4583 3691 l 4583 3690 l 4583 3689 l 4583 3688 l 4587 3654 l 4588 3653 l 4588 3652 l 4588 3651 l 4588 3650 l 4588 3649 l 4592 3616 l 4593 3615 l 4593 3614 l 4593 3613 l 4593 3612 l 4593 3611 l 4597 3580 l 4598 3579 l 4598 3578 l 4598 3577 l 4598 3576 l 4598 3575 l 4602 3544 l 4602 3543 l 4603 3542 l 4603 3541 l 4603 3540 l 4603 3539 l 4729 2762 l 4729 2761 l 4729 2760 l 4729 2759 l 4732 2744 l 4732 2743 l 4733 2743 l 4733 2742 l 4736 2727 l 4736 2726 l 4736 2725 l 4736 2724 l 4739 2709 l 4739 2708 l 4740 2707 l 4742 2693 l 4743 2692 l 4743 2691 l 4743 2690 l 4746 2676 l 4746 2675 l 4746 2674 l 4836 2276 l 4837 2276 l 4837 2275 l 4839 2266 l 4839 2265 l 4842 2256 l 4842 2255 l 4844 2247 l 4844 2246 l 4845 2245 l 4847 2237 l 4847 2236 l 4849 2227 l 4850 2226 l 4853 2212 l 4857 2198 l 4861 2184 l 4865 2170 l 4868 2157 l 4872 2144 l 4876 2131 l 4879 2119 l 4883 2106 l 4886 2094 l 4890 2082 l 4893 2071 l 4897 2059 l 4900 2048 l 4904 2037 l 4907 2026 l 4910 2016 l 4913 2005 l 4917 1995 l 4920 1985 l 4920 1984 l 4922 1979 l 4922 1978 l 4924 1972 l 4924 1971 l 4926 1966 l 4926 1965 l 4928 1960 l 4928 1959 l 4930 1954 l 4930 1953 l 4988 1791 l 5074 2982 l 5143 3769 l 5202 4328 l 5253 4745 l 5297 5068 l 5337 5325 l 5373 5536 l 5406 5711 l 5436 5859 l 5464 5985 l 5490 6095 l 5515 6190 l 5538 6275 l 5559 6350 l 5579 6417 l 5599 6477 l 5617 6532 l 5635 6581 l 5651 6626 l 5658 6645 l 5659 6641 l 5659 6640 l 5659 6636 l 5659 6632 l 5659 6631 l 5660 6627 l 5660 6623 l 5660 6622 l 5669 6488 l 5669 6487 l 5669 6484 l 5669 6483 l 5669 6479 l 5670 6475 l 5670 6471 l 5670 6470 l 5670 6467 l 5670 6466 l 5679 6339 l 5679 6338 l 5679 6335 l 5679 6334 l 5679 6331 l 5679 6330 l 5680 6327 l 5680 6326 l 5680 6323 l 5680 6322 l 5680 6319 l 5680 6318 l 5688 6197 l 5689 6193 l 5689 6190 l 5689 6189 l 5689 6186 l 5689 6185 l 5689 6182 l 5689 6181 l 5690 6178 l 5698 6063 l 5698 6062 l 5698 6059 l 5698 6055 l 5699 6052 l 5699 6051 l 5699 6048 l 5699 6045 l 5699 6044 l 5707 5935 l 5707 5934 l 5707 5931 l 5708 5928 l 5708 5927 l 5708 5924 l 5708 5921 l 5708 5918 l 5708 5917 l 5918 3733 l 5918 3732 l 5918 3731 l 5918 3730 l 5918 3729 l 5918 3728 l 5918 3727 l 5919 3727 l 5923 3693 l 5923 3692 l 5923 3691 l 5923 3690 l 5923 3689 l 5923 3688 l 5924 3688 l 5924 3687 l 5928 3654 l 5928 3653 l 5928 3652 l 5928 3651 l 5928 3650 l 5929 3650 l 5929 3649 l 5929 3648 l 5933 3616 l 5933 3615 l 5933 3614 l 5933 3613 l 5934 3612 l 5934 3611 l 5938 3579 l 5938 3578 l 5938 3577 l 5938 3576 l 5939 3575 l 5939 3574 l 5943 3543 l 5943 3542 l 5943 3541 l 5943 3540 l 5943 3539 l 5944 3539 l 5944 3538 l 5951 3485 l 5958 3434 l 5965 3384 l 5972 3337 l 5979 3291 l 5986 3247 l 5993 3205 l 5999 3163 l 6005 3124 l 6012 3085 l 6018 3048 l 6024 3012 l 6030 2977 l 6036 2944 l 6042 2911 l 6047 2879 l 6053 2849 l 6059 2819 l 6064 2790 l 6069 2762 l 6070 2761 l 6070 2760 l 6070 2759 l 6073 2744 l 6073 2743 l 6073 2742 l 6073 2741 l 6076 2726 l 6077 2725 l 6077 2724 l 6080 2709 l 6080 2708 l 6080 2707 l 6083 2692 l 6083 2691 l 6083 2690 l 6084 2690 l 6086 2676 l 6086 2675 l 6087 2675 l 6087 2674 l stroke 14 setlinewidth newpath 6087 2674 moveto 6087 2674 l 6087 2673 l 6177 2276 l 6177 2275 l 6180 2266 l 6180 2265 l 6182 2256 l 6182 2255 l 6183 2255 l 6185 2246 l 6185 2245 l 6187 2237 l 6187 2236 l 6188 2236 l 6188 2235 l 6190 2227 l 6190 2226 l 6261 1985 l 6261 1984 l 6263 1978 l 6265 1972 l 6265 1971 l 6267 1966 l 6267 1965 l 6269 1960 l 6269 1959 l 6271 1953 l 6329 1791 l 6414 2982 l 6484 3769 l 6543 4328 l 6593 4744 l 6638 5068 l 6678 5325 l 6714 5536 l 6747 5711 l 6777 5859 l 6805 5985 l 6831 6095 l 6855 6190 l 6878 6275 l 6900 6350 l 6920 6417 l 6939 6477 l 6958 6532 l 6975 6581 l 6992 6626 l 6999 6645 l 6999 6641 l 6999 6640 l 7000 6636 l 7000 6632 l 7000 6631 l 7000 6627 l 7001 6623 l 7001 6622 l 7009 6488 l 7009 6487 l 7010 6484 l 7010 6483 l 7010 6479 l 7010 6475 l 7010 6474 l 7010 6471 l 7010 6470 l 7011 6467 l 7011 6466 l 7019 6339 l 7019 6338 l 7020 6335 l 7020 6334 l 7020 6331 l 7020 6330 l stroke 14 setlinewidth newpath 7020 6330 moveto 7020 6330 l 7020 6327 l 7020 6326 l 7020 6323 l 7020 6322 l 7021 6319 l 7021 6318 l 7029 6197 l 7029 6193 l 7030 6190 l 7030 6189 l 7030 6186 l 7030 6185 l 7030 6182 l 7030 6181 l 7030 6178 l 7038 6063 l 7038 6062 l 7039 6062 l 7039 6059 l 7039 6055 l 7039 6052 l 7039 6051 l 7040 6048 l 7040 6045 l 7040 6044 l 7048 5935 l 7048 5934 l 7048 5931 l 7048 5928 l 7048 5927 l 7048 5924 l 7049 5924 l 7049 5921 l 7049 5918 l 7049 5917 l 7258 3733 l 7258 3732 l 7259 3731 l 7259 3730 l 7259 3729 l 7259 3728 l 7259 3727 l 7264 3693 l stroke 14 setlinewidth newpath 7264 3693 moveto 7264 3693 l 7264 3692 l 7264 3691 l 7264 3690 l 7264 3689 l 7264 3688 l 7264 3687 l 7269 3654 l 7269 3653 l 7269 3652 l 7269 3651 l 7269 3650 l 7269 3649 l 7269 3648 l 7274 3616 l 7274 3615 l 7274 3614 l 7274 3613 l 7274 3612 l 7274 3611 l 7279 3579 l 7279 3578 l 7279 3577 l 7279 3576 l 7279 3575 l 7279 3574 l 7283 3543 l 7284 3543 l 7284 3542 l 7284 3541 l 7284 3540 l 7284 3539 l stroke 14 setlinewidth newpath 7284 3539 moveto 7284 3539 l 7284 3538 l 7410 2762 l 7410 2761 l 7410 2760 l 7411 2759 l 7414 2744 l 7414 2743 l 7414 2742 l 7414 2741 l 7417 2726 l 7417 2725 l 7417 2724 l 7420 2709 l 7420 2708 l 7421 2708 l 7421 2707 l 7424 2692 l 7424 2691 l stroke 14 setlinewidth newpath 7424 2691 moveto 7424 2691 l 7424 2690 l 7427 2676 l 7427 2675 l 7427 2674 l 7427 2673 l 7518 2276 l 7518 2275 l 7520 2266 l 7520 2265 l 7521 2265 l 7523 2256 l 7523 2255 l 7525 2246 l stroke 14 setlinewidth newpath 7525 2246 moveto 7525 2246 l 7526 2246 l 7526 2245 l 7528 2237 l 7528 2236 l 7528 2235 l 7530 2227 l 7531 2227 l 7531 2226 l 7601 1985 l 7601 1984 l 7603 1978 l stroke 14 setlinewidth newpath 7603 1978 moveto 7603 1978 l 7604 1978 l 7605 1972 l 7606 1971 l 7607 1966 l 7607 1965 l 7608 1965 l 7609 1960 l 7609 1959 l 7610 1959 l 7611 1953 l 7612 1953 l stroke 14 setlinewidth newpath 1637 1791 moveto 7669 1791 l stroke 14 setlinewidth 1 0 0 sss [] 0 setdash newpath 2510 4218 moveto 2519 4135 l 2528 4056 l 2537 3981 l 2545 3909 l 2554 3840 l 2562 3775 l 2570 3712 l 2578 3652 l 2586 3594 l 2593 3538 l 2601 3485 l 2608 3434 l 2615 3384 l 2622 3337 l 2629 3291 l 2636 3247 l 2642 3205 l 2649 3163 l 2655 3124 l 2661 3085 l 2769 2519 l 2853 2179 l 2924 3123 l 2984 3787 l 3035 4281 l 3081 4661 l 3121 4964 l 3158 5210 l 3191 5415 l 3221 5587 l 3250 5735 l 3276 5862 l 3300 5974 l 3323 6072 l 3345 6159 l 3366 6236 l 3385 6306 l 3403 6369 l 3421 6426 l 3438 6479 l 3454 6527 l 3455 6529 l 3469 6309 l 3483 6106 l 3497 5917 l 3510 5741 l 3714 3674 l 3721 3618 l 3729 3565 l 3736 3513 l 3743 3463 l 3750 3412 l 3757 3364 l 3764 3317 l 3771 3272 l 3778 3229 l 3784 3187 l 3791 3146 l 3797 3107 l 3803 3069 l 3809 3033 l 3815 2997 l 3821 2963 l 3827 2930 l 3833 2897 l 3839 2866 l 3844 2836 l 3850 2806 l 3855 2778 l 3860 2750 l 3866 2723 l 3871 2698 l 3876 2673 l 3880 2650 l 3885 2626 l 3973 2247 l 3977 2233 l 3981 2219 l 3984 2205 l 3988 2192 l 4057 1962 l 4128 2937 l 4187 3625 l 4238 4137 l 4284 4532 l 4324 4847 l 4360 5103 l 4393 5316 l 4424 5496 l 4452 5650 l 4478 5783 l 4503 5899 l 4526 6001 l 4547 6092 l 4568 6173 l 4587 6246 l 4605 6312 l 4623 6371 l 4640 6426 l 4656 6476 l 4659 6485 l 4659 6476 l 4660 6467 l 4661 6458 l 4661 6449 l 4674 6267 l 4674 6259 l 4675 6251 l 4675 6242 l 4676 6234 l 4688 6066 l 4688 6058 l 4689 6050 l 4689 6042 l 4690 6035 l 4701 5878 l 4702 5871 l 4702 5864 l 4703 5857 l 4703 5850 l 4714 5704 l 4715 5697 l 4715 5690 l 4716 5684 l 4716 5677 l 4730 5509 l 4742 5353 l 4755 5207 l 4766 5070 l 4778 4941 l 4789 4819 l 4800 4705 l 4811 4597 l 4821 4495 l 4831 4398 l 4840 4307 l 4850 4219 l 4859 4136 l 4868 4057 l 4877 3982 l 4886 3910 l 4894 3841 l 4902 3776 l 4910 3713 l 4918 3653 l 4918 3650 l 4919 3648 l 4919 3646 l 4919 3644 l 4926 3597 l 4926 3595 l 4926 3593 l 4926 3590 l 4927 3588 l 4933 3544 l 4933 3541 l 4933 3539 l 4934 3537 l 4934 3535 l 4940 3492 l 4940 3490 l 4941 3488 l 4941 3486 l 4941 3484 l 4947 3443 l 4947 3441 l 4948 3439 l 4948 3437 l 4948 3435 l 5070 2708 l 5070 2707 l 5070 2706 l 5070 2705 l 5071 2704 l 5075 2683 l 5075 2682 l 5075 2681 l 5076 2680 l 5080 2659 l 5080 2658 l 5080 2657 l 5080 2656 l 5081 2655 l 5085 2636 l 5085 2635 l 5085 2634 l 5085 2633 l 5085 2632 l 5089 2613 l 5090 2612 l 5090 2611 l 5090 2610 l 5090 2609 l 5178 2236 l 5178 2235 l 5178 2234 l 5181 2222 l 5182 2221 l 5182 2220 l 5185 2208 l 5185 2207 l 5185 2206 l 5189 2195 l 5189 2194 l 5189 2193 l 5192 2182 l 5192 2181 l 5193 2180 l 5261 1953 l 5332 2929 l 5391 3619 l 5443 4131 l 5488 4527 l 5528 4842 l 5564 5099 l 5597 5312 l 5628 5492 l 5656 5646 l 5682 5779 l 5707 5896 l 5730 5998 l 5751 6089 l 5772 6170 l 5791 6243 l 5810 6309 l 5827 6369 l 5844 6424 l 5860 6474 l 5863 6484 l 5863 6483 l 5863 6482 l 5864 6474 l 5864 6473 l 5864 6465 l 5864 6464 l 5865 6456 l 5865 6455 l 5865 6448 l 5865 6447 l 5866 6447 l 5866 6446 l 5878 6266 l 5878 6265 l 5878 6264 l 5878 6257 l 5878 6256 l 5879 6249 l 5879 6248 l 5879 6241 l 5879 6240 l 5880 6240 l 5880 6239 l 5880 6232 l 5880 6231 l 5892 6064 l 5892 6063 l 5892 6056 l 5892 6055 l 5893 6055 l 5893 6049 l 5893 6048 l 5893 6047 l 5894 6041 l 5894 6040 l 5894 6033 l 5894 6032 l 5905 5877 l 5905 5876 l 5906 5876 l 5906 5869 l 5906 5868 l 5907 5862 l 5907 5861 l 5907 5855 l 5907 5854 l 5908 5848 l 5908 5847 l 5919 5702 l 5919 5701 l 5919 5696 l 5919 5695 l 5920 5689 l 5920 5688 l 5920 5682 l 5920 5681 l 5921 5676 l 5921 5675 l 5921 5674 l 6122 3652 l 6122 3651 l 6123 3649 l 6123 3647 l 6123 3645 l 6124 3643 l 6124 3642 l 6130 3596 l 6130 3594 l 6130 3592 l 6130 3591 l 6131 3590 l 6131 3589 l 6131 3587 l 6137 3543 l 6137 3542 l 6137 3541 l 6137 3540 l 6138 3539 l 6138 3538 l 6138 3536 l 6138 3534 l 6144 3491 l 6144 3489 l 6145 3489 l 6145 3487 l 6145 3485 l 6145 3483 l 6151 3442 l 6151 3440 l 6152 3438 l 6152 3436 l 6152 3434 l 6274 2708 l 6274 2707 l 6274 2706 l 6275 2706 l 6275 2705 l 6275 2704 l 6279 2683 l 6279 2682 l 6279 2681 l 6280 2680 l 6280 2679 l 6284 2659 l 6284 2658 l 6284 2657 l 6284 2656 l 6285 2656 l 6285 2655 l 6289 2635 l 6289 2634 l 6289 2633 l 6289 2632 l 6290 2631 l 6294 2612 l 6294 2611 l 6294 2610 l 6294 2609 l 6294 2608 l 6382 2236 l 6382 2235 l 6382 2234 l 6382 2233 l 6385 2222 l 6386 2221 l 6386 2220 l 6386 2219 l 6389 2208 l 6389 2207 l 6390 2206 l 6393 2195 l 6393 2194 l 6393 2193 l 6393 2192 l 6396 2181 l 6397 2181 l 6397 2180 l 6397 2179 l 6465 1953 l 6536 2929 l 6595 3618 l 6647 4131 l 6692 4527 l 6732 4842 l 6769 5099 l 6802 5312 l 6832 5492 l 6860 5646 l 6886 5779 l 6911 5896 l 6934 5998 l 6955 6089 l 6976 6170 l 6995 6243 l 7014 6309 l 7031 6369 l 7048 6424 l 7064 6474 l 7067 6483 l 7067 6482 l 7068 6474 l 7068 6473 l 7068 6465 l 7069 6465 l 7069 6464 l 7069 6456 l 7069 6455 l 7070 6447 l 7070 6446 l 7082 6266 l 7082 6265 l 7082 6264 l 7083 6257 l 7083 6256 l 7083 6249 l 7083 6248 l 7084 6241 l 7084 6240 l 7084 6239 l 7084 6232 l stroke 14 setlinewidth newpath 7084 6232 moveto 7084 6232 l 7084 6231 l 7096 6064 l 7096 6063 l 7097 6056 l 7097 6055 l 7097 6048 l 7097 6047 l 7098 6041 l 7098 6040 l 7098 6039 l 7098 6033 l 7098 6032 l 7110 5877 l 7110 5876 l 7110 5875 l 7110 5869 l 7110 5868 l 7111 5862 l 7111 5861 l 7111 5855 l 7111 5854 l 7112 5848 l 7112 5847 l 7123 5702 l 7123 5701 l 7123 5696 l 7123 5695 l 7123 5694 l 7124 5689 l 7124 5688 l 7124 5682 l 7124 5681 l 7125 5675 l 7125 5674 l 7327 3652 l 7327 3651 l 7327 3649 l 7327 3647 l 7327 3645 l 7328 3645 l 7328 3644 l 7328 3643 l 7328 3642 l 7334 3596 l 7334 3594 l 7335 3592 l 7335 3591 l 7335 3590 l 7335 3589 l 7335 3587 l 7341 3543 l 7341 3542 l 7342 3541 l 7342 3540 l 7342 3539 l 7342 3538 l 7342 3536 l 7342 3534 l 7348 3491 l 7349 3489 l 7349 3487 l 7349 3485 l 7350 3483 l 7355 3442 l 7356 3440 l 7356 3438 l 7356 3436 l 7356 3434 l stroke 14 setlinewidth newpath 7356 3434 moveto 7356 3434 l 7357 3434 l 7364 3384 l 7371 3337 l 7378 3291 l 7384 3247 l 7391 3205 l 7397 3163 l 7404 3124 l 7410 3085 l 7416 3048 l 7422 3012 l 7428 2977 l 7434 2944 l 7440 2911 l 7446 2879 l 7451 2849 l 7457 2819 l 7462 2790 l 7468 2762 l 7473 2734 l 7478 2708 l 7478 2707 l 7479 2706 l 7479 2705 l 7479 2704 l 7483 2683 l 7483 2682 l 7484 2681 l 7484 2680 l 7484 2679 l 7488 2659 l 7488 2658 l 7488 2657 l 7489 2657 l 7489 2656 l 7489 2655 l 7493 2635 l 7493 2634 l 7493 2633 l 7494 2632 l 7494 2631 l 7498 2612 l 7498 2611 l 7498 2610 l 7498 2609 l 7498 2608 l 7586 2236 l 7586 2235 l 7586 2234 l 7586 2233 l 7587 2233 l 7590 2222 l 7590 2221 l 7590 2220 l 7590 2219 l 7593 2208 l 7593 2207 l 7594 2207 l 7594 2206 l stroke 14 setlinewidth newpath 7594 2206 moveto 7594 2206 l 7597 2194 l 7597 2193 l 7598 2192 l 7601 2181 l 7601 2180 l 7601 2179 l stroke 14 setlinewidth newpath 1637 1953 moveto 7669 1953 l stroke 14 setlinewidth 0 1 0 sss [] 0 setdash newpath 3261 4218 moveto 3413 3085 l 3486 2682 l 3520 2519 l 3577 3231 l 3626 3769 l 3669 4189 l 3708 4527 l 3743 4804 l 3775 5035 l 3805 5231 l 3832 5400 l 3858 5546 l 3881 5674 l 3904 5788 l 3925 5888 l 3945 5979 l 3964 6060 l 3982 6133 l 4000 6200 l 4016 6261 l 4032 6318 l 4039 6342 l 4062 6017 l 4083 5728 l 4103 5470 l 4298 3581 l 4310 3497 l 4321 3418 l 4332 3344 l 4450 2661 l 4458 2623 l 4465 2587 l 4473 2552 l 4541 2262 l 4558 2200 l 4613 2948 l 4662 3514 l 4704 3958 l 4743 4315 l 4778 4609 l 4810 4854 l 4839 5063 l 4866 5242 l 4892 5398 l 4915 5534 l 4938 5655 l 4959 5763 l 4979 5859 l 4998 5946 l 5016 6024 l 5033 6096 l 5049 6161 l 5065 6221 l 5076 6262 l 5078 6241 l 5079 6220 l 5081 6199 l 5099 5942 l 5100 5923 l 5102 5905 l 5103 5886 l 5120 5658 l 5122 5641 l 5123 5624 l 5124 5608 l 5141 5403 l 5142 5388 l 5143 5373 l 5144 5358 l 5336 3541 l 5336 3536 l 5337 3530 l 5338 3525 l 5347 3459 l 5348 3454 l 5349 3449 l 5349 3444 l 5358 3381 l 5359 3376 l 5360 3372 l 5360 3367 l 5369 3308 l 5370 3303 l 5370 3299 l 5371 3295 l 5487 2634 l 5488 2632 l 5488 2629 l 5489 2627 l 5495 2597 l 5495 2595 l 5496 2592 l 5496 2590 l 5503 2561 l 5503 2559 l 5503 2557 l 5504 2555 l 5510 2527 l 5510 2525 l 5511 2523 l 5511 2521 l 5580 2237 l 5595 2180 l 5650 2930 l 5699 3498 l 5742 3943 l 5780 4302 l 5815 4596 l 5847 4843 l 5876 5052 l 5903 5232 l 5929 5388 l 5952 5526 l 5975 5647 l 5996 5755 l 6016 5851 l 6035 5939 l 6053 6017 l 6070 6089 l 6086 6155 l 6102 6215 l 6114 6257 l 6114 6256 l 6114 6255 l 6114 6253 l 6115 6236 l 6115 6235 l 6115 6233 l 6115 6232 l 6116 6215 l 6117 6214 l 6117 6212 l 6117 6211 l 6118 6194 l 6118 6193 l 6118 6192 l 6118 6190 l 6136 5937 l 6136 5936 l 6136 5935 l 6136 5934 l 6138 5919 l 6138 5917 l 6138 5916 l 6138 5915 l 6139 5900 l 6139 5899 l 6139 5898 l 6139 5897 l 6140 5882 l 6140 5880 l 6140 5879 l 6141 5878 l 6158 5653 l 6158 5652 l 6158 5651 l 6158 5650 l 6159 5636 l 6159 5635 l 6159 5634 l 6159 5633 l 6160 5620 l 6160 5619 l 6160 5618 l 6160 5617 l 6161 5603 l 6162 5602 l 6162 5601 l 6162 5600 l 6178 5399 l 6178 5398 l 6178 5397 l 6178 5396 l 6179 5384 l 6179 5383 l 6179 5382 l 6179 5381 l 6180 5369 l 6180 5368 l 6180 5367 l 6180 5366 l 6181 5354 l 6182 5353 l 6182 5352 l 6182 5351 l 6373 3539 l 6373 3538 l 6374 3533 l 6374 3532 l 6374 3528 l 6374 3527 l 6375 3523 l 6375 3522 l 6384 3456 l 6384 3455 l 6385 3451 l 6385 3450 l 6386 3446 l 6386 3445 l 6386 3441 l 6387 3441 l 6387 3440 l 6396 3379 l 6396 3378 l 6396 3374 l 6396 3373 l 6397 3369 l 6397 3368 l 6398 3365 l 6398 3364 l 6406 3305 l 6407 3301 l 6407 3300 l 6408 3297 l 6408 3296 l 6408 3292 l 6408 3291 l 6525 2632 l 6525 2630 l 6525 2629 l 6525 2628 l 6526 2627 l 6526 2625 l 6532 2595 l 6533 2593 l 6533 2591 l 6533 2590 l 6534 2589 l 6534 2588 l 6540 2560 l 6540 2559 l 6540 2558 l 6540 2557 l 6541 2555 l 6541 2553 l 6547 2526 l 6547 2525 l 6548 2524 l 6548 2523 l 6548 2522 l 6548 2521 l 6549 2519 l 6617 2236 l 6632 2179 l 6688 2929 l 6736 3497 l 6779 3943 l 6817 4301 l 6852 4596 l 6884 4842 l 6913 5052 l 6940 5232 l 6966 5388 l 6990 5525 l 7012 5646 l 7033 5754 l 7053 5851 l 7072 5938 l 7090 6017 l 7107 6089 l 7124 6155 l 7139 6215 l 7151 6257 l 7151 6256 l 7151 6255 l 7151 6254 l 7151 6253 l 7152 6236 l 7152 6234 l 7152 6233 l 7152 6232 l 7154 6215 l 7154 6213 l 7154 6212 l 7154 6211 l 7155 6194 l 7155 6193 l 7155 6192 l 7155 6191 l 7155 6190 l 7173 5937 l 7174 5936 l 7174 5935 l 7174 5934 l 7174 5933 l 7175 5918 l 7175 5917 l 7175 5916 l 7175 5915 l 7176 5900 l 7176 5899 l 7176 5898 l 7176 5897 l 7176 5896 l 7178 5881 l 7178 5880 l 7178 5879 l 7178 5878 l 7195 5653 l 7195 5652 l 7195 5651 l 7195 5650 l 7195 5649 l 7196 5636 l 7196 5635 l 7196 5634 l 7196 5633 l 7197 5619 l 7197 5618 l 7198 5617 l 7198 5616 l 7199 5603 l 7199 5602 l 7199 5601 l 7199 5600 l 7215 5398 l 7215 5397 l 7215 5396 l 7215 5395 l 7216 5383 l 7216 5382 l 7216 5381 l 7217 5381 l 7217 5380 l 7217 5369 l 7217 5368 l 7218 5368 l 7218 5367 l 7218 5366 l 7219 5354 l 7219 5353 l 7219 5352 l 7219 5351 l 7410 3538 l 7410 3537 l 7411 3533 l 7411 3532 l 7412 3528 l 7412 3527 l 7412 3523 l 7412 3522 l 7412 3521 l 7422 3456 l 7422 3455 l 7422 3451 l 7422 3450 l 7423 3446 l 7423 3445 l 7424 3441 l 7424 3440 l 7433 3378 l 7433 3374 l 7433 3373 l 7434 3373 l 7434 3369 l 7434 3368 l 7435 3364 l 7435 3363 l 7444 3305 l 7444 3304 l 7444 3301 l 7444 3300 l 7445 3297 l 7445 3296 l 7446 3292 l 7446 3291 l 7562 2632 l 7562 2630 l 7562 2629 l 7563 2627 l 7563 2625 l 7569 2595 l 7570 2595 l 7570 2593 l 7570 2592 l 7570 2591 l 7570 2590 l 7571 2590 l 7571 2588 l 7577 2560 l 7577 2559 l 7577 2558 l 7577 2557 l 7578 2557 l stroke 14 setlinewidth newpath 7578 2557 moveto 7578 2557 l 7578 2555 l 7578 2553 l 7579 2553 l 7584 2526 l 7584 2525 l 7585 2525 l 7585 2523 l 7585 2521 l 7586 2519 l stroke 14 setlinewidth newpath 1637 2179 moveto 7669 2179 l stroke 14 setlinewidth 0 0 1 sss [] 0 setdash newpath 4230 4218 moveto 4290 3712 l 4342 3337 l 4381 3085 l 4423 3559 l 4460 3942 l 4494 4260 l 4525 4527 l 4553 4755 l 4580 4951 l 4605 5122 l 4628 5273 l 4650 5406 l 4671 5525 l 4690 5632 l 4709 5728 l 4727 5816 l 4744 5896 l 4760 5969 l 4775 6036 l 4790 6097 l 4792 6106 l 4832 5577 l 4867 5145 l 5052 3463 l 5072 3324 l 5091 3199 l 5144 2884 l 5191 2643 l 5203 2582 l 5243 3093 l 5280 3510 l 5313 3856 l 5343 4149 l 5371 4399 l 5397 4616 l 5421 4805 l 5444 4972 l 5466 5120 l 5486 5252 l 5506 5371 l 5524 5479 l 5541 5577 l 5558 5666 l 5574 5748 l 5590 5823 l 5604 5892 l 5614 5938 l 5619 5876 l 5624 5815 l 5654 5426 l 5658 5376 l 5662 5326 l 5689 5007 l 5693 4966 l 5697 4925 l 5874 3379 l 5876 3363 l 5878 3348 l 5894 3244 l 5896 3230 l 5898 3216 l 5913 3123 l 5915 3110 l 5917 3098 l 5970 2800 l 6016 2571 l 6025 2526 l 6065 3041 l 6101 3461 l 6134 3811 l 6164 4106 l 6192 4359 l 6218 4578 l 6243 4769 l 6266 4938 l 6287 5087 l 6307 5221 l 6327 5342 l 6345 5451 l 6363 5550 l 6379 5640 l 6395 5723 l 6411 5799 l 6425 5869 l 6436 5919 l 6437 5912 l 6437 5905 l 6441 5857 l 6441 5850 l 6442 5844 l 6446 5797 l 6446 5790 l 6447 5783 l 6476 5409 l 6476 5404 l 6477 5398 l 6480 5359 l 6480 5353 l 6481 5348 l 6484 5310 l 6484 5304 l 6485 5299 l 6511 4992 l 6512 4988 l 6512 4983 l 6515 4950 l 6516 4946 l 6516 4941 l 6519 4910 l 6519 4905 l 6520 4901 l 6696 3370 l 6696 3368 l 6696 3366 l 6698 3354 l 6698 3352 l 6699 3351 l 6700 3339 l 6701 3337 l 6701 3335 l 6716 3235 l 6716 3234 l 6716 3232 l 6718 3221 l 6718 3220 l 6719 3218 l 6720 3207 l 6721 3206 l 6721 3204 l 6735 3115 l 6735 3113 l 6736 3112 l 6737 3102 l 6737 3101 l 6738 3099 l 6739 3089 l 6739 3088 l 6740 3087 l 6792 2791 l 6838 2563 l 6847 2520 l 6887 3035 l 6923 3456 l 6956 3806 l 6986 4101 l 7014 4354 l 7040 4574 l 7065 4765 l 7088 4934 l 7109 5084 l 7129 5218 l 7149 5338 l 7167 5448 l 7185 5547 l 7201 5637 l 7217 5720 l 7233 5796 l 7247 5866 l 7258 5917 l 7258 5916 l 7259 5910 l 7259 5909 l 7259 5903 l 7260 5902 l 7263 5855 l 7263 5854 l 7263 5848 l 7264 5848 l 7264 5847 l 7264 5842 l 7264 5841 l 7264 5840 l 7267 5795 l 7268 5794 l 7268 5793 l 7268 5788 l 7268 5787 l 7268 5786 l 7268 5781 l 7269 5780 l 7298 5408 l 7298 5407 l 7298 5406 l 7298 5402 l 7298 5401 l 7299 5396 l 7299 5395 l 7302 5357 l 7302 5356 l 7302 5352 l 7302 5351 l 7302 5350 l 7303 5346 l 7303 5345 l 7306 5308 l 7306 5307 l 7306 5302 l 7307 5302 l 7307 5301 l 7307 5297 l 7307 5296 l 7333 4991 l 7333 4990 l 7334 4989 l 7334 4986 l 7334 4985 l 7334 4981 l 7334 4980 l 7337 4949 l 7337 4948 l 7338 4944 l 7338 4943 l 7338 4940 l 7338 4939 l 7341 4908 l 7341 4907 l 7341 4903 l 7341 4902 l 7342 4899 l 7342 4898 l 7518 3369 l 7518 3368 l 7518 3367 l 7518 3366 l 7518 3365 l 7520 3353 l 7520 3351 l 7520 3349 l 7521 3349 l 7522 3338 l 7522 3337 l 7523 3336 l 7523 3335 l 7523 3334 l 7538 3234 l 7538 3233 l 7538 3231 l 7540 3220 l 7540 3219 l 7541 3217 l 7542 3207 l 7542 3206 l 7542 3205 l 7543 3205 l 7543 3203 l 7557 3114 l 7557 3113 l 7557 3112 l 7558 3111 l 7559 3101 l 7559 3100 l 7559 3099 l 7560 3098 l 7561 3088 l 7561 3087 l 7562 3087 l 7562 3086 l 7562 3085 l stroke 14 setlinewidth newpath 1637 2519 moveto 7669 2519 l stroke %%EndDocument @endspecial 432 697 a Fg(max\()p Fh(p)p Fg(\()p Fh(r)r Fg(\)\))2579 2823 y Fh(r)2596 815 y(n)19 b Fg(=)g(6)2596 921 y Fh(n)g Fg(=)g(5)2596 1028 y Fh(n)g Fg(=)g(4)2596 1134 y Fh(n)g Fg(=)g(3)469 3373 y Fn(Figure)d(3.)34 b(The)17 b(oscillations)g(of)g Fg(p)r(or\()p Fh(r)r Fg(\))h Fn(as)f(a)h(function)e(of)h Fh(r)h Fn(\(on)f(a)g(logarithmic)f (scale\))i(for)e(the)h(sets)h Fh(A)2998 3349 y Fa(\()p Fd(n)p Fa(\))3088 3373 y Fn(,)f(with)g Fh(n)j Fg(=)f(3)p Fh(;)12 b(:)f(:)h(:)f(;)g Fg(6)p Fn(.)315 3733 y Fm(The)34 b(proof)e(of)h(our)g(main)h(result)f(is)g (based)g(on)g(comparing)h(the)f(porosity)h(of)e(a)i(measure)f(with)g(the)165 3849 y(porosity)f(of)f(subsets)g(with)h(positi)m(v)n(e)h(measure.)56 b(F)n(or)31 b(this,)j(we)d(use)g(the)h(quantity)h Fj(\014)6 b Fe(\()p Fj(\026)p Fe(\))30 b Fm(introduced)j(in)165 3965 y([8]:)914 4221 y Fj(\014)6 b Fe(\()p Fj(\026)p Fe(\))44 b(=)g(sup)q Fk(f)p Fe(p)s(or)o(\()p Fj(A)p Fe(\))28 b(:)g Fj(A)d Fm(is)h(a)f(Borel)h(set) g(with)g Fj(\026)p Fe(\()p Fj(A)p Fe(\))h Fj(>)h Fe(0)p Fk(g)d Fj(:)165 4477 y Fm(The)34 b(inequality)g Fj(\014)6 b Fe(\()p Fj(\026)p Fe(\))37 b Fk(\024)h Fe(p)s(or)o(\()p Fj(\026)p Fe(\))33 b Fm(holds)f(for)h(an)n(y)g(Borel)h(probability)g(measure)f Fj(\026)p Fm(,)i(b)n(ut)e(the)g(con)l(v)n(erse)165 4593 y(inequality)f(does)f (not)f(need)h(to)f(be)h(true.)52 b(W)-8 b(e)30 b(sho)m(w)g(that)h(it)g(holds) f(when)h(the)f(measure)h Fj(\026)f Fm(satis\256es)g(the)165 4709 y(doubling)22 b(condition.)37 b(W)-8 b(e)21 b(ha)n(v)n(e)h(sho)m(wn)f (in)g([2])g(that)h(the)g(doubling)f(condition)i(implies)f Fj(\014)6 b Fe(\()p Fj(\026)p Fe(\))27 b(=)g(p)s(or\()p Fj(\026)p Fe(\))p Fm(.)165 4826 y(T)-8 b(ogether)32 b(with)f(the)g(results)f(of)h([11])f(it)h (implies)h(our)e(main)i(bound)e(on)h(the)g(dimension)g(\(see)g(Theorem)165 4942 y(4\).)89 b(W)-8 b(e)43 b(also)g(sho)m(wed)f(that)i(there)f(are)g (measures)g(violating)h(the)f(doubling)g(condition)h(for)f(which)165 5058 y Fj(\014)6 b Fe(\()p Fj(\026)p Fe(\))27 b Fk(6)p Fe(=)h(p)s(or)o(\()p Fj(\026)p Fe(\))p Fm(.)37 b(In)25 b(the)g(case)g(of)g(measures)g(on)g(the)g (line)h Ff(R)p Fm(,)f Fl(i.e)n(.,)j Fm(in)d(1)f(dimension,)j(we)d(also)h(sho) m(w)g(that)165 5174 y(a)h(some)m(what)g(weak)o(er)g(condition)h(than)f(the)g (doubling)g(condition)h(implies)g(Theorem)f(4.)3894 5554 y Fq(9)p eop %%Page: 10 10 bop 402 2862 a @beginspecial 5 @llx 1 @lly 534 @urx 537 @ury 2856 @rwi @setspecial %%BeginDocument: lacunnodivp.ps /X-CENTER-PLOT 0 def /Y-CENTER-PLOT 0 def clippath pathbbox /page-height exch def /page-width exch def pop pop 538.583 page-width gt { % Rotation required Y-CENTER-PLOT 1 eq { % Center in y page-height 538.583 sub 2 div } { % Don't center in y 0 } ifelse X-CENTER-PLOT 1 eq { % Center in x page-width 538.583 sub 2 div } { % Don't center in x 0 } ifelse translate } { % No rotation - just handle centering X-CENTER-PLOT 1 eq { % Center in x page-width 538.583 sub 2 div } { % Don't center in x 0 } ifelse Y-CENTER-PLOT 1 eq { % Center in y page-height 538.583 sub 2 div } { % Don't center in y 0 } ifelse translate } ifelse 0.06 0.06 scale /choose-font % stack: fontsize fontname => --- { findfont exch scalefont setfont newpath 0 0 moveto (0) true charpath flattenpath pathbbox /top exch def pop /bottom exch def pop bottom top bottom top add 2 div /center-font-val exch def /upper-font-val exch def /lower-font-val exch def } def /center-x-just % stack: (string) x y => (string) newx y { exch 2 index stringwidth pop 2 div sub exch } def /left-x-just % stack: (string) x y => (string) newx y { } def /right-x-just % stack: (string) x y => (string) newx y { exch 2 index stringwidth pop sub exch } def /center-y-just % stack: (string) x y => (string) x newy { center-font-val sub } def /lower-y-just % stack: (string) x y => (string) x newy { lower-font-val sub } def /upper-y-just % stack: (string) x y => (string) x newy { upper-font-val sub } def /just-string % stack: (string) x y just => --- { dup 0 eq { pop center-x-just center-y-just } if dup 1 eq { pop left-x-just center-y-just } if dup 2 eq { pop left-x-just upper-y-just } if dup 3 eq { pop center-x-just upper-y-just } if dup 4 eq { pop right-x-just upper-y-just } if dup 5 eq { pop right-x-just center-y-just } if dup 6 eq { pop right-x-just lower-y-just } if dup 7 eq { pop center-x-just lower-y-just } if dup 8 eq { pop left-x-just lower-y-just } if moveto show } def /mark0 {/size exch def /y exch def /x exch def newpath x size sub y size sub moveto size size add 0 rlineto 0 size size add rlineto 0 size size add sub 0 rlineto closepath fill} def /mark1 {/size exch def /y exch def /x exch def newpath x size sub y size sub moveto size size add 0 rlineto 0 size size add rlineto 0 size size add sub 0 rlineto closepath stroke} def /mark2 {/size exch def /y exch def /x exch def newpath x y moveto x y size 0 360 arc stroke} def /mark3 {/size exch def /y exch def /x exch def newpath x size sub y size sub moveto x size add y size add lineto x size sub y size add moveto x size add y size sub lineto stroke} def /mark4 {/size exch def /y exch def /x exch def newpath x size sub y moveto x y size add lineto x size add y lineto x y size sub lineto closepath stroke} def /mark5 {/size exch def /y exch def /x exch def x y size mark1 newpath x size sub y moveto size size add 0 rlineto stroke} def /mark6 {/size exch def /y exch def /x exch def newpath x y moveto x y size 0 360 arc fill} def /mark7 {/size exch def /y exch def /x exch def newpath x y moveto x size sub y size sub lineto x size add y size sub lineto closepath fill newpath x y moveto x size add y size add lineto x size sub y size add lineto closepath fill} def 0 setgray 300 /Times-Bold choose-font () 4488 8856 3 just-string 300 /Times-Roman choose-font /sss{setrgbcolor}def /l{lineto}def 14 setlinewidth [] 0 setdash 1 0 1 sss newpath 8061 7986 moveto 8886 7986 lineto stroke 14 setlinewidth 1 0 0 sss [] 0 setdash newpath 8061 7594 moveto 8886 7594 lineto stroke 14 setlinewidth 0 1 0 sss [] 0 setdash newpath 8061 7202 moveto 8886 7202 lineto stroke 14 setlinewidth 0 0 1 sss [] 0 setdash newpath 8061 6810 moveto 8886 6810 lineto stroke 0 0 0 sss () 90 8344 2 just-string () 8886 380 5 just-string (0.00) 860 724 5 just-string (0.10) 860 1413 5 just-string (0.20) 860 2101 5 just-string (0.30) 860 2790 5 just-string (0.40) 860 3478 5 just-string (0.50) 860 4167 5 just-string (0.60) 860 4856 5 just-string (0.70) 860 5544 5 just-string (0.80) 860 6233 5 just-string (0.90) 860 6921 5 just-string (1.00) 860 7610 5 just-string (0.00) 1267 90 7 just-string (0.20) 2533 90 7 just-string (0.40) 3800 90 7 just-string (0.60) 5066 90 7 just-string (0.80) 6333 90 7 just-string (1.00) 7599 90 7 just-string [] 0 setdash 8 setlinewidth 2 setlinewidth newpath 950 724 moveto 7916 724 lineto stroke newpath 950 1413 moveto 7916 1413 lineto stroke newpath 950 2101 moveto 7916 2101 lineto stroke newpath 950 2790 moveto 7916 2790 lineto stroke newpath 950 3478 moveto 7916 3478 lineto stroke newpath 950 4167 moveto 7916 4167 lineto stroke 950 4856 moveto 7916 4856 lineto stroke newpath 950 5544 moveto 7916 5544 lineto stroke newpath 950 6233 moveto 7916 6233 lineto stroke newpath 950 6921 moveto 7916 6921 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 7610 moveto 7916 7610 lineto stroke [] 0 setdash 8 setlinewidth newpath 1267 7954 moveto 1267 380 lineto stroke [] 0 setdash 2 setlinewidth newpath 2533 7954 moveto 2533 380 lineto stroke [] 0 setdash 2 setlinewidth newpath 3800 7954 moveto 3800 380 lineto stroke [] 0 setdash 2 setlinewidth newpath 5066 7954 moveto 5066 380 lineto stroke [] 0 setdash 2 setlinewidth newpath 6333 7954 moveto 6333 380 lineto stroke [] 0 setdash 2 setlinewidth newpath 7599 7954 moveto 7599 380 lineto stroke 14 setlinewidth [] 0 setdash 1 0 1 sss newpath 1267 724 moveto 1267 724 lineto 1267 725 lineto 1268 725 lineto 1269 725 lineto 1270 725 lineto 1271 725 lineto 1272 725 lineto 1273 726 lineto 1274 728 lineto 1275 729 lineto 1277 730 lineto 1278 732 lineto 1279 733 lineto 1280 735 lineto 1282 736 lineto 1283 737 lineto 1284 739 lineto 1285 740 lineto 1287 741 lineto 1288 743 lineto 1289 744 lineto 1290 746 lineto 1292 747 lineto 1293 748 lineto 1294 750 lineto 1295 751 lineto 1296 752 lineto 1297 752 lineto 1298 752 lineto 1299 752 lineto 1300 752 lineto 1325 752 lineto 1326 752 lineto 1327 752 lineto 1328 752 lineto 1329 752 lineto 1331 752 lineto 1332 752 lineto 1333 752 lineto 1334 752 lineto 1336 752 lineto 1337 752 lineto 1338 752 lineto 1340 752 lineto 1341 752 lineto 1342 752 lineto 1343 752 lineto 1345 752 lineto 1346 752 lineto 1347 752 lineto 1348 752 lineto 1350 752 lineto 1351 752 lineto 1352 752 lineto 1353 752 lineto 1355 752 lineto 1356 752 lineto 1357 752 lineto 1358 752 lineto 1359 752 lineto stroke 14 setlinewidth newpath 1359 752 moveto 1359 752 lineto 1384 752 lineto 1385 752 lineto 1386 752 lineto 1387 752 lineto 1388 752 lineto 1413 752 lineto 1414 752 lineto 1415 752 lineto 1416 752 lineto 1417 752 lineto 1443 752 lineto 1488 801 lineto 1533 850 lineto 1578 899 lineto 1623 948 lineto 1669 998 lineto 1714 1047 lineto 1759 1096 lineto 1804 1145 lineto 1850 1194 lineto 1895 1243 lineto 1940 1293 lineto 1985 1342 lineto 2031 1391 lineto 2076 1440 lineto 2121 1489 lineto 2166 1539 lineto 2212 1588 lineto 2257 1637 lineto 2302 1686 lineto 2322 1708 lineto 2323 1708 lineto 2324 1708 lineto 2325 1708 lineto 2326 1708 lineto 2351 1708 lineto 2352 1708 lineto 2353 1708 lineto 2354 1708 lineto 2355 1708 lineto 2356 1708 lineto 2381 1708 lineto 2382 1708 lineto stroke 14 setlinewidth newpath 2382 1708 moveto 2382 1708 lineto 2383 1708 lineto 2384 1708 lineto 2385 1708 lineto 2410 1708 lineto 2411 1708 lineto 2412 1708 lineto 2413 1708 lineto 2414 1708 lineto 2439 1708 lineto 2440 1708 lineto 2441 1708 lineto 2442 1708 lineto 2443 1708 lineto 2444 1708 lineto 2469 1708 lineto 2470 1708 lineto 2471 1708 lineto 2472 1708 lineto 2473 1708 lineto 3378 1708 lineto 3379 1708 lineto 3380 1708 lineto 3381 1708 lineto 3382 1708 lineto 3407 1708 lineto stroke 14 setlinewidth newpath 3407 1708 moveto 3407 1708 lineto 3408 1708 lineto 3409 1708 lineto 3410 1708 lineto 3411 1708 lineto 3436 1708 lineto 3437 1708 lineto 3438 1708 lineto 3439 1708 lineto 3440 1708 lineto 3465 1708 lineto 3466 1708 lineto 3467 1708 lineto 3468 1708 lineto 3469 1708 lineto 3470 1708 lineto 3495 1708 lineto 3496 1708 lineto 3497 1708 lineto 3498 1708 lineto 3499 1708 lineto 3524 1708 lineto 3525 1708 lineto 3526 1708 lineto 3527 1708 lineto stroke 14 setlinewidth newpath 3527 1708 moveto 3527 1708 lineto 3528 1708 lineto 4433 1708 lineto 4434 1708 lineto 4435 1708 lineto 4436 1708 lineto 4437 1708 lineto 4462 1708 lineto 4463 1708 lineto 4464 1708 lineto 4465 1708 lineto 4466 1708 lineto 4467 1708 lineto 4492 1708 lineto 4493 1708 lineto 4494 1708 lineto 4495 1708 lineto 4496 1708 lineto 4521 1708 lineto 4522 1708 lineto 4523 1708 lineto 4524 1708 lineto 4525 1708 lineto 4550 1708 lineto 4551 1708 lineto 4552 1708 lineto stroke 14 setlinewidth newpath 4552 1708 moveto 4552 1708 lineto 4553 1708 lineto 4554 1708 lineto 4580 1708 lineto 4581 1708 lineto 4582 1708 lineto 4583 1708 lineto 4584 1708 lineto 5488 1708 lineto 5489 1708 lineto 5490 1708 lineto 5491 1708 lineto 5492 1708 lineto 5493 1708 lineto 5518 1708 lineto 5519 1708 lineto 5520 1708 lineto 5521 1708 lineto 5522 1708 lineto 5547 1708 lineto 5548 1708 lineto 5549 1708 lineto 5550 1708 lineto 5551 1708 lineto 5576 1708 lineto stroke 14 setlinewidth newpath 5576 1708 moveto 5576 1708 lineto 5577 1708 lineto 5578 1708 lineto 5579 1708 lineto 5580 1708 lineto 5581 1708 lineto 5606 1708 lineto 5607 1708 lineto 5608 1708 lineto 5609 1708 lineto 5610 1708 lineto 5635 1708 lineto 5636 1708 lineto 5637 1708 lineto 5638 1708 lineto 5639 1708 lineto 6544 1708 lineto 6545 1708 lineto 6546 1708 lineto 6547 1708 lineto 6548 1708 lineto 6573 1708 lineto 6574 1708 lineto 6575 1708 lineto 6576 1708 lineto stroke 14 setlinewidth newpath 6576 1708 moveto 6576 1708 lineto 6577 1708 lineto 6603 1708 lineto 6604 1708 lineto 6605 1708 lineto 6606 1708 lineto 6607 1708 lineto 6632 1708 lineto 6633 1708 lineto 6634 1708 lineto 6635 1708 lineto 6636 1708 lineto 6661 1708 lineto 6662 1708 lineto 6663 1708 lineto 6664 1708 lineto 6665 1708 lineto 6690 1708 lineto 6691 1708 lineto 6692 1708 lineto 6693 1708 lineto 6694 1708 lineto 6695 1708 lineto stroke 14 setlinewidth 1 0 0 sss newpath 1267 724 moveto 1267 724 lineto 1267 725 lineto 1268 725 lineto 1268 726 lineto 1269 726 lineto 1270 726 lineto 1271 726 lineto 1273 726 lineto 1275 726 lineto 1277 726 lineto 1279 728 lineto 1281 731 lineto 1283 733 lineto 1285 735 lineto 1287 738 lineto 1289 740 lineto 1292 742 lineto 1294 744 lineto 1296 747 lineto 1298 749 lineto 1300 751 lineto 1302 754 lineto 1304 756 lineto 1306 758 lineto 1308 761 lineto 1311 763 lineto 1313 765 lineto 1315 767 lineto 1317 770 lineto 1318 770 lineto 1319 770 lineto 1320 770 lineto 1321 770 lineto 1322 770 lineto 1323 770 lineto 1324 770 lineto 1325 770 lineto 1326 770 lineto 1368 770 lineto 1370 770 lineto 1372 770 lineto 1374 770 lineto 1376 770 lineto 1419 770 lineto 1421 770 lineto 1423 770 lineto 1425 770 lineto 1427 770 lineto 1469 770 lineto 1470 770 lineto 1471 770 lineto 1472 770 lineto 1473 770 lineto 1474 770 lineto 1475 770 lineto 1476 770 lineto 1477 770 lineto 1478 770 lineto 1520 770 lineto 1573 828 lineto 1625 885 lineto 1678 942 lineto 1731 1000 lineto 1784 1057 lineto 1837 1114 lineto 1889 1172 lineto 1942 1229 lineto 1995 1287 lineto 2048 1344 lineto 2100 1401 lineto 2153 1459 lineto 2206 1516 lineto 2259 1573 lineto 2312 1631 lineto 2364 1688 lineto 2417 1746 lineto 2470 1803 lineto 2523 1860 lineto 2533 1872 lineto 2534 1872 lineto 2535 1872 lineto 2536 1872 lineto 2537 1872 lineto 2538 1872 lineto 2539 1872 lineto 2540 1872 lineto 2541 1872 lineto 2542 1872 lineto 2584 1872 lineto 2586 1872 lineto 2588 1872 lineto 2590 1872 lineto 2592 1872 lineto stroke 14 setlinewidth newpath 2592 1872 moveto 2592 1872 lineto 2635 1872 lineto 2637 1872 lineto 2639 1872 lineto 2641 1872 lineto 2643 1872 lineto 2685 1872 lineto 2686 1872 lineto 2687 1872 lineto 2688 1872 lineto 2689 1872 lineto 2690 1872 lineto 2691 1872 lineto 2692 1872 lineto 2693 1872 lineto 2694 1872 lineto 2736 1872 lineto 2738 1872 lineto 2740 1872 lineto 2742 1872 lineto 2744 1872 lineto 3800 1872 lineto 3802 1872 lineto 3804 1872 lineto 3806 1872 lineto 3808 1872 lineto 3850 1872 lineto 3851 1872 lineto 3852 1872 lineto 3853 1872 lineto 3854 1872 lineto 3855 1872 lineto 3856 1872 lineto 3857 1872 lineto 3858 1872 lineto 3859 1872 lineto 3901 1872 lineto 3903 1872 lineto 3905 1872 lineto 3907 1872 lineto 3909 1872 lineto 3952 1872 lineto 3954 1872 lineto 3956 1872 lineto 3958 1872 lineto 3960 1872 lineto 4002 1872 lineto 4003 1872 lineto 4004 1872 lineto 4005 1872 lineto 4006 1872 lineto 4007 1872 lineto 4008 1872 lineto 4009 1872 lineto 4010 1872 lineto 4011 1872 lineto stroke 14 setlinewidth newpath 4011 1872 moveto 4011 1872 lineto 4064 1872 lineto 4116 1872 lineto 4169 1872 lineto 4222 1872 lineto 4275 1872 lineto 4327 1872 lineto 4380 1872 lineto 4433 1872 lineto 4486 1872 lineto 4539 1872 lineto 4591 1872 lineto 4644 1872 lineto 4697 1872 lineto 4750 1872 lineto 4802 1872 lineto 4855 1872 lineto 4908 1872 lineto 4961 1872 lineto 5013 1872 lineto 5066 1872 lineto 5067 1872 lineto 5068 1872 lineto 5069 1872 lineto 5070 1872 lineto 5071 1872 lineto 5072 1872 lineto 5073 1872 lineto 5074 1872 lineto 5075 1872 lineto 5117 1872 lineto 5119 1872 lineto 5121 1872 lineto 5123 1872 lineto 5125 1872 lineto 5168 1872 lineto 5170 1872 lineto 5172 1872 lineto 5174 1872 lineto 5176 1872 lineto 5218 1872 lineto 5219 1872 lineto 5220 1872 lineto 5221 1872 lineto 5222 1872 lineto 5223 1872 lineto 5224 1872 lineto 5225 1872 lineto 5226 1872 lineto 5227 1872 lineto 5269 1872 lineto 5271 1872 lineto 5273 1872 lineto 5275 1872 lineto 5277 1872 lineto 6333 1872 lineto 6335 1872 lineto 6337 1872 lineto 6339 1872 lineto 6341 1872 lineto 6383 1872 lineto 6384 1872 lineto 6386 1872 lineto 6388 1872 lineto 6390 1872 lineto 6392 1872 lineto 6434 1872 lineto 6436 1872 lineto 6437 1872 lineto 6438 1872 lineto 6439 1872 lineto 6440 1872 lineto stroke 14 setlinewidth newpath 6440 1872 moveto 6440 1872 lineto 6441 1872 lineto 6442 1872 lineto 6443 1872 lineto 6485 1872 lineto 6487 1872 lineto 6489 1872 lineto 6491 1872 lineto 6493 1872 lineto 6535 1872 lineto 6536 1872 lineto 6537 1872 lineto 6538 1872 lineto 6540 1872 lineto 6542 1872 lineto 6544 1872 lineto stroke 14 setlinewidth 0 1 0 sss newpath 1267 724 moveto 1267 724 lineto 1267 725 lineto 1268 725 lineto 1269 725 lineto 1269 726 lineto 1270 726 lineto 1270 727 lineto 1271 727 lineto 1271 728 lineto 1272 728 lineto 1272 729 lineto 1273 729 lineto 1273 730 lineto 1274 730 lineto 1279 730 lineto 1280 730 lineto 1285 730 lineto 1286 730 lineto 1290 730 lineto 1291 730 lineto 1295 734 lineto 1299 738 lineto 1303 743 lineto 1307 747 lineto 1311 751 lineto 1315 755 lineto 1319 760 lineto 1323 764 lineto 1327 768 lineto 1331 773 lineto 1335 777 lineto 1339 781 lineto 1343 786 lineto 1347 790 lineto 1351 794 lineto 1355 799 lineto 1359 803 lineto 1363 807 lineto 1366 810 lineto 1367 810 lineto 1372 810 lineto 1373 810 lineto 1378 810 lineto 1379 810 lineto 1384 810 lineto 1385 810 lineto 1465 810 lineto 1466 810 lineto 1471 810 lineto 1472 810 lineto 1477 810 lineto 1478 810 lineto 1483 810 lineto 1484 810 lineto 1563 810 lineto 1564 810 lineto 1565 810 lineto 1570 810 lineto 1571 810 lineto 1576 810 lineto 1577 810 lineto 1582 810 lineto 1583 810 lineto 1647 810 lineto 1662 810 lineto 1726 879 lineto 1789 948 lineto 1852 1017 lineto 1916 1086 lineto 1979 1155 lineto 2042 1223 lineto 2106 1292 lineto 2169 1361 lineto 2232 1430 lineto 2296 1499 lineto 2359 1568 lineto 2422 1637 lineto 2486 1705 lineto 2549 1774 lineto 2612 1843 lineto 2676 1912 lineto 2739 1981 lineto 2802 2050 lineto 2850 2101 lineto 2851 2101 lineto 2856 2101 lineto 2857 2101 lineto 2862 2101 lineto 2863 2101 lineto 2868 2101 lineto 2869 2101 lineto 2870 2101 lineto 2949 2101 lineto 2950 2101 lineto 2955 2101 lineto 2956 2101 lineto 2961 2101 lineto 2962 2101 lineto 2967 2101 lineto 2968 2101 lineto 2969 2101 lineto 3048 2101 lineto 3049 2101 lineto 3054 2101 lineto 3055 2101 lineto 3060 2101 lineto 3061 2101 lineto 3066 2101 lineto 3067 2101 lineto 3147 2101 lineto 3148 2101 lineto 3153 2101 lineto 3154 2101 lineto 3159 2101 lineto 3160 2101 lineto 3165 2101 lineto 3166 2101 lineto 4433 2101 lineto 4434 2101 lineto 4439 2101 lineto 4440 2101 lineto 4445 2101 lineto 4446 2101 lineto 4447 2101 lineto 4452 2101 lineto 4453 2101 lineto 4532 2101 lineto 4533 2101 lineto 4538 2101 lineto 4539 2101 lineto 4544 2101 lineto 4545 2101 lineto 4546 2101 lineto 4551 2101 lineto 4552 2101 lineto 4631 2101 lineto 4632 2101 lineto 4637 2101 lineto 4638 2101 lineto 4643 2101 lineto 4644 2101 lineto 4649 2101 lineto 4650 2101 lineto 4651 2101 lineto 4730 2101 lineto 4731 2101 lineto 4736 2101 lineto 4737 2101 lineto 4742 2101 lineto 4743 2101 lineto 4748 2101 lineto 4749 2101 lineto 4750 2101 lineto 6016 2101 lineto 6017 2101 lineto 6022 2101 lineto 6023 2101 lineto 6024 2101 lineto 6029 2101 lineto 6030 2101 lineto 6035 2101 lineto 6036 2101 lineto 6115 2101 lineto 6116 2101 lineto 6121 2101 lineto 6122 2101 lineto 6123 2101 lineto 6127 2101 lineto 6128 2101 lineto 6129 2101 lineto 6134 2101 lineto 6135 2101 lineto 6214 2101 lineto 6215 2101 lineto 6220 2101 lineto 6221 2101 lineto stroke 14 setlinewidth newpath 6221 2101 moveto 6221 2101 lineto 6226 2101 lineto 6227 2101 lineto 6228 2101 lineto 6233 2101 lineto 6234 2101 lineto 6313 2101 lineto 6314 2101 lineto 6319 2101 lineto 6320 2101 lineto 6325 2101 lineto 6326 2101 lineto 6327 2101 lineto 6332 2101 lineto 6333 2101 lineto stroke 14 setlinewidth 0 0 1 sss newpath 1267 725 moveto 1267 725 lineto 1268 725 lineto 1269 726 lineto 1269 727 lineto 1270 727 lineto 1272 727 lineto 1273 727 lineto 1274 727 lineto 1275 727 lineto 1276 728 lineto 1277 729 lineto 1278 730 lineto 1279 731 lineto 1280 732 lineto 1281 733 lineto 1282 734 lineto 1283 735 lineto 1284 736 lineto 1285 737 lineto 1286 738 lineto 1287 740 lineto 1288 741 lineto 1289 742 lineto 1290 743 lineto 1291 744 lineto 1292 745 lineto 1293 746 lineto 1296 746 lineto 1298 746 lineto 1299 746 lineto 1319 746 lineto 1322 746 lineto 1325 746 lineto 1334 746 lineto 1343 746 lineto 1345 746 lineto 1354 755 lineto 1362 765 lineto 1371 774 lineto 1380 784 lineto 1389 793 lineto 1398 803 lineto 1406 813 lineto 1415 822 lineto 1424 832 lineto 1433 841 lineto 1442 851 lineto 1450 860 lineto 1459 870 lineto 1468 880 lineto 1477 889 lineto 1486 899 lineto 1494 908 lineto 1501 916 lineto 1502 916 lineto 1504 916 lineto 1505 916 lineto 1507 916 lineto 1508 916 lineto 1527 916 lineto 1528 916 lineto 1530 916 lineto 1531 916 lineto 1533 916 lineto 1534 916 lineto 1553 916 lineto 1554 916 lineto 1556 916 lineto 1557 916 lineto 1559 916 lineto 1560 916 lineto 1736 916 lineto 1739 916 lineto 1742 916 lineto 1762 916 lineto 1765 916 lineto 1768 916 lineto 1788 916 lineto 1791 916 lineto 1794 916 lineto 1873 916 lineto 1953 916 lineto 1970 916 lineto 2049 1002 lineto 2129 1088 lineto 2208 1174 lineto 2287 1260 lineto 2366 1346 lineto 2445 1432 lineto 2524 1518 lineto 2604 1604 lineto 2683 1690 lineto 2762 1776 lineto 2841 1862 lineto 2920 1948 lineto 2999 2035 lineto 3079 2121 lineto 3158 2207 lineto 3237 2293 lineto 3316 2379 lineto 3378 2446 lineto 3380 2446 lineto 3381 2446 lineto 3383 2446 lineto 3384 2446 lineto 3404 2446 lineto 3407 2446 lineto 3409 2446 lineto 3410 2446 lineto 3430 2446 lineto 3433 2446 lineto 3435 2446 lineto 3436 2446 lineto 3612 2446 lineto 3613 2446 lineto 3615 2446 lineto 3616 2446 lineto 3618 2446 lineto 3619 2446 lineto 3638 2446 lineto 3639 2446 lineto 3641 2446 lineto 3642 2446 lineto 3644 2446 lineto 3645 2446 lineto 3664 2446 lineto 3665 2446 lineto 3667 2446 lineto 3668 2446 lineto 3670 2446 lineto 3671 2446 lineto 3847 2446 lineto 3850 2446 lineto 3852 2446 lineto 3853 2446 lineto 3873 2446 lineto 3876 2446 lineto 3878 2446 lineto 3879 2446 lineto 3899 2446 lineto 3902 2446 lineto 3905 2446 lineto 5488 2446 lineto 5489 2446 lineto 5491 2446 lineto 5492 2446 lineto 5494 2446 lineto 5495 2446 lineto 5515 2446 lineto 5517 2446 lineto 5518 2446 lineto 5520 2446 lineto 5521 2446 lineto 5541 2446 lineto 5543 2446 lineto 5544 2446 lineto 5546 2446 lineto 5547 2446 lineto 5723 2446 lineto 5724 2446 lineto 5726 2446 lineto 5727 2446 lineto 5729 2446 lineto 5749 2446 lineto 5750 2446 lineto 5752 2446 lineto 5753 2446 lineto 5755 2446 lineto 5775 2446 lineto 5776 2446 lineto 5778 2446 lineto 5779 2446 lineto 5781 2446 lineto 5782 2446 lineto 5958 2446 lineto 5960 2446 lineto 5961 2446 lineto 5963 2446 lineto 5964 2446 lineto 5984 2446 lineto 5987 2446 lineto 5989 2446 lineto 5990 2446 lineto 6010 2446 lineto 6013 2446 lineto 6015 2446 lineto 6016 2446 lineto stroke showpage %%EndDocument @endspecial 570 697 a Fh(\032)2579 2823 y(r)2596 815 y(n)19 b Fg(=)g(6)2596 921 y Fh(n)g Fg(=)g(5)2596 1028 y Fh(n)g Fg(=)g(4)2596 1134 y Fh(n)g Fg(=)g(3)584 3373 y Fn(Figure)d(4.)34 b(The)17 b(lar)o(gest)g(v)o(oid)g Fh(\032)g Fn(as)g(a)g(function)g(of)g Fh(r)r Fn(,)f(measured)i(from)e(the)h(left)g(most)g(point)g(in)f Fh(A)2941 3349 y Fa(\()p Fd(n)p Fa(\))3049 3373 y Fn(for)g Fh(n)j Fg(=)h(3)p Fh(;)12 b(:)f(:)g(:)h(;)f Fg(6)p Fn(.)165 3753 y Fq(Refer)n(ences)235 4012 y Fm(1.)49 b(F)-8 b(.)45 b(Da)n(vis,)50 b(Is)44 b(the)i(uni)m(v)n(erse)g(homogeneous)f(on)g(lar)n(ge)h(scales?)95 b(In:)76 b Fl(Critical)46 b(Dialo)o(gues)f(in)360 4128 y(Cosmolo)o(gy)p Fm(,)26 b(N.)g(T)-5 b(urok,)26 b(ed.)38 b(\(Sing)o(apore,)27 b(W)-8 b(orld)26 b(Scienti\256c,)h(1997\))235 4245 y(2.)49 b(J.-P)-11 b(.)29 b(Eckmann,)h(E.)f(Ja)-40 b(\310)6 b(rv)n(enpa)-40 b(\310)6 b(a)-40 b(\310)35 b(and)29 b(M.)f(Ja)-40 b(\310)6 b(rv)n(enpa)-40 b(\310)6 b(a)-40 b(\310)6 b(,)30 b(Porosities)f(and)f (dimensions)h(of)f(measures.)360 4361 y(Nonlinearity)-7 b(,)28 b(submitted.)235 4477 y(3.)49 b(J.-P)-11 b(.)32 b(Eckmann,)i(D.)d(Ruelle,)j (The)e(er)n(godic)g(theory)g(of)f(strange)g(attractors)i(Re)m(v)-7 b(.)55 b(Mod.)f(Ph)o(ysics)360 4593 y Fq(57)p Fm(,)26 b(617\261656)g (\(1985\))235 4709 y(4.)49 b(K.)26 b(J.)g(F)n(alconer)l(,)h Fl(T)-9 b(ec)n(hniques)26 b(in)g(fr)n(actal)g(g)o(eometry)g Fm(\(John)f(W)l(ile)n(y)i(&)f(Sons,)g(1997\))235 4826 y(5.)49 b(P)-11 b(.)27 b(Grassber)n(ger)f(and)h(I.)g(Procaccia,)h(On)e(the)h (characterization)j(of)c(strange)h(attractors)h(Ph)o(ys.)40 b(Re)m(v)-7 b(.)360 4942 y(Lett.)39 b Fq(50)p Fm(,)26 b(346)f(\(1983\))235 5058 y(6.)49 b(P)-11 b(.)23 b(Mattila,)i(Distrib)n(ution)e(of)g(sets)f(and)h (measures)g(along)g(planes.)36 b(J.)23 b(London)g(Math.)36 b(Soc.)h(\(2\))44 b Fq(38)p Fm(,)360 5174 y(125\261132)26 b(\(1988\))3843 5554 y Fq(10)p eop %%Page: 11 11 bop 235 533 a Fm(7.)49 b(P)-11 b(.)28 b(Mattila,)j(Geometry)e(of)f(Sets)g (and)g(Measures)g(in)g(Euclidean)i(Spaces)e(\(Cambridge)h(Uni)m(v)n(ersity) 360 649 y(Press,)d(1995\))235 765 y(8.)49 b(M.)25 b(E.)f(Mera)g(and)h(M.)f (Mora)-40 b(\302)6 b(n,)25 b(Porosity)e(of)h(measures)h(and)f(the)h(doubling) f(condition)h(\(to)g(appear\).)235 882 y(9.)49 b(G.)42 b(Murante,)k(A.)c(Pro) n(v)n(enzale,)48 b(E.A.)42 b(Spie)n(gel)h(and)f(R.)g(Thieber)n(ger)l(,)47 b(Scaling)c(re)n(gimes)f(in)g(the)360 998 y(distrib)n(ution)27 b(of)f(g)o(alaxies.)38 b(Ann.)f(NY)25 b(Academy)h(of)g(Sciences)g Fq(867)p Fm(,)g(258\261267)f(\(1998\))184 1114 y(10.)49 b(L.)32 b(Pietronero,)h(M.)e(Montuori,)h(and)f(F)-8 b(.)31 b(Sylos)f(Labini,)j(On)e (the)g(fractal)h(structure)f(of)g(the)g(visible)360 1230 y(uni)m(v)n(erse.)97 b(In:)75 b Fl(Critical)46 b(Dialo)o(gues)f(in)g(Cosmolo)o(gy)p Fm(,)k(N.)c(T)-5 b(urok,)51 b(ed.)95 b(\(Sing)o(apore,)51 b(W)-8 b(orld)360 1347 y(Scienti\256c,)28 b(1997\))184 1463 y(11.)49 b(A.)42 b(Salli,)k(On)41 b(the)g(Mink)o(o)m(wski)g(dimension)h(of)f(strongly) h(porous)e(fractal)j(sets)e(in)g Ff(R)3577 1427 y Fh(n)3623 1463 y Fm(.)85 b(Proc.)360 1579 y(London)26 b(Math.)38 b(Soc.)f(\(3\))25 b Fq(62)p Fm(,)h(353\261372)g(\(1991\))3843 5554 y Fq(11)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9904211331363--