Content-Type: multipart/mixed; boundary="-------------0209120831959" This is a multi-part message in MIME format. ---------------0209120831959 Content-Type: text/plain; name="02-375.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-375.keywords" Random operator, Anderson, Landau, multiscale analysis, localization ---------------0209120831959 Content-Type: application/postscript; name="finvolmparc.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="finvolmparc.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: finvolmparc.dvi %%Pages: 28 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips finvolmparc.dvi -o %DVIPSParameters: dpi=1200, compressed %DVIPSSource: TeX output 2002.09.12:1149 %%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 TeXDict begin 39158280 55380996 1000 1200 1200 (finvolmparc.dvi) @start %DVIPSBitmapFont: Fa cmtt8 8 19 /Fa 19 119 df<001FB812F84817FC4817FEBAFCA56C17FE6C17FC6C17F8380B79AD47> 45 DI49 D64 D<91B5FC010F14F0013F14FE90B77E48824816F0834882488283DAC00380 ED003F040F7F82707F6C497F6C90C87F6C5AD80078157F90C9FCA2EE0FFF033FB5FC0207 B6FC147F0103B7FC131F137F48B8FC5A5A489138FC007F4891C7FC4813F014804848C8FC 5BEAFFF0A25BA317FF7F5E6C6C5C6D5C6D141F6C6C6C137FDAF007B612FC6C90B812FE6C 18FF7E7E6C16BF6C161F6DECFC07011FDAE00313FE010791C7EA7FFC010001F091C7FC40 3C7ABA47>97 D<92B5FC020714F0023F14FC91B7FC010316804916C0131F4916E04916F0 90B8FC48ECC003ECFE004813F84813E05C48496D13E04890C813C049ED7F804848ED1E00 94C7FC5B127FA25B12FFA25BA87FA2127F7FA36C6CED0FE06DED1FF0001FEE3FF87F6C01 C0147F806C6DECFFF06C01FC5B02FF4913E06CECF00F6C91B612C06D16807F6D16006D5D 010315F801005D023F14C0020749C7FC020013F0353C77BA47>99 D<93383FFFF04C7F93B57EA68282EE001FADEC03FF021F13F091B512FC010314FF4915DF 011F15FF5B5B90B8FC5A48EC807F9138FC001F4801F013074849130102807F4890C8127F 5B003F163F5B007F161F5BA25B12FFA25BA97FA2007F163F7FA2177F6C7E17FF6C7E6D5C 6C6D5B6E5B6C6D131F6C01F85BDAFF01B612FE6C91B8FC6C19807F6D15DF6D159F6D151F 010314FE6DDAF80F1400D9003FD9C0075BDA07FECAFC41527CD047>III I105 D<381FFFE0007F7F80B5FCA47EA2121FEA000FAE047FB512F893B67E4B81A66F5D705C04 0301F0C7FC4C5B4C5B4C90C8FC4C5A4C5A4C5A4B5B4B5B030F5B4B5B4B90C9FC4B5A4B5A 14F902FB7F91B6FC83A28303FD7F03F87F4B6C7E15E04B6C7E9238001FFE4A6D7E4A6D7F 5C707F707F707F717EA2717E717E717E001FB500FC017F13FF007FDAFF01B612801AC0B7 1283A46C15011A80001F9126FC007F140042517ED047>107 D<003FB512FE4880B77EA6 7E7EC71203B3B3B3A7003FB812F84817FCB912FEA66C17FC6C17F8375178D047>I<03FF EC7F80263FFC079039C003FFE0267FFE1FD9F00F13F8B5007FD9F83F7F91B55B70B57E93 B7FCA2867E6C020714830001D9FC03D9FE017F4A6C497E02F05C4A6C49137FA24A5CA24A 5CA391C75BB3A9003FD9F8079039FC03FFFE486D486D487FB56C486D481480A66C496C49 6C14006C496C496C5B493A81B947>II114 D116 D<263FFFF0903807FFFE486D497FB56C4980A66C816C81D800 1FEC0003B3AC5FA35F5F6E147F6E49B5FC6DD9C00FECFFC092B812E01AF07F7FA27F6D15 F3023F02C114E0020F49C614C0020101F090C8FC443A7FB847>I<001FB5D8C003B512F8 486E4814FC486E4814FEA2B616FFA26C18FEA26C4A6C14FC6C4A6C14F826003FF0C7380F FC00A2011F5E6E141FA2010F5E6E143FA201075E6E147FA201035E6E14FFA26D5E6F5AA2 6D93C7FC6F5AA2027F5CA2EDE007023F5CA2EDF00F021F5CA2EDF81F020F5CA2EDFC3F02 075CA2EDFE7F02035CA292B5FC6E5CA36E91C8FCA36F5AA26F5AED0FF0403A7DB847>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb cmbx8 8 14 /Fb 14 112 df<13FC3807FF804813C04813E04813F04813F8A3B512FCA66C13F8A36C13 F06C13E06C13C06C13803800FC00161675952D>46 D<92381FFF804AB512F8020F14FF02 7F15E091B77E010316FC4982011FD9FC036D7E499026E0007F7F49496D7F92C7120F4948 6E7F48496E7FA248496E7F48844A804884A3481980A348496F13C0A44819E0A7B518F0B3 A56C19E0A76C19C0A26E92B5FCA26C1980A36C1900A26C6D4A5BA26C606E5C6C6D4A5B6C 6D4A5B6F5B6D6D495B6D01F090B55A6DD9FC035C6D90B7C7FC010316FC6D5E6D6C15E002 0F92C8FC020114F8DA001F1380445A7AD751>48 D<16FCED03FE1507151F157FEC03FF14 1F49B5FC0007B6FCB7FCA514E7EBFE07EAF800C7FCB3B3B3A9003FB912C0A83A5875D751 >I<913803FFF8027FEBFFC00103B612F8010F15FF013F16C090B812F04883488348D9F8 0314FF489026C0007F14804849011F14C04801E0010714E06E7F486F14F06E6D14F8B581 6E16FC8319FE83A319FF6C5B836C5BA26C5B6C5B000390C8FCD800785D90C9FC19FEA34D 13FCA24D13F8A294B512F019E04C14C0A24C14804C14004C13FC604C5B4C5B4C5B93B5C7 FC4B5B4B13F84B5B4B13C04B5B4B90C712FFED7FFC4B5A4A5B4A01C0EB01FE4A5B4A90C7 FCEC1FFC4A5A4A481403ECFFC0494914074990C8EA0FFC4990B7FC131F5B5B90B9FC5A5A 4818F85A5A5A5ABAFCA319F0A4405878D751>I<92B5FC020F14F8023F14FF91B712C001 0316F0010F16FC498249D9C01F7F90277FFC000380D9FFF06D804801C06D806E6E7F4813 F86E82486D806E82A25A81A64D5B6C91C7FCA26C495E6C4991B5FC6C495ED93FE05B0107 C75D90C8485C4C91C7FC4C5B4C5B4C5B0303B55A0207B612C04A92C8FC17FC5F17FF18E0 6E15F891C7003F13FE04077F040114C07080717F85717FA2717F85A2013818802603FF80 80487F001F01F017C0487FA2487FA2B57EA51A80A36C494A1400A24A5E6C495C4A5E6C49 5C91C85C6C6D91B55A6C01E0495C6C01FC01075C6C9026FFC03F5C6C91B648C7FC013F5E 010F16F0010316C0010093C8FC021F14F0020049C9FC425A79D751>IIIII<92383FFF800203B512FC021FECFF80027F15E049 B712F8010716FE498249D9C00380D93FFCC7003F7FD97FF0020F7F4948804A02017F485B 48707F91C9127FA24884183FA2487F80808014FC6E157F6E6C5D15E003F814FF03FE5D6C 6E6C5ADCE0035B70485B6CEDFC0FDCFF3F5B6C93B5C7FC18FC6C5F606D16C06D826D16F8 6D826D16FF01018385010783011F8349835B2701FFFE3F8148D9F80F8148497EDAE00181 48496C7E4849011F158091C71207481501496E6C14C0007F8249150F170300FF16004982 848484A2841A80A27FA2007F1900A26D4C5A6C7E4E5A6C6D153F6C6D4B5A02F04B5A6C6D 02035B6C01FF020F5B6CDAF001B55A6C91B7C7FC013F16FC010F5E010316E0010093C8FC 021F14F8DA007F1380425A79D751>II86 D<903807FF80B6FCA812016C7EB3B3B3B3A2 B712C0A8225C79DB2D>108 D<923807FFE092B6FC020715E0023F15FC91B8FC010317C0 4983011F49C66C13F84901F0010F7F4901C001037F90B5C87F48496F138048496F13C04A 814819E04819F04A814819F8A24819FC4A81A24819FEA4B518FFAC6C19FEA36E5D6C19FC A26C19F8A26C6D4B13F0A26C6D4B13E06C6D4B13C06E92B5FC6C6E4914806C02E0010714 006D01FE017F5B011F90B712F86D5F010317C06D5F6D6C4BC7FC020F15F002011580DA00 0F01F0C8FC483E7CBC51>111 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc cmex7 5 1 /Fc 1 103 df<912601FFF01840023FD9FF80EE03C00103B600F8EE0FE0011FDBFF80ED 7F80017F04F0913807FE002803FFC0003FD9FF80EB7FF8D80FFCC7000191B612C0D83FC0 DA003F93C7FC00FEC9000315F80078DC003F14800040050101F0C8FC5B0B7FBD5F>102 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd cmex7 7 1 /Fd 1 103 df<93261FFFC01A0C0303B500FE1A3C037FDAFFE019FC0203B700FCF007FE 021FDCFF80EF3FF891B900E0933801FFE00107DAC01F02FC040F1380011F01E0C7003FD9 FFC092387FFC00D97FFCC8000702F891380FFFF02603FFE0DB007F9026FFF007B512C000 0F90CA000F91B648C7FCD83FF8050317F0D8FFC0DD007F1680007ECC000F03FCC8FC0078 070015800060080701F0C9FC7F107ED684>102 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe cmr5 5 13 /Fe 13 57 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Ff cmsy5 5 4 /Ff 4 50 df<007FB912F8BA12FCA36C18F83E0572965A>0 D<147014F8497E6D5AA400 30156000FCEC01F86C1403D8FF80130F01E0133FD87FF0EB7FF03A1FFC71FFC02607FF77 1300C6EBFFF8013F13E0010790C7FCA2013F13E090B512F80007EB77FF261FFC7113C03A 7FF0F87FF0D8FFE0EB3FF80180130FD8FE0013034814010030EC0060C71400A4497E6D5A 1470252474A63D>3 D48 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg cmti10 10 55 /Fg 55 122 df11 D<973801FFFE087FEBFFF00703B612FE070FEDFF80073F16C04F D9000713E0952601FFF09038003FF04E0180EC0FF84E90C8EA03FCF00FFC4E48150F4E48 ED1FFE067F163F4F157F18FF4D49EDFFFCA25F4F16F85FF57FF096C9EA3FE04DEF1F809A C7FCA24D5AA4173F60A4177F60A417FF60A45E6092BB12FC5CA480DB000301C0C86C5AA2 1CFF5E4E5EA263A24C6095C8FC63A2655E4D5DA265A2043F5E5F9AC7FCA263167F4D5EA2 1B3FA26416FF4D157FA264A24B17FF5F64A2625D4D9338E001F8A25014031EF04B18C05F 5014071EE01C805D94C8150F1EC01C001D1F4B1A804C183F1E00745C1D7E4B486FEB80FE F481FC74EBC3F8F37FFF755B4B48705B09071380E101FEC7FC98C9FC5E15FFA25EA25C5E A35E3807E003EA0FF0D81FF891CEFCEA3FFC007F495A12FF5DA24A5A13F849485A01E05B EBC03F01805B6C48485AD9E1FFCFFC6CB45A6C5B6C5B000313E038007F80679784F45D> I<19F0F001F8F003F0F00FE0F01FC0F03F80F07F0018FEEF03FC4D5A4D5A4D5A4D5A4D5A 4DC7FC4C5A16035F4C5A4C5A4C5A163F4C5A5F4CC8FC5D4B5A4B5AA24B5A4B5AA24B5A15 7F5E4B5A5C93C9FC5C5D14075D140F4A5AA24A5AA24A5AA24A5AA25B5DA24990CAFCA25B 5C130F5CA2131F5C133FA25C137FA25C13FFA25C5AA25C5AA391CBFC5AA35B120FA35B12 1FA35B123FA35BA3127FA35BA412FFA25BA95BB2127FA27FA3123FA47F121FA36C7EA312 077F1203A26C7EA26C7EA2137FA26D7E131F80130F6D7E6D7EA26D5A3DA767FC44>40 D<1778177C177E8384171F717E84170784717EA284170184170084A2841980A3F03FC0A3 19E0A2181FA319F0A5180F19F8AD181FAA183FA219F0A5187FA319E0A218FFA319C0A25F A319805FA319005FA360170FA360171FA260173FA260177FA26017FFA2605EA2605E95C7 FCA25E5F160F5FA24C5AA24C5AA24C5AA24C5AA24B5BA24B90C8FCA24B5A5E150F5E151F 5E4B5AA24B5A4B5AA24A90C9FC4A5AA24A5A4A5A5D4A5A143F4A5A5D4ACAFC495A495A49 5A5C495A495A495A49CBFC13FE485A485A485A485A485A007FCCFC12FE5A5A3DA77EFC44 >I44 D<001FB712804816C05A A21780A2B8FCA217007E6C5D2A0B72A83B>I<13FE3803FF804813C04813E05A5A4813F0 A3B512E0A314C0A26C138014006C5AEA1FF8EA07E014136E9233>I49 D<943801FFC0051F13F8057F13FF4CB612C00407 814C01017F933A3FF0003FFCDC7FC0EB0FFE4CC76C7E4B4880DB03F86E13804B4816C04B 486E13E04B5A4B48ED7FF04BC9FC157E4B17F814014A48163F9226F0038015FC02071307 4B487E140FEDC007DA1F807F023F0103157F1500A25C147E14FE4A010715FF5F13015C04 0F5C01034B15F84A131F94C7FC6101074A16F04A133E047E5C4C16E04C4A13C015014B48 4A13804B5A902703F01FC04A1300DB7F804A5A91B5C85B6D494B5A6D494A5BDA7FF04A5B DA3FC04A5B91C95C4E90C7FC4E5A4E5AF0FFF84D5B4D13C04D5BDD1FFEC8FC4D5AEF7FF0 4C485A4C1380040F90C9FCEE3FFCEE7FF04B485A030713804B48CAFCED1FF8ED7FF0EDFF C04A90CBFC4A5AEC0FF84A5A4A5ADA7F80EE1F804ACA123F494818005C49485F495A4948 177E011F18FE4A160149485F49CA120301FE4D5AA248484D5AD9FFE0161F48D9FFC0ED7F E003FE15FF48DAFFF001035B9338FF801FD80FFC92B65AD9F00F94C7FC261FE0015E496C 7E48486D5D030F5D90C76C5D4802015D007E6E5D043F91C8FC705B00FE030313F8007C92 38007FC04E7373EE55>II<1A F8F103FCF107FE190FA2191FA3193F1AFCA3197F1AF8A3F1FFF0A3601AE0A34E13C0A21A 8060A21A0060A261181F61183FA261187F61A24E5AA25F61A24D5BA24D90C7FCA24D5AA2 4D5AA24D5A60177F6017FF605E604C90C8FCA24C5A5F160F5F161F4C5A5F4C5A04FFEB01 F094388007F84B49487E4B4A7E5E4B48133F4B485CA24B5A4B48137F4B485C5E15FF4A90 C712FF4A485D4A5A5D4A485B4A485D4A5A4A5A4AC75A49485E495A495A49485C49485E49 5A90387FFFF890B65B000303E091C7FC000F15FC4815FF4893B5FC277FF8001F5CD8FF80 0101163890C8003FEC03FC48030F90B512FE007815030030ED007FCA6C14F84D14C0F1FC 0006F8C7FCA217FF60A35E60A35E60A35E60A35E95C8FCA35EA25FA2705A705AEE03E047 8F79EE55>I<030617E0DB0F80150304F0153FDB1FFEEC01FFDCFFF0017F13C094B71280 1B004B5E62621AE04B5E97C7FC19FC19F092B712C04EC8FC03FD14F09226FC0FFEC9FC02 0190CBFC5DA314035DA314075DA3140F5DA3141F5DA3143F92CCFCEE07FE93387FFFC0DA 7F03B512F0DA7E0F14FC4B8092397FF807FF9127FEFFC0017F6E48C77F03F86E7E03E081 4949143F4B814AC8121F4A82495AA24A826D5A90CAFCA385A660A44E5AA418FFD803F85F EA0FFC487E003F5E6D5F5AA2494B5B12FF61495D495F495D01C05F00FCC9FC4D90C8FC60 173F4D5A606C16FF4C5B60007F4B5B4C5B6D4A90C9FC003F4B5A6DEC7FFC6C6C4A5A6C6C 010313E001FC010F5B2707FF807F90CAFC6C90B55A6C15F86C15E0013F1480010F01FCCB FC010113C04B736EEE55>II<023ED907F8ED0F80023FD97FFC151F4A48486C 153FDA7E076D1600DAFE1F6E5CDAFC7F17FE010190B64A5A14F9D903FB6F495A91B74A5A 49180F63494E5ADCE03F143FDC001FECFF804901F8010F5B03E06D6C4890C7FC490180ED 0FFE92C73803F07FD97FFC6EB55A5C49486E5C4A8148018092381FCFF09538000FE04890 CA121F494D5A485A494D5A485A4FC8FC495E48485F180348484C5A90CAFC484D5A127E00 FE4D5A48173F61CB127F6118FF615F5F96C9FC5F60170F60171FA24D5AA24D5A17FFA24C 5BA24C5BA25E605E95CAFCA25E5F163FA24C5AA216FF5F5DA25F5DA24B5BA34B5BA25DA2 5F5DA294CBFC5DA25E15FFA34A5BA44A5BA35C5EA35EA25E6E5B6E90CCFCEC00FC517369 EE55>I58 DI65 D<030FBA12C04B19FCF5FF801E E01EF86F86DB000391C800037F9326007FFE0300EBFF804E043F13C08905FF7113E01FF0 4E821FF84C841FFC60894C1AFEA260A25EA260A24C601FFC60A24C601FF895CAFC5313F0 5E5313E04D19C065047F4E13801F004D4D5A6404FF4D5B5213F04D4C5B525B4B067F5B52 48C7FC4D03035B090F13F04B053F13C00803B5C8FC94B812F81CC04B18F01CFEF4FFC005 C0C8000F7F4B050313F809007F4D707E767E4B727F8A94CA6C7FA24B727F8A5E88037F86 A25EA215FFA25EA25CA25E645C665E644A63644C61644A63644C6199B5FC4A4E91C7FC51 5B93CA5C515B4A60515B4B057F13C098B55A027F4D91C8FC08075B4B041F5B02FF057F13 F0010F0507B512C0007FBC5ABC48C9FC1BF81BC050CAFC6C19C06F7176F075>I<030FBA 12C04B19FE777E1EF08A6F1AFEDB00030280C7000F7FDC007F90C914C04E043F7F0B0F7F 05FF05037F777F4E82787E4C737E8A4E71138020C04C8520E04E8320F05E8A4E19F8A25E 7813FC60A25EA26020FE5EA295CCFCA25EA25FA204FF61A25FA25DA24D6020FC5DA25F66 5D20F85F665D20F05F664B1CE0A24D6020C05DA294CC481380A24B1C009BB5FC4C626503 FF63A24C4E5BA24A505BA24C4E5B674A62674C4E90C7FC654A63535A4C4D5B664A4F5B64 4C4D5B525B4A4F90C8FC525A4C4D5A515B4A060713E0515B93CA485B097F90C9FC4A9438 01FFFC08075B4B041F13E002FF94B51280010F051F91CAFC007FBB12FCBC12F01B8050CB FC1AE06C06FCCCFC777177F07D>68 D<030FBDFC4B1C80A3200081DB00030280C8123FDC 007F90C912014E707E1E1F05FF848A4E715AA25E1E0160A25E6760A25EA260A25E6760A2 5EA24EEC01E0517E4C04071403525C95C8FC785A4C040F91C8FC645F1B1F16FF644D153F 1B7F5D51CAFC4D14031A0F4B167F94B75AA35D63A205E0C7FC4B161F745A4D1407A25D74 5A4D1407A25D6394C8FC1E1F4B040F4B7E5192C7FC5E6603FF1A7E744814FE4C92C95AA2 4A1A01664C1803664A1A07A24C611D0F4A631D1F5E535A4A1A7F9BC8FC4C60A24A4F5A1C 034C4D5AA24A190F525A93CB127F525A4A18031B0F4B057F5B02FF0503B5FC010F95B65A 007FBDFCBEFC65A36C99C9FC717176F071>I<030FBC12FE4B1BFFA31FFE81DB00030280 C87EDC007F90C912034E16001E3F05FF181F1E0F4EEF07FCA24C1903A260A25E1FF8601E 014C1903A260A25E1FF060A25EA260A24CDC07801307516C14E095C8FCF603C04C041F91 C7FC645FA204FF163F99C9FC5F635D1BFE4D14011A034B1607F21FFC4DEB01FF94B7FC5D A263A25D9426E000015B4DEB003F1A1F4B160F635FA25D6394C8FCA24B161F635EA203FF 163F98CAFC5E1A1E4A94CBFCA25EA25CA25EA25CA25EA25CA25EA25CA25EA25CA293CFFC A25CA391B5FC010F14E0007FB8FCB97EA46C94CDFC707177F06C>I<962601FFF0153807 3F01FF153C0603B600C0147C061F03F014F8067F03FC13014DB76C1303050F9126C00FFF 13074D01FCC79038800FF0057F01E091383FC01F4CB5C8380FE03F4C01FC923807F07F04 0F01F0923903F8FFE04C01C0ED01FD047F496FB5FC4C48CAFC4B497113C04B13F0030F49 834B5B4B497113804B90CBFC4B48844A5B4A491A004A5B4C844A5B4A634A5B4A90CCFC91 B5FC4B62495B491B035D4964495BA24949190749645D5B5D90B56392CDFC5AA24849639A C8FC5C5AA25A5CA25A5CA35A5CA35A5CA45CB5FCA2070FB712FC4F82A291CA5FA2735E96 C76C01F8C7FC090F13E0A265A263A26C64A263A29AC8FCA26C6280647E1BFF806C63A26C 6D5F646C7F626C6D5F6E616C616D6C5F6D6C5F6F4C5B6D6DEEFF7F6D6D923803FE3F6D01 F8ED07FC6D01FE92391FF81FC001006D6CEC7FE06E01E0903903FFC00F6E01FF013FEB00 07020F91B5486D5A020304F81301020004E090CAFC033F92CCFC030714F8DB003F90CDFC 6E7968F480>I<030FB700FE91B812E04B704917F0A46F4C6D17E0DB000302F0C9003F91 C7FCDC007F90CA000713F0A24E6105FF60A24E61A24C616860A24C619DC8FC60A24C6167 60A24C19FF6760A24C606760A24C606795CAFCA24C60675FA204FF60675FA24B61675FA2 4B6194BBC9FCA35D66A205E0CA127F4B19FF665FA24B60665FA24B606694CAFCA24B6066 5EA203FF60665EA24A61665EA24A619BCAFC5EA24A61655EA24A19FF655EA24A60655EA2 4A606593CAFCA24A60A24B6191B54D7F010F02C093B512FC007FB700F80107B87EB86C49 83A24D61A26C4C6D94C9FC847177F07B>I<030FB812804B17C0A46F1780DB000102F8C7 FCDC003F13C06196C8FC5FA260A217FFA260A25EA260A25EA260A25EA260A25EA260A25E A260A25EA295C9FCA25EA25FA216FFA25FA25DA25FA25DA25FA25DA25FA25DA25FA25DA2 5FA25DA294CAFCA25DA25EA215FFA25EA25CA25EA25CA25EA25CA25EA25CA25EA25CA25E A25CA293CBFC4A7F0107B512E0003FB712FC4882B8FCA36C5E4A7177F040>I<030FB812 E04B83A46F5FDB000302F8C9FCDC007F138096CAFC6017FFA260A25EA260A25EA260A25E A260A25EA260A25EA260A25EA295CBFCA25EA25FA216FFA25FA25DA25FA25DA25FA25DA2 5FA25DA25FA25DA25FA25DA294CCFC1D384B197C1DFC5E1C0103FF19F8A24C17031DF05C 1C074C18E01C0F5C1DC04C171FA24AF13F80A24C177F1D004A61634C5F1B034A4E5AA24C 160F515A4A183F1B7F93C948485A624A170F083F5B4B93B5FC02FF1607010F94B65A007F BCFCBDFC64A36C98C7FC5E7177F068>76 D<030FB600C00703B612E04B0B0F15F0A27361 A26F5315E0DB00035291C7FCDC007F1DF09EB5FC6B94B561F803F705FD6505FC6DF007E7 04011CEFE60FCF5B4DF21F8FA20403515AA0C8FC4D1A7E20FC040764726CDD01F85B17E0 F703F0040F973807E07F6A05C0F10FC0A2041F97381F80FF55485B1780726C177E043F63 0DFC5C0500F001F8A24C4F485A6A047EF107E0F60FC004FE63726CDC1F805C5EF63F0003 01087E5B6A4C61A203034F485B534892C9FC5E726C4B5A03076453485C4C4E5AA2030F4F C7127F694C187E65031F1CFF726D49485D5E525A033F4E485B6993C74C5AA24B4E485B52 C75C157E726D137E03FE63525E4B4D5AA202014E485C694B4D5A515A020364726D484893 CAFC5D51C8FC0207067E5D684B5FA2020FDDE1F8157FE0E3F05E5D72EBF7E0021F1CFFE0 FFC05E4B5FA2023F95C85A68027F5F6202FF6349715A496D64011F6D4C4B7F90B500FC4C 037F13FC007FDAFFF80503B87EB76C4B4983A24C4B6197C7FC6C4B023E6E94C9FC9C7176 F095>I<030FB694B712E04B6F4B16F0A285A26F7316E092C76E0403ECF00071DE007F13 80737148C7FCF71FF894B56173715AA205FD6204016E171F17FC4D6D60A20403017F183F 7395C8FC17F08404076F5E72187E05E080A2040F6D18FE745E17C084041F6F1501726017 8086043F6D1703745E1700844C6F1507077F5F167E8604FE6E160F755D5E85030171141F 735F5E8703036F163F7592C9FC5E850307715C73167E5E87030F6F16FE755C5E1A7F031F 711301745D5E1C80033F7014030AC05B93C9FC864BF0E007745D157E1CF003FE70140F0A F85B5D860201F1FC1F745D5D1CFE0203067F133F0AFF90CAFC5D8702071AFF755B5DA202 0F84655D87141F755B5DA2023F8465147F8714FF49745A497F011F7F90B500FC183F007F DAFFF860B77E1C1F5EA26C4B715A847177F07B>I<030FBAFC4B19F01DFE777E1EE06F1A F8DB00030280C7003F7FDC007F90C800037F4E03007F0B3F138005FF7113C0894E7013E0 1FF04C841FF860A24C1AFCA260A25EA260A25E6560A25E5313F860A24C1AF06595CAFC1F E04C601FC04D19806504FF1A00535A4D60644B4E5B525B4D4C5B525B4B4E90C7FCF4FFFE 4D4B5B090713F04B053F5B0803B5128094B848C8FC1CF84B18C099C9FC1BF005C0CDFC5D A25FA25DA294CEFCA25DA25EA215FFA25EA25CA25EA25CA25EA25CA25EA25CA25EA25CA2 5EA25CA293CFFCA25CA25D91B5FC010F14C0007FB712F8B87EA25FA26C5E6E7176F071> 80 D<030FB912E04B18FF1DF01DFC1DFF6F1AC0DB0003912680000180DC007F90C8000F 13F84E03037F0A007F05FF717E7713804E7013C0894C1AE0894E18F0A24C1AF8A260A25E A260A25E6560A25E5313F060A24C4E13E0A295CA14C05313805E9AB512004D606404FF61 525B4D4C5B525B4B4E13805290C7FC4D4C5A983801FFF84B05075B091F13C04D92B5C8FC 081F13FC4B90B812F01C8009FCC9FC1BF04B17FC05E0C76C7E4DEC0FFF08037F4B707F74 7F4D82757E5D757E94C9FC885DA25EA203FF84A25EA24A60A25EA24A95B5FC645EA24A5F A25EA24A5FA25EA25C50163C4C1A7EA24A1CFE1FFC93C9FC1E014A1CF8864BF203F091B5 1A07010F02C06F16E0007FB700F870130FB86C6EED1FC075EC3F804D6F9038807F0075EB C1FE6C4C6FEBFFFCCE6C5C09015C756C13C00A0790C7FC6F7577F079>82 D<952603FFC01470063F01FC14784DB614F805079238C001F0051FEDE0034DEDF0079426 FFFE00EBF80F4C01E090391FFC1FE004070180EB03FE4C48C73801FF3F4C486E13FFDC3F F06F13C04C48814C48814B5B4B90C96C13805E4B4882150F4C18004B4882153F5E037F60 5E15FFA24A615EA34A61A4645CA21B07705F1B037093C8FCA2826E7F8282707E17F86E14 FF18F06E15FEF0FFE06F15FC6FEDFF801AE06F826F826F826F82030082043F82040F8216 01DC001F811703DD003F801807060080191F190785858785A21A7FA21A3FA2133E133F49 61A2137EA213FE63A34848187F63A263486C18FFA24F5BA2000797C8FC4F5A6D17076248 180F6E4C5A626E4C5A486D4C5A4F5A6E4B90C9FC6EED07FE263FF7FE4B5AD9F3FF4B5A01 E101C0EC7FF001C001F8903803FFE0297F807FFF801F5BD9001F90B6CAFC007E6D15FC00 7C01035D00FC010015E048021F91CBFC0070020113F05D7975F45D>I<0107BE1280491D C05B1F805BA203FCC7003F9038C000034901C04CEB003F4AC891C8120F02F84B701300D9 FFE0864A4C815C4890C9834916FF4F5E485AA2495D0007605B6648485D615B001F1C015F 494D5E123F90C9FC5F4860007E1C036600FE5E4860A20078765ACA4894C8FC96CCFCA35F 60A317FF60A35E60A35E60A35E60A35E60A35E60A35E95CDFCA35E5FA316FF5FA35D5FA3 5D5FA35D5FA35D5FA35D5FA25DA25D92B5FC021F14F8000FB912F04884A25AA26C606A70 64EF77>I<003FB7023FB76C49B612FE487049704915FFB86366A112FE6C93C76C93C76C 15FCC66C91C9003F91C9001F1480011F01F8040F01F80403EBFC006D01E006E07013F04B 775B52606BA0C7FC765FA2744E5A6A21036A21075061210F5061575A626F53C8FC62217E 6D95B517FE4F6120014F6108F7170307076108E74D5AF10FC7565ADF1F877F565AF13F07 4F6C4DC9FC077E6007FE187E07FC18FE06016107F81701DE03F060555AF007E0555A6FEC 0FC0555A6DDC1F80171F063F610700173F4E96CAFC067E6006FE187E4E18FE4D4860545A 4D5A76495A4D5A545A4D486D150F051F614E171F053F6195C7163F4D96CBFC057E187E5F 666F485A535A4C5A6D02074E5A4D1707040F614D170F041F614D171F4CC85E53CCFC167E 1D7E4C17FC65DBF9F86F13FD03FB614C17FF03FF615E655E4C606593CAFC9ACDFC5D646D 5B645D645D644B605D645D6492CBFC75CEFC5C027C187E98755EF0A6>87 D97 D<157F0103B57E90B6FC5AA393C9FC7E1300141F5DA2143FA25DA2147FA25DA214FFA25D A25BA25DA25BA25DA25BA25DA25BA292CAFCA25BA24AEB7FC0923803FFF8013F010F13FE 4B7FDAFC7F80DBFFC17F903B7FFDFE007FE0DAFFFC6D7E03F06D7E4B8090B548130F4B80 92C712075C4849815CA24A1680485BA25CA25AA25C5F5AA291C8FCA2485EA25BA2003F5E 19005BA25F007F5F5BA217FF605B00FF5F5EA260495C60A24C5BA24C5B007F94C7FCA24C 5A4C5A5F003F157F5F6D4A5A001F4A5B4B5B6C6C4990C8FC6D495A00074A5A6C6CEB7FF0 3A01FF81FFE06C90B55A6D91C9FC6D13FC010F13F0010190CAFC39756EF24C>IIIIIII<16FC4B7E4B7E4B13805D5DA417005E6F5A 6F5AED03E092C8FCB3A5EC7FC0903801FFF001077F4913FE497FEB3FC1D97F017FEBFE00 4848805B485A4848805CEA0FC0A2EA1F804A5B123F13005C007E5DA25C00FE92C7FC485B 5DA2C7123F5DA2147F5DA214FF5D5B5DA25B5DA25B5D5B92C8FCA25B5CEE0FC0133F4A13 1F1780137F5C01FF143F4A1400A2167E485B16FE5E150102C05B4B5AA26C4A5A4B5A151F 017F495ADAE1FFC7FC6DB45A6D5B6D13F001035B010090C8FC2A7073ED33>I107 DIIIII114 DIIIII II E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fh cmti8 8 29 /Fh 29 128 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Fi cmmi5 5 27 /Fi 27 121 df11 D<157015F0A6EDF7FE92B512C0A25C1407021F148091393F87FC00027EC8FC5CEB03F849 5A495A495A5C49C9FC137E5B485AA2485A485AA2485AA2485AA2123F90CAFCA35A127EA5 12FEA27E7E7FA27F7F6C7E13FC6CB4FC14F06C13FF6C14E06C14FC6C14FF6C6C80011F80 010380EB003F1407EC007F151F150FA21507A29138C00FC0903903F81F8091B5FC6D91C7 FC9038003FFCEC07F02A4C77BA37>16 D<15FE913803FFC0020F7F023F7F91387F83F890 3901FE01FC49486C7E495A4948137F495A5C4948EB3F80137F49C7FCA2485A000316C05B 120749147F120FA25B001F168016FF123F5B90B7FCA2481600A43AFFC00003FE5BA24B5A A290C7FC4B5AA25E151F5E4B5AA24B5AA26C4AC7FC4A5A5D6C6C485A4A5A6C6C485A4A5A 390FF07F802607FFFEC8FC6C5BC613F0EB3F802A3C77BA38>18 D21 D<01FF151E007F6D147FA217FFB5C8FC7E00015D17FE5B1603000316FC160749EC0FF817 F00007151FEE3FE04915C0EE7F80000F15FF4B130049495A4B5A001F4A5AED3FE049495A 4B5A003FD903FEC7FCEC0FFC9038C01FF0ECFFC0D87FC390C8FCEBDFFCEBFFF01480D8FF FCC9FC13E0007ECAFC302577A43B>23 D<0107B712FE011F16FF137F48B9FC5A4817FE48 17FC281FE007801FC7FCD83F00141E007C130F4814004849133EC7FCA2143EA2167E147E 147C14FCA20101147F5C1303A2494880A2010F81A2494880163F013F815C161F4A5C705A 011EC76C5A382579A345>25 D33 D<121FEA3F80EA7FC0EAFFE0A213F0A3 127F123F121F1200A3120113E0A2120313C01207EA0F80A2EA1F00123E5AA212300C1B72 8A26>59 D<0103B8FC19F04917FC19FF6D188090260003FEC7000713C0060113E0020792 38007FF04B153F191F1AF8140F5DA3021F17F05DF13FE0197F023FEEFFC04B4A13804E13 00F00FFE027FED3FF84B903801FFE092B712804EC7FC91B7FC727E922680000313E09438 003FF849EE0FFC92C86C7E727EA2497013805CA28413074A5DA3010F4C13005C4E5A4E5A 011F161F4AED7FF84E5A013F030713C0053F5B007FB848C7FC60B912E018806C04F8C8FC 453976B856>66 D<0103B912F81AFC5BA27FD9000390C8127FF107F84A16034B15011900 A2140F5DA21A78021F15704B14F01AF8050114F0023F5D4B1670050314001707027F4A5A 4B137F92B6FCA291B75AA39238C0007F49031FC8FC5D1AE0F101F049031E14E092C7FC19 031AC049031C13074A91C71380190FF11F00130F4A163E197E19FE011F4C5A4A1503F00F F8013F163F943807FFF0007FB9FC61BAFCA26C60463977B854>69 D<0103B612F0A25BA26D5DD900030180C8FC93C9FC5C5DA3140F5DA3141F5DA3143F5DA3 147F5DA314FF5DA35B5D1938197C49177892C9FC19F819F04916014A16E01803A2010FEE 07C04A150F181FF03F80011F167F4AEDFF001703013F150FEFFFFE007FB8FC60B9FCA26C 5F3E3977B84F>76 D<0103B500C00407B5FC704C14804961A26D4F1400D90003067B90C7 FCDBDFF0EE7FFE020718F7039F933801E7FCA2F203CF91260F8FF8ED078F030F93380F0F F8A2F21E1F021F173C91261E07FC5F1A78F2F03F023EEE01E0023C616F6CEC03C0963807 807F147C0278DC0F005B6F6C141E1BFF02F85E02F04C5C616F6D5C0101EE01E04ADB03C0 91C8FCA24E485A010391397FC00F004A031E5CA24E130701075E4AD93FE05D60DDE1E013 0F130F91C7D9E3C05C93381FF78005FFC7121F5B011E4B5D013E5D017E020F153F48B46C 5C007F01FC4A017FB512E06E4A82B594B6FC705A6C494A6D5C613976B86B>I<0103B56C 020FB512807017C0496E5CA26D6E6E1480D900036D9139007FE000F21F804A6D93C7FC03 9F161E82DB8FFF153EEC0F8703076D143C8171147C4A6C7FDA1E00167883047F15F8023E 80023C6D6C5C161F711301027C130F02786E5C70138070140302F815C04A6D5D18E070EB F0070101157F4A03F85B173FF0FC0F0103ED1FFE4A020F91C8FC18FF715B0107179F4A6E 13DE8319FE010F8191C95B187F183F5B011E705A133E017E160F48B46C1507007F01FC5E 6E1503B5FC18016C496F5A523977B859>I83 D<007FB500F00103B512F06F16F8B65CA26C4A6D14F026007FF0C8381F FC004948ED07F0614A5EA248170F615CA248171F96C7FC91C9FCA2485F183E5BA2000F17 7E187C5BA2001F17FC605BA2003F1601605BA2007F1603605BA2170700FF5F5B170F6017 1F4DC8FC127F177E6D5D003F4B5A6DEC07F0001F4B5AD80FFCEC3FC06C6C49B45A2803FF C01FFEC9FC6C90B55A6C15F0013F14C0010F91CAFC010013F0453B75B84E>85 D<007FB500E0020FB512806F5CB6FCA26C4A6E140026007FFCC8000113E0013F4DC7FCF0 03F86D6C4B5A4E5A6D6CED1F804EC8FC6D6D147E606D6D495A4D5A6D6D495A4D5A6D6D49 5A4DC9FC6E6C137E5F6E6C485A4C5A6E6C485A4C5A6E6C485A4CCAFC6E13FE5E6E5B5E6E 5B5E6E5B5CA393CBFC5CA35D1407A35D140FA34A5AA2013FB512F8825B5E7F493977B844 >89 D96 DII100 D107 D<01FCDAFF80EB07FC2703FF800701F090383FFF8048D9C01F01FC90B512E0486D486D48 803F1F0FF0FF01FF07F80FF8001E9026F9F80090380FC007273C07FBF0903A7F9F8003FC DAFFE0ECBF00007C028014FC007891C75B5B00F8495D00F0495D4A5DD8001F03FF14074A 4B5C4A92C7FCA2013F4A150F634A5C1A1F017F02035E1A3F4A4AEDC01EA201FF0207037F 133EF3803C91C74902FF137CF3007848030F17F8F301F0494B150397387F87E0F3FFC049 4B6E1300F20FFED800F0DA0780EC03F8572678A466>109 D<01FCECFF802703FF800713 F048D9C01F13FC486D487F3B1F0FF0FF01FF001E9038F9F800273E07FBF06D7E003CEBFF E0007C1480007891C7FC5B00F85B00F05B5CD8001F15FF4A92C7FC5CA2013F5C5F5C1603 017F5D16074A9138F803C0A201FF020F130705F0138091C7001F130F05E01300485F183E 49167E706C5AEFFFF8496E13E004015BD800F06E6CC7FC3A2678A449>I112 D115 D<143814FE1301A31303A25CA21307A2 5CA2130FA25C007FB512FEB7FC15FEA27E39003FE000A25CA2137FA25CA213FFA291C7FC A25AA25BA21203A249130FA20007141F153E49133C157C15F8EC01F09038FC07E00003EB 1FC06CB512806CEBFE00EB7FF8EB0FE0203579B32F>I120 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fj msbm7 7 2 /Fj 2 91 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Fk cmsy7 7 17 /Fk 17 113 df<007FBB12F0BC12F8A46C1AF04D06739F68>0 DI3 D<177017F883B3B3A2007FBC12F0BD12F8A46C1BF0CA00FCCAFCB3B0007FBC12F0 BD12F8A46C1BF0555677D268>6 D14 D<93B812F0031F17F892B9FC1403140F023F18F04A0180CAFC902601FFF0CBFC49138049 48CCFCEB0FF8495AEB3FC0495A49CDFC5B485A485AA2485A485AA2485A5BA2123F90CEFC A25A127EA312FE5AAC7E127EA3127F7EA27F121FA27F6C7EA26C7E6C7EA26C7E6C7E7F6D 7E6D7EEB1FF06D7EEB07FE903803FFC06D13F06D6CB47E6E90B812F0020F18F81403EC00 7F151F030017F04D4E73C368>26 D<1C781CFC881C7EA31C7F88A2891C1F891C0F89767E A2767E767EA2767E1D7F777E1EE0777E777EF507FE9A3801FF807713E0007FBF12F8C1FC 2080A220006C1EF8D113E0531380E307FEC7FCF50FF8535A535A1E8053C8FC1DFE525AA2 525A525AA2525A651C1F651C3F9AC9FCA2641C7EA31CFE641C78713E77BB84>33 D48 DI<047FB512FE0307B7FC153F4AB8FC14074A16FE023F01C0C8FCDA7FFCC9FC903801FF E04990CAFC495AEB0FF8495AEB3FC0495A49CBFC5B485A485AA2485A485AA2485A5BA212 3F90CCFC5A127EA412FE5AA2BA12FE19FFA419FE00FCCCFCA27E127EA4127F7E7F121FA2 7F6C7EA26C7E6C7EA26C7E6C7E7F6D7E6D7EEB1FF06D7EEB07FE6D7E6D13E09038007FF8 91383FFFC0020F90B612FE6E16FF1401EC003F150FDB007F14FE404E74C359>I<041FB8 12C00307B912E0037F18C00203BAFC020F1980023FF0FE0091BA5A4919E0010701F0C7D8 7F80C8FCD90FFCC812FFD91FE04A90C9FC49485D017F150349484A5AA24890C8120F495E 4848151F01F05ED80180153FCAFC60177FA24D5AA35E60A25EA295CAFCA25EA25FA2160F A25FA2161F5FA3163F5FA2167F5FA34C5AA25F5DA294CBFC5D5E15075E1A074B48ED1F80 4C037FC7FC031F4B5A4B484A5A5E4B484A5A4BC8485A90BA5A000361000F19804896C8FC 4818FC4818F0BA12806C05F8C9FC534F80CE4B>73 D<92381FFFE00203B6FC021F15E091 B712FC010316FF010F17C0499026F0003F7FD97FFEC7000113F8D9FFF09138003FFC4801 80ED07FE4848C9EA01FF4848701380D80FF0EF3FC04848171F49EF0FE04848EF07F090CB FC48F003F8007E1801A200FE19FC481800B3B3AD007C197C00381938464877C559>92 D102 DI<127812FCB3B3B3B3B3B3A61278067470D6 27>106 D<0038140E007C141F00FC1580B3B3B3B3B3B3A4007C15000038140E21746FD6 44>I<1E381E7C1EFC1D01A2F503F81EF01D071EE01D0F1EC01D1F1E801D3FF57F00A21D FE651C01651C03651C07651C0F651C1F525AA252C7FC1C7E1CFE641B01641B03641B0764 1B0F515AA2515A99C8FC631B7E1BFE631A01631A03505AA2505A631A1F6302C0173FD903 E095C9FC010760D91FF0177E013F18FED9FFF85F48180100076D4C5A5A263FDFFE4C5AD8 7F0F6000FE6D160FD8F8076000706E151FC66C606F153F6D95CAFC6F5D6D177E6F15FE02 7F4B5A81023F4B5A6F5D021F15076F5D020F150F6F5D6E151F705C6E153F7049CBFC8070 13FE6E5DEEF001037F5CEEF803033F5CEEFC07031F5CEEFE0F030F5CEEFF1F6F495A17BF 6F01FFCCFC5F815F815F167F5F163F5F161F5F160F5F667576856D>112 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fl cmex10 10 36 /Fl 36 113 df0 D<127812FC7E127F6C7E6C7E6C7E6C7E7F 6C7E6C7E12007F6D7E6D7E80131F6D7E806D7EA26D7EA26D7E817F81147F816E7EA26E7E A281140F811407A2818082A28082A28280A282157FA282A3153F82A46F7EA48281A51780 A381A317C0A881A217E0B3AB17C0A25DA81780A35DA31700A55D5EA44B5AA45E157FA35E A215FF5EA25C5EA25E5CA293C7FC5C5DA2140F5D141F5DA24A5AA24A5A5D14FF5D5B92C8 FC495AA2495AA2495A5C495A133F5C495A49C9FC5B1201485A485A5B485A485A485A48CA FC12FE5A12782BC779864C>III8 D<127CEAFFC013F813FF6C13C0001F13F8000713FEC66D7E013F7F010F13 F001037F6D7F6D6C7E6E7E6E7F6E7F6E7FA26E7F80A28280B3B3B3A282157FA282153F82 816F7F836F7F6F7F6F7F6F7FEE3FFF707F040713E0040113F89338007FFE94381FFFC005 0713F01700181F18FF1707051F13C094387FFE00933801FFF8040713E0041F13804C90C7 FCEEFFFC4B5B4B5B4B5B4B5B5F4B90C8FC5D5E157F5EA215FF5EB3B3B3A25C5EA25C4A5B A24A5B4A5B4A90C9FC4A5A49485A495B010F5B013F13C090B55A000749CAFC001F13F800 7F13C0B5CBFC13F813C0007CCCFC3CC76E8661>I<1238127C12FEB3B3B3B3B3AE127C12 38076C688337>12 D[<19FE18011803F007FCF00FF8F01FF0F03FE0F07FC0F0FF804D13 004D5A4D5A170F4D5A604D5A4D5A17FF4C5B4C90C7FC5F16074C5A4C5AA24C5A4C5AA24C 5A5D5F4B90C8FC5D5E150F4B5AA24B5AA24B5AA24B5A5C5E5C5E5C93C9FC5CA24A5AA25D 143F5D147FA24A5AA3495BA25B5DA25B5DA25B92CAFCA25B5CA2133FA25C137FA35C13FF A3485BA5485BA45A5CA45AA25CA45AA491CBFCA25AA75BA2127FAB5B12FFB3B2127F7FAB 123FA27FA77EA280A47EA480A27EA4807EA46C7FA56C7FA3137F80A3133F80A2131FA280 7FA2817FA2817FA2817FA26D7FA36E7EA2143F81141F81A26E7EA2808280828082806F7E A26F7EA26F7EA26F7E150782816F7F8381707EA2707E707EA2707E707E160383707F707F 177F717E717E84717E1707717E717E711380F07FC0F03FE0F01FF0F00FF8F007FCF003FE 18011800>63 298 99 134 99 16 D[<127EB4FC7F6C7E6C7E6C7E6C7E6C7E6C7E6C7E6C 7F6D7E806D7E131F6D7E6D7E806D7E6D7F7F816E7E6E7EA26E7E6E7EA26E7E81806E7F82 80826F7EA26F7EA26F7EA26F7E82818381838183A26F7FA2167F83163F83A2707EA3707E A28482A28482A28482A28482A284A2177F84A3173F84A3717EA5711380A419C083A419E0 A283A419F0A483A219F8A783A219FCAB187F19FEB3B219FC18FFAB19F8A25FA719F0A25F A419E0A45FA219C0A45F1980A44D1300A54D5AA360177FA36017FFA260A25E60A25E60A2 5E60A25E95C7FCA24C5AA34C5AA25F167F5F16FFA24B5BA25F5D5F5D94C8FC5D5E4B5AA2 4B5AA24B5AA24B5A5E5C5E4A90C9FC5C5D4A5AA24A5A4A5AA24A5A4A5A5D5B4990CAFC49 5A5C495A495A133F495A5C495A4890CBFC485A485A485A485A485A485A485A90CCFC127E >63 298 122 134 99 I[81 398 94 134 122 I[<127EB4FC7F6C7E6C7E6C7E6C7E6C7E6C7E6C7E6C7F6D7E6D7E6D7E 806D7E13076D7E6D7E816D7F6E7E6E7E141F816E7E6E7E81806E7F826E7F157F826F7E15 1F826F7E1507826F7FA26F7F8381707EA2707E83161F83160F838284707FA2707FA2707F A2717EA284173F84171FA2717EA28583858385A28385A2717FA2187F85A2183F85A2727E A38684A28684A28684A286A284A286A28486A3197F86A3193F86A4737FA5737FA58785A4 87A285A487A385A487A585A387A71A7FA387AD861C80B3B3A61C0062AD63A31AFFA763A3 61A563A461A363A461A263A46163A54F5BA54F90C7FCA462197FA36219FFA36260A262A2 60A262A26062A26062A26097C8FCA34E5AA261187FA26118FFA24D5BA2615FA2615F615F 96C9FCA24D5AA2173F60177F60A24D5AA24C5BA24C5BA24C5B95CAFC5E5F161F5F163F5F 4C5AA24C5A5D5F4B5BA24B90CBFC5E150F4B5A5E153F4B5A5E15FF4A5B5E4A90CCFC5C5D 4A5A4A5A5D143F4A5A4A5A495B92CDFC495A495A130F495A5C495A495A495A4890CEFC48 5A485A485A485A485A485A485A90CFFC127E>81 398 122 134 122 I[ 44 398 87 134 88 I[44 398 126 134 88 I[<1B7FF201FF1A0F1A3F624FB5FC19074F13FC073F 13F04F13C04EB512004E5B4E13F84E5B4E5B067F138095B5C7FC4D5B4D5B614D5B4D5B5F 4D5B4D5B96C8FC94B5FC4C5B605E605E605E60A25E60A25E60A35E95C9FCB3B3B3B3B3B3 A34C5AA54B5BA34B5BA34B5BA24B5BA24B5BA24B5B94CAFC5D4B5A5E4A5B5C4A5B5E4A5B 4A5B4A90CBFC4A5A4A5A495B01075B4913C0495B017F90CCFCEBFFFC00035B485B001F13 C0007F90CDFCEAFFFC5B13C0A213F87F6CB4FC001F13C0000713F06C7FC67F6DB4FC011F 7F6D7F6D13F001017F6D7F6E7E6E7E6E7F6E7F6E7F826E7F806E7F826F7E81836F7FA26F 7FA26F7FA26F7FA36F7FA36F7FA5707EB3B3B3B3B3B3A38482A38482A28482A284828482 848284707F8385717F717F83717F717F85717F717F727F061F13E0727F727F7213FE727F 726C13C07313F0070F13FC7313FF1901737E861A0F1A01F2007F>80 398 106 134 125 26 D[<127EEAFFC013F013FC13FF806C13E06C13F8000F7F000313FF C6806D7F011F13F06D7F01037F6D7F6D7F6E7F6E7F6E7F6E7F6E7F6E7F6E7F806F7E8381 6F7F838183818381A28381A28381A38382B3B3B3B3B3B3A3707FA5707FA3707FA3707FA2 707FA2707FA2707FA2717E83717F8583717F717F717F717F727E727E727F727F727F727F 727F9538007FFE737E7313C0070713E07313F8070013FEF23FFF1A0F1A03A21A0F1A3FF2 FFFE070313F84F13E0071F13C04F13004F5A953801FFF84E5B4E5B4E5B4E5B4E90C7FC4E 5A4E5A4D5B4D5B4D5B4D5B5F614D90C8FC5F4D5AA24C5BA24C5BA24C5BA24C5BA34C5BA3 4C5BA54C90C9FCB3B3B3B3B3B3A393B5FC5FA35D5FA25D5FA25D5F5D5F5D5F4B5B5D94CA FC4B5A5C4A5B4A5B4A5B4A5B4A5B4A5B91B5CBFC495B495B010F5B495B017F13C090B55A 000391CCFC000F13FC003F5B4813E0B5128091CDFC13FC13F013C0007ECEFC>80 398 106 134 125 I[86 497 89 134 131 32 D[<127F5A7F6C7E6C7E6C7E6C7E7F6C7E6C7E6C7F7E806D7E6D7E 6D7EA26D7E6D7E806D7F7F816D7F6E7EA26E7E6E7EA26E7E6E7EA26E7F82806E7F82157F 826F7EA26F7EA26F7E6F7FA26F7FA26F7FA26F7F83167F83163F83161F838284A2707FA2 707FA2707FA2848284177FA2717EA2848385A2717FA28385A2717FA3717FA3717FA28518 7FA28584A28684A286A28486A28486A38486A3727FA4727FA38685A38785A387A285A387 A285A287A385A287A48587A58587A6737FA61C80A286A61CC0A386A51CE0A586A51CF0A8 86A41CF8B186A21CFCB3B3B11CF8A262B11CF0A462A81CE0A562A51CC0A562A31C80A697 B5FCA21C00A64F5BA66361A56361A463A261A363A261A263A361A263A36198C7FCA396B5 FC62A34E5BA44E5BA36260A36260A26260A262A26097C8FCA26061A218FF61A24D5BA34D 5BA34D5BA2615FA24D5BA296C9FC5F60A24D5AA217FF605E60A24C5BA24C5BA24C5BA295 CAFC5E5F163F5F167F5F16FF5F4B5BA24B5BA24B5BA24B90CBFC4B5AA24B5AA24B5A5E15 FF5E4A5B5C5E4A90CCFCA24A5A4A5AA24A5A4A5AA24A5A495B5D5B4990CDFC5C495A495A A2495A495A495A5C5A4890CEFC485A485A5B485A485A485A485A90CFFC7E>86 497 123 134 131 I[50 497 83 134 97 I[50 497 127 134 97 I[85 497 104 134 134 40 D[<127FEAFFC013F013FC7F6D7E806C13F06C7F000F13FE6C7F00 01806C806D7F011F7F6D13FC6D7F6D7F6D807F6E7F6E7F6E7F6E7F6E7FA26E7F6E7F8380 6F7FA26F7F838183818381A26F7FA28381A28481A38482A5707FB3B3B3B3B3B3B3AC707F A6707FA4707FA38284A28284828583A2717FA2717FA2717F717FA2717F8385717F84727F 86727F84727F727F727F727F737E737F737F7313F007037F737F737F97383FFF807413C0 7413F0080313F81A001B7F1B1F1B7F1BFF1A03080F13F05013C05013809738FFFE004F5B 4F5B070F5B4F13C04F5B4F90C7FC4F5A4E5B4E5B4E5B4E5B604E5B624E90C8FC95B5FC4D 5B615F4D5BA24D5B4D5BA24D5BA24D5BA294B5FC96C9FC5E605EA2605EA34C5BA44C5BA6 4C5BB3B3B3B3B3B3B3AC4C5BA593B5FC60A35D95CAFCA25D5FA24B5BA25D5F5D5F5D5F4B 5BA292B55A5C94CBFC4A5B4A5BA24A5B4A5B4A5B4A5B91B55A5B4991CCFC495B495B4913 F0017F5B90B55A485C000791CDFC485B003F13F8485BB512C05C49CEFC5B13F013C06CCF FC>85 497 104 134 134 I56 D<387FFFE0B57EB3B3B3B3 AB6C7FA76C7FA47E80A37E80A27E817EA2817E817E817FA26D7F7F816D7FA26D7F7F826D 807F826E7F806E7F6E7F6E7F82806E806E806F7F6F7F6F7F030713FE6F7F6F14806F14C0 7013F0041F13F87013FE82160182173F171F170717013796408194>58 D[<933807FFFE4C13FFB3B3B3B3AB4C13FEA618FCA25EA318F8A25EA218F0A293B5FC18 E0A24B14C0A34B1480A218005D5FA24B5BA24B5BA24B5BA24B5B5F92B5FC5F4A91C7FC5C 5E4A5B5E5C4A5B4A5B5E4A5B91B5C8FC495B495B5D495B495B495B495B4990C9FC48485A 4813F8485B485B003F13804890CAFC485A13F813E05B7F13F813FE6C7E6C7F000F13E06C 7F6C7F6C13FE6C6C7E6D7F6D7F6D7F6D7F6D7F816D7F6D7F6E7F6E7F826E7F6E7F80826E 7F82806E808381836F7FA26F7FA26F7FA26F7FA283811880A26F14C0A36F14E0A218F082 A218F8A282A218FCA382A218FEA67013FFB3B3B3B3AB7013FE>56 303 100 129 148 60 D<387FFFE0B512F0B3B3B06C13E01436408194>62 D<007FC312FCC47EA38FA27E02F0CE6C836CF5001F6C6DF4007F6C6D0B07816EF5007F6C 201F6C6D0C03806C6E1D006F0C3F7F6C210F6D6D8A6F0C017F6D8C6D6D1F7F6D6DF71FFC 6F1F0F6D21076D6D797E6D6E1E0170787E6D8E6E7F70F73F806E211F6E7F6E6DF70FC070 1F07806E6DF703E0836E806EF901F06F6D9BC7FC83816F7F6F7F83816F7F846F8081707F 8482707F707F8482707F85708082717F8583717F717F8583717F86837180727F8684727F 727F8684727F878472808563857390D0FC735A62735A4F5A191F4F5A4F5A624FD1FC604E 5A4E5A4E5A614E5A183F4E5A4E5A4D90D2FC6017034D5A4D5A4D5A604D5A177F4D5A4C90 D3FC4C5A5F16074C5A4C5A4C5A5F4C5A04FFF701F04B90D2EA03E04B5A4B5A4CF707C015 0F4B481F0F4B4820804B481F1F4C1F3F4BD313004A694A48684A48555A4A481F034B1F07 021F565A4A481F1F4A481F3F4A48555A92D2EA01FF49486701030E0F5B49486749481F7F 494854B55A4A1E0F013F1F7F49480B03B65A49481D3F4890D0000FB7FC490A3FB85A48C4 FC5A48A0C7FC5A5A6B5AC4FC6B7E9CA6777FAF>80 D82 D<007FC5B97EC5BA7EA3A185A2A1857E4BD26C856C6EF700036CFB00036C6FE6000F8270 F9007F6C6F1007816CA114006C6F111F807023036C6F23006DA1033F8070A1120F6D6EA1 0003806DA115006D6FA1003F7F71A17E6D6FA112076DA1707F6D6FA1120071A16D7E6D6F A1133F6EA1707E6E6EA1130F71A113076E6FA16C7E6EA1160172A16C7E6E6FA17F6EA1EF 3F806E6FA1131F846E6FA1EB0FC06FA116076F6EA114E072A113036F6FA113F06FA11601 856F6FA1EA00F86FA117006F81856F818270808570818286708182708186708183718086 718183877181718183877181847280877281847281887281848872818573808873818573 81897381858973818674808974818674818A7481868A74818775808A75818775818B7581 878B87765C765C6788765C7691D4FC66765B765B765BA2525B525B525B5290D5FC525A65 1C3F525A525A515B515B515B9AD6FC63515A515A515A515A505B6462505B5090D7FC505A 505A505A631AFF4F5B4F5B4F5B4F90D7C6FC4F5A4F5A62197F4F5A4E5B4E5B4E5B4E90D7 C7FC61181F4E5A4E5A4E5A4D5B4D5B615F4D90D7C712F84D48A1EB01F04D5A4D5A4D48A1 EB03E0604CA115074C49A1EB0FC04C5B4C90D7C7EA1F804C48A1143F4C5A4DA1EC7F0004 7FA15D4C48A15B4B49A1495A4B49A113074B49A1130F4B90D7C7485A4CA1143F031FA14B 5A4B48A1EB01FF4B48A15B4B48A1495B4A49A1131F4A49A1137F4CA190B55A4AA114034A 90D7C6000F5C4A48A1133F4A48A148B6FC4A48A100075D4A48A1123F49491103B7FC4B11 3F93C7FC49A10003B8FC49499FB85A4990D5B9FC49480E1FBAFC49C5B85A5BA290C5B95A 5A5A48A1605A48A160A25A48A160C5BAFCA26CA160DDE8777FF0>88 D[ 147 369 119 127 92 90 DI<9426 01FFFC1E2094B600F01DF0040F03FF1C03047F04E0F31FF80307B800FC1B7F033FDDFF80 963801FFE092BA00F0070F1380020707FE96383FFE00021FDAE0006F6C943801FFF891B5 00F0C7000303F0050F13E0010391C9003F02FE057F1380010F01F00403DAFFE0922607FF FEC7FC013F0180DC007F02FE037F13F8D9FFFCCB000FDAFFF8013FB512C0000301E00603 92B8C8FC000F0180DE007F18F8D83FFCCD000F18E0D8FFF0080195C9FC01C0E0003F16F0 007ECF000716800078E2007F02F8CAFC00200B0101FCCBFCA51680FCA6>102 D[39 298 92 134 78 104 D[39 298 126 134 78 I[<1A7FF103FF191F197F4EB5FC1807181F067F13FC95B512E0 050314804DEBFE004D13F8053F5B4D13C094B55A4C49C7FC4C5B4C5B4C5B4C5B604C5B4C 90C8FCA24C5A5F5D5F5DA25FA25D5FB3B3B3B3AB5D5FA34B5BA25D94C9FCA24B5A15FF5E 5C4A5B5E4A5B4A5B4A5B4A90CAFC4A5A4A5A495B01075B495B495B017F90CBFC48485A00 0713F8001F13E0007F13804848CCFC13F813E0A213F813FE6C6C7E001F13E0000713F800 0113FE6C6C7E011F13C06D7F6D7F01017F6D7F6E7E6E7E6E7F6E7F6E7F6E7F826E7F8082 157F6F7EA28381A26F7FA38381B3B3B3B3AB8381A283A281838183707EA2707F707F8470 7F707F707F707F706D7E717F7113F0050F7F7113FE71EBFF80050014E07213FC061F13FF 18071801727E191F1903F1007F>72 298 109 134 111 110 D[<127EEAFFC013F813FE 6D7E14E06C13F86C13FE00077F000114C06C6C7F011F7F6D13FC01037F6D7F6D6C7F6E7F 6E7F6E7F6E7F806E7F6E7FA26F7E81838183A281A28381B3B3B3B3AB8381A36F7FA28381 A2707E838284707F82707F707F707F707F717E717E717F7113E0717F717F050013FE9538 7FFF80061F13E0060713F8060113FE9538007FFF191F1907A2191F197F953801FFFE0607 13F8061F13E0067F13809538FFFE00050313F84D5B4D5B4D13804D90C7FC4D5A4D5A4C5B 4C5B4C5B4C5B5E4C5B95C8FC5E5F4C5AA25D5FA24B5BA35D5FB3B3B3B3AB5D5FA25DA25F 5D94C9FC5D4B5AA24A5B4A5B5C4A5B4A5B4A5B4A5B49B5CAFC495B010F5B4913F0017F5B 48B55A000791CBFC003F5B4813F8B512E0148049CCFC13F813C0007ECDFC>72 298 109 134 111 II E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fm msbm10 10 6 /Fm 6 91 df67 D<003FBEFC4889BFFCA27E7E29001FF00007F0C7ECE01F01034A48020313F86E4A913800 7FFC0101F10FFE041FEE03FF4D0300139FF43FDFF41FFF8888888888A21D7FA21D3FA31D 1F62F203C0507EA277C7FC9AC8FCA21A0FA51A1FA21A3FA21A7FA21AFF6161190FF11FF7 F17FE7953803FFC795B5128794B5EAFE0719FC19F019F819FEF1FF87DD803F13C7060113 E79538003FF7F10FFF190385851A7FA21A3F1A1FA21A0FA41A07A5745A97CBFC1E0E1E1F 1F801E3F1F00A266A366535AA21D031D071D0F1D1FF57FFC1DFF64F407FEF41FFCF47FF1 71923901FFE1F8040F040713C10103073F13014A6E913803FFFC011F6E6C027F13F0003F BEFC4865BFFCA27E6C6569727EF14C>69 D<007FB600FE057FB61280B894B712C0715D83 876C7070158000039027800007F00400ECC000C66C6C6EDD1FFEC7FC6D6C0103725AD90F F06D6C17076D6C6D7E01036F606D6C6E6C16036E143F6E6E7E717E6F806F13076F6D7E6F 6D7E846F1300DAFDFC147F6F6E7E02FC82037F6E7E6F6C130F706D7E6F6C6D7E030F816F 6C13016F6C6D7E70147F6F6C810300153F047F6E7E706C6D7E7180041F6E7E706C130370 6C6D7E716D7E706C80040181706C6E7E057F6E7E7280716C130F051F6E7E716C6D7E716C 80726D7E05031400716C147F716CEC3F807215C0726C131F063FEC0FE0726CEB07F0726C 14F8731303726CEB01FC0603EC00FE726C14FF726CEC7F8373143F73EC1FC3736CEB0FE3 736C14F3741307736CEB03FB0707EC01FF737E736C7F74147F0700153F1A7F746C131F75 130F746C1307080F1403747E746C130187747E1A001B7FF33F801CC01B1FF30FE0F307F0 1CF8F303FC1B01F300FE1C7F1D81F43FC11C1FF40FE1F407F16E1AF901031A03F401FD49 6CF100FF497F013F01F0197F001FB600F0183F486F181F481C0FA26C1C076C4B1803D012 011D001E7072757DF155>78 D<003FB912FE48F0FFF8BC7E1BF06C1AFE6C747E2A003FE0 0007FC0015E00107DA0FF0010F806E02C0D907F813FC0103021F913901FC1FFE4D0100EB 03FF74C67F087E6D7E087FEB1FE0746D7E1C07756C7E081F6D7EA2767E1D7EA275137F08 0F80A31E801D1FAA1D3F1E00A3081F5C51137E1DFE651C01083F495AF30007525A50495A 087E495A525A9726FE01FFC7FC963901FC07FE0703EB1FFC4F48B45ADF0FF75B077FB512 C094B8C8FC1BFC1BF01B8008FCC9FC1A800580CCFCB3B3A50107814A130F013F15F8003F B812FE4883BAFCA27E6C5F61727EF157>80 D<003FBA12C048F1FF80BC12F81BFF6C1BC0 6C1BF82B001FF00003FE003F14FE0103DA07F80103806E02E09026007FCF13C00101020F DB3FC17F4D91390FE07FF8973907F00FFC767E973903F801FF08016D7F75EB3FC0080014 1F777E756D7E1B7E777E1D01A2097F80751300A28A1E7EAA1EFE66A3511301097E5CA21D 03535A1BFE51495A535A0801147F51495A0803010390C8FC973907F00FFE080FEB7FFC97 393FE3FFF04FB65A071F158005C1B748C9FC94B812F81CC051CAFC1BC0639539FE001FC0 DDC07F6D7E1980063F6D7E726C6C7EF00FE0747E726C6C7E726C7F87726C6D7E726C6D7E 85736D7E736C6C7EDF1FC07F1B03736C6C7E736C6C7E860703147F736C6D7E736C801C1F 087F6D7E746C6C7E8997381FC003746C6C7E756C7E080780746C6D7E746C6D7E1D1F746C 6D7E097F6D7E8A756C6C7E756C6C7E766C7E090F6E7E756C6D7E01036F6F6C6D7E4A0107 F10FF8011F03FC6F6C6D7E003FB96EB712C0487218E0BA81886C876C95C86C15C073727E F159>82 D<013FBC12FE498790BEFCA365D9FC070280C739FE0001FC0001011F01E0C748 5A494848C849495ADA7FF003035D9026F9FFC04B48130701FB90C94B5AD9FFFC4C5A4A4D 495A4A041F4A5A4801C04C5A4A4FC7FC91CA48C7127E4905FE14FE515A494C5A494D495A 06034A5A494C5A0007505A4E485C494C48131F515A4E5A96C748C8FC6C484C147ECB00FE 14FE505A4D5A4E495A05034A5A4D5A505A4D485C4D48131F505A4D5A95C748C9FC4D14FE 17FE4F5A4C485C4C4813034F5A4C5A4D495A040F4A5A4C5A4F5A4C4891CAFC4CC75A19FE 16FE4C495A03015D4B4813034E5A4B5A4C495A030F4A5A4B5A4E5A4B4891CBFC4BC75A18 FE15FE4B495A02014A5A4A5A4D5A4A484A17704A48010F18F84D4816014A5A4B495A023F 92CA12034AC75A05FE18F014FE4A4948170701014A5A49481A0F4C5A49484AEF1FE04948 130F4C48173F495A4A4948177F013F4ACB12FF49C76004FE4D13C001FE4A5F4848010160 4B48EF1FEF4848F23FCF494948EF7F8F00074A48933801FF1F4848DF07FE13804B48EE0F FC48484AEE7FF84848013F933801FFF04BC9001F13C048C74CB5EA003F90BE1200BFFCA4 6C6465727BF161>90 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fn cmr7 7 55 /Fn 55 127 df3 D7 D<00F0151EA46C153E0078153CA2007C157C003C 1578003E15F8003F14016C6CEB03F06D13076C6CEB0FE0D807F8EB3FC03A03FF01FF806C 90B512006C5C6D5B6D5B010F13E0010190C7FC271673D042>21 D24 D<153C157C15FCEC01F0EC03E0EC0FC0EC1F80A2EC3F 00147E5C1301495A495AA2495A495AA2495AA249C7FC5B5B1201A2485AA25B1207A2120F 5BA3485AA3123FA25BA3127FA35BA312FFB3A2127FA37FA3123FA37FA2121FA36C7EA37F 1207A212037FA26C7EA212007F7F6D7EA26D7EA26D7E6D7EA26D7E6D7E1300147E80EC1F 80A2EC0FC0EC03E0EC01F0EC00FC157C153C1E7473D634>40 D<127012F8127C7E7EEA0F C06C7EA26C7E6C7E6C7E7F137F6D7EA26D7E6D7EA26D7EA26D7E80130180A26D7EA28015 80A215C0143FA3EC1FE0A315F0A2140FA315F8A31407A315FCB3A215F8A3140FA315F0A3 141FA215E0A3EC3FC0A3147F1580A215005CA2495AA25C13035C495AA2495AA2495A495A A249C7FC13FE5B485A485A485AA2485A48C8FC123E5A5A12701E7478D634>I<177817FC B3B3A3007FBC12F0BD12F8A46C1BF0CA00FCCAFCB3B3A31778555678C766>43 D48 DIIII<486C151CD803F015FC01FE1407D9FFF013 FF91B65A5F5F5F94C7FC5E16F85E168001EF01FCC8FC9038E03F8091CAFCAEEC07FF027F 13E001E1B512FC01E780903AEFFC03FF809026FFC0007F91C7EA3FE001FC6E7E4981496E 7E496E7E6C5AC96C7EA21880A27013C0A318E0A4EA1F80EA3FE0487E12FF7FA318C0495C A24916806C5A90C8481300127E6C4B5A5F6C6C141F6C6C4A5A01F04A5A6C6C4A5A6CB401 035B6CD9E01F90C7FC6C6CB55A6D14F8010F14E0010314809026003FF8C8FC334F79CC42 >IIIII<007FBC12F0BD12F8A4003F1BF0D0FCB3A200 3FBC12F0BD12F8A46C1BF0552078AC66>61 D<177C17FEA24C7EA34C7FA34C7FA24C7FA3 4C7FA24C7FEE3E7FA2047E7FEE7C3F04FC7FEEF81FA20301804C7EA2DB03E08083030781 4C7EA2030F814C7E031F8193C7FCA24B81033E147F037E81037C143FA203FC814B141F02 01825D840203834B80A20207834B804AB87EA34A83A24AC97F023E167FA2027E83027C16 3F02FC835C191F0101844A82A20103854A8201078585130F011F85013F83D97FF84C7F26 03FFFE040F13FCB600C00207B612FEA557527BD162>65 DI<932607FFE0141C93B500FC143C0307DAFF 80137C033F15E092B700F813FC02039139801FFE01020F903AF00001FF034A0180903800 7F83DA7FFEC8EA1FC7DAFFF8ED0FEF4901E0ED03FF4949814949814948CA127F4948173F 4948171F137F4948170F484917075C481903485B1A014890CBFCA24819005BA2123F4919 7CA2127FA21B005B12FFAD127F7FA21B7C123FA27F121FA26D19FC6C1AF8A26C7F1A016C 6D18F06C6D17031BE06C6D17076D6C18C0013F180F6D6CEF1F806D6CEF3F006D6C6C5E6D 6D16FE6D6D4B5A6D01F8ED07F8DA7FFE4B5A91261FFF80EC3FE06E01F0903801FF800203 9026FF801F90C7FC020091B512FC033F15F0030715C003004AC8FC040713E04E5378D05F >II II<932607FFE0141C93 B500FC143C0307DAFF80137C033F15E092B700F813FC02039139801FFE01020F903AF000 01FF034A01809038007F83DA7FFEC8EA1FC7DAFFF8ED0FEF4901E0ED03FF494981494981 4948CA127F4948173F4948171F137F4948170F484917075C481903485B1A014890CBFCA2 4819005BA2123F49197CA2127FA298C7FC5B12FFAD007F053FB612FE7FA3123FDE0001EB FE006D716C5A001F193FA27F7E807E807E6C7FA26C7F6D7E6D7E6D7E6D7E6D6D167F6D7F 6D01F016FF6D01FC5D6EB45D021F01C0EC0FF36E01F8EC7FE10203903AFFC007FFC00200 91B6EA007C033F4B133C030703F0131C0300038090C7FC040701F0C9FC575378D067>I< B712F0A5D8003FEBC000010F90C7FCB3B3B3AD013F13C0B712F0A5244F7ACE30>73 D75 DIII<93380FFF804BB512FC030FECFF80037F15F0912701FFF80013FC020701C0 EB1FFF4A48C700037FDA3FF8020013E0DA7FE0ED3FF04A486F7E010349ED0FFE4990C96C 7E4948707F4948707F4948707F4A177F4948717E01FF854849717E4A170F48864890CB6C 7EA2481B804984001F1BC0A24984003F1BE0A3007F1BF04984A400FF1BF8AC6C6C4E13F0 A5003F1BE06D60A2001F1BC06D606C1B80A26C6D4D1300A26C6D4D5AA26C6D4D5A6C6D4D 5AA26D6C4D5A6D6C4C5B6D6C4C5B6D6C4C5B6D6D4B90C7FC6D6D4B5A6D01F0ED7FFC6D6C 6CEDFFF0DA3FFE02035B91260FFFC0011F13806E01F890B5C8FC020190B612FC6E6C15F0 030F1580030102FCC9FCDB000F1380555378D066>II82 DI<003FBC FCA5912680003F90C77E01FCC76C48140FD87FF00603138001C0180049197F90C8173FA2 007E1A1FA4007C1A0FA500FC1BC0481A07A6C994C7FCB3B3A74C7E93B512C0023FB8FCA5 524E7ACD5F>III<003FB600F00103B612E0A5D8000F91C86C01F8C7FC010301 FC031F13806D496F48C8FC6D18F86F6F5A6E6C4B5A6E6C5E6E4C5A70143F6E6D92C9FC6E 6D147E616E6D13016E6D5C6E6D495A4E5A6F6C130F6F6C5C6F6D485A4ECAFC6F6D5A6FEB E07E6F5CEFF1F86F13FB6FEBFFF0705B6082705BA2707F707F707FA2844C7F4C7FA29338 1FBFFFDC3F1F7F4C7E047E804C6C7F4B486C7F03037F04F0804B486C7F4B486D7E151F4C 6D7E4BC76C7F037E6E7F15FE4B6E7F4A486E7F4A486E7F14074B6E7F4A486F7E4A48153F 023F8392C96C7F027E707F02FE82010185D907FF707F011F6D4B7F48B56C031F13FEB600 F84AB71280A5594F7CCE62>88 D<001FB912FEA592C813FC02E04A13F8028016F049C85A 494B13E04917C0495D494B1380491700003F5E494B5A6017FF90C8485B604C5B5E003E4B 5B605E4C90C7FC5F167FC9485A5F5D4B5B5F4B5B5D4B5B94C8FC5D4B5A5E15FF4A5B5E5C 4A5B5E4A5B4A161F4A90C8FC5D147F4A5A5D49173F495B4B153E5B495B5D4990C9127E5B 495A4A16FE13FF485B4A1501481703484915075C48171F4849153F91C9EAFFFC48160348 48157FBAFCA5404F78CE50>90 DI93 D97 D100 D103 D<137E13FF487F487F487FA66C5B6C5B6C90C7FC137E90C8FCAEEB3FE0EA7FFFA51203C6 FC137FB3B3497EB612E0A51B4F7BCE25>105 D108 D<90287FC001FFE0EC3FFCB5010F01FC49B57E033F01 FF010714E092B6D8801F8002C10101903AC03FE03FF8913EC3F8007FE07F000FFC0003D9 C7E090263FF0FC6D7EC6D9CF80ECF1F0D97FDFC7261FFBE06D7E02FEEDFFC04A6E496D7F A24A93C7FCA24A5DA34A5DB3AA496C4A6C4A7FB6D8F01FB5D8FE03B612C0A562337AB26D >II<913801FFC0021F13FC027F13FF49B612C001079038007FF0D91FF8EB0F FCD93FE0EB03FE49486D7E49486D7F4890C86C7E48486F7E48486F7EA2000F8348486F7E A2003F83491507A2007F83A300FF1880AA007F1800A36D5D003F5FA2001F5F6D151F000F 5F6C6C4B5AA26C6C4B5A6C6D4A5A6C6D495BD93FE04948C7FCD91FF8EB0FFC6DB4EB7FF8 010390B512E001001580021F01FCC8FC020113C039357CB342>I<903A3FE001FFC0B501 1F13FC92B6FC02E315C09126E7FE0313F0913AFFF0007FF8000302C0EB1FFCC691C76C7E D97FFC6E7E4A6E13804A16C07113E04A8019F0187F19F8A2F03FFCA4F01FFEAAF03FFCA4 F07FF8A2F0FFF0A26E4A13E019C06E4A13806E5C6E4A13006E4A5A6FEB3FF8DAEFE0EBFF F09126E7FC075B02E3B6128002E04AC7FC033F13F80303138092CAFCB0497EB612F0A53F 4A7CB249>I<903903FFE00E013FEBFC1E90B6127E000315FE3907FE007FD80FF0130FD8 1FC013034848130190C8FC127E167E12FE163EA27E7F7F01F091C7FC13FE387FFFF0ECFF C06C14FC6C14FF6C15C06C816C15F86C816C6C80130F01008002031480EC001F15070078 020113C000F8EC007FA26C153FA2161F7EA26C1680163F7F6D15006D147E6DEB01FE01FC 495A3AFEFF803FF8486CB512E0486C5CD8F00F49C7FC48C613F02A357BB335>115 DII120 D126 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fo cmsy10 10 30 /Fo 30 115 df<003FBE12C0481DE0BF12F0A36C1DE06C1DC0640772AC81>0 DI6 D15 D20 D<1238127EB47E13E013F8EA7FFE6C6C7E000F13E0 000313FCC613FF013F13C0010F13F0010313FC010013FF023F13C0020713F0020113FC6E 6CB4FC031F13C0030713F0030113FC6F6CB4FC041F13C0040713F8040113FE706C6C7E05 1F13E0050713F8050113FE716C6C7E060F13E0060313F8060013FE96383FFF80070F13E0 070313F8070013FE97383FFF80080F13F0080313FC080013FF093F13C0090F13F0090313 FC090013FF0A1F13C00A0713E00A0113F0F4007FF401FF0A0713E00A1F13C00A7F130098 3803FFFC090F13F0093F13C098B5C7FC080313FC080F13F0083F13C0E0FFFEC8FC070313 F8070F13E0073F1380DFFFFEC9FC060313F8060F13E0063F13804DB448CAFC050713F805 1F13E0057F13804C4848CBFC040713F8041F13E0047F90CCFC923801FFFC030713F0031F 13C0037F90CDFC913801FFFC020713F0021F13C091B5CEFC010313FC010F13F0013F13C0 90B5CFFC000313FC000F13F0003F13804848D0FCEAFFF813E01380007ED1FC1238D2FCB3 A9003FBE12C0481DE0BF12F0A36C1DE06C1DC0648372E981>I<912603FF801A18021F01 F81A3891B500FE1A7C01036E7E4915E0011F15F84915FE496F7E90B87E4805F018FC4890 26FC007F6D18F84801E0010701FE170191C700017F48486E6C6D160301F8031F13E04848 03076DEE07F049030101FC160F48486F6DEE1FE04993263FFF80153F48CA6C01E0ED7FC0 060701F8EC01FF007E7101FE4A1380726D6C131F007C716C01F890B5120000FC7290B65A 48060F5E735E07015E736C5D081F1580080792C7FC080114FC0070736C13E0090790C8FC D3FCA6912603FF801A10021F01F81A3891B500FE1A7C01036E7E4915E0011F15F84915FE 496F7E90B87E4805F018FC489026FC007F6D18F84801E0010701FE170191C700017F4848 6E6C6D160301F8031F13E0484803076DEE07F049030101FC160F48486F6DEE1FE0499326 3FFF80153F48CA6C01E0ED7FC0060701F8EC01FF007E7101FE4A1380726D6C131F007C71 6C01F890B5120000FC7290B65A48060F5E735E07015E736C5D081F1580080792C7FC0801 14FC0070736C13E0090790C8FC6E4A77CF81>25 D<053FB912C00407BA12E0043F19F04B BBFC1507031F1AE0037F1AC04AB500E0CCFC020701FCCDFC4A13E04A90CEFCEC7FFCECFF F04913C0495B4990CFFCEB0FFC495A495A5C495A495A91D0FC485A12035B485AA2485A5B A2121F5B123F5BA2127F90D1FCA45A5AAD7E7EA47F123FA27F121F7F120FA27F6C7EA26C 7E7F12016C7E806D7E6D7E806D7E6D7E6DB4FC6D7F6D7F6D13F0EC7FFCEC1FFF6E13E06E 13FC0201EBFFE06E6C90BA12C0031F1AE003071AF01501ED003F040719E0DC003F18C064 6172D981>I<1F3E1F7FA48C1F3FA28C1F1FA28C1F0FA28C1F078C1F038C1F018C797EA2 7A7E7A7E8D201F7A7E7A7E8D7A7E7A7E7A7FF97FE07B7E7B7EF90FFE7B6C7E7B13E00F00 13F8003FC312FE48FAFFC0C512E0A36C23806CFAFE00D513F80F0313E05713805748C7FC F91FF8575A575AF9FF805690C8FC565A565A69565A565A203F69565A56C9FCA2555A681F 03681F07681F0F68A21F1F68A21F3F68A21F7F9DCAFCA41F3E935777D4A6>33 D40 DI47 D49 D<053FB7FC0407B81280043F17C04BB9FC1507031F1880037F18004AB500E0CAFC020701 FCCBFC4A13E04A90CCFCEC7FFCECFFF04913C0495B4990CDFCEB0FFC495A495A5C495A49 5A91CEFC485A12035B485AA2485A5BA2121F5B123F5BA2127F90CFFCA45A5AA3BDFC1C80 1CC0A31C801C0048CFFCA37E7EA47F123FA27F121F7F120FA27F6C7EA26C7E7F12016C7E 806D7E6D7E806D7E6D7E6DB4FC6D7F6D13E06D7FEC7FFCEC1FFF6E13E0020313FC6EEBFF F06E6C90B8FC031F1880030718C01501ED003F04071780DC001F1600526172D96F>I<00 3FB612FE48EDFFF8B9FC18E018F86C17FE6C717EC9000314E0DC000F7F050113FC943800 3FFE95380FFF8006037F727F726C7E737EF10FFC737E737E85737F747E1A3F747E871A0F 747EA2747E1A01A2871A008787A21C801B3FA41CC01B1FA3003FBCFC5ABDFCA37E7ECE12 1FA31B3F1C80A41B7F1C00A263631A0163A21A03505AA2505A1A1F63505A1A7F505A4F90 C7FC614F5A4F5AF13FF84F5A4F5A06035B060F5BDE3FFEC8FC4DB45A050F5B4CB512E000 3FB91280484DC9FCB912F818E095CAFC6C16F86C93CBFC526172D96F>I<1B07F30F80F3 1FC0A21B3F1C801B7F1C0063631A01631A03631A07631A0F631A1F631A3F631A7F98C7FC 6262190162190362190762190F62191F62193F62197F4FC8FCA24E5AA24E5A6118076118 0F61181F61183F61187F96C9FC6060170160170360170760170F60171F60173F60177F95 CAFC5F5F16015F16035F16075F160F5F161F5F163F5F167F94CBFC5E5E15015E15035E15 075E150F5E151F5E153F5E157F93CCFC5D5D14015D14035D14075D140F4A5AA24A5AA24A 5A92CDFC5C5C13015C13035C13075C130F5C131F5C133F5C137F91CEFC5B5B12015B1203 5B12075B120F5B121F5B123F5B127F90CFFC5A5AA2127C1238529B69F600>54 D66 D<063FB912FC053FBAFC0403BBFC043F1AF84BBC12F003071BE0031F1BC0037F1B004ABC 12FC02071BF04A91C8D81FF8C9FC023F90C9123FDA7FF04C5ADAFFC04C5A495B494D5B49 5F4990C95C61494D90CAFC495A4A4C5A5C4948163F4A5F0280167F017ECAFC01204D5A90 CBFC6260A34E5BA26062A34E5BA36097CBFCA260A261A2187FA261A218FFA261A25FA261 A25F61A35F61A25FA261A25F96CCFCA34D5AA34D5AA26017FFA2605E605E60A24C5B1D3C 4C90C9EA01FC4D1607041F170F4DEE3FF84C48167F525A4C5A4C484B5B4D5F4B494B5B4B 90C94890C8FC011FBB5A90BC5A00031BF0000F63481B804850C9FC481AF8BC12C050CAFC 6C19E06E7184F05A>73 D75 DI79 D<0038F301C0007CF303E000FEF307F0B3B3B3B16C1B0F6C1CE0A36D1A 1F003F1CC0A26D1A3F6C6CF27F80A26C6CF2FF006D616C6C4F5A6C6C4F5A6D190F6C6D4E 5A6C01E0F07FF06D6C4E5AD93FFC05035B6DB4050F5B6D01C0043F90C7FC010301F89238 01FFFC6DD9FF80021F5B6D6C01FE0107B512E0021F90B8128002074DC8FC020117F86E6C 16E0030F93C9FC030015F0040301FCCAFC5C6877E36F>91 D<933803FFFC93B612F0030F 15FF037F16E04AB812F8020717FE021F717E027F9026FE000714E049B50080D9001F13F8 4901F8C800017F010F01C09238003FFF4990CA000F7FD93FFC05037FD97FF005007F4948 727E480180F01FF84890CC6C7E4919074848737E4848737E49854848F27F80A24848F23F C0491A1FA2007F1CE090CE120FA3481CF0481B07B3B3B3B1007CF303E00038F301C05C68 77E36F>I102 DI<1238127C12FEB3B3B3B3B3B3B3B3B3127C123807A76DFC2E>106 D<00381507007EED1F80A200FE16C0B3B3B3B3B3B3B3B3B1007E1680A20038ED07002AA7 6CFC53>I<1238127C12FEA27E7EA27F123FA27F121FA27F120FA27F12077F1203A27F12 01A27F1200A27F7FA280133F80131FA280130FA2801307A2801303A2801301A280130080 80A281143FA281141FA281140FA2811407811403A2811401A2811400A28181A282153F82 151FA282150FA2821507A2821503A2821501821500A28282A283163FA283161FA283160F 831607A2831603A2831601A2831600A2838384173FA284171FA284170FA2841707A28417 03A2841701841700A28484A21980183FA219C0181FA219E0180F19F01807A219F81803A2 19FC1801A219FE1800A219FF197FA2193E191C40A777FC53>110 D<2107F90F80F91FC0A2213F2280217F22006969200169200369200769200F69201F69A2 203F69207F9EC7FC68681F01681F03681F07681F0F681F1F681F3F681F7F9DC8FC67671E 01671E03671E07671E0F67A21E1F671E3F671E7F9CC9FC66661D01661D03661D07661D0F 661D1F661D3F661D7F9BCAFC65651C01651C03651C07020C62021E190F027E6214FF491A 1F01076D61491A3F496D61017F1A7F90B56C96CBFC486300076E60D80FF31901D81FE362 D87F836D1703D8FF016200FE6E1707D878006200306E170FC7007F616F171F6E6170163F 6E6170167F6E96CCFC63705E6E1701705E6E1703705E6E1707705E8070150F037F5F1A1F 705E6F163F715D6F167F7192CDFC6F5E715C6F1501715C6F1503627113076F5E71130F6F 5E71131F047F5D71133F705DF0807F7092CEFC61725A7013C1F0E1FC7013E3F0F3F88218 FF705CA2705CA2715BA2618396CFFC8360170F60170760170360715A82A675868A>112 D114 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fp cmmi7 7 45 /Fp 45 122 df11 DII<16E04B7EA315035EA59338E7FF806FB5 12F018F85D923807FC01031FB5FC4B14F0DBFF3F13E0912601FE0F1300DA03F8C8FC4A5A 4A5A4A5A4A5A4AC9FC14FE495A495A13075C495A131F495A5C137F49CAFC5B12015B1203 5B1207A2485AA2485AA3123F5BA3127F5BA412FFA77FA37F127F7F7F7F6C7E7F6C13E014 F86CEBFF806C14F06C14FE6CECFFC06C15F86D80011F14FF6D81010181EB003F02078014 00151F03017FED007F163F161F5FA2160F161F5F1430027C495A027F91C7FC91383FE0FE 6EB45A6E5B02035B9138003FC0356A79D13C>16 D<16FF030713C0031F13F04B7F9238FF 83FC4A48C67E4A487F4A487F4A4814804A48133F4A4814C04A5A02FF15E04990C7121F49 5AA2494815F0130F5C495A133FA2495AA2495A173F5A5C5AA24890C8FC177FA2485AA2EF FFE0121F5BA290B8FC4817C0A4D87FF8C7000313805BA25E1800485AA24C5AA2495D161F 5FA2163F5FA2494A5A5F16FF5F5D007F93C7FC4B5AA24B5A6D495A003F5D4B5A153F001F 5D6D495A000F4AC8FC3907F003FE9038F807F83903FC1FF06CB55A6C1480D93FFEC9FCEB 0FF0345379D13F>18 D20 D<49B4FC4913F015FC6D7F90 38003FFF020F7F806E7FA280828082A26F7EA2153F82A26F7EA2150F82A26F7FA28183A2 6F7FA28183A2707EA2163F83A2707EA2160F8382041F7F5E4C7F93B5FC5DDB03FD7FED07 F9DB0FF07FED3FE092387FC07FDBFF807F4A13004A486D7E4A5A4A48131FDA3FF0804A5A 4A486D7E495B49814990C71480495A49486E13C0EB7FF8494880484916E0485B48496E13 F04890C9FC4848167F484817F84848163F484817FC5B49EE1FFE49EE0FFF6CCA7E003E83 405278D04F>III<49B812 F0010717F8131F5B5B48B9FC5A4818F019E0280FFC000FC0C8FCEA1FE0485A48C7485A12 7E5A48143F5AC8FC4BC9FCA35DA25DA21401A34A5AA31407A25D140FA3141FA25DA2143F A34A5AA45D6E5A6ECAFC3D337AB13D>28 D<0106187C010F18FE496C4C7E013F4D138001 7F4D13C049CAFC5B485A85485A484817001A7F5B4848183FA24848181FA248C8000F1680 4C7E003E4B7E167F007E03FF1600127CA26200FC4A49143E5A1A7E94C8FC62A26C4B4A5A A24C14034F5A6C0203150F4B6C5D6C021F153F6D496C4A5A6D90B500C0495A267FF003EC E003D9FE3FDAFE3F90C7FC90B66CB55A6C02FE5D4B7E6C02F85D4B6C5C6C4A6C14C06C4A 6C5C6CDA000391C8FCC601FC010013FCD93FE0EC3FE04A347CB254>33 D<4BB4FC031F13E0037F13F84AB512FE0207EB03FF91260FFC001380DA1FF0EB7FC0DA3F C0EB3FE04A48131F4AC713F0495A0103ED0FF8495A495A011F16FC5C133F495AA2495A17 1FA2485BA2173F4817F891C8FCA2177F4817F05BA2EFFFE0120F4C13C01880A2001F4B13 004C5AA24C5A003F4B5AD83EFF5D4C5A6D6CEBFFC0013F495B007E9026E007FEC7FC9039 1FF81FFCD87C07B512F06D14C0010091C8FCEC1FF000FC90CAFCA87E7E13E090B612C06C 15FC827E827E12071201D8003FEBE03F90C87E161E364A75B245>37 D39 D59 D<1707EF0F80171FA2173F18005F177EA217FE5F16015FA216035FA216075F 160F5FA2161F5F163F94C7FCA25E167E16FE5EA215015E15035EA215075EA2150F5E151F 5EA2153F93C8FC5D157EA215FE5D14015DA214035D14075DA2140F5DA2141F5D143F92C9 FCA25C147E14FE5CA213015C13035CA213075CA2130F5C131F5CA2133F91CAFC5B137EA2 13FE5B12015BA212035B12075BA2120F5BA2121F5B123F90CBFCA25A127E12FE5AA25A12 78317477D644>61 D<127012FCB4FC13C013F0EA7FFCEA1FFF000713C0000113F038007F FCEB1FFF010713C0010113F09038007FFCEC1FFF020713C0020113F09138007FFCED1FFF 030713C0030113F09238007FFCEE1FFF040713C0040113F09338007FFCEF1FFF050713C0 050113F09438007FFCF01FFF060713C0060113F09538007FFCF11FFF070713C0070113F0 9638007FF81A1FA21A7F963801FFF0070713C0071F1300F17FFC953801FFF0060713C006 1F90C7FCF07FFC943801FFF0050713C0051F90C8FCEF7FFC933801FFF0040713C0041F90 C9FCEE7FFC923801FFF0030713C0031F90CAFCED7FFC913801FFF0020713C0021F90CBFC EC7FFC903801FFF0010713C0011F90CCFCEB7FFC3801FFF0000713C0001F90CDFCEA3FFC EAFFF013C090CEFC12FC12704D4E73C368>I65 D<021FB812FC4AEFFFC01BF81B FE6EF0FF80DA000F0180C7001F13C06F90C8000713E0080113F04B7013F81B7F4CEE3FFC A2031F18FE1B1F5EA2153FA25EA2037F173F1CFC5EF37FF815FFF3FFF04C4B13E0624A19 C0080F13804C92381FFE00505A4AEFFFF8070313E04C020F1380DFFFFEC7FC4A90B712F0 1A801AF81AFE4A90C8380FFF80070113E04B6F7FF23FF8021F84747E4B707EA2023F7113 80A25D1CC0147FA25DA214FF1C805D62491A00624B5F1A3F494E5A505A4B5D4F5B494D5B 4F5B92C9003F90C7FCF1FFFE4904035B013F041F13F0007FBA12C0BBC8FC19FC19E006FC C9FC574F77CE64>II<021FBB1280A25CA280DA000F01C0C712016F49EC000F1B034B8375130094 CA7EA25D885EA2033F183EA25EA2157FF103C04C14071C7E03FF030F147C625E1C3C4A04 1F91C7FC97C9FC4C5C614A5E4E5A4C131F93B6FC5C61A34A903880003FF007F893C71203 18015C615DA2023F030315F04F13015D1B03027F614F13074B91C85B1B0F14FF644B171F 51C7FC5B1B7E5D1BFE494E5AA24B4C5A1A07494E5A1A3F4B16FF07035B49171F013F4CB5 5A007FBBFCBC5AA398C8FC594F78CE61>69 D<021FB600FC010FB612FEA24A5FA26E4B6D 15FCDA000F01F0C80007EBF8006F01C06F13E04D5F4B5F6594C9FCA24B5F9AC7FC5EA203 3F5F645EA2037F173F645EA203FF177F645EA24A18FF645EA24A5F645E93B9FC5C64A34A 0180C812076493C9FCA24A5F99C8FC5DA2023F5F635DA2027F173F635DA202FF177F635D A24918FF635DA2495F635DA2495F635DA2495F013F6D031F7F007FB600E0013FB612F0B7 6C49814C5FA3674F78CE6D>72 D<020FB612FE5CA318FCDA0007EBF8006F13E05F5DA25F A25DA294C7FCA25DA25EA2153FA25EA2157FA25EA215FFA25EA25CA25EA25CA25EA25CA2 5EA25CA293C8FCA25CA25DA2143FA25DA2147FA25DA214FFA25DA25BA25DA25BA25DA25B 011F7F007FB612F082B7FCA26C5D374F79CE3B>I<021FB600FC0203B6FC635CA26E4B17 FEDA000F01F0C914C06F01C093387FFC004D17F04BF0FFC05190C7FC94C9EA03FC515A4B EF1FE0F33F804C4CC8FCF201FE033FEE03F8505A4CED1FE0F23F80037F04FFC9FCF101FC 4C4A5AF10FF003FFED1FC0F17F804C02FECAFC4E5A4AED07F8F00FE04C495A187F4A15FF 05037F4C5A4D7F4A143F4D7FEE81FEDC83FC7F4A903887F07FDC9FE07F93383F803FDC7F 007F4A01FE7F04F8814C7F04C0814A497F93C78003FC8086027F81864B808602FF167F86 4B153F864983874B81874983874B81874983874B81874985013F6D03036D7E007FB600E0 027F14FEB76C91B7FC5EA2735C684F78CE70>75 D<021FB77EA25CA26E93C8FCDA000F01 F0C9FC6F13C05F5DA294CAFCA25DA25EA2153FA25EA2157FA25EA215FFA25EA25CA25EA2 5CA25EA25CA25EA25CA293CBFCA25CA25DA2023FEF01E01A035D1A07027F18C0A24B160F 1B8002FF171FA24BEE3F00A249601A7E4B16FEA2494D5A19034B4B5A190F49171FF17FF0 4B15FF180349041F5B013F4BB5FC007FBAFCBB5AA3624B4F78CE5B>I<021FB500E0061F B512F071605C666E98B612E0DA000FF4C0006F4F91C7FCDCDFF8943803EFFE030F1AFF9A 3807DFFC049FF00F9FA2031FF11F3FDC8FFC053E5B160F1D7C033FF1F87F67033EF001F0 706CEE03E0037E1AFF52485B037C4E5AA203FCF01F01706C043E5C15F81C7C02014F5A67 03F0EF01F0706DEC03E0020362E107C091C8FC03E0EF0F80A202074E485A706D023E5C5D 63020F4E131F664B4C5A716C495A021F1A3F50485C92C74B5AA24A4DC7127F716C013E5D 143E62027E4D14FF66027C4C5A716C485A02FC614F485D4A4C5AA201014DC75A94260FFC 3E5E5C6101034D5C9BC9FC4AEDFDF071B45A0107624F5D010F5F131F496C93C8121F2601 FFFC6E48ED7FFF007FD9FFF094B77EB66C4A49824B4A60A2EF01E07C4F78CE7F>I<021F B500E0031FB512FEA24A6E5DA26E6E6F14FCDA00070500140071EE1FF8F40FE04B6D5FA2 DC9FFF5FA2031F6E151F048F95C7FC040F7F82033F6E5D70173E033E8082037E6E157E70 177C157C717E03FC18FC716C5D5D717E02011801716D5C5D717F02031803716D5C5D717F 02071807716D5C5D727E020F180F726C5C5D727E021F181F72018090C8FC92C8FC7213C0 4A60F2E03E023E811AF0027E6F147EF2F87C027C811AFC02FC047F13FC745A4A163FA201 01EF1FFF634A82A2010383634A82A201078363010F83131F496C177F2601FFFC60007FD9 FFF0163FB67E4B161F98C9FC86674F78CE69>I<932601FFE01303041F01FEEB078093B6 EA800F03039238E01F00030F6F5A4B9039007FF87FDB3FF0903807FCFFDBFFC0903801FF FE4A90C8127F4A48153FDA07F8151F4A485E4A48150F4B1507143F4A485E19034AC9FCA2 49605CA3010360A280626F92C8FCA28115F06D13FCEDFF8016F86DECFFC017FC6EECFFC0 6E15F06E15FC6E816E810201826E6C81031F81030181ED000F040080170F1701716C7E18 3F181FA2180FA21807EA03C07F1207615BA2000F170F61A24E5A487E4E5A616D167F003F 4DC8FC6D4B5A6D4B5A6D4B5A486CED1FF002C0EC7FE0D9BFF849485A90271FFF800F90C9 FCD8FE0790B55A486C15F848C615E048011F148048010001F8CAFC495377D052>83 D<003FB600F8023FB512FCA24860A26C4B6E14F8D8001F01E0C80001EBFE006D01809238 003FF092CAEA1FC04961A24A95C7FCA2013F601A3E5CA2017F187E1A7C5CA201FF18FC62 5CA2481801625CA2481803625CA2481807625CA248180F6291CAFCA248181F97C8FC5BA2 003F60193E5BA2007F187E197C5BA219FC00FF605BA21801611803495F180761180F4E5A 6C6C4CC9FC183E187E003F5F6D4B5A001FEE07F06C6C4B5A6DED3FC06C6CEDFF806C6DD9 03FECAFC6C01E0EB0FFC6C01FCEBFFF8013FB612E06D158001074ACBFC010014F0020F90 CCFC565176CE5A>85 D<007FB600C0033FB512E0B719F0621DE04C81C66C01E0C90007EB F800011F01807013C099C7FC6D6D17FCF207F06D6D5F505A6F4C5A6D4EC8FC1A7E6D6D5E 4F5A6D6D4B5A4F5A6E6C4B5A191F6F5E6E4CC9FC197E6E6D5C4E5A6E6D495A4E5A6E6D49 5A4E5A7049CAFC6E157E606E6D485A4D5A6EEBFC07606F6C485A4D5A4DCBFC92383FFF7E 5F6F5B5F6F5B5F6F5B94CCFCA25DA25EA2151FA25EA2153FA25EA2157FA25EA215FFA25E A25CA34A5B020F7F011FB612F85BA35F5C4F79CE4E>89 D96 DI<49B4FC0003B57E5AA292C8FCA2EA000F7F5CA2130FA25CA2 131FA25CA2133FA25CA2137FA25CA213FFA25CA248EC3FE09138C1FFFC028713FF029F14 8048D9BFE013C09139FF003FE002FCEB0FF04A14F84801E013074A14FC4A130391C713FE 5A5B4915FFA2121FA25BA2003F5DA25BA2007F5D17FE5BA2161F17FC485AA2EE3FF8A217 F0EE7FE0127FEEFFC017805D003F4A13005E6D495A001F4A5A6C6C495AED7FE03A07F801 FF802603FE0790C7FC6CB512FC6C6C5B6D13C0D907FEC8FC30527AD03A>III103 D<157CEC01FE4A7E5CA25C A35D6E5A6E5A6E5A91C8FCAFEB07F8EB1FFEEB7FFF90B57E3901FC3FC0D803F07FEA07E0 01C07FEA0F80121F13005A003E137F007E5C007C13FF5DA2EAFC0100785C12005B92C7FC 5B5CA2130F5C131F5CA2133F5CA2017FEB07C014E013FFECC00F16805AEC801F16005DEC 003E157E5D5D6CEB03F0EC8FE06DB45A6D5B010F90C7FCEB03F8224F7BCD2F>105 D107 D109 DI<02FEEC1FF0902603FFC0EBFFFE49D9E0036D7E49D9F00F80903C 1F8FF81FF07FE0903C3F07FC7F801FF0903C3E03FEFE0007F8017E6D4880017C02F01303 4C8001F84A13014C805CD801F091C7FC4B1680A20003130F13E000015CC7FC021F5DA25D A2023F5D1A005DA2027F5D615DA202FF4B5AA24B5D4E5A5B4E5A4B5D6F14FF494B5B96C7 FC6F495A4D5A496D495A6FEB3FF04A6CEBFFC0DB7F035B010F6DB448C8FC6F5B4A6C13E0 030190C9FC011F90CBFCA25CA2133FA25CA2137FA25CA213FFA25CA2487F007FEBFFC0B6 7E5DA3414A81B244>112 D115 DII120 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fq cmmi10 10 63 /Fq 63 123 df<933803FFC0043F13F84BB6FC030F81033F15E04B01017F913B01FFF000 3FF84A01C06D7E020F90C7D807FEEC01E04A486F15F0DA3FF86E6D13034A486E6D14E04A 5A49496E7F49497013074990C9007F15C049844948190F4948043F1580017F844A191F01 FF1B004849715B071F143E485B481B7E4A197C481BFC644849EFFF01641B034890CA5D1B 0764481A0F494F5A99C7FC6300FF1A7E5B6363A249616363A26398C8FCA25BA285A2127F 6D5F61003F95B5FC606D5E001FDD0FF7EC0780DE1FE7EC0FC06C6CDC7FC7138000079326 01FF0315806DDB07FE151F6C6DDA1FF816006C6D9126FFF0016D5A6C01F00107D9C00014 FED97FFE90B5C7EBE1FC011FB600FC6EB45A6D03F06E5B010303806E5B010002FCC80007 5B020F0180DB01FEC7FC5C4B79C86A>11 D<963801FFE0070F13FE077F6D7E4EB612E006 07814E9038007FF8DE3FF0EB0FFCDE7FC06D7E4EC7EA01FFDD03FC6E13804D4816C0DD0F E0157F4D48ED3FE04D5A95C913F0177E4D161F4C4817F84C5A5F16074C5A5F041F173F4C CAFC163E167E047C18F004FC177F5E15014C18E0030318FF5E03074D13C05E1D80030F5F 4C1800515A031F170F93CA5B515A4B4D5A033E4D5A515A037E91B5485B037C010702E390 C7FC4DECFFFE4D5D03FC17F04B90263F800F5B95B67E8702016E814B6DECC7FE05019038 FC03FF94C86C7F14034B707F1B7F8814074B717EA288140F4B171FA3021F8592CBFCA35C 143E1B3FA2147E147CA302FC187F4A61A3010119FF645CA201036064A2505B1307505BA2 5090C7FC010F611A1F6E60505A011F4E5A6E4D5A63027E4C5B013F4D90C8FCD93E3FEE0F FE6F4B5A6E6C4B5A90267E0FE0EDFFF0017C6D020313C0DA07FC020F5B6EB4DA7FFEC9FC D9FC019039F007FFF8496C6CB65A6F1580030F4ACAFC0001020314E0499026003FFECBFC 93CDFCA212035BA312075BA3120F5BA3121F90D0FCA35A123EA3127E127CA312FC5A1278 5D957EF45E>II<93 3801FFF0040FEBFF80043F14FC4CECFFC093B712E04B16F0923803FE0F922607F00114F8 4C6C7E4B486D13F04C130F031F807213E093C813C070EC7F80F10C0096C7FCA282A382A2 6F7EA28282150782A26F7E838381836F7FA283707E83707EA2707F84707FA24C7F93B57E 1503030F80033F80EDFFF94A01E07F020713804A48486C7F4A5ADA3FF07F4A48814A487F 495B4990C76C7F5B495A494880495A8549488013FF5C5A4A805A485BA291C8FC5AA2485A 61A2123F5BA3007F605BA34D5B5BA212FF4D90C7FCA3007F5F170F60A2171F604D5A123F 606D157F001F5F4D5A6C6C5C95C8FC6C6C4A5A00034B5A6D5D6C6DEB1FF06C01E0495AD9 7FF8495A903A3FFE03FF80010FB6C9FC6D5C010114F86D6C13E0020790CAFC45797AF54A >I<170E171E173EA6177EA3173EA295381FFF80053FB512F01AF87114FC177F9438FFF0 00040390B5FC5EDC0FFB14F8DC3FE114E093267FC03F13004CC9FC4B5A4B5A4B5A4B5A4B 5A4B5A4B5A4BCAFC4A5A4A5A4A5A4A5AA24A5A4A5A4A5AA24ACBFC495A13035C13075C49 5A131F5C133F5C137F5C13FF91CCFC5A5B1203A2485AA3485AA2121F5BA3123F5BA3127F 5BA412FFAC7FA47FA27F127F7F7F6C7E80806C7F14F86C13FE6CEBFFC015F86C14FF6C15 E06C15FC6DECFF806D15F06D15FC010715FF6D82010082021F81020781020081031F8015 03DB007F7F160F1601706C7E83170FA283A283A260A360170760EC03C04A6C495A03F849 5A03FE495A913903FFC0FF020090B5C8FC6F5B031F5B030713F0030013C0469678F349> 16 DII20 DIII26 D<0407B812FC047F17FE0303BAFC150F153F92BBFC5C020719FE5C4A19FC4A19F04AD9F0 0749C8FC91B5C7FC4901FCEC3FFF4901F0804901C06E7F4949804990C8FC49486F7F5C49 5A494881A2485B484983A2485BA24890C9FCA348486060123F5BA260007F615BA26000FF 96C8FC5BA24E5AA24E5AA2495F187F6118FF007F604D5BA26D4B5B003F4C90C9FC604D5A 6C6C4B5A4D5A6C6C4B5A4D5A6C6C02035B6C6C4A90CAFC6C6DEB1FFC6C01E0EB7FF8903A 7FFC07FFE06DB65A010F92CBFC6D14FC010014E0DA1FFECCFC58497AC65F>I<0207B912 FC021F18FE027F18FF49BBFC5B5B5B5B4919FE4919FC90BB12F0489027F00003F0C9FC48 138049C7485AEA07F8485A01C0140F485A48C85B161F127E5A5A4C5AC9FCA2167FA34CCA FCA35DA25EA21503A34B5AA3150FA25E151FA44B5AA3157FA34B5AA35CA25EA25CA34A5B A493CBFC5D6E5A6E5A50497CC649>I31 D33 D37 D39 D58 DII<191C193E197FA219FF19FEA2180119FCA2180319 F8A2180719F0A2180F19E0181F19C0A2183F1980A2187F1900A26060A2170160170360A2 170760A2170F60A2171F60A2173F60A2177F95C7FC5F5FA216015FA216035FA216075FA2 160F5F161F5FA2163F5FA2167F94C8FCA25E5EA215015E15035EA215075EA2150F5EA215 1F5EA2153F5E157F93C9FCA25D5DA214015DA214035DA214075D140F5DA2141F5DA2143F 5DA2147F92CAFCA25C5C13015CA213035CA213075CA2130F5CA2131F5CA2133F5C137F91 CBFCA25B5BA212015BA212035BA212075B120F5BA2121F5BA2123F5BA2127F90CCFCA25A 5AA2127C123840A777FC53>I<1238127EB47E13E013F8EA7FFE6C6C7E000F13F0000313 FCC613FF013F13C0010F13F0010313FC010013FF021F13C0020713F0020113FC6E6CB4FC 031F13C0030713F0030113FC6F6CB4FC041F13E0040713F8040113FE706C6C7E051F13E0 050713F8050113FE9439003FFF80060F13E0060313F8060013FE96383FFF80070F13E007 0313F8070013FE97383FFFC0080F13F0080313FC080013FF093F13C0090F13F0090313FC 9838007FFF0A1F13C00A0713E00A0113F0F4007FF401FF0A0713E00A1F13C00A7F130098 3803FFFC090F13F0093F13C098B5C7FC080313FC080F13F0083F13C0E0FFFEC8FC070313 F8070F13E0073F1380DFFFFEC9FC060313F8060F13E0063F13804DB448CAFC050713F805 1F13E0057F13804C4848CBFC040713F8041F13E0047F90CCFC923801FFFC030713F0031F 13C0037F90CDFC913801FFFC020713F0021F13C091B5CEFC010313FC010F13F0013F13C0 90B5CFFC000313FC000F13F0003F13804848D0FCEAFFF813E01380007ED1FC1238646172 D981>I<1C1F527E1C7FA21CFF638963A26363A263A26363A298B57EA26262A2621BEFF2 0FCFA2F21F8F1A3F090F7F1A7EA21AFC07017F1AF819031AF0F107E0190F1AC0071F811A 80F13F0061077E7F19FE614E5AA24E5A18074F814E5AA24E5A063F8096C7FC60187E6017 0160050383604D5A170F4E80171F604DC9FCA2177E17FE4D834C5AA24C5A4CBAFCA25EA2 5E5E94CA7E4C85167E5E15014C8315035E4B5AA24B5A151F5E4BCB80A2157E03FE845D14 015D4A5A14075D140F021F87143F147FEB01FF496D60011F6D4E7F90B500FE0507B512FC 007FDAFFF00307B81280B76C4B17C0622080A26C4B6F170072777AF67D>65 D<0307BBFC4B1AF04B1AFEF6FFC08B6F1BF892C702E0C8001F7F051F49030313FF4F0300 14807813C04D7213E08A96CA6C13F08A4D1AF88A4E19FCA205FF8420FE60A25EA260A24C 6120FC60A24C4F13F8A2605413F05E5413E04E4D13C0A24C4F13809BB512004E4C5B535B 4C4E5B535B95CA485B5313804C95B5C7FC5213FC4D04075B0A1F13E004FF94B51280091F 49C8FC94B912F09AC9FC4B19E01DFC1DFF05F8C9003F13C04B060713F0767F4D040013FE 777E4B737F777F5F777F4B87894D85A24B858B5FA25DA294CBFCA25DA25E6515FF675E65 4A64654C62654A64654C95B55A9CC7FC4A61525B4C4D5B525B4A4F5B525B4C4D5B51B55A 4A4E91C8FC090F13FC4C043F5B4A4DB512E00103B5041F5C003FBDC9FC481BFC1CE0BDCA FC1BF86C97CBFC77717AF07E>I<97261FFF8015380707B500F8153C077F02FF157C0603 B700C014F8061F04F01301067F7013034DB527FC007FFC130705070280D907FEEB0FF005 1F01FCC7D801FF131F057F01E09139007F803F4CB5008092381FC07F4C49C9390FE0FFE0 040F01F8EE07F14C01E0EE03FB4C0180160193B5CA90B512C04B01FC834B49844B5B031F 01C07213804B5B4B90CC7E4B5A4A491B004A49854A5B4A5B674A4919074A5B4A90CDFC91 B563495B5D5B494963A2495B495B67495BA2495B90B564A24891CEFCA24A64489AC8FCA2 485BA25A5CA25A5CA35A5CA35A5CA45CB5FCA6F501E091CE1203A21D0766A21D0F6C65A2 1D1F9BC8FC656E1A3E6C1C7E1D7C1DFC6C515A6E621C036C515A6E626C1B0F525A6C6D4F C9FC1C7E6C6D611B016C6D4E5A6D6C4E5A6E4E5A6DF13FC06D6D4D5A6D01E005FECAFC6D 6DEE03FC6D6DEE0FF86D01FEEE3FF06D6D6CEDFFC0023F01E002035B6E01FCDA1FFECBFC 0207903AFFE003FFF86E91B612E002001780033F4BCCFC030715F003001580040701F0CD FC767978F477>I<030FBA12FE4BF1FFF01EFE4B747E1FE06F1BF8DB000102E0C86C7FDC 003F49030713FF4F0300800C3F7F4D060F7F787F96CA6C7F787F94B5717F1F7F4E727E79 13805E7913C04E8421E05E7913F060A24C7413F8A260A24C1CFCA24E84A25E21FE60A25E A24E60A25EA295CCFCA293B5FCA24D6121FC5DA25F675D21F85F675D21F05F674B1DE0A2 4D6121C05D674D1B80A24B98B51200A24D62664B646694CC5C6692B563545B5E545B4A65 664C4F90C7FC674A515A535B5E535B4A505B535B4C4E5B5390C8FC4A505A535A4C4D5B0A 0713E04A4F5B0A3F5B4C4D48C9FC51485A4A06075B091F13E04C93B512804A050791CAFC 0107B593B512FC007FBC12F0BD128051CBFC1BF098CCFC6C19E07F717AF089>I<030FBD 12FE4B1CFFA25DA26F1CFEDB000102E0C9FCDC003F4916034F16001F3F4D190F1F0796CB FC1F0394B5FCF701FC60A24C1A00A260A25EA24E19F8A25EA260A25EA24E150E0A1F1401 4C4D15F01C3E60F700E04C057E15001C7C601CFC5E515A95C8FC1B0393B5FC515A4D151F 1B7F4BEE07FF94B85AA35D64A205F8C7120F4B1600755A4D153FA25D99CAFC5FA25D1B3E 5F1F1E4B057E153F097C153E5F1F7E4B1B7C093815FC94CD5A1E0192B5FC674C1903674A 1B07674C190FA24A515AA24C4FC7FC664A1B7E1EFE4C1801664A1A03535A4C180F664A1A 1F1D7F4C4E5A1C034A4F5B1C3F4C94B55A4A180F0107B50403B6FC007FBEC8FCBFFC65A3 6C6478717AF07B>I<0307B800F00103B812F84B714917FC4B61A25119F86F4D6D17F092 C74ACA6C91C7FC051F01F0050F13F84F614F614D61A24F61A24D616996CBFCA294B5609E C8FC60A24C97B5FC6860A24C616860A24C616860A24C616860A24C616860A24C616860A2 4C616895CBFCA293B56095BBC9FCA35D67A205FCCBFC4B61675FA24B61675FA24B61675F A24B61675FA24B61675FA24B616794CBFCA292B5609CCAFC5EA24A97B5FC665EA24A6166 5EA24A61665EA24A61665EA24A61A24C614A6D4D7F0103B500F84CB512FC003FB86C011F B812C048714983A2B95CA26C4D6D5F8E717BF08A>72 D<030FB812F84B17FCA24B17F8A2 6F17F092C792C7FC051F13F061615FA261A25FA296C8FCA294B5FCA260A25EA260A25EA2 60A25EA260A25EA260A25EA260A25EA260A25EA295C9FCA293B5FCA25FA25DA25FA25DA2 5FA25DA25FA25DA25FA25DA25FA25DA25FA25DA294CAFCA292B5FCA25EA25CA25EA25CA2 5EA25CA25EA25CA25EA25CA25E027F7F0103B512FC007FB812C0B97EA3606C5F4E717BF0 49>I<030FB800E0033FB612F04B714B15F8A24B4D92B7FC23F06F4D6F15E0DB000102FC CA0007ECFC00DC003F01E0050114C04F7149C7FC4F19F84D4F13E0218096CB4848C8FCF7 07F894B54E5AF71FC04E4E5A55C9FC4CF101FC545A4EEF07E0F61FC04C4F5A0C7ECAFC4E 5FF503F84C4E5AF50FC04EEE3F8053CBFC4C18FCF403F84E4B5AF40FC04C4D5A0A7FCCFC 4E15FEF301F84CEE07F0515A4EEC1F80097FCDFC4C16FE505A95C712031A0F93B5141F50 7E4D14FF4F7F4B5D614D011F80614BDB7E7F7F4E487E9426F803F880953807E01F4BEC1F C04E486C7F9438F07E004E814BD9F3F87FDDF7F081DDEFC07FEFFF804B91C76C7F17FC4D 824D804B49834D815F757F5D8994C97E8992B582A24C707FA24A86874C84875C757F5E8A 4A858A4C83A24A737FA24C85884A87885E8A5C8A4C854A6D4D800107B500F0057F14F000 7FB8033FB712E0B96C4A82A395C8FC6C4C6F5E8D717AF08D>75 D<030FB812FE4B83A25D 636F5FDB000192CAFCDC003F13E061615FA296CBFCA294B5FCA260A25EA260A25EA260A2 5EA260A25EA260A25EA260A25EA260A25EA295CCFCA293B5FCA25FA25DA25FA25DA25FA2 5DA25FA25DA25FA25DA25F1E604B1AF01D015F1D034B1AE0A294CB12071EC092B5180FA2 4C19801D1F4A1B00655E1D7E5C1DFE4C601C014A4F5AA24C1707525A5C1C1F4C4D5A1C7F 4A4F5A1B034C5E091F5B4A6050B5FC4C030791C7FC4A173F0107B50307B55A007FBCFCBD FC64A36C6364717AF071>I<030FB600E00807B612F84B6F5015FC6A4B22F86A6F5415F0 DB00016E9AC7FCDC003F99B512F8053EE101F75BA15A057E515AF907CF057C66726CF10F 8F05FC1C9FE71F1F5B05F81B3EA20401525A0FF891C8FC17F0726CF001F0040365E603E0 5B05E0F207C0A2040798380F80FF56485B17C0726C183E040F640E7C5C05801AF8A2041F 50485AE503E05C4D6C7FF707C04C50485A6B043EF21F00A2047E083E5B555D047C6D7F67 04FC4F485B6B4C4F5AA2030150485B544892C9FC4C6D7F54C7FC0303083E5C6A4C616603 071DFF726D4A485D5E535A030F4F485B6A4C4E5A53C7FC031F64726D023E5E93C8FC654B 4F5C6A033E4E5A525A037E64736C49485E157C525A03FC4EC85A6A4B183E640201706C5F 5294CAFC4B4D5AA202034E485D694B4D5A515A0207DC1FFF17FF51C95B4B173EA2020F4E 5D694B5F63021F705F515F023F60A2027F4E5D02FF95C9FC01036D65010F01F06F484C7F 90B500FE98B512FC007FDAFFFC4B020FB812C0B76C4B4A83A2624C4B626C4B6E486E5FA6 7179F0A1>I<0307B600E094B712F84B6F4C16FC4B6F5EA2731BF86F7416F092C76C6DDC 0003ECF800051FDF007F1380E61FFEC7FC73F00FF84D63726D715A173E726D60057E1A0F 86DD7C1F62A2DDFC0F6D171F9EC8FC4D6C7FA204016D6D5F203E17F0727F04031B7E726D 177C5F737F04071BFC755E4D7FA2040F6E6D1501684D6D7FA2041F6E6D15036894C78085 4C1A07736D5E163E737F047E1A0F765D047C81A204FC6F6D141F9DC9FC4C6F7FA2030170 6D5C1F3E4C838603031B7E746D147C5E747F03071BFC765C4C81A2030F71EB8001674C70 13C0A2031F71EBE0036793CA14F0874B1A077501F85B153E7513FC037E1A0F0BFE5B037C 83A203FC71EBFF1F9CCAFC4B72139FA202017313FF665D881403765B5D881407664B84A2 020F8566021F85A2023F1A7F027F6349487E010701F8193F017F13FF003FB600FE725A48 811D0FB8FCA26C4B725A8E717BF085>I81 D<030FB912FE4BF0FFF01D FF4B1AE01EF86F1AFEDB000102E0C7001F7FDC003F49020014C04F031F7F0B077F4D717F 777F96CA7F787E94B5832080607813C05E20E060A25EA260A25E6660A25E5413C060A24C 96B51280A24E1900535B5E535B4E60535B4C4E5B6795CA485B5390C7FC93B54D5A52485A 4D4C13F00A0F5B4B063F1380E2FFFEC8FC4D030F13F850B512E04B90B912800AFCC9FC1C E0644B18F005F8C7000713FC4D02007FF33FFF4B050F7F757F4D6F7F894B83894D81895D A24D83A25DA294CAFCA292B55EA24C60A24A60A25EA24A60A25E654A60A25EA25C51161C 4C1B3EA24A1D7E53147C5E20FC4A7216F81F014C1BF04A6D7213030107B500F070ED07E0 007FB8F10FC0B96C6E151F756DEB3F8075ED7F0095C9397FFF03FE6C4C70EBFFFCCF000F 5C7614E00A005CE30FFEC7FC77757AF07E>I I<000FB800E00203B712F048714A16F84862A24E19F06C4D6E16E0D8000102FCCA000FEC F000D9003F01E0050191C7FC4C9438007FFC4CF03FF04A63775A93CCFC6691B5193FA24B 97C8FCA249631D7E5DA2491BFE655DA2491A01655DA2491A03655DA2491A07655DA2491A 0F655DA2491A1F6592CCFCA290B5193F9AC9FC5CA248631C7E5CA2481BFE645CA2481A01 645CA2481A03645CA2481A07A24A61A2481A0FA24A61A2481A1FA291CC5BA21B3F99CAFC 485A631B7E1BFE631A0163491803631A07007F621A0F505A6D611A3F003F4FCBFC1AFE00 1F4E5A6D4D5A4F5A6C180F6C6D4C5A4F5A6C6D04FFCCFC6C6D4B5A6C6DED07FC6EED1FF8 6D6CED7FF06DB4913801FFC0010F01C0010F5B6D01FCD9FFFECDFC6D90B65A010016F002 3F15C0020F4ACEFC020114F0DA001F90CFFC757576F071>85 D<003FB700F8040FB612FC 48704C15FE64B81BFCA26C4C7015F8D8001F02F0CBECFE0001030280063F13F04BCC000F 13804B98C7FC6D1CFC66666F62535AA2535A6D99C8FC651D7E656F180165525A1C076E62 1C0F65525A70173F9AC9FC1C7E1CFE646E180164515A701607641B0F515A646E183F99CA FC1B7E1BFE705E1A01505A636E170763505A1A1F705E1A3F98CBFC1A7E6E17FE624F5A19 03705D1907624F5A6E161F624FCCFC6170147E19FE614E5A6E1503614E5A180F705C181F 614ECDFC6E5D187E60170104FF5B1703604D5A6F130F604D5A173F05BFCEFC17FF5F5F81 5F5FA25FA25F5F8194CFFCA25E5EA25E5EA26F5A5E777577F061>I<003FB700FC0407B7 FC48704C1680A2B81C00A26C4C705DD8001F02FCCB003F1480010102C0060F01F8C7FC4C 1AE06D1D8054C8FC6E6D19F866704E5A6E505A666E6D4EC9FC1DFE525A6E6D4D5A525A1C 0F6E6D60525A704D5A6E4FCAFC1CFE6E6D4C5A1B03646E6E4B5A515A515A6F6D4B5A51CB FC715D6F5F505A6F6D4A5A505A505A6F6D4A5A1A3F636F6D4ACCFC1AFE6F6D495A4F5A4F 5A6F6D130F624F5A6F6E485A4FCDFCF0C0FE70EBC1FC18C370EBE3F8F0E7F0F0EFE070EB FFC06196CEFC826060705B60A260A25EA260A25EA260A25EA295CFFCA25EA25FA216FFA2 5FA25DA25FA25DA25FA25DA25F5DA35D4AB512FC011FB812C04983A2495FA26D5F79717B F060>89 D96 DI<157F0103B57E90B6FC5AA393C9FC7E1300141F5DA2143F A25DA2147FA25DA214FFA25DA25BA25DA25BA25DA25BA25DA25BA292CAFCA25BA24AEB1F F0EEFFFE013F01036D7E030F14E0DAFC3F8092397FF03FF8017F9039FF800FFC913AFDFE 0007FEDAFBF86D7EDAFFF06D138090B512C04B6D13C092C813E05C485B4AED7FF0A25C48 4916F8A25CA25AA25C18FF5AA291C9FCA2485EA25BA2003F5E19F05BA25F007F18E05BA2 5F19C05B00FF18805FA219005F495E60177F007F5F17FF604C5BA2003F4B5B6D4A5B95C7 FC001F4B5A4C5A6D4A5A000F4B5A6C6C4A5A6D495B000302075B6C6C6C4890C8FC6C9038 E07FFC6DB55A6D14E0010F1480010301FCC9FC9038007FE03D7579F247>I<933803FFC0 043F13F84BB6FC03071580031F15E0037F010013F0912601FFF0EB0FF84A01C01303020F 90C7EA01FC4A5ADA3FF8EC0FFE4A48141F4A48143F4949147F494914FF4990C75A5B495A 495A017F17FC5C01FF17F8484916F07113E04849ED7FC04894C7FC5C5AA2485BA34890CB FCA35A5BA312FF5BA45BA75BA2190C191E6C6C173E197E19FC003F1701F003F86DEE07F0 001FEF0FE0F03FC06C6CEE7F800007933801FF006C6CED07FC6EEC1FF86C6DECFFF06C01 F0010713C0D93FFE90B5C7FC6DB612FC6D15F001031580D9007F01FCC8FC020F13803F4B 79C848>II<933803FF80047F13F00303B512FC030F14FF037F15809226FFFE0113C002039039F0 003FE0020F0180130F4A90C7EA07F0DA3FFCEC03F84A5A902601FFE01401495B495B4990 C8FC5B495A5C017F1603494816F05A4A150748EF0FE05C48EF3FC04AED7F8048933801FF 00EF07FE4849EC7FFC93380FFFF091B75A48178005FCC7FC17C004F8C8FC4890CBFC5BA4 5BA312FFA35BA2127FA5190C191E003F183E197E6D17FC001F1701F003F8000FEF07F06D EE0FE00007EF3FC06DEE7F806C933801FF006C6DEC07FC6EEC1FF86C6DECFFF0D97FF801 0713C0D93FFE90B5C7FC010FB612FC6D15F0010115806D6C01FCC8FC020F13803F4B79C8 4D>IIII<167E4BB4FC4B13804B13C05D4B13E0A25D17C0A31780 6F13005E6F5AED03F092C8FCB3A4EC7FC0903801FFF0010713FC497F497FD93FC17F49C6 FC01FC80485A49801203485A5B120F5B001F5B13005A003E5B5E127E007C5B5E00FC5B48 92C7FCA25CC75BA2143F5D147F5DA214FF5DA25B5D5B5DA25B5DA25B92C8FC5B4AEB03E0 A2013F14074A14C0A2137F4A130F01FF15804A131F1700A24A5B163E167E167C16FC4B5A 4A485A14E0017FEB0FE04B5A90393FF07F806DB5C7FC6D5B6D5B010113F09038007F802B 717BEE39>I107 D109 DI<933803FFC0 043F13F84BB6FC030781031F15E0037F01017F913B01FFF0003FF84A01C06D7E020F90C7 EA07FE4A486E7EDA3FF816804A486E13C04A5A49496E13E0495B4990C9EA7FF05B494817 F8495A137F5C01FF18FC485BA2485B5A5C5AA2485B19FFA24890CAFCA2604819F85BA260 00FF19F05BA24E13E0A25B4E13C0A21A80601A00A2494C5AA24E5A616C6C4C5A5F61003F 4C5B4D5B6D4B90C7FC001F5F4D5A6C6C4B5A0007EEFFF06C6C4A5B6E010713806C6D4990 C8FC6C01F0EB7FFE903A3FFE03FFF86DB612E06D158001034AC9FC010014F0020F90CAFC 464B79C850>II114 DIII119 D III E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fr cmcsc10 10 34 /Fr 34 123 df46 D50 D52 D<0160171C01FC17FCD9FF80150702F8157FDAFFE0EB1FFF92B75A61616196C7FC606018 F06018804DC8FC17F817C0D9FC7F01FCC9FCDA007FCAFC92CBFCB3A3923807FF80037F13 F84AB512FE02076E7E021F15E091263FFC0113F8913AFFC0007FFC01FD90C7EA1FFED9FF FC6E7E02F06E7F4A6E7F02806E7F91C97F8549167F4983727E137890CA7F841A80A21AC0 A37213E0A51AF0A47FEA0FF8487E487E487EA2B57EA31AE0A391C95A1AC05B6C5A491880 EA7E8090CA5A003E1900123F4E5A6C6C5F6D16FF000F606C6C4B5B6D4B5B6C6C4B5B6C6C 4B5B6CB44B90C7FC6D6C4A5AD93FE0ECFFFCD91FFC01035B90270FFFC03F5B6D90B612C0 6D93C8FC010015FC023F14F002071480020001F8C9FC447375EE5C>I<933801FFF0041F 13FF93B612C0030315F0030F81033F9038801FFC923AFFFC0003FE4A01E0EB00FF4A0180 EC3F80020F90C8EA1FC0DA1FFC150F4A48ED7FE04A48EC01FF4A485C49494A13F0495B60 4990C8FC495A131F5C133F49486F13E0A249486F13C0721380489438007E004A93C7FC5A A2485BA25AA25C5AA35AEE7FF80303B57E030F14E048D9003F14F84B14FE9239FF807FFF 912701FC00077FDA03F0010113E04A486D7FB5D80F806E7E4AC86C7E181F023E6F7E4A82 7213804A17C04A811AE01AF04A811AF85C1AFCA2844A17FEA51AFF6C90CAFCA87EA280A3 7E1AFEA37EA21AFC6C6D5DA26C19F8A26E17F06C5F1AE06C6D17C0017F5E6E1780013F4C 13006D6C4B5A806D6C4B5A6D6D4A5A6D6D4A5A6D6D01035B6D01F8010F5B6EB4017F90C7 FC021F90B55A6E5D020315F0020015C0033F49C8FC030313E0487377EE5C>II<19F84E7E4E7EA34E 7EA34E7FA34E7FA24E7FA34E7FA395B57EA34D8018FB05038018F318F105078018E118E0 050F814E7EA2051F814E7E053F81A24E7E4D81177E8505FE814D7FA20401824D7FA20403 824D7F0407825F85040F835F86041F834D80A2043F8394C87E4C83167E8604FE835E8603 01844C81A20303844C81A20307844C81030F855E93BAFC4B85A34B85A293CA121F4B8503 7E8303FE855D870201865D870203864B83A20207864B83020F875D88021F875D88023F87 92CC7E5C8A4A85497F8A01077F496D4E7F013F01F84E7F0003B64DB612C0B700F04BB812 F8A6757778F687>65 D69 D72 DI76 D80 D82 D<001FBF12FCA64891C7000102E0C7003F7F02E06E4A140302806F49 EC007F49C9183F01F81C0F491C07491C03491C01A2491C00A290CA197EA2481E7F007E8A A4007C8AA848F60F80A6CB96C7FCB3B3B3AB95B57E4D80051F14FE037FBA7EA6717078EF 83>84 D<173E177FA24D7EA34C7FA34C7FA24C7FA34C7FA34C7F177F043F7FEE3E3FA204 7E7FEE7C1F04FC804C7EA20301814C7EA20303814C7E0307814C7EA2030F814C7E031F81 93C7FC187F4B81033E143FA2037E81037C8003FC824B80A20201834B80A24A4882840207 834B80A24AB87EA24A83A34AC96C7E023E163FA24A838502FC844A82A20101854A820103 855C8501078585130F011F85133F496C8401FF8500036D4C7F000F01FE040F14C0B600C0 4AB71280A559587AD766>97 D<932601FFF8EC0380043FD9FF8013074BB600E0130F030F 03F8131F033F15FE92B53AE007FF803F020349C7387FC07F4A01F091381FE0FF021F01C0 EC07F14A90C8EA01FBDA7FFC6FB5FC902601FFF0824949824949824949824990CA7E4948 835C017F84495A4849831B7F485B5A4A183F5AA24849181FA35A91CC120FA25AA298C7FC 5BA212FFAD127FA27FA36CF20F80A2807EA36C6D181F1C007E806C626C6D183E1B7E6C7F 6D6C60133F6E4D5A6D6C4D5A6D7F6D6D4C5A6D6D4C5A6D01F84C5A6D6C6C047FC7FC6EB4 16FE6E01C0EC03FC020701F0EC0FF86E01FEEC7FF00200903AFFE003FFC0033F90B6C8FC 030F5D030115F0DB003F1480040101F8C9FC515978D662>99 DIII<932601FFF01407043F01FF5C4BB600E05B030F03F85B033F 8192B526C00FFF5B020301FCC76D5A020F01E0EC3FC14A0180EC0FF34A48C80003B5FCDA FFFC814901F0814949824949824949824990CA7E494883495A86494883485B86485B5A4A 845AA2484984A35A91CC7EA25AA298C8FC5BA212FFAD127F061FB712807FA37EDE0003EC E0006EDC007F13806C7390C7FC86A26C7FA27E807E6C7FA26C7F6D7EA26D7E6D7E6D7F6D 6D5E6D7F6D01F85E6D6D93B5FCDA3FFF5D6E01C0EC03FB6E01F0EC0FF1020301FEEC3FE0 0200903BFFE003FFC07F033F90B6EA801F030F4B487E030103F87FDB003F02C090C8FC04 0101F8CAFC595978D66A>III108 DII<933803FFE0047F13FF0303B612E0031F15FC037F15FF912801FFFE003F13C0 020701F0010713F04A018001007FDA3FFEC8EA3FFE4A486F7EDAFFF003077F49496F7F49 496F7F49496F7F4990CA6C7E4948717E4948717E017F854A834948717F48874A83488748 49717FA248874A834887A291CC127F4887A34887A24985A300FF1C80AE6C6C4F1300A46C 636E18FFA36C636E5F6C63A26C6D4D5BA26C6D4D5BA26C6D4D5B6C636E5F6D6C4D90C7FC 013F616D6C4D5A6F16FF6D6D4B5B6D6D4B5B010101F8030F13C06D6D4B5B6EB4037F90C8 FC6E6D4A5A020F01F0010713F86E01FE013F5B020190B712C06E6C93C9FC031F15FC0303 15E0DB007F91CAFC040313E0595978D66A>II114 DI<003FBC12E0A59126C0 003F9038E0001F01FCC76C491301D87FF06E499038007FF049193F49191F49190F90C817 07A2007E1A03A3007C1A01A600FC1BF8481A00A6C91800B3B3AB4C7F4C7F4BB512FC021F B812C0A555547AD362>II119 D<003FB600F091B612FC A5C66C02C0023F1480010F91C8000F01F8C7FC01034917E06D71138098C8FC6D18FC6E6C 5E6E5F705D6E6D4A5A6E6D141F626E6D4AC9FC6E6D147E80705C6E6D495A6F6C1303616F 6D485A6F6D485A6F141F05E05B6F6D48CAFC6FEBF87E18FE6F6D5A6FEBFFF8705BA2705B 705BA2707F707F8284707F845E4C7F85EE0FDFDC1F9F7F058F7FDC3F077F167EDCFE037F 4C6C7F4B487E4B488003076E7E4C6D7F4B5A4B486D7F033F6E7F93C77E037E824B6E7F72 7F4A5A4A486E7F0207707E4B814A48834A486F7F023F707F147F4AC96C7F49717F5B496D 83011F8590B56C4B6D7E00076E031F14E0B600F8037FECFFE0A55B557BD466>I<001FBA 12801AC0A40380C7003F138002F0C8140002C05D91C9485A01FC5F495D494B5B495F5F48 484B5B61495D4D5B96C7FC90C95A4D5A605E003E4B5B605E4C5B605EC9485B95C8FC5E4C 5A5F5D4B5B5F5D4B5B5F5D4B5B94C9FC5D4B5A5E5C4A5B5E5C4A5B4CEC07C05C4A5B93C8 FC5C4A5A5D49170F494916805D5B495B5D49171F495B92C9FC49173F495A5C48187F4849 16FF5C485F48495D4A5D48051F13004849157F91C848B5FC48163FBBFCA47E425578D453 >122 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fs cmr10 10 86 /Fs 86 128 df0 D<193EF1FF80A24E7FA24E7FA24E7FA34E7FA24E7F A24E7FA24E7FA295B67E18FC0501814E7E0503814E7E0507814E7E050F8118C0051F6D7F 1880053F6D7F18004D6D7F177E05FE6D805F04016E805F04036F7F5F04076F7FA24C486E 7FA24C486E7FA24C486E7FA24CC86C7FA204FE6F80A24B486F80A24B48707FA24B48707F A24B48707FA24C82031F854C82033F8593CA7E4B85037E8303FE864B830201874B840203 874B840207874B84020F874B84021F875D023F737F92CCFC4A737F147E02FE73805C0101 74805C0103757F5C0107757F5C010F757FA24948747FA24948747FA249CE6C7FA201FE75 80A248487580A24848767FA248C07EA2488BA3488BA2488BA2488BA2C21280A3797778F6 8A>I<187E18FFA34D7FA34D7FA44D7FA34D7FA34D7FA34D7FA34D7FA494B6FC17FD17FC 0401815F8404038117F08404078117E0A2040F6D7F17C0A2041F6D7F1780A284043F8117 00844C81167E8404FE815E840301835EA203036F7F5EA2850307835E85030F835E85031F 835E85033F8393C8FCA24B6F7F157EA203FE6F7F5DA2850201855D860203855D86020785 5D86020F855DA2021F717F5DA286023F8592CAFC864A85147E8602FE85A2010184891303 497E496D85496D5E017F01F04C800003B500FE041F14FCB700E0023FB8FCA668777BF673 >3 D<007FBEFCBFFCA31E807E02F8CA121F6CF1003F6C6D18036EF0007F6C6D191F6C09 0713C06C6E846F846C876C6E197F816D1B3F6D6D191F6F1AE06D1B0F6D7F6D7F7018077F 6D807018036D1CF06E7F701801806E7F6E7F826E1BF86E6E170083806E6E180083816F7F 6F7F83816F7F84816F808481707F84707F82707FA282705B60705B705B60715AA24DCCFC 4C5A4C5A4C5A5F4C5A4C5A4C5A047F19F84CCCFC4B5A4B5A4B48F001F05E4B5A4B5A4B5A 037F19034BCCFC4A5A4A481AE04A4819074A5A4B190F4A5A4A5A027F1A1F4ACDFC4948F2 3FC049481A7F495A49481AFF4A6149486149486149CD121F49087F138048480703B5FC48 48193F4848061FB6FC48BEFCA25A5A481D00BFFCA27E657177F078>6 DI<94380FFFF84CB6 12C0041F15FC93B87E030717F0031F9026FE003F13FC037F01E0010313FF92B5C86C7F02 0301FC031F13E0020F01F0030713F84A01C003017F4A496F7F4A90CA6C7E4A48717F4949 717F4949717F4949717F4949717F4949717F49874949717F90B58792CC7E48894A854889 4849737FA348894A854889A44A854889AC6C656E61A46C65A26E616C65A36C6D4F5BA26C 65A26C6D4F5BA26D99C7FC6F95B5FC6D63A26D6D4D5B6D63A26D6D4D5B6D63A26D6D4D5B A26D636E6C4D90C8FCA2023F616E6C4D5AA2020F616E6C4D5AA26E6C4D5A6E61A200F86D 4FEB0F807016FF037F95C7FC007CF51F006F6C4B5AA2031F5F7015036C020F4D143EA203 075FA2003F6F0307157E03035F6C6C6401E01B0390B694B6FCA2705DA26C65A56C65A24C 81697579F478>10 D<94260FFF80EC3FF00403B500F8903803FFFE041F02FE011FEBFF80 93B7D8807F14C0030370B612E0030FD9F80101E39038F03FF0033F902780001FF7EB807F 4B48C76CB53800FFF8912601FFF84A5B4A01E04A494813FC4A494A13F84A4991B512F04A 90C8FC4A4817E04A5A4A484D6C13F8A249496F49EB7FF0F53FE049496FED07007392C7FC 98C9FC495BB3A6007FBD12C0A6D8000701C0C8001F90C9FCB3B3B3A2496D4B7F87013F01 F892B512F0B7D8FE03B8FCA666757DF461>I<94380FFF800403B512F0041F14FC93B7FC 030382030FD9F8037F033F903980003FE04B48C7EA0FF0912601FFF8EC03F84A01E04A7E 4A49141F4A494A7E4A90C8127F4A4815FF4A5A4A5AA249494A7F725B495BA2735A495B73 5AF10FE096C9FCB3F11FFF0503B6FC007FBBFCA6D8000701C0C8FC858585B3B3B1496D4B 7FA2013F01F892B512E0B7D8FE03B712F8A655757DF45C>I<94380FFFC00403B512FC04 1FECFF8793B8FC1503030F9038F8007F033F01807F4B48C7FC912601FFF85C4A01E091B5 FC4A5B4A5B4A90C8FC4A5A4A5A4A4881A249498185495BA3495BB3A6007FBBFCA6D80007 01C0C8121FB3B3B3A2496D4B7FA2013F01F892B512E0B7D8FE03B712F8A655757DF45C> I<94260FFF80ED3FFE0403B500F0020FB512C0041F02FC027F14F093B66C0103B612FC03 039226FF800F81030FD9F803D9C03F9038E00FFF033F902980007FE0FFFEC77F4B48C7D8 0FF101F8EC3FC0912601FFF8DA07F701E0EC0FE04A01E04AB500804A7E4A49023F91C812 7F4A494A494B7E4A90C8B5485C4A484D5C4A484A5C4A485FA249494D4A7F785B49495F84 785B49496F90C8FC73705B73EF3F809CC9FCB3F77FFC0B0FB5FC007FC0FCA6D8000701C0 C8001F90C812038A8A1F7FB3B3B1496D4B6D4B7EA2013F01F892B500E002036D7EB7D8FE 03B7D8F80FB712E0A683757DF48A>I<00F8EE07C0A56C160F007C1780A2007E161FA26C EE3F00A26D5D6C6C15FE6D1401000F5E01F814076C6C4A5A6CB4EC3FF002C013FF6CD9F8 075B6C90B65A6D5D6D92C7FC6D5C010714F8010114E06D6C1380DA0FFCC8FC321D70F253 >21 DI33 DI39 D<163E167E16FEED01FCED03F8ED07F0ED0FE0ED1FC0ED3F80ED7F0015FE14014A5A5D14 074A5A4A5AA24A5A147F5D4AC7FC5B5C13035C1307495AA2131F5CA2495AA2137F5C13FF A25C5AA291C8FC5AA3485AA3120F5BA3121FA25BA3123FA35BA2127FA85B12FFB3A7127F 7FA8123FA27FA3121FA37FA2120FA37F1207A36C7EA37E80A27E80A2137F80133FA26D7E A280130FA26D7E1303801301807F6E7E81143F6E7EA26E7E6E7E1403816E7E1400157FED 3F80ED1FC0ED0FE0ED07F0ED03F8ED01FCED00FE167E163E27A770FC41>I<127812FC12 7E127F6C7E6C7E6C7E6C7E6C7E6C7E6C7E7F6D7E133F806D7E6D7EA26D7E8013036D7E80 7F81147F816E7EA281141FA26E7EA281140781A2140381A2801680A36E13C0A316E0157F A316F0A2153FA316F8A3151FA216FCA8150F16FEB3A716FC151FA816F8A2153FA316F0A3 157FA216E0A315FF16C0A34A1380A316005CA25D1407A25D140F5DA24A5AA2143F5DA24A 5A5D14FF92C7FC5B5C495A13075C495AA2495A495A5C137F49C8FC5B485A485A485A485A 485A485A48C9FC127E5A127827A777FC41>I43 DIII<92 3807FF80037F13F84AB512FE02076E7E021F15E091267FFE0113F8913AFFF0003FFC4901 C0EB0FFE49496D7E4990C76C7FD90FFC02007F011F8349486F7E4A153F49486F7E01FF83 4A150F4884A248496F7EA2481980A34890C96C13C0A34819E0A44819F0A24982A2007F19 F8A900FF19FCB3AA007F19F8A86C6C4C13F0A56C19E0A46C19C06E5DA26C1980A26C1900 6E5D6C60A26C6D4B5AA26D6C4B5AA26D6C4B5A6D6C4B5A6D6C4A5B6D6C4A5B6D6D4990C7 FC6D6D495A6D01F0EB3FFC913A7FFE01FFF8021FB612E00207158002014AC8FC6E6C13F8 0307138046737AEE53>48 DIII<181F4E7E187FA218FF5FA25FA25F5FA25F5FA25F94B5FCA25E5EA2EE07EF17CF16 0FEE1F8FA2EE3F0F167EA216FCED01F8A2ED03F0ED07E0A2ED0FC0ED1F80A2ED3F00153E 157E5DA24A5A4A5AA24A5A4A5AA24A5A4AC7FCA2147E5CA2495A5C1303495AA2495A495A A249C8FC137EA25B485AA2485A485AA2485AA2485A48C9FCA2127E5ABCFCA6CA001FEB80 00B34D7F5F4CB512F8031FB8FCA648717BF053>I<016017E001FC1607D9FF80153F02F8 EC03FFDAFFC0137F92B75A6196C7FC60606018E06095C8FC5F17F817E094C9FC01FC14F8 DA00FCCAFC92CBFCB3A3ED0FFF037F13F00203B512FC020F14FF4A15C091267FF8037F91 26FFC0007FD9FDFEC7EA3FFCD9FFF86E7E4A6E7E02C06E7F5C91C86C7F717F49835B717F 137890CA7F187F85A285A2841A80A51AC0A47FEA0FF8487E487E487EA2B57EA31A80A391 C9FC604918006C5A5BD87E805F90CA12FF003E60123F4D5B6C7E6D4B5B000F606D4B5B6C 7E6C6C4B5B6C6C4B90C7FC01FF4B5A6C6DECFFFCD97FE0495BD93FF8010713E090271FFF 807F5B010790B65A6D4BC8FC010015F8023F14E0020F91C9FC020013F0427378EE53>I< 933807FF80047F13F00303B512FE030F80033F15C04B48C67F912601FFF0EB1FF04A01C0 13074A90C7EA01F8DA1FFC6E7E4A5A4A48EC07FE4A48141F4949143F49494A7E92C8FC49 93B5FC495A131F5C133F495A725A495A725A48496F5AF007E04894C8FCA2485BA25AA25C 5AA35AA2923801FFC0DA000F13FC033F13FF484A14C092B612F0912601FE017F913A03F0 003FFC4A48EB0FFEDA0F806D7EB548C76C7F023E6E7F717F4A8202786F7E14F84A6F7E85 5C727EA24A1780A21AC0845C1AE0A591C914F0A27EA87EA380A27E1AE0A27EA21AC0607E 6E17807E1A007E6E4B5A7E6E5E017F4C5A80013F4C5A6D6C4A5B6E5E6D6C4A5B6D6D4990 C7FC6D6DEB1FFE6D01F0EB7FFC6D9039FE03FFF8023FB65A6E5D0207158002014AC8FC6E 6C13F003071380447379EE53>IIIIII<003FBF 12F0481EF8C012FCA36C1EF86C1EF0D3FCB3A9003FBF12F0481EF8C012FCA36C1EF86C1E F06E2977BD81>61 D<913807FFF0027FEBFF8049B612F0010715FE011F6F7E90277FFC00 7F7FD9FFE001077F4890C700017FD803FC6E7FD807F06F7E48486F7E48486F7E49178048 C9FC123ED87FF06F13C07F7F486C17E07FA76C5A6C5A6C484B13C06C5AEA03C0CA481380 A24D13006017FF4C5B4C5B4C5B4C13C0604C90C7FCEE3FFC5F4C5A4C5A5F4B5B94C8FC5D 5E4B5A5EA24B5A5EA24B5AA25EA2153F93C9FCA3153E157EA3157CAF92CAFCAE15FE9138 03FF804A7F4A7F4A7FA24A7FA76E5BA26E5B6E5B6E5BDA00FEC9FC3B7577F44E>63 D65 DII< BB12F0F2FF801BF01BFE757E1CE0D8000F02E0C7000114F801014ADA001F7F6D4A030313 FF6E050080093F7F090F13F009037F757F757F767E767F767F767F88767F8A767F888A1D 7F8A771380A27713C0A21FE089A21FF0A27713F8A41FFC89A31FFEA51FFFB01FFEA61FFC A265A21FF8A35313F0A21FE0A2651FC0A2531380A2531300661DFF66525B6466525B525B 525B5290C7FC525A51485A515B515B091F5B097F5B91B54CB5C8FC496E030F5B010F6E4A B512F8BD12E01C8051C9FC1BF01B8008F0CAFC70717BF07F>IIIIII75 DIIIII82 DI<001FBFFC A64849C7000F91C7000F148002E06E49140091C86C49151F01FC1B07498749874987491C 7F491C3FA21E1F90C9FCA2481EC0007E1D0FA4007C1D07A848F503E0A6CA96C7FCB3B3B3 AB4D7F4D7F94B612F00203BA12FCA66B707AEF78>II87 D91 D<0103ED0180496CEC03C0496CEC07E0011F150F4948EC1FC091C81380017EED3F004915 7E000116FE495D48484A5A495D0007150348484A5AA2495D001F150F90C85B48151F003E 93C7FCA2007E5DA2007C153EA300FC157E48157CA501FE157F26FBFF8090387DFFC0B56C 017F13E06E15F06E15F8A26E15FC6C81A46C81A26C496D13F8A26C496D13F06C496D13E0 6C496D13C0C648C8EA7F00363369F253>II<913803FFF8027FEBFF800103B612F0010F15FC013F15FF90277FFC003F 7FD9FF80010713E0486D01017F486D6D7F6E6E7E48707E6E6E7E717FA2717FA2717F6C5B A26C496E7F6C5B013FC8FC90C9FCA8173F047FB5FC031FB6FC92B7FC140F023F140191B5 12E00103EBFE00010F13F8013F13C04990C7FC495A4813F8485B485B485B5A5C4890C8FC A24848F003E0A312FF5BA25FA35FA26D5D127F6D5D6C043E9038F807C06EEC7E7F6C6D02 FCEBFC0F6C6D902601F83F14806C6DD907F090B5FC6C01FC90261FE01F14006C9027FF80 FFC05C6C91B5486C5B013F9126FE00035B010F02F86D13E0010102E06D6CC7FCD9001F90 CBFC4B4C7AC953>97 DI<923803FFF8033FEBFFC04AB612F8020715FE021FEDFF80027FD9000713C0DAFFF8 9038003FE04901E0EC7FF04949ECFFF8010F495B4948C714FC013F5D5C495A495A5A485B 7113F8485B7113F048EF7FE04AED1F804894C7FCA24890CBFCA35AA35BA212FFAD127F7F A47E807E193E807E197E6C6D167C6C18FC6E16F87E6C6D15016EED03F06D6C16E0013F16 076D6C6CEC0FC06D6DEC1F806D6DEC3F006D01F814FE010001FEEB03FC913A7FFFC01FF8 021F90B55A020715C0020192C7FCDA003F13FC030313C03F4C7BC94A>I<1AFCF01FFF05 0FB5FCA6EF000F180184A2197FB3AC923803FFC0033F13FC4AB6FC020715C0021F15F002 7F9038C03FF8913AFFFC0007FC010301F0EB00FF4901C0EC7FFF4949141F4948C87E4948 81017F824948814A81485B48187F485BA25A5C5AA291CAFC5AA25AA25BA312FFAD127FA4 7F7EA37E807EA26C7F19FF7E6E5D6C7F6C5F6D6C5D6D6C4B7F6D6C5D6D6C92387F7FFF6D 6D02FE14E06D01E0D903FCECFFE06D01F8EB0FF86D01FFEBFFF0023F90B512C0020F1500 020314FC020002F0EDF000030F90C7007EC8FC53757BF25C>IIIIIIIIIII<923803FFC0033F13FC4AB67E020715E0021F15F84A01007FDAFFF0EB0FFF49 01C001037F49496D7FD90FFEC8EA7FF049486F7E49486F7E4A150F017F8349486F7E4849 6F1380A248496F13C04819E0A24890CA13F0A24819F8A34848EF7FFCA3007F19FEA400FF 19FFAD007F19FEA56C6CEFFFFCA36C19F8A26C6D4B13F0A26C19E06E5D6C19C06C6D4B13 80A26C6D4B13006D6C4B5A013F5F6D6C4B5A6DB4EDFFF06D6D495B6D6D495B010001F001 0F90C7FC6EB4EBFFFE021F90B512F8020715E002011580DA003F01FCC8FC030313C0484C 7BC953>II<922603FF80 14F8037F01F013014AB512FC020702FF1303021F15C0027F9039C07FE007913AFFFE000F F0010301F8EB03F84901E0903800FC0F4949147E4949EC3F1F4990C8121F494816BF4948 ED0FFF84485B4849815A4A815A4A815AA2485BA34890CAFCA5485AAD6C7EA57E80A27E80 7EA26C6D5DA26C6D5DA26C6D5D6C6D5D017F5E6D6C5D6E157E6D6D14FE6D6D495A010301 F0EB07F86D6DEB1FE06D01FFEBFFC0023F90B55A6E4A5A020314F8020014E0DB0FFEC7FC 92C9FCB3A34E7FA2060713FF051FB712C0A652697AC858>I<91391F8001FF0003B5010F 13E0B6497F047F13FC93B57E0381130F923983FC1FFF922687F03F1380C615C0011FEB8F 806D139F16006D13BEA203FC6D1300715A4B6D5A715A94C8FC5DA35DA55DB3B3497F8101 3F13FCB812C0A639497DC841>I<913A1FFF8001C049B5EAF807010FECFE0F013FECFF1F 4915FF3901FFF800480180131F4848C712074848140148488049157F4848153FA2007F16 1F5B170F12FFA217077FA27FA27F01FE92C7FC7F6C13C014FC6CEBFFE015FF6C15F016FE 6C6F7E6C16E06C826C826C82013F816D8101071680010016C0021F15E01400030714F0ED 003F040F13F8160300788100F86F13FC177F6C163FA2171FA26C160FA37E18F87FA26D15 1F18F07F6DED3FE06D157F6D16C06D913801FF806D6C491300D9BFE0EB0FFED91FFCEBFF FC486CB65AD8FC0315E0486C158026F0003F01FCC7FC48010713C0364C7BC941>I IIII<007FB600F0017FB6FCA6D8003F0280010F14C0010791C7 6C01FCC7FC0101496E13F06D4916C097C8FC6E6C5D6E6C5D6E6D5C4E5A6E6D5C6E6D495A 6E4B5A70133F6E6D49C9FC6E6D137E037F5C70485A92383FFF036FEB83F06FEB87E0EFCF C06F13FF6F5C6F91CAFC5F6F5B707E163F83707F707F5E844C7F4C7F93B5FCDB01FC7F4C 6C7E4B486C7EED07E0030F6D7F4C6C7F4B486C7FED3F004B6D7F03FE6D7F4B6D7F4A5A4A 486E7E02076F7E4B6E7F4A5A021F6F7F027F6F7F02FF6F7F1303496D4A7F013F4C13FE00 03B500F84AEBFFC0B600FE91B712E0A653477EC658>II<000FB912F019F8A402F8 C7000713F002804A13E04848C84813C013F8494B1380494B1300604915FF4C5B494A5BA2 4C5B90C8485B4C5BA24C5B484B90C7FC003E5E16FF4B5B4B5B5F5DC8485B4B5BA24B5B4B 90C8FC5E15FF4A5B4A5B5E5C4A4914F84A5BA24A5B4A90C8FC4BEC01F014FF495B495B5D 5B49491403495BA2495B4990C81207494816E0A24849150F485B4A151F48173F4849157F 484915FF17034849140F4890C8B5FCBA12C0A47E3D477BC64A>I126 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ft cmmi6 6 2 /Ft 2 113 df<027FB8FC91B912F01AFC1AFF1BC09126007FF0C7000F13E0070113F04C 6E13F81A7F03FFEE3FFC1A1F4C16FEA24A170FA25EA24A171F1BFC93C9FCF23FF84A177F 1BF04BEEFFE04F13C0020F4C1380070F13004B4B5AF17FF8021F923803FFE0067F138092 B700FCC7FC614A16FF1AC003F0C813F0F13FFC027F707EF107FF4B17808502FF7013C0A2 4B17E0A25BA25DA25B1BC092C95A1B80495F4F13005C4F5A010FEF7FFC4F5A4A4B5B0607 5B011F043F1380013F4BB5C7FC007FB912FCBA12F019C04EC8FC18C04F4478C35C>66 D112 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fu cmr6 6 3 /Fu 3 112 df<177F913A7FF003FFC0902703FFFE0F13E0010FD9FF9F13F0013FECFFCF 903A7FE03FFE0F9039FF800FFC3901FE000300039238FE07E0490101EB03C000076FC7FC 497F000F82A7000793C7FC6D5B00035D6D130300015D3A00FF800FF89138E03FF04890B5 5A1680D803E349C8FC9038E07FF091CAFC1207A37F6C7E7F90B612E016FF6C16C06C16F0 6D15FC90B77E000382481780D81FF8C7123FD83FE0020313C00180EC007F48C9EA3FE048 161F48160FA46C161FA26C6CED3FC06C6CED7F806C6CEDFF00D80FF8EC03FE6CB4EC1FFC 6C9039F001FFF8C690B612E0013F1580010702FCC7FCD9003F138034427CAC3D>103 D108 D111 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fv cmmi8 8 3 /Fv 3 77 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Fw cmsy8 8 1 /Fw 1 80 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Fx cmcsc10 8 47 /Fx 47 122 df<1408147E14FF15805B4913C05B491380A2491300EB3FFC495A495A14C0 485B4890C7FC485AEA0FF8485A5BEA3F8048C8FC12FE127C127012201A1A62DC49>19 D39 D44 DII<157815F814011407140F143FEB01FF131F 000FB5FCB6FCA2EBFE3F13E0EAF000C7FCB3B3B3AE4A7E0103B57E007FB712FCA52E5973 D849>49 DI<913807FF80027F13FC49B6FC010715C090261FFC0313F0903A 3FC0007FF849C7EA1FFE01FC6E7ED801F06E1380484816C04848804916E0D80FFC6E13F0 7F7F486D15F8A282805C5EA27E6C90C7FC6C4816F0EA01F8C9FC4C13E0A218C05E188018 005E4C5A5F4C5A4C5A923801FFC04B5BDB7FFEC7FC91387FFFFC16E016F816FFDA000313 C0030013F0EE3FF8EE1FFE707E7013807013C018E07013F0A27013F818FCA218FE177FA2 18FFA2EA07E0EA1FF8487E487EA2B5FCA318FEA217FF4916FCA26C5A494A13F84916F06C C85A6C6C16E06D4A13C06C6C4A1380D807F84A1300D803FE4A5A2601FFC0495A6C9039FC 03FFF8013FB612E0010F5D010392C7FC010014F802071380385C78D849>I<173E177E17 FEA216011603A21607160FA2161F163FA2167F16FF16F71501ED03E716C71507ED0F8716 07151F153E153C157C15F8EC01F0A2EC03E0EC07C0A2EC0F80EC1F00A2143E5C147814F8 495A5C1303495A5C130F49C7FC131E133E5B137813F8485A485AA2485A485AA248C8FC12 3EA25A5ABA12F0A5C9D80FFEC7FCB04C7E047F13C04AB712F0A53C5A7AD949>II<923803FF80033F13F092B512FC020314FF020F01001380DA1FF8EB1FC0 DA7FE0EB07E0DAFF8013034990C713F04948141F4948143F4948EC7FF8494814FF495AA2 495A13FF5C48EE7FF04890C8EA3FE0EF1FC04893C7FC5B120FA3485AA2123FA315043A7F F801FFF8020713FF4A804A14E091397F003FF002FCEB0FF800FF496D7ED9F9F06D7ED9FB E06D7E4A6D1380D9FF8015C0EF7FE091C8FCEF3FF04916F8A218FCA249151F18FEA44916 FFA3127FA7123F7FA218FE121FA3000F17FC173F6C7E18F8A26C6CED7FF06C17E06E14FF 6C17C06E4913806D6C15006D6C495A6D6CEB0FFC6D6C495A6DB46CB45A6D90B55A010015 80023F91C7FC020F13FC020113C0385C78D849>II<913803FFC0021F13FC91B6FC4915C0902607FE00 13F0D90FF0EB1FF8D91F80EB03FE017FC76C7E017C9138007F8049153F4848ED1FC00003 17E049150F000717F0491507120FEF03F8A2121FA27FA37F6D15076D16F07F6C6CED0FE0 14C002F0EC1FC06E143F6C01FEEC7F806E15006C02C013FE6C9138E003FC9238F807F86C 9138FE0FF06D9038FF3FC06DECFF806D4AC7FC010714F86D14FE0100806E14C06E8091B6 12F8010381D907F88090261FF07F7F90263FC01F148049486C14C09026FE000314E04848 6D14F048486D7E4848021F13F8000F6F13FC491403484880003F9238007FFE49151F007F 160F90C9EA07FF83A200FE8283A3187FA3187E7E7E18FE6D16FCA26C6CED01F8A26C6CED 03F06C6CED07E06C6C150F6DED1FC06CB4ED7F806C01C0903801FF0026007FF0EB07FED9 3FFEEB7FF86DB65A010715C0010192C7FCD9003F13FC020313C0385C78D849>I<913801 FFC0021F13F8027F13FE49B67E010701807F903A0FFC003FE0D91FF0EB0FF049486D7E49 486D7E494880486F7E4890C81380485A18C0000F167F484816E0A2003FEE3FF0A25B007F 17F8A318FC12FF171FA218FEA718FFA3007F5EA4123F6D5DA2121F94B5FC120F6C7E5E6C 7E6CED03DF6C6D13076D6CEB0F9F6D6CEB1F1F6D6C013E13FED90FFC13FE6DB512F80101 14F06D14E0021F1380913A0020003FFC92C7FCA318F8A3EF7FF0A218E017FFD803F816C0 487E486C4A1380487E4C1300A24C5A5F4C5A49141F6C484A5A494A5A01C04A5A6C6C495B D803F8D907FEC7FC3A01FF803FFC6C90B55A6D14E06D1480010749C8FC010013F0385C78 D849>I65 D67 DI70 D73 D76 DII82 D<912607FF80EB0380027F01F8130749B500FE130F0107ECFF80011FEDE0 1F4948C6EBF03FD97FF0EB0FFCD9FFC0903803FE7F4890C8B5FC4848153F484881484815 0783485A83484881187F127F49163FA300FF171FA2180FA27FA36D1607A27FA26C6C93C7 FC7F7F6C13C014F06C13FEECFFE06C14FEEDFFE06C15FE6CEDFFE06C16F86C16FE6D816D 826D16E06D820103821300021F81020181DA000F801500040F14801601706C13C0171F71 13E083838319F083127000F0177FA3183FA37EA319E07EA26C177F19C07E6DEEFF80A26D 4B13007F6D4B5A01FC4B5A6D5E6D6C141FD93FE04A5A26FE1FFEECFFE0902707FFE00713 80486C90B6C7FC48C65D021F14F848010314C0489026003FFEC8FC3C6176DD51>I85 DI<167816FCA34B7EA24B7EA34B7FA34B7FA24B7F151EA2033E7FED3C7FA203 787F163F03F87FEDF01FA2020180EDE00FA24A486C7EA20207814B7EA2020F814B7EA202 1E6D7FA2023E81023C147FA2027C810278143F02F8814A141F91B7FC4982A3D903E0C7EA 0FFF4A800107834A80A2010F8391C87EA24983011E81013E83187F137E01FE83487E486D 4B7E001F6D4A13FFB500F8023F14FCA446467BC551>97 DI<922603FFC01307033F01FC130F4AB6FC0207EDC01F021F9039803FE03F913B7FFC00 07F07FDAFFE0903801FCFF01030180EB007E4990C87ED90FFC8149481507495A49488149 488148835C4890CA127F5A49173F120F5B001F181FA2123F5B190F127FA349170012FFAC 127F7F190FA2123FA27F121F191F000F181E7F0007183E6D173C7E6C6D167C6E16786C18 F86D6CED01F06D6CED03E06D6C16C06D6C150F6DB4ED1F806D6DEC3F00010001E014FEDA 7FFCEB03F8913A1FFF803FF0020790B512C002015DDA003F01FCC7FC030313C0404879C5 4F>IIII104 DI108 DII< ED07FF037F13F00203B512FE020F6E7E91263FFC0113E0913AFFE0003FF8490180EB0FFC 4948C7EA03FED90FFC6E6C7E49486E7F49486F7E49486F7E49486F7E4A150F48844890C9 6C7EA24848707E000F19804982001F19C0A24982003F19E0A3007F19F049177FA300FF19 F8AC007F19F06D17FFA4003F19E06D5EA2001F19C06D5E000F1980A26C6C4C1300A26C6D 4B5A6C6D4B5AA26C6D4B5A6D6C4B5A6D6C4B5A6D6C4A5B6DB402075B6D6D4990C7FC0101 01E0EB3FFC6D9039FC01FFF8023FB612E0020F158002034AC8FCDA007F13F0030790C9FC 454879C554>IIII<9026 01FFC013E0010FEBF801013F13FE49EBFF833A01FF807FC73A03FC000FEF4848EB03FFD8 0FE01300167F4848143F4848141FA2007F150F90C8FC16075A1603A37F16017FA27F6C6C 91C7FC7F13FE383FFFE014FE6CEBFFE015FE6CECFFC06C15F06C816C15FE6C816D158013 1F010315C0D9007F14E01403DA003F13F01503030013F8167F163FEE1FFC160F127000F0 1507A21603A37EA217F87E16076C16F0A26CED0FE07F6DEC1FC001F0EC3F8001FCEC7F00 48B4EB01FE3AFC7FF007FCD8F81FB55AD8F00714E001015C27E0001FFEC7FC2E4879C53D >I<003FBA12C0A49026FC000FEB800301E06D90C7127FD87F80EF1FE090C7160F007E18 07A2007C1803A219011278A400F819F0481800A5C81700B3B3A24B7F033F13E091B712F8 A444437BC24F>III<007FB500FC010FB512F0A4C602E00103EBFE00013F496D13F06D496D 13C06D94C7FC6D5E6D6D5C6F5C6D4B5A6D7F6F495A6E6C495A023F5D6F130F6E6C49C8FC 6E143E16806E6D5A6E5C6E6D5A16F16EEBF3E06FB45AA26F5B6F90C9FCA26F7F81836F7F 814B7F835D92380FBFFC92381F3FFE161F92383E0FFF037C80707F4B7EDA01F0804A486C 7F824A48804A486D7E173F4AC76C7E023E814A80717F4A8249488001036F7F010783010F 6F7F131F013F83D9FFF84A7F000F6D4AEBFF80B6020F14FFA448447CC351>120 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fy cmr8 8 75 /Fy 75 128 df11 DI<1206EA1F80487E487E7F487EA2 6C7E7F6C7E6C7F7E6C7F00017F6C7F6D7E133FEB0FFC6D7E6D7E01001380147FEC3FC0EC 1F80EC070014021A1A71DC47>18 D<1418147E14FF4913805B4913C0A24913805B491300 495A5C495A4813E0485B485B91C7FCEA0FFC485A485AEA7FC05B48C8FC127E123812101A 1A63DC47>I24 D40 D<127812FC127E7E6C7E7F6C7E6C 7E6C7E6C7E6C7E7F7F6D7E806D7E130F806D7EA26D7EA26D7EA26D7EA281147F81143F81 A2141F81A2140F81A36E7EA4811403A481A380A51680B3A41600A55CA35DA414075DA44A 5AA35D141FA25D143FA25D147F5D14FF92C7FCA2495AA2495AA2495AA2495A5C131F495A 5C49C8FC5B5B485A485A485A485A485A5B48C9FC127E5A1278218678E337>I44 DII48 D<157815FC14011407140F 147FEB01FF131F001FB5FCB6FC14BFEBFE3F13E0EAE000C7FCB3B3B3AD4A7EA20103B512 C0007FB8FCA5305974D847>I<91380FFF80027F13F80103B6FC4915C0011F8190263FF0 0F13F89026FF80017F4848C76C7ED803F8EC1FFF496E138048486E13C048486E13E04848 16F090C87E4817F8003E6F13FC127E007C167FD87FE016FE487E7FEF3FFF7FA66C5A6C5A 6C5AEA0780CA13FE177FA3EFFFFCA218F85E18F04C13E0A24C13C04C138018004C5A4C5A 5F4C5A4C5A4B5B4B5B4B90C7FC5EED0FF84B5A4B5A4B5A4B5A4A48C8FC4A5A4A5A4A5A4A 48141F4A5A4A5A4AC8123E5C495A495A495AD90FC0157E4948157C49C9FC017E16FC4915 0148B8FC5A5A4817F85A5A5AB9FCA218F0A3385979D847>I<913803FFE0023F13FE91B6 12C0010315F090260FFE017F903A1FF0003FFED93F80EB0FFF49C76C7F01FC6E7F484882 48486E7F5BD807FE826D80486D81A280A66C5B6C90C85B6C5AD800785C90C8FC605E60A2 4C5B604C90C7FC4C5A4C5A4C5A4C5A03035B037F5B027FB5C8FC16FC16F016FEEEFFC0DA 000113F09238003FFCEE0FFE706C7E707F707F707F84717EA284717EA21980A28319C0A2 EA07C0EA1FF0487E487EA2487EA319805FA24917006C5A495D495E6CC95B17FF6C6C5E6C 6C4A5B01F04A5BD807FC4A5B6CB44A5BC601C0013F90C7FC903A7FFE01FFFE6DB612F801 0F15E001031580D9007F01FCC8FC020713C03A5C7AD847>I<171F4D7E177F17FFA25E5E A25E5EA25E5EA25E93B5FC5D16FBED03F3150716E3ED0FC3ED1F83A2ED3F03157EA215FC EC01F8140315F0EC07E0140F15C0EC1F80EC3F00A2147E5CA2495A495A13075C495A131F 5C49C7FC137EA25B485AA2485A485A120F5B485A123F90C8FC127E5ABA12FCA5C90007EB 8000B04C7F043F13F04AB712FCA53E5A7CD947>II<923801FFE0031F13FC037F13FF4AB67E02079038807FE0913A1FFC000FF0DA 3FF01303DA7FC06D7E4A486D7E010390C712074948140F49484A7E4A143F495A133F495A 13FF5C48705A4A6E5A48705A4893C8FC91CBFCA25A5B121FA3123FA2ED3FF8484890B5FC 020314C04A14F0021F8091393FC03FFE91397F000FFF00FF017C01037F4A6D7FD9FDF06D 7F4A147F01FF824A6E7E844A141F8491C8FC84A2496F1380A419C05BA2127FA7123FA37F 001F1880A3120F19005F6C7E607E4D5A6C6D5DA26C6D4A5A6D6C4A5A013F5E6E495BD91F FC4990C7FC6D6CEB0FFE903A03FFC07FFC6D90B512F06D5D023F1480020F49C8FC020013 E03A5C7AD847>II<913801FFE0020F13FE027F6D7E91B612E001039038807FF8903A07F8 000FFCD90FE0EB03FE4948903800FF8049C8127F017E6F7E496F7E00018349150F000383 1707485A841703120FA37FA27F17076D5E7F6E140F6C01E05D6E141F02FC4A5A6C6D5D6E 6C495A6C02E049C7FC6C9138F003FE9238FC07FC6D9038FF0FF86DECBFF06DECFFE06D15 806D4AC8FC0101806D81023F14E017F891B67E010315FF49486C8090261FF83F8090263F F00F8090267FC007809026FF8001804890C76C7F4848804848020F7F48481403496E7F48 486E7E003F70138049150F007F82496F13C0838348CAFC187FA3183FA319807F127F187F 6D1700123F6D16FE121F6D4B5A6C6C4B5A6C6C4B5A6C6C4B5A6C6C6CEC3FC06C01E0ECFF 80D97FF8010390C7FC6DB4EB3FFE010F90B55A6D15F0010015C0023F49C8FC020113E03A 5C7AD847>I<913801FFE0020F13FC027F13FF49B612C0499038C07FF0903A0FFE001FF8 D91FF8EB07FC49486D7E49486D7E49486D7F48498148167F91C87F48707E485A001F8317 1F003F83A25B007F83A38400FF82A31980A719C0A3007F5EA4123F6D5DA2121F5F120F6D 92B5FC7E6C6D5B6C16EF6C6D13036EEB07CF6D6CEB0F8FD93FFC013F14806DB4EBFF0F01 07EBFFFE6D14F8010014F0023F13C00207D9001F130091C8FCA260A360173FA260A2D803 F84B5A487E486C4B5A487E604C5B5E95C7FC4C5A494A5A6C484A5A494A5A01E04A5A6C6C 4A5AD803FC01035B2701FFC01F90C8FC6C90B512FC6D5C011F14E0010791C9FC9038007F F83A5C7AD847>III65 DIIIIIIII<023FB712C0 A5DA0001ECC000DB003F5B95C7FC82B3B3B3A6EA0380EA1FF0487E487EA2487EA34C5AA3 495D6C48147F495D4914FF6CC85B6C6C495B6D495BD80FF0495BD807FC011F90C8FC3A03 FFC07FFEC690B55A6D14F0011F14C0010791C9FC9038007FF03A5E7ADA48>IIIIIII82 D<912607FF80EB0380027F01F8130749B6130F 010715C049EDE01F90263FFE00EBF83FD97FF0EB0FFCD9FF80903803FE7F4890C8B5FC48 48153FD807F881000F824981484881A248488183127F49167FA2183F12FFA2181FA27FA2 180F7FA27F127F6D93C7FC7F7F6C6C7E14E06C13F8ECFF806C14F86CECFF8016FC6CEDFF C06C16F86C16FE6D816D16C0010F826D82010182D9003F81020381DA003F801503DB003F 1480040114C0707E051F13E01707837113F083187FA2F03FF8127800F8171FA3180FA37E A319F07E181F7E19E07FF03FC07F6DEE7F806D16FF6D170001FE4B5A6D4B5A02C04A5AD9 3FF0EC1FF026FE1FFEEC7FE0903B07FFF003FFC0486C90B65A48C64BC7FC021F5C480103 14E048D9001F90C8FC3D6178DD4E>I<001FBDFCA5489026F000079026FC0001148091C7 000101F0EB001F01FC190701F019014985491A7F491A3FA290C8181FA2003E1B0F007E1C C0A3007C1B07A748F303E0A5C995C7FCB3B3AF4C7FA2041F13FF020FB812FEA55B5A7BD9 66>IIII89 D<000FBA12F01AF8A403E0C7000713F002FCC814E002E05D48 01804B13C091C91480495E491800494C5A4916FF495F4D5B495D61494B5B5F6190C9485B 5F4895C7FC4D5A17FF003E5F5E4C5B605EC9485B605E4C5B95C8FC5E5F4C5A5D5F4B5B5D 5F4B5B5D5F4B90C9FC5D5E4B5A5C5E4A5B5C4C157C5C4A5B5E5C4A90C9FC5D02FF17FC49 5B4B16F85B5D495B5B4B1501495B5B92C91203495A13FF4A16074849160F5A4A161F4849 163F48187F4A16FF48050313F04849150F91C9127F480407B5FCBBFCA47E465B78DA56> II93 D97 DIII<913801FFE0020F13FE027F6D7E49B612E04981010F01C07F903A1F FE001FFCD93FF8EB07FE49486D7E49486D138048496D13C0485B91C8EA7FE04817F04848 153FA2001F17F849151F123FA2007F17FCA249150FA2B9FCA418F801F8CAFCA7127F7FA2 123FA36C6C1638187C6C7E18FC6C6D15F86C16016C6D15F06C6D14036D6CEC07E06D6CEC 1FC0D91FFEEC3F806D6C6CEBFF006D9038F00FFE010190B55A6D15F0023F14C0020791C7 FC9138007FF0363C7CBA3F>IIIII107 DIIIIII<90393FF001FEB590380FFFC04B13F04B13F84B 13FCEDFF1F00019039F1FC3FFE39007FF3F0ECF7E0013F13C0ECFF80A29238001FFCEE0F F84AEB07F093C7FCA25CA45CB3AC497E90B5FCB77EA52F3A7CB937>I<903A01FFF001C0 011FEBFF07017F14CF48B7FC5A48EB001FD80FF81303D81FE013004848147F49143F007F 151F90C8120F5AA21607A27F7F7F6D91C7FC13FC387FFFC014FE6CEBFFF015FF6C15C06C 15F06C816C15FEC6816D1580131F010315C0D9003F14E01400030F13F015019238007FF8 0078153F00F8151F160F7E1607A27EA26C16F0A26D140F6D15E06D141F6DEC3FC06D147F 01FE903801FF809026FFC00F1300486CB55A486C14F8D8F80F14E0D8F003148027E0007F F8C7FC2D3C7BBA38>I<147CA614FCA51301A31303A21307A2130F131F133F137F13FF12 03120FB81280A526001FFCC8FCB3AAEE03E0AC16076D6C14C0A2160F6D6C1480ED801F6D 9038C03F006DEBF0FE6DEBFFFC6E5B6E5B020F5B020190C7FC2B537DD137>IIII<007FB500F8013FB512C0A5C6 6C01E06DEBF800011F01806D13C06D6F90C7FC6D16FC6D6D5C6D5E6F495A6D6D495A6E6C 5C6E6C49C8FC021F147E6F13FE6E6C485A6EEB83F86EEBC3F06EEBC7E0EEEFC06E13FF6F 5B6F90C9FC6F5AA26F7E6F7F835D834B7F4B6C7E4B6C7EEDFE1FDA01FC7F4B6C7E4A486C 7F4A486C7F020F7FDA1FC0804B6C7F4AC76C7E027E6E7E02FE141F49488101036F7E010F 6F7F011F83017F4B7F2607FFFE4A13F8B66C017FEBFFF8A545397EB84A>II<000FB8 12E018F0A318E09138C000034848C74813C001F84A138001E04A1300A2494A5A494A5A4C 5A4B5B90C7FC4B5B4B5B484A5B003E4A90C7FCA24B5A4B5A4B5AC7485BA24A5B4A5B4A5B 93C8FC5C4A48EB01F04A5A4A5AA2495B495B49491303494914E0A24990C7FC495A494814 07495AA24849140F485B4849141F485B177F4890C812FF4848020313C04848143F90B8FC B9FCA37E34397CB83F>II127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fz cmbx10 10 46 /Fz 46 118 df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ndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 1200dpi TeXDict begin %%PaperSize: A4 %%EndSetup %%Page: 1 1 1 0 bop 890 1790 a Fz(EXPLICIT)84 b(FINITE)g(V)-5 b(OLUME)84 b(CRITERIA)g(F)-5 b(OR)84 b(LOCALIZA)-16 b(TION)1023 2023 y(IN)84 b(CONTINUOUS)h(RANDOM)g(MEDIA)f(AND)g(APPLICA)-16 b(TIONS)2273 2521 y Fy(FRANC)-82 b(\030)19 b(OIS)61 b(GERMINET)f(AND)g (ABEL)h(KLEIN)1355 2974 y Fx(Abstra)m(ct.)83 b Fy(W)-12 b(e)45 b(giv)l(e)f(\014nite)h(v)l(olume)f(criteria)f(for)i(lo)t (calization)40 b(of)k(quan)l(tum)j(or)d(clas-)1355 3140 y(sical)50 b(w)l(a)l(v)l(es)i(in)f(con)l(tin)l(uous)h(random)g(media.) 77 b(W)-12 b(e)52 b(pro)l(vide)g(explicit)e(conditions,)h(de-)1355 3306 y(p)t(ending)45 b(on)f(the)g(parameters)e(of)i(the)g(mo)t(del,)f (for)h(starting)e(the)i(b)t(o)t(otstrap)f(m)l(ultiscale)1355 3473 y(analysis.)79 b(A)53 b(simple)e(application)g(yields)h(lo)t (calization)c(for)k(Anderson)h(Hamiltonians)1355 3639 y(on)40 b(the)g(con)l(tin)l(uum)f(at)g(the)h(b)t(ottom)f(of)g(the)h(sp) t(ectrum)f(in)g(an)h(in)l(terv)-8 b(al)38 b(of)h(size)f Fw(O)t Fy(\()p Fv(\025)p Fy(\))i(for)1355 3805 y(large)32 b Fv(\025)p Fy(,)37 b(where)d Fv(\025)g Fy(stands)g(for)f(the)h (disorder)f(parameter.)57 b(A)34 b(more)e(sophisticated)g(appli-)1355 3971 y(cation)53 b(pro)l(v)l(es)g(lo)t(calization)c(for)k(t)l(w)l (o-dimensional)e(random)i(Sc)l(hr\177)-71 b(odinger)53 b(op)t(erators)1355 4137 y(in)c(a)f(constan)l(t)g(magnetic)e(\014eld)j (\(random)f(Landau)h(Hamiltonians\))c(up)k(to)g(a)f(distance)1355 4317 y Fw(O)t Fy(\()1546 4256 y Fu(log)23 b Ft(B)p 1546 4287 277 6 v 1636 4369 a(B)1843 4317 y Fy(\))37 b(from)f(the)h(Landau)h (lev)l(els,)f(where)g Fv(B)44 b Fy(is)36 b(the)h(strength)g(of)g(the)g (magnetic)e(\014eld.)3072 5135 y Fs(1.)92 b Fr(Intr)l(oduction)957 5434 y Fs(W)-14 b(e)65 b(giv)-5 b(e)66 b(\014nite)f(v)-5 b(olume)67 b(criteria)e(for)g(lo)5 b(calization)65 b(of)h(w)-5 b(a)g(v)g(es)66 b(in)g(random)h(media.)105 b(The)758 5633 y(emphasis)58 b(is)g(on)f(pro)-5 b(viding)58 b(explicit)e (conditions,)j(dep)5 b(ending)57 b(on)g(the)g(v)-9 b(arious)57 b(parameters)758 5833 y(of)k(the)h(mo)5 b(del,)64 b(for)e(starting)g (the)f(b)5 b(o)g(otstrap)62 b(m)-5 b(ultiscale)63 b(analysis)g([GK1)q (].)94 b(These)62 b(criteria)758 6032 y(th)-5 b(us)68 b(yield)f(Anderson)g(lo)5 b(calization,)70 b(strong)e(dynamical)f(lo)5 b(calization,)70 b(SULE,)d(etc.)108 b(\(See)758 6231 y(also)40 b([GK3)q(])g(for)f(a)h(discussion)i(of)e(the)f(consequences)h (of)g(the)f(b)5 b(o)g(otstrap)40 b(m)-5 b(ultiscale)40 b(analysis.\))957 6430 y(In)60 b(the)g(lattice)f(\014nite)h(v)-5 b(olume)61 b(criteria)e(for)h(lo)5 b(calization)60 b(w)-5 b(ere)60 b(pro)-5 b(vided)61 b(b)-5 b(y)60 b(Aizenman)758 6630 y(et)e(al)h([ASFH].)84 b(Although)59 b(w)-5 b(e)59 b(w)-5 b(ere)59 b(motiv)-9 b(ated)58 b(b)-5 b(y)59 b(con)-5 b(tin)g(uum)60 b(mo)5 b(dels,)61 b(our)e(criteria)f(are)758 6829 y(also)66 b(v)-9 b(alid)66 b(on)g(the)f(lattice;)71 b(in)66 b(particular,)j(they)c(can)h(b)5 b(e)65 b(used)i(to)e(satisfy)h (the)g(criteria)f(in)758 7028 y([ASFH)o(].)957 7227 y(The)57 b(explicit)g(\014nite)g(v)-5 b(olume)58 b(criteria)f(giv)-5 b(en)57 b(in)h(this)g(article)f(yield)g(lo)5 b(calization)57 b(in)g(situ-)758 7427 y(ations)h(where)h(the)e(crucial)i(quan)-5 b(tities)58 b(of)h(the)e(mo)5 b(del)59 b(that)f(en)-5 b(ter)58 b(the)g(m)-5 b(ultiscale)59 b(analysis)758 7626 y(\(the)52 b(constan)-5 b(t)53 b(in)g(W)-14 b(egner's)53 b(estimate)f(and)h(the)f(constan)-5 b(t)53 b(in)h(the)e(Simon-Lieb)i (inequalit)-5 b(y\))758 7825 y(dep)5 b(end)64 b(on)h(the)f(parameters)h (of)g(the)f(mo)5 b(del)64 b(\(e.g.,)j(the)d(disorder)i(parameter,)h (the)d(energy)758 8024 y(where)55 b(lo)5 b(calization)55 b(is)h(to)f(b)5 b(e)55 b(pro)-5 b(v)g(en,)56 b(the)f(strength)g(of)g (the)g(magnetic)h(\014eld\).)957 8224 y(T)-14 b(o)57 b(illustrate)g(the)f(need)h(for)g(suc)-5 b(h)57 b(explicit)g(criteria,) f(let)h(us)g(consider)h(the)e(simplest)i(An-)758 8423 y(derson)e(Hamiltonian)g(in)f(the)g(con)-5 b(tin)g(uum,)2434 8703 y Fq(H)2572 8728 y Fp(\025;!)2832 8703 y Fs(=)46 b Fo(\000)p Fs(\001)38 b(+)f Fq(\025V)3672 8728 y Fp(!)3906 8703 y Fs(on)139 b(L)4324 8635 y Fn(2)4398 8703 y Fs(\()p Fm(R)4573 8635 y Fp(d)4660 8703 y Fq(;)28 b Fs(d)p Fq(x)p Fs(\))p Fq(;)1362 b Fs(\(1.1\))758 8985 y(where)55 b Fq(\025)46 b(>)g Fs(0)55 b(is)h(the)f(disorder)i(parameter)e(and)h(the) f(random)h(p)5 b(oten)-5 b(tial)55 b Fq(V)5596 9010 y Fp(!)5747 8985 y Fs(is)h(of)g(the)f(form)2852 9289 y Fq(V)2949 9314 y Fp(!)3045 9289 y Fs(\()p Fq(x)p Fs(\))45 b(=)3512 9131 y Fl(X)3490 9499 y Fp(i)p Fk(2)p Fj(Z)3711 9466 y Fi(d)3802 9289 y Fq(!)3905 9314 y Fp(i)3960 9289 y Fq(u)p Fs(\()p Fq(x)37 b Fo(\000)g Fq(i)p Fs(\))28 b Fq(;)1780 b Fs(\(1.2\))758 9747 y(with)74 b Fq(u)p Fs(\()p Fq(x)p Fs(\))i Fo(\025)h Fs(0)d(a)g(b)5 b(ounded)75 b(measurable)g(function)f(with)g(compact)g(supp)5 b(ort)75 b(suc)-5 b(h)75 b(that)758 9946 y(0)46 b Fq(<)g(U)1175 9971 y Fk(\000)1334 9946 y Fo(\024)1509 9821 y Fl(P)1684 9996 y Fp(i)p Fk(2)p Fj(Z)1905 9962 y Fi(d)2004 9946 y Fq(u)p Fs(\()p Fq(x)15 b Fo(\000)g Fq(i)p Fs(\),)45 b(and)g Fq(!)52 b Fs(=)46 b Fo(f)p Fq(!)3459 9971 y Fp(i)3515 9946 y Fs(;)72 b Fq(i)46 b Fo(2)g Fm(Z)3995 9886 y Fp(d)4081 9946 y Fo(g)f Fs(a)f(family)g(of)g(indep)5 b(enden)-5 b(t,)47 b(iden)-5 b(tically)758 10146 y(distributed)76 b(random)h(v)-9 b(ariables)76 b(taking)g(v)-9 b(alues)76 b(in)g(the)g(in)-5 b(terv)c(al)76 b([0)p Fq(;)28 b Fs(1],)81 b(whose)76 b(common)758 10345 y(probabilit)-5 b(y)64 b(distribution)g Fq(\026)f Fs(has)h(a)f(b)5 b(ounded)64 b(densit)-5 b(y)64 b Fq(g)h(>)59 b Fs(0)64 b(a.e.)97 b(in)64 b([0)p Fq(;)28 b Fs(1].)97 b(This)64 b(mo)5 b(del)p 758 10491 997 7 v 957 10674 a Fy(2000)45 b Fh(Mathematics)50 b(Subje)-7 b(ct)49 b(Classi\014c)-7 b(ation.)63 b Fy(Primary)46 b(82B44;)f(Secondary)j(47B80,)d(60H25.)957 10840 y(A.K.)i(w)l(as)f (supp)t(orted)h(in)g(part)g(b)l(y)h(NSF)f(Gran)l(t)g(DMS-0200710.)3713 11039 y Fn(1)p eop %%Page: 2 2 2 1 bop 758 861 a Fn(2)1551 b(FRANC)-77 b(\030)17 b(OIS)59 b(GERMINET)g(AND)f(ABEL)g(KLEIN)758 1259 y Fs(w)-5 b(as)74 b(studied)h(b)-5 b(y)74 b(Com)-5 b(b)5 b(es)74 b(and)g(Hislop)h([CH1)o (])f(and)g(Kirsc)-5 b(h)75 b([Ki],)k(who)74 b(pro)-5 b(v)g(ed)74 b(that)f(for)758 1458 y(an)-5 b(y)65 b(\014xed)f(energy)g Fq(E)2154 1483 y Fn(1)2290 1458 y Fq(>)e Fs(0)j(w)-5 b(e)64 b(ha)-5 b(v)g(e)65 b(Anderson)h(lo)5 b(calization)64 b(in)h(the)f(in)-5 b(terv)c(al)65 b([0)p Fq(;)28 b(E)6352 1483 y Fn(1)6426 1458 y Fs(])64 b(for)758 1658 y(su\016cien)-5 b(tly)56 b(large)f(disorder.)75 b(\(Note)54 b(that)h(0)g(is)h(the)f(b)5 b(ottom)55 b(of)g(the)g(sp)5 b(ectrum.\))957 1857 y(But)59 b(supp)5 b(ose)61 b(w)-5 b(e)60 b(ask)g(a)g(di\013eren)-5 b(t)59 b(question:)83 b(If)60 b([0)p Fq(;)28 b(E)4498 1882 y Fp(\025)4584 1857 y Fs(])60 b(is)h(the)e(largest)h(in)-5 b(terv)c(al)59 b(at)h(the)758 2056 y(b)5 b(ottom)56 b(of)g(the)g(sp)5 b(ectrum)57 b(in)g(whic)-5 b(h)57 b Fq(H)3358 2081 y Fp(\025;!)3629 2056 y Fs(is)g(lo)5 b(calized,)56 b(ho)-5 b(w)57 b(do)5 b(es)57 b Fq(E)5368 2081 y Fp(\025)5511 2056 y Fs(gro)-5 b(w)57 b(with)g Fq(\025)p Fs(?)77 b(T)-14 b(o)758 2255 y(pro)-5 b(vide)38 b(a)h(lo)-5 b(w)g(er)39 b(b)5 b(ound)39 b(on)f Fq(E)2701 2280 y Fp(\025)2826 2255 y Fs(w)-5 b(e)39 b(need)f(explicit)g(criteria,)j(b)5 b(ecause)38 b(b)5 b(oth)38 b(the)g(constan)-5 b(t)39 b Fq(Q)6623 2280 y Fp(E)758 2455 y Fs(in)53 b(W)-14 b(egner's)53 b(estimate)f(and)i(the)e(constan)-5 b(t)53 b Fq(\015)3677 2480 y Fp(E)3842 2455 y Fs(in)g(the)g(Simon-Lieb)h(inequalit)-5 b(y)53 b(increase)g(as)758 2654 y(the)43 b(energy)g Fq(E)53 b Fs(increases,)47 b(so)e(as)f(w)-5 b(e)44 b(increase)g Fq(E)53 b Fs(the)43 b(initial)h(length)g(scale)g(for)g(the)f(m)-5 b(ultiscale)758 2853 y(analysis)57 b(also)g(increases.)77 b(Th)-5 b(us)58 b(if)e Fq(E)66 b Fs(is)56 b(increasing)i(with)e Fq(\025)g Fs(w)-5 b(e)56 b(cannot)g(\014x)g(a)h(length)f(scale)758 3052 y(and)81 b(satisfy)g(the)g(initial)g(condition)h(for)f(the)f(m)-5 b(ultiscale)83 b(analysis)f(at)e(the)h Fq(\025)g Fs(dep)5 b(enden)-5 b(t)758 3252 y(energy)66 b Fq(E)76 b Fs(b)-5 b(y)67 b(taking)f(the)g(disorder)i(large)e(enough;)73 b(the)67 b(initial)g(length)f(scale)h(also)g(gro)-5 b(ws)758 3451 y(with)59 b Fq(\025)g Fs(and)h(hence)g(is)g(a)f(mo)-5 b(ving)61 b(target.)85 b(Nev)-5 b(ertheless,)60 b(b)-5 b(y)60 b(using)h(our)f(explicit)e(criterion)758 3650 y(w)-5 b(e)53 b(will)g(sho)-5 b(w)53 b(that)g(there)f(is)h(a)g(constan) -5 b(t)53 b Fq(C)64 b Fs(suc)-5 b(h)54 b(that)f(w)-5 b(e)52 b(ha)-5 b(v)g(e)54 b Fq(E)5124 3675 y Fp(\025)5256 3650 y Fo(\025)47 b Fq(C)12 b(\025)52 b Fs(for)g(large)h Fq(\025)p Fs(.)73 b(In)758 3849 y(addition,)51 b(w)-5 b(e)50 b(sho)-5 b(w)51 b(that)e(eigenfunctions)h(with)g(eigen)-5 b(v)c(alues)50 b(in)g(the)f(in)-5 b(terv)c(al)50 b([0)p Fq(;)28 b(E)6147 3874 y Fp(\025)6234 3849 y Fs(])49 b(deca)-5 b(y)758 4062 y(exp)5 b(onen)-5 b(tially)61 b(with)h(a)g(rate)f Fo(\025)c Fq(C)2983 4002 y Fk(0)3030 3921 y Fo(p)p 3168 3921 97 7 v 141 x Fq(\025)62 b Fs(for)f(some)i(constan)-5 b(t)62 b Fq(C)4817 4002 y Fk(0)4863 4062 y Fs(.)94 b(T)-14 b(o)62 b(our)g(kno)-5 b(wledge)62 b(these)758 4261 y(are)55 b(the)g(\014rst)h(results)g(of)f(this)h(kind)f(in)h(the)f(large)g (disorder)i(regime..)957 4461 y(In)52 b([GK4)q(])g(w)-5 b(e)52 b(extend)g(this)g(result)g(b)-5 b(y)53 b(ha)-5 b(ving)52 b(the)g(random)h(v)-9 b(ariables)53 b Fq(!)5642 4486 y Fp(i)5749 4461 y Fs(tak)-5 b(e)52 b(v)-9 b(alues)52 b(in)758 4660 y(the)k(in)-5 b(terv)c(al)56 b([)p Fo(\000)p Fs(1)p Fq(;)28 b Fs(1],)57 b(with)f(a)h(b)5 b(ounded)57 b(densit)-5 b(y)57 b Fq(\032)48 b(>)g Fs(0)57 b(a.e.)77 b(in)57 b([)p Fo(\000)p Fs(1)p Fq(;)28 b Fs(1].)77 b(No)-5 b(w)57 b(the)f(b)5 b(ottom)758 4859 y(of)53 b(the)g(sp)5 b(ectrum)53 b(of)g Fq(H)2277 4884 y Fp(\025)2418 4859 y Fs(is)g(at)g(ab)5 b(out)53 b Fo(\000)p Fq(C)12 b(\025)53 b Fs(for)g(some)h(constan)-5 b(t)54 b Fq(C)12 b Fs(,)53 b(and)h(the)f Fq(\015)5921 4884 y Fp(E)6086 4859 y Fs(gro)-5 b(w)54 b(lik)-5 b(e)758 4921 y Fo(p)p 896 4921 V 142 x Fq(\025)68 b Fs(for)h(\014xed)f Fq(E)10 b Fs(.)113 b(It)68 b(follo)-5 b(ws)70 b(that)e(ev)-5 b(en)68 b(to)g(pro)-5 b(v)g(e)70 b(lo)5 b(calization)68 b(in)h(an)g(in)-5 b(terv)c(al)68 b(of)g(\014xed)758 5262 y(length)73 b(at)g(the)h(b)5 b(ottom)73 b(of)g(the)g(sp)5 b(ectrum)74 b(w)-5 b(e)73 b(need)h(explicit)f(criteria,)78 b(since)c(the)f(initial)758 5461 y(length)49 b(scale)g(no)-5 b(w)49 b(gro)-5 b(ws)50 b(with)f Fq(\025)p Fs(.)72 b(Our)49 b(explicit)g(criteria)f(can)h(b)5 b(e)49 b(used)h(to)e(pro)-5 b(v)g(e)50 b(Anderson)758 5661 y(lo)5 b(calization)59 b(in)i(an)f(in)-5 b(terv)c(al)60 b(of)g(size)g Fo(O)5 b Fs(\()p Fq(\025)p Fs(\))59 b(for)h(large)g Fq(\025)g Fs(at)g(the)g(b)5 b(ottom)59 b(of)h(the)g(sp)5 b(ectrum.)758 5865 y(The)55 b(result)h(is)g(still)f(true)g(if)h Fq(u)46 b Fo(2)g Fq(L)2967 5805 y Fp(p)3099 5865 y Fs(with)55 b Fq(p)46 b(>)3802 5800 y Fp(d)p 3802 5827 70 7 v 3803 5922 a Fn(2)3946 5865 y Fs(instead)56 b(of)f(b)5 b(ounded.)957 6064 y(F)-14 b(or)62 b(a)g(more)h(sophisticated)f(application)h(of)e (our)i(explicit)e(criteria)g(w)-5 b(e)62 b(revisit)g(the)g(t)-5 b(w)g(o-)758 6263 y(dimensional)77 b(random)g(Sc)-5 b(hr\177)-83 b(odinger)77 b(op)5 b(erator)75 b(in)i(a)e(constan)-5 b(t)77 b(magnetic)e(\014eld)i(\(random)758 6463 y(Landau)62 b(Hamiltonian\).)92 b(This)63 b(mo)5 b(del)61 b(w)-5 b(as)63 b(studied)f(b)-5 b(y)62 b(Com)-5 b(b)5 b(es)62 b(and)g(Hislop)g([CH2])g(and)758 6662 y(W)-14 b(ang)43 b([W1],)j(who)e(obtained)f(W)-14 b(egner)43 b(estimates)h(and)g (probabilit)-5 b(y)44 b(estimates)g(on)f(the)g(deca)-5 b(y)758 6861 y(of)56 b(lo)5 b(cal)56 b(resolv)-5 b(en)g(ts)57 b(using)g(p)5 b(ercolation)56 b(argumen)-5 b(ts.)78 b(\(See)56 b([DMP1,)h(DMP2])f(for)g(a)h(related)758 7060 y(mo)5 b(del.\))121 b(Ho)-5 b(w)g(ev)g(er)71 b(the)g(e\013ect)f(of)h(the)g (dep)5 b(endency)71 b(of)g(the)f(initial)i(length)f(scale)h(for)f(the) 758 7260 y(m)-5 b(ultiscale)58 b(analysis)h(on)f(the)f(strength)h Fq(B)66 b Fs(of)57 b(the)g(magnetic)h(\014eld)g(and)g(on)g(the)f (distance)h(to)758 7459 y(the)47 b(Landau)i(lev)-5 b(el)48 b(w)-5 b(as)49 b(o)-5 b(v)g(erlo)5 b(ok)-5 b(ed.)72 b(It)47 b(is)i(fair)f(to)g(sa)-5 b(y)48 b(that)g(the)g(m)-5 b(ultiscale)49 b(analyses)g(a)-5 b(v)c(ail-)758 7658 y(able)62 b(at)f(the)g(time)h (did)g(not)f(pro)-5 b(vide)62 b(the)g(to)5 b(ols)61 b(to)g(handle)i (this)f(dep)5 b(endency)-14 b(.)93 b(Com)-5 b(bining)758 7857 y(the)58 b(W)-14 b(egner)58 b(estimate)g(and)h(the)f(probabilit)-5 b(y)59 b(estimate)f(on)h(the)f(deca)-5 b(y)58 b(of)g(lo)5 b(cal)58 b(resolv)-5 b(en)g(ts)758 8057 y(obtained)49 b(in)g([CH2])g(with)g(our)g(explicit)f(criterion,)j(w)-5 b(e)49 b(giv)-5 b(e)49 b(a)g(complete)f(pro)5 b(of)49 b(of)g(Anderson)758 8270 y(lo)5 b(calization)59 b(up)i(to)e(a)h (distance)h Fo(O)5 b Fs(\()3112 8196 y Fn(log)24 b Fp(B)p 3111 8231 301 7 v 3208 8327 a(B)3431 8270 y Fs(\))60 b(from)g(the)f(Landau)i(lev)-5 b(els)60 b(for)g(large)g Fq(B)8 b Fs(.)88 b(It)60 b(tuns)758 8469 y(out)h(that)g(the)h(initial)g (length)g(scale)f(for)h(this)g(problem)h(is)f(a)g(\\Goldilo)5 b(c)-5 b(k")62 b(scale:)87 b(it)61 b(cannot)758 8668 y(b)5 b(e)51 b(to)5 b(o)52 b(big)g(or)g(to)5 b(o)51 b(small,)j(it)e(m) -5 b(ust)53 b(b)5 b(e)52 b(just)g(righ)-5 b(t)52 b(for)g(the)f (strength)h(of)g(the)g(magnetic)g(\014eld.)957 8867 y(W)-14 b(e)64 b(also)g(obtain)h(estimates)f(on)h(the)e(rate)h(of)g(exp)5 b(onen)-5 b(tial)64 b(deca)-5 b(y)64 b(of)g(eigenfunctions)g(of)758 9071 y(random)41 b(Landau)g(Hamiltonians:)67 b(for)40 b(eigen)-5 b(v)c(alues)41 b(at)f(a)g(distance)g Fo(O)5 b Fs(\()5263 8997 y Fn(log)25 b Fp(B)p 5263 9033 V 5360 9128 a(B)5583 9071 y Fs(\))40 b(from)g(a)h(Landau)758 9345 y(lev)-5 b(el)43 b(the)g(rate)g(of)g(deca)-5 b(y)43 b(is)h Fo(\025)j(O)2849 9161 y Fl(\020\020)3067 9271 y Fn(log)26 b Fp(B)p 3067 9307 V 3165 9402 a(B)3388 9161 y Fl(\021)3487 9195 y Fp(\027)3569 9161 y Fl(\021)3669 9345 y Fs(,)46 b(where)d(the)g(exp)5 b(onen)-5 b(t)43 b Fq(\027)56 b(>)47 b Fs(0)c(comes)h(from)g(the)758 9585 y(probabilit)-5 b(y)62 b(distribution)g(of)e(the)h(random)h(v)-9 b(ariables)61 b(and)h(t)-5 b(w)g(o-dimensional)64 b(p)5 b(ercolation,)758 9784 y(t)-5 b(ypically)58 b Fq(\027)63 b(>)52 b Fs(1;)61 b(at)e(a)g(\014xed)g(distance)g(of)g(a)g(Landau)h (lev)-5 b(el,)59 b(i.e.,)i Fo(O)5 b Fs(\(1\),)59 b(the)f(rate)h(of)f (deca)-5 b(y)758 10033 y(is)56 b Fo(\025)46 b(O)1264 9849 y Fl(\020)1364 9892 y Fo(p)p 1502 9892 135 7 v 141 x Fq(B)1636 9849 y Fl(\021)1735 10033 y Fs(.)957 10273 y(The)i(issue)i(of)e(the)g(dep)5 b(endency)48 b(of)g(the)g(initial)h (length)g(scale)f(for)h(the)f(m)-5 b(ultiscale)49 b(analysis)758 10472 y(in)42 b(terms)g(of)f(the)h(parameters)g(of)f(the)h(mo)5 b(del)42 b(also)g(app)5 b(ears)42 b(when)g(in)-5 b(v)g(estigating)43 b(lo)5 b(calization)758 10671 y(at)62 b(the)g(b)5 b(ottom)62 b(of)g(the)g(sp)5 b(ectrum)63 b(for)f(the)g(Anderson)i(mo)5 b(del)62 b(at)g(lo)-5 b(w)63 b(disorder,)j(as)d(in)g(the)p eop %%Page: 3 3 3 2 bop 1773 861 a Fn(EXPLICIT)58 b(FINITE)g(V)l(OLUME)h(CRITERIA)g(F)l (OR)g(LOCALIZA)-11 b(TION)949 b(3)758 1259 y Fs(w)-5 b(ork)63 b(of)g(W)-14 b(ang)63 b([W2])g(and)g(Klopp)h([Klo2)q(],)h (since)e(the)g(constan)-5 b(t)64 b(in)f(the)g(W)-14 b(egner)62 b(estimate)758 1458 y(blo)-5 b(ws)57 b(up)g(as)g(the)g(disorder)g(go)5 b(es)57 b(to)f(zero.)77 b(Our)57 b(explicit)f(criteria)g(ma)-5 b(y)57 b(b)5 b(e)56 b(used)h(to)g(handle)758 1658 y(this)e(dep)5 b(endency)55 b(and)h(p)5 b(erform)56 b(a)f(m)-5 b(ultiscale)56 b(analysis.)957 1857 y(Lo)5 b(calization)85 b(for)g(con)-5 b(tin)g(uous)87 b(random)f(op)5 b(erators)85 b(has)h(b)5 b(een)85 b(usually)i(pro)-5 b(v)g(en)86 b(b)-5 b(y)85 b(a)758 2056 y(m)-5 b(ultiscale)84 b(analysis.)159 b(\(But)82 b(note)h(that)f(the)h(Aizenman-Molc)-5 b(hano)g(v)85 b(metho)5 b(d)83 b(has)h(just)758 2255 y(b)5 b(een)70 b(extended)g(to)g(the)g(con)-5 b(tin)g(uum)73 b([AENSS)o(].\))120 b(In)71 b(this)g(con)-5 b(text)70 b(the)g(m)-5 b(ultiscale)71 b(anal-)758 2455 y(ysis)60 b(is)g(a)f(tec)-5 b(hnique,)60 b(initially)g(dev)-5 b(elop)5 b(ed)59 b(b)-5 b(y)60 b(F)-14 b(r\177)-83 b(ohlic)-5 b(h)61 b(and)e(Sp)5 b(encer)60 b([FS])g(and)g(F)-14 b(r\177)-83 b(ohlic)-5 b(h,)758 2654 y(Martinelli,)94 b(Sp)5 b(encer)87 b(and)g(Scopp)5 b(olla)86 b([FMSS)r(],)94 b(and)87 b(simpli\014ed)h(b)-5 b(y)87 b(v)-5 b(on)87 b(Dreifus)g([vD)o(])758 2853 y(and)69 b(v)-5 b(on)70 b(Dreifus)f(and)h(Klein)g([vDK],)j(for)c(the)f(purp)5 b(ose)70 b(of)f(pro)-5 b(ving)70 b(Anderson)g(lo)5 b(caliza-)758 3052 y(tion.)87 b(i.e.,)62 b(pure)e(p)5 b(oin)-5 b(t)60 b(sp)5 b(ectrum)60 b(and)h(exp)5 b(onen)-5 b(tial)60 b(deca)-5 b(y)59 b(of)h(eigenfunctions.)88 b(\(See)60 b(also)758 3252 y([HM,)68 b(CKM)q(,)g(Sp)q(,)h(KLS,)g(vDK2)o(,)g(K,)g (CH1)o(,)g(Klo1,)g(CH2,)f(FK1)q(,)h(FK2,)g(W1,)g(BCH)o(,)f(KSS1)q(,)758 3451 y(KSS2,)60 b(CHT)o(,)g(FLM,)g(W2,)f(St,)h(KK2,)f(U].\))86 b(It)59 b(w)-5 b(as)60 b(later)f(sho)-5 b(wn)61 b(to)e(also)h(yield)g (dynamical)758 3650 y(lo)5 b(calization)55 b(\(non)g(spreading)i(of)e (the)g(w)-5 b(a)g(v)g(e)56 b(pac)-5 b(k)g(ets\))55 b([GDB,)g(DS,)h (GK1].)957 3849 y(Recen)-5 b(tly)-14 b(,)78 b(the)73 b(authors)i(dev)-5 b(elop)5 b(ed)73 b(a)h(more)g(in)-5 b(v)g(olv)g(ed)74 b(pro)5 b(cedure,)78 b(built)c(out)g(of)f(four)758 4049 y(di\013eren)-5 b(t)47 b(m)-5 b(ultiscale)48 b(analyses,)i(called) d(a)h(b)5 b(o)g(otstrap)46 b(m)-5 b(ultiscale)49 b(analysis)f([GK1)q (].)71 b(It)47 b(yields)758 4248 y(Anderson)53 b(lo)5 b(calization,)54 b(SULE-lik)-5 b(e)53 b(estimates,)h(sub-exp)5 b(onen)-5 b(tial)54 b(deca)-5 b(y)53 b(of)g(the)g(k)-5 b(ernel)53 b(of)758 4447 y(the)h(ev)-5 b(olution)54 b(op)5 b(erator,)54 b(and)h(strong)g(Hilb)5 b(ert-Sc)-5 b(hmidt)55 b(dynamical)g(lo)5 b(calization.)73 b(A)54 b(new)758 4646 y(feature)62 b(of)i(this)g(b)5 b(o)g(otstrap)63 b(m)-5 b(ultiscale)65 b(analysis)f(is)g(that,)i(unlik)-5 b(e)64 b(the)f(formerly)g(standard)758 4846 y(m)-5 b(ultiscale)82 b(analysis)g(\(e.g.)152 b([FS,)82 b(FMSS)q(,)f(vD,)g(vDK,)g(CH1,)g (St]\),)87 b(the)81 b(strength)h(of)f(the)758 5045 y(conclusions)57 b(is)f(not)f(a\013ected)f(b)-5 b(y)56 b(the)f(rate)g(of)g(deca)-5 b(y)55 b(in)h(the)f(starting)g(condition.)957 5244 y(W)-14 b(e)48 b(giv)-5 b(e)49 b(three)g(explicit)f(\014nite)h(v)-5 b(olume)49 b(criteria)g(for)f(starting)h(the)g(b)5 b(o)g(otstrap)48 b(m)-5 b(ultiscale)758 5443 y(analysis.)70 b(Theorem)40 b(2.4)g(giv)-5 b(es)41 b(a)f(criterion)g(for)g(starting)g(the)f(b)5 b(o)g(otstrap)40 b(m)-5 b(ultiscale)41 b(analysis)758 5643 y(at)f(a)i(\\a)f(priori")g(sp)5 b(eci\014ed)42 b(initial)f(length) h(scale;)k(w)-5 b(e)41 b(ma)-5 b(y)42 b(th)-5 b(us)42 b(pla)-5 b(y)41 b(with)g(the)g(parameters)g(of)758 5842 y(the)d(mo)5 b(del)40 b(to)f(satisfy)g(the)g(starting)g(condition)h(at) e(the)h(sp)5 b(eci\014ed)40 b(length)f(scale.)69 b(Theorem)39 b(2.5)758 6041 y(pro)-5 b(vides)43 b(an)f(explicit)f(expression)i(for)f (the)g(minim)-5 b(um)44 b(initial)f(length)f(scale)g(required)g(to)g (start)758 6240 y(the)80 b(b)5 b(o)g(otstrap)80 b(m)-5 b(ultiscale)81 b(analysis)h(with)e(a)h(\\a)f(priori")h(sp)5 b(eci\014ed)81 b(deca)-5 b(y;)93 b(it)81 b(t)-5 b(ypically)758 6440 y(requires)54 b(large)f(scales,)i(but)f(not)g Fg(lar)-8 b(ge)59 b(enough)54 b Fs(scales:)74 b(ho)-5 b(w)55 b(large)f(the)f (initial)h(length)g(scale)758 6639 y(has)46 b(to)g(b)5 b(e)45 b(is)i(made)f(precise)h(b)-5 b(y)46 b(the)f(criterion.)71 b(Theorem)46 b(2.6)g(is)h(an)f(analog)g(of)g(Theorem)g(2.5)758 6838 y(but)54 b(for)f(the)h(second)g(m)-5 b(ultiscale)55 b(analysis)g(of)f(the)f(b)5 b(o)g(otstrap)53 b(sc)-5 b(heme)55 b(describ)5 b(ed)54 b(in)g([GK1)q(];)758 7037 y(it)63 b(also)i(giv)-5 b(es)64 b(an)h(estimate)e(of)h(the)g(exp)5 b(onen)-5 b(tial)63 b(rate)h(of)f(deca)-5 b(y)64 b(of)g(the)f (eigenfunctions)i(in)758 7237 y(terms)60 b(of)g(the)f(constan)-5 b(ts)61 b(of)f(the)f(problems.)90 b(These)60 b(criteria)f(reduce)h(the) g(requiremen)-5 b(ts)60 b(of)758 7436 y(the)73 b(m)-5 b(ultiscale)75 b(analysis)f(to)g(explicit)f(and)h(transparen)-5 b(t)74 b(conditions.)130 b(\(The)73 b(criterion)g(of)758 7635 y(Theorem)55 b(2.4)h(is)g(sp)5 b(ecially)55 b(simple.\))957 7834 y(This)39 b(article)f(is)h(organized)g(as)g(follo)-5 b(ws:)67 b(Our)39 b(explicit)f(criteria)g(for)g(lo)5 b(calization)38 b(are)h(stated)758 8034 y(in)57 b(Section)h(2;)g(their) g(pro)5 b(ofs)57 b(are)h(giv)-5 b(en)57 b(in)h(Section)f(5)h (\(Theorems)g(2.4)f(and)h(2.5\),)g(Section)f(6)758 8233 y(\(Theorem)48 b(2.7,)i(a)f(general)g(criterion)f(that)g(giv)-5 b(es)49 b(Theorems)g(2.4)g(and)g(2.5)g(as)g(sp)5 b(ecial)49 b(cases\),)758 8432 y(and)58 b(Section)f(7)h(\(Theorem)g(2.6\).)80 b(Section)58 b(3)g(is)g(dedicated)g(to)f(lo)5 b(calization)57 b(at)g(the)h(b)5 b(ottom)758 8631 y(of)63 b(the)g(sp)5 b(ectrum)64 b(of)f(the)g(Anderson)h(mo)5 b(del)64 b(\(in)f(the)h(con)-5 b(tin)g(uum\))64 b(at)f(large)h(disorder;)69 b(the)758 8831 y(main)g(result)g(is)g(Theorem)g(3.1.)114 b(Section)69 b(4)f(con)-5 b(tains)70 b(our)f(discussion)i(of)d(lo)5 b(calization)68 b(for)758 9030 y(random)k(Landau)h(Hamiltonians,)k(the) 71 b(detailed)h(description)g(of)g(our)g(results)g(is)h(giv)-5 b(en)72 b(in)758 9229 y(Theorem)55 b(4.1.)2173 9744 y(2.)92 b Fr(Explicit)63 b(criteria)g(f)l(or)g(localiza)-11 b(tion)957 10043 y Fs(Our)38 b(random)h(medium)g(is)f(mo)5 b(deled)38 b(b)-5 b(y)39 b(a)e Fm(Z)3719 9983 y Fp(d)3806 10043 y Fs(-ergo)5 b(dic)37 b(random)i(self-adjoin)-5 b(t)39 b(op)5 b(erator)37 b Fq(H)6639 10068 y Fp(!)758 10242 y Fs(on)h(L)1075 10182 y Fn(2)1149 10242 y Fs(\()p Fm(R)1324 10182 y Fp(d)1411 10242 y Fq(;)28 b Fs(d)p Fq(x)p Fs(;)g Fm(C)1856 10182 y Fp(n)1956 10242 y Fs(\),)41 b(where)c Fq(!)44 b Fs(b)5 b(elongs)38 b(to)f(a)h(probabilit)-5 b(y)38 b(set)g(\012)f(with)h(a)f(probabilit)-5 b(y)38 b(measure)758 10441 y Fm(P)87 b Fs(and)74 b(exp)5 b(ectation)71 b Fm(E)26 b Fs(.)136 b(Our)73 b(results)h(apply)f(to)g(random)g(Sc)-5 b(hr\177)-83 b(odinger)75 b(op)5 b(erators)72 b(\(e.g.,)758 10641 y([HM,)63 b(CH1)o(,)g(Klo1)q(,)g(KSS1)q(,)g(GK3,)h(AENSS)o(]\),)h (random)f(magnetic)f(Sc)-5 b(hr\177)-83 b(odinger)64 b(op)5 b(erators)758 10840 y(\(e.g.,)41 b([CH2,)d(W1]\),)k(and)d (classical)g(w)-5 b(a)g(v)g(e)39 b(op)5 b(erators)38 b(\(e.g.,)k([FK1)q(,)c(FK2)q(,)g(CHT,)g(KK1)q(,)g(KK2]\).)p eop %%Page: 4 4 4 3 bop 758 861 a Fn(4)1551 b(FRANC)-77 b(\030)17 b(OIS)59 b(GERMINET)g(AND)f(ABEL)g(KLEIN)758 1259 y Fs(W)-14 b(e)61 b(recall)h(that)f(it)h(follo)-5 b(ws)62 b(from)g(the)g(ergo)5 b(dicit)-5 b(y)61 b(that)g(there)h(exists)f(a)h(nonrandom)i(set)d(\006) 758 1458 y(suc)-5 b(h)62 b(that)f Fq(\033)6 b Fs(\()p Fq(H)1808 1483 y Fp(!)1904 1458 y Fs(\))56 b(=)g(\006)61 b(with)h(probabilit)-5 b(y)62 b(one,)h(and)f(that)f(the)g(decomp)5 b(osition)62 b(of)f Fq(\033)6 b Fs(\()p Fq(H)6575 1483 y Fp(!)6671 1458 y Fs(\))758 1658 y(in)-5 b(to)48 b(pure)g(p)5 b(oin)-5 b(t)49 b(sp)5 b(ectrum,)50 b(absolutely)e(con)-5 b(tin)g(uous)50 b(sp)5 b(ectrum,)50 b(and)e(singular)h(con)-5 b(tin)g(uous)758 1857 y(sp)5 b(ectrum)55 b(is)h(also)g(indep)5 b(enden)-5 b(t)56 b(of)f(the)g(c)-5 b(hoice)56 b(of)f Fq(!)61 b Fs(with)56 b(probabilit)-5 b(y)56 b(one.)957 2056 y(As)f(in)g([GK1)q(],)g(w)-5 b(e)55 b(tak)-5 b(e)55 b(as)g(assumptions)j(the)c(prop)5 b(erties)55 b(of)g(the)g(random)h(op) 5 b(erator)54 b Fq(H)6639 2081 y Fp(!)758 2255 y Fs(that)46 b(are)g(required)g(for)g(the)h(m)-5 b(ultiscale)47 b(analysis.)72 b(These)47 b(prop)5 b(erties)46 b(are)h(routinely)f(v)-5 b(eri\014ed)758 2455 y(for)49 b(the)g(op)5 b(erators)50 b(of)f(in)-5 b(terest,)51 b(except)d(for)i(the)f(W)-14 b(egner)49 b(estimate)g(whic)-5 b(h)51 b(usually)f(requires)758 2654 y(more)55 b(h)-5 b(yp)5 b(otheses)56 b(on)g(the)f(parameters)g(of) h(the)f(random)h(op)5 b(erator.)957 2853 y(Throughout)56 b(this)g(pap)5 b(er)55 b(w)-5 b(e)55 b(use)h(the)f(sup)i(norm)f(in)f Fm(R)4531 2793 y Fp(d)4618 2853 y Fs(:)2403 3146 y Fo(j)p Fq(x)p Fo(j)47 b Fs(=)f Fo(j)p Fq(x)p Fo(j)2999 3171 y Fk(1)3186 3146 y Fs(=)g(max)p Fo(fj)p Fq(x)3894 3171 y Fp(i)3950 3146 y Fo(j)p Fq(;)74 b(i)46 b Fs(=)g(1)p Fq(;)28 b(:)g(:)g(:)55 b(;)28 b(d)p Fo(g)p Fq(:)758 3439 y Fs(By)54 b(\003)1133 3464 y Fp(L)1233 3439 y Fs(\()p Fq(x)p Fs(\))g(w)-5 b(e)56 b(denote)f(the)g(op)5 b(en)55 b(b)5 b(o)-5 b(x)55 b(\(or)g(cub)5 b(e\))55 b(of)g(side)h Fq(L)46 b(>)g Fs(0:)2532 3740 y(\003)2647 3765 y Fp(L)2746 3740 y Fs(\()p Fq(x)p Fs(\))g(=)3192 3606 y Fl(\010)3289 3740 y Fq(y)52 b Fo(2)46 b Fm(R)3690 3671 y Fp(d)3776 3740 y Fs(;)111 b Fo(j)p Fq(y)44 b Fo(\000)37 b Fq(x)p Fo(j)46 b Fq(<)4652 3675 y Fp(L)p 4652 3702 92 7 v 4664 3797 a Fn(2)4763 3606 y Fl(\011)4887 3740 y Fq(;)1461 b Fs(\(2.1\))758 4065 y(and)44 b(b)-5 b(y)p 1288 3932 116 7 v 44 w(\003)1403 4090 y Fp(L)1502 4065 y Fs(\()p Fq(x)p Fs(\))43 b(the)g(closed)h(b)5 b(o)-5 b(x.)70 b(The)44 b(c)-5 b(haracteristic)43 b(function)h(of)f(a)h(set)f(\003)k Fo(\032)f Fm(R)5878 4005 y Fp(d)6008 4065 y Fs(is)e(denoted)758 4264 y(b)-5 b(y)55 b Fq(\037)1092 4289 y Fn(\003)1191 4264 y Fs(;)h(w)-5 b(e)55 b(set)2550 4557 y Fq(\037)2654 4582 y Fp(x;L)2914 4557 y Fs(=)46 b Fq(\037)3193 4587 y Fn(\003)3284 4604 y Fi(L)3371 4587 y Fn(\()p Fp(x)p Fn(\))3586 4557 y Fq(;)120 b Fs(with)93 b Fq(\037)4272 4582 y Fp(x)4401 4557 y Fs(=)46 b Fq(\037)4680 4582 y Fp(x;)p Fn(1)4897 4557 y Fq(:)1451 b Fs(\(2.2\))758 4877 y Fg(In)69 b(this)g(article)i(we)e(wil)8 b(l)71 b(take)f(b)-8 b(oxes)69 b(c)-8 b(enter)g(e)g(d)70 b(at)f(sites)h Fq(x)63 b Fo(2)h Fm(Z)4907 4816 y Fp(d)5062 4877 y Fg(with)70 b(side)f Fq(L)64 b Fo(2)f Fs(2)p Fm(N)26 b Fg(.)115 b Fs(F)-14 b(or)758 5076 y(suc)-5 b(h)56 b(a)g(b)5 b(o)-5 b(x)55 b(w)-5 b(e)55 b(set)2400 5369 y(\007)2529 5394 y Fp(L)2629 5369 y Fs(\()p Fq(x)p Fs(\))45 b(=)3074 5234 y Fl(\010)3171 5369 y Fq(y)53 b Fo(2)45 b Fm(Z)3563 5300 y Fp(d)3649 5369 y Fs(;)84 b Fo(j)p Fq(y)43 b Fo(\000)37 b Fq(x)p Fo(j)46 b Fs(=)4497 5303 y Fp(L)p 4497 5330 92 7 v 4509 5426 a Fn(2)4645 5369 y Fo(\000)37 b Fs(1)4894 5234 y Fl(\011)5019 5369 y Fq(;)1329 b Fs(\(2.3\))758 5662 y(and)56 b(de\014ne)f(its)h(b)5 b(oundary)55 b(b)5 b(elt)55 b(b)-5 b(y)2125 5937 y(~)2102 5979 y(\007)2231 6004 y Fp(L)2331 5979 y Fs(\()p Fq(x)p Fs(\))45 b(=)p 2776 5846 116 7 v 46 w(\003)2891 6004 y Fp(L)p Fk(\000)p Fn(1)3161 5979 y Fs(\()p Fq(x)p Fs(\))p Fo(n)p Fs(\003)3584 6004 y Fp(L)p Fk(\000)p Fn(3)3853 5979 y Fs(\()p Fq(x)p Fs(\))g(=)4471 5822 y Fl([)4298 6186 y Fp(y)t Fk(2)p Fn(\007)4560 6203 y Fi(L)4649 6186 y Fn(\()p Fp(x)p Fn(\))p 4855 5846 V 4855 5979 a Fs(\003)4971 6004 y Fn(1)5045 5979 y Fs(\()p Fq(y)6 b Fs(\))55 b(;)1031 b(\(2.4\))758 6457 y(it)55 b(has)h(the)f(c)-5 b(haracteristic)55 b(function)2580 6750 y(\000)2684 6775 y Fp(x;L)2944 6750 y Fs(=)47 b Fq(\037)3242 6771 y Fn(~)3224 6800 y(\007)3326 6817 y Fi(L)3413 6800 y Fn(\()p Fp(x)p Fn(\))3647 6750 y Fs(=)3967 6592 y Fl(X)3822 6956 y Fp(y)t Fk(2)p Fn(\007)4084 6973 y Fi(L)4172 6956 y Fn(\()p Fp(x)p Fn(\))4379 6750 y Fq(\037)4483 6775 y Fp(y)4655 6750 y Fq(a:e:)1482 b Fs(\(2.5\))758 7253 y(Note)54 b(that)h Fo(j)p Fs(\007)1694 7278 y Fp(L)1793 7253 y Fs(\()p Fq(x)p Fs(\))p Fo(j)46 b(\024)g Fq(d)p Fs(\()p Fq(L)37 b Fo(\000)g Fs(1\))2900 7193 y Fp(d)p Fk(\000)p Fn(1)3147 7253 y Fs(.)74 b(W)-14 b(e)55 b(will)g(also)h (write)2610 7546 y([)p Fq(K)12 b Fs(])2855 7571 y Fn(6)p Fj(N)3060 7546 y Fs(=)46 b(max)p Fo(f)p Fq(L)g Fo(2)g Fs(6)p Fm(N)26 b Fs(;)83 b Fq(L)46 b Fo(\024)h Fq(K)12 b Fo(g)p Fq(:)1511 b Fs(\(2.6\))957 7839 y(W)-14 b(e)70 b(need)h(a)f(notion)h(of)f Fg(\014nite)75 b(volume)f(op)-8 b(er)g(ator)p Fs(,)74 b(i.e.)119 b(a)71 b(\\restriction")f Fq(H)5885 7864 y Fp(!)t(;x;L)6297 7839 y Fs(of)g Fq(H)6639 7864 y Fp(!)758 8039 y Fs(where)d(the)h(\\randomness)i(based)f(outside) f(the)f(b)5 b(o)-5 b(x)68 b(\003)4355 8064 y Fp(L)4455 8039 y Fs(\()p Fq(x)p Fs(\)")f(is)h(not)g(tak)-5 b(en)68 b(in)-5 b(to)68 b(accoun)-5 b(t.)758 8238 y(Usually)42 b Fq(H)1483 8263 y Fp(!)t(;x;L)1866 8238 y Fs(is)g(de\014ned)g(as)g (the)g(restriction)f(of)g Fq(H)4134 8263 y Fp(!)4231 8238 y Fs(,)j(either)d(to)h(the)f(op)5 b(en)41 b(b)5 b(o)-5 b(x)42 b(\003)6047 8263 y Fp(L)6147 8238 y Fs(\()p Fq(x)p Fs(\))e(with)758 8449 y(Diric)-5 b(hlet)61 b(b)5 b(oundary)62 b(condition,)h(or)f(to)f(the)g(closed)g(b)5 b(o)-5 b(x)p 4516 8315 V 62 w(\003)4631 8474 y Fp(L)4730 8449 y Fs(\()p Fq(x)p Fs(\))61 b(with)g(p)5 b(erio)g(dic)61 b(b)5 b(oundary)758 8648 y(condition.)97 b(The)63 b(op)5 b(erator)62 b Fq(H)2738 8673 y Fp(!)t(;x;L)3143 8648 y Fs(then)h(acts)g(on)g(L)4221 8588 y Fn(2)4295 8648 y Fs(\(\003)4475 8673 y Fp(L)4574 8648 y Fs(\()p Fq(x)p Fs(\))p Fq(;)28 b Fs(d)p Fq(x)p Fs(;)g Fm(C)5244 8588 y Fp(n)5343 8648 y Fs(\).)97 b(But)62 b Fq(H)6026 8673 y Fp(!)t(;x;L)6431 8648 y Fs(ma)-5 b(y)758 8847 y(also)62 b(b)5 b(e)61 b(de\014ned)h(as)g(acting)g(on)g(the)f(whole)g(space,)j(b) -5 b(y)62 b(thro)-5 b(wing)62 b(a)-5 b(w)g(a)g(y)62 b(the)f(random)i (co)5 b(ef-)758 9047 y(\014cien)-5 b(ts)65 b(\\based)h(outside)f(the)f (b)5 b(o)-5 b(x)65 b(\003)3201 9072 y Fp(L)3300 9047 y Fs(\()p Fq(x)p Fs(\)";)k(this)c(is)g(usually)h(used)f(for)g(random)g (magnetic)758 9246 y(Sc)-5 b(hr\177)-83 b(odinger)75 b(op)5 b(erators)73 b([CH2,)g(W1].)129 b(In)74 b(all)g(cases)g(the)f (\014nite)h(v)-5 b(olume)74 b(op)5 b(erators)74 b(ha)-5 b(v)g(e)758 9445 y(either)48 b(compact)h(resolv)-5 b(en)g(t)50 b(or)f(are)f(p)5 b(erturbations)50 b(of)f(the)f(free)h(\014nite)g(v)-5 b(olume)49 b(op)5 b(erators)49 b(b)-5 b(y)758 9644 y(relativ)g(ely)54 b(compact)h(op)5 b(erators.)957 9844 y(W)-14 b(e)77 b(assume)i(that)e (an)g(appropriate)h(c)-5 b(hoice)78 b(of)f(\014nite)g(v)-5 b(olume)78 b(op)5 b(erators)78 b Fq(H)6076 9869 y Fp(!)t(;x;L)6494 9844 y Fs(has)758 10043 y(b)5 b(een)68 b(made.)115 b(\(The)69 b(m)-5 b(ultiscale)70 b(analysis)g(is)g(not)e(sensitiv)-5 b(e)70 b(to)f(suc)-5 b(h)70 b(a)f(c)-5 b(hoice)69 b(as)g(long)h(as)758 10242 y(the)57 b(required)h(prop)5 b(erties)58 b(can)h(b)5 b(e)57 b(pro)-5 b(v)g(en.\))83 b(W)-14 b(e)57 b(write)h Fq(R)4540 10267 y Fp(!)t(;x;L)4932 10242 y Fs(=)51 b(\()p Fq(H)5315 10267 y Fp(!)t(;x;L)5694 10242 y Fo(\000)39 b Fq(z)7 b Fs(\))6011 10182 y Fk(\000)p Fn(1)6248 10242 y Fs(for)58 b(the)758 10441 y(resolv)-5 b(en)g(t)56 b(of)f Fq(H)1783 10466 y Fp(!)t(;x;L)2124 10441 y Fs(.)957 10641 y(W)-14 b(e)49 b(assume)i(that)f(the)f(random)i(op)5 b(erator)49 b Fq(H)3846 10666 y Fp(!)3992 10641 y Fs(satisfy)h(the)f (requiremen)-5 b(ts)50 b(for)g(the)f(m)-5 b(ulti-)758 10840 y(scale)43 b(analysis)g(as)g(in)g([GK1)q(],)i(i.e.,)h(it)c (satis\014es)i(Assumptions)h(SLI)d(\(Simon-Lieb)i(inequalit)-5 b(y\),)p eop %%Page: 5 5 5 4 bop 1773 861 a Fn(EXPLICIT)58 b(FINITE)g(V)l(OLUME)h(CRITERIA)g(F)l (OR)g(LOCALIZA)-11 b(TION)949 b(5)758 1259 y Fs(EDI)47 b(\(eigenfunction)g(deca)-5 b(y)47 b(inequalit)-5 b(y\),)49 b(IAD)e(\(indep)5 b(endence)47 b(at)g(a)h(distance\),)h(NE)e(\(a)-5 b(v)g(er-)758 1458 y(age)60 b(n)-5 b(um)g(b)5 b(er)61 b(of)f(eigen)-5 b(v)c(alues\),)61 b(W)f(\(W)-14 b(egner)59 b(estimate\),)i(and)g(SGEE)f(\(strong)g(generalized)758 1658 y(eigenfunction)55 b(expansion\))h(in)f(a)h(giv)-5 b(en)55 b(op)5 b(en)56 b(in)-5 b(terv)c(al)55 b Fo(I)12 b Fs(.)957 1857 y(W)-14 b(e)72 b(will)g(no)-5 b(w)72 b(state)g(Assumptions)i(SLI,)e(IAD,)f(NE)h(and)g(W,)g(since)g(the)g (constan)-5 b(ts)73 b(in)758 2056 y(these)j(assumptions)j(will)e(b)5 b(e)76 b(used)i(explicitly)d(in)i(this)h(article.)137 b(W)-14 b(e)76 b(refer)g(to)g([GK1)q(])g(for)758 2255 y(Assumptions)57 b(EDI)e(and)h(SGEE)g(and)g(for)f(a)g(discussion)j(of)d (all)h(these)f(assumptions.)957 2455 y(In)g(what)h(follo)-5 b(ws)56 b Fo(I)68 b Fs(will)55 b(b)5 b(e)55 b(a)h(\014xed)f(op)5 b(en)55 b(in)-5 b(terv)c(al.)758 2843 y Fz(Assumption)58 b(SLI.)153 b Fg(Ther)-8 b(e)55 b(exists)f(a)g(c)-8 b(onstant)55 b Fq(\015)4113 2868 y Fk(I)4250 2843 y Fo(\025)46 b Fs(1)p Fg(,)55 b(such)f(that)h(given)g Fq(L;)28 b(`)6004 2783 y Fk(0)6051 2843 y Fq(;)g(`)6194 2783 y Fk(00)6325 2843 y Fo(2)45 b Fs(2)p Fm(N)26 b Fg(,)758 3042 y Fq(x;)i(y)6 b(;)28 b(y)1175 2982 y Fk(0)1295 3042 y Fo(2)74 b Fm(Z)1581 2982 y Fp(d)1742 3042 y Fg(with)h Fs(\003)2233 3067 y Fp(`)2289 3034 y Ff(00)2379 3042 y Fs(\()p Fq(y)6 b Fs(\))73 b Fo(\032)h Fs(\003)2987 3067 y Fp(`)3043 3034 y Ff(0)3088 3067 y Fk(\000)p Fn(3)3266 3042 y Fs(\()p Fq(y)3418 2982 y Fk(0)3464 3042 y Fs(\))h Fg(and)f Fs(\003)4056 3067 y Fp(`)4112 3034 y Ff(0)4165 3042 y Fs(\()p Fq(y)4317 2982 y Fk(0)4364 3042 y Fs(\))f Fo(\032)h Fs(\003)4820 3067 y Fp(L)p Fk(\000)p Fn(3)5090 3042 y Fs(\()p Fq(x)p Fs(\))p Fg(,)j(then)f(for)e(a.e)h Fq(!)6 b Fg(,)78 b(if)758 3241 y Fq(E)55 b Fo(2)46 b(I)72 b Fg(with)60 b Fq(E)73 b(=)-101 b Fo(2)46 b Fq(\033)6 b Fs(\()p Fq(H)2255 3266 y Fp(!)t(;x;L)2595 3241 y Fs(\))37 b Fo([)f Fq(\033)6 b Fs(\()p Fq(H)3148 3266 y Fp(!)t(;y)3346 3233 y Ff(0)3392 3266 y Fp(;`)3487 3233 y Ff(0)3540 3241 y Fs(\))p Fg(,)59 b(we)h(have)898 3535 y Fo(k)p Fs(\000)1085 3560 y Fp(x;L)1299 3535 y Fq(R)1425 3560 y Fp(!)t(;x;L)1766 3535 y Fs(\()p Fq(E)10 b Fs(\))p Fq(\037)2133 3560 y Fp(y)t(;`)2299 3527 y Ff(0)o(0)2388 3535 y Fo(k)46 b(\024)g Fq(\015)2778 3560 y Fk(I)2897 3535 y Fo(k)p Fs(\000)3084 3560 y Fp(y)3155 3527 y Ff(0)3200 3560 y Fp(;`)3295 3527 y Ff(0)3348 3535 y Fq(R)3474 3560 y Fp(!)t(;y)3672 3527 y Ff(0)3718 3560 y Fp(;`)3813 3527 y Ff(0)3866 3535 y Fs(\()p Fq(E)10 b Fs(\))p Fq(\037)4233 3560 y Fp(y)t(;`)4399 3527 y Ff(0)o(0)4488 3535 y Fo(k)28 b(k)p Fs(\000)4786 3560 y Fp(x;L)5000 3535 y Fq(R)5126 3560 y Fp(!)t(;x;L)5467 3535 y Fs(\()p Fq(E)10 b Fs(\)\000)5834 3560 y Fp(y)5905 3527 y Ff(0)5949 3560 y Fp(;`)6044 3527 y Ff(0)6097 3535 y Fo(k)28 b Fq(:)140 b Fs(\(2.7\))957 4028 y(W)-14 b(e)48 b(sa)-5 b(y)50 b(that)e(an)h(ev)-5 b(en)g(t)49 b(is)g(based)h(on)f(the)g(b)5 b(o)-5 b(x)49 b(\003)4073 4053 y Fp(L)4172 4028 y Fs(\()p Fq(x)p Fs(\))f(if)h(it)g (is)g(determined)g(b)-5 b(y)49 b(conditions)758 4227 y(on)40 b(the)g(\014nite)g(v)-5 b(olume)40 b(op)5 b(erator)40 b Fq(H)3002 4252 y Fp(!)t(;x;L)3343 4227 y Fs(.)69 b(Giv)-5 b(en)41 b Fq(\045)46 b Fo(\025)g Fs(0,)d(w)-5 b(e)40 b(sa)-5 b(y)41 b(that)e(t)-5 b(w)g(o)41 b(b)5 b(o)-5 b(xes)40 b(\003)6104 4252 y Fp(L)6204 4227 y Fs(\()p Fq(x)p Fs(\))f(and)758 4453 y(\003)873 4478 y Fp(L)964 4445 y Ff(0)1017 4453 y Fs(\()p Fq(x)1177 4393 y Fk(0)1223 4453 y Fs(\))i(are)h Fq(\045)p Fg(-nonoverlapping)58 b Fs(if)42 b Fo(j)p Fq(x)9 b Fo(\000)g Fq(x)3413 4393 y Fk(0)3460 4453 y Fo(j)46 b Fq(>)3747 4388 y Fp(L)p Fn(+)p Fp(L)4031 4338 y Ff(0)p 3747 4415 330 7 v 3879 4510 a Fn(2)4106 4453 y Fs(+)9 b Fq(\045)41 b Fs(\(i.e.,)k(if)c (dist\(\003)5325 4478 y Fp(L)5425 4453 y Fs(\()p Fq(x)p Fs(\))p Fq(;)28 b Fs(\003)5839 4478 y Fp(L)5930 4445 y Ff(0)5982 4453 y Fs(\()p Fq(x)6142 4393 y Fk(0)6188 4453 y Fs(\)\))45 b Fq(>)i(\045)p Fs(\).)758 4842 y Fz(Assumption)k (IAD.)88 b Fg(Ther)-8 b(e)49 b(exists)f Fq(\045)e Fo(\025)h Fs(0)h Fg(such)h(that)f(events)i(b)-8 b(ase)g(d)48 b(on)h Fq(\045)p Fg(-nonoverlapping)758 5041 y(b)-8 b(oxes)59 b(ar)-8 b(e)59 b(indep)-8 b(endent.)758 5429 y Fz(Assumption)64 b(NE.)172 b Fg(Ther)-8 b(e)59 b(exists)h(a)f(c)-8 b(onstant)60 b Fq(C)4172 5454 y Fk(I)4322 5429 y Fg(such)g(that)2823 5723 y Fm(E)2953 5588 y Fl(\000)3038 5723 y Fs(tr)27 b Fq(E)3318 5748 y Fp(H)3427 5765 y Fi(!)s(;x;L)3740 5723 y Fs(\()p Fo(I)12 b Fs(\))3972 5588 y Fl(\001)4095 5723 y Fo(\024)46 b Fq(C)4389 5748 y Fk(I)4479 5723 y Fq(L)4592 5654 y Fp(d)6394 5723 y Fs(\(2.8\))758 6043 y Fg(for)59 b(al)8 b(l)60 b Fq(x)47 b Fo(2)e Fm(Z)1660 5983 y Fp(d)1806 6043 y Fg(and)59 b Fq(L)46 b Fo(2)g Fs(2)p Fm(N)26 b Fg(.)758 6431 y Fz(Assumption)64 b(W.)112 b Fg(Ther)-8 b(e)59 b(exist)h(c)-8 b(onstants)60 b Fq(b)46 b Fo(\025)g Fs(1)p Fg(,)60 b Fs(0)46 b Fq(<)g(\021)4644 6456 y Fk(I)4782 6431 y Fo(\024)g Fs(1)p Fg(,)59 b(and)h Fq(Q)5604 6456 y Fk(I)5695 6431 y Fg(,)f(such)h(that)2330 6725 y Fm(P)34 b Fo(f)9 b Fs(dist\()p Fq(\033)d Fs(\()p Fq(H)3180 6750 y Fp(!)t(;x;L)3520 6725 y Fs(\))p Fq(;)28 b(E)10 b Fs(\))45 b Fo(\024)h Fq(\021)6 b Fo(g)46 b(\024)h Fq(Q)4601 6750 y Fk(I)4692 6725 y Fq(\021)6 b(L)4893 6657 y Fp(bd)5089 6725 y Fq(;)1259 b Fs(\(2.9\))758 7045 y Fg(for)59 b(al)8 b(l)60 b Fq(E)c Fo(2)46 b(I)12 b Fg(,)60 b Fs(0)46 b Fq(<)g(\021)53 b Fo(\024)46 b Fq(\021)2505 7070 y Fk(I)2596 7045 y Fg(,)60 b Fq(x)46 b Fo(2)f Fm(Z)3106 6985 y Fp(d)3193 7045 y Fg(,)59 b(and)g Fq(L)46 b Fo(2)g Fs(2)p Fm(N)26 b Fg(.)957 7245 y Fs(In)51 b(practice)f(w)-5 b(e)51 b(ha)-5 b(v)g(e)51 b(either)g Fq(b)46 b Fs(=)g(1)51 b(or)g Fq(b)46 b Fs(=)h(2)j(in)i(the)e(W)-14 b(egner)50 b(estimate)h(\(2.9\).)72 b(Recen)-5 b(tly)758 7444 y(the)77 b(correct)f(v)-5 b(olume)78 b(dep)5 b(endency)77 b(\(i.e.,)82 b Fq(b)h Fs(=)g(1\))77 b(w)-5 b(as)78 b(obtained)f(for)h(certain)e(op)5 b(erators)758 7643 y([CHN)o(,)51 b(CHKN)o(,)h(HK],)g(at)f(the)g(price)h (of)f(losing)i(a)e(bit)g(in)h(the)f Fq(\021)58 b Fs(dep)5 b(endency)-14 b(.)73 b(In)51 b(this)h(pap)5 b(er,)758 7842 y(w)-5 b(e)46 b(shall)i(use)f(\(2.9\))e(as)i(stated,)h(the)e(mo)5 b(di\014cations)48 b(in)e(our)h(metho)5 b(ds)47 b(required)f(for)h(the) f(other)758 8042 y(forms)56 b(of)g(\(2.9\))f(b)5 b(eing)55 b(ob)-5 b(vious.)77 b(\(F)-14 b(or)56 b(a)g(discussion)i(of)d(p)5 b(ossible)57 b(mo)5 b(di\014cations)57 b(see)f([GK1,)758 8241 y(Remark)71 b(2.4].\))123 b(Note)70 b(also)j(that)e(usually)h Fq(\021)3700 8266 y Fk(I)3865 8241 y Fs(=)h(1,)j(but)71 b(not)h(alw)-5 b(a)g(ys,)77 b(as)72 b(in)g(the)f(case)g(of)758 8440 y(random)56 b(Landau)g(Hamiltonians)g(\(see)f(Section)g(4\).)957 8639 y(W)-14 b(e)48 b(will)h(lo)5 b(ok)48 b(for)h(lo)5 b(calization)48 b(b)-5 b(y)49 b(studying)g(the)g(deca)-5 b(y)48 b(of)h(the)f(\014nite)g(v)-5 b(olume)50 b(resolv)-5 b(en)g(t)758 8839 y(from)55 b(the)g(cen)-5 b(ter)55 b(of)g(a)h(b)5 b(o)-5 b(x)55 b(\003)2691 8864 y Fp(L)2791 8839 y Fs(\()p Fq(x)p Fs(\))f(to)h(its)g(b)5 b(oundary)56 b(as)g(measured)g(b)-5 b(y)2877 9132 y Fo(k)p Fs(\000)3064 9157 y Fp(x;L)3278 9132 y Fq(R)3404 9157 y Fp(!)t(;x;L)3745 9132 y Fs(\()p Fq(E)10 b Fs(\))p Fq(\037)4112 9162 y Fp(x;L=)p Fn(3)4459 9132 y Fo(k)28 b Fq(:)1695 b Fs(\(2.10\))758 9436 y(W)-14 b(e)55 b(use)h(the)e(con)-5 b(v)g(en)g(tion)57 b(that)d Fo(k)p Fs(\000)2993 9461 y Fp(x;L)3207 9436 y Fq(R)3333 9461 y Fp(!)t(;x;L)3675 9436 y Fs(\()p Fq(E)10 b Fs(\))p Fq(\037)4042 9466 y Fp(x;L=)p Fn(3)4388 9436 y Fo(k)47 b Fs(=)f Fo(1)55 b Fs(if)g Fq(E)h Fo(2)45 b Fq(\033)6 b Fs(\()p Fq(H)5705 9461 y Fp(!)t(;x;L)6046 9436 y Fs(\).)957 9636 y(W)-14 b(e)62 b(start)h(with)f(t)-5 b(w)g(o)63 b(deterministic)g(\(i.e.,)i(for)d(a)h(giv)-5 b(en)63 b Fq(!)6 b Fs(,)65 b(whic)-5 b(h)63 b(is)g(omitted)g(from)g(the)758 9836 y(notation\))54 b(de\014nitions.)758 10149 y Fz(De\014nition)62 b(2.1.)84 b Fg(Given)70 b Fq(\022)d(>)62 b Fs(0)p Fg(,)71 b Fq(E)g Fo(2)62 b Fm(R)22 b Fg(,)79 b Fq(x)62 b Fo(2)g Fm(Z)4104 10088 y Fp(d)4190 10149 y Fg(,)71 b(and)d Fq(L)62 b Fo(2)g Fs(6)p Fm(N)26 b Fg(,)80 b(we)69 b(say)e(that)i(the)f(b)-8 b(ox)758 10348 y Fs(\003)873 10373 y Fp(L)972 10348 y Fs(\()p Fq(x)p Fs(\))59 b Fg(is)g Fs(\()p Fq(\022)5 b(;)28 b(E)10 b Fs(\))p Fg(-suitable)59 b(if)g Fq(E)74 b(=)-102 b Fo(2)46 b Fq(\033)6 b Fs(\()p Fq(H)3314 10373 y Fp(x;L)3528 10348 y Fs(\))59 b Fg(and)2700 10726 y Fo(k)p Fs(\000)2887 10751 y Fp(x;L)3101 10726 y Fq(R)3227 10751 y Fp(x;L)3441 10726 y Fs(\()p Fq(E)10 b Fs(\))p Fq(\037)3808 10756 y Fp(x;L=)p Fn(3)4155 10726 y Fo(k)46 b(\024)4532 10614 y Fs(1)p 4479 10688 189 7 v 4479 10840 a Fq(L)4592 10792 y Fp(\022)4747 10726 y Fq(:)1518 b Fs(\(2.11\))p eop %%Page: 6 6 6 5 bop 758 861 a Fn(6)1551 b(FRANC)-77 b(\030)17 b(OIS)59 b(GERMINET)g(AND)f(ABEL)g(KLEIN)758 1259 y Fz(De\014nition)k(2.2.)84 b Fg(Given)68 b Fq(m)57 b(>)h Fs(0)p Fg(,)67 b Fq(E)g Fo(2)58 b Fm(R)22 b Fg(,)76 b Fq(x)58 b Fo(2)f Fm(Z)4132 1199 y Fp(d)4218 1259 y Fg(,)67 b(and)f Fq(L)58 b Fo(2)f Fs(6)p Fm(N)26 b Fg(,)77 b(we)66 b(say)f(that)h(the)g(b)-8 b(ox)758 1458 y Fs(\003)873 1483 y Fp(L)972 1458 y Fs(\()p Fq(x)p Fs(\))59 b Fg(is)g Fs(\()p Fq(m;)28 b(E)10 b Fs(\))p Fg(-r)-8 b(e)g(gular)58 b(if)h Fq(E)74 b(=)-102 b Fo(2)46 b Fq(\033)6 b Fs(\()p Fq(H)3326 1483 y Fp(x;L)3540 1458 y Fs(\))59 b Fg(and)2632 1774 y Fo(k)p Fs(\000)2819 1799 y Fp(x;L)3033 1774 y Fq(R)3159 1799 y Fp(x;L)3374 1774 y Fs(\()p Fq(E)10 b Fs(\))p Fq(\037)3741 1804 y Fp(x;L=)p Fn(3)4087 1774 y Fo(k)46 b(\024)g Fs(e)4465 1706 y Fk(\000)p Fp(m)4707 1661 y Fi(L)p 4707 1679 80 6 v 4718 1746 a Fe(2)4814 1774 y Fq(:)1451 b Fs(\(2.12\))957 2081 y(W)-14 b(e)52 b(de\014ne)h(the)f(m)-5 b(ultiscale)53 b(analysis)h(region)e (\006)4014 2106 y Fn(MSA)4360 2081 y Fo(\032)46 b Fs(\006)52 b(b)-5 b(y)53 b(requiring)g(the)f(conclusions)758 2280 y(of)j(the)g(b)5 b(o)g(otstrap)55 b(m)-5 b(ultiscale)56 b(analysis)g([GK1)q(,)g(Theorem)f(3.4].)758 2586 y Fz(De\014nition)62 b(2.3.)84 b Fg(The)55 b(multisc)-8 b(ale)55 b(analysis)g(r)-8 b(e)g(gion)54 b Fs(\006)4368 2611 y Fn(MSA)4722 2586 y Fg(for)g(the)h(r)-8 b(andom)54 b(op)-8 b(er)g(ator)53 b Fq(H)6639 2611 y Fp(!)758 2786 y Fg(is)75 b(the)h(set)f(of)g Fq(E)84 b Fo(2)75 b Fs(\006)g Fg(for)g(which)h(ther)-8 b(e)76 b(exists)f(some)h(op)-8 b(en)75 b(interval)i Fq(I)88 b Fo(3)74 b Fq(E)10 b Fg(,)79 b(such)c(that)758 2985 y(Assumptions)62 b(SLI,)h(EDI,)g(IAD,)g(NE,)g(W,)f(and)h(SGEE)f(hold)h (in)g Fq(I)13 b Fg(,)63 b(and)g(given)h(any)e Fq(\020)12 b Fg(,)63 b Fs(0)52 b Fq(<)758 3184 y(\020)58 b(<)46 b Fs(1)p Fg(,)58 b(and)h Fq(\013)q Fg(,)f Fs(1)46 b Fq(<)g(\013)h(<)g (\020)2512 3124 y Fk(\000)p Fn(1)2690 3184 y Fg(,)58 b(ther)-8 b(e)59 b(is)f(a)g(length)i(sc)-8 b(ale)59 b Fq(L)4530 3209 y Fn(0)4662 3184 y Fg(and)g(a)f(mass)g Fq(m)5688 3209 y Fp(\020)5810 3184 y Fq(>)46 b Fs(0)p Fg(,)58 b(so)g(if)h(we)758 3383 y(set)h Fq(L)1130 3408 y Fp(k)s Fn(+1)1425 3383 y Fs(=)47 b([)p Fq(L)1760 3323 y Fp(\013)1760 3430 y(k)1854 3383 y Fs(])1900 3408 y Fn(6)p Fj(N)2059 3383 y Fg(,)59 b Fq(k)52 b Fs(=)46 b(0)p Fq(;)28 b Fs(1)p Fq(;)g(:)g(:)g(:)f Fg(,)60 b(we)g(have)2524 3711 y Fm(P)70 b Fo(f)p Fq(R)29 b Fs(\()p Fq(m)3130 3736 y Fp(\020)3206 3711 y Fq(;)f(L)3393 3736 y Fp(k)3474 3711 y Fq(;)g(I)13 b(;)28 b(x;)g(y)6 b Fs(\))p Fo(g)46 b(\025)g Fs(1)37 b Fo(\000)g Fs(e)4693 3643 y Fk(\000)p Fp(L)4888 3587 y Fi(\020)4888 3683 y(k)6311 3711 y Fs(\(2.13\))758 4019 y Fg(for)59 b(al)8 b(l)60 b Fq(k)52 b Fs(=)47 b(0)p Fq(;)28 b Fs(1)p Fq(;)g(:)g(:)g(:)f Fg(,)59 b(and)g Fq(x;)28 b(y)53 b Fo(2)45 b Fm(Z)3102 3959 y Fp(d)3248 4019 y Fg(with)60 b Fo(j)p Fq(x)37 b Fo(\000)g Fq(y)6 b Fo(j)46 b Fq(>)g(L)4420 4044 y Fp(k)4539 4019 y Fs(+)37 b Fq(\045)p Fg(,)59 b(wher)-8 b(e)1332 4300 y Fq(R)q Fs(\()p Fq(m;)28 b(L;)g(I)13 b(;)28 b(x;)g(y)6 b Fs(\))45 b(=)3725 b(\(2.14\))1664 4549 y Fo(f)p Fg(for)59 b(every)47 b Fq(E)2571 4481 y Fk(0)2663 4549 y Fo(2)f Fq(I)13 b(;)87 b Fg(either)60 b Fs(\003)3627 4574 y Fp(L)3727 4549 y Fs(\()p Fq(x)p Fs(\))e Fg(or)i Fs(\003)4340 4574 y Fp(L)4439 4549 y Fs(\()p Fq(y)6 b Fs(\))59 b Fg(is)g Fs(\()p Fq(m;)28 b(E)5311 4489 y Fk(0)5356 4549 y Fs(\))p Fg(-r)-8 b(e)g(gular)p Fo(g)60 b Fq(:)957 4855 y Fs(On)88 b(\006)1386 4880 y Fn(MSA)1772 4855 y Fs(w)-5 b(e)87 b(ha)-5 b(v)g(e)88 b(all)f(desired)h(prop)5 b(erties)87 b(of)g(lo)5 b(calization:)137 b(Anderson)88 b(lo)5 b(caliza-)758 5055 y(tion,)42 b(SULE-lik)-5 b(e)39 b(estimates,)k(strong)d(Hilb)5 b(ert-Sc)-5 b(hmidt)39 b(dynamical)h(lo)5 b(calization,)42 b(and)e(more)758 5254 y([GK1].)74 b(W)-14 b(e)55 b(will)h(giv)-5 b(e)55 b(explicit)g(criteria)g(for)g Fq(E)g Fo(2)46 b Fs(\006)4121 5279 y Fn(MSA)4420 5254 y Fs(.)957 5453 y(In)55 b([GK3)q(])h(w)-5 b(e)55 b(de\014ned)h(the)f(strong)h(insulator)g(region)f(\006)4577 5478 y Fn(SI)4762 5453 y Fs(for)g Fq(H)5154 5478 y Fp(!)5306 5453 y Fs(b)-5 b(y)1024 5735 y(\006)1144 5760 y Fn(SI)1319 5735 y Fs(=)47 b Fo(f)p Fq(E)55 b Fo(2)46 b Fs(\006)28 b(;)73 b Fq(H)2318 5760 y Fp(!)2470 5735 y Fs(exhibits)55 b(strong)h(HS-dynamical)g(lo)5 b(calization)55 b(at)g Fq(E)9 b Fo(g)28 b Fq(:)266 b Fs(\(2.15\))758 6017 y(The)41 b(op)5 b(erator)41 b Fq(H)1879 6042 y Fp(!)2017 6017 y Fs(is)h(said)h(to)e(exhibit)g(strong)h(Hilb)5 b(ert-Sc)-5 b(hmidt)42 b(dynamical)g(lo)5 b(calization)42 b(at)758 6216 y Fq(E)53 b Fs(if)45 b(it)f(exhibits)h(strong)g(Hilb)5 b(ert-Sc)-5 b(hmidt)45 b(dynamical)g(lo)5 b(calization)45 b(in)g(some)g(op)5 b(en)45 b(in)-5 b(terv)c(al)758 6415 y(around)48 b Fq(E)10 b Fs(.)70 b(\(W)-14 b(e)46 b(refer)h(to)g([GK3])g (for)g(precise)h(de\014nitions.\))72 b(Note)46 b(that)g(\006)5534 6440 y Fn(SI)5711 6415 y Fs(is)i(an)f(op)5 b(en)47 b(set,)758 6614 y(and)59 b Fq(H)1222 6639 y Fp(!)1378 6614 y Fs(has)h(pure)f(p)5 b(oin)-5 b(t)60 b(sp)5 b(ectrum)59 b(in)h(\006)3545 6639 y Fn(SI)3674 6614 y Fs(.)86 b(W)-14 b(e)59 b(ha)-5 b(v)g(e)59 b(\006)4602 6639 y Fn(MSA)4954 6614 y Fo(\032)53 b Fs(\006)5256 6639 y Fn(SI)5445 6614 y Fs(from)59 b([GK1,)i(Pro)5 b(of)758 6814 y(of)57 b(Theorem)h(3.8)g(and)g(Corollary)g(3.10].)81 b(\(Note)57 b(that)g(\006)4404 6839 y Fn(MSA)4760 6814 y Fs(in)i([GK3])f(is)g(the)g(same)g(set)g(as)758 7013 y(in)e(this)h(article,)f(although)h(the)f(t)-5 b(w)g(o)57 b(de\014nitions)g(are)g(somewhat)f(di\013eren)-5 b(t.\))77 b(If)56 b(in)h(addition)758 7212 y(w)-5 b(e)50 b(ha)-5 b(v)g(e)50 b(the)g(deca)-5 b(y)50 b(estimates)g(of)g([GK2)q(,)h (Theorem)f(2],)i(uniformly)e(for)g(a.e.)72 b Fq(!)56 b Fs(\(true)50 b(under)758 7411 y(v)-5 b(ery)50 b(natural)h(h)-5 b(yp)5 b(otheses\),)52 b(it)e(is)h(sho)-5 b(wn)53 b(in)e([GK3])g(that)f (if)h(Assumptions)h(SLI,)f(EDI,)g(IAD,)758 7611 y(NE,)44 b(W,)g(and)i(SGEE)f(hold)g(in)g(an)g(op)5 b(en)45 b(in)-5 b(terv)c(al)44 b Fo(I)12 b Fs(,)48 b(then)c(\006)4577 7550 y Fk(I)4577 7656 y Fn(MSA)4923 7611 y Fs(=)i(\006)5218 7550 y Fk(I)5218 7656 y Fn(SI)5347 7611 y Fs(,)h(where)e Fq(B)6044 7550 y Fk(I)6181 7611 y Fs(=)h Fq(B)24 b Fo(\\)16 b(I)758 7813 y Fs(\(see)45 b(also)i([GK3,)h(Theorem)e(4.2]\),)i(and)e (w)-5 b(e)46 b(m)-5 b(ust)47 b(ha)-5 b(v)g(e)46 b(non)-5 b(trivial)46 b(transp)5 b(ort)46 b(in)g(\006)6096 7752 y Fk(I)6187 7813 y Fo(n)p Fs(\006)6390 7752 y Fk(I)6390 7858 y Fn(MSA)6689 7813 y Fs(.)957 8012 y(In)74 b(this)f(article)g(w)-5 b(e)74 b(pro)-5 b(vide)74 b(three)e(explicit)h(\014nite)h(v)-5 b(olume)73 b(criteria)g(for)h(lo)5 b(calization:)758 8211 y(Theorems)65 b(2.4)f(and)g(2.5)h(whic)-5 b(h)65 b(corresp)5 b(ond)64 b(to)g(the)g(\014rst)g(step)h(\(i.e.,)h(the)e (\014rst)g(m)-5 b(ultiscale)758 8410 y(analysis)82 b([GK1,)87 b(Theorem)81 b(5.1]\))g(in)g(the)f(Bo)5 b(otstrap)80 b(Multiscale)i(Analysis,)88 b(and)81 b(Theo-)758 8610 y(rem)62 b(2.6)h(whic)-5 b(h)63 b(corresp)5 b(onds)63 b(to)f(the)g(second)h(step)f(\([GK1)q(,)i(Theorem)e(5.2]\))g(in)h(the)f (Bo)5 b(ot-)758 8809 y(strap)78 b(Multiscale)h(Analysis.)142 b(Theorems)79 b(2.4)g(and)f(2.5)g(are)g(sp)5 b(ecial)79 b(cases)f(of)g(a)g(general)758 9008 y(explicit)54 b(criterion,)i (Theorem)f(2.7.)957 9207 y(The)i(\014rst)h(criterion)g(w)-5 b(orks)58 b(for)f(a)h Fg(pr)-8 b(escrib)g(e)g(d)74 b Fs(v)-9 b(alue)57 b(of)h(the)f(initial)h(length)f(scale)h Fq(L)6382 9232 y Fn(0)6456 9207 y Fs(.)81 b(In)758 9407 y(Section)45 b(3)h(w)-5 b(e)47 b(illustrate)f(its)g(use)g(b)-5 b(y)46 b(pro)-5 b(ving)47 b(lo)5 b(calization)46 b(for)g(Anderson)g (Hamiltonians)h(on)758 9606 y(the)58 b(con)-5 b(tin)g(uum)59 b(at)f(the)g(b)5 b(ottom)58 b(of)g(the)g(sp)5 b(ectrum)58 b(in)h(an)g(in)-5 b(terv)c(al)58 b(of)g(size)g Fo(O)5 b Fs(\()p Fq(\025)p Fs(\))57 b(for)h(large)758 9805 y Fq(\025)p Fs(,)d(where)g Fq(\025)g Fs(stands)i(for)e(the)g(disorder)h (parameter.)758 10111 y Fz(Theorem)63 b(2.4.)84 b Fg(L)-8 b(et)77 b Fq(H)2391 10136 y Fp(!)2564 10111 y Fg(b)-8 b(e)78 b(a)f(r)-8 b(andom)76 b(op)-8 b(er)g(ator)77 b(such)g(that)g (Assumptions)h(SLI,)f(EDI,)758 10310 y(IAD,)60 b(NE,)f(W,)g(and)h(SGEE) f(hold)h(in)f(an)h(op)-8 b(en)59 b(interval)i Fo(I)12 b Fg(.)77 b(Fix)60 b(a)f(length)i(sc)-8 b(ale)60 b Fq(L)6099 10335 y Fn(0)6220 10310 y Fo(2)46 b Fs(6)p Fm(N)26 b Fg(,)2547 10682 y Fq(L)2660 10707 y Fn(0)2781 10682 y Fo(\025)46 b Fs(max)3293 10448 y Fl(\032)3418 10682 y Fs(6)p Fq(;)28 b Fs(3)p Fq(\045;)g(\021)3906 10571 y Fk(\000)4010 10583 y Fs([)o(\()4140 10526 y Fe(5)p 4140 10544 57 6 v 4140 10611 a(3)4216 10571 y Fn(+)p Fp(b)4376 10583 y Fs(\))4441 10571 y Fp(d)4510 10583 y Fs(])4556 10500 y Ff(\000)p Fe(1)3900 10731 y Fk(I)4719 10448 y Fl(\033)4899 10682 y Fq(:)1366 b Fs(\(2.16\))p eop %%Page: 7 7 7 6 bop 1773 861 a Fn(EXPLICIT)58 b(FINITE)g(V)l(OLUME)h(CRITERIA)g(F)l (OR)g(LOCALIZA)-11 b(TION)949 b(7)758 1259 y Fg(L)-8 b(et)59 b Fq(E)1167 1284 y Fn(0)1287 1259 y Fo(2)46 b Fs(\006)37 b Fo(\\)g(I)72 b Fg(and)59 b(supp)-8 b(ose)1444 1645 y Fm(P)1564 1411 y Fl(\032)1697 1645 y Fq(D)1834 1670 y Fk(I)1926 1645 y Fq(L)2039 1546 y Fs(\()2123 1489 y Fe(5)p 2123 1507 57 6 v 2123 1574 a(3)2200 1534 y Fn(+)p Fp(b)2360 1546 y Fs(\))2425 1534 y Fp(d)2039 1689 y Fn(0)2502 1645 y Fo(k)p Fs(\000)2689 1670 y Fn(0)p Fp(;L)2885 1687 y Fe(0)2959 1645 y Fq(R)3085 1670 y Fp(!)t(;)p Fn(0)p Fp(;L)3408 1687 y Fe(0)3482 1645 y Fs(\()p Fq(E)3670 1670 y Fn(0)3743 1645 y Fs(\))p Fq(\037)3912 1675 y Fn(0)p Fp(;L)4108 1692 y Fe(0)4173 1675 y Fp(=)p Fn(3)4316 1645 y Fo(k)46 b Fq(<)g Fs(1)4703 1411 y Fl(\033)4874 1645 y Fo(\025)g Fs(1)37 b Fo(\000)5456 1579 y Fn(2)p 5355 1606 269 7 v 5355 1704 a(368)5553 1671 y Fi(d)5671 1645 y Fq(;)594 b Fs(\(2.17\))758 2031 y Fg(with)2196 2326 y Fq(D)2333 2351 y Fk(I)2471 2326 y Fs(=)46 b(39)2812 2257 y Fn(\(3+)p Fp(b)p Fn(\))p Fp(d)3247 2326 y Fs(max)3584 2191 y Fl(\010)3681 2326 y Fs(16)37 b Fo(\001)g Fs(60)4133 2257 y Fp(d)4210 2326 y Fq(Q)4341 2351 y Fk(I)4433 2326 y Fq(;)28 b Fs(1)4590 2191 y Fl(\011)4714 2326 y Fq(\015)4809 2257 y Fn(2)4800 2367 y Fk(I)4919 2326 y Fq(:)1346 b Fs(\(2.18\))758 2621 y Fg(Then)59 b Fq(E)1313 2646 y Fn(0)1434 2621 y Fo(2)45 b Fs(\006)1710 2646 y Fp(M)12 b(S)6 b(A)2046 2621 y Fg(.)957 2935 y Fs(The)53 b(follo)-5 b(wing)55 b(criterion)e(is)h(for)f Fg(lar)-8 b(ge)54 b Fs(initial)g(length)g(scale)f Fq(L)4963 2960 y Fn(0)5091 2935 y Fs(and)h Fg(we)-8 b(ak)54 b Fs(initial)g(deca)-5 b(y:)758 3134 y(an)g(y)53 b(rate)f(that)h(is)h(faster)e(than)h(the)g(v) -5 b(olume)54 b(\(if)e Fq(b)46 b Fs(=)h(1)53 b(in)g(W)-14 b(egner\))52 b(or)i(the)e(v)-5 b(olume)54 b(squared)758 3334 y(\(if)70 b Fq(b)i Fs(=)h(2\))d(is)i(allo)-5 b(w)g(ed.)121 b(It)71 b(pro)-5 b(vides)72 b(a)e(precise)i(estimate)e(on)h(ho)-5 b(w)72 b(large)f Fq(L)5843 3359 y Fn(0)5988 3334 y Fs(has)h(to)e(b)5 b(e,)758 3533 y(dep)g(ending)58 b(on)f(the)h(parameters)f Fq(Q)3071 3558 y Fk(I)3220 3533 y Fs(and)h Fq(\015)3631 3558 y Fk(I)3780 3533 y Fs(of)f(the)g(mo)5 b(del,)58 b(and)g(on)g(the)f(prescrib)5 b(ed)58 b(rate)758 3732 y(of)d(deca)-5 b(y)55 b(of)g(the)g(resolv)-5 b(en)g(t.)758 4046 y Fz(Theorem)63 b(2.5.)84 b Fg(L)-8 b(et)77 b Fq(H)2391 4071 y Fp(!)2564 4046 y Fg(b)-8 b(e)78 b(a)f(r)-8 b(andom)76 b(op)-8 b(er)g(ator)77 b(such)g(that)g(Assumptions)h(SLI,)f(EDI,)758 4245 y(IAD,)60 b(NE,)f(W,)g(and)h(SGEE)f(hold)h(in)f(an)h(op)-8 b(en)59 b(interval)i Fo(I)12 b Fg(.)77 b(Given)61 b Fq(s)46 b(>)g(bd)p Fg(,)60 b(set)1377 4638 y Fo(L)46 b Fs(=)g(max)2050 4404 y Fl(\032)2174 4638 y Fs(3)p Fq(\045;)28 b Fs(42)p Fq(;)g Fs(3)2768 4454 y Fl(\020)2887 4573 y Fn(107)3085 4523 y Fi(d)p 2887 4600 V 2988 4696 a Fn(2)3175 4454 y Fl(\021)3395 4444 y Fe(2)p 3294 4462 260 6 v 3294 4528 a Fi(s)p Ff(\000)p Fi(bd)3609 4638 y Fq(;)3736 4573 y Fn(1)p 3703 4600 133 7 v 3703 4696 a(37)3883 4504 y Fl(\000)3959 4638 y Fs(16)37 b Fo(\001)g Fs(60)4411 4578 y Fp(d)4489 4638 y Fq(Q)4620 4663 y Fk(I)4711 4504 y Fl(\001)4908 4494 y Fe(2)p 4807 4512 260 6 v 4807 4578 a Fi(s)p Ff(\000)p Fi(bd)5122 4638 y Fq(;)28 b(\021)5284 4541 y Fk(\000)5408 4496 y Fe(1)p 5408 4514 57 6 v 5408 4581 a Fi(s)5278 4687 y Fk(I)5493 4404 y Fl(\033)5645 4638 y Fq(:)620 b Fs(\(2.19\))758 5024 y Fg(Supp)-8 b(ose)59 b(that)g(for)h(some)f Fq(L)2532 5049 y Fn(0)2653 5024 y Fo(\025)46 b(L)p Fg(,)59 b Fq(L)3166 5049 y Fn(0)3286 5024 y Fo(2)46 b Fs(6)p Fm(N)35 b Fg(,)59 b(and)h Fq(E)4202 5049 y Fn(0)4322 5024 y Fo(2)46 b Fs(\006)37 b Fo(\\)f(I)12 b Fg(,)1341 5319 y Fm(P)1461 5185 y Fl(\010)1567 5319 y Fs(90)1733 5251 y Fp(d)1810 5319 y Fq(\015)1905 5251 y Fn(2)1896 5360 y Fk(I)1987 5319 y Fs(\(37)p Fq(L)2331 5344 y Fn(0)2405 5319 y Fs(\))2470 5251 y Fp(s)2541 5319 y Fo(k)p Fs(\000)2728 5344 y Fn(0)p Fp(;L)2924 5361 y Fe(0)2997 5319 y Fq(R)3123 5344 y Fp(!)t(;)p Fn(0)p Fp(;L)3446 5361 y Fe(0)3520 5319 y Fs(\()p Fq(E)3708 5344 y Fn(0)3782 5319 y Fs(\))p Fq(\037)3951 5349 y Fn(0)p Fp(;L)4147 5366 y Fe(0)4212 5349 y Fp(=)p Fn(3)4355 5319 y Fo(k)46 b Fq(<)g Fs(1)4742 5185 y Fl(\011)4885 5319 y Fo(\025)g Fs(1)37 b Fo(\000)5467 5254 y Fn(2)p 5366 5281 269 7 v 5366 5379 a(344)5564 5345 y Fi(d)5682 5319 y Fq(:)583 b Fs(\(2.20\))758 5615 y Fg(Then)59 b Fq(E)1313 5640 y Fn(0)1434 5615 y Fo(2)45 b Fs(\006)1710 5640 y Fp(M)12 b(S)6 b(A)2046 5615 y Fg(.)957 5929 y Fs(The)71 b(next)g(criterion)h(is)g(an)g(analog)g(of)f(Theorem)h (2.5)g(but)f(for)h(the)f(second)h(m)-5 b(ultiscale)758 6129 y(analysis)46 b(of)f(the)g(b)5 b(o)g(otstrap)45 b(sc)-5 b(heme)46 b(describ)5 b(ed)46 b(in)g([GK1],)i(i.e.,)g(for)d (the)g(m)-5 b(ultiscale)46 b(analysis)758 6328 y(with)60 b(exp)5 b(onen)-5 b(tial)61 b(deca)-5 b(y)61 b(of)g(the)f(resolv)-5 b(en)g(t)62 b(and)f(p)5 b(olynomial)62 b(deca)-5 b(y)61 b(of)f(the)h(probabilities)758 6527 y(as)45 b(in)g(v)-5 b(on)45 b(Dreifus)f(and)h(Klein)g([vDK],)i(mo)5 b(di\014ed)45 b(as)g(in)g(Figotin)g(and)h(Klein)f([FK1,)i(Theorem)758 6726 y(32])67 b(to)f(allo)-5 b(w)67 b(the)f(mass)i(in)f(the)g(starting) g(h)-5 b(yp)5 b(otheses)67 b(to)f(decrease)h(as)g(the)f(initial)h (length)758 6926 y(scale)51 b(increases.)74 b(Theorem)52 b(2.6)f(k)-5 b(eeps)52 b(trac)-5 b(k)51 b(of)g(the)h(dep)5 b(endency)51 b(of)g(ho)-5 b(w)52 b(large)g(the)f(initial)758 7125 y(length)c(scale)h(has)g(to)f(b)5 b(e)47 b(in)h(terms)g(of)f(the)g (constan)-5 b(ts)49 b(coming)f(from)g(Assumptions)h(SLI,)f(NE,)758 7324 y(and)64 b(W.)g(It)g(will)g(b)5 b(e)64 b(used)g(in)h(Section)e(4)i (to)e(pro)-5 b(v)g(e)65 b(lo)5 b(calization)63 b(for)h(random)h(Sc)-5 b(hr\177)-83 b(odinger)758 7523 y(op)5 b(erators)81 b(in)g(a)g(constan) -5 b(t)82 b(magnetic)f(\014eld)h(\(random)f(Landau)h(Hamiltonians\))g (up)f(to)g(a)758 7736 y(distance)55 b Fo(O)5 b Fs(\()1626 7662 y Fn(log)24 b Fp(B)p 1625 7698 301 7 v 1722 7794 a(B)1945 7736 y Fs(\))54 b(from)h(the)f(Landau)h(lev)-5 b(els,)56 b(where)e Fq(B)63 b Fs(is)55 b(the)f(strength)h(of)f(the)g (magnetic)758 7936 y(\014eld.)957 8135 y(While)d(Theorem)h(2.6)g(has)g (stronger)g(h)-5 b(yp)5 b(otheses)52 b(than)g(Theorem)g(2.5,)g(it)g (deliv)-5 b(ers)52 b(more:)758 8334 y(an)62 b(estimate)f(of)g(the)h (exp)5 b(onen)-5 b(tial)61 b(rate)g(of)h(deca)-5 b(y)61 b(of)h(the)f(eigenfunctions)h(in)g(terms)g(of)g(the)758 8533 y(constan)-5 b(ts)54 b(of)f(the)h(problems.)74 b(This)55 b(is)f(of)f(great)g(in)-5 b(terest)54 b(in)g(the)f(large)g(disorder)i (regime)e(for)758 8733 y(the)48 b(Anderson)h(mo)5 b(del)48 b(as)h(discussed)h(in)f(Section)f(3,)i(for)f(the)f(random)h(Landau)g (Hamiltonian)758 8932 y(as)k(in)g(Section)f(4,)h(and)g(also)g(in)g(the) f(regime)h(of)f(small)i(disorder)f(for)g(the)f(Anderson)h(mo)5 b(del)53 b(as)758 9131 y(studied)j(in)f([W2,)h(Klo2].)957 9330 y(If)46 b(the)h(probabilit)-5 b(y)47 b(estimate)f(in)h(the)g (initial)g(length)f(scale)h(is)h(su\016cien)-5 b(tly)47 b(go)5 b(o)g(d,)48 b(one)f(ma)-5 b(y)758 9530 y(use)47 b(Theorem)g(2.6)g(to)f(start)h(the)f(b)5 b(o)g(otstrap)46 b(m)-5 b(ultiscale)48 b(analysis,)i(instead)d(of)f(applying)i(\014rst) 758 9729 y(Theorem)j(2.4)g(or)g(Theorem)g(2.5)g(and)g(then)g(b)5 b(o)g(otstrapping)51 b(to)f(Theorem)h(2.6)g(to)g(obtain)g(the)758 9928 y(exp)5 b(onen)-5 b(tial)45 b(rate)g(of)g(deca)-5 b(y)45 b(of)g(eigenfunctions.)71 b(This)47 b(is)f(the)f(case)g(for)g (the)g(random)i(Landau)758 10127 y(Hamiltonians)73 b(as)g(in)f(Section) g(4)h(and)f(in)h(the)f(w)-5 b(eak)72 b(disorder)h(regime)g(for)f(the)g (Anderson)758 10327 y(mo)5 b(del)55 b(as)h(in)g([Klo2].)758 10641 y Fz(Theorem)63 b(2.6.)84 b Fg(L)-8 b(et)77 b Fq(H)2391 10666 y Fp(!)2564 10641 y Fg(b)-8 b(e)78 b(a)f(r)-8 b(andom)76 b(op)-8 b(er)g(ator)77 b(such)g(that)g(Assumptions)h(SLI,)f(EDI,)758 10840 y(IAD,)72 b(NE,)f(W)g(and)h(SGEE)f(hold)g(in)h(an)g(op)-8 b(en)71 b(interval)i Fo(I)12 b Fg(.)112 b(Given)73 b Fq(E)5383 10865 y Fn(0)5525 10840 y Fo(2)68 b Fs(\006)45 b Fo(\\)g(I)12 b Fg(,)75 b Fq(p)67 b(>)h Fs(0)p Fg(,)p eop %%Page: 8 8 8 7 bop 758 861 a Fn(8)1551 b(FRANC)-77 b(\030)17 b(OIS)59 b(GERMINET)g(AND)f(ABEL)g(KLEIN)758 1261 y Fq(\013)47 b Fs(=)f(1)37 b(+)1443 1180 y Fp(p=)p Fn(2)p 1392 1223 306 7 v 1392 1318 a Fp(p)p Fn(+2)p Fp(d)1717 1261 y Fg(,)60 b Fq(s)46 b(>)g Fs(2)p Fq(p)36 b Fs(+)h(\()p Fq(b)g Fs(+)g(2\))p Fq(d)p Fg(,)60 b(and)f Fq(\022)51 b(>)46 b Fs(4)p Fq(\013)q(s)p Fg(,)59 b(set)1174 1709 y Fo(L)45 b Fs(=)i(max)1846 1475 y Fl(\032)1971 1709 y Fs(3)p Fq(\045;)28 b Fs(417)2565 1593 y Fe(1)p 2483 1611 223 6 v 2483 1678 a Fi(\013)p Ff(\000)p Fe(1)2733 1709 y Fq(;)g(\021)2895 1612 y Fk(\000)3019 1567 y Fe(1)p 3019 1585 57 6 v 3019 1652 a Fi(s)2889 1758 y Fk(I)3104 1709 y Fq(;)g Fs(dist\()p Fq(E)3634 1734 y Fn(0)3708 1709 y Fq(;)g Fm(R)21 b Fo(nI)13 b Fs(\))4143 1641 y Fk(\000)4275 1596 y Fe(1)p 4275 1614 V 4275 1681 a Fi(s)4360 1709 y Fq(;)28 b(Q)4762 1551 y Fe(1)p 4585 1569 411 6 v 4585 1636 a Fi(s)p Ff(\000)p Fi(p)p Ff(\000)p Fi(bd)4565 1758 y Fk(I)5024 1709 y Fq(;)1241 b Fs(\(2.21\))1646 2005 y Fl(\000)1722 2140 y Fs(16)37 b Fo(\001)g Fs(36)2174 2071 y Fp(d)2251 2140 y Fs(2)2334 2071 y Fp(s)p Fk(\000)p Fn(2)p Fp(p)2644 2140 y Fq(C)2763 2165 y Fk(I)2853 2140 y Fq(Q)2984 2165 y Fk(I)3076 2005 y Fl(\001)3576 1987 y Fe(1)p 3172 2005 865 6 v 3172 2073 a Fi(\013)p Fe(\()p Fi(s)p Ff(\000)p Fe(2)p Fi(p)p Ff(\000)p Fe(\()p Fi(b)p Fe(+2\))p Fi(d)p Fe(\))4093 2140 y Fq(;)4167 2005 y Fl(\000)4242 2140 y Fs(69)4408 2071 y Fp(d)4486 2140 y Fq(\015)4581 2071 y Fn(2)4572 2181 y Fk(I)4663 2005 y Fl(\001)4899 1987 y Fe(4)p 4759 2005 338 6 v 4759 2072 a Fi(\022)s Ff(\000)p Fe(4)p Fi(\013s)5152 2140 y Fq(;)5226 2005 y Fl(\000)5302 2140 y Fs(3)5385 2071 y Fp(d)5462 2140 y Fq(\015)5548 2165 y Fk(I)5639 2005 y Fl(\001)5826 1987 y Fe(16)p Fi(\013)p 5735 2005 372 6 v 5735 2073 a(\022)s Fe(\()p Fi(\013)p Ff(\000)p Fe(1\))6135 1955 y Fl(o)6273 2140 y Fq(:)758 2491 y Fg(If)59 b(for)g(some)h Fq(L)1735 2516 y Fn(0)1855 2491 y Fo(\025)46 b(L)p Fg(,)59 b Fq(L)2368 2516 y Fn(0)2489 2491 y Fo(2)46 b Fs(6)p Fm(N)35 b Fg(,)59 b(we)h(have)2286 2870 y Fm(P)34 b Fo(f)8 b Fs(\003)2613 2895 y Fp(L)2704 2912 y Fe(0)2778 2870 y Fs(\(0\))59 b Fg(is)g Fs(\()p Fq(\022)5 b(;)28 b(E)3573 2895 y Fn(0)3646 2870 y Fs(\))p Fg(-suitable)q Fo(g)46 b Fq(>)g Fs(1)37 b Fo(\000)4977 2758 y Fs(1)p 4923 2832 190 7 v 4923 2984 a Fq(L)5036 2905 y Fp(p)5036 3029 y Fn(0)5133 2870 y Fq(;)1132 b Fs(\(2.22\))758 3281 y Fg(then)61 b(ther)-8 b(e)61 b(exists)g(an)g(op)-8 b(en)60 b(interval)i Fq(I)g Fs(=)48 b Fq(I)13 b Fs(\()p Fq(\022)5 b(;)28 b(s;)g(L)4124 3306 y Fn(0)4197 3281 y Fs(\))p Fg(,)61 b(with)g Fq(E)4859 3306 y Fn(0)4981 3281 y Fo(2)48 b Fq(I)62 b Fo(\032)48 b(I)12 b Fg(,)62 b(such)e(that)h(if)g(we)758 3494 y(set)f Fq(m)1163 3519 y Fn(0)1283 3494 y Fs(=)46 b(2)p Fq(\022)1644 3420 y Fn(log)25 b Fp(L)1929 3437 y Fe(0)p 1644 3456 351 7 v 1741 3551 a Fp(L)1832 3568 y Fe(0)2014 3494 y Fg(,)59 b(and)h Fq(L)2560 3519 y Fp(k)s Fn(+1)2856 3494 y Fs(=)46 b([)p Fq(L)3190 3434 y Fp(\013)3190 3541 y(k)3285 3494 y Fs(])3331 3519 y Fn(6)p Fj(N)3489 3494 y Fg(,)60 b Fq(k)51 b Fs(=)c(0)p Fq(;)28 b Fs(1)p Fq(;)g(:)g(:)g(:)f Fg(,)59 b(we)h(have)1695 3915 y Fm(P)5 b Fo(f)p Fs(\003)1985 3940 y Fp(L)2077 3958 y Fi(k)2166 3915 y Fs(\(0\))92 b Fg(is)59 b Fs(\()2734 3847 y Fp(m)2852 3864 y Fe(0)p 2734 3877 183 7 v 2792 3972 a Fn(4)2936 3915 y Fq(;)28 b(E)10 b Fs(\))p Fg(-r)-8 b(e)g(gular)o Fo(g)46 b(\025)h Fs(1)37 b Fo(\000)4424 3803 y Fs(2)p 4369 3877 195 7 v 4369 4029 a Fq(L)4482 3949 y Fp(p)4482 4079 y(k)4768 3915 y Fg(for)59 b(al)8 b(l)93 b Fq(E)56 b Fo(2)45 b Fq(I)13 b(;)541 b Fs(\(2.23\))758 4322 y Fg(and)1160 4670 y Fm(P)1317 4535 y Fl(\002)1386 4670 y Fq(R)1541 4535 y Fl(\000)1637 4601 y Fp(m)1755 4618 y Fe(0)p 1637 4631 183 7 v 1695 4727 a Fn(4)1840 4670 y Fq(;)28 b(L)2027 4695 y Fp(k)2108 4670 y Fq(;)g(I)13 b(;)28 b(x;)g(y)2598 4535 y Fl(\001\003)2789 4670 y Fo(\025)47 b Fs(1)36 b Fo(\000)3357 4557 y Fs(4)p 3270 4631 256 7 v 3270 4806 a Fq(L)3383 4726 y Fn(2)p Fp(p)3383 4856 y(k)3592 4670 y Fg(for)46 b Fq(x;)28 b(y)53 b Fo(2)45 b Fm(Z)4405 4601 y Fp(d)4519 4670 y Fq(;)28 b Fo(j)p Fq(x)37 b Fo(\000)g Fq(y)6 b Fo(j)47 b Fq(>)f(L)5405 4695 y Fp(k)5523 4670 y Fs(+)37 b Fq(\045)60 b(;)430 b Fs(\(2.24\))758 5099 y Fg(for)59 b(al)8 b(l)60 b Fq(k)52 b Fs(=)47 b(0)p Fq(;)28 b Fs(1)p Fq(;)g(:)g(:)g(:)f Fg(,)59 b(wher)-8 b(e)1378 5410 y Fq(R)q Fs(\()p Fq(m;)28 b(L;)g(I)13 b(;)28 b(x;)g(y)6 b Fs(\))45 b(=)3679 b(\(2.25\))1711 5659 y Fo(f)p Fg(for)59 b(every)47 b Fq(E)55 b Fo(2)46 b Fq(I)13 b(;)87 b Fg(either)60 b Fs(\003)3627 5684 y Fp(L)3727 5659 y Fs(\()p Fq(x)p Fs(\))e Fg(or)i Fs(\003)4340 5684 y Fp(L)4439 5659 y Fs(\()p Fq(y)6 b Fs(\))59 b Fg(is)g Fs(\()p Fq(m;)28 b(E)10 b Fs(\))p Fg(-r)-8 b(e)g(gular)o Fo(g)59 b Fq(:)758 5969 y Fg(Mor)-8 b(e)g(over,)970 6224 y Fz(\(i\):)82 b Fq(I)50 b Fo(\\)37 b Fs(\006)46 b Fo(\032)g Fs(\006)2039 6249 y Fp(M)12 b(S)6 b(A)2402 6224 y Fg(.)970 6424 y Fz(\(ii\):)81 b Fg(F)-13 b(or)69 b(almost)f(every)h Fq(!)6 b Fg(,)71 b(an)e(eigenfunction)h Fq(')4288 6449 y Fp(!)t(;E)4596 6424 y Fg(of)e Fq(H)4938 6449 y Fp(!)5102 6424 y Fg(with)g(eigenvalue)j Fq(E)g Fo(2)62 b Fq(I)1169 6643 y Fg(de)-8 b(c)g(ays)59 b(exp)-8 b(onential)8 b(ly)61 b(\(in)e(the)h Fq(L)3345 6582 y Fn(2)3419 6643 y Fg(-sense\))h(with)f(a)f(r)-8 b(ate)59 b Fo(\025)5026 6577 y Fp(\022)p 5026 6604 67 7 v 5026 6700 a Fn(2)5132 6568 y(log)26 b Fp(L)5418 6585 y Fe(0)p 5132 6604 351 7 v 5229 6700 a Fp(L)5320 6717 y Fe(0)5502 6643 y Fg(,)60 b(i.e.,)2499 7081 y Fs(lim)28 b(inf)2520 7196 y Fk(j)p Fp(x)p Fk(j!1)2987 7081 y Fo(\000)3136 6969 y Fs(log)j Fo(k)p Fq(\037)3566 6994 y Fp(x)3649 6969 y Fq(')3758 6994 y Fp(!)t(;E)3997 6969 y Fo(k)p 3136 7043 944 7 v 3515 7195 a(j)p Fq(x)p Fo(j)4146 7081 y(\025)4341 6969 y Fq(\022)i Fs(log)d Fq(L)4807 6994 y Fn(0)p 4341 7043 540 7 v 4476 7195 a Fs(2)p Fq(L)4672 7220 y Fn(0)4947 7081 y Fq(:)1318 b Fs(\(2.26\))957 7497 y(Theorems)50 b(2.5)f(and)h(2.4)g(are)f(corollaries)h(of)f(a)g(general) g(criterion)g(w)-5 b(e)50 b(will)f(no)-5 b(w)50 b(state.)72 b(W)-14 b(e)758 7696 y(set)1291 8077 y Fq(\014)9 b Fs(\()p Fq(Y)g(;)28 b(S)10 b Fs(\))165 b(=)2295 7965 y(\(3)p Fq(Y)74 b Fo(\000)37 b Fs(4\))2927 7905 y Fp(d)3004 7965 y Fs(\(\(3)p Fq(Y)73 b Fo(\000)37 b Fs(4\))3700 7905 y Fp(d)3814 7965 y Fo(\000)g Fs(4)4063 7905 y Fp(d)4140 7965 y Fs(\))28 b Fo(\001)g(\001)g(\001)f Fs(\(\(3)p Fq(Y)73 b Fo(\000)38 b Fs(4\))5151 7905 y Fp(d)5264 7965 y Fo(\000)f Fs(\()p Fq(S)46 b Fo(\000)37 b Fs(1\)4)6040 7905 y Fp(d)6117 7965 y Fs(\))p 2295 8039 3888 7 v 4160 8191 a Fq(S)10 b Fs(!)1980 8511 y Fq(<)2295 8399 y Fs(\(3)p Fq(Y)74 b Fo(\000)37 b Fs(4\))2927 8338 y Fp(S)6 b(d)p 2295 8473 797 7 v 2614 8625 a Fq(S)k Fs(!)3139 8511 y Fq(:)3126 b Fs(\(2.27\))758 8880 y Fz(Theorem)63 b(2.7.)84 b Fg(L)-8 b(et)77 b Fq(H)2391 8905 y Fp(!)2564 8880 y Fg(b)-8 b(e)78 b(a)f(r)-8 b(andom)76 b(op)-8 b(er)g(ator)77 b(such)g(that)g(Assumptions)h(SLI,)f(EDI,)758 9079 y(IAD,)g(NE,)g(W,)g (and)f(SGEE)h(hold)g(in)g(an)g(op)-8 b(en)77 b(interval)h Fo(I)12 b Fg(.)129 b(Pick)77 b Fq(p)h(>)f Fs(0)p Fg(,)k Fq(S)87 b Fo(\025)78 b Fs(2)p Fg(,)j(and)758 9324 y Fq(S)55 b Fo(2)46 b Fm(N)26 b Fg(.)84 b(T)-13 b(ake)55 b Fq(s)46 b(>)g(p)26 b Fs(+)g Fq(bd)p Fg(,)56 b Fq(Y)83 b Fo(\025)46 b Fs(8)p Fq(S)36 b Fs(+)26 b(5)p Fg(,)56 b Fq(Y)91 b Fg(o)-8 b(dd,)55 b(and)f Fq(L)4434 9349 y Fn(0)4555 9324 y Fo(2)46 b Fs(6)p Fm(N)26 b Fg(,)64 b Fq(L)5134 9349 y Fn(0)5255 9324 y Fo(\025)46 b Fs(max)5767 9139 y Fl(n)5877 9324 y Fs(6)p Fq(;)28 b Fs(3)p Fq(\045;)g(\021)6365 9226 y Fk(\000)6489 9182 y Fe(1)p 6489 9200 57 6 v 6489 9267 a Fi(s)6359 9372 y Fk(I)6574 9139 y Fl(o)6684 9324 y Fg(,)758 9564 y(such)59 b(that)2754 9911 y Fq(Y)74 b Fo(\000)3284 9798 y Fs(2)p 3110 9873 430 7 v 3110 10025 a(log)31 b Fq(L)3466 10050 y Fn(0)3588 9911 y Fs(log)f Fq(Y)83 b Fo(\025)46 b Fs(8)p Fq(S)h Fs(+)37 b(3)p Fq(;)1600 b Fs(\(2.28\))758 10300 y Fg(and)2427 10681 y Fs(\()p Fq(Y)37 b(L)2738 10706 y Fn(0)2812 10681 y Fs(\))2877 10613 y Fp(s)p Fk(\000)p Fp(p)p Fk(\000)p Fp(bd)3397 10681 y Fo(\025)47 b Fs(2)3656 10613 y Fp(d)p Fn(+2)3901 10681 y Fq(Q)4032 10706 y Fk(I)4143 10569 y Fq(S)10 b Fs(\(7)p Fq(S)46 b Fs(+)37 b(2\))4865 10509 y Fp(bd)p 4143 10643 857 7 v 4194 10798 a Fs(\(3)p Fq(Y)g Fs(\))4540 10750 y Fn(\()p Fp(b)p Fk(\000)p Fn(1\))p Fp(d)5020 10681 y Fq(:)1245 b Fs(\(2.29\))p eop %%Page: 9 9 9 8 bop 1773 861 a Fn(EXPLICIT)58 b(FINITE)g(V)l(OLUME)h(CRITERIA)g(F)l (OR)g(LOCALIZA)-11 b(TION)949 b(9)758 1259 y Fg(In)59 b(addition)h(assume)f(that)2306 1531 y Fq(either)281 b(L)3142 1556 y Fn(0)3263 1531 y Fo(\025)46 b Fs(\(2)p Fq(\014)9 b Fs(\()p Fq(Y)g(;)28 b(S)46 b Fs(+)37 b(1\))p Fq(Y)4528 1463 y Fp(p)4605 1531 y Fs(\))4845 1406 y Fe(1)p 4689 1424 369 6 v 4689 1493 a(\()p Fi(S)t Fe(+1\))p Fi(p)5113 1531 y Fq(;)1152 b Fs(\(2.30\))2593 1867 y Fq(or)281 b(p)46 b(>)g(d;)75 b(S)55 b Fo(\025)3893 1754 y Fq(p)37 b Fs(+)g Fq(d)p 3893 1829 373 7 v 3893 1981 a(p)g Fo(\000)g Fq(d)4314 1867 y(:)1951 b Fs(\(2.31\))758 2218 y Fg(De\014ne)61 b Fq(\022)1360 2243 y Fn(0)1480 2218 y Fq(>)47 b(s)59 b Fg(by)2715 2494 y Fq(L)2828 2420 y Fp(\022)2891 2437 y Fe(0)2828 2538 y Fn(0)3011 2494 y Fs(=)46 b Fq(\015)3281 2425 y Fn(2)3272 2535 y Fk(I)3363 2494 y Fs(3)3446 2425 y Fp(d)3524 2494 y Fs(\(7)p Fq(S)f Fs(+)37 b(2\))4133 2425 y Fp(d)4210 2494 y Fs(\()p Fq(Y)g(L)4521 2519 y Fn(0)4596 2494 y Fs(\))4661 2425 y Fp(s)4731 2494 y Fq(:)1534 b Fs(\(2.32\))758 2767 y Fg(Then,)59 b(if)h(for)f(some)g Fq(E)2215 2792 y Fn(0)2335 2767 y Fo(2)46 b Fs(\006)37 b Fo(\\)g(I)72 b Fg(and)59 b Fq(\022)51 b(>)46 b(\022)3663 2792 y Fn(0)3765 2767 y Fg(,)1777 3070 y Fm(P)6 b Fo(f)p Fs(\003)2068 3095 y Fp(L)2159 3112 y Fe(0)2241 3070 y Fs(\(0\))59 b Fg(is)g Fs(\()p Fq(\022)5 b(;)28 b(E)3036 3095 y Fn(0)3109 3070 y Fs(\))p Fg(-suitable)q Fo(g)46 b Fq(>)g Fs(1)37 b Fo(\000)g Fs(\(2)p Fq(\014)9 b Fs(\()p Fq(Y)g(;)28 b(S)46 b Fs(+)37 b(1\)\))5387 2987 y Fk(\000)5520 2942 y Fe(1)p 5511 2960 76 6 v 5511 3027 a Fi(S)5642 3070 y Fq(;)623 b Fs(\(2.33\))758 3352 y Fg(wher)-8 b(e)54 b Fq(\014)9 b Fs(\()p Fq(Y)g(;)28 b(S)34 b Fs(+)25 b(1\))53 b Fg(is)h(de\014ne)-8 b(d)54 b(in)67 b Fs(\(2.27\))o Fg(,)55 b(then)g(setting)g Fq(L)4533 3377 y Fp(k)s Fn(+1)4829 3352 y Fs(=)46 b Fq(Y)37 b(L)5250 3377 y Fp(k)5332 3352 y Fg(,)55 b Fq(k)d Fs(=)46 b(0)p Fq(;)28 b Fs(1)p Fq(;)g Fs(2)p Fq(;)g(:)g(:)g(:)f Fg(,)55 b(we)758 3551 y(have)2248 3860 y Fm(P)5 b Fo(f)p Fs(\003)2538 3885 y Fp(L)2630 3903 y Fi(k)2719 3860 y Fs(\(0\))92 b Fg(is)59 b Fs(\()p Fq(\022)5 b(;)28 b(E)3547 3885 y Fn(0)3620 3860 y Fs(\))p Fg(-suitable)q Fo(g)46 b(\025)h Fs(1)36 b Fo(\000)4953 3748 y Fs(1)p 4897 3822 195 7 v 4897 3974 a Fq(L)5010 3894 y Fp(p)5010 4024 y(k)5171 3860 y Fq(;)1094 b Fs(\(2.34\))758 4238 y Fg(for)59 b(al)8 b(l)60 b Fq(k)52 b Fo(\025)47 b(K)r Fg(,)59 b(wher)-8 b(e)60 b Fo(K)48 b Fs(=)e Fo(K)r Fs(\()p Fq(d;)28 b(p;)g(Y)9 b(;)28 b(S)10 b Fs(\))45 b Fq(<)h Fo(1)p Fg(.)76 b(In)60 b(p)-8 b(articular)59 b Fq(E)5112 4263 y Fn(0)5233 4238 y Fo(2)45 b Fs(\006)5509 4263 y Fp(M)12 b(S)6 b(A)5845 4238 y Fg(.)957 4539 y Fs(Theorems)59 b(2.4)f(and)h(2.5)f(are)g(deriv)-5 b(ed)58 b(from)h(Theorem)f(2.7)g(in)h(Section)f(5.)82 b(Theorem)58 b(2.7)758 4738 y(is)e(pro)-5 b(v)g(en)56 b(in)f(Section)h(6,)f(and)h (Theorem)g(2.6)f(is)h(pro)-5 b(v)g(en)56 b(in)g(Section)f(7.)758 5039 y Fz(Remark)62 b(2.8.)84 b Fg(The)73 b(initial)g(pr)-8 b(ob)g(abilities)73 b Fq(p)3729 5064 y Fn(0)3873 5039 y Fs(=)4192 4973 y Fn(2)p 4091 5000 269 7 v 4091 5098 a(368)4289 5065 y Fi(d)4452 5039 y Fg(in)g(The)-8 b(or)g(em)72 b(2.4)g(and)g Fq(p)6084 5064 y Fn(0)6228 5039 y Fs(=)6548 4973 y Fn(2)p 6447 5000 V 6447 5098 a(344)6645 5065 y Fi(d)758 5238 y Fg(in)79 b(The)-8 b(or)g(em)79 b(2.5)g(ar)-8 b(e)80 b(mostly)f(indic)-8 b(ative.)137 b(We)80 b(ar)-8 b(e)80 b(applying)f(The)-8 b(or)g(em)79 b(2.7)g(with)h Fq(S)92 b Fs(=)758 5437 y(4)p Fq(;)55 b(Y)116 b Fs(=)78 b(39)f Fg(and)h Fq(S)87 b Fs(=)79 b(4)p Fq(;)55 b(Y)116 b Fs(=)78 b(37)p Fg(,)k(r)-8 b(esp)g(e)g(ctively,)82 b(which)77 b(ar)-8 b(e)77 b(not)h(optimal)f(in)g(terms)g(of)758 5637 y(pr)-8 b(ob)g(abilities.)74 b(Inde)-8 b(e)g(d,)53 b(fr)-8 b(om)51 b(The)-8 b(or)g(em)51 b(2.7)h(one)h(gets)f(that)h Fq(p)4696 5662 y Fn(0)4816 5637 y Fs(=)46 b(\(2)p Fq(\014)9 b Fs(\()p Fq(Y)g(;)28 b(S)i Fs(+)21 b(1\)\))5981 5576 y Fk(\000)p Fn(1)p Fp(=S)6366 5637 y Fg(\(with)758 5836 y Fq(\014)9 b Fs(\()p Fq(Y)g(;)28 b(S)10 b Fs(\))66 b Fg(de\014ne)-8 b(d)69 b(in)81 b Fs(\(2.27\))o Fg(\),)70 b(with)e Fq(Y)99 b Fs(=)62 b(8)p Fq(S)53 b Fs(+)43 b(5)68 b Fg(as)g(the)g(b)-8 b(est)68 b(choic)-8 b(e.)102 b(By)68 b(c)-8 b(omputing)68 b Fq(p)6661 5861 y Fn(0)758 6035 y Fg(for)59 b(di\013er)-8 b(ent)61 b(values)f(of)g Fq(S)69 b Fg(\()p Fq(S)56 b Fs(=)47 b(2)p Fq(;)28 b Fs(3)p Fq(;)g Fo(\001)g(\001)g(\001)f Fg(\),)60 b(one)h(c)-8 b(an)59 b(che)-8 b(ck)61 b(\(by)f(numeric)-8 b(al)61 b(c)-8 b(omputation\))758 6234 y(that)59 b(the)h(optimal)f(pr)-8 b(ob)g(abilities)60 b(ar)-8 b(e:)1003 6470 y Fo(\017)83 b Fq(d)47 b Fs(=)f(1)p Fg(;)59 b Fq(p)1754 6495 y Fn(0)1874 6470 y Fs(=)47 b(\(24e\))2420 6410 y Fk(\000)p Fn(1)2643 6470 y Fo(\031)f Fs(0)p Fq(:)p Fs(015)60 b Fg(for)f Fq(S)d Fs(=)46 b(+)p Fo(1)p Fg(,)59 b Fq(p)4343 6495 y Fn(0)4464 6470 y Fo(\031)46 b Fs(0)p Fq(:)p Fs(013)59 b Fg(for)h Fq(S)55 b Fs(=)46 b(13)p Fg(.)1003 6669 y Fo(\017)83 b Fq(d)47 b Fs(=)f(2)p Fg(;)59 b Fq(p)1754 6694 y Fn(0)1874 6669 y Fo(\031)47 b Fs(2)p Fq(:)p Fs(9)37 b Fo(\001)g Fs(10)2548 6609 y Fk(\000)p Fn(5)2785 6669 y Fg(for)60 b Fq(S)55 b Fs(=)46 b(7)p Fg(,)1003 6869 y Fo(\017)83 b Fq(d)47 b Fs(=)f(3)p Fg(;)59 b Fq(p)1754 6894 y Fn(0)1874 6869 y Fo(\031)47 b Fs(7)p Fq(:)p Fs(9)37 b Fo(\001)g Fs(10)2548 6808 y Fk(\000)p Fn(8)2785 6869 y Fg(for)60 b Fq(S)55 b Fs(=)46 b(6)p Fg(.)758 7104 y(These)64 b(pr)-8 b(ob)g(abilities)65 b(ar)-8 b(e)64 b(far)g(fr)-8 b(om)64 b(b)-8 b(eing)65 b(optimal,)g(but)g(one)g(c)-8 b(an)65 b(c)-8 b(omp)g(ar)g(e)63 b(them)i(to)f Fq(p)6477 7129 y Fn(0)6606 7104 y Fs(=)879 7238 y Fn(1)p 778 7265 V 778 7363 a(841)976 7330 y Fi(d)1125 7303 y Fg(given)d(in)f Fs([GK1,)c(Theorem)f(3.4])p Fg(.)2298 7651 y Fs(3.)92 b Fr(Localiza)-11 b(tion)63 b(a)-11 b(t)63 b(lar)l(ge)g(disorder)957 7950 y Fs(In)38 b(this)g(section)g(w)-5 b(e)38 b(fo)5 b(cus)37 b(our)h(atten)-5 b(tion)38 b(on)g(the)f (simplest)i(con)-5 b(tin)g(uous)40 b(Anderson)e(mo)5 b(del,)2434 8222 y Fq(H)2572 8247 y Fp(\025;!)2832 8222 y Fs(=)46 b Fo(\000)p Fs(\001)38 b(+)f Fq(\025V)3672 8247 y Fp(!)3906 8222 y Fs(on)139 b(L)4324 8153 y Fn(2)4398 8222 y Fs(\()p Fm(R)4573 8153 y Fp(d)4660 8222 y Fq(;)28 b Fs(d)p Fq(x)p Fs(\))p Fq(;)1362 b Fs(\(3.1\))758 8495 y(where)55 b Fq(\025)46 b(>)g Fs(0)55 b(is)h(the)f(disorder)i (parameter)e(and)h(the)f(random)h(p)5 b(oten)-5 b(tial)55 b Fq(V)5596 8520 y Fp(!)5747 8495 y Fs(is)h(of)g(the)f(form)2852 8791 y Fq(V)2949 8816 y Fp(!)3045 8791 y Fs(\()p Fq(x)p Fs(\))45 b(=)3512 8633 y Fl(X)3490 9001 y Fp(i)p Fk(2)p Fj(Z)3711 8968 y Fi(d)3802 8791 y Fq(!)3905 8816 y Fp(i)3960 8791 y Fq(u)p Fs(\()p Fq(x)37 b Fo(\000)g Fq(i)p Fs(\))28 b Fq(;)1780 b Fs(\(3.2\))758 9240 y(with)50 b Fq(u)p Fs(\()p Fq(x)p Fs(\))45 b Fo(\025)i Fs(0)k(a)f(b)5 b(ounded)52 b(measurable)g(function)e(with)h(compact)f(supp)5 b(ort,)53 b(sa)-5 b(y)51 b(supp)29 b Fq(u)46 b Fo(\032)758 9439 y Fs(\003)873 9464 y Fp(\045)950 9439 y Fs(\(0\),)77 b(suc)-5 b(h)73 b(that)g(0)i Fq(<)g(U)2530 9464 y Fk(\000)2717 9439 y Fo(\024)2922 9315 y Fl(P)3097 9489 y Fp(i)p Fk(2)p Fj(Z)3318 9456 y Fi(d)3417 9439 y Fq(u)p Fs(\()p Fq(x)48 b Fo(\000)g Fq(i)p Fs(\),)77 b(and)c Fq(!)82 b Fs(=)75 b Fo(f)p Fq(!)5057 9464 y Fp(i)5113 9439 y Fs(;)100 b Fq(i)75 b Fo(2)g Fm(Z)5679 9379 y Fp(d)5765 9439 y Fo(g)e Fs(a)g(family)f(of)758 9639 y(indep)5 b(enden)-5 b(t,)57 b(iden)-5 b(tically)56 b(distributed)i(random)f(v)-9 b(ariables)56 b(taking)g(v)-9 b(alues)57 b(in)g(the)f(in)-5 b(terv)c(al)758 9838 y([0)p Fq(;)28 b Fs(1],)48 b(whose)e(common)g (probabilit)-5 b(y)47 b(distribution)g Fq(\026)e Fs(has)i(a)f(b)5 b(ounded)46 b(densit)-5 b(y)46 b Fq(g)52 b Fs(with)46 b Fq(g)52 b(>)46 b Fs(0)758 10037 y(a.e.)111 b(in)69 b([0)p Fq(;)28 b(a)p Fs(\))67 b(for)h(some)h Fq(a)e(>)g Fs(0.)112 b(Note)67 b(w)-5 b(e)68 b(ha)-5 b(v)g(e)68 b(\006)4222 10062 y Fp(\025)4376 10037 y Fs(=)g([0)p Fq(;)28 b Fo(1)p Fs(\),)71 b(where)c(\006)5736 10062 y Fp(\025)5891 10037 y Fs(denotes)i(the)758 10237 y(almost-sure)57 b(sp)5 b(ectrum)55 b(of)g Fq(H)2698 10262 y Fp(\025;!)2968 10237 y Fs([KM,)h(Theorem)f(3].)957 10436 y(The)f(\014nite)f(v)-5 b(olume)54 b(op)5 b(erators)54 b Fq(H)3161 10461 y Fp(\025;!)t(;x;L) 3674 10436 y Fs(are)g(tak)-5 b(en)53 b(as)h(the)f(restriction)h(of)f Fq(H)6005 10461 y Fp(\025;!)6220 10436 y Fs(,)h(either)758 10635 y(to)73 b(the)g(op)5 b(en)74 b(b)5 b(o)-5 b(x)74 b(\003)2155 10660 y Fp(L)2254 10635 y Fs(\()p Fq(x)p Fs(\))f(with)h(Diric)-5 b(hlet)74 b(b)5 b(oundary)74 b(condition,)79 b(or)73 b(to)h(the)f(closed)h(b)5 b(o)-5 b(x)p 758 10706 116 7 v 758 10840 a(\003)873 10865 y Fp(L)972 10840 y Fs(\()p Fq(x)p Fs(\))55 b(with)g(p)5 b(erio)g(dic)55 b(b)5 b(oundary)56 b(condition.)p eop %%Page: 10 10 10 9 bop 758 861 a Fn(10)1485 b(FRANC)-77 b(\030)17 b(OIS)59 b(GERMINET)g(AND)f(ABEL)g(KLEIN)957 1259 y Fs(F)-14 b(or)55 b(eac)-5 b(h)55 b Fq(\025)46 b(>)h Fs(0,)55 b Fq(H)2268 1284 y Fp(\025;!)2537 1259 y Fs(is)g(kno)-5 b(wn)56 b(to)e(b)5 b(e)54 b(a)h(random)h(op)5 b(erator)54 b(suc)-5 b(h)56 b(that)e(Assumptions)758 1458 y(SLI,)63 b(EDI,)g(IAD,)g(NE,)g(W,)g(and) h(SGEE)f(hold)h(in)g(an)-5 b(y)64 b(op)5 b(en)63 b(in)-5 b(terv)c(al)63 b(\(e.g.,)i([CH1)o(,)f(Ki,)f(St,)758 1658 y(GK3]\).)70 b(In)43 b(particular,)j([GK3)q(,)g(Theorem)d(A.1])g(pro)-5 b(v)g(es)44 b(Asumption)h(SLI)e(with)g(the)g(constan)-5 b(t)758 1857 y Fq(\015)844 1882 y Fk(I)990 1857 y Fs(in)56 b(\(2.7\))e(giv)-5 b(en)56 b(b)-5 b(y)1620 2157 y Fq(\015)1706 2182 y Fp(\025;)p Fk(I)1961 2157 y Fs(=)59 b(sup)2136 2297 y Fp(E)6 b Fk(2I)2440 2157 y Fq(\015)2526 2182 y Fp(\025;E)2803 2157 y Fq(;)120 b Fs(with)138 b Fq(\015)3516 2182 y Fp(\025;E)3792 2157 y Fs(=)47 b(6)4051 2008 y Fo(p)p 4189 2008 170 7 v 149 x Fs(2)p Fq(d)4386 2007 y Fl(p)p 4552 2007 1220 7 v 150 x Fs(max)p Fo(f)p Fq(E)10 b(;)28 b Fs(0)p Fo(g)36 b Fs(+)h(50)p Fq(d)56 b(:)521 b Fs(\(3.3\))758 2542 y(The)53 b(W)-14 b(egner)52 b(estimate)h(\(2.9\)) f(can)h(b)5 b(e)53 b(deriv)-5 b(ed)53 b(as)h(in)f([Ki,)h(Prop)5 b(osition)54 b(1])f(or)g([FK1,)h(Theo-)758 2741 y(rem)h(2.3])h(with)f Fq(b)46 b Fs(=)h(2,)55 b Fq(\021)2341 2766 y Fk(I)2479 2741 y Fs(=)46 b(1,)56 b(and)1219 3067 y Fq(Q)1350 3092 y Fp(\025;)p Fk(I)1606 3067 y Fs(=)j(sup)1781 3207 y Fp(E)6 b Fk(2I)2085 3067 y Fq(Q)2216 3092 y Fp(\025;E)2493 3067 y Fq(;)120 b Fs(with)138 b Fq(Q)3251 3092 y Fp(\025;E)3528 3067 y Fs(=)3723 2958 y Fp(C)3818 2976 y Fi(d;U)3995 3003 y Ff(\000)p 3723 3029 380 7 v 3873 3124 a Fp(\025)4122 3067 y Fo(k)p Fq(g)6 b Fo(k)4373 3092 y Fk(1)4542 3067 y Fs(\(max)p Fo(f)p Fq(E)46 b Fs(+)37 b(1)p Fq(;)28 b Fs(0)p Fo(g)p Fs(\))5741 2939 y Fi(d)p 5741 2957 62 6 v 5743 3024 a Fe(2)5858 3067 y Fq(;)490 b Fs(\(3.4\))758 3443 y(where)55 b Fq(C)1357 3468 y Fp(d;U)1555 3485 y Ff(\000)1717 3443 y Fs(is)h(a)g(constan)-5 b(t)55 b(dep)5 b(ending)56 b(only)g(on)f Fq(d)h Fs(and)g Fq(U)4663 3468 y Fk(\000)4775 3443 y Fs(.)957 3642 y(Assumption)h(\(NE\))d(is)i (satis\014ed)g(with)f(a)h(constan)-5 b(t)1603 3965 y Fq(C)1722 3990 y Fp(\025;)p Fk(I)1976 3965 y Fs(=)60 b(sup)2152 4104 y Fp(E)6 b Fk(2I)2455 3965 y Fq(C)2574 3990 y Fp(\025;E)2851 3965 y Fq(;)120 b Fs(with)138 b Fq(C)3597 3990 y Fp(\025;E)3873 3965 y Fs(=)46 b Fq(C)4167 3990 y Fp(d;U)4365 4007 y Ff(\000)4500 3965 y Fs(\(max)p Fo(f)p Fq(E)g Fs(+)37 b(1)p Fq(;)28 b Fs(0)p Fo(g)p Fs(\))5699 3837 y Fi(d)p 5699 3855 V 5701 3921 a Fe(2)5817 3965 y Fq(;)531 b Fs(\(3.5\))957 4349 y(Note)72 b(that)h(this)h(random)g(op) 5 b(erator)73 b(satis\014es)i(the)e(conditions)h(of)f([GK3)q(],)78 b(so)c(w)-5 b(e)74 b(ha)-5 b(v)g(e)758 4549 y(\006)878 4574 y Fn(MSA)1223 4549 y Fs(=)46 b(\006)1518 4574 y Fn(SI)1648 4549 y Fs(.)957 4748 y(Our)56 b(result)f(is)758 5052 y Fz(Theorem)63 b(3.1.)84 b Fg(L)-8 b(et)59 b Fq(H)2373 5077 y Fp(\025;!)2647 5052 y Fg(b)-8 b(e)60 b(the)f(A)l(nderson)i (Hamiltonian)f(as)f(in)73 b Fs(\(3.1\))o Fg(,)59 b(and)h(set)2077 5358 y Fq(L)2190 5383 y Fn(0)2310 5358 y Fs(=)47 b(min)p Fo(f)p Fq(L)g Fo(2)e Fs(6)p Fm(N)26 b Fs(;)83 b Fq(L)46 b Fo(\025)h Fs(max)p Fo(f)p Fs(3)p Fq(\045;)28 b Fs(3\(2)36 b(+)4887 5208 y Fo(p)p 5025 5208 87 7 v 150 x Fq(d)q Fs(\))p Fo(gg)28 b Fq(:)977 b Fs(\(3.6\))758 5646 y Fg(Then)69 b(ther)-8 b(e)70 b(exists)f Fq(\025)2185 5586 y Fk(\003)2261 5646 y Fg(,)j(dep)-8 b(ending)70 b(only)f(on)h Fq(d)p Fg(,)i Fq(U)4106 5671 y Fk(\000)4218 5646 y Fg(,)g Fo(k)p Fq(g)6 b Fo(k)4592 5671 y Fk(1)4733 5646 y Fg(,)71 b(and)e Fq(\045)g Fg(,)j(such)d(that)g(for)g(any)758 5845 y Fq(\025)46 b Fo(\025)g Fq(\025)1173 5785 y Fk(\003)1309 5845 y Fg(we)60 b(have)2310 6125 y Fs([0)p Fq(;)28 b(c\025)p Fs(])46 b Fo(\032)g Fs(\006)3069 6150 y Fp(M)12 b(S)6 b(A)3496 6125 y Fg(with)93 b Fq(c)46 b Fs(=)4536 6043 y Fp(U)4626 6060 y Ff(\000)p 4203 6086 855 7 v 4203 6184 a Fn(\(368)p Fp(L)4544 6201 y Fe(0)4610 6184 y Fn(\))4662 6151 y Fi(d)4731 6184 y Fk(k)p Fp(g)t Fk(k)4936 6201 y Ff(1)5137 6125 y Fq(:)1211 b Fs(\(3.7\))758 6433 y Fg(Mor)-8 b(e)g(over)53 b(ther)-8 b(e)55 b(exists)f Fq(c)2412 6373 y Fk(0)2505 6433 y Fq(>)46 b Fs(0)p Fg(,)55 b(dep)-8 b(ending)55 b(on)f Fq(d)p Fg(,)h Fq(U)4173 6458 y Fk(\000)4286 6433 y Fg(,)g Fo(k)p Fq(g)6 b Fo(k)4643 6458 y Fk(1)4784 6433 y Fg(,)55 b(and)f Fq(\045)p Fg(,)h(but)f(not)h(on)f Fq(\025)46 b Fo(\025)g Fq(\025)6608 6373 y Fk(\003)6684 6433 y Fg(,)758 6632 y(such)72 b(that)g(if)g Fq(\025)d Fo(\025)g Fq(\025)2135 6572 y Fk(\003)2211 6632 y Fg(,)75 b(then)e(for)f(a.e.)114 b Fq(!)6 b Fg(,)75 b(if)d Fq(')3892 6657 y Fp(\025;!)t(;E)4321 6632 y Fg(is)g(an)g(eigenfunction)j(of)d Fq(H)6148 6657 y Fp(\025;!)6434 6632 y Fg(with)758 6832 y(eigenvalue)62 b Fq(E)55 b Fo(2)46 b Fs([0)p Fq(;)28 b(c\025)p Fs(])p Fg(,)59 b(then)2636 7206 y Fs(lim)28 b(inf)2656 7320 y Fk(j)p Fp(x)p Fk(j!1)3123 7206 y Fo(\000)3272 7093 y Fs(log)j Fo(k)p Fq(\037)3702 7118 y Fp(x)3785 7093 y Fq(')3894 7118 y Fp(!)t(;E)4133 7093 y Fo(k)p 3272 7167 944 7 v 3651 7319 a(j)p Fq(x)p Fo(j)4282 7206 y(\025)47 b Fq(c)4530 7137 y Fk(0)4576 7056 y Fo(p)p 4714 7056 97 7 v 150 x Fq(\025:)1537 b Fs(\(3.8\))758 7603 y Fg(Pr)-8 b(o)g(of.)81 b Fs(F)-14 b(or)56 b(an)-5 b(y)56 b Fq(\016)c(>)46 b Fs(0)55 b(and)h Fq(L)2765 7628 y Fn(0)2886 7603 y Fo(2)46 b Fs(2)p Fm(N)90 b Fs(w)-5 b(e)55 b(ha)-5 b(v)g(e)1122 7881 y Fm(P)34 b Fo(f)8 b Fq(!)1437 7906 y Fp(i)1539 7881 y Fq(>)46 b(\016)53 b Fs(for)i(all)46 b Fq(i)g Fo(2)g Fs(\003)2691 7906 y Fp(L)2782 7923 y Fe(0)2856 7881 y Fs(\(0\))p Fo(g)165 b Fs(=)h(1)37 b Fo(\000)g Fm(P)d Fo(f)8 b Fq(!)4213 7906 y Fp(i)4315 7881 y Fo(2)46 b Fs([0)p Fq(;)28 b(\016)6 b Fs(])46 b(for)55 b(some)47 b Fq(i)f Fo(2)g Fs(\003)5883 7906 y Fp(L)5974 7923 y Fe(0)6048 7881 y Fs(\(0\))p Fo(g)3317 8138 y(\025)166 b Fs(1)37 b Fo(\000)g Fq(\016)6 b Fo(k)p Fq(g)g Fo(k)4229 8163 y Fk(1)4370 8138 y Fq(L)4483 8070 y Fp(d)4483 8179 y Fn(0)4561 8138 y Fq(:)1787 b Fs(\(3.9\))758 8416 y(W)-14 b(e)55 b(\014x)g Fq(\016)61 b Fs(and)56 b Fq(L)1850 8441 y Fn(0)1925 8416 y Fs(,)f(and)h(set)3240 8694 y Fq(E)3363 8719 y Fp(\025)3496 8694 y Fs(=)3691 8629 y Fn(1)p 3691 8656 67 7 v 3691 8752 a(2)3777 8694 y Fq(\016)6 b(U)3970 8719 y Fk(\000)4082 8694 y Fq(\025)28 b(:)2058 b Fs(\(3.10\))758 8973 y(It)55 b(follo)-5 b(ws)56 b(that)2871 9251 y Fm(P)6 b Fo(f)p Fq(A)3172 9276 y Fp(\025)3267 9251 y Fo(g)47 b(\025)f Fs(1)37 b Fo(\000)g Fq(\016)6 b Fo(k)p Fq(g)g Fo(k)4189 9276 y Fk(1)4330 9251 y Fq(L)4443 9183 y Fp(d)4443 9292 y Fn(0)4548 9251 y Fq(;)1717 b Fs(\(3.11\))758 9530 y(where)55 b Fq(A)1363 9555 y Fp(\025)1505 9530 y Fs(denotes)g(the)g (ev)-5 b(en)g(t)2584 9808 y Fq(A)2709 9833 y Fp(\025;L)2972 9808 y Fs(=)47 b Fo(f)p Fs(inf)40 b Fq(\033)6 b Fs(\()p Fq(H)3764 9833 y Fp(\025;!)t(;L)4100 9850 y Fe(0)4173 9808 y Fs(\))46 b Fo(\025)g Fs(2)p Fq(E)4665 9833 y Fp(\025)4752 9808 y Fo(g)28 b Fq(:)1402 b Fs(\(3.12\))957 10087 y(W)-14 b(e)59 b(no)-5 b(w)61 b(use)f(Theorem)g(2.4.)87 b(T)-14 b(o)60 b(maximize)g Fq(\016)6 b Fs(,)61 b(and)f(hence)g Fq(E)5039 10112 y Fp(\025)5125 10087 y Fs(,)h(w)-5 b(e)60 b(need)g(to)f(minimize)758 10286 y Fq(L)871 10311 y Fn(0)945 10286 y Fs(;)e(hence)g(w)-5 b(e)56 b(pic)-5 b(k)57 b Fq(L)2221 10311 y Fn(0)2352 10286 y Fs(as)g(in)g(\(3.6\))f(and)h(c)-5 b(ho)5 b(ose)57 b Fq(\016)62 b Fs(b)-5 b(y)57 b(matc)-5 b(hing)57 b(the)g(righ)-5 b(t)57 b(hand)g(sides)g(of)758 10485 y(\(3.11\))d(and)i(\(2.17\),)f(i.e.,)3114 10772 y Fq(\016)6 b Fo(k)p Fq(g)g Fo(k)3445 10797 y Fk(1)3586 10772 y Fq(L)3699 10703 y Fp(d)3699 10813 y Fn(0)3822 10772 y Fs(=)4118 10706 y Fn(2)p 4017 10733 269 7 v 4017 10831 a(368)4215 10798 y Fi(d)4333 10772 y Fq(:)1932 b Fs(\(3.13\))p eop %%Page: 11 11 11 10 bop 1773 861 a Fn(EXPLICIT)58 b(FINITE)g(V)l(OLUME)h(CRITERIA)g (F)l(OR)g(LOCALIZA)-11 b(TION)883 b(11)758 1259 y Fs(If)55 b Fq(!)d Fo(2)46 b Fq(A)1361 1284 y Fp(\025)1448 1259 y Fs(,)55 b(it)g(follo)-5 b(ws)56 b(that)2109 1614 y(dist)q(\()p Fq(E)10 b(;)28 b(\033)6 b Fs(\()p Fq(H)2954 1639 y Fp(\025;!)t(;L)3296 1614 y Fs(\)\))46 b Fo(\025)g Fq(E)3770 1639 y Fp(\025)3949 1614 y Fs(fo)55 b(an)-5 b(y)92 b Fq(E)56 b Fo(2)46 b Fs([0)p Fq(;)28 b(E)5150 1639 y Fp(\025)5236 1614 y Fs(])g Fq(;)955 b Fs(\(3.14\))758 1969 y(so)64 b(w)-5 b(e)65 b(can)f(use)g(the)g(Com)-5 b(b)5 b(es-Thomas)66 b(estimate)e(to)g(get)g (the)f(deca)-5 b(y)64 b(of)g(the)g(resolv)-5 b(en)g(t)64 b(for)758 2168 y(an)-5 b(y)59 b Fq(E)h Fo(2)51 b Fs([0)p Fq(;)28 b(E)1746 2193 y Fp(\015)1831 2168 y Fs(].)84 b(Note)57 b(that)h(the)h(exact)e(dep)5 b(endency)58 b(of)h(the)f(exp)5 b(onen)-5 b(tial)58 b(deca)-5 b(y)59 b(rate)f(in)758 2367 y(the)46 b(Com)-5 b(b)5 b(es-Thomas)50 b(estimate)d(in)g(terms)h (of)e(the)h(energy)g(parameter)g(and)g(the)g(distance)h(to)758 2567 y(the)f(sp)5 b(ectrum)48 b(is)h(crucial)f(in)g(our)g(argumen)-5 b(t,)50 b(since)e(w)-5 b(e)48 b(deal)g(with)g(large)g(energies)g(and)g (large)758 2766 y(distances)f(from)f(the)g(sp)5 b(ectrum.)71 b(Suc)-5 b(h)48 b(a)e(precise)h(estimate)f(is)h(pro)-5 b(vided)47 b(in)g([GK2,)h(Eq.)71 b(\(19\))758 2965 y(in)d(Theorem)h (1].)112 b(Although)68 b(these)g(estimates)h(are)f(for)g(op)5 b(erators)68 b(on)g Fm(R)5553 2905 y Fp(d)5640 2965 y Fs(,)j(the)d(results)h(in)758 3164 y([GK2])56 b(can)g(adapted)h(for)e (\014nite)h(v)-5 b(olume.)77 b(In)56 b(a)g(b)5 b(o)-5 b(x)56 b(\003)4290 3189 y Fp(L)4381 3206 y Fe(0)4454 3164 y Fs(\(0\),)g(with)g(either)f(Diric)-5 b(hlet)56 b(or)g(p)5 b(e-)758 3364 y(rio)g(dic)53 b(b)5 b(oundary)53 b(condition,)h(the)f(same)h(estimates)f(as)h(in)f([GK2)q(,)g(Eq.)73 b(\(19\))52 b(in)i(Theorem)f(1])758 3563 y(hold)g(for)g Fq(x;)28 b(y)52 b Fo(2)1851 3521 y Fs(\026)1835 3563 y(\003)1950 3588 y Fp(L)2041 3605 y Fe(0)2106 3588 y Fk(\000)p Fn(2)2284 3563 y Fs(\(0\))g(\()h(i.e.,)g(at)g(a)g(distance)g Fo(\025)46 b Fs(1)53 b(from)g(the)f(b)5 b(oundary\))53 b(but)g(the)g(exp)5 b(o-)758 3776 y(nen)-5 b(tial)48 b(rates)g(of)g(deca)-5 b(y)48 b(get)f(divided)h(b)-5 b(y)49 b(1)22 b(+)g(2)3745 3634 y Fo(p)p 3884 3634 87 7 v 3884 3776 a Fq(dL)4083 3801 y Fn(0)4157 3776 y Fs(.)72 b(This)49 b(can)f(b)5 b(e)47 b(seen)i(as)f(follo)-5 b(ws:)71 b(in)49 b(the)758 3975 y(pro)5 b(of)50 b(of)g([GK2)q(,)h(Theorem)g(1])g (w)-5 b(e)50 b(replace)h(e)3606 3915 y Fk(\006)p Fp(\013)p Fk(\001)p Fp(x)3969 3975 y Fs(b)-5 b(y)51 b(e)4269 3915 y Fk(\006)p Fn(\()p Fp(\013)p Fk(\001)p Fp(x)p Fn(\))p Fp(')p Fn(\()p Fp(x)p Fn(\))4952 3975 y Fs(,)h(where)e Fq(')p Fs(\()p Fq(x)p Fs(\))f(is)i(a)g(smo)5 b(oth)758 4174 y(function)53 b(suc)-5 b(h)54 b(that)e(0)46 b Fo(\024)g Fq(')p Fs(\()p Fq(x)p Fs(\))f Fo(\024)h Fs(1,)54 b Fq(')p Fs(\()p Fq(x)p Fs(\))45 b(=)h(1)53 b(for)g Fq(x)46 b Fo(2)4432 4132 y Fs(\026)4415 4174 y(\003)4530 4199 y Fp(L)4621 4216 y Fe(0)4687 4199 y Fk(\000)p Fn(2)4865 4174 y Fs(\(0\),)53 b(supp)29 b Fq(')45 b Fo(\032)i Fs(\003)5992 4199 y Fp(L)6083 4216 y Fe(0)6156 4174 y Fs(\(0\),)53 b(and)758 4387 y Fo(jr)p Fq(')p Fs(\()p Fq(x)p Fs(\))p Fo(j)46 b(\024)g Fs(3)1626 4246 y Fo(p)p 1765 4246 V 1765 4387 a Fq(d)p Fs(.)75 b(It)55 b(follo)-5 b(ws)57 b(that)e Fo(jr)p Fs(\(\()p Fq(\013)37 b Fo(\001)h Fq(x)p Fs(\))p Fq(')p Fs(\()p Fq(x)p Fs(\)\))p Fo(j)44 b(\024)j(j)p Fq(\013)q Fo(j)p Fs(\(1)37 b(+)g(3)5057 4246 y Fo(p)p 5195 4246 V 141 x Fq(d)5302 4322 y Fp(L)p 5302 4349 92 7 v 5314 4444 a Fn(2)5413 4387 y Fs(\))46 b Fo(\024)h(j)p Fq(\013)q Fo(j)p Fs(\(1)37 b(+)g(2)6333 4246 y Fo(p)p 6471 4246 87 7 v 141 x Fq(d)q(L)p Fs(\))758 4586 y(for)51 b(all)h Fq(x)46 b Fo(2)g Fs(\003)1648 4611 y Fp(L)1739 4628 y Fe(0)1813 4586 y Fs(\(0\).)72 b(The)51 b(pro)5 b(of)52 b(then)g(pro)5 b(ceeds)52 b(as)g(in)g([GK2)q(,)g(Pro)5 b(of)52 b(of)f(Theorem)h(1])g(using)758 4786 y(this)e(estimate.)72 b(\(W)-14 b(e)48 b(can)i(do)g(b)5 b(etter)49 b(if)g(w)-5 b(e)50 b(sp)5 b(ecify)49 b(p)5 b(erio)g(dic)49 b(b)5 b(oundary)50 b(condition)g(and)h(use)758 4985 y(the)k(distance)g(on)h (the)f(torus,)h(see[FK1,)g(Lemma)g(18])f(and)h([KK1,)g(Theorem)f (3.6].\))957 5184 y(Th)-5 b(us,)80 b(using)c(the)d(adaptation)i(of)f ([GK2,)79 b(Eq.)130 b(\(19\))74 b(in)g(Theorem)h(1])f(to)g(\014nite)g (b)5 b(o)-5 b(xes)758 5416 y(with)63 b Fq(\015)68 b Fs(=)1506 5254 y Fk(p)p 1615 5254 67 6 v 96 x Fn(2)p 1506 5378 176 7 v 1561 5473 a(2)1701 5416 y Fs(,)d(w)-5 b(e)64 b(conclude)f(that,)i(if)e Fq(!)i Fo(2)59 b Fq(A)3794 5441 y Fp(\025)3943 5416 y Fs(and)64 b Fq(E)k Fo(2)59 b Fs([0)p Fq(;)28 b(E)4961 5441 y Fp(\025)5047 5416 y Fs(],)66 b(w)-5 b(e)63 b(ha)-5 b(v)g(e)64 b(that)e(for)h(an)-5 b(y)758 5630 y Fq(x;)28 b(y)52 b Fo(2)46 b Fm(Z)1318 5570 y Fp(d)1442 5630 y Fo(\\)36 b Fs(\003)1704 5655 y Fp(L)1795 5672 y Fe(0)1869 5630 y Fs(\(0\))45 b(=)h Fm(Z)2404 5570 y Fp(d)2527 5630 y Fo(\\)2691 5589 y Fs(\026)2675 5630 y(\003)2790 5655 y Fp(L)2881 5672 y Fe(0)2946 5655 y Fk(\000)p Fn(2)3124 5630 y Fs(\(0\),)1311 6042 y Fo(k)p Fq(\037)1498 6067 y Fp(x)1581 6042 y Fq(R)1707 6067 y Fp(\025;!)t(;L)2043 6084 y Fe(0)2117 6042 y Fs(\()p Fq(E)10 b Fs(\))p Fq(\037)2484 6067 y Fp(y)2562 6042 y Fo(k)166 b(\024)3318 5977 y Fn(2)p 3126 6004 450 7 v 3126 6099 a(2)p Fp(E)3289 6117 y Fi(\025)3368 6099 y Fk(\000)p Fp(E)3623 6042 y Fs(e)3697 5970 y Fk(\000)3801 5858 y Fo(p)p 3939 5858 620 7 v 112 x Fn(2\(2)p Fp(E)4220 5988 y Fi(\025)4299 5970 y Fk(\000)p Fp(E)6 b Fn(\))4581 5982 y Fs(\()4646 5970 y Fn(1+2)4880 5871 y Fk(p)p 4989 5871 70 6 v 99 x Fp(dL)5149 5987 y Fe(0)5214 5982 y Fs(\))5279 5899 y Ff(\000)p Fe(1)5433 5982 y Fs(\()5498 5970 y Fk(j)p Fp(x)p Fk(\000)p Fp(y)t Fk(j\000)5930 5871 y(p)p 6040 5871 V 6040 5970 a Fp(d)6109 5982 y Fs(\))2811 6382 y Fo(\024)3181 6316 y Fn(2)p 3126 6343 176 7 v 3126 6439 a Fp(E)3223 6457 y Fi(\025)3349 6382 y Fs(e)3423 6310 y Fk(\000)3527 6220 y(p)p 3636 6220 242 6 v 90 x Fn(2)p Fp(E)3799 6328 y Fi(\025)3900 6322 y Fs(\()3965 6310 y Fn(1+2)4199 6211 y Fk(p)p 4308 6211 70 6 v 99 x Fp(dL)4468 6327 y Fe(0)4533 6322 y Fs(\))4598 6239 y Ff(\000)p Fe(1)4753 6322 y Fs(\()4817 6310 y Fk(j)p Fp(x)p Fk(\000)p Fp(y)t Fk(j\000)5249 6211 y(p)p 5359 6211 V 5359 6310 a Fp(d)5429 6322 y Fs(\))5529 6382 y Fq(:)736 b Fs(\(3.15\))758 6753 y(Summing)65 b(o)-5 b(v)g(er)64 b(the)g(supp)5 b(ort)64 b(of)g(\000)3088 6778 y Fp(L)3179 6795 y Fe(0)3315 6753 y Fs(and)h(of)e Fq(\037)3948 6783 y Fp(L)4039 6800 y Fe(0)4104 6783 y Fp(=)p Fn(3)4247 6753 y Fs(,)j(noting)e(that)f(if)h Fq(x)c Fo(2)g Fs(\003)5854 6783 y Fp(L)5945 6800 y Fe(0)6010 6783 y Fp(=)p Fn(3)6153 6753 y Fs(\(0\),)65 b Fq(y)i Fo(2)758 6976 y Fs(\007)887 7001 y Fp(L)978 7018 y Fe(0)1051 6976 y Fs(\(0\))55 b(w)-5 b(e)55 b(ha)-5 b(v)g(e)56 b Fo(j)p Fq(x)37 b Fo(\000)g Fq(y)6 b Fo(j)46 b(\025)2665 6908 y Fp(L)2756 6925 y Fe(0)p 2665 6938 156 7 v 2709 7033 a Fn(3)2877 6976 y Fo(\000)37 b Fs(1,)56 b(and)g(using)i(\(3.6\))d (yields)1742 7403 y Fo(k)p Fs(\000)1929 7428 y Fp(L)2020 7445 y Fe(0)2093 7403 y Fq(R)2219 7428 y Fp(\025;!)t(;L)2555 7445 y Fe(0)2628 7403 y Fs(\()p Fq(E)10 b Fs(\))p Fq(\037)2995 7433 y Fp(L)3086 7450 y Fe(0)3149 7433 y Fp(=)p Fn(3)3292 7403 y Fo(k)46 b(\024)3636 7338 y Fn(2)p Fp(d)p 3616 7365 176 7 v 3616 7461 a(E)3713 7479 y Fi(\025)3839 7403 y Fq(L)3952 7332 y Fn(2)p Fp(d)p Fk(\000)p Fn(1)3952 7448 y(0)4266 7403 y Fs(e)4340 7332 y Fk(\000)4444 7242 y(p)p 4553 7242 242 6 v 90 x Fn(2)p Fp(E)4716 7350 y Fi(\025)4817 7344 y Fs(\()4882 7332 y Fn(1+2)5116 7232 y Fk(p)p 5225 7232 70 6 v 100 x Fp(dL)5385 7349 y Fe(0)5450 7344 y Fs(\))5515 7261 y Ff(\000)p Fe(1)5705 7403 y Fq(:)560 b Fs(\(3.16\))957 7775 y(W)-14 b(e)56 b(will)h(sho)-5 b(w)58 b(that)f(\(2.17\))f(of)g(Theorem)h(2.4)g(is)h(satis\014ed)f(for) g(all)g Fq(E)h Fo(2)48 b Fs([0)p Fq(;)28 b(E)5939 7800 y Fp(\025)6026 7775 y Fs(])57 b(for)f(large)758 7974 y(disorder.)89 b(Let)59 b Fo(I)1867 7999 y Fp(\025)2008 7974 y Fs(=)54 b(\()p Fo(\0001)p Fq(;)28 b Fs(2)p Fq(E)2831 7999 y Fp(\025)2958 7974 y Fo(\000)40 b Fs(1\),)61 b(w)-5 b(e)60 b(ha)-5 b(v)g(e,)62 b(using)h(\(2.18\))o(,)f(\(3.3\))o(,)f (\(3.4\),)g(\(3.10\))o(,)h(and)758 8173 y(\(3.16\))o(,)1211 8610 y Fq(D)1348 8635 y Fk(I)1423 8653 y Fi(\025)1510 8610 y Fq(L)1623 8511 y Fs(\()1707 8454 y Fe(5)p 1707 8472 57 6 v 1707 8539 a(3)1784 8499 y Fn(+)p Fp(b)1944 8511 y Fs(\))2008 8499 y Fp(d)1623 8654 y Fn(0)2086 8610 y Fo(k)p Fs(\000)2273 8635 y Fp(L)2364 8652 y Fe(0)2437 8610 y Fq(R)2563 8635 y Fp(\025;!)t(;L)2899 8652 y Fe(0)2973 8610 y Fs(\()p Fq(E)10 b Fs(\))p Fq(\037)3340 8640 y Fp(L)3431 8657 y Fe(0)3494 8640 y Fp(=)p Fn(3)3636 8610 y Fo(k)2592 b Fs(\(3.17\))1849 8931 y(=)166 b(39)2310 8862 y Fn(5)p Fp(d)2482 8931 y Fs(max)2818 8796 y Fl(\010)2915 8931 y Fs(16)37 b Fo(\001)g Fs(60)3367 8862 y Fp(d)3445 8931 y Fq(Q)3576 8956 y Fk(I)3651 8974 y Fi(\025)3737 8931 y Fq(;)28 b Fs(1)3894 8796 y Fl(\011)4019 8931 y Fq(\015)4114 8862 y Fn(2)4105 8972 y Fk(I)4180 8990 y Fi(\025)4266 8931 y Fq(L)4399 8789 y Fe(11)p 4399 8807 114 6 v 4427 8873 a(3)4532 8833 y Fp(d)4379 8975 y Fn(0)4610 8931 y Fo(k)p Fs(\000)4797 8956 y Fp(L)4888 8973 y Fe(0)4961 8931 y Fq(R)5087 8956 y Fp(\025;!)t(;;L)5462 8973 y Fe(0)5536 8931 y Fs(\()p Fq(E)10 b Fs(\))p Fq(\037)5903 8961 y Fp(L)5994 8978 y Fe(0)6057 8961 y Fp(=)p Fn(3)6199 8931 y Fo(k)1849 9240 y(\024)166 b Fq(c)2216 9265 y Fn(1)2291 9240 y Fq(\025)2388 9168 y Fn(max)2633 9181 y Fo(f)2735 9123 y Fi(d)p 2735 9141 62 6 v 2737 9208 a Fe(2)2817 9168 y Fk(\000)p Fn(1)p Fp(;)p Fn(0)3092 9180 y Fo(g)3184 9240 y Fs(e)3258 9171 y Fk(\000)p Fp(c)3421 9188 y Fe(2)3485 9072 y Fk(p)p 3595 9072 79 6 v 3595 9171 a Fp(\025)6311 9240 y Fs(\(3.18\))1849 9489 y Fq(<)g Fs(1)28 b Fq(;)4010 b Fs(\(3.19\))758 9844 y(where)58 b(\(3.18\))g(holds)i(for)e Fq(!)g Fo(2)52 b Fq(A)2868 9869 y Fp(\025)3013 9844 y Fs(with)58 b Fq(c)3466 9869 y Fn(1)3599 9844 y Fs(and)h Fq(c)3997 9869 y Fn(2)4130 9844 y Fs(constan)-5 b(ts)60 b(dep)5 b(ending)59 b(only)g(on)g Fq(d)p Fs(,)h Fq(U)6577 9869 y Fk(\000)6689 9844 y Fs(,)758 10043 y Fo(k)p Fq(g)6 b Fo(k)1009 10068 y Fk(1)1150 10043 y Fs(,)61 b(and)g Fq(\045)f Fs(\(note)g(that)f(w)-5 b(e)61 b Fg(\014xe)-8 b(d)77 b Fq(L)3297 10068 y Fn(0)3432 10043 y Fs(and)61 b Fq(\016)66 b Fs(in)61 b(\(3.6\))f(and)h(\(3.13\))o(\).)88 b(W)-14 b(e)60 b(conclude)h(that)758 10242 y(there)h(exists)h Fq(\025)1753 10182 y Fk(\003)1829 10242 y Fs(,)i(dep)5 b(ending)64 b(only)e(on)i Fq(d)p Fs(,)h Fq(U)3666 10267 y Fk(\000)3778 10242 y Fs(,)g Fo(k)p Fq(g)6 b Fo(k)4140 10267 y Fk(1)4281 10242 y Fs(,)65 b(and)e Fq(\045)p Fs(,)i(suc)-5 b(h)64 b(that)f(w)-5 b(e)63 b(ha)-5 b(v)g(e)63 b(\(3.19\))758 10441 y(for)55 b(all)g Fq(\025)46 b(>)h(\025)1658 10381 y Fk(\003)1789 10441 y Fs(and)56 b Fq(!)d Fo(2)45 b Fq(A)2549 10466 y Fp(\025)2636 10441 y Fs(.)957 10641 y(Th)-5 b(us)62 b(condition)f(\(2.17\))f(holds)i(for)f Fq(\025)55 b(>)g(\025)3735 10580 y Fk(\003)3872 10641 y Fs(and)62 b Fq(E)i Fo(2)55 b Fs([0)p Fq(;)28 b(E)4880 10666 y Fp(\025)4966 10641 y Fs(])61 b(b)-5 b(y)64 b(\(3.11\))o(,)e(hence)f(Theo-)758 10840 y(rem)55 b(2.4)h(implies)g(that)f([0)p Fq(;)2504 10774 y Fn(1)p 2504 10801 67 7 v 2504 10897 a(2)2590 10840 y Fq(\016)6 b(U)2783 10865 y Fk(\000)2896 10840 y Fq(\025)p Fs(])46 b Fo(\032)g Fs(\006)3380 10865 y Fn(MSA)3679 10840 y Fs(.)p eop %%Page: 12 12 12 11 bop 758 861 a Fn(12)1485 b(FRANC)-77 b(\030)17 b(OIS)59 b(GERMINET)g(AND)f(ABEL)g(KLEIN)957 1259 y Fs(Let)82 b(us)i(turn)f(to)f(the)h(rate)f(of)h(the)f(exp)5 b(onen)-5 b(tial)83 b(deca)-5 b(y)83 b(of)f(the)h(eigenfunctions)g(with)758 1458 y(energies)55 b(in)h([0)p Fq(;)1803 1393 y Fn(1)p 1803 1420 67 7 v 1803 1516 a(2)1889 1458 y Fq(\016)6 b(U)2082 1483 y Fk(\000)2195 1458 y Fq(\025)p Fs(].)73 b(Let)55 b Fq(\022)2833 1483 y Fp(\025)2975 1458 y Fs(b)5 b(e)55 b(de\014ned)h(b)-5 b(y)2375 1830 y Fq(L)2488 1755 y Fk(\000)p Fp(\022)2655 1773 y Fi(\025)2488 1874 y Fn(0)2788 1830 y Fs(=)3003 1765 y Fn(2)p Fp(d)p 2983 1792 176 7 v 2983 1887 a(E)3080 1905 y Fi(\025)3206 1830 y Fq(L)3319 1759 y Fn(2)p Fp(d)p Fk(\000)p Fn(1)3319 1874 y(0)3632 1830 y Fs(e)3706 1758 y Fk(\000)3810 1669 y(p)p 3919 1669 242 6 v 89 x Fn(2)p Fp(E)4082 1776 y Fi(\025)4184 1770 y Fs(\()4248 1758 y Fn(1+2)4482 1659 y Fk(p)p 4591 1659 70 6 v 99 x Fp(d)q(L)4752 1775 y Fe(0)4816 1770 y Fs(\))4881 1687 y Ff(\000)p Fe(1)5072 1830 y Fq(:)1193 b Fs(\(3.20\))758 2146 y(Then)70 b(\(3.16\))g(sa)-5 b(ys)71 b(\003)2183 2171 y Fp(L)2274 2188 y Fe(0)2348 2146 y Fs(\(0\))e(is)i(\()p Fq(\022)2955 2171 y Fp(\025)3042 2146 y Fq(;)28 b(E)10 b Fs(\)-suitable)69 b(for)h(an)-5 b(y)71 b Fq(E)80 b Fo(2)71 b Fs([0)p Fq(;)28 b(E)5310 2171 y Fp(\025)5396 2146 y Fs(].)119 b(Note)69 b(that)h(from)758 2345 y(\(3.17\))61 b(and)i(\(2.18\))f(\(see)g(\(5.4\)\))g(one)g(c)-5 b(hec)g(ks)63 b(that)f Fq(\022)4146 2370 y Fp(\025)4291 2345 y Fq(>)c(\022)4556 2370 y Fn(0)p Fp(;\025)4748 2345 y Fs(,)65 b(where)d Fq(\022)5424 2370 y Fn(0)p Fp(;\025)5679 2345 y Fs(is)h(as)g(in)g(\(2.32\))o(.)758 2545 y(Moreo)-5 b(v)g(er,)55 b(it)h(follo)-5 b(ws)56 b(from)f(\(3.20\))g(that)2713 2865 y Fq(\022)2791 2890 y Fp(\025)2924 2865 y Fo(\025)46 b Fq(c)3171 2890 y Fn(0)3246 2716 y Fo(p)p 3384 2716 97 7 v 149 x Fq(\025)138 b Fs(for)55 b(all)139 b Fq(\025)46 b Fo(\025)g Fq(\025)4602 2797 y Fk(\003)4706 2865 y Fq(;)1559 b Fs(\(3.21\))758 3174 y(where)62 b Fq(c)1317 3199 y Fn(0)1454 3174 y Fs(is)h(some)g(constan)-5 b(t)63 b(and)g Fq(\025)3155 3114 y Fk(\003)3294 3174 y Fs(is)g(tak)-5 b(en)62 b(large)g(enough)i(\(b)5 b(oth)61 b(dep)5 b(ending)63 b(only)g(on)758 3374 y Fq(d)p Fs(,)56 b Fq(U)1059 3399 y Fk(\000)1171 3374 y Fs(,)f Fo(k)p Fq(g)6 b Fo(k)1523 3399 y Fk(1)1664 3374 y Fs(,)56 b(and)g Fq(\045)p Fs(\).)957 3573 y(As)63 b(in)h([GK1,)i(Pro)5 b(of)62 b(of)h(Theorem)h(3.4],)i(w)-5 b(e)63 b(b)5 b(o)g(otstrap)63 b(from)g(Theorem)h(2.7)f(\(of)g(whic)-5 b(h)758 3772 y(Theorem)57 b(2.4)f(is)i(a)f(sp)5 b(ecial)57 b(case\))f(to)g(Theorem)h(2.6.)78 b(Note)56 b(that)g(w)-5 b(e)57 b(ha)-5 b(v)g(e)57 b(the)g(conclusions)758 3971 y(of)78 b(Theorem)g(2.7)g(with)g Fq(p)84 b Fs(=)2786 3906 y Fn(5)p 2786 3933 67 7 v 2786 4029 a(3)2872 3971 y Fs(,)g Fq(S)93 b Fs(=)85 b(4,)e(and)c Fq(Y)121 b Fs(=)85 b(39,)f(i.e.,)g(if)78 b(w)-5 b(e)78 b(set)g Fq(L)5875 3996 y Fp(k)6041 3971 y Fs(=)84 b(39)6420 3911 y Fp(k)6502 3971 y Fq(L)6615 3996 y Fn(0)6689 3971 y Fs(,)758 4171 y Fq(k)51 b Fs(=)c(0)p Fq(;)28 b Fs(1)p Fq(;)g Fs(2)p Fq(;)g(:)g(:)g(:)f Fs(,)55 b(where)g Fq(L)2457 4196 y Fn(0)2587 4171 y Fs(is)h(as)g(in)g(\(3.6\))o(,)f(then)h(for)f(all)g Fq(\025)46 b(>)h(\025)4873 4110 y Fk(\003)5004 4171 y Fs(w)-5 b(e)56 b(ha)-5 b(v)g(e)2227 4548 y Fm(P)5 b Fo(f)p Fs(\003)2518 4573 y Fp(L)2609 4591 y Fi(k)2699 4548 y Fs(\(0\))91 b(is)56 b(\()p Fq(\022)3313 4573 y Fp(\025)3400 4548 y Fq(;)28 b(E)10 b Fs(\)-suitable)o Fo(g)46 b(\025)g Fs(1)37 b Fo(\000)4967 4436 y Fs(1)p 4899 4510 218 7 v 4899 4725 a Fq(L)5032 4583 y Fe(5)p 5032 4601 57 6 v 5032 4668 a(3)5012 4775 y Fp(k)5192 4548 y Fq(;)1073 b Fs(\(3.22\))758 5016 y(for)47 b(all)h Fq(E)56 b Fo(2)45 b Fs([0)p Fq(;)28 b(E)1888 5041 y Fp(\025)1975 5016 y Fs(])48 b(and)g Fq(k)k Fo(\025)46 b(K)r Fs(,)j(where)f Fo(K)g Fs(=)e Fo(K)r Fs(\()p Fq(d;)28 b(\045)p Fs(\))46 b Fq(<)g Fo(1)i Fs(is)g(a)g(constan)-5 b(t)48 b(dep)5 b(ending)48 b(only)758 5215 y(on)55 b Fq(d)h Fs(and)g Fq(\045)p Fs(.)74 b(A)55 b(k)-5 b(ey)54 b(fact)h(is)h(that)f Fo(K)i Fs(do)5 b(es)55 b(not)h(dep)5 b(end)55 b(on)h Fq(\025)p Fs(.)957 5415 y(W)-14 b(e)41 b(no)-5 b(w)43 b(w)-5 b(an)g(t)42 b(to)g(feed)f(\(3.22\))h(in)-5 b(to)42 b(the)g(h)-5 b(yp)5 b(otheses)42 b(of)g(Theorem)g(2.6.)69 b(W)-14 b(e)42 b(ha)-5 b(v)g(e)42 b(already)758 5614 y(\014xed)54 b Fq(p)46 b Fs(=)1483 5548 y Fn(5)p 1483 5575 67 7 v 1483 5671 a(3)1569 5614 y Fs(,)56 b(and)f(hence)g Fq(\013)h Fs(is)f(\014xed.)74 b(On)55 b(the)g(other)f(hand)i(for)f(eac) -5 b(h)55 b Fq(\025)f Fs(w)-5 b(e)55 b(ha)-5 b(v)g(e)56 b Fq(\022)51 b Fs(=)46 b Fq(\022)6445 5639 y Fp(\025)6587 5614 y Fs(as)758 5813 y(in)i(\(3.20\),)i(and)e Fq(Q)1911 5838 y Fk(I)1986 5856 y Fi(\025)2073 5813 y Fs(,)i Fq(C)2288 5838 y Fk(I)2363 5856 y Fi(\025)2449 5813 y Fs(,)g(and)f Fq(\015)2947 5838 y Fk(I)3022 5856 y Fi(\025)3157 5813 y Fs(gro)-5 b(w)48 b(p)5 b(olynomially)49 b(with)f Fq(\025)p Fs(.)71 b(T)-14 b(o)49 b(con)-5 b(trol)48 b Fo(L)6025 5838 y Fp(\025)6160 5813 y Fs(as)h(giv)-5 b(en)758 6012 y(in)51 b(\(2.21\))f(with)g(all)h(this)g(dep)5 b(endence)51 b(in)g Fq(\025)p Fs(,)g(w)-5 b(e)51 b(pic)-5 b(k)51 b(the)f(remaining)i (parameter,)g Fq(s)p Fs(,)f(to)g(also)758 6212 y(dep)5 b(end)67 b(on)h Fq(\025)f Fs(b)-5 b(y)68 b Fq(s)2074 6237 y Fp(\025)2227 6212 y Fs(=)f(log)30 b Fq(\025)p Fs(.)110 b(By)66 b(taking)i Fq(\025)3812 6151 y Fk(\003)3955 6212 y Fs(su\016cien)-5 b(tly)68 b(large,)j(as)d(b)5 b(efore)67 b(dep)5 b(ending)758 6411 y(only)65 b(on)i Fq(d)p Fs(,)i Fq(U)1688 6436 y Fk(\000)1800 6411 y Fs(,)g Fo(k)p Fq(g)6 b Fo(k)2166 6436 y Fk(1)2306 6411 y Fs(,)69 b(and)d Fq(\045)p Fs(,)j(and)d(using)j(\(3.21\),)f(w)-5 b(e)66 b(can)g(also)g(guaran)-5 b(tee)66 b Fq(\022)6037 6436 y Fp(\025)6188 6411 y Fq(>)e Fs(4)p Fq(\013)q(s)6649 6436 y Fp(\025)758 6610 y Fs(and)f Fq(s)1166 6635 y Fp(\025)1311 6610 y Fq(>)58 b Fs(2)p Fq(p)41 b Fs(+)h(\()p Fq(b)f Fs(+)h(2\))p Fq(d)63 b Fs(for)f(all)h Fq(\025)57 b(>)i(\025)3460 6550 y Fk(\003)3536 6610 y Fs(.)95 b(It)62 b(follo)-5 b(ws)63 b(from)g(the)f(explicit)g(form)h(of)75 b(\(2.21\))758 6809 y(that)2564 7108 y Fo(L)2679 7133 y Fk(1)2866 7108 y Fs(=)86 b(sup)3041 7249 y Fp(\025>\025)3303 7216 y Ff(\003)3549 7108 y Fs(sup)3399 7256 y Fp(E)6 b Fk(2)p Fn([0)p Fp(;E)3831 7274 y Fi(\025)3911 7256 y Fn(])3977 7108 y Fo(L)4092 7133 y Fp(\025)4178 7108 y Fs(\()p Fq(E)k Fs(\))45 b Fq(<)h Fo(1)28 b Fq(;)1410 b Fs(\(3.23\))758 7558 y(where)59 b Fo(L)1357 7583 y Fp(\025)1443 7558 y Fs(\()p Fq(E)10 b Fs(\))58 b(is)i(giv)-5 b(en)59 b(b)-5 b(y)62 b(\(2.21\))d(with)g Fq(E)3598 7583 y Fn(0)3725 7558 y Fs(=)53 b Fq(E)10 b Fs(,)59 b Fo(I)65 b Fs(=)53 b Fo(I)4572 7583 y Fp(\025)4660 7558 y Fs(,)60 b Fq(p)52 b Fs(=)5104 7493 y Fn(5)p 5104 7520 V 5104 7615 a(3)5190 7558 y Fs(,)61 b Fq(\022)c Fs(=)c Fq(\022)5692 7583 y Fp(\025)5779 7558 y Fs(,)60 b(and)g Fq(s)52 b Fs(=)h Fq(s)6602 7583 y Fp(\025)6689 7558 y Fs(.)758 7757 y(Note)h(that)h Fo(L)1634 7782 y Fk(1)1829 7757 y Fs(dep)5 b(ends)56 b(only)g(on)f Fq(d)p Fs(,)h Fq(U)3367 7782 y Fk(\000)3480 7757 y Fs(,)f Fo(k)p Fq(g)6 b Fo(k)3832 7782 y Fk(1)3973 7757 y Fs(,)55 b(and)h Fq(\045)p Fs(.)957 7956 y(W)-14 b(e)54 b(no)-5 b(w)54 b(\014x)g Fq(\024)g Fs(to)g(b)5 b(e)53 b(that)h(smallest)h Fq(k)d Fo(\025)46 b(K)56 b Fs(suc)-5 b(h)55 b(that)e Fq(L)4732 7981 y Fp(k)4860 7956 y Fs(=)47 b(39)5202 7896 y Fp(k)5283 7956 y Fq(L)5396 7981 y Fn(0)5517 7956 y Fo(\025)f(L)5807 7981 y Fk(1)5947 7956 y Fs(.)74 b(It)53 b(follo)-5 b(ws)758 8156 y(from)44 b(Theorem)g(2.6)g(that)f(if)h Fq(\025)i(>)g(\025)2989 8095 y Fk(\003)3066 8156 y Fs(,)g(then)e(for)g(almost)g(ev)-5 b(ery)43 b Fq(!)6 b Fs(,)47 b(the)d(eigenfunctions)g Fq(')6378 8181 y Fp(\025;!)t(;E)758 8355 y Fs(with)55 b(energy)g Fq(E)g Fo(2)46 b Fs([0)p Fq(;)28 b(E)2328 8380 y Fp(\025)2414 8355 y Fs(])56 b(deca)-5 b(y)55 b(exp)5 b(onen)-5 b(tially)55 b(\(in)g(the)g Fq(L)4658 8295 y Fn(2)4733 8355 y Fs(-sense\))h(with)2393 8750 y(lim)28 b(inf)2414 8865 y Fk(j)p Fp(x)p Fk(j!1)2881 8750 y Fo(\000)3030 8637 y Fs(log)j Fo(k)p Fq(\037)3460 8662 y Fp(x)3543 8637 y Fq(')3652 8662 y Fp(\025;!)t(;E)4009 8637 y Fo(k)p 3030 8712 1062 7 v 3468 8864 a(j)p Fq(x)p Fo(j)4158 8750 y(\025)4354 8637 y Fq(\022)4432 8662 y Fp(\025)4546 8637 y Fs(log)f Fq(L)4901 8662 y Fp(\024)p 4354 8712 634 7 v 4529 8864 a Fs(2)p Fq(L)4725 8889 y Fp(\024)5053 8750 y Fq(:)1212 b Fs(\(3.24\))758 9151 y(so)56 b(\(3.8\))e(follo)-5 b(ws)56 b(from)g(\(3.21\))o(.)p 6614 9151 7 113 v 6621 9046 100 7 v 6621 9151 V 6720 9151 7 113 v 1013 9576 a(4.)92 b Fr(Applica)-11 b(tion)62 b(to)g(Land)l(a)l(u)h(Hamil)-11 b(tonians)63 b(with)f(random)g(potentials)957 9875 y Fs(In)57 b(this)f(section)h(w)-5 b(e)56 b(study)h(a)f(t)-5 b(w)g(o-dimensional)59 b(random)f(Sc)-5 b(hr\177)-83 b(odinger)57 b(op)5 b(erator)56 b(in)h(the)758 10074 y(presence)k(of)f(a)h(constan)-5 b(t)61 b(transv)-5 b(erse)61 b(magnetic)g(\014eld,)h(the)f(Landau)g(Hamiltonian,)i(with)d(a)758 10273 y(random)c(p)5 b(oten)-5 b(tial:)2478 10772 y Fq(H)2616 10797 y Fp(B)6 b(;!)2903 10772 y Fs(=)47 b Fq(H)3217 10797 y Fp(B)3368 10772 y Fs(+)37 b Fq(V)3631 10797 y Fp(!)3865 10772 y Fs(on)139 b(L)4283 10703 y Fn(2)4357 10772 y Fs(\()p Fm(R)4533 10703 y Fn(2)4616 10772 y Fq(;)28 b Fs(d)p Fq(x)p Fs(\))p Fq(;)1406 b Fs(\(4.1\))p eop %%Page: 13 13 13 12 bop 1773 861 a Fn(EXPLICIT)58 b(FINITE)g(V)l(OLUME)h(CRITERIA)g (F)l(OR)g(LOCALIZA)-11 b(TION)883 b(13)758 1259 y Fs(with)2399 1626 y Fq(H)2537 1651 y Fp(B)2697 1626 y Fs(=)46 b(\()p Fq(i)p Fo(r)37 b Fs(+)g Fq(A)p Fs(\))3525 1557 y Fn(2)3691 1626 y Fq(;)166 b(A)46 b Fs(=)4269 1513 y Fq(B)p 4269 1587 135 7 v 4295 1739 a Fs(2)4423 1626 y(\()p Fq(x)4583 1651 y Fn(2)4657 1626 y Fq(;)28 b Fo(\000)p Fq(x)4955 1651 y Fn(1)5030 1626 y Fs(\))1299 b(\(4.2\))758 1995 y(where)64 b Fq(A)f Fs(is)i(the)f(v)-5 b(ector)63 b(p)5 b(oten)-5 b(tial,)66 b Fq(B)j(>)61 b Fs(0)j(is)h(the)f(strength)g(of)g (the)g(magnetic)g(\014eld,)j(and)758 2195 y(the)55 b(random)h(p)5 b(oten)-5 b(tial)55 b Fq(V)2451 2220 y Fp(!)2602 2195 y Fs(is)h(of)f(the)g(form)2841 2542 y Fq(V)2938 2567 y Fp(!)3034 2542 y Fs(\()p Fq(x)p Fs(\))45 b(=)3498 2384 y Fl(X)3479 2748 y Fp(i)p Fk(2)p Fj(Z)3700 2715 y Fe(2)3785 2542 y Fq(!)3888 2567 y Fp(i)3972 2542 y Fq(u)p Fs(\()p Fq(x)36 b Fo(\000)h Fq(i)p Fs(\))28 b Fq(;)1769 b Fs(\(4.3\))758 3038 y(where,)55 b(follo)-5 b(wing)56 b(Com)-5 b(b)5 b(es)56 b(and)g(Hislop)g([CH2],)f(w)-5 b(e)56 b(require)f(conditions)h (\(R1\))f(and)g(\(R2\):)970 3300 y Fz(\(R1\):)84 b Fs(0)70 b Fo(\024)h Fq(u)p Fs(\()p Fq(x)p Fs(\))e Fo(2)h Fq(C)2547 3240 y Fn(2)2621 3300 y Fs(,)j(with)d Fq(u)p Fs(\(0\))f Fq(>)h Fs(0)g(and)g(supp)29 b Fq(u)70 b Fo(\032)h Fq(B)36 b Fs(\()o(0)p Fq(;)28 b(r)5392 3325 y Fp(u)5479 3300 y Fs(\))o(,)74 b(0)c Fq(<)g(r)6090 3325 y Fp(u)6247 3300 y Fq(<)6521 3234 y Fn(1)p 6466 3261 176 7 v 6466 3280 a Fk(p)p 6575 3280 67 6 v 96 x Fn(2)6689 3300 y Fs(.)1169 3598 y(\(Note)54 b Fq(B)36 b Fs(\()o(0)p Fq(;)28 b(r)5 b Fs(\))45 b(=)2384 3413 y Fl(n)2494 3598 y Fq(x)h Fo(2)g Fm(R)2903 3537 y Fn(2)2987 3598 y Fs(;)74 b Fo(j)p Fq(x)p Fo(j)3294 3623 y Fn(2)3414 3598 y Fs(=)47 b(\()p Fq(x)3750 3537 y Fn(2)3750 3639 y(1)3861 3598 y Fs(+)37 b Fq(x)4122 3537 y Fn(2)4122 3639 y(2)4196 3598 y Fs(\))4281 3493 y Fe(1)p 4280 3511 57 6 v 4280 3577 a(2)4411 3598 y Fq(<)47 b(r)4667 3413 y Fl(o)4777 3598 y Fs(.\))970 3863 y Fz(\(R2\):)84 b Fq(!)53 b Fs(=)46 b Fo(f)p Fq(!)2010 3888 y Fp(i)2065 3863 y Fs(;)74 b Fq(i)46 b Fo(2)g Fm(Z)2547 3803 y Fp(d)2633 3863 y Fo(g)g Fs(a)g(family)g(of)f(indep)5 b(enden)-5 b(t,)49 b(iden)-5 b(tically)46 b(distributed)g(random)1169 4062 y(v)-9 b(ariables,)52 b(whose)f(common)g(probabilit)-5 b(y)51 b(distribution)g Fq(\026)g Fs(has)g(a)f(b)5 b(ounded)51 b(densit)-5 b(y)51 b Fq(g)6 b Fs(,)1169 4262 y(ev)-5 b(en)41 b(\(i.e.,)j Fq(g)6 b Fs(\()p Fq(t)p Fs(\))45 b(=)i Fq(g)6 b Fs(\()p Fo(\000)p Fq(t)p Fs(\)\),)43 b(with)e Fq(g)52 b(>)46 b Fs(0)c(a.e.)69 b(in)41 b([)p Fo(\000)p Fq(M)9 b(;)28 b(M)18 b Fs(])42 b(and)g Fq(g)52 b Fs(=)46 b(0)41 b(on)h Fm(R)22 b Fo(n)p Fs([)p Fo(\000)p Fq(M)9 b(;)28 b(M)18 b Fs(])1169 4461 y(for)55 b(some)h(\014nite)g Fq(M)64 b(>)46 b Fs(0.)957 4722 y(W)-14 b(e)60 b(set)h Fo(k)p Fq(u)p Fo(k)1774 4747 y Fk(1)1969 4722 y Fs(=)55 b(1)61 b(without)f(loss)i(of)e(generalit)-5 b(y)-14 b(.)90 b(\(W)-14 b(e)60 b(absorb)i(it)e(in)h Fq(M)79 b Fs(and)61 b Fq(g)6 b Fs(.\))89 b(Note)758 4921 y(that)54 b Fq(V)1214 4946 y Fp(!)1366 4921 y Fs(is)i(uniformly)g(b)5 b(ounded:)2273 5245 y Fo(j)p Fq(V)2416 5270 y Fp(!)2512 5245 y Fs(\()p Fq(x)p Fs(\))p Fo(j)46 b(\024)g Fs(2)p Fq(M)110 b Fs(for)56 b(a.e.)73 b Fq(!)62 b Fs(and)56 b(all)g Fq(x)46 b Fo(2)f Fm(R)5062 5185 y Fn(2)5173 5245 y Fq(:)1175 b Fs(\(4.4\))957 5569 y(In)55 b(addition,)i(w)-5 b(e)55 b(mak)-5 b(e)56 b(a)f(further)g(requiremen)-5 b(t)55 b(in)h(Theorem)g(4.1:)970 5830 y Fz(\(R3\):)84 b Fs(F)-14 b(or)56 b(some)g Fq(\020)i(>)46 b Fs(0)56 b(and)g(some)g(\014nite)f(constan)-5 b(t)56 b Fq(C)4604 5855 y Fn(1)4724 5830 y Fq(>)46 b Fs(0)56 b(w)-5 b(e)55 b(ha)-5 b(v)g(e)1786 6269 y Fq(\026)p Fs(\([0)p Fq(;)28 b(t)p Fs(]\))46 b(=)2546 6043 y Fl(Z)2712 6084 y Fp(t)2638 6420 y Fn(0)2798 6269 y Fq(g)6 b Fs(\()p Fq(s)p Fs(\))28 b(d)p Fq(s)45 b Fo(\025)h Fq(C)3628 6294 y Fn(1)3758 6269 y Fs(min)q Fo(f)p Fq(t;)28 b(M)18 b Fo(g)4514 6200 y Fp(\020)4728 6269 y Fs(for)55 b(all)139 b Fq(t)46 b(>)g Fs(0)p Fq(:)688 b Fs(\(4.5\))957 6685 y(The)37 b(h)-5 b(yp)5 b(othesis)38 b(\(R3\))e(is)i(new;)43 b(it)36 b(allo)-5 b(ws)38 b(the)f(use)h(of)e(non)-5 b(trivial)38 b(results)g(from)f(p)5 b(ercolation)758 6911 y(theory)-14 b(,)69 b(as)e(in)h(\(4.24\))e(and)h(\(4.27\).)108 b(It)67 b(is)g(equiv)-9 b(alen)k(t)67 b(to)f(lim)28 b(inf)4996 6936 y Fp(t)p Fk(!)p Fn(0)5301 6831 y Fp(\026)p Fn(\([0)p Fp(;t)p Fn(]\))p 5301 6873 416 7 v 5450 6971 a Fp(t)5500 6937 y Fi(\020)5802 6911 y Fq(>)65 b Fs(0.)109 b(W)-14 b(e)66 b(set)758 7111 y Fq(\027)90 b Fs(=)80 b Fq(\027)1221 7136 y Fn(0)1295 7111 y Fq(\020)92 b(>)79 b Fs(0,)i Fq(C)1997 7136 y Fn(3)2151 7111 y Fs(=)f Fq(C)2479 7136 y Fn(2)2553 7111 y Fq(C)2684 7037 y Fp(\027)2751 7054 y Fe(0)2672 7155 y Fn(1)2823 7111 y Fs(,)h(where)75 b Fq(\027)3532 7136 y Fn(0)3686 7111 y Fo(\025)80 b Fs(1)75 b(and)h Fq(C)4515 7136 y Fn(2)4665 7111 y Fs(are)f(constan)-5 b(ts)76 b(giv)-5 b(en)76 b(b)-5 b(y)76 b(t)-5 b(w)g(o-)758 7310 y(dimensional)69 b(p)5 b(ercolation)67 b(theory)g(\(see)h (\(4.24\))o(\).)110 b(The)68 b(exp)5 b(onen)-5 b(t)67 b Fq(\027)79 b Fs(pla)-5 b(ys)68 b(an)g(imp)5 b(ortan)-5 b(t)758 7510 y(role)60 b(in)g(our)g(analysis)h(as,)h(e.g.,)f(in)f (\(4.9\))o(.)88 b(Note)59 b(that)g(if)h(the)f(densit)-5 b(y)61 b Fq(g)66 b Fs(is)60 b(con)-5 b(tin)g(uous)62 b(at)d(0)758 7709 y(with)c Fq(g)6 b Fs(\(0\))45 b Fq(>)i Fs(0,)55 b(then)g(w)-5 b(e)56 b(ma)-5 b(y)56 b(tak)-5 b(e)54 b Fq(\020)k Fs(=)47 b(1)55 b(and)h(hence)f Fq(\027)i Fs(=)46 b Fq(\027)4806 7734 y Fn(0)4927 7709 y Fo(\025)g Fs(1.)957 7908 y Fq(H)1095 7933 y Fp(B)6 b(;!)1389 7908 y Fs(is)53 b(a)f Fm(Z)1789 7848 y Fp(d)1876 7908 y Fs(-ergo)5 b(dic)52 b(random)h(self-adjoin)-5 b(t)53 b(op)5 b(erator)52 b(on)g(L)4960 7848 y Fn(2)5035 7908 y Fs(\()p Fm(R)5210 7848 y Fn(2)5294 7908 y Fq(;)28 b Fs(d)p Fq(x)p Fs(\))51 b([CH2];)j(\006)6311 7933 y Fp(B)6477 7908 y Fs(will)758 8107 y(denote)78 b(its)g(almost)h(sure)g(sp)5 b(ectrum.)143 b(Recall)78 b(that)g(the)g(sp)5 b(ectrum)78 b(of)g(the)g(free)g(Landau) 758 8307 y(Hamiltonian)g Fq(H)1873 8332 y Fp(B)2066 8307 y Fs(consists)i(of)e(a)g(sequence)h(of)f(in\014nitely)g(degenerate)g (eigen)-5 b(v)c(alues,)85 b(the)758 8506 y(Landau)56 b(lev)-5 b(els:)2521 8829 y Fq(B)2647 8854 y Fp(n)2784 8829 y Fs(=)46 b(\(2)p Fq(n)36 b Fs(+)h(1\))p Fq(B)f(;)120 b(n)46 b Fs(=)g(0)p Fq(;)28 b Fs(1)p Fq(;)g Fs(2)p Fq(;)g(:)g(:)g(:)54 b(:)1423 b Fs(\(4.6\))758 9153 y(It)55 b(follo)-5 b(ws)56 b(that)1918 9568 y(\006)2038 9593 y Fp(B)2198 9568 y Fo(\032)2432 9361 y Fk(1)2406 9410 y Fl([)2373 9762 y Fp(n)p Fn(=0)2651 9568 y Fo(B)2760 9593 y Fp(n)2878 9568 y Fq(;)166 b Fs(with)139 b Fo(B)3661 9593 y Fp(n)3797 9568 y Fs(=)46 b([)p Fq(B)4144 9593 y Fp(n)4272 9568 y Fo(\000)37 b Fs(2)p Fq(M)18 b(;)28 b(B)4900 9593 y Fp(n)5027 9568 y Fs(+)37 b(2)p Fq(M)18 b Fs(])28 b Fq(:)819 b Fs(\(4.7\))758 10035 y(W)-14 b(e)55 b(will)h(assume)i Fq(B)d(>)47 b Fs(2)p Fq(M)74 b Fs(so)56 b(the)g(bands)h Fo(B)3689 10060 y Fp(n)3835 10035 y Fs(are)f(disjoin)-5 b(t.)77 b(W)-14 b(e)55 b(recall)h(that)f(the)h(size)g(of)f(a)758 10242 y(p)5 b(ossible)56 b(sp)5 b(ectral)55 b(gap)h(in)g(\006)2634 10267 y Fp(B)2785 10242 y Fo(\\)36 b(B)3041 10267 y Fp(n)3187 10242 y Fs(is)56 b(at)f(most)h Fo(O)5 b Fs(\()p Fq(B)4300 10182 y Fk(\000)4423 10137 y Fe(1)p 4423 10155 57 6 v 4423 10222 a(2)4508 10242 y Fs(\))55 b(for)g(large)g Fq(B)64 b Fs(\(dep)5 b(ending)55 b(on)h Fq(n)p Fs(\))758 10441 y([CH2)o(,)i(Theorem)f(7.1].)80 b(\(Note)56 b(that)g(if)h Fq(u)49 b Fo(2)g Fq(C)3740 10381 y Fk(1)3880 10441 y Fs(,)58 b(then)f(the)g(size)g(of)g(p)5 b(ossible)59 b(sp)5 b(ectral)56 b(gaps)758 10641 y(is)64 b(at)g(most)g Fo(O)5 b Fs(\()p Fq(B)1896 10580 y Fk(\0001)2141 10641 y Fs(\),)65 b(see)g([W1)o(,)i(p.)100 b(3].\))g(This)65 b(is)f(enough)h(to)e(ensure) i(that)f(the)f(result)h(of)758 10840 y(Theorem)55 b(4.1)h(b)5 b(elo)-5 b(w)55 b(is)h(not)f(empt)-5 b(y)-14 b(.)p eop %%Page: 14 14 14 13 bop 758 861 a Fn(14)1485 b(FRANC)-77 b(\030)17 b(OIS)59 b(GERMINET)g(AND)f(ABEL)g(KLEIN)758 1259 y Fz(Theorem)63 b(4.1.)84 b Fg(L)-8 b(et)73 b Fq(H)2387 1284 y Fp(B)6 b(;!)2702 1259 y Fg(b)-8 b(e)73 b(the)h(r)-8 b(andom)73 b(L)-8 b(andau)73 b(Hamiltonian)h(\(as)f(in)86 b Fs(\(4.1\))o Fg(\),)77 b(sat-)758 1458 y(isfying)f(\(R1\),)k(\(R2\),)g(and)c (\(R3\).)127 b(F)-13 b(or)76 b(any)g Fq(n)h Fs(=)f(0)p Fq(;)28 b Fs(1)p Fq(;)g Fs(2)p Fq(;)g(:)g(:)g(:)103 b Fg(ther)-8 b(e)77 b(exists)f(a)g(\014nite)i(c)-8 b(on-)758 1658 y(stant)63 b Fq(\014)1271 1683 y Fp(n)1412 1658 y Fq(>)52 b Fs(0)p Fg(,)63 b(dep)-8 b(ending)64 b(only)e(on)h Fq(n)p Fg(,)g Fq(M)18 b Fg(,)63 b Fq(r)3752 1683 y Fp(u)3839 1658 y Fg(,)g Fo(kr)p Fq(u)p Fo(k)4352 1683 y Fk(1)4493 1658 y Fg(,)g(and)g Fo(k)p Fs(\001)p Fq(u)p Fo(k)5332 1683 y Fk(1)5473 1658 y Fg(,)g(such)g(that,)g(given)758 1882 y Fs(0)46 b Fq(<)1082 1840 y Fl(f)1062 1882 y Fq(M)64 b(<)47 b(M)18 b Fg(,)59 b(ther)-8 b(e)59 b(exists)h(a)e(\014nite)j(p)-8 b(ositive)59 b(c)-8 b(onstant)59 b Fz(B)p Fs(\()p Fq(n)p Fs(\))p Fg(,)f(dep)-8 b(ending)60 b(only)f(on)g Fq(n)p Fg(,)g Fq(M)18 b Fg(,)758 2107 y Fq(r)833 2132 y Fp(u)919 2107 y Fg(,)60 b Fo(kr)p Fq(u)p Fo(k)1429 2132 y Fk(1)1570 2107 y Fg(,)f Fo(k)p Fs(\001)p Fq(u)p Fo(k)2079 2132 y Fk(1)2220 2107 y Fg(,)h Fo(k)p Fq(g)6 b Fo(k)2582 2132 y Fk(1)2723 2107 y Fg(,)59 b Fq(C)2952 2132 y Fn(3)3026 2107 y Fg(,)g Fq(\027)11 b Fg(,)59 b(and)3682 2065 y Fl(f)3662 2107 y Fq(M)18 b Fg(,)59 b(such)h(that)f(for)h(al)8 b(l)60 b Fq(B)55 b Fo(\025)46 b Fz(B)p Fs(\()p Fq(n)p Fs(\))58 b Fg(we)i(have)1680 2459 y Fs(\006)1800 2484 y Fp(B)6 b(;n)2082 2459 y Fs(=)46 b(\006)2377 2484 y Fp(B)2528 2459 y Fo(\\)2676 2274 y Fl(n)2786 2459 y Fq(E)56 b Fo(2)45 b(B)3230 2484 y Fp(n)3321 2459 y Fq(;)74 b Fo(j)p Fq(E)46 b Fo(\000)37 b Fq(B)3948 2484 y Fp(n)4038 2459 y Fo(j)47 b(\025)f Fs(2)p Fq(K)4530 2484 y Fp(n)4640 2385 y Fn(log)26 b Fp(B)p 4640 2420 301 7 v 4737 2516 a(B)4961 2274 y Fl(o)5117 2459 y Fo(\032)47 b Fs(\006)5413 2484 y Fn(MSA)5739 2459 y Fq(;)609 b Fs(\(4.8\))758 2854 y Fg(with)71 b Fq(K)1271 2879 y Fp(n)1430 2854 y Fs(=)1647 2773 y Fn(\(43+20)p Fp(\027)8 b Fn(\))p 1647 2815 545 7 v 1841 2911 a Fp(\014)1916 2928 y Fi(n)2211 2854 y Fg(,)75 b(and)c(for)h(a.e.)112 b Fq(!)6 b Fg(,)75 b(if)c Fq(')3841 2879 y Fp(B)6 b(;!)t(;E)4297 2854 y Fg(is)71 b(an)h(eigenfunction)i(of)d Fq(H)6121 2879 y Fp(B)6 b(;!)6434 2854 y Fg(with)758 3065 y(eigenvalue)62 b Fq(E)55 b Fo(2)46 b Fs(\006)2015 3090 y Fp(B)6 b(;n)2250 3065 y Fg(,)60 b(then)1024 3469 y Fs(lim)28 b(inf)1084 3569 y Fp(x)p Fk(!1)1512 3469 y Fo(\000)1661 3357 y Fs(log)i Fo(k)p Fq(\037)2090 3382 y Fp(x)2174 3357 y Fq(')2283 3382 y Fp(B)6 b(;!)t(;E)2667 3357 y Fo(k)p 1661 3431 1090 7 v 2112 3583 a(j)p Fq(x)p Fo(j)6394 3469 y Fs(\(4.9\))1695 4148 y Fo(\025)1871 3708 y Fl(8)1871 3857 y(>)1871 3907 y(>)1871 3957 y(<)1871 4256 y(>)1871 4305 y(>)1871 4355 y(:)2121 3797 y Fp(\034)2182 3814 y Fi(n)2264 3797 y Fk(j)p Fp(E)g Fk(\000)p Fp(B)2610 3814 y Fi(n)2693 3797 y Fk(j)2732 3746 y Fe(1+)p Fi(\027)2947 3797 y Fp(B)p 2121 3839 933 7 v 2216 3934 a Fk(j)q Fn(log)25 b Fk(j)p Fp(E)6 b Fk(\000)p Fp(B)2796 3951 y Fi(n)2879 3934 y Fk(jj)3119 3877 y Fo(\025)3315 3792 y Fp(\034)3389 3742 y Ff(0)3376 3826 y Fi(n)3457 3792 y Fn(\(log)25 b Fp(B)6 b Fn(\))3861 3742 y Fi(\027)p 3315 3839 621 7 v 3535 3934 a Fp(B)3641 3901 y Fi(\027)4121 3877 y Fg(if)46 b Fs(2)p Fq(K)4493 3902 y Fp(n)4603 3803 y Fn(log)25 b Fp(B)p 4603 3839 301 7 v 4700 3934 a(B)4969 3877 y Fo(\024)47 b(j)p Fq(E)f Fo(\000)37 b Fq(B)5652 3902 y Fp(n)5742 3877 y Fo(j)46 b(\024)6104 3812 y Fn(2)p 6030 3839 216 7 v 6030 3857 a Fk(p)p 6139 3857 106 6 v 98 x Fp(B)2121 4124 y(\034)2182 4141 y Fi(n)2264 4124 y Fk(j)p Fp(E)6 b Fk(\000)p Fp(B)2610 4141 y Fi(n)2693 4124 y Fk(j)2732 4074 y Fi(\027)2805 4026 y Fk(p)p 2915 4026 V 2915 4124 a Fp(B)p 2121 4167 900 7 v 2200 4262 a Fk(j)q Fn(log)25 b Fk(j)p Fp(E)6 b Fk(\000)p Fp(B)2780 4279 y Fi(n)2863 4262 y Fk(jj)3087 4205 y Fo(\025)3282 4120 y Fp(\034)3356 4070 y Ff(0)3343 4153 y Fi(n)3424 4120 y Fp(B)3550 4023 y Fe(1)p Ff(\000)p Fi(\027)p 3550 4045 212 5 v 3628 4108 a Fe(2)p 3282 4167 509 7 v 3386 4262 a Fn(log)25 b Fp(B)4121 4205 y Fg(if)4363 4139 y Fn(2)p 4288 4166 216 7 v 4288 4184 a Fk(p)p 4398 4184 106 6 v 4398 4283 a Fp(B)4570 4205 y Fo(\024)46 b(j)p Fq(E)g Fo(\000)37 b Fq(B)5252 4230 y Fp(n)5342 4205 y Fo(j)47 b(\024)f Fs(2)5713 4163 y Fl(f)5693 4205 y Fq(M)2101 4462 y(\034)2174 4487 y Fp(n)2264 4321 y Fo(p)p 2403 4321 135 7 v 2403 4462 a Fq(B)1592 b Fg(if)46 b Fs(2)4372 4420 y Fl(f)4352 4462 y Fq(M)64 b Fo(\024)46 b(j)p Fq(E)g Fo(\000)37 b Fq(B)5259 4487 y Fp(n)5350 4462 y Fo(j)46 b(\024)g Fs(2)p Fq(M)6423 4148 y(;)758 4807 y Fg(wher)-8 b(e)60 b Fq(\034)1300 4832 y Fp(n)1390 4807 y Fq(;)28 b(\034)1556 4747 y Fk(0)1537 4848 y Fp(n)1672 4807 y Fq(>)47 b Fs(0)59 b Fg(ar)-8 b(e)59 b(c)-8 b(onstants)60 b(dep)-8 b(ending)60 b(only)g(on)g Fq(M)18 b Fg(,)4683 4765 y Fl(f)4662 4807 y Fq(M)g Fg(,)60 b Fq(\014)5046 4832 y Fp(n)5136 4807 y Fg(,)f Fq(C)5365 4832 y Fn(3)5440 4807 y Fg(,)g(and)g Fq(\027)11 b Fg(.)957 5125 y Fs(Theorem)68 b(4.1)g(is)g(pro)-5 b(v)g(en)69 b(b)-5 b(y)68 b(using)g(the)g(results)g (of)g([CH2)o(])g(to)g(satisfy)f(the)h(h)-5 b(yp)5 b(otheses)758 5324 y(of)57 b(Theorem)g(2.6.)80 b(Note)57 b(that)g(w)-5 b(e)57 b(could)h(also)g(w)-5 b(ork)57 b(with)g(the)g(h)-5 b(yp)5 b(otheses)58 b(used)h(b)-5 b(y)57 b(W)-14 b(ang)758 5523 y([W1)o(],)73 b(in)c(whic)-5 b(h)70 b(case)f(w)-5 b(e)69 b(obtain)g(the)g(results)g(of)g(Theorem)g(4.1)g(for)g(energies)g (at)f(a)h(\014xed)758 5748 y(distance)57 b(from)g(a)g(Landau)g(lev)-5 b(el,)57 b(i.e.,)h(2)3388 5706 y Fl(f)3368 5748 y Fq(M)67 b Fo(\024)48 b(j)p Fq(E)g Fo(\000)38 b Fq(B)4283 5773 y Fp(n)4373 5748 y Fo(j)49 b(\024)f Fs(2)p Fq(M)75 b Fs(for)57 b(a)g(giv)-5 b(en)57 b(0)49 b Fq(<)6125 5706 y Fl(f)6104 5748 y Fq(M)67 b(<)49 b(M)18 b Fs(,)758 5960 y(including)56 b(a)g(rate)e(of)h(exp)5 b(onen)-5 b(tial)56 b(deca)-5 b(y)55 b(for)g(the)g(eigenfunctions)h Fo(/)5298 5820 y(p)p 5436 5820 V 140 x Fq(B)9 b Fs(.)957 6159 y(F)-14 b(ollo)-5 b(wing)57 b([CH2)o(],)f(w)-5 b(e)55 b(in)-5 b(tro)5 b(duce)56 b(the)f(lattice)2875 6479 y(\000)46 b(=)3200 6344 y Fl(\000)3296 6414 y Fn(1)p 3296 6441 67 7 v 3296 6536 a(2)3382 6479 y Fq(;)3476 6414 y Fn(1)p 3476 6441 V 3476 6536 a(2)3562 6344 y Fl(\001)3675 6479 y Fs(+)37 b(e)3915 6410 y Fp(i)3982 6366 y Fi(\031)p 3982 6384 72 6 v 3990 6450 a Fe(4)4082 6334 y Fo(p)p 4221 6334 84 7 v 4221 6479 a Fs(2)27 b Fm(Z)4433 6410 y Fn(2)4544 6479 y Fq(;)1721 b Fs(\(4.10\))758 6808 y(whic)-5 b(h)64 b(is)h(just)f(the)f(lattice)g Fm(Z)2666 6747 y Fn(2)2813 6808 y Fs(with)h(the)f(origin)h(shifted)g(to)4734 6673 y Fl(\000)4830 6742 y Fn(1)p 4830 6769 67 7 v 4830 6865 a(2)4916 6808 y Fq(;)5010 6742 y Fn(1)p 5010 6769 V 5010 6865 a(2)5096 6673 y Fl(\001)5172 6808 y Fs(,)i(rotated)d(b)-5 b(y)64 b(45)6279 6747 y Fk(\016)6356 6808 y Fs(,)i(and)758 7033 y(rescaled)50 b(b)-5 b(y)1607 6896 y Fo(p)p 1745 6896 84 7 v 137 x Fs(2.)73 b(W)-14 b(e)50 b(will)g(denote)h(sites)f(in) h(\000)f(b)-5 b(y)62 b(\025)-94 b Fq(x;)39 b Fs(\025)-94 b Fq(y)6 b(;)28 b(:)g(:)g(:)f Fs(.)73 b(Eac)-5 b(h)51 b(site)f(in)h Fm(Z)5669 6973 y Fn(2)5803 7033 y Fs(is)g(the)f(middle) 758 7232 y(p)5 b(oin)-5 b(t)55 b(of)g(a)h(b)5 b(ond)55 b(in)h(\000.)74 b(W)-14 b(e)55 b(will)g(w)-5 b(ork)56 b(with)f(op)5 b(en)55 b(b)5 b(o)-5 b(xes)56 b(in)g(the)f(lattice)f (\000:)2506 7499 y(\025)2490 7541 y(\003)2605 7566 y Fp(L)2704 7541 y Fs(\()11 b(\025)-94 b Fq(x)p Fs(\))45 b(=)3150 7407 y Fl(\010)3247 7541 y Fq(y)52 b Fo(2)46 b Fm(R)3648 7473 y Fp(d)3734 7541 y Fs(;)111 b Fo(j)p Fq(y)43 b Fo(\000)48 b Fs(\025)-94 b Fq(x)p Fo(j)4368 7566 y Fn(1)4489 7541 y Fq(<)46 b(L)4777 7407 y Fl(\011)4930 7541 y Fq(;)1335 b Fs(\(4.11\))758 7886 y(where)64 b(\025)-94 b Fq(x)46 b Fo(2)g Fs(\000,)54 b Fq(L)47 b Fo(2)e Fs(4)p Fm(N)26 b Fs(,)64 b(and)54 b Fo(j)p Fq(x)p Fo(j)2866 7911 y Fn(1)2987 7886 y Fs(=)46 b Fo(j)p Fq(x)3303 7911 y Fn(1)3378 7886 y Fo(j)34 b Fs(+)h Fo(j)p Fq(x)3763 7911 y Fn(2)3837 7886 y Fo(j)p Fs(,)55 b(so)f(the)g(the)f(b)5 b(oundary)55 b(of)5706 7844 y(\025)5689 7886 y(\003)5804 7911 y Fp(L)5904 7886 y Fs(\()11 b(\025)-94 b Fq(x)p Fs(\))53 b(is)i(made)758 8085 y(out)47 b(of)h(b)5 b(onds)49 b(in)f(\000.)71 b(Note)47 b(that)2878 8020 y Fn(1)p 2878 8047 67 7 v 2878 8142 a(2)2992 7944 y Fl(\014)2992 8044 y(\014)3047 8085 y Fq(x)37 b Fo(\000)3345 7951 y Fl(\000)3441 8020 y Fn(1)p 3441 8047 V 3441 8142 a(2)3527 8085 y Fq(;)3621 8020 y Fn(1)p 3621 8047 V 3621 8142 a(2)3707 7951 y Fl(\001)3783 7944 y(\014)3783 8044 y(\014)3838 8152 y Fn(1)3961 8085 y Fs(is)48 b(just)g(the)f(sup)i(norm)g(in)f(the)f(co)5 b(ordinates)758 8309 y(giv)-5 b(en)43 b(b)-5 b(y)44 b(axes)f(parallel)h (to)f(the)g(b)5 b(onds)44 b(of)f(\000,)i(cen)-5 b(tered)43 b(at)4479 8175 y Fl(\000)4575 8244 y Fn(1)p 4575 8271 V 4575 8367 a(2)4661 8309 y Fq(;)4755 8244 y Fn(1)p 4755 8271 V 4755 8367 a(2)4841 8175 y Fl(\001)4917 8309 y Fs(,)j(and)e(rescaled)g(so)f(of)g(b)5 b(onds)758 8529 y(in)62 b(\000)g(ha)-5 b(v)g(e)62 b(length)g(one.)94 b(\(That's)61 b(wh)-5 b(y)63 b(w)-5 b(e)62 b(ha)-5 b(v)g(e)62 b Fo(j)p Fq(y)48 b Fo(\000)k Fs(\025)-94 b Fq(x)p Fo(j)4497 8554 y Fn(1)4629 8529 y Fq(<)57 b(L)62 b Fs(and)g(not)g Fq(<)5827 8463 y Fp(L)p 5827 8490 92 7 v 5839 8586 a Fn(2)6000 8529 y Fs(in)g(\(4.11\).\))758 8728 y(W)-14 b(e)55 b(also)h(adapt)f(\(2.2\))o(-\(2.5\))g(to)g(the)g(lattice)g (\000:)2536 9030 y Fq(\037)2648 9055 y Fn(\025)-74 b Fp(x;L)2900 9030 y Fs(=)46 b Fq(\037)3191 9054 y Fn(\025)3179 9083 y(\003)3270 9100 y Fi(L)3358 9083 y Fn(\()8 b(\025)-74 b Fp(x)p Fn(\))3573 9030 y Fq(;)120 b Fs(with)92 b Fq(\037)4266 9055 y Fn(\025)-74 b Fp(x)4387 9030 y Fs(=)47 b Fq(\037)4675 9055 y Fn(\025)-74 b Fp(x;)p Fn(1)4883 9030 y Fq(;)1382 b Fs(\(4.12\))2456 9459 y(\007)2585 9484 y Fp(L)2684 9459 y Fs(\()11 b(\025)-94 b Fq(x)p Fs(\))45 b(=)i Fo(f)11 b Fs(\025)-94 b Fq(y)52 b Fo(2)46 b Fs(\000;)28 b Fo(j)11 b Fs(\025)-94 b Fq(y)43 b Fo(\000)k Fs(\025)-93 b Fq(x)p Fo(j)4158 9484 y Fn(1)4279 9459 y Fs(=)46 b Fq(L)37 b Fo(\000)g Fs(2)p Fo(g)27 b Fq(;)1302 b Fs(\(4.13\))2125 9898 y(~)2102 9940 y(\007)2231 9965 y Fp(L)2331 9940 y Fs(\()11 b(\025)-94 b Fq(x)o Fs(\))46 b(=)p 2776 9762 116 7 v 2792 9898 a(\025)2776 9940 y(\003)2891 9965 y Fp(L)p Fk(\000)p Fn(1)3161 9940 y Fs(\()11 b(\025)-94 b Fq(x)o Fs(\))p Fo(n)3484 9898 y Fs(\025)3468 9940 y(\003)3583 9965 y Fp(L)p Fk(\000)p Fn(2)3853 9940 y Fs(\()11 b(\025)-94 b Fq(x)o Fs(\))46 b(=)4471 9782 y Fl([)4308 10146 y Fn(\025)-76 b Fp(y)t Fk(2)p Fn(\007)4560 10163 y Fi(L)4649 10146 y Fn(\()8 b(\025)-74 b Fp(x)p Fn(\))p 4855 9762 V 4871 9898 a Fs(\025)4855 9940 y(\003)4971 9965 y Fn(1)5045 9940 y Fs(\()11 b(\025)-94 b Fq(y)6 b Fs(\))55 b Fq(;)948 b Fs(\(4.14\))2580 10588 y(\000)2692 10613 y Fn(\025)-74 b Fp(x;L)2944 10588 y Fs(=)47 b Fq(\037)3242 10609 y Fn(~)3224 10638 y(\007)3326 10655 y Fi(L)3413 10638 y Fn(\()8 b(\025)-74 b Fp(x)p Fn(\))3647 10588 y Fs(=)3967 10430 y Fl(X)3832 10794 y Fn(\025)e Fp(y)t Fk(2)p Fn(\007)4084 10811 y Fi(L)4172 10794 y Fn(\()8 b(\025)-74 b Fp(x)q Fn(\))4379 10588 y Fq(\037)4493 10613 y Fn(\025)e Fp(y)4655 10588 y Fq(a:e:)1399 b Fs(\(4.15\))p eop %%Page: 15 15 15 14 bop 1773 861 a Fn(EXPLICIT)58 b(FINITE)g(V)l(OLUME)h(CRITERIA)g (F)l(OR)g(LOCALIZA)-11 b(TION)883 b(15)758 1259 y Fs(Note)64 b(that)g(all)i(of)f(our)g(results)h(are)f(v)-9 b(alid)65 b(with)g(this)g(c)-5 b(hoice)66 b(of)f(b)5 b(o)-5 b(xes)65 b(with)g(no)g(c)-5 b(hange)66 b(in)758 1458 y(the)59 b(constan)-5 b(ts)61 b(in)f(our)g(theorems.)87 b(\(The)59 b(b)5 b(o)-5 b(xes)60 b(and)g(the)g(corresp)5 b(onding)60 b(b)5 b(oundary)60 b(b)5 b(elts)758 1658 y(are)55 b(just)g(the)g(usual) i(ones)f(\(e.g.,)f(\(2.1\))o(-\(2.5\)\))g(in)g(the)g(rescaled)h (lattice)e(\000.\))957 1857 y(W)-14 b(e)55 b(de\014ne)h(\(as)f(in)h ([CH2)o(]\))f(the)g(\014nite)h(v)-5 b(olume)56 b(op)5 b(erators)55 b(b)-5 b(y)2826 2149 y Fq(H)2964 2174 y Fp(B)6 b(;!)t(;)i Fn(\025)-74 b Fp(x)q(;L)3497 2149 y Fs(=)46 b Fq(H)3810 2174 y Fp(B)3961 2149 y Fs(+)37 b Fq(V)4224 2174 y Fp(!)t(;)8 b Fn(\025)-74 b Fp(x)q(;L)4593 2149 y Fq(;)1672 b Fs(\(4.16\))758 2442 y(with)65 b(\025)-93 b Fq(x)46 b Fo(2)g Fs(\000,)55 b Fq(L)46 b Fo(2)g Fs(4)p Fm(N)26 b Fs(,)64 b(and)1251 2758 y Fq(V)1348 2783 y Fp(!)t(;)8 b Fn(\025)-74 b Fp(x;L)1735 2758 y Fs(=)2154 2600 y Fl(X)1911 2987 y Fp(i)p Fk(2)p Fj(Z)2132 2954 y Fe(2)2189 2987 y Fk(\\)2290 2958 y Fn(\025)2278 2987 y(\003)2369 3004 y Fi(L)2457 2987 y Fn(\()8 b(\025)-74 b Fp(x)q Fn(\))2664 2758 y Fq(!)2767 2783 y Fp(i)2850 2758 y Fq(u)p Fs(\()p Fq(x)37 b Fo(\000)g Fq(i)p Fs(\))46 b(=)4292 2600 y Fl(X)3651 2987 y Fp(i)p Fk(2)p Fj(Z)3872 2954 y Fe(2)3930 2987 y Fp(;)38 b Fn(supp)23 b Fp(u)p Fn(\()p Fp(x)p Fk(\000)p Fp(i)p Fn(\))p Fk(\032)4827 2958 y Fn(\025)4815 2987 y(\003)4906 3004 y Fi(L)4993 2987 y Fn(\()8 b(\025)-74 b Fp(x)p Fn(\))4978 2758 y Fq(!)5081 2783 y Fp(i)5165 2758 y Fq(u)p Fs(\()p Fq(x)36 b Fo(\000)h Fq(i)p Fs(\))28 b Fq(:)493 b Fs(\(4.17\))758 3299 y(Note)66 b(that)g Fq(V)1639 3324 y Fp(!)t(;)8 b Fn(\025)-74 b Fp(x;L)2047 3299 y Fs(has)68 b(supp)5 b(ort)67 b(in)3203 3257 y(\025)3187 3299 y(\003)3302 3324 y Fp(L)3401 3299 y Fs(\()11 b(\025)-94 b Fq(x)p Fs(\))66 b(and)i(is)f(relativ)-5 b(ely)67 b(compact)f(with)h(resp)5 b(ect)67 b(to)758 3498 y Fq(H)896 3523 y Fp(B)1010 3498 y Fs(,)50 b(so)e(the)g(essen)-5 b(tial)49 b(sp)5 b(ectrum)48 b(of)g(the)g(\014nite)g(v)-5 b(olume)49 b(op)5 b(erators)48 b(consists)h(of)f(the)g(Landau)758 3698 y(lev)-5 b(els)55 b(giv)-5 b(en)56 b(in)g(\(4.6\))o(.)74 b(Note)54 b(that)h(\(4.7\))g(also)h(holds)g(for)f(these)g(\014nite)h(v) -5 b(olume)56 b(op)5 b(erators.)957 3897 y(By)68 b(the)g(de\014nition)i (of)e Fq(V)2581 3922 y Fp(!)t(;)8 b Fn(\025)-74 b Fp(x)q(;L)2991 3897 y Fs(w)-5 b(e)69 b(clearly)g(ha)-5 b(v)g(e)69 b(Assumption)h(IAD)e (with)h Fq(\045)f Fs(=)h(0.)114 b(As-)758 4096 y(sumption)74 b(SLI)g(inequalit)-5 b(y)73 b(can)g(b)5 b(e)73 b(pro)-5 b(v)g(en)74 b(as)g(in)g([GK3,)k(Theorem)c(A.1])f(for)g(an)-5 b(y)73 b(op)5 b(en)758 4295 y(in)-5 b(terv)c(al)55 b Fo(I)12 b Fs(,)56 b(with)f(the)g(constan)-5 b(t)56 b Fq(\015)2982 4320 y Fk(I)3128 4295 y Fs(in)g(\(2.7\))e(giv)-5 b(en)56 b(b)-5 b(y)2030 4610 y Fq(\015)2116 4635 y Fk(I)2253 4610 y Fs(=)59 b(sup)2428 4749 y Fp(E)6 b Fk(2I)2732 4610 y Fq(\015)2818 4635 y Fp(E)2976 4610 y Fq(;)120 b Fs(with)138 b Fq(\015)3689 4635 y Fp(E)3848 4610 y Fs(=)52 b(~)-89 b Fq(\015)4118 4460 y Fl(p)p 4284 4460 1050 7 v 150 x Fs(max)p Fo(f)p Fq(E)10 b(;)28 b Fs(0)p Fo(g)36 b Fs(+)h(1)55 b Fq(;)876 b Fs(\(4.18\))758 5009 y(where)54 b(the)f(constan)-5 b(t)61 b(~)-89 b Fq(\015)63 b Fs(dep)5 b(ends)55 b(only)f(on)h Fq(M)18 b Fs(;)55 b(it)f(do)5 b(es)54 b(not)g(dep)5 b(end)54 b(on)h Fq(B)8 b Fs(.)74 b(\(Although,)54 b(as)758 5208 y(noted)h(in)h([CH2)o(],)g(w) -5 b(e)55 b(ha)-5 b(v)g(e)2140 5516 y Fq(\037)2254 5541 y Fn(\025)-76 b Fp(y)t(;`)2418 5516 y Fq(V)2515 5541 y Fp(!)t(;)8 b Fn(\025)-74 b Fp(x)q(;L)2893 5516 y Fo(\000)38 b Fq(V)3157 5541 y Fp(!)t(;)10 b Fn(\025)-76 b Fp(y)t(;`)3505 5516 y Fo(6)p Fs(=)46 b(0)138 b Fq(if)4212 5474 y Fs(\025)4196 5516 y(\003)4311 5541 y Fp(`)4375 5516 y Fs(\()11 b(\025)-94 b Fq(y)6 b Fs(\))46 b Fo(\032)4829 5474 y Fs(\025)4813 5516 y(\003)4928 5541 y Fp(L)5028 5516 y Fs(\()11 b(\025)-94 b Fq(x)o Fs(\))28 b Fq(;)985 b Fs(\(4.19\))758 5809 y(w)-5 b(e)55 b(also)h(ha)-5 b(v)g(e)1990 6101 y Fq(\037)2104 6126 y Fn(\025)-76 b Fp(y)t(;`)p Fk(\000)p Fn(1)2439 6101 y Fs(\()p Fq(V)2601 6126 y Fp(!)t(;)8 b Fn(\025)-74 b Fp(x;L)2978 6101 y Fo(\000)38 b Fq(V)3242 6126 y Fp(!)t(;)10 b Fn(\025)-76 b Fp(y)t(;`)3543 6101 y Fs(\))46 b(=)g(0)139 b Fq(if)4362 6059 y Fs(\025)4346 6101 y(\003)4461 6126 y Fp(`)4525 6101 y Fs(\()11 b(\025)-94 b Fq(y)6 b Fs(\))45 b Fo(\032)4979 6059 y Fs(\025)4963 6101 y(\003)5078 6126 y Fp(L)5177 6101 y Fs(\()11 b(\025)-94 b Fq(x)p Fs(\))28 b Fq(;)835 b Fs(\(4.20\))758 6403 y(so)65 b(the)f(pro)5 b(of)64 b(of)g(Assumption)i(SLI)f(is)g(not)f(a\013ected.\))100 b(Assumption)66 b(EDI)e(also)i(follo)-5 b(ws)65 b(in)758 6602 y(the)55 b(same)h(w)-5 b(a)g(y)-14 b(.)957 6801 y(Assumption)38 b(SGEE)g(follo)-5 b(ws)37 b(from)g([Si)q(,)k(Theorem)c (B.13.2])g(and)g(prop)5 b(erties)37 b(of)g(Sc)-5 b(hr\177)-83 b(odinger)758 7001 y(op)5 b(erators,)55 b(as)h(in)g([CH2)o(,)g(Section) f(5].)957 7200 y(Com)-5 b(b)5 b(es)63 b(and)g(Hislop)h([CH2)o(,)h (Theorem)e(3.1])f(pro)-5 b(v)g(ed)64 b(the)e(follo)-5 b(wing)63 b(form)g(of)g(Assump-)758 7399 y(tions)55 b(NE)g(and)h(W)f (\(note)g(that)g Fq(b)46 b Fs(=)g(1\):)758 7711 y Fz(Theorem)63 b(4.2)h Fs(\([CH2)o(]\))p Fz(.)83 b Fg(L)-8 b(et)44 b Fq(H)2970 7736 y Fp(B)6 b(;!)3255 7711 y Fg(b)-8 b(e)43 b(the)h(r)-8 b(andom)43 b(L)-8 b(andau)43 b(Hamiltonian)i(\(as)e(in)56 b Fs(\(4.1\))o Fg(\),)758 7911 y(satisfying)71 b(\(R1\))g(and)h (\(R2\).)112 b(F)-13 b(or)71 b(any)h Fq(n)67 b Fs(=)h(0)p Fq(;)28 b Fs(1)p Fq(;)g Fs(2)p Fq(;)g(:)g(:)g(:)f Fg(,)74 b(ther)-8 b(e)72 b(exists)g(a)f(c)-8 b(onstant)72 b Fq(Q)6448 7936 y Fp(n)6606 7911 y Fs(=)796 8080 y(~)758 8122 y Fq(Q)889 8147 y Fp(n)979 8122 y Fo(k)p Fq(g)6 b Fo(k)1230 8147 y Fk(1)1371 8122 y Fg(,)79 b(with)d(the)f(c)-8 b(onstant)2885 8080 y Fs(~)2847 8122 y Fq(Q)2978 8147 y Fp(n)3144 8122 y Fg(dep)g(ending)76 b(only)g(on)f Fq(r)4637 8147 y Fp(u)4799 8122 y Fg(and)g Fq(M)18 b Fg(,)80 b(such)75 b(that)g(for)g(any)758 8321 y(close)-8 b(d)59 b(interval)i Fq(I)e Fo(\032)47 b(B)2267 8346 y Fp(n)2357 8321 y Fo(nf)p Fq(B)2649 8346 y Fp(n)2740 8321 y Fo(g)59 b Fg(we)h(have)2176 8617 y Fm(E)2305 8482 y Fl(\010)2411 8617 y Fq(E)2534 8642 y Fp(H)2643 8660 y Fi(B)t(;!)s(;)7 b Fe(\025)-64 b Fi(x)o(;L)3085 8617 y Fs(\()p Fq(I)13 b Fs(\))3301 8482 y Fl(\011)3443 8617 y Fo(\024)3638 8551 y Fn(1)p 3638 8578 67 7 v 3638 8674 a(2)3724 8617 y Fq(Q)3855 8642 y Fp(n)4317 8551 y(B)p 3966 8578 810 7 v 3966 8685 a Fk(f)p Fn(dist\()p Fp(I)c(;B)4508 8702 y Fi(n)4590 8685 y Fn(\))p Fk(g)4710 8638 y Fe(2)4822 8617 y Fo(j)p Fq(I)k Fo(j)28 b Fq(L)5141 8556 y Fn(2)5243 8617 y Fq(;)1022 b Fs(\(4.21\))758 8938 y Fg(for)59 b(al)8 b(l)60 b Fq(B)55 b(>)46 b Fs(2)p Fq(M)18 b Fg(,)70 b Fs(\025)-93 b Fq(x)46 b Fo(2)g Fs(\000)p Fg(,)59 b(and)g Fq(L)46 b Fo(2)g Fs(2)p Fm(N)26 b Fg(.)957 9138 y(In)60 b(p)-8 b(articular,)59 b(for)g(any)h Fq(E)55 b Fo(2)46 b(B)3012 9163 y Fp(n)3102 9138 y Fo(nf)p Fq(B)3394 9163 y Fp(n)3485 9138 y Fo(g)59 b Fg(we)h(have)1688 9514 y Fm(P)33 b Fo(f)9 b Fs(dist\()p Fq(\033)d Fs(\()p Fq(H)2537 9539 y Fp(B)g(;!)t(;)i Fn(\025)-74 b Fp(x;L)3023 9514 y Fs(\))p Fq(;)28 b(E)10 b Fs(\))44 b Fo(\024)j Fq(\021)6 b Fo(g)46 b(\024)g Fq(Q)4103 9539 y Fp(n)4716 9402 y Fq(B)p 4214 9476 1139 7 v 4214 9654 a Fs(\()p Fo(j)p Fq(E)g Fo(\000)37 b Fq(B)4786 9679 y Fp(n)4876 9654 y Fo(j)g(\000)g Fq(\021)6 b Fs(\))5278 9571 y Fn(2)5400 9514 y Fq(\021)34 b(L)5629 9446 y Fn(2)5731 9514 y Fq(;)534 b Fs(\(4.22\))758 9930 y Fg(for)59 b(al)8 b(l)60 b Fs(0)47 b Fq(<)f(\021)52 b(<)47 b Fo(j)p Fq(E)f Fo(\000)37 b Fq(B)2382 9955 y Fp(n)2472 9930 y Fo(j)p Fg(,)60 b Fq(B)54 b(>)46 b Fs(2)p Fq(M)19 b Fg(,)69 b Fs(\025)-93 b Fq(x)46 b Fo(2)g Fs(\000)p Fg(,)59 b(and)g Fq(L)46 b Fo(2)g Fs(2)p Fm(N)26 b Fg(.)957 10242 y Fs(In)74 b(addition,)k(Com)-5 b(b)5 b(es)75 b(and)f(Hislop)g([CH2,)k(Prop)5 b(osition)74 b(5.1])g(obtained)f(length)h(scale)758 10441 y(estimates.)e(Their)49 b(argumen)-5 b(t)49 b(relies)g(on)h(indep)5 b(enden)-5 b(t)49 b(b)5 b(ond)49 b(p)5 b(ercolation)49 b(on)g(the)f(lattice)g (\000.)957 10641 y(W)-14 b(e)48 b(will)g(need)g(some)h(notions)g(of)f (indep)5 b(enden)-5 b(t)48 b(b)5 b(ond)49 b(p)5 b(ercolation)47 b(in)i Fm(Z)5578 10580 y Fn(2)5709 10641 y Fs(\(see)f([Gr]\).)72 b(W)-14 b(e)758 10840 y(denote)63 b(b)-5 b(y)64 b Fq(p)g Fs(the)f(probabilit)-5 b(y)65 b(of)e(a)h(b)5 b(ond)64 b(b)5 b(eing)64 b(o)5 b(ccupied.)100 b(The)63 b(connectivit)-5 b(y)63 b(function)p eop %%Page: 16 16 16 15 bop 758 861 a Fn(16)1485 b(FRANC)-77 b(\030)17 b(OIS)59 b(GERMINET)g(AND)f(ABEL)g(KLEIN)758 1259 y Fq(\034)831 1284 y Fp(p)907 1259 y Fs(\()p Fq(x)p Fs(\))d(is)h(giv)-5 b(en)56 b(b)-5 b(y)56 b(the)g(probabilit)-5 b(y)56 b(that)f(0)h(is)h (connected)e(to)g Fq(x)47 b Fo(2)g Fm(Z)5201 1199 y Fn(2)5341 1259 y Fs(b)-5 b(y)56 b(o)5 b(ccupied)55 b(b)5 b(onds.)758 1458 y(W)-14 b(e)55 b(ha)-5 b(v)g(e)55 b([Gr)q(,)g(Prop)5 b(osition)56 b(5.47\)])1746 1759 y Fq(\034)1819 1784 y Fp(p)1895 1759 y Fs(\()p Fq(x)p Fs(\))45 b Fo(\024)i Fs(e)2415 1690 y Fk(\000)p Fp(')p Fn(\()p Fp(p)p Fn(\))p Fk(j)p Fp(x)p Fk(j)2968 1759 y Fq(;)166 b Fs(with)138 b Fq(')p Fs(\()p Fq(p)p Fs(\))45 b(=)h Fo(\000)4341 1693 y Fn(1)p 4333 1720 83 7 v 4333 1816 a Fp(n)4463 1759 y Fs(log)30 b Fq(\034)4778 1784 y Fp(p)4854 1759 y Fs(\(\()p Fq(n;)e Fs(0\)\))44 b Fo(\025)i Fs(0)28 b Fq(:)564 b Fs(\(4.23\))758 2070 y(The)71 b(function)g Fq(')p Fs(\()p Fq(p)p Fs(\))f(is)h(con)-5 b(tin)g(uous)73 b(and)f(nonincreasing)h(on)e ([0)p Fq(;)28 b Fs(1],)75 b(with)d Fq(')p Fs(\()5890 2005 y Fn(1)p 5889 2032 67 7 v 5889 2127 a(2)5975 2070 y Fs(\))g(=)h(0)e(and)758 2269 y Fq(')p Fs(\(0\))57 b(=)i Fo(1)k Fs([Gr)q(,)i(Theorem)e(5.14],)i(where)e(w)-5 b(e)63 b(used)h(that)e(in)i(t)-5 b(w)g(o)63 b(dimensions)i(the)e(critical)758 2469 y(probabilit)-5 b(y)56 b Fq(p)1691 2494 y Fp(c)1805 2469 y Fs(=)2002 2403 y Fn(1)p 2002 2430 V 2002 2526 a(2)2144 2469 y Fs([Gr,)g(Theorem)g(9.11].)76 b(Moreo)-5 b(v)g(er,)56 b(it)g(follo)-5 b(ws)57 b(from)f([Gr,)g(Eqs.)76 b(\(5.51\),)758 2668 y(\(9.95\),)54 b(\(9.97\)])h(that)g(there)g(are)g (\014nite)g(constan)-5 b(ts)57 b Fq(C)4150 2693 y Fn(2)4270 2668 y Fq(>)46 b Fs(0)56 b(and)f Fq(\027)4988 2693 y Fn(0)5109 2668 y Fo(\025)46 b Fs(1)56 b(suc)-5 b(h)56 b(that)2573 2963 y Fq(')p Fs(\()p Fq(p)p Fs(\))44 b Fo(\025)i Fq(C)3234 2988 y Fn(2)3336 2828 y Fl(\000)3432 2897 y Fn(1)p 3432 2924 V 3432 3020 a(2)3555 2963 y Fo(\000)37 b Fq(p)3805 2828 y Fl(\001)3881 2862 y Fp(\027)3948 2879 y Fe(0)4141 2963 y Fq(if)156 b(p)46 b(<)4760 2897 y Fn(1)p 4760 2924 V 4760 3020 a(2)4874 2963 y Fq(:)1391 b Fs(\(4.24\))957 3258 y(T)-14 b(o)71 b(mak)-5 b(e)70 b(the)h(connection)f(to)g(random)i (Landau)f(Hamiltonians)h(satisfying)f(\(R1\))f(and)758 3457 y(\(R2\),)55 b(giv)-5 b(en)55 b Fq(a)46 b Fo(\025)g Fs(0)56 b(w)-5 b(e)55 b(let)2129 3837 y Fq(p)p Fs(\()p Fq(a)p Fs(\))44 b(=)j(1)37 b Fo(\000)2937 3611 y Fl(Z)3103 3652 y Fp(a)3029 3988 y Fk(\0001)3301 3837 y Fq(g)6 b Fs(\()p Fq(t)p Fs(\))28 b(d)p Fq(t)45 b Fs(=)3996 3772 y Fn(1)p 3996 3799 V 3996 3894 a(2)4119 3837 y Fo(\000)4285 3703 y Fl(R)4396 3744 y Fp(a)4364 3896 y Fn(0)4504 3837 y Fq(g)6 b Fs(\()p Fq(t)p Fs(\))28 b(d)p Fq(t)g(;)1278 b Fs(\(4.25\))758 4221 y(and)56 b(set)3148 4506 y Fq(m)p Fs(\()p Fq(a)p Fs(\))45 b(=)h Fq(')p Fs(\()p Fq(p)p Fs(\()p Fq(a)p Fs(\)\))28 b Fq(:)1964 b Fs(\(4.26\))758 4802 y(\(Note)59 b(that)h Fq(p)p Fs(\()p Fq(a)p Fs(\))53 b(=)i(0)61 b(if)f Fq(a)55 b Fo(\025)g Fq(M)18 b Fs(.\))89 b(If)60 b(in)h(addition)h(w)-5 b(e)60 b(ha)-5 b(v)g(e)61 b(h)-5 b(yp)5 b(othesis)62 b(\(R3\),)f(it)f(follo)-5 b(ws)758 5001 y(from)55 b(\(4.5\))g(and)h(\(4.24\))e(that)1113 5287 y Fq(m)p Fs(\()p Fq(a)p Fs(\))44 b Fo(\025)j Fq(C)1816 5312 y Fn(3)1945 5287 y Fs(min)q Fo(f)p Fq(a;)28 b(M)18 b Fo(g)2729 5218 y Fp(\027)2858 5287 y Fs(for)45 b Fq(a)h Fo(\025)h Fs(0)28 b Fq(;)73 b(w)t(ith)46 b(\027)57 b Fs(=)46 b Fq(\027)4420 5312 y Fn(0)4495 5287 y Fq(\020)58 b(>)46 b Fs(0)28 b Fq(;)74 b(C)5151 5312 y Fn(3)5271 5287 y Fs(=)46 b Fq(C)5565 5312 y Fn(2)5640 5287 y Fq(C)5771 5213 y Fp(\027)5838 5230 y Fe(0)5759 5331 y Fn(1)5910 5287 y Fq(:)355 b Fs(\(4.27\))758 5583 y(\(In)52 b(fact)g Fq(m)p Fs(\()p Fq(a)p Fs(\))45 b(=)h Fo(1)53 b Fs(if)g Fq(a)46 b Fo(\025)g Fq(M)18 b Fs(.\))73 b(This)54 b(lo)-5 b(w)g(er)53 b(b)5 b(ound)54 b(on)f Fq(m)p Fs(\()p Fq(a)p Fs(\))f(will)h(pla)-5 b(y)53 b(a)g(crucial)g(role)g(in)758 5808 y(the)i(pro)5 b(of)55 b(of)g(Theorem)g(4.1)h(as)g(w)-5 b(e)55 b(let)g Fq(a)46 b Fs(=)3653 5728 y Fk(j)p Fp(E)6 b Fk(\000)p Fp(B)3999 5745 y Fi(n)4082 5728 y Fk(j)p 3653 5770 469 7 v 3854 5866 a Fn(2)4187 5808 y Fo(!)47 b Fs(0)55 b(as)h Fq(B)e Fo(!)46 b(1)p Fs(.)957 6008 y(W)-14 b(e)55 b(ma)-5 b(y)56 b(no)-5 b(w)56 b(restate)e([CH2,)h(Prop)5 b(osition)56 b(5.1])g(as)g(follo)-5 b(ws:)758 6316 y Fz(Theorem)63 b(4.3)h Fs(\([CH2)o(]\))p Fz(.)83 b Fg(L)-8 b(et)44 b Fq(H)2970 6341 y Fp(B)6 b(;!)3255 6316 y Fg(b)-8 b(e)43 b(the)h(r)-8 b(andom)43 b(L)-8 b(andau)43 b(Hamiltonian)i(\(as)e (in)56 b Fs(\(4.1\))o Fg(\),)758 6525 y(satisfying)42 b(\(R1\))h(and)f(\(R2\).)71 b(L)-8 b(et)42 b Fq(E)56 b Fs(=)46 b Fq(B)3382 6550 y Fp(n)3473 6525 y Fo(\006)p Fs(2)p Fq(a)c Fg(for)h(some)f Fq(n)k Fs(=)g(0)p Fq(;)28 b Fs(1)p Fq(;)g Fs(2)p Fq(;)g(:)g(:)g(:)f Fg(,)46 b(with)d Fs(0)j Fq(<)g(a)g(<)6558 6460 y Fp(B)p 6558 6487 106 7 v 6578 6582 a Fn(2)6684 6525 y Fg(.)758 6724 y(Ther)-8 b(e)82 b(exist)g(a)g(ge)-8 b(ometric)83 b(c)-8 b(onstant)83 b Fq(C)3414 6749 y Fn(4)3576 6724 y Fq(>)k Fs(0)p Fg(,)h(and)82 b(c)-8 b(onstants)83 b Fq(Y)5208 6749 y Fp(n)5386 6724 y Fq(>)88 b Fs(0)82 b Fg(and)g Fq(\014)6207 6749 y Fp(n)6385 6724 y Fq(>)87 b Fs(0)p Fg(,)758 6924 y(dep)-8 b(ending)71 b(only)g(on)g Fq(n)p Fg(,)i Fq(M)18 b Fg(,)73 b Fq(r)2764 6949 y Fp(u)2851 6924 y Fg(,)p Fo(kr)p Fq(u)p Fo(k)3301 6949 y Fk(1)3442 6924 y Fg(,)g(and)e Fo(k)p Fs(\001)p Fq(u)p Fo(k)4299 6949 y Fk(1)4440 6924 y Fg(,)i(such)e(that)g(for)f (any)h Fs(0)66 b Fq(<)h(\016)72 b Fo(\024)67 b Fq(a)p Fg(,)768 7123 y Fs(\025)-93 b Fq(x)46 b Fo(2)g Fs(\000)p Fg(,)59 b(and)g Fq(L)46 b Fo(2)g Fs(6)p Fm(N)26 b Fg(,)1228 7499 y Fm(P)1348 7265 y Fl(\032)1481 7499 y Fo(k)p Fs(\000)1676 7524 y Fn(\025)-74 b Fp(x;L)1882 7499 y Fq(R)2008 7524 y Fp(B)6 b(;!)t(;)i Fn(\025)-74 b Fp(x)q(;L)2495 7499 y Fs(\()p Fq(E)10 b Fs(\))p Fq(\037)2870 7529 y Fn(\025)-74 b Fp(x)n(;L=)p Fn(3)3209 7499 y Fo(k)46 b(\024)g Fq(Y)3609 7524 y Fp(n)3774 7387 y Fq(B)p 3720 7461 243 7 v 3720 7613 a(a\016)3888 7565 y Fn(2)4010 7499 y Fs(e)4084 7427 y Fk(\000)p Fp(\014)4263 7444 y Fi(n)4367 7427 y Fn(min)4587 7439 y Fo(f)4670 7427 y Fp(aB)6 b(;)4887 7329 y Fk(p)p 4996 7329 106 6 v 98 x Fp(B)5102 7439 y Fo(g)5193 7265 y Fl(\033)6311 7499 y Fs(\(4.28\))3960 7951 y Fo(\025)74 b Fs(1)36 b Fo(\000)i Fq(C)4568 7976 y Fn(4)4642 7951 y Fq(L)p Fs(e)4829 7882 y Fk(\000)p Fp(m)p Fn(\()p Fp(a)p Fn(\))p Fp(L)5362 7951 y Fo(\000)f Fq(Q)5659 7976 y Fp(n)5770 7838 y Fq(\016)6 b(B)p 5770 7913 215 7 v 5796 8065 a(a)5884 8017 y Fn(2)6004 7951 y Fq(L)6117 7882 y Fn(2)6219 7951 y Fq(;)758 8283 y Fg(wher)-8 b(e)60 b(the)f(c)-8 b(onstant)60 b Fq(Q)2295 8308 y Fp(n)2445 8283 y Fg(is)g(as)f(in)g(The)-8 b(or)g(em)59 b(4.2.)957 8591 y Fs(W)-14 b(e)74 b(are)h(no)-5 b(w)75 b(ready)f(to)h(pro)-5 b(v)g(e)75 b(Theorem)g(4.1.)132 b(As)75 b(in)g([CH2)o(],)80 b(Theorem)75 b(4.3)g(will)g(b)5 b(e)758 8791 y(used)69 b(to)f(obtain)h(the)f(initial)h(length)f (estimate)g(for)h(the)f(m)-5 b(ultiscale)69 b(analysis.)115 b(The)68 b(deca)-5 b(y)758 8990 y(giv)g(en)62 b(in)g(\(4.28\))f(is)i (indep)5 b(enden)-5 b(t)62 b(of)g(the)g(scale,)h(so)g(the)e(rate)h(of)f (exp)5 b(onen)-5 b(tial)62 b(deca)-5 b(y)62 b(giv)-5 b(en)758 9266 y(at)53 b(the)h(length)g(scale)h Fq(L)2260 9291 y Fn(0)2388 9266 y Fs(is)g Fo(\031)2749 9160 y Fn(2)p Fp(\014)2890 9177 y Fi(n)2995 9160 y Fn(min)3215 9172 y Fo(f)3298 9160 y Fp(aB)6 b(;)3515 9062 y Fk(p)p 3624 9062 106 6 v 98 x Fp(B)3730 9172 y Fo(g)p 2749 9227 1065 7 v 3203 9323 a Fp(L)3294 9340 y Fe(0)3833 9266 y Fs(.)74 b(Th)-5 b(us)55 b Fq(L)4485 9291 y Fn(0)4614 9266 y Fs(cannot)f(b)5 b(e)53 b(to)5 b(o)54 b(big,)h(dep)5 b(ending)758 9471 y(on)71 b(the)g(strength)f(of)h Fq(B)80 b Fs(through)71 b(the)f(constan)-5 b(ts)72 b(of)f(the)g(mo)5 b(del)71 b(and)g(on)h(the)e(distance)h Fq(a)758 9671 y Fs(to)55 b(the)g(Landau)i(lev)-5 b(el.)75 b(On)56 b(the)f(other)h(hand)g(the)g (m)-5 b(ultiscale)57 b(analysis)g(requires)f(the)f(initial)758 9870 y(length)44 b(scale)g Fq(L)1753 9895 y Fn(0)1872 9870 y Fs(to)g(b)5 b(e)44 b(large)g(enough,)j(dep)5 b(ending)45 b(also)g(on)f(the)g(strength)h(of)e Fq(B)53 b Fs(through)45 b(the)758 10069 y(constan)-5 b(ts)46 b(of)f(the)h(mo)5 b(del)45 b(and)h(on)g(the)f(distance)h Fq(a)g Fs(to)f(the)g(Landau)h (lev)-5 b(el.)71 b(F)-14 b(or)46 b(these)f(reasons)758 10268 y(Theorem)56 b(4.1)g(cannot)g(b)5 b(e)56 b(pro)-5 b(v)g(en)57 b(b)-5 b(y)56 b(a)g(naiv)-5 b(e)56 b(application)h(of)f (the)f(m)-5 b(ultiscale)58 b(analysis)f(as)758 10468 y(stated)50 b(in)i([FS,)f(vDK,)g(CH1)o(,)g(GK1)q(,)g(St].)72 b(Theorem)51 b(2.6)h(will)f(tak)-5 b(e)50 b(care)h(of)f(these)h (con\015icting)758 10667 y(demands.)p eop %%Page: 17 17 17 16 bop 1773 861 a Fn(EXPLICIT)58 b(FINITE)g(V)l(OLUME)h(CRITERIA)g (F)l(OR)g(LOCALIZA)-11 b(TION)883 b(17)758 1259 y Fg(Pr)-8 b(o)g(of)57 b(of)j(The)-8 b(or)g(em)58 b(4.1.)83 b Fs(Let)54 b(us)i(\014x)g Fq(n)45 b Fs(=)i(0)p Fq(;)28 b Fs(1)p Fq(;)g Fs(2)p Fq(;)g(:)g(:)g(:)e Fs(.)74 b(F)-14 b(or)56 b(eac)-5 b(h)56 b Fq(E)f Fo(2)46 b(B)5437 1284 y Fp(n)5527 1259 y Fo(nf)p Fq(B)5819 1284 y Fp(n)5910 1259 y Fo(g)55 b Fs(w)-5 b(e)56 b(set)2158 1656 y Fq(a)46 b Fs(=)g Fq(a)p Fs(\()p Fq(E)10 b Fs(\))44 b(=)3058 1544 y Fo(j)p Fq(E)i Fo(\000)37 b Fq(B)3565 1569 y Fp(n)3655 1544 y Fo(j)p 3058 1618 644 7 v 3338 1770 a Fs(2)3906 1656 y(\(note)91 b(0)46 b Fq(<)g(a)g Fo(\024)h Fq(M)18 b Fs(\))28 b Fq(;)1003 b Fs(\(4.29\))758 2005 y(and)56 b(de\014ne)f(op)5 b(en)55 b(in)-5 b(terv)c(als)2973 2306 y Fo(I)3063 2331 y Fp(E)3222 2306 y Fs(=)46 b(\()p Fq(E)g Fo(\000)37 b Fq(a;)28 b(E)45 b Fs(+)38 b Fq(a)p Fs(\))p Fq(;)1818 b Fs(\(4.30\))758 2609 y(where)55 b(w)-5 b(e)55 b(ha)-5 b(v)g(e)56 b(Assumptions)h(SLI,)f (NE,)f(and)h(W)f(with)g(constan)-5 b(ts)2193 2923 y Fq(\015)2279 2948 y Fk(I)2354 2965 y Fi(E)2625 2923 y Fs(=)166 b Fq(\015)3006 2948 y Fp(n)3096 2775 y Fo(p)p 3235 2775 135 7 v 3235 2923 a Fq(B)36 b(;)120 b Fs(with)101 b Fq(\015)4073 2948 y Fp(n)4209 2923 y Fs(=)53 b(~)-89 b Fq(\015)4480 2773 y Fl(p)p 4646 2773 598 7 v 150 x Fs(2\()p Fq(n)36 b Fs(+)h(1\))27 b(;)994 b(\(4.31\))2160 3258 y Fq(C)2279 3283 y Fk(I)2354 3300 y Fi(E)2625 3258 y Fs(=)166 b Fq(Q)3051 3283 y Fp(n)3161 3146 y Fq(B)p 3161 3220 135 7 v 3184 3372 a(a)3343 3258 y Fs(;)2922 b(\(4.32\))2148 3664 y Fq(Q)2279 3689 y Fk(I)2354 3706 y Fi(E)2625 3664 y Fs(=)166 b(4)p Fq(Q)3134 3689 y Fp(n)3258 3552 y Fq(B)p 3244 3626 163 7 v 3244 3778 a(a)3332 3730 y Fn(2)3454 3664 y Fq(;)120 b(\021)3702 3689 y Fk(I)3777 3706 y Fi(E)3929 3664 y Fs(=)4124 3599 y Fp(a)p 4124 3626 73 7 v 4127 3722 a Fn(2)4216 3664 y Fq(;)g(b)46 b Fs(=)g(1)28 b(;)1480 b(\(4.33\))758 4022 y(where)55 b(\(4.31\))f(follo)-5 b(ws)57 b(from)e(\(4.18\),)g(\(4.32\)) g(from)g(\(4.21\),)g(and)h(\(4.33\))f(from)g(\(4.22\))o(.)957 4221 y(T)-14 b(o)43 b(apply)g(Theorem)g(2.6)h(at)e(a)h(giv)-5 b(en)43 b(energy)g Fq(E)55 b Fo(2)46 b(B)4278 4246 y Fp(n)4369 4221 y Fo(nf)p Fq(B)4661 4246 y Fp(n)4751 4221 y Fo(g)p Fs(,)g(w)-5 b(e)43 b(m)-5 b(ust)44 b(pic)-5 b(k)43 b(appropriate)758 4447 y Fq(p)i(>)i Fs(0,)k Fq(\013)c Fs(=)g(1)28 b(+)1876 4366 y Fp(p=)p Fn(2)p 1860 4409 237 7 v 1860 4504 a Fp(p)p Fn(+4)2116 4447 y Fs(,)52 b Fq(s)46 b(>)g Fs(2)p Fq(p)28 b Fs(+)g(6,)51 b(and)h Fq(\022)e(>)d Fs(4)p Fq(\013)q(s)p Fs(,)k(and)h(use)f(the)f(estimates)h (of)g(Theorem)g(4.3)758 4668 y(to)64 b(satisfy)k(\(2.22\))c(at)g(some)i (length)f(scale)g Fq(L)3665 4693 y Fn(0)3802 4668 y Fo(\025)d(L)p Fs(,)67 b(with)e Fo(L)c Fs(=)h Fo(L)p Fs(\()p Fq(E)10 b(;)28 b(B)8 b Fs(\))63 b(giv)-5 b(en)65 b(b)-5 b(y)68 b(\(2.21\))758 4867 y(with)55 b Fo(I)k Fs(=)46 b Fo(I)1550 4892 y Fp(E)1663 4867 y Fs(.)957 5076 y(W)-14 b(e)65 b(start)g(b)-5 b(y)66 b(\014xing)f Fq(p)e(>)g Fs(0,)68 b(to)d(b)5 b(e)65 b(sp)5 b(eci\014ed)66 b(later,)h(and)f(setting)g Fq(\013)d Fs(=)g(1)44 b(+)6066 4996 y Fp(p=)p Fn(2)p 6049 5038 V 6049 5134 a Fp(p)p Fn(+4)6305 5076 y Fs(;)71 b(note)758 5313 y(that)54 b(1)47 b Fq(<)f(\013)h(<)1770 5248 y Fn(3)p 1770 5275 67 7 v 1770 5370 a(2)1856 5313 y Fs(.)74 b(W)-14 b(e)55 b(will)h(c)-5 b(ho)5 b(ose)55 b Fq(s)h Fs(and)f Fq(\022)60 b Fs(later)55 b(dep)5 b(ending)56 b(on)g Fq(a;)28 b(B)8 b Fs(,)55 b(and)h Fq(p)p Fs(.)957 5512 y(Giv)-5 b(en)56 b Fq(E)f Fo(2)46 b(B)1882 5537 y Fp(n)1972 5512 y Fo(nf)p Fq(B)2264 5537 y Fp(n)2355 5512 y Fo(g)p Fs(,)55 b(to)g(satisfy)j(\(2.22\))d(from)g(\(4.28\))o(,)h (it)f(su\016ces)h(to)f(v)-5 b(erify)2484 5899 y Fq(Y)2580 5924 y Fp(n)2744 5786 y Fq(B)p 2690 5860 243 7 v 2690 6013 a(a\016)2858 5965 y Fn(2)2980 5899 y Fs(e)3054 5830 y Fk(\000)p Fp(\014)3233 5847 y Fi(n)3337 5830 y Fn(min)q Fk(f)p Fp(aB)6 b(;)3842 5732 y Fk(p)p 3952 5732 106 6 v 3952 5830 a Fp(B)g Fk(g)4300 5899 y Fo(\024)4668 5786 y Fs(1)p 4615 5860 189 7 v 4615 6018 a Fq(L)4728 5961 y Fp(\022)4728 6063 y Fn(0)4851 5899 y Fq(;)1414 b Fs(\(4.34\))3090 6348 y Fq(C)3209 6373 y Fn(4)3283 6348 y Fq(L)3396 6373 y Fn(0)3499 6348 y Fs(e)3573 6280 y Fk(\000)p Fp(m)p Fn(\()o Fp(a)p Fn(\))p Fp(L)4061 6297 y Fe(0)4300 6348 y Fq(<)4710 6236 y Fs(1)p 4615 6310 273 7 v 4615 6462 a(2)p Fq(L)4811 6382 y Fp(p)4811 6506 y Fn(0)4936 6348 y Fq(;)1329 b Fs(\(4.35\))3471 6801 y Fq(Q)3602 6826 y Fp(n)3712 6688 y Fq(\016)6 b(B)p 3712 6762 215 7 v 3738 6914 a(a)3826 6866 y Fn(2)3947 6801 y Fq(L)4060 6732 y Fn(2)4060 6842 y(0)4300 6801 y Fo(\024)4710 6688 y Fs(1)p 4615 6762 273 7 v 4615 6914 a(2)p Fq(L)4811 6835 y Fp(p)4811 6959 y Fn(0)4936 6801 y Fq(;)1329 b Fs(\(4.36\))758 7193 y(for)55 b(some)h(0)46 b Fq(<)g(\016)52 b(<)47 b(a)55 b Fs(and)h(an)f(appropriate)h(c)-5 b(hoice)56 b(of)f(the)g(initial)g(length)h(scale)f Fq(L)6092 7218 y Fn(0)6194 7193 y Fs(.)957 7392 y(W)-14 b(e)55 b(start)g(with)g (condition)h(\(4.35\))o(.)74 b(Using)58 b(\(4.27\),)d(it)g(su\016ces)i (to)e(establish)2784 7772 y Fq(C)2903 7797 y Fn(4)2977 7772 y Fq(L)3090 7797 y Fn(0)3192 7772 y Fs(e)3266 7704 y Fk(\000)p Fp(C)3465 7721 y Fe(3)3552 7704 y Fp(a)3624 7653 y Fi(\027)3697 7704 y Fp(L)3788 7721 y Fe(0)4028 7772 y Fq(<)4438 7660 y Fs(1)p 4343 7734 V 4343 7886 a(2)p Fq(L)4539 7806 y Fp(p)4539 7930 y Fn(0)4663 7772 y Fq(:)1602 b Fs(\(4.37\))758 8216 y(T)-14 b(o)55 b(do)h(so,)g(w)-5 b(e)55 b(pic)-5 b(k)56 b(0)46 b Fq(<)2403 8174 y Fl(f)2382 8216 y Fq(M)64 b(<)47 b(M)73 b Fs(and)56 b(c)-5 b(ho)5 b(ose)56 b(\(k)-5 b(eeping)55 b(in)h(mind)g(\(2.21\))o(\))1175 8671 y Fq(L)1288 8696 y Fn(0)1408 8671 y Fs(=)47 b Fq(L)1697 8696 y Fn(0)1771 8671 y Fs(\()p Fq(a)p Fs(\))e(=)2209 8387 y Fl(")2306 8671 y Fs(max)2643 8387 y Fl(\()2797 8549 y Fq(C)2916 8574 y Fn(5)3018 8549 y Fs(log)3280 8484 y Fp(M)p 3280 8511 140 7 v 3313 8606 a(a)p 2797 8633 642 7 v 3032 8785 a Fq(a)3120 8737 y Fp(\027)3459 8671 y Fq(;)3552 8521 y(C)3671 8546 y Fn(5)3773 8521 y Fs(log)4035 8456 y Fp(M)p 4035 8483 140 7 v 4049 8577 a Fd(f)4035 8606 y Fp(M)p 3552 8633 642 7 v 3763 8777 a Fl(f)3742 8819 y Fq(M)3921 8771 y Fp(\027)4214 8387 y Fl(\))4385 8671 y Fs(+)37 b(6)4634 8387 y Fl(#)4731 8887 y Fn(6)p Fj(N)4935 8671 y Fo(\025)5130 8549 y Fq(C)5249 8574 y Fn(5)5351 8549 y Fs(log)5613 8484 y Fp(M)p 5613 8511 140 7 v 5647 8606 a(a)p 5130 8633 642 7 v 5366 8785 a Fq(a)5454 8737 y Fp(\027)5820 8671 y Fq(;)445 b Fs(\(4.38\))758 9172 y(where)42 b(the)f(constan)-5 b(t)43 b Fq(C)2273 9197 y Fn(5)2389 9172 y Fs(is)g(c)-5 b(hosen)43 b(large)f(enough)h(so)g (that)e(\(4.37\))h(holds)h(for)f(all)g(0)47 b Fq(<)f(a)g Fo(\024)6530 9130 y Fl(f)6510 9172 y Fq(M)18 b Fs(,)758 9471 y(and)54 b(hence)f(for)g(all)h(0)46 b Fq(<)g(a)g Fo(\024)g Fq(M)18 b Fs(,)54 b(and)g(w)-5 b(e)54 b(also)g(ha)-5 b(v)g(e)53 b Fq(C)4305 9496 y Fn(5)4426 9471 y Fq(>)46 b Fs(max)4938 9237 y Fl(\032)5082 9391 y Fn(\()p Fp(p)p Fn(+1\)\()p Fp(\027)8 b Fn(+1\))p 5082 9433 687 7 v 5345 9528 a Fp(C)5440 9545 y Fe(3)5789 9471 y Fq(;)5882 9406 y Fn(417)6080 9333 y Fe(2+)6245 9301 y(8)p 6243 9314 62 5 v 6243 9377 a Fi(p)42 b Fd(f)6332 9406 y Fp(M)6471 9356 y Fi(\027)p 5882 9433 663 7 v 6039 9541 a Fn(log)6253 9496 y Fi(M)p 6253 9514 116 6 v 6272 9583 a Fc(f)6253 9604 y Fi(M)6565 9237 y Fl(\033)6689 9471 y Fs(.)758 9812 y(The)k(constan)-5 b(t)47 b Fq(C)1870 9837 y Fn(5)1991 9812 y Fs(dep)5 b(ends)48 b(only)e(on)h Fq(C)3321 9837 y Fn(3)3396 9812 y Fs(,)h Fq(M)18 b Fs(,)3784 9770 y Fl(f)3764 9812 y Fq(M)g Fs(,)49 b Fq(\027)11 b Fs(,)48 b(and)f Fq(p)f Fs(\()p Fq(C)4853 9837 y Fn(4)4974 9812 y Fs(is)h(a)g(geometric)f(constan)-5 b(t\).)758 10011 y(By)54 b(construction)2719 10457 y Fq(L)2832 10482 y Fn(0)2906 10457 y Fs(\()p Fq(a)p Fs(\))45 b Fo(\025)3364 10307 y Fq(C)3483 10332 y Fn(5)3585 10307 y Fs(log)3847 10242 y Fp(M)p 3847 10269 140 7 v 3861 10363 a Fd(f)3847 10392 y Fp(M)p 3364 10419 642 7 v 3575 10563 a Fl(f)3554 10605 y Fq(M)3733 10557 y Fp(\027)4072 10457 y Fo(\025)i Fs(417)4497 10383 y Fn(2+)4687 10339 y Fe(8)p 4685 10357 62 6 v 4685 10423 a Fi(p)6311 10457 y Fs(\(4.39\))758 10840 y(for)55 b(all)g(0)47 b Fq(<)f(a)g Fo(\024)g Fq(M)18 b Fs(.)p eop %%Page: 18 18 18 17 bop 758 861 a Fn(18)1485 b(FRANC)-77 b(\030)17 b(OIS)59 b(GERMINET)g(AND)f(ABEL)g(KLEIN)957 1259 y Fs(W)-14 b(e)55 b(no)-5 b(w)56 b(\014x)f Fq(K)1965 1284 y Fp(n)2102 1259 y Fq(>)46 b Fs(0,)55 b(to)g(b)5 b(e)55 b(sp)5 b(eci\014ed)56 b(later,)f(and)h(require)2865 1645 y Fq(K)3006 1670 y Fp(n)3116 1532 y Fs(log)30 b Fq(B)p 3116 1607 377 7 v 3237 1759 a(B)3559 1645 y Fo(\024)46 b Fq(a)g Fo(\024)g Fq(M)g(:)2015 b Fs(\(4.40\))758 2038 y(F)-14 b(or)56 b Fq(B)63 b Fs(large)56 b(enough)g(\(so)f Fq(K)2633 2063 y Fp(n)2751 2038 y Fs(log)30 b Fq(B)55 b Fo(\024)3349 1898 y(p)p 3487 1898 135 7 v 140 x Fq(B)9 b Fs(\),)55 b(it)g(follo)-5 b(ws)56 b(from)f(\(4.40\))g(that)2775 2368 y(min)q Fo(f)p Fq(aB)8 b(;)3431 2219 y Fo(p)p 3569 2219 V 149 x Fq(B)h Fo(g)46 b(\025)g Fq(K)4149 2393 y Fp(n)4267 2368 y Fs(log)30 b Fq(B)36 b(:)1594 b Fs(\(4.41\))758 2677 y(W)-14 b(e)55 b(ha)-5 b(v)g(e)1033 3293 y Fq(L)1146 3318 y Fn(0)1220 3293 y Fs(\()p Fq(a)p Fs(\))45 b Fo(\024)1659 2803 y Fl(8)1659 2953 y(>)1659 3003 y(>)1659 3052 y(>)1659 3102 y(<)1659 3401 y(>)1659 3451 y(>)1659 3501 y(>)1659 3551 y(:)1889 2994 y Fq(L)2002 3019 y Fn(0)2104 2809 y Fl(\020)2203 2994 y Fq(K)2344 3019 y Fp(n)2455 2919 y Fn(log)25 b Fp(B)p 2455 2955 301 7 v 2552 3051 a(B)2775 2809 y Fl(\021)2920 2994 y Fo(\024)3132 2925 y Fp(C)3227 2942 y Fe(5)p 3116 2955 194 7 v 3116 3051 a Fp(K)3236 3018 y Fi(\027)3228 3084 y(n)3356 2994 y Fs(\(log)30 b Fq(B)8 b Fs(\))3862 2910 y Fn(1)p Fk(\000)p Fp(\027)4142 2994 y Fq(B)4276 2933 y Fp(\027)4525 2994 y Fs(if)46 b Fq(K)4809 3019 y Fp(n)4919 2919 y Fn(log)26 b Fp(B)p 4919 2955 301 7 v 5017 3051 a(B)5286 2994 y Fo(\024)46 b Fq(a)g Fo(\024)5865 2928 y Fn(1)p 5790 2955 216 7 v 5790 2973 a Fk(p)p 5899 2973 106 6 v 99 x Fp(B)1889 3292 y Fq(L)2002 3317 y Fn(0)2104 3108 y Fl(\020)2298 3227 y Fn(1)p 2223 3254 216 7 v 2223 3272 a Fk(p)p 2332 3272 106 6 v 99 x Fp(B)2458 3108 y Fl(\021)2604 3292 y Fo(\024)g Fq(C)2898 3317 y Fn(5)3000 3292 y Fs(\()o(log)31 b Fq(B)8 b Fs(\))27 b Fq(B)3687 3188 y Fi(\027)p 3687 3206 65 6 v 3691 3272 a Fe(2)4525 3292 y Fs(if)4762 3227 y Fn(1)p 4688 3254 216 7 v 4688 3272 a Fk(p)p 4797 3272 106 6 v 99 x Fp(B)4969 3292 y Fo(\024)46 b Fq(a)g Fo(\024)5474 3250 y Fl(f)5453 3292 y Fq(M)1889 3616 y(L)2002 3641 y Fn(0)2077 3616 y Fs(\()2162 3574 y Fl(f)2142 3616 y Fq(M)17 b Fs(\))46 b Fo(\024)2626 3502 y Fp(C)2721 3519 y Fe(5)2809 3502 y Fn(log)3023 3457 y Fi(M)p 3023 3475 116 6 v 3043 3543 a Fc(f)3023 3564 y Fi(M)p 2626 3578 533 7 v 2801 3672 a Fd(f)2786 3701 y Fp(M)2925 3668 y Fi(\027)3216 3616 y Fs(+)37 b(6)1060 b(if)4688 3574 y Fl(f)4668 3616 y Fq(M)64 b Fo(\024)47 b Fq(a)e Fo(\024)i Fq(M)6311 3293 y Fs(\(4.42\))758 3993 y(for)57 b Fq(B)i Fo(\025)51 b Fz(B)1515 4018 y Fn(1)1589 3993 y Fs(\()p Fq(n)p Fs(\),)58 b(with)g Fz(B)2440 4018 y Fn(1)2514 3993 y Fs(\()p Fq(n)p Fs(\))f(dep)5 b(ending)58 b(only)g(only)g(on)g Fq(C)4687 4018 y Fn(3)4762 3993 y Fs(,)g Fq(M)18 b Fs(,)5171 3951 y Fl(f)5150 3993 y Fq(M)h Fs(,)58 b Fq(\027)11 b Fs(,)59 b Fq(p)p Fs(,)f(and)g Fq(K)6286 4018 y Fp(n)6377 3993 y Fs(.)82 b(W)-14 b(e)758 4192 y(tak)-5 b(e)54 b Fz(B)1253 4217 y Fn(1)1328 4192 y Fs(\()p Fq(n)p Fs(\))g(su\016cien)-5 b(tly)56 b(large)f(so)h(w)-5 b(e)56 b(also)g(ha)-5 b(v)g(e)2164 4501 y(log)30 b Fq(L)2519 4526 y Fn(0)2594 4501 y Fs(\()p Fq(a)p Fs(\))45 b Fo(\024)h Fs(2)p Fq(\027)38 b Fs(log)30 b Fq(B)202 b Fs(for)55 b(all)h Fq(a)f Fs(as)h(in)g(\(4.40\))27 b Fq(:)982 b Fs(\(4.43\))957 4819 y(T)-14 b(o)55 b(satisfy)j(\(4.36\))o (,)e(w)-5 b(e)55 b(set)2463 5234 y Fq(\016)d Fs(=)47 b Fq(\016)6 b Fs(\()p Fq(a;)28 b(B)8 b Fs(\))45 b(=)3974 5122 y Fq(a)4062 5061 y Fn(2)p 3511 5196 1089 7 v 3511 5348 a Fs(2)p Fq(Q)3725 5373 y Fp(n)3816 5348 y Fq(B)8 b(L)4063 5373 y Fn(0)4137 5348 y Fs(\()p Fq(a)p Fs(\))4355 5300 y Fp(p)p Fn(+2)4665 5234 y Fq(<)47 b(a)28 b(;)1308 b Fs(\(4.44\))758 5640 y(the)59 b(last)i(inequalit)-5 b(y)60 b(holding)h(for)f(all)h Fq(a)f Fs(as)h(in)f(\(4.40\))g(and)g Fq(B)j Fo(\025)54 b Fz(B)5108 5665 y Fn(2)5183 5640 y Fs(\()p Fq(n)p Fs(\))59 b(in)h(view)g(of)73 b(\(4.39\))o(,)758 5864 y(where)55 b(the)g(constan)-5 b(t)55 b Fz(B)2329 5889 y Fn(2)2404 5864 y Fs(\()p Fq(n)p Fs(\))f(dep)5 b(ends)56 b(only)f(on)h Fq(M)18 b Fs(,)4226 5822 y Fl(f)4206 5864 y Fq(M)g Fs(,)55 b Fq(\027)11 b Fs(,)55 b Fq(C)4799 5889 y Fn(3)4874 5864 y Fs(,)g Fq(Q)5106 5889 y Fp(n)5197 5864 y Fs(,)g Fq(p)p Fs(,)g(and)h Fq(K)5947 5889 y Fp(n)6037 5864 y Fs(.)957 6064 y(It)f(remains)h(to)f(v)-5 b(erify)55 b(condition)h(\(4.34\))e(for)h(this)h(c)-5 b(hoice)56 b(of)f Fq(\016)6 b Fs(,)55 b(whic)-5 b(h)56 b(follo)-5 b(ws)57 b(if)1845 6450 y(4)p Fq(Y)2024 6475 y Fp(n)2115 6450 y Fq(Q)2246 6382 y Fn(2)2246 6491 y Fp(n)2336 6450 y Fq(B)2470 6382 y Fn(3)2545 6450 y Fq(a)2633 6382 y Fk(\000)p Fn(5)2839 6450 y Fs(e)2913 6382 y Fk(\000)p Fp(\014)3092 6399 y Fi(n)3196 6382 y Fn(min)q Fk(f)p Fp(aB)6 b(;)3701 6283 y Fk(p)p 3810 6283 106 6 v 99 x Fp(B)g Fk(g)4039 6450 y Fo(\024)4687 6338 y Fs(1)p 4234 6412 988 7 v 4234 6567 a Fq(L)4347 6592 y Fn(0)4422 6567 y Fs(\()p Fq(a)p Fs(\))4640 6519 y Fp(\022)t Fn(+2\()p Fp(p)p Fn(+2\))5270 6450 y Fq(;)995 b Fs(\(4.45\))758 6845 y(i.e.,)1586 7280 y Fq(\022)51 b Fo(\024)1910 7157 y Fq(\014)2004 7182 y Fp(n)2122 7157 y Fs(min)p Fo(f)p Fq(aB)8 b(;)2777 7017 y Fo(p)p 2916 7017 135 7 v 2916 7157 a Fq(B)g Fo(g)37 b(\000)g Fs(log)3578 7023 y Fl(\000)3654 7157 y Fs(4)p Fq(Y)3833 7182 y Fp(n)3924 7157 y Fq(Q)4055 7097 y Fn(2)4055 7198 y Fp(n)4146 7157 y Fq(B)4280 7097 y Fn(3)4354 7157 y Fq(a)4442 7097 y Fk(\000)p Fn(5)4620 7023 y Fl(\001)p 1910 7242 2787 7 v 2980 7394 a Fs(log)30 b Fq(L)3335 7419 y Fn(0)3410 7394 y Fs(\()p Fq(a)p Fs(\))4753 7280 y Fo(\000)37 b Fs(2\()p Fq(p)f Fs(+)i(2\))28 b Fq(:)735 b Fs(\(4.46\))758 7677 y(W)-14 b(e)55 b(tak)-5 b(e)2396 8069 y Fq(\022)50 b Fs(=)c Fq(\022)5 b Fs(\()p Fq(a;)28 b(B)8 b Fs(\))45 b(=)3448 7956 y Fq(\014)3551 7896 y Fk(0)3542 7997 y Fp(n)3660 7956 y Fs(min)q Fo(f)p Fq(aB)8 b(;)4316 7816 y Fo(p)p 4454 7816 135 7 v 140 x Fq(B)h Fo(g)p 3448 8031 1224 7 v 3737 8183 a Fs(log)30 b Fq(L)4092 8208 y Fn(0)4166 8183 y Fs(\()p Fq(a)p Fs(\))4719 8069 y Fq(;)1546 b Fs(\(4.47\))758 8465 y(where)55 b(w)-5 b(e)55 b(require)2647 8860 y Fq(\014)2750 8792 y Fk(0)2741 8901 y Fp(n)2877 8860 y Fs(=)47 b Fq(\014)3147 8885 y Fp(n)3274 8860 y Fo(\000)3460 8748 y Fs(8)37 b(+)g(4)p Fq(\027)11 b Fs(\()p Fq(p)35 b Fs(+)j(2\))p 3460 8822 961 7 v 3824 8974 a Fq(K)3965 8999 y Fp(n)4486 8860 y Fq(>)46 b Fs(0)28 b Fq(;)1493 b Fs(\(4.48\))758 9249 y(and,)49 b(using)i(\(4.43\))c(and)h(\(4.41\),)h(c)-5 b(hec)g(k)48 b(that)f Fq(\022)5 b Fs(\()p Fq(a;)28 b(B)8 b Fs(\))46 b(satis\014es)i(\(4.46\))f(for)h(all)f Fq(a)h Fs(as)g(in)g(\(4.40\))o(,)758 9474 y(as)62 b(long)h(as)g(and)f Fq(B)k Fo(\025)58 b Fz(B)2390 9499 y Fn(3)2464 9474 y Fs(\()p Fq(n)p Fs(\),)63 b(where)f(the)g(constan)-5 b(t)62 b Fz(B)4395 9499 y Fn(3)4470 9474 y Fs(\()p Fq(n)p Fs(\))f(dep)5 b(ends)62 b(only)h(on)f Fq(M)18 b Fs(,)6328 9432 y Fl(f)6307 9474 y Fq(M)h Fs(,)64 b Fq(\027)11 b Fs(,)758 9673 y Fq(C)877 9698 y Fn(3)951 9673 y Fs(,)55 b Fq(Y)1148 9698 y Fp(n)1239 9673 y Fs(,)h Fq(Q)1472 9698 y Fp(n)1562 9673 y Fs(,)g Fq(p)p Fs(,)f(and)h Fq(K)2313 9698 y Fp(n)2403 9673 y Fs(.)957 9872 y(W)-14 b(e)60 b(no)-5 b(w)61 b(c)-5 b(ho)5 b(ose)62 b Fq(s)54 b Fs(=)i Fq(s)p Fs(\()p Fq(a;)28 b(B)8 b Fs(\))53 b Fq(>)j Fs(2)p Fq(p)40 b Fs(+)g(6,)62 b(satisfying)f Fq(\022)5 b Fs(\()p Fq(a;)28 b(B)8 b Fs(\))54 b Fq(>)h Fs(4)p Fq(\013)q(s)p Fs(\()p Fq(a;)28 b(B)8 b Fs(\),)61 b(and)g(sho)-5 b(w)758 10072 y(that)64 b(for)h(su\016cien) -5 b(tly)66 b(large)f Fq(B)74 b Fs(w)-5 b(e)65 b(ha)-5 b(v)g(e,)68 b(for)d(all)h Fq(a)f Fs(as)g(in)h(\(4.40\))o(,)i(that)d Fq(L)5639 10097 y Fn(0)5713 10072 y Fs(\()p Fq(a)p Fs(\))c Fo(\025)i(L)p Fs(,)k(with)758 10271 y Fo(L)45 b Fs(=)i Fo(L)p Fs(\()p Fq(E)10 b(;)28 b(B)8 b Fs(\))53 b(giv)-5 b(en)55 b(b)-5 b(y)58 b(\(2.21\))d(with)g Fo(I)k Fs(=)46 b Fo(I)3671 10296 y Fp(E)3784 10271 y Fs(.)957 10470 y(T)-14 b(o)55 b(satisfy)1296 10840 y Fq(Q)1596 10682 y Fe(1)p 1447 10700 354 6 v 1447 10766 a Fi(s)p Ff(\000)p Fi(p)p Ff(\000)p Fe(2)1427 10889 y Fk(I)1502 10906 y Fi(E)1875 10840 y Fo(\024)47 b Fq(L)2164 10865 y Fn(0)2238 10840 y Fs(\()p Fq(a)p Fs(\))91 b(and)2935 10705 y Fl(\000)3011 10840 y Fs(16)37 b Fo(\001)g Fs(36)3463 10771 y Fn(2)3537 10840 y Fs(2)3620 10771 y Fp(s)p Fk(\000)p Fn(2)p Fp(p)3930 10840 y Fq(C)4049 10865 y Fk(I)4124 10882 y Fi(E)4228 10840 y Fq(Q)4359 10865 y Fk(I)4434 10882 y Fi(E)4539 10705 y Fl(\001)4894 10693 y Fe(1)p 4635 10711 576 6 v 4635 10779 a Fi(\013)p Fe(\()p Fi(s)p Ff(\000)p Fe(2)p Fi(p)p Ff(\000)p Fe(6\))5285 10840 y Fo(\024)46 b Fq(L)5573 10865 y Fn(0)5648 10840 y Fs(\()p Fq(a)p Fs(\))445 b(\(4.49\))p eop %%Page: 19 19 19 18 bop 1773 861 a Fn(EXPLICIT)58 b(FINITE)g(V)l(OLUME)h(CRITERIA)g (F)l(OR)g(LOCALIZA)-11 b(TION)883 b(19)758 1259 y Fs(for)48 b(large)g Fq(B)57 b Fs(and)48 b(all)h Fq(a)f Fs(as)h(in)f(\(4.40\),)h (it)f(su\016ces)i(to)e(require)g(\()p Fq(s)23 b Fo(\000)g Fq(p)g Fo(\000)g Fs(2\))28 b(log)f Fq(L)5735 1284 y Fn(0)5810 1259 y Fs(\()p Fq(a)p Fs(\))45 b Fo(\025)h Fs(3)28 b(log)i Fq(B)758 1458 y Fs(and)56 b(\()p Fq(s)36 b Fo(\000)h Fs(2)p Fq(p)f Fo(\000)h Fs(6\))28 b(log)i Fq(L)2326 1483 y Fn(0)2400 1458 y Fs(\()p Fq(a)p Fs(\))45 b Fo(\025)i Fs(5)28 b(log)h Fq(B)8 b Fs(,)56 b(resp)5 b(ectiv)-5 b(ely)-14 b(.)73 b(Th)-5 b(us)57 b(w)-5 b(e)55 b(set)2513 1882 y Fq(s)46 b Fs(=)g Fq(s)p Fs(\()p Fq(a;)28 b(B)8 b Fs(\))45 b(=)3636 1770 y(5)28 b(log)i Fq(B)p 3556 1844 647 7 v 3556 1996 a Fs(log)h Fq(L)3912 2021 y Fn(0)3986 1996 y Fs(\()p Fq(a)p Fs(\))4260 1882 y(+)37 b(2)p Fq(p)f Fs(+)h(6)28 b Fq(;)1359 b Fs(\(4.50\))758 2316 y(and)62 b(satisfy)j(\(4.49\))d(for)g(all)g Fq(a)g Fs(as)h(in)f(\(4.40\))g(if)g Fq(B)k Fo(\025)58 b Fz(B)4322 2341 y Fn(4)4396 2316 y Fs(\()p Fq(n)p Fs(\),)63 b(with)f Fz(B)5256 2341 y Fn(4)5330 2316 y Fs(\()p Fq(n)p Fs(\))f(dep)5 b(ending)63 b(only)758 2541 y(only)c(on)h Fq(M)18 b Fs(,)1667 2499 y Fl(f)1646 2541 y Fq(M)h Fs(,)60 b Fq(C)2051 2566 y Fn(3)2125 2541 y Fs(,)h Fq(\027)11 b Fs(,)60 b Fq(p)p Fs(,)g Fq(Q)2752 2566 y Fp(n)2843 2541 y Fs(,)g(and)g Fq(K)3417 2566 y Fp(n)3508 2541 y Fs(.)86 b(Moreo)-5 b(v)g(er,)60 b(since)g(it)f(follo) -5 b(ws)61 b(from)e(\(4.43\))g(that)758 2740 y Fq(s)p Fs(\()p Fq(a;)28 b(B)8 b Fs(\))45 b Fo(\025)1539 2675 y Fn(5)p 1502 2702 141 7 v 1502 2797 a(2)p Fp(\027)1708 2740 y Fq(>)1908 2675 y Fn(1)p 1904 2702 75 7 v 1904 2797 a Fp(\027)2052 2740 y Fs(for)54 b Fq(B)g Fo(\025)47 b Fz(B)2797 2765 y Fn(1)2871 2740 y Fs(\()p Fq(n)p Fs(\),)53 b(w)-5 b(e)55 b(ma)-5 b(y)54 b(tak)-5 b(e)54 b Fz(B)4297 2765 y Fn(4)4371 2740 y Fs(\()p Fq(n)p Fs(\))f(large)h(enough)h(so)g(w) -5 b(e)54 b(also)h(ha)-5 b(v)g(e)758 3009 y Fq(\021)846 2912 y Fk(\000)970 2867 y Fe(1)p 970 2885 57 6 v 970 2952 a Fi(s)840 3058 y Fk(I)915 3075 y Fi(E)1055 3009 y Fq(;)28 b Fs(dist\()p Fq(E)10 b(;)28 b Fm(R)20 b Fo(nI)1952 3034 y Fp(E)2073 3009 y Fs(\))2138 2949 y Fk(\000)2261 2904 y Fe(1)p 2261 2922 V 2261 2989 a Fi(s)2392 3009 y Fq(<)46 b(L)2680 3034 y Fn(0)2755 3009 y Fs(\()p Fq(a)p Fs(\))54 b(for)h Fq(B)f Fo(\025)47 b Fz(B)3773 3034 y Fn(4)3847 3009 y Fs(\()p Fq(n)p Fs(\).)957 3232 y(In)55 b(view)g(of)68 b(\(4.31\),)55 b Fq(\013)47 b(<)2623 3166 y Fn(3)p 2623 3193 67 7 v 2623 3289 a(2)2709 3232 y Fs(,)56 b(\(4.47\))o(,)g(\(4.50\))o(,)f(and)h(\(4.43\),)f(if)g(w)-5 b(e)56 b(require)2799 3680 y Fq(\014)2902 3612 y Fk(0)2893 3721 y Fp(n)3030 3680 y Fq(>)3225 3568 y Fs(34)37 b(+)g(4)p Fq(\027)11 b Fs(\()p Fq(p)35 b Fs(+)i(3\))p 3225 3642 1044 7 v 3631 3794 a Fq(K)3772 3819 y Fp(n)4334 3680 y Fq(>)46 b Fs(0)28 b Fq(;)1645 b Fs(\(4.51\))758 4089 y(w)-5 b(e)55 b(ha)-5 b(v)g(e)2898 4309 y Fl(\000)2974 4444 y Fs(69)3140 4375 y Fn(2)3215 4444 y Fq(\015)3310 4375 y Fn(2)3301 4485 y Fk(I)3376 4502 y Fi(E)3480 4309 y Fl(\001)3716 4297 y Fe(4)p 3576 4315 338 6 v 3576 4381 a Fi(\022)s Ff(\000)p Fe(4)p Fi(\013s)3988 4444 y Fo(\024)46 b Fq(L)4276 4469 y Fn(0)4350 4444 y Fs(\()p Fq(a)p Fs(\))1743 b(\(4.52\))758 4833 y(if)55 b Fq(B)f Fo(\025)47 b Fz(B)1402 4858 y Fn(5)1476 4833 y Fs(\()p Fq(n)p Fs(\),)54 b(with)h Fz(B)2320 4858 y Fn(5)2395 4833 y Fs(\()p Fq(n)p Fs(\))f(dep)5 b(ending)56 b(only)f(only)h(on)f Fq(M)18 b Fs(,)4737 4791 y Fl(f)4717 4833 y Fq(M)g Fs(,)55 b Fq(C)5116 4858 y Fn(3)5191 4833 y Fs(,)g Fq(\027)11 b Fs(,)55 b Fq(p)p Fs(,)g(and)h Fq(K)6135 4858 y Fp(n)6225 4833 y Fs(.)957 5032 y(Recalling)i(\(4.39\))o(,)e(it)f(only)g(remains)i(to)e(satisfy) 2885 5294 y Fl(\000)2962 5428 y Fs(3)3045 5360 y Fn(2)3119 5428 y Fq(\015)3205 5453 y Fk(I)3280 5470 y Fi(E)3385 5294 y Fl(\001)3572 5281 y Fe(16)p Fi(\013)p 3481 5299 372 6 v 3481 5368 a(\022)s Fe(\()p Fi(\013)p Ff(\000)p Fe(1\))3926 5428 y Fo(\024)47 b Fq(L)4215 5453 y Fn(0)4289 5428 y Fs(\()p Fq(a)p Fs(\))28 b Fq(;)1730 b Fs(\(4.53\))758 5791 y(whic)-5 b(h,)56 b(again)g(using)g Fq(\013)47 b(<)2504 5726 y Fn(3)p 2504 5753 67 7 v 2504 5849 a(2)2590 5791 y Fs(,)56 b(follo)-5 b(ws)56 b(from)3141 6240 y Fq(\014)3244 6171 y Fk(0)3235 6281 y Fp(n)3371 6240 y Fq(>)3566 6127 y Fs(24\()p Fq(p)36 b Fs(+)h(4\))p 3566 6202 665 7 v 3741 6354 a Fq(pK)3966 6379 y Fp(n)4279 6240 y Fq(;)1986 b Fs(\(4.54\))758 6707 y(and)56 b Fq(B)e Fo(\025)46 b Fz(B)1572 6732 y Fn(6)1647 6707 y Fs(\()p Fq(n)p Fs(\),)54 b(with)h Fz(B)2491 6732 y Fn(6)2565 6707 y Fs(\()p Fq(n)p Fs(\))f(dep)5 b(ending)56 b(only)g(only)f(on)h Fq(M)18 b Fs(,)4908 6665 y Fl(f)4887 6707 y Fq(M)h Fs(,)55 b Fq(\027)11 b Fs(,)55 b Fq(C)5481 6732 y Fn(3)5555 6707 y Fs(,)h Fq(p)p Fs(,)f(and)h Fq(K)6306 6732 y Fp(n)6396 6707 y Fs(.)957 6906 y(In)f(pursuit)i(of)e(simpli\014cation,)i(w)-5 b(e)55 b(no)-5 b(w)56 b(c)-5 b(ho)5 b(ose)56 b Fq(p)46 b Fs(=)4444 6841 y Fn(8)p 4440 6868 75 7 v 4440 6963 a Fp(\027)4589 6906 y Fs(and)56 b(set)3136 7337 y Fq(K)3277 7362 y Fp(n)3413 7337 y Fs(=)3608 7225 y(43)37 b(+)g(20)p Fq(\027)p 3608 7299 628 7 v 3830 7451 a(\014)3924 7476 y Fp(n)4284 7337 y Fq(;)1981 b Fs(\(4.55\))758 7762 y(so)37 b(conditions)h(\(4.48\))o(,)j(\(4.51\))o(,)g(and)c(\(4.54\))g(are)f (satis\014ed.)69 b(W)-14 b(e)37 b(also)g(set)g Fz(B)p Fs(\()p Fq(n)p Fs(\))45 b(=)h(max)p Fo(f)p Fz(B)6399 7787 y Fp(i)6454 7762 y Fs(\()p Fq(n)p Fs(\))p Fq(;)73 b(i)46 b Fs(=)758 7987 y(1)p Fq(;)28 b Fs(2)p Fq(;)g(:)g(:)g(:)54 b(;)28 b Fs(6)p Fo(g)p Fs(,)55 b(whic)-5 b(h)56 b(dep)5 b(ends)56 b(only)f(on)h Fq(n)p Fs(,)e Fq(M)18 b Fs(,)3874 7945 y Fl(f)3854 7987 y Fq(M)g Fs(,)55 b Fq(r)4209 8012 y Fp(u)4351 7987 y Fs(,)p Fo(kr)p Fq(u)p Fo(k)4796 8012 y Fk(1)4937 7987 y Fs(,)h Fo(k)p Fs(\001)p Fq(u)p Fo(k)5438 8012 y Fk(1)5579 7987 y Fs(,)f Fo(k)p Fq(g)6 b Fo(k)5931 8012 y Fk(1)6072 7987 y Fs(,)55 b Fq(C)6292 8012 y Fn(3)6366 7987 y Fs(,)h(and)758 8186 y Fq(\027)11 b Fs(,)71 b(and)d(tak)-5 b(e)68 b Fq(B)76 b Fo(\025)67 b Fz(B)p Fs(\()p Fq(n)p Fs(\).)111 b(F)-14 b(or)69 b(an)-5 b(y)69 b Fq(E)76 b Fo(2)67 b Fs(\006)3732 8211 y Fp(B)6 b(;n)4036 8186 y Fs(w)-5 b(e)69 b(can)f(no)-5 b(w)68 b(v)-5 b(erify)68 b(the)g(h)-5 b(yp)5 b(otheses)68 b(of)758 8385 y(Theorem)56 b(2.6)h(with)f Fq(L)2221 8410 y Fn(0)2351 8385 y Fs(and)h Fq(\022)k Fs(as)c(in)f(\(4.38\))g(and)h(\(4.47\))o(,)g(and)f(conclude)h (that)e(\(4.8\))h(holds,)758 8585 y(i.e.,)49 b(\006)1185 8610 y Fp(B)6 b(;n)1467 8585 y Fo(\032)47 b Fs(\006)1763 8610 y Fn(MSA)2062 8585 y Fs(.)71 b(Moreo)-5 b(v)g(er,)50 b(for)e(a.e.)71 b Fq(!)6 b Fs(,)50 b(if)e Fq(')3975 8610 y Fp(B)6 b(;!)t(;E)4408 8585 y Fs(is)49 b(an)f(eigenfunction)g(of)g Fq(H)6123 8610 y Fp(B)6 b(;!)6412 8585 y Fs(with)758 8784 y(eigen)-5 b(v)c(alue)55 b Fq(E)g Fo(2)46 b Fs(\006)2006 8809 y Fp(B)6 b(;n)2242 8784 y Fs(,)55 b(then)1314 9252 y(lim)28 b(inf)1382 9361 y Fn(\025)-74 b Fp(x)q Fk(!1)1802 9252 y Fo(\000)1951 9140 y Fs(log)30 b Fo(k)p Fq(\037)2388 9165 y Fn(\025)-74 b Fp(x)2464 9140 y Fq(')2573 9165 y Fp(B)6 b(;!)t(;E)2957 9140 y Fo(k)p 1951 9214 1090 7 v 2365 9366 a(j)11 b Fs(\025)-94 b Fq(x)p Fo(j)2552 9391 y Fn(1)3106 9252 y Fo(\025)3302 9140 y Fq(\014)3405 9080 y Fk(0)3396 9181 y Fp(n)3514 9140 y Fs(min)p Fo(f)p Fq(aB)8 b(;)4169 8999 y Fo(p)p 4307 8999 135 7 v 141 x Fq(B)h Fo(g)p 3302 9214 1224 7 v 3669 9366 a Fs(2)p Fq(L)3865 9391 y Fn(0)3940 9366 y Fs(\()p Fq(a)p Fs(\))6311 9252 y(\(4.56\))1646 10038 y Fo(\025)1821 9498 y Fl(8)1821 9648 y(>)1821 9698 y(>)1821 9748 y(>)1821 9797 y(>)1821 9847 y(<)1821 10146 y(>)1821 10196 y(>)1821 10246 y(>)1821 10296 y(>)1821 10345 y(:)2318 9579 y Fp(\014)2399 9529 y Ff(0)2393 9613 y Fi(n)2474 9579 y Fp(a)2546 9529 y Fe(1+)p Fi(\027)2762 9579 y Fp(B)p 2072 9626 1042 7 v 2072 9745 a Fn(2)2138 9757 y Fs(\()2203 9745 y Fp(C)2298 9762 y Fe(5)2385 9745 y Fn(log)2600 9700 y Fi(M)p 2600 9718 116 6 v 2626 9785 a(a)2735 9745 y Fn(+6)p Fp(a)2975 9711 y Fi(\027)3049 9757 y Fs(\))3179 9664 y Fo(\025)3375 9579 y Fp(\014)3456 9529 y Ff(0)3450 9613 y Fi(n)3531 9579 y Fp(K)3651 9529 y Fe(1+)p Fi(\027)3643 9613 y(n)3866 9579 y Fn(\(log)26 b Fp(B)6 b Fn(\))4271 9529 y Fi(\027)p 3375 9626 970 7 v 3656 9721 a Fn(2)p Fp(C)3817 9738 y Fe(5)3883 9721 y Fp(B)3989 9688 y Fi(\027)4530 9664 y Fs(if)92 b Fq(K)4860 9689 y Fp(n)4970 9590 y Fn(log)25 b Fp(B)p 4970 9626 301 7 v 5067 9721 a(B)5337 9664 y Fo(\024)46 b Fq(a)g Fo(\024)5915 9599 y Fn(1)p 5841 9626 216 7 v 5841 9644 a Fk(p)p 5950 9644 106 6 v 98 x Fp(B)2334 9951 y(\014)2415 9901 y Ff(0)2409 9984 y Fi(n)2491 9951 y Fp(a)2563 9901 y Fi(\027)2636 9853 y Fk(p)p 2745 9853 V 98 x Fp(B)p 2072 9998 1042 7 v 2072 10116 a Fn(2)2138 10128 y Fs(\()2203 10116 y Fp(C)2298 10133 y Fe(5)2385 10116 y Fn(log)2600 10072 y Fi(M)p 2600 10090 116 6 v 2626 10156 a(a)2735 10116 y Fn(+6)p Fp(a)2975 10083 y Fi(\027)3049 10128 y Fs(\))3179 10036 y Fo(\025)3388 9951 y Fp(\014)3469 9901 y Ff(0)3463 9984 y Fi(n)3545 9951 y Fp(B)3671 9854 y Fe(1)p Ff(\000)p Fi(\027)p 3671 9876 212 5 v 3748 9939 a Fe(2)p 3375 9998 550 7 v 3375 10093 a Fn(2)p Fp(C)3536 10110 y Fe(5)3624 10093 y Fn(log)25 b Fp(B)4530 10036 y Fs(if)4813 9971 y Fn(1)p 4739 9998 216 7 v 4739 10016 a Fk(p)p 4848 10016 106 6 v 98 x Fp(B)5020 10036 y Fo(\024)46 b Fq(a)g Fo(\024)5525 9994 y Fl(f)5504 10036 y Fq(M)2334 10287 y Fp(\014)2415 10237 y Ff(0)2409 10320 y Fi(n)2505 10258 y Fd(f)2491 10287 y Fp(M)2630 10237 y Fi(\027)2703 10188 y Fk(p)p 2812 10188 V 99 x Fp(B)p 2072 10333 1109 7 v 2072 10457 a Fn(2)2138 10469 y Fs(\()2203 10457 y Fp(C)2298 10474 y Fe(5)2385 10457 y Fn(log)2600 10412 y Fi(M)p 2600 10430 116 6 v 2619 10498 a Fc(f)2600 10519 y Fi(M)2735 10457 y Fn(+6)2917 10428 y Fd(f)2903 10457 y Fp(M)3042 10423 y Fi(\027)3116 10469 y Fs(\))4530 10372 y(if)4739 10330 y Fl(f)4719 10372 y Fq(M)64 b Fo(\024)46 b Fq(a)g Fo(\024)g Fq(M)6311 10038 y Fs(\(4.57\))758 10798 y(where)55 b(w)-5 b(e)55 b(used)h(\(4.42\).)74 b(The)55 b(estimate)g(\(4.9\))g(follo)-5 b(ws.)p 6614 10798 7 113 v 6621 10693 100 7 v 6621 10798 V 6720 10798 7 113 v eop %%Page: 20 20 20 19 bop 758 861 a Fn(20)1485 b(FRANC)-77 b(\030)17 b(OIS)59 b(GERMINET)g(AND)f(ABEL)g(KLEIN)2320 1259 y Fs(5.)92 b Fr(Pr)l(oof)62 b(of)h(Theorems)d(2.4)j(and)f(2.5)957 1558 y Fs(The)45 b(follo)-5 b(wing)45 b(lemma)h(lists)f(explicit)f (scenarios)i(under)g(whic)-5 b(h)45 b(condition)g(\(2.28\))f(of)h(The-) 758 1757 y(orem)55 b(2.7)h(holds.)758 2060 y Fz(Lemma)63 b(5.1.)84 b Fg(L)-8 b(et)59 b Fq(S)d Fo(\025)46 b Fs(2)p Fg(.)77 b(If)59 b(one)h(of)f(the)h(fol)8 b(lowing)61 b(c)-8 b(onditions)60 b(holds:)758 2259 y(\(i\))f Fq(Y)83 b Fo(\025)46 b Fs(8)p Fq(S)h Fs(+)37 b(5)p Fg(,)59 b Fq(L)2062 2284 y Fn(0)2183 2259 y Fo(\025)46 b Fs(8)p Fq(S)g Fs(+)37 b(5)p Fg(;)758 2499 y(\(ii\))59 b Fq(Y)83 b Fo(\025)46 b Fs(9)p Fq(S)h Fs(+)37 b(3)p Fg(,)59 b Fq(L)2113 2524 y Fn(0)2234 2499 y Fo(\025)46 b Fs(\(9)p Fq(S)g Fs(+)37 b(3\))3048 2371 y Fe(2)p 3038 2389 76 6 v 3038 2455 a Fi(S)3142 2499 y Fg(;)758 2698 y(then)60 b(c)-8 b(ondition)72 b Fs(\(2.28\))59 b Fg(of)g(The)-8 b(or)g(em)59 b(2.7)g(is)g(satis\014e)-8 b(d,)59 b(i.e.)2768 3052 y Fq(Y)74 b Fo(\000)3298 2939 y Fs(2)p 3124 3013 430 7 v 3124 3165 a(log)30 b Fq(L)3479 3190 y Fn(0)3601 3052 y Fs(log)h Fq(Y)83 b Fo(\025)46 b Fs(8)p Fq(S)h Fs(+)37 b(3)p Fq(:)758 3433 y Fg(Pr)-8 b(o)g(of.)81 b Fs(The)38 b(lemma)h(follo)-5 b(ws)39 b(from)g(the)f(fact)f(that)h(the)g (function)h Fq(t)46 b Fo(!)g Fq(t)s Fo(\000)s Fq(t)5422 3458 y Fn(0)5524 3433 y Fs(log)30 b Fq(t)38 b Fs(is)h(increasing)758 3632 y(for)55 b Fq(t)46 b Fo(\025)g Fq(t)1353 3657 y Fn(0)1428 3632 y Fs(.)957 3831 y(If)53 b Fq(Y)84 b Fo(\025)46 b Fs(8)p Fq(S)d Fs(+)33 b(5)54 b(and)g(log)r(\(8)p Fq(S)43 b Fs(+)33 b(5\))p Fq(=)28 b Fs(log)i Fq(L)3607 3856 y Fn(0)3727 3831 y Fo(\024)47 b Fs(1,)54 b(condition)g(\(2.28\))f(follo) -5 b(ws.)74 b(This)54 b(pro)-5 b(v)g(es)758 4031 y(\(i\).)957 4230 y(T)-14 b(o)55 b(pro)-5 b(v)g(e)56 b(\(ii\),)f(tak)-5 b(e)55 b Fq(Y)83 b Fo(\025)47 b Fs(9)p Fq(S)f Fs(+)37 b(3)55 b(and)h(note)f(that)g(\(2.28\))g(will)g(hold)h(if)2753 4584 y Fq(S)47 b Fo(\000)3261 4471 y Fs(2)p 3088 4545 V 3088 4697 a(log)30 b Fq(L)3443 4722 y Fn(0)3565 4584 y Fs(log)r(\(9)p Fq(S)46 b Fs(+)37 b(3\))46 b Fo(\025)g Fs(0)p Fq(:)1655 b Fs(\(5.1\))p 6614 4935 7 113 v 6621 4829 100 7 v 6621 4935 V 6720 4935 7 113 v 758 5243 a Fg(Pr)-8 b(o)g(of)57 b(of)j(The)-8 b(or)g(em)58 b(2.4.)83 b Fs(W)-14 b(e)68 b(tak)-5 b(e)67 b Fq(p)g Fs(=)3482 5177 y Fn(5)p 3482 5204 67 7 v 3482 5300 a(3)3568 5243 y Fq(d)28 b Fs(,)71 b Fq(S)77 b Fs(=)68 b(4,)j(and)e Fq(Y)105 b Fs(=)68 b(9)p Fq(S)54 b Fs(+)46 b(3)68 b(=)f(39)h(in)h (Theo-)758 5442 y(rem)55 b(2.7,)h(so)g(w)-5 b(e)55 b(satisfy)j (\(2.31\))o(,)e(and)g(obtain)f(\(2.28\))g(from)g(Lemma)h(5.1)g(\(ii\))e (if)i Fq(L)6013 5467 y Fn(0)6133 5442 y Fo(\025)46 b Fs(6.)957 5741 y(Giv)-5 b(en)56 b Fq(L)1551 5766 y Fn(0)1671 5741 y Fo(\025)47 b Fs(max)2183 5507 y Fl(\032)2308 5741 y Fs(6)p Fq(;)28 b Fs(3)p Fq(\045;)g(\021)2796 5630 y Fk(\000)2900 5642 y Fs([\()3030 5585 y Fe(5)p 3030 5603 57 6 v 3030 5670 a(3)3107 5630 y Fn(+)p Fp(b)3267 5642 y Fs(\))3331 5630 y Fp(d)3400 5642 y Fs(])3447 5559 y Ff(\000)p Fe(1)2790 5790 y Fk(I)3610 5507 y Fl(\033)3734 5741 y Fs(,)56 b Fq(L)3949 5766 y Fn(0)4069 5741 y Fo(2)46 b Fs(6)p Fm(N)35 b Fs(,)56 b(w)-5 b(e)55 b(de\014ne)h Fq(s)5333 5766 y Fn(0)5462 5741 y Fs(b)-5 b(y)1811 6175 y(\(39)p Fq(L)2155 6200 y Fn(0)2229 6175 y Fs(\))2294 6103 y Fp(s)2357 6120 y Fe(0)2421 6103 y Fk(\000)2525 6115 y Fs(\()2609 6059 y Fe(5)p 2609 6077 V 2609 6143 a(3)2686 6103 y Fn(+)p Fp(b)2846 6115 y Fs(\))2911 6103 y Fp(d)3154 6175 y Fs(=)166 b(max)3786 6041 y Fl(\010)3883 6175 y Fs(16)37 b Fo(\001)g Fs(60)4335 6107 y Fp(d)4412 6175 y Fq(Q)4543 6200 y Fk(I)4634 6175 y Fq(;)28 b Fs(1)4791 6041 y Fl(\011)6394 6175 y Fs(\(5.2\))3154 6491 y Fo(\025)166 b Fs(max)3786 6306 y Fl(n)3897 6491 y Fs(16)37 b Fo(\001)g Fs(234)4432 6422 y Fp(d)4537 6356 y Fl(\000)4666 6425 y Fn(30)p 4633 6452 199 7 v 4633 6548 a(117)4851 6356 y Fl(\001)4927 6391 y Fp(bd)5091 6491 y Fq(Q)5222 6516 y Fk(I)5313 6491 y Fq(;)28 b Fs(1)5470 6306 y Fl(o)5636 6491 y Fq(;)712 b Fs(\(5.3\))758 6832 y(so)50 b Fq(s)1034 6857 y Fn(0)1154 6832 y Fo(\025)1329 6698 y Fl(\000)1425 6767 y Fn(5)p 1425 6794 67 7 v 1425 6890 a(3)1548 6832 y Fs(+)37 b Fq(b)1785 6698 y Fl(\001)1889 6832 y Fq(d)50 b Fs(and)g(\(2.29\))f(is)h(satis\014ed)h(for)e Fq(s)d Fo(\025)g Fq(s)4235 6857 y Fn(0)4309 6832 y Fs(.)72 b(W)-14 b(e)49 b(also)i(de\014ne)e Fq(\022)5586 6857 y Fn(0)5707 6832 y Fq(>)d(s)5960 6857 y Fn(0)6084 6832 y Fs(b)-5 b(y)52 b(\(2.32\))758 7032 y(with)j Fq(s)46 b Fs(=)g Fq(s)1513 7057 y Fn(0)1587 7032 y Fs(;)56 b(note)f(that)g Fq(D)2555 7057 y Fk(I)2701 7032 y Fs(is)h(de\014ned)g(in)g(\(2.18\))f (so)3072 7389 y Fq(L)3185 7315 y Fp(\022)3248 7332 y Fe(0)3185 7434 y Fn(0)3367 7389 y Fo(\024)47 b Fq(D)3680 7414 y Fk(I)3771 7389 y Fq(L)3884 7290 y Fs(\()3969 7233 y Fe(5)p 3969 7251 57 6 v 3969 7318 a(3)4045 7278 y Fn(+)p Fp(b)4205 7290 y Fs(\))4270 7278 y Fp(d)3884 7434 y Fn(0)4375 7389 y Fq(:)1973 b Fs(\(5.4\))957 7675 y(Th)-5 b(us)57 b(condition)e(\(2.17\))g(in)h(Theorem)f(2.4)h(then)f(implies)1688 8050 y Fm(P)1809 7816 y Fl(\032)1942 8050 y Fo(k)p Fs(\000)2129 8075 y Fn(0)p Fp(;L)2334 8050 y Fq(R)2460 8075 y Fn(0)p Fp(;L)2665 8050 y Fs(\()p Fq(E)2853 8075 y Fn(0)2927 8050 y Fs(\))p Fq(\037)3096 8080 y Fn(0)p Fp(;L=)p Fn(3)3434 8050 y Fo(k)3517 8075 y Fn(0)p Fp(;L)3768 8050 y Fq(<)4047 7937 y Fs(1)p 3964 8011 250 7 v 3964 8164 a Fq(L)4077 8116 y Fp(\022)4140 8133 y Fe(0)4233 7816 y Fl(\033)4431 8050 y Fo(\025)74 b Fs(1)37 b Fo(\000)5041 7984 y Fn(2)p 4940 8011 269 7 v 4940 8109 a(368)5138 8076 y Fi(d)6394 8050 y Fs(\(5.5\))2860 8455 y Fq(>)47 b Fs(1)37 b Fo(\000)g Fs(60)3508 8342 y Fe(1)p 3508 8360 57 6 v 3508 8427 a(4)3592 8455 y Fs(\(113\))3971 8387 y Fk(\000)4094 8342 y Fe(5)p 4094 8360 V 4094 8427 a(4)4171 8387 y Fp(d)4294 8455 y Fq(>)47 b Fs(1)36 b Fo(\000)h Fs(\(2)p Fq(\014)9 b Fs(\(39)p Fq(;)28 b Fs(5\)\))5522 8372 y Fk(\000)5646 8327 y Fe(1)p 5646 8345 V 5646 8412 a(4)5758 8455 y Fq(;)758 8731 y Fs(and)58 b(hence)g(condition)g(\(2.33\))f(in)i(Theorem)f(2.7)g (holds)h(for)e(some)i Fq(\022)c(>)c(\022)5434 8756 y Fn(0)5508 8731 y Fs(,)59 b(so)f(Theorem)g(2.4)758 8930 y(follo)-5 b(ws)56 b(from)f(Theorem)h(2.7.)p 6614 8930 7 113 v 6621 8825 100 7 v 6621 8930 V 6720 8930 7 113 v 758 9238 a Fg(Pr)-8 b(o)g(of)57 b(of)j(The)-8 b(or)g(em)58 b(2.5.)83 b Fs(W)-14 b(e)78 b(tak)-5 b(e)77 b Fq(S)94 b Fs(=)85 b(4)78 b(and)h Fq(Y)122 b Fs(=)84 b(8)p Fq(S)62 b Fs(+)52 b(5)85 b(=)f(37)79 b(in)f(Theorem)h(2.7;)758 9437 y(condition)55 b(\(2.27\))g(holds)i(for)e Fq(L)2768 9462 y Fn(0)2888 9437 y Fo(\025)47 b Fs(37)55 b(b)-5 b(y)56 b(Lemma)f(5.1\(i\).)74 b(W)-14 b(e)55 b(will)g(use)h(\(see)f (\(2.27\))o(\))3105 9825 y Fq(\014)9 b Fs(\(37)p Fq(;)28 b Fs(5\))45 b Fo(\024)3901 9713 y Fs(107)4150 9652 y Fn(5)p Fp(d)p 3901 9787 393 7 v 3973 9939 a Fs(120)4342 9825 y Fq(:)2006 b Fs(\(5.6\))957 10172 y(Giv)-5 b(en)56 b Fq(s)46 b(>)g(bd)p Fs(,)56 b(w)-5 b(e)56 b(set)f Fq(p)45 b Fs(=)2825 10106 y Fn(1)p 2825 10133 67 7 v 2825 10229 a(2)2911 10172 y Fs(\()p Fq(s)36 b Fo(\000)h Fq(bd)p Fs(\))56 b(and)1000 10632 y Fo(L)1115 10564 y Fk(0)1207 10632 y Fs(=)47 b(max)1719 10348 y Fl(\()1853 10632 y Fs(3)p Fq(\045;)28 b Fs(37)p Fq(;)g Fs(37)2522 10519 y Fe(1)p 2521 10537 57 6 v 2521 10604 a(5)2634 10398 y Fl(\022)2776 10520 y Fs(107)3025 10460 y Fp(d)p 2776 10594 327 7 v 2804 10770 a Fs(60)2990 10667 y Fe(1)p 2990 10685 57 6 v 2990 10752 a(5)3122 10398 y Fl(\023)3366 10375 y Fe(2)p 3265 10393 260 6 v 3265 10460 a Fi(s)p Ff(\000)p Fi(bd)3580 10632 y Fq(;)3715 10520 y Fs(1)p 3674 10594 167 7 v 3674 10746 a(37)3887 10498 y Fl(\000)3964 10632 y Fs(16)36 b Fo(\001)i Fs(222)4499 10564 y Fp(d)4576 10632 y Fq(Q)4707 10657 y Fk(I)4798 10498 y Fl(\001)4996 10479 y Fe(2)p 4894 10497 260 6 v 4894 10564 a Fi(s)p Ff(\000)p Fi(bd)5210 10632 y Fq(;)28 b(\021)5372 10535 y Fk(\000)5496 10490 y Fe(1)p 5496 10508 57 6 v 5496 10575 a Fi(s)5366 10681 y Fk(I)5580 10348 y Fl(\))5760 10632 y Fo(\024)47 b(L)28 b Fq(;)269 b Fs(\(5.7\))p eop %%Page: 21 21 21 20 bop 1773 861 a Fn(EXPLICIT)58 b(FINITE)g(V)l(OLUME)h(CRITERIA)g (F)l(OR)g(LOCALIZA)-11 b(TION)883 b(21)758 1259 y Fs(where)52 b Fo(L)g Fs(is)g(as)h(in)g(\(2.19\))o(.)73 b(.)g(Conditions)53 b(\(2.29\))f(and)h(\(2.30\))e(of)h(Theorem)h(2.7)f(are)g(satis\014ed) 758 1458 y(for)j Fq(L)1125 1483 y Fn(0)1245 1458 y Fo(\025)47 b(L)1536 1398 y Fk(0)1637 1458 y Fs(b)-5 b(y)56 b(its)f(de\014nition.) 957 1662 y(No)-5 b(w)68 b(set)f Fq(\022)1697 1687 y Fn(0)1838 1662 y Fq(>)g(s)h Fs(\(recall)f Fq(\015)2786 1687 y Fk(I)2944 1662 y Fo(\025)g Fs(1\))g(b)-5 b(y)68 b Fq(L)3711 1588 y Fp(\022)3774 1605 y Fe(0)3711 1706 y Fn(0)3914 1662 y Fs(=)f(90)4276 1601 y Fp(d)4354 1662 y Fq(\015)4449 1601 y Fn(2)4440 1707 y Fk(I)4531 1662 y Fs(\(37)p Fq(L)4875 1687 y Fn(0)4949 1662 y Fs(\))5014 1601 y Fp(s)5084 1662 y Fs(,)k(i.e.,)g(b)-5 b(y)71 b(\(2.32\))o(.)111 b(Then)758 1861 y(condition)55 b(\(2.20\))g(is)h(that)f(same)h(as)1568 2274 y Fm(P)1689 2040 y Fl(\032)1822 2274 y Fo(k)p Fs(\000)2009 2299 y Fn(0)p Fp(;L)2214 2274 y Fq(R)2340 2299 y Fn(0)p Fp(;L)2545 2274 y Fs(\()p Fq(E)2733 2299 y Fn(0)2807 2274 y Fs(\))p Fq(\037)2976 2304 y Fn(0)p Fp(;L=)p Fn(3)3314 2274 y Fo(k)3397 2299 y Fn(0)p Fp(;L)3648 2274 y Fq(<)3927 2161 y Fs(1)p 3844 2235 250 7 v 3844 2388 a Fq(L)3957 2340 y Fp(\022)4020 2357 y Fe(0)4113 2040 y Fl(\033)4311 2274 y Fo(\025)74 b Fs(1)37 b Fo(\000)4921 2208 y Fn(2)p 4820 2235 269 7 v 4820 2333 a(344)5018 2300 y Fi(d)6394 2274 y Fs(\(5.8\))2980 2679 y Fq(>)47 b Fs(1)37 b Fo(\000)g Fs(60)3628 2566 y Fe(1)p 3628 2584 57 6 v 3628 2651 a(4)3712 2679 y Fs(\(107\))4091 2611 y Fk(\000)4214 2566 y Fe(5)p 4214 2584 V 4214 2651 a(4)4291 2611 y Fp(d)4414 2679 y Fq(>)47 b Fs(1)36 b Fo(\000)h Fs(\(2)p Fq(\014)9 b Fs(\(37)p Fq(;)28 b Fs(5\)\))5642 2596 y Fk(\000)5766 2551 y Fe(1)p 5766 2569 V 5766 2636 a(4)5878 2679 y Fq(;)758 2992 y Fs(and)58 b(hence)g(condition)g(\(2.33\))f(in)i(Theorem)f(2.7)g (holds)h(for)e(some)i Fq(\022)c(>)c(\022)5434 3017 y Fn(0)5508 2992 y Fs(,)59 b(so)f(Theorem)g(2.5)758 3192 y(follo)-5 b(ws)56 b(from)f(Theorem)h(2.7.)p 6614 3192 7 113 v 6621 3086 100 7 v 6621 3192 V 6720 3192 7 113 v 2693 3656 a(6.)92 b Fr(Pr)l(oof)62 b(of)g(Theorem)f(2.7)957 3955 y Fs(Theorem)55 b(2.7)g(is)h(a)f(re\014nemen)-5 b(t)55 b(of)g([GK1,)g(Theorem)h(5.1],)f(the)g(main)g(di\013erence)g (are)g(the)758 4154 y(explicit)f(conditions.)758 4516 y Fg(Pr)-8 b(o)g(of)57 b(of)j(The)-8 b(or)g(em)58 b(2.7.)83 b Fs(W)-14 b(e)78 b(pro)5 b(ceed)78 b(as)h(in)g(the)g(pro)5 b(of)78 b(of)g([GK1)q(,)85 b(Theorem)78 b(5.1].)145 b(W)-14 b(e)758 4715 y(start)69 b(b)-5 b(y)70 b(pic)-5 b(king)71 b Fq(E)2125 4740 y Fn(0)2269 4715 y Fo(2)f(I)12 b Fs(,)74 b Fq(p)c(>)g Fs(0,)k Fq(S)79 b Fo(\025)71 b Fs(2)e(with)h Fq(S)80 b Fo(2)70 b Fm(N)25 b Fs(,)83 b Fq(s)70 b(>)g(p)46 b Fs(+)h Fq(bd)p Fs(,)74 b Fq(\022)h(>)70 b(\022)6116 4740 y Fn(0)6191 4715 y Fs(,)j(where)758 4915 y Fq(\022)836 4940 y Fn(0)984 4915 y Fs(is)h(de\014ned)g(b)-5 b(y)74 b(condition)h(\(2.32\))e(\(note)g Fq(\022)81 b(>)c(s)p Fs(\),)h Fq(Y)114 b Fo(\025)76 b Fs(8)p Fq(S)59 b Fs(+)49 b(5,)79 b Fq(Y)111 b Fs(o)5 b(dd,)78 b(and)d(scales)758 5168 y Fq(L)46 b Fo(\025)g Fs(max)1429 4984 y Fl(n)1539 5168 y Fs(6)p Fq(;)28 b Fs(3)p Fq(\045;)g(\021)2027 5071 y Fk(\000)2151 5026 y Fe(1)p 2151 5044 57 6 v 2151 5111 a Fi(s)2021 5217 y Fk(I)2236 4984 y Fl(o)2402 5168 y Fs(with)55 b Fq(L)46 b Fo(2)g Fs(6)p Fm(N)26 b Fs(.)83 b(W)-14 b(e)55 b(set)2346 5531 y Fq(p)2430 5556 y Fp(L)2575 5531 y Fs(=)46 b Fm(P)6 b Fo(f)p Fs(\003)3041 5556 y Fp(L)3150 5531 y Fs(\(0\))91 b(is)56 b(not)f(\()p Fq(\022)5 b(;)28 b(E)4261 5556 y Fn(0)4334 5531 y Fs(\)-suitable)q Fo(g)p Fq(:)1247 b Fs(\(6.1\))957 5845 y(The)83 b(pro)5 b(of)82 b(pro)5 b(ceeds)83 b(b)-5 b(y)84 b(induction.)157 b(F)-14 b(or)83 b(the)g(induction)g(step,)91 b(let)82 b Fq(`)92 b Fo(2)g Fs(6)p Fm(N)26 b Fs(,)99 b Fq(`)92 b Fo(\025)758 6089 y Fs(max)1094 5905 y Fl(n)1205 6089 y Fs(6)p Fq(;)28 b Fs(3)p Fq(\045;)g(\021)1693 5992 y Fk(\000)1817 5947 y Fe(1)p 1817 5965 V 1817 6032 a Fi(s)1687 6138 y Fk(I)1901 5905 y Fl(o)2012 6089 y Fs(,)60 b(and)f Fq(L)52 b Fs(=)g Fq(Y)37 b(`)p Fs(.)85 b(W)-14 b(e)58 b(estimate)h Fq(p)4165 6114 y Fp(L)4322 6089 y Fs(from)g Fq(p)4802 6114 y Fp(`)4924 6089 y Fs(as)h(in)f([GK1,)h(\(Eq.)84 b(\(5.22\)],)758 6338 y(but)55 b(b)5 b(eing)55 b(sligh)-5 b(tly)57 b(more)e(careful)g(in)h(our)g(use)f(of)h(the)f(W)-14 b(egner)55 b(estimate)g(\(2.9\))o(,)h(obtaining)1353 6800 y Fq(p)1437 6825 y Fp(L)1702 6800 y Fo(\024)166 b Fq(\014)9 b Fs(\()p Fq(Y)g(;)28 b(S)46 b Fs(+)37 b(1\))p Fq(p)2890 6729 y Fp(S)6 b Fn(+1)2890 6851 y Fp(`)3190 6800 y Fs(+)37 b Fq(Q)3487 6825 y Fk(I)3606 6517 y Fl(")3703 6800 y Fq(S)3842 6566 y Fl(\022)3984 6688 y Fs(7)p Fq(S)47 b Fs(+)37 b(2)p 3984 6762 481 7 v 4116 6914 a(3)p Fq(Y)4485 6566 y Fl(\023)4607 6601 y Fp(bd)4770 6800 y Fs(\(6)p Fq(Y)g Fs(\))5116 6732 y Fp(d)5230 6800 y Fs(+)g(1)5479 6517 y Fl(#)5604 6800 y Fq(L)5717 6732 y Fp(bd)5852 6800 y Fq(L)5965 6732 y Fk(\000)p Fp(s)1702 7328 y Fo(\024)166 b Fq(\014)9 b Fs(\()p Fq(Y)g(;)28 b(S)46 b Fs(+)37 b(1\))p Fq(p)2890 7257 y Fp(S)6 b Fn(+1)2890 7378 y Fp(`)3190 7328 y Fs(+)37 b(2)3439 7259 y Fp(d)p Fn(+1)3685 7328 y Fq(Q)3816 7353 y Fk(I)3907 7328 y Fq(S)4043 7215 y Fs(\(7)p Fq(S)46 b Fs(+)37 b(2\))4653 7155 y Fp(bd)p 4038 7290 756 7 v 4038 7445 a Fs(\(3)p Fq(Y)g Fs(\))4384 7397 y Fn(\()p Fp(b)p Fk(\000)p Fn(1\))p Fp(d)4813 7328 y Fq(L)4926 7259 y Fk(\000)p Fp(s)p Fn(+)p Fp(bd)p Fn(+)p Fp(p)5574 7215 y Fs(1)p 5521 7290 190 7 v 5521 7442 a Fq(L)5634 7394 y Fp(p)6394 7328 y Fs(\(6.2\))1702 7772 y Fo(\024)166 b Fq(\014)9 b Fs(\()p Fq(Y)g(;)28 b(S)46 b Fs(+)37 b(1\))p Fq(p)2890 7701 y Fp(S)6 b Fn(+1)2890 7822 y Fp(`)3190 7772 y Fs(+)3376 7660 y(1)p 3376 7734 84 7 v 3376 7886 a(2)3552 7660 y(1)p 3499 7734 190 7 v 3499 7886 a Fq(L)3612 7838 y Fp(p)3709 7772 y Fq(;)2639 b Fs(\(6.3\))758 8141 y(where)63 b Fq(\014)9 b Fs(\()p Fq(Y)g(;)28 b(S)10 b Fs(\))62 b(is)i(as)f(in)h(\(2.27\))o(.)98 b(\(Note)62 b(that)g(\(2.27\))h(is)h(a)f(sligh)-5 b(tly)64 b(b)5 b(etter)62 b(estimate)h(than)758 8340 y([GK1,)56 b(\(Eq.)73 b(\(5.20\)].\))g(T)-14 b(o)55 b(obtain)h(\(6.3\))f(w)-5 b(e)55 b(assume)2445 8765 y(\()p Fq(Y)37 b(L)2756 8790 y Fn(0)2830 8765 y Fs(\))2895 8697 y Fp(s)p Fk(\000)p Fp(bd)p Fk(\000)p Fp(p)3416 8765 y Fo(\025)46 b Fs(2)3674 8697 y Fp(d)p Fn(+2)3919 8765 y Fq(Q)4050 8790 y Fk(I)4142 8765 y Fq(S)4278 8653 y Fs(\(7)p Fq(S)g Fs(+)37 b(2\))4888 8593 y Fp(bd)p 4273 8727 756 7 v 4273 8882 a Fs(\(3)p Fq(Y)g Fs(\))4619 8834 y Fn(\()p Fp(b)p Fk(\000)p Fn(1\))p Fp(d)6394 8765 y Fs(\(6.4\))758 9178 y(whic)-5 b(h)56 b(is)g(exactly)e(condition)h(\(2.29\))g(in)h(Theorem)f(2.7.)957 9378 y(As)c(in)f(the)h(pro)5 b(of)50 b(of)g([GK1)q(,)h(Theorem)g(5.1],) h(w)-5 b(e)51 b(next)f(m)-5 b(ust)51 b(satisfy)g([GK1,)h(\(Eq.)72 b(\(5.24\)].)758 9577 y(i.e.,)55 b(w)-5 b(e)56 b(m)-5 b(ust)56 b(sho)-5 b(w)57 b(that)2648 9890 y([)p Fq(\015)2780 9915 y Fk(I)2872 9890 y Fs(\(7)p Fq(S)45 b Fs(+)37 b(2\))3481 9821 y Fp(d)3558 9890 y Fq(L)3671 9821 y Fp(s)3742 9890 y Fs(][)p Fq(\015)3920 9915 y Fk(I)4012 9890 y Fs(3)4095 9821 y Fp(d)4172 9890 y Fq(`)4241 9821 y Fk(\000)p Fp(\022)4420 9890 y Fs(])46 b Fq(<)h Fs(1)p Fq(;)1577 b Fs(\(6.5\))758 10204 y(whic)-5 b(h)56 b(follo)-5 b(ws)56 b(if)2807 10517 y Fq(L)2920 10446 y Fp(\022)t Fk(\000)p Fp(s)2920 10561 y Fn(0)3208 10517 y Fq(>)46 b(\015)3478 10448 y Fn(2)3469 10558 y Fk(I)3560 10517 y Fs(3)3643 10448 y Fp(d)3721 10517 y Fs(\(7)p Fq(S)g Fs(+)37 b(2\))4331 10448 y Fp(d)4408 10517 y Fq(Y)4541 10448 y Fp(s)4612 10517 y Fq(;)1736 b Fs(\(6.6\))758 10840 y(whic)-5 b(h)56 b(is)g(true)f(if)g Fq(\022)c(>)46 b(\022)2285 10865 y Fn(0)2359 10840 y Fs(,)56 b(where)f Fq(\022)3019 10865 y Fn(0)3149 10840 y Fs(is)h(de\014ned)g(b)-5 b(y)55 b(condition)h(\(2.32\))o(.)p eop %%Page: 22 22 22 21 bop 758 861 a Fn(22)1485 b(FRANC)-77 b(\030)17 b(OIS)59 b(GERMINET)g(AND)f(ABEL)g(KLEIN)957 1259 y Fs(Next,)77 b(w)-5 b(e)74 b(de\014ne)h Fq(N)18 b Fs(\()p Fq(Y)37 b Fs(\))74 b(as)g(in)h([GK1,)k(\(Eq.)129 b(\(5.26\)],)78 b(note)c(that)f(since)i(w)-5 b(e)74 b(sp)5 b(eci\014ed)758 1458 y Fq(Y)83 b Fo(\025)46 b Fs(8)p Fq(S)h Fs(+)37 b(5,)55 b(w)-5 b(e)56 b(ha)-5 b(v)g(e)2831 1739 y Fq(N)18 b Fs(\()p Fq(Y)38 b Fs(\))45 b(=)i Fq(Y)74 b Fo(\000)37 b Fs(8)p Fq(S)46 b Fo(\000)37 b Fs(1)46 b Fo(\025)h Fs(4)28 b Fq(:)1732 b Fs(\(6.7\))758 2029 y(W)-14 b(e)55 b(no)-5 b(w)56 b(need)f(to)g(satisfy)g([GK1)q(,)g(\(Eq.)74 b(\(5.27\)]:)2838 2468 y Fq(Y)2972 2400 y Fp(d)3077 2234 y Fl(\024)3184 2356 y Fq(\015)3270 2381 y Fk(I)3361 2356 y Fs(3)3444 2296 y Fp(d)p 3184 2430 338 7 v 3281 2582 a Fq(`)3350 2534 y Fp(\022)3542 2234 y Fl(\025)3629 2256 y Fp(N)12 b Fn(\()p Fp(Y)30 b Fn(\))3993 2468 y Fq(L)4106 2400 y Fp(s)4224 2468 y Fq(<)4471 2356 y Fs(1)p 4419 2430 189 7 v 4419 2582 a Fq(L)4532 2534 y Fp(\022)6394 2468 y Fs(\(6.8\))758 2835 y(i.e.,)2449 3121 y(\(3)2597 3052 y Fp(d)2674 3121 y Fq(\015)2760 3146 y Fp(I)2819 3163 y Fe(0)2892 3121 y Fs(\))2957 3052 y Fp(N)12 b Fn(\()p Fp(Y)28 b Fn(\))3292 3121 y Fq(Y)3426 3052 y Fp(s)p Fn(+)p Fp(\022)t Fn(+)p Fp(d)3883 3121 y Fq(<)46 b(`)4127 3052 y Fp(\022)t Fn(\()p Fp(N)12 b Fn(\()p Fp(Y)29 b Fn(\))p Fk(\000)p Fn(1\))p Fk(\000)p Fp(s)4998 3121 y Fq(:)1350 b Fs(\(6.9\))758 3411 y(But)58 b(\(6.9\))h(is)h(automatically)e (ful\014lled)i(for)f(an)-5 b(y)60 b Fq(`)52 b Fo(\025)h Fq(L)4304 3436 y Fn(0)4431 3411 y Fo(\025)g Fs(6)59 b(in)h(view)e(of)h (condition)h(\(2.28\))o(.)758 3610 y(First)77 b(note)g(that)g Fq(\022)87 b(>)c(s)77 b Fs(and)h Fq(N)18 b Fs(\()p Fq(Y)37 b Fs(\))82 b Fo(\025)h Fs(4)77 b(imply)h(that)e Fq(\022)5 b Fs(\()p Fq(N)18 b Fs(\()p Fq(Y)37 b Fs(\))51 b Fo(\000)h Fs(1\))f Fo(\000)g Fq(s)83 b(>)f Fs(0.)140 b(As)77 b(a)758 3810 y(consequence)55 b(\(6.9\))g(certainly)f(holds)j(if)2347 3995 y Fl( )2498 4167 y Fq(L)2611 4081 y Fp(N)12 b Fn(\()p Fp(Y)29 b Fn(\))p Fk(\000)p Fn(1)2611 4211 y(0)p 2498 4241 620 7 v 2741 4393 a Fq(Y)3137 3995 y Fl(!)3269 4030 y Fp(\022)3390 4279 y Fo(\025)46 b Fs(\(3)3713 4211 y Fp(d)3790 4279 y Fq(\015)3876 4304 y Fk(I)3967 4279 y Fs(\))4032 4211 y Fp(N)12 b Fn(\()p Fp(Y)29 b Fn(\))4368 4279 y Fq(Y)4502 4211 y Fp(d)4579 4279 y Fs(\()p Fq(Y)37 b(L)4890 4304 y Fn(0)4964 4279 y Fs(\))5029 4211 y Fp(s)5100 4279 y Fq(:)1165 b Fs(\(6.10\))758 4700 y(Second,)55 b(w)-5 b(e)56 b(alw)-5 b(a)g(ys)56 b(ha)-5 b(v)g(e)2997 5003 y Fq(L)3110 4917 y Fp(N)12 b Fn(\()p Fp(Y)30 b Fn(\))p Fk(\000)p Fn(1)3110 5048 y(0)p 2997 5078 V 3240 5230 a Fq(Y)3683 5116 y Fo(\025)46 b Fq(L)3971 5030 y Fp(N)12 b Fn(\()p Fp(Y)29 b Fn(\))p Fp(=)p Fn(2)3971 5160 y(0)4442 5116 y Fq(;)1823 b Fs(\(6.11\))758 5452 y(pro)-5 b(vided)50 b Fq(N)18 b Fs(\()p Fq(Y)37 b Fs(\))46 b Fo(\025)g Fs(\(2)28 b(log)i Fq(Y)18 b(=)28 b Fs(log)i Fq(L)3068 5477 y Fn(0)3143 5452 y Fs(\))25 b(+)g(2,)52 b(whic)-5 b(h)50 b(is)g(exactly)h(\(2.28\)) o(.)72 b(Th)-5 b(us)51 b(plugging)i(\(6.6\))758 5651 y(\(whic)-5 b(h)55 b(is)h(the)f(same)h(as)g(\(2.32\))o(\))f(in)-5 b(to)56 b(\(6.10\))f(leads)h(to)1518 5837 y Fl( )1670 6009 y Fq(L)1783 5922 y Fp(N)12 b Fn(\()p Fp(Y)29 b Fn(\))p Fk(\000)p Fn(1)1783 6053 y(0)p 1670 6083 V 1913 6235 a Fq(Y)2309 5837 y Fl(!)2440 5871 y Fp(\022)2681 6121 y Fo(\025)2977 5986 y Fl(\000)3053 6121 y Fq(L)3166 6052 y Fp(\022)3166 6162 y Fn(0)3241 5986 y Fl(\001)3337 5954 y Fi(N)9 b Fe(\()p Fi(Y)24 b Fe(\))p 3337 5986 280 6 v 3449 6053 a(2)3691 6121 y Fo(\025)3922 5986 y Fl(\000)3998 6121 y Fq(\015)4093 6052 y Fn(2)4084 6162 y Fk(I)4175 6121 y Fs(3)4258 6052 y Fp(d)4335 6121 y Fs(\(7)p Fq(S)46 b Fs(+)37 b(2\))4945 6052 y Fp(d)5022 6121 y Fs(\()p Fq(Y)g(L)5333 6146 y Fn(0)5407 6121 y Fs(\))5472 6052 y Fp(s)5543 5986 y Fl(\001)5639 5954 y Fi(N)9 b Fe(\()p Fi(Y)24 b Fe(\))p 5639 5986 V 5751 6053 a(2)2681 6590 y Fo(\025)2977 6456 y Fl(\000)3053 6590 y Fs(3)3136 6522 y Fp(d)3213 6590 y Fq(\015)3299 6615 y Fk(I)3390 6456 y Fl(\001)3466 6482 y Fp(N)12 b Fn(\()p Fp(Y)29 b Fn(\))3830 6590 y Fq(Y)3964 6522 y Fp(d)4041 6590 y Fs(\()p Fq(Y)37 b(L)4352 6615 y Fn(0)4426 6590 y Fs(\))4491 6522 y Fp(s)4562 6590 y Fq(;)758 6871 y Fs(where)55 b(w)-5 b(e)55 b(used)h Fq(N)18 b Fs(\()p Fq(Y)38 b Fs(\))45 b Fo(\025)i Fs(2)55 b(and)h Fq(s)46 b(>)g(d)p Fs(.)75 b(Th)-5 b(us)56 b(w)-5 b(e)56 b(ha)-5 b(v)g(e)55 b(\(6.10\),)g(and)h(hence)f(\(6.8\).)957 7070 y(W)-14 b(e)55 b(no)-5 b(w)56 b(set)f Fq(L)1961 7095 y Fp(k)s Fn(+1)2257 7070 y Fs(=)46 b Fq(Y)38 b(L)2679 7095 y Fp(k)2760 7070 y Fs(,)56 b Fq(k)51 b Fs(=)c(0)p Fq(;)28 b Fs(1)p Fq(;)g Fs(2)p Fq(;)g Fo(\001)g(\001)g(\001)f Fs(,)55 b Fq(p)4052 7095 y Fp(k)4179 7070 y Fs(=)47 b Fq(p)4439 7095 y Fp(L)4530 7113 y Fi(k)4610 7070 y Fs(.)74 b(It)55 b(follo)-5 b(ws)56 b(from)g(\(6.3\))e(that)1806 7430 y Fq(p)1890 7455 y Fp(k)s Fn(+1)2186 7430 y Fo(\024)46 b Fq(\014)9 b Fs(\()p Fq(Y)g(;)28 b(S)46 b Fs(+)37 b(1\))p Fq(p)3254 7359 y Fp(S)6 b Fn(+1)3254 7480 y Fp(k)3554 7430 y Fs(+)3740 7318 y(1)p 3740 7392 84 7 v 3740 7544 a(2)3843 7430 y Fq(L)3956 7350 y Fk(\000)p Fp(p)3956 7480 y(k)s Fn(+1)4317 7430 y Fs(for)92 b Fq(k)52 b Fs(=)46 b(0)p Fq(;)28 b Fs(1)p Fq(;)g Fs(2)p Fq(;)g(:)g(:)g(:)54 b(:)625 b Fs(\(6.12\))957 7793 y(If)50 b(condition)g(\(2.30\))g(holds,) i(w)-5 b(e)50 b(ma)-5 b(y)51 b(\014nish)g(the)f(pro)5 b(of)50 b(as)h(in)f([GK1)q(].)72 b(If)50 b Fq(p)5719 7818 y Fp(k)5846 7793 y Fq(<)d(L)6135 7714 y Fk(\000)p Fp(p)6135 7843 y(k)6315 7793 y Fs(,)k(then)2164 8161 y Fq(p)2248 8186 y Fp(k)s Fn(+1)2663 8161 y Fo(\024)166 b Fq(\014)9 b Fs(\()p Fq(Y)g(;)28 b(S)46 b Fs(+)37 b(1\))4020 8048 y(1)p 3787 8122 550 7 v 3787 8316 a Fq(L)3900 8230 y Fn(\()p Fp(S)6 b Fn(+1\))p Fp(p)3900 8366 y(k)4393 8161 y Fs(+)4760 8048 y(1)p 4579 8122 446 7 v 4579 8275 a(2)p Fq(L)4775 8195 y Fp(p)4775 8325 y(k)s Fn(+1)2663 8669 y Fo(\024)3033 8557 y Fs(2)p Fq(\014)j Fs(\()p Fq(Y)g(;)28 b(S)46 b Fs(+)37 b(1\))p Fq(Y)4058 8497 y Fp(p)p 3033 8631 1102 7 v 3309 8824 a Fq(L)3422 8738 y Fn(\()p Fp(S)6 b Fn(+1\))p Fp(p)3422 8869 y Fn(0)4356 8557 y Fs(1)p 4175 8631 446 7 v 4175 8783 a(2)p Fq(L)4371 8703 y Fp(p)4371 8833 y(k)s Fn(+1)4678 8669 y Fs(+)5045 8557 y(1)p 4864 8631 V 4864 8783 a(2)p Fq(L)5060 8703 y Fp(p)5060 8833 y(k)s Fn(+1)2663 9154 y Fo(\024)3118 9042 y Fs(1)p 2978 9116 363 7 v 2978 9268 a Fq(L)3091 9188 y Fp(p)3091 9318 y(k)s Fn(+1)3361 9154 y Fq(;)2904 b Fs(\(6.13\))758 9582 y(where)60 b(w)-5 b(e)60 b(used)h(\(2.30\))o(.)88 b(If)60 b(not,)h(w)-5 b(e)60 b(m)-5 b(ust)62 b(ha)-5 b(v)g(e)60 b Fq(p)4089 9607 y Fp(k)s Fn(+1)4392 9582 y Fo(\025)55 b Fq(L)4689 9502 y Fk(\000)p Fp(p)4689 9632 y(k)s Fn(+1)4999 9582 y Fs(for)k Fq(k)h Fs(=)54 b(0)p Fq(;)28 b Fs(1)p Fq(;)g Fs(2)p Fq(;)g(:)g(:)g(:)55 b(;)28 b(n)p Fs(,)61 b(so)758 9787 y(\(6.12\))54 b(implies)1989 10033 y(1)p 1989 10107 84 7 v 1989 10259 a(2)2092 10146 y Fq(L)2205 10066 y Fk(\000)p Fp(p)2205 10196 y(k)s Fn(+1)2501 10146 y Fo(\024)46 b Fq(\014)9 b Fs(\()p Fq(Y)g(;)28 b(S)46 b Fs(+)37 b(1\))p Fq(p)3569 10075 y Fp(S)6 b Fn(+1)3569 10196 y Fp(k)3925 10146 y Fs(for)92 b Fq(k)52 b Fs(=)46 b(0)p Fq(;)28 b Fs(1)p Fq(;)g Fs(2)p Fq(;)g(:)g(:)g(:)55 b(;)28 b(n)g(;)814 b Fs(\(6.14\))758 10482 y(whic)-5 b(h)56 b(w)-5 b(e)55 b(plug)h(bac)-5 b(k)56 b(in)-5 b(to)55 b(\(6.12\))g(to)g(get)2018 10772 y Fq(p)2102 10797 y Fp(k)s Fn(+1)2397 10772 y Fo(\024)47 b Fs(2)p Fq(\014)9 b Fs(\()p Fq(Y)g(;)28 b(S)46 b Fs(+)37 b(1\))p Fq(p)3549 10701 y Fp(S)6 b Fn(+1)3549 10822 y Fp(k)3904 10772 y Fs(for)92 b Fq(k)52 b Fs(=)46 b(0)p Fq(;)28 b Fs(1)p Fq(;)g Fs(2)p Fq(;)g(:)g(:)g(:)55 b(;)28 b(n)g(:)835 b Fs(\(6.15\))p eop %%Page: 23 23 23 22 bop 1773 861 a Fn(EXPLICIT)58 b(FINITE)g(V)l(OLUME)h(CRITERIA)g (F)l(OR)g(LOCALIZA)-11 b(TION)883 b(23)758 1259 y Fs(Hence)1380 1542 y(1)p 1044 1616 754 7 v 1044 1771 a Fq(Y)1178 1723 y Fn(\()p Fp(n)p Fn(+1\))p Fp(p)1608 1771 y Fq(L)1721 1691 y Fp(p)1721 1815 y Fn(0)1864 1654 y Fo(\024)46 b Fq(p)2123 1679 y Fp(n)p Fn(+1)2428 1654 y Fo(\024)g Fs(\(2)p Fq(\014)9 b Fs(\()p Fq(Y)g(;)28 b(S)45 b Fs(+)37 b(1\)\))3624 1586 y Fk(\000)3757 1541 y Fe(1)p 3747 1559 76 6 v 3747 1626 a Fi(S)3878 1470 y Fl(\020)3978 1654 y Fs(\(2)p Fq(\014)9 b Fs(\()p Fq(Y)g(;)28 b(S)45 b Fs(+)37 b(1\)\))5028 1541 y Fe(1)p 5018 1559 V 5018 1626 a Fi(S)5122 1654 y Fq(p)5206 1679 y Fn(0)5280 1470 y Fl(\021)5379 1504 y Fn(\()p Fp(S)6 b Fn(+1\))5739 1454 y Fi(n)p Fe(+1)5998 1654 y Fq(:)267 b Fs(\(6.16\))758 2080 y(If)55 b(condition)g(\(2.33\))g (holds,)i(i.e.,)2896 2428 y Fq(p)2980 2453 y Fn(0)3100 2428 y Fq(<)46 b Fs(\(2)p Fq(\014)9 b Fs(\()p Fq(Y)g(;)28 b(S)46 b Fs(+)37 b(1\)\))4297 2359 y Fk(\000)4429 2315 y Fe(1)p 4420 2333 V 4420 2399 a Fi(S)4523 2428 y Fq(;)1742 b Fs(\(6.17\))758 2751 y(w)-5 b(e)55 b(get)g(a)g(con)-5 b(tradiction)56 b(for)f Fq(n)g Fs(large,)g(uniformly)h(in)g Fq(L)4353 2776 y Fn(0)4474 2751 y Fo(\025)46 b Fs(6,)55 b(whic)-5 b(h)56 b(\014nishes)h(the)e(pro)5 b(of.)957 2951 y(If)60 b(If)h(condition)g(\(2.30\))f(do)5 b(es)61 b(not)g(hold,)i(w)-5 b(e)61 b(w)-5 b(ork)60 b(with)h(the)g(scale)g Fq(L)5475 2976 y Fn(1)5604 2951 y Fs(=)56 b Fq(Y)37 b(L)6035 2976 y Fn(0)6109 2951 y Fs(,)63 b(instead)758 3150 y(of)81 b Fq(L)1086 3175 y Fn(0)1161 3150 y Fs(.)153 b(That)81 b(pro)-5 b(vides)83 b(an)f(extra)f(term)h(in)g Fq(Y)119 b Fs(that)81 b(will)h(supply)h(the)e(needed)h(con)-5 b(trol)758 3349 y(uniformly)56 b(in)h Fq(L)1823 3374 y Fn(0)1945 3349 y Fo(\025)47 b Fs(6.)76 b(The)56 b(price)g(w)-5 b(e)57 b(pa)-5 b(y)56 b(is)h(expressed)f(in)h(condition)f(\(2.31\))o(:) 76 b Fq(p)47 b(>)g(d)57 b Fs(and)758 3548 y Fq(S)e Fo(\025)46 b Fs(\()p Fq(p)36 b Fs(+)h Fq(d)p Fs(\))p Fq(=)p Fs(\()p Fq(p)g Fo(\000)g Fq(d)p Fs(\).)957 3758 y(W)-14 b(e)55 b(no)-5 b(w)56 b(supp)5 b(ose)56 b(condition)g(\(2.31\))f(holds.)75 b(If)55 b Fq(p)4171 3783 y Fp(k)4298 3758 y Fq(<)46 b(L)4586 3678 y Fk(\000)p Fp(p)4586 3808 y(k)4822 3758 y Fs(for)55 b(some)i Fq(k)51 b Fo(\025)c Fs(1,)55 b(w)-5 b(e)56 b(ha)-5 b(v)g(e)2130 4168 y Fq(p)2214 4193 y Fp(k)s Fn(+1)2629 4168 y Fo(\024)166 b Fq(\014)9 b Fs(\()p Fq(Y)g(;)28 b(S)46 b Fs(+)37 b(1\))3986 4055 y(1)p 3753 4129 550 7 v 3753 4323 a Fq(L)3866 4236 y Fn(\()p Fp(S)6 b Fn(+1\))p Fp(p)3866 4373 y(k)4359 4168 y Fs(+)4727 4055 y(1)p 4545 4129 446 7 v 4545 4281 a(2)p Fq(L)4741 4202 y Fp(p)4741 4332 y(k)s Fn(+1)2629 4665 y Fo(\024)222 b Fs(2)p Fq(\014)9 b Fs(\()p Fq(Y)g(;)28 b(S)46 b Fs(+)37 b(1\))3925 4553 y Fq(Y)4058 4492 y Fp(p)p 3891 4627 278 7 v 3891 4806 a Fq(L)4004 4726 y Fp(S)6 b(p)4004 4850 y Fn(1)4390 4553 y Fs(1)p 4209 4627 446 7 v 4209 4779 a(2)p Fq(L)4405 4699 y Fp(p)4405 4829 y(k)s Fn(+1)4711 4665 y Fs(+)5079 4553 y(1)p 4897 4627 V 4897 4779 a(2)p Fq(L)5093 4699 y Fp(p)5093 4829 y(k)s Fn(+1)6311 4665 y Fs(\(6.18\))2629 5136 y Fo(\024)3084 5024 y Fs(1)p 2944 5098 363 7 v 2944 5250 a Fq(L)3057 5170 y Fp(p)3057 5300 y(k)s Fn(+1)3355 5136 y Fq(;)2910 b Fs(\(6.19\))758 5609 y(since)55 b Fq(p)46 b(>)g(d)56 b Fs(and)g Fq(S)g Fo(\025)2288 5534 y Fp(p)p Fn(+)p Fp(d)p 2287 5570 242 7 v 2287 5666 a(p)p Fk(\000)p Fp(d)2548 5609 y Fs(.)74 b(Indeed,)56 b(in)f(this)h(case)g(w) -5 b(e)55 b(ha)-5 b(v)g(e)1479 6088 y(2)p Fq(\014)9 b Fs(\()p Fq(Y)g(;)28 b(S)46 b Fs(+)37 b(1\))2425 5976 y Fq(Y)2558 5915 y Fp(p)p 2391 6050 278 7 v 2391 6229 a Fq(L)2504 6149 y Fp(S)6 b(p)2504 6273 y Fn(1)2734 6088 y Fo(\024)102 b Fs(2)3068 5976 y(\(3)p Fq(Y)37 b Fs(\))3414 5915 y Fn(\()p Fp(S)6 b Fn(+1\))p Fp(d)p 3068 6050 783 7 v 3381 6202 a Fq(S)k Fs(!)4093 5976 y Fq(Y)4226 5915 y Fp(p)p 3891 6050 615 7 v 3891 6202 a Fs(\()p Fq(Y)36 b(L)4201 6227 y Fn(0)4276 6202 y Fs(\))4341 6154 y Fp(S)6 b(p)6311 6088 y Fs(\(6.20\))1645 6620 y Fo(\024)1960 6508 y Fs(2)47 b(3)2173 6448 y Fn(\()p Fp(S)6 b Fn(+1\))p Fp(d)p 1960 6582 650 7 v 2082 6734 a Fs(6)2165 6686 y Fp(S)g(p)2330 6734 y Fq(S)k Fs(!)2650 6508 y Fq(Y)2783 6448 y Fn(\()p Fp(S)c Fn(+1\))p Fp(d)p Fn(+)p Fp(p)p 2650 6582 741 7 v 2871 6734 a Fq(Y)3004 6686 y Fp(S)g(p)3457 6620 y Fo(\024)3696 6508 y Fs(3)3779 6448 y Fp(d)p 3652 6582 249 7 v 3652 6734 a Fs(2)3735 6686 y Fp(S)g(d)3920 6620 y Fq(Y)4053 6552 y Fn(\()p Fp(p)p Fn(+)p Fp(d)p Fn(\))p Fk(\000)p Fp(S)g Fn(\()p Fp(p)p Fk(\000)p Fp(d)p Fn(\))4988 6620 y Fo(\024)5163 6386 y Fl(\022)5305 6508 y Fs(3)p 5305 6582 84 7 v 5305 6734 a(4)5408 6386 y Fl(\023)5530 6420 y Fp(d)5654 6620 y Fq(<)46 b Fs(1)56 b Fq(;)758 7070 y Fs(where)70 b(w)-5 b(e)70 b(also)h(used)g Fq(L)2367 7095 y Fn(0)2513 7070 y Fo(\025)g Fs(6)f(and)h Fq(S)80 b Fo(\025)71 b Fs(2.)119 b(Otherwise,)74 b(w)-5 b(e)71 b(m)-5 b(ust)71 b(ha)-5 b(v)g(e)71 b Fq(p)5852 7095 y Fp(k)s Fn(+1)6172 7070 y Fo(\025)g Fq(L)6485 6990 y Fk(\000)p Fp(p)6485 7120 y(k)s Fn(+1)758 7302 y Fs(for)j Fq(k)83 b Fs(=)77 b(1)p Fq(;)28 b Fs(2)p Fq(;)g(:)g(:)g(:)55 b(;)28 b(n)p Fs(,)79 b(so)74 b(it)g(follo)-5 b(ws)75 b(from)g(\(6.12\))e(that)h Fq(\014)9 b Fs(\()p Fq(Y)g(;)28 b(S)58 b Fs(+)50 b(1\))p Fq(p)5446 7231 y Fp(S)6 b Fn(+1)5446 7352 y Fp(k)5787 7302 y Fo(\025)6013 7236 y Fn(1)p 6013 7263 67 7 v 6013 7359 a(2)6099 7302 y Fq(L)6212 7222 y Fk(\000)p Fp(p)6212 7352 y(k)s Fn(+1)6536 7302 y Fs(for)758 7507 y Fq(k)55 b Fs(=)c(1)p Fq(;)28 b Fs(2)p Fq(;)g(:)g(:)g(:)54 b(;)28 b(n)p Fs(,)58 b(so)g(w)-5 b(e)58 b(again)g(get)f(\(6.14\))g(and) i(\(6.15\))o(,)f(but)g(w)-5 b(e)58 b(start)f(from)h Fq(L)5894 7532 y Fn(1)6026 7507 y Fs(instead)g(of)758 7706 y Fq(L)871 7731 y Fn(0)945 7706 y Fs(,)e(so)f(instead)h(of)68 b(\(6.16\))55 b(w)-5 b(e)56 b(get)1451 8012 y(1)p 1115 8086 754 7 v 1115 8241 a Fq(Y)1249 8194 y Fn(\()p Fp(n)p Fn(+1\))p Fp(p)1679 8241 y Fq(L)1792 8162 y Fp(p)1792 8286 y Fn(0)1935 8125 y Fo(\024)46 b Fq(p)2194 8150 y Fp(n)p Fn(+1)2498 8125 y Fo(\024)h Fs(\(2)p Fq(\014)9 b Fs(\()p Fq(Y)g(;)28 b(S)45 b Fs(+)37 b(1\)\))3695 8056 y Fk(\000)3828 8011 y Fe(1)p 3818 8029 76 6 v 3818 8096 a Fi(S)3949 7940 y Fl(\020)4049 8125 y Fs(\(2)p Fq(\014)9 b Fs(\()p Fq(Y)g(;)28 b(S)45 b Fs(+)37 b(1\)\))5099 8011 y Fe(1)p 5089 8029 V 5089 8096 a Fi(S)5193 8125 y Fq(p)5277 8150 y Fn(1)5351 7940 y Fl(\021)5450 7975 y Fn(\()p Fp(S)6 b Fn(+1\))5810 7925 y Fi(n)5927 8125 y Fq(:)338 b Fs(\(6.21\))758 8541 y(W)-14 b(e)55 b(will)g(get)g(a)g(con)-5 b(tradiction)56 b(for)f(large)g Fq(n)p Fs(,)h(uniformly)f(in)h Fq(L)4708 8566 y Fn(0)4829 8541 y Fo(\025)46 b Fs(6,)56 b(if)f(w)-5 b(e)55 b(ha)-5 b(v)g(e)2906 8880 y(\(2)p Fq(\014)9 b Fs(\()p Fq(Y)g(;)28 b(S)46 b Fs(+)37 b(1\)\))3956 8767 y Fe(1)p 3947 8785 V 3947 8851 a Fi(S)4050 8880 y Fq(p)4134 8905 y Fn(1)4255 8880 y Fq(<)46 b Fs(1)28 b Fq(:)1724 b Fs(\(6.22\))758 9212 y(In)58 b(other)g(terms,)i(w)-5 b(e)59 b(need)f(condition)h(\(6.17\))f(but)g(with)h Fq(p)4544 9237 y Fn(1)4676 9212 y Fs(instead)g(of)g Fq(p)5529 9237 y Fn(0)5603 9212 y Fs(.)83 b(Th)-5 b(us)60 b(w)-5 b(e)58 b(need)758 9412 y(to)d(estimate)g Fq(p)1710 9437 y Fn(1)1839 9412 y Fs(in)h(terms)f(of)h Fq(p)2769 9437 y Fn(0)2843 9412 y Fs(.)73 b(Note)55 b(that)g(\(6.12\))f(holds)j(for)e Fq(k)d Fs(=)46 b(0)55 b(as)h(w)-5 b(ell,)56 b(so)2515 9812 y Fq(p)2599 9837 y Fn(1)2720 9812 y Fo(\024)46 b Fq(\014)9 b Fs(\()p Fq(Y)g(;)28 b(S)46 b Fs(+)37 b(1\))p Fq(p)3788 9741 y Fp(S)6 b Fn(+1)3788 9857 y(0)4088 9812 y Fs(+)4537 9700 y(1)p 4274 9774 610 7 v 4274 9926 a(2\()p Fq(Y)37 b(L)4668 9951 y Fn(0)4742 9926 y Fs(\))4807 9878 y Fp(p)4931 9812 y Fq(:)1334 b Fs(\(6.23\))758 10232 y(Hence)54 b(\(6.22\))h(is)h(certainly)f(true)g(if)2087 10684 y(2)2199 10571 y Fe(1)p 2190 10589 76 6 v 2190 10656 a Fi(S)2293 10684 y Fq(\014)9 b Fs(\()p Fq(Y)g(;)28 b(S)46 b Fs(+)37 b(1\))3122 10564 y Fi(S)t Fe(+1)p 3122 10589 218 6 v 3193 10656 a Fi(S)3367 10684 y Fq(p)3451 10613 y Fp(S)6 b Fn(+1)3451 10729 y(0)3752 10684 y Fs(+)3938 10572 y(\(2)p Fq(\014)j Fs(\()p Fq(Y)g(;)28 b(S)45 b Fs(+)37 b(1\)\))4988 10467 y Fe(1)p 4978 10485 76 6 v 4978 10552 a Fi(S)p 3938 10646 1145 7 v 4205 10798 a Fs(2\()p Fq(Y)g(L)4599 10823 y Fn(0)4673 10798 y Fs(\))4738 10750 y Fp(p)5148 10684 y Fq(<)46 b Fs(1)905 b(\(6.24\))p eop %%Page: 24 24 24 23 bop 758 861 a Fn(24)1485 b(FRANC)-77 b(\030)17 b(OIS)59 b(GERMINET)g(AND)f(ABEL)g(KLEIN)758 1259 y Fs(for)47 b(all)g Fq(L)1339 1284 y Fn(0)1460 1259 y Fo(\025)f Fs(6.)71 b(Since)48 b(the)e(\014rst)i(term)f(on)g(the)g(righ)-5 b(t)48 b(hand)g(side)f(of)60 b(\(6.24\))47 b(is)h Fo(\024)5907 1194 y Fn(1)p 5907 1221 67 7 v 5907 1316 a(2)6040 1259 y Fs(b)-5 b(y)50 b(\(6.17\))o(,)758 1473 y(i.e.,)69 b(condition)e (\(2.33\))o(,)j(it)c(su\016ces)i(to)e(sho)-5 b(w)67 b(that)f(the)h (second)g(term)f(is)h(also)g Fo(\024)6114 1408 y Fn(1)p 6114 1435 V 6114 1530 a(2)6200 1473 y Fs(,)j(whic)-5 b(h)758 1672 y(can)55 b(b)5 b(e)55 b(seen)h(as)g(follo)-5 b(ws:)2197 1978 y(\(2)p Fq(\014)9 b Fs(\()p Fq(Y)g(;)28 b(S)46 b Fs(+)37 b(1\)\))3247 1874 y Fe(1)p 3238 1892 76 6 v 3238 1958 a Fi(S)p 2197 2053 1145 7 v 2506 2205 a Fs(\()p Fq(Y)g(L)2817 2230 y Fn(0)2891 2205 y Fs(\))2956 2157 y Fp(p)3407 2091 y Fs(=)3583 1857 y Fl(\022)3725 1978 y Fs(2)p Fq(\014)9 b Fs(\()p Fq(Y)g(;)28 b(S)46 b Fs(+)37 b(1\))p 3725 2053 892 7 v 3863 2205 a(\()p Fq(Y)g(L)4174 2230 y Fn(0)4249 2205 y Fs(\))4314 2157 y Fp(S)6 b(p)4636 1857 y Fl(\023)4788 1846 y Fe(1)p 4779 1864 76 6 v 4779 1931 a Fi(S)4928 2091 y Fq(<)46 b Fs(1)56 b Fq(;)1023 b Fs(\(6.25\))758 2493 y(using)58 b(\(6.20\).)74 b(Hence)54 b(\(6.22\))h(holds,)i(so)e(w)-5 b(e)56 b(get)f(in)g (\(6.21\))g(a)g(con)-5 b(tradiction)56 b(for)f(large)g Fq(n)p Fs(.)957 2693 y(The)h(con)-5 b(tradictions)58 b(to)e(either)g(\(6.16\))g(or)g(\(6.21\))g(happ)5 b(en)57 b(at)f(some)h(large)g Fq(n)p Fs(,)f(ho)-5 b(w)57 b(large)758 2892 y(dep)5 b(ending)60 b(only)g(on)g Fq(d;)28 b(p;)g(Y)9 b(;)28 b(S)70 b Fs(and)60 b Fq(L)3273 2917 y Fn(0)3348 2892 y Fs(.)87 b(Th)-5 b(us)62 b(there)d(is)i Fo(K)55 b Fs(=)f Fo(K)r Fs(\()p Fq(d;)28 b(p;)g(Y)9 b(;)28 b(S;)g(L)5853 2917 y Fn(0)5927 2892 y Fs(\))60 b(suc)-5 b(h)61 b(that)758 3102 y Fq(p)842 3127 y Fp(k)969 3102 y Fq(<)46 b(L)1257 3022 y Fk(\000)p Fp(p)1257 3152 y(k)1493 3102 y Fs(for)55 b(all)h Fq(k)c Fo(\025)46 b(K)r Fs(.)957 3301 y(That)55 b Fq(E)1495 3326 y Fn(0)1615 3301 y Fo(2)46 b Fs(\006)1892 3326 y Fp(M)12 b(S)6 b(A)2282 3301 y Fs(no)-5 b(w)56 b(follo)-5 b(ws)57 b(from)e([GK1)q(].)p 6614 3301 7 113 v 6621 3196 100 7 v 6621 3301 V 6720 3301 7 113 v 2693 3737 a(7.)92 b Fr(Pr)l(oof)62 b(of)g(Theorem)f(2.6)957 4036 y Fs(Theorem)42 b(2.6)g(is)g(a)g(re\014nemen)-5 b(t)42 b(of)g([GK1,)j(Theorem)d(5.2],)j(again)d(the)g(the)f(main)i (di\013erence)758 4235 y(b)5 b(eing)55 b(the)g(explicit)g(conditions.) 758 4583 y Fg(Pr)-8 b(o)g(of)57 b(of)j(The)-8 b(or)g(em)58 b(2.6.)83 b Fs(W)-14 b(e)37 b(pro)5 b(ceed)38 b(as)h(in)f(the)g(pro)5 b(of)37 b(of)h([GK1)q(,)j(Theorem)e(5.2])f(\(see)g([GK1,)758 4809 y(Section)54 b(5.4]\),)g(where)g(w)-5 b(e)54 b(\014x)g Fq(S)i Fs(=)46 b(3)54 b(and)h(tak)-5 b(e)54 b Fq(\013)47 b Fs(=)f(1)34 b(+)4549 4728 y Fp(p=)p Fn(2)p 4497 4771 306 7 v 4497 4866 a Fp(p)p Fn(+2)p Fp(d)4869 4809 y Fo(2)46 b Fs(\(1)p Fq(;)28 b Fs(1)34 b(+)5667 4735 y Fp(p)p 5548 4771 V 5548 4866 a(p)p Fn(+2)p Fp(d)5874 4809 y Fs(\).)73 b(W)-14 b(e)54 b(pro)-5 b(v)g(e)758 5030 y(\(2.24\))o(,)55 b(note)g(that)g(\(2.23\))g(is)h(easier)f(and)h(ma)-5 b(y)56 b(b)5 b(e)55 b(pro)-5 b(v)g(en)56 b(in)g(a)f(similar)i(w)-5 b(a)g(y)-14 b(.)957 5229 y(W)g(e)55 b(start)g(b)-5 b(y)56 b(deriving)f(from)h(\(2.22\))e(the)h(initial)h(step)f(of)h(the)e (inductiv)-5 b(e)56 b(pro)5 b(cess,)56 b(i.e.,)2557 5610 y Fm(P)2714 5476 y Fl(\002)2783 5610 y Fq(R)2938 5476 y Fl(\000)3034 5542 y Fp(m)3152 5559 y Fe(0)p 3034 5572 183 7 v 3092 5668 a Fn(2)3237 5610 y Fq(;)28 b(L)3424 5635 y Fn(0)3498 5610 y Fq(;)g(I)13 b(;)28 b(x;)g(y)3988 5476 y Fl(\001\003)4179 5610 y Fo(\025)46 b Fs(1)37 b Fo(\000)4747 5498 y Fs(4)p 4660 5572 256 7 v 4660 5747 a Fq(L)4773 5667 y Fn(2)p Fp(p)4773 5791 y Fn(0)6394 5610 y Fs(\(7.1\))758 6052 y(for)67 b(all)h Fq(x;)28 b(y)73 b Fo(2)67 b Fm(Z)1870 5992 y Fp(d)2024 6052 y Fs(with)g Fo(j)p Fq(x)46 b Fo(\000)f Fq(y)6 b Fo(j)67 b Fq(>)g(L)3284 6077 y Fn(0)3404 6052 y Fs(+)46 b Fq(\045)p Fs(,)70 b(whic)-5 b(h)69 b(is)f(just)g(\(2.24\))f(with)h Fq(k)73 b Fs(=)67 b(0)g(and)6533 5984 y Fp(m)6651 6001 y Fe(0)p 6533 6014 183 7 v 6591 6110 a Fn(2)758 6252 y Fs(substituted)56 b(for)1901 6184 y Fp(m)2019 6201 y Fe(0)p 1901 6214 V 1959 6309 a Fn(4)2103 6252 y Fs(.)74 b(T)-14 b(o)56 b(do)f(that,)g(w)-5 b(e)56 b(recall)f(that)1591 6584 y(\003)1706 6609 y Fp(L)1806 6584 y Fs(\()p Fq(x)p Fs(\))f(is)i(\()p Fq(\022)5 b(;)28 b(E)10 b Fs(\)-suitable)91 b Fo(\()-28 b(\))92 b Fs(\003)3892 6609 y Fp(L)3992 6584 y Fs(\()p Fq(x)p Fs(\))55 b(is)h(\(2)p Fq(\022)4690 6510 y Fn(log)24 b Fp(L)p 4689 6546 286 7 v 4786 6641 a(L)4994 6584 y Fq(;)k(E)10 b Fs(\)-regular)27 b Fq(:)492 b Fs(\(7.2\))758 6897 y(As)55 b(in)h([GK1,)g(p.)74 b(440],)56 b(w)-5 b(e)55 b(set)2037 7279 y Fq(\016)2111 7304 y Fn(2)2231 7279 y Fs(=)47 b Fq(\016)2481 7304 y Fn(2)2555 7279 y Fs(\()p Fq(m)2766 7304 y Fn(0)2840 7279 y Fq(;)28 b(L)3027 7304 y Fn(0)3101 7279 y Fq(;)g(s)p Fs(\))45 b(=)3684 7166 y(1)p 3559 7240 334 7 v 3559 7392 a(2)p Fq(L)3755 7335 y Fn(2)p Fp(s)3755 7437 y Fn(0)3939 7094 y Fl(h)4018 7279 y Fs(e)4092 7210 y Fk(\000)4215 7148 y Fi(m)4317 7173 y Fe(0)p 4215 7184 167 6 v 4270 7250 a(2)4422 7148 y Fi(L)4501 7173 y Fe(0)p 4422 7184 145 6 v 4466 7250 a(2)4631 7279 y Fo(\000)37 b Fs(e)4871 7210 y Fk(\000)p Fp(m)5093 7227 y Fe(0)5177 7148 y Fi(L)5256 7173 y Fe(0)p 5177 7184 V 5221 7250 a(2)5350 7094 y Fl(i)6394 7279 y Fs(\(7.3\))758 7700 y(and)49 b Fq(I)13 b Fs(\()p Fq(\016)1299 7725 y Fn(2)1373 7700 y Fs(\))46 b(=)g(\()p Fq(E)1847 7725 y Fn(0)1945 7700 y Fo(\000)24 b Fq(\016)2172 7725 y Fn(2)2246 7700 y Fq(;)k(E)2443 7725 y Fn(0)2541 7700 y Fs(+)c Fq(\016)2768 7725 y Fn(2)2842 7700 y Fs(\))g Fo(\\)g(I)12 b Fs(.)72 b(Using)52 b(\(2.22\))o(,)e(Assumption)h(W)d (with)h Fq(\021)j Fs(=)47 b Fq(L)6197 7629 y Fk(\000)p Fp(s)6197 7744 y Fn(0)6371 7700 y Fs(,)k(and)758 7899 y(requiring)3196 8264 y Fq(L)3309 8289 y Fn(0)3430 8264 y Fo(\025)46 b Fq(Q)3933 8106 y Fe(1)p 3756 8124 411 6 v 3756 8190 a Fi(s)p Ff(\000)p Fi(p)p Ff(\000)p Fi(bd)3736 8312 y Fk(I)4223 8264 y Fq(;)2125 b Fs(\(7.4\))758 8577 y(obtain)55 b(the)g(equiv)-9 b(alen)k(t)55 b(to)g([GK1,)h(Eq)f (\(5.37\)]:)1809 8880 y Fm(P)6 b Fo(f)p Fs(for)64 b(ev)-5 b(ery)55 b Fq(E)g Fo(2)46 b Fq(I)13 b Fs(\()p Fq(\016)3247 8905 y Fn(2)3321 8880 y Fs(\))o Fq(;)74 b Fs(\003)3620 8905 y Fp(L)3711 8922 y Fe(0)3785 8880 y Fs(\(0\))91 b(is)56 b(\()4341 8812 y Fp(m)4459 8829 y Fe(0)p 4340 8842 183 7 v 4399 8937 a Fn(2)4543 8880 y Fq(;)28 b(E)10 b Fs(\)-regular)o Fo(g)934 b Fs(\(7.5\))3341 9223 y Fo(\025)47 b Fs(1)37 b Fo(\000)3876 9110 y Fs(1)p 3823 9185 190 7 v 3823 9337 a Fq(L)3936 9257 y Fp(p)3936 9381 y Fn(0)4069 9223 y Fo(\000)g Fq(Q)4366 9248 y Fk(I)4458 9223 y Fq(L)4571 9152 y Fp(bd)p Fk(\000)p Fp(s)4571 9267 y Fn(0)4919 9223 y Fo(\025)46 b Fs(1)37 b Fo(\000)5453 9110 y Fs(2)p 5400 9185 V 5400 9337 a Fq(L)5513 9257 y Fp(p)5513 9381 y Fn(0)5637 9223 y Fq(:)758 9626 y Fs(Using)55 b(Assumption)i(IAD,)e (\(7.1\))g(follo)-5 b(ws)56 b(from)f(\(7.5\).)957 9825 y(F)-14 b(or)45 b(reasons)g(that)e(will)i(b)5 b(ecome)44 b(clear)g(later)f(w)-5 b(e)45 b(will)f(w)-5 b(ork)44 b(with)g(a)g(smaller)h(op)5 b(en)45 b(in)-5 b(terv)c(al)758 10024 y Fq(I)13 b Fs(,)55 b(with)g Fq(E)1446 10049 y Fn(0)1567 10024 y Fo(2)45 b Fq(I)60 b Fo(\032)46 b(I)12 b Fs(,)56 b(de\014ned)g(as)2174 10327 y Fq(I)j Fs(=)46 b Fq(I)13 b Fs(\()p Fq(\016)2706 10352 y Fn(2)2780 10327 y Fs(\))37 b Fo(\\)f(f)p Fq(E)56 b Fo(2)45 b(I)c Fs(;)120 b(dist\()p Fq(E)10 b(;)28 b Fm(R)21 b Fo(nI)12 b Fs(\))55 b Fq(>)46 b(L)4987 10256 y Fk(\000)p Fp(s)4987 10372 y Fn(0)5162 10327 y Fo(g)28 b Fq(:)1075 b Fs(\(7.6\))957 10641 y(The)69 b(induction)g(no)-5 b(w)70 b(pro)5 b(ceeds)69 b(as)h(in)f([GK1].)115 b(The)69 b(induction)h(step)f(go)5 b(es)69 b(from)g(scale)758 10840 y Fq(`)46 b Fo(\025)g Fq(L)1161 10865 y Fn(0)1337 10840 y Fo(\025)g Fs(3)p Fq(\045)55 b Fs(to)f(scale)h Fq(L)46 b Fs(=)g([)p Fq(`)2784 10780 y Fp(\013)2879 10840 y Fs(])2925 10865 y Fn(6)p Fj(N)3084 10840 y Fs(,)55 b(using)j(\(2.9\))c(with)h Fq(\021)d Fs(=)47 b Fq(L)4818 10780 y Fk(\000)p Fp(s)4992 10840 y Fs(.)74 b(W)-14 b(e)55 b(need)f(to)h(satisfy)g(the)p eop %%Page: 25 25 25 24 bop 1773 861 a Fn(EXPLICIT)58 b(FINITE)g(V)l(OLUME)h(CRITERIA)g (F)l(OR)g(LOCALIZA)-11 b(TION)883 b(25)758 1259 y Fs(equiv)-9 b(alen)k(t)54 b(to)h([GK1)q(,)g(Eq.)74 b(\(5.42\)])55 b(\(recall)g Fq(S)g Fs(=)46 b(3\),)2517 1481 y Fl(\002)2587 1616 y Fq(\015)2673 1641 y Fk(I)2764 1616 y Fs(\(21)36 b(+)h(2\))3345 1547 y Fp(d)3422 1616 y Fq(L)3535 1547 y Fp(s)3606 1481 y Fl(\003)3703 1432 y(h)3781 1616 y Fq(\015)3867 1641 y Fk(I)3958 1616 y Fs(3)4041 1547 y Fp(d)4119 1616 y Fs(e)4193 1547 y Fk(\000)p Fp(m)4438 1503 y Fi(`)p 4434 1521 57 6 v 4434 1587 a Fe(2)4519 1432 y Fl(i)4643 1616 y Fq(<)47 b Fs(1)p Fq(;)1446 b Fs(\(7.7\))758 2003 y(for)55 b(all)g Fq(`)47 b Fo(\025)f Fq(L)1646 2028 y Fn(0)1776 2003 y Fs(and)2119 1935 y Fp(m)2237 1952 y Fe(0)p 2119 1965 183 7 v 2177 2060 a Fn(4)2367 2003 y Fo(\024)h Fq(m)e Fo(\024)2930 1935 y Fp(m)3048 1952 y Fe(0)p 2930 1965 V 2988 2060 a Fn(2)3132 2003 y Fs(,)56 b Fq(m)3380 2028 y Fn(0)3500 2003 y Fs(=)46 b(2)p Fq(\022)3861 1929 y Fn(log)25 b Fp(L)4146 1946 y Fe(0)p 3861 1965 351 7 v 3958 2060 a Fp(L)4049 2077 y Fe(0)4231 2003 y Fs(.)74 b(T)-14 b(o)56 b(do)f(so,)h(it)f(su\016ces)i (to)e(require)2442 2372 y(69)2608 2304 y Fp(d)2685 2372 y Fq(\015)2780 2304 y Fn(2)2771 2413 y Fk(I)2862 2372 y Fq(`)2931 2304 y Fp(\013s)3089 2372 y Fs(e)3163 2304 y Fk(\000)p Fp(m)3385 2321 y Fe(0)3472 2259 y Fi(`)p 3469 2277 57 6 v 3469 2344 a Fe(8)3600 2372 y Fq(<)46 b Fs(1)92 b(for)55 b(all)93 b Fq(`)46 b Fo(\025)g Fq(L)4875 2397 y Fn(0)4977 2372 y Fq(;)1371 b Fs(\(7.8\))758 2679 y(i.e.,)2425 3042 y(69)2591 2973 y Fp(d)2668 3042 y Fq(\015)2763 2973 y Fn(2)2754 3083 y Fk(I)2845 3042 y Fq(t)2905 2973 y Fp(\013s)3063 3042 y Fq(L)3176 2945 y Fk(\000)3299 2893 y Fi(\022)s(t)p Ff(\000)p Fe(4)p Fi(\013s)p 3299 2918 384 6 v 3463 2985 a Fe(4)3176 3086 y Fn(0)3758 3042 y Fq(<)46 b Fs(1)92 b(for)55 b(all)93 b Fq(t)46 b Fo(\025)g Fs(1)28 b Fq(:)1326 b Fs(\(7.9\))758 3350 y(T)-14 b(o)55 b(satisfy)h(the)f(inequalit)-5 b(y)55 b(at)g Fq(t)46 b Fs(=)g(1)55 b(w)-5 b(e)56 b(require)2501 3713 y Fq(\022)50 b(>)d Fs(4)p Fq(\013)q(s)91 b Fs(and)i Fq(L)3637 3738 y Fn(0)3758 3713 y Fo(\025)3933 3578 y Fl(\000)4009 3713 y Fs(69)4175 3644 y Fp(d)4253 3713 y Fq(\015)4348 3644 y Fn(2)4339 3754 y Fk(I)4430 3578 y Fl(\001)4666 3560 y Fe(4)p 4526 3578 338 6 v 4526 3645 a Fi(\022)s Ff(\000)p Fe(4)p Fi(\013s)4946 3713 y Fq(:)1319 b Fs(\(7.10\))758 4030 y(Since)51 b(log)30 b Fq(L)1542 4055 y Fn(0)1663 4030 y Fo(\025)46 b Fs(1)51 b(as)h Fq(L)2285 4055 y Fn(0)2406 4030 y Fo(\025)46 b Fs(6,)52 b(it)f(follo)-5 b(ws)52 b(that)f(\(7.9\))f(then)i(holds)g(for)f(all)g Fq(t)c Fo(\025)f Fs(1.)72 b(Th)-5 b(us)53 b(\(7.7\))758 4229 y(is)j(satis\014ed)g(if)68 b(\(7.10\))55 b(holds.)957 4428 y(No)-5 b(w,)55 b(the)g(equiv)-9 b(alen)k(t)55 b(of)g([GK1)q(,)g (Eq.)74 b(\(5.47\)])54 b(is)1748 4820 y Fq(N)1881 4845 y Fp(`)2111 4820 y Fo(\025)2426 4708 y Fq(L)p 2426 4782 114 7 v 2448 4934 a(`)2596 4820 y Fo(\000)37 b Fs(8)g Fo(\001)g Fs(3)g Fo(\000)g Fs(1)46 b(=)3576 4708 y Fq(L)p 3576 4782 V 3598 4934 a(`)3745 4820 y Fo(\000)37 b Fs(25)47 b Fo(\025)4319 4708 y Fq(L)p 4319 4782 V 4341 4934 a(`)4479 4636 y Fl(\020)4579 4820 y Fs(1)37 b Fo(\000)g Fs(50)p Fq(`)5100 4751 y Fk(\000)p Fn(\()p Fp(\013)p Fk(\000)p Fn(1\))5572 4636 y Fl(\021)5699 4820 y Fq(;)566 b Fs(\(7.11\))758 5174 y(where)55 b(w)-5 b(e)55 b(used,)i(and)e(hence)h(require,)3314 5481 y Fq(`)46 b Fo(\025)g Fq(L)3717 5506 y Fn(0)3838 5481 y Fo(\025)g Fs(12)2132 b(\(7.12\))758 5798 y(Th)-5 b(us)56 b([GK1)q(,)f(Eq.)74 b(\(5.48\)])55 b(follo)-5 b(ws)56 b(with)1256 6217 y Fq(m)1402 6148 y Fk(0)1615 6217 y Fo(\025)166 b Fq(m)2084 5983 y Fl(\022)2206 6217 y Fs(1)36 b Fo(\000)2595 6104 y Fs(50)p 2511 6179 334 7 v 2511 6331 a Fq(`)2580 6283 y Fp(\013)p Fk(\000)p Fn(1)2865 5983 y Fl(\023)3024 6217 y Fo(\000)h Fs(2)3301 5983 y Fl(\024)3409 6104 y Fs(\()p Fq(\013)g Fo(\000)g Fs(1\))p Fq(d)28 b Fs(log)i Fq(`)p 3409 6179 948 7 v 3657 6331 a(`)3726 6283 y Fp(\013)3858 6331 y Fo(\000)37 b Fs(6)4413 6217 y(+)4599 6104 y(log)r(\(3)4961 6044 y Fp(d)5038 6104 y Fq(\015)5124 6129 y Fk(I)5215 6104 y Fs(\))p 4599 6179 682 7 v 4905 6331 a Fq(`)5337 6217 y Fs(+)5550 6104 y Fq(\013)q(s)28 b Fs(log)i Fq(`)p 5523 6179 579 7 v 5523 6331 a Fs(\()o Fq(`)5656 6283 y Fp(\013)5788 6331 y Fo(\000)37 b Fs(6\))6121 5983 y Fl(\025)1615 6694 y Fo(\025)166 b Fq(m)2084 6460 y Fl(\024)2171 6694 y Fs(1)37 b Fo(\000)2561 6582 y Fs(50)p 2477 6656 334 7 v 2477 6808 a Fq(`)2546 6760 y Fp(\013)p Fk(\000)p Fn(1)2867 6694 y Fo(\000)3053 6582 y Fs(4)28 b(log)s(\(3)3527 6522 y Fp(d)3604 6582 y Fq(\015)3690 6607 y Fk(I)3781 6582 y Fs(\))p 3053 6656 792 7 v 3238 6808 a Fq(\022)33 b Fs(log)d Fq(`)3902 6694 y Fo(\000)4088 6582 y Fs(4\()p Fq(\013)37 b Fo(\000)g Fs(1\))p Fq(d)g Fs(+)g(4)p Fq(\013)q(s)p 4088 6656 1163 7 v 4461 6808 a(\022)5 b(`)4613 6760 y Fp(\013)p Fk(\000)p Fn(1)5270 6460 y Fl(\025)6311 6694 y Fs(\(7.13\))1615 7172 y Fo(\025)166 b Fq(m)2084 6938 y Fl(\024)2171 7172 y Fs(1)37 b Fo(\000)2561 7059 y Fs(52)p 2477 7134 334 7 v 2477 7286 a Fq(`)2546 7238 y Fp(\013)p Fk(\000)p Fn(1)2867 7172 y Fo(\000)3053 7059 y Fs(4)28 b(log)s(\(3)3527 6999 y Fp(d)3604 7059 y Fq(\015)3690 7084 y Fk(I)3781 7059 y Fs(\))p 3053 7134 792 7 v 3238 7286 a Fq(\022)33 b Fs(log)d Fq(`)3865 6938 y Fl(\025)3980 7172 y Fq(:)2285 b Fs(\(7.14\))758 7609 y(where)54 b(w)-5 b(e)55 b(used)g(the)g(fact)e(that)2850 7535 y Fn(log)26 b Fp(L)p 2850 7570 286 7 v 2947 7666 a(L)3210 7609 y Fs(is)56 b(decreasing)f(in)g Fq(L)f Fs(for)h Fq(L)46 b Fo(\025)g Fs(3,)55 b(1)46 b Fq(<)h(\013)g(<)f Fs(2,)55 b Fq(d;)28 b(s)p Fs(,)55 b(and)758 7808 y Fq(\022)50 b(>)d Fs(4)p Fq(\013)q(s)p Fs(.)957 8007 y(Con)-5 b(tin)g(uing)83 b(as)f(in)g([GK1)q(],)89 b(w)-5 b(e)81 b(construct)h(the)f(sequence)h (of)f(length)h(scales)g Fq(L)6266 8032 y Fp(k)s Fn(+1)6606 8007 y Fs(=)758 8206 y([)p Fq(L)917 8146 y Fp(\013)917 8253 y(k)1011 8206 y Fs(])1057 8231 y Fn(6)p Fj(N)1216 8206 y Fs(,)61 b Fq(k)e Fs(=)54 b(0)p Fq(;)28 b Fs(1)p Fq(;)g(:)g(:)g(:)f Fs(.)87 b(Applying)60 b(the)g(inductiv)-5 b(e)60 b(step)f(from)h(scale)g Fq(L)5325 8231 y Fp(k)5467 8206 y Fs(to)f(scale)h Fq(L)6189 8231 y Fp(k)s Fn(+1)6439 8206 y Fs(,)h(w)-5 b(e)758 8406 y(obtain)65 b(a)g(decreasing)g (sequence)g(of)g(masses)i Fq(m)3845 8345 y Fk(0)3845 8453 y Fp(k)3926 8406 y Fs(,)g(with)e Fq(m)4573 8345 y Fk(0)4573 8447 y Fn(0)4710 8406 y Fs(=)4921 8338 y Fp(m)5039 8355 y Fe(0)p 4921 8368 183 7 v 4979 8463 a Fn(2)5123 8406 y Fs(,)j(satisfying)d([GK1)q(,)i(Eq.)758 8605 y(\(5.48\)])54 b(and)i(\(7.14\))f(at)g(scale)h Fq(L)2800 8630 y Fp(k)2881 8605 y Fs(.)74 b(W)-14 b(e)55 b(th)-5 b(us)56 b(need)g(to)f(v)-5 b(erify)55 b([GK1,)h(\(5.51\)],)e(i.e.,)2552 9045 y(0)46 b Fo(\024)2911 8837 y Fk(1)2857 8887 y Fl(X)2857 9245 y Fp(k)s Fn(=0)3098 9045 y Fs(\()p Fq(m)3309 8976 y Fk(0)3309 9086 y Fp(k)3427 9045 y Fo(\000)37 b Fq(m)3739 8976 y Fk(0)3739 9086 y Fp(k)s Fn(+1)3989 9045 y Fs(\))45 b Fq(<)4295 8933 y(m)4441 8958 y Fn(0)p 4295 9007 221 7 v 4363 9159 a Fs(4)4562 9045 y Fq(:)1703 b Fs(\(7.15\))758 9510 y(T)-14 b(o)59 b(do)g(so,)h(note)f(that)f(it)h(follo)-5 b(ws)60 b(from)f(\(7.14\))f(that)g(it)h(su\016ces)h(to)f(sho)-5 b(w)60 b(that)e(for)h(suitable)758 9709 y Fq(L)871 9734 y Fn(0)1001 9709 y Fs(w)-5 b(e)55 b(ha)-5 b(v)g(e)3322 9925 y Fk(1)3268 9975 y Fl(X)3267 10333 y Fp(k)s Fn(=0)3662 10021 y Fs(52)p 3557 10095 378 7 v 3557 10261 a Fq(L)3670 10190 y Fp(\013)p Fk(\000)p Fn(1)3670 10311 y Fp(k)4120 10133 y Fq(<)4435 10021 y Fs(1)p 4435 10095 84 7 v 4435 10247 a(4)4566 10133 y Fq(;)1699 b Fs(\(7.16\))2908 10482 y Fk(1)2854 10532 y Fl(X)2853 10890 y Fp(k)s Fn(=0)3142 10578 y Fs(4)28 b(log)s(\(3)3616 10517 y Fp(d)3693 10578 y Fq(\015)3779 10603 y Fk(I)3870 10578 y Fs(\))p 3142 10652 792 7 v 3265 10804 a Fq(\022)k Fs(log)e Fq(L)3730 10829 y Fp(k)4120 10690 y Fq(<)4435 10578 y Fs(1)p 4435 10652 84 7 v 4435 10804 a(4)4566 10690 y Fq(:)1699 b Fs(\(7.17\))p eop %%Page: 26 26 26 25 bop 758 861 a Fn(26)1485 b(FRANC)-77 b(\030)17 b(OIS)59 b(GERMINET)g(AND)f(ABEL)g(KLEIN)758 1261 y Fs(Recall)d Fq(L)1371 1286 y Fp(k)1499 1261 y Fs(=)46 b Fq(L)1787 1201 y Fp(\013)1873 1151 y Fi(k)1787 1302 y Fn(0)1954 1261 y Fs(.)74 b(F)-14 b(or)56 b(\(7.16\),)f(w)-5 b(e)56 b(use)f Fq(\013)3536 1201 y Fp(k)3664 1261 y Fo(\025)46 b Fq(k)5 b(\013)q Fs(,)56 b(so)1529 1490 y Fk(1)1475 1540 y Fl(X)1474 1897 y Fp(k)s Fn(=0)2042 1585 y Fs(1)p 1764 1659 641 7 v 1764 1856 a Fq(L)1877 1770 y Fp(\013)1963 1737 y Fi(k)2035 1770 y Fn(\()p Fp(\013)p Fk(\000)p Fn(1\))1877 1901 y(0)2590 1698 y Fo(\024)3052 1585 y Fs(1)p 2905 1659 378 7 v 2905 1825 a Fq(L)3018 1754 y Fp(\013)p Fk(\000)p Fn(1)3018 1869 y(0)3339 1698 y Fs(+)3560 1490 y Fk(1)3506 1540 y Fl(X)3505 1897 y Fp(k)s Fn(=1)4073 1585 y Fs(1)p 3794 1659 642 7 v 3794 1853 a Fq(L)3907 1766 y Fp(\013)p Fn(\()p Fp(\013)p Fk(\000)p Fn(1\))p Fp(k)3907 1897 y Fn(0)6311 1698 y Fs(\(7.18\))2590 2200 y(=)3052 2087 y(1)p 2905 2161 378 7 v 2905 2327 a Fq(L)3018 2256 y Fp(\013)p Fk(\000)p Fn(1)3018 2372 y(0)3339 2200 y Fs(+)4221 2087 y(1)p 3525 2161 1475 7 v 3525 2372 a Fq(L)3638 2301 y Fp(\013)p Fk(\000)p Fn(1)3638 2417 y(0)3931 2188 y Fl(\020)4030 2372 y Fs(1)37 b Fo(\000)g Fq(L)4429 2286 y Fk(\000)p Fn(\()p Fp(\013)p Fk(\000)p Fn(1\))4429 2417 y(0)4901 2188 y Fl(\021)5066 2200 y Fo(\024)5551 2087 y Fs(2)p 5261 2161 664 7 v 5261 2327 a Fq(L)5374 2256 y Fp(\013)p Fk(\000)p Fn(1)5374 2372 y(0)5676 2327 y Fo(\000)g Fs(1)5972 2200 y Fq(;)758 2726 y Fs(and)56 b(hence)f(\(7.16\))f(holds)j(if)e(w)-5 b(e)56 b(require)3245 3058 y Fq(L)3358 3083 y Fn(0)3479 3058 y Fo(\025)46 b Fs(417)4006 2941 y Fe(1)p 3923 2959 223 6 v 3923 3026 a Fi(\013)p Ff(\000)p Fe(1)4201 3058 y Fq(:)2064 b Fs(\(7.19\))758 3388 y(Since)1191 3263 y Fl(P)1366 3304 y Fk(1)1366 3437 y Fp(k)s Fn(=0)1644 3388 y Fq(\013)1751 3327 y Fk(\000)p Fp(k)1982 3388 y Fs(=)2263 3322 y Fp(\013)p 2178 3349 257 7 v 2178 3445 a(\013)p Fk(\000)p Fn(1)2454 3388 y Fs(,)55 b(\(7.17\))g(holds)h(if)g(w)-5 b(e)55 b(require)3037 3786 y Fq(L)3150 3811 y Fn(0)3270 3786 y Fo(\025)3446 3651 y Fl(\000)3522 3786 y Fs(3)3605 3717 y Fp(d)3682 3786 y Fq(\015)3768 3811 y Fk(I)3859 3651 y Fl(\001)4046 3633 y Fe(16)p Fi(\013)p 3955 3651 372 6 v 3955 3719 a(\022)s Fe(\()p Fi(\013)p Ff(\000)p Fe(1\))4410 3786 y Fq(:)1855 b Fs(\(7.20\))957 4145 y(Note)54 b(that)h(417)2070 4028 y Fe(1)p 1987 4046 223 6 v 1987 4113 a Fi(\013)p Ff(\000)p Fe(1)2284 4145 y Fo(\025)47 b Fs(417)2729 4040 y Fe(1)p 2729 4058 57 6 v 2729 4124 a(2)2860 4145 y Fo(\025)f Fs(20)h Fq(>)f Fs(12)56 b(since)g Fq(\013)47 b Fo(\024)4401 4079 y Fn(3)p 4401 4106 67 7 v 4401 4202 a(2)4487 4145 y Fs(,)55 b(and)h(hence)g(\(7.12\))f(follo)-5 b(ws)56 b(from)758 4344 y(\(7.19\))o(.)957 4555 y(W)-14 b(e)58 b(con)-5 b(tin)g(ue)60 b(as)g(in)f([GK1,)h(Eqs.)85 b (\(5.54\)-\(5.55\)],)59 b(with)4668 4513 y(~)4648 4555 y Fq(I)4721 4580 y Fn(0)4847 4555 y Fs(=)52 b Fo(I)72 b Fs(and)59 b Fq(I)5589 4580 y Fn(0)5716 4555 y Fs(=)52 b Fq(I)13 b Fs(.)85 b(W)-14 b(e)58 b(need)758 4754 y(the)d(equiv)-9 b(alen)k(t)54 b(of)h([GK1)q(,)h(Eq.)73 b(\(5.55\)],)55 b(i.e.,)1997 5197 y(2)p Fq(C)2199 5222 y Fk(I)2290 5197 y Fq(Q)2421 5222 y Fk(I)2512 5197 y Fs(\(3)37 b(+)g(1\))3011 5128 y Fn(2)3113 4963 y Fl(\022)3255 5084 y Fs(6)p Fq(L)p 3255 5159 197 7 v 3318 5311 a(`)3471 4963 y Fl(\023)3593 4997 y Fn(2)p Fp(d)3764 5197 y Fq(L)3877 5128 y Fn(\()p Fp(b)p Fn(+1\))p Fp(d)4285 5197 y Fq(L)4398 5128 y Fk(\000)p Fp(s)4619 5197 y Fo(\024)4900 5084 y Fs(2)p 4814 5159 256 7 v 4814 5311 a Fq(L)4927 5263 y Fn(2)p Fp(p)5117 5197 y Fq(;)1148 b Fs(\(7.21\))758 5591 y(so)56 b(it)f(su\016ces)h(to)f (establish)i(the)e(\014rst)g(inequalit)-5 b(y)55 b(in)1116 6018 y(16)37 b Fo(\001)g Fs(36)1568 5950 y Fp(d)1646 6018 y Fq(C)1765 6043 y Fk(I)1855 6018 y Fq(Q)1986 6043 y Fk(I)2078 6018 y Fq(`)2147 5950 y Fn(2)p Fp(d)p Fn(\()p Fp(\013)p Fk(\000)p Fn(1\)+)p Fp(\013)p Fn(\()p Fp(b)p Fn(+1\))p Fp(d)3284 6018 y Fo(\024)3460 5784 y Fl(\022)3602 5906 y Fq(`)3671 5846 y Fp(\013)p 3602 5980 164 7 v 3642 6132 a Fs(2)3786 5784 y Fl(\023)3908 5819 y Fp(s)p Fk(\000)p Fn(2)p Fp(p)4263 6018 y Fo(\024)47 b Fs(\()o Fq(`)4572 5950 y Fp(\013)4704 6018 y Fo(\000)37 b Fs(6\))5017 5935 y Fp(s)p Fk(\000)p Fn(2)p Fp(p)5373 6018 y Fo(\024)46 b Fq(L)5661 5950 y Fp(s)p Fk(\000)p Fn(2)p Fp(p)5998 6018 y Fq(:)267 b Fs(\(7.22\))758 6412 y(Th)-5 b(us)56 b(w)-5 b(e)56 b(m)-5 b(ust)56 b(sho)-5 b(w)57 b(that)d(for)i(all)f Fq(`)46 b Fo(\025)h Fq(L)3494 6437 y Fn(0)3624 6412 y Fs(w)-5 b(e)55 b(ha)-5 b(v)g(e)2254 6716 y(16)37 b Fo(\001)g Fs(36)2706 6647 y Fp(d)2783 6716 y Fs(2)2866 6647 y Fp(s)p Fk(\000)p Fn(2)p Fp(p)3175 6716 y Fq(C)3294 6741 y Fk(I)3385 6716 y Fq(Q)3516 6741 y Fk(I)3653 6716 y Fo(\024)47 b Fq(`)3898 6647 y Fp(\013)p Fn(\()p Fp(s)p Fk(\000)p Fn(2)p Fp(p)p Fk(\000)p Fn(\()p Fp(b)p Fn(+3\))p Fp(d)p Fn(\)+2)p Fp(d)5166 6716 y Fq(;)1099 b Fs(\(7.23\))758 7020 y(whic)-5 b(h)56 b(is)g(true)f(if)2338 7368 y Fq(s)46 b(>)g Fs(2)p Fq(p)36 b Fs(+)h(\()p Fq(b)g Fs(+)g(2\))p Fq(d)46 b(>)h Fs(2)p Fq(p)36 b Fs(+)h(\()p Fq(b)g Fs(+)g(3\))p Fq(d)g Fo(\000)4966 7256 y Fs(2)p Fq(d)p 4966 7330 170 7 v 4997 7482 a(\013)6311 7368 y Fs(\(7.24\))758 7718 y(and)2282 8045 y Fq(L)2395 8070 y Fn(0)2515 8045 y Fo(\025)2691 7911 y Fl(\000)2767 8045 y Fs(16)g Fo(\001)g Fs(36)3219 7977 y Fp(d)3296 8045 y Fs(2)3379 7977 y Fp(s)p Fk(\000)p Fn(2)p Fp(p)3689 8045 y Fq(C)3808 8070 y Fk(I)3898 8045 y Fq(Q)4029 8070 y Fk(I)4120 7911 y Fl(\001)4621 7893 y Fe(1)p 4217 7911 865 6 v 4217 7979 a Fi(\013)p Fe(\()p Fi(s)p Ff(\000)p Fe(2)p Fi(p)p Ff(\000)p Fe(\()p Fi(b)p Fe(+2\))p Fi(d)p Fe(\))5165 8045 y Fq(:)1100 b Fs(\(7.25\))957 8359 y(Using)52 b(the)g(de\014nition)g(of)g(the)g(in)-5 b(terv)c(al)51 b Fq(I)13 b Fs(,)53 b(giv)-5 b(en)52 b(in)h(\(7.6\))o(,) g(the)e(equiv)-9 b(alen)k(t)51 b(of)h([GK1)q(,)g(Eq.)758 8558 y(\(5.56\)])59 b(follo)-5 b(ws)61 b(from)f(\(7.21\).)87 b(T)-14 b(o)61 b(ensure)f(the)g(equiv)-9 b(alen)k(t)59 b(of)h([GK1)q(,)h(Eqs.)88 b(\(5.57\)-\(5.58\)],)758 8757 y(w)-5 b(e)55 b(need)2613 8950 y Fl(")2710 9000 y(\022)2852 9122 y Fs(3)p Fq(L)p 2852 9196 197 7 v 2915 9348 a(`)3068 9000 y Fl(\023)3190 9035 y Fn(2)p Fp(d)3446 9122 y Fs(4)p 3381 9196 213 7 v 3381 9348 a Fq(`)3450 9300 y Fn(2)p Fp(p)3613 8950 y Fl(#)3730 8933 y Fe(3+1)p 3730 8958 199 6 v 3801 9025 a(2)4003 9234 y Fo(\024)4285 9122 y Fs(1)p 4198 9196 256 7 v 4198 9348 a Fq(L)4311 9300 y Fn(2)p Fp(p)4502 9234 y Fq(;)1763 b Fs(\(7.26\))758 9678 y(so)56 b(it)f(su\016ces)h(to)f(ensure)2206 9839 y Fl(\024)2294 10073 y Fs(9)2377 10005 y Fp(d)2454 10073 y Fq(`)2523 10005 y Fn(2\()p Fp(\013)p Fk(\000)p Fn(1\))p Fp(d)3112 9961 y Fs(4)p 3047 10035 213 7 v 3047 10187 a Fq(`)3116 10139 y Fn(2)p Fp(p)3279 9839 y Fl(\025)3367 9874 y Fn(2)3487 10073 y Fo(\024)3790 9961 y Fs(1)p 3683 10035 299 7 v 3683 10187 a Fq(`)3752 10139 y Fn(2)p Fp(\013p)4139 10073 y Fs(for)g(all)139 b Fq(`)46 b Fo(\025)h Fq(L)5111 10098 y Fn(0)5213 10073 y Fq(;)1052 b Fs(\(7.27\))758 10463 y(i.e.,)2290 10772 y(4)37 b Fo(\001)g Fs(9)2576 10703 y Fp(d)2700 10772 y Fo(\024)46 b Fq(`)2944 10703 y Fn(2)p Fp(p)p Fk(\000)p Fp(\013p)p Fk(\000)p Fn(2\()p Fp(\013)p Fk(\000)p Fn(1\))p Fp(d)4083 10772 y Fs(for)55 b(all)139 b Fq(`)46 b Fo(\025)g Fq(L)5054 10797 y Fn(0)5157 10772 y Fq(:)1108 b Fs(\(7.28\))p eop %%Page: 27 27 27 26 bop 1773 861 a Fn(EXPLICIT)58 b(FINITE)g(V)l(OLUME)h(CRITERIA)g (F)l(OR)g(LOCALIZA)-11 b(TION)883 b(27)758 1259 y Fs(Since)63 b(2)p Fq(p)41 b Fo(\000)h Fq(\013)q(p)f Fo(\000)h Fs(2\()p Fq(\013)g Fo(\000)g Fs(1\))p Fq(d)59 b Fs(=)2948 1194 y Fn(7)p 2948 1221 67 7 v 2948 1316 a(2)3034 1259 y Fq(p)k Fs(b)-5 b(y)63 b(the)f(de\014nition)h(of)g Fq(\013)q Fs(,)h(\(7.28\))e(follo)-5 b(ws)64 b(from)f(\(7.19\))758 1458 y(since)1699 1760 y(\(4)37 b Fo(\001)g Fs(9)2050 1692 y Fp(d)2127 1760 y Fs(\))2242 1641 y Fe(2)p 2212 1659 118 6 v 2212 1726 a(7)p Fi(p)2404 1760 y Fs(=)46 b(\(4)37 b Fo(\001)g Fs(9)2930 1692 y Fp(d)3007 1760 y Fs(\))3425 1635 y Fe(1)p 3092 1653 724 6 v 3092 1721 a(7\()p Fi(\013)p Ff(\000)p Fe(1\)\()p Fi(p)p Fe(+2)p Fi(d)p Fe(\))3889 1760 y Fo(\024)47 b Fs(\(4)36 b Fo(\001)h Fs(9)4415 1692 y Fp(d)4493 1760 y Fs(\))4736 1635 y Fe(1)p 4577 1653 374 6 v 4577 1721 a Fi(d)p Fe(\()p Fi(\013)p Ff(\000)p Fe(1\))5025 1760 y Fo(\024)46 b Fs(417)5552 1644 y Fe(1)p 5469 1662 223 6 v 5469 1728 a Fi(\013)p Ff(\000)p Fe(1)5748 1760 y Fq(:)517 b Fs(\(7.29\))957 2072 y(The)49 b(desired)h(conclusion)g(\(2.24\))e(no)-5 b(w)50 b(follo)-5 b(ws)49 b(as)h(in)f([GK1)q(].)72 b(The)49 b(fact)f(that)g Fq(I)59 b Fo(\032)47 b Fs(\006)6373 2097 y Fp(M)12 b(S)6 b(A)758 2271 y Fs(no)-5 b(w)56 b(follo)-5 b(ws)56 b(from)f(\(2.24\))g(and)h([GK1,)g(Theorem)f(3.4].)957 2470 y(The)60 b(exp)5 b(onen)-5 b(tial)61 b(deca)-5 b(y)60 b(of)g(the)h(eigenfunctions)f(giv)-5 b(en)61 b(in)g(\(2.26\))f(follo)-5 b(ws)61 b(from)g(\(2.24\))758 2670 y(as)56 b(in)f([vDK,)g(Theorem)h (2.3].)p 6614 2670 7 113 v 6621 2564 100 7 v 6621 2670 V 6720 2670 7 113 v 3262 3102 a Fr(References)758 3368 y Fy([AM])259 b(Aizenman,)52 b(M.,)i(Molc)l(hano)l(v,)f(S.:)74 b(Lo)t(calization)47 b(at)52 b(large)e(disorder)h(and)i(extreme)e (energies:)71 b(an)1330 3534 y(elemen)l(tary)46 b(deriv)-8 b(ation.)45 b(Comm)l(un.)i(Math.)h(Ph)l(ys.)g Fb(157)p Fy(,)g(245-278)c(\(1993\))758 3700 y([ASFH])112 b(Aizenman,)45 b(M.,)h(Sc)l(henk)l(er,)i(J.,)f(F)-12 b(riedric)l(h,)45 b(R.)h(,)g(Hundertmark,)h(D.:)62 b(Finite-v)l(olume)43 b(criteria)h(for)1330 3866 y(Anderson)j(lo)t(calization.)42 b(Comm)l(un.)47 b(Math.)g(Ph)l(ys.)h Fb(224)p Fy(,)h(219-253)44 b(\(2001\))758 4032 y([AENSS])71 b(Aizenman,)46 b(M.,)h(Elgart,)f(A.,)h (Nab)t(ok)l(o,)f(S.,)i(Sc)l(henk)l(er,)h(J.,)f(Stolz,)e(G.:)62 b(In)48 b(preparation)758 4198 y([BCH])186 b(Barbaroux,)55 b(J.M,)f(Com)l(b)t(es,)g(J.M.,)i(Hislop,)d(P)-12 b(.D.:)150 b(Landau)54 b(Hamiltonians)49 b(with)k(Un)l(b)t(ounded)1330 4364 y(Random)47 b(P)l(oten)l(tials,)d(Lett.)j(Math.)g(Ph)l(ys.)h Fb(40)p Fy(,)g(355-369)d(\(1997\).)758 4530 y([CKM])153 b(Carmona,)60 b(R.,)h(Klein,)f(A.,)h(Martinelli,)d(F.:)86 b(Anderson)58 b(lo)t(calization)53 b(for)58 b(Bernoulli)e(and)j(other) 1330 4696 y(singular)45 b(p)t(oten)l(tials.)g(Comm)l(un.)i(Math.)g(Ph)l (ys.)h Fb(108)p Fy(,)g(41-66)e(\(1987\))758 4862 y([CH1])215 b(Com)l(b)t(es,)52 b(J.M.,)j(Hislop,)c(P)-12 b(.D.:)73 b(Lo)t(calization)48 b(for)j(some)g(con)l(tin)l(uous,)i(random)e (Hamiltonian)e(in)1330 5028 y(d-dimension.)d(J.)i(F)-12 b(unct.)48 b(Anal.)e Fb(124)p Fy(,)i(149-180)d(\(1994\))758 5194 y([CH2])215 b(Com)l(b)t(es,)45 b(J.M.,)j(Hislop,)c(P)-12 b(.D.:)63 b(Landau)46 b(Hamiltonians)c(with)k(random)g(p)t(oten)l (tials:)59 b(lo)t(calization)1330 5360 y(and)48 b(the)f(densit)l(y)g (of)f(states.)g(Comm)l(un.)h(Math.)g(Ph)l(ys.)h Fb(177)p Fy(,)g(603-629)d(\(1996\))758 5526 y([CHN])180 b(Com)l(b)t(es,)43 b(J.M.,)h(Hislop,)d(P)-12 b(.D.,)44 b(Nak)-8 b(am)l(ura,)43 b(S.:)61 b(The)43 b Fv(L)4399 5480 y Ft(p)4471 5526 y Fy(-theory)g(of)f(the)g(sp)t(ectral)f(shift)g(function,)1330 5692 y(the)59 b(W)-12 b(egner)59 b(estimate)e(and)i(the)h(in)l (tegrated)d(densit)l(y)i(of)g(states)f(for)h(some)f(random)h(op)t (erators.)1330 5858 y(Comm)l(un.)47 b(Math.)g(Ph)l(ys.)h Fb(218)p Fy(,)h(113-130)44 b(\(2001\))758 6025 y([CHKN])70 b(Com)l(b)t(es,)62 b(J.M.,)h(Hislop,)e(P)-12 b(.D.,)63 b(Klopp,)f(F.,)h(Nak)-8 b(am)l(ura,)61 b(S.:)88 b(The)60 b(W)-12 b(egner)59 b(estimate)d(and)k(the)1330 6191 y(in)l(tegrated)45 b(densit)l(y)i(of)f(states)g(for)g(some)f(random)i(op)t(erators.)e(Pro) t(c.)h(Indean)h(Acad.)g(Sci.)f(\(Math.)1330 6357 y(Sci.\))h Fb(112)p Fy(,)h(31-53)d(\(2002\))758 6523 y([CHT])184 b(Com)l(b)t(es,)56 b(J.M.,)i(Hislop,)c(P)-12 b(.D.,)58 b(Tip,)e(A.:)78 b(Band)55 b(edge)f(lo)t(calization)c(and)55 b(the)g(densit)l(y)f(of)h(states)1330 6689 y(for)50 b(acoustic)e(and)j (electromagnetic)46 b(w)l(a)l(v)l(es)k(in)f(random)h(media.)f(Ann.)i (Inst.)g(H.)f(P)l(oincare)e(Ph)l(ys.)1330 6855 y(Theor.)f Fb(70)h Fy(,)f(381-428)d(\(1999\))758 7021 y([DS])308 b(Damanik,)46 b(D.,)h(Stollmann,)f(P)-12 b(.:)64 b(Multi-scale)44 b(analysis)g(implies)h(strong)h(dynamical)f(lo)t(calization.)1330 7187 y(Geom.)h(F)-12 b(unct.)48 b(Anal.)e Fb(11)p Fy(,)i(11-29)d (\(2001\))758 7353 y([DMP1])90 b(Dorlas)62 b(T.C.)j(,)k(Macris)63 b(N.)i(,)k(Pul)l(\023)-67 b(e)64 b(J.V.)h(,)138 b(The)65 b(Nature)e(of)i(the)f(Sp)t(ectrum)h(for)f(the)h(Landau)1330 7519 y(Hamiltonian)43 b(with)k(delta)f(impurities,)e(J.)k(Stat.)f(Ph)l (ys.)h Fb(87)p Fy(,)g(847-875)d(\(1997\))758 7685 y([DMP2])90 b(Dorlas)48 b(T.C.)j(,)h(Macris)d(N.)h(,)i(Pul)l(\023)-67 b(e)50 b(J.V.)h(,)103 b(Characterization)47 b(of)k(the)f(Sp)t(ectrum)h (of)f(the)h(Landau)1330 7851 y(Hamiltonian)43 b(with)k(delta)f (impurities,)e(Comm)l(un.)j(Math.)h(Ph)l(ys.)g Fb(204)p Fy(,)g(367-396)c(\(1999\))758 8017 y([vD])312 b(v)l(on)42 b(Dreifus,)g(H.:)60 b Fh(On)47 b(the)e(e\013e)-7 b(cts)46 b(of)f(r)-7 b(andomness)46 b(in)g(ferr)-7 b(omagnetic)43 b(mo)-7 b(dels)46 b(and)g(Schr\177)-73 b(odinger)1330 8183 y(op)-7 b(er)g(ators)p Fy(.)47 b(Ph.D.)g(thesis,)f(New)h(Y)-12 b(ork)47 b(Univ)l(ersit)l(y)f(\(1987\))758 8349 y([vDK])202 b(v)l(on)57 b(Dreifus,)i(H.,)g(Klein,)e(A.:)83 b(A)56 b(new)h(pro)t(of)f(of)h(lo)t(calization)51 b(in)56 b(the)h(Anderson)g (tigh)l(t)f(binding)1330 8515 y(mo)t(del.)46 b(Comm)l(un.)h(Math.)g(Ph) l(ys.)h Fb(124)p Fy(,)g(285-299)d(\(1989\))758 8681 y([vDK2])131 b(v)l(on)33 b(Dreifus,)h(H.,)h(Klein,)e(A.:)56 b(Lo)t(calization)28 b(for)j(random)h(Sc)l(hro)t(dinger)g(op)t(erators)e(with)h(correlated) 1330 8847 y(p)t(oten)l(tials.)45 b(Comm)l(un.)i(Math.)g(Ph)l(ys.)h Fb(140)p Fy(,)g(133-147)d(\(1991\))758 9013 y([FK1])221 b(Figotin,)38 b(A.,)j(Klein,)d(A.:)59 b(Lo)t(calization)34 b(of)k(classical)c(w)l(a)l(v)l(es)k(I:)g(Acoustic)e(w)l(a)l(v)l(es.)j (Comm)l(un.)f(Math.)1330 9179 y(Ph)l(ys.)48 b Fb(180)p Fy(,)g(439-482)d(\(1996\).)758 9345 y([FK2])221 b(Figotin,)52 b(A.,)h(Klein,)f(A.:)73 b(Lo)t(calization)48 b(of)k(classical)47 b(w)l(a)l(v)l(es)52 b(I)t(I:)g(Electromagnetic)c(w)l(a)l(v)l(es.)k (Com-)1330 9511 y(m)l(un.)c(Math.)f(Ph)l(ys.)h Fb(184)p Fy(,)g(411-441)d(\(1997\).)758 9677 y([FMSS])117 b(F)-12 b(r\177)-71 b(ohlic)l(h,)44 b(J.,)i(Martinelli,)41 b(F.,)46 b(Scopp)t(ola,)d(E.,)j(Sp)t(encer,)f(T.:)62 b(Constructiv)l(e)43 b(pro)t(of)h(of)f(lo)t(calization)1330 9844 y(in)k(the)g(Anderson)g (tigh)l(t)f(binding)h(mo)t(del.)e(Comm)l(un.)i(Math.)g(Ph)l(ys.)h Fb(101)p Fy(,)h(21-46)c(\(1985\))758 10010 y([FS])324 b(F)-12 b(r\177)-71 b(ohlic)l(h,)58 b(J.,)i(Sp)t(encer,)g(T.:)83 b(Absence)56 b(of)g(di\013usion)g(with)g(Anderson)h(tigh)l(t)f(binding) g(mo)t(del)g(for)1330 10176 y(large)45 b(disorder)h(or)h(lo)l(w)f (energy.)h(Comm)l(un.)g(Math.)g(Ph)l(ys.)h Fb(88)p Fy(,)g(151-184)c (\(1983\))758 10342 y([FLM])185 b(Fisc)l(her,)35 b(W.,)i(Lesc)l(hk)l (e,)f(H.,)g(M)t(\177)-74 b(uller,)35 b(P)-12 b(.:)56 b(Sp)t(ectral)32 b(lo)t(calization)d(b)l(y)34 b(Gaussian)e(random)h(p)t (oten)l(tials)1330 10508 y(in)47 b(m)l(ulti-dimensional)42 b(con)l(tin)l(uous)47 b(space.)f(J.)i(Statist.)e(Ph)l(ys.)h Fb(101)i Fy(935{985)44 b(\(2000\))758 10674 y([GDB])175 b(Germinet,)46 b(F,)h(De)f(Bi)l(\022)-67 b(evre,)47 b(S.:)63 b(Dynamical)45 b(lo)t(calization)d(for)k(discrete)g(and)h(con)l(tin)l (uous)g(random)1330 10840 y(Sc)l(hr\177)-71 b(odinger)47 b(op)t(erators.)e(Comm)l(un.)i(Math.)g(Ph)l(ys.)h Fb(194)p Fy(,)h(323-341)44 b(\(1998\))p eop %%Page: 28 28 28 27 bop 758 861 a Fn(28)1485 b(FRANC)-77 b(\030)17 b(OIS)59 b(GERMINET)g(AND)f(ABEL)g(KLEIN)758 1259 y Fy([GK1])202 b(Germinet,)71 b(F,)d(Klein,)j(A.:)103 b(Bo)t(otstrap)66 b(Multiscale)e(Analysis)i(and)i(Lo)t(calization)63 b(in)k(Random)1330 1425 y(Media.)46 b(Comm)l(un.)h(Math.)h(Ph)l(ys.)g Fb(222)p Fy(,)g(415-448)c(\(2001\).)758 1591 y([GK2])202 b(Germinet,)36 b(F,)g(Klein,)g(A.:)57 b(Op)t(erator)35 b(k)l(ernel)g(estimates)d(for)j (functions)g(of)h(generalized)c(Sc)l(hrdinger)1330 1757 y(op)t(erators.)46 b(Pro)t(c.)g(Amer.)g(Mat.)h(So)t(c.)g(\(2002\))758 1923 y([GK3])202 b(Germinet,)59 b(F,)e(Klein,)i(A.:)83 b(A)57 b(c)l(haracterization)d(of)j(the)g(Anderson)h(metal-insulator)53 b(transp)t(ort)1330 2089 y(transition.)45 b(Submitted.)758 2255 y([GK4])202 b(Germinet,)46 b(F,)h(Klein,)e(A.:)63 b(In)48 b(preparation)758 2421 y([Gr])328 b(Grimmet,)45 b(G.:)63 b(P)l(ercolation,)44 b(Springer-V)-12 b(erlag)45 b(1989.)758 2587 y([HK])278 b(Hislop,)37 b(P)-12 b(.D.,)40 b(Klopp,)e(F.:)59 b(The)37 b(in)l(tegrated)f(densit)l(y)h(of)f(states)g (for)h(some)f(random)g(op)t(erators)g(with)1330 2753 y(nonsign)46 b(de\014nite)h(p)t(oten)l(tials.)e(J.)j(F)-12 b(unct.)48 b(Anal.)e(T)-12 b(o)47 b(app)t(ear)758 2919 y([HM])259 b(Holden,)46 b(H.,)h(Martinelli,)d(F.:)63 b(On)47 b(absence)g(of)f(di\013usion)g(near)h(the)g(b)t(ottom)f(of)g (the)h(sp)t(ectrum)g(for)1330 3086 y(a)g(random)f(Sc)l(hr\177)-71 b(odinger)47 b(op)t(erator.)f(Comm)l(un.)h(Math.)g(Ph)l(ys.)h Fb(93)p Fy(,)g(197-217)c(\(1984\))758 3252 y([Ki])344 b(Kirsc)l(h,)37 b(W.:)58 b(W)-12 b(egner)36 b(estimates)d(and)k(lo)t (calization)31 b(for)36 b(allo)l(y-t)l(yp)t(e)f(p)t(oten)l(tials.)f (Math.)i(Zeit.)g Fb(221)p Fy(,)1330 3418 y(507-512)45 b(\(1996\))758 3584 y([KM])255 b(W.)40 b(Kirsc)l(h)f(and)i(F.)f (Martinelli,)e(On)i(the)h(Sp)t(ectrum)f(of)f(Sc)l(hr\177)-71 b(odinger)40 b(Op)t(erators)f(with)g(a)h(Random)1330 3750 y(P)l(oten)l(tial.)45 b(Comm)l(un.)i(Math.)g(Ph)l(ys.)h Fb(85)p Fy(,)g(329-350)d(\(1982\).)758 3916 y([KSS1])157 b(Kirsc)l(h,)55 b(W.,)h(Stolz,)f(G.,)h(Stollman,)e(P)-12 b(.:)78 b(Lo)t(calization)50 b(for)j(random)h(p)t(erturbations)f(of)h (p)t(erio)t(dic)1330 4082 y(Sc)l(hr\177)-71 b(odinger)47 b(op)t(erators.)e(Random)i(Op)t(er.)g(Sto)t(c)l(hastic)e(Equations)h Fb(6)p Fy(,)i(241-268)c(\(1998\))758 4248 y([KSS2])157 b(Kirsc)l(h,)43 b(W.,)g(Stolz,)g(G.,)h(Stollman,)d(P)-12 b(.:)62 b(Anderson)42 b(lo)t(calization)c(for)k(random)g(Sc)l(hr\177) -71 b(odinger)43 b(op)t(er-)1330 4414 y(ators)j(with)g(long)g(range)g (in)l(teractions.)f(Comm)l(un.)i(Math.)g(Ph)l(ys.)h Fb(195)p Fy(,)h(495-507)44 b(\(1998\))758 4580 y([K])384 b(Klein,)33 b(A.:)56 b(Lo)t(calization)27 b(in)k(the)h(Anderson)g(mo)t(del)e(with)h (long)g(range)g(hopping.)h(Brazilian)d(Journal)1330 4746 y(of)47 b(Ph)l(ysics)f Fb(23)p Fy(,)i(363-371)d(\(1993\))758 4912 y([KK1])203 b(Klein,)53 b(A.,)h(Koines,)f(A.:)74 b(A)52 b(general)f(framew)l(ork)h(for)h(lo)t(calization)47 b(of)53 b(classical)48 b(w)l(a)l(v)l(es:)74 b(I.)53 b(Inho-)1330 5078 y(mogeneous)45 b(media)h(and)h(defect)g(eigenmo)t(des.)e(Math.)i (Ph)l(ys.)h(Anal.)e(Geom.)g Fb(4)p Fy(,)i(97-130)d(\(2001\))758 5244 y([KK2])203 b(Klein,)34 b(A.,)i(Koines,)f(A.:)56 b(A)32 b(general)g(framew)l(ork)g(for)h(lo)t(calization)28 b(of)33 b(classical)28 b(w)l(a)l(v)l(es:)56 b(I)t(I.)33 b(Random)1330 5410 y(media.)46 b(In)h(preparation.)758 5576 y([KLS])218 b(Klein.,)48 b(A.,)i(Lacroix,)e(J.,)i(Sp)t(eis,)f(A.:) 133 b(Lo)t(calization)45 b(for)j(the)h(Anderson)g(mo)t(del)e(on)i(a)g (strip)f(with)1330 5742 y(singular)d(p)t(oten)l(tials.)g(J.)j(F)-12 b(unct.)48 b(Anal.)e Fb(94)p Fy(,)i(135-155)c(\(1990\))758 5908 y([Klo1])202 b(Klopp,)49 b(F.:)66 b(Lo)t(calization)44 b(for)49 b(con)l(tin)l(uous)f(random)g(Sc)l(hr\177)-71 b(odinger)48 b(op)t(erators.)f(Comm)l(un.)i(Math.)1330 6074 y(Ph)l(ys.)f Fb(167)p Fy(,)g(553-569)d(\(1995\))758 6240 y([Klo2])202 b(Klopp)79 b(F.:)129 b(W)-12 b(eak)79 b(disorder)g(lo)t(calization)c(and)80 b(Lifshitz)e(tails:)126 b(con)l(tin)l(uous)79 b(Hamiltonians,)1330 6406 y(Preprin)l(t.)758 6572 y([Si])376 b(Simon,)47 b(B.:)63 b(Sc)l(hr\177)-71 b(odinger)46 b(semi-groups.)f(Bull.)h(Amer.)h(Math.)g(So)t(c.)g Fb(V)-14 b(ol.7)p Fy(,)48 b(447-526)d(\(1982\))758 6739 y([Sp])338 b(Sp)t(encer,)57 b(T.)d(:)78 b(Lo)t(calization)50 b(for)55 b(random)f(and)g(quasip)t(erio)t(dic)e(p)t(oten)l(tials.)h(J.) i(Stat.)f(Ph)l(ys.)h Fb(51)p Fy(,)1330 6905 y(1009-1019)44 b(\(1988\))758 7071 y([St])361 b(Stollmann,)46 b(P)-12 b(.:)63 b(Caugh)l(t)47 b(b)l(y)h(disorder.)e(Bound)i(States)e(in)h (Random)f(Media.)h(Birk)-8 b(a)t(\177)-74 b(user)45 b(2001.)758 7237 y([U])388 b(Ueki,)46 b(N.:)63 b(W)-12 b(egner)47 b(estimates)d(and)j(lo)t(calization)42 b(for)47 b(Gaussian)e(random)i (p)t(oten)l(tials.)e(Preprin)l(t)758 7403 y([W1])278 b(W)-12 b(ang,)35 b(W.-M.:)55 b(Microlo)t(calization,)29 b(p)t(ercolation,)j(and)f(Anderson)h(lo)t(calization)26 b(for)31 b(the)h(magnetic)1330 7569 y(Sc)l(hr\177)-71 b(odinger)47 b(op)t(erator)e(with)i(a)f(random)h(p)t(oten)l(tial.)e(J.) j(F)-12 b(unct.)48 b(Anal.)e Fb(146)p Fy(,)i(1-26)e(\(1997\))758 7735 y([W2])278 b(W)-12 b(ang,)35 b(W.-M.:)56 b(Lo)t(calization)27 b(and)32 b(univ)l(ersalit)l(y)e(of)h(P)l(oisson)f(statistics)e(for)j (the)h(m)l(ultidimensional)1330 7901 y(Anderson)47 b(mo)t(del)f(at)g(w) l(eak)h(disorder.)f(In)l(v)l(en)l(t.)i(Math.)g Fb(146)p Fy(,)g(365-398)d(\(2001\))957 8212 y Fx(Universit)1601 8201 y(\023)1601 8212 y(e)52 b(de)f(Lille)g(1,)i(UMR)e(8524)i(CNRS,)g (UFR)e(de)g(Ma)-9 b(th)4661 8201 y(\023)4661 8212 y(ema)g(tiques,)54 b(F-59655)f(Villeneuve)758 8378 y(d'Ascq)d(C)1369 8367 y(\023)1369 8378 y(edex,)i(France)957 8544 y Fh(E-mail)e(addr)-7 b(ess)11 b Fy(:)62 b Fa(germinet@agat.univ-lille)o(1.)o(fr)957 8827 y Fx(University)44 b(of)g(Calif)m(ornia,)k(Ir)-12 b(vine,)47 b(Dep)-9 b(ar)g(tment)44 b(of)g(Ma)-9 b(thema)g(tics,)46 b(Ir)-12 b(vine,)46 b(CA)e(92697-3875,)758 8993 y(USA)957 9159 y Fh(E-mail)50 b(addr)-7 b(ess)11 b Fy(:)62 b Fa(aklein@uci.edu)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0209120831959--