This is a multi-part message in MIME format. ---------------0210290650366 Content-Type: text/plain; name="02-439.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-439.keywords" essential spectrum, graph, tree, free monoid, C*-algebra, shrodinger operator, anisotropic potential, hyperbolic, compatification, ultrametric ---------------0210290650366 Content-Type: application/postscript; name="graphe2.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="graphe2.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.90a Copyright 2002 Radical Eye Software %%Title: graphe2.dvi %%Pages: 27 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips graphe2.dvi -o graphe2.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.10.29:1529 %%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 600 600 (graphe2.dvi) @start %DVIPSBitmapFont: Fa cmcsc10 12 43 /Fa 43 123 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Fb cmex10 17.28 1 /Fb 1 102 df101 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc eufm10 12 1 /Fc 1 68 df<17181778EE03FC160F163F16FF923803EFFEED078FED1F0791390C3F03FF EC3FFF9126FFFE01138E01036E13FED90FE315FCD91F03EC7FF0017EED3F8001F8ED1C00 00036E90C7FCEA07F0EA0FE0A2EA1FC082EA3F8080A2007F811300A280A25AA55EA393C8 FC6D485A5D4A5A4A5A397FC01FC0EC7F8002FCC9FCEBE04091CAFC6C7EA27F121F7F7F6C 7E806C7F6C7F02F8150C6C01FE153C6C6D6C14F86D01F0EB03F06D01FEEB0FC06D90B612 0001075D6D15F801005D023F14C002075C9126003FF8C7FC37477AC43D>67 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd cmr17 17.28 3 /Fd 3 42 df0 D<150E151E153C157815F0EC01E0EC03C01407EC0F80EC1F00143EA25C5C13015C 495A13075C130F5C131F91C7FC5B133E137E137C13FCA2485AA3485AA3485AA3120F5BA3 121F5BA3123FA390C8FCA25AA5127EA312FEB3A7127EA3127FA57EA27FA3121FA37F120F A37F1207A36C7EA36C7EA36C7EA2137C137E133E133F7F80130F8013078013036D7E8013 00147C80A280EC0F80EC07C01403EC01E0EC00F01578153C151E150E1F8F73EA33>40 D<12E07E12787E7E7E6C7E7F6C7E6C7E6C7EA2137C7F133F7F6D7E801307801303801301 80130080147C147EA280A3EC1F80A3EC0FC0A315E01407A315F01403A315F8A31401A215 FCA51400A315FEB3A715FCA31401A515F8A21403A315F0A3140715E0A3140F15C0A3EC1F 80A3EC3F00A3147EA2147C14FC5C13015C13035C13075C130F5C49C7FC5B133E5B5BA248 5A485A485A5B48C8FC121E5A5A5A5A1F8F7AEA33>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe cmmi12 17.28 1 /Fe 1 97 df96 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff msam10 12 1 /Ff 1 4 df<007FBA1280BB12C0A300F0CB1203B3B3B3A6BBFCA36C198042447BC34D>3 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg stmary8 8 2 /Fg 2 76 df74 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fh cmr6 6 5 /Fh 5 51 df<130C1338137013E0EA01C0EA038013005A120EA25AA25AA312781270A312 F0AB1270A312781238A37EA27EA27E7E1380EA01C0EA00E013701338130C0E317AA418> 40 D<12C012707E7E7E7E7E1380EA01C0A2EA00E0A21370A313781338A3133CAB1338A3 13781370A313E0A2EA01C0A2EA038013005A120E5A5A5A12C00E317CA418>I<13FF0003 13C0380781E0380F00F0001E137848133CA248131EA400F8131FAD0078131EA2007C133E 003C133CA26C13786C13F0380781E03803FFC0C6130018227DA01E>48 D<13E01201120712FF12F91201B3A7487EB512C0A212217AA01E>II E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fi cmsy7 7 1 /Fi 1 4 df<1338A50060130C00F8133E00FC137E00FE13FE383FBBF83807FFC0000113 00EA007C48B4FC000713C0383FBBF838FE38FE00FC137E00F8133E0060130C00001300A5 17197B9A22>3 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fj cmsy10 10 4 /Fj 4 111 df<126012F812FEEA7F80EA3FE0EA0FF8EA03FEC66C7EEB3FE0EB0FF8EB03 FE903800FF80EC3FE0EC0FF8EC03FE913800FF80ED3FE0ED0FF8ED03FE923800FF80EE3F E0EE0FF8EE03FE933800FF80EF3FC0171FEF7F80933801FF00EE07FCEE1FF0EE7FC04B48 C7FCED07FCED1FF0ED7FC04A48C8FCEC07FCEC1FF0EC7FC04948C9FCEB07FCEB1FF0EB7F C04848CAFCEA07FCEA1FF0EA7FC048CBFC12FC1270CCFCAE007FB81280B912C0A26C1780 324479B441>21 D102 D<12FCEAFFC0EA07F0EA01FC EA007E7F80131F80130FB3A7801307806D7E6D7EEB007EEC1FF0EC07F8EC1FF0EC7E0049 5A495A495A5C130F5CB3A7131F5C133F91C7FC137E485AEA07F0EAFFC000FCC8FC1D537A BD2A>I<126012F07EA21278127CA2123C123EA2121E121FA27E7FA212077FA212037FA2 12017FA212007FA21378137CA2133C133EA2131E131FA27F80A2130780A26D7EA2130180 A2130080A21478147CA2143C143EA2141E141FA2801580A2140715C0A2140315E0A21401 15F0A2140015F8A21578157CA2153C153EA2151E150C1F537BBD2A>110 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fk msbm10 10 1 /Fk 1 79 df78 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fl cmr10 10 16 /Fl 16 117 df<121C127FEAFF80A5EA7F00121C0909798817>46 D48 D<007FB812F8B912FCA26C17F8CCFCAE007FB812F8B912FCA26C17F836167B9F41> 61 D87 D97 D100 DI<147E903803FF8090380FC1E0EB1F8790383F0FF0137EA213FC A23901F803C091C7FCADB512FCA3D801F8C7FCB3AB487E387FFFF8A31C3B7FBA19>IIII<3903F00FF000FFEB3F FCECF03F9039F1C01F803A0FF3800FC03803F70013FE496D7EA25BA35BB3A3486C497EB5 00C1B51280A329257EA42E>110 DI<3807E01F00FFEB7FC09038E1E3 E09038E387F0380FE707EA03E613EE9038EC03E09038FC0080491300A45BB3A2487EB512 F0A31C257EA421>114 DI<1318A51338A31378A313F8120112031207001FB5FCB6FCA2D801F8C7FC B215C0A93800FC011580EB7C03017E13006D5AEB0FFEEB01F81A347FB220>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fm cmr7 7 1 /Fm 1 50 df<13381378EA01F8121F12FE12E01200B3AB487EB512F8A215267BA521>49 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fn stmary10 12 2 /Fn 2 76 df74 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fo cmmi6 6 9 /Fo 9 122 df11 DI99 D<1338137CA2137813701300A7EA 0780EA1FC0EA38E01230EA60F0EAC1E0A3EA03C0A3EA0780A2EA0F0013041306EA1E0CA2 1318121CEA1E70EA0FE0EA07800F237DA116>105 D<1418143C147CA214381400A7EB07 80EB1FE01338EB60F013C0A2EA0180A2380001E0A4EB03C0A4EB0780A4EB0F00A4131EA2 1238EA783CEAF8381378EA70F0EA7FC0001FC7FC162D81A119>I<13F8EA0FF0A21200A2 485AA4485AA43807801E147FEB81C3EB8387380F060F495A1318EB700E4848C7FCA213FC EA1E7EEA3C0F80EB0781158039780F0300A21402EB070600F0138CEB03F8386000F01924 7CA221>I<000F13FC381FC3FF3931C707803861EC0301F813C0EAC1F0A213E03903C007 80A3EC0F00EA0780A2EC1E041506D80F00130C143C15181538001EEB1C70EC1FE0000CEB 07801F177D9526>110 D<3801F01E3907FC7F80390E1CE1C038180F8100301383007013 071260EC0380D8001EC7FCA45BA21580003014C0397878018012F8EC030038F0FC0638E1 9C1C387F0FF8381E03E01A177D9523>120 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fp cmex8 8 2 /Fp 2 102 df<143014FCEB03FF010F13C0013F13F090387F03F83901FC00FED807E0EB 1F80D81F80EB07E0007EC7EA01F800F0EC003C00C0150C260C80B027>98 D<017F14082601FFC0131C000701F01378489038FC01F0D83E00B512C00078013F138000 E090380FFE000040EB03F8260880AF27>101 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fq cmti12 12 50 /Fq 50 128 df12 D<123C127C12FEA27E7EEA3F80121F13C0120FEA07E0120313F0120112001378A2133813 100D1364C432>18 DI<13F0EA03F8EA07FC120FA6EA 03CCEA001C1318A213381330A2137013E013C0120113801203EA0700120E5A5A5A5A5A0E 1D6BC41E>39 D<13F0EA03FC1207A2EA0FFEA4EA07FCEA03CCEA000C131C1318A2133813 301370136013E0EA01C013801203EA0700120E5A5A5A5A5A0F1D7A891E>44 D<007FB5FCB6FCA214FEA21805789723>I<120FEA3FC0127FA212FFA31380EA7F00123C 0A0A76891E>I<16C01501A215031507ED0F80151F153F157F913801FF005C140F147F90 3807FCFEEB0FF0EB0700EB00015DA314035DA314075DA3140F5DA3141F5DA3143F5DA314 7F92C7FCA35C5CA313015CA313035CA313075CA2130FA2131F133FB612FCA25D224276C1 32>49 D51 D<130FEB1FC0133FEB7FE013FFA214C0 EB7F801400131E90C7FCB3A5120FEA3FC0127FA212FFA35B6CC7FC123C132B76AA1E>58 D65 D67 D<91B912C0A30201902680000313806E90C8127F4A163F191F4B150FA30203EE07005DA3 14074B5D190EA2140F4B1307A25F021F020E90C7FC5DA2171E023F141C4B133C177C17FC 027FEB03F892B5FCA39139FF8003F0ED00011600A2495D5CA2160101034B13705C19F061 010791C8FC4A1501611803010F5F4A150796C7FC60131F4A151E183E183C013F167C4A15 FC4D5A017F1503EF0FF04A143F01FF913803FFE0B9FCA26042447AC342>69 D<91B91280A30201902680000713006E90C8FC4A163FA24B81A30203160E5DA314074B15 1E191CA2140F5D17075F021F020E90C7FC5DA2171E023F141C4B133CA2177C027F5CED80 0392B5FCA291B65AED00071601A2496E5A5CA2160101035D5CA2160301075D4A90CAFCA3 130F5CA3131F5CA3133F5CA2137FA313FFB612E0A341447AC340>II<91B6D8803FB512E0A302 010180C7387FE0006E90C86C5A4A167FA24B5EA219FF14034B93C7FCA26014074B5DA218 03140F4B5DA21807141F4B5DA2180F143F4B5DA2181F147F92B75AA3DAFF80C7123F92C8 5BA2187F5B4A5EA218FF13034A93C8FCA25F13074A5DA21703130F4A5DA21707131F4A5D A2170F133F4A5DA2017F151FA24A5D496C4A7EB6D8803FB512E0A34B447AC348>I<027F B512E091B6FCA20200EBE000ED7F8015FFA293C7FCA35C5DA314035DA314075DA3140F5D A3141F5DA3143F5DA3147F5DA314FF92C8FCA35B5CA313035CA313075CA3130F5CA3131F 5CA2133FA25CEBFFE0B612E0A25D2B447BC326>I<91B612F0A25F020101C0C7FC6E5B4A 90C8FCA25DA314035DA314075DA3140F5DA3141F5DA3143F5DA3147F5DA314FF92C9FCA3 5B5CA3010316104A1538A21878010716705C18F018E0010F15015C18C01703011F15074A 1580170FA2013FED1F004A5C5F017F15FE16034A130F01FFEC7FFCB8FCA25F35447AC33D >76 D<91B56C93387FFFC08298B5FC02014DEBC0006E614A5FA203DF4C6CC7FC1A0E6391 2603CFE05D038F5F1A381A711407030FEEE1FCA2F101C3020FEE0383020E60F107036F6C 1507021E160E021C60191CF1380F143C023804705BA2F1E01F0278ED01C091267003F85E F003801A3F02F0ED070002E0030E5CA24E137F130102C04B91C8FC606201036D6C5B0280 5F4D5A943803800113070200DA07005BA2050E1303495D010E606F6C5A1907011E5D011C 4B5CA27048130F133C01384B5C017892C7FC191F01F85C486C027E5DD807FE027C4A7EB5 00F00178013FB512C0A216705A447AC357>I79 D83 D<48B912F85AA2913B0007FC001FF0D807F84A130701E0010F 140349160148485C90C71500A2001E021F15E05E121C123C0038143F4C1301007818C012 7000F0147F485DA3C800FF91C7FC93C9FCA35C5DA314035DA314075DA3140F5DA3141F5D A3143F5DA3147F5DA314FF92CAFCA35B5CA21303A21307497E007FB612C0A25E3D446FC3 46>I87 D97 DIIIII<15FCEC03FF91390F83838091393E01CFC091387C00EF4A13 FF4948137F010315804948133F495A131F4A1400133F91C75A5B167E13FE16FE1201495C A215011203495CA21503A2495CA21507A25EA2150F151F5E0001143F157F6C6C13FF9138 01DF8090387C039F90383E0F3FEB0FFCD903F090C7FC90C7FC5DA2157EA215FEA25DA200 1C495A127F48495A14074A5A485C023FC8FC00F8137E387C01F8381FFFE0000390C9FC2A 407BAB2D>I<14FE137FA3EB01FC13001301A25CA21303A25CA21307A25CA2130FA25CA2 131FA25C157F90393F83FFC091388F81F091381E00F802387F4948137C5C4A137EA2495A 91C7FCA25B484814FE5E5BA2000314015E5BA2000714035E5B1507000F5DA249130F5E00 1F1678031F1370491480A2003F023F13F0EE00E090C7FC160148023E13C01603007E1680 EE070000FEEC1E0FED1F1E48EC0FF80038EC03E02D467AC432>I<143C147E14FE1301A3 EB00FC14701400AE137C48B4FC3803C780380703C0000F13E0120E121C13071238A21278 EA700F14C0131F00F0138012E0EA003F1400A25B137EA213FE5B12015BA212035B141E00 07131C13E0A2000F133CEBC038A21478EB807014F014E0EB81C0EA0783EBC7803803FE00 EA00F8174378C11E>I<16F0ED03F8A21507A316F0ED01C092C7FCAEEC01F0EC07FCEC1E 1EEC380F0270138014E0130114C0EB03800107131F1400A2130E153F131E011C140090C7 FC5DA2157EA215FEA25DA21401A25DA21403A25DA21407A25DA2140FA25DA2141FA25DA2 143FA292C7FCA25C147EA214FE001C5B127F48485A495AA248485A495AD8F81FC8FCEA70 7EEA3FF8EA0FC0255683C11E>I<14FE137FA3EB01FC13001301A25CA21303A25CA21307 A25CA2130FA25CA2131FA25C167E013F49B4FC92380783C09138000E07ED3C1F491370ED 603F017E13E0EC01C09026FE03801380913907000E00D9FC0E90C7FC5C00015B5C495AEB F9C03803FB8001FFC9FCA214F03807F3FCEBF07F9038E01FC06E7E000F130781EBC003A2 001F150FA20180140EA2003F151E161C010013E0A2485DA2007E1578167000FE01015B15 F1489038007F800038021FC7FC2A467AC42D>I II< D801F0EB0FE0D803FCEB7FF83A071E01F03E3A0E0F03C01F001ED987001380001C018E13 0F003C139C003801B814C014F838781FF000705BA25C00F049131FD8E03F158091C7FC12 00163F491500137EA25E01FE147E5B16FE5E12014913015E170F00030203130E4914F0A2 0307131E0007EDE01C5B173CEEC038000F167849157017E0ED03C1001FEDE3C049903801 FF000007C8127C302D78AB37>III<91381F800C91 387FE01C903901F0703C903907C0387890390F801CF890381F001D013E130F017E14F05B 48481307A2484814E012075B000F140F16C0485AA2003F141F491480A3007F143F90C713 00A35D00FE147EA315FE5DA2007E1301A24A5A1407003E130FA26C495A143B380F80F338 07C3E73901FF87E038007E071300140F5DA3141F5DA3143F92C7FCA25CA25C017F13FEA2 5D263F76AB2D>II I<1470EB01F8A313035CA313075CA3130F5CA3131F5CA2007FB512E0B6FC15C0D8003FC7 FCA25B137EA313FE5BA312015BA312035BA312075BA3120F5BA2EC0780001F140013805C 140E003F131EEB001C143C14385C6C13F0495A6C485AEB8780D807FEC7FCEA01F81B3F78 BD20>I<137C48B414072603C780EB1F80380703C0000F7F000E153F121C010715001238 5E1278D8700F147E5C011F14FE00F05B00E05DEA003FEC0001A2495C137E150313FE495C A215071201495CA2030F13380003167849ECC070A3031F13F0EE80E0153F00011581037F 13C06DEBEF8300000101148090397C03C787903A3E0F07C70090391FFE01FE903903F000 782D2D78AB34>I<017C143848B414FC3A03C78001FE380703C0000F13E0120E001C1400 0107147E1238163E1278D8700F141E5C131F00F049131C12E0EA003F91C7123C16385B13 7E167801FE14705BA216F0000115E05B150116C0A24848EB0380A2ED0700A2150E12015D 6D5B000014786D5B90387C01E090383F0780D90FFFC7FCEB03F8272D78AB2D>I<017CEE 038048B4020EEB0FC02603C780013FEB1FE0380703C0000E7F5E001C037E130F01071607 123804FE130300785DEA700F4A1501011F130100F001804914C012E0EA003FDA00031403 4C14805B137E0307140701FE1700495CA2030F5C0001170E495CA260A24848495A60A260 1201033F5C7F4B6C485A000002F713036D9039E7E0078090267E01C349C7FC903A1F0781 F81E903A0FFF007FF8D901FCEB0FE03B2D78AB41>I<02F8133FD907FEEBFFE0903A0F0F 83C0F0903A1C07C780F890393803CF03017013EE01E0EBFC07120101C013F8000316F001 80EC01C000074AC7FC13001407485C120EC7FC140F5DA3141F5DA3143F92C8FCA34AEB03 C01780147EA202FEEB0700121E003F5D267F81FC130E6E5BD8FF83143CD903BE5B26FE07 9E5B3A7C0F1F01E03A3C1E0F83C0271FF803FFC7FC3907E000FC2D2D7CAB2D>I<137C48 B414072603C780EB1F80380703C0000F7F000E153F001C1600130712385E0078157EEA70 0F5C011F14FE00F0495B12E0EA003FEC00015E5B137E150301FE5C5BA2150700015D5BA2 150F00035D5BA2151F5EA2153F12014BC7FC6D5B00005BEB7C0390383E0F7EEB1FFEEB03 F090C712FE5DA214015D121F397F8003F0A24A5A4848485A5D48131F00F049C8FC007013 7E007813F8383801F0381E07C06CB4C9FCEA01FC294078AB2F>I<027C130749B4130F49 EB800E010F141E49EBC03CEDE03890393F03F07890397C00FDF00178EB3FE00170EB03C0 01F0148049130790C7EA0F00151E5D5D5D4A5A4A5A4A5A4AC7FC141E5C5C5C495A495A49 5A49C8FC011E14F04914E05B491301485A4848EB03C0D807B0130701FEEB0F80390FCF80 1F3A1F07E07F00393E03FFFED83C015B486C5B00705C00F0EB7FC048011FC7FC282D7BAB 28>I<000FEB0780393F801FC0397FC03FE000FF137FA4018013C0397F003F80003CEB1E 001B0A66C232>127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fr cmbx12 12 49 /Fr 49 128 df12 D16 D45 DI48 D III<163FA25E5E5D5DA25D5D5D5D A25D92B5FCEC01F7EC03E7140715C7EC0F87EC1F07143E147E147C14F8EB01F0EB03E013 0714C0EB0F80EB1F00133E5BA25B485A485A485A120F5B48C7FC123E5A12FCB91280A5C8 000F90C7FCAC027FB61280A531417DC038>I<0007150301E0143F01FFEB07FF91B6FC5E 5E5E5E5E16804BC7FC5D15E092C8FC01C0C9FCAAEC3FF001C1B5FC01C714C001DF14F090 39FFE03FFC9138000FFE01FC6D7E01F06D13804915C0497F6C4815E0C8FC6F13F0A317F8 A4EA0F80EA3FE0487E12FF7FA317F05B5D6C4815E05B007EC74813C0123E003F4A1380D8 1FC0491300D80FF0495AD807FEEBFFFC6CB612F0C65D013F1480010F01FCC7FC010113C0 2D427BC038>I<4AB47E021F13F0027F13FC49B6FC01079038807F8090390FFC001FD93F F014C04948137F4948EBFFE048495A5A1400485A120FA248486D13C0EE7F80EE1E00003F 92C7FCA25B127FA2EC07FC91381FFF8000FF017F13E091B512F89039F9F01FFC9039FBC0 07FE9039FF8003FF17804A6C13C05B6F13E0A24915F0A317F85BA4127FA5123FA217F07F 121FA2000F4A13E0A26C6C15C06D4913806C018014006C6D485A6C9038E01FFC6DB55A01 1F5C010714C0010191C7FC9038003FF02D427BC038>I<121E121F13FC90B712FEA45A17 FC17F817F017E017C0A2481680007EC8EA3F00007C157E5E00785D15014B5A00F84A5A48 4A5A5E151FC848C7FC157E5DA24A5A14035D14074A5AA2141F5D143FA2147F5D14FFA25B A35B92C8FCA35BA55BAA6D5A6D5A6D5A2F447AC238>I III66 D68 DI76 DI<923807FFC092B5 12FE0207ECFFC0021F15F091267FFE0013FC902601FFF0EB1FFF01070180010313C04990 C76C7FD91FFC6E6C7E49486F7E49486F7E01FF8348496F7E48496F1380A248496F13C0A2 4890C96C13E0A24819F04982003F19F8A3007F19FC49177FA400FF19FEAD007F19FC6D17 FFA3003F19F8A26D5E6C19F0A26E5D6C19E0A26C6D4B13C06C19806E5D6C6D4B13006C6D 4B5A6D6C4B5A6D6C4B5A6D6C4A5B6D01C001075B6D01F0011F5B010101FE90B5C7FC6D90 B65A023F15F8020715C002004AC8FC030713C047467AC454>79 DI82 DI<003FBA12E0A59026FE 000FEB8003D87FE09338003FF049171F90C71607A2007E1803007C1801A300781800A400 F819F8481978A5C81700B3B3A20107B8FCA545437CC24E>I<903801FFE0011F13FE017F 6D7E48B612E03A03FE007FF84848EB1FFC6D6D7E486C6D7EA26F7FA36F7F6C5A6C5AEA00 F090C7FCA40203B5FC91B6FC1307013F13F19038FFFC01000313E0000F1380381FFE0048 5A5B127F5B12FF5BA35DA26D5B6C6C5B4B13F0D83FFE013EEBFFC03A1FFF80FC7F0007EB FFF86CECE01FC66CEB8007D90FFCC9FC322F7DAD36>97 DIIIIIII<137C48B4FC4813804813C0A24813E0A5 6C13C0A26C13806C1300EA007C90C7FCAAEB7FC0EA7FFFA512037EB3AFB6FCA518467CC5 20>I 107 DI<90277F8007FEEC0FFC B590263FFFC090387FFF8092B5D8F001B512E002816E4880913D87F01FFC0FE03FF8913D 8FC00FFE1F801FFC0003D99F009026FF3E007F6C019E6D013C130F02BC5D02F86D496D7E A24A5D4A5DA34A5DB3A7B60081B60003B512FEA5572D7CAC5E>I<90397F8007FEB59038 3FFF8092B512E0028114F8913987F03FFC91388F801F000390399F000FFE6C139E14BC02 F86D7E5CA25CA35CB3A7B60083B512FEA5372D7CAC3E>II<90397FC00FF8B590B57E02C314E002CF14F89139DFC03F FC9139FF001FFE000301FCEB07FF6C496D13804A15C04A6D13E05C7013F0A2EF7FF8A4EF 3FFCACEF7FF8A318F017FFA24C13E06E15C06E5B6E4913806E4913006E495A9139DFC07F FC02CFB512F002C314C002C091C7FCED1FF092C9FCADB67EA536407DAC3E>I<90387F80 7FB53881FFE0028313F0028F13F8ED8FFC91389F1FFE000313BE6C13BC14F8A214F0ED0F FC9138E007F8ED01E092C7FCA35CB3A5B612E0A5272D7DAC2E>114 D<90391FFC038090B51287000314FF120F381FF003383FC00049133F48C7121F127E00FE 140FA215077EA27F01E090C7FC13FE387FFFF014FF6C14C015F06C14FC6C800003806C15 806C7E010F14C0EB003F020313E0140000F0143FA26C141F150FA27EA26C15C06C141FA2 6DEB3F8001E0EB7F009038F803FE90B55A00FC5CD8F03F13E026E007FEC7FC232F7CAD2C >IIII120 DI<001FB71280A4 9026FC001F130001E0495A5B49495A90C7485A48495B123E4A5B4A5B003C495BA24A90C7 FC4A5A4A5AC7FC4A5A495B495BA2495B499038800780491300A2495A4948130F49481400 A2485B48495B485BA248495B4890C75A48485C15034848EB1FFEB7FCA4292C7DAB32>I< D80FC0137E486C13FF486C481380486C4813C000FF15E06D5AA4497E007F15C06C486C13 806C486C13006C48137E230E76C538>127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fs msbm10 12 8 /Fs 8 91 df<1838183C187CA260A24D5AA2601703A24D5AA260170FA2037FB612E00207 B712F0143F91B812E001039026C0003EC7FCD907FCC75AEB0FE0D93F805C49C8FC01FC5D 48481401485A48484A5A5B48485D001F150790C8FC003E4B5AA2003C93C8FC007C5DA200 78153E12F8163C48157CA25EA24B5AA25E15037E4B5A1278007C5D150F123C003E4AC9FC A26C141E6D133E120F6C6C5B7F6C6C13786C6C13F86C7E90387E01F0EB3F8190381FE3E0 EB07FF6D5B010090B712E0023F16F0140F19E092CAFC5CA2143EA2143C147CA25CA25C13 01A2495AA2495A001FB912E04818F0A26C18E0D8001FCBFCA2131E133EA25BA2137813F8 A2485AA2485AA25B12013C6E78CF4D>42 D<007FB77EB812F817FF6C832800FC03F83F13 E0017C9039E00F8FF0013C9039C007C1F8EFC0FC933803E07C841601EFF01E181F160084 A560A2181E0401133E187E4D5A933803E1F8EFE3F0933807CFE04CB45A043F5B9226C3FF FEC7FC92B55A17FF18C018F09239C00FF7F8933801F1FE933800F87FF01F80EF7C0FF007 C0173C94383E03E0A294381E01F0A21800A61801A2053E13E01803EF3C0719C094387C0F 80181F9438F87F00933801F9FE017C9039E003FFF801FCD9F81F5B007FB812C0B9C7FC17 F86C93C8FC3C447FC32D>66 D<922601FFE01330033F01FC13784AB6FC020FEDC0F8023F 9038C07FF8913AFFFE000FFF4901F81303902607FBF07F90260FE7E0EB007F90261F87C0 EC3F78494848141FD97E1FED0FF801FC90C8FC2601F83E1507D803F0160348485A01C016 01380F807802F81500EA1F004A1678EA3E01A2003C5B007C183019001278130312F85C12 F0AC12F8A200787FA2EA7C01A2123C003E7FA2EA1F00A26C6C7E19062607C07C160F01E0 171F6C6C6C163FD801F8177E6C6C6C167C017E6D15FCD93F0FED01F890261F87C0EC07F0 90260FE7E0EC0FE0902607FBF8EC3FC0902601FFFE903801FF806D903AFFC00FFE00023F 90B55A020F15F0020115C0DA003F49C7FC030113F040487CC52E>I<007FB6D8803FB512 E0B76C4880A26C4B6C5C2A00F807C00007E090C7FC943803E3F8017849903801E7F0F0EF C0F0FF8096C8FC6060EF03F04D5A4D5A4D5A4DC9FC177E5F4C5A4C5A4C5A4C5A4C5A4CCA FC167E16FEED81FF03837FED87EF92388FC7C092389F87E09238BF03F09238FE01F81600 6F137CEE807E9238EFC03F9239C7E01F800383130F923981F007C004F87F923980FC03F0 93387E01F8EE3E0070137CEF807E706C7E706C6C7E04036D7E933801F007716C7EDC00FC 7F94387E01F894383E00FC71137C727E94380FC03F716C6C7E05036D7E943801F007726C 7E01F86D01008095387801F8007FB6D88003B6FCB76C481580A26C4B6C150049447DC336 >75 D<007FB54AB512C0B66C4914E0816C6E6D14C02707F003F09039000FF8002601F801 ED03F000006D7E017C6D6E5A017E137E017F133E8102807FECC00F6E6C7E017B80903979 F003F0ECF801903978FC00F8027C7F6E137E023F133E6E6C7E020F1480913907C00FC0ED E007913903F003E0020114F0913900F801F8EDFC00037E137C033E137E6F133EEE801FDB 0FC013810307EB0FC1923803E0079338F003E1DB01F813F1923900FC01F9EE7C0070137D 043F137F93381F803F040F131F933807C00F17E0933803F00704011303933800F80117FC 177E173E171F1881EF0FC11707EF03E118F1EF01F91700187D01FC167F183FD803FF161F 007F01F8150FB57E18076C491503CB1201725A43467DC339>78 D<007FB712C0B812FCEF FF806C17E02800F807F00F13F8DBC00113FE017890398000FCFF94387C3F8094383E0FC0 727E94381E03F0EF1F011800717FA21978A519F8A24D5B1801EF1E034E5A94383E0FC094 387E3F80DDFDFFC7FC933807FFFE92B612F818E095C8FC17F0ED87C1EEC0F8923883E0FC 177C923881F03EA2923880F81F84EE7C0F717E163E93383F03E0041F7FEE0F81EF80F8EE 07C0187C933803E07E183E706C7E85706C6C7E180794387C03E0057E7F94383E01F8716C 7E197C01F86D6D6C7EF13F80007FB66C6CB512E0B700C015F0836C4B6C14E044447EC33D >82 D<007FB912E0BA12F0A33CF07FE3C03C7FE026F1FE03EC07F8D8F3F8ED01FCD8F7E0 ED007ED8FFC0163F0180161F0100160F481707A2481703481701A3481700A400601860C7 1700B3B3A40207133E020F133F0107B612FE4981A26D5D3C447DC33F>84 D<0003B812FE4883A301879039000F803ED98FF0011F137ED9BFC0EC007C01FFC7003E5B 13FC48484A485A49ECFC034902F85B4B48485A5B494948485A0307131F04C090C7FC90C7 380F803EA24B485A00064A13FCC8003E5B4B485AA24B485A0201130703F05B4A48485AA2 4A4848C8FC5E91380F803E021F5B1500023E5B1501027C5B9138FC03E014F84948485AA2 4948485A0107011F156002C090C812F090380F803EA249484814014913FC013E5B494848 EC03E0A249484814070001130701F049140F48484848141FA2484848C8EA3FC0000F4915 7FD9803E15FF484848EC01FBEF07F3003E49EC0FE7D87E01ED7F87007C49903903FF0780 BAFCA36C18003C447DC345>90 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ft msbm8 8 2 /Ft 2 91 df78 D<001FB612FCA23A183E00701801F0EB6038D81BC0EBE030001FC7EAC070003E010113E0 003C90380380C01501003801071380EC0603003090380E0700EC1C06EC180EC7EA380CEC 301CEC7038ECE030ECC07001015B4A5AEB03819038070180EB0603D90E07C7FCEB0C06EB 1C0EEB380CEB301CD970381303EBE030EBC070000101601307EB80E0260381C013062607 0180130ED80603141E000E90C7FCD80C07143ED81C0E1476D8380C14E6D8301CEB01CED8 7018EB078CD86038EB3E0CB712FCA2282E7EAD38>90 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fu rsfs10 12 9 /Fu 9 86 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Fv cmsy6 6 4 /Fv 4 107 df0 D<136013701360A20040132000E0137038F861 F0387E67E0381FFF803807FE00EA00F0EA07FE381FFF80387E67E038F861F038E0607000 40132000001300A21370136014157B9620>3 D48 D<12E0B3B3AD033179A413>106 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fw cmex10 12 13 /Fw 13 102 df0 D<12E07E12787E7E7E7F6C7E6C7E7F12016C7E7F137C137E7FA26D7EA26D7EA26D7EA36D 7EA2801301A2801300A280A2147EA2147FA4801580A7EC1FC0B3A5EC3F80A715005CA414 7EA214FEA25CA213015CA213035CA2495AA3495AA2495AA249C7FCA2137E137C13FC5B48 5A12035B485A485A90C8FC121E5A5A5A5A1A777C832E>I<1A381A7CA21AFC1AF8A21901 1AF0A219031AE019071AC0A2190F1A80A2191F1A0061193EA2197E197CA219FC61180161 A2180361A2180761180F61A2181F96C7FCA260183E187E187CA218FC60A2170160170360 A2170760A2170F60171F95C8FCA25F173EA2177E177C17FC5FA216015FA216035F16075F A2160F5FA2161F94C9FC5E163EA2167E167C16FC5EA215015EA215035E15075EA2150F5E A2151F93CAFC5D153EA2157E157CA215FC5D14015DA214035DA214075D140F5DA2141F92 CBFCA25C143E147E147CA214FC5CA213015C13035CA213075CA2130F5C131F91CCFCA25B 133EA2137E137C13FC5BA212015BA212035B12075BA2120F5BA2121F90CDFC5A123EA212 7E127CA212FC5AA2127046B27B8351>46 D56 D58 D<913807FF80B3B3B04A1300A55D141FA35D14 3F5DA2147F5D14FF5DA2495B5D5B4990C7FC5C130F5C495A495A495AA2495A485B4890C8 FCEA07FC485A485AEA7FE0EAFF8090C9FC12FCB4FC7FEA7FE0EA1FF06C7E6C7E6CB4FC6C 7F6C7F6D7EA26D7E6D7E6D7E801307806D7F7F816D7FA281147F81143FA281141F81A314 0F81A56E1380B3B3B021B56F8059>60 D80 D<007C193EA200FE197FB3B3B3AE6C19FFA26C19FEA26D1701A26C6CEF03FCA2001F19F8 6D17076D170F000F19F06C6CEF1FE06D173F6C6CEF7FC06C6CEFFF806E5D6C01E0030713 006D6C4B5AD93FFCED3FFC6DB4EDFFF86D01E001075B6D01FE017F5B010190B712806D94 C7FC023F15FC020F15F002011580DA003F01FCC8FC030313C048647B7F53>83 D88 D<003EF407C0007FF40FE0486CF31FF0B3B3B3 B3B3A56D1B3F007F1DE0A46D1B7F003F1DC0A26D1BFF001F1D806D62A26C6C501300A26C 6C505A6D1A0F6C6D4F5AA26C6D4F5A6E197F6C6D4F5A6D6C4E5B6D6C4E5B6E606D6C4E5B 6D01C0053F90C7FC6D6D4D5A6D01F84C485A6D01FE04075B6D6D6C031F5B6E01E0037F5B 021F01FE0207B512806ED9FFF090B6C8FC020391B712FC6E606E6C17E0031F178003074C C9FC030116F8DB003F15C0040302FCCAFCDC001F1380648B7B7F6F>91 D<153015FC4A7E913807FF80021F13E0027F13F89138FFCFFC0103EB03FF90260FFC0013 C0D93FF0EB3FF0D97FC0EB0FF84848C7EA03FED807F89138007F80D81FE0ED1FE0D87F80 ED07F800FEC9EA01FC00F8EE007C00E0171C361280C937>98 D<18E0EF07FC94383FFF80 4CB512F0040F14FE047FECFFC04BB5001F13F0030FD9F80313FE037F903A80003FFFC091 2603FFFCC7000713F8021F01C09138007FFFDAFFFEC9000F13E0010701E0040013FC013F 90CB381FFF802601FFF0060113F0000F90CDEA1FFED87FF8973803FFC0D8FF809738003F E000FCCF120700C0F40060631480CC64>I101 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fx cmmi8 8 34 /Fx 34 123 df11 DI<131FD9 FFC013304801F0137000076D13604815E0D9807C13C0391E003C0148011E13800038EB0E 03480107130000605CEC030612E0C7138EEC018C159C159815B815B0A215F05DA35DA25D A21403A44AC7FCA4140EA45CA31418242C7F9D24>I23 D<123C127E12FFA4127E123C08087A8714>58 D<123C127EB4FCA21380A2127F123D1201 A312031300A25A1206120E5A5A5A126009157A8714>I<147F903801FFE090380780F890 380E003C497F497F49148001781307017C14C001FC130316E0A2137090C7FC16F0A314FE 903807FF8390381F01C390397C00E7E049137748481337D807E0133F49131F484814C012 1F48C7FCA2481580127EA2ED3F0012FE48147EA2157C15FC5D4A5A007C495AA26C495A00 1E49C7FC6C133E3807C0F83803FFE038007F8024307DAE25>64 D<1670A216F01501A24B 7EA21507150DA2151915391531ED61FC156015C0EC0180A2EC03005C14064A7F167E5C5C A25C14E05C4948137F91B6FC5B0106C7123FA25B131C1318491580161F5B5B1201120312 07000FED3FC0D8FFF8903807FFFEA22F2F7DAE35>I<013FB6FC17C0903A00FE0007F0EE 01F84AEB00FC177E1301177F5CA21303177E4A14FEA20107EC01FC17F84AEB03F0EE07E0 010FEC1FC0EE7F009138C003FC91B55A4914FE9139C0003F804AEB0FC017E0013F140717 F091C7FC16035BA2017E1407A201FE15E0160F4915C0161F0001ED3F80EE7F004914FEED 03F80003EC0FF0B712C003FCC7FC302D7CAC35>I<90273FFFFC0FB5FCA2D900FEC7EA3F 80A24A1500A201015D177E5CA2010315FE5F5CA2010714015F5CA2010F14035F5C91B6FC 5B9139C00007E05CA2013F140F5F91C7FCA249141F5F137EA201FE143F94C7FC5BA20001 5D167E5BA2000315FEB539E03FFFF8A2382D7CAC3A>72 D<90383FFFFCA2903800FE00A2 5CA21301A25CA21303A25CA21307A25CA2130FA25CA2131FA25CA2133FA291C7FCA25BA2 137EA213FEA25BA21201A25BA21203B512E0A21E2D7DAC1F>I<91383FFFF8A29138007F 00A2157EA215FE5DA314015DA314035DA314075DA3140F5DA3141F5DA3143FA292C7FCA2 003C5B127E00FE137E14FE5CEAFC0100F05B48485A386007E038781F80D81FFEC8FCEA07 F0252E7BAC27>I78 DI81 D<913807F00691383FFE0E9138F80F9E903903E001FE903807800049C7127C131E49 143CA2491438A313F81630A26D1400A27FEB7F8014F86DB47E15F06D13FC01077F01007F 141F02011380EC003F151F150FA215071218A3150F00381500A2151EA2007C5C007E5C00 7F5C397B8003E039F1F00F8026E07FFEC7FC38C00FF0272F7CAD2B>83 D<3B7FFFF801FFFEA2D801FCC7EA0FC0178049EC070016060003150E160C5BA20007151C 16185BA2000F153816305BA2001F157016605BA2003F15E05E90C8FCA24814015E127EA2 150300FE92C7FC5A5D1506150E007C5C151815386C5C5D6CEB03C0260F800FC8FC3803E0 3C3801FFF038003FC02F2E7BAC30>85 D97 D<13F8121FA21201A25BA21203A25BA21207A25BA2120FEBC7E0 EB9FF8EBB83C381FF01EEBE01F13C09038800F80EA3F00A2123EA2007E131FA2127CA214 3F00FC14005AA2147EA2147C14FC5C387801F01303495A383C0F806C48C7FCEA0FFCEA03 F0192F7DAD1E>II101 D<14FCEB03FF90380F839C90381F01BC013E13FC EB7C005B1201485A15F8485A1401120F01C013F0A21403121F018013E0A21407A215C0A2 000F130F141F0007EB3F80EBC07F3803E1FF3800FF9F90383E1F0013005CA2143EA2147E 0038137C00FC13FC5C495A38F807E038F00F80D87FFEC7FCEA1FF81E2C7E9D22>103 D<1307EB0F80EB1FC0A2EB0F80EB070090C7FCA9EA01E0EA07F8EA0E3CEA1C3E12381230 1270EA607EEAE07C12C013FC485A120012015B12035BA21207EBC04014C0120F13801381 381F01801303EB0700EA0F06131EEA07F8EA01F0122E7EAC18>105 D<15E0EC01F01403A3EC01C091C7FCA9147CEB03FE9038078F80EB0E07131C013813C013 30EB700F0160138013E013C0EB801F13001500A25CA2143EA2147EA2147CA214FCA25CA2 1301A25CA21303A25CA2130700385BEAFC0F5C49C7FCEAF83EEAF0F8EA7FF0EA1F801C3B 81AC1D>I<131FEA03FFA2EA003FA2133EA2137EA2137CA213FCA25BA2120115F89038F0 03FCEC0F0E0003EB1C1EEC387EEBE07014E03807E1C09038E3803849C7FC13CEEA0FDC13 F8A2EBFF80381F9FE0EB83F0EB01F81300481404150C123EA2007E141C1518007CEBF038 ECF83000FC1470EC78E048EB3FC00070EB0F801F2F7DAD25>I<137CEA0FFCA21200A213 F8A21201A213F0A21203A213E0A21207A213C0A2120FA21380A2121FA21300A25AA2123E A2127EA2127CA2EAFC08131812F8A21338133012F01370EAF860EA78E0EA3FC0EA0F000E 2F7DAD15>I<27078007F0137E3C1FE01FFC03FF803C18F0781F0783E03B3878E00F1E01 263079C001B87F26707F8013B00060010013F001FE14E000E015C0485A4914800081021F 130300015F491400A200034A13076049133E170F0007027EEC8080188149017C131F1801 000F02FCEB3F03053E130049495C180E001F0101EC1E0C183C010049EB0FF0000E6D48EB 03E0391F7E9D3E>I<3907C007E0391FE03FF83918F8783E393879E01E39307B801F3870 7F00126013FEEAE0FC12C05B00815C0001143E5BA20003147E157C5B15FC0007ECF80816 18EBC00115F0000F1538913803E0300180147016E0001F010113C015E390C7EAFF00000E 143E251F7E9D2B>I<90387C01F89038FE07FE3901CF8E0F3A03879C0780D907B813C000 0713F000069038E003E0EB0FC0000E1380120CA2D8081F130712001400A249130F16C013 3EA2017EEB1F80A2017C14005D01FC133E5D15FC6D485A3901FF03E09038FB87C0D9F1FF C7FCEBF0FC000390C8FCA25BA21207A25BA2120FA2EAFFFCA2232B829D24>112 D<3807C01F390FF07FC0391CF8E0E0383879C138307B8738707F07EA607E13FC00E0EB03 804848C7FCA2128112015BA21203A25BA21207A25BA2120FA25BA2121FA290C8FC120E1B 1F7E9D20>114 DI<013F137C90 38FFC1FF3A01C1E383803A0380F703C0390700F60F000E13FE4813FC12180038EC070000 3049C7FCA2EA200100005BA313035CA301075B5D14C000385CD87C0F130600FC140E011F 130C011B131C39F03BE038D8707113F0393FE0FFC0260F803FC7FC221F7E9D28>120 DI<011E13 30EB3F809038FFC07048EBE0E0ECF1C03803C0FF9038803F80903800070048130EC75A5C 5C5C495A495A49C7FC131E13385B491340484813C0485A38070001000EEB0380380FE007 391FF81F0038387FFF486C5A38601FFC38E00FF038C003C01C1F7D9D21>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fy cmr8 8 14 /Fy 14 116 df0 D31 D<13031307130E131C1338137013F0EA01E013C01203EA0780A2 EA0F00A2121EA35AA45AA512F8A25AAB7EA21278A57EA47EA37EA2EA0780A2EA03C01201 13E0EA00F013701338131C130E1307130310437AB11B>40 D<12C07E12707E7E7E120FEA 0780120313C0EA01E0A2EA00F0A21378A3133CA4131EA5131FA2130FAB131FA2131EA513 3CA41378A313F0A2EA01E0A2EA03C013801207EA0F00120E5A5A5A5A5A10437CB11B>I< EC0380B3A4B812FCA3C7D80380C7FCB3A42E2F7CA737>43 D 48 D<130C133C137CEA03FC12FFEAFC7C1200B3B113FE387FFFFEA2172C7AAB23>II61 D<13FF000713C0380F01F0381C00F8 003F137C80A2143F001E7FC7FCA4EB07FF137F3801FE1FEA07F0EA1FC0EA3F80EA7F0012 7E00FE14065AA3143F7E007E137F007FEBEF8C391F83C7FC390FFF03F83901FC01E01F20 7D9E23>97 D101 D<013F13F89038FFC3FE3903 E1FF1E3807807C000F140C391F003E00A2003E7FA76C133EA26C6C5A00071378380FE1F0 380CFFC0D81C3FC7FC90C8FCA3121E121F380FFFF814FF6C14C04814F0391E0007F84813 0048147C12F848143CA46C147C007C14F86CEB01F06CEB03E03907E01F803901FFFE0038 003FF01F2D7E9D23>103 D108 D<3801FE183807FFB8381E01F8EA3C00481378481338A21418A27E7EB41300EA 7FF06CB4FC6C13C06C13F0000113F838001FFC130138C0007E143EA26C131EA27EA26C13 3CA26C137838FF01F038E3FFC000C0130017207E9E1C>115 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fz cmsy10 12 33 /Fz 33 121 df<007FB912E0BA12F0A26C18E03C04789A4D>0 D<121FEA3F80EA7FC0EA FFE0A5EA7FC0EA3F80EA1F000B0B789E1C>I<0060160600F8160F6C161F007E163F6C16 7E6C6C15FC6C6CEC01F86C6CEC03F06C6CEC07E06C6CEC0FC06C6CEC1F80017EEC3F006D 147E6D6C5B6D6C485A6D6C485A6D6C485A6D6C485A6D6C485ADA7E3FC7FCEC3F7E6E5A6E 5A6E5AA24A7E4A7EEC3F7EEC7E3F4A6C7E49486C7E49486C7E49486C7E49486C7E49486C 7E49C7127E017E8049EC1F804848EC0FC04848EC07E04848EC03F04848EC01F84848EC00 FC48C9127E007E163F48161F48160F00601606303072B04D>I<147014F8A81470007815 F0007C1401B4EC07F8D87F80EB0FF0D83FE0EB3FE0D80FF0EB7F80D803F8EBFE003900FE 73F890383F77E090380FFF80D903FEC7FCEB00F8EB03FE90380FFF8090383F77E09038FE 73F83903F870FED80FF0EB7F80D83FE0EB3FE0D87F80EB0FF0D8FF00EB07F8007CEC01F0 00781400C7140014F8A81470252B7AAD32>I10 D<49B4FC010F13E0013F13F8497F3901FF01FF3A03F8003F80 D807E0EB0FC04848EB07E04848EB03F090C71201003EEC00F8A248157CA20078153C00F8 153EA248151EA56C153EA20078153C007C157CA26C15F8A26CEC01F06D13036C6CEB07E0 6C6CEB0FC0D803F8EB3F803A01FF01FF0039007FFFFC6D5B010F13E0010190C7FC27277B AB32>14 D<007FBA1280BB12C0A26C1980CEFCB0007FBA1280BB12C0A26C1980CEFCB000 7FBA1280BB12C0A26C1980422C7BAE4D>17 D<19E0F003F0180FF03FE0F0FF80943803FE 00EF0FF8EF3FE0EFFF80DC03FEC7FCEE0FF8EE3FE0EEFF80DB03FEC8FCED1FF8ED7FE091 3801FF80DA07FEC9FCEC1FF0EC7FC04948CAFCEB07FCEB1FF0EB7FC04848CBFCEA07FCEA 1FF0EA7FC048CCFCA2EA7FC0EA1FF0EA07FCEA01FF38007FC0EB1FF0EB07FCEB01FF9038 007FC0EC1FF0EC07FC913801FF809138007FE0ED1FF8ED07FE923800FF80EE3FE0EE0FF8 EE03FE933800FF80EF3FE0EF0FF8EF03FE943800FF80F03FE0F00FF01803F000E01900B0 007FB912E0BA12F0A26C18E03C4E78BE4D>20 D<127012FCB4FCEA7FC0EA1FF0EA07FCEA 01FF38007FC0EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC07FC913801FF809138007FE0 ED1FF8ED07FE923800FF80EE3FE0EE0FF8EE03FE933800FF80EF3FE0EF0FF8EF03FE9438 00FF80F03FE0F00FF0A2F03FE0F0FF80943803FE00EF0FF8EF3FE0EFFF80DC03FEC7FCEE 0FF8EE3FE0EEFF80DB03FEC8FCED1FF8ED7FE0913801FF80DA07FEC9FCEC1FF0EC7FC049 48CAFCEB07FCEB1FF0EB7FC04848CBFCEA07FCEA1FF0EA7FC048CCFC12FC1270CDFCB000 7FB912E0BA12F0A26C18E03C4E78BE4D>I24 D<037FB612E00207B712F0143F91B812E0010301C0C9FCD907FCCAFCEB0FE0EB3F8049CB FC13FC485A485A485A5B485A121F90CCFC123EA2123C127CA2127812F8A25AA87EA21278 127CA2123C123EA27E7F120F6C7E7F6C7E6C7E6C7E137E6D7EEB1FE0EB07FC6DB47E0100 90B712E0023F16F01407020016E03C3A78B54D>26 D<007FB612F0B712FEEEFFC06C16F0 C9EA1FFCEE03FE9338007F80EF1FC0EF07E0717E717E717E187E183E841980180FF007C0 A2180319E0A2180119F0A21800A81801A219E01803A219C01807A2F00F80181F1900183E 187E604D5A4D5AEF0FE04D5A057FC7FCEE03FEEE3FFC007FB712F0B812C04CC8FC6C15E0 3C3A78B54D>I<1AF0A3861A78A21A7C1A3CA21A3E1A1E1A1F747EA2747E747E87747E74 7E1B7E87757EF30FE0F303F8007FBC12FEBE1280A26CF3FE00CEEA03F8F30FE0F31F8051 C7FC1B7E63505A505A63505A505AA250C8FC1A1E1A3E1A3CA21A7C1A78A21AF862A35934 7BB264>33 D39 D<18034E7E8518038518018572 7E1978197C8585737E86737E737E007FBA7EBB7E866C85CDEA0FC0747EF203F8F200FEF3 7F80F31FE0F307FC983801FF80A2983807FC00F31FE0F37F8009FEC7FCF203F8F207E050 5A007FBBC8FCBB5A626C61CCEA03F04F5A4F5A624FC9FC193E61197819F84E5A61180361 18076172CAFC59387BB464>41 D<031CED01C0033E4B7E033C1501037C820378150003F8 824B16780201177C4B163C0203173E4A48824B82020F844ACA6C7E023E717E027E8491BA 7E498549854985D90FC0CBEA1F804948727E017FCCEA07F001FCF101F8D803F8F100FED8 0FE0F23F80D83FC0F21FE0B4CEEA07F8A2D83FC0F21FE0D80FE0F23F80D803F8F2FE00C6 6CF101F8017FF107F0D91F80F00FC06D6C4E5A6DBBC7FC6D616D616D61027ECAEA03F002 3E606E4D5A6E6C4C5A020795C8FC6F5E6E6C163E0201173C6F167C020017786F16F80378 5E037C1501033C5E033E1503031C6F5A5D387DB464>44 D<49B4EF3FC0010F01E0923803 FFF8013F01FC030F13FE4901FF92383FE01F48B66C91397E0007C02603F80301E0D901F8 EB01E02807E0007FF049486D7E01806D6CD907C0147048C76C6C494880001EDA07FE49C8 7E001C6E6C013E150C486E6D48150E71481506486E01E0160793387FF1F0006092263FF3 E08193381FFBC000E004FF1780486F4915017090C9FC82707F8482717E844D7E6C4B6D15 03006004EF1700933803E7FE0070922607C7FF5DDC0F837F003004816D140E00384BC6FC 0018033E6D6C5C001C4B6D6C143C6C4BD91FFC5C6C4A486D6C5C6DD907E06D6C13036C6C 49486D9038E00FE0D801F0013FC890B55A27007C03FE6F91C7FC90263FFFF8031F5B010F 01E0030313F8D901FECAEA7FC0592D7BAB64>49 D<92B6FC02071580143F91B712000103 0180C8FCD907FCC9FCEB1FE0EB3F80017ECAFC5B485A485A485A5B485A121F90CBFC123E A2123C127CA2127812F8A25AA2B9FC1880A2180000F0CBFCA27EA21278127CA2123C123E A27E7F120F6C7E7F6C7E6C7E6C7E137E6D7EEB1FE0EB07FC6DB47E010090B6FC023F1580 140702001500313A78B542>I<1706170F171FA2173EA2177CA217F8A2EE01F0A2EE03E0 A2EE07C0A2EE0F80A2EE1F00A2163EA25EA25EA24B5AA24B5AA24B5AA24B5AA24BC7FCA2 153EA25DA25DA24A5AA24A5AA24A5AA24A5AA24AC8FCA2143EA25CA25CA2495AA2495AA2 495AA2495AA249C9FCA2133EA25BA25BA2485AA2485AA2485AA2485AA248CAFCA2123EA2 5AA25AA25A1260305C72C600>54 D<126012F0B012FC12FEA212FC12F0B0126007267BAB 00>I<0060171800F0173C6C177CA200781778007C17F8A2003C17F0003E1601A26CEE03 E0A26C17C06D1507A2000717806D150FA26C6CED1F00A20001161E6D153EA20000163C90 B712FCA26D5DA2013CC85A013E1401A2011E5D011F1403A26D5D6E1307A26D6C495AA201 0392C7FC6E5BA20101141E6E133EA26D6C5BA202781378027C13F8A2023C5BEC3E01A26E 485AA2020F5B1587A202075B15CFA26EB4C8FCA26E5AA36E5AA315781530364780C437> I<007FB712E0B812F0A27ECAFCB3AA001FB7FC127FA3CAFCB3AB007FB7FCB8FCA26C16E0 2C457BC437>I<0060170C00F0171EB3B3A66C173EA20078173C007C177C007E17FC003E 17F86CEE01F06D15036C6CED07E06C6CED0FC0D803F8ED3F80D801FEEDFF0026007FC0EB 07FCD93FFCEB7FF8010FB612E001031580D9007F01FCC7FC020713C0373D7BBA42>91 D<913807FFC0027F13FC0103B67E010F15E0903A3FFC007FF8D97FC0EB07FCD801FEC8B4 FCD803F8ED3F80D807E0ED0FC04848ED07E04848ED03F090C91201003EEE00F8007E17FC 007C177C0078173C00F8173EA248171EB3B3A60060170C373D7BBA42>I102 D<12FEEAFFE0EA07F8EA 00FEEB7F806D7E6D7E130F6D7EA26D7EB3AD6D7EA26D7E806E7E6E7EEC0FE0EC03FC9138 00FFE0A2913803FC00EC0FE0EC3FC04A5A4AC7FC5C495AA2495AB3AD495AA2495A131F49 5A495A01FEC8FCEA07F8EAFFE048C9FC236479CA32>I<140C141E143EA2143C147CA214 F8A214F01301A2EB03E0A214C01307A2EB0F80A214005BA2133EA2133C137CA2137813F8 A2485AA25B1203A2485AA25B120FA248C7FCA2121E123EA25AA2127812F8A41278127CA2 7EA2121E121FA26C7EA212077FA26C7EA212017FA26C7EA21378137CA2133C133EA27FA2 7F1480A2EB07C0A2130314E0A2EB01F0A2130014F8A2147CA2143C143EA2141E140C1764 76CA27>I<126012F07EA21278127CA27EA2121E121FA26C7EA212077FA26C7EA212017F A26C7EA21378137CA2133C133EA27FA27F1480A2EB07C0A2130314E0A2EB01F0A2130014 F8A2147CA2143C143EA4143C147CA214F8A214F01301A2EB03E0A214C01307A2EB0F80A2 14005BA2133EA2133C137CA2137813F8A2485AA25B1203A2485AA25B120FA248C7FCA212 1E123EA25AA2127812F8A25A126017647BCA27>I<126012F0B3B3B3B3B3A81260046474 CA1C>I<0070130700F01480B3B3B3B3B3A800701400196474CA32>I<126012F07EA21278 127CA2123C123EA2121E121FA26C7EA212077FA212037FA212017FA26C7EA21378137CA2 133C133EA2131E131FA26D7EA2130780A2130380A2130180A26D7EA21478147CA2143C14 3EA280A28081A2140781A2140381A26E7EA2140081A21578157CA2153C153EA281A28116 80A2150716C0A2150316E0A2ED01F0A2150016F8A21678167CA2163C163EA2161E160C27 647BCA32>110 D<1B0C1B1E1B3EA21B7CA21BF8A2F201F0A2F203E0A2F207C0A2F20F80 A2F21F00A21A3EA262A262A24F5AA2621903A24F5AA24F5AA24FC7FCA2193EA261A261A2 4E5AA24E5AA24E5AA24E5AA2010C4CC8FC133C017C163EEA01FE00035F487E001E5F0038 7FD8707F4B5A00E07FD8003F4B5A80011F4B5AA26E4A5A130F6E4AC9FC13076E143E1303 6E5C13016E5C7F6F5B027F1301A26F485A143F6F485A141F6F485A140F6F48CAFC1407ED FC3E14035E15FE02015B15FF6E5BA26F5AA26F5AA26F5AA26FCBFC150E4F647A8353> 112 D120 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FA cmmi12 12 66 /FA 66 123 df11 DI I<1578913807FFE0021F13FC91383C7FFEEC7007EC6003ECE0004A13381600A280A380A2 80147CA2147E143E143F816E7EA26E7E81140781EC3FFC14FF903803E1FEEB07C190381F 00FF133E49EB7F805B0001143F485A484814C049131F120F485AA248C7FC150F5A127EA3 00FEEC1F805AA316005A5DA2153E157E157CA26C5C127C4A5A6C495AA26C495A6C6C485A 6C6C48C7FC3803E07C3800FFF0EB1FC027487CC62B>I<130E011FEC03E049EC1FF8167F 16E749EB03C792380707F0017E130E92381C03C001FE0178C7FC15E049485A4A5A000149 C8FC141EEBF8385C3803F9E0EBFF8014F8ECFFC03907F07FF0EC03FC49C6B4FCED3F8000 0F6E7EA249130FA2001F160717065BA2003F160E170C90C71380171C4816181738007E16 30923807C07000FE16E0923803E1C048913800FF800038ED3E00302D7BAB38>20 DI23 D26 D<0203B612E0021F15F091B7FC4916E0010716C0 90270FF80FF8C7FC90381FC00349486C7E017EC7FC49147E485A4848143E0007153F5B48 5AA2485AA2123F90C8FC5E48157E127EA216FE00FE5D5A15015EA24B5A007C5D15074B5A 5E6C4AC8FC153E6C5C5D390F8003F03907C007C02601F03FC9FC38007FFCEB1FE0342C7D AA37>I<161CA21618A21638A21630A21670A21660A216E0A25EA21501A25EA21503A293 C8FCA25DED7FE0913807FFFE91391FC63F809139FE0E07C0D901F8EB03F0903A07E00C00 F8D91FC08090263F001C137E017E814913184848ED1F8000031438485A4848013014C0A2 48481370A248481360A248C712E0A24B133F481780481301A24B137F180014034816FE92 C7FC4C5A6C49495AA2007E0106495A4C5A6C010E495A4C5A261F800C49C7FC000F15FC3A 07C01C01F8D803E0EB07E03A01F8181F80D8007E01FEC8FC90381FFFF801011380D90030 C9FCA21470A21460A214E0A25CA21301A25CA21303A291CAFCA332597BC43A>30 D<137E48B46C150626078FE0150E260607F0151C260E03F81538000C6D1570D81C0116E0 00006D15C0010015016EEC03806EEC0700170E6E6C5B5F5F6E6C136017E04C5A6E6C485A 4CC7FC0207130E6F5A5E1630913803F8705EEDF9C06EB45A93C8FC5D6E5A81A2157E15FF 5C5C9138073F80140E141C9138181FC014381470ECE00FD901C07FEB038049486C7E130E 130C011C6D7E5B5B496D7E485A48488048C8FC000681000E6F137048EE806048033F13E0 4892381FC0C048ED0FE348923803FF00CA12FC37407DAB3D>I<1730A317701760A317E0 5FA316015FA3160394C8FCA35E1606A3160E160C013E1607D9FF80ED1F802603C3C0011C EB3FC0260703E01318260601F0157F000E173F001C1538D818030230131F0038170F0030 170700701570D86007026013035CA2D8E00F02E0148000C049491301EA001F4A15030301 1500013F5C1400604901031406017E91C7FC180E180C01FE49141C490106141818386003 0E1460030C14E04D5A4D5A031C49C7FC0318130E017E5D5F6D01385B90261F80305BD90F C0EB03C0D907F0010FC8FC903901FE707C9039003FFFF002031380DA0060C9FC15E05DA3 14015DA3140392CAFCA35C1406A3140E140C3A597DC43F>I34 D<177F0130913803FFC00170020F13E0494A13F0 48484A13F84848EC7F0190C838F8007C484A48133C00064A48131C000E4B131E000C4A48 130E001C92C7FC0018140E150C0038021C1406003014185D180E00704A140C1260154003 C0141C00E017184A5A4817386C17304AC8127018E01260007049EC01C0EF038000780106 14070038EE0F00003C010E141E6C167CD81F805D6C6C48EB03F0D807F0EC0FE0D803FEEC 3FC02801FFFC03FFC7FC6C6CB55A6D14F8010F14E0010114809026007FF8C8FC02F8C9FC A25CA21301A3495AA31307A25C130FA4131F5C6DCAFC37417BAB40>39 D<133EEA01FF5AEA0FFEEA1FE0EA3F00127E127C5AA25AA47EA2127C127E7EEA1FE0EA0F FEEA03FF7EEA003E10187BAE1B>44 D<121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A0A78 891B>58 D<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413E013C0A31201138012 0313005A1206120E5A5A5A12600B1D78891B>II<16 18163C167CA2167816F8A216F01501A216E01503A216C01507A21680150FA2ED1F00A215 1E153EA2153C157CA2157815F8A25D1401A24A5AA25D1407A25D140FA292C7FC5CA2141E 143EA2143C147CA25CA25C1301A25C1303A25C1307A25C130FA291C8FC5BA2133EA2133C 137CA2137813F8A25B1201A25B1203A2485AA25B120FA290C9FC5AA2121E123EA2123C12 7CA2127812F8A25A126026647BCA31>I<127012FCB4FCEA7FC0EA1FF0EA07FCEA01FF38 007FC0EB1FF0EB07FE903801FF809038007FE0EC1FF8EC03FE913800FF80ED3FE0ED0FF8 ED03FF030013C0EE3FF0EE0FFCEE01FF9338007FC0EF1FF0EF07FCEF01FF9438007FC0F0 1FE0A2F07FC0943801FF00EF07FCEF1FF0EF7FC04C48C7FCEE0FFCEE3FF0EEFFC0030390 C8FCED0FF8ED3FE0EDFF80DA03FEC9FCEC1FF8EC7FE0903801FF80D907FECAFCEB1FF0EB 7FC04848CBFCEA07FCEA1FF0EA7FC048CCFC12FC12703B3878B44C>I64 D<1830187018F0A21701 1703A24D7EA2170F171FA21737A2176717E717C793380187FCA2EE0307EE07031606160C A216181638163004607FA216C0030113011680ED0300A21506150E150C5D845D03707F15 605DA24A5A4AB7FCA25C0206C87F5C021C157F14185CA25C14E05C495A8549C9FC49163F 1306130E5B133C137C01FE4C7ED807FFED01FF007F01F0027FEBFFC0B5FC5C42477DC649 >I<91B87E19F019FC02009039C00003FF6F480100138003FFED3FC01AE093C8121FF10F F0A24A17F84B1507A314035D190FA2020717F04B151F1AE0193F020F17C04BED7F80F1FF 004E5A021F4B5A4B4A5AF01FF0F03FC0023F4AB4C7FC4BEB1FFC92B612F018FEDA7FC0C7 EA7F804BEC1FC0F00FF0727E02FF6F7E92C8FC727EA249835CA313035CA301075F4A1503 A24E5A130F4A4B5A4E5AA2011F4C5A4A4B5A4D485A013F4B48C7FCEF0FFC4AEC3FF801FF 913801FFE0B9128005FCC8FC17C045447CC34A>I<4CB46C1318043F01F013384BB512FC 0307D9007E1378DB1FF090380F80F0DB7F80EB03C1DA01FEC7EA01C34A48EC00E7DA0FF0 ED7FE04A48153F4A5A02FFC9121F494817C04948160F495A130F4A178049481607495A13 7F4948170091CAFC5A485A1906485AA2485A96C7FC121F5BA2123F5BA3127F5BA4485AA4 19C0A2180161127F180396C7FC6018066C6C160E601818001F17386D5E000F5F6D4B5A6C 6C4B5A00034CC8FC6C6C150E6C6C153C017F5DD93FC0EB01E0D91FF0EB0FC0D907FE017F C9FC0101B512FCD9003F13E0020790CAFC45487CC546>I<91B87E19F019FC02009039C0 0007FF6F489038007FC003FFED1FE0737E93C86C7E737E19014A707E5D1A7FA20203EF3F 805DA21BC014075DA3140F4B17E0A3141F4B17C0A3143F4B167FA3027F18804B16FFA302 FF180092C95A62A24917034A5F19076201034D5A5C4F5A620107173F4A5F4FC7FC19FE01 0F4C5A4A15034E5AF00FE0011F4C5A4A4B5A06FFC8FC013FED01FCEF0FF84AEC3FE001FF 913803FF80B848C9FC17F094CAFC4B447CC351>I<91B912FCA3020001C0C7123F6F48EC 03F803FF1501190093C91278A21A385C5DA3020317305DA314074B1460A218E0020F4B13 005DA21701021F5D4B13031707170F023F027FC8FC92B6FCA391397FC0007E4B131EA217 0E02FF140C92C7FCA2171C49031813035C611906010392C7FC4A160E190C191C01071718 4A163819301970130F4A5E180161011F16034A15074E5A013F163F4EC7FC4AEC03FF01FF ED3FFEB9FCA26046447CC348>I<91B912F8A3020001C0C7123F6F48EC07F003FF150319 0193C9FCA21A705C5DA3020317605DA314075D18C01701020F4B13005DA21703021F92C8 FC4B5BA25F023F141E4B13FE92B5FCA24A5CED8000173CA202FF141892C7FCA217384915 305CA21770010315604A91C9FCA313075CA3130F5CA3131F5CA2133FA313FFB612F8A345 447CC33F>I<4CB46C1318043F01F013384BB512FC0307D9007E1378DB1FF090380F80F0 DB7F80EB03C1DA01FEC7EA01C34A48EC00E7DA0FF0ED7FE04A48153F4A5A02FFC9121F49 4817C04948160F495A130F4A178049481607495A137F4948170091CAFC5A485A1906485A A2485A96C7FC121F5BA2123F5BA3127F5BA4485A4CB612805EA293C7EBE000725AA3007F 60A218FF96C7FCA26C7E5F606C7EA2000F16036D5E6C6C15070003160F6C6C151F6C6CED 3DF8D97F8014786D6CEB01E0D91FF0903807C078D907FE90387F00700101B500FC1330D9 003F01F090C8FC020790CAFC45487CC54D>I<91B6D8E003B61280A3020001E0C70003EB 8000DB7F806E48C7FC03FF1503A293C85BA219075C4B5EA2190F14034B5EA2191F14074B 5EA2193F140F4B5EA2197F141F4B5EA219FF143F92B8C8FCA3DA7FC0C712014B5DA21803 14FF92C85BA218075B4A5EA2180F13034A5EA2181F13074A5EA2183F130F4A5EA2187F13 1F4A5EA2013F16FFA24A93C9FCD9FFE002037FB6D8E003B67EA351447CC351>I<027FB5 12F8A217F09139007FF000ED3FC0157FA25EA315FF93C7FCA35C5DA314035DA314075DA3 140F5DA3141F5DA3143F5DA3147F5DA314FF92C8FCA35B5CA313035CA313075CA3130F5C A2131FA25CEB7FF0007FB512F0B6FCA22D447DC32B>I<031FB512FC5D18F89239000FFE 00705AA35FA2160FA25FA2161FA25FA2163FA25FA2167FA25FA216FFA294C7FCA25DA25E A21503A25EA21507A25EA2150FA25EA2151FA25EA2153FA25EEA0F80D83FE0137F5E127F A24BC8FC485A4A5A1300006C495A0060495A0070495A0030495A0038EB3F806C49C9FC38 0F81FC3803FFF038007F80364679C336>I<91B600E049B512C0A3020001E0C8383FF800 DB7F80ED1FE003FF94C7FC1A3E93C9127862F101C04A4C5A4B4BC8FC191C6102035E4B5D F003804EC9FC0207150E4B14386060020F4A5A4B0107CAFC170E5F021F14784B13F84C7E 1603023F130F4B487E163BEEE1FF91387FC1C1DB83807FED8700159CDAFFB86D7E5D03C0 6D7E5D4990C7FC4A6E7EA2717E13034A811707A201076F7E5C717EA2130F4A6E7FA2727E 131F5C727E133F854A82D9FFE04B7EB600E0010FB512E05FA252447CC353>I<91B612F8 A3020001E0C8FC6F5A4B5AA293C9FCA35C5DA314035DA314075DA3140F5DA3141F5DA314 3F5DA3147F5DA314FF92CAFCA35B4A16C0A21801010317804A15031900A201075E4A1506 180E181E010F161C4A153C18381878011F16F84A4A5A1703013F150F4D5A4A14FF01FF02 075BB9FCA2603A447CC342>I<91B500C0933803FFFE63630200F1FE00DB6FE0EE1BF803 EF171F1B3703CFEF67F0A21BCF0201EF018F038F60DB87F0ED030F1B1F02031706030704 0C5BA2F2183F020717300206616F6C15601B7F020E17C0020CDC018090C7FCA24F485A02 1C16060218606F6C5C1A0102381618023004305BA2F16003027016C00260606F6CEB0180 1A0702E0ED03004A03065CA24E130F01015E4A60047F5B1A1F01035E91C74A5CA24D4813 3F494BC7FC010661EE3F861A7F010E158C010C039892C8FCA205B05C011C15E001186001 386E5A190101785D01FC92C75BD803FFEF07FEB500F8011E0107B512FE161C160C5F447B C35E>I<91B500C0020FB5128082A2DA007F9239007FE00070ED1F8074C7FCDBEFF8150E 15CF03C7160C70151C1401DB83FE1518A2DB81FF1538140303001630831A704A6D7E0206 1760163F7114E0140E020C6D6C5CA2706C1301141C021801075D83190302386D7E023094 C8FC1601715B147002606DEB8006A294387FC00E14E04A023F130C18E0191C0101ED1FF0 4A1618170FF0F838130391C83807FC30A2943803FE705B01060301136018FF19E0010E81 010C5F187FA2131C0118705A1338181F137801FC70C9FCEA03FFB512F884180651447CC3 4E>II<91B712FEF0FFE019F802009039C0000FFE6F48EB01FF03FF9138 007F80F13FC093C8EA1FE0A24AEE0FF0A25D1AF81403A25DA21407F11FF05DA2020FEE3F E0A24B16C0197F021F1780F1FF004B4A5A4E5A023F4B5A4E5A4BEC3FC006FFC7FC027FEC 07FC92B612F018800380CAFC14FFA292CBFCA25BA25CA21303A25CA21307A25CA2130FA2 5CA2131FA25CA2133FA25CEBFFE0B612E0A345447CC33F>II<91B712F018FF19E002009039C0003FF86F48EB07FC03FFEC01FEF0007F 93C8EA3F801AC0F11FE05C5D1AF0A214035DA30207EE3FE05DA2F17FC0020F17804B15FF 1A004E5A021F4B5A4B4A5AF00FE04E5A023F037FC7FC4BEB03FCEF1FF092B612804A4AC8 FC923980007F80EF0FC0EF07F002FF6E7E92C77F1701845B4A1400A2170113035CA21703 13075CA24D5A130F5CA3011F18185CA2013F4C13381A304A6F1370D9FFE0020314E0B600 E0ED01C00501EB0380943900FE0F00CBEA3FFEF007F045467CC34A>I<9339FF80018003 07EBF003033F13FC9239FF007E07DA01F8EB0F0FDA07E09038079F004A486DB4FC4AC77E 023E804A5D187E5C495A183C495AA213074A1538A3130F183080A295C7FC806D7E8014FF 6D13E015FC6DEBFFC06D14FC6E13FF6E14C0020F80020314F8EC003F03077F9238007FFE 160F1603707E8283A283A21206A4000E163EA2120C177E001E167CA25F5F003F15014C5A 6D4A5A4C5A486C4AC8FC6D143ED87CF85CD8787E495A3AF01FC00FE0D8E007B512800101 49C9FC39C0003FF039487BC53C>I<48BA12C05AA291C7D980001380D807F092C7121F49 49150F0180170748C75B1903120E48020316005E12181238003014074C5C007018061260 00E0140F485DA3C8001F92C7FC5EA3153F5EA3157F5EA315FF93CAFCA35C5DA314035DA3 14075DA3140F5DA3141F5DA3143F5DA2147FA214FF01037F001FB612FCA25E42447EC339 >I<003FB500F80103B512E0A326003FF8C8381FF800D91FE0ED07E0013F705A615C96C7 FC60017F16065CA2180E01FF160C91C9FCA2181C4817185BA21838000317305BA2187000 0717605BA218E0120F495EA21701121F495EA21703123F4993C8FCA25F127F491506A217 0E00FF160C90C9FC171CA21718173817304816705F6C5E6C15014C5A4CC9FC6C150E6D14 1E001F5D6D5C6C6CEB01E06C6C495A6C6CEB1F80C6B401FECAFC90387FFFF8011F13E001 0190CBFC43467AC342>I<007FB56C91381FFFF8B65DA2000101E0C8000313006C0180ED 01FCF000F0614E5AA2017F4C5A96C7FC1806A2606E5DA2013F5E1870186060A24D5A6E4A C8FCA2011F1506170E170C5FA26E5C5FA2010F5D16015F4CC9FCA26E13065EA201075C5E A25E16E06E5B4B5A13034BCAFC1506A25D151CECFE185D13015D5DA26E5AA292CBFC5C13 005C5CA25CA25C45467BC339>I96 DIIIIII<157E 913803FF8091390FC1E0E091391F0073F0027E13334A133F4948131F010315E04948130F 495AA2494814C0133F4A131F137F91C713805B163F5A491500A25E120349147EA216FEA2 495CA21501A25EA21503150700015D150F0000141F6D133F017CEB77E090383E01E79038 1F078F903807FE0FD901F85B90C7FC151FA25EA2153FA293C7FCA2001C147E007F14FE48 5C4A5A140348495AEC0FC000F8495A007C01FEC8FC381FFFF8000313C02C407EAB2F>I< 14FE137FA3EB01FC13001301A25CA21303A25CA21307A25CA2130FA25CA2131FA25CED3F C090393F81FFF0913887C0FC91380E007E023C133ED97F70133F4A7F4A14805C13FF91C7 FC5BA24848143F17005BA200035D167E5BA2000715FE5E5B1501000F5DA24913035E001F 1607030713064914E0150F003FEDC00E170C90C7141CEE80184816381730007E167017E0 00FE91380781C0EEC38048913801FF000038EC007C30467BC438>I<141E143F5C5CA314 7E143891C7FCAE133EEBFF803801C3C0380781E0380601F0120E121CEA180312381230A2 EA700700605BA2EAE00F00C05BEA001F5CA2133F91C7FCA25B137E13FE5BA212015BEC03 800003140013F01207495A1406140E140CEBC01C141814385C00035BEBE1C0C6B45A013E C7FC19437DC121>I<163C16FEA21501A316FCED00701600AE15FCEC03FF91380F078002 1C13C091383803E0147014E014C01301EC8007130314005B0106130F130E010C14C090C7 FC151FA21680A2153FA21600A25DA2157EA215FEA25DA21401A25DA21403A25DA21407A2 5DA2140FA25DA2141F5DA2143F001C91C7FC127F48137E5CA248485AEB03E038F807C038 781F80D83FFEC8FCEA07F0275681C128>I<14FE137FA3EB01FC13001301A25CA21303A2 5CA21307A25CA2130FA25CA2131FA25C163F013FECFFC0923803C0E09138000703ED1E0F 491338ED701F017E13E0EC01C001FE018013C00203EB07004948C8FC140E00015B5C495A 5C3803FBC001FFC9FC8014F83807F1FE9038F03F809038E00FE06E7E000F130381EBC001 A2001FED01C017801380A2003F15031700010013F05E481506160E007E150C161C00FE01 005BED787048EC3FE00038EC0F802B467BC433>II<01F8D903FCEC7F80D803FED91FFF903803FFE0D8071F903B7C0FC00F81F83E0E0F 80E007E01C00FC001C9026C3C0030178137C271807C700D9F0E0137E02CE902601F1C013 3E003801DCDAFB80133F003001D892C7FCD90FF814FF0070495C0060495CA200E0494948 5CD8C01F187E4A5C1200040715FE013F6091C75BA2040F14014960017E5D1903041F5D13 FE494B130762043F160E0001060F130C4992C713C0191F4CED801C00031A1849027E1638 F2003004FE167000071A60494A16E0F201C0030192380F0380000FF18700494AEC03FED8 0380D90070EC00F84F2D7DAB55>I<01F8EB03FCD803FEEB1FFFD8071F90387C0FC03B0E 0F80E007E03A0C07C3C003001CD9C7007F001801CE1301003801DC80003013D8EB0FF800 705B00605BA200E0491303D8C01F5D5C12001607013F5D91C7FCA2160F495D137E161F5F 13FE49143F94C7FC187000014B136049147E16FE4C13E0000317C049150104F813801703 00071700495D170EEE781C000FED7C3849EC1FF0D80380EC07C0342D7DAB3A>III<01F8EB0FC0D803FEEB7FF0D807 0FEBF038000E903883C07C3A0C07C701FC001C13CE0018EBDC03003813D8003013F8D90F F013F800709038E000E0006015005C12E0EAC01F5C1200A2133F91C8FCA35B137EA313FE 5BA312015BA312035BA312075BA3120F5BEA0380262D7DAB2C>114 D<133ED9FF8014E02603C3C0EB03F0380703E0380601F0000E1507121CD818035D123800 30150FA2D870075D00605B161FEAE00F00C0495CEA001F4A133FA2013F92C7FC91C7FC5E 5B017E147EA216FE13FE495CA20301EB01801703484802F81300A25F0303130616F00000 1407030F130E6D010D130C017C011D131C033913186D9038F0F838903A1F03C07870903A 07FF803FE0903A01FC000F80312D7DAB38>117 D<013E140ED9FF80EB3F802603C3C013 7F380703E0380601F0120E121CD81803143F0038151F0030150FA2D87007140700605BA2 D8E00F150000C0497FEA001F4A5B1606133F91C7FC160E49140C137EA2161C01FE14185B 1638163016704848146016E05E150100005D15036D49C7FC1506017C130E017E5B6D1378 90380F81E06DB45AD900FEC8FC292D7DAB2F>I<02FCEB07E0903A03FF801FFC903A0F07 C0781E903A1C03E0E01F903A3801F1C07FD9700013804901FB13FF4848EBFF00495B0003 16FE90C71438484A130012061401000E5C120CC7FC14035DA314075DA3140F5DA3021F14 3817305D1770023F1460121E003F16E0267F807FEB01C0026F148000FF01EF1303D901CF EB070000FE903887C00E267C03835B3A3C0F01E0783A1FFC00FFE0D803F0EB3F80302D7E AB37>120 D<133ED9FF8014E02603C3C0EB03F0380703E0380601F0000E1507001C16E0 EA180312380030150F007016C0EA60075C161FD8E00F158000C05BEA001F4A133F170013 3F91C7FC5E49147E137EA216FE01FE5C5BA215015E485AA215035EA200001407150F6D5C 017C131F153F6D13FF90391F03CFC0903807FF8F903801FC0F90C7121F5EA2153F93C7FC D807C05BD81FE0137E5DA24848485A4A5A01805B39380007C00018495A001C49C8FC6C13 7C380781F83803FFE0C66CC9FC2C407DAB30>I<027CEB018049B413034901801300010F 6D5A49EBE00E6F5A90393F03F838903978007EF80170EB1FF00160EB01E001E05C49495A 90C748C7FC150E5D5D5D5D4A5A4A5A4AC8FC140E5C5C5C5CEB03C049C9FC130E49141C49 14185B49143848481430491470D8039014F048B4495A3A0FEFC007C0391E03F01FD81C01 B55A486C91C7FC485C00606D5A00E0EB3FF048EB0FC0292D7CAB2D>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: FB cmr12 12 81 /FB 81 128 df0 D<1618163CA2167EA216FFA24B7FA24B6C7EA29238063FE0A24B6C7EA24B6C7EA2923838 07FC153092387003FE15609238E001FF15C002016D7F5D02036E7E92C7FC4A6E7E140602 0E6E7E140C021C6E7E141802386E7E143002706E7E146002E06E7E5C01016F7F5C010370 7E91C9FC183F010683181F4983180F49831807498318034983A249707EA24848701380A2 48CBEA7FC0A20006F03FE0A248F01FF0A2001FBA12F8A24819FCA24819FEA2BCFC48477C C651>I<0103B612FCA390C701F0C8FC6F5A6F5AA8913801FFF0023FEBFF80903A01FF3F DFF0D907F0EBC1FCD91FC0EBC07FD93F00EC1F8001FEED0FE048486F7E48486F7E48486F 7E48486F7E001F834982003F1880007F18C0A249163F00FF18E0A8007F18C06D167FA200 3F1880001F18006D5E000F5F6C6C4B5A6C6C4B5A6C6C4B5A6C6C4B5A013FED1F80D91FC0 027FC7FCD907F0EBC1FCD901FFEBDFF0D9003FB51280020101F0C8FC9138003FC0A84B7E 4B7E0103B612FCA33B447BC346>8 D<9239FFC001FC020F9038F80FFF913B3F803E3F03 C0913BFC00077E07E0D903F890390FFC0FF0494890383FF81F4948EB7FF0495A494814E0 49C7FCF00FE04991393FC0038049021F90C7FCAFB912F0A3C648C7D81FC0C7FCB3B2486C EC3FF0007FD9FC0FB512E0A33C467EC539>11 D<4AB4FC020F13E091387F80F8903901FC 001C49487FD907E0130F4948137F011FECFF80495A49C7FCA25B49EC7F00163E93C7FCAC EE3F80B8FCA3C648C7FC167F163FB3B0486CEC7FC0007FD9FC1FB5FCA330467EC536>I< DBFF80EB3FE0020F9039F001FFFC913B3F807C0FF01F913CFC000E3F800380D903F86D48 486C7E4948D90FFC804948D93FF8130F4948017F4A7E49485C49C75BA25B494B6D5A041F 6E5A96C8FCACF107F0BBFCA3C648C7391FC0001F190F1907B3B0486C4A6C497E007FD9FC 0FB50083B512E0A34B467EC551>14 D16 D<131F1480133F137FA2EBFF00485A485A5B485A485A138048C7FC123E 123C5A12E0124011126CC431>19 D<1970196019E0DB1FFC495A4AB500C05B912807F007 F003C7FC913A1F8000FC07027EC7EA3F06D901F8EC0FCCD903E0EC03FC49486E5A494814 0049C9127C013E167E4916DF01FC03017F4848EE8FC000039338030FE049ED0707000704 067F49ED0C03000F041C7F49ED1801001F04307F177048484B6C7E5F1601007F4B487F94 C7FC160690C8000E80160C484B1580163816305E16E05E4B5A150393C8FC1506150E150C 5D6C0238160015306D495D15E0003F49485D5D14036C6C48C8485A1406000F495ED9E01C 1503000701185E5C2603F0704B5A14606C6C484B5AD800FD4C5AD97D8093C7FC017FC95A 6D167E6D6C5D6D6CEC01F0496C4A5AD919F8EC0FC0D9307C4AC8FC9026701F8013FC903A 6007F007F0D9C001B512C000019026001FFCC9FC484890CBFC90CDFC5A414E7BC84C>31 D<001EEB03C0397F800FF000FF131F01C013F8A201E013FCA3007F130F391E6003CC0000 EB000CA401E0131C491318A3000114384913300003147090C712604814E0000614C0000E 130148EB038048EB070048130E0060130C1E1D7DC431>34 D<140C141C1438147014E0EB 01C01303EB0780EB0F00A2131E5BA25B13F85B12015B1203A2485AA3485AA348C7FCA35A A2123EA2127EA4127CA312FCB3A2127CA3127EA4123EA2123FA27EA36C7EA36C7EA36C7E A212017F12007F13787FA27F7FA2EB0780EB03C01301EB00E014701438141C140C166476 CA26>40 D<12C07E12707E7E7E120F6C7E6C7EA26C7E6C7EA21378137C133C133E131E13 1FA2EB0F80A3EB07C0A3EB03E0A314F0A21301A214F8A41300A314FCB3A214F8A31301A4 14F0A21303A214E0A3EB07C0A3EB0F80A3EB1F00A2131E133E133C137C13785BA2485A48 5AA2485A48C7FC120E5A5A5A5A5A16647BCA26>I<16C04B7EB3AB007FBAFCBB1280A26C 1900C8D801E0C9FCB3AB6F5A41407BB84C>43 D<121EEA7F8012FF13C0A213E0A3127FEA 1E601200A413E013C0A312011380120313005A1206120E5A5A5A12600B1D78891B>II<121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A0A78891B>I<14FF01 0713E090381F81F890383E007C01FC133F4848EB1F8049130F4848EB07C04848EB03E0A2 000F15F0491301001F15F8A2003F15FCA390C8FC4815FEA54815FFB3A46C15FEA56D1301 003F15FCA3001F15F8A26C6CEB03F0A36C6CEB07E0000315C06D130F6C6CEB1F806C6CEB 3F00013E137C90381F81F8903807FFE0010090C7FC28447CC131>48 D<143014F013011303131F13FFB5FC13E713071200B3B3B0497E497E007FB6FCA3204278 C131>II<49B4FC010F13E0013F13FC 9038FE01FE3A01F0007F80D803C0EB3FC048C7EA1FE0120EED0FF0EA0FE0486C14F8A215 077F5BA26C48130FEA03C0C813F0A3ED1FE0A2ED3FC01680ED7F0015FE4A5AEC03F0EC1F C0D90FFFC7FC15F090380001FCEC007FED3F80ED1FC0ED0FE016F0ED07F816FC150316FE A2150116FFA3121EEA7F80487EA416FE491303A2007EC713FC00701407003015F8003814 0F6C15F06CEC1FE06C6CEB3FC0D803E0EB7F803A01FE01FE0039007FFFF8010F13E00101 90C7FC28447CC131>II<000615C0D807C0130701FCEB7F8090B612005D5D5D 15E0158026063FFCC7FC90C9FCAE14FF010713C090381F01F090383800FC01F0137ED807 C07F49EB1F8016C090C7120F000615E0C8EA07F0A316F81503A216FCA5123E127F487EA4 16F890C712075A006015F0A20070140F003015E00038EC1FC07E001EEC3F806CEC7F006C 6C13FE6C6C485A3901F807F039007FFFE0011F90C7FCEB07F826447BC131>II<121CA2EA1F8090B712C0A3481680A217005E0038C8120C0030 151C00705D0060153016705E5E4814014B5A4BC7FCC81206150E5D151815385D156015E0 4A5AA24A5A140792C8FC5CA25C141E143EA2147E147CA214FCA21301A3495AA41307A613 0FAA6D5AEB01C02A457BC231>I<14FF010713E0011F13F890387F00FE01FC133FD801F0 EB1F804848EB0FC049EB07E00007EC03F048481301A290C713F8481400A47FA26D130116 F07F6C6CEB03E013FC6C6CEB07C09039FF800F806C9038C01F006CEBF03EECF87839007F FEF090383FFFC07F01077F6D13F8497F90381E7FFFD97C1F1380496C13C02601E00313E0 48486C13F000079038007FF84848EB3FFC48C7120F003EEC07FE150148140016FF167F48 153FA2161FA56C151E007C153EA2007E153C003E157C6C15F86DEB01F06C6CEB03E06C6C EB07C0D803F8EB1F80C6B4EBFF0090383FFFFC010F13F00101138028447CC131>I<14FF 010713E0011F13F890387F80FC9038FC007E48487F4848EB1F804848EB0FC0000FEC07E0 485AED03F0485A16F8007F140190C713FCA25AA216FE1500A516FFA46C5CA36C7E5D121F 7F000F5C6C6C130E150C6C6C131C6C6C5BD8007C5B90383F01E090390FFF80FE903801FE 0090C8FC150116FCA4ED03F8A216F0D80F801307486C14E0486C130F16C0ED1F80A249EB 3F0049137E001EC75A001C495A000F495A3907E01FE06CB51280C649C7FCEB1FF028447C C131>I<121EEA7F80A2EAFFC0A4EA7F80A2EA1E00C7FCB3A5121EEA7F80A2EAFFC0A4EA 7F80A2EA1E000A2B78AA1B>I<121EEA7F80A2EAFFC0A4EA7F80A2EA1E00C7FCB3A5121E 127FEAFF80A213C0A4127F121E1200A512011380A3120313005A1206120E120C121C5A5A 12600A3E78AA1B>I<007FBAFCBB1280A26C1900CEFCB0007FBAFCBB1280A26C19004118 7BA44C>61 D<16C04B7EA34B7EA34B7EA34B7EA3ED19FEA3ED30FFA203707FED607FA203 E07FEDC03FA2020180ED801FA2DA03007F160FA20206801607A24A6D7EA34A6D7EA34A6D 7EA20270810260147FA202E08191B7FCA249820280C7121FA249C87F170FA20106821707 A2496F7EA3496F7EA3496F7EA201788313F8486C83D80FFF03037FB500E0027FEBFFC0A3 42477DC649>65 DIIIIIIII75 DIIIII82 D<49B41303010FEBE007013F13F89039FE00FE0FD801F8 131FD807E0EB079F49EB03DF48486DB4FC48C8FC4881003E81127E82127C00FC81A282A3 7E82A27EA26C6C91C7FC7F7FEA3FF813FE381FFFE06C13FE6CEBFFE06C14FC6C14FF6C15 C0013F14F0010F80010180D9001F7F14019138001FFF03031380816F13C0167F163F161F 17E000C0150FA31607A37EA36C16C0160F7E17806C151F6C16006C5D6D147ED8FBC05CD8 F9F0495AD8F07C495A90393FC00FE0D8E00FB51280010149C7FC39C0003FF02B487BC536 >I<003FB912F8A3903BF0001FF8001F01806D481303003EC7150048187C0078183CA200 70181CA30060180CA5481806A5C81600B3B3A54B7EED7FFE49B77EA33F447DC346>IIII91 D<01C01318000114384848137048C712E0 000EEB01C0000C1480001C13030018140000385B003013060070130E0060130CA300E013 1C481318A400CFEB19E039FFC01FF801E013FCA3007F130FA2003F130701C013F8390F00 01E01E1D71C431>II<130C131E 133F497EEBF3C03801E1E03803C0F03807807848487E001E7F487F0070EB038048EB01C0 0040EB00801A0E75C331>I97 DII<167FED3FFFA315018182B3EC7F809038 03FFF090380FC07C90383F000E017E1307496D5AD803F87F48487F5B000F81485AA2485A A2127FA290C8FC5AAB7E7FA2123FA26C7EA2000F5D7F6C6C5B00035C6C6C9038077F806C 6C010E13C0013F011C13FE90380FC0F8903803FFE09026007F0013002F467DC436>IIIIII<143C14FFA2491380A46D1300A2143C 91C7FCADEC7F80EB3FFFA31300147F143FB3B3AA123E127F39FF807F00A2147EA25C6C48 5A383C01F06C485A3807FF80D801FEC7FC195785C21E>IIII<3901FC01FE00FF903807FFC091381E07F091 383801F8000701707F0003EBE0002601FDC07F5C01FF147F91C7FCA25BA35BB3A8486CEC FF80B5D8F83F13FEA32F2C7DAB36>II<39 01FC03FC00FF90380FFF8091383C07E091387001F83A07FDE000FE00030180137FD801FF EC3F8091C7EA1FC04915E049140F17F0160717F8160317FCA3EE01FEABEE03FCA3EE07F8 A217F0160F6D15E0EE1FC06D143F17806EEB7E00D9FDC05B9039FCF003F891383C0FE091 381FFF80DA03FCC7FC91C9FCAE487EB512F8A32F3F7DAB36>I<91387F8003903903FFE0 0790380FE07890393F801C0F90387E000E496D5AD803F8EB039F0007EC01BF4914FF4848 7F121F5B003F81A2485AA348C8FCAB6C7EA3123F7F121F6D5C120F6D5B12076C6C5B6C6C 497E6C6C130E013F131C90380FC0F8903803FFE09038007F0091C7FCAEEEFF80033F13FE A32F3F7DAB33>I<3903F803F000FFEB1FFCEC3C3EEC707F0007EBE0FF3803F9C000015B 13FBEC007E153C01FF13005BA45BB3A748B4FCB512FEA3202C7DAB26>I<90383FE01839 01FFFC383907E01F78390F0003F8001E1301481300007C1478127800F81438A21518A27E A27E6C6C13006C7E13FC383FFFE06C13FC6C13FF6C14C06C14E0C614F0011F13F81300EC 0FFC140300C0EB01FE1400157E7E153EA27EA36C143C6C147C15786C14F86CEB01F039F3 8003E039F1F00F8039E07FFE0038C00FF01F2E7DAC26>I<1306A5130EA4131EA3133E13 7EA213FE12011207001FB512F0B6FCA2C648C7FCB3A4150CAA017E131C017F1318A26D13 3890381F8030ECC070903807E0E0903801FFC09038007F001E3E7EBC26>IIIIII<003FB612E0A29038C0003F90C713C0 003CEC7F800038ECFF00A20030495A0070495AA24A5A0060495AA24A5A4A5AA2C7485A4A C7FC5B5C495A13075C495A131F4A1360495A495AA249C712C0485AA2485A485A1501485A 48481303A24848EB07804848131F00FF14FF90B6FCA2232B7DAA2B>I<001EEB0780007F EB0FE039FF801FF0EBC03FA4EB801F397F000FE0001EEB07801C0A76C231>127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FC cmbx12 17.28 28 /FC 28 121 df<16F04B7E1507151F153FEC01FF1407147F010FB5FCB7FCA41487EBF007 C7FCB3B3B3B3007FB91280A6395E74DD51>49 D<913801FFF8021FEBFFC091B612F80103 15FF010F16C0013F8290267FFC0114F89027FFE0003F7F4890C7000F7F48486E7FD807F8 6E148048486E14C048486E14E048486F13F001FC17F8486C816D17FC6E80B56C16FE8380 A219FFA283A36C5BA26C5B6C90C8FCD807FC5DEA01F0CA14FEA34D13FCA219F85F19F04D 13E0A294B512C019804C14004C5B604C5B4C5B604C13804C90C7FC4C5A4C5A4B13F05F4B 13804B90C8FC4B5AED1FF84B5A4B5A4B48143F4A5B4A48C8FC4A5A4A48157E4A5A4A5AEC 7F8092C9FC02FE16FE495A495A4948ED01FCD90FC0150749B8FC5B5B90B9FC5A4818F85A 5A5A5A5ABAFCA219F0A4405E78DD51>I<92B5FC020F14F8023F14FF49B712C04916F001 0FD9C01F13FC90271FFC00077FD93FE001017F49486D8049C86C7F484883486C6F7F14C0 486D826E806E82487FA4805CA36C5E4A5E6C5B6C5B6C495E011FC85A90C95CA294B55A61 4C91C7FC604C5B4C5B4C5B4C5B047F138092260FFFFEC8FC020FB512F817E094C9FC17F8 17FF91C7003F13E0040713F8040113FE707F717F7113E085717FA2717F85A285831A80A3 1AC0EA03FCEA0FFF487F487F487FA2B57EA31A80A34D14005C7E4A5E5F6C495E49C8485B D81FF85F000F5ED807FE92B55A6C6C6C4914806C01F0010791C7FC6C9026FF803F5B6D90 B65A011F16F0010716C001014BC8FCD9001F14F0020149C9FC426079DD51>II<01C0EE01C0D801F816 0F01FF167F02F0EC07FFDAFF8090B5FC92B7128019006060606060606095C7FC17FC5F17 E0178004FCC8FC16E09026FC3FFCC9FC91CBFCADED3FFE0203B512F0020F14FE023F6E7E 91B712E001FDD9E00F7F9027FFFE00037F02F801007F02E06EB4FC02806E138091C8FC49 6F13C04917E07113F0EA00F090C914F8A219FC83A219FEA419FFA3EA03F0EA0FFC487E48 7E487FA2B57EA319FEA35C4D13FC6C90C8FC5B4917F8EA3FF001804B13F06D17E0001F5E 6C6C17C06D4B1380D807FC92B512006C6C4A5B6C6C6C01075B6C01E0011F5BD97FFE90B5 5A6DB712C0010F93C7FC6D15FC010115F0D9003F1480020301F0C8FC406078DD51>I69 D73 D77 D<94381FFFE00407B67E043F 15F04BB712FE030FEEFFC0033FD9FC0014F092B500C0010F13FC020349C7000113FF4A01 F86E6C7F021F496F13E04A01C0030F7F4A496F7F91B5C96C7F0103497013FF494970804B 834949717F49874949717F49874B8390B586484A717FA24891CB6C7FA2481D804A84481D C0A348497214E0A3481DF0A34A85481DF8A5B51CFCB06C1DF8A36E96B5FCA36C1DF0A46C 6D4E14E0A36C1DC06E606C1D80A26C6E4D1400A26C6E4D5BA26C6E4D5BA26D6D4D5B6D63 6D6D4D5B6F94B5FC6D636D6D4C5C6D6D4C91C7FC6D6E4B5B6D02E0031F5B023F6D4B13F0 6E01FC92B55A6E01FF02035C020302C0010F91C8FC020002FC90B512FC033F90B712F003 0F17C0030394C9FCDB007F15F804071580DC001F01E0CAFC666677E379>79 D82 D<001FBEFCA64849C79126E0000F148002E0180091 C8171F498601F81A0349864986A2491B7FA2491B3F007F1DC090C9181FA4007E1C0FA600 FE1DE0481C07A5CA95C7FCB3B3B3A3021FBAFCA663617AE070>84 D<913803FFFE027FEBFFF00103B612FE010F6F7E4916E090273FFE001F7FD97FE001077F D9FFF801017F486D6D7F717E486D6E7F85717FA2717FA36C496E7FA26C5B6D5AEB1FC090 C9FCA74BB6FC157F0207B7FC147F49B61207010F14C0013FEBFE004913F048B512C04891 C7FC485B4813F85A5C485B5A5CA2B55AA45FA25F806C5E806C047D7F6EEB01F96C6DD903 F1EBFF806C01FED90FE114FF6C9027FFC07FC01580000191B5487E6C6C4B7E011F02FC13 0F010302F001011400D9001F90CBFC49437CC14E>97 D<903807FF80B6FCA6C6FC7F7FB3 A8EFFFF8040FEBFF80047F14F00381B612FC038715FF038F010014C0DBBFF0011F7FDBFF C001077F93C76C7F4B02007F03F8824B6F7E4B6F13804B17C0851BE0A27313F0A21BF8A3 7313FCA41BFEAE1BFCA44F13F8A31BF0A24F13E0A24F13C06F17804F1300816F4B5A6F4A 5B4AB402075B4A6C6C495B9126F83FE0013F13C09127F00FFC03B55A4A6CB648C7FCDAC0 0115F84A6C15E091C7001F91C8FC90C8000313E04F657BE35A>I<92380FFFF04AB67E02 0F15F0023F15FC91B77E01039039FE001FFF4901F8010113804901E0010713C049018049 13E0017F90C7FC49484A13F0A2485B485B5A5C5A7113E0485B7113C048701380943800FE 0095C7FC485BA4B5FCAE7EA280A27EA2806C18FCA26C6D150119F87E6C6D15036EED07F0 6C18E06C6D150F6D6DEC1FC06D01E0EC7F806D6DECFF00010701FCEB03FE6D9039FFC03F FC010091B512F0023F5D020F1580020102FCC7FCDA000F13C03E437BC148>II<92380FFFC0 4AB512FC020FECFF80023F15E091B712F80103D9FE037F499039F0007FFF011F01C0011F 7F49496D7F4990C76C7F49486E7F48498048844A804884485B727E5A5C48717EA35A5C72 1380A2B5FCA391B9FCA41A0002C0CBFCA67EA380A27EA27E6E160FF11F806C183F6C7FF1 7F006C7F6C6D16FE6C17016D6C4B5A6D6D4A5A6D01E04A5A6D6DEC3FE0010301FC49B45A 6D9026FFC01F90C7FC6D6C90B55A021F15F8020715E0020092C8FC030713F041437CC14A >II105 DI<903807FF80B6FCA6C6FC7F7FB3 B3B3B3ADB712E0A623647BE32C>108 D<902607FF80EB1FFFB691B512F0040714FC041F 14FF4C8193267FE07F7F922781FE001F7FC6DA83F86D7F6DD987F07F6DD98FC0814C7F03 9FC78015BE03BC8003FC825DA25DA25DA45DB3B2B7D8F007B71280A651417BC05A>110 D<923807FFE092B6FC020715E0021F15F8027F15FE494848C66C6C7E010701F0010F13E0 4901C001037F49496D7F4990C87F49486F7E49486F7E48496F13804819C04A814819E048 496F13F0A24819F8A348496F13FCA34819FEA4B518FFAD6C19FEA46C6D4B13FCA36C19F8 A26C6D4B13F0A26C19E06C6D4B13C0A26C6D4B13806C6D4B13006D6C4B5A6D6D495B6D6D 495B010701F0010F13E06D01FE017F5B010090B7C7FC023F15FC020715E0020092C8FC03 0713E048437CC151>I<902607FF80EBFFF8B6010FEBFF80047F14F00381B612FC038715 FF038F010114C09227BFF0003F7FC6DAFFC0010F7F6D91C76C7F6D496E7F03F86E7F4B6E 7F4B17804B6F13C0A27313E0A27313F0A21BF885A21BFCA3851BFEAE4F13FCA41BF861A2 1BF0611BE0611BC06F92B512801B006F5C6F4A5B6F4A5B03FF4A5B70495B04E0017F13C0 9226CFFC03B55A03C7B648C7FC03C115F803C015E0041F91C8FC040313E093CBFCB3A3B7 12F0A64F5D7BC05A>I114 D<913A3FFF8007800107B5EAF81F011FECFE7F017F91B5FC48B8FC48EBE0014890C7121F D80FFC1407D81FF0801600485A007F167F49153FA212FF171FA27F7F7F6D92C7FC13FF14 E014FF6C14F8EDFFC06C15FC16FF6C16C06C16F06C826C826C826C82013F1680010F16C0 1303D9007F15E0020315F0EC001F1500041F13F81607007C150100FC81177F6C163FA217 1F7EA26D16F0A27F173F6D16E06D157F6D16C001FEEDFF806D0203130002C0EB0FFE02FC EB7FFC01DFB65A010F5DD8FE0315C026F8007F49C7FC48010F13E035437BC140>II<902607FFC0ED3F FEB60207B5FCA6C6EE00076D826D82B3B3A260A360A2607F60183E6D6D147E4E7F6D6D49 48806D6DD907F0ECFF806D01FFEB3FE06D91B55A6E1500021F5C020314F8DA003F018002 F0C7FC51427BC05A>I<007FB600C0017FB512F8A6D8001F01F8C70007EBF0006D040190 C7FC6D6D5D6D6D4A5A6D6D4A5A70495A6D4C5A6E7F6E6D495A6E6D495A7049C8FC6E4A5A 6E6D485A6E6D485A6E13FFEF8FF06EEC9FE06FEBFFC06F5C6F91C9FC5F6F5B816F7F6F7F 8481707F8493B57E4B805D4B80DB0FF37FDB1FE17F04C080153F4B486C7F4B486C7F4A48 6D7F4A486D7F4A5A4B6D7F020F6E7F4A486D7F4A486D804A5A4AC86C7F49486F7F4A6F7F 0107707FEB3FFFB600F049B7FCA650407EBF55>120 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FD cmsy8 8 14 /FD 14 111 df0 D<130C131EA50060EB01800078130739FC0C 0FC0007FEB3F80393F8C7F003807CCF83801FFE038007F80011EC7FCEB7F803801FFE038 07CCF8383F8C7F397F0C3F8000FCEB0FC039781E078000601301000090C7FCA5130C1A1D 7C9E23>3 D20 D<12E012F812FEEA3F80EA0FE0EA03F8EA00FEEB3F80EB0FE0EB03F8EB 00FC143FEC0FC0EC07F0EC01FCEC007FED1FC0ED07F0ED01FCED007FEE1FC01607161FEE 7F00ED01FCED07F0ED1FC0037FC7FCEC01FCEC07F0EC0FC0023FC8FC14FCEB03F8EB0FE0 EB3F8001FEC9FCEA03F8EA0FE0EA3F80007ECAFC12F812E0CBFCAD007FB71280B812C0A2 2A3B7AAB37>I<01FE150C3803FF804813E0487F381F83F8263E00FC141C003C133F486D 6C131800706D6C133800606D6C137800E0D901F013F0913800FC014891387F07E092383F FFC06F138003071300ED01FC2E117C9837>24 D<170EA3170F8384170384170184717E18 78187C84180FF007C0BA12F819FC19F8CBEA07C0F00F00183E601878604D5A6017036017 0795C7FC5F170EA33E237CA147>33 D<137813FE1201A3120313FCA3EA07F8A313F0A2EA 0FE0A313C0121F1380A3EA3F00A3123E127E127CA35AA35A0F227EA413>48 DI<91B512C01307131FD97F80C7FC01FCC8FCEA01F0EA03 C0485A48C9FC120E121E5A123812781270A212F05AA3B712C0A300E0C9FCA37E1270A212 781238123C7E120E120F6C7E6C7EEA01F0EA00FCEB7F80011FB512C013071300222B7AA5 2F>I54 D<141F14FFEB03F0EB0FC0EB1F8014005B133EB3A2137E137C 13FC485A485AEA7FC048C7FCEA7FC0EA03F06C7E6C7E137C137E133EB3A2133F7F1480EB 0FC0EB03F0EB00FF141F18437BB123>102 D<12FCB47EEA0FE0EA01F0EA00FC137C137E 133EB3A37F1480130FEB07E0EB01FEEB007FEB01FEEB07E0EB0F80131F1400133EB3A313 7E137C13FCEA01F0EA0FE0EAFF8000FCC7FC18437BB123>I<12E0B3B3B3AD034378B114> 106 D<12E0A27E1270A212781238A2123C121CA2121E120EA2120F7E7F1203A27F1201A2 7F1200A27F137013781338A2133C131CA2131E130EA2130F7FA2801303801301A2801300 A2801470A214781438143C141CA2141E140EA2140F80A215801403A215C0140114001A43 7CB123>110 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FE cmmi10 10.95 1 /FE 1 68 df67 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FF cmbx10 10.95 17 /FF 17 120 df 58 D<16FCA24B7EA24B7EA34B7FA24B7FA34B7FA24B7FA34B7F157C03FC7FEDF87FA202 0180EDF03F0203804B7E02078115C082020F814B7E021F811500824A81023E7F027E8102 7C7FA202FC814A147F49B77EA34982A2D907E0C7001F7F4A80010F835C83011F8391C87E 4983133E83017E83017C81B500FC91B612FCA5463F7CBE4F>65 D<903807FFC0013F13F8 48B6FC48812607FE037F260FF8007F6DEB3FF0486C806F7EA36F7EA26C5A6C5AEA01E0C8 FC153F91B5FC130F137F3901FFFE0F4813E0000F1380381FFE00485A5B485A12FF5BA415 1F7F007F143F6D90387BFF806C6C01FB13FE391FFF07F36CEBFFE100031480C6EC003FD9 1FF890C7FC2F2B7DA933>97 D<13FFB5FCA512077EAFEDFFE0020713FC021FEBFF80027F 80DAFF8113F09139FC003FF802F06D7E4A6D7E4A13074A80701380A218C082A318E0AA18 C0A25E1880A218005E6E5C6E495A6E495A02FCEB7FF0903AFCFF01FFE0496CB55AD9F01F 91C7FCD9E00713FCC7000113C033407DBE3A>I III<903A03 FF8007F0013F9038F83FF8499038FCFFFC48B712FE48018313F93A07FC007FC34848EB3F E1001FEDF1FC4990381FF0F81700003F81A7001F5DA26D133F000F5D6C6C495A3A03FF83 FF8091B5C7FC4814FC01BF5BD80F03138090CAFCA2487EA27F13F06CB6FC16F016FC6C15 FF17806C16C06C16E01207001F16F0393FE000034848EB003F49EC1FF800FF150F90C812 07A56C6CEC0FF06D141F003F16E001F0147FD81FFC903801FFC02707FF800F13006C90B5 5AC615F8013F14E0010101FCC7FC2F3D7DA834>103 D<13FFB5FCA512077EB092380FFF FEA5DB01FEC7FC4B5AED07F0ED1FE04B5A4B5A4BC8FCEC03FC4A5A4A5A141F4A7EECFFFC A2818102E77F02C37F148102007F826F7E6F7E151F6F7E826F7F6F7F816F7FB5D8FC07EB FFC0A5323F7DBE37>107 D<13FFB5FCA512077EB3B3AFB512FCA5163F7CBE1D>I<01FFD9 1FF8ECFFC0B590B5010713F80203DAC01F13FE4A6E487FDA0FE09026F07F077F91261F00 3FEBF8010007013EDAF9F0806C0178ECFBC04A6DB4486C7FA24A92C7FC4A5CA34A5CB3A4 B5D8FE07B5D8F03FEBFF80A551297CA858>I<01FFEB1FF8B5EBFFFE02036D7E4A80DA0F E07F91381F007F0007013C806C5B4A6D7E5CA25CA35CB3A4B5D8FE0FB512E0A533297CA8 3A>II<3901FE01FE00FF903807FF 804A13E04A13F0EC3F1F91387C3FF8000713F8000313F0EBFFE0A29138C01FF0ED0FE091 388007C092C7FCA391C8FCB3A2B6FCA525297DA82B>114 D<90383FFC1E48B512BE0007 14FE5A381FF00F383F800148C7FC007E147EA200FE143EA27E7F6D90C7FC13F8EBFFE06C 13FF15C06C14F06C806C806C806C80C61580131F1300020713C014000078147F00F8143F 151F7EA27E16806C143F6D140001E013FF9038F803FE90B55A15F0D8F87F13C026E00FFE C7FC222B7DA929>II119 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FG cmtt10 10 21 /FG 21 122 df<007FB6FCB71280A46C150021067B9B2C>45 D<121FEA3F80EA7FC0EAFF E0A5EA7FC0EA3F80EA1F000B0B708A2C>I64 D<903901FC038090390FFF87C04913EF017F13FF90B6FC4813073803FC01497E4848137F 4848133F49131F121F5B003F140F90C7FCA2127EED078092C7FCA212FE5AA8913803FFF8 4A13FCA27E007E6D13F89138000FC0A36C141FA27F121F6D133F120F6D137F6C7E6C6C13 FF6D5A3801FF076C90B5FC6D13EF011F13CF6DEB0780D901FCC7FC26357DB32C>71 D<90381FF80790B5EA0F804814CF000714FF5A381FF01F383FC003497E48C7FC007E147F 00FE143F5A151FA46CEC0F00007E91C7FC127F7FEA3FE0EA1FFCEBFFC06C13FC0003EBFF C06C14F06C6C7F01077F9038007FFEEC07FF02001380153FED1FC0A2ED0FE0A200781407 12FCA56CEC0FC0A26CEC1F806D133F01E0EB7F009038FE01FF90B55A5D00F914F0D8F83F 13C0D8700790C7FC23357CB32C>83 D<3801FFF0000713FE001F6D7E15E048809038C01F F81407EC01FC381F80000006C77EC8127EA3ECFFFE131F90B5FC1203120F48EB807E383F F800EA7FC090C7FC12FE5AA47E007F14FEEB8003383FE01F6CB612FC6C15FE6C14BF0001 EBFE1F3A003FF007FC27247CA32C>97 D<903803FFE0011F13F8017F13FE48B5FC488048 48C6FCEA0FF0485A49137E4848131890C9FC5A127EA25AA8127EA2127F6C140F6DEB1F80 6C7E6D133F6C6CEB7F003907FE03FF6CB55A6C5C6C6C5B011F13E0010390C7FC21247AA3 2C>99 D101 DIII<1307EB1FC0A2497EA36D5AA20107C7 FC90C8FCA7387FFFC080B5FC7EA2EA0007B3A8007FB512FCB612FEA36C14FC1F3479B32C >I<387FFFE0B57EA37EEA0003B3B3A5007FB61280B712C0A36C158022337BB22C>108 D<3A7F83F007E09039CFFC1FF83AFFDFFE3FFCD87FFF13FF91B57E3A07FE1FFC3E01FCEB F83F496C487E01F013E001E013C0A301C01380B33B7FFC3FF87FF0027F13FFD8FFFE6D13 F8D87FFC4913F0023F137F2D2481A32C>I<397FF01FE039FFF87FFC9038F9FFFE01FB7F 6CB6FC00019038F03F80ECC01F02807FEC000F5B5BA25BB3267FFFE0B5FCB500F11480A3 6C01E0140029247FA32C>II114 D<131E133FA9007FB6FCB71280A36C1500D8003FC8FCB1ED03C0 ED07E0A5EC800F011FEB1FC0ECE07F6DB51280160001035B6D13F89038003FE0232E7EAD 2C>116 D<3A7FF003FF80486C487FA3007F7F0001EB000FB3A3151FA2153F6D137F3900 FE03FF90B7FC6D15807F6D13CF902603FE07130029247FA32C>I<3A7FFF01FFFCB514FE 148314016C15FC3A03E0000F80A26D131F00011500A26D5B0000143EA26D137E017C137C A2017E13FC013E5BA2EB3F01011F5BA21483010F5BA214C701075BA214EF01035BA214FF 6D90C7FCA26D5A147C27247EA32C>I<3A7FFF01FFFCB5008113FE148314816C010113FC 3A03E0000F806C7E151F6D140012005D6D133E137C017E137E013E137CA2013F13FC6D5B A2EB0F815DA2EB07C1ECC3E0A2EB03E3ECE7C0130114F75DEB00FFA292C7FC80A2143EA2 147E147CA214FC5CA2EA0C01003F5BEA7F83EB87E0EA7E0F495A387FFF806C90C8FC6C5A 6C5AEA07E027367EA32C>121 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FH cmr10 10.95 54 /FH 54 128 df<4AB4EB0FE0021F9038E03FFC913A7F00F8FC1ED901FC90383FF03FD907 F090397FE07F80494801FF13FF4948485BD93F805C137F0200ED7F00EF003E01FE6D91C7 FC82ADB97EA3C648C76CC8FCB3AE486C4A7E007FD9FC3FEBFF80A339407FBF35>11 D14 D<133E133F137F13FFA2EA01FEEA03FCEA07F813F0EA0FE0EA1F C01380EA3E005A5A1270122010116EBE2D>19 D<001E130F397F803FC000FF137F01C013 E0A201E013F0A3007F133F391E600F3000001300A401E01370491360A3000114E04913C0 0003130101001380481303000EEB070048130E0018130C0038131C003013181C1C7DBE2D >34 D<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413E013C0A312011380120313 005A120E5A1218123812300B1C798919>44 DI<121EEA7F80A2 EAFFC0A4EA7F80A2EA1E000A0A798919>I48 D50 DI<00061403D807 80131F01F813FE90B5FC5D5D5D15C092C7FC14FCEB3FE090C9FCACEB01FE90380FFF8090 383E03E090387001F8496C7E49137E497F90C713800006141FC813C0A216E0150FA316F0 A3120C127F7F12FFA416E090C7121F12FC007015C012780038EC3F80123C6CEC7F00001F 14FE6C6C485A6C6C485A3903F80FE0C6B55A013F90C7FCEB07F8243F7CBC2D>53 D57 D<121EEA7F80A2EAFFC0A4EA7F 80A2EA1E00C7FCB3121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A2779A619>I<15074B7E A34B7EA34B7EA34B7EA34B7E15E7A2913801C7FC15C3A291380381FEA34AC67EA3020E6D 7EA34A6D7EA34A6D7EA34A6D7EA34A6D7EA349486D7E91B6FCA249819138800001A249C8 7EA24982010E157FA2011E82011C153FA2013C820138151FA2017882170F13FC00034C7E D80FFF4B7EB500F0010FB512F8A33D417DC044>65 D67 DIIII 73 D80 D82 DI<003FB91280A3903AF0007FE00101 8090393FC0003F48C7ED1FC0007E1707127C00781703A300701701A548EF00E0A5C81600 B3B14B7E4B7E0107B612FEA33B3D7DBC42>IIII<486C13C0000313 0101001380481303000EEB070048130E0018130C0038131C003013180070133800601330 A300E01370481360A400CFEB678039FFC07FE001E013F0A3007F133FA2003F131F01C013 E0390F0007801C1C73BE2D>92 D97 DI<49B4FC010F13E090383F00F8017C131E4848131F 4848137F0007ECFF80485A5B121FA24848EB7F00151C007F91C7FCA290C9FC5AAB6C7EA3 003FEC01C07F001F140316806C6C13076C6C14000003140E6C6C131E6C6C137890383F01 F090380FFFC0D901FEC7FC222A7DA828>II II<167C903903F801 FF903A1FFF078F8090397E0FDE1F9038F803F83803F001A23B07E000FC0600000F6EC7FC 49137E001F147FA8000F147E6D13FE00075C6C6C485AA23901F803E03903FE0FC026071F FFC8FCEB03F80006CAFC120EA3120FA27F7F6CB512E015FE6C6E7E6C15E06C810003813A 0FC0001FFC48C7EA01FE003E140048157E825A82A46C5D007C153E007E157E6C5D6C6C49 5A6C6C495AD803F0EB0FC0D800FE017FC7FC90383FFFFC010313C0293D7EA82D>III<1478EB01FEA2EB03FFA4EB01FEA2EB00781400AC147FEB7FFFA313 017F147FB3B3A5123E127F38FF807E14FEA214FCEB81F8EA7F01387C03F0381E07C0380F FF803801FC00185185BD1C>II I<2701F801FE14FF00FF902707FFC00313E0913B1E07E00F03F0913B7803F03C01F80007 903BE001F87000FC2603F9C06D487F000101805C01FBD900FF147F91C75B13FF4992C7FC A2495CB3A6486C496CECFF80B5D8F87FD9FC3F13FEA347287DA74C>I<3901F801FE00FF 903807FFC091381E07E091387803F000079038E001F82603F9C07F0001138001FB6D7E91 C7FC13FF5BA25BB3A6486C497EB5D8F87F13FCA32E287DA733>I<14FF010713E090381F 81F890387E007E01F8131F4848EB0F804848EB07C04848EB03E0000F15F04848EB01F8A2 003F15FCA248C812FEA44815FFA96C15FEA36C6CEB01FCA3001F15F86C6CEB03F0A26C6C EB07E06C6CEB0FC06C6CEB1F80D8007EEB7E0090383F81FC90380FFFF0010090C7FC282A 7EA82D>I<3901FC03FC00FF90381FFF8091387C0FE09039FDE003F03A07FFC001FC6C49 6C7E6C90C7127F49EC3F805BEE1FC017E0A2EE0FF0A3EE07F8AAEE0FF0A4EE1FE0A2EE3F C06D1580EE7F007F6E13FE9138C001F89039FDE007F09039FC780FC0DA3FFFC7FCEC07F8 91C9FCAD487EB512F8A32D3A7EA733>I<02FF131C0107EBC03C90381F80F090397F0038 7C01FC131CD803F8130E4848EB0FFC150748481303121F485A1501485AA448C7FCAA6C7E A36C7EA2001F14036C7E15076C6C130F6C7E6C6C133DD8007E137990383F81F190380FFF C1903801FE0190C7FCAD4B7E92B512F8A32D3A7DA730>I<3901F807E000FFEB1FF8EC78 7CECE1FE3807F9C100031381EA01FB1401EC00FC01FF1330491300A35BB3A5487EB512FE A31F287EA724>I<90383FC0603901FFF8E03807C03F381F000F003E1307003C1303127C 0078130112F81400A27E7E7E6D1300EA7FF8EBFFC06C13F86C13FE6C7F6C1480000114C0 D8003F13E0010313F0EB001FEC0FF800E01303A214017E1400A27E15F07E14016C14E06C EB03C0903880078039F3E01F0038E0FFFC38C01FE01D2A7DA824>I<131CA6133CA4137C A213FCA2120112031207001FB512C0B6FCA2D801FCC7FCB3A215E0A912009038FE01C0A2 EB7F03013F138090381F8700EB07FEEB01F81B397EB723>IIIIII<001C130E007FEB3F8039FF807FC0A5397F003F80001CEB0E001A 0977BD2D>127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FI cmcsc10 14.4 11 /FI 11 122 df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ndDVIPSBitmapFont %DVIPSBitmapFont: FJ cmr17 20.74 19 /FJ 19 128 df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ndDVIPSBitmapFont %DVIPSBitmapFont: FK cmsy10 14.4 1 /FK 1 4 df<140E141FAA0030ED018000F8ED03E000FE150F6C151F01C0147FD87FE0EC FFC0D83FF8010313803B0FFC0E07FE00D803FFEB1FF8C690388E3FE090393FCE7F809026 0FFFFEC7FC010313F8010013E0EC3F80ECFFE0010313F8010F13FE90393FCE7F809039FF 8E3FE0000390380E1FF8D80FFCEB07FE3B3FF81F03FF80D87FE0010013C0D8FFC0EC7FE0 0100141F48150F00F815030030ED0180C791C7FCAA140E2B3378B73C>3 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FL cmmi12 20.74 1 /FL 1 68 df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ndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: A4 end %%EndSetup %%Page: 1 1 TeXDict begin 1 0 bop 615 951 a FL(C)748 888 y FK(\003)812 951 y FJ(-algebras)53 b(of)f(anisotropic)i(Sc)l(hr\177)-79 b(odinger)1323 1158 y(op)t(erators)52 b(on)g(trees)1455 1454 y FI(Syl)-11 b(v)g(ain)47 b(Gol)2181 1444 y(\023)2181 1454 y(enia)1307 1753 y FH(D)m(\023)-43 b(epartemen)m(t)33 b(de)d(math)m(\023)-43 b(ematiques)1366 1902 y(Univ)m(ersit)m(\023)g(e) 30 b(de)g(Cergy-P)m(on)m(toise)1382 2052 y(2,)h(Av)m(en)m(ue)g(Adolphe) e(Chauvin)1381 2201 y(95302)k(Cergy)d(Cedex)g(-)h(F)-8 b(rance)1095 2351 y(E-mail:)40 b FG(Sylvain.Golenia)o(@)9 b(math.u-cergy.fr)1743 2680 y FF(Abstract)704 2808 y FH(W)-8 b(e)33 b(study)d(a)i FE(C)1262 2775 y FD(\003)1301 2808 y FH(-algebra)g(generated)h(b)m(y)e(\\di\013eren)m(tial")g(op)s (erators)h(on)f(a)568 2920 y(tree.)59 b(W)-8 b(e)37 b(giv)m(e)g(a)f (complete)h(description)d(of)j(its)e(quotien)m(t)i(with)e(resp)s(ect)h (to)568 3033 y(the)30 b(compact)i(op)s(erators.)40 b(This)29 b(allo)m(ws)g(us)h(to)h(compute)f(the)h(essen)m(tial)f(sp)s(ec-)568 3146 y(trum)i(of)h(self-adjoin)m(t)f(op)s(erators)h(a\016liated)g(to)g (the)g(algebra.)49 b(Results)32 b(co)m(v)m(er)568 3259 y(Sc)m(hr\177)-45 b(odinger)39 b(op)s(erators)h(with)f(highly)f (anisotropic,)43 b(p)s(ossibly)37 b(un)m(b)s(ounded,)568 3372 y(p)s(oten)m(tials.)324 3700 y FC(1)161 b(In)l(tro)t(duction)324 3919 y FB(Giv)m(en)38 b(a)f FA(\027)6 b FB(-fold)37 b(tree)h(\000)g(of) f(origin)f FA(e)i FB(with)f(its)g(canonical)g(metric)f FA(d)p FB(,)j(w)m(e)g(denote)g(b)m(y)324 4039 y FA(x)46 b Fz(\030)h FA(y)f FB(when)f FA(x)e FB(and)h FA(y)i FB(are)e(connected) h(b)m(y)f(an)f(edge)h(and)g(w)m(e)g(set)g Fz(j)p FA(x)p Fz(j)i FB(=)g FA(d)p FB(\()p FA(x;)17 b(e)p FB(\).)324 4159 y(F)-8 b(or)42 b(eac)m(h)j FA(x)h Fz(2)g FB(\000)30 b Fz(n)f(f)p FA(e)p Fz(g)p FB(,)46 b(w)m(e)e(denote)g(b)m(y)h FA(x)2022 4123 y FD(0)2091 4159 y Fz(\021)i FA(x)2270 4123 y Fy(\(1\))2408 4159 y FB(the)d(unique)f(elemen)m(t)h FA(y)49 b Fz(\030)d FA(x)324 4280 y FB(suc)m(h)f(that)e Fz(j)p FA(y)t Fz(j)h FB(=)i Fz(j)p FA(x)p Fz(j)29 b(\000)h FB(1)44 b(and)f(w)m(e)h(set)g FA(x)1963 4244 y Fy(\()p Fx(p)p Fy(\))2105 4280 y FB(=)h(\()p FA(x)2319 4244 y Fy(\()p Fx(p)p FD(\000)p Fy(1\))2505 4280 y FB(\))2543 4244 y FD(0)2609 4280 y FB(for)e(1)j Fz(\024)g FA(p)g Fz(\024)h(j)p FA(x)p Fz(j)p FB(.)75 b(Let)324 4400 y FA(x)p FB(\000)28 b(=)f Fz(f)p FA(y)k Fz(2)d FB(\000)g Fz(j)f FA(y)990 4364 y Fy(\()p FD(j)p Fx(y)r FD(j\000j)p Fx(x)p FD(j)p Fy(\))1286 4400 y FB(=)h FA(x)p Fz(g)p FB(.)470 4521 y(On)22 b FA(`)663 4484 y Fy(2)702 4521 y FB(\(\000\))g(w)m(e)h(de\014ne)f(the)h(b)s(ounded)f(op)s(erator)f FA(@)28 b FB(giv)m(en)22 b(b)m(y)g(\()p FA(@)5 b(f)11 b FB(\)\()p FA(x)p FB(\))29 b(=)3092 4446 y Fw(P)3198 4550 y Fx(y)3235 4531 y Fv(0)3257 4550 y Fy(=)p Fx(x)3373 4521 y FA(f)11 b FB(\()p FA(y)t FB(\).)324 4641 y(Its)33 b(adjoin)m(t)f(is)h(giv)m(en)g(b)m(y)h(\()p FA(@)1383 4605 y FD(\003)1423 4641 y FA(f)11 b FB(\)\()p FA(e)p FB(\))28 b(=)g(0)33 b(and)g(\()p FA(@)2139 4605 y FD(\003)2179 4641 y FA(f)11 b FB(\)\()p FA(x)p FB(\))28 b(=)g FA(f)11 b FB(\()p FA(x)2691 4605 y FD(0)2715 4641 y FB(\))33 b(for)f Fz(j)p FA(x)p Fz(j)c(\025)h FB(1.)44 b(Let)33 b Fu(D)324 4761 y FB(b)s(e)g(the)g FA(C)702 4725 y FD(\003)741 4761 y FB(-algebra)e(generated)i(b)m(y)h FA(@)5 b FB(.)470 4882 y(In)29 b(order)f(to)h(obtain)e(our)h(algebra)f(of)h(p)s(oten)m (tials,)h(w)m(e)g(consider)g(the)g(\\h)m(yp)s(erb)s(olic")324 5002 y(compacti\014cation)1064 4977 y Fw(b)1061 5002 y FB(\000)38 b(=)f(\000)26 b Fz([)g FA(@)5 b FB(\000)40 b(of)d(\000)i(constructed)h(as)e(follo)m(ws.)60 b(An)38 b(elemen)m(t)g FA(x)h FB(of)1918 5251 y(1)p eop end %%Page: 2 2 TeXDict begin 2 1 bop 324 548 a FB(the)40 b(b)s(oundary)g(at)g (in\014nit)m(y)f FA(@)5 b FB(\000)40 b(is)g(a)f(\000-v)-5 b(alued)39 b(sequence)j FA(x)f FB(=)f(\()p FA(x)2889 563 y Fx(n)2936 548 y FB(\))2974 563 y Fx(n)p FD(2)p Ft(N)3156 548 y FB(suc)m(h)h(that)324 668 y Fz(j)p FA(x)407 683 y Fx(n)454 668 y Fz(j)27 b FB(=)h FA(n)h FB(and)g FA(x)941 683 y Fx(n)p Fy(+1)1107 668 y Fz(\030)f FA(x)1267 683 y Fx(n)1343 668 y FB(for)h(all)e FA(n)h Fz(2)g Fs(N)9 b FB(.)48 b(W)-8 b(e)30 b(set)g Fz(j)p FA(x)p Fz(j)d FB(=)h Fz(1)g FB(for)h FA(x)f Fz(2)g FA(@)5 b FB(\000.)43 b(The)30 b(space)327 773 y Fw(b)324 799 y FB(\000)38 b(is)g(equipp)s(ed)i(with)e(a)g(natural)g(ultrametric)e(space)k (structure.)63 b(F)-8 b(or)37 b FA(x)i Fz(2)f FA(@)5 b FB(\000)39 b(and)324 919 y(\()p FA(y)410 934 y Fx(n)456 919 y FB(\))494 934 y Fx(n)p FD(2)p Ft(N)673 919 y FB(a)e(sequence)j (in)d(\000)g(w)m(e)h(ha)m(v)m(e)h(lim)1899 934 y Fx(n)p FD(!1)2103 919 y FA(y)2151 934 y Fx(n)2233 919 y FB(=)d FA(x)i FB(if)e(for)h(eac)m(h)h FA(m)e Fz(2)g Fs(N)52 b FB(there)38 b(is)324 1049 y FA(N)g Fz(2)28 b Fs(N)45 b FB(suc)m(h)32 b(that)e(for)g(eac)m(h)i FA(n)c Fz(\025)g FA(N)41 b FB(w)m(e)31 b(ha)m(v)m(e)h FA(y)2150 1064 y Fx(n)2224 1049 y Fz(2)c FA(x)2373 1064 y Fx(m)2440 1049 y FB(\000.)43 b(W)-8 b(e)31 b(denote)g(b)m(y)g FA(C)7 b FB(\()3300 1024 y Fw(b)3297 1049 y FB(\000\))30 b(the)324 1180 y(set)h(of)g(complex-v)-5 b(alued)29 b(con)m(tin)m(uous)j (functions)f(de\014ned)h(on)2647 1155 y Fw(b)2644 1180 y FB(\000.)43 b(Since)31 b(\000)g(is)f(dense)i(in)327 1275 y Fw(b)324 1300 y FB(\000,)40 b(w)m(e)h(can)e(view)g FA(C)7 b FB(\()1137 1275 y Fw(b)1134 1300 y FB(\000\))39 b(as)g(a)g FA(C)1563 1264 y FD(\003)1602 1300 y FB(-subalgebra)g(of)f FA(C)2317 1315 y Fx(b)2351 1300 y FB(\(\000\),)j(the)e(algebra)f(of)h (b)s(ounded)324 1430 y(complex-v)-5 b(alued)34 b(functions)i(de\014ned) i(on)e(\000.)53 b(F)-8 b(or)35 b FA(V)55 b Fz(2)34 b FA(C)7 b FB(\()2567 1405 y Fw(b)2564 1430 y FB(\000\),)36 b(w)m(e)h(denote)g(b)m(y)g FA(V)22 b FB(\()p FA(Q)p FB(\))324 1551 y(the)33 b(op)s(erator)f(of)g(m)m(ultiplication)c(b)m(y)33 b FA(V)54 b FB(in)32 b FA(`)2023 1515 y Fy(2)2062 1551 y FB(\(\000\).)470 1681 y(Let)f(us)h(no)m(w)g(denote)g(b)m(y)g Fu(C)17 b FB(\()1540 1656 y Fw(b)1537 1681 y FB(\000\))31 b(the)g FA(C)1910 1645 y FD(\003)1950 1681 y FB(-algebra)e(generated)j (b)m(y)g Fu(D)41 b FB(and)31 b FA(C)7 b FB(\()3323 1656 y Fw(b)3320 1681 y FB(\000\).)43 b(It)324 1802 y(con)m(tains)35 b(the)h(compact)f(op)s(erators)g(of)g FA(`)1862 1765 y Fy(2)1902 1802 y FB(\(\000\).)51 b(F)-8 b(ollo)m(wing)33 b(the)i(strategy)h(exp)s(osed)h(in)324 1922 y([GI],)31 b(w)m(e)h(shall)d(\014rst)j(compute)f(its)f(quotien)m(t)h(with)f(resp)s (ect)i(to)f(the)g(ideal)f(of)g(compact)324 2042 y(op)s(erators.)42 b(Then,)31 b(w)m(e)e(shall)f(use)h(the)h(follo)m(wing)c(theorem)i(in)g (order)h(to)g(calculate)e(the)324 2163 y(essen)m(tial)40 b(sp)s(ectrum)h(of)f(self-adjoin)m(t)f(op)s(erators)i(related)f(to)g Fu(C)17 b FB(\()2807 2138 y Fw(b)2804 2163 y FB(\000\).)68 b(In)41 b(this)f(in)m(tro-)324 2283 y(duction)32 b(w)m(e)i(restrict)e (ourselv)m(es)i(to)e(the)h(most)f(imp)s(ortan)m(t)f(case)j(when)f FA(\027)h(>)28 b FB(1.)324 2487 y Fr(Theorem)37 b(1.1)49 b Fq(L)-5 b(et)42 b FA(\027)47 b(>)40 b FB(1)p Fq(.)65 b(Ther)-5 b(e)41 b(is)g(a)g(unique)h(morphism)e FB(\010)h(:)f Fu(C)18 b FB(\()3095 2461 y Fw(b)3092 2487 y FB(\000)o(\))41 b Fz(!)f Fu(D)c Fz(\012)324 2607 y FA(C)7 b FB(\()p FA(@)e FB(\000\))40 b Fq(such)f(that)h FB(\010\()p FA(D)s FB(\))c(=)f FA(D)28 b Fz(\012)e FB(1)39 b Fq(for)h(al)5 b(l)38 b FA(D)h Fz(2)d Fu(D)49 b Fq(and)39 b FB(\010\()p FA(')p FB(\()p FA(Q)p FB(\)\))d(=)g(1)25 b Fz(\012)h FB(\()p FA(')p Fz(j)3405 2622 y Fx(@)t Fy(\000)3494 2607 y FB(\))p Fq(.)324 2727 y(This)34 b(morphism)f(is)i(surje)-5 b(ctive)35 b(and)f(its)h(kernel)f(is)h Fs(K)17 b FB(\(\000\))p Fq(.)470 2931 y FB(W)-8 b(e)35 b(no)m(w)g(giv)m(e)f(examples.)49 b(Let)34 b FA(\027)j(>)31 b FB(1)j(and)g(let)g FA(H)k FB(=)2522 2856 y Fw(P)2627 2960 y Fx(\013;\014)2756 2931 y FA(a)2807 2946 y Fx(\013;\014)2919 2931 y FB(\()p FA(Q)p FB(\))p FA(@)3128 2895 y FD(\003)3168 2888 y Fx(\013)3218 2931 y FA(@)3274 2895 y Fx(\014)3345 2931 y FB(+)23 b FA(K)7 b FB(,)324 3078 y(where)37 b FA(a)660 3093 y Fx(\013;\014)805 3078 y Fz(2)c FA(C)7 b FB(\()1022 3053 y Fw(b)1019 3078 y FB(\000\),)36 b FA(a)1232 3093 y Fx(\013;\014)1377 3078 y FB(=)d(0)i(for)g(almost)f(all)f(\()p FA(\013)q(;)17 b(\014)6 b FB(\))32 b Fz(2)h Fs(N)2619 3042 y Fy(2)2700 3078 y FB(and)j FA(K)43 b FB(is)35 b(a)g(compact)324 3211 y(op)s(erator.)80 b(Clearly)44 b FA(H)56 b Fz(2)49 b Fu(C)17 b FB(\()1519 3186 y Fw(b)1516 3211 y FB(\000\))45 b(and)g(\010\()p FA(H)8 b FB(\))49 b(=)2270 3137 y Fw(P)2375 3240 y Fx(\013;\014)2504 3211 y FA(@)2560 3175 y FD(\003)2601 3168 y Fx(\013)2650 3211 y FA(@)2706 3175 y Fx(\014)2785 3211 y Fz(\012)31 b FB(\()p FA(a)2982 3226 y Fx(\013;\014)3094 3211 y FB(\))p Fz(j)3160 3226 y Fx(@)t Fy(\000)3249 3211 y FB(.)80 b(As)46 b(a)324 3332 y(consequence)40 b(of)d(the)h(Theorem)g (1.1,)g(if)f FA(H)45 b FB(self-adjoin)m(t,)37 b(its)g(essen)m(tial)g (sp)s(ectrum)h(is)324 3452 y(giv)m(en)32 b(b)m(y:)1184 3584 y FA(\033)1239 3599 y Fy(ess)1331 3584 y FB(\()p FA(H)8 b FB(\))27 b(=)1657 3490 y Fw([)1626 3702 y Fx(\015)t FD(2)p Fx(@)t Fy(\000)1815 3584 y FA(\033)1874 3504 y Fw(\000)1949 3510 y(P)1948 3672 y Fx(\013;\014)2083 3584 y FA(a)2134 3599 y Fx(\013;\014)2247 3584 y FB(\()p FA(\015)5 b FB(\))p FA(@)2435 3543 y FD(\003)2475 3539 y Fx(\013)2524 3584 y FA(@)2580 3543 y Fx(\014)2629 3504 y Fw(\001)2674 3584 y FA(:)324 3856 y FB(This)43 b(result)f(can)i(made)e(quite)h (explicit)e(in)h(the)h(particular)e(case)j(of)e(a)h(Shr\177)-49 b(odinger)324 3977 y(op)s(erator)43 b FA(H)54 b FB(=)46 b(\001)30 b(+)g FA(V)22 b FB(\()p FA(Q)p FB(\))43 b(with)h(p)s(oten)m (tial)e FA(V)65 b FB(in)43 b FA(C)7 b FB(\()2499 3951 y Fw(b)2496 3977 y FB(\000\).)76 b(W)-8 b(e)44 b(recall)f(that)g(\001)h (is)324 4097 y(a)37 b(b)s(ounded)h(op)s(erator)f(on)g FA(`)1392 4061 y Fy(2)1432 4097 y FB(\(\000\))g(de\014ned)i(b)m(y)f (\(\001)p FA(f)11 b FB(\)\()p FA(x)p FB(\))36 b(=)2582 4022 y Fw(P)2687 4126 y Fx(y)r FD(\030)p Fx(x)2823 4097 y FB(\()p FA(f)11 b FB(\()p FA(y)t FB(\))25 b Fz(\000)g FA(f)11 b FB(\()p FA(x)p FB(\)\).)59 b(It)324 4244 y(b)s(elongs)32 b(to)h Fu(C)18 b FB(\()919 4219 y Fw(b)916 4244 y FB(\000)o(\).)46 b(W)-8 b(e)33 b(then)h(set)g(\001)1712 4259 y Fy(0)1780 4244 y FB(=)29 b FA(@)f FB(+)23 b FA(@)2119 4208 y FD(\003)2182 4244 y Fz(\000)g FA(\027)6 b FB(Id)34 b(\(whic)m(h)f(b)s(elongs)g(to)g Fu(D)9 b FB(\))33 b(and)324 4365 y(notice)f(that)g(\001)23 b Fz(\000)f FB(\001)1106 4380 y Fy(0)1179 4365 y FB(is)32 b(compact.)43 b(W)-8 b(e)33 b(get)f(\(see)i([AF])f(for)f(instance\):) 754 4585 y FA(\033)809 4600 y Fx(ess)912 4585 y FB(\()p FA(@)c FB(+)22 b FA(@)1183 4544 y FD(\003)1223 4585 y FB(\))28 b(=)f FA(\033)1447 4600 y Fx(ac)1520 4585 y FB(\()p FA(@)h FB(+)22 b FA(@)1791 4544 y FD(\003)1831 4585 y FB(\))28 b(=)f FA(\033)t FB(\()p FA(@)h FB(+)22 b FA(@)2330 4544 y FD(\003)2371 4585 y FB(\))27 b(=)h([)p Fz(\000)p FB(2)2693 4508 y Fz(p)p 2776 4508 55 4 v 77 x FA(\027)7 b(;)17 b FB(2)2924 4508 y Fz(p)p 3006 4508 V 3006 4585 a FA(\027)23 b FB(])p FA(;)324 4805 y FB(where)50 b FA(\033)677 4820 y Fx(ac)749 4805 y FB(\()p FA(T)14 b FB(\))49 b(denotes)h(the)f(absolute)g(con)m(tin)m(uous)g(part)g(of)f (the)h(sp)s(ectrum)h(of)e(a)324 4925 y(giv)m(en)29 b(self-adjoin)m(t)e (op)s(erator)h FA(T)14 b FB(.)42 b(Notice)29 b(that)g(Theorem)g(1.1)f (giv)m(es)i(us)f(directly)g(that)1918 5251 y(2)p eop end %%Page: 3 3 TeXDict begin 3 2 bop 324 548 a FA(\033)379 563 y Fy(e)p Fx(ss)480 548 y FB(\()p FA(@)574 512 y FD(\003)636 548 y FB(+)22 b FA(@)5 b FB(\))29 b(=)f FA(\033)t FB(\()p FA(@)1114 512 y FD(\003)1176 548 y FB(+)22 b FA(@)5 b FB(\).)44 b(Finally)-8 b(,)30 b(w)m(e)k(obtain)782 768 y FA(\033)837 783 y Fx(ess)940 768 y FB(\(\001)22 b(+)g FA(V)g FB(\()p FA(Q)p FB(\)\))83 b(=)g FA(\033)t FB(\(\001)1869 783 y Fy(0)1908 768 y FB(\))23 b(+)f FA(V)f FB(\()p FA(@)5 b FB(\000\))1532 913 y(=)83 b([)p Fz(\000)p FA(\027)29 b Fz(\000)23 b FB(2)2021 837 y Fz(p)p 2103 837 55 4 v 2103 913 a FA(\027)7 b(;)17 b Fz(\000)p FA(\027)29 b FB(+)22 b(2)2503 837 y Fz(p)p 2585 837 V 2585 913 a FA(\027)i FB(])e(+)g FA(V)f FB(\()p FA(@)5 b FB(\000\))p FA(:)324 1133 y FB(In)33 b(fact)f(this)g(result)h(holds)f(\(and)h(is)f (trivial\))e(in)h(the)i(case)h(of)e FA(\027)i FB(=)28 b(1,)k(when)i(\000)27 b(=)h Fs(N)9 b FB(.)470 1254 y(Giv)m(en)36 b(a)f(con)m(tin)m(uous)h(function)f(on)h FA(@)5 b FB(\000,)37 b(the)f(Tietze)g(theorem)f(allo)m(ws)g(us)h(to)f(ex-)324 1374 y(tend)43 b(it)e(to)h(a)f(con)m(tin)m(uous)i(function)f(on)1920 1349 y Fw(b)1917 1374 y FB(\000,)j(so)d(one)g(ma)m(y)g(construct)i(a)d (large)g(class)324 1494 y(of)h(Hamiltonians)e(with)i(giv)m(en)h(essen)m (tial)g(sp)s(ectra.)76 b(Nev)m(ertheless)45 b(w)m(e)f(are)f(able)f(to) 324 1615 y(p)s(oin)m(t)37 b(out)h(a)f(concrete)i(class)f(of)g (non-trivial)d(p)s(oten)m(tials)i FA(V)58 b Fz(2)37 b FA(C)7 b FB(\()2867 1590 y Fw(b)2864 1615 y FB(\000\))38 b(with)f(uniform)324 1745 y(b)s(eha)m(viour)g(at)g(in\014nit)m(y)g (whic)m(h)h(form)e(a)h(dense)h(family)d(of)i FA(C)7 b FB(\()2658 1720 y Fw(b)2655 1745 y FB(\000\).)57 b(Namely)-8 b(,)38 b(for)f(eac)m(h)324 1866 y(b)s(ounded)c(function)f FA(f)39 b FB(:)27 b(\000)h Fz(!)f Fs(R)44 b FB(and)32 b(eac)m(h)i(real)e FA(\013)c(>)f FB(1)33 b(let)324 2185 y(\(1.1\))1027 b FA(V)22 b FB(\()p FA(x)p FB(\))28 b(=)1926 2053 y FD(j)p Fx(x)p FD(j)1893 2090 y Fw(X)1901 2302 y Fx(k)r Fy(=1)2064 2117 y FA(f)11 b FB(\()p FA(x)2216 2132 y Fx(k)2258 2117 y FB(\))p 2064 2162 233 4 v 2128 2253 a FA(k)2182 2224 y Fx(\013)2306 2185 y FA(;)324 2517 y FB(where)34 b FA(x)661 2532 y Fx(k)731 2517 y FB(=)28 b FA(x)890 2481 y FD(j)p Fx(x)p FD(j\000)p Fx(k)1099 2517 y FB(for)k FA(x)c Fz(2)h FB(\000)j(\()p FA(V)54 b FB(b)s(elongs)32 b(to)g FA(C)7 b FB(\()2255 2492 y Fw(b)2252 2517 y FB(\000\))33 b(b)s(ecause)h(of)e(Lemma)f(2.3\).)470 2638 y(Concerning)e(\014ner)h(sp)s(ectral)f(features,)h(based)g(mainly) d(on)i(the)g(Mourre)g(estimate,)324 2758 y(w)m(e)j(men)m(tion)f(that)g (in)f(the)i(case)g FA(H)k FB(=)27 b(\001)20 b(+)g FA(V)h FB(\()p FA(Q)p FB(\),)32 b(with)f FA(V)53 b FB(as)31 b(in)g(\(1.1\))g(where)h FA(\013)d Fz(\025)f FB(3)324 2879 y(and)i(suc)m(h)h(that)f FA(V)21 b FB(\()p FA(@)5 b FB(\000\))29 b(=)e(0,)k(the)f(results)g(of)g([AF])g(can)g(b)s(e)g (applied)f(\(the)h(h)m(yp)s(otheses)324 2999 y(of)f(the)i(Lemmas)e(6)g (and)i(7)e(from)g([AF])h(are)g(v)m(eri\014ed)h(since)f FA(V)22 b FB(\()p FA(x)p FB(\))28 b(=)f FA(O)s FB(\()p Fz(j)p FA(x)p Fz(j)3078 2963 y FD(\000)p Fx(\013)p Fy(+1)3272 2999 y FB(\))i(when)324 3119 y Fz(j)p FA(x)p Fz(j)43 b(!)g(1)p FB(\).)72 b(The)42 b(aim)e(of)i(our)g(w)m(ork)h(in)e (preparation)g([Go)o(])h(is)g(to)f(pro)m(v)m(e)i(that)f(the)324 3240 y(Mourre)34 b(estimate)f(holds)g(for)g(more)g(general)g(classes)h (of)g(Hamiltonians)c(a\016liated)i(to)324 3360 y Fu(C)17 b FB(\()448 3335 y Fw(b)445 3360 y FB(\000\))32 b(and)h(to)f(dev)m (elop)h(a)g(scattering)f(theory)h(for)f(them.)470 3480 y(The)f(preceding)f(results)h(on)e(trees)i(allo)m(w)e(us)h(to)g(treat)g (more)f(general)g(graphs.)43 b(W)-8 b(e)324 3601 y(recall)27 b(that)h(a)g(graph)g(is)f(said)h(to)g(b)s(e)g Fq(c)-5 b(onne)g(cte)g(d)27 b FB(if)g(t)m(w)m(o)i(of)f(its)g(elemen)m(ts)g(can) h(b)s(e)f(joined)324 3721 y(b)m(y)36 b(a)e(sequence)k(of)c(neigh)m(b)s (ours.)51 b(Let)35 b FA(G)g FB(b)s(e)g(a)f(\014nite)h(disjoin)m(t)e (union)i(of)f(\000)3160 3736 y Fx(i)3188 3721 y FB(,)i(eac)m(h)g(\000) 3534 3736 y Fx(i)324 3842 y FB(b)s(eing)h(a)h FA(\027)727 3857 y Fx(i)756 3842 y FB(-fold)e(branc)m(hing)j(tree)f(with)g FA(\027)1921 3857 y Fx(i)1987 3842 y Fz(\025)g FB(1)g(and)h(of)e FA(G)2578 3857 y Fy(0)2656 3842 y FB(a)h(compact)g(connected)324 3962 y(graph:)1599 4153 y FA(G)27 b FB(=)1843 4028 y Fx(n)1809 4058 y Fw([)1807 4268 y Fx(i)p Fy(=1)1938 4153 y FB(\000)1999 4168 y Fx(i)2044 4078 y Fw(S)2143 4153 y FA(G)2220 4168 y Fy(0)2260 4153 y FA(:)324 4409 y FB(W)-8 b(e)27 b(endo)m(w)i FA(G)e FB(with)f(a)h(connected)i(graph)e(structure) h(that)f(resp)s(ects)i(the)e(graph)g(struc-)324 4530 y(ture)j(of)g(\000)697 4545 y Fx(i)755 4530 y FB(and)g(the)h(one)f(of)f FA(G)1469 4545 y Fy(0)1539 4530 y FB(and)h(suc)m(h)h(that)f(\000)2213 4545 y Fx(i)2271 4530 y FB(is)g(connected)i(to)d(\000)2997 4545 y Fx(j)3064 4530 y FB(\()p FA(i)f Fz(6)p FB(=)f FA(j)6 b FB(\))30 b(only)324 4650 y(through)k FA(G)771 4665 y Fy(0)845 4650 y FB(and)h(suc)m(h)h(that)e(\000)1533 4665 y Fx(i)1596 4650 y FB(is)g(connected)i(to)f FA(G)2352 4665 y Fy(0)2426 4650 y FB(only)f(through)g FA(e)3057 4665 y Fx(i)3086 4650 y FB(,)h(the)g(origin)324 4771 y(of)44 b(\000)508 4786 y Fx(i)536 4771 y FB(.)80 b(The)46 b(graph)e FA(G)h FB(is)f(h)m(yp)s(erb)s(olic)g(and)h(its)f(b)s(oundary) h(at)f(in\014nit)m(y)h FA(@)5 b(G)45 b FB(is)f(ex-)324 4891 y(actly)34 b(the)h(disjoin)m(t)e(union)h Fz([)1425 4855 y Fx(n)1425 4916 y(i)p Fy(=1)1544 4891 y FA(@)5 b FB(\000)1661 4906 y Fx(i)1690 4891 y FB(.)49 b(W)-8 b(e)35 b(no)m(w)g(c)m(ho)s(ose)g FA(V)53 b Fz(2)31 b FA(C)7 b FB(\()p FA(G)23 b Fz([)h FA(@)5 b(G)p FB(\).)50 b(One)35 b(has)1918 5251 y(3)p eop end %%Page: 4 4 TeXDict begin 4 3 bop 324 548 a FA(V)21 b Fz(j)432 562 y Fp(b)430 578 y Fy(\000)474 588 y Fo(i)532 548 y Fz(2)28 b FA(C)7 b FB(\()744 523 y Fw(b)741 548 y FB(\000)802 563 y Fx(i)830 548 y FB(\))33 b(for)f(all)e FA(i)e FB(=)g(1)p FA(;)17 b(:)g(:)g(:)e(;)i(n)33 b FB(and)g(w)m(e)g(easily)f(obtain)f (the)i(next)h(result:)660 832 y FA(\033)715 847 y Fx(ess)818 832 y FB(\(\001)22 b(+)g FA(V)g FB(\()p FA(Q)p FB(\)\))28 b(=)1494 707 y Fx(n)1460 737 y Fw([)1458 947 y Fx(i)p Fy(=1)1589 751 y Fw(\000)1651 832 y FB([)p Fz(\000)p FA(\027)1803 847 y Fx(i)1854 832 y Fz(\000)23 b FB(2)2003 763 y Fz(p)p 2086 763 77 4 v 69 x FA(\027)2134 847 y Fx(i)2162 832 y FA(;)17 b Fz(\000)p FA(\027)2331 847 y Fx(i)2382 832 y FB(+)22 b(2)2529 763 y Fz(p)p 2612 763 V 69 x FA(\027)2660 847 y Fx(i)2705 832 y FB(])g(+)g FA(V)f FB(\()p FA(@)5 b FB(\000)3085 847 y Fx(i)3114 832 y FB(\))3152 751 y Fw(\001)3198 832 y FA(:)470 1106 y FB(This)44 b(co)m(v)m(ers)h(in)d(particular)g(the)h(case)i(of)d(the)i (Ca)m(yley)g(graph)f(of)g(a)g(free)h(group)324 1227 y(with)30 b(\014nite)h(system)h(of)e(generators.)44 b(W)-8 b(e)31 b(recall)e(that)i(the)g(Ca)m(yley)h(graph)f(of)f(a)h(group)324 1347 y FA(G)45 b FB(with)f(a)h(system)h(of)f(generators)g FA(S)51 b FB(is)44 b(the)i(graph)f(de\014ned)h(on)f(the)g(set)h FA(G)f FB(with)324 1468 y(the)c(relation)d FA(x)k Fz(\030)g FA(y)h FB(if)d FA(xy)1377 1431 y FD(\000)p Fy(1)1512 1468 y Fz(2)h FA(S)47 b FB(or)40 b FA(y)t(x)1960 1431 y FD(\000)p Fy(1)2095 1468 y Fz(2)i FA(S)6 b FB(.)67 b(Let)40 b FA(G)h FB(b)s(e)f(a)h(free)g(group)f(with)324 1588 y(a)h(system)h(of)e(generators)i FA(S)47 b FB(suc)m(h)42 b(that)f FA(S)48 b FB(=)43 b FA(S)2194 1552 y FD(\000)p Fy(1)2288 1588 y FB(.)69 b(W)-8 b(e)41 b(denote)h(b)m(y)g FA(e)g FB(its)e(neutral)324 1708 y(elemen)m(t)g(and)g(w)m(e)i(set)f Fz(j)p FA(S)6 b Fz(j)40 b FB(=)g FA(\027)34 b FB(+)27 b(1.)67 b(One)40 b(ma)m(y)g(asso)s(ciate)g(the)h(restriction)e(of)h (the)324 1829 y(Ca)m(yley)26 b(graph)g(to)f(the)h(set)h(of)e(w)m(ords)h (starting)f(with)g(a)h(giv)m(en)g(generator)f(with)g(a)h FA(\027)6 b FB(-fold)324 1949 y(branc)m(hing)29 b(tree)i(ha)m(ving)e (as)h(origin)e(the)i(generator.)42 b(Hence,)32 b(the)e(Ca)m(yley)h (graph)e(of)h FA(G)324 2069 y FB(will)c(b)s(e)k Fz([)700 2033 y Fx(\027)700 2094 y(i)p Fy(=1)818 2069 y FB(\000)879 2084 y Fx(i)922 2069 y Fz([)15 b(f)p FA(e)p Fz(g)29 b FB(where)i(\000)1517 2084 y Fx(i)1574 2069 y FB(is)d(a)h FA(\027)6 b FB(-fold)28 b(branc)m(hing)g(tree)i(with)f(the)g(ab)s(o)m (v)m(e)h(graph)324 2190 y(structure.)470 2310 y(W)-8 b(e)39 b(no)m(w)g(go)g(further)g(b)m(y)g(taking)f FA(V)59 b Fz(2)39 b FA(C)7 b FB(\()2111 2285 y Fw(b)2108 2310 y FB(\000)p FA(;)p 2213 2230 72 4 v 17 w Fs(R)j FB(\))39 b(suc)m(h)h(that)e FA(V)22 b FB(\(\000\))38 b Fz(\032)g Fs(R)5 b FB(,)46 b(where)p 324 2350 V 324 2431 a Fs(R)63 b FB(=)52 b Fs(R)43 b Fz([)32 b(f1g)46 b FB(is)g(the)i(Alexandro)m(v)f (compacti\014cation)e(of)i Fs(R)5 b FB(.)92 b(More)47 b(precisely)-8 b(,)324 2561 y FA(V)55 b Fz(2)34 b FA(C)7 b FB(\()654 2536 y Fw(b)651 2561 y FB(\000)p FA(;)p 756 2480 V 17 w Fs(R)j FB(\))36 b(if)f(and)h(only)f(if)g(for)h(eac)m(h)h FA(\015)h Fz(2)c FA(@)5 b FB(\000)37 b(w)m(e)g(ha)m(v)m(e)g(either)f (lim)3006 2576 y Fx(x)p FD(!)p Fx(\015)3178 2561 y FA(V)21 b FB(\()p FA(x)p FB(\))34 b(=)f FA(l)324 2681 y FB(where)h FA(l)d Fz(2)d Fs(R)44 b FB(or)33 b(for)f(eac)m(h)i FA(M)39 b Fz(\025)29 b FB(0)k(there)h(is)e FA(N)39 b Fz(2)29 b Fs(N)48 b FB(suc)m(h)34 b(that)f Fz(j)p FA(V)21 b FB(\()p FA(x)p FB(\))p Fz(j)29 b(\025)g FA(M)43 b FB(for)33 b(all)324 2802 y FA(n)28 b Fz(\025)g FA(N)43 b FB(and)33 b FA(x)28 b Fz(2)g FA(\015)1054 2817 y Fx(n)1100 2802 y FB(\000)33 b(\(see)g(Lemma)f(2.3\).)43 b(W)-8 b(e)33 b(set)1124 2994 y FA(D)s FB(\()p FA(V)21 b FB(\))28 b(=)g Fz(f)p FA(f)38 b Fz(2)28 b FA(`)1765 2953 y Fy(2)1804 2994 y FB(\(\000\))g Fz(j)f(k)p FA(V)21 b FB(\()p FA(Q)p FB(\))p FA(f)11 b Fz(k)2414 2953 y Fy(2)2481 2994 y FA(<)28 b Fz(1g)p FA(:)324 3186 y FB(Let)i FA(T)41 b Fz(2)28 b Fu(D)39 b FB(and)30 b FA(T)1048 3201 y Fy(0)1115 3186 y FB(=)e(\010\()p FA(T)14 b FB(\).)42 b(Since)30 b FA(T)44 b FB(is)29 b(b)s(ounded,)i(the)f(op)s(erator)f FA(H)35 b FB(=)28 b FA(T)i FB(+)16 b FA(V)22 b FB(\()p FA(Q)p FB(\))324 3317 y(with)k(domain)e FA(D)s FB(\()p FA(V)d FB(\))26 b(is)g(self-adjoin)m(t)e(and)j(it)e(is)g(a\016liated)g(to)h Fu(C)17 b FB(\()2729 3291 y Fw(b)2726 3317 y FB(\000\))26 b(\(i.e.)41 b(its)25 b(resolv)m(en)m(t)324 3447 y(b)s(elongs)32 b(to)h Fu(C)17 b FB(\()918 3422 y Fw(b)915 3447 y FB(\000\)\).)45 b(Indeed,)35 b(for)d(eac)m(h)i FA(z)f Fz(2)c Fs(C)48 b Fz(n)23 b Fs(R)44 b FB(w)m(e)34 b(ha)m(v)m(e)g(\()p FA(V)22 b FB(\()p FA(Q)p FB(\))g(+)h FA(z)t FB(\))3130 3411 y FD(\000)p Fy(1)3253 3447 y Fz(2)29 b FA(C)7 b FB(\()3466 3422 y Fw(b)3463 3447 y FB(\000\))324 3567 y(and)32 b(for)h(large)e FA(z)h Fz(2)c Fs(C)49 b Fz(n)22 b Fs(R)43 b FB(one)33 b(has)900 3777 y(\()p FA(H)d FB(+)22 b FA(z)t FB(\))1234 3736 y FD(\000)p Fy(1)1357 3777 y FB(=)27 b(\()p FA(V)22 b FB(\()p FA(Q)p FB(\))g(+)g FA(z)t FB(\))1937 3736 y FD(\000)p Fy(1)2049 3682 y Fw(X)2054 3893 y Fx(n)p FD(\025)p Fy(0)2193 3777 y FB(\()p FA(T)14 b FB(\()p FA(V)21 b FB(\()p FA(Q)p FB(\))h(+)g FA(z)t FB(\))2778 3736 y FD(\000)p Fy(1)2873 3777 y FB(\))2911 3736 y Fx(n)2958 3777 y FA(;)324 4063 y FB(where)47 b(the)f(series)h (is)e(norm)h(con)m(v)m(ergen)m(t.)85 b(No)m(w,)50 b(with)c(the)g(same)g FA(z)t FB(,)k(w)m(e)d(use)g(the)324 4184 y(Theorem)33 b(1.1)f(and)h(the)g(fact)f(that)g Fu(D)g Fz(\012)23 b FA(C)7 b FB(\()p FA(@)e FB(\000\))28 b Fz(')g FA(C)7 b FB(\()p FA(@)e FB(\000)p FA(;)17 b Fu(D)9 b FB(\))33 b(to)g(obtain)394 4393 y(\010)464 4408 y Fx(\015)508 4393 y FB(\(\()p FA(H)d FB(+)22 b FA(z)t FB(\))880 4352 y FD(\000)p Fy(1)975 4393 y FB(\))28 b Fz(\021)g FB(\010\(\()p FA(H)i FB(+)22 b FA(z)t FB(\))1588 4352 y FD(\000)p Fy(1)1683 4393 y FB(\)\()p FA(\015)5 b FB(\))27 b(=)h(\()p FA(V)21 b FB(\()p FA(\015)5 b FB(\))22 b(+)g FA(z)t FB(\))2439 4352 y FD(\000)p Fy(1)2551 4299 y Fw(X)2556 4509 y Fx(n)p FD(\025)p Fy(0)2695 4393 y FB(\()p FA(T)2790 4408 y Fy(0)2829 4393 y FB(\()p FA(V)g FB(\()p FA(\015)5 b FB(\))22 b(+)g FA(z)t FB(\))3285 4352 y FD(\000)p Fy(1)3380 4393 y FB(\))3418 4352 y Fx(n)3465 4393 y FA(:)324 4689 y FB(Note)36 b(that)g(\()p FA(V)21 b FB(\()p FA(\015)5 b FB(\))24 b(+)h FA(z)t FB(\))1238 4653 y FD(\000)p Fy(1)1366 4689 y FB(=)34 b(0)h(if)g FA(V)22 b FB(\()p FA(\015)5 b FB(\))33 b(=)g Fz(1)p FB(.)53 b(By)37 b(analytic)e(con)m(tin)m(uation)g(w)m(e)i(get)324 4810 y(that)32 b(for)g(all)f FA(z)h Fz(2)c Fs(C)48 b Fz(n)22 b Fs(R)1041 5002 y FB(\010)1111 5017 y Fx(\015)1156 5002 y FB(\()p FA(T)36 b FB(+)22 b FA(V)f FB(\()p FA(Q)p FB(\))i(+)f FA(z)t FB(\))1824 4961 y FD(\000)p Fy(1)1946 5002 y FB(=)28 b(\()p FA(T)2145 5017 y Fy(0)2207 5002 y FB(+)22 b FA(V)f FB(\()p FA(\015)5 b FB(\))22 b(+)g FA(z)t FB(\))2722 4961 y FD(\000)p Fy(1)2817 5002 y FA(;)1918 5251 y FB(4)p eop end %%Page: 5 5 TeXDict begin 5 4 bop 324 548 a FB(with)32 b(the)h(con)m(v)m(en)m(tion) h(that)e(\()p FA(T)1508 563 y Fy(0)1570 548 y FB(+)22 b FA(V)f FB(\()p FA(\015)5 b FB(\))22 b(+)g FA(z)t FB(\))2085 512 y FD(\000)p Fy(1)2208 548 y FB(=)28 b(0)k(if)f FA(V)22 b FB(\()p FA(\015)5 b FB(\))27 b(=)h Fz(1)p FB(.)470 668 y(W)-8 b(e)41 b(no)m(w)h(compute)f(the)g(essen)m(tial)g(sp)s (ectrum)g(of)f FA(H)8 b FB(.)68 b(If)41 b FA(V)22 b FB(\()p FA(\015)5 b FB(\))41 b(=)h Fz(1)p FB(,)g(w)m(e)g(ha)m(v)m(e)324 789 y FA(\033)t FB(\(\010)491 804 y Fx(\015)536 789 y FB(\()p FA(H)8 b FB(\)\))39 b(=)h(\037.)66 b(Otherwise,)42 b(one)f(has)f FA(\033)t FB(\(\010)2095 804 y Fx(\015)2140 789 y FB(\()p FA(H)8 b FB(\)\))39 b(=)i FA(\033)t FB(\()p FA(T)2653 804 y Fy(0)2719 789 y FB(+)27 b FA(V)22 b FB(\()p FA(\015)5 b FB(\)\))40 b(=)g FA(\033)t FB(\()p FA(T)3381 804 y Fy(0)3421 789 y FB(\))27 b(+)324 909 y FA(V)21 b FB(\()p FA(\015)5 b FB(\).)44 b(Hence)34 b(w)m(e)f(obtain:)1209 1129 y FA(\033)1264 1144 y Fy(ess)1356 1129 y FB(\()p FA(T)j FB(+)22 b FA(V)f FB(\()p FA(Q)p FB(\)\))28 b(=)g FA(\033)t FB(\()p FA(T)2140 1144 y Fy(0)2179 1129 y FB(\))22 b(+)g FA(V)g FB(\()p FA(@)5 b FB(\000)2571 1144 y Fy(0)2611 1129 y FB(\))p FA(;)324 1349 y FB(where)34 b FA(@)5 b FB(\000)723 1364 y Fy(0)795 1349 y FB(is)33 b(the)g(set)g(of)f FA(\015)h Fz(2)28 b FA(@)5 b FB(\000)33 b(suc)m(h)h(that)e FA(V)22 b FB(\()p FA(\015)5 b FB(\))28 b Fz(2)g Fs(R)5 b FB(.)324 1590 y FF(Ac)m(kno)m(wledgemen)m(ts:)36 b FH(I)22 b(tak)m(e)i(this)e(opp)s(ortunit)m(y)e(to)k(express)e(m)m(y)g (gratitude)g(to)h(Vladimir)324 1710 y(Georgescu)34 b(for)e(suggesting)h (me)g(the)f(sub)5 b(ject)33 b(of)g(this)e(w)m(ork)i(and)f(for)g (helpful)e(discussions.)324 1831 y(I)g(am)g(also)h(indebted)d(to)k (Andrei)c(Iftimo)m(vici)h(for)i(commen)m(ts)g(and)f(suggestions.)324 2163 y FC(2)161 b(T)-13 b(rees)52 b(and)i(related)g(ob)9 b(jects)324 2382 y Fr(2.1.)34 b(The)g(free)g(mono)-12 b(\177)-44 b(\020d)33 b FB(\000)p Fr(.)42 b FB(Let)30 b Fu(A)52 b FB(b)s(e)30 b(a)f(\014nite)g(set)h(consisting)f(of)g FA(\027)36 b FB(ob)5 b(jects.)43 b(Let)324 2503 y(\000)32 b(b)s(e)g(the)g(free)g(mono)-11 b(\177)-38 b(\020d)30 b(o)m(v)m(er)j Fu(A)23 b FB(;)32 b(its)g(elemen)m(ts)g(are)g Fq(wor)-5 b(ds)31 b FB(and)h(those)g(of)g Fu(A)54 b Fq(letters)p FB(.)324 2623 y(W)-8 b(e)30 b(refer)g(to)g([Bo,)g(Chapter)h(I,)f Fz(x)p FB(7])f(for)h(a)f(detailed)g(discussion)h(of)f(these)i(notions,) f(but)324 2744 y(w)m(e)42 b(recall)e(that)h(a)g(w)m(ord)h FA(x)f FB(is)g(an)g Fu(A)23 b FB(-v)-5 b(alued)40 b(map)h(de\014ned)h (on)g(a)e(set)i(of)f(the)h(form)324 2864 y Fn(J)p FB(1)p FA(;)17 b(n)p Fn(K)36 b FB(with)h FA(n)e Fz(2)g Fs(N)52 b FB(\(w)m(e)38 b(use)f(the)g(notation)2010 2828 y Fy(1)2085 2864 y Fn(J)p FB(1)p FA(;)17 b(n)p Fn(K)35 b FB(=)g([1)p FA(;)17 b(n)p FB(])25 b Fz(\\)g Fs(N)9 b FB(\),)44 b FA(x)p FB(\()p FA(i)p FB(\))37 b(b)s(eing)f(the)324 2984 y FA(i)p FB(-th)h(letter)g(of)g(the)h(w)m(ord)g FA(x)p FB(.)58 b(The)38 b(in)m(teger)f FA(n)h FB(\(the)g(n)m(um)m(b)s(er)f(of) g(letters)h(of)f FA(x)p FB(\))g(is)g(the)324 3105 y(length)43 b(of)g(the)h(w)m(ord)g(and)g(will)d(b)s(e)j(denoted)h Fz(j)p FA(x)p Fz(j)p FB(.)76 b(There)45 b(is)e(a)g(unique)h(w)m(ord)g FA(e)g FB(of)324 3225 y(length)32 b(0,)h(its)f(domain)f(b)s(eing)h(the) h(empt)m(y)g(set.)45 b(This)33 b(is)f(the)i(neutral)e(elemen)m(t)g(of)h Fu(A)23 b FB(.)324 3346 y(W)-8 b(e)33 b(will)d(also)i(iden)m(tify)g Fu(A)55 b FB(with)32 b(the)h(set)g(of)g(w)m(ords)g(of)f(length)g(1.)470 3466 y(The)37 b(mono)-11 b(\177)-38 b(\020d)33 b(\000)j(will)d(b)s(e)j (endo)m(w)m(ed)h(with)f(the)g(discrete)g(top)s(ology)-8 b(.)50 b(If)36 b FA(x)d Fz(2)g FB(\000,)k(w)m(e)324 3586 y(denote)d FA(x)p FB(\000)h(and)f(\000)p FA(x)g FB(the)g(righ)m(t)f (and)h(left)f(ideals)g(generated)i(b)m(y)g FA(x)p FB(.)48 b(W)-8 b(e)34 b(ha)m(v)m(e)h(on)f(\000)g(a)324 3707 y(canonical)d (order)i(relation)d(whic)m(h)j(is)g(b)m(y)g(de\014nition:)1587 3927 y FA(x)28 b Fz(\024)h FA(y)h Fz(,)e FA(y)j Fz(2)d FA(x)p FB(\000)p FA(:)324 4147 y FB(W)-8 b(e)33 b(sa)m(y)h(that)e FA(x)i Fq(c)-5 b(overs)32 b FA(y)k FB(if)31 b FA(x)e Fz(\024)f FA(y)36 b FB(and)d Fz(j)p FA(y)t Fz(j)27 b FB(=)h Fz(j)p FA(x)p Fz(j)22 b FB(+)g(1.)44 b(Eac)m(h)34 b(elemen)m(t)e FA(x)d Fz(2)f FB(\000)23 b Fz(n)f(f)p FA(e)p Fz(g)324 4267 y FB(is)34 b(co)m(v)m(ered)k(b)m(y)e(a)f(unique)g (elemen)m(t)g FA(x)1736 4231 y FD(0)1760 4267 y FB(,)h(its)e Fq(father)h FB(and)h(eac)m(h)g(elemen)m(t)f FA(x)d Fz(2)h FB(\000)i(co)m(v)m(ers)324 4387 y FA(\027)40 b FB(elemen)m(ts,)35 b(its)f Fq(sons)p FB(.)47 b(Observ)m(e)36 b(that)e(the)h(set)g(of)e (sons)i(of)f FA(x)g FB(is)j Fw(e)-58 b FA(x)31 b FB(=)f Fz(f)p FA(x")g Fz(j)g FA(")g Fz(2)h Fu(A)23 b Fz(g)p FB(.)324 4508 y(W)-8 b(e)33 b(notice)f(that:)1314 4628 y FA(x)i FB(co)m(v)m(ers)h FA(y)c Fz(,)c FA(y)1952 4587 y FD(0)2002 4628 y FB(=)h FA(x)g Fz(,)f FA(y)k Fz(2)g Fw(e)-58 b FA(x:)p 324 4716 1296 4 v 436 4777 a Fm(1)473 4807 y Fl(W)-7 b(e)28 b(stress)f(that)h Fk(N)37 b Fl(is)28 b(the)g(set)f(of)h(in)n(tegers)e Fj(\025)d Fl(0)k(and)h Fk(N)2242 4777 y Fi(\003)2309 4807 y Fl(=)22 b Fk(N)29 b Fj(n)18 b(f)p Fl(0)p Fj(g)p Fl(.)1918 5251 y FB(5)p eop end %%Page: 6 6 TeXDict begin 6 5 bop 324 548 a FB(F)-8 b(or)21 b Fz(j)p FA(x)p Fz(j)27 b(\025)i FA(n)p FB(,)24 b(w)m(e)f(de\014ne)g FA(x)1300 512 y Fy(\()p Fx(n)p Fy(\))1424 548 y FB(inductiv)m(ely)f(b)m (y)h(setting)f FA(x)2405 512 y Fy(\(0\))2527 548 y FB(=)28 b FA(x)22 b FB(and)g FA(x)2942 512 y Fy(\()p Fx(m)p Fy(+1\))3182 548 y FB(=)28 b(\()p FA(x)3379 512 y Fy(\()p Fx(m)p Fy(\))3500 548 y FB(\))3538 512 y FD(0)324 668 y FB(for)k FA(m)c Fz(\024)g FA(n)22 b Fz(\000)h FB(1.)43 b(One)33 b(ma)m(y)g(notice)f (that:)1394 865 y Fz(j)p FA(x)1477 824 y Fy(\()p Fx(\013)p Fy(\))1582 865 y Fz(j)27 b FB(=)h Fz(j)p FA(x)p Fz(j)22 b(\000)g FA(\013)q FB(,)32 b(if)g FA(\013)c Fz(\024)g(j)p FA(x)p Fz(j)324 1062 y FB(and)k(for)h FA(\013)28 b Fz(\024)g(j)p FA(ab)p Fz(j)1311 1260 y FB(\()p FA(ab)p FB(\))1479 1218 y Fy(\()p Fx(\013)p Fy(\))1667 1260 y FB(=)83 b FA(ab)1918 1218 y Fy(\()p Fx(\013)p Fy(\))2023 1260 y FB(,)33 b(if)e FA(\013)d Fz(\024)h(j)p FA(b)p Fz(j)1667 1405 y FB(=)83 b FA(a)1877 1364 y Fy(\()p Fx(\013)p FD(\000j)p Fx(b)p FD(j)p Fy(\))2106 1405 y FB(,)32 b(if)g FA(\013)c Fz(\025)g(j)p FA(b)p Fz(j)p FA(:)324 1602 y FB(W)-8 b(e)27 b(remark)f(that)h(if)e FA(\027)34 b FB(=)28 b(1)e(then)i(\000)f(=)h Fs(N)42 b FB(and)26 b(if)g FA(\027)34 b(>)28 b FB(1)e(then)h(\000)g(is)f(the)h (set)h(of)e(monoms)324 1722 y(in)32 b FA(\027)39 b FB(non-comm)m (utativ)m(e)31 b(v)-5 b(ariables.)324 1886 y Fr(2.2.)48 b(The)h(tree)f FB(\000)p Fr(.)72 b FB(A)42 b(graph)f(is)h(a)g(couple)g FA(G)h FB(=)h(\()p FA(V)5 b(;)17 b(E)6 b FB(\),)44 b(where)g FA(V)63 b FB(is)42 b(a)f(set)i(\(of)324 2006 y Fq(vertic)-5 b(es)p FB(\))32 b(and)i FA(E)39 b FB(is)33 b(a)g(set)h(of)e(pairs)h(of) g(elemen)m(ts)g(of)g FA(V)55 b FB(\(the)34 b Fq(e)-5 b(dges)p FB(\).)44 b(If)33 b FA(x)h FB(and)f FA(y)k FB(are)324 2126 y(joined)29 b(b)m(y)h(an)g(edge,)h(w)m(e)g(sa)m(y)f(that)g(they)g (are)g Fq(neighb)-5 b(ours)29 b FB(and)h(w)m(e)g(abbreviate)g FA(x)e Fz(\030)g FA(y)t FB(.)324 2247 y(The)h(graph)g(structure)g(allo) m(ws)f(one)h(to)f(endo)m(w)h FA(V)50 b FB(with)29 b(a)f(canonical)f (metric)g FA(d)p FB(,)i(where)324 2367 y FA(d)p FB(\()p FA(x;)17 b(y)t FB(\))31 b(is)h(the)h(length)f(of)g(the)h(shortest)h (path)f(in)e FA(G)i FB(joining)d FA(x)j FB(to)g FA(y)t FB(.)470 2488 y(The)39 b(graph)e FA(G)1034 2503 y Fy(\000)1120 2488 y FB(asso)s(ciated)g(to)g(the)h(mono)-11 b(\177)-38 b(\020d)36 b(\000)i(is)f(de\014ned)i(as)f(follo)m(ws:)52 b FA(V)58 b FB(=)36 b(\000)324 2608 y(and)f FA(x)e Fz(\030)h FA(y)k FB(if)c FA(x)i FB(co)m(v)m(ers)i FA(y)g FB(or)d FA(y)k FB(co)m(v)m(ers)e FA(x)p FB(.)53 b(It)36 b(is)f(usual)g(to)g (iden)m(tify)g(\000)g(and)h FA(G)3315 2623 y Fy(\000)3363 2608 y FB(,)g(the)324 2728 y(so-called)31 b FA(\027)6 b FB(-fold)31 b(branc)m(hing)h(tree.)44 b(F)-8 b(or)31 b(all)g FA(x)d Fz(2)g FB(\000,)k(w)m(e)h(ha)m(v)m(e)h Fz(j)p FA(x)p Fz(j)27 b FB(=)h FA(d)p FB(\()p FA(e;)17 b(x)p FB(\).)43 b(W)-8 b(e)33 b(will)324 2849 y(set)g FA(B)5 b FB(\()p FA(x;)17 b(r)s FB(\))28 b(=)f Fz(f)p FA(y)k Fz(2)d FB(\000)f Fz(j)h FA(d)p FB(\()p FA(x;)17 b(y)t FB(\))26 b FA(<)i(r)s Fz(g)k FB(and)h FA(S)2068 2813 y Fx(n)2142 2849 y FB(=)28 b Fz(f)p FA(x)f Fz(2)i FB(\000)e Fz(j)g(j)p FA(x)p Fz(j)h FB(=)f FA(n)p Fz(g)p FB(.)324 3012 y Fr(2.3.)40 b(Extended)f(tree)h(asso)s(ciated)f(to)h Fu(A)22 b Fr(.)49 b FB(W)-8 b(e)35 b(shall)e(de\014ne)j(the)f(extended) h(tree)324 3132 y(b)m(y)g(mimic)m(king)d(the)j(de\014nition)e(of)h(the) h(free)g(mono)-11 b(\177)-38 b(\020d)34 b(o)m(v)m(er)i Fu(A)23 b FB(.)52 b(F)-8 b(or)35 b(eac)m(h)h(in)m(teger)g FA(r)s FB(,)324 3253 y(w)m(e)f(set)f Fs(Z)691 3268 y Fx(r)757 3253 y FB(=)c Fz(f)p FA(i)g Fz(2)g Fs(Z)e Fz(j)i FA(i)g Fz(\024)h FA(r)s Fz(g)p FB(.)47 b(The)35 b Fq(extende)-5 b(d)35 b(tr)-5 b(e)g(e)2358 3228 y Fw(e)2355 3253 y FB(\000)34 b(asso)s(ciated)g(to)f Fu(A)57 b FB(is)33 b(the)i(set)324 3383 y(of)30 b Fu(A)23 b FB(-v)-5 b(alued)30 b(maps)g(de\014ned)i(on)f (sets)h(of)e(the)h(form)e Fs(Z)2355 3398 y Fx(r)2391 3383 y FB(.)42 b(Hence)32 b FA(x)d Fz(2)2928 3358 y Fw(e)2926 3383 y FB(\000)h(if)g(and)h(only)f(if)324 3503 y(there)j(is)e FA(r)g Fz(2)d Fs(Z)h FB(suc)m(h)34 b(that)e FA(x)h FB(is)e(a)h(map)g Fs(Z)1920 3518 y Fx(r)1983 3503 y Fz(!)27 b Fu(A)c FB(;)32 b(this)g(uniquely)g(de\014ned)i(in)m(teger)e FA(r)324 3624 y FB(will)e(b)s(e)j(denoted)g Fz(j)p FA(x)p Fz(j)g FB(and)f(will)f(b)s(e)h(called)g Fq(length)g FB(of)g FA(x)p FB(.)470 3754 y(In)i(order)g(to)f(em)m(b)s(ed)h(\000)g(in)m(to) 1576 3729 y Fw(e)1573 3754 y FB(\000,)g(w)m(e)g(c)m(ho)s(ose)h FA(o)29 b Fz(2)h Fu(A)23 b FB(;)34 b(this)g(elemen)m(t)f(will)f(b)s(e)h (\014xed)324 3875 y(from)25 b(no)m(w)j(on.)42 b(W)-8 b(e)27 b(set)h(\000)1288 3890 y Fy(0)1355 3875 y FB(=)f Fz(f)p FA(x)h Fz(j)f(j)p FA(x)p Fz(j)h(\025)g FB(0)k(and)h FA(x)p FB(\()p FA(i)p FB(\))28 b(=)g FA(o)k FB(if)g FA(i)c Fz(\024)g FB(0)p Fz(g)e FB(and)h(w)m(e)h(iden)m(tify)324 3995 y(\000)34 b(with)g(\000)704 4010 y Fy(0)744 3995 y FB(.)49 b(More)34 b(precisely)-8 b(,)36 b(if)d FA(x)e Fz(2)g FB(\000)k(then)g(the)g(corresp)s(onding)f(elemen)m(t)g(of)g (\000)3422 4010 y Fy(0)3496 3995 y FB(is)324 4115 y(de\014ned)h(on)e Fs(Z)866 4131 y FD(j)p Fx(x)p FD(j)980 4115 y FB(b)m(y)h(extending)g FA(x)g FB(with)f FA(x)p FB(\()p FA(i)p FB(\))c(=)g FA(o)k FB(if)g FA(i)c Fz(\024)h FB(0.)45 b(The)35 b(elemen)m(t)e FA(e)g FB(will)e(b)s(e)324 4253 y(iden)m(ti\014ed)h(with)h(the)g(map)f FA(e)c Fz(2)1527 4228 y Fw(e)1524 4253 y FB(\000)33 b(suc)m(h)h(that)f Fz(j)p FA(e)p Fz(j)27 b FB(=)h(0)33 b(and)f FA(e)p FB(\()p FA(i)p FB(\))d(=)f FA(o)p FB(,)k Fz(8)p FA(i)d Fz(\024)f FB(0.)44 b(Notice)324 4373 y(that)32 b(the)h(t)m(w)m(o)g(notions)f(of)g (length)h(are)f(consisten)m(t)i(on)e(\000.)470 4494 y(There)h(is)e(a)g (natural)f(righ)m(t)h(action)g(of)g(\000)g(on)2134 4468 y Fw(e)2132 4494 y FB(\000)g(b)m(y)h(concatenation,)g(i.e.)43 b(for)30 b FA(x)f Fz(2)f FB(\000)324 4614 y(and)36 b FA(y)g Fz(2)704 4589 y Fw(e)701 4614 y FB(\000,)h FA(y)t(x)e FB(will)f(b)s(e)i(the)g(function)g FA(z)k FB(de\014ned)e(on)e Fs(Z)2482 4629 y FD(j)p Fx(y)r FD(j)p Fy(+)p FD(j)p Fx(x)p FD(j)2729 4614 y FB(suc)m(h)i(that)d FA(z)t Fz(j)3244 4629 y Ft(Z)3295 4643 y Fv(j)p Fo(y)q Fv(j)3404 4614 y FB(and)324 4761 y FA(z)t FB(\()p Fz(j)p FA(y)t Fz(j)23 b FB(+)i FA(i)p FB(\))34 b(=)f FA(x)p FB(\()p FA(i)p FB(\))k(for)e FA(i)f FB(=)f(1)p FA(;)17 b(:)g(:)g(:)f(;)h Fz(j)p FA(x)p Fz(j)p FB(.)53 b(Then)37 b(w)m(e)g(equip)2520 4736 y Fw(e)2517 4761 y FB(\000)f(with)f(an)h(order)g(relation)324 4882 y(b)m(y)d(setting:)1587 5002 y FA(x)28 b Fz(\024)h FA(y)h Fz(,)e FA(y)j Fz(2)d FA(x)p FB(\000)p FA(:)1918 5251 y FB(6)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 324 548 a FB(Clearly)29 b FA(x)i FB(co)m(v)m(ers)h FA(y)i FB(if)29 b(and)i(only)e(if)h FA(x)e Fz(\024)g FA(y)33 b FB(and)e Fz(j)p FA(y)t Fz(j)26 b FB(=)i Fz(j)p FA(x)p Fz(j)17 b FB(+)g(1.)43 b(Eac)m(h)31 b(elemen)m(t)f FA(x)f Fz(2)3503 523 y Fw(e)3501 548 y FB(\000)324 668 y(is)k(co)m(v)m(ered)i(b)m(y)g(a)e(unique)h(elemen)m(t) f FA(x)1726 632 y FD(0)1750 668 y FB(,)h(its)f Fq(father)p FB(,)g(and)h(eac)m(h)h(elemen)m(t)e FA(x)d Fz(2)f FB(\000)34 b(co)m(v)m(ers)324 789 y FA(\027)h FB(elemen)m(ts,)c(its)d Fq(sons)p FB(,)i(namely)e(the)i(elemen)m(ts)f(of)j Fw(e)-58 b FA(x)28 b FB(=)g Fz(f)p FA(x")f Fz(j)g FA(")h Fz(2)g Fu(A)23 b Fz(g)p FB(.)42 b(W)-8 b(e)29 b(still)e(ha)m(v)m(e)324 909 y(that:)1314 1029 y FA(x)34 b FB(co)m(v)m(ers)h FA(y)c Fz(,)c FA(y)1952 988 y FD(0)2002 1029 y FB(=)h FA(x)g Fz(,)f FA(y)k Fz(2)g Fw(e)-58 b FA(x:)324 1196 y FB(Observ)m(e)34 b(that)f FA(x)965 1160 y FD(0)1016 1196 y FB(=)28 b FA(x)p Fz(j)1203 1211 y Ft(Z)1253 1225 y Fv(j)p Fo(x)p Fv(j\000)p Fh(1)1406 1196 y FB(.)44 b(W)-8 b(e)33 b(will)d(set)j FA(x)2036 1160 y Fy(\()p Fx(\013)p Fy(\))2169 1196 y FB(=)27 b FA(x)p Fz(j)2355 1211 y Ft(Z)2406 1225 y Fv(j)p Fo(x)o Fv(j\000)p Fo(\013)2569 1196 y FB(.)470 1344 y(Let)41 b FA(G)732 1358 y Fp(e)730 1374 y Fy(\000)819 1344 y FB(b)s(e)h(the)f(graph)g(suc)m(h)h(that)1873 1318 y Fw(e)1870 1344 y FB(\000)f(is)f(the)i(set)g(of)e(v)m(ertices)i(and)f(there)h(is)f (an)324 1480 y(edge)d(b)s(et)m(w)m(een)i FA(x;)34 b(y)40 b Fz(2)1243 1454 y Fw(e)1240 1480 y FB(\000)e(if)e FA(x)j FB(co)m(v)m(ers)g FA(y)i FB(or)d FA(y)j FB(co)m(v)m(ers)f FA(x)p FB(.)60 b(As)38 b(b)s(efore,)h(w)m(e)g(shall)e(not)324 1600 y(distinguish)31 b(b)s(et)m(w)m(een)k FA(G)1274 1614 y Fp(e)1272 1630 y Fy(\000)1352 1600 y FB(and)1545 1575 y Fw(e)1542 1600 y FB(\000.)324 1763 y Fr(2.4.)42 b(The)f(b)s(oundary)j(at)d(in\014nit)m(y)f(of)i FB(\000)p Fr(.)54 b FB(W)-8 b(e)37 b(shall)e(see)i(in)e(the)i(ending)f(remark)324 1884 y(of)g(this)g(subsection)h(that)f(the)h(b)s(oundary)g(at)f (in\014nit)m(y)g(of)g(\000)g(can)g(b)s(e)h(though)m(t)g(as)f(the)324 2004 y(b)s(oundary)23 b(of)f(a)g(0-h)m(yp)s(erb)s(olic)g(space)h(in)f (the)h(sense)i(of)d(Gromo)m(v.)39 b(W)-8 b(e)23 b(prefer,)i(ho)m(w)m (ev)m(er,)324 2124 y(to)32 b(giv)m(e)h(a)g(simpler)e(presen)m(tation)j (that)e(is)h(closer)g(to)f(the)i(theory)f(of)g FA(p)p FB(-adic)f(n)m(um)m(b)s(ers)324 2245 y(\(see)i([Ro])f(for)g (instance\).)46 b(In)34 b(fact,)g(if)e FA(\027)40 b FB(is)33 b(prime)f(the)i(b)s(oundary)f(will)f(b)s(e)h(the)h(set)g(of)324 2365 y FA(\027)6 b FB(-adic)32 b(in)m(tegers.)324 2554 y Fr(De\014nition)k(2.1)49 b Fq(The)27 b(b)-5 b(oundary)27 b(at)h(in\014nity)f(of)g FB(\000)h Fq(is)f(the)g(set)h FA(@)5 b FB(\000)29 b(=)e Fz(f)p FA(x)h FB(:)g Fs(N)3179 2517 y FD(\003)3252 2554 y Fz(!)f Fu(A)c Fz(g)p Fq(.)324 2674 y(F)-7 b(or)34 b FA(x)28 b Fz(2)g FA(@)5 b FB(\000)p Fq(,)36 b(we)e(set)h Fz(j)p FA(x)p Fz(j)27 b FB(=)h Fz(1)p Fq(.)324 2872 y FB(Let)508 2847 y Fw(b)506 2872 y FB(\000)39 b(b)s(e)h(\000)27 b Fz([)g FA(@)5 b FB(\000.)66 b(F)-8 b(or)39 b FA(x)h Fz(2)1523 2847 y Fw(b)1520 2872 y FB(\000,)i(w)m(e)e (de\014ne)h(the)f(sequence)j(\()p FA(x)2769 2887 y Fx(n)2816 2872 y FB(\))2854 2888 y Fx(n)p FD(\024j)p Fx(x)p FD(j)3074 2872 y FB(with)d(v)-5 b(alues)324 2993 y(in)39 b(\000)h(b)m(y)g (setting)g FA(x)1071 3008 y Fy(0)1151 2993 y FB(=)g FA(e)g FB(and)g FA(x)1604 3008 y Fx(n)1691 2993 y FB(=)g FA(x)p Fz(j)1890 3008 y Fg(J)p Fy(1)p Fx(;n)p Fg(K)2089 2993 y FB(for)f FA(n)h Fz(\025)h FB(1.)65 b(Observ)m(e)41 b(that)f(the)g(map)324 3113 y FA(x)28 b Fz(7!)f FB(\()p FA(x)627 3128 y Fx(n)675 3113 y FB(\))713 3129 y Fx(n)p FD(\024j)p Fx(x)p FD(j)926 3113 y FB(is)32 b(injectiv)m(e.)470 3251 y(There)i(is)e(a)h(natural)e(left)h(action)g(of)g(\000)g(on)2080 3225 y Fw(b)2077 3251 y FB(\000.)43 b(F)-8 b(or)32 b FA(x)d Fz(2)f FB(\000)k(and)h FA(y)e Fz(2)3020 3225 y Fw(b)3017 3251 y FB(\000,)i FA(xy)j FB(will)30 b(b)s(e)324 3371 y(de\014ned)37 b(on)f Fn(J)p FB(1)p FA(;)17 b Fz(j)p FA(x)p Fz(j)24 b FB(+)h Fz(j)p FA(y)t Fz(j)p Fn(K)34 b FB(b)m(y)j FA(x)p FB(\()p FA(i)p FB(\))g(for)f FA(i)e Fz(\024)g(j)p FA(x)p Fz(j)i FB(and)g(b)m(y)h FA(y)t FB(\()p FA(i)24 b Fz(\000)h(j)p FA(x)p Fz(j)p FB(\))36 b(for)f FA(i)f(>)g Fz(j)p FA(x)p Fz(j)i FB(\(w)m(e)324 3491 y(use)d(the)g(con)m (v)m(en)m(tion)h Fn(J)p FB(1)p FA(;)17 b Fz(1)p Fn(K)27 b FB(=)h(\([1)p FA(;)17 b Fz(1)p FB([)p Fz(\\)p Fs(N)8 b FB(\))28 b Fz([)23 b(f1g)p FB(\).)470 3622 y(W)-8 b(e)36 b(will)d(no)m(w)j(equip)1302 3597 y Fw(b)1299 3622 y FB(\000)f(with)g(a)g(structure)i(of)e(ultrametric)e(space.)53 b(W)-8 b(e)35 b(de\014ne)i(a)324 3742 y(kind)32 b(of)g(v)-5 b(aluation)31 b FA(v)36 b FB(on)1301 3717 y Fw(b)1299 3742 y FB(\000)22 b Fz(\002)1484 3717 y Fw(b)1481 3742 y FB(\000)33 b(b)m(y)324 3945 y(\(2.2\))441 b FA(v)t FB(\()p FA(x;)17 b(y)t FB(\))27 b(=)h(max)o Fz(f)p FA(n)g Fz(2)g Fn(J)p FB(0)p FA(;)17 b FB(min)n(\()p Fz(j)p FA(x)p Fz(j)p FA(;)g Fz(j)p FA(y)t Fz(j)p FB(\))p Fn(K)26 b Fz(j)h FA(x)2596 3960 y Fx(n)2671 3945 y FB(=)h FA(y)2823 3960 y Fx(n)2869 3945 y Fz(g)324 4157 y FB(with)k(the)h(con)m(v)m(en)m (tion)h(that)e FA(v)t FB(\()p FA(x;)17 b(x)p FB(\))28 b(=)f Fz(1)p FB(.)44 b(If)32 b FA(x;)17 b(y)t(;)g(z)31 b Fz(2)2461 4132 y Fw(b)2459 4157 y FB(\000)h(it)g(is)g(easy)h(to)g (see)g(that:)324 4360 y(\(2.3\))785 b FA(v)t FB(\()p FA(x;)17 b(y)t FB(\))26 b Fz(\025)j FB(min)n(\()p FA(v)t FB(\()p FA(x;)17 b(z)t FB(\))p FA(;)g(v)t FB(\()p FA(z)t(;)g(y)t FB(\)\))p FA(:)324 4572 y FB(Let)33 b(us)g(set)g(on)914 4547 y Fw(b)912 4572 y FB(\000)o(:)1449 4666 y Fw(b)1436 4693 y FA(d)o FB(\()p FA(x;)17 b(y)t FB(\))27 b(=)h(exp)q(\()p Fz(\000)p FA(v)t FB(\()p FA(x;)17 b(y)t FB(\)\))p FA(:)324 4871 y FB(The)39 b(relation)d(\(2.3\))i(clearly)f(implies)f(that)i(\() 2044 4845 y Fw(b)2041 4871 y FB(\000)p FA(;)2160 4844 y Fw(b)2146 4871 y FA(d)p FB(\))g(is)g(an)g(ultrametric)e(space,)41 b(i.e.)60 b(a)324 5002 y(metric)26 b(space)i(suc)m(h)g(that)1311 4976 y Fw(b)1297 5002 y FA(d)o FB(\()p FA(x;)17 b(y)t FB(\))27 b Fz(\024)h FB(max\()1940 4976 y Fw(b)1926 5002 y FA(d)p FB(\()p FA(x;)17 b(z)t FB(\))p FA(;)2259 4976 y Fw(b)2245 5002 y FA(d)p FB(\()p FA(z)t(;)g(y)t FB(\)\),)27 b(for)g FA(x;)17 b(y)t(;)g(z)31 b Fz(2)3121 4977 y Fw(b)3118 5002 y FB(\000.)42 b(W)-8 b(e)27 b(will)1918 5251 y(7)p eop end %%Page: 8 8 TeXDict begin 8 7 bop 324 548 a FB(denote)28 b(b)m(y)784 523 y Fw(b)764 548 y FA(B)5 b FB(\()p FA(x;)17 b(r)s FB(\))28 b(=)f Fz(f)p FA(y)k Fz(2)1422 523 y Fw(b)1419 548 y FB(\000)d Fz(j)1577 522 y Fw(b)1563 548 y FA(d)p FB(\()p FA(x;)17 b(y)t FB(\))26 b FA(<)i(r)s Fz(g)p FB(.)41 b(Notice)28 b(that)f(ultrametricit)m(y)f(implies)324 678 y(that)555 653 y Fw(b)535 678 y FA(B)5 b FB(\()p FA(x;)17 b(r)s FB(\))32 b(is)h(closed)f(for)g(all)f FA(x)d Fz(2)1719 653 y Fw(b)1716 678 y FB(\000)33 b(and)f FA(r)f(>)c FB(0.)470 809 y(The)32 b(top)s(ology)e(induced)i(b)m(y)1568 783 y Fw(b)1565 809 y FB(\000)g(on)f(\000)g(coincides)g(with)g(the)h (initial)c(top)s(ology)i(of)h(\000,)324 929 y(the)i(discrete)g(one.)44 b(F)-8 b(or)32 b FA(x)c Fz(2)g FA(@)5 b FB(\000)33 b(and)g FA(n)28 b Fz(2)g Fs(N)9 b FB(,)38 b(w)m(e)c(ha)m(v)m(e)g(that)840 1142 y FA(x)895 1157 y Fx(n)946 1116 y Fw(b)943 1142 y FB(\000)83 b(=)g Fz(f)p FA(y)30 b Fz(2)1471 1116 y Fw(b)1468 1142 y FB(\000)e Fz(j)f FA(v)t FB(\()p FA(x;)17 b(y)t FB(\))27 b Fz(\025)h FA(n)p Fz(g)g FB(=)2281 1116 y Fw(b)2261 1142 y FA(B)5 b FB(\()p FA(x;)17 b FB(exp)r(\()p Fz(\000)p FA(n)22 b FB(+)g(1\)\))324 1364 y(is)35 b(the)h(closure)g(of) f FA(x)1094 1379 y Fx(n)1142 1364 y FB(\000)g(in)1358 1339 y Fw(b)1355 1364 y FB(\000.)53 b(Hence)37 b Fz(f)p FA(x)1894 1379 y Fx(n)1944 1339 y Fw(b)1941 1364 y FB(\000)p Fz(g)2052 1379 y Fx(n)p FD(2)p Ft(N)2230 1364 y FB(is)e(a)g(basis)h(of) f(neigh)m(b)s(ourho)s(o)s(ds)g(in)327 1469 y Fw(b)324 1494 y FB(\000)j(of)f FA(x)h FB(for)g FA(x)f Fz(2)g FA(@)5 b FB(\000.)61 b(Observ)m(e)40 b(that)e FA(x@)5 b FB(\000)38 b(=)e FA(x)2164 1469 y Fw(b)2161 1494 y FB(\000)27 b Fz(\\)f FA(@)5 b FB(\000)q(,)39 b(for)e FA(x)h Fz(2)f FB(\000.)60 b(W)-8 b(e)38 b(ha)m(v)m(e)h(the)324 1615 y(follo)m(wing)30 b(prop)s(ert)m(y)-8 b(.)324 1812 y Fr(Prop)s(osition)35 b(2.2)49 b FB(\000)35 b Fq(and)f FA(@)5 b FB(\000)36 b Fq(ar)-5 b(e)35 b(c)-5 b(omp)g(act)34 b(sp)-5 b(ac)g(es.)324 2009 y Fr(Pro)s(of:)81 b FB(W)-8 b(e)46 b(ha)m(v)m(e)i FA(@)5 b FB(\000)51 b(=)g Fu(A)1526 1973 y Ft(N)1570 1949 y Fv(\003)1614 2009 y FB(,)e(so)d FA(@)5 b FB(\000)47 b(endo)m(w)m(ed)i(with)c(the)i(pro)s(duct)f(top)s (ology)324 2129 y(is)d(compact.)78 b(In)44 b(fact,)j(this)d(top)s (ology)e(coincides)i(with)g(the)g(one)h(induced)f(b)m(y)h(the)324 2250 y(restriction)31 b(of)915 2223 y Fw(b)901 2250 y FA(d)h FB(on)g FA(@)5 b FB(\000.)45 b(Indeed)34 b(for)d FA(x)e Fz(2)f FA(@)5 b FB(\000,)33 b(the)g(pro)s(duct)f(top)s(ology)f (giv)m(es)i(us)g(the)324 2370 y(same)f(basis)h(of)f(neigh)m(b)s(ourho)s (o)s(ds)g Fz(f)p FA(x)1729 2385 y Fx(n)1776 2370 y FA(@)5 b FB(\000)p Fz(g)1943 2385 y Fx(n)p FD(2)p Ft(N)2118 2370 y FB(as)2252 2344 y Fw(b)2238 2370 y FA(d)p Fz(j)2317 2385 y Fx(@)t Fy(\000)2406 2370 y FB(.)470 2500 y(It)40 b(remains)f(to)g(sho)m(w)i(that)1551 2475 y Fw(b)1548 2500 y FB(\000)e(is)h(compact.)65 b(In)40 b(fact,)h(since)f FA(@)5 b FB(\000)41 b(is)e(compact,)j(it)324 2621 y(su\016ces)34 b(to)e(remark)g(that)h Fz([)1385 2636 y Fx(x)p FD(2)p Fx(@)t Fy(\000)1581 2596 y Fw(b)1561 2621 y FA(B)5 b FB(\()p FA(x;)17 b FB(exp)q(\()p Fz(\000)p FA(k)s FB(\)\))28 b(=)g Fz(f)p FA(y)2407 2596 y Fw(b)2405 2621 y FB(\000)f Fz(j)g(j)p FA(y)t Fz(j)f FB(=)i FA(k)c FB(+)e(1)p Fz(g)32 b FB(has)h(a)f(\014nite)324 2751 y(complemen)m(tary)g(in)1123 2726 y Fw(b)1120 2751 y FB(\000,)h(for)f(all)f FA(k)f Fz(2)e Fs(N)9 b FB(.)50 b(This)33 b(ends)g(our)g(pro)s(of.)140 b Ff(\003)470 2871 y FB(Since)35 b(\000)f(is)g(dense)h(in)1308 2846 y Fw(b)1305 2871 y FB(\000,)1431 2846 y Fw(b)1428 2871 y FB(\000)f(is)g(a)g(compacti\014cation)e(of)i(\000.)49 b(One)34 b(ma)m(y)h(also)e(notice)324 2992 y(that)f FA(@)5 b FB(\000)34 b(is)e(a)g(p)s(erfect)h(top)s(ological)c(space)34 b(if)d FA(\027)j(>)28 b FB(1.)470 3112 y(The)34 b FA(C)748 3076 y FD(\003)787 3112 y FB(-algebra)d FA(C)7 b FB(\()1279 3087 y Fw(b)1276 3112 y FB(\000\))32 b(of)g(con)m(tin)m(uous)i (complex-v)-5 b(alued)31 b(functions)h(on)3254 3087 y Fw(b)3251 3112 y FB(\000)g(pla)m(ys)324 3233 y(an)41 b(imp)s(ortan)m(t)f(r^)-49 b(ole)41 b(in)g(our)h(dev)m(elopmen)m(ts.)72 b(Notice)41 b(that)h(the)g(dense)h(em)m(b)s(edding)324 3353 y(\000)27 b Fz(\032)520 3328 y Fw(b)518 3353 y FB(\000)k(giv)m(es) h(a)f(canonical)f(inclusion)g FA(C)7 b FB(\()1879 3328 y Fw(b)1876 3353 y FB(\000)o(\))28 b Fz(\032)g FA(C)2177 3368 y Fx(b)2211 3353 y FB(\(\000\),)k(where)g FA(C)2757 3368 y Fx(b)2792 3353 y FB(\(\000\))f(is)g(the)h(space)g(of)324 3473 y(b)s(ounded)h(complex-v)-5 b(alued)31 b(functions)i(on)f(\000.)44 b(Moreo)m(v)m(er,)34 b(w)m(e)g(will)c(ha)m(v)m(e)324 3686 y(\(2.4\))749 b FA(C)1344 3701 y Fy(0)1383 3686 y FB(\(\000\))28 b(=)f Fz(f)p FA(f)39 b Fz(2)28 b FA(C)7 b FB(\()2000 3661 y Fw(b)1997 3686 y FB(\000\))27 b Fz(j)h FA(f)11 b Fz(j)2266 3701 y Fx(@)t Fy(\000)2382 3686 y FB(=)28 b(0)p Fz(g)p FA(;)324 3898 y FB(where)590 4111 y FA(C)660 4126 y Fy(0)700 4111 y FB(\(\000\))f(=)h Fz(f)p FA(f)38 b FB(:)28 b(\000)f Fz(!)h Fs(C)53 b Fz(j)28 b(8)p FA(")f(>)h FB(0)p FA(;)33 b Fz(9)p FA(M)39 b(>)27 b FB(0)h Fz(j)f(j)p FA(x)p Fz(j)h FA(>)f(M)39 b Fz(\))27 b(j)p FA(f)11 b FB(\()p FA(x)p FB(\))p Fz(j)27 b FA(<)g(")p Fz(g)p FA(:)324 4324 y FB(W)-8 b(e)33 b(shall)e(often)h(abbreviate)h FA(C)1516 4339 y Fy(0)1555 4324 y FB(\(\000\))g(b)m(y)g FA(C)1930 4339 y Fy(0)1969 4324 y FB(.)470 4444 y(The)26 b(follo)m(wing)c(prop)s(osition)h(giv)m(es)j(us)f(a)g(b)s(etter)g (understanding)h(of)e(the)i(functions)324 4564 y(in)32 b FA(C)7 b FB(\()556 4539 y Fw(b)553 4564 y FB(\000)o(\).)324 4761 y Fr(Prop)s(osition)35 b(2.3)49 b Fq(L)-5 b(et)43 b FA(E)49 b Fq(b)-5 b(e)42 b(a)h(metrisable)e(top)-5 b(olo)g(gic)g(al)42 b(sp)-5 b(ac)g(e.)67 b(A)43 b(function)f FA(V)64 b FB(:)324 4882 y(\000)33 b Fz(!)f FA(E)44 b Fq(extends)37 b(to)h(a)g(c)-5 b(ontinuous)37 b(function)2114 4857 y Fw(b)2103 4882 y FA(V)54 b FB(:)2277 4857 y Fw(b)2275 4882 y FB(\000)32 b Fz(!)h FA(E)44 b Fq(if)37 b(and)h(only)f(if)h(for)f (e)-5 b(ach)324 5002 y FA(x)28 b Fz(2)g FA(@)5 b FB(\000)36 b Fq(the)f(limit)f(of)h FA(V)21 b FB(\()p FA(y)t FB(\))p Fq(,)34 b(when)g FA(y)d Fz(2)d FB(\000)35 b Fq(c)-5 b(onver)g(ges)34 b(to)h FA(x)p Fq(,)g(exists.)1918 5251 y FB(8)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 324 548 a Fr(Pro)s(of:)56 b FB(Let)35 b FA(x)e Fz(2)g FA(@)5 b FB(\000)36 b(and)1404 523 y Fw(b)1392 548 y FA(V)22 b FB(\()p FA(x)p FB(\))35 b(b)s(e)h(the)g(ab)s (o)m(v)m(e)g(limit.)48 b(Let)36 b FA(F)48 b FB(b)s(e)36 b(a)f(closed)h(neigh-)324 678 y(b)s(ourho)s(o)s(d)i(of)902 653 y Fw(b)891 678 y FA(V)21 b FB(\()p FA(x)p FB(\))40 b(in)f FA(E)6 b FB(;)43 b(there)d(is)f FA(k)k FB(suc)m(h)e(that)e FA(V)22 b FB(\()p FA(x)2481 693 y Fx(k)2524 678 y FB(\000\))39 b Fz(\032)h FA(F)14 b FB(.)65 b(Then)40 b FA(x)3264 693 y Fx(k)3310 653 y Fw(b)3307 678 y FB(\000)g(is)f(a)324 809 y(neigh)m(b)s(ourho)s(o)s(d)31 b(of)h FA(x)h FB(in)1306 783 y Fw(b)1303 809 y FB(\000)g(and,)f(since)h FA(F)47 b FB(is)32 b(closed,)h(w)m(e)g(ha)m(v)m(e)2755 783 y Fw(b)2743 809 y FA(V)22 b FB(\()p FA(x)2915 824 y Fx(k)2961 783 y Fw(b)2958 809 y FB(\000\))27 b Fz(\032)h FA(F)14 b FB(.)141 b Ff(\003)470 979 y FB(Later)39 b(on,)i(w)m(e)f(will)d(need) j(the)f(next)h(ultrametricit)m(y)d(result.)63 b(W)-8 b(e)39 b(will)e(sa)m(y)j(that)324 1108 y Fu(U)53 b FB(=)27 b Fz(f)p FA(x)665 1123 y Fx(i)694 1108 y FB(\000)p Fz(g)32 b FB(is)g(a)h Fq(c)-5 b(overing)33 b(of)i FA(@)5 b FB(\000)33 b(if)1783 1081 y Fw(c)1756 1108 y Fu(U)53 b FB(=)27 b Fz(f)p FA(x)2097 1123 y Fx(i)2129 1083 y Fw(b)2126 1108 y FB(\000)p Fz(g)32 b FB(is)g(a)g(co)m(v)m(ering)h(of)g FA(@)5 b FB(\000.)324 1311 y Fr(Prop)s(osition)35 b(2.4)49 b Fq(F)-7 b(or)37 b(e)-5 b(ach)38 b(op)-5 b(en)37 b(c)-5 b(overing)37 b Fz(f)p Fu(O)2273 1326 y Fx(i)2301 1311 y Fz(g)2351 1326 y Fx(i)p FD(2)p Fx(I)2499 1311 y Fq(of)h FA(@)5 b FB(\000)p Fq(,)40 b(ther)-5 b(e)38 b(is)f(a)h(disjoint)324 1432 y(and)i(\014nite)g(c)-5 b(overing)39 b Fz(f)p FA(x)1271 1447 y Fx(j)1308 1432 y FB(\000)p Fz(g)1419 1447 y Fx(j)t FD(2)p Fx(J)1588 1432 y Fq(of)h FA(@)5 b FB(\000)41 b Fq(such)f(that)h(for)f(e)-5 b(ach)40 b FA(j)k Fz(2)39 b FA(J)49 b Fq(ther)-5 b(e)41 b(is)f FA(i)e Fz(2)h FA(I)324 1565 y Fq(such)34 b(that)i FA(x)800 1580 y Fx(j)839 1540 y Fw(b)837 1565 y FB(\000)27 b Fz(\032)h Fu(O)1104 1580 y Fx(i)1132 1565 y Fq(.)324 1769 y Fr(Pro)s(of:)62 b FB(F)-8 b(or)37 b(eac)m(h)h FA(x)f Fz(2)h FA(@)5 b FB(\000)38 b(there)h(is)e FA(i)h FB(suc)m(h)h(that)f FA(x)g FB(b)s(elongs)g(to)f (the)h(op)s(en)g(set)h Fu(O)3534 1784 y Fx(i)324 1889 y FB(and)d(there)h(is)e FA(n)f FB(=)f FA(n)p FB(\()p FA(x;)17 b(i)p FB(\))37 b(suc)m(h)g(that)f FA(x)1868 1904 y Fx(n)1918 1864 y Fw(b)1915 1889 y FB(\000)e Fz(\032)g Fu(O)2195 1904 y Fx(i)2222 1889 y FB(.)54 b(F)-8 b(rom)35 b(the)h(compactness)h(of)f FA(@)5 b FB(\000,)324 2019 y(w)m(e)39 b(obtain)d(a)i(\014nite)f(co)m(v)m(ering)h(of)f FA(@)5 b FB(\000)39 b(b)m(y)g(sets)g(of)e(the)h(form)e Fz(f)p FA(y)2740 2034 y Fx(j)2779 1994 y Fw(b)2776 2019 y FB(\000)p Fz(g)2887 2034 y Fx(j)t Fy(=1)p Fx(;:::)n(;m)3212 2019 y FB(whic)m(h)i(is)324 2140 y(a)c(sub-co)m(v)m(er)j(of)d Fz(f)p Fu(O)1078 2155 y Fx(i)1105 2140 y Fz(g)1155 2155 y Fx(i)p FD(2)p Fx(I)1266 2140 y FB(.)50 b(No)m(w)35 b(since)g(in)f(ultrametric)e(spaces)37 b(t)m(w)m(o)e(balls)e(are)i (either)324 2270 y(disjoin)m(t)25 b(or)h(one)h(of)f(them)g(is)g (included)g(in)g(the)h(other)g(one,)h(and)e(since)h Fz(f)p FA(y)3009 2285 y Fx(j)3048 2245 y Fw(b)3045 2270 y FB(\000)p Fz(g)f FB(are)h(balls,)324 2403 y(w)m(e)38 b(get)f(the)h(result.)57 b(One)38 b(ma)m(y)f(also)f(c)m(ho)s(ose)i Fz(f)p FA(y)2180 2378 y Fw(b)2178 2403 y FB(\000)c Fz(j)i(j)p FA(y)t Fz(j)e FB(=)h(max)2772 2418 y Fx(j)t Fy(=1)p Fx(;:::)n(;m)3075 2403 y Fz(j)p FA(y)3151 2418 y Fx(j)3187 2403 y Fz(jg)i FB(as)g(the)324 2524 y(required)c(co)m(v)m(ering.)98 b Ff(\003)324 2764 y Fr(Remark:)73 b FB(This)43 b(section)f(can)h(b)s (e)g(reread)h(from)d(the)i(p)s(ersp)s(ectiv)m(e)h(of)f(h)m(yp)s(erb)s (olic-)324 2885 y(it)m(y)33 b(in)g(the)h(sense)h(of)e(Gromo)m(v,)g(see) i([An,)f(Chapter)h(V])e(\(a)g(deep)s(er)i(in)m(v)m(estigation)d(can)324 3005 y(b)s(e)k(found)g(in)g([CDP])g(and)h([GH)o(]\).)55 b(Let)36 b(\()p FA(M)5 b(;)17 b(d)p FB(\))36 b(b)s(e)g(a)g(metric)f (space.)55 b(F)-8 b(or)36 b FA(x;)17 b(y)37 b Fz(2)d FA(M)324 3126 y FB(and)e(a)h(giv)m(en)f FA(O)e Fz(2)e FA(M)10 b FB(,)34 b(w)m(e)g(de\014ne)f(the)g Fq(Gr)-5 b(omov)35 b(pr)-5 b(o)g(duct)32 b FB(as)h(b)s(eing:)324 3384 y(\(2.5\))563 b(\()p FA(x;)17 b(y)t FB(\))1315 3399 y Fx(O)1402 3384 y FB(=)1515 3317 y(1)p 1515 3362 49 4 v 1515 3453 a(2)1574 3384 y(\()p FA(d)p FB(\()p FA(O)s(;)g(x)p FB(\))k(+)h FA(d)p FB(\()p FA(O)s(;)17 b(y)t FB(\))j Fz(\000)i FA(d)p FB(\()p FA(x;)17 b(y)t FB(\)\))p FA(:)324 3635 y FB(The)33 b(metric)e(space)j(\()p FA(M)5 b(;)17 b(d)p FB(\))32 b(is)g(said)g(to)g(b)s(e)g FA(\016)t Fq(-hyp)-5 b(erb)g(olic)32 b FB(if)f(there)i(is)f FA(\016)k FB(suc)m(h)e(that,)e (for)324 3756 y(all)e FA(x;)17 b(y)t(;)g(z)t(;)g(O)30 b Fz(2)e FA(M)10 b FB(,)324 3976 y(\(2.6\))690 b(\()p FA(x;)17 b(y)t FB(\))1442 3991 y Fx(O)1528 3976 y Fz(\025)28 b FB(min)o(\(\()p FA(x;)17 b(z)t FB(\))2058 3991 y Fx(O)2118 3976 y FA(;)g FB(\()p FA(z)t(;)g(y)t FB(\))2383 3991 y Fx(O)2442 3976 y FB(\))22 b Fz(\000)h FA(\016)n(:)324 4196 y FB(A)45 b(metric)f(space)i(is)e Fq(hyp)-5 b(erb)g(olic)44 b FB(if)g(it)g(is)g FA(\016)t FB(-h)m(yp)s(erb)s(olic)g(for)g(a)h (certain)f FA(\016)t FB(.)80 b(In)46 b(fact,)324 4316 y(if)c(there)h(is)f FA(\016)47 b FB(suc)m(h)e(that)d(\(2.6\))h(holds)f (for)g(all)f FA(x;)17 b(y)t(;)g(z)49 b Fz(2)d FA(M)53 b FB(and)43 b(a)g(giv)m(en)g FA(O)i FB(then)324 4437 y(\()p FA(M)5 b(;)17 b(d)p FB(\))36 b(is)f(2)p FA(\016)t FB(-h)m(yp)s(erb)s(olic.)53 b(Classical)35 b(examples)h(of)g(0-h)m(yp)s (erb)s(olic)f(spaces)i(are)g(trees)324 4557 y(\(connected)j(graphs)f (with)f(no)h(cycle\))g(and)g(real)e(trees)j(\(see)g([GH])e(for)g(this)h (notion\).)324 4677 y(Cartan-Hadamard)i(manifolds,)i(the)g(P)m(oincar)m (\023)-46 b(e)43 b(half-plane)e(and,)46 b(more)c(generally)-8 b(,)324 4798 y(complete)39 b(simply)f(connected)j(manifolds)c(with)i (sectional)f(curv)-5 b(ature)40 b(b)s(ounded)h(b)m(y)324 4918 y FA(\024)28 b(<)f FB(0)32 b(are)h FA(\016)t FB(-h)m(yp)s(erb)s (olic)e(spaces)k(with)d FA(\016)f(>)d FB(0.)1918 5251 y(9)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 470 548 a FB(W)-8 b(e)46 b(equip)f(the)g(set)h (of)f(sequences)j(with)d(v)-5 b(alues)45 b(in)f FA(M)56 b FB(with)45 b(the)g(equiv)-5 b(alence)324 668 y(relation)45 b(\()p FA(u)790 683 y Fx(n)837 668 y FB(\))p FA(R)q FB(\()p FA(v)1035 683 y Fx(n)1082 668 y FB(\))i(de\014ned)i(b)m(y)f(the)g (condition)d(if)i(lim)2533 684 y Fy(\()p Fx(n;m)p Fy(\))p FD(!1)2858 668 y FB(\()p FA(u)2952 683 y Fx(n)2999 668 y FA(;)17 b(v)3090 683 y Fx(m)3156 668 y FB(\))3194 683 y Fx(O)3306 668 y FB(=)53 b Fz(1)p FB(.)324 789 y(The)42 b(b)s(oundary)g(at)f(in\014nit)m(y)g FA(@)5 b(M)53 b FB(is)41 b(the)h(set)g(of)f(equiv)-5 b(alence)41 b(classes.)71 b(A)42 b(basis)f(of)324 909 y(op)s(en)33 b(set)g(of)f FA(@)5 b(M)44 b FB(is)32 b(giv)m(en)h(b)m(y)g(the)g(sets)630 1099 y Fw(e)600 1125 y Fu(O)i FB(=)28 b Fz(f)p FA(\015)k Fz(2)c FA(@)5 b(M)40 b Fz(j)27 b FA(\015)37 b FB(is)c(not)f(asso)s (ciated)g(with)h(a)f(sequence)j(of)d FA(M)h Fz(n)22 b Fu(O)8 b Fz(g)p FA(;)324 1342 y FB(where)27 b Fu(O)33 b FB(is)25 b(an)h(op)s(en)f(of)h FA(M)10 b FB(.)42 b(The)26 b(b)s(oundary)g(of)f(0-h)m(yp)s(erb)s(olic)g(space)h(is)g(ultrametric.) 470 1462 y(In)45 b(our)g(con)m(text,)k(if)43 b(w)m(e)j(drop)e(the)i (con)m(v)m(en)m(tion)f FA(v)t FB(\()p FA(x;)17 b(x)p FB(\))49 b(=)f Fz(1)p FB(,)f(our)e(v)-5 b(aluation)324 1583 y(\(2.2\))34 b(is)g(exactly)g(\(2.5\).)49 b(Hence)36 b(\(2.3\))e(implies)d(that)k(\000)f(is)g(0-h)m(yp)s(erb)s(olic.)48 b(W)-8 b(e)34 b(de\014ne)324 1703 y(a)42 b Fq(ge)-5 b(o)g(desic)43 b(r)-5 b(ay)43 b FB(as)h(b)s(eing)e FA(\015)50 b FB(:)45 b Fs(N)60 b Fz(!)45 b FB(\000)e(suc)m(h)h(that)f Fz(j)p FA(\015)5 b FB(\()p FA(n)p FB(\))p Fz(j)44 b FB(=)h FA(n)e FB(and)g FA(\015)3132 1718 y Fx(n)p Fy(+1)3314 1703 y Fz(\030)j FA(\015)3488 1718 y Fx(n)3535 1703 y FB(.)324 1823 y(Geo)s(desic)39 b(ra)m(ys)g(are)g(represen)m(tativ)m(e)i(elemen)m (ts)f(of)e(the)h(ab)s(o)m(v)m(e)h(equiv)-5 b(alence)40 b(classes.)324 1944 y(The)33 b(t)m(w)m(o)h(notions)e(of)g(b)s(oundary)h (at)f(in\014nit)m(y)g(are)g(iden)m(ti\014ed)h(b)m(y)g(setting)g FA(x)3155 1959 y Fx(n)3230 1944 y FB(=)27 b FA(\015)5 b FB(\()p FA(n)p FB(\).)324 2276 y FC(3)161 b(Op)t(erators)53 b(in)h Fe(`)1573 2224 y FB(2)1626 2276 y Fd(\(\000\))324 2495 y Fr(3.1.)44 b(Bounded)g(and)h(compact)d(op)s(erators.)60 b FB(W)-8 b(e)39 b(are)f(in)m(terested)h(in)e(op)s(erators)324 2615 y(acting)31 b(on)i(the)g(Hilb)s(ert)e(space)1125 2844 y FA(`)1166 2802 y Fy(2)1205 2844 y FB(\(\000\))c(=)h Fz(f)p FA(f)38 b FB(:)28 b(\000)g Fz(!)f Fs(C)54 b Fz(j)2035 2749 y Fw(X)2041 2960 y Fx(x)p FD(2)p Fy(\000)2195 2844 y Fz(j)p FA(f)11 b FB(\()p FA(x)p FB(\))p Fz(j)2441 2802 y Fy(2)2508 2844 y FA(<)27 b Fz(1g)324 3173 y FB(endo)m(w)m(ed)34 b(with)f(the)g(inner)f(pro)s(duct:)44 b Fz(h)p FA(f)5 b(;)17 b(g)t Fz(i)26 b FB(=)2119 3099 y Fw(P)2224 3202 y Fx(x)p FD(2)p Fy(\000)p 2375 3087 191 4 v 2375 3173 a FA(f)11 b FB(\()p FA(x)p FB(\))p FA(g)t FB(\()p FA(x)p FB(\).)470 3294 y(W)-8 b(e)40 b(em)m(b)s(ed)h(\000)f Fz(\032)g FA(`)1221 3258 y Fy(2)1260 3294 y FB(\(\000\))g(b)m(y)h(iden) m(tifying)d FA(x)j FB(=)f FA(\037)2345 3309 y FD(f)p Fx(x)p FD(g)2459 3294 y FB(,)i(where)f FA(\037)2878 3309 y Fx(A)2975 3294 y FB(is)e(the)h(c)m(harac-)324 3414 y(teristic)35 b(function)i(of)f(the)h(set)g FA(A)p FB(.)55 b(Observ)m(e)39 b(that)d(\000)h(is)f(the)h(canonical)e(orthonormal)324 3535 y(basis)d(in)g FA(`)718 3498 y Fy(2)758 3535 y FB(\(\000\).)470 3655 y(W)-8 b(e)44 b(denote)g(b)m(y)h Fs(B)15 b FB(\(\000\))49 b(\(resp.)77 b Fs(K)18 b FB(\(\000\)\))49 b(the)44 b(set)g(of)f(b)s (ounded)h(\(resp.)77 b(compact\))324 3775 y(op)s(erators)37 b(in)g FA(`)920 3739 y Fy(2)960 3775 y FB(\(\000\).)58 b(F)-8 b(or)37 b FA(T)50 b Fz(2)37 b Fs(B)15 b FB(\(\000\),)45 b(w)m(e)39 b(will)c(denote)j(b)m(y)h FA(T)2710 3739 y FD(\003)2787 3775 y FB(its)e(adjoin)m(t.)58 b(Giv)m(en)324 3896 y FA(A)33 b Fz(\032)h FB(\000)i(w)m(e)h(denote)g(b)m(y)g Fr(1)1298 3911 y Fx(A)1391 3896 y FB(the)f(op)s(erator)g(of)f(m)m (ultiplication)d(b)m(y)37 b FA(\037)2903 3911 y Fx(A)2996 3896 y FB(in)e FA(`)3154 3860 y Fy(2)3193 3896 y FB(\(\000\))h(\(this) 324 4016 y(is)c(an)g(orthogonal)f(pro)5 b(jection\).)43 b(W)-8 b(e)33 b(ha)m(v)m(e)h(the)f(follo)m(wing)d(compacit)m(y)i (criterion:)324 4216 y Fr(Prop)s(osition)j(3.1)49 b Fq(L)-5 b(et)36 b FA(T)41 b Fz(2)28 b Fs(B)15 b FB(\(\000\))p Fq(,)41 b(then:)961 4433 y FA(T)h Fz(2)28 b Fs(K)17 b FB(\(\000\))34 b Fz(,)27 b(jj)p Fr(1)1635 4448 y FD(\025)p Fx(r)1727 4433 y FA(T)14 b Fz(jj)34 b(\000)-16 b(!)1881 4492 y Fx(r)r FD(!1)2084 4433 y FB(0)27 b Fz(,)h(jj)p FA(T)14 b Fr(1)2471 4448 y FD(\025)p Fx(r)2562 4433 y Fz(jj)35 b(\000)-16 b(!)2646 4492 y Fx(r)r FD(!1)2848 4433 y FB(0)p FA(;)324 4681 y Fq(wher)-5 b(e)30 b Fr(1)651 4696 y FD(\025)p Fx(r)774 4681 y Fq(is)g(the)h(ortho)-5 b(gonal)29 b(pr)-5 b(oje)g(ction)30 b(asso)-5 b(ciate)g(d)29 b(to)i(the)f(set)h Fz(f)p FA(x)d Fz(2)g FB(\000)f Fz(j)h(j)p FA(x)p Fz(j)f(\025)h FA(r)s Fz(g)p Fq(.)324 4882 y Fr(Pro)s(of:)52 b FB(The)35 b(sequence)h(of)d(op)s(erators)g Fr(1)1886 4897 y FD(\025)p Fx(r)2013 4882 y FB(con)m(v)m(erges)j(strongly)d(to)g (zero,)i(so)f(if)e FA(T)43 b Fz(2)324 5002 y Fs(K)17 b FB(\(\000\))54 b(then)48 b Fz(jj)p Fr(1)935 5017 y FD(\025)p Fx(r)1027 5002 y FA(T)14 b Fz(jj)52 b(!)h FB(0)47 b(and)h Fz(jj)p FA(T)14 b Fr(1)1843 5017 y FD(\025)p Fx(r)1935 5002 y Fz(jj)53 b(!)f FB(0.)89 b(Recipro)s(cally)-8 b(,)49 b(if)d(one)i(has)g(for)1894 5251 y(10)p eop end %%Page: 11 11 TeXDict begin 11 10 bop 324 548 a FB(example)28 b Fz(jj)p Fr(1)814 563 y FD(\025)p Fx(r)906 548 y FA(T)14 b Fz(jj)27 b(!)g FB(0,)i(then)g FA(T)42 b FB(is)28 b(the)h(norm)e(limit)e(of)j (the)h(sequence)i(of)d(\014nite)g(rank)324 668 y(op)s(erators)k Fr(1)811 684 y Fx(B)s Fy(\()p Fx(e;r)r Fy(\))1013 668 y FA(T)14 b FB(,)32 b(hence)i(is)e(compact.)141 b Ff(\003)324 832 y Fr(3.2.)36 b(The)h(op)s(erator)f FA(@)5 b Fr(.)44 b FB(W)-8 b(e)32 b(no)m(w)g(extend)h FA(x)28 b Fz(7!)g FA(x)2298 795 y FD(0)2353 832 y FB(to)j(a)g(map)g FA(`)2808 795 y Fy(2)2847 832 y FB(\(\000\))d Fz(!)f FA(`)3180 795 y Fy(2)3219 832 y FB(\(\000\).)43 b(W)-8 b(e)324 952 y(set)34 b FA(e)522 916 y FD(0)575 952 y FB(=)29 b(0.)46 b(F)-8 b(or)32 b FA(f)40 b Fz(2)30 b FA(`)1202 916 y Fy(2)1241 952 y FB(\(\000\),)k(w)m(e)h(ha)m(v)m(e)f FA(f)40 b FB(=)2002 877 y Fw(P)2108 981 y Fx(x)p FD(2)p Fy(\000)2259 952 y FA(f)11 b FB(\()p FA(x)p FB(\))p FA(x)p FB(,)34 b(and)g(w)m(e)h(de\014ne)f FA(@)5 b(f)45 b FB(or)34 b FA(f)3512 916 y FD(0)3535 952 y FB(,)324 1072 y(the)f(deriv)-5 b(ativ)m(e)32 b(of)g FA(f)11 b FB(,)32 b(as)h(follo)m(ws:)832 1304 y(\()p FA(@)5 b(f)11 b FB(\)\()p FA(x)p FB(\))28 b Fz(\021)g FA(f)1346 1263 y FD(0)1369 1304 y FB(\()p FA(x)p FB(\))g(=)1632 1209 y Fw(X)1639 1421 y Fx(x)p FD(2)p Fy(\000)1793 1304 y FA(f)11 b FB(\()p FA(x)p FB(\))p FA(x)2038 1263 y FD(0)2089 1304 y FB(=)2198 1209 y Fw(X)2192 1422 y Fx(y)2229 1403 y Fv(0)2252 1422 y Fy(=)p Fx(x)2363 1304 y FA(f)g FB(\()p FA(y)t FB(\))27 b(=)2680 1209 y Fw(X)2690 1423 y Fx(y)r FD(2)r Fp(e)-41 b Fx(x)2841 1304 y FA(f)11 b FB(\()p FA(y)t FB(\))p FA(:)324 1633 y FB(This)33 b(giv)m(es)g(a)f(b)s(ounded)h(op)s(erator)f FA(@)i FB(:)27 b FA(l)1828 1597 y Fy(2)1868 1633 y FB(\(\000\))h Fz(!)f FA(l)2191 1597 y Fy(2)2231 1633 y FB(\(\000\).)43 b(Indeed,)844 1870 y Fz(k)p FA(f)953 1829 y FD(0)976 1870 y Fz(k)1026 1829 y Fy(2)1149 1870 y FB(=)1309 1776 y Fw(X)1315 1987 y Fx(x)p FD(2)p Fy(\000)1469 1870 y Fz(j)p FA(f)1556 1829 y FD(0)1579 1870 y FB(\()p FA(x)p FB(\))p Fz(j)1738 1829 y Fy(2)1805 1870 y FB(=)1909 1776 y Fw(X)1915 1987 y Fx(x)p FD(2)p Fy(\000)2069 1870 y Fz(j)2114 1776 y Fw(X)2123 1990 y Fx(y)r FD(2)r Fp(e)-41 b Fx(x)2274 1870 y FA(f)11 b FB(\()p FA(y)t FB(\))p Fz(j)2489 1829 y Fy(2)1148 2150 y Fz(\024)84 b FA(\027)1380 2056 y Fw(X)1386 2267 y Fx(x)p FD(2)p Fy(\000)1540 2056 y Fw(X)1550 2270 y Fx(y)r FD(2)r Fp(e)-41 b Fx(x)1701 2150 y Fz(j)p FA(f)11 b FB(\()p FA(y)t FB(\))p Fz(j)1944 2109 y Fy(2)2009 2150 y Fz(\024)28 b FA(\027)2185 2056 y Fw(X)2192 2267 y Fx(x)p FD(2)p Fy(\000)2346 2150 y Fz(j)p FA(f)11 b FB(\()p FA(x)p FB(\))p Fz(j)2592 2109 y Fy(2)2658 2150 y FB(=)28 b FA(\027)6 b Fz(k)p FA(f)11 b Fz(k)2975 2109 y Fy(2)3014 2150 y FA(:)324 2479 y FB(W)-8 b(e)33 b(pro)m(v)m(e)h(that)e(the)h(adjoin)m(t) f FA(@)1521 2443 y FD(\003)1593 2479 y FB(acts)h(on)g FA(f)38 b Fz(2)28 b FA(`)2151 2443 y Fy(2)2191 2479 y FB(\(\000\))k(as)h(follo)m(ws:)1424 2699 y FA(@)1480 2658 y FD(\003)1520 2699 y FA(f)11 b FB(\()p FA(x)p FB(\))28 b(=)g FA(\037)1903 2714 y Fy(\000)p FD(nf)p Fx(e)p FD(g)2090 2699 y FB(\()p FA(x)p FB(\))p FA(f)11 b FB(\()p FA(x)2373 2658 y FD(0)2396 2699 y FB(\))p FA(:)324 2919 y FB(In)33 b(fact)926 3151 y Fz(h)p FA(@)5 b(f)g(;)17 b(f)11 b Fz(i)83 b FB(=)1458 3056 y Fw(X)1465 3268 y Fx(x)p FD(2)p Fy(\000)p 1619 3064 323 4 v 1619 3151 a FB(\()p FA(@)5 b(f)11 b FB(\)\()p FA(x)p FB(\))p FA(f)g FB(\()p FA(x)p FB(\))28 b(=)2263 3056 y Fw(X)2269 3268 y Fx(x)p FD(2)p Fy(\000)p 2423 3034 347 4 v 2423 3056 a Fw(X)2433 3270 y Fx(y)r FD(2)r Fp(e)-41 b Fx(x)2584 3151 y FA(f)11 b FB(\()p FA(y)t FB(\))n FA(f)g FB(\()p FA(x)p FB(\))1299 3431 y(=)1458 3336 y Fw(X)1465 3548 y Fx(x)p FD(2)p Fy(\000)p 1619 3344 570 4 v 1619 3431 a FA(f)g FB(\()p FA(x)p FB(\))p FA(\037)1870 3446 y Fy(\000)p FD(nf)p Fx(e)p FD(g)2057 3431 y FB(\()p FA(x)p FB(\))p FA(f)g FB(\()p FA(x)2340 3390 y FD(0)2363 3431 y FB(\))28 b(=)f Fz(h)p FA(f)5 b(;)17 b(@)2724 3390 y FD(\003)2764 3431 y FA(f)11 b Fz(i)p FA(:)324 3739 y FB(Observ)m(e)34 b(that)837 3959 y Fz(k)p FA(@)943 3917 y FD(\003)983 3959 y FA(f)11 b Fz(k)1092 3917 y Fy(2)1214 3959 y FB(=)1373 3864 y Fw(X)1380 4075 y Fx(x)p FD(2)p Fy(\000)p 1534 3872 363 4 v 1534 3959 a FB(\()p FA(@)1628 3930 y FD(\003)1668 3959 y FA(f)g FB(\)\()p FA(x)p FB(\))p FA(@)1952 3917 y FD(\003)1992 3959 y FA(f)g FB(\()p FA(x)p FB(\))28 b(=)2387 3864 y Fw(X)2314 4080 y Fx(x)p FD(2)p Fy(\000)p FD(nf)p Fy(\037)p FD(g)p 2622 3872 214 4 v 2622 3959 a FA(f)11 b FB(\()p FA(x)2774 3930 y FD(0)2797 3959 y FB(\))p FA(f)g FB(\()p FA(x)2987 3917 y FD(0)3010 3959 y FB(\))1214 4244 y(=)83 b FA(\027)1444 4150 y Fw(X)1451 4361 y Fx(x)p FD(2)p Fy(\000)p 1605 4157 191 4 v 1605 4244 a FA(f)11 b FB(\()p FA(x)p FB(\))p FA(f)g FB(\()p FA(x)p FB(\))27 b(=)h FA(\027)6 b Fz(k)p FA(f)11 b Fz(k)2329 4203 y Fy(2)2368 4244 y FA(;)324 4563 y FB(hence)34 b FA(@)651 4526 y FD(\003)691 4563 y FA(=)740 4491 y Fz(p)p 823 4491 55 4 v 72 x FA(\027)39 b FB(is)32 b(isometric)f(on)h FA(`)1602 4526 y Fy(2)1642 4563 y FB(\(\000\),)g(in)g(other)h(terms:)324 4782 y(\(3.7\))1190 b FA(@)5 b(@)1827 4741 y FD(\003)1896 4782 y FB(=)27 b FA(\027)6 b FB(Id)q FA(:)324 5002 y FB(This)33 b(giv)m(es)g(us)g Fz(jj)p FA(@)5 b Fz(jj)27 b FB(=)h Fz(jj)p FA(@)1322 4966 y FD(\003)1361 5002 y Fz(jj)f FB(=)1548 4930 y Fz(p)p 1631 4930 V 72 x FA(\027)6 b FB(.)1894 5251 y(11)p eop end %%Page: 12 12 TeXDict begin 12 11 bop 470 548 a FB(F)-8 b(or)33 b FA(\013)e Fz(2)g Fs(N)49 b FB(w)m(e)35 b(set)g FA(f)1300 512 y Fy(\()p Fx(\013)p Fy(\))1435 548 y FB(=)30 b FA(@)1597 512 y Fx(\013)1647 548 y FA(f)11 b FB(.)48 b(Th)m(us,)36 b(if)d FA(x)e Fz(2)g FB(\000)j(then)h FA(x)2705 512 y Fy(\()p Fx(\013)p Fy(\))2844 548 y FB(is)e(w)m(ell)h(de\014ned)h(in)324 668 y FA(`)365 632 y Fy(2)404 668 y FB(\(\000\))i(and)h FA(x)828 632 y Fy(\()p Fx(\013)p Fy(\))968 668 y FB(=)e(0)f Fz(,)g FA(\013)i(>)e Fz(j)p FA(x)p Fz(j)p FB(.)57 b(F)-8 b(or)37 b Fz(j)p FA(x)p Fz(j)e(\025)h FA(\013)i FB(the)g(notation)e(is) h(consisten)m(t)h(with)324 789 y(our)32 b(old)g(de\014nition.)324 952 y Fr(3.3.)45 b FA(C)620 916 y FD(\003)659 952 y Fr(-algebras)h(of)f (energy)h(observ)-6 b(ables)45 b(related)g(to)f FB(\000)p Fr(.)63 b FB(W)-8 b(e)40 b(summarize)324 1072 y(the)35 b(metho)s(d)f(used)h(in)f([GI])h(to)f(study)i(the)f(essen)m(tial)f(sp)s (ectrum)h(of)f(large)g(families)d(of)324 1193 y(op)s(erators.)324 1313 y(Let)26 b Fu(H)54 b FB(b)s(e)26 b(a)g(Hilb)s(ert)e(space)j(and)f FA(H)33 b FB(a)25 b(b)s(ounded)i(self-adjoin)m(t)d(op)s(erator)h(on)g Fu(H)k FB(.)42 b(The)324 1434 y(essen)m(tial)33 b(sp)s(ectrum)g FA(\033)1194 1449 y Fx(ess)1297 1434 y FB(\()p FA(H)8 b FB(\))33 b(of)f FA(H)41 b FB(is)32 b(the)i(set)g(of)f FA(\025)28 b Fz(2)h FA(\033)t FB(\()p FA(H)8 b FB(\))33 b(suc)m(h)h(that)f(either)g FA(\025)g FB(is)324 1554 y(not)38 b(isolated)g(from)f(the)j(rest)f(of)f(the)i(sp)s(ectrum)f(or)f (it)g(is)g(an)h(eigen)m(v)-5 b(alue)38 b(of)h(in\014nite)324 1674 y(m)m(ultiplicit)m(y)-8 b(.)75 b(Let)44 b FA(C)7 b FB(\()p Fu(H)29 b FB(\))48 b(=)g FA(B)5 b FB(\()p Fu(H)29 b FB(\))p FA(=K)7 b FB(\()p Fu(H)29 b FB(\))44 b(b)s(e)g(the)h(Calkin)f FA(C)2940 1638 y FD(\003)2979 1674 y FB(-algebra.)77 b(W)-8 b(e)324 1805 y(denote)45 b(b)m(y)g FA(S)53 b Fz(7!)1071 1779 y Fw(b)1058 1805 y FA(S)d FB(the)44 b(canonical)f(surjection)i(of) e FA(B)5 b FB(\()p Fu(H)30 b FB(\))44 b(on)m(to)g FA(C)7 b FB(\()p Fu(H)29 b FB(\))44 b(and)h(w)m(e)324 1935 y(recall)27 b(that)i FA(\033)843 1950 y Fx(ess)946 1935 y FB(\()p FA(H)8 b FB(\))27 b(=)h FA(\033)t FB(\()1361 1910 y Fw(b)1339 1935 y FA(H)7 b FB(\))29 b(\(this)g(is)f(a)h(v)m(ersion)h(of)e(W)-8 b(eyl)30 b(Theorem\).)42 b(If)29 b Fc(C)g FB(is)g(a)g FA(C)3490 1899 y FD(\003)3529 1935 y FB(-)324 2055 y(subalgebra)i(of)g FA(B)5 b FB(\()p Fu(H)29 b FB(\))i(whic)m(h)h(con)m(tains)g(the)g (compacts,)f(then)h(one)g(has)g(a)f(canonical)324 2176 y(em)m(b)s(edding)43 b Fc(C)p FA(=K)7 b FB(\()p Fu(H)29 b FB(\))46 b Fz(\032)h FA(C)7 b FB(\()p Fu(H)29 b FB(\).)76 b(Th)m(us)45 b(in)e(order)g(to)h(determine)f(the)h(essen)m(tial)324 2296 y(sp)s(ectrum)36 b(of)f(an)h(op)s(erator)f FA(H)41 b Fz(2)34 b Fc(C)i FB(it)f(su\016ces)j(to)d(giv)m(e)h(a)g(go)s(o)s(d)e (description)i(of)f(the)324 2416 y(quotien)m(t)29 b Fc(C)p FA(=K)7 b FB(\()p Fu(H)29 b FB(\))g(and)h(to)f(compute)1853 2391 y Fw(b)1831 2416 y FA(H)37 b FB(as)29 b(elemen)m(t)h(of)e(this)h (quotien)m(t.)43 b(In)30 b(fact,)f(as)324 2537 y(explained)j(in)h([GI)o (],)h(w)m(e)g(can)f(go)f(further)h(b)m(y)h(taking)e FA(H)41 b FB(as)33 b(an)g(un)m(b)s(ounded)h(op)s(erator)324 2657 y(o)m(v)m(er)28 b Fu(H)56 b FB(suc)m(h)28 b(that)f(\()p FA(H)18 b FB(+)11 b FA(i)p FB(\))1395 2621 y FD(\000)p Fy(1)1517 2657 y Fz(2)28 b Fc(C)p FB(.)42 b(W)-8 b(e)27 b(shall)e(apply)i(this)g(strategy)g(in)f(our)h(con)m(text.)470 2803 y(Let)43 b Fu(D)732 2818 y Fy(alg)871 2803 y FB(b)s(e)g(the)g Fz(\003)p FB(-algebra)f(of)g(op)s(erators)h(in)f FA(`)2355 2766 y Fy(2)2395 2803 y FB(\(\000\))g(generated)i(b)m(y)g FA(@)49 b FB(and)43 b Fu(D)324 2923 y FB(the)36 b FA(C)572 2887 y FD(\003)612 2923 y FB(-algebra)e(of)i(op)s(erators)f(in)h FA(`)1697 2887 y Fy(2)1736 2923 y FB(\(\000\))g(generated)h(b)m(y)g FA(@)5 b FB(.)55 b(Because)37 b(of)f(\(3.7\),)g Fu(D)3466 2938 y Fy(alg)324 3043 y FB(is)j(unital.)64 b(W)-8 b(e)41 b(will)c(denote)k(b)m(y)g FA(')p FB(\()p FA(Q)p FB(\))f(the)g(op)s (erator)g(on)g(m)m(ultiplication)35 b(b)m(y)41 b FA(')f FB(on)324 3164 y FA(`)365 3128 y Fy(2)404 3164 y FB(\(\000\).)59 b(If)37 b FA(C)45 b FB(is)37 b(a)g FA(C)1110 3128 y FD(\003)1150 3164 y FB(-algebra)f(of)h(b)s(ounded)h(function)f(on)h(\000)f(\(i.e.)59 b FA(C)43 b Fz(\032)37 b FA(`)3164 3128 y FD(1)3238 3164 y FB(\(\000\))h FA(C)3490 3128 y FD(\003)3529 3164 y FB(-)324 3284 y(subalgebra\),)d(then)h FA(C)42 b FB(is)34 b(em)m(b)s(edded)i(in)e Fs(B)16 b FB(\(\000\))41 b(b)m(y)35 b FA(')d Fz(7!)g FA(')p FB(\()p FA(Q)p FB(\).)50 b(Let)35 b Fz(h)p Fu(D)9 b FA(;)17 b(C)7 b Fz(i)35 b FB(b)s(e)g(the)324 3404 y FA(C)401 3368 y FD(\003)440 3404 y FB(-algebra)40 b(generated)i(b)m(y)h Fu(D)38 b Fz([)28 b FA(C)7 b FB(.)71 b(In)41 b(this)h(pap)s(er)f(w)m(e)i(shall)d(tak)m(e)i Fc(C)h FB(=)g Fz(h)p Fu(D)9 b FA(;)17 b(C)7 b Fz(i)p FB(.)324 3525 y(This)40 b(algebra)e(con)m(tains)i(man)m(y)g (Hamiltonians)d(of)i(ph)m(ysical)h(in)m(terest,)i(for)d(instance)324 3645 y(Shr\177)-49 b(odinger)36 b(op)s(erators)h(with)g(p)s(oten)m (tial)e(in)h FA(C)7 b FB(.)57 b(W)-8 b(e)38 b(recall)d(that)i(giv)m(en) g(a)g(graph)g FA(G)p FB(,)324 3766 y(the)c(Laplace)f(op)s(erator)g (acts)h(on)f FA(`)1621 3729 y Fy(2)1661 3766 y FB(\()p FA(G)p FB(\))g(as)h(follo)m(ws:)1331 4003 y(\(\001)p FA(f)11 b FB(\)\()p FA(x)p FB(\))28 b(=)1810 3908 y Fw(X)1816 4118 y Fx(y)r FD(\030)p Fx(x)1954 4003 y FB(\()p FA(f)11 b FB(\()p FA(y)t FB(\))20 b Fz(\000)j FA(f)11 b FB(\()p FA(x)p FB(\)\))p FA(:)324 4318 y FB(In)42 b(our)g(case)h(\001)h(=)f FA(@)35 b FB(+)28 b FA(@)1344 4282 y FD(\003)1413 4318 y Fz(\000)h FA(\027)6 b FB(Id)29 b(+)g FA(\037)1857 4334 y FD(f)p Fx(e)p FD(g)1964 4318 y FB(.)72 b(If)42 b FA(\027)50 b(>)43 b FB(1)f(then)g Fu(D)52 b FB(do)s(es)42 b(not)g(con)m(tain)324 4439 y(an)m(y)30 b(compact)g(op)s(erator)f(\(see)i(b)s(elo)m(w\),)g(so) f(\001)g(do)s(es)h(not)e(b)s(elong)g(to)h Fu(D)9 b FB(.)43 b(On)30 b(the)g(other)324 4559 y(hand,)h(if)e FA(C)34 b Fz(\033)28 b FA(C)959 4574 y Fy(0)1029 4559 y FB(and)i FA(V)49 b Fz(2)28 b FA(C)37 b FB(then)31 b(the)f(Sc)m(hr\177)-49 b(odinger)30 b(op)s(erator)g(\001)17 b(+)g FA(V)k FB(\()p FA(Q)p FB(\))31 b(clearly)324 4679 y(b)s(elongs)h(to)g Fz(h)p Fu(D)9 b FA(;)17 b(C)7 b Fz(i)p FB(.)470 4800 y(W)-8 b(e)33 b(no)m(w)g(giv)m(e)g(a)f(new)i(description)e(of)g Fs(K)17 b FB(\(\000\).)1894 5251 y(12)p eop end %%Page: 13 13 TeXDict begin 13 12 bop 324 548 a Fr(Prop)s(osition)35 b(3.2)49 b Fq(L)-5 b(et)36 b Fu(C)1356 563 y Fy(0)1430 548 y Fq(b)-5 b(e)35 b(the)g FA(C)1789 512 y FD(\003)1828 548 y Fq(-algebr)-5 b(a)34 b(gener)-5 b(ate)g(d)34 b(by)h Fu(D)d Fz(\001)22 b FA(C)2981 563 y Fy(0)3020 548 y Fq(.)44 b(One)35 b(has:)324 768 y FB(\(3.8\))1178 b Fu(C)1769 783 y Fy(0)1837 768 y FB(=)28 b Fs(K)17 b FB(\(\000\))p FA(:)324 988 y Fr(Pro)s(of:)61 b FB(W)-8 b(e)38 b(ha)m(v)m(e)h Fu(C)1157 1003 y Fy(0)1233 988 y Fz(\032)d Fs(K)18 b FB(\(\000\))43 b(b)s(ecause)c FA(')p FB(\()p FA(Q)p FB(\))f(is)f (compact)g(for)g FA(')f Fz(2)g FA(C)3143 1003 y Fy(0)3220 988 y FB(\(indeed,)324 1108 y(w)m(e)d(ha)m(v)m(e)h Fz(k)p Fr(1)798 1123 y FD(\025)p Fx(n)900 1108 y FA(')p FB(\()p FA(Q)p FB(\))p Fz(k)28 b(!)f FB(0)32 b(so)h(w)m(e)h(can)e(use)i(Prop)s (osition)d(3.1\).)470 1229 y(T)-8 b(o)25 b(sho)m(w)h(the)g Fu(C)1067 1244 y Fy(0)1134 1229 y Fz(\033)j Fs(K)17 b FB(\(\000\),)32 b(w)m(e)27 b(tak)m(e)e FA(T)42 b Fz(2)28 b Fs(K)17 b FB(\(\000\))31 b(and)25 b(w)m(e)i(\014x)e FA(")j(>)f FB(0.)41 b(F)-8 b(rom)23 b(Prop)s(o-)324 1349 y(sition)28 b(\(3.1\),)h(it)g(follo)m(ws)f(that)h(there)h(is)f(an)h(op) s(erator)f FA(T)2399 1313 y FD(0)2451 1349 y FB(with)g(compactly)g (supp)s(orted)324 1469 y(k)m(ernel)43 b(suc)m(h)h(that)e Fz(jj)p FA(T)g Fz(\000)29 b FA(T)1404 1433 y FD(0)1427 1469 y Fz(jj)44 b(\024)h FA(")p FB(.)72 b(De\014ne)43 b FA(\016)2149 1484 y Fx(x;y)2294 1469 y Fz(2)i Fs(K)18 b FB(\(\000\))48 b(b)m(y)43 b(\()p FA(\016)2888 1484 y Fx(x;y)2989 1469 y FA(f)11 b FB(\)\()p FA(z)t FB(\))45 b(=)f FA(f)11 b FB(\()p FA(y)t FB(\))324 1590 y(if)41 b FA(z)48 b FB(=)43 b FA(x)g FB(and)f(0)f(elsewhere.)73 b(W)-8 b(e)43 b(ha)m(v)m(e)g FA(\016)1977 1605 y Fx(x;x)2124 1590 y FB(=)g FA(\037)2304 1605 y FD(f)p Fx(x)p FD(g)2419 1590 y FB(\()p FA(Q)p FB(\))h Fz(2)g FA(C)2796 1605 y Fy(0)2835 1590 y FB(.)71 b(As)43 b FA(T)3158 1554 y FD(0)3223 1590 y FB(is)e(a)h(lin-)324 1710 y(ear)d(com)m(binaison)e(of)h FA(\016)1215 1725 y Fx(x;y)1316 1710 y FB(,)j(it)d(su\016ces)j(to)d (sho)m(w)i(that)f FA(\016)2460 1725 y Fx(x;y)2600 1710 y FB(is)f(in)g Fu(C)2890 1725 y Fy(0)2930 1710 y FB(.)62 b(But)39 b(w)m(e)h(ha)m(v)m(e)324 1831 y FA(\016)367 1846 y Fx(x;y)495 1831 y FB(=)28 b FA(\016)642 1846 y Fx(x;x)745 1831 y FB(\()p FA(@)839 1794 y FD(\003)880 1831 y FB(\))918 1794 y FD(j)p Fx(x)p FD(j)1001 1831 y FA(@)1057 1794 y FD(j)p Fx(y)r FD(j)1138 1831 y FA(\016)1181 1846 y Fx(y)r(;y)1280 1831 y FB(,)k(whic)m(h)h(\014nishes)h(the)f(pro)s (of.)140 b Ff(\003)470 2071 y FB(Th)m(us,)29 b(if)c FA(C)32 b FB(is)26 b(a)f FA(C)1167 2035 y FD(\003)1207 2071 y FB(-subalgebra)g(of)g FA(`)1866 2035 y FD(1)1941 2071 y FB(\(\000\))g(that)h(con)m(tain)f FA(C)2715 2086 y Fy(0)2754 2071 y FB(,)j(w)m(e)e(obtain)f Fs(K)18 b FB(\(\000\))33 b Fz(\032)324 2192 y(h)p Fu(D)9 b FA(;)17 b(C)7 b Fz(i)p FB(.)52 b(Hence,)37 b(in)e(order)h(to)f(apply)g(the)h(tec)m(hnique)h (describ)s(ed)f(ab)s(o)m(v)m(e,)h(it)e(remains)324 2312 y(to)d(giv)m(e)h(a)f(su\016cien)m(tly)h(explicit)e(description)h(of) 1656 2515 y Fz(h)p Fu(D)9 b FA(;)17 b(C)7 b Fz(i)1924 2443 y Fw(.)1988 2601 y Fs(K)18 b FB(\(\000\))p FA(:)324 2831 y FB(In)33 b(this)f(pap)s(er,)i(w)m(e)f(shall)f(concen)m(trate)i (on)f(the)g(case)h FA(C)7 b FB(\()2462 2806 y Fw(b)2459 2831 y FB(\000\))32 b(whic)m(h)i(is,)e(geometrically)324 2952 y(sp)s(eaking,)g(the)h(most)f(in)m(teresting)h(one)f(\(see)i(the)f (last)f(Remark)g(in)g(2.4\).)324 3165 y Fr(De\014nition)k(3.3)49 b Fq(The)29 b FA(C)1310 3129 y FD(\003)1349 3165 y Fq(-algebr)-5 b(a)29 b(gener)-5 b(ate)g(d)28 b(by)i FA(@)35 b Fq(and)29 b FA(C)7 b FB(\()2645 3140 y Fw(b)2642 3165 y FB(\000\))29 b Fq(is)g(denote)-5 b(d)29 b(by)h Fu(C)17 b FB(\()3466 3140 y Fw(b)3463 3165 y FB(\000\))324 3295 y Fq(and)34 b(the)h Fz(\003)p Fq(-sub)-5 b(algebr)g(a)34 b(gener)-5 b(ate)g(d)34 b(by)h FA(@)41 b Fq(and)34 b FA(C)7 b FB(\()2180 3270 y Fw(b)2177 3295 y FB(\000\))35 b Fq(is)f(denote)-5 b(d)34 b(by)h Fu(C)18 b FB(\()3023 3270 y Fw(b)3020 3295 y FB(\000)o(\))3118 3310 y Fy(alg)3214 3295 y Fq(.)470 3499 y FB(W)-8 b(e)33 b(will)d(need)k(the)f(next)g(fundamen)m(tal)f (prop)s(ert)m(y)-8 b(.)324 3702 y Fr(Prop)s(osition)35 b(3.4)49 b FB([)p FA(@)5 b(;)17 b(C)7 b FB(\()1366 3677 y Fw(b)1363 3702 y FB(\000)q(\)])28 b Fz(\032)g Fs(K)17 b FB(\(\000\))p Fq(.)324 3915 y Fr(Pro)s(of:)50 b FB(If)32 b FA(')c Fz(2)g FA(C)7 b FB(\()1078 3890 y Fw(b)1075 3915 y FB(\000\))32 b(then)699 4153 y(\([)p FA(@)5 b(;)17 b(')p FB(\()p FA(Q)p FB(\)])p FA(f)11 b FB(\)\()p FA(x)p FB(\))28 b(=)1473 4058 y Fw(X)1468 4271 y Fx(y)1505 4252 y Fv(0)1527 4271 y Fy(=)p Fx(x)1622 4153 y FB(\()p FA(')p FB(\()p FA(y)t FB(\))21 b Fz(\000)i FA(')p FB(\()p FA(x)p FB(\)\))p FA(f)11 b FB(\()p FA(y)t FB(\))26 b(=)i(\()p FA(@)g Fz(\016)22 b FA( )t FB(\()p FA(Q)p FB(\))p FA(f)11 b FB(\)\()p FA(x)p FB(\))p FA(;)324 4499 y FB(where)31 b FA( )i FB(b)s(elongs)c(to)g FA(C)7 b FB(\()1280 4474 y Fw(b)1277 4499 y FB(\000\))30 b(and)f(is)g(de\014ned)i(b)m(y)g FA( )t FB(\()p FA(y)t FB(\))26 b(=)i FA(')p FB(\()p FA(y)t FB(\))16 b Fz(\000)g FA(')p FB(\()p FA(y)2933 4463 y FD(0)2954 4499 y FB(\))30 b(when)h Fz(j)p FA(y)t Fz(j)26 b(\025)i FB(1)324 4619 y(and)38 b FA( )t FB(\()p FA(e)p FB(\))e(=)g(0.)59 b(Observ)m(e)39 b(that)f(for)f FA(\015)k Fz(2)c FA(@)5 b FB(\000)39 b(w)m(e)f(ha)m(v)m(e)h FA( )t FB(\()p FA(\015)5 b FB(\))37 b(=)f FA(')p FB(\()p FA(\015)5 b FB(\))25 b Fz(\000)h FA(')p FB(\()p FA(\015)5 b FB(\))37 b(=)f(0.)324 4740 y(Hence)i(the)f(relation)e(\(2.4\))h(giv)m(es)h(us)h (that)e FA( )j Fz(2)c FA(C)2250 4755 y Fy(0)2326 4740 y FB(and)i(the)g(Prop)s(osition)e(3.2)i(that)324 4860 y FA( )t FB(\()p FA(Q)p FB(\))32 b(is)h(a)f(compact)g(op)s(erator.)140 b Ff(\003)1894 5251 y FB(13)p eop end %%Page: 14 14 TeXDict begin 14 13 bop 324 548 a Fr(Remark:)79 b FB(The)46 b(algebra)e Fu(D)55 b FB(is)45 b(the)h(tree)g(analog)d(of)i(the)h (algebra)e(generated)j(b)m(y)324 668 y(the)31 b(momen)m(tum)d(op)s (erator)i(on)g(the)h(real)e(line.)42 b(Ho)m(w)m(ev)m(er,)33 b(these)f(algebras)d(are)i(rather)324 789 y(di\013eren)m(t.)42 b(First,)27 b Fu(D)37 b FB(is)26 b(not)h(comm)m(utativ)m(e.)41 b(Then,)30 b(the)d(sp)s(ectrum)h(and)f(the)h(essen)m(tial)324 909 y(sp)s(ectrum)k(of)f(the)h(op)s(erators)f(from)f Fu(D)41 b FB(is)31 b(not)h(connected)h(in)e(general.)42 b(F)-8 b(or)31 b(instance,)324 1029 y(one)36 b(has)f FA(\033)t FB(\()p FA(@)835 993 y FD(\003)876 1029 y FA(@)5 b FB(\))33 b(=)g FA(\033)1167 1044 y Fy(ess)1258 1029 y FB(\()p FA(@)1352 993 y FD(\003)1393 1029 y FA(@)5 b FB(\))33 b(=)g Fz(f)p FB(0)p FA(;)17 b(\027)6 b Fz(g)35 b FB(if)f FA(\027)39 b(>)33 b FB(1.)52 b(Indeed,)38 b(w)m(e)e(remind)f (that)g(if)f FA(A)p FB(,)324 1150 y FA(B)43 b FB(are)c(elemen)m(ts)g (of)f(a)g(Banac)m(h)h(algebra)f(w)m(e)h(ha)m(v)m(e)h FA(\033)t FB(\()p FA(AB)5 b FB(\))27 b Fz([)f(f)p FB(0)p Fz(g)38 b FB(=)f FA(\033)t FB(\()p FA(B)5 b(A)p FB(\))27 b Fz([)f(f)p FB(0)p Fz(g)324 1270 y FB(and)32 b(that,)h(as)g(noticed)f (b)s(elo)m(w,)h(dim)15 b(k)m(er)q(\()p FA(@)5 b FB(\))33 b(is)f(in\014nite)g(for)g FA(\027)i(>)28 b FB(1.)324 1434 y Fr(3.4.)40 b(T)-9 b(ranslations)39 b(in)g FA(`)1350 1397 y Fy(2)1389 1434 y FB(\(\000\))p Fr(.)49 b FB(\000)35 b(acts)g(on)f(itself)f(to)i(the)g(left)e(and)i(to)f(the)h(righ)m(t:)47 b(if)324 1554 y FA(a)37 b Fz(2)h FB(\000)g(then)h FA(\025)900 1569 y Fx(a)979 1554 y FB(:)f(\000)f Fz(!)g FB(\000)h(is)g FA(x)g Fz(7!)f FA(ax)i FB(and)f FA(\032)2102 1569 y Fx(a)2182 1554 y FB(:)f(\000)h Fz(!)f FB(\000)h(is)g FA(x)g Fz(7!)f FA(xa)p FB(.)61 b(W)-8 b(e)39 b(clearly)324 1674 y(ha)m(v)m(e,)34 b(for)e FA(a;)17 b(b)28 b Fz(2)g FB(\000,)33 b(the)g(follo)m(wing)c (prop)s(erties:)1101 1884 y FA(\025)1158 1899 y Fx(ab)1258 1884 y FB(=)e FA(\025)1418 1899 y Fx(a)1460 1884 y FA(\025)1517 1899 y Fx(b)1551 1884 y FA(;)33 b(\025)1668 1899 y Fx(e)1733 1884 y FB(=)27 b(Id)q FA(;)33 b(\032)2036 1899 y Fx(ab)2136 1884 y FB(=)27 b FA(\032)2289 1899 y Fx(b)2324 1884 y FA(\032)2374 1899 y Fx(a)2416 1884 y FA(;)33 b(\032)2526 1899 y Fx(e)2591 1884 y FB(=)28 b(Id)324 2095 y(and)1680 2215 y FA(\025)1737 2230 y Fx(a)1779 2215 y FA(\032)1829 2230 y Fx(b)1891 2215 y FB(=)g FA(\032)2045 2230 y Fx(b)2079 2215 y FA(\025)2136 2230 y Fx(a)2178 2215 y FA(:)324 2385 y FB(F)-8 b(or)34 b FA(x)g Fz(2)f FA(a)p FB(\000,)j(w)m(e)h (de\014ne)f FA(a)1346 2349 y FD(\000)p Fy(1)1441 2385 y FA(x)g FB(as)f(b)s(eing)g FA(y)k FB(where)d FA(x)d FB(=)g FA(ay)t FB(,)j(and)f(for)g FA(x)e Fz(2)g FB(\000)p FA(a)p FB(,)k FA(xa)3467 2349 y FD(\000)p Fy(1)324 2505 y FB(will)30 b(b)s(e)j FA(y)i FB(where)f FA(x)28 b FB(=)g FA(y)t(a)p FB(.)470 2626 y(W)-8 b(e)41 b(no)m(w)f(giv)m(e)h(the)f (natural)f(extensions)i(of)f(these)i(translations)c(to)i FA(`)3150 2590 y Fy(2)3189 2626 y FB(\(\000\).)66 b(Let)324 2746 y FA(f)56 b FB(=)550 2671 y Fw(P)655 2775 y Fx(x)p FD(2)p Fy(\000)806 2746 y FA(f)11 b FB(\()p FA(x)p FB(\))p FA(x)44 b FB(in)e FA(`)1260 2710 y Fy(2)1300 2746 y FB(\(\000\).)74 b(The)44 b(translation)e FA(\025)2311 2761 y Fx(a)2395 2746 y FB(on)h FA(f)54 b FB(will)41 b(b)s(e)2981 2671 y Fw(P)3086 2775 y Fx(x)p FD(2)p Fy(\000)3238 2746 y FA(f)11 b FB(\()p FA(x)p FB(\))p FA(ax)p FB(,)324 2867 y(more)32 b(precisely)1374 2987 y(\()p FA(\025)1469 3002 y Fx(a)1511 2987 y FA(f)11 b FB(\)\()p FA(x)p FB(\))28 b(=)f FA(\037)1931 3002 y Fx(a)p Fy(\000)2017 2987 y FB(\()p FA(x)p FB(\))p FA(f)11 b FB(\()p FA(a)2296 2946 y FD(\000)p Fy(1)2390 2987 y FA(x)p FB(\))p FA(:)324 3157 y FB(By)33 b(the)g(same)f(w)m(a)m(y)-8 b(,)34 b(w)m(e)g(obtain:) 1378 3367 y(\()p FA(\032)1466 3382 y Fx(a)1508 3367 y FA(f)11 b FB(\)\()p FA(x)p FB(\))27 b(=)h FA(\037)1928 3382 y Fy(\000)p Fx(a)2014 3367 y FB(\()p FA(x)p FB(\))p FA(f)11 b FB(\()p FA(xa)2348 3326 y FD(\000)p Fy(1)2443 3367 y FB(\))p FA(:)324 3577 y FB(The)33 b(op)s(erators)g FA(\025)1013 3592 y Fx(a)1087 3577 y FB(and)f FA(\032)1326 3592 y Fx(a)1401 3577 y FB(are)g(isometries)g(on)g FA(`)2196 3541 y Fy(2)2236 3577 y FB(\(\000\).)43 b(F)-8 b(or)32 b(instance:)889 3805 y Fz(jj)p FA(\025)1002 3820 y Fx(a)1042 3805 y FA(f)11 b Fz(jj)1157 3764 y Fy(2)1279 3805 y FB(=)1438 3710 y Fw(X)1444 3922 y Fx(x)p FD(2)p Fy(\000)1598 3805 y Fz(j)p FA(\037)1687 3820 y Fx(a)p Fy(\000)1773 3805 y FB(\()p FA(x)p FB(\))p FA(f)g FB(\()p FA(a)2052 3764 y FD(\000)p Fy(1)2146 3805 y FA(x)p FB(\))p Fz(j)2267 3764 y Fy(2)1279 4074 y FB(=)1450 3979 y Fw(X)1438 4190 y Fx(x)p FD(2)p Fx(a)p Fy(\000)1623 4074 y Fz(j)p FA(f)g FB(\()p FA(a)1799 4032 y FD(\000)p Fy(1)1893 4074 y FA(x)p FB(\))p Fz(j)2014 4032 y Fy(2)2081 4074 y FB(=)2184 3979 y Fw(X)2191 4190 y Fx(x)p FD(2)p Fy(\000)2345 4074 y Fz(j)p FA(f)g FB(\()p FA(x)p FB(\))p Fz(j)2591 4032 y Fy(2)2657 4074 y FB(=)28 b Fz(jj)p FA(f)11 b Fz(jj)2932 4032 y Fy(2)2970 4074 y FA(:)324 4372 y FB(Th)m(us,)34 b(w)m(e)g(obtain:)324 4582 y(\(3.9\))882 b FA(\025)1464 4541 y FD(\003)1464 4606 y Fx(a)1506 4582 y FA(\025)1563 4597 y Fx(a)1632 4582 y FB(=)27 b(Id)33 b(and)g FA(\032)2097 4541 y FD(\003)2097 4606 y Fx(a)2139 4582 y FA(\032)2189 4597 y Fx(a)2258 4582 y FB(=)28 b(Id)p FA(:)324 4792 y FB(W)-8 b(e)33 b(no)m(w)g(compute)g(their)f(adjoin)m(ts.)43 b(They)34 b(act)e(as)h(follo)m(ws)e(on)i FA(f)38 b Fz(2)28 b FA(`)2942 4756 y Fy(2)2982 4792 y FB(\(\000\):)1165 5002 y(\()p FA(\025)1260 4961 y FD(\003)1260 5027 y Fx(a)1302 5002 y FA(f)11 b FB(\)\()p FA(x)p FB(\))28 b(=)f FA(f)11 b FB(\()p FA(ax)p FB(\))p FA(;)34 b FB(\()p FA(\032)2051 4961 y FD(\003)2051 5027 y Fx(a)2092 5002 y FA(f)11 b FB(\)\()p FA(x)p FB(\))28 b(=)g FA(f)11 b FB(\()p FA(xa)p FB(\))p FA(:)1894 5251 y FB(14)p eop end %%Page: 15 15 TeXDict begin 15 14 bop 324 548 a FB(Indeed,)34 b(for)e FA(\025)874 563 y Fx(a)915 548 y FB(,)h(one)g(has:)753 772 y Fz(h)p FA(f)5 b(;)17 b(\025)946 787 y Fx(a)987 772 y FA(f)11 b Fz(i)83 b FB(=)1327 677 y Fw(X)1333 888 y Fx(x)p FD(2)p Fy(\000)p 1487 685 191 4 v 1487 772 a FA(f)11 b FB(\()p FA(x)p FB(\))p FA(\037)1738 787 y Fx(a)p Fy(\000)1824 772 y FB(\()p FA(x)p FB(\))p FA(f)g FB(\()p FA(a)2103 730 y FD(\000)p Fy(1)2197 772 y FA(x)p FB(\))28 b(=)2434 677 y Fw(X)2422 888 y Fx(x)p FD(2)p Fx(a)p Fy(\000)p 2607 685 V 2607 772 a FA(f)11 b FB(\()p FA(x)p FB(\))p FA(f)g FB(\()p FA(a)2945 730 y FD(\000)p Fy(1)3039 772 y FA(x)p FB(\))1168 1040 y(=)1327 946 y Fw(X)1333 1157 y Fx(x)p FD(2)p Fy(\000)p 1487 954 242 4 v 1487 1040 a FA(f)g FB(\()p FA(ax)p FB(\))p FA(f)g FB(\()p FA(x)p FB(\))28 b(=)g Fz(h)p FA(\025)2146 999 y FD(\003)2146 1065 y Fx(a)2187 1040 y FA(f)5 b(;)17 b(f)11 b Fz(i)p FA(:)324 1339 y FB(Moreo)m(v)m(er,)920 1563 y Fz(jj)p FA(\025)1033 1522 y FD(\003)1033 1587 y Fx(a)1074 1563 y FA(f)g Fz(jj)1189 1522 y Fy(2)1255 1563 y FB(=)1359 1468 y Fw(X)1365 1679 y Fx(x)p FD(2)p Fy(\000)1519 1563 y Fz(j)p FA(f)g FB(\()p FA(ax)p FB(\))p Fz(j)1816 1522 y Fy(2)1883 1563 y FB(=)1999 1468 y Fw(X)1987 1679 y Fx(x)p FD(2)p Fx(a)p Fy(\000)2171 1563 y Fz(j)p FA(f)g FB(\()p FA(x)p FB(\))p Fz(j)2417 1522 y Fy(2)2484 1563 y FB(=)28 b Fz(jj)p Fr(1)2700 1578 y Fx(a)p Fy(\000)2785 1563 y FA(f)11 b Fz(jj)2900 1522 y Fy(2)2938 1563 y FA(:)324 1862 y FB(Th)m(us,)34 b(w)m(e)g(obtain:)324 2074 y(\(3.10\))781 b FA(\025)1412 2089 y Fx(a)1453 2074 y FA(\025)1510 2033 y FD(\003)1510 2099 y Fx(a)1579 2074 y FB(=)28 b Fr(1)1739 2089 y Fx(a)p Fy(\000)1857 2074 y FB(and)33 b FA(\032)2097 2089 y Fx(a)2139 2074 y FA(\032)2189 2033 y FD(\003)2189 2099 y Fx(a)2258 2074 y FB(=)28 b Fr(1)2418 2089 y Fy(\000)p Fx(a)2504 2074 y FA(:)324 2286 y FB(It)k(is)h(easy)g(to)f(c)m(hec)m(k)j (that)1335 2510 y FA(@)1391 2468 y FD(\003)1459 2510 y FB(=)1574 2415 y Fw(X)1562 2631 y FD(j)p Fx(a)p FD(j)p Fy(=1)1746 2510 y FA(\032)1796 2525 y Fx(a)1870 2510 y FB(and)e FA(@)h FB(=)2260 2415 y Fw(X)2248 2631 y FD(j)p Fx(a)p FD(j)p Fy(=1)2432 2510 y FA(\032)2482 2468 y FD(\003)2482 2534 y Fx(a)2524 2510 y FA(:)324 2854 y Fr(3.5.)43 b(Lo)s(calizations)f (at)i(in\014nit)m(y)-9 b(.)57 b FB(In)38 b(order)g(to)f(study)i Fu(C)17 b FB(\()2699 2829 y Fw(b)2696 2854 y FB(\000\))p FA(=)p Fs(K)h FB(\(\000\))43 b(w)m(e)c(ha)m(v)m(e)g(to)324 2985 y(de\014ne)32 b(the)f(lo)s(calizations)d(at)i(in\014nit)m(y)g(of)h FA(T)41 b Fz(2)28 b Fu(C)17 b FB(\()2202 2959 y Fw(b)2199 2985 y FB(\000\))31 b(b)m(y)h(lo)s(oking)d(at)h(the)h(b)s(eha)m(vior)g (of)324 3115 y(the)f(translated)f(op)s(erator)f FA(\025)1393 3079 y FD(\003)1393 3139 y Fx(a)1435 3115 y FA(T)14 b(\025)1563 3130 y Fx(a)1633 3115 y FB(as)30 b FA(a)f FB(con)m(v)m(erges)j(to)d FA(\015)34 b FB(in)2578 3090 y Fw(b)2576 3115 y FB(\000)29 b(\(abbreviated)g FA(a)f Fz(!)g FA(\015)5 b FB(\),)324 3235 y(for)32 b(eac)m(h)h FA(\015)g Fz(2)28 b FA(@)5 b FB(\000.)470 3356 y(Let)39 b FA(\015)j Fz(2)37 b FB(\000.)61 b(W)-8 b(e)38 b(notice)g(that)g(u-lim)1901 3371 y Fx(a)p FD(!)p Fx(\015)2071 3356 y FA(\025)2128 3319 y FD(\003)2128 3380 y Fx(a)2169 3356 y Fs(K)18 b FB(\(\000\))p FA(\025)2435 3371 y Fx(a)2519 3356 y FB(=)37 b(0,)j(where)f(u-lim)d(means)324 3476 y(con)m(v)m(ergence)43 b(in)d(norm.)66 b(Indeed)42 b(for)e FA(T)55 b Fz(2)41 b Fs(K)18 b FB(\(\000\),)48 b(the)41 b(relations)e(\(3.9\))h(and)g(\(3.10\))324 3596 y(imply)28 b(that)i Fz(jj)p FA(\025)917 3560 y FD(\003)917 3621 y Fx(a)958 3596 y FA(T)14 b(\025)1086 3611 y Fx(a)1127 3596 y Fz(jj)27 b FB(=)h Fz(jj)p Fr(1)1426 3611 y Fx(a)p Fy(\000)1511 3596 y FA(T)14 b Fr(1)1638 3611 y Fx(a)p Fy(\000)1723 3596 y Fz(jj)p FB(,)30 b(and)h(then)f(the)h(Prop)s (osition)d(3.1)i(\014nishes)h(the)324 3727 y(pro)s(of.)79 b(No)m(w,)48 b(w)m(e)e(compute)e(the)h(uniform)e(limit)e(of)k FA(\025)2464 3691 y FD(\003)2464 3751 y Fx(a)2505 3727 y FA(T)14 b(\025)2633 3742 y Fx(a)2719 3727 y FB(where)46 b FA(T)62 b Fz(2)48 b Fu(C)18 b FB(\()3371 3702 y Fw(b)3368 3727 y FB(\000)o(\))3466 3742 y Fy(alg)324 3847 y FB(There)43 b(is)e FA(P)55 b FB(b)s(e)42 b(a)g(non-comm)m(utativ)m(e)e(complex)h(p) s(olynomial)d(in)j FA(m)29 b FB(+)f(2)42 b(v)-5 b(ariables)324 3967 y(and)32 b FA(')577 3982 y Fx(i)633 3967 y Fz(2)c FA(C)7 b FB(\()845 3942 y Fw(b)842 3967 y FB(\000\))33 b(for)f FA(i)c FB(=)f(1)p FA(;)17 b(:)g(:)g(:)f(;)h(m)32 b FB(suc)m(h)i(that)1346 4179 y FA(T)41 b FB(=)28 b FA(P)14 b FB(\()p FA(')1727 4194 y Fy(1)1765 4179 y FA(;)j(')1873 4194 y Fy(2)1912 4179 y FA(;)g(:)g(:)g(:)f(;)h(')2195 4194 y Fx(m)2261 4179 y FA(;)g(@)5 b(;)17 b(@)2461 4138 y FD(\003)2502 4179 y FB(\))324 4391 y(and)32 b(w)m(e)i(set)1068 4511 y FA(T)14 b FB(\()p FA(\015)5 b FB(\))28 b(=)f FA(P)14 b FB(\()p FA(')1581 4526 y Fy(1)1620 4511 y FB(\()p FA(\015)5 b FB(\))p FA(;)17 b(')1860 4526 y Fy(2)1899 4511 y FB(\()p FA(\015)5 b FB(\))p FA(;)17 b(:)g(:)g(:)e(;)i(')2313 4526 y Fx(m)2380 4511 y FB(\()p FA(\015)5 b FB(\))p FA(;)17 b(@)5 b(;)17 b(@)2712 4470 y FD(\003)2752 4511 y FB(\))p FA(:)324 4708 y Fr(Lemma)37 b(3.5)49 b Fq(Ther)-5 b(e)34 b(is)h FA(a)1349 4723 y Fy(0)1416 4708 y Fz(2)28 b FB(\000)35 b Fq(such)g(that)1369 4919 y FB(u-)7 b(lim)1456 4979 y Fx(a)p FD(!)p Fx(\015)1621 4919 y FA(\025)1678 4878 y FD(\003)1678 4944 y Fx(a)1720 4919 y FA(T)14 b(\025)1848 4934 y Fx(a)1917 4919 y FB(=)27 b FA(\025)2077 4878 y FD(\003)2077 4944 y Fx(a)2114 4953 y Fh(0)2153 4919 y FA(T)14 b FB(\()p FA(\015)5 b FB(\))p FA(\025)2413 4934 y Fx(a)2450 4943 y Fh(0)2489 4919 y FA(:)1894 5251 y FB(15)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 324 548 a Fr(Pro)s(of:)56 b FB(The)36 b(Prop)s(osition)e(3.4)h(and)g(the)h(relation)d(\(3.7\))i(giv)m(e)g(us) h FA(\036)2922 563 y Fx(k)2997 548 y Fz(2)d FA(C)7 b FB(\()3214 523 y Fw(b)3211 548 y FB(\000\),)36 b FA(K)j Fz(2)324 668 y Fs(K)17 b FB(\(\000\))39 b(and)32 b(in)m(tegers)h FA(\013)1183 683 y Fx(k)1226 668 y FA(;)17 b(\014)1325 683 y Fx(k)1395 668 y Fz(\025)28 b FB(0)33 b(suc)m(h)h(that)1368 936 y FA(T)42 b FB(=)1621 812 y Fx(n)1570 841 y Fw(X)1578 1054 y Fx(k)r Fy(=1)1731 936 y FA(\036)1789 951 y Fx(k)1831 936 y FB(\()p FA(Q)p FB(\))p FA(@)2040 895 y FD(\003)2081 890 y Fx(\013)2126 902 y Fo(k)2168 936 y FA(@)2224 895 y Fx(\014)2264 907 y Fo(k)2329 936 y FB(+)22 b FA(K)324 1212 y FB(and)1404 1384 y FA(T)14 b FB(\()p FA(\015)5 b FB(\))28 b(=)1789 1259 y Fx(n)1738 1289 y Fw(X)1746 1501 y Fx(k)r Fy(=1)1899 1384 y FA(\036)1957 1399 y Fx(k)1999 1384 y FB(\()p FA(\015)5 b FB(\))p FA(@)2187 1343 y FD(\003)2227 1338 y Fx(\013)2272 1350 y Fo(k)2315 1384 y FA(@)2371 1343 y Fx(\014)2411 1355 y Fo(k)2454 1384 y FA(:)324 1659 y FB(Th)m(us,)43 b(it)38 b(su\016xes)k(to)d(compute)h(u-lim)1809 1674 y Fx(a)p FD(!)p Fx(\015)1978 1659 y FA(\025)2035 1623 y FD(\003)2035 1683 y Fx(a)2077 1659 y FA(')p FB(\()p FA(Q)p FB(\))p FA(@)2350 1623 y FD(\003)2390 1616 y Fx(\013)2440 1659 y FA(@)2496 1623 y Fx(\014)2544 1659 y FA(\025)2601 1674 y Fx(a)2682 1659 y FB(with)f FA(')h Fz(2)g FA(C)7 b FB(\()3239 1634 y Fw(b)3236 1659 y FB(\000\).)64 b(W)-8 b(e)324 1779 y(supp)s(ose)34 b Fz(j)p FA(a)p Fz(j)28 b(\025)h FA(\013)34 b FB(and)f(tak)m(e)h FA(f)39 b Fz(2)29 b FA(`)1653 1743 y Fy(2)1693 1779 y FB(\(\000\).)45 b(W)-8 b(e)33 b(will)e(\014rst)j(sho)m(w)g(the)f(result)g(for)g FA(')28 b FB(=)h(1.)324 1899 y(Since)525 2091 y(\()p FA(\025)620 2050 y FD(\003)620 2116 y Fx(a)662 2091 y FA(@)718 2050 y FD(\003)758 2046 y Fx(\013)808 2091 y FA(@)864 2050 y Fx(\014)912 2091 y FA(\025)969 2106 y Fx(a)1010 2091 y FA(f)11 b FB(\)\()p FA(x)p FB(\))83 b(=)g(\()p FA(@)1574 2050 y FD(\003)1614 2046 y Fx(\013)1664 2091 y FA(@)1720 2050 y Fx(\014)1768 2091 y FA(\025)1825 2106 y Fx(a)1867 2091 y FA(f)11 b FB(\)\()p FA(ax)p FB(\))27 b(=)h(\()p FA(@)2371 2050 y Fx(\014)2419 2091 y FA(\025)2476 2106 y Fx(a)2518 2091 y FA(f)11 b FB(\)\(\()p FA(ax)p FB(\))2835 2050 y Fy(\()p Fx(\013)p Fy(\))2939 2091 y FB(\))1321 2254 y(=)1675 2159 y Fw(X)1480 2382 y FD(f)p Fx(y)r FD(j)p Fx(y)1609 2363 y Fh(\()p Fo(\014)s Fh(\))1699 2382 y Fy(=\()p Fx(ax)p Fy(\))1885 2363 y Fh(\()p Fo(\013)p Fh(\))1978 2382 y FD(g)2014 2254 y FB(\()p FA(\025)2109 2269 y Fx(a)2150 2254 y FA(f)g FB(\)\()p FA(y)t FB(\))26 b(=)2746 2159 y Fw(X)2505 2382 y FD(f)p Fx(y)r FD(j)p Fy(\()p Fx(ay)r Fy(\))2725 2363 y Fh(\()p Fo(\014)s Fh(\))2816 2382 y Fy(=\()p Fx(ax)p Fy(\))3002 2363 y Fh(\()p Fo(\013)p Fh(\))3095 2382 y FD(g)3147 2254 y FA(f)11 b FB(\()p FA(y)t FB(\))p FA(;)-3037 b FB(\(3.11\))324 2575 y(it)34 b(su\016ces)j(to)e(sho)m(w)h(that)f(the)h(set)g Fz(f)p FA(y)f Fz(j)d FB(\()p FA(ay)t FB(\))2036 2539 y Fy(\()p Fx(\014)s Fy(\))2169 2575 y FB(=)g(\()p FA(ax)p FB(\))2459 2539 y Fy(\()p Fx(\013)p Fy(\))2564 2575 y Fz(g)j FB(is)f(indep)s (enden)m(t)j(of)d FA(a)i FB(if)324 2695 y Fz(j)p FA(a)p Fz(j)27 b(\025)h FA(\013)q FB(.)43 b(This)33 b(is)f(the)h(aim)e(of)h (our)g(next)i(lemma.)324 2874 y Fr(Lemma)j(3.6)49 b Fq(F)-7 b(or)34 b Fz(j)p FA(a)p Fz(j)27 b(\025)h FA(\013)36 b Fq(we)e(have:)420 3186 y Fz(f)p FA(y)c Fz(j)e FB(\()p FA(ay)t FB(\))783 3145 y Fy(\()p Fx(\014)s Fy(\))912 3186 y FB(=)f(\()p FA(ax)p FB(\))1197 3145 y Fy(\()p Fx(\013)p Fy(\))1302 3186 y Fz(g)h FB(=)1483 2982 y Fw(8)1483 3072 y(<)1483 3251 y(:)1613 3063 y FB(\037)354 b Fq(for)35 b Fz(j)p FA(x)p Fz(j)22 b FB(+)g FA(\014)28 b Fz(\000)22 b FA(\013)29 b(<)e FB(0)p FA(;)1613 3185 y(S)1679 3149 y FD(j)p Fx(x)p FD(j)p Fy(+)p Fx(\014)s FD(\000)p Fx(\013)2043 3185 y Fq(for)35 b Fz(j)p FA(x)p Fz(j)27 b FA(<)h(\013)36 b Fq(and)e Fz(j)p FA(x)p Fz(j)22 b FB(+)g FA(\014)27 b Fz(\000)c FA(\013)28 b Fz(\025)h FB(0)p FA(;)1613 3307 y(x)1668 3271 y Fy(\()p Fx(\013)p Fy(\))1773 3307 y FA(S)1839 3271 y Fx(\014)2043 3307 y Fq(for)35 b Fz(j)p FA(x)p Fz(j)27 b(\025)i FA(\013)35 b Fq(and)f Fz(j)p FA(x)p Fz(j)22 b FB(+)g FA(\014)28 b Fz(\000)23 b FA(\013)28 b Fz(\025)g FB(0)p FA(:)324 3509 y Fr(Pro)s(of:)50 b FB(Let)32 b FA(J)905 3524 y Fx(x)977 3509 y FB(=)27 b Fz(f)p FA(y)k Fz(j)c FB(\()p FA(ay)t FB(\))1443 3473 y Fy(\()p Fx(\014)s Fy(\))1572 3509 y FB(=)h(\()p FA(ax)p FB(\))1858 3473 y Fy(\()p Fx(\013)p Fy(\))1963 3509 y Fz(g)p FB(.)43 b(Then)1233 3701 y FA(aJ)1338 3716 y Fx(x)1465 3701 y FB(=)83 b Fz(f)p FA(ay)31 b Fz(j)c FB(\()p FA(ay)t FB(\))2038 3660 y Fy(\()p Fx(\014)s Fy(\))2167 3701 y FB(=)h(\()p FA(ax)p FB(\))2453 3660 y Fy(\()p Fx(\013)p Fy(\))2558 3701 y Fz(g)1465 3846 y FB(=)83 b Fz(f)p FA(y)31 b Fz(j)c FA(y)1860 3805 y Fy(\()p Fx(\014)s Fy(\))1989 3846 y FB(=)h(\()p FA(ax)p FB(\))2275 3805 y Fy(\()p Fx(\013)p Fy(\))2379 3846 y Fz(g)22 b(\\)h FA(a)p FB(\000)1465 3992 y(=)83 b(\(\()p FA(ax)p FB(\))1844 3950 y Fy(\()p Fx(\013)p Fy(\))1949 3992 y FA(S)2015 3950 y Fx(\014)2062 3992 y FB(\(\000\)\))22 b Fz(\\)g FA(a)p FB(\000)p FA(:)324 4183 y FB(W)-8 b(e)33 b(notice)f(that)g(\()p FA(ax)p FB(\))1172 4147 y Fy(\()p Fx(\013)p Fy(\))1277 4183 y FA(S)1343 4147 y Fx(\014)1418 4183 y Fz(\032)c FA(S)1589 4147 y FD(j)p Fx(a)p FD(j)p Fy(+)p FD(j)p Fx(x)p FD(j\000)p Fx(\013)p Fy(+)p Fx(\014)2002 4183 y FB(.)470 4304 y(If)36 b Fz(j)p FA(x)p Fz(j)24 b(\000)g FA(\013)h FB(+)f FA(\014)38 b(<)33 b FB(0)i(then)h(\(\()p FA(ax)p FB(\))1725 4268 y Fy(\()p Fx(\013)p Fy(\))1830 4304 y FA(S)1896 4268 y Fx(\014)1943 4304 y FB(\))24 b Fz(\\)h FA(a)p FB(\000)33 b(=)f(\037,)37 b(so)e FA(aJ)2716 4319 y Fx(x)2793 4304 y FB(=)e(\037.)52 b(This)36 b(clearly)324 4424 y(implies)30 b FA(J)709 4439 y Fx(x)780 4424 y FB(=)e(\037.)470 4545 y(If)i Fz(j)p FA(x)p Fz(j)17 b(\000)h FA(\013)g FB(+)g FA(\014)33 b Fz(\025)28 b FB(0)i(then)h(\(\()p FA(ax)p FB(\))1674 4508 y Fy(\()p Fx(\013)p Fy(\))1779 4545 y FA(S)1845 4508 y Fx(\014)1892 4545 y FB(\))17 b Fz(\\)h FA(a)p FB(\000)28 b Fz(6)p FB(=)f(\037.)43 b(If)30 b(w)m(e)h(supp)s (ose)h(that)e Fz(j)p FA(x)p Fz(j)d FA(<)h(\013)q FB(,)324 4665 y(i.e.)43 b Fz(j)p FB(\()p FA(ax)p FB(\))702 4629 y Fy(\()p Fx(\013)p Fy(\))806 4665 y Fz(j)27 b FA(<)h Fz(j)p FA(a)p Fz(j)p FB(,)k(w)m(e)h(ha)m(v)m(e)g FA(a)28 b Fz(2)g FB(\()p FA(ax)p FB(\))1853 4629 y Fy(\()p Fx(\013)p Fy(\))1958 4665 y FB(\000.)43 b(Let)32 b FA(b)g FB(suc)m(h)h(that)f FA(a)c FB(=)f(\()p FA(ax)p FB(\))3130 4629 y Fy(\()p Fx(\013)p Fy(\))3235 4665 y FA(b)p FB(.)44 b(Th)m(us)575 4857 y(\(\()p FA(ax)p FB(\))795 4816 y Fy(\()p Fx(\013)p Fy(\))899 4857 y FA(S)965 4816 y Fx(\014)1012 4857 y FB(\))22 b Fz(\\)h FA(a)p FB(\000)83 b(=)g(\(\()p FA(ax)p FB(\))1735 4816 y Fy(\()p Fx(\013)p Fy(\))1840 4857 y FA(S)1906 4816 y Fx(\014)1953 4857 y FB(\))22 b Fz(\\)g FB(\()p FA(ax)p FB(\))2283 4816 y Fy(\()p Fx(\013)p Fy(\))2388 4857 y FA(b)p FB(\000)28 b(=)g(\()p FA(ax)p FB(\))2804 4816 y Fy(\()p Fx(\013)p Fy(\))2909 4857 y FB(\()p FA(S)3013 4816 y Fx(\014)3082 4857 y Fz(\\)22 b FA(b)p FB(\000\))1356 5002 y(=)83 b(\()p FA(ax)p FB(\))1697 4961 y Fy(\()p Fx(\013)p Fy(\))1802 5002 y FA(bS)1909 4961 y Fx(\014)s FD(\000j)p Fx(b)p FD(j)2108 5002 y FB(=)28 b FA(aS)2329 4961 y Fx(\014)s FD(\000j)p Fx(b)p FD(j)2528 5002 y FB(=)g FA(aS)2749 4961 y Fx(\014)s Fy(+)p FD(j)p Fx(x)p FD(j\000)p Fx(\013)3030 5002 y FA(:)1894 5251 y FB(16)p eop end %%Page: 17 17 TeXDict begin 17 16 bop 324 548 a FB(So)32 b(w)m(e)i(ha)m(v)m(e)g FA(aJ)933 563 y Fx(x)1004 548 y FB(=)28 b FA(aS)1225 512 y Fx(\014)s Fy(+)p FD(j)p Fx(x)p FD(j\000)p Fx(\013)1506 548 y FB(,)33 b(hence)h FA(J)1891 563 y Fx(x)1962 548 y FB(=)28 b FA(S)2132 512 y Fx(\014)s Fy(+)p FD(j)p Fx(x)p FD(j\000)p Fx(\013)2413 548 y FB(.)470 668 y(Finally)-8 b(,)38 b(if)f Fz(j)p FA(x)p Fz(j)h(\025)g FA(\013)q FB(,)i(i.e.)61 b Fz(j)p FB(\()p FA(ax)p FB(\))1714 632 y Fy(\()p Fx(\013)p Fy(\))1819 668 y Fz(j)38 b(\025)g(j)p FA(a)p Fz(j)p FB(,)i(one)f(has)g (\()p FA(ax)p FB(\))2721 632 y Fy(\()p Fx(\013)p Fy(\))2864 668 y Fz(2)f FA(a)p FB(\000.)62 b(Th)m(us,)42 b(w)m(e)324 789 y(obtain)31 b(that)i FA(aJ)944 804 y Fx(x)1015 789 y FB(=)28 b(\()p FA(ax)p FB(\))1301 753 y Fy(\()p Fx(\013)p Fy(\))1406 789 y FA(S)1472 753 y Fx(\014)1546 789 y FB(=)g FA(ax)1756 753 y Fy(\()p Fx(\013)p Fy(\))1861 789 y FA(S)1927 753 y Fx(\014)1974 789 y FB(,)k(hence)i FA(J)2358 804 y Fx(x)2430 789 y FB(=)27 b FA(x)2588 753 y Fy(\()p Fx(\013)p Fy(\))2693 789 y FA(S)2759 753 y Fx(\014)2806 789 y FB(.)141 b Ff(\003)470 1029 y FB(No)m(w)33 b(w)m(e)h(treat)e(the)h(case)h(of)e (a)g(general)g FA(')c Fz(2)g FA(C)7 b FB(\()2281 1004 y Fw(b)2278 1029 y FB(\000\).)43 b(W)-8 b(e)33 b(ha)m(v)m(e)919 1249 y(\()p FA(\025)1014 1208 y FD(\003)1014 1274 y Fx(a)1055 1249 y FA(')p FB(\()p FA(Q)p FB(\))p FA(@)1328 1208 y FD(\003)1369 1204 y Fx(\013)1418 1249 y FA(@)1474 1208 y Fx(\014)1522 1249 y FA(\025)1579 1264 y Fx(a)1621 1249 y FA(f)11 b FB(\)\()p FA(x)p FB(\))28 b(=)f FA(')p FB(\()p FA(ax)p FB(\)\()p FA(\025)2321 1208 y FD(\003)2321 1274 y Fx(a)2363 1249 y FA(@)2419 1208 y FD(\003)2459 1204 y Fx(\013)2509 1249 y FA(@)2565 1208 y Fx(\014)2613 1249 y FA(\025)2670 1264 y Fx(a)2711 1249 y FA(f)11 b FB(\)\()p FA(x)p FB(\))p FA(:)324 1469 y FB(Hence,)542 1689 y Fz(jj)p FA(\025)655 1648 y FD(\003)655 1714 y Fx(a)696 1689 y FA(')p FB(\()p FA(Q)p FB(\))p FA(@)969 1648 y FD(\003)1009 1644 y Fx(\013)1059 1689 y FA(@)1115 1648 y Fx(\014)1163 1689 y FA(\025)1220 1704 y Fx(a)1284 1689 y Fz(\000)22 b FA(')p FB(\()p FA(\015)5 b FB(\))p FA(\025)1636 1648 y FD(\003)1636 1714 y Fx(a)1678 1689 y FA(@)1734 1648 y FD(\003)1774 1644 y Fx(\013)1823 1689 y FA(@)1879 1648 y Fx(\014)1928 1689 y FA(\025)1985 1704 y Fx(a)2026 1689 y Fz(jj)27 b(\024)h(jj)p FA(')p FB(\()p FA(aQ)p FB(\))22 b Fz(\000)h FA(')p FB(\()p FA(\015)5 b FB(\))p Fz(jj)21 b(\001)h(jj)p FA(@)3095 1648 y FD(\003)3134 1644 y Fx(\013)3184 1689 y FA(@)3240 1648 y Fx(\014)3288 1689 y Fz(jj)324 1910 y FB(The)40 b(righ)m(t)e(hand)h(side)g(tends)h(to)e(zero)i(when)g FA(a)f FB(tends)h(to)e FA(\015)5 b FB(.)63 b(On)39 b(the)g(other)g (hand,)324 2030 y FA(')p FB(\()p FA(\015)5 b FB(\))p FA(\025)577 1994 y FD(\003)577 2055 y Fx(a)618 2030 y FA(@)674 1994 y FD(\003)714 1987 y Fx(\013)764 2030 y FA(@)820 1994 y Fx(\014)868 2030 y FA(\025)925 2045 y Fx(a)1009 2030 y FB(is)41 b(constan)m(t)i(for)f Fz(j)p FA(a)p Fz(j)h(\025)i FA(\013)q FB(.)72 b(Th)m(us,)46 b(it)41 b(su\016ces)j(to)e(c)m(ho)s(ose)h Fz(j)p FA(a)3373 2045 y Fy(0)3413 2030 y Fz(j)g(\025)324 2150 y FB(max)o Fz(f)p FA(\013)617 2165 y Fx(k)688 2150 y Fz(j)27 b FA(k)k FB(=)c(1)p FA(;)17 b(:)g(:)g(:)f(;)h(n)p Fz(g)28 b FB(in)g(the)h (statemen)m(t)g(of)f(the)h(lemma)d(to)i(end)h(the)g(pro)s(of.)139 b Ff(\003)324 2391 y Fr(Remark:)72 b FB(As)43 b(seen)h(in)d(the)i(pro)s (of,)h(one)f(ma)m(y)f(c)m(ho)s(ose)i(an)m(y)f FA(a)2761 2406 y Fy(0)2843 2391 y FB(suc)m(h)h(that)e Fz(j)p FA(a)3373 2406 y Fy(0)3412 2391 y Fz(j)i(\025)324 2511 y FB(deg\()p FA(P)14 b FB(\).)53 b(On)37 b(the)f(other)g(hand,)i(w)m(e)f(stress)h (that)e(the)g(limit)d(is)i(not)h(a)g(m)m(ultiplicativ)m(e)324 2632 y(function)c(of)g FA(T)14 b FB(.)43 b(F)-8 b(or)32 b(instance,)925 2852 y(u-)7 b(lim)1012 2912 y Fx(a)p FD(!)p Fx(\015)1177 2852 y FA(\025)1234 2811 y FD(\003)1234 2876 y Fx(a)1276 2852 y FA(@)1332 2811 y FD(\003)1372 2852 y FA(@)e(\025)1485 2867 y Fx(a)1555 2852 y Fz(6)p FB(=)28 b(\(u-)6 b(lim)1783 2912 y Fx(a)p FD(!)p Fx(\015)1948 2852 y FA(\025)2005 2811 y FD(\003)2005 2876 y Fx(a)2047 2852 y FA(@)2103 2811 y FD(\003)2143 2852 y FA(\025)2200 2867 y Fx(a)2242 2852 y FB(\))22 b Fz(\001)f FB(\(u-)7 b(lim)2476 2912 y Fx(a)p FD(!)p Fx(\015)2641 2852 y FA(\025)2698 2811 y FD(\003)2698 2876 y Fx(a)2740 2852 y FA(@)e(\025)2853 2867 y Fx(a)2895 2852 y FB(\))p FA(:)324 3140 y FB(Therefore)42 b(to)f(describ)s(e)h(the)f(morphism)f(of)h(the)g(algebra)f Fu(C)18 b FB(\()2700 3114 y Fw(b)2697 3140 y FB(\000)o(\))42 b(on)m(to)f(its)f(quotien)m(t)324 3270 y Fu(C)17 b FB(\()448 3245 y Fw(b)445 3270 y FB(\000\))p FA(=)p Fs(K)g FB(\(\000\))37 b(w)m(e)31 b(ha)m(v)m(e)h(to)f(impro)m(v)m(e)f(our)h(de\014nition)e(of) i(the)g(lo)s(calizations)c(at)j(in\014nit)m(y)-8 b(.)324 3603 y FC(4)161 b(Extensions)52 b(to)1578 3566 y Fb(e)1577 3603 y Fd(\000)324 3822 y Fr(4.1.)e(Op)s(erators)g(on)g FA(`)1297 3786 y Fy(2)1336 3822 y FB(\()1377 3797 y Fw(e)1374 3822 y FB(\000\))p Fr(.)76 b FB(The)44 b(space)g FA(`)2103 3786 y Fy(2)2142 3822 y FB(\()2183 3797 y Fw(e)2180 3822 y FB(\000\))f(is)g(de\014ned)i(similarly)39 b(to)k FA(`)3358 3786 y Fy(2)3398 3822 y FB(\(\000\).)324 3952 y(Since)d(\000)h Fz(\032)809 3927 y Fw(e)806 3952 y FB(\000,)h(w)m(e)f(ha)m(v)m(e)h FA(`)1361 3916 y Fy(2)1400 3952 y FB(\(\000\))f FA(,)-17 b Fz(!)41 b FA(l)1760 3916 y Fy(2)1799 3952 y FB(\()1840 3927 y Fw(e)1837 3952 y FB(\000\).)67 b(As)41 b(b)s(efore,)h(w)m(e)f (em)m(b)s(ed)2981 3927 y Fw(e)2979 3952 y FB(\000)f(in)f FA(`)3242 3916 y Fy(2)3282 3952 y FB(\()3323 3927 y Fw(e)3320 3952 y FB(\000\))h(b)m(y)324 4082 y(sending)30 b FA(x)g FB(on)g FA(\037)953 4098 y FD(f)p Fx(x)p FD(g)1098 4082 y FB(and)g(w)m(e)h(notice)e(that)1922 4057 y Fw(e)1919 4082 y FB(\000)h(is)g(an)f(orthonormal)f(basis)i(of)f FA(`)3180 4046 y Fy(2)3220 4082 y FB(\()3261 4057 y Fw(e)3258 4082 y FB(\000\).)42 b(W)-8 b(e)324 4221 y(no)m(w)33 b(de\014ne)818 4195 y Fw(e)809 4221 y FA(@)g FB(:)28 b FA(l)979 4185 y Fy(2)1019 4221 y FB(\()1060 4196 y Fw(e)1057 4221 y FB(\000\))f Fz(!)h FA(l)1342 4185 y Fy(2)1381 4221 y FB(\()1422 4196 y Fw(e)1419 4221 y FB(\000\))33 b(b)m(y)1352 4458 y(\()1399 4432 y Fw(e)1390 4458 y FA(@)5 b(f)11 b FB(\)\()p FA(x)p FB(\))28 b(=)g FA(f)1865 4417 y FD(0)1888 4458 y FB(\()p FA(x)p FB(\))g(=)2155 4364 y Fw(X)2150 4576 y Fx(y)2187 4558 y Fv(0)2210 4576 y Fy(=)p Fx(x)2321 4458 y FA(f)11 b FB(\()p FA(y)t FB(\))p FA(:)324 4805 y FB(F)-8 b(or)43 b FA(\013)48 b Fz(2)g Fs(N)9 b FB(,)53 b(w)m(e)45 b(set)g FA(f)1258 4769 y Fy(\()p Fx(\013)p Fy(\))1409 4805 y FB(=)1541 4779 y Fw(e)1532 4805 y FA(@)1588 4769 y Fx(\013)1639 4805 y FA(f)11 b FB(.)77 b(This)44 b(notation)f(is)h(consisten)m(t)h(with)f (our)g(old)324 4926 y(de\014nition)31 b(of)h FA(x)923 4890 y Fy(\()p Fx(\013)p Fy(\))1028 4926 y FB(,)h(the)g(restriction)e (of)i FA(x)g FB(to)f Fs(Z)2110 4941 y FD(j)p Fx(x)p FD(j\000)p Fx(\013)2290 4926 y FB(.)1894 5251 y(17)p eop end %%Page: 18 18 TeXDict begin 18 17 bop 470 548 a FB(W)-8 b(e)35 b(notice)f(that)1151 522 y Fw(e)1142 548 y FA(@)41 b FB(is)33 b(a)i(b)s(ounded)g(op)s (erator)f(on)g FA(`)2390 512 y Fy(2)2429 548 y FB(\()2470 523 y Fw(e)2467 548 y FB(\000\))h(and)f(its)g(adjoin)m(t)3271 522 y Fw(e)3262 548 y FA(@)3318 512 y FD(\003)3393 548 y FB(acts)324 668 y(as)f(follo)m(ws:)1576 789 y(\()1623 762 y Fw(e)1614 789 y FA(@)1670 748 y FD(\003)1710 789 y FA(f)11 b FB(\)\()p FA(x)p FB(\))28 b(=)f FA(f)11 b FB(\()p FA(x)2221 748 y FD(0)2245 789 y FB(\))p FA(:)324 969 y FB(Moreo)m(v)m(er,)788 943 y Fw(e)779 969 y FA(@)835 933 y FD(\003)875 969 y FA(=)924 898 y Fz(p)p 1007 898 55 4 v 71 x FA(\027)39 b FB(is)32 b(an)h(isometry)f(on)g FA(`)1903 933 y Fy(2)1942 969 y FB(\()1983 944 y Fw(e)1980 969 y FB(\000\),)h(in)f(other)g(terms:)324 1190 y(\(4.12\))1724 1163 y Fw(e)1715 1190 y FA(@)1781 1163 y Fw(e)1772 1190 y FA(@)1828 1149 y FD(\003)1896 1190 y FB(=)27 b FA(\027)6 b FB(Id)q FA(:)324 1410 y FB(This)33 b(implies)d Fz(jj)943 1384 y Fw(e)934 1410 y FA(@)5 b Fz(jj)27 b FB(=)g Fz(jj)1241 1384 y Fw(e)1232 1410 y FA(@)1288 1374 y FD(\003)1328 1410 y Fz(jj)g FB(=)h FA(\027)6 b FB(.)470 1542 y(W)-8 b(e)28 b(denote)h(b)m(y)1100 1515 y Fw(e)1074 1542 y Fu(D)37 b FB(the)28 b FA(C)1428 1506 y FD(\003)1467 1542 y FB(-algebra)e(generated)j(b)m(y)2416 1516 y Fw(e)2407 1542 y FA(@)k FB(and)28 b(b)m(y)2834 1515 y Fw(e)2807 1542 y Fu(D)2884 1557 y Fy(alg)3007 1542 y FB(the)g Fz(\003)p FB(-algebra)324 1676 y(generated)33 b(b)m(y)913 1650 y Fw(e)904 1676 y FA(@)6 b FB(;)32 b(b)s(oth)h(of)f(them)g(are)g (unital.)324 1840 y Fr(4.2.)56 b(Some)f(tec)m(hnicalities.)89 b FB(The)49 b(next)h(lemma)c(will)g(allo)m(w)h(us)i(to)f(mak)m(e)h(the) 324 1960 y(connection)33 b(b)s(et)m(w)m(een)h Fu(D)1265 1975 y Fy(a)p Fx(l)q(g)1395 1960 y FB(and)1611 1933 y Fw(e)1585 1960 y Fu(D)1662 1975 y Fy(a)p Fx(l)q(g)1759 1960 y FB(.)324 2154 y Fr(Lemma)j(4.1)49 b Fq(We)35 b(have)f(for)h Fz(j)p FA(a)p Fz(j)27 b(\025)h FA(\013)q Fq(:)1307 2377 y Fr(1)1363 2392 y Fy(\000)1411 2377 y FA(\025)1468 2336 y FD(\003)1468 2401 y Fx(a)1509 2377 y FA(@)1565 2336 y FD(\003)1606 2331 y Fx(\013)1655 2377 y FA(@)1711 2336 y Fx(\014)1759 2377 y FA(\025)1816 2392 y Fx(a)1858 2377 y Fr(1)1914 2392 y Fy(\000)1990 2377 y FB(=)f Fr(1)2149 2392 y Fy(\000)2218 2351 y Fw(e)2198 2377 y FA(@)2254 2348 y FD(\003)2294 2305 y Fx(\013)2352 2351 y Fw(e)2343 2377 y FA(@)2399 2336 y Fx(\014)2447 2377 y Fr(1)2503 2392 y Fy(\000)2552 2377 y FA(:)324 2596 y Fr(Pro)s(of:)50 b FB(F)-8 b(or)31 b FA(f)39 b Fz(2)28 b FA(`)1073 2560 y Fy(2)1112 2596 y FB(\()1153 2571 y Fw(e)1150 2596 y FB(\000\),)33 b(w)m(e)g(ha)m(v)m(e:)384 2822 y(\()p Fr(1)478 2837 y Fy(\000)546 2796 y Fw(e)526 2822 y FA(@)582 2794 y FD(\003)622 2750 y Fx(\013)681 2796 y Fw(e)672 2822 y FA(@)728 2781 y Fx(\014)776 2822 y Fr(1)832 2837 y Fy(\000)880 2822 y FA(f)11 b FB(\)\()p FA(x)p FB(\))83 b(=)g Fr(1)1406 2837 y Fy(\000)1454 2822 y FB(\()p FA(x)p FB(\))1642 2728 y Fw(X)1602 2951 y Fx(y)r Fy(=)p Fx(x)1734 2931 y Fh(\()p Fo(\013)p Fh(\))1826 2822 y FB(\()1873 2796 y Fw(e)1864 2822 y FA(@)1920 2781 y Fx(\014)1968 2822 y Fr(1)2024 2837 y Fy(\000)2072 2822 y FA(f)11 b FB(\)\()p FA(y)t FB(\))27 b(=)g Fr(1)2483 2837 y Fy(\000)2532 2822 y FB(\()p FA(x)p FB(\))2720 2728 y Fw(X)2680 2951 y Fx(y)r Fy(=)p Fx(x)2812 2931 y Fh(\()p Fo(\013)p Fh(\))2957 2728 y Fw(X)2920 2951 y Fx(z)2956 2931 y Fh(\()p Fo(\014)s Fh(\))3045 2951 y Fy(=)p Fx(y)3137 2822 y FB(\()p Fr(1)3231 2837 y Fy(\000)3280 2822 y FA(f)11 b FB(\)\()p FA(z)t FB(\))1191 3111 y(=)83 b Fr(1)1406 3126 y Fy(\000)1454 3111 y FB(\()p FA(x)p FB(\))1687 3017 y Fw(X)1602 3240 y Fx(y)1639 3220 y Fh(\()p Fo(\014)s Fh(\))1729 3240 y Fy(=)p Fx(x)1824 3220 y Fh(\()p Fo(\013)p Fh(\))1932 3111 y Fr(1)1988 3126 y Fy(\000)2036 3111 y FB(\()p FA(y)t FB(\))p FA(f)11 b FB(\()p FA(y)t FB(\))p FA(:)324 3434 y FB(The)44 b(pro)s(of)e(of)g(the)h(lemma)e(is)h(\014nished)i(b)m(y)f (comparing)f(this)g(with)h(\(3.11\))f(and)h(b)m(y)324 3555 y(taking)24 b(in)m(to)g(accoun)m(t)i(the)f(next)h(result,)g(whose) h(pro)s(of)d(is)g(iden)m(tical)f(to)i(that)g(of)f(Lemma)324 3675 y(3.6.)324 3869 y Fr(Lemma)37 b(4.2)49 b Fq(On)1080 3844 y Fw(e)1077 3869 y FB(\000)p Fq(,)35 b(let)g FA(x)28 b Fz(2)g FB(\000)35 b Fq(we)f(have:)456 4193 y Fz(f)p FA(y)c Fz(2)e FB(\000)g Fz(j)f FA(y)874 4152 y Fy(\()p Fx(\014)s Fy(\))1003 4193 y FB(=)h FA(x)1162 4152 y Fy(\()p Fx(\013)p Fy(\))1266 4193 y Fz(g)g FB(=)1448 3989 y Fw(8)1448 4078 y(<)1448 4258 y(:)1578 4070 y FB(\037)354 b Fq(for)34 b Fz(j)p FA(x)p Fz(j)22 b FB(+)g FA(\014)28 b Fz(\000)23 b FA(\013)28 b(<)f FB(0)p FA(;)1578 4192 y(S)1644 4156 y FD(j)p Fx(x)p FD(j)p Fy(+)p Fx(\014)s FD(\000)p Fx(\013)2008 4192 y Fq(for)34 b Fz(j)p FA(x)p Fz(j)28 b FA(<)f(\013)36 b Fq(and)e Fz(j)p FA(x)p Fz(j)22 b FB(+)g FA(\014)28 b Fz(\000)22 b FA(\013)29 b Fz(\025)f FB(0)p FA(;)1578 4314 y(x)1633 4278 y Fy(\()p Fx(\013)p Fy(\))1737 4314 y FA(S)1803 4278 y Fx(\014)2008 4314 y Fq(for)34 b Fz(j)p FA(x)p Fz(j)28 b(\025)g FA(\013)35 b Fq(and)g Fz(j)p FA(x)p Fz(j)22 b FB(+)g FA(\014)27 b Fz(\000)c FA(\013)28 b Fz(\025)h FB(0)p FA(:)324 4546 y FB(W)-8 b(e)33 b(will)d(also)i(need) h(a)g(result)f(concerning)h(the)g(lo)s(calization)28 b(of)k(the)h(norm)f(on)3321 4519 y Fw(e)3295 4546 y Fu(D)3372 4561 y Fy(a)p Fx(l)q(g)3469 4546 y FB(.)324 4755 y Fr(Lemma)37 b(4.3)49 b Fq(L)-5 b(et)1097 4730 y Fw(e)1081 4755 y FA(T)42 b Fz(2)1300 4728 y Fw(e)1274 4755 y Fu(D)1351 4770 y Fx(al)q(g)1450 4755 y Fq(.)j(We)35 b(have:)1578 4977 y Fz(jj)1650 4952 y Fw(e)1634 4977 y FA(T)13 b Fz(jj)27 b FB(=)h Fz(jj)p Fr(1)2003 4992 y Fy(\000)2066 4952 y Fw(e)2050 4977 y FA(T)14 b Fr(1)2177 4992 y Fy(\000)2225 4977 y Fz(jj)p FA(:)1894 5251 y FB(18)p eop end %%Page: 19 19 TeXDict begin 19 18 bop 324 549 a Fr(Pro)s(of:)70 b FB(Because)43 b(of)e(\(4.12\),)j(w)m(e)e(can)g(supp)s(ose)h(that)2472 524 y Fw(e)2456 549 y FA(T)57 b FB(=)2689 474 y Fw(P)2794 500 y Fx(n)2794 578 y(k)r Fy(=1)2943 549 y FA(c)2985 564 y Fx(k)3049 523 y Fw(e)3028 549 y FA(@)3084 520 y FD(\003)3124 477 y Fx(\013)3169 489 y Fo(k)3221 523 y Fw(e)3212 549 y FA(@)3268 513 y Fx(\014)3308 525 y Fo(k)3351 549 y FA(:)41 b FB(Let)324 683 y FA(\013)28 b FB(=)g(max)o Fz(f)p FA(\013)811 698 y Fx(k)881 683 y Fz(j)g FA(k)j FB(=)c(1)p FA(;)17 b(:)g(:)g(:)f(;)h(n)p Fz(g)p FB(.)41 b(F)-8 b(or)27 b(eac)m(h)i FA(")e(>)h FB(0,)g(there)h(is)e FA(g)k Fz(2)d FA(`)2783 647 y Fy(2)2822 683 y FB(\()2863 658 y Fw(e)2860 683 y FB(\000\))g(with)f(compact)324 804 y(supp)s(ort)33 b(suc)m(h)h(that)e Fz(jj)p FA(g)t Fz(jj)26 b FB(=)h(1)33 b(and)1572 1034 y Fz(jj)1644 1009 y Fw(e)1628 1034 y FA(T)14 b(g)t Fz(jj)26 b(\025)i(jj)2009 1009 y Fw(e)1993 1034 y FA(T)13 b Fz(jj)22 b(\000)g FA(":)324 1254 y FB(Note)47 b(that)g(if)f FA(y)952 1269 y Fy(1)991 1254 y FA(;)17 b(y)1083 1269 y Fy(2)1122 1254 y FA(;)g(:)g(:)g(:)f(;)h (y)1389 1269 y Fx(m)1502 1254 y FB(are)47 b(distinct)f(p)s(oin)m(ts)h (of)g(\000,)k(and)c FA(a)2874 1269 y Fy(1)2914 1254 y FA(;)17 b(a)3009 1269 y Fy(2)3048 1254 y FA(;)g(:)g(:)g(:)f(;)h(a)3318 1269 y Fx(m)3432 1254 y FB(are)324 1374 y(complex)32 b(n)m(um)m(b)s(ers)h(and)g FA(x)1344 1389 y Fy(1)1384 1374 y FA(;)17 b(x)1483 1389 y Fy(2)1550 1374 y Fz(2)1647 1349 y Fw(e)1644 1374 y FB(\000,)33 b(w)m(e)g(ha)m(v)m(e)324 1664 y(\(4.13\))479 b Fz(jj)1166 1540 y Fx(m)1125 1570 y Fw(X)1140 1780 y Fx(i)p Fy(=1)1286 1664 y FA(a)1337 1679 y Fx(i)1365 1664 y FA(x)1420 1679 y Fy(1)1460 1664 y FA(y)1508 1679 y Fx(i)1536 1664 y Fz(jj)1592 1623 y Fy(2)1658 1664 y FB(=)1803 1540 y Fx(m)1762 1570 y Fw(X)1777 1780 y Fx(i)p Fy(=1)1922 1664 y Fz(j)p FA(a)2001 1679 y Fx(i)2029 1664 y Fz(j)2057 1623 y Fy(2)2124 1664 y FB(=)28 b Fz(jj)2340 1540 y Fx(m)2300 1570 y Fw(X)2315 1780 y Fx(i)p Fy(=1)2460 1664 y FA(a)2511 1679 y Fx(i)2540 1664 y FA(x)2595 1679 y Fy(2)2634 1664 y FA(y)2682 1679 y Fx(i)2710 1664 y Fz(jj)2766 1623 y Fy(2)2805 1664 y FA(:)324 1995 y FB(Th)m(us,)42 b(since)d FA(g)j FB(has)e(compact)e (supp)s(ort,)j(there)f(are)e FA(x)h Fz(2)2539 1970 y Fw(e)2536 1995 y FB(\000,)i(\()p FA(y)2751 2010 y Fx(i)2778 1995 y FB(\))2816 2010 y Fx(i)p Fy(=1)p Fx(;:::)o(;m)3133 1995 y Fz(2)e FB(\000)3299 1959 y Fx(m)3404 1995 y FB(and)324 2115 y(\()p FA(a)413 2130 y Fx(i)441 2115 y FB(\))479 2130 y Fx(i)p Fy(=1)p Fx(;:::)o(;m)811 2115 y Fz(2)54 b Fs(C)997 2079 y Fx(m)1117 2115 y FB(suc)m(h)49 b(that)f Fz(j)p FA(y)1655 2130 y Fx(i)1682 2115 y Fz(j)53 b(\025)h FA(\013)48 b FB(and)g FA(g)57 b FB(=)2442 2040 y Fw(P)2547 2067 y Fx(m)2547 2144 y(k)r Fy(=1)2697 2115 y FA(a)2748 2130 y Fx(i)2776 2115 y FA(xy)2879 2130 y Fx(i)2907 2115 y FB(.)89 b(W)-8 b(e)48 b(set)h FA(f)64 b FB(=)324 2161 y Fw(P)429 2187 y Fx(m)429 2265 y(k)r Fy(=1)578 2236 y FA(a)629 2251 y Fx(i)658 2236 y FA(ey)751 2251 y Fx(i)779 2236 y FB(.)73 b(Then)44 b(\(4.13\))e(giv)m(es)h(us)g Fz(jj)p FA(f)11 b Fz(jj)44 b FB(=)g Fz(jj)p FA(g)t Fz(jj)f FB(=)i(1.)73 b(Since)43 b Fz(j)p FA(y)2972 2251 y Fx(i)2999 2236 y Fz(j)i(\025)g FA(\013)f FB(for)e(all)324 2370 y FA(i)28 b FB(=)f(1)17 b FA(:)g(:)g(:)f(m)p FB(,)33 b(w)m(e)g(obtain)f(that)g FA(f)39 b Fz(2)28 b FA(`)1710 2334 y Fy(2)1749 2370 y FB(\(\000\))k(and)h(that)2335 2345 y Fw(e)2319 2370 y FA(T)14 b(f)39 b Fz(2)28 b FA(`)2612 2334 y Fy(2)2651 2370 y FB(\(\000\).)470 2490 y(Moreo)m(v)m(er,)45 b(using)40 b(once)h(more)g(the)g(fact)f(that)h Fz(j)p FA(y)2354 2505 y Fx(i)2381 2490 y Fz(j)h(\025)g FA(\013)f FB(and)g(\(4.13\),)h(w)m(e)g(ha)m(v)m(e,)324 2611 y(with)32 b FA(z)g Fz(2)d FB(\000,)324 2901 y Fz(jj)396 2876 y Fw(e)380 2901 y FA(T)13 b(g)t Fz(jj)82 b FB(=)g Fz(jj)920 2777 y Fx(n)869 2807 y Fw(X)877 3019 y Fx(k)r Fy(=1)1071 2777 y Fx(m)1030 2807 y Fw(X)1045 3017 y Fx(i)p Fy(=1)1190 2901 y FA(c)1232 2916 y Fx(k)1275 2901 y FA(a)1326 2916 y Fx(i)1375 2875 y Fw(e)1354 2901 y FA(@)1410 2872 y FD(\003)1451 2829 y Fx(\013)1496 2841 y Fo(k)1547 2875 y Fw(e)1538 2901 y FA(@)1594 2860 y Fx(\014)1634 2872 y Fo(k)1677 2901 y FA(xy)1780 2916 y Fx(i)1808 2901 y Fz(jj)27 b FB(=)h Fz(jj)2117 2777 y Fx(n)2067 2807 y Fw(X)2074 3019 y Fx(k)r Fy(=1)2268 2777 y Fx(m)2227 2807 y Fw(X)2242 3017 y Fx(i)p Fy(=1)2420 2807 y Fw(X)2388 3022 y FD(j)p Fx(z)s FD(j)p Fy(=)p Fx(\014)2559 3034 y Fo(k)2612 2901 y FA(c)2654 2916 y Fx(k)2697 2901 y FA(a)2748 2916 y Fx(i)2776 2901 y FB(\()p FA(xy)2917 2916 y Fx(i)2945 2901 y FB(\))2983 2860 y Fy(\()p Fx(\013)3055 2872 y Fo(k)3094 2860 y Fy(\))3125 2901 y FA(z)t Fz(jj)639 3245 y FB(=)82 b Fz(jj)920 3121 y Fx(n)869 3151 y Fw(X)877 3363 y Fx(k)r Fy(=1)1071 3121 y Fx(m)1030 3151 y Fw(X)1045 3361 y Fx(i)p Fy(=1)1222 3151 y Fw(X)1190 3367 y FD(j)p Fx(z)s FD(j)p Fy(=)p Fx(\014)1361 3379 y Fo(k)1414 3245 y FA(c)1456 3260 y Fx(k)1499 3245 y FA(a)1550 3260 y Fx(i)1578 3245 y FA(x)p FB(\()p FA(y)1719 3260 y Fx(i)1748 3245 y FB(\))1786 3204 y Fy(\()p Fx(\013)1858 3216 y Fo(k)1896 3204 y Fy(\))1928 3245 y FA(z)t Fz(jj)28 b FB(=)f Fz(jj)2287 3121 y Fx(n)2236 3151 y Fw(X)2244 3363 y Fx(k)r Fy(=1)2437 3121 y Fx(m)2397 3151 y Fw(X)2412 3361 y Fx(i)p Fy(=1)2589 3151 y Fw(X)2557 3367 y FD(j)p Fx(z)s FD(j)p Fy(=)p Fx(\014)2728 3379 y Fo(k)2781 3245 y FA(c)2823 3260 y Fx(k)2866 3245 y FA(a)2917 3260 y Fx(i)2945 3245 y FA(e)p FB(\()p FA(y)3076 3260 y Fx(i)3104 3245 y FB(\))3142 3204 y Fy(\()p Fx(\013)3214 3216 y Fo(k)3253 3204 y Fy(\))3285 3245 y FA(z)t Fz(jj)639 3589 y FB(=)82 b Fz(jj)920 3465 y Fx(n)869 3495 y Fw(X)877 3707 y Fx(k)r Fy(=1)1071 3465 y Fx(m)1030 3495 y Fw(X)1045 3705 y Fx(i)p Fy(=1)1222 3495 y Fw(X)1190 3711 y FD(j)p Fx(z)s FD(j)p Fy(=)p Fx(\014)1361 3723 y Fo(k)1414 3589 y FA(c)1456 3604 y Fx(k)1499 3589 y FA(a)1550 3604 y Fx(i)1578 3589 y FB(\()p FA(ey)1709 3604 y Fx(i)1737 3589 y FB(\))1775 3548 y Fy(\()p Fx(\013)1847 3560 y Fo(k)1886 3548 y Fy(\))1918 3589 y FA(z)t Fz(jj)28 b FB(=)f Fz(jj)2277 3465 y Fx(n)2226 3495 y Fw(X)2234 3707 y Fx(k)r Fy(=1)2427 3465 y Fx(m)2387 3495 y Fw(X)2401 3705 y Fx(i)p Fy(=1)2547 3589 y FA(c)2589 3604 y Fx(k)2632 3589 y FA(a)2683 3604 y Fx(i)2732 3563 y Fw(e)2711 3589 y FA(@)2767 3561 y FD(\003)2807 3517 y Fx(\013)2852 3529 y Fo(k)2904 3563 y Fw(e)2895 3589 y FA(@)2951 3548 y Fx(\014)2991 3560 y Fo(k)3034 3589 y FA(ey)3127 3604 y Fx(i)3155 3589 y Fz(jj)g FB(=)g Fz(jj)3413 3564 y Fw(e)3397 3589 y FA(T)13 b(f)e Fz(jj)p FA(:)324 3942 y FB(Hence,)34 b(there)f(is)f FA(f)39 b Fz(2)28 b FA(`)1210 3906 y Fy(2)1249 3942 y FB(\()1290 3917 y Fw(e)1287 3942 y FB(\000\))k(suc)m(h)i(that) 859 4172 y Fz(jj)p Fr(1)971 4187 y Fy(\000)1034 4147 y Fw(e)1019 4172 y FA(T)13 b Fr(1)1145 4187 y Fy(\000)1194 4172 y FA(f)e Fz(jj)26 b FB(=)i Fz(jj)p Fr(1)1551 4187 y Fy(\000)1614 4147 y Fw(e)1599 4172 y FA(T)13 b(f)e Fz(jj)27 b FB(=)h Fz(jj)1987 4147 y Fw(e)1971 4172 y FA(T)13 b(f)e Fz(jj)26 b FB(=)i Fz(jj)2358 4147 y Fw(e)2342 4172 y FA(T)13 b(g)t Fz(jj)26 b(\025)i(jj)2722 4147 y Fw(e)2706 4172 y FA(T)14 b Fz(jj)21 b(\000)i FA(":)324 4392 y FB(This)33 b(sho)m(ws)h(us)f(that)1577 4513 y Fz(jj)p Fr(1)1689 4528 y Fy(\000)1752 4487 y Fw(e)1737 4513 y FA(T)13 b Fr(1)1863 4528 y Fy(\000)1912 4513 y Fz(jj)27 b(\025)h(jj)2172 4487 y Fw(e)2156 4513 y FA(T)13 b Fz(jj)p FA(:)324 4687 y FB(The)33 b(rev)m(erse)i(inequalit)m(y)d(is)g (ob)m(vious.)98 b Ff(\003)1894 5251 y FB(19)p eop end %%Page: 20 20 TeXDict begin 20 19 bop 324 548 a FC(5)161 b(Main)55 b(results)324 767 y Fr(5.1.)43 b(Morphism.)58 b FB(In)37 b(the)h(follo)m(wing,)e(a)i(morphism)d(will)g(b)s(e)j(a)f(morphism)f (of)h FA(C)3490 731 y FD(\003)3529 767 y FB(-)324 887 y(algebras.)63 b(T)-8 b(o)39 b(describ)s(e)h(to)f(quotien)m(t)h Fu(C)17 b FB(\()1938 862 y Fw(b)1935 887 y FB(\000\))p FA(=)p Fs(K)g FB(\(\000\),)47 b(w)m(e)41 b(need)f(to)f(\014nd)h(an)f (adapted)324 1008 y(morphism.)i(W)-8 b(e)33 b(ha)m(v)m(e)h(the)f(follo) m(wing)d(result.)324 1211 y Fr(Theorem)37 b(5.1)49 b Fq(F)-7 b(or)31 b(e)-5 b(ach)32 b FA(\015)h Fz(2)28 b FA(@)5 b FB(\000)33 b Fq(ther)-5 b(e)32 b(is)g(a)g(unique)h(morphism)e FB(\010)2973 1226 y Fx(\015)3045 1211 y FB(:)d Fu(C)17 b FB(\()3224 1186 y Fw(b)3221 1211 y FB(\000\))28 b Fz(!)3502 1184 y Fw(e)3475 1211 y Fu(D)324 1346 y Fq(such)45 b(that)i FB(\010)837 1361 y Fx(\015)881 1346 y FB(\()p FA(@)5 b FB(\))49 b(=)1195 1319 y Fw(e)1186 1346 y FA(@)j Fq(and)45 b FB(\010)1559 1361 y Fx(\015)1604 1346 y FB(\()p FA(')p FB(\()p FA(Q)p FB(\)\))j(=)g FA(')p FB(\()p FA(\015)5 b FB(\))p FA(;)45 b Fq(for)h(al)5 b(l)45 b FA(')j Fz(2)h FA(C)7 b FB(\()2999 1320 y Fw(b)2996 1346 y FB(\000)o(\))p Fq(.)78 b(One)45 b(has)324 1466 y FB(\010)394 1481 y Fx(\015)439 1466 y FB(\()p Fs(K)17 b FB(\(\000\)\))34 b(=)27 b(0)p FA(:)324 1682 y Fr(Pro)s(of:)57 b FB(Let)36 b FA(T)46 b Fz(2)34 b Fu(C)17 b FB(\()1189 1657 y Fw(b)1186 1682 y FB(\000\))1285 1697 y Fy(alg)1380 1682 y FB(.)53 b(W)-8 b(e)36 b(k)m(eep)h(the)g(notations)d(in)m(tro)s(duced)i(b)s (efore)g(Lemma)324 1803 y(3.5.)53 b(Then)37 b(b)m(y)f(Lemma)f(3.5)g(w)m (e)i(ha)m(v)m(e)g(u-lim)2034 1818 y Fx(a)p FD(!)p Fx(\015)2204 1803 y FA(\025)2261 1766 y FD(\003)2261 1827 y Fx(a)2302 1803 y FA(T)14 b(\025)2430 1818 y Fx(a)2505 1803 y FB(=)33 b FA(\025)2671 1766 y FD(\003)2671 1827 y Fx(a)2708 1836 y Fh(0)2747 1803 y FA(T)14 b FB(\()p FA(\015)5 b FB(\))p FA(\025)3007 1818 y Fx(a)3044 1827 y Fh(0)3083 1803 y FB(.)53 b(W)-8 b(e)36 b(recall)324 1923 y(that)1068 2043 y FA(T)14 b FB(\()p FA(\015)5 b FB(\))28 b(=)f FA(P)14 b FB(\()p FA(')1581 2058 y Fy(1)1620 2043 y FB(\()p FA(\015)5 b FB(\))p FA(;)17 b(')1860 2058 y Fy(2)1899 2043 y FB(\()p FA(\015)5 b FB(\))p FA(;)17 b(:)g(:)g(:)e(;)i(')2313 2058 y Fx(m)2380 2043 y FB(\()p FA(\015)5 b FB(\))p FA(;)17 b(@)5 b(;)17 b(@)2712 2002 y FD(\003)2752 2043 y FB(\))p FA(:)324 2218 y FB(No)m(w)33 b(let)1084 2313 y Fw(e)1068 2338 y FA(T)14 b FB(\()p FA(\015)5 b FB(\))28 b(=)f FA(P)14 b FB(\()p FA(')1581 2353 y Fy(1)1620 2338 y FB(\()p FA(\015)5 b FB(\))p FA(;)17 b(')1860 2353 y Fy(2)1899 2338 y FB(\()p FA(\015)5 b FB(\))p FA(;)17 b(:)g(:)g(:)e(;)i(')2313 2353 y Fx(m)2380 2338 y FB(\()p FA(\015)5 b FB(\))p FA(;)2564 2312 y Fw(e)2556 2338 y FA(@)g(;)2665 2312 y Fw(e)2656 2338 y FA(@)2712 2297 y FD(\003)2752 2338 y FB(\))p FA(:)324 2512 y FB(F)-8 b(rom)31 b(Lemma)g(4.1)h(and)h(\(4.12\))f(it)f(follo)m (ws)h(that)g(one)h(can)g(c)m(ho)s(ose)g FA(a)2904 2527 y Fy(0)2976 2512 y FB(suc)m(h)h(that)1424 2742 y FA(\025)1481 2701 y FD(\003)1481 2767 y Fx(a)1518 2776 y Fh(0)1557 2742 y FA(T)14 b FB(\()p FA(\015)5 b FB(\))p FA(\025)1817 2757 y Fx(a)1854 2766 y Fh(0)1920 2742 y FB(=)28 b Fr(1)2080 2757 y Fy(\000)2144 2717 y Fw(e)2128 2742 y FA(T)14 b FB(\()p FA(\015)5 b FB(\))p Fr(1)2387 2757 y Fy(\000)2435 2742 y FA(:)324 2962 y FB(Using)32 b(Lemma)f(4.3)h(w)m(e)i(obtain:)1049 3182 y Fz(jj)1121 3157 y Fw(e)1105 3182 y FA(T)13 b FB(\()p FA(\015)5 b FB(\))p Fz(jj)82 b FB(=)h Fz(jj)p Fr(1)1716 3197 y Fy(\000)1779 3157 y Fw(e)1764 3182 y FA(T)13 b FB(\()p FA(\015)5 b FB(\))p Fr(1)2022 3197 y Fy(\000)2071 3182 y Fz(jj)27 b FB(=)g Fz(jj)p FA(\025)2370 3141 y FD(\003)2370 3207 y Fx(a)2407 3216 y Fh(0)2446 3182 y FA(T)14 b FB(\()p FA(\015)5 b FB(\))p FA(\025)2706 3197 y Fx(a)2743 3206 y Fh(0)2781 3182 y Fz(jj)1445 3328 y FB(=)83 b Fz(jj)p FB(u-)5 b(lim)1746 3388 y Fx(a)p FD(!)p Fx(\015)1911 3328 y FA(\025)1968 3287 y FD(\003)1968 3352 y Fx(a)2010 3328 y FA(T)14 b(\025)2138 3343 y Fx(a)2179 3328 y Fz(jj)27 b(\024)h(jj)p FA(T)14 b Fz(jj)p FA(:)324 3617 y FB(This)47 b(sho)m(ws)h(that)e(there)i(is)e(a)g(linear)g (mapping)f(\010)2322 3581 y Fy(0)2322 3642 y Fx(\015)2418 3617 y FB(:)52 b Fu(C)17 b FB(\()2621 3592 y Fw(b)2618 3617 y FB(\000\))2717 3632 y Fy(alg)2864 3617 y Fz(!)3042 3590 y Fw(e)3016 3617 y Fu(D)56 b FB(suc)m(h)48 b(that)324 3738 y(\010)394 3701 y Fy(0)394 3762 y Fx(\015)439 3738 y FB(\()p FA(T)14 b FB(\))51 b(=)h FA(T)14 b FB(\()p FA(\015)5 b FB(\))46 b(and)h(that)g(this)f(map)g(is)g(a)h(con)m (traction.)85 b(It)47 b(is)g(clear)f(that)g(it)g(is)324 3880 y(also)36 b(m)m(ultiplicativ)m(e.)53 b(The)37 b(densit)m(y)h(of)f Fu(C)17 b FB(\()1975 3855 y Fw(b)1972 3880 y FB(\000\))2071 3895 y Fy(alg)2203 3880 y FB(in)36 b Fu(C)17 b FB(\()2445 3855 y Fw(b)2442 3880 y FB(\000\))37 b(allo)m(ws)e(us)j(to)e(extend)j (\010)3517 3844 y Fy(0)3517 3905 y Fx(\015)324 4025 y FB(to)46 b(a)h(morphism)e(\010)1093 4040 y Fx(\015)1190 4025 y FB(:)52 b Fu(C)18 b FB(\()1394 4000 y Fw(b)1391 4025 y FB(\000)o(\))53 b Fz(!)1720 3998 y Fw(e)1693 4025 y Fu(D)k FB(whic)m(h)47 b(clearly)f(satis\014es)i(the)f(conditions)f (of)324 4146 y(the)36 b(theorem.)55 b(The)37 b(uniqueness)h(of)e(\010) 1808 4161 y Fx(\015)1889 4146 y FB(is)f(ob)m(vious)i(and)f(the)h(last)e (assertion)h(of)g(the)324 4266 y(theorem)c(follo)m(ws)f(from)h(the)h (Prop)s(osition)e(3.2.)140 b Ff(\003)324 4429 y Fr(5.2.)41 b(The)f(case)h FA(\027)e(>)32 b FB(1)p Fr(.)51 b FB(W)-8 b(e)36 b(shall)d(impro)m(v)m(e)i(the)h(theorem)f(under)h(the)f(h)m(yp)s (othesis)324 4550 y FA(\027)f(>)27 b FB(1,)33 b(whic)m(h)g(will)d(b)s (e)j(assumed)g(in)f(all)f(this)h(subsection.)470 4670 y(W)-8 b(e)37 b(recall)e(that)h(an)h(isometry)f(is)g(said)g(to)g(b)s(e) h Fq(pr)-5 b(op)g(er)36 b FB(if)f(it)g(is)h(not)h(unitary)-8 b(.)55 b(The)324 4790 y(op)s(erators)32 b FA(@)811 4754 y FD(\003)884 4790 y FB(and)1082 4764 y Fw(e)1073 4790 y FA(@)1129 4754 y FD(\003)1202 4790 y FB(are)h(prop)s(er)f(isometries) f(and)i(the)g(dimensions)f(of)g(the)g(k)m(ernels)324 4911 y(of)41 b FA(@)47 b FB(and)749 4884 y Fw(e)741 4911 y FA(@)g FB(are)41 b(in\014nite.)69 b(F)-8 b(or)41 b(instance)h(in)e (the)i(case)h(of)e FA(@)5 b FB(,)44 b(let)d FA(a;)17 b(b)42 b FB(b)s(e)f(di\013eren)m(t)1894 5251 y(20)p eop end %%Page: 21 21 TeXDict begin 21 20 bop 324 548 a FB(letters)33 b(of)f Fu(A)23 b FB(.)45 b(Cho)s(ose)34 b FA(g)e Fz(2)c FA(`)1463 512 y Fy(2)1503 548 y FB(\(\000)p FA(a)p FB(\))33 b(and)g FA(h)c Fz(2)g FA(`)2135 512 y Fy(2)2174 548 y FB(\(\000)p FA(b)p FB(\))k(suc)m(h)i(that)e FA(h)p FB(\()p FA(xb)p FB(\))c(=)f FA(g)t FB(\()p FA(xa)p FB(\))33 b(for)324 668 y(all)d FA(x)e Fz(2)g FB(\000,)33 b(then)g FA(g)26 b Fz(\000)c FA(h)33 b FB(is)f(in)g(Ker\()p FA(@)5 b FB(\).)470 789 y(Let)28 b Fs(T)j FB(b)s(e)c(the)h(unit)f(circle)g(of)g Fs(R)1647 753 y Fy(2)1719 789 y FB(and)h FA(H)1993 753 y Fy(2)2059 789 y FB(the)g(closure)g(of)f(the)h(subspace)h(spanned)324 909 y(b)m(y)h Fz(f)p FA(e)551 873 y Fx(inQ)678 909 y FA(;)17 b(n)27 b Fz(2)i Fs(N)9 b Fz(g)35 b FB(in)29 b FA(`)1205 873 y Fy(2)1244 909 y FB(\()p Fs(T)p FB(\).)46 b(F)-8 b(or)29 b FA(g)i Fz(2)d FA(L)1866 873 y FD(1)1941 909 y FB(\()p Fs(T)p FB(\),)33 b(w)m(e)e(de\014ne)g(the)f Fq(T)-7 b(o)i(eplitz)31 b(op)-5 b(er)g(ator)29 b FA(T)3521 924 y Fx(g)324 1029 y FB(on)36 b FA(H)552 993 y Fy(2)627 1029 y FB(b)m(y)h FA(T)823 1044 y Fx(g)863 1029 y FA(h)d FB(=)f FA(P)1125 1046 y Fx(H)1188 1027 y Fh(2)1227 1029 y FA(g)t(h;)i FB(where)i FA(P)1744 1046 y Fx(H)1807 1027 y Fh(2)1882 1029 y FB(is)f(the)g(pro)5 b(jection)36 b(on)g FA(H)2850 993 y Fy(2)2889 1029 y FB(.)53 b(W)-8 b(e)37 b(denote)g(b)m(y)324 1150 y Fu(T)60 b FB(the)31 b FA(C)701 1114 y FD(\003)741 1150 y FB(-algebra)e(generated)j(b)m(y)g FA(T)1747 1165 y Fx(z)1787 1150 y FB(.)43 b(The)32 b(next)f(theorem)g (is)g(due)g(to)g(Coburn)h(\(see)324 1270 y([Da)o(])h(for)f(a)g(pro)s (of)7 b(\).)324 1474 y Fr(Theorem)37 b(5.2)49 b Fq(If)38 b(S)h(is)g(pr)-5 b(op)g(er)38 b(isometry,)i(then)e(ther)-5 b(e)39 b(is)g(a)g(unique)f(isomorphism)324 1594 y FA(')d Fq(of)f Fu(T)64 b Fq(onto)35 b Fu(S)20 b Fq(,)34 b(the)h FA(C)1306 1558 y FD(\003)1346 1594 y Fq(-algebr)-5 b(a)34 b(gener)-5 b(ate)g(d)34 b(by)h FA(S)6 b Fq(,)34 b(such)h(that)g FA(')p FB(\()p FA(T)2980 1609 y Fx(z)3020 1594 y FB(\))28 b(=)f FA(S)6 b Fq(.)324 1809 y FB(Th)m(us,)40 b(w)m(e)e(see)h(that)e (there)h(is)e(a)h(unique)h(isomorphism)d FA(')i FB(of)g Fu(D)46 b FB(on)m(to)3061 1782 y Fw(e)3034 1809 y Fu(D)h FB(suc)m(h)39 b(that)324 1929 y FA(')p FB(\()p FA(@)5 b FB(\))28 b(=)g FA(')p FB(\()763 1903 y Fw(e)754 1929 y FA(@)5 b FB(\).)44 b(W)-8 b(e)33 b(can)g(rewrite)f(our)h(theorem)f (as)h(follo)m(ws.)324 2133 y Fr(Theorem)k(5.3)49 b Fq(L)-5 b(et)48 b FA(\027)57 b(>)51 b FB(1)d Fq(and)f(let)g FA(\015)57 b Fz(2)51 b FA(@)5 b FB(\000)p Fq(.)84 b(Ther)-5 b(e)47 b(is)g(a)g(unique)h(morphism)324 2253 y FB(\010)394 2268 y Fx(\015)477 2253 y FB(:)38 b Fu(C)17 b FB(\()666 2228 y Fw(b)663 2253 y FB(\000\))37 b Fz(!)h Fu(D)50 b Fq(such)40 b(that)h FB(\010)1566 2268 y Fx(\015)1611 2253 y FB(\()p FA(')p FB(\()p FA(Q)p FB(\)\))d(=)f FA(')p FB(\()p FA(\015)5 b FB(\))p FA(;)33 b Fz(8)p FA(')39 b Fz(2)f FA(C)7 b FB(\()2691 2228 y Fw(b)2688 2253 y FB(\000\))40 b Fq(and)g FB(\010)3092 2268 y Fx(\015)3137 2253 y FB(\()p FA(D)s FB(\))d(=)h FA(D)s Fq(,)324 2374 y Fz(8)p FA(D)31 b Fz(2)d Fu(D)9 b Fq(.)324 2577 y FB(On)34 b(the)g(other)g(hand,)g(when)h FA(\027)h FB(=)30 b(1,)j(there)i(is)e(no)h(isomorphism)d Fu(J)49 b FB(:)29 b Fu(D)40 b Fz(!)3281 2550 y Fw(e)3254 2577 y Fu(D)j FB(suc)m(h)324 2712 y(that)37 b Fu(J)18 b FB(\()p FA(@)5 b FB(\))36 b(=)951 2685 y Fw(e)942 2712 y FA(@)43 b FB(b)s(ecause)1428 2685 y Fw(e)1402 2712 y Fu(D)j FB(is)36 b(comm)m(utativ)m(e.)56 b(Th)m(us,)40 b(one)d(cannot)g(hop)s(e)g(suc)m(h)i(a)324 2832 y(theorem)32 b(in)g(that)g(case.)470 2953 y(Another)25 b(w)m(a)m(y)h(of)f(pro)m (ving)f(Theorem)h(5.3)f(is)h(through)f(the)i(use)f(of)g(the)g(next)h (prop)s(o-)324 3073 y(sition.)324 3276 y Fr(Prop)s(osition)35 b(5.4)49 b Fq(If)33 b FA(\027)h Fz(\025)28 b FB(1)33 b Fq(then)g Fz(f)p FA(@)1812 3240 y FD(\003)1852 3233 y Fx(\013)1901 3276 y FA(@)1957 3240 y Fx(\014)2005 3276 y Fz(g)2055 3292 y FD(f)p Fx(\013;\014)s FD(2)p Ft(N)p FD(g)2366 3276 y Fq(is)g(a)g(b)-5 b(asis)32 b(of)h(the)g(ve)-5 b(ctor)32 b(sp)-5 b(ac)g(e)324 3417 y Fu(D)401 3432 y Fy(alg)496 3417 y Fq(.)45 b(One)34 b(has)h FA(\027)f(>)27 b FB(1)35 b Fq(if)g(and)f(only)h(if)f Fz(f)1884 3391 y Fw(e)1864 3417 y FA(@)1920 3388 y FD(\003)1960 3345 y Fx(\013)2019 3391 y Fw(e)2010 3417 y FA(@)2066 3381 y Fx(\014)2114 3417 y Fz(g)2164 3433 y FD(f)p Fx(\013;\014)s FD(2)p Ft(N)p FD(g)2476 3417 y Fq(is)h(a)g(b)-5 b(asis)34 b(of)g(sp)-5 b(ac)g(e)3295 3390 y Fw(e)3268 3417 y Fu(D)3345 3432 y Fy(alg)3441 3417 y Fq(.)324 3621 y Fr(Pro)s(of:)52 b FB(Assume)35 b(that)1255 3546 y Fw(P)1360 3572 y Fx(n)1360 3650 y(i)p Fy(=1)1495 3621 y FA(\025)1552 3636 y Fx(i)1580 3621 y FA(@)1636 3585 y FD(\003)1676 3578 y Fx(\013)1721 3588 y Fo(i)1752 3621 y FA(@)1808 3585 y Fx(\014)1848 3595 y Fo(i)1909 3621 y FB(=)30 b(0,)j(where)j(\()p FA(\013)2508 3636 y Fx(i)2536 3621 y FA(;)17 b(\014)2635 3636 y Fx(i)2663 3621 y FB(\))33 b(are)h(distinct)f(couples)324 3741 y(and)g FA(\025)571 3756 y Fx(i)627 3741 y Fz(6)p FB(=)c(0.)44 b(W)-8 b(e)34 b(set)f FA(\013)d FB(=)e(min)n(\()p FA(\013)1631 3756 y Fx(i)1660 3741 y FA(;)17 b(i)28 b FB(=)g(1)p FA(;)17 b(:)g(:)g(:)f(;)h(n)p FB(\))33 b(and)g FA(I)j FB(=)28 b Fz(f)p FA(i)h Fz(j)f FA(\013)2869 3756 y Fx(i)2926 3741 y FB(=)g FA(\013)q Fz(g)p FB(.)44 b(W)-8 b(e)34 b(tak)m(e)324 3861 y FA(x)k Fz(2)g FB(\000)h(suc)m(h)h(that)e Fz(j)p FA(x)p Fz(j)f FB(=)h FA(\013)h FB(and)g(w)m(e)g(obtain)2082 3787 y Fw(P)2187 3891 y Fx(i)p FD(2)p Fx(I)2315 3861 y FA(\025)2372 3876 y Fx(i)2400 3861 y FB(\()p FA(@)2494 3825 y Fx(\014)2534 3835 y Fo(i)2565 3861 y FA(f)11 b FB(\)\()p FA(e)p FB(\))37 b(=)h(0.)61 b(Notice)38 b(that)324 3982 y Fz(f)p FA(\014)429 3997 y Fx(i)457 3982 y Fz(g)507 3997 y Fx(i)p FD(2)p Fx(I)647 3982 y FB(are)30 b(pairwise)f(distinct)f (b)m(y)j(h)m(yp)s(otheses.)45 b(No)m(w,)30 b(b)m(y)h(taking)e FA(i)2910 3997 y Fy(0)2977 3982 y Fz(2)f FA(I)37 b FB(and)30 b FA(f)40 b FB(the)324 4102 y(c)m(haracteristic)32 b(function)g(of)g FA(S)1476 4117 y Fx(\014)1516 4127 y Fo(i)1538 4142 y Fh(0)1581 4102 y FB(,)h(w)m(e)g(get)g(that)f FA(\025)2215 4117 y Fx(i)2239 4126 y Fh(0)2306 4102 y FB(=)27 b(0)32 b(whic)m(h)h(is)g(a)f(con)m(tradiction.)470 4251 y(No)m(w)45 b(assume)g FA(\027)54 b(>)47 b FB(1)d(and)g(let)1725 4176 y Fw(P)1830 4203 y Fx(n)1830 4280 y(i)p Fy(=1)1965 4251 y FA(\025)2022 4266 y Fx(i)2071 4225 y Fw(e)2050 4251 y FA(@)2106 4222 y FD(\003)2146 4179 y Fx(\013)2191 4189 y Fo(i)2231 4225 y Fw(e)2222 4251 y FA(@)2278 4215 y Fx(\014)2318 4225 y Fo(i)2397 4251 y FB(=)j(0,)g(with)d(\()p FA(\013)2977 4266 y Fx(i)3005 4251 y FA(;)17 b(\014)3104 4266 y Fx(i)3132 4251 y FB(\))44 b(pairwise)324 4386 y(distinct)32 b(and)i FA(\025)925 4401 y Fx(i)982 4386 y Fz(6)p FB(=)29 b(0.)46 b(W)-8 b(e)34 b(\014x)g FA(x)c Fz(2)1700 4360 y Fw(e)1697 4386 y FB(\000)k(on)f(and)h(set)g FA(\013)c FB(=)f(max)o(\()p FA(\013)2750 4401 y Fx(i)2778 4386 y FA(;)17 b(i)30 b FB(=)f(1)p FA(;)17 b(:)g(:)g(:)e(;)i(n)p FB(\).)46 b(W)-8 b(e)324 4506 y(ha)m(v)m(e:)1397 4652 y Fx(n)1347 4682 y Fw(X)1361 4892 y Fx(i)p Fy(=1)1507 4777 y FA(\025)1564 4792 y Fx(i)1714 4682 y Fw(X)1609 4909 y Fx(y)r FD(2)p Fx(x)1733 4886 y Fh(\()p Fo(\013)1797 4902 y(i)1823 4886 y Fh(\))1851 4909 y Fx(S)1898 4886 y Fo(\014)1933 4902 y(i)1980 4777 y FA(f)11 b FB(\()p FA(y)t FB(\))26 b(=)i(0)p FA(:)1894 5251 y FB(21)p eop end %%Page: 22 22 TeXDict begin 22 21 bop 324 548 a FB(Noticing)35 b(that)i FA(x)992 512 y Fy(\()p Fx(\013)p Fy(\))1097 548 y FA(S)1163 512 y Fx(\014)1235 548 y Fz(\\)26 b FA(x)1382 512 y Fy(\()p Fx(\013)1454 488 y Fv(0)1478 512 y Fy(\))1509 548 y FA(S)1575 512 y Fx(\014)1618 488 y Fv(0)1680 548 y FB(=)35 b(\037)i(if)f(and)i (only)e(if)h FA(\013)2569 512 y FD(0)2617 548 y Fz(\000)26 b FA(\013)36 b Fz(6)p FB(=)f FA(\014)2990 512 y FD(0)3038 548 y Fz(\000)26 b FA(\014)6 b FB(,)38 b(w)m(e)g(can)324 668 y(supp)s(ose)c(that)f(there)h(is)f FA(k)j FB(suc)m(h)f(that)e FA(\013)1834 683 y Fx(i)1885 668 y Fz(\000)23 b FA(\014)2040 683 y Fx(i)2097 668 y FB(=)28 b FA(k)37 b FB(for)32 b(all)f FA(i)e FB(=)g(1)p FA(;)17 b(:)g(:)g(:)f(;)h(n)p FB(.)45 b(Moreo)m(v)m(er,)324 789 y(since)33 b(for)f(all)e FA(l)g FB(=)e(1)p FA(;)17 b(:)g(:)g(:)e(;)i FB(\()p FA(\013)23 b Fz(\000)f FA(k)s FB(\),)1077 1009 y FA(x)1132 968 y Fy(\()p Fx(\013)p FD(\000)p Fx(l)q Fy(\))1313 1009 y FA(S)1379 968 y Fx(\013)p FD(\000)p Fx(k)r FD(\000)p Fx(l)1626 1009 y Fz(\032)28 b FA(x)1786 968 y Fy(\()p Fx(\013)p FD(\000)p Fy(1\))1981 1009 y FA(S)2047 968 y Fx(\013)p FD(\000)p Fx(k)r FD(\000)p Fy(1)2308 1009 y Fs(*)g FA(x)2468 968 y Fy(\()p Fx(\013)p Fy(\))2573 1009 y FA(S)2639 968 y Fx(\013)p FD(\000)p Fx(k)2782 1009 y FA(;)324 1229 y FB(b)m(y)33 b(the)g(de\014nition)f(of)g FA(\013)h FB(w)m(e)h(obtain:)1444 1449 y Fz([)1510 1464 y Fx(\013)1555 1474 y Fo(i)1582 1464 y FD(6)p Fy(=)p Fx(\013)1687 1449 y FA(x)1742 1408 y Fy(\()p Fx(\013)1814 1418 y Fo(i)1841 1408 y Fy(\))1873 1449 y FA(S)1939 1408 y Fx(\014)1979 1418 y Fo(i)2036 1449 y Fs(*)28 b FA(x)2196 1408 y Fy(\()p Fx(\013)p Fy(\))2301 1449 y FA(S)2367 1408 y Fx(\014)2414 1449 y FA(:)324 1669 y FB(W)-8 b(e)42 b(no)m(w)g(c)m(ho)s(ose)g FA(y)1079 1684 y Fy(0)1160 1669 y FB(strictly)e(in)h(the)h FA(x)1857 1633 y Fy(\()p Fx(\013)p Fy(\))1962 1669 y FA(S)2028 1633 y Fx(\014)2075 1669 y FB(.)70 b(Hence,)45 b(b)m(y)d(taking)f FA(f)54 b FB(=)42 b FA(\037)3235 1684 y FD(f)p Fx(y)3305 1693 y Fh(0)3340 1684 y FD(g)3380 1669 y FB(,)i(w)m(e)324 1810 y(obtain)32 b FA(\025)685 1825 y Fx(i)709 1834 y Fh(0)777 1810 y FB(=)c(0)34 b(whic)m(h)f(is)g(a)g(con)m(tradiction.)45 b(Finally)-8 b(,)31 b Fz(f)2480 1783 y Fw(e)2460 1810 y FA(@)2516 1781 y FD(\003)2556 1738 y Fx(\013)2614 1783 y Fw(e)2605 1810 y FA(@)2661 1774 y Fx(\014)2709 1810 y Fz(g)2759 1825 y Fx(\013;\014)s FD(2)p Ft(N)3000 1810 y FB(is)h(not)i(a)f(basis)324 1944 y(if)e FA(\027)j FB(=)28 b(1)k(b)s(ecause)1050 1918 y Fw(e)1041 1944 y FA(@)1107 1918 y Fw(e)1098 1944 y FA(@)1154 1908 y FD(\003)1222 1944 y FB(=)1335 1918 y Fw(e)1326 1944 y FA(@)1382 1908 y FD(\003)1431 1918 y Fw(e)1422 1944 y FA(@)i FB(=)27 b(1.)141 b Ff(\003)324 2107 y Fr(5.3.)33 b(Description)d(of)j Fu(C)17 b FB(\()1373 2082 y Fw(b)1370 2107 y FB(\000\))p FA(=)p Fs(K)g FB(\(\000\))p Fr(.)48 b FB(W)-8 b(e)29 b(no)m(w)g(compute)f(the)h(quotien)m(t)g(of)e Fu(C)18 b FB(\()3335 2082 y Fw(b)3332 2107 y FB(\000)o(\))29 b(b)m(y)324 2228 y Fs(K)17 b FB(\(\000\).)324 2443 y Fr(Theorem)37 b(5.5)49 b Fq(Ther)-5 b(e)41 b(is)h(a)f(unique)h (morphism)f FB(\010)g(:)g Fu(C)17 b FB(\()2567 2418 y Fw(b)2564 2443 y FB(\000\))40 b Fz(!)2870 2416 y Fw(e)2844 2443 y Fu(D)c Fz(\012)28 b FA(C)7 b FB(\()p FA(@)e FB(\000\))43 b Fq(such)324 2574 y(that)c FB(\010\()p FA(@)5 b FB(\))36 b(=)884 2548 y Fw(e)875 2574 y FA(@)31 b Fz(\012)25 b FB(1)39 b Fq(and)f FB(\010\()p FA(')p FB(\()p FA(Q)p FB(\)\))d(=)f(1)25 b Fz(\012)h FB(\()p FA(')p Fz(j)2155 2589 y Fx(@)t Fy(\000)2243 2574 y FB(\))p Fq(.)56 b(This)38 b(morphism)g(is)g(surje)-5 b(ctive)324 2695 y(and)34 b(its)h(kernel)f(is)h Fs(K)17 b FB(\(\000\))p Fq(.)324 2910 y Fr(Pro)s(of:)48 b FB(According)30 b(to)g(the)h(Theorem)f(5.1)g (there)h(is)f(a)g(morphism)e(\010)g(:)g Fu(C)17 b FB(\()3135 2884 y Fw(b)3132 2910 y FB(\000\))28 b Fz(!)3412 2883 y Fw(e)3386 2910 y Fu(D)3472 2873 y Fx(@)t Fy(\000)324 3041 y FB(suc)m(h)45 b(that)f(\(\010\()p FA(@)5 b FB(\)\)\()p FA(\015)g FB(\))49 b(=)1369 3015 y Fw(e)1360 3041 y FA(@)h FB(and)44 b(\(\010\()p FA(')p FB(\()p FA(Q)p FB(\)\)\)\()p FA(\015)5 b FB(\))48 b(=)f FA(')p FB(\()p FA(\015)5 b FB(\))44 b(,)j(for)c(all)f FA(\015)53 b Fz(2)48 b FA(@)5 b FB(\000)44 b(and)324 3171 y(all)33 b FA(')f Fz(2)h FA(C)7 b FB(\()775 3146 y Fw(b)772 3171 y FB(\000)o(\).)52 b(Since)35 b(the)h(images)e(of)h FA(@)41 b FB(and)35 b FA(')p FB(\()p FA(Q)p FB(\))g(b)s(elong)f(to)h(the)h FA(C)3034 3135 y FD(\003)3073 3171 y FB(-subalgebra)324 3303 y FA(C)7 b FB(\()p FA(@)e FB(\000)p FA(;)627 3277 y Fw(e)600 3303 y Fu(D)10 b FB(\),)25 b(and)f(since)g Fu(C)17 b FB(\()1312 3278 y Fw(b)1309 3303 y FB(\000\))23 b(is)g(generated)i(b)m(y)f FA(@)30 b FB(and)23 b(suc)m(h)i FA(')p FB(\()p FA(Q)p FB(\),)h(it)c(follo)m(ws)h(that)g(the)324 3435 y(range)32 b(of)f(\010)h(is)f(included)h(in)f FA(C)7 b FB(\()p FA(@)e FB(\000)p FA(;)1704 3409 y Fw(e)1677 3435 y Fu(D)10 b FB(\).)43 b(W)-8 b(e)32 b(ha)m(v)m(e)h FA(C)7 b FB(\()p FA(@)e FB(\000)p FA(;)2566 3409 y Fw(e)2539 3435 y Fu(D)10 b FB(\))2692 3408 y Fz(\030)2693 3440 y FB(=)2823 3409 y Fw(e)2797 3435 y Fu(D)30 b Fz(\012)21 b FA(C)7 b FB(\()p FA(@)e FB(\000\),)33 b(so)f(w)m(e)324 3567 y(get)e(the)h(required)g(morphism)e(\010)f(:)g Fu(C)17 b FB(\()1761 3542 y Fw(b)1758 3567 y FB(\000\))27 b Fz(!)2038 3541 y Fw(e)2012 3567 y Fu(D)g Fz(\012)18 b FA(C)7 b FB(\()p FA(@)e FB(\000\).)44 b(No)m(w)31 b(since)f(\010\()p FA(@)5 b FB(\))29 b(=)3352 3541 y Fw(e)3343 3567 y FA(@)24 b Fz(\012)18 b FB(1)324 3688 y(and)29 b(\010\()p FA(')p FB(\()p FA(Q)p FB(\)\))f(=)g(1)15 b Fz(\012)g FB(\()p FA(')p Fz(j)1291 3703 y Fx(@)t Fy(\000)1381 3688 y FB(\),)30 b(and)f(since)h(an)m(y)g(function)f(in)f FA(C)7 b FB(\()p FA(@)e FB(\000\))30 b(is)f(the)h(restriction)324 3818 y(of)h(some)h(function)f(from)g FA(C)7 b FB(\()1407 3793 y Fw(b)1404 3818 y FB(\000)o(\),)32 b(it)f(follo)m(ws)f(that)i(\010)g (is)g(surjectiv)m(e.)44 b(Its)32 b(uniqueness)i(is)324 3939 y(clear.)43 b(W)-8 b(e)33 b(no)m(w)g(compute)g(the)g(k)m(ernel.) 470 4059 y(As)c(seen)h(in)e(the)h(Theorem)g(5.1)f Fs(K)17 b FB(\(\000\))34 b Fz(\032)28 b FB(Ker\(\010\).)42 b(In)29 b(the)g(remainder)f(of)g(this)g(sec-)324 4179 y(tion)j(w)m(e)i(shall)d (pro)m(v)m(e)j(the)g(rev)m(erse)h(inclusion.)42 b(F)-8 b(or)31 b(this)g(w)m(e)i(need)g(some)f(preliminary)324 4300 y(lemmas.)324 4503 y Fr(Lemma)37 b(5.6)49 b Fq(L)-5 b(et)44 b FA(R)i FB(=)f FA(')p FB(\()p FA(Q)p FB(\))p FA(@)1604 4467 y FD(\003)1644 4460 y Fx(\013)1694 4503 y FA(@)1750 4467 y Fx(\014)1842 4503 y Fq(and)e Fu(U)70 b FB(=)45 b Fz(f)p FA(a)2412 4518 y Fx(i)2440 4503 y FB(\000)p Fz(g)2551 4518 y Fx(i)p Fy(=1)p Fx(;:::)n(;n)2854 4503 y Fq(b)-5 b(e)44 b(a)g(disjoint)f(c)-5 b(o-)324 4623 y(vering)42 b(of)h FA(@)5 b FB(\000)p Fq(.)71 b(F)-7 b(or)43 b(e)-5 b(ach)42 b FA(")h(>)g FB(0)g Fq(ther)-5 b(e)44 b(ar)-5 b(e)43 b FA(c)2154 4638 y Fy(1)2193 4623 y FA(;)17 b(c)2279 4638 y Fy(2)2318 4623 y FA(;)g(:)g(:)g(:)f(;)h(c) 2579 4638 y Fx(m)2688 4623 y Fz(2)44 b FB(Ran\()p FA(')p FB(\))f Fq(and)g(ther)-5 b(e)324 4744 y(is)51 b(a)h(disjoint)f(c)-5 b(overing)50 b Fu(U)1422 4708 y FD(0)1504 4744 y FB(=)58 b Fz(f)p FA(b)1729 4759 y Fx(j)1766 4744 y FB(\000)p Fz(g)1877 4759 y Fx(j)t Fy(=1)p Fx(;:::)n(;m)2215 4744 y Fq(of)52 b FA(@)5 b FB(\000)52 b Fq(\014ner)f(than)h Fu(U)77 b Fq(such)51 b(that)324 4864 y Fz(k)p Fr(1)430 4879 y Fx(U)485 4860 y Fv(0)511 4864 y FA(R)23 b Fz(\000)g FA(R)783 4828 y FD(0)806 4864 y Fz(k)28 b(\024)g FA(")p Fq(,)34 b(wher)-5 b(e)34 b FA(R)1449 4828 y FD(0)1500 4864 y FB(=)1604 4790 y Fw(P)1709 4816 y Fx(m)1709 4893 y(j)t Fy(=1)1852 4864 y Fr(1)1908 4879 y Fx(b)1938 4889 y Fo(j)1971 4879 y Fy(\000)2019 4864 y FA(c)2061 4879 y Fx(j)2098 4864 y FA(@)2154 4828 y FD(\003)2194 4821 y Fx(\013)2244 4864 y FA(@)2300 4828 y Fx(\014)2383 4864 y Fq(and)g FA(U)2648 4828 y FD(0)2700 4864 y FB(=)27 b Fz([)2869 4828 y Fx(m)2869 4889 y(j)t Fy(=1)2996 4864 y FA(b)3037 4879 y Fx(j)3074 4864 y FB(\000)p Fq(.)1894 5251 y FB(22)p eop end %%Page: 23 23 TeXDict begin 23 22 bop 324 548 a Fr(Pro)s(of:)46 b FB(Let)27 b FA(")g(>)h FB(0,)g(w)m(e)f(denote)h FA("=)p Fz(k)p FA(@)1770 512 y FD(\003)1810 505 y Fx(\013)1859 548 y FA(@)1915 512 y Fx(\014)1963 548 y Fz(k)f FB(b)m(y)g FA(")2215 512 y FD(0)2238 548 y FB(.)42 b(Since)27 b FA(')p FB(\()p FA(@)5 b FB(\000\))27 b(is)f(compact,)i(there)324 668 y(are)37 b FA(\015)542 683 y Fy(1)581 668 y FA(;)17 b(\015)676 683 y Fy(2)715 668 y FA(;)g(:)g(:)g(:)f(;)h(\015)985 683 y Fx(N)1087 668 y Fz(\032)37 b FA(@)5 b FB(\000)38 b(suc)m(h)h(that)f FA(')p FB(\()p FA(@)5 b FB(\000\))36 b Fz(\032)h([)2271 632 y Fx(N)2271 694 y(k)r Fy(=1)2404 668 y FA(D)s FB(\()p FA(')p FB(\()p FA(\015)2679 683 y Fx(k)2721 668 y FB(\))p FA(;)17 b(")2849 632 y FD(0)2871 668 y FB(\),)39 b(where)g FA(D)s FB(\()p FA(z)t(;)17 b(r)s FB(\))324 789 y(is)32 b(the)h(complex)f(op)s(en)h(disk)f(of)h (cen)m(ter)h FA(z)j FB(and)32 b(ra)m(y)h FA(r)s FB(.)44 b(The)33 b(op)s(en)g(sets)1302 972 y Fu(O)1376 987 y Fx(i;k)1489 972 y FB(=)28 b FA(a)1644 987 y Fx(i)1675 947 y Fw(b)1672 972 y FB(\000)23 b Fz(\\)f FA(')1908 931 y FD(\000)p Fy(1)2002 972 y FB(\()p FA(D)s FB(\()p FA(')p FB(\()p FA(\015)2315 987 y Fx(k)2357 972 y FB(\))p FA(;)17 b(")2485 931 y FD(0)2508 972 y FB(\)\))324 1156 y(co)m(v)m(er)30 b FA(@)5 b FB(\000.)43 b(The)29 b(Prop)s(osition)e (2.4)h(giv)m(es)g(us)i(a)e(disjoin)m(t)f(co)m(v)m(ering)i Fz(f)p FA(b)2874 1171 y Fx(j)2911 1156 y FB(\000)p Fz(g)3022 1171 y Fx(j)t Fy(=1)p Fx(;:::)m(;m)3337 1156 y FB(of)f FA(@)5 b FB(\000)324 1289 y(suc)m(h)33 b(that)e(for)g(eac)m(h)h FA(j)i Fz(2)28 b(f)p FB(1)p FA(;)17 b(:)g(:)g(:)f(;)h(m)p Fz(g)31 b FB(there)h(are)g FA(i)f FB(and)h FA(k)j FB(suc)m(h)e(that)e FA(b)2990 1304 y Fx(j)3030 1264 y Fw(b)3027 1289 y FB(\000)c Fz(\032)i Fu(O)3295 1304 y Fx(i;k)3380 1289 y FB(.)44 b(T)-8 b(o)324 1410 y(simplify)28 b(the)j(notations,)f(w)m(e)h(will)e (denote)i(b)m(y)g FA(\015)2131 1425 y Fx(j)2198 1410 y FB(the)g FA(\015)2415 1425 y Fx(k)2488 1410 y FB(asso)s(ciated)f(to)g FA(b)3108 1425 y Fx(j)3145 1410 y FB(\000.)43 b(W)-8 b(e)31 b(set)324 1530 y Fu(U)429 1494 y FD(0)480 1530 y FB(=)d Fz(f)p FA(b)675 1545 y Fx(j)712 1530 y FB(\000)p Fz(g)823 1545 y Fx(j)t Fy(=1)p Fx(;:::)m(;m)1142 1530 y FB(and)1412 1793 y FA(R)1487 1752 y FD(0)1538 1793 y FB(=)1692 1668 y Fx(n)1642 1698 y Fw(X)1653 1908 y Fx(j)t Fy(=1)1802 1793 y Fr(1)1858 1808 y Fx(b)1888 1818 y Fo(j)1921 1808 y Fy(\000)1969 1793 y FA(')p FB(\()p FA(\015)2122 1808 y Fx(j)2158 1793 y FB(\))p FA(@)2252 1752 y FD(\003)2293 1747 y Fx(\013)2342 1793 y FA(@)2398 1752 y Fx(\014)2446 1793 y FA(:)324 2077 y FB(Remem)m(b)s(ering)j(that) h(sup)1301 2101 y Fx(x)p FD(2)p Fx(b)1418 2111 y Fo(j)1451 2101 y Fy(\000)1515 2077 y Fz(j)p FA(')p FB(\()p FA(\015)1696 2092 y Fx(j)1732 2077 y FB(\))22 b Fz(\000)h FA(')p FB(\()p FA(x)p FB(\))p Fz(j)k(\024)i FA(")2294 2041 y FD(0)2316 2077 y FB(,)k(w)m(e)h(obtain:)535 2356 y Fz(k)p FB(\()p FA(R)698 2314 y FD(0)743 2356 y Fz(\000)23 b Fr(1)899 2371 y Fx(U)954 2352 y Fv(0)980 2356 y FA(R)q FB(\))p FA(f)11 b Fz(k)1202 2314 y Fy(2)1325 2356 y FB(=)1484 2261 y Fw(X)1491 2472 y Fx(x)p FD(2)p Fy(\000)1645 2356 y Fz(j)1730 2231 y Fx(m)1689 2261 y Fw(X)1700 2471 y Fx(j)t Fy(=1)1850 2356 y Fr(1)1906 2371 y Fx(b)1936 2381 y Fo(j)1969 2371 y Fy(\000)2017 2356 y FB(\()p FA(x)p FB(\)\()p FA(')p FB(\()p FA(\015)2339 2371 y Fx(j)2375 2356 y FB(\))22 b Fz(\000)h FA(')p FB(\()p FA(x)p FB(\)\)\()p FA(@)2862 2314 y FD(\003)2902 2310 y Fx(\013)2952 2356 y FA(@)3008 2314 y Fx(\014)3056 2356 y FA(f)11 b FB(\)\()p FA(x)p FB(\))p Fz(j)3312 2314 y Fy(2)1325 2690 y FB(=)1525 2565 y Fx(m)1484 2595 y Fw(X)1495 2805 y Fx(j)t Fy(=1)1670 2595 y Fw(X)1645 2808 y Fx(x)p FD(2)p Fx(b)1762 2818 y Fo(j)1794 2808 y Fy(\000)1855 2690 y Fz(j)p FB(\()p FA(')p FB(\()p FA(\015)2074 2705 y Fx(j)2110 2690 y FB(\))22 b Fz(\000)g FA(')p FB(\()p FA(x)p FB(\)\)\()p FA(@)2596 2649 y FD(\003)2637 2644 y Fx(\013)2686 2690 y FA(@)2742 2649 y Fx(\014)2790 2690 y FA(f)11 b FB(\)\()p FA(x)p FB(\))p Fz(j)3046 2649 y Fy(2)1324 3034 y Fz(\024)1525 2909 y Fx(m)1484 2939 y Fw(X)1495 3149 y Fx(j)t Fy(=1)1668 3034 y FB(sup)1645 3116 y Fx(x)p FD(2)p Fx(b)1762 3126 y Fo(j)1794 3116 y Fy(\000)1855 3034 y Fz(j)p FA(')p FB(\()p FA(\015)2036 3049 y Fx(j)2072 3034 y FB(\))22 b Fz(\000)g FA(')p FB(\()p FA(x)p FB(\))p Fz(j)2454 2992 y Fy(2)2535 2939 y Fw(X)2510 3151 y Fx(x)p FD(2)p Fx(b)2627 3161 y Fo(j)2660 3151 y Fy(\000)2721 3034 y Fz(j)p FB(\()p FA(@)2843 2992 y FD(\003)2883 2988 y Fx(\013)2932 3034 y FA(@)2988 2992 y Fx(\014)3036 3034 y FA(f)11 b FB(\)\()p FA(x)p FB(\))p Fz(j)3292 2992 y Fy(2)1324 3377 y Fz(\024)83 b FA(")1530 3336 y FD(0)1553 3325 y Fy(2)1650 3253 y Fx(m)1609 3283 y Fw(X)1620 3493 y Fx(j)t Fy(=1)1795 3283 y Fw(X)1770 3495 y Fx(x)p FD(2)p Fx(b)1887 3505 y Fo(j)1919 3495 y Fy(\000)1980 3377 y Fz(j)p FB(\()p FA(@)2102 3336 y FD(\003)2142 3331 y Fx(\013)2192 3377 y FA(@)2248 3336 y Fx(\014)2296 3377 y FA(f)11 b FB(\)\()p FA(x)p FB(\))p Fz(j)2552 3336 y Fy(2)1324 3645 y Fz(\024)83 b FA(")1530 3604 y Fy(2)1570 3645 y Fz(k)p FA(@)1676 3604 y FD(\003)1716 3599 y Fx(\013)1765 3645 y FA(@)1821 3604 y Fx(\014)1869 3645 y Fz(k)1919 3604 y FD(\000)p Fy(2)2036 3645 y Fz(\001)21 b(k)p FA(@)2191 3604 y FD(\003)2232 3599 y Fx(\013)2281 3645 y FA(@)2337 3604 y Fx(\014)2385 3645 y Fz(k)2435 3604 y Fy(2)2496 3645 y Fz(\001)h(k)p FA(f)11 b Fz(k)2705 3604 y Fy(2)2772 3645 y FB(=)27 b FA(")2921 3604 y Fy(2)2960 3645 y Fz(k)p FA(f)11 b Fz(k)3119 3604 y Fy(2)3158 3645 y FA(:)324 3829 y FB(By)33 b(denoting)f FA(')p FB(\()p FA(\025)1037 3844 y Fx(j)1073 3829 y FB(\))h(b)m(y)g FA(c)1321 3844 y Fx(j)1390 3829 y FB(w)m(e)h(obtain)d(the)i(result.)98 b Ff(\003)324 4014 y Fr(Lemma)37 b(5.7)49 b Fq(L)-5 b(et)43 b FA(T)56 b FB(=)1320 3939 y Fw(P)1426 3966 y Fx(n)1426 4043 y(k)r Fy(=1)1575 4014 y FA(')1639 4029 y Fx(k)1682 4014 y FB(\()p FA(Q)p FB(\))p FA(@)1891 3978 y FD(\003)1931 3971 y Fx(\013)1976 3983 y Fo(k)2019 4014 y FA(@)2075 3978 y Fx(\014)2115 3990 y Fo(k)2200 4014 y Fq(with)43 b FA(')2484 4029 y Fx(k)2569 4014 y Fz(2)g FA(C)7 b FB(\()2796 3989 y Fw(b)2793 4014 y FB(\000\))42 b Fq(and)h(let)g FA(")f(>)g FB(0)p Fq(.)324 4134 y(Ther)-5 b(e)32 b(ar)-5 b(e)33 b(a)g(c)-5 b(omp)g(act)33 b(op)-5 b(er)g(ator)33 b FA(K)7 b Fq(,)33 b(a)g(disjoint)g(c)-5 b(overing)32 b Fz(f)p FA(a)2676 4149 y Fx(j)2712 4134 y FB(\000)p Fz(g)2823 4149 y Fx(j)t Fy(=1)p Fx(;:::)n(;m)3143 4134 y Fq(of)h FA(@)5 b FB(\000)34 b Fq(and)324 4255 y(an)g(op)-5 b(er)g(ator)35 b FA(S)40 b Fq(of)35 b(the)g(form)324 4509 y FB(\(5.14\))686 b FA(S)33 b FB(=)1507 4384 y Fx(n)1457 4414 y Fw(X)1464 4626 y Fx(k)r Fy(=1)1658 4384 y Fx(m)1617 4414 y Fw(X)1628 4624 y Fx(j)t Fy(=1)1778 4509 y Fr(1)1834 4524 y Fx(a)1871 4534 y Fo(j)1904 4524 y Fy(\000)1952 4509 y FA(')2016 4524 y Fx(k)2059 4509 y FB(\()p FA(\015)2148 4524 y Fx(j;k)2238 4509 y FB(\))p FA(@)2332 4468 y FD(\003)2372 4463 y Fx(\013)2417 4475 y Fo(k)2460 4509 y FA(@)2516 4468 y Fx(\014)2556 4480 y Fo(k)2599 4509 y FA(;)324 4793 y Fq(such)h(that)i FB(min)n Fz(fj)p FA(a)1036 4808 y Fx(j)1073 4793 y Fz(j)27 b(j)g FA(j)34 b FB(=)27 b(1)p FA(;)17 b(:)g(:)g(:)f(;)h(m)p Fz(g)27 b(\025)i FB(max)o Fz(f)p FA(\013)2189 4808 y Fx(k)2259 4793 y Fz(j)f FA(k)i FB(=)e(1)p FA(;)17 b(:)g(:)g(:)f(;)h(n)p Fz(g)p Fq(,)34 b FA(\015)2991 4808 y Fx(j;k)3109 4793 y Fz(2)28 b FA(@)5 b FB(\000)36 b Fq(and)1555 4977 y Fz(k)p FA(T)g Fz(\000)22 b FA(S)28 b Fz(\000)23 b FA(K)7 b Fz(k)28 b(\024)g FA(":)1894 5251 y FB(23)p eop end %%Page: 24 24 TeXDict begin 24 23 bop 324 548 a Fr(Pro)s(of:)43 b FB(W)-8 b(e)22 b(denote)h(b)m(y)g FA(\013)28 b FB(=)f(max)p Fz(f)p FA(\013)1743 563 y Fx(k)1813 548 y Fz(j)h FA(k)i FB(=)e(1)p FA(;)17 b(:)g(:)g(:)e(;)i(n)p Fz(g)p FB(.)40 b(W)-8 b(e)22 b(set)h FA(T)2852 563 y Fx(k)2922 548 y FB(=)28 b FA(')3090 563 y Fx(k)3132 548 y FB(\()p FA(Q)p FB(\))p FA(@)3341 512 y FD(\003)3382 505 y Fx(\013)3427 517 y Fo(k)3469 548 y FA(@)3525 512 y Fx(\014)3565 524 y Fo(k)3608 548 y FA(:)324 668 y FB(Setting)40 b Fu(U)748 683 y Fy(0)828 668 y FB(=)g Fz([)1010 684 y FD(f)p Fx(a)p FD(jj)p Fx(a)p FD(j)p Fy(=)p Fx(\013)p FD(g)1320 668 y Fz(f)p FA(a)p FB(\000)p Fz(g)p FB(,)i(w)m(e)f(no)m(w)g(apply)f(the)h(Lemma)e(5.6)h (inductiv)m(ely)g(for)324 789 y FA(k)33 b FB(=)d(1)p FA(;)17 b(:)g(:)g(:)f(;)h(n)34 b FB(with)f FA("=n)h FB(instead)g(of)f FA(")p FB(,)i Fu(U)55 b FB(=)30 b Fu(U)2163 804 y Fx(k)r FD(\000)p Fy(1)2330 789 y FB(and)k FA(R)d FB(=)f FA(T)2789 804 y Fx(k)2832 789 y FB(,)k(denoting)g Fu(U)3401 753 y FD(0)3459 789 y FB(b)m(y)324 909 y Fu(U)404 924 y Fx(k)479 909 y FB(and)e FA(R)743 873 y FD(0)799 909 y FB(b)m(y)i FA(S)995 924 y Fx(k)1037 909 y FB(.)44 b(Then,)34 b(w)m(e)f(obtain)f (for)g FA(k)f Fz(2)d(f)p FB(1)p FA(;)17 b(:)g(:)g(:)e(;)i(n)p Fz(g)p FB(:)1554 1152 y Fz(k)p Fr(1)1660 1167 y Fx(U)1708 1179 y Fo(k)1750 1152 y FA(T)1807 1167 y Fx(k)1872 1152 y Fz(\000)23 b FA(S)2032 1167 y Fx(k)2074 1152 y Fz(k)28 b(\024)2271 1085 y FA(")p 2267 1129 55 4 v 2267 1221 a(k)324 1398 y FB(Since)k Fu(U)658 1413 y Fx(k)r Fy(+1)824 1398 y FB(is)g(\014ner)h(than)f Fu(U)1451 1413 y Fx(k)1526 1398 y FB(for)g FA(k)f FB(=)d(1)p FA(;)17 b(:)g(:)g(:)e(;)i(n)22 b Fz(\000)h FB(1,)32 b(w)m(e)i(obtain:)1435 1691 y Fz(k)p Fr(1)1541 1706 y Fx(U)1589 1714 y Fo(n)1702 1567 y Fx(n)1651 1596 y Fw(X)1659 1809 y Fx(k)r Fy(=1)1795 1691 y FB(\()p FA(T)1890 1706 y Fx(k)1955 1691 y Fz(\000)23 b FA(S)2115 1706 y Fx(k)2158 1691 y FB(\))p Fz(k)k(\024)h FA(":)324 1995 y FB(Hence,)1307 2185 y Fz(k)p FA(T)36 b Fz(\000)23 b Fr(1)1606 2200 y Fx(U)1661 2181 y Fo(c)1654 2217 y(n)1700 2185 y FA(T)36 b Fz(\000)23 b Fr(1)1949 2200 y Fx(U)1997 2208 y Fo(n)2110 2061 y Fx(n)2060 2091 y Fw(X)2067 2303 y Fx(k)r Fy(=1)2220 2185 y FA(S)2280 2200 y Fx(k)2323 2185 y Fz(k)k(\024)h FA(":)324 2452 y FB(Finally)-8 b(,)41 b(w)m(e)i(denote)f(the)g(compact)g(op)s(erator)f Fr(1)2197 2467 y Fx(U)2252 2448 y Fo(c)2245 2484 y(n)2291 2452 y FA(T)56 b FB(b)m(y)42 b FA(K)7 b FB(,)44 b(\()p Fr(1)2803 2467 y Fx(U)2851 2475 y Fo(n)2914 2377 y Fw(P)3020 2404 y Fx(n)3020 2481 y(k)r Fy(=1)3169 2452 y FA(S)3229 2467 y Fx(k)3272 2452 y FB(\))d(b)m(y)i FA(S)324 2573 y FB(and)32 b Fu(U)593 2588 y Fx(n)673 2573 y FB(b)m(y)h Fz(f)p FA(a)909 2588 y Fx(j)946 2573 y FB(\000)p Fz(g)1057 2588 y Fx(j)t Fy(=1)p Fx(;:::)m(;m)1376 2573 y FB(to)f(\014nish)h(the)g(pro)s(of.)97 b Ff(\003)470 2693 y FB(W)-8 b(e)33 b(no)m(w)g(go)f(bac)m(k)i(to)e(the) h(pro)s(of)f(of)g(Theorem)h(5.5.)43 b(Let)33 b FA(T)41 b Fz(2)28 b FB(Ker\(\010\).)44 b(F)-8 b(or)32 b(eac)m(h)324 2823 y FA(")i(>)h FB(0)i(there)g(is)g FA(T)1028 2787 y FD(0)1086 2823 y Fz(2)e Fu(C)17 b FB(\()1311 2798 y Fw(b)1308 2823 y FB(\000\))1407 2838 y Fy(alg)1539 2823 y FB(suc)m(h)39 b(that)d Fz(k)p FA(T)j Fz(\000)26 b FA(T)2299 2787 y FD(0)2322 2823 y Fz(k)34 b(\024)i FA(")p FB(.)56 b(By)37 b(relation)e(\(3.7\))h(and)324 2944 y(Prop)s(osition)31 b(3.4,)h(w)m(e)i(can)e(write:)1343 3234 y FA(T)1414 3193 y FD(0)1464 3234 y FB(=)1618 3109 y Fx(n)1568 3139 y Fw(X)1575 3352 y Fx(k)r Fy(=1)1728 3234 y FA(')1792 3249 y Fx(k)1835 3234 y FB(\()p FA(Q)p FB(\))p FA(@)2044 3193 y FD(\003)2084 3188 y Fx(\013)2129 3200 y Fo(k)2172 3234 y FA(@)2228 3193 y Fx(\014)2268 3205 y Fo(k)2333 3234 y FB(+)22 b FA(K)r(;)324 3567 y FB(where)35 b FA(K)j Fz(2)31 b Fs(K)18 b FB(\(\000\))40 b(and)34 b FA(')1329 3582 y Fx(k)1403 3567 y Fz(2)d FA(C)7 b FB(\()1618 3541 y Fw(b)1615 3567 y FB(\000)o(\).)49 b(Th)m(us)36 b Fz(k)p FB(\010\()p FA(T)2267 3530 y FD(0)2290 3567 y FB(\))p Fz(k)31 b(\024)g FA(")p FB(.)48 b(Using)34 b(Lemma)f(5.7,)i(w)m(e)324 3687 y(get)h(an)g(op)s(erator)f FA(S)42 b FB(of)36 b(the)g(form)f (\(5.14\))h(and)g(a)g(compact)f(op)s(erator)h FA(K)3084 3702 y Fy(1)3159 3687 y FB(suc)m(h)i(that)324 3807 y Fz(k)p FA(T)445 3771 y FD(0)490 3807 y Fz(\000)22 b FA(S)28 b Fz(\000)23 b FA(K)860 3822 y Fy(1)899 3807 y Fz(k)28 b(\024)g FA(")p FB(.)43 b(This)33 b(implies)d(that)i Fz(k)p FB(\010\()p FA(S)6 b FB(\))p Fz(k)28 b(\024)g FB(2)p FA(")p FB(.)324 4011 y Fr(Lemma)37 b(5.8)49 b Fq(Ther)-5 b(e)34 b(is)h FA(K)1381 4026 y Fy(2)1448 4011 y Fz(2)28 b Fs(K)17 b FB(\(\000\))41 b Fq(such)35 b(that:)1388 4231 y Fz(k)p FA(S)28 b Fz(\000)23 b FA(K)1709 4246 y Fy(2)1748 4231 y Fz(k)28 b(\024)g(k)p FB(\010\()p FA(S)6 b FB(\))p Fz(k)27 b(\024)h FB(2)p FA(":)324 4451 y FB(The)33 b(pro)s(of)f(will)e(b)s(e)j(giv)m(en)g(b)s(elo)m(w.)43 b(By)33 b(using)f(this)h(w)m(e)g(get)661 4671 y Fz(k)p FA(T)j Fz(\000)22 b FA(K)986 4686 y Fy(1)1048 4671 y Fz(\000)g FA(K)1230 4686 y Fy(2)1270 4671 y Fz(k)83 b FB(=)h Fz(k)p FA(T)36 b Fz(\000)22 b FA(T)1876 4630 y FD(0)1922 4671 y FB(+)g FA(T)2091 4630 y FD(0)2136 4671 y Fz(\000)g FA(S)28 b Fz(\000)23 b FA(K)2506 4686 y Fy(1)2568 4671 y FB(+)f FA(S)27 b Fz(\000)c FA(K)2936 4686 y Fy(2)2976 4671 y Fz(k)1403 4816 y(\024)83 b(k)p FA(T)36 b Fz(\000)22 b FA(T)1876 4775 y FD(0)1899 4816 y Fz(k)g FB(+)g Fz(k)p FA(T)2190 4775 y FD(0)2235 4816 y Fz(\000)h FA(S)28 b Fz(\000)23 b FA(K)2606 4831 y Fy(1)2645 4816 y Fz(k)f FB(+)g Fz(k)p FA(S)28 b Fz(\000)22 b FA(K)3135 4831 y Fy(2)3175 4816 y Fz(k)1403 4961 y(\024)83 b FB(4)p FA(":)1894 5251 y FB(24)p eop end %%Page: 25 25 TeXDict begin 25 24 bop 324 548 a FB(Finally)-8 b(,)32 b(the)k(space)f Fs(K)18 b FB(\(\000\))40 b(b)s(eing)34 b(closed,)i(w)m(e)g(conclude)f(that)g FA(T)45 b Fz(2)32 b Fs(K)17 b FB(\(\000\).)56 b(This)35 b(\014n-)324 668 y(ishes)e(the)g(pro)s(of)f(of)g(the)h(Theorem.)324 909 y Fr(Pro)s(of)46 b(of)g(Lemma)g(5.8:)59 b FB(First,)42 b(w)m(e)f(should)f(remark)g(that)g(for)g(eac)m(h)h FA(a)g Fz(2)g FB(\000)f(and)324 1029 y FA(\013)q(;)33 b(\014)53 b Fz(\025)47 b FB(0,)g(the)e(Prop)s(osition)d(3.4)i(giv)m(es)g(us)h (that)f Fr(1)2353 1044 y Fx(a)p Fy(\000)2439 1029 y FA(@)2495 993 y FD(\003)2535 987 y Fx(\013)2584 1029 y FA(@)2640 993 y Fx(\014)2718 1029 y Fz(\000)31 b Fr(1)2882 1044 y Fx(a)p Fy(\000)2968 1029 y FA(@)3024 993 y FD(\003)3064 987 y Fx(\013)3113 1029 y FA(@)3169 993 y Fx(\014)3217 1029 y Fr(1)3273 1044 y Fx(a)p Fy(\000)3403 1029 y FB(is)44 b(a)324 1150 y(compact)32 b(op)s(erator.)43 b(W)-8 b(e)33 b(in)m(tro)s(duce)1175 1399 y FA(S)1241 1358 y FD(0)1291 1399 y FB(=)1445 1275 y Fx(n)1395 1305 y Fw(X)1402 1517 y Fx(k)r Fy(=1)1596 1275 y Fx(m)1555 1305 y Fw(X)1566 1515 y Fx(j)t Fy(=1)1716 1399 y Fr(1)1772 1414 y Fx(a)1809 1424 y Fo(j)1842 1414 y Fy(\000)1890 1399 y FA(')1954 1414 y Fx(k)1997 1399 y FB(\()p FA(\015)2086 1414 y Fx(j;k)2176 1399 y FB(\))p FA(@)2270 1358 y FD(\003)2310 1353 y Fx(\013)2355 1365 y Fo(k)2398 1399 y FA(@)2454 1358 y Fx(\014)2494 1370 y Fo(k)2537 1399 y Fr(1)2593 1414 y Fx(a)2630 1424 y Fo(j)2663 1414 y Fy(\000)324 1683 y FB(and)h(w)m(e)i(set)f FA(K)898 1698 y Fy(2)969 1683 y FB(=)30 b FA(S)g Fz(\000)24 b FA(S)1332 1647 y FD(0)1389 1683 y FB(whic)m(h)35 b(is)f(a)g(compact)h (op)s(erator.)48 b(Let)35 b FA(f)42 b Fz(2)31 b FA(`)3087 1647 y Fy(2)3126 1683 y FB(\(\000\).)49 b(Since)324 1804 y Fz(f)p FA(a)425 1819 y Fx(j)461 1804 y FB(\000)p Fz(g)572 1819 y Fx(j)t Fy(=1)p Fx(;:::)n(;m)892 1804 y FB(is)32 b(a)g(disjoin)m(t)g(co)m(v)m(ering)h(of)f FA(@)5 b FB(\000,)33 b(w)m(e)h(ha)m(v)m(e:)706 2061 y Fz(k)p FA(S)822 2020 y FD(0)845 2061 y FA(f)11 b Fz(k)954 2020 y Fy(2)1077 2061 y FB(=)1237 1967 y Fw(X)1243 2178 y Fx(x)p FD(2)p Fy(\000)1397 2061 y Fz(j)1492 1937 y Fx(n)1441 1967 y Fw(X)1449 2179 y Fx(k)r Fy(=1)1643 1937 y Fx(m)1602 1967 y Fw(X)1613 2177 y Fx(j)t Fy(=1)1746 2061 y FB(\()p Fr(1)1840 2076 y Fx(a)1877 2086 y Fo(j)1910 2076 y Fy(\000)1958 2061 y FA(')2022 2076 y Fx(k)2064 2061 y FB(\()p FA(\015)2153 2076 y Fx(j;k)2244 2061 y FB(\))p FA(@)2338 2020 y FD(\003)2378 2016 y Fx(\013)2423 2028 y Fo(k)2466 2061 y FA(@)2522 2020 y Fx(\014)2562 2032 y Fo(k)2604 2061 y Fr(1)2660 2076 y Fx(a)2697 2086 y Fo(j)2730 2076 y Fy(\000)2779 2061 y FA(f)g FB(\)\()p FA(x)p FB(\))p Fz(j)3035 2020 y Fy(2)1077 2396 y FB(=)1277 2271 y Fx(m)1237 2301 y Fw(X)1247 2511 y Fx(j)t Fy(=1)1397 2301 y Fw(X)1404 2513 y Fx(x)p FD(2)p Fy(\000)1558 2396 y Fz(j)1652 2271 y Fx(n)1602 2301 y Fw(X)1609 2513 y Fx(k)r Fy(=1)1746 2396 y FB(\()p Fr(1)1840 2411 y Fx(a)1877 2421 y Fo(j)1910 2411 y Fy(\000)1958 2396 y FA(')2022 2411 y Fx(k)2064 2396 y FB(\()p FA(\015)2153 2411 y Fx(j;k)2244 2396 y FB(\))p FA(@)2338 2355 y FD(\003)2378 2350 y Fx(\013)2423 2362 y Fo(k)2466 2396 y FA(@)2522 2355 y Fx(\014)2562 2367 y Fo(k)2604 2396 y Fr(1)2660 2411 y Fx(a)2697 2421 y Fo(j)2730 2411 y Fy(\000)2779 2396 y FA(f)g FB(\)\()p FA(x)p FB(\))p Fz(j)3035 2355 y Fy(2)1076 2730 y Fz(\024)1277 2606 y Fx(m)1237 2636 y Fw(X)1247 2846 y Fx(j)t Fy(=1)1397 2730 y Fz(k)1514 2606 y Fx(n)1463 2636 y Fw(X)1471 2848 y Fx(k)r Fy(=1)1624 2730 y Fr(1)1680 2745 y Fx(a)1717 2755 y Fo(j)1750 2745 y Fy(\000)1798 2730 y FA(')1862 2745 y Fx(k)1905 2730 y FB(\()p FA(\015)1994 2745 y Fx(j;k)2084 2730 y FB(\))p FA(@)2178 2689 y FD(\003)2218 2685 y Fx(\013)2263 2697 y Fo(k)2306 2730 y FA(@)2362 2689 y Fx(\014)2402 2701 y Fo(k)2445 2730 y Fr(1)2501 2745 y Fx(a)2538 2755 y Fo(j)2571 2745 y Fy(\000)2619 2730 y Fz(k)2669 2689 y Fy(2)2730 2730 y Fz(\001)22 b(k)p Fr(1)2886 2745 y Fx(a)2923 2755 y Fo(j)2956 2745 y Fy(\000)3004 2730 y FA(f)11 b Fz(k)3113 2689 y Fy(2)3152 2730 y FA(:)324 3010 y FB(No)m(w)33 b(w)m(e)h(use)f(\(3.9\))f(and)h(\(3.10\))f(and)g (get:)576 3265 y Fz(k)p Fr(1)682 3280 y Fx(a)719 3290 y Fo(j)752 3280 y Fy(\000)800 3265 y FB(\()888 3141 y Fx(n)838 3170 y Fw(X)846 3383 y Fx(k)r Fy(=1)998 3265 y FA(')1062 3280 y Fx(k)1105 3265 y FB(\()p FA(\015)1194 3280 y Fx(j;k)1284 3265 y FB(\))p FA(@)1378 3224 y FD(\003)1418 3219 y Fx(\013)1463 3231 y Fo(k)1506 3265 y FA(@)1562 3224 y Fx(\014)1602 3236 y Fo(k)1645 3265 y FB(\))p Fr(1)1739 3280 y Fx(a)1776 3290 y Fo(j)1809 3280 y Fy(\000)1857 3265 y Fz(k)27 b FB(=)h Fz(k)p FA(\025)2145 3224 y FD(\003)2145 3290 y Fx(a)2182 3300 y Fo(j)2219 3265 y FB(\()2307 3141 y Fx(n)2257 3170 y Fw(X)2265 3383 y Fx(k)r Fy(=1)2417 3265 y FA(')2481 3280 y Fx(k)2524 3265 y FB(\()p FA(\015)2613 3280 y Fx(j;k)2703 3265 y FB(\))p FA(@)2797 3224 y FD(\003)2837 3219 y Fx(\013)2882 3231 y Fo(k)2925 3265 y FA(@)2981 3224 y Fx(\014)3021 3236 y Fo(k)3064 3265 y FB(\))p FA(\025)3159 3280 y Fx(a)3196 3290 y Fo(j)3233 3265 y Fz(k)p FA(:)324 3534 y FB(Since)k Fz(j)p FA(a)657 3549 y Fx(j)694 3534 y Fz(j)27 b(\025)h FB(max)p Fz(f)p FA(\013)1148 3549 y Fx(k)1218 3534 y Fz(j)g FA(k)i FB(=)e(1)p FA(;)17 b(:)g(:)g(:)e(;)i (n)p Fz(g)p FB(,)33 b(the)g(Lemmas)e(4.1)i(and)f(4.3)g(giv)m(e)h(us:) 604 3792 y Fz(k)p FA(\025)711 3751 y FD(\003)711 3817 y Fx(a)748 3827 y Fo(j)785 3792 y FB(\()873 3668 y Fx(n)823 3697 y Fw(X)831 3910 y Fx(k)r Fy(=1)983 3792 y FA(')1047 3807 y Fx(k)1090 3792 y FB(\()p FA(\015)1179 3807 y Fx(j;k)1269 3792 y FB(\))p FA(@)1363 3751 y FD(\003)1403 3746 y Fx(\013)1448 3758 y Fo(k)1491 3792 y FA(@)1547 3751 y Fx(\014)1587 3763 y Fo(k)1630 3792 y FB(\))p FA(\025)1725 3807 y Fx(a)1762 3817 y Fo(j)1799 3792 y Fz(k)83 b FB(=)f Fz(k)p Fr(1)2196 3807 y Fy(\000)2245 3792 y FB(\()2333 3668 y Fx(n)2283 3697 y Fw(X)2291 3910 y Fx(k)r Fy(=1)2443 3792 y FA(')2507 3807 y Fx(k)2550 3792 y FB(\()p FA(\015)2639 3807 y Fx(j;k)2729 3792 y FB(\))2787 3766 y Fw(e)2767 3792 y FA(@)2823 3763 y FD(\003)2863 3720 y Fx(\013)2908 3732 y Fo(k)2960 3766 y Fw(e)2951 3792 y FA(@)3007 3751 y Fx(\014)3047 3763 y Fo(k)3089 3792 y FB(\))p Fr(1)3183 3807 y Fy(\000)3232 3792 y Fz(k)1932 4116 y FB(=)g Fz(k)2207 3991 y Fx(n)2157 4021 y Fw(X)2164 4233 y Fx(k)r Fy(=1)2317 4116 y FA(')2381 4131 y Fx(k)2424 4116 y FB(\()p FA(\015)2513 4131 y Fx(j;k)2603 4116 y FB(\))2662 4090 y Fw(e)2641 4116 y FA(@)2697 4087 y FD(\003)2738 4044 y Fx(\013)2783 4056 y Fo(k)2834 4090 y Fw(e)2825 4116 y FA(@)2881 4075 y Fx(\014)2921 4087 y Fo(k)2964 4116 y Fz(k)p FA(:)324 4385 y FB(F)-8 b(or)48 b(eac)m(h)i FA(j)k FB(w)m(e)c(c)m(ho)s(ose)g FA(\015)1382 4400 y Fx(j)1473 4385 y Fz(2)56 b FA(a)1646 4400 y Fx(j)1683 4385 y FA(@)5 b FB(\000.)93 b(The)49 b(family)e Fz(f)p FA(a)2552 4400 y Fx(j)2588 4385 y FB(\000)p Fz(g)2699 4400 y Fx(j)t Fy(=1)p Fx(;:::)n(;m)3035 4385 y FB(is)h(a)h(disjoin)m(t) 324 4505 y(co)m(v)m(ering)37 b(of)f FA(@)5 b FB(\000,)38 b(so)f(w)m(e)g(ha)m(v)m(e)h(lim)1642 4520 y Fx(x)p FD(!)p Fx(\015)1789 4530 y Fo(j)1842 4505 y FA(\037)1903 4520 y Fx(a)1940 4530 y Fo(j)1973 4520 y Fy(\000)2022 4505 y FB(\()p FA(x)p FB(\))c(=)g(1)i(and)h(lim)2711 4520 y Fx(x)p FD(!)p Fx(\015)2858 4530 y Fo(j)2911 4505 y FA(\037)2972 4520 y Fx(a)3009 4530 y Fo(i)3036 4520 y Fy(\000)3084 4505 y FB(\()p FA(x)p FB(\))e(=)f(0)i(for)324 4649 y FA(i)28 b Fz(6)p FB(=)f FA(j)6 b FB(.)44 b(Hence)34 b(\010)965 4664 y Fx(\015)1001 4674 y Fo(j)1038 4649 y FB(\()p FA(S)1142 4613 y FD(0)1165 4649 y FB(\))28 b(=)1334 4574 y Fw(P)1439 4601 y Fx(n)1439 4678 y(k)r Fy(=1)1589 4649 y FA(')1653 4664 y Fx(k)1695 4649 y FB(\()p FA(\015)1784 4664 y Fx(j;k)1875 4649 y FB(\))1933 4623 y Fw(e)1913 4649 y FA(@)1969 4620 y FD(\003)2009 4577 y Fx(\013)2054 4589 y Fo(k)2105 4623 y Fw(e)2097 4649 y FA(@)2153 4613 y Fx(\014)2193 4625 y Fo(k)2235 4649 y FB(.)44 b(W)-8 b(e)33 b(obtain)633 4914 y Fz(k)p FA(S)749 4873 y FD(0)772 4914 y FA(f)11 b Fz(k)881 4873 y Fy(2)1003 4914 y Fz(\024)1204 4789 y Fx(m)1163 4819 y Fw(X)1174 5029 y Fx(j)t Fy(=1)1324 4914 y Fz(k)p FB(\010)1444 4929 y Fx(\015)1480 4939 y Fo(j)1517 4914 y FB(\()p FA(S)1621 4873 y FD(0)1644 4914 y FB(\))p Fz(k)1732 4873 y Fy(2)1793 4914 y Fz(\001)22 b(k)p Fr(1)1949 4929 y Fx(a)1986 4939 y Fo(j)2019 4929 y Fy(\000)2067 4914 y FA(f)11 b Fz(k)2176 4873 y Fy(2)2243 4914 y Fz(\024)41 b FB(sup)2348 4996 y Fx(\015)t FD(2)p Fx(@)t Fy(\000)2537 4914 y Fz(k)p FB(\010)2657 4929 y Fx(\015)2702 4914 y FB(\()p FA(S)2806 4873 y FD(0)2829 4914 y FB(\))p Fz(k)2917 4873 y Fy(2)2978 4914 y Fz(\001)22 b(k)p FA(f)11 b Fz(k)3187 4873 y Fy(2)3226 4914 y FA(:)1894 5251 y FB(25)p eop end %%Page: 26 26 TeXDict begin 26 25 bop 324 548 a FB(Finally)-8 b(,)33 b(since)j Fs(K)18 b FB(\(\000\))38 b Fz(\032)33 b FB(Ker\(\010\),)k Fz(k)p FB(\010\()p FA(S)6 b FB(\))p Fz(k)32 b FB(=)h Fz(k)p FB(\010\()p FA(S)2321 512 y FD(0)2344 548 y FB(\))p Fz(k)f FB(=)h(sup)2720 571 y Fx(\015)t FD(2)p Fx(@)t Fy(\000)2913 548 y Fz(k)p FB(\010)3033 563 y Fx(\015)3078 548 y FB(\()p FA(S)3182 512 y FD(0)3205 548 y FB(\))p Fz(k)p FB(.)51 b(This)324 668 y(\014nishes)33 b(the)g(pro)s(of.)97 b Ff(\003)470 789 y FB(In)33 b(the)g(case)g FA(\027)i(>)27 b FB(1,)33 b(w)m(e)g(can)g(reform)m(ulate)e(the)i(theorem)g(as)f(follo) m(ws:)324 974 y Fr(Theorem)37 b(5.9)49 b Fq(L)-5 b(et)42 b FA(\027)47 b(>)40 b FB(1)p Fq(.)65 b(Ther)-5 b(e)41 b(is)g(a)g(unique)h(morphism)e FB(\010)h(:)f Fu(C)18 b FB(\()3095 949 y Fw(b)3092 974 y FB(\000)o(\))41 b Fz(!)f Fu(D)c Fz(\012)324 1095 y FA(C)7 b FB(\()p FA(@)e FB(\000\))40 b Fq(such)f(that)h FB(\010\()p FA(D)s FB(\))c(=)f FA(D)28 b Fz(\012)e FB(1)39 b Fq(for)h(al)5 b(l)38 b FA(D)h Fz(2)d Fu(D)49 b Fq(and)39 b FB(\010\()p FA(')p FB(\()p FA(Q)p FB(\)\))d(=)g(1)25 b Fz(\012)h FB(\()p FA(')p Fz(j)3405 1110 y Fx(@)t Fy(\000)3494 1095 y FB(\))p Fq(.)324 1215 y(This)34 b(morphism)f(is)i(surje)-5 b(ctive)35 b(and)f(its)h(kernel)f(is)h Fs(K)17 b FB(\(\000\))p Fq(.)324 1401 y FB(Once)44 b(again)e(this)h(Theorem)h(is)f(false)g(if)g FA(\027)53 b FB(=)46 b(1.)76 b(As)44 b(a)f(corollary)f(w)m(e)j(obtain)d (the)324 1521 y(follo)m(wing)30 b(result.)324 1707 y Fr(Prop)s(osition)35 b(5.10)49 b Fq(If)44 b FA(\027)52 b(>)44 b FB(1)g Fq(then)g Fu(D)39 b Fz(\\)30 b Fs(K)17 b FB(\(\000\))51 b(=)45 b Fz(f)p FB(0)p Fz(g)e Fq(and)h(if)g FA(\027)52 b FB(=)44 b(1)g Fq(one)g(has)324 1827 y Fs(K)17 b FB(\(\000\))34 b Fz(\032)28 b Fu(D)9 b Fq(.)324 2013 y Fr(Pro)s(of:)44 b FB(Let)24 b FA(\027)34 b(>)28 b FB(1)23 b(and)h FA(T)41 b Fz(2)28 b Fu(D)13 b Fz(\\)t Fs(K)18 b FB(\(\000\).)46 b(Theorem)24 b(5.9)f(giv)m(es)h(us)g(that)f(\010\()p FA(T)14 b FB(\))28 b(=)f FA(T)18 b Fz(\012)t FB(1)324 2133 y(and)38 b(that)g(\010\()p FA(T)14 b FB(\))36 b(=)h(0)h(since)g FA(T)52 b FB(is)37 b(compact.)59 b(F)-8 b(or)37 b FA(\027)43 b FB(=)37 b(1,)i(as)f(seen)i(in)d(the)h(pro)s(of)f(of)324 2254 y(Prop)s(osition)c(3.2,)i(it)e(su\016ces)k(to)e(pro)m(v)m(e)h (that)e FA(\016)2114 2269 y Fx(x;x)2252 2254 y FB(is)g(in)g Fu(D)9 b FB(.)51 b(But)34 b(this)h(is)f(clear)g(since)324 2382 y FA(\016)367 2397 y Fx(x;x)498 2382 y FB(=)27 b FA(@)657 2346 y FD(\003)698 2340 y(j)p Fx(x)p Fy(+1)p FD(j)871 2382 y FA(@)927 2346 y FD(j)p Fx(x)p Fy(+1)p FD(j)1123 2382 y Fz(\000)c FA(@)1279 2346 y FD(\003)1319 2340 y(j)p Fx(x)p FD(j)1402 2382 y FA(@)1458 2346 y FD(j)p Fx(x)p FD(j)1542 2382 y FB(.)141 b Ff(\003)324 2712 y FC(References)324 2931 y FB([AF])131 b Fa(C.)48 b(Allard)h(&)e(R.)h(Fr) n(oese)p FB(,)43 b Fq(A)h(Mourr)-5 b(e)44 b(estimate)f(for)h(a)f (Schr\177)-50 b(odinger)646 3052 y(op)-5 b(er)g(ator)47 b(on)f(a)h(binary)f(tr)-5 b(e)g(e)p FB(,)49 b(Rev.)d(Math.)g(Ph)m(ys)i Fr(12)p FB(,)h(No.12,)f(1655-1667)646 3172 y(\(2000\).)324 3369 y([An])141 b Fa(A.)54 b(Ancona)p FB(,)f Fq(Th)n(\023)-47 b(eorie)47 b(du)i(p)-5 b(otentiel)49 b(sur)g(les)g(gr)-5 b(aphes)48 b(et)h(les)g(vari)n(\022)-47 b(et)n(\023)g(es)p FB(,)646 3490 y(Lecture)34 b(Notes)f(in)f(Mathematics)g Fr(1427)p FB(,)h(Springer-V)-8 b(erlag.)324 3687 y([Bo])150 b Fa(N.)46 b(Bourbaki)p FB(,)d Fq(El)n(\023)-47 b(ements)40 b(de)h(math)n(\023)-47 b(ematiques,)42 b(A)n(lg)n(\022)-47 b(ebr)-5 b(e,)42 b(chapitr)-5 b(es)41 b(1)h(\022)-50 b(a)646 3808 y(3)p FB(,)33 b(Di\013usion)d(C.C.L.S.)k(P)m(aris)f (\(1970\).)324 4005 y([CDP])57 b Fa(M.)27 b(Coornaer)-7 b(t,)31 b(T.)d(Delzant)g(&)g(A.)g(P)-9 b(ap)i(adopoulos)p FB(,)26 b Fq(G)n(\023)-47 b(eom)n(\023)g(etrie)24 b(et)646 4126 y(th)n(\023)-47 b(eorie)38 b(des)g(gr)-5 b(oup)g(es)p FB(,)38 b(Springer-V)-8 b(erlag,)37 b(Lecture)h(Notes)g(in)e (Mathematics)646 4246 y Fr(1441)p FB(.)324 4443 y([Da])144 b Fa(K.)48 b(R.D)m(a)-9 b(vidson)p FB(,)44 b FA(C)1525 4407 y FD(\003)1564 4443 y Fq(-algebr)-5 b(a)42 b(by)i(examples)p FB(,)f(American)e(Mathematical)646 4564 y(So)s(ciet)m(y)33 b(\(1996\).)324 4761 y([GI])156 b Fa(V.)38 b(Geor)n(gescu)f(&)i(A.)f (Iftimo)n(vici)p FB(,)c Fq(Cr)-5 b(osse)g(d)35 b(pr)-5 b(o)g(ducts)35 b(of)g FA(C)3152 4725 y FD(\003)3192 4761 y Fq(-algebr)-5 b(as)646 4882 y(and)56 b(sp)-5 b(e)g(ctr)g(al)56 b(analysis)f(of)h(quantum)h(Hamiltonians)p FB(,)k(Comm)m(un.)55 b(Math.)646 5002 y(Ph)m(ys)34 b Fr(228)p FB(,)f(519-560)e(\(2002\).) 1894 5251 y(26)p eop end %%Page: 27 27 TeXDict begin 27 26 bop 324 548 a FB([GH])118 b Fa(E.)32 b(Ghys)h(&)f(P.)h(De)f(la)g(Harpe)p FB(,)d Fq(Sur)h(les)h(gr)-5 b(oup)g(es)30 b(hyp)-5 b(erb)g(oliques)30 b(d'apr)n(\022)-47 b(es)646 668 y(Mikhael)34 b(Gr)-5 b(omov)p FB(,)33 b(Birkh\177)-49 b(auser)33 b(\(1990\).)324 872 y([Go])142 b Fa(S.)35 b(Gol)973 864 y(\023)973 872 y(enia)p FB(,)30 b Fq(Mourr)-5 b(e)33 b(estimates)f(for)h(anisotr)-5 b(opic)31 b(op)-5 b(er)g(ators)32 b(on)g(tr)-5 b(e)g(es)30 b FB(\(in)646 992 y(preparation\).)324 1196 y([Ro])147 b Fa(A.)39 b(M.R)m(ober)-7 b(t)p FB(,)33 b Fq(A)k(Course)e(in)h(p-adic)f(A)n(nalysis)p FB(,)e(Springer-V)-8 b(erlag)32 b(GTM)646 1316 y Fr(198)p FB(.)1894 5251 y(27)p eop end %%Trailer userdict /end-hook known{end-hook}if %%EOF ---------------0210290650366--