Content-Type: multipart/mixed; boundary="-------------0202080611434" This is a multi-part message in MIME format. ---------------0202080611434 Content-Type: text/plain; name="02-62.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-62.keywords" Thomas-Fermi Theory, Scott correction, Multiscale Analysis, Estability of Matter ---------------0202080611434 Content-Type: application/postscript; name="Scott.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Scott.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: Scott.dvi %%Pages: 60 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips Scott.dvi -o Scott.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.02.07:1224 %%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 (Scott.dvi) @start %DVIPSBitmapFont: Fa cmsy10 10 2 /Fa 2 91 df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ndDVIPSBitmapFont %DVIPSBitmapFont: Fb cmr7 7 2 /Fb 2 52 df<13FF000313E0380E03F0381800F848137C48137E00787F12FC6CEB1F80A4 127CC7FC15005C143E147E147C5C495A495A5C495A010EC7FC5B5B903870018013E0EA01 80390300030012065A001FB5FC5A485BB5FCA219267DA521>50 D<13FF000313E0380F01 F8381C007C0030137E003C133E007E133FA4123CC7123E147E147C5C495AEB07E03801FF 8091C7FC380001E06D7E147C80143F801580A21238127C12FEA21500485B0078133E0070 5B6C5B381F01F03807FFC0C690C7FC19277DA521>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc cmmi10 10 4 /Fc 4 105 df<013FB612E090B712F05A120717E0270F807006C7FC391E00600E48140C 003813E04813C048141CEAC0011200148001035BA213071400A25B1578011E137CA3133E 133C137C157E13FC5B1201157F1203497FA3D801C0131C2C257EA32F>25 D<121C127FEAFF80A213C0A3127F121C1200A412011380A2120313005A1206120E5A5A5A 12600A19798817>59 D<150C151E153EA2153C157CA2157815F8A215F01401A215E01403 A215C01407A21580140FA215005CA2141E143EA2143C147CA2147814F8A25C1301A25C13 03A2495AA25C130FA291C7FC5BA2131E133EA2133C137CA2137813F8A25B1201A25B1203 A25B1207A25B120FA290C8FC5AA2121E123EA2123C127CA2127812F8A25A12601F537BBD 2A>61 D104 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd cmsy10 14.4 1 /Fd 1 1 df<007FBA12FEBCFCA36C19FE4805769F5D>0 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe cmr10 10 7 /Fe 7 85 df0 D49 DII54 D70 D<003FB812E0A3D9C003EB001F273E0001FE130348EE01F00078 160000701770A300601730A400E01738481718A4C71600B3B0913807FF80011FB612E0A3 35397DB83C>84 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff cmmi12 14.4 1 /Ff 1 27 df26 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg msbm8 8 2 /Fg 2 83 df<91390FFC018091B5FC903903FC07E390390F7001FF90393CE0007FD9F1C0 13392601C380131DD80383C7120D4848140FD80E061407D80C0E1403EA1C0C00381501EA 301C13181270006092C7FC1338EAE03012C0AA12E0EA6038131812701230A2EA381CEA1C 0CD80C0E1570D80E0615F0D80707EC01E026038380EB03C0D801C114072600F1E0EB0F00 D93C70133E90390F3E01FC0103B512F0010014C0DA0FFEC7FC2C307EAE21>67 D82 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fh cmsy6 6 2 /Fh 2 49 df0 D48 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fi cmmi6 6 15 /Fi 15 121 df18 D21 D<13C03901E00180EC03C0A23903C00780A439 07800F00A4380F001EA21508150C48EB3C18A2147CECFC30393F839E609038FF0FE0393C FC078090C8FC5AA45AA45A1E217D9525>I<4A7EA34AC7FCA41406A45CA2D807C01307D8 0FE0EB0F803818F01812300060140712C0EC3003EAC1E000011500EA03C01460A2D80780 1306A24A5A5D15385D3903C180E09038E181C02601F98FC7FC38007FFEEB0FF00103C8FC A31306A45BA3212E7DA229>32 D<140C141C143C1438A21478147014F014E0130114C0A2 1303148013071400A25B130E131E131CA2133C13381378137013F05BA212015B12035BA2 120790C7FC5A120EA2121E121C123C123812781270A212F05AA216317CA420>61 D78 D<013FB512F0A291388007E0D97C 0013C00178EB0F800170EB1F0049133E495B5D4848485A4A5A4A5AC7485A4AC7FC143E5C 14FC495A495A495A90380FC006EB1F80EB3F00017E5B5B4848131C484813184848133848 5A48485B393F0001F0007E130FB65AA224227CA12A>90 D99 DI<13F8EA0FF0A21200A2485AA4485AA4380787E0EB9FF8EBB83CEBE01C380FC01E1380 A21300001E5BA35C5AA2ECF0201530481460EB01E015C0A239F000E380ECFF000060133C 1C247CA224>104 D<1338137CA2137813701300A7EA0780EA1FC0EA38E01230EA60F0EA C1E0A3EA03C0A3EA0780A2EA0F0013041306EA1E0CA21318121CEA1E70EA0FE0EA07800F 237DA116>I<1418143C147CA214381400A7EB0780EB1FE01338EB60F013C0A2EA0180A2 380001E0A4EB03C0A4EB0780A4EB0F00A4131EA21238EA783CEAF8381378EA70F0EA7FC0 001FC7FC162D81A119>I108 D115 D<3801F01E3907FC7F80390E1CE1C038180F8100301383007013071260 EC0380D8001EC7FCA45BA21580003014C0397878018012F8EC030038F0FC0638E19C1C38 7F0FF8381E03E01A177D9523>120 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fj cmr6 6 9 /Fj 9 57 df<130C1338137013E0EA01C0EA038013005A120EA25AA25AA312781270A312 F0AB1270A312781238A37EA27EA27E7E1380EA01C0EA00E013701338130C0E317AA418> 40 D<12C012707E7E7E7E7E1380EA01C0A2EA00E0A21370A313781338A3133CAB1338A3 13781370A313E0A2EA01C0A2EA038013005A120E5A5A5A12C00E317CA418>I<13FF0003 13C0380781E0380F00F0001E137848133CA248131EA400F8131FAD0078131EA2007C133E 003C133CA26C13786C13F0380781E03803FFC0C6130018227DA01E>48 D<13E01201120712FF12F91201B3A7487EB512C0A212217AA01E>II<13FF000313C0380F03E0381C00F014F8003E13FC147CA2001E13 FC120CC712F8A2EB01F0EB03E0EB0FC03801FF00A2380003E0EB00F01478147C143E143F 1230127812FCA2143E48137E0060137C003813F8381E03F0380FFFC00001130018227DA0 1E>I<14E01301A213031307A2130D131D13391331136113E113C1EA01811203EA070112 06120C121C12181230127012E0B6FCA2380001E0A6EB03F0EB3FFFA218227DA11E>I<00 101330381E01F0381FFFE014C01480EBFE00EA1BF00018C7FCA513FE381BFF80381F03C0 381C01E0381800F014F8C71278A2147CA21230127812F8A214784813F8006013F0387001 E01238381E07803807FF00EA01F816227CA01E>I<137F3803FFC0380781E0380E007048 13380018131C1238A3123C003F1338381FC078EBE0F0380FF9E03807FF80120114C00007 13F0380F0FF8381C03FC383801FE3870007E141F48130F1407A314060070130E0078130C 6C1338001F13F03807FFC0C6130018227DA01E>56 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fk cmex10 12 42 /Fk 42 112 df0 D<12E07E12787E7E7E7F6C7E6C7E7F12016C7E7F137C137E7FA26D7EA26D7EA26D7EA36D 7EA2801301A2801300A280A2147EA2147FA4801580A7EC1FC0B3A5EC3F80A715005CA414 7EA214FEA25CA213015CA213035CA2495AA3495AA2495AA249C7FCA2137E137C13FC5B48 5A12035B485A485A90C8FC121E5A5A5A5A1A777C832E>II I8 D<12F012FEEAFFC0EA3FF0EA07FCEA01FE6C6C7EEB3FC06D7E130F801307801303B3B080 1301A26D7E806E7E6E7E6E7EEC07F8EC01FC9138007F80ED1FE01503151FED7F80913801 FC00EC07F8EC1FE04A5A4A5A4AC7FC5C495AA213035CB3B013075C130F5C131F495AEBFF 804848C8FCEA07FCEA3FF0EAFFC048C9FC12F0237775833A>I<12F0B3B3B3AA04407281 21>12 D<16F01501ED03E0ED07C0ED0F80ED1F005D157E5D5D14014A5A4A5A4A5AA24A5A 143F92C7FC147EA25C13015C13035C13075C130F5C131FA2495AA349C8FCA213FEA31201 5BA212035BA21207A25BA2120FA25BA2121FA45BA2123FA55B127FA990C9FC5AB3AA7E7F A9123F7FA5121FA27FA4120FA27FA21207A27FA21203A27F1201A27F1200A3137FA26D7E A36D7EA2130F801307801303801301801300147EA28081141F6E7EA26E7E6E7E6E7E1400 81157E8181ED0F80ED07C0ED03E0ED01F0150024B26E833B>16 D<12F07E127C7E7E6C7E 7F6C7E6C7E12017F6C7E137E7FA26D7E80130F6D7EA26D7E80130180130080147E147F80 81A26E7EA36E7EA26E7EA3811403A2811401A281A21400A281A281A21680A4153FA216C0 A5151F16E0A9150F16F0B3AA16E0151FA916C0153FA51680A2157FA41600A25DA25DA214 01A25DA214035DA214075DA34A5AA24A5AA34A5AA292C7FC5C147E14FE5C13015C13035C 495AA2495A131F5C49C8FCA2137E5B485A5B1203485A485A5B48C9FC123E5A5A5A24B27C 833B>I<171E173E177C17F8EE01F0EE03E0EE07C0160FEE1F80EE3F00167E167C16FC4B 5A4B5A15075E4B5A4B5A153F93C7FC5D15FE5D14015D14034A5AA24A5AA24A5AA24A5AA2 4AC8FCA214FEA213015C13035C1307A25C130F5C131FA25C133FA3495AA349C9FCA35A5B A312035BA31207A25BA2120FA35BA3121FA35BA3123FA55BA2127FAB485AB3B06C7EAB12 3FA27FA5121FA37FA3120FA37FA31207A27FA21203A37F1201A37F7EA36D7EA36D7EA313 1F80A2130F80130780A21303801301801300A2147FA26E7EA26E7EA26E7EA26E7EA26E7E 140181140081157F8182151F6F7E6F7E8215036F7E6F7E167C167E82EE1F80EE0FC01607 EE03E0EE01F0EE00F8177C173E171E2FEE6B8349>I<12F07E127C7E7E6C7E6C7E7F6C7E 6C7E6C7E137C137E7F6D7E80130F6D7E6D7E801301806D7E147E147F80816E7EA26E7EA2 6E7EA26E7EA26E7EA26E7EA2818182153F82A2151F82150F82A2150782A36F7EA36F7EA3 8281A31780167FA317C0A2163FA217E0A3161FA317F0A3160FA317F8A51607A217FCABEE 03FEB3B0EE07FCAB17F8A2160FA517F0A3161FA317E0A3163FA317C0A2167FA21780A316 FF1700A35D5EA34B5AA34B5AA35E150FA25E151F5E153FA25E157F93C7FC5D5DA24A5AA2 4A5AA24A5AA24A5AA24A5AA24A5A92C8FC5C147E14FE495A5C13035C495A495A131F5C49 C9FC137E137C13FC485A485A485A5B485A48CAFC123E5A5A5A2FEE7C8349>III<170F173F17FF1603EE0FFCEE1FF0 EE7FE0EEFF804B13004B5A4B5A4B5A4B5A4B5A4B5A15FF5E5C93C7FC5C5D14075DA3140F 5DB3B3B3AE4A5AA3143F5DA24A5AA24A5AA24990C8FC495AA2495A495A495A495A495A49 C9FC485AEA07FCEA0FF0EA3FC0B4CAFC12FCA2B4FCEA3FC0EA0FF0EA07FCEA01FE6C7EEB 7FC06D7E6D7E6D7E6D7E6D7EA26D7E6D7FA26E7EA26E7EA281141FA36E7EB3B3B3AE8114 07A38114038180828082157F6F7E6F7E6F7E6F7E6F7E6F7E6F1380EE7FE0EE1FF0EE0FFC EE03FF1600173F170F30EE73834B>26 D<12F012FE6C7E7FEA3FF0EA0FFC6C7E3801FF80 6C7F6D7EEB1FF06D7E6D7E806D7E7F6D7F81147F81143F81141FA381140FB3B3B3AE6E7E A46E7EA26E7EA26E7FA26F7EA26F7E6F7E6F7E15076F7E6F7E923800FF80EE7FC0EE1FE0 EE0FF8EE03FCEE00FF173FA217FFEE03FCEE0FF8EE1FE0EE7FC0EEFF80923801FE004B5A 4B5A150F4B5A4B5A4B5AA24B5AA24A90C7FCA24A5AA24A5AA44A5AB3B3B3AE141F5DA314 3F5D147F5D14FF5D4990C8FC5B495A5C495A495AEB7FE0495A485BD807FEC9FC485AEA3F F0EAFFC05B48CAFC12F030EE73834B>I[51 298 104 131 79 32 D[<12F87E127E7E7F121F6C7E6C7E7F6C7E12016C7E7F137F7F806D7E6D7EA26D7E80 13036D7EA26D7E808081143F81141F816E7EA26E7EA26E7EA26E7EA36E7EA26F7EA28215 3FA282151F82150FA2821507A282150382A3150182A28183A3707EA4707EA483161FA483 160FA383A31607A383A31603A383A582A21880A882A218C0AD177F18E0B3B3A618C017FF AD1880A25EA81800A25EA55FA31607A35FA3160FA35FA3161F5FA4163F5FA44C5AA44C5A A394C7FC5DA25E1503A35E15075EA2150F5EA2151F5E153F5EA2157F5EA24BC8FCA24A5A A34A5AA24A5AA24A5AA24A5A5D143F5D147F92C9FC5C5C495AA2495A13075C495AA2495A 495A91CAFC5B13FE5B485A1203485A5B485A485A123F90CBFC127E5A5A>51 298 125 131 79 I[29 298 101 131 58 I[29 298 127 131 58 I[51 298 114 131 80 40 D[<12F812FE6C7E13E07FEA3FF8EA1FFEEA07FF6C7F6C7F6C7FEB 3FF06D7E806D7E6D7E7F6D7F817F81147F81143F81141FA281140FA3811407B3B3B3B3AD 8180A46E7FA36E7FA36F7EA26F7E151F82150F826F7E6F7E816F7F707E707E707E707E70 7E707E933801FF809338007FC0EF3FE0170FA2173FEF7FC0933801FF80933803FE004C5A 4C5A4C5A4C5A4C5A4C5A4B90C7FC5D4B5A4B5A5E151F5E153F4B5AA24B5AA34A5BA34A90 C8FCA45C5DB3B3B3B3AD140F5DA3141F5DA2143F5D147F5D14FF5D5B5D4990C9FC5B495A 495A5C495AEBFFE0485B485B4890CAFCEA1FFEEA3FF8B45A5B138048CBFC12F8>51 298 114 131 80 I[<181E183FA2187FA2187E18FEA218FC1701A218F81703A218F01707 A218E0170FA218C0171FA21880173FA218005FA2177E17FEA25F1601A25F1603A25F1607 A24C5AA25F161FA25F163FA294C7FC5EA2167E16FEA25E1501A25E1503A25E1507A25E15 0FA25E151FA25E153FA293C8FC5DA215FEA25D1401A25D1403A25D1407A25D140FA25D14 1FA25D143FA292C9FC5CA2147E14FEA25C1301A25C1303A25C1307A2495AA25C131FA25C 133FA291CAFC5BA2137E13FEA25B1201A25B1203A25B1207A25B120FA25B121FA25B123F A290CBFC5AA2127E12FEA4127E127FA27E7FA2121F7FA2120F7FA212077FA212037FA212 017FA212007FA2137E137FA27F80A2131F80A2130F80A26D7EA2130380A2130180A21300 80A2147E147FA28081A2141F81A2140F81A2140781A2140381A2140181A2140081A2157F A28182A2151F82A2150F82A2150782A2150382A2150182A2150082A2167E167FA28283A2 161F83A2160F83A2707EA2160383A2160183A2160083A2177E177FA2831880A2171F18C0 A2170F18E0A2170718F0A2170318F8A2170118FCA2170018FEA2187E187FA2183FA2181E >56 298 115 131 80 I[<127812FCA27EA2127E127FA27E7FA2121F7FA2120F7FA21207 7FA212037FA212017FA212007FA2137E137FA27F80A2131F80A2130F80A26D7EA2130380 A2130180A2130080A2147E147FA28081A2141F81A2140F81A2140781A2140381A2140181 A2140081A2157FA28182A2151F82A2150F82A2150782A2150382A2150182A2150082A216 7E167FA28283A2161F83A2160F83A2707EA2160383A2160183A2160083A2177E177FA283 1880A2171F18C0A2170F18E0A2170718F0A2170318F8A2170118FCA2170018FEA2187E18 7FA4187E18FEA218FC1701A218F81703A218F01707A218E0170FA218C0171FA21880173F A218005FA2177E17FEA25F1601A25F1603A25F1607A24C5AA25F161FA25F163FA294C7FC 5EA2167E16FEA25E1501A25E1503A25E1507A25E150FA25E151FA25E153FA293C8FC5DA2 15FEA25D1401A25D1403A25D1407A25D140FA25D141FA25D143FA292C9FC5CA2147E14FE A25C1301A25C1303A25C1307A2495AA25C131FA25C133FA291CAFC5BA2137E13FEA25B12 01A25B1203A25B1207A25B120FA25B121FA25B123FA290CBFC5AA2127E12FEA25AA21278 >56 298 118 131 80 I50 DI<12FEB3B3B3B3B3B3B3B3B3A9 B7FCA720B2608342>I<157FB3B3B3B3B3B3B3B3B3A9B7FCA720B27F8342>I56 D58 D<913807FF80B3B3B04A1300A55D141FA35D143F5DA2147F5D14FF5DA2495B5D5B4990C7 FC5C130F5C495A495A495AA2495A485B4890C8FCEA07FC485A485AEA7FE0EAFF8090C9FC 12FCB4FC7FEA7FE0EA1FF06C7E6C7E6CB4FC6C7F6C7F6D7EA26D7E6D7E6D7E801307806D 7F7F816D7FA281147F81143FA281141F81A3140F81A56E1380B3B3B021B56F8059>60 D62 D80 D<17FF040313C093380F81 F093381E0078043E137C93387C01FC9338F803FEA2150116F01503EF01FC9338E0007003 071400A3150FA45E151FA7153FA74B5AA715FFA85C93C8FCA95C5DA85DA74A5AA75DA75D 140FA45DA3001C5C007F131FEAFF8092C9FC5C143EA26C485A007C5B003C5B381F03E038 07FF80D801FECAFC376F7B7F2F>82 D<007C193EA200FE197FB3B3B3AE6C19FFA26C19FE A26D1701A26C6CEF03FCA2001F19F86D17076D170F000F19F06C6CEF1FE06D173F6C6CEF 7FC06C6CEFFF806E5D6C01E0030713006D6C4B5AD93FFCED3FFC6DB4EDFFF86D01E00107 5B6D01FE017F5B010190B712806D94C7FC023F15FC020F15F002011580DA003F01FCC8FC 030313C048647B7F53>I<163C167E167F5EA24B7FA34B7FA34B7FA34B7FA216E7031F7F A216C3033F7FA21681037F7FA24BC67EA24B7F020181A24B133F020381A24B131F020781 A24B130F020F81A24B1307021F81A24B1303023F81A24A486D7EA292C8FC4A81A24A8101 0183A24A153F010383A24A151F010783A24A150F010F83A24A1507011F83A249486F7EA2 4A1501017F83A291CAFC4983A2498300011980A249173F000319C0A249171F000719E0A2 49170F000F19F0A24848EF07F8A2491703003F19FCA2491701007F19FEA290CCFC4819FF A248197FA2007C193E1A1E48647B7F53>86 D88 D<1B3FF3FFC0973803E0F0973807C03897380F801C081F137E97383F01FEF303FF1A7E1A FE1AFC1901A2963903F801FEF300781C004F5AA2190FA262191FA34F5AA44F5AA319FF97 C8FCA360A261A21803A3611807A461180FA44E5AA4183FA261A2187FA361A218FFA44D5B A55F96C9FCA35FA360A2170FA460171FA460173FA54D5AA54D5AA45E60A55E60A44C90CA FCA54C5AA55F161FA45F163FA45FA2167FA35FA316FF5FA54B5BA494CBFCA25DA35EA215 07A25EA44B5AA45E151FA45E153FA35EA2157FA25EA315FF93CCFCA34A5AA44A5AA35D14 075DA2140F5D121E397F801FC0EAFFC05D143F92CDFC5C147E6C485AEA7E00383801F86C 485A380F07C03803FF80D800FCCEFC58DD7B7F37>90 D<003EF407C0007FF40FE0486CF3 1FF0B3B3B3B3B3A56D1B3F007F1DE0A46D1B7F003F1DC0A26D1BFF001F1D806D62A26C6C 501300A26C6C505A6D1A0F6C6D4F5AA26C6D4F5A6E197F6C6D4F5A6D6C4E5B6D6C4E5B6E 606D6C4E5B6D01C0053F90C7FC6D6D4D5A6D01F84C485A6D01FE04075B6D6D6C031F5B6E 01E0037F5B021F01FE0207B512806ED9FFF090B6C8FC020391B712FC6E606E6C17E0031F 178003074CC9FC030116F8DB003F15C0040302FCCAFCDC001F1380648B7B7F6F>I104 DI110 D<12F012FE6C7E13E0EA 3FF8EA0FFCEA03FFC67F6D7E6D7E6D7E6D7E6D7E6D7EA26D7E7FA281147FB3B3AF81143F A281141F81140F8114076E7E6E7E6E7E6F7E6F7EED1FF0ED07F8ED01FE923800FF80163F A216FF923801FE00ED07F8ED1FF0ED3FC04B5A4BC7FC4A5A4A5A4A5A140F5D141F5D143F 5DA2147F5DB3B3AF14FF92C8FCA25B495AA2495A495A495A495A495A495A000390C9FCEA 0FFCEA3FF8EAFFE0138048CAFC12F029B2748342>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fl msbm10 12 3 /Fl 3 83 df<922601FFE01330033F01FC13784AB6FC020FEDC0F8023F9038C07FF8913A FFFE000FFF4901F81303902607FBF07F90260FE7E0EB007F90261F87C0EC3F7849484814 1FD97E1FED0FF801FC90C8FC2601F83E1507D803F0160348485A01C01601380F807802F8 1500EA1F004A1678EA3E01A2003C5B007C183019001278130312F85C12F0AC12F8A20078 7FA2EA7C01A2123C003E7FA2EA1F00A26C6C7E19062607C07C160F01E0171F6C6C6C163F D801F8177E6C6C6C167C017E6D15FCD93F0FED01F890261F87C0EC07F090260FE7E0EC0F E0902607FBF8EC3FC0902601FFFE903801FF806D903AFFC00FFE00023F90B55A020F15F0 020115C0DA003F49C7FC030113F040487CC52E>67 D<007FB54AB512C0B66C4914E0816C 6E6D14C02707F003F09039000FF8002601F801ED03F000006D7E017C6D6E5A017E137E01 7F133E8102807FECC00F6E6C7E017B80903979F003F0ECF801903978FC00F8027C7F6E13 7E023F133E6E6C7E020F1480913907C00FC0EDE007913903F003E0020114F0913900F801 F8EDFC00037E137C033E137E6F133EEE801FDB0FC013810307EB0FC1923803E0079338F0 03E1DB01F813F1923900FC01F9EE7C0070137D043F137F93381F803F040F131F933807C0 0F17E0933803F00704011303933800F80117FC177E173E171F1881EF0FC11707EF03E118 F1EF01F91700187D01FC167F183FD803FF161F007F01F8150FB57E18076C491503CB1201 725A43467DC339>78 D<007FB712C0B812FCEFFF806C17E02800F807F00F13F8DBC00113 FE017890398000FCFF94387C3F8094383E0FC0727E94381E03F0EF1F011800717FA21978 A519F8A24D5B1801EF1E034E5A94383E0FC094387E3F80DDFDFFC7FC933807FFFE92B612 F818E095C8FC17F0ED87C1EEC0F8923883E0FC177C923881F03EA2923880F81F84EE7C0F 717E163E93383F03E0041F7FEE0F81EF80F8EE07C0187C933803E07E183E706C7E85706C 6C7E180794387C03E0057E7F94383E01F8716C7E197C01F86D6D6C7EF13F80007FB66C6C B512E0B700C015F0836C4B6C14E044447EC33D>82 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fm cmmi8 8 51 /Fm 51 122 df11 DI<131FD9 FFC013304801F0137000076D13604815E0D9807C13C0391E003C0148011E13800038EB0E 03480107130000605CEC030612E0C7138EEC018C159C159815B815B0A215F05DA35DA25D A21403A44AC7FCA4140EA45CA31418242C7F9D24>II< EB03FE131FEB7E0013F8EA03F0485A485A485AA248C7FCA25AEBFFF0A2007EC7FCA25AA3 127CA47EA26C1308380F801C3807C0F83801FFE038007F00171E7D9C1D>I<1460A5EC7F F0EC3FF814FF903803CFE0903807800049C7FC131E5B5B5BA2485A485AA2485A48C8FCA2 5A121E123E123CA3127C1278A412F8A4127CA2127E127F6C7E13E0EA1FFC6CB47E6C13F0 00017F38003FFCEB07FE1300143F80A280141EA2EB601CEB7838EB1FF0EB07C01D3C7DAD 1F>I<147C49B4FC903803C78090380783C090381F03E0EB1E01133E017C13F013F8A2EA 01F0120313E01207A2EA0FC01403A2EA1F80A21407003F14E0130090B5FCA2397F000FC0 127EA2141F1580127C00FC14005CA2147EA248137C14FC00785B495AA2387C03E0383C07 C0495A001C90C7FCEA1E3EEA0FF8EA03E01C307DAE21>18 D<13E0486C133C15FF000313 03EC0F7F9038E01C7EEC381C000749C7FC5CEBC3C001C7C8FCEA0FDE13F8EBFF8014F838 1F83FEEB803F496C7E140F48154016C0123EA2007E14811680007C1403ED830000FC1487 EC078E48EB03FC0070EB00F0221F7D9D29>20 D<13FC13FFEB1FC0130F6D7EA36D7EA213 0180A26D7EA3147EA280A36E7EA2140F81A24A7E143F147FECF3F0EB01E3EB03C1903807 81F8130F49C67E133E5B49137E485A48487F1207485A4848EB1F8048C7FC127E48EC0FC0 48EC07E000701403232F7DAD29>I<131C013EEB0380ED07C0017E130F1680137CA201FC 131F16005BA200015C153E5BA20003147E157C5BA20007ECFC08EDF8185BA2000F010113 3816309038E003F002071370001F90380EF8609039F83C78E090397FF03FC090391FC00F 0048C9FCA2123EA2127EA2127CA212FCA25AA21270252C7E9D2A>I<90B612F812035A48 15F03A1E0380C000003C130000701301130700E05CEAC00638000E03A3131CA2133C1407 13381378A201F07FA21201A2D803E07FA20007130313C0A26C486C5A251E7E9C29>25 DI<0103B512F0131F137F90B612E03A01FC1F80003903F00FC03807C00748486C 7E121F1300123EA25AA2140700FC5C5AA2140F5D141F92C7FC143E0078133C147C007C5B 383C01E0381F07C0D807FFC8FCEA01F8241E7D9C28>I29 D<1506A3150E150CA3151C1518A31538 1530A31570D801E0EB6007D807F8EC1F80EA0E3CD81C3E01E013C0003814C00030150F00 70150726607E011480D8E07CEB800312C013FC3880F803000002001300120113F04A5B00 030106130601E0140E160C020E131C020C131801C0143801E05C021C5B91381801C0D801 F0495A030FC7FC3900FC381C90383F30F890380FFFE0010190C8FCEB00701460A314E05C A313015CA42A3C7EAD2E>32 D<123C127E12FFA4127E123C08087A8714>58 D<123C127EB4FCA21380A2127F123D1201A312031300A25A1206120E5A5A5A126009157A 8714>II<15C0140114031580 A214071500A25C140EA2141E141CA2143C143814781470A214F05CA213015CA213035C13 0791C7FCA25B130EA2131E131CA2133C1338A21378137013F05BA212015BA212035BA212 0790C8FC5A120EA2121E121CA2123C1238A212781270A212F05AA21A437CB123>I<12E0 12F812FEEA3F80EA0FE0EA03F8EA00FEEB3F80EB0FE0EB03F8EB00FC143FEC0FC0EC07F0 EC01FCEC007FED1FC0ED07F0ED01FCED007FEE1FC01607161FEE7F00ED01FCED07F0ED1F C0037FC7FCEC01FCEC07F0EC0FC0023FC8FC14FCEB03F8EB0FE0EB3F8001FEC9FCEA03F8 EA0FE0EA3F8000FECAFC12F812E02A2B7AA537>I<147F903801FFE090380780F890380E 003C497F497F49148001781307017C14C001FC130316E0A2137090C7FC16F0A314FE9038 07FF8390381F01C390397C00E7E049137748481337D807E0133F49131F484814C0121F48 C7FCA2481580127EA2ED3F0012FE48147EA2157C15FC5D4A5A007C495AA26C495A001E49 C7FC6C133E3807C0F83803FFE038007F8024307DAE25>64 D<013FB6FC17C0903A00FE00 07F0EE01F84AEB00FC177E1301177F5CA21303177E4A14FEA20107EC01FC17F84AEB03F0 EE07E0010FEC1FC0EE7F009138C003FC91B55A4914FE9139C0003F804AEB0FC017E0013F 140717F091C7FC16035BA2017E1407A201FE15E0160F4915C0161F0001ED3F80EE7F0049 14FEED03F80003EC0FF0B712C003FCC7FC302D7CAC35>66 D<92387FC003913903FFF807 91391FC03E0F91397E00071FD901F8EB03BF4948EB01FED90FC013004948147E49C8FC01 7E157C49153C485A120348481538485AA2485A173048481500A2127F90CAFCA35A5AA5EE 018016031700A2007E5D1606160E6C5D5E6C6C5C000F5D6C6C495A6C6CEB0780D801F801 1EC7FCD8007E13F890381FFFE0010390C8FC302F7CAD32>I<013FB512FEEEFFC0903A00 FE0007F0EE01F84AEB007E8301018118804A140F18C00103150718E05CA21307A25CA213 0FA24A140FA2131F18C04A141FA2013F1680173F91C81300A249157EA2017E5D5F01FE14 014C5A494A5A4C5A00014BC7FC163E4914FCED03F00003EC1FC0B7C8FC15F8332D7CAC3A >I<92387FC003913903FFF80791391FC03E0F91397E00071FD901F8EB03BF4948EB01FE D90FC013004948147E49C8FC017E157C49153C485A120348481538485AA2485A17304848 1500A2127F90CAFCA35A5AA292381FFFFCA29238003FC0EE1F80163F1700A2127E5E167E 7EA26C6C14FE000F4A5A6C7E6C6C1307D801F8EB1E3CD8007EEBFC3890391FFFE0180103 90C8FC302F7CAD37>71 D<90263FFFFC90381FFF80A2D900FEC73803F80018E04AEC0780 4DC7FC0101151C5F4A14E04C5A01034A5A040EC8FC4A5B5E010714E04B5A9138E0078003 0EC9FC010F131F157F4A487E14C190391FC71FC014CEEC9C0F02F07F90383FE00702C07F EC0003825B6F7E137E6F7E13FE167F5B707E1201161F4981831203B539E001FFFEA2392D 7CAC3C>75 D<90383FFFFEA2010090C8FC5C5CA21301A25CA21303A25CA21307A25CA213 0FA25CA2131FA25CA2133FA291C7EA0180A24914031700017E5C160601FE140EA2495C16 3C12015E49EB01F84B5A0003141FB7FC5E292D7DAC30>III81 D<013FB512F816FF903A00FE001FC0EE07E04A6D7E707E01016E7EA24A80A213034C5A5C A201074A5A5F4A495A4C5A010F4A5A047EC7FC9138C003F891B512E04991C8FC9138C007 C04A6C7E6F7E013F80150091C77EA2491301A2017E5CA201FE1303A25BA20001EE038018 005B5F0003913801FC0EB5D8E000133CEE7FF0C9EA0FC0312E7CAC35>I<913807F00691 383FFE0E9138F80F9E903903E001FE903807800049C7127C131E49143CA2491438A313F8 1630A26D1400A27FEB7F8014F86DB47E15F06D13FC01077F01007F141F02011380EC003F 151F150FA215071218A3150F00381500A2151EA2007C5C007E5C007F5C397B8003E039F1 F00F8026E07FFEC7FC38C00FF0272F7CAD2B>I86 DI<0107B612F8A2903A0FFC00 07F002E0EB0FE00280EB1FC049C71380011E143F011CEC7F004914FE4B5A0130495A0170 495A0160495A4B5A4B5A90C748C7FCA215FE4A5A4A5A4A5A4A5A4A5A4A5A4AC8FC14FE5C 130149481306495A4948130E4948130C495A49C7121C01FE141848481438485A5E484814 F048481301484813034848495A48C7127FB7FC5E2D2D7CAC30>90 D99 D<151FEC03FFA2EC003FA2153EA2157EA2157CA2 15FCA215F8A21401EB07E190381FF9F0EB7C1DEBF80FEA01F03903E007E0EA07C0120FEA 1F8015C0EA3F00140F5A007E1480A2141F12FE481400A2EC3F021506143E5AEC7E0E007C EBFE0C14FC0101131C393E07BE18391F0E1E38390FFC0FF03903F003C0202F7DAD24>I< 157C4AB4FC913807C380EC0F87150FEC1F1FA391383E0E0092C7FCA3147E147CA414FC90 383FFFF8A2D900F8C7FCA313015CA413035CA413075CA5130F5CA4131F91C8FCA4133EA3 EA383C12FC5BA25B12F0EAE1E0EA7FC0001FC9FC213D7CAE22>102 D<131FEA03FFA2EA003FA2133EA2137EA2137CA213FCA25BA21201143F9038F1FFC09038 F3C1F03803FF0001FC7F5BA2485A5BA25B000F13015D1380A2001F13035D1300140748EC C04016C0003E130F1580007E148191381F0180007C1403ED070000FCEB0F06151E48EB07 F80070EB01E0222F7DAD29>104 D<1307EB0F80EB1FC0A2EB0F80EB070090C7FCA9EA01 E0EA07F8EA0E3CEA1C3E123812301270EA607EEAE07C12C013FC485A120012015B12035B A21207EBC04014C0120F13801381381F01801303EB0700EA0F06131EEA07F8EA01F0122E 7EAC18>I<15E0EC01F01403A3EC01C091C7FCA9147CEB03FE9038078F80EB0E07131C01 3813C01330EB700F0160138013E013C0EB801F13001500A25CA2143EA2147EA2147CA214 FCA25CA21301A25CA21303A25CA2130700385BEAFC0F5C49C7FCEAF83EEAF0F8EA7FF0EA 1F801C3B81AC1D>I<131FEA03FFA2EA003FA2133EA2137EA2137CA213FCA25BA2120115 F89038F003FCEC0F0E0003EB1C1EEC387EEBE07014E03807E1C09038E3803849C7FC13CE EA0FDC13F8A2EBFF80381F9FE0EB83F0EB01F81300481404150C123EA2007E141C151800 7CEBF038ECF83000FC1470EC78E048EB3FC00070EB0F801F2F7DAD25>I<137CEA0FFCA2 1200A213F8A21201A213F0A21203A213E0A21207A213C0A2120FA21380A2121FA21300A2 5AA2123EA2127EA2127CA2EAFC08131812F8A21338133012F01370EAF860EA78E0EA3FC0 EA0F000E2F7DAD15>I<27078007F0137E3C1FE01FFC03FF803C18F0781F0783E03B3878 E00F1E01263079C001B87F26707F8013B00060010013F001FE14E000E015C0485A491480 0081021F130300015F491400A200034A13076049133E170F0007027EEC8080188149017C 131F1801000F02FCEB3F03053E130049495C180E001F0101EC1E0C183C010049EB0FF000 0E6D48EB03E0391F7E9D3E>I<3907C007E0391FE03FF83918F8783E393879E01E39307B 801F38707F00126013FEEAE0FC12C05B00815C0001143E5BA20003147E157C5B15FC0007 ECF8081618EBC00115F0000F1538913803E0300180147016E0001F010113C015E390C7EA FF00000E143E251F7E9D2B>I<90387C01F89038FE07FE3901CF8E0F3A03879C0780D907 B813C0000713F000069038E003E0EB0FC0000E1380120CA2D8081F130712001400A24913 0F16C0133EA2017EEB1F80A2017C14005D01FC133E5D15FC6D485A3901FF03E09038FB87 C0D9F1FFC7FCEBF0FC000390C8FCA25BA21207A25BA2120FA2EAFFFCA2232B829D24> 112 D<903807E03090381FF87090387C1CF0EBF80D3801F00F3903E007E0EA07C0000F13 03381F800715C0EA3F00A248130F007E1480A300FE131F481400A35C143E5A147E007C13 FE5C1301EA3E07EA1F0E380FFCF8EA03F0C7FC13015CA313035CA21307A2EBFFFEA21C2B 7D9D20>I<3807C01F390FF07FC0391CF8E0E0383879C138307B8738707F07EA607E13FC 00E0EB03804848C7FCA2128112015BA21203A25BA21207A25BA2120FA25BA2121FA290C8 FC120E1B1F7E9D20>II<013F13 7C9038FFC1FF3A01C1E383803A0380F703C0390700F60F000E13FE4813FC12180038EC07 00003049C7FCA2EA200100005BA313035CA301075B5D14C000385CD87C0F130600FC140E 011F130C011B131C39F03BE038D8707113F0393FE0FFC0260F803FC7FC221F7E9D28> 120 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fn cmr8 8 33 /Fn 33 127 df0 D7 D<010FB512C0A29026001FE0C7FC6E5AA500FEED01FCD8FF801407001F16E001C0140F00 0F16C001E0141F00071680A9D803F0EC3F00A4D801F8147EA2D800FC5C017EEBC1F8013F EBC3F090391F8FC7E0903907EFDF80902601FFFEC7FC9038003FF0EC0FC0A64A7E010FB5 12C0A22E2D7CAC37>9 DI<13031307130E131C1338137013F0EA01E013C01203EA0780A2EA0F00A2 121EA35AA45AA512F8A25AAB7EA21278A57EA47EA37EA2EA0780A2EA03C0120113E0EA00 F013701338131C130E1307130310437AB11B>40 D<12C07E12707E7E7E120FEA07801203 13C0EA01E0A2EA00F0A21378A3133CA4131EA5131FA2130FAB131FA2131EA5133CA41378 A313F0A2EA01E0A2EA03C013801207EA0F00120E5A5A5A5A5A10437CB11B>I43 D48 D<130C133C137CEA03FC12FFEAFC7C1200B3B113FE387FFFFEA2172C7AAB23>III<140EA2141E143EA2147E14FEA2EB01BE1303143E13 06130E130C131813381330136013E013C0EA0180120313001206120E120C5A123812305A 12E0B612FCA2C7EA3E00A9147F90381FFFFCA21E2D7EAC23>I<000CEB0180380FC01F90 B512005C5C14F014C0D80C7EC7FC90C8FCA8EB1FC0EB7FF8380DE07C380F801F01001380 000E130F000CEB07C0C713E0A2140315F0A4127812FCA448EB07E012E0006014C0007013 0F6C14806CEB1F006C133E380780F83801FFE038007F801C2D7DAB23>II<1230123C003FB512F8A215F05A15E039700001C00060 1480140348EB0700140E140CC7121C5C143014705C495AA2495AA249C7FCA25B130E131E A2133EA3133C137CA413FCA913781D2E7CAC23>III<123C127E12FFA4127E123C1200AD123C127E12FFA4127E123C081D7A9C14>I<123C 127E12FFA4127E123C1200AD123C127E12FE12FFA3127F123F1203A312071206A2120E12 0C121C1218123812701260082A7A9C14>I61 D68 D70 D72 D82 D<90383F80303901FFF0703807C07C390F000EF0001E13074813034813011400127000F0 1470A315307EA26C1400127E127FEA3FE013FE381FFFE06C13FC6C13FF00011480D8003F 13E013039038003FF0EC07F81401140015FC157C12C0153CA37EA215787E6C14706C14F0 6CEB01E039F78003C039E3F00F0038E07FFE38C00FF01E2F7CAD27>I<007FB712F8A290 39000FC003007C150000701638A200601618A200E0161CA248160CA5C71500B3A94A7E01 1FB512E0A22E2D7EAC33>I93 D104 D<3807C0FE39FFC7FF809038CF03 E0390FDC01F03907F800FC49137E49133E49133FED1F80A3ED0FC0A8151F1680A2ED3F00 A26D137E6D137C5D9038FC01F09038CE07E09038C7FF80D9C1FCC7FC01C0C8FCA9487EEA FFFEA2222B7E9D27>112 D<3801FE183807FFB8381E01F8EA3C00481378481338A21418 A27E7EB41300EA7FF06CB4FC6C13C06C13F0000113F838001FFC130138C0007E143EA26C 131EA27EA26C133CA26C137838FF01F038E3FFC000C0130017207E9E1C>115 D117 D<3AFFFC01FFC0A23A0FE0007E000007 147C1538000314306D137000011460A26C6C5BA2EBFC01017C5BEB7E03013E90C7FCA2EB 1F06A2148EEB0F8CA2EB07D8A2EB03F0A36D5AA26D5AA2495AA2130391C8FC1278EAFC06 A25B131CEA7838EA7070EA3FE0EA0F80222B7F9C25>121 D<38078008380FE01C381FF8 38383FFFF038707FE038E01FC03840078016077AAC23>126 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fo cmsy8 8 22 /Fo 22 108 df0 D<123C127E12FFA4127E123C08087A9414>I< 006015C000E014016C14030078EC07806CEC0F006C141E6C5C6C6C5B6C6C5B6C6C485A6C 6C485A90387807806D48C7FCEB1E1E6D5AEB07F86D5A6D5A497E497EEB0F3CEB1E1E497E 496C7E496C7E48486C7E48486C7E4848137848C77E001E80488048EC078048EC03C04814 0100601400222376A137>I<130C131EA50060EB01800078130739FC0C0FC0007FEB3F80 393F8C7F003807CCF83801FFE038007F80011EC7FCEB7F803801FFE03807CCF8383F8C7F 397F0C3F8000FCEB0FC039781E078000601301000090C7FCA5130C1A1D7C9E23>I20 D<12E012F812FEEA3F80EA0FE0EA03F8EA00FEEB3F80EB0FE0EB03F8EB00FC143FEC0FC0 EC07F0EC01FCEC007FED1FC0ED07F0ED01FCED007FEE1FC01607161FEE7F00ED01FCED07 F0ED1FC0037FC7FCEC01FCEC07F0EC0FC0023FC8FC14FCEB03F8EB0FE0EB3F8001FEC9FC EA03F8EA0FE0EA3F80007ECAFC12F812E0CBFCAD007FB71280B812C0A22A3B7AAB37>I< 170EA3170F8384170384170184717E1878187C84180FF007C0BA12F819FC19F8CBEA07C0 F00F00183E601878604D5A60170360170795C7FC5F170EA33E237CA147>33 D<137813FE1201A3120313FCA3EA07F8A313F0A2EA0FE0A313C0121F1380A3EA3F00A312 3E127E127CA35AA35A0F227EA413>48 DI<91B512C01307 131FD97F80C7FC01FCC8FCEA01F0EA03C0485A48C9FC120E121E5A123812781270A212F0 5AA3B712C0A300E0C9FCA37E1270A212781238123C7E120E120F6C7E6C7EEA01F0EA00FC EB7F80011FB512C013071300222B7AA52F>I54 D<1406140EA2141EEB1F9CEBFFFC3801E07C3803803C3807003E48133F001EEB7F80001C 1373003C14C014F314E3007C14E0007813E1130114C100F814F013031481A313071401A2 130F130EA3131E131CA3133C1338007814E013781370007C1303003C14C013F013E0001E EB0780121F390FC00F000007131E143E3803E078EBFFF0EB9F80D80780C7FC90C8FCA31C 387DB223>59 D<91383FFFF00107B6FC011F15E0017F812701F83E0313FCD803C0903800 3FFED80F00EC07FF001E017E6D1380481500007CEE7FC00078163F48EE1FE048137CC715 0FA202FC1407A25CA3010116C05CA2EF0F8013034A15005F171E49485C177C177849485C 4C5A4C5A49C7485A040EC7FC163C013E14F0ED03E0013CEB0F80017C01FEC8FC90387FFF F848B512E04849C9FC4813E0332D7EAC37>68 D<027F1518902607FF801478013F6D14F0 90B514012601F03F15E02607801F1403380F000F001EEE07C05A007C5C0078EE0F805A48 131FC7ED1F0092C7FCA2173E5CA2023E5CA34A5C010FB6FC5B137F5F903900F80001A249 4813035FA2495A1607A2495A5F5C130F040F130249C7140E183C011E167C013EEDE078EF FFF04916C0017016000140EC07F837307EAC3C>72 D<023FB57E0107B612F8011F15FE01 7F813A01F83E001FD803C002017FD80F00EC007F001E017E143F5A007C161F12784894C7 FC48137CC7FC173E02FC143C177C4A14785F4C5A01014A5A4A010FC8FC167EED1FF80103 EB7FE09138E0FF8002E17FECE07F49486C7E6F7E150F010F80EC80076F7E1400496D7EF0 0180013E6D6CEB07006093387F801E49EDC03893383FE0F00178EDFFC001F86E5B01E06E 48C7FC4848EC07F0392E7EAC3C>82 D<9239FFC001C002079038FF0780021FECFF00027F 5C902601F01F5B903A03C0007FF849481300010F4A5A49C75B011E4A5A494A5A01304AC7 FC90C8121E5E5E5E4B5A4B5A4B5A91381FFFF84A7F5C9138007878EDF0304A48C8FCEC03 80020FC9FC141E5C5C5C4948140C4948143C49C812F8011E1401495D5B494A5AD801C05D 2607FFF8495A489026FFF00FC7FC48ECFFFE4815F8D8700314E026C000071380322D7DAC 33>90 D<0060150C00E0151CB3AA6C153C00701538A2007815786C15F06CEC01E06C6CEB 07C0D807E0EB1F803A03FE01FF00C6B512FC013F13F0010390C7FC26297CA72F>II<141F14FFEB03F0EB 0FC0EB1F8014005B133EB3A2137E137C13FC485A485AEA7FC048C7FCEA7FC0EA03F06C7E 6C7E137C137E133EB3A2133F7F1480EB0FC0EB03F0EB00FF141F18437BB123>102 D<12FCB47EEA0FE0EA01F0EA00FC137C137E133EB3A37F1480130FEB07E0EB01FEEB007F EB01FEEB07E0EB0F80131F1400133EB3A3137E137C13FCEA01F0EA0FE0EAFF8000FCC7FC 18437BB123>I<12E0B3B3B3AD034378B114>106 D<0060136000E01370B3B3B3AB006013 60144379B123>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fp cmmi12 12 73 /Fp 73 123 df11 DI I<1578913807FFE0021F13FC91383C7FFEEC7007EC6003ECE0004A13381600A280A380A2 80147CA2147E143E143F816E7EA26E7E81140781EC3FFC14FF903803E1FEEB07C190381F 00FF133E49EB7F805B0001143F485A484814C049131F120F485AA248C7FC150F5A127EA3 00FEEC1F805AA316005A5DA2153E157E157CA26C5C127C4A5A6C495AA26C495A6C6C485A 6C6C48C7FC3803E07C3800FFF0EB1FC027487CC62B>II<1530A7ED33FF033F13C0ED7C0092B5FC DA03C31300DA0780C7FC4AC8FC141C14785C495A5C495A130749C9FC131E131C133C5B5B A2485AA2485AA2485AA248CAFCA25A121EA2123E123CA3127C1278A412F8A67EA2127C12 7EA2127F6C7E7F6C7E13F8EA0FFE3807FFC06C13F86C13FF6C6C13E0011F7F010313FC90 38007FFEEC1FFF14039138007F80153F151FA2150FA393C7FCA20102131E13076D6C5A90 3801E0789038007FE0EC1F802A597CC42B>I<01F8EB03FCD803FEEB1FFFD8071F90387C 0FC03B0E0F80E007E0001C9038C3C003271807C70013F002CE1301003801DC14F8003013 D8EB0FF800705B00605BA200E0491303D8C01F15F05C12001607133F91C713E0A2160F5B 017E15C0A2161F13FE491580A2163F1201491500A25E120349147EA216FE1207495CA215 01120F495CEA0380C81203A25EA21507A25EA2150FA25EA2151FA25EA2153FA293C7FC15 0E2D417DAB30>I<157E913801FF80913807C3E091381F01F0EC3E004A13F814FC494813 7C495A5C0107147E495A131F5C133F49C7127FA213FEA212015B12034914FF1207A25B00 0F15FE1501A2485AA21503003F15FC5B90B6FCA24815F89038800007A2150F00FF15F090 C7FCA2ED1FE0A25AED3FC0A21680157F16005A15FEA24A5AA25D14035D4A5A007C495AA2 4A5A007E49C7FC003E133E5C001E5B6C485A380783C06CB4C8FCEA00FC28477CC52D>I< 130E011FEC03E049EC1FF8167F16E749EB03C792380707F0017E130E92381C03C001FE01 78C7FC15E049485A4A5A000149C8FC141EEBF8385C3803F9E0EBFF8014F8ECFFC03907F0 7FF0EC03FC49C6B4FCED3F80000F6E7EA249130FA2001F160717065BA2003F160E170C90 C71380171C4816181738007E1630923807C07000FE16E0923803E1C048913800FF800038 ED3E00302D7BAB38>20 DI<147002F8140E0101153FA301035DA24A147EA2010715FE A24A5CA2010F1401A24A5CA2011F1403A24A5CA2013F1407A291C75BA249140FA2017E5D A201FE021F1318183849ED8030A20001033F13701860EE7F005E486C16E0DB01DF13C092 38039F016DD9071F1380489039801E0F83903BF7C078078700903AE1FFE003FE903AE07F 8000F8000F90CAFCA25BA2121FA25BA2123FA290CBFCA25AA2127EA212FEA25A12383541 7DAB3B>II<1560A7ED7FFF92B512C0 913807F80191381FDFFF91397F87FE004AC8FCEB03FC495A130F5C495A133F5C137F5C13 FFA291C9FCA57F80A2133F6D7E90390FE3FF806DB512E0903901FC006049B512E0D90F8F 1380011EC9FC5B13F8485A5B485A485A48CAFCA2121E123E123C127C1278A312F8A47EA2 7E127F7FEA3FE013F86CB4FC6C13C0000313F86C13FE39007FFFC0010F13F0010313FC90 38007FFF021F7F02037F1400151F150F1507A401025C903803800FD901C090C7FC903800 F83EEC3FF8EC07E02A597EC42B>I<010FB712E0013F16F05B48B812E04817C02807E006 0030C7FCEB800EEA0F00001E010C13705A0038011C13605A0060011813E000E013381240 C7FC5C4B5AA214F014E01301150314C01303A3EB078082130FA2EB1F00A34980133E137E A24980A2000114015BA26C48EB00E0342C7EAA37>II<0203B612E0021F15F091B7FC4916E0010716C090270FF8 0FF8C7FC90381FC00349486C7E017EC7FC49147E485A4848143E0007153F5B485AA2485A A2123F90C8FC5E48157E127EA216FE00FE5D5A15015EA24B5A007C5D15074B5A5E6C4AC8 FC153E6C5C5D390F8003F03907C007C02601F03FC9FC38007FFCEB1FE0342C7DAA37>I< 010FB612FC013F15FE5B48B712FC4816F82707E001C0C7FC01805B380F0003121E121C5A 4849C8FC126012E000405BC7FC140E141EA45CA3147CA2147814F8A4495AA31303A25C13 07A3130FA25C6D5A2F2C7EAA2A>I<013E1538D9FF8014FC2603C3C0EB01FE380703E038 0601F0120E121CD8180314000038167E0030163E1270D86007151E5CA2D8E00F151C00C0 49140CEA001F4A141C1718133F91C8FC1738491530137E1770176013FE4915E017C01601 1780160349EC07001606160E6D5C16181638017E5C013E5C013F495A90391F8007809026 07E01EC7FC903801FFF89038003FE02F2D7DAB35>I<161CA21618A21638A21630A21670 A21660A216E0A25EA21501A25EA21503A293C8FCA25DED7FE0913807FFFE91391FC63F80 9139FE0E07C0D901F8EB03F0903A07E00C00F8D91FC08090263F001C137E017E81491318 4848ED1F8000031438485A4848013014C0A248481370A248481360A248C712E0A24B133F 481780481301A24B137F180014034816FE92C7FC4C5A6C49495AA2007E0106495A4C5A6C 010E495A4C5A261F800C49C7FC000F15FC3A07C01C01F8D803E0EB07E03A01F8181F80D8 007E01FEC8FC90381FFFF801011380D90030C9FCA21470A21460A214E0A25CA21301A25C A21303A291CAFCA332597BC43A>I<137E48B46C150626078FE0150E260607F0151C260E 03F81538000C6D1570D81C0116E000006D15C0010015016EEC03806EEC0700170E6E6C5B 5F5F6E6C136017E04C5A6E6C485A4CC7FC0207130E6F5A5E1630913803F8705EEDF9C06E B45A93C8FC5D6E5A81A2157E15FF5C5C9138073F80140E141C9138181FC014381470ECE0 0FD901C07FEB038049486C7E130E130C011C6D7E5B5B496D7E485A48488048C8FC000681 000E6F137048EE806048033F13E04892381FC0C048ED0FE348923803FF00CA12FC37407D AB3D>I<1730A317701760A317E05FA316015FA3160394C8FCA35E1606A3160E160C013E 1607D9FF80ED1F802603C3C0011CEB3FC0260703E01318260601F0157F000E173F001C15 38D818030230131F0038170F0030170700701570D86007026013035CA2D8E00F02E01480 00C049491301EA001F4A150303011500013F5C1400604901031406017E91C7FC180E180C 01FE49141C4901061418183860030E1460030C14E04D5A4D5A031C49C7FC0318130E017E 5D5F6D01385B90261F80305BD90FC0EB03C0D907F0010FC8FC903901FE707C9039003FFF F002031380DA0060C9FC15E05DA314015DA3140392CAFCA35C1406A3140E140C3A597DC4 3F>I<0110160E0138163F0178EE7F80137001F016FF4848167F5B0003173F49161F1207 90CA120FA2000E1707A248180015060018140FA20038021E14061230A2180E00704A140C 1260181CA203381418183800E05C6015F86C01015D170114030078D907BC495ADA0FBE13 07007CD91F3E495A007ED97E3F013FC7FC3B7F83FE1FE0FF263FFFFCEBFFFE4A6C5B6C01 F05C6CD9C0075B6CD9000113C0D801FC6D6CC8FC392D7FAB3D>I<177F0130913803FFC0 0170020F13E0494A13F048484A13F84848EC7F0190C838F8007C484A48133C00064A4813 1C000E4B131E000C4A48130E001C92C7FC0018140E150C0038021C1406003014185D180E 00704A140C1260154003C0141C00E017184A5A4817386C17304AC8127018E01260007049 EC01C0EF03800078010614070038EE0F00003C010E141E6C167CD81F805D6C6C48EB03F0 D807F0EC0FE0D803FEEC3FC02801FFFC03FFC7FC6C6CB55A6D14F8010F14E00101148090 26007FF8C8FC02F8C9FCA25CA21301A3495AA31307A25C130FA4131F5C6DCAFC37417BAB 40>39 D<121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A0A78891B>58 D<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413E013C0A312011380120313005A 1206120E5A5A5A12600B1D78891B>II<1618163C16 7CA2167816F8A216F01501A216E01503A216C01507A21680150FA2ED1F00A2151E153EA2 153C157CA2157815F8A25D1401A24A5AA25D1407A25D140FA292C7FC5CA2141E143EA214 3C147CA25CA25C1301A25C1303A25C1307A25C130FA291C8FC5BA2133EA2133C137CA213 7813F8A25B1201A25B1203A2485AA25B120FA290C9FC5AA2121E123EA2123C127CA21278 12F8A25A126026647BCA31>I<127012FCB4FCEA7FC0EA1FF0EA07FCEA01FF38007FC0EB 1FF0EB07FE903801FF809038007FE0EC1FF8EC03FE913800FF80ED3FE0ED0FF8ED03FF03 0013C0EE3FF0EE0FFCEE01FF9338007FC0EF1FF0EF07FCEF01FF9438007FC0F01FE0A2F0 7FC0943801FF00EF07FCEF1FF0EF7FC04C48C7FCEE0FFCEE3FF0EEFFC0030390C8FCED0F F8ED3FE0EDFF80DA03FEC9FCEC1FF8EC7FE0903801FF80D907FECAFCEB1FF0EB7FC04848 CBFCEA07FCEA1FF0EA7FC048CCFC12FC12703B3878B44C>I64 D<1830187018F0A217011703A24D 7EA2170F171FA21737A2176717E717C793380187FCA2EE0307EE07031606160CA2161816 38163004607FA216C0030113011680ED0300A21506150E150C5D845D03707F15605DA24A 5A4AB7FCA25C0206C87F5C021C157F14185CA25C14E05C495A8549C9FC49163F1306130E 5B133C137C01FE4C7ED807FFED01FF007F01F0027FEBFFC0B5FC5C42477DC649>I<91B8 7E19F019FC02009039C00003FF6F480100138003FFED3FC01AE093C8121FF10FF0A24A17 F84B1507A314035D190FA2020717F04B151F1AE0193F020F17C04BED7F80F1FF004E5A02 1F4B5A4B4A5AF01FF0F03FC0023F4AB4C7FC4BEB1FFC92B612F018FEDA7FC0C7EA7F804B EC1FC0F00FF0727E02FF6F7E92C8FC727EA249835CA313035CA301075F4A1503A24E5A13 0F4A4B5A4E5AA2011F4C5A4A4B5A4D485A013F4B48C7FCEF0FFC4AEC3FF801FF913801FF E0B9128005FCC8FC17C045447CC34A>I<4CB46C1318043F01F013384BB512FC0307D900 7E1378DB1FF090380F80F0DB7F80EB03C1DA01FEC7EA01C34A48EC00E7DA0FF0ED7FE04A 48153F4A5A02FFC9121F494817C04948160F495A130F4A178049481607495A137F494817 0091CAFC5A485A1906485AA2485A96C7FC121F5BA2123F5BA3127F5BA4485AA419C0A218 0161127F180396C7FC6018066C6C160E601818001F17386D5E000F5F6D4B5A6C6C4B5A00 034CC8FC6C6C150E6C6C153C017F5DD93FC0EB01E0D91FF0EB0FC0D907FE017FC9FC0101 B512FCD9003F13E0020790CAFC45487CC546>I<91B87E19F019FC02009039C00007FF6F 489038007FC003FFED1FE0737E93C86C7E737E19014A707E5D1A7FA20203EF3F805DA21B C014075DA3140F4B17E0A3141F4B17C0A3143F4B167FA3027F18804B16FFA302FF180092 C95A62A24917034A5F19076201034D5A5C4F5A620107173F4A5F4FC7FC19FE010F4C5A4A 15034E5AF00FE0011F4C5A4A4B5A06FFC8FC013FED01FCEF0FF84AEC3FE001FF913803FF 80B848C9FC17F094CAFC4B447CC351>I<91B912FCA3020001C0C7123F6F48EC03F803FF 1501190093C91278A21A385C5DA3020317305DA314074B1460A218E0020F4B13005DA217 01021F5D4B13031707170F023F027FC8FC92B6FCA391397FC0007E4B131EA2170E02FF14 0C92C7FCA2171C49031813035C611906010392C7FC4A160E190C191C010717184A163819 301970130F4A5E180161011F16034A15074E5A013F163F4EC7FC4AEC03FF01FFED3FFEB9 FCA26046447CC348>I<91B912F8A3020001C0C7123F6F48EC07F003FF1503190193C9FC A21A705C5DA3020317605DA314075D18C01701020F4B13005DA21703021F92C8FC4B5BA2 5F023F141E4B13FE92B5FCA24A5CED8000173CA202FF141892C7FCA217384915305CA217 70010315604A91C9FCA313075CA3130F5CA3131F5CA2133FA313FFB612F8A345447CC33F >I<4CB46C1318043F01F013384BB512FC0307D9007E1378DB1FF090380F80F0DB7F80EB 03C1DA01FEC7EA01C34A48EC00E7DA0FF0ED7FE04A48153F4A5A02FFC9121F494817C049 48160F495A130F4A178049481607495A137F4948170091CAFC5A485A1906485AA2485A96 C7FC121F5BA2123F5BA3127F5BA4485A4CB612805EA293C7EBE000725AA3007F60A218FF 96C7FCA26C7E5F606C7EA2000F16036D5E6C6C15070003160F6C6C151F6C6CED3DF8D97F 8014786D6CEB01E0D91FF0903807C078D907FE90387F00700101B500FC1330D9003F01F0 90C8FC020790CAFC45487CC54D>I<91B6D8E003B61280A3020001E0C70003EB8000DB7F 806E48C7FC03FF1503A293C85BA219075C4B5EA2190F14034B5EA2191F14074B5EA2193F 140F4B5EA2197F141F4B5EA219FF143F92B8C8FCA3DA7FC0C712014B5DA2180314FF92C8 5BA218075B4A5EA2180F13034A5EA2181F13074A5EA2183F130F4A5EA2187F131F4A5EA2 013F16FFA24A93C9FCD9FFE002037FB6D8E003B67EA351447CC351>I<027FB512F8A217 F09139007FF000ED3FC0157FA25EA315FF93C7FCA35C5DA314035DA314075DA3140F5DA3 141F5DA3143F5DA3147F5DA314FF92C8FCA35B5CA313035CA313075CA3130F5CA2131FA2 5CEB7FF0007FB512F0B6FCA22D447DC32B>I<91B600E049B512C0A3020001E0C8383FF8 00DB7F80ED1FE003FF94C7FC1A3E93C9127862F101C04A4C5A4B4BC8FC191C6102035E4B 5DF003804EC9FC0207150E4B14386060020F4A5A4B0107CAFC170E5F021F14784B13F84C 7E1603023F130F4B487E163BEEE1FF91387FC1C1DB83807FED8700159CDAFFB86D7E5D03 C06D7E5D4990C7FC4A6E7EA2717E13034A811707A201076F7E5C717EA2130F4A6E7FA272 7E131F5C727E133F854A82D9FFE04B7EB600E0010FB512E05FA252447CC353>75 D<91B612F8A3020001E0C8FC6F5A4B5AA293C9FCA35C5DA314035DA314075DA3140F5DA3 141F5DA3143F5DA3147F5DA314FF92CAFCA35B4A16C0A21801010317804A15031900A201 075E4A1506180E181E010F161C4A153C18381878011F16F84A4A5A1703013F150F4D5A4A 14FF01FF02075BB9FCA2603A447CC342>I<91B500C0933803FFFE63630200F1FE00DB6F E0EE1BF803EF171F1B3703CFEF67F0A21BCF0201EF018F038F60DB87F0ED030F1B1F0203 17060307040C5BA2F2183F020717300206616F6C15601B7F020E17C0020CDC018090C7FC A24F485A021C16060218606F6C5C1A0102381618023004305BA2F16003027016C0026060 6F6CEB01801A0702E0ED03004A03065CA24E130F01015E4A60047F5B1A1F01035E91C74A 5CA24D48133F494BC7FC010661EE3F861A7F010E158C010C039892C8FCA205B05C011C15 E001186001386E5A190101785D01FC92C75BD803FFEF07FEB500F8011E0107B512FE161C 160C5F447BC35E>I<91B500C0020FB5128082A2DA007F9239007FE00070ED1F8074C7FC DBEFF8150E15CF03C7160C70151C1401DB83FE1518A2DB81FF1538140303001630831A70 4A6D7E02061760163F7114E0140E020C6D6C5CA2706C1301141C021801075D8319030238 6D7E023094C8FC1601715B147002606DEB8006A294387FC00E14E04A023F130C18E0191C 0101ED1FF04A1618170FF0F838130391C83807FC30A2943803FE705B01060301136018FF 19E0010E81010C5F187FA2131C0118705A1338181F137801FC70C9FCEA03FFB512F88418 0651447CC34E>I<91B712FEF0FFE019F802009039C0000FFE6F48EB01FF03FF9138007F 80F13FC093C8EA1FE0A24AEE0FF0A25D1AF81403A25DA21407F11FF05DA2020FEE3FE0A2 4B16C0197F021F1780F1FF004B4A5A4E5A023F4B5A4E5A4BEC3FC006FFC7FC027FEC07FC 92B612F018800380CAFC14FFA292CBFCA25BA25CA21303A25CA21307A25CA2130FA25CA2 131FA25CA2133FA25CEBFFE0B612E0A345447CC33F>80 D<91B712F018FF19E002009039 C0003FF86F48EB07FC03FFEC01FEF0007F93C8EA3F801AC0F11FE05C5D1AF0A214035DA3 0207EE3FE05DA2F17FC0020F17804B15FF1A004E5A021F4B5A4B4A5AF00FE04E5A023F03 7FC7FC4BEB03FCEF1FF092B612804A4AC8FC923980007F80EF0FC0EF07F002FF6E7E92C7 7F1701845B4A1400A2170113035CA2170313075CA24D5A130F5CA3011F18185CA2013F4C 13381A304A6F1370D9FFE0020314E0B600E0ED01C00501EB0380943900FE0F00CBEA3FFE F007F045467CC34A>82 D<9339FF8001800307EBF003033F13FC9239FF007E07DA01F8EB 0F0FDA07E09038079F004A486DB4FC4AC77E023E804A5D187E5C495A183C495AA213074A 1538A3130F183080A295C7FC806D7E8014FF6D13E015FC6DEBFFC06D14FC6E13FF6E14C0 020F80020314F8EC003F03077F9238007FFE160F1603707E8283A283A21206A4000E163E A2120C177E001E167CA25F5F003F15014C5A6D4A5A4C5A486C4AC8FC6D143ED87CF85CD8 787E495A3AF01FC00FE0D8E007B51280010149C9FC39C0003FF039487BC53C>I<003FB5 00F80103B512E0A326003FF8C8381FF800D91FE0ED07E0013F705A615C96C7FC60017F16 065CA2180E01FF160C91C9FCA2181C4817185BA21838000317305BA21870000717605BA2 18E0120F495EA21701121F495EA21703123F4993C8FCA25F127F491506A2170E00FF160C 90C9FC171CA21718173817304816705F6C5E6C15014C5A4CC9FC6C150E6D141E001F5D6D 5C6C6CEB01E06C6C495A6C6CEB1F80C6B401FECAFC90387FFFF8011F13E0010190CBFC43 467AC342>85 D<007FB56C91381FFFF8B65DA2000101E0C8000313006C0180ED01FCF000 F0614E5AA2017F4C5A96C7FC1806A2606E5DA2013F5E1870186060A24D5A6E4AC8FCA201 1F1506170E170C5FA26E5C5FA2010F5D16015F4CC9FCA26E13065EA201075C5EA25E16E0 6E5B4B5A13034BCAFC1506A25D151CECFE185D13015D5DA26E5AA292CBFC5C13005C5CA2 5CA25C45467BC339>II<020FB812C05C1A809326800001130003F8C7FCDA3FE04A 5A03804A5A92C8485A027E4B5A027C4B5A02784B5A4A4B5AA24A4A90C7FC4A4A5A01014B 5A4D5A4A4A5A01034B5A91C8485A4D5AA290C84890C8FC4C5A4C5A4C5A4C5A4C5A4C5A4C 5AA24B90C9FC4B5A4B5A4B5A4B5A4B5A4B5AA24B5A4A90CAFC4A5A4A4814064A5A4A5A4A 48140E4A48140CA24A48141C4990C8121849481538495A49485D495A494815F049485D17 01494814034890C8485A4848150F4848151F48484B5A484815FF48481403043F90C8FC48 B8FCB9FC5F42447BC343>90 D97 DIII102 D<157E913803FF8091390FC1E0E091391F 0073F0027E13334A133F4948131F010315E04948130F495AA2494814C0133F4A131F137F 91C713805B163F5A491500A25E120349147EA216FEA2495CA21501A25EA2150315070001 5D150F0000141F6D133F017CEB77E090383E01E790381F078F903807FE0FD901F85B90C7 FC151FA25EA2153FA293C7FCA2001C147E007F14FE485C4A5A140348495AEC0FC000F849 5A007C01FEC8FC381FFFF8000313C02C407EAB2F>I<14FE137FA3EB01FC13001301A25C A21303A25CA21307A25CA2130FA25CA2131FA25CED3FC090393F81FFF0913887C0FC9138 0E007E023C133ED97F70133F4A7F4A14805C13FF91C7FC5BA24848143F17005BA200035D 167E5BA2000715FE5E5B1501000F5DA24913035E001F1607030713064914E0150F003FED C00E170C90C7141CEE80184816381730007E167017E000FE91380781C0EEC38048913801 FF000038EC007C30467BC438>I<141E143F5C5CA3147E143891C7FCAE133EEBFF803801 C3C0380781E0380601F0120E121CEA180312381230A2EA700700605BA2EAE00F00C05BEA 001F5CA2133F91C7FCA25B137E13FE5BA212015BEC03800003140013F01207495A140614 0E140CEBC01C141814385C00035BEBE1C0C6B45A013EC7FC19437DC121>I<163C16FEA2 1501A316FCED00701600AE15FCEC03FF91380F0780021C13C091383803E0147014E014C0 1301EC8007130314005B0106130F130E010C14C090C7FC151FA21680A2153FA21600A25D A2157EA215FEA25DA21401A25DA21403A25DA21407A25DA2140FA25DA2141F5DA2143F00 1C91C7FC127F48137E5CA248485AEB03E038F807C038781F80D83FFEC8FCEA07F0275681 C128>I<14FE137FA3EB01FC13001301A25CA21303A25CA21307A25CA2130FA25CA2131F A25C163F013FECFFC0923803C0E09138000703ED1E0F491338ED701F017E13E0EC01C001 FE018013C00203EB07004948C8FC140E00015B5C495A5C3803FBC001FFC9FC8014F83807 F1FE9038F03F809038E00FE06E7E000F130381EBC001A2001FED01C017801380A2003F15 031700010013F05E481506160E007E150C161C00FE01005BED787048EC3FE00038EC0F80 2B467BC433>II<01F8D903FCEC7F80D803FE D91FFF903803FFE0D8071F903B7C0FC00F81F83E0E0F80E007E01C00FC001C9026C3C003 0178137C271807C700D9F0E0137E02CE902601F1C0133E003801DCDAFB80133F003001D8 92C7FCD90FF814FF0070495C0060495CA200E04949485CD8C01F187E4A5C1200040715FE 013F6091C75BA2040F14014960017E5D1903041F5D13FE494B130762043F160E0001060F 130C4992C713C0191F4CED801C00031A1849027E1638F2003004FE167000071A60494A16 E0F201C0030192380F0380000FF18700494AEC03FED80380D90070EC00F84F2D7DAB55> I<01F8EB03FCD803FEEB1FFFD8071F90387C0FC03B0E0F80E007E03A0C07C3C003001CD9 C7007F001801CE1301003801DC80003013D8EB0FF800705B00605BA200E0491303D8C01F 5D5C12001607013F5D91C7FCA2160F495D137E161F5F13FE49143F94C7FC187000014B13 6049147E16FE4C13E0000317C049150104F81380170300071700495D170EEE781C000FED 7C3849EC1FF0D80380EC07C0342D7DAB3A>III<91380FC00391383FF0079138F83C0F903903E00E1E90390FC0063E 90381F800790393F00037E4914FC01FE1301485AA2484814F812075B000F140316F0485A A2003F14074914E0A3007F140F4914C0A3151F90C713805AA2153F6C1500A2127E5D007F 14FE6C1301A214036C6C485A000F131E3807C0383803E0F13901FFC1F838003F01130014 035DA314075DA3140F5DA2141FA2143F011FB51280A21600283F7DAB2B>I<01F8EB0FC0 D803FEEB7FF0D8070FEBF038000E903883C07C3A0C07C701FC001C13CE0018EBDC030038 13D8003013F8D90FF013F800709038E000E0006015005C12E0EAC01F5C1200A2133F91C8 FCA35B137EA313FE5BA312015BA312035BA312075BA3120F5BEA0380262D7DAB2C>II<141C147EA314FE5CA313015CA313 035CA313075CA2007FB512FCB6FC15F839000FC000A2131F5CA3133F91C7FCA35B137EA3 13FE5BA312015BA312035BA21570000714605B15E015C0000F130101C013801403EC0700 00071306140E5C6C6C5A000113F03800FFC0013FC7FC1E3F7EBD23>I<133ED9FF8014E0 2603C3C0EB03F0380703E0380601F0000E1507121CD818035D12380030150FA2D870075D 00605B161FEAE00F00C0495CEA001F4A133FA2013F92C7FC91C7FC5E5B017E147EA216FE 13FE495CA20301EB01801703484802F81300A25F0303130616F000001407030F130E6D01 0D130C017C011D131C033913186D9038F0F838903A1F03C07870903A07FF803FE0903A01 FC000F80312D7DAB38>I<013E1738D9FF80D901C013FC2603C3C0903907E001FE380703 E0380601F0000E150F001C16C0D8180316000038187E0030031F143E00705ED86007171E 5C163FD8E00F92C7121C00C049160CEA001F4A49141C047E1418133F91C7FC04FE143849 1730017E5CA20301157001FE1760495C19E019C0A24949481301198018031900606D0107 140670130E017C010F5C017E010C1418013ED91CFC13386DD9387E13F0903B0FC0F01F01 C0903B03FFC00FFF809028007F0001FEC7FC3F2D7DAB46>119 D<02FCEB07E0903A03FF 801FFC903A0F07C0781E903A1C03E0E01F903A3801F1C07FD9700013804901FB13FF4848 EBFF00495B000316FE90C71438484A130012061401000E5C120CC7FC14035DA314075DA3 140F5DA3021F143817305D1770023F1460121E003F16E0267F807FEB01C0026F148000FF 01EF1303D901CFEB070000FE903887C00E267C03835B3A3C0F01E0783A1FFC00FFE0D803 F0EB3F80302D7EAB37>I<133ED9FF8014E02603C3C0EB03F0380703E0380601F0000E15 07001C16E0EA180312380030150F007016C0EA60075C161FD8E00F158000C05BEA001F4A 133F1700133F91C7FC5E49147E137EA216FE01FE5C5BA215015E485AA215035EA2000014 07150F6D5C017C131F153F6D13FF90391F03CFC0903807FF8F903801FC0F90C7121F5EA2 153F93C7FCD807C05BD81FE0137E5DA24848485A4A5A01805B39380007C00018495A001C 49C8FC6C137C380781F83803FFE0C66CC9FC2C407DAB30>I<027CEB018049B413034901 801300010F6D5A49EBE00E6F5A90393F03F838903978007EF80170EB1FF00160EB01E001 E05C49495A90C748C7FC150E5D5D5D5D4A5A4A5A4AC8FC140E5C5C5C5CEB03C049C9FC13 0E49141C4914185B49143848481430491470D8039014F048B4495A3A0FEFC007C0391E03 F01FD81C01B55A486C91C7FC485C00606D5A00E0EB3FF048EB0FC0292D7CAB2D>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fq cmsy10 12 41 /Fq 41 115 df<007FB912E0BA12F0A26C18E03C04789A4D>0 D<121FEA3F80EA7FC0EA FFE0A5EA7FC0EA3F80EA1F000B0B789E1C>I<0060160600F8160F6C161F007E163F6C16 7E6C6C15FC6C6CEC01F86C6CEC03F06C6CEC07E06C6CEC0FC06C6CEC1F80017EEC3F006D 147E6D6C5B6D6C485A6D6C485A6D6C485A6D6C485A6D6C485ADA7E3FC7FCEC3F7E6E5A6E 5A6E5AA24A7E4A7EEC3F7EEC7E3F4A6C7E49486C7E49486C7E49486C7E49486C7E49486C 7E49C7127E017E8049EC1F804848EC0FC04848EC07E04848EC03F04848EC01F84848EC00 FC48C9127E007E163F48161F48160F00601606303072B04D>I<147014F8A81470007815 F0007C1401B4EC07F8D87F80EB0FF0D83FE0EB3FE0D80FF0EB7F80D803F8EBFE003900FE 73F890383F77E090380FFF80D903FEC7FCEB00F8EB03FE90380FFF8090383F77E09038FE 73F83903F870FED80FF0EB7F80D83FE0EB3FE0D87F80EB0FF0D8FF00EB07F8007CEC01F0 00781400C7140014F8A81470252B7AAD32>I10 D<037FB612E00207B712F0143F91B812E0010301C0C9FCD907 FCCAFCEB0FE0EB3F8049CBFC13FC485A485A485A5B485A121F90CCFC123EA2123C127CA2 127812F8A25AA87EA21278127CA2123C123EA27E7F120F6C7E7F6C7E6C7E6C7E137E6D7E EB1FE0EB07FC6DB47E010090B712E0023F16F01407020016E092CAFCB0001FB912E04818 F0A26C18E03C4E78BE4D>18 D<19E0F003F0180FF03FE0F0FF80943803FE00EF0FF8EF3F E0EFFF80DC03FEC7FCEE0FF8EE3FE0EEFF80DB03FEC8FCED1FF8ED7FE0913801FF80DA07 FEC9FCEC1FF0EC7FC04948CAFCEB07FCEB1FF0EB7FC04848CBFCEA07FCEA1FF0EA7FC048 CCFCA2EA7FC0EA1FF0EA07FCEA01FF38007FC0EB1FF0EB07FCEB01FF9038007FC0EC1FF0 EC07FC913801FF809138007FE0ED1FF8ED07FE923800FF80EE3FE0EE0FF8EE03FE933800 FF80EF3FE0EF0FF8EF03FE943800FF80F03FE0F00FF01803F000E01900B0007FB912E0BA 12F0A26C18E03C4E78BE4D>20 D<127012FCB4FCEA7FC0EA1FF0EA07FCEA01FF38007FC0 EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC07FC913801FF809138007FE0ED1FF8ED07FE 923800FF80EE3FE0EE0FF8EE03FE933800FF80EF3FE0EF0FF8EF03FE943800FF80F03FE0 F00FF0A2F03FE0F0FF80943803FE00EF0FF8EF3FE0EFFF80DC03FEC7FCEE0FF8EE3FE0EE FF80DB03FEC8FCED1FF8ED7FE0913801FF80DA07FEC9FCEC1FF0EC7FC04948CAFCEB07FC EB1FF0EB7FC04848CBFCEA07FCEA1FF0EA7FC048CCFC12FC1270CDFCB0007FB912E0BA12 F0A26C18E03C4E78BE4D>I24 D<037FB612E00207B712F0143F91B812E0010301C0C9FCD907FCCAFCEB0FE0EB3F8049CB FC13FC485A485A485A5B485A121F90CCFC123EA2123C127CA2127812F8A25AA87EA21278 127CA2123C123EA27E7F120F6C7E7F6C7E6C7E6C7E137E6D7EEB1FE0EB07FC6DB47E0100 90B712E0023F16F01407020016E03C3A78B54D>26 D<07C01403DE03E0EC0F80060F153F DE3FC0ECFF00DEFF80EB03FEDD03FEC7EA0FF8DD07F8EC1FE0DD1FE0EC7F80DD7F80D901 FEC7FCDC01FEC7EA07F8DC07FCEC1FF0DC1FF0EC7FC0DC3FC04AC8FC04FFC7EA03FCDB03 FCEC0FF0DB0FF0EC3FC0DB3FE0ECFF80DBFF80D903FEC9FC4A48C7EA07F8DA07F8EC1FE0 DA1FE0EC7F80DA7F80D901FECAFC4948C7EA07FCD907FCEC1FF0D90FF0EC3FC0D93FC002 FFCBFC01FFC7EA03FCD803FCEC0FF0D80FF8EC3FE0D83FE0ECFF80D87F804948CCFC00FE C7EA03F8A2D87F80EB01FED83FE06D6C7ED80FF8EC3FE0D803FCEC0FF0C6B4EC03FCD93F C0EB00FFD90FF0EC3FC0D907FCEC1FF0D901FFEC07FC6D6C6CEB01FEDA1FE09038007F80 DA07F8EC1FE0DA01FEEC07F86E6C6CEB03FEDB3FE0903800FF80DB0FF0EC3FC0DB03FCEC 0FF0DB00FFEC03FCDC3FC0EB00FFDC1FF0EC7FC0DC07FCEC1FF0DC01FEEC07F89326007F 80EB01FEDD1FE09038007F80DD07F8EC1FE0DD03FEEC0FF8942600FF80EB03FEDE3FC0EB 00FFDE0FE0EC3F800603150FDE00C0EC030059407BB864>28 D<1AF0A3861A78A21A7C1A 3CA21A3E1A1E1A1F747EA2747E747E87747E747E1B7E87757EF30FE0F303F8007FBC12FE BE1280A26CF3FE00CEEA03F8F30FE0F31F8051C7FC1B7E63505A505A63505A505AA250C8 FC1A1E1A3E1A3CA21A7C1A78A21AF862A359347BB264>33 D<14034A7EB3B3B3A5006016 1800FC16FCB4150301C0140FD81FF0EC3FE0D803F8EC7F00D800FC14FC017FEB83F89039 1F8787E090390FC78FC001075C902603E79FC7FC903801F7BE903800FFFC6E5AA26E5A6E 5AA26E5AA26E5AA46EC8FCA32E587EC432>35 D<166016F015015E15035E15074B5A93CD FC5D153E5D5D14014A5A4A5A4ABBFC4A1A805C4A1A00D901F8CEFC495AEB0FE0EB3F8001 FFCFFCEA03FCEA1FF0EAFFC0A2EA1FF0EA03FCC6B4FCEB3F80EB0FE0EB03F06D7ED9007F BBFC6E1A80806E1A00DA07E0CDFC6E7E6E7E1400157C818181826F7E1503821501821500 166059387BB464>40 D<18034E7E85180385180185727E1978197C8585737E86737E737E 007FBA7EBB7E866C85CDEA0FC0747EF203F8F200FEF37F80F31FE0F307FC983801FF80A2 983807FC00F31FE0F37F8009FEC7FCF203F8F207E0505A007FBBC8FCBB5A626C61CCEA03 F04F5A4F5A624FC9FC193E61197819F84E5A6118036118076172CAFC59387BB464>I<03 1CED01C0033E4B7E033C1501037C820378150003F8824B16780201177C4B163C0203173E 4A48824B82020F844ACA6C7E023E717E027E8491BA7E498549854985D90FC0CBEA1F8049 48727E017FCCEA07F001FCF101F8D803F8F100FED80FE0F23F80D83FC0F21FE0B4CEEA07 F8A2D83FC0F21FE0D80FE0F23F80D803F8F2FE00C66CF101F8017FF107F0D91F80F00FC0 6D6C4E5A6DBBC7FC6D616D616D61027ECAEA03F0023E606E4D5A6E6C4C5A020795C8FC6F 5E6E6C163E0201173C6F167C020017786F16F803785E037C1501033C5E033E1503031C6F 5A5D387DB464>44 D<49B4EF3FC0010F01E0923803FFF8013F01FC030F13FE4901FF9238 3FE01F48B66C91397E0007C02603F80301E0D901F8EB01E02807E0007FF049486D7E0180 6D6CD907C0147048C76C6C494880001EDA07FE49C87E001C6E6C013E150C486E6D48150E 71481506486E01E0160793387FF1F0006092263FF3E08193381FFBC000E004FF1780486F 4915017090C9FC82707F8482717E844D7E6C4B6D1503006004EF1700933803E7FE007092 2607C7FF5DDC0F837F003004816D140E00384BC6FC0018033E6D6C5C001C4B6D6C143C6C 4BD91FFC5C6C4A486D6C5C6DD907E06D6C13036C6C49486D9038E00FE0D801F0013FC890 B55A27007C03FE6F91C7FC90263FFFF8031F5B010F01E0030313F8D901FECAEA7FC0592D 7BAB64>49 D<92B6FC02071580143F91B7120001030180C8FCD907FCC9FCEB1FE0EB3F80 017ECAFC5B485A485A485A5B485A121F90CBFC123EA2123C127CA2127812F8A25AA2B9FC 1880A2180000F0CBFCA27EA21278127CA2123C123EA27E7F120F6C7E7F6C7E6C7E6C7E13 7E6D7EEB1FE0EB07FC6DB47E010090B6FC023F1580140702001500313A78B542>I<1706 170F171FA2173EA2177CA217F8A2EE01F0A2EE03E0A2EE07C0A2EE0F80A2EE1F00A2163E A25EA25EA24B5AA24B5AA24B5AA24B5AA24BC7FCA2153EA25DA25DA24A5AA24A5AA24A5A A24A5AA24AC8FCA2143EA25CA25CA2495AA2495AA2495AA2495AA249C9FCA2133EA25BA2 5BA2485AA2485AA2485AA2485AA248CAFCA2123EA25AA25AA25A1260305C72C600>54 D<126012F0B012FC12FEA212FC12F0B0126007267BAB00>I<0060171800F0173C6C177C A200781778007C17F8A2003C17F0003E1601A26CEE03E0A26C17C06D1507A2000717806D 150FA26C6CED1F00A20001161E6D153EA20000163C90B712FCA26D5DA2013CC85A013E14 01A2011E5D011F1403A26D5D6E1307A26D6C495AA2010392C7FC6E5BA20101141E6E133E A26D6C5BA202781378027C13F8A2023C5BEC3E01A26E485AA2020F5B1587A202075B15CF A26EB4C8FCA26E5AA36E5AA315781530364780C437>I<007FB712E0B812F0A27ECAFCB3 AA001FB7FC127FA3CAFCB3AB007FB7FCB8FCA26C16E02C457BC437>I<4B7E4B7EA21507 A25EECFF8F010313EF90260F80FFC7FC90383E003F497F49804848804848497E5B0007EC 3DF049133C000FEC7CF8A248C7EA787C15F848157E15F0A2140148157F007E4A7E1403A2 15C0A200FE01071480A21580140FA21500A25CA2141E143EA2143CA2147CA21478A214F8 A25C1301A2007E491400A21303A2007F495B1307003F157E5CA2130F001F157C018FC712 FCD80F9F5CA201DE130100075DD803FE495AA26C48495A00004A5A017C49C7FC017E133E 90387F80F89038FBFFE001F8138049C9FC1201A25BA26C5A29557CCC32>59 D<031FB512C00203B7FC021F16E091B812F8010317FE010F717E90283FE07FC03F80D9FE 00020080D801F8041F7FD803E04A01077F48481601000F716C7E4848717E003F02FF151F 007F180F90C7707E00FE92C8FC488400F01A80008084C75AA24B81A414035DA21B00A24A 5AA24F5AA24A5A621903624A5A4F5AA24B4B5A023F5F191F4B5E027F4CC7FC197E92C912 7C4A5E4E5A4A4B5A01014C5AF01F804A033EC8FC01035E4A4A5AEF07E00107ED1FC04A02 FFC9FC010FEC07FC4AEBFFF091B612C0017F4ACAFC90B612F04815804802F8CBFC4891CC FC49447EC34D>68 D I72 D77 D79 D<031FB512FC0203B712E0021F16FC91B9FC010318C0010F8490283F E07FC00380D9FE00EC001FD801F804037FD803E04A13004848EF3FFC000F181F4848170F 003F14FF007F180790C7FC00FE92C8FC486112F01280C7485F190F4B5E62191F6202034C C7FC4B157E197C614E5A4A48EC07E0F00F80063FC8FCEF03FC4A4848B45A040F13E04C13 804C48C9FC4A48487EA2041F7FEDC007023F6D7F824B6C7F147F717E92C7FC4A6E7EA24A 141F010182170F4A8101031907716C141F4A183E01076F6D137C4A18F8719038C001F001 0F9438E003E04A6E9038F007C0011F9438FC1F804A92397FFFFE006249486F13F091C96C 13C0017C7048C7FC0170EE03F050467EC354>82 D<932607FF801307047F01FC131F4BB6 EAF07E0307EDFFFC031F16F8037F16F0DBFE0115E0913903F0000F4A48D9003F13C04A48 9138003F80021F17004A48157E027F5E92C8485A02FE4B5A4A4B5AD901F0150F02C04B5A 90CA485A96C7FC187E604D5A4D5A4D5A4D5A4D5A4DC8FC177E030FB512F8033F14FE5DA2 9238000FC193381F807C93383F0070047E90C8FC5E4B5A4B5A4B5A4B5A4B5A4BCAFC157C 5DEC03F04A4816C04A4815074A48150F4AC9EA3F80027E167F4A16FF49481700495AD907 C04B5A49485E013FC9FC017E4C5A90B500F04A5A48DAFFE05C489239FFE00FC04893B5C7 FC485F003F5F267E003F15F000F8D9007F14C000E09126007FFEC8FC48447CC348>90 D<0060170C00F0171EB3B3A66C173EA20078173C007C177C007E17FC003E17F86CEE01F0 6D15036C6CED07E06C6CED0FC0D803F8ED3F80D801FEEDFF0026007FC0EB07FCD93FFCEB 7FF8010FB612E001031580D9007F01FCC7FC020713C0373D7BBA42>I<913807FFC0027F 13FC0103B67E010F15E0903A3FFC007FF8D97FC0EB07FCD801FEC8B4FCD803F8ED3F80D8 07E0ED0FC04848ED07E04848ED03F090C91201003EEE00F8007E17FC007C177C0078173C 00F8173EA248171EB3B3A60060170C373D7BBA42>I<1538157CA215FEA24A7EA215EF02 037FA2913807C7C0A291380F83E0A291381F01F0A2EC1E00023E7FA24A137CA24A7FA249 487FA24A7F010381A249486D7EA249486D7EA249C76C7EA2011E1400013E81A249157CA2 4981A2484881A24848ED0F80A2491507000717C0A24848ED03E0A248C9EA01F0A2003EEE 00F8A2003C1778007C177CA248173EA248171E0060170C373D7BBA42>94 D102 D<12FEEAFFE0EA07F8EA00FEEB7F806D7E6D7E130F6D7EA26D7EB3AD6D7EA26D7E806E7E 6E7EEC0FE0EC03FC913800FFE0A2913803FC00EC0FE0EC3FC04A5A4AC7FC5C495AA2495A B3AD495AA2495A131F495A495A01FEC8FCEA07F8EAFFE048C9FC236479CA32>I<140C14 1E143EA2143C147CA214F8A214F01301A2EB03E0A214C01307A2EB0F80A214005BA2133E A2133C137CA2137813F8A2485AA25B1203A2485AA25B120FA248C7FCA2121E123EA25AA2 127812F8A41278127CA27EA2121E121FA26C7EA212077FA26C7EA212017FA26C7EA21378 137CA2133C133EA27FA27F1480A2EB07C0A2130314E0A2EB01F0A2130014F8A2147CA214 3C143EA2141E140C176476CA27>I<126012F07EA21278127CA27EA2121E121FA26C7EA2 12077FA26C7EA212017FA26C7EA21378137CA2133C133EA27FA27F1480A2EB07C0A21303 14E0A2EB01F0A2130014F8A2147CA2143C143EA4143C147CA214F8A214F01301A2EB03E0 A214C01307A2EB0F80A214005BA2133EA2133C137CA2137813F8A2485AA25B1203A2485A A25B120FA248C7FCA2121E123EA25AA2127812F8A25A126017647BCA27>I<126012F0B3 B3B3B3B3A81260046474CA1C>I<0070130700F01480B3B3B3B3B3A800701400196474CA 32>I<126012F07EA21278127CA2123C123EA2121E121FA26C7EA212077FA212037FA212 017FA26C7EA21378137CA2133C133EA2131E131FA26D7EA2130780A2130380A2130180A2 6D7EA21478147CA2143C143EA280A28081A2140781A2140381A26E7EA2140081A2157815 7CA2153C153EA281A2811680A2150716C0A2150316E0A2ED01F0A2150016F8A21678167C A2163C163EA2161E160C27647BCA32>110 D114 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fr cmti12 12 68 /Fr 68 123 df12 D19 D<13F0EA03F8EA07FC120FA6EA03CCEA001C1318 A213381330A2137013E013C0120113801203EA0700120E5A5A5A5A5A0E1D6BC41E>39 D<167016E0ED01C0ED0380ED0700150E153C5D15F85D4A5A4A5A4A5A140F4AC7FC141E14 3E5C147814F8495A5C1303495AA2495AA249C8FCA25B133E137E137CA25BA212015BA212 035BA212075BA2120FA25BA2121FA290C9FCA25AA2123EA3127EA2127CA65AAB1278A612 7C123CA47EA2120E120FA27E6C7EA26C7EA26C7E1360246472CA28>I<1560A2157081A2 81151E150E150FA2811680A3ED03C0A516E0A21501A71503A91507A216C0A4150FA21680 A2151FA21600A25DA2153EA2157EA2157C15FCA25D1401A25D14035DA214075D140F5DA2 4AC7FCA2143EA25C147814F8495AA2495A5C1307495A91C8FC131E133E5B13785B485A48 5A485A48C9FC121E5A5A12E05A23647FCA28>I<13F0EA03FC1207A2EA0FFEA4EA07FCEA 03CCEA000C131C1318A2133813301370136013E0EA01C013801203EA0700120E5A5A5A5A 5A0F1D7A891E>44 D<007FB5FCB6FCA214FEA21805789723>I<120FEA3FC0127FA212FF A31380EA7F00123C0A0A76891E>I48 D<16C01501A215031507ED0F80151F153F157F913801FF005C140F147F903807FCFEEB0F F0EB0700EB00015DA314035DA314075DA3140F5DA3141F5DA3143F5DA3147F92C7FCA35C 5CA313015CA313035CA313075CA2130FA2131F133FB612FCA25D224276C132>IIII<0260143002 78EB01F091397F801FE091B612C01780170016FC4914F016C0DACFFEC7FC02C0C8FC1303 5CA3130791C9FCA35B130EA3131E90381C07F8EC3FFE9138F80F8090393DC007C0D93F80 7F90383E0003013C80496D7E1370A290C7FC82A41503A415075E120E123F486C130F00FF 5DA390C7485A5A00F84A5A12E04B5A93C7FC15FE14016C5C0070495A0078495A6CEB1FC0 003E495A261F80FEC8FC6CB45A000313E0C66CC9FC2C4476C132>II<9026380FC0131C9038787FE0902671FFF0133C01F3157801EF15F090B5FC48 01E0EB01E09139007003C04848EB380701F8EC1F804848EB1C3F4990381FFF004848EB07 DF49EB001E48C85A121E5E4815F800385D0078140100705D00F01403485D1507C8485AA2 4BC7FCA25D153E157E157C15FC5D1401A24A5AA214075D140F5D141FA25D143FA24AC8FC A314FEA21301A25C1303A25C1307A35C130FA35C5C6D5A2E4472C132>I57 D<130FEB1FC0133FEB7FE013FFA214C0EB7F801400131E90C7FCB3A5120FEA3FC0127FA2 12FFA35B6CC7FC123C132B76AA1E>I<14F0EB01FC1303EB07FE130FA214FCEB07F814F0 EB01E090C7FCB3A513F0EA03F8487EA2120FA46C5AEA03D8EA001813381330A213701360 13E05B12015B120348C7FC1206120E5A5A5A5A5A173E7AAA1E>I65 D<91B712FCF0FF8019E00201903980001FF06E90C7EA07F84A6F7E727E4B81841A800203 167F5DA314075D19FFA2020F17004B5C611803021F5E4B4A5A180F4E5A023F4B5A4BEC7F 804EC7FCEF03FC027FEC0FF84BEBFFC092B6C8FC18E0913AFF800007F892C7EA01FC717E 187F49834A6F7EA30103835CA313075CA3010F5F4A157FA24E5A131F4A4A90C7FC601703 013F4B5A4A4A5A4D5A017F4B5A4D5A4A4948C8FC01FFEC0FFEB812F817C04CC9FC41447A C345>II<91 B712F818FF19C00201903980003FF06E90C7EA0FF84AED03FCF000FE4B157FA2F13F8002 03EE1FC05DF10FE0A214074B16F01907A2140F5D1AF8A2141F5DA2190F143F5D1AF0A214 7F4B151FA302FF17E092C9123FA34918C04A167F1A80A2010317FF4A1700A24E5A13074A 4B5A611807010F5F4A4B5A181F61011F4C5A4A4BC7FC18FE4D5A013F4B5A4A4A5A4D5A01 7FED3FC005FFC8FC4AEB03FE01FFEC1FF8B812E094C9FC16F845447AC34A>I<91B912C0 A30201902680000313806E90C8127F4A163F191F4B150FA30203EE07005DA314074B5D19 0EA2140F4B1307A25F021F020E90C7FC5DA2171E023F141C4B133C177C17FC027FEB03F8 92B5FCA39139FF8003F0ED00011600A2495D5CA2160101034B13705C19F061010791C8FC 4A1501611803010F5F4A150796C7FC60131F4A151E183E183C013F167C4A15FC4D5A017F 1503EF0FF04A143F01FF913803FFE0B9FCA26042447AC342>I<91B91280A30201902680 000713006E90C8FC4A163FA24B81A30203160E5DA314074B151E191CA2140F5D17075F02 1F020E90C7FC5DA2171E023F141C4B133CA2177C027F5CED800392B5FCA291B65AED0007 1601A2496E5A5CA2160101035D5CA2160301075D4A90CAFCA3130F5CA3131F5CA3133F5C A2137FA313FFB612E0A341447AC340>II<91B6D8803FB512E0A302010180C7387FE0006E90C8 6C5A4A167FA24B5EA219FF14034B93C7FCA26014074B5DA21803140F4B5DA21807141F4B 5DA2180F143F4B5DA2181F147F92B75AA3DAFF80C7123F92C85BA2187F5B4A5EA218FF13 034A93C8FCA25F13074A5DA21703130F4A5DA21707131F4A5DA2170F133F4A5DA2017F15 1FA24A5D496C4A7EB6D8803FB512E0A34B447AC348>I<027FB512E091B6FCA20200EBE0 00ED7F8015FFA293C7FCA35C5DA314035DA314075DA3140F5DA3141F5DA3143F5DA3147F 5DA314FF92C8FCA35B5CA313035CA313075CA3130F5CA3131F5CA2133FA25CEBFFE0B612 E0A25D2B447BC326>I<031FB512F05DA29239000FFC005FA35FA2161FA25FA2163FA25F A2167FA25FA216FFA294C7FCA25DA25EA21503A25EA21507A25EA2150FA25EA2151FA25E A2153FA25EA2157FA25EEA0F80D83FE013FF93C8FC127FA24A5AEAFFC04A5A1300007C49 5A0070495A4A5A6C5C003C495A6C01FEC9FC380F81F83803FFE0C690CAFC344679C333> I<91B612F0A25F020101C0C7FC6E5B4A90C8FCA25DA314035DA314075DA3140F5DA3141F 5DA3143F5DA3147F5DA314FF92C9FCA35B5CA3010316104A1538A21878010716705C18F0 18E0010F15015C18C01703011F15074A1580170FA2013FED1F004A5C5F017F15FE16034A 130F01FFEC7FFCB8FCA25F35447AC33D>76 D<91B56C93387FFFC08298B5FC02014DEBC0 006E614A5FA203DF4C6CC7FC1A0E63912603CFE05D038F5F1A381A711407030FEEE1FCA2 F101C3020FEE0383020E60F107036F6C1507021E160E021C60191CF1380F143C02380470 5BA2F1E01F0278ED01C091267003F85EF003801A3F02F0ED070002E0030E5CA24E137F13 0102C04B91C8FC606201036D6C5B02805F4D5A943803800113070200DA07005BA2050E13 03495D010E606F6C5A1907011E5D011C4B5CA27048130F133C01384B5C017892C7FC191F 01F85C486C027E5DD807FE027C4A7EB500F00178013FB512C0A216705A447AC357>I<91 B56C49B512E0A28202009239000FFC00F107F0706E5A4A5F15DF705D1907EC03CFDB8FF8 92C7FCA203875D02077F0303150EA270141EEC0F01020E161C826F153C141E021C6E1338 167F1978023C800238013F1470A27113F00278131F02705E83040F130102F014F84A5E16 07EFFC0313014A01035C17FE1807010314014A02FF90C8FCA2705B0107168F91C8138E17 7F18DE5B010EED3FDC18FCA2011E151F011C5EA2170F133C01386F5A1378A201F8150348 6C5EEA07FEB500F01401A2604B447AC348>I<91B712F018FEF0FF800201903980007FE0 6E90C7EA1FF04AED07F818034B15FCF001FE1403A24B15FFA21407A25DA2140FF003FE5D A2021F16FC18074B15F8180F023F16F0F01FE04B15C0F03F80027FED7F0018FE4BEB03FC EF0FF002FFEC7FC092B6C7FC17F892CAFC5BA25CA21303A25CA21307A25CA2130FA25CA2 131FA25CA2133FA25CA2137FA25C497EB67EA340447AC342>80 DI<91B77E18F818FE0201 90398001FF806E90C7EA3FC04AED1FE0F00FF04BEC07F8180319FC14034B15FEA314075D A3020FED07FC5DA2F00FF8141F4B15F0F01FE0F03FC0023F16804BEC7F0018FEEF03F802 7F4A5A4BEB1FC04CB4C7FC92B512F891B612E092380003F8EE00FE177F496F7E4A6E7EA2 8413034A140FA2171F13075CA2173F130F5CA24D5A131F5CA3013F170E5CA2017FEE801E 191C4A163C496C1638B66C90383FC070051F13F094380FE1E0CA3803FF80943800FE003F 467AC347>II<48B912F85AA2 913B0007FC001FF0D807F84A130701E0010F140349160148485C90C71500A2001E021F15 E05E121C123C0038143F4C1301007818C0127000F0147F485DA3C800FF91C7FC93C9FCA3 5C5DA314035DA314075DA3140F5DA3141F5DA3143F5DA3147F5DA314FF92CAFCA35B5CA2 1303A21307497E007FB612C0A25E3D446FC346>I<001FB500F090383FFFFCA326003FF0 C7000113806D48913800FE00013F167C18785C187018F0017F5E5CA2170101FF5E91C8FC A21703485F5BA21707000394C7FC5BA25F0007160E5BA2171E120F49151CA2173C121F49 1538A21778123F491570A217F0127F495DA2160100FF5E90C8FCA216035F16074893C8FC 5E160E161E5E007E1538007F15785E6C4A5A6D495A001F4A5A6D49C9FC6C6C133E6C6C13 F83903FC07F0C6B512C0013F90CAFCEB07F83E466DC348>III91 D93 D97 DIIIII<15FCEC03FF91390F838380 91393E01CFC091387C00EF4A13FF4948137F010315804948133F495A131F4A1400133F91 C75A5B167E13FE16FE1201495CA215011203495CA21503A2495CA21507A25EA2150F151F 5E0001143F157F6C6C13FF913801DF8090387C039F90383E0F3FEB0FFCD903F090C7FC90 C7FC5DA2157EA215FEA25DA2001C495A127F48495A14074A5A485C023FC8FC00F8137E38 7C01F8381FFFE0000390C9FC2A407BAB2D>I<14FE137FA3EB01FC13001301A25CA21303 A25CA21307A25CA2130FA25CA2131FA25C157F90393F83FFC091388F81F091381E00F802 387F4948137C5C4A137EA2495A91C7FCA25B484814FE5E5BA2000314015E5BA200071403 5E5B1507000F5DA249130F5E001F1678031F1370491480A2003F023F13F0EE00E090C7FC 160148023E13C01603007E1680EE070000FEEC1E0FED1F1E48EC0FF80038EC03E02D467A C432>I<143C147E14FE1301A3EB00FC14701400AE137C48B4FC3803C780380703C0000F 13E0120E121C13071238A21278EA700F14C0131F00F0138012E0EA003F1400A25B137EA2 13FE5B12015BA212035B141E0007131C13E0A2000F133CEBC038A21478EB807014F014E0 EB81C0EA0783EBC7803803FE00EA00F8174378C11E>I<16F0ED03F8A21507A316F0ED01 C092C7FCAEEC01F0EC07FCEC1E1EEC380F0270138014E0130114C0EB03800107131F1400 A2130E153F131E011C140090C7FC5DA2157EA215FEA25DA21401A25DA21403A25DA21407 A25DA2140FA25DA2141FA25DA2143FA292C7FCA25C147EA214FE001C5B127F48485A495A A248485A495AD8F81FC8FCEA707EEA3FF8EA0FC0255683C11E>I<14FE137FA3EB01FC13 001301A25CA21303A25CA21307A25CA2130FA25CA2131FA25C167E013F49B4FC92380783 C09138000E07ED3C1F491370ED603F017E13E0EC01C09026FE03801380913907000E00D9 FC0E90C7FC5C00015B5C495AEBF9C03803FB8001FFC9FCA214F03807F3FCEBF07F9038E0 1FC06E7E000F130781EBC003A2001F150FA20180140EA2003F151E161C010013E0A2485D A2007E1578167000FE01015B15F1489038007F800038021FC7FC2A467AC42D>IIIIII<91381F800C91387FE01C903901F0703C903907C0387890390F801CF890 381F001D013E130F017E14F05B48481307A2484814E012075B000F140F16C0485AA2003F 141F491480A3007F143F90C71300A35D00FE147EA315FE5DA2007E1301A24A5A1407003E 130FA26C495A143B380F80F33807C3E73901FF87E038007E071300140F5DA3141F5DA314 3F92C7FCA25CA25C017F13FEA25D263F76AB2D>III<1470EB01F8A313035CA313075CA3130F5CA3131F5CA2 007FB512E0B6FC15C0D8003FC7FCA25B137EA313FE5BA312015BA312035BA312075BA312 0F5BA2EC0780001F140013805C140E003F131EEB001C143C14385C6C13F0495A6C485AEB 8780D807FEC7FCEA01F81B3F78BD20>I<137C48B414072603C780EB1F80380703C0000F 7F000E153F121C0107150012385E1278D8700F147E5C011F14FE00F05B00E05DEA003FEC 0001A2495C137E150313FE495CA215071201495CA2030F13380003167849ECC070A3031F 13F0EE80E0153F00011581037F13C06DEBEF8300000101148090397C03C787903A3E0F07 C70090391FFE01FE903903F000782D2D78AB34>I<017C143848B414FC3A03C78001FE38 0703C0000F13E0120E001C14000107147E1238163E1278D8700F141E5C131F00F049131C 12E0EA003F91C7123C16385B137E167801FE14705BA216F0000115E05B150116C0A24848 EB0380A2ED0700A2150E12015D6D5B000014786D5B90387C01E090383F0780D90FFFC7FC EB03F8272D78AB2D>I<017CEE038048B4020EEB0FC02603C780013FEB1FE0380703C000 0E7F5E001C037E130F01071607123804FE130300785DEA700F4A1501011F130100F00180 4914C012E0EA003FDA000314034C14805B137E0307140701FE1700495CA2030F5C000117 0E495CA260A24848495A60A2601201033F5C7F4B6C485A000002F713036D9039E7E00780 90267E01C349C7FC903A1F0781F81E903A0FFF007FF8D901FCEB0FE03B2D78AB41>I<02 F8133FD907FEEBFFE0903A0F0F83C0F0903A1C07C780F890393803CF03017013EE01E0EB FC07120101C013F8000316F00180EC01C000074AC7FC13001407485C120EC7FC140F5DA3 141F5DA3143F92C8FCA34AEB03C01780147EA202FEEB0700121E003F5D267F81FC130E6E 5BD8FF83143CD903BE5B26FE079E5B3A7C0F1F01E03A3C1E0F83C0271FF803FFC7FC3907 E000FC2D2D7CAB2D>I<137C48B414072603C780EB1F80380703C0000F7F000E153F001C 1600130712385E0078157EEA700F5C011F14FE00F0495B12E0EA003FEC00015E5B137E15 0301FE5C5BA2150700015D5BA2150F00035D5BA2151F5EA2153F12014BC7FC6D5B00005B EB7C0390383E0F7EEB1FFEEB03F090C712FE5DA214015D121F397F8003F0A24A5A484848 5A5D48131F00F049C8FC0070137E007813F8383801F0381E07C06CB4C9FCEA01FC294078 AB2F>I<027C130749B4130F49EB800E010F141E49EBC03CEDE03890393F03F07890397C 00FDF00178EB3FE00170EB03C001F0148049130790C7EA0F00151E5D5D5D4A5A4A5A4A5A 4AC7FC141E5C5C5C495A495A495A49C8FC011E14F04914E05B491301485A4848EB03C0D8 07B0130701FEEB0F80390FCF801F3A1F07E07F00393E03FFFED83C015B486C5B00705C00 F0EB7FC048011FC7FC282D7BAB28>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fs cmbx12 17.28 35 /Fs 35 122 df45 DI<16F04B7E 1507151F153FEC01FF1407147F010FB5FCB7FCA41487EBF007C7FCB3B3B3B3007FB91280 A6395E74DD51>49 D<913801FFF8021FEBFFC091B612F8010315FF010F16C0013F829026 7FFC0114F89027FFE0003F7F4890C7000F7F48486E7FD807F86E148048486E14C048486E 14E048486F13F001FC17F8486C816D17FC6E80B56C16FE8380A219FFA283A36C5BA26C5B 6C90C8FCD807FC5DEA01F0CA14FEA34D13FCA219F85F19F04D13E0A294B512C019804C14 004C5B604C5B4C5B604C13804C90C7FC4C5A4C5A4B13F05F4B13804B90C8FC4B5AED1FF8 4B5A4B5A4B48143F4A5B4A48C8FC4A5A4A48157E4A5A4A5AEC7F8092C9FC02FE16FE495A 495A4948ED01FCD90FC0150749B8FC5B5B90B9FC5A4818F85A5A5A5A5ABAFCA219F0A440 5E78DD51>I<92B5FC020F14F8023F14FF49B712C04916F0010FD9C01F13FC90271FFC00 077FD93FE001017F49486D8049C86C7F484883486C6F7F14C0486D826E806E82487FA480 5CA36C5E4A5E6C5B6C5B6C495E011FC85A90C95CA294B55A614C91C7FC604C5B4C5B4C5B 4C5B047F138092260FFFFEC8FC020FB512F817E094C9FC17F817FF91C7003F13E0040713 F8040113FE707F717F7113E085717FA2717F85A285831A80A31AC0EA03FCEA0FFF487F48 7F487FA2B57EA31A80A34D14005C7E4A5E5F6C495E49C8485BD81FF85F000F5ED807FE92 B55A6C6C6C4914806C01F0010791C7FC6C9026FF803F5B6D90B65A011F16F0010716C001 014BC8FCD9001F14F0020149C9FC426079DD51>II<01C0EE01C0D801F8160F01FF167F02F0EC07FFDA FF8090B5FC92B7128019006060606060606095C7FC17FC5F17E0178004FCC8FC16E09026 FC3FFCC9FC91CBFCADED3FFE0203B512F0020F14FE023F6E7E91B712E001FDD9E00F7F90 27FFFE00037F02F801007F02E06EB4FC02806E138091C8FC496F13C04917E07113F0EA00 F090C914F8A219FC83A219FEA419FFA3EA03F0EA0FFC487E487E487FA2B57EA319FEA35C 4D13FC6C90C8FC5B4917F8EA3FF001804B13F06D17E0001F5E6C6C17C06D4B1380D807FC 92B512006C6C4A5B6C6C6C01075B6C01E0011F5BD97FFE90B55A6DB712C0010F93C7FC6D 15FC010115F0D9003F1480020301F0C8FC406078DD51>I58 D65 D69 DI73 D77 D82 DI< 001FBEFCA64849C79126E0000F148002E0180091C8171F498601F81A0349864986A2491B 7FA2491B3F007F1DC090C9181FA4007E1C0FA600FE1DE0481C07A5CA95C7FCB3B3B3A302 1FBAFCA663617AE070>I<913803FFFE027FEBFFF00103B612FE010F6F7E4916E090273F FE001F7FD97FE001077FD9FFF801017F486D6D7F717E486D6E7F85717FA2717FA36C496E 7FA26C5B6D5AEB1FC090C9FCA74BB6FC157F0207B7FC147F49B61207010F14C0013FEBFE 004913F048B512C04891C7FC485B4813F85A5C485B5A5CA2B55AA45FA25F806C5E806C04 7D7F6EEB01F96C6DD903F1EBFF806C01FED90FE114FF6C9027FFC07FC01580000191B548 7E6C6C4B7E011F02FC130F010302F001011400D9001F90CBFC49437CC14E>97 D<92380FFFF04AB67E020F15F0023F15FC91B77E01039039FE001FFF4901F80101138049 01E0010713C04901804913E0017F90C7FC49484A13F0A2485B485B5A5C5A7113E0485B71 13C048701380943800FE0095C7FC485BA4B5FCAE7EA280A27EA2806C18FCA26C6D150119 F87E6C6D15036EED07F06C18E06C6D150F6D6DEC1FC06D01E0EC7F806D6DECFF00010701 FCEB03FE6D9039FFC03FFC010091B512F0023F5D020F1580020102FCC7FCDA000F13C03E 437BC148>99 DI<92380FFFC04AB512FC020FECFF80023F15E091B712F80103D9FE037F4990 39F0007FFF011F01C0011F7F49496D7F4990C76C7F49486E7F48498048844A804884485B 727E5A5C48717EA35A5C721380A2B5FCA391B9FCA41A0002C0CBFCA67EA380A27EA27E6E 160FF11F806C183F6C7FF17F006C7F6C6D16FE6C17016D6C4B5A6D6D4A5A6D01E04A5A6D 6DEC3FE0010301FC49B45A6D9026FFC01F90C7FC6D6C90B55A021F15F8020715E0020092 C8FC030713F041437CC14A>III<903807FF80B6FCA6C6FC7F7FB3 A8EF1FFF94B512F0040714FC041F14FF4C8193267FE07F7F922781FE001F7FDB83F86D7F DB87F07FDB8FC0814C7F039FC78015BE03BC8003FC825DA25DA25DA45DB3B2B7D8F007B7 1280A651647BE35A>II<903807FF80B6FCA6C6FC7F7FB3B3B3B3ADB712E0A623647BE32C>108 D<902607FF80D91FFFEEFFF8B691B500F00207EBFF80040702FC023F14E0041F02FF91B6 12F84C6F488193267FE07F6D4801037F922781FE001F9027E00FF0007FC6DA83F86D9026 F01FC06D7F6DD987F06D4A487F6DD98FC0DBF87EC7804C6D027C80039FC76E488203BEEE FDF003BC6E4A8003FC04FF834B5FA24B5FA24B94C8FCA44B5EB3B2B7D8F007B7D8803FB6 12FCA67E417BC087>I<902607FF80EB1FFFB691B512F0040714FC041F14FF4C8193267F E07F7F922781FE001F7FC6DA83F86D7F6DD987F07F6DD98FC0814C7F039FC78015BE03BC 8003FC825DA25DA25DA45DB3B2B7D8F007B71280A651417BC05A>I<923807FFE092B6FC 020715E0021F15F8027F15FE494848C66C6C7E010701F0010F13E04901C001037F49496D 7F4990C87F49486F7E49486F7E48496F13804819C04A814819E048496F13F0A24819F8A3 48496F13FCA34819FEA4B518FFAD6C19FEA46C6D4B13FCA36C19F8A26C6D4B13F0A26C19 E06C6D4B13C0A26C6D4B13806C6D4B13006D6C4B5A6D6D495B6D6D495B010701F0010F13 E06D01FE017F5B010090B7C7FC023F15FC020715E0020092C8FC030713E048437CC151> I<902607FF80EBFFF8B6010FEBFF80047F14F00381B612FC038715FF038F010114C09227 BFF0003F7FC6DAFFC0010F7F6D91C76C7F6D496E7F03F86E7F4B6E7F4B17804B6F13C0A2 7313E0A27313F0A21BF885A21BFCA3851BFEAE4F13FCA41BF861A21BF0611BE0611BC06F 92B512801B006F5C6F4A5B6F4A5B03FF4A5B70495B04E0017F13C09226CFFC03B55A03C7 B648C7FC03C115F803C015E0041F91C8FC040313E093CBFCB3A3B712F0A64F5D7BC05A> I114 D<913A3FFF8007800107B5EAF81F011FEC FE7F017F91B5FC48B8FC48EBE0014890C7121FD80FFC1407D81FF0801600485A007F167F 49153FA212FF171FA27F7F7F6D92C7FC13FF14E014FF6C14F8EDFFC06C15FC16FF6C16C0 6C16F06C826C826C826C82013F1680010F16C01303D9007F15E0020315F0EC001F150004 1F13F81607007C150100FC81177F6C163FA2171F7EA26D16F0A27F173F6D16E06D157F6D 16C001FEEDFF806D0203130002C0EB0FFE02FCEB7FFC01DFB65A010F5DD8FE0315C026F8 007F49C7FC48010F13E035437BC140>II<902607FFC0ED3FFEB60207B5FCA6C6EE00076D826D82B3B3 A260A360A2607F60183E6D6D147E4E7F6D6D4948806D6DD907F0ECFF806D01FFEB3FE06D 91B55A6E1500021F5C020314F8DA003F018002F0C7FC51427BC05A>I119 D121 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ft cmbx12 12 54 /Ft 54 123 df12 D40 D<12F07E127E7E6C7E6C7E6C7E7F6C7E6C7E12 007F137F80133F806D7EA26D7EA26D7EA2801303A2801301A280A27F1580A4EC7FC0A615 E0A2143FAE147FA215C0A6ECFF80A415005BA25CA213035CA213075CA2495AA2495AA249 5A5C137F91C7FC13FE5B1201485A485A5B485A485A48C8FC127E12F85A1B647ACA2C>I< EA07C0EA1FF0EA3FF8EA7FFCEAFFFEA7EA7FFCEA3FF8EA1FF0EA07C00F0F788E1F>46 D48 DIII<163FA25E5E 5D5DA25D5D5D5DA25D92B5FCEC01F7EC03E7140715C7EC0F87EC1F07143E147E147C14F8 EB01F0EB03E0130714C0EB0F80EB1F00133E5BA25B485A485A485A120F5B48C7FC123E5A 12FCB91280A5C8000F90C7FCAC027FB61280A531417DC038>I<0007150301E0143F01FF EB07FF91B6FC5E5E5E5E5E16804BC7FC5D15E092C8FC01C0C9FCAAEC3FF001C1B5FC01C7 14C001DF14F09039FFE03FFC9138000FFE01FC6D7E01F06D13804915C0497F6C4815E0C8 FC6F13F0A317F8A4EA0F80EA3FE0487E12FF7FA317F05B5D6C4815E05B007EC74813C012 3E003F4A1380D81FC0491300D80FF0495AD807FEEBFFFC6CB612F0C65D013F1480010F01 FCC7FC010113C02D427BC038>I<4AB47E021F13F0027F13FC49B6FC01079038807F8090 390FFC001FD93FF014C04948137F4948EBFFE048495A5A1400485A120FA248486D13C0EE 7F80EE1E00003F92C7FCA25B127FA2EC07FC91381FFF8000FF017F13E091B512F89039F9 F01FFC9039FBC007FE9039FF8003FF17804A6C13C05B6F13E0A24915F0A317F85BA4127F A5123FA217F07F121FA2000F4A13E0A26C6C15C06D4913806C018014006C6D485A6C9038 E01FFC6DB55A011F5C010714C0010191C7FC9038003FF02D427BC038>I<121E121F13FC 90B712FEA45A17FC17F817F017E017C0A2481680007EC8EA3F00007C157E5E00785D1501 4B5A00F84A5A484A5A5E151FC848C7FC157E5DA24A5A14035D14074A5AA2141F5D143FA2 147F5D14FFA25BA35B92C8FCA35BA55BAA6D5A6D5A6D5A2F447AC238>IIII< EE1F80A24C7EA24C7EA34C7EA24B7FA34B7FA24B7FA34B7F169F031F80161F82033F80ED 3E07037E80157C8203FC804B7E02018115F0820203814B137F0207815D173F020F814B7F 021F8292C77EA24A82023E80027E82027FB7FCA291B87EA2498302F0C8FCA20103834A15 7F0107834A153FA249488284011F8491C97E4984133E017E82B6020FB612F0A54C457CC4 55>65 DII69 DIIII76 D I<923807FFC092B512FE0207ECFFC0021F15F091267FFE0013FC902601FFF0EB1FFF0107 01C0010713C04990C700017F49486E7F49486F7E49486F7E49486F7E48496F7E48496F13 80A248496F13C0A24819E091C97E4819F0A248487013F8A3007F19FCA249177FA300FF19 FEAD007F19FCA36D17FF003F19F8A3001F19F06D5EA26C19E06E01FE5B6C912603FF8014 C06C6D486D4813804B13E06C9028E01F83F00F13006C903BF01E00F81FFE90267FF83E90 387C3FFC90263FFC3C6D485AD91FFE91381EFFF0D90FFF021F5B6D01FE5D010194C7FC6D 6D6CB45A023F90B512F8020703E0130202006F1307030713C792C7EA07F8716C130F7213 1F9538FF80FF96B5FC7114FEA3831AFCA27213F81AF0847213E07213C0721300F001FC48 587AC454>81 DII<003FBA12E0A59026 FE000FEB8003D87FE09338003FF049171F90C71607A2007E1803007C1801A300781800A4 00F819F8481978A5C81700B3B3A20107B8FCA545437CC24E>I<903801FFE0011F13FE01 7F6D7E48B612E03A03FE007FF84848EB1FFC6D6D7E486C6D7EA26F7FA36F7F6C5A6C5AEA 00F090C7FCA40203B5FC91B6FC1307013F13F19038FFFC01000313E0000F1380381FFE00 485A5B127F5B12FF5BA35DA26D5B6C6C5B4B13F0D83FFE013EEBFFC03A1FFF80FC7F0007 EBFFF86CECE01FC66CEB8007D90FFCC9FC322F7DAD36>97 DIIIIIII<137C48B4FC4813804813C0A24813E0 A56C13C0A26C13806C1300EA007C90C7FCAAEB7FC0EA7FFFA512037EB3AFB6FCA518467C C520>II< EB7FC0B5FCA512037EB293387FFFE0A593380FE0004C5A4CC7FC167E5EED03F8ED07E04B 5A4B5A037FC8FC15FEECC1FCECC3FE14C7ECDFFF91B57E82A202F97F02E17F02C07FEC80 7F6F7E826F7E816F7F836F7F816F7F83707E163FB60003B512F8A535457DC43B>II<90277F8007FEEC0FFCB590263FFF C090387FFF8092B5D8F001B512E002816E4880913D87F01FFC0FE03FF8913D8FC00FFE1F 801FFC0003D99F009026FF3E007F6C019E6D013C130F02BC5D02F86D496D7EA24A5D4A5D A34A5DB3A7B60081B60003B512FEA5572D7CAC5E>I<90397F8007FEB590383FFF8092B5 12E0028114F8913987F03FFC91388F801F000390399F000FFE6C139E14BC02F86D7E5CA2 5CA35CB3A7B60083B512FEA5372D7CAC3E>II<90397FC00FF8B590B57E02C314E002CF14F89139DFC03FFC9139FF00 1FFE000301FCEB07FF6C496D13804A15C04A6D13E05C7013F0A2EF7FF8A4EF3FFCACEF7F F8A318F017FFA24C13E06E15C06E5B6E4913806E4913006E495A9139DFC07FFC02CFB512 F002C314C002C091C7FCED1FF092C9FCADB67EA536407DAC3E>I<90387F807FB53881FF E0028313F0028F13F8ED8FFC91389F1FFE000313BE6C13BC14F8A214F0ED0FFC9138E007 F8ED01E092C7FCA35CB3A5B612E0A5272D7DAC2E>114 D<90391FFC038090B512870003 14FF120F381FF003383FC00049133F48C7121F127E00FE140FA215077EA27F01E090C7FC 13FE387FFFF014FF6C14C015F06C14FC6C800003806C15806C7E010F14C0EB003F020313 E0140000F0143FA26C141F150FA27EA26C15C06C141FA26DEB3F8001E0EB7F009038F803 FE90B55A00FC5CD8F03F13E026E007FEC7FC232F7CAD2C>IIIIIII<00 1FB71280A49026FC001F130001E0495A5B49495A90C7485A48495B123E4A5B4A5B003C49 5BA24A90C7FC4A5A4A5AC7FC4A5A495B495BA2495B499038800780491300A2495A494813 0F49481400A2485B48495B485BA248495B4890C75A48485C15034848EB1FFEB7FCA4292C 7DAB32>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fu cmbx12 14.4 7 /Fu 7 117 df58 D<932601FFFCEC01C0047FD9FFC013030307B600F81307033F03FE131F92B8EA803F0203 DAE003EBC07F020F01FCC7383FF0FF023F01E0EC0FF94A01800203B5FC494848C9FC4901 F8824949824949824949824949824990CA7E494883A2484983485B1B7F485B481A3FA248 49181FA3485B1B0FA25AA298C7FC5CA2B5FCAE7EA280A2F307C07EA36C7FA21B0F6C6D19 80A26C1A1F6C7F1C006C6D606C6D187EA26D6C606D6D4C5A6D6D16036D6D4C5A6D6D4C5A 6D01FC4C5A6D6DEE7F806D6C6C6C4BC7FC6E01E0EC07FE020F01FEEC1FF80203903AFFE0 01FFF0020091B612C0033F93C8FC030715FCDB007F14E0040101FCC9FC525479D261>67 D<913803FFC0023F13FC49B6FC010715C04901817F903A3FFC007FF849486D7E49486D7E 4849130F48496D7E48178048497F18C0488191C7FC4817E0A248815B18F0A212FFA490B8 FCA318E049CAFCA6127FA27F7EA218E06CEE01F06E14037E6C6DEC07E0A26C6DEC0FC06C 6D141F6C6DEC3F806D6CECFF00D91FFEEB03FE903A0FFFC03FF8010390B55A010015C002 1F49C7FC020113F034387CB63D>101 D110 D<913801FFE0021F13FE91B612 C0010315F0010F9038807FFC903A1FFC000FFED97FF86D6C7E49486D7F48496D7F48496D 7F4A147F48834890C86C7EA24883A248486F7EA3007F1880A400FF18C0AC007F1880A300 3F18006D5DA26C5FA26C5F6E147F6C5F6C6D4A5A6C6D495B6C6D495B6D6C495BD93FFE01 1F90C7FC903A0FFF807FFC6D90B55A010015C0023F91C8FC020113E03A387CB643>I<90 3903FFF00F013FEBFE1F90B7FC120348EB003FD80FF81307D81FE0130148487F4980127F 90C87EA24881A27FA27F01F091C7FC13FCEBFFC06C13FF15F86C14FF16C06C15F06C816C 816C81C681013F1580010F15C01300020714E0EC003F030713F015010078EC007F00F815 3F161F7E160FA27E17E07E6D141F17C07F6DEC3F8001F8EC7F0001FEEB01FE9039FFC00F FC6DB55AD8FC1F14E0D8F807148048C601F8C7FC2C387CB635>115 D<143EA6147EA414FEA21301A313031307A2130F131F133F13FF5A000F90B6FCB8FCA426 003FFEC8FCB3A9EE07C0AB011FEC0F8080A26DEC1F0015806DEBC03E6DEBF0FC6DEBFFF8 6D6C5B021F5B020313802A4D7ECB34>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fv cmr12 12 92 /Fv 92 128 df0 D<1618163CA2167EA216FFA24B7FA24B6C7EA29238063FE0A24B6C7EA24B6C7EA2923838 07FC153092387003FE15609238E001FF15C002016D7F5D02036E7E92C7FC4A6E7E140602 0E6E7E140C021C6E7E141802386E7E143002706E7E146002E06E7E5C01016F7F5C010370 7E91C9FC183F010683181F4983180F49831807498318034983A249707EA24848701380A2 48CBEA7FC0A20006F03FE0A248F01FF0A2001FBA12F8A24819FCA24819FEA2BCFC48477C C651>II<003FB812FCA60038CA121C0030170C0070170EA2 00601706A5CCFCAA01601506A5017FB612FEA60160C81206A590CBFCAB00C01703A66C17 07A2006017060070170E007FB812FEA638447CC341>4 D7 D<0103B612FCA390C701F0C8FC6F5A6F5AA8913801FFF0023F EBFF80903A01FF3FDFF0D907F0EBC1FCD91FC0EBC07FD93F00EC1F8001FEED0FE048486F 7E48486F7E48486F7E48486F7E001F834982003F1880007F18C0A249163F00FF18E0A800 7F18C06D167FA2003F1880001F18006D5E000F5F6C6C4B5A6C6C4B5A6C6C4B5A6C6C4B5A 013FED1F80D91FC0027FC7FCD907F0EBC1FCD901FFEBDFF0D9003FB51280020101F0C8FC 9138003FC0A84B7E4B7E0103B612FCA33B447BC346>I<027FB67EA39126001FFEC9FC6F 5A6F5AA8B46CEFFF8001E01603D81FF0933807FC006C6C4C5A0007606D161F000360A26D 163F000160AC6C6C5F187FA4D97F804BC7FCA2013F5E02C01401131F02E04A5A010F5ED9 07F01407D903F85DD901FC4A5AD900FE4A5A027F027FC8FCDA1FC713FE0207B512F80201 14C09126001FFCC9FCED07F8A84B7E4B7E027FB67EA341447BC34C>II<9239FFC001FC020F9038F80F FF913B3F803E3F03C0913BFC00077E07E0D903F890390FFC0FF0494890383FF81F4948EB 7FF0495A494814E049C7FCF00FE04991393FC0038049021F90C7FCAFB912F0A3C648C7D8 1FC0C7FCB3B2486CEC3FF0007FD9FC0FB512E0A33C467EC539>I<4AB4FC020F13E09138 7F80F8903901FC001C49487FD907E0130F4948137F011FECFF80495A49C7FCA25B49EC7F 00163E93C7FCACEE3F80B8FCA3C648C7FC167F163FB3B0486CEC7FC0007FD9FC1FB5FCA3 30467EC536>I14 D<131F1480133F137FA2EBFF0048 5A485A5B485A485A138048C7FC123E123C5A12E0124011126CC431>19 D<121EEA7F80EAFFC0A9EA7F80ACEA3F00AB121EAC120CA5C7FCAA121EEA7F80A2EAFFC0 A4EA7F80A2EA1E000A4778C61B>33 D<001EEB03C0397F800FF000FF131F01C013F8A201 E013FCA3007F130F391E6003CC0000EB000CA401E0131C491318A3000114384913300003 147090C712604814E0000614C0000E130148EB038048EB070048130E0060130C1E1D7DC4 31>I38 D<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413E013C0A312011380 120313005A1206120E5A5A5A12600B1D78C41B>I<140C141C1438147014E0EB01C01303 EB0780EB0F00A2131E5BA25B13F85B12015B1203A2485AA3485AA348C7FCA35AA2123EA2 127EA4127CA312FCB3A2127CA3127EA4123EA2123FA27EA36C7EA36C7EA36C7EA212017F 12007F13787FA27F7FA2EB0780EB03C01301EB00E014701438141C140C166476CA26>I< 12C07E12707E7E7E120F6C7E6C7EA26C7E6C7EA21378137C133C133E131E131FA2EB0F80 A3EB07C0A3EB03E0A314F0A21301A214F8A41300A314FCB3A214F8A31301A414F0A21303 A214E0A3EB07C0A3EB0F80A3EB1F00A2131E133E133C137C13785BA2485A485AA2485A48 C7FC120E5A5A5A5A5A16647BCA26>I<16C04B7EB3AB007FBAFCBB1280A26C1900C8D801 E0C9FCB3AB6F5A41407BB84C>43 D<121EEA7F8012FF13C0A213E0A3127FEA1E601200A4 13E013C0A312011380120313005A1206120E5A5A5A12600B1D78891B>II<121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A0A78891B>I<1618163C167CA216 7816F8A216F01501A216E01503A216C01507A21680150FA2ED1F00A2151E153EA2153C15 7CA2157815F8A25D1401A24A5AA25D1407A25D140FA292C7FC5CA2141E143EA2143C147C A25CA25C1301A25C1303A25C1307A25C130FA291C8FC5BA2133EA2133C137CA2137813F8 A25B1201A25B1203A2485AA25B120FA290C9FC5AA2121E123EA2123C127CA2127812F8A2 5A126026647BCA31>I<14FF010713E090381F81F890383E007C01FC133F4848EB1F8049 130F4848EB07C04848EB03E0A2000F15F0491301001F15F8A2003F15FCA390C8FC4815FE A54815FFB3A46C15FEA56D1301003F15FCA3001F15F8A26C6CEB03F0A36C6CEB07E00003 15C06D130F6C6CEB1F806C6CEB3F00013E137C90381F81F8903807FFE0010090C7FC2844 7CC131>I<143014F013011303131F13FFB5FC13E713071200B3B3B0497E497E007FB6FC A3204278C131>II<49B4FC010F13E0 013F13FC9038FE01FE3A01F0007F80D803C0EB3FC048C7EA1FE0120EED0FF0EA0FE0486C 14F8A215077F5BA26C48130FEA03C0C813F0A3ED1FE0A2ED3FC01680ED7F0015FE4A5AEC 03F0EC1FC0D90FFFC7FC15F090380001FCEC007FED3F80ED1FC0ED0FE016F0ED07F816FC 150316FEA2150116FFA3121EEA7F80487EA416FE491303A2007EC713FC00701407003015 F80038140F6C15F06CEC1FE06C6CEB3FC0D803E0EB7F803A01FE01FE0039007FFFF8010F 13E0010190C7FC28447CC131>II<000615C0D807C0130701FCEB7F8090B612 005D5D5D15E0158026063FFCC7FC90C9FCAE14FF010713C090381F01F090383800FC01F0 137ED807C07F49EB1F8016C090C7120F000615E0C8EA07F0A316F81503A216FCA5123E12 7F487EA416F890C712075A006015F0A20070140F003015E00038EC1FC07E001EEC3F806C EC7F006C6C13FE6C6C485A3901F807F039007FFFE0011F90C7FCEB07F826447BC131>I< EC07FCEC3FFF91B512C0903903FC03E0903907E000F0D91FC0133849C71258017EEB01FC 01FE1303491307485A485AA24848EB03F8000FEC01F092C7FC485AA3485AA3127FA29038 007F80903801FFF090380780FC39FF0E003E49EB1F8049EB0FC049EB07E0136001E0EB03 F04914F8150116FC5BED00FEA390C812FFA47EA57F123FA216FE121F15016D14FC120FED 03F86C7EED07F06C6C14E06C6CEB0FC06C6CEB1F80017EEB3F0090383F80FE90380FFFF8 010313E00100138028447CC131>I<121CA2EA1F8090B712C0A3481680A217005E0038C8 120C0030151C00705D0060153016705E5E4814014B5A4BC7FCC81206150E5D151815385D 156015E04A5AA24A5A140792C8FC5CA25C141E143EA2147E147CA214FCA21301A3495AA4 1307A6130FAA6D5AEB01C02A457BC231>I<14FF010713E0011F13F890387F00FE01FC13 3FD801F0EB1F804848EB0FC049EB07E00007EC03F048481301A290C713F8481400A47FA2 6D130116F07F6C6CEB03E013FC6C6CEB07C09039FF800F806C9038C01F006CEBF03EECF8 7839007FFEF090383FFFC07F01077F6D13F8497F90381E7FFFD97C1F1380496C13C02601 E00313E048486C13F000079038007FF84848EB3FFC48C7120F003EEC07FE150148140016 FF167F48153FA2161FA56C151E007C153EA2007E153C003E157C6C15F86DEB01F06C6CEB 03E06C6CEB07C0D803F8EB1F80C6B4EBFF0090383FFFFC010F13F00101138028447CC131 >I<14FF010713E0011F13F890387F80FC9038FC007E48487F4848EB1F804848EB0FC000 0FEC07E0485AED03F0485A16F8007F140190C713FCA25AA216FE1500A516FFA46C5CA36C 7E5D121F7F000F5C6C6C130E150C6C6C131C6C6C5BD8007C5B90383F01E090390FFF80FE 903801FE0090C8FC150116FCA4ED03F8A216F0D80F801307486C14E0486C130F16C0ED1F 80A249EB3F0049137E001EC75A001C495A000F495A3907E01FE06CB51280C649C7FCEB1F F028447CC131>I<121EEA7F80A2EAFFC0A4EA7F80A2EA1E00C7FCB3A5121EEA7F80A2EA FFC0A4EA7F80A2EA1E000A2B78AA1B>I<121EEA7F80A2EAFFC0A4EA7F80A2EA1E00C7FC B3A5121E127FEAFF80A213C0A4127F121E1200A512011380A3120313005A1206120E120C 121C5A5A12600A3E78AA1B>I<007FBAFCBB1280A26C1900CEFCB0007FBAFCBB1280A26C 190041187BA44C>61 D<16C04B7EA34B7EA34B7EA34B7EA3ED19FEA3ED30FFA203707FED 607FA203E07FEDC03FA2020180ED801FA2DA03007F160FA20206801607A24A6D7EA34A6D 7EA34A6D7EA20270810260147FA202E08191B7FCA249820280C7121FA249C87F170FA201 06821707A2496F7EA3496F7EA3496F7EA201788313F8486C83D80FFF03037FB500E0027F EBFFC0A342477DC649>65 DIIIII< DB0FFE146092B500C013E0020314F0913A0FFC01FC0191393FC0003E02FFC7EA0F83D903 FCEC03C74948EC01E74948EC00FF4948157F4948153F4948151F49C9120F485A49160712 0348481603A248481601A248481600A2123FA2491760127FA396C7FC485AAD4CB612C06C 7EA293C7387FF000725A003F171F7FA2121F7F120FA26C7EA26C7E6C7EA26C7E6D7E6D6C 153F6D7E6D6C157F6D6C15E7D903FEEC01C7D900FFEC0383DA3FE0EB0F01DA0FFCEBFE00 0203B500F81360020002E090C7FCDB0FFEC9FC42487BC54D>III<010FB512FEA3D9000313806E130080B3B3AB123F487E487EA44A 5A13801300006C495A00705C6C13076C5C6C495A6CEB1F802603E07FC7FC3800FFFCEB1F E027467BC332>I76 DIIIIII<49B41303010FEBE007013F13 F89039FE00FE0FD801F8131FD807E0EB079F49EB03DF48486DB4FC48C8FC4881003E8112 7E82127C00FC81A282A37E82A27EA26C6C91C7FC7F7FEA3FF813FE381FFFE06C13FE6CEB FFE06C14FC6C14FF6C15C0013F14F0010F80010180D9001F7F14019138001FFF03031380 816F13C0167F163F161F17E000C0150FA31607A37EA36C16C0160F7E17806C151F6C1600 6C5D6D147ED8FBC05CD8F9F0495AD8F07C495A90393FC00FE0D8E00FB51280010149C7FC 39C0003FF02B487BC536>I<003FB912F8A3903BF0001FF8001F01806D481303003EC715 0048187C0078183CA20070181CA30060180CA5481806A5C81600B3B3A54B7EED7FFE49B7 7EA33F447DC346>IIII89 D91 D<01C01318000114384848137048C712E0000EEB01C0000C1480001C1303001814000038 5B003013060070130E0060130CA300E0131C481318A400CFEB19E039FFC01FF801E013FC A3007F130FA2003F130701C013F8390F0001E01E1D71C431>II<13C01201EA0380EA0700120E120C121C121812381230 12701260A312E05AA412CFEAFFC013E0A3127FA2123F13C0EA0F000B1D79C41B>96 DIII<167FED3FFFA315018182B3EC7F80903803FFF090380FC07C90383F00 0E017E1307496D5AD803F87F48487F5B000F81485AA2485AA2127FA290C8FC5AAB7E7FA2 123FA26C7EA2000F5D7F6C6C5B00035C6C6C9038077F806C6C010E13C0013F011C13FE90 380FC0F8903803FFE09026007F0013002F467DC436>IIIIII<143C14FFA2491380A46D1300A2143C91C7FCADEC7F80EB3FFFA313 00147F143FB3B3AA123E127F39FF807F00A2147EA25C6C485A383C01F06C485A3807FF80 D801FEC7FC195785C21E>IIII<3901FC01FE00FF903807FFC091381E07F091383801F8000701707F0003EB E0002601FDC07F5C01FF147F91C7FCA25BA35BB3A8486CECFF80B5D8F83F13FEA32F2C7D AB36>II<3901FC03FC00FF90380FFF8091 383C07E091387001F83A07FDE000FE00030180137FD801FFEC3F8091C7EA1FC04915E049 140F17F0160717F8160317FCA3EE01FEABEE03FCA3EE07F8A217F0160F6D15E0EE1FC06D 143F17806EEB7E00D9FDC05B9039FCF003F891383C0FE091381FFF80DA03FCC7FC91C9FC AE487EB512F8A32F3F7DAB36>I<91387F8003903903FFE00790380FE07890393F801C0F 90387E000E496D5AD803F8EB039F0007EC01BF4914FF48487F121F5B003F81A2485AA348 C8FCAB6C7EA3123F7F121F6D5C120F6D5B12076C6C5B6C6C497E6C6C130E013F131C9038 0FC0F8903803FFE09038007F0091C7FCAEEEFF80033F13FEA32F3F7DAB33>I<3903F803 F000FFEB1FFCEC3C3EEC707F0007EBE0FF3803F9C000015B13FBEC007E153C01FF13005B A45BB3A748B4FCB512FEA3202C7DAB26>I<90383FE0183901FFFC383907E01F78390F00 03F8001E1301481300007C1478127800F81438A21518A27EA27E6C6C13006C7E13FC383F FFE06C13FC6C13FF6C14C06C14E0C614F0011F13F81300EC0FFC140300C0EB01FE140015 7E7E153EA27EA36C143C6C147C15786C14F86CEB01F039F38003E039F1F00F8039E07FFE 0038C00FF01F2E7DAC26>I<1306A5130EA4131EA3133E137EA213FE12011207001FB512 F0B6FCA2C648C7FCB3A4150CAA017E131C017F1318A26D133890381F8030ECC070903807 E0E0903801FFC09038007F001E3E7EBC26>IIII II<003FB612E0A29038C0003F90C713C0003CEC7F800038ECFF00A200 30495A0070495AA24A5A0060495AA24A5A4A5AA2C7485A4AC7FC5B5C495A13075C495A13 1F4A1360495A495AA249C712C0485AA2485A485A1501485A48481303A24848EB07804848 131F00FF14FF90B6FCA2232B7DAA2B>II<01F81302D803FE130739 07FF800E48EBE01C391F1FF8F8393807FFF0D8700113E039E0007FC00040EB1F00200978 C131>126 D<001EEB0780007FEB0FE039FF801FF0EBC03FA4EB801F397F000FE0001EEB 07801C0A76C231>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fw cmti10 10.95 14 /Fw 14 122 df<49B5933807FFFC496062D90003F0FC00505ADBBF805E1A771AEF140703 3F923801CFE0A2F1039F020FEE071F020E606F6C140E1A3F021E161C021C04385BA2F170 7F143C023804E090C7FCF001C0629126780FE0495A02705FF00700F00E0114F002E0031C 5BA2F03803010116704A6C6C5D18E019070103ED01C00280DA03805BA2943807000F1307 0200020E5C5FDB03F8141F495D010E4B5CA24D133F131E011CDAF9C05CEEFB80197F013C 6DB4C7FC013895C8FC5E01784A5C13F8486C4A5CD807FE4C7EB500F04948B512FE16E015 00563E7BBD52>77 D<92391FE00380ED7FFC913A01FFFE0700913907F01F8F91390FC007 DF4AC66CB4FC023E6D5A4A130014FC495A4948147CA2495AA2010F15785CA3011F1570A4 6E91C7FCA2808014FE90380FFFE015FC6DEBFF8016E06D806D806D6C7F141F02037FEC00 3FED07FF1501A281A282A212075A167E120EA2001E15FE5EA25E003E14015E003F14034B 5A486C5C150F6D495A6D49C8FCD8F9F0137C39F8FE01F839F03FFFF0D8E00F13C026C001 FEC9FC314279BF33>83 D<147E49B47E903907C1C38090391F80EFC090383F00FF017E13 7F4914804848133F485AA248481400120F5B001F5C157E485AA215FE007F5C90C7FCA214 01485C5AA21403EDF0385AA21407EDE078020F1370127C021F13F0007E013F13E0003E13 7FECF3E1261F01E313C03A0F8781E3803A03FF00FF00D800FC133E252977A72E>97 DII101 D<167C4BB4FC923807C78092380F83C0ED1F 87161FED3F3FA2157EA21780EE0E004BC7FCA414015DA414035DA30103B512F8A3902600 07E0C7FCA3140F5DA5141F5DA4143F92C8FCA45C147EA414FE5CA413015CA4495AA4495A A4495A121E127F5C12FF49C9FCA2EAFE1EEAF83C1270EA7878EA3FE0EA0F802A5383BF1C >I<1478EB01FCA21303A314F8EB00E01400AD137C48B4FC38038F80EA0707000E13C012 1E121CEA3C0F1238A2EA781F00701380A2EAF03F140012005B137E13FE5BA212015BA212 035B1438120713E0000F1378EBC070A214F0EB80E0A2EB81C01383148038078700EA03FE EA00F8163E79BC1C>105 D108 D110 DI114 D116 D<137C48B4143826038F8013FCEA 0707000E7F001E1401001C15F8EA3C0F12381503D8781F14F000701380A2D8F03F130702 0013E012005B017E130F16C013FE5B151F1201491480A2153F000315005BA25D157EA315 FE5D00011301EBF8030000130790387C1FF8EB3FF9EB07E1EB00035DA21407000E5CEA3F 80007F495AA24A5AD8FF0090C7FC143E007C137E00705B387801F0383803E0381E0FC06C B4C8FCEA03F8263B79A72C>121 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fx cmr10 10.95 31 /Fx 31 122 df<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413E013C0A3120113 80120313005A120E5A1218123812300B1C798919>44 DI<121E EA7F80A2EAFFC0A4EA7F80A2EA1E000A0A798919>I<121EEA7F80A2EAFFC0A4EA7F80A2 EA1E00C7FCB3121E127FEAFF80A213C0A4127F121E1200A412011380A3120313005A1206 120E120C121C5A1230A20A3979A619>59 D70 D73 D77 DI<003FB91280A3903AF0007FE001018090393FC0003F48C7ED1FC000 7E1707127C00781703A300701701A548EF00E0A5C81600B3B14B7E4B7E0107B612FEA33B 3D7DBC42>84 D97 DI<49B4FC010F13E090383F00F8017C131E4848131F 4848137F0007ECFF80485A5B121FA24848EB7F00151C007F91C7FCA290C9FC5AAB6C7EA3 003FEC01C07F001F140316806C6C13076C6C14000003140E6C6C131E6C6C137890383F01 F090380FFFC0D901FEC7FC222A7DA828>II II<167C903903F801 FF903A1FFF078F8090397E0FDE1F9038F803F83803F001A23B07E000FC0600000F6EC7FC 49137E001F147FA8000F147E6D13FE00075C6C6C485AA23901F803E03903FE0FC026071F FFC8FCEB03F80006CAFC120EA3120FA27F7F6CB512E015FE6C6E7E6C15E06C810003813A 0FC0001FFC48C7EA01FE003E140048157E825A82A46C5D007C153E007E157E6C5D6C6C49 5A6C6C495AD803F0EB0FC0D800FE017FC7FC90383FFFFC010313C0293D7EA82D>III107 DI<27 01F801FE14FF00FF902707FFC00313E0913B1E07E00F03F0913B7803F03C01F80007903B E001F87000FC2603F9C06D487F000101805C01FBD900FF147F91C75B13FF4992C7FCA249 5CB3A6486C496CECFF80B5D8F87FD9FC3F13FEA347287DA74C>I<3901F801FE00FF9038 07FFC091381E07E091387803F000079038E001F82603F9C07F0001138001FB6D7E91C7FC 13FF5BA25BB3A6486C497EB5D8F87F13FCA32E287DA733>I<14FF010713E090381F81F8 90387E007E01F8131F4848EB0F804848EB07C04848EB03E0000F15F04848EB01F8A2003F 15FCA248C812FEA44815FFA96C15FEA36C6CEB01FCA3001F15F86C6CEB03F0A26C6CEB07 E06C6CEB0FC06C6CEB1F80D8007EEB7E0090383F81FC90380FFFF0010090C7FC282A7EA8 2D>I<3901FC03FC00FF90381FFF8091387C0FE09039FDE003F03A07FFC001FC6C496C7E 6C90C7127F49EC3F805BEE1FC017E0A2EE0FF0A3EE07F8AAEE0FF0A4EE1FE0A2EE3FC06D 1580EE7F007F6E13FE9138C001F89039FDE007F09039FC780FC0DA3FFFC7FCEC07F891C9 FCAD487EB512F8A32D3A7EA733>I<3901F807E000FFEB1FF8EC787CECE1FE3807F9C100 031381EA01FB1401EC00FC01FF1330491300A35BB3A5487EB512FEA31F287EA724>114 D<90383FC0603901FFF8E03807C03F381F000F003E1307003C1303127C0078130112F814 00A27E7E7E6D1300EA7FF8EBFFC06C13F86C13FE6C7F6C1480000114C0D8003F13E00103 13F0EB001FEC0FF800E01303A214017E1400A27E15F07E14016C14E06CEB03C090388007 8039F3E01F0038E0FFFC38C01FE01D2A7DA824>I<131CA6133CA4137CA213FCA2120112 031207001FB512C0B6FCA2D801FCC7FCB3A215E0A912009038FE01C0A2EB7F03013F1380 90381F8700EB07FEEB01F81B397EB723>IIII121 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fy cmbx10 10.95 7 /Fy 7 117 df<16FCA24B7EA24B7EA34B7FA24B7FA34B7FA24B7FA34B7F157C03FC7FED F87FA2020180EDF03F0203804B7E02078115C082020F814B7E021F811500824A81023E7F 027E81027C7FA202FC814A147F49B77EA34982A2D907E0C7001F7F4A80010F835C83011F 8391C87E4983133E83017E83017C81B500FC91B612FCA5463F7CBE4F>65 D<903807FFC0013F13F848B6FC48812607FE037F260FF8007F6DEB3FF0486C806F7EA36F 7EA26C5A6C5AEA01E0C8FC153F91B5FC130F137F3901FFFE0F4813E0000F1380381FFE00 485A5B485A12FF5BA4151F7F007F143F6D90387BFF806C6C01FB13FE391FFF07F36CEBFF E100031480C6EC003FD91FF890C7FC2F2B7DA933>97 D<13FFB5FCA512077EAFEDFFE002 0713FC021FEBFF80027F80DAFF8113F09139FC003FF802F06D7E4A6D7E4A13074A807013 80A218C082A318E0AA18C0A25E1880A218005E6E5C6E495A6E495A02FCEB7FF0903AFCFF 01FFE0496CB55AD9F01F91C7FCD9E00713FCC7000113C033407DBE3A>II<3901FE01FE00FF903807FF804A13E04A13F0EC3F1F91387C3F F8000713F8000313F0EBFFE0A29138C01FF0ED0FE091388007C092C7FCA391C8FCB3A2B6 FCA525297DA82B>114 D<90383FFC1E48B512BE000714FE5A381FF00F383F800148C7FC 007E147EA200FE143EA27E7F6D90C7FC13F8EBFFE06C13FF15C06C14F06C806C806C806C 80C61580131F1300020713C014000078147F00F8143F151F7EA27E16806C143F6D140001 E013FF9038F803FE90B55A15F0D8F87F13C026E00FFEC7FC222B7DA929>II E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fz cmr12 14.4 35 /Fz 35 123 df<023FB612FEA4DA000701F0C8FC030113C06F5BAA030713E092B6FC020F 15F0DA3FFCEBBFFC902601FF80903881FF80902607FE009038807FE0D90FF8ED1FF0D93F E0ED07FC49486F7E49486F7E4890C76E13804848EF7FC04848EF3FE0000F19F04848EF1F F8A24848EF0FFCA2007F19FE491707A200FF19FFA8007F19FEA26D170F003F19FCA26C6C EF1FF8A26C6CEF3FF0000719E06C6CEF7FC06C6CEFFF806C6D4B13006D6C4B5A6D6C4B5A D90FF8ED1FF0D907FEED7FE0902601FF80903881FF809026003FFCD9BFFCC7FC020FB612 F0020092C8FC030713E003001380AA4B7F030713F0023FB612FEA448527AD155>8 D<15E01401EC03C0EC0780EC0F00141E5C147C5C495A13035C495A130F5C131F91C7FC13 3E137EA25BA2485AA25B1203A2485AA3120F5BA2121FA25BA2123FA290C8FCA35AA5127E A312FEB3A3127EA3127FA57EA37FA2121FA27FA2120FA27F1207A36C7EA212017FA26C7E A2137EA2133E7F80130F8013076D7E8013016D7E147C143C8080EC0780EC03C0EC01E014 001B7974D92E>40 D<12E07E12787E7E7E6C7E7F6C7E6C7E7F1200137C137E133E133F7F 6D7E80A26D7EA26D7EA2130180A26D7EA380147EA2147FA280A21580A2141FA315C0A514 0FA315E0B3A315C0A3141FA51580A3143FA21500A25CA2147EA214FE5CA3495AA25C1303 A2495AA2495AA25C49C7FC5B133E137E137C5B12015B485A485A5B48C8FC121E5A5A5A5A 1B797AD92E>I<120FEA3FC0EA7FE0EAFFF0A6EA7FE0EA3FC0EA0F000C0C768B21>46 D48 D50 D<120FEA3FC0EA7FE0EAFFF0A6EA7FE0EA3FC0EA0F00C7FCB3A9120FEA3FC0EA7FE0 EAFFF0A6EA7FE0EA3FC0EA0F000C3376B221>58 D66 DI69 D<49B612FEA490C7003F138092380FFE00 1507B3B3B3A21206EA3FC0487E487EA44B5AA25B007F5D0180131F0078C75B6C143F003E 4A5A6C5D6C6C495A2707E003FEC7FC3901FC07FC6CB512F0013F13C0D907FCC8FC2F547B D13C>74 D77 D80 D97 DII<17FF4BB5FCA4ED0007160182B3A6EC0FF8EC7FFF49 B512E0903907FC03F090391FE0007C49487F49C7120F01FE80484880485A000781484880 A2485AA2485AA2127FA35B12FFAB127FA27FA2123FA27F121FA26C6C5C00075D7F6C6C5C 6C6C5C6C6C021E7F6D6C017C13E0D91FC049EBFF8090390FF807E00103B512800100495A DA1FF091C7FC39547CD241>II<157F913803FFE002 0F13F091383FC0F891387F01FC903901FE03FE903803FC0714F81307EB0FF0A290391FE0 03FCED01F892C7FC495AB3B612FEA426003FC0C7FCB3B3A580EBFFF0007FEBFFF8A42754 7DD324>III<1378 EA01FE487E487FA66C90C7FC6C5AEA007890C8FCB0EB7F80B5FCA41203C6FC137FB3B3A4 3801FFE0B61280A419507CCF21>I108 D<01FFD907FEEC03FFB590261FFFC0010F13E0037F01F0013F 13F8912701F80FFC9038FC07FE913D03C003FE01E001FF000390260700019038038000C6 010E6D6C48C76C7E6D48DA7F8E6E7E4A159CA24ADA3FF86E7E02605D14E04A5DA34A5DB3 AD2601FFE0DAFFF0EC7FF8B6D8C07F9026FFE03FB512F0A45C347CB363>I<01FFEB07FC B590383FFF8092B512E0913901F00FF8913903C007FC000349C66C7EC6010E13016D486D 7E5C143002706E7E146014E05CA35CB3AD2601FFE0903801FFE0B600C0B612C0A43A347C B341>II<90397F8007FCB590387FFF800281B512E09139 87F00FF891398F8003FC000390399E0001FFC601BC6D7FD97FF86E7E4A6E7E4A6E7E4A14 0F844A6E7EA2717EA3717EA4711380AB4D1300A44D5AA24D5AA2606E140F4D5A6E5D6E4A 5A6E4A5A02BC4AC7FC029E495A028FEB07FC913987E01FF00281B512C0DA807F90C8FCED 0FF892CAFCB13801FFE0B612C0A4394B7DB341>I<01FFEB1F80B5EB7FF0913801FFF891 3803E1FC91380783FE0003EB0F07C6131EEB7F1C1438143091387003FC91386000F01600 14E05CA45CB3AA8048487EB612F0A427347DB32E>114 DII< D97F80EC7F80B591B5FCA400031503C61500017F157FB3AC17FFA35EA3013F5C6EEB077F A2011F020E7F6D6C011C13F06E0178EBFFC0903903FC01F06DB512E06D6C1380912607FC 00EB80003A357CB341>II< 007FB5D8800FB51280A4C69026FC0003EBF000D93FF86D1380011F4BC7FC010F15F80107 5D6D6C5C6E495A6D6C5C6D14076E6C48C8FCEDC01E6E6C5A021F133891380FF0786F5A91 3807FDE002035BEC01FF5E80157F6F7E824B7E15FFEDE7F802017F913803C3FEEC07814A C67E020E80021E6D7E4A133F4A6D7E4A80707E4948130749486D7E010781010F6E7E013F 8201FF8200076D010713F0B500F8011FEBFFE0A43B337FB23E>120 DI<001FB712E0A301FCC7EA7FC001E014FF018049138090C714004B5A 001E14074B5A485D4B5A153F4B5A00385D15FF4A5B93C7FC4A5A1407C7485A5D4A5A143F 4A5A5D14FF495B92C8FC494814E01307495A5C495A013FEC01C0495A5C13FF485B91C712 03485A12074848140749140F485A003FED3F80484814FF491307B8FCA32B337DB234>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: FA cmr17 20.74 15 /FA 15 117 df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ndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: A4 %%EndSetup %%Page: 1 1 1 0 bop 466 809 a FA(A)53 b(pro)t(of)f(of)h(Scott's)e(correction)i(for) g(Matter)1210 1105 y Fz(P)m(edro)37 b(Balo)s(dis)h(Matesanz)1488 1338 y(Jan)m(uary)f(2002)1630 1705 y Fy(Abstract)500 1873 y Fx(In)44 b(this)f(pap)s(er)h(is)f(giv)m(en)i(a)g(strengh)m (tening)f(of)g Fw(Stability)j(of)f(Matter)p Fx(,)i(whic)m(h)364 1986 y(in)28 b(particular)f(sho)m(ws)i(that)h(Thomas-F)-8 b(ermi)29 b(theory;)h(whic)m(h)e(is)g(already)h(kno)m(wn)f(to)364 2099 y(giv)m(e)21 b(the)h(leading)e(order)g(con)m(tribution)g(to)i(the) f(energy)h(of)f(Matter,)k(if)20 b(supplemen)m(ted)364 2212 y(with)36 b(the)h(so-called)g Fw(Sc)-5 b(ott)40 b(c)-5 b(orr)g(e)g(ction)40 b Fx(is)c(correct)i(uniformly)d(in)h(the)h (n)m(um)m(b)s(er)f(of)364 2325 y(n)m(uclei.)j(New)31 b(more)f(precise)g(estimates)h(of)g(the)f(v)m(olume)g(of)h(Matter)h (also)e(follo)m(ws.)1806 5712 y Fv(1)p eop %%Page: 2 2 2 1 bop 266 407 a Fu(Con)l(ten)l(ts:)120 610 y Ft(1.)49 b Fv(In)m(tro)s(duction.)58 b(.)15 b(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f (.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.) g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)e(2)120 814 y Ft(2.)49 b Fv(Estimates)32 b(on)h(the)g(Thomas-F)-8 b(ermi)30 b(p)s(oten)m(tial.)52 b(.)16 b(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.) g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g (.)g(.)g(.)f(.)h(.)g(.)10 b(5)120 1017 y Ft(3.)49 b Fv(Reduction)32 b(to)h(the)g(case)g(of)f(uniformly)f(separated)i(n)m(uclei.)57 b(.)16 b(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.) g(.)g(.)g(.)g(.)d(14)120 1220 y Ft(4.)49 b Fv(Semiclassics)32 b(in)f(the)i(go)s(o)s(d)f(region:)42 b(Multiscale)32 b(Analysis.)68 b(.)16 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h (.)g(.)g(.)g(.)g(.)g(.)g(.)25 b(21)120 1424 y Ft(5.)49 b Fv(Reconstruction)33 b(of)f(the)h(whole)f(system)i(and)f(pro)s(of)e (of)h(the)h(main)e(results.)61 b(.)16 b(.)g(.)g(.)g(.)g(.)g(.)g(52)169 1627 y(References)27 b(.)16 b(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g (.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.) g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g (.)f(.)h(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)g(.)f(.)h(.)g(.)g(.) g(.)25 b(58)120 1960 y Fs(1)161 b(In)l(tro)t(duction.)120 2179 y Fr(Stability)39 b(of)g(Matter)f Fv(is)f(one)g(the)h (cornerstones)h(of)d(our)h(presen)m(t)i(understanding)f(of)e(ho)m(w)120 2299 y(Quan)m(tum)41 b(Mec)m(hanics)h(\(QM)f(in)g(what)g(follo)m(ws\))f (describ)s(es)i(the)f(real)f(w)m(orld,)j(and)f(since)120 2420 y(the)35 b(\014rst)g(rigurous)e(pro)s(of)h(of)g(that)g(fact)g(b)m (y)h(F.Dyson)g(and)f(E.L)m(\023)-46 b(enard)35 b(in)f(the)h(late)e (sixties)120 2540 y(m)m(uc)m(h)f(e\013ort)f(has)h(b)s(een)g(dev)m(oted) h(to)e(impro)m(v)m(e)g(and)h(to)f(generalize)g(their)g(results)g (either)h(in)120 2661 y(the)37 b(direction)e(of)h(getting)f(sharp)s(er) i(and)g(sharp)s(er)g(estimates)f(and/or)g(considering)g(more)120 2781 y(general)26 b(quan)m(tum)g(mo)s(dels)f(than)h(that)g(of)g(the)h (standard)f(non-relativistic)e(of)i(Sc)m(hr\177)-49 b(odinger)120 2901 y(b)m(y)33 b(including)e(relativit)m(y)-8 b(,)31 b(arbitrary)h(magnetic)f(\014elds)i(or)f(quan)m(tized)i(\014elds.)266 3022 y(In)d(this)f(pap)s(er)h(I)g(am)e(concerned)j(in)e(a)g(kind)h(of)f (strengh)m(tening)h(of)f Fr(Stability)j(of)g(Matter)120 3142 y Fv(whic)m(h)43 b(tries)f(to)g(obtain)f(the)i(righ)m(t)e(dep)s (endence)k(of)d(the)h(total)d(energy)k(in)d(the)i(ph)m(ysical)120 3262 y(parameters.)79 b(It)44 b(is)g(also)g(related)g(to)g(the)g (remark)-5 b(able)43 b(fact)i(that)f(exp)s(erimen)m(tally)f(one)120 3383 y(\014nds)34 b(that)f(mean)g(n)m(uclear)g(spacing)g(is)f(nearly)h (indep)s(enden)m(t)h(of)f(the)g(kind)g(of)g(atoms)f(that)120 3503 y(matter)g(is)g(made)g(of.)266 3624 y(In)41 b(the)f(Thomas-F)-8 b(ermi)38 b(theory)i(\(TF)g(in)g(the)g(next\),)j(distances)e(scale)f (as)g(the)g Fq(\000)p Fv(1)p Fp(=)p Fv(3)120 3744 y(p)s(o)m(w)m(er)45 b(of)e(the)h(n)m(uclear)f(a)m(v)m(erage)i(c)m(harge)f Fp(Z)7 b Fv(,)46 b(i.e,)g(w)m(e)f(w)m(ould)f(exp)s(ect)g(that)g(v)m (olume)f(p)s(er)120 3864 y(particle)24 b(should)g(b)s(eha)m(v)m(e)j (lik)m(e)d Fp(Z)1329 3828 y Fo(\000)p Fn(1)p Fm(=)p Fn(3)1494 3864 y Fv(.)41 b(As)25 b(I)g(ha)m(v)m(e)h(p)s(oin)m(ted)f(out,)h(one)f (\014nds)h(exp)s(erimen)m(tally)120 3985 y(that)47 b(the)g(real)f(b)s (eha)m(viour)h(seems)h(to)f(b)s(e)g(almost)f(indep)s(enden)m(t)i(of)e Fp(Z)7 b Fv(,)51 b(and)c(is)g(a)f(v)m(ery)120 4105 y(c)m(hallenging)31 b(problem)g(to)h(try)h(to)f(explain)g(rigurously)g(that)g(fact.)266 4226 y(The)39 b(usual)f(Lieb-Thirring's)e(pro)s(of)h(of)g(Stabilit)m(y) f(of)h(Matter)h(indeed)g(imply)e(a)i(lo)m(w)m(er)120 4346 y(b)s(ound)43 b(of)f(the)h(v)m(olume)f(p)s(er)g(particle)f(of)h (order)h Fp(Z)2078 4310 y Fo(\000)p Fn(1)2172 4346 y Fv(.)73 b(By)43 b(the)g(results)g(I)g(am)e(going)g(to)120 4466 y(explain,)g(a)e(b)s(ound)i(of)e(the)h(v)m(olume)f(p)s(er)h(atom)f (of)g(the)i(order)f Fp(Z)2593 4430 y Fo(\000)p Fn(1+)p Fm(\016)2815 4466 y Fv(follo)m(ws,)h(for)e(some)120 4587 y(suitable)25 b Fp(\016)32 b(>)c Fv(0)e(\(see)h(Theorem)g(3\),)g(whic)m (h)g(is)f(still)e(far)i(of)g(the)h(b)s(ound)f(of)g(order)h Fp(Z)3148 4551 y Fn(0)3215 4587 y Fv(=)h(1)e(one)120 4707 y(migh)m(t)j(hop)s(e,)i(but)f(nev)m(ertheless)j(is)d(the)h(b)s (est)g(one)f(already)g(giv)m(en)h(as)f(far)g(as)h(the)f(author)g(is)120 4827 y(a)m(w)m(are)j(of.)266 4948 y(W)-8 b(e)34 b(consider)g(matter)f (formed)g(b)m(y)i Fp(M)44 b Fv(n)m(uclei)33 b(of)g(c)m(harges)i Fp(Z)2546 4963 y Fm(j)2612 4948 y Fq(\025)30 b Fv(1,)k Fp(j)h Fv(=)29 b(1)p Fp(;)17 b(:)g(:)g(:)33 b(;)17 b(M)44 b Fv(lo-)120 5068 y(cated)22 b(at)g(p)s(ositions)e Fp(R)951 5083 y Fm(j)1016 5068 y Fq(2)28 b Fl(R)1176 5032 y Fn(3)1221 5068 y Fv(.)40 b(W)-8 b(e)22 b(denote)g Fq(R)29 b Fv(=)e Fq(f)p Fp(R)2088 5083 y Fn(1)2128 5068 y Fp(;)17 b(:)g(:)g(:)32 b(;)17 b(R)2437 5083 y Fm(M)2516 5068 y Fq(g)k Fv(and)h Fq(Z)36 b Fv(=)28 b(\()p Fp(Z)3083 5083 y Fn(1)3122 5068 y Fp(;)17 b(:)g(:)g(:)32 b(;)17 b(Z)3424 5083 y Fm(M)3503 5068 y Fv(\).)120 5189 y(W)-8 b(e)35 b(consider)g(these)h(n)m(uclei)e (as)h(static)g(and)f(consider)h(the)h(non-relativistic)c(Hamiltonian) 1806 5712 y(2)p eop %%Page: 3 3 3 2 bop 120 407 a Fv(of)31 b Fp(N)42 b Fv(electrons)32 b(mo)m(ving)e(in)h(the)h(electric)f(p)s(oten)m(tial)f(of)h(the)h(n)m (uclei.)42 b(The)33 b(Hamiltonian)27 b(is)168 714 y Fp(H)249 729 y Fo(R)p Fm(;)p Fo(Z)5 b Fm(;N)500 714 y Fv(=)643 589 y Fm(N)603 619 y Fk(X)618 829 y Fm(i)p Fn(=1)764 544 y Fk(\()844 714 y Fq(\000)p Fv(\001)1002 729 y Fm(i)1053 714 y Fq(\000)1187 589 y Fm(M)1153 619 y Fk(X)1163 829 y Fm(j)t Fn(=1)1457 647 y Fp(Z)1524 662 y Fm(j)p 1323 691 372 4 v 1323 782 a Fq(j)p Fp(x)1406 797 y Fm(i)1456 782 y Fq(\000)23 b Fp(R)1630 797 y Fm(j)1667 782 y Fq(j)1704 544 y Fk(\))1807 714 y Fv(+)1993 619 y Fk(X)1905 831 y Fn(1)p Fo(\024)p Fm(i)g Fv(0)35 b Fr(\014nite)g(c)-5 b(onstants)35 b(only)g(dep)-5 b(ending)34 b(on)h Fp(A)g Fr(and)g(some)f(absolute)i Fp(\016)c(>)c Fv(0)120 1724 y Fr(\()p Fp(\016)j Fv(=)d(8)p Fp(=)p Fv(51)34 b Fr(c)-5 b(ould)35 b(do\))f(such)h(that)120 2037 y Fp(E)198 1989 y Fm(Q)192 2064 y Fo(R)p Fm(;)p Fo(Z)5 b Fm(;N)443 2037 y Fv(:=)173 b(inf)573 2103 y Fn(\011)p Fo(2H)p Fm(;)12 b Fo(k)p Fn(\011)p Fo(k)p Fn(=1)999 2037 y Fq(h)p Fv(\011)p Fp(;)17 b(H)1239 2052 y Fo(R)p Fm(;)p Fo(Z)t Fm(;N)1461 2037 y Fv(\011)p Fq(i)1575 2069 y Fo(H)1667 2037 y Fq(\025)1807 1912 y Fm(M)1773 1942 y Fk(X)1783 2152 y Fm(j)t Fn(=1)1933 2037 y Fp(E)2011 1996 y Fm(Q)2071 2037 y Fv(\()p Fp(Z)2176 2052 y Fm(j)2212 2037 y Fv(\))22 b(+)g Fp(cZ)2486 1996 y Fn(7)p Fm(=)p Fn(3)2647 1912 y Fm(M)2613 1942 y Fk(X)2623 2152 y Fm(j)t Fn(=1)2773 2037 y Fv(\000)2851 1956 y Fk(\000)2896 2037 y Fp(Z)2970 1996 y Fn(1)p Fm(=)p Fn(3)3080 2037 y Fp(\016)3123 2052 y Fm(j)3160 1956 y Fk(\001)3228 2037 y Fq(\000)g Fp(C)7 b(M)j(Z)3582 1996 y Fn(2)p Fo(\000)p Fm(\016)3415 2267 y Fv(\(2\))120 2493 y Fr(wher)-5 b(e)41 b Fp(E)480 2457 y Fm(Q)539 2493 y Fv(\()p Fp(Z)7 b Fv(\))42 b Fr(stands)e(for)i(the)f(quantum)g(ener)-5 b(gy)41 b(of)g(an)g(atom)g (of)g(char)-5 b(ge)41 b Fp(Z)7 b Fr(,)43 b(which)d(is)120 2613 y(given)34 b(by)1155 2833 y Fp(E)1233 2792 y Fm(Q)1293 2833 y Fv(\()p Fp(Z)7 b Fv(\))27 b(:=)46 b(inf)1601 2895 y Fm(N)7 b Fo(\025)p Fn(1)1771 2833 y Fv(inf)23 b(sp)s(ec)17 b Fp(H)2185 2849 y Fo(f)p Fn(0)p Fo(g)p Fm(;)p Fn(\()p Fm(Z)5 b Fn(\))p Fm(;N)3415 2833 y Fv(\(3\))120 3175 y Fr(Her)-5 b(e)30 b(the)g(Hilb)-5 b(ert)30 b(sp)-5 b(ac)g(e)29 b(wher)-5 b(e)29 b(the)h(Hamiltonian)f(acts)g(is)h Fq(H)f Fv(=)2576 3080 y Fm(N)2566 3100 y Fk(V)2666 3175 y Fp(L)2732 3138 y Fn(2)2772 3175 y Fv(\()p Fl(R)2876 3138 y Fn(3)2921 3175 y Fv(;)17 b Fl(C)3031 3138 y Fn(2)3076 3175 y Fv(\))30 b Fr(\(F)-7 b(ermion)120 3295 y(sp)i(ac)g(e\),)40 b Fv(\000\()p Fp(t)p Fv(\))d(=)f(min)o Fq(f)p Fp(t)1019 3259 y Fo(\000)p Fn(1)1113 3295 y Fp(;)17 b(t)1192 3259 y Fo(\000)p Fn(7)1286 3295 y Fq(g)40 b Fr(and)f Fp(\016)1613 3310 y Fm(j)1686 3295 y Fv(=)e(min)1961 3310 y Fm(i)p Fo(6)p Fn(=)p Fm(j)2094 3295 y Fq(j)p Fp(R)2196 3310 y Fm(i)2249 3295 y Fq(\000)27 b Fp(R)2427 3310 y Fm(j)2463 3295 y Fq(j)40 b Fr(is)f(the)h(distanc)-5 b(e)39 b(of)g Fp(R)3381 3310 y Fm(j)3458 3295 y Fr(to)120 3415 y(its)c(ne)-5 b(ar)g(est)34 b(neighb)-5 b(or.)266 3619 y Fv(As)33 b(the)g(QM)g(energy)h Fp(E)1166 3583 y Fm(Q)1226 3619 y Fv(\()p Fp(Z)7 b Fv(\))32 b(is)g(giv)m(en)h(b)m(y)g (the)g(three-term)f(asymptotics)759 3883 y Fp(E)837 3842 y Fm(Q)897 3883 y Fv(\()p Fp(Z)7 b Fv(\))27 b(=)h Fq(\000)p Fp(C)1325 3898 y Fn(TF)1426 3883 y Fp(Z)1500 3842 y Fn(7)p Fm(=)p Fn(3)1632 3883 y Fv(+)1740 3816 y(1)p 1740 3860 49 4 v 1740 3951 a(4)1799 3883 y Fp(Z)1873 3842 y Fn(2)1935 3883 y Fq(\000)22 b Fp(C)2104 3898 y Fn(DS)2201 3883 y Fp(Z)2275 3842 y Fn(5)p Fm(=)p Fn(3)2407 3883 y Fv(+)g Fp(o)p Fv(\()p Fp(Z)2664 3842 y Fn(5)p Fm(=)p Fn(3)p Fo(\000)p Fm(\016)2863 3883 y Fv(\))514 b(\(4\))120 4138 y(where)32 b(in)e(\(4\))h Fp(\016)h(>)27 b Fv(0)k(is)f(some)h(absolute) g(constan)m(t)h(and)f(the)g(second)i(term,)d(of)h(order)g Fp(Z)3404 4102 y Fn(2)3474 4138 y Fv(is)120 4259 y(kno)m(wn)d(as)f(the) g Fr(Sc)-5 b(ott)29 b(term)p Fv(.)42 b(In)27 b(Theorem)g(1,)h(the)f (constan)m(ts)h Fp(C)2488 4274 y Fn(TF)2589 4259 y Fv(,)f Fp(C)2713 4274 y Fn(DS)2837 4259 y Fv(are,)h(resp)s(ectiv)m(ely)-8 b(,)120 4379 y(the)36 b(corresp)s(onding)f Fr(Thomas-F)-7 b(ermi)32 b Fv(constan)m(t)k(\(see)h(Sect.)f(2)f(b)s(elo)m(w)g(for)f (the)i(de\014nition\))120 4499 y(and)d(the)g Fr(Dir)-5 b(ac-Schwinger)30 b Fv(one.)44 b(This)32 b(w)m(ork)i(shall)d(not)h (address)i(this)f(one.)266 4620 y(Theorem)i(1)f(in)f(particular)f(sho)m (ws)k(that)e(adding)f(up)i(the)f(con)m(tributions)g(of)g(Thomas-)120 4740 y(F)-8 b(ermi)31 b(\(the)h(main)f(one)h(of)g(order)h Fp(Z)1465 4704 y Fn(7)p Fm(=)p Fn(3)1575 4740 y Fv(\))f(and)g(the)h (Scott's)g(o)m(v)m(er)h(all)c(atoms)i(giv)m(es)g(us)h(abso-)120 4861 y(lute)d(b)s(ounds)h(for)e(the)i(total)d(QM)j(energy)g(up)g(to)e (an)h(error)h(of)e(smaller)f(order)j(uniformly)d(in)120 4981 y(the)38 b(n)m(um)m(b)s(er)g(of)f(n)m(uclei.)58 b(Unfortunately)37 b(the)h(metho)s(d)f(do)s(es)h(not)f(giv)m(e)g(the)h (next)h(Dirac-)120 5101 y(Sc)m(h)m(winger)47 b(term,)h(not)d(ev)m(en)i (a)e(remainder)g(of)g(its)g(order.)82 b(This)46 b(dra)m(wbac)m(k)h (migh)m(t)d(b)s(e)120 5222 y(hop)s(efully)24 b(o)m(v)m(ercome)h(if)f (one)h(uses)i(instead)e(of)f(the)h Fr(r)-5 b(e)g(duc)g(e)g(d)28 b(Hartr)-5 b(e)g(e-F)e(o)i(ck)26 b(mo)-5 b(del)p Fv(,)26 b(whic)m(h)f(is)120 5342 y(merely)i(a)g(1-particle)e(apro)m(ximation)g (to)j(the)f(full)f(quan)m(tum)i(problem)e(the)i(whole)f(Hartree-)120 5463 y(F)-8 b(o)s(c)m(k)22 b(mo)s(del,)g(whic)m(h)h(ev)m(en)g(if)e(it)g (is)g(also)g(another)h(apro)m(ximation)d(to)j(the)g(quan)m(tum)g (problem,)1806 5712 y(3)p eop %%Page: 4 4 4 3 bop 120 407 a Fv(can)35 b(b)s(e)g(sho)m(wn)h(to)e(yield)g(an)g (energy)i(m)m(uc)m(h)f(closer)g(to)f(the)h(quan)m(tum)g(one)g(than)g (its)f(bab)m(y)120 527 y(brother,)d(whic)m(h)f(is)f(nev)m(ertheless)j (easier)e(to)g(handle)f(b)s(ecause)i(it)e(a)m(v)m(oids)h(the)g(subtle)g (corre-)120 648 y(lation)j(e\013ects.)54 b(Despite)36 b(this,)g(the)g(reasons)g(I)g(c)m(hose)h(to)e(use)i(this)e Fr(r)-5 b(e)g(duc)g(e)g(d)37 b(Hartr)-5 b(e)g(e-F)e(o)i(ck)120 768 y(mo)g(del)33 b Fv(are)g(t)m(w)m(ofold:)46 b(The)34 b(\014rst)h(is)e(b)s(ecause)i(it)e(can)h(b)s(e)f(sho)m(wn)i(to)f(appro) m(ximate)f(the)h(QM)120 888 y(energy)c(up)f(to)f(energies)h(of)f(order) h Fp(Z)1477 852 y Fn(5)p Fm(=)p Fn(3)1615 888 y Fv(\(at)g(least)f(for)g (isolated)f(atoms\))h(and)g(the)i(second)f(is)120 1009 y(its)g(simplicit)m(y)-8 b(,)28 b(whic)m(h)i(is)f(mainly)f(due)i(to)g (the)g(fact)f(it)g(is)g(a)h(1-particle)d(mo)s(del)h(\(a)i(complete)120 1129 y(treatmen)m(t)40 b(of)g(the)g(quan)m(tum)h(Hamiltonian)36 b(w)m(ould)k(require)h(a)f(2-particle)e(form)m(ulation,)120 1249 y(due)33 b(to)f(the)h(fact)g(the)g(electronic)f(in)m(teractions)g (are)g(of)g(2-particle)f(t)m(yp)s(e\).)266 1370 y(W)-8 b(e)45 b(can)f(write)g(do)m(wn)h(a)e(full)f(t)m(w)m(o-term)i (asymptotics)g(for)f(the)i(QM)f(energy)h(if)e(one)120 1490 y(restricts)c(atten)m(tion)f(to)h(con\014gurations)f(of)g (uniformly)f(separated)i(n)m(uclei,)h(whic)m(h)f(is)g(the)120 1611 y(con)m(ten)m(t)34 b(of)e(next)h(Theorem:)120 1814 y Ft(Theorem)k(2)g(\(Asymptotics)e(for)j(the)f(QM)h(energy)g(of)g (uniformly)d(separated)k(n)m(uclei\).)120 1934 y Fr(Assume)30 b(the)f(c)-5 b(onditions)29 b(of)g(Th.)g(1)h(ar)-5 b(e)29 b(ful\014l)5 b(le)-5 b(d,)30 b(and)f(b)-5 b(esides)28 b(that)i(al)5 b(l)30 b(nucle)-5 b(ar)29 b(distanc)-5 b(es)120 2055 y Fp(\016)163 2070 y Fm(j)227 2055 y Fq(\025)29 b Fp(\026Z)466 2019 y Fo(\000)p Fn(1)p Fm(=)p Fn(3)665 2055 y Fr(for)35 b(some)f(\014xe)-5 b(d)34 b Fp(\026)27 b(>)h Fv(0)p Fr(.)44 b(Then)34 b(the)h(fol)5 b(lowing)34 b(asymptotics)g(holds)713 2370 y Fp(C)783 2385 y Fn(1)822 2370 y Fp(M)10 b(Z)1000 2329 y Fn(2)p Fo(\000)p Fm(\016)1121 2338 y Fj(1)1189 2370 y Fq(\025)28 b Fp(E)1372 2323 y Fm(Q)1366 2397 y Fo(R)p Fm(;)p Fo(Z)1528 2370 y Fq(\000)23 b Fp(E)1706 2329 y Fn(TF)1700 2395 y Fo(R)p Fm(;)p Fo(Z)1862 2370 y Fq(\000)1972 2303 y Fv(1)p 1972 2348 49 4 v 1972 2439 a(4)2082 2246 y Fm(M)2047 2276 y Fk(X)2058 2486 y Fm(j)t Fn(=1)2208 2370 y Fp(Z)2282 2329 y Fn(2)2275 2395 y Fm(j)2349 2370 y Fq(\025)28 b(\000)p Fp(C)2601 2385 y Fn(2)2640 2370 y Fp(M)10 b(Z)2818 2329 y Fn(2)p Fo(\000)p Fm(\016)3415 2370 y Fv(\(5\))120 2691 y Fr(\(wher)-5 b(e)38 b Fp(C)509 2706 y Fn(1)548 2691 y Fp(;)17 b(C)662 2706 y Fn(2)740 2691 y Fr(ar)-5 b(e)39 b(two)g(p)-5 b(ositive)38 b(and)h(\014nite)f(c)-5 b(onstants)39 b(only)f(dep)-5 b(end)38 b(on)h Fp(\026)f Fr(ab)-5 b(ove)39 b(and)120 2812 y Fp(A)p Fr(\).)82 b Fp(E)423 2776 y Fn(TF)417 2837 y Fo(R)p Fm(;)p Fo(Z)605 2812 y Fr(stands)46 b(for)i(the)f(Thomas-F)-7 b(ermi)44 b(ener)-5 b(gy)48 b(of)f(the)g(ful)5 b(l)47 b(c)-5 b(on\014gur)g(ation)47 b(\(lo)-5 b(ok)120 2958 y(at)47 b(se)-5 b(ction)46 b(2)g(for)g(de\014nition\))g(with)g(p)-5 b(ar)g(ameter)46 b Fp(\014)55 b Fv(=)2276 2919 y Fn(3)2311 2895 y Fj(5)p Fi(=)p Fj(3)2408 2919 y Fm(\031)2451 2895 y Fj(4)p Fi(=)p Fj(3)p 2276 2935 272 4 v 2394 2992 a Fn(5)2558 2958 y Fr(.)79 b(and)46 b(the)h(unr)-5 b(estricte)g(d)120 3093 y(quantum)35 b(ener)-5 b(gy)35 b Fp(E)907 3046 y Fm(Q)901 3120 y Fo(R)p Fm(;)p Fo(Z)1076 3093 y Fr(is)f(given)g(by)1412 3334 y Fp(E)1490 3287 y Fm(Q)1484 3361 y Fo(R)p Fm(;)p Fo(Z)1652 3334 y Fv(:=)45 b(inf)1783 3396 y Fm(N)7 b Fo(\025)p Fn(0)1953 3334 y Fp(E)2031 3287 y Fm(Q)2025 3361 y Fo(R)p Fm(;)p Fo(Z)e Fm(;N)3415 3334 y Fv(\(6\))120 3600 y Fr(i.e.)44 b(it)34 b(is)g(the)g(quantum)g(ener)-5 b(gy)34 b(when)f(one)g(arrives)h(at)g(ele)-5 b(ctr)g(onic)33 b(satur)-5 b(ation)34 b(\(and)g(thus)120 3720 y(adding)g(up)h(mor)-5 b(e)35 b(of)g(them)g(do)-5 b(es)34 b(not)h(lower)g(the)g(ener)-5 b(gy\).)45 b(The)34 b(exp)-5 b(onent)35 b Fp(\016)k Fr(is)c(the)g(same) 120 3840 y(as)g(in)f(Th.)g(1,)h(and)f(we)h(c)-5 b(an)34 b(take)h Fp(\016)1426 3855 y Fn(1)1493 3840 y Fv(=)28 b(5)p Fp(=)p Fv(27)p Fr(.)364 4069 y Ft(Remark:)43 b Fv(As)33 b(the)g(Thomas-F)-8 b(ermi)30 b(energy)364 4376 y Fp(E)442 4335 y Fn(TF)436 4401 y Fo(R)p Fm(;)p Fo(Z)604 4376 y Fq(\025)e(\000)p Fp(C)856 4391 y Fn(TF)1009 4251 y Fm(M)974 4281 y Fk(X)985 4491 y Fm(j)t Fn(=1)1134 4376 y Fp(Z)1208 4325 y Fn(7)p Fm(=)p Fn(3)1201 4401 y Fm(j)1318 4376 y Fv(+)p Fp(cZ)1510 4335 y Fn(7)p Fm(=)p Fn(3)1671 4251 y Fm(M)1637 4281 y Fk(X)1647 4491 y Fm(j)t Fn(=1)1797 4376 y Fv(\000\()p Fp(Z)1970 4335 y Fn(1)p Fm(=)p Fn(3)2080 4376 y Fp(\016)2123 4391 y Fm(j)2160 4376 y Fv(\))65 b(\(Strong)32 b(form)f(of)i(T)-8 b(eller's)32 b(non-binding)f (Theorem\))364 4697 y(the)39 b(asymptotics)g(giv)m(en)g(b)m(y)h(\(5\))e (indeed)h(mak)m(es)h(more)e(precise)i(the)f(energy)364 4817 y(estimate)32 b(giv)m(en)g(in)g(Thm)h(1.)266 5045 y(As)41 b(Corollary)d(of)h(our)h(main)e(Theorem)i(1,)h(it)e(follo)m(ws) g(that)g(if)g(w)m(e)i(are)f(giv)m(en)g(a)f(con-)120 5166 y(\014guration)i(of)h(n)m(uclei)g(of)g(total)f(energy)i(close)g(to)f (the)h(minim)m(un)d(p)s(ossible)i(one,)j(n)m(uclear)120 5286 y(distances)31 b(should)g(b)s(e)g(bigger)e(in)h(a)m(v)m(erage)i (than)e Fp(Z)2024 5250 y Fo(\000)p Fn(1+)p Fm(\016)2202 5227 y Fh(0)2229 5286 y Fv(;)i Fp(\016)2335 5250 y Fo(0)2386 5286 y Fv(=)2499 5247 y Fn(1+3)p Fm(\016)p 2499 5263 160 4 v 2561 5321 a Fn(7)2668 5286 y Fv(.)43 b(T)-8 b(o)31 b(b)s(e)f(more)g(precise)120 5407 y(w)m(e)k(ha)m(v)m(e)g(the)f(follo)m (wing)c(result:)1806 5712 y(4)p eop %%Page: 5 5 5 4 bop 120 407 a Ft(Theorem)37 b(3)g(\(The)h(v)m(olume)e(of)i (matter\).)47 b Fr(With)31 b(the)f(c)-5 b(onditions)29 b(in)h(The)-5 b(or)g(em)30 b(1,)g(as-)120 527 y(sume)35 b(mor)-5 b(e)g(over)33 b(that)j(for)e(the)h(c)-5 b(on\014gur)g(ation)34 b Fq(R)885 828 y Fv(inf)740 894 y Fn(\011)p Fo(2H)p Fm(;)11 b Fo(k)p Fn(\011)p Fo(k)p Fn(=1)1165 828 y Fq(h)p Fv(\011)p Fp(;)17 b(H)1405 843 y Fo(R)p Fm(;)p Fo(Z)t Fm(;N)1627 828 y Fv(\011)p Fq(i)1742 860 y Fo(H)1834 828 y Fq(\024)1974 703 y Fm(M)1939 733 y Fk(X)1950 943 y Fm(j)t Fn(=1)2100 828 y Fp(E)2178 787 y Fm(Q)2237 828 y Fv(\()p Fp(Z)2342 843 y Fm(j)2378 828 y Fv(\))23 b(+)f Fp(C)7 b(M)j(Z)2792 787 y Fn(2)p Fo(\000)p Fm(\016)3415 828 y Fv(\(7\))120 1142 y Fr(\(observe)37 b(that)h(b)-5 b(e)g(c)g(ause)37 b(of)g(The)-5 b(or)g(em)37 b(2\))g(such)g(c)-5 b(on\014gur)g(ations)37 b(inde)-5 b(e)g(d)37 b(exist\).)53 b(Then)36 b(we)120 1262 y(have)e(that)h Fq(R)h Fr(should)e(satisfy,)901 1563 y Fq(h)p Fp(\016)t Fq(i)27 b Fv(:=)1222 1495 y(1)p 1194 1540 105 4 v 1194 1631 a Fp(M)1360 1438 y Fm(M)1325 1468 y Fk(X)1336 1678 y Fm(j)t Fn(=1)1486 1563 y Fp(\016)1529 1578 y Fm(j)1593 1563 y Fq(\025)h Fp(\024Z)1828 1521 y Fn(1)p Fm(=)p Fn(3\()p Fo(\000)p Fn(1+)p Fm(\016)2138 1498 y Fh(0)2163 1521 y Fn(\))2194 1563 y Fv(;)52 b Fp(\016)2320 1521 y Fo(0)2371 1563 y Fv(=)2484 1495 y(1)22 b(+)g(3)p Fp(\016)p 2484 1540 265 4 v 2592 1631 a Fv(7)3415 1563 y(\(8\))120 1871 y Fr(A)n(nother)35 b(interpr)-5 b(etation)34 b(of)h(this)g(r)-5 b(esult)35 b(is)f(as)h(fol)5 b(lows:)44 b(L)-5 b(et)35 b(us)g(c)-5 b(al)5 b(l)577 2172 y Fv(v)m(olume)27 b(=)1046 2047 y Fm(M)1011 2077 y Fk(X)1022 2287 y Fm(j)t Fn(=1)1172 2057 y Fk(\014)1172 2117 y(\014)1172 2177 y(\014)1205 2172 y Fp(B)1289 2170 y Fj(1)p 1289 2182 31 3 v 1289 2223 a(2)1330 2197 y Fm(\016)1361 2207 y Fi(j)1414 2172 y Fv(\()p Fp(R)1526 2187 y Fm(j)1563 2172 y Fv(\)\))1639 2057 y Fk(\014)1639 2117 y(\014)1639 2177 y(\014)1688 2172 y Fv(;)52 b Fp(B)1841 2187 y Fm(r)1879 2172 y Fv(\(x\))28 b(=)2138 2091 y Fk(\010)2196 2172 y Fv(y)h Fq(2)f Fl(R)2436 2130 y Fn(3)2509 2172 y Fv(:)g Fq(j)p Fv(x)23 b Fq(\000)f Fv(y)q Fq(j)28 b(\024)h Fp(r)3026 2091 y Fk(\011)120 2486 y Fr(\(A)n(b)-5 b(ove,)33 b(and)g(her)-5 b(e)g(after,)33 b Fq(j)p Fp(G)p Fq(j)g Fr(stands)g(for)g(L)-5 b(eb)g(esgue)33 b(me)-5 b(asur)g(e)33 b(of)g Fp(G)p Fr(\).)44 b(Then)33 b(we)g(have)g(the)120 2606 y(estimate)1380 2819 y Fv(v)m(olume)27 b Fq(\025)h Fp(\024)1872 2778 y Fo(0)1896 2819 y Fp(M)10 b(Z)2074 2778 y Fo(\000)p Fn(1+)p Fm(\016)2252 2755 y Fh(0)3415 2819 y Fv(\(9\))120 3033 y Fr(wher)-5 b(e)34 b Fp(\024;)17 b(\024)551 2997 y Fo(0)602 3033 y Fp(>)28 b Fv(0)34 b Fr(ar)-5 b(e)35 b(universal)f(c)-5 b(onstants.)120 3364 y Fs(2)161 b(Estimates)53 b(on)g(the)g(Thomas-F)-13 b(ermi)53 b(p)t(oten)l(tial.)120 3584 y Fv(In)25 b(this)e(section)i(w)m(e)g(are)f(going)f(to)h(deriv)m (e)h(t)m(w)m(o)g(results)g(whic)m(h)f(will)e(b)s(e)j(essen)m(tial)f(to) g(general-)120 3704 y(ize)j(the)h(previously)f(kno)m(wn)i(results)f(of) f(V.Ivrii)f(and)i(I.M.Sigal)e([16])h(on)g(the)h(correctedness)120 3824 y(of)35 b(Scott)h(correction)f(for)g(molecules)g(to)g(the)h(case)h (of)e(man)m(y)g(n)m(uclei)h(and)f(therefore)h(to)g(b)s(e)120 3945 y(able)30 b(to)f(go)h(b)s(ey)m(ond)h(their)f(analysis,)g(whic)m(h) h(w)m(as)g(based)g(on)f(the)h(assumption)f(of)f(ha)m(ving)h(a)120 4065 y(\014xed)k(n)m(um)m(b)s(er)f(of)f(them.)266 4185 y(Let)g(us)h(call)c(\010)812 4149 y Fn(TF)812 4211 y Fo(R)p Fm(;)p Fo(Z)985 4185 y Fv(the)j Fr(Thomas-F)-7 b(ermi)29 b Fv(p)s(oten)m(tial)h(of)h(a)g(con\014guration)g Fq(R)p Fp(;)17 b Fq(Z)8 b Fv(.)44 b(Recall)120 4306 y(that)22 b(the)g(Thomas-F)-8 b(ermi)20 b(\(TF)h(in)h(the)g(next\))h(mo)s(del)d (is)h(giv)m(en)h(b)m(y)h(the)f(follo)m(wing)e(functional)611 4568 y Fq(E)673 4520 y Fn(TF)p Fm(;\014)664 4595 y Fo(R)p Fm(;)p Fo(Z)837 4568 y Fv(\()p Fp(\032)p Fv(\))83 b(=)g Fp(\014)1282 4432 y Fk(Z)1337 4657 y Fg(R)1385 4639 y Fj(3)1441 4568 y Fp(\032)p Fv(\()p Fp(x)p Fv(\))1622 4526 y Fn(5)p Fm(=)p Fn(3)1732 4568 y Fp(dx)23 b Fq(\000)1960 4432 y Fk(Z)2016 4657 y Fg(R)2064 4639 y Fj(3)2119 4568 y Fp(V)e Fv(\()p Fp(x)p Fv(\))p Fp(\032)p Fv(\()p Fp(x)p Fv(\))p Fp(dx)1205 4871 y Fv(+)1291 4803 y(1)p 1291 4848 49 4 v 1291 4939 a(2)1366 4735 y Fk(Z)11 b(Z)1532 4960 y Fg(R)1580 4942 y Fj(3)1615 4960 y Fo(\002)p Fg(R)1718 4942 y Fj(3)1783 4803 y Fp(\032)p Fv(\()p Fp(x)p Fv(\))p Fp(\032)p Fv(\()p Fp(y)t Fv(\))p 1783 4848 359 4 v 1820 4939 a Fq(j)p Fp(x)23 b Fq(\000)f Fp(y)t Fq(j)2152 4871 y Fp(dxdy)j Fv(+)2514 4746 y Fm(M)2480 4776 y Fk(X)2497 4988 y Fm(l)q()45 b Fv(0)e(is)f(a)h(parameter.)74 b(In)43 b(our)120 5342 y(units)36 b(the)h(ph)m(ysical)f(v)-5 b(alue)35 b(of)h(this)g(parameter)g(is)g Fp(\014)2109 5357 y Fn(ph)n(ys)2288 5342 y Fv(=)2408 5303 y Fn(3)p 2408 5320 36 4 v 2408 5377 a(5)2453 5342 y Fv(\(3)p Fp(\031)2599 5306 y Fn(2)2638 5342 y Fv(\))2676 5306 y Fn(2)p Fm(=)p Fn(3)2822 5342 y Fv(\(taking)g(spin)g(in)m(to)120 5463 y(accoun)m(t\).)72 b(W)-8 b(e)42 b(shall)e(ho)m(w)m(ev)m(er)k(use)f (also)d(Thomas-F)-8 b(ermi)40 b(theory)i(for)f(other)h(v)-5 b(alues)41 b(of)1806 5712 y(5)p eop %%Page: 6 6 6 5 bop 120 407 a Fv(this)39 b(parameter)g(and)h(w)m(e)h(therefore)f (allo)m(w)e(it)h(to)g(b)s(e)h(arbitrary)e(for)h(the)h(momen)m(t.)64 b(The)120 527 y(Thomas-F)-8 b(ermi)30 b(energy)k(is)e(de\014ned)i(b)m (y)1085 710 y Fp(E)1163 663 y Fn(TF)p Fm(;\014)1157 737 y Fo(R)p Fm(;)p Fo(Z)1354 710 y Fv(:=)330 b(inf)1485 784 y Fn(0)p Fo(\024)p Fm(\032)p Fo(2)p Fm(L)1706 765 y Fj(1)1741 784 y Fn(\()p Fg(R)1816 765 y Fj(3)1851 784 y Fn(\))p Fo(\\)p Fm(L)1973 764 y Fj(5)p Fi(=)p Fj(3)2070 784 y Fn(\()p Fg(R)2145 765 y Fj(3)2180 784 y Fn(\))2224 710 y Fq(E)2286 663 y Fn(TF)p Fm(;\014)2277 737 y Fo(R)p Fm(;)p Fo(Z)2449 710 y Fv(\()p Fp(\032)p Fv(\))792 b(\(11\))266 975 y(When)38 b Fp(\014)i Fv(=)34 b Fp(\014)815 990 y Fn(ph)n(ys)997 975 y Fv(w)m(e)k(shall)d(write)h Fp(E)1708 928 y Fn(TF)p Fm(;\014)1702 1002 y Fo(R)p Fm(;)p Fo(Z)1908 975 y Fv(simply)g(as)g Fp(E)2426 939 y Fn(TF)2420 1000 y Fo(R)p Fm(;)p Fo(Z)2561 975 y Fv(.)55 b(If)36 b Fp(M)45 b Fv(=)35 b(1)h(and)h Fp(Z)k Fv(=)34 b(1)120 1117 y(w)m(e)48 b(shall)e(write)h Fp(C)862 1070 y Fm(\014)855 1144 y Fn(TF)1003 1117 y Fv(instead)g(of)g Fq(j)p Fp(E)1586 1070 y Fn(TF)p Fm(;\014)1580 1144 y Fo(R)p Fm(;)p Fo(Z)1749 1117 y Fq(j)g Fv(and)g(if)f Fp(\014)59 b Fv(=)52 b Fp(\014)2429 1132 y Fn(ph)n(ys)2622 1117 y Fv(simply)46 b Fp(C)3019 1132 y Fn(TF)3167 1117 y Fv(\(see)i(also)120 1237 y(Theorems)33 b(1{3\).)266 1358 y(The)h(energy)f(satis\014es)h(the)f(scaling)e(prop)s (erties)1171 1541 y Fp(E)1249 1494 y Fn(TF)p Fm(;\014)1243 1568 y Fo(R)p Fm(;)p Fo(Z)1441 1541 y Fv(=)c Fp(\025E)1679 1494 y Fn(TF)p Fm(;\025\014)1673 1569 y(\025)p Fo(R)p Fm(;)p Fo(Z)1912 1541 y Fv(=)g Fp(\025)2072 1500 y Fn(7)2112 1541 y Fp(E)2190 1494 y Fn(TF)p Fm(;\014)2184 1573 y(\025)p Fo(R)p Fm(;\025)2346 1555 y Fh(\000)p Fj(3)2429 1573 y Fo(Z)3367 1541 y Fv(\(12\))120 1724 y(for)32 b(an)m(y)h Fp(\025)28 b(>)f Fv(0.)266 1844 y(It)34 b(is)f(kno)m(wn)h(that)g(there) g(is)f(a)g(unique)h(function)f Fp(\032)g Fv(whic)m(h)h(minimizes)d(the) j(functional)120 1965 y(\(11\).)43 b(This)33 b(function)f(ful\014ls)f (the)i(Thomas-F)-8 b(ermi)30 b(equations)1163 2108 y Fn(5)p 1163 2124 36 4 v 1163 2181 a(3)1209 2147 y Fp(\014)6 b(\032)1320 2111 y Fn(2)p Fm(=)p Fn(3)1430 2147 y Fv(\()p Fp(x)p Fv(\))83 b(=)g Fp(\036)p Fv(\()p Fp(x)p Fv(\))1372 2267 y Fp(\036)p Fv(\()p Fp(x)p Fv(\))g(=)g Fp(V)21 b Fv(\()p Fp(x)p Fv(\))i Fq(\000)2165 2228 y Fn(1)p 2144 2244 77 4 v 2144 2302 a Fo(j)p Fm(y)r Fo(j)2253 2267 y Fq(\003)f Fp(\032)p Fv(\()p Fp(x)p Fv(\))3367 2215 y(\(13\))120 2471 y(and)40 b(is)f(moreo)m(v)m(er)i(the)f(unique)g (non-negativ)m(e)g(solution)e Fp(\032)i Fv(of)f(\(13\).)65 b(This)40 b Fp(\032)g Fv(:=)g Fp(\032)3294 2435 y Fn(TF)3294 2496 y Fo(R)p Fm(;)p Fo(Z)3474 2471 y Fv(is)120 2591 y(called)h(the)h Fr(TF)g(density)g Fv(and)g(then)g(\010)1600 2555 y Fn(TF)1600 2617 y Fo(R)p Fm(;)p Fo(Z)1784 2591 y Fv(:=)h Fp(V)50 b Fq(\000)29 b(j)f(\001)g(j)2283 2555 y Fo(\000)p Fn(1)2405 2591 y Fq(\003)g Fp(\032)2533 2555 y Fn(TF)2533 2617 y Fo(R)p Fm(;)p Fo(Z)2715 2591 y Fv(is)42 b(the)g Fr(TF)g(p)-5 b(otential)120 2712 y Fv(men)m(tioned)32 b(ab)s(o)m(v)m(e.)266 2832 y(Our)40 b(aim)f(is)g(to)h(deriv)m(e)h(go)s (o)s(d)e(b)s(ounds)i(for)e(it)h(and)g(its)f(deriv)-5 b(ativ)m(es,)43 b(but)d(as)g(the)h(TF)120 2953 y(p)s(oten)m(tial)31 b(\010)602 2916 y Fn(TF)602 2978 y Fo(R)p Fm(;)p Fo(Z)775 2953 y Fv(satis\014es)i(the)g(next)h(scaling)d(relations:)914 3066 y Fk(\()1100 3161 y Fv(\010)1170 3125 y Fn(TF)1170 3186 y Fo(R)p Fm(;)p Fo(Z)1311 3161 y Fv(\()p Fp(x)p Fv(\))83 b(=)g Fp(Z)1758 3125 y Fn(4)p Fm(=)p Fn(3)1868 3161 y Fv(\010)1938 3125 y Fn(TF)1938 3198 y Fm(Z)1991 3178 y Fj(1)p Fi(=)p Fj(3)2088 3198 y Fo(R)p Fm(;Z)2221 3179 y Fh(\000)p Fj(1)2303 3198 y Fo(Z)2363 3161 y Fv(\()p Fp(Z)2475 3125 y Fn(1)p Fm(=)p Fn(3)2585 3161 y Fp(x)p Fv(\))1035 3304 y(\010)1105 3257 y Fn(TF)q Fm(;\025\014)1105 3331 y Fo(R)p Fm(;)p Fo(Z)1311 3304 y Fv(\()p Fp(x)p Fv(\))g(=)g Fp(\025)1741 3268 y Fo(\000)p Fn(1)1835 3304 y Fv(\010)1905 3268 y Fn(TF)1905 3335 y Fm(\025)1946 3316 y Fh(\000)p Fj(1)2030 3335 y Fo(R)p Fm(;)p Fo(Z)2170 3304 y Fv(\()p Fp(\025)2265 3268 y Fo(\000)p Fn(1)2359 3304 y Fp(x)p Fv(\))3367 3236 y(\(14\))120 3501 y(w)m(e)36 b(are)f(led)g(to)g(the)g(case)h(where)h(all)c(c)m(harges)j(are)f(of)g (the)h(same)f(order.)51 b(Moreo)m(v)m(er,)37 b(using)120 3622 y(the)j Fr(Thomas-F)-7 b(ermi)38 b(e)-5 b(quation)39 b Fv(next)h(giv)m(en)f(that)g(the)h(TF)f(p)s(oten)m(tial)e(\010)2895 3585 y Fn(TF)2895 3647 y Fo(R)p Fm(;)p Fo(Z)3075 3622 y Fv(ful\014ls)h(\(and)120 3742 y(whic)m(h)d(shall)e(allo)m(w)g(us)i (to)g(express)h(an)m(y)g(deriv)-5 b(ativ)m(e)34 b(of)g(\010)2294 3706 y Fn(TF)2294 3767 y Fo(R)p Fm(;)p Fo(Z)2469 3742 y Fv(in)g(terms)g(of)g(some)h(in)m(tegral)120 3862 y(op)s(erators)41 b(applied)f(to)g(itself)7 b(\))40 b(and)h Fr(T)-7 b(el)5 b(ler's)41 b(L)-5 b(emma)p Fv(,)42 b(whic)m(h)g(sa)m(ys)g(that)f(the)g (total)f(TF)120 3983 y(p)s(oten)m(tial)45 b(increases)j(ev)m(erywhere)h (as)e(long)f(as)h(an)m(y)g(of)f(its)g(c)m(harges)i(increases)g(w)m(e)g (can)120 4103 y(assume)33 b(in)f(the)h(follo)m(wing)d(that)i(all)f(c)m (harges)i(are)g(equal)f(to)h(one.)266 4223 y(No)m(w)g(w)m(e)h(are)f (going)e(to)h(in)m(tro)s(duce)h(the)g(fundamen)m(tal)e(estimate)h(of)g (this)g(section:)120 4395 y Ft(Lemma)37 b(1)h(\(The)f(size)f(of)i(TF)g (p)s(oten)m(tial)e(of)i(uniformly)d(separated)k(n)m(uclei\).)47 b Fr(L)-5 b(et)120 4516 y(us)47 b(b)-5 b(e)46 b(given)f(a)h(set)h Fq(f)p Fp(R)1039 4531 y Fm(j)1075 4516 y Fq(g)1125 4480 y Fm(M)1125 4540 y(j)t Fn(=1)1301 4516 y Fq(\032)i Fl(R)1493 4480 y Fn(3)1585 4516 y Fr(and)d(assume)g(that)h Fp(\016)2393 4531 y Fm(j)2478 4516 y Fq(\025)j Fv(1)17 b Fq(8)p Fp(j)52 b Fr(\(r)-5 b(e)g(c)g(al)5 b(l)46 b(that)g Fp(\016)3378 4531 y Fm(j)3464 4516 y Fv(=)120 4636 y(min)283 4651 y Fm(i)p Fn(:)p Fm(i)p Fo(6)p Fn(=)p Fm(j)458 4636 y Fq(j)p Fp(R)560 4651 y Fm(i)600 4636 y Fq(\000)11 b Fp(R)762 4651 y Fm(j)799 4636 y Fq(j)p Fr(\).)43 b(Then)29 b(ther)-5 b(e)30 b(exists)g(two)g(absolute)g(p)-5 b(ositive)29 b(c)-5 b(onstants)30 b Fv(0)d Fp(<)h(c)f Fq(\024)i Fp(C)34 b(<)120 4756 y Fq(1)25 b Fr(such)g(that)g(the)h(fol)5 b(lowing)23 b(estimate)i(holds)g(for)g(any)g Fp(x)j Fq(2)g Fl(R)2387 4720 y Fn(3)2458 4756 y Fr(\(b)-5 b(elow)24 b Fq(h)p Fp(x)p Fq(i)j Fv(:=)h(\(1)q(+)q Fq(j)p Fp(x)p Fq(j)3314 4720 y Fn(2)3352 4756 y Fv(\))3390 4720 y Fn(1)p Fm(=)p Fn(2)3500 4756 y Fr(\))859 4940 y Fp(c)17 b(\032)p Fv(\()p Fp(x)p Fv(\))1099 4899 y Fo(\000)p Fn(1)1193 4940 y Fq(h)p Fp(\032)p Fv(\()p Fp(x)p Fv(\))p Fq(i)1452 4899 y Fo(\000)p Fn(3)1574 4940 y Fq(\024)28 b Fv(\010)1749 4955 y Fo(R)1814 4940 y Fv(\()p Fp(x)p Fv(\))g Fq(\024)g Fp(C)c(\032)p Fv(\()p Fp(x)p Fv(\))2353 4899 y Fo(\000)p Fn(1)2448 4940 y Fq(h)p Fp(\032)p Fv(\()p Fp(x)p Fv(\))p Fq(i)2707 4899 y Fo(\000)p Fn(3)3367 4940 y Fv(\(15\))120 5123 y Fr(wher)-5 b(e)38 b Fv(\010)469 5138 y Fo(R)568 5123 y Fv(:=)c(\010)775 5087 y Fn(TF)775 5148 y Fo(R)p Fm(;)p Fo(Z)916 5123 y Fv(;)f Fp(Z)1043 5138 y Fn(1)1117 5123 y Fv(=)h Fp(:)17 b(:)g(:)34 b Fv(=)g Fp(Z)1553 5138 y Fm(M)1666 5123 y Fv(=)g(1)39 b Fr(and)f(the)g(distanc)-5 b(e)38 b(function)g Fp(\032)p Fv(\()p Fp(x)p Fv(\))h Fr(in)f(15)g(is)120 5243 y(given)c(by)1025 5427 y Fp(\032)p Fv(\()p Fp(x)p Fv(\))28 b(:=)g(min)n Fq(fj)p Fp(x)22 b Fq(\000)h Fp(R)1856 5442 y Fm(j)1893 5427 y Fq(j)k Fv(:)45 b Fp(j)33 b Fv(=)28 b(1)p Fp(;)17 b(:)g(:)g(:)32 b(;)17 b(M)10 b Fq(g)732 b Fv(\(16\))1806 5712 y(6)p eop %%Page: 7 7 7 6 bop 120 407 a Fr(Pr)-5 b(o)g(of.)48 b Fv(First)39 b(w)m(e)j(note)e(that)g(w)m(e)h(can)f(reform)m(ulate)f(Lemma)f(1)i(b)m (y)h(in)m(tro)s(ducing)e(the)h(so-)120 527 y(called)31 b Fr(V)-7 b(or)i(onoi)34 b(c)-5 b(el)5 b(ls)32 b Fq(f)p Fp(D)1109 542 y Fm(j)1146 527 y Fq(g)1196 491 y Fm(M)1196 552 y(j)t Fn(=1)1355 527 y Fv(asso)s(ciated)g(to)g(the)h(set)h Fq(R)f Fv(and)f(whic)m(h)h(are)g(giv)m(en)g(b)m(y)860 738 y Fp(D)941 753 y Fm(j)1006 738 y Fv(=)27 b Fq(f)p Fp(x)h Fv(:)g Fq(j)p Fp(x)22 b Fq(\000)h Fp(R)1576 753 y Fm(i)1604 738 y Fq(j)28 b(\025)g(j)p Fp(x)22 b Fq(\000)h Fp(R)2044 753 y Fm(j)2080 738 y Fq(j)17 b(8)g Fp(i)28 b Fv(=)f(1)p Fp(;)17 b(:)g(:)g(:)32 b(;)17 b(M)10 b Fq(g)568 b Fv(\(17\))120 938 y(Then)34 b(w)m(e)f(ha)m(v)m(e)h Fp(x)28 b Fq(2)h Fp(D)1002 953 y Fm(j)1066 938 y Fq(,)e Fp(\032)p Fv(\()p Fp(x)p Fv(\))h(=)g Fq(j)p Fp(x)22 b Fq(\000)h Fp(R)1785 953 y Fm(j)1822 938 y Fq(j)p Fv(,)32 b(so)h(the)g(pro)s(of)e(of)i(Lemma)e(1)h(is)g(reduced)i(to)120 1059 y(pro)m(v)m(e)g(the)f(next)g(estimate)f(for)g(an)m(y)h(V)-8 b(oronoi)31 b(cell:)263 1259 y Fp(c)17 b Fq(j)p Fp(x)22 b Fq(\000)h Fp(R)601 1274 y Fm(j)637 1259 y Fq(j)665 1218 y Fo(\000)p Fn(1)759 1259 y Fq(hj)p Fp(x)f Fq(\000)h Fp(R)1077 1274 y Fm(j)1114 1259 y Fq(ji)1181 1218 y Fo(\000)p Fn(3)1302 1259 y Fq(\024)28 b Fv(\010)1477 1274 y Fo(R)1542 1259 y Fv(\()p Fp(x)p Fv(\))g Fq(\024)g Fp(C)c Fq(j)p Fp(x)e Fq(\000)h Fp(R)2179 1274 y Fm(j)2215 1259 y Fq(j)2243 1218 y Fo(\000)p Fn(1)2337 1259 y Fq(hj)p Fp(x)f Fq(\000)h Fp(R)2655 1274 y Fm(j)2692 1259 y Fq(ji)2759 1218 y Fo(\000)p Fn(3)2853 1259 y Fv(;)49 b Fp(x)28 b Fq(2)g Fp(D)3187 1274 y Fm(j)3367 1259 y Fv(\(18\))120 1460 y(W)-8 b(e)33 b(ha)m(v)m(e)h(t)m(w)m(o)f(cases:)266 1580 y(a\))g(Assume)h Fp(x)29 b Fq(2)g Fp(B)1001 1596 y Fn(1)p Fm(=)p Fn(2)1111 1580 y Fv(\()p Fp(R)1223 1595 y Fm(j)1260 1580 y Fv(\)\()p Fq(\032)g Fp(D)1523 1595 y Fm(j)1560 1580 y Fv(\).)45 b(Then)34 b Fq(j)p Fp(x)23 b Fq(\000)g Fp(R)2205 1595 y Fm(i)2233 1580 y Fq(j)28 b(\025)i Fv(1)p Fp(=)p Fv(2)p Fq(j)p Fp(R)2645 1595 y Fm(i)2695 1580 y Fq(\000)23 b Fp(R)2869 1595 y Fm(j)2905 1580 y Fq(j)33 b Fv(for)g(an)m(y)g Fp(i)c Fq(6)p Fv(=)g Fp(j)6 b Fv(.)120 1701 y(The)34 b(TF)e(p)s(oten)m(tial)f(satis\014es)884 1989 y(\010)954 1947 y Fn(TF)954 2013 y Fo(R)p Fm(;)p Fo(Z)1094 1989 y Fv(\()p Fp(x)p Fv(\))d Fq(\024)1393 1864 y Fm(M)1359 1894 y Fk(X)1369 2104 y Fm(j)t Fn(=1)1519 1989 y Fv(\010)1589 1947 y Fn(TF)1589 2013 y Fm(R)1642 2023 y Fi(j)1675 2013 y Fm(;Z)1743 2023 y Fi(j)1779 1989 y Fv(\()p Fp(x)p Fv(\))g(=)2076 1864 y Fm(M)2042 1894 y Fk(X)2053 2104 y Fm(j)t Fn(=1)2202 1989 y Fv(\010)2272 1947 y Fn(TF)2272 2013 y(0)p Fm(;Z)2375 2023 y Fi(j)2412 1989 y Fv(\()p Fp(x)23 b Fq(\000)f Fp(R)2701 2004 y Fm(j)2738 1989 y Fv(\))591 b(\(19\))120 2297 y(and)33 b(for)f Fp(')27 b Fv(=)h(\010)724 2261 y Fn(TF)724 2322 y(0)p Fm(;)p Fn(1)858 2297 y Fv(w)m(e)34 b(ha)m(v)m(e)f(the)g(2-sided)g (estimate)1130 2507 y Fp(c)p Fq(j)p Fp(x)p Fq(jh)p Fp(x)p Fq(i)1416 2466 y Fo(\000)p Fn(3)1538 2507 y Fq(\024)28 b Fp(')p Fv(\()p Fp(x)p Fv(\))g Fq(\024)g Fp(C)7 b Fq(j)p Fp(x)p Fq(jh)p Fp(x)p Fq(i)2292 2466 y Fo(\000)p Fn(3)2419 2507 y Fq(8)p Fp(x)838 b Fv(\(20\))120 2708 y(where)33 b(in)f(\(20\))f Fp(c;)17 b(C)39 b Fv(are)32 b(some)g(p)s(ositiv)m(e)f (absolute)h(constan)m(ts.)45 b(Therefore,)33 b(if)e Fq(j)p Fp(x)22 b Fq(\000)f Fp(R)3370 2723 y Fm(j)3407 2708 y Fq(j)28 b(\024)120 2828 y Fv(1)p Fp(=)p Fv(2,)k(from)f(\(19\),)h (\(20\))g(w)m(e)i(get)1014 3122 y(\010\()p Fp(x)p Fv(\))28 b Fq(\024)g Fp(')p Fv(\()p Fp(x)23 b Fq(\000)g Fp(R)1702 3137 y Fm(j)1738 3122 y Fv(\))f(+)g Fp(C)2030 2997 y Fm(M)1995 3027 y Fk(X)1990 3239 y Fm(i)p Fn(:)p Fm(i)p Fo(6)p Fn(=)p Fm(j)2161 3122 y Fq(j)p Fp(R)2263 3137 y Fm(i)2314 3122 y Fq(\000)g Fp(R)2487 3137 y Fm(j)2524 3122 y Fq(j)2552 3081 y Fo(\000)p Fn(4)3367 3122 y Fv(\(21\))120 3442 y(Let)36 b(us)g(call)630 3417 y(~)608 3442 y Fp(R)682 3457 y Fm(i)744 3442 y Fv(=)d Fp(R)927 3457 y Fm(i)980 3442 y Fq(\000)24 b Fp(R)1155 3457 y Fm(j)1228 3442 y Fv(for)35 b Fp(i)f Fq(6)p Fv(=)f Fp(j)6 b Fv(.)53 b(W)-8 b(e)36 b(can)g(assume)g(1)d Fq(\024)h(j)2616 3417 y Fv(~)2596 3442 y Fp(R)2670 3457 y Fn(1)2709 3442 y Fq(j)f(\024)g Fp(:)17 b(:)g(:)33 b Fq(\024)h(j)3187 3417 y Fv(~)3167 3442 y Fp(R)3241 3457 y Fm(M)3319 3442 y Fq(j)i Fv(and)120 3572 y(since)g Fq(j)411 3547 y Fv(~)390 3572 y Fp(R)464 3587 y Fm(k)531 3572 y Fq(\000)653 3547 y Fv(~)632 3572 y Fp(R)706 3587 y Fm(l)732 3572 y Fq(j)d(\025)g Fv(1)i(the)h(measure)f (of)g(the)h(set)1980 3497 y Fk(S)2063 3524 y Fm(N)2063 3601 y(i)p Fn(:)p Fm(i)p Fo(6)p Fn(=)p Fm(j)2239 3572 y Fp(B)2313 3587 y Fn(1)p Fm(=)p Fn(2)2423 3572 y Fv(\()2482 3547 y(~)2461 3572 y Fp(R)2535 3587 y Fm(i)2563 3572 y Fv(\))g(equals)f Fp(N)10 b Fq(j)p Fp(B)3125 3587 y Fn(1)p Fm(=)p Fn(2)3236 3572 y Fv(\(0\))p Fq(j)34 b Fv(for)120 3692 y(an)m(y)f(1)28 b Fq(\024)g Fp(N)38 b Fq(\024)28 b Fp(M)10 b Fv(.)44 b(But)1286 3856 y Fm(N)1262 3886 y Fk([)1240 4098 y Fm(i)p Fn(:)p Fm(i)p Fo(6)p Fn(=)p Fm(j)1411 3980 y Fp(B)1485 3996 y Fn(1)p Fm(=)p Fn(2)1595 3980 y Fv(\()1655 3955 y(~)1633 3980 y Fp(R)1707 3995 y Fm(i)1736 3980 y Fv(\))27 b Fq(\032)i Fp(B)1981 4007 y Fo(j)2015 3990 y Fn(~)2001 4007 y Fm(R)2054 4018 y Fi(N)2111 4007 y Fo(j)p Fn(+1)p Fm(=)p Fn(2)2296 3980 y Fv(\(0\))946 b(\(22\))120 4278 y(and)33 b(therefore)g(w)m(e)g(get) 1146 4531 y Fq(j)1195 4506 y Fv(~)1174 4531 y Fp(R)1248 4546 y Fm(N)1315 4531 y Fq(j)27 b(\025)i Fv(max)1674 4390 y Fk(\032)1748 4531 y Fv(1)p Fp(;)1851 4463 y Fv(1)p 1851 4508 49 4 v 1851 4599 a(2)1910 4531 y Fp(N)1998 4490 y Fn(1)p Fm(=)p Fn(3)2108 4390 y Fk(\033)2210 4531 y Fq(\030)g Fp(N)2404 4490 y Fn(1)p Fm(=)p Fn(3)3367 4531 y Fv(\(23\))120 4789 y(Using)j(\(20\),)g(\(21\),)g(\(23\))g(w)m(e) i(get)e(for)g Fq(j)p Fp(x)23 b Fq(\000)f Fp(R)1800 4804 y Fm(j)1837 4789 y Fq(j)27 b(\024)i Fv(1)p Fp(=)p Fv(2)727 5085 y(\010\()p Fp(x)p Fv(\))84 b Fq(\024)f Fp(')p Fv(\()p Fp(x)23 b Fq(\000)f Fp(R)1525 5100 y Fm(j)1562 5085 y Fv(\))g(+)g Fp(C)1814 4961 y Fm(M)7 b Fo(\000)p Fn(1)1824 4990 y Fk(X)1830 5200 y Fm(n)p Fn(=1)1995 5085 y Fp(n)2053 5044 y Fo(\000)p Fn(4)p Fm(=)p Fn(3)1012 5318 y Fq(\024)83 b Fp(')p Fv(\()p Fp(x)23 b Fq(\000)f Fp(R)1525 5333 y Fm(j)1562 5318 y Fv(\))g(+)g Fp(C)1577 b Fv(\(24\))1012 5463 y Fq(\024)83 b Fp(C)7 b(')p Fv(\()p Fp(x)23 b Fq(\000)f Fp(R)1602 5478 y Fm(j)1639 5463 y Fv(\);)49 b Fp(')p Fv(\()p Fp(x)23 b Fq(\000)f Fp(R)2106 5478 y Fm(j)2143 5463 y Fv(\))28 b Fq(\030)g(j)p Fp(x)22 b Fq(\000)h Fp(R)2593 5478 y Fm(j)2629 5463 y Fq(j)2657 5422 y Fo(\000)p Fn(1)2779 5463 y Fq(\025)28 b Fv(1)1806 5712 y(7)p eop %%Page: 8 8 8 7 bop 120 407 a Fv(W)-8 b(e)34 b(consider)h(no)m(w)f(the)g(case)h Fp(x)c Fq(2)f Fp(D)1515 422 y Fm(j)1574 407 y Fq(n)23 b Fp(B)1721 422 y Fn(1)p Fm(=)p Fn(2)1831 407 y Fv(\()p Fp(R)1943 422 y Fm(j)1980 407 y Fv(\))34 b(and)g(w)m(e)h(note)f(that)g (\011\()p Fp(x)p Fv(\))c(:=)g(144)o Fq(j)p Fp(x)p Fq(j)3446 371 y Fo(\000)p Fn(4)120 527 y Fv(is)f(a)h(solution)e(\(the)i Fr(Sommerfeld)h(solution)p Fv(\))e(of)h(the)g(PDE)g(\001\011)e(=)f (\011)2671 491 y Fn(3)p Fm(=)p Fn(2)2811 527 y Fv(in)i Fl(R)2988 491 y Fn(3)3050 527 y Fq(n)16 b(f)p Fv(0)p Fq(g)p Fv(.)42 b(Next)120 648 y(w)m(e)34 b(consider)750 868 y(\011)826 883 y Fm(\025)871 868 y Fv(\()p Fp(x)p Fv(\))83 b(:=)g Fp(\025)p Fq(j)p Fp(x)22 b Fq(\000)h Fp(R)1607 883 y Fm(j)1644 868 y Fq(j)1672 826 y Fo(\000)p Fn(4)945 1013 y Fp(\025)83 b Fv(:=)g(max)16 b Fq(f)p Fv(144)p Fp(;)h Fv(sup)p Fq(f)p Fv(\010\()p Fp(x)p Fv(\))28 b(:)g Fq(j)p Fp(x)22 b Fq(\000)h Fp(R)2469 1028 y Fm(j)2506 1013 y Fq(j)k Fv(=)g(1)p Fp(=)p Fv(2)p Fq(gg)456 b Fv(\(25\))120 1233 y(So)38 b(w)m(e)i(ha)m(v)m(e)f(\011)717 1248 y Fm(\025)800 1233 y Fq(\025)f Fv(\010)h(if)e Fq(j)p Fp(x)26 b Fq(\000)h Fp(R)1406 1248 y Fm(j)1443 1233 y Fq(j)37 b Fv(=)g(1)p Fp(=)p Fv(2)h(and)g(consider)h(\000)f(:=)f(\011)2702 1248 y Fm(\025)2773 1233 y Fq(\000)27 b Fv(\010.)61 b(W)-8 b(e)39 b(w)m(an)m(t)g(to)120 1353 y(sho)m(w)33 b(that)e(\000)d Fq(\025)g Fv(0)k(on)f Fp(D)1061 1368 y Fm(j)1118 1353 y Fq(n)20 b Fp(B)1262 1369 y Fn(1)p Fm(=)p Fn(2)1373 1353 y Fv(\()p Fp(R)1485 1368 y Fm(j)1521 1353 y Fv(\).)43 b(Let)32 b(\012)c(=)g Fq(f)p Fp(x)g Fq(2)g Fv(in)m(t\()p Fp(D)2468 1368 y Fm(j)2504 1353 y Fv(\))20 b Fq(n)p 2633 1273 80 4 v 21 w Fp(B)2712 1369 y Fn(1)p Fm(=)p Fn(2)2822 1353 y Fv(\()p Fp(R)2934 1368 y Fm(j)2970 1353 y Fv(\))28 b(:)g(\000\()p Fp(x)p Fv(\))g Fp(<)f Fv(0)p Fq(g)p Fv(.)120 1474 y(Since)35 b(there)g(are)g(no)f(singularities)e(of)i(the)i(p)s (oten)m(tial)c(on)j(the)g(set)g(in)m(t\()p Fp(D)2870 1489 y Fm(j)2906 1474 y Fv(\))24 b Fq(n)p 3041 1394 V 23 w Fp(B)3120 1489 y Fn(1)p Fm(=)p Fn(2)3230 1474 y Fv(\()p Fp(R)3342 1489 y Fm(j)3379 1474 y Fv(\),)35 b(\000)120 1594 y(is)d(con)m(tin)m(uous,)h(\012)g(is)g(op)s(en)f(and)h(\000)28 b(=)f(0)32 b(on)h Fp(@)5 b Fv(\012.)45 b(F)-8 b(or)31 b Fp(x)e Fq(2)f Fv(\012)33 b(w)m(e)g(\014nd)694 1814 y(\001\000\()p Fp(x)p Fv(\))85 b(=)f(12)p Fp(\025)p Fq(j)p Fp(x)21 b Fq(\000)i Fp(R)1645 1829 y Fm(j)1682 1814 y Fq(j)1710 1773 y Fo(\000)p Fn(6)1826 1814 y Fq(\000)f Fv(\010\()p Fp(x)p Fv(\))2126 1773 y Fn(3)p Fm(=)p Fn(2)1051 1959 y Fq(\024)84 b Fp(\025)1269 1918 y Fn(3)p Fm(=)p Fn(2)1378 1959 y Fq(j)p Fp(x)23 b Fq(\000)f Fp(R)1657 1974 y Fm(j)1694 1959 y Fq(j)1722 1918 y Fo(\000)p Fn(6)1838 1959 y Fq(\000)h Fv(\010\()p Fp(x)p Fv(\))2139 1918 y Fn(3)p Fm(=)p Fn(2)2282 1959 y Fv(\()33 b(since)g Fp(\025)27 b Fq(\025)h Fv(144\))1052 2105 y(=)84 b(\010)1282 2120 y Fm(\025)1327 2105 y Fv(\()p Fp(x)p Fv(\))1458 2063 y Fn(3)p Fm(=)p Fn(2)1591 2105 y Fq(\000)22 b Fv(\010\()p Fp(x)p Fv(\))1891 2063 y Fn(3)p Fm(=)p Fn(2)3367 2105 y Fv(\(26\))1052 2250 y Fp(<)84 b Fv(0)32 b(\(since)h(\000\()p Fp(x)p Fv(\))28 b Fp(<)f Fv(0\))120 2470 y(meaning)33 b(that)h(\000)g(is)g Fr(sup)-5 b(erharmonic)33 b Fv(in)h(\012,)h(whic)m (h)g(implies)d(that)i(\000)g(tak)m(es)h(its)f(minim)m(um)120 2590 y(on)c Fp(@)5 b Fv(\012)18 b Fq([)g(f1g)p Fv(,)30 b(but)g(as)41 b(\000\()p Fp(x)p Fv(\))p Fq(j)1261 2619 y Fm(@)t Fn(\012)p Fo([f1g)1574 2590 y Fv(=)27 b(0,)j(\012)f(=)e Fq(;)j Fv(and)g(\000)e Fq(\025)g Fv(0,)i(whic)m(h)h(pro)m(v)m(es)h(the) e(upp)s(er)120 2711 y(b)s(ound)h(in)f(\(15\).)42 b(The)32 b(lo)m(w)m(er)e(b)s(ound)h(is)f(an)h(inmediate)e(consequence)34 b(of)c(T)-8 b(eller's)30 b(Lemma,)120 2831 y(whic)m(h)j(in)f (particular)f(implies)f(that)i(\010)1575 2846 y Fo(R)1640 2831 y Fv(\()p Fp(x)p Fv(\))c Fq(\025)g Fp(')p Fv(\()p Fp(x)23 b Fq(\000)f Fp(R)2257 2846 y Fm(j)2294 2831 y Fv(\).)p 3469 2831 4 66 v 3473 2769 59 4 v 3473 2831 V 3531 2831 4 66 v 364 3059 a Ft(Remark:)47 b Fv(It)34 b(should)h(b)s(e)f(noted)h(that)g(the)f(lo)m(w)m(er)h(b)s(ound)g(in)f (\(15\))f(holds)i(for)364 3180 y Fr(any)28 b Fv(n)m(uclear)g (con\014guration,)h(and)f(this)g(is)f(again)g(just)i(a)e(simple)g (consequence)364 3300 y(of)e(T)-8 b(eller's)25 b(Lemma.)39 b(So)26 b(this)f(Lemma)f(pro)m(vides)i(a)f(condition)f(to)h(ensure)h (that)364 3421 y(w)m(e)31 b(ha)m(v)m(e)h(an)e(upp)s(er)h(b)s(ound)g (for)f(the)h(Thomas-F)-8 b(ermi)28 b(p)s(oten)m(tial)g(of)i(the)h(same) 364 3541 y(kind)h(that)h(the)g(lo)m(w)m(er)f(one.)120 3769 y(Our)i(next)g(Lemma)e(essen)m(tially)h(claims)f(that)h Fr(arbitr)-5 b(ary)34 b Fv(n)m(uclear)g(con\014gurations)f(b)s(eha)m(v) m(e)120 3890 y(essen)m(tially)f(as)g(if)f(their)h(n)m(uclei)f(where)i (uniformly)d(separated)j(in)f(those)h(p)s(oin)m(ts)e(whic)m(h)i(are)120 4010 y(separated)h(b)m(y)f(distance)g(one)g(or)f(more)g(to)g(the)h(n)m (uclear)g(con\014guration:)120 4213 y Ft(Lemma)k(2)h(\(The)f(size)f(of) i(TF)g(p)s(oten)m(tial)e(a)m(w)m(a)m(y)i(from)f(the)g(n)m(uclei\).)47 b Fr(L)-5 b(et)26 b(us)f(b)-5 b(e)24 b(given)120 4334 y(an)h Ft(arbitrary)h Fr(c)-5 b(on\014gur)g(ation)25 b Fq(R)j(\032)g Fl(R)1577 4298 y Fn(3)1648 4334 y Fr(and)d(assume)g Fp(\032)p Fv(\()p Fp(x)p Fv(\))k Fq(\025)f Fv(1)p Fp(=)p Fv(2)d Fr(\()p Fp(\032)h Fr(is)g(as)f(in)g(L)-5 b(emma)25 b(1\).)120 4454 y(Then)34 b(the)h(estimate)f(\(15\))h(stil)5 b(l)34 b(holds)120 4658 y(Pr)-5 b(o)g(of.)48 b Fv(Let)c(us)h(assume)f Fp(\016)50 b Fv(=)d(min)n Fq(fj)p Fp(R)1626 4673 y Fm(i)1684 4658 y Fq(\000)30 b Fp(R)1865 4673 y Fm(j)1902 4658 y Fq(j)46 b Fv(:)h Fp(i;)17 b(j)53 b Fv(:)46 b(1)p Fp(;)17 b(:)g(:)g(:)33 b(;)17 b(M)10 b Fv(;)33 b Fp(i)47 b Fq(6)p Fv(=)g Fp(j)6 b Fq(g)46 b(\024)h Fv(1,)g(since)120 4778 y(otherwise)33 b(b)m(y)h(Lemma)d(1)h(there)h(w)m(ould)g(b)s(e)g (nothing)e(to)i(pro)m(v)m(e.)44 b(Then)34 b(w)m(e)g(ha)m(v)m(e)630 4998 y(\010)700 5013 y Fo(R)765 4998 y Fv(\()p Fp(x)p Fv(\))84 b(=)g Fp(\016)1187 4957 y Fo(\000)p Fn(4)1281 4998 y Fv(\010)1351 4957 y Fn(TF)1351 5024 y Fm(\016)1384 5005 y Fh(\000)p Fj(1)1468 5024 y Fo(R)p Fm(;)p Fn(\()p Fm(\016)1608 5005 y Fj(3)1643 5024 y Fm(;:::)11 b(;\016)1787 5005 y Fj(3)1821 5024 y Fn(\))1853 4998 y Fv(\()p Fp(\016)1938 4957 y Fo(\000)p Fn(1)2032 4998 y Fp(x)p Fv(\))33 b(\(b)m(y)h(TF)e (scaling)g(\(14\)\))979 5160 y Fq(\024)84 b Fp(\016)1187 5119 y Fo(\000)p Fn(4)1281 5160 y Fv(\010)1351 5119 y Fn(TF)1351 5186 y Fm(\016)1384 5167 y Fh(\000)p Fj(1)1468 5186 y Fo(R)p Fm(;)p Fn(\(1)p Fm(;:::)11 b(;)p Fn(1\))1787 5160 y Fv(\()p Fp(\016)1872 5119 y Fo(\000)p Fn(1)1966 5160 y Fp(x)p Fv(\))33 b(\(b)m(y)h(T)-8 b(eller's)32 b(Lemma\))980 5320 y(=)84 b Fp(\016)1187 5279 y Fo(\000)p Fn(4)1281 5320 y Fv(\010)1351 5336 y Fm(\016)1384 5317 y Fh(\000)p Fj(1)1468 5336 y Fo(R)1532 5320 y Fv(\()p Fp(\016)1617 5279 y Fo(\000)p Fn(1)1711 5320 y Fp(x)p Fv(\))1563 b(\(27\))1806 5712 y(8)p eop %%Page: 9 9 9 8 bop 120 407 a Fv(Note)36 b(that)f Fp(x)f Fq(2)f Fp(D)842 422 y Fm(R)933 407 y Fq(,)f Fp(\016)1112 371 y Fo(\000)p Fn(1)1207 407 y Fp(x)h Fq(2)g Fp(D)1475 424 y Fm(\016)1508 405 y Fh(\000)p Fj(1)1592 424 y Fm(R)1685 407 y Fv(and)j(the)g(new)h (con\014guration)d Fp(\016)2893 371 y Fo(\000)p Fn(1)2988 407 y Fq(R)i Fv(is)f(suitable)120 527 y(to)d(apply)g(Lemma)g(1)g(to)g (it.)43 b(So)32 b(w)m(e)i(get)692 740 y Fp(\016)739 699 y Fo(\000)p Fn(4)833 740 y Fv(\010)903 757 y Fm(\016)936 738 y Fh(\000)p Fj(1)1020 757 y Fo(R)1084 740 y Fv(\()p Fp(\016)1169 699 y Fo(\000)p Fn(1)1263 740 y Fp(x)p Fv(\))28 b Fq(\024)h Fp(C)7 b(\016)1614 699 y Fo(\000)p Fn(4)1708 740 y Fq(j)p Fp(\016)1783 699 y Fo(\000)p Fn(1)1877 740 y Fv(\()p Fp(x)22 b Fq(\000)h Fp(R)q Fv(\))p Fq(j)2233 699 y Fo(\000)p Fn(1)2327 740 y Fq(h)p Fp(\016)2413 699 y Fo(\000)p Fn(1)2507 740 y Fv(\()p Fp(x)f Fq(\000)h Fp(R)q Fv(\))p Fq(i)2874 699 y Fo(\000)p Fn(3)3367 740 y Fv(\(28\))120 953 y(If)k Fp(\025)h Fv(=)g Fp(\016)448 917 y Fo(\000)p Fn(1)569 953 y Fq(\025)h Fv(1)e(and)g Fq(j)p Fp(x)p Fq(j)h(\025)g Fv(1)p Fp(=)p Fv(2,)g(w)m(e)g(ha)m(v)m(e)h Fp(\026\025)p Fq(h)p Fp(x)p Fq(i)e(\024)h(h)p Fp(\025x)p Fq(i)g(\024)g Fp(\025)p Fq(h)p Fp(x)p Fq(i)f Fv(for)g(some)h(\014xed)g Fp(\026)g(>)f Fv(0,)120 1073 y(and)33 b(from)e(\(28\))h(w)m(e)i(get)e (then)980 1286 y Fp(\016)1027 1245 y Fo(\000)p Fn(4)1121 1286 y Fv(\010)1191 1303 y Fm(\016)1224 1284 y Fh(\000)p Fj(1)1308 1303 y Fo(R)1372 1286 y Fv(\()p Fp(\016)1457 1245 y Fo(\000)p Fn(1)1551 1286 y Fp(x)p Fv(\))c Fq(\024)g Fp(C)7 b Fq(j)p Fp(x)23 b Fq(\000)f Fp(R)q Fq(j)2162 1245 y Fo(\000)p Fn(1)2256 1286 y Fq(h)p Fp(x)h Fq(\000)f Fp(R)q Fq(i)2586 1245 y Fo(\000)p Fn(3)3367 1286 y Fv(\(29\))120 1499 y(whic)m(h)33 b(concludes)h(the)f(pro)s(of.)p 3469 1499 4 66 v 3473 1437 59 4 v 3473 1499 V 3531 1499 4 66 v 120 1692 a(Next)43 b(result)e(is)h(the)g(main)e(result)i(of)f (this)h(section)g(and)g(explains)f(that)h(w)m(e)h(can)f(obtain)120 1813 y(b)s(ounds)f(for)e(the)h(deriv)-5 b(ativ)m(es)40 b(of)g(the)g(TF)g(p)s(oten)m(tial)e(from)h(the)h(b)s(ounds)h(for)e(the) h(size)g(of)120 1933 y(the)33 b(p)s(oten)m(tial)e(just)i(deriv)m(ed.)44 b(It)33 b(shall)e(b)s(e)i(essen)m(tial)g(in)e(section)i(4)f(in)g(order) h(to)f(apply)h(the)120 2053 y(metho)s(ds)f(of)h Fr(Multisc)-5 b(ale)35 b(A)n(nalysis)120 2251 y Ft(Theorem)i(4)g(\(Size)g(of)g(deriv) -6 b(ativ)m(es)37 b(of)g(TF)h(p)s(oten)m(tial\).)47 b Fr(L)-5 b(et)44 b(us)g(b)-5 b(e)44 b(given)f(char)-5 b(ges)120 2371 y Fv(\()p Fp(Z)225 2386 y Fn(1)264 2371 y Fp(;)17 b(:)g(:)g(:)33 b(;)17 b(Z)567 2386 y Fm(M)645 2371 y Fv(\))34 b Fr(such)g(that)h Fv(1)27 b Fq(\024)h Fp(Z)1385 2386 y Fm(i)1441 2371 y Fq(\024)g Fp(A)34 b Fr(for)g(some)g(\014xe)-5 b(d)33 b(p)-5 b(ositive)34 b Fp(A)g Fr(and)g(assume)f(either:)266 2491 y(i.)44 b(Ther)-5 b(e)30 b(is)g(given)g(a)h(nucle)-5 b(ar)31 b(c)-5 b(on\014gur)g(ation) 30 b Fq(R)h Fr(such)g(that)g Fp(\016)2567 2506 y Fm(i)2623 2491 y Fq(\025)d Fv(1)j Fq(8)17 b Fp(i)p Fr(.)43 b(Then)30 b(for)h(any)120 2612 y(multiindex)36 b Fp(\013)31 b Fq(2)g Fl(N)856 2576 y Fn(3)938 2612 y Fr(the)37 b(next)f(b)-5 b(ound)36 b(holds)g(for)g(any)g Fp(x)31 b Fq(2)g Fl(R)2443 2576 y Fn(3)2525 2612 y Fr(for)36 b(some)g(\014nite)g(c)-5 b(onstant)120 2732 y Fp(C)42 b Fr(only)35 b(dep)-5 b(ending)33 b(on)h(the)h(numb)-5 b(er)35 b(of)f(derivatives)g Fq(j)p Fp(\013)q Fq(j)g Fr(and)g Fp(A)p Fr(:)1089 2945 y Fq(j)p Fp(@)1173 2904 y Fm(\013)1223 2945 y Fv(\010)1293 2904 y Fn(TF)1293 2970 y Fo(R)p Fm(;)p Fo(Z)1433 2945 y Fv(\()p Fp(x)p Fv(\))p Fq(j)28 b(\024)g Fp(C)7 b(\032)p Fv(\()p Fp(x)p Fv(\))1983 2904 y Fo(\000)p Fn(1)p Fo(\000j)p Fm(\013)p Fo(j)2218 2945 y Fq(h)p Fp(\032)p Fv(\()p Fp(x)p Fv(\))p Fq(i)2477 2904 y Fo(\000)p Fn(3)3367 2945 y Fv(\(30\))266 3158 y Fr(ii.)42 b(Ther)-5 b(e)27 b(is)g(given)f(an)h Ft(arbitrary)g Fr(nucle)-5 b(ar)27 b(c)-5 b(on\014gur)g(ation)26 b Fq(R)i Fr(and)e(assume)h(mor)-5 b(e)g(over)120 3278 y Fp(\032)p Fv(\()p Fp(x)p Fv(\))28 b Fq(\025)g Fv(1)p Fp(=)p Fv(2)p Fr(.)44 b(Then)35 b(the)f(b)-5 b(ound)35 b(\(30\))f(stil)5 b(l)35 b(holds.)120 3476 y(Pr)-5 b(o)g(of.)48 b Fv(W)-8 b(e)41 b(are)e(going)g(to)g(giv)m(e)h(a)f(pro)s(of)g(b)m(y)i (induction)e(on)g Fq(j)p Fp(\013)q Fq(j)p Fv(.)64 b(By)41 b(lemmas)d(1)h(and)h(2,)120 3596 y(Theorem)i(4)g(is)g(v)-5 b(alid)40 b(if)h Fp(\013)j Fv(=)g(0)e(and)g(all)e(c)m(harges)j(are)f (equal)g(to)g(one.)72 b(By)43 b(the)f(scaling)120 3717 y(relation)32 b(\(12\))h(and)h(T)-8 b(eller's)33 b(Lemma,)g(lemmas)f(1) h(and)h(2)g(inmediately)d(generalize)i(to)h(the)120 3837 y(case)j Fp(Z)j Fq(\024)34 b Fp(Z)615 3852 y Fm(j)684 3837 y Fq(\024)g Fp(AZ)43 b Fq(8)17 b Fp(j)42 b Fv(and)36 b(the)g(b)s(ounds)h(pro)m(vided)f(b)m(y)h(these)g(lemmas)d(are)i(the)g (same)120 3957 y(except)e(the)f(constan)m(ts)h(no)m(w)f(dep)s(end)h(on) f Fp(A)p Fv(.)266 4078 y(1.-)38 b(Starting)f(with)h Fq(j)p Fp(\013)q Fq(j)f Fv(=)g(0,)j(\014x)f Fp(x)1610 4093 y Fn(0)1687 4078 y Fq(2)f Fp(D)1872 4093 y Fm(j)1935 4078 y Fq(n)25 b Fp(B)2084 4093 y Fn(1)p Fm(=)p Fn(4)2195 4078 y Fv(\()p Fp(R)2307 4093 y Fm(j)2343 4078 y Fv(\).)61 b(W)-8 b(e)39 b(call)d Fp(D)2908 4093 y Fm(j)2982 4078 y Fv(=)i Fp(D)3177 4093 y Fm(R)3230 4103 y Fi(j)3267 4078 y Fv(.)60 b(Pic)m(k)120 4198 y Fp(\020)35 b Fq(2)28 b Fp(C)369 4162 y Fo(1)362 4223 y Fm(c)444 4198 y Fv(\()p Fl(R)547 4162 y Fn(3)593 4198 y Fv(\))k(suc)m(h)i(that)1375 4329 y Fk(\032)1491 4408 y Fp(\020)h Fv(=)27 b(1)p Fp(;)83 b Fq(j)p Fp(y)t Fq(j)27 b(\024)h Fv(1)p Fp(=)p Fv(2)1491 4528 y Fp(\020)35 b Fv(=)27 b(0)p Fp(;)83 b Fq(j)p Fp(y)t Fq(j)27 b(\025)h Fv(1)p Fp(=)p Fv(4)3367 4469 y(\(31\))120 4740 y(and)38 b(assume)h(\(30\))f(is)g(already)g(pro)m(v)m(en)h(for)f (an)m(y)h(m)m(ultiindex)e Fp(\013)i Fv(suc)m(h)g(that)f Fq(j)p Fp(\013)q Fq(j)f(\024)h Fp(N)10 b Fv(.)61 b(If)120 4860 y Fq(j)p Fp(\013)211 4824 y Fo(0)234 4860 y Fq(j)41 b Fv(=)h Fp(N)d Fv(+)27 b(1,)43 b Fp(@)816 4824 y Fm(\013)861 4801 y Fh(0)931 4860 y Fv(=)f Fp(@)1105 4824 y Fm(\013)1155 4860 y Fp(@)1211 4824 y Fm(j)1290 4860 y Fv(for)e(some)h Fp(j)48 b Fv(=)42 b(1)p Fp(;)17 b Fv(2)p Fp(;)g Fv(3)40 b(and)h Fq(j)p Fp(\013)q Fq(j)g Fv(=)h Fp(N)51 b Fv(W)-8 b(e)42 b(in)m(tro)s(duce)f(the)120 4981 y(functions)1284 5093 y Fk(8)1284 5183 y(<)1284 5362 y(:)1414 5166 y Fp(\020)1457 5181 y Fm(x)1497 5190 y Fj(0)1535 5166 y Fv(\()p Fp(x)p Fv(\))83 b(=)g Fp(\020)8 b Fv(\()2049 5126 y Fm(x)p Fo(\000)p Fm(x)2184 5135 y Fj(0)p 2006 5143 255 4 v 2006 5200 a Fo(j)p Fm(R)2079 5210 y Fi(j)2111 5200 y Fo(\000)p Fm(x)2206 5209 y Fj(0)2241 5200 y Fo(j)2270 5166 y Fv(\))1517 5425 y(\010)1587 5440 y Fm(x)1627 5449 y Fj(0)1749 5425 y Fv(=)83 b(\010)1978 5389 y Fn(TF)1978 5450 y Fo(R)p Fm(;)p Fo(Z)2119 5425 y Fp(\020)2162 5440 y Fm(x)2202 5449 y Fj(0)3367 5297 y Fv(\(32\))1806 5712 y(9)p eop %%Page: 10 10 10 9 bop 120 407 a Fv(If)29 b Fp(x)g Fv(b)s(elongs)f(to)g(the)h(supp)s (ort)g(of)f(\010)1457 422 y Fm(x)1497 431 y Fj(0)1536 407 y Fv(,)i(a)e(computation)f(giv)m(es)i Fq(j)p Fp(R)2572 422 y Fm(i)2614 407 y Fq(\000)14 b Fp(x)p Fq(j)29 b(\025)f Fv(1)p Fp(=)p Fv(2)p Fq(j)p Fp(R)3171 422 y Fm(j)3221 407 y Fq(\000)14 b Fp(x)3367 422 y Fn(0)3407 407 y Fq(j)28 b(\025)120 527 y Fv(1)p Fp(=)p Fv(4)40 b Fq(8)17 b Fp(i)p Fv(.)69 b(Therefore,)44 b(the)e(function)e(\010)1624 542 y Fm(x)1664 551 y Fj(0)1744 527 y Fv(has)h(no)g(singularities.)66 b(By)41 b(standard)h(elliptic)120 648 y(estimates,)35 b(w)m(e)g(get)g(\010)959 663 y Fm(x)999 672 y Fj(0)1069 648 y Fq(2)c Fp(C)1243 611 y Fo(1)1236 672 y Fm(c)1318 648 y Fv(.)49 b(Also,)35 b(since)g(\010)1955 611 y Fn(TF)1955 673 y Fo(R)p Fm(;)p Fo(Z)2129 648 y Fv(and)g(\010)2391 663 y Fm(x)2431 672 y Fj(0)2504 648 y Fv(agree)g(in)f(a)g(neigh)m(b)s (orho)s(o)s(d)120 768 y(of)40 b Fp(x)294 783 y Fn(0)334 768 y Fv(,)i(all)c(their)i(deriv)-5 b(ativ)m(es)40 b(coincide)g(at)g (that)g(p)s(oin)m(t.)66 b(A)40 b(calculation,)g(using)g(the)h(TF)120 888 y(equation)32 b(\(13\))g(yields)701 1086 y(\001\010)852 1101 y Fm(x)892 1110 y Fj(0)959 1086 y Fv(=)27 b(4)p Fp(\031)t Fv(\(\010)1278 1045 y Fn(TF)1278 1110 y Fo(R)p Fm(;)p Fo(Z)1419 1086 y Fv(\))1457 1045 y Fn(3)p Fm(=)p Fn(2)1566 1086 y Fp(\020)1609 1101 y Fm(x)1649 1110 y Fj(0)1710 1086 y Fv(+)22 b(2)p Fq(r)p Fp(\020)1983 1101 y Fm(x)2023 1110 y Fj(0)2083 1086 y Fq(\001)g(r)p Fv(\010)2286 1045 y Fn(TF)2286 1110 y Fo(R)p Fm(;)p Fo(Z)2448 1086 y Fv(+)g(\010)2616 1045 y Fn(TF)2616 1110 y Fo(R)p Fm(;)p Fo(Z)2757 1086 y Fv(\001)p Fp(\020)2881 1101 y Fm(x)2921 1110 y Fj(0)3367 1086 y Fv(\(33\))120 1283 y(whic)m(h)33 b(could)f(b)s(e)h(written)f(as)h(follo)m(ws:)576 1500 y(\010)646 1515 y Fm(x)686 1524 y Fj(0)808 1500 y Fv(=)82 b Fq(j)22 b(\001)g(j)1094 1459 y Fo(\000)p Fn(1)1210 1500 y Fq(\003)g Fv(\(\(\010)1428 1459 y Fn(TF)1428 1525 y Fo(R)p Fm(;)p Fo(Z)1569 1500 y Fv(\))1607 1459 y Fn(3)p Fm(=)p Fn(2)1716 1500 y Fp(\020)1759 1515 y Fm(x)1799 1524 y Fj(0)1838 1500 y Fv(\))g(+)2035 1433 y(1)p 2006 1477 108 4 v 2006 1568 a(2)p Fp(\031)2123 1500 y Fq(j)g(\001)g(j)2251 1459 y Fo(\000)p Fn(1)2367 1500 y Fq(\003)g Fv(\()p Fq(r)p Fv(\010)2630 1459 y Fn(TF)2630 1525 y Fo(R)p Fm(;)p Fo(Z)2793 1500 y Fq(\001)f(r)p Fp(\020)2968 1515 y Fm(x)3008 1524 y Fj(0)3046 1500 y Fv(\))966 1733 y(+)1082 1666 y(1)p 1052 1710 V 1052 1801 a(2)p Fp(\031)1170 1733 y Fq(j)h(\001)g(j)1298 1692 y Fo(\000)p Fn(1)1414 1733 y Fq(\003)g Fv(\(\010)1594 1692 y Fn(TF)1594 1758 y Fo(R)p Fm(;)p Fo(Z)1734 1733 y Fv(\001)p Fp(\020)1858 1748 y Fm(x)1898 1757 y Fj(0)1936 1733 y Fv(\))808 1966 y(=)82 b Fq(j)22 b(\001)g(j)1094 1925 y Fo(\000)p Fn(1)1210 1966 y Fq(\003)g Fv(\(\(\010)1428 1925 y Fn(TF)1428 1991 y Fo(R)p Fm(;)p Fo(Z)1569 1966 y Fv(\))1607 1925 y Fn(3)p Fm(=)p Fn(2)1716 1966 y Fp(\020)1759 1981 y Fm(x)1799 1990 y Fj(0)1838 1966 y Fv(\))g Fq(\000)2037 1899 y Fv(1)p 2007 1943 V 2007 2035 a(4)p Fp(\031)2125 1966 y Fq(j)g(\001)f(j)2252 1925 y Fo(\000)p Fn(1)2369 1966 y Fq(\003)h Fv(\(\010)2549 1925 y Fn(TF)2549 1991 y Fo(R)p Fm(;)p Fo(Z)2689 1966 y Fv(\001)p Fp(\020)2813 1981 y Fm(x)2853 1990 y Fj(0)2891 1966 y Fv(\))438 b(\(34\))966 2245 y Fq(\000)1083 2178 y Fv(1)p 1053 2222 V 1053 2313 a(2)p Fp(\031)1242 2121 y Fn(3)1188 2150 y Fk(X)1203 2360 y Fm(i)p Fn(=1)1394 2178 y Fp(y)1442 2193 y Fm(i)p 1359 2222 146 4 v 1359 2313 a Fq(j)p Fp(y)t Fq(j)1467 2285 y Fn(3)1537 2245 y Fq(\003)22 b Fv(\(\010)1717 2204 y Fn(TF)1717 2270 y Fo(R)p Fm(;)p Fo(Z)1857 2245 y Fp(@)1908 2260 y Fm(i)1937 2245 y Fp(\020)1980 2260 y Fm(x)2020 2269 y Fj(0)2058 2245 y Fv(\))32 b(\(in)m(tegrating)f(b)m(y)j(parts\)) 120 2525 y(whic)m(h)f(implies)873 2807 y Fp(@)929 2766 y Fm(\013)979 2807 y Fv(\010)1049 2822 y Fm(x)1089 2831 y Fj(0)1156 2807 y Fv(=)28 b Fp(h)1316 2766 y Fm(i)1366 2807 y Fq(\003)22 b Fp(f)1497 2766 y Fm(\013)1486 2831 y Fn(1)1568 2807 y Fq(\000)1707 2739 y Fv(1)p 1678 2784 108 4 v 1678 2875 a(4)p Fp(\031)1795 2807 y(g)1842 2822 y Fm(\013)1914 2807 y Fq(\003)g Fp(f)2034 2822 y Fn(2)2095 2807 y Fq(\000)2234 2739 y Fv(1)p 2205 2784 V 2205 2875 a(2)p Fp(\031)2394 2682 y Fn(3)2339 2712 y Fk(X)2354 2922 y Fm(i)p Fn(=1)2500 2807 y Fp(h)2556 2766 y Fm(i)2556 2831 y(\013)2628 2807 y Fq(\003)g Fp(f)2759 2766 y Fm(i)2748 2831 y Fn(3)3367 2807 y Fv(\(35\))120 3087 y(where)1194 3178 y Fk(8)1194 3268 y(>)1194 3298 y(>)1194 3328 y(>)1194 3357 y(>)1194 3387 y(>)1194 3417 y(>)1194 3447 y(<)1194 3626 y(>)1194 3656 y(>)1194 3686 y(>)1194 3716 y(>)1194 3746 y(>)1194 3776 y(>)1194 3806 y(:)1324 3258 y Fp(f)1383 3222 y Fm(\013)1372 3282 y Fn(1)1515 3258 y Fv(=)83 b Fp(@)1730 3222 y Fm(\013)1797 3177 y Fk(\002)1839 3258 y Fv(\(\010)1947 3222 y Fn(TF)1947 3283 y Fo(R)p Fm(;)p Fo(Z)2087 3258 y Fv(\))2125 3222 y Fn(3)p Fm(=)p Fn(2)2235 3258 y Fp(\020)2278 3273 y Fm(x)2318 3282 y Fj(0)2356 3177 y Fk(\003)1324 3380 y Fp(f)1372 3395 y Fn(2)1515 3380 y Fv(=)g(\010)1744 3344 y Fn(TF)1744 3405 y Fo(R)p Fm(;)p Fo(Z)1885 3380 y Fv(\001)p Fp(\020)2009 3395 y Fm(x)2049 3404 y Fj(0)1324 3502 y Fp(f)1383 3466 y Fm(i)1372 3527 y Fn(3)1515 3502 y Fv(=)g(\010)1744 3466 y Fn(TF)1744 3528 y Fo(R)p Fm(;)p Fo(Z)1885 3502 y Fp(@)1936 3517 y Fm(i)1964 3502 y Fp(\020)2007 3517 y Fm(x)2047 3526 y Fj(0)1324 3625 y Fp(g)1371 3640 y Fm(\013)1515 3625 y Fv(=)g Fp(@)1730 3589 y Fm(\013)1781 3625 y Fq(j)22 b(\001)f(j)1908 3589 y Fo(\000)p Fn(1)1324 3745 y Fp(h)1380 3709 y Fm(i)1380 3770 y(\013)1515 3745 y Fv(=)83 b Fp(@)1730 3709 y Fm(\013)1781 3745 y Fp(@)1837 3709 y Fm(i)1866 3745 y Fq(j)22 b(\001)f(j)1993 3709 y Fo(\000)p Fn(1)1324 3866 y Fp(h)1380 3829 y Fm(i)1515 3866 y Fv(=)83 b Fp(h)1730 3829 y Fm(i)1730 3890 y Fn(0)3367 3562 y Fv(\(36\))120 4056 y(It)33 b(is)f(elemen)m(tary)g(to)h(sho)m(w)g(b)m(y)h(induction)d (that)i Fp(g)2011 4071 y Fm(\013)2092 4056 y Fv(has)g(alw)m(a)m(ys)g (the)g(form)804 4303 y Fp(g)851 4318 y Fm(\013)900 4303 y Fv(\()p Fp(x)p Fv(\))28 b(=)1213 4236 y Fp(P)1276 4251 y Fm(\013)1326 4236 y Fv(\()p Fp(x)p Fv(\))p 1173 4280 325 4 v 1173 4371 a Fq(j)p Fp(x)p Fq(j)1284 4343 y Fn(2)p Fo(j)p Fm(\013)p Fo(j)p Fn(+1)1508 4303 y Fv(;)49 b Fp(P)1647 4318 y Fm(\013)1724 4303 y Fv(:)60 b(p)s(olinomial)g(of)32 b(degree)i Fq(j)p Fp(\013)q Fq(j)120 4556 y Fv(Recall)j Fq(j)p Fp(R)522 4571 y Fm(i)577 4556 y Fq(\000)27 b Fp(x)p Fq(j)39 b(\025)g Fv(1)p Fp(=)p Fv(4)f(for)g Fp(x)i Fv(in)e(the)h(supp)s (ort)g(of)g Fp(\020)2176 4571 y Fm(x)2216 4580 y Fj(0)2254 4556 y Fv(,)i(so)e(the)g(estimate)f(of)h(Lemma)e(2)120 4677 y(applies)h(\(ma)m(yb)s(e)g(with)g(mo)s(di\014ed)f(constan)m(ts\)) j(and)f(no)m(w)g(w)m(e)g(estimate)f(an)m(y)h(of)f(pieces)h(of)120 4797 y(\(35\))32 b(at)g Fp(x)500 4812 y Fn(0)540 4797 y Fv(:)266 4917 y(First)39 b(w)m(e)h(estimate)e(p)s(oin)m(t)m(wise)i Fp(f)1556 4881 y Fm(\013)1545 4942 y(i)1605 4917 y Fv(.)63 b(Call)38 b Fp(F)53 b Fv(:=)39 b(\010)2236 4881 y Fn(TF)2236 4943 y Fo(R)p Fm(;)p Fo(Z)2376 4917 y Fv(.)64 b(Then)40 b Fp(F)53 b(>)39 b Fv(0)g(and)g(w)m(e)i(can)120 5038 y(write)468 5315 y Fp(@)524 5274 y Fm(\013)574 5315 y Fp(F)651 5274 y Fm(p)718 5315 y Fv(=)851 5183 y Fo(j)p Fm(\013)p Fo(j)822 5220 y Fk(X)827 5430 y Fm(n)p Fn(=1)982 5315 y Fp(p)p Fv(\()p Fp(p)22 b Fq(\000)h Fv(1\))17 b Fp(:)g(:)g(:)e Fv(\()p Fp(p)22 b Fq(\000)h Fp(n)f Fv(+)g(1\))p Fp(F)2025 5274 y Fm(p)p Fo(\000)p Fm(N)7 b Fn(+)p Fm(n)2433 5220 y Fk(X)2297 5436 y Fm(\014)s Fn(:)p Fm(\014)s Fo(\024)p Fm(\013;)k Fo(j)p Fm(\014)s Fo(j)p Fn(=)p Fm(n)2730 5315 y Fp(@)2786 5274 y Fm(\014)2835 5315 y Fp(F)j(@)2968 5274 y Fm(\013)p Fo(\000)p Fm(\014)3116 5315 y Fp(F)188 b Fv(\(37\))1781 5712 y(10)p eop %%Page: 11 11 11 10 bop 1099 412 a Fp(f)1158 371 y Fm(\013)1147 436 y(i)1235 412 y Fv(=)1339 317 y Fk(X)1339 529 y Fm(\014)s Fo(\024)p Fm(\013)1499 412 y Fp(C)1569 427 y Fm(\013;\014)1681 412 y Fp(@)1737 371 y Fm(\014)1785 412 y Fv([\(\010)1920 371 y Fn(TF)1920 436 y Fo(R)p Fm(;)p Fo(Z)2060 412 y Fv(\))2098 371 y Fn(3)p Fm(=)p Fn(2)2208 412 y Fv(])p Fp(@)2291 371 y Fm(\013)p Fo(\000)p Fm(\014)2440 412 y Fp(\020)2483 427 y Fm(x)2523 436 y Fj(0)3367 412 y Fv(\(38\))120 703 y(F)-8 b(rom)34 b(Lemma)g(2)h(\(notice)g(that)g(w)m (e)h(ha)m(v)m(e)h(to)e(use)h(the)g(2-sided)f(estimate)f(\(15\))h (\(37\),)h(\(38\))120 823 y(and)d(recalling)d Fq(j)p Fp(R)802 838 y Fm(i)853 823 y Fq(\000)23 b Fp(x)p Fq(j)28 b(\025)g Fv(1)p Fp(=)p Fv(4)33 b(for)f Fp(x)h Fv(in)f(the)h(supp)s(ort) h(of)e Fp(\020)2384 838 y Fm(x)2424 847 y Fj(0)2462 823 y Fv(,)h(it)f(follo)m(ws)f(that)i Fq(k)p Fp(f)3261 787 y Fm(\013)3250 848 y(i)3310 823 y Fq(k)3360 838 y Fo(1)3463 823 y Fq(\024)120 944 y Fp(C)190 959 y Fm(\013)239 944 y Fq(j)p Fp(x)322 959 y Fn(0)384 944 y Fq(\000)23 b Fp(R)558 959 y Fm(j)594 944 y Fq(j)622 908 y Fo(\000)p Fn(6)p Fo(\000j)p Fm(\013)p Fo(j)856 944 y Fv(.)43 b(Then)34 b(w)m(e)g(apply)e(the)h(induction)e(h)m(yp)s(othesis)j(and)f(w)m(e)g (get:)692 1192 y Fq(j)p Fp(h)776 1151 y Fm(i)826 1192 y Fq(\003)22 b Fp(f)957 1151 y Fm(\013)946 1217 y Fn(1)1007 1192 y Fv(\()p Fp(x)1100 1207 y Fn(0)1139 1192 y Fv(\))p Fq(j)84 b Fv(=)1449 1048 y Fk(\014)1449 1108 y(\014)1449 1167 y(\014)1449 1227 y(\014)1482 1057 y(Z)1598 1192 y Fp(d)1649 1151 y Fn(3)1688 1192 y Fp(x)1817 1125 y(f)1876 1089 y Fm(\013)1865 1150 y Fn(1)1925 1125 y Fv(\()p Fp(x)p Fv(\))p 1753 1169 367 4 v 1753 1261 a Fq(j)p Fp(x)23 b Fq(\000)f Fp(x)2013 1276 y Fn(0)2053 1261 y Fq(j)2081 1232 y Fn(2)2130 1048 y Fk(\014)2130 1108 y(\014)2130 1167 y(\014)2130 1227 y(\014)1288 1462 y Fq(\024)84 b Fp(C)1542 1327 y Fk(Z)1597 1552 y Fo(j)p Fm(x)p Fo(\000)p Fm(x)1752 1561 y Fj(0)1786 1552 y Fo(j\024)1871 1525 y Fj(1)p 1870 1537 31 3 v 1870 1578 a(2)1911 1552 y Fo(j)p Fm(x)1971 1561 y Fj(0)2005 1552 y Fo(\000)p Fm(R)2113 1562 y Fi(j)2145 1552 y Fo(j)2186 1462 y Fp(d)2237 1421 y Fn(3)2276 1462 y Fp(x)2377 1395 y Fq(j)p Fp(f)2464 1359 y Fm(\013)2453 1420 y Fn(1)2513 1395 y Fv(\()p Fp(x)p Fv(\))p Fq(j)p 2341 1439 367 4 v 2341 1531 a(j)p Fp(x)22 b Fq(\000)h Fp(x)2601 1546 y Fn(0)2641 1531 y Fq(j)2669 1502 y Fn(2)1288 1765 y Fq(\024)84 b Fp(C)1542 1629 y Fk(Z)1597 1855 y Fo(j)p Fm(x)p Fo(\000)p Fm(x)1752 1864 y Fj(0)1786 1855 y Fo(j\024)1871 1828 y Fj(1)p 1870 1840 31 3 v 1870 1881 a(2)1911 1855 y Fo(j)p Fm(x)1971 1864 y Fj(0)2005 1855 y Fo(\000)p Fm(R)2113 1865 y Fi(j)2145 1855 y Fo(j)2186 1765 y Fp(d)2237 1724 y Fn(3)2276 1765 y Fp(x)2341 1697 y Fq(j)p Fp(x)2424 1712 y Fn(0)2486 1697 y Fq(\000)23 b Fp(R)2660 1712 y Fm(j)2696 1697 y Fq(j)2724 1661 y Fo(\000)p Fn(6)p Fo(\000j)p Fm(\013)p Fo(j)p 2341 1742 617 4 v 2466 1833 a Fq(j)p Fp(x)f Fq(\000)h Fp(x)2726 1848 y Fn(0)2766 1833 y Fq(j)2794 1804 y Fn(2)3367 1765 y Fv(\(39\))1289 2005 y(=)84 b Fp(C)7 b Fq(j)p Fp(x)1609 2020 y Fn(0)1670 2005 y Fq(\000)23 b Fp(R)1844 2020 y Fm(j)1881 2005 y Fq(j)1909 1964 y Fo(\000)p Fn(4)p Fo(\000j)p Fm(\013)p Fo(j)120 2193 y Fv(Analogously)315 2447 y Fq(j)p Fp(g)390 2462 y Fm(\013)461 2447 y Fq(\003)f Fp(f)581 2462 y Fn(2)621 2447 y Fv(\()p Fp(x)714 2462 y Fn(0)753 2447 y Fv(\))p Fq(j)83 b(\024)h Fp(C)1179 2380 y Fv(1)p 1150 2424 108 4 v 1150 2516 a(4)p Fp(\031)1284 2312 y Fk(Z)1400 2447 y Fp(d)1451 2406 y Fn(3)1490 2447 y Fp(x)1555 2370 y Fv(\010)1625 2334 y Fn(TF)1625 2396 y Fo(R)p Fm(;)p Fo(Z)1766 2370 y Fv(\()p Fp(x)p Fv(\))p Fq(j)p Fv(\001)p Fp(\020)2049 2385 y Fm(x)2089 2394 y Fj(0)2127 2370 y Fq(j)p 1555 2424 600 4 v 1602 2516 a(j)p Fp(x)22 b Fq(\000)h Fp(x)1862 2531 y Fn(0)1902 2516 y Fq(j)1930 2487 y Fo(j)p Fm(\013)p Fo(j)p Fn(+1)902 2720 y Fq(\024)84 b Fp(C)1179 2653 y Fv(1)p 1150 2697 108 4 v 1150 2788 a(4)p Fp(\031)1267 2720 y Fq(j)p Fp(x)1350 2735 y Fn(0)1412 2720 y Fq(\000)22 b Fp(R)1585 2735 y Fm(j)1622 2720 y Fq(j)1650 2679 y Fo(\000)p Fn(6)1761 2585 y Fk(Z)1826 2783 y Fj(1)p 1826 2795 31 3 v 1826 2836 a(4)1867 2810 y Fo(j)p Fm(x)1927 2819 y Fj(0)1960 2810 y Fo(\000)p Fm(R)2068 2820 y Fi(j)2101 2810 y Fo(j\024j)p Fm(x)p Fo(\000)p Fm(x)2331 2819 y Fj(0)2364 2810 y Fo(j\024)2449 2783 y Fj(1)p 2448 2795 V 2448 2836 a(2)2489 2810 y Fo(j)p Fm(x)2549 2819 y Fj(0)2583 2810 y Fo(\000)p Fm(R)2691 2820 y Fi(j)2723 2810 y Fo(j)2981 2653 y Fp(d)3032 2617 y Fn(3)3072 2653 y Fp(x)p 2774 2697 562 4 v 2774 2788 a Fq(j)p Fp(x)g Fq(\000)h Fp(x)3034 2803 y Fn(0)3073 2788 y Fq(j)3101 2760 y Fo(\000j)p Fm(\013)p Fo(j\000)p Fn(1)3367 2720 y Fv(\(40\))902 2960 y Fq(\024)84 b Fp(C)7 b Fq(j)p Fp(x)1223 2975 y Fn(0)1284 2960 y Fq(\000)23 b Fp(R)1458 2975 y Fm(j)1495 2960 y Fq(j)1523 2919 y Fo(\000)p Fn(4)p Fo(\000j)p Fm(\013)p Fo(j)120 3149 y Fv(since)33 b Fq(k)p Fv(\001)p Fp(\020)533 3164 y Fm(x)573 3173 y Fj(0)611 3149 y Fq(k)661 3164 y Fo(1)763 3149 y Fq(\024)c Fp(C)7 b Fq(j)p Fp(x)1029 3164 y Fn(0)1090 3149 y Fq(\000)23 b Fp(R)1264 3164 y Fm(j)1301 3149 y Fq(j)1329 3112 y Fo(\000)p Fn(2)1423 3149 y Fv(.)43 b(Finally)311 3411 y Fq(j)p Fp(h)395 3370 y Fm(i)395 3436 y(\013)467 3411 y Fq(\003)22 b Fp(f)598 3370 y Fm(i)587 3436 y Fn(3)626 3411 y Fv(\()p Fp(x)719 3426 y Fn(0)759 3411 y Fv(\))p Fq(j)83 b(\024)g Fp(C)1162 3275 y Fk(Z)1227 3474 y Fj(1)p 1227 3486 31 3 v 1227 3527 a(4)1268 3501 y Fo(j)p Fm(x)1328 3510 y Fj(0)1361 3501 y Fo(\000)p Fm(R)1469 3511 y Fi(j)1502 3501 y Fo(j\024j)p Fm(x)p Fo(\000)p Fm(x)1732 3510 y Fj(0)1765 3501 y Fo(j\024)1850 3474 y Fj(1)p 1850 3486 V 1850 3527 a(2)1890 3501 y Fo(j)p Fm(x)1950 3510 y Fj(0)1984 3501 y Fo(\000)p Fm(R)2092 3511 y Fi(j)2124 3501 y Fo(j)2165 3411 y Fp(d)2216 3370 y Fn(3)2255 3411 y Fp(x)2320 3334 y Fv(\010)2390 3298 y Fn(TF)2390 3359 y Fo(R)p Fm(;)p Fo(Z)2531 3334 y Fv(\()p Fp(x)p Fv(\))p Fq(j)p Fp(@)2746 3298 y Fm(i)2775 3334 y Fp(\020)2818 3349 y Fm(x)2858 3358 y Fj(0)2896 3334 y Fq(j)p 2320 3388 604 4 v 2369 3479 a(j)p Fp(x)22 b Fq(\000)h Fp(x)2629 3494 y Fn(0)2669 3479 y Fq(j)2697 3451 y Fo(j)p Fm(\013)p Fo(j)p Fn(+2)908 3696 y Fq(\024)83 b Fp(C)7 b Fq(j)p Fp(x)1228 3711 y Fn(0)1290 3696 y Fq(\000)23 b Fp(R)1464 3711 y Fm(j)1500 3696 y Fq(j)1528 3655 y Fo(\000)p Fn(5)1639 3560 y Fk(Z)1704 3759 y Fj(1)p 1704 3771 31 3 v 1704 3812 a(4)1745 3786 y Fo(j)p Fm(x)1805 3795 y Fj(0)1839 3786 y Fo(\000)p Fm(R)1947 3796 y Fi(j)1979 3786 y Fo(j\024j)p Fm(x)p Fo(\000)p Fm(x)2209 3795 y Fj(0)2242 3786 y Fo(j\024)2327 3759 y Fj(1)p 2327 3771 V 2327 3812 a(2)2367 3786 y Fo(j)p Fm(x)2427 3795 y Fj(0)2461 3786 y Fo(\000)p Fm(R)2569 3796 y Fi(j)2601 3786 y Fo(j)2642 3696 y Fp(d)2693 3655 y Fn(3)2732 3696 y Fp(x)p Fq(j)p Fp(x)g Fq(\000)f Fp(x)3047 3711 y Fn(0)3087 3696 y Fq(j)3115 3655 y Fo(\000j)p Fm(\013)p Fo(j\000)p Fn(2)3367 3696 y Fv(\(41\))909 3936 y(=)83 b Fp(C)7 b Fq(j)p Fp(x)1228 3951 y Fn(0)1290 3936 y Fq(\000)23 b Fp(R)1464 3951 y Fm(j)1500 3936 y Fq(j)1528 3895 y Fo(\000)p Fn(4)p Fo(\000j)p Fm(\013)p Fo(j)120 4124 y Fv(since)33 b Fq(k)p Fp(@)465 4088 y Fm(i)494 4124 y Fp(\020)537 4139 y Fm(x)577 4148 y Fj(0)615 4124 y Fq(k)665 4139 y Fo(1)767 4124 y Fq(\024)28 b Fp(C)7 b Fq(j)p Fp(x)1032 4139 y Fn(0)1094 4124 y Fq(\000)23 b Fp(R)1268 4139 y Fm(j)1304 4124 y Fq(j)1332 4088 y Fo(\000)p Fn(1)1426 4124 y Fv(.)266 4245 y(2.-)34 b(W)-8 b(e)35 b(assume)f(no)m(w)h(that)f (all)e Fp(\016)1515 4260 y Fm(i)1574 4245 y Fq(\025)f Fv(1)j(and)h(pic)m(k)f Fp(x)d Fq(2)g Fp(D)2428 4260 y Fm(j)2499 4245 y Fv(with)j Fp(\032)p Fv(\()p Fp(x)p Fv(\))d Fq(\024)g Fv(1)p Fp(=)p Fv(2.)48 b(W)-8 b(e)34 b(\014x)120 4365 y Fp(\021)d Fq(2)d Fp(C)370 4329 y Fo(1)445 4365 y Fv(\()p Fl(R)549 4329 y Fn(3)594 4365 y Fv(\))33 b(with)f(the)h(prop) s(erties)1374 4471 y Fk(\032)1490 4550 y Fp(\021)f Fv(=)27 b(0)p Fp(;)83 b Fq(j)p Fp(y)t Fq(j)26 b(\024)i Fv(1)p Fp(=)p Fv(2)1490 4671 y Fp(\021)k Fv(=)27 b(1)p Fp(;)83 b Fq(j)p Fp(y)t Fq(j)26 b(\025)i Fv(3)p Fp(=)p Fv(4)3367 4611 y(\(42\))120 4852 y(F)-8 b(or)32 b(\014xed)i Fp(j)f Fv(=)28 b(1)p Fp(;)17 b(:)g(:)g(:)32 b(;)17 b(M)43 b Fv(w)m(e)33 b(consider)641 4966 y Fk(8)641 5056 y(>)641 5086 y(>)641 5116 y(<)641 5295 y(>)641 5325 y(>)641 5355 y(:)771 5044 y Fv(\010)841 5008 y Fn(1)841 5068 y Fm(j)881 5044 y Fv(\()p Fp(x)p Fv(\))83 b(=)g Fp(V)22 b Fv(\()p Fp(x)p Fv(\))g Fq(\000)1665 4993 y Fm(Z)1713 5003 y Fi(j)p 1596 5021 220 4 v 1596 5078 a Fo(j)p Fm(x)p Fo(\000)p Fm(R)1764 5088 y Fi(j)1796 5078 y Fo(j)1847 5044 y Fq(\000)h(j)f(\001)g (j)2075 5008 y Fo(\000)p Fn(1)2191 5044 y Fq(\003)g Fv(\()p Fp(\021)2349 5059 y Fm(j)2385 5044 y Fv(\(\010)2493 5008 y Fn(TF)2493 5069 y Fo(R)p Fm(;)p Fo(Z)2634 5044 y Fv(\))2672 5008 y Fn(3)p Fm(=)p Fn(2)2782 5044 y Fv(\)\()p Fp(x)p Fv(\))771 5184 y(\010)841 5148 y Fn(2)841 5209 y Fm(j)881 5184 y Fv(\()p Fp(x)p Fv(\))83 b(=)g Fq(j)22 b(\001)g(j)1382 5148 y Fo(\000)p Fn(1)1498 5184 y Fq(\003)g Fv(\()p Fp( )1671 5199 y Fm(j)1708 5184 y Fv(\(\010)1816 5148 y Fn(TF)1816 5209 y Fo(R)p Fm(;)p Fo(Z)1956 5184 y Fv(\))1994 5148 y Fn(3)p Fm(=)p Fn(2)2104 5184 y Fv(\)\()p Fp(x)p Fv(\))771 5306 y Fp(\021)819 5321 y Fm(j)1095 5306 y Fv(=)83 b Fp(\021)t Fv(\()p Fq(\001)22 b(\000)g Fp(R)1567 5321 y Fm(j)1604 5306 y Fv(\))771 5427 y Fp( )834 5442 y Fm(j)1095 5427 y Fv(=)83 b(1)22 b Fq(\000)h Fp(\021)1473 5442 y Fm(j)3367 5230 y Fv(\(43\))1781 5712 y(11)p eop %%Page: 12 12 12 11 bop 120 407 a Fv(so)23 b(that)f(\010)501 371 y Fn(TF)501 432 y Fo(R)p Fm(;)p Fo(Z)641 407 y Fv(\()p Fp(x)p Fv(\))28 b(=)g(\010)974 371 y Fn(1)974 431 y Fm(j)1014 407 y Fv(\()p Fp(x)p Fv(\))r Fq(\000)r Fv(\010)1296 371 y Fn(2)1296 431 y Fm(j)1336 407 y Fv(\()p Fp(x)p Fv(\))r Fq(\000)1627 356 y Fm(Z)1675 366 y Fi(j)p 1558 384 220 4 v 1558 441 a Fo(j)p Fm(x)p Fo(\000)p Fm(R)1726 451 y Fi(j)1757 441 y Fo(j)1787 407 y Fv(.)40 b(The)24 b(functions)e(\010) 2525 371 y Fn(2)2525 431 y Fm(j)2587 407 y Fv(are)h(uniformly)d(b)s (ounded)120 538 y(b)m(y)33 b(some)g(\014xed)h(constan)m(t,)f(b)s (ecause)h(b)m(y)g(Lemma)d(1)h(w)m(e)i(ha)m(v)m(e)528 788 y(\010)598 747 y Fn(2)598 813 y Fm(j)666 788 y Fq(\024)28 b Fp(C)7 b Fq(j)22 b(\001)f(j)975 747 y Fo(\000)p Fn(1)1092 788 y Fq(\003)g Fp(G)1240 803 y Fm(j)1277 788 y Fv(;)49 b Fp(G)1430 803 y Fm(j)1467 788 y Fv(\()p Fp(x)p Fv(\))28 b(=)f Fp(G)p Fv(\()p Fp(x)c Fq(\000)f Fp(R)2095 803 y Fm(j)2132 788 y Fv(\);)49 b Fp(G)p Fv(\()p Fp(x)p Fv(\))28 b(=)2682 721 y(1)p 2596 765 221 4 v 2596 857 a Fq(j)p Fp(x)p Fq(j)2707 828 y Fn(3)p Fm(=)p Fn(2)2826 788 y Fp(\037)2887 804 y Fm(B)2940 818 y Fj(3)p Fi(=)p Fj(4)3038 804 y Fn(\(0\))3367 788 y Fv(\(44\))120 1078 y(The)47 b(function)g Fp(G)f Fv(is)g(radial,)i(therefore)f(b)m(y)g(Newton's)h (Theorem,)j Fp(G)p Fv(\()p Fp(x)p Fv(\))g Fq(\024)3144 997 y Fk(R)3237 1030 y Fm(G)p Fn(\()p Fm(y)r Fn(\))p 3237 1055 148 4 v 3273 1112 a Fo(j)p Fm(y)r Fo(j)3411 1078 y Fp(dy)t Fv(,)120 1198 y(whic)m(h)33 b(is)f(\014nite.)266 1318 y(W)-8 b(e)36 b(can)g(easily)e(c)m(hec)m(k)k(that)d(\001\010)1521 1282 y Fn(1)1521 1343 y Fm(j)1593 1318 y Fv(=)d(0)j(in)g(the)h(ball)d Fp(B)2339 1334 y Fn(1)p Fm(=)p Fn(2)2449 1318 y Fv(\()p Fp(R)2561 1333 y Fm(j)2598 1318 y Fv(\).)51 b(As)36 b(a)f(result,)h(w)m (e)h(can)120 1439 y(write)32 b(that)h(function)f(using)g(the)h(P)m (oisson)g(k)m(ernel:)757 1712 y(\010)827 1671 y Fn(1)827 1737 y Fm(j)867 1712 y Fv(\()p Fp(x)p Fv(\))28 b(=)1140 1645 y Fp(r)1187 1609 y Fn(2)1248 1645 y Fq(\000)22 b(j)p Fp(x)p Fq(j)1458 1609 y Fn(2)p 1140 1690 359 4 v 1241 1781 a Fv(4)p Fp(\031)t(r)1524 1577 y Fk(Z)1580 1802 y Fo(j)p Fm(y)r Fo(\000)p Fm(R)1745 1812 y Fi(j)1777 1802 y Fo(j)p Fn(=)p Fm(r)1959 1636 y Fv(\010)2029 1600 y Fn(1)2029 1661 y Fm(j)2069 1636 y Fv(\()p Fp(y)t Fv(\))p 1916 1690 324 4 v 1916 1781 a Fq(j)p Fp(x)g Fq(\000)h Fp(y)t Fq(j)2201 1752 y Fn(3)2266 1712 y Fp(d\033)t Fv(\()p Fp(y)t Fv(\);)48 b Fp(r)30 b Fv(=)e(1)p Fp(=)p Fv(2)463 b(\(45\))120 2006 y(and)34 b(since)h(b)m(y)h(Lemma)d(1)h(and)g(\(44\),) h(sup)p Fq(f)10 b Fv(\010)1826 1970 y Fn(1)1826 2030 y Fm(j)1866 1921 y Fk(\014)1866 1981 y(\014)1899 2047 y Fo(j)p Fm(x)p Fo(\000)p Fm(R)2067 2057 y Fi(j)2099 2047 y Fo(j)p Fn(=1)p Fm(=)p Fn(4)2284 2006 y Fq(g)30 b(\024)h Fp(C)7 b Fv(,)35 b(w)m(e)h(get)e Fq(j)p Fp(@)3005 1970 y Fm(\013)3055 2006 y Fv(\010)3125 1970 y Fn(1)3125 2030 y Fm(j)3165 2006 y Fv(\()p Fp(x)p Fv(\))p Fq(j)d(\024)g Fp(C)120 2150 y Fv(if)g Fq(j)p Fp(x)23 b Fq(\000)f Fp(R)488 2165 y Fm(j)525 2150 y Fq(j)27 b(\024)i Fv(1)p Fp(=)p Fv(2)i(for)h(all)f(m)m(ultiindex)g Fp(\013)q Fv(.)266 2271 y(W)-8 b(e)35 b(are)f(going)e(no)m(w)j(to)e(estimate)g(the)h (deriv)-5 b(ativ)m(es)34 b(of)g(\010)2417 2234 y Fn(1)2417 2295 y Fm(j)2457 2271 y Fv(.)47 b(T)-8 b(o)34 b(do)g(so,)g(assume)h(it) e(has)120 2402 y(already)h(b)s(een)i(pro)m(v)m(en)g(that)e Fp(\032)p Fv(\()p Fp(x)p Fv(\))e Fq(\024)g Fv(1)p Fp(=)p Fv(2)i(implies)e Fq(j)p Fp(@)2151 2366 y Fm(\013)2201 2402 y Fv(\010)2271 2366 y Fn(TF)2271 2428 y Fo(R)p Fm(;)p Fo(Z)2411 2402 y Fv(\()p Fp(x)p Fv(\))p Fq(j)g(\024)f Fp(C)7 b(\032)p Fv(\()p Fp(x)p Fv(\))2968 2366 y Fo(\000)p Fn(1)p Fo(\000j)p Fm(\013)p Fo(j)3237 2402 y Fv(for)34 b(an)m(y)120 2536 y(m)m(ultiindex)g Fp(\013)j Fv(with)e Fq(j)p Fp(\013)q Fq(j)e(\024)h Fp(n)p Fv(.)54 b(Then)37 b(consider)f(again)f Fp(@)2298 2500 y Fm(\013)2343 2476 y Fh(0)2404 2536 y Fv(=)e Fp(@)2569 2500 y Fm(j)2607 2536 y Fp(@)2663 2500 y Fm(\013)2749 2536 y Fv(for)i Fp(j)40 b Fv(=)33 b(1)p Fp(;)17 b Fv(2)p Fp(;)g Fv(3.)53 b(W)-8 b(e)120 2656 y(ha)m(v)m(e:)850 2867 y Fq(j)p Fp(@)934 2826 y Fm(\013)979 2803 y Fh(0)1006 2867 y Fv(\010)1076 2826 y Fn(1)1076 2892 y Fm(j)1116 2867 y Fv(\()p Fp(x)p Fv(\))p Fq(j)84 b Fv(=)g Fq(j)p Fp(@)1603 2826 y Fm(j)1640 2867 y Fp(@)1696 2826 y Fm(\013)1746 2867 y Fv(\010)1816 2826 y Fm(i)1816 2892 y(j)1853 2867 y Fv(\()p Fp(x)p Fv(\))p Fq(j)1359 3029 y Fv(=)g Fq(j)p Fp(h)1603 2988 y Fm(k)1667 3029 y Fq(\003)22 b Fv(\()p Fp(@)1833 2988 y Fm(\013)1884 3029 y Fv(\()p Fp( )1985 3044 y Fm(j)2021 3029 y Fv(\()p Fp(x)p Fv(\)\010)2222 2988 y Fn(TF)2222 3054 y Fo(R)p Fm(;)p Fo(Z)2363 3029 y Fv(\()p Fp(x)p Fv(\))2494 2988 y Fn(3)p Fm(=)p Fn(2)2604 3029 y Fv(\)\))p Fq(j)659 b Fv(\(46\))1358 3263 y Fq(\024)84 b Fp(C)1612 3127 y Fk(Z)1728 3263 y Fp(d)1779 3222 y Fn(3)1819 3263 y Fp(y)1881 3186 y Fq(j)p Fp(@)1965 3150 y Fm(\013)2014 3186 y Fv(\()p Fp( )2115 3201 y Fm(j)2152 3186 y Fv(\()p Fp(x)p Fv(\)\010)2353 3150 y Fn(TF)2353 3211 y Fo(R)p Fm(;)p Fo(Z)2493 3186 y Fv(\()p Fp(x)p Fv(\))2624 3150 y Fn(3)p Fm(=)p Fn(2)2735 3186 y Fv(\))p Fq(j)p 1881 3240 921 4 v 2178 3331 a(j)p Fp(x)23 b Fq(\000)f Fp(y)t Fq(j)2463 3302 y Fn(2)120 3524 y Fv(Reasoning)32 b(as)h(in)f(the)h (previous)g(part,)f(w)m(e)i(get)e(the)h(estimate)941 3735 y Fq(j)p Fp(@)1025 3694 y Fm(\013)1075 3735 y Fv(\()p Fp( )1176 3750 y Fm(j)1213 3735 y Fv(\()p Fp(x)p Fv(\)\010)1414 3694 y Fn(TF)1414 3760 y Fo(R)p Fm(;)p Fo(Z)1555 3735 y Fv(\()p Fp(x)p Fv(\))1686 3694 y Fn(3)p Fm(=)p Fn(2)1796 3735 y Fv(\))p Fq(j)27 b(\024)h Fp(C)7 b Fq(j)p Fp(x)23 b Fq(\000)f Fp(R)2350 3750 y Fm(j)2387 3735 y Fq(j)2415 3694 y Fo(\000)p Fn(3)p Fm(=)p Fn(2)p Fo(\000j)p Fm(\013)p Fo(j)3367 3735 y Fv(\(47\))120 3946 y(Putting)32 b(together)h (estimates)f(\(46\))g(and)h(\(47\))f(follo)m(ws)599 4157 y Fq(j)p Fp(@)683 4116 y Fm(\013)728 4092 y Fh(0)755 4157 y Fv(\010)825 4116 y Fn(1)825 4181 y Fm(j)865 4157 y Fv(\()p Fp(x)p Fv(\))p Fq(j)84 b Fv(=)g Fq(j)p Fp(@)1352 4116 y Fm(j)1389 4157 y Fp(@)1445 4116 y Fm(\013)1495 4157 y Fv(\010)1565 4116 y Fm(i)1565 4181 y(j)1602 4157 y Fv(\()p Fp(x)p Fv(\))p Fq(j)1107 4381 y(\024)g Fp(C)1361 4245 y Fk(Z)1477 4381 y Fp(d)1528 4340 y Fn(3)1567 4381 y Fp(y)1629 4314 y Fq(j)p Fp(x)22 b Fq(\000)g Fp(R)1907 4329 y Fm(j)1944 4314 y Fq(j)1972 4277 y Fo(\000)p Fn(3)p Fm(=)p Fn(2)p Fo(\000j)p Fm(\013)p Fo(j)p 1629 4358 648 4 v 1791 4449 a Fq(j)p Fp(x)g Fq(\000)h Fp(y)t Fq(j)2076 4421 y Fn(2)1107 4643 y Fq(\024)84 b Fp(C)1361 4507 y Fk(Z)1416 4733 y Fo(j)p Fm(y)r Fo(\000)p Fm(R)1581 4743 y Fi(j)1614 4733 y Fo(j\024)p Fn(3)p Fm(=)p Fn(4)1815 4643 y Fp(d)1866 4602 y Fn(3)1905 4643 y Fp(y)t Fq(j)p Fp(x)21 b Fq(\000)i Fp(y)t Fq(j)2241 4602 y Fo(\000)p Fn(2)2334 4643 y Fq(j)p Fp(y)i Fq(\000)e Fp(R)2609 4658 y Fm(j)2646 4643 y Fq(j)2674 4602 y Fo(\000)p Fn(3)p Fm(=)p Fn(2)p Fo(\000j)p Fm(\013)p Fo(j)3367 4643 y Fv(\(48\))1107 4922 y Fq(\024)84 b Fp(C)7 b Fq(j)p Fp(x)22 b Fq(\000)g Fp(R)1623 4937 y Fm(j)1660 4922 y Fq(j)1688 4880 y Fo(\000)p Fn(1)p Fm(=)p Fn(2)p Fo(\000j)p Fm(\013)p Fo(j)2009 4786 y Fk(Z)2125 4922 y Fp(d)2176 4880 y Fn(3)2215 4922 y Fp(x)p Fq(j)p Fv(1)g Fq(\000)h Fp(x)p Fq(j)2552 4880 y Fo(\000)p Fn(2)2646 4922 y Fq(j)p Fp(x)p Fq(j)2757 4880 y Fo(\000)p Fn(3)p Fm(=)p Fn(2)p Fo(\000j)p Fm(\013)p Fo(j)1108 5132 y Fv(=)84 b Fp(C)7 b Fq(j)p Fp(x)22 b Fq(\000)g Fp(R)1623 5147 y Fm(j)1660 5132 y Fq(j)1688 5090 y Fo(\000)p Fn(1)p Fm(=)p Fn(2)p Fo(\000j)p Fm(\013)p Fo(j)120 5342 y Fv(Since)32 b(the)g(coulom)m(bic)e(term)h Fp(Z)1297 5357 y Fm(j)1333 5342 y Fp(=)p Fq(j)p Fp(x)21 b Fq(\000)f Fp(R)1657 5357 y Fm(j)1694 5342 y Fq(j)31 b Fv(ob)m(viously)h(satis\014es)g(the)g(conclusion)g(of)f(Theo-)120 5463 y(rem)h(4,)g(pro)s(of)g(of)g(Theorem)h(4)f(is)h(complete.)p 3469 5463 4 66 v 3473 5400 59 4 v 3473 5463 V 3531 5463 4 66 v 1781 5712 a(12)p eop %%Page: 13 13 13 12 bop 266 407 a Fv(Another)29 b(result)f(ab)s(out)f(the)i(TF)f(p)s (oten)m(tial)e(w)m(e)j(shall)e(need)i(is)f(to)f(isolate)g(near)h(the)h (sin-)120 527 y(gularities)24 b(of)h(a)h(con\014guration)f(of)g (uniformly)f(separated)j(n)m(uclei)e(the)i(asymptotic)e(coulom-)120 648 y(bic)32 b(term)g(and)h(the)g(rest.)44 b(The)34 b(result)e(is)g(as) h(follo)m(ws:)120 835 y Ft(Lemma)k(3)h(\(regular)e(part)i(of)f(TF)h(p)s (oten)m(tial)e(near)i(singularities\).)46 b Fr(Assume)120 955 y Fp(\016)163 970 y Fm(i)219 955 y Fq(\025)28 b Fv(1)17 b Fq(8)p Fp(i)p Fr(,)35 b Fv(1)27 b Fq(\024)i Fp(Z)792 970 y Fm(j)856 955 y Fq(\024)f Fp(A)266 1076 y Fr(1.-)35 b(Write)194 1430 y Fv(\010)264 1388 y Fn(TF)264 1454 y Fo(R)p Fm(;)p Fo(Z)405 1430 y Fv(\()p Fp(x)p Fv(\))28 b(=)797 1362 y Fp(Z)864 1377 y Fm(j)p 677 1407 344 4 v 677 1498 a Fq(j)p Fp(x)22 b Fq(\000)h Fp(R)956 1513 y Fm(j)993 1498 y Fq(j)1053 1430 y Fv(+)k(lim)1151 1489 y Fm(x)p Fo(!)p Fn(0)1313 1289 y Fk(\024)1365 1430 y Fv(\010)1435 1388 y Fn(TF)1435 1454 y Fo(R)p Fm(;)p Fo(Z)1576 1430 y Fv(\()p Fp(x)p Fv(\))22 b Fq(\000)1959 1362 y Fp(Z)2026 1377 y Fm(j)p 1839 1407 V 1839 1498 a Fq(j)p Fp(x)g Fq(\000)h Fp(R)2118 1513 y Fm(j)2154 1498 y Fq(j)2192 1289 y Fk(\025)2267 1430 y Fv(+)f Fp(\026)2424 1445 y Fm(j)2460 1430 y Fv(\()p Fp(x)p Fv(\);)52 b Fq(j)p Fp(x)22 b Fq(\000)h Fp(R)2949 1445 y Fm(j)2986 1430 y Fq(j)k(\024)h Fv(1)p Fp(=)p Fv(2)74 b(\(49\))120 1684 y Fr(We)35 b(have)f(the)h (estimate)748 1885 y Fq(j)p Fp(@)832 1844 y Fm(\013)882 1885 y Fp(\026)941 1900 y Fm(j)978 1885 y Fv(\()p Fp(x)p Fv(\))p Fq(j)27 b(\024)h Fp(C)1339 1900 y Fm(\013)1389 1885 y Fp(\032)p Fv(\()p Fp(x)p Fv(\))1570 1844 y Fn(1)p Fm(=)p Fn(2)p Fo(\000j)p Fm(\013)p Fo(j)1820 1885 y Fv(;)51 b Fq(j)p Fp(x)22 b Fq(\000)h Fp(R)2177 1900 y Fm(j)2214 1885 y Fq(j)k(\024)h Fv(1)p Fp(=)p Fv(2)p Fp(;)33 b(\013)28 b Fq(2)g Fl(N)2831 1844 y Fn(3)3367 1885 y Fv(\(50\))266 2086 y Fr(2.-)35 b(Now)g(write)628 2288 y Fv(\010)698 2247 y Fn(TF)698 2312 y Fo(R)p Fm(;)p Fo(Z)839 2288 y Fv(\()p Fp(x)p Fv(\))28 b(=)f Fp(Z)1175 2237 y Fn(4)p Fm(=)p Fn(3)1168 2313 y Fm(j)1285 2288 y Fp(')p Fv(\()p Fp(Z)1461 2237 y Fn(1)p Fm(=)p Fn(3)1454 2313 y Fm(j)1571 2288 y Fv(\()p Fp(x)c Fq(\000)f Fp(R)1860 2303 y Fm(j)1897 2288 y Fv(\)\))g(+)g Fp(w)2163 2303 y Fm(j)2199 2288 y Fv(\()p Fp(x)p Fv(\);)52 b Fq(j)p Fp(x)22 b Fq(\000)h Fp(R)2688 2303 y Fm(j)2725 2288 y Fq(j)k(\024)h Fv(1)p Fp(=)p Fv(2)335 b(\(51\))120 2489 y Fr(Then)34 b(we)h(have)f(the)h (estimates)1076 2690 y Fq(k)p Fp(@)1182 2649 y Fm(\013)1233 2690 y Fp(w)1303 2705 y Fm(j)1339 2690 y Fq(k)1389 2705 y Fo(1)1491 2690 y Fq(\024)28 b Fp(C)1666 2705 y Fm(\013)1715 2690 y Fv(;)34 b Fp(\013)28 b Fq(2)g Fl(N)2026 2649 y Fn(3)2106 2690 y Fr(and)35 b Fp(w)2366 2705 y Fm(j)2430 2690 y Fq(\025)28 b Fv(0)783 b(\(52\))266 2891 y Fr(3.-)35 b(Write)1140 3125 y Fp(k)1191 3140 y Fm(j)1255 3125 y Fv(=)58 b(lim)1359 3187 y Fm(x)p Fo(!)p Fm(R)1523 3197 y Fi(j)1571 2984 y Fk(\024)1624 3125 y Fv(\010)1694 3084 y Fn(TF)1694 3149 y Fo(R)p Fm(;)p Fo(Z)1834 3125 y Fv(\()p Fp(x)p Fv(\))23 b Fq(\000)2217 3057 y Fp(Z)2284 3072 y Fm(j)p 2097 3102 V 2097 3193 a Fq(j)p Fp(x)g Fq(\000)f Fp(R)2376 3208 y Fm(j)2413 3193 y Fq(j)2451 2984 y Fk(\025)3367 3125 y Fv(\(53\))120 3379 y Fr(Then)34 b(we)h(have)f(the)h(estimate) 1654 3580 y Fq(j)p Fp(k)1733 3595 y Fm(j)1769 3580 y Fq(j)27 b(\024)h Fp(C)1368 b Fv(\(54\))120 3782 y Fr(Pr)-5 b(o)g(of.)48 b Fv(An)33 b(easy)h(calculation)c(using)i(the)h(TF)g (equation)f(\(13\))g(yields)g(the)h(result)392 4076 y(lim)362 4138 y Fm(x)p Fo(!)p Fm(R)526 4148 y Fi(j)575 3935 y Fk(\024)627 4076 y Fv(\010)697 4034 y Fn(TF)697 4100 y Fo(R)p Fm(;)p Fo(Z)838 4076 y Fv(\()p Fp(x)p Fv(\))22 b Fq(\000)1221 4008 y Fp(Z)1288 4023 y Fm(j)p 1101 4053 V 1101 4144 a Fq(j)p Fp(x)g Fq(\000)h Fp(R)1380 4159 y Fm(j)1417 4144 y Fq(j)1454 3935 y Fk(\025)1534 4076 y Fv(=)1723 3951 y Fm(M)1689 3981 y Fk(X)1638 4193 y Fm(i)p Fn(=1)p Fm(;i)p Fo(6)p Fn(=)p Fm(j)2053 4008 y Fp(Z)2120 4023 y Fm(j)p 1910 4053 391 4 v 1910 4144 a Fq(j)p Fp(R)2012 4159 y Fm(i)2062 4144 y Fq(\000)f Fp(R)2235 4159 y Fm(j)2272 4144 y Fq(j)2332 4076 y(\000)h(j)e(\001)h(j)2559 4034 y Fo(\000)p Fn(1)2675 4076 y Fq(\003)g Fp(\032)2797 4034 y Fn(TF)2797 4100 y Fo(R)p Fm(;)p Fo(Z)2938 4076 y Fv(\()p Fp(R)3050 4091 y Fm(j)3086 4076 y Fv(\))243 b(\(55\))120 4387 y(whic)m(h)23 b(is)e(\014nite)h(giv)m(en)g(that)g Fp(\032)1210 4350 y Fn(TF)1210 4412 y Fo(R)p Fm(;)p Fo(Z)1373 4387 y Fv(b)s(elongs)f(to)h Fp(L)1887 4350 y Fn(1)1949 4387 y Fv(and)g(therefore)h Fq(j)q(\001)q(j)2613 4350 y Fo(\000)p Fn(1)2707 4387 y Fq(\003)q Fp(\032)2808 4350 y Fn(TF)2808 4412 y Fo(R)p Fm(;)p Fo(Z)2969 4387 y Fv(is)f(con)m(tin)m (uous.)120 4507 y(Then,)42 b(reasoning)d(lik)m(e)f(in)g(the)i(second)g (part)f(of)g(the)g(pro)s(of)f(of)h(Theorem)h(4)e(w)m(e)i(get)g(\(50\)) 120 4627 y(\(see)34 b(ecs.)44 b(\(46\),)32 b(\(47\),)g(\(48\)\).)43 b(The)34 b(second)g(assertion)e(can)h(b)s(e)g(justi\014ed)f(as)h(follo) m(ws:)528 4921 y Fq(\000)645 4854 y Fv(1)p 615 4898 108 4 v 615 4990 a(4)p Fp(\031)733 4921 y Fv(\001)p Fp(w)884 4936 y Fm(j)948 4921 y Fv(=)1092 4797 y Fm(M)1057 4827 y Fk(X)1052 5039 y Fm(i)p Fn(:)p Fm(i)p Fo(6)p Fn(=)p Fm(j)1223 4921 y Fp(Z)1290 4936 y Fm(i)1318 4921 y Fp(\016)1361 4936 y Fm(R)1414 4946 y Fi(i)1467 4921 y Fv(+)22 b Fp(Z)1639 4880 y Fn(2)1632 4946 y Fm(j)1678 4921 y Fp(')p Fv(\()p Fp(Z)1854 4870 y Fn(1)p Fm(=)p Fn(3)1847 4947 y Fm(j)1964 4921 y Fv(\()p Fq(\001)g(\000)h Fp(R)2226 4936 y Fm(j)2262 4921 y Fv(\)\))2338 4880 y Fn(3)p Fm(=)p Fn(2)2470 4921 y Fq(\000)g Fv(\(\010)2678 4880 y Fn(TF)2678 4946 y Fo(R)p Fm(;)p Fo(Z)2818 4921 y Fv(\))2856 4880 y Fn(3)p Fm(=)p Fn(2)3367 4921 y Fv(\(56\))120 5226 y(F)-8 b(rom)31 b(eq.)44 b(\(56\))32 b(and)h(Lemma)e(4\))h(follo)m(ws)g(that)1208 5427 y Fq(k)p Fp(@)1314 5386 y Fm(\013)1364 5427 y Fv(\(\011\001)p Fp(w)1629 5442 y Fm(j)1665 5427 y Fv(\))p Fq(k)1753 5442 y Fo(1)1856 5427 y Fq(\024)c Fp(C)2031 5442 y Fm(\013)2080 5427 y Fv(;)49 b Fp(\013)29 b Fq(2)f Fl(N)2407 5386 y Fn(3)3367 5427 y Fv(\(57\))1781 5712 y(13)p eop %%Page: 14 14 14 13 bop 120 407 a Fv(where)49 b(0)k Fq(\024)g Fv(\011)g Fq(2)g Fp(C)974 371 y Fo(1)967 431 y Fm(c)1049 407 y Fv(\()p Fp(B)1161 422 y Fn(3)p Fm(=)p Fn(4)1271 407 y Fv(\()p Fp(R)1383 422 y Fm(j)1420 407 y Fv(\)\),)e(\011)i(=)g(1)47 b(if)f Fq(j)p Fp(x)33 b Fq(\000)f Fp(R)2331 422 y Fm(j)2368 407 y Fq(j)53 b(\024)g Fv(1)p Fp(=)p Fv(2)47 b(and)h(follo)m(wing)d (the)120 527 y(pro)s(cedure)34 b(ab)s(o)m(v)m(e)f(the)g(estimates)f (holds:)1249 747 y Fq(k)p Fp(@)1355 706 y Fm(\013)1405 747 y Fv(\(\011)p Fp(w)1589 762 y Fm(j)1625 747 y Fv(\))p Fq(k)1713 762 y Fo(1)1815 747 y Fq(\024)c Fp(C)1990 762 y Fm(\013)2039 747 y Fv(;)50 b Fp(\013)28 b Fq(2)g Fl(N)2366 706 y Fn(3)3367 747 y Fv(\(58\))120 967 y(The)34 b(fact)e(that)g Fp(w)794 982 y Fm(j)863 967 y Fv(is)g(p)s(ositiv)m(e)g(is)g(a)g(simple) f(consequence)36 b(of)c(T)-8 b(eller's)32 b(Lemma.)266 1088 y(Finally)e(w)m(e)k(ha)m(v)m(e,)g(again)d(b)m(y)i(T)-8 b(eller's)33 b(Lemma)e(and)i(\(19\))120 1308 y Fp(Z)194 1266 y Fn(4)p Fm(=)p Fn(3)304 1308 y Fp(')p Fv(\()p Fp(Z)480 1266 y Fn(1)p Fm(=)p Fn(3)590 1308 y Fv(\()p Fp(x)22 b Fq(\000)h Fp(R)879 1323 y Fm(j)916 1308 y Fv(\)\))k Fq(\024)h Fv(\010)1194 1266 y Fn(TF)1194 1332 y Fo(R)p Fm(;)p Fo(Z)1335 1308 y Fv(\()p Fp(x)p Fv(\))g Fq(\024)g Fp(Z)1673 1266 y Fn(4)p Fm(=)p Fn(3)1783 1308 y Fp(')p Fv(\()p Fp(Z)1959 1266 y Fn(1)p Fm(=)p Fn(3)2069 1308 y Fv(\()p Fp(x)22 b Fq(\000)h Fp(R)2358 1323 y Fm(j)2395 1308 y Fv(\)\))f(+)g(\010)2661 1266 y Fn(TF)2661 1332 y Fo(R)2721 1342 y Fi(j)2754 1332 y Fm(;)p Fo(Z)2825 1342 y Fi(j)2861 1308 y Fv(\()p Fp(x)p Fv(\);)49 b Fq(R)3152 1323 y Fm(j)3217 1308 y Fv(=)27 b Fq(R)c(n)f(f)p Fp(R)3623 1323 y Fm(j)3660 1308 y Fq(g)3367 1445 y Fv(\(59\))120 1663 y(and)33 b(from)e(\(59\))h(follo)m(ws)f(that)186 1941 y Fp(C)256 1956 y Fn(1)323 1941 y Fq(\024)d Fp(Z)502 1900 y Fn(4)p Fm(=)p Fn(3)612 1941 y Fp(k)i Fq(\024)f Fp(k)850 1956 y Fm(j)914 1941 y Fq(\024)f Fp(C)1089 1956 y Fn(1)1156 1941 y Fq(\024)g Fp(Z)1335 1900 y Fn(4)p Fm(=)p Fn(3)1445 1941 y Fp(k)d Fv(+)d(\010)1689 1900 y Fn(TF)1689 1965 y Fo(R)1749 1975 y Fi(j)1782 1965 y Fm(;)p Fo(Z)1853 1975 y Fi(j)1889 1941 y Fv(\()p Fp(R)2001 1956 y Fm(j)2038 1941 y Fv(\))27 b Fq(\024)i Fp(C)2279 1956 y Fn(2)2318 1941 y Fv(;)49 b Fp(k)31 b Fv(=)c(lim)2731 1800 y Fk(\024)2784 1941 y Fp(')p Fv(\()p Fp(x)p Fv(\))22 b Fq(\000)3142 1873 y Fv(1)p 3111 1918 111 4 v 3111 2009 a Fq(j)p Fp(x)p Fq(j)3232 1800 y Fk(\025)3367 1941 y Fv(\(60\))120 2212 y(where)34 b(w)m(e)f(ha)m(v)m(e)h(applied)e(again)f (Theorem)i(1.)p 3469 2212 4 66 v 3473 2150 59 4 v 3473 2212 V 3531 2212 4 66 v 364 2441 a Ft(Remarks:)364 2603 y Fv(i.)65 b(Lo)s(oking)39 b(at)h(the)g(case)h(of)f(an)g(isolated)f (atom,)i(the)f(constan)m(t)h(correction)364 2745 y Fp(k)31 b Fv(=)c(lim)685 2760 y Fm(x)p Fo(!)p Fn(0)851 2634 y Fk(h)898 2745 y Fp(Z)972 2709 y Fn(4)p Fm(=)p Fn(3)1082 2745 y Fp(')p Fv(\()p Fp(Z)1258 2709 y Fn(1)p Fm(=)p Fn(3)1368 2745 y Fp(x)p Fv(\))22 b Fq(\000)1606 2706 y Fm(Z)p 1593 2722 79 4 v 1593 2779 a Fo(j)p Fm(x)p Fo(j)1682 2634 y Fk(i)1761 2745 y Fv(can)33 b(b)s(e)g(written)f(as)728 3054 y Fp(k)f Fv(=)958 2987 y Fp(d)p 924 3031 120 4 v 924 3122 a(d)-5 b(Z)1053 3054 y(E)1131 3013 y Fn(TF)1232 3054 y Fv(\()p Fp(Z)7 b Fv(\))27 b(=)h Fq(\000)1600 2987 y Fv(7)p 1600 3031 49 4 v 1600 3122 a(3)1659 3054 y Fp(C)1729 3069 y Fn(TF)1830 3054 y Fp(Z)1904 3013 y Fn(4)p Fm(=)p Fn(3)2014 3054 y Fv(;)49 b Fp(C)2160 3069 y Fn(TF)2289 3054 y Fv(=)27 b Fq(j)p Fp(E)2498 3013 y Fn(TF)2599 3054 y Fv(\(1\))p Fq(j)g Fp(>)h Fv(0)191 b(\(61\))364 3300 y(so)33 b(the)g(correction)f(is)g Fr(strictly)j(ne)-5 b(gative)32 b Fv(in)g(that)g(case.)364 3461 y(ii.)38 b(F)-8 b(or)23 b(the)h(case)g(of)f(an)g(isolated)f(atom,)i(the)g Fp(\026)f Fv(app)s(earing)g(in)f(\(55\))h(is)g Fr(ne)-5 b(gative)p Fv(.)364 3582 y(This)33 b(follo)m(ws)e(from)g(the)i (estimate)f Fp(')p Fv(\()p Fp(x)p Fv(\))c Fq(\024)2056 3543 y Fn(1)p 2034 3559 79 4 v 2034 3616 a Fo(j)p Fm(x)p Fo(j)2123 3582 y Fv(.)120 3915 y Fs(3)161 b(Reduction)50 b(to)h(the)f(case)h(of)g(uniformly)h(separated)362 4097 y(n)l(uclei.)120 4316 y Fv(In)47 b(this)f(section)h(our)g(ob)5 b(ject)47 b(of)f(study)i(will)c(b)s(e)j(the)g(so-called)f Fr(r)-5 b(e)g(duc)g(e)g(d)47 b(Hartr)-5 b(e)g(e-F)e(o)i(ck)120 4437 y(mo)g(del)40 b Fv(\(RHF)h(in)g(the)h(next\),)i(whic)m(h)e(is)f (one)g(in)m(termediate)g(in)f(sophistication)g(b)s(et)m(w)m(een)120 4557 y(the)30 b(purely)g(semiclassical)d(TF)j(mo)s(del)e(and)i(the)g (fully)f(quan)m(tum)h(Sc)m(hr\177)-49 b(odinger)30 b(one.)42 b(I)30 b(will)120 4677 y(also)36 b(ignore)g(the)h(spin,)h(b)s(ecause)g (its)f(e\013ect)h(tourns)f(to)f(b)s(e)i(just)f(a)f(mo)s(di\014cation)f (of)h(some)120 4798 y(constan)m(ts)28 b(\(something)e(whic)m(h)h (already)f(happ)s(ens)h(when)h(one)f(compares)g(the)g(Sc)m(hr\177)-49 b(odinger)120 4918 y(mo)s(del)31 b(with)h(the)h(TF)g(one\).)1781 5712 y(14)p eop %%Page: 15 15 15 14 bop 266 407 a Fv(The)34 b(RHF)e(mo)s(del)f(is)h(giv)m(en)h(b)m(y) g(the)g(functional)323 578 y Fq(E)385 537 y Fn(RHF)376 603 y Fo(R)p Fm(;)p Fo(Z)539 578 y Fv(\()p Fp(\015)5 b Fv(\))83 b(:=)g(T)-8 b(r)17 b([\()p Fq(\000)p Fv(\001)23 b Fq(\000)g Fp(V)1460 593 y Fo(R)p Fm(;)p Fo(Z)1600 578 y Fv(\))p Fp(\015)5 b Fv(])22 b(+)g Fp(D)s Fv(\()p Fp(\032)2013 593 y Fm(\015)2057 578 y Fv(\))g(+)g Fp(U)2281 593 y Fo(R)p Fm(;)p Fo(Z)343 829 y Fp(V)400 844 y Fo(R)p Fm(;)p Fo(Z)540 829 y Fv(\()p Fp(x)p Fv(\))83 b(:=)975 704 y Fm(M)940 734 y Fk(X)951 944 y Fm(j)t Fn(=1)1231 761 y Fp(Z)1298 776 y Fm(j)p 1111 806 344 4 v 1111 897 a Fq(j)p Fp(x)22 b Fq(\000)h Fp(R)1390 912 y Fm(j)1426 897 y Fq(j)364 1162 y Fp(D)s Fv(\()p Fp(f)5 b(;)17 b(g)t Fv(\))82 b(:=)950 1095 y(1)p 950 1139 49 4 v 950 1230 a(2)1026 1026 y Fk(Z)11 b(Z)1192 1252 y Fg(R)1240 1233 y Fj(3)1274 1252 y Fo(\002)p Fg(R)1377 1233 y Fj(3)1442 1095 y Fp(f)g Fv(\()p Fp(x)p Fv(\))p 1632 1008 178 4 v Fp(g)t Fv(\()p Fp(y)t Fv(\))p 1442 1139 368 4 v 1484 1230 a Fq(j)p Fp(x)22 b Fq(\000)h Fp(y)t Fq(j)1820 1162 y Fp(dxdy)t Fv(;)48 b Fp(f)5 b(;)17 b(g)31 b Fq(2)d Fp(L)2439 1121 y Fn(6)p Fm(=)p Fn(5)2549 1162 y Fv(\()p Fl(R)2653 1121 y Fn(3)2699 1162 y Fv(\))630 b(\(62\))576 1376 y Fp(\032)626 1391 y Fm(\015)754 1376 y Fv(:=)116 b(densit)m(y)33 b(of)f Fp(\015)h Fq(2)28 b Fp(S)6 b(p)1713 1335 y Fn(1)1752 1376 y Fv(\()p Fp(L)1856 1335 y Fn(2)1896 1376 y Fv(\()p Fl(R)2000 1335 y Fn(3)2045 1376 y Fv(\)\);)34 b(0)27 b Fq(\024)h Fp(\015)33 b Fq(\024)28 b Fv(1;)33 b(T)-8 b(r[\()p Fq(\000)p Fv(\001\))p Fp(\015)5 b Fv(])29 b Fp(<)e Fq(1)465 1573 y Fp(U)531 1588 y Fo(R)p Fm(;)p Fo(Z)754 1573 y Fv(:=)1033 1478 y Fk(X)940 1690 y Fn(1)p Fo(\024)p Fm(l)q()g Fv(0)k(to)g(b)s(e)h(c)m(hosen)h (later.)45 b(W)-8 b(e)34 b(assume)g(the)f(n)m(uclei)120 3627 y(are)g(ordered)g(in)f(suc)m(h)i(a)e(w)m(a)m(y)i(that)e(0)c Fp(<)f(\016)1682 3642 y Fn(1)1749 3627 y Fq(\024)i Fp(\016)1898 3642 y Fn(2)1965 3627 y Fq(\024)f Fp(:)17 b(:)g(:)27 b Fq(\024)h Fp(\016)2360 3642 y Fm(M)2472 3627 y Fv(and)33 b(de\014ne)409 3850 y Fp(M)503 3865 y Fn(1)625 3850 y Fv(=)784 3710 y Fk(\032)901 3789 y Fv(max)o Fq(f)p Fp(j)h Fq(2)28 b(f)p Fv(1)p Fp(;)17 b(:)g(:)g(:)32 b(;)17 b(M)10 b Fq(g)28 b Fv(:)f Fp(\016)1913 3804 y Fm(i)1969 3789 y Fq(\024)i Fp(\025)p Fq(g)115 b Fv(if)31 b Fq(9)p Fp(i)i Fv(suc)m(h)h(that)f Fp(\016)2982 3804 y Fm(i)3038 3789 y Fq(\024)28 b Fp(\025)2133 3910 y Fv(0)115 b(otherwise)419 4015 y(~)409 4040 y(\012)479 4055 y Fn(1)625 4040 y Fv(=)784 3965 y Fk(S)867 3991 y Fm(M)935 4000 y Fj(1)867 4069 y Fm(j)t Fn(=1)p 1011 3960 80 4 v 1011 4040 a Fp(B)15 b Fi(\025)p 1100 4052 37 3 v 1103 4093 a Fj(2)1151 4040 y Fv(\()p Fp(R)1263 4055 y Fm(i)1291 4040 y Fv(\))120 4244 y(In)40 b(this)f(w)m(a)m(y)i(the)e(set)995 4218 y(~)985 4244 y(\012)1055 4259 y Fn(1)1134 4244 y Fv(con)m(tains)g(the)h (`bad')g(set)g Fq(R)2192 4259 y Fn(1)2272 4244 y Fv(=)f Fq(f)p Fp(R)2511 4259 y Fm(i)2539 4244 y Fq(g)2589 4200 y Fm(M)2657 4209 y Fj(1)2589 4269 y Fm(i)p Fn(=1)2707 4244 y Fv(.)64 b(Fix)39 b(\011)g Fq(2)h Fp(C)3278 4207 y Fo(1)3271 4268 y Fm(c)3353 4244 y Fv(\()p Fl(R)3457 4207 y Fn(3)3502 4244 y Fv(\))120 4364 y(suc)m(h)34 b(that:)266 4484 y(i.)364 4404 y Fk(R)447 4484 y Fv(\011)27 b(=)h(1.)266 4605 y(ii.)42 b(\011)28 b Fq(\025)g Fv(0)k(and)h(\011)27 b(=)h(0)k(if)f Fq(j)p Fp(x)p Fq(j)d(\025)1502 4566 y Fm(\025)p 1502 4582 42 4 v 1505 4639 a Fn(4)1553 4605 y Fv(.)266 4725 y(iii.)41 b Fq(kr)p Fv(\011)p Fq(k)677 4740 y Fo(1)779 4725 y Fq(\024)894 4686 y Fm(C)p 894 4702 55 4 v 901 4760 a(\025)992 4725 y Fv(with)32 b Fp(C)39 b Fv(indep)s(enden)m(t)34 b(of)e Fp(\025)p Fv(.)266 4846 y(Let)46 b Fp(\020)497 4861 y Fn(1)586 4846 y Fv(:=)j(\011)31 b Fq(\003)g Fp(\037)987 4861 y Fn(\012)1042 4846 y Fv(.)82 b(As)46 b(it)e(is)h(easy)i(to)e(c)m(hec)m(k,)51 b(this)45 b(function)g(has)h(the)g(follo)m(wing)120 4966 y(prop)s(erties:)661 5044 y Fk(8)661 5134 y(>)661 5164 y(<)661 5343 y(>)661 5373 y(:)791 5140 y Fv(supp)18 b Fp(\020)1052 5155 y Fn(1)1118 5140 y Fq(\032)p 1223 5060 80 4 v 28 w Fp(B)e Fi(\025)p 1313 5152 37 3 v 1316 5193 a Fj(4)1386 5140 y Fv(+)22 b(\012)83 b(=)1798 5065 y Fk(S)1881 5092 y Fm(M)1949 5101 y Fj(1)1881 5169 y Fm(j)t Fn(=1)p 2024 5060 80 4 v 2024 5140 a Fp(B)15 b Fj(3)p Fi(\025)p 2113 5152 68 3 v 2131 5193 a Fj(4)2194 5140 y Fv(\()p Fp(R)2306 5155 y Fm(i)2335 5140 y Fv(\))27 b(=)h(\012)2574 5155 y Fn(1)1341 5287 y Fp(\020)1384 5302 y Fn(1)1423 5287 y Fv(\()p Fp(x)p Fv(\))83 b(=)i(1)32 b(if)f Fp(x)d Fq(2)2145 5213 y Fk(S)2228 5239 y Fm(M)2296 5248 y Fj(1)2228 5316 y Fm(j)t Fn(=1)p 2372 5207 80 4 v 2372 5287 a Fp(B)15 b Fi(\025)p 2461 5299 37 3 v 2464 5341 a Fj(4)2512 5287 y Fv(\()p Fp(R)2624 5302 y Fm(i)2652 5287 y Fv(\))28 b(=)2832 5262 y(~)2821 5287 y(\012)2891 5302 y Fn(2)1215 5425 y Fq(kr)p Fp(\020)1391 5440 y Fn(1)1429 5425 y Fq(k)1479 5440 y Fo(1)1637 5425 y Fq(\024)84 b(kr)p Fv(\011)p Fq(k)2057 5440 y Fo(1)3367 5279 y Fv(\(64\))1781 5712 y(15)p eop %%Page: 16 16 16 15 bop 120 407 a Fv(By)33 b(means)g(of)f(function)g Fp(\020)1108 422 y Fn(1)1179 407 y Fv(w)m(e)i(de\014ne)g(also:)1285 507 y Fk(\()1407 593 y Fv(\002)1483 608 y Fn(1)1606 593 y Fv(=)2003 549 y Fm(\020)2034 558 y Fj(1)p 1774 570 523 4 v 1774 631 a Fn(\()p Fm(\020)1837 608 y Fj(2)1832 652 y(1)1872 631 y Fn(+\(1)p Fo(\000)p Fm(\020)2075 640 y Fj(1)2110 631 y Fn(\))2137 612 y Fj(2)2172 631 y Fn(\))2199 611 y Fj(1)p Fi(=)p Fj(2)1407 742 y Fv(\002)1483 757 y Fn(2)1606 742 y Fv(=)1958 698 y Fn(1)p Fo(\000)p Fm(\020)2079 707 y Fj(1)p 1774 719 V 1774 780 a Fn(\()p Fm(\020)1837 757 y Fj(2)1832 801 y(1)1872 780 y Fn(+\(1)p Fo(\000)p Fm(\020)2075 789 y Fj(1)2110 780 y Fn(\))2137 761 y Fj(2)2172 780 y Fn(\))2199 760 y Fj(1)p Fi(=)p Fj(2)3367 677 y Fv(\(65\))120 957 y(So,)f Fq(f)p Fv(\002)409 921 y Fn(2)409 982 y(1)448 957 y Fp(;)17 b Fv(\002)568 921 y Fn(2)568 982 y(2)607 957 y Fq(g)32 b Fv(is)g(a)g(partition)f(of)h(unit)m(y)h (with)f(the)h(prop)s(erties:)849 1063 y Fk(8)849 1153 y(<)849 1332 y(:)979 1143 y Fv(supp)17 b(\002)1272 1158 y Fn(1)1423 1143 y Fq(\032)84 b Fv(\012)1654 1158 y Fn(1)979 1270 y Fv(supp)17 b(\002)1272 1285 y Fn(2)1423 1270 y Fq(\032)84 b Fv(\012)1654 1285 y Fn(2)1721 1270 y Fv(:=)28 b Fl(R)1918 1234 y Fn(3)1985 1270 y Fq(n)22 b Fv(in)m(t)2201 1245 y(~)2190 1270 y(\012)2260 1285 y Fn(2)979 1391 y Fq(kr)p Fv(\002)1188 1406 y Fm(i)1215 1391 y Fq(k)1265 1406 y Fo(1)1423 1391 y Fq(\024)1593 1351 y Fm(C)p 1593 1368 55 4 v 1600 1425 a(\025)1691 1391 y Fv(with)32 b Fp(C)40 b Fv(indep)s(enden)m(t)33 b(of)g Fp(\025)3367 1267 y Fv(\(66\))266 1572 y(The)f(n)m(uclei)d(in)h(the)h(\014rst)f (con\014guration)g Fq(R)1890 1587 y Fn(1)1960 1572 y Fv(shall)f(b)s(e)h(treated)h(b)m(y)g(direct)f(comparison)120 1693 y(to)c(the)g(energy)i(of)d(the)i(corresp)s(onding)f(TF)g(mo)s (del,)g(with)g(a)g(suitably)f(mo)s(di\014ed)g(parameter)120 1813 y Fp(\014)39 b Fv(\(cf.)45 b(10\).)f(Those)34 b(`go)s(o)s(d')f(n)m (uclei)f(of)h(the)g(second)i(con\014guration)d(shall)g(b)s(e)h(treated) g(with)120 1934 y(the)f(Multiscale)e(metho)s(ds)i(in)m(tro)s(duced)g(b) m(y)g(Ivrii&Sigal.)40 b(In)32 b(order)g(to)f(get)h(rid)f(of)g(the)h (bad)120 2054 y(n)m(uclei,)g(w)m(e)i(write)661 2242 y(0)83 b Fq(\024)h Fp(D)s Fv(\()p Fp(\032)1126 2257 y Fm(\015)1192 2242 y Fq(\000)23 b Fp(\032)1342 2195 y Fn(TF)p Fm(;\014)1342 2269 y Fo(R)1402 2278 y Fj(1)1437 2269 y Fm(;)p Fo(Z)1508 2278 y Fj(1)1568 2242 y Fq(\000)g Fp(\032)1718 2201 y Fn(TF)1718 2267 y Fo(R)1778 2276 y Fj(2)1813 2267 y Fm(;)p Fo(Z)1884 2276 y Fj(2)1922 2242 y Fv(\))794 2409 y(=)84 b Fp(D)s Fv(\()p Fp(\032)1126 2424 y Fm(\015)1170 2409 y Fv(\))22 b(+)g Fp(D)s Fv(\()p Fp(\032)1500 2362 y Fn(TF)p Fm(;\014)1500 2436 y Fo(R)1560 2445 y Fj(1)1595 2436 y Fm(;)p Fo(Z)1666 2445 y Fj(1)1704 2409 y Fv(\))g(+)g Fp(D)s Fv(\()p Fp(\032)2034 2368 y Fn(TF)2034 2434 y Fo(R)2094 2443 y Fj(2)2129 2434 y Fm(;)p Fo(Z)2200 2443 y Fj(2)2238 2409 y Fv(\))g Fq(\000)h Fv(2)p Fp(D)s Fv(\()p Fp(\032)2619 2424 y Fm(\015)2663 2409 y Fp(;)17 b(\032)2757 2362 y Fn(TF)p Fm(;\014)2757 2436 y Fo(R)2817 2445 y Fj(1)2852 2436 y Fm(;)p Fo(Z)2923 2445 y Fj(1)2961 2409 y Fv(\))368 b(\(67\))954 2575 y Fq(\000)p Fv(2)p Fp(D)s Fv(\()p Fp(\032)1252 2590 y Fm(\015)1296 2575 y Fp(;)17 b(\032)1390 2534 y Fn(TF)1390 2600 y Fo(R)1450 2609 y Fj(2)1485 2600 y Fm(;)p Fo(Z)1556 2609 y Fj(2)1594 2575 y Fv(\))22 b(+)g(2)p Fp(D)s Fv(\()p Fp(\032)1973 2528 y Fn(TF)p Fm(;\014)1973 2602 y Fo(R)2033 2611 y Fj(1)2068 2602 y Fm(;)p Fo(Z)2139 2611 y Fj(1)2177 2575 y Fp(;)17 b(\032)2271 2534 y Fn(TF)2271 2600 y Fo(R)2331 2609 y Fj(2)2366 2600 y Fm(;)p Fo(Z)2437 2609 y Fj(2)2475 2575 y Fv(\))120 2764 y(In)39 b(\(67\),)i Fp(\014)j(>)39 b Fv(0)f(is)h(some)g(\014xed)h(parameter)f(to)f(b)s(e)i(c)m(hosen)g (later.)62 b(Then)41 b(a)d(calculation)120 2884 y(using)32 b(the)h(TF)g(equation)f(\(13\))g(and)h(\(67\))f(yields)359 3073 y Fq(E)421 3032 y Fn(RHF)412 3098 y Fo(R)p Fm(;)p Fo(Z)575 3073 y Fv(\()p Fp(\015)5 b Fv(\))96 b Fq(\025)g Fv(T)-8 b(r)17 b([\()p Fq(\000)p Fv(\001)23 b Fq(\000)g Fv(\010)1509 3026 y Fn(TF)p Fm(;\014)1509 3100 y Fo(R)1569 3109 y Fj(1)1604 3100 y Fm(;)p Fo(Z)1675 3109 y Fj(1)1735 3073 y Fq(\000)g Fv(\010)1905 3032 y Fn(TF)1905 3098 y Fo(R)1965 3107 y Fj(2)2000 3098 y Fm(;)p Fo(Z)2071 3107 y Fj(2)2109 3073 y Fv(\))p Fp(\015)5 b Fv(])22 b Fq(\000)h Fp(D)s Fv(\()p Fp(\032)2524 3026 y Fn(TF)p Fm(;\014)2524 3100 y Fo(R)2584 3109 y Fj(1)2619 3100 y Fm(;)p Fo(Z)2690 3109 y Fj(1)2728 3073 y Fv(\))f Fq(\000)g Fp(D)s Fv(\()p Fp(\032)3059 3032 y Fn(TF)3059 3098 y Fo(R)3119 3107 y Fj(2)3154 3098 y Fm(;)p Fo(Z)3225 3107 y Fj(2)3263 3073 y Fv(\))976 3218 y(+)p Fp(U)1118 3233 y Fo(R)1178 3242 y Fj(1)1213 3233 y Fm(;)p Fo(Z)1284 3242 y Fj(1)1344 3218 y Fv(+)g Fp(U)1508 3233 y Fo(R)1568 3242 y Fj(2)1603 3233 y Fm(;)p Fo(Z)1674 3242 y Fj(2)1735 3218 y Fv(+)g(\001)p Fp(U)1980 3233 y Fn(1)p Fm(;)p Fn(2)437 3415 y Fp(U)503 3430 y Fo(R)563 3439 y Fj(1)598 3430 y Fm(;)p Fo(Z)669 3439 y Fj(1)790 3415 y Fv(:=)1083 3320 y Fk(X)976 3532 y Fn(1)p Fo(\024)p Fm(l)q()h Fv(0)35 b Fr(only)f(dep)-5 b(ending)34 b(on)g Fp(\025;)17 b(\014)6 b Fr(.)44 b(The)34 b(function)h Fv(\000)g Fr(is)f(as)h(The)-5 b(or)g(em)34 b(1.)266 5012 y Fv(If)48 b Fp(x)54 b Fq(2)g Fv(supp)17 b(\002)901 5027 y Fn(1)941 5012 y Fv(,)51 b(b)m(y)e(Lemma)d(1)h(and)h(the)g(scaling)f (relation)f(\(14\),)51 b(the)d(follo)m(wing)120 5132 y(estimate)22 b(holds:)38 b(\(here)23 b Fp(\014)33 b Fv(=)28 b Fp(\024\014)1331 5147 y Fm(f)1399 5132 y Fv(where)c Fp(\014)1726 5147 y Fm(f)1794 5132 y Fv(is)e(some)g(\014xed)i (parameter)e(tak)m(en)i(as)e(reference)120 5252 y(and)33 b(0)27 b Fp(<)h(\024)g Fq(\024)g Fv(1\))566 5463 y(\010)636 5416 y Fn(TF)p Fm(;\014)636 5490 y Fo(R)696 5499 y Fj(1)731 5490 y Fm(;)p Fo(Z)802 5499 y Fj(1)840 5463 y Fv(\()p Fp(x)p Fv(\))22 b(+)g(\010)1161 5422 y Fn(TF)1161 5487 y Fo(R)1221 5496 y Fj(2)1257 5487 y Fm(;)p Fo(Z)1328 5496 y Fj(2)1366 5463 y Fv(\()p Fp(x)p Fv(\))83 b(=)g Fp(\024)1795 5422 y Fo(\000)p Fn(1)1889 5463 y Fv(\010)1959 5422 y Fn(TF)1959 5489 y Fm(\024)2000 5470 y Fh(\000)p Fj(1)2083 5489 y Fo(R)2143 5498 y Fj(1)2178 5489 y Fm(;)p Fo(Z)2249 5498 y Fj(1)2287 5463 y Fv(\()p Fp(\024)2381 5422 y Fo(\000)p Fn(1)2475 5463 y Fp(x)p Fv(\))23 b(+)f(\010)2759 5422 y Fn(TF)2759 5487 y Fo(R)2819 5496 y Fj(2)2854 5487 y Fm(;)p Fo(Z)2925 5496 y Fj(2)2963 5463 y Fv(\()p Fp(x)p Fv(\))273 b(\(75\))1781 5712 y(17)p eop %%Page: 18 18 18 17 bop 120 407 a Fv(But)36 b(with)g Fp(\032)p Fv(\()p Fp(x)p Fv(\))d(=)h(min)n Fq(fj)p Fp(x)24 b Fq(\000)h Fp(R)1362 422 y Fm(j)1399 407 y Fq(j)33 b Fv(:)50 b Fp(j)39 b Fv(=)33 b(1)p Fp(;)17 b(:)g(:)g(:)33 b(;)17 b(M)2104 422 y Fn(1)2143 407 y Fq(g)36 b Fv(and)g Fp(\032)2472 371 y Fo(0)2495 407 y Fv(\()p Fp(x)p Fv(\))e(=)f(min)o Fq(fj)p Fp(x)24 b Fq(\000)h Fp(\024)3247 371 y Fo(\000)p Fn(1)3341 407 y Fp(R)3415 422 y Fm(j)3452 407 y Fq(j)33 b Fv(:)120 527 y Fp(j)h Fv(=)27 b(1)p Fp(;)17 b(:)g(:)g(:)32 b(;)17 b(M)675 542 y Fn(1)715 527 y Fq(g)p Fv(,)32 b(using)h(Lemma)e(1) h(and)h(again)e(\(14\))h(w)m(e)h(get)672 723 y Fp(\024)728 682 y Fo(\000)p Fn(1)822 723 y Fv(\010)892 682 y Fn(TF)892 749 y Fm(\024)933 730 y Fh(\000)p Fj(1)1016 749 y Fo(R)1076 758 y Fj(1)1111 749 y Fm(;)p Fo(Z)1182 758 y Fj(1)1220 723 y Fv(\()p Fp(\024)1314 682 y Fo(\000)p Fn(1)1408 723 y Fp(x)p Fv(\))84 b Fq(\025)f Fp(c\024)1843 682 y Fo(\000)p Fn(1)1937 723 y Fp(Z)7 b(\032)2061 682 y Fo(0)2085 723 y Fv(\()p Fp(\024)2179 682 y Fo(\000)p Fn(1)2273 723 y Fp(x)p Fv(\))2366 682 y Fo(\000)p Fn(1)2461 723 y Fq(h)p Fp(\032)2550 682 y Fo(0)2573 723 y Fv(\()p Fp(\024)2667 682 y Fo(\000)p Fn(1)2762 723 y Fp(x)p Fv(\))p Fq(i)2894 682 y Fo(\000)p Fn(3)1585 886 y Fv(=)84 b Fp(cZ)7 b(\032)p Fv(\()p Fp(x)p Fv(\))2042 845 y Fo(\000)p Fn(1)2137 886 y Fq(h)p Fp(\024)2232 845 y Fo(\000)p Fn(1)2326 886 y Fp(Z)2400 845 y Fn(1)p Fm(=)p Fn(3)2510 886 y Fp(\032)p Fv(\()p Fp(x)p Fv(\))p Fq(i)2730 845 y Fo(\000)p Fn(3)3367 886 y Fv(\(76\))1585 1031 y Fq(\025)83 b Fp(cZ)7 b(\025)1918 990 y Fo(\000)p Fn(1)2012 1031 y Fq(h)p Fp(\024)2107 990 y Fo(\000)p Fn(1)2201 1031 y Fp(Z)2275 990 y Fn(1)p Fm(=)p Fn(3)2385 1031 y Fp(\025)p Fq(i)2481 990 y Fo(\000)p Fn(3)1585 1177 y Fq(\025)83 b Fp(cZ)7 b(\025)1918 1136 y Fo(\000)p Fn(1)2012 1177 y Fq(h)p Fp(\024)2107 1136 y Fo(\000)p Fn(1)2201 1177 y Fp(\026)p Fq(i)2299 1136 y Fo(\000)p Fn(3)120 1373 y Fv(A)33 b(similar)c(estimate)j(giv)m(es) 1333 1569 y(\010)1403 1528 y Fn(TF)1403 1593 y Fo(R)1463 1602 y Fj(2)1498 1593 y Fm(;)p Fo(Z)1569 1602 y Fj(2)1608 1569 y Fv(\()p Fp(x)p Fv(\))c Fq(\024)g Fp(C)7 b(\026)2008 1528 y Fo(\000)p Fn(3)2102 1569 y Fp(Z)g(\025)2233 1528 y Fo(\000)p Fn(1)3367 1569 y Fv(\(77\))120 1765 y(and)33 b(therefore)562 1961 y(\010)632 1914 y Fn(TF)q Fm(;\014)632 1988 y Fo(R)692 1997 y Fj(1)728 1988 y Fm(;)p Fo(Z)799 1997 y Fj(1)837 1961 y Fv(\()p Fp(x)p Fv(\))22 b(+)g(\010)1158 1920 y Fn(TF)1158 1986 y Fo(R)1218 1995 y Fj(2)1253 1986 y Fm(;)p Fo(Z)1324 1995 y Fj(2)1363 1961 y Fv(\()p Fp(x)p Fv(\))83 b Fq(\024)g Fv(\(1)22 b(+)g Fp(C)7 b(\026)2080 1920 y Fo(\000)p Fn(3)2174 1961 y Fq(h)p Fp(\024)2269 1920 y Fo(\000)p Fn(1)2363 1961 y Fp(\026)p Fq(i)2461 1920 y Fn(3)2500 1961 y Fv(\)\010)2608 1914 y Fn(TF)q Fm(;\014)2608 1988 y Fo(R)2668 1997 y Fj(1)2704 1988 y Fm(;)p Fo(Z)2775 1997 y Fj(1)2813 1961 y Fv(\()p Fp(x)p Fv(\))1577 2127 y Fq(\024)83 b Fp(C)7 b(\026)1873 2086 y Fo(\000)p Fn(3)1967 2127 y Fq(h)p Fp(\024)2062 2086 y Fo(\000)p Fn(1)2157 2127 y Fp(\026)p Fq(i)2255 2086 y Fn(3)2293 2127 y Fv(\010)2363 2080 y Fn(TF)q Fm(;\014)2363 2154 y Fo(R)2423 2163 y Fj(1)2459 2154 y Fm(;)p Fo(Z)2530 2163 y Fj(1)2568 2127 y Fv(\()p Fp(x)p Fv(\))668 b(\(78\))1577 2294 y Fq(\024)83 b Fp(C)7 b(\026)1873 2253 y Fo(\000)p Fn(3)1967 2294 y Fv(\010)2037 2247 y Fn(TF)q Fm(;\014)2037 2321 y Fo(R)2097 2330 y Fj(1)2133 2321 y Fm(;)p Fo(Z)2204 2330 y Fj(1)2242 2294 y Fv(\()p Fp(x)p Fv(\))33 b(if)e(w)m(e)j(tak)m(e) f Fp(\026)27 b Fq(\024)h Fp(\024)120 2511 y Fv(No)m(w)35 b(w)m(e)g(are)f(in)f(p)s(osition)g(to)h(estimate)f(T)-8 b(r)17 b([\()p Fq(\000)p Fv(\001)24 b Fq(\000)g Fv(\(\010)2228 2464 y Fn(TF)p Fm(;\014)2228 2538 y Fo(R)2288 2547 y Fj(1)2323 2538 y Fm(;)p Fo(Z)2394 2547 y Fj(1)2455 2511 y Fv(+)f(\010)2624 2475 y Fn(TF)2624 2536 y Fo(R)2684 2545 y Fj(2)2719 2536 y Fm(;)p Fo(Z)2790 2545 y Fj(2)2828 2511 y Fv(\)\002)2942 2475 y Fn(2)2942 2536 y(1)2982 2511 y Fv(\))p Fp(\015)5 b Fv(])34 b(b)m(y)h(means)120 2631 y(of)d(the)h(Lieb-Thirring)d(inequalit)m(y:)429 2828 y(T)-8 b(r)17 b([\()p Fq(\000)p Fv(\001)23 b Fq(\000)g Fv(\(\010)1000 2780 y Fn(TF)p Fm(;\014)1000 2855 y Fo(R)1060 2864 y Fj(1)1095 2855 y Fm(;)p Fo(Z)1166 2864 y Fj(1)1226 2828 y Fv(+)f(\010)1394 2786 y Fn(TF)1394 2852 y Fo(R)1454 2861 y Fj(2)1490 2852 y Fm(;)p Fo(Z)1561 2861 y Fj(2)1599 2828 y Fv(\)\002)1713 2786 y Fn(2)1713 2852 y(1)1752 2828 y Fv(\))p Fp(\015)5 b Fv(])83 b Fq(\025)g(\000)p Fv(T)-8 b(r)18 b(\()p Fq(\000)p Fv(\001)23 b Fq(\000)f Fp(C)7 b(\026)2765 2786 y Fo(\000)p Fn(3)2859 2828 y Fv(\010)2929 2780 y Fn(TF)q Fm(;\014)2929 2855 y Fo(R)2989 2864 y Fj(1)3025 2855 y Fm(;)p Fo(Z)3096 2864 y Fj(1)3134 2828 y Fv(\))3172 2843 y Fo(\000)1956 3036 y Fq(\025)83 b(\000)p Fp(K)7 b(\026)2342 2995 y Fo(\000)p Fn(5)2454 2901 y Fk(Z)2553 3036 y Fv(\(\010)2661 2989 y Fn(TF)q Fm(;\014)2661 3063 y Fo(R)2721 3072 y Fj(1)2757 3063 y Fm(;)p Fo(Z)2828 3072 y Fj(1)2866 3036 y Fv(\))2904 2995 y Fn(5)p Fm(=)p Fn(2)3367 3036 y Fv(\(79\))266 3275 y(It)31 b(will)d(b)s(e)i(useful)h(for)f(us)h(to)f(b)s(e)g(able)g(to)g (express)j(the)d(TF)h(energy)g(only)f(using)g(the)h(TF)120 3396 y(p)s(oten)m(tial,)d(the)g(Coulom)m(b)g(self-energy)h(of)e(the)i (densit)m(y)h(and)e(the)h(n)m(uclear)f(repulsion.)42 b(This)120 3516 y(is)32 b(an)h(easy)g(calculation)d(using)j(the)g(TF)f (equation)h(and)f(is)g(as)h(follo)m(ws:)120 3699 y Ft(Lemma)k(6)h (\(Expression)e(of)i(the)f(TF)h(energy\).)49 b Fr(It)43 b(is)h(p)-5 b(ossible)42 b(to)i(expr)-5 b(ess)42 b(the)i(TF)120 3819 y(ener)-5 b(gy)35 b Fp(E)506 3772 y Fn(TF)p Fm(;\014)500 3846 y Fo(R)p Fm(;)p Fo(Z)704 3819 y Fr(in)g(the)g(fol)5 b(lowing)33 b(way,)i(wher)-5 b(e)34 b Fp(\014)f Fv(=)28 b Fp(\024\014)2204 3834 y Fm(f)2284 3819 y Fr(:)642 4111 y Fp(E)720 4064 y Fn(TF)p Fm(;\014)714 4138 y Fo(R)p Fm(;)p Fo(Z)911 4111 y Fv(=)g Fq(\000)1102 4044 y Fv(2)p 1102 4088 49 4 v 1102 4179 a(5)1178 3971 y Fk(\022)1339 4044 y Fv(3)p 1261 4088 206 4 v 1261 4179 a(5)p Fp(\024\014)1421 4194 y Fm(f)1476 3971 y Fk(\023)1550 3993 y Fn(3)p Fm(=)p Fn(2)1676 3975 y Fk(Z)1776 4111 y Fv(\(\010)1884 4064 y Fn(TF)p Fm(;\014)1884 4138 y Fo(R)p Fm(;)p Fo(Z)2048 4111 y Fv(\))2086 4070 y Fn(5)p Fm(=)p Fn(2)2218 4111 y Fq(\000)23 b Fp(D)s Fv(\()p Fp(\032)2490 4064 y Fn(TF)p Fm(;\014)2490 4138 y Fo(R)p Fm(;)p Fo(Z)2654 4111 y Fv(\))f(+)g Fp(U)2878 4126 y Fo(R)p Fm(;)p Fo(Z)3367 4111 y Fv(\(80\))266 4366 y(F)-8 b(rom)31 b(Lemma)h(6)g(and)h(\(79\))e(w)m(e)j(get)f(the)g (estimate)1514 4562 y Fq(E)1567 4577 y Fn(1)1606 4562 y Fv(\()p Fp(\015)5 b Fv(\))27 b Fq(\025)h Fp(E)1948 4515 y Fn(TF)p Fm(;\014)1942 4589 y Fo(R)2002 4598 y Fj(1)2037 4589 y Fm(;)p Fo(Z)2108 4598 y Fj(1)3367 4562 y Fv(\(81\))120 4758 y(assuming)k(w)m(e)h(ha)m(v)m(e)h(c)m(hosen)g Fp(\026)28 b Fq(\024)g Fp(\024)33 b Fv(in)e(suc)m(h)k(w)m(a)m(y)e(that) 319 5059 y Fp(K)7 b(\026)468 5018 y Fo(\000)p Fn(5)589 5059 y Fv(=)703 4992 y(2)p 703 5036 49 4 v 703 5128 a(5)778 4919 y Fk(\022)940 4992 y Fv(3)p 862 5036 206 4 v 862 5128 a(5)p Fp(\024\014)1022 5143 y Fm(f)1077 4919 y Fk(\023)1150 4941 y Fn(3)p Fm(=)p Fn(2)1288 5059 y Fq(\()-17 b(\))27 b Fp(\026)h Fv(=)f Fp(K)1778 5018 y Fo(0)1802 5059 y Fp(\024)1858 5018 y Fn(3)p Fm(=)p Fn(10)2003 5059 y Fv(;)49 b Fp(K)2169 5018 y Fo(0)2220 5059 y Fv(=)2324 4889 y Fk(")2392 4992 y Fv(5)p Fp(K)p 2392 5036 139 4 v 2437 5128 a Fv(2)2557 4919 y Fk(\022)2641 4992 y Fv(5)p Fp(\014)2745 5007 y Fm(f)p 2641 5036 150 4 v 2691 5128 a Fv(3)2800 4919 y Fk(\023)2873 4941 y Fn(3)p Fm(=)p Fn(2)2983 4889 y Fk(#)3041 4911 y Fn(1)p Fm(=)p Fn(5)3367 5059 y Fv(\(82\))120 5342 y(As)34 b(the)g(exp)s(onen)m(t)h(3)p Fp(=)p Fv(10)29 b Fp(<)h Fv(1,)j(the)h(c)m(hoice)h(of)e Fp(\026)g Fv(is)g(consisten)m (t.)48 b(No)m(w)34 b(w)m(e)h(are)f(in)f(p)s(osition)120 5463 y(to)f(pro)m(v)m(e)i(Lemma)d(5:)1781 5712 y(18)p eop %%Page: 19 19 19 18 bop 120 407 a Fr(Pr)-5 b(o)g(of)35 b(of)f(L)-5 b(emma)34 b(5.)49 b Fv(By)35 b(the)g(scaling)e(relation)g(\(12\))g(of)h (the)h(TF)g(energy)g(and)g(Lemma)e(2)120 527 y(of)f([2],)h(w)m(e)g(get) 566 747 y Fp(E)644 700 y Fn(TF)p Fm(;\014)638 774 y Fo(R)698 783 y Fj(1)733 774 y Fm(;)p Fo(Z)804 783 y Fj(1)925 747 y Fq(\025)84 b Fp(\024)1142 706 y Fo(\000)p Fn(1)1236 747 y Fp(E)1314 706 y Fn(TF)1308 773 y Fm(\024)1349 754 y Fh(\000)p Fj(1)1432 773 y Fo(R)1492 782 y Fj(1)1526 773 y Fm(;)p Fo(Z)1597 782 y Fj(1)925 1001 y Fq(\025)g Fp(\024)1142 959 y Fo(\000)p Fn(1)1253 830 y Fk( )1332 1001 y Fq(\000)p Fp(C)1479 1016 y Fn(TF)1617 876 y Fm(M)1685 885 y Fj(1)1597 906 y Fk(X)1607 1116 y Fm(j)t Fn(=1)1757 1001 y Fp(Z)1831 950 y Fn(7)p Fm(=)p Fn(3)1824 1026 y Fm(j)1963 1001 y Fv(+)22 b Fp(cZ)2177 959 y Fn(7)p Fm(=)p Fn(3)2324 876 y Fm(M)2392 885 y Fj(1)2304 906 y Fk(X)2314 1116 y Fm(j)t Fn(=1)2464 1001 y Fv(\000\()p Fp(\024)2619 959 y Fo(\000)p Fn(1)2714 1001 y Fp(Z)2788 959 y Fn(1)p Fm(=)p Fn(3)2897 1001 y Fp(\016)2940 1016 y Fm(j)2977 1001 y Fv(\))3015 830 y Fk(!)3367 1001 y Fv(\(83\))120 1333 y(But)41 b Fp(\016)365 1348 y Fm(j)444 1333 y Fq(\024)i Fp(\026Z)697 1297 y Fo(\000)p Fn(1)p Fm(=)p Fn(3)861 1333 y Fv(,)h(so)d(that)g Fp(\024)1336 1297 y Fo(\000)p Fn(1)1430 1333 y Fp(Z)1504 1297 y Fn(1)p Fm(=)p Fn(3)1614 1333 y Fp(\016)1657 1348 y Fm(j)1736 1333 y Fq(\024)i Fp(K)1946 1297 y Fo(0)1969 1333 y Fp(\024)2025 1297 y Fn(3)p Fm(=)p Fn(10)2213 1333 y Fq(#)f Fv(0)e(\()p Fp(\024)j Fq(#)f Fv(0\))e(so)i(that,)h(b)m(y)f(taking)120 1453 y Fp(\024)28 b(>)f Fv(0)33 b(small)d(enough)j(w)m(e)h(get)191 1761 y Fp(E)269 1713 y Fn(TF)p Fm(;\014)263 1788 y Fo(R)323 1797 y Fj(1)358 1788 y Fm(;)p Fo(Z)429 1797 y Fj(1)550 1761 y Fq(\025)84 b Fp(\024)767 1719 y Fo(\000)p Fn(1)861 1761 y Fp(Z)935 1719 y Fn(7)p Fm(=)p Fn(3)1062 1680 y Fk(\000)1107 1761 y Fq(\000)p Fp(C)1254 1776 y Fn(TF)1356 1761 y Fp(A)1429 1719 y Fn(7)p Fm(=)p Fn(3)1561 1761 y Fv(+)22 b Fp(K)1749 1719 y Fo(00)1792 1761 y Fp(\024)1848 1719 y Fo(\000)p Fn(3)p Fm(=)p Fn(10)2048 1680 y Fk(\001)2110 1761 y Fp(M)2204 1776 y Fn(1)2266 1761 y Fv(+)g Fp(c\024)2462 1719 y Fo(\000)p Fn(1)2557 1761 y Fp(Z)2631 1719 y Fn(7)p Fm(=)p Fn(3)2778 1636 y Fm(M)2846 1645 y Fj(1)2757 1666 y Fk(X)2768 1876 y Fm(j)t Fn(=1)2918 1761 y Fv(\000\()p Fp(\024)3073 1719 y Fo(\000)p Fn(1)3167 1761 y Fp(Z)3241 1719 y Fn(1)p Fm(=)p Fn(3)3351 1761 y Fp(\016)3394 1776 y Fm(j)3431 1761 y Fv(\))550 2112 y Fq(\025)84 b Fp(c\024)809 2071 y Fo(\000)p Fn(1)903 2112 y Fp(Z)977 2071 y Fn(7)p Fm(=)p Fn(3)1125 1987 y Fm(M)1193 1996 y Fj(1)1104 2018 y Fk(X)1114 2228 y Fm(j)t Fn(=1)1264 2112 y Fv(\000\()p Fp(\024)1419 2071 y Fo(\000)p Fn(1)1514 2112 y Fp(Z)1588 2071 y Fn(1)p Fm(=)p Fn(3)1698 2112 y Fp(\016)1741 2127 y Fm(j)1777 2112 y Fv(\))33 b(for)f Fp(\024)c(>)f Fv(0)32 b(small)f(enough)550 2464 y Fq(\025)84 b Fp(cZ)827 2423 y Fn(7)p Fm(=)p Fn(3)974 2339 y Fm(M)1042 2348 y Fj(1)953 2369 y Fk(X)964 2579 y Fm(j)t Fn(=1)1114 2464 y Fv(\000\()p Fp(Z)1287 2423 y Fn(1)p Fm(=)p Fn(3)1397 2464 y Fp(\016)1440 2479 y Fm(j)1476 2464 y Fv(\))1853 b(\(84\))120 2779 y(whic)m(h)33 b(concludes)h(the)f(pro)s(of.)p 3469 2779 4 66 v 3473 2717 59 4 v 3473 2779 V 3531 2779 4 66 v 266 2974 a(T)-8 b(o)33 b(end)g(this)f(section)h(w)m(e)g(are)f(going)f (to)h(analyze)h(the)g(TF)f(energy)i(of)e(a)g(con\014guration)120 3095 y(of)g(uniformly)e(separated)k(n)m(uclei.)43 b(This)33 b(is)f(the)h(con)m(ten)m(t)h(of)e(next)h(Lemma:)120 3298 y Ft(Lemma)k(7)h(\(Bound)f(of)h(the)f(energy)h(of)g(uniformly)d (separated)k(n)m(uclei\).)47 b Fr(L)-5 b(et)25 b(us)g(as-)120 3418 y(sume)50 b(we)g(ar)-5 b(e)51 b(given)e(a)h(nucle)-5 b(ar)51 b(c)-5 b(on\014gur)g(ation)49 b Fq(R)p Fp(;)17 b Fq(Z)59 b Fr(such)50 b(that)h Fp(\016)2810 3433 y Fm(j)2903 3418 y Fq(\025)58 b Fp(\026Z)3171 3382 y Fo(\000)p Fn(1)p Fm(=)p Fn(3)3386 3418 y Fr(and)120 3539 y Fp(Z)f Fq(\024)52 b Fp(Z)440 3554 y Fm(j)526 3539 y Fq(\024)f Fp(AZ)k Fr(for)47 b(al)5 b(l)47 b(j.)81 b(Then)47 b(the)g(next)g(upp)-5 b(er)47 b(b)-5 b(ound)47 b(of)g(the)g(total)h(TF)e(ener)-5 b(gy)120 3659 y Fp(E)198 3623 y Fn(TF)192 3684 y Fo(R)p Fm(;)p Fo(Z)367 3659 y Fr(holds:)1033 3985 y Fp(E)1111 3944 y Fn(TF)1105 4010 y Fo(R)p Fm(;)p Fo(Z)1273 3985 y Fq(\024)28 b(\000)p Fp(C)1525 4000 y Fn(TF)1678 3861 y Fm(M)1643 3890 y Fk(X)1654 4100 y Fm(j)t Fn(=1)1804 3985 y Fp(Z)1878 3934 y Fn(7)p Fm(=)p Fn(3)1871 4011 y Fm(j)2010 3985 y Fv(+)22 b Fp(C)7 b(Z)2259 3944 y Fn(7)p Fm(=)p Fn(3)2369 3985 y Fp(\026)2428 3944 y Fo(\000)p Fn(3)2522 3985 y Fp(M)751 b Fv(\(85\))120 4306 y Fr(Pr)-5 b(o)g(of.)48 b Fv(T)-8 b(o)34 b(pro)m(v)m(e)i(estimate)d(\(85\))g(is)h (enough)g(to)g(construct)h(a)f(trial)d(densit)m(y)k Fp(\032)g Fv(suc)m(h)g(that)120 4440 y Fq(E)182 4404 y Fn(TF)173 4465 y Fo(R)p Fm(;)p Fo(Z)313 4440 y Fv(\()p Fp(\032)p Fv(\))27 b Fq(\024)i(\000)p Fp(C)719 4455 y Fn(TF)837 4365 y Fk(P)942 4391 y Fm(M)942 4469 y(j)t Fn(=1)1085 4440 y Fp(Z)1159 4389 y Fn(7)p Fm(=)p Fn(3)1152 4465 y Fm(j)1291 4440 y Fv(+)22 b Fp(C)7 b(Z)1540 4404 y Fn(7)p Fm(=)p Fn(3)1650 4440 y Fp(\026)1709 4404 y Fo(\000)p Fn(4)1803 4440 y Fp(M)j Fv(.)44 b(T)-8 b(o)33 b(do)f(so,)h(tak)m(e)392 4770 y Fp(\032)28 b Fv(=)608 4645 y Fm(M)573 4675 y Fk(X)584 4885 y Fm(j)t Fn(=1)734 4770 y Fp(\025)791 4785 y Fm(j)827 4770 y Fp(\032)877 4729 y Fn(TF)877 4794 y Fm(Z)925 4804 y Fi(j)978 4770 y Fv(\()p Fq(\001)22 b(\000)h Fp(R)1240 4785 y Fm(j)1277 4770 y Fv(\))p Fp(\037)p 1376 4741 57 3 v 26 x Fm(B)1442 4808 y Fj(1)p 1442 4816 31 3 v 1442 4851 a(2)1483 4828 y Fi(\026Z)1565 4810 y Fh(\000)p Fj(1)p Fi(=)p Fj(3)1746 4770 y Fv(=)1884 4645 y Fm(M)1849 4675 y Fk(X)1860 4885 y Fm(j)t Fn(=1)2010 4770 y Fp(\032)2060 4785 y Fm(j)2097 4770 y Fv(;)49 b Fp(\025)2230 4785 y Fm(j)2294 4770 y Fv(=)2398 4629 y Fk(\024)2450 4634 y(Z)2566 4770 y Fp(\032)2616 4729 y Fn(TF)2616 4794 y Fm(Z)2664 4804 y Fi(j)2718 4770 y Fp(\037)p 2779 4741 57 3 v 26 x Fm(B)2845 4808 y Fj(1)p 2845 4816 31 3 v 2845 4851 a(2)2886 4828 y Fi(\026Z)2968 4810 y Fh(\000)p Fj(1)p Fi(=)p Fj(3)3122 4629 y Fk(\025)3174 4651 y Fo(\000)p Fn(1)120 5102 y Fv(Since)390 5021 y Fk(R)473 5102 y Fp(\032)523 5066 y Fn(TF)523 5127 y Fm(Z)678 5102 y Fv(=)54 b Fp(Z)h Fv(and)48 b Fp(\032)1185 5066 y Fn(TF)1185 5127 y Fm(Z)1287 5102 y Fv(\()p Fp(x)p Fv(\))54 b(=)g Fp(Z)1676 5066 y Fn(2)1715 5102 y Fp(\032)1765 5066 y Fn(TF)1765 5126 y(1)1866 5102 y Fv(\()p Fp(Z)1978 5066 y Fn(1)p Fm(=)p Fn(3)2088 5102 y Fp(x)p Fv(\),)e(w)m(e)e(get,)h(using)d(that)g Fp(\032)3176 5066 y Fn(TF)3176 5126 y(1)3277 5102 y Fv(\()p Fp(x)p Fv(\))55 b Fq(\030)120 5222 y(j)p Fp(x)p Fq(j)231 5186 y Fo(\000)p Fn(6)325 5222 y Fv(;)33 b Fq(j)p Fp(x)p Fq(j)28 b(\025)g Fp(\026)p Fv(,)k Fp(\025)804 5237 y Fm(j)863 5222 y Fq(\000)23 b Fv(1)k Fq(\030)h Fp(\026)1203 5186 y Fo(\000)p Fn(3)1297 5222 y Fv(.)266 5343 y(With)k(this)h(c)m (hoice)f(of)h Fp(\032)p Fv(,)f(w)m(e)i(get:)1781 5712 y(19)p eop %%Page: 20 20 20 19 bop 266 407 a Fv(i.-)1122 679 y Fk(Z)1238 815 y Fp(\032)1288 774 y Fn(5)p Fm(=)p Fn(3)1426 815 y Fq(\024)1566 690 y Fm(M)1531 720 y Fk(X)1542 930 y Fm(j)t Fn(=1)1692 815 y Fp(\025)1749 764 y Fn(5)p Fm(=)p Fn(3)1749 841 y Fm(j)1875 679 y Fk(Z)1992 704 y(\020)2051 815 y Fp(\032)2101 774 y Fn(TF)2101 840 y Fm(Z)2149 850 y Fi(j)2203 704 y Fk(\021)2262 727 y Fn(5)p Fm(=)p Fn(3)266 1127 y Fv(ii.-)396 1400 y Fk(Z)512 1535 y Fp(V)569 1550 y Fo(R)p Fm(;)p Fo(Z)709 1535 y Fp(\032)84 b Fv(=)1038 1411 y Fm(M)1003 1441 y Fk(X)1014 1651 y Fm(j)t Fn(=1)1163 1535 y Fp(Z)1230 1550 y Fm(j)1267 1535 y Fq(j)22 b(\001)f(j)1394 1494 y Fo(\000)p Fn(1)1511 1535 y Fq(\003)g Fp(\032)1632 1550 y Fm(j)1669 1535 y Fv(\()p Fp(R)1781 1550 y Fm(j)1818 1535 y Fv(\))h(+)g(2)p Fp(U)2091 1550 y Fo(R)p Fm(;)p Fo(Z)843 1887 y Fv(=)1038 1762 y Fm(M)1003 1792 y Fk(X)1014 2002 y Fm(j)t Fn(=1)1163 1887 y Fp(\025)1220 1902 y Fm(j)1274 1746 y Fk(\024)1326 1751 y(Z)1456 1819 y Fp(Z)1523 1834 y Fm(j)p 1452 1864 111 4 v 1452 1955 a Fq(j)p Fp(x)p Fq(j)1573 1887 y Fp(\032)1623 1846 y Fn(TF)1623 1911 y Fm(Z)1671 1921 y Fi(j)1747 1887 y Fq(\000)g(j)g(\001)g(j)1974 1846 y Fo(\000)p Fn(1)2090 1887 y Fq(\003)g Fv([)p Fp(\032)2239 1902 y Fm(j)2276 1887 y Fp(\037)2337 1902 y Fm(B)2393 1879 y Fi(c)2400 1926 y Fj(1)p 2400 1934 31 3 v 2400 1969 a(2)2441 1946 y Fi(\026Z)2523 1928 y Fh(\000)p Fj(1)p Fi(=)p Fj(3)2676 1887 y Fv(]\(0\))2828 1746 y Fk(\025)2903 1887 y Fv(+)g(2)p Fp(U)3116 1902 y Fo(R)p Fm(;)p Fo(Z)842 2238 y Fq(\025)1038 2113 y Fm(M)1003 2143 y Fk(X)1014 2353 y Fm(j)t Fn(=1)1163 2102 y Fk(Z)1293 2171 y Fp(Z)1360 2186 y Fm(j)p 1290 2215 111 4 v 1290 2306 a Fq(j)p Fp(x)p Fq(j)1410 2238 y Fp(\032)1460 2197 y Fn(TF)1460 2263 y Fm(Z)1508 2273 y Fi(j)1584 2238 y Fq(\000)1718 2113 y Fm(M)1684 2143 y Fk(X)1694 2353 y Fm(j)t Fn(=1)1844 2238 y Fp(Z)1918 2197 y Fn(3)1911 2263 y Fm(j)1958 2238 y Fp(Z)2032 2197 y Fo(\000)p Fn(2)p Fm(=)p Fn(3)2213 2102 y Fk(Z)2268 2328 y Fo(j)p Fm(x)p Fo(j\025)2413 2301 y Fj(1)p 2412 2313 31 3 v 2412 2354 a(2)2452 2328 y Fm(\026)2516 2238 y Fp(\032)2566 2197 y Fn(TF)2667 2238 y Fq(j)g(\001)g(j)2795 2197 y Fo(\000)p Fn(1)2911 2238 y Fv(+)g(2)p Fp(U)3124 2253 y Fo(R)p Fm(;)p Fo(Z)842 2589 y Fq(\025)1038 2465 y Fm(M)1003 2495 y Fk(X)1014 2705 y Fm(j)t Fn(=1)1163 2454 y Fk(Z)1293 2522 y Fp(Z)1360 2537 y Fm(j)p 1290 2566 111 4 v 1290 2658 a Fq(j)p Fp(x)p Fq(j)1410 2589 y Fp(\032)1460 2548 y Fn(TF)1460 2614 y Fm(Z)1508 2624 y Fi(j)1584 2589 y Fq(\000)h Fp(C)7 b(Z)1835 2548 y Fn(7)p Fm(=)p Fn(3)1945 2589 y Fp(\026)2004 2548 y Fo(\000)p Fn(4)2098 2589 y Fp(M)32 b Fv(+)22 b(2)p Fp(U)2437 2604 y Fo(R)p Fm(;)p Fo(Z)120 2910 y Fv(\(w)m(e)34 b(ha)m(v)m(e)f(used)h (Newton's)g(Theorem)f(to)f(get)h(the)g(\014rst)g(line)e(on)i(the)g (expression)g(ab)s(o)m(v)m(e\))266 3030 y(iii.-)1156 3439 y Fp(D)s Fv(\()p Fp(\032)p Fv(\))84 b(=)g Fp(U)1676 3454 y Fo(R)p Fm(;)p Fo(Z)1838 3439 y Fv(+)1971 3314 y Fm(M)1936 3344 y Fk(X)1947 3554 y Fm(j)t Fn(=1)2096 3439 y Fp(D)s Fv(\()p Fp(\032)2268 3454 y Fm(j)2305 3439 y Fv(\))1449 3790 y Fq(\024)g Fp(U)1676 3805 y Fo(R)p Fm(;)p Fo(Z)1838 3790 y Fv(+)1971 3665 y Fm(M)1936 3695 y Fk(X)1947 3905 y Fm(j)t Fn(=1)2096 3790 y Fp(\025)2153 3749 y Fn(2)2153 3815 y Fm(j)2193 3790 y Fp(D)s Fv(\()p Fp(\032)2365 3749 y Fn(TF)2365 3815 y Fm(Z)2413 3825 y Fi(j)2466 3790 y Fv(\))120 4111 y(\(again,)31 b(b)m(y)j(Newton's)g (Theorem\))266 4231 y(Putting)e(the)h(estimates)g(ab)s(o)m(v)m(e)g (together)g(w)m(e)g(get)365 4538 y Fq(E)427 4497 y Fn(TF)418 4563 y Fo(R)p Fm(;)p Fo(Z)557 4538 y Fv(\()p Fp(\032)p Fv(\))83 b Fq(\024)h(\000)p Fp(C)1074 4553 y Fn(TF)1227 4414 y Fm(M)1192 4444 y Fk(X)1203 4654 y Fm(j)t Fn(=1)1353 4538 y Fp(Z)1427 4487 y Fn(7)p Fm(=)p Fn(3)1420 4564 y Fm(j)1559 4538 y Fv(+)1691 4414 y Fm(M)1657 4444 y Fk(X)1667 4654 y Fm(j)t Fn(=1)1800 4538 y Fv(\()p Fp(\025)1895 4497 y Fn(3)1895 4563 y Fm(j)1957 4538 y Fq(\000)23 b Fv(1\))2161 4403 y Fk(Z)2276 4428 y(\020)2336 4538 y Fp(\032)2386 4497 y Fn(TF)2386 4563 y Fm(Z)2434 4573 y Fi(j)2487 4428 y Fk(\021)2547 4450 y Fn(5)p Fm(=)p Fn(3)2679 4538 y Fv(+)f Fp(C)7 b(Z)2928 4497 y Fn(7)p Fm(=)p Fn(3)3038 4538 y Fp(\026)3097 4497 y Fo(\000)p Fn(4)3191 4538 y Fp(M)766 4890 y Fq(\024)84 b(\000)p Fp(C)1074 4905 y Fn(TF)1227 4765 y Fm(M)1192 4795 y Fk(X)1203 5005 y Fm(j)t Fn(=1)1353 4890 y Fp(Z)1427 4839 y Fn(7)p Fm(=)p Fn(3)1420 4915 y Fm(j)1559 4890 y Fv(+)22 b Fp(C)7 b(Z)1808 4848 y Fn(7)p Fm(=)p Fn(3)1917 4890 y Fp(\026)1976 4848 y Fo(\000)p Fn(3)2070 4890 y Fp(M)1203 b Fv(\(86\))120 5210 y(whic)m(h)33 b(is)f(\(85\).)p 3469 5210 4 66 v 3473 5148 59 4 v 3473 5210 V 3531 5210 4 66 v 1781 5712 a(20)p eop %%Page: 21 21 21 20 bop 120 407 a Fs(4)161 b(Semiclassics)77 b(in)g(the)f(go)t(o)t(d) g(region:)116 b(Multiscale)362 589 y(Analysis.)120 808 y Fv(In)28 b(this)g(section)g(w)m(e)h(are)f(going)e(to)i(analize)e(the) i(functional)f Fq(E)2401 823 y Fn(2)2440 808 y Fv(\()p Fp(\015)5 b Fv(\))27 b(b)m(y)i(semiclassical)d(meth-)120 929 y(o)s(ds)45 b(taking)f(adv)-5 b(an)m(tage)45 b(of)g(the)g(fact)g(w) m(e)h(ha)m(v)m(e)g(remo)m(v)m(ed)g(all)d(the)j(singularities)c(of)j (the)120 1049 y(p)s(oten)m(tial)31 b(whic)m(h)i(can)g(get)f(almost)f (in)h(top)g(of)h(eac)m(h)g(other.)44 b(Let)32 b(us)i(call)958 1261 y Fq(E)9 b Fv(\()p Fp(\015)c Fv(\))27 b(:=)h(T)-8 b(r)17 b([\()p Fq(\000)p Fv(\001)1650 1220 y Fm(D)1650 1286 y Fn(2)1737 1261 y Fq(\000)22 b Fv(\(\010)1944 1214 y Fn(TF)q Fm(;\014)1944 1288 y Fo(R)2004 1297 y Fj(1)2039 1288 y Fm(;)p Fo(Z)2110 1297 y Fj(1)2171 1261 y Fv(+)g(\010)2339 1220 y Fn(TF)2339 1286 y Fo(R)2399 1295 y Fj(2)2434 1286 y Fm(;)p Fo(Z)2505 1295 y Fj(2)2543 1261 y Fv(\)\))p Fp(\015)5 b Fv(])665 b(\(87\))120 1473 y(with)32 b Fp(\015)38 b Fv(diagonalizable)29 b(in)j Fp(H)1268 1437 y Fn(1)1260 1498 y(0)1307 1473 y Fv(\(\012)1415 1488 y Fn(1)1454 1473 y Fv(\).)44 b(The)33 b(follo)m(wing)d(trace)j(estimate)f(holds:) 273 1685 y(T)-8 b(r)16 b([\()p Fq(\000)p Fv(\001)612 1644 y Fm(D)612 1709 y Fn(2)699 1685 y Fq(\000)23 b Fv(\(\010)907 1638 y Fn(TF)p Fm(;\014)907 1712 y Fo(R)967 1721 y Fj(1)1002 1712 y Fm(;)p Fo(Z)1073 1721 y Fj(1)1133 1685 y Fv(+)f(\010)1301 1644 y Fn(TF)1301 1709 y Fo(R)1361 1718 y Fj(2)1397 1709 y Fm(;)p Fo(Z)1468 1718 y Fj(2)1506 1685 y Fv(\)\))p Fp(\015)5 b Fv(])83 b Fq(\025)g Fv(T)-8 b(r)17 b([\()p Fq(\000)p Fv(\001)23 b Fq(\000)g Fv(\(\010)2479 1638 y Fn(TF)p Fm(;\014)2479 1712 y Fo(R)2539 1721 y Fj(1)2574 1712 y Fm(;)p Fo(Z)2645 1721 y Fj(1)2705 1685 y Fv(+)f(\010)2873 1644 y Fn(TF)2873 1709 y Fo(R)2933 1718 y Fj(2)2969 1709 y Fm(;)p Fo(Z)3040 1718 y Fj(2)3078 1685 y Fv(\))p Fp(\037)3177 1700 y Fn(\012)3228 1709 y Fj(2)3266 1685 y Fv(\))p Fp(\015)5 b Fv(])1748 1851 y Fq(\025)83 b(\000)p Fv(T)-8 b(r)18 b([)p Fq(\000)p Fv(\001)23 b Fq(\000)f Fv(\(\010)2518 1804 y Fn(TF)q Fm(;\014)2518 1878 y Fo(R)2578 1887 y Fj(1)2614 1878 y Fm(;)p Fo(Z)2685 1887 y Fj(1)2745 1851 y Fv(+)g(\010)2913 1810 y Fn(TF)2913 1876 y Fo(R)2973 1885 y Fj(2)3008 1876 y Fm(;)p Fo(Z)3079 1885 y Fj(2)3117 1851 y Fv(\)\004])3247 1866 y Fo(\000)3367 1851 y Fv(\(88\))120 2063 y(where)34 b(in)e(\(88\))g(w)m(e)h(tak)m(e)h(\004)e(with)g(the)h (follo)m(wing)d(prop)s(erties:)1081 2191 y Fk(8)1081 2281 y(>)1081 2311 y(>)1081 2341 y(>)1081 2371 y(>)1081 2401 y(>)1081 2431 y(<)1081 2610 y(>)1081 2640 y(>)1081 2670 y(>)1081 2700 y(>)1081 2729 y(>)1081 2759 y(:)1211 2272 y Fv(\004)e Fq(2)g Fp(C)1475 2236 y Fo(1)1550 2272 y Fv(\()p Fl(R)1654 2236 y Fn(3)1699 2272 y Fv(\))1211 2392 y(\004)g(=)f(1)33 b(in)f(\012)1673 2407 y Fn(2)1211 2545 y Fv(\004)c(=)f(0)33 b(in)f(\012)c(=)1804 2435 y Fk(\020)1864 2471 y(S)1947 2497 y Fm(M)2015 2506 y Fj(1)1947 2574 y Fm(j)t Fn(=1)p 2090 2465 80 4 v 2090 2545 a Fp(B)15 b Fi(\025)p 2179 2557 37 3 v 2182 2599 a Fj(8)2230 2545 y Fv(\()p Fp(R)2342 2560 y Fm(j)2379 2545 y Fv(\))2417 2435 y Fk(\021)2476 2457 y Fm(c)1211 2694 y Fv(0)28 b Fq(\024)g Fv(\004)g Fq(\024)g Fv(1)1211 2817 y Fq(jr)p Fv(\004)p Fq(j)f(\024)h Fp(C)7 b(Z)1698 2780 y Fo(\000)p Fn(1)p Fm(=)p Fn(3)3367 2545 y Fv(\(89\))120 3022 y(T)-8 b(o)30 b(do)f(semiclassics,)h(w)m(e)h(are)e(going)g(to)g(use)i(the)f (scaling)e(prop)s(erties)i(of)f(the)h(TF)g(p)s(oten)m(tial)120 3142 y(whic)m(h)40 b(are)g(giv)m(en)h(b)m(y)f(\(14\))g(and)g(rescaling) f(ev)m(erything)i(conjugating)d(with)i(the)g(unitary)120 3262 y(op)s(erator)32 b Fp(U)579 3277 y Fm(l)605 3262 y Fp(\036)p Fv(\()p Fp(x)p Fv(\))c(:=)g Fp(l)984 3226 y Fo(\000)p Fn(3)p Fm(=)p Fn(2)1149 3262 y Fp(\036)p Fv(\()p Fp(l)r(x)p Fv(\).)43 b(Then)34 b(a)e(computation)f(yields)202 3474 y(T)-8 b(r)17 b([)p Fq(\000)p Fv(\001)23 b Fq(\000)g Fv(\(\010)735 3427 y Fn(TF)p Fm(;\014)735 3501 y Fo(R)795 3510 y Fj(1)830 3501 y Fm(;)p Fo(Z)901 3510 y Fj(1)961 3474 y Fv(+)f(\010)1129 3433 y Fn(TF)1129 3499 y Fo(R)1189 3508 y Fj(2)1224 3499 y Fm(;)p Fo(Z)1295 3508 y Fj(2)1334 3474 y Fv(\)\004])1464 3489 y Fo(\000)1619 3474 y Fv(=)97 b Fp(Z)1866 3433 y Fn(4)p Fm(=)p Fn(3)1976 3474 y Fv(T)-8 b(r[)p Fq(\000)p Fp(Z)2254 3433 y Fo(\000)p Fn(2)p Fm(=)p Fn(3)2419 3474 y Fv(\001)23 b Fq(\000)f Fp(W)14 b Fv(])2755 3489 y Fo(\000)3367 3474 y Fv(\(90\))1417 3641 y Fp(W)97 b Fv(:=)83 b(\(\010)1900 3600 y Fn(TF)1900 3672 y Fm(Z)1953 3653 y Fj(1)p Fi(=)p Fj(3)2049 3672 y Fo(R)2109 3681 y Fj(2)2144 3672 y Fm(;Z)2217 3654 y Fh(\000)p Fj(1)2299 3672 y Fo(Z)2350 3681 y Fj(2)2411 3641 y Fv(+)22 b(\010)2579 3593 y Fn(TF)p Fm(;\014)2579 3679 y(Z)2632 3660 y Fj(1)p Fi(=)p Fj(3)2728 3679 y Fo(R)2788 3688 y Fj(1)2823 3679 y Fm(;Z)2896 3660 y Fh(\000)p Fj(1)2978 3679 y Fo(Z)3029 3688 y Fj(1)3067 3641 y Fv(\)\004\()p Fp(Z)3282 3600 y Fn(1)p Fm(=)p Fn(3)3392 3641 y Fq(\001)p Fv(\))120 3861 y(so)39 b(w)m(e)h(are)e(led)h(to)f(consider)h(the)g(follo)m(wing)d (problem)i(of)g(singular)f(p)s(erturbation)h(of)h(the)120 3982 y(Laplacian:)951 4193 y Fq(M)p Fv(\()p Fp(h;)17 b(W)d Fv(\))27 b(:=)h(T)-8 b(r[)p Fq(\000)p Fp(h)1771 4152 y Fn(2)1811 4193 y Fv(\001)23 b Fq(\000)f Fp(W)14 b Fv(])2147 4208 y Fo(\000)2206 4193 y Fv(;)49 b Fp(h)28 b Fv(=)g Fp(Z)2544 4152 y Fo(\000)p Fn(1)p Fm(=)p Fn(3)3367 4193 y Fv(\(91\))120 4405 y(W)-8 b(e)33 b(no)m(w)g(giv)m(e)g(the)g (fundamen)m(tal)e(theorem)i(on)f(this)g(section:)120 4602 y Ft(Theorem)37 b(5)g(\(Go)s(o)s(d)h(part)g(semiclasssics\).)46 b Fr(The)41 b(quantity)h Fq(M)p Fv(\()p Fp(h;)17 b(W)d Fv(\))40 b Fr(has)h(the)h(fol-)120 4722 y(lowing)34 b(asymptotic)-5 b(al)34 b(b)-5 b(ehaviour:)489 5021 y Fq(jM)p Fv(\()p Fp(h;)17 b(W)d Fv(\))21 b Fq(\000)1104 4954 y Fv(1)p 1050 4998 157 4 v 1050 5089 a(15)p Fp(\031)1216 5021 y(h)1272 4980 y Fo(\000)p Fn(3)1383 4885 y Fk(Z)1499 5021 y Fp(W)1605 4980 y Fn(5)p Fm(=)p Fn(2)1737 5021 y Fv(+)1845 4954 y(1)p 1845 4998 49 4 v 1845 5089 a(8)1904 5021 y Fp(h)1960 4980 y Fo(\000)p Fn(2)2174 4896 y Fm(M)2139 4926 y Fk(X)2071 5138 y Fm(j)t Fn(=)p Fm(M)2227 5147 y Fj(1)2261 5138 y Fn(+1)2351 5021 y Fv(\()2399 4954 y Fp(Z)2466 4969 y Fm(j)p 2399 4998 104 4 v 2413 5089 a Fp(Z)2512 5021 y Fv(\))2550 4980 y Fn(2)2589 5021 y Fq(j)28 b(\024)g Fp(C)7 b(h)2883 4980 y Fo(\000)p Fn(1)p Fo(\000)p Fm(\016)3066 5021 y Fp(M)207 b Fv(\(92\))120 5342 y Fr(wher)-5 b(e)41 b Fp(\016)46 b Fr(is)41 b(some)g(absolute)g(p) -5 b(ositive)41 b(c)-5 b(onstant)41 b(\()p Fp(\016)j Fv(=)2262 5303 y Fn(9)p 2244 5320 71 4 v 2244 5377 a(17)2366 5342 y Fr(c)-5 b(ould)42 b(do\))f(and)g Fp(C)49 b Fr(is)41 b(a)g(\014nite)120 5463 y(c)-5 b(onstant)35 b(only)f(dep)-5 b(ending)34 b(on)g Fp(A)p Fr(.)1781 5712 y Fv(21)p eop %%Page: 22 22 22 21 bop 266 407 a Fv(T)-8 b(o)34 b(pro)m(v)m(e)h(Theorem)g(5)e(w)m(e) i(shall)e(distinguish)f(b)s(et)m(w)m(een)k(the)e(region)f(of)h(the)g(p) s(oin)m(ts)g(in)120 527 y Fl(R)186 491 y Fn(3)266 527 y Fv(whic)m(h)h(are)f(far)g(a)m(w)m(a)m(y)h(from)f(an)m(y)h(n)m(uclei,) f(whic)m(h,)i(giv)m(en)e(the)h(rescaling)e(w)m(e)j(just)e(made)120 648 y(and)c(the)h(fact)f(w)m(e)h(got)f(rid)g(of)f(the)i(`bad')g(n)m (uclei,)f(means)g(w)m(e)i(are)e(lo)s(oking)e(at)i(p)s(oin)m(ts)g(whose) 120 768 y(distance)39 b(to)g(the)g(n)m(uclei)g(is)g(b)s(ounded)g(a)m(w) m(a)m(y)i(zero)e(b)m(y)h(\014xed)g(p)s(ositiv)m(e)e(n)m(um)m(b)s(er)i (\(namely)-8 b(,)120 888 y(the)34 b(n)m(um)m(b)s(er)f Fp(\026)g Fv(obtained)g(in)f(the)i(previous)g(section\))f(and)g(its)g (complemen)m(t.)45 b(Also,)33 b(since)120 1009 y(most)f(of)h(the)g (semiclassical)e(estimates)i(are)g(already)f(a)m(v)-5 b(aliable)31 b(in)h(the)h(litterature,)f(I)h(will)120 1129 y(rep)s(eteadly)44 b(refer)f(to)h(the)f(pap)s(ers)h(of)f(A.V.Sob)s (olev)h([30)o(,)g(31)o(,)g(32)o(])g(and)f(to)g(the)h(w)m(ork)g(of)120 1249 y(V.Ivrii)32 b(and)g(I.M.Sigal)f([16].)120 1438 y Fr(Pr)-5 b(o)g(of)35 b(of)f(The)-5 b(or)g(em)34 b(5.)48 b Fv(The)38 b(p)s(oten)m(tial)c Fp(W)50 b Fv(is)36 b(ev)m(erywhere)k (regular)35 b(with)h(the)h(exception)120 1558 y(of)d(the)g(set)h Fq(f)p Fp(Z)680 1522 y Fn(1)p Fm(=)p Fn(3)790 1558 y Fp(R)864 1573 y Fm(j)901 1558 y Fq(g)951 1522 y Fm(M)951 1584 y(M)1019 1593 y Fj(1)1053 1584 y Fn(+1)1182 1558 y Fv(where)g(it)e(has)i(coulom)m(bic)e(singularities)e(of)j(strengh)m (t)h Fp(Z)3315 1522 y Fo(\000)p Fn(1)3410 1558 y Fp(Z)3477 1573 y Fm(j)3513 1558 y Fv(.)120 1679 y(Moreo)m(v)m(er,)28 b(b)m(y)f(the)e(reduction)h(made)f(in)f(section)i(3,)g(all)e(those)i (singularities)d(are)i(separated)120 1799 y(one)32 b(another)f(b)m(y)i (a)e(distance)h Fp(\026)f Fv(\(at)g(least\).)43 b(Our)31 b(\014rst)h(step)h(shall)d(b)s(e)i(to)f(isolate)f(those)i(b)m(y)120 1920 y(means)37 b(of)e(a)i(suitable)e(partition)g(of)h(unit)m(y)g(and)h (then)g(to)f(use)h(`lo)s(cal)e(trace)h(estimates')h(in)120 2040 y(the)c(spirit)e(of)h(the)h(w)m(ork)h(of)e(V.Ivrii&I.M.Sigal)e ([16])i(and)h(A.V.Sob)s(olev)g([30)o(,)g(31)o(,)g(32].)266 2160 y(-.)43 b Fr(Step)35 b(1)g(\(Construction)f(of)h(a)f(p)-5 b(artition)35 b(of)g(unity\):)266 2281 y Fv(This)f(step)g(will)d(b)s(e) j(v)m(ery)h(similar)30 b(to)j(the)h(one)f(in)g(section)g(3)g(of)g (constructing)h(another)120 2401 y(partition)c(of)j(unit)m(y:)43 b(w)m(e)34 b(\014x)f(a)f(function)g(\011)c Fq(2)g Fp(C)1950 2365 y Fo(1)1943 2426 y Fm(c)2025 2401 y Fv(\()p Fl(R)2129 2365 y Fn(3)2174 2401 y Fv(\))k(suc)m(h)i(that:)266 2521 y(i.)43 b(0)27 b Fq(\024)h Fv(\011)g Fq(\024)g Fv(1,)33 b(\011)27 b(=)h(1)k(if)f Fq(j)p Fp(x)p Fq(j)d(\024)1494 2478 y Fm(\026)p 1494 2499 43 4 v 1498 2556 a Fn(8)1579 2521 y Fv(and)33 b(\011)27 b(=)h(0)k(if)f Fq(j)p Fp(x)p Fq(j)d(\025)2400 2478 y Fm(\026)p 2400 2499 V 2404 2556 a Fn(4)2452 2521 y Fv(.)266 2642 y(ii.)42 b Fq(kr)p Fv(\011)p Fq(k)650 2657 y Fo(1)752 2642 y Fq(\024)867 2603 y Fm(C)p 867 2619 55 4 v 873 2676 a(\026)964 2642 y Fv(with)33 b Fp(C)39 b Fv(indep)s(enden)m(t)34 b(of)e Fp(\026)p Fv(.)266 2762 y(No)m(w)h(the)g(partition)e(of)h(unit)m(y)h(w)m(e)g (shall)f(tak)m(e)h(is)f(giv)m(en)h(b)m(y)734 2880 y Fk(\032)850 2952 y Fv(\007)926 2967 y Fm(j)1048 2952 y Fv(=)83 b(\011\()p Fq(\001)22 b(\000)h Fp(Z)1545 2916 y Fn(1)p Fm(=)p Fn(3)1654 2952 y Fp(R)1728 2967 y Fm(j)t Fn(+)p Fm(M)1884 2976 y Fj(1)1922 2952 y Fv(\);)50 b Fp(j)33 b Fv(=)28 b(1)p Fp(;)17 b(:)g(:)g(:)32 b(;)17 b(M)33 b Fq(\000)22 b Fp(M)2818 2967 y Fn(1)850 3082 y Fv(\007)926 3097 y Fn(0)1048 3082 y Fv(=)83 b(1)22 b Fq(\000)1378 3008 y Fk(P)1483 3034 y Fm(M)1483 3111 y(j)t Fn(=1)1626 3082 y Fv(\007)1702 3097 y Fm(j)3367 3020 y Fv(\(93\))266 3284 y(-.)43 b Fr(Step)35 b(2)g(\(L)-5 b(o)g(c)g(al)34 b(tr)-5 b(ac)g(e)35 b(estimates\):)266 3404 y Fv(Notice)f(that)f(the)h(supp)s(orts)h(of)e (\007)1544 3419 y Fn(1)1583 3404 y Fp(;)17 b(:)g(:)g(:)33 b(;)17 b Fv(\007)1895 3419 y Fm(M)7 b Fo(\000)p Fm(M)2093 3428 y Fj(1)2164 3404 y Fv(are)34 b(m)m(utually)e(disjoin)m(t.)45 b(With)33 b(this)120 3525 y(partition)d(of)j(unit)m(y)f(w)m(e)i(just)f (constructed)h(w)m(e)g(can)e(split)g Fq(M)p Fv(\()p Fp(h;)17 b(W)d Fv(\):)883 3647 y Fk(\032)1140 3728 y Fq(M)p Fv(\()p Fp(h;)j(W)d Fv(\))82 b(=)1810 3654 y Fk(P)1915 3680 y Fm(M)7 b Fo(\000)p Fm(M)2113 3689 y Fj(1)1915 3757 y Fm(j)t Fn(=0)2168 3728 y Fq(M)p Fv(\()p Fp(h;)17 b(W)m(;)g Fv(\007)2635 3743 y Fm(j)2670 3728 y Fv(\))1000 3854 y Fq(M)p Fv(\()p Fp(h;)g(W)m(;)g Fv(\007)1467 3869 y Fm(j)1503 3854 y Fv(\))83 b(:=)g(T)-8 b(r[\007)2013 3869 y Fm(j)2049 3854 y Fv(\()p Fq(\000)p Fp(h)2220 3818 y Fn(2)2260 3854 y Fv(\001)23 b Fq(\000)f Fp(W)14 b Fv(\)])2634 3869 y Fo(\000)3367 3787 y Fv(\(94\))120 4061 y(In)26 b(the)g(supp)s(ort)f(of)g(\004\()p Fp(Z)1030 4025 y Fn(1)p Fm(=)p Fn(3)1140 4061 y Fq(\001)p Fv(\))g(w)m(e)i(ha)m(v)m(e,)h(b)m(y)e (the)g(results)g(of)f(section)g(2)g(the)h(global)d(estimates)704 4175 y Fk(\()826 4274 y Fq(j)p Fp(@)910 4238 y Fm(\013)960 4274 y Fv(\010)1030 4227 y Fn(TF)p Fm(;\014)1030 4312 y(Z)1083 4293 y Fj(1)p Fi(=)p Fj(3)1180 4312 y Fo(R)1240 4321 y Fj(1)1274 4312 y Fm(;Z)1347 4293 y Fh(\000)p Fj(1)1429 4312 y Fo(Z)1480 4321 y Fj(1)1519 4274 y Fv(\()p Fp(x)p Fv(\))p Fq(j)83 b(\024)g Fp(C)1991 4289 y Fm(\013)2040 4274 y Fp(\032)2090 4289 y Fn(1)2130 4274 y Fv(\()p Fp(x)p Fv(\))2261 4238 y Fo(\000)p Fn(1)p Fo(\000j)p Fm(\013)p Fo(j)2495 4274 y Fq(h)p Fp(\032)2584 4289 y Fn(1)2624 4274 y Fv(\()p Fp(x)p Fv(\))p Fq(i)2794 4238 y Fo(\000)p Fn(3)826 4411 y Fq(j)p Fp(@)910 4375 y Fm(\013)960 4411 y Fv(\010)1030 4375 y Fn(TF)1030 4448 y Fm(Z)1083 4428 y Fj(1)p Fi(=)p Fj(3)1180 4448 y Fo(R)1240 4457 y Fj(2)1274 4448 y Fm(;Z)1347 4429 y Fh(\000)p Fj(1)1429 4448 y Fo(Z)1480 4457 y Fj(2)1519 4411 y Fv(\()p Fp(x)p Fv(\))p Fq(j)g(\024)g Fp(C)1991 4426 y Fm(\013)2040 4411 y Fp(\032)2090 4426 y Fn(2)2130 4411 y Fv(\()p Fp(x)p Fv(\))2261 4375 y Fo(\000)p Fn(1)p Fo(\000j)p Fm(\013)p Fo(j)2495 4411 y Fq(h)p Fp(\032)2584 4426 y Fn(2)2624 4411 y Fv(\()p Fp(x)p Fv(\))p Fq(i)2794 4375 y Fo(\000)p Fn(3)3367 4346 y Fv(\(95\))120 4624 y(with)732 4710 y Fk(8)732 4800 y(<)732 4979 y(:)862 4793 y Fp(\032)912 4808 y Fn(1)952 4793 y Fv(\()p Fp(x)p Fv(\))g(:=)g(min)o Fq(fj)p Fp(x)22 b Fq(\000)g Fp(R)1843 4808 y Fm(j)1880 4793 y Fq(j)28 b Fv(:)f Fp(j)34 b Fv(=)27 b(1)p Fp(;)17 b(:)g(:)g(:)33 b(;)17 b(M)2546 4808 y Fn(1)2585 4793 y Fq(g)862 4914 y Fp(\032)912 4929 y Fn(2)952 4914 y Fv(\()p Fp(x)p Fv(\))83 b(:=)g(min)o Fq(fj)p Fp(x)22 b Fq(\000)g Fp(R)1843 4929 y Fm(j)1880 4914 y Fq(j)28 b Fv(:)f Fp(j)34 b Fv(=)27 b Fp(M)2261 4929 y Fn(1)2323 4914 y Fv(+)22 b(1)p Fp(;)17 b(:)g(:)g(:)33 b(;)17 b(M)10 b Fq(g)902 5034 y Fp(\032)p Fv(\()p Fp(x)p Fv(\))83 b(:=)g(min)o Fq(fj)p Fp(x)22 b Fq(\000)g Fp(R)1843 5049 y Fm(j)1880 5034 y Fq(j)28 b Fv(:)f Fp(j)34 b Fv(=)27 b(1)p Fp(;)17 b(:)g(:)g(:)33 b(;)17 b(M)10 b Fq(g)3367 4915 y Fv(\(96\))120 5230 y(Notice)32 b(that)h Fp(\032)p Fv(\()p Fp(x)p Fv(\))28 b(=)f(min)o Fq(f)p Fp(\032)1213 5245 y Fn(1)1253 5230 y Fv(\()p Fp(x)p Fv(\))p Fp(;)17 b(\032)1478 5245 y Fn(2)1517 5230 y Fv(\()p Fp(x)p Fv(\))p Fq(g)33 b Fv(As)g(a)f(result)h(w)m(e)g (get)g(the)g(global)d(estimate)922 5427 y Fq(j)p Fp(@)1006 5386 y Fm(\013)1056 5427 y Fp(W)14 b Fv(\()p Fp(x)p Fv(\))p Fq(j)27 b(\024)h Fp(C)1523 5442 y Fm(\013)1589 5427 y Fv(\004\()p Fp(Z)1766 5386 y Fn(1)p Fm(=)p Fn(3)1876 5427 y Fp(x)p Fv(\))p Fp(\032)p Fv(\()p Fp(x)p Fv(\))2150 5386 y Fo(\000)p Fn(1)p Fo(\000j)p Fm(\013)p Fo(j)2385 5427 y Fq(h)p Fp(\032)p Fv(\()p Fp(x)p Fv(\))p Fq(i)2644 5386 y Fo(\000)p Fn(3)3367 5427 y Fv(\(97\))1781 5712 y(22)p eop %%Page: 23 23 23 22 bop 120 407 a Fv(whic)m(h)39 b(means)g(w)m(e)g(are)g(in)f(p)s (osition)e(to)j(apply)f(the)h Fr(Multisc)-5 b(ale)40 b(Metho)-5 b(d)39 b Fv(\014rst)g(dev)m(elop)s(ed)120 527 y(b)m(y)33 b(V.Ivrii)f(and)h(extended)h(in)e(the)h(references)i(of) d(A.V.Sob)s(olev.)266 648 y Fr(Step)j(3:)45 b(Estimating)34 b(the)h(r)-5 b(e)g(gular)35 b(p)-5 b(art)35 b(of)f Fp(W)14 b Fr(:)266 768 y Fv(The)27 b(zeroth)f(term)g(in)f(\(94\))g(is)g(the)i (easiest)f(to)f(handle,)i(b)s(ecause)h(it)c(has)j(no)e(singularities) 120 888 y(of)46 b(the)h(p)s(oten)m(tial.)84 b(Therefore)48 b(the)f(results)g(\(Theorem)g(5.2)f(of)g([31]\))h(can)g(b)s(e)g (directly)120 1009 y(applied)31 b(to)i(it:)42 b(Call)871 1135 y Fk(\032)987 1215 y Fp(f)11 b Fv(\()p Fp(x)p Fv(\))83 b(:=)g(\()p Fp(\032)p Fv(\()p Fp(x)p Fv(\))1665 1179 y Fo(\000)p Fn(2)1760 1215 y Fq(h)p Fp(\032)p Fv(\()p Fp(x)p Fv(\))p Fq(i)2019 1179 y Fo(\000)p Fn(6)2136 1215 y Fv(+)22 b Fp(h)2290 1179 y Fn(2)2329 1215 y Fv(\))2367 1179 y Fn(1)p Fm(=)p Fn(4)1015 1337 y Fp(l)r Fv(\()p Fp(x)p Fv(\))83 b(:=)g(\()p Fp(\033)1543 1301 y Fn(2)1583 1337 y Fp(\032)p Fv(\()p Fp(x)p Fv(\))1764 1301 y Fn(2)1826 1337 y Fv(+)22 b Fp(h)p Fv(\))2018 1301 y Fn(1)p Fm(=)p Fn(2)2128 1337 y Fv(;)49 b(0)28 b Fp(<)f(\033)32 b(<)c Fv(1)p Fp(=)p Fv(2)3367 1276 y(\(98\))120 1548 y(to)45 b(b)s(e)g(c)m(hosen)h(later.)80 b(W)-8 b(e)46 b(ha)m(v)m(e)g(to)f (break)g(up)h(again)d(this)i(piece)g(as)h Fq(M)p Fv(\()p Fp(h;)17 b(W)m(;)g Fv(\007)3339 1563 y Fn(0)3377 1548 y Fv(\))49 b(=)120 1669 y Fq(M)p Fv(\()p Fp(h;)17 b(W)m(;)g Fv(\007)587 1684 y Fn(0)p Fm(;h)686 1669 y Fv(\))g(+)f Fq(M)p Fv(\()p Fp(h;)h(W)m(;)g(\035)1277 1684 y Fn(0)1316 1669 y Fv(\))29 b(where)j(\007)1739 1684 y Fn(0)p Fm(;h)1868 1669 y Fv(is)d(supp)s(orted)i(in)e Fq(\\)2596 1633 y Fm(M)2596 1693 y(j)t Fn(=1)2723 1669 y Fq(f)p Fp(x)f Fv(:)g Fp(\026)f Fq(\024)i(j)p Fp(x)16 b Fq(\000)h Fp(R)3370 1684 y Fm(j)3407 1669 y Fq(j)28 b(\024)120 1789 y Fp(h)176 1753 y Fo(\000)p Fn(1)298 1789 y Fv(:)g Fp(j)34 b Fv(=)27 b(1)p Fp(;)17 b(:)g(:)g(:)32 b(;)17 b(M)10 b Fq(g)p Fv(,)30 b(equals)f(one)h(in)e Fq(\\)1669 1753 y Fm(M)1669 1814 y(j)t Fn(=1)1796 1789 y Fq(f)p Fp(x)g Fv(:)g Fp(\026)f Fq(\024)h(j)p Fp(x)15 b Fq(\000)g Fp(R)2439 1804 y Fm(j)2476 1789 y Fq(j)27 b(\024)2646 1750 y Fn(1)p 2646 1766 36 4 v 2646 1823 a(2)2691 1789 y Fp(h)2747 1753 y Fo(\000)p Fn(1)2870 1789 y Fv(:)g Fp(j)34 b Fv(=)28 b(1)p Fp(;)17 b(:)g(:)g(:)32 b(;)17 b(M)10 b Fq(g)120 1909 y Fv(and)26 b(is)f(b)s(ounded)h(along)e(together)i(with)f(all)e(its)i(deriv)-5 b(ativ)m(es)26 b(uniformly)d(in)i Fp(M)36 b Fv(\(this)26 b(will)d(b)s(e)120 2030 y(justi\014ed)i(later)g(on\).)41 b(As)25 b Fp(\032)p Fv(\()p Fp(x)p Fv(\))h(is)f(giv)m(en)g(as)h(the)g (minim)m(un)c(of)j(the)h(Lipsc)m(hitz)f(functions)g Fp(x)j Fq(7!)120 2150 y(j)p Fp(x)t Fq(\000)t Fp(Z)362 2114 y Fn(1)p Fm(=)p Fn(3)472 2150 y Fp(R)546 2165 y Fm(j)583 2150 y Fq(j)23 b Fv(all)f(ha)m(ving)h(the)h(same)f(Lipsc)m(hitz)h (constan)m(t)g(\(namely)-8 b(,)25 b(1\),)g(the)f(same)f(prop)s(ert)m(y) 120 2271 y(holds)34 b(for)f Fp(\032)i Fv(and)f(an)g(easy)h(calculation) d(yields)h(the)i(b)s(ounds)g(for)e(an)m(y)i Fp(x)c Fq(2)g Fv(supp)17 b(\007)3249 2286 y Fn(0)p Fm(;h)3383 2271 y Fv(and)120 2391 y(0)27 b Fp(<)h(h)g Fq(\024)g Fp(h)545 2406 y Fn(0)612 2391 y Fv(for)g Fp(h)813 2406 y Fn(0)880 2391 y Fp(>)g Fv(0)f(small)f(enough)i(\(but)g(indep)s(enden)m(t)h(of)e Fp(M)10 b Fv(,)30 b(so)e(this)g(only)f(amoun)m(ts)120 2511 y(to)32 b(tak)m(e)i(the)f(parameter)f Fp(Z)39 b Fv(big)32 b(enough\))573 2642 y Fk(\032)903 2721 y Fv(\(1)22 b Fq(\000)h Fv(2)p Fp(\033)t Fv(\))p Fp(\032)p Fv(\()p Fp(y)t Fv(\))k Fq(\024)h Fp(\032)p Fv(\()p Fp(x)p Fv(\))g Fq(\024)g Fv(\(1)22 b(+)g(2)p Fp(\033)t Fv(\))p Fp(\032)p Fv(\()p Fp(y)t Fv(\);)82 b Fp(x)28 b Fq(2)g Fp(B)2773 2737 y Fm(l)q Fn(\()p Fm(y)r Fn(\))2892 2721 y Fv(\()p Fp(y)t Fv(\))689 2842 y(\(1)22 b(+)g(2)p Fp(\033)t Fv(\))1042 2805 y Fo(\000)p Fn(2)1136 2842 y Fp(f)11 b Fv(\()p Fp(y)t Fv(\))27 b Fq(\024)h Fp(f)11 b Fv(\()p Fp(x)p Fv(\))28 b Fq(\024)g Fv(\(1)22 b Fq(\000)g Fv(2)p Fp(\033)t Fv(\))2132 2805 y Fo(\000)p Fn(2)2226 2842 y Fp(f)11 b Fv(\()p Fp(y)t Fv(\);)82 b Fp(x)28 b Fq(2)g Fp(B)2773 2857 y Fm(l)q Fn(\()p Fm(y)r Fn(\))2892 2842 y Fv(\()p Fp(y)t Fv(\))3367 2782 y(\(99\))120 3064 y(\(where)40 b(in)f(\(99\))f(w)m(e)i(ha)m(v)m(e) h(tak)m(en)f(in)m(to)e(accoun)m(t)i(that)2276 3025 y Fm(h)p 2232 3041 129 4 v 2232 3098 a(\032)p Fn(\()p Fm(y)r Fn(\))2409 3064 y Fq(\024)2535 3025 y Fn(8)p Fm(h)p 2535 3041 76 4 v 2552 3098 a(\026)2660 3064 y Fv(for)f Fp(y)j Fq(2)d Fv(supp)17 b(\007)3304 3079 y Fn(0)3383 3064 y Fv(and)120 3198 y Fp(h)37 b Fq(7!)f Fv(0+\))h(meaning)g(that)h Fp(\032)p Fv(,)h Fp(f)49 b Fv(are)37 b(functions)h(of)g Fr(slow)h(variation)p Fv(.)58 b(Moreo)m(v)m(er,)41 b(for)c(an)m(y)120 3318 y Fq(j)p Fp(\013)q Fq(j)29 b(\025)g Fv(1,)34 b(the)g(deriv)-5 b(ativ)m(es)34 b Fp(@)1196 3282 y Fm(\013)1246 3318 y Fv(\007)1322 3333 y Fn(0)p Fm(;h)1455 3318 y Fv(are)g(in)f Fp(L)1800 3282 y Fo(1)1875 3318 y Fv(\()p Fl(R)1979 3282 y Fn(3)2024 3318 y Fv(\))h(uniformly)d(in)i(the)h(n)m(um)m(b)s(er)g(of) f(n)m(uclei)120 3439 y Fp(M)43 b Fv(and)33 b(are)g(supp)s(orted)g(in) 228 3614 y Fm(M)210 3644 y Fk([)204 3854 y Fm(j)t Fn(=1)343 3598 y Fk(\032)418 3738 y Fp(x)28 b Fv(:)566 3671 y Fp(\026)p 566 3716 59 4 v 571 3807 a Fv(8)662 3738 y Fq(\024)g(j)p Fp(x)23 b Fq(\000)f Fp(Z)1046 3697 y Fn(1)p Fm(=)p Fn(3)1156 3738 y Fp(R)1230 3753 y Fm(j)1267 3738 y Fq(j)27 b(\024)1437 3671 y Fp(\026)p 1437 3716 V 1442 3807 a Fv(4)1539 3738 y(or)1668 3671 y Fp(h)1724 3635 y Fo(\000)p Fn(1)p 1668 3716 151 4 v 1719 3807 a Fv(2)1856 3738 y Fq(\024)h(j)p Fp(x)22 b Fq(\000)h Fp(Z)2240 3697 y Fn(1)p Fm(=)p Fn(3)2350 3738 y Fp(R)2424 3753 y Fm(j)2461 3738 y Fq(j)k(\024)h Fp(h)2677 3697 y Fo(\000)p Fn(1)2772 3738 y Fv(;)17 b Fp(j)33 b Fv(=)27 b(1)p Fp(;)17 b(:)g(:)g(:)33 b(;)17 b(M)3381 3598 y Fk(\033)120 4047 y Fv(Therefore,)34 b(the)f(b)s(ounds) 1347 4259 y Fq(j)p Fp(@)1431 4218 y Fm(\013)1481 4259 y Fv(\007)1557 4274 y Fn(0)1596 4259 y Fv(\()p Fp(x)p Fv(\))p Fq(j)28 b(\024)g Fp(C)1958 4274 y Fm(\013)2007 4259 y Fp(l)r Fv(\()p Fp(x)p Fv(\))2169 4218 y Fo(\000j)p Fm(\013)p Fo(j)3318 4259 y Fv(\(100\))120 4472 y(trivially)i(holds.)43 b(No)m(w,)33 b(w)m(e)h(ha)m(v)m(e)f(to)g(tak)m(e)g Fp(\015)38 b Fv(su\016cien)m(tly)33 b(small)d(suc)m(h)k(that)f(if)e(w)m(e)j(tak)m (e)1032 4772 y Fq(D)d Fv(=)1267 4647 y Fm(M)1249 4677 y Fk([)1243 4887 y Fm(j)t Fn(=1)1366 4772 y Fq(f)p Fp(x)d Fv(:)f Fp(\015)5 b(\026)28 b Fq(\024)g(j)p Fp(x)22 b Fq(\000)h Fp(R)2080 4787 y Fm(j)2117 4772 y Fq(j)k(\024)h Fp(\015)2333 4731 y Fo(\000)p Fn(1)2427 4772 y Fp(h)2483 4731 y Fo(\000)p Fn(1)2578 4772 y Fq(g)690 b Fv(\(101\))120 5080 y(the)33 b(follo)m(wing)d(inclusion)h(holds:)1479 5210 y Fk([)1134 5426 y Fm(x)p Fo(2D)r Fn(:)p Fm(B)1351 5440 y Fi(l)p Fj(\()p Fi(x)p Fj(\))1457 5426 y Fn(\()p Fm(x)p Fn(\))p Fo(\\)p Fn(supp)13 b(\007)1811 5435 y Fj(0)1845 5426 y Fo(6)p Fn(=)p Fo(;)1952 5305 y Fp(B)2026 5320 y Fn(8)p Fm(l)q Fn(\()p Fm(x)p Fn(\))2182 5305 y Fv(\()p Fp(x)p Fv(\))28 b Fq(\032)g(D)795 b Fv(\(102\))1781 5712 y(23)p eop %%Page: 24 24 24 23 bop 120 407 a Fv(The)34 b(c)m(hoices)f Fp(\033)f Fv(=)867 368 y Fn(7)p 849 384 71 4 v 849 441 a(56)946 407 y Fv(\()p Fp(<)1098 368 y Fn(1)p 1098 384 36 4 v 1098 441 a(8)1143 407 y Fv(\))g(and)h Fp(\015)g Fv(=)27 b(1)22 b Fq(\000)h Fv(8)p Fp(\033)36 b Fv(are)d(\014ne,)g(since)266 527 y(i.)43 b(If)32 b Fp(x)c Fq(2)g(D)i Fv(:)e Fp(B)874 543 y Fm(l)q Fn(\()p Fm(x)p Fn(\))995 527 y Fv(\()p Fp(x)p Fv(\))22 b Fq(\\)f Fv(supp)d(\007)1529 542 y Fn(0)1596 527 y Fq(6)p Fv(=)28 b Fq(;)p Fv(,)k Fp(x)c Fq(2)g Fv(supp)18 b(\007)2280 542 y Fn(0)2319 527 y Fv(,)32 b Fq(j)p Fp(x)2461 491 y Fo(0)2506 527 y Fq(\000)22 b Fp(x)p Fq(j)28 b(\024)g Fv(8)p Fp(l)r Fv(\()p Fp(x)p Fv(\))k(and)h Fp(x)28 b Fq(2)g Fp(D)3512 542 y Fm(i)120 648 y Fv(\(V)-8 b(oronoi)31 b(cell)g(of)i(the)g(p)s(oin)m(t)e Fp(Z)1302 611 y Fn(1)p Fm(=)p Fn(3)1412 648 y Fp(Z)1479 663 y Fm(i)1507 648 y Fv(,)i(w)m(e)g(ha)m(v)m(e,)h(for)e Fp(j)i Fv(=)28 b(1)p Fp(;)17 b(:)g(:)g(:)32 b(;)17 b(M)120 868 y Fq(j)p Fp(x)203 826 y Fo(0)249 868 y Fq(\000)22 b Fp(Z)422 826 y Fn(1)p Fm(=)p Fn(3)532 868 y Fp(R)606 883 y Fm(j)643 868 y Fq(j)83 b(\025)g(j)p Fp(x)22 b Fq(\000)h Fp(Z)1193 826 y Fn(1)p Fm(=)p Fn(3)1303 868 y Fp(R)1377 883 y Fm(j)1414 868 y Fq(j)e(\000)i(j)p Fp(x)f Fq(\000)h Fp(x)1823 826 y Fo(0)1847 868 y Fq(j)754 1013 y(\025)83 b(j)p Fp(x)22 b Fq(\000)h Fp(Z)1193 972 y Fn(1)p Fm(=)p Fn(3)1303 1013 y Fp(R)1377 1028 y Fm(i)1405 1013 y Fq(j)f(\000)h Fv(8)p Fp(\033)t Fq(j)p Fp(x)f Fq(\000)g Fp(Z)1941 972 y Fn(1)p Fm(=)p Fn(3)2051 1013 y Fp(R)2125 1028 y Fm(i)2154 1013 y Fq(j)32 b Fv(since)h Fq(j)p Fp(x)22 b Fq(\000)h Fp(x)2713 972 y Fo(0)2737 1013 y Fq(j)k(\024)h Fv(8)p Fp(l)r Fv(\()p Fp(x)p Fv(\))g(=)g(8)p Fp(\033)t Fq(j)p Fp(x)22 b Fq(\000)g Fp(Z)3626 972 y Fn(1)p Fm(=)p Fn(3)3736 1013 y Fp(R)3810 1028 y Fm(i)3839 1013 y Fq(j)754 1158 y(\025)83 b Fv(\(1)22 b Fq(\000)h Fv(8)p Fp(\033)t Fv(\))16 b Fq(j)p Fp(x)22 b Fq(\000)h Fp(R)1564 1173 y Fm(i)1592 1158 y Fq(j)2024 b Fv(\(103\))754 1327 y Fq(\025)83 b Fv(\(1)22 b Fq(\000)h Fv(8)p Fp(\033)t Fv(\))1295 1259 y Fp(\026)p 1295 1304 59 4 v 1300 1395 a Fv(8)1391 1327 y(=)28 b Fp(\015)5 b(\026)120 1572 y Fv(Similarly)954 1792 y Fq(j)p Fp(x)1037 1751 y Fo(0)1082 1792 y Fq(\000)23 b Fp(Z)1256 1751 y Fn(1)p Fm(=)p Fn(3)1366 1792 y Fp(R)1440 1807 y Fm(j)1476 1792 y Fq(j)83 b(\024)h(j)p Fp(x)22 b Fq(\000)h Fp(Z)2027 1751 y Fn(1)p Fm(=)p Fn(3)2136 1792 y Fp(R)2210 1807 y Fm(j)2247 1792 y Fq(j)f Fv(+)g Fq(j)p Fp(x)g Fq(\000)h Fp(x)2655 1751 y Fo(0)2679 1792 y Fq(j)1587 1937 y(\024)84 b Fv(\(1)22 b(+)g(8)p Fp(\033)t(\015)2119 1896 y Fo(\000)p Fn(1)2213 1937 y Fv(\))p Fp(h)2307 1896 y Fo(\000)p Fn(1)3318 1937 y Fv(\(104\))1588 2127 y(=)1758 2060 y(1)g(+)g(56)p Fp(\033)p 1758 2104 326 4 v 1781 2195 a Fv(1)g Fq(\000)h Fv(8)p Fp(\033)2093 2127 y(h)2149 2086 y Fo(\000)p Fn(1)2271 2127 y Fp(<)28 b(\015)2431 2086 y Fo(\000)p Fn(1)2525 2127 y Fp(h)2581 2086 y Fo(\000)p Fn(1)266 2386 y Fv(ii.)40 b(If)28 b Fp(x)h Fq(2)f(D)i Fv(:)e Fp(B)896 2402 y Fm(l)q Fn(\()p Fm(x)p Fn(\))1016 2386 y Fv(\()p Fp(x)p Fv(\))13 b Fq(\\)g Fv(supp)19 b(\007)1534 2401 y Fn(0)1601 2386 y Fq(6)p Fv(=)27 b Fq(;)p Fv(,)i Fp(x)40 b(=)-61 b Fq(2)28 b Fv(supp)18 b(\007)2281 2401 y Fn(0)2320 2386 y Fv(,)29 b Fq(j)p Fp(x)2459 2350 y Fo(0)2496 2386 y Fq(\000)13 b Fp(x)p Fq(j)28 b(\024)g Fv(8)p Fp(l)r Fv(\()p Fp(x)p Fv(\))g(and)h Fp(x)f Fq(2)g Fp(D)3485 2401 y Fm(i)3513 2386 y Fv(,)120 2507 y(w)m(e)k(ha)m(v)m(e,)g(for)e Fq(j)p Fp(x)742 2471 y Fo(0)783 2507 y Fq(\000)18 b Fp(x)p Fq(j)28 b(\024)g Fv(8)p Fp(l)r Fv(\()p Fp(x)p Fv(\))j(and)g Fp(j)i Fv(=)28 b(1)p Fp(;)17 b(:)g(:)g(:)32 b(;)17 b(M)41 b Fv(and)30 b(taking)g Fp(y)h Fq(2)d Fp(B)2853 2522 y Fm(l)q Fn(\()p Fm(x)p Fn(\))2974 2507 y Fv(\()p Fp(x)p Fv(\))18 b Fq(\\)g Fv(supp)g(\007)3501 2522 y Fn(0)741 2727 y Fq(j)p Fp(x)824 2686 y Fo(0)870 2727 y Fq(\000)k Fp(R)1043 2742 y Fm(j)1080 2727 y Fq(j)83 b(\025)g(j)p Fp(x)22 b Fq(\000)h Fp(R)1630 2742 y Fm(j)1667 2727 y Fq(j)f(\000)g(j)p Fp(y)j Fq(\000)e Fp(x)p Fq(j)f(\000)h(j)p Fp(x)f Fq(\000)h Fp(x)2482 2686 y Fo(0)2505 2727 y Fq(j)1191 2898 y(\025)1361 2831 y Fp(\026)p 1361 2875 59 4 v 1366 2966 a Fv(8)1452 2898 y Fq(\000)g Fv(2)p Fp(l)r Fv(\()p Fp(x)p Fv(\))1191 3110 y Fq(\025)1361 3043 y Fp(\026)p 1361 3087 V 1366 3178 a Fv(8)1452 3110 y Fq(\000)1562 3043 y Fp(\033)p 1562 3087 60 4 v 1567 3178 a Fv(4)1631 3110 y Fp(\026)32 b Fv(since)h Fp(l)r Fv(\()p Fp(x)p Fv(\))28 b Fq(\024)g Fp(\033)t Fq(j)p Fp(x)22 b Fq(\000)h Fp(R)2594 3125 y Fm(i)2622 3110 y Fq(j)28 b Fp(<)2791 3043 y(\033)p 2791 3087 V 2796 3178 a Fv(8)2860 3110 y Fp(\026)399 b Fv(\(105\))1191 3343 y Fp(>)1361 3276 y Fv(3)p 1361 3320 49 4 v 1361 3411 a(4)1420 3343 y Fp(\026)27 b(>)h(\015)5 b(\026)266 3598 y Fv(iii.)40 b(Let)30 b Fp(F)41 b Fv(:)28 b Fl(R)21 b Fq(\002)16 b Fl(R)995 3562 y Fn(3)1068 3598 y Fq(7!)27 b Fl(R)1261 3613 y Fn(+)1326 3598 y Fv(;)46 b(\()p Fp(t;)17 b(x)p Fv(\))28 b Fq(7!)f Fp(t)16 b Fv(+)g Fq(j)p Fp(x)p Fq(j)2018 3562 y Fn(2)2057 3598 y Fv(.)43 b(There)30 b(is)f(a)g(`non-critical')e(condition)120 3719 y(that)32 b(lo)s(oks)g(as)h(follo)m(ws:)42 b(F)-8 b(or)32 b(some)g Fp(!)f(>)d Fv(0)k(the)h(next)h(condition)d(is)h(satis\014ed:)871 3974 y Fp(F)964 3833 y Fk(\022)1037 3974 y Fq(j)p Fp(W)14 b Fv(\()p Fp(x)p Fv(\))p Fq(j)22 b Fv(+)g Fp(h)1516 3906 y(f)11 b Fv(\()p Fp(x)p Fv(\))p 1516 3951 191 4 v 1530 4042 a Fp(l)r Fv(\()p Fp(x)p Fv(\))1716 3974 y Fp(;)1783 3906 y(l)r Fv(\()p Fp(x)p Fv(\))p 1769 3951 V 1769 4042 a Fp(f)g Fv(\()p Fp(x)p Fv(\))1969 3974 y Fq(r)p Fp(W)j Fv(\()p Fp(x)p Fv(\))2289 3833 y Fk(\023)2390 3974 y Fq(\025)29 b Fp(!)t(f)11 b Fv(\()p Fp(x)p Fv(\))2751 3933 y Fn(2)3318 3974 y Fv(\(106\))120 4251 y(Condition)30 b(\(106\))h(is)g(ful\014lled)f(b)s(ecause)i(of)g(the)g(estimate)e(from) h(b)s(elo)m(w)g(pro)m(vided)h(b)m(y)g(The-)120 4371 y(orem)d(4.)42 b(Finally)27 b(the)k(supplemen)m(tary)f(condition)e(on)i(the)g(scale)g (functions)f Fp(f)11 b Fv(\()p Fp(x)p Fv(\))p Fp(l)r Fv(\()p Fp(x)p Fv(\))29 b Fq(\025)f Fp(h)120 4492 y Fv(is)k(also)g (trivially)d(satis\014ed.)266 4612 y(As)35 b(a)e(consequence,)k(all)32 b(the)i(h)m(yp)s(othesis)h(of)f(Theorem)g(5.2)f(of)g([31])h(are)g (satis\014ed)g(and)1781 5712 y(24)p eop %%Page: 25 25 25 24 bop 120 407 a Fv(applying)31 b(that)i(Theorem)g(w)m(e)g(get)304 637 y Fq(jM)p Fv(\()p Fp(h;)17 b(W)m(;)g Fv(\007)799 652 y Fn(0)p Fm(;h)898 637 y Fv(\))83 b Fq(\000)1263 570 y Fv(1)p 1189 614 196 4 v 1189 706 a(15)p Fp(\031)1346 677 y Fn(2)1395 637 y Fp(h)1451 596 y Fo(\000)p Fn(3)1562 502 y Fk(Z)1678 637 y Fv(\007)1754 652 y Fn(0)p Fm(;h)1854 637 y Fp(W)1960 596 y Fn(5)p Fm(=)p Fn(2)2070 637 y Fq(j)1019 898 y(\024)g Fp(C)1289 763 y Fk(Z)1345 988 y Fo(D)1422 898 y Fp(f)11 b Fv(\()p Fp(x)p Fv(\))1612 857 y Fn(2)1662 831 y Fp(f)g Fv(\()p Fp(x)p Fv(\))p Fp(l)r Fv(\()p Fp(x)p Fv(\))p 1662 875 353 4 v 1810 967 a Fp(h)2024 898 y(l)r Fv(\()p Fp(x)p Fv(\))2186 857 y Fo(\000)p Fn(3)2298 898 y Fp(dx)1019 1157 y Fq(\024)83 b Fp(C)24 b(h)1329 1116 y Fo(\000)p Fn(1)1440 1022 y Fk(Z)1495 1247 y Fo(D)1556 1157 y Fv(\()p Fp(\032)p Fv(\()p Fp(x)p Fv(\))1775 1116 y Fo(\000)p Fn(2)1870 1157 y Fq(h)p Fp(\032)p Fv(\()p Fp(x)p Fv(\))p Fq(i)2129 1116 y Fo(\000)p Fn(6)2245 1157 y Fv(+)e Fp(h)2399 1116 y Fn(2)2439 1157 y Fv(\))2477 1116 y Fn(3)p Fm(=)p Fn(4)2587 1157 y Fv(\(\()p Fp(\032)p Fv(\()p Fp(x)p Fv(\))2844 1116 y Fn(2)2906 1157 y Fv(+)g Fp(h)3060 1116 y Fn(2)3100 1157 y Fv(\))3138 1116 y Fo(\000)p Fn(1)3248 1157 y Fp(dx)1019 1460 y Fq(\024)83 b Fp(C)24 b(h)1329 1419 y Fo(\000)p Fn(1)1474 1336 y Fm(M)1440 1366 y Fk(X)1451 1575 y Fm(j)t Fn(=1)1600 1325 y Fk(Z)1656 1550 y Fm(D)1714 1560 y Fi(j)1746 1550 y Fo(\\D)1854 1460 y Fv(\()p Fp(\032)p Fv(\()p Fp(x)p Fv(\))2073 1419 y Fo(\000)p Fn(3)p Fm(=)p Fn(2)2238 1460 y Fq(h)p Fp(\032)p Fv(\()p Fp(x)p Fv(\))p Fq(i)2497 1419 y Fo(\000)p Fn(9)p Fm(=)p Fn(2)2685 1460 y Fv(+)e Fp(h)p Fv(\))p Fp(\032)p Fv(\()p Fp(x)p Fv(\))3058 1419 y Fo(\000)p Fn(2)3169 1460 y Fp(dx)43 b Fv(\(107\))1019 1811 y Fq(\024)83 b Fp(C)24 b(h)1329 1770 y Fo(\000)p Fn(1)1474 1687 y Fm(M)1440 1717 y Fk(X)1451 1927 y Fm(j)t Fn(=1)1600 1676 y Fk(Z)1656 1901 y Fm(\015)t(\026)p Fo(\024j)p Fm(x)p Fo(j)1893 1811 y Fq(j)p Fp(x)p Fq(j)2004 1770 y Fo(\000)p Fn(7)p Fm(=)p Fn(2)2185 1811 y Fp(dx)f Fv(+)2446 1687 y Fm(M)2412 1717 y Fk(X)2422 1927 y Fm(j)t Fn(=1)2572 1676 y Fk(Z)2627 1901 y Fo(j)p Fm(x)p Fo(j\024)p Fm(\015)2802 1882 y Fh(\000)p Fj(1)2884 1901 y Fm(h)2925 1882 y Fh(\000)p Fj(1)3028 1811 y Fq(j)p Fp(x)p Fq(j)3139 1770 y Fo(\000)p Fn(2)3250 1811 y Fp(dx)1019 2067 y Fq(\024)83 b Fp(C)7 b(h)1312 2026 y Fo(\000)p Fn(1)1407 2067 y Fp(M)33 b Fv(+)22 b Fp(C)7 b(h)1765 2026 y Fo(\000)p Fn(1)1859 2067 y Fp(M)1019 2212 y Fq(\024)83 b Fp(C)7 b(h)1312 2171 y Fo(\000)p Fn(1)1407 2212 y Fp(M)j Fv(;)50 b(0)27 b Fp(<)h(h)g Fq(\024)g Fv(1)120 2400 y(for)d(some)g(\014nite)h(absolute)f Fp(C)32 b Fv(indep)s(enden)m(t)27 b(of)e Fp(h)j(>)f Fv(0)e(and.)42 b(T)-8 b(o)25 b(con)m(trol)g(the)h(other)g(piece)f(of)120 2520 y Fq(M)p Fv(\()p Fp(h;)17 b(W)m(;)g Fv(\007)587 2535 y Fn(0)626 2520 y Fv(\),)28 b(namely)-8 b(,)26 b Fq(M)p Fv(\()p Fp(h;)17 b(W)m(;)g(\035)1515 2535 y Fn(0)1554 2520 y Fv(\),)28 b(w)m(e)f(shall)e(just)i(use)h(the)f(Lieb-Thirring)d (inequalit)m(y)120 2640 y(and)33 b(w)m(e)g(get)551 2828 y Fq(M)p Fv(\()p Fp(h;)17 b(W)m(;)g(\035)995 2843 y Fn(0)1033 2828 y Fv(\))83 b Fq(\024)h Fv(T)-8 b(r)17 b([)p Fp(\035)1512 2843 y Fn(0)1551 2828 y Fv(\()p Fq(\000)p Fp(h)1722 2787 y Fn(2)1762 2828 y Fv(\001)22 b Fq(\000)h Fp(W)14 b Fv(\)])2136 2843 y Fo(\000)1154 2973 y Fq(\024)84 b Fp(h)1371 2932 y Fn(2)1410 2973 y Fv(T)-8 b(r)17 b(\()p Fq(\000)p Fv(\001)23 b Fq(\000)g Fp(h)1902 2932 y Fo(\000)p Fn(2)1996 2973 y Fp(\037)2057 2988 y Fn(supp)12 b Fm(\035)2252 2997 y Fj(0)2292 2973 y Fp(W)i Fv(\))2436 2988 y Fo(\000)1154 3170 y Fq(\024)84 b Fp(K)7 b(h)1461 3129 y Fo(\000)p Fn(3)1572 3034 y Fk(Z)1627 3260 y Fm(x)p Fn(:)p Fm(\032)p Fn(\()p Fm(x)p Fn(\))p Fo(\025)1900 3233 y Fj(1)p 1882 3245 67 3 v 1882 3286 a(2)p Fi(h)1980 3170 y Fp(\032)p Fv(\()p Fp(x)p Fv(\))2161 3129 y Fn(5)p Fm(=)p Fn(2)2288 3170 y Fp(dx)924 b Fv(\(108\))1154 3499 y Fq(\024)84 b Fp(C)7 b(h)1448 3458 y Fo(\000)p Fn(3)1593 3374 y Fm(M)1559 3404 y Fk(X)1569 3614 y Fm(j)t Fn(=1)1719 3363 y Fk(Z)1775 3589 y Fm(x)p Fn(:)p Fo(j)p Fm(x)p Fo(\000)p Fm(Z)2003 3569 y Fj(1)p Fi(=)p Fj(3)2098 3589 y Fm(R)2151 3599 y Fi(j)2183 3589 y Fo(j\025)2286 3562 y Fj(1)p 2268 3574 V 2268 3615 a(2)p Fi(h)2365 3499 y Fq(j)p Fp(x)22 b Fq(\000)h Fp(Z)2644 3458 y Fn(1)p Fm(=)p Fn(3)2754 3499 y Fp(R)2828 3514 y Fm(j)2865 3499 y Fq(j)2893 3458 y Fo(\000)p Fn(6)3003 3499 y Fp(dx)1154 3738 y Fq(\024)84 b Fp(C)7 b(M)120 3926 y Fv(and)52 b(therefore)h(it)d(is)i(v)m(ery)h(small)d(compared)h (to)h(the)g(error)g(coming)e(from)h(the)h(term)120 4046 y Fq(M)p Fv(\()p Fp(h;)17 b(W)m(;)g Fv(\007)587 4061 y Fn(0)p Fm(;h)686 4046 y Fv(\),)33 b(and)f(so,)h(putting)f(together)h (estimates)f(\(107\))g(and)g(\(108\))g(w)m(e)i(get)822 4285 y Fq(jM)p Fv(\()p Fp(h;)17 b(W)m(;)g Fv(\007)1317 4300 y Fn(0)1355 4285 y Fv(\))23 b Fq(\000)1599 4218 y Fv(1)p 1525 4262 196 4 v 1525 4354 a(15)p Fp(\031)1682 4325 y Fn(2)1731 4285 y Fp(h)1787 4244 y Fo(\000)p Fn(3)1898 4150 y Fk(Z)2014 4285 y Fv(\007)2090 4300 y Fn(0)2130 4285 y Fp(W)2236 4244 y Fn(5)p Fm(=)p Fn(2)2345 4285 y Fq(j)28 b(\024)g Fp(C)7 b(h)2639 4244 y Fo(\000)p Fn(1)2733 4285 y Fp(M)491 b Fv(\(109\))266 4521 y(It)38 b(remains)e(to)h (consider)g(the)h(other)f(terms)g(of)g(the)g(sum)h(in)e(\(94\).)57 b(The)38 b(Scott)f(terms)120 4642 y(shall)e(come)i(en)m(tirely)f(from)f (them.)56 b(In)37 b(order)f(to)h(study)h(those)f(terms,)h(w)m(e)f(note) g(that)f(for)120 4762 y Fp(x)28 b Fq(2)g Fv(supp)18 b(\007)591 4777 y Fm(i)619 4762 y Fv(;)49 b Fp(M)789 4777 y Fn(1)857 4762 y Fq(\024)28 b Fp(i)g Fq(\024)g Fp(M)43 b Fv(w)m(e)34 b(can)f(write)872 4992 y Fp(W)14 b Fv(\()p Fp(x)p Fv(\))27 b(=)h Fp(')p Fv(\()p Fp(\025)1399 5007 y Fm(i)1427 4992 y Fv(\()p Fp(x)22 b Fq(\000)h Fp(Z)1716 4951 y Fn(1)p Fm(=)p Fn(3)1826 4992 y Fp(R)1900 5007 y Fm(i)1928 4992 y Fv(\)\))f(+)g Fp(w)2194 5007 y Fm(i)2222 4992 y Fv(\()p Fp(x)p Fv(\);)50 b Fp(\025)2487 5007 y Fm(i)2543 4992 y Fv(:=)2683 4925 y Fp(Z)2750 4940 y Fm(i)p 2683 4969 95 4 v 2694 5061 a Fp(Z)3318 4992 y Fv(\(110\))120 5222 y(where)37 b Fp(')d Fv(=)g(\010)683 5186 y Fn(TF)683 5247 y(0)p Fm(;)p Fn(1)820 5222 y Fv(and)j Fp(w)1084 5237 y Fm(i)1145 5222 y Fq(2)d Fp(C)1322 5186 y Fo(1)1397 5222 y Fv(\()p Fp(B)1519 5209 y Fi(\026)p 1519 5225 38 3 v 1522 5266 a Fj(4)1571 5222 y Fv(\()p Fp(Z)1683 5186 y Fn(1)p Fm(=)p Fn(3)1792 5222 y Fp(R)1866 5237 y Fm(i)1895 5222 y Fv(\)\))i(is)g(b)s(ounded)g(along)f(together)h(with)g(an)m(y)120 5342 y(of)f(its)g(deriv)-5 b(ativ)m(es)35 b(uniformly)e(in)i Fp(M)10 b Fv(.)52 b(W)-8 b(e)36 b(can)g(use)g(this)f(information)d(to)j (get)g(estimates)120 5463 y(on)d(lo)s(cal)f(traces)i(\014rstly)g(for)f (a)g(purely)h(coulom)m(bic)e(p)s(oten)m(tial)g(near)i(its)f(singularit) m(y)1781 5712 y(25)p eop %%Page: 26 26 26 25 bop 266 407 a Fr(Step)35 b(4:)45 b(Estimating)34 b(the)h(singularities)f(of)h Fp(W)14 b Fr(:)266 527 y Fv(Here)45 b(w)m(e)f(shall)f(consider)h(\014rst)g(sp)s(ectral)f(c)m (haracteristics)h(of)f(coulom)m(bic)f(p)s(oten)m(tials)120 648 y(and)28 b(the)h(analysis)f(of)g(this)g(problem,)g(suitably)f(mo)s (di\014ed,)h(w)m(ould)g(yield)g(the)h(result)f(for)g(the)120 768 y(full)35 b(TF)i(p)s(oten)m(tial)f(of)g(a)h(con\014guration)f(of)h (uniformly)d(separated)k(n)m(uclei.)57 b(So,)38 b(to)f(b)s(egin)120 888 y(with,)32 b(w)m(e)i(consider)f(the)g(follo)m(wing)d(result:)120 1059 y Ft(Lemma)37 b(8)h(\(Semiclassics)c(for)k(the)f(Coulom)m(b)g(p)s (oten)m(tial\).)47 b Fr(Consider)23 b(we)i(ar)-5 b(e)25 b(given)120 1179 y Fp( )32 b Fq(2)c Fp(C)386 1143 y Fo(1)379 1204 y Fm(c)460 1179 y Fv(\()p Fp(B)572 1194 y Fm(r)610 1179 y Fv(\(0\)\))f Fr(\()p Fv(0)h Fp(<)f(r)j Fq(\024)f Fv(1)p Fr(\))e(and)f Fp(s)i(>)f Fv(0)h Fr(such)f(that)g Fp( )32 b Fv(=)27 b(1)h Fr(if)f Fq(j)p Fp(x)p Fq(j)g(\024)h Fp(r)s(=)p Fv(2)f Fr(and)f Fp( )32 b Fr(satis\014es)120 1299 y(estimates)i(of)h(the)g(kind)f Fq(k)p Fp(@)1145 1263 y Fm(\013)1195 1299 y Fp( )t Fq(k)1312 1314 y Fo(1)1414 1299 y Fq(\024)29 b Fp(C)1590 1314 y Fm(\013)1639 1299 y Fp(r)1686 1263 y Fo(\000j)p Fm(\013)p Fo(j)1829 1299 y Fp(;)17 b Fq(8)p Fp(\013)29 b Fq(2)f Fl(N)2179 1263 y Fn(3)2224 1299 y Fr(.)45 b(Then)34 b(we)g(have)559 1588 y Fq(jM)p Fv(\()p Fp(h;)17 b( )t Fv(\))82 b Fq(\000)1276 1521 y Fv(1)p 1203 1566 196 4 v 1203 1657 a(15)p Fp(\031)1360 1628 y Fn(2)1408 1588 y Fp(h)1464 1547 y Fo(\000)p Fn(3)1576 1453 y Fk(Z)1692 1588 y Fp( )t Fv(\()p Fp(x)p Fv(\))1907 1448 y Fk(\022)2021 1521 y Fv(1)p 1990 1566 111 4 v 1990 1657 a Fq(j)p Fp(x)p Fq(j)2133 1588 y(\000)22 b Fp(s)2278 1448 y Fk(\023)2352 1470 y Fn(5)p Fm(=)p Fn(2)2352 1687 y(+)2495 1588 y Fp(dx)g Fv(+)2731 1521 y(1)p 2731 1566 49 4 v 2731 1657 a(8)2790 1588 y Fp(h)2846 1547 y Fo(\000)p Fn(2)2940 1588 y Fq(j)28 b(\024)1193 1818 y Fp(C)7 b Fv(\()p Fp(s)1354 1777 y Fo(\000)p Fn(1)p Fm(=)p Fn(2)1540 1818 y Fv(+)23 b Fp(r)1686 1777 y Fo(\000)p Fn(1)p Fm(=)p Fn(2)1850 1818 y Fv(\))p Fp(h)1944 1777 y Fo(\000)p Fn(1)2038 1818 y Fv(;)52 b(0)28 b Fp(<)f(h)h Fq(\024)g Fv(1)783 b(\(111\))120 2027 y Fr(wher)-5 b(e)30 b Fq(M)p Fv(\()p Fp(h;)17 b( )t Fv(\))27 b(:=)g(T)-8 b(r)17 b([)p Fp( )t Fv(\()p Fq(\000)p Fp(h)1293 1991 y Fn(2)1333 2027 y Fv(\001)12 b Fq(\000)1548 1988 y Fn(1)p 1526 2004 79 4 v 1526 2061 a Fo(j)p Fm(x)p Fo(j)1627 2027 y Fv(+)g Fp(s)p Fv(\))1799 2042 y Fo(\000)1858 2027 y Fv(])30 b Fr(and)g(the)g(r)-5 b(emainder)29 b(estimate)h(is)h(uniform)120 2147 y(in)k Fp( )j Fr(in)d(the)g(sense)f(that)h(only)g(dep)-5 b(ends)33 b(on)i(the)g(c)-5 b(onstants)34 b Fp(C)2456 2162 y Fm(\013)2540 2147 y Fr(ab)-5 b(ove.)120 2339 y(Pr)g(o)g(of.)48 b Fv(Call)31 b Fq(M)p Fv(\()p Fp(h)p Fv(\))d(:=)f Fq(M)p Fv(\()p Fp(h;)17 b Fv(1\).)43 b(Then,)1130 2547 y Fq(M)p Fv(\()p Fp(h;)17 b( )t Fv(\))27 b(=)h Fq(M)p Fv(\()p Fp(h)p Fv(\))21 b Fq(\000)i(M)p Fv(\()p Fp(h;)17 b Fv(1)k Fq(\000)i Fp( )t Fv(\))788 b(\(112\))120 2756 y(Using)32 b(Multiscale)f(Analysis)i(lik)m (e)f(in)f(Step)j(2)e(ab)s(o)m(v)m(e,)h(w)m(e)h(get)465 3039 y Fq(jM)p Fv(\()p Fp(h;)17 b Fv(1)k Fq(\000)h Fp( )t Fv(\))83 b Fq(\000)1352 2972 y Fv(1)p 1279 3016 196 4 v 1279 3108 a(15)p Fp(\031)1436 3079 y Fn(2)1485 3039 y Fp(h)1541 2998 y Fo(\000)p Fn(3)1652 2904 y Fk(Z)1751 3039 y Fv(\(1)22 b Fq(\000)h Fp( )t Fv(\()p Fp(x)p Fv(\)\))2213 2899 y Fk(\022)2327 2972 y Fv(1)p 2296 3016 111 4 v 2296 3108 a Fq(j)p Fp(x)p Fq(j)2439 3039 y(\000)f Fp(s)2584 2899 y Fk(\023)2658 2921 y Fn(5)p Fm(=)p Fn(2)2658 3138 y(+)2801 3039 y Fp(dx)p Fq(j)27 b(\024)i Fp(C)7 b(I)974 3314 y(I)92 b Fv(=)1269 3178 y Fk(Z)1324 3404 y Fm(\015)t(r)r Fo(\024j)p Fm(x)p Fo(j\024)p Fm(\015)1628 3385 y Fh(\000)p Fj(1)1730 3314 y Fq(j)p Fp(x)p Fq(j)1841 3273 y Fo(\000)p Fn(7)p Fm(=)p Fn(2)2023 3314 y Fp(dx)1189 b Fv(\(113\))1108 3545 y Fq(\024)84 b Fp(C)7 b(r)1393 3503 y Fo(\000)p Fn(1)p Fm(=)p Fn(2)1557 3545 y Fp(h)1613 3503 y Fo(\000)p Fn(1)1708 3545 y Fv(;)49 b(0)27 b Fp(<)h(h)g Fq(\024)g Fv(1)120 3753 y(The)37 b(p)s(oin)m(t)e(no)m(w)i(is)e(that)h(the)h(quan) m(tit)m(y)f Fq(M)p Fv(\()p Fp(h)p Fv(\))g(\(i.e,)g(the)h(sum)f(of)f (eigen)m(v)-5 b(alues)37 b(less)f(than)120 3873 y Fq(\000)p Fp(s)d Fv(of)f(a)g(h)m(ydrogenic)i(atom\))d(can)i(b)s(e)f(exactly)h (computed:)266 4021 y(Consider)g(the)g(Hamiltonian)c Fp(H)35 b Fv(:=)28 b Fq(\000)p Fp(h)1782 3985 y Fn(2)1822 4021 y Fv(\001)23 b Fq(\000)2025 3911 y Fk(\020)2116 3982 y Fn(1)p 2095 3998 79 4 v 2095 4055 a Fo(j)p Fm(x)p Fo(j)2206 4021 y Fq(\000)f Fp(s)2351 3911 y Fk(\021)2411 4021 y Fv(.)43 b(Its)33 b(eigen)m(v)-5 b(alues)33 b(are)771 4314 y Fp(\025)828 4329 y Fm(n)902 4314 y Fv(=)28 b Fq(\000)1190 4246 y Fv(1)p 1093 4291 243 4 v 1093 4382 a(4)p Fp(h)1198 4353 y Fn(2)1238 4382 y Fp(n)1296 4353 y Fn(2)1367 4314 y Fv(+)22 b Fp(s)p Fv(;)34 b Fp(n)28 b Fv(=)f(1)p Fp(;)17 b Fv(2)p Fp(;)g(:)g(:)g(:)32 b Fv(;)49 b(m)m(ultiplicit)m(y)24 b(=)k Fp(n)2850 4273 y Fn(2)3318 4314 y Fv(\(114\))120 4548 y(W)-8 b(e)33 b(ha)m(v)m(e)h(the)f(form)m(ula)1102 4807 y Fp(S)1162 4822 y Fm(m)1256 4807 y Fv(:=)1428 4682 y Fm(m)1387 4712 y Fk(X)1393 4922 y Fm(n)p Fn(=1)1548 4807 y Fp(n)1606 4766 y Fn(2)1673 4807 y Fv(=)1786 4740 y(1)p 1786 4784 49 4 v 1786 4875 a(3)1845 4807 y Fp(m)1930 4766 y Fn(3)1992 4807 y Fv(+)2100 4740 y(1)p 2100 4784 V 2100 4875 a(2)2159 4807 y Fp(m)2244 4766 y Fn(2)2306 4807 y Fv(+)2414 4740 y(1)p 2414 4784 V 2414 4875 a(6)2472 4807 y Fp(m)761 b Fv(\(115\))120 5103 y(and)33 b(from)e(\(114\))h (follo)m(ws)1205 5368 y Fp(\025)1262 5383 y Fm(n)1336 5368 y Fq(\024)c Fv(0)g Fq(,)f Fp(n)h Fq(\024)g Fp(m)g Fv(=)2053 5228 y Fk(\024)2221 5301 y Fv(1)p 2115 5345 261 4 v 2115 5437 a(2)p Fp(hs)2266 5408 y Fn(1)p Fm(=)p Fn(2)2386 5228 y Fk(\025)3318 5368 y Fv(\(116\))1781 5712 y(26)p eop %%Page: 27 27 27 26 bop 120 407 a Fv(W)-8 b(rite)32 b Fp(\013)c Fv(=)g(\(2)p Fp(hs)774 371 y Fn(1)p Fm(=)p Fn(2)884 407 y Fv(\))922 371 y Fo(\000)p Fn(1)1038 407 y Fq(\000)23 b Fp(m)28 b Fq(2)g Fv([0)p Fp(;)17 b Fv(1\).)43 b(Then)410 694 y(T)-8 b(r)543 554 y Fk(\024)596 694 y Fq(\000)p Fp(h)729 653 y Fn(2)769 694 y Fv(\001)22 b Fq(\000)972 554 y Fk(\022)1086 627 y Fv(1)p 1055 671 111 4 v 1055 763 a Fq(j)p Fp(x)p Fq(j)1198 694 y(\000)h Fp(s)1344 554 y Fk(\023\025)1469 793 y Fo(\000)1611 694 y Fv(=)1811 570 y Fm(m)1770 600 y Fk(X)1776 809 y Fm(n)p Fn(=1)1931 554 y Fk(\022)2111 627 y Fv(1)p 2014 671 243 4 v 2014 763 a(4)p Fp(h)2119 734 y Fn(2)2159 763 y Fp(n)2217 734 y Fn(2)2288 694 y Fq(\000)g Fp(s)2434 554 y Fk(\023)2524 694 y Fp(n)2582 653 y Fn(2)1611 984 y Fv(=)1828 916 y(1)p 1780 961 145 4 v 1780 1052 a(4)p Fp(h)1885 1023 y Fn(2)1935 984 y Fp(m)f Fq(\000)h Fp(sS)2248 999 y Fm(m)1611 1217 y Fv(=)1828 1150 y(1)p 1780 1194 V 1780 1285 a(4)p Fp(h)1885 1256 y Fn(2)1935 1217 y Fv(\(\(2)p Fp(hs)2162 1176 y Fn(1)p Fm(=)p Fn(2)2271 1217 y Fv(\))2309 1176 y Fo(\000)p Fn(1)2426 1217 y Fq(\000)f Fp(\013)q Fv(\))g Fq(\000)2860 1150 y Fv(1)p 2758 1194 254 4 v 2758 1285 a(24)p Fp(s)2902 1256 y Fn(1)p Fm(=)p Fn(2)3021 1217 y Fp(h)3077 1176 y Fo(\000)p Fn(3)3318 1217 y Fv(\(117\))1770 1463 y Fq(\000)1864 1322 y Fk(\022)1948 1396 y Fv(1)p 1948 1440 49 4 v 1948 1531 a(8)2029 1463 y Fq(\000)2138 1396 y Fp(\013)p 2138 1440 63 4 v 2145 1531 a Fv(4)2211 1322 y Fk(\023)2301 1463 y Fp(h)2357 1422 y Fo(\000)p Fn(2)2473 1463 y Fv(+)g Fq(O)s Fv(\()p Fp(s)2737 1422 y Fo(\000)p Fn(1)p Fm(=)p Fn(2)2902 1463 y Fp(h)2958 1422 y Fo(\000)p Fn(1)3053 1463 y Fv(\))1611 1722 y(=)83 b Fq(\000)1960 1655 y Fv(1)p 1857 1699 254 4 v 1857 1791 a(12)p Fp(s)2001 1762 y Fn(1)p Fm(=)p Fn(2)2121 1722 y Fp(h)2177 1681 y Fo(\000)p Fn(3)2294 1722 y Fv(+)2402 1655 y(1)p 2402 1699 49 4 v 2402 1791 a(8)2461 1722 y Fp(h)2517 1681 y Fo(\000)p Fn(2)2633 1722 y Fv(+)22 b Fq(O)s Fv(\()p Fp(s)2897 1681 y Fo(\000)p Fn(1)p Fm(=)p Fn(2)3062 1722 y Fp(h)3118 1681 y Fo(\000)p Fn(1)3213 1722 y Fv(\))120 1959 y(On)33 b(the)g(other)f(hand,)h(w)m(e)h (can)f(compute)711 2111 y Fk(Z)827 2106 y(\022)942 2179 y Fv(1)p 911 2224 111 4 v 911 2315 a Fq(j)p Fp(x)p Fq(j)1054 2246 y(\000)22 b Fp(s)1199 2106 y Fk(\023)1272 2128 y Fn(5)p Fm(=)p Fn(2)1272 2345 y(+)1416 2246 y Fp(dx)83 b Fv(=)g(4)p Fp(\031)t(s)1918 2205 y Fo(\000)p Fn(1)p Fm(=)p Fn(2)2099 2111 y Fk(Z)2199 2137 y Fn(1)2154 2336 y(0)2255 2246 y Fp(t)2290 2205 y Fo(\000)p Fn(1)p Fm(=)p Fn(2)2455 2246 y Fv(\(1)22 b Fq(\000)g Fp(t)p Fv(\))2736 2205 y Fn(5)p Fm(=)p Fn(2)2863 2246 y Fp(dt)1605 2530 y Fv(=)83 b(4)p Fp(\031)t(s)1918 2489 y Fo(\000)p Fn(1)p Fm(=)p Fn(2)2082 2530 y Fp(B)2178 2389 y Fk(\022)2261 2462 y Fv(1)p 2261 2507 49 4 v 2261 2598 a(2)2320 2530 y Fp(;)2374 2462 y Fv(7)p 2374 2507 V 2374 2598 a(2)2432 2389 y Fk(\023)3318 2530 y Fv(\(118\))1605 2816 y(=)g(4)p Fp(\031)t(s)1918 2775 y Fo(\000)p Fn(1)p Fm(=)p Fn(2)2092 2742 y Fv(\000)2170 2662 y Fk(\000)2225 2703 y Fn(1)p 2225 2719 36 4 v 2225 2777 a(2)2271 2662 y Fk(\001)2333 2742 y Fv(\000)2411 2662 y Fk(\000)2466 2703 y Fn(7)p 2466 2719 V 2466 2777 a(2)2511 2662 y Fk(\001)p 2092 2793 465 4 v 2232 2884 a Fv(\000\(4\))1605 3089 y(=)1803 3021 y(15)p Fp(\031)1960 2985 y Fn(2)p 1774 3066 254 4 v 1774 3157 a Fv(12)p Fp(s)1918 3128 y Fn(1)p Fm(=)p Fn(2)120 3323 y Fv(so)33 b(w)m(e)g(can)g(write)514 3591 y(T)-8 b(r)648 3450 y Fk(\024)700 3591 y Fq(\000)p Fp(h)833 3549 y Fn(2)873 3591 y Fv(\001)23 b Fq(\000)1076 3450 y Fk(\022)1191 3523 y Fv(1)p 1160 3568 111 4 v 1160 3659 a Fq(j)p Fp(x)p Fq(j)1302 3591 y(\000)g Fp(s)1448 3450 y Fk(\023\025)1574 3689 y Fo(\000)1716 3591 y Fv(=)83 b Fq(\000)2036 3523 y Fv(1)p 1962 3568 196 4 v 1962 3659 a(15)p Fp(\031)2119 3630 y Fn(2)2168 3591 y Fp(h)2224 3549 y Fo(\000)p Fn(3)2335 3455 y Fk(Z)2451 3450 y(\022)2566 3523 y Fv(1)p 2535 3568 111 4 v 2535 3659 a Fq(j)p Fp(x)p Fq(j)2678 3591 y(\000)22 b Fp(s)2823 3450 y Fk(\023)2897 3472 y Fn(5)p Fm(=)p Fn(2)2897 3689 y(+)3040 3591 y Fp(dx)1875 3861 y Fv(+)1961 3794 y(1)p 1961 3838 49 4 v 1961 3929 a(8)2019 3861 y Fp(h)2075 3820 y Fo(\000)p Fn(2)2192 3861 y Fv(+)g Fq(O)s Fv(\()p Fp(s)2456 3820 y Fo(\000)p Fn(1)p Fm(=)p Fn(2)2621 3861 y Fp(h)2677 3820 y Fo(\000)p Fn(1)2771 3861 y Fv(\))509 b(\(119\))120 4104 y(Substracting)33 b(\(119\))e(from)h(\(113\))f(w)m(e)j(get)f(\(111\).)p 3469 4104 4 66 v 3473 4041 59 4 v 3473 4104 V 3531 4104 4 66 v 120 4296 a Ft(Lemma)k(9)h(\(Semiclassics)c(for)k(a)g(regular)f (p)s(erturbation)g(of)g(TF)h(p)s(oten)m(tial\).)47 b Fr(Consider)120 4417 y(we)38 b(ar)-5 b(e)39 b(given)f Fp( )g Fq(2)e Fp(C)976 4381 y Fo(1)969 4441 y Fm(c)1050 4417 y Fv(\()p Fp(B)1162 4432 y Fn(1)1202 4417 y Fv(\(0\)\))i Fr(such)h(that)g Fp( )f Fv(=)d(1)j Fr(if)h Fq(j)p Fp(x)p Fq(j)34 b(\024)i Fv(1)p Fp(=)p Fv(2)i Fr(and)g Fp(w)f Fq(2)e Fp(C)3151 4381 y Fo(1)3144 4441 y Fm(c)3226 4417 y Fv(\()p Fp(B)3338 4432 y Fn(1)3377 4417 y Fv(\(0\)\))120 4537 y Fr(such)g(that)g Fq(8)p Fp(\013)29 b Fq(2)f Fl(N)847 4501 y Fn(3)892 4537 y Fp(;)33 b Fq(k)p Fp(@)1058 4501 y Fm(\013)1109 4537 y Fp( )t Fq(k)1226 4552 y Fo(1)1300 4537 y Fp(;)g Fq(k)p Fp(@)1466 4501 y Fm(\013)1517 4537 y Fp(w)s Fq(k)1640 4552 y Fo(1)1741 4537 y Fq(\024)28 b Fp(C)1916 4552 y Fm(\013)1966 4537 y Fr(.)44 b(Then)34 b(we)h(have,)f(for)h Fp(\016)c Fv(=)d(9)p Fp(=)p Fv(17)290 4797 y Fq(jM)p Fv(\()p Fp(h;)17 b(w)s(;)g( )t Fv(\))j Fq(\000)1002 4730 y Fv(1)p 928 4774 196 4 v 928 4866 a(15)p Fp(\031)1085 4837 y Fn(2)1134 4797 y Fp(h)1190 4756 y Fo(\000)p Fn(3)1301 4662 y Fk(Z)1417 4797 y Fp( )t Fv(\()p Fp(')i Fv(+)g Fp(w)s Fv(\))1817 4756 y Fn(5)p Fm(=)p Fn(2)1949 4797 y Fv(+)2057 4730 y(1)p 2057 4774 49 4 v 2057 4866 a(8)2115 4797 y Fp(h)2171 4756 y Fo(\000)p Fn(2)2266 4797 y Fq(j)27 b(\024)h Fp(C)7 b(h)2559 4756 y Fo(\000)p Fn(2+)p Fm(\016)2742 4797 y Fv(;)17 b(0)28 b Fp(<)f(h)h Fq(\024)g Fv(1)114 b(\(120\))120 5052 y Fr(wher)-5 b(e)35 b(the)g(r)-5 b(emainder)34 b(estimate)h(is)g(uniform) g(in)f Fp(w)k Fr(in)d(the)g(sense)g(that)g(only)g(dep)-5 b(ends)34 b(on)120 5172 y(the)h(c)-5 b(onstants)34 b Fp(C)780 5187 y Fm(\013)864 5172 y Fr(ab)-5 b(ove.)266 5342 y Fv(W)d(e)31 b(shall)f(defer)h(for)f(a)g(momen)m(t)g(the)h(pro)s (of)f(of)g(Lemma)f(9,)i(but)g(note)g(that)f(Theorem)h(5)120 5463 y(inmediately)23 b(follo)m(ws)h(from)g(it)g(and)h(the)h(estimate)e (\(109\),)i(since)f(b)m(y)h(a)f(shift)g(and)g(a)g(rescaling)1781 5712 y(27)p eop %%Page: 28 28 28 27 bop 120 407 a Fv(\(whic)m(h)33 b(do)s(es)g(not)g(a\013ect)f(the)h (uniformit)m(y)e(of)h(the)h(estimates,)g(giv)m(en)f(that)h(1)27 b Fq(\024)h Fp(\025)3181 422 y Fm(i)3237 407 y Fv(=)3351 366 y Fm(Z)3399 376 y Fi(i)p 3351 384 75 4 v 3361 441 a Fm(Z)3463 407 y Fq(\024)120 527 y Fp(A)p Fv(\))33 b(w)m(e)g(get)179 792 y Fq(jM)p Fv(\()p Fp(h;)17 b(W)m(;)g Fv(\007)674 807 y Fm(i)701 792 y Fv(\))83 b Fq(\000)1066 724 y Fv(1)p 993 769 196 4 v 993 860 a(15)p Fp(\031)1150 831 y Fn(2)1199 792 y Fp(h)1255 751 y Fo(\000)p Fn(3)1366 656 y Fk(Z)1482 792 y Fv(\007)1558 807 y Fm(i)1586 792 y Fv(\()p Fp(x)p Fv(\)\()p Fp(')p Fv(\()p Fp(\025)1914 807 y Fm(i)1942 792 y Fv(\()p Fp(x)23 b Fq(\000)f Fp(Z)2231 751 y Fn(1)p Fm(=)p Fn(3)2341 792 y Fp(R)2415 807 y Fm(i)2444 792 y Fv(\)\))g(+)g Fp(w)s Fv(\()p Fp(x)p Fv(\)\))2882 751 y Fn(5)p Fm(=)p Fn(2)3008 792 y Fp(dx)g Fv(+)3244 724 y(1)p 3244 769 49 4 v 3244 860 a(8)3303 792 y Fp(h)3359 751 y Fo(\000)p Fn(2)3453 792 y Fq(j)822 998 y(\024)84 b Fp(C)7 b(h)1116 957 y Fo(\000)p Fn(2+)p Fm(\016)1299 998 y Fv(;)49 b(0)27 b Fp(<)h(h)g Fq(\024)g Fv(1)1525 b(\(121\))120 1214 y(and)31 b(putting)g(together)h(estimate)e(\(121\),) h(the)h(decomp)s(osition)d(of)i Fp(W)45 b Fv(giv)m(en)32 b(in)e(\(110\))h(and)120 1334 y(the)39 b(decomp)s(osition)d(in)h(lo)s (cal)f(traces)j(\(94\))e(w)m(e)i(get)g(the)f(global)e(trace)i(estimate) g(\(92\))f(b)m(y)120 1455 y(collecting)31 b(estimates.)p 3469 1455 4 66 v 3473 1392 59 4 v 3473 1455 V 3531 1455 4 66 v 120 1649 a Fr(Pr)-5 b(o)g(of)35 b(of)f(L)-5 b(emma)34 b(9.)49 b Fv(Giv)m(en)34 b(0)d Fq(\024)g Fp( )k Fq(2)c Fp(C)1714 1613 y Fo(1)1707 1673 y Fm(c)1789 1649 y Fv(\()p Fp(B)1901 1664 y Fn(1)1941 1649 y Fv(\(0\)\))j(suc)m(h)i(that)e Fp( )h Fv(=)30 b(1)35 b(if)e Fq(j)p Fp(x)p Fq(j)e(\024)g Fv(1)p Fp(=)p Fv(2)j(and)120 1769 y(call)f Fp( )364 1784 y Fm(r)433 1769 y Fv(:=)e Fp( )t Fv(\()p Fp(r)719 1733 y Fo(\000)p Fn(1)813 1769 y Fq(\001)p Fv(\),)k(so)g(that)f Fp( )1339 1784 y Fm(r)1412 1769 y Fv(satis\014es)h(estimates)f(of)g (the)h(kind)g Fq(k)p Fp(@)2815 1733 y Fm(\013)2865 1769 y Fp( )2928 1784 y Fm(r)2966 1769 y Fq(k)3016 1784 y Fo(1)3122 1769 y Fq(\024)c Fp(C)3300 1784 y Fm(\013)3350 1769 y Fp(r)3397 1733 y Fo(\000j)p Fm(\013)p Fo(j)120 1889 y Fv(and)i Fp(h)366 1853 y Fn(2)433 1889 y Fq(\024)28 b Fp(r)i Fq(\024)f Fv(1)j(is)g(to)g(b)s(e)h(c)m(hosen)h(later.)266 2010 y(First)e(w)m(e)i(write)e Fp( )g Fv(=)27 b Fp( )1158 2025 y Fm(r)1219 2010 y Fv(+)22 b Fp( )t Fv(\(1)f Fq(\000)i Fp( )1655 2025 y Fm(r)1693 2010 y Fv(\))33 b(and)g(split)e(the)i(lo)s (cal)d(trace)j Fq(M)p Fv(\()p Fp(h;)17 b(w)s(;)g( )t Fv(\))31 b(as)594 2226 y Fq(M)p Fv(\()p Fp(h;)17 b(w)s(;)g( )t Fv(\))26 b(=)h Fq(M)p Fv(\()p Fp(h;)17 b(w)s(;)g( )t Fv(\(1)k Fq(\000)h Fp( )1915 2241 y Fm(r)1954 2226 y Fv(\)\))g(+)g Fq(M)p Fv(\()p Fp(h;)17 b(w)s(;)g( )2588 2241 y Fm(r)2625 2226 y Fv(\))27 b(=)h Fp(I)i Fv(+)22 b Fp(I)8 b(I)259 b Fv(\(122\))120 2442 y(The)49 b(estimate)d(for)h(the) h(\014rst)g(piece)g(follo)m(ws)f(directly)g(from)f(the)i(Multiscale)e (Analysis)120 2562 y(mac)m(hinery)32 b(and)h(yields)374 2821 y Fq(j)p Fp(I)d Fq(\000)658 2754 y Fv(1)p 585 2798 196 4 v 585 2890 a(15)p Fp(\031)742 2861 y Fn(2)791 2821 y Fp(h)847 2780 y Fo(\000)p Fn(3)958 2686 y Fk(Z)1074 2821 y Fp( )t Fv(\(1)22 b Fq(\000)g Fp( )1412 2836 y Fm(r)1451 2821 y Fv(\)\()p Fp(')g Fv(+)g Fp(w)s Fv(\))1822 2780 y Fn(5)p Fm(=)p Fn(2)1931 2821 y Fq(j)27 b(\024)h Fp(C)7 b(r)2215 2780 y Fo(\000)p Fn(1)p Fm(=)p Fn(2)2380 2821 y Fp(h)2436 2780 y Fo(\000)p Fn(1)2530 2821 y Fv(;)50 b Fp(h)2663 2780 y Fn(2)2730 2821 y Fq(\024)28 b Fp(r)i Fq(\024)f Fv(1)254 b(\(123\))266 3086 y(T)-8 b(o)27 b(estimate)f(the)h (second)h(piece)f(w)m(e)h(use)g(the)f(fact)f(that)h(for)f Fq(j)p Fp(x)p Fq(j)h(\024)h Fp(r)j Fq(\034)c Fv(1)g(the)g(total)e(TF) 120 3206 y(p)s(oten)m(tial)37 b(is)h(w)m(ell)f(apro)m(ximated)h(b)m(y)h (the)g(Coulom)m(b)e(one.)61 b(Recall)37 b(that)h(from)f(Lemma)g(3)120 3327 y(w)m(e)d(can)e(write,)h(recalling)d(the)j(remark)f(after)h(the)g (pro)s(of)e(of)h(Lemma)g(3,)g(for)g Fq(j)p Fp(x)p Fq(j)c(\024)3234 3287 y Fn(1)p 3234 3304 36 4 v 3234 3361 a(2)905 3609 y Fp(')p Fv(\()p Fp(x)p Fv(\))22 b(+)g Fp(w)s Fv(\()p Fp(x)p Fv(\))84 b(=)1668 3469 y Fk(\022)1782 3542 y Fv(1)p 1751 3586 111 4 v 1751 3678 a Fq(j)p Fp(x)p Fq(j)1894 3609 y(\000)22 b Fp(k)2047 3469 y Fk(\023)2143 3609 y Fv(+)g Fp(\026)p Fv(\()p Fp(x)p Fv(\))g(+)g Fp(w)s Fv(\()p Fp(x)p Fv(\))1072 3828 y Fq(j)p Fp(@)1156 3787 y Fm(\013)1206 3828 y Fp(\026)p Fv(\()p Fp(x)p Fv(\))p Fq(j)83 b(\024)h Fp(C)1738 3843 y Fm(\013)1787 3828 y Fq(j)p Fp(x)p Fq(j)1898 3787 y Fn(1)p Fm(=)p Fn(2)p Fo(\000j)p Fm(\013)p Fo(j)1234 3973 y Fp(\026)p Fv(\()p Fp(x)p Fv(\))f Fq(\024)h Fv(0)1601 b(\(124\))1071 4119 y Fq(k)p Fp(@)1177 4078 y Fm(\013)1227 4119 y Fp(w)s Fq(k)1350 4134 y Fo(1)1507 4119 y Fq(\024)84 b Fp(C)1738 4134 y Fm(\013)1375 4264 y Fv(0)f Fq(\024)h Fp(w)30 b Fq(\024)e Fp(c)g(<)f Fq(1)p Fv(;)49 b Fp(c)32 b Fv(absolute)120 4480 y(Let)h(us)g(de\014ne)1201 4593 y Fk(\032)1317 4668 y Fp(w)1387 4683 y Fm(r)1508 4668 y Fv(:=)83 b(sup)17 b(ess)1978 4692 y Fo(j)p Fm(x)p Fo(j\024)p Fm(r)2150 4668 y Fp(w)s Fv(\()p Fp(x)p Fv(\))1339 4793 y Fp(\027)1387 4808 y Fm(r)1508 4793 y Fv(:=)83 b(sup)17 b(ess)1978 4816 y Fo(j)p Fm(x)p Fo(j\024)p Fm(r)2150 4793 y Fq(j)p Fp(\027)6 b Fv(\()p Fp(x)p Fv(\))p Fq(j)3318 4733 y Fv(\(125\))120 5006 y(F)-8 b(or)43 b(0)j Fp(<)g(r)j Fq(\024)d Fv(1,)g(w)m(e)f(ha)m(v)m(e)g(0)h Fq(\024)g Fp(w)1540 5021 y Fm(r)1578 5006 y Fp(;)17 b(\027)1670 5021 y Fm(r)1754 5006 y Fq(\024)47 b Fp(C)j Fv(for)43 b(some)h(absolute)f Fp(C)7 b Fv(.)76 b(W)-8 b(e)44 b(ha)m(v)m(e)h(the) 120 5126 y(op)s(erator)32 b(inequalities)120 5342 y Fp( )187 5301 y Fn(1)p Fm(=)p Fn(2)183 5367 y Fm(r)297 5342 y Fv(\()p Fq(\000)p Fp(h)468 5301 y Fn(2)508 5342 y Fv(\001)22 b Fq(\000)h Fp(V)44 b Fq(\000)22 b Fp(w)981 5357 y Fm(r)1019 5342 y Fv(\))p Fp( )1124 5301 y Fn(1)p Fm(=)p Fn(2)1120 5367 y Fm(r)1262 5342 y Fq(\024)28 b Fp( )1434 5301 y Fn(1)p Fm(=)p Fn(2)1430 5367 y Fm(r)1544 5342 y Fv(\()p Fq(\000)p Fp(h)1715 5301 y Fn(2)1755 5342 y Fv(\001)22 b Fq(\000)h Fp(')f Fq(\000)h Fp(w)s Fv(\))p Fp( )2322 5301 y Fn(1)p Fm(=)p Fn(2)2318 5367 y Fm(r)2459 5342 y Fq(\024)28 b Fp( )2631 5301 y Fn(1)p Fm(=)p Fn(2)2627 5367 y Fm(r)2741 5342 y Fv(\()p Fq(\000)p Fp(h)2912 5301 y Fn(2)2952 5342 y Fv(\001)22 b Fq(\000)h Fp(V)44 b Fv(+)22 b Fp(\027)3402 5357 y Fm(r)3440 5342 y Fv(\))p Fp( )3545 5301 y Fn(1)p Fm(=)p Fn(2)3541 5367 y Fm(r)3318 5463 y Fv(\(126\))1781 5712 y(28)p eop %%Page: 29 29 29 28 bop 120 407 a Fv(Since)23 b Fp(A)28 b Fq(\024)g Fp(B)g Fv(implies)21 b([)p Fp(A)p Fv(])1122 422 y Fo(\000)1209 407 y Fq(\025)28 b Fv([)p Fp(B)5 b Fv(])1447 422 y Fo(\000)1530 407 y Fv(and)23 b(the)h(trace)f(is)g(a)g(functional)e(op)s(erator)i (monotone,)120 527 y(w)m(e)34 b(get)879 727 y Fp(M)10 b Fv(\()p Fp(h;)17 b Fq(\000)p Fp(w)1268 742 y Fm(r)1307 727 y Fp(;)g( )1414 742 y Fm(r)1452 727 y Fv(\))28 b Fq(\025)g(M)p Fv(\()p Fp(h;)17 b(w)s(;)g( )2061 742 y Fm(r)2098 727 y Fv(\))28 b Fq(\025)g Fp(M)10 b Fv(\()p Fp(h;)17 b(\027)2559 742 y Fm(r)2597 727 y Fp(;)g( )2704 742 y Fm(r)2743 727 y Fv(\))537 b(\(127\))120 928 y(where)668 1147 y Fp(M)10 b Fv(\()p Fp(h;)17 b(\027)q(;)g( )t Fv(\))28 b(:=)f(T)-8 b(r)17 b([)p Fp( )t Fv(\()p Fq(\000)p Fp(h)1648 1106 y Fn(2)1688 1147 y Fv(\001)23 b Fq(\000)f Fp(V)44 b Fv(+)22 b Fp(\027)6 b Fv(\))2182 1162 y Fo(\000)2241 1147 y Fv(];)50 b Fp(V)21 b Fv(\()p Fp(x)p Fv(\))28 b(=)2727 1080 y(1)p 2696 1124 111 4 v 2696 1216 a Fq(j)p Fp(x)p Fq(j)2839 1147 y(\000)22 b Fp(k)329 b Fv(\(128\))120 1403 y(W)-8 b(e)33 b(w)m(an)m(t)g(to)g(compute)f Fp(M)10 b Fv(\()p Fp(h;)17 b(\027)q(;)g( )1439 1418 y Fm(r)1478 1403 y Fv(\))32 b(for)g Fq(j)p Fp(\027)6 b Fq(j)28 b(\024)g Fp(C)7 b Fv(.)44 b(By)33 b(the)g(Lieb-Thirring)d(inequalit)m(y)134 1604 y Fp(M)10 b Fv(\()p Fp(h;)17 b(\027)q(;)g( )532 1619 y Fm(r)571 1604 y Fv(\))83 b Fq(\024)g Fp(M)10 b Fv(\()p Fp(h;)17 b(\027)1142 1619 y Fo(\000)1202 1604 y Fp(;)g( )1309 1619 y Fm(r)1347 1604 y Fv(\))692 1749 y Fq(\024)83 b Fv(T)-8 b(r)17 b([)p Fp( )1059 1764 y Fm(r)1098 1749 y Fv(\()p Fq(\000)p Fp(h)1269 1708 y Fn(2)1309 1749 y Fv(\(1)k Fq(\000)i Fp(\017)p Fv(\)\001)g Fq(\000)f Fp(V)g Fv(\))1914 1764 y Fo(\000)1973 1749 y Fv(])g(+)g(T)-8 b(r)17 b([)p Fp( )2327 1764 y Fm(r)2366 1749 y Fv(\()p Fq(\000)p Fp(h)2537 1708 y Fn(2)2577 1749 y Fp(\017)p Fv(\001)22 b Fq(\000)h Fp(\027)2867 1764 y Fo(\000)2926 1749 y Fv(\))2964 1764 y Fo(\000)3023 1749 y Fv(];)50 b(0)27 b Fp(<)h(\017)g(<)f Fv(1)693 1894 y(=)83 b Fp(M)10 b Fv(\()p Fp(h)p Fv(\(1)23 b Fq(\000)g Fp(\017)p Fv(\))1337 1853 y Fn(1)p Fm(=)p Fn(2)1447 1894 y Fp(;)17 b Fv(0)p Fp(;)g( )1647 1909 y Fm(r)1684 1894 y Fv(\))23 b(+)f(T)-8 b(r)16 b([)p Fp( )t Fv(\()p Fq(\000)p Fp(h)2224 1853 y Fn(2)2265 1894 y Fp(\017)p Fv(\001)22 b Fq(\000)h Fp(\027)2555 1909 y Fo(\000)2614 1894 y Fp( )2677 1909 y Fm(r)2716 1894 y Fv(\))2754 1909 y Fo(\000)2813 1894 y Fv(])478 b(\(129\))692 2053 y Fq(\024)83 b Fp(M)10 b Fv(\()p Fp(h)p Fv(\(1)23 b Fq(\000)g Fp(\017)p Fv(\))1337 2012 y Fn(1)p Fm(=)p Fn(2)1447 2053 y Fp(;)17 b Fv(0)p Fp(;)g( )1647 2068 y Fm(r)1684 2053 y Fv(\))23 b(+)f Fp(K)7 b(h)1989 2012 y Fo(\000)p Fn(3)2083 2053 y Fp(\017)2122 2012 y Fo(\000)p Fn(3)p Fm(=)p Fn(2)2287 2053 y Fp(r)2334 2012 y Fn(3)2374 2053 y Fp(\027)2428 2002 y Fn(5)p Fm(=)p Fn(2)2422 2075 y Fo(\000)120 2253 y Fv(W)-8 b(e)33 b(ha)m(v)m(e,)h(b)m (y)f(Lemma)f(8)189 2529 y Fp(M)10 b Fv(\()p Fp(h;)17 b(\027)479 2544 y Fm(r)517 2529 y Fp(;)g( )624 2544 y Fm(r)662 2529 y Fv(\))28 b(=)915 2461 y(1)p 841 2506 196 4 v 841 2597 a(15)p Fp(\031)998 2568 y Fn(2)1047 2529 y Fp(h)1103 2488 y Fo(\000)p Fn(3)1214 2393 y Fk(Z)1331 2529 y Fp( )1394 2544 y Fm(r)1432 2529 y Fv(\()p Fp(x)p Fv(\))1580 2388 y Fk(\022)1694 2461 y Fv(1)p 1663 2506 111 4 v 1663 2597 a Fq(j)p Fp(x)p Fq(j)1806 2529 y(\000)23 b Fp(k)i Fq(\000)e Fp(\027)2130 2544 y Fm(r)2168 2388 y Fk(\023)2241 2410 y Fn(5)p Fm(=)p Fn(2)2241 2627 y(+)2384 2529 y Fp(dx)g Fq(\000)2622 2461 y Fv(1)p 2622 2506 49 4 v 2622 2597 a(8)2681 2529 y Fp(h)2737 2488 y Fo(\000)p Fn(2)2853 2529 y Fv(+)f Fq(O)s Fv(\()p Fp(r)3118 2488 y Fo(\000)p Fn(1)p Fm(=)p Fn(2)3283 2529 y Fp(h)3339 2488 y Fo(\000)p Fn(1)3434 2529 y Fv(\))3318 2727 y(\(130\))120 2920 y(Also)32 b(w)m(e)i(can)e(estimate)451 3176 y(0)83 b Fq(\024)743 3041 y Fk(Z)859 3176 y Fp( )922 3191 y Fm(r)961 3176 y Fv(\()p Fp(x)p Fv(\))1109 3036 y Fk(\022)1223 3109 y Fv(1)p 1192 3154 111 4 v 1192 3245 a Fq(j)p Fp(x)p Fq(j)1335 3176 y(\000)23 b Fp(k)1489 3036 y Fk(\023)1562 3058 y Fn(5)p Fm(=)p Fn(2)1562 3275 y(+)1705 3176 y Fp(dx)f Fq(\000)1933 3041 y Fk(Z)2049 3176 y Fp( )2112 3191 y Fm(r)2151 3176 y Fv(\()p Fp(x)p Fv(\))2299 3036 y Fk(\022)2413 3109 y Fv(1)p 2382 3154 V 2382 3245 a Fq(j)p Fp(x)p Fq(j)2525 3176 y(\000)g Fp(k)k Fq(\000)c Fp(\027)2848 3191 y Fm(r)2886 3036 y Fk(\023)2960 3058 y Fn(5)p Fm(=)p Fn(2)2960 3275 y(+)3103 3176 y Fp(dx)583 3520 y Fq(\024)743 3384 y Fk(Z)799 3610 y Fo(j)p Fm(x)p Fo(j\024)p Fm(r)987 3379 y Fk(\022)1101 3452 y Fv(1)p 1070 3497 V 1070 3588 a Fq(j)p Fp(x)p Fq(j)1213 3520 y(\000)h Fp(k)i Fq(\000)e Fp(\027)1537 3535 y Fm(r)1575 3379 y Fk(\023)1648 3401 y Fn(5)p Fm(=)p Fn(2)1648 3618 y(+)1775 3319 y Fk(2)1775 3499 y(4)1841 3349 y( )1920 3520 y Fv(1)f(+)2297 3452 y Fp(\027)2345 3467 y Fm(r)p 2099 3497 483 4 v 2131 3551 a Fn(1)p 2109 3568 79 4 v 2109 3625 a Fo(j)p Fm(x)p Fo(j)2220 3591 y Fq(\000)h Fp(k)i Fq(\000)d Fp(\027)2543 3606 y Fm(r)2591 3349 y Fk(!)2670 3372 y Fn(5)p Fm(=)p Fn(2)2802 3520 y Fq(\000)h Fv(1)2951 3319 y Fk(3)2951 3499 y(5)3051 3520 y Fp(dx)161 b Fv(\(131\))583 3843 y Fq(\024)83 b Fp(K)7 b(\027)881 3858 y Fm(r)936 3707 y Fk(Z)991 3933 y Fo(j)p Fm(x)p Fo(j\024)p Fm(r)1180 3843 y Fq(j)p Fp(x)p Fq(j)1291 3802 y Fo(\000)p Fn(3)p Fm(=)p Fn(2)1472 3843 y Fp(dx)583 4074 y Fq(\024)83 b Fp(C)7 b(r)867 4032 y Fn(3)p Fm(=)p Fn(2)120 4274 y Fv(F)-8 b(rom)31 b(\(130\))h(and)g(\(131\))g(w)m(e)i (get)120 4555 y Fp(M)10 b Fv(\()p Fp(h;)17 b(\027)410 4570 y Fm(r)449 4555 y Fp(;)g( )556 4570 y Fm(r)594 4555 y Fv(\))27 b(=)846 4487 y(1)p 773 4532 196 4 v 773 4623 a(15)p Fp(\031)930 4594 y Fn(2)979 4555 y Fp(h)1035 4514 y Fo(\000)p Fn(3)1146 4419 y Fk(Z)1262 4555 y Fp( )1325 4570 y Fm(r)1363 4555 y Fv(\()p Fp(x)p Fv(\))1511 4414 y Fk(\022)1626 4487 y Fv(1)p 1595 4532 111 4 v 1595 4623 a Fq(j)p Fp(x)p Fq(j)1738 4555 y(\000)22 b Fp(k)1891 4414 y Fk(\023)1965 4437 y Fn(5)p Fm(=)p Fn(2)1965 4654 y(+)2108 4555 y Fp(dx)g Fq(\000)2346 4487 y Fv(1)p 2346 4532 49 4 v 2346 4623 a(8)2404 4555 y Fp(h)2460 4514 y Fo(\000)p Fn(2)2577 4555 y Fv(+)g Fq(O)s Fv(\()p Fp(r)2842 4514 y Fo(\000)p Fn(1)p Fm(=)p Fn(2)3007 4555 y Fp(h)3063 4514 y Fo(\000)p Fn(1)3179 4555 y Fv(+)g Fp(r)3324 4514 y Fn(3)p Fm(=)p Fn(2)3434 4555 y Fp(h)3490 4514 y Fo(\000)p Fn(3)3584 4555 y Fv(\))3318 4753 y(\(132\))120 4952 y(Analogously)-8 b(,)31 b(b)m(y)j(\(129\))d(w)m(e)j(can)f(write)f(the)h(estimate)546 5233 y Fp(M)10 b Fv(\()p Fp(h;)17 b(w)858 5248 y Fm(r)897 5233 y Fp(;)g( )1004 5248 y Fm(r)1042 5233 y Fv(\))83 b(=)1405 5166 y(1)p 1332 5210 196 4 v 1332 5301 a(15)p Fp(\031)1489 5273 y Fn(2)1537 5233 y Fp(h)1593 5192 y Fo(\000)p Fn(3)1705 5097 y Fk(Z)1821 5233 y Fp( )1884 5248 y Fm(r)1922 5233 y Fv(\()p Fp(x)p Fv(\))2070 5093 y Fk(\022)2184 5166 y Fv(1)p 2153 5210 111 4 v 2153 5301 a Fq(j)p Fp(x)p Fq(j)2296 5233 y(\000)23 b Fp(k)2450 5093 y Fk(\023)2523 5115 y Fn(5)p Fm(=)p Fn(2)2523 5332 y(+)2666 5233 y Fp(dx)g Fq(\000)2904 5166 y Fv(1)p 2904 5210 49 4 v 2904 5301 a(8)2963 5233 y Fp(h)3019 5192 y Fo(\000)p Fn(2)1322 5463 y Fv(+)p Fq(O)s Fv(\()p Fp(r)1565 5422 y Fo(\000)p Fn(1)p Fm(=)p Fn(2)1729 5463 y Fp(h)1785 5422 y Fo(\000)p Fn(1)1902 5463 y Fv(+)f Fp(\017)2039 5422 y Fo(\000)p Fn(3)p Fm(=)p Fn(2)2204 5463 y Fp(r)2251 5422 y Fn(3)2290 5463 y Fp(h)2346 5422 y Fo(\000)p Fn(3)2463 5463 y Fv(+)g Fp(\017h)2656 5422 y Fo(\000)p Fn(3)2751 5463 y Fv(\))529 b(\(133\))1781 5712 y(29)p eop %%Page: 30 30 30 29 bop 120 407 a Fv(w)m(e)38 b(can)f(estimate)f(the)h(di\013erence)h (of)e(w)m(eylian)h(terms)g(of)f Fp(M)10 b Fv(\()p Fp(h;)17 b Fv(0)p Fp(;)g( )2720 422 y Fm(r)2758 407 y Fv(\))37 b(and)g Fq(M)p Fv(\()p Fp(h;)17 b(w)s(;)g( )3465 422 y Fm(r)3502 407 y Fv(\))120 527 y(as)33 b(in)f(\(131\):)577 790 y Fq(j)622 654 y Fk(Z)738 790 y Fp( )801 805 y Fm(r)839 790 y Fv(\()p Fp(x)p Fv(\)\()p Fp(')22 b Fv(+)g Fp(w)s Fv(\))1303 749 y Fn(5)p Fm(=)p Fn(2)1429 790 y Fp(dx)84 b Fq(\000)1779 654 y Fk(Z)1895 790 y Fp( )1958 805 y Fm(r)1997 790 y Fv(\()p Fp(x)p Fv(\))2145 650 y Fk(\022)2259 723 y Fv(1)p 2228 767 111 4 v 2228 858 a Fq(j)p Fp(x)p Fq(j)2371 790 y(\000)22 b Fp(k)k Fq(\000)c Fp(\027)2694 805 y Fm(r)2732 650 y Fk(\023)2806 672 y Fn(5)p Fm(=)p Fn(2)2806 889 y(+)2949 790 y Fp(dx)p Fq(j)1619 1020 y(\024)83 b Fp(K)7 b(r)1916 979 y Fn(3)p Fm(=)p Fn(2)3318 1020 y Fv(\(134\))120 1202 y(Putting)50 b(together)i(the)f(decomp)s(osition) e(\(122\))h(and)h(estimates)g(\(123\);)59 b(\(127\),)c(\(132\),)120 1322 y(\(133\),)32 b(\(134\))g(w)m(e)h(get)523 1553 y Fq(M)p Fv(\()p Fp(h;)17 b(w)s(;)g( )t Fv(\))81 b(=)1327 1485 y(1)p 1253 1530 196 4 v 1253 1621 a(15)p Fp(\031)1410 1592 y Fn(2)1459 1553 y Fp(h)1515 1512 y Fo(\000)p Fn(3)1626 1417 y Fk(Z)1742 1553 y Fp( )t Fv(\()p Fp(x)p Fv(\)\()p Fp(')23 b Fv(+)f Fp(w)s Fv(\))2274 1512 y Fn(5)p Fm(=)p Fn(2)2399 1553 y Fp(dx)h Fq(\000)2637 1485 y Fv(1)p 2637 1530 49 4 v 2637 1621 a(8)2696 1553 y Fp(h)2752 1512 y Fo(\000)p Fn(2)1243 1763 y Fv(+)p Fq(O)s Fv(\()p Fp(r)1486 1722 y Fo(\000)p Fn(1)p Fm(=)p Fn(2)1651 1763 y Fp(h)1707 1722 y Fo(\000)p Fn(1)1823 1763 y Fv(+)f Fp(\017)1960 1722 y Fo(\000)p Fn(3)p Fm(=)p Fn(2)2125 1763 y Fp(r)2172 1722 y Fn(3)2212 1763 y Fp(h)2268 1722 y Fo(\000)p Fn(3)2384 1763 y Fv(+)g Fp(\017h)2577 1722 y Fo(\000)p Fn(3)2694 1763 y Fv(+)g Fp(r)2839 1722 y Fn(3)p Fm(=)p Fn(2)2949 1763 y Fp(h)3005 1722 y Fo(\000)p Fn(3)3099 1763 y Fv(\))181 b(\(135\))120 1945 y(T)-8 b(aking)37 b Fp(\017)f Fv(=)g Fp(h)693 1909 y Fm(\013)743 1945 y Fv(,)j Fp(r)f Fv(=)e Fp(h)1059 1909 y Fm(\014)1144 1945 y Fv(and)h Fp(\017)1377 1909 y Fo(\000)p Fn(3)p Fm(=)p Fn(2)1543 1945 y Fp(r)1590 1909 y Fn(3)1629 1945 y Fp(h)1685 1909 y Fo(\000)p Fn(3)1815 1945 y Fv(=)f Fp(r)1974 1909 y Fo(\000)p Fn(1)p Fm(=)p Fn(2)2139 1945 y Fp(h)2195 1909 y Fo(\000)p Fn(1)2325 1945 y Fv(=)g Fp(\017h)2532 1909 y Fo(\000)p Fn(3)2664 1945 y Fv(w)m(e)j(get)e Fp(\017)g Fv(=)f Fp(h)3224 1909 y Fn(24)p Fm(=)p Fn(17)3440 1945 y Fq(\034)120 2065 y Fp(r)30 b Fv(=)e Fp(h)354 2029 y Fn(20)p Fm(=)p Fn(17)567 2065 y Fv(and)33 b(the)g(\014nal)f(estimate)356 2271 y Fq(M)p Fv(\()p Fp(h;)17 b(w)s(;)g( )t Fv(\))81 b(=)1160 2203 y(1)p 1086 2248 196 4 v 1086 2339 a(15)p Fp(\031)1243 2310 y Fn(2)1292 2271 y Fp(h)1348 2230 y Fo(\000)p Fn(3)1459 2135 y Fk(Z)1575 2271 y Fp( )t Fv(\()p Fp(x)p Fv(\)\()p Fp(')23 b Fv(+)f Fp(w)s Fv(\))2107 2230 y Fn(5)p Fm(=)p Fn(2)2233 2271 y Fp(dx)g Fq(\000)2470 2203 y Fv(1)p 2470 2248 49 4 v 2470 2339 a(8)2529 2271 y Fp(h)2585 2230 y Fo(\000)p Fn(2)2702 2271 y Fv(+)g Fq(O)s Fv(\()p Fp(h)2976 2230 y Fo(\000)p Fn(1)p Fo(\000)p Fn(9)p Fm(=)p Fn(17)3266 2271 y Fv(\))14 b(\(136\))120 2496 y(whic)m(h)33 b(concludes)h(the)f (pro)s(of.)p 3469 2496 4 66 v 3473 2433 59 4 v 3473 2496 V 3531 2496 4 66 v 266 2680 a(As)f(a)e(Corollary)f(of)h(Theorem)h(5)g (w)m(e)h(can)f(write)f(the)h(next)h(lo)m(w)m(er)f(b)s(ound)g(for)f(the) h(RHF)120 2800 y(energy:)120 2971 y Ft(Corollary)36 b(1)h(\(RHF)g (energy)h(of)f(a)h(go)s(o)s(d)g(con\014guration:)50 b(lo)m(w)m(er)36 b(b)s(ound\).)50 b Fr(Assume)120 3092 y(that)35 b Fq(R)p Fr(,)g Fq(Z)43 b Fr(is)34 b(a)g(c)-5 b(on\014gur)g(ation)34 b(of)h(nucle)-5 b(ar)34 b(char)-5 b(ges)33 b(ful\014l)5 b(ling)34 b(the)h(hyp)-5 b(othesis)34 b(of)g(Th.)g(2.)120 3212 y(Then)g(the)h(fol)5 b(lowing)33 b(lower)i(b)-5 b(ound)34 b(for)h(the)g(RHF)g(ener)-5 b(gy)34 b(holds:)537 3394 y Fp(E)615 3353 y Fn(RHF)609 3419 y Fo(R)p Fm(;)p Fo(Z)853 3394 y Fq(\025)84 b(\000)p Fv(T)-8 b(r)17 b([)p Fq(\000)p Fv(\001)23 b Fq(\000)g Fv(\010)1586 3353 y Fn(TF)1586 3419 y Fo(R)p Fm(;)p Fo(Z)1726 3394 y Fv(])1753 3409 y Fo(\000)1834 3394 y Fq(\000)g Fp(D)s Fv(\()p Fp(\032)2106 3353 y Fn(TF)2106 3419 y Fo(R)p Fm(;)p Fo(Z)2246 3394 y Fv(\))f(+)g Fp(U)2470 3409 y Fo(R)p Fm(;)p Fo(Z)3318 3394 y Fv(\(137\))853 3600 y Fq(\025)84 b(\000)1175 3533 y Fv(1)p 1101 3577 196 4 v 1101 3669 a(15)p Fp(\031)1258 3640 y Fn(2)1323 3465 y Fk(Z)1423 3600 y Fv(\(\010)1531 3559 y Fn(TF)1531 3625 y Fo(R)p Fm(;)p Fo(Z)1671 3600 y Fv(\))1709 3559 y Fn(5)p Fm(=)p Fn(2)1842 3600 y Fq(\000)22 b Fp(D)s Fv(\()p Fp(\032)2113 3559 y Fn(TF)2113 3625 y Fo(R)p Fm(;)p Fo(Z)2253 3600 y Fv(\))g(+)g Fp(U)2477 3615 y Fo(R)p Fm(;)p Fo(Z)2640 3600 y Fq(\000)g Fp(C)7 b(M)j(Z)2994 3559 y Fn(2)p Fo(\000)p Fm(\016)120 3838 y Fr(\()p Fp(\016)31 b Fv(=)366 3799 y Fn(9)p 348 3815 71 4 v 348 3873 a(51)463 3838 y Fr(c)-5 b(ould)35 b(do\).)120 4009 y(Pr)-5 b(o)g(of.)48 b Fv(Pro)s(of)35 b(of)h(Corollary)e(1)h (follo)m(ws)f(inmediately)g(from)g(Theorem)i(2)g(b)m(y)g(taking)f(in)g (it)120 4129 y Fq(R)204 4144 y Fn(1)274 4129 y Fv(=)29 b Fq(;)p Fv(,)35 b Fp(h)30 b Fv(=)f Fp(Z)756 4093 y Fo(\000)p Fn(1)p Fm(=)p Fn(3)921 4129 y Fv(,)34 b Fp(W)44 b Fv(=)30 b(\010)1294 4093 y Fn(TF)1294 4166 y Fm(Z)1347 4147 y Fj(1)p Fi(=)p Fj(3)1443 4166 y Fo(R)p Fm(;Z)1576 4147 y Fh(\000)p Fj(1)1658 4166 y Fo(Z)1752 4129 y Fv(and)k(applying)f(the)h (scaling)f(prop)s(erties)g(of)h(TF)120 4262 y(p)s(oten)m(tial)d(and)i (densit)m(y)g(as)g(giv)m(en)g(b)m(y)g(eq.)44 b(\(12\))32 b(to)h(the)g(\014nal)e(result.)p 3469 4262 4 66 v 3473 4200 59 4 v 3473 4262 V 3531 4262 4 66 v 266 4446 a(I)g(am)f(going)f (to)h(conclude)h(this)f(section)h(with)f(an)h(upp)s(er)g(b)s(ound)g (for)f(the)h(RHF)f(energy)120 4567 y(of)48 b(uniformly)f(separated)j(n) m(uclei)e(whic)m(h)h(tak)m(en)h(together)f(with)f(Theorem)h(5)g(yields) f(a)120 4687 y(complete)32 b(t)m(w)m(o-term)g(asymptotics)g(for)h(the)g (RHF)f(energy)h(in)f(suc)m(h)i(a)e(case.)120 4858 y Ft(Theorem)37 b(6)g(\(RHF)g(energy)h(of)g(a)f(go)s(o)s(d)h(con\014guration:)50 b(upp)s(er)39 b(b)s(ound\).)49 b Fr(Assume)120 4978 y(that)35 b Fq(R)p Fr(,)g Fq(Z)43 b Fr(is)34 b(a)g(c)-5 b(on\014gur)g(ation)34 b(of)h(nucle)-5 b(ar)34 b(char)-5 b(ges)33 b(ful\014l)5 b(ling)34 b(the)h(hyp)-5 b(othesis)34 b(of)g(Th.)g(2.)120 5099 y(Then)g(the)h(fol)5 b(lowing)33 b(upp)-5 b(er)35 b(and)f(lower)h(b)-5 b(ounds)34 b(for)h(the)g(RHF)g(ener)-5 b(gy)34 b(holds:)380 5281 y Fv(0)27 b Fq(\024)i Fp(E)640 5240 y Fn(RHF)634 5305 y Fo(R)p Fm(;)p Fo(Z)817 5281 y Fv(+)22 b(T)-8 b(r)17 b([)p Fq(\000)p Fv(\001)23 b Fq(\000)f Fv(\010)1409 5240 y Fn(TF)1409 5305 y Fo(R)p Fm(;)p Fo(Z)1550 5281 y Fv(])1577 5296 y Fo(\000)1658 5281 y Fv(+)g Fp(D)s Fv(\()p Fp(\032)1928 5240 y Fn(TF)1928 5305 y Fo(R)p Fm(;)p Fo(Z)2068 5281 y Fv(\))g Fq(\000)h Fp(U)2294 5296 y Fo(R)p Fm(;)p Fo(Z)2462 5281 y Fq(\024)28 b Fp(C)7 b(M)j(Z)2822 5240 y Fn(2)p Fo(\000)p Fn(5)p Fm(=)p Fn(27)3318 5281 y Fv(\(138\))120 5463 y Fr(wher)-5 b(e)34 b Fp(C)42 b Fr(is)35 b(a)f(\014nite)h(p)-5 b(ositive)34 b(c)-5 b(onstant)35 b(only)f(dep)-5 b(ending)34 b(on)g Fp(\026)h Fr(and)f Fp(A)p Fr(.)1781 5712 y Fv(30)p eop %%Page: 31 31 31 30 bop 120 407 a Fr(Pr)-5 b(o)g(of.)48 b Fv(The)28 b(lo)m(w)m(er)f(b)s(ound)g(\(137\))f(is)h(trivial)d(and)j(holds)f(for)h (an)m(y)g(con\014guration.)41 b(T)-8 b(o)27 b(pro)m(v)m(e)120 527 y(the)33 b(upp)s(er)g(b)s(ound.)44 b(w)m(e)33 b(write)g(do)m(wn)g (the)g(iden)m(tit)m(y)-8 b(,)33 b(for)f(an)m(y)h(densit)m(y)g(matrix)e Fp(\015)310 718 y Fq(E)372 676 y Fn(RHF)363 742 y Fo(R)p Fm(;)p Fo(Z)527 718 y Fv(\()p Fp(\015)5 b Fv(\))27 b(=)h(T)-8 b(r)17 b([\()p Fq(\000)p Fv(\001)23 b Fq(\000)f Fv(\010)1322 676 y Fn(TF)1322 742 y Fo(R)p Fm(;)p Fo(Z)1463 718 y Fv(\))p Fp(\015)5 b Fv(])1584 733 y Fo(\000)1665 718 y Fq(\000)23 b Fp(D)s Fv(\()p Fp(\032)1937 676 y Fn(TF)1937 742 y Fo(R)p Fm(;)p Fo(Z)2077 718 y Fv(\))f(+)g Fp(U)2301 733 y Fo(R)p Fm(;)p Fo(Z)2463 718 y Fv(+)g Fp(D)s Fv(\()p Fp(\032)2733 733 y Fm(\015)2800 718 y Fq(\000)g Fp(\032)2949 676 y Fn(TF)2949 742 y Fo(R)p Fm(;)p Fo(Z)3090 718 y Fv(\))190 b(\(139\))120 908 y(Our)32 b(aim)e(w)m(ould)i(b)s(e)g(to)g (sho)m(w)h(it)e(is)g(p)s(ossible)h(to)f(c)m(ho)s(ose)i Fp(\015)k Fv(prop)s(erly)32 b(in)f(suc)m(h)i(a)f(w)m(a)m(y)h(that)1235 1098 y Fp(D)s Fv(\()p Fp(\032)1407 1113 y Fm(\015)1473 1098 y Fq(\000)23 b Fp(\032)1623 1057 y Fn(TF)1623 1123 y Fo(R)p Fm(;)p Fo(Z)1763 1098 y Fv(\))28 b Fq(\024)g Fp(C)7 b(M)j(Z)2189 1057 y Fn(2)p Fo(\000)p Fn(5)p Fm(=)p Fn(27)3318 1098 y Fv(\(140\))120 1289 y(Since)42 b Fq(E)446 1252 y Fn(RHF)437 1314 y Fo(R)p Fm(;)p Fo(Z)601 1289 y Fv(\()p Fp(\015)5 b Fv(\))44 b Fq(\025)h Fp(E)977 1252 y Fn(RHF)971 1314 y Fo(R)p Fm(;)p Fo(Z)1132 1289 y Fv(,)f(assuming)e (\(140\))f(is)h(righ)m(t)f(the)i(estimate)e(\(138\))h(will)e(follo)m(w) 120 1409 y(from)45 b(Lemmas)h(3,)k(11,)g(13)c(and)h(14)f(b)s(elo)m(w,)k (together)d(with)f(the)h(pletora)f(of)g(auxiliary)120 1529 y(results)33 b(hanging)f(around)g(them)g(whic)m(h)h(are)g(presen)m (ted)i(in)m(to)d(the)h(next)g(sections.)266 1650 y(The)d(ma)5 b(jor)27 b(di\016cult)m(y)h(in)g(pro)m(ving)g(\(138\))g(is)g(to)g(con)m (trol)g(the)h(Coulom)m(b)e(norm)h(squared)120 1770 y(of)39 b Fp(\032)288 1785 y Fm(\015)360 1770 y Fq(\000)28 b Fp(\032)515 1734 y Fn(TF)515 1795 y Fo(R)p Fm(;)p Fo(Z)655 1770 y Fv(,)42 b(since)e(it)f(is)g(not)h(in)f(principle)f(linear)h(in)g (the)h(n)m(um)m(b)s(er)g(of)f(n)m(uclei)h Fp(M)10 b Fv(.)66 b(T)-8 b(o)120 1891 y(o)m(v)m(ercome)27 b(this)e(problem)g(w)m(e)i(m)m (ust)e(use)i(cancellation)d(prop)s(erties)i(of)f Fp(\032)2764 1906 y Fm(\015)2817 1891 y Fq(\000)8 b Fp(\032)2952 1854 y Fn(TF)2952 1916 y Fo(R)p Fm(;)p Fo(Z)3093 1891 y Fv(,)27 b(since)g(just)120 2011 y(b)m(y)33 b(using)f(size)h(estimates)f(of)g (it)f(w)m(ould)h(not)h(b)s(e)f(su\016cien)m(t)i(to)e(obtain)f(our)h (result)h(\(b)m(y-the-)120 2131 y(w)m(a)m(y)-8 b(,)36 b(this)f(is)f(the)h(main)e(reason)i(I)f(am)g(not)g(just)h(follo)m(wing) d(here)k(the)f(pro)s(cedure)g(of)f([16]\).)120 2252 y(T)-8 b(o)33 b(pro)s(duce)h(a)f(trial)e(densit)m(y)j(matrix)e Fp(\015)38 b Fv(whic)m(h)c(appro)m(ximates)f(w)m(ell)g(enough)g(the)h (energy)120 2372 y(and)h(b)s(esides)g(mak)m(es)h(the)f(Coulom)m(b)f (norm)f(of)i(the)g(error)f(term)h(small)d(w)m(e)k(shall)d(tak)m(e)j (one)120 2492 y(made)h(up)g(of)g(sev)m(eral)h(pieces.)59 b(This)37 b(will)e(b)s(e)j(done)f(in)g(sev)m(eral)h(steps:)54 b(\(here)38 b(w)m(e)g(assume)120 2613 y(again)30 b(w)m(e)j(ha)m(v)m(e)h (made)d(a)h(rescaling)f(of)g(our)h(con\014guration)f(to)h(one)g(suc)m (h)h(that)f Fp(\016)3131 2628 y Fm(i)3187 2613 y Fq(\025)c Fp(\026)k Fv(and)120 2733 y(1)27 b Fq(\024)i Fp(Z)369 2748 y Fm(i)424 2733 y Fq(\024)f Fp(A)p Fv(,)33 b(all)e Fp(i)p Fv(.\))120 2972 y Fz(i.)51 b(Construction)36 b(of)i(a)h (partition)d(of)i(unit)m(y:)120 3163 y Fv(Pic)m(k)c(0)28 b Fq(\024)h Fp(\021)j Fq(2)e Fp(C)775 3127 y Fo(1)768 3188 y Fm(c)849 3163 y Fv(\()p 887 3083 80 4 v Fp(B)966 3178 y Fn(1)1006 3163 y Fv(\(0\)\))i(with)1424 3083 y Fk(R)1507 3163 y Fp(\021)g Fv(=)d(1.)45 b(No)m(w)34 b(mak)m(e)f(out)g (of)f(it)g(an)i(appro)m(ximation)c(of)120 3283 y(iden)m(tit)m(y)k(\()p Fp(\021)568 3298 y Fm(r)606 3283 y Fv(\))644 3298 y Fm(r)r(>)p Fn(0)806 3283 y Fv(b)m(y)h(the)g(recip)s(e)f Fp(\021)1447 3298 y Fm(r)1516 3283 y Fv(:=)c Fp(r)1696 3247 y Fo(\000)p Fn(3)1790 3283 y Fp(\021)t Fv(\()p Fp(r)1927 3247 y Fo(\000)p Fn(1)2021 3283 y Fq(\001)p Fv(\))k(\()p Fp(r)i Fv(shall)d(b)s(e)h(some) h(small)c(parameter)120 3404 y(w)m(e)j(shall)d(c)m(ho)s(ose)j(later,)d (dep)s(ending)i(on)g Fp(h)p Fv(\).)44 b(T)-8 b(ak)m(e)34 b Fp(\020)2105 3419 y Fm(j)2169 3404 y Fv(:=)27 b Fp(\021)2347 3419 y Fm(r)2408 3404 y Fq(\003)22 b Fp(\037)2541 3419 y Fm(D)2599 3429 y Fi(j)2668 3404 y Fv(so)33 b(that)2999 3329 y Fk(P)3105 3355 y Fm(M)3105 3433 y(j)t Fn(=1)3248 3404 y Fp(\021)3296 3419 y Fm(j)3360 3404 y Fv(=)28 b(1.)120 3524 y(Moreo)m(v)m(er)34 b(the)f(functions)g Fp(\021)1185 3539 y Fm(j)1254 3524 y Fv(ha)m(v)m(e)h(the)f(prop)s(erties:)880 3635 y Fk(\032)996 3714 y Fp(\020)1039 3729 y Fm(j)1075 3714 y Fv(\()p Fp(x)p Fv(\))28 b(=)g(1)k(if)f Fp(x)e Fq(2)f Fp(D)1767 3729 y Fm(j)1836 3714 y Fv(and)k(dist\()p Fp(x;)17 b(@)5 b(D)2457 3729 y Fm(j)2495 3714 y Fv(\))27 b Fq(\025)h Fp(r)996 3835 y(\020)1039 3850 y Fm(j)1075 3835 y Fv(\()p Fp(x)p Fv(\))g(=)g(0)k(if)f Fp(x)40 b(=)-60 b Fq(2)28 b Fp(D)1767 3850 y Fm(j)1836 3835 y Fv(and)k(dist\()p Fp(x;)17 b(@)5 b(D)2457 3850 y Fm(j)2495 3835 y Fv(\))27 b Fq(\025)h Fp(r)3318 3775 y Fv(\(141\))266 4024 y(Once)e(w)m(e)h(ha)m (v)m(e)f(constructed)h(the)f(partition)d Fq(f)p Fp(\021)2039 4039 y Fm(j)2076 4024 y Fq(g)2126 4039 y Fn(1)p Fo(\024)p Fm(j)t Fo(\024)p Fm(M)2407 4024 y Fv(w)m(e)j(shall)e(re\014ne)i(it)f (as)g(follo)m(ws:)120 4144 y(\014x)33 b(some)g(radial)d(function)i(0)c Fq(\024)g Fv(\007)f Fq(2)h Fp(C)1617 4108 y Fo(1)1610 4169 y Fm(c)1692 4144 y Fv(\()p 1730 4064 V Fp(B)1809 4159 y Fn(1)1848 4144 y Fv(\(0\)\))33 b(with)f(the)h(prop)s(erties)1321 4252 y Fk(\032)1437 4332 y Fv(\007)28 b(=)f(1)115 b(if)32 b Fq(j)p Fp(x)p Fq(j)27 b(\024)h Fv(1)p Fp(=)p Fv(2)1437 4452 y(\007)g(=)f(0)115 b(if)32 b Fq(j)p Fp(x)p Fq(j)27 b(\025)h Fv(3)p Fp(=)p Fv(4)120 4636 y(More)k(imp)s(ortan)m(tly)-8 b(,)29 b(w)m(e)j(shall)e(imp)s(ose)g(the)i(follo)m(wing)c(sort)k(of)f (orthogonalit)m(y)e(conditions)120 4756 y(on)j(\007:)760 4834 y Fk(Z)877 4970 y Fv(\001\007)p Fq(j)22 b(\001)f(j)1161 4929 y Fo(\000)p Fn(5)p Fm(=)p Fn(2)1354 4970 y Fv(=)1457 4834 y Fk(Z)1573 4970 y Fv(\001\007)p Fq(j)h(\001)g(j)1858 4929 y Fo(\000)p Fn(3)p Fm(=)p Fn(2)2050 4970 y Fv(=)2154 4834 y Fk(Z)2270 4970 y Fv(\001\007)p Fq(j)g(\001)g(j)2555 4929 y Fo(\000)p Fn(1)p Fm(=)p Fn(2)2747 4970 y Fv(=)28 b(0)418 b(\(142\))120 5222 y(Consider)35 b(then)p 754 5142 76 4 v 36 w(\007)830 5237 y Fm(j)898 5222 y Fv(:=)d(\007\()p Fp(r)s Fv(\()p Fq(\001)23 b(\000)h Fp(R)1458 5237 y Fm(j)1495 5222 y Fv(\)\);)51 b Fp(j)38 b Fv(=)32 b(1)17 b Fp(:)g(:)g(:)32 b(;)17 b(M)45 b Fv(\()p Fp(r)38 b Fv(is)c(the)i(same)f(as)g(ab)s(o)m(v) m(e\).)51 b(No)m(w)120 5342 y(tak)m(e,)38 b(for)d Fp(M)h Fv(+)24 b(1)34 b Fq(\024)g Fp(j)39 b Fq(\024)c Fv(2)p Fp(M)10 b Fv(,)37 b(\007)1423 5357 y Fm(j)1493 5342 y Fv(=)d Fp(\020)1646 5357 y Fm(j)1706 5342 y Fq(\000)p 1809 5262 V 26 w Fv(\007)1884 5357 y Fm(j)t Fo(\000)p Fm(M)2051 5342 y Fv(.)54 b(Next,)38 b(follo)m(wing)33 b(an)j(standard)h(pro-)120 5463 y(cedure,)f(w)m(e)f(mak)m(e)f(out)f (the)i(partition)c(of)j(unit)m(y)g Fq(f)p 2029 5383 V Fv(\007)2104 5478 y Fm(j)2141 5463 y Fq(g)2191 5478 y Fn(1)p Fo(\024)p Fm(j)t Fo(\024)p Fn(2)p Fm(M)2516 5463 y Fv(another)g(one,)h(denoted)g(as)1781 5712 y(31)p eop %%Page: 32 32 32 31 bop 120 407 a Fq(f)p Fv(\007)246 422 y Fm(j)282 407 y Fq(g)332 422 y Fn(1)p Fo(\024)p Fm(j)t Fo(\024)p Fn(2)p Fm(M)659 407 y Fv(that)34 b Fr(squar)-5 b(es)p Fv(,)36 b(suc)m(h)g(that)f(supp)17 b(\007)1972 422 y Fm(j)2040 407 y Fv(=)32 b(supp)p 2366 327 76 4 v 18 w(\007)2442 422 y Fm(j)2513 407 y Fv(and)j Fq(kr)p Fv(\007)2914 422 y Fm(j)2950 407 y Fq(k)d(\024)g Fp(C)7 b Fq(kr)p 3351 327 V Fv(\007)3427 422 y Fm(j)3463 407 y Fq(k)p Fv(.)120 554 y(In)25 b(addition,)f(for)g(su\016cien)m(tly)h(small)d Fp(r)30 b(>)d Fv(0,)f(\007)1862 569 y Fm(j)1926 554 y Fv(=)p 2030 474 V 28 w(\007)2106 493 y Fn(1)p Fm(=)p Fn(2)2106 579 y Fm(j)2216 554 y Fv(;)33 b Fp(j)h Fv(=)27 b(1)p Fp(;)17 b(:)g(:)g(:)32 b(;)17 b(M)10 b Fv(,)27 b(so)d(the)h(partition)120 675 y(of)32 b(unit)m(y)h Fq(f)p Fv(\007)612 690 y Fm(j)648 675 y Fq(g)698 690 y Fn(1)p Fo(\024)p Fm(j)t Fo(\024)p Fn(2)p Fm(M)1022 675 y Fv(is)f(suc)m(h)i (that)701 785 y Fk(Z)817 921 y Fv(\001\007)974 880 y Fn(2)974 945 y Fm(j)1014 921 y Fq(j)22 b(\001)g(j)1142 880 y Fo(\000)p Fn(5)p Fm(=)p Fn(2)1334 921 y Fv(=)1437 785 y Fk(Z)1554 921 y Fv(\001\007)1711 880 y Fn(2)1711 945 y Fm(j)1750 921 y Fq(j)g(\001)g(j)1878 880 y Fo(\000)p Fn(3)p Fm(=)p Fn(2)2070 921 y Fv(=)2174 785 y Fk(Z)2290 921 y Fv(\001\007)2447 880 y Fn(2)2447 945 y Fm(j)2487 921 y Fq(j)f(\001)h(j)2614 880 y Fo(\000)p Fn(1)p Fm(=)p Fn(2)2806 921 y Fv(=)28 b(0)359 b(\(143\))266 1175 y(Moreorer,)38 b(with)e Fp(\030)987 1190 y Fm(j)1057 1175 y Fv(:=)e(\(\007)1308 1139 y Fn(2)1308 1200 y Fm(j)1371 1175 y Fv(+)25 b(\007)1548 1139 y Fn(2)1548 1200 y Fm(j)t Fn(+)p Fm(M)1714 1175 y Fv(\))1752 1139 y Fn(1)p Fm(=)p Fn(2)1862 1175 y Fv(;)33 b Fp(j)40 b Fv(=)34 b(1)p Fp(;)17 b(:)g(:)g(:)32 b(;)17 b(M)10 b Fv(,)38 b Fq(f)p Fp(\030)2663 1139 y Fn(2)2658 1200 y Fm(j)2701 1175 y Fq(g)2751 1139 y Fm(M)2751 1200 y(j)t Fn(=1)2914 1175 y Fv(is)e(also)f(another)120 1295 y(partition)30 b(of)j(unit)m(y)-8 b(.)43 b(This)33 b(latter)e(one)i(is) f(suc)m(h)i(that)888 1402 y Fk(\032)1004 1482 y Fp(\030)1047 1497 y Fm(j)1084 1482 y Fv(\()p Fp(x)p Fv(\))28 b(=)f(1)32 b(if)g Fp(x)c Fq(2)g Fp(D)1775 1497 y Fm(j)1844 1482 y Fv(and)33 b(dist)o(\()p Fp(x;)17 b(@)5 b(D)2465 1497 y Fm(j)2503 1482 y Fv(\))28 b Fq(\025)g Fp(r)1004 1602 y(\030)1047 1617 y Fm(j)1084 1602 y Fv(\()p Fp(x)p Fv(\))g(=)f(0)32 b(if)g Fp(x)39 b(=)-60 b Fq(2)28 b Fp(D)1775 1617 y Fm(j)1844 1602 y Fv(and)33 b(dist)o(\()p Fp(x;)17 b(@)5 b(D)2465 1617 y Fm(j)2503 1602 y Fv(\))28 b Fq(\025)g Fp(r)120 1795 y Fv(Only)37 b(some)g(w)m(ords)i(ab)s(out)d(the)i(conditions)f (\(142\).)57 b(W)-8 b(e)37 b(shall)f(only)h(con)m(vince)i(ourselv)m(es) 120 1916 y(that)31 b(those)h(orthogonalit)m(y)d(conditions)h(lea)m(v)m (es)j(us)e(a)g(lot)f(of)h(freedom.)43 b(I)31 b(am)f(not)i(going)d(to) 120 2036 y(presen)m(t)35 b(here)g(an)f(explicit)e(example)h(of)g(suc)m (h)j(a)d(function,)h(but)g(lea)m(v)m(e)g(instead)g(this)f(as)h(an)120 2157 y(instructiv)m(e)f(exercise)h(for)e(the)h(in)m(terested)g(reader.) 120 2395 y Fz(ii.)51 b(Coheren)m(t)37 b(states:)120 2586 y Fv(W)-8 b(e)37 b(shall)d(construct)k(our)e(trial)d(densit)m(y)38 b(matrix)c(b)m(y)j(patc)m(hing)f(together)h(the)f(pro)5 b(jection)120 2707 y(in)m(to)28 b(the)g(negativ)m(e)h(part)f(of)g(the)h (sp)s(ectrum)g(of)f Fq(\000)p Fp(h)1998 2670 y Fn(2)2038 2707 y Fv(\001)14 b Fq(\000)g Fv(\010)2294 2670 y Fn(TF)2294 2732 y Fo(R)p Fm(;)p Fo(Z)2463 2707 y Fv(with)28 b(another)g(pieces)h (made)120 2827 y(up)k(of)f(coheren)m(t)i(states,)f(These)i(coheren)m(t) f(states)f(are)g(giv)m(en)g(as)f(follo)m(ws:)266 2947 y(Pic)m(k)49 b(some)f(function)f Fp(g)57 b Fq(2)e Fp(H)1471 2911 y Fn(1)1510 2947 y Fv(\()p Fl(R)1614 2911 y Fm(d)1660 2947 y Fv(\))48 b(whic)m(h)h(is)e(spherically)g(symmetric)g(and)h(with) 120 2988 y Fk(R)167 3102 y Fg(R)215 3083 y Fi(d)272 3068 y Fq(j)p Fp(g)t Fq(j)379 3032 y Fn(2)463 3068 y Fv(=)e(1.)76 b(In)m(tro)s(duce)44 b(a)f(parameter)g Fp(r)49 b(>)d Fv(0)d(and)g(the)h(family)d(of)i(functions)g Fp(g)3353 3083 y Fm(r)3437 3068 y Fv(:=)120 3197 y Fp(r)167 3160 y Fo(\000)p Fm(d=)p Fn(2)333 3197 y Fp(g)t Fv(\()p Fp(r)469 3160 y Fo(\000)p Fn(1)562 3197 y Fq(\001)p Fv(\).)g(The)33 b(coheren)m(t)h(states)g(w)m(e)f(shall)f(use)h(are)g(then)g(giv)m(en)g (b)m(y)1102 3391 y Fp(f)1150 3406 y Fm(p;s)p Fn(;)p Fm(r)1296 3391 y Fv(\()p Fp(x)p Fv(\))28 b(=)f Fp(g)1605 3406 y Fm(r)1643 3391 y Fv(\()p Fp(x)c Fq(\000)f Fp(s)p Fv(\)e)1985 3350 y Fm(ip)p Fo(\001)p Fm(x)2108 3391 y Fv(;)50 b Fp(p;)17 b(s)27 b Fq(2)h Fl(R)2511 3350 y Fm(d)3318 3391 y Fv(\(144\))120 3586 y(and)33 b(let)f(us)h(in)m(tro)s(duce)f(the)h(pro)5 b(jections)1396 3780 y Fp(\031)1451 3795 y Fm(p;s)p Fn(;)p Fm(r)1625 3780 y Fv(=)27 b Fq(h)p Fp(f)1815 3795 y Fm(p;s)p Fn(;)p Fm(r)1960 3780 y Fp(;)17 b Fq(\001i)p Fp(f)2119 3795 y Fm(p;s)p Fn(;)p Fm(r)3318 3780 y Fv(\(145\))120 3975 y(Then,)34 b(for)e(an)m(y)h Fp(m)28 b Fq(2)g Fp(L)1008 3939 y Fn(2)1048 3975 y Fv(\()p Fl(R)1152 3939 y Fm(d)1198 3975 y Fv(\))33 b(a)f(computation)f(giv)m(es)1015 4218 y Fq(k)p Fp(m)p Fq(k)1200 4177 y Fn(2)1322 4218 y Fv(=)83 b(\(2)p Fp(\031)t Fv(\))1665 4177 y Fo(\000)p Fm(d)1776 4082 y Fk(Z)12 b(Z)1942 4308 y Fg(R)1990 4289 y Fi(d)2027 4308 y Fo(\002)p Fg(R)2130 4289 y Fi(d)2170 4218 y Fq(h)p Fp(m;)17 b(\031)2393 4233 y Fm(p;s)p Fn(;)p Fm(r)2538 4218 y Fp(m)p Fq(i)p Fp(dp)g(ds)580 4348 y Fk(Z)635 4574 y Fg(R)683 4555 y Fj(3)738 4484 y Fq(jr)p Fp(m)p Fv(\()p Fp(x)p Fv(\))p Fq(j)1093 4443 y Fn(2)1133 4484 y Fp(dx)83 b Fv(=)g(\(2)p Fp(\031)t Fv(\))1665 4443 y Fo(\000)p Fm(d)1776 4348 y Fk(Z)12 b(Z)1942 4574 y Fg(R)1990 4555 y Fi(d)2027 4574 y Fo(\002)p Fg(R)2130 4555 y Fi(d)2186 4484 y Fq(j)p Fp(p)p Fq(j)2291 4443 y Fn(2)2330 4484 y Fq(h)p Fp(m;)17 b(\031)2553 4499 y Fm(p;s)p Fn(;)p Fm(r)2699 4484 y Fp(m)p Fq(i)p Fp(dp)g(ds)k Fq(\000)h Fp(s)3203 4443 y Fo(\000)p Fn(2)3298 4484 y Fq(k)p Fp(m)p Fq(k)3483 4443 y Fn(2)3522 4484 y Fq(k)p Fp(g)3623 4443 y Fn(2)3661 4484 y Fq(k)3711 4443 y Fn(2)436 4614 y Fk(Z)491 4839 y Fg(R)539 4820 y Fj(3)594 4749 y Fq(j)p Fp(m)p Fv(\()p Fp(x)p Fv(\))p Fq(j)866 4708 y Fn(2)906 4749 y Fp(\036)964 4764 y Fm(r)1002 4749 y Fv(\()p Fp(x)p Fv(\))p Fp(dx)83 b Fv(=)g(\(2)p Fp(\031)t Fv(\))1665 4708 y Fo(\000)p Fm(d)1776 4614 y Fk(Z)12 b(Z)1942 4839 y Fg(R)1990 4820 y Fi(d)2027 4839 y Fo(\002)p Fg(R)2130 4820 y Fi(d)2186 4749 y Fp(\036)p Fv(\()p Fp(s)p Fv(\))p Fq(h)p Fp(m;)17 b(\031)2589 4764 y Fm(p;s)p Fn(;)p Fm(r)2734 4749 y Fp(m)p Fq(i)p Fp(dp)g(ds)456 b Fv(\(146\))120 5015 y(\(2)p Fp(\031)t Fv(\))304 4974 y Fo(\000)p Fm(d)416 4879 y Fk(Z)11 b(Z)582 5105 y Fg(R)630 5086 y Fi(d)666 5105 y Fo(\002)p Fg(R)769 5086 y Fi(d)825 5015 y Fp(\031)880 5030 y Fm(p;s)p Fn(;)p Fm(r)1026 5015 y Fp(dp)17 b(ds)82 b Fv(=)h Ft(I)32 b Fv(on)h Fp(L)1757 4974 y Fn(2)1796 5015 y Fv(\()p Fl(R)1900 4974 y Fm(d)1947 5015 y Fv(\))f(\(closure)h (relation\))120 5264 y(where)1441 5277 y Fk(\032)1557 5356 y Fp(\036)1615 5371 y Fm(r)1736 5356 y Fv(=)110 b Fp(h)1978 5371 y Fm(r)2038 5356 y Fq(\003)22 b Fp(\036)1557 5477 y(h)1613 5492 y Fm(r)1736 5477 y Fv(:=)83 b Fq(j)p Fp(g)1997 5492 y Fm(r)2034 5477 y Fq(j)2062 5440 y Fn(2)1781 5712 y Fv(32)p eop %%Page: 33 33 33 32 bop 120 407 a Fv(\(\()p Fp(h)252 422 y Fm(r)290 407 y Fv(\))328 422 y Fm(r)r(>)p Fn(0)486 407 y Fv(is)30 b(an)h(standard)g(appro)m(ximation)d(of)i(iden)m(tit)m(y\).)42 b(F)-8 b(or)30 b(all)e(densit)m(y)k(matrices)d Fp(\015)36 b Fv(w)m(e)120 527 y(ha)m(v)m(e)e(the)f(follo)m(wing)d(form)m(ulas)h (generalizing)f(\(146\):)447 795 y(T)-8 b(r)16 b([\()p Fq(\000)p Fv(\001\))p Fp(\015)5 b Fv(])84 b(=)f(\(2)p Fp(\031)t Fv(\))1334 754 y Fo(\000)p Fm(d)1446 659 y Fk(Z)11 b(Z)1612 885 y Fg(R)1660 866 y Fi(d)1696 885 y Fo(\002)p Fg(R)1799 866 y Fi(d)1856 795 y Fv(T)-8 b(r)16 b([)p Fp(\031)2054 810 y Fm(p;s)p Fn(;)p Fm(r)2200 795 y Fp(\015)5 b Fv(])p Fq(j)p Fp(p)p Fq(j)2388 754 y Fn(2)2444 795 y Fp(dp)17 b(ds)k Fq(\000)h Fp(s)2824 754 y Fo(\000)p Fn(2)2919 795 y Fq(k)p Fp(g)3020 754 y Fn(2)3058 795 y Fq(k)3108 754 y Fn(2)3147 795 y Fv(T)-8 b(r)17 b Fp(\015)340 1060 y Fv(T)-8 b(r)17 b([\()p Fp(h)578 1075 y Fm(s)637 1060 y Fq(\003)22 b Fp(V)f Fv(\))p Fp(\015)5 b Fv(])83 b(=)g(\(2)p Fp(\031)t Fv(\))1334 1019 y Fo(\000)p Fm(d)1446 925 y Fk(Z)11 b(Z)1612 1150 y Fg(R)1660 1131 y Fi(d)1696 1150 y Fo(\002)p Fg(R)1799 1131 y Fi(d)1856 1060 y Fv(T)-8 b(r)16 b([)p Fp(\031)2054 1075 y Fm(p;s)p Fn(;)p Fm(r)2200 1060 y Fp(\015)5 b Fv(])p Fp(V)22 b Fv(\()p Fp(s)p Fv(\))17 b Fp(dp)g(ds)603 b Fv(\(147\))120 1518 y Fz(iii.)50 b(Construction)36 b(of)i(a)h(trial)e(densit)m(y)g(matrix)266 1638 y Fv(Consider,)c(for)f Fp(d)c Fv(=)f(3,)1030 1900 y Fp(\015)33 b Fv(:=)27 b(\(2)p Fp(\031)t Fv(\))1428 1859 y Fo(\000)p Fn(3)1539 1765 y Fk(Z)11 b(Z)1705 1990 y Fm(h)1746 1971 y Fj(2)1780 1990 y Fo(j)p Fm(p)p Fo(j)1856 1971 y Fj(2)1890 1990 y Fo(\000)p Fm(V)k Fn(\()p Fm(s)p Fn(\))p Fo(\024)p Fn(0)2200 1900 y Fp(\031)2255 1915 y Fm(p;s)p Fn(;)p Fm(r)2417 1900 y Fp(dp)i(ds)120 2188 y Fv(By)34 b(the)g(closure)f(relation)f(giv) m(en)h(in)g(eq.)46 b(\(146\),)33 b(w)m(e)i(see)f(that)f Fp(\015)39 b Fv(is)33 b(a)g(densit)m(y)h(matrix.)44 b(Its)120 2309 y(densit)m(y)34 b(is)e(obtained)g(b)m(y)h(restricting)f(its)g(Sc)m (h)m(w)m(arz)i(k)m(ernel)g(to)e(the)h(diagonal:)788 2571 y Fp(\032)838 2586 y Fm(\015)882 2571 y Fv(\()p Fp(x)p Fv(\))84 b(=)f(\(2)p Fp(\031)t Fv(\))1440 2529 y Fo(\000)p Fm(d)1551 2435 y Fk(Z)11 b(Z)1717 2661 y Fm(h)1758 2642 y Fj(2)1792 2661 y Fo(j)p Fm(p)p Fo(j)1868 2642 y Fj(2)1902 2661 y Fo(\000)p Fm(V)k Fn(\()p Fm(s)p Fn(\))p Fo(\024)p Fn(0)2212 2571 y Fp(\031)2267 2586 y Fm(p;s)p Fn(;)p Fm(r)2412 2571 y Fv(\()p Fp(x;)i(x)p Fv(\))g Fp(dp)g(ds)1097 2846 y Fv(=)83 b(\(2)p Fp(\031)t Fv(\))1440 2805 y Fo(\000)p Fm(d)1551 2710 y Fk(Z)11 b(Z)1717 2936 y Fm(h)1758 2917 y Fj(2)1792 2936 y Fo(j)p Fm(p)p Fo(j)1868 2917 y Fj(2)1902 2936 y Fo(\000)p Fm(V)k Fn(\()p Fm(s)p Fn(\))p Fo(\024)p Fn(0)2212 2846 y Fp(h)2268 2861 y Fm(r)2306 2846 y Fv(\()p Fp(x)22 b Fq(\000)h Fp(s)p Fv(\))17 b Fp(dp)g(ds)1097 3135 y Fv(=)1305 3068 y Fq(j)p Fp(S)1399 3032 y Fm(d)p Fo(\000)p Fn(1)1529 3068 y Fq(j)p 1266 3113 331 4 v 1266 3204 a Fp(d)p Fv(\(2)p Fp(\031)t(h)p Fv(\))1557 3175 y Fm(d)1606 3135 y Fp(h)1662 3150 y Fm(r)1723 3135 y Fq(\003)22 b Fp(V)1852 3150 y Fn(+)1910 3135 y Fv(\()p Fp(x)p Fv(\))2041 3094 y Fm(d=)p Fn(2)3318 3135 y Fv(\(148\))1097 3393 y(=)1362 3326 y(1)p 1266 3371 243 4 v 1266 3462 a(6)p Fp(\031)1374 3433 y Fn(2)1413 3462 y Fp(h)1469 3433 y Fn(3)1518 3393 y Fp(h)1574 3408 y Fm(r)1634 3393 y Fq(\003)g Fp(V)1763 3408 y Fn(+)1822 3393 y Fv(\()p Fp(x)p Fv(\))1953 3352 y Fn(3)p Fm(=)p Fn(2)2096 3393 y Fv(\(for)32 b Fp(d)27 b Fv(=)h(3\))120 3638 y(W)-8 b(e)33 b(can)g(compute)f(also)1013 3900 y(T)-8 b(r)17 b([\()p Fq(\000)p Fv(\001\))p Fp(\031)1446 3915 y Fm(p;s)p Fn(;)p Fm(r)1592 3900 y Fv(])84 b(=)1861 3764 y Fk(Z)1917 3990 y Fg(R)1965 3971 y Fi(d)2022 3900 y Fq(jr)p Fp(f)2181 3915 y Fm(p;s)p Fn(;)p Fm(r)2326 3900 y Fv(\()p Fp(x)p Fv(\))p Fq(j)2485 3859 y Fn(2)2541 3900 y Fp(dx)1703 4107 y Fv(=)e Fq(j)p Fp(p)p Fq(j)1966 4066 y Fn(2)2027 4107 y Fv(+)22 b Fp(r)2172 4066 y Fo(\000)p Fn(2)2266 4107 y Fq(kr)p Fp(g)t Fq(k)2500 4066 y Fn(2)3318 4107 y Fv(\(149\))120 4326 y(Using)32 b(\(149\))g(w)m(e)i(can)e(compute)h (\(in)f(the)h(next,)g Fp(d)28 b Fv(=)f(3\))349 4604 y(T)-8 b(r)17 b([\()p Fq(\000)p Fp(h)664 4563 y Fn(2)704 4604 y Fv(\001\))p Fp(\015)5 b Fv(])28 b(=)1169 4537 y(1)p 1047 4581 292 4 v 1047 4672 a(10)p Fp(\031)1204 4644 y Fn(2)1243 4672 y Fp(h)1299 4644 y Fn(3)1366 4469 y Fk(Z)1421 4694 y Fg(R)1469 4675 y Fj(3)1524 4604 y Fp(V)1581 4619 y Fn(+)1640 4604 y Fv(\()p Fp(x)p Fv(\))1771 4563 y Fn(5)p Fm(=)p Fn(2)1898 4604 y Fp(dx)22 b Fv(+)2143 4537 y Fq(kr)p Fp(g)t Fq(k)2377 4501 y Fn(2)p 2134 4581 290 4 v 2134 4672 a Fv(6)p Fp(\031)2242 4644 y Fn(2)2281 4672 y Fp(r)2328 4644 y Fn(2)2367 4672 y Fp(h)2450 4469 y Fk(Z)2506 4694 y Fg(R)2554 4675 y Fj(3)2609 4604 y Fp(V)2666 4619 y Fn(+)2725 4604 y Fv(\()p Fp(x)p Fv(\))2856 4563 y Fn(3)p Fm(=)p Fn(2)2983 4604 y Fp(dx)229 b Fv(\(150\))120 4870 y(Giv)m(en)31 b(a)f(function)h Fp(\022)j Fv(suc)m(h)e(that)f(0)d Fq(\024)g Fp(\022)j Fq(\024)d Fv(1,)j(consider)g Fp(\015)2267 4885 y Fm(\022)2333 4870 y Fv(:=)d Fp(\022)s(\015)5 b(\022)s Fv(.)43 b(Then)32 b Fp(\015)2990 4885 y Fm(\022)3060 4870 y Fv(is)e(a)h(densit)m(y)120 4991 y(matrix)41 b(if)g Fp(\015)47 b Fv(is.)73 b(Its)43 b(densit)m(y)g(is)f Fp(\032)1468 5006 y Fm(\015)1504 5018 y Fi(\022)1588 4991 y Fv(=)j Fp(\022)1757 4954 y Fn(2)1796 4991 y Fp(\032)1846 5006 y Fm(\015)1891 4991 y Fv(.)73 b(Then)43 b(w)m(e)h(easily)d(compute,)k (with)d Fp(\015)48 b Fv(as)120 5111 y(ab)s(o)m(v)m(e,)893 5373 y(T)-8 b(r)17 b([)p Fp(V)22 b(\015)1167 5388 y Fm(\022)1205 5373 y Fv(])28 b(=)1470 5306 y(1)p 1373 5350 243 4 v 1373 5441 a(6)p Fp(\031)1481 5412 y Fn(2)1521 5441 y Fp(h)1577 5412 y Fn(3)1643 5237 y Fk(Z)1698 5463 y Fg(R)1746 5444 y Fj(3)1801 5373 y Fp(\022)1849 5332 y Fn(2)1889 5373 y Fp(V)22 b Fv(\()p Fp(x)p Fv(\))p Fp(h)2155 5388 y Fm(r)2215 5373 y Fq(\003)g Fp(V)2344 5388 y Fn(+)2403 5373 y Fv(\()p Fp(x)p Fv(\))2534 5332 y Fn(3)p Fm(=)p Fn(2)2661 5373 y Fp(dx)551 b Fv(\(151\))1781 5712 y(33)p eop %%Page: 34 34 34 33 bop 120 407 a Fv(Similarly)-8 b(,)573 670 y(T)g(r)17 b([\()p Fq(\000)p Fv(\001\))p Fp(\015)1002 685 y Fm(\022)1041 670 y Fv(])28 b(=)g(\(2)p Fp(\031)t Fv(\))1384 629 y Fo(\000)p Fn(3)1494 534 y Fk(Z)11 b(Z)1660 760 y Fm(h)1701 741 y Fj(2)1735 760 y Fo(j)p Fm(p)p Fo(j)1811 741 y Fj(2)1845 760 y Fo(\000)p Fm(V)k Fn(\()p Fm(s)p Fn(\))p Fo(\024)p Fn(0)2155 670 y Fv(T)-8 b(r)17 b([\()p Fq(\000)p Fv(\001\))p Fp(\022)s(\031)2636 685 y Fm(p;s)p Fn(;)p Fm(r)2782 670 y Fp(\022)s Fv(])g Fp(dp)g(ds)120 953 y Fv(But)33 b(a)f(calculation)e (yields)335 1173 y(T)-8 b(r)17 b([\()p Fq(\000)p Fv(\001\))p Fp(\022)s(\031)816 1188 y Fm(p;s)p Fn(;)p Fm(r)963 1173 y Fp(\022)s Fv(])96 b(=)h Fq(j)p Fp(p)p Fq(j)1412 1132 y Fn(2)1451 1173 y Fp(h)1507 1188 y Fm(r)1567 1173 y Fq(\003)22 b Fp(\022)1687 1132 y Fn(2)1726 1173 y Fv(\()p Fp(s)p Fv(\))g(+)g Fp(h)2024 1188 y Fm(r)2085 1173 y Fq(\003)g Fv([)p Fp(\022)s Fv(\()p Fq(\000)p Fv(\001)p Fp(\022)s Fv(\)]\()p Fp(s)p Fv(\))h(+)f Fp(r)2831 1132 y Fo(\000)p Fn(2)2925 1173 y Fp(\026)2984 1188 y Fm(r)3043 1173 y Fq(\003)g Fp(\022)3163 1132 y Fn(2)3203 1173 y Fv(\()p Fp(s)p Fv(\))941 1319 y Fp(\026)1000 1334 y Fm(r)1121 1319 y Fv(:=)83 b Fq(j)p Fv(\()p Fq(r)p Fp(g)t Fv(\))1545 1334 y Fm(r)1582 1319 y Fq(j)1610 1278 y Fn(2)1676 1319 y Fq(\024)29 b Fp(C)7 b(h)1915 1334 y Fm(r)3318 1319 y Fv(\(152\))120 1539 y(so)33 b(w)m(e)g(obtain)564 1782 y(T)-8 b(r)17 b([\()p Fq(\000)p Fp(h)879 1741 y Fn(2)919 1782 y Fv(\001\))p Fp(\015)1089 1797 y Fm(\022)1128 1782 y Fv(])83 b(=)1528 1715 y(1)p 1407 1759 292 4 v 1407 1851 a(10)p Fp(\031)1564 1822 y Fn(2)1602 1851 y Fp(h)1658 1822 y Fn(3)1725 1647 y Fk(Z)1780 1872 y Fg(R)1828 1853 y Fj(3)1883 1782 y Fp(h)1939 1797 y Fm(r)2000 1782 y Fq(\003)21 b Fp(\022)2119 1741 y Fn(2)2159 1782 y Fv(\()p Fp(x)p Fv(\))p Fp(V)2347 1797 y Fn(+)2406 1782 y Fv(\()p Fp(x)p Fv(\))2537 1741 y Fn(5)p Fm(=)p Fn(2)2664 1782 y Fp(dx)1397 2041 y Fv(+)1560 1974 y(1)p 1483 2018 204 4 v 1483 2109 a(6)p Fp(\031)1591 2081 y Fn(2)1630 2109 y Fp(h)1712 1905 y Fk(Z)1768 2131 y Fg(R)1816 2112 y Fj(3)1850 2131 y Fm(h)1912 2041 y Fp(h)1968 2056 y Fm(r)2028 2041 y Fq(\003)h Fv([)p Fp(\022)s Fv(\()p Fq(\000)p Fv(\001)p Fp(\022)s Fv(\)]\()p Fp(x)p Fv(\))p Fp(V)2672 2056 y Fn(+)2732 2041 y Fv(\()p Fp(x)p Fv(\))2863 2000 y Fn(3)p Fm(=)p Fn(2)2990 2041 y Fp(dx)222 b Fv(\(153\))1397 2300 y(+)1603 2232 y(1)p 1483 2277 290 4 v 1483 2368 a(6)p Fp(\031)1591 2339 y Fn(2)1630 2368 y Fp(r)1677 2339 y Fn(2)1716 2368 y Fp(h)1798 2164 y Fk(Z)1854 2390 y Fg(R)1902 2371 y Fj(3)1957 2300 y Fp(\026)2016 2315 y Fm(r)2076 2300 y Fq(\003)22 b Fp(\022)2196 2259 y Fn(2)2235 2300 y Fv(\()p Fp(x)p Fv(\))p Fp(V)2423 2315 y Fn(+)2483 2300 y Fv(\()p Fp(x)p Fv(\))2614 2259 y Fn(3)p Fm(=)p Fn(2)2741 2300 y Fp(dx)120 2572 y Fv(Equations)h(\(151\))g(and)g(\(153\))f(ha)m (v)m(e)j(b)s(oth)e(semiclassical)d(c)m(haracter,)26 b(since)e(the)g (main)d(terms)120 2693 y(on)38 b(the)g(r.h.s)g(of)g(b)s(oth)f(iden)m (tities)g(con)m(v)m(erges)j(to)d(w)m(eylian)h(expressions)h(as)f Fp(r)h Fq(!)e Fv(0.)58 b(No)m(w)120 2813 y(w)m(e)38 b(mak)m(e)g(this)f (more)g(explicit)f(when)i Fp(V)58 b Fv(=)36 b(\010)1877 2777 y Fn(TF)1877 2838 y Fo(R)p Fm(;)p Fo(Z)2017 2813 y Fv(.)58 b(Next)38 b(Lemma)e(is)h(a)h(con)m(v)m(enien)m(t)h(to)s(ol) 120 2933 y(for)32 b(that)g(business:)120 3104 y Ft(Lemma)37 b(10.)49 b Fr([R)-5 b(e)g(gularization)38 b(of)h(a)f Fp(C)1667 3067 y Fn(3+1)1836 3104 y Fr(function])g(Consider)g(an)g (appr)-5 b(oximation)37 b(of)120 3224 y(identity)e Fv(\()p Fp(h)570 3239 y Fm(r)608 3224 y Fv(\))646 3239 y Fm(r)r(>)p Fn(0)774 3224 y Fv(;)f Fp(h)891 3239 y Fm(r)956 3224 y Fv(=)28 b Fp(r)1107 3188 y Fo(\000)p Fm(d)1202 3224 y Fp(h)p Fv(\()p Fp(r)1343 3188 y Fo(\000)p Fn(1)1437 3224 y Fq(\001)p Fv(\))34 b Fr(with)h(the)g(pr)-5 b(op)g(erties:)266 3344 y(i.)45 b Fp(h)28 b Fq(\025)g Fv(0)35 b Fr(is)f(r)-5 b(adial.)266 3465 y(ii.)45 b Fp(h)35 b Fr(is)f(supp)-5 b(orte)g(d)35 b(in)p 1151 3385 80 4 v 35 w Fp(B)1230 3480 y Fn(1)1269 3465 y Fv(\(0\))p Fr(.)266 3585 y(iii.)431 3505 y Fk(R)514 3585 y Fp(h)28 b Fv(=)f(1)p Fr(.)266 3706 y(Now,)g(given)d(a)h(function)g Fp(V)49 b Fq(2)28 b Fp(C)1480 3669 y Fn(3+1)1609 3706 y Fv(\()p Fl(R)1713 3669 y Fm(d)1760 3706 y Fv(\))p Fr(.)41 b(Then,)26 b(with)f Fp(K)35 b Fv(:=)27 b Fp(K)7 b Fv(\()p Fp(h)p Fv(\))28 b(=)2959 3666 y Fn(1)p 2959 3683 36 4 v 2959 3740 a(2)3020 3625 y Fk(R)3103 3706 y Fp(h)p Fv(\()p Fp(x)p Fv(\))p Fq(j)p Fp(x)p Fq(j)3401 3669 y Fn(2)3458 3706 y Fp(dx)p Fr(,)120 3826 y(the)35 b(next)g(appr)-5 b(oximation)33 b(hols:)787 4046 y Fq(k)p Fp(V)44 b Fq(\000)22 b Fp(h)1093 4061 y Fm(r)1154 4046 y Fq(\003)g Fp(V)43 b Fq(\000)23 b Fp(K)7 b(r)1563 4005 y Fn(2)1602 4046 y Fv(\001)p Fp(V)22 b Fq(k)1812 4061 y Fo(1)1914 4046 y Fq(\024)28 b Fp(C)7 b(r)2143 4005 y Fn(4)2199 4046 y Fv(max)2202 4112 y Fo(j)p Fm(\013)p Fo(j)p Fn(=4)2397 4046 y Fq(fk)p Fp(@)2553 4005 y Fm(\013)2603 4046 y Fp(V)22 b Fq(k)2732 4061 y Fo(1)2806 4046 y Fq(g)462 b Fv(\(154\))120 4327 y Fr(Pr)-5 b(o)g(of.)48 b Fv(Since)680 4247 y Fk(R)763 4327 y Fp(h)819 4342 y Fm(s)884 4327 y Fv(=)27 b(1,)33 b(w)m(e)g(can)g(write)760 4601 y Fp(h)816 4616 y Fm(r)876 4601 y Fq(\003)22 b Fp(V)g Fv(\()p Fp(x)p Fv(\))g Fq(\000)h Fp(V)e Fv(\()p Fp(x)p Fv(\))28 b(=)1621 4465 y Fk(Z)1737 4601 y Fp(h)1793 4616 y Fm(r)1831 4601 y Fv(\()p Fp(x)23 b Fq(\000)f Fp(y)t Fv(\))17 b(\()o Fp(V)k Fv(\()p Fp(y)t Fv(\))h Fq(\000)g Fp(V)g Fv(\()p Fp(x)p Fv(\)\))33 b Fp(dy)421 b Fv(\(155\))120 4864 y(W)-8 b(e)33 b(can)g(write)f(do)m(wn)i(also)d(T)-8 b(a)m(ylor's)33 b(form)m(ula)e(in)g(the)i(follo)m(wing)d(w)m(a)m(y:)120 5169 y Fp(V)22 b Fv(\()p Fp(y)t Fv(\))p Fq(\000)p Fp(V)f Fv(\()p Fp(x)p Fv(\))28 b(=)798 5044 y Fn(3)744 5074 y Fk(X)755 5284 y Fm(j)t Fn(=1)919 5074 y Fk(X)905 5290 y Fo(j)p Fm(\013)p Fo(j)p Fn(=)p Fm(j)1103 5101 y Fp(@)1159 5065 y Fm(\013)1209 5101 y Fp(V)22 b Fv(\()p Fp(x)p Fv(\))p 1103 5146 317 4 v 1225 5237 a Fp(j)6 b Fv(!)1429 5169 y(\()p Fp(y)t Fq(\000)p Fp(x)p Fv(\))1689 5127 y Fm(\013)1739 5169 y Fv(+)1838 5101 y(1)p 1825 5146 76 4 v 1825 5237 a(3!)1942 5074 y Fk(X)1927 5290 y Fo(j)p Fm(\013)p Fo(j)p Fn(=4)2102 5169 y Fv(\()p Fp(y)t Fq(\000)p Fp(x)p Fv(\))2362 5127 y Fm(\013)2428 5033 y Fk(Z)2527 5059 y Fn(1)2483 5259 y(0)2583 5169 y Fp(@)2639 5127 y Fm(\013)2690 5169 y Fp(V)21 b Fv(\(\(1)p Fq(\000)p Fp(t)p Fv(\))p Fp(x)p Fv(+)p Fp(ty)t Fv(\)\(1)p Fq(\000)p Fp(t)p Fv(\))3536 5127 y Fn(3)3592 5169 y Fp(dt)1781 5712 y Fv(34)p eop %%Page: 35 35 35 34 bop 120 407 a Fv(Because)27 b(of)d(the)i(radial)d(symmetry)-8 b(,)1488 327 y Fk(R)1571 407 y Fp(h)1627 422 y Fm(r)1665 407 y Fv(\()p Fp(x)p Fv(\))p Fp(x)1851 371 y Fm(\013)1918 407 y Fp(dx)27 b Fv(=)h(0)d(if)f(either)h Fq(j)p Fp(\013)q Fq(j)i Fv(=)g(1)p Fp(;)33 b Fv(3)25 b(or)g Fp(@)3180 371 y Fm(\013)3258 407 y Fv(=)i Fp(@)3417 371 y Fm(i)3447 407 y Fp(@)3503 371 y Fm(j)120 527 y Fv(with)33 b Fp(i)d Fq(6)p Fv(=)f Fp(j)6 b Fv(.)47 b(Therefore,)35 b(plugging)c(the)j(T)-8 b(a)m(ylor)34 b(expansion)g(in)m(to)e(\(155\))h(and)h(taking)f(in)m(to) 120 648 y(accoun)m(t)g(the)g(cancellations)e(just)i(men)m(tioned)g(w)m (e)g(get)120 850 y Fp(h)176 865 y Fm(r)236 850 y Fq(\003)22 b Fp(V)g Fv(\()p Fp(x)p Fv(\))83 b(=)g Fp(V)22 b Fv(\()p Fp(x)p Fv(\))g(+)g Fp(K)7 b(r)1227 809 y Fn(2)1266 850 y Fv(\001)p Fp(V)22 b Fv(\()p Fp(x)p Fv(\))1769 b(\(156\))760 1060 y(+)859 993 y(1)p 846 1037 76 4 v 846 1128 a(3!)964 965 y Fk(X)948 1181 y Fo(j)p Fm(\013)p Fo(j)p Fn(=4)1140 924 y Fk(Z)1239 951 y Fn(1)1195 1150 y(0)1279 1060 y Fv(\(1)21 b Fq(\000)i Fp(t)p Fv(\))1560 1019 y Fn(3)1616 1060 y Fp(dt)1719 924 y Fk(Z)1835 1060 y Fp(dy)c(h)2009 1075 y Fm(r)2048 1060 y Fv(\()p Fp(x)j Fq(\000)h Fp(y)t Fv(\)\()p Fp(y)h Fq(\000)f Fp(x)p Fv(\))2656 1019 y Fm(\013)2705 1060 y Fp(@)2761 1019 y Fm(\013)2812 1060 y Fp(V)e Fv(\(\(1)h Fq(\000)h Fp(t)p Fv(\))p Fp(x)g Fv(+)f Fp(ty)t Fv(\)\))120 1367 y(The)34 b(next)f(estimate)f(holds:)510 1465 y Fk(\014)510 1525 y(\014)510 1585 y(\014)510 1644 y(\014)543 1474 y(Z)642 1500 y Fn(1)598 1700 y(0)682 1610 y Fv(\(1)22 b Fq(\000)g Fp(t)p Fv(\))963 1568 y Fn(3)1020 1610 y Fp(dt)1123 1474 y Fk(Z)1238 1610 y Fp(dy)e(h)1413 1625 y Fm(r)1451 1610 y Fv(\()p Fp(x)i Fq(\000)h Fp(y)t Fv(\)\()p Fp(y)h Fq(\000)f Fp(x)p Fv(\))2059 1568 y Fm(\013)2109 1610 y Fp(@)2165 1568 y Fm(\013)2215 1610 y Fp(V)f Fv(\(\(1)f Fq(\000)i Fp(t)p Fv(\))p Fp(x)g Fv(+)f Fp(ty)t Fv(\)\))2952 1465 y Fk(\014)2952 1525 y(\014)2952 1585 y(\014)2952 1644 y(\014)1450 1882 y Fq(\024)28 b(k)p Fp(@)1661 1841 y Fm(\013)1712 1882 y Fp(V)21 b Fq(k)1840 1897 y Fo(1)1931 1737 y Fk(\014)1931 1797 y(\014)1931 1857 y(\014)1931 1917 y(\014)1964 1746 y(Z)2081 1882 y Fp(dy)e(h)2255 1897 y Fm(r)2293 1882 y Fv(\()p Fp(x)k Fq(\000)f Fp(y)t Fv(\)\()p Fp(y)j Fq(\000)d Fp(x)p Fv(\))2901 1841 y Fm(\013)2951 1737 y Fk(\014)2951 1797 y(\014)2951 1857 y(\014)2951 1917 y(\014)1755 2145 y Fv(=)28 b Fp(C)7 b(r)1983 2104 y Fn(4)2038 2010 y Fk(Z)2155 2145 y Fp(h)p Fv(\()p Fp(x)p Fv(\))p Fq(j)p Fp(x)2425 2104 y Fm(\013)2475 2145 y Fq(j)17 b Fp(dx)p Fq(k)p Fp(@)2732 2104 y Fm(\013)2781 2145 y Fp(V)22 b Fq(k)2910 2160 y Fo(1)2357 2348 y Fq(\024)28 b Fp(C)7 b(r)2586 2307 y Fn(4)2625 2348 y Fq(k)p Fp(@)2731 2307 y Fm(\013)2781 2348 y Fp(V)22 b Fq(k)2910 2363 y Fo(1)120 2551 y Fv(and)33 b(plugging)d(it)i(in)m(to)g(\(156\))f(w)m(e)j (get)f(\(154\))p 3469 2551 4 66 v 3473 2488 59 4 v 3473 2551 V 3531 2551 4 66 v 120 2741 a(W)-8 b(e)28 b(can)g(also)e(lo)s (calize)f(the)j(conclusion)f(of)g(Lemma)f(10.)41 b(This)28 b(is)f(giv)m(en)h(in)e(next)j(Corollary:)120 2911 y Ft(Corollary)36 b(2)h(\(Lo)s(cal)g(regularization)f(of)h(a)h Fp(C)2063 2875 y Fn(3+1)2230 2911 y Ft(function\).)48 b Fr(Assume)35 b(the)g(c)-5 b(ondi-)120 3031 y(tions)35 b(of)f(L)-5 b(emma)34 b(10.)45 b(Then,)34 b(at)h(a)f(given)g Fp(x)28 b Fq(2)g Fl(R)1978 2995 y Fm(d)2060 3031 y Fr(the)35 b(next)f(appr)-5 b(oximation)34 b(holds:)210 3255 y Fq(j)p Fp(h)294 3270 y Fm(r)354 3255 y Fq(\003)22 b Fp(V)f Fv(\()p Fp(x)p Fv(\))i Fq(\000)f Fp(V)g Fv(\()p Fp(x)p Fv(\))g Fq(\000)h Fp(K)7 b(r)1226 3214 y Fn(2)1265 3255 y Fv(\001)p Fp(V)22 b Fv(\()p Fp(x)p Fv(\))p Fq(j)28 b(\024)g Fp(C)7 b(r)1841 3214 y Fn(4)1880 3255 y Fv(\(1)22 b(+)g Fp(s)2133 3214 y Fo(\000)p Fn(4)2227 3255 y Fv(\))17 b(max)2285 3322 y Fo(j)p Fm(\013)p Fo(j\024)p Fn(4)2480 3145 y Fk(n)2546 3255 y Fq(k)p Fp(@)2652 3214 y Fm(\013)2712 3255 y Fp(V)22 b Fq(j)2819 3284 y Fm(B)2872 3292 y Fi(s)2905 3284 y Fn(\()p Fm(x)p Fn(\))3020 3255 y Fq(k)3070 3270 y Fo(1)3145 3145 y Fk(o)3318 3255 y Fv(\(157\))120 3527 y Fr(Pr)-5 b(o)g(of.)48 b Fv(Fix)41 b(some)f(function)h Fp(\022)k Fq(2)e Fp(C)1528 3490 y Fo(1)1521 3551 y Fm(c)1602 3527 y Fv(\()p 1640 3446 80 4 v Fp(B)1719 3542 y Fn(1)1759 3527 y Fv(\(0\)\))d(suc)m(h)j(that)e Fp(\022)k Fv(=)d(1)e(if)g Fq(j)p Fp(x)p Fq(j)i(\024)g Fv(1)p Fp(=)p Fv(2.)69 b(With)120 3647 y Fp(\022)165 3662 y Fm(s;x)304 3647 y Fv(:=)42 b Fp(\022)s Fv(\()p Fp(s)581 3611 y Fo(\000)p Fn(1)675 3647 y Fv(\()p Fp(y)31 b Fq(\000)d Fp(x)p Fv(\)\),)44 b Fp(h)1155 3662 y Fm(r)1221 3647 y Fq(\003)28 b Fp(V)21 b Fv(\()p Fp(x)p Fv(\))43 b(=)f Fp(h)1725 3662 y Fm(r)1791 3647 y Fq(\003)27 b Fv(\()p Fp(f)11 b(\022)2010 3662 y Fm(s;x)2107 3647 y Fv(\))16 b(\()p Fp(x)p Fv(\))41 b(as)h(long)d(as)j Fp(r)i Fq(\024)f Fv(2)p Fp(s)p Fv(.)69 b(Also)40 b(all)120 3767 y(the)d(deriv)-5 b(ativ)m(es)36 b(of)g Fp(V)22 b(\022)1020 3782 y Fm(r)n(;x)1150 3767 y Fv(coincide)36 b(with)g(those)h(of)f Fp(V)58 b Fv(at)36 b Fp(x)p Fv(.)55 b(W)-8 b(riting)35 b(do)m(wn)i(Leibnitz's)120 3888 y(form)m(ula)1057 4008 y Fp(@)1113 3967 y Fm(\013)1180 4008 y Fv(\()p Fp(V)22 b(\022)1342 4023 y Fm(s;x)1438 4008 y Fv(\))28 b(=)1607 3913 y Fk(X)1608 4126 y Fm(\014)s Fo(\024)p Fm(\013)1768 4008 y Fp(C)1838 4023 y Fm(\013;\014)1966 3927 y Fk(\000)2012 4008 y Fp(@)2068 3967 y Fm(\014)2116 4008 y Fp(\022)2161 4023 y Fm(s;x)2258 3927 y Fk(\001)2320 4008 y Fp(@)2376 3967 y Fm(\013)p Fo(\000)p Fm(\014)2524 4008 y Fp(V)120 4272 y Fv(so,)33 b(if)e Fp(\013)e Fv(=)e(4)32 b(w)m(e)i(can)f(estimate)735 4486 y Fq(j)p Fp(@)819 4445 y Fm(\013)886 4486 y Fv(\()p Fp(V)21 b(\022)1047 4501 y Fm(s;x)1144 4486 y Fv(\))16 b(\()p Fp(x)p Fv(\))p Fq(j)28 b(\024)g Fp(C)7 b Fv(\(1)22 b(+)g Fp(s)1820 4445 y Fo(\000)p Fn(4)1914 4486 y Fv(\))1984 4392 y Fk(X)1969 4608 y Fo(j)p Fm(\013)p Fo(j\024)p Fn(4)2160 4486 y Fq(k)p Fp(@)2266 4445 y Fm(\013)2326 4486 y Fp(V)g Fq(j)2432 4516 y Fm(B)2485 4524 y Fi(s)2519 4516 y Fn(\()p Fm(x)p Fn(\))2634 4486 y Fq(k)2684 4501 y Fo(1)120 4794 y Fv(and)33 b(the)g(result)f(follo)m (ws)f(b)m(y)j(Lemma)d(10.)p 3469 4794 4 66 v 3473 4731 59 4 v 3473 4794 V 3531 4794 4 66 v 266 4984 a(No)m(w)i(consider)g(the) g(follo)m(wing)d(problem:)266 5104 y(W)-8 b(e)33 b(are)f(giv)m(en)h(a)f (p)s(oten)m(tial)e Fp(W)46 b Fv(and)33 b(a)f(function)f Fp(\022)g Fq(2)d Fp(C)2380 5068 y Fo(1)2373 5129 y Fm(c)2455 5104 y Fv(\()p 2493 5024 80 4 v Fp(B)2572 5119 y Fm(r)2610 5104 y Fv(\(0\)\),)k Fp(\022)f Fv(=)c(1)32 b(on)p 3228 5024 V 33 w Fp(B)3307 5120 y Fm(r)r(=)p Fn(2)3415 5104 y Fv(\(0\))120 5224 y(suc)m(h)i(that)e Fp(\022)f Fv(=)d(1)k(on)p 948 5144 V 33 w Fp(B)1027 5240 y Fm(r)r(=)p Fn(2)1135 5224 y Fv(\(0\))g(and)h Fq(k)p Fp(@)1588 5188 y Fm(\013)1638 5224 y Fp(\022)s Fq(k)1736 5239 y Fo(1)1839 5224 y Fq(\024)28 b Fp(C)2014 5239 y Fm(\013)2063 5224 y Fp(r)2110 5188 y Fo(\000j)p Fm(\013)p Fo(j)2253 5224 y Fv(.)44 b(No)m(w)33 b(consider)g(the)g(functional)749 5427 y Fq(E)802 5442 y Fm(h;W)n(;\022)985 5427 y Fv(\()p Fp(\015)5 b Fv(\))27 b(:=)h(T)-8 b(r)17 b([\()p Fq(\000)p Fp(h)1590 5386 y Fn(2)1630 5427 y Fv(\001)22 b Fq(\000)h Fp(W)14 b Fv(\))p Fp(\015)2028 5442 y Fm(\022)2066 5427 y Fv(];)49 b Fp(\015)70 b Fv(densit)m(y)34 b(matrix)405 b(\(158\))1781 5712 y(35)p eop %%Page: 36 36 36 35 bop 120 407 a Fv(De\014ne)1067 601 y Fp(E)1139 616 y Fm(h;W)n(;\022)1351 601 y Fv(:=)27 b(inf)6 b Fq(fE)1702 616 y Fm(h;W)n(;\022)1885 601 y Fv(\()p Fp(\015)f Fv(\))28 b(:)44 b(0)28 b Fq(\024)g Fp(\015)33 b Fq(\024)28 b Ft(1)p Fq(g)725 b Fv(\(159\))120 795 y(No)m(w)33 b(the)g(follo)m(wing)d (result)j(holds:)120 966 y Ft(Lemma)k(11)h(\(A)e(lo)s(calization)f (estimate\).)47 b Fr(L)-5 b(et)34 b Fp(E)2199 981 y Fm(W)n(;\022)2323 966 y Fp(;)f(W)m(;)h(\022)j Fr(b)-5 b(e)33 b(as)h(ab)-5 b(ove)33 b(with)h Fp(W)41 b Fv(=)120 1086 y Fp(')184 1050 y Fn(TF)184 1111 y(1)314 1086 y Fv(+)28 b Fp(w)s Fr(,)45 b(wher)-5 b(e)43 b Fp(w)j Fq(2)e Fp(C)1153 1050 y Fo(1)1227 1086 y Fv(\()p 1265 1006 80 4 v Fp(B)1345 1101 y Fm(\026)1391 1086 y Fv(\(0\)\))f Fr(\()p Fp(\026)g Fr(is)g(some)g(\014xe)-5 b(d)43 b(numb)-5 b(er,)45 b(i.e,)g(indep)-5 b(endent)42 b(of)120 1206 y Fp(h)35 b(>)g Fv(0)p Fr(\))k(and)f Fq(k)p Fp(@)749 1170 y Fm(\013)800 1206 y Fp(w)s Fq(k)923 1221 y Fo(1)1032 1206 y Fq(\024)d Fp(C)1214 1221 y Fm(\013)1264 1206 y Fr(.)56 b(Fix)39 b(some)f Fp(\030)h Fq(2)d Fp(C)2042 1170 y Fo(1)2035 1231 y Fm(c)2117 1206 y Fv(\()p Fp(B)2229 1221 y Fm(\026)2275 1206 y Fv(\(0\)\))j Fr(such)g(that)g Fv(0)c Fq(\024)h Fp(\030)j Fq(\024)d Fv(1)j Fr(and)120 1327 y Fp(\030)h Fv(=)35 b(1)k Fr(if)g Fq(j)p Fp(x)p Fq(j)c(\024)h Fp(\026=)p Fv(2)i Fr(also)h(ful\014l)5 b(ling)38 b(the)h(same)f(kind)h(of)f(estimates)h(as)g Fp(w)i Fr(do)-5 b(es.)57 b(Final)5 b(ly)120 1447 y(assume)41 b Fp( )j Fv(=)c(\(1)27 b Fq(\000)h Fp(\022)958 1411 y Fn(2)997 1447 y Fv(\))1035 1411 y Fn(1)p Fm(=)p Fn(2)1187 1447 y Fr(ful\014ls)41 b(the)h(same)e(kind)h(of)h(estimates)f(as)g Fp(\022)k Fr(do)-5 b(es.)64 b(Then)41 b(the)120 1567 y(next)35 b(estimate)f(holds)g(for)h Fv(0)27 b Fp(<)h(r)i Fq(\024)e Fp(\026=)p Fv(2)p Fr(:)742 1810 y Fp(E)814 1825 y Fm(h;W)n(;\022)1026 1810 y Fv(=)f Fq(\000)1290 1743 y Fv(1)p 1216 1787 196 4 v 1216 1878 a(15)p Fp(\031)1373 1850 y Fn(2)1422 1810 y Fp(h)1478 1769 y Fo(\000)p Fn(3)1589 1675 y Fk(Z)1706 1810 y Fp(\022)1754 1769 y Fn(2)1793 1810 y Fp(W)1899 1769 y Fn(5)p Fm(=)p Fn(2)2031 1810 y Fv(+)2139 1743 y(1)p 2139 1787 49 4 v 2139 1878 a(8)2198 1810 y Fp(h)2254 1769 y Fo(\000)p Fn(2)2370 1810 y Fv(+)22 b Fq(O)s Fv(\()p Fp(r)2635 1769 y Fo(\000)p Fn(2)2729 1810 y Fp(h)2785 1769 y Fo(\000)p Fn(1)2880 1810 y Fv(\))400 b(\(160\))120 2047 y Fr(wher)-5 b(e)39 b Fp(C)46 b Fr(only)39 b(dep)-5 b(ens)38 b(on)h(the)g(c)-5 b(onstants)39 b Fp(C)1863 2062 y Fm(\013)1951 2047 y Fr(ab)-5 b(ove.)57 b(Mor)-5 b(e)g(over,)40 b(we)f(c)-5 b(an)39 b(achieve)f(the)120 2168 y(estimate)d(\(160\))f(by)h(cho)-5 b(osing)1069 2362 y Fp(\015)33 b Fv(=)27 b(\000)1317 2377 y Fm(\022)1356 2362 y Fv(;)52 b(\000)27 b(:=)h Fp(\037)1715 2377 y Fn(\()p Fo(\0001)p Fm(;)p Fn(0])1947 2362 y Fv(\()p Fq(\000)p Fp(h)2118 2321 y Fn(2)2158 2362 y Fv(\001)22 b Fq(\000)h Fp(\030)2409 2321 y Fn(2)2448 2362 y Fp(W)14 b Fv(\))726 b(\(161\))364 2556 y Ft(Remarks:)364 2714 y Fr(i.-)36 b(The)g(\014nal)g(asymptotics)g(in)g(L)-5 b(emma)36 b(11)g(do)-5 b(es)36 b(not)h(dep)-5 b(end)35 b(at)i(al)5 b(l)36 b(on)g(the)364 2835 y(choic)-5 b(e)34 b(of)g(the)h(auxiliar)g(function)f Fp(\030)364 2993 y Fr(ii.-)d(The)g(pr)-5 b(esenc)g(e)31 b(of)g(function)g Fp(\030)36 b Fr(estr)-5 b(ess)31 b(the)h(fact)g(that) g(in)f(the)g(asymptotics)364 3113 y(given)47 b(by)g(e)-5 b(q.)83 b(\(160\))46 b(only)i(dep)-5 b(ends)46 b(on)h(the)h(b)-5 b(ehaviour)47 b(of)g Fp(W)61 b Fr(on)47 b(a)h(b)-5 b(al)5 b(l)364 3234 y(ar)-5 b(ound)35 b(the)g(origin)g(of)g(r)-5 b(adius)35 b Fp(r)1595 3198 y Fo(0)1646 3234 y Fv(=)29 b(const)17 b Fp(r)s Fr(.)46 b(The)35 b(only)g(glob)-5 b(al)34 b(information)364 3354 y(we)g(ne)-5 b(e)g(d)35 b(is)f(semib)-5 b(ounde)g(dness)33 b(fr)-5 b(om)34 b(b)-5 b(elow)34 b(of)h(our)g(functional.)120 3536 y(Pr)-5 b(o)g(of.)48 b Fv(As)36 b(0)c Fp(<)g(r)i Fq(\024)f Fp(\026=)p Fv(2,)i Fp(\030)h Fv(=)c(1)j(on)g(the)g(supp)s(ort)h(of)f Fp(\022)s Fv(,)g Fp(\030)5 b(\022)35 b Fv(=)d Fp(\022)38 b Fv(and)d(so)h(w)m(e)g (can)f(write,)120 3656 y(for)d(an)m(y)h(densit)m(y)h(matrix)d Fp(\015)5 b Fv(,)1114 3850 y Fq(E)1167 3865 y Fm(h;W)n(;\022)1350 3850 y Fv(\()p Fp(\015)g Fv(\))28 b(=)f(T)-8 b(r)17 b([\()p Fq(\000)p Fp(h)1928 3809 y Fn(2)1968 3850 y Fv(\001)23 b Fq(\000)f Fp(\030)2219 3809 y Fn(2)2258 3850 y Fp(W)14 b Fv(\))p Fp(\015)2453 3865 y Fm(\022)2492 3850 y Fv(])p Fp(:)120 4044 y Fv(By)33 b(the)g(cyclicit)m(y)f(of)g(the)h(trace)g(w)m (e)h(can)f(therefore)g(write)1000 4239 y Fq(E)1053 4254 y Fm(h;W)n(;\022)1236 4239 y Fv(\()p Fp(\015)5 b Fv(\))84 b(=)g(T)-8 b(r)16 b([)p Fp(\022)s Fv(\()p Fq(\000)p Fp(h)1974 4198 y Fn(2)2015 4239 y Fv(\001)22 b Fq(\000)h Fp(\030)2266 4198 y Fn(2)2305 4239 y Fp(W)14 b Fv(\))p Fp(\022)s(\015)5 b Fv(])1451 4384 y Fq(\025)84 b(\000)p Fv(T)-8 b(r)17 b([)p Fp(\022)s Fv(\()p Fq(\000)p Fp(h)2052 4343 y Fn(2)2092 4384 y Fv(\001)23 b Fq(\000)f Fp(\030)2343 4343 y Fn(2)2382 4384 y Fp(W)14 b Fv(\))p Fp(\022)s Fv(])2601 4399 y Fo(\000)3318 4384 y Fv(\(162\))1452 4529 y(=)84 b Fq(\000)p Fv(T)-8 b(r)17 b([)p Fp(\022)1881 4488 y Fn(2)1921 4529 y Fv(\()p Fq(\000)p Fp(h)2092 4488 y Fn(2)2132 4529 y Fv(\001)22 b Fq(\000)h Fp(\030)2383 4488 y Fn(2)2422 4529 y Fp(W)14 b Fv(\)])2593 4544 y Fo(\000)120 4724 y Fv(Also)32 b(w)m(e)i(can)e (decomp)s(ose,)i(with)e Fp(A)c Fv(:=)f Fq(\000)p Fp(h)1767 4687 y Fn(2)1807 4724 y Fv(\001)c Fq(\000)f Fp(\030)2058 4687 y Fn(2)2097 4724 y Fp(W)14 b Fv(,)1171 4918 y Fp(\022)s(A\022)31 b Fv(=)d Fp(\022)s(A)1593 4933 y Fn(+)1652 4918 y Fp(\022)d Fq(\000)e Fp(\022)s(A)1943 4933 y Fo(\000)2002 4918 y Fp(\022)31 b Fq(\025)d(\000)p Fp(\022)s(A)2381 4933 y Fo(\000)2441 4918 y Fp(\022)120 5112 y Fv(since)35 b(is)e(easily)h(v)m (eri\014ed)h(that)f Fp(\022)s(A)1408 5128 y Fn(+)p Fm(=)p Fo(\000)1558 5112 y Fp(\022)f Fq(\025)e Fv(0.)49 b(As)34 b(a)g(consequence,)k(\()p Fp(\022)s(A\022)s Fv(\))2922 5127 y Fo(\000)3012 5112 y Fq(\024)31 b Fp(\022)s(A)3241 5127 y Fo(\000)3300 5112 y Fp(\022)38 b Fv(and)120 5232 y(then)987 5427 y Fq(\000)p Fv(T)-8 b(r)17 b([)p Fp(\022)s(A\022)s Fv(])1404 5442 y Fo(\000)1491 5427 y Fq(\025)29 b(\000)p Fv(T)-8 b(r)17 b([)p Fp(\022)s(A)1939 5442 y Fo(\000)1998 5427 y Fp(\022)s Fv(])28 b(=)g Fq(\000)p Fv(T)-8 b(r)17 b([)p Fp(\022)2474 5386 y Fn(2)2514 5427 y Fp(A)2587 5442 y Fo(\000)2646 5427 y Fv(])645 b(\(163\))1781 5712 y(36)p eop %%Page: 37 37 37 36 bop 120 407 a Fv(Putting)32 b(together)h(\(162\))f(and)g(\(163\)) g(w)m(e)i(\014nally)d(obtain)g(the)i(lo)m(w)m(er)g(b)s(ound)1102 611 y Fp(E)1174 626 y Fm(h;W)n(;\022)1386 611 y Fq(\025)28 b(\000)p Fv(T)-8 b(r)17 b([)p Fp(\022)1760 570 y Fn(2)1800 611 y Fv(\()p Fq(\000)p Fp(h)1971 570 y Fn(2)2011 611 y Fv(\001)22 b Fq(\000)h Fp(\030)2262 570 y Fn(2)2301 611 y Fp(W)14 b Fv(\))2445 626 y Fo(\000)2503 611 y Fv(])p Fp(:)761 b Fv(\(164\))120 815 y(In)33 b(order)g(to)f(obtain)f(a)i (similar)c(upp)s(er)k(b)s(ound)g(w)m(e)g(pro)s(ceed)h(as)f(follo)m(ws:) 266 935 y(W)-8 b(e)36 b(write)g(do)m(wn)g Fq(\000)p Fv(T)-8 b(r)17 b([\()p Fq(\000)p Fp(h)1342 899 y Fn(2)1383 935 y Fv(\001)24 b Fq(\000)h Fp(\030)1638 899 y Fn(2)1677 935 y Fp(W)14 b Fv(\))1821 950 y Fo(\000)1879 935 y Fv(])33 b(=)g(T)-8 b(r)17 b([\()p Fq(\000)p Fp(h)2363 899 y Fn(2)2403 935 y Fv(\001)24 b Fq(\000)h Fp(\030)2658 899 y Fn(2)2697 935 y Fp(W)14 b Fv(\)\000].)52 b(Then)37 b(b)m(y)g(the)120 1056 y(IMS)29 b(form)m(ula)d(and)i(taking)f(in)m(to)g(accoun)m(t)i (that)f(T)-8 b(r)16 b(\000)28 b(=)g Fp(N)2304 1071 y Fo(\000)2363 1056 y Fv(\()p Fq(\000)p Fp(h)2534 1020 y Fn(2)2574 1056 y Fv(\001)13 b Fq(\000)g Fp(\030)2806 1020 y Fn(2)2845 1056 y Fp(W)h Fv(\))27 b(\(the)i(n)m(um)m(b)s(er)120 1176 y(of)j(negativ)m(e)h(eigen)m(v)-5 b(alues\),)32 b(w)m(e)i(can)f(write)302 1380 y Fq(\000)p Fv(T)-8 b(r)18 b([\()p Fq(\000)p Fp(h)695 1339 y Fn(2)735 1380 y Fv(\001)66 b Fq(\000)h Fp(\030)1074 1339 y Fn(2)1113 1380 y Fp(W)14 b Fv(\))1257 1395 y Fo(\000)1316 1380 y Fv(])28 b(=)f(T)-8 b(r)17 b([\()p Fq(\000)p Fp(h)1789 1339 y Fn(2)1829 1380 y Fv(\001)23 b Fq(\000)f Fp(\030)2080 1339 y Fn(2)2119 1380 y Fp(W)14 b Fv(\)\000])883 1525 y(=)84 b(T)-8 b(r)17 b([\()p Fq(\000)p Fp(h)1358 1484 y Fn(2)1398 1525 y Fv(\001)22 b Fq(\000)h Fp(\030)1649 1484 y Fn(2)1688 1525 y Fp(W)14 b Fv(\)\000)1893 1540 y Fm(\022)1931 1525 y Fv(])23 b(+)f(T)-8 b(r)17 b([\()p Fq(\000)p Fp(h)2394 1484 y Fn(2)2434 1525 y Fv(\001)22 b Fq(\000)h Fp(\030)2685 1484 y Fn(2)2724 1525 y Fp(W)14 b Fv(\)\000)2929 1540 y Fm( )2981 1525 y Fv(])1043 1722 y Fq(\000)p Fp(h)1176 1681 y Fn(2)1233 1586 y Fk(Z)1349 1641 y(\000)1395 1722 y Fq(jr)p Fp(\022)s Fq(j)1582 1681 y Fn(2)1643 1722 y Fv(+)22 b Fq(jr)p Fp( )t Fq(j)1947 1681 y Fn(2)1985 1641 y Fk(\001)2048 1722 y Fp(\032)2098 1737 y Fn(\000)882 1925 y Fq(\025)84 b Fv(T)-8 b(r)17 b([\()p Fq(\000)p Fp(h)1358 1884 y Fn(2)1398 1925 y Fv(\001)22 b Fq(\000)h Fp(\030)1649 1884 y Fn(2)1688 1925 y Fp(W)14 b Fv(\)\000)1893 1940 y Fm(\022)1931 1925 y Fv(])23 b(+)f(T)-8 b(r)17 b([\()p Fq(\000)p Fp(h)2394 1884 y Fn(2)2434 1925 y Fv(\001)22 b Fq(\000)h Fp(\030)2685 1884 y Fn(2)2724 1925 y Fp(W)14 b Fv(\)\000)2929 1940 y Fm( )2981 1925 y Fv(])1043 2070 y Fq(\000)p Fp(C)7 b(r)1244 2029 y Fo(\000)p Fn(2)1338 2070 y Fp(h)1394 2029 y Fn(2)1434 2070 y Fp(N)1512 2085 y Fo(\000)1571 2070 y Fv(\()p Fq(\000)p Fp(h)1742 2029 y Fn(2)1782 2070 y Fv(\001)23 b Fq(\000)f Fp(\030)2033 2029 y Fn(2)2072 2070 y Fp(W)14 b Fv(\))882 2216 y Fq(\025)84 b Fv(T)-8 b(r)17 b([\()p Fq(\000)p Fp(h)1358 2174 y Fn(2)1398 2216 y Fv(\001)22 b Fq(\000)h Fp(\030)1649 2174 y Fn(2)1688 2216 y Fp(W)14 b Fv(\)\000)1893 2231 y Fm(\022)1931 2216 y Fv(])23 b(+)f(T)-8 b(r)17 b([\()p Fq(\000)p Fp(h)2394 2174 y Fn(2)2434 2216 y Fv(\001)22 b Fq(\000)h Fp(\030)2685 2174 y Fn(2)2724 2216 y Fp(W)14 b Fv(\)\000)2929 2231 y Fm( )2981 2216 y Fv(])310 b(\(165\))1043 2412 y Fq(\000)p Fp(C)7 b(r)1244 2371 y Fo(\000)p Fn(2)1338 2412 y Fp(h)1394 2371 y Fo(\000)p Fn(1)1505 2277 y Fk(Z)1622 2412 y Fp(\030)1670 2371 y Fn(3)1709 2412 y Fp(W)1815 2371 y Fn(3)p Fm(=)p Fn(2)1957 2412 y Fv(\(b)m(y)33 b(the)g(CLR)g(estimate\))882 2667 y Fq(\025)84 b Fp(E)1115 2682 y Fm(h;W)n(;\022)1321 2667 y Fv(+)22 b Fp(E)1491 2682 y Fm(h;W)n(; )1710 2667 y Fq(\000)h Fp(C)7 b(r)1934 2626 y Fo(\000)p Fn(2)2028 2667 y Fp(h)2084 2626 y Fo(\000)p Fn(1)2195 2531 y Fk(Z)2311 2667 y Fp(\030)2359 2626 y Fn(3)2398 2667 y Fp(W)2504 2626 y Fn(3)p Fm(=)p Fn(2)882 2870 y Fq(\025)84 b Fp(E)1115 2885 y Fm(h;W)n(;\022)1321 2870 y Fq(\000)22 b Fv(T)-8 b(r)17 b([)p Fp( )1631 2829 y Fn(2)1671 2870 y Fv(\()p Fq(\000)p Fp(h)1842 2829 y Fn(2)1882 2870 y Fv(\001)22 b Fq(\000)h Fp(\030)2133 2829 y Fn(2)2172 2870 y Fp(W)14 b Fv(\))2316 2885 y Fo(\000)2374 2870 y Fv(])23 b Fq(\000)f Fp(C)7 b(r)2647 2829 y Fo(\000)p Fn(2)2741 2870 y Fp(h)2797 2829 y Fo(\000)p Fn(1)2924 2870 y Fv(\(b)m(y)34 b(\(164\)\))120 3074 y(F)-8 b(rom)31 b(estimate)h(\(165\))g(w)m(e)h(obtain)f(the)h(upp) s(er)g(b)s(ound)240 3278 y Fp(E)312 3293 y Fm(h;W)n(;\022)524 3278 y Fq(\024)28 b(\000)p Fv(T)-8 b(r)17 b([\()p Fq(\000)p Fp(h)1021 3237 y Fn(2)1061 3278 y Fv(\001)23 b Fq(\000)f Fp(\030)1312 3237 y Fn(2)1351 3278 y Fp(W)14 b Fv(\))1495 3293 y Fo(\000)1554 3278 y Fv(])22 b(+)g(T)-8 b(r)17 b([)p Fp( )1912 3237 y Fn(2)1951 3278 y Fv(\()p Fq(\000)p Fp(h)2122 3237 y Fn(2)2162 3278 y Fv(\001)23 b Fq(\000)f Fp(\030)2413 3237 y Fn(2)2452 3278 y Fp(W)14 b Fv(\))2596 3293 y Fo(\000)2655 3278 y Fv(])22 b(+)g Fp(C)7 b(r)2926 3237 y Fo(\000)p Fn(2)3020 3278 y Fp(h)3076 3237 y Fo(\000)p Fn(1)3171 3278 y Fp(:)120 b Fv(\(166\))120 3482 y(No)m(w)37 b(w)m(e)h(are)f(in)f(p)s(osition)f(to)i(use)h(the)f(semiclassical)d (asymptotics)j(estimates)f(pro)m(vided)120 3602 y(b)m(y)h(lo)s(cal)c (trace)j(estimates)f(similar)e(to)i(those)h(of)f(eq.)54 b(\(94\))35 b(and)h(Lemma)e(9.)52 b(Using)36 b(these)120 3723 y(results)d(w)m(e)h(get)606 3841 y Fk(\032)722 3920 y Fp(E)794 3935 y Fm(h;W)n(;\022)1061 3920 y Fq(\025)83 b(\000)1365 3881 y Fn(1)p 1308 3897 149 4 v 1308 3954 a(15)p Fm(\031)1421 3936 y Fj(2)1467 3920 y Fp(h)1523 3884 y Fo(\000)p Fn(3)1634 3840 y Fk(R)1717 3920 y Fp(\022)1765 3884 y Fn(2)1804 3920 y Fp(W)1910 3884 y Fn(5)p Fm(=)p Fn(2)2042 3920 y Fv(+)2150 3881 y Fn(1)p 2150 3897 36 4 v 2150 3954 a(8)2195 3920 y Fp(h)2251 3884 y Fo(\000)p Fn(2)2368 3920 y Fv(+)22 b Fq(O)s Fv(\()p Fp(r)2633 3884 y Fo(\000)p Fn(1)p Fm(=)p Fn(2)2798 3920 y Fp(h)2854 3884 y Fo(\000)p Fn(1)2948 3920 y Fv(\))722 4042 y Fp(E)794 4057 y Fm(h;W)n(;\022)1061 4042 y Fq(\024)83 b(\000)1365 4003 y Fn(1)p 1308 4019 149 4 v 1308 4077 a(15)p Fm(\031)1421 4058 y Fj(2)1467 4042 y Fp(h)1523 4006 y Fo(\000)p Fn(3)1634 3962 y Fk(R)1717 4042 y Fp(\022)1765 4006 y Fn(2)1804 4042 y Fp(W)1910 4006 y Fn(5)p Fm(=)p Fn(2)2042 4042 y Fv(+)2150 4003 y Fn(1)p 2150 4019 36 4 v 2150 4077 a(8)2195 4042 y Fp(h)2251 4006 y Fo(\000)p Fn(2)2368 4042 y Fv(+)22 b Fq(O)s Fv(\()p Fp(r)2633 4006 y Fo(\000)p Fn(2)2727 4042 y Fp(h)2783 4006 y Fo(\000)p Fn(1)2878 4042 y Fv(\))3318 3981 y(\(167\))120 4239 y(whic)m(h)33 b(concludes)h(the)f(pro)s(of.)p 3469 4239 4 66 v 3473 4177 59 4 v 3473 4239 V 3531 4239 4 66 v 120 4430 a Ft(Lemma)k(12)h (\(Semiclassics)c(of)k(the)f(pro)6 b(jection)37 b(in)m(to)g(the)g (negativ)m(e)g(estates\).)49 b Fr(Assume,)120 4550 y(for)28 b(a)h(c)-5 b(on\014gur)g(ation)28 b(of)g(nucle)-5 b(ar)28 b(char)-5 b(ges)28 b Fq(R)p Fr(,)i Fq(Z)37 b Fr(such)29 b(that)g Fp(\016)2414 4565 y Fm(i)2470 4550 y Fq(\025)f Fp(\026)p Fr(,)h Fv(1)f Fq(\024)g Fp(Z)2942 4565 y Fm(i)2998 4550 y Fq(\024)g Fp(Z)42 b Fr(for)34 b(al)5 b(l)35 b Fp(i)1218 4754 y(\015)e Fv(:=)28 b Fp(\037)1494 4770 y Fn(\()p Fo(\0001)p Fm(;)p Fn(0])1742 4754 y Fv([)p Fq(\000)p Fp(h)1902 4713 y Fn(2)1942 4754 y Fv(\001)22 b Fq(\000)h Fv(\010)2215 4713 y Fn(TF)2215 4779 y Fo(R)p Fm(;)p Fo(Z)2356 4754 y Fv(])2383 4769 y Fo(\000)3318 4754 y Fv(\(168\))120 4970 y Fr(Then,)34 b(with)h Fp(R)q Fv(\()p Fp(h;)17 b(x)p Fv(\))28 b(:=)f Fp(\032)1130 4985 y Fm(\015)1175 4970 y Fv(\()p Fp(x)p Fv(\))22 b Fq(\000)1514 4931 y Fn(1)p 1438 4947 189 4 v 1438 5004 a(6)p Fm(\031)1516 4985 y Fj(2)1551 5004 y Fm(h)1592 4985 y Fj(3)1636 4970 y Fv(\010)1706 4934 y Fn(TF)1706 4995 y Fo(R)p Fm(;)p Fo(Z)1846 4970 y Fv(\()p Fp(x)p Fv(\))1977 4934 y Fn(3)p Fm(=)p Fn(2)2088 4970 y Fr(,)35 b(the)f(estimate)h(holds:)682 5306 y Fq(j)p Fp(R)q Fv(\()p Fp(h;)17 b(x)p Fv(\))p Fq(j)27 b(\024)1177 5102 y Fk(8)1177 5192 y(<)1177 5371 y(:)1570 5186 y Fp(C)7 b(h)1703 5150 y Fo(\000)p Fn(3)1798 5186 y Fp(l)r Fv(\()p Fp(x)p Fv(\))1960 5150 y Fo(\000)p Fn(3)p Fm(=)p Fn(2)2125 5186 y Fv(;)83 b Fp(l)r Fv(\()p Fp(x)p Fv(\))29 b Fq(\024)f Fp(h)2587 5150 y Fo(\000)p Fn(2)1307 5306 y Fp(C)7 b(h)1440 5270 y Fo(\000)p Fn(2)1534 5306 y Fp(l)r Fv(\()p Fp(x)p Fv(\))1696 5270 y Fo(\000)p Fn(2)1791 5306 y Fq(h)p Fp(l)r Fv(\()p Fp(x)p Fv(\))p Fq(i)2031 5270 y Fo(\000)p Fn(3)2125 5306 y Fv(;)83 b Fp(h)2291 5270 y Fn(2)2359 5306 y Fq(\024)28 b Fp(l)r Fv(\()p Fp(x)p Fv(\))g Fq(\024)g Fp(h)2815 5270 y Fo(\000)p Fn(1)1792 5427 y Fp(C)7 b(l)r Fv(\()p Fp(x)p Fv(\))2031 5391 y Fo(\000)p Fn(3)2125 5427 y Fv(;)83 b Fp(l)r Fv(\()p Fp(x)p Fv(\))29 b Fq(\025)f Fp(h)2587 5391 y Fo(\000)p Fn(1)3318 5306 y Fv(\(169\))1781 5712 y(37)p eop %%Page: 38 38 38 37 bop 120 407 a Fr(wher)-5 b(e)34 b(we)h(take)816 520 y Fk(\032)951 599 y Fp(l)r Fv(\()p Fp(x)p Fv(\))83 b(=)g Fp(\024\032)p Fv(\()p Fp(x)p Fv(\);)53 b(0)27 b Fp(<)h(\024)35 b Fr(smal)5 b(l)34 b(enough)932 721 y Fp(\032)p Fv(\()p Fp(x)p Fv(\))83 b(=)g(min)o Fq(fj)p Fp(x)22 b Fq(\000)h Fp(Z)1847 685 y Fn(1)p Fm(=)p Fn(3)1956 721 y Fp(R)2030 736 y Fm(j)2067 721 y Fq(j)28 b Fv(:)f Fp(j)34 b Fv(=)28 b(1)p Fp(;)17 b(:)g(:)g(:)32 b(;)17 b(M)10 b Fq(g)120 934 y Fr(and)34 b(with)h(the)g(c)-5 b(onstant)34 b Fp(C)42 b Fr(uniform)35 b(in)f(al)5 b(l)35 b(the)g(p)-5 b(ar)g(ameters)34 b(involve)-5 b(d)120 1129 y(Pr)g(o)g(of.)48 b Fv(Lemma)21 b(12)g(is)g(pro)m(v)m(en)i(also)e(b)m (y)i(the)f(Multiscale)e(Analysis)i(mac)m(hinery)g(of)f(Ivrii&Sigal.)120 1250 y(More)37 b(sp)s(eci\014cally)-8 b(,)36 b(w)m(e)h(use)g(Theorem)g (10.4)f(of)f([16])h(whic)m(h)h(in)f(our)g(case)h(at)f(hand)g(yields)120 1370 y(the)k(result)g(\(nev)m(ertheless)j(notice)c(that)h(what)g(is)g (stated)g(in)f(Lemma)g(12)h(is)f(what)h(really)120 1491 y(follo)m(ws)31 b(from)h(their)g(pro)s(of,)f(since)i(their)f(result)h (w)m(as)h(stated)f(mistak)m(enly\).)p 3469 1491 4 66 v 3473 1428 59 4 v 3473 1491 V 3531 1491 4 66 v 120 1686 a(F)-8 b(rom)31 b(Lemma)g(12)h(w)m(e)i(can)f(obtain)e(the)i(follo)m (wing)d(Corollary:)120 1856 y Ft(Corollary)36 b(3)h(\(Lo)s(cal)g(trace) g(near)h(singularities\).)46 b Fr(T)-7 b(ake)45 b Fp( )52 b Fq(2)c Fp(C)2853 1820 y Fo(1)2846 1880 y Fm(c)2928 1856 y Fv(\()p 2966 1776 80 4 v Fp(B)3045 1871 y Fm(r)3083 1856 y Fv(\()p Fp(R)3195 1871 y Fm(j)3232 1856 y Fv(\)\))d Fr(such)120 1976 y(that)34 b Fv(0)27 b Fq(\024)i Fp( )i Fq(\024)d Fv(1)34 b Fr(and)f Fq(k)p Fp(@)1076 1940 y Fm(\013)1126 1976 y Fp( )t Fq(k)1243 1991 y Fo(1)1345 1976 y Fq(\024)28 b Fp(C)1520 1991 y Fm(\013)1570 1976 y Fp(r)1617 1940 y Fo(\000j)p Fm(\013)p Fo(j)1793 1976 y Fr(and)34 b Fp(h)2038 1940 y Fn(2)2105 1976 y Fq(\024)28 b Fp(r)i Fq(\024)e Fp(\026)p Fr(.)44 b(With)34 b Fp(\015)39 b Fr(as)33 b(in)h(L)-5 b(emma)32 b(12)120 2097 y(the)j(fol)5 b(lowing)33 b(asymptotics)i(holds)f(with)h(some)f(absolute)g Fv(0)28 b Fq(\024)g Fp(\016)k(<)27 b Fv(1)p Fr(:)266 2217 y(a.-)198 2461 y Fv(T)-8 b(r)17 b([\()p Fq(\000)p Fp(h)513 2419 y Fn(2)553 2461 y Fv(\001)23 b Fq(\000)f Fv(\010)826 2419 y Fn(TF)826 2485 y Fo(R)p Fm(;)p Fo(Z)967 2461 y Fv(\))p Fp(\015)1056 2476 y Fm( )1108 2461 y Fv(])28 b(=)1350 2393 y(1)p 1276 2438 196 4 v 1276 2529 a(15)p Fp(\031)1433 2500 y Fn(2)1482 2461 y Fp(h)1538 2419 y Fo(\000)p Fn(3)1649 2325 y Fk(Z)1765 2461 y Fp( )1832 2419 y Fn(2)1872 2461 y Fv(\(\010)1980 2419 y Fn(TF)1980 2485 y Fo(R)p Fm(;)p Fo(Z)2120 2461 y Fv(\))2158 2419 y Fn(5)p Fm(=)p Fn(2)2290 2461 y Fq(\000)2400 2393 y Fv(1)p 2400 2438 49 4 v 2400 2529 a(8)2459 2461 y Fp(h)2515 2419 y Fo(\000)p Fn(2)2609 2461 y Fp(Z)2683 2419 y Fn(2)2676 2485 y Fm(j)2745 2461 y Fv(+)22 b Fq(O)s Fv(\()p Fp(h)3019 2419 y Fo(\000)p Fn(1)p Fo(\000)p Fm(\016)3202 2461 y Fv(\))78 b(\(170\))266 2723 y Fr(b.-)913 2967 y Fv(T)-8 b(r)17 b([)p Fp(\015)1108 2982 y Fm( )1160 2967 y Fv(])28 b(=)1378 2900 y(1)p 1329 2944 148 4 v 1329 3035 a(6)p Fp(\031)1437 3007 y Fn(2)1486 2967 y Fp(h)1542 2926 y Fo(\000)p Fn(3)1653 2831 y Fk(Z)1769 2967 y Fp( )1836 2926 y Fn(2)1875 2967 y Fv(\(\010)1983 2926 y Fn(TF)1983 2992 y Fo(R)p Fm(;)p Fo(Z)2124 2967 y Fv(\))2162 2926 y Fn(3)p Fm(=)p Fn(2)2294 2967 y Fv(+)22 b Fq(O)s Fv(\()p Fp(h)2568 2926 y Fo(\000)p Fn(2)2662 2967 y Fp(r)s Fv(\))571 b(\(171\))120 3230 y Fr(Pr)-5 b(o)g(of.)48 b Fv(P)m(art)22 b(a.-)f(follo)m(ws)g(inmediately)e(from)i(Lemma)f(11.)40 b(T)-8 b(o)21 b(pro)m(v)m(e)i(b.-)f(w)m(e)h(use)f(Lemma)f(12)120 3350 y(as)33 b(follo)m(ws:)42 b(\(recall)31 b(that)h Fp(\032)1157 3365 y Fm(\015)1193 3377 y Fi( )1273 3350 y Fv(=)27 b Fp( )1443 3314 y Fn(2)1483 3350 y Fp(\032)1533 3365 y Fm(\015)1577 3350 y Fv(\))449 3630 y(T)-8 b(r)17 b([)p Fp(\015)644 3645 y Fm( )697 3630 y Fv(])83 b(=)966 3495 y Fk(Z)1082 3630 y Fp( )1149 3589 y Fn(2)1188 3630 y Fp(\032)1238 3645 y Fm(\015)807 3894 y Fv(=)966 3758 y Fk(Z)1082 3894 y Fp( )1149 3852 y Fn(2)1205 3753 y Fk(\032)1339 3826 y Fv(1)p 1290 3871 V 1290 3962 a(6)p Fp(\031)1398 3933 y Fn(2)1447 3894 y Fp(h)1503 3852 y Fo(\000)p Fn(3)1597 3894 y Fv(\(\010)1705 3852 y Fn(TF)1705 3918 y Fo(R)p Fm(;)p Fo(Z)1846 3894 y Fv(\))1884 3852 y Fn(3)p Fm(=)p Fn(2)2016 3894 y Fv(+)2114 3753 y Fk(\022)2187 3894 y Fp(\032)2237 3909 y Fm(\015)2304 3894 y Fq(\000)2510 3826 y Fv(1)p 2413 3871 243 4 v 2413 3962 a(6)p Fp(\031)2521 3933 y Fn(2)2561 3962 y Fp(h)2617 3933 y Fn(3)2666 3894 y Fv(\(\010)2774 3852 y Fn(TF)2774 3918 y Fo(R)p Fm(;)p Fo(Z)2915 3894 y Fv(\))2953 3852 y Fn(3)p Fm(=)p Fn(2)3063 3753 y Fk(\023\033)120 4165 y Fv(and)33 b(therefore,)g(b)m(y)g(Lemma)f (12)120 4293 y Fk(\014)120 4352 y(\014)120 4412 y(\014)120 4472 y(\014)153 4437 y Fv(T)-8 b(r)17 b([)p Fp(\015)348 4452 y Fm( )400 4437 y Fv(])23 b Fq(\000)608 4370 y Fv(1)p 559 4414 148 4 v 559 4505 a(6)p Fp(\031)667 4477 y Fn(2)716 4437 y Fp(h)772 4396 y Fo(\000)p Fn(3)883 4302 y Fk(Z)1000 4437 y Fp( )1067 4396 y Fn(2)1106 4437 y Fv(\(\010)1214 4396 y Fn(TF)1214 4462 y Fo(R)p Fm(;)p Fo(Z)1354 4437 y Fv(\))1392 4396 y Fn(3)p Fm(=)p Fn(2)1502 4293 y Fk(\014)1502 4352 y(\014)1502 4412 y(\014)1502 4472 y(\014)1619 4437 y Fq(\024)83 b Fp(C)7 b(h)1912 4396 y Fo(\000)p Fn(3)2023 4302 y Fk(Z)2079 4527 y Fo(j)p Fm(x)p Fo(j\024)p Fm(h)2255 4508 y Fj(2)2308 4437 y Fq(j)p Fp(x)p Fq(j)2419 4396 y Fo(\000)p Fn(3)p Fm(=)p Fn(2)2601 4437 y Fp(dx)22 b Fv(+)g Fp(C)7 b(h)2960 4396 y Fo(\000)p Fn(2)3071 4302 y Fk(Z)3126 4527 y Fo(j)p Fm(x)p Fo(j\024)p Fm(r)3315 4437 y Fq(j)p Fp(x)p Fq(j)3426 4396 y Fo(\000)p Fn(2)3536 4437 y Fp(dx)1619 4661 y Fq(\024)83 b Fp(C)7 b(h)1912 4620 y Fo(\000)p Fn(2)2007 4661 y Fp(r)p 3469 4881 4 66 v 3473 4818 59 4 v 3473 4881 V 3531 4881 4 66 v 120 5076 a Fv(No)m(w)32 b(w)m(e)h(pro)s(ceed)f(to)g(construct)g(e\013ectiv) m(ely)h(a)e(go)s(o)s(d)g(trial)e(densit)m(y)k(matrix)d(w)m(e)i(shall)f (call)120 5196 y(\000.)43 b(This)33 b(is)f(the)h(con)m(ten)m(t)h(of)e (our)h(next)g(Lemma:)1781 5712 y(38)p eop %%Page: 39 39 39 38 bop 120 407 a Ft(Lemma)37 b(13)h(\(Choice)e(of)h(a)h(go)s(o)s(d)g (trial)e(densit)m(y)h(matrix\).)47 b Fr(Given)32 b(a)h(nucle)-5 b(ar)33 b(c)-5 b(on-)120 527 y(\014gur)g(ation)45 b Fq(R)p Fp(;)17 b Fq(Z)53 b Fr(such)45 b(that)h Fv(1)h Fq(\024)g Fp(Z)1552 542 y Fm(j)1635 527 y Fq(\024)g Fp(A)f Fr(and)e Fp(\016)2120 542 y Fm(j)2204 527 y Fq(\025)j Fp(\026)e Fr(for)g(al)5 b(l)45 b Fp(j)6 b Fr(,)47 b(it)f(is)f(p)-5 b(ossible)44 b(to)120 648 y(c)-5 b(onstruct)35 b(a)g(density)g(matrix)f Fv(\000)h Fr(with)g(the)g(fol)5 b(lowing)33 b(pr)-5 b(op)g(erties:)266 768 y(-.i)336 895 y Fk(\014)336 955 y(\014)336 1015 y(\014)336 1074 y(\014)369 1040 y Fp(\032)419 1055 y Fn(\000)468 1040 y Fv(\()p Fp(x)p Fv(\))22 b Fq(\000)828 972 y Fv(1)p 731 1017 243 4 v 731 1108 a(6)p Fp(\031)839 1079 y Fn(2)878 1108 y Fp(h)934 1079 y Fn(3)983 1040 y Fv(\010)1053 998 y Fn(TF)1053 1064 y Fo(R)p Fm(;)p Fo(Z)1194 1040 y Fv(\()p Fp(x)p Fv(\))1325 998 y Fn(3)p Fm(=)p Fn(2)1435 895 y Fk(\014)1435 955 y(\014)1435 1015 y(\014)1435 1074 y(\014)1496 1040 y Fq(\024)1601 835 y Fk(8)1601 925 y(<)1601 1104 y(:)1731 919 y Fp(C)7 b(h)1864 883 y Fo(\000)p Fn(3)1959 919 y Fp(l)r Fv(\()p Fp(x)p Fv(\))2121 883 y Fo(\000)p Fn(3)p Fm(=)p Fn(2)2286 919 y Fv(;)83 b Fp(l)r Fv(\()p Fp(x)p Fv(\))29 b Fq(\024)f Fp(h)2748 883 y Fo(\000)p Fn(2)1802 1040 y Fp(C)7 b(h)1935 1003 y Fo(\000)p Fn(2)2029 1040 y Fp(l)r Fv(\()p Fp(x)p Fv(\))2191 1003 y Fo(\000)p Fn(2)2286 1040 y Fv(;)83 b Fp(h)2452 1003 y Fn(2)2520 1040 y Fq(\024)28 b Fp(l)r Fv(\()p Fp(x)p Fv(\))g Fq(\024)g Fp(\026)1802 1160 y(C)7 b(h)1935 1124 y Fo(\000)p Fn(2)2029 1160 y Fp(l)r Fv(\()p Fp(x)p Fv(\))2191 1124 y Fo(\000)p Fn(6)2286 1160 y Fv(;)83 b Fp(\026)28 b Fq(\024)g Fp(l)r Fv(\()p Fp(x)p Fv(\))g Fq(\024)g Fp(h)2939 1124 y Fo(\000)p Fn(1)3318 1040 y Fv(\(172\))266 1326 y Fr(-.ii)266 1446 y(We)35 b(c)-5 b(an)35 b(write)f Fp(\032)920 1461 y Fn(\000)997 1446 y Fv(=)1100 1372 y Fk(P)1205 1398 y Fm(M)1205 1475 y(j)t Fn(=1)1349 1446 y Fp(G)1426 1461 y Fm(j)1462 1446 y Fr(,)h(wher)-5 b(e)34 b Fv(supp)18 b Fp(G)2097 1461 y Fm(j)2161 1446 y Fq(\032)28 b Fv(supp)18 b Fp(\030)2527 1461 y Fm(j)2598 1446 y Fr(and)410 1572 y Fk(8)410 1662 y(>)410 1691 y(>)410 1721 y(>)410 1751 y(>)410 1781 y(<)410 1960 y(>)410 1990 y(>)410 2020 y(>)410 2050 y(>)410 2080 y(:)1065 1559 y(R)1148 1640 y Fp(G)1225 1655 y Fm(j)1345 1640 y Fv(=)1590 1600 y Fn(1)p 1513 1617 189 4 v 1513 1674 a(6)p Fm(\031)1591 1655 y Fj(2)1626 1674 y Fm(h)1667 1655 y Fj(3)1728 1559 y Fk(R)1811 1640 y Fp(\030)1859 1604 y Fn(2)1854 1664 y Fm(j)1898 1640 y Fv(\(\010)2006 1604 y Fn(TF)2006 1665 y Fo(R)p Fm(;)p Fo(Z)2146 1640 y Fv(\))2184 1604 y Fn(3)p Fm(=)p Fn(2)1503 1773 y Fv(+)1618 1733 y Fm(K)5 b(s)1716 1710 y Fj(2)p 1589 1750 V 1589 1807 a Fn(6)p Fm(\031)1667 1788 y Fj(2)1702 1807 y Fm(h)1743 1788 y Fj(3)1804 1692 y Fk(R)1870 1773 y Fv(\(\010)1978 1736 y Fn(TF)1978 1798 y Fo(R)p Fm(;)p Fo(Z)2119 1773 y Fv(\))2157 1736 y Fn(3)p Fm(=)p Fn(2)2267 1773 y Fv(\001)p Fp(\030)2396 1736 y Fn(2)2391 1797 y Fm(j)540 1937 y Fk(R)623 2017 y Fp(G)700 2032 y Fm(j)736 2017 y Fv(\()p Fp(x)829 2032 y Fm(k)895 2017 y Fq(\000)22 b Fv(\()p Fp(R)1106 2032 y Fm(j)1143 2017 y Fv(\))1181 2032 y Fm(k)1224 2017 y Fv(\))83 b(=)1590 1978 y Fn(1)p 1513 1994 V 1513 2051 a(6)p Fm(\031)1591 2032 y Fj(2)1626 2051 y Fm(h)1667 2032 y Fj(3)1728 1937 y Fk(R)1811 2017 y Fp(\030)1859 1981 y Fn(2)1854 2042 y Fm(j)1898 2017 y Fv(\(\010)2006 1981 y Fn(TF)2006 2042 y Fo(R)p Fm(;)p Fo(Z)2146 2017 y Fv(\))2184 1981 y Fn(3)p Fm(=)p Fn(2)2294 2017 y Fv(\()p Fp(x)2387 2032 y Fm(k)2452 2017 y Fq(\000)23 b Fv(\()p Fp(R)2664 2032 y Fm(j)2701 2017 y Fv(\))2739 2032 y Fm(k)2781 2017 y Fv(\))1503 2150 y(+)1618 2111 y Fm(K)5 b(s)1716 2087 y Fj(2)p 1589 2127 V 1589 2184 a Fn(6)p Fm(\031)1667 2165 y Fj(2)1702 2184 y Fm(h)1743 2165 y Fj(3)1804 2070 y Fk(R)1870 2150 y Fv(\(\010)1978 2114 y Fn(TF)1978 2175 y Fo(R)p Fm(;)p Fo(Z)2119 2150 y Fv(\))2157 2114 y Fn(3)p Fm(=)p Fn(2)2267 2150 y Fv(\001)p Fp(\030)2396 2114 y Fn(2)2391 2174 y Fm(j)2435 2150 y Fv(\()p Fp(x)2528 2165 y Fm(k)2593 2150 y Fq(\000)23 b Fv(\()p Fp(R)2805 2165 y Fm(j)2841 2150 y Fv(\))2879 2165 y Fm(k)2922 2150 y Fv(\))3318 1896 y(\(173\))120 2322 y Fq(8)p Fp(j)34 b Fv(=)28 b(1)p Fp(;)17 b(:)g(:)g(:)32 b(;)17 b(M)27 b Fq(8)p Fp(k)k Fv(=)c(1)p Fp(;)17 b Fv(2)p Fp(;)g Fv(3)266 2443 y Fr(\()p Fq(f)p Fp(\030)404 2407 y Fn(2)399 2467 y Fm(j)443 2443 y Fq(g)493 2407 y Fm(M)493 2467 y(j)t Fn(=1)657 2443 y Fr(is)38 b(the)f(p)-5 b(artition)38 b(of)f(unity)i(c)-5 b(onstructe)g(d)38 b(at)f(our)h(\014rst)g(step)g (of)f(the)h(pr)-5 b(o)g(of)37 b(of)120 2563 y(The)-5 b(or)g(em)34 b(6)g(and)h Fp(K)42 b Fr(is)34 b(the)h(c)-5 b(onstant)35 b(app)-5 b(e)g(aring)33 b(in)i(L)-5 b(emma)34 b(10\).)266 2684 y(-.iii)185 2929 y Fv(T)-8 b(r)17 b([\()p Fq(\000)p Fp(h)500 2888 y Fn(2)540 2929 y Fv(\001)23 b Fq(\000)f Fv(\010)813 2888 y Fn(TF)813 2953 y Fo(R)p Fm(;)p Fo(Z)954 2929 y Fv(\)\000])28 b Fq(\024)g(\000)1422 2861 y Fv(1)p 1300 2906 292 4 v 1300 2997 a(15)p Fp(\031)1457 2968 y Fn(2)1496 2997 y Fp(h)1552 2968 y Fn(3)1618 2793 y Fk(Z)1718 2929 y Fv(\(\010)1826 2888 y Fn(TF)1826 2953 y Fo(R)p Fm(;)p Fo(Z)1966 2929 y Fv(\))2004 2888 y Fn(5)p Fm(=)p Fn(2)2136 2929 y Fv(+)2244 2861 y(1)p 2244 2906 49 4 v 2244 2997 a(8)2303 2929 y Fp(h)2359 2888 y Fo(\000)p Fn(2)2505 2804 y Fm(M)2470 2834 y Fk(X)2481 3044 y Fm(j)t Fn(=1)2631 2929 y Fp(Z)2705 2888 y Fn(2)2698 2953 y Fm(j)2766 2929 y Fv(+)22 b Fp(C)7 b(h)2997 2888 y Fo(\000)p Fn(1)p Fo(\000)p Fn(4)p Fm(=)p Fn(9)3318 2929 y Fv(\(174\))120 3201 y Fr(Pr)-5 b(o)g(of.)48 b Fv(W)-8 b(e)38 b(shall)f(assume)h(from)e (the)i(outset)h(that)e Fp(s)g Fv(and)h Fp(r)i Fv(are)e(c)m(hosen)h(in)e (suc)m(h)i(a)f(w)m(a)m(y)120 3322 y(that)1297 3336 y Fk(\032)1415 3416 y Fp(s)28 b Fv(=)83 b Fp(o)p Fv(\()p Fp(h)1789 3380 y Fn(1)p Fm(=)p Fn(2+)p Fm(\017)1982 3416 y Fv(\))17 b Fq(8)p Fp(\017)28 b(>)g Fv(0)1413 3537 y Fp(r)i Fq(\024)84 b Fp(C)7 b(h)1781 3500 y Fm(\026)1828 3537 y Fv(;)49 b Fp(\026)27 b(>)h Fv(0)120 3686 y(Then,)51 b(assuming)46 b(this)g(is)g(satis\014ed,)51 b(w)m(e)c(shall)e(\014rst)i (construct)h(\000)e(satisfying)g(ii.)83 b(The)120 3806 y(condition)40 b(i.)71 b(w)m(ould)42 b(then)g(b)s(e)g(ful\014lled)e (more)h(or)g(less)i(automatically)-8 b(,)40 b(and)i(\014nally)f(w)m(e) 120 3927 y(shall)31 b(mak)m(e)i(concrete)h(c)m(hoices)f(of)f(these)i (parameters)f(trying)e(to)i(satisfy)f(condition)f(iii.)266 4047 y(W)-8 b(e)34 b(\014rst)g(observ)m(e)h(the)f(follo)m(wing)d(fact:) 45 b(assume)34 b(w)m(e)g(are)g(giv)m(en)f(a)g(collection)f Fq(f)p Fp(\015)3318 4062 y Fn(1)3356 4047 y Fp(;)17 b(\015)3451 4062 y Fn(2)3490 4047 y Fq(g)120 4167 y Fv(of)32 b(densit)m(y)i (matrices)e(and)h Fq(f)p Fp(\022)1247 4131 y Fn(2)1244 4192 y Fm(j)1287 4167 y Fq(g)1337 4131 y Fm(N)1337 4192 y(j)t Fn(=1)1496 4167 y Fv(a)g(partition)e(of)h(unit)m(y)-8 b(.)45 b(Then)34 b(w)m(e)g(can)f(mak)m(e)g(another)120 4307 y(densit)m(y)g(matrix)d(out)i(of)g(them)f(b)m(y)i(the)f(recip)s(e) g(\000)c(=)2082 4232 y Fk(P)2188 4259 y Fm(N)2244 4268 y Fj(1)2188 4336 y Fm(j)t Fn(=1)2314 4307 y Fv(\()p Fp(\015)2403 4322 y Fn(1)2442 4307 y Fv(\))2480 4322 y Fm(\022)2513 4332 y Fi(j)2571 4307 y Fv(+)2667 4232 y Fk(P)2772 4259 y Fm(M)2772 4336 y(j)t Fn(=)p Fm(N)2916 4345 y Fj(1)2950 4336 y Fn(+1)3045 4307 y Fv(\()p Fp(\015)3134 4322 y Fn(2)3173 4307 y Fv(\))3211 4322 y Fm(\022)3244 4332 y Fi(j)3280 4307 y Fv(.)43 b(No)m(w)120 4427 y(w)m(e)34 b(use)f(this)f(with)299 4507 y Fk(8)299 4597 y(<)299 4776 y(:)429 4577 y Fp(\015)480 4592 y Fn(1)602 4577 y Fv(:=)83 b Fp(\037)849 4592 y Fn(\()p Fo(\0001)p Fm(;)p Fn(0])1080 4577 y Fv(\()p Fq(\000)p Fp(h)1251 4541 y Fn(2)1291 4577 y Fv(\001)23 b Fq(\000)f Fv(\010)1564 4541 y Fn(TF)1564 4602 y Fo(R)p Fm(;)p Fo(Z)1705 4577 y Fv(\))429 4829 y Fp(\015)480 4844 y Fn(2)602 4829 y Fv(:=)83 b(\(2)p Fp(\031)t Fv(\))972 4793 y Fo(\000)p Fn(3)1082 4749 y Fk(R)1129 4864 y Fm(h)1170 4845 y Fj(2)1204 4864 y Fm(p)1240 4845 y Fj(2)1275 4864 y Fo(\000)p Fm(W)10 b Fn(\()p Fm(q)r Fn(\))p Fo(\024)p Fn(0)1606 4829 y Fp(\031)1661 4844 y Fm(p;q)r Fn(;)p Fm(s)1823 4829 y Fp(dp)17 b(dp)p Fv(;)82 b Fp(W)41 b Fv(=)28 b(\010)2456 4793 y Fn(TF)2456 4854 y Fo(R)p Fm(;)p Fo(Z)2618 4829 y Fv(+)2716 4755 y Fk(P)2822 4781 y Fm(M)2822 4858 y(j)t Fn(=1)2965 4829 y Fp(w)3035 4844 y Fm(j)3318 4711 y Fv(\(175\))120 5020 y(where)38 b(in)d(\(175\))h(the)h(functions)f Fp(w)1448 5035 y Fm(j)1484 5020 y Fv(;)d Fp(j)41 b Fv(=)34 b(1)p Fp(;)17 b(:)g(:)g(:)32 b(;)17 b(M)47 b Fv(are)36 b(c)m(hosen)i(sub)5 b(ject)38 b(to)e(the)h(condi-)120 5141 y(tions:)740 5152 y Fk(8)740 5241 y(<)740 5421 y(:)870 5236 y Fv(supp)17 b Fp(w)1157 5251 y Fm(j)1277 5236 y Fq(\032)83 b Fp(A)22 b Fv(+)g Fp(R)1704 5251 y Fm(j)1741 5236 y Fv(;)83 b Fp(A)28 b Fv(=)2056 5155 y Fk(\010)2114 5236 y Fp(x)g Fv(:)2262 5192 y Fm(\026)p 2262 5213 43 4 v 2266 5270 a Fn(9)2342 5236 y Fq(\024)g(j)p Fp(x)p Fq(j)f(\024)2700 5192 y Fn(2)p Fm(\026)p 2700 5213 78 4 v 2722 5270 a Fn(9)2788 5155 y Fk(\011)913 5477 y Fq(k)p Fp(w)1033 5492 y Fm(j)1069 5477 y Fq(k)1119 5492 y Fo(1)1277 5477 y Fq(\024)83 b Fp(C)7 b(h)1570 5440 y Fn(1+)p Fm(\033)1707 5477 y Fv(;)117 b Fp(\033)32 b(>)27 b Fv(0)33 b(absolute)f(constan)m(t) 1781 5712 y(39)p eop %%Page: 40 40 40 39 bop 120 407 a Fv(The)34 b(partition)c(of)i(unit)m(y)h(w)m(e)g (shall)f(use)h(is)f Fq(f)p Fv(\007)1857 371 y Fn(2)1857 431 y Fm(j)1896 407 y Fq(g)1946 371 y Fn(2)p Fm(M)1946 431 y(j)t Fn(=1)2073 407 y Fv(,)g(so)h(that)g(w)m(e)g(tak)m(e)1205 720 y(\000)27 b(:=)1459 595 y Fm(M)1424 625 y Fk(X)1435 835 y Fm(j)t Fn(=1)1568 720 y Fv(\()p Fp(\015)1657 735 y Fn(1)1696 720 y Fv(\))1734 735 y Fn(\007)1789 745 y Fi(j)1848 720 y Fv(+)2017 595 y Fn(2)p Fm(M)2000 625 y Fk(X)1946 836 y Fm(j)t Fn(=)p Fm(M)7 b Fn(+1)2198 720 y Fv(\()p Fp(\015)2287 735 y Fn(2)2326 720 y Fv(\))2364 735 y Fn(\007)2419 745 y Fi(j)3318 720 y Fv(\(176\))266 1142 y(and)33 b(then)238 1417 y Fp(\032)288 1432 y Fn(\000)364 1417 y Fv(=)502 1293 y Fm(M)468 1323 y Fk(X)478 1533 y Fm(j)t Fn(=1)628 1417 y Fv(\007)704 1376 y Fn(2)704 1442 y Fm(j)743 1417 y Fp(\032)793 1432 y Fm(\015)829 1441 y Fj(1)891 1417 y Fv(+)1060 1293 y Fn(2)p Fm(M)1043 1323 y Fk(X)989 1534 y Fm(j)t Fn(=)p Fm(M)7 b Fn(+1)1258 1417 y Fv(\007)1334 1376 y Fn(2)1334 1442 y Fm(j)1373 1417 y Fp(\032)1423 1432 y Fm(\015)1459 1441 y Fj(2)1526 1417 y Fv(=)1664 1293 y Fm(M)1630 1323 y Fk(X)1640 1533 y Fm(j)t Fn(=1)1790 1337 y Fk(\000)1836 1417 y Fv(\007)1912 1376 y Fn(2)1912 1442 y Fm(j)1951 1417 y Fp(\032)2001 1432 y Fm(\015)2037 1441 y Fj(1)2099 1417 y Fv(+)22 b(\()p Fp(\030)2283 1376 y Fn(2)2278 1442 y Fm(j)2344 1417 y Fq(\000)g Fv(\007)2519 1376 y Fn(2)2519 1442 y Fm(j)2559 1417 y Fv(\))p Fp(\032)2647 1432 y Fm(\015)2683 1441 y Fj(2)2722 1337 y Fk(\001)2795 1417 y Fv(:=)2961 1293 y Fm(M)2926 1323 y Fk(X)2937 1533 y Fm(j)t Fn(=1)3086 1417 y Fp(G)3163 1432 y Fm(j)3318 1417 y Fv(\(177\))120 1722 y(Then)34 b(w)m(e)f(ha)m(v)m(e:)286 1817 y Fk(Z)402 1953 y Fp(G)479 1968 y Fm(j)543 1953 y Fv(=)286 2275 y(=)390 2140 y Fk(Z)506 2275 y Fv(\007)582 2234 y Fn(2)582 2300 y Fm(j)638 2135 y Fk(\022)818 2208 y Fv(1)p 721 2252 243 4 v 721 2344 a(6)p Fp(\031)829 2315 y Fn(2)868 2344 y Fp(h)924 2315 y Fn(3)974 2275 y Fv(\(\010)1082 2234 y Fn(TF)1082 2300 y Fo(R)p Fm(;)p Fo(Z)1222 2275 y Fv(\))1260 2234 y Fn(3)p Fm(=)p Fn(2)1392 2275 y Fv(+)22 b Fp(R)q Fv(\()p Fp(h;)17 b Fq(\001)p Fv(\))1769 2135 y Fk(\023)1864 2275 y Fv(+)2069 2208 y(1)p 1972 2252 V 1972 2344 a(6)p Fp(\031)2080 2315 y Fn(2)2119 2344 y Fp(h)2175 2315 y Fn(3)2241 2140 y Fk(Z)2341 2275 y Fv(\()p Fp(\030)2427 2234 y Fn(2)2422 2300 y Fm(j)2488 2275 y Fq(\000)23 b Fv(\007)2664 2234 y Fn(2)2664 2300 y Fm(j)2703 2275 y Fv(\))p Fp(h)2797 2290 y Fm(s)2856 2275 y Fq(\003)2928 2105 y Fk( )3007 2275 y Fv(\010)3077 2234 y Fn(TF)3077 2300 y Fo(R)p Fm(;)p Fo(Z)3239 2275 y Fv(+)3372 2151 y Fm(M)3337 2181 y Fk(X)3345 2393 y Fm(k)r Fn(=1)3498 2275 y Fp(w)3568 2290 y Fm(k)3610 2105 y Fk(!)3689 2122 y Fn(3)p Fm(=)p Fn(2)286 2639 y Fv(=)497 2572 y(1)p 400 2616 V 400 2708 a(6)p Fp(\031)508 2679 y Fn(2)547 2708 y Fp(h)603 2679 y Fn(3)669 2504 y Fk(Z)785 2639 y Fv(\007)861 2598 y Fn(2)861 2664 y Fm(j)900 2639 y Fv(\(\010)1008 2598 y Fn(TF)1008 2664 y Fo(R)p Fm(;)p Fo(Z)1149 2639 y Fv(\))1187 2598 y Fn(3)p Fm(=)p Fn(2)1319 2639 y Fv(+)1524 2572 y(1)p 1427 2616 V 1427 2708 a(6)p Fp(\031)1535 2679 y Fn(2)1574 2708 y Fp(h)1630 2679 y Fn(3)1696 2504 y Fk(Z)1812 2639 y Fp(h)1868 2654 y Fm(s)1927 2639 y Fq(\003)1999 2559 y Fk(\002)2041 2639 y Fv(\()p Fp(\030)2127 2598 y Fn(2)2122 2664 y Fm(j)2188 2639 y Fq(\000)f Fv(\007)2363 2598 y Fn(2)2363 2664 y Fm(j)2403 2639 y Fv(\))2441 2559 y Fk(\003)2499 2469 y( )2578 2639 y Fv(\010)2648 2598 y Fn(TF)2648 2664 y Fo(R)p Fm(;)p Fo(Z)2810 2639 y Fv(+)2943 2515 y Fm(M)2908 2545 y Fk(X)2916 2757 y Fm(k)r Fn(=1)3069 2639 y Fp(w)3139 2654 y Fm(k)3181 2469 y Fk(!)3260 2486 y Fn(3)p Fm(=)p Fn(2)3594 2639 y Fv(\(178\))286 2936 y(+)469 2868 y(1)p 372 2913 V 372 3004 a(6)p Fp(\031)480 2975 y Fn(2)519 3004 y Fp(h)575 2975 y Fn(3)641 2800 y Fk(Z)757 2936 y Fv(\007)833 2894 y Fn(2)833 2960 y Fm(j)873 2936 y Fv(\(\010)981 2894 y Fn(TF)981 2960 y Fo(R)p Fm(;)p Fo(Z)1121 2936 y Fv(\))1159 2894 y Fn(3)p Fm(=)p Fn(2)1291 2936 y Fv(+)1496 2868 y(1)p 1399 2913 V 1399 3004 a(6)p Fp(\031)1507 2975 y Fn(2)1546 3004 y Fp(h)1602 2975 y Fn(3)1668 2800 y Fk(Z)1785 2936 y Fp(h)1841 2951 y Fm(s)1900 2936 y Fq(\003)1972 2855 y Fk(\002)2013 2936 y Fv(\()p Fp(\030)2099 2894 y Fn(2)2094 2960 y Fm(j)2160 2936 y Fq(\000)h Fv(\007)2336 2894 y Fn(2)2336 2960 y Fm(j)2375 2936 y Fv(\))2413 2855 y Fk(\003)16 b(\000)2517 2936 y Fv(\010)2587 2894 y Fn(TF)2587 2960 y Fo(R)p Fm(;)p Fo(Z)2749 2936 y Fv(+)22 b Fp(w)2917 2951 y Fm(j)2954 2855 y Fk(\001)2999 2875 y Fn(3)p Fm(=)p Fn(2)3131 2936 y Fv(+)g Fq(O)s Fv(\()p Fp(h)3405 2894 y Fo(\000)p Fn(2)3500 2936 y Fp(r)s Fv(\))120 3196 y(\(w)m(e)28 b(ha)m(v)m(e)h(used)f(that)f(supp)18 b Fp(h)1213 3211 y Fm(s)1261 3196 y Fq(\003)11 b Fv([\()p Fp(\030)1435 3160 y Fn(2)1430 3220 y Fm(j)1485 3196 y Fq(\000)g Fv(\007)1649 3160 y Fn(2)1649 3220 y Fm(j)1689 3196 y Fv(\)])27 b(is)g(only)g(sligh) m(t)m(y)g(bigger)f(than)h(that)g(of)g Fp(\030)3259 3160 y Fn(2)3254 3220 y Fm(j)3309 3196 y Fq(\000)11 b Fv(\007)3473 3160 y Fn(2)3473 3220 y Fm(j)3513 3196 y Fv(,)120 3316 y(and)40 b(therefore)h(only)e Fp(w)1025 3331 y Fm(j)1102 3316 y Fv(con)m(tributes)h(in)g(the)g(sum)g(app)s(earing)f(in)g (\(178\).)65 b(No)m(w)41 b(w)m(e)g(use)120 3437 y(the)h(estimate)f(on)g (the)h(regularization)d(of)i(a)h(function)f(giv)m(en)h(b)m(y)g(Lemma)e (10)i(and)f(then)120 3557 y(w)m(e)46 b(can)g(write,)i(for)c(an)m(y)i Fp(s)j Fq(\024)h Fp(r)s(=)p Fv(6,)d(and)f(with)e Fp(B)2070 3572 y Fm(j)2156 3557 y Fv(=)49 b Fq(f)p Fp(x)g Fv(:)h Fp(r)s(=)p Fv(3)e Fq(\024)i(j)p Fp(x)30 b Fq(\000)i Fp(R)3128 3572 y Fm(j)3164 3557 y Fq(j)49 b(\024)h Fv(2)p Fp(r)s Fq(g)p Fv(,)120 3677 y Fp(C)190 3692 y Fm(j)254 3677 y Fv(=)28 b Fq(f)p Fp(x)f Fv(:)h Fp(d)p Fv(\()p Fp(x;)17 b(@)5 b(D)870 3692 y Fm(j)907 3677 y Fv(\))28 b Fq(\024)g Fv(2)p Fp(r)s Fq(g)p Fv(,)594 3885 y Fp(h)650 3900 y Fm(s)709 3885 y Fq(\003)22 b Fv([\()p Fp(\030)894 3844 y Fn(2)889 3910 y Fm(j)955 3885 y Fq(\000)g Fv(\007)1130 3844 y Fn(2)1130 3910 y Fm(j)1170 3885 y Fv(\)])84 b(=)f Fp(\030)1526 3844 y Fn(2)1521 3910 y Fm(j)1587 3885 y Fq(\000)23 b Fv(\007)1763 3844 y Fn(2)1763 3910 y Fm(j)1824 3885 y Fv(+)f Fp(K)7 b(s)2058 3844 y Fn(2)2098 3885 y Fv(\(\001)p Fp(\030)2265 3844 y Fn(2)2260 3910 y Fm(j)2326 3885 y Fq(\000)23 b Fv(\001\007)2583 3844 y Fn(2)2583 3910 y Fm(j)2622 3885 y Fv(\))f(+)g Fp(\024)2836 3900 y Fm(j)2873 3885 y Fv(\()p Fp(s;)17 b Fq(\001)p Fv(\))894 4040 y Fq(j)p Fp(\024)978 4055 y Fm(j)1014 4040 y Fv(\()p Fp(s;)g Fq(\001)p Fv(\))p Fq(j)82 b(\024)h Fp(C)7 b(s)1601 3999 y Fn(4)1641 4040 y Fp(r)1688 3999 y Fo(\000)p Fn(4)1798 3960 y Fk(\000)1844 4040 y Fp(\037)1905 4055 y Fm(B)1958 4065 y Fi(j)2017 4040 y Fv(+)22 b Fp(\037)2176 4055 y Fm(C)2226 4065 y Fi(j)2263 3960 y Fk(\001)3318 4040 y Fv(\(179\))120 4248 y(W)-8 b(e)33 b(can)g(estimate)720 4343 y Fk(\014)720 4403 y(\014)720 4463 y(\014)720 4523 y(\014)860 4420 y Fv(1)p 763 4465 V 763 4556 a(6)p Fp(\031)871 4527 y Fn(2)910 4556 y Fp(h)966 4527 y Fn(3)1032 4352 y Fk(Z)1149 4488 y Fp(\024)1205 4503 y Fm(j)1241 4488 y Fv(\()p Fp(s;)17 b Fq(\001)p Fv(\))1452 4407 y Fk(\000)1497 4488 y Fv(\010)1567 4447 y Fn(TF)1567 4512 y Fo(R)p Fm(;)p Fo(Z)1729 4488 y Fv(+)22 b Fp(w)1897 4503 y Fm(j)1934 4407 y Fk(\001)1979 4428 y Fn(3)p Fm(=)p Fn(2)2089 4343 y Fk(\014)2089 4403 y(\014)2089 4463 y(\014)2089 4523 y(\014)2150 4488 y Fq(\024)752 4751 y(\024)29 b Fp(C)7 b(h)991 4710 y Fo(\000)p Fn(3)1085 4751 y Fp(s)1131 4710 y Fn(4)1170 4751 y Fp(r)1217 4710 y Fo(\000)p Fn(4)1328 4616 y Fk(Z)1383 4841 y Fm(B)1436 4851 y Fi(j)1490 4751 y Fq(j)p Fp(x)p Fq(j)1601 4710 y Fo(\000)p Fn(3)p Fm(=)p Fn(2)1782 4751 y Fp(dx)22 b Fv(+)g Fp(C)7 b(h)2141 4710 y Fo(\000)p Fn(3)2236 4751 y Fp(s)2282 4710 y Fn(4)2321 4751 y Fp(r)2368 4710 y Fo(\000)p Fn(4)2479 4616 y Fk(Z)2534 4841 y Fm(C)2584 4851 y Fi(j)2617 4841 y Fo(\000)p Fm(R)2725 4851 y Fi(j)2778 4751 y Fq(j)p Fp(x)p Fq(j)2889 4710 y Fo(\000)p Fn(6)3000 4751 y Fp(dx)752 5030 y Fq(\024)29 b Fp(C)7 b(h)991 4989 y Fo(\000)p Fn(3)1085 5030 y Fp(s)1131 4989 y Fn(4)1170 5030 y Fp(r)1217 4989 y Fo(\000)p Fn(5)p Fm(=)p Fn(2)1404 5030 y Fv(+)22 b Fp(C)7 b(h)1635 4989 y Fo(\000)p Fn(3)1730 5030 y Fp(s)1776 4989 y Fn(4)1815 5030 y Fp(r)1862 4989 y Fo(\000)p Fn(4)1973 4894 y Fk(Z)2028 5120 y Fm(C)2078 5130 y Fi(j)2111 5120 y Fo(\000)p Fm(R)2219 5130 y Fi(j)2272 5030 y Fq(j)p Fp(x)p Fq(j)2383 4989 y Fo(\000)p Fn(6)2494 5030 y Fp(dx)120 5315 y Fv(T)-8 b(o)46 b(estimate)f(the)h(second)h(in)m(tegral)d(ab)s(o)m(v)m(e,)50 b(w)m(e)d(write)e Fp(F)14 b Fv(\()p Fp(r)s Fv(\))49 b(:=)2699 5235 y Fk(R)2747 5350 y Fm(C)2797 5360 y Fi(j)2829 5350 y Fo(\000)p Fm(R)2937 5360 y Fi(j)2991 5315 y Fq(j)p Fp(x)p Fq(j)3102 5279 y Fo(\000)p Fn(6)3212 5315 y Fp(dx)p Fv(.)83 b(Its)120 5458 y(deriv)-5 b(ativ)m(e)30 b(is)g Fp(F)738 5422 y Fo(0)761 5458 y Fv(\()p Fp(r)s Fv(\))d(=)1015 5378 y Fk(R)1062 5493 y Fo(f)p Fm(x)p Fn(:)p Fm(d)p Fn(\()p Fm(x;@)t Fn(\()p Fm(D)1406 5503 y Fi(j)1439 5493 y Fo(\000)p Fm(R)1547 5503 y Fi(j)1579 5493 y Fn(\)=2)p Fm(r)r Fn(\))p Fo(g)1814 5458 y Fq(j)p Fp(x)p Fq(j)1925 5422 y Fo(\000)p Fn(6)2036 5458 y Fp(d\033)2142 5473 y Fm(r)2180 5458 y Fv(,)k(where)h Fp(d\033)2624 5473 y Fm(r)2692 5458 y Fv(is)e(the)h(2-dimensional)1781 5712 y(40)p eop %%Page: 41 41 41 40 bop 120 407 a Fv(measure)31 b(induced)h(b)m(y)g(Leb)s(esgue's)h (measure)e(in)m(to)f Fq(f)p Fp(x)e Fv(:)g Fp(d)p Fv(\()p Fp(x;)17 b(@)5 b Fv(\()p Fp(D)2605 422 y Fm(j)2661 407 y Fq(\000)20 b Fp(R)2832 422 y Fm(j)2868 407 y Fv(\))28 b(=)g(2)p Fp(r)s Fv(\))p Fq(g)p Fv(.)42 b(Then,)120 540 y Fp(F)197 504 y Fo(0)220 540 y Fv(\()p Fp(r)s Fv(\))29 b(=)h Fq(O)s Fv(\(1\))k(if)f Fp(r)f Fq(\025)f Fv(1)p Fp(=)p Fv(2.)46 b(Since)35 b Fp(F)14 b Fv(\()p Fp(r)s Fv(\))29 b(=)1806 459 y Fk(R)1872 486 y Fn(2)p Fm(r)1853 574 y Fn(0)1962 540 y Fp(F)2039 504 y Fo(0)2062 540 y Fv(\()p Fp(r)s Fv(\))17 b Fp(dr)s Fv(,)33 b(w)m(e)i(obtain)d Fp(F)14 b Fv(\()p Fp(r)s Fv(\))30 b(=)f Fq(O)s Fv(\()p Fp(r)s Fv(\))34 b(and)120 660 y(the)f(estimate)835 768 y Fk(\014)835 828 y(\014)835 888 y(\014)835 947 y(\014)976 845 y Fv(1)p 879 890 243 4 v 879 981 a(6)p Fp(\031)987 952 y Fn(2)1026 981 y Fp(h)1082 952 y Fn(3)1148 777 y Fk(Z)1264 913 y Fp(\024)1320 928 y Fm(j)1357 913 y Fv(\()p Fp(s;)17 b Fq(\001)p Fv(\))1568 832 y Fk(\000)1612 913 y Fv(\010)1682 871 y Fn(TF)1682 937 y Fo(R)p Fm(;)p Fo(Z)1845 913 y Fv(+)22 b Fp(w)2013 928 y Fm(j)2049 832 y Fk(\001)2095 852 y Fn(3)p Fm(=)p Fn(2)2205 768 y Fk(\014)2205 828 y(\014)2205 888 y(\014)2205 947 y(\014)2266 913 y Fq(\024)28 b Fp(C)7 b(h)2504 871 y Fo(\000)p Fn(3)2598 913 y Fp(s)2644 871 y Fn(4)2684 913 y Fp(r)2731 871 y Fo(\000)p Fn(3)3318 913 y Fv(\(180\))120 1190 y(and)33 b(therefore)g(the)g(estimate)f (\(178\))f(can)i(also)f(b)s(e)h(written)f(as)461 1333 y Fk(Z)577 1469 y Fp(G)654 1484 y Fm(j)774 1469 y Fv(=)1040 1402 y(1)p 943 1446 V 943 1537 a(6)p Fp(\031)1051 1509 y Fn(2)1090 1537 y Fp(h)1146 1509 y Fn(3)1212 1333 y Fk(Z)1328 1469 y Fp(\030)1376 1428 y Fn(2)1371 1494 y Fm(j)1432 1388 y Fk(\000)1477 1469 y Fv(\010)1547 1428 y Fn(TF)1547 1494 y Fo(R)p Fm(;)p Fo(Z)1710 1469 y Fv(+)22 b Fp(w)1878 1484 y Fm(j)1914 1388 y Fk(\001)1960 1409 y Fn(3)p Fm(=)p Fn(2)2092 1469 y Fv(+)2234 1402 y Fp(K)7 b(s)2370 1365 y Fn(2)p 2200 1446 V 2200 1537 a Fv(6)p Fp(\031)2308 1509 y Fn(2)2347 1537 y Fp(h)2403 1509 y Fn(3)2469 1333 y Fk(Z)2586 1469 y Fv(\001)p Fp(\030)2715 1428 y Fn(2)2710 1494 y Fm(j)2770 1388 y Fk(\000)2816 1469 y Fv(\010)2886 1428 y Fn(TF)2886 1494 y Fo(R)p Fm(;)p Fo(Z)3026 1388 y Fk(\001)3072 1409 y Fn(3)p Fm(=)p Fn(2)933 1734 y Fq(\000)1054 1667 y Fp(K)g(s)1190 1631 y Fn(2)p 1020 1711 V 1020 1803 a Fv(6)p Fp(\031)1128 1774 y Fn(2)1167 1803 y Fp(h)1223 1774 y Fn(3)1290 1599 y Fk(Z)1406 1734 y Fv(\001\007)1563 1693 y Fn(2)1563 1759 y Fm(j)1619 1654 y Fk(\000)1665 1734 y Fv(\010)1735 1693 y Fn(TF)1735 1759 y Fo(R)p Fm(;)p Fo(Z)1875 1654 y Fk(\001)1921 1674 y Fn(3)p Fm(=)p Fn(2)2053 1734 y Fv(+)22 b Fq(O)s Fv(\()p Fp(h)2327 1693 y Fo(\000)p Fn(2)2421 1734 y Fp(r)j Fv(+)d Fp(h)2644 1693 y Fo(\000)p Fn(3)2739 1734 y Fp(s)2785 1693 y Fn(4)2824 1734 y Fp(r)2871 1693 y Fo(\000)p Fn(3)2965 1734 y Fv(\))315 b(\(181\))120 2014 y(Since)27 b(\007)445 2029 y Fm(j)509 2014 y Fv(are)g(c)m(hosen)i(suc)m(h)g(that)1394 1934 y Fk(R)1477 2014 y Fv(\001\007)1634 1978 y Fn(2)1634 2039 y Fm(j)1674 2014 y Fq(j)11 b(\001)g(j)1780 1978 y Fo(\000)p Fn(3)p Fm(=)p Fn(2)1972 2014 y Fv(=)2076 1934 y Fk(R)2159 2014 y Fv(\001\007)2316 1978 y Fn(2)2316 2039 y Fm(j)2356 2014 y Fq(j)g(\001)g(j)2462 1978 y Fo(\000)p Fn(1)p Fm(=)p Fn(2)2654 2014 y Fv(=)28 b(0)f(and)g(b)m(y)h(Lemma)e(3) 120 2134 y(w)m(e)34 b(can)e(write)636 2374 y(\010)706 2333 y Fn(TF)706 2399 y Fo(R)p Fm(;)p Fo(Z)846 2374 y Fv(\()p Fp(x)p Fv(\))977 2333 y Fn(3)p Fm(=)p Fn(2)1115 2374 y Fv(=)c Fp(Z)1293 2323 y Fn(3)p Fm(=)p Fn(2)1286 2399 y Fm(j)1403 2374 y Fq(j)p Fp(x)22 b Fq(\000)h Fp(R)1682 2389 y Fm(j)1719 2374 y Fq(j)1747 2333 y Fo(\000)p Fn(3)p Fm(=)p Fn(2)1933 2374 y Fq(\000)2043 2307 y Fv(3)p 2043 2351 49 4 v 2043 2442 a(2)2102 2374 y Fp(k)2153 2389 y Fm(j)2189 2374 y Fq(j)p Fp(x)f Fq(\000)h Fp(R)2468 2389 y Fm(j)2505 2374 y Fq(j)2533 2333 y Fo(\000)p Fn(1)p Fm(=)p Fn(2)2719 2374 y Fv(+)f Fq(O)s Fv(\(1\))120 2619 y(w)m(e)34 b(obtain)d(the)i(estimate)431 2729 y Fk(\014)431 2789 y(\014)431 2849 y(\014)431 2909 y(\014)508 2806 y Fp(K)7 b(s)644 2770 y Fn(2)p 474 2851 243 4 v 474 2942 a Fv(6)p Fp(\031)582 2913 y Fn(2)622 2942 y Fp(h)678 2913 y Fn(3)744 2738 y Fk(Z)860 2874 y Fv(\001\007)1017 2833 y Fn(2)1017 2898 y Fm(j)1073 2793 y Fk(\000)1119 2874 y Fv(\010)1189 2833 y Fn(TF)1189 2898 y Fo(R)p Fm(;)p Fo(Z)1329 2793 y Fk(\001)1375 2813 y Fn(3)p Fm(=)p Fn(2)1485 2729 y Fk(\014)1485 2789 y(\014)1485 2849 y(\014)1485 2909 y(\014)1546 2874 y Fq(\024)28 b Fp(C)7 b Fq(k)p Fv(\001\007)1935 2833 y Fn(2)1935 2898 y Fm(j)1974 2874 y Fq(k)2024 2889 y Fo(1)2099 2874 y Fp(s)2145 2833 y Fn(2)2184 2874 y Fp(h)2240 2833 y Fo(\000)p Fn(3)2351 2738 y Fk(Z)2407 2964 y Fo(j)p Fm(x)p Fo(j\024)p Fn(2)p Fm(r)2630 2874 y Fp(dx)28 b Fq(\024)g Fp(C)7 b(r)s(s)3039 2833 y Fn(2)3078 2874 y Fp(h)3134 2833 y Fo(\000)p Fn(3)120 3157 y Fv(Let)33 b(us)g(de\014ne)979 3470 y Fp(W)1071 3485 y Fm(j)1135 3470 y Fv(:=)1266 3389 y Fk(\000)1311 3470 y Fv(\010)1381 3428 y Fn(TF)1381 3494 y Fo(R)p Fm(;)p Fo(Z)1522 3389 y Fk(\001)1568 3409 y Fn(3)p Fm(=)p Fn(2)1694 3269 y Fk(2)1694 3449 y(4)1761 3299 y( )1839 3470 y Fv(1)22 b(+)2070 3402 y Fp(w)2140 3417 y Fm(j)p 2018 3447 211 4 v 2018 3538 a Fv(\010)2088 3504 y Fn(TF)2088 3565 y Fo(R)p Fm(;)p Fo(Z)2239 3299 y Fk(!)2318 3321 y Fn(3)p Fm(=)p Fn(2)2450 3470 y Fq(\000)g Fv(1)2598 3269 y Fk(3)2598 3449 y(5)3318 3470 y Fv(\(182\))120 3801 y(Then)34 b(is)e(easy)h(to)g (see)g(that)727 3997 y(1)p 630 4041 243 4 v 630 4132 a(6)p Fp(\031)738 4103 y Fn(2)777 4132 y Fp(h)833 4103 y Fn(3)899 3928 y Fk(Z)1015 4064 y Fp(\030)1063 4023 y Fn(2)1058 4089 y Fm(j)1119 3953 y Fk(h)1166 3983 y(\000)1212 4064 y Fv(\010)1282 4023 y Fn(TF)1282 4089 y Fo(R)p Fm(;)p Fo(Z)1444 4064 y Fv(+)22 b Fp(w)1612 4079 y Fm(j)1648 3983 y Fk(\001)1694 4004 y Fn(3)p Fm(=)p Fn(2)1826 4064 y Fq(\000)1926 3983 y Fk(\000)1972 4064 y Fv(\010)2042 4023 y Fn(TF)2042 4089 y Fo(R)p Fm(;)p Fo(Z)2182 3983 y Fk(\001)2228 4004 y Fn(3)p Fm(=)p Fn(2)2338 3953 y Fk(i)2412 4064 y Fv(=)2623 3997 y(1)p 2526 4041 V 2526 4132 a(6)p Fp(\031)2634 4103 y Fn(2)2673 4132 y Fp(h)2729 4103 y Fn(3)2795 3928 y Fk(Z)2911 4064 y Fp(W)3003 4079 y Fm(j)120 4332 y Fv(By)39 b(the)g(conditions)e(imp)s(osed)h(on)g Fp(w)1525 4347 y Fm(j)1600 4332 y Fv(w)m(e)h(ha)m(v)m(e)h(that)e Fp(W)2289 4347 y Fm(j)2363 4332 y Fq(\030)g Fp(w)2548 4347 y Fm(j)2585 4332 y Fv(.)61 b(Moreo)m(v)m(er,)41 b(eq.)62 b(\(182\))120 4453 y(de\014nes)41 b(implicitly)35 b Fp(w)960 4468 y Fm(j)1036 4453 y Fv(once)40 b Fp(W)1357 4468 y Fm(j)1433 4453 y Fv(are)g(giv)m(en.)65 b(So,)41 b(in)e(terms)g(of)g(the)h(functions)g Fp(W)3300 4468 y Fm(j)3336 4453 y Fv(,)i(the)120 4573 y(\014rst)33 b(condition)e(on)i (ii.)41 b(of)32 b(Lemma)g(13)g(is)982 4749 y(1)p 885 4794 V 885 4885 a(6)p Fp(\031)993 4856 y Fn(2)1032 4885 y Fp(h)1088 4856 y Fn(3)1154 4681 y Fk(Z)1270 4817 y Fp(W)1362 4832 y Fm(j)1427 4817 y Fv(=)27 b Fq(O)s Fv(\()p Fp(h)1706 4776 y Fo(\000)p Fn(2)1801 4817 y Fp(r)e Fv(+)d Fp(h)2024 4776 y Fo(\000)p Fn(3)2118 4817 y Fp(s)2164 4776 y Fn(4)2203 4817 y Fp(r)2250 4776 y Fo(\000)p Fn(3)2367 4817 y Fv(+)g Fp(r)s(s)2558 4776 y Fn(2)2597 4817 y Fp(h)2653 4776 y Fo(\000)p Fn(3)2747 4817 y Fv(\))533 b(\(183\))1781 5712 y(41)p eop %%Page: 42 42 42 41 bop 120 407 a Fv(No)m(w)33 b(w)m(e)h(pro)s(ceed)f(to)f(compute) 1361 327 y Fk(R)1444 407 y Fp(G)1521 422 y Fm(j)1558 407 y Fv(\()p Fp(x)1651 422 y Fm(k)1716 407 y Fq(\000)22 b Fv(\()p Fp(R)1927 422 y Fm(j)1964 407 y Fv(\))2002 422 y Fm(k)2045 407 y Fv(\):)533 522 y Fk(Z)649 657 y Fp(G)726 672 y Fm(j)763 657 y Fv(\()p Fp(x)856 672 y Fm(k)921 657 y Fq(\000)h Fv(\()p Fp(R)1133 672 y Fm(j)1169 657 y Fv(\))1207 672 y Fm(k)1250 657 y Fv(\))k(=)374 921 y(=)533 785 y Fk(Z)649 921 y Fv(\007)725 879 y Fn(2)725 945 y Fm(j)781 780 y Fk(\022)961 853 y Fv(1)p 865 898 243 4 v 865 989 a(6)p Fp(\031)973 960 y Fn(2)1012 989 y Fp(h)1068 960 y Fn(3)1117 921 y Fv(\(\010)1225 879 y Fn(TF)1225 945 y Fo(R)p Fm(;)p Fo(Z)1366 921 y Fv(\))1404 879 y Fn(3)p Fm(=)p Fn(2)1536 921 y Fv(+)22 b Fp(R)q Fv(\()p Fp(h;)17 b Fq(\001)p Fv(\))1913 780 y Fk(\023)2002 921 y Fv(\()p Fp(x)2095 936 y Fm(k)2160 921 y Fq(\000)23 b Fv(\()p Fp(R)2372 936 y Fm(j)2409 921 y Fv(\))2447 936 y Fm(k)2489 921 y Fv(\))533 1252 y(+)716 1184 y(1)p 619 1229 V 619 1320 a(6)p Fp(\031)727 1291 y Fn(2)766 1320 y Fp(h)822 1291 y Fn(3)888 1116 y Fk(Z)988 1252 y Fv(\()p Fp(\030)1074 1211 y Fn(2)1069 1276 y Fm(j)1135 1252 y Fq(\000)f Fv(\007)1310 1211 y Fn(2)1310 1276 y Fm(j)1350 1252 y Fv(\))p Fp(h)1444 1267 y Fm(s)1503 1252 y Fq(\003)1575 1082 y Fk( )1654 1252 y Fv(\010)1724 1211 y Fn(TF)1724 1276 y Fo(R)p Fm(;)p Fo(Z)1886 1252 y Fv(+)2019 1127 y Fm(M)1984 1157 y Fk(X)1992 1369 y Fm(k)r Fn(=1)2145 1252 y Fp(w)2215 1267 y Fm(k)2257 1082 y Fk(!)2336 1098 y Fn(3)p Fm(=)p Fn(2)2463 1252 y Fv(\()p Fp(x)2556 1267 y Fm(k)2621 1252 y Fq(\000)g Fv(\()p Fp(R)2832 1267 y Fm(j)2869 1252 y Fv(\))2907 1267 y Fm(k)2950 1252 y Fv(\))374 1548 y(=)640 1481 y(1)p 543 1525 V 543 1617 a(6)p Fp(\031)651 1588 y Fn(2)690 1617 y Fp(h)746 1588 y Fn(3)812 1413 y Fk(Z)929 1548 y Fv(\007)1005 1507 y Fn(2)1005 1573 y Fm(j)1044 1548 y Fv(\(\010)1152 1507 y Fn(TF)1152 1573 y Fo(R)p Fm(;)p Fo(Z)1292 1548 y Fv(\))1330 1507 y Fn(3)p Fm(=)p Fn(2)1440 1548 y Fv(\()p Fp(x)1533 1563 y Fm(k)1598 1548 y Fq(\000)h Fv(\()p Fp(R)1810 1563 y Fm(j)1847 1548 y Fv(\))1885 1563 y Fm(k)1927 1548 y Fv(\))f(+)g Fq(O)s Fv(\()p Fp(h)2261 1507 y Fo(\000)p Fn(2)2356 1548 y Fp(r)2403 1507 y Fn(2)2442 1548 y Fv(\))838 b(\(184\))533 1871 y(+)716 1803 y(1)p 619 1848 V 619 1939 a(6)p Fp(\031)727 1910 y Fn(2)766 1939 y Fp(h)822 1910 y Fn(3)888 1735 y Fk(Z)1004 1871 y Fp(h)1060 1886 y Fm(s)1120 1871 y Fq(\003)1191 1790 y Fk(\002)1233 1871 y Fv(\()p Fp(\030)1319 1829 y Fn(2)1314 1895 y Fm(j)1380 1871 y Fq(\000)23 b Fv(\007)1556 1829 y Fn(2)1556 1895 y Fm(j)1595 1871 y Fv(\)\()p Fp(x)1726 1886 y Fm(k)1791 1871 y Fq(\000)g Fv(\()p Fp(R)2003 1886 y Fm(j)2039 1871 y Fv(\))2077 1886 y Fm(k)2120 1871 y Fv(\))2158 1790 y Fk(\003)2216 1700 y( )2295 1871 y Fv(\010)2365 1829 y Fn(TF)2365 1895 y Fo(R)p Fm(;)p Fo(Z)2527 1871 y Fv(+)2660 1746 y Fm(M)2625 1776 y Fk(X)2633 1988 y Fm(k)r Fn(=1)2786 1871 y Fp(w)2856 1886 y Fm(k)2898 1700 y Fk(!)2977 1717 y Fn(3)p Fm(=)p Fn(2)374 2167 y Fv(=)640 2099 y(1)p 543 2144 V 543 2235 a(6)p Fp(\031)651 2206 y Fn(2)690 2235 y Fp(h)746 2206 y Fn(3)812 2031 y Fk(Z)929 2167 y Fv(\007)1005 2126 y Fn(2)1005 2191 y Fm(j)1044 2167 y Fv(\(\010)1152 2126 y Fn(TF)1152 2191 y Fo(R)p Fm(;)p Fo(Z)1292 2167 y Fv(\))1330 2126 y Fn(3)p Fm(=)p Fn(2)1440 2167 y Fv(\()p Fp(x)1533 2182 y Fm(k)1598 2167 y Fq(\000)g Fv(\()p Fp(R)1810 2182 y Fm(j)1847 2167 y Fv(\))1885 2182 y Fm(k)1927 2167 y Fv(\))533 2421 y(+)716 2354 y(1)p 619 2399 V 619 2490 a(6)p Fp(\031)727 2461 y Fn(2)766 2490 y Fp(h)822 2461 y Fn(3)888 2286 y Fk(Z)1004 2421 y Fp(h)1060 2436 y Fm(s)1120 2421 y Fq(\003)1191 2341 y Fk(\002)1233 2421 y Fv(\()p Fp(\030)1319 2380 y Fn(2)1314 2446 y Fm(j)1380 2421 y Fq(\000)g Fv(\007)1556 2380 y Fn(2)1556 2446 y Fm(j)1595 2421 y Fv(\)\()p Fp(x)1726 2436 y Fm(k)1791 2421 y Fq(\000)g Fv(\()p Fp(R)2003 2436 y Fm(j)2039 2421 y Fv(\))2077 2436 y Fm(k)2120 2421 y Fv(\))2158 2341 y Fk(\003)16 b(\000)2262 2421 y Fv(\010)2332 2380 y Fn(TF)2332 2446 y Fo(R)p Fm(;)p Fo(Z)2494 2421 y Fv(+)22 b Fp(w)2662 2436 y Fm(j)2698 2341 y Fk(\001)2744 2361 y Fn(3)p Fm(=)p Fn(2)2876 2421 y Fv(+)g Fq(O)s Fv(\()p Fp(h)3150 2380 y Fo(\000)p Fn(2)3245 2421 y Fp(r)3292 2380 y Fn(2)3331 2421 y Fv(\))120 2666 y(No)m(w)33 b(w)m(e)h(can)f(write)f(a)g(form)m (ula)f(similar)e(to)j(\(179\):)168 2863 y Fp(h)224 2878 y Fm(s)283 2863 y Fq(\003)22 b Fv([\()p Fp(\030)468 2822 y Fn(2)463 2887 y Fm(j)529 2863 y Fq(\000)h Fv(\007)705 2822 y Fn(2)705 2887 y Fm(j)744 2863 y Fv(\)\()p Fp(x)875 2878 y Fm(k)940 2863 y Fq(\000)g Fv(\()p Fp(R)1152 2878 y Fm(j)1189 2863 y Fv(\))1227 2878 y Fm(k)1269 2863 y Fv(\)])84 b(=)g(\()p Fp(\030)1664 2822 y Fn(2)1659 2887 y Fm(j)1725 2863 y Fq(\000)22 b Fv(\007)1900 2822 y Fn(2)1900 2887 y Fm(j)1940 2863 y Fv(\)\()p Fp(x)2071 2878 y Fm(k)2136 2863 y Fq(\000)h Fv(\()p Fp(R)2348 2878 y Fm(j)2384 2863 y Fv(\))2422 2878 y Fm(k)2465 2863 y Fv(\))f(+)g Fp(K)7 b(s)2759 2822 y Fn(2)2798 2863 y Fv(\001)p Fp(\030)2927 2822 y Fn(2)2922 2887 y Fm(j)2967 2863 y Fv(\()p Fp(x)3060 2878 y Fm(k)3125 2863 y Fq(\000)22 b Fv(\()p Fp(R)3336 2878 y Fm(j)3373 2863 y Fv(\))3411 2878 y Fm(k)3454 2863 y Fv(\))1578 3017 y Fq(\000)p Fp(K)7 b(s)1791 2976 y Fn(2)1831 3017 y Fv(\001\007)1988 2976 y Fn(2)1988 3042 y Fm(j)2028 3017 y Fv(\()p Fp(x)2121 3032 y Fm(k)2186 3017 y Fq(\000)22 b Fv(\()p Fp(R)2397 3032 y Fm(j)2434 3017 y Fv(\))2472 3032 y Fm(k)2515 3017 y Fv(\))g(+)g Fp(K)7 b(s)2809 2976 y Fn(2)2848 3017 y Fv(\()p Fp(@)2937 3032 y Fm(k)2980 3017 y Fp(\030)3028 2976 y Fn(2)3023 3042 y Fm(j)3089 3017 y Fq(\000)23 b Fp(@)3240 3032 y Fm(k)3283 3017 y Fv(\007)3359 2976 y Fn(2)3359 3042 y Fm(j)3398 3017 y Fv(\))1578 3163 y(+)p Fp(\027)1702 3178 y Fm(j)1739 3163 y Fv(\()p Fp(s;)17 b Fq(\001)p Fv(\))1385 b(\(185\))1001 3308 y Fq(j)p Fp(\027)1077 3323 y Fm(j)1114 3308 y Fv(\()p Fp(s;)17 b Fq(\001)p Fv(\))p Fq(j)81 b(\024)j Fp(C)7 b(s)1701 3267 y Fn(4)1740 3308 y Fp(r)1787 3267 y Fo(\000)p Fn(4)1898 3227 y Fk(\000)1944 3308 y Fp(\037)2005 3323 y Fm(B)2058 3333 y Fi(j)2117 3308 y Fv(+)22 b Fp(\037)2276 3323 y Fm(C)2326 3333 y Fi(j)2363 3227 y Fk(\001)266 3504 y Fv(As)33 b(b)s(efore,)g(w)m(e)h(can)f(estimate)756 3608 y Fk(\014)756 3668 y(\014)756 3727 y(\014)756 3787 y(\014)896 3685 y Fv(1)p 800 3729 V 800 3821 a(6)p Fp(\031)908 3792 y Fn(2)947 3821 y Fp(h)1003 3792 y Fn(3)1069 3617 y Fk(Z)1185 3752 y Fp(\027)1233 3767 y Fm(j)1270 3752 y Fv(\()p Fp(s;)17 b Fq(\001)p Fv(\))1481 3672 y Fk(\000)1525 3752 y Fv(\010)1595 3711 y Fn(TF)1595 3777 y Fo(R)p Fm(;)p Fo(Z)1758 3752 y Fv(+)22 b Fp(w)1926 3767 y Fm(j)1962 3672 y Fk(\001)2008 3692 y Fn(3)p Fm(=)p Fn(2)2118 3608 y Fk(\014)2118 3668 y(\014)2118 3727 y(\014)2118 3787 y(\014)2179 3752 y Fq(\024)28 b Fp(C)7 b(h)2417 3711 y Fo(\000)p Fn(3)2511 3752 y Fp(s)2557 3711 y Fn(4)2597 3752 y Fp(r)2644 3711 y Fo(\000)p Fn(3)120 4001 y Fv(and)965 4087 y Fk(\014)965 4146 y(\014)965 4206 y(\014)965 4266 y(\014)1042 4164 y Fp(K)g(s)1178 4128 y Fn(2)p 1008 4208 V 1008 4300 a Fv(6)p Fp(\031)1116 4271 y Fn(2)1155 4300 y Fp(h)1211 4271 y Fn(3)1278 4096 y Fk(Z)1394 4231 y Fp(@)1445 4246 y Fm(k)1488 4231 y Fp(\030)1536 4190 y Fn(2)1531 4256 y Fm(j)1591 4150 y Fk(\000)1637 4231 y Fv(\010)1707 4190 y Fn(TF)1707 4256 y Fo(R)p Fm(;)p Fo(Z)1848 4150 y Fk(\001)1893 4171 y Fn(3)p Fm(=)p Fn(2)2003 4087 y Fk(\014)2003 4146 y(\014)2003 4206 y(\014)2003 4266 y(\014)2064 4231 y Fq(\024)28 b Fp(C)7 b(h)2302 4190 y Fo(\000)p Fn(3)2397 4231 y Fp(s)2443 4190 y Fn(2)2482 4231 y Fp(r)120 4496 y Fv(Also,)56 b(since)d Fp(\030)698 4460 y Fn(2)693 4521 y Fm(j)789 4496 y Fv(is)e(radial)f(around)i Fp(R)1626 4511 y Fm(j)1663 4496 y Fv(,)1747 4416 y Fk(R)1830 4496 y Fv(\001\007)1987 4460 y Fn(2)1987 4521 y Fm(j)2026 4496 y Fv(\()p Fp(x)2119 4511 y Fm(k)2198 4496 y Fq(\000)36 b Fv(\()p Fp(R)2423 4511 y Fm(j)2459 4496 y Fv(\))2497 4511 y Fm(k)2540 4496 y Fv(\))p Fq(j)f(\001)g(\000)p Fp(R)2855 4511 y Fm(j)2892 4496 y Fq(j)2920 4460 y Fo(\000)p Fn(3)p Fm(=)p Fn(2)3145 4496 y Fv(=)61 b(0)52 b(and)120 4548 y Fk(R)203 4628 y Fv(\001\007)360 4592 y Fn(2)360 4653 y Fm(j)400 4628 y Fv(\()p Fp(x)493 4643 y Fm(k)558 4628 y Fq(\000)22 b Fv(\()p Fp(R)769 4643 y Fm(j)806 4628 y Fv(\))844 4643 y Fm(k)887 4628 y Fv(\))p Fq(j)f(\001)h(\000)p Fp(R)1175 4643 y Fm(j)1213 4628 y Fq(j)1241 4592 y Fo(\000)p Fn(3)p Fm(=)p Fn(2)1433 4628 y Fv(=)27 b(0,)33 b(so)f(w)m(e)i(obtain)e (the)h(estimates)149 4752 y Fk(\014)149 4811 y(\014)149 4871 y(\014)149 4931 y(\014)226 4829 y Fp(K)7 b(s)362 4793 y Fn(2)p 192 4873 V 192 4964 a Fv(6)p Fp(\031)300 4936 y Fn(2)339 4964 y Fp(h)395 4936 y Fn(3)461 4760 y Fk(Z)578 4896 y Fv(\001\007)735 4855 y Fn(2)735 4921 y Fm(j)774 4896 y Fv(\()p Fp(x)867 4911 y Fm(k)932 4896 y Fq(\000)23 b Fv(\()p Fp(R)1144 4911 y Fm(j)1181 4896 y Fv(\))1219 4911 y Fm(k)1261 4896 y Fv(\))1316 4815 y Fk(\000)1362 4896 y Fv(\010)1432 4855 y Fn(TF)1432 4921 y Fo(R)p Fm(;)p Fo(Z)1572 4815 y Fk(\001)1618 4836 y Fn(3)p Fm(=)p Fn(2)1728 4752 y Fk(\014)1728 4811 y(\014)1728 4871 y(\014)1728 4931 y(\014)1789 4896 y Fq(\024)28 b Fp(C)7 b Fq(k)p Fv(\001\007)2178 4855 y Fn(2)2178 4921 y Fm(j)2217 4896 y Fq(k)2267 4911 y Fo(1)2342 4896 y Fp(s)2388 4855 y Fn(2)2427 4896 y Fp(h)2483 4855 y Fo(\000)p Fn(3)2594 4760 y Fk(Z)2649 4986 y Fo(j)p Fm(x)p Fo(j\024)p Fn(2)p Fm(r)2873 4896 y Fp(dx)28 b Fq(\024)g Fp(C)7 b(r)3236 4855 y Fn(2)3275 4896 y Fp(s)3321 4855 y Fn(2)3361 4896 y Fp(h)3417 4855 y Fo(\000)p Fn(3)120 5156 y Fv(and)33 b(similarly)155 5262 y Fk(\014)155 5321 y(\014)155 5381 y(\014)155 5441 y(\014)231 5339 y Fp(K)7 b(s)367 5303 y Fn(2)p 198 5383 V 198 5474 a Fv(6)p Fp(\031)306 5446 y Fn(2)345 5474 y Fp(h)401 5446 y Fn(3)467 5271 y Fk(Z)583 5406 y Fp(@)634 5421 y Fm(k)678 5406 y Fv(\007)754 5365 y Fn(2)754 5431 y Fm(j)793 5406 y Fv(\()p Fp(x)886 5421 y Fm(k)951 5406 y Fq(\000)23 b Fv(\()p Fp(R)1163 5421 y Fm(j)1199 5406 y Fv(\))1237 5421 y Fm(k)1280 5406 y Fv(\))1335 5325 y Fk(\000)1380 5406 y Fv(\010)1450 5365 y Fn(TF)1450 5431 y Fo(R)p Fm(;)p Fo(Z)1591 5325 y Fk(\001)1636 5346 y Fn(3)p Fm(=)p Fn(2)1746 5262 y Fk(\014)1746 5321 y(\014)1746 5381 y(\014)1746 5441 y(\014)1807 5406 y Fq(\024)28 b Fp(C)7 b Fq(k)p Fp(@)e Fv(\007)2171 5365 y Fn(2)2171 5431 y Fm(j)2211 5406 y Fq(k)2261 5421 y Fo(1)2336 5406 y Fp(s)2382 5365 y Fn(2)2421 5406 y Fp(h)2477 5365 y Fo(\000)p Fn(3)2588 5271 y Fk(Z)2644 5496 y Fo(j)p Fm(x)p Fo(j\024)p Fn(2)p Fm(r)2867 5406 y Fp(dx)28 b Fq(\024)g Fp(C)7 b(r)3230 5365 y Fn(2)3269 5406 y Fp(s)3315 5365 y Fn(2)3355 5406 y Fp(h)3411 5365 y Fo(\000)p Fn(3)1781 5712 y Fv(42)p eop %%Page: 43 43 43 42 bop 120 407 a Fv(and)33 b(therefore)g(w)m(e)g(get)362 533 y Fk(Z)478 669 y Fp(G)555 684 y Fm(j)592 669 y Fv(\()p Fp(x)685 684 y Fm(k)750 669 y Fq(\000)22 b Fv(\()p Fp(R)961 684 y Fm(j)998 669 y Fv(\))1036 684 y Fm(k)1079 669 y Fv(\))27 b(=)203 934 y(=)469 866 y(1)p 372 911 243 4 v 372 1002 a(6)p Fp(\031)480 973 y Fn(2)519 1002 y Fp(h)575 973 y Fn(3)641 798 y Fk(Z)757 934 y Fp(\030)805 893 y Fn(2)800 958 y Fm(j)861 853 y Fk(\000)907 934 y Fv(\010)977 893 y Fn(TF)977 958 y Fo(R)p Fm(;)p Fo(Z)1139 934 y Fv(+)22 b Fp(w)1307 949 y Fm(j)1343 853 y Fk(\001)1389 874 y Fn(3)p Fm(=)p Fn(2)1516 934 y Fv(\()p Fp(x)1609 949 y Fm(k)1674 934 y Fq(\000)g Fv(\()p Fp(R)1885 949 y Fm(j)1922 934 y Fv(\))1960 949 y Fm(k)2003 934 y Fv(\))g(+)2204 866 y Fp(K)7 b(s)2340 830 y Fn(2)p 2171 911 V 2171 1002 a Fv(6)p Fp(\031)2279 973 y Fn(2)2318 1002 y Fp(h)2374 973 y Fn(3)2440 798 y Fk(Z)2556 934 y Fv(\001)p Fp(\030)2685 893 y Fn(2)2680 958 y Fm(j)2741 853 y Fk(\000)2787 934 y Fv(\010)2857 893 y Fn(TF)2857 958 y Fo(R)p Fm(;)p Fo(Z)2997 853 y Fk(\001)3043 874 y Fn(3)p Fm(=)p Fn(2)3169 934 y Fv(\()p Fp(x)3262 949 y Fm(k)3328 934 y Fq(\000)22 b Fv(\()p Fp(R)3539 949 y Fm(j)3576 934 y Fv(\))3614 949 y Fm(k)3657 934 y Fv(\))362 1137 y(+)p Fq(O)s Fv(\()p Fp(h)614 1096 y Fo(\000)p Fn(2)708 1137 y Fp(r)755 1096 y Fn(2)817 1137 y Fv(+)g Fp(h)971 1096 y Fo(\000)p Fn(3)1065 1137 y Fp(s)1111 1096 y Fn(4)1150 1137 y Fp(r)1197 1096 y Fo(\000)p Fn(3)1314 1137 y Fv(+)g Fp(r)1459 1096 y Fn(2)1498 1137 y Fp(s)1544 1096 y Fn(2)1583 1137 y Fp(h)1639 1096 y Fo(\000)p Fn(3)1734 1137 y Fv(\))1700 b(\(186\))120 1356 y(So,)33 b(the)g(second)g(condition)f(on)g(ii.)42 b(of)32 b(Lemma)f(13)h(is)680 1550 y(1)p 583 1594 V 583 1686 a(6)p Fp(\031)691 1657 y Fn(2)730 1686 y Fp(h)786 1657 y Fn(3)852 1482 y Fk(Z)968 1617 y Fp(W)1060 1632 y Fm(j)1097 1617 y Fv(\()p Fp(x)1190 1632 y Fm(k)1255 1617 y Fq(\000)23 b Fv(\()p Fp(R)1467 1632 y Fm(j)1504 1617 y Fv(\))1542 1632 y Fm(k)1584 1617 y Fv(\))28 b(=)f Fq(O)s Fv(\()p Fp(h)1929 1576 y Fo(\000)p Fn(2)2024 1617 y Fp(r)2071 1576 y Fn(2)2132 1617 y Fv(+)22 b Fp(h)2286 1576 y Fo(\000)p Fn(3)2381 1617 y Fp(s)2427 1576 y Fn(4)2466 1617 y Fp(r)2513 1576 y Fo(\000)p Fn(3)2629 1617 y Fv(+)g Fp(r)2774 1576 y Fn(2)2813 1617 y Fp(s)2859 1576 y Fn(2)2899 1617 y Fp(h)2955 1576 y Fo(\000)p Fn(3)3049 1617 y Fv(\))231 b(\(187\))120 1892 y(No)m(w,)30 b(if)e(the)h(conditions)f(assumed)i(on) e Fp(s;)17 b(r)31 b Fv(are)e(satis\014ed,)h(is)f(easy)h(to)e(see)i (that)3190 1853 y Fn(1)p 3114 1869 189 4 v 3114 1926 a(6)p Fm(\031)3192 1907 y Fj(2)3227 1926 y Fm(h)3268 1907 y Fj(3)3328 1812 y Fk(R)3411 1892 y Fp(W)3503 1907 y Fm(j)120 2012 y Fv(and)399 1973 y Fn(1)p 322 1989 V 322 2047 a(6)p Fm(\031)400 2028 y Fj(2)435 2047 y Fm(h)476 2028 y Fj(3)537 1932 y Fk(R)620 2012 y Fp(W)712 2027 y Fm(j)749 2012 y Fv(\()p Fp(x)842 2027 y Fm(k)909 2012 y Fq(\000)24 b Fv(\()p Fp(R)1122 2027 y Fm(j)1159 2012 y Fv(\))1197 2027 y Fm(k)1239 2012 y Fv(\))36 b(are)f(b)s(oth)g Fp(o)p Fv(\()p Fp(h)1852 1976 y Fo(\000)p Fn(2+)p Fm(\033)2044 2012 y Fv(\))g(for)g(su\016cien)m(tly)h(small)d Fp(\033)j(>)c Fv(0.)51 b(As)36 b(a)120 2133 y(result,)28 b(the)e(restrictions)g(p)s (osed)h(on)g(the)g(functions)f Fp(w)2118 2148 y Fm(j)2180 2133 y Fv(can)h(also)f(b)s(e)g(ful\014lled.)40 b(This)26 b(pro)m(v)m(es)120 2253 y(i.)43 b(and)32 b(ii.)266 2373 y(T)-8 b(o)33 b(pro)m(v)m(e)h(iii.,)c(w)m(e)j(pro)s(ceed)h(similarly)29 b(computing)i(the)i(energy:)585 2592 y(T)-8 b(r)17 b([\()p Fq(\000)p Fp(h)900 2551 y Fn(2)940 2592 y Fv(\001)23 b Fq(\000)f Fv(\010)1213 2551 y Fn(TF)1213 2617 y Fo(R)p Fm(;)p Fo(Z)1354 2592 y Fv(\)\000])1838 b(\(188\))426 2843 y(=)620 2718 y Fm(M)585 2748 y Fk(X)596 2958 y Fm(j)t Fn(=1)746 2843 y Fv(T)-8 b(r)17 b([\()p Fq(\000)p Fp(h)1061 2802 y Fn(2)1101 2843 y Fv(\001)22 b Fq(\000)h Fv(\010)1374 2802 y Fn(TF)1374 2867 y Fo(R)p Fm(;)p Fo(Z)1514 2843 y Fv(\)\()p Fp(\015)1641 2858 y Fn(1)1680 2843 y Fv(\))1718 2858 y Fn(\007)1773 2868 y Fi(j)1810 2843 y Fv(])f(+)2028 2718 y Fn(2)p Fm(M)2011 2748 y Fk(X)1957 2960 y Fm(j)t Fn(=)p Fm(M)7 b Fn(+1)2226 2843 y Fv(T)-8 b(r)17 b([\()p Fq(\000)p Fp(h)2541 2802 y Fn(2)2581 2843 y Fv(\001)22 b Fq(\000)h Fv(\010)2854 2802 y Fn(TF)2854 2867 y Fo(R)p Fm(;)p Fo(Z)2994 2843 y Fv(\)\()p Fp(\015)3121 2858 y Fn(2)3160 2843 y Fv(\))3198 2858 y Fn(\007)3253 2868 y Fi(j)3290 2843 y Fv(])120 3164 y(No)m(w)33 b(w)m(e)h(analyze)e(an)m (y)i(of)e(the)h(terms)f(on)h(\(188\):)266 3284 y(i.-)266 3405 y(By)g(Lemma)f(11)g(w)m(e)h(ha)m(v)m(e)120 3767 y(T)-8 b(r)17 b([\()p Fq(\000)p Fp(h)435 3726 y Fn(2)475 3767 y Fv(\001)22 b Fq(\000)h Fv(\010)748 3726 y Fn(TF)748 3792 y Fo(R)p Fm(;)p Fo(Z)889 3767 y Fv(\)\()p Fp(\015)1016 3782 y Fn(1)1054 3767 y Fv(\))1092 3782 y Fn(\007)1147 3792 y Fi(j)1184 3767 y Fv(])28 b(=)f Fq(\000)1551 3700 y Fv(1)p 1429 3744 292 4 v 1429 3836 a(15)p Fp(\031)1586 3807 y Fn(2)1626 3836 y Fp(h)1682 3807 y Fn(3)1748 3632 y Fk(Z)1864 3767 y Fv(\007)1940 3726 y Fn(2)1940 3792 y Fm(j)1979 3767 y Fv(\(\010)2087 3726 y Fn(TF)2087 3792 y Fo(R)p Fm(;)p Fo(Z)2228 3767 y Fv(\))2266 3726 y Fn(5)p Fm(=)p Fn(2)2398 3767 y Fv(+)2554 3700 y(1)p 2506 3744 145 4 v 2506 3836 a(8)p Fp(h)2611 3807 y Fn(2)2660 3767 y Fp(Z)2734 3726 y Fn(2)2727 3792 y Fm(j)2796 3767 y Fv(+)22 b Fq(O)s Fv(\()p Fp(r)3061 3726 y Fo(\000)p Fn(2)3155 3767 y Fp(h)3211 3726 y Fo(\000)p Fn(1)3305 3767 y Fv(\);)49 b Fp(j)34 b Fv(=)27 b(1)p Fp(;)17 b(:)g(:)g(:)33 b(;)17 b(M)3318 3946 y Fv(\(189\))266 4155 y(ii.-)266 4275 y(Using)33 b(eqs.)44 b(\(151\))32 b(and)h(\(153\))e(w)m(e)j(can)f(write,)f(for)g Fp(j)i Fv(=)27 b Fp(M)33 b Fv(+)22 b(1)p Fp(;)17 b(:)g(:)g(:)33 b(;)17 b Fv(2)p Fp(M)10 b Fv(,)544 4494 y(T)-8 b(r)17 b([\()p Fq(\000)p Fp(h)859 4453 y Fn(2)899 4494 y Fv(\001)23 b Fq(\000)f Fv(\010)1172 4453 y Fn(TF)1172 4519 y Fo(R)p Fm(;)p Fo(Z)1313 4494 y Fv(\)\()p Fp(\015)1440 4509 y Fn(2)1479 4494 y Fv(\))1517 4509 y Fn(\007)1572 4519 y Fi(j)1608 4494 y Fv(])385 4700 y(=)675 4633 y(1)p 554 4677 292 4 v 554 4769 a(10)p Fp(\031)711 4740 y Fn(2)750 4769 y Fp(h)806 4740 y Fn(3)872 4565 y Fk(Z)988 4700 y Fp(h)1044 4715 y Fm(s)1103 4700 y Fq(\003)g Fv(\(\007)1289 4659 y Fn(2)1289 4725 y Fm(j)1329 4700 y Fv(\)\(\010)1475 4659 y Fn(TF)1475 4725 y Fo(R)p Fm(;)p Fo(Z)1615 4700 y Fv(\))1653 4659 y Fn(5)p Fm(=)p Fn(2)1785 4700 y Fv(+)1970 4633 y(1)p 1893 4677 204 4 v 1893 4769 a(6)p Fp(\031)2001 4740 y Fn(2)2040 4769 y Fp(h)2123 4565 y Fk(Z)2239 4700 y Fp(h)2295 4715 y Fm(s)2354 4700 y Fq(\003)g Fv(\(\007)2540 4715 y Fm(j)2576 4700 y Fv(\()p Fq(\000)p Fv(\001\007)2848 4715 y Fm(j)2886 4700 y Fv(\)\)\(\010)3070 4659 y Fn(TF)3070 4725 y Fo(R)p Fm(;)p Fo(Z)3210 4700 y Fv(\))3248 4659 y Fn(3)p Fm(=)p Fn(2)544 4955 y Fv(+)750 4887 y(1)p 630 4932 289 4 v 630 5023 a(6)p Fp(\031)738 4994 y Fn(2)777 5023 y Fp(hs)879 4994 y Fn(2)945 4819 y Fk(Z)1061 4955 y Fp(\026)1120 4970 y Fm(s)1179 4955 y Fq(\003)g Fv(\(\007)1365 4914 y Fn(2)1365 4979 y Fm(j)1404 4955 y Fv(\)\(\010)1550 4914 y Fn(TF)1550 4979 y Fo(R)p Fm(;)p Fo(Z)1691 4955 y Fv(\))1729 4914 y Fn(3)p Fm(=)p Fn(2)1861 4955 y Fq(\000)2067 4887 y Fv(1)p 1970 4932 243 4 v 1970 5023 a(6)p Fp(\031)2078 4994 y Fn(2)2117 5023 y Fp(h)2173 4994 y Fn(3)2240 4819 y Fk(Z)2356 4955 y Fv(\007)2432 4914 y Fn(2)2432 4979 y Fm(j)2471 4955 y Fv(\010)2541 4914 y Fn(TF)2541 4979 y Fo(R)p Fm(;)p Fo(Z)2682 4955 y Fp(h)2738 4970 y Fm(s)2797 4955 y Fq(\003)g Fv(\(\010)2977 4914 y Fn(TF)2977 4979 y Fo(R)p Fm(;)p Fo(Z)3117 4955 y Fv(\))3155 4914 y Fn(3)p Fm(=)p Fn(2)3318 4955 y Fv(\(190\))266 5222 y(No)m(w)46 b(w)m(e)h(analyze)f(the)g(terms)f(of)g(\(190\))g(\(w)m(e)h(assume)g(no) m(w)h Fp(j)55 b Fv(=)50 b Fp(M)42 b Fv(+)31 b(1)p Fp(;)17 b(:)g(:)g(:)32 b(;)17 b Fv(2)p Fp(M)120 5342 y Fv(without)32 b(further)h(notice;)f Fp(i)c Fv(=)g Fp(j)g Fq(\000)23 b Fp(M)10 b Fv(\):)266 5463 y(ii.1.-)1781 5712 y(43)p eop %%Page: 44 44 44 43 bop 266 407 a Fv(Using)33 b(\(179\))e(w)m(e)j(can)f(write)493 608 y(1)p 372 652 292 4 v 372 744 a(10)p Fp(\031)529 715 y Fn(2)568 744 y Fp(h)624 715 y Fn(3)690 540 y Fk(Z)806 675 y Fp(h)862 690 y Fm(s)921 675 y Fq(\003)22 b Fv(\(\007)1107 634 y Fn(2)1107 700 y Fm(j)1146 675 y Fv(\)\(\010)1292 634 y Fn(TF)1292 700 y Fo(R)p Fm(;)p Fo(Z)1433 675 y Fv(\))1471 634 y Fn(5)p Fm(=)p Fn(2)203 941 y Fv(=)493 873 y(1)p 372 918 V 372 1009 a(10)p Fp(\031)529 980 y Fn(2)568 1009 y Fp(h)624 980 y Fn(3)690 805 y Fk(Z)806 941 y Fv(\007)882 899 y Fn(2)882 965 y Fm(j)921 941 y Fv(\(\010)1029 899 y Fn(TF)1029 965 y Fo(R)p Fm(;)p Fo(Z)1170 941 y Fv(\))1208 899 y Fn(5)p Fm(=)p Fn(2)1340 941 y Fv(+)1506 873 y Fp(K)7 b(s)1642 837 y Fn(2)p 1448 918 V 1448 1009 a Fv(10)p Fp(\031)1605 980 y Fn(2)1644 1009 y Fp(h)1700 980 y Fn(3)1766 805 y Fk(Z)1882 941 y Fv(\001)p Fp(\030)2011 899 y Fn(2)2006 965 y Fm(i)2050 941 y Fv(\(\010)2158 899 y Fn(TF)2158 965 y Fo(R)p Fm(;)p Fo(Z)2299 941 y Fv(\))2337 899 y Fn(5)p Fm(=)p Fn(2)2469 941 y Fq(\000)2636 873 y Fp(K)g(s)2772 837 y Fn(2)p 2578 918 V 2578 1009 a Fv(10)p Fp(\031)2735 980 y Fn(2)2774 1009 y Fp(h)2830 980 y Fn(3)2897 805 y Fk(Z)3013 941 y Fv(\001\007)3170 899 y Fn(2)3170 965 y Fm(i)3209 941 y Fv(\(\010)3317 899 y Fn(TF)3317 965 y Fo(R)p Fm(;)p Fo(Z)3458 941 y Fv(\))3496 899 y Fn(5)p Fm(=)p Fn(2)362 1195 y Fv(+)569 1128 y(1)p 448 1172 V 448 1264 a(10)p Fp(\031)605 1235 y Fn(2)644 1264 y Fp(h)700 1235 y Fn(3)766 1060 y Fk(Z)882 1195 y Fp(\024)938 1210 y Fm(j)975 1195 y Fv(\()p Fp(s;)17 b Fq(\001)p Fv(\)\(\010)1277 1154 y Fn(TF)1277 1220 y Fo(R)p Fm(;)p Fo(Z)1416 1195 y Fv(\))1454 1154 y Fn(5)p Fm(=)p Fn(2)120 1475 y Fv(Since)375 1395 y Fk(R)458 1475 y Fv(\001\007)615 1439 y Fn(2)615 1500 y Fm(i)654 1475 y Fq(j)22 b(\001)g(\000)p Fp(R)905 1490 y Fm(j)942 1475 y Fq(j)970 1439 y Fo(\000)p Fn(5)p Fm(=)p Fn(2)1162 1475 y Fv(=)1266 1395 y Fk(R)1349 1475 y Fv(\001\007)1506 1439 y Fn(2)1506 1500 y Fm(i)1546 1475 y Fq(j)f(\001)h(\000)p Fp(R)1796 1490 y Fm(j)1834 1475 y Fq(j)1862 1439 y Fo(\000)p Fn(3)p Fm(=)p Fn(2)2054 1475 y Fv(=)27 b(0)33 b(w)m(e)g(obtain)1032 1615 y Fk(\014)1032 1675 y(\014)1032 1735 y(\014)1032 1794 y(\014)1133 1692 y Fp(K)7 b(s)1269 1656 y Fn(2)p 1075 1737 V 1075 1828 a Fv(10)p Fp(\031)1232 1799 y Fn(2)1271 1828 y Fp(h)1327 1799 y Fn(3)1393 1624 y Fk(Z)1509 1760 y Fv(\001\007)1666 1718 y Fn(2)1666 1784 y Fm(i)1706 1760 y Fv(\(\010)1814 1718 y Fn(TF)1814 1784 y Fo(R)p Fm(;)p Fo(Z)1954 1760 y Fv(\))1992 1718 y Fn(5)p Fm(=)p Fn(2)2102 1615 y Fk(\014)2102 1675 y(\014)2102 1735 y(\014)2102 1794 y(\014)2163 1760 y Fq(\024)28 b Fp(C)7 b(s)2391 1718 y Fn(2)2431 1760 y Fp(h)2487 1718 y Fo(\000)p Fn(3)2581 1760 y Fp(r)120 2047 y Fv(and)33 b(using)f(that)g Fq(j)p Fp(\024)860 2062 y Fm(j)897 2047 y Fv(\()p Fp(s;)17 b Fq(\001)p Fv(\))p Fq(j)26 b(\024)i Fp(C)7 b(s)1373 2011 y Fn(4)1413 2047 y Fp(r)1460 2011 y Fo(\000)p Fn(4)1570 1966 y Fk(\000)1616 2047 y Fp(\037)1677 2062 y Fm(B)1730 2072 y Fi(j)1789 2047 y Fv(+)22 b Fp(\037)1948 2062 y Fm(C)1998 2072 y Fi(j)2035 1966 y Fk(\001)2113 2047 y Fv(w)m(e)34 b(get)905 2190 y Fk(\014)905 2249 y(\014)905 2309 y(\014)905 2369 y(\014)1069 2267 y Fv(1)p 948 2311 V 948 2402 a(10)p Fp(\031)1105 2374 y Fn(2)1144 2402 y Fp(h)1200 2374 y Fn(3)1266 2199 y Fk(Z)1382 2334 y Fp(\024)1438 2349 y Fm(j)1475 2334 y Fv(\()p Fp(s;)17 b Fq(\001)p Fv(\)\(\010)1777 2293 y Fn(TF)1777 2359 y Fo(R)p Fm(;)p Fo(Z)1917 2334 y Fv(\))1955 2293 y Fn(5)p Fm(=)p Fn(2)2065 2190 y Fk(\014)2065 2249 y(\014)2065 2309 y(\014)2065 2369 y(\014)2126 2334 y Fq(\024)28 b Fp(C)7 b(h)2364 2293 y Fo(\000)p Fn(3)2458 2334 y Fp(s)2504 2293 y Fn(4)2543 2334 y Fp(r)2590 2293 y Fo(\000)p Fn(7)p Fm(=)p Fn(2)120 2606 y Fv(and)33 b(then)251 2793 y(1)p 130 2837 V 130 2929 a(10)p Fp(\031)287 2900 y Fn(2)326 2929 y Fp(h)382 2900 y Fn(3)448 2725 y Fk(Z)564 2860 y Fp(h)620 2875 y Fm(s)679 2860 y Fq(\003)22 b Fv(\(\007)865 2819 y Fn(2)865 2885 y Fm(j)905 2860 y Fv(\)\(\010)1051 2819 y Fn(TF)1051 2885 y Fo(R)p Fm(;)p Fo(Z)1191 2860 y Fv(\))1229 2819 y Fn(5)p Fm(=)p Fn(2)1422 2860 y Fv(=)1712 2793 y(1)p 1591 2837 V 1591 2929 a(10)p Fp(\031)1748 2900 y Fn(2)1787 2929 y Fp(h)1843 2900 y Fn(3)1909 2725 y Fk(Z)2025 2860 y Fv(\007)2101 2819 y Fn(2)2101 2885 y Fm(i)2140 2860 y Fv(\(\010)2248 2819 y Fn(TF)2248 2885 y Fo(R)p Fm(;)p Fo(Z)2389 2860 y Fv(\))2427 2819 y Fn(5)p Fm(=)p Fn(2)2559 2860 y Fv(+)2725 2793 y Fp(K)7 b(s)2861 2757 y Fn(2)p 2667 2837 V 2667 2929 a Fv(10)p Fp(\031)2824 2900 y Fn(2)2863 2929 y Fp(h)2919 2900 y Fn(3)2985 2725 y Fk(Z)3101 2860 y Fv(\001)p Fp(\030)3230 2819 y Fn(2)3225 2885 y Fm(i)3269 2860 y Fv(\(\010)3377 2819 y Fn(TF)3377 2885 y Fo(R)p Fm(;)p Fo(Z)3518 2860 y Fv(\))3556 2819 y Fn(5)p Fm(=)p Fn(2)1581 3070 y Fv(+)p Fq(O)s Fv(\()p Fp(s)1823 3029 y Fn(2)1862 3070 y Fp(h)1918 3029 y Fo(\000)p Fn(3)2012 3070 y Fp(r)25 b Fv(+)d Fp(h)2235 3029 y Fo(\000)p Fn(3)2330 3070 y Fp(s)2376 3029 y Fn(4)2415 3070 y Fp(r)2462 3029 y Fo(\000)p Fn(7)p Fm(=)p Fn(2)2627 3070 y Fv(\))266 3290 y(ii.2.-)266 3411 y(Using)33 b(that)f Fp(h)808 3426 y Fm(s)867 3411 y Fq(\003)22 b Fv([\007)1042 3426 y Fm(j)1078 3411 y Fv(\()p Fq(\000)p Fv(\001\007)1350 3426 y Fm(j)1387 3411 y Fv(\)])28 b Fq(\024)g Fp(C)7 b Fv([\007)1765 3426 y Fm(j)1802 3411 y Fv(\()p Fq(\000)p Fv(\001\007)2074 3426 y Fm(j)2111 3411 y Fv(\)])32 b(is)h(easy)g(to)f(see)i(that)809 3546 y Fk(\014)809 3606 y(\014)809 3666 y(\014)809 3726 y(\014)929 3623 y Fv(1)p 852 3668 204 4 v 852 3759 a(6)p Fp(\031)960 3730 y Fn(2)999 3759 y Fp(h)1082 3555 y Fk(Z)1198 3691 y Fp(h)1254 3706 y Fm(s)1313 3691 y Fq(\003)22 b Fv(\(\007)1499 3706 y Fm(j)1535 3691 y Fv(\()p Fq(\000)p Fv(\001\007)1807 3706 y Fm(j)1845 3691 y Fv(\)\)\(\010)2029 3650 y Fn(TF)2029 3715 y Fo(R)p Fm(;)p Fo(Z)2169 3691 y Fv(\))2207 3650 y Fn(3)p Fm(=)p Fn(2)2317 3546 y Fk(\014)2317 3606 y(\014)2317 3666 y(\014)2317 3726 y(\014)2378 3691 y Fq(\024)28 b Fp(C)7 b(h)2616 3650 y Fo(\000)p Fn(1)2710 3691 y Fp(r)2757 3650 y Fo(\000)p Fn(1)266 3959 y Fv(ii.3.-)266 4080 y(W)-8 b(e)43 b(can)f(write)g Fp(\026)950 4095 y Fm(s)1016 4080 y Fq(\003)28 b Fv(\007)1170 4044 y Fn(2)1170 4104 y Fm(j)1254 4080 y Fv(=)44 b Fp(k)s Fv(\007)1504 4044 y Fn(2)1504 4104 y Fm(j)1572 4080 y Fv(+)28 b Fp(\017)1715 4095 y Fm(j)1752 4080 y Fv(\()p Fp(s;)17 b Fq(\001)p Fv(\),)44 b(where)g Fp(k)j(>)d Fv(0)e(is)g(some)g(constan)m(t)h(and)120 4210 y Fq(j)p Fp(\017)187 4225 y Fm(j)224 4210 y Fv(\()p Fp(s;)17 b Fq(\001)p Fv(\))p Fq(j)26 b(\024)i Fp(C)7 b(s)700 4174 y Fn(2)740 4210 y Fp(r)787 4174 y Fo(\000)p Fn(2)897 4130 y Fk(\000)943 4210 y Fp(\037)1004 4225 y Fm(B)1057 4235 y Fi(j)1116 4210 y Fv(+)22 b Fp(\037)1275 4225 y Fm(C)1325 4235 y Fi(j)1362 4130 y Fk(\001)1408 4210 y Fv(.)43 b(As)33 b(a)g(result)f(w)m(e)i(can)f(write)524 4422 y(1)p 404 4467 289 4 v 404 4558 a(6)p Fp(\031)512 4529 y Fn(2)551 4558 y Fp(hs)653 4529 y Fn(2)719 4354 y Fk(Z)835 4490 y Fp(\026)894 4505 y Fm(s)953 4490 y Fq(\003)22 b Fv(\(\007)1139 4449 y Fn(2)1139 4514 y Fm(j)1178 4490 y Fv(\)\(\010)1324 4449 y Fn(TF)1324 4514 y Fo(R)p Fm(;)p Fo(Z)1464 4490 y Fv(\))1502 4449 y Fn(3)p Fm(=)p Fn(2)1640 4490 y Fv(=)1871 4422 y Fp(k)p 1754 4467 V 1754 4558 a Fv(6)p Fp(\031)1862 4529 y Fn(2)1901 4558 y Fp(hs)2003 4529 y Fn(2)2069 4354 y Fk(Z)2185 4490 y Fv(\007)2261 4449 y Fn(2)2261 4514 y Fm(j)2300 4490 y Fv(\(\010)2408 4449 y Fn(TF)2408 4514 y Fo(R)p Fm(;)p Fo(Z)2549 4490 y Fv(\))2587 4449 y Fn(3)p Fm(=)p Fn(2)2719 4490 y Fv(+)g Fq(O)s Fv(\()p Fp(h)2993 4449 y Fo(\000)p Fn(1)3087 4490 y Fp(r)3134 4449 y Fo(\000)p Fn(1)3228 4490 y Fv(\))120 4753 y(Moreo)m(v)m(er,)34 b(is)e(easy)i(to)e(see)i (that)1269 4949 y Fp(k)p 1152 4994 V 1152 5085 a Fv(6)p Fp(\031)1260 5056 y Fn(2)1299 5085 y Fp(hs)1401 5056 y Fn(2)1467 4881 y Fk(Z)1583 5017 y Fv(\007)1659 4975 y Fn(2)1659 5041 y Fm(j)1698 5017 y Fv(\(\010)1806 4975 y Fn(TF)1806 5041 y Fo(R)p Fm(;)p Fo(Z)1947 5017 y Fv(\))1985 4975 y Fn(3)p Fm(=)p Fn(2)2122 5017 y Fq(\030)29 b Fp(h)2284 4975 y Fo(\000)p Fn(1)2378 5017 y Fp(s)2424 4975 y Fo(\000)p Fn(2)3318 5017 y Fv(\(191\))120 5279 y(uniformly)h(in)i Fp(j)6 b Fv(.)266 5400 y(ii.4.-)1781 5712 y(44)p eop %%Page: 45 45 45 44 bop 266 407 a Fv(Finally)28 b(w)m(e)k(analyze)f(the)g(term)1569 368 y Fn(1)p 1493 384 189 4 v 1493 441 a(6)p Fm(\031)1571 422 y Fj(2)1605 441 y Fm(h)1646 422 y Fj(3)1707 327 y Fk(R)1790 407 y Fv(\007)1866 371 y Fn(2)1866 431 y Fm(j)1906 407 y Fv(\010)1976 371 y Fn(TF)1976 432 y Fo(R)p Fm(;)p Fo(Z)2116 407 y Fp(h)2172 422 y Fm(s)2227 407 y Fq(\003)18 b Fv(\(\010)2403 371 y Fn(TF)2403 432 y Fo(R)p Fm(;)p Fo(Z)2543 407 y Fv(\))2581 371 y Fn(3)p Fm(=)p Fn(2)2691 407 y Fv(.)43 b(T)-8 b(o)31 b(do)f(so,)i(w)m(e)f(write)120 636 y Fp(h)176 651 y Fm(s)213 636 y Fq(\003)p Fv(\(\010)371 595 y Fn(TF)371 661 y Fo(R)p Fm(;)p Fo(Z)511 636 y Fv(\))549 595 y Fn(3)p Fm(=)p Fn(2)687 636 y Fv(=)c(\(\010)898 595 y Fn(TF)898 661 y Fo(R)p Fm(;)p Fo(Z)1039 636 y Fv(\))1077 595 y Fn(3)p Fm(=)p Fn(2)1187 636 y Fv(+)p Fp(K)7 b(s)1399 595 y Fn(2)1438 636 y Fv(\001\(\010)1627 595 y Fn(TF)1627 661 y Fo(R)p Fm(;)p Fo(Z)1768 636 y Fv(\))1806 595 y Fn(3)p Fm(=)p Fn(2)1916 636 y Fv(+)p Fp(\025)p Fv(\()p Fp(s;)17 b Fq(\001)p Fv(\);)37 b Fq(j)p Fp(\025)p Fv(\()p Fp(s;)17 b Fq(\001)p Fv(\))p Fq(j)26 b(\024)i Fp(C)7 b(s)2868 595 y Fn(4)2908 636 y Fp(l)r Fv(\()p Fp(x)p Fv(\))3070 595 y Fo(\000)p Fn(11)p Fm(=)p Fn(2)3270 636 y Fq(h)p Fp(l)r Fv(\()p Fp(x)p Fv(\))p Fq(i)3510 595 y Fo(\000)p Fn(9)p Fm(=)p Fn(2)120 848 y Fv(Also,)32 b(outside)h(the)g(set)g Fq(R)g Fv(w)m(e)g(can)g(write,)g(using)f(that)g (for)g Fp(f)39 b(>)27 b Fv(0)33 b(and)f Fp(p)c(>)f Fv(0,)1041 1061 y(\001)p Fp(f)1181 1020 y Fm(p)1248 1061 y Fv(=)h Fp(p)p Fv(\()p Fp(p)22 b Fq(\000)h Fv(1\))p Fp(f)1756 1020 y Fm(p)p Fo(\000)p Fn(2)1885 1061 y Fq(jr)p Fp(f)11 b Fq(j)2083 1020 y Fn(2)2143 1061 y Fv(+)22 b Fp(pf)2349 1020 y Fm(p)p Fo(\000)p Fn(1)2479 1061 y Fv(\001)p Fp(f)120 1273 y Fv(and)33 b(the)g(TF)f(equation)h(\(13\))f(the)h(expression)822 1530 y(\001\(\010)1011 1489 y Fn(TF)1011 1554 y Fo(R)p Fm(;)p Fo(Z)1151 1530 y Fv(\))1189 1489 y Fn(3)p Fm(=)p Fn(2)1327 1530 y Fv(=)1441 1462 y(1)p 1441 1507 49 4 v 1441 1598 a(4)1499 1530 y(\(\010)1607 1489 y Fn(TF)1607 1554 y Fo(R)p Fm(;)p Fo(Z)1748 1530 y Fv(\))1786 1489 y Fo(\000)p Fn(1)p Fm(=)p Fn(2)1951 1530 y Fq(jr)p Fv(\010)2132 1489 y Fn(TF)2132 1554 y Fo(R)p Fm(;)p Fo(Z)2272 1530 y Fq(j)2300 1489 y Fn(2)2361 1530 y Fv(+)22 b Fp(\034)11 b Fv(\(\010)2620 1489 y Fn(TF)2620 1554 y Fo(R)p Fm(;)p Fo(Z)2761 1530 y Fv(\))2799 1489 y Fn(2)3318 1530 y Fv(\(192\))120 1768 y(where)34 b Fp(\034)39 b(>)28 b Fv(0)k(is)g(some)g(constan)m(t.) 45 b(No)m(w)33 b(w)m(e)g(estimate)308 1870 y Fk(\014)308 1930 y(\014)308 1990 y(\014)308 2050 y(\014)448 1948 y Fv(1)p 351 1992 243 4 v 351 2083 a(6)p Fp(\031)459 2054 y Fn(2)499 2083 y Fp(h)555 2054 y Fn(3)621 1879 y Fk(Z)737 2015 y Fv(\007)813 1974 y Fn(2)813 2040 y Fm(j)852 2015 y Fv(\010)922 1974 y Fn(TF)922 2040 y Fo(R)p Fm(;)p Fo(Z)1063 2015 y Fp(\025)p Fv(\()p Fp(s;)17 b Fq(\001)p Fv(\))1314 1870 y Fk(\014)1314 1930 y(\014)1314 1990 y(\014)1314 2050 y(\014)1429 2015 y Fq(\024)84 b Fp(C)7 b(h)1723 1974 y Fo(\000)p Fn(3)1817 2015 y Fp(s)1863 1974 y Fn(4)1919 1879 y Fk(Z)1974 2105 y Fm(x)p Fo(2)p Fm(D)2119 2115 y Fi(i)2145 2105 y Fn(:)p Fm(l)q Fn(\()p Fm(x)p Fn(\))p Fo(\025)p Fm(r)r(=)p Fn(3)2461 2015 y Fp(l)r Fv(\()p Fp(x)p Fv(\))2623 1974 y Fo(\000)p Fn(11)p Fm(=)p Fn(2)2824 2015 y Fq(h)p Fp(l)r Fv(\()p Fp(x)p Fv(\))p Fq(i)3064 1974 y Fo(\000)p Fn(9)p Fm(=)p Fn(2)3246 2015 y Fp(dx)1429 2246 y Fq(\024)84 b Fp(C)7 b(h)1723 2205 y Fo(\000)p Fn(3)1817 2246 y Fp(s)1863 2205 y Fn(4)1902 2246 y Fp(r)1949 2205 y Fo(\000)p Fn(5)p Fm(=)p Fn(2)120 2458 y Fv(So)32 b(w)m(e)i(can)f(write,)f(with)606 2724 y Fp(R)680 2739 y Fm(j)716 2724 y Fv(\()p Fp(s;)17 b(h)p Fv(\))28 b(=)1113 2657 y Fp(K)7 b(s)1249 2620 y Fn(2)p 1079 2701 V 1079 2792 a Fv(6)p Fp(\031)1187 2763 y Fn(2)1226 2792 y Fp(h)1282 2763 y Fn(3)1348 2588 y Fk(Z)1465 2724 y Fv(\007)1541 2683 y Fn(2)1541 2749 y Fm(j)1597 2583 y Fk(\024)1659 2657 y Fv(1)p 1659 2701 49 4 v 1659 2792 a(4)1718 2724 y(\(\010)1826 2683 y Fn(TF)1826 2749 y Fo(R)p Fm(;)p Fo(Z)1966 2724 y Fv(\))2004 2683 y Fn(1)p Fm(=)p Fn(2)2114 2724 y Fq(jr)p Fv(\010)2295 2683 y Fn(TF)2295 2749 y Fo(R)p Fm(;)p Fo(Z)2435 2724 y Fq(j)2463 2683 y Fn(2)2525 2724 y Fv(+)22 b Fp(\034)11 b Fv(\(\010)2784 2683 y Fn(TF)2784 2749 y Fo(R)p Fm(;)p Fo(Z)2925 2724 y Fv(\))2963 2683 y Fn(3)3002 2583 y Fk(\025)484 3079 y Fv(1)p 387 3123 243 4 v 387 3215 a(6)p Fp(\031)495 3186 y Fn(2)534 3215 y Fp(h)590 3186 y Fn(3)656 3011 y Fk(Z)772 3146 y Fv(\007)848 3105 y Fn(2)848 3171 y Fm(j)888 3146 y Fv(\010)958 3105 y Fn(TF)958 3171 y Fo(R)p Fm(;)p Fo(Z)1098 3146 y Fp(h)1154 3161 y Fm(s)1213 3146 y Fq(\003)22 b Fv(\(\010)1393 3105 y Fn(TF)1393 3171 y Fo(R)p Fm(;)p Fo(Z)1534 3146 y Fv(\))1572 3105 y Fn(3)p Fm(=)p Fn(2)1765 3146 y Fv(=)2030 3079 y(1)p 1933 3123 V 1933 3215 a(6)p Fp(\031)2041 3186 y Fn(2)2080 3215 y Fp(h)2136 3186 y Fn(3)2203 3011 y Fk(Z)2319 3146 y Fv(\007)2395 3105 y Fn(2)2395 3171 y Fm(j)2434 3146 y Fv(\(\010)2542 3105 y Fn(TF)2542 3171 y Fo(R)p Fm(;)p Fo(Z)2683 3146 y Fv(\))2721 3105 y Fn(5)p Fm(=)p Fn(2)2853 3146 y Fv(+)g Fp(R)3025 3161 y Fm(j)3061 3146 y Fv(\()p Fp(s;)17 b(h)p Fv(\))1923 3356 y(+)p Fq(O)s Fv(\()p Fp(h)2175 3315 y Fo(\000)p Fn(3)2270 3356 y Fp(s)2316 3315 y Fn(4)2355 3356 y Fp(r)2402 3315 y Fo(\000)p Fn(5)p Fm(=)p Fn(2)2567 3356 y Fv(\))120 3569 y(Using)32 b(the)h(expression)h(\(192\))e(w)m(e)h (get)1374 3781 y Fp(R)1448 3796 y Fm(j)1484 3781 y Fv(\()p Fp(s;)17 b(h)p Fv(\))28 b Fq(\030)g Fp(s)1885 3740 y Fn(2)1924 3781 y Fp(h)1980 3740 y Fo(\000)p Fn(3)2075 3781 y Fp(r)2122 3740 y Fo(\000)p Fn(3)p Fm(=)p Fn(2)3318 3781 y Fv(\(193\))120 4003 y(uniformly)43 b(in)i Fp(j)51 b Fv(\(since)46 b(\001\010)1236 3967 y Fn(TF)1236 4028 y Fo(R)p Fm(;)p Fo(Z)1377 4003 y Fv(\()p Fp(x)p Fv(\))1508 3967 y Fo(\000)p Fn(3)p Fm(=)p Fn(2)1723 4003 y Fq(\030)k(j)p Fp(x)31 b Fq(\000)h Fp(R)2147 4018 y Fm(j)2183 4003 y Fq(j)2211 3967 y Fo(\000)p Fn(7)p Fm(=)p Fn(2)2376 4003 y Fv(;)h Fq(j)p Fp(x)e Fq(\000)h Fp(R)2733 4018 y Fm(j)2769 4003 y Fq(j)50 b(\024)g Fp(\026=)p Fv(2\))44 b(Putting)120 4124 y(together)33 b(the)g(results)g(of)f(ii.1,...,ii.4)d(w)m(e)34 b(obtain)591 4336 y(T)-8 b(r)17 b([\()p Fq(\000)p Fp(h)906 4295 y Fn(2)946 4336 y Fv(\001)22 b Fq(\000)h Fv(\010)1219 4295 y Fn(TF)1219 4361 y Fo(R)p Fm(;)p Fo(Z)1360 4336 y Fv(\)\()p Fp(\015)1487 4351 y Fn(2)1525 4336 y Fv(\))1563 4351 y Fn(\007)1618 4361 y Fi(j)1655 4336 y Fv(])624 4553 y(=)k Fq(\000)936 4486 y Fv(1)p 814 4530 292 4 v 814 4621 a(15)p Fp(\031)971 4593 y Fn(2)1010 4621 y Fp(h)1066 4593 y Fn(3)1133 4418 y Fk(Z)1232 4553 y Fv(\()p Fp(\030)1318 4512 y Fn(2)1313 4578 y Fm(i)1379 4553 y Fq(\000)c Fv(\007)1555 4512 y Fn(2)1555 4578 y Fm(i)1594 4553 y Fv(\)\(\010)1740 4512 y Fn(TF)1740 4578 y Fo(R)p Fm(;)p Fo(Z)1880 4553 y Fv(\))1918 4512 y Fn(5)p Fm(=)p Fn(2)2051 4553 y Fv(+)2217 4486 y Fp(K)7 b(s)2353 4450 y Fn(2)p 2159 4530 V 2159 4621 a Fv(10)p Fp(\031)2316 4593 y Fn(2)2354 4621 y Fp(h)2410 4593 y Fn(3)2477 4418 y Fk(Z)2593 4553 y Fv(\001)p Fp(\030)2722 4512 y Fn(2)2717 4578 y Fm(i)2761 4553 y Fv(\(\010)2869 4512 y Fn(TF)2869 4578 y Fo(R)p Fm(;)p Fo(Z)3009 4553 y Fv(\))3047 4512 y Fn(5)p Fm(=)p Fn(2)3318 4553 y Fv(\(194\))624 4809 y(+)827 4741 y Fp(k)p 710 4786 289 4 v 710 4877 a Fv(6)p Fp(\031)818 4848 y Fn(2)857 4877 y Fp(hs)959 4848 y Fn(2)1025 4673 y Fk(Z)1141 4809 y Fv(\007)1217 4768 y Fn(2)1217 4833 y Fm(j)1256 4809 y Fv(\(\010)1364 4768 y Fn(TF)1364 4833 y Fo(R)p Fm(;)p Fo(Z)1505 4809 y Fv(\))1543 4768 y Fn(3)p Fm(=)p Fn(2)1675 4809 y Fq(\000)22 b Fp(R)1848 4824 y Fm(j)1885 4809 y Fv(\()p Fp(s;)17 b(h)p Fv(\))22 b(+)g Fq(O)s Fv(\()p Fp(s)2393 4768 y Fn(2)2432 4809 y Fp(h)2488 4768 y Fo(\000)p Fn(3)2583 4809 y Fp(r)j Fv(+)d Fp(h)2806 4768 y Fo(\000)p Fn(3)2900 4809 y Fp(s)2946 4768 y Fn(4)2985 4809 y Fp(r)3032 4768 y Fo(\000)p Fn(5)p Fm(=)p Fn(2)3197 4809 y Fv(\))120 5069 y(Using)32 b(eqs.)45 b(\(189\),)32 b(\(190\),)g(\(194\),)f(the)i (estimates)g(\(191\),)e(\(193\))h(and)h(noting)e(that)465 5250 y Fm(M)430 5280 y Fk(X)441 5490 y Fm(j)t Fn(=1)659 5307 y Fp(K)7 b(s)795 5271 y Fn(2)p 601 5352 292 4 v 601 5443 a Fv(10)p Fp(\031)758 5414 y Fn(2)797 5443 y Fp(h)853 5414 y Fn(3)919 5239 y Fk(Z)1035 5374 y Fv(\001)p Fp(\030)1164 5333 y Fn(2)1159 5399 y Fm(j)1203 5374 y Fv(\(\010)1311 5333 y Fn(TF)1311 5399 y Fo(R)p Fm(;)p Fo(Z)1452 5374 y Fv(\))1490 5333 y Fn(5)p Fm(=)p Fn(2)1627 5374 y Fv(=)1799 5307 y Fp(K)g(s)1935 5271 y Fn(2)p 1741 5352 V 1741 5443 a Fv(10)p Fp(\031)1898 5414 y Fn(2)1937 5443 y Fp(h)1993 5414 y Fn(3)2059 5239 y Fk(Z)2175 5374 y Fv(\001)2273 5204 y Fk(")2366 5250 y Fm(M)2331 5280 y Fk(X)2342 5490 y Fm(j)t Fn(=1)2492 5374 y Fp(\030)2540 5333 y Fn(2)2535 5399 y Fm(j)2579 5204 y Fk(#)2653 5374 y Fv(\(\010)2761 5333 y Fn(TF)2761 5399 y Fo(R)p Fm(;)p Fo(Z)2902 5374 y Fv(\))2940 5333 y Fn(5)p Fm(=)p Fn(2)3077 5374 y Fv(=)28 b(0)1781 5712 y(45)p eop %%Page: 46 46 46 45 bop 120 407 a Fv(\(since)397 332 y Fk(P)502 358 y Fm(M)502 436 y(j)t Fn(=1)645 407 y Fp(\030)693 371 y Fn(2)688 431 y Fm(j)760 407 y Fv(=)27 b(1\))33 b(w)m(e)g(obtain)281 754 y(T)-8 b(r)17 b([\()p Fq(\000)p Fp(h)596 712 y Fn(2)636 754 y Fv(\001)22 b Fq(\000)h Fv(\010)909 712 y Fn(TF)909 778 y Fo(R)p Fm(;)p Fo(Z)1049 754 y Fv(\)\000])83 b(=)g Fq(\000)1626 686 y Fv(1)p 1504 731 292 4 v 1504 822 a(15)p Fp(\031)1661 793 y Fn(2)1701 822 y Fp(h)1757 793 y Fn(3)1823 618 y Fk(Z)1939 583 y( )2018 754 y Fv(\010)2088 712 y Fn(TF)2088 778 y Fo(R)p Fm(;)p Fo(Z)2251 754 y Fv(+)2383 629 y Fm(M)2349 659 y Fk(X)2359 869 y Fm(j)t Fn(=1)2509 754 y Fp(w)2579 769 y Fm(j)2615 583 y Fk(!)2694 600 y Fn(5)p Fm(=)p Fn(2)2826 754 y Fv(+)2982 686 y(1)p 2934 731 145 4 v 2934 822 a(8)p Fp(h)3039 793 y Fn(2)3140 629 y Fm(M)3105 659 y Fk(X)3116 869 y Fm(j)t Fn(=1)3266 754 y Fp(Z)3340 712 y Fn(2)3333 778 y Fm(j)1450 1105 y Fv(+)1577 980 y Fm(M)1542 1010 y Fk(X)1553 1220 y Fm(j)t Fn(=1)1830 1037 y Fp(k)p 1713 1082 289 4 v 1713 1173 a Fv(6)p Fp(\031)1821 1144 y Fn(2)1860 1173 y Fp(hs)1962 1144 y Fn(2)2028 969 y Fk(Z)2144 1105 y Fv(\007)2220 1064 y Fn(2)2220 1129 y Fm(j)2260 1105 y Fv(\(\010)2368 1064 y Fn(TF)2368 1129 y Fo(R)p Fm(;)p Fo(Z)2508 1105 y Fv(\))2546 1064 y Fn(3)p Fm(=)p Fn(2)2678 1105 y Fq(\000)2812 980 y Fm(M)2778 1010 y Fk(X)2788 1220 y Fm(j)t Fn(=1)2938 1105 y Fp(R)3012 1120 y Fm(j)3049 1105 y Fv(\()p Fp(s;)17 b(h)p Fv(\))1450 1367 y(+)p Fq(O)s Fv(\(\()p Fp(s)1730 1326 y Fn(2)1769 1367 y Fp(h)1825 1326 y Fo(\000)p Fn(3)1920 1367 y Fp(r)24 b Fv(+)e Fp(h)2142 1326 y Fo(\000)p Fn(3)2237 1367 y Fp(s)2283 1326 y Fn(4)2322 1367 y Fp(r)2369 1326 y Fo(\000)p Fn(5)p Fm(=)p Fn(2)2534 1367 y Fv(\))p Fp(M)10 b Fv(\))604 b(\(195\))120 1581 y(No)m(w)47 b(w)m(e)g(ha)m(v)m(e)751 1506 y Fk(P)856 1532 y Fm(M)856 1610 y(j)t Fn(=1)999 1581 y Fp(R)1073 1596 y Fm(j)1110 1581 y Fv(\()p Fp(s;)17 b(h)p Fv(\))50 b Fq(\030)h Fp(s)1556 1544 y Fn(2)1596 1581 y Fp(h)1652 1544 y Fo(\000)p Fn(3)1746 1581 y Fp(r)1793 1544 y Fo(\000)p Fn(3)p Fm(=)p Fn(2)1958 1581 y Fp(M)57 b Fv(and)2312 1506 y Fk(P)2417 1532 y Fm(M)2417 1610 y(j)t Fn(=1)2661 1541 y Fm(k)p 2570 1558 221 4 v 2570 1615 a Fn(6)p Fm(\031)2648 1596 y Fj(2)2683 1615 y Fm(hs)2757 1596 y Fj(2)2817 1500 y Fk(R)2900 1581 y Fv(\007)2976 1544 y Fn(2)2976 1605 y Fm(j)3016 1581 y Fv(\(\010)3124 1544 y Fn(TF)3124 1606 y Fo(R)p Fm(;)p Fo(Z)3264 1581 y Fv(\))3302 1544 y Fn(3)p Fm(=)p Fn(2)3463 1581 y Fq(\030)120 1717 y Fp(h)176 1681 y Fo(\000)p Fn(1)270 1717 y Fp(s)316 1681 y Fo(\000)p Fn(2)411 1717 y Fp(M)10 b Fv(,)34 b(so,)f(these)i (terms)e(cancel)g(out)g(b)m(y)h(taking)f Fp(s)28 b Fq(\030)h Fp(h)2387 1681 y Fn(1)p Fm(=)p Fn(2)2498 1717 y Fp(r)2545 1681 y Fn(3)p Fm(=)p Fn(8)2687 1717 y Fv(and)34 b(then)f(obtain)g(the) 120 1837 y(next)g(estimate)f(for)g(the)h(energy)281 2161 y(T)-8 b(r)17 b([\()p Fq(\000)p Fp(h)596 2120 y Fn(2)636 2161 y Fv(\001)22 b Fq(\000)h Fv(\010)909 2120 y Fn(TF)909 2186 y Fo(R)p Fm(;)p Fo(Z)1049 2161 y Fv(\)\000])83 b(=)g Fq(\000)1626 2094 y Fv(1)p 1504 2138 292 4 v 1504 2230 a(15)p Fp(\031)1661 2201 y Fn(2)1701 2230 y Fp(h)1757 2201 y Fn(3)1823 2026 y Fk(Z)1939 1991 y( )2018 2161 y Fv(\010)2088 2120 y Fn(TF)2088 2186 y Fo(R)p Fm(;)p Fo(Z)2251 2161 y Fv(+)2383 2037 y Fm(M)2349 2067 y Fk(X)2359 2277 y Fm(j)t Fn(=1)2509 2161 y Fp(w)2579 2176 y Fm(j)2615 1991 y Fk(!)2694 2008 y Fn(5)p Fm(=)p Fn(2)2826 2161 y Fv(+)2982 2094 y(1)p 2934 2138 145 4 v 2934 2230 a(8)p Fp(h)3039 2201 y Fn(2)3140 2037 y Fm(M)3105 2067 y Fk(X)3116 2277 y Fm(j)t Fn(=1)3266 2161 y Fp(Z)3340 2120 y Fn(2)3333 2186 y Fm(j)1450 2424 y Fv(+)p Fq(O)s Fv(\(\()p Fp(s)1730 2382 y Fn(2)1769 2424 y Fp(h)1825 2382 y Fo(\000)p Fn(3)1920 2424 y Fp(r)24 b Fv(+)e Fp(h)2142 2382 y Fo(\000)p Fn(3)2237 2424 y Fp(s)2283 2382 y Fn(4)2322 2424 y Fp(r)2369 2382 y Fo(\000)p Fn(5)p Fm(=)p Fn(2)2534 2424 y Fv(\))p Fp(M)10 b Fv(\))604 b(\(196\))120 2637 y(W)-8 b(e)36 b(conclude)g(our)f (optimization)d(b)m(y)37 b(imp)s(osing)c Fp(h)2056 2601 y Fo(\000)p Fn(3)2150 2637 y Fp(s)2196 2601 y Fn(4)2236 2637 y Fp(r)2283 2601 y Fo(\000)p Fn(3)2410 2637 y Fv(=)f Fp(r)s(s)2611 2601 y Fn(2)2650 2637 y Fp(h)2706 2601 y Fo(\000)p Fn(3)2801 2637 y Fv(.)52 b(This)36 b(yields)f Fp(r)h Fq(\030)120 2757 y Fp(h)176 2721 y Fn(4)p Fm(=)p Fn(9)319 2757 y Fv(and)c Fp(s)c Fq(\030)g Fp(h)743 2721 y Fn(2)p Fm(=)p Fn(3)853 2757 y Fv(.)44 b(Plugging)31 b(this)h(in)m(to)g(the)h(energy)g(estimate)f(\(196\))g(w)m(e)h(obtain) 281 3087 y(T)-8 b(r)17 b([\()p Fq(\000)p Fp(h)596 3046 y Fn(2)636 3087 y Fv(\001)22 b Fq(\000)h Fv(\010)909 3046 y Fn(TF)909 3112 y Fo(R)p Fm(;)p Fo(Z)1049 3087 y Fv(\)\000])83 b(=)g Fq(\000)1626 3020 y Fv(1)p 1504 3064 292 4 v 1504 3155 a(15)p Fp(\031)1661 3127 y Fn(2)1701 3155 y Fp(h)1757 3127 y Fn(3)1823 2951 y Fk(Z)1939 2917 y( )2018 3087 y Fv(\010)2088 3046 y Fn(TF)2088 3112 y Fo(R)p Fm(;)p Fo(Z)2251 3087 y Fv(+)2383 2963 y Fm(M)2349 2992 y Fk(X)2359 3202 y Fm(j)t Fn(=1)2509 3087 y Fp(w)2579 3102 y Fm(j)2615 2917 y Fk(!)2694 2933 y Fn(5)p Fm(=)p Fn(2)2826 3087 y Fv(+)2982 3020 y(1)p 2934 3064 145 4 v 2934 3155 a(8)p Fp(h)3039 3127 y Fn(2)3140 2963 y Fm(M)3105 2992 y Fk(X)3116 3202 y Fm(j)t Fn(=1)3266 3087 y Fp(Z)3340 3046 y Fn(2)3333 3112 y Fm(j)1417 3349 y Fv(+)p Fq(O)s Fv(\()p Fp(h)1669 3308 y Fo(\000)p Fn(11)p Fm(=)p Fn(9)1870 3349 y Fp(M)10 b Fv(\))1306 b(\(197\))120 3563 y(and)33 b(the)g(estimates)1402 3732 y(1)p 1305 3777 243 4 v 1305 3868 a(6)p Fp(\031)1413 3839 y Fn(2)1452 3868 y Fp(h)1508 3839 y Fn(3)1574 3664 y Fk(Z)1691 3800 y Fp(W)1783 3815 y Fm(j)1847 3800 y Fv(=)28 b Fq(O)s Fv(\()p Fp(h)2127 3759 y Fo(\000)p Fn(14)p Fm(=)p Fn(9)2327 3800 y Fv(\))953 b(\(198\))120 4056 y(and)1140 4226 y(1)p 1043 4270 V 1043 4361 a(6)p Fp(\031)1151 4333 y Fn(2)1190 4361 y Fp(h)1246 4333 y Fn(3)1312 4157 y Fk(Z)1428 4293 y Fp(W)1520 4308 y Fm(j)1557 4293 y Fv(\()p Fp(x)1650 4308 y Fm(k)1715 4293 y Fq(\000)23 b Fv(\()p Fp(R)1927 4308 y Fm(j)1963 4293 y Fv(\))2001 4308 y Fm(k)2044 4293 y Fv(\))k(=)h Fq(O)s Fv(\()p Fp(h)2389 4252 y Fo(\000)p Fn(11)p Fm(=)p Fn(9)2589 4293 y Fv(\))691 b(\(199\))120 4566 y(Since)34 b(the)g(functions)f Fp(W)1058 4581 y Fm(j)1128 4566 y Fv(no)m(w)h(can)g(b)s(e)g(tak)m(en)g(in)f(suc)m(h)i(a)e(w)m(a)m(y)i (that)e Fq(k)p Fp(W)2883 4581 y Fm(j)2919 4566 y Fq(k)2969 4581 y Fo(1)3073 4566 y Fq(\024)d Fp(C)7 b(h)3313 4530 y Fn(1+4)p Fm(=)p Fn(9)3513 4566 y Fv(,)120 4687 y(w)m(e)34 b(also)d(obtain)h(the)h(\014nal)f(estimate)f(for)h(the)h(energy)508 4987 y(T)-8 b(r)16 b([\()p Fq(\000)p Fp(h)822 4946 y Fn(2)863 4987 y Fv(\001)22 b Fq(\000)h Fv(\010)1136 4946 y Fn(TF)1136 5012 y Fo(R)p Fm(;)p Fo(Z)1276 4987 y Fv(\)\000])83 b(=)g Fq(\000)1853 4920 y Fv(1)p 1731 4964 292 4 v 1731 5055 a(15)p Fp(\031)1888 5027 y Fn(2)1927 5055 y Fp(h)1983 5027 y Fn(3)2049 4851 y Fk(Z)2166 4906 y(\000)2211 4987 y Fv(\010)2281 4946 y Fn(TF)2281 5012 y Fo(R)p Fm(;)p Fo(Z)2422 4906 y Fk(\001)2468 4927 y Fn(5)p Fm(=)p Fn(2)2600 4987 y Fv(+)2755 4920 y(1)p 2708 4964 145 4 v 2708 5055 a(8)p Fp(h)2813 5027 y Fn(2)2913 4863 y Fm(M)2879 4892 y Fk(X)2889 5102 y Fm(j)t Fn(=1)3039 4987 y Fp(Z)3113 4946 y Fn(2)3106 5012 y Fm(j)1676 5249 y Fv(+)p Fq(O)s Fv(\()p Fp(h)1928 5208 y Fo(\000)p Fn(14)p Fm(=)p Fn(9)2129 5249 y Fp(M)10 b Fv(\))1047 b(\(200\))p 3469 5463 4 66 v 3473 5400 59 4 v 3473 5463 V 3531 5463 4 66 v 1781 5712 a(46)p eop %%Page: 47 47 47 46 bop 120 413 a Fz(iv.)51 b(Estimating)34 b(the)39 b(Coulom)m(b)d(norm)g(of)i Ff(\032)2097 431 y Fe(\000)2180 413 y Fd(\000)2396 366 y Fe(1)p 2310 385 215 5 v 2310 454 a(6)p Fc(\031)2402 430 y Fb(2)2439 454 y Fc(h)2487 430 y Fb(3)2534 413 y Fz(\(\010)2665 369 y Fe(TF)2665 445 y Fa(R)p Fc(;)p Fa(Z)2828 413 y Fz(\))2874 369 y Fe(3)p Fc(=)p Fe(2)3003 413 y Fz(.)120 594 y Ft(Lemma)f(14)h (\(Estimate)d(on)i(Coulom)m(b)g(in)m(teraction\).)46 b Fr(A.-)d(L)-5 b(et)44 b Fp(f)5 b(;)17 b(g)46 b Fr(b)-5 b(e)42 b(supp)-5 b(orte)g(d)120 715 y(in)p 240 635 80 4 v 35 w Fp(B)319 730 y Fn(1)358 715 y Fv(\(0\))35 b Fr(and)f(inte)-5 b(gr)g(able;)34 b Fq(j)p Fp(R)q Fq(j)27 b(\025)h Fv(1)p Fr(.)45 b(Cal)5 b(l)910 843 y Fk(8)910 933 y(<)910 1112 y(:)1040 926 y Fq(h)p Fp(f)11 b Fq(i)82 b Fv(:=)1445 846 y Fk(R)1528 926 y Fp(f)11 b Fv(\()p Fp(x)p Fv(\))17 b Fp(dx)1085 1047 y(F)1148 1062 y Fm(i)1259 1047 y Fv(:=)1445 966 y Fk(R)1528 1047 y Fp(f)11 b Fv(\()p Fp(x)p Fv(\))p Fp(x)1773 1062 y Fm(i)1819 1047 y Fp(dx)p Fv(;)51 b Fp(i)28 b Fv(=)g(1)p Fp(;)17 b(:)g(:)g(:)32 b(;)17 b Fv(3)1053 1167 y Fp(F)1116 1182 y Fm(ij)1259 1167 y Fv(:=)1445 1087 y Fk(R)1528 1167 y Fp(f)11 b Fv(\()p Fp(x)p Fv(\))p Fp(x)1773 1182 y Fm(i)1802 1167 y Fp(x)1857 1182 y Fm(j)1910 1167 y Fp(dx)p Fv(;)52 b Fp(i;)17 b(j)34 b Fv(=)27 b(1)p Fp(;)17 b(:)g(:)g(:)32 b(;)17 b Fv(3)3318 1048 y(\(201\))120 1385 y Fr(Then)34 b(the)h(fol)5 b(lowing)33 b(estimates)i(holds,)f(with)g Fp(u)1924 1400 y Fm(i)1980 1385 y Fv(=)27 b Fq(j)p Fp(R)q Fq(j)2214 1349 y Fo(\000)p Fn(1)2308 1385 y Fp(R)2382 1400 y Fm(i)2411 1385 y Fr(:)266 1505 y(i.-)560 1750 y Fp(D)19 b Fv(\()p Fp(f)5 b(;)17 b(g)t Fv(\()p Fq(\001)k Fv(+)h Fp(R)q Fv(\)\))83 b(=)1434 1682 y Fq(h)p Fp(f)11 b Fq(ih)p Fp(g)t Fq(i)p 1434 1727 264 4 v 1477 1818 a Fv(2)p Fq(j)p Fp(R)q Fq(j)1730 1750 y Fv(+)1881 1682 y Fp(u)1937 1697 y Fm(i)p 1838 1727 170 4 v 1838 1818 a Fq(j)p Fp(R)q Fq(j)1969 1789 y Fn(2)2035 1750 y Fv(\()p Fq(h)p Fp(f)g Fq(i)p Fp(G)2287 1765 y Fm(i)2336 1750 y Fv(+)22 b Fq(h)p Fp(g)t Fq(i)p Fp(F)2626 1765 y Fm(i)2653 1750 y Fv(\))1424 2013 y(+)1510 1945 y(3)p Fp(u)1615 1960 y Fm(i)1643 1945 y Fp(u)1699 1960 y Fm(j)1757 1945 y Fq(\000)g Fp(\016)1899 1960 y Fm(ij)p 1510 1990 451 4 v 1650 2081 a Fq(j)p Fp(R)q Fq(j)1781 2052 y Fn(3)1987 2013 y Fv(\()p Fp(F)2088 2028 y Fm(ij)2148 2013 y Fq(h)p Fp(g)t Fq(i)f(\000)i Fv(2)p Fp(F)2510 2028 y Fm(i)2538 2013 y Fp(G)2615 2028 y Fm(j)2674 2013 y Fv(+)f Fp(G)2849 2028 y Fm(ij)2909 2013 y Fq(h)p Fp(f)11 b Fq(i)p Fv(\))234 b(\(202\))1424 2284 y(+)p Fq(O)1599 2143 y Fk(\022)1682 2216 y Fq(k)p Fp(f)11 b Fq(k)1841 2231 y Fn(1)1880 2216 y Fq(k)p Fp(g)t Fq(k)2031 2231 y Fn(1)p 1682 2261 388 4 v 1791 2352 a Fq(j)p Fp(R)q Fq(j)1922 2323 y Fn(4)2079 2143 y Fk(\023)266 2545 y Fr(ii.-)433 2794 y Fq(j)22 b(\001)g(j)561 2753 y Fo(\000)p Fn(1)677 2794 y Fq(\003)g Fp(f)11 b Fv(\()p Fp(x)p Fv(\))27 b(=)1080 2727 y Fq(h)p Fp(f)11 b Fq(i)p 1080 2771 137 4 v 1093 2863 a(j)p Fp(x)p Fq(j)1248 2794 y Fv(+)1356 2727 y Fp(x)1411 2742 y Fm(i)1440 2727 y Fp(F)1503 2742 y Fm(i)p 1356 2771 175 4 v 1369 2863 a Fq(j)p Fp(x)p Fq(j)1480 2834 y Fn(3)1563 2794 y Fv(+)1671 2727 y(3)p Fp(x)1775 2742 y Fm(i)1804 2727 y Fp(x)1859 2742 y Fm(j)1918 2727 y Fq(\000)22 b Fp(\016)2060 2742 y Fm(ij)2121 2727 y Fq(j)p Fp(x)p Fq(j)2232 2691 y Fn(2)p 1671 2771 601 4 v 1896 2863 a Fq(j)p Fp(x)p Fq(j)2007 2834 y Fn(5)2281 2794 y Fp(F)2344 2809 y Fm(ij)2427 2794 y Fv(+)g Fq(O)2624 2654 y Fk(\022)2707 2727 y Fq(k)p Fp(f)11 b Fq(k)2866 2742 y Fn(1)p 2707 2771 198 4 v 2731 2863 a Fq(j)p Fp(x)p Fq(j)2842 2834 y Fn(4)2915 2654 y Fk(\023)3318 2794 y Fv(\(203\))266 3070 y Fr(B.-)43 b(Assume)h(now)f(that)i Fp(f)5 b(;)17 b(g)47 b Fr(ar)-5 b(e)43 b(inte)-5 b(gr)g(able,)46 b(that)e Fq(j)p Fp(f)11 b Fv(\()p Fp(x)p Fv(\))p Fq(j)p Fp(;)32 b Fq(j)p Fp(g)t Fv(\()p Fp(x)p Fv(\))p Fq(j)44 b(\024)g Fp(C)3068 3085 y Fn(0)3108 3070 y Fq(j)p Fp(x)p Fq(j)3219 3034 y Fo(\000)p Fn(3)3313 3070 y Fq(h)p Fp(x)p Fq(i)3446 3034 y Fo(\000)p Fn(3)120 3191 y Fr(and)35 b(\014nal)5 b(ly)35 b(the)h(c)-5 b(anc)g(el)5 b(lation)34 b(c)-5 b(onditions)35 b Fq(h)p Fp(f)11 b Fq(i)28 b Fv(=)h Fq(h)p Fp(g)t Fq(i)f Fv(=)h(0)36 b Fr(and)f Fp(f)2628 3206 y Fm(i)2686 3191 y Fv(=)29 b Fp(g)2838 3206 y Fm(i)2895 3191 y Fv(=)g(0;)k Fp(i)c Fv(=)g(1)p Fp(;)17 b Fv(2)p Fp(;)g Fv(3)p Fr(.)120 3311 y(Then,)34 b(with)h Fp(')p Fv(\()p Fp(x)p Fv(\))28 b(:=)f(\(1)22 b(+)g(log)1302 3335 y Fn(+)1378 3311 y Fq(j)p Fp(x)p Fq(j)p Fv(\))p Fq(j)p Fp(x)p Fq(j)1638 3275 y Fo(\000)p Fn(1)1731 3311 y Fq(h)p Fp(x)p Fq(i)1864 3275 y Fo(\000)p Fn(3)1959 3311 y Fr(,)35 b(the)f(next)h(estimate)g(holds:)266 3431 y(iv.-)856 3646 y Fq(j)p Fp(D)19 b Fv(\()p Fp(f)5 b(;)17 b(g)t Fv(\()p Fq(\001)k Fv(+)h Fp(R)q Fv(\)\))p Fq(j)28 b(\024)g Fp(C)23 b Fv(max)p Fq(fk)p Fp(f)11 b Fq(k)2151 3661 y Fn(1)2190 3646 y Fq(k)p Fp(g)t Fq(k)2341 3661 y Fn(1)2379 3646 y Fp(;)17 b(C)2500 3605 y Fn(2)2493 3671 y(0)2539 3646 y Fq(g)p Fp(')p Fv(\()p Fp(R)q Fv(\))514 b(\(204\))266 3861 y Fr(\(Her)-5 b(e)28 b(we)g(ar)-5 b(e)28 b(using)g(Einstein)-10 b('s)27 b(c)-5 b(onvention)27 b(of)h(summation)f(over)h(r)-5 b(ep)g(e)g(ate)g(d)27 b(indic)-5 b(es\).)120 4054 y(Pr)g(o)g(of.)48 b Fv(A.-)266 4175 y(i.-)41 b(F)-8 b(or)42 b Fq(j)p Fp(x)p Fq(j)i(\024)h Fv(1)d(and)h Fq(j)p Fp(z)49 b Fv(=)44 b Fp(y)32 b Fq(\000)e Fp(R)q Fq(j)44 b(\024)h Fv(1)d(w)m(e)h(write)g(do)m(wn)g(T)-8 b(a)m(ylor's)42 b(expansion)h(of)120 4295 y Fq(j)p Fp(x)22 b Fq(\000)h Fp(y)t Fq(j)405 4259 y Fo(\000)p Fn(1)531 4295 y Fv(up)32 b(to)h(order)f(three:)619 4492 y(1)p 502 4536 284 4 v 502 4627 a Fq(j)p Fp(x)22 b Fq(\000)g Fp(y)t Fq(j)878 4559 y Fv(=)1088 4492 y(1)p 1047 4536 131 4 v 1047 4627 a Fq(j)p Fp(R)q Fq(j)1210 4559 y Fv(+)1361 4492 y Fp(u)1417 4507 y Fm(i)p 1318 4536 170 4 v 1318 4627 a Fq(j)p Fp(R)q Fq(j)1449 4598 y Fn(2)1497 4559 y Fv(\()p Fp(x)1590 4574 y Fm(i)1641 4559 y Fq(\000)h Fp(z)1786 4574 y Fm(i)1814 4559 y Fv(\))f(+)1982 4492 y(3)p Fp(u)2087 4507 y Fm(i)2115 4492 y Fp(u)2171 4507 y Fm(j)2229 4492 y Fq(\000)h Fp(\016)2372 4507 y Fm(ij)p 1982 4536 451 4 v 2122 4627 a Fq(j)p Fp(R)q Fq(j)2253 4598 y Fn(3)2442 4559 y Fv(\()p Fp(x)2535 4574 y Fm(i)2586 4559 y Fq(\000)g Fp(z)2731 4574 y Fm(i)2759 4559 y Fv(\)\()p Fp(x)2890 4574 y Fm(j)2949 4559 y Fq(\000)g Fp(z)3094 4574 y Fm(j)3130 4559 y Fv(\))1037 4830 y(+)p Fq(O)1212 4689 y Fk(\022)1356 4762 y Fv(1)p 1295 4807 170 4 v 1295 4898 a Fq(j)p Fp(R)q Fq(j)1426 4869 y Fn(4)1475 4689 y Fk(\023)3318 4830 y Fv(\(205\))120 5102 y(Plugging)31 b(form)m(ula)g(\(205\))g(in)m(to)h(the)h(de\014nition)f(of)g Fp(D)s Fv(\()p Fp(f)5 b(;)17 b(g)t Fv(\))32 b(giv)m(en)g(b)m(y)i(eq.)44 b(\(62\))32 b(w)m(e)i(arriv)m(e)120 5222 y(at)e(\(202\).)266 5342 y(The)i(pro)s(of)e(of)g(ii.-)e(is)i(completely)g(analogous.)266 5463 y(B.-)1781 5712 y(47)p eop %%Page: 48 48 48 47 bop 266 407 a Fv(W)-8 b(e)33 b(\014rst)g(estimate)f Fq(j)22 b(\001)g(j)1154 371 y Fo(\000)p Fn(1)1270 407 y Fq(\003)g Fp(f)11 b Fv(\()p Fp(x)p Fv(\):)201 682 y Fq(j)22 b(\001)g(j)329 641 y Fo(\000)p Fn(1)445 682 y Fq(\003)g Fp(f)11 b Fv(\()p Fp(x)p Fv(\))27 b(=)838 546 y Fk(Z)1013 614 y Fp(f)11 b Fv(\()p Fp(y)t Fv(\))p 964 659 284 4 v 964 750 a Fq(j)p Fp(x)22 b Fq(\000)h Fp(y)t Fq(j)1274 682 y Fp(dy)31 b Fv(=)1507 546 y Fk(Z)1563 772 y Fo(j)p Fm(y)r Fo(j\024)1705 745 y Fj(1)p 1704 757 31 3 v 1704 798 a(2)1745 772 y Fo(j)p Fm(x)p Fo(j)1844 682 y Fv(+)1937 546 y Fk(Z)2002 745 y Fj(1)p 2002 757 V 2002 798 a(2)2042 772 y Fo(j)p Fm(x)p Fo(j\024j)p Fm(y)r Fo(j\024)p Fn(2)p Fo(j)p Fm(x)p Fo(j)2443 682 y Fv(+)2536 546 y Fk(Z)2590 772 y Fo(j)p Fm(y)r Fo(j\025)p Fn(2)p Fo(j)p Fm(x)p Fo(j)2868 682 y Fv(=)c Fp(I)3014 697 y Fn(1)3076 682 y Fv(+)22 b Fp(I)3217 697 y Fn(2)3279 682 y Fv(+)g Fp(I)3420 697 y Fn(3)3318 890 y Fv(\(206\))120 1113 y(Is)33 b(inmediate)e(to)h(estimate)g(for)g Fp(f)43 b Fv(ful\014lling)29 b(the)k(h)m(yp)s(othesis)h(B.-)e(and)h(for)f(an)m (y)h Fp(x)28 b Fq(2)g Fl(R)3384 1077 y Fn(3)3430 1113 y Fv(:)708 1245 y Fk(\()830 1339 y Fq(j)p Fp(I)901 1354 y Fn(1)940 1339 y Fq(j)83 b(\024)1243 1300 y Fn(2)p 1221 1316 79 4 v 1221 1373 a Fo(j)p Fm(x)p Fo(j)1310 1339 y Fq(k)p Fp(f)11 b Fq(k)1469 1354 y Fn(1)830 1474 y Fq(j)p Fp(I)901 1489 y Fn(3)940 1474 y Fq(j)83 b(\024)g Fv(2)1277 1394 y Fk(R)1324 1509 y Fo(j)p Fm(y)r Fo(j\025)p Fn(2)p Fo(j)p Fm(x)p Fo(j)1590 1474 y Fq(j)p Fp(y)t Fq(j)1698 1438 y Fo(\000)p Fn(4)1791 1474 y Fq(h)p Fp(y)t Fq(i)1921 1438 y Fo(\000)p Fn(3)2030 1474 y Fp(dy)31 b Fq(\024)d Fp(C)c(C)2429 1489 y Fn(0)2468 1474 y Fq(j)p Fp(x)p Fq(j)2579 1438 y Fo(\000)p Fn(1)2673 1474 y Fq(h)p Fp(x)p Fq(i)2806 1438 y Fo(\000)p Fn(3)120 1717 y Fv(Similarly)-8 b(,)762 1980 y Fq(j)p Fp(I)833 1995 y Fn(2)872 1980 y Fq(j)82 b(\024)i Fp(C)1213 1995 y Fn(0)1252 1980 y Fq(j)p Fp(x=)p Fv(2)p Fq(j)1461 1938 y Fo(\000)p Fn(3)1555 1980 y Fq(h)p Fp(x=)p Fv(2)p Fq(i)1786 1938 y Fo(\000)p Fn(3)1896 1844 y Fk(Z)1961 2042 y Fj(1)p 1961 2054 31 3 v 1961 2096 a(2)2002 2069 y Fo(j)p Fm(x)p Fo(j\024j)p Fm(y)r Fo(j\024)p Fn(2)p Fo(j)p Fm(x)p Fo(j)2402 1980 y Fq(j)p Fp(x)22 b Fq(\000)h Fp(y)t Fq(j)2687 1938 y Fo(\000)p Fn(1)2797 1980 y Fp(dy)982 2290 y Fq(\024)84 b Fp(C)23 b(C)1306 2305 y Fn(0)1346 2290 y Fq(j)p Fp(x)p Fq(j)1457 2249 y Fo(\000)p Fn(1)1551 2290 y Fq(h)p Fp(x)p Fq(i)1684 2249 y Fo(\000)p Fn(3)1795 2154 y Fk(Z)1860 2352 y Fj(1)p 1860 2364 V 1860 2406 a(2)1900 2380 y Fo(\024j)p Fm(y)r Fo(j\024)p Fn(2)2143 2145 y Fk(\014)2143 2205 y(\014)2143 2265 y(\014)2143 2325 y(\014)2213 2222 y Fp(x)p 2186 2267 111 4 v 2186 2358 a Fq(j)p Fp(x)p Fq(j)2329 2290 y(\000)f Fp(y)2480 2145 y Fk(\014)2480 2205 y(\014)2480 2265 y(\014)2480 2325 y(\014)2513 2172 y Fo(\000)p Fn(1)2640 2290 y Fp(dy)983 2523 y Fv(=)84 b Fp(C)23 b(C)1306 2538 y Fn(0)1346 2523 y Fq(j)p Fp(x)p Fq(j)1457 2482 y Fo(\000)p Fn(1)1551 2523 y Fq(h)p Fp(x)p Fq(i)1684 2482 y Fo(\000)p Fn(3)266 2743 y Fv(No)m(w)51 b(w)m(e)h(estimate)d(more)h(carefully)f (the)i(term)e Fp(I)2230 2758 y Fn(1)2320 2743 y Fv(for)h Fq(j)p Fp(x)p Fq(j)58 b(\025)g Fv(1,)c(using)c(that)g(for)120 2863 y Fq(j)p Fp(y)t Fq(j)26 b(\024)i Fv(1)p Fp(=)p Fv(2)p Fq(j)p Fp(x)p Fq(j)k Fv(w)m(e)i(can)e(write)724 3075 y(1)p 607 3119 284 4 v 607 3211 a Fq(j)p Fp(x)22 b Fq(\000)h Fp(y)t Fq(j)928 3142 y Fv(=)1072 3075 y(1)p 1041 3119 111 4 v 1041 3211 a Fq(j)p Fp(x)p Fq(j)1184 3142 y Fv(+)1292 3075 y Fp(x)1347 3090 y Fm(i)1376 3075 y Fp(y)1424 3090 y Fm(i)p 1292 3119 160 4 v 1297 3211 a Fq(j)p Fp(x)p Fq(j)1408 3182 y Fn(3)1484 3142 y Fv(+)1592 3075 y(3)p Fp(x)1696 3090 y Fm(i)1724 3075 y Fp(x)1779 3090 y Fm(j)1838 3075 y Fq(\000)g Fp(\016)1981 3090 y Fm(ij)2042 3075 y Fq(j)p Fp(x)p Fq(j)2153 3039 y Fn(2)p 1592 3119 601 4 v 1817 3211 a Fq(j)p Fp(x)p Fq(j)1928 3182 y Fn(5)2202 3142 y Fp(y)2250 3157 y Fm(i)2278 3142 y Fp(y)2326 3157 y Fm(j)2384 3142 y Fv(+)f Fq(O)2580 3002 y Fk(\022)2666 3075 y Fq(j)p Fp(y)t Fq(j)2774 3039 y Fn(3)p 2664 3119 151 4 v 2664 3211 a Fq(j)p Fp(x)p Fq(j)2775 3182 y Fn(4)2824 3002 y Fk(\023)120 3414 y Fv(Then)34 b(w)m(e)f(get)g Fp(I)724 3429 y Fn(1)791 3414 y Fv(=)28 b Fp(a)22 b Fv(+)g Fp(b)h Fv(+)f Fp(c)g Fv(+)g Fp(d)32 b Fv(where)908 3554 y Fk(8)908 3644 y(>)908 3674 y(>)908 3704 y(>)908 3734 y(<)908 3913 y(>)908 3943 y(>)908 3973 y(>)908 4003 y(:)1093 3626 y Fp(a)83 b Fv(=)1420 3587 y Fn(1)p 1398 3603 79 4 v 1398 3660 a Fo(j)p Fm(x)p Fo(j)1503 3545 y Fk(R)1550 3660 y Fo(j)p Fm(y)r Fo(j\024)p Fn(1)p Fm(=)p Fn(2)p Fo(j)p Fm(x)p Fo(j)1887 3626 y Fp(f)11 b Fv(\()p Fp(y)t Fv(\))17 b Fp(dy)1103 3761 y(b)83 b Fv(=)1422 3720 y Fm(x)1462 3730 y Fi(i)p 1398 3738 114 4 v 1398 3796 a Fo(j)p Fm(x)p Fo(j)1478 3777 y Fj(3)1538 3681 y Fk(R)1585 3796 y Fo(j)p Fm(y)r Fo(j\024)p Fn(1)p Fm(=)p Fn(2)p Fo(j)p Fm(x)p Fo(j)1922 3761 y Fp(y)1970 3776 y Fm(i)1998 3761 y Fp(f)11 b Fv(\()p Fp(y)t Fv(\))17 b Fp(dy)1102 3919 y(c)83 b Fv(=)1398 3868 y Fn(3)p Fm(x)1473 3878 y Fi(i)1499 3868 y Fm(x)1539 3878 y Fi(j)1571 3868 y Fo(\000)p Fm(\016)1657 3878 y Fi(ij)1712 3868 y Fo(j)p Fm(x)p Fo(j)1792 3844 y Fj(2)p 1398 3896 428 4 v 1555 3953 a Fo(j)p Fm(x)p Fo(j)1635 3934 y Fj(5)1852 3838 y Fk(R)1899 3953 y Fo(j)p Fm(y)r Fo(j\024)p Fn(1)p Fm(=)p Fn(2)p Fo(j)p Fm(x)p Fo(j)2236 3919 y Fp(y)2284 3934 y Fm(i)2312 3919 y Fp(y)2360 3934 y Fm(j)2396 3919 y Fp(f)11 b Fv(\()p Fp(y)t Fv(\))17 b Fp(dy)1038 4054 y Fq(j)p Fp(d)p Fq(j)82 b(\024)i Fp(C)1514 4015 y Fn(1)p 1475 4031 114 4 v 1475 4088 a Fo(j)p Fm(x)p Fo(j)1555 4070 y Fj(4)1615 3974 y Fk(R)1662 4089 y Fo(j)p Fm(y)r Fo(j\024)p Fn(1)p Fm(=)p Fn(2)p Fo(j)p Fm(x)p Fo(j)1999 4054 y Fq(j)p Fp(y)t Fq(j)2107 4018 y Fn(3)2145 4054 y Fp(f)11 b Fv(\()p Fp(y)t Fv(\))17 b Fp(dy)120 4282 y Fv(No)m(w)33 b(w)m(e)h(use)f(the)g(cancellation)e(conditions)g (satis\014ed)i(b)m(y)h Fp(f)43 b Fv(to)32 b(get:)266 4403 y(a.-)986 4676 y Fq(j)p Fp(a)p Fq(j)27 b Fv(=)1264 4609 y(1)p 1233 4653 111 4 v 1233 4744 a Fq(j)p Fp(x)p Fq(j)1371 4531 y Fk(\014)1371 4591 y(\014)1371 4651 y(\014)1371 4711 y(\014)1404 4540 y(Z)1459 4766 y Fo(j)p Fm(y)r Fo(j\025)p Fn(1)p Fm(=)p Fn(2)p Fo(j)p Fm(x)p Fo(j)1796 4676 y Fp(f)11 b Fv(\()p Fp(y)t Fv(\))17 b Fp(dy)2103 4531 y Fk(\014)2103 4591 y(\014)2103 4651 y(\014)2103 4711 y(\014)2161 4676 y Fq(\024)28 b Fp(C)c(C)2430 4691 y Fn(0)2469 4676 y Fq(j)p Fp(x)p Fq(j)2580 4635 y Fo(\000)p Fn(4)266 4914 y Fv(b.-)933 5187 y Fq(j)p Fp(b)p Fq(j)j Fv(=)1161 5043 y Fk(\014)1161 5102 y(\014)1161 5162 y(\014)1161 5222 y(\014)1237 5120 y Fp(x)1292 5135 y Fm(i)p 1204 5164 151 4 v 1204 5255 a Fq(j)p Fp(x)p Fq(j)1315 5227 y Fn(3)1381 5051 y Fk(Z)1436 5277 y Fo(j)p Fm(y)r Fo(j\025)p Fn(1)p Fm(=)p Fn(2)p Fo(j)p Fm(x)p Fo(j)1773 5187 y Fp(y)1821 5202 y Fm(i)1849 5187 y Fp(f)11 b Fv(\()p Fp(y)t Fv(\))17 b Fp(dy)2156 5043 y Fk(\014)2156 5102 y(\014)2156 5162 y(\014)2156 5222 y(\014)2214 5187 y Fq(\024)28 b Fp(C)c(C)2483 5202 y Fn(0)2522 5187 y Fq(j)p Fp(x)p Fq(j)2633 5146 y Fo(\000)p Fn(4)266 5399 y Fv(c.-)1781 5712 y(48)p eop %%Page: 49 49 49 48 bop 380 624 a Fp(c)28 b Fv(=)563 557 y(3)p Fp(x)667 572 y Fm(i)695 557 y Fp(x)750 572 y Fm(j)810 557 y Fq(\000)22 b Fp(\016)952 572 y Fm(ij)1013 557 y Fq(j)p Fp(x)p Fq(j)1124 521 y Fn(2)p 563 601 601 4 v 788 692 a Fq(j)p Fp(x)p Fq(j)899 664 y Fn(5)1173 624 y Fp(F)1236 639 y Fm(ij)1319 624 y Fv(+)g Fp(c)1459 639 y Fn(1)1498 624 y Fv(;)50 b Fp(c)1617 639 y Fn(1)1684 624 y Fv(=)1797 557 y(3)p Fp(x)1901 572 y Fm(i)1930 557 y Fp(x)1985 572 y Fm(j)2044 557 y Fq(\000)22 b Fp(\016)2186 572 y Fm(ij)2247 557 y Fq(j)p Fp(x)p Fq(j)2358 521 y Fn(2)p 1797 601 V 2022 692 a Fq(j)p Fp(x)p Fq(j)2133 664 y Fn(5)2424 489 y Fk(Z)2479 714 y Fo(j)p Fm(y)r Fo(j\025)p Fn(1)p Fm(=)p Fn(2)p Fo(j)p Fm(x)p Fo(j)2816 624 y Fp(y)2864 639 y Fm(i)2892 624 y Fp(y)2940 639 y Fm(j)2976 624 y Fp(f)11 b Fv(\()p Fp(y)t Fv(\))17 b Fp(dy)120 871 y Fv(and)33 b(similarly)645 1108 y Fq(j)p Fp(c)715 1123 y Fn(1)754 1108 y Fq(j)28 b(\024)915 963 y Fk(\014)915 1023 y(\014)915 1083 y(\014)915 1142 y(\014)958 1040 y Fv(3)p Fp(x)1062 1055 y Fm(i)1090 1040 y Fp(x)1145 1055 y Fm(j)1204 1040 y Fq(\000)23 b Fp(\016)1347 1055 y Fm(ij)1408 1040 y Fq(j)p Fp(x)p Fq(j)1519 1004 y Fn(2)p 958 1085 V 1183 1176 a Fq(j)p Fp(x)p Fq(j)1294 1147 y Fn(5)1585 972 y Fk(Z)1640 1198 y Fo(j)p Fm(y)r Fo(j\025)p Fn(1)p Fm(=)p Fn(2)p Fo(j)p Fm(x)p Fo(j)1977 1108 y Fp(y)2025 1123 y Fm(i)2053 1108 y Fp(y)2101 1123 y Fm(j)2137 1108 y Fp(f)11 b Fv(\()p Fp(y)t Fv(\))17 b Fp(dy)2444 963 y Fk(\014)2444 1023 y(\014)2444 1083 y(\014)2444 1142 y(\014)2502 1108 y Fq(\024)28 b Fp(C)23 b(C)2770 1123 y Fn(0)2810 1108 y Fq(j)p Fp(x)p Fq(j)2921 1066 y Fo(\000)p Fn(4)266 1354 y Fv(d.-)558 1627 y Fq(j)p Fp(d)p Fq(j)j(\024)i Fp(C)c(C)960 1642 y Fn(0)999 1627 y Fq(j)p Fp(x)p Fq(j)1110 1586 y Fo(\000)p Fn(4)1221 1483 y Fk(\014)1221 1543 y(\014)1221 1603 y(\014)1221 1662 y(\014)1254 1492 y(Z)1310 1717 y Fo(j)p Fm(y)r Fo(j\024)p Fn(1)p Fm(=)p Fn(2)p Fo(j)p Fm(x)p Fo(j)1630 1627 y Fq(h)p Fp(y)t Fq(i)1760 1586 y Fo(\000)p Fn(3)1869 1627 y Fp(dy)1972 1483 y Fk(\014)1972 1543 y(\014)1972 1603 y(\014)1972 1662 y(\014)2032 1627 y Fq(\024)k Fp(C)c(C)2301 1642 y Fn(0)2340 1627 y Fq(j)p Fp(x)p Fq(j)2451 1586 y Fo(\000)p Fn(4)2545 1627 y Fv(\(1)e(+)g(log)2878 1651 y Fn(+)2954 1627 y Fq(j)p Fp(x)p Fq(j)p Fv(\))120 1850 y(and)33 b(therefore)g(w)m (e)g(obtain)f(the)h(estimate)387 1925 y Fk(\()1213 2005 y Fq(jj)22 b(\001)f(j)1368 1969 y Fo(\000)p Fn(1)1485 2005 y Fq(\003)g Fp(f)11 b Fv(\()p Fp(x)p Fv(\))p Fq(j)83 b(\024)h Fp(C)23 b Fv(max)p Fq(fk)p Fp(f)11 b Fq(k)2502 2020 y Fn(1)2540 2005 y Fp(;)17 b(C)2654 2020 y Fn(0)2693 2005 y Fq(gj)p Fp(x)p Fq(j)2854 1969 y Fo(\000)p Fn(1)509 2041 y Fk(\014)509 2101 y(\014)509 2160 y(\014)542 2156 y Fq(j)22 b(\001)g(j)670 2119 y Fo(\000)p Fn(1)786 2156 y Fq(\003)g Fp(f)11 b Fv(\()p Fp(x)p Fv(\))22 b Fq(\000)1179 2105 y Fn(3)p Fm(x)1254 2115 y Fi(i)1281 2105 y Fm(x)1321 2115 y Fi(j)1353 2105 y Fo(\000)p Fm(\016)1439 2115 y Fi(ij)1494 2105 y Fo(j)p Fm(x)p Fo(j)1574 2081 y Fj(2)p 1179 2133 428 4 v 1337 2190 a Fo(j)p Fm(x)p Fo(j)1417 2171 y Fj(5)1617 2156 y Fp(F)1680 2171 y Fm(ij)1741 2041 y Fk(\014)1741 2101 y(\014)1741 2160 y(\014)1857 2156 y Fq(\024)84 b Fp(C)23 b(C)2181 2171 y Fn(0)2220 2156 y Fq(j)p Fp(x)p Fq(j)2331 2119 y Fo(\000)p Fn(4)2426 2156 y Fv(\(1)e(+)h(log)2758 2179 y Fn(+)2834 2156 y Fq(j)p Fp(x)p Fq(j)p Fv(\))3318 2095 y(\(207\))120 2359 y(No)m(w)33 b(w)m(e)h(write)239 2566 y(2)p Fp(D)s Fv(\()p Fp(f)5 b(;)17 b(g)t Fv(\()p Fq(\001)j Fv(+)i Fp(R)q Fv(\)\))83 b(=)1135 2430 y Fk(Z)1251 2566 y Fp(f)11 b Fv(\()p Fp(x)p Fv(\))p Fq(j)22 b(\001)g(j)1569 2525 y Fo(\000)p Fn(1)1685 2566 y Fq(\003)g Fp(g)t Fv(\()p Fp(x)g Fv(+)g Fp(R)q Fv(\))17 b Fp(dx)1061 b Fv(\(208\))976 2821 y(=)1135 2685 y Fk(Z)1190 2910 y Fo(j)p Fm(x)p Fo(j\024)p Fn(1)p Fm(=)p Fn(2)p Fo(j)p Fm(R)p Fo(j)1544 2821 y Fv(+)1637 2685 y Fk(Z)1691 2910 y Fn(1)p Fm(=)p Fn(2)p Fo(j)p Fm(R)p Fo(j\024j)p Fm(x)p Fo(j\024)p Fn(2)p Fo(j)p Fm(R)p Fo(j)2228 2821 y Fv(+)2321 2685 y Fk(Z)2375 2910 y Fo(j)p Fm(x)p Fo(j\024)p Fn(1)p Fm(=)p Fn(2)p Fo(j)p Fm(R)p Fo(j)2740 2821 y Fv(=)27 b Fp(A)2916 2836 y Fn(1)2978 2821 y Fv(+)22 b Fp(A)3149 2836 y Fn(2)3211 2821 y Fv(+)g Fp(A)3382 2836 y Fn(3)120 3073 y Fv(and)37 b(w)m(e)h(estimate)e(an)m(y)i(of)e (the)h(terms)g(of)g(\(208\))f(using)g(the)i(estimate)e(\(207\))g (\(applied)f(to)120 3193 y Fp(g)t Fv(\):)266 3314 y Fp(A)339 3329 y Fn(1)379 3314 y Fv(.-)266 3434 y(If)j Fq(j)p Fp(x)p Fq(j)d(\024)h Fv(1)p Fp(=)p Fv(2)p Fq(j)p Fp(R)q Fq(j)p Fv(,)h Fq(jj)22 b(\001)g(j)1126 3398 y Fo(\000)p Fn(1)1242 3434 y Fq(\003)g Fp(g)t Fv(\()p Fp(x)p Fv(\))p Fq(j)35 b(\024)h Fp(C)23 b Fv(max)p Fq(fk)p Fp(g)t Fq(k)2148 3449 y Fn(1)2186 3434 y Fp(;)17 b(C)2300 3449 y Fn(0)2339 3434 y Fq(gj)p Fp(R)q Fq(j)2520 3398 y Fo(\000)p Fn(1)2651 3434 y Fv(and)37 b(therefore)h Fq(j)p Fp(A)3360 3449 y Fn(1)3399 3434 y Fq(j)e(\024)120 3554 y Fp(C)24 b Fv(max)o Fq(fk)p Fp(g)t Fq(k)596 3569 y Fn(1)634 3554 y Fp(;)17 b(C)748 3569 y Fn(0)787 3554 y Fq(gj)p Fp(R)q Fq(j)968 3518 y Fo(\000)p Fn(1)1062 3554 y Fq(k)p Fp(f)11 b Fq(k)1221 3569 y Fn(1)1260 3554 y Fv(.)43 b(If)32 b(b)s(esides)i Fq(j)p Fp(R)q Fq(j)27 b(\025)h Fv(1,)33 b(w)m(e)g(can)g(write)722 3787 y Fq(j)22 b(\001)f(j)849 3746 y Fo(\000)p Fn(1)966 3787 y Fq(\003)h Fp(g)t Fv(\()p Fp(x)p Fv(\))27 b(=)1360 3719 y(6)p Fp(x)1464 3734 y Fm(i)1493 3719 y Fp(R)1567 3734 y Fm(j)1626 3719 y Fq(\000)22 b Fv(2)p Fp(x)1829 3734 y Fm(i)1858 3719 y Fp(R)1932 3734 y Fm(i)1960 3719 y Fp(\016)2003 3734 y Fm(ij)p 1360 3764 704 4 v 1627 3855 a Fq(j)p Fp(R)q Fq(j)1758 3826 y Fn(5)2074 3787 y Fp(G)2151 3802 y Fm(ij)2233 3787 y Fv(+)2362 3719 y Fp(G)2439 3734 y Fm(ii)p 2341 3764 170 4 v 2341 3855 a Fq(j)p Fp(R)q Fq(j)2472 3826 y Fn(3)2543 3787 y Fv(+)g Fp(r)s Fv(\()p Fp(x;)17 b(R)q Fv(\))380 b(\(209\))120 4030 y(where)28 b(in)f(\(209\),)g Fq(j)p Fp(r)s Fv(\()p Fp(x;)17 b(R)q Fv(\))p Fq(j)27 b(\024)h Fp(C)c Fv(max)o Fq(fk)p Fp(g)t Fq(k)1743 4045 y Fn(1)1781 4030 y Fp(;)17 b(C)1895 4045 y Fn(0)1934 4030 y Fq(g)p Fv(\()p Fq(j)p Fp(R)q Fq(j)2153 3994 y Fo(\000)p Fn(4)2247 4030 y Fq(j)p Fp(x)p Fq(j)11 b Fv(+)g Fq(j)p Fp(R)q Fq(j)2587 3994 y Fo(\000)p Fn(4)2697 4030 y Fv(log)2823 4053 y Fn(+)2899 4030 y Fq(j)p Fp(R)q Fq(j)p Fv(\).)41 b(Using)27 b(the)120 4150 y(cancellation)k(conditions)g Fp(f)43 b Fv(satisfy)-8 b(,)33 b(w)m(e)h(get,)e(reasoning)g(as)h(ab)s(o)m(v)m(e,)1111 4333 y Fq(j)p Fp(A)1212 4348 y Fn(1)1251 4333 y Fq(j)28 b(\024)g Fp(C)c Fv(max)o Fq(fk)p Fp(f)11 b Fq(k)1896 4348 y Fn(1)1935 4333 y Fq(k)p Fp(g)t Fq(k)2086 4348 y Fn(1)2124 4333 y Fp(;)17 b(C)2245 4292 y Fn(2)2238 4358 y(0)2284 4333 y Fq(g)p Fp(')p Fv(\()p Fp(R)q Fv(\))120 4516 y(The)34 b(other)e(t)m(w)m(o)h(terms)g(are)g(estimated)f (similarly:)266 4637 y Fp(A)339 4652 y Fn(2)379 4637 y Fv(:-)151 4964 y Fq(j)p Fp(A)252 4979 y Fn(2)291 4964 y Fq(j)83 b(\024)g Fp(C)24 b(C)726 4979 y Fn(0)765 4964 y Fq(j)p Fp(R)q Fq(j)896 4923 y Fo(\000)p Fn(3)990 4964 y Fq(h)p Fp(R)q Fq(i)1143 4923 y Fo(\000)p Fn(3)1253 4828 y Fk(Z)1309 5054 y Fn(1)p Fm(=)p Fn(2)p Fo(j)p Fm(R)p Fo(j\024j)p Fm(y)r Fo(\000)p Fm(R)p Fo(j\024)p Fn(2)p Fo(j)p Fm(R)p Fo(j)1951 4879 y Fk(\014)1951 4939 y(\014)1984 4964 y Fq(j)e(\001)g(j)2112 4923 y Fo(\000)p Fn(1)2228 4964 y Fq(\003)g Fp(g)t Fv(\()p Fp(x)p Fv(\))2482 4879 y Fk(\014)2482 4939 y(\014)2548 4964 y Fp(dx)402 5239 y Fq(\024)83 b Fp(C)24 b Fv(max)p Fq(fk)p Fp(f)11 b Fq(k)1047 5254 y Fn(1)1085 5239 y Fq(k)p Fp(g)t Fq(k)1236 5254 y Fn(1)1275 5239 y Fp(;)17 b(C)1396 5198 y Fn(2)1389 5264 y(0)1435 5239 y Fq(gj)p Fp(R)q Fq(j)1616 5198 y Fo(\000)p Fn(3)1709 5239 y Fq(h)p Fp(R)q Fq(i)1862 5198 y Fo(\000)p Fn(3)1973 5104 y Fk(Z)2028 5329 y Fn(1)p Fm(=)p Fn(2)p Fo(j)p Fm(R)p Fo(j\024j)p Fm(y)r Fo(\000)p Fm(R)p Fo(j\024)p Fn(2)p Fo(j)p Fm(R)p Fo(j)2670 5239 y Fv(min)o Fq(fj)p Fp(x)p Fq(j)2994 5198 y Fo(\000)p Fn(1)3088 5239 y Fp(;)g Fq(j)p Fp(x)p Fq(j)3243 5198 y Fo(\000)p Fn(3)3336 5239 y Fq(g)g Fp(dx)402 5463 y Fq(\024)83 b Fp(C)24 b Fv(max)p Fq(fk)p Fp(f)11 b Fq(k)1047 5478 y Fn(1)1085 5463 y Fq(k)p Fp(g)t Fq(k)1236 5478 y Fn(1)1275 5463 y Fp(;)17 b(C)1396 5422 y Fn(2)1389 5487 y(0)1435 5463 y Fq(gj)p Fp(R)q Fq(j)1616 5422 y Fo(\000)p Fn(1)1709 5463 y Fq(h)p Fp(R)q Fq(i)1862 5422 y Fo(\000)p Fn(3)1781 5712 y Fv(49)p eop %%Page: 50 50 50 49 bop 266 407 a Fp(A)339 422 y Fn(3)379 407 y Fv(:-)453 715 y Fq(j)p Fp(A)554 730 y Fn(3)593 715 y Fq(j)83 b(\024)g Fp(C)24 b Fv(max)o Fq(fk)p Fp(g)t Fq(k)1340 730 y Fn(1)1379 715 y Fp(;)17 b(C)1493 730 y Fn(0)1532 715 y Fq(g)1599 580 y Fk(Z)1653 805 y Fo(j)p Fm(x)p Fo(j\025)p Fn(2)p Fo(j)p Fm(R)p Fo(j)1936 715 y Fv(min)n Fq(fj)p Fp(x)p Fq(j)2259 674 y Fo(\000)p Fn(1)2353 715 y Fp(;)g Fq(j)p Fp(x)p Fq(j)2508 674 y Fo(\000)p Fn(3)2602 715 y Fq(gj)p Fp(x)p Fq(j)2763 674 y Fo(\000)p Fn(3)2857 715 y Fq(h)p Fp(x)p Fq(i)2990 674 y Fo(\000)p Fn(3)3101 715 y Fp(dx)704 939 y Fq(\024)83 b Fp(C)24 b Fv(max)o Fq(fk)p Fp(f)11 b Fq(k)1348 954 y Fn(1)1387 939 y Fq(k)p Fp(g)t Fq(k)1538 954 y Fn(1)1576 939 y Fp(;)17 b(C)1697 898 y Fn(2)1690 964 y(0)1736 939 y Fq(gj)p Fp(R)q Fq(j)1917 898 y Fo(\000)p Fn(1)2011 939 y Fq(h)p Fp(R)q Fq(i)2164 898 y Fo(\000)p Fn(3)120 1104 y Fv(so)33 b(w)m(e)g(obtain)f(\(204\))g(b)m(y)h (collecting)e(estimates.)p 3469 1104 4 66 v 3473 1041 59 4 v 3473 1104 V 3531 1104 4 66 v 266 1283 a(W)-8 b(e)41 b(claim)c(that)j(under)h(the)f(conditions)f(giv)m(en)h(b)m(y)h(eqs.)67 b(\(172\))39 b(and)i(\(173\),)g(w)m(e)g(can)120 1403 y(estimate)954 1600 y Fp(D)1054 1460 y Fk(\022)1128 1600 y Fp(\032)1178 1615 y Fn(\000)1248 1600 y Fq(\000)1455 1533 y Fv(1)p 1358 1578 243 4 v 1358 1669 a(6)p Fp(\031)1466 1640 y Fn(2)1505 1669 y Fp(h)1561 1640 y Fn(3)1611 1600 y Fv(\(\010)1719 1559 y Fn(TF)1719 1625 y Fo(R)p Fm(;)p Fo(Z)1859 1600 y Fv(\))1897 1559 y Fn(3)p Fm(=)p Fn(2)2007 1460 y Fk(\023)2108 1600 y Fq(\024)28 b Fp(C)7 b(h)2346 1559 y Fo(\000)p Fn(4)p Fo(\000)p Fn(2)p Fm(=)p Fn(9)2601 1600 y Fp(M)623 b Fv(\(210\))266 1834 y(T)-8 b(o)33 b(pro)m(v)m(e)h (this)e(fact,)g(w)m(e)i(write,)e(with)h Fp(F)41 b Fv(:=)28 b Fp(\032)2010 1849 y Fn(\000)2080 1834 y Fq(\000)2249 1794 y Fn(1)p 2190 1811 154 4 v 2190 1868 a(6)p Fm(\031)r(h)2309 1849 y Fj(3)2353 1834 y Fv(\(\010)2461 1797 y Fn(TF)2461 1859 y Fo(R)p Fm(;)p Fo(Z)2602 1834 y Fv(\))2640 1797 y Fn(3)p Fm(=)p Fn(2)2782 1834 y Fv(and)120 1988 y Fp(F)183 2003 y Fm(j)247 1988 y Fv(:=)g Fp(G)455 2003 y Fm(j)514 1988 y Fq(\000)651 1949 y Fm(K)5 b(s)749 1926 y Fj(2)p 623 1965 189 4 v 623 2023 a Fn(6)p Fm(\031)701 2004 y Fj(2)736 2023 y Fm(h)777 2004 y Fj(3)838 1908 y Fk(\000)883 1988 y Fv(\010)953 1952 y Fn(TF)953 2013 y Fo(R)p Fm(;)p Fo(Z)1094 1908 y Fk(\001)1140 1930 y Fn(3)p Fm(=)p Fn(2)1266 1988 y Fv(\001\007)1423 1952 y Fn(2)1423 2013 y Fm(j)1485 1988 y Fq(\000)1585 1908 y Fk(\000)1630 1988 y Fv(\010)1700 1952 y Fn(TF)1700 2013 y Fo(R)p Fm(;)p Fo(Z)1841 1908 y Fk(\001)1886 1930 y Fn(3)p Fm(=)p Fn(2)2013 1988 y Fv(\007)2089 1952 y Fn(2)2089 2013 y Fm(j)2128 1988 y Fv(;)34 b Fp(j)f Fv(=)28 b(1)p Fp(;)17 b(:)g(:)g(:)32 b(;)17 b(M)32 b Fv(\(so,)24 b Fp(F)42 b Fv(=)3160 1913 y Fk(P)3265 1940 y Fm(M)3265 2017 y(j)t Fn(=1)3409 1988 y Fp(F)3472 2003 y Fm(j)3508 1988 y Fv(\).)354 2187 y Fp(D)s Fv(\()p Fp(F)14 b Fv(\))27 b(=)876 2093 y Fk(X)722 2309 y Fm(i;j)t Fn(:)p Fo(j)p Fm(R)892 2319 y Fi(i)916 2309 y Fo(\000)p Fm(R)1024 2319 y Fi(j)1057 2309 y Fo(j\024)p Fm(\026)1190 2187 y Fp(D)s Fv(\()p Fp(F)1375 2202 y Fm(i)1403 2187 y Fp(;)17 b(F)1510 2202 y Fm(j)1547 2187 y Fv(\))22 b(+)g(2)1924 2093 y Fk(X)1771 2309 y Fm(i;j)t Fn(:)p Fo(j)p Fm(R)1941 2319 y Fi(i)1965 2309 y Fo(\000)p Fm(R)2073 2319 y Fi(j)2106 2309 y Fo(j)p Fm(>\026)2239 2187 y Fp(D)s Fv(\()p Fp(F)2424 2202 y Fm(i)2452 2187 y Fp(;)17 b(F)2559 2202 y Fm(j)2595 2187 y Fv(\))28 b(=)g Fp(S)2825 2202 y Fn(1)2886 2187 y Fv(+)22 b Fp(S)3044 2202 y Fn(2)3318 2187 y Fv(\(211\))120 2460 y(The)48 b(terms)e Fp(F)683 2475 y Fm(j)766 2460 y Fv(satisfy)h(the)g(cancellation)e(conditions)g (needed)k(to)d(apply)g(Lemma)f(14.)120 2580 y(Also,)31 b(giv)m(en)g(the)h(c)m(hoices)g(of)e(the)i(parameters)f Fp(r)m(;)17 b(s)31 b Fv(obtained)g(in)f(the)i(pro)s(of)e(of)h(Lemma)e (13)120 2701 y(\(namely)-8 b(,)46 b Fp(r)k Fq(\030)e Fp(h)805 2665 y Fn(4)p Fm(=)p Fn(9)959 2701 y Fv(and)c Fp(s)k Fq(\030)f Fp(h)1434 2665 y Fn(2)p Fm(=)p Fn(3)1545 2701 y Fv(\),)g Fp(s)1703 2665 y Fn(2)1742 2701 y Fp(r)1789 2665 y Fo(\000)p Fn(2)1883 2701 y Fp(h)1939 2665 y Fo(\000)p Fn(3)2081 2701 y Fq(6)p Fv(=)g Fq(O)s Fv(\()p Fp(h)2380 2665 y Fo(\000)p Fn(2)2474 2701 y Fv(\))d(meaning)f(that)h(the)h(sub-) 120 2821 y(terms)437 2782 y Fm(K)5 b(s)535 2759 y Fj(2)p 409 2798 V 409 2856 a Fn(6)p Fm(\031)487 2837 y Fj(2)522 2856 y Fm(h)563 2837 y Fj(3)607 2821 y Fv(\001)p Fp(\030)736 2785 y Fn(2)731 2846 y Fm(j)815 2821 y Fv(app)s(earing)40 b(in)f(the)i Fp(F)1650 2785 y Fo(0)1636 2846 y Fm(j)1673 2821 y Fp(s)f Fv(cannot)g(b)s(e)h(absorv)m(ed)g(in)m(to)f(the)g(main)f (terms,)120 2942 y(so)46 b(Lemma)e(14)g(is)h(not)g(directly)g (applicable.)80 b(Nev)m(ertheless,)51 b(this)45 b(trouble)g(is)g(easy)h (to)120 3062 y(o)m(v)m(ercome)34 b(if)e(w)m(e)i(notice)e(that)h(the)g (bad)g(subterms)2094 3023 y Fm(K)5 b(s)2192 2999 y Fj(2)p 2066 3039 V 2066 3096 a Fn(6)p Fm(\031)2144 3077 y Fj(2)2179 3096 y Fm(h)2220 3077 y Fj(3)2264 3062 y Fv(\001)p Fp(\030)2393 3026 y Fn(2)2388 3087 y Fm(j)2465 3062 y Fv(are)33 b(supp)s(orted)h(in) m(to)e(a)h(thin)120 3194 y(neigh)m(b)s(orho)s(o)s(d)e(of)h Fp(@)5 b(D)980 3209 y Fm(j)1050 3194 y Fv(of)33 b(thic)m(kness)h Fp(r)c Fq(\030)e Fp(h)1815 3158 y Fn(4)p Fm(=)p Fn(9)1926 3194 y Fv(.)43 b(This)33 b(giv)m(es)g(us)1560 3330 y Fm(K)5 b(s)1658 3306 y Fj(2)p 1532 3346 V 1532 3403 a Fn(6)p Fm(\031)1610 3384 y Fj(2)1645 3403 y Fm(h)1686 3384 y Fj(3)1747 3289 y Fk(R)1830 3369 y Fv(\001)p Fp(\030)1959 3333 y Fn(2)1954 3394 y Fm(j)1035 3462 y(K)g(s)1133 3439 y Fj(2)p 1007 3478 V 1007 3536 a Fn(6)p Fm(\031)1085 3517 y Fj(2)1120 3536 y Fm(h)1161 3517 y Fj(3)1222 3421 y Fk(R)1305 3501 y Fv(\001)p Fp(\030)1434 3465 y Fn(2)1429 3526 y Fm(j)1473 3501 y Fv(\()p Fp(x)1566 3516 y Fm(k)1631 3501 y Fq(\000)23 b Fv(\()p Fp(R)1843 3516 y Fm(j)1879 3501 y Fv(\))1917 3516 y Fm(k)1960 3501 y Fv(\))2039 3291 y Fk(\033)2142 3431 y Fv(=)k Fq(O)s Fv(\()p Fp(h)2421 3390 y Fo(\000)p Fn(2)p Fo(\000)p Fn(1)p Fm(=)p Fn(9)2677 3431 y Fv(\))120 3666 y(Moreo)m(v)m(er,)k(b)m(y)f(mimic)m(king)25 b(the)k(pro)s(of)f(of)g(Lemma)f(14)i(w)m(e)g(can)g(obtain)f(that)g Fq(j)p Fp(D)s Fv(\()p Fp(F)3198 3681 y Fm(i)3226 3666 y Fp(;)17 b(F)3333 3681 y Fm(j)3369 3666 y Fv(\))p Fq(j)28 b(\024)120 3787 y Fp(C)7 b(h)253 3750 y Fo(\000)p Fn(2)p Fo(\000)p Fn(1)p Fm(=)p Fn(9)508 3787 y Fp(')p Fv(\()p Fp(R)684 3802 y Fm(i)733 3787 y Fq(\000)21 b Fp(R)905 3802 y Fm(j)942 3787 y Fv(\).)43 b(I)32 b(shall)e(omit)g(to)h(c)m(hec)m (k)j(this)e(since)g(only)f(rep)s(eats)i(the)f(argumen)m(ts)120 3907 y(leading)f(to)h(Lemma)f(14)h(with)h(some)f(minor)f(mo)s (di\014cations.)266 4027 y(The)g(\014rst)f(sum)f(in)g(\(211\))f(con)m (tains)i(no)f(more)g(than)g(2)2253 3991 y Fn(3)p Fm(=)p Fn(2)2363 4027 y Fp(M)40 b Fv(summands)29 b(\(giv)m(en)h(that)f(it)120 4148 y(is)d(not)g(p)s(ossible)g(to)g(\014t)g(more)g(than)g(2)1479 4112 y Fn(3)p Fm(=)p Fn(2)1615 4148 y Fv(disjoin)m(t)f(balls)g(of)h (radius)g(1)g(inside)g(a)g(ball)e(of)i(radius)120 4268 y(2\).)43 b(Since)33 b Fq(j)p Fp(D)s Fv(\()p Fp(F)745 4283 y Fm(i)773 4268 y Fp(;)17 b(F)880 4283 y Fm(j)916 4268 y Fv(\))p Fq(j)27 b(\024)h Fv(2\()p Fp(D)s Fv(\()p Fp(F)1386 4283 y Fm(i)1414 4268 y Fv(\))22 b(+)g Fp(D)s Fv(\()p Fp(F)1757 4283 y Fm(j)1793 4268 y Fv(\)\),)33 b(w)m(e)g(can)g(estimate,)f(using)g(the)h(inequalit)m(y)1475 4433 y Fp(D)s Fv(\()p Fp(f)11 b Fv(\))27 b Fq(\024)h Fp(K)7 b Fq(k)p Fp(f)k Fq(k)2075 4392 y Fn(2)2075 4457 y(6)p Fm(=)p Fn(5)120 4597 y Fv(whic)m(h)33 b(follo)m(ws)e(from)h(the)h (Hardy-Littlew)m(o)s(o)s(d-Sob)s(olev)d(inequalit)m(y)1333 4849 y Fq(j)p Fp(S)1421 4864 y Fn(1)1460 4849 y Fq(j)83 b(\024)h Fv(2)1781 4808 y Fn(5)p Fm(=)p Fn(2)1941 4725 y Fm(M)1907 4755 y Fk(X)1918 4965 y Fm(j)t Fn(=1)2067 4849 y Fp(D)s Fv(\()p Fp(F)2252 4864 y Fm(j)2289 4849 y Fv(\))1571 5201 y Fq(\024)g Fp(C)1860 5076 y Fm(M)1825 5106 y Fk(X)1836 5316 y Fm(j)t Fn(=1)1986 5201 y Fq(k)p Fp(F)2099 5216 y Fm(j)2135 5201 y Fq(k)2185 5159 y Fn(2)2185 5225 y(6)p Fm(=)p Fn(5)3318 5201 y Fv(\(212\))1571 5463 y Fq(\024)g Fp(C)7 b(h)1865 5422 y Fo(\000)p Fn(4)p Fo(\000)p Fn(2)p Fm(=)p Fn(9)2120 5463 y Fp(M)1781 5712 y Fv(50)p eop %%Page: 51 51 51 50 bop 120 407 a Fv(since)44 b(it)f(is)h(easily)f(v)m(eri\014ed)h (that)g(the)g Fp(F)1700 371 y Fo(0)1686 431 y Fm(j)1723 407 y Fp(s)g Fv(satify)f Fq(k)p Fp(F)2203 422 y Fm(j)2240 407 y Fq(k)2290 422 y Fn(6)p Fm(=)p Fn(5)2447 407 y Fq(\024)k Fp(C)7 b(h)2704 371 y Fo(\000)p Fn(2)p Fo(\000)p Fn(1)p Fm(=)p Fn(9)3003 407 y Fv(uniformly)42 b(in)120 527 y Fp(h)28 b(>)f Fv(0.)44 b(T)-8 b(o)32 b(estimate)f(the)i(second)h(sum)e (in)g(\(211\),)f(w)m(e)j(use)f(P)m(art)f(B.)h(of)f(Lemma)f(14)g(whic)m (h)120 648 y(yields,)h(with)h Fp(')p Fv(\()p Fp(x)p Fv(\))27 b(:=)h Fq(j)p Fp(x)p Fq(j)1108 611 y Fo(\000)p Fn(4)1202 648 y Fv(\(1)22 b(+)g(log)1535 671 y Fn(+)1611 648 y Fq(j)p Fp(x)p Fq(j)p Fv(\))893 859 y Fq(j)p Fp(S)981 874 y Fn(2)1020 859 y Fq(j)27 b(\024)i Fp(C)7 b(h)1314 818 y Fo(\000)p Fn(4)p Fo(\000)p Fn(2)p Fm(=)p Fn(9)1740 764 y Fk(X)1585 980 y Fm(i;j)t Fn(:)p Fo(j)p Fm(R)1755 990 y Fi(i)1780 980 y Fo(\000)p Fm(R)1888 990 y Fi(j)1921 980 y Fo(j)p Fm(>\026)2054 859 y Fp(')17 b Fv(\()p Fq(j)p Fp(R)2275 874 y Fm(i)2325 859 y Fq(\000)23 b Fp(R)2499 874 y Fm(j)2535 859 y Fq(j)p Fv(\))717 b(\(213\))120 1161 y(T)-8 b(o)41 b(estimate)e(the)i(sum)g(in)f(the)h(r.h.s)g(of)f (\(213\))g(w)m(e)i(pro)s(ceed)f(in)f(a)g(v)m(ery)i(similar)37 b(w)m(a)m(y)42 b(to)120 1281 y(what)28 b(w)m(e)i(ha)m(v)m(e)f(done)g (to)e(pro)m(v)m(e)j(the)e(main)f(estimate)g(ab)s(out)g(the)i(size)f(of) g(the)g(TF)h(p)s(oten)m(tial)120 1402 y(\(Theorem)36 b(1\).)53 b(W)-8 b(e)36 b(shall)f(not)h(rep)s(eat)g(the)g(details)f (here,)i(but)f(w)m(e)h(p)s(oin)m(t)e(out)h(that)g(what)120 1522 y(w)m(e)e(obtain)d(is)h(the)h(b)s(ound)981 1778 y Fq(j)p Fp(S)1069 1793 y Fn(2)1108 1778 y Fq(j)82 b(\024)i Fp(C)7 b(h)1512 1737 y Fo(\000)p Fn(4)p Fo(\000)p Fn(2)p Fm(=)p Fn(9)1818 1654 y Fm(M)1784 1684 y Fk(X)1794 1893 y Fm(j)t Fn(=1)1944 1684 y Fk(X)1950 1894 y Fm(n)p Fo(\025)p Fn(1)2105 1778 y Fp(')2186 1697 y Fk(\000)2231 1778 y Fp(cn)2331 1737 y Fn(1)p Fm(=)p Fn(3)2441 1778 y Fp(\026)2500 1697 y Fk(\001)1218 2054 y Fq(\024)84 b Fp(C)7 b(h)1512 2013 y Fo(\000)p Fn(4)p Fo(\000)p Fn(2)p Fm(=)p Fn(9)1767 2054 y Fp(')17 b Fv(\()p Fp(\026)p Fv(\))f Fp(M)2120 1960 y Fk(X)2126 2170 y Fm(n)p Fo(\025)p Fn(1)2281 2054 y Fp(')2362 1973 y Fk(\000)2407 2054 y Fp(cn)2507 2013 y Fn(1)p Fm(=)p Fn(3)2617 1973 y Fk(\001)3318 2054 y Fv(\(214\))1218 2315 y Fq(\024)84 b Fp(C)1449 2330 y Fm(\026)1495 2315 y Fp(h)1551 2274 y Fo(\000)p Fn(4)p Fo(\000)p Fn(2)p Fm(=)p Fn(9)1807 2315 y Fp(M)120 2504 y Fv(whic)m(h,)43 b(together)e(with)g(estimate)f(\(212\))g(establishes) h(eq.)69 b(\(210\).)e(T)-8 b(o)41 b(\014nish)g(the)g(argu-)120 2624 y(men)m(t,)30 b(assume)g(no)m(w)g(that)f(w)m(e)h(ha)m(v)m(e)h(a)e (n)m(uclear)h(con\014guration)e Fq(R)p Fp(;)17 b Fq(Z)38 b Fv(whic)m(h)29 b(falls)f(in)m(to)h(the)120 2744 y(h)m(yp)s(othesis)34 b(of)f(Thm.)g(2.)44 b(W)-8 b(e)33 b(rescal)g(it)f(to)h(the)g(new)h (con\014guration)e Fp(Z)2791 2708 y Fn(1)p Fm(=)p Fn(3)2901 2744 y Fq(Z)8 b Fp(;)17 b(Z)3099 2708 y Fo(\000)p Fn(1)3193 2744 y Fq(Z)8 b Fv(.)45 b(T)-8 b(ak-)120 2865 y(ing)33 b(the)i(parameter)e Fp(h)28 b Fv(=)g Fp(Z)1183 2829 y Fo(\000)p Fn(1)p Fm(=)p Fn(3)1348 2865 y Fv(,)34 b(the)h(Hamiltonian)30 b Fp(H)42 b Fv(app)s(earing)33 b(at)h(the)g Fp(R)q(H)8 b(F)48 b Fv(mo)s(del)120 2985 y(is)33 b Fp(H)j Fv(=)29 b Fq(\000)p Fv(\001)24 b Fq(\000)f Fv(\010)793 2949 y Fn(TF)793 3010 y Fo(R)p Fm(;)p Fo(Z)966 2985 y Fv(whic)m(h)34 b(is)f(unitarily)e(equiv)-5 b(alen)m(t,)34 b(via)e(conjugation)g(with)h Fp(U)3214 3000 y Fm(h)3288 2985 y Fv(:)c Fp( )k Fq(!)120 3118 y Fp(h)176 3081 y Fo(\000)p Fn(3)p Fm(=)p Fn(2)341 3118 y Fp( )t Fv(\()p Fp(h)502 3081 y Fo(\000)p Fn(1)596 3118 y Fq(\001)p Fv(\))45 b(to)g Fp(h)895 3081 y Fo(\000)p Fn(4)990 3118 y Fv(\()p Fq(\000)p Fp(h)1161 3081 y Fn(2)1201 3118 y Fv(\001)22 b Fq(\000)h Fv(\010)1474 3081 y Fn(TF)1474 3154 y Fm(Z)1527 3135 y Fj(1)p Fi(=)p Fj(3)1623 3154 y Fo(Z)5 b Fm(;Z)1752 3135 y Fh(\000)p Fj(1)1834 3154 y Fo(Z)1894 3118 y Fv(\))45 b(\(in)g(that)g(case,)k Fp(H)57 b Fv(=)49 b Fp(h)2934 3081 y Fo(\000)p Fn(4)3028 3118 y Fp(U)3094 3133 y Fm(h)3139 3118 y Fv(\()p Fq(\000)p Fp(h)3310 3081 y Fn(2)3350 3118 y Fv(\001)32 b Fq(\000)120 3264 y Fv(\010)190 3227 y Fn(TF)190 3300 y Fm(Z)243 3281 y Fj(1)p Fi(=)p Fj(3)340 3300 y Fo(Z)5 b Fm(;Z)469 3281 y Fh(\000)p Fj(1)551 3300 y Fo(Z)611 3264 y Fv(\))p Fp(U)725 3227 y Fo(\003)715 3289 y Fm(h)765 3264 y Fv(\).)49 b(By)35 b(cyclit)m(y)g(of)f(the)h(trace,)g(T)-8 b(r)17 b([\()p Fq(\000)p Fp(h)2219 3227 y Fn(2)2259 3264 y Fv(\001)24 b Fq(\000)g Fv(\010)2535 3227 y Fn(TF)2535 3300 y Fm(Z)2588 3281 y Fj(1)p Fi(=)p Fj(3)2685 3300 y Fo(Z)5 b Fm(;Z)2814 3281 y Fh(\000)p Fj(1)2895 3300 y Fo(Z)2956 3264 y Fv(\)\000])31 b(=)f(T)-8 b(r)17 b([)p Fp(H)p 3452 3183 61 4 v 8 w Fv(\000])120 3409 y(with)p 343 3329 V 33 w(\000)30 b(:=)g Fp(U)633 3424 y Fm(h)678 3409 y Fv(\000)p Fp(U)815 3373 y Fo(\003)805 3435 y Fm(h)855 3409 y Fv(.)47 b(But)34 b(is)f(easily)g(v)m(eri\014ed)i (that)f(if)e Fp(K)2222 3424 y Fm(\015)2267 3409 y Fv(\()p Fp(x;)17 b(y)t Fv(\))33 b(is)g(the)h(Sc)m(h)m(w)m(arz)i(k)m(ernel)e(of) 120 3530 y(a)e(densit)m(y)i(matrix)d Fp(\015)5 b Fv(,)33 b(with)p 1193 3475 57 4 v 32 w Fp(\015)k Fv(as)c(ab)s(o)m(v)m(e)g(w)m (e)h(ha)m(v)m(e)g(the)f(form)m(ulas)1038 3630 y Fk(\032)1154 3710 y Fp(K)p 1237 3687 45 3 v 15 x Fm(\015)1281 3710 y Fv(\()p Fp(x;)17 b(y)t Fv(\))82 b(=)h Fp(h)1805 3674 y Fo(\000)p Fn(3)1900 3710 y Fp(K)1983 3725 y Fm(\015)2027 3710 y Fv(\()p Fp(h)2121 3674 y Fo(\000)p Fn(1)2216 3710 y Fp(x;)17 b(h)2371 3674 y Fo(\000)p Fn(1)2465 3710 y Fp(y)t Fv(\))1281 3830 y Fp(\032)p 1331 3808 V 15 x Fm(\015)1376 3830 y Fv(\()p Fp(x)p Fv(\))83 b(=)g Fp(h)1805 3794 y Fo(\000)p Fn(3)1900 3830 y Fp(\032)1950 3845 y Fm(\015)1995 3830 y Fv(\()p Fp(h)2089 3794 y Fo(\000)p Fn(1)2183 3830 y Fp(x)p Fv(\))3318 3771 y(\(215\))266 4012 y(No)m(w,)32 b(what)f(w)m(e)h(do)e(obtain)g(are)h(the)g(follo)m(wing)d(estimates)i (with)g(\000)h(b)s(eing)f(the)h(densit)m(y)120 4132 y(matrix)g (constructed)j(ab)s(o)m(v)m(e:)120 4388 y(T)-8 b(r)17 b([\(\001)22 b Fq(\000)h Fv(\(\010)613 4347 y Fn(TF)613 4413 y Fo(R)p Fm(;)p Fo(Z)753 4388 y Fv(\))791 4347 y Fn(5)p Fm(=)p Fn(2)901 4388 y Fv(\))p 939 4308 61 4 v(\000])28 b Fq(\024)g(\000)1369 4321 y Fv(1)p 1247 4365 292 4 v 1247 4457 a(15)p Fp(\031)1404 4428 y Fn(2)1443 4457 y Fp(h)1499 4428 y Fn(7)1566 4253 y Fk(Z)1665 4388 y Fv(\()p Fp(')1767 4347 y Fn(TF)1767 4420 y Fm(Z)1820 4400 y Fj(1)p Fi(=)p Fj(3)1916 4420 y Fo(R)p Fm(;Z)2049 4401 y Fh(\000)p Fj(1)2131 4420 y Fo(Z)2192 4388 y Fv(\))2230 4347 y Fn(5)p Fm(=)p Fn(2)2362 4388 y Fv(+)2517 4321 y(1)p 2470 4365 145 4 v 2470 4457 a(8)p Fp(h)2575 4428 y Fn(6)2675 4264 y Fm(M)2641 4294 y Fk(X)2651 4504 y Fm(j)t Fn(=1)2811 4312 y Fp(Z)2885 4276 y Fn(2)2878 4337 y Fm(j)p 2811 4365 114 4 v 2811 4457 a Fp(Z)2885 4428 y Fn(2)2956 4388 y Fq(\000)23 b Fp(C)7 b(M)j(h)3293 4347 y Fo(\000)p Fn(5)p Fo(\000)p Fn(4)p Fm(=)p Fn(9)3318 4619 y Fv(\(216\))120 4819 y(By)38 b(the)g(TF)g(scaling)e(\(14\))h(and)h(the)g(relation)e(T) -8 b(r)16 b([\()p Fq(\000)p Fp(h)2204 4783 y Fn(2)2245 4819 y Fv(\001)25 b Fq(\000)h Fv(\010)2524 4783 y Fn(TF)2524 4856 y Fm(Z)2577 4837 y Fj(1)p Fi(=)p Fj(3)2674 4856 y Fo(Z)5 b Fm(;Z)2803 4837 y Fh(\000)p Fj(1)2885 4856 y Fo(Z)2945 4819 y Fv(\)\000])36 b(=)g(T)-8 b(r)17 b([)p Fp(H)p 3452 4739 61 4 v 8 w Fv(\000])120 4940 y(w)m(e)34 b(obtain)248 5196 y(T)-8 b(r)16 b([\(\001)23 b Fq(\000)g Fv(\(\010)741 5155 y Fn(TF)741 5221 y Fo(R)p Fm(;)p Fo(Z)881 5196 y Fv(\))919 5155 y Fn(5)p Fm(=)p Fn(2)p 1029 5116 V 1029 5196 a Fv(\000])28 b Fq(\024)g(\000)1411 5129 y Fv(1)p 1337 5173 196 4 v 1337 5264 a(15)p Fp(\031)1494 5236 y Fn(2)1560 5060 y Fk(Z)1659 5196 y Fv(\(\010)1767 5155 y Fn(TF)1767 5221 y Fo(R)p Fm(;)p Fo(Z)1908 5196 y Fv(\))1946 5155 y Fn(5)p Fm(=)p Fn(2)2078 5196 y Fv(+)2186 5129 y(1)p 2186 5173 49 4 v 2186 5264 a(8)2296 5071 y Fm(M)2261 5101 y Fk(X)2272 5311 y Fm(j)t Fn(=1)2422 5196 y Fp(Z)2496 5155 y Fn(2)2489 5221 y Fm(j)2557 5196 y Fq(\000)23 b Fp(C)7 b(M)j(Z)2912 5155 y Fn(2)p Fo(\000)p Fm(\016)3041 5196 y Fv(;)49 b Fp(\016)31 b Fv(=)3329 5129 y(5)p 3305 5173 98 4 v 3305 5264 a(27)3318 5427 y(\(217\))1781 5712 y(51)p eop %%Page: 52 52 52 51 bop 120 407 a Fv(and)33 b(using)f(the)h(estimate)f(\(210\))f(w)m (e)j(get)651 655 y Fp(D)752 575 y Fk(\000)797 655 y Fp(\032)p 847 627 49 3 v 26 x Fn(\000)918 655 y Fq(\000)23 b Fp(\032)1068 614 y Fn(TF)1068 680 y Fo(R)p Fm(;)p Fo(Z)1208 575 y Fk(\001)1337 655 y Fv(=)84 b Fp(Z)1571 614 y Fn(1)p Fm(=)p Fn(3)1681 655 y Fp(D)1781 515 y Fk(\022)1855 655 y Fp(\032)1905 670 y Fn(\000)1975 655 y Fq(\000)2122 588 y Fp(Z)p 2085 632 148 4 v 2085 724 a Fv(6)p Fp(\031)2193 695 y Fn(2)2242 655 y Fv(\(\010)2350 614 y Fn(TF)2350 687 y Fm(Z)2403 667 y Fj(1)p Fi(=)p Fj(3)2500 687 y Fo(Z)5 b Fm(;Z)2629 668 y Fh(\000)p Fj(1)2711 687 y Fo(Z)2771 655 y Fv(\))2809 614 y Fn(3)p Fm(=)p Fn(2)2919 515 y Fk(\023)1337 874 y Fq(\024)83 b Fp(C)7 b(M)j(Z)1752 833 y Fn(5)p Fm(=)p Fn(3+2)p Fm(=)p Fn(27)3318 874 y Fv(\(218\))120 1065 y(Ev)-5 b(aluating)31 b(the)i Fp(R)q(H)8 b(F)46 b Fv(functional)31 b(at)h(the)h(densiy)g(matrix)p 2416 985 61 4 v 31 w(\000)g(w)m(e)g (\014nally)f(obtain)240 1344 y Fq(E)302 1303 y Fn(RHF)293 1368 y Fo(R)p Fm(;)p Fo(Z)456 1344 y Fv(\()p 494 1264 V(\000\))c Fq(\024)g(\000)887 1276 y Fv(1)p 813 1321 196 4 v 813 1412 a(15)p Fp(\031)970 1383 y Fn(2)1036 1208 y Fk(Z)1135 1344 y Fv(\(\010)1243 1303 y Fn(TF)1243 1368 y Fo(R)p Fm(;)p Fo(Z)1384 1344 y Fv(\))1422 1303 y Fn(5)p Fm(=)p Fn(2)1554 1344 y Fv(+)1662 1276 y(1)p 1662 1321 49 4 v 1662 1412 a(8)1772 1219 y Fm(M)1737 1249 y Fk(X)1748 1459 y Fm(j)t Fn(=1)1898 1344 y Fp(Z)1972 1303 y Fn(2)1965 1368 y Fm(j)2033 1344 y Fq(\000)23 b Fp(D)s Fv(\()p Fp(\032)2305 1303 y Fn(TF)2305 1368 y Fo(R)p Fm(;)p Fo(Z)2445 1344 y Fv(\))f(+)g Fp(U)2669 1359 y Fo(R)p Fm(;)p Fo(Z)2831 1344 y Fv(+)g Fp(C)7 b(M)j(Z)3184 1303 y Fn(2)p Fo(\000)p Fn(5)p Fm(=)p Fn(27)3318 1575 y Fv(\(219\))120 1759 y(whic)m(h)33 b(concludes)h(the)f(pro)s(of)e(of)h (Theorem)h(6)p 3469 1759 4 66 v 3473 1696 59 4 v 3473 1759 V 3531 1759 4 66 v 120 2087 a Fs(5)161 b(Reconstruction)37 b(of)j(the)e(whole)h(system)f(and)h(pro)t(of)362 2270 y(of)54 b(the)f(main)h(results.)120 2489 y Fv(Putting)31 b(together)h(the)h(estimates)e(of)h(sections)g(3)g(and)g(4)g(w)m(e)h (ha)m(v)m(e)g(pro)m(v)m(en)g(the)g(follo)m(wing:)266 2610 y Ft(Claim:)39 b Fv(Let)26 b(us)g(lo)s(ok)f(at)h(the)g(RHF)f (energy)i Fp(E)2011 2573 y Fn(RHF)2005 2635 y Fo(R)p Fm(;)p Fo(Z)2166 2610 y Fv(.)42 b(Then,)28 b(decomp)s(osing)d(our)h (system)120 2730 y(in)32 b(its)g(`go)s(o)s(d')f(and)i(`bad')g(parts)g (w)m(e)g(ha)m(v)m(e)h(the)f(estimates)335 3009 y Fp(E)413 2968 y Fn(RHF)407 3033 y Fo(R)p Fm(;)p Fo(Z)651 3009 y Fq(\025)84 b Fp(cZ)928 2968 y Fn(7)p Fm(=)p Fn(3)1075 2884 y Fm(M)1143 2893 y Fj(1)1054 2914 y Fk(X)1065 3124 y Fm(j)t Fn(=1)1215 3009 y Fv(\000\()p Fp(Z)1388 2968 y Fn(1)p Fm(=)p Fn(3)1498 3009 y Fp(\016)1541 3024 y Fm(j)1577 3009 y Fv(\))22 b Fq(\000)1820 2941 y Fv(1)p 1747 2986 196 4 v 1747 3077 a(15)p Fp(\031)1904 3048 y Fn(2)1969 2873 y Fk(Z)2086 2898 y(\020)2145 3009 y Fv(\010)2215 2961 y Fn(TF)q Fm(;\014)2215 3036 y Fo(R)2275 3045 y Fj(1)2310 3036 y Fm(;)p Fo(Z)2381 3045 y Fj(1)2442 3009 y Fv(+)g(\010)2610 2968 y Fn(TF)2610 3033 y Fo(R)2670 3042 y Fj(2)2705 3033 y Fm(;)p Fo(Z)2776 3042 y Fj(2)2814 2898 y Fk(\021)2873 2920 y Fn(5)p Fm(=)p Fn(2)3017 3009 y Fp(dx)812 3360 y Fq(\000)p Fp(D)s Fv(\()p Fp(\032)1061 3319 y Fn(TF)1061 3385 y Fo(R)1121 3394 y Fj(2)1156 3385 y Fm(;)p Fo(Z)1227 3394 y Fj(2)1265 3360 y Fv(\))g(+)1433 3293 y(1)p 1433 3337 49 4 v 1433 3428 a(8)1611 3235 y Fm(M)1576 3265 y Fk(X)1508 3477 y Fm(j)t Fn(=)p Fm(M)1664 3486 y Fj(1)1698 3477 y Fn(+1)1805 3360 y Fp(Z)1879 3319 y Fn(2)1872 3385 y Fm(j)1940 3360 y Fv(+)g Fq(O)s Fv(\()p Fp(M)10 b(Z)2336 3319 y Fn(2)p Fo(\000)p Fm(\016)2465 3360 y Fv(\))22 b(+)g(\001)p Fp(U)2770 3375 y Fn(12)3318 3360 y Fv(\(220\))335 3713 y Fp(E)413 3672 y Fn(RHF)407 3737 y Fo(R)p Fm(;)p Fo(Z)651 3713 y Fq(\025)84 b(\000)973 3645 y Fv(1)p 899 3690 196 4 v 899 3781 a(15)p Fp(\031)1056 3752 y Fn(2)1122 3577 y Fk(Z)1238 3632 y(\000)1284 3713 y Fv(\010)1354 3672 y Fn(TF)1354 3737 y Fo(R)p Fm(;)p Fo(Z)1494 3632 y Fk(\001)1540 3653 y Fn(5)p Fm(=)p Fn(2)1683 3713 y Fp(dx)22 b Fq(\000)h Fp(D)s Fv(\()p Fp(\032)2083 3672 y Fn(TF)2083 3737 y Fo(R)p Fm(;)p Fo(Z)2223 3713 y Fv(\))f(+)2391 3645 y(1)p 2391 3690 49 4 v 2391 3781 a(8)2501 3588 y Fm(M)2466 3618 y Fk(X)2477 3828 y Fm(j)t Fn(=1)2627 3713 y Fp(Z)2701 3672 y Fn(2)2694 3737 y Fm(j)2762 3713 y Fv(+)g Fq(O)s Fv(\()p Fp(M)10 b(Z)3158 3672 y Fn(2)p Fo(\000)p Fm(\016)3287 3713 y Fv(\))-7 b(\(221\))120 4005 y(The)35 b(estimate)f(\(220\))f(holds)h(without)g(an)m(y)h (restriction)e(on)i(the)f(p)s(ositions)f Fq(R)p Fv(.)49 b(The)36 b(more)120 4125 y(precise)h(estimate)e(\(221\))g(holds)h(if)e Fq(R)1531 4140 y Fn(1)1605 4125 y Fv(=)f Fq(;)p Fv(.)54 b(Using)35 b(the)i(expression)g(for)e(the)i(TF)f(energy)120 4246 y(giv)m(en)d(b)m(y)g(Lemma)e(6)i(w)m(e)g(obtain)1009 4524 y Fp(E)1087 4483 y Fn(RHF)1081 4549 y Fo(R)p Fm(;)p Fo(Z)1270 4524 y Fq(\025)28 b Fp(E)1453 4483 y Fn(TF)1447 4549 y Fo(R)p Fm(;)p Fo(Z)1609 4524 y Fv(+)1717 4457 y(1)p 1717 4501 V 1717 4592 a(8)1827 4399 y Fm(M)1792 4429 y Fk(X)1803 4639 y Fm(j)t Fn(=1)1953 4524 y Fp(Z)2027 4483 y Fn(2)2020 4549 y Fm(j)2088 4524 y Fv(+)22 b Fq(O)s Fv(\()p Fp(M)10 b(Z)2484 4483 y Fn(2)p Fo(\000)p Fm(\016)2613 4524 y Fv(\))667 b(\(222\))120 4816 y(under)37 b(the)f(h)m(yp)s (othesis)h Fq(R)1139 4831 y Fn(1)1213 4816 y Fv(=)c Fq(;)p Fv(.)53 b(The)37 b(other)f(situation)f(could)g(b)s(e)h(tak)m(en)h(care) g(of)e(using)120 4936 y(next)e(Lemma,)f(whose)i(easy)f(pro)s(of)f(I)h (will)d(omit:)120 5115 y Ft(Lemma)37 b(15)h(\()p Fp(p)e Ft(norm)h(estimate)f(of)i(a)g(sum\).)48 b Fr(F)-7 b(or)40 b Fv(1)f Fq(\024)h Fp(p)f(<)g Fq(1)i Fr(and)g Fp(x;)17 b(y)42 b Fq(\025)e Fv(0)h Fr(the)120 5236 y(fol)5 b(lowing)33 b(estimate)i(holds:)736 5427 y Fp(x)791 5386 y Fm(p)853 5427 y Fv(+)22 b Fp(y)1003 5386 y Fm(p)1070 5427 y Fq(\024)28 b Fv(\()p Fp(x)23 b Fv(+)f Fp(y)t Fv(\))1479 5386 y Fm(p)1545 5427 y Fq(\024)28 b Fp(x)1705 5386 y Fm(p)1767 5427 y Fv(+)22 b Fp(y)1917 5386 y Fm(p)1978 5427 y Fv(+)g(2)2125 5386 y Fm(p)p Fo(\000)p Fn(1)2255 5427 y Fv(\()p Fp(x)2348 5386 y Fm(p)p Fo(\000)p Fn(1)2478 5427 y Fp(y)j Fv(+)d Fp(xy)2756 5386 y Fm(p)p Fo(\000)p Fn(1)2886 5427 y Fv(\))394 b(\(223\))1781 5712 y(52)p eop %%Page: 53 53 53 52 bop 266 407 a Fv(Using)33 b(Lemma)e(15)h(and)g(estimate)g (\(220\))g(w)m(e)i(obtain)486 689 y Fp(E)564 648 y Fn(RHF)558 713 y Fo(R)p Fm(;)p Fo(Z)802 689 y Fq(\025)84 b Fp(cZ)1079 648 y Fn(7)p Fm(=)p Fn(3)1226 564 y Fm(M)1294 573 y Fj(1)1205 594 y Fk(X)1216 804 y Fm(j)t Fn(=1)1366 689 y Fv(\000\()p Fp(Z)1539 648 y Fn(1)p Fm(=)p Fn(3)1648 689 y Fp(\016)1691 704 y Fm(j)1728 689 y Fv(\))22 b Fq(\000)1971 621 y Fv(1)p 1898 666 196 4 v 1898 757 a(15)p Fp(\031)2055 728 y Fn(2)2120 553 y Fk(Z)2236 578 y(\020)2296 689 y Fv(\010)2366 642 y Fn(TF)p Fm(;\014)2366 716 y Fo(R)2426 725 y Fj(1)2461 716 y Fm(;)p Fo(Z)2532 725 y Fj(1)2570 578 y Fk(\021)2630 601 y Fn(5)p Fm(=)p Fn(2)2773 689 y Fp(dx)963 1040 y Fq(\000)p Fp(D)s Fv(\()p Fp(\032)1212 999 y Fn(TF)1212 1065 y Fo(R)1272 1074 y Fj(2)1307 1065 y Fm(;)p Fo(Z)1378 1074 y Fj(2)1416 1040 y Fv(\))g(+)1584 973 y(1)p 1584 1017 49 4 v 1584 1108 a(8)1762 916 y Fm(M)1727 945 y Fk(X)1659 1157 y Fm(j)t Fn(=)p Fm(M)1815 1166 y Fj(1)1849 1157 y Fn(+1)1956 1040 y Fp(Z)2030 999 y Fn(2)2023 1065 y Fm(j)2091 1040 y Fv(+)g Fq(O)s Fv(\()p Fp(M)10 b(Z)2487 999 y Fn(2)p Fo(\000)p Fm(\016)2616 1040 y Fv(\))664 b(\(224\))963 1366 y(+\001)p Fp(U)1186 1381 y Fn(12)1283 1366 y Fq(\000)1411 1299 y Fv(2)1460 1263 y Fn(5)p Fm(=)p Fn(2)p 1392 1343 196 4 v 1392 1435 a Fv(15)p Fp(\031)1549 1406 y Fn(2)1615 1226 y Fk(\032)1690 1231 y(Z)1806 1366 y Fp(\032)1856 1319 y Fn(TF)p Fm(;\014)1856 1393 y Fo(R)1916 1402 y Fj(1)1951 1393 y Fm(;)p Fo(Z)2022 1402 y Fj(1)2060 1366 y Fv(\010)2130 1325 y Fn(TF)2130 1391 y Fo(R)2190 1400 y Fj(2)2225 1391 y Fm(;)p Fo(Z)2296 1400 y Fj(2)2356 1366 y Fv(+)2455 1231 y Fk(Z)2571 1366 y Fp(\032)2621 1325 y Fn(TF)2621 1391 y Fo(R)2681 1400 y Fj(2)2716 1391 y Fm(;)p Fo(Z)2787 1400 y Fj(2)2825 1366 y Fv(\010)2895 1319 y Fn(TF)p Fm(;\014)2895 1393 y Fo(R)2955 1402 y Fj(1)2990 1393 y Fm(;)p Fo(Z)3061 1402 y Fj(1)3099 1226 y Fk(\033)120 1638 y Fv(But)33 b(since)g(\001)p Fp(U)700 1653 y Fn(1)p Fm(;)p Fn(2)822 1638 y Fv(=)926 1558 y Fk(R)1009 1638 y Fp(\032)1059 1591 y Fn(TF)p Fm(;\014)1059 1665 y Fo(R)1119 1674 y Fj(1)1154 1665 y Fm(;)p Fo(Z)1225 1674 y Fj(1)1263 1638 y Fv(\010)1333 1602 y Fn(TF)1333 1663 y Fo(R)1393 1672 y Fj(2)1428 1663 y Fm(;)p Fo(Z)1499 1672 y Fj(2)1560 1638 y Fv(+)1658 1558 y Fk(R)1741 1638 y Fp(\032)1791 1602 y Fn(TF)1791 1663 y Fo(R)1851 1672 y Fj(2)1886 1663 y Fm(;)p Fo(Z)1957 1672 y Fj(2)1995 1638 y Fv(\010)2065 1591 y Fn(TF)p Fm(;\014)2065 1665 y Fo(R)2125 1674 y Fj(1)2160 1665 y Fm(;)p Fo(Z)2231 1674 y Fj(1)2297 1638 y Fq(\025)28 b Fv(0)k(and)2691 1599 y Fn(2)2726 1575 y Fj(5)p Fi(=)p Fj(2)p 2683 1615 149 4 v 2683 1672 a Fn(15)p Fm(\031)2796 1653 y Fj(2)2869 1638 y Fp(<)27 b Fv(1,)33 b(w)m(e)g(get)336 1935 y Fp(E)414 1894 y Fn(RHF)408 1960 y Fo(R)p Fm(;)p Fo(Z)597 1935 y Fq(\025)28 b Fp(cZ)818 1894 y Fn(7)p Fm(=)p Fn(3)965 1810 y Fm(M)1033 1819 y Fj(1)944 1840 y Fk(X)955 2050 y Fm(j)t Fn(=1)1105 1935 y Fv(\000\()p Fp(Z)1278 1894 y Fn(1)p Fm(=)p Fn(3)1388 1935 y Fp(\016)1431 1950 y Fm(j)1467 1935 y Fv(\))23 b(+)f Fp(E)1704 1894 y Fn(TF)1698 1960 y Fo(R)1758 1969 y Fj(2)1793 1960 y Fm(;)p Fo(Z)1864 1969 y Fj(2)1924 1935 y Fv(+)2032 1868 y(1)p 2032 1912 49 4 v 2032 2003 a(8)2210 1810 y Fm(M)2175 1840 y Fk(X)2107 2052 y Fm(j)t Fn(=)p Fm(M)2263 2061 y Fj(1)2297 2052 y Fn(+1)2404 1935 y Fp(Z)2478 1894 y Fn(2)2471 1960 y Fm(j)2539 1935 y Fv(+)g Fq(O)s Fv(\()p Fp(M)10 b(Z)2935 1894 y Fn(2)p Fo(\000)p Fm(\016)3064 1935 y Fv(\))216 b(\(225\))266 2221 y Ft(Inclusion)39 b(of)h(spin:)48 b Fv(The)35 b(full)e(quan)m(tum)i(system)h(describ)s(ed)f(b)m(y)h(the)f (Hamiltonian)120 2398 y(\(1\))h(acts)h(in)f(the)h(F)-8 b(ermion)34 b(space)k Fq(H)d Fv(=)1664 2304 y Fm(N)1654 2324 y Fk(V)1754 2398 y Fp(L)1820 2362 y Fn(2)1860 2398 y Fv(\()p Fl(R)1963 2362 y Fn(3)2009 2398 y Fv(;)17 b Fl(C)2119 2362 y Fn(2)2164 2398 y Fv(\).)55 b(The)37 b(analogous)f(of)g(this)g(for)g(the)120 2519 y(RHF)c(mo)s(del)f(is)i (to)f(lo)s(ok)g(at)g(the)h(RHF)g(functional,)e(but)i(no)m(w)g(the)h(Sc) m(hr\177)-49 b(odinger)33 b(op)s(erator)120 2639 y Fq(\000)p Fv(\001)17 b Fq(\000)g Fp(V)446 2654 y Fo(R)p Fm(;)p Fo(Z)616 2639 y Fv(acts)30 b(on)g Fp(L)1013 2603 y Fn(2)1053 2639 y Fv(\()p Fl(R)1157 2603 y Fn(3)1202 2639 y Fv(;)17 b Fl(C)1312 2603 y Fn(2)1357 2639 y Fv(\))30 b(instead)g(of)f Fp(L)1933 2603 y Fn(2)1973 2639 y Fv(\()p Fl(R)2077 2603 y Fn(3)2122 2639 y Fv(\).)43 b(But)30 b(b)m(y)h(the)f(elemen)m(tary)g (form)m(ula)120 2759 y(T)-8 b(r)17 b(\()p Fp(A)e Fq(\012)g Fl(C)520 2723 y Fm(q)564 2759 y Fv(\))28 b(=)g Fp(q)t Fv(T)-8 b(r)16 b Fp(A)p Fv(,)30 b(the)f(e\013ect)h(of)e(considering)h (spin)f(in)g(the)i(semiclassics)e(of)g(the)h(RHF)120 2880 y(mo)s(del)23 b(only)h(amoun)m(ts)g(to)g(get)h(constan)m(ts)h(t)m (wice)f(bigger,)g(so,)h(including)d(spin,)j(the)f(estimate)120 3000 y(\(225\))32 b(tak)m(es)i(the)f(form)336 3282 y Fp(E)414 3241 y Fn(RHF)408 3307 y Fo(R)p Fm(;)p Fo(Z)597 3282 y Fq(\025)28 b Fp(cZ)818 3241 y Fn(7)p Fm(=)p Fn(3)965 3157 y Fm(M)1033 3166 y Fj(1)944 3188 y Fk(X)955 3398 y Fm(j)t Fn(=1)1105 3282 y Fv(\000\()p Fp(Z)1278 3241 y Fn(1)p Fm(=)p Fn(3)1388 3282 y Fp(\016)1431 3297 y Fm(j)1467 3282 y Fv(\))23 b(+)f Fp(E)1704 3241 y Fn(TF)1698 3307 y Fo(R)1758 3316 y Fj(2)1793 3307 y Fm(;)p Fo(Z)1864 3316 y Fj(2)1924 3282 y Fv(+)2032 3215 y(1)p 2032 3259 V 2032 3351 a(4)2210 3158 y Fm(M)2175 3188 y Fk(X)2107 3399 y Fm(j)t Fn(=)p Fm(M)2263 3408 y Fj(1)2297 3399 y Fn(+1)2404 3282 y Fp(Z)2478 3241 y Fn(2)2471 3307 y Fm(j)2539 3282 y Fv(+)g Fq(O)s Fv(\()p Fp(M)10 b(Z)2935 3241 y Fn(2)p Fo(\000)p Fm(\016)3064 3282 y Fv(\))216 b(\(226\))120 3607 y(for)42 b(the)g(concrete)i(c)m(hoice)e(of)g (parameter)1757 3568 y Fn(3)p 1757 3584 36 4 v 1757 3641 a(5)1819 3526 y Fk(\000)1874 3568 y Fn(2)p 1874 3584 V 1874 3641 a(5)1920 3526 y Fk(\001)1965 3549 y Fn(2)p Fm(=)p Fn(3)2092 3607 y Fp(q)2139 3571 y Fo(\000)p Fn(2)p Fm(=)p Fn(3)2320 3526 y Fk(\000)2432 3568 y Fn(1)p 2376 3584 149 4 v 2376 3641 a(15)p Fm(\031)2489 3622 y Fj(2)2534 3526 y Fk(\001)2579 3549 y Fo(\000)p Fn(2)p Fm(=)p Fn(3)2788 3607 y Fv(=)2918 3568 y Fn(3)2953 3544 y Fj(5)p Fi(=)p Fj(3)3050 3568 y Fm(\031)3093 3544 y Fj(4)p Fi(=)p Fj(3)p 2918 3584 272 4 v 3037 3641 a Fn(5)3242 3607 y Fv(\()p Fp(q)48 b Fv(=)c(2)120 3727 y(in)33 b(our)h(case\))h(,)g(whic)m(h)g (follo)m(ws)d(from)h(matc)m(hing)g(the)i(asymptotic)e(expression)j(of)d (the)i(TF)120 3848 y(energy)f(w)m(e)f(obtained)f(and)h(the)g (expression)h(of)e(it)f(as)i(giv)m(en)g(b)m(y)g(Lemma)f(6.)266 3968 y(No)m(w)h(w)m(e)h(are)f(in)e(p)s(osition)g(to)h(pro)m(v)m(e)i (our)f(main)e(result:)120 4212 y Fr(Pr)-5 b(o)g(of)35 b(of)f(The)-5 b(or)g(em)34 b(1.)48 b Fv(Pic)m(k)31 b Fp( )g Fq(2)1443 4117 y Fm(N)1433 4137 y Fk(V)1533 4212 y Fp(L)1599 4175 y Fn(2)1639 4212 y Fv(\()p Fl(R)1743 4175 y Fn(3)1788 4212 y Fv(;)17 b Fl(C)1898 4175 y Fn(2)1943 4212 y Fv(\))30 b(normalized)e(and)i(consider)g(the)g(densit)m(y)120 4332 y(matrix)h Fp(\015)488 4347 y Fm( )573 4332 y Fv(on)h Fp(L)774 4296 y Fn(2)814 4332 y Fv(\()p Fl(R)918 4296 y Fn(3)963 4332 y Fp(;)17 b Fl(C)1073 4296 y Fn(2)1119 4332 y Fv(\))32 b(whose)i(in)m(tegral)d(k)m(ernel)i(is)f(giv)m(en)h(b)m (y)g(\(notice)f(that)130 4423 y Fm(N)120 4443 y Fk(V)220 4518 y Fp(L)286 4482 y Fn(2)325 4518 y Fv(\()p Fl(R)429 4482 y Fn(3)475 4518 y Fv(;)17 b Fl(C)584 4482 y Fn(2)630 4518 y Fv(\))27 b Fq(\022)811 4423 y Fm(N)801 4443 y Fk(V)900 4518 y Fp(L)966 4482 y Fn(2)1006 4518 y Fv(\()p Fl(R)1110 4482 y Fn(3)1155 4518 y Fv(;)17 b Fl(C)1265 4482 y Fn(2)1300 4459 y Fi(N)1368 4518 y Fv(\)\))195 4756 y Fp(\015)246 4771 y Fm( )298 4756 y Fv(\()p Fp(x;)g(y)t Fv(\))27 b(:=)g Fp(N)787 4620 y Fk(Z)843 4845 y Fg(R)891 4826 y Fj(3\()p Fi(N)5 b Fh(\000)p Fj(1\))1109 4756 y Fq(h)p Fp( )t Fv(\()p Fp(y)t(;)17 b(x)1404 4771 y Fn(2)1443 4756 y Fp(;)g(:)g(:)g(:)32 b(;)17 b(x)1733 4771 y Fm(N)1801 4756 y Fv(\))p Fp(;)g( )t Fv(\()p Fp(x;)g(x)2142 4771 y Fn(2)2181 4756 y Fp(;)g(:)g(:)g(:)33 b(;)17 b(x)2472 4771 y Fm(N)2539 4756 y Fv(\))p Fq(i)2616 4788 y Fg(C)2658 4770 y Fj(2)2688 4756 y Fi(N)41 b Fp(dx)2883 4771 y Fn(2)2939 4756 y Fp(:)17 b(:)g(:)f(dx)3176 4771 y Fm(N)3318 4756 y Fv(\(227\))120 4997 y(\(That)35 b(this)f(indeed)g(de\014nes)i(a)e(w)m (ell-normalized)d(densit)m(y)36 b(matrix)d(is)g(an)i(instructiv)m(e)f (cal-)120 5117 y(culation)26 b(that)h(uses)i(strongly)e(the)h(an)m (tisymmetry)f(of)h Fp(\015)2206 5132 y Fm( )2258 5117 y Fv(.)42 b(Without)26 b(it)h(one)h(w)m(ould)f(simply)120 5238 y(obtain)k(the)i(op)s(erator)f(inequalities)f(0)d Fq(\024)g Fp(\015)1730 5253 y Fm( )1810 5238 y Fq(\024)g Fp(N)10 b Fv(\).)44 b(No)m(w)33 b(w)m(e)g(can)g(compute)781 5427 y Fq(h)p Fp( )t(;)17 b Fv(\()p Fq(\000)p Fv(\001)22 b Fq(\000)h Fp(V)1306 5442 y Fo(R)p Fm(;)p Fo(Z)1446 5427 y Fv(\))p Fp( )t Fq(i)k Fv(=)h(T)-8 b(r)17 b(\()p Fp(H)8 b(\015)2016 5442 y Fm( )2067 5427 y Fv(\);)50 b Fp(H)35 b Fv(=)27 b Fq(\000)p Fv(\001)c Fq(\000)g Fp(V)2739 5442 y Fo(R)p Fm(;)p Fo(Z)3318 5427 y Fv(\(228\))1781 5712 y(53)p eop %%Page: 54 54 54 53 bop 120 407 a Fv(By)33 b(the)g(Lieb-Oxford)f(inequalit)m(y)f(w)m (e)j(ha)m(v)m(e)g(the)f(Coulom)m(b)e(estimate)684 499 y Fk(*)764 669 y Fp( )t(;)962 574 y Fk(X)875 786 y Fn(1)p Fo(\024)p Fm(i)d Fv(0)120 5101 y(By)33 b(Theorem)g(1)f(the)h(follo)m(wing)d(lo)m(w)m(er)j(b)s(ound)g(holds:) 704 5374 y Fp(E)782 5327 y Fm(Q)776 5401 y Fo(R)p Fm(;)p Fo(Z)944 5374 y Fq(\025)1084 5250 y Fm(M)1049 5280 y Fk(X)1060 5490 y Fm(j)t Fn(=1)1210 5374 y Fp(E)1288 5333 y Fm(Q)1348 5374 y Fv(\()p Fp(Z)1453 5389 y Fm(j)1489 5374 y Fv(\))22 b(+)g Fp(cZ)1763 5333 y Fn(7)p Fm(=)p Fn(3)1924 5250 y Fm(M)1889 5280 y Fk(X)1900 5490 y Fm(j)t Fn(=1)2050 5374 y Fv(\000\()p Fp(Z)2223 5333 y Fn(1)p Fm(=)p Fn(3)2333 5374 y Fp(\016)2376 5389 y Fm(j)2412 5374 y Fv(\))g Fq(\000)h Fp(C)7 b(M)j(Z)2827 5333 y Fn(2)p Fo(\000)p Fm(\016)1781 5712 y Fv(56)p eop %%Page: 57 57 57 56 bop 120 407 a Fv(with)32 b(\000\()p Fp(t)p Fv(\))c(:=)g(min)n Fq(f)p Fp(t)920 371 y Fo(\000)p Fn(1)1015 407 y Fp(;)17 b(t)1094 371 y Fo(\000)p Fn(7)1188 407 y Fq(g)32 b Fv(so)h(w)m(e)g(get) 1191 720 y Fp(Z)1265 678 y Fn(7)p Fm(=)p Fn(3)1426 595 y Fm(M)1392 625 y Fk(X)1402 835 y Fm(j)t Fn(=1)1552 720 y Fv(\000\()p Fp(Z)1725 678 y Fn(1)p Fm(=)p Fn(3)1835 720 y Fp(\016)1878 735 y Fm(j)1915 720 y Fv(\))27 b Fq(\024)h Fp(C)7 b(M)j(Z)2340 678 y Fn(2)p Fo(\000)p Fm(\016)3318 720 y Fv(\(245\))120 1035 y(Assume)48 b Fp(\016)539 1050 y Fn(1)631 1035 y Fq(\025)53 b Fp(:)17 b(:)g(:)51 b Fq(\025)i Fp(\016)1100 1050 y Fm(M)1232 1035 y Fp(>)f Fv(0.)86 b(Since)47 b(\000)g(is)g(decreasing)g(and)h(its)e(in)m(v)m(erse)i (function)120 1155 y Fp(\015)33 b Fv(:=)27 b(\000)395 1119 y Fn(\()p Fo(\000)p Fn(1\))572 1155 y Fv(:)h Fp(s)g Fq(!)f Fv(min)n Fq(f)p Fp(s)1086 1119 y Fo(\000)p Fn(1)p Fm(=)p Fn(7)1251 1155 y Fp(;)17 b(s)1341 1119 y Fo(\000)p Fn(1)1435 1155 y Fq(g)p Fv(,)33 b(from)e(\(245\))h(w)m(e)h(get)1125 1432 y Fp(\016)1168 1447 y Fm(j)1233 1432 y Fq(\025)28 b Fp(Z)1412 1391 y Fo(\000)p Fn(1)p Fm(=)p Fn(3)1577 1432 y Fp(\015)1649 1292 y Fk(\022)1723 1432 y Fp(C)7 b(Z)1874 1391 y Fn(1)p Fm(=)p Fn(3)2162 1365 y Fp(M)p 1993 1410 442 4 v 1993 1501 a(M)33 b Fq(\000)23 b Fp(j)28 b Fv(+)22 b(1)2445 1292 y Fk(\023)3318 1432 y Fv(\(246\))120 1720 y(No)m(w,)33 b(if)f(1)27 b Fq(\024)h Fp(j)34 b Fq(\024)819 1639 y Fk(\002)871 1681 y Fm(M)p 871 1697 75 4 v 890 1754 a Fn(2)955 1639 y Fk(\003)997 1720 y Fv(,)e(eq.)45 b(\(246\))31 b(yields)995 2007 y Fp(\016)1038 2022 y Fm(j)1102 2007 y Fq(\025)d Fp(cZ)1323 1966 y Fo(\000)p Fn(1)p Fm(=)p Fn(3+1)p Fm(=)p Fn(7\(1)p Fm(=)p Fn(3+)p Fm(\016)r Fn(\))1898 2007 y Fv(;)49 b Fp(j)34 b Fv(=)28 b(1)p Fp(;)17 b(:)g(:)g(:)32 b(;)2436 1867 y Fk(\024)2498 1940 y Fp(M)p 2498 1984 105 4 v 2526 2075 a Fv(2)2613 1867 y Fk(\025)120 2284 y Fv(and)h(this)f(clearly)g(implies)e(the)j(a)m (v)m(erage)g(estimate)f(giv)m(en)h(b)m(y)g(eq.)44 b(\(8\).)p 3469 2284 4 66 v 3473 2222 59 4 v 3473 2284 V 3531 2284 4 66 v 1781 5712 a(57)p eop %%Page: 58 58 58 57 bop 120 407 a Fs(References)342 626 y Fv([1])49 b(V.)58 b(Bac)m(h,)66 b(Error)58 b(b)s(ound)g(for)g(the)h(Hartree-F)-8 b(o)s(c)m(k)58 b(Energy)h(of)f(A)m(toms)g(and)494 746 y(Molecules.)33 b Fr(Comm.Math.Phys)p Fv(,)e Ft(147\(3\))h Fv(527-548,)f(\(1992\))342 950 y([2])49 b(P)-8 b(.)24 b(Balo)s(dis)d(&)i(P)-8 b(.)24 b(Solo)m(v)m(ej,)h(On)f(the)f (Asymptotic)g(Exacteness)j(of)d(Thomas-F)-8 b(ermi)494 1070 y(Theory)41 b(in)e(the)i(Thermo)s(dinamic)c(Limit.)h Fr(A)n(nn.Henri)j(Poinc)-5 b(ar)n(\023)-47 b(e)p Fv(,)40 b Ft(1)g Fv(281-306,)494 1190 y(\(2000\))342 1394 y([3])49 b(H.)33 b(Brezis,)g Fr(A)n(nalyse)h(F)-7 b(onctionel)5 b(le)p Fv(.)30 b(Masson,)k(P)m(aris,)f(\(1983\))342 1597 y([4])49 b(H.)c(Brezis)g(&)g(E.)h(Lieb,)h(Long)e(range)g(atomic)e(p)s (oten)m(tials)h(in)g(Thomas-F)-8 b(ermi)494 1718 y(Theory)g(.)34 b Fr(Commun.Math.Phys.)p Fv(,)d Ft(65)p Fv(,)i(231{246)e(\(1979\))342 1921 y([5])49 b(G.)78 b(Conlon,)89 b(The)79 b(ground)f(state)h(energy)g (of)f(a)g(classical)f(gas.)h Fr(Com-)494 2041 y(mun.Math.Phys.)p Fv(,)32 b Ft(94)p Fv(,)h(439{458)e(\(1984\))342 2245 y([6])49 b(M.)e(Cwik)m(el,)k(W)-8 b(eak)47 b(t)m(yp)s(e)g(estimates)g (for)f(singular)f(v)-5 b(alues)47 b(and)f(the)i(n)m(um)m(b)s(er)494 2365 y(of)41 b(b)s(ound)h(states)h(of)e(Sc)m(hr\177)-49 b(odinger)42 b(op)s(erators.)g Fr(A)n(nn.)g(of)h(Math.)f Ft(106)p Fv(,)i(93{100)494 2486 y(\(1977\))342 2689 y([7])49 b(F.)40 b(Dyson)g(&)g(A.)g(Lenard,)i(Stabilit)m(y)c(of)h(Matter)i(I)f (and)g(I)s(I,)g Fr(J.Math.Phys.)p Fv(,)i Ft(8)p Fv(,)494 2809 y(423{434)31 b(\(1967\);)g(ibid)h Ft(9)p Fv(,)g(698-711)f (\(1968\))342 3013 y([8])49 b(P)-8 b(.)33 b(F)-8 b(ederbush,)35 b(A)e(new)i(approac)m(h)e(to)g(the)h(stabilit)m(y)d(of)i(matter)f (problem.)g(I,)i(I)s(I.)494 3133 y Fr(J.Math.Phys.)p Fv(,)f Ft(16)p Fv(,)g(347{351)d(\(1975\);)i(ibid.)f Ft(16)p Fv(,)i(706{709)e(\(1975\))342 3337 y([9])49 b(C.)66 b(F)-8 b(e\013erman,)73 b(J.)66 b(F)-8 b(r\177)-49 b(ohlic)m(h,)73 b(&)65 b(G-M.)g(Graf,)74 b(Stabilit)m(y)63 b(of)j(ultra)m(violet-)494 3457 y(cuto\013)g(quan)m(tum)g(electro)s(dynamics)f(with)g (non-relativistic)e(matter)i Fr(Com-)494 3577 y(mun.Math.Phys.)p Fv(,)32 b Ft(190)h Fv(309{330)e(\(1997\))293 3781 y([10])49 b(C.)69 b(F)-8 b(e\013erman,)77 b(Stabilit)m(y)67 b(of)h(Coulom)m(b)f (systems)k(in)c(a)i(magnetic)e(\014eld,)494 3901 y Fr(Pr)-5 b(o)g(c.Nat.A)g(c)g(ad.Sci.)34 b(U.S.A.)p Fv(,)f Ft(92)p Fv(,)g(5006{5007)d(\(1995\))293 4104 y([11])49 b(C.)31 b(F)-8 b(e\013erman)31 b(&)g(L.Seco,)h(On)f(the)g(energy)h(of)f(a)f (large)g(atom,)h Fr(Bul)5 b(l.)33 b(A.M.S)p Fv(,)e Ft(23)494 4225 y Fv(\(1990\),)h(525-530)e(\(1990\))293 4428 y([12])49 b(C.)58 b(F)-8 b(e\013erman)57 b(&)h(R.)g(de)g(la)f(Lla)m(v)m(e,)64 b(Relativistic)55 b(stabilit)m(y)h(of)i(matter.)f(I.)494 4549 y Fr(R)-5 b(ev.Mat.Ib)g(er)g(o)g(americ)g(ana)p Fv(,)30 b Ft(2)p Fv(,)j(119{213)e(\(1986\))293 4752 y([13])49 b(G.M.)60 b(Graf,)65 b(Stabilit)m(y)58 b(of)h(matter)g(through)g(an)h (electrostatic)f(inequalit)m(y)-8 b(.)494 4872 y Fr(Helv.Phys.A)j(cta)p Fv(,)33 b Ft(70)p Fv(,)g(72{79)e(\(1997\))293 5076 y([14])49 b(G.M.)39 b(Graf)e(&)i(D.)f(Sc)m(henk)m(er,)43 b(On)c(the)g(Molecular)e (Limit)f(of)i(Coulom)m(b)g(gases.)494 5196 y Fr(Commun.Math.Phys.)p Fv(,)32 b Ft(174)p Fv(,)g(215{227)f(\(1995\))1781 5712 y(58)p eop %%Page: 59 59 59 58 bop 293 407 a Fv([15])49 b(M.)34 b(de)h(Guzm\023)-49 b(an,)33 b(A)h(co)m(v)m(ering)h(lemma)d(with)h(applications)f(to)i (di\013eren)m(tiabilit)m(y)494 527 y(of)g(measures)h(and)g(singular)e (in)m(tegral)g(op)s(erators,)i Fr(Studia)h(Math.)p Fv(,)f Ft(34)p Fv(,)h(299{317)494 648 y(1970)293 845 y([16])49 b(V.)39 b(Ivrii)e(&)i(I.M.)g(Sigal,)f(Asym)m(totics)h(of)f(the)h (ground)g(state)g(energies)g(of)f(large)494 966 y(Coulom)m(b)32 b(systems.)i Fr(A)n(nnals)g(of)g(Mathematics)p Fv(,)e Ft(128)p Fv(,)h(243-335)e(\(1993\))293 1163 y([17])49 b(E.H.)42 b(Lieb,)g(Bounds)g(on)f(the)g(eigen)m(v)-5 b(alues)41 b(of)g(the)g(Laplace)g(and)g(Sc)m(hr\177)-49 b(odinger)494 1284 y(op)s(erators.)32 b Fr(Bul)5 b(l.)35 b(A)n(mer.)f(Math.)h(So)-5 b(c.)p Fv(,)p Ft(82)32 b Fv(751{753)f (\(1976\))293 1482 y([18])49 b(E.H.)33 b(Lieb,)f(The)i(stabilit)m(y)d (of)h(matter.)g Fr(R)-5 b(ev.Mo)g(d.Phys.)p Fv(,)32 b Ft(48)p Fv(,)h(553{569)d(\(1981\))293 1679 y([19])49 b(E.H.)35 b(Lieb,)h(Thomas-F)-8 b(ermi)32 b(and)j(related)g(theories)f (of)h(atoms)f(and)h(molecules.)494 1800 y Fr(R)-5 b(ev.Mo)g(d.Phys.)p Fv(,)32 b Ft(53)p Fv(,)h(603{641)e(\(1981\))293 1998 y([20])49 b(E.H.)d(Lieb,)j(Bound)e(of)e(the)h(maxim)m(um)e(negativ)m(e) i(ionization)d(of)j(atoms)f(and)494 2118 y(molecules,)32 b Fr(Phys.)i(R)-5 b(ev.)35 b(A)p Fv(,)e Ft(29)p Fv(,)g(3018{3028)d (\(1984\))293 2316 y([21])49 b(E.H.)35 b(Lieb,)f(M.)h(Loss,)g (Analysis.)g Fr(Gr)-5 b(aduate)36 b(Studies)h(in)f(Mathematics,)g(vol)g (14)494 2436 y Fv(A.M.S)d(\(1997\))293 2634 y([22])49 b(E.H.)26 b(Lieb,)g(M.)g(Loss,)h(&)e(J.P)-8 b(.)26 b(Solo)m(v)m(ej,)h (Stabilit)m(y)c(of)i(matter)f(in)h(magnetic)f(\014elds.)494 2754 y Fr(Phys.R)-5 b(ev.L)g(ett.)p Fv(,)33 b Ft(75)p Fv(,)f(985{989)f(\(1995\))293 2952 y([23])49 b(E.H.)58 b(Lieb)e(&)h(S.)h(Oxford,)63 b(An)58 b(impro)m(v)m(ed)f(lo)m(w)m(er)g (b)s(ound)h(of)e(the)i(indirect)494 3072 y(Coulom)m(b)32 b(energy)-8 b(.)33 b Fr(Int.J.Quamtum.Chem.)p Fv(,)e Ft(19)p Fv(,)i(427{439)e(\(1981\))293 3270 y([24])49 b(E.H.)38 b(Lieb)e(&)h(B.)g(Simon,)g(The)h(Thomas-F)-8 b(ermi)35 b(theory)j(of)e(atoms,)i(molecules)494 3391 y(and)33 b(solids.)e Fr(A)-5 b(dvanc)g(es)34 b(in)h(Math.)p Fv(,)e Ft(23)p Fv(,)g(22{116,)e(\(1977\))293 3588 y([25])49 b(E.H.)29 b(Lieb,)g(H.)g(Sieden)m(top,)h(&)f(J.P)-8 b(.)29 b(Solo)m(v)m(ej,)g(Stabilit)m(y)e(and)i(instabilit)m(y)d(of)i(rela-)494 3709 y(tivistic)i(electrons)j(in)e(classical)f(electromagnetic)h (\014elds.)h Fr(Jour.Stat.Phys.)p Fv(,)h Ft(89)p Fv(,)494 3829 y(37{59)e(\(1997\))293 4027 y([26])49 b(E.H.)41 b(Lieb)g(&)f(W.E.)i(Thirring,)f(Bound)g(for)g(the)g(kinetic)f(energy)i (of)e(fermions)494 4147 y(whic)m(h)29 b(pro)m(v)m(es)i(the)f(stabilit)m (y)d(of)i(matter.)f Fr(Phys.R)-5 b(ev.L)g(ett.)p Fv(,)30 b Ft(35)p Fv(,)g(687{689)d(\(1975\))293 4345 y([27])49 b(E.H.)26 b(Lieb)f(&)g(W.E.)h(Thirring,)g(Inequalities)e(for)h(the)h (momen)m(ts)f(of)g(the)g(eigen)m(v)-5 b(al-)494 4465 y(ues)24 b(of)e(the)h(Sc)m(hr\177)-49 b(odinger)23 b(Hamiltonian)c(and) k(their)g(relation)d(to)j(Sob)s(olev)f(inequal-)494 4586 y(ities.)37 b(In)i Fr(Studies)h(in)g(Mathematic)-5 b(al)40 b(Physics,)h(Essays)f(in)g(honor)g(of)g(V)-7 b(alentin)494 4706 y(Bar)i(gmann)p Fv(,)31 b(E.H.)j(Lieb,)e(B.)h(Simon,)e(and)i(A.S.) g(Wigh)m(tman)f(\(Eds.\),)i(Princeton)494 4827 y(Univ)m(ersit)m(y)f (Press,)h(Princeton,)f(New)h(Jersey)-8 b(,)34 b(1976)293 5024 y([28])49 b(E.H.)24 b(Lieb)e(&)h(H.T.)h(Y)-8 b(au,)25 b(The)g(stabilit)m(y)c(and)i(instabilit)m(y)e(of)i(relativistic)e (matter.)494 5145 y Fr(Commun.Math.Phys.)p Fv(,)32 b Ft(118)p Fv(,)g(177{213)f(\(1988\))293 5342 y([29])49 b(G.)24 b(Rozen)m(blum,)i(Distribution)c(of)i(the)h(discrete)g(sp)s (ectrum)g(of)f(singular)f(di\013eren-)494 5463 y(tial)29 b(op)s(eraqors.)j Fr(Doklady)h(A)n(kademii)g(Nauk)h(SSSR)p Fv(,)d Ft(202)h Fv(5,)g(1012{1015)d(\(1972\))1781 5712 y(59)p eop %%Page: 60 60 60 59 bop 293 407 a Fv([30])49 b(A.V.)37 b(Sob)s(olev,)h(The)g (quasi-classical)d(asymptotics)i(of)g(lo)s(cal)d(Riesz)j(means)g(for) 494 527 y(the)g(Sc)m(hr\177)-49 b(odinger)37 b(op)s(erator)f(in)h(a)f (strong)h(homogeneus)g(magnetic)f(\014eld.)g Fr(Duke)494 648 y(Math.)f(J.)p Fv(,)e Ft(74\(2\))p Fv(,)f(319-429)e(\(1994\))293 851 y([31])49 b(A.V.)38 b(Sob)s(olev,)h(Quasi-classical)c(asymptotics)j (of)g(lo)s(cal)d(Riesz)j(means)g(for)g(the)494 971 y(Sc)m(hr\177)-49 b(odinger)44 b(op)s(erator)g(in)g(a)g(mo)s(derate)f(magnetic)g (\014eld.)h Fr(A)n(nn.)h(Inst.)h(Henry)494 1092 y(Poinc)-5 b(ar)n(\023)-47 b(e.)p Fv(,)31 b Ft(62\(4\))p Fv(,)h(325-359)e (\(1995\))293 1295 y([32])49 b(A.V.)34 b(Sob)s(olev,)f(Discrete)h(sp)s (ectrum)g(asymptotics)f(for)g(the)h(Sc)m(hr\177)-49 b(odinger)34 b(op)s(er-)494 1416 y(ator)i(with)g(a)g(singular)e(p)s(oten)m(tial)h (and)h(a)g(magnetic)f(\014eld.)i Fr(R)-5 b(eviews)37 b(in)h(Mathe-)494 1536 y(matic)-5 b(al)34 b(Physics.)e Ft(8\(6\))p Fv(,)g(861-903)f(\(1996\))1781 5712 y(60)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0202080611434--