Content-Type: multipart/mixed; boundary="-------------0207160616603" This is a multi-part message in MIME format. ---------------0207160616603 Content-Type: text/plain; name="02-310.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-310.keywords" periodic elliptic operators, band gap structure, spectral gaps ---------------0207160616603 Content-Type: application/postscript; name="isaac41.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="isaac41.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: isaac.dvi %%Pages: 12 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips isaac.dvi -o isaac.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.07.12:1806 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N /vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N /rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N /@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{ /hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B /@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{ /urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known {userdict/md get type/dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{ itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack} if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{ noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{ Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale }if 0 setgray}N/psfts{S 65781.76 div N}N/startTexFig{/psf$SavedState save N userdict maxlength dict begin/magscale true def normalscale currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts /psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR/showpage{}N/erasepage{}N/copypage{}N/p 3 def @MacSetUp}N/doclip{ psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N/endTexFig{end psf$SavedState restore}N/@beginspecial{SDict begin/SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count/ocount X/dcount countdictstack N}N/@setspecial{ CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR }{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if/showpage{}N/erasepage{}N/copypage{}N newpath}N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{end} repeat grestore SpecialSave restore end}N/@defspecial{SDict begin}N /@fedspecial{end}B/li{lineto}B/rl{rlineto}B/rc{rcurveto}B/np{/SaveX currentpoint/SaveY X N 1 setlinecap newpath}N/st{stroke SaveX SaveY moveto}N/fil{fill SaveX SaveY moveto}N/ellipse{/endangle X/startangle X /yrad X/xrad X/savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet TeXDict begin 39158280 55380996 1000 600 600 (isaac.dvi) @start %DVIPSBitmapFont: Fa cmtt10 10 20 /Fa 20 120 df<007FB6FCB71280A46C150021067B9B2C>45 D<121FEA3F80EA7FC0EAFF E0A5EA7FC0EA3F80EA1F000B0B708A2C>I64 D<3801FFF0000713FE001F6D7E15E048809038C01FF81407EC01FC381F80000006C77EC8 127EA3ECFFFE131F90B5FC1203120F48EB807E383FF800EA7FC090C7FC12FE5AA47E007F 14FEEB8003383FE01F6CB612FC6C15FE6C14BF0001EBFE1F3A003FF007FC27247CA32C> 97 DI<903803FFE0011F13F8017F13FE48B5FC48804848C6FCEA0FF0485A49137E 4848131890C9FC5A127EA25AA8127EA2127F6C140F6DEB1F806C7E6D133F6C6CEB7F0039 07FE03FF6CB55A6C5C6C6C5B011F13E0010390C7FC21247AA32C>III104 D<1307EB1FC0A2497EA36D5AA20107C7FC90C8FCA7387FFFC080B5FC7EA2EA0007B3A800 7FB512FCB612FEA36C14FC1F3479B32C>I<387FFFE0B57EA37EEA0003B3B3A5007FB612 80B712C0A36C158022337BB22C>108 D<3A7F83F007E09039CFFC1FF83AFFDFFE3FFCD8 7FFF13FF91B57E3A07FE1FFC3E01FCEBF83F496C487E01F013E001E013C0A301C01380B3 3B7FFC3FF87FF0027F13FFD8FFFE6D13F8D87FFC4913F0023F137F2D2481A32C>I<397F F01FE039FFF87FFC9038F9FFFE01FB7F6CB6FC00019038F03F80ECC01F02807FEC000F5B 5BA25BB3267FFFE0B5FCB500F11480A36C01E0140029247FA32C>II<397FF01FE0 39FFF8FFF801FB13FE90B6FC6C158000019038F07FC09138801FE091380007F049EB03F8 5BED01FC491300A216FE167EA816FE6D14FCA2ED01F86D13036DEB07F0150F9138801FE0 9138E07FC091B51280160001FB5B01F813F8EC3FC091C8FCAD387FFFE0B57EA36C5B2736 7FA32C>I114 D<90387FF8700003B512F8120F5A5A387FC00F387E00034813015AA36CEB00F0007F1400 13F0383FFFC06C13FE6CEBFF80000314E0C66C13F8010113FCEB0007EC00FE0078147F00 FC143F151F7EA26C143F6D133E6D13FE9038F007FC90B5FC15F815E000F8148039701FFC 0020247AA32C>I<131E133FA9007FB6FCB71280A36C1500D8003FC8FCB1ED03C0ED07E0 A5EC800F011FEB1FC0ECE07F6DB51280160001035B6D13F89038003FE0232E7EAD2C>I< 3A7FF003FF80486C487FA3007F7F0001EB000FB3A3151FA2153F6D137F3900FE03FF90B7 FC6D15807F6D13CF902603FE07130029247FA32C>I119 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb cmbx10 10 10 /Fb 10 58 df<49B4FC010F13E0017F13FC9038FF83FE4848C67E4848EB7F804848EB3F C04848EB1FE0A2001F15F0A24848EB0FF8A3007F15FCA500FF15FEB3007F15FCA4003F15 F8A26D131F001F15F0A2000F15E06D133F000715C06C6CEB7F806C6CEBFF003900FF83FE 6DB45A011F13F0010190C7FC27387CB630>48 D<141E143E14FE1307133FB5FCA313CFEA 000FB3B3A6007FB61280A4213779B630>IIII<001C15C0D81F80130701F8137F90B61280A216005D5D15F05D15 804AC7FC14F090C9FCA8EB07FE90383FFFE090B512F89038FC07FC9038E003FFD9800113 8090C713C0120EC813E0157F16F0A216F8A21206EA3F80EA7FE012FF7FA44914F0A26C48 13FF90C713E0007C15C06C5B6C491380D9C0071300390FF01FFE6CB512F8000114E06C6C 1380D90FF8C7FC25387BB630>II<12 3C123EEA3FE090B71280A41700485D5E5E5EA25E007CC7EA0FC000784A5A4BC7FC00F814 7E48147C15FC4A5A4A5AC7485A5D140F4A5A143F92C8FC5C147E14FE1301A2495AA31307 A2130F5CA2131FA5133FA96D5A6D5A6D5A293A7BB830>I<49B47E010F13F0013F13FC90 38FE01FF3A01F8007F804848EB3FC04848EB1FE0150F485AED07F0121FA27FA27F7F01FE EB0FE0EBFF809138E01FC06CEBF03F02FC13809138FF7F006C14FC6C5C7E6C14FE6D7F6D 14C04914E048B612F0EA07F848486C13F8261FE01F13FC383FC007EB8001007F6D13FE90 C7123F48140F48140715031501A21500A216FC7E6C14016D14F86C6CEB03F06D13076C6C EB0FE0D80FFEEB7FC00003B61200C614FC013F13F00103138027387CB630>II E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc cmr6 6 1 /Fc 1 49 df<13FF000313C0380781E0380F00F0001E137848133CA248131EA400F8131F AD0078131EA2007C133E003C133CA26C13786C13F0380781E03803FFC0C6130018227DA0 1E>48 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd cmex10 12 4 /Fd 4 91 df0 D<12E07E12787E7E7E7F6C7E6C7E7F12016C7E7F137C137E7FA26D7EA26D7EA26D7EA36D 7EA2801301A2801300A280A2147EA2147FA4801580A7EC1FC0B3A5EC3F80A715005CA414 7EA214FEA25CA213015CA213035CA2495AA3495AA2495AA249C7FCA2137E137C13FC5B48 5A12035B485A485A90C8FC121E5A5A5A5A1A777C832E>I<17FF040313C093380F81F093 381E0078043E137C93387C01FC9338F803FEA2150116F01503EF01FC9338E00070030714 00A3150FA45E151FA7153FA74B5AA715FFA85C93C8FCA95C5DA85DA74A5AA75DA75D140F A45DA3001C5C007F131FEAFF8092C9FC5C143EA26C485A007C5B003C5B381F03E03807FF 80D801FECAFC376F7B7F2F>82 D<1B3FF3FFC0973803E0F0973807C03897380F801C081F 137E97383F01FEF303FF1A7E1AFE1AFC1901A2963903F801FEF300781C004F5AA2190FA2 62191FA34F5AA44F5AA319FF97C8FCA360A261A21803A3611807A461180FA44E5AA4183F A261A2187FA361A218FFA44D5BA55F96C9FCA35FA360A2170FA460171FA460173FA54D5A A54D5AA45E60A55E60A44C90CAFCA54C5AA55F161FA45F163FA45FA2167FA35FA316FF5F A54B5BA494CBFCA25DA35EA21507A25EA44B5AA45E151FA45E153FA35EA2157FA25EA315 FF93CCFCA34A5AA44A5AA35D14075DA2140F5D121E397F801FC0EAFFC05D143F92CDFC5C 147E6C485AEA7E00383801F86C485A380F07C03803FF80D800FCCEFC58DD7B7F37>90 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe cmti12 12 46 /Fe 46 123 df12 DI<167016E0ED01C0ED0380ED0700150E153C5D15F85D4A5A4A5A4A5A14 0F4AC7FC141E143E5C147814F8495A5C1303495AA2495AA249C8FCA25B133E137E137CA2 5BA212015BA212035BA212075BA2120FA25BA2121FA290C9FCA25AA2123EA3127EA2127C A65AAB1278A6127C123CA47EA2120E120FA27E6C7EA26C7EA26C7E1360246472CA28>40 D<1560A2157081A281151E150E150FA2811680A3ED03C0A516E0A21501A71503A91507A2 16C0A4150FA21680A2151FA21600A25DA2153EA2157EA2157C15FCA25D1401A25D14035D A214075D140F5DA24AC7FCA2143EA25C147814F8495AA2495A5C1307495A91C8FC131E13 3E5B13785B485A485A485A48C9FC121E5A5A12E05A23647FCA28>I<13F0EA03FC1207A2 EA0FFEA4EA07FCEA03CCEA000C131C1318A2133813301370136013E0EA01C013801203EA 0700120E5A5A5A5A5A0F1D7A891E>44 D<007FB5FCB6FCA214FEA21805789723>I<120F EA3FC0127FA212FFA31380EA7F00123C0A0A76891E>I<16C01501A215031507ED0F8015 1F153F157F913801FF005C140F147F903807FCFEEB0FF0EB0700EB00015DA314035DA314 075DA3140F5DA3141F5DA3143F5DA3147F92C7FCA35C5CA313015CA313035CA313075CA2 130FA2131F133FB612FCA25D224276C132>49 DI<130FEB1FC0133FEB7FE013FFA214C0EB7F801400131E90C7FCB3A5120FEA 3FC0127FA212FFA35B6CC7FC123C132B76AA1E>58 D<91B712F818FF19C0020190398000 3FF06E90C7EA0FF84AED03FCF000FE4B157FA2F13F800203EE1FC05DF10FE0A214074B16 F01907A2140F5D1AF8A2141F5DA2190F143F5D1AF0A2147F4B151FA302FF17E092C9123F A34918C04A167F1A80A2010317FF4A1700A24E5A13074A4B5A611807010F5F4A4B5A181F 61011F4C5A4A4BC7FC18FE4D5A013F4B5A4A4A5A4D5A017FED3FC005FFC8FC4AEB03FE01 FFEC1FF8B812E094C9FC16F845447AC34A>68 D<91B91280A30201902680000713006E90 C8FC4A163FA24B81A30203160E5DA314074B151E191CA2140F5D17075F021F020E90C7FC 5DA2171E023F141C4B133CA2177C027F5CED800392B5FCA291B65AED00071601A2496E5A 5CA2160101035D5CA2160301075D4A90CAFCA3130F5CA3131F5CA3133F5CA2137FA313FF B612E0A341447AC340>70 D<027FB512E091B6FCA20200EBE000ED7F8015FFA293C7FCA3 5C5DA314035DA314075DA3140F5DA3141F5DA3143F5DA3147F5DA314FF92C8FCA35B5CA3 13035CA313075CA3130F5CA3131F5CA2133FA25CEBFFE0B612E0A25D2B447BC326>73 D<91B612F0A25F020101C0C7FC6E5B4A90C8FCA25DA314035DA314075DA3140F5DA3141F 5DA3143F5DA3147F5DA314FF92C9FCA35B5CA3010316104A1538A21878010716705C18F0 18E0010F15015C18C01703011F15074A1580170FA2013FED1F004A5C5F017F15FE16034A 130F01FFEC7FFCB8FCA25F35447AC33D>76 D<91B56C93387FFFC08298B5FC02014DEBC0 006E614A5FA203DF4C6CC7FC1A0E63912603CFE05D038F5F1A381A711407030FEEE1FCA2 F101C3020FEE0383020E60F107036F6C1507021E160E021C60191CF1380F143C02380470 5BA2F1E01F0278ED01C091267003F85EF003801A3F02F0ED070002E0030E5CA24E137F13 0102C04B91C8FC606201036D6C5B02805F4D5A943803800113070200DA07005BA2050E13 03495D010E606F6C5A1907011E5D011C4B5CA27048130F133C01384B5C017892C7FC191F 01F85C486C027E5DD807FE027C4A7EB500F00178013FB512C0A216705A447AC357>I<91 B56C49B512E0A28202009239000FFC00F107F0706E5A4A5F15DF705D1907EC03CFDB8FF8 92C7FCA203875D02077F0303150EA270141EEC0F01020E161C826F153C141E021C6E1338 167F1978023C800238013F1470A27113F00278131F02705E83040F130102F014F84A5E16 07EFFC0313014A01035C17FE1807010314014A02FF90C8FCA2705B0107168F91C8138E17 7F18DE5B010EED3FDC18FCA2011E151F011C5EA2170F133C01386F5A1378A201F8150348 6C5EEA07FEB500F01401A2604B447AC348>I<91B77E18F818FE020190398001FF806E90 C7EA3FC04AED1FE0F00FF04BEC07F8180319FC14034B15FEA314075DA3020FED07FC5DA2 F00FF8141F4B15F0F01FE0F03FC0023F16804BEC7F0018FEEF03F8027F4A5A4BEB1FC04C B4C7FC92B512F891B612E092380003F8EE00FE177F496F7E4A6E7EA28413034A140FA217 1F13075CA2173F130F5CA24D5A131F5CA3013F170E5CA2017FEE801E191C4A163C496C16 38B66C90383FC070051F13F094380FE1E0CA3803FF80943800FE003F467AC347>82 DI<48B912F85AA2913B0007FC 001FF0D807F84A130701E0010F140349160148485C90C71500A2001E021F15E05E121C12 3C0038143F4C1301007818C0127000F0147F485DA3C800FF91C7FC93C9FCA35C5DA31403 5DA314075DA3140F5DA3141F5DA3143F5DA3147F5DA314FF92CAFCA35B5CA21303A21307 497E007FB612C0A25E3D446FC346>I87 D97 DIIIII<15FCEC03FF91390F838380 91393E01CFC091387C00EF4A13FF4948137F010315804948133F495A131F4A1400133F91 C75A5B167E13FE16FE1201495CA215011203495CA21503A2495CA21507A25EA2150F151F 5E0001143F157F6C6C13FF913801DF8090387C039F90383E0F3FEB0FFCD903F090C7FC90 C7FC5DA2157EA215FEA25DA2001C495A127F48495A14074A5A485C023FC8FC00F8137E38 7C01F8381FFFE0000390C9FC2A407BAB2D>I<14FE137FA3EB01FC13001301A25CA21303 A25CA21307A25CA2130FA25CA2131FA25C157F90393F83FFC091388F81F091381E00F802 387F4948137C5C4A137EA2495A91C7FCA25B484814FE5E5BA2000314015E5BA200071403 5E5B1507000F5DA249130F5E001F1678031F1370491480A2003F023F13F0EE00E090C7FC 160148023E13C01603007E1680EE070000FEEC1E0FED1F1E48EC0FF80038EC03E02D467A C432>I<143C147E14FE1301A3EB00FC14701400AE137C48B4FC3803C780380703C0000F 13E0120E121C13071238A21278EA700F14C0131F00F0138012E0EA003F1400A25B137EA2 13FE5B12015BA212035B141E0007131C13E0A2000F133CEBC038A21478EB807014F014E0 EB81C0EA0783EBC7803803FE00EA00F8174378C11E>I<16F0ED03F8A21507A316F0ED01 C092C7FCAEEC01F0EC07FCEC1E1EEC380F0270138014E0130114C0EB03800107131F1400 A2130E153F131E011C140090C7FC5DA2157EA215FEA25DA21401A25DA21403A25DA21407 A25DA2140FA25DA2141FA25DA2143FA292C7FCA25C147EA214FE001C5B127F48485A495A A248485A495AD8F81FC8FCEA707EEA3FF8EA0FC0255683C11E>I<14FE137FA3EB01FC13 001301A25CA21303A25CA21307A25CA2130FA25CA2131FA25C167E013F49B4FC92380783 C09138000E07ED3C1F491370ED603F017E13E0EC01C09026FE03801380913907000E00D9 FC0E90C7FC5C00015B5C495AEBF9C03803FB8001FFC9FCA214F03807F3FCEBF07F9038E0 1FC06E7E000F130781EBC003A2001F150FA20180140EA2003F151E161C010013E0A2485D A2007E1578167000FE01015B15F1489038007F800038021FC7FC2A467AC42D>IIIIII<91381F800C91387FE01C903901F0703C903907C0387890390F801CF890 381F001D013E130F017E14F05B48481307A2484814E012075B000F140F16C0485AA2003F 141F491480A3007F143F90C71300A35D00FE147EA315FE5DA2007E1301A24A5A1407003E 130FA26C495A143B380F80F33807C3E73901FF87E038007E071300140F5DA3141F5DA314 3F92C7FCA25CA25C017F13FEA25D263F76AB2D>III<1470EB01F8A313035CA313075CA3130F5CA3131F5CA2 007FB512E0B6FC15C0D8003FC7FCA25B137EA313FE5BA312015BA312035BA312075BA312 0F5BA2EC0780001F140013805C140E003F131EEB001C143C14385C6C13F0495A6C485AEB 8780D807FEC7FCEA01F81B3F78BD20>I<137C48B414072603C780EB1F80380703C0000F 7F000E153F121C0107150012385E1278D8700F147E5C011F14FE00F05B00E05DEA003FEC 0001A2495C137E150313FE495CA215071201495CA2030F13380003167849ECC070A3031F 13F0EE80E0153F00011581037F13C06DEBEF8300000101148090397C03C787903A3E0F07 C70090391FFE01FE903903F000782D2D78AB34>I<017C143848B414FC3A03C78001FE38 0703C0000F13E0120E001C14000107147E1238163E1278D8700F141E5C131F00F049131C 12E0EA003F91C7123C16385B137E167801FE14705BA216F0000115E05B150116C0A24848 EB0380A2ED0700A2150E12015D6D5B000014786D5B90387C01E090383F0780D90FFFC7FC EB03F8272D78AB2D>I<017CEE038048B4020EEB0FC02603C780013FEB1FE0380703C000 0E7F5E001C037E130F01071607123804FE130300785DEA700F4A1501011F130100F00180 4914C012E0EA003FDA000314034C14805B137E0307140701FE1700495CA2030F5C000117 0E495CA260A24848495A60A2601201033F5C7F4B6C485A000002F713036D9039E7E00780 90267E01C349C7FC903A1F0781F81E903A0FFF007FF8D901FCEB0FE03B2D78AB41>I<02 F8133FD907FEEBFFE0903A0F0F83C0F0903A1C07C780F890393803CF03017013EE01E0EB FC07120101C013F8000316F00180EC01C000074AC7FC13001407485C120EC7FC140F5DA3 141F5DA3143F92C8FCA34AEB03C01780147EA202FEEB0700121E003F5D267F81FC130E6E 5BD8FF83143CD903BE5B26FE079E5B3A7C0F1F01E03A3C1E0F83C0271FF803FFC7FC3907 E000FC2D2D7CAB2D>I<137C48B414072603C780EB1F80380703C0000F7F000E153F001C 1600130712385E0078157EEA700F5C011F14FE00F0495B12E0EA003FEC00015E5B137E15 0301FE5C5BA2150700015D5BA2150F00035D5BA2151F5EA2153F12014BC7FC6D5B00005B EB7C0390383E0F7EEB1FFEEB03F090C712FE5DA214015D121F397F8003F0A24A5A484848 5A5D48131F00F049C8FC0070137E007813F8383801F0381E07C06CB4C9FCEA01FC294078 AB2F>I<027C130749B4130F49EB800E010F141E49EBC03CEDE03890393F03F07890397C 00FDF00178EB3FE00170EB03C001F0148049130790C7EA0F00151E5D5D5D4A5A4A5A4A5A 4AC7FC141E5C5C5C495A495A495A49C8FC011E14F04914E05B491301485A4848EB03C0D8 07B0130701FEEB0F80390FCF801F3A1F07E07F00393E03FFFED83C015B486C5B00705C00 F0EB7FC048011FC7FC282D7BAB28>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff cmsy8 8 8 /Ff 8 93 df0 D<123C127E12FFA4127E123C08087A9414>I<13 0C131EA50060EB01800078130739FC0C0FC0007FEB3F80393F8C7F003807CCF83801FFE0 38007F80011EC7FCEB7F803801FFE03807CCF8383F8C7F397F0C3F8000FCEB0FC039781E 078000601301000090C7FCA5130C1A1D7C9E23>3 D<140381B3A3B812FCA3C7D80380C7 FCB3B812FCA32E2F7CAD37>6 D14 D<12E012F812FEEA3F80EA0FE0EA03F8EA00FEEB3F80EB0FE0EB03F8EB00FC143FEC0FC0 EC07F0EC01FCEC007FED1FC0ED07F0ED01FCED007FEE1FC01607161FEE7F00ED01FCED07 F0ED1FC0037FC7FCEC01FCEC07F0EC0FC0023FC8FC14FCEB03F8EB0FE0EB3F8001FEC9FC EA03F8EA0FE0EA3F80007ECAFC12F812E0CBFCAD007FB71280B812C0A22A3B7AAB37>21 D49 D92 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg eufm10 12 4 /Fg 4 117 df<14C0EAC003495AD8E01FC8FC133C137CEAE1F8EA63F0127F5BA25BA45B A65CEC07C04A7EEC1FF8EC7FFE91B51280018114E01383138FEB9F87EBBE019038F8007F 01E0131F1380B116C0A400FF143F7F6D14807F6DEB7E006D137C6D13F06C6C485A6CEBC7 806C01FEC7FC6C5B6C13F0000113C06C90C8FC137C1330234777C333>98 D104 D<1403EC07C0DA0FF01340DA3FF813C091397FFE038049B5EA0F004914FE010F5C D91F9F5BD97F075B49C66C5AED078092C8FCA616C0ED03F09138800FF8ED3FFC9138C1FF FE14C791B5FCECFE0F90387FF807ECF00390383FC0011400131C90C7FCA6017F5C3801FF E04813F04813FC48EBFE03489038FF83F8D83C1FEBC3E0D87007EBF780D8C00101FEC7FC 486C13F8C7EA3FE0EC1F80020EC8FC2A3281AF2C>115 D<14385C5C1303495A130F133F 137FA25CAF001FB512E04814C05AB6128039007F8000B3A58080ECF001ECF807ECFE1FEC FFFC6D13F86D13F06D13C06D13806D1300EB00FC1478142020417DBE21>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fh cmr9 9 29 /Fh 29 88 df48 D<13075B5B137FEA07FFB5FC13BFEAF83F1200B3B3A2497E007FB51280A319327AB126> IIII<000C14C0380FC00F90B5128015005C5C14F014C0D80C18C7FC90C8FCA9 EB0FC0EB7FF8EBF07C380FC03F9038001F80EC0FC0120E000CEB07E0A2C713F01403A215 F8A41218127E12FEA315F0140712F8006014E01270EC0FC06C131F003C14806CEB7F0038 0F80FE3807FFF8000113E038003F801D347CB126>I<14FE903807FF80011F13E090383F 00F0017C13703901F801F8EBF003EA03E01207EA0FC0EC01F04848C7FCA248C8FCA35A12 7EEB07F0EB1FFC38FE381F9038700F809038E007C039FFC003E0018013F0EC01F8130015 FC1400A24814FEA5127EA4127F6C14FCA26C1301018013F8000F14F0EBC0030007EB07E0 3903E00FC03901F81F806CB51200EB3FFCEB0FE01F347DB126>I<1230123C003FB6FCA3 4814FEA215FC0070C7123800601430157015E04814C01401EC0380C7EA07001406140E5C 141814385CA25CA2495A1303A3495AA2130FA3131F91C7FCA25BA55BA9131C20347CB126 >III<123C127E12FFA4127E123C1200B0123C127E12FFA4127E123C08207A9F15>I<15E0A3 4A7EA24A7EA34A7EA3EC0DFE140CA2EC187FA34A6C7EA202707FEC601FA202E07FECC00F A2D901807F1507A249486C7EA301066D7EA2010E80010FB5FCA249800118C77EA2498116 3FA2496E7EA3496E7EA20001821607487ED81FF04A7ED8FFFE49B512E0A333367DB53A> 65 D67 DIIIIII76 DIIII82 D<90381FE00390387FFC0748B5FC3907F01FCF390F8003FF48C7FC 003E80814880A200788000F880A46C80A27E92C7FC127F13C0EA3FF013FF6C13F06C13FF 6C14C06C14F0C680013F7F01037F9038003FFF140302001380157F153FED1FC0150F12C0 A21507A37EA26CEC0F80A26C15006C5C6C143E6C147E01C05B39F1FC03F800E0B512E001 1F138026C003FEC7FC22377CB42B>I<007FB712FEA390398007F001D87C00EC003E0078 161E0070160EA20060160600E01607A3481603A6C71500B3AB4A7E011FB512FCA330337D B237>I86 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fi cmti10 10 53 /Fi 53 128 df<04FFEB03F003039038E00FFC923A0FC0F01F1E923A3F00783E0F923A7E 01F87C3FDB7C03EBFC7F03FC14F8DA01F813F905F1137EDC01E1133C913B03F00003F000 A314074B130760A3140F4B130F60A3010FB812C0A3903C001F80001F8000A3023F143F92 C790C7FCA44A5C027E147EA402FE14FE4A5CA413014A13015FA313034A13035FA313074A 495AA44948495AA44948495AA3001CD9038090C8FC007E90380FC03F013E143E00FE011F 5B133C017C5C3AF8780F01E0D878F0EB07C0273FE003FFC9FC390F8000FC404C82BA33> 11 DI< EE7FE0923903FFFC7E92380FC03E92381F000F033EEB3FFE4B137F03FC14FC5D1401173D 4A48EB01F8A21703A24A4814F0A21707A2020F15E05D170FA218C0010FB7FCA3903B001F 80001F80A2173F143F92C71300A25FA24A147E147E17FEA25F14FE4A1301A25FA2010114 035CEFF070A21607010316F04AECE0E0A3EFE1C013074A14C3933803E380EE01E7933800 FF004948143C94C7FCA3495AA3001C90CAFC127E133E12FE133C137CEAF878EA78F0EA3F E0EA0F80374C82BA31>I<130FEB1F80133F137FEBFF00485A5BEA03F0485A485A485A00 3EC7FC5A5A12E05A111064B92A>19 D39 D44 D<387FFFF8A2B5FCA214F0150579941E>I<12 0EEA3F80127F12FFA31300127E123C0909778819>I<133C137E13FF5AA313FE13FCEA00 701300B2120EEA3F80127F12FFA31300127E123C102477A319>58 D65 D<0107B612FCEFFF8018C0903B000F F0001FF04BEB07F81703021F15FC17014B14FEA2023F1400A24B1301A2147F18FC92C712 0318F84A140718F04AEC0FE0EF1FC00101ED3F80EF7F004AEB01FEEE07F849B612E05F91 39F80007F0EE01FC01076E7E177F4AEC3F80A2010F16C0171F5CA2131F173F5CA2133FEF 7F805C1800017F5D4C5A91C7485A5F49140FEE1FE0494A5A00014AB45AB748C7FC16F816 C037397BB83A>II<0103B612FEEFFFC018F0903B0007F8000FF84BEB03FCEF00FE020F157FF0 3F804B141F19C0021F150F19E05D1807143F19F05DA2147FA292C8FCA25C180F5CA21301 19E04A151FA2130319C04A153FA201071780187F4A1600A2010F16FEA24A4A5A60011F15 034D5A4A5D4D5A013F4B5A173F4A4AC7FC17FC017FEC03F84C5A91C7EA1FC04949B45A00 7F90B548C8FCB712F016803C397CB83F>I<0107B8FCA3903A000FF000034BEB007F183E 141F181E5DA2143FA25D181C147FA29238000380A24A130718004A91C7FC5E13015E4A13 3E167E49B512FEA25EECF8000107147C163C4A1338A2010F147818E04A13701701011F16 C016004A14031880013F150718004A5CA2017F151E173E91C8123C177C4915FC4C5A4914 070001ED7FF0B8FCA25F38397BB838>I<0107B712FEA3903A000FF000074B1300187C02 1F153CA25DA2143FA25D1838147FA292C8FCEE03804A130718004A91C7FCA201015CA24A 131E163E010314FE91B5FC5EA2903807F800167C4A1378A2130FA24A1370A2011F14F0A2 4A90C8FCA2133FA25CA2137FA291CAFCA25BA25B487EB6FCA337397BB836>II<01 03B5D8F80FB512E0A390260007F8C7381FE0004B5DA2020F153F615DA2021F157F96C7FC 5DA2023F5D605DA2027F14016092C7FCA24A1403605CA249B7FC60A202FCC71207010315 0F605CA20107151F605CA2010F153F605CA2011F157F95C8FC5CA2013F5D5F5CA2017F14 015F91C7FC491403007FD9FE01B512F8B55BA243397CB83E>I<0103B512F8A390390007 F8005DA2140FA25DA2141FA25DA2143FA25DA2147FA292C7FCA25CA25CA21301A25CA213 03A25CA21307A25CA2130FA25CA2131FA25CA2133FA25CA2137FA291C8FC497EB6FCA25C 25397CB820>I<0103B500F890387FFFE0A21AC090260007F8C7380FFC004B15E061020F 4BC7FC183E4B5C18F0021F4A5A4D5A4BEB0F804DC8FC023F143C5F4B5B4C5A027FEB07C0 4CC9FCED001E5E4A5BED01FCECFE0315070101497E151FECFC7C4B7E903903FDE07FDAFF C07F1580ED003F49488014F84A131F83130F160F4A801607011F81A24A130383133F1601 4A80A2017F6E7EA291C8FC494A7F007F01FE011F13FCB55CA243397CB840>75 D<0107B512FCA25E9026000FF8C7FC5D5D141FA25DA2143FA25DA2147FA292C8FCA25CA2 5CA21301A25CA21303A25CA21307A25CA2130F170C4A141CA2011F153C17384A1478A201 3F157017F04A14E01601017F140317C091C71207160F49EC1F80163F4914FF0001020713 00B8FCA25E2E397BB834>I<902607FFF8923807FFF0614F13E0D9000FEFF0004F5AA202 1F167FF1EFC0141DDA1CFCEC01CF023C16DF9538039F800238ED071FA20278ED0E3F97C7 FC0270151CA202F04B5AF0707E14E0037E14E0010117FE4D485A02C0EC0380A20103ED07 01610280140EA20107ED1C0305385B14006F137049160705E05B010EEC01C0A2011E9138 03800F61011CEC0700A2013C020E131F4C5C1338ED1FB80178163F04F091C8FC01705CA2 01F04A5B187E00015DD807F816FEB500C09039007FFFFC151E150E4C397AB84A>I79 D<0107B612F817FF1880903B000FF0003FE04BEB0FF0EF03F8141FEF01FC5DA2023F15FE A25DA2147FEF03FC92C7FCA24A15F817074A15F0EF0FE01301EF1FC04AEC3F80EFFE0001 034A5AEE0FF091B612C04CC7FCD907F8C9FCA25CA2130FA25CA2131FA25CA2133FA25CA2 137FA291CAFCA25BA25B1201B512FCA337397BB838>II<0103B612F017FEEFFF80903B0007F8003FC04BEB0FF01707020FEC03F8EF01 FC5DA2021F15FEA25DA2143FEF03FC5DA2027FEC07F818F092C7120F18E04AEC1FC0EF3F 004A14FEEE01F80101EC0FE091B6128004FCC7FC9138FC003F0103EC0F80834A6D7E8301 071403A25C83010F14075F5CA2011F140FA25CA2133F161F4AECE007A2017F160F180E91 C7FC49020F131C007F01FE153CB5913807F078040313F0CAEAFFE0EF3F80383B7CB83D> I<92383FC00E913901FFF01C020713FC91391FC07E3C91393F001F7C027CEB0FF84A1307 49481303495A4948EB01F0A2495AA2011F15E091C7FCA34915C0A36E90C7FCA2806D7E14 FCECFF806D13F015FE6D6D7E6D14E0010080023F7F14079138007FFC150F15031501A215 00A2167C120EA3001E15FC5EA3003E4A5AA24B5AA2007F4A5A4B5A6D49C7FC6D133ED8F9 F013FC39F8FC03F839F07FFFE0D8E01F138026C003FCC8FC2F3D7ABA2F>I<0007B812E0 A25AD9F800EB001F01C049EB07C0485AD900011403121E001C5C003C1780140312380078 5C00701607140700F01700485CA2140FC792C7FC5DA2141FA25DA2143FA25DA2147FA292 C9FCA25CA25CA21301A25CA21303A25CA21307A25CA2130FA25CEB3FF0007FB512F8B6FC A2333971B83B>I86 D<14F8EB07FE90381F871C90383E 03FE137CEBF801120148486C5A485A120FEBC001001F5CA2EA3F801403007F5C1300A214 07485C5AA2140F5D48ECC1C0A2141F15831680143F1587007C017F1300ECFF076C485B90 38038F8E391F0F079E3907FE03FC3901F000F0222677A42A>97 D<133FEA1FFFA3C67E13 7EA313FE5BA312015BA312035BA31207EBE0F8EBE7FE9038EF0F80390FFC07C013F89038 F003E013E0D81FC013F0A21380A2123F1300A214075A127EA2140F12FE4814E0A2141F15 C05AEC3F80A215005C147E5C387801F8007C5B383C03E0383E07C0381E1F80D80FFEC7FC EA01F01C3B77B926>I<147F903803FFC090380FC1E090381F0070017E13784913383901 F801F83803F003120713E0120FD81FC013F091C7FC485AA2127F90C8FCA35A5AA45AA315 3015381578007C14F0007EEB01E0003EEB03C0EC0F806CEB3E00380F81F83803FFE0C690 C7FC1D2677A426>II<147F903803FFC090380FC1E090383F00F0017E13785B485A 485A485A120F4913F8001F14F0383F8001EC07E0EC1F80397F81FF00EBFFF891C7FC90C8 FC5A5AA55AA21530007C14381578007E14F0003EEB01E0EC03C06CEB0F806CEB3E003807 81F83803FFE0C690C7FC1D2677A426>III II107 DIII<147F903803FFC09038 0FC1F090381F00F8017E137C5B4848137E4848133E0007143F5B120F485AA2485A157F12 7F90C7FCA215FF5A4814FEA2140115FC5AEC03F8A2EC07F015E0140F007C14C0007EEB1F 80003EEB3F00147E6C13F8380F83F03803FFC0C648C7FC202677A42A>I<9039078007C0 90391FE03FF090393CF0787C903938F8E03E9038787FC00170497EECFF00D9F0FE148013 E05CEA01E113C15CA2D80003143FA25CA20107147FA24A1400A2010F5C5E5C4B5A131F5E EC80035E013F495A6E485A5E6E48C7FC017F133EEC70FC90387E3FF0EC0F8001FEC9FCA2 5BA21201A25BA21203A25B1207B512C0A3293580A42A>II<3903C003F0390FF01FFC391E783C0F381C7C703A3C3EE03F8038383FC0 EB7F800078150000701300151CD8F07E90C7FCEAE0FE5BA2120012015BA312035BA31207 5BA3120F5BA3121F5BA3123F90C9FC120E212679A423>I<14FE903807FF8090380F83C0 90383E00E04913F00178137001F813F00001130313F0A215E00003EB01C06DC7FC7FEBFF C06C13F814FE6C7F6D13807F010F13C01300143F141F140F123E127E00FE1480A348EB1F 0012E06C133E00705B6C5B381E03E06CB45AD801FEC7FC1C267AA422>II<13F8D803FEEB01C0D8078FEB03E0390E 0F8007121E121C0038140F131F007815C01270013F131F00F0130000E015805BD8007E13 3FA201FE14005B5D120149137EA215FE120349EBFC0EA20201131E161C15F813E0163CD9 F003133814070001ECF07091381EF8F03A00F83C78E090393FF03FC090390FC00F002726 79A42D>I<01F0130ED803FC133FD8071EEB7F80EA0E1F121C123C0038143F49131F0070 140FA25BD8F07E140000E08013FEC6485B150E12015B151E0003141C5BA2153C00071438 5B5DA35DA24A5A140300035C6D48C7FC0001130E3800F83CEB7FF8EB0FC0212679A426> I<01F01507D803FC903903801F80D8071E903907C03FC0D80E1F130F121C123C0038021F 131F49EC800F00701607A249133FD8F07E168000E0ED000313FEC6484913071800000114 7E5B03FE5B0003160E495BA2171E00070101141C01E05B173C1738A217781770020314F0 5F0003010713016D486C485A000190391E7C07802800FC3C3E0FC7FC90393FF81FFE9039 0FE003F0322679A437>I<903907E007C090391FF81FF89039787C383C9038F03E703A01 E01EE0FE3803C01F018013C0D8070014FC481480000E1570023F1300001E91C7FC121CA2 C75AA2147EA214FEA25CA21301A24A1370A2010314F016E0001C5B007E1401010714C000 FEEC0380010F1307010EEB0F0039781CF81E9038387C3C393FF03FF03907C00FC027267C A427>I<13F0D803FCEB01C0D8071EEB03E0D80E1F1307121C123C0038140F4914C01270 A249131FD8F07E148012E013FEC648133F160012015B5D0003147E5BA215FE00075C5BA2 14015DA314035D14070003130FEBF01F3901F87FE038007FF7EB1FC7EB000F5DA2141F00 3F5C48133F92C7FC147E147C007E13FC387001F8EB03E06C485A383C1F80D80FFEC8FCEA 03F0233679A428>I<001E1338007F13FEEAFF811383A3EB03FC00FE13F8383800F01709 6AB72A>127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fj cmmi8 8 19 /Fj 19 111 df<131FD9FFC013304801F0137000076D13604815E0D9807C13C0391E003C 0148011E13800038EB0E03480107130000605CEC030612E0C7138EEC018C159C159815B8 15B0A215F05DA35DA25DA21403A44AC7FCA4140EA45CA31418242C7F9D24>13 D<147C49B4FC903803C78090380783C090381F03E0EB1E01133E017C13F013F8A2EA01F0 120313E01207A2EA0FC01403A2EA1F80A21407003F14E0130090B5FCA2397F000FC0127E A2141F1580127C00FC14005CA2147EA248137C14FC00785B495AA2387C03E0383C07C049 5A001C90C7FCEA1E3EEA0FF8EA03E01C307DAE21>18 D<13FC13FFEB1FC0130F6D7EA36D 7EA2130180A26D7EA3147EA280A36E7EA2140F81A24A7E143F147FECF3F0EB01E3EB03C1 90380781F8130F49C67E133E5B49137E485A48487F1207485A4848EB1F8048C7FC127E48 EC0FC048EC07E000701403232F7DAD29>21 D 34 D<15F8EC03FCEC0F0EEC1C0F913838078014301470026013C0A214E014C0A514E0EC 700F3807C030380FE038381CF81E393878079F0030903801FF8039707C007F0060EC1FF0 01FCEB3FF8D8E0F8140012C0A200815C0001147E5BA200035CA2495B14015D0007130301 C05B4A5AA24A5A4AC7FC3803E01E5C6C6C5A3800FFE0EB1F8025307EAE2A>I<15C01401 14031580A214071500A25C140EA2141E141CA2143C143814781470A214F05CA213015CA2 13035C130791C7FCA25B130EA2131E131CA2133C1338A21378137013F05BA212015BA212 035BA2120790C8FC5A120EA2121E121CA2123C1238A212781270A212F05AA21A437CB123 >61 D<013FB6FC17C0903A00FE0007F0EE01F84AEB00FC177E1301177F5CA21303177E4A 14FEA20107EC01FC17F84AEB03F0EE07E0010FEC1FC0EE7F009138C003FC91B55A4914FE 9139C0003F804AEB0FC017E0013F140717F091C7FC16035BA2017E1407A201FE15E0160F 4915C0161F0001ED3F80EE7F004914FEED03F80003EC0FF0B712C003FCC7FC302D7CAC35 >66 D<92387FC003913903FFF80791391FC03E0F91397E00071FD901F8EB03BF4948EB01 FED90FC013004948147E49C8FC017E157C49153C485A120348481538485AA2485A173048 481500A2127F90CAFCA35A5AA5EE018016031700A2007E5D1606160E6C5D5E6C6C5C000F 5D6C6C495A6C6CEB0780D801F8011EC7FCD8007E13F890381FFFE0010390C8FC302F7CAD 32>I<013FB512FEEEFFC0903A00FE0007F0EE01F84AEB007E8301018118804A140F18C0 0103150718E05CA21307A25CA2130FA24A140FA2131F18C04A141FA2013F1680173F91C8 1300A249157EA2017E5D5F01FE14014C5A494A5A4C5A00014BC7FC163E4914FCED03F000 03EC1FC0B7C8FC15F8332D7CAC3A>I77 DI81 D<3B7FFFF801FFFEA2D801FCC7EA0FC0178049EC070016060003150E160C5BA20007151C 16185BA2000F153816305BA2001F157016605BA2003F15E05E90C8FCA24814015E127EA2 150300FE92C7FC5A5D1506150E007C5C151815386C5C5D6CEB03C0260F800FC8FC3803E0 3C3801FFF038003FC02F2E7BAC30>85 D99 D<1307EB0F80EB1FC0A2EB0F80EB070090C7FCA9EA01E0EA07F8EA0E3CEA1C3E12381230 1270EA607EEAE07C12C013FC485A120012015B12035BA21207EBC04014C0120F13801381 381F01801303EB0700EA0F06131EEA07F8EA01F0122E7EAC18>105 D<15E0EC01F01403A3EC01C091C7FCA9147CEB03FE9038078F80EB0E07131C013813C013 30EB700F0160138013E013C0EB801F13001500A25CA2143EA2147EA2147CA214FCA25CA2 1301A25CA21303A25CA2130700385BEAFC0F5C49C7FCEAF83EEAF0F8EA7FF0EA1F801C3B 81AC1D>I<131FEA03FFA2EA003FA2133EA2137EA2137CA213FCA25BA2120115F89038F0 03FCEC0F0E0003EB1C1EEC387EEBE07014E03807E1C09038E3803849C7FC13CEEA0FDC13 F8A2EBFF80381F9FE0EB83F0EB01F81300481404150C123EA2007E141C1518007CEBF038 ECF83000FC1470EC78E048EB3FC00070EB0F801F2F7DAD25>I<27078007F0137E3C1FE0 1FFC03FF803C18F0781F0783E03B3878E00F1E01263079C001B87F26707F8013B0006001 0013F001FE14E000E015C0485A4914800081021F130300015F491400A200034A13076049 133E170F0007027EEC8080188149017C131F1801000F02FCEB3F03053E130049495C180E 001F0101EC1E0C183C010049EB0FF0000E6D48EB03E0391F7E9D3E>109 D<3907C007E0391FE03FF83918F8783E393879E01E39307B801F38707F00126013FEEAE0 FC12C05B00815C0001143E5BA20003147E157C5B15FC0007ECF8081618EBC00115F0000F 1538913803E0300180147016E0001F010113C015E390C7EAFF00000E143E251F7E9D2B> I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fk msbm10 12 3 /Fk 3 91 df<007FB54AB512C0B66C4914E0816C6E6D14C02707F003F09039000FF80026 01F801ED03F000006D7E017C6D6E5A017E137E017F133E8102807FECC00F6E6C7E017B80 903979F003F0ECF801903978FC00F8027C7F6E137E023F133E6E6C7E020F1480913907C0 0FC0EDE007913903F003E0020114F0913900F801F8EDFC00037E137C033E137E6F133EEE 801FDB0FC013810307EB0FC1923803E0079338F003E1DB01F813F1923900FC01F9EE7C00 70137D043F137F93381F803F040F131F933807C00F17E0933803F00704011303933800F8 0117FC177E173E171F1881EF0FC11707EF03E118F1EF01F91700187D01FC167F183FD803 FF161F007F01F8150FB57E18076C491503CB1201725A43467DC339>78 D<007FB712C0B812FCEFFF806C17E02800F807F00F13F8DBC00113FE017890398000FCFF 94387C3F8094383E0FC0727E94381E03F0EF1F011800717FA21978A519F8A24D5B1801EF 1E034E5A94383E0FC094387E3F80DDFDFFC7FC933807FFFE92B612F818E095C8FC17F0ED 87C1EEC0F8923883E0FC177C923881F03EA2923880F81F84EE7C0F717E163E93383F03E0 041F7FEE0F81EF80F8EE07C0187C933803E07E183E706C7E85706C6C7E180794387C03E0 057E7F94383E01F8716C7E197C01F86D6D6C7EF13F80007FB66C6CB512E0B700C015F083 6C4B6C14E044447EC33D>82 D<0003B812FE4883A301879039000F803ED98FF0011F137E D9BFC0EC007C01FFC7003E5B13FC48484A485A49ECFC034902F85B4B48485A5B49494848 5A0307131F04C090C7FC90C7380F803EA24B485A00064A13FCC8003E5B4B485AA24B485A 0201130703F05B4A48485AA24A4848C8FC5E91380F803E021F5B1500023E5B1501027C5B 9138FC03E014F84948485AA24948485A0107011F156002C090C812F090380F803EA24948 4814014913FC013E5B494848EC03E0A249484814070001130701F049140F48484848141F A2484848C8EA3FC0000F49157FD9803E15FF484848EC01FBEF07F3003E49EC0FE7D87E01 ED7F87007C49903903FF0780BAFCA36C18003C447DC345>90 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fl cmr8 8 9 /Fl 9 62 df10 D<13031307130E131C1338137013F0EA01E013C01203EA0780A2EA0F00A2121EA35A A45AA512F8A25AAB7EA21278A57EA47EA37EA2EA0780A2EA03C0120113E0EA00F0137013 38131C130E1307130310437AB11B>40 D<12C07E12707E7E7E120FEA0780120313C0EA01 E0A2EA00F0A21378A3133CA4131EA5131FA2130FAB131FA2131EA5133CA41378A313F0A2 EA01E0A2EA03C013801207EA0F00120E5A5A5A5A5A10437CB11B>I43 D48 D<130C133C137CEA03FC12FFEAFC7C1200B3B113FE387FFFFEA2172C7AAB23>III61 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fm cmmi12 12 50 /Fm 50 127 df13 D<1578913807FFE0021F13FC91383C7FFEEC7007EC6003 ECE0004A13381600A280A380A280147CA2147E143E143F816E7EA26E7E81140781EC3FFC 14FF903803E1FEEB07C190381F00FF133E49EB7F805B0001143F485A484814C049131F12 0F485AA248C7FC150F5A127EA300FEEC1F805AA316005A5DA2153E157E157CA26C5C127C 4A5A6C495AA26C495A6C6C485A6C6C48C7FC3803E07C3800FFF0EB1FC027487CC62B>I< 157E913801FF80913807C3E091381F01F0EC3E004A13F814FC4948137C495A5C0107147E 495A131F5C133F49C7127FA213FEA212015B12034914FF1207A25B000F15FE1501A2485A A21503003F15FC5B90B6FCA24815F89038800007A2150F00FF15F090C7FCA2ED1FE0A25A ED3FC0A21680157F16005A15FEA24A5AA25D14035D4A5A007C495AA24A5A007E49C7FC00 3E133E5C001E5B6C485A380783C06CB4C8FCEA00FC28477CC52D>18 D21 D<147002F8140E0101153FA301035DA24A147EA2010715FEA24A5CA2010F1401A2 4A5CA2011F1403A24A5CA2013F1407A291C75BA249140FA2017E5DA201FE021F13181838 49ED8030A20001033F13701860EE7F005E486C16E0DB01DF13C09238039F016DD9071F13 80489039801E0F83903BF7C078078700903AE1FFE003FE903AE07F8000F8000F90CAFCA2 5BA2121FA25BA2123FA290CBFCA25AA2127EA212FEA25A123835417DAB3B>II<010FB712E0013F16F05B48B812E04817C02807 E0060030C7FCEB800EEA0F00001E010C13705A0038011C13605A0060011813E000E01338 1240C7FC5C4B5AA214F014E01301150314C01303A3EB078082130FA2EB1F00A34980133E 137EA24980A2000114015BA26C48EB00E0342C7EAA37>25 DI<0203B612E0021F15F091B7FC4916E0010716C090 270FF80FF8C7FC90381FC00349486C7E017EC7FC49147E485A4848143E0007153F5B485A A2485AA2123F90C8FC5E48157E127EA216FE00FE5D5A15015EA24B5A007C5D15074B5A5E 6C4AC8FC153E6C5C5D390F8003F03907C007C02601F03FC9FC38007FFCEB1FE0342C7DAA 37>I<137E48B46C150626078FE0150E260607F0151C260E03F81538000C6D1570D81C01 16E000006D15C0010015016EEC03806EEC0700170E6E6C5B5F5F6E6C136017E04C5A6E6C 485A4CC7FC0207130E6F5A5E1630913803F8705EEDF9C06EB45A93C8FC5D6E5A81A2157E 15FF5C5C9138073F80140E141C9138181FC014381470ECE00FD901C07FEB038049486C7E 130E130C011C6D7E5B5B496D7E485A48488048C8FC000681000E6F137048EE806048033F 13E04892381FC0C048ED0FE348923803FF00CA12FC37407DAB3D>31 D34 DI<15FE913803FFC0 91380F83E091383E01F89138F800FC4948137C4948137E49487F130F495A91C713805B13 7EA25BA2485A167F12035BA216FF000716005BA25D000F5D5B5E1503001F5D15075E150F 003F5D00374A5A00335D6D49C7FC157ED871F05B0060495A90387803E090383C0FC0D91F FFC8FC38E003F890CAFCA25AA47EA312F812FE007FB512C015F0001F800007801201C812 1CA2293F78AB32>37 D<121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A0A78891B>58 D<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413E013C0A312011380120313005A 1206120E5A5A5A12600B1D78891B>II<1618163C16 7CA2167816F8A216F01501A216E01503A216C01507A21680150FA2ED1F00A2151E153EA2 153C157CA2157815F8A25D1401A24A5AA25D1407A25D140FA292C7FC5CA2141E143EA214 3C147CA25CA25C1301A25C1303A25C1307A25C130FA291C8FC5BA2133EA2133C137CA213 7813F8A25B1201A25B1203A2485AA25B120FA290C9FC5AA2121E123EA2123C127CA21278 12F8A25A126026647BCA31>I<127012FCB4FCEA7FC0EA1FF0EA07FCEA01FF38007FC0EB 1FF0EB07FE903801FF809038007FE0EC1FF8EC03FE913800FF80ED3FE0ED0FF8ED03FF03 0013C0EE3FF0EE0FFCEE01FF9338007FC0EF1FF0EF07FCEF01FF9438007FC0F01FE0A2F0 7FC0943801FF00EF07FCEF1FF0EF7FC04C48C7FCEE0FFCEE3FF0EEFFC0030390C8FCED0F F8ED3FE0EDFF80DA03FEC9FCEC1FF8EC7FE0903801FF80D907FECAFCEB1FF0EB7FC04848 CBFCEA07FCEA1FF0EA7FC048CCFC12FC12703B3878B44C>I64 D<91B87E19F019FC02009039C000 03FF6F480100138003FFED3FC01AE093C8121FF10FF0A24A17F84B1507A314035D190FA2 020717F04B151F1AE0193F020F17C04BED7F80F1FF004E5A021F4B5A4B4A5AF01FF0F03F C0023F4AB4C7FC4BEB1FFC92B612F018FEDA7FC0C7EA7F804BEC1FC0F00FF0727E02FF6F 7E92C8FC727EA249835CA313035CA301075F4A1503A24E5A130F4A4B5A4E5AA2011F4C5A 4A4B5A4D485A013F4B48C7FCEF0FFC4AEC3FF801FF913801FFE0B9128005FCC8FC17C045 447CC34A>66 D<4CB46C1318043F01F013384BB512FC0307D9007E1378DB1FF090380F80 F0DB7F80EB03C1DA01FEC7EA01C34A48EC00E7DA0FF0ED7FE04A48153F4A5A02FFC9121F 494817C04948160F495A130F4A178049481607495A137F4948170091CAFC5A485A190648 5AA2485A96C7FC121F5BA2123F5BA3127F5BA4485AA419C0A2180161127F180396C7FC60 18066C6C160E601818001F17386D5E000F5F6D4B5A6C6C4B5A00034CC8FC6C6C150E6C6C 153C017F5DD93FC0EB01E0D91FF0EB0FC0D907FE017FC9FC0101B512FCD9003F13E00207 90CAFC45487CC546>I<91B87E19F019FC02009039C00007FF6F489038007FC003FFED1F E0737E93C86C7E737E19014A707E5D1A7FA20203EF3F805DA21BC014075DA3140F4B17E0 A3141F4B17C0A3143F4B167FA3027F18804B16FFA302FF180092C95A62A24917034A5F19 076201034D5A5C4F5A620107173F4A5F4FC7FC19FE010F4C5A4A15034E5AF00FE0011F4C 5A4A4B5A06FFC8FC013FED01FCEF0FF84AEC3FE001FF913803FF80B848C9FC17F094CAFC 4B447CC351>I<91B912FCA3020001C0C7123F6F48EC03F803FF1501190093C91278A21A 385C5DA3020317305DA314074B1460A218E0020F4B13005DA21701021F5D4B1303170717 0F023F027FC8FC92B6FCA391397FC0007E4B131EA2170E02FF140C92C7FCA2171C490318 13035C611906010392C7FC4A160E190C191C010717184A163819301970130F4A5E180161 011F16034A15074E5A013F163F4EC7FC4AEC03FF01FFED3FFEB9FCA26046447CC348>I< 91B912F8A3020001C0C7123F6F48EC07F003FF1503190193C9FCA21A705C5DA302031760 5DA314075D18C01701020F4B13005DA21703021F92C8FC4B5BA25F023F141E4B13FE92B5 FCA24A5CED8000173CA202FF141892C7FCA217384915305CA21770010315604A91C9FCA3 13075CA3130F5CA3131F5CA2133FA313FFB612F8A345447CC33F>I<91B6D8E003B61280 A3020001E0C70003EB8000DB7F806E48C7FC03FF1503A293C85BA219075C4B5EA2190F14 034B5EA2191F14074B5EA2193F140F4B5EA2197F141F4B5EA219FF143F92B8C8FCA3DA7F C0C712014B5DA2180314FF92C85BA218075B4A5EA2180F13034A5EA2181F13074A5EA218 3F130F4A5EA2187F131F4A5EA2013F16FFA24A93C9FCD9FFE002037FB6D8E003B67EA351 447CC351>72 D<027FB512F8A217F09139007FF000ED3FC0157FA25EA315FF93C7FCA35C 5DA314035DA314075DA3140F5DA3141F5DA3143F5DA3147F5DA314FF92C8FCA35B5CA313 035CA313075CA3130F5CA2131FA25CEB7FF0007FB512F0B6FCA22D447DC32B>I<91B612 F8A3020001E0C8FC6F5A4B5AA293C9FCA35C5DA314035DA314075DA3140F5DA3141F5DA3 143F5DA3147F5DA314FF92CAFCA35B4A16C0A21801010317804A15031900A201075E4A15 06180E181E010F161C4A153C18381878011F16F84A4A5A1703013F150F4D5A4A14FF01FF 02075BB9FCA2603A447CC342>76 D<91B500C0933803FFFE63630200F1FE00DB6FE0EE1B F803EF171F1B3703CFEF67F0A21BCF0201EF018F038F60DB87F0ED030F1B1F0203170603 07040C5BA2F2183F020717300206616F6C15601B7F020E17C0020CDC018090C7FCA24F48 5A021C16060218606F6C5C1A0102381618023004305BA2F16003027016C00260606F6CEB 01801A0702E0ED03004A03065CA24E130F01015E4A60047F5B1A1F01035E91C74A5CA24D 48133F494BC7FC010661EE3F861A7F010E158C010C039892C8FCA205B05C011C15E00118 6001386E5A190101785D01FC92C75BD803FFEF07FEB500F8011E0107B512FE161C160C5F 447BC35E>I81 D<9339FF8001800307EBF003 033F13FC9239FF007E07DA01F8EB0F0FDA07E09038079F004A486DB4FC4AC77E023E804A 5D187E5C495A183C495AA213074A1538A3130F183080A295C7FC806D7E8014FF6D13E015 FC6DEBFFC06D14FC6E13FF6E14C0020F80020314F8EC003F03077F9238007FFE160F1603 707E8283A283A21206A4000E163EA2120C177E001E167CA25F5F003F15014C5A6D4A5A4C 5A486C4AC8FC6D143ED87CF85CD8787E495A3AF01FC00FE0D8E007B51280010149C9FC39 C0003FF039487BC53C>83 D<48BA12C05AA291C7D980001380D807F092C7121F4949150F 0180170748C75B1903120E48020316005E12181238003014074C5C00701806126000E014 0F485DA3C8001F92C7FC5EA3153F5EA3157F5EA315FF93CAFCA35C5DA314035DA314075D A3140F5DA3141F5DA3143F5DA2147FA214FF01037F001FB612FCA25E42447EC339>I<00 3FB500F80103B512E0A326003FF8C8381FF800D91FE0ED07E0013F705A615C96C7FC6001 7F16065CA2180E01FF160C91C9FCA2181C4817185BA21838000317305BA2187000071760 5BA218E0120F495EA21701121F495EA21703123F4993C8FCA25F127F491506A2170E00FF 160C90C9FC171CA21718173817304816705F6C5E6C15014C5A4CC9FC6C150E6D141E001F 5D6D5C6C6CEB01E06C6C495A6C6CEB1F80C6B401FECAFC90387FFFF8011F13E0010190CB FC43467AC342>I<007FB56C91381FFFF8B65DA2000101E0C8000313006C0180ED01FCF0 00F0614E5AA2017F4C5A96C7FC1806A2606E5DA2013F5E1870186060A24D5A6E4AC8FCA2 011F1506170E170C5FA26E5C5FA2010F5D16015F4CC9FCA26E13065EA201075C5EA25E16 E06E5B4B5A13034BCAFC1506A25D151CECFE185D13015D5DA26E5AA292CBFC5C13005C5C A25CA25C45467BC339>I97 DI 100 DII<157E913803FF8091390FC1E0E091391F0073F0027E13334A133F4948131F010315 E04948130F495AA2494814C0133F4A131F137F91C713805B163F5A491500A25E12034914 7EA216FEA2495CA21501A25EA21503150700015D150F0000141F6D133F017CEB77E09038 3E01E790381F078F903807FE0FD901F85B90C7FC151FA25EA2153FA293C7FCA2001C147E 007F14FE485C4A5A140348495AEC0FC000F8495A007C01FEC8FC381FFFF8000313C02C40 7EAB2F>I<141E143F5C5CA3147E143891C7FCAE133EEBFF803801C3C0380781E0380601 F0120E121CEA180312381230A2EA700700605BA2EAE00F00C05BEA001F5CA2133F91C7FC A25B137E13FE5BA212015BEC03800003140013F01207495A1406140E140CEBC01C141814 385C00035BEBE1C0C6B45A013EC7FC19437DC121>105 D<14FE137FA3EB01FC13001301 A25CA21303A25CA21307A25CA2130FA25CA2131FA25C163F013FECFFC0923803C0E09138 000703ED1E0F491338ED701F017E13E0EC01C001FE018013C00203EB07004948C8FC140E 00015B5C495A5C3803FBC001FFC9FC8014F83807F1FE9038F03F809038E00FE06E7E000F 130381EBC001A2001FED01C017801380A2003F15031700010013F05E481506160E007E15 0C161C00FE01005BED787048EC3FE00038EC0F802B467BC433>107 D<01F8D903FCEC7F80D803FED91FFF903803FFE0D8071F903B7C0FC00F81F83E0E0F80E0 07E01C00FC001C9026C3C0030178137C271807C700D9F0E0137E02CE902601F1C0133E00 3801DCDAFB80133F003001D892C7FCD90FF814FF0070495C0060495CA200E04949485CD8 C01F187E4A5C1200040715FE013F6091C75BA2040F14014960017E5D1903041F5D13FE49 4B130762043F160E0001060F130C4992C713C0191F4CED801C00031A1849027E1638F200 3004FE167000071A60494A16E0F201C0030192380F0380000FF18700494AEC03FED80380 D90070EC00F84F2D7DAB55>109 D<01F8EB03FCD803FEEB1FFFD8071F90387C0FC03B0E 0F80E007E03A0C07C3C003001CD9C7007F001801CE1301003801DC80003013D8EB0FF800 705B00605BA200E0491303D8C01F5D5C12001607013F5D91C7FCA2160F495D137E161F5F 13FE49143F94C7FC187000014B136049147E16FE4C13E0000317C049150104F813801703 00071700495D170EEE781C000FED7C3849EC1FF0D80380EC07C0342D7DAB3A>I<01F8EB 0FC0D803FEEB7FF0D8070FEBF038000E903883C07C3A0C07C701FC001C13CE0018EBDC03 003813D8003013F8D90FF013F800709038E000E0006015005C12E0EAC01F5C1200A2133F 91C8FCA35B137EA313FE5BA312015BA312035BA312075BA3120F5BEA0380262D7DAB2C> 114 D<141C147EA314FE5CA313015CA313035CA313075CA2007FB512FCB6FC15F839000F C000A2131F5CA3133F91C7FCA35B137EA313FE5BA312015BA312035BA21570000714605B 15E015C0000F130101C013801403EC070000071306140E5C6C6C5A000113F03800FFC001 3FC7FC1E3F7EBD23>116 D<133ED9FF8014E02603C3C0EB03F0380703E0380601F0000E 1507121CD818035D12380030150FA2D870075D00605B161FEAE00F00C0495CEA001F4A13 3FA2013F92C7FC91C7FC5E5B017E147EA216FE13FE495CA20301EB01801703484802F813 00A25F0303130616F000001407030F130E6D010D130C017C011D131C033913186D9038F0 F838903A1F03C07870903A07FF803FE0903A01FC000F80312D7DAB38>I<013E140ED9FF 80EB3F802603C3C0137F380703E0380601F0120E121CD81803143F0038151F0030150FA2 D87007140700605BA2D8E00F150000C0497FEA001F4A5B1606133F91C7FC160E49140C13 7EA2161C01FE14185B1638163016704848146016E05E150100005D15036D49C7FC150601 7C130E017E5B6D137890380F81E06DB45AD900FEC8FC292D7DAB2F>I<02FCEB07E0903A 03FF801FFC903A0F07C0781E903A1C03E0E01F903A3801F1C07FD9700013804901FB13FF 4848EBFF00495B000316FE90C71438484A130012061401000E5C120CC7FC14035DA31407 5DA3140F5DA3021F143817305D1770023F1460121E003F16E0267F807FEB01C0026F1480 00FF01EF1303D901CFEB070000FE903887C00E267C03835B3A3C0F01E0783A1FFC00FFE0 D803F0EB3F80302D7EAB37>120 D<133ED9FF8014E02603C3C0EB03F0380703E0380601 F0000E1507001C16E0EA180312380030150F007016C0EA60075C161FD8E00F158000C05B EA001F4A133F1700133F91C7FC5E49147E137EA216FE01FE5C5BA215015E485AA215035E A200001407150F6D5C017C131F153F6D13FF90391F03CFC0903807FF8F903801FC0F90C7 121F5EA2153F93C7FCD807C05BD81FE0137E5DA24848485A4A5A01805B39380007C00018 495A001C49C8FC6C137C380781F83803FFE0C66CC9FC2C407DAB30>I<16301678167CA2 163C163E163F82007FB712C0B8FCA27EC9EA7F0016FC4B5A4B5A5E4B5AA26FC7FC2A146E C631>126 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fn cmsy10 12 26 /Fn 26 115 df<007FB912E0BA12F0A26C18E03C04789A4D>0 D<121FEA3F80EA7FC0EA FFE0A5EA7FC0EA3F80EA1F000B0B789E1C>I<007FBA1280BB12C0A26C1980C8D801E0C9 FCB3A9007FBA1280BB12C0A26C1980C8D801E0C9FCB3AC6F5A42427BB14D>7 DI<19E0F003F0180FF03FE0F0FF80 943803FE00EF0FF8EF3FE0EFFF80DC03FEC7FCEE0FF8EE3FE0EEFF80DB03FEC8FCED1FF8 ED7FE0913801FF80DA07FEC9FCEC1FF0EC7FC04948CAFCEB07FCEB1FF0EB7FC04848CBFC EA07FCEA1FF0EA7FC048CCFCA2EA7FC0EA1FF0EA07FCEA01FF38007FC0EB1FF0EB07FCEB 01FF9038007FC0EC1FF0EC07FC913801FF809138007FE0ED1FF8ED07FE923800FF80EE3F E0EE0FF8EE03FE933800FF80EF3FE0EF0FF8EF03FE943800FF80F03FE0F00FF01803F000 E01900B0007FB912E0BA12F0A26C18E03C4E78BE4D>20 D<127012FCB4FCEA7FC0EA1FF0 EA07FCEA01FF38007FC0EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC07FC913801FF8091 38007FE0ED1FF8ED07FE923800FF80EE3FE0EE0FF8EE03FE933800FF80EF3FE0EF0FF8EF 03FE943800FF80F03FE0F00FF0A2F03FE0F0FF80943803FE00EF0FF8EF3FE0EFFF80DC03 FEC7FCEE0FF8EE3FE0EEFF80DB03FEC8FCED1FF8ED7FE0913801FF80DA07FEC9FCEC1FF0 EC7FC04948CAFCEB07FCEB1FF0EB7FC04848CBFCEA07FCEA1FF0EA7FC048CCFC12FC1270 CDFCB0007FB912E0BA12F0A26C18E03C4E78BE4D>I<037FB612E00207B712F0143F91B8 12E0010301C0C9FCD907FCCAFCEB0FE0EB3F8049CBFC13FC485A485A485A5B485A121F90 CCFC123EA2123C127CA2127812F8A25AA87EA21278127CA2123C123EA27E7F120F6C7E7F 6C7E6C7E6C7E137E6D7EEB1FE0EB07FC6DB47E010090B712E0023F16F01407020016E03C 3A78B54D>26 D<00604A7E00F8EC03E000FE15F8D87F80EB01FED83FE06D6C7ED80FF8EC 3FE0D803FCEC0FF0C6B4EC03FCD93FC0EB00FFD90FF0EC3FC0D907FCEC1FF0D901FFEC07 FC6D6C6CEB01FEDA1FE09038007F80DA07F8EC1FE0DA01FEEC07F86E6C6CEB03FEDB3FE0 903800FF80DB0FF0EC3FC0DB03FCEC0FF0DB00FFEC03FCDC3FC0EB00FFDC1FF0EC7FC0DC 07FCEC1FF0DC01FEEC07F89326007F80EB01FEDD1FE09038007F80DD07F8EC1FE0DD03FE EC0FF8942600FF80EB03FEDE3FC0EB00FFDE0FE0EC3F80A2DE3FC0ECFF00DEFF80EB03FE DD03FEC7EA0FF8DD07F8EC1FE0DD1FE0EC7F80DD7F80D901FEC7FCDC01FEC7EA07F8DC07 FCEC1FF0DC1FF0EC7FC0DC3FC04AC8FC04FFC7EA03FCDB03FCEC0FF0DB0FF0EC3FC0DB3F E0ECFF80DBFF80D903FEC9FC4A48C7EA07F8DA07F8EC1FE0DA1FE0EC7F80DA7F80D901FE CAFC4948C7EA07FCD907FCEC1FF0D90FF0EC3FC0D93FC002FFCBFC01FFC7EA03FCD803FC EC0FF0D80FF8EC3FE0D83FE0ECFF80D87F804948CCFC00FEC7EA03F800F815E00060EC01 8059407BB864>29 D<1AF0A3861A78A21A7C1A3CA21A3E1A1E1A1F747EA2747E747E8774 7E747E1B7E87757EF30FE0F303F8007FBC12FEBE1280A26CF3FE00CEEA03F8F30FE0F31F 8051C7FC1B7E63505A505A63505A505AA250C8FC1A1E1A3E1A3CA21A7C1A78A21AF862A3 59347BB264>33 D<1403A34A7EA44A7EA24A7EA24A7E4A7EA24A7E903801F7BE903803E7 9F903907C78F80010F8090391F8787E090397F0783F801FCEB80FCD803F8147FD81FF0EC 3FE0D8FFC0EC0FFC0100140300FC150000601618C71500B3B3B3A56EC8FC2E587EC432> I<49B4EF3FC0010F01E0923803FFF8013F01FC030F13FE4901FF92383FE01F48B66C9139 7E0007C02603F80301E0D901F8EB01E02807E0007FF049486D7E01806D6CD907C0147048 C76C6C494880001EDA07FE49C87E001C6E6C013E150C486E6D48150E71481506486E01E0 160793387FF1F0006092263FF3E08193381FFBC000E004FF1780486F4915017090C9FC82 707F8482717E844D7E6C4B6D1503006004EF1700933803E7FE0070922607C7FF5DDC0F83 7F003004816D140E00384BC6FC0018033E6D6C5C001C4B6D6C143C6C4BD91FFC5C6C4A48 6D6C5C6DD907E06D6C13036C6C49486D9038E00FE0D801F0013FC890B55A27007C03FE6F 91C7FC90263FFFF8031F5B010F01E0030313F8D901FECAEA7FC0592D7BAB64>49 D<92B6FC02071580143F91B7120001030180C8FCD907FCC9FCEB1FE0EB3F80017ECAFC5B 485A485A485A5B485A121F90CBFC123EA2123C127CA2127812F8A25AA2B9FC1880A21800 00F0CBFCA27EA21278127CA2123C123EA27E7F120F6C7E7F6C7E6C7E6C7E137E6D7EEB1F E0EB07FC6DB47E010090B6FC023F1580140702001500313A78B542>I<1706170F171FA2 173EA2177CA217F8A2EE01F0A2EE03E0A2EE07C0A2EE0F80A2EE1F00A2163EA25EA25EA2 4B5AA24B5AA24B5AA24B5AA24BC7FCA2153EA25DA25DA24A5AA24A5AA24A5AA24A5AA24A C8FCA2143EA25CA25CA2495AA2495AA2495AA2495AA249C9FCA2133EA25BA25BA2485AA2 485AA2485AA2485AA248CAFCA2123EA25AA25AA25A1260305C72C600>54 D<4B7E4B7EA21507A25EECFF8F010313EF90260F80FFC7FC90383E003F497F4980484880 4848497E5B0007EC3DF049133C000FEC7CF8A248C7EA787C15F848157E15F0A214014815 7F007E4A7E1403A215C0A200FE01071480A21580140FA21500A25CA2141E143EA2143CA2 147CA21478A214F8A25C1301A2007E491400A21303A2007F495B1307003F157E5CA2130F 001F157C018FC712FCD80F9F5CA201DE130100075DD803FE495AA26C48495A00004A5A01 7C49C7FC017E133E90387F80F89038FBFFE001F8138049C9FC1201A25BA26C5A29557CCC 32>59 D72 D77 D<933801FFC0041F13F893 B512FE03076E7E031F81037F81912701FE003F7FDA03F001077FDA0FC013014AC86C7E02 7E6F7E02F8151F49486F7E494881495A49486F1380011F8249C9FC137E01FE7013C05B48 5A120349177F1207120F5B121F5BA2123F1A80485AA3F1FF0012FFA2611801A2616D1603 61A24E5A6C7E4E5A6D5F6D4C5A6C6C14066D023E49C7FC6C6C6C01FC137E9126F003F85B 6C90B500E05B6C92388001F06C4A48485A6C02F8495A6C6C01C0495AD90FFCC7003FC8FC 90C9127C5FEE03F0EE0FC0EE3F80DB01FEC9FCEDFFF8017FB500E015E00003B600801403 4802FCC8EA0FC04802F0151F4802FE153F486E6C1580C66C02F0EC7F00010702FE147ED9 007FD9FFC0137C020F02FE5B020191B55A6E6C15C0030F5D03014AC7FCDB003F13F00401 1380435375C551>81 D<913807FFC0027F13FC0103B67E010F15E0903A3FFC007FF8D97F C0EB07FCD801FEC8B4FCD803F8ED3F80D807E0ED0FC04848ED07E04848ED03F090C91201 003EEE00F8007E17FC007C177C0078173C00F8173EA248171EB3B3A60060170C373D7BBA 42>92 D102 D<12FEEAFFE0EA07F8EA00FEEB7F806D7E6D7E130F6D7EA26D7EB3AD6D7EA26D7E806E7E 6E7EEC0FE0EC03FC913800FFE0A2913803FC00EC0FE0EC3FC04A5A4AC7FC5C495AA2495A B3AD495AA2495A131F495A495A01FEC8FCEA07F8EAFFE048C9FC236479CA32>I<140C14 1E143EA2143C147CA214F8A214F01301A2EB03E0A214C01307A2EB0F80A214005BA2133E A2133C137CA2137813F8A2485AA25B1203A2485AA25B120FA248C7FCA2121E123EA25AA2 127812F8A41278127CA27EA2121E121FA26C7EA212077FA26C7EA212017FA26C7EA21378 137CA2133C133EA27FA27F1480A2EB07C0A2130314E0A2EB01F0A2130014F8A2147CA214 3C143EA2141E140C176476CA27>I<126012F07EA21278127CA27EA2121E121FA26C7EA2 12077FA26C7EA212017FA26C7EA21378137CA2133C133EA27FA27F1480A2EB07C0A21303 14E0A2EB01F0A2130014F8A2147CA2143C143EA4143C147CA214F8A214F01301A2EB03E0 A214C01307A2EB0F80A214005BA2133EA2133C137CA2137813F8A2485AA25B1203A2485A A25B120FA248C7FCA2121E123EA25AA2127812F8A25A126017647BCA27>I<126012F0B3 B3B3B3B3A81260046474CA1C>I<126012F07EA21278127CA2123C123EA2121E121FA26C 7EA212077FA212037FA212017FA26C7EA21378137CA2133C133EA2131E131FA26D7EA213 0780A2130380A2130180A26D7EA21478147CA2143C143EA280A28081A2140781A2140381 A26E7EA2140081A21578157CA2153C153EA281A2811680A2150716C0A2150316E0A2ED01 F0A2150016F8A21678167CA2163C163EA2161E160C27647BCA32>110 D<1B0C1B1E1B3EA21B7CA21BF8A2F201F0A2F203E0A2F207C0A2F20F80A2F21F00A21A3E A262A262A24F5AA2621903A24F5AA24F5AA24FC7FCA2193EA261A261A24E5AA24E5AA24E 5AA24E5AA2010C4CC8FC133C017C163EEA01FE00035F487E001E5F00387FD8707F4B5A00 E07FD8003F4B5A80011F4B5AA26E4A5A130F6E4AC9FC13076E143E13036E5C13016E5C7F 6F5B027F1301A26F485A143F6F485A141F6F485A140F6F48CAFC1407EDFC3E14035E15FE 02015B15FF6E5BA26F5AA26F5AA26F5AA26FCBFC150E4F647A8353>112 D114 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fo cmr12 12 87 /Fo 87 128 df0 D<1618163CA2167EA216FFA24B7FA24B6C7EA29238063FE0A24B6C7EA24B6C7EA2923838 07FC153092387003FE15609238E001FF15C002016D7F5D02036E7E92C7FC4A6E7E140602 0E6E7E140C021C6E7E141802386E7E143002706E7E146002E06E7E5C01016F7F5C010370 7E91C9FC183F010683181F4983180F49831807498318034983A249707EA24848701380A2 48CBEA7FC0A20006F03FE0A248F01FF0A2001FBA12F8A24819FCA24819FEA2BCFC48477C C651>I10 D<9239FFC001FC020F9038F80FFF913B3F803E3F03C0913BFC00077E07E0D903F890390F FC0FF0494890383FF81F4948EB7FF0495A494814E049C7FCF00FE04991393FC003804902 1F90C7FCAFB912F0A3C648C7D81FC0C7FCB3B2486CEC3FF0007FD9FC0FB512E0A33C467E C539>I<4AB4FC020F13E091387F80F8903901FC001C49487FD907E0130F4948137F011F ECFF80495A49C7FCA25B49EC7F00163E93C7FCACEE3F80B8FCA3C648C7FC167F163FB3B0 486CEC7FC0007FD9FC1FB5FCA330467EC536>I<913801FFC0020FEBFB8091387F803F90 3801FC00494813FFEB07E0EB1FC0A2495A49C7FC167F49143F5BAFB8FCA3C648C7123FB3 B2486CEC7FC0007FD9FC1FB5FCA330467EC536>II< EA01FC12FFA3120712031201B3B0487EB512F8A3152C7DAB1C>16 D<131F1480133F137FA2EBFF00485A485A5B485A485A138048C7FC123E123C5A12E01240 11126CC431>19 D<121EEA7F80EAFFC0A9EA7F80ACEA3F00AB121EAC120CA5C7FCAA121E EA7F80A2EAFFC0A4EA7F80A2EA1E000A4778C61B>33 D<001EEB03C0397F800FF000FF13 1F01C013F8A201E013FCA3007F130F391E6003CC0000EB000CA401E0131C491318A30001 14384913300003147090C712604814E0000614C0000E130148EB038048EB070048130E00 60130C1E1D7DC431>I<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413E013C0A3 12011380120313005A1206120E5A5A5A12600B1D78C41B>39 D<140C141C1438147014E0 EB01C01303EB0780EB0F00A2131E5BA25B13F85B12015B1203A2485AA3485AA348C7FCA3 5AA2123EA2127EA4127CA312FCB3A2127CA3127EA4123EA2123FA27EA36C7EA36C7EA36C 7EA212017F12007F13787FA27F7FA2EB0780EB03C01301EB00E014701438141C140C1664 76CA26>I<12C07E12707E7E7E120F6C7E6C7EA26C7E6C7EA21378137C133C133E131E13 1FA2EB0F80A3EB07C0A3EB03E0A314F0A21301A214F8A41300A314FCB3A214F8A31301A4 14F0A21303A214E0A3EB07C0A3EB0F80A3EB1F00A2131E133E133C137C13785BA2485A48 5AA2485A48C7FC120E5A5A5A5A5A16647BCA26>I<16C04B7EB3AB007FBAFCBB1280A26C 1900C8D801E0C9FCB3AB6F5A41407BB84C>43 D<121EEA7F8012FF13C0A213E0A3127FEA 1E601200A413E013C0A312011380120313005A1206120E5A5A5A12600B1D78891B>II<121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A0A78891B>I<161816 3C167CA2167816F8A216F01501A216E01503A216C01507A21680150FA2ED1F00A2151E15 3EA2153C157CA2157815F8A25D1401A24A5AA25D1407A25D140FA292C7FC5CA2141E143E A2143C147CA25CA25C1301A25C1303A25C1307A25C130FA291C8FC5BA2133EA2133C137C A2137813F8A25B1201A25B1203A2485AA25B120FA290C9FC5AA2121E123EA2123C127CA2 127812F8A25A126026647BCA31>I<14FF010713E090381F81F890383E007C01FC133F48 48EB1F8049130F4848EB07C04848EB03E0A2000F15F0491301001F15F8A2003F15FCA390 C8FC4815FEA54815FFB3A46C15FEA56D1301003F15FCA3001F15F8A26C6CEB03F0A36C6C EB07E0000315C06D130F6C6CEB1F806C6CEB3F00013E137C90381F81F8903807FFE00100 90C7FC28447CC131>I<143014F013011303131F13FFB5FC13E713071200B3B3B0497E49 7E007FB6FCA3204278C131>II<49B4 FC010F13E0013F13FC9038FE01FE3A01F0007F80D803C0EB3FC048C7EA1FE0120EED0FF0 EA0FE0486C14F8A215077F5BA26C48130FEA03C0C813F0A3ED1FE0A2ED3FC01680ED7F00 15FE4A5AEC03F0EC1FC0D90FFFC7FC15F090380001FCEC007FED3F80ED1FC0ED0FE016F0 ED07F816FC150316FEA2150116FFA3121EEA7F80487EA416FE491303A2007EC713FC0070 1407003015F80038140F6C15F06CEC1FE06C6CEB3FC0D803E0EB7F803A01FE01FE003900 7FFFF8010F13E0010190C7FC28447CC131>II<000615C0D807C0130701FCEB 7F8090B612005D5D5D15E0158026063FFCC7FC90C9FCAE14FF010713C090381F01F09038 3800FC01F0137ED807C07F49EB1F8016C090C7120F000615E0C8EA07F0A316F81503A216 FCA5123E127F487EA416F890C712075A006015F0A20070140F003015E00038EC1FC07E00 1EEC3F806CEC7F006C6C13FE6C6C485A3901F807F039007FFFE0011F90C7FCEB07F82644 7BC131>II<121CA2EA1F8090B712C0A3481680A217 005E0038C8120C0030151C00705D0060153016705E5E4814014B5A4BC7FCC81206150E5D 151815385D156015E04A5AA24A5A140792C8FC5CA25C141E143EA2147E147CA214FCA213 01A3495AA41307A6130FAA6D5AEB01C02A457BC231>I<14FF010713E0011F13F890387F 00FE01FC133FD801F0EB1F804848EB0FC049EB07E00007EC03F048481301A290C713F848 1400A47FA26D130116F07F6C6CEB03E013FC6C6CEB07C09039FF800F806C9038C01F006C EBF03EECF87839007FFEF090383FFFC07F01077F6D13F8497F90381E7FFFD97C1F138049 6C13C02601E00313E048486C13F000079038007FF84848EB3FFC48C7120F003EEC07FE15 0148140016FF167F48153FA2161FA56C151E007C153EA2007E153C003E157C6C15F86DEB 01F06C6CEB03E06C6CEB07C0D803F8EB1F80C6B4EBFF0090383FFFFC010F13F001011380 28447CC131>I<121EEA7F80A2EAFFC0A4EA7F80A2EA1E00C7FCB3A5121EEA7F80A2EAFF C0A4EA7F80A2EA1E000A2B78AA1B>58 D<121EEA7F80A2EAFFC0A4EA7F80A2EA1E00C7FC B3A5121E127FEAFF80A213C0A4127F121E1200A512011380A3120313005A1206120E120C 121C5A5A12600A3E78AA1B>I<007FBAFCBB1280A26C1900CEFCB0007FBAFCBB1280A26C 190041187BA44C>61 D<16C04B7EA34B7EA34B7EA34B7EA3ED19FEA3ED30FFA203707FED 607FA203E07FEDC03FA2020180ED801FA2DA03007F160FA20206801607A24A6D7EA34A6D 7EA34A6D7EA20270810260147FA202E08191B7FCA249820280C7121FA249C87F170FA201 06821707A2496F7EA3496F7EA3496F7EA201788313F8486C83D80FFF03037FB500E0027F EBFFC0A342477DC649>65 DIIIII< DB0FFE146092B500C013E0020314F0913A0FFC01FC0191393FC0003E02FFC7EA0F83D903 FCEC03C74948EC01E74948EC00FF4948157F4948153F4948151F49C9120F485A49160712 0348481603A248481601A248481600A2123FA2491760127FA396C7FC485AAD4CB612C06C 7EA293C7387FF000725A003F171F7FA2121F7F120FA26C7EA26C7E6C7EA26C7E6D7E6D6C 153F6D7E6D6C157F6D6C15E7D903FEEC01C7D900FFEC0383DA3FE0EB0F01DA0FFCEBFE00 0203B500F81360020002E090C7FCDB0FFEC9FC42487BC54D>III75 DIIIII82 D<49B41303010FEBE007013F13F89039FE00FE0FD801F8 131FD807E0EB079F49EB03DF48486DB4FC48C8FC4881003E81127E82127C00FC81A282A3 7E82A27EA26C6C91C7FC7F7FEA3FF813FE381FFFE06C13FE6CEBFFE06C14FC6C14FF6C15 C0013F14F0010F80010180D9001F7F14019138001FFF03031380816F13C0167F163F161F 17E000C0150FA31607A37EA36C16C0160F7E17806C151F6C16006C5D6D147ED8FBC05CD8 F9F0495AD8F07C495A90393FC00FE0D8E00FB51280010149C7FC39C0003FF02B487BC536 >I<003FB912F8A3903BF0001FF8001F01806D481303003EC7150048187C0078183CA200 70181CA30060180CA5481806A5C81600B3B3A54B7EED7FFE49B77EA33F447DC346>IIII89 D<001FB81280A39126800001130001FCC7FC01F04A5A01C04A5A5B90C8485A121E4C5A48 4B5AA200384B5A4C5AA24B90C7FC00304A5AA24B5AA24B5AC8485AA24B5A4B5AA24B5A5C 93C8FC4A5AA24A5A4A5AA24A5A4A5AA24A5A14FF5D4990C9FCEF0180495A495AA2495A49 4814031800495AA2495A495A5F4890C8FC485A5F485A48485D5F48485D17FE4848140348 48140F16FFB8FCA331447BC33C>II<01C01318000114384848137048C712E0000EEB01C0000C1480001C130300181400 00385B003013060070130E0060130CA300E0131C481318A400CFEB19E039FFC01FF801E0 13FCA3007F130FA2003F130701C013F8390F0001E01E1D71C431>II<130C131E133F497EEBF3C03801E1E03803C0F038 07807848487E001E7F487F0070EB038048EB01C00040EB00801A0E75C331>I97 DII<167FED3FFFA315018182B3EC7F80903803FFF090380FC07C 90383F000E017E1307496D5AD803F87F48487F5B000F81485AA2485AA2127FA290C8FC5A AB7E7FA2123FA26C7EA2000F5D7F6C6C5B00035C6C6C9038077F806C6C010E13C0013F01 1C13FE90380FC0F8903803FFE09026007F0013002F467DC436>III III<143C14FFA2491380A46D1300A2143C91C7FCADEC7F80EB 3FFFA31300147F143FB3B3AA123E127F39FF807F00A2147EA25C6C485A383C01F06C485A 3807FF80D801FEC7FC195785C21E>IIII<3901FC01FE00FF903807FFC091381E07F091383801F800070170 7F0003EBE0002601FDC07F5C01FF147F91C7FCA25BA35BB3A8486CECFF80B5D8F83F13FE A32F2C7DAB36>II<3901FC03FC00FF9038 0FFF8091383C07E091387001F83A07FDE000FE00030180137FD801FFEC3F8091C7EA1FC0 4915E049140F17F0160717F8160317FCA3EE01FEABEE03FCA3EE07F8A217F0160F6D15E0 EE1FC06D143F17806EEB7E00D9FDC05B9039FCF003F891383C0FE091381FFF80DA03FCC7 FC91C9FCAE487EB512F8A32F3F7DAB36>I<91387F8003903903FFE00790380FE0789039 3F801C0F90387E000E496D5AD803F8EB039F0007EC01BF4914FF48487F121F5B003F81A2 485AA348C8FCAB6C7EA3123F7F121F6D5C120F6D5B12076C6C5B6C6C497E6C6C130E013F 131C90380FC0F8903803FFE09038007F0091C7FCAEEEFF80033F13FEA32F3F7DAB33>I< 3903F803F000FFEB1FFCEC3C3EEC707F0007EBE0FF3803F9C000015B13FBEC007E153C01 FF13005BA45BB3A748B4FCB512FEA3202C7DAB26>I<90383FE0183901FFFC383907E01F 78390F0003F8001E1301481300007C1478127800F81438A21518A27EA27E6C6C13006C7E 13FC383FFFE06C13FC6C13FF6C14C06C14E0C614F0011F13F81300EC0FFC140300C0EB01 FE1400157E7E153EA27EA36C143C6C147C15786C14F86CEB01F039F38003E039F1F00F80 39E07FFE0038C00FF01F2E7DAC26>I<1306A5130EA4131EA3133E137EA213FE12011207 001FB512F0B6FCA2C648C7FCB3A4150CAA017E131C017F1318A26D133890381F8030ECC0 70903807E0E0903801FFC09038007F001E3E7EBC26>IIIIII<003FB612E0A29038C0003F90C713C0003CEC7F800038EC FF00A20030495A0070495AA24A5A0060495AA24A5A4A5AA2C7485A4AC7FC5B5C495A1307 5C495A131F4A1360495A495AA249C712C0485AA2485A485A1501485A48481303A24848EB 07804848131F00FF14FF90B6FCA2232B7DAA2B>I124 D<01F81302D803FE13073907FF800E48EBE01C391F1FF8F8393807FFF0D8700113E039E0 007FC00040EB1F00200978C131>126 D<001EEB0780007FEB0FE039FF801FF0EBC03FA4 EB801F397F000FE0001EEB07801C0A76C231>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fp cmcsc10 12 34 /Fp 34 128 df45 D<121FEA3F80EA7FC0EAFFE0A5EA7FC0EA3F 80EA1F000B0B768A20>I<1438147814F81303130F13FFB5FC13F713071200B3B3B0497E 497EB712C0A3224276C137>49 D66 D70 D73 D76 DI80 D82 D<49B46C13C0010FEBF001 013FEBFC039038FF007FD801F8EB0F874848EB03E7D807C0EB01FF48487F001F157F90C8 123F003E151FA2007E150F127C160712FC1603A37E16017EA27F6C6C91C7FC7F7FEA3FFC EBFFC06C13FC6CEBFFC015FC6CECFF806C15E0C615F86D80011F80010380D9003F148002 0314C0EC003F030313E01500EE7FF0161FA2EE0FF8160712E01603A21601A37EA217F07E 16037E17E06C15076C16C06DEC0F806D141F6DEC3F00D8F8F8147E017F5C3AF01FE007F0 0107B55AD8E00191C7FC39C0001FFC2D4879C53D>I<157015F8A34A7EA24A7EA34A7E81 A291380E3F80A2021E7FEC1C1FA24A6C7EA34A6C7EA202F07FECE003A249486C7EA34948 6C7EA201078091C77EA249B67EA24981011CC7121FA2013C810138140FA2496E7EA201F0 81491403120183486C140100074B7ED81FF84A7EB5027F13F8A335357CB43D>97 DI<4AB4EB0180021FEBF00391B5EA FC0701039038007E0FD907F8EB0F9FD91FE0EB03DF4948EB01FF01FFC8FC4848157F4848 153FA24848151F4848150F121F491507123F5BA2007F1603A3484892C7FCAB6C7EEF0380 A2123FA27F001F16076D1600000F5E6C6C150E6C6C151E171C6C6C153C6C6C5DD93FC05C 6D6CEB03E0D907F8495A902703FF807FC7FC0100EBFFFC021F13F00201138031357BB33B >IIIIIII108 DIIII114 D<90390FF0018090387FFE0348B512873907F00FEF390FC001FF48C7FC003E143F151F5A 150F5A1507A36C1403A27E6C91C7FC6C7E7FEA3FF8EBFF806C13FC6CEBFFC06C14F06C80 C614FE011F7F01031480D9001F13C014019138003FE0151F150FED07F0150312E01501A3 7EA216E06C1403A26CEC07C06CEC0F806C6CEB1F0001E0133ED8FBFE13FC00F0B55AD8E0 1F13E0D8C00390C7FC24357BB32E>I<007FB812C0A3903A8007FC003F277E0003F8130F 007C16070078160300701601A200F017E0A2481600A6C71600B3AA4A7E4A7E010FB512FE A333327CB13B>IIII<267FFFF890383FFFF0A3000101E0010F13006C49EB07F8017F 5D013F15C06D6C5C6D6C49C7FC160E6D6C131E6D6C5B6D6C133816786D6C5B91387F81E0 023F5B15C391381FE7806EB4C8FC5D14076E5A6E7EA26E7E4A7FA24A7F9138079FE09138 0F0FF0140E91381E07F84A6C7E4A6C7E14709138F000FF49486D7E4948133F4A8001076E 7E49C76C7E131E013E6E7E017E6E7E01FE81000182D80FFF4A1380B500C0011F13FEA337 337DB23D>II<001EEB03C0007FEB07F039FF800FF8A2EBC01FA2 EB800FA2397F0007F0001EEB03C01D0A74C337>127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fq cmsy10 10 1 /Fq 1 22 df<126012F812FEEA7F80EA3FE0EA0FF8EA03FEC66C7EEB3FE0EB0FF8EB03FE 903800FF80EC3FE0EC0FF8EC03FE913800FF80ED3FE0ED0FF8ED03FE923800FF80EE3FE0 EE0FF8EE03FE933800FF80EF3FC0171FEF7F80933801FF00EE07FCEE1FF0EE7FC04B48C7 FCED07FCED1FF0ED7FC04A48C8FCEC07FCEC1FF0EC7FC04948C9FCEB07FCEB1FF0EB7FC0 4848CAFCEA07FCEA1FF0EA7FC048CBFC12FC1270CCFCAE007FB81280B912C0A26C178032 4479B441>21 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fr cmmi7 7 1 /Fr 1 110 df<3B07801FC007E03B0FE07FF01FF83B18F0E0F8783C3B30F1807CE03E90 3AFB007D801ED860FEEB3F005B49133E00C14A133E5B1201A24848495BA35F4848485A18 30EE01F0A23C0F8003E003E060A218C0933801E180271F0007C013E3933800FF00000E6D 48137C341B7D993B>109 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fs msbm10 10 1 /Fs 1 83 df<007FB612E0B712FE6C6F7E2703C01E0313E0000190393C00F3F00238EB70 F8EE783CEE381E83EE3C07161C18801703A617071800EE3C0FEE380E173EEE78FCEEF7F8 92380FFFE0023FB5128004FCC7FC16B8913838F03CED701CED781EED380EED3C0FED1C07 031E7FED0E03030F7FED0701EE81E0ED0380707E030113701778EEE0380300133C707EEE 700EEE780F9338380780EE3C03041C13C093381E01E00003013C90380E00F0007FB539F0 0FFFFEB67F6C8137397DB836>82 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ft cmr7 7 1 /Ft 1 51 df<13FF000313E0380E03F0381800F848137C48137E00787F12FC6CEB1F80A4 127CC7FC15005C143E147E147C5C495A495A5C495A010EC7FC5B5B903870018013E0EA01 80390300030012065A001FB5FC5A485BB5FCA219267DA521>50 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fu cmmi10 10 2 /Fu 2 110 df<0103B6FC5B5E90260007FCC8FC5D5D140FA25DA2141FA25DA2143FA25D A2147FA292C9FCA25CA25CA21301A25CA21303A25CA2130718404A15C0A2010F15011880 4A1403A2011F16005F4A1406170E013F151E171C4A143C177C017F5D160391C7120F49EC 7FF0B8FCA25F32397DB839>76 D109 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fv cmcsc10 10 46 /Fv 46 128 df<017F0160903807F0062701FFE0E090381FFE0E260780F99138780F9E26 0E001F9138E001FE480107903901C0007E4801034948133E007815070070010191C7121E 00F0010049140EA3036015066C82A26C02006D13006C82007F6F7E01C015FCD83FFE9138 03FFE0D9FFE015FE6C01F86DEBFF80000701FE6D6C13E06C6D6E13F0C66E010F13F8010F 6D010013FC0100160FDA0FE0EC00FE0207167E6E6C153F0201161FA200C0150C0200160F A36C150E03E0150E6C0101010F141EA26CD903C06D133C6CD907806D1378B4D90F0001F0 13F026F3E01E91383E01E026E0FFF890390E0FFF8026C01FE090390C01FE00402D7BAB4A >25 D<121C127FEAFF80A213C0A3127F121C1200A412011380A2120313005A1206120E5A 5A5A12600A1977881B>44 DI<121C127FEAFF80A5EA7F00121C 090977881B>I48 DIII<151C153CA2157C15FCA214011403A21407140F141D 141914311471146114C11301EB038114011307130E130C131813381330136013E0EA01C0 1380EA03005A12065A121C5A123012705AB712FEA3C73801FC00AB4A7E49B512FCA32739 7DB82E>I<00061406D80780131E9038F801FC90B5FC5D5D15C05D4AC7FC38067FF090C9 FCABEB03FC90381FFF8090387C07E09038E001F03907C000F8497F90C7127E0006147FC8 EA3F80A216C0151FA216E0A4123E127F487EA490C713C048143F126016800070147F6C15 0015FE6C5C000F495A39078007F03903F01FE06CB512806C6C48C7FCEB0FF0233A7BB72E >II56 D<150EA3151FA24B7EA34B7EA3EDDFE0A202017F158FA29138030FF81507A202067F1503 020E7FEC0C01A2021C7FEC1800A24A80167FA24A6D7EA202E0804A131FA2494880160FA2 49B67EA249810106C71203A249811601A2498182A2496F7EA20170820160153F13E06D82 1203D80FFCED7FF8B56C010FB512E0A33B3C7CBB44>65 DI71 DII77 D80 D82 D<003FB812FCA3D9C001EB800390C7 90C7FC007C173E0078171E0070170EA300601706A400E01707481703A4C81500B3B00203 13C0010FB612F0A338397CB841>84 DI< B500FC90B500FE90380FFFF8A300070180010301C0010013C06C90C76C90C8EA3F006C48 191E6D6E161C6C1A18717E6E1838017F1930846E1870013F70146017DF6E18E0011F705C EE018F6E1701010F705CEE03076E170301077091C7FCEE06036E5F0103701306EE0C016E 170E010170130C4C7E6D6C5F043814800430137FDA7F805E047014C00460133FDA3FC05E 04E014E04C131FDA1FE05E03E115F04C130FDA0FF1EDF18003F315F993C71207DA07FB03 FBC8FC03FF15FF4B8002035EA24B140102015E4B1400A202005E4B1578A2037015700360 1530553B7DB85C>87 D<1407A24A7EA34A7EA3EC37E0A2EC77F01463A2ECC1F8A201017F 1480A2903803007EA301067FA2010E80010C131FA2496D7EA2013FB57EA2903830000749 6D7EA3496D7EA200018149130012036D801207D81FE0903801FF80D8FFF8010F13F8A22D 2C7DAB33>97 DI<91383FC006903901FFF80E90390FE03E1E90381F0007017EEB03BE01F8EB 01FE484813004848147E0007153E485A001F151E5B003F150E90C8FC5A1606A212FE1600 AA007F1506A37E6D140E001F150C7F000F151C6C6C1418000315386C6C14706C6C14E001 7EEB01C0011FEB078090390FE03E00903801FFF89038003FC0272D7BAB31>IIII<91383FE003903901FFF807903907E01E0F90391F00078F017EEB01DF496D B4FC484880484880484880485A001F815B003F8190C8FC5A82A212FE93C7FCA892383FFF F8A2007F02001380EE3F00A27E7F121F7F120F6C7E6C7E6C6C5C6C7E017E5C011FEB01CF 903907E00F87903901FFFE039026003FF0C7FC2D2D7BAB35>I II 107 DIIIII114 D<017F13603901FFE0E0380780F9380E001F48130748130312780070130100F01300A315 607EA26C14007E127F13C0EA3FFEEBFFE06C13F8000713FE6C7FC61480010F13C01300EC 0FE01407EC03F01401A212C01400A37E15E06C1301A26CEB03C06CEB0780B4EB0F0038F3 E01E38E0FFF838C01FE01C2D7BAB26>I<007FB712C0A23A7E003FC00F007890381F8003 007015011600126000E016E0A2481660A5C71500B3A8EC7FE0011FB57EA22B2B7DAA31> II< B56CEB3FFEA2D80FFCC7EA0FF06C48EC07E00003ED03C01780000116006D5C00001506A2 017E5CA2017F141C6D141880011F5CA26D6C5BA28001075CA26D6C485AA2ECF803010191 C7FCA2903800FC06A2ECFE0EEC7E0C147F6E5AA2EC1FB0A215F06E5AA26E5AA36E5AA22F 2C7EAA33>II 121 D<003C131E007F137F481480EB80FFA3EB007F6C1400003C131E190976B72E>127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fw cmr10 10 74 /Fw 74 128 df11 DI14 D<133C137EA213FE1201EA03FC13F0EA07E0EA 0FC0EA1F80EA1E005A5A5A12C00F0F6FB92A>19 D<001C131C007F137F39FF80FF80A26D 13C0A3007F137F001C131C00001300A40001130101801380A20003130301001300485B00 061306000E130E485B485B485B006013601A197DB92A>34 D<146014E0EB01C0EB0380EB 0700130E131E5B5BA25B485AA2485AA212075B120F90C7FCA25A121EA2123EA35AA65AB2 127CA67EA3121EA2121F7EA27F12077F1203A26C7EA26C7E1378A27F7F130E7FEB0380EB 01C0EB00E01460135278BD20>40 D<12C07E12707E7E7E120F6C7E6C7EA26C7E6C7EA213 78A2137C133C133E131EA2131F7FA21480A3EB07C0A6EB03E0B2EB07C0A6EB0F80A31400 A25B131EA2133E133C137C1378A25BA2485A485AA2485A48C7FC120E5A5A5A5A5A13527C BD20>I<121C127FEAFF80A213C0A3127F121C1200A412011380A2120313005A1206120E 5A5A5A12600A19798817>44 DI<121C127FEAFF80A5EA7F0012 1C0909798817>I48 DIII<1538A2157815F8A2140114031407A2140F141F141B14331473146314C313011483EB 030313071306130C131C131813301370136013C01201EA038013005A120E120C5A123812 305A12E0B712F8A3C73803F800AB4A7E0103B512F8A325397EB82A>I<0006140CD80780 133C9038F003F890B5FC5D5D158092C7FC14FC38067FE090C9FCABEB07F8EB3FFE903878 0F803907E007E090388003F0496C7E12066E7EC87EA28181A21680A4123E127F487EA490 C71300485C12E000605C12700030495A00385C6C1303001E495A6C6C485A3907E03F8000 01B5C7FC38007FFCEB1FE0213A7CB72A>II<123012 38123E003FB612E0A316C05A168016000070C712060060140E5D151800E01438485C5D5D C712014A5A92C7FC5C140E140C141C5CA25CA214F0495AA21303A25C1307A2130FA3495A A3133FA5137FA96DC8FC131E233B7BB82A>III<121C127FEAFF80A5EA7F00121CC7FCB2121C127FEAFF80A5EA7F00121C09 2479A317>I<121C127FEAFF80A5EA7F00121CC7FCB2121C127F5A1380A4127F121D1201 A412031300A25A1206A2120E5A121812385A1260093479A317>I<1538A3157CA315FEA3 4A7EA34A6C7EA202077FEC063FA2020E7FEC0C1FA2021C7FEC180FA202387FEC3007A202 707FEC6003A202C07F1501A2D901807F81A249C77F167FA20106810107B6FCA24981010C C7121FA2496E7EA3496E7EA3496E7EA213E0707E1201486C81D80FFC02071380B56C90B5 12FEA3373C7DBB3E>65 DI<913A01FF800180020FEBE003027F13F8 903A01FF807E07903A03FC000F0FD90FF0EB039F4948EB01DFD93F80EB00FF49C8127F01 FE153F12014848151F4848150FA248481507A2485A1703123F5B007F1601A35B00FF93C7 FCAD127F6DED0180A3123F7F001F160318006C7E5F6C7E17066C6C150E6C6C5D00001618 017F15386D6C5CD91FE05C6D6CEB03C0D903FCEB0F80902701FF803FC7FC9039007FFFFC 020F13F002011380313D7BBA3C>IIIIIII<013FB512E0A39039001FFC00EC07F8B3B3A3123FEA7F80EA FFC0A44A5A1380D87F005B0070131F6C5C6C495A6C49C7FC380781FC3801FFF038007F80 233B7DB82B>IIIIIII82 DI<003FB812E0A3D9C003EB001F273E0001FE130348EE01F0007816 0000701770A300601730A400E01738481718A4C71600B3B0913807FF80011FB612E0A335 397DB83C>IIII89 D<003FB7FCA39039FC0001FE01C0130349495A003EC7FC003C4A5A5E0038141F00784A5A 12704B5A5E006014FF4A90C7FCA24A5A5DC712074A5AA24A5A5D143F4A5AA24A5A92C8FC 5B495AA2495A5C130F4948EB0180A2495A5C137F495A16034890C7FC5B1203485AEE0700 485A495C001F5D48485C5E4848495A49130FB8FCA329397BB833>II 93 D97 DIIII<147E903803FF8090380FC1E0EB1F 8790383F0FF0137EA213FCA23901F803C091C7FCADB512FCA3D801F8C7FCB3AB487E387F FFF8A31C3B7FBA19>IIII107 DI<2703F00FF0EB1FE000 FFD93FFCEB7FF8913AF03F01E07E903BF1C01F83803F3D0FF3800FC7001F802603F70013 CE01FE14DC49D907F8EB0FC0A2495CA3495CB3A3486C496CEB1FE0B500C1B50083B5FCA3 40257EA445>I<3903F00FF000FFEB3FFCECF03F9039F1C01F803A0FF3800FC03803F700 13FE496D7EA25BA35BB3A3486C497EB500C1B51280A329257EA42E>I I<3903F01FE000FFEB7FF89038F1E07E9039F3801F803A0FF7000FC0D803FEEB07E049EB 03F04914F849130116FC150016FEA3167FAA16FEA3ED01FCA26DEB03F816F06D13076DEB 0FE001F614C09039F7803F009038F1E07E9038F0FFF8EC1FC091C8FCAB487EB512C0A328 357EA42E>II<3807E01F00FFEB7FC09038E1E3E09038E387F0380FE707EA03E613EE 9038EC03E09038FC0080491300A45BB3A2487EB512F0A31C257EA421>II<1318A51338A31378A313 F8120112031207001FB5FCB6FCA2D801F8C7FCB215C0A93800FC011580EB7C03017E1300 6D5AEB0FFEEB01F81A347FB220>IIIIII123 D<001C131C007F137F39FF80FF80A5397F007F00001C131C190978B72A>127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fx cmbx12 12 34 /Fx 34 118 df46 D48 D58 D65 D67 DIIIIII76 D78 D<923807FFC092B512FE0207ECFFC0021F15 F091267FFE0013FC902601FFF0EB1FFF01070180010313C04990C76C7FD91FFC6E6C7E49 486F7E49486F7E01FF8348496F7E48496F1380A248496F13C0A24890C96C13E0A24819F0 4982003F19F8A3007F19FC49177FA400FF19FEAD007F19FC6D17FFA3003F19F8A26D5E6C 19F0A26E5D6C19E0A26C6D4B13C06C19806E5D6C6D4B13006C6D4B5A6D6C4B5A6D6C4B5A 6D6C4A5B6D01C001075B6D01F0011F5B010101FE90B5C7FC6D90B65A023F15F8020715C0 02004AC8FC030713C047467AC454>II82 DI<003FBA12E0A59026FE000FEB 8003D87FE09338003FF049171F90C71607A2007E1803007C1801A300781800A400F819F8 481978A5C81700B3B3A20107B8FCA545437CC24E>I86 DI<903801FFE0011F13FE017F6D7E48 B612E03A03FE007FF84848EB1FFC6D6D7E486C6D7EA26F7FA36F7F6C5A6C5AEA00F090C7 FCA40203B5FC91B6FC1307013F13F19038FFFC01000313E0000F1380381FFE00485A5B12 7F5B12FF5BA35DA26D5B6C6C5B4B13F0D83FFE013EEBFFC03A1FFF80FC7F0007EBFFF86C ECE01FC66CEB8007D90FFCC9FC322F7DAD36>97 DI 101 D104 D<137C48B4FC4813804813C0A24813E0A56C13C0A26C13806C1300EA007C90C7FC AAEB7FC0EA7FFFA512037EB3AFB6FCA518467CC520>I108 D<90277F8007FEEC0FFCB590263FFFC090387FFF8092B5D8 F001B512E002816E4880913D87F01FFC0FE03FF8913D8FC00FFE1F801FFC0003D99F0090 26FF3E007F6C019E6D013C130F02BC5D02F86D496D7EA24A5D4A5DA34A5DB3A7B60081B6 0003B512FEA5572D7CAC5E>I<90397F8007FEB590383FFF8092B512E0028114F8913987 F03FFC91388F801F000390399F000FFE6C139E14BC02F86D7E5CA25CA35CB3A7B60083B5 12FEA5372D7CAC3E>II<90 397FC00FF8B590B57E02C314E002CF14F89139DFC03FFC9139FF001FFE000301FCEB07FF 6C496D13804A15C04A6D13E05C7013F0A2EF7FF8A4EF3FFCACEF7FF8A318F017FFA24C13 E06E15C06E5B6E4913806E4913006E495A9139DFC07FFC02CFB512F002C314C002C091C7 FCED1FF092C9FCADB67EA536407DAC3E>I<90387F807FB53881FFE0028313F0028F13F8 ED8FFC91389F1FFE000313BE6C13BC14F8A214F0ED0FFC9138E007F8ED01E092C7FCA35C B3A5B612E0A5272D7DAC2E>114 D<90391FFC038090B51287000314FF120F381FF00338 3FC00049133F48C7121F127E00FE140FA215077EA27F01E090C7FC13FE387FFFF014FF6C 14C015F06C14FC6C800003806C15806C7E010F14C0EB003F020313E0140000F0143FA26C 141F150FA27EA26C15C06C141FA26DEB3F8001E0EB7F009038F803FE90B55A00FC5CD8F0 3F13E026E007FEC7FC232F7CAD2C>III E %EndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%BeginPaperSize: a4 a4 %%EndPaperSize %%EndSetup %%Page: 1 1 1 0 bop 233 531 a Fx(SPECTRAL)48 b(GAPS)g(F)m(OR)h(PERIODIC)e(ELLIPTIC) g(OPERA)-9 b(TORS)684 648 y(WITH)48 b(HIGH)g(CONTRAST:)g(AN)f(O)m(VER) -12 b(VIEW)1104 897 y Fw(RAINER)28 b(HEMPEL)e(AND)j(OLAF)f(POST)419 1138 y Fv(Abstra)n(ct.)41 b Fw(W)-7 b(e)30 b(discuss)f(the)h(band-gap)e (structure)h(and)g(the)h(in)n(tegrated)f(densit)n(y)g(of)419 1237 y(states)h(for)g(p)r(erio)r(dic)h(elliptic)g(op)r(erators)e(in)i (the)g(Hilb)r(ert)g(space)f Fu(L)2573 1249 y Ft(2)2610 1237 y Fw(\()p Fs(R)2696 1207 y Fr(m)2765 1237 y Fw(\),)i(for)f Fu(m)d Fq(\025)g Fw(2.)419 1337 y(W)-7 b(e)25 b(sp)r(eci\014cally)f (consider)g(situations)g(where)g(high)g(con)n(trast)g(in)h(the)g(co)r (e\016cien)n(ts)f(leads)419 1437 y(to)j(w)n(eak)e(coupling)i(b)r(et)n (w)n(een)g(the)g(p)r(erio)r(d)g(cells.)36 b(W)-7 b(eak)27 b(coupling)f(of)h(p)r(erio)r(dic)g(systems)419 1536 y(frequen)n(tly)38 b(pro)r(duces)h(sp)r(ectral)f(gaps)g(or)g(sp)r(ectral)g(concen)n (tration.)70 b(Our)38 b(examples)419 1636 y(include)31 b(Sc)n(hr\177)-42 b(odinger)28 b(op)r(erators,)h(elliptic)i(op)r (erators)d(in)j(div)n(ergence)e(form,)i(Laplace-)419 1736 y(Beltrami-op)r(erators,)j(Sc)n(hr\177)-42 b(odinger)32 b(and)j(P)n(auli)f(op)r(erators)f(with)i(p)r(erio)r(dic)g(magnetic)419 1835 y(\014elds.)64 b(There)36 b(are)g(corresp)r(onding)f(applications) h(in)h(heat)g(and)f(w)n(a)n(v)n(e)f(propagation,)419 1935 y(quan)n(tum)27 b(mec)n(hanics,)g(and)h(photonic)f(crystals.)1491 2413 y Fp(Intr)n(oduction)220 2588 y Fo(W)-8 b(e)36 b(consider)f(p)s (erio)s(dic)f(elliptic)f(partial)g(di\013eren)m(tial)h(op)s(erators)h (of)g(the)h(second)h(order)120 2704 y(with)24 b(high)g(con)m(trast)g (in)g(\(some)g(of)7 b(\))24 b(the)h(co)s(e\016cien)m(ts.)42 b(The)25 b(classes)g(of)f(op)s(erators)g(discussed)120 2820 y(here)41 b(include)e(Sc)m(hr\177)-49 b(odinger)40 b(op)s(erators,)i(magnetic)c(Sc)m(hr\177)-49 b(odinger)40 b(and)g(P)m(auli)f(op)s(erators,)120 2936 y(the)f(acoustic)g(op)s (erator,)g(Laplace-Beltrami)c(op)s(erators,)39 b(and)f(Maxw)m(ell)f(op) s(erators.)59 b(W)-8 b(e)120 3053 y(are)42 b(mainly)e(in)m(terested)k (in)d(the)i(question)g(whether)g(high)f(con)m(trast)h(in)e(the)i(co)s (e\016cien)m(ts)120 3169 y(leads)f(to)h(w)m(eak)g(coupling)f(b)s(et)m (w)m(een)i(the)f(p)s(erio)s(d)f(cells)g(and)g(to)g(sp)s(ectral)h (concen)m(tration)120 3285 y(and/or)35 b(sp)s(ectral)f(gaps.)51 b(F)-8 b(or)34 b(most)h(of)f(the)i(pap)s(er,)f(w)m(e)h(assume)g(a)e (simple)g(geometry)h(lik)m(e)120 3401 y(the)45 b(one)g(sho)m(wn)h(in)e (Figure)g(1,)k(for)c(whic)m(h)h(the)g(strongest)h(results)f(are)g (obtained.)80 b(W)-8 b(e)120 3518 y(include)39 b(some)g(remarks)h(on)f (more)g(general)g(situations,)h(lik)m(e)f(sp)s(onges)h(in)f(dimensions) f(3)120 3634 y(and)29 b(higher.)42 b(Our)29 b(main)e(theme)i(is)g(to)f (sho)m(w)i(that)f(simple)e(top)s(ological)f(assumptions)j(lead)120 3750 y(to)j(sp)s(ectral)f(consequences)36 b(for)31 b(a)h(wide)g(class)g (of|rather)f(di\013eren)m(t|elliptic)e(op)s(erators.)120 3866 y(W)-8 b(e)33 b(do)f(not)h(discuss)h(the)f(question)g(of)f (absolute)g(con)m(tin)m(uit)m(y)h(here.)220 3983 y(The)39 b(pap)s(er)f(is)f(organized)h(as)g(follo)m(ws.)58 b(Section)38 b(1)g(con)m(tains)g(some)g(basic)f(de\014nitions,)120 4099 y(assumptions)k(and)f(bac)m(kground)i(results)f(concerning)g (monotone)f(con)m(v)m(ergence)k(of)c(qua-)120 4215 y(dratic)d(forms.)60 b(In)39 b(Section)f(2,)h(w)m(e)h(brie\015y)e(rep)s(ort)g(on)g(some)g (\\classical")f(results)h(on)h(the)120 4331 y(con)m(v)m(ergence)25 b(of)e(Sc)m(hr\177)-49 b(odinger)23 b(op)s(erators)f(with)h(high)e (barriers.)40 b(Similar)20 b(results)j(for)f(strong)120 4447 y(magnetic)36 b(barriers)i(in)f(the)h(Sc)m(hr\177)-49 b(odinger)38 b(and)f(P)m(auli)g(case)i(are)e(discussed)j(in)d(Section)g (3.)120 4564 y(Section)f(4)f(is)h(dev)m(oted)h(to)e(div)m(ergence)j(t)m (yp)s(e)e(op)s(erators)g Fn(\000r)25 b(\001)f Fo(\(1)g(+)g Fm(\025\037)2843 4579 y Fl(\012)2898 4564 y Fo(\))p Fn(r)p Fo(,)37 b(where)g(\012)f(is)120 4680 y(a)f(p)s(erio)s(dic,)f(op)s(en)h (subset)i(of)d Fk(R)1327 4644 y Fj(m)1400 4680 y Fo(,)h(with)g Fm(m)d Fn(\025)h Fo(2.)50 b(There)37 b(is)d(some)h(sup)s(er\014cial)f (similarit)m(y)120 4796 y(b)s(et)m(w)m(een)45 b(div)m(ergence)e(t)m(yp) s(e)h(op)s(erators)e(and)h(Laplace-Beltrami)c(op)s(erators.)73 b(The)43 b(exis-)120 4912 y(tence)36 b(of)e(sp)s(ectral)h(gaps)g(for)g (p)s(erio)s(dic)e(Laplace-Beltrami)e(op)s(erators)k(with)f(a)h (conformal)p 120 5054 499 4 v 220 5147 a Fi(Date)6 b Fw(:)27 b(21.10.2001.)1811 5247 y Fh(1)p eop %%Page: 2 2 2 1 bop 240 66 a Fh(2)1001 b(RAINER)23 b(HEMPEL)k(AND)d(OLAF)h(POST) 1117 1870 y @beginspecial 70 @llx 559 @lly 270 @urx 759 @ury 2000 @rwi @setspecial %%BeginDocument: figure-1.ps %!PS-Adobe-2.0 EPSF-2.0 %%Title: pic1-2.sdd %%Creator: Presentation Manager 2.40 %%CreationDate: 10:31 11/09/1998 %%BoundingBox: 70 559 270 759 /epsSave save def /dw .11 .11 dtransform pop def /dpi 600 def /dpiY 600 def 14 13 translate 72 dpi div 72 dpiY div scale /mdevice matrix currentmatrix def 200 dict begin /B {bind def} bind def /L {load def} B /w /setlinewidth L /wc {dw dup idtransform pop w} B /wcx {dup dw mul exch dw mul idtransform pop w} B /_snap { transform .25 sub round .25 add exch .25 sub round .25 add exch itransform} B /c {_snap curveto} B /d /setdash L /e /eofill L /f /fill L /g /setgray L /gr /grestore L /gs /gsave L /j /setlinejoin L /l {_snap lineto} B /m {_snap moveto} B /mx /matrix L /n /newpath L /r /setrgbcolor L /s /stroke L /sm /setmatrix L /t /show L /cm /currentmatrix L /cp /closepath L /cw /currentlinewidth L /cf /currentfont L /sd {1 1 idtransform pop abs} B /ic /initclip L /SF {findfont exch scalefont setfont} B /MF {findfont exch makefont setfont} B /ms mx B /SM /setmatrix L /scm {/msclp mx cm def} B /md mx cm def /setmx {md sm concat} B /st {exch def} B /AS /awidthshow L /box {/by2 st /bx2 st /by1 st /bx1 st bx1 by1 m bx2 by1 l bx2 by2 l bx1 by2 l cp} B /CP /currentpoint L /K1 {{pop pop 3 1 roll add exch 2 copy moveto} exch kshow} B /K {K1 3 1 roll add exch moveto} B /eject {ms cm currentlinewidth showpage w 0 g sm} B /is ( ) def /u {{is 0 3 -1 roll put is stringwidth pop dup is show neg 0 rmoveto round cvi 0 rmoveto} forall} B /ip {/ip_inv st neg /ip_y st neg /ip_x st /ip_w st} B /id {ip_w 1 ip_inv [1 0 0 1 ip_x ip_y] 5 -1 roll imagemask /ip_y ip_y 1 add def} B %%EndProcSet %%EndProlog %%BeginSetup wc 0 g 2 setlinejoin 0 0 0 true ip 0 0 m [1 0 0 1 0 0] setmx wc %%EndSetup %%Page: 1 1 /SavedState0 save def [] 0 d [] 0 d [] 0 d scm n 472 5976 m 472 5976 l 472 5969 l 508 5969 l 508 5976 l 472 5976 l 472 5976 l e n 472 5976 m scm n 472 5976 m 543 5976 m 543 5976 l 543 5969 l 579 5969 l 579 5976 l 543 5976 l 543 5976 l e n 543 5976 m scm n 543 5976 m 614 5976 m 614 5976 l 614 5969 l 650 5969 l 650 5976 l 614 5976 l 614 5976 l e n 614 5976 m scm n 614 5976 m 685 5976 m 685 5976 l 685 5969 l 720 5969 l 720 5976 l 685 5976 l 685 5976 l e n 685 5976 m scm n 685 5976 m 756 5976 m 756 5976 l 756 5969 l 791 5969 l 791 5976 l 756 5976 l 756 5976 l e n 756 5976 m scm n 756 5976 m 827 5976 m 827 5976 l 827 5969 l 862 5969 l 862 5976 l 827 5976 l 827 5976 l e n 827 5976 m scm n 827 5976 m 898 5976 m 898 5976 l 898 5969 l 933 5969 l 933 5976 l 898 5976 l 898 5976 l e n 898 5976 m scm n 898 5976 m 969 5976 m 969 5976 l 969 5969 l 1004 5969 l 1004 5976 l 969 5976 l 969 5976 l e n 969 5976 m scm n 969 5976 m 1039 5976 m 1039 5976 l 1039 5969 l 1075 5969 l 1075 5976 l 1039 5976 l 1039 5976 l e n 1039 5976 m scm n 1039 5976 m 1110 5976 m 1110 5976 l 1110 5969 l 1146 5969 l 1146 5976 l 1110 5976 l 1110 5976 l e n 1110 5976 m scm n 1110 5976 m 1181 5976 m 1181 5976 l 1181 5969 l 1217 5969 l 1217 5976 l 1181 5976 l 1181 5976 l e n 1181 5976 m scm n 1181 5976 m 1252 5976 m 1252 5976 l 1252 5969 l 1287 5969 l 1287 5976 l 1252 5976 l 1252 5976 l e n 1252 5976 m scm n 1252 5976 m 1323 5976 m 1323 5976 l 1323 5969 l 1358 5969 l 1358 5976 l 1323 5976 l 1323 5976 l e n 1323 5976 m scm n 1323 5976 m 1394 5976 m 1394 5976 l 1394 5969 l 1429 5969 l 1429 5976 l 1394 5976 l 1394 5976 l e n 1394 5976 m scm n 1394 5976 m 1465 5976 m 1465 5976 l 1465 5969 l 1500 5969 l 1500 5976 l 1465 5976 l 1465 5976 l e n 1465 5976 m scm n 1465 5976 m 1535 5976 m 1535 5976 l 1535 5969 l 1571 5969 l 1571 5976 l 1535 5976 l 1535 5976 l e n 1535 5976 m scm n 1535 5976 m 1606 5976 m 1606 5976 l 1606 5969 l 1642 5969 l 1642 5976 l 1606 5976 l 1606 5976 l e n 1606 5976 m scm n 1606 5976 m 1677 5976 m 1677 5976 l 1677 5969 l 1713 5969 l 1713 5976 l 1677 5976 l 1677 5976 l e n 1677 5976 m scm n 1677 5976 m 1748 5976 m 1748 5976 l 1748 5969 l 1783 5969 l 1783 5976 l 1748 5976 l 1748 5976 l e n 1748 5976 m scm n 1748 5976 m 1819 5976 m 1819 5976 l 1819 5969 l 1854 5969 l 1854 5976 l 1819 5976 l 1819 5976 l e n 1819 5976 m scm n 1819 5976 m 1890 5976 m 1890 5976 l 1890 5969 l 1925 5969 l 1925 5976 l 1890 5976 l 1890 5976 l e n 1890 5976 m scm n 1890 5976 m 1961 5976 m 1961 5976 l 1961 5969 l 1996 5969 l 1996 5976 l 1961 5976 l 1961 5976 l e n 1961 5976 m scm n 1961 5976 m 2031 5976 m 2031 5976 l 2031 5969 l 2067 5969 l 2067 5976 l 2031 5976 l 2031 5976 l e n 2031 5976 m scm n 2031 5976 m 2102 5976 m 2102 5976 l 2102 5969 l 2126 5969 l 2126 5976 l 2102 5976 l 2102 5976 l e n 2102 5976 m scm n 2102 5976 m 472 4795 m 472 4795 l 472 4788 l 508 4788 l 508 4795 l 472 4795 l 472 4795 l e n 472 4795 m scm n 472 4795 m 543 4795 m 543 4795 l 543 4788 l 579 4788 l 579 4795 l 543 4795 l 543 4795 l e n 543 4795 m scm n 543 4795 m 614 4795 m 614 4795 l 614 4788 l 650 4788 l 650 4795 l 614 4795 l 614 4795 l e n 614 4795 m scm n 614 4795 m 685 4795 m 685 4795 l 685 4788 l 720 4788 l 720 4795 l 685 4795 l 685 4795 l e n 685 4795 m scm n 685 4795 m 756 4795 m 756 4795 l 756 4788 l 791 4788 l 791 4795 l 756 4795 l 756 4795 l e n 756 4795 m scm n 756 4795 m 827 4795 m 827 4795 l 827 4788 l 862 4788 l 862 4795 l 827 4795 l 827 4795 l e n 827 4795 m scm n 827 4795 m 898 4795 m 898 4795 l 898 4788 l 933 4788 l 933 4795 l 898 4795 l 898 4795 l e n 898 4795 m scm n 898 4795 m 969 4795 m 969 4795 l 969 4788 l 1004 4788 l 1004 4795 l 969 4795 l 969 4795 l e n 969 4795 m scm n 969 4795 m 1039 4795 m 1039 4795 l 1039 4788 l 1075 4788 l 1075 4795 l 1039 4795 l 1039 4795 l e n 1039 4795 m scm n 1039 4795 m 1110 4795 m 1110 4795 l 1110 4788 l 1146 4788 l 1146 4795 l 1110 4795 l 1110 4795 l e n 1110 4795 m scm n 1110 4795 m 1181 4795 m 1181 4795 l 1181 4788 l 1217 4788 l 1217 4795 l 1181 4795 l 1181 4795 l e n 1181 4795 m scm n 1181 4795 m 1252 4795 m 1252 4795 l 1252 4788 l 1287 4788 l 1287 4795 l 1252 4795 l 1252 4795 l e n 1252 4795 m scm n 1252 4795 m 1323 4795 m 1323 4795 l 1323 4788 l 1358 4788 l 1358 4795 l 1323 4795 l 1323 4795 l e n 1323 4795 m scm n 1323 4795 m 1394 4795 m 1394 4795 l 1394 4788 l 1429 4788 l 1429 4795 l 1394 4795 l 1394 4795 l e n 1394 4795 m scm n 1394 4795 m 1465 4795 m 1465 4795 l 1465 4788 l 1500 4788 l 1500 4795 l 1465 4795 l 1465 4795 l e n 1465 4795 m scm n 1465 4795 m 1535 4795 m 1535 4795 l 1535 4788 l 1571 4788 l 1571 4795 l 1535 4795 l 1535 4795 l e n 1535 4795 m scm n 1535 4795 m 1606 4795 m 1606 4795 l 1606 4788 l 1642 4788 l 1642 4795 l 1606 4795 l 1606 4795 l e n 1606 4795 m scm n 1606 4795 m 1677 4795 m 1677 4795 l 1677 4788 l 1713 4788 l 1713 4795 l 1677 4795 l 1677 4795 l e n 1677 4795 m scm n 1677 4795 m 1748 4795 m 1748 4795 l 1748 4788 l 1783 4788 l 1783 4795 l 1748 4795 l 1748 4795 l e n 1748 4795 m scm n 1748 4795 m 1819 4795 m 1819 4795 l 1819 4788 l 1854 4788 l 1854 4795 l 1819 4795 l 1819 4795 l e n 1819 4795 m scm n 1819 4795 m 1890 4795 m 1890 4795 l 1890 4788 l 1925 4788 l 1925 4795 l 1890 4795 l 1890 4795 l e n 1890 4795 m scm n 1890 4795 m 1961 4795 m 1961 4795 l 1961 4788 l 1996 4788 l 1996 4795 l 1961 4795 l 1961 4795 l e n 1961 4795 m scm n 1961 4795 m 2031 4795 m 2031 4795 l 2031 4788 l 2067 4788 l 2067 4795 l 2031 4795 l 2031 4795 l e n 2031 4795 m scm n 2031 4795 m 2102 4795 m 2102 4795 l 2102 4788 l 2126 4788 l 2126 4795 l 2102 4795 l 2102 4795 l e n 2102 4795 m scm n 2102 4795 m 712 6208 m 712 6208 l 705 6208 l 705 6173 l 712 6173 l 712 6208 l 712 6208 l e n 712 6208 m scm n 712 6208 m 712 6138 m 712 6138 l 705 6138 l 705 6102 l 712 6102 l 712 6138 l 712 6138 l e n 712 6138 m scm n 712 6138 m 712 6067 m 712 6067 l 705 6067 l 705 6031 l 712 6031 l 712 6067 l 712 6067 l e n 712 6067 m scm n 712 6067 m 712 5996 m 712 5996 l 705 5996 l 705 5960 l 712 5960 l 712 5996 l 712 5996 l e n 712 5996 m scm n 712 5996 m 712 5925 m 712 5925 l 705 5925 l 705 5890 l 712 5890 l 712 5925 l 712 5925 l e n 712 5925 m scm n 712 5925 m 712 5854 m 712 5854 l 705 5854 l 705 5819 l 712 5819 l 712 5854 l 712 5854 l e n 712 5854 m scm n 712 5854 m 712 5783 m 712 5783 l 705 5783 l 705 5748 l 712 5748 l 712 5783 l 712 5783 l e n 712 5783 m scm n 712 5783 m 712 5712 m 712 5712 l 705 5712 l 705 5677 l 712 5677 l 712 5712 l 712 5712 l e n 712 5712 m scm n 712 5712 m 712 5642 m 712 5642 l 705 5642 l 705 5606 l 712 5606 l 712 5642 l 712 5642 l e n 712 5642 m scm n 712 5642 m 712 5571 m 712 5571 l 705 5571 l 705 5535 l 712 5535 l 712 5571 l 712 5571 l e n 712 5571 m scm n 712 5571 m 712 5500 m 712 5500 l 705 5500 l 705 5464 l 712 5464 l 712 5500 l 712 5500 l e n 712 5500 m scm n 712 5500 m 712 5429 m 712 5429 l 705 5429 l 705 5393 l 712 5393 l 712 5429 l 712 5429 l e n 712 5429 m scm n 712 5429 m 712 5358 m 712 5358 l 705 5358 l 705 5323 l 712 5323 l 712 5358 l 712 5358 l e n 712 5358 m scm n 712 5358 m 712 5287 m 712 5287 l 705 5287 l 705 5252 l 712 5252 l 712 5287 l 712 5287 l e n 712 5287 m scm n 712 5287 m 712 5216 m 712 5216 l 705 5216 l 705 5181 l 712 5181 l 712 5216 l 712 5216 l e n 712 5216 m scm n 712 5216 m 712 5145 m 712 5145 l 705 5145 l 705 5110 l 712 5110 l 712 5145 l 712 5145 l e n 712 5145 m scm n 712 5145 m 712 5075 m 712 5075 l 705 5075 l 705 5039 l 712 5039 l 712 5075 l 712 5075 l e n 712 5075 m scm n 712 5075 m 712 5004 m 712 5004 l 705 5004 l 705 4968 l 712 4968 l 712 5004 l 712 5004 l e n 712 5004 m scm n 712 5004 m 712 4933 m 712 4933 l 705 4933 l 705 4897 l 712 4897 l 712 4933 l 712 4933 l e n 712 4933 m scm n 712 4933 m 712 4862 m 712 4862 l 705 4862 l 705 4827 l 712 4827 l 712 4862 l 712 4862 l e n 712 4862 m scm n 712 4862 m 712 4791 m 712 4791 l 705 4791 l 705 4756 l 712 4756 l 712 4791 l 712 4791 l e n 712 4791 m scm n 712 4791 m 712 4720 m 712 4720 l 705 4720 l 705 4685 l 712 4685 l 712 4720 l 712 4720 l e n 712 4720 m scm n 712 4720 m 712 4649 m 712 4649 l 705 4649 l 705 4614 l 712 4614 l 712 4649 l 712 4649 l e n 712 4649 m scm n 712 4649 m 712 4579 m 712 4579 l 705 4579 l 705 4555 l 712 4555 l 712 4579 l 712 4579 l e n 712 4579 m scm n 712 4579 m 1893 6208 m 1893 6208 l 1886 6208 l 1886 6173 l 1893 6173 l 1893 6208 l 1893 6208 l e n 1893 6208 m scm n 1893 6208 m 1893 6138 m 1893 6138 l 1886 6138 l 1886 6102 l 1893 6102 l 1893 6138 l 1893 6138 l e n 1893 6138 m scm n 1893 6138 m 1893 6067 m 1893 6067 l 1886 6067 l 1886 6031 l 1893 6031 l 1893 6067 l 1893 6067 l e n 1893 6067 m scm n 1893 6067 m 1893 5996 m 1893 5996 l 1886 5996 l 1886 5960 l 1893 5960 l 1893 5996 l 1893 5996 l e n 1893 5996 m scm n 1893 5996 m 1893 5925 m 1893 5925 l 1886 5925 l 1886 5890 l 1893 5890 l 1893 5925 l 1893 5925 l e n 1893 5925 m scm n 1893 5925 m 1893 5854 m 1893 5854 l 1886 5854 l 1886 5819 l 1893 5819 l 1893 5854 l 1893 5854 l e n 1893 5854 m scm n 1893 5854 m 1893 5783 m 1893 5783 l 1886 5783 l 1886 5748 l 1893 5748 l 1893 5783 l 1893 5783 l e n 1893 5783 m scm n 1893 5783 m 1893 5712 m 1893 5712 l 1886 5712 l 1886 5677 l 1893 5677 l 1893 5712 l 1893 5712 l e n 1893 5712 m scm n 1893 5712 m 1893 5642 m 1893 5642 l 1886 5642 l 1886 5606 l 1893 5606 l 1893 5642 l 1893 5642 l e n 1893 5642 m scm n 1893 5642 m 1893 5571 m 1893 5571 l 1886 5571 l 1886 5535 l 1893 5535 l 1893 5571 l 1893 5571 l e n 1893 5571 m scm n 1893 5571 m 1893 5500 m 1893 5500 l 1886 5500 l 1886 5464 l 1893 5464 l 1893 5500 l 1893 5500 l e n 1893 5500 m scm n 1893 5500 m 1893 5429 m 1893 5429 l 1886 5429 l 1886 5393 l 1893 5393 l 1893 5429 l 1893 5429 l e n 1893 5429 m scm n 1893 5429 m 1893 5358 m 1893 5358 l 1886 5358 l 1886 5323 l 1893 5323 l 1893 5358 l 1893 5358 l e n 1893 5358 m scm n 1893 5358 m 1893 5287 m 1893 5287 l 1886 5287 l 1886 5252 l 1893 5252 l 1893 5287 l 1893 5287 l e n 1893 5287 m scm n 1893 5287 m 1893 5216 m 1893 5216 l 1886 5216 l 1886 5181 l 1893 5181 l 1893 5216 l 1893 5216 l e n 1893 5216 m scm n 1893 5216 m 1893 5145 m 1893 5145 l 1886 5145 l 1886 5110 l 1893 5110 l 1893 5145 l 1893 5145 l e n 1893 5145 m scm n 1893 5145 m 1893 5075 m 1893 5075 l 1886 5075 l 1886 5039 l 1893 5039 l 1893 5075 l 1893 5075 l e n 1893 5075 m scm n 1893 5075 m 1893 5004 m 1893 5004 l 1886 5004 l 1886 4968 l 1893 4968 l 1893 5004 l 1893 5004 l e n 1893 5004 m scm n 1893 5004 m 1893 4933 m 1893 4933 l 1886 4933 l 1886 4897 l 1893 4897 l 1893 4933 l 1893 4933 l e n 1893 4933 m scm n 1893 4933 m 1893 4862 m 1893 4862 l 1886 4862 l 1886 4827 l 1893 4827 l 1893 4862 l 1893 4862 l e n 1893 4862 m scm n 1893 4862 m 1893 4791 m 1893 4791 l 1886 4791 l 1886 4756 l 1893 4756 l 1893 4791 l 1893 4791 l e n 1893 4791 m scm n 1893 4791 m 1893 4720 m 1893 4720 l 1886 4720 l 1886 4685 l 1893 4685 l 1893 4720 l 1893 4720 l e n 1893 4720 m scm n 1893 4720 m 1893 4649 m 1893 4649 l 1886 4649 l 1886 4614 l 1893 4614 l 1893 4649 l 1893 4649 l e n 1893 4649 m scm n 1893 4649 m 1893 4579 m 1893 4579 l 1886 4579 l 1886 4555 l 1893 4555 l 1893 4579 l 1893 4579 l e n 1893 4579 m scm n 1893 4579 m 709 5972 m 709 5972 l 1890 5972 l 1890 4791 l 709 4791 l 709 5972 l 709 5972 l 0.70193 g e 0 g n 709 5972 m [] 0 d 709 5972 l 1890 5972 l 1890 4791 l 709 4791 l 709 5972 l 709 5972 l s [] 0 d scm 709 5972 m 1420 5736 m 1420 5736 l 1389 5734 l 1359 5731 l 1329 5726 l 1302 5721 l 1276 5716 l 1251 5710 l 1226 5703 l 1204 5696 l 1182 5688 l 1162 5680 l 1143 5672 l 1125 5663 l 1107 5654 l 1092 5645 l 1077 5636 l 1063 5626 l 1050 5617 l 1038 5607 l 1027 5598 l 1017 5588 l 1008 5579 l 1000 5570 l 993 5561 l 986 5552 l 981 5544 l 967 5521 l 954 5496 l 943 5470 l 933 5443 l 925 5415 l 919 5387 l 914 5358 l 913 5330 l 913 5301 l 916 5273 l 922 5246 l 931 5220 l 943 5194 l 958 5172 l 973 5153 l 991 5136 l 1010 5123 l 1029 5113 l 1049 5107 l 1070 5103 l 1091 5103 l 1112 5106 l 1132 5113 l 1153 5122 l 1172 5135 l 1180 5142 l 1189 5150 l 1198 5159 l 1206 5168 l 1214 5179 l 1221 5190 l 1227 5203 l 1233 5218 l 1237 5234 l 1239 5252 l 1240 5271 l 1239 5293 l 1237 5316 l 1232 5342 l 1260 5329 l 1287 5317 l 1313 5307 l 1339 5299 l 1364 5294 l 1389 5291 l 1413 5290 l 1436 5290 l 1459 5292 l 1482 5296 l 1505 5302 l 1527 5309 l 1549 5318 l 1572 5328 l 1594 5339 l 1613 5352 l 1629 5366 l 1644 5381 l 1656 5398 l 1666 5417 l 1673 5437 l 1678 5457 l 1681 5478 l 1682 5500 l 1681 5522 l 1678 5544 l 1672 5565 l 1665 5587 l 1655 5608 l 1643 5628 l 1630 5646 l 1614 5664 l 1597 5680 l 1577 5695 l 1556 5708 l 1532 5719 l 1507 5727 l 1480 5733 l 1451 5736 l 1420 5736 l 1420 5736 l 1420 5736 l 1 g e 0 g n 1420 5736 m scm n 1420 5736 m 1420 5734 m 1420 5734 l 1420 5739 l 1388 5736 l 1389 5732 l 1420 5734 l 1420 5734 l e n 1420 5734 m scm n 1420 5734 m 1389 5732 m 1389 5732 l 1388 5736 l 1358 5733 l 1359 5728 l 1389 5732 l 1389 5732 l e n 1389 5732 m scm n 1389 5732 m 1359 5728 m 1359 5728 l 1358 5733 l 1329 5729 l 1330 5724 l 1359 5728 l 1359 5728 l e n 1359 5728 m scm n 1359 5728 m 1330 5724 m 1330 5724 l 1329 5729 l 1301 5724 l 1302 5719 l 1330 5724 l 1330 5724 l e n 1330 5724 m scm n 1330 5724 m 1302 5719 m 1302 5719 l 1301 5724 l 1275 5718 l 1276 5714 l 1302 5719 l 1302 5719 l e n 1302 5719 m scm n 1302 5719 m 1276 5714 m 1276 5714 l 1275 5718 l 1250 5712 l 1251 5708 l 1276 5714 l 1276 5714 l e n 1276 5714 m scm n 1276 5714 m 1251 5708 m 1251 5708 l 1250 5712 l 1226 5706 l 1227 5701 l 1251 5708 l 1251 5708 l e n 1251 5708 m scm n 1251 5708 m 1227 5701 m 1227 5701 l 1226 5706 l 1203 5698 l 1204 5694 l 1227 5701 l 1227 5701 l e n 1227 5701 m scm n 1227 5701 m 1204 5694 m 1204 5694 l 1203 5698 l 1181 5691 l 1183 5686 l 1204 5694 l 1204 5694 l e n 1204 5694 m scm n 1204 5694 m 1183 5686 m 1183 5686 l 1181 5691 l 1161 5683 l 1163 5678 l 1183 5686 l 1183 5686 l e n 1183 5686 m scm n 1183 5686 m 1163 5678 m 1163 5678 l 1161 5683 l 1142 5674 l 1144 5670 l 1163 5678 l 1163 5678 l e n 1163 5678 m scm n 1163 5678 m 1144 5670 m 1144 5670 l 1142 5674 l 1123 5665 l 1126 5661 l 1144 5670 l 1144 5670 l e n 1144 5670 m scm n 1144 5670 m 1126 5661 m 1126 5661 l 1123 5665 l 1106 5656 l 1108 5652 l 1126 5661 l 1126 5661 l e n 1126 5661 m scm n 1126 5661 m 1108 5652 m 1108 5652 l 1106 5656 l 1090 5647 l 1093 5643 l 1108 5652 l 1108 5652 l e n 1108 5652 m scm n 1108 5652 m 1093 5643 m 1093 5643 l 1090 5647 l 1075 5638 l 1078 5634 l 1093 5643 l 1093 5643 l e n 1093 5643 m scm n 1093 5643 m 1078 5634 m 1078 5634 l 1075 5638 l 1061 5628 l 1064 5624 l 1078 5634 l 1078 5634 l e n 1078 5634 m scm n 1078 5634 m 1064 5624 m 1064 5624 l 1061 5628 l 1048 5619 l 1051 5615 l 1064 5624 l 1064 5624 l e n 1064 5624 m scm n 1064 5624 m 1051 5615 m 1051 5615 l 1048 5619 l 1037 5609 l 1039 5605 l 1051 5615 l 1051 5615 l e n 1051 5615 m scm n 1051 5615 m 1039 5605 m 1039 5605 l 1037 5609 l 1025 5600 l 1029 5596 l 1039 5605 l 1039 5605 l e n 1039 5605 m scm n 1039 5605 m 1029 5596 m 1029 5596 l 1025 5600 l 1016 5590 l 1019 5587 l 1029 5596 l 1029 5596 l e n 1029 5596 m scm n 1029 5596 m 1019 5587 m 1019 5587 l 1016 5590 l 1007 5581 l 1010 5578 l 1019 5587 l 1019 5587 l e n 1019 5587 m scm n 1019 5587 m 1010 5578 m 1010 5578 l 1007 5581 l 998 5572 l 1002 5569 l 1010 5578 l 1010 5578 l e n 1010 5578 m scm n 1010 5578 m 1002 5569 m 1002 5569 l 998 5572 l 991 5563 l 995 5560 l 1002 5569 l 1002 5569 l e n 1002 5569 m scm n 1002 5569 m 995 5560 m 995 5560 l 991 5563 l 985 5554 l 988 5551 l 995 5560 l 995 5560 l e n 995 5560 m scm n 995 5560 m 988 5551 m 988 5551 l 985 5554 l 979 5546 l 983 5543 l 988 5551 l 988 5551 l e n 988 5551 m scm n 988 5551 m 983 5543 m 983 5543 l 979 5546 l 965 5522 l 969 5520 l 983 5543 l 983 5543 l e n 983 5543 m scm n 983 5543 m 969 5520 m 969 5520 l 965 5522 l 952 5497 l 956 5495 l 969 5520 l 969 5520 l e n 969 5520 m scm n 969 5520 m 956 5495 m 956 5495 l 952 5497 l 941 5471 l 945 5469 l 956 5495 l 956 5495 l e n 956 5495 m scm n 956 5495 m 945 5469 m 945 5469 l 941 5471 l 931 5444 l 935 5442 l 945 5469 l 945 5469 l e n 945 5469 m scm n 945 5469 m 935 5442 m 935 5442 l 931 5444 l 923 5415 l 927 5415 l 935 5442 l 935 5442 l e n 935 5442 m scm n 935 5442 m 927 5415 m 927 5415 l 923 5415 l 916 5387 l 921 5386 l 927 5415 l 927 5415 l e n 927 5415 m scm n 927 5415 m 921 5386 m 921 5386 l 916 5387 l 912 5359 l 917 5358 l 921 5386 l 921 5386 l e n 921 5386 m scm n 921 5386 m 917 5358 m 917 5358 l 912 5359 l 910 5330 l 915 5330 l 917 5358 l 917 5358 l e n 917 5358 m scm n 917 5358 m 915 5330 m 915 5330 l 910 5330 l 911 5301 l 915 5301 l 915 5330 l 915 5330 l e n 915 5330 m scm n 915 5330 m 915 5301 m 915 5301 l 911 5301 l 914 5273 l 919 5274 l 915 5301 l 915 5301 l e n 915 5301 m scm n 915 5301 m 919 5274 m 919 5274 l 914 5273 l 920 5246 l 925 5247 l 919 5274 l 919 5274 l e n 919 5274 m scm n 919 5274 m 925 5247 m 925 5247 l 920 5246 l 929 5219 l 934 5221 l 925 5247 l 925 5247 l e n 925 5247 m scm n 925 5247 m 934 5221 m 934 5221 l 929 5219 l 941 5193 l 946 5195 l 934 5221 l 934 5221 l e n 934 5221 m scm n 934 5221 m 946 5195 m 946 5195 l 941 5193 l 956 5170 l 960 5173 l 946 5195 l 946 5195 l e n 946 5195 m scm n 946 5195 m 960 5173 m 960 5173 l 956 5170 l 972 5151 l 975 5154 l 960 5173 l 960 5173 l e n 960 5173 m scm n 960 5173 m 975 5154 m 975 5154 l 972 5151 l 989 5135 l 993 5138 l 975 5154 l 975 5154 l e n 975 5154 m scm n 975 5154 m 993 5138 m 993 5138 l 989 5135 l 1008 5121 l 1011 5125 l 993 5138 l 993 5138 l e n 993 5138 m scm n 993 5138 m 1011 5125 m 1011 5125 l 1008 5121 l 1028 5111 l 1030 5115 l 1011 5125 l 1011 5125 l e n 1011 5125 m scm n 1011 5125 m 1030 5115 m 1030 5115 l 1028 5111 l 1049 5104 l 1050 5109 l 1030 5115 l 1030 5115 l e n 1030 5115 m scm n 1030 5115 m 1050 5109 m 1050 5109 l 1049 5104 l 1070 5101 l 1070 5106 l 1050 5109 l 1050 5109 l e n 1050 5109 m scm n 1050 5109 m 1070 5106 m 1070 5106 l 1070 5101 l 1091 5101 l 1091 5105 l 1070 5106 l 1070 5106 l e n 1070 5106 m scm n 1070 5106 m 1091 5105 m 1091 5105 l 1091 5101 l 1113 5104 l 1111 5109 l 1091 5105 l 1091 5105 l e n 1091 5105 m scm n 1091 5105 m 1111 5109 m 1111 5109 l 1113 5104 l 1133 5110 l 1131 5115 l 1111 5109 l 1111 5109 l e n 1111 5109 m scm n 1111 5109 m 1131 5115 m 1131 5115 l 1133 5110 l 1154 5120 l 1152 5124 l 1131 5115 l 1131 5115 l e n 1131 5115 m scm n 1131 5115 m 1152 5124 m 1152 5124 l 1154 5120 l 1174 5133 l 1170 5137 l 1152 5124 l 1152 5124 l e n 1152 5124 m scm n 1152 5124 m 1170 5137 m 1170 5137 l 1174 5133 l 1182 5140 l 1179 5144 l 1170 5137 l 1170 5137 l e n 1170 5137 m scm n 1170 5137 m 1179 5144 m 1179 5144 l 1182 5140 l 1191 5148 l 1188 5152 l 1179 5144 l 1179 5144 l e n 1179 5144 m scm n 1179 5144 m 1188 5152 m 1188 5152 l 1191 5148 l 1199 5157 l 1196 5160 l 1188 5152 l 1188 5152 l e n 1188 5152 m scm n 1188 5152 m 1196 5160 m 1196 5160 l 1199 5157 l 1208 5166 l 1204 5170 l 1196 5160 l 1196 5160 l e n 1196 5160 m scm n 1196 5160 m 1204 5170 m 1204 5170 l 1208 5166 l 1216 5177 l 1212 5180 l 1204 5170 l 1204 5170 l e n 1204 5170 m scm n 1204 5170 m 1212 5180 m 1212 5180 l 1216 5177 l 1223 5189 l 1219 5191 l 1212 5180 l 1212 5180 l e n 1212 5180 m scm n 1212 5180 m 1219 5191 m 1219 5191 l 1223 5189 l 1230 5202 l 1225 5204 l 1219 5191 l 1219 5191 l e n 1219 5191 m scm n 1219 5191 m 1225 5204 m 1225 5204 l 1230 5202 l 1235 5217 l 1230 5218 l 1225 5204 l 1225 5204 l e n 1225 5204 m scm n 1225 5204 m 1230 5218 m 1230 5218 l 1235 5217 l 1239 5233 l 1234 5234 l 1230 5218 l 1230 5218 l e n 1230 5218 m scm n 1230 5218 m 1234 5234 m 1234 5234 l 1239 5233 l 1242 5251 l 1237 5252 l 1234 5234 l 1234 5234 l e n 1234 5234 m scm n 1234 5234 m 1237 5252 m 1237 5252 l 1242 5251 l 1243 5271 l 1238 5271 l 1237 5252 l 1237 5252 l e n 1237 5252 m scm n 1237 5252 m 1238 5271 m 1238 5271 l 1243 5271 l 1242 5293 l 1237 5293 l 1238 5271 l 1238 5271 l e n 1238 5271 m scm n 1238 5271 m 1237 5293 m 1237 5293 l 1242 5293 l 1239 5317 l 1234 5316 l 1237 5293 l 1237 5293 l e n 1237 5293 m scm n 1237 5293 m 1234 5316 m 1234 5316 l 1239 5317 l 1235 5338 l 1229 5342 l 1234 5316 l 1234 5316 l e n 1234 5316 m scm n 1234 5316 m 1229 5342 m 1229 5342 l 1235 5338 l 1259 5326 l 1261 5331 l 1233 5345 l 1229 5342 l e n 1229 5342 m scm n 1229 5342 m 1261 5331 m 1261 5331 l 1259 5326 l 1286 5315 l 1288 5319 l 1261 5331 l 1261 5331 l e n 1261 5331 m scm n 1261 5331 m 1288 5319 m 1288 5319 l 1286 5315 l 1312 5305 l 1314 5309 l 1288 5319 l 1288 5319 l e n 1288 5319 m scm n 1288 5319 m 1314 5309 m 1314 5309 l 1312 5305 l 1339 5297 l 1340 5302 l 1314 5309 l 1314 5309 l e n 1314 5309 m scm n 1314 5309 m 1340 5302 m 1340 5302 l 1339 5297 l 1364 5292 l 1365 5297 l 1340 5302 l 1340 5302 l e n 1340 5302 m scm n 1340 5302 m 1365 5297 m 1365 5297 l 1364 5292 l 1388 5289 l 1389 5293 l 1365 5297 l 1365 5297 l e n 1365 5297 m scm n 1365 5297 m 1389 5293 m 1389 5293 l 1388 5289 l 1412 5287 l 1413 5292 l 1389 5293 l 1389 5293 l e n 1389 5293 m scm n 1389 5293 m 1413 5292 m 1413 5292 l 1412 5287 l 1436 5288 l 1436 5292 l 1413 5292 l 1413 5292 l e n 1413 5292 m scm n 1413 5292 m 1436 5292 m 1436 5292 l 1436 5288 l 1460 5290 l 1459 5295 l 1436 5292 l 1436 5292 l e n 1436 5292 m scm n 1436 5292 m 1459 5295 m 1459 5295 l 1460 5290 l 1483 5294 l 1482 5299 l 1459 5295 l 1459 5295 l e n 1459 5295 m scm n 1459 5295 m 1482 5299 m 1482 5299 l 1483 5294 l 1505 5299 l 1504 5304 l 1482 5299 l 1482 5299 l e n 1482 5299 m scm n 1482 5299 m 1504 5304 m 1504 5304 l 1505 5299 l 1528 5307 l 1526 5311 l 1504 5304 l 1504 5304 l e n 1504 5304 m scm n 1504 5304 m 1526 5311 m 1526 5311 l 1528 5307 l 1550 5316 l 1549 5320 l 1526 5311 l 1526 5311 l e n 1526 5311 m scm n 1526 5311 m 1549 5320 m 1549 5320 l 1550 5316 l 1573 5326 l 1571 5330 l 1549 5320 l 1549 5320 l e n 1549 5320 m scm n 1549 5320 m 1571 5330 m 1571 5330 l 1573 5326 l 1595 5337 l 1593 5342 l 1571 5330 l 1571 5330 l e n 1571 5330 m scm n 1571 5330 m 1593 5342 m 1593 5342 l 1595 5337 l 1615 5350 l 1611 5354 l 1593 5342 l 1593 5342 l e n 1593 5342 m scm n 1593 5342 m 1611 5354 m 1611 5354 l 1615 5350 l 1631 5364 l 1628 5367 l 1611 5354 l 1611 5354 l e n 1611 5354 m scm n 1611 5354 m 1628 5367 m 1628 5367 l 1631 5364 l 1646 5380 l 1642 5383 l 1628 5367 l 1628 5367 l e n 1628 5367 m scm n 1628 5367 m 1642 5383 m 1642 5383 l 1646 5380 l 1658 5397 l 1654 5399 l 1642 5383 l 1642 5383 l e n 1642 5383 m scm n 1642 5383 m 1654 5399 m 1654 5399 l 1658 5397 l 1668 5416 l 1663 5418 l 1654 5399 l 1654 5399 l e n 1654 5399 m scm n 1654 5399 m 1663 5418 m 1663 5418 l 1668 5416 l 1676 5436 l 1671 5437 l 1663 5418 l 1663 5418 l e n 1663 5418 m scm n 1663 5418 m 1671 5437 m 1671 5437 l 1676 5436 l 1681 5457 l 1676 5458 l 1671 5437 l 1671 5437 l e n 1671 5437 m scm n 1671 5437 m 1676 5458 m 1676 5458 l 1681 5457 l 1684 5478 l 1679 5479 l 1676 5458 l 1676 5458 l e n 1676 5458 m scm n 1676 5458 m 1679 5479 m 1679 5479 l 1684 5478 l 1685 5500 l 1680 5500 l 1679 5479 l 1679 5479 l e n 1679 5479 m scm n 1679 5479 m 1680 5500 m 1680 5500 l 1685 5500 l 1684 5522 l 1679 5522 l 1680 5500 l 1680 5500 l e n 1680 5500 m scm n 1680 5500 m 1679 5522 m 1679 5522 l 1684 5522 l 1680 5544 l 1676 5543 l 1679 5522 l 1679 5522 l e n 1679 5522 m scm n 1679 5522 m 1676 5543 m 1676 5543 l 1680 5544 l 1675 5566 l 1670 5565 l 1676 5543 l 1676 5543 l e n 1676 5543 m scm n 1676 5543 m 1670 5565 m 1670 5565 l 1675 5566 l 1667 5588 l 1663 5586 l 1670 5565 l 1670 5565 l e n 1670 5565 m scm n 1670 5565 m 1663 5586 m 1663 5586 l 1667 5588 l 1657 5609 l 1653 5607 l 1663 5586 l 1663 5586 l e n 1663 5586 m scm n 1663 5586 m 1653 5607 m 1653 5607 l 1657 5609 l 1646 5629 l 1641 5626 l 1653 5607 l 1653 5607 l e n 1653 5607 m scm n 1653 5607 m 1641 5626 m 1641 5626 l 1646 5629 l 1632 5648 l 1628 5645 l 1641 5626 l 1641 5626 l e n 1641 5626 m scm n 1641 5626 m 1628 5645 m 1628 5645 l 1632 5648 l 1616 5666 l 1612 5663 l 1628 5645 l 1628 5645 l e n 1628 5645 m scm n 1628 5645 m 1612 5663 m 1612 5663 l 1616 5666 l 1598 5682 l 1595 5679 l 1612 5663 l 1612 5663 l e n 1612 5663 m scm n 1612 5663 m 1595 5679 m 1595 5679 l 1598 5682 l 1578 5697 l 1576 5693 l 1595 5679 l 1595 5679 l e n 1595 5679 m scm n 1595 5679 m 1576 5693 m 1576 5693 l 1578 5697 l 1557 5710 l 1554 5706 l 1576 5693 l 1576 5693 l e n 1576 5693 m scm n 1576 5693 m 1554 5706 m 1554 5706 l 1557 5710 l 1533 5721 l 1531 5716 l 1554 5706 l 1554 5706 l e n 1554 5706 m scm n 1554 5706 m 1531 5716 m 1531 5716 l 1533 5721 l 1507 5729 l 1506 5725 l 1531 5716 l 1531 5716 l e n 1531 5716 m scm n 1531 5716 m 1506 5725 m 1506 5725 l 1507 5729 l 1480 5735 l 1479 5731 l 1506 5725 l 1506 5725 l e n 1506 5725 m scm n 1506 5725 m 1479 5731 m 1479 5731 l 1480 5735 l 1451 5739 l 1451 5734 l 1479 5731 l 1479 5731 l e n 1479 5731 m scm n 1479 5731 m 1451 5734 m 1451 5734 l 1451 5739 l 1420 5739 l 1420 5734 l 1451 5734 l 1451 5734 l e n 1451 5734 m ic /msbeforeclp mx cm def n [1 0 0 1 0 0] setmx 1313 5431 m 1461 5431 l 1461 5591 l 1313 5591 l cp clip n msbeforeclp SM 125 /Times-Italic SF 1313 5476 m (M) t 87 /Times-Roman SF 1417 5451 m (0) t ic /msbeforeclp mx cm def n [1 0 0 1 0 0] setmx 0 0 m 4724 0 l 4724 6799 l 0 6799 l cp clip n msbeforeclp SM ic /msbeforeclp mx cm def n [1 0 0 1 0 0] setmx 1418 4950 m 1558 4950 l 1558 5120 l 1418 5120 l cp clip n msbeforeclp SM 125 /Symbol SF 1418 4995 m (W) t 87 /Times-Roman SF 1514 4970 m (0) t ic /msbeforeclp mx cm def n [1 0 0 1 0 0] setmx 0 0 m 4724 0 l 4724 6799 l 0 6799 l cp clip n msbeforeclp SM ic /msbeforeclp mx cm def n [1 0 0 1 0 0] setmx 527 5638 m 662 5638 l 662 5798 l 527 5798 l cp clip n msbeforeclp SM 125 /Times-Italic SF 527 5683 m (Q) t 87 /Times-Roman SF 617 5658 m %(0) t ic /msbeforeclp mx cm def n [1 0 0 1 0 0] setmx 0 0 m 4724 0 l 4724 6799 l 0 6799 l cp clip n msbeforeclp SM scm n 700 5980 m 700 5980 l 717 5964 l 1881 5964 l 1898 5980 l 700 5980 l 700 5980 l e n 700 5980 m scm n 700 5980 m 1898 5980 m 1898 5980 l 1881 5964 l 1881 4799 l 1898 4783 l 1898 5980 l 1898 5980 l e n 1898 5980 m scm n 1898 5980 m 1898 4783 m 1898 4783 l 1881 4799 l 717 4799 l 700 4783 l 1898 4783 l 1898 4783 l e n 1898 4783 m scm n 1898 4783 m 700 4783 m 700 4783 l 717 4799 l 717 5964 l 700 5980 l 700 4783 l 700 4783 l e n 700 4783 m [] 0 d %%Trailer end epsSave restore %%EndDocument @endspecial 448 2086 a Fp(Figure)37 b(1.)49 b Fo(The)33 b(fundamen)m(tal)f(p)s(erio)s(d)g(cell)f Fm(Q)i Fo(and)f(the)h(set)h Fm(M)2918 2101 y Fl(0)2985 2086 y Fo(=)28 b Fm(Q)22 b Fn(\\)h Fm(M)10 b Fo(.)240 2387 y(factor)36 b(in)g(the)h(metric)f (tensor)i(is)e(discussed)i(in)e(Section)h(5.)56 b(W)-8 b(e)37 b(also)f(refer)h(to)f(construc-)240 2503 y(tions)d(of)f(a)h (purely)g(\\geometric")f(nature)h(that)g(yield)f(Laplace-Beltrami-op)s (erators)d(with)240 2619 y(gaps.)50 b(W)-8 b(e)35 b(conclude)h(in)e (Section)g(6)h(with)f(some)h(references)i(to)d(the)h(w)m(ork)h(of)e(A.) h(Figotin,)240 2735 y(P)-8 b(.)37 b(Kuc)m(hmen)m(t)g(and)g(others)g(on) g(the)g(Maxw)m(ell)f(op)s(erator)g(and)h(applications)d(to)i(photonic) 240 2852 y(crystals.)967 3143 y(1.)48 b Fp(Preliminaries)38 b(and)g(basic)f(assumptions)340 3318 y Fo(Throughout)25 b(the)h(presen)m(t)i(pap)s(er,)f(w)m(e)f(will)d(alw)m(a)m(ys)k(assume)f (that)f(\012)j Fn(\032)g Fk(R)3085 3281 y Fj(m)3183 3318 y Fo(is)d(op)s(en)h(and)240 3434 y(p)s(erio)s(dic)j(w.r.t.)i(some)f (discrete)h(lattice)e(\000;)j(for)d(most)h(purp)s(oses)i(it)e(is)g (enough)g(to)h(consider)240 3550 y(\000)40 b(=)h Fk(Z)527 3514 y Fj(m)591 3550 y Fo(.)66 b Fm(Q)40 b Fo(denotes)i(a)d(fundamen)m (tal)h(p)s(erio)s(d)f(cell)g(of)g(\000;)44 b(w)m(e)d(ma)m(y)f(tak)m(e)h Fm(Q)g Fo(=)f(\(0)p Fm(;)17 b Fo(1])3594 3514 y Fj(m)240 3666 y Fo(if)39 b(\000)h(=)g Fk(Z)623 3630 y Fj(m)687 3666 y Fo(.)66 b(The)41 b(complemen)m(t)e(of)g(\012)i(is)e(denoted)i (as)f Fm(M)10 b Fo(,)43 b(a)d(closed)g(and)g(p)s(erio)s(dic)e(set.)240 3783 y(W)-8 b(e)35 b(will)d(study)k(t)m(w)m(o-comp)s(onen)m(t)f(media)e (with)h(material)e(parameter)i Fm(\025)d Fo(=)g(1)j(on)h Fm(M)10 b Fo(,)36 b(and)240 3899 y Fm(\025)47 b Fn(\035)h Fo(1)c(on)g(\012.)79 b(The)45 b(strongest)g(results)f(are)h(obtained)e (in)h(the)h(\\mo)s(del)d(situation")g(of)240 4015 y(Figure)32 b(1)g(where)i(\012)f(con)m(tains)f(the)h(b)s(oundary)g(of)f(the)h(p)s (erio)s(d)f(cell)f Fm(Q)p Fo(.)340 4131 y(Our)36 b(basic)f(Hilb)s(ert)g (spaces)i(are)f Fn(H)f Fo(=)e Fm(L)1893 4146 y Fl(2)1933 4131 y Fo(\()p Fk(R)2037 4095 y Fj(m)2110 4131 y Fo(\))i(and)i Fn(H)2462 4095 y Fl(1)2501 4131 y Fo(\()p Fk(R)2605 4095 y Fj(m)2678 4131 y Fo(\),)f(the)h(Sob)s(olev)e(space)i(of)240 4247 y Fm(L)306 4262 y Fl(2)346 4247 y Fo(-functions)32 b(that)g(ha)m(v)m(e)i(w)m(eak)g(deriv)-5 b(ativ)m(es)33 b(in)e Fm(L)2144 4262 y Fl(2)2184 4247 y Fo(.)340 4364 y(All)39 b(our)i(op)s(erators)h(are)f(semi-b)s(ounded,)i(self-adjoin)m (t)d(op)s(erators)h Fm(T)55 b Fo(acting)40 b(in)h(some)240 4480 y Fm(L)306 4495 y Fl(2)346 4480 y Fo(-space;)47 b(they)c(are)f(obtained)f(from)g(some)h(closed,)i(semi-b)s(ounded)e (quadratic)f(form)g Fg(t)240 4596 y Fo(via)d(the)i(usual)f(represen)m (tation)h(theorem)f(\([K,)g(RS-I]\))g(and)g(satisfy)g Fm(D)s Fo(\()p Fm(T)14 b Fo(\))39 b Fn(\032)g Fm(D)s Fo(\()p Fg(t)p Fo(\))g(and)240 4712 y(the)33 b(relation)1018 4862 y Fg(t)p Fo([)p Fm(u;)17 b(v)t Fo(])27 b(=)h Fn(h)p Fm(T)14 b(u)o(;)j(v)t Fn(i)f Fm(;)211 b(u)28 b Fn(2)g Fm(D)s Fo(\()p Fm(T)14 b Fo(\))p Fm(;)113 b(v)32 b Fn(2)c Fm(D)s Fo(\()p Fg(t)p Fo(\))p Fm(:)577 b Fo(\(1)p Fm(:)p Fo(1\))240 5031 y(As)30 b(usual,)g(w)m(e)h(sa)m(y)f Fm(T)1024 5046 y Fl(2)1091 5031 y Fn(\025)e Fm(T)1253 5046 y Fl(1)1323 5031 y Fo(i\013)g(the)i(asso)s(ciated)f(quadratic)h(forms)e(satisfy)i Fm(D)s Fo(\()p Fg(t)3218 5046 y Fl(2)3257 5031 y Fo(\))e Fn(\032)g Fm(D)s Fo(\()p Fg(t)3583 5046 y Fl(1)3622 5031 y Fo(\))240 5147 y(and)g Fg(t)458 5162 y Fl(2)498 5147 y Fo([)p Fm(u;)17 b(u)p Fo(])26 b Fn(\025)j Fg(t)873 5162 y Fl(1)912 5147 y Fo([)p Fm(u;)17 b(u)p Fo(],)28 b(for)g(all)e Fm(u)h Fn(2)h Fm(D)s Fo(\()p Fg(t)1785 5162 y Fl(2)1824 5147 y Fo(\).)42 b(W)-8 b(e)29 b(also)e(write)h Fg(t)p Fo([)p Fm(u)p Fo(])f(:=)h Fg(t)p Fo([)p Fm(u;)17 b(u)p Fo(])27 b(in)g(the)i(sequel.)p eop %%Page: 3 3 3 2 bop 120 66 a Fh(SPECTRAL)33 b(GAPS)h(F)n(OR)e(PERIODIC)h(ELLIPTIC)h (OPERA)-6 b(TORS)31 b(WITH)i(HIGH)g(CONTRAST:)g(AN)f(O)n(VER)-9 b(VIEW)-38 b(3)220 266 y Fo(F)-8 b(or)33 b(a)g(sequence)k(of)d (non-negativ)m(e,)g(self-adjoin)m(t)e(op)s(erators)i Fm(T)2593 281 y Fj(n)2674 266 y Fo(in)f Fn(H)i Fo(w)m(e)g(sa)m(y)g (that)f Fm(T)3493 281 y Fj(n)120 382 y Fo(con)m(v)m(erges)h(to)d(a)g (limit)d Fm(T)1047 397 y Ff(1)1155 382 y Fo(in)j(the)h Fe(str)-5 b(ong)34 b(r)-5 b(esolvent)34 b(sense)40 b Fo(i\013)972 540 y(\()p Fm(T)1067 555 y Fj(n)1136 540 y Fo(+)22 b Fm(I)8 b Fo(\))1323 499 y Ff(\000)p Fl(1)1417 540 y Fm(f)38 b Fn(!)28 b Fo(\()p Fm(T)1726 555 y Ff(1)1823 540 y Fo(+)22 b Fm(I)8 b Fo(\))2010 499 y Ff(\000)p Fl(1)2104 540 y Fm(f)d(;)212 b(f)38 b Fn(2)28 b(H)q Fo(;)651 b(\(1)p Fm(:)p Fo(2\))120 699 y(for)34 b(con)m(v)m(ergence)k(in)c(the)h Fe(norm)i(r)-5 b(esolvent)36 b(sense)42 b Fo(w)m(e)36 b(require)f(uniform)e(con)m(v)m(ergence)k(for)120 815 y(all)30 b Fm(f)39 b Fn(2)28 b(H)34 b Fo(s.th.)f Fn(j)-17 b(j)p Fm(f)11 b Fn(j)-18 b(j)28 b(\024)g Fo(1.)220 931 y(Most)e(of)e(our)i(results)f(hinge)g(up)s(on)h(the)g(follo)m(wing)c (con)m(v)m(ergence)28 b(theorem)d(for)g(quadratic)120 1048 y(forms)32 b(\([RS-I],)g([S]\):)120 1223 y(1.1)p Fx(.)45 b(Theorem.)h Fe(L)-5 b(et)41 b Fo(\()p Fm(T)1105 1238 y Fj(n)1152 1223 y Fo(\))h Fe(b)-5 b(e)41 b(a)g(monotonic)g(se)-5 b(quenc)g(e)40 b(of)i(non-ne)-5 b(gative,)41 b(self-adjoint)120 1339 y(op)-5 b(er)g(ators.)63 b(Then)40 b(ther)-5 b(e)41 b(exists)g(a)g(self-adjoint)f(op)-5 b(er)g(ator)41 b Fm(T)2434 1354 y Ff(1)2548 1339 y Fn(\025)f Fo(0)h Fe(s.th.)63 b Fm(T)3057 1354 y Fj(n)3143 1339 y Fn(!)39 b Fm(T)3339 1354 y Ff(1)3455 1339 y Fe(in)120 1455 y(str)-5 b(ong)35 b(r)-5 b(esolvent)34 b(sense.)220 1631 y Fo(F)-8 b(or)30 b(an)i Fe(incr)-5 b(e)g(asing)39 b Fo(sequence)34 b(of)d(quadratic)h (forms)f(0)c Fn(\024)h Fg(t)2424 1646 y Fj(n)2499 1631 y Fn(\024)g Fg(t)2637 1646 y Fj(n)p Fl(+1)2806 1631 y Fo(with)j(domains)g Fn(Q)3493 1646 y Fj(n)120 1747 y Fo(the)k(limiting)29 b(form)k(domain)f(is)i(giv)m(en)g(b)m(y)h Fn(Q)1808 1762 y Ff(1)1914 1747 y Fo(=)30 b Fn(f)p Fm(u)g Fn(2)g(\\)2318 1711 y Ff(1)2318 1772 y Fj(n)p Fl(=1)2456 1747 y Fn(Q)2537 1762 y Fj(n)2584 1747 y Fo(;)17 b(sup)g Fg(t)2824 1762 y Fj(n)2872 1747 y Fo([)p Fm(u)p Fo(])30 b Fm(<)g Fn(1g)p Fo(.)47 b(Here)120 1863 y(it)36 b(ma)m(y)g(happ)s(en)h (that)f Fm(T)1057 1878 y Ff(1)1169 1863 y Fo(acts)h(in)f(a)g(strictly)g (smaller)f(Hilb)s(ert)g(space)i(than)g(the)g Fm(T)3321 1878 y Fj(n)3368 1863 y Fo(.)56 b(In)120 1979 y(this)31 b(case,)h(\()p Fm(T)636 1994 y Ff(1)729 1979 y Fo(+)19 b Fm(I)8 b Fo(\))913 1943 y Ff(\000)p Fl(1)1038 1979 y Fo(has)31 b(to)g(b)s(e)g(complemen)m(ted)g(b)m(y)g(the)h(zero)f(op)s (erator)f(on)h(a)g(suitable)120 2096 y(subspace)e(of)d Fn(H)q Fo(.)42 b(F)-8 b(or)26 b(a)g Fe(de)-5 b(cr)g(e)g(asing)34 b Fo(sequence)c(it)c(is)g(sho)m(wn)i(in)e([RS-I,)h(S])g(that)g(there)g (exists)120 2212 y(a)32 b(largest)g(closable)g(form)f(that)h(is)h (smaller)d(than)j Fg(t)2017 2227 y Fj(n)2064 2212 y Fo(,)g(for)f(all)e Fm(n)p Fo(.)220 2328 y(In)g(a)g(p)s(erio)s(dic)e(situation,)h(strong)h (resolv)m(en)m(t)h(con)m(v)m(ergence)i(often)d(implies)d(con)m(v)m (ergence)120 2444 y(of)32 b(sp)s(ectral)h(densities.)44 b(F)-8 b(or)32 b(the)h(de\014nition)f(of)g(the)i Fe(inte)-5 b(gr)g(ate)g(d)34 b(density)h(of)g(states)41 b Fo(\(i.d.s.\))120 2561 y(w)m(e)e(refer)f(to)f([RS-IV],)h([PF])g(or)f([DIT].)h(In)g(our)f (con)m(text,)k(the)d(i.d.s.)g(is)f(a)g(monotonically)120 2677 y(increasing)f(function)g Fm(F)48 b Fo(:)35 b Fk(R)45 b Fn(!)34 b Fk(R)47 b Fo(that)37 b(tends)g(to)g(zero)g(at)f Fn(\0001)p Fo(.)55 b(Suc)m(h)38 b(functions)f(ha)m(v)m(e)120 2793 y(at)e(most)f(a)h(coun)m(table)g(n)m(um)m(b)s(er)g(of)g(discon)m (tin)m(uities.)50 b(The)36 b(asso)s(ciated)f(Borel)f(measure)h Fm(\026)120 2909 y Fo(is)e(then)g(called)f(the)i Fe(density)h(of)g (states)h(me)-5 b(asur)g(e)p Fo(.)45 b(W)-8 b(e)33 b(will)e(sa)m(y)j (that)f(a)g(sequence)j(\()p Fm(\026)3344 2924 y Fj(n)3390 2909 y Fo(\))d(of)120 3025 y(densit)m(y)e(of)f(states)h(measures)f(con) m(v)m(erges)j(w)m(eakly)e(to)e Fm(\026)2183 3040 y Ff(1)2288 3025 y Fo(if)g Fm(F)2438 3040 y Fj(n)2485 3025 y Fo(\()p Fm(t)p Fo(\))e Fn(!)h Fm(F)2814 3040 y Ff(1)2889 3025 y Fo(\()p Fm(t)p Fo(\))i(at)f(all)f(p)s(oin)m(ts)120 3142 y(of)36 b(con)m(tin)m(uit)m(y)h Fm(t)g Fo(of)f Fm(F)950 3157 y Ff(1)1025 3142 y Fo(.)55 b(The)38 b(pro)s(of)e(of)g(the)h (subsequen)m(t)j(prop)s(osition)34 b(is)i(elemen)m(tary)h(if)120 3258 y(one)c(uses)h(the)f(de\014nition)e(of)i([RS-IV])f(for)g(the)h (i.d.s.)120 3433 y(1.2)p Fx(.)38 b(Prop)s(osition.)j Fe(In)35 b(addition)g(to)h(the)g(assumptions)f(of)h(The)-5 b(or)g(em)35 b(1.1,)g(supp)-5 b(ose)36 b(that)120 3549 y(the)g(op)-5 b(er)g(ators)35 b Fm(T)764 3564 y Fj(n)847 3549 y Fe(ar)-5 b(e)35 b(p)-5 b(erio)g(dic)35 b(\(with)h(r)-5 b(esp)g(e)g(ct)35 b(to)h(the)g(the)g(same)e(lattic)-5 b(e)36 b Fo(\000)p Fe(\),)g(and)f(that)h(a)120 3666 y (\(\014nitely-value)-5 b(d\))34 b(i.d.s.)g Fm(F)1127 3681 y Fj(n)1210 3666 y Fe(exist)g(for)h Fm(T)1649 3681 y Fj(n)1731 3666 y Fe(as)g(wel)5 b(l)35 b(as)f(for)h Fm(T)2393 3681 y Ff(1)2468 3666 y Fe(.)45 b(Then)35 b(the)g(c)-5 b(orr)g(esp)g(onding)120 3782 y(me)g(asur)g(es)34 b Fm(\026)596 3797 y Fj(n)678 3782 y Fe(c)-5 b(onver)g(ge)33 b(we)-5 b(akly)35 b(to)g Fm(\026)1555 3797 y Ff(1)1629 3782 y Fe(.)220 3957 y Fo(It)c(is)g(w)m(ell-kno)m(wn)g(that)g Fe(norm)i(r)-5 b(esolvent)33 b(c)-5 b(onver)g(genc)g(e)37 b Fo(implies)28 b(sp)s(ectral)j(con)m(v)m(ergence)120 4073 y(\(cf.)k([RS-I;)g(Ch.)h(VI)s(I)s(I]\))f(on)g(an)m(y)h(compact)e (in)m(terv)-5 b(al)34 b(of)g(the)i(real)e(line.)50 b(In)35 b(particular,)f(w)m(e)120 4190 y(ha)m(v)m(e)c(the)f(follo)m(wing)c (simple)i(consequence)k(for)d(the)h(existence)h(of)e(sp)s(ectral)g (gaps:)42 b(supp)s(ose)120 4306 y Fm(T)177 4321 y Fj(n)269 4306 y Fn(!)i Fm(T)470 4321 y Ff(1)587 4306 y Fo(in)e(norm-resolv)m(en) m(t)g(sense.)75 b(If)43 b Fm(T)1868 4321 y Ff(1)1985 4306 y Fo(has)g(a)f(gap)g(around)h Fm(E)50 b Fn(2)45 b Fk(R)5 b Fo(,)51 b(then)44 b(the)120 4422 y Fm(T)177 4437 y Fj(n)270 4422 y Fo(will)f(also)h(ha)m(v)m(e)j(a)e(gap)g(around)h Fm(E)6 b Fo(,)49 b(for)c(large)f Fm(n)p Fo(.)82 b(More)46 b(precisely)-8 b(,)49 b(supp)s(ose)e(that)120 4538 y(\()p Fm(a;)17 b(b)p Fo(\))22 b Fn(\\)h Fm(\033)t Fo(\()p Fm(T)597 4553 y Ff(1)672 4538 y Fo(\))k(=)h Fn(;)k Fo(and)h(let)f Fm(")27 b(>)h Fo(0.)43 b(Then)34 b(\()p Fm(a)22 b Fo(+)g Fm(";)17 b(b)22 b Fn(\000)h Fm(")p Fo(\))e Fn(\\)i Fm(\033)t Fo(\()p Fm(T)2615 4553 y Fj(n)2662 4538 y Fo(\))28 b(=)f Fn(;)p Fo(,)33 b(for)f(all)e Fm(n)e Fn(\025)g Fm(n)3474 4553 y Fj(")3511 4538 y Fo(.)1172 4741 y(2.)49 b Fp(Schr)1542 4733 y(\177)1538 4741 y(odinger)37 b(opera)-7 b(tors)220 4915 y Fo(As)25 b(a)f(w)m(arm-up,)h(w)m(e)h(illustrate)c(the)j(ab)s(o)m (v)m(e)g(con)m(v)m(ergence)i(theorems)e(b)m(y)g(some)f(simple)f(re-)120 5031 y(sults)k(on)g(Sc)m(hr\177)-49 b(odinger)27 b(op)s(erators)g(of)f (the)i(form)d Fn(\000)p Fo(\001)11 b(+)g Fm(\025V)22 b Fo(\()p Fm(x)p Fo(\),)28 b(where)g Fm(V)49 b Fo(is)26 b(non-negativ)m(e.)120 5147 y(It)38 b(has)h(b)s(een)g(a)f(\\folk)f (theorem")g(for)h(quite)g(some)g(time)f(that)h Fn(\000)p Fo(\001)27 b(+)f Fm(\025\037)2918 5162 y Fj(B)3016 5147 y Fo(con)m(v)m(erges)41 b(in)p eop %%Page: 4 4 4 3 bop 240 66 a Fh(4)1001 b(RAINER)23 b(HEMPEL)k(AND)d(OLAF)h(POST)240 266 y Fo(strong)45 b(resolv)m(en)m(t)h(sense)g(to)f(the)g(Diric)m(hlet) e(Laplacian)g(on)h(the)i(complemen)m(t)d Fk(R)3370 230 y Fj(m)3473 266 y Fn(n)31 b Fm(B)5 b Fo(,)240 382 y(under)33 b(suitable)f(assumptions)g(on)h Fm(B)5 b Fo(.)340 498 y(Supp)s(ose)43 b Fm(V)22 b Fo(\()p Fm(x)p Fo(\))44 b Fn(\025)h Fo(0)e(is)f(a)g(p)s(erio)s(dic,)h(real-v)-5 b(alued)41 b(\(con)m(tin)m(uous)i(or)f(measurable)g(and)240 614 y(b)s(ounded\))33 b(p)s(oten)m(tial)e(s.th.)i Fn(f)p Fm(x)28 b Fn(2)g Fk(R)1598 578 y Fj(m)1699 614 y Fn(j)f Fm(V)22 b Fo(\()p Fm(x)p Fo(\))28 b Fn(6)p Fo(=)f(0)p Fn(g)g Fo(=)h(\012.)44 b(W)-8 b(e)33 b(let)1296 773 y Fm(H)1377 788 y Fj(\025)1450 773 y Fo(=)28 b Fn(\000)p Fo(\001)23 b(+)f Fm(\025V)f Fo(\()p Fm(x)p Fo(\))p Fm(;)212 b(\025)28 b Fn(\025)g Fo(0)p Fm(;)855 b Fo(\(2)p Fm(:)p Fo(1\))240 940 y(with)32 b(asso)s(ciated)h(quadratic)f(forms)g Fg(h)1684 955 y Fj(\025)1729 940 y Fo([)p Fm(u)p Fo(])27 b(=)h Fn(j)-17 b(jr)p Fm(u)o Fn(j)g(j)2186 891 y Fl(2)2247 940 y Fo(+)23 b Fm(\025)2420 859 y Fd(R)2502 940 y Fm(V)f Fn(j)p Fm(u)p Fn(j)2693 904 y Fl(2)2731 940 y Fo(,)33 b(for)f Fm(u)27 b Fn(2)h(H)3202 904 y Fl(1)3241 940 y Fo(.)340 1056 y(T)-8 b(o)43 b(describ)s(e)h(the)g(limit,)f(w)m(e)i (will)c(need)k(here)f(\(and)g(in)f(the)h(follo)m(wing)c(sections\))45 b(the)240 1172 y Fe(Dirichlet)30 b(L)-5 b(aplacian)35 b Fo(on)29 b(the)g(closed)g(set)g Fm(M)39 b Fo(=)27 b Fk(R)2118 1136 y Fj(m)2205 1172 y Fn(n)14 b Fo(\012,)30 b(denoted)g(as)f Fn(\000)p Fo(\001)3035 1187 y Fj(M)3143 1172 y Fo(\([Hed,)h(HZ]\).)240 1297 y(The)k(op)s(erator)d Fn(\000)p Fo(\001)991 1312 y Fj(M)1104 1297 y Fo(is)h(constructed)i (from)d(the)i(quadratic)g(form)e Fn(j)-17 b(jr)p Fm(u)o Fn(j)g(j)3009 1248 y Fl(2)3081 1297 y Fo(with)32 b(domain)957 1438 y(~)927 1463 y Fn(H)1012 1422 y Fl(1)1011 1488 y(0)1052 1463 y Fo(\()p Fm(M)10 b Fo(\))28 b(:=)g Fn(f)p Fm(u)f Fn(2)h(H)1703 1422 y Fl(1)1742 1463 y Fo(\()p Fk(R)1846 1422 y Fj(m)1919 1463 y Fo(\))g Fn(j)f Fm(u)p Fo(\()p Fm(x)p Fo(\))g(=)h(0)k(a.e.)h(in)f Fm(M)2836 1422 y Fj(C)2896 1463 y Fn(g)p Fm(:)486 b Fo(\(2)p Fm(:)p Fo(3\))240 1622 y(F)-8 b(or)44 b(su\016cien)m(tly)h(regular)e(b)s(oundary)i(of)f Fm(M)10 b Fo(,)48 b Fn(\000)p Fo(\001)2180 1637 y Fj(M)2304 1622 y Fo(agrees)d(with)f(the)h(usual)f(Diric)m(hlet)240 1738 y(Laplacian.)81 b(In)46 b(the)h(general)e(case,)50 b(it)45 b(is)g(sho)m(wn)i(in)e([HZ])h(that)f(there)i(exists)f(a)g (Borel)240 1854 y(set)38 b Fm(M)501 1818 y Ff(\003)542 1854 y Fo(,)g Fm(M)711 1818 y Ff(\016)788 1854 y Fn(\032)f Fm(M)1006 1818 y Ff(\003)1082 1854 y Fn(\032)g Fm(M)10 b Fo(,)40 b(suc)m(h)f(that)e Fn(\000)p Fo(\001)1966 1869 y Fj(M)2084 1854 y Fo(acts)h(as)f(a)h(self-adjoin)m(t)e(op)s(erator)h (in)g(the)240 1971 y(Hilb)s(ert-space)32 b Fm(L)903 1986 y Fl(2)942 1971 y Fo(\()p Fm(M)1084 1935 y Ff(\003)1125 1971 y Fo(\).)43 b(It)33 b(is)f(no)m(w)h(easy)h(to)e(see)h(that)1068 2137 y Fn(f)p Fm(u)27 b Fn(2)h(H)1380 2096 y Fl(1)1420 2137 y Fo(\()p Fk(R)1524 2096 y Fj(m)1596 2137 y Fo(\))g Fn(j)f Fo(sup)1725 2220 y Fj(\025)p Ff(\025)p Fl(0)1881 2137 y Fg(h)1933 2152 y Fj(\025)1978 2137 y Fo([)p Fm(u)p Fo(])g Fm(<)h Fn(1g)f Fo(=)2529 2112 y(~)2500 2137 y Fn(H)2585 2096 y Fl(1)2584 2162 y(0)2624 2137 y Fo(\()p Fm(M)10 b Fo(\))p Fm(;)628 b Fo(\(2)p Fm(:)p Fo(4\))240 2361 y(and)29 b(so)f(Theorem)h(1.1)f(implies)e(that)i Fm(H)1717 2376 y Fj(\025)1791 2361 y Fo(con)m(v)m(erges)j(in)d(strong)g (resolv)m(en)m(t)i(sense)g(to)e Fn(\000)p Fo(\001)3553 2376 y Fj(M)3633 2361 y Fo(.)240 2477 y(\(T)-8 b(o)37 b(b)s(e)f(more)g(precise,)i(the)f(resolv)m(en)m(t)h(of)e Fn(\000)p Fo(\001)2021 2492 y Fj(M)2137 2477 y Fo(acts)h(in)f Fm(L)2526 2492 y Fl(2)2566 2477 y Fo(\()p Fm(M)2708 2441 y Ff(\003)2748 2477 y Fo(\))h(and)f(has)h(to)f(b)s(e)h(com-)240 2593 y(plemen)m(ted)30 b(b)m(y)g(the)g(zero)g(op)s(erator)f(on)g Fm(L)1794 2608 y Fl(2)1834 2593 y Fo(\()p Fk(R)1938 2557 y Fj(m)2026 2593 y Fn(n)16 b Fm(M)2196 2557 y Ff(\003)2236 2593 y Fo(\))29 b(in)g(Eqn.)i(\(1.1\).\))42 b(Up)29 b(to)h(this)f(p)s (oin)m(t,)240 2709 y(p)s(erio)s(dicit)m(y)36 b(w)m(as)i(not)g (required.)59 b(If)38 b(w)m(e)g(no)m(w)h(assume)f(that)f(\012)h(is)f(p) s(erio)s(dic,)h(then)g(for)f(all)240 2825 y(op)s(erators)g(in)m(v)m (olv)m(ed)h(the)g(i.d.s.)f(is)g(w)m(ell-de\014ned,)i(and)e(w)m(e)h(ma)m (y)f(also)g(conclude)g(that)h(the)240 2942 y(i.d.s.)33 b(con)m(v)m(erges.)340 3058 y(As)38 b(w)m(as)h(observ)m(ed)h(in)e ([HH-1],)h(one)f(can)g(actually)f(do)h(b)s(etter)h(and)f(upgrade)g(to)g (norm)240 3174 y(resolv)m(en)m(t)c(con)m(v)m(ergence.)46 b(The)34 b(pro)s(of)e(com)m(bines)g(uniform)f(exp)s(onen)m(tial)h(deca) m(y)i(estimates)240 3290 y(for)e(the)h(resolv)m(en)m(t)h(k)m(ernels)f (with)g(Sc)m(h)m(ur's)h(Lemma)d(and)i(lo)s(cal)d(compactness)k(of)e Fn(\000)p Fo(\001.)340 3428 y(The)26 b(ab)s(o)m(v)m(e)g(con)m(v)m (ergence)h(results)f(do)f(not)g(require)g(\012)h(to)e(b)s(e)i (connected.)42 b(In)26 b(our)f(\\mo)s(del)240 3544 y(case")36 b(of)f(Figure)f(1)h(where)i(\012)f(con)m(tains)f(a)g(neigh)m(b)s(orho)s (o)s(d)g(of)f Fm(@)5 b(Q)p Fo(,)38 b Fn(\000)p Fo(\001)2951 3559 y Fj(M)3066 3544 y Fo(is)d(a)g(coun)m(table)240 3660 y(direct)i(sum)g(of)f(copies)h(of)g Fn(\000)p Fo(\001)1413 3675 y Fj(M)7 b Ff(\\)p Fj(Q)1595 3660 y Fo(.)57 b(Therefore,)39 b(the)f(sp)s(ectrum)f(of)f Fn(\000)p Fo(\001)3029 3675 y Fj(M)3146 3660 y Fo(is)h(a)f(discrete)240 3776 y(set)43 b(with)e(eac)m(h)i(p)s(oin)m(t)e(in)h(the)g(sp)s(ectrum)g(an)g(eigen)m (v)-5 b(alue)42 b(of)f(in\014nite)g(m)m(ultiplicit)m(y)-8 b(.)69 b(W)-8 b(e)240 3893 y(therefore)41 b(see)h(that)f(the)g(sp)s (ectrum)g(of)g Fm(H)1854 3908 y Fj(\025)1939 3893 y Fo(concen)m(trates) i(at)d(a)h(discrete)g(set)h(of)e(p)s(oin)m(ts,)240 4009 y(and)33 b(an)f(arbitrarily)e(large)h(n)m(um)m(b)s(er)i(of)f(sp)s (ectral)h(gaps)g(op)s(ens)g(up,)g(as)g Fm(\025)27 b Fn(!)g(1)p Fo(.)767 4211 y(3.)49 b Fp(Ma)n(gnetic)36 b(Schr)1660 4203 y(\177)1656 4211 y(odinger)i(and)g(P)-9 b(a)n(uli)38 b(opera)-7 b(tors)340 4386 y Fo(Here)45 b(w)m(e)g(brie\015y)g(men)m (tion)e(some)i(results)f(of)g([HH-1,2],)j([HN],)e([B-1,2])f(on)g (strongly)240 4502 y(coupled)33 b(p)s(erio)s(dic)e(magnetic)g (\014elds.)340 4639 y(\(1\))48 b(Let)42 b Fm(~)-43 b(a)55 b Fn(2)f Fm(C)1006 4603 y Fl(1)1046 4639 y Fo(\()p Fk(R)1150 4603 y Fj(m)1222 4639 y Fm(;)17 b Fk(R)1332 4603 y Fj(m)1405 4639 y Fo(\))48 b(b)s(e)g(a)g(magnetic)f(v)m(ector)j(p)s(oten)m(tial)c (and)j Fm(B)59 b Fo(=)54 b Fm(d)-6 b(~)-43 b(a)49 b Fo(the)240 4755 y(asso)s(ciated)31 b(magnetic)f(\014eld.)43 b(W)-8 b(e)32 b(assume)g(that)f Fm(B)36 b Fo(is)31 b(p)s(erio)s(dic)f(w.r.t.)i (the)f(lattice)f(\000,)i(and)240 4872 y(that)e Fm(B)5 b Fo(\()p Fm(x)p Fo(\))28 b(=)g(0)i(for)g(all)f Fm(x)f Fn(2)g Fm(M)10 b Fo(,)31 b(while)f Fm(B)5 b Fo(\()p Fm(x)p Fo(\))28 b Fn(6)p Fo(=)g(0)i(for)g(all)e Fm(x)h Fn(2)f Fo(\012.)43 b(Note)30 b(that)h(p)s(erio)s(dicit)m(y)240 4988 y(of)23 b Fm(~)-42 b(a)30 b Fo(is)g Fe(not)39 b Fo(required.)44 b(The)31 b(magnetic)e(Sc)m(hr\177)-49 b(odinger)30 b(op)s(erator)f(is)h(then)h(giv)m(en)f(as)h([CFKS])1468 5147 y Fm(H)8 b Fo(\()p Fm(\025)-6 b(~)-43 b(a)p Fo(\))27 b(=)h(\()p Fn(\000)p Fo(i)p Fn(r)22 b(\000)g Fm(\025)-6 b(~)-43 b(a)p Fo(\))2365 5106 y Fl(2)2405 5147 y Fm(;)1027 b Fo(\(3)p Fm(:)p Fo(1\))p eop %%Page: 5 5 5 4 bop 120 66 a Fh(SPECTRAL)33 b(GAPS)h(F)n(OR)e(PERIODIC)h(ELLIPTIC)h (OPERA)-6 b(TORS)31 b(WITH)i(HIGH)g(CONTRAST:)g(AN)f(O)n(VER)-9 b(VIEW)-38 b(5)120 266 y Fo(and)33 b(obtained)f(via)g(the)h(closure)f (of)g(the)h(forms)701 424 y Fn(h)o Fm(H)8 b Fo(\()p Fm(\025)-6 b(~)-43 b(a)p Fo(\))p Fm(u;)17 b(u)p Fn(i)26 b Fo(=)h Fn(j)-16 b(j)o Fo(\()p Fn(\000)p Fo(i)p Fn(r)22 b(\000)h Fm(\025)-6 b(~)-43 b(a)p Fo(\()p Fm(x)p Fo(\)\))p Fm(u)o Fn(j)-17 b(j)2094 375 y Fl(2)2150 424 y Fm(;)212 b(u)27 b Fn(2)h Fm(C)2643 383 y Ff(1)2636 448 y Fj(c)2718 424 y Fo(\()p Fk(R)2822 383 y Fj(m)2894 424 y Fo(\))p Fm(:)120 582 y Fo(Eviden)m(tly)-8 b(,)47 b(there)d(is)f Fe(no)49 b Fo(monotonicit)m(y)42 b(w.r.t.)i Fm(\025)p Fo(.)77 b(Ho)m(w)m(ev)m(er,)48 b(in)43 b(an)m(y)h(op)s(en)g(subset)h Fm(U)120 698 y Fo(of)c(\012)h(where)h(a)e(\014xed)i(en)m(try)g Fm(B)1316 713 y Fj(ij)1376 698 y Fo(\()p Fm(x)p Fo(\))f(of)f(the)h (magnetic)f(tensor)h(is)f(b)s(ounded)h(a)m(w)m(a)m(y)h(from)120 814 y(zero,)k(the)e(Avron-Herbst-Simon)e(b)s(ound)i([AHS])f(giv)m(es)h (a)f Fe(lo)-5 b(c)g(al)53 b Fo(lo)m(w)m(er)45 b(b)s(ound)f(for)g(the) 120 930 y(quadratic)d(form,)g(expressing)h(the)g(barrier-lik)m(e)d (e\013ect)i(of)g(a)f(strong)h(magnetic)f(\014eld.)68 b(If)120 1047 y(the)34 b(v)m(ector)h(p)s(oten)m(tial)25 b Fm(~)-42 b(a)34 b Fo(v)-5 b(anishes)34 b(on)g Fm(M)10 b Fo(,)35 b(the)f(ab)s(o)m(v)m(e)g(results)g(for)f Fn(\000)p Fo(\001)24 b(+)f Fm(\026\037)3076 1062 y Fj(U)3135 1047 y Fo(,)34 b Fm(\026)29 b Fn(!)g(1)p Fo(,)120 1163 y(can)38 b(b)s(e)h(com)m(bined)e(with)h(the)h(F)-8 b(eynman-Kac-It^)-49 b(o-form)m(ula)34 b(to)k(obtain)f(strong)h(resolv)m(en)m(t)120 1279 y(con)m(v)m(ergence)c(of)e Fm(H)8 b Fo(\()p Fm(\025)-6 b(~)-43 b(a)o Fo(\))32 b(to)g Fn(\000)p Fo(\001)1350 1294 y Fj(M)1429 1279 y Fo(,)g(as)g Fm(\025)c Fn(!)f(1)32 b Fo(\([HH-1]\).)43 b(F)-8 b(or)31 b(giv)m(en)h(magnetic)e(\014eld)i Fm(B)5 b Fo(,)120 1395 y(it)33 b(dep)s(ends)j(on)e(the)g(top)s(ology)f (of)h(the)g(set)h(\012)f(and)h(on)f(\015ux)h(conditions)e(whether)i(a)f (gauge)120 1512 y(exists)f(s.th.)h Fm(d)-6 b(~)-43 b(a)27 b Fo(=)h Fm(B)37 b Fo(and)27 b Fm(~)-43 b(a)p Fo(\()p Fm(x)p Fo(\))28 b(=)g(0)k(a.e.)h(on)f Fm(M)10 b Fo(.)220 1628 y(Under)30 b(the)h(ab)s(o)m(v)m(e)g(conditions,)e(w)m(e)i(also)e (obtain)g(an)h(upgrade)h(to)e(norm-resolv)m(en)m(t)h(con-)120 1744 y(v)m(ergence,)35 b(as)d(b)s(efore,)h(and)g(get)f(the)h(same)g(sp) s(ectral)f(consequences)k(as)d(for)f Fn(\000)p Fo(\001)23 b(+)f Fm(\025\037)3368 1759 y Fl(\012)3423 1744 y Fo(.)220 1881 y(\(2\))27 b(The)j(situation)c(where)d Fm(~)-42 b(a)p Fo(\()p Fm(x)p Fo(\))28 b(is)g(allo)m(w)m(ed)g(to)g(ha)m(v)m(e)h (non-zero)f(v)-5 b(alues)28 b(on)h Fm(M)39 b Fo(has)28 b(b)s(een)120 1997 y(analyzed)43 b(in)f(detail)g(in)g([HN],)h(using)g (some)g(basic)f(homology)f(theory)j(and)f(de)g(Rham's)120 2114 y(Theorem.)71 b(It)42 b(can)g(b)s(e)f(sho)m(wn)i(that,)h(under)f (suitable)d(assumptions,)k(the)e(norm)f(di\013er-)120 2230 y(ence)h(of)f(the)g(resolv)m(en)m(ts)h(of)f Fm(H)8 b Fo(\()p Fm(\025)-6 b(~)-43 b(a)o Fo(\))41 b(and)g Fm(H)1809 2245 y Fj(M)1888 2230 y Fo(\()p Fm(\025)-6 b(~)-43 b(a)p Fo(\))41 b(tends)h(to)e(zero,)k(as)d Fm(\025)h Fn(!)f(1)p Fo(,)i(where)120 2346 y Fm(H)201 2361 y Fj(M)280 2346 y Fo(\()p Fm(\025)-6 b(~)-43 b(a)p Fo(\))36 b(=)h(\()p Fn(\000)p Fo(i)p Fn(r)25 b(\000)h Fm(\025)-6 b(~)-43 b(a)p Fo(\))1113 2310 y Fl(2)1153 2346 y Fo(,)39 b(acting)e(in)g Fm(L)1702 2361 y Fl(2)1742 2346 y Fo(\()p Fm(M)1884 2310 y Ff(\003)1924 2346 y Fo(\))h(with)f(Diric)m(hlet)f(b)s(oundary)i (conditions.)120 2462 y(As)i(a)f(consequence,)45 b(one)40 b(can)g(conclude)g(that)f(the)h(sp)s(ectrum)g(of)f Fm(H)8 b Fo(\()p Fm(\025)-6 b(~)-43 b(a)p Fo(\))39 b(approac)m(hes)i(a)120 2579 y(p)s(erio)s(dic)29 b(or)i(a)f(quasi-p)s(erio)s(dic)f(function)i (of)f Fm(\025)p Fo(,)h(as)g Fm(\025)d Fn(!)f(1)p Fo(.)43 b(The)32 b(precise)f(result)g(in)m(v)m(olv)m(es)120 2695 y(the)38 b(\015ux)h(of)f Fm(B)43 b Fo(through)38 b(the)g(connected)i (comp)s(onen)m(ts)f(of)e(\()p 2430 2615 145 4 v Fm(M)2534 2666 y Ff(\016)2574 2695 y Fo(\))2612 2659 y Fj(C)2671 2695 y Fo(,)j(the)e(complemen)m(t)f(of)120 2811 y(the)c(closure)g(of)f (the)h(in)m(terior)e(of)h Fm(M)10 b Fo(.)220 2970 y(\(3\))32 b(F)-8 b(or)32 b(the)h Fx(P)m(auli)i(op)s(erator)e Fo(in)f Fk(R)1640 2934 y Fl(2)1685 2970 y Fo(,)h(w)m(e)h(ha)m(v)m(e)f(the)g (pair)f(of)g(op)s(erators)1352 3128 y Fm(H)1433 3143 y Ff(\006)1492 3128 y Fo(\()p Fm(\025)-6 b(~)-43 b(a)p Fo(\))28 b(=)f Fm(H)8 b Fo(\()p Fm(\025)-6 b(~)-43 b(a)p Fo(\))22 b Fn(\007)h Fm(B)5 b(;)1031 b Fo(\(3)p Fm(:)p Fo(2\))120 3286 y(b)s(oth)32 b(acting)g(in)g Fm(L)823 3301 y Fl(2)863 3286 y Fo(\()p Fk(R)966 3250 y Fl(2)1012 3286 y Fo(\).)43 b(Here)34 b(sup)s(ersymmetry)f(\(cf.)g([CFKS]\))g (implies)1050 3444 y Fm(\033)t Fo(\()p Fm(H)1228 3459 y Fl(+)1286 3444 y Fo(\()p Fm(\025)-6 b(~)-43 b(a)p Fo(\)\))22 b Fn(n)g(f)p Fo(0)p Fn(g)28 b Fo(=)f Fm(\033)t Fo(\()p Fm(H)2060 3459 y Ff(\000)2119 3444 y Fo(\()p Fm(\025)-6 b(~)-43 b(a)p Fo(\)\))22 b Fn(n)g(f)p Fo(0)p Fn(g)p Fm(:)728 b Fo(\(3)p Fm(:)p Fo(3\))120 3602 y(Note)37 b(that)f(the)i(results)f (in)f(\(1\))g(and)h(\(2\))g(remain)e(unc)m(hanged)j(if)e(w)m(e)i (include)e(an)h(electric)120 3718 y(bac)m(kground)27 b(p)s(oten)m(tial)e Fm(V)47 b Fo(\(e.g.,)28 b Fm(V)48 b Fo(b)s(ounded)27 b(and)f(measurable\),)h(while)e(sup)s(ersymmetry)120 3835 y(is)32 b(destro)m(y)m(ed)j(up)s(on)d(addition)f(of)h(an)h (electric)f(p)s(oten)m(tial.)220 3951 y(Under)d(suitable)e (assumptions,)i(one)f(can)h(sho)m(w)g(\([B-1,2]\))f(that)g(the)g(op)s (erators)g Fm(H)3297 3966 y Ff(\006)3356 3951 y Fo(\()p Fm(\025)-6 b(~)-43 b(a)p Fo(\))120 4067 y(ha)m(v)m(e)40 b(a)e(common)g(gap:)55 b(in)38 b(the)h(simplest)e(case,)k(w)m(e)f (assume)f Fm(B)k Fn(\025)c Fo(0.)61 b(A)39 b(more)f(delicate)120 4183 y(result)c(requires)i(that)e Fm(B)40 b Fo(is)34 b(p)s(ositiv)m(e)g(in)g(a)g(neigh)m(b)s(orho)s(o)s(d)f(of)h Fm(@)5 b(Q)36 b Fo(and)f(that)f(there)h(exists)120 4300 y(a)c(p)s(erio)s(dic)e(gauge)i(s.th.)h Fm(d)-6 b(~)-43 b(a)27 b Fo(=)h Fm(B)5 b Fo(.)43 b(\(By)31 b(the)g(div)m(ergence)i (theorem,)e(this)g(gauge)f(condition)120 4416 y(is)i(equiv)-5 b(alen)m(t)32 b(to)h(the)g Fe(\015ux)i(c)-5 b(ondition)1589 4336 y Fd(R)1636 4450 y Fj(Q)1712 4416 y Fm(B)5 b Fo(\()p Fm(x)p Fo(\))p Fm(dx)29 b Fo(=)e(0\).)839 4624 y(4.)49 b Fp(Periodic)38 b(diver)n(gence)e(type)h(opera)-7 b(tors)220 4799 y Fo(Here)29 b(w)m(e)h(rep)s(ort)e(on)g(some)h(results)g(of)f ([HL-1,2])g(concerning)h(p)s(erio)s(dic)e(t)m(w)m(o-comp)s(onen)m(t)120 4915 y(media)46 b(with)i(high)f(con)m(trast.)89 b(In)48 b(the)g(follo)m(wing)d(w)m(e)k(alw)m(a)m(ys)f(assume)h Fm(m)k Fn(\025)h Fo(2.)89 b(The)120 5031 y(op)s(erators)28 b(dicussed)h(b)s(elo)m(w)f(o)s(ccur)h(in)e(acoustics,)i(heat)g (conduction,)g(and)f(propagation)e(of)120 5147 y(electro-magnetic)31 b(w)m(a)m(v)m(es)k(in)c(photonic)h(crystals.)p eop %%Page: 6 6 6 5 bop 240 66 a Fh(6)1001 b(RAINER)23 b(HEMPEL)k(AND)d(OLAF)h(POST)340 266 y Fo(The)33 b(quadratic)f(form)g(is)g(giv)m(en)h(b)m(y)g(the)g Fe(Dirichlet)h(inte)-5 b(gr)g(al)743 470 y Fg(t)776 485 y Fj(\025)822 470 y Fo([)p Fm(u)p Fo(])27 b(:=)1090 334 y Fd(Z)1189 470 y Fo(\(1)22 b(+)g Fm(\025\037)1514 485 y Fl(\012)1569 470 y Fo(\()p Fm(x)p Fo(\)\))p Fn(jr)p Fm(u)p Fo(\()p Fm(x)p Fo(\))p Fn(j)2064 428 y Fl(2)2103 470 y Fm(dx;)212 b(u)27 b Fn(2)h(H)2710 428 y Fl(1)2750 470 y Fm(;)114 b(\025)28 b Fn(\025)g Fo(0;)302 b(\(4)p Fm(:)p Fo(1\))240 674 y(this)48 b(de\014nes)i(a)e(monotonically)d (increasing)j(family)d(of)j(forms.)90 b(The)50 b(asso)s(ciated)e(self-) 240 790 y(adjoin)m(t)32 b(op)s(erators)g Fm(T)1059 805 y Fj(\025)1137 790 y Fo(can)h(formally)d(b)s(e)i(written)h(as)g Fm(T)2352 805 y Fj(\025)2425 790 y Fo(=)27 b Fn(\000r)c(\001)f Fo(\(1)g(+)g Fm(\025\037)3086 805 y Fl(\012)3141 790 y Fo(\))p Fn(r)p Fo(.)340 906 y(It)44 b(is)g(clear)f(that)i(only)e(suc) m(h)j(functions)e Fm(u)g Fo(from)f(the)i(form)e(domain)f(surviv)m(e)k (taking)240 1022 y Fm(\025)28 b Fn(!)f(1)32 b Fo(that)g(are)h(constan)m (t)g(on)g(eac)m(h)g(connected)h(comp)s(onen)m(t)f(of)f(\012.)44 b(If)32 b(\012)h(is)f(connected,)240 1139 y(this)g(implies)e(that)j (only)f(suc)m(h)i Fm(u)p Fo('s)f(will)d(remain)h(in)h(the)h(limiting)28 b(form)k(domain)f Fn(Q)3374 1154 y Ff(1)3481 1139 y Fo(that)240 1255 y(are)i(iden)m(tically)d(zero)j(on)f(\012.)44 b(W)-8 b(e)33 b(ha)m(v)m(e:)240 1420 y(4.1)p Fx(.)51 b(Theorem.)d Fo([HL-1])e Fe(Supp)-5 b(ose)45 b Fo(\012)i Fe(is)f(op)-5 b(en,)48 b(p)-5 b(erio)g(dic,)48 b(and)d(c)-5 b(onne)g(cte)g(d.)78 b(Then)46 b Fm(T)3615 1435 y Fj(\025)240 1537 y Fe(c)-5 b(onver)g(ges)40 b(to)h(the)g(Dirichlet-L)-5 b(aplacian)38 b Fn(\000)p Fo(\001)1986 1552 y Fj(M)2107 1537 y Fe(on)i Fm(M)50 b Fo(=)39 b Fk(R)2577 1501 y Fj(m)2676 1537 y Fn(n)26 b Fo(\012)41 b Fe(in)g(str)-5 b(ong)41 b(r)-5 b(esolvent)240 1653 y(sense.)44 b(F)-7 b(urthermor)i(e,)34 b(the)h(asso)-5 b(ciate)g(d)33 b(density)i(of)g(states)g(me)-5 b(asur)g(es)34 b(c)-5 b(onver)g(ge)34 b(we)-5 b(akly.)340 1831 y Fo(The)39 b(standard)g(Flo)s(quet-decomp)s(osition)d(giv)m(es)j (a)g(more)f(detailed)f(picture.)62 b(Let)39 b Fm(T)3566 1780 y Fl(\()p Fj(\022)r Fl(\))3552 1859 y Fj(\025)240 1947 y Fo(denote)30 b(the)g(op)s(erator)e Fn(\000r)15 b(\001)g Fo(\(1)g(+)g Fm(\025\037)1634 1962 y Fl(\012)1690 1947 y Fo(\))p Fn(r)p Fo(,)30 b(acting)e(in)h Fm(L)2334 1962 y Fl(2)2373 1947 y Fo(\()p Fm(Q)p Fo(\),)h(with)f Fm(\022)s Fo(-p)s(erio)s(dic)f(b)s(oundary)240 2063 y(conditions)35 b(on)h Fm(@)5 b(Q)p Fo(,)37 b(for)f Fm(\022)g Fn(2)d Fo(\()p Fn(\000)p Fm(\031)t(;)17 b(\031)t Fo(])1683 2027 y Fj(m)1750 2063 y Fo(;)37 b(cf.)f([RS-IV,)g(Ku].)53 b(Then)37 b(Theorem)f(1.1)g(can)g(b)s(e)240 2197 y(applied)f(in)g(eac)m (h)i(\014b)s(er,)g(i.e.,)g(for)e(eac)m(h)i(\014xed)h Fm(\022)s Fo(.)54 b(As)36 b(eac)m(h)h Fm(T)2555 2146 y Fl(\()p Fj(\022)r Fl(\))2541 2225 y Fj(\025)2685 2197 y Fo(has)f(compact)g(resolv)m(en)m(t,)240 2313 y(w)m(e)e(ev)m(en)g (obtain)e(norm)f(resolv)m(en)m(t)j(con)m(v)m(ergence)i(in)c(the)h (\014b)s(ers.)44 b(The)34 b(limiting)28 b(op)s(erators)240 2442 y Fm(T)311 2391 y Fl(\()p Fj(\022)r Fl(\))297 2452 y Ff(1)445 2442 y Fo(ha)m(v)m(e)41 b(form)e(domains)g Fn(Q)1389 2391 y Fl(\()p Fj(\022)r Fl(\))1389 2452 y Ff(1)1523 2442 y Fo(consisting)g(of)h(those)h(functions)f Fm(u)g Fn(2)g(H)3078 2406 y Fl(1)3118 2442 y Fo(\()p Fm(Q)p Fo(\))g(that)g(are)240 2558 y(constan)m(t)32 b(in)f(eac)m(h)h (connected)h(comp)s(onen)m(t)e(of)g(\012)20 b Fn(\\)g Fm(Q)31 b Fo(and)h(satisfy)f Fm(\022)s Fo(-p)s(erio)s(dic)f(b)s (oundary)240 2674 y(conditions.)73 b(If)43 b(\012)g(is)f(connected)i (and)f(in)m(tersects)h(eac)m(h)g(face)f(of)f(the)h(p)s(erio)s(d)f(cell) f Fm(Q)p Fo(,)46 b(it)240 2791 y(follo)m(ws)29 b(that)h(suc)m(h)i Fm(u)e Fo(m)m(ust)h(v)-5 b(anish)30 b(on)g(\012)18 b Fn(\\)h Fm(Q)p Fo(,)31 b(pro)m(vided)g Fm(\022)f Fn(6)p Fo(=)e(\(0)p Fm(;)17 b(:)g(:)g(:)f(;)h Fo(0\).)42 b(Pursuing)30 b(this)240 2907 y(observ)-5 b(ation)32 b(in)g(detail,)f(one)i(obtains) 240 3072 y(4.2)p Fx(.)39 b(Theorem.)j Fo(\([HL-1]\))36 b Fe(Supp)-5 b(ose)35 b Fo(\012)c Fn(\032)g Fk(R)1975 3036 y Fj(m)2084 3072 y Fe(is)36 b(op)-5 b(en,)36 b(p)-5 b(erio)g(dic,)36 b(c)-5 b(onne)g(cte)g(d)35 b(and)h(c)-5 b(on-)240 3189 y(tains)31 b Fm(@)5 b(Q)p Fe(.)45 b(We)32 b(also)e(assume)h(that)h Fm(M)1693 3153 y Ff(\003)1765 3189 y Fe(\(de\014ne)-5 b(d)30 b(in)h(Se)-5 b(ction)31 b(2\))g(is)g(not)g(of)h(me)-5 b(asur)g(e)30 b(zer)-5 b(o.)240 3305 y(L)g(et)35 b Fo(0)28 b Fm(<)f(\016)631 3320 y Fl(1)699 3305 y Fn(\024)h Fm(\016)847 3320 y Fl(2)914 3305 y Fn(\024)g Fm(:)17 b(:)g(:)35 b Fe(denote)f(the)h(eigenvalues)e (of)i Fn(\000)p Fo(\001)2425 3320 y Fj(M)7 b Ff(\\)p Fj(Q)2607 3305 y Fe(.)45 b(We)35 b(then)g(have:)340 3421 y Fo(\()p Fm(i)p Fo(\))g Fe(The)f(\014rst)h(sp)-5 b(e)g(ctr)g(al)34 b(b)-5 b(and)34 b(of)h Fm(T)1634 3436 y Fj(\025)1714 3421 y Fe(c)-5 b(onver)g(ges)34 b(to)h(the)g(interval)f Fo([0)p Fm(;)17 b(\016)2948 3436 y Fl(1)2988 3421 y Fo(])p Fe(.)340 3537 y Fo(\()p Fm(ii)p Fo(\))35 b Fe(F)-7 b(or)34 b Fm(\025)h Fe(lar)-5 b(ge,)34 b Fm(T)1113 3552 y Fj(\025)1193 3537 y Fe(has)h(a)f(sp)-5 b(e)g(ctr)g(al)35 b(gap)f(fol)5 b(lowing)33 b(the)i(\014rst)g(b)-5 b(and.)340 3654 y Fo(\()p Fm(iii)p Fo(\))35 b Fe(The)f(i.d.s.)h(me)-5 b(asur)g(e)34 b(of)g Fm(T)1542 3669 y Fj(\025)1623 3654 y Fe(c)-5 b(onc)g(entr)g (ates)34 b(at)h(the)g(set)g Fn(f)p Fm(\016)2701 3669 y Fj(k)2744 3654 y Fo(;)17 b Fm(k)30 b Fn(2)e Fk(N)9 b Fn(g)p Fe(,)41 b(as)34 b Fm(\025)28 b Fn(!)f(1)p Fe(.)340 3819 y Fo(It)38 b(follo)m(ws)f(from)g(\()p Fm(i)p Fo(\))h(that)g(there) h(is)e Fe(no)44 b Fo(norm)37 b(resolv)m(en)m(t)i(con)m(v)m(ergence)i (of)d(the)g(family)240 3936 y(\()p Fm(T)335 3951 y Fj(\025)380 3936 y Fo(\))e(\(The)h(only)f(p)s(ossible)f(candidate)h(for)f(the)i (limit)32 b(is)k Fn(\000)p Fo(\001)2552 3951 y Fj(M)2668 3936 y Fo(whic)m(h)g(has)h(no)f(sp)s(ectrum)240 4052 y(b)s(elo)m(w)c Fm(\016)559 4067 y Fl(1)627 4052 y Fm(>)27 b Fo(0.)43 b(But)33 b(norm)f(resolv)m(en)m(t)i(con)m(v)m(ergence)h (implies)30 b(sp)s(ectral)i(con)m(v)m(ergence.\))340 4168 y(F)-8 b(or)28 b(smo)s(othly)g(b)s(ounded)i Fm(M)40 b Fo(a)29 b(m)m(uc)m(h)h(more)f(detailed)f(picture)h(of)g(the)h(pro)s (cess)h(of)d(sp)s(ec-)240 4284 y(tral)j(concen)m(tration)i(is)f(giv)m (en)h(in)f([F].)240 4450 y(4.3)p Fe(.)h(R)-5 b(emark.)39 b Fo(More)32 b(general)e(results)h(can)h(b)s(e)f(found)g(in)f([HL].)h (First)f(of)h(all,)e(without)i(ad-)240 4566 y(ditional)25 b(e\013ort)i(one)h(can)g(handle)f(elliptic)e(op)s(erators)i(of)g(the)h (t)m(yp)s(e)g Fn(\000r)12 b(\001)g Fo(\()p Fx(a)p Fo(\()p Fm(x)p Fo(\))g(+)g Fm(\025)p Fx(b)p Fo(\()p Fm(x)p Fo(\)\))p Fn(r)p Fo(,)240 4682 y(under)50 b(suitable)f(assumptions)g(on)h(the)f (\(symmetric\))g(matrices)g Fx(a)p Fo(\()p Fm(x)p Fo(\))h(and)f Fx(b)p Fo(\()p Fm(x)p Fo(\),)55 b(lik)m(e)240 4799 y Fm(a)291 4814 y Fj(ij)352 4799 y Fm(;)17 b(b)437 4814 y Fj(ij)540 4799 y Fn(2)44 b Fm(L)716 4814 y Ff(1)791 4799 y Fo(\()p Fk(R)895 4763 y Fj(m)967 4799 y Fo(\),)g Fx(a)p Fo(\()p Fm(x)p Fo(\))e(uniformly)e(p)s(ositiv)m(e)h(de\014nite,) j Fx(b)p Fo(\()p Fm(x)p Fo(\))f(=)g Fx(0)f Fo(on)f Fm(M)10 b Fo(,)45 b(and)c(p)s(os-)240 4915 y(itiv)m(e)32 b(de\014nite)h(for)f Fm(x)c Fn(2)g Fo(\012.)340 5031 y(Second,)d([HL])d(also)e(giv)m(es)j (criteria)d(for)h(the)h(existence)h(of)f(higher)f(gaps:)38 b(if)21 b(the)h(eigenspace)240 5147 y(of)35 b(the)g Fm(k)s Fo(-th)g(Diric)m(hlet)e(eigen)m(v)-5 b(alue)35 b Fm(\016)1655 5162 y Fj(k)1733 5147 y Fo(of)g Fn(\000)p Fo(\001)2005 5162 y Fj(M)7 b Ff(\\)p Fj(Q)2222 5147 y Fo(con)m(tains)35 b(an)g(eigenfunction)g Fm(d)3398 5162 y Fj(k)3475 5147 y Fo(s.th.)p eop %%Page: 7 7 7 6 bop 120 66 a Fh(SPECTRAL)33 b(GAPS)h(F)n(OR)e(PERIODIC)h(ELLIPTIC)h (OPERA)-6 b(TORS)31 b(WITH)i(HIGH)g(CONTRAST:)g(AN)f(O)n(VER)-9 b(VIEW)-38 b(7)120 185 y Fd(R)203 266 y Fm(d)254 281 y Fj(k)338 266 y Fn(6)p Fo(=)42 b(0)f(then)g Fm(\033)t Fo(\()p Fm(T)930 281 y Fj(\025)976 266 y Fo(\))g(will)d(ha)m(v)m(e)k(a) f(gap)g(for)f Fm(\025)h Fo(large)f(that)g(con)m(v)m(erges)k(to)c(\()p Fm(\016)3138 281 y Fj(k)3181 266 y Fm(;)17 b(\027)3273 281 y Fj(k)3315 266 y Fo(\),)43 b(for)120 382 y(some)32 b Fm(\027)412 397 y Fj(k)483 382 y Fm(>)27 b(\016)629 397 y Fj(k)672 382 y Fo(.)120 551 y(4.4)p Fe(.)i(R)-5 b(emark.)36 b Fo(There)28 b(are)f(natural)e(and)i(simple)e(situations)h (where)i(one)f(is)f(led)g(to)h(consider)120 667 y(p)s(erio)s(d)32 b(cells)h(that)g(are)g(more)g(general)g(than)g(\(0)p Fm(;)17 b Fo(1])2029 631 y Fj(m)2095 667 y Fo(.)46 b(In)34 b(fact,)f(all)e(of)i(the)h(ab)s(o)m(v)m(e)g(w)m(orks)h(if)120 783 y(\000)29 b(is)f(a)g(discrete)h(sublattice)f(of)g Fk(R)1349 747 y Fj(m)1451 783 y Fo(that)g(spans)i Fk(R)1987 747 y Fj(m)2088 783 y Fo(and)f(if)e Fm(Q)i Fo(is)f(a)g(con)m(tractible) g(subset)i(of)120 899 y Fk(R)186 863 y Fj(m)294 899 y Fo(with)35 b(piecewise)h(smo)s(oth)f(b)s(oundary)h(tessalating)e Fk(R)2287 863 y Fj(m)2395 899 y Fo(via)h(the)h(translations)e Fm(\015)k Fn(2)33 b Fo(\000.)120 1016 y(Here)26 b(w)m(e)h(think,)g(in)e (particular,)g(of)g(the)h(class)g(of)f(results)h(that)f(dep)s(end)i(on) f(the)g(assumption)120 1132 y(that)32 b(\012)h(con)m(tains)g(the)g(b)s (oundary)g(of)f(the)h(p)s(erio)s(d)f(cell.)120 1301 y(4.5)p Fe(.)37 b(R)-5 b(emark.)42 b Fo(In)35 b Fk(R)901 1264 y Fj(m)974 1301 y Fo(,)h Fm(m)c Fn(\025)h Fo(3,)i(it)g(is)f(p)s (ossible)h(to)g(ha)m(v)m(e)h(b)s(oth)f(\012)h(and)f Fm(M)46 b Fo(connected.)53 b(A)120 1417 y Fe(sp)-5 b(onge)43 b Fo(is)36 b(a)h(t)m(ypical)f(example)g(\(alb)s(eit,)g(in)g(realit)m(y) -8 b(,)36 b(not)h(p)s(erio)s(dic)e(but)i(random!\).)55 b(Here)120 1533 y(our)31 b(analysis)g(w)m(ould)g(yield)g(that,)h(for)e (high)h(con)m(trast,)h(the)g(i.d.s.)g(is)f(appro)m(ximately)f(giv)m(en) 120 1649 y(b)m(y)f(the)g(i.d.s.)g(of)e Fm(T)808 1664 y Ff(1)883 1649 y Fo(.)42 b(The)30 b(latter)d(can)i(b)s(e)f(further)h (analyzed)g(b)m(y)g(Flo)s(quet-decomp)s(osition)120 1766 y(if)44 b(w)m(e)i(assume)g(that)e(\012)i(in)m(tersects)g(eac)m(h)g (face)g(of)e Fm(Q)h Fo(\(think)g(of)g(the)g(case)h(where)h Fm(M)55 b Fo(is)120 1882 y(connected)34 b(from)e(one)g(cell)g(to)g(the) h(neigh)m(b)s(oring)e(cells)h(through)h(thin)f(\014lamen)m(ts)g (only\).)120 2051 y(4.6)p Fe(.)48 b(R)-5 b(emark.)49 b Fo(It)e(is)g(natural)g(to)g(ask)h(what)g(happ)s(ens)h(for)e(a)g Fe(de)-5 b(cr)g(e)g(asing)55 b Fo(sequence)50 b(of)120 2167 y(op)s(erators)32 b(lik)m(e)1006 2326 y Fm(S)1066 2341 y Fj(\025)1139 2326 y Fo(=)c Fn(\000r)23 b(\001)1644 2259 y Fo(1)p 1485 2303 366 4 v 1485 2394 a(1)f(+)g Fm(\025\037)1772 2409 y Fj(M)1861 2326 y Fn(r)p Fm(;)212 b Fo(0)28 b Fn(\024)g Fm(\025)f Fn(")h(1)p Fm(:)685 b Fo(\(4)p Fm(:)p Fo(2\))120 2523 y(The)35 b(asso)s(ciated)f(quadratic)g(forms)g Fg(s)1541 2538 y Fj(\025)1620 2523 y Fo(are)h(de\014ned)g(b)m(y)h Fg(s)2304 2538 y Fj(\025)2349 2523 y Fo([)p Fm(u)p Fo(])30 b(=)2596 2442 y Fd(R)2662 2523 y Fo(\(1)23 b(+)g Fm(\025\037)2989 2538 y Fj(M)3068 2523 y Fo(\))3106 2487 y Ff(\000)p Fl(1)3201 2523 y Fn(jr)p Fm(u)p Fn(j)3396 2487 y Fl(2)3434 2523 y Fm(dx)120 2640 y Fo(on)29 b Fn(H)337 2604 y Fl(1)376 2640 y Fo(\()p Fk(R)480 2604 y Fj(m)553 2640 y Fo(\).)42 b(Eviden)m(tly)-8 b(,)30 b(the)f(largest)g(form)e(that)i(is)f(smaller)f (than)i Fg(s)2724 2655 y Fj(\025)2770 2640 y Fo(,)g(for)g(all)d Fm(\025)i Fn(\025)g Fo(0,)i(has)120 2757 y(domain)i Fn(H)553 2720 y Fl(1)592 2757 y Fo(\()p Fk(R)696 2720 y Fj(m)769 2757 y Fo(\))h(and)h(tak)m(es)g(the)g(v)-5 b(alues)1741 2676 y Fd(R)1788 2791 y Fl(\012)1860 2757 y Fn(jr)p Fm(u)p Fn(j)2055 2720 y Fl(2)2093 2757 y Fm(dx)p Fo(.)47 b(Let)33 b(us)h(call)e(this)h(form)f Fg(s)3220 2772 y Ff(1)3295 2757 y Fo(.)46 b(It)33 b(is)120 2873 y(easy)27 b(to)e(see)i(that)e Fg(s)839 2888 y Ff(1)940 2873 y Fo(is)g(closable.)40 b(Let)26 b Fm(S)1658 2888 y Ff(1)1759 2873 y Fo(denote)g(the)g (self-adjoin)m(t)e(op)s(erator)h(asso)s(ciated)120 2989 y(with)33 b(the)g(closure)h(of)e Fg(s)993 3004 y Ff(1)1068 2989 y Fo(.)45 b(It)33 b(follo)m(ws)f(from)g(Simon's)g(theorem)h ([RS-I;)g(Thm.)g(S.)g(16])g(that)120 3105 y(the)g(op)s(erators)f Fm(S)779 3120 y Fj(\025)857 3105 y Fo(con)m(v)m(erge)j(to)d Fm(S)1435 3120 y Ff(1)1542 3105 y Fo(in)g(strong)g(resolv)m(en)m(t)i (sense.)220 3222 y(Let)42 b(us)h(assume,)i(for)d(the)h(momen)m(t,)h (that)e(an)m(y)h Fm(u)h Fn(2)h(H)2379 3185 y Fl(1)2418 3222 y Fo(\(\012\))e(can)f(b)s(e)h(extended)h(to)e(a)120 3338 y(function)29 b(in)g Fn(H)695 3302 y Fl(1)735 3338 y Fo(\()p Fk(R)839 3302 y Fj(m)911 3338 y Fo(\))h(and)g(that)g Fm(C)1452 3302 y Ff(1)1445 3362 y Fj(c)1526 3338 y Fo(\()p Fm(M)1668 3302 y Ff(\016)1708 3338 y Fo(\))g(is)g(dense)h(in)e Fm(L)2312 3353 y Fl(2)2352 3338 y Fo(\()p Fm(M)10 b Fo(\).)43 b(Then)31 b(it)e(is)g(not)h(di\016cult)120 3454 y(to)k(sho)m(w)i(that)e (the)h(closure)f(of)g(the)h(form)e Fg(s)1754 3469 y Ff(1)1863 3454 y Fo(has)i(domain)e Fn(H)2473 3418 y Fl(1)2512 3454 y Fo(\(\012\))24 b Fn(\010)g Fm(L)2849 3469 y Fl(2)2889 3454 y Fo(\()p Fm(M)10 b Fo(\),)35 b(and)g(tak)m(es)120 3570 y(the)29 b(v)-5 b(alues)28 b Fg(s)614 3585 y Ff(1)688 3570 y Fo([)p Fm(u)p Fo(])g(=)929 3490 y Fd(R)976 3605 y Fl(\012)1048 3570 y Fn(jr)p Fm(u)p Fo(\()p Fm(x)p Fo(\))p Fn(j)1374 3534 y Fl(2)1413 3570 y Fm(dx)p Fo(.)42 b(Under)29 b(these)g(assumptions)f Fm(S)2734 3585 y Ff(1)2837 3570 y Fo(is)g(the)g(direct)g(sum)120 3688 y(of)k(the)h(Neumann)g(Laplacian) d(on)j(\012)g(and)g(the)g(zero)f(op)s(erator)g(on)h Fm(L)2683 3703 y Fl(2)2723 3688 y Fo(\()p Fm(M)10 b Fo(\).)44 b(In)33 b(particular,)120 3804 y(the)40 b(i.d.s.)f(of)g Fm(S)713 3819 y Ff(1)827 3804 y Fo(w)m(ould)g(b)s(e)g(in\014nite)g(on)g(the)g(p) s(ositiv)m(e)g(real)f(line,)i(and)f(it)g(seems)h(rather)120 3920 y(hop)s(eless)d(to)e(extract)i(an)m(y)g(useful)f(sp)s(ectral)g (information)c(on)k Fm(S)2537 3935 y Fj(\025)2583 3920 y Fo(.)53 b(\(Of)36 b(course,)i(the)e(r^)-49 b(oles)120 4037 y(of)32 b Fm(M)43 b Fo(and)33 b(\012)g(are)g(in)m(terc)m (hangeable)f(here)i(if)d(the)i(b)s(oundary)g(of)f Fm(M)43 b Fo(is)33 b(smo)s(oth)e(enough.\))120 4205 y(4.7)p Fe(.)40 b(R)-5 b(emark.)44 b Fo(In)39 b(the)h(theory)f(of)f(photonic)h (crystals,)i(one)e(is)f(led)h(to)f(study)i(the)f(\\high-)120 4322 y(con)m(trast/thin-w)m(all)32 b(limit")e(of)j(the)h(acoustic)g(op) s(erator)f(in)g Fk(R)2439 4285 y Fl(2)2485 4322 y Fo(.)47 b(In)34 b(t)m(ypical)f(applications,)120 4438 y(the)39 b(optically)e(dense)j(medium)d(o)s(ccupies)i(a)f(connected,)k(p)s(erio) s(dic)c(set)h Fm(M)49 b Fo(consisting)38 b(of)120 4554 y(thin)43 b(w)m(alls,)i(cf.)f([FK-1,2].)75 b(In)44 b(our)f(notation,)i (w)m(e)f(w)m(ould)g(then)g(consider)g(the)f(limit)d(of)120 4670 y Fm(T)177 4685 y Fj(\025)250 4670 y Fo(=)28 b Fn(\000r)p Fo(\(1)16 b(+)g Fm(\025\037)827 4685 y Fl(\012)883 4670 y Fo(\))p Fn(r)p Fo(,)30 b(as)g Fm(\025)e Fn(!)f(1)p Fo(.)42 b(Note)30 b(that)g(no)m(w)g(\012)g(is)f Fe(not)40 b Fo(connected.)k(Our)30 b(results)120 4786 y(yield)e(a)h(strong)g (resolv)m(en)m(t)h(limit)c(and)j(a)g(Flo)s(quet-decomp)s(osition)d (analogous)i(to)h(what)g(w)m(e)120 4915 y(ha)m(v)m(e)35 b(discussed)g(ab)s(o)m(v)m(e.)47 b(The)34 b(limiting)29 b(op)s(erators)k Fm(T)2156 4864 y Fl(\()p Fj(\022)r Fl(\))2142 4925 y Ff(1)2283 4915 y Fo(in)f(the)i(\014b)s(ers)g(no)m(w)g(ha)m(v)m (e)h(a)e(form)120 5031 y(domain)j(consisting)g(of)h(functions)g(that)g (are)g(constan)m(t)i(on)e(\012)25 b Fn(\\)h Fm(Q)p Fo(,)39 b(where)f(the)g(constan)m(t)120 5147 y(v)-5 b(alue)32 b(will)e(b)s(e)i(non-zero,)h(in)e(general.)43 b(Therefore,)34 b(it)d(is)h(m)m(uc)m(h)h(harder)g(to)f(determine)g(the)p eop %%Page: 8 8 8 7 bop 240 66 a Fh(8)1001 b(RAINER)23 b(HEMPEL)k(AND)d(OLAF)h(POST)240 283 y Fo(sp)s(ectrum)32 b(of)f(the)h Fm(T)1013 232 y Fl(\()p Fj(\022)r Fl(\))999 294 y Ff(1)1137 283 y Fo(than)g(in)f(the)g (case)i(of)e(Eqn.)h(\(4.1\).)43 b(If)31 b(\012)20 b Fn(\\)g Fm(Q)32 b Fo(exhausts)h(\\most)e(of)7 b(")240 400 y Fm(Q)33 b Fo(\(thin)f(w)m(all)f(asymptotics\),)h(then)h(the)g(analysis)f(can)h (pro)s(ceed)h(again)d([FK-1,2],)g([KK].)240 586 y(4.8)p Fe(.)36 b(R)-5 b(emark.)42 b Fo(Random)33 b(Media)h(and)h(the)f(con)m (v)m(ergence)j(of)d(the)h(i.d.s.)g(for)e(the)i(op)s(erators)240 702 y(of)30 b(Eqn.)i(\(4.1\))e(with)g(a)g(random)g(geometry)g(ha)m(v)m (e)i(b)s(een)f(studied)g(b)m(y)h(K.)e(Lienau)g([L].)h(Since)240 819 y(w)m(e)38 b(need)f(a)g(connected)h(\012)f(for)g(our)f(approac)m(h) h(to)g(w)m(ork)g(w)m(ell,)g(the)g(ergo)s(dic)f(pro)s(cess)i(that)240 935 y(generates)c(the)f(random)g(set)g Fm(M)44 b Fo(has)33 b(to)g(resp)s(ect)h(this)f(prop)s(ert)m(y)-8 b(.)45 b(Here)34 b(v)-5 b(arious)32 b(metho)s(ds)240 1051 y(from)f(sto)s(c)m(hastic)i (geometry)g(\([SKM]\))g(can)f(b)s(e)h(emplo)m(y)m(ed.)921 1310 y(5.)48 b Fp(Periodic)38 b(Lapla)n(ce-Bel)-7 b(trami)37 b(opera)-7 b(tors)340 1485 y Fo(In)31 b(this)g(section)h(w)m(e)h(rep)s (ort)e(on)g(some)g(related)g(results)h(on)g(p)s(erio)s(dic)d (Laplace-Beltrami)240 1601 y(op)s(erators,)k(cf.)h([P-1,2].)45 b(Let)33 b Fn(M)g Fo(b)s(e)h(a)f(non-compact)f(Riemannian)f(manifold)g (of)h(dimen-)240 1717 y(sion)h Fm(d)c Fn(\025)g Fo(2.)46 b(W)-8 b(e)34 b(call)e Fn(M)h Fo(a)g Fe(p)-5 b(erio)g(dic)38 b Fo(or)33 b Fe(c)-5 b(overing)35 b(manifold)42 b Fo(i\013)33 b(\000)c(=)g Fk(Z)3003 1681 y Fj(m)3100 1717 y Fo(acts)34 b(prop)s(erly)240 1833 y(discon)m(tin)m(uously)40 b(and)h (isometrically)c(on)j Fn(M)p Fo(.)66 b(Under)41 b(these)h(conditions,)f (the)g(quotien)m(t)240 1949 y Fn(M)p Fm(=)p Fo(\000)32 b(is)g(again)f(a)h(Riemannian)f(manifold)e(whic)m(h)34 b(w)m(e)f(assume)g(to)f(b)s(e)h(compact.)340 2066 y(W)-8 b(e)41 b(call)e Fm(Q)i Fo(a)f Fe(p)-5 b(erio)g(d)42 b(c)-5 b(el)5 b(l)50 b Fo(or)41 b Fe(fundamental)g(domain)47 b Fo(of)40 b Fn(M)h Fo(i\013)e Fm(Q)i Fo(is)g(op)s(en,)i Fm(Q)d Fo(do)s(es)240 2182 y(not)g(in)m(tersect)g(an)m(y)g(other)g (translate)f Fm(\015)32 b Fo(+)27 b Fm(Q)40 b Fo(except)h(for)e Fm(\015)45 b Fo(=)40 b(0,)h(and)f(the)g(union)f(of)g(all)240 2298 y(translates)32 b(of)p 796 2218 78 4 v 32 w Fm(Q)h Fo(co)m(v)m(ers)h(the)f(whole)f(manifold)d Fn(M)p Fo(.)44 b(Note)32 b(that)g(w)m(e)i(assume)e(here)i(that)e Fm(Q)240 2414 y Fo(is)e(an)g(op)s(en)h(set.)44 b(One)30 b(can)h(apply)f(Flo)s (quet)g(decomp)s(osition)e(in)i(the)h(same)f(w)m(a)m(y)i(as)f(b)s (efore;)240 2531 y(here,)i Fm(#)p Fo(-p)s(erio)s(dicit)m(y)c(means)j (that)f Fm(u)p Fo(\()p Fm(\015)25 b Fo(+)19 b Fm(x)p Fo(\))28 b(=)g Fm(e)2097 2494 y Fj(i#)p Ff(\001)p Fj(\015)2227 2531 y Fm(u)p Fo(\()p Fm(x)p Fo(\))j(for)g(all)f Fm(x)e Fn(2)p 2905 2451 V 28 w Fm(Q)k Fo(and)f Fm(\015)i Fn(2)28 b Fo(\000)j(suc)m(h)240 2652 y(that)h Fm(\015)c Fo(+)22 b Fm(x)28 b Fn(2)p 805 2572 V 28 w Fm(Q)k Fo(\(plus)h(an)f(analogous)g (condition)f(for)h(the)h(\014rst)g(deriv)-5 b(ativ)m(es\).)340 2768 y(Denote)31 b(b)m(y)i(\()p Fm(g)893 2783 y Fj(ij)953 2768 y Fo(\))f(the)g(metric)f(tensor)i(of)e Fn(M)g Fo(in)g(some)h(c)m (hart.)44 b(W)-8 b(e)32 b(consider)g(conformal)240 2884 y(deformations)f(of)h(the)h(giv)m(en)g(p)s(erio)s(dic)e(metric,)g (i.e.,)i(w)m(e)g(set)1460 3063 y Fm(g)1507 3078 y Fj(ij)1567 3063 y Fo(\()p Fm(x)p Fo(;)17 b Fm(\025)p Fo(\))28 b(=)g Fm(\045)1981 3022 y Fl(2)1981 3087 y Fj(\025)2026 3063 y Fo(\()p Fm(x)p Fo(\))17 b Fm(g)2221 3078 y Fj(ij)2281 3063 y Fo(\()p Fm(x)p Fo(\))p Fm(:)240 3241 y Fo(Here,)34 b Fm(\045)548 3256 y Fj(\025)626 3241 y Fo(is)e(a)h(strictly)f(p)s (ositiv)m(e,)g(p)s(erio)s(dic,)f(smo)s(oth)h(function)g(con)m(v)m (erging)i(p)s(oin)m(t)m(wise)e(to)240 3357 y(the)39 b(indicator)d (function)h(of)h(a)g(closed)g(p)s(erio)s(dic)f(set)h Fm(M)49 b Fo(as)38 b Fm(\025)f Fn(!)g(1)p Fo(.)60 b(In)38 b(particular,)g(w)m(e)240 3474 y(assume)33 b(that)437 3621 y(\(1\))f Fm(M)688 3636 y Fl(0)756 3621 y Fo(:=)27 b Fm(M)33 b Fn(\\)23 b Fm(Q)33 b Fo(is)f(compactly)g(con)m(tained)g(in) g Fm(Q)h Fo(with)f(smo)s(oth)g Fm(@)5 b(M)10 b Fo(;)437 3759 y(\(2\))32 b Fm(\045)644 3774 y Fj(\025)690 3759 y Fo(\()p Fm(x)p Fo(\))c(=)f(1)33 b(for)f(all)e Fm(x)e Fn(2)h Fm(M)10 b Fo(;)437 3896 y(\(3\))32 b Fm(\045)644 3911 y Fj(\025)690 3896 y Fo(\()p Fm(x)p Fo(\))c(=)965 3857 y Fl(1)p 962 3873 42 4 v 962 3931 a Fj(\025)1014 3896 y Fo(,)k(for)g(all)f Fm(x)d Fn(2)g Fo(\012)33 b(with)f(dist\()p Fm(x;)17 b(M)10 b Fo(\))28 b Fn(\025)g Fm(\025)2487 3860 y Ff(\000)p Fj(m)2609 3896 y Fo(.)340 4044 y(T)-8 b(o)28 b(simplify)e(the)j(notation,)f(w)m(e)h(restrict)g(ourselv)m(es)h(to)e (the)h(conformally)d(\015at)i(case,)i(i.e.,)240 4160 y(w)m(e)35 b(assume)g(that)g Fn(M)30 b Fo(=)h Fk(R)1261 4124 y Fj(m)1368 4160 y Fo(with)j(the)g(Euclidean)g(metric)g(tensor)h Fm(g)2862 4175 y Fj(ij)2922 4160 y Fo(\()p Fm(x)p Fo(\))c(=)g Fm(\016)3234 4175 y Fj(ij)3295 4160 y Fo(,)k(de\014ned)240 4276 y(on)k Fk(R)448 4240 y Fj(m)521 4276 y Fo(.)63 b(W)-8 b(e)40 b(assume)g(that)f(the)h(b)s(oundary)f Fm(@)5 b(M)51 b Fo(has)40 b(a)f(tubular)g(neigh)m(b)s(orho)s(o)s(d)f(where)240 4392 y(w)m(e)h(can)g(in)m(tro)s(duce)f(lo)s(cal)e(co)s(ordinates)h(\()p Fm(r)m(;)17 b(y)t Fo(\))37 b(giv)m(en)h(b)m(y)h(the)g(distance)f Fm(r)j Fo(from)c Fm(@)5 b(M)50 b Fo(and)240 4509 y Fm(y)d Fn(2)d Fm(@)5 b(M)10 b Fo(.)73 b(F)-8 b(or)41 b(tec)m(hnical)h(reasons) g(w)m(e)h(need)g(the)g(additional)c(assumption)i(that)h(these)240 4625 y(co)s(ordinates)c(can)g(b)s(e)g(extended)j(to)c(\012)27 b Fn(\\)f Fm(Q)p Fo(;)41 b(e.g.,)f(think)e(of)g(a)g(ball)e Fm(M)2934 4640 y Fl(0)3012 4625 y Fo(cen)m(tered)k(in)e(the)240 4741 y(p)s(erio)s(d)30 b(cell)g Fm(Q)e Fo(=)g(\(0)p Fm(;)17 b Fo(1\))1141 4705 y Fj(m)1206 4741 y Fo(.)43 b(The)32 b(Laplace-Beltrami-op)s(erator)27 b Fm(B)2705 4756 y Fj(\025)2782 4741 y Fo(corresp)s(onding)k(to)g(the)240 4857 y(deformed)h(metric)g(is)g(giv)m(en)h(b)m(y)1297 5079 y Fm(B)1371 5094 y Fj(\025)1444 5079 y Fo(=)28 b Fn(\000)1699 5011 y Fo(1)p 1635 5056 176 4 v 1635 5085 a Fn(p)p 1718 5085 93 4 v 62 x Fm(g)1765 5162 y Fj(\025)1820 5079 y Fm(@)1871 5094 y Fj(j)1925 4998 y Fd(\000)1971 5079 y Fm(g)2022 5038 y Fj(ij)2082 5079 y Fo(\()p Fn(\001)p Fo(;)17 b Fm(\025)p Fo(\))2287 5012 y Fn(p)p 2369 5012 V 2369 5079 a Fm(g)2416 5094 y Fj(\025)2461 4998 y Fd(\001)2523 5079 y Fm(@)2574 5094 y Fj(i)p eop %%Page: 9 9 9 8 bop 120 66 a Fh(SPECTRAL)33 b(GAPS)h(F)n(OR)e(PERIODIC)h(ELLIPTIC)h (OPERA)-6 b(TORS)31 b(WITH)i(HIGH)g(CONTRAST:)g(AN)f(O)n(VER)-9 b(VIEW)-38 b(9)120 266 y Fo(\(using)41 b(the)i(summation)c(con)m(v)m (en)m(tion\),)45 b(where)e Fm(g)2015 281 y Fj(\025)2104 266 y Fo(=)g(det)17 b Fm(g)2422 281 y Fj(ij)2482 266 y Fo(\()p Fn(\001)p Fm(;)g(\025)p Fo(\))43 b(=)g Fm(\032)2899 230 y Fl(2)p Fj(m)2899 292 y(\025)3043 266 y Fo(denotes)g(the)120 382 y(square)h(of)e(the)h(v)m(olume)f(densit)m(y)i(of)e(the)h(deformed) g(metric)e(tensor.)75 b(This)42 b(op)s(erator)g(is)120 498 y(de\014ned)34 b(as)f(a)f(non-negativ)m(e,)h(self-adjoin)m(t)d(op)s (erator)i(via)g(the)h(quadratic)g(form)244 703 y Fg(b)295 718 y Fj(\025)341 703 y Fo([)p Fm(u)p Fo(])27 b(:=)609 567 y Fd(Z)725 703 y Fm(g)776 662 y Fj(ij)836 703 y Fo(\()p Fn(\001)p Fm(;)17 b(\025)p Fo(\))p Fm(@)1092 718 y Fj(i)1120 703 y Fm(u)g(@)1244 718 y Fj(j)p 1280 648 56 4 v 1280 703 a Fm(u)1335 636 y Fn(p)p 1418 636 93 4 v 67 x Fm(g)1465 718 y Fj(\025)1527 703 y Fm(dx)28 b Fo(=)1764 567 y Fd(Z)1880 703 y Fn(jr)p Fm(u)p Fn(j)2075 662 y Fl(2)2130 703 y Fm(\045)2180 662 y Fj(m)p Ff(\000)p Fl(2)2180 731 y Fj(\025)2354 703 y Fm(dx;)212 b(u)27 b Fn(2)h(H)2961 662 y Fl(1)3001 703 y Fo(\()p Fk(R)3105 662 y Fj(m)3177 703 y Fo(\))124 b(\(5)p Fm(:)p Fo(1\))120 919 y(in)29 b(the)g(Hilb)s(ert)f(space)j Fn(H)1070 934 y Fj(\025)1143 919 y Fo(:=)c Fm(L)1339 934 y Fl(2)1379 919 y Fo(\()p Fk(R)1483 883 y Fj(m)1556 919 y Fo(\))i(with)g(inner)g(pro)s(duct)2451 839 y Fd(R)2534 919 y Fm(u)p 2590 864 51 4 v(v)2640 857 y Fn(p)p 2723 857 93 4 v 62 x Fm(g)2770 934 y Fj(\025)2815 919 y Fm(dx)p Fo(.)42 b(Note)30 b(that,)g(in)120 1035 y(general,)25 b(b)s(oth)e(the)h(Diric)m(hlet)e(in)m(tegral)g Fe(and)33 b Fo(the)24 b(v)m(olume)f(form)f(dep)s(end)j(on)e(the)h(parameter)120 1151 y Fm(\025)p Fo(.)67 b(The)41 b(fundamen)m(tal)f(di\013erence)h(to) f(the)h(case)g(of)f(div)m(ergence)i(t)m(yp)s(e)f(op)s(erators)f(lies)g (in)120 1268 y(the)35 b(dep)s(endence)j(of)c(the)i(inner)e(pro)s(duct)h (on)g Fm(\025)p Fo(.)51 b(In)35 b(particular,)f(for)g Fm(m)e Fn(\025)h Fo(3,)i(there)h(is)e(no)120 1384 y(monotonicit)m(y)-8 b(.)220 1500 y(Flo)s(quet-decomp)s(osition)38 b(implies)i(that)h(it)g (is)g(enough)i(to)e(study)i Fm(#)p Fo(-p)s(erio)s(dic)d(b.v.p.'s)120 1616 y(on)32 b Fm(Q)p Fo(.)43 b(F)-8 b(urthermore,)31 b(Diric)m(hlet-Neumann-brac)m(k)m(eting)f(yields)h(enclosures)i(of)e (bands)i(b)m(y)120 1745 y(Neumann)f(and)f(Diric)m(hlet)f(eigen)m(v)-5 b(alues)32 b Fm(E)1733 1694 y Fj(N)q(=D)1727 1772 y(k)1889 1745 y Fo(\()p Fm(\025)p Fo(\))f(\(ordered)i(b)m(y)f(min-max)d(and)j (rep)s(eated)120 1862 y(according)37 b(to)h(m)m(ultiplicities\).)55 b(Similarly)-8 b(,)35 b(denote)k(b)m(y)g Fm(E)2343 1826 y Fj(N)2337 1888 y(k)2448 1862 y Fo(the)f(Neumann)g(eigen)m(v)-5 b(alues)120 1978 y(of)32 b Fm(M)325 1993 y Fl(0)393 1978 y Fo(=)27 b Fm(M)33 b Fn(\\)23 b Fm(Q)p Fo(.)220 2095 y(Since)44 b(the)h(norm)f(of)g(the)g(Hilb)s(ert)f(space)j(dep)s(ends)g (on)e Fm(\025)p Fo(,)k(w)m(e)d(cannot)g(apply)f(Theo-)120 2211 y(rem)27 b(1.1.)42 b(Instead,)30 b(the)e(con)m(v)m(ergence)j(of)c (the)i(eigen)m(v)-5 b(alues)28 b(on)g Fm(Q)g Fo(to)f(the)i(Neumann)f (eigen-)120 2327 y(v)-5 b(alues)24 b(on)g Fm(M)623 2342 y Fl(0)687 2327 y Fo(is)g(obtained)g(directly)g(through)g(an)g (application)e(of)i(the)g(min-max-principle)120 2443 y(and)33 b(a)f(careful)g(comparison)f(of)h(the)h(corresp)s(onding)g (Raleigh)e(quotien)m(ts.)220 2560 y(In)38 b(particular,)g(w)m(e)h(sho)m (w)g(that)f Fm(E)1522 2523 y Fj(N)1516 2585 y(k)1627 2560 y Fo(is)g(\(appro)m(ximately\))e(a)i(lo)m(w)m(er)g(b)s(ound)g(for) g Fm(E)3340 2523 y Fj(N)3334 2585 y(k)3407 2560 y Fo(\()p Fm(\025)p Fo(\))120 2676 y(\(using)j(the)i(restriction)e Fm(u)p Fn(j)1160 2691 y Fj(M)1228 2700 y Fc(0)1307 2676 y Fo(as)h(transition)e(op)s(erator)i(from)e Fn(H)2617 2640 y Fl(1)2657 2676 y Fo(\()p Fm(Q)p Fo(\))i(to)f Fn(H)3065 2640 y Fl(1)3105 2676 y Fo(\()p Fm(M)3237 2691 y Fl(0)3276 2676 y Fo(\)\).)72 b(In)120 2792 y(the)33 b(same)f(w)m(a)m(y)-8 b(,)34 b(w)m(e)f(pro)m(v)m(e)h(that)e Fm(E)1444 2756 y Fj(N)1438 2818 y(k)1544 2792 y Fo(is)g(also)g(an)g(appro)m(ximate)g (upp)s(er)h(b)s(ound)f(for)g Fm(E)3343 2756 y Fj(D)3337 2818 y(k)3407 2792 y Fo(\()p Fm(\025)p Fo(\))120 2908 y(pro)m(vided)39 b Fm(m)f Fn(\025)h Fo(3)f(\(using)g(an)h(extension)g (op)s(erator)f(from)f Fn(H)2451 2872 y Fl(1)2491 2908 y Fo(\()p Fm(M)2623 2923 y Fl(0)2662 2908 y Fo(\))i(to)f Fn(H)2949 2872 y Fl(1)2948 2933 y(0)2989 2908 y Fo(\()p Fm(Q)p Fo(\),)i(the)f Fn(H)3468 2872 y Fl(1)3507 2908 y Fo(-)120 3025 y(closure)33 b(of)f Fm(C)634 2988 y Ff(1)627 3049 y Fj(c)708 3025 y Fo(\()p Fm(Q)p Fo(\),)h(as)g(transition)e(op)s (erator\).)43 b(Therefore,)33 b(w)m(e)h(ha)m(v)m(e)120 3199 y(5.1)p Fx(.)27 b(Theorem.)35 b Fo(\([P-1,2]\))27 b Fe(L)-5 b(et)27 b Fm(m)h Fn(\025)g Fo(3)p Fe(.)42 b(Then)26 b Fm(\033)t Fo(\()p Fm(B)2104 3214 y Fj(\025)2150 3199 y Fo(\))h Fe(c)-5 b(onver)g(ges)25 b(to)i(the)g(sp)-5 b(e)g(ctrum)27 b(of)g(the)120 3316 y(Neumann)35 b(L)-5 b(aplacian)33 b(on)i Fm(M)10 b Fe(,)35 b(as)g Fm(\025)27 b Fn(!)h(1)p Fe(.)44 b(In)34 b(p)-5 b(articular,)35 b(gaps)f(op)-5 b(en)34 b(up,)h(as)g Fm(\025)27 b Fn(!)h(1)p Fe(.)120 3490 y Fo(5.2)p Fe(.)38 b(R)-5 b(emark.)43 b Fo(The)38 b(case)g Fm(m)e Fo(=)f(2)h(is)h(sp)s(ecial.)56 b(Here,)39 b(only)d(the)h(inner)g(pro)s(duct)g(dep)s(ends)120 3607 y(on)k Fm(\025)p Fo(.)69 b(In)m(v)m(erting)42 b(the)g(r^)-49 b(ole)40 b(of)h(the)h(norm)e(and)h(the)h(quadratic)f(form)f(yields)g(a) h(limiting)120 3723 y(op)s(erators)h(in)f(eac)m(h)i(\014b)s(er)g(whic)m (h,)i(ho)m(w)m(ev)m(er,)h(dep)s(end)e(on)e Fm(\022)s Fo(.)72 b(In)42 b(the)h(sp)s(ecial)e(case)i(of)e(a)120 3839 y(2-dimensional)31 b(cylinder)j(one)g(can)h(nev)m(ertheless)h(pro) m(v)m(e)g(the)e(existence)i(of)e(sp)s(ectral)g(gaps)120 3955 y(b)m(y)f(direct)g(calculations)e(\(separation)g(of)i(v)-5 b(ariables\),)31 b(cf.)i([P-1,2].)120 4130 y(5.3)p Fe(.)42 b(R)-5 b(emark.)45 b Fo(In)d(con)m(trast)g(to)f(the)g(Euclidean)g (case,)j(w)m(e)f(ma)m(y)e(c)m(ho)s(ose)h(the)f(dimension)120 4246 y Fm(d)35 b Fo(of)g(the)h(manifold)c Fn(M)j Fo(indep)s(enden)m (tly)i(of)d(the)i(dimension)e(of)h(p)s(erio)s(dicit)m(y)f Fm(m)p Fo(:)49 b(think)35 b(of)120 4363 y(the)h(surface)g(of)f(a)h (cylinder)f(\()p Fm(d)d Fo(=)h(2)p Fm(;)17 b(m)32 b Fo(=)h(1\))i(or)g (the)h(surface)g(of)f(a)h(p)s(erio)s(dic)e(jungle)g(gym)120 4479 y(\()p Fm(d)27 b Fo(=)h(2)p Fm(;)17 b(m)27 b Fo(=)h(3\).)42 b(F)-8 b(urthermore,)29 b(one)g(can)g(allo)m(w)e(an)m(y)i(Ab)s(elian)f (\014nitely)g(generated)h(group)120 4595 y(\000)j(of)h(in\014nite)e (order,)i(i.e.,)f(pro)s(ducts)i(of)e Fk(Z)e Fo(and)i(cyclic)g(groups.) 120 4770 y(5.4)p Fe(.)25 b(R)-5 b(emark.)33 b Fo(In)24 b(some)e(sense,)27 b(the)c(case)g(of)f(Laplace-Beltrami)e(op)s(erators) i(is)g(the)h(opp)s(osite)120 4886 y(of)h(the)i(case)g(of)e(div)m (ergence)i(t)m(yp)s(e)g(op)s(erators:)40 b(while)24 b(the)h(div)m (ergence)h(t)m(yp)s(e)g(op)s(erator)e(needs)120 5002 y(the)44 b(factor)f(\(1)29 b(+)g Fm(\025\037)928 5017 y Fl(\012)983 5002 y Fo(\))44 b(to)f(b)s(e)g Fe(lar)-5 b(ge)51 b Fo(on)43 b(\012)h(to)f(pro)s(duce)h(sp)s(ectral)f(gaps,)j (the)e(Laplace-)120 5120 y(Beltrami)30 b(op)s(erator)i(needs)i(the)f (factor)f Fm(\045)1681 5078 y Fj(m)p Ff(\000)p Fl(2)1681 5147 y Fj(\025)1871 5120 y Fo(to)g(b)s(e)h Fe(smal)5 b(l)42 b Fo(on)32 b(\012.)p eop %%Page: 10 10 10 9 bop 240 66 a Fh(10)963 b(RAINER)23 b(HEMPEL)k(AND)d(OLAF)h(POST) 240 266 y Fo(5.5)p Fe(.)32 b(R)-5 b(emark.)39 b Fo(Da)m(vies)30 b(and)g(Harrell)e([DH])i(pro)m(v)m(ed)h(the)g(existence)g(of)f(at)f (least)h(one)g(gap)g(in)240 382 y(the)37 b(p)s(erio)s(dic)e (conformally)f(\015at)i(case)h(\(transforming)e(the)i(conformal)d (Laplacian)h(in)m(to)g(a)240 498 y(corresp)s(onding)h(Sc)m(hr\177)-49 b(odinger)36 b(op)s(erator\).)52 b(This)36 b(result)g(is)g(a)f(sp)s (ecial)g(case)i(of)e(our)h(result.)240 614 y(Using)24 b(similar)d(metho)s(ds)k(as)f([DH],)h(Green)f([G])h(obtained)f (examples)g(with)g(a)g(\014nite)h(n)m(um)m(b)s(er)240 731 y(of)e(gaps)h(in)f(the)h(2-dimensional)d(conformally)g(\015at)i (case.)42 b(Recen)m(tly)-8 b(,)26 b(Y)-8 b(oshitomi)22 b([Y])h(pro)m(v)m(ed)240 847 y(the)28 b(existence)i(of)e(sp)s(ectral)f (gaps)i(for)e(the)i(\(Diric)m(hlet\))c(Laplacian)h(on)i(p)s(erio)s (dically)d(curv)m(ed)240 963 y(quan)m(tum)44 b(w)m(a)m(v)m(e)h(guides.) 76 b(In)44 b([DH,)f(G,)g(Y],)h(the)g(existence)h(of)e(gaps)h(is)e (established)i(b)m(y)240 1079 y(analyzing)39 b(a)g(one-dimensional)e (problem.)64 b(Here,)43 b(in)c(con)m(trast,)j(w)m(e)f(directly)e(study) j(the)240 1196 y(m)m(ulti-dimensional)27 b(problem.)240 1374 y(5.6)p Fe(.)j(R)-5 b(emark.)37 b Fo(More)28 b(\\geometric")e (constructions)j(of)e(p)s(erio)s(dic)f(manifolds)g(with)h(sp)s(ectral) 240 1490 y(gaps)38 b(can)g(b)s(e)g(found)g(in)f([P-1,2],)i(think)e(for) g(example)h(of)f(an)h(in\014nite)e(n)m(um)m(b)s(er)i(of)g(copies)240 1606 y(of)45 b(a)g(compact)h(manifold,)g(joined)f(b)m(y)h(thin)f (cylinders.)83 b(If)45 b(these)i(cylinders)f(are)f(small)240 1722 y(or)c(thin)f(enough,)k(one)e(again)e(obtains)g(sp)s(ectral)h (gaps)g(of)g(the)h(corresp)s(onding)f(Laplace-)240 1839 y(Beltrami)d(op)s(erator.)65 b(These)41 b(results)g(are)f(closely)f (related)h(to)g(earlier)e(w)m(orks)k(of)d(Cha)m(v)m(el)240 1955 y(and)33 b(F)-8 b(eldman)31 b([CF])h(and)h(Ann)m(\023)-46 b(e)34 b([A].)1380 2172 y(6.)48 b Fp(Maxwell)38 b(opera)-7 b(tors)340 2346 y Fo(Another)33 b(imp)s(ortan)m(t)d(elliptic)g(op)s (erator)i(is)g(the)h Fe(Maxwel)5 b(l)34 b(op)-5 b(er)g(ator)43 b Fo(giv)m(en)33 b(as)1519 2553 y Fm(M)38 b Fo(=)28 b Fn(\000)p Fo(curl)2069 2485 y(1)p 2005 2530 177 4 v 2005 2621 a Fm(")p Fo(\()p Fm(x)p Fo(\))2192 2553 y(curl)o Fm(;)1078 b Fo(\(6)p Fm(:)p Fo(1\))240 2780 y(acting)35 b(on)i(soleno)-11 b(\177)-38 b(\020dal)34 b(v)m(ector)j(\014elds)g(in)f Fk(R)1864 2744 y Fl(2)1945 2780 y Fo(or)g(in)g Fk(R)2252 2744 y Fl(3)2297 2780 y Fo(,)i(where)f Fm(")p Fo(\()p Fm(x)p Fo(\))g(denotes)g(the)g(dielec-)240 2897 y(tricit)m(y)32 b(of)g(the)h(medium.)340 3013 y(The)f(existence)h(of)f(gaps)g(for)f(p)s (erio)s(dic)f(structures)j(is)e(of)h(fundamen)m(tal)e(imp)s(ortance)g (for)240 3129 y(photonic)h(crystals.)44 b(In)31 b(man)m(y)h(cases,)h (one)f(is)f(satis\014ed)h(to)f(\014nd)h(in)m(terv)-5 b(als)31 b(of)g(lo)m(w)g(sp)s(ectral)240 3245 y(densit)m(y)f (\(quasi-gaps\).)42 b(This)29 b(class)g(of)f(problems)g(has)h(b)s(een)h (studied)f(in)f(depth)i(in)e(a)g(series)240 3362 y(of)g(pap)s(ers)h(b)m (y)g(A.)g(Figotin,)e(P)-8 b(.)28 b(Kuc)m(hmen)m(t)i(and)e(others)h (\(cf.,)h(e.g.,)f([FK-1,2],)g([KK]\))f(where)240 3478 y(a)k(complete)g(analysis)g(for)g(high-con)m(trast)g(and/or)g(thin-w)m (all)e(asymptotics)j(is)f(giv)m(en.)1660 3694 y Fp(References)240 3852 y Fw([A])207 b(C.)25 b(Ann)n(\023)-39 b(e,)26 b Fi(Sp)l(e)l(ctr)l(e)h(du)h(L)l(aplacien)i(et)25 b(\023)-40 b(ecr)l(asement)28 b(d'anses)p Fw(,)f(Ann.)f(Sci.)2874 3831 y(\023)2866 3852 y(Ec.)f(Norm.)g(Sup)r(er.,)i(IV.)555 3952 y(S)n(\023)-39 b(er.)26 b Fb(20)h Fw(\(1987\),)g(271{280.)240 4051 y([AHS])99 b(J.)23 b(Avron,)h(I.)g(Herbst)f(and)h(B.)f(Simon,)i Fi(Schr\177)-42 b(odinger)27 b(op)l(er)l(ators)g(with)g(singular)f (magnetic)h(\014elds,)555 4151 y(I.)j(Gener)l(al)g(Inter)l(actions)p Fw(,)e(Duk)n(e)g(Math.)f(J)h Fb(45)f Fw(\(1978\),)f(847{883.)240 4251 y([B-1])140 b(A.)31 b(Besc)n(h,)f Fi(Eigenwerte)j(in)g(Sp)l(ektr)l (al)t(l)q(\177)-43 b(ucken)33 b(des)g(Pauli-Op)l(er)l(ators)g(im)g Fs(R)2948 4221 y Ft(2)2991 4251 y Fw(,)e(Dissertation,)g(TU)555 4350 y(Braunsc)n(h)n(w)n(eig,)25 b(2000.)240 4450 y([B-2])140 b(A.)24 b(Besc)n(h,)g Fi(Eigenvalues)k(in)f(sp)l(e)l(ctr)l(al)g(gaps)g (of)h(the)e(two-dimensional)j(Pauli)e(op)l(er)l(ator)p Fw(,)f(J.)e(Math.)555 4550 y(Ph)n(ys)i(\(to)i(app)r(ear\).)240 4649 y([CF])155 b(I.)40 b(Cha)n(v)n(el)e(and)h(E.)h(A.)g(F)-7 b(eldman,)43 b Fi(Sp)l(e)l(ctr)l(a)d(of)i(manifolds)h(with)e(smal)t(l)h (hand)t(les)p Fw(,)i(Commen)n(t.)555 4749 y(Math.)28 b(Helv.)f Fb(56)g Fw(\(1981\),)g(83{102.)240 4848 y([CFKS])44 b(H.)23 b(Cycon,)f(R.)h(F)-7 b(ro)r(ese,)23 b(W.)f(Kirsc)n(h)f(and)i (B.)f(Simon,)h Fi(Schr\177)-42 b(odinger)27 b(op)l(er)l(ators)p Fw(,)e(Springer-V)-7 b(erlag,)555 4948 y(New)28 b(Y)-7 b(ork)26 b(\(1987\))240 5048 y([DH])144 b(E.)26 b(B.)g(Da)n(vies)f(and) h(E.)g(M.)h(I)r(I)f(Harrell,)g Fi(Conformal)t(ly)31 b(\015at)d(R)n (iemannian)h(metrics,)g(Schr\177)-42 b(odinger)555 5147 y(op)l(er)l(ators,)31 b(and)f(semiclassic)l(al)i(appr)l(oximation)p Fw(,)e(J.)d(Di\013.)i(Equations)d Fb(66)h Fw(\(1987\),)g(165{188.)p eop %%Page: 11 11 11 10 bop 120 66 a Fh(SPECTRAL)33 b(GAPS)h(F)n(OR)e(PERIODIC)h (ELLIPTIC)h(OPERA)-6 b(TORS)31 b(WITH)i(HIGH)g(CONTRAST:)g(AN)f(O)n (VER)-9 b(VIEW)-78 b(11)120 266 y Fw([DIM])100 b(Sh.)28 b(Doi,)g(A.)g(Iw)n(atsuk)-5 b(a,)28 b(and)f(T.)h(Mine,)h Fi(The)i(uniqueness)e(of)i(the)f(inte)l(gr)l(ate)l(d)g(density)g(of)h (states)435 365 y(for)f(the)g(Schr\177)-42 b(odinger)32 b(op)l(er)l(ators)e(with)h(magnetic)f(\014elds)p Fw(,)e(Math.)g(Z.)f Fb(237)g Fw(\(2001\),)g(335{371.)120 465 y([FK-1])80 b(A.)22 b(Figotin)g(and)g(P)-7 b(.)21 b(Kuc)n(hmen)n(t,)i Fi(Band-Gap)k(Structur)l(e)22 b(of)k(Sp)l(e)l(ctr)l(a)e(of)h(Perio)l (dic)i(Diele)l(ctric)e(and)435 565 y(A)l(c)l(oustic)h(Me)l(dia.)i Fw(I.)f Fi(Sc)l(alar)g(mo)l(del)p Fw(,)e(68{888;)e(I)r(I.)h Fi(Two-dimensional)29 b(photonic)f(crystals)p Fw(,)e(SIAM)435 664 y(J.)h(Appl.)i(Math.)e Fb(56)g Fw(\(1996\),)g(1561{1620.)120 764 y([FK-2])80 b(A.)37 b(Figotin)g(and)g(P)-7 b(.)37 b(Kuc)n(hmen)n(t,)i Fi(Sp)l(e)l(ctr)l(al)g(pr)l(op)l(erties)h(of)f (classic)l(al)h(waves)g(in)e(high-c)l(ontr)l(ast)435 863 y(p)l(erio)l(dic)32 b(me)l(dia)p Fw(,)d(SIAM)f(J.)f(Appl.)h(Math.)g Fb(58)f Fw(\(1998\),)g(683{702.)120 963 y([F])215 b(L.)30 b(F)-7 b(riedlander,)31 b Fi(On)h(the)h(density)g(of)g(states)f(of)i(p) l(erio)l(dic)h(me)l(dia)e(in)g(the)g(lar)l(ge)g(c)l(oupling)g(limit)p Fw(,)435 1063 y(Preprin)n(t,)26 b(2001.)120 1162 y([G])204 b(E.)36 b(L.)g(Green,)i Fi(Sp)l(e)l(ctr)l(al)g(the)l(ory)g(of)h(L)l (aplac)l(e-Beltr)l(ami)h(op)l(er)l(ators)e(with)h(p)l(erio)l(dic)g (metrics)p Fw(,)g(J.)435 1262 y(Di\013.)28 b(Equations)f Fb(133)g Fw(\(1997\),)f(no.)h(1,)h(15{29.)120 1362 y([Hed])124 b(L.)38 b(Hedb)r(erg,)k Fi(Sp)l(e)l(ctr)l(al)d(synthesis)i(and)f (stability)h(in)f(Sob)l(olev)h(sp)l(ac)l(es)p Fw(,)h(Euclidean)c (Harmonic)435 1461 y(Analysis,)57 b(Conf.)52 b(Pro)r(ceedings)e (Maryland.)h(Lecture)h(Notes)f(in)i(Mathematics,)k(v)n(o.)51 b(779,)435 1561 y(Springer-V)-7 b(erlag,)25 b(Berlin,)i(1980,)f(pp.)i (73{103.)120 1660 y([H])207 b(R.)23 b(Hemp)r(el,)h Fi(Se)l(c)l(ond)h (or)l(der)h(p)l(erturb)l(ations)f(of)h(diver)l(genc)l(e)g(typ)l(e)f(op) l(er)l(ators)h(with)g(a)g(sp)l(e)l(ctr)l(al)f(gap)p Fw(,)435 1760 y(Op)r(erator)i(Theory:)37 b(Adv)-5 b(ances)29 b(and)f (Applications)g Fb(57)p Fw(,)h(Birkh"auser,)d(Basel,)i(1992,)f(pp.)i (117{)435 1860 y(126.)120 1959 y([HH-1])75 b(R.)35 b(Hemp)r(el)h(and)f (I.)h(Herbst,)h Fi(Str)l(ong)f(magnetic)h(\014elds,)j(Dirichlet)d(b)l (oundaries,)j(and)e(sp)l(e)l(ctr)l(al)435 2059 y(gaps)p Fw(,)29 b(Comm)n(un.)e(Math.)h(Ph)n(ys.)e Fb(169)h Fw(\(1995\),)g (237{259.)120 2159 y([HH-2])75 b(R.)28 b(Hemp)r(el)i(and)e(I.)g (Herbst,)h Fi(Bands)i(and)g(gaps)g(for)h(p)l(erio)l(dic)g(magnetic)f (Hamiltonians)p Fw(,)f(Op)r(er-)435 2258 y(ator)c(Theory:)36 b(Adv)-5 b(ances)28 b(and)f(Applications)g Fb(78)p Fw(,)h(Birkh\177)-42 b(auser,)25 b(Basel,)i(1995,)f(pp.)i(175{184.)120 2358 y([HL-1])85 b(R.)31 b(Hemp)r(el)g(and)g(K.)f(Lienau,)i Fi(Sp)l(e)l(ctr)l(al)g(pr)l(op)l(erties)i(of)g(p)l(erio)l(dic)g(me)l (dia)g(in)f(the)g(lar)l(ge)g(c)l(oupling)435 2457 y(limit)p Fw(,)28 b(Comm)n(un.)g(P)n(artial)d(Di\013er.)k(Equations)d Fb(25)h Fw(\(2000\),)f(1445{1470.)120 2557 y([HL-2])85 b(R.)32 b(Hemp)r(el)g(and)g(K.)f(Lienau,)h Fi(Genericity)j(of)f(the)g (b)l(and-gap)h(structur)l(e)c(of)k(p)l(erio)l(dic)g(me)l(dia)g(in)435 2657 y(the)24 b(lar)l(ge)g(c)l(oupling)h(limit)p Fw(,)e(Di\013eren)n (tial)e(equations)f(and)h(mathematical)f(ph)n(ysics)h(\(Birmingham,)435 2756 y(AL,)28 b(1999\),)e(197{205.)120 2856 y([HN])145 b(I.)21 b(Herbst)g(and)g(Sh.)h(Nak)-5 b(am)n(ura,)21 b Fi(Schr\177)-42 b(odinger)25 b(op)l(er)l(ators)g(with)f(str)l(ong)g (magnetic)g(\014elds:)36 b(Quasi-)435 2956 y(p)l(erio)l(dicity)h(of)e (sp)l(e)l(ctr)l(al)g(orbits)g(and)g(top)l(olo)l(gy)p Fw(,)h(Di\013eren)n(tial)d(op)r(erators)e(and)h(sp)r(ectral)h(theory)-7 b(,)435 3055 y(Amer.)18 b(Math.)h(So)r(c.)f(T)-7 b(ransl.)18 b(Ser.)g(2,)i Fb(189)p Fw(,)g(Amer.)e(Math.)h(So)r(c.,)h(Pro)n (vidence,)f(RI,)g(1999,)f(pp.)h(105{)435 3155 y(123.)120 3254 y([HZ])156 b(I.)26 b(Herbst)g(and)g(Z.)h(Zhao,)e Fi(Sob)l(olev)30 b(sp)l(ac)l(es,)g(Kac-r)l(e)l(gularity,)f(and)g(the)g (F)-6 b(eynman-Kac-formula)p Fw(,)435 3354 y(Seminar)39 b(on)h(Sto)r(c)n(hastic)f(Pro)r(cesses,)i(1987)d(\(Princeton,)k(NJ,)e (1987\),)i(Birkh\177)-42 b(auser,)41 b(Boston,)435 3454 y(1988,)26 b(pp.)i(171{191.)120 3553 y([K])204 b(T.)27 b(Kato,)g Fi(Perturb)l(ation)j(the)l(ory)g(for)h(line)l(ar)f(op)l(er)l (ators)p Fw(,)f(Springer-V)-7 b(erlag,)25 b(Berlin,)j(1966.)120 3653 y([Ku])158 b(P)-7 b(.)36 b(Kuc)n(hmen)n(t,)j Fi(Flo)l(quet)f(The)l (ory)h(for)g(Partial)h(Di\013er)l(ential)e(Equations)p Fw(,)i(Birkh\177)-42 b(auser,)37 b(Basel,)435 3753 y(1993.)120 3852 y([KK])139 b(P)-7 b(.)20 b(Kuc)n(hmen)n(t)g(and)h(L.)g(Kun)n(y)n (ansky)-7 b(,)20 b Fi(Sp)l(e)l(ctr)l(al)j(pr)l(op)l(erties)i(of)f(high) h(c)l(ontr)l(ast)d(b)l(and-gap)j(materials)435 3952 y(and)30 b(op)l(er)l(ators)h(on)f(gr)l(aphs)p Fw(,)e(Exp)r(erimen)n(t.)g(Math.)f Fb(8)h Fw(\(1999\),)e(1{28.)120 4051 y([L])217 b(K.)33 b(Lienau,)h Fi(Sp)l(e)l(ctr)l(al)h(c)l(onc)l(entr)l(ation)f(for)i(high) g(c)l(ontr)l(ast)f(r)l(andom)g(me)l(dia)p Fw(,)h(Ph.D.)d(thesis,)i(TU) 435 4151 y(Braunsc)n(h)n(w)n(eig,)25 b(1999.)120 4251 y([PF])158 b(L.)20 b(P)n(astur)f(and)i(A.)g(Figotin,)g Fi(Sp)l(e)l(ctr)l(a)i(of)h(R)l(andom)g(and)f(A)n(lmost-Perio)l(dic)i (Op)l(er)l(ators)p Fw(,)d(Springer-)435 4350 y(V)-7 b(erlag,)26 b(New)i(Y)-7 b(ork,)27 b(1992.)120 4450 y([P-1])142 b(O.)34 b(P)n(ost,)h Fi(Perio)l(dic)j(Manifolds,)i(Sp)l(e)l(ctr)l(al)c(Gaps,)j (and)d(Eigenvalues)h(in)f(Gaps)p Fw(,)i(Ph.D.)c(thesis,)435 4550 y(TU)28 b(Braunsc)n(h)n(w)n(eig,)d(2000.)120 4649 y([P-2])142 b(O.)27 b(P)n(ost,)g Fi(Perio)l(dic)32 b(Manifolds)g(with)e (Sp)l(e)l(ctr)l(al)g(Gaps)p Fw(,)f(Preprin)n(t,)d(R)-9 b(WTH)28 b(Aac)n(hen)f(\(2001\).)120 4749 y([RS-I])104 b(M.)29 b(Reed)h(and)f(B.)h(Simon,)g Fi(Metho)l(ds)i(of)h(mo)l(dern)e (mathematic)l(al)i(physics)g(I:)f(F)-6 b(unctional)32 b(anal-)435 4848 y(ysis.)f(R)l(evise)l(d)f(and)g(enlar)l(ge)l(d)h(e)l (dition)p Fw(,)d(Academic)g(Press,)e(New)i(Y)-7 b(ork,)27 b(1980.)120 4948 y([RS-IV])42 b(M.)f(Reed)g(and)f(B.)h(Simon,)j Fi(Metho)l(ds)f(of)g(mo)l(dern)f(mathematic)l(al)h(physics)h(IV:)e(A)n (nalysis)g(of)435 5048 y(op)l(er)l(ators)p Fw(,)29 b(Academic)e(Press,) f(New)i(Y)-7 b(ork,)27 b(1978.)p eop %%Page: 12 12 12 11 bop 240 66 a Fh(12)963 b(RAINER)23 b(HEMPEL)k(AND)d(OLAF)h(POST) 240 266 y Fw([S])223 b(B.)18 b(Simon,)j Fi(A)g(c)l(anonic)l(al)i(de)l (c)l(omp)l(osition)h(for)e(quadr)l(atic)h(forms)f(with)g(applic)l (ations)i(to)e(monotone)555 365 y(c)l(onver)l(genc)l(e)30 b(the)l(or)l(ems)p Fw(,)e(J.)f(F)-7 b(unct.)29 b(Anal.)e Fb(28)g Fw(\(1978\),)g(377{385.)240 465 y([SKM])82 b(D.)37 b(Sto)n(y)n(an,)h(W.S.)g(Kendall,)g(and)f(J.)g(Mec)n(k)n(e,)h Fi(Sto)l(chastic)h(Ge)l(ometry)g(and)g(its)f(Applic)l(ations.)555 565 y Fw(J.)27 b(Wiley)-7 b(,)28 b(Chic)n(hester)f(1995.)240 664 y([Y])207 b(K.)24 b(Y)-7 b(oshitomi,)24 b Fi(Band)k(gap)g(of)f(the) g(sp)l(e)l(ctrum)f(in)g(p)l(erio)l(dic)l(al)t(ly)k(curve)l(d)d(quantum) f(waveguides)p Fw(,)g(J.)555 764 y(Di\013.)i(Equations)f Fb(142)g Fw(\(1998\),)f(no.)h(1,)h(123{166.)340 951 y Fv(Institut)j(f)764 944 y(\177)761 951 y(ur)i(Anal)-6 b(ysis,)31 b(Technische)i(Universit)2222 944 y(\177)2219 951 y(at)f(Bra)n(unschweig,)h(Pockelsstra\031e)240 1050 y(14,)e(38106)f(Bra)n(unschweig,)i(Germany)340 1150 y Fi(E-mail)e(addr)l(ess)7 b Fw(:)38 b Fa(r.hempel@tu-bs.de)340 1319 y Fv(Institut)44 b(f)777 1312 y(\177)774 1319 y(ur)h(Reine)h(und)f (Angew)-7 b(andte)44 b(Ma)-6 b(thema)g(tik,)49 b(R)-10 b(WTH)45 b(Aa)n(chen,)j(Templer-)240 1419 y(graben)32 b(55,)f(52062)f(Aa)n(chen,)h(Germany)340 1518 y Fi(E-mail)f(addr)l(ess) 7 b Fw(:)38 b Fa(post@iram.rwth-aa)o(ch)o(en.)o(de)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0207160616603--