Content-Type: multipart/mixed; boundary="-------------0511070752145" This is a multi-part message in MIME format. ---------------0511070752145 Content-Type: text/plain; name="05-377.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-377.keywords" 70H20, 37M25, 37J50 ---------------0511070752145 Content-Type: application/postscript; name="scheme_fin.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="scheme_fin.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: scheme_fin.dvi %%Pages: 64 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips scheme_fin %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2005.11.07:1251 %%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 (scheme_fin.dvi) @start %DVIPSBitmapFont: Fa cmti10 10 49 /Fa 49 122 df<04FFEB03F003039038E00FFC923A0FC0F01F1E923A3F00783E0F923A7E 01F87C3FDB7C03EBFC7F03FC14F8DA01F813F905F1137EDC01E1133C913B03F00003F000 A314074B130760A3140F4B130F60A3010FB812C0A3903C001F80001F8000A3023F143F92 C790C7FCA44A5C027E147EA402FE14FE4A5CA413014A13015FA313034A13035FA313074A 495AA44948495AA44948495AA3001CD9038090C8FC007E90380FC03F013E143E00FE011F 5B133C017C5C3AF8780F01E0D878F0EB07C0273FE003FFC9FC390F8000FC404C82BA33> 11 DI< 150C151C153815F0EC01E0EC03C0EC0780EC0F00141E5C147C5C5C495A1303495A5C130F 49C7FCA2133EA25BA25BA2485AA212035B12075BA2120F5BA2121FA290C8FCA25AA2123E A2127EA2127CA412FC5AAD1278A57EA3121C121EA2120E7EA26C7E6C7EA212001E5274BD 22>40 D<140C140E80EC0380A2EC01C015E0A2140015F0A21578A4157C153CAB157CA715 FCA215F8A21401A215F0A21403A215E0A21407A215C0140F1580A2141F1500A2143EA25C A25CA2495AA2495A5C1307495A91C7FC5B133E133C5B5B485A12035B48C8FC120E5A1278 5A12C01E527FBD22>I44 D<387FFFF8A2B5FCA214 F0150579941E>I<120EEA3F80127F12FFA31300127E123C0909778819>I<151815381578 15F0140114031407EC0FE0141F147FEB03FF90383FEFC0148FEB1C1F13001580A2143FA2 1500A25CA2147EA214FEA25CA21301A25CA21303A25CA21307A25CA2130FA25CA2131FA2 5CA2133FA291C7FC497EB61280A31D3877B72A>49 D<16E0ED01F01503A3150716E0A315 0F16C0A2151F1680A2ED3F00A3157EA2157C15FC5D14015D14035D14075D140F5D141F92 C7FC143EA25CECF81C153E903801F07EEB03E014C090380780FE130F49485A133EEB7C01 137801F05BEA01E03803C003EA0FFE391FFFC3F04813FB267C01FF13403AF0003FFFE000 601307C71400EC0FE05DA3141F5DA3143F92C7FCA4143E141C24487DB72A>52 D56 DI65 D<0107B612FCEFFF8018C0903B000FF0001F F04BEB07F81703021F15FC17014B14FEA2023F1400A24B1301A2147F18FC92C7120318F8 4A140718F04AEC0FE0EF1FC00101ED3F80EF7F004AEB01FEEE07F849B612E05F9139F800 07F0EE01FC01076E7E177F4AEC3F80A2010F16C0171F5CA2131F173F5CA2133FEF7F805C 1800017F5D4C5A91C7485A5F49140FEE1FE0494A5A00014AB45AB748C7FC16F816C03739 7BB83A>II<0103B612FEEFFFC018F0903B0007F8000FF84BEB03FCEF00FE020F157FF03F804B 141F19C0021F150F19E05D1807143F19F05DA2147FA292C8FCA25C180F5CA2130119E04A 151FA2130319C04A153FA201071780187F4A1600A2010F16FEA24A4A5A60011F15034D5A 4A5D4D5A013F4B5A173F4A4AC7FC17FC017FEC03F84C5A91C7EA1FC04949B45A007F90B5 48C8FCB712F016803C397CB83F>I<0107B8FCA3903A000FF000034BEB007F183E141F18 1E5DA2143FA25D181C147FA29238000380A24A130718004A91C7FC5E13015E4A133E167E 49B512FEA25EECF8000107147C163C4A1338A2010F147818E04A13701701011F16C01600 4A14031880013F150718004A5CA2017F151E173E91C8123C177C4915FC4C5A4914070001 ED7FF0B8FCA25F38397BB838>I<0107B712FEA3903A000FF000074B1300187C021F153C A25DA2143FA25D1838147FA292C8FCEE03804A130718004A91C7FCA201015CA24A131E16 3E010314FE91B5FC5EA2903807F800167C4A1378A2130FA24A1370A2011F14F0A24A90C8 FCA2133FA25CA2137FA291CAFCA25BA25B487EB6FCA337397BB836>II<0103B5D8 F80FB512E0A390260007F8C7381FE0004B5DA2020F153F615DA2021F157F96C7FC5DA202 3F5D605DA2027F14016092C7FCA24A1403605CA249B7FC60A202FCC712070103150F605C A20107151F605CA2010F153F605CA2011F157F95C8FC5CA2013F5D5F5CA2017F14015F91 C7FC491403007FD9FE01B512F8B55BA243397CB83E>I<0103B512F8A390390007F8005D A2140FA25DA2141FA25DA2143FA25DA2147FA292C7FCA25CA25CA21301A25CA21303A25C A21307A25CA2130FA25CA2131FA25CA2133FA25CA2137FA291C8FC497EB6FCA25C25397C B820>I<0207B512F0A391390007FC006F5AA215075EA3150F5EA3151F5EA3153F5EA315 7F93C7FCA35D5DA314015DA314035DA31407A25DA2140FA2003F5C5A141F485CA24A5A12 FC00E049C8FC14FE00705B495A6C485A381E0FC06CB4C9FCEA01F82C3B78B82C>I<0107 B512FCA25E9026000FF8C7FC5D5D141FA25DA2143FA25DA2147FA292C8FCA25CA25CA213 01A25CA21303A25CA21307A25CA2130F170C4A141CA2011F153C17384A1478A2013F1570 17F04A14E01601017F140317C091C71207160F49EC1F80163F4914FF000102071300B8FC A25E2E397BB834>76 D<902607FFF8923807FFF0614F13E0D9000FEFF0004F5AA2021F16 7FF1EFC0141DDA1CFCEC01CF023C16DF9538039F800238ED071FA20278ED0E3F97C7FC02 70151CA202F04B5AF0707E14E0037E14E0010117FE4D485A02C0EC0380A20103ED070161 0280140EA20107ED1C0305385B14006F137049160705E05B010EEC01C0A2011E91380380 0F61011CEC0700A2013C020E131F4C5C1338ED1FB80178163F04F091C8FC01705CA201F0 4A5B187E00015DD807F816FEB500C09039007FFFFC151E150E4C397AB84A>I<902603FF F891B512E0A281D90007923807F8006F6E5A61020F5E81DA0E7F5DA2021E6D1307033F92 C7FC141C82DA3C1F5C70130EEC380FA202786D131E0307141C147082DAF003143C701338 14E0150101016E1378030014705C8201036E13F0604A1480163F010715C1041F5B91C7FC 17E149EC0FE360010E15F31607011E15FF95C8FC011C80A2013C805F1338160013785F01 F8157CEA03FC267FFFE0143CB51538A243397CB83E>II<0107B612F817FF1880903B000FF000 3FE04BEB0FF0EF03F8141FEF01FC5DA2023F15FEA25DA2147FEF03FC92C7FCA24A15F817 074A15F0EF0FE01301EF1FC04AEC3F80EFFE0001034A5AEE0FF091B612C04CC7FCD907F8 C9FCA25CA2130FA25CA2131FA25CA2133FA25CA2137FA291CAFCA25BA25B1201B512FCA3 37397BB838>I<92383FC00E913901FFF01C020713FC91391FC07E3C91393F001F7C027C EB0FF84A130749481303495A4948EB01F0A2495AA2011F15E091C7FCA34915C0A36E90C7 FCA2806D7E14FCECFF806D13F015FE6D6D7E6D14E0010080023F7F14079138007FFC150F 15031501A21500A2167C120EA3001E15FC5EA3003E4A5AA24B5AA2007F4A5A4B5A6D49C7 FC6D133ED8F9F013FC39F8FC03F839F07FFFE0D8E01F138026C003FCC8FC2F3D7ABA2F> 83 D<0007B812E0A25AD9F800EB001F01C049EB07C0485AD900011403121E001C5C003C 17801403123800785C00701607140700F01700485CA2140FC792C7FC5DA2141FA25DA214 3FA25DA2147FA292C9FCA25CA25CA21301A25CA21303A25CA21307A25CA2130FA25CEB3F F0007FB512F8B6FCA2333971B83B>I<91B712F0A25B9239E0001FE092C7EA3FC0D903FC EC7F8002F015004A14FE16014948495A4A495A4C5A49C75B4C5A010E143F011E4A5A011C 4AC7FC4B5A5E90C7485A15074B5A4B5A4B5A5E157F4BC8FC4A5A4A5A4A5A5D140F4A5A4A 5A4A5A4AC712E05C13014948130149485C495A494813034A5C013F1407495A49C7FC4848 4AC7FC48485C5B0007153E4848147E4848EB01FE4848EB07FC4848133F90B6FCB7FC5E34 397AB833>90 D<14F8EB07FE90381F871C90383E03FE137CEBF801120148486C5A485A12 0FEBC001001F5CA2EA3F801403007F5C1300A21407485C5AA2140F5D48ECC1C0A2141F15 831680143F1587007C017F1300ECFF076C485B9038038F8E391F0F079E3907FE03FC3901 F000F0222677A42A>97 D<133FEA1FFFA3C67E137EA313FE5BA312015BA312035BA31207 EBE0F8EBE7FE9038EF0F80390FFC07C013F89038F003E013E0D81FC013F0A21380A2123F 1300A214075A127EA2140F12FE4814E0A2141F15C05AEC3F80A215005C147E5C387801F8 007C5B383C03E0383E07C0381E1F80D80FFEC7FCEA01F01C3B77B926>I<147F903803FF C090380FC1E090381F0070017E13784913383901F801F83803F003120713E0120FD81FC0 13F091C7FC485AA2127F90C8FCA35A5AA45AA3153015381578007C14F0007EEB01E0003E EB03C0EC0F806CEB3E00380F81F83803FFE0C690C7FC1D2677A426>II<147F9038 03FFC090380FC1E090383F00F0017E13785B485A485A485A120F4913F8001F14F0383F80 01EC07E0EC1F80397F81FF00EBFFF891C7FC90C8FC5A5AA55AA21530007C14381578007E 14F0003EEB01E0EC03C06CEB0F806CEB3E00380781F83803FFE0C690C7FC1D2677A426> II104 DI108 DII<147F903803FFC090380FC1F0 90381F00F8017E137C5B4848137E4848133E0007143F5B120F485AA2485A157F127F90C7 FCA215FF5A4814FEA2140115FC5AEC03F8A2EC07F015E0140F007C14C0007EEB1F80003E EB3F00147E6C13F8380F83F03803FFC0C648C7FC202677A42A>I<9039078007C090391F E03FF090393CF0787C903938F8E03E9038787FC00170497EECFF00D9F0FE148013E05CEA 01E113C15CA2D80003143FA25CA20107147FA24A1400A2010F5C5E5C4B5A131F5EEC8003 5E013F495A6E485A5E6E48C7FC017F133EEC70FC90387E3FF0EC0F8001FEC9FCA25BA212 01A25BA21203A25B1207B512C0A3293580A42A>II<3903C003F0390FF01FFC391E783C0F381C7C703A3C3EE03F8038383FC0EB7F80 0078150000701300151CD8F07E90C7FCEAE0FE5BA2120012015BA312035BA312075BA312 0F5BA3121F5BA3123F90C9FC120E212679A423>I<14FE903807FF8090380F83C090383E 00E04913F00178137001F813F00001130313F0A215E00003EB01C06DC7FC7FEBFFC06C13 F814FE6C7F6D13807F010F13C01300143F141F140F123E127E00FE1480A348EB1F0012E0 6C133E00705B6C5B381E03E06CB45AD801FEC7FC1C267AA422>II<13F8D803FEEB01C0D8078FEB03E0390E0F8007 121E121C0038140F131F007815C01270013F131F00F0130000E015805BD8007E133FA201 FE14005B5D120149137EA215FE120349EBFC0EA20201131E161C15F813E0163CD9F00313 3814070001ECF07091381EF8F03A00F83C78E090393FF03FC090390FC00F00272679A42D >I<01F0130ED803FC133FD8071EEB7F80EA0E1F121C123C0038143F49131F0070140FA2 5BD8F07E140000E08013FEC6485B150E12015B151E0003141C5BA2153C000714385B5DA3 5DA24A5A140300035C6D48C7FC0001130E3800F83CEB7FF8EB0FC0212679A426>I<13F0 D803FCEB01C0D8071EEB03E0D80E1F1307121C123C0038140F4914C01270A249131FD8F0 7E148012E013FEC648133F160012015B5D0003147E5BA215FE00075C5BA214015DA31403 5D14070003130FEBF01F3901F87FE038007FF7EB1FC7EB000F5DA2141F003F5C48133F92 C7FC147E147C007E13FC387001F8EB03E06C485A383C1F80D80FFEC8FCEA03F0233679A4 28>121 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fb cmcsc10 9 7 /Fb 7 116 df82 D<02FF13100107EBE03090391FC0787090397E 001CF001F813074848130348481301485A000F1400485A167048C8FCA2481530127EA200 FE1500A8127E1630127F7EA26C6C1460A26C7E000715C06C6CEB01806C6C13036C6CEB07 00017E130E90381FC078903807FFE00100138024287DA62C>99 D101 DI110 D114 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fc cmsy5 5 1 /Fc 1 107 df<12E0B3B3A50329799E13>106 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fd cmsy7 7 5 /Fd 5 107 df0 D<160E163E16FEED03F8ED0FE0ED3F80EDFE00 EC03F8EC0FE0EC3F8002FEC7FCEB03F8EB0FE0EB3F8001FEC8FCEA03F8EA0FE0EA3F8000 FEC9FC12F812FEEA3F80EA0FE0EA03F8EA00FEEB3F80EB0FE0EB03F8EB00FEEC3F80EC0F E0EC03F8EC00FEED3F80ED0FE0ED03F8ED00FE163E160E1600AB007FB612FCB712FEA227 357AA734>20 D<12E012F812FEEA3F80EA0FE0EA03F8EA00FEEB3F80EB0FE0EB03F8EB00 FEEC3F80EC0FE0EC03F8EC00FEED3F80ED0FE0ED03F8ED00FE163E16FEED03F8ED0FE0ED 3F80EDFE00EC03F8EC0FE0EC3F8002FEC7FCEB03F8EB0FE0EB3F8001FEC8FCEA03F8EA0F E0EA3F80007EC9FC12F812E0CAFCAB007FB612FCB712FEA227357AA734>I<017F157F26 01FFE0903803FFC0000701F890380FF1F0260F83FC90381F0038261E00FF013C7F001890 263F8078130C4890261FC0E07F007090260FE1C07F0060EB07E3913803F780486DB4C7EA 01806E5A157E157F81824B7E0060DAF7E0EB0300913801E3F0DBC3F85B6C90260381FC13 066C90260F00FE5B001C011E90387F803C6C017C90381FE0F82607C7F86DB45A2601FFE0 010313C06C6CC86CC7FC391B7C9942>49 D<12E0B3B3B3A5033B78AB14>106 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fe msbm7 7 1 /Fe 1 83 df82 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ff msbm10 10 3 /Ff 3 85 df78 D<007FB612E0B712FE6C6F7E2703C01E0313 E0000190393C00F3F00238EB70F8EE783CEE381E83EE3C07161C18801703A617071800EE 3C0FEE380E173EEE78FCEEF7F892380FFFE0023FB5128004FCC7FC16B8913838F03CED70 1CED781EED380EED3C0FED1C07031E7FED0E03030F7FED0701EE81E0ED0380707E030113 701778EEE0380300133C707EEE700EEE780F9338380780EE3C03041C13C093381E01E000 03013C90380E00F0007FB539F00FFFFEB67F6C8137397DB836>82 D<007FB812C0B9FCA23BE1FE38071FE1D8E3F0EC03F1D8E7C0EC00F9D8EF00153D00FE16 1F48160F481607A2481603A2481601A400601600C71600B3B102F813C0011FB6FC5B7F32 397DB838>84 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fg cmr5 5 8 /Fg 8 111 df<13301360EA01C0EA038013001206120E5AA25AA35AA312F0AB1270A37E A37EA27E12067E1380EA01C0EA006013300C297B9E16>40 D<12C0126012387E120C7E12 07EA0380A2EA01C0A3EA00E0A313F0AB13E0A3EA01C0A3EA0380A2EA070012065A121C5A 12605A0C297C9E16>I<14E0B0B712C0A3C700E0C7FCB022237C9B2B>43 D<1360EA01E0120F12FF12F11201B3A3387FFF80A2111C7B9B1C>49 DI<131FEBFF803801E7C0EA03C7138738078380EB8000A5EAFFFCA2EA07 80AEEA3FF8A2121D7F9C12>102 D<121C123EA3121CC7FCA612FEA2121EAEEAFFC0A20A 1D7D9C11>105 D<38FE1FC0EB7FE0381EC0F0381F0078A2121EAB38FFC3FFA218127D91 1F>110 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fh cmmi5 5 10 /Fh 10 119 df<137EEBFFE0EA01C71383EB80C014007F7F12007F1378137C137E48B4FC EA07DF380F0F80121CEA3C071278A212F0A414005BEA700E6C5AEA1FF86C5A131E7C9D1B >14 D<137FEA03FFEA0780000EC7FC5A5A1278EA7FFC12FF00F0C7FCA312701278EA3801 381E0780380FFE00EA03F811127C9119>I<3A03FFF007FF16FE3A003E0003E01680ED0E 005D4913705DEC03804AC7FCEBF81C143C147EEBF9FE3801F33FEBFE1F9038F80F8001F0 7F3803E007811403813807C001811400813AFFFC07FFC0A2281C7C9B2F>75 D78 D<0003B512E015FC39003E003FED0F80 ED07C0A25BA3ED0F8049EB1F00153E15F890B512E05A9038F003F0EC00F8A2485AA44848 485AA2163016603AFFFC00F8E0ED7FC0C8EA1F00241D7C9B2B>82 D<137F3801FFC0EA07C3380F03E0381C07C0EA3C0348C7FCA25AA500701340007813E038 3C03C0381FFF00EA07F813127C911B>99 D<14F8EB03FCEB073CEB0F7CA2EB0E38EB1E00 A53803FFF05A38003C00A35BA65BA5485AA4EA71C012FB5B00F3C7FC12FE123C16257B9C 1C>102 D 104 D<380F03F0383F87FC3833DC1EEA63F8EAC3F013E0EA03C0A248485AA3EC7820D80F 00136014F015C014F1001EEB7F80000CEB3E001B127D9125>110 D<380F8038381FC07CEA39E00061133C141C00C1130CEA03C0A21418EA0780A214301470 146014C03803C3803801FF00EA00FC16127D911E>118 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fi cmmi7 7 14 /Fi 14 119 df<1403A21406A45CA45CA4903807FF80011F13E090387C30F0D801F0133C 3803C060D80780131ED80F00131F48140F003E13C0A25AA239F801801FA3151E90380300 3E153C157C1578D8780613F0EC01E0003CEB03C0001EEB0F80390F0C3E003807FFF80001 13E0D8000CC7FC5BA45BA45BA220347CA728>30 D<1238127C12FE12FFA2127F123B1203 A31206A3120C121812381270122008127A8614>59 D<013FB512F816FF903A01FC001FC0 4AEB07E0EE03F001031401A24A14F8A2130717F04A130317E0010F1407EE0FC04AEB1F80 EE7E00011F495A91B512F0A291388001FC013FEB007E8291C7EA1F80160F4915C0A2137E A213FEEE1F805BEE3F000001153E16FE49EB01F84B5A0003EC1FC0B7C7FC15F82D287DA7 32>66 D<90263FFFF0EB7FF8A2D901FCC7EA1FC04AEC1E005F010315704C5A4AEB03804C C7FC0107141C5E4A13E04B5A010FEB0780030EC8FC4A5A157C011F13FE14C3EC877F149E 90393FB83F8014F09138C01FC0148049486C7EA2017E6D7EA201FE6D7EA2496D7EA20001 6E7EA249147FA2000382B539C007FFF8A235287DA738>75 D78 D<013FB512E016FC903901FC007F4AEB0F80 EE07C0010315E016035C17F01307EE07E05CA2010FEC0FC017804AEB1F00163E011F14F8 ED07F091B51280A290393F800FE0ED03F002007F15015BA2137EA201FE1303A2495CA200 01160817184914E017380003EDF070B5D8C00113E0923800FFC0C9EA3F002D297DA732> 82 D<000FB712E05A9039800FE007D81E009038C001C05A0038011F1300123000705C00 601501023F148012E0481400A2C74890C7FCA2147EA214FEA25CA21301A25CA21303A25C A21307A25CA2130FA25CA2131F001FB57EA22B287DA727>84 D<3B7FFFE003FFF0A2D803 F8C7EA3E0049143C16180007153816305BA2000F157016605BA2001F15E05E5BA2003F14 015E90C7FCA248140393C7FC127EA200FE5C15065AA2150E150C151C5D007C5C5D6C495A 003F495A261F800FC8FC3807C07C3801FFF038007F802C297BA72D>I97 D<130E131F5BA2133E131C90C7FCA7EA03E0487EEA0C78EA187C1230A212605B12C0A2EA 01F0A3485AA2485AA2EBC180EA0F81A2381F0300A213066C5A131CEA07F06C5A11287DA6 17>105 D<1407EC0F80141FA21500140E91C7FCA7EB03E0EB07F8EB0C3C1318EB303E13 6013C0A248485AA2C7FCA25CA4495AA4495AA4495AA4495AA21238D87C1FC7FC12FC133E 485AEA70F8EA7FE0EA1F80193380A61B>I<3907801FC0390FE07FF03918F0E0F83930F1 807CEBFB00D860FE133C5B5B00C1147C5B1201A248485BA34A5AEA07C01660EC03E0A23A 0F8007C0C0A2EDC180913803C300D81F0013C7EC01FE000EEB00F8231B7D9929>110 D<131C133EA25BA45BA4485AB512E0A23801F000485AA4485AA4485AA448C7FC1460A214 C0123EEB0180EB0300EA1E06EA1F1CEA0FF8EA03E013267EA419>116 D<3903E001C03907F003E0380C7807EA187C0030130314011260EBF80000C014C0A2EA01 F0A2EC0180EA03E0A2EC0300EA07C0A21406A25CA200035B6D5A3801F0E06CB45A013FC7 FC1B1B7D9921>118 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fj msam10 12 1 /Fj 1 4 df<007FBA1280BB12C0A300F0CB1203B3B3B3A6BBFCA36C198042447BC34D>3 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fk msbm6 6 1 /Fk 1 83 df82 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fl cmr8 8 22 /Fl 22 111 df<1406140FA34A7EA34A7EA3EC6FE01467A2ECC7F014C3A290380183F814 8101037F1400A2497F0106137EA249137F81A24980151FA24980150FA249801507A24980 1503000181491301A2000381A2487ED81FE0497ED8FFF890383FFFF0A22C2F7EAE31>3 D<007FB612FEA50060C8120600E01507481503A4CAFCA5486CEB0180A390B6FCA5903880 0001A3CAFCA700C01503A46C150700601506007FB612FEA5282D7DAC2F>I<1303130713 0E131C1338137013F0EA01E013C01203EA0780A2EA0F00A2121EA35AA45AA512F8A25AAB 7EA21278A57EA47EA37EA2EA0780A2EA03C0120113E0EA00F013701338131C130E130713 0310437AB11B>40 D<12C07E12707E7E7E120FEA0780120313C0EA01E0A2EA00F0A21378 A3133CA4131EA5131FA2130FAB131FA2131EA5133CA41378A313F0A2EA01E0A2EA03C013 801207EA0F00120E5A5A5A5A5A10437CB11B>I43 D48 D<130C133C137CEA03FC12FFEAFC7C 1200B3B113FE387FFFFEA2172C7AAB23>III< 140EA2141E143EA2147E14FEA2EB01BE1303143E1306130E130C131813381330136013E0 13C0EA0180120313001206120E120C5A123812305A12E0B612FCA2C7EA3E00A9147F9038 1FFFFCA21E2D7EAC23>I<000CEB0180380FC01F90B512005C5C14F014C0D80C7EC7FC90 C8FCA8EB1FC0EB7FF8380DE07C380F801F01001380000E130F000CEB07C0C713E0A21403 15F0A4127812FCA448EB07E012E0006014C00070130F6C14806CEB1F006C133E380780F8 3801FFE038007F801C2D7DAB23>II<12 30123C003FB512F8A215F05A15E039700001C000601480140348EB0700140E140CC7121C 5C143014705C495AA2495AA249C7FCA25B130E131EA2133EA3133C137CA413FCA913781D 2E7CAC23>III61 D91 D93 D<13C0487E487E487EEA0F3CEA1E1E 487E3870038038E001C0EAC000120A78AD23>I102 D105 D<3807C0FE39FFC3FF809038C703E0390FDE01F0EA07F8496C 7EA25BA25BB2486C487E3AFFFE1FFFC0A2221E7E9D27>110 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fm cmsy6 6 7 /Fm 7 108 df0 D<136013701360A20040132000E0137038F861 F0387E67E0381FFF803807FE00EA00F0EA07FE381FFF80387E67E038F861F038E0607000 40132000001300A21370136014157B9620>3 D<01FEEC0FE02603FFC0EB3FF8000F01F0 EBFE3E3B1F0FF801F0073C3C01FC07C003803B3000FE0F00010070D93F1EEB00C00060EB 1F9C00E0D90FF81460485C14076E7E6E7E81020315E00060D9073F14C091390F1F80016C 90261E0FE01380003890397C07F0073C1C01F003FE1F003B0F8FE001FFFE3B03FF80007F F8C648C7EA0FE033177C953D>49 D<903807FFF0133F5BD801FCC7FCEA03E0485A48C8FC 121E121C123C5A1270A212F05AA2B612F0A300E0C8FCA27E1270A212787E121C121E7E6C 7EEA03E0EA01FC6CB512F0133F13071C237A9D2A>I<1570A215F015E01401EC03C01580 140715005C140E141E5C14381478147014F05C1301495A5C130791C7FC5B130E131E131C 133C5B137013F05B12015B1203485A90C8FC5A120E121E121C123C5A127012F05A12601C 2F77A300>54 D<12E0B3B3AD033179A413>106 DI E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fn msbm10 12 4 /Fn 4 91 df<007FB54AB512C0B66C4914E0816C6E6D14C02707F003F09039000FF80026 01F801ED03F000006D7E017C6D6E5A017E137E017F133E8102807FECC00F6E6C7E017B80 903979F003F0ECF801903978FC00F8027C7F6E137E023F133E6E6C7E020F1480913907C0 0FC0EDE007913903F003E0020114F0913900F801F8EDFC00037E137C033E137E6F133EEE 801FDB0FC013810307EB0FC1923803E0079338F003E1DB01F813F1923900FC01F9EE7C00 70137D043F137F93381F803F040F131F933807C00F17E0933803F00704011303933800F8 0117FC177E173E171F1881EF0FC11707EF03E118F1EF01F91700187D01FC167F183FD803 FF161F007F01F8150FB57E18076C491503CB1201725A43467DC339>78 D<007FB712C0B812FCEFFF806C17E02800F807F00F13F8DBC00113FE017890398000FCFF 94387C3F8094383E0FC0727E94381E03F0EF1F011800717FA21978A519F8A24D5B1801EF 1E034E5A94383E0FC094387E3F80DDFDFFC7FC933807FFFE92B612F818E095C8FC17F0ED 87C1EEC0F8923883E0FC177C923881F03EA2923880F81F84EE7C0F717E163E93383F03E0 041F7FEE0F81EF80F8EE07C0187C933803E07E183E706C7E85706C6C7E180794387C03E0 057E7F94383E01F8716C7E197C01F86D6D6C7EF13F80007FB66C6CB512E0B700C015F083 6C4B6C14E044447EC33D>82 D<007FB912E0BA12F0A33CF07FE3C03C7FE026F1FE03EC07 F8D8F3F8ED01FCD8F7E0ED007ED8FFC0163F0180161F0100160F481707A2481703481701 A3481700A400601860C71700B3B3A40207133E020F133F0107B612FE4981A26D5D3C447D C33F>84 D<0003B812FE4883A301879039000F803ED98FF0011F137ED9BFC0EC007C01FF C7003E5B13FC48484A485A49ECFC034902F85B4B48485A5B494948485A0307131F04C090 C7FC90C7380F803EA24B485A00064A13FCC8003E5B4B485AA24B485A0201130703F05B4A 48485AA24A4848C8FC5E91380F803E021F5B1500023E5B1501027C5B9138FC03E014F849 48485AA24948485A0107011F156002C090C812F090380F803EA249484814014913FC013E 5B494848EC03E0A249484814070001130701F049140F48484848141FA2484848C8EA3FC0 000F49157FD9803E15FF484848EC01FBEF07F3003E49EC0FE7D87E01ED7F87007C499039 03FF0780BAFCA36C18003C447DC345>90 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fo cmex10 10 5 /Fo 5 114 df<151E153E157C15F8EC01F0EC03E01407EC0FC0EC1F8015005C147E5CA2 495A495AA2495AA2495AA2495AA249C7FCA2137EA213FE5B12015BA212035BA21207A25B 120FA35B121FA45B123FA548C8FCA912FEB3A8127FA96C7EA5121F7FA4120F7FA312077F A21203A27F1201A27F12007F137EA27FA26D7EA26D7EA26D7EA26D7EA26D7E6D7EA2147E 80801580EC0FC0EC07E01403EC01F0EC00F8157C153E151E1F94718232>16 D<12F07E127C7E7E6C7E7F6C7E6C7E12017F6C7E137EA27F6D7EA26D7EA26D7EA26D7EA2 6D7EA26D7EA280147E147F80A21580141FA215C0A2140F15E0A3140715F0A4140315F8A5 EC01FCA9EC00FEB3A8EC01FCA9EC03F8A515F01407A415E0140FA315C0141FA21580A214 3F1500A25C147E14FE5CA2495AA2495AA2495AA2495AA2495AA249C7FC137EA25B485A5B 1203485A485A5B48C8FC123E5A5A5A1F947D8232>I<167F923801FFC0923803C0F09238 07803892380F007892381F01FC151E153EA2157E92387C0070170015FCA44A5AA81403A4 5DA41407A94A5AAA4A5AA95DA4143FA492C8FCA7143E147EA4147C123800FE13FC5CA249 5A5CEA7803387007C0383C0F80D80FFEC9FCEA03F82E5C7C7F27>82 D90 D<1B301B78A21BF8A21BF01A01A21BE01A03A21BC01A07A21B 801A0FA21B0062A21A1E1A3EA21A3C1A7CA21A781AF8A262A21901A2621903A2621907A2 62190FA297C7FC61A2191E193EA2193C197CA2197819F8A2611801A2611803A261A21807 A261180FA296C8FC60A2181E183EA2183C187C131001301678017016F813F86000011601 1203486C5E000F1603121DD838FE5E00701607126000C05FEA407F0000160FA26D6C92C9 FC5FA2171E6D6C143EA2173C6D6C147CA2177817F86D7E5F16016D7E5F1603A26D6C5C16 07A26D6C5C160FA294CAFC027F5BA2161EEC3F80163EA2163C91381FC07CA2167891380F E0F8A25E15E1EC07F15E15F3EC03FB5E15FFA26E5BA36E90CBFCA35D157EA2157C153C15 384D96788353>113 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fp cmmi10 10 42 /Fp 42 121 df<1403EC3FF891387FFF80D901E313C014800103133F9138001F80ED0700 92C7FC80A280A2808013018080130080147F81143F8149B47E130790380F8FF0EB3E0F49 6C7E13F83801F003D803E07F1207380FC0011380121FEA3F0014005A127EA212FE5D4813 01A35DA24813035D6C13075D127C4A5A6C91C7FC5C6C133E6C6C5A3807C0F03801FFE0D8 003FC8FC223D7DBB25>14 DI18 D<133F14C0EB07F06D7E80 1301A26D7EA3147FA36E7EA36E7EA36E7EA36E7EA36E7EA36E7EA26E7EA214014A7E5C4A 7E91381E3F80143C14784A6C7E1301EB03E049486C7EEB0F80EB1F00496D7E137E5B4848 6D7E485A485A000F6E7E485A485A48C87E12FE167F4816800070151F293B7CB930>21 DI<027FB512C00103B612E0130F5B017F15C09026FF81FEC7FC3901FC007E48487F 485A497F484880485AA248C7FCA2127EA2153F00FE92C7FC5AA25D157E5A5DA24A5AA24A 5A007C495A5D003C495A003E013FC8FC6C137C380F81F83803FFE0C66CC9FC2B257DA32F >27 D<013FB512FE90B7FC5A5A4815FE260F801CC7FCEA1E005A00385B5A5A481378C7FC 147014F0A4495AA31303A3495AA3130FA25C131FA3133FA291C8FC131E28257EA324>I< 1503A35DA21506A2150EA2150CA2151CA21518A21538A21530A21570A2EC07FE91383FFF C0903901FCE3F0903907E0E0F890391F80C03ED93E007FEB7C01D801F8EC0F80D803F001 8013C0D807E014071403D80FC015E0D81F801300A248485AA2007E1306A2020E130F12FE 48010C14C0A2021CEB1F80A20218EB3F00A20238137E007C5D1430007E4A5A003E903870 03F06CEC07C09138600F80D80F80013FC7FC3903E0E0FC3901F8E7F039007FFF80D90FFC C8FCEB01C0A25CA21303A291C9FCA25BA21306A2130EA2130CA22B4B7CB931>30 D<121C127FEAFF80A5EA7F00121C0909798817>58 D<121C127FEAFF80A213C0A3127F12 1C1200A412011380A2120313005A1206120E5A5A5A12600A19798817>I<150C151E153E A2153C157CA2157815F8A215F01401A215E01403A215C01407A21580140FA215005CA214 1E143EA2143C147CA2147814F8A25C1301A25C1303A2495AA25C130FA291C7FC5BA2131E 133EA2133C137CA2137813F8A25B1201A25B1203A25B1207A25B120FA290C8FC5AA2121E 123EA2123C127CA2127812F8A25A12601F537BBD2A>61 D<126012FCB4FCEA7FC0EA1FF0 EA07FCEA01FF38007FC0EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC07FCEC01FF913800 7FC0ED1FF0ED07FCED01FF9238007FC0EE1FF0EE07FCEE01FF9338007F80EF1FC0A2EF7F 80933801FF00EE07FCEE1FF0EE7FC04B48C7FCED07FCED1FF0ED7FC04A48C8FCEC07FCEC 1FF0EC7FC04948C9FCEB07FCEB1FF0EB7FC04848CAFCEA07FCEA3FF0EA7FC048CBFC12FC 1270323279AD41>I64 D<0103B77E4916F018FC903B0007F80003FE4B EB00FFF07F80020FED3FC0181F4B15E0A2141FA25DA2143F19C04B143F1980027F157F19 0092C812FE4D5A4A4A5AEF0FF04AEC1FC005FFC7FC49B612FC5F02FCC7B4FCEF3FC00103 ED0FE0717E5C717E1307844A1401A2130F17035CA2131F4D5A5C4D5A133F4D5A4A4A5A4D 5A017F4BC7FC4C5A91C7EA07FC49EC3FF0B812C094C8FC16F83B397DB83F>66 D<9339FF8001C0030F13E0037F9038F80380913A01FF807E07913A07F8000F0FDA1FE0EB 079FDA3F80903803BF0002FFC76CB4FCD901FC80495A4948157E495A495A4948153E017F 163C49C9FC5B1201484816385B1207485A1830121F4993C7FCA2485AA3127F5BA312FF90 CCFCA41703A25F1706A26C160E170C171C5F6C7E5F001F5E6D4A5A6C6C4A5A16076C6C02 0EC8FC6C6C143C6C6C5C6CB4495A90393FE00FC0010FB5C9FC010313FC9038007FC03A3D 7CBA3B>I<0103B5D8F803B512F8495DA290260007F8C73807F8004B5DA2020F150F615D A2021F151F615DA2023F153F615DA2027F157F96C7FC92C8FCA24A5D605CA249B7FC60A2 02FCC7120101031503605CA201071507605CA2010F150F605CA2011F151F605CA2013F15 3F605CA2017F157F95C8FC91C8FC496C4A7EB690B6FCA345397DB845>72 D<0103B500F8903807FFFC5BA290260007F8C813804BEDFC0019F0020F4B5AF003804B4A C7FC180E021F1538604B5CEF0380023F4AC8FC170E4B133C1770027F5C4C5ADB0007C9FC 160E4A5B167E4A13FE4B7E01015B92380E7F80ECFC1CED383F010301E07FECFDC04A486C 7EECFF00D907FC6D7E5C4A130783130F707E5C1601011F81A24A6D7EA2013F6F7EA24A14 3F84137F717E91C8123F496C81B60107B512C0A26146397DB847>75 D<0103B6FC5B5E90260007FCC8FC5D5D140FA25DA2141FA25DA2143FA25DA2147FA292C9 FCA25CA25CA21301A25CA21303A25CA2130718404A15C0A2010F150118804A1403A2011F 16005F4A1406170E013F151E171C4A143C177C017F5D160391C7120F49EC7FF0B8FCA25F 32397DB839>I<902603FFF893383FFF80496081D900079438FF80000206DC01BFC7FCA2 020E4C5A1A7E020C1606190CDA1C7E16FE4F5A02181630A20238166162023016C1F00181 DA703F158395380303F002601506A202E0ED0C076202C01518183001016D6C140F06605B 028015C0A20103923801801FDD03005B140092380FC00649173F4D91C8FC01065DA2010E 4B5B4D137E130C6F6C5A011C17FEDCE1805B011802E3C7FCA2013802E6130104EC5C1330 ED03F8017016034C5C01F05CD807FC4C7EB500E0D9C007B512F01680150151397CB851> I<902603FFF891381FFFF8496D5CA2D90007030113006FEC007C02061678DA0EFF157081 020C6D1460A2DA1C3F15E0705CEC181F82023815016F6C5C1430150702706D1303030392 C7FC02607FA2DAE0015C701306ECC0008201016E130EEF800C5C163F0103EDC01C041F13 1891C713E0160F49EDF03818300106140717F8010E02031370EFFC60130CEE01FE011C16 E004005B011815FF177F1338600130153FA20170151F95C8FC01F081EA07FCB512E01706 A245397DB843>I<0103B612F849EDFF8018E0903B0007F8001FF84BEB03FCEF00FE020F 157FA24BEC3F80A2021F16C0A25DA2143FF07F805DA2027FEDFF006092C7485A4D5A4A4A 5A4D5A4AEC1F80057FC7FC0101EC07F891B612E094C8FC9139FC000FC00103EC03F0707E 4A6D7E831307177E5C177F010F5D5F5CA2011F1401A25CA2133F16034A4A1360A2017F17 E019C091C71401496C01011480B61503933900FE0700EF7E0ECAEA1FFCEF07F03B3B7DB8 3F>82 D<92391FE00380DBFFFC130002036D5A91390FE01F8F91393F0007DF027EEB01FE 02F81300495A4948147E177C4948143C495AA2011F153891C8FCA3491530A28094C7FC80 806D7E14FEECFFE06D13FE6DEBFFC06D14F06D806D80021F7F02037FEC003F03037F1500 167F163F161FA3120C160FA2001C151F94C7FCA3003C153EA25E003E5D127E007F4A5A6D 495A6DEB0FC0D8F9F0495AD8F0FE01FEC8FC39E03FFFF8010F13E0D8C00190C9FC313D7C BA33>I<0003B812FEA25A903AF8003FC00101C0913880007E4848163C90C7007F141C12 1E001C92C7FCA2485CA200305C007017180060130112E0485CA21403C716005DA21407A2 5DA2140FA25DA2141FA25DA2143FA25DA2147FA292C9FCA25CA25CA21301A25CA21303A2 5CEB0FFC003FB6FC5AA237397EB831>I<003FB56C48B51280485DA226007F80C7381FF0 0091C8EA07C0604993C7FCA2491506A20001160E170C5BA20003161C17185BA200071638 17305BA2000F167017605BA2001F16E05F5BA2003F15015F5BA2007F150394C8FC90C8FC A25E4815065A160E160C161C161816385E127E5E4B5A6C4A5A4BC9FC6C6C131E6C6C5B6C 6C13F83903F807E06CB55A6C6C48CAFCEB0FF0393B7BB839>I<277FFFFC01B500F890B5 1280B5FC60000390C7D807FCC7380FF80001FC4BEC03E000016204035E98C7FC621A0604 075DA2040F5DA2041B5D6216336D02735D1663000003C34A5A83DB01834AC8FC04815CDB 0301140603075D1506030C5DA203185D1970033015606115606D01E04A5A15C090267F01 804AC9FC17FEDA030014060400130E0206150C020E5D140C4A5DA24A5D18E04A5D715A5C 4A92CAFCA26DC85AA2013E157C1778133C1770133801301560513B7CB84E>87 D<1578EC01FEEC07C6EC0F861507EC1E03143E147C1507ECF806A2EB01F00103130EECE0 0C1307A2ECC01C010F1318153890381F80301570156090383F00E015C01401017F1380EB 7E03EC07001406EBFE0E495A5C143000011370495AEBF9C0EBFB8001FFC7FC5B5B485AA2 5BA4485A120F121DEA39F0127100E1140C0080143C0000147015E090387801C0EC078090 383C1E00EB1FF8EB07E0203C7FBA23>96 D<147E903803FF8090390FC1C38090391F00EF C0017E137F49133F485A4848EB1F8012075B000F143F48481400A2485A5D007F147E90C7 FCA215FE485C5AA214015D48150CA21403EDF01C16181407007C1538007E010F1330003E 131F027B13706C01E113E03A0F83C0F9C03A03FF007F80D800FCEB1F0026267DA42C>I< 133FEA1FFFA3C67E137EA313FE5BA312015BA312035BA31207EBE0FCEBE3FF9038E707C0 390FFE03E09038F801F001F013F8EBE000485A15FC5BA2123F90C7FCA214015A127EA214 0312FE4814F8A2140715F05AEC0FE0A215C0EC1F80143F00781400007C137E5C383C01F8 6C485A380F07C06CB4C7FCEA01FC1E3B7CB924>II<163FED1FFFA3ED007F167EA216FE A216FCA21501A216F8A21503A216F0A21507A2027E13E0903803FF8790380FC1CF90381F 00EF017EEB7FC049133F485A4848131F000715805B000F143F485A1600485A5D127F90C7 127EA215FE5A485CA21401A248ECF80CA21403161CEDF0181407007C1538007E010F1330 003E131F027B13706C01E113E03A0F83C0F9C03A03FF007F80D800FCEB1F00283B7DB92B >II<16F8ED03FEED0F8792381F0F80ED3E3F167F157CA215FC1700161C4A48C7 FCA414035DA414075DA20107B512F0A39026000FE0C7FC5DA4141F5DA4143F92C8FCA45C 147EA514FE5CA413015CA4495AA45C1307A25C121E123F387F8F80A200FF90C9FC131E12 FEEA7C3CEA7878EA1FF0EA07C0294C7CBA29>I104 D<14E0EB03F8A21307A314F0EB01C090C7 FCAB13F8EA03FEEA070F000E1380121C121812381230EA701F1260133F00E0130012C05B EA007EA213FE5B1201A25B12035BA20007131813E01438000F133013C01470EB806014E0 14C01381EB838038078700EA03FEEA00F815397EB71D>I110 D<90390F8003F090391FE00FFC903939F03C1F903A70F8700F8090 3AE0FDE007C09038C0FF80030013E00001491303018015F05CEA038113015CA2D8000314 07A25CA20107140FA24A14E0A2010F141F17C05CEE3F80131FEE7F004A137E16FE013F5C 6E485A4B5A6E485A90397F700F80DA383FC7FC90387E1FFCEC07E001FEC9FCA25BA21201 A25BA21203A25B1207B512C0A32C3583A42A>112 D<02FC13C0903803FF0190380F8383 90383F01C790397E00EF8049137F485A4848133F000715005B485A001F5C157E485AA200 7F14FE90C75AA3481301485CA31403485CA314075D140F127C141F007E495A003E137F38 1F01EF380F839F3903FF1F80EA00FC1300143F92C7FCA35C147EA314FE5C130190387FFF F0A322357DA425>I116 D<13F8D803FE1438D8070F147C000E6D13FC121C1218003814011230D8701F5C12601503 EAE03F00C001005B5BD8007E1307A201FE5C5B150F1201495CA2151F120349EC80C0A215 3F1681EE0180A2ED7F0303FF130012014A5B3A00F8079F0E90397C0E0F1C90393FFC07F8 903907F001F02A267EA430>I<01F8EB03C0D803FEEB07E0D8070F130F000E018013F012 1C12180038140700301403D8701F130112601500D8E03F14E000C090C7FC5BEA007E16C0 13FE5B1501000115805B150316001203495B1506150E150C151C151815385D00015C6D48 5A6C6C485AD97E0FC7FCEB1FFEEB07F024267EA428>I<01F816F0D803FE9138E001F8D8 070F903801F003000ED9800314FC121C12180038020713010030EDE000D8701F167C1260 030F143CD8E03F163800C001005B5BD8007E131F183001FE5C5B033F1470000117604991 C7FCA218E000034A14C049137E17011880170318005F03FE1306170E000101015C01F801 BF5B3B00FC039F8070903A7E0F0FC0E0903A1FFC03FFC0902703F0007FC7FC36267EA43B >I<903907E001F090391FF807FC9039783E0E0F9039E01F1C1FD801C09038383F803A03 800FF07F0100EBE0FF5A000E4A1300000C157E021F133C001C4AC7FC1218A2C7123FA292 C8FCA25CA2147EA214FEA24A130CA20101141C001E1518003F5BD87F81143801835C00FF 1560010714E03AFE0E7C01C0D87C1C495A2778383E0FC7FC391FF00FFC3907C003F02926 7EA42F>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fq cmmi8 8 42 /Fq 42 122 df12 D<131FD9FFC013304801F01370 00076D13604815E0D9807C13C0391E003C0148011E13800038EB0E03480107130000605C EC030612E0C7138EEC018C159C159815B815B0A215F05DA35DA25DA21403A44AC7FCA414 0EA45CA31418242C7F9D24>I15 D<1460A5EC7FF0EC3FF814FF903803CFE0903807800049C7FC131E5B 5B5BA2485A485AA2485A48C8FCA25A121E123E123CA3127C1278A412F8A4127CA2127E12 7F6C7E13E0EA1FFC6CB47E6C13F000017F38003FFCEB07FE1300143F80A280141EA2EB60 1CEB7838EB1FF0EB07C01D3C7DAD1F>I<3907C007E0390FE03FF8391CF8783E393879E0 1E39307B801F38707F00126013FEEAE0FC12C05B0081143F120149133EA20003147EA249 137CA2000714FCA24913F8A2000F1301A2018013F0A2001F1303A2010013E0120EC71207 A215C0A2140FA21580A2141FA21500A2140E202C7E9D23>I<147C49B4FC903803C78090 380783C090381F03E0EB1E01133E017C13F013F8A2EA01F0120313E01207A2EA0FC01403 A2EA1F80A21407003F14E0130090B5FCA2397F000FC0127EA2141F1580127C00FC14005C A2147EA248137C14FC00785B495AA2387C03E0383C07C0495A001C90C7FCEA1E3EEA0FF8 EA03E01C307DAE21>I<13E0486C133C15FF00031303EC0F7F9038E01C7EEC381C000749 C7FC5CEBC3C001C7C8FCEA0FDE13F8EBFF8014F8381F83FEEB803F496C7E140F48154016 C0123EA2007E14811680007C1403ED830000FC1487EC078E48EB03FC0070EB00F0221F7D 9D29>20 D<13FC13FFEB1FC0130F6D7EA36D7EA2130180A26D7EA3147EA280A36E7EA214 0F81A24A7E143F147FECF3F0EB01E3EB03C190380781F8130F49C67E133E5B49137E485A 48487F1207485A4848EB1F8048C7FC127E48EC0FC048EC07E000701403232F7DAD29>I< 131C013EEB0380ED07C0017E130F1680137CA201FC131F16005BA200015C153E5BA20003 147E157C5BA20007ECFC08EDF8185BA2000F0101133816309038E003F002071370001F90 380EF8609039F83C78E090397FF03FC090391FC00F0048C9FCA2123EA2127EA2127CA212 FCA25AA21270252C7E9D2A>I26 D<90B6128012035A481500261E00E0C7FC5A00 705B130112E012C0EA0003A25CA21307A349C8FCA35BA2131E133EA45BA21338211E7E9C 1F>28 D<1560A315E0A25DA21401A25DA21403A292C7FCA25CEC3FE0903803FFF890380F C63E90393E0E0F80017CEB07C03A01F00C03E0D803E0EB01F03807C01CD80F801300001F 011813F81300003E1338A2481330A2EC700100FC15F0481360150302E013E01507007801 C013C0007CEC0F800101EB1F00003C143E003E495A001F5C390F8383E03903E39F802600 FFFEC7FCEB1FF00107C8FCA21306A2130EA2130CA2131CA21318A3253C7DAD2A>30 D<1506A3150E150CA3151C1518A315381530A31570D801E0EB6007D807F8EC1F80EA0E3C D81C3E01E013C0003814C00030150F0070150726607E011480D8E07CEB800312C013FC38 80F803000002001300120113F04A5B00030106130601E0140E160C020E131C020C131801 C0143801E05C021C5B91381801C0D801F0495A030FC7FC3900FC381C90383F30F890380F FFE0010190C8FCEB00701460A314E05CA313015CA42A3C7EAD2E>32 D<123C127EB4FCA21380A2127F123D1201A312031300A25A1206120E5A5A5A126009157A 8714>59 D<15C0140114031580A214071500A25C140EA2141E141CA2143C143814781470 A214F05CA213015CA213035C130791C7FCA25B130EA2131E131CA2133C1338A213781370 13F05BA212015BA212035BA2120790C8FC5A120EA2121E121CA2123C1238A212781270A2 12F05AA21A437CB123>61 D<14C0A5497EA700F0EC03C039FF83F07F003FB61200000F14 FC000114E06C6C1380D91FFEC7FCEB07F8497EA2497EEB3F3FEB3E1F496C7EEB7807496C 7EA248486C7E48486C7E49137090C71230222180A023>63 D<013FB6FC17C0903A00FE00 07F0EE01F84AEB00FC177E1301177F5CA21303177E4A14FEA20107EC01FC17F84AEB03F0 EE07E0010FEC1FC0EE7F009138C003FC91B55A4914FE9139C0003F804AEB0FC017E0013F 140717F091C7FC16035BA2017E1407A201FE15E0160F4915C0161F0001ED3F80EE7F0049 14FEED03F80003EC0FF0B712C003FCC7FC302D7CAC35>66 D<92387FC003913903FFF807 91391FC03E0F91397E00071FD901F8EB03BF4948EB01FED90FC013004948147E49C8FC01 7E157C49153C485A120348481538485AA2485A173048481500A2127F90CAFCA35A5AA5EE 018016031700A2007E5D1606160E6C5D5E6C6C5C000F5D6C6C495A6C6CEB0780D801F801 1EC7FCD8007E13F890381FFFE0010390C8FC302F7CAD32>I<90263FFFFC90381FFF80A2 D900FEC73803F80018E04AEC07804DC7FC0101151C5F4A14E04C5A01034A5A040EC8FC4A 5B5E010714E04B5A9138E00780030EC9FC010F131F157F4A487E14C190391FC71FC014CE EC9C0F02F07F90383FE00702C07FEC0003825B6F7E137E6F7E13FE167F5B707E1201161F 4981831203B539E001FFFEA2392D7CAC3C>75 D<90383FFFFEA2010090C8FC5C5CA21301 A25CA21303A25CA21307A25CA2130FA25CA2131FA25CA2133FA291C7EA0180A249140317 00017E5C160601FE140EA2495C163C12015E49EB01F84B5A0003141FB7FC5E292D7DAC30 >I78 D<013FB512F816FF903A00FE001FC0EE07E04A6D7E707E01016E7E A24A80A213034C5A5CA201074A5A5F4A495A4C5A010F4A5A047EC7FC9138C003F891B512 E04991C8FC9138C007C04A6C7E6F7E013F80150091C77EA2491301A2017E5CA201FE1303 A25BA20001EE038018005B5F0003913801FC0EB5D8E000133CEE7FF0C9EA0FC0312E7CAC 35>82 D<000FB8FCA23B1FC003F8003F0100151F001C4A130E123C003801071406123000 704A130EA20060010F140C12E0485CA2141FC715005DA2143FA292C8FCA25CA2147EA214 FEA25CA21301A25CA21303A25CA21307A25C130F131F001FB512F0A2302D7FAC29>84 D<3B7FFFF801FFFEA2D801FCC7EA0FC0178049EC070016060003150E160C5BA20007151C 16185BA2000F153816305BA2001F157016605BA2003F15E05E90C8FCA24814015E127EA2 150300FE92C7FC5A5D1506150E007C5C151815386C5C5D6CEB03C0260F800FC8FC3803E0 3C3801FFF038003FC02F2E7BAC30>III<1303A51230A714C01307133F13FF1237 123FB5120013FB13C3130312F812F01230B114C01307133F13FF1237123FB5120013FB13 C3130312F812F01230A390C7FCA5123D7CAE1B>93 D<157C4AB4FC913807C380EC0F8715 0FEC1F1FA391383E0E0092C7FCA3147E147CA414FC90383FFFF8A2D900F8C7FCA313015C A413035CA413075CA5130F5CA4131F91C8FCA4133EA3EA383C12FC5BA25B12F0EAE1E0EA 7FC0001FC9FC213D7CAE22>102 D<131FEA03FFA2EA003FA2133EA2137EA2137CA213FC A25BA21201143F9038F1FFC09038F3C1F03803FF0001FC7F5BA2485A5BA25B000F13015D 1380A2001F13035D1300140748ECC04016C0003E130F1580007E148191381F0180007C14 03ED070000FCEB0F06151E48EB07F80070EB01E0222F7DAD29>104 D<1307EB0F80EB1FC0A2EB0F80EB070090C7FCA9EA01E0EA07F8EA0E3CEA1C3E12381230 1270EA607EEAE07C12C013FC485A120012015B12035BA21207EBC04014C0120F13801381 381F01801303EB0700EA0F06131EEA07F8EA01F0122E7EAC18>I<15E0EC01F01403A3EC 01C091C7FCA9147CEB03FE9038078F80EB0E07131C013813C01330EB700F0160138013E0 13C0EB801F13001500A25CA2143EA2147EA2147CA214FCA25CA21301A25CA21303A25CA2 130700385BEAFC0F5C49C7FCEAF83EEAF0F8EA7FF0EA1F801C3B81AC1D>I<131FEA03FF A2EA003FA2133EA2137EA2137CA213FCA25BA2120115F89038F003FCEC0F0E0003EB1C1E EC387EEBE07014E03807E1C09038E3803849C7FC13CEEA0FDC13F8A2EBFF80381F9FE0EB 83F0EB01F81300481404150C123EA2007E141C1518007CEBF038ECF83000FC1470EC78E0 48EB3FC00070EB0F801F2F7DAD25>I<27078007F0137E3C1FE01FFC03FF803C18F0781F 0783E03B3878E00F1E01263079C001B87F26707F8013B00060010013F001FE14E000E015 C0485A4914800081021F130300015F491400A200034A13076049133E170F0007027EEC80 80188149017C131F1801000F02FCEB3F03053E130049495C180E001F0101EC1E0C183C01 0049EB0FF0000E6D48EB03E0391F7E9D3E>109 D<3907C007E0391FE03FF83918F8783E 393879E01E39307B801F38707F00126013FEEAE0FC12C05B00815C0001143E5BA2000314 7E157C5B15FC0007ECF8081618EBC00115F0000F1538913803E0300180147016E0001F01 0113C015E390C7EAFF00000E143E251F7E9D2B>I<90387C01F89038FE07FE3901CF8E0F 3A03879C0780D907B813C0000713F000069038E003E0EB0FC0000E1380120CA2D8081F13 0712001400A249130F16C0133EA2017EEB1F80A2017C14005D01FC133E5D15FC6D485A39 01FF03E09038FB87C0D9F1FFC7FCEBF0FC000390C8FCA25BA21207A25BA2120FA2EAFFFC A2232B829D24>112 D115 D<130E131FA25BA2133EA2137EA2137CA213FCA2B512F8A23801F800A25BA21203A25BA2 1207A25BA2120FA25BA2001F1310143013001470146014E0381E01C0EB0380381F0700EA 0F0EEA07FCEA01F0152B7EA919>IIII<013F137C9038FFC1FF3A01C1E383803A0380F703C0390700F60F000E13FE48 13FC12180038EC0700003049C7FCA2EA200100005BA313035CA301075B5D14C000385CD8 7C0F130600FC140E011F130C011B131C39F03BE038D8707113F0393FE0FFC0260F803FC7 FC221F7E9D28>II E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fr cmr6 6 12 /Fr 12 127 df<130C1338137013E0EA01C0EA038013005A120EA25AA25AA312781270A3 12F0AB1270A312781238A37EA27EA27E7E1380EA01C0EA00E013701338130C0E317AA418 >40 D<12C012707E7E7E7E7E1380EA01C0A2EA00E0A21370A313781338A3133CAB1338A3 13781370A313E0A2EA01C0A2EA038013005A120E5A5A5A12C00E317CA418>I<1438B2B7 12FEA3C70038C7FCB227277C9F2F>43 D<13FF000313C0380781E0380F00F0001E137848 133CA248131EA400F8131FAD0078131EA2007C133E003C133CA26C13786C13F0380781E0 3803FFC0C6130018227DA01E>48 D<13E01201120712FF12F91201B3A7487EB512C0A212 217AA01E>II<13FF000313C0380F03E0381C 00F014F8003E13FC147CA2001E13FC120CC712F8A2EB01F0EB03E0EB0FC03801FF00A238 0003E0EB00F01478147C143E143F1230127812FCA2143E48137E0060137C003813F8381E 03F0380FFFC00001130018227DA01E>I<14E01301A213031307A2130D131D1339133113 6113E113C1EA01811203EA07011206120C121C12181230127012E0B6FCA2380001E0A6EB 03F0EB3FFFA218227DA11E>I<137F3803FFC0380781E0380E00704813380018131C1238 A3123C003F1338381FC078EBE0F0380FF9E03807FF80120114C0000713F0380F0FF8381C 03FC383801FE3870007E141F48130F1407A314060070130E0078130C6C1338001F13F038 07FFC0C6130018227DA01E>56 D61 D<1203EA0780EA0FC0EA1FE0EA3CF0EA7878EAE01CEAC00C0E0878A21E>94 D<000F1380381F81C0383FF780387BFF00EAE07EEA403C12067AA11E>126 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fs cmsy8 8 16 /Fs 16 111 df0 D<006015C000E014016C14030078EC07806C EC0F006C141E6C5C6C6C5B6C6C5B6C6C485A6C6C485A90387807806D48C7FCEB1E1E6D5A EB07F86D5A6D5A497E497EEB0F3CEB1E1E497E496C7E496C7E48486C7E48486C7E484813 7848C77E001E80488048EC078048EC03C048140100601400222376A137>2 D<130C131EA50060EB01800078130739FC0C0FC0007FEB3F80393F8C7F003807CCF83801 FFE038007F80011EC7FCEB7F803801FFE03807CCF8383F8C7F397F0C3F8000FCEB0FC039 781E078000601301000090C7FCA5130C1A1D7C9E23>I<140381B3A3B812FCA3C7D80380 C7FCB3B812FCA32E2F7CAD37>6 DI20 D<170EA3170F8384170384170184717E1878187C84180F F007C0BA12F819FC19F8CBEA07C0F00F00183E601878604D5A60170360170795C7FC5F17 0EA33E237CA147>33 D<137813FE1201A3120313FCA3EA07F8A313F0A2EA0FE0A313C012 1F1380A3EA3F00A3123E127E127CA35AA35A0F227EA413>48 DI<91B512C01307131FD97F80C7FC01FCC8FCEA01F0EA03C0485A48C9FC120E121E5A 123812781270A212F05AA3B712C0A300E0C9FCA37E1270A212781238123C7E120E120F6C 7E6C7EEA01F0EA00FCEB7F80011FB512C013071300222B7AA52F>I<91383FFFF00107B6 FC011F15E0017F812701F83E0313FCD803C09038003FFED80F00EC07FF001E017E6D1380 481500007CEE7FC00078163F48EE1FE048137CC7150FA202FC1407A25CA3010116C05CA2 EF0F8013034A15005F171E49485C177C177849485C4C5A4C5A49C7485A040EC7FC163C01 3E14F0ED03E0013CEB0F80017C01FEC8FC90387FFFF848B512E04849C9FC4813E0332D7E AC37>68 D78 D<91383FFFFC0107B612C0011F15F0017F15FC2701F83E007FD803C0EC0FFFD80F001403 001E017E0100138048167F007C163F127848017C141F5AC7160014FC171E173E4A143C17 7C177801015D4A495A4C5A4CC7FC0103141E4A137CED03F09138E1FFC0D907E790C8FCEC CFF8ECDF8002C0C9FC495AA349CAFCA3133EA2133C137CA25BA25B485A138031307EAC31 >80 D<12E0B3B3B3AD034378B114>106 D<0060136000E01370B3B3B3AB006013601443 79B123>I<12E0A27E1270A212781238A2123C121CA2121E120EA2120F7E7F1203A27F12 01A27F1200A27F137013781338A2133C131CA2131E130EA2130F7FA2801303801301A280 1300A2801470A214781438143C141CA2141E140EA2140F80A215801403A215C014011400 1A437CB123>110 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Ft cmmi6 6 33 /Ft 33 122 df14 DI21 D<90381FFFFC90B5FC5A4814F8 3907C07C00380F003C001E131C48131E12381278A2485BA35C1470007013F0495A6C485A D81C0FC7FCEA0FFEEA03F01E167E9424>27 D<140C5CA45CA45CA4EB0FFE90387FFF8090 38F0C3C03903C0C1E039078180F0D80F011370001E1478123CEB03001278A3D8F00613F0 A215E0140139700C03C0007814800038EB0700003C131E381E187C3807FFF06C13C0D800 18C7FC5BA45BA45B1D2E7DA225>30 D<4A7EA34AC7FCA41406A45CA2D807C01307D80FE0 EB0F803818F01812300060140712C0EC3003EAC1E000011500EA03C01460A2D807801306 A24A5A5D15385D3903C180E09038E181C02601F98FC7FC38007FFEEB0FF00103C8FCA313 06A45BA3212E7DA229>32 D38 DI<127812FCA212FEA2127E1206A3120CA2121C121812301260124007107A8513>59 D<140C141C143C1438A21478147014F014E0130114C0A21303148013071400A25B130E13 1E131CA2133C13381378137013F05BA212015B12035BA2120790C7FC5A120EA2121E121C 123C123812781270A212F05AA216317CA420>61 D<1308A4131CA500C0EB0180397FBEFF 00381FFFFC000713F0000113C06C6CC7FCA2497E13F73801E3C013C1380380E0EB006000 067F00041310487F19197D9820>63 D<91380FF0039138FFFE06903903F8070E90390FC0 019E013FC712FE017C147C5B4848143C485A48481438485A121F90C8FC481530123E007E 1500A25AA516C0A2ED0180A2007CEC0300A2003C1406003E5C6C5C6C6C13706C6C5B3903 F807802600FFFEC7FCEB1FF028247CA22C>67 D<90B6FC16E0903907C003F8ED00FC4948 133E161E161FEE0F8049C7FCA317C0133EA4491580161FA21700495CA2163E5E485A5E4B 5A4B5A4848495A4B5A033EC7FC0007EB01F8B612E092C8FC2A227CA132>I<903AFFFE07 FFF0A2903A07C0003E00A249485BA449C75AA4013E495AA3013FB5FC495C90387C0003A3 49495AA44848495AA4484849C7FCA300075C3AFFFE07FFF0A22C227CA132>72 D75 D<90B57E92C7FCEB07C0A2495AA449C8FCA4133EA45BA45BED0180A2ED0300485A1506A2 150E48485B153C15F800071303B6FC5D21227CA12A>I78 D<90B512FCEDFF80903907C00FC0ED03F090380F800116F81500A290 381F0001A3ED03F0013E14E0ED07C0ED1F80ED7E0090387FFFF815E090387C01F0EC0078 49137C153C153EA24848137EA448485BEDFE06A20007EC7E0CD8FFFE1418ED3FF0C8EA07 E027237CA12E>82 D85 D<1318A31230A7131E133EEA31FE1237123FEAFFF813D8131812F812 F01230AB131E133EEA31FE1237123FEAFFF813D8131812F812F01230A31300A30F2F7CA3 19>93 D99 D<140FEC3FC0EC71E014E3A2010113C0ECE180ECE000495AA5495AA2EBFFFEA2EB0780A2 49C7FCA5131EA65BA55BA31370A2EA38F0EA78E012F8EAF9C0EA7180007FC8FC121E1B2F 7CA31E>102 D<13F8EA0FF0A21200A2485AA4485AA4380787E0EB9FF8EBB83CEBE01C38 0FC01E1380A21300001E5BA35C5AA2ECF0201530481460EB01E015C0A239F000E380ECFF 000060133C1C247CA224>104 D<1338137CA2137813701300A7EA0780EA1FC0EA38E012 30EA60F0EAC1E0A3EA03C0A3EA0780A2EA0F0013041306EA1E0CA21318121CEA1E70EA0F E0EA07800F237DA116>I<1418143C147CA214381400A7EB0780EB1FE01338EB60F013C0 A2EA0180A2380001E0A4EB03C0A4EB0780A4EB0F00A4131EA21238EA783CEAF8381378EA 70F0EA7FC0001FC7FC162D81A119>I<13F8EA0FF0A21200A2485AA4485AA43807801E14 7FEB81C3EB8387380F060F495A1318EB700E4848C7FCA213FCEA1E7EEA3C0F80EB078115 8039780F0300A21402EB070600F0138CEB03F8386000F019247CA221>I<000F13FC381F C3FF3931C707803861EC0301F813C0EAC1F0A213E03903C00780A3EC0F00EA0780A2EC1E 041506D80F00130C143C15181538001EEB1C70EC1FE0000CEB07801F177D9526>110 D115 D117 D<3807800E380FE01FEA38F012300060130F12C01407EAC1E000 011306EA03C0A33807800CA214081418A21430146014C0EA03C13801FF00EA007E18177D 951F>II<3801F01E3907FC7F80390E1CE1C038180F8100 301383007013071260EC0380D8001EC7FCA45BA21580003014C0397878018012F8EC0300 38F0FC0638E19C1C387F0FF8381E03E01A177D9523>II E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fu msbm8 8 3 /Fu 3 91 df82 D84 D<001FB612FCA23A183E00701801F0EB6038D81BC0EBE030001FC7EAC070003E01 0113E0003C90380380C01501003801071380EC0603003090380E0700EC1C06EC180EC7EA 380CEC301CEC7038ECE030ECC07001015B4A5AEB03819038070180EB0603D90E07C7FCEB 0C06EB1C0EEB380CEB301CD970381303EBE030EBC070000101601307EB80E0260381C013 0626070180130ED80603141E000E90C7FCD80C07143ED81C0E1476D8380C14E6D8301CEB 01CED87018EB078CD86038EB3E0CB712FCA2282E7EAD38>90 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fv cmex10 12 26 /Fv 26 113 df0 D<12E07E12787E7E7E7F6C7E6C7E7F12016C7E7F137C137E7FA26D7EA26D7EA26D7EA36D 7EA2801301A2801300A280A2147EA2147FA4801580A7EC1FC0B3A5EC3F80A715005CA414 7EA214FEA25CA213015CA213035CA2495AA3495AA2495AA249C7FCA2137E137C13FC5B48 5A12035B485A485A90C8FC121E5A5A5A5A1A777C832E>II I8 D<12F012FEEAFFC0EA3FF0EA07FCEA01FE6C6C7EEB3FC06D7E130F801307801303B3B080 1301A26D7E806E7E6E7E6E7EEC07F8EC01FC9138007F80ED1FE01503151FED7F80913801 FC00EC07F8EC1FE04A5A4A5A4AC7FC5C495AA213035CB3B013075C130F5C131F495AEBFF 804848C8FCEA07FCEA3FF0EAFFC048C9FC12F0237775833A>I<12F0B3B3B3AA04407281 21>12 D<00F0EB03C0B3B3B3AA1A40728137>I<16F01501ED03E0ED07C0ED0F80ED1F00 5D157E5D5D14014A5A4A5A4A5AA24A5A143F92C7FC147EA25C13015C13035C13075C130F 5C131FA2495AA349C8FCA213FEA312015BA212035BA21207A25BA2120FA25BA2121FA45B A2123FA55B127FA990C9FC5AB3AA7E7FA9123F7FA5121FA27FA4120FA27FA21207A27FA2 1203A27F1201A27F1200A3137FA26D7EA36D7EA2130F801307801303801301801300147E A28081141F6E7EA26E7E6E7E6E7E140081157E8181ED0F80ED07C0ED03E0ED01F0150024 B26E833B>16 D<12F07E127C7E7E6C7E7F6C7E6C7E12017F6C7E137E7FA26D7E80130F6D 7EA26D7E80130180130080147E147F8081A26E7EA36E7EA26E7EA3811403A2811401A281 A21400A281A281A21680A4153FA216C0A5151F16E0A9150F16F0B3AA16E0151FA916C015 3FA51680A2157FA41600A25DA25DA21401A25DA214035DA214075DA34A5AA24A5AA34A5A A292C7FC5C147E14FE5C13015C13035C495AA2495A131F5C49C8FCA2137E5B485A5B1203 485A485A5B48C9FC123E5A5A5A24B27C833B>I<171E173E177C17F8EE01F0EE03E0EE07 C0160FEE1F80EE3F00167E167C16FC4B5A4B5A15075E4B5A4B5A153F93C7FC5D15FE5D14 015D14034A5AA24A5AA24A5AA24A5AA24AC8FCA214FEA213015C13035C1307A25C130F5C 131FA25C133FA3495AA349C9FCA35A5BA312035BA31207A25BA2120FA35BA3121FA35BA3 123FA55BA2127FAB485AB3B06C7EAB123FA27FA5121FA37FA3120FA37FA31207A27FA212 03A37F1201A37F7EA36D7EA36D7EA3131F80A2130F80130780A21303801301801300A214 7FA26E7EA26E7EA26E7EA26E7EA26E7E140181140081157F8182151F6F7E6F7E8215036F 7E6F7E167C167E82EE1F80EE0FC01607EE03E0EE01F0EE00F8177C173E171E2FEE6B8349 >I<12F07E127C7E7E6C7E6C7E7F6C7E6C7E6C7E137C137E7F6D7E80130F6D7E6D7E8013 01806D7E147E147F80816E7EA26E7EA26E7EA26E7EA26E7EA26E7EA2818182153F82A215 1F82150F82A2150782A36F7EA36F7EA38281A31780167FA317C0A2163FA217E0A3161FA3 17F0A3160FA317F8A51607A217FCABEE03FEB3B0EE07FCAB17F8A2160FA517F0A3161FA3 17E0A3163FA317C0A2167FA21780A316FF1700A35D5EA34B5AA34B5AA35E150FA25E151F 5E153FA25E157F93C7FC5D5DA24A5AA24A5AA24A5AA24A5AA24A5AA24A5A92C8FC5C147E 14FE495A5C13035C495A495A131F5C49C9FC137E137C13FC485A485A485A5B485A48CAFC 123E5A5A5A2FEE7C8349>II I<170F173F17FF1603EE0FFCEE1FF0EE7FE0EEFF804B13004B5A4B5A4B5A4B5A4B5A4B5A 15FF5E5C93C7FC5C5D14075DA3140F5DB3B3B3AE4A5AA3143F5DA24A5AA24A5AA24990C8 FC495AA2495A495A495A495A495A49C9FC485AEA07FCEA0FF0EA3FC0B4CAFC12FCA2B4FC EA3FC0EA0FF0EA07FCEA01FE6C7EEB7FC06D7E6D7E6D7E6D7E6D7EA26D7E6D7FA26E7EA2 6E7EA281141FA36E7EB3B3B3AE811407A38114038180828082157F6F7E6F7E6F7E6F7E6F 7E6F7E6F1380EE7FE0EE1FF0EE0FFCEE03FF1600173F170F30EE73834B>26 D<12F012FE6C7E7FEA3FF0EA0FFC6C7E3801FF806C7F6D7EEB1FF06D7E6D7E806D7E7F6D 7F81147F81143F81141FA381140FB3B3B3AE6E7EA46E7EA26E7EA26E7FA26F7EA26F7E6F 7E6F7E15076F7E6F7E923800FF80EE7FC0EE1FE0EE0FF8EE03FCEE00FF173FA217FFEE03 FCEE0FF8EE1FE0EE7FC0EEFF80923801FE004B5A4B5A150F4B5A4B5A4B5AA24B5AA24A90 C7FCA24A5AA24A5AA44A5AB3B3B3AE141F5DA3143F5D147F5D14FF5D4990C8FC5B495A5C 495A495AEB7FE0495A485BD807FEC9FC485AEA3FF0EAFFC05B48CAFC12F030EE73834B> I[29 298 101 131 58 34 D[29 298 127 131 58 I<17FF040313C093380F81F093381E0078043E137C93387C01 FC9338F803FEA2150116F01503EF01FC9338E0007003071400A3150FA45E151FA7153FA7 4B5AA715FFA85C93C8FCA95C5DA85DA74A5AA75DA75D140FA45DA3001C5C007F131FEAFF 8092C9FC5C143EA26C485A007C5B003C5B381F03E03807FF80D801FECAFC376F7B7F2F> 82 D88 D<1B3FF3FFC0973803E0F0973807C03897 380F801C081F137E97383F01FEF303FF1A7E1AFE1AFC1901A2963903F801FEF300781C00 4F5AA2190FA262191FA34F5AA44F5AA319FF97C8FCA360A261A21803A3611807A461180F A44E5AA4183FA261A2187FA361A218FFA44D5BA55F96C9FCA35FA360A2170FA460171FA4 60173FA54D5AA54D5AA45E60A55E60A44C90CAFCA54C5AA55F161FA45F163FA45FA2167F A35FA316FF5FA54B5BA494CBFCA25DA35EA21507A25EA44B5AA45E151FA45E153FA35EA2 157FA25EA315FF93CCFCA34A5AA44A5AA35D14075DA2140F5D121E397F801FC0EAFFC05D 143F92CDFC5C147E6C485AEA7E00383801F86C485A380F07C03803FF80D800FCCEFC58DD 7B7F37>90 D104 DI110 D<12F012FE6C7E13E0EA 3FF8EA0FFCEA03FFC67F6D7E6D7E6D7E6D7E6D7E6D7EA26D7E7FA281147FB3B3AF81143F A281141F81140F8114076E7E6E7E6E7E6F7E6F7EED1FF0ED07F8ED01FE923800FF80163F A216FF923801FE00ED07F8ED1FF0ED3FC04B5A4BC7FC4A5A4A5A4A5A140F5D141F5D143F 5DA2147F5DB3B3AF14FF92C8FCA25B495AA2495A495A495A495A495A495A000390C9FCEA 0FFCEA3FF8EAFFE0138048CAFC12F029B2748342>I<1DC0F401E01C03A2F407C0A2F40F 80A2F41F00A21C3EA264A264A2641B01A2515AA2515AA2515AA251C7FCA21B3EA263A263 A2505AA2505AA2505AA2505AA250C8FCA21A3EA21A3C1A7CA262A24F5AA24F5AA24F5AA2 4F5AA24FC9FCA20104173E130E011E5F137F495F5A486D4B5A120F261C7FC04B5A123826 F03FE04B5A124000004D5A6D7E96CAFC6D6C5DA26D6C153EA2606D7E606D7E4D5A6D7F4D 5AA26E6C495AA26E6C495AA26E6C49CBFCA26E6C133EA25F6E7E5F6E7E4C5AEC01FF4C5A A26EEB83C01687ED7FC7EECF80ED3FEF04FFCCFCA26F5AA26F5AA26F5AA25E15035E6F5A 5B78758364>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fw cmmi12 12 70 /Fw 70 123 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>I<1560A7ED7FFF92B512C0913807F80191381FDFFF91397F87FE004AC8FCEB03 FC495A130F5C495A133F5C137F5C13FFA291C9FCA57F80A2133F6D7E90390FE3FF806DB5 12E0903901FC006049B512E0D90F8F1380011EC9FC5B13F8485A5B485A485A48CAFCA212 1E123E123C127C1278A312F8A47EA27E127F7FEA3FE013F86CB4FC6C13C0000313F86C13 FE39007FFFC0010F13F0010313FC9038007FFF021F7F02037F1400151F150F1507A40102 5C903803800FD901C090C7FC903800F83EEC3FF8EC07E02A597EC42B>24 D<010FB712E0013F16F05B48B812E04817C02807E0060030C7FCEB800EEA0F00001E010C 13705A0038011C13605A0060011813E000E013381240C7FC5C4B5AA214F014E013011503 14C01303A3EB078082130FA2EB1F00A34980133E137EA24980A2000114015BA26C48EB00 E0342C7EAA37>II<02 03B612E0021F15F091B7FC4916E0010716C090270FF80FF8C7FC90381FC00349486C7E01 7EC7FC49147E485A4848143E0007153F5B485AA2485AA2123F90C8FC5E48157E127EA216 FE00FE5D5A15015EA24B5A007C5D15074B5A5E6C4AC8FC153E6C5C5D390F8003F03907C0 07C02601F03FC9FC38007FFCEB1FE0342C7DAA37>I<010FB612FC013F15FE5B48B712FC 4816F82707E001C0C7FC01805B380F0003121E121C5A4849C8FC126012E000405BC7FC14 0E141EA45CA3147CA2147814F8A4495AA31303A25C1307A3130FA25C6D5A2F2C7EAA2A> I<161CA21618A21638A21630A21670A21660A216E0A25EA21501A25EA21503A293C8FCA2 5DED7FE0913807FFFE91391FC63F809139FE0E07C0D901F8EB03F0903A07E00C00F8D91F C08090263F001C137E017E814913184848ED1F8000031438485A4848013014C0A2484813 70A248481360A248C712E0A24B133F481780481301A24B137F180014034816FE92C7FC4C 5A6C49495AA2007E0106495A4C5A6C010E495A4C5A261F800C49C7FC000F15FC3A07C01C 01F8D803E0EB07E03A01F8181F80D8007E01FEC8FC90381FFFF801011380D90030C9FCA2 1470A21460A214E0A25CA21301A25CA21303A291CAFCA332597BC43A>30 D<1730A317701760A317E05FA316015FA3160394C8FCA35E1606A3160E160C013E1607D9 FF80ED1F802603C3C0011CEB3FC0260703E01318260601F0157F000E173F001C1538D818 030230131F0038170F0030170700701570D86007026013035CA2D8E00F02E0148000C049 491301EA001F4A150303011500013F5C1400604901031406017E91C7FC180E180C01FE49 141C4901061418183860030E1460030C14E04D5A4D5A031C49C7FC0318130E017E5D5F6D 01385B90261F80305BD90FC0EB03C0D907F0010FC8FC903901FE707C9039003FFFF00203 1380DA0060C9FC15E05DA314015DA3140392CAFCA35C1406A3140E140C3A597DC43F>32 D35 D38 D<177F0130913803FFC00170020F13E0494A13F048 484A13F84848EC7F0190C838F8007C484A48133C00064A48131C000E4B131E000C4A4813 0E001C92C7FC0018140E150C0038021C1406003014185D180E00704A140C1260154003C0 141C00E017184A5A4817386C17304AC8127018E01260007049EC01C0EF03800078010614 070038EE0F00003C010E141E6C167CD81F805D6C6C48EB03F0D807F0EC0FE0D803FEEC3F C02801FFFC03FFC7FC6C6CB55A6D14F8010F14E0010114809026007FF8C8FC02F8C9FCA2 5CA21301A3495AA31307A25C130FA4131F5C6DCAFC37417BAB40>I42 D<121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A0A 78891B>58 D<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413E013C0A312011380 120313005A1206120E5A5A5A12600B1D78891B>II< 1618163C167CA2167816F8A216F01501A216E01503A216C01507A21680150FA2ED1F00A2 151E153EA2153C157CA2157815F8A25D1401A24A5AA25D1407A25D140FA292C7FC5CA214 1E143EA2143C147CA25CA25C1301A25C1303A25C1307A25C130FA291C8FC5BA2133EA213 3C137CA2137813F8A25B1201A25B1203A2485AA25B120FA290C9FC5AA2121E123EA2123C 127CA2127812F8A25A126026647BCA31>I<127012FCB4FCEA7FC0EA1FF0EA07FCEA01FF 38007FC0EB1FF0EB07FE903801FF809038007FE0EC1FF8EC03FE913800FF80ED3FE0ED0F F8ED03FF030013C0EE3FF0EE0FFCEE01FF9338007FC0EF1FF0EF07FCEF01FF9438007FC0 F01FE0A2F07FC0943801FF00EF07FCEF1FF0EF7FC04C48C7FCEE0FFCEE3FF0EEFFC00303 90C8FCED0FF8ED3FE0EDFF80DA03FEC9FCEC1FF8EC7FE0903801FF80D907FECAFCEB1FF0 EB7FC04848CBFCEA07FCEA1FF0EA7FC048CCFC12FC12703B3878B44C>I64 D<1830187018F0A217011703A24D7EA2170F171FA21737A2176717E717C793380187FCA2 EE0307EE07031606160CA216181638163004607FA216C0030113011680ED0300A2150615 0E150C5D845D03707F15605DA24A5A4AB7FCA25C0206C87F5C021C157F14185CA25C14E0 5C495A8549C9FC49163F1306130E5B133C137C01FE4C7ED807FFED01FF007F01F0027FEB FFC0B5FC5C42477DC649>I<91B87E19F019FC02009039C00003FF6F480100138003FFED 3FC01AE093C8121FF10FF0A24A17F84B1507A314035D190FA2020717F04B151F1AE0193F 020F17C04BED7F80F1FF004E5A021F4B5A4B4A5AF01FF0F03FC0023F4AB4C7FC4BEB1FFC 92B612F018FEDA7FC0C7EA7F804BEC1FC0F00FF0727E02FF6F7E92C8FC727EA249835CA3 13035CA301075F4A1503A24E5A130F4A4B5A4E5AA2011F4C5A4A4B5A4D485A013F4B48C7 FCEF0FFC4AEC3FF801FF913801FFE0B9128005FCC8FC17C045447CC34A>I<4CB46C1318 043F01F013384BB512FC0307D9007E1378DB1FF090380F80F0DB7F80EB03C1DA01FEC7EA 01C34A48EC00E7DA0FF0ED7FE04A48153F4A5A02FFC9121F494817C04948160F495A130F 4A178049481607495A137F4948170091CAFC5A485A1906485AA2485A96C7FC121F5BA212 3F5BA3127F5BA4485AA419C0A2180161127F180396C7FC6018066C6C160E601818001F17 386D5E000F5F6D4B5A6C6C4B5A00034CC8FC6C6C150E6C6C153C017F5DD93FC0EB01E0D9 1FF0EB0FC0D907FE017FC9FC0101B512FCD9003F13E0020790CAFC45487CC546>I<91B8 7E19F019FC02009039C00007FF6F489038007FC003FFED1FE0737E93C86C7E737E19014A 707E5D1A7FA20203EF3F805DA21BC014075DA3140F4B17E0A3141F4B17C0A3143F4B167F A3027F18804B16FFA302FF180092C95A62A24917034A5F19076201034D5A5C4F5A620107 173F4A5F4FC7FC19FE010F4C5A4A15034E5AF00FE0011F4C5A4A4B5A06FFC8FC013FED01 FCEF0FF84AEC3FE001FF913803FF80B848C9FC17F094CAFC4B447CC351>I<91B6D8E003 B61280A3020001E0C70003EB8000DB7F806E48C7FC03FF1503A293C85BA219075C4B5EA2 190F14034B5EA2191F14074B5EA2193F140F4B5EA2197F141F4B5EA219FF143F92B8C8FC A3DA7FC0C712014B5DA2180314FF92C85BA218075B4A5EA2180F13034A5EA2181F13074A 5EA2183F130F4A5EA2187F131F4A5EA2013F16FFA24A93C9FCD9FFE002037FB6D8E003B6 7EA351447CC351>72 D<027FB512F8A217F09139007FF000ED3FC0157FA25EA315FF93C7 FCA35C5DA314035DA314075DA3140F5DA3141F5DA3143F5DA3147F5DA314FF92C8FCA35B 5CA313035CA313075CA3130F5CA2131FA25CEB7FF0007FB512F0B6FCA22D447DC32B>I< 91B600E049B512C0A3020001E0C8383FF800DB7F80ED1FE003FF94C7FC1A3E93C9127862 F101C04A4C5A4B4BC8FC191C6102035E4B5DF003804EC9FC0207150E4B14386060020F4A 5A4B0107CAFC170E5F021F14784B13F84C7E1603023F130F4B487E163BEEE1FF91387FC1 C1DB83807FED8700159CDAFFB86D7E5D03C06D7E5D4990C7FC4A6E7EA2717E13034A8117 07A201076F7E5C717EA2130F4A6E7FA2727E131F5C727E133F854A82D9FFE04B7EB600E0 010FB512E05FA252447CC353>75 D<91B612F8A3020001E0C8FC6F5A4B5AA293C9FCA35C 5DA314035DA314075DA3140F5DA3141F5DA3143F5DA3147F5DA314FF92CAFCA35B4A16C0 A21801010317804A15031900A201075E4A1506180E181E010F161C4A153C18381878011F 16F84A4A5A1703013F150F4D5A4A14FF01FF02075BB9FCA2603A447CC342>I<91B500C0 020FB5128082A2DA007F9239007FE00070ED1F8074C7FCDBEFF8150E15CF03C7160C7015 1C1401DB83FE1518A2DB81FF1538140303001630831A704A6D7E02061760163F7114E014 0E020C6D6C5CA2706C1301141C021801075D83190302386D7E023094C8FC1601715B1470 02606DEB8006A294387FC00E14E04A023F130C18E0191C0101ED1FF04A1618170FF0F838 130391C83807FC30A2943803FE705B01060301136018FF19E0010E81010C5F187FA2131C 0118705A1338181F137801FC70C9FCEA03FFB512F884180651447CC34E>78 DI<91B712F018FF19E002009039C0003FF86F48EB07FC03FFEC01FEF000 7F93C8EA3F801AC0F11FE05C5D1AF0A214035DA30207EE3FE05DA2F17FC0020F17804B15 FF1A004E5A021F4B5A4B4A5AF00FE04E5A023F037FC7FC4BEB03FCEF1FF092B612804A4A C8FC923980007F80EF0FC0EF07F002FF6E7E92C77F1701845B4A1400A2170113035CA217 0313075CA24D5A130F5CA3011F18185CA2013F4C13381A304A6F1370D9FFE0020314E0B6 00E0ED01C00501EB0380943900FE0F00CBEA3FFEF007F045467CC34A>82 D<9339FF8001800307EBF003033F13FC9239FF007E07DA01F8EB0F0FDA07E09038079F00 4A486DB4FC4AC77E023E804A5D187E5C495A183C495AA213074A1538A3130F183080A295 C7FC806D7E8014FF6D13E015FC6DEBFFC06D14FC6E13FF6E14C0020F80020314F8EC003F 03077F9238007FFE160F1603707E8283A283A21206A4000E163EA2120C177E001E167CA2 5F5F003F15014C5A6D4A5A4C5A486C4AC8FC6D143ED87CF85CD8787E495A3AF01FC00FE0 D8E007B51280010149C9FC39C0003FF039487BC53C>I<48BA12C05AA291C7D980001380 D807F092C7121F4949150F0180170748C75B1903120E48020316005E1218123800301407 4C5C00701806126000E0140F485DA3C8001F92C7FC5EA3153F5EA3157F5EA315FF93CAFC A35C5DA314035DA314075DA3140F5DA3141F5DA3143F5DA2147FA214FF01037F001FB612 FCA25E42447EC339>I<003FB500F80103B512E0A326003FF8C8381FF800D91FE0ED07E0 013F705A615C96C7FC60017F16065CA2180E01FF160C91C9FCA2181C4817185BA2183800 0317305BA21870000717605BA218E0120F495EA21701121F495EA21703123F4993C8FCA2 5F127F491506A2170E00FF160C90C9FC171CA21718173817304816705F6C5E6C15014C5A 4CC9FC6C150E6D141E001F5D6D5C6C6CEB01E06C6C495A6C6CEB1F80C6B401FECAFC9038 7FFFF8011F13E0010190CBFC43467AC342>I<007FB56C91381FFFF8B65DA2000101E0C8 000313006C0180ED01FCF000F0614E5AA2017F4C5A96C7FC1806A2606E5DA2013F5E1870 186060A24D5A6E4AC8FCA2011F1506170E170C5FA26E5C5FA2010F5D16015F4CC9FCA26E 13065EA201075C5EA25E16E06E5B4B5A13034BCAFC1506A25D151CECFE185D13015D5DA2 6E5AA292CBFC5C13005C5CA25CA25C45467BC339>II<023FB500E0011FB5FCA391 26007FFEC7000313E0DB3FF8913801FE006F486E5A1AF06F6C4A5A626F6C4A5A0706C7FC 190E6F6C5C616F6C5C6171485A6F5D4EC8FC93387FC00660706C5A6060706C5A17F19338 0FFB8005FFC9FC5F705AA2707EA3707E5E04067F5E93381C7FC0163816704C6C7EED01C0 4B486C7E160015064B6D7E5D4B6D7E5D5D4A486D7E14034AC76C7E140E5C4A6E7F143002 E06F7E495A0103707E495A131F496C4B7E2603FFE04A487E007F01FC021FEBFFF0B5FCA2 50447EC351>II97 DIIIII<157E913803FF8091 390FC1E0E091391F0073F0027E13334A133F4948131F010315E04948130F495AA2494814 C0133F4A131F137F91C713805B163F5A491500A25E120349147EA216FEA2495CA21501A2 5EA21503150700015D150F0000141F6D133F017CEB77E090383E01E790381F078F903807 FE0FD901F85B90C7FC151FA25EA2153FA293C7FCA2001C147E007F14FE485C4A5A140348 495AEC0FC000F8495A007C01FEC8FC381FFFF8000313C02C407EAB2F>I<14FE137FA3EB 01FC13001301A25CA21303A25CA21307A25CA2130FA25CA2131FA25CED3FC090393F81FF F0913887C0FC91380E007E023C133ED97F70133F4A7F4A14805C13FF91C7FC5BA2484814 3F17005BA200035D167E5BA2000715FE5E5B1501000F5DA24913035E001F160703071306 4914E0150F003FEDC00E170C90C7141CEE80184816381730007E167017E000FE91380781 C0EEC38048913801FF000038EC007C30467BC438>I<141E143F5C5CA3147E143891C7FC AE133EEBFF803801C3C0380781E0380601F0120E121CEA180312381230A2EA700700605B A2EAE00F00C05BEA001F5CA2133F91C7FCA25B137E13FE5BA212015BEC03800003140013 F01207495A1406140E140CEBC01C141814385C00035BEBE1C0C6B45A013EC7FC19437DC1 21>I<163C16FEA21501A316FCED00701600AE15FCEC03FF91380F0780021C13C0913838 03E0147014E014C01301EC8007130314005B0106130F130E010C14C090C7FC151FA21680 A2153FA21600A25DA2157EA215FEA25DA21401A25DA21403A25DA21407A25DA2140FA25D A2141F5DA2143F001C91C7FC127F48137E5CA248485AEB03E038F807C038781F80D83FFE C8FCEA07F0275681C128>I<14FE137FA3EB01FC13001301A25CA21303A25CA21307A25C A2130FA25CA2131FA25C163F013FECFFC0923803C0E09138000703ED1E0F491338ED701F 017E13E0EC01C001FE018013C00203EB07004948C8FC140E00015B5C495A5C3803FBC001 FFC9FC8014F83807F1FE9038F03F809038E00FE06E7E000F130381EBC001A2001FED01C0 17801380A2003F15031700010013F05E481506160E007E150C161C00FE01005BED787048 EC3FE00038EC0F802B467BC433>I<01F8D903FCEC7F80D803FED91FFF903803FFE0D807 1F903B7C0FC00F81F83E0E0F80E007E01C00FC001C9026C3C0030178137C271807C700D9 F0E0137E02CE902601F1C0133E003801DCDAFB80133F003001D892C7FCD90FF814FF0070 495C0060495CA200E04949485CD8C01F187E4A5C1200040715FE013F6091C75BA2040F14 014960017E5D1903041F5D13FE494B130762043F160E0001060F130C4992C713C0191F4C ED801C00031A1849027E1638F2003004FE167000071A60494A16E0F201C0030192380F03 80000FF18700494AEC03FED80380D90070EC00F84F2D7DAB55>109 D<01F8EB03FCD803FEEB1FFFD8071F90387C0FC03B0E0F80E007E03A0C07C3C003001CD9 C7007F001801CE1301003801DC80003013D8EB0FF800705B00605BA200E0491303D8C01F 5D5C12001607013F5D91C7FCA2160F495D137E161F5F13FE49143F94C7FC187000014B13 6049147E16FE4C13E0000317C049150104F81380170300071700495D170EEE781C000FED 7C3849EC1FF0D80380EC07C0342D7DAB3A>I112 D<91380FC00391383FF0079138 F83C0F903903E00E1E90390FC0063E90381F800790393F00037E4914FC01FE1301485AA2 484814F812075B000F140316F0485AA2003F14074914E0A3007F140F4914C0A3151F90C7 13805AA2153F6C1500A2127E5D007F14FE6C1301A214036C6C485A000F131E3807C03838 03E0F13901FFC1F838003F01130014035DA314075DA3140F5DA2141FA2143F011FB51280 A21600283F7DAB2B>I<01F8EB0FC0D803FEEB7FF0D8070FEBF038000E903883C07C3A0C 07C701FC001C13CE0018EBDC03003813D8003013F8D90FF013F800709038E000E0006015 005C12E0EAC01F5C1200A2133F91C8FCA35B137EA313FE5BA312015BA312035BA312075B A3120F5BEA0380262D7DAB2C>I I<141C147EA314FE5CA313015CA313035CA313075CA2007FB512FCB6FC15F839000FC000 A2131F5CA3133F91C7FCA35B137EA313FE5BA312015BA312035BA21570000714605B15E0 15C0000F130101C013801403EC070000071306140E5C6C6C5A000113F03800FFC0013FC7 FC1E3F7EBD23>I<133ED9FF8014E02603C3C0EB03F0380703E0380601F0000E1507121C D818035D12380030150FA2D870075D00605B161FEAE00F00C0495CEA001F4A133FA2013F 92C7FC91C7FC5E5B017E147EA216FE13FE495CA20301EB01801703484802F81300A25F03 03130616F000001407030F130E6D010D130C017C011D131C033913186D9038F0F838903A 1F03C07870903A07FF803FE0903A01FC000F80312D7DAB38>I<013E140ED9FF80EB3F80 2603C3C0137F380703E0380601F0120E121CD81803143F0038151F0030150FA2D8700714 0700605BA2D8E00F150000C0497FEA001F4A5B1606133F91C7FC160E49140C137EA2161C 01FE14185B1638163016704848146016E05E150100005D15036D49C7FC1506017C130E01 7E5B6D137890380F81E06DB45AD900FEC8FC292D7DAB2F>I<013E1738D9FF80D901C013 FC2603C3C0903907E001FE380703E0380601F0000E150F001C16C0D8180316000038187E 0030031F143E00705ED86007171E5C163FD8E00F92C7121C00C049160CEA001F4A49141C 047E1418133F91C7FC04FE1438491730017E5CA20301157001FE1760495C19E019C0A249 49481301198018031900606D0107140670130E017C010F5C017E010C1418013ED91CFC13 386DD9387E13F0903B0FC0F01F01C0903B03FFC00FFF809028007F0001FEC7FC3F2D7DAB 46>I<02FCEB07E0903A03FF801FFC903A0F07C0781E903A1C03E0E01F903A3801F1C07F D9700013804901FB13FF4848EBFF00495B000316FE90C71438484A130012061401000E5C 120CC7FC14035DA314075DA3140F5DA3021F143817305D1770023F1460121E003F16E026 7F807FEB01C0026F148000FF01EF1303D901CFEB070000FE903887C00E267C03835B3A3C 0F01E0783A1FFC00FFE0D803F0EB3F80302D7EAB37>I<133ED9FF8014E02603C3C0EB03 F0380703E0380601F0000E1507001C16E0EA180312380030150F007016C0EA60075C161F D8E00F158000C05BEA001F4A133F1700133F91C7FC5E49147E137EA216FE01FE5C5BA215 015E485AA215035EA200001407150F6D5C017C131F153F6D13FF90391F03CFC0903807FF 8F903801FC0F90C7121F5EA2153F93C7FCD807C05BD81FE0137E5DA24848485A4A5A0180 5B39380007C00018495A001C49C8FC6C137C380781F83803FFE0C66CC9FC2C407DAB30> I<027CEB018049B413034901801300010F6D5A49EBE00E6F5A90393F03F838903978007E F80170EB1FF00160EB01E001E05C49495A90C748C7FC150E5D5D5D5D4A5A4A5A4AC8FC14 0E5C5C5C5CEB03C049C9FC130E49141C4914185B49143848481430491470D8039014F048 B4495A3A0FEFC007C0391E03F01FD81C01B55A486C91C7FC485C00606D5A00E0EB3FF048 EB0FC0292D7CAB2D>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fx cmsy10 12 32 /Fx 32 121 df<007FB912E0BA12F0A26C18E03C04789A4D>0 D<121FEA3F80EA7FC0EA FFE0A5EA7FC0EA3F80EA1F000B0B789E1C>I<0060160600F8160F6C161F007E163F6C16 7E6C6C15FC6C6CEC01F86C6CEC03F06C6CEC07E06C6CEC0FC06C6CEC1F80017EEC3F006D 147E6D6C5B6D6C485A6D6C485A6D6C485A6D6C485A6D6C485ADA7E3FC7FCEC3F7E6E5A6E 5A6E5AA24A7E4A7EEC3F7EEC7E3F4A6C7E49486C7E49486C7E49486C7E49486C7E49486C 7E49C7127E017E8049EC1F804848EC0FC04848EC07E04848EC03F04848EC01F84848EC00 FC48C9127E007E163F48161F48160F00601606303072B04D>I<16C04B7EB3AC007FBA12 80BB12C0A26C1980C8D801E0C9FCB3A9007FBA1280BB12C0A26C198042427BC14D>6 D<49B4FC010F13E0013F13F8497F3901FF01FF3A03F8003F80D807E0EB0FC04848EB07E0 4848EB03F090C71201003EEC00F8A248157CA20078153C00F8153EA248151EA56C153EA2 0078153C007C157CA26C15F8A26CEC01F06D13036C6CEB07E06C6CEB0FC0D803F8EB3F80 3A01FF01FF0039007FFFFC6D5B010F13E0010190C7FC27277BAB32>14 D<037FB612E00207B712F0143F91B812E0010301C0C9FCD907FCCAFCEB0FE0EB3F8049CB FC13FC485A485A485A5B485A121F90CCFC123EA2123C127CA2127812F8A25AA87EA21278 127CA2123C123EA27E7F120F6C7E7F6C7E6C7E6C7E137E6D7EEB1FE0EB07FC6DB47E0100 90B712E0023F16F01407020016E092CAFCB0001FB912E04818F0A26C18E03C4E78BE4D> 18 D<19E0F003F0180FF03FE0F0FF80943803FE00EF0FF8EF3FE0EFFF80DC03FEC7FCEE 0FF8EE3FE0EEFF80DB03FEC8FCED1FF8ED7FE0913801FF80DA07FEC9FCEC1FF0EC7FC049 48CAFCEB07FCEB1FF0EB7FC04848CBFCEA07FCEA1FF0EA7FC048CCFCA2EA7FC0EA1FF0EA 07FCEA01FF38007FC0EB1FF0EB07FCEB01FF9038007FC0EC1FF0EC07FC913801FF809138 007FE0ED1FF8ED07FE923800FF80EE3FE0EE0FF8EE03FE933800FF80EF3FE0EF0FF8EF03 FE943800FF80F03FE0F00FF01803F000E01900B0007FB912E0BA12F0A26C18E03C4E78BE 4D>20 D<127012FCB4FCEA7FC0EA1FF0EA07FCEA01FF38007FC0EB1FF0EB07FCEB01FF90 38007FC0EC1FF0EC07FC913801FF809138007FE0ED1FF8ED07FE923800FF80EE3FE0EE0F F8EE03FE933800FF80EF3FE0EF0FF8EF03FE943800FF80F03FE0F00FF0A2F03FE0F0FF80 943803FE00EF0FF8EF3FE0EFFF80DC03FEC7FCEE0FF8EE3FE0EEFF80DB03FEC8FCED1FF8 ED7FE0913801FF80DA07FEC9FCEC1FF0EC7FC04948CAFCEB07FCEB1FF0EB7FC04848CBFC EA07FCEA1FF0EA7FC048CCFC12FC1270CDFCB0007FB912E0BA12F0A26C18E03C4E78BE4D >I<037FB612E00207B712F0143F91B812E0010301C0C9FCD907FCCAFCEB0FE0EB3F8049 CBFC13FC485A485A485A5B485A121F90CCFC123EA2123C127CA2127812F8A25AA87EA212 78127CA2123C123EA27E7F120F6C7E7F6C7E6C7E6C7E137E6D7EEB1FE0EB07FC6DB47E01 0090B712E0023F16F01407020016E03C3A78B54D>26 D<1AF0A3861A78A21A7C1A3CA21A 3E1A1E1A1F747EA2747E747E87747E747E1B7E87757EF30FE0F303F8007FBC12FEBE1280 A26CF3FE00CEEA03F8F30FE0F31F8051C7FC1B7E63505A505A63505A505AA250C8FC1A1E 1A3E1A3CA21A7C1A78A21AF862A359347BB264>33 D39 D<49B4EF3FC0010F01E0923803FFF8013F01FC030F13FE4901FF92383FE01F48B6 6C91397E0007C02603F80301E0D901F8EB01E02807E0007FF049486D7E01806D6CD907C0 147048C76C6C494880001EDA07FE49C87E001C6E6C013E150C486E6D48150E7148150648 6E01E0160793387FF1F0006092263FF3E08193381FFBC000E004FF1780486F4915017090 C9FC82707F8482717E844D7E6C4B6D1503006004EF1700933803E7FE0070922607C7FF5D DC0F837F003004816D140E00384BC6FC0018033E6D6C5C001C4B6D6C143C6C4BD91FFC5C 6C4A486D6C5C6DD907E06D6C13036C6C49486D9038E00FE0D801F0013FC890B55A27007C 03FE6F91C7FC90263FFFF8031F5B010F01E0030313F8D901FECAEA7FC0592D7BAB64>49 D<92B6FC02071580143F91B7120001030180C8FCD907FCC9FCEB1FE0EB3F80017ECAFC5B 485A485A485A5B485A121F90CBFC123EA2123C127CA2127812F8A25AA2B9FC1880A21800 00F0CBFCA27EA21278127CA2123C123EA27E7F120F6C7E7F6C7E6C7E6C7E137E6D7EEB1F E0EB07FC6DB47E010090B6FC023F1580140702001500313A78B542>I<1706170F171FA2 173EA2177CA217F8A2EE01F0A2EE03E0A2EE07C0A2EE0F80A2EE1F00A2163EA25EA25EA2 4B5AA24B5AA24B5AA24B5AA24BC7FCA2153EA25DA25DA24A5AA24A5AA24A5AA24A5AA24A C8FCA2143EA25CA25CA2495AA2495AA2495AA2495AA249C9FCA2133EA25BA25BA2485AA2 485AA2485AA2485AA248CAFCA2123EA25AA25AA25A1260305C72C600>54 D<126012F0B012FC12FEA212FC12F0B0126007267BAB00>I<0060171800F0173C6C177C A200781778007C17F8A2003C17F0003E1601A26CEE03E0A26C17C06D1507A2000717806D 150FA26C6CED1F00A20001161E6D153EA20000163C90B712FCA26D5DA2013CC85A013E14 01A2011E5D011F1403A26D5D6E1307A26D6C495AA2010392C7FC6E5BA20101141E6E133E A26D6C5BA202781378027C13F8A2023C5BEC3E01A26E485AA2020F5B1587A202075B15CF A26EB4C8FCA26E5AA36E5AA315781530364780C437>I<031FB512C00203B7FC021F16E0 91B812F8010317FE010F717E90283FE07FC03F80D9FE00020080D801F8041F7FD803E04A 01077F48481601000F716C7E4848717E003F02FF151F007F180F90C7707E00FE92C8FC48 8400F01A80008084C75AA24B81A414035DA21B00A24A5AA24F5AA24A5A621903624A5A4F 5AA24B4B5A023F5F191F4B5E027F4CC7FC197E92C9127C4A5E4E5A4A4B5A01014C5AF01F 804A033EC8FC01035E4A4A5AEF07E00107ED1FC04A02FFC9FC010FEC07FC4AEBFFF091B6 12C0017F4ACAFC90B612F04815804802F8CBFC4891CCFC49447EC34D>68 DI<0403B712F8043F 16FE4BB9FC1507151F157F912601FC0090C7EA07FE912603F001ED01FCDA07C04915F0DA 0F80EE0080021F1800EC3F004A495A5C5C495A4A495A5C495A6DC7FC90C8485AA35F161F A34C5AA35F167F94B612C0A293B7FC624B93C7FC19FC04FCC71270030392C8FC5EA24B5A A2150F5E151F5EA24B5AA24BCBFCA215FEA24A5AA24A5AEA0180000F495AEA1FC0486C48 5AD87FF05B39FFFC1F80D87FFF90CCFC14FE6C5B6C13F06C5B00031380D800FCCDFC5047 7EC348>II76 D78 D<031FB512F00203B77E021F16 F091B812FC010317FF010F188090283FE07FC00F14C0D9FE00DA007F13E0D801F84A010F 13F0D803E016034848040013F8000F187F484801FF153F003FF01FFC007F180F90C7FC00 FE92C8FC48180712F01280C74817F85DA21AF0190F020317E05DF11FC01A80193F020717 004B157E61614E5A4A484A5A4E5AF01F80063EC7FC4A4814FCEF07F0EF7FE09239C07FFF 8091273FC1FFFEC8FC03C713F003CF138091267F9FFCC9FC16800380CAFC92CBFC5CA25C 1301A25C1303A25C13075CA2130F5C131FA25C133F5C91CCFC137E137C136046497EC345 >80 D<913807FFC0027F13FC0103B67E010F15E0903A3FFC007FF8D97FC0EB07FCD801FE C8B4FCD803F8ED3F80D807E0ED0FC04848ED07E04848ED03F090C91201003EEE00F8007E 17FC007C177C0078173C00F8173EA248171EB3B3A60060170C373D7BBA42>92 D102 D<12FEEAFFE0EA07F8EA00FEEB7F806D7E6D7E130F6D7EA26D7EB3AD6D7EA26D7E806E7E 6E7EEC0FE0EC03FC913800FFE0A2913803FC00EC0FE0EC3FC04A5A4AC7FC5C495AA2495A B3AD495AA2495A131F495A495A01FEC8FCEA07F8EAFFE048C9FC236479CA32>I<126012 F0B3B3B3B3B3A81260046474CA1C>106 D<0070130700F01480B3B3B3B3B3A800701400 196474CA32>I<126012F07EA21278127CA2123C123EA2121E121FA26C7EA212077FA212 037FA212017FA26C7EA21378137CA2133C133EA2131E131FA26D7EA2130780A2130380A2 130180A26D7EA21478147CA2143C143EA280A28081A2140781A2140381A26E7EA2140081 A21578157CA2153C153EA281A2811680A2150716C0A2150316E0A2ED01F0A2150016F8A2 1678167CA2163C163EA2161E160C27647BCA32>110 D<1B0C1B1E1B3EA21B7CA21BF8A2 F201F0A2F203E0A2F207C0A2F20F80A2F21F00A21A3EA262A262A24F5AA2621903A24F5A A24F5AA24FC7FCA2193EA261A261A24E5AA24E5AA24E5AA24E5AA2010C4CC8FC133C017C 163EEA01FE00035F487E001E5F00387FD8707F4B5A00E07FD8003F4B5A80011F4B5AA26E 4A5A130F6E4AC9FC13076E143E13036E5C13016E5C7F6F5B027F1301A26F485A143F6F48 5A141F6F485A140F6F48CAFC1407EDFC3E14035E15FE02015B15FF6E5BA26F5AA26F5AA2 6F5AA26FCBFC150E4F647A8353>112 D114 D 120 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fy cmr7 7 3 /Fy 3 51 df48 D<13381378EA01F8121F12FE12E01200B3AB487EB512F8A2 15267BA521>I<13FF000313E0380E03F0381800F848137C48137E00787F12FC6CEB1F80 A4127CC7FC15005C143E147E147C5C495A495A5C495A010EC7FC5B5B903870018013E0EA 0180390300030012065A001FB5FC5A485BB5FCA219267DA521>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: Fz cmcsc10 12 43 /Fz 43 123 df<1406140E141C143814F014E01301EB03C0EB0780EB0F005B131E133E5B 137813F85B1201A2485AA2485AA2120F5BA2121FA290C7FCA25AA3123E127EA65AB3A212 7EA6123E123FA37EA27FA2120FA27F1207A26C7EA26C7EA212007F1378137C7F131E131F 7FEB0780EB03C0EB01E0130014F01438141C140E1406176475CA2B>40 D<12C07E12707E121E120E120F6C7E6C7E6C7E7F12007F137C133C133E131E131FA2EB0F 80A2EB07C0A214E01303A214F0A21301A214F8A3130014FCA6147EB3A214FCA614F81301 A314F0A21303A214E0A2130714C0A2EB0F80A2EB1F00A2131E133E133C137C5B5B12015B 485A485A48C7FC120E121E12385A5A5A176479CA2B>I45 D<121FEA3F80EA7FC0EAFFE0A5EA7FC0EA3F80EA1F000B0B768A20>I<16E015011503A2 1507150FA2151F153FA2157F15EF15CF1401EC038F150F1407140E141CA214381470A214 E0EB01C0A2EB0380EB0700A2130E5B131813385B136013E0485A485AA248C7FC120EA25A 5AA25A5AB812F8A3C8381FE000AC4B7E4B7E027FB512F8A32D437CC237>52 D<121C121EEA1FC090B712F0A34816E0A217C017801700003CC8120F48151E5E00701538 16785E4B5A485D15034B5AC848C7FC150E151E5D5D157015F04A5AA24A5AA24A5A140FA2 4AC8FCA25C143E147EA35CA21301A31303A25CA21307A6130FAA6D5AEB01C02C4579C237 >55 D<1638167CA316FEA34B7EA24B7FA34B7F167FA2030E7F163FA24B6C7EA2033C7FED 380FA203787FED7007A203E07F1603A24A486C7EA20203814B7EA202078192C7127FA202 0E81173FA24A6E7EA2023C810238140FA2027FB67EA302E0C7EA07FE17030101824A80A2 0103834A80A249C97F187FA2010E707EA2011E83181F133E85137E48B483000701C0ED7F FFB500FC021FB512FEA347477CC651>65 DII69 DI<922603FF801306033F01F0130E4A B500FE131E02079038003F80DA1FF0903807C03EDA7FC0903801F07E4948C812F8D903FC ED3CFE4948151F4948150F4948150749481503495A49C91201481700485A49177E120748 48173EA2001F181E5BA2123F49170E127FA396C7FC5B12FFAC007F047FB512FC7FA2DD00 1F1300F007FE003F17037F121FA27F120FA26C7E12037F6C7E7E6D7E6D7E6D6C15076D7E 6D6C150FD903FE151E6D6CED3C7E9026007FC0ECF83EDA1FF8903803F01EDA07FF90381F C00E020190B5EA0006DA003F01FC90C7FC030313C0464879C555>III77 D82 D<49B46C13C0010FEBF001013FEBFC039038FF007FD801 F8EB0F874848EB03E7D807C0EB01FF48487F001F157F90C8123F003E151FA2007E150F12 7C160712FC1603A37E16017EA27F6C6C91C7FC7F7FEA3FFCEBFFC06C13FC6CEBFFC015FC 6CECFF806C15E0C615F86D80011F80010380D9003F1480020314C0EC003F030313E01500 EE7FF0161FA2EE0FF8160712E01603A21601A37EA217F07E16037E17E06C15076C16C06D EC0F806D141F6DEC3F00D8F8F8147E017F5C3AF01FE007F00107B55AD8E00191C7FC39C0 001FFC2D4879C53D>I<003FBAFCA3903BF8000FFE000701C06D48130090C7163F007EF0 1F80007C180FA200781807A300701803A548F001C0A5C893C7FCB3B3A44B7E92383FFF80 49B712F0A342437BC24E>I<157015F8A34A7EA24A7EA34A7E81A291380E3F80A2021E7F EC1C1FA24A6C7EA34A6C7EA202F07FECE003A249486C7EA349486C7EA201078091C77EA2 49B67EA24981011CC7121FA2013C810138140FA2496E7EA201F081491403120183486C14 0100074B7ED81FF84A7EB5027F13F8A335357CB43D>97 DI<4AB4EB0180021FEBF00391B5EAFC0701039038007E0FD907F8EB 0F9FD91FE0EB03DF4948EB01FF01FFC8FC4848157F4848153FA24848151F4848150F121F 491507123F5BA2007F1603A3484892C7FCAB6C7EEF0380A2123FA27F001F16076D160000 0F5E6C6C150E6C6C151E171C6C6C153C6C6C5DD93FC05C6D6CEB03E0D907F8495A902703 FF807FC7FC0100EBFFFC021F13F00201138031357BB33B>IIII< DA03FF1303021FEBE00791B5EAF80F0103903800FE1FD90FF8EB1F3FD91FE0EB07BFD97F 806DB4FC49C77E484880484881484881A2484881121F4981123F5BA2007F82A25B00FF93 C7FCAA4BB512F86C7EA2DB00011380003F6F1300837F121F7F120F6C7E7F12036C7E6C6C 5DEB7FC0D91FE05BD90FF8EB07DF903A03FF803F8F01009038FFFE07021FEBF803020301 80C7FC35357BB340>III107 DIII< EC07FF023F13E0903901FE03FC903907F0007FD90FC0EB1F80D93F80EB0FE049C76C7E01 FE6E7E48486E7E48486E7E4848157FA24848ED3F80001F17C0A24848ED1FE0A3007F17F0 49150FA300FF17F8AA007F17F06D151FA2003F17E0A26D153F001F17C0A26C6CED7F8000 0717006D5D00035E6C6C4A5A6C6C4A5A017F4A5A6D6C495AD90FC0EB1F80D907F0017FC7 FC903901FE03FC9039003FFFE0020790C8FC35357BB33F>IIII<90390FF0018090387FFE0348B51287 3907F00FEF390FC001FF48C7FC003E143F151F5A150F5A1507A36C1403A27E6C91C7FC6C 7E7FEA3FF8EBFF806C13FC6CEBFFC06C14F06C80C614FE011F7F01031480D9001F13C014 019138003FE0151F150FED07F0150312E01501A37EA216E06C1403A26CEC07C06CEC0F80 6C6CEB1F0001E0133ED8FBFE13FC00F0B55AD8E01F13E0D8C00390C7FC24357BB32E>I< 007FB812C0A3903A8007FC003F277E0003F8130F007C16070078160300701601A200F017 E0A2481600A6C71600B3AA4A7E4A7E010FB512FEA333327CB13B>IIII<267FFFF890 383FFFF0A3000101E0010F13006C49EB07F8017F5D013F15C06D6C5C6D6C49C7FC160E6D 6C131E6D6C5B6D6C133816786D6C5B91387F81E0023F5B15C391381FE7806EB4C8FC5D14 076E5A6E7EA26E7E4A7FA24A7F9138079FE091380F0FF0140E91381E07F84A6C7E4A6C7E 14709138F000FF49486D7E4948133F4A8001076E7E49C76C7E131E013E6E7E017E6E7E01 FE81000182D80FFF4A1380B500C0011F13FEA337337DB23D>II< 003FB7FCA39039FC0001FE01E01303018014FC90C7EA07F8003E140F003C15F0007CEC1F E00078EC3FC0A2ED7F800070ECFF00A24A5A4A5AC712075D4A5A141F5D4A5A4A5AA24AC7 FC495AA2495A495A130F4A1307495A133F5C495A49C7FC160F485A485AA24848141E485A 001F153E49147E484814FE007F140349131FB7FCA328337BB232>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: FA cmitt10 9 8 /FA 8 117 df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ndDVIPSBitmapFont %DVIPSBitmapFont: FB cmtt9 9 38 /FB 38 119 df<383FFF80487FA37EEA000FB3A4007FB512F0B6FC15F815F07E1D207B9F 27>16 D<1278B4FC13C013F813FEEA7FFF120F1203EA007F130E100A78AD27>18 D<123C127E12FFAF127EAE123C1200A7123C127E12FFA4127E123C082F71AE27>33 D<00101320007C13F838FE01FCAAEAFC00007C13F8A900381370161778AE27>I<120FEA 1FC0123F13E0A213F0121F120F1201A4120313E01207EA0FC0A2EA3F80EA7F005A5A12F8 12700C1773AD27>39 DI<127012F812FE7E6C7E6C7EEA 0FE06C7E12037F6C7E1200137EA27FA2EB1F80A3EB0FC0A4EB07E0ACEB0FC0A4EB1F80A3 EB3F00A2137EA25B1201485A5B1207485AEA3FC0485A48C7FC5A12F81270133A7AB327> I<120FEA3FC013E0EA7FF0A213F8A2123FA2120F120113F01203EA07E0121FEA7FC0EAFF 8013005A12700D14738927>44 D<007FB512F8B612FCA46C14F81E067C9927>I<121EEA 7F80A2EAFFC0A4EA7F80A2EA1E000A0A728927>I<121EEA7F80A2EAFFC0A4EA7F80A2EA 1E00C7FCAC121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A20729F27>58 D65 D<007FB5FCB612C08115F87E3907E003FCEC00FE157E157F81A615 7EA25D1403EC0FF890B55A15C015F081819038E000FE157FED3F80151FA2ED0FC0A6151F 1680153FED7F004A5A007FB55AB65A5D15E06C1480222E7FAD27>I<903803F80E90381F FE1F90383FFFBF90B6FC5A3803FE0F3807F803497E48487E485A49137FA248C7123FA25A 127E151E150012FE5AAA7E127EA2151E007F143F7EA26C7E157F6D137E6C6C13FE3907F0 01FCEBF8033903FE0FF86CB512F06C14E0013F13C06D1300EB03F820307DAE27>I<387F FFFC14FFB612C06C80813907E00FF81407EC01FC6E7EA2157E157F811680151FA316C015 0FABED1F80A3153F1600A25D15FEA24A5A4A5A140F007FB55A5DB65A6C91C7FC14FC222E 7FAD27>I<007FB61280B712C0A37E3907E0000FA6ED078092C7FCA4EC07804A7EA390B5 FCA5EBE00FA36E5A91C8FCAC387FFF80B57EA36C5B222E7EAD27>70 D<007FB512E0B612F0A36C14E039001F8000B3B2007FB512E0B612F0A36C14E01C2E7BAD 27>73 D<387FFFC080B5FC7E5CD803F0C8FCB3AAED0780ED0FC0A7007FB6FCA2B7FC7E16 80222E7FAD27>76 D<13E0EA01F01207120F13E0EA1FC0EA3F00A2127E127C12FC5AA4B4 FC138013C0127FA2123F1380EA0F000C1773B227>96 D<3803FFC0000F13F04813FC4813 FF811380EC1FC0381F000F000480C71207A2EB0FFF137F0003B5FC120F5A383FFC07EA7F C0130012FE5AA46C130F007F131FEBC0FF6CB612806C15C07E000313F1C69038807F8022 207C9F27>IIIIII104 D<130F497E497EA46D5A6DC7FC90C8FCA7383FFF80487FA37EEA00 0FB3A4007FB512F0B6FC15F815F07E1D2F7BAE27>I<387FFF80B57EA37EEA000FB3B200 7FB512F8B612FCA36C14F81E2E7CAD27>108 D<397F07C01F3AFF9FF07FC09039FFF9FF E091B57E7E3A0FFC7FF1F89038F03FC001E0138001C01300A3EB803EB03A7FF0FFC3FF48 6C01E3138001F913E701F813E36C4801C313002920819F27>I<387FE07F39FFF1FFC001 F713F090B5FC6C80000313C1EC01FCEBFE005B5BA25BB03A7FFF83FFE0B500C713F0A36C 018313E024207F9F27>II< 387FE0FFD8FFF313C090B512F0816C800003EB81FE49C67E49EB3F8049131F16C049130F A216E01507A6150F16C07F151F6DEB3F80157F6DEBFF009038FF83FEECFFFC5D5D01F313 C0D9F0FEC7FC91C8FCAC387FFF80B57EA36C5B23317F9F27>I<90380FF03C90383FFE7E 90B5FC000314FE5A380FFC1F381FE007EBC003383F800148C7FC127EA200FE147E5AA67E 007E14FEA2007F1301EA3F80EBC003381FE007380FF81F6CB5FC7E6C147E38007FFCEB0F F090C7FCAC91381FFFF8A24A13FC6E13F8A226317E9F27>I<397FFC03FC39FFFE0FFF02 3F13804A13C0007F90B5FC39007FFE1F14F89138F00F809138E002004AC7FC5CA291C8FC A2137EAD007FB57EB67EA36C5C22207E9F27>I<9038FFF3800007EBFFC0121F5A5AEB80 3F38FC000F5AA2EC07806C90C7FCEA7F8013FC383FFFF06C13FC000713FF00011480D800 0F13C09038003FE014070078EB03F000FC1301A27E14036CEB07E0EBE01F90B512C01580 150000FB13FC38707FF01C207B9F27>I<133C137EA8007FB512F0B612F8A36C14F0D800 7EC7FCAE1518157EA415FE6D13FC1483ECFFF86D13F06D13E0010313C0010013001F297E A827>I<397FE01FF8486C487EA3007F131F00031300B21401A21403EBFC0F6CB612E016 F07EEB3FFE90390FF87FE024207F9F27>I<3A7FFC0FFF80486C4813C0A36C486C13803A 07C000F800EBE00100035CA2EBF00300015CA2EBF80700005CA390387C0F80A36D48C7FC A3EB3F3FEB1F3EA214FE6D5AA36D5AA26D5A22207E9F27>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: FC cmr9 9 58 /FC 58 122 df<127812FCA27E7EEA7F80121FEA0FC0EA07E01203EA00F01378133C1308 0E0E78B326>18 D<137813FCA212011203EA07F813E0EA0FC0EA1F801300123C5A5A1240 0E0E71B326>I<123C127EB4FCA21380A2127F123D1201A412031300A25A1206120E120C 121C5A5A126009177A8715>44 DI<123C127E12FFA4127E123C 08087A8715>I48 D<13075B5B137FEA07FFB5FC13BFEAF83F1200B3B3A2497E007FB51280A319327AB126> IIII<000C14C0380FC00F90B5128015005C5C14F014C0D80C18C7FC90C8FCA9 EB0FC0EB7FF8EBF07C380FC03F9038001F80EC0FC0120E000CEB07E0A2C713F01403A215 F8A41218127E12FEA315F0140712F8006014E01270EC0FC06C131F003C14806CEB7F0038 0F80FE3807FFF8000113E038003F801D347CB126>I<14FE903807FF80011F13E090383F 00F0017C13703901F801F8EBF003EA03E01207EA0FC0EC01F04848C7FCA248C8FCA35A12 7EEB07F0EB1FFC38FE381F9038700F809038E007C039FFC003E0018013F0EC01F8130015 FC1400A24814FEA5127EA4127F6C14FCA26C1301018013F8000F14F0EBC0030007EB07E0 3903E00FC03901F81F806CB51200EB3FFCEB0FE01F347DB126>I<1230123C003FB6FCA3 4814FEA215FC0070C7123800601430157015E04814C01401EC0380C7EA07001406140E5C 141814385CA25CA2495A1303A3495AA2130FA3131F91C7FCA25BA55BA9131C20347CB126 >III<123C127E12FFA4127E123C1200B0123C127E12FFA4127E123C08207A9F15>I64 D<15E0A34A7EA24A7EA34A7EA3EC 0DFE140CA2EC187FA34A6C7EA202707FEC601FA202E07FECC00FA2D901807F1507A24948 6C7EA301066D7EA2010E80010FB5FCA249800118C77EA24981163FA2496E7EA3496E7EA2 0001821607487ED81FF04A7ED8FFFE49B512E0A333367DB53A>IIIIIIIII<017FB5FCA39038003FE0EC1FC0B3B1127EB4FC A4EC3F805A0060140000705B6C13FE6C485A380F03F03803FFC0C690C7FC20357DB227> I76 DIIII82 D<90381FE00390387FFC0748B5FC3907F01F CF390F8003FF48C7FC003E80814880A200788000F880A46C80A27E92C7FC127F13C0EA3F F013FF6C13F06C13FF6C14C06C14F0C680013F7F01037F9038003FFF140302001380157F 153FED1FC0150F12C0A21507A37EA26CEC0F80A26C15006C5C6C143E6C147E01C05B39F1 FC03F800E0B512E0011F138026C003FEC7FC22377CB42B>I<007FB712FEA390398007F0 01D87C00EC003E0078161E0070160EA20060160600E01607A3481603A6C71500B3AB4A7E 011FB512FCA330337DB237>III89 D<003FB612FCA39039F80007F813C090C7 EA0FF0003EEC1FE0123C0038EC3FC00078EC7F801270EDFF004A5AA20060495AA24A5A4A 5AC7FC4A5A4A5AA24A5A4AC7FCA2495A495AA2495A495AA24948130C495AA2495A49C7FC A24848141CA2485A485A1638485A4848147816F84848130148481307153FB7FCA326337C B22F>I97 DII<153FEC0FFFA3EC007F81AEEB07F0EB3FFCEBFC0F3901F003BF39 07E001FF48487E48487F8148C7FCA25A127E12FEAA127E127FA27E6C6C5BA26C6C5B6C6C 4813803A03F007BFFC3900F81E3FEB3FFCD90FE0130026357DB32B>II<151F90391FC07F809039FFF8E3C03901F0 7FC73907E03F033A0FC01F83809039800F8000001F80EB00074880A66C5CEB800F000F5C EBC01F6C6C48C7FCEBF07C380EFFF8380C1FC0001CC9FCA3121EA2121F380FFFFEECFFC0 6C14F06C14FC4880381F0001003EEB007F4880ED1F8048140FA56C141F007C15006C143E 6C5C390FC001F83903F007E0C6B51280D91FFCC7FC22337EA126>103 DII108 D<2703F01FE013FF00FF90267F F80313C0903BF1E07C0F03E0903BF3803E1C01F02807F7003F387FD803FE1470496D486C 7EA2495CA2495CB3486C496C487EB53BC7FFFE3FFFF0A33C217EA041>I<3903F01FC000 FFEB7FF09038F1E0FC9038F3807C3907F7007EEA03FE497FA25BA25BB3486CEB7F80B538 C7FFFCA326217EA02B>II< 3903F03F8000FFEBFFE09038F3C0F89038F7007ED807FE7F6C48EB1F804914C049130F16 E0ED07F0A3ED03F8A9150716F0A216E0150F16C06D131F6DEB3F80160001FF13FC9038F3 81F89038F1FFE0D9F07FC7FC91C8FCAA487EB512C0A325307EA02B>I<3803E07C38FFE1 FF9038E38F809038E71FC0EA07EEEA03ECA29038FC0F8049C7FCA35BB2487EB512E0A31A 217FA01E>114 DI<1330A513 70A313F0A21201A212031207381FFFFEB5FCA23803F000AF1403A814073801F806A23800 FC0EEB7E1CEB1FF8EB07E0182F7FAD1E>III<3A7FFF807FF8A33A07F8001FC00003EC0F800001EC07 0015066C6C5BA26D131C017E1318A26D5BA2EC8070011F1360ECC0E0010F5BA2903807E1 80A214F3010390C7FC14FBEB01FEA26D5AA31478A21430A25CA214E05CA2495A1278D8FC 03C8FCA21306130EEA701CEA7838EA1FF0EA0FC025307F9F29>121 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FD cmti12 12 56 /FD 56 123 df12 D14 D<167016E0ED01C0ED0380ED0700150E153C5D15F85D4A5A4A5A4A5A140F4AC7FC 141E143E5C147814F8495A5C1303495AA2495AA249C8FCA25B133E137E137CA25BA21201 5BA212035BA212075BA2120FA25BA2121FA290C9FCA25AA2123EA3127EA2127CA65AAB12 78A6127C123CA47EA2120E120FA27E6C7EA26C7EA26C7E1360246472CA28>40 D<1560A2157081A281151E150E150FA2811680A3ED03C0A516E0A21501A71503A91507A2 16C0A4150FA21680A2151FA21600A25DA2153EA2157EA2157C15FCA25D1401A25D14035D A214075D140F5DA24AC7FCA2143EA25C147814F8495AA2495A5C1307495A91C8FC131E13 3E5B13785B485A485A485A48C9FC121E5A5A12E05A23647FCA28>I<13F0EA03FC1207A2 EA0FFEA4EA07FCEA03CCEA000C131C1318A2133813301370136013E0EA01C013801203EA 0700120E5A5A5A5A5A0F1D7A891E>44 D<007FB5FCB6FCA214FEA21805789723>I<120F EA3FC0127FA212FFA31380EA7F00123C0A0A76891E>I 48 D<16C01501A215031507ED0F80151F153F157F913801FF005C140F147F903807FCFE EB0FF0EB0700EB00015DA314035DA314075DA3140F5DA3141F5DA3143F5DA3147F92C7FC A35C5CA313015CA313035CA313075CA2130FA2131F133FB612FCA25D224276C132>II II<026014 300278EB01F091397F801FE091B612C01780170016FC4914F016C0DACFFEC7FC02C0C8FC 13035CA3130791C9FCA35B130EA3131E90381C07F8EC3FFE9138F80F8090393DC007C0D9 3F807F90383E0003013C80496D7E1370A290C7FC82A41503A415075E120E123F486C130F 00FF5DA390C7485A5A00F84A5A12E04B5A93C7FC15FE14016C5C0070495A0078495A6CEB 1FC0003E495A261F80FEC8FC6CB45A000313E0C66CC9FC2C4476C132>II<9026380FC0131C9038787FE0902671FFF0133C01F3157801EF15F090B5 FC4801E0EB01E09139007003C04848EB380701F8EC1F804848EB1C3F4990381FFF004848 EB07DF49EB001E48C85A121E5E4815F800385D0078140100705D00F01403485D1507C848 5AA24BC7FCA25D153E157E157C15FC5D1401A24A5AA214075D140F5D141FA25D143FA24A C8FCA314FEA21301A25C1303A25C1307A35C130FA35C5C6D5A2E4472C132>III<130FEB1FC013 3FEB7FE013FFA214C0EB7F801400131E90C7FCB3A5120FEA3FC0127FA212FFA35B6CC7FC 123C132B76AA1E>I65 D<91B712FCF0FF8019E00201903980001FF06E90C7EA07F8 4A6F7E727E4B81841A800203167F5DA314075D19FFA2020F17004B5C611803021F5E4B4A 5A180F4E5A023F4B5A4BEC7F804EC7FCEF03FC027FEC0FF84BEBFFC092B6C8FC18E0913A FF800007F892C7EA01FC717E187F49834A6F7EA30103835CA313075CA3010F5F4A157FA2 4E5A131F4A4A90C7FC601703013F4B5A4A4A5A4D5A017F4B5A4D5A4A4948C8FC01FFEC0F FEB812F817C04CC9FC41447AC345>II<91B91280A30201902680000713006E90C8FC4A163FA24B81A3 0203160E5DA314074B151E191CA2140F5D17075F021F020E90C7FC5DA2171E023F141C4B 133CA2177C027F5CED800392B5FCA291B65AED00071601A2496E5A5CA2160101035D5CA2 160301075D4A90CAFCA3130F5CA3131F5CA3133F5CA2137FA313FFB612E0A341447AC340 >70 D<027FB512E091B6FCA20200EBE000ED7F8015FFA293C7FCA35C5DA314035DA31407 5DA3140F5DA3141F5DA3143F5DA3147F5DA314FF92C8FCA35B5CA313035CA313075CA313 0F5CA3131F5CA2133FA25CEBFFE0B612E0A25D2B447BC326>73 D<91B612F0A25F020101 C0C7FC6E5B4A90C8FCA25DA314035DA314075DA3140F5DA3141F5DA3143F5DA3147F5DA3 14FF92C9FCA35B5CA3010316104A1538A21878010716705C18F018E0010F15015C18C017 03011F15074A1580170FA2013FED1F004A5C5F017F15FE16034A130F01FFEC7FFCB8FCA2 5F35447AC33D>76 D<91B56C93387FFFC08298B5FC02014DEBC0006E614A5FA203DF4C6C C7FC1A0E63912603CFE05D038F5F1A381A711407030FEEE1FCA2F101C3020FEE0383020E 60F107036F6C1507021E160E021C60191CF1380F143C023804705BA2F1E01F0278ED01C0 91267003F85EF003801A3F02F0ED070002E0030E5CA24E137F130102C04B91C8FC606201 036D6C5B02805F4D5A943803800113070200DA07005BA2050E1303495D010E606F6C5A19 07011E5D011C4B5CA27048130F133C01384B5C017892C7FC191F01F85C486C027E5DD807 FE027C4A7EB500F00178013FB512C0A216705A447AC357>I<91B712F018FEF0FF800201 903980007FE06E90C7EA1FF04AED07F818034B15FCF001FE1403A24B15FFA21407A25DA2 140FF003FE5DA2021F16FC18074B15F8180F023F16F0F01FE04B15C0F03F80027FED7F00 18FE4BEB03FCEF0FF002FFEC7FC092B6C7FC17F892CAFC5BA25CA21303A25CA21307A25C A2130FA25CA2131FA25CA2133FA25CA2137FA25C497EB67EA340447AC342>80 D83 D<48B912F85AA2913B0007 FC001FF0D807F84A130701E0010F140349160148485C90C71500A2001E021F15E05E121C 123C0038143F4C1301007818C0127000F0147F485DA3C800FF91C7FC93C9FCA35C5DA314 035DA314075DA3140F5DA3141F5DA3143F5DA3147F5DA314FF92CAFCA35B5CA21303A213 07497E007FB612C0A25E3D446FC346>I<001FB500F090383FFFFCA326003FF0C7000113 806D48913800FE00013F167C18785C187018F0017F5E5CA2170101FF5E91C8FCA2170348 5F5BA21707000394C7FC5BA25F0007160E5BA2171E120F49151CA2173C121F491538A217 78123F491570A217F0127F495DA2160100FF5E90C8FCA216035F16074893C8FC5E160E16 1E5E007E1538007F15785E6C4A5A6D495A001F4A5A6D49C9FC6C6C133E6C6C13F83903FC 07F0C6B512C0013F90CAFCEB07F83E466DC348>I87 D97 DIIIII<15FCEC03FF91 390F83838091393E01CFC091387C00EF4A13FF4948137F010315804948133F495A131F4A 1400133F91C75A5B167E13FE16FE1201495CA215011203495CA21503A2495CA21507A25E A2150F151F5E0001143F157F6C6C13FF913801DF8090387C039F90383E0F3FEB0FFCD903 F090C7FC90C7FC5DA2157EA215FEA25DA2001C495A127F48495A14074A5A485C023FC8FC 00F8137E387C01F8381FFFE0000390C9FC2A407BAB2D>I<14FE137FA3EB01FC13001301 A25CA21303A25CA21307A25CA2130FA25CA2131FA25C157F90393F83FFC091388F81F091 381E00F802387F4948137C5C4A137EA2495A91C7FCA25B484814FE5E5BA2000314015E5B A2000714035E5B1507000F5DA249130F5E001F1678031F1370491480A2003F023F13F0EE 00E090C7FC160148023E13C01603007E1680EE070000FEEC1E0FED1F1E48EC0FF80038EC 03E02D467AC432>I<143C147E14FE1301A3EB00FC14701400AE137C48B4FC3803C78038 0703C0000F13E0120E121C13071238A21278EA700F14C0131F00F0138012E0EA003F1400 A25B137EA213FE5B12015BA212035B141E0007131C13E0A2000F133CEBC038A21478EB80 7014F014E0EB81C0EA0783EBC7803803FE00EA00F8174378C11E>I<16F0ED03F8A21507 A316F0ED01C092C7FCAEEC01F0EC07FCEC1E1EEC380F0270138014E0130114C0EB038001 07131F1400A2130E153F131E011C140090C7FC5DA2157EA215FEA25DA21401A25DA21403 A25DA21407A25DA2140FA25DA2141FA25DA2143FA292C7FCA25C147EA214FE001C5B127F 48485A495AA248485A495AD8F81FC8FCEA707EEA3FF8EA0FC0255683C11E>I<14FE137F A3EB01FC13001301A25CA21303A25CA21307A25CA2130FA25CA2131FA25C167E013F49B4 FC92380783C09138000E07ED3C1F491370ED603F017E13E0EC01C09026FE038013809139 07000E00D9FC0E90C7FC5C00015B5C495AEBF9C03803FB8001FFC9FCA214F03807F3FCEB F07F9038E01FC06E7E000F130781EBC003A2001F150FA20180140EA2003F151E161C0100 13E0A2485DA2007E1578167000FE01015B15F1489038007F800038021FC7FC2A467AC42D >IIIIII<91381F800C91387FE01C903901F0703C903907C038789039 0F801CF890381F001D013E130F017E14F05B48481307A2484814E012075B000F140F16C0 485AA2003F141F491480A3007F143F90C71300A35D00FE147EA315FE5DA2007E1301A24A 5A1407003E130FA26C495A143B380F80F33807C3E73901FF87E038007E071300140F5DA3 141F5DA3143F92C7FCA25CA25C017F13FEA25D263F76AB2D>III<1470EB01F8A313035CA313075CA3130F5C A3131F5CA2007FB512E0B6FC15C0D8003FC7FCA25B137EA313FE5BA312015BA312035BA3 12075BA3120F5BA2EC0780001F140013805C140E003F131EEB001C143C14385C6C13F049 5A6C485AEB8780D807FEC7FCEA01F81B3F78BD20>I<137C48B414072603C780EB1F8038 0703C0000F7F000E153F121C0107150012385E1278D8700F147E5C011F14FE00F05B00E0 5DEA003FEC0001A2495C137E150313FE495CA215071201495CA2030F13380003167849EC C070A3031F13F0EE80E0153F00011581037F13C06DEBEF8300000101148090397C03C787 903A3E0F07C70090391FFE01FE903903F000782D2D78AB34>I<017C143848B414FC3A03 C78001FE380703C0000F13E0120E001C14000107147E1238163E1278D8700F141E5C131F 00F049131C12E0EA003F91C7123C16385B137E167801FE14705BA216F0000115E05B1501 16C0A24848EB0380A2ED0700A2150E12015D6D5B000014786D5B90387C01E090383F0780 D90FFFC7FCEB03F8272D78AB2D>I<017CEE038048B4020EEB0FC02603C780013FEB1FE0 380703C0000E7F5E001C037E130F01071607123804FE130300785DEA700F4A1501011F13 0100F001804914C012E0EA003FDA000314034C14805B137E0307140701FE1700495CA203 0F5C0001170E495CA260A24848495A60A2601201033F5C7F4B6C485A000002F713036D90 39E7E0078090267E01C349C7FC903A1F0781F81E903A0FFF007FF8D901FCEB0FE03B2D78 AB41>I<02F8133FD907FEEBFFE0903A0F0F83C0F0903A1C07C780F890393803CF030170 13EE01E0EBFC07120101C013F8000316F00180EC01C000074AC7FC13001407485C120EC7 FC140F5DA3141F5DA3143F92C8FCA34AEB03C01780147EA202FEEB0700121E003F5D267F 81FC130E6E5BD8FF83143CD903BE5B26FE079E5B3A7C0F1F01E03A3C1E0F83C0271FF803 FFC7FC3907E000FC2D2D7CAB2D>I<137C48B414072603C780EB1F80380703C0000F7F00 0E153F001C1600130712385E0078157EEA700F5C011F14FE00F0495B12E0EA003FEC0001 5E5B137E150301FE5C5BA2150700015D5BA2150F00035D5BA2151F5EA2153F12014BC7FC 6D5B00005BEB7C0390383E0F7EEB1FFEEB03F090C712FE5DA214015D121F397F8003F0A2 4A5A4848485A5D48131F00F049C8FC0070137E007813F8383801F0381E07C06CB4C9FCEA 01FC294078AB2F>I<027C130749B4130F49EB800E010F141E49EBC03CEDE03890393F03 F07890397C00FDF00178EB3FE00170EB03C001F0148049130790C7EA0F00151E5D5D5D4A 5A4A5A4A5A4AC7FC141E5C5C5C495A495A495A49C8FC011E14F04914E05B491301485A48 48EB03C0D807B0130701FEEB0F80390FCF801F3A1F07E07F00393E03FFFED83C015B486C 5B00705C00F0EB7FC048011FC7FC282D7BAB28>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: FE cmr12 12 86 /FE 86 127 df<1618163CA2167EA216FFA24B7FA24B6C7EA29238063FE0A24B6C7EA24B 6C7EA292383807FC153092387003FE15609238E001FF15C002016D7F5D02036E7E92C7FC 4A6E7E1406020E6E7E140C021C6E7E141802386E7E143002706E7E146002E06E7E5C0101 6F7F5C0103707E91C9FC183F010683181F4983180F49831807498318034983A249707EA2 4848701380A248CBEA7FC0A20006F03FE0A248F01FF0A2001FBA12F8A24819FCA24819FE A2BCFC48477CC651>1 DI<1507A34B7EA34B7EA34B7EA34B 7E156FA2EDEFF815C7A202017F1587158302037F1503150102077F140681020E80140C16 7FA24A80163FA24A80161FA24A80160FA24A801607A24948801603A249C77F1601A20106 8182A24982177FA24982173FA24982171FA2017082136001E0150F6D821201486C82D80F FEED3FFEB500C00107B512F8A33D477DC644>I<003FB812FCA60038CA121C0030170C00 70170EA200601706A5CCFCAA01601506A5017FB612FEA60160C81206A590CBFCAB00C017 03A66C1707A2006017060070170E007FB812FEA638447CC341>I6 D<9239FFC001FC020F9038F80F FF913B3F803E3F03C0913BFC00077E07E0D903F890390FFC0FF0494890383FF81F4948EB 7FF0495A494814E049C7FCF00FE04991393FC0038049021F90C7FCAFB912F0A3C648C7D8 1FC0C7FCB3B2486CEC3FF0007FD9FC0FB512E0A33C467EC539>11 D<4AB4FC020F13E091387F80F8903901FC001C49487FD907E0130F4948137F011FECFF80 495A49C7FCA25B49EC7F00163E93C7FCACEE3F80B8FCA3C648C7FC167F163FB3B0486CEC 7FC0007FD9FC1FB5FCA330467EC536>I<913801FFC0020FEBFB8091387F803F903801FC 00494813FFEB07E0EB1FC0A2495A49C7FC167F49143F5BAFB8FCA3C648C7123FB3B2486C EC7FC0007FD9FC1FB5FCA330467EC536>II22 D<121EEA7F80EAFFC0A9EA7F80ACEA3F00AB121EAC120CA5C7FCAA12 1EEA7F80A2EAFFC0A4EA7F80A2EA1E000A4778C61B>33 D<001EEB03C0397F800FF000FF 131F01C013F8A201E013FCA3007F130F391E6003CC0000EB000CA401E0131C491318A300 0114384913300003147090C712604814E0000614C0000E130148EB038048EB070048130E 0060130C1E1D7DC431>I<121EEA7F8012FF13C0A213E0A3127FEA1E601200A413E013C0 A312011380120313005A1206120E5A5A5A12600B1D78C41B>39 D<140C141C1438147014 E0EB01C01303EB0780EB0F00A2131E5BA25B13F85B12015B1203A2485AA3485AA348C7FC A35AA2123EA2127EA4127CA312FCB3A2127CA3127EA4123EA2123FA27EA36C7EA36C7EA3 6C7EA212017F12007F13787FA27F7FA2EB0780EB03C01301EB00E014701438141C140C16 6476CA26>I<12C07E12707E7E7E120F6C7E6C7EA26C7E6C7EA21378137C133C133E131E 131FA2EB0F80A3EB07C0A3EB03E0A314F0A21301A214F8A41300A314FCB3A214F8A31301 A414F0A21303A214E0A3EB07C0A3EB0F80A3EB1F00A2131E133E133C137C13785BA2485A 485AA2485A48C7FC120E5A5A5A5A5A16647BCA26>I<16C04B7EB3AB007FBAFCBB1280A2 6C1900C8D801E0C9FCB3AB6F5A41407BB84C>43 D<121EEA7F8012FF13C0A213E0A3127F EA1E601200A413E013C0A312011380120313005A1206120E5A5A5A12600B1D78891B>I< B612C0A61A067F9721>I<121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A0A78891B>I<1618 163C167CA2167816F8A216F01501A216E01503A216C01507A21680150FA2ED1F00A2151E 153EA2153C157CA2157815F8A25D1401A24A5AA25D1407A25D140FA292C7FC5CA2141E14 3EA2143C147CA25CA25C1301A25C1303A25C1307A25C130FA291C8FC5BA2133EA2133C13 7CA2137813F8A25B1201A25B1203A2485AA25B120FA290C9FC5AA2121E123EA2123C127C A2127812F8A25A126026647BCA31>I<14FF010713E090381F81F890383E007C01FC133F 4848EB1F8049130F4848EB07C04848EB03E0A2000F15F0491301001F15F8A2003F15FCA3 90C8FC4815FEA54815FFB3A46C15FEA56D1301003F15FCA3001F15F8A26C6CEB03F0A36C 6CEB07E0000315C06D130F6C6CEB1F806C6CEB3F00013E137C90381F81F8903807FFE001 0090C7FC28447CC131>I<143014F013011303131F13FFB5FC13E713071200B3B3B0497E 497E007FB6FCA3204278C131>II<49 B4FC010F13E0013F13FC9038FE01FE3A01F0007F80D803C0EB3FC048C7EA1FE0120EED0F F0EA0FE0486C14F8A215077F5BA26C48130FEA03C0C813F0A3ED1FE0A2ED3FC01680ED7F 0015FE4A5AEC03F0EC1FC0D90FFFC7FC15F090380001FCEC007FED3F80ED1FC0ED0FE016 F0ED07F816FC150316FEA2150116FFA3121EEA7F80487EA416FE491303A2007EC713FC00 701407003015F80038140F6C15F06CEC1FE06C6CEB3FC0D803E0EB7F803A01FE01FE0039 007FFFF8010F13E0010190C7FC28447CC131>II<000615C0D807C0130701FC EB7F8090B612005D5D5D15E0158026063FFCC7FC90C9FCAE14FF010713C090381F01F090 383800FC01F0137ED807C07F49EB1F8016C090C7120F000615E0C8EA07F0A316F81503A2 16FCA5123E127F487EA416F890C712075A006015F0A20070140F003015E00038EC1FC07E 001EEC3F806CEC7F006C6C13FE6C6C485A3901F807F039007FFFE0011F90C7FCEB07F826 447BC131>II<121CA2EA1F8090B712C0A3481680A2 17005E0038C8120C0030151C00705D0060153016705E5E4814014B5A4BC7FCC81206150E 5D151815385D156015E04A5AA24A5A140792C8FC5CA25C141E143EA2147E147CA214FCA2 1301A3495AA41307A6130FAA6D5AEB01C02A457BC231>I<14FF010713E0011F13F89038 7F00FE01FC133FD801F0EB1F804848EB0FC049EB07E00007EC03F048481301A290C713F8 481400A47FA26D130116F07F6C6CEB03E013FC6C6CEB07C09039FF800F806C9038C01F00 6CEBF03EECF87839007FFEF090383FFFC07F01077F6D13F8497F90381E7FFFD97C1F1380 496C13C02601E00313E048486C13F000079038007FF84848EB3FFC48C7120F003EEC07FE 150148140016FF167F48153FA2161FA56C151E007C153EA2007E153C003E157C6C15F86D EB01F06C6CEB03E06C6CEB07C0D803F8EB1F80C6B4EBFF0090383FFFFC010F13F0010113 8028447CC131>I<14FF010713E0011F13F890387F80FC9038FC007E48487F4848EB1F80 4848EB0FC0000FEC07E0485AED03F0485A16F8007F140190C713FCA25AA216FE1500A516 FFA46C5CA36C7E5D121F7F000F5C6C6C130E150C6C6C131C6C6C5BD8007C5B90383F01E0 90390FFF80FE903801FE0090C8FC150116FCA4ED03F8A216F0D80F801307486C14E0486C 130F16C0ED1F80A249EB3F0049137E001EC75A001C495A000F495A3907E01FE06CB51280 C649C7FCEB1FF028447CC131>I<121EEA7F80A2EAFFC0A4EA7F80A2EA1E00C7FCB3A512 1EEA7F80A2EAFFC0A4EA7F80A2EA1E000A2B78AA1B>I<121EEA7F80A2EAFFC0A4EA7F80 A2EA1E00C7FCB3A5121E127FEAFF80A213C0A4127F121E1200A512011380A3120313005A 1206120E120C121C5A5A12600A3E78AA1B>I<007FBAFCBB1280A26C1900CEFCB0007FBA FCBB1280A26C190041187BA44C>61 D<16C04B7EA34B7EA34B7EA34B7EA3ED19FEA3ED30 FFA203707FED607FA203E07FEDC03FA2020180ED801FA2DA03007F160FA20206801607A2 4A6D7EA34A6D7EA34A6D7EA20270810260147FA202E08191B7FCA249820280C7121FA249 C87F170FA20106821707A2496F7EA3496F7EA3496F7EA201788313F8486C83D80FFF0303 7FB500E0027FEBFFC0A342477DC649>65 DI IIIIIII<010FB512FEA3D9000313806E130080B3B3AB123F 487E487EA44A5A13801300006C495A00705C6C13076C5C6C495A6CEB1F802603E07FC7FC 3800FFFCEB1FE027467BC332>I76 DIIII82 D<49B41303010FEBE007013F13F89039FE00FE0FD801F8131FD807E0EB079F49EB03DF48 486DB4FC48C8FC4881003E81127E82127C00FC81A282A37E82A27EA26C6C91C7FC7F7FEA 3FF813FE381FFFE06C13FE6CEBFFE06C14FC6C14FF6C15C0013F14F0010F80010180D900 1F7F14019138001FFF03031380816F13C0167F163F161F17E000C0150FA31607A37EA36C 16C0160F7E17806C151F6C16006C5D6D147ED8FBC05CD8F9F0495AD8F07C495A90393FC0 0FE0D8E00FB51280010149C7FC39C0003FF02B487BC536>I<003FB912F8A3903BF0001F F8001F01806D481303003EC7150048187C0078183CA20070181CA30060180CA5481806A5 C81600B3B3A54B7EED7FFE49B77EA33F447DC346>IIII91 D<01C01318000114384848137048C712E0000EEB01C0000C1480001C1303 0018140000385B003013060070130E0060130CA300E0131C481318A400CFEB19E039FFC0 1FF801E013FCA3007F130FA2003F130701C013F8390F0001E01E1D71C431>II<130C131E133F497EEBF3C03801E1E038 03C0F03807807848487E001E7F487F0070EB038048EB01C00040EB00801A0E75C331>I< 121EEA7F80A2EAFFC0A4EA7F80A2EA1E000A0A78C21B>I97 DII<167FED3F FFA315018182B3EC7F80903803FFF090380FC07C90383F000E017E1307496D5AD803F87F 48487F5B000F81485AA2485AA2127FA290C8FC5AAB7E7FA2123FA26C7EA2000F5D7F6C6C 5B00035C6C6C9038077F806C6C010E13C0013F011C13FE90380FC0F8903803FFE0902600 7F0013002F467DC436>IIIIII<143C14 FFA2491380A46D1300A2143C91C7FCADEC7F80EB3FFFA31300147F143FB3B3AA123E127F 39FF807F00A2147EA25C6C485A383C01F06C485A3807FF80D801FEC7FC195785C21E>I< EA01FC12FFA3120712031201B3A292381FFFE0A36F1300ED07F816E05E5E030EC7FC5D5D 5D5D4A5A4A5A4AC8FC5CEC3F804A7E14FF9038FDCFE09038FF8FF01407496C7E01FC7F14 016E7E81816F7E82151F6F7E821507826F7E8282486C491380B5D8F81F13F8A32D457DC4 33>III<3901FC01FE00FF 903807FFC091381E07F091383801F8000701707F0003EBE0002601FDC07F5C01FF147F91 C7FCA25BA35BB3A8486CECFF80B5D8F83F13FEA32F2C7DAB36>II<3901FC03FC00FF90380FFF8091383C07E091387001F83A07FDE0 00FE00030180137FD801FFEC3F8091C7EA1FC04915E049140F17F0160717F8160317FCA3 EE01FEABEE03FCA3EE07F8A217F0160F6D15E0EE1FC06D143F17806EEB7E00D9FDC05B90 39FCF003F891383C0FE091381FFF80DA03FCC7FC91C9FCAE487EB512F8A32F3F7DAB36> I<91387F8003903903FFE00790380FE07890393F801C0F90387E000E496D5AD803F8EB03 9F0007EC01BF4914FF48487F121F5B003F81A2485AA348C8FCAB6C7EA3123F7F121F6D5C 120F6D5B12076C6C5B6C6C497E6C6C130E013F131C90380FC0F8903803FFE09038007F00 91C7FCAEEEFF80033F13FEA32F3F7DAB33>I<3903F803F000FFEB1FFCEC3C3EEC707F00 07EBE0FF3803F9C000015B13FBEC007E153C01FF13005BA45BB3A748B4FCB512FEA3202C 7DAB26>I<90383FE0183901FFFC383907E01F78390F0003F8001E1301481300007C1478 127800F81438A21518A27EA27E6C6C13006C7E13FC383FFFE06C13FC6C13FF6C14C06C14 E0C614F0011F13F81300EC0FFC140300C0EB01FE1400157E7E153EA27EA36C143C6C147C 15786C14F86CEB01F039F38003E039F1F00F8039E07FFE0038C00FF01F2E7DAC26>I<13 06A5130EA4131EA3133E137EA213FE12011207001FB512F0B6FCA2C648C7FCB3A4150CAA 017E131C017F1318A26D133890381F8030ECC070903807E0E0903801FFC09038007F001E 3E7EBC26>IIIIII<003FB612 E0A29038C0003F90C713C0003CEC7F800038ECFF00A20030495A0070495AA24A5A006049 5AA24A5A4A5AA2C7485A4AC7FC5B5C495A13075C495A131F4A1360495A495AA249C712C0 485AA2485A485A1501485A48481303A24848EB07804848131F00FF14FF90B6FCA2232B7D AA2B>I<01F81302D803FE13073907FF800E48EBE01C391F1FF8F8393807FFF0D8700113 E039E0007FC00040EB1F00200978C131>126 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FF cmsy10 10 11 /FF 11 107 df<121C127FEAFF80A5EA7F00121C0909799917>1 D<0060150600F8150F6C151F007E153F6C157E6C6C14FC6C6CEB01F86C6CEB03F06C6CEB 07E06C6CEB0FC06C6CEB1F80017EEB3F006D137E6D6C5A90380FC1F8903807E3F0903803 F7E06DB45A6D5B6EC7FCA24A7E497F903803F7E0903807E3F090380FC1F890381F80FC90 383F007E017E7F49EB1F804848EB0FC04848EB07E04848EB03F04848EB01F84848EB00FC 48C8127E007E153F48151F48150F00601506282874A841>I20 D<126012F812FEEA7F80EA3FE0EA0FF8EA03FEC66C7EEB3FE0EB0FF8EB03FE903800FF80 EC3FE0EC0FF8EC03FE913800FF80ED3FE0ED0FF8ED03FE923800FF80EE3FE0EE0FF8EE03 FE933800FF80EF3FC0171FEF7F80933801FF00EE07FCEE1FF0EE7FC04B48C7FCED07FCED 1FF0ED7FC04A48C8FCEC07FCEC1FF0EC7FC04948C9FCEB07FCEB1FF0EB7FC04848CAFCEA 07FCEA1FF0EA7FC048CBFC12FC1270CCFCAE007FB81280B912C0A26C1780324479B441> I<181EA4181F84A285180785727EA2727E727E85197E85F11F80F10FC0F107F0007FBA12 FCBCFCA26C19FCCCEA07F0F10FC0F11F80F13F00197E61614E5A4E5AA24E5A61180F96C7 FCA260181EA4482C7BAA53>33 D<91381FFFFE91B6FC1303010F14FED91FF0C7FCEB7F80 01FEC8FCEA01F8485A485A485A5B48C9FCA2123EA25AA2127812F8A25AA2B712FE16FFA2 16FE00F0C9FCA27EA21278127CA27EA27EA26C7E7F6C7E6C7E6C7EEA00FEEB7F80EB1FF0 6DB512FE010314FF1300021F13FE283279AD37>50 D71 D76 D102 D<12FCEAFFC0EA07F0EA01FCEA007E7F80131F 80130FB3A7801307806D7E6D7EEB007EEC1FF0EC07F8EC1FF0EC7E00495A495A495A5C13 0F5CB3A7131F5C133F91C7FC137E485AEA07F0EAFFC000FCC8FC1D537ABD2A>I<126012 F0B3B3B3B3A91260045377BD17>106 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FG cmcsc10 10 12 /FG 12 117 df<121C127FEAFF80A5EA7F00121C090977881B>46 D<150EA3151FA24B7EA34B7EA3EDDFE0A202017F158FA29138030FF81507A202067F1503 020E7FEC0C01A2021C7FEC1800A24A80167FA24A6D7EA202E0804A131FA2494880160FA2 49B67EA249810106C71203A249811601A2498182A2496F7EA20170820160153F13E06D82 1203D80FFCED7FF8B56C010FB512E0A33B3C7CBB44>65 D67 D<1407A24A7EA34A7EA3EC37E0A2EC77 F01463A2ECC1F8A201017F1480A2903803007EA301067FA2010E80010C131FA2496D7EA2 013FB57EA29038300007496D7EA3496D7EA200018149130012036D801207D81FE0903801 FF80D8FFF8010F13F8A22D2C7DAB33>97 DI<91383FC006903901FFF80E90390FE03E1E9038 1F0007017EEB03BE01F8EB01FE484813004848147E0007153E485A001F151E5B003F150E 90C8FC5A1606A212FE1600AA007F1506A37E6D140E001F150C7F000F151C6C6C14180003 15386C6C14706C6C14E0017EEB01C0011FEB078090390FE03E00903801FFF89038003FC0 272D7BAB31>I101 D110 DI 114 D<017F13603901FFE0E0380780F9380E001F48130748130312780070130100F01300 A315607EA26C14007E127F13C0EA3FFEEBFFE06C13F8000713FE6C7FC61480010F13C013 00EC0FE01407EC03F01401A212C01400A37E15E06C1301A26CEB03C06CEB0780B4EB0F00 38F3E01E38E0FFF838C01FE01C2D7BAB26>I<007FB712C0A23A7E003FC00F007890381F 8003007015011600126000E016E0A2481660A5C71500B3A8EC7FE0011FB57EA22B2B7DAA 31>I E %EndDVIPSBitmapFont %DVIPSBitmapFont: FH cmr10 10 81 /FH 81 128 df 6 D11 DI<133C13 7EA213FE1201EA03FC13F0EA07E0EA0FC0EA1F80EA1E005A5A5A12C00F0F6FB92A>19 D22 D<001C131C007F137F39FF80FF80A26D13C0A3007F137F00 1C131C00001300A40001130101801380A20003130301001300485B00061306000E130E48 5B485B485B006013601A197DB92A>34 D<141FEC7FC0903801F0E0903803C06001071370 90380F803090381F00381518A25BA2133E133F15381530A215705D5D140190381F838092 CAFC1487148E02DC49B51280EB0FF85C4A9039003FF8000107ED0FC06E5D71C7FC6E140E 010F150CD91DFC141C01391518D970FE143801E015302601C07F1470D803805D00076D6C 5BD80F00EBC00148011F5C4890380FE003003E6E48C8FC007E903807F8060203130E00FE 6E5A6E6C5A1400ED7F706C4B13036F5A6F7E6C6C6D6C5B7013066C6C496C130E6DD979FE 5B281FF001F07F133C3C07F80FE03FC0F86CB539800FFFF0C69026FE000313C0D91FF0D9 007FC7FC393E7DBB41>38 D<121C127FEAFF80A213C0A3127F121C1200A412011380A212 0313005A1206120E5A5A5A12600A1979B917>I<146014E0EB01C0EB0380EB0700130E13 1E5B5BA25B485AA2485AA212075B120F90C7FCA25A121EA2123EA35AA65AB2127CA67EA3 121EA2121F7EA27F12077F1203A26C7EA26C7E1378A27F7F130E7FEB0380EB01C0EB00E0 1460135278BD20>I<12C07E12707E7E7E120F6C7E6C7EA26C7E6C7EA21378A2137C133C 133E131EA2131F7FA21480A3EB07C0A6EB03E0B2EB07C0A6EB0F80A31400A25B131EA213 3E133C137C1378A25BA2485A485AA2485A48C7FC120E5A5A5A5A5A13527CBD20>I<1530 1578B3A6007FB812F8B912FCA26C17F8C80078C8FCB3A6153036367BAF41>43 D<121C127FEAFF80A213C0A3127F121C1200A412011380A2120313005A1206120E5A5A5A 12600A19798817>II<121C127FEAFF80A5EA7F00121C09097988 17>I<150C151E153EA2153C157CA2157815F8A215F01401A215E01403A215C01407A215 80140FA215005CA2141E143EA2143C147CA2147814F8A25C1301A25C1303A2495AA25C13 0FA291C7FC5BA2131E133EA2133C137CA2137813F8A25B1201A25B1203A25B1207A25B12 0FA290C8FC5AA2121E123EA2123C127CA2127812F8A25A12601F537BBD2A>IIIII<1538A2157815F8A2140114 031407A2140F141F141B14331473146314C313011483EB030313071306130C131C131813 301370136013C01201EA038013005A120E120C5A123812305A12E0B712F8A3C73803F800 AB4A7E0103B512F8A325397EB82A>I<0006140CD80780133C9038F003F890B5FC5D5D15 8092C7FC14FC38067FE090C9FCABEB07F8EB3FFE9038780F803907E007E090388003F049 6C7E12066E7EC87EA28181A21680A4123E127F487EA490C71300485C12E000605C127000 30495A00385C6C1303001E495A6C6C485A3907E03F800001B5C7FC38007FFCEB1FE0213A 7CB72A>II<12301238123E003FB612E0A316C05A16 8016000070C712060060140E5D151800E01438485C5D5DC712014A5A92C7FC5C140E140C 141C5CA25CA214F0495AA21303A25C1307A2130FA3495AA3133FA5137FA96DC8FC131E23 3B7BB82A>III<121C127FEAFF80 A5EA7F00121CC7FCB2121C127FEAFF80A5EA7F00121C092479A317>I<007FB812F8B912 FCA26C17F8CCFCAE007FB812F8B912FCA26C17F836167B9F41>61 D<1538A3157CA315FEA34A7EA34A6C7EA202077FEC063FA2020E7FEC0C1FA2021C7FEC18 0FA202387FEC3007A202707FEC6003A202C07F1501A2D901807F81A249C77F167FA20106 810107B6FCA24981010CC7121FA2496E7EA3496E7EA3496E7EA213E0707E1201486C81D8 0FFC02071380B56C90B512FEA3373C7DBB3E>65 DI<913A01FF8001 80020FEBE003027F13F8903A01FF807E07903A03FC000F0FD90FF0EB039F4948EB01DFD9 3F80EB00FF49C8127F01FE153F12014848151F4848150FA248481507A2485A1703123F5B 007F1601A35B00FF93C7FCAD127F6DED0180A3123F7F001F160318006C7E5F6C7E17066C 6C150E6C6C5D00001618017F15386D6C5CD91FE05C6D6CEB03C0D903FCEB0F80902701FF 803FC7FC9039007FFFFC020F13F002011380313D7BBA3C>IIIIIII<013FB512E0A39039001FFC00EC07 F8B3B3A3123FEA7F80EAFFC0A44A5A1380D87F005B0070131F6C5C6C495A6C49C7FC3807 81FC3801FFF038007F80233B7DB82B>I76 DIIII82 D I<003FB812E0A3D9C003EB001F273E0001FE130348EE01F00078160000701770A3006017 30A400E01738481718A4C71600B3B0913807FF80011FB612E0A335397DB83C>IIII89 D91 D<3901800180000313033907000700000E130E485B0018 131800381338003013300070137000601360A200E013E0485BA400CE13CE39FF80FF806D 13C0A3007F137FA2393F803F80390E000E001A1974B92A>II<13101338137C13FE487E3803C780380783C0380F01E0381E 00F04813780070131C48130E00401304170D77B92A>I<121E123FEA7F80A2EAFFC0EA7F 80A2EA3F00121E0A097AB717>I97 DIIII<147E90 3803FF8090380FC1E0EB1F8790383F0FF0137EA213FCA23901F803C091C7FCADB512FCA3 D801F8C7FCB3AB487E387FFFF8A31C3B7FBA19>IIII107 DI<2703F00FF0 EB1FE000FFD93FFCEB7FF8913AF03F01E07E903BF1C01F83803F3D0FF3800FC7001F8026 03F70013CE01FE14DC49D907F8EB0FC0A2495CA3495CB3A3486C496CEB1FE0B500C1B500 83B5FCA340257EA445>I<3903F00FF000FFEB3FFCECF03F9039F1C01F803A0FF3800FC0 3803F70013FE496D7EA25BA35BB3A3486C497EB500C1B51280A329257EA42E>II<3903F01FE000FFEB7FF89038F1E07E9039F3801F803A0FF7000FC0D803FEEB 07E049EB03F04914F849130116FC150016FEA3167FAA16FEA3ED01FCA26DEB03F816F06D 13076DEB0FE001F614C09039F7803F009038F1E07E9038F0FFF8EC1FC091C8FCAB487EB5 12C0A328357EA42E>II<3807E01F00FFEB7FC09038E1E3E09038E387F0380FE707EA 03E613EE9038EC03E09038FC0080491300A45BB3A2487EB512F0A31C257EA421>II<1318A51338A3 1378A313F8120112031207001FB5FCB6FCA2D801F8C7FCB215C0A93800FC011580EB7C03 017E13006D5AEB0FFEEB01F81A347FB220>II IIII<003FB512FCA2EB8003D83E0013F8 003CEB07F00038EB0FE012300070EB1FC0EC3F800060137F150014FE495AA2C6485A495A A2495A495A495AA290387F000613FEA2485A485A0007140E5B4848130C4848131CA24848 133C48C7127C48EB03FC90B5FCA21F247EA325>II<001C131C00 7F137F39FF80FF80A5397F007F00001C131C190978B72A>127 D E %EndDVIPSBitmapFont %DVIPSBitmapFont: FI cmbx12 12 46 /FI 46 122 df45 DI48 DIII<163FA25E5E5D5DA25D5D5D5DA25D92B5FCEC01F7EC03E714 0715C7EC0F87EC1F07143E147E147C14F8EB01F0EB03E0130714C0EB0F80EB1F00133E5B A25B485A485A485A120F5B48C7FC123E5A12FCB91280A5C8000F90C7FCAC027FB61280A5 31417DC038>I<0007150301E0143F01FFEB07FF91B6FC5E5E5E5E5E16804BC7FC5D15E0 92C8FC01C0C9FCAAEC3FF001C1B5FC01C714C001DF14F09039FFE03FFC9138000FFE01FC 6D7E01F06D13804915C0497F6C4815E0C8FC6F13F0A317F8A4EA0F80EA3FE0487E12FF7F A317F05B5D6C4815E05B007EC74813C0123E003F4A1380D81FC0491300D80FF0495AD807 FEEBFFFC6CB612F0C65D013F1480010F01FCC7FC010113C02D427BC038>I<4AB47E021F 13F0027F13FC49B6FC01079038807F8090390FFC001FD93FF014C04948137F4948EBFFE0 48495A5A1400485A120FA248486D13C0EE7F80EE1E00003F92C7FCA25B127FA2EC07FC91 381FFF8000FF017F13E091B512F89039F9F01FFC9039FBC007FE9039FF8003FF17804A6C 13C05B6F13E0A24915F0A317F85BA4127FA5123FA217F07F121FA2000F4A13E0A26C6C15 C06D4913806C018014006C6D485A6C9038E01FFC6DB55A011F5C010714C0010191C7FC90 38003FF02D427BC038>I<121E121F13FC90B712FEA45A17FC17F817F017E017C0A24816 80007EC8EA3F00007C157E5E00785D15014B5A00F84A5A484A5A5E151FC848C7FC157E5D A24A5A14035D14074A5AA2141F5D143FA2147F5D14FFA25BA35B92C8FCA35BA55BAA6D5A 6D5A6D5A2F447AC238>III65 DIIIII72 D I<0107B7FCA590C7001F1300B3B3A9EA1FE0487E487EA2487EA44B5AA26C48495A495C6C 4813FF6C48485B260FFC0713C06CB65A6C4AC7FCC66C13F8010F138030457DC33A>I76 DII<923807FFC092B512FE0207ECFFC0021F15F091267FFE0013FC902601FFF0EB1FFF01 070180010313C04990C76C7FD91FFC6E6C7E49486F7E49486F7E01FF8348496F7E48496F 1380A248496F13C0A24890C96C13E0A24819F04982003F19F8A3007F19FC49177FA400FF 19FEAD007F19FC6D17FFA3003F19F8A26D5E6C19F0A26E5D6C19E0A26C6D4B13C06C1980 6E5D6C6D4B13006C6D4B5A6D6C4B5A6D6C4B5A6D6C4A5B6D01C001075B6D01F0011F5B01 0101FE90B5C7FC6D90B65A023F15F8020715C002004AC8FC030713C047467AC454>II<923807FFC092B512FE0207ECFFC0021F15F091267FFE00 13FC902601FFF0EB1FFF010701C0010713C04990C700017F49486E7F49486F7E49486F7E 49486F7E48496F7E48496F1380A248496F13C0A24819E091C97E4819F0A248487013F8A3 007F19FCA249177FA300FF19FEAD007F19FCA36D17FF003F19F8A3001F19F06D5EA26C19 E06E01FE5B6C912603FF8014C06C6D486D4813804B13E06C9028E01F83F00F13006C903B F01E00F81FFE90267FF83E90387C3FFC90263FFC3C6D485AD91FFE91381EFFF0D90FFF02 1F5B6D01FE5D010194C7FC6D6D6CB45A023F90B512F8020703E0130202006F1307030713 C792C7EA07F8716C130F72131F9538FF80FF96B5FC7114FEA3831AFCA27213F81AF08472 13E07213C0721300F001FC48587AC454>III<003FBA12E0A59026FE000FEB8003D87FE09338003FF049171F90C71607A2007E18 03007C1801A300781800A400F819F8481978A5C81700B3B3A20107B8FCA545437CC24E> II89 D<001FB812FEA59126F8000113FC028015F801FCC75A494A13F04916E0 495C494A13C0484816805E90C84813005F003E15FF4B5B5F003C5C4B5B5F5D4B5BC85C4B 90C7FC5D5E4B5A5C5E4A5B5C5E4A5B5C5E4A90C8FC5C5D4A48140F5B5D495B5B4949141F 5D49161E495B92C8FC49163E495A5C48177E485B4A15FE481601484914034A140748160F 4849143F91C8EAFFFC48150FB9FCA538447AC344>I<903801FFE0011F13FE017F6D7E48 B612E03A03FE007FF84848EB1FFC6D6D7E486C6D7EA26F7FA36F7F6C5A6C5AEA00F090C7 FCA40203B5FC91B6FC1307013F13F19038FFFC01000313E0000F1380381FFE00485A5B12 7F5B12FF5BA35DA26D5B6C6C5B4B13F0D83FFE013EEBFFC03A1FFF80FC7F0007EBFFF86C ECE01FC66CEB8007D90FFCC9FC322F7DAD36>97 D101 D104 D<137C48B4FC4813804813C0A24813E0 A56C13C0A26C13806C1300EA007C90C7FCAAEB7FC0EA7FFFA512037EB3AFB6FCA518467C C520>I108 D<90277F8007FEEC0FFCB590263FFFC090387FFF8092B5D8F001B512E002816E4880913D 87F01FFC0FE03FF8913D8FC00FFE1F801FFC0003D99F009026FF3E007F6C019E6D013C13 0F02BC5D02F86D496D7EA24A5D4A5DA34A5DB3A7B60081B60003B512FEA5572D7CAC5E> I<90397F8007FEB590383FFF8092B512E0028114F8913987F03FFC91388F801F00039039 9F000FFE6C139E14BC02F86D7E5CA25CA35CB3A7B60083B512FEA5372D7CAC3E>II<90397FC00FF8B590B57E02C314 E002CF14F89139DFC03FFC9139FF001FFE000301FCEB07FF6C496D13804A15C04A6D13E0 5C7013F0A2EF7FF8A4EF3FFCACEF7FF8A318F017FFA24C13E06E15C06E5B6E4913806E49 13006E495A9139DFC07FFC02CFB512F002C314C002C091C7FCED1FF092C9FCADB67EA536 407DAC3E>I<90387F807FB53881FFE0028313F0028F13F8ED8FFC91389F1FFE000313BE 6C13BC14F8A214F0ED0FFC9138E007F8ED01E092C7FCA35CB3A5B612E0A5272D7DAC2E> 114 D<90391FFC038090B51287000314FF120F381FF003383FC00049133F48C7121F127E 00FE140FA215077EA27F01E090C7FC13FE387FFFF014FF6C14C015F06C14FC6C80000380 6C15806C7E010F14C0EB003F020313E0140000F0143FA26C141F150FA27EA26C15C06C14 1FA26DEB3F8001E0EB7F009038F803FE90B55A00FC5CD8F03F13E026E007FEC7FC232F7C AD2C>II121 D E %EndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: A4 %%EndSetup %%Page: 1 1 1 0 bop 755 716 a FI(ENTR)m(OPY)46 b(PENALIZA)-9 b(TION)47 b(METHODS)h(F)m(OR)1044 832 y(HAMIL)-9 b(TON-JA)m(COBI)47 b(EQUA)-9 b(TIONS)1136 1081 y FH(DIOGO)28 b(A.)g(GOMES,)f(ENRICO)g(V)-9 b(ALDINOCI)754 1324 y FG(Abstra)n(ct.)42 b FH(In)31 b(this)h(pap)r(er)f (w)n(e)g(presen)n(t)f(a)h(new)h(en)n(trop)n(y)e(p)r(enalization)754 1434 y(problem)24 b(and)g(w)n(e)g(discuss)g(its)h(relations)e(with)h (appro)n(ximate)f(solutions)h(of)754 1543 y(Hamilton-Jacobi)j (equations,)g(the)h(con)n(v)n(ergence)d(of)j(asso)r(ciated)e(discrete) 754 1653 y(sc)n(hemes,)c(as)d(w)n(ell)h(as)g(sev)n(eral)e (applications,)j(suc)n(h)f(as:)33 b(a)19 b(generalization)g(of)754 1762 y(the)k(Hopf-Cole)f(transformation)e(whic)n(h)i(con)n(v)n(erts)f (non-linear)g(Hamilton-)754 1872 y(Jacobi)i(equations)g(in)n(to)h (linear)f(ev)n(olution)g(equations,)h(the)g(study)g(of)g(\014xed)754 1981 y(p)r(oin)n(t)40 b(problems,)h(appro)n(ximation)c(of)i(certain)g (linear)f(ev)n(olution)g(equa-)754 2091 y(tions,)25 b(and)g(the)g (construction)e(of)i(en)n(trop)n(y)e(p)r(enalized)i(Mather)f(measures.) 1749 2507 y FG(Contents)555 2674 y FH(1.)83 b(In)n(tro)r(duction)2245 b(2)555 2784 y(2.)83 b(Motiv)-5 b(ation)27 b(for)g(the)h(sc)n(heme)1749 b(5)555 2894 y(3.)83 b(Existence)26 b(of)h(a)f(minimizing)h(measure)f (and)g(equiv)-5 b(alency)26 b(b)r(et)n(w)n(een)h(\(4\))g(and)f(\(7\))92 b(7)555 3003 y(4.)83 b(Generalized)27 b(Hopf-Cole)g(transformation)1315 b(8)555 3113 y(5.)83 b(Some)27 b(prop)r(erties)g(of)g FF(G)1903 b FH(10)555 3222 y(6.)83 b(Some)27 b(prop)r(erties)g(of)g FF(L)h FH(and)1675 3201 y(^)1656 3222 y FF(L)1648 b FH(19)555 3332 y(7.)83 b(F)-7 b(ormal)27 b(Asymptotics)1922 b(21)555 3441 y(8.)83 b(Fixed)28 b(p)r(oin)n(t)f(problems)1881 b(29)555 3551 y(9.)83 b(Con)n(v)n(ergence)25 b(issues)1971 b(31)555 3661 y(10.)82 b(En)n(trop)n(y)26 b(p)r(enalized)i(Mather)f (measures)1305 b(38)555 3770 y(11.)82 b(Con)n(v)n(ergence)26 b(to)h(Mather)g(measures)1410 b(50)555 3880 y(App)r(endix)29 b(A.)83 b(A)28 b(Banac)n(h-Cacciopp)r(oli-t)n(yp)r(e)d(Theorem)910 b(58)555 3989 y(App)r(endix)29 b(B.)83 b(Some)27 b(elemen)n(tary)g (prop)r(erties)f(of)i(the)g(semiconca)n(v)n(e)d(functions)188 b(61)555 4099 y(References)2423 b(63)983 4493 y FE(Supp)s(orted)34 b(in)d(part)i(b)m(y)g(F)m(CT,)h(POCTI/FEDER,)1130 4620 y(POCI/FEDER/MA)-8 b(T/55745/2004)31 b(and)560 4748 y(MIUR)i(V)-8 b(ariational)29 b(Metho)s(ds)34 b(and)e(Nonlinear)f(Di\013eren)m(tial)g (Equations)456 4986 y FD(Subje)-5 b(ct)34 b(Classi\014c)-5 b(ation:)42 b FE(70H20,)32 b(37M25,)g(37J50)1931 5315 y FC(1)p eop %%Page: 2 2 2 1 bop 456 251 a FC(2)704 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)555 450 y FB(``A)40 b(quel)g(tempo,)h(di)f(numeri)g(ce)g (n'erano)h(soltanto)g(due:)79 b(il)40 b(numero)g FA(e)g FB(e)f(il)h(numero)456 551 y FA(pi)f(greco)p FB(.)80 b(Il)40 b(Decano)g(fa)g(un)g(calcolo)h(a)e(occhio)i(e)e(croce,)i(e)e (risponde:)81 b(-)39 b(Cresce)i(di)456 651 y FA(e)e FB(elevato)i(a)e FA(ti)p FB(.)79 b(Bravo)41 b(furbo!)80 b(Fin)40 b(l\022)-39 b(\020)39 b(ci)h(arrivano)h(tutti.")555 779 y(\(Italo)g(Calvino,)g (``Le)f(Cosmicomiche"\).)1549 1452 y FE(1.)49 b Fz(Intr)n(oduction)555 1638 y FE(The)35 b(ob)5 b(jectiv)m(e)36 b(of)d(this)h(pap)s(er)g(is)g (to)g(study)i(a)d(new)j(appro)m(ximation)c(pro)s(ce-)456 1766 y(dure)d(for)g(viscosit)m(y)g(solutions)f(of)h(the)g (Hamilton-Jacobi)c(equations)30 b(arising)d(in)456 1894 y(optimal)38 b(con)m(trol.)71 b(This)41 b(pro)s(cedure)i(consists)f(in) f(the)h(discretization)f(of)g(the)456 2022 y(con)m(trol)d(problem)f (and)i(in)f(the)h(addition)e(of)h(a)h(suitable)f(en)m(trop)m(y)i (functional.)456 2149 y(This)32 b(functional)e(regularizes)i(the)g(ev)m (olution)f(of)h(the)h(discrete)f(v)-5 b(alue)32 b(function)456 2277 y(b)m(y)47 b(acting)f(as)h(a)f(viscosit)m(y)h(term.)85 b(The)48 b(en)m(trop)m(y)g(p)s(enalization)c(metho)s(d)i(is)456 2405 y(motiv)-5 b(ated)31 b(b)m(y)i(considerations)g(in)f(statistical)e (mec)m(hanics,)j(as)g(w)m(ell)f(as)h(b)m(y)h(the)456 2533 y(w)m(orks)g(of)e([Ana04)o(],)h([FS86])f(and)h([Ev)-5 b(a04].)2063 2494 y Fy(1)555 2661 y FE(In)49 b(this)g(pap)s(er,)k(w)m (e)d(in)m(tro)s(duce)f(an)g(en)m(trop)m(y)h(p)s(enalized)f(sc)m(heme)h (related)456 2789 y(to)27 b(Lagrangian/Hamiltonian)22 b(dynamics)27 b(and)h(Hamilton-Jacobi-t)m(yp)s(e)c(equa-)456 2917 y(tions,)53 b(w)m(e)d(pro)m(v)m(e)g(its)f(con)m(v)m(ergence,)56 b(w)m(e)51 b(in)m(tro)s(duce)e(a)g(related)g(generalized)456 3044 y(Hopf-Cole)28 b(transformation,)g(and)i(w)m(e)h(in)m(v)m (estigate)e(the)h(connections)g(b)s(et)m(w)m(een)456 3172 y(\014xed)45 b(p)s(oin)m(ts)e(of)h(the)g(sc)m(heme,)k(the)d(v)-5 b(anishing)42 b(viscosit)m(y)j(metho)s(d,)h(and)e(the)456 3300 y(theory)33 b(of)f(Mather)h(measures.)555 3428 y(In)f(further)h (detail,)d(the)j(organization)c(of)j(this)f(pap)s(er)h(and)g(its)g (main)e(results)456 3556 y(are)i(the)h(follo)m(wing.)555 3684 y(In)g Fx(x)g FE(2,)f(w)m(e)i(in)m(tro)s(duce)e(and)h(motiv)-5 b(ate)31 b(the)i(en)m(trop)m(y)h(sc)m(heme)519 3923 y Fw(\036)44 b Fx(7\000)-16 b(!)44 b(G)6 b FE([)p Fw(\036)p FE(])83 b(=)g Fx(\000)p Fw(\017)17 b FE(ln)1476 3783 y Fv(\024)1529 3788 y(Z)1584 4013 y Fu(R)1632 3994 y Ft(N)1710 3923 y Fw(e)1755 3882 y Fs(\000)1820 3849 y Ft(hL)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p 1821 3867 495 3 v 2055 3908 a Ft(\017)2346 3923 y Fw(dv)2448 3783 y Fv(\025)1086 4194 y FE(=)83 b(inf)1379 4058 y Fv(Z)1435 4284 y Fu(R)1483 4265 y Ft(N)1561 4194 y FE([)p Fw(h)17 b(L)p FE(\()p Fw(x;)g(v)t FE(\))22 b(+)g Fw(\036)2131 4209 y Fq(n)2178 4194 y FE(\()p Fw(x)h FE(+)f Fw(hv)t FE(\))f(+)h Fw(\017)17 b FE(ln)f Fw(\015)5 b FE(\()p Fw(v)t FE(\)])17 b Fw(\015)5 b FE(\()p Fw(v)t FE(\))17 b Fw(dv)i(;)p 456 4355 499 4 v 555 4416 a Fy(1)588 4448 y FH(It)i(ma)n(y)e(b)r(e)i(useful)f(to)g (giv)n(e)f(a)h(short)f(sk)n(etc)n(h)g(ab)r(out)h(the)h(motiv)-5 b(ations)19 b(of)h(en)n(trop)n(y)f(metho)r(ds)456 4558 y(in)j(mathematical)g(ph)n(ysics.)34 b(Rougly)21 b(sp)r(eaking,)i(in)f (genetral,)g(an)g(en)n(trop)n(y)f(p)r(enalized)h(meto)r(d)456 4668 y(consists)33 b(in)h(p)r(erturbing)f(a)g(giv)n(en)g(problem)h(b)n (y)f(adding)h(an)f(\\en)n(trop)n(y)f(term")h(of)h(the)g(t)n(yp)r(e)456 4777 y Fp(S)5 b FH(\()p Fp(f)k FH(\))23 b(=)736 4710 y Fo(R)805 4777 y Fp(f)g FH(log)14 b Fp(f)9 b FH(.)35 b(In)25 b(statistical)f(mec)n(hanics,)h(suc)n(h)f(en)n(trop)n(y)g Fp(S)29 b FH(\\coun)n(ts",)24 b(in)h(a)f(logaritmic)456 4887 y(scale,)42 b(the)e(n)n(um)n(b)r(er)f(of)h(microscopic)e (con\014gurations)g(of)i(a)f(ph)n(ysical)g(system)h(whic)n(h)f(are)456 4996 y(compatible)f(with)h(a)f(giv)n(en)g(macroscopic)e(b)r(eha)n (vior,)k(while)f Fp(f)47 b FH(has)38 b(the)h(meaning)f(of)g(the)456 5106 y(frequency)22 b(of)g(an)n(y)g(state.)35 b(W)-7 b(e)23 b(refer)f(to)g([Vil03])g(for)g(a)g(more)g(detailed)h (statistical)f(motiv)-5 b(ation)456 5216 y(of)27 b(the)h(en)n(trop)n(y) e(functional.)p eop %%Page: 3 3 3 2 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)458 b(3)456 450 y FE(where)48 b(the)f(in\014m)m(um)e(ab)s(o)m(v)m(e)j(is)e(tak)m(en)i(o)m (v)m(er)g(all)c(the)j(probabilit)m(y)e(densities)456 578 y Fw(\015)52 b FE(on)47 b Fn(R)775 542 y Fq(N)848 578 y FE(.)87 b(Here,)51 b Fw(h)c FE(pla)m(ys)h(the)f(r^)-49 b(ole)47 b(of)f(a)h(time)f(step)i(discretization)d(and)456 706 y Fw(\017)g FE(is)f(the)h(en)m(trop)m(y)h(p)s(enalization)c (parameter.)79 b(W)-8 b(e)45 b(notice)f(from)f(the)j(ab)s(o)m(v)m(e)456 834 y(form)m(ulas)32 b(that)i(there)h(are)g FD(two)h(e)-5 b(quivalent)34 b FE(w)m(a)m(ys)i(of)d(de\014ning)h(suc)m(h)i(a)e(sc)m (heme)456 962 y(\(see)h(form)m(ulas)e(\(4\))h(and)g(\(7\))g(here)h(b)s (elo)m(w\).)48 b(The)35 b(equiv)-5 b(alence)35 b(b)s(et)m(w)m(een)h (these)456 1089 y(t)m(w)m(o)d(de\014nitions)f(is)g(sho)m(wn)i(in)e Fx(x)g FE(3.)555 1217 y(Then,)i(a)e FD(line)-5 b(ar)32 b FE(sc)m(heme)1095 1482 y Fw( )48 b Fx(7\000)-16 b(!)44 b(L)p FE([)p Fw( )t FE(])28 b(=)1732 1346 y Fv(Z)1788 1572 y Fu(R)1836 1553 y Ft(N)1914 1482 y Fw(e)1959 1441 y Fs(\000)2024 1407 y Ft(h)11 b(L)p Fr(\()p Ft(x;v)r Fr(\))p 2024 1426 224 3 v 2123 1467 a Ft(\017)2262 1482 y Fw( )t FE(\()p Fw(x)22 b FE(+)g Fw(hv)t FE(\))17 b Fw(dv)456 1756 y FE(is)23 b(in)m(tro)s(duced)h(in)f Fx(x)i FE(4,)g(whic)m(h)g(turns)f(out)g(to)g(b)s(e)g(equiv)-5 b(alen)m(t)23 b(to)h(the)g(\(non-linear\))456 1884 y(en)m(trop)m(y)31 b(p)s(enalized)e(metho)s(d,)i(via)e(a)h(generalized)f(Hopf-Cole)g (transformation)456 2012 y(of)j(the)h(t)m(yp)s(e)1670 2196 y Fw(\036)44 b Fx(7\000)-16 b(!)44 b Fw(e)2022 2155 y Fs(\000)p Fq(\036=\017)2203 2196 y Fw(:)456 2403 y FE(W)-8 b(e)28 b(dev)m(ote)i Fx(x)e FE(5)g(to)g(the)g(study)i(of)d (some)h(prop)s(erties)h(of)e(the)i(non-linear)d(sc)m(heme)456 2531 y Fx(G)6 b FE(.)49 b(In)34 b(particular,)g(the)g(sc)m(heme)i(is)e (pro)m(v)m(ed)i(to)e(enjo)m(y)h(a)f FD(c)-5 b(ontr)g(action)36 b(pr)-5 b(op)g(erty)456 2658 y FE(in)44 b(the)h(space)h(of)e(functions) h(\\where)g(constan)m(ts)i(are)d(quotien)m(ted)i(out")e(\(see)456 2786 y(Theorem)36 b(9)h(for)f(details\).)54 b(Though)37 b(this)f(feature)h(is)f(w)m(eak)m(er)j(than)d(the)h(stan-)456 2914 y(dard)h(strict)h(con)m(traction)f(prop)s(ert)m(y)-8 b(,)41 b(it)c(will)g(b)s(e)h(still)f(su\016cien)m(t)j(to)e(solv)m(e)h (the)456 3042 y(related)d(\014xed)j(p)s(oin)m(t)d(problem)g(b)m(y)i (iteration.)55 b(W)-8 b(e)38 b(also)e(study)j(the)e(semicon-)456 3170 y(ca)m(vit)m(y)27 b(prop)s(erties)h(of)e(the)i(sc)m(heme.)43 b(In)28 b(particular,)e(a)h FD(uniform)j(semic)-5 b(onc)g(avity)456 3298 y(estimate)40 b FE(is)g(obtained)g(in)f(Theorem)i(13,)h(from)d (whic)m(h)i(a)f FD(uniform)i(Lipschitz)456 3426 y(b)-5 b(ound)39 b FE(follo)m(ws.)64 b(More)40 b(precisely)-8 b(,)42 b(w)m(e)e(sho)m(w)h(that)f(the)g(Lipsc)m(hitz)g(and)g(semi-)456 3553 y(conca)m(vit)m(y)h(mo)s(duli)c(of)j Fx(G)6 b FE([)p Fw(\036)p FE(])40 b(are)g(b)s(ounded)h(b)m(y)g(the)f(maxim)m(um)e(b)s (et)m(w)m(een)k(the)456 3681 y(semiconca)m(vit)m(y)32 b(mo)s(dulus)g(of)g Fw(\036)g FE(and)h(a)f(univ)m(ersal)h(constan)m(t.) 555 3809 y(Some)f(elemen)m(tary)h(prop)s(erties)f(of)h(linear)e(sc)m (heme)i Fx(L)g FE(are)f(outlined)g(in)f Fx(x)i FE(6.)555 3937 y(W)-8 b(e)23 b(carry)f(out)g(some)f FD(asymptotics)h FE(in)f Fx(x)h FE(7,)i(according)d(to)h(di\013eren)m(t)g(c)m(hoices)h (of)456 4065 y(the)g(parameters)g(in)m(v)m(olv)m(ed.)40 b(Namely)-8 b(,)24 b(the)f(linear)f(sc)m(heme)i(is)e(related)g(to)h(a)f (para-)456 4193 y(b)s(olic)e(equation)h(\(see)j(Prop)s(ositions)c(23)i (and)g(24\),)h(while)f(the)g(non-linear)e(sc)m(heme)456 4321 y(asymptotic)44 b(is)h(related)g(to)g(the)h(Hamilton-Jacobi)41 b(equation,)49 b(as)c(sho)m(wn)i(in)456 4448 y(Prop)s(osition)30 b(25.)555 4576 y(In)23 b Fx(x)f FE(8,)i(the)f Fw(n)p FE(th)g(iteration)d(of)i(the)g(non-linear)f(sc)m(heme)i(is)f(pro)m(v)m (ed)i(to)d FD(c)-5 b(onver)g(ge)456 4704 y(uniformly)p FE(.)70 b(This)42 b(result)f(is)h(obtained)f(b)m(y)h(a)g(\014xed)g(p)s (oin)m(t)f(pro)s(cedure)i(whic)m(h)456 4832 y(mak)m(es)28 b(use)h(of)e(the)h(non-standard)g(con)m(traction)f(prop)s(ert)m(y)h(of) g(the)g(sc)m(heme.)43 b(W)-8 b(e)456 4960 y(sho)m(w)38 b(that)f Fx(G)983 4924 y Fq(n)1030 4960 y FE([)p Fw(\036)p FE(])h(con)m(v)m(erges,)j(as)c Fw(n)f Fx(!)g FE(+)p Fx(1)p FE(,)i(to)f(a)g(\014xed)i(p)s(oin)m(t)d(of)h Fx(G)43 b FE(\\when)456 5088 y(constan)m(ts)35 b(are)f(quotien)m(ted)h(out")e (\(w)m(e)i(refer)f(to)g(Theorem)g(27)g(for)f(further)h(de-)456 5216 y(tails\).)p eop %%Page: 4 4 4 3 bop 456 251 a FC(4)704 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)555 450 y FE(The)24 b(con)m(v)m(ergence)i(of)d(the)g(sc)m (heme)i(for)d FD(smal)5 b(l)26 b(time)g(steps)c FE(is)h(discussed)i(in) d Fx(x)i FE(9,)456 578 y(b)m(y)32 b(follo)m(wing)d([Sou85])i(and)h ([BS91].)43 b(W)-8 b(e)32 b(pro)m(v)m(e)h(that,)e(when)i(the)f(time)e (step)i Fw(h)456 706 y FE(go)s(es)i(to)g(zero,)h(the)g(solutions)e(of)h (the)g(suitably)g(scaled)g(linear)f(and)h(non-linear)456 834 y(sc)m(hemes)42 b(con)m(v)m(erge)h(uniformly)c(to)i(solutions)f(of) g(the)i(heat)f(equation)g(and)g(of)456 962 y(the)c(viscous)h (Hamilton-Jacobi)c(equation,)k(resp)s(ectiv)m(ely)-8 b(.)58 b(The)38 b(elliptic)d(term)456 1089 y(in)23 b(these)j(equations) f(is)f(pro)m(vided)h(b)m(y)g(the)g(en)m(trop)m(y)h(\(see)f(Theorems)g (29)f(and)h(30\).)555 1217 y(In)i Fx(x)f FE(10,)h(w)m(e)g(in)m(v)m (estigate)f(the)g(connection)h(b)s(et)m(w)m(een)h(the)e(\014xed)h(p)s (oin)m(ts)f(of)f(our)456 1345 y(sc)m(heme)h(and)f(suitably)g FD(p)-5 b(enalize)g(d)27 b(Mather)h(me)-5 b(asur)g(es)p FE(.)40 b(In)26 b(particular,)f(the)h(gen-)456 1473 y(eralized)34 b(Hopf-Cole)g(transformation)f(pro)m(vides)i(a)g(corresp)s(ondence)i(b) s(et)m(w)m(een)456 1601 y(\014xed)30 b(p)s(oin)m(ts)f(of)f(the)i(sc)m (heme)g(and)g(minimizing)24 b(measures)30 b(for)f(an)g(en)m(trop)m(y)i (p)s(e-)456 1729 y(nalized)f(action)h(\(see)i(Theorem)f(32\),)f(whic)m (h,)i(in)e(turn,)h(con)m(v)m(erge)h(to)f(the)g(usual)456 1857 y(Mather)23 b(measures)h(when)h(the)e(p)s(enalization)e(v)-5 b(anishes)23 b(\(see)i(Theorem)e(40\).)40 b(W)-8 b(e)456 1984 y(will)37 b(also)h(pro)m(v)m(e)i FD(uniqueness)f FE(of)g(the)g(en)m(trop)m(y)i(p)s(enalized)e(Mather)h(measures)456 2112 y(\(see)30 b(Theorem)h(36\).)42 b(W)-8 b(e)30 b(also)f(remark)g (that)h(an)f(\\explicit")f(represen)m(tation)j(of)456 2240 y(en)m(trop)m(y)e(p)s(enalized)e(Mather)h(measures)g(in)f(terms)h (of)f(\014xed)i(p)s(oin)m(ts)e(of)h(the)g(non-)456 2368 y(linear)39 b(sc)m(heme)j(holds)e(\(namely)-8 b(,)42 b(form)m(ula)d(\(105\)\).)66 b(In)41 b(particular,)g(w)m(e)h(sho)m(w) 456 2496 y(that,)32 b(if)f Fw(\036)i FE(is)f(the)h(unique)g(\(up)g(to)f (constan)m(ts\))i(solution)d(of)1581 2704 y Fx(G)6 b FE([)p Fw(\036)p FE(])22 b Fx(\000)h Fw(\036)k FE(=)h Fw(\025)f Fx(2)h Fn(R)456 2912 y FE(and)k Fw(\022)k FE(is)c(the)h (unique)g(probabilit)m(y)e(densit)m(y)i(solving)907 3020 y Fv(Z)963 3245 y Fu(R)1011 3226 y Ft(N)1089 3155 y Fw(\022)s FE(\()p Fw(x)23 b Fx(\000)f Fw(hv)t FE(\))p Fw(e)1542 3114 y Fs(\000)1607 3081 y Ft(hL)p Fr(\()p Ft(x)p Fm(\000)p Ft(hv)r(;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(x)p Fr(\))p Fm(\000)p Ft(\036)p Fr(\()p Ft(x)p Fm(\000)p Ft(hv)r Fr(\))p Fm(\000)p Ft(\025)p 1607 3099 866 3 v 2027 3140 a(\017)2504 3155 y Fw(dv)47 b FE(=)d Fw(\022)s FE(\()p Fw(x)p FE(\))17 b Fw(;)456 3403 y FE(then)1200 3566 y Fw(\026)p FE(\()p Fw(x;)g(v)t FE(\))44 b(=)g Fw(\022)s FE(\()p Fw(x)p FE(\))p Fw(e)1873 3525 y Fs(\000)1938 3492 y Ft(hL)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p Fm(\000)p Ft(\036)p Fr(\()p Ft(x)p Fr(\))p Fm(\000)p Ft(\025)p 1939 3510 748 3 v 2299 3551 a(\017)456 3752 y FE(is)32 b(the)h(unique)g(minimizer)c(for)j (the)h(en)m(trop)m(y)h(p)s(enalized)e(action.)555 3880 y(W)-8 b(e)38 b(dev)m(ote)g Fx(x)g FE(11)e(to)h(the)h(analysis)e(of)h (the)h(con)m(v)m(ergence)i(of)c(the)i(p)s(enalized)456 4008 y(problem)i(to)h(the)h(classical)e(problem)h(of)g(Mather.)71 b(In)41 b(particular,)i(b)s(oth)e(the)456 4136 y(p)s(enalized)30 b(measure)h(and)g(the)h(p)s(enalized)e(\\e\013ectiv)m(e)i(Hamiltonian") 27 b(con)m(v)m(erge)456 4264 y(to)46 b(the)h(ones)h(of)e(Mather's)i (problem)e(\(see)i(Theorems)f(38)g(and)g(40)f(for)g(pre-)456 4391 y(cise)31 b(statemen)m(ts\).)44 b(In)32 b(particular,)f(b)s(oth)g (the)h(minimizing)27 b(measures)32 b(and)g(the)456 4519 y(minimal)e(v)-5 b(alues)34 b(of)f(the)i(en)m(trop)m(y)g(p)s(enalized)f (action)f(con)m(v)m(erge)j(to)e(the)g(analo-)456 4647 y(gous)k(ob)5 b(jects)40 b(in)e(the)h(non-p)s(enalized)f(setting.)61 b(More)39 b(precisely)-8 b(,)40 b(if)e Fw(L)h FE(is)f(the)456 4775 y(Lagrangian,)i(the)h(en)m(trop)m(y)h(p)s(enalized)d(action)h(of)g (a)g(\\holonomic")d(\(or)j(\\\015o)m(w)456 4903 y(in)m(v)-5 b(arian)m(t",)31 b(see)i(\(88\)\))f(measure)h Fw(\026)f FE(is)g(giv)m(en)h(b)m(y)1236 5016 y Fv(Z)1291 5242 y Fu(T)1341 5223 y Ft(N)1393 5242 y Fs(\002)p Fu(R)1496 5223 y Ft(N)1575 5152 y Fw(L)p FE(\()p Fw(x;)17 b(v)t FE(\))p Fw(d\026)p FE(\()p Fw(x;)g(v)t FE(\))k(+)2340 5085 y Fw(\017)p 2332 5129 57 4 v 2332 5220 a(h)2398 5152 y(S)6 b FE([)p Fw(\026)p FE(])p Fw(;)33 b(:)p eop %%Page: 5 5 5 4 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)458 b(5)456 450 y FE(Then,)35 b(w)m(e)f(construct)h(the)f(densities)f(whic)m(h)h (minimize)d(suc)m(h)k(p)s(enalized)d(func-)456 578 y(tional)39 b(and)j(w)m(e)g(pro)m(v)m(e)h(their)e(con)m(v)m(ergence)k(to)c(Mather)h (measures)h(for)e(small)456 706 y(en)m(trop)m(y)-8 b(.)555 834 y(The)43 b(pap)s(er)f(ends)i(with)d(t)m(w)m(o)i(app)s(endices.)73 b(The)43 b(\014rst)f(one)h(is)e(dev)m(oted)j(to)456 962 y(the)30 b(\\abstract")g(\014xed)h(p)s(oin)m(t)e(argumen)m(t)h(whic)m (h)g(is)g(used)h(in)e(the)h(pro)s(of)f(of)h(The-)456 1089 y(orem)44 b(27.)82 b(The)46 b(second)h(app)s(endix)e(collects)g (some)g(elemen)m(tary)h(results)f(on)456 1217 y(semiconca)m(v)m(e)33 b(functions.)555 1345 y(The)j(pap)s(er)f(is,)g(essen)m(tially)-8 b(,)35 b(self-con)m(tained,)f(except)j(for)d(some)g(basic)h(facts)456 1473 y(ab)s(out)21 b(Mather)h(theory)-8 b(,)25 b(for)c(whic)m(h)h(w)m (e)h(refer)f(to)f([Mat91],)j([MF94])e(and)f([Gom05)o(],)456 1601 y(and)50 b(for)f(some)h(general)g(fact)f(ab)s(out)h(viscosit)m(y)g (solutions,)k(whic)m(h)c(ma)m(y)g(b)s(e)456 1729 y(found,)f(for)c (instance,)50 b(in)45 b([CIL92].)83 b(In)46 b(general,)j(w)m(e)e(dev)m (oted)g(some)e(e\013ort)456 1857 y(to)40 b(using)g(elemen)m(tary)g (argumen)m(ts)g(in)g(the)h(pro)s(ofs,)g(rather)g(than)f(other)h(ones) 456 1984 y(in)m(v)m(olving)31 b(\014ner)i(tec)m(hnologies.)1195 2281 y(2.)49 b Fz(Motiv)-9 b(a)i(tion)38 b(f)n(or)h(the)e(scheme)555 2467 y FE(Let)f Fw(H)8 b FE(\()p Fw(p;)17 b(x)p FE(\))35 b(b)s(e)h(a)f(Hamiltonian,)e(whic)m(h)k(has)f Fw(L)p FE(\()p Fw(x;)17 b(v)t FE(\))35 b(as)h(conjugated)g(La-)456 2595 y(grangian)31 b(\(the)i(precise)g(assumptions)f(on)h(whic)m(h)g (will)d(b)s(e)j(listed)f(in)f Fx(x)i FE(3\):)456 2801 y(\(1\))720 b Fw(L)p FE(\()p Fw(x;)17 b(v)t FE(\))28 b(=)g(sup)1780 2880 y Fq(p)1888 2801 y Fx(\000)p Fw(p)23 b Fx(\001)f Fw(v)j Fx(\000)e Fw(H)8 b FE(\()p Fw(p;)17 b(x)p FE(\))p Fw(:)456 3065 y FE(Let)32 b Fw(\036)p FE(\()p Fw(x;)17 b(t)p FE(\))33 b(b)s(e)g(a)f(viscosit)m(y)h(solution)e(of)h (the)h(Hamilton-Jacobi)c(equation)1502 3271 y Fx(\000)p Fw(\036)1637 3286 y Fq(t)1689 3271 y FE(+)22 b Fw(H)8 b FE(\()p Fx(r)p Fw(\036;)17 b(x)p FE(\))27 b(=)g(0)p Fw(;)456 3477 y FE(with)40 b(prescrib)s(ed)h(terminal)d(condition)h Fw(\036)p FE(\()p Fw(x;)17 b(T)d FE(\).)67 b(The)42 b(solution)d (admits)g(the)456 3605 y(follo)m(wing)29 b(represen)m(tation)34 b(form)m(ula:)456 3866 y(\(2\))384 b Fw(\036)p FE(\()p Fw(x;)17 b(t)p FE(\))28 b(=)f(inf)1499 3731 y Fv(Z)1599 3757 y Fq(T)1554 3956 y(t)1670 3866 y Fw(L)p FE(\()p Fw(q)t FE(\()p Fw(\034)11 b FE(\))p Fw(;)36 b FE(_)-46 b Fw(q)t FE(\()p Fw(\034)11 b FE(\)\))17 b Fw(d\034)34 b FE(+)22 b Fw(\036)p FE(\()p Fw(q)t FE(\()p Fw(T)14 b FE(\))p Fw(;)j(T)d FE(\))j Fw(;)456 4112 y FE(for)42 b(an)m(y)j Fw(T)59 b(>)46 b(t)p FE(,)h(where)d(the)g(ab)s(o)m(v)m(e)g (in\014m)m(um)e(is)h(tak)m(en)i(o)m(v)m(er)f(all)d(Lipsc)m(hitz)456 4240 y(curv)m(es)i Fw(q)j FE(:)d([)p Fw(t;)17 b(T)d FE(])43 b Fx(!)f Fn(R)1377 4204 y Fq(N)1491 4240 y FE(satisfying)f Fw(q)t FE(\()p Fw(t)p FE(\))h(=)h Fw(x)e FE(\(see,)k(for)c(instance,) 3116 4201 y Fl(2)3195 4240 y FE(Theo-)456 4368 y(rem)32 b(6.4.5)g(in)g([CS04]\).)555 4496 y(A)d(discrete)h(sc)m(heme)g(with)e (time)f(step)j Fw(h)e(>)f FE(0)i(to)f(appro)m(ximate)g Fw(\036)h FE(consists)g(in)456 4623 y(the)k(iteration)1082 4830 y Fw(\036)1140 4845 y Fq(n)p Fl(+1)1277 4830 y FE(\()p Fw(x)p FE(\))28 b(:=)63 b(inf)1566 4899 y Fq(v)r Fs(2)p Fu(R)1697 4880 y Ft(N)1772 4830 y FE([)p Fw(h)17 b(L)p FE(\()p Fw(x;)g(v)t FE(\))22 b(+)g Fw(\036)2342 4845 y Fq(n)2389 4830 y FE(\()p Fw(x)h FE(+)f Fw(hv)t FE(\)])16 b Fw(:)p 456 5007 499 4 v 555 5074 a Fl(2)591 5106 y FH(Bew)n(are)36 b(of)i(a)g(sign)g(c)n(hange)f(b)r(oth)i(in)f(the)h (time)f(direction)g(and)g(in)h(\(1\))f(b)r(et)n(w)n(een)g(our)456 5216 y(notation)27 b(and)g(the)h(one)f(in)h([CS04)o(].)p eop %%Page: 6 6 6 5 bop 456 251 a FC(6)704 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)456 450 y FE(By)33 b(elemen)m(tary)f(considerations,)h(the)g (previous)g(form)m(ula)d(can)j(b)s(e)g(written)f(as)456 700 y(\(3\))295 b Fw(\036)934 715 y Fq(n)p Fl(+1)1071 700 y FE(\()p Fw(x)p FE(\))44 b(=)58 b(inf)1366 762 y Fq(\026)p Fs(2P)1527 564 y Fv(Z)1583 790 y Fu(R)1631 771 y Ft(N)1709 700 y FE([)p Fw(h)17 b(L)p FE(\()p Fw(x;)g(v)t FE(\))22 b(+)g Fw(\036)2279 715 y Fq(n)2326 700 y FE(\()p Fw(x)g FE(+)g Fw(hv)t FE(\))17 b(])g Fw(d\026)p FE(\()p Fw(v)t FE(\))g Fw(;)456 966 y FE(where)46 b Fx(P)54 b FE(denotes)46 b(the)f(space)i(of)d(the)i(probabilit)m(y)d(measures)j (on)f Fn(R)3152 929 y Fq(N)3225 966 y FE(.)81 b(T)-8 b(o)456 1093 y(mak)m(e)32 b(the)g(sc)m(heme)h(smo)s(other,)f(w)m(e)h (replace)f(\(3\))f(b)m(y)i(the)f(\\en)m(trop)m(y)h(p)s(enalized")456 1221 y(sc)m(heme)456 1451 y(\(4\))116 b Fw(\036)755 1466 y Fq(n)p Fl(+1)892 1451 y FE(\()p Fw(x)p FE(\))28 b(:=)41 b(inf)1182 1513 y Fq(\015)t Fs(2D)1342 1316 y Fv(Z)1398 1541 y Fu(R)1446 1522 y Ft(N)1524 1451 y FE([)p Fw(h)17 b(L)p FE(\()p Fw(x;)g(v)t FE(\))22 b(+)g Fw(\036)2094 1466 y Fq(n)2141 1451 y FE(\()p Fw(x)g FE(+)g Fw(hv)t FE(\))g(+)g Fw(\017)17 b FE(ln)f Fw(\015)5 b FE(\()p Fw(v)t FE(\)])17 b Fw(\015)5 b FE(\()p Fw(v)t FE(\))17 b Fw(dv)t(;)456 1711 y FE(where)39 b Fw(\017)e(>)f FE(0)h(\(here)i(and) f(in)f(the)h(rest)g(of)g(the)g(pap)s(er\))g(is)f(a)g(small)f(parameter) 456 1838 y(and)c(w)m(e)i(denoted)f(b)m(y)h Fx(D)h FE(the)e(set)g(of)f (probabilit)m(y)f(densities)i(on)f Fn(R)2933 1802 y Fq(N)3006 1838 y FE(,)h(i.e.)456 2097 y(\(5\))205 b Fx(D)30 b FE(:=)1024 1956 y Fv(\032)1099 2097 y Fw(\015)i Fx(2)d Fw(L)1343 2056 y Fl(1)1382 2097 y FE(\()p Fn(R)1486 2056 y Fq(N)1559 2097 y FE(\))17 b Fx(j)27 b Fw(\015)5 b FE(\()p Fw(v)t FE(\))28 b Fx(\025)g FE(0)44 b Fw(a:e:)17 b(;)2316 1961 y Fv(Z)2372 2187 y Fu(R)2420 2168 y Ft(N)2498 2097 y Fw(\015)5 b FE(\()p Fw(v)t FE(\))17 b Fw(dv)31 b FE(=)c(1)2979 1956 y Fv(\033)3087 2097 y Fw(:)555 2367 y FE(F)-8 b(or)32 b(a)g(giv)m(en)h(\(measurable\))f(function)g Fw(\036)27 b FE(:)h Fn(R)2240 2331 y Fq(N)2341 2367 y Fx(!)f Fn(R)5 b FE(,)39 b(w)m(e)33 b(de\014ne)h Fx(G)6 b FE([)p Fw(\036)p FE(])33 b(as)456 2631 y(\(6\))486 b Fx(G)6 b FE([)p Fw(\036)p FE(]\()p Fw(x)p FE(\))28 b(:=)g Fx(\000)p Fw(\017)17 b FE(ln)1765 2491 y Fv(\024)1818 2496 y(Z)1873 2721 y Fu(R)1921 2702 y Ft(N)2000 2631 y Fw(e)2045 2590 y Fs(\000)2110 2557 y Ft(hL)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p 2110 2575 495 3 v 2344 2616 a Ft(\017)2635 2631 y Fw(dv)2737 2491 y Fv(\025)2806 2631 y Fw(:)456 2895 y FE(Note)38 b(that)h Fx(G)45 b FE(is)38 b(w)m(ell)g(de\014ned,)j(for)d(instance,)j(if)c Fw(\036)i FE(is)f(b)s(ounded)h(and)g Fx(G)6 b FE([)p Fw(\036)p FE(])39 b(is)456 3023 y(also)31 b(b)s(ounded.)555 3151 y(In)38 b(Theorem)f(2)g(b)s(elo)m(w,)h(w)m(e)h(sho)m(w)f(that)f (\(4\))g(is)f(equiv)-5 b(alen)m(t)37 b(to)g(the)h(explicit)456 3279 y(iteration)30 b(sc)m(heme)456 3493 y(\(7\))1059 b Fw(\036)1698 3508 y Fq(n)p Fl(+1)1862 3493 y FE(:=)28 b Fx(G)6 b FE([)p Fw(\036)2143 3508 y Fq(n)2190 3493 y FE(])17 b Fw(:)456 3707 y FE(Then,)46 b(in)c(Theorem)h(3,)i(w)m(e)f (will)c(establish)i(the)h(equiv)-5 b(alence)43 b(with)g(a)f FD(line)-5 b(ar)456 3835 y FE(sc)m(heme,)45 b(whic)m(h)d(generalizes)f (the)h(Hopf-Cole)e(transformation.)68 b(In)42 b(relation)456 3963 y(with)48 b(that,)k(w)m(e)e(recall)d(that)i(if)e Fw(u)h FE(is)g(a)h(solution)e(to)h(the)h(Hamilton-Jacobi)456 4090 y(equation)1553 4306 y Fw(u)1609 4321 y Fq(t)1660 4306 y FE(+)1768 4239 y Fx(j)p Fw(D)s(u)p Fx(j)1964 4202 y Fl(2)p 1768 4283 235 4 v 1861 4374 a FE(2)2040 4306 y(=)28 b Fw(\017)p FE(\001)p Fw(u;)456 4532 y FE(then)23 b(the)h(Hopf-Cole)e(transform)g Fw(v)32 b FE(=)27 b Fw(e)1943 4495 y Fs(\000)2008 4468 y Ft(u)p 2009 4480 37 3 v 2014 4521 a(\017)2082 4532 y FE(is)c(a)g(solution)f(to)h(the)g(heat)h (equation)1738 4746 y Fw(v)1785 4761 y Fq(t)1843 4746 y FE(=)j Fw(\017)p FE(\001)p Fw(v)21 b(;)456 4960 y FE(th)m(us)33 b(it)e(is)h(conceiv)-5 b(able)32 b(that)g(similar)d(exp)s(onen)m(tial)i (transformations)g(happ)s(en)456 5088 y(to)f(b)s(e)h(useful)g(in)g(our) f(framew)m(ork.)43 b(W)-8 b(e)32 b(no)m(w)f(pro)m(vide)h(a)e(formal)f (setting)h(of)h(the)456 5216 y(ab)s(o)m(v)m(e)i(discussions.)p eop %%Page: 7 7 7 6 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)458 b(7)578 450 y FE(3.)49 b Fz(Existence)36 b(of)j(a)e(minimizing)i(measure)e(and)h (equiv)-9 b(alency)1508 578 y(between)36 b(\(4\))j(and)f(\(7\))555 764 y FE(F)-8 b(rom)34 b(no)m(w)i(on,)h(w)m(e)f(assume)g(that)g(the)g (Lagrangian)d(is)i(a)h(suitably)e(smo)s(oth)456 892 y(\(e.g.,)e(Lipsc)m (hitz\))h(function)f(that)g(has)h(the)g(form)864 1110 y Fw(L)p FE(\()p Fw(x;)17 b(v)t FE(\))28 b(=)g Fw(K)7 b FE(\()p Fw(v)t FE(\))22 b Fx(\000)g Fw(U)10 b FE(\()p Fw(x)p FE(\))17 b Fw(;)213 b FE(for)97 b Fw(v)31 b Fx(2)d Fn(R)2542 1069 y Fq(N)2615 1110 y Fw(;)34 b(x)28 b Fx(2)g Fn(R)2919 1069 y Fq(N)3009 1110 y Fw(;)456 1328 y FE(in)g(whic)m(h)j Fw(K)7 b FE(,)30 b(the)g(\\kinetic)f(enery",)i(is)e(strictly)g(con)m(v) m(ex)j(in)d Fw(v)k FE(and)d(sup)s(erlinear)456 1456 y(at)46 b(in\014nit)m(y)-8 b(,)49 b(and)e Fw(U)57 b FE(is)46 b(the)h(\\p)s(oten)m(tial)d(energy")j(whic)m(h)g(is)f(b)s(ounded,)51 b Fn(Z)3347 1420 y Fq(N)3412 1456 y FE(-)456 1584 y(p)s(erio)s(dic)31 b(and)h(semicon)m(v)m(ex,)i(that)f(is,)f(there)h(exists)g Fw(C)2481 1599 y Fq(U)2568 1584 y Fw(>)27 b FE(0)33 b(suc)m(h)h(that) 456 1844 y(\(8\))465 b(sup)1008 1928 y Ft(x;y)r Fm(2)p Fk(R)1179 1914 y Ft(N)1064 1985 y(y)r Fm(6)p Fr(=0)1267 1776 y Fw(U)10 b FE(\()p Fw(x)23 b FE(+)f Fw(y)t FE(\))f(+)h Fw(U)10 b FE(\()p Fw(x)23 b Fx(\000)g Fw(y)t FE(\))e Fx(\000)i FE(2)p Fw(U)10 b FE(\()p Fw(x)p FE(\))p 1267 1821 1259 4 v 1823 1912 a Fx(j)p Fw(y)t Fx(j)1931 1883 y Fl(2)2563 1844 y Fx(\025)28 b(\000)p Fw(C)2815 1859 y Fq(U)2875 1844 y Fw(:)456 2172 y FE(W)-8 b(e)38 b(supp)s(ose)i (further)e(that)g Fw(K)45 b FE(is)38 b(semiconca)m(v)m(e,)i(i.e.,)g (that)e(there)g(exists)h Fw(C)3376 2187 y Fq(K)456 2300 y FE(suc)m(h)34 b(that)456 2540 y(\(9\))464 b(sup)1002 2625 y Ft(v)r(;w)q Fm(2)p Fk(R)1184 2611 y Ft(N)1057 2681 y(w)q Fm(6)p Fr(=0)1272 2473 y Fw(K)7 b FE(\()p Fw(v)26 b FE(+)c Fw(w)s FE(\))f(+)h Fw(K)7 b FE(\()p Fw(v)26 b Fx(\000)d Fw(w)s FE(\))e Fx(\000)i FE(2)p Fw(K)7 b FE(\()p Fw(v)t FE(\))p 1272 2518 1328 4 v 1852 2609 a Fx(j)p Fw(w)s Fx(j)1981 2580 y Fl(2)2637 2540 y Fx(\024)28 b Fw(C)2812 2555 y Fq(K)2880 2540 y Fw(:)456 2881 y FI(Prop)s(osition)d (1.)35 b FD(L)-5 b(et)27 b Fx(D)i FD(as)e(in)f(\(5\).)42 b(Fix)26 b Fw(x)i Fx(2)g Fn(R)2277 2845 y Fq(N)2350 2881 y FD(.)42 b(Then,)28 b(ther)-5 b(e)26 b(exist)h Fw(\015)3204 2896 y Fq(?)3271 2881 y Fx(2)h(D)456 3009 y FD(r)-5 b(e)g(alizing)456 3262 y FE(\(10\))284 b(inf)901 3324 y Fq(\015)t Fs(2D)1062 3127 y Fv(Z)1178 3152 y(\020)1237 3262 y Fw(h)17 b(L)p FE(\()p Fw(x;)g(v)t FE(\))22 b(+)g Fw(\036)p FE(\()p Fw(x)h FE(+)f Fw(hv)t FE(\))g(+)g Fw(\017)17 b FE(ln)e Fw(\015)5 b FE(\()p Fw(v)t FE(\))2595 3152 y Fv(\021)2655 3262 y Fw(\015)g FE(\()p Fw(v)t FE(\))17 b Fw(dv)i(:)456 3516 y FD(Mor)-5 b(e)35 b(explicitly,)f Fw(\015)1198 3531 y Fq(?)1272 3516 y FD(is)g(given)h(by)1620 3756 y Fw(\015)1671 3771 y Fq(?)1738 3756 y FE(:=)28 b Fw(e)1914 3715 y Fs(\000)1986 3668 y Fr(^)1979 3681 y Ft(L)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft(\025)p 1979 3699 260 3 v 2096 3741 a(\017)2253 3756 y Fw(;)456 3974 y FD(with)1280 4122 y FE(^)1269 4147 y Fw(L)p FE(\()p Fw(x;)17 b(v)t FE(\))28 b(:=)f Fw(hL)p FE(\()p Fw(x;)17 b(v)t FE(\))23 b(+)f Fw(\036)p FE(\()p Fw(x)g FE(+)g Fw(hv)t FE(\))p Fw(;)456 4343 y FD(and)1484 4532 y Fw(\025)28 b FE(:=)g Fw(\017)17 b FE(ln)1853 4397 y Fv(Z)1970 4532 y Fw(e)2015 4491 y Fs(\000)2087 4445 y Fr(^)2080 4458 y Ft(L)p Fr(\()p Ft(x;v)r Fr(\))p 2080 4476 177 3 v 2155 4517 a Ft(\017)2271 4532 y Fw(dv)j(:)456 4769 y FD(A)n(lso,)34 b(the)h(quantity)g(in)g(\(10\))f (is)h(e)-5 b(qual)34 b(to)h Fx(\000)p Fw(\025)p FD(.)456 4997 y(Pr)-5 b(o)g(of.)41 b FE(By)33 b(de\014nition,)e Fw(\015)1418 5012 y Fq(?)1490 4997 y FE(is)h(a)g(probabilit)m(y)f (densit)m(y)j(satisfying)456 5216 y(\(11\))478 b Fw(hL)p FE(\()p Fw(x;)17 b(v)t FE(\))22 b(+)g Fw(\036)p FE(\()p Fw(x)g FE(+)g Fw(hv)t FE(\))g(+)g Fw(\017)17 b FE(ln)f Fw(\015)2317 5231 y Fq(?)2356 5216 y FE(\()p Fw(v)t FE(\))28 b(=)f Fx(\000)p Fw(\025)17 b(:)p eop %%Page: 8 8 8 7 bop 456 251 a FC(8)704 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)456 450 y FE(W)h(e)34 b(claim)e(that)i Fw(\015)1151 465 y Fq(?)1224 450 y FE(is)g(optimal.)45 b(Indeed,)37 b(b)m(y)e(the)f(con)m(v)m(exit)m(y)i(of)e(the)h(function)456 578 y Fw(t)28 b Fx(7!)f Fw(t)17 b FE(ln)f Fw(t)p FE(,)32 b(w)m(e)i(ha)m(v)m(e,)g(for)e(an)m(y)h(other)g(probabilit)m(y)e(densit) m(y)i Fw(\015)5 b FE(,)33 b(that)697 671 y Fv(Z)753 897 y Fu(R)801 878 y Ft(N)879 807 y FE(\()p Fw(h)17 b(L)p FE(\()p Fw(x;)g(v)t FE(\))22 b(+)g Fw(\036)p FE(\()p Fw(x)g FE(+)g Fw(hv)t FE(\))g(+)g Fw(\017)17 b FE(ln)f Fw(\015)5 b FE(\))17 b Fw(\015)5 b FE(\()p Fw(v)t FE(\))p Fw(dv)30 b Fx(\025)795 937 y Fv(Z)850 1163 y Fu(R)898 1144 y Ft(N)977 1073 y FE(\()p Fw(h)17 b(L)p FE(\()p Fw(x;)g(v)t FE(\))22 b(+)g Fw(\036)p FE(\()p Fw(x)g FE(+)g Fw(hv)t FE(\))g(+)g Fw(\017)17 b FE(ln)f Fw(\015)2241 1088 y Fq(?)2280 1073 y FE(\))g Fw(\015)2385 1088 y Fq(?)2424 1073 y FE(\()p Fw(v)t FE(\))p Fw(dv)k FE(+)795 1203 y Fv(Z)850 1428 y Fu(R)898 1410 y Ft(N)977 1339 y FE(\()p Fw(h)d(L)p FE(\()p Fw(x;)g(v)t FE(\))22 b(+)g Fw(\036)p FE(\()p Fw(x)g FE(+)g Fw(hv)t FE(\))g(+)g Fw(\017)17 b FE(ln)f Fw(\015)2241 1354 y Fq(?)2302 1339 y FE(+)22 b Fw(\017)p FE(\))17 b(\()p Fw(\015)5 b FE(\()p Fw(v)t FE(\))22 b Fx(\000)g Fw(\015)2887 1354 y Fq(?)2926 1339 y FE(\()p Fw(v)t FE(\)\))p Fw(dv)t(:)456 1577 y FE(F)-8 b(rom)27 b(\(11\))h(and)h(the)g(fact)g(that)g(b)s(oth)f Fw(\015)5 b FE(\()p Fw(v)t FE(\))29 b(and)g Fw(\015)2332 1592 y Fq(?)2399 1577 y FE(are)g(probabilit)m(y)e(densities,)456 1705 y(w)m(e)33 b(conclude)g(that)g(the)g(last)e(in)m(tegral)g(v)-5 b(anishes,)34 b(and)e(so)h Fw(\015)2690 1720 y Fq(?)2762 1705 y FE(is)f(optimal.)555 1833 y(The)37 b(last)e(claim)e(in)i(Prop)s (osition)f(1)i(follo)m(ws)e(from)h(\(11\))g(and)h(the)g(fact)g(that)456 1961 y Fw(\015)507 1976 y Fq(?)578 1961 y FE(is)c(a)h(probabilit)m(y)d (densit)m(y)-8 b(.)1789 b Fj(\003)555 2162 y FE(F)-8 b(rom)31 b(Prop)s(osition)g(1,)i(w)m(e)g(easily)f(conclude)h(that:)456 2319 y FI(Theorem)k(2.)k FD(We)35 b(have)g(that)1081 2529 y FE(inf)1068 2591 y Fq(\015)t Fs(2D)1228 2393 y Fv(Z)1345 2529 y FE([)p Fw(h)17 b(L)p FE(\()p Fw(x;)g(v)t FE(\))22 b(+)g Fw(\036)p FE(\()p Fw(x)g FE(+)g Fw(hv)t FE(\))g(+)g Fw(\017)17 b FE(ln)f Fw(\015)5 b FE(\()p Fw(v)t FE(\)])16 b Fw(\015)5 b FE(\()p Fw(v)t FE(\))17 b Fw(dv)909 2799 y FE(=)83 b Fx(\000)p Fw(\017)17 b FE(ln)1299 2659 y Fv(\024)1352 2664 y(Z)1407 2889 y Fu(R)1455 2870 y Ft(N)1534 2799 y Fw(e)1579 2758 y Fs(\000)1644 2725 y Ft(h)11 b(L)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p 1644 2743 506 3 v 1883 2784 a Ft(\017)2180 2799 y Fw(dv)2282 2659 y Fv(\025)2367 2799 y Fw(:)456 3043 y FD(In)34 b(p)-5 b(articular,)35 b(the)f(schemes)g(in)h(\(4\))f(and)g(\(7\))h(ar)-5 b(e)34 b(e)-5 b(quivalent.)865 3306 y FE(4.)49 b Fz(Generalized)37 b(Hopf-Cole)h(transf)n(orma)-7 b(tion)555 3492 y FE(In)43 b(this)f(section,)j(w)m(e)f(discuss)g(a)e(c)m(hange)h(of)f(v)-5 b(ariables)42 b(whic)m(h)h(transforms)456 3620 y(the)31 b(non-linear)e(ev)m(olution)i(op)s(erator)f Fx(G)37 b FE(in)m(to)31 b(a)f(linear)g(ev)m(olution)g(op)s(erator)h Fx(L)p FE(.)456 3748 y(This)36 b(pro)s(cedure)i(is)e(related)g(with)h (the)g(classical)e(Hopf-Cole)g(transformation)456 3876 y(\(see,)e(for)f(instance,)h Fx(x)p FE(4.4.1)g(of)f([Ev)-5 b(a98]\).)555 4004 y(F)d(or)32 b(this,)g(w)m(e)i(de\014ne)g(the)f FD(line)-5 b(ar)32 b FE(op)s(erator)456 4232 y(\(12\))536 b Fx(L)p FE([)p Fw( )t FE(]\()p Fw(x)p FE(\))27 b(=)1618 4097 y Fv(Z)1673 4322 y Fu(R)1721 4303 y Ft(N)1800 4232 y Fw(e)1845 4191 y Fs(\000)1910 4158 y Ft(h)11 b(L)p Fr(\()p Ft(x;v)r Fr(\))p 1910 4176 224 3 v 2009 4217 a Ft(\017)2147 4232 y Fw( )t FE(\()p Fw(x)23 b FE(+)f Fw(hv)t FE(\))17 b Fw(dv)i(:)456 4466 y FE(W)-8 b(e)33 b(will)d(relate)i Fx(L)g FE(and)h Fx(G)38 b FE(through)33 b(the)g(exp)s(onen)m(tial)f(transformation:)456 4671 y(\(13\))928 b Fx(E)9 b FE([)p Fw(\036)p FE(]\()p Fw(x)p FE(\))44 b(:=)g Fw(e)2099 4630 y Fs(\000)2164 4596 y Ft(\036)p Fr(\()p Ft(x)p Fr(\))p 2164 4615 120 3 v 2211 4656 a Ft(\017)2315 4671 y Fw(:)456 4864 y FE(Then,)33 b(it)f(is)g(easy)i(to)e(see)i(that:)456 5022 y FI(Theorem)j(3.)k FD(We)35 b(have)456 5216 y FE(\(14\))993 b Fx(L)22 b(\016)g(E)53 b FE(=)44 b Fx(E)31 b(\016)22 b(G)g Fw(:)p eop %%Page: 9 9 9 8 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)458 b(9)456 450 y FD(In)34 b(p)-5 b(articular,)35 b(if)1585 604 y Fw( )1648 619 y Fq(n)1695 604 y FE(\()p Fw(x)p FE(\))28 b(:=)g Fw(e)2030 563 y Fs(\000)2095 530 y Ft(\036)2132 538 y(n)2174 530 y Fr(\()p Ft(x)p Fr(\))p 2095 548 163 3 v 2163 589 a Ft(\017)2288 604 y Fw(;)456 769 y FD(the)34 b(schemes)g(in)h(\(4\))f (and)g(\(7\))h(ar)-5 b(e)34 b(e)-5 b(quivalent)35 b(to)g(the)g(scheme) 1646 956 y Fw( )1709 971 y Fq(n)p Fl(+1)1874 956 y FE(=)27 b Fx(L)p FE([)p Fw( )2136 971 y Fq(n)2184 956 y FE(])17 b Fw(:)555 1143 y FE(W)-8 b(e)22 b(omit)e(the)i(details)f(of)g(the)h (elemen)m(tary)g(calculation)d(needed)24 b(to)d(pro)m(v)m(e)i(\(14\)) 456 1271 y(and)32 b(hence)i(Theorem)f(3.)555 1399 y(It)g(is)f(also)g (con)m(v)m(enien)m(t)i(to)e(in)m(tro)s(duce)h(the)g(rescaled)g(linear)e (op)s(erator)1135 1603 y(^)1111 1628 y Fx(L)p FE([)p Fw( )t FE(]\()p Fw(x)p FE(\))84 b(:=)1871 1561 y Fx(L)p FE([)p Fw( )t FE(]\()p Fw(x)p FE(\))p 1712 1605 640 4 v 1712 1644 a Fv(R)1759 1759 y Fu(R)1807 1740 y Ft(N)1885 1725 y Fw(e)1930 1688 y Fs(\000)1995 1654 y Ft(h)11 b(L)p Fr(\()p Ft(x;v)r Fr(\))p 1995 1673 224 3 v 2094 1714 a Ft(\017)2249 1725 y Fw(dv)1529 1981 y FE(=)1712 1827 y Fv(R)1759 1942 y Fu(R)1807 1923 y Ft(N)1885 1907 y Fw(e)1930 1870 y Fs(\000)1995 1837 y Ft(h)g(K)s Fr(\()p Ft(v)r Fr(\))p 1995 1855 183 3 v 2073 1897 a Ft(\017)2191 1907 y Fw( )t FE(\()p Fw(x)23 b FE(+)f Fw(hv)t FE(\))17 b Fw(dv)p 1712 1958 1024 4 v 1924 1997 a Fv(R)1971 2112 y Fu(R)2019 2093 y Ft(N)2098 2077 y Fw(e)2143 2041 y Fs(\000)2208 2007 y Ft(h)11 b(K)s Fr(\()p Ft(v)r Fr(\))p 2208 2026 183 3 v 2286 2067 a Ft(\017)2421 2077 y Fw(dv)2761 1981 y(:)456 2249 y FE(Accordingly)-8 b(,)31 b(w)m(e)j(ma)m(y)e (consider)h(the)g(iteration)456 2436 y(\(15\))1666 2410 y(^)1646 2436 y Fw( )1709 2451 y Fq(n)p Fl(+1)1874 2436 y FE(=)2001 2411 y(^)1977 2436 y Fx(L)p FE([)2093 2410 y(^)2073 2436 y Fw( )2136 2451 y Fq(n)2184 2436 y FE(])17 b Fw(:)456 2623 y FE(Notice)32 b(that)997 2598 y(^)973 2623 y Fx(L)g FE(is)g(indep)s(enden)m(t)i(on)f(the)g(p)s(oten)m(tial)e Fw(U)10 b FE(.)555 2751 y(As)44 b(an)f(application)e(of)i(the)h (generalized)f(Hopf-Cole)f(transformation)f(w)m(e)456 2879 y(no)m(w)f(sho)m(w)g(the)g(follo)m(wing)d(prop)s(ert)m(y)-8 b(,)42 b(whic)m(h)e(is)f(someho)m(w)i(related)e(with)g(the)456 3007 y(fact)32 b(that)g(the)h(minim)m(um)d(of)i(viscosit)m(y)h (solutions)e(is)h(a)h(viscosit)m(y)g(solution.)456 3161 y FI(Prop)s(osition)k(4.)43 b FD(Assume)37 b(that)g Fw(\036)1818 3176 y Fq(n)1901 3161 y FD(and)f Fw(')2156 3176 y Fq(n)2239 3161 y FD(ar)-5 b(e)37 b(solutions)f(to)h(the)f(iter)-5 b(ative)456 3289 y(scheme)1272 3432 y Fw(\036)1330 3447 y Fq(n)p Fl(+1)1495 3432 y FE(=)27 b Fx(G)6 b FE([)p Fw(\036)1748 3447 y Fq(n)1795 3432 y FE(])200 b Fw(')2086 3447 y Fq(n)p Fl(+1)2250 3432 y FE(=)28 b Fx(G)6 b FE([)p Fw(')2510 3447 y Fq(n)2557 3432 y FE(])17 b Fw(:)456 3596 y FD(Then)34 b(so)g(is)1374 3739 y Fw(\021)1422 3754 y Fq(n)1497 3739 y FE(=)28 b Fx(\000)p Fw(\017)17 b FE(ln)1832 3628 y Fv(\020)1892 3739 y Fw(e)1937 3697 y Fs(\000)2002 3667 y Ft(\036)2039 3675 y(n)p 2002 3682 80 3 v 2028 3724 a(\017)2117 3739 y FE(+)22 b Fw(e)2260 3697 y Fs(\000)2325 3667 y Ft(')2365 3675 y(n)p 2325 3682 83 3 v 2354 3724 a(\017)2422 3628 y Fv(\021)2499 3739 y Fw(:)456 3952 y FD(Pr)-5 b(o)g(of.)41 b FE(This)32 b(follo)m(ws)g(from)f(\(14\))h(and)h(the)g(linearit)m(y)e(of)h(the)h (sc)m(heme)g Fx(L)p FE(.)183 b Fj(\003)555 4146 y FE(W)-8 b(e)33 b(observ)m(e)i(that)1158 4333 y Fx(\000)p Fw(\017)17 b FE(ln)1389 4223 y Fv(\020)1448 4333 y Fw(e)1493 4292 y Fs(\000)1558 4261 y Ft(\036)1595 4269 y(n)p 1559 4277 80 3 v 1585 4318 a(\017)1674 4333 y FE(+)22 b Fw(e)1817 4292 y Fs(\000)1882 4261 y Ft(')1922 4269 y(n)p 1882 4277 83 3 v 1911 4318 a(\017)1979 4223 y Fv(\021)2067 4333 y Fx(!)27 b FE(min)n Fx(f)p Fw(\036)2464 4348 y Fq(n)2511 4333 y Fw(;)17 b(')2619 4348 y Fq(n)2665 4333 y Fx(g)p Fw(;)456 4542 y FE(as)46 b Fw(\017)51 b Fx(!)g FE(0)879 4506 y Fl(+)938 4542 y FE(,)e(whic)m(h)e(is)f(consisten)m(t)h (with)f(the)g(fact)g(that)g(the)h(minim)m(um)c(of)456 4670 y(viscosit)m(y)24 b(solutions)f(of)h(the)g(terminal)e(v)-5 b(alue)23 b(problem)g(is)g(a)h(viscosit)m(y)3021 4631 y Fl(3)3081 4670 y FE(solution.)p 456 4757 499 4 v 555 4824 a Fl(3)591 4856 y FH(Indeed,)34 b(if)g Fp(\036)1021 4868 y Fy(1)1092 4856 y FH(and)f Fp(\036)1308 4868 y Fy(2)1378 4856 y FH(are)f(solutions)h(of)g(\(2\),)h(so)f(is)g Fp(\036)f FH(:=)g(min)q FF(f)p Fp(\036)2766 4868 y Fy(1)2803 4856 y Fp(;)14 b(\036)2889 4868 y Fy(2)2926 4856 y FF(g)p FH(.)53 b(T)-7 b(o)33 b(con\014rm)456 4966 y(this,)28 b(note)f(that)675 5060 y Fo(Z)758 5081 y Fi(T)721 5249 y(t)824 5173 y Fp(L)p FH(\()p Fp(q)s FH(\()p Fp(\034)9 b FH(\))p Fp(;)30 b FH(_)-39 b Fp(q)t FH(\()p Fp(\034)9 b FH(\)\))14 b Fp(d\034)30 b FH(+)18 b Fp(\036)1536 5185 y Fi(i)1564 5173 y FH(\()p Fp(q)s FH(\()p Fp(T)12 b FH(\))p Fp(;)i(T)e FH(\))22 b FF(\025)2001 5060 y Fo(Z)2084 5081 y Fi(T)2047 5249 y(t)2150 5173 y Fp(L)p FH(\()p Fp(q)s FH(\()p Fp(\034)9 b FH(\))p Fp(;)30 b FH(_)-39 b Fp(q)t FH(\()p Fp(\034)9 b FH(\)\))14 b Fp(d\034)30 b FH(+)18 b Fp(\036)p FH(\()p Fp(q)s FH(\()p Fp(T)12 b FH(\))p Fp(;)i(T)e FH(\))i Fp(;)p eop %%Page: 10 10 10 9 bop 456 251 a FC(10)666 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)555 450 y FE(Note)28 b(that)e(the)i(fact)f(that)g Fx(L)p FE([)p Fw(\025 )t FE(])g(=)h Fw(\025)p Fx(L)p FE([)p Fw( )t FE(])f(re\015ects)i(in)m(to)d Fx(G)6 b FE([)p Fw(\036)11 b FE(+)g Fw(\025)p FE(])28 b(=)f Fx(G)6 b FE([)p Fw(\036)p FE(])11 b(+)456 578 y Fw(\025)p FE(,)27 b(that)f(is)g Fx(G)33 b FE(comm)m(utes)26 b(with)g(addition)f(of)h (constan)m(ts.)43 b(This)26 b(elemen)m(tary)h(fact)456 706 y(will)j(b)s(e)j(further)g(stressed)h(and)f(used)h(in)d(the)i (sequel.)1348 926 y(5.)48 b Fz(Some)38 b(pr)n(oper)-7 b(ties)36 b(of)i Fx(G)555 1112 y FE(W)-8 b(e)37 b(no)m(w)f(p)s(oin)m(t) f(out)h(some)g(features)h(of)e Fx(G)6 b FE(,)37 b(suc)m(h)h(as)e (algebraic)e(prop)s(erties,)456 1240 y(smo)s(othness,)39 b(analytic)e(b)s(ounds)h(and)g(semiconca)m(vit)m(y)g(estimates.)59 b(In)38 b(partic-)456 1368 y(ular,)50 b(a)d(\014ne)h FD(c)-5 b(ontr)g(action)47 b(pr)-5 b(op)g(erty)48 b FE(is)f (established)g(in)g(Theorem)g(9)g(and)h(a)456 1496 y(uniform)35 b FD(b)-5 b(ound)38 b(on)h(the)g(semic)-5 b(onc)g(avity)38 b(mo)-5 b(dulus)36 b FE(is)h(giv)m(en)g(in)f(Theorem)i(13.)456 1624 y(The)i(b)s(ounds)g(for)f(the)h(semiconca)m(vit)m(y)f(mo)s(dulus)f (are)i(essen)m(tial)f(to)g(pro)m(v)m(e)i(re-)456 1751 y(sults)35 b(suc)m(h)i(as)e(existence)i(of)d(\014xed)j(p)s(oin)m(ts)e (and)g(con)m(v)m(ergence)j(of)c(the)i(sc)m(heme,)456 1879 y(among)31 b(others.)555 2007 y(The)k(\014rst)f(three)g(prop)s (erties)g(follo)m(w)e(directly)h(from)f(the)i(de\014nition)f(and)g(w)m (e)456 2135 y(th)m(us)g(omit)e(the)i(pro)s(of:)456 2285 y FI(Prop)s(osition)38 b(5.)43 b FD(If)37 b Fw(\036)1357 2300 y Fl(0)1434 2285 y FD(is)g Fn(Z)1610 2249 y Fq(N)1675 2285 y FD(-p)-5 b(erio)g(dic,)36 b(then)i(so)f(is)g Fw(\036)2619 2300 y Fq(n)2698 2285 y FE(:=)32 b Fx(G)2898 2249 y Fq(n)2945 2285 y FE([)p Fw(\036)3030 2300 y Fl(0)3070 2285 y FE(])37 b FD(for)g(any)456 2413 y Fw(n)p FD(.)45 b(If)34 b Fw(U)45 b FD(and)35 b Fw(\036)1050 2428 y Fl(0)1124 2413 y FD(b)-5 b(elong)34 b(to)h Fw(C)1612 2377 y Fl(1)1651 2413 y FE(\()p Fn(R)1755 2377 y Fq(N)1828 2413 y FE(\))p FD(,)g(then)g(so)f(do)-5 b(es)34 b Fw(\036)2544 2428 y Fq(n)2626 2413 y FD(for)h(any)f Fw(n)p FD(.)456 2563 y FI(Prop)s(osition)h(6)j FE(\(Monotonicit)m(y\))p FI(.)i FD(If)35 b Fw(\036)27 b Fx(\025)h Fw(\036)2220 2578 y Fs(\003)2259 2563 y FD(,)35 b(then)g Fx(G)6 b FE([)p Fw(\036)p FE(])28 b Fx(\025)g(G)6 b FE([)p Fw(\036)3001 2578 y Fs(\003)3040 2563 y FE(])p FD(.)456 2713 y FI(Prop)s(osition)35 b(7.)42 b Fx(G)6 b FE([)p Fw(\036)22 b FE(+)g Fw(r)s FE(])27 b(=)h Fx(G)6 b FE([)p Fw(\036)p FE(])22 b(+)g Fw(r)s FD(,)35 b(for)f(any)h Fw(r)30 b Fx(2)f Fn(R)t FD(.)555 2863 y FE(W)-8 b(e)33 b(no)m(w)g(p)s(oin)m(t)f(out)h(that)f Fx(G)39 b FE(is)32 b(a)g(\(non-strict\))g(con)m(traction)g(on)g Fw(L)3077 2827 y Fs(1)3153 2863 y FE(\()p Fn(R)3256 2827 y Fq(N)3330 2863 y FE(\):)456 3014 y FI(Prop)s(osition)j(8.)42 b FD(F)-7 b(or)34 b(any)h Fw(\036;)17 b(\036)1719 3029 y Fs(\003)1785 3014 y Fx(2)28 b Fw(L)1945 2977 y Fs(1)2020 3014 y FE(\()p Fn(R)2124 2977 y Fq(N)2197 3014 y FE(\))p FD(,)1088 3190 y Fx(kG)6 b FE([)p Fw(\036)p FE(])22 b Fx(\000)h(G)6 b FE([)p Fw(\036)1587 3205 y Fs(\003)1626 3190 y FE(])p Fx(k)1703 3209 y Fq(L)1751 3190 y Fm(1)1816 3209 y Fl(\()p Fu(R)1891 3190 y Ft(N)1950 3209 y Fl(\))2009 3190 y Fx(\024)28 b(k)p Fw(\036)22 b Fx(\000)g Fw(\036)2401 3205 y Fs(\003)2440 3190 y Fx(k)2490 3209 y Fq(L)2538 3190 y Fm(1)2603 3209 y Fl(\()p Fu(R)2678 3190 y Ft(N)2736 3209 y Fl(\))2785 3190 y Fw(:)456 3373 y FD(Pr)-5 b(o)g(of.)41 b FE(Fix)32 b Fw(x)c Fx(2)g Fn(R)1172 3337 y Fq(N)1245 3373 y FE(.)43 b(Without)32 b(loss)h(of)f(generalit)m(y)f(w)m(e)j(ma)m (y)e(assume)i(that)1534 3550 y Fx(G)6 b FE([)p Fw(\036)p FE(]\()p Fw(x)p FE(\))28 b Fx(\025)g(G)6 b FE([)p Fw(\036)2125 3565 y Fs(\003)2164 3550 y FE(]\()p Fw(x)p FE(\))17 b Fw(;)p 456 3617 499 4 v 456 3708 a FH(for)27 b Fp(i)22 b FH(=)h(1)p Fp(;)14 b FH(2)27 b(and)g(th)n(us,)h(b)n(y)f(taking)g(the) h(in\014m)n(um)g(as)f(in)h(\(2\),)1097 3911 y Fp(\036)1146 3923 y Fi(i)1174 3911 y FH(\()p Fp(x;)14 b(t)p FH(\))24 b FF(\025)f FH(inf)1579 3798 y Fo(Z)1662 3819 y Fi(T)1625 3987 y(t)1728 3911 y Fp(L)p FH(\()p Fp(q)s FH(\()p Fp(\034)9 b FH(\))p Fp(;)30 b FH(_)-39 b Fp(q)t FH(\()p Fp(\034)9 b FH(\)\))14 b Fp(d\034)29 b FH(+)18 b Fp(\036)p FH(\()p Fp(q)s FH(\()p Fp(T)12 b FH(\))p Fp(;)i(T)e FH(\))i Fp(;)456 4095 y FH(for)27 b Fp(i)c FH(=)g(1)p Fp(;)14 b FH(2.)37 b(On)28 b(the)g(other)f(hand,)h(\014xed)g(an)n(y)f Fp(a)d(>)f FH(0,)k(there)h(exists)f(a)h(suitable)g Fp(q)3099 4107 y Fi(a)3167 4095 y FH(so)f(that,)456 4205 y(p)r(ossibly)g(in)n(tec)n (hanging)f(the)i(indexes)f(of)h Fp(\036)1839 4217 y Fi(i)1867 4205 y FH(,)1143 4403 y Fp(a)18 b FH(+)g(inf)1403 4290 y Fo(Z)1486 4311 y Fi(T)1449 4479 y(t)1552 4403 y Fp(L)p FH(\()p Fp(q)s FH(\()p Fp(\034)9 b FH(\))p Fp(;)31 b FH(_)-40 b Fp(q)t FH(\()p Fp(\034)9 b FH(\)\))14 b Fp(d\034)30 b FH(+)18 b Fp(\036)p FH(\()p Fp(q)s FH(\()p Fp(T)12 b FH(\))p Fp(;)i(T)e FH(\))1204 4647 y FF(\025)1292 4534 y Fo(Z)1375 4554 y Fi(T)1338 4722 y(t)1441 4647 y Fp(L)p FH(\()p Fp(q)1567 4659 y Fi(a)1607 4647 y FH(\()p Fp(\034)d FH(\))p Fp(;)30 b FH(_)-39 b Fp(q)1790 4659 y Fi(a)1831 4647 y FH(\()p Fp(\034)9 b FH(\)\))14 b Fp(d\034)30 b FH(+)18 b Fp(\036)p FH(\()p Fp(q)2296 4659 y Fi(a)2337 4647 y FH(\()p Fp(T)12 b FH(\))p Fp(;)i(T)e FH(\))1167 4890 y(=)1254 4777 y Fo(Z)1338 4798 y Fi(T)1301 4966 y(t)1404 4890 y Fp(L)p FH(\()p Fp(q)1530 4902 y Fi(a)1570 4890 y FH(\()p Fp(\034)d FH(\))p Fp(;)30 b FH(_)-39 b Fp(q)1753 4902 y Fi(a)1794 4890 y FH(\()p Fp(\034)9 b FH(\)\))14 b Fp(d\034)30 b FH(+)18 b Fp(\036)2190 4902 y Fy(1)2228 4890 y FH(\()p Fp(q)2297 4902 y Fi(a)2337 4890 y FH(\()p Fp(T)12 b FH(\))p Fp(;)i(T)e FH(\))1834 5064 y FF(\025)23 b Fp(\036)1971 5076 y Fy(1)2008 5064 y FH(\()p Fp(x;)14 b(t)p FH(\))38 b FF(\025)f Fp(\036)p FH(\()p Fp(x;)14 b(t)p FH(\))g Fp(:)456 5216 y FH(These)27 b(computations)g(sho)n(w)g(that)h Fp(\036)g FH(is)f(a)g(solution)g(of)h (\(2\).)p eop %%Page: 11 11 11 10 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)420 b(11)456 450 y FE(th)m(us)1009 709 y Fx(jG)6 b FE([)p Fw(\036)p FE(]\()p Fw(x)p FE(\))22 b Fx(\000)h(G)6 b FE([)p Fw(\036)1617 724 y Fs(\003)1656 709 y FE(]\()p Fw(x)p FE(\))p Fx(j)28 b FE(=)f Fw(\017)17 b FE(ln)2137 556 y Fv(R)2184 671 y Fu(R)2232 652 y Ft(N)2311 636 y Fw(e)2356 599 y Fs(\000)2421 566 y Ft(L)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft(\036)2682 574 y Fm(\003)2717 566 y Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p 2421 584 495 3 v 2655 625 a Ft(\017)2946 636 y Fw(dv)p 2137 686 911 4 v 2155 726 a Fv(R)2202 841 y Fu(R)2250 822 y Ft(N)2329 806 y Fw(e)2374 769 y Fs(\000)2439 736 y Ft(L)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p 2439 754 459 3 v 2655 795 a Ft(\017)2928 806 y Fw(dv)1009 1093 y Fx(\024)28 b Fw(\017)17 b FE(ln)1278 940 y Fv(R)1325 1054 y Fu(R)1373 1035 y Ft(N)1451 1020 y Fw(e)1496 983 y Fs(\000)1561 949 y Ft(L)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p 1561 968 V 1778 1009 a Ft(\017)2051 1020 y Fw(e)2106 918 y Fm(k)p Ft(\036)p Fm(\000)p Ft(\036)2260 926 y Fm(\003)2295 918 y(k)2327 949 y Ft(L)2369 935 y Fm(1)2434 949 y Fr(\()p Fk(R)2500 935 y Ft(N)2552 949 y Fr(\))p 2106 968 475 3 v 2330 1009 a Ft(\017)2611 1020 y Fw(dv)p 1278 1070 1435 4 v 1558 1110 a Fv(R)1605 1225 y Fu(R)1653 1206 y Ft(N)1731 1190 y Fw(e)1776 1153 y Fs(\000)1841 1120 y Ft(L)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p 1841 1138 459 3 v 2058 1179 a Ft(\017)2331 1190 y Fw(dv)1009 1340 y FE(=)27 b Fx(k)p Fw(\036)22 b Fx(\000)h Fw(\036)1400 1355 y Fs(\003)1439 1340 y Fx(k)1489 1358 y Fq(L)1537 1339 y Fm(1)1602 1358 y Fl(\()p Fu(R)1677 1339 y Ft(N)1735 1358 y Fl(\))1783 1340 y Fw(:)3367 1537 y Fj(\003)555 1743 y FE(The)34 b(con)m(traction)e(prop)s(ert)m(y)h(can)g(b)s(e)g (impro)m(v)m(ed)f(in)g(the)h(follo)m(wing)d(w)m(a)m(y:)456 1902 y FI(Theorem)f(9.)36 b FD(Supp)-5 b(ose)28 b(that)g(b)-5 b(oth)29 b Fw(\036)f FD(and)f Fw(\036)2130 1917 y Fs(\003)2198 1902 y FD(ar)-5 b(e)28 b Fn(Z)2426 1866 y Fq(N)2491 1902 y FD(-p)-5 b(erio)g(dic)27 b(and)h(Lipschitz)456 2030 y(functions,)38 b(with)g(Lipschitz)f(c)-5 b(onstant)38 b(b)-5 b(ounde)g(d)37 b(by)h FE(\003)p FD(.)54 b(Then,)38 b(ther)-5 b(e)38 b(exists)g(a)456 2158 y(c)-5 b(onstant)46 b Fw(C)58 b(>)50 b FE(0)p FD(,)g(p)-5 b(ossibly)46 b(dep)-5 b(ending)46 b(on)h Fw(h)p FD(,)j Fw(\017)p FD(,)h FE(\003)46 b FD(and)h(other)g(universal)456 2286 y(quantities,)34 b(such)h(that)456 2484 y FE(\(16\))288 b Fx(kG)6 b FE([)p Fw(\036)p FE(])22 b Fx(\000)g(G)6 b FE([)p Fw(\036)1416 2499 y Fs(\003)1456 2484 y FE(])p Fx(k)1533 2499 y Fq(])1609 2484 y Fx(\024)44 b FE(\(1)22 b Fx(\000)h Fw(C)h Fx(k)p Fw(\036)d Fx(\000)i Fw(\036)2320 2499 y Fs(\003)2359 2484 y Fx(k)2409 2443 y Fq(N)2409 2508 y(])2476 2484 y FE(\))17 b Fx(k)p Fw(\036)k Fx(\000)i Fw(\036)2818 2499 y Fs(\003)2857 2484 y Fx(k)2907 2499 y Fq(])2955 2484 y Fw(;)456 2681 y FD(in)34 b(which)g(we)h(use)-5 b(d)34 b(the)h(norm)456 2879 y FE(\(17\))747 b Fx(k)p Fw( )t Fx(k)1544 2894 y Fq(])1603 2879 y FE(:=)34 b(inf)1734 2942 y Fq(\020)5 b Fs(2)p Fu(R)1881 2879 y Fx(k)p Fw( )26 b FE(+)c Fw(\020)8 b Fx(k)2219 2898 y Fq(L)2267 2879 y Fm(1)2331 2898 y Fl(\()p Fu(R)2406 2879 y Ft(N)2464 2898 y Fl(\))2496 2879 y Fw(:)456 3120 y FD(Pr)-5 b(o)g(of.)41 b FE(Let)33 b(us)g(observ)m(e)h(that)456 3318 y(\(18\))932 b Fx(k)p Fw( )t Fx(k)1729 3333 y Fq(])1788 3318 y Fx(\024)28 b(k)p Fw( )t Fx(k)2060 3336 y Fq(L)2108 3317 y Fm(1)2173 3336 y Fl(\()p Fu(R)2248 3317 y Ft(N)2306 3336 y Fl(\))456 3515 y FE(and)456 3713 y(\(19\))951 b Fx(k)p Fw( )t Fx(k)1748 3728 y Fq(])1807 3713 y FE(=)27 b Fx(k)p Fw( )f FE(+)c Fw(r)s Fx(k)2244 3728 y Fq(])2292 3713 y Fw(;)456 3911 y FE(for)32 b(an)m(y)h(function)g Fw( )j FE(and)d(an)m(y)h Fw(r)d Fx(2)d Fn(R)5 b FE(.)50 b(Also,)32 b(there)i(exists)g(a)e (constan)m(t)i Fw(\020)h Fx(2)28 b Fn(R)456 4039 y FE(suc)m(h)34 b(that)1246 4191 y Fx(k)p Fw(\036)22 b FE(+)g Fw(\020)29 b Fx(\000)23 b Fw(\036)1704 4206 y Fs(\003)1743 4191 y Fx(k)1793 4210 y Fq(L)1841 4191 y Fm(1)1906 4210 y Fl(\()p Fu(R)1981 4191 y Ft(N)2039 4210 y Fl(\))2098 4191 y FE(=)28 b Fx(k)p Fw(\036)22 b Fx(\000)g Fw(\036)2489 4206 y Fs(\003)2528 4191 y Fx(k)2578 4206 y Fq(])2626 4191 y Fw(:)456 4367 y FE(By)28 b(\(19\))f(and)h(the)h(fact)e(that)h Fx(G)34 b FE(comm)m(utes)28 b(with)g(constan)m(ts)h(\(recall)d(Prop)s (opsi-)456 4495 y(tion)h(7\),)i(p)s(ossibly)e(replacing)g Fw(\036)h FE(with)g Fw(\036)13 b FE(+)g Fw(\020)20 b Fx(\000)13 b FE(inf)24 b Fw(\036)2389 4510 y Fs(\003)2456 4495 y FE(and)k Fw(\036)2699 4510 y Fs(\003)2767 4495 y FE(with)f Fw(\036)3042 4510 y Fs(\003)3095 4495 y Fx(\000)13 b FE(inf)23 b Fw(\036)3378 4510 y Fs(\003)3417 4495 y FE(,)456 4622 y(w)m(e)33 b(ma)m(y)g(assume,)g(without)f(loss)g(of)g (generalit)m(y)-8 b(,)32 b(that)456 4820 y(\(20\))702 b Fx(k)p Fw(\036)21 b Fx(\000)i Fw(\036)1619 4835 y Fs(\003)1658 4820 y Fx(k)1708 4839 y Fq(L)1756 4820 y Fm(1)1821 4839 y Fl(\()p Fu(R)1896 4820 y Ft(N)1954 4839 y Fl(\))2013 4820 y FE(=)28 b Fx(k)p Fw(\036)22 b Fx(\000)g Fw(\036)2404 4835 y Fs(\003)2443 4820 y Fx(k)2493 4835 y Fq(])2541 4820 y Fw(:)456 5018 y FE(and)32 b(that)456 5216 y(\(21\))1092 b(inf)23 b Fw(\036)1915 5231 y Fs(\003)1982 5216 y FE(=)k(0)17 b Fw(:)p eop %%Page: 12 12 12 11 bop 456 251 a FC(12)666 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)456 450 y FE(Notice)32 b(that,)g(b)m(y)i(\(21\))e(and)g(the) h(p)s(erio)s(dicit)m(y)e(assumption,)1403 611 y Fx(k)p Fw(\036)22 b Fx(\000)g Fw(\036)1690 626 y Fs(\003)1729 611 y Fx(k)1779 626 y Fq(])1838 611 y Fx(\024)29 b(k)p Fw(\036)21 b Fx(\000)i Fw(\036)p FE(\(0\))e Fx(\000)i Fw(\036)2535 626 y Fs(\003)2574 611 y Fx(k)2624 626 y Fq(L)2672 607 y Fm(1)1159 791 y Fx(\024)28 b(k)p Fw(\036)22 b Fx(\000)g Fw(\036)p FE(\(0\))p Fx(k)1726 806 y Fq(L)1774 787 y Fm(1)1865 791 y FE(+)g Fx(k)p Fw(\036)2071 806 y Fs(\003)2110 791 y Fx(k)2160 806 y Fq(L)2208 787 y Fm(1)2304 791 y Fx(\024)29 b FE(2\003)2527 702 y Fx(p)p 2609 702 89 4 v 2609 791 a Fw(N)e(:)456 709 y FE(\(22\))456 962 y(Let)32 b Fw(x)685 977 y Fs(\003)758 962 y FE(b)s(e)h(a)f(p)s(oin) m(t)g(maximizing)456 1141 y(\(23\))909 b Fx(jG)6 b FE([)p Fw(\036)p FE(]\()p Fx(\001)p FE(\))22 b Fx(\000)g(G)6 b FE([)p Fw(\036)2119 1156 y Fs(\003)2159 1141 y FE(]\()p Fx(\001)p FE(\))p Fx(j)17 b Fw(;)456 1320 y FE(and)32 b(let)g Fw(\015)837 1335 y Fs(\003)909 1320 y FE(the)h(optimal)c (probabilit)m(y)i(measure)i(on)f Fn(R)2514 1284 y Fq(N)2620 1320 y FE(that)g(yields:)456 1535 y(\(24\))188 b Fx(G)6 b FE([)p Fw(\036)968 1550 y Fs(\003)1007 1535 y FE(]\()p Fw(x)1127 1550 y Fs(\003)1167 1535 y FE(\))28 b(=)1336 1399 y Fv(Z)1392 1625 y Fu(R)1440 1606 y Ft(N)1518 1424 y Fv(\020)1578 1535 y Fw(hL)p FE(\()p Fw(x)1793 1550 y Fs(\003)1833 1535 y Fw(;)17 b(v)t FE(\))k(+)h Fw(\036)2143 1550 y Fs(\003)2183 1535 y FE(\()p Fw(x)2276 1550 y Fs(\003)2337 1535 y FE(+)h Fw(hv)t FE(\))e(+)h Fw(\017)17 b FE(ln)f Fw(\015)2905 1550 y Fs(\003)2944 1424 y Fv(\021)3004 1535 y Fw(\015)3055 1550 y Fs(\003)3111 1535 y Fw(dv)k(:)456 1753 y FE(Recall)31 b(indeed)h(that,)h(b)m(y)g(Prop)s(osition)e(1,)1394 1944 y Fw(\015)1445 1959 y Fs(\003)1512 1944 y FE(=)d Fw(e)1661 1903 y Fs(\000)1726 1869 y Ft(hL)p Fr(\()p Ft(x)1863 1877 y Fm(\003)1899 1869 y Ft(;v)r Fr(\)+)p Ft(\036)2059 1877 y Fm(\003)2094 1869 y Fr(\()p Ft(x)2153 1877 y Fm(\003)2189 1869 y Fr(+)p Ft(hv)r Fr(\)+)p Ft(\025)2413 1877 y Fm(\003)p 1726 1888 722 3 v 2074 1929 a Ft(\017)2478 1944 y Fw(;)456 2123 y FE(with)456 2318 y(\(25\))594 b Fw(\025)1281 2333 y Fs(\003)1348 2318 y FE(=)28 b Fw(\017)17 b FE(ln)1606 2182 y Fv(Z)1661 2408 y Fu(R)1709 2389 y Ft(N)1787 2318 y Fw(e)1832 2277 y Fs(\000)1897 2243 y Ft(hL)p Fr(\()p Ft(x)2034 2251 y Fm(\003)2070 2243 y Ft(;v)r Fr(\)+)p Ft(\036)2230 2251 y Fm(\003)2266 2243 y Fr(\()p Ft(x)2325 2251 y Fm(\003)2360 2243 y Fr(+)p Ft(hv)r Fr(\))p 1897 2262 603 3 v 2186 2303 a Ft(\017)2531 2318 y Fw(dv)j(:)456 2542 y FE(Also,)32 b(due)h(to)f(\(4\),)456 2762 y(\(26\))226 b Fx(G)6 b FE([)p Fw(\036)p FE(]\()p Fw(x)1126 2777 y Fs(\003)1166 2762 y FE(\))28 b Fx(\024)1337 2626 y Fv(Z)1392 2852 y Fu(R)1440 2833 y Ft(N)1519 2651 y Fv(\020)1578 2762 y Fw(hL)p FE(\()p Fw(x)1793 2777 y Fs(\003)1834 2762 y Fw(;)17 b(v)t FE(\))k(+)h Fw(\036)p FE(\()p Fw(x)2237 2777 y Fs(\003)2299 2762 y FE(+)g Fw(hv)t FE(\))g(+)g Fw(\017)17 b FE(ln)f Fw(\015)2867 2777 y Fs(\003)2906 2651 y Fv(\021)2965 2762 y Fw(\015)3016 2777 y Fs(\003)3072 2762 y Fw(dv)k(:)456 2993 y FE(De\014ne,)33 b(for)f Fw(v)f Fx(2)1119 2954 y Fl(1)p 1116 2970 41 4 v 1116 3028 a Fq(h)1167 2993 y Fn(T)1230 2957 y Fq(N)1301 2993 y FE(,)456 3199 y(\(27\))409 b Fw(\021)1087 3214 y Fs(\003)1127 3199 y FE(\()p Fw(x;)17 b(v)t FE(\))27 b(:=)1532 3104 y Fv(X)1511 3323 y Fq(k)r Fs(2)p Fu(Z)1646 3304 y Ft(N)1715 3199 y Fw(e)1760 3158 y Fs(\000)1825 3124 y Ft(hL)p Fr(\()p Ft(x;v)r Fr(+)p Ft(k)q(=h)p Fr(\)+)p Ft(\036)2271 3132 y Fm(\003)2306 3124 y Fr(\()p Ft(x)p Fr(+)p Ft(h)p Fr(\()p Ft(v)r Fr(+)p Ft(k)q(=h)p Fr(\)\)+)p Ft(\025)2786 3132 y Fm(\003)p 1825 3143 996 3 v 2309 3184 a Ft(\017)2834 3199 y Fw(:)456 3466 y FE(Note)36 b(that,)i(since)f Fw(\036)1240 3481 y Fs(\003)1316 3466 y FE(is)f(b)s(ounded)h(and)g Fw(L)g FE(gro)m(ws)g(sup)s(erlinearly)f (in)g Fw(v)k FE(b)m(y)e(as-)456 3594 y(sumption,)31 b(the)i(series)h (ab)s(o)m(v)m(e)f(con)m(v)m(erges.)46 b(F)-8 b(urthermore,)456 3808 y(\(28\))1489 3673 y Fv(Z)1557 3871 y Fr(1)p 1554 3883 37 3 v 1554 3924 a Ft(h)1600 3898 y Fu(T)1650 3879 y Ft(N)1723 3808 y Fw(\021)1771 3823 y Fs(\003)1811 3808 y FE(\()p Fw(x;)17 b(v)t FE(\))g Fw(dv)47 b FE(=)d(1)17 b Fw(;)456 4065 y FE(since)33 b Fw(\015)746 4080 y Fs(\003)817 4065 y FE(is)f(a)h(probabilit)m(y)d(measure)j(on)g Fn(R)2076 4029 y Fq(N)2150 4065 y FE(.)555 4193 y(W)-8 b(e)33 b(claim)d(that)j (there)g(exists)g(a)g(p)s(ositiv)m(e)f(constan)m(t)h Fw(\022)j FE(suc)m(h)e(that)456 4372 y(\(29\))1056 b Fw(\022)48 b Fx(\024)c Fw(\021)1948 4387 y Fs(\003)1988 4372 y FE(\()p Fw(x;)17 b(v)t FE(\))456 4551 y(for)31 b(an)m(y)h Fw(x)c Fx(2)g Fn(R)1030 4515 y Fq(N)1135 4551 y FE(and)k Fw(v)g Fx(2)1509 4512 y Fl(1)p 1507 4528 41 4 v 1507 4585 a Fq(h)1557 4551 y Fn(T)1620 4515 y Fq(N)1691 4551 y FE(.)43 b(Indeed,)34 b(\014rst)e(note)g(that,)g(b)m(y)g(the)g (Lipsc)m(hitz)456 4679 y(prop)s(erties)g(of)g Fw(\036)1083 4694 y Fs(\003)1155 4679 y FE(and)h(\(21\),)f(w)m(e)h(ha)m(v)m(e)h (that)456 4858 y(\(30\))1060 b(0)27 b Fx(\024)h Fw(\036)1929 4873 y Fs(\003)1996 4858 y Fx(\024)g Fw(C)2171 4873 y Fl(1)456 5037 y FE(and)k(therefore,)i(b)m(y)f(\(25\),)456 5216 y(\(31\))1082 b Fw(e)1757 5174 y Fq(\025)1798 5182 y Fm(\003)1834 5174 y Fq(=\017)1930 5216 y Fx(\025)28 b Fw(C)2105 5231 y Fl(2)2161 5216 y Fw(;)p eop %%Page: 13 13 13 12 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)420 b(13)456 450 y FE(where)29 b Fw(C)803 465 y Fq(i)859 450 y Fw(>)e FE(0)h(denote)i(here)f(suitable)e(quan)m(tities)h(p)s(ossibly)g(dep)s (ending)h(on)f Fw(N)10 b FE(,)456 578 y(\003,)32 b Fw(h)h FE(and)f Fw(\017)p FE(.)555 706 y(Also,)38 b(if)e Fw(v)j Fx(2)1103 667 y Fl(1)p 1100 683 41 4 v 1100 740 a Fq(h)1151 706 y Fn(T)1214 670 y Fq(N)1322 706 y FE(\(and)e(th)m(us)h Fx(j)p Fw(v)t Fx(j)d(\024)2028 621 y(p)p 2111 621 89 4 v 85 x Fw(N)10 b(=h)p FE(\),)39 b(using)d(again)g(\(30\),)i(w)m(e)g (see)456 834 y(that)456 1030 y(\(32\))850 b Fw(e)1525 989 y Fs(\000)1590 955 y Ft(hL)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft(\036)1887 963 y Fm(\003)1922 955 y Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p 1590 974 531 3 v 1842 1015 a Ft(\017)2162 1030 y Fx(\025)28 b Fw(C)2337 1045 y Fl(3)2393 1030 y Fw(:)456 1226 y FE(Then,)33 b(from)f(\(27\),)g(\(31\))g(and)g(\(32\),)g (w)m(e)i(ha)m(v)m(e)g(that)1197 1440 y Fw(\021)1245 1455 y Fs(\003)1285 1440 y FE(\()p Fw(x;)17 b(v)t FE(\))27 b Fx(\025)h Fw(e)1688 1399 y Fs(\000)1753 1365 y Ft(hL)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft(\036)2050 1373 y Fm(\003)2086 1365 y Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\)+)p Ft(\025)2369 1373 y Fm(\003)p 1753 1384 651 3 v 2065 1425 a Ft(\017)2445 1440 y Fx(\025)g Fw(C)2620 1455 y Fl(4)2676 1440 y Fw(;)456 1636 y FE(for)k(an)m(y)h(\()p Fw(x;)17 b(v)t FE(\))27 b Fx(2)h Fn(R)1202 1600 y Fq(N)1298 1636 y Fx(\002)1410 1597 y Fl(1)p 1407 1613 41 4 v 1407 1671 a Fq(h)1458 1636 y Fn(T)1521 1600 y Fq(N)1592 1636 y FE(.)43 b(This)33 b(pro)m(v)m(es)h(\(29\).)555 1764 y(No)m(w,)j(w)m(e)f(use)g(the)g Fn(Z)1365 1728 y Fq(N)1430 1764 y FE(-p)s(erio)s(dicit)m(y)d(of)h Fw(\036)h FE(and)h Fw(\036)2410 1779 y Fs(\003)2449 1764 y FE(,)g(\(24\),)f(\(26\))g(and)g(\(27\))g(to)456 1892 y(deduce)f(that)1058 2088 y Fx(G)6 b FE([)p Fw(\036)p FE(]\()p Fw(x)1328 2103 y Fs(\003)1368 2088 y FE(\))22 b Fx(\000)h(G)6 b FE([)p Fw(\036)1678 2103 y Fs(\003)1717 2088 y FE(]\()p Fw(x)1837 2103 y Fs(\003)1877 2088 y FE(\))897 2289 y Fx(\024)1058 2154 y Fv(Z)1113 2379 y Fu(R)1161 2360 y Ft(N)1240 2289 y FE([)p Fw(\036)p FE(\()p Fw(x)1418 2304 y Fs(\003)1479 2289 y FE(+)23 b Fw(hv)t FE(\))e Fx(\000)i Fw(\036)1902 2304 y Fs(\003)1941 2289 y FE(\()p Fw(x)2034 2304 y Fs(\003)2096 2289 y FE(+)f Fw(hv)t FE(\)])17 b Fw(\015)2434 2304 y Fs(\003)2473 2289 y FE(\()p Fw(v)t FE(\))g Fw(dv)898 2555 y FE(=)1058 2419 y Fv(Z)1126 2618 y Fr(1)p 1123 2630 37 3 v 1123 2671 a Ft(h)1169 2645 y Fu(T)1219 2626 y Ft(N)1292 2555 y FE([)p Fw(\036)p FE(\()p Fw(x)1470 2570 y Fs(\003)1532 2555 y FE(+)22 b Fw(hv)t FE(\))g Fx(\000)g Fw(\036)1954 2570 y Fs(\003)1994 2555 y FE(\()p Fw(x)2087 2570 y Fs(\003)2149 2555 y FE(+)g Fw(hv)t FE(\)])16 b Fw(\021)2483 2570 y Fs(\003)2523 2555 y FE(\()p Fw(x)2616 2570 y Fs(\003)2656 2555 y Fw(;)h(v)t FE(\))g Fw(dv)-2448 b FE(\(33\))898 2856 y(=)1105 2788 y(1)p 1068 2833 124 4 v 1068 2924 a Fw(h)1124 2895 y Fq(N)1218 2720 y Fv(Z)1273 2946 y Fq(x)1313 2954 y Fm(\003)1349 2946 y Fl(+)p Fu(T)1454 2927 y Ft(N)1527 2856 y FE([)p Fw(\036)p FE(\()p Fw(w)s FE(\))21 b Fx(\000)h Fw(\036)1939 2871 y Fs(\003)1979 2856 y FE(\()p Fw(w)s FE(\)])32 b Fw(\021)2235 2871 y Fs(\003)2291 2715 y Fv(\022)2365 2856 y Fw(x)2420 2871 y Fs(\003)2460 2856 y Fw(;)2513 2788 y(w)25 b Fx(\000)d Fw(x)2762 2803 y Fs(\003)p 2513 2833 290 4 v 2630 2924 a Fw(h)2812 2715 y Fv(\023)2919 2856 y Fw(dw)c(:)456 3100 y FE(Note)32 b(also)g(that)456 3296 y(\(34\))451 b(inf)23 b Fw(\036)f Fx(\000)h Fw(\036)1454 3311 y Fs(\003)1520 3296 y FE(=)28 b Fx(\000)17 b FE(sup)h Fw(\036)j Fx(\000)i Fw(\036)2119 3311 y Fs(\003)2186 3296 y FE(=)k Fx(\000k)p Fw(\036)c Fx(\000)f Fw(\036)2654 3311 y Fs(\003)2693 3296 y Fx(k)2743 3311 y Fq(])2791 3296 y Fw(;)456 3493 y FE(thanks)33 b(to)f(\(20\).)555 3620 y(W)-8 b(e)28 b(claim)d(that)j(there)g(exists)g(a)g(set)g Fx(N)42 b(\032)28 b Fw(x)2196 3635 y Fs(\003)2248 3620 y FE(+)12 b Fn(T)2399 3584 y Fq(N)2497 3620 y FE(of)27 b(measure)h(larger)e(than)478 3723 y(^)456 3748 y Fw(C)7 b Fx(k)p Fw(\036)24 b Fx(\000)i Fw(\036)826 3763 y Fs(\003)865 3748 y Fx(k)915 3712 y Fq(N)915 3774 y(])982 3748 y FE(,)39 b(for)e(a)g(suitable)1678 3723 y(^)1656 3748 y Fw(C)42 b(>)36 b FE(0,)i(in)f(whic)m(h)g Fw(\036)25 b Fx(\000)h Fw(\036)2640 3763 y Fs(\003)2715 3748 y Fx(\024)36 b(\000k)p Fw(\036)25 b Fx(\000)h Fw(\036)3199 3763 y Fs(\003)3238 3748 y Fx(k)3288 3763 y Fq(])3320 3748 y Fw(=)p FE(8.)456 3876 y(F)-8 b(or)36 b(pro)m(ving)h(this,)i (note)e(that)h(if)e Fw(\036)f FE(=)h Fw(\036)2011 3891 y Fs(\003)2088 3876 y FE(the)i(claim)d(is)i(ob)m(vious;)j(otherwise,) 456 4004 y(b)m(y)j(\(34\),)h(there)f(is)f(a)g(p)s(oin)m(t)48 b(\026)-55 b Fw(x)45 b Fx(2)f Fw(x)1834 4019 y Fs(\003)1903 4004 y FE(+)29 b Fn(T)2071 3968 y Fq(N)2184 4004 y FE(so)42 b(that)g(\()p Fw(\036)29 b Fx(\000)g Fw(\036)2823 4019 y Fs(\003)2862 4004 y FE(\)\()6 b(\026)-55 b Fw(x)q FE(\))44 b Fw(<)g FE(0)e(and)456 4132 y Fx(k)p Fw(\036)19 b Fx(\000)i Fw(\036)739 4147 y Fs(\003)778 4132 y Fx(k)828 4147 y Fq(])887 4132 y Fx(\024)29 b FE(2)p Fx(j)p FE(\()p Fw(\036)19 b Fx(\000)i Fw(\036)1341 4147 y Fs(\003)1380 4132 y FE(\)\()6 b(\026)-55 b Fw(x)p FE(\))p Fx(j)p FE(.)43 b(By)32 b(the)g(Lipsc)m (hitz)g(prop)s(ert)m(y)g(of)f(the)h(functions)456 4260 y(in)m(v)m(olv)m(ed,)39 b(it)e(follo)m(ws)g(that)h Fw(\036)25 b Fx(\000)h Fw(\036)1761 4275 y Fs(\003)1837 4260 y Fx(\024)38 b(\000k)p Fw(\036)25 b Fx(\000)i Fw(\036)2324 4275 y Fs(\003)2363 4260 y Fx(k)2413 4275 y Fq(])2444 4260 y Fw(=)p FE(8)38 b(in)f(the)h(ball)e(of)i(radius)456 4387 y Fx(k)p Fw(\036)21 b Fx(\000)i Fw(\036)743 4402 y Fs(\003)782 4387 y Fx(k)832 4402 y Fq(])863 4387 y Fw(=)p FE(\(4\003\))32 b(cen)m(tered)j(at)j(\026)-55 b Fw(x)p FE(.)44 b(Th)m(us,)34 b(if)d(w)m(e)j(tak)m(e)1187 4584 y Fx(N)42 b FE(:=)28 b Fw(B)1516 4599 y Fs(k)p Fq(\036)p Fs(\000)p Fq(\036)1690 4607 y Fm(\003)1726 4599 y Fs(k)1761 4611 y Ft(])1791 4599 y Fq(=)p Fl(\(4\003\))1969 4584 y FE(\()6 b(\026)-55 b Fw(x)q FE(\))38 b Fx(\\)i FE(\()p Fw(x)2338 4599 y Fs(\003)2400 4584 y FE(+)22 b Fn(T)2561 4543 y Fq(N)2631 4584 y FE(\))17 b Fw(;)456 4780 y FE(the)33 b(ab)s(o)m(v)m(e)g(claim)d (follo)m(ws)h(from)h(\(22\).)555 4908 y(Then,)i(b)m(y)g(suc)m(h)g(a)e (claim)e(and)j(\(29\),)f(w)m(e)h(deduce)h(that)1094 5019 y Fv(Z)1149 5244 y Fs(N)1238 5154 y Fw(\021)1286 5169 y Fs(\003)1342 5014 y Fv(\022)1415 5154 y Fw(x)1470 5169 y Fs(\003)1510 5154 y Fw(;)1564 5087 y(w)24 b Fx(\000)f Fw(x)1813 5102 y Fs(\003)p 1564 5131 V 1680 5222 a Fw(h)1863 5014 y Fv(\023)1969 5154 y Fw(dw)30 b Fx(\025)e Fw(C)p 2225 5170 77 4 v 24 w Fx(k)p Fw(\036)21 b Fx(\000)i Fw(\036)2606 5169 y Fs(\003)2645 5154 y Fx(k)2695 5113 y Fq(N)2695 5179 y(])2779 5154 y Fw(;)p eop %%Page: 14 14 14 13 bop 456 251 a FC(14)666 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)456 450 y FE(for)33 b(a)g(suitable)g Fw(C)p 1053 466 77 4 v 36 w(>)d FE(0.)46 b(Com)m(bining)32 b(this)i(with)f (\(33\))g(and)h(\(28\),)f(w)m(e)i(conclude)456 578 y(that)568 794 y Fx(G)6 b FE([)p Fw(\036)p FE(]\()p Fw(x)838 809 y Fs(\003)878 794 y FE(\))p Fx(\000G)g FE([)p Fw(\036)1143 809 y Fs(\003)1182 794 y FE(]\()p Fw(x)1302 809 y Fs(\003)1342 794 y FE(\))1021 1021 y Fx(\024)1173 954 y FE(1)p 1136 998 124 4 v 1136 1089 a Fw(h)1192 1061 y Fq(N)1302 885 y Fv(Z)1358 1111 y Fl(\()p Fq(x)1425 1119 y Fm(\003)1461 1111 y Fl(+)p Fu(T)1566 1092 y Ft(N)1618 1111 y Fl(\))p Fs(nN)1769 910 y Fv(\020)1828 1021 y Fw(\036)p FE(\()p Fw(w)s FE(\))21 b Fx(\000)i Fw(\036)2214 1036 y Fs(\003)2253 1021 y FE(\()p Fw(w)s FE(\))2402 910 y Fv(\021)2478 1021 y Fw(\021)2526 1036 y Fs(\003)2582 881 y Fv(\022)2655 1021 y Fw(x)2710 1036 y Fs(\003)2750 1021 y Fw(;)2804 954 y(w)h Fx(\000)f Fw(x)3053 969 y Fs(\003)p 2804 998 290 4 v 2920 1089 a Fw(h)3103 881 y Fv(\023)3209 1021 y Fw(dw)1021 1312 y Fx(\024)1136 1245 y(k)p Fw(\036)e Fx(\000)i Fw(\036)1423 1260 y Fs(\003)1462 1245 y Fx(k)1512 1260 y Fq(])p 1136 1289 408 4 v 1278 1381 a Fw(h)1334 1352 y Fq(N)1587 1177 y Fv(Z)1642 1402 y Fl(\()p Fq(x)1709 1410 y Fm(\003)1745 1402 y Fl(+)p Fu(T)1850 1383 y Ft(N)1902 1402 y Fl(\))p Fs(nN)2053 1312 y Fw(\021)2101 1327 y Fs(\003)2157 1172 y Fv(\022)2231 1312 y Fw(x)2286 1327 y Fs(\003)2326 1312 y Fw(;)2379 1245 y(w)i Fx(\000)e Fw(x)2629 1260 y Fs(\003)p 2379 1289 290 4 v 2496 1381 a Fw(h)2678 1172 y Fv(\023)2785 1312 y Fw(dw)1021 1633 y Fx(\024)1136 1566 y(k)p Fw(\036)e Fx(\000)i Fw(\036)1423 1581 y Fs(\003)1462 1566 y Fx(k)1512 1581 y Fq(])p 1136 1611 408 4 v 1278 1702 a Fw(h)1334 1673 y Fq(N)1587 1463 y Fv(")1645 1633 y Fw(h)1701 1592 y Fq(N)1785 1498 y Fv(Z)1853 1696 y Fr(1)p 1850 1708 37 3 v 1850 1749 a Ft(h)1896 1723 y Fu(T)1946 1705 y Ft(N)2019 1633 y Fw(\021)2067 1648 y Fs(\003)2123 1633 y FE(\()p Fw(x)2216 1648 y Fs(\003)2256 1633 y Fw(;)17 b(v)t FE(\))33 b Fw(dv)25 b Fx(\000)e Fw(C)p 2645 1649 77 4 v 23 w Fx(k)p Fw(\036)f Fx(\000)h Fw(\036)3026 1648 y Fs(\003)3065 1633 y Fx(k)3115 1592 y Fq(N)3115 1658 y(])3182 1463 y Fv(#)1021 1885 y FE(=)k(\(1)22 b Fx(\000)h Fw(C)7 b Fx(k)p Fw(\036)21 b Fx(\000)i Fw(\036)1697 1900 y Fs(\003)1736 1885 y Fx(k)1786 1844 y Fq(N)1786 1909 y(])1853 1885 y FE(\))17 b Fx(k)p Fw(\036)k Fx(\000)i Fw(\036)2195 1900 y Fs(\003)2234 1885 y Fx(k)2284 1900 y Fq(])2332 1885 y Fw(;)456 2101 y FE(for)41 b(a)i(suitable)e Fw(C)51 b(>)45 b FE(0.)72 b(Then,)46 b(b)m(y)e(the)e(c)m(hoice)h(of)f Fw(x)2562 2116 y Fs(\003)2644 2101 y FE(p)s(erformed)g(in)g(\(23\),)456 2229 y(p)s(ossibly)32 b(in)m(terc)m(hanging)g(the)h(r^)-49 b(oles)32 b(of)g Fw(\036)g FE(and)h Fw(\036)2283 2244 y Fs(\003)2322 2229 y FE(,)g(w)m(e)g(gather)g(that)803 2444 y Fx(kG)6 b FE([)p Fw(\036)p FE(])22 b Fx(\000)h(G)6 b FE([)p Fw(\036)1302 2459 y Fs(\003)1341 2444 y FE(])p Fx(k)1418 2463 y Fq(L)1466 2444 y Fm(1)1531 2463 y Fl(\()p Fu(R)1606 2444 y Ft(N)1664 2463 y Fl(\))1740 2444 y Fx(\024)45 b FE(\(1)22 b Fx(\000)g Fw(C)7 b Fx(k)p Fw(\036)22 b Fx(\000)h Fw(\036)2435 2459 y Fs(\003)2474 2444 y Fx(k)2524 2403 y Fq(N)2524 2469 y(])2591 2444 y FE(\))17 b Fx(k)p Fw(\036)k Fx(\000)i Fw(\036)2933 2459 y Fs(\003)2972 2444 y Fx(k)3022 2459 y Fq(])3070 2444 y Fw(:)456 2660 y FE(This)32 b(and)h(\(18\))f(imply)f(\(16\).)1818 b Fj(\003)555 2886 y FE(W)-8 b(e)33 b(no)m(w)g(de\014ne)1239 3118 y Fw(\024)1295 3133 y Fq(\017;h)1433 3118 y FE(:=)67 b(sup)1580 3207 y Fq(x)p Fs(2)p Fu(R)1715 3188 y Ft(N)1789 3003 y Fv(\014)1789 3063 y(\014)1789 3123 y(\014)1839 3118 y FE(ln)1937 2982 y Fv(Z)1992 3208 y Fu(R)2040 3189 y Ft(N)2118 3118 y Fw(e)2163 3077 y Fs(\000)2228 3043 y Ft(h)11 b(L)p Fr(\()p Ft(x;v)r Fr(\))p 2228 3062 224 3 v 2327 3103 a Ft(\017)2483 3118 y Fw(dv)2585 3003 y Fv(\014)2585 3063 y(\014)2585 3123 y(\014)2634 3118 y Fw(:)456 3387 y FE(Note)47 b(that,)k(in)46 b(man)m(y)i(cases,)k Fw(\024)1718 3402 y Fq(\017;h)1858 3387 y FE(ma)m(y)47 b(b)s(e)h(estimated)e(explicitly;)53 b(e.g.,)e(if)456 3514 y Fw(L)p FE(\()p Fw(x;)17 b(v)t FE(\))27 b(=)h Fw(K)7 b FE(\()p Fw(v)t FE(\))27 b(=)h Fx(j)p Fw(v)t Fx(j)1334 3478 y Fl(2)1372 3514 y FE(,)33 b(then)g Fw(\024)1710 3529 y Fq(\017;h)1831 3514 y FE(=)28 b(\()p Fw(N)r(=)p FE(2\))p Fx(j)17 b FE(ln)n(\()p Fw(\017\031)t(=h)p FE(\))p Fx(j)p FE(.)555 3642 y(By)33 b(taking)f Fw(\036)1067 3657 y Fs(\003)1134 3642 y FE(=)27 b(0)33 b(in)f(Prop)s(osition)e(8,)j (w)m(e)g(gather)g(the)g(follo)m(wing)456 3810 y FI(Prop)s(osition)i(10) j FE(\()p Fw(L)1315 3774 y Fs(1)1390 3810 y FE(-b)s(ound\))p FI(.)1204 4026 y Fx(kG)6 b FE([)p Fw(\036)p FE(])p Fx(k)1481 4045 y Fq(L)1529 4026 y Fm(1)1594 4045 y Fl(\()p Fu(R)1669 4026 y Ft(N)1727 4045 y Fl(\))1786 4026 y Fx(\024)29 b(k)p Fw(\036)p Fx(k)2050 4045 y Fq(L)2098 4026 y Fm(1)2162 4045 y Fl(\()p Fu(R)2237 4026 y Ft(N)2295 4045 y Fl(\))2349 4026 y FE(+)22 b Fw(\017)17 b(\024)2559 4041 y Fq(\017;h)2669 4026 y Fw(:)456 4242 y FD(In)34 b(p)-5 b(articular,)1192 4458 y Fx(k)p Fw(\036)1300 4473 y Fq(n)1346 4458 y Fx(k)1396 4476 y Fq(L)1444 4457 y Fm(1)1509 4476 y Fl(\()p Fu(R)1584 4457 y Ft(N)1642 4476 y Fl(\))1701 4458 y Fx(\024)29 b(k)p Fw(\036)1915 4473 y Fl(0)1954 4458 y Fx(k)2004 4476 y Fq(L)2052 4457 y Fm(1)2116 4476 y Fl(\()p Fu(R)2191 4457 y Ft(N)2250 4476 y Fl(\))2303 4458 y FE(+)22 b Fw(\017n)17 b(\024)2571 4473 y Fq(\017;h)2681 4458 y Fw(:)555 4674 y FE(W)-8 b(e)30 b(p)s(oin)m(t)f(out)g(that,)h(in)f(general,)h(it)e(is) h(not)g(p)s(ossible)g(to)g(obtain)g(a)g(b)s(ound)h(of)456 4801 y Fx(k)p Fw(\036)564 4816 y Fq(n)610 4801 y Fx(k)660 4816 y Fq(L)708 4797 y Fm(1)805 4801 y FE(whic)m(h)e(is)f(uniform)e(on) j Fw(n)p FE(:)41 b(ev)m(en)29 b(in)e(the)h(case)g(of)f Fw(L)h FE(=)g Fx(j)p Fw(v)t Fx(j)2890 4765 y Fl(2)2956 4801 y FE(and)f Fw(\036)3198 4816 y Fl(0)3265 4801 y FE(=)h(0,)456 4929 y(whic)m(h)33 b(can)g(b)s(e)f(w)m(ork)m(ed)j(out)d (explicitly)-8 b(,)31 b(one)i(obtains)456 5180 y(\(35\))816 b Fx(k)p Fw(\036)1554 5195 y Fq(n)1601 5180 y Fx(k)1651 5195 y Fq(L)1699 5176 y Fm(1)1796 5180 y FE(=)1909 5113 y Fw(\017N)10 b(n)p 1909 5158 187 4 v 1978 5249 a FE(2)2122 5180 y Fx(j)17 b FE(ln)2274 5113 y Fw(\017\031)p 2274 5158 99 4 v 2295 5249 a(h)2382 5180 y Fx(j)g Fw(:)p eop %%Page: 15 15 15 14 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)420 b(15)555 450 y FE(W)-8 b(e)32 b(no)m(w)f(study)h(the)f(semiconca)m(vit)m(y)h(of)e Fw(\036)2146 465 y Fq(n)2193 450 y FE(.)43 b(The)31 b(semiconca)m(vit)m (y)g(mo)s(dulus)456 578 y(of)h(a)g(function)g Fw(\036)g FE(is)456 1364 y(\(36\))404 b(\006)1104 1379 y Fq(\036)1195 1364 y FE(:=)91 b(sup)1352 1448 y Ft(x;y)r Fm(2)p Fk(R)1522 1434 y Ft(N)1408 1505 y(y)r Fm(6)p Fr(=0)1611 1296 y Fw(\036)p FE(\()p Fw(x)22 b FE(+)g Fw(y)t FE(\))f(+)h Fw(\036)p FE(\()p Fw(x)g Fx(\000)h Fw(y)t FE(\))e Fx(\000)i FE(2)p Fw(\036)p FE(\()p Fw(x)p FE(\))p 1611 1341 1203 4 v 2139 1432 a Fx(j)p Fw(y)t Fx(j)2247 1403 y Fl(2)2839 1364 y Fw(:)456 2218 y FE(W)-8 b(e)33 b(sa)m(y)g(that)f(a)h(con)m(tin)m (uous)g(function)f Fw(\036)g FE(is)g(semiconca)m(v)m(e)i(if)e(\006)2854 2233 y Fq(\036)2928 2218 y Fw(<)27 b FE(+)p Fx(1)p FE(.)555 2346 y(W)-8 b(e)23 b(will)d(pro)m(v)m(e)k(that)e(\006)1411 2361 y Fq(\036)1453 2369 y Ft(n)1522 2346 y FE(is)g(uniformly)e(b)s (ounded)k(\(see)f(Theorem)g(13\).)39 b(Since)456 2473 y Fw(\036)514 2488 y Fq(n)585 2473 y FE(is)24 b(con)m(tin)m(uous)i(if)e (so)g(is)h Fw(\036)1498 2488 y Fl(0)1537 2473 y FE(,)h(this)f(yields)f (that)h Fw(\036)2301 2488 y Fq(n)2372 2473 y FE(is)g(uniformly)d (semiconca)m(v)m(e.)555 2601 y(The)44 b(estimate)e(on)g(\006)1382 2616 y Fq(\036)1424 2624 y Ft(n)1514 2601 y FE(will)f(b)s(e)i(obtained) f(in)g(three)h(steps:)65 b(w)m(e)44 b(\014rst)f(get)456 2729 y(a)j(\014rst)h(rough)g(estimate)e(on)i(\006)1681 2744 y Fq(\036)1723 2752 y Ft(n)1770 2729 y FE(;)54 b(w)m(e)47 b(then)h(pro)m(v)m(e)g(a)e(semiconca)m(vit)m(y)h(\\im-)456 2857 y(pro)m(v)m(emen)m(t")34 b(estimate)f(\(that)h(is,)f(kno)m(wing)h (a)g(b)s(ound)f(on)h(\006)2728 2872 y Fq(\036)2770 2880 y Ft(n)2817 2857 y FE(,)g(w)m(e)h(sho)m(w)g(that)456 2985 y(actually)k(a)i(sligh)m(tly)f(b)s(etter)i(b)s(ound)f(is)g(p)s (ossible\);)j(\014nally)-8 b(,)43 b(b)m(y)f(iterating)d(the)456 3113 y(pro)s(cedure,)d(w)m(e)h(will)c(b)s(e)i(able)f(to)h(obtain)f(a)h (uniform)f(b)s(ound)h(on)g(\006)3007 3128 y Fq(\036)3049 3136 y Ft(n)3096 3113 y FE(.)51 b(Let)36 b(us)456 3241 y(no)m(w)d(\014x)g(a)f(semiconca)m(v)m(e)i(function)e Fw(\036)1869 3256 y Fl(0)1941 3241 y FE(and)g(w)m(ork)i(out)e(the)h (details:)456 3648 y FI(Lemma)k(11.)k FD(L)-5 b(et)36 b Fw(C)1276 3663 y Fq(U)1369 3648 y FD(and)f Fw(C)1629 3663 y Fq(K)1732 3648 y FD(b)-5 b(e)34 b(as)h(in)f(\(8\))h(and)f (\(9\).)44 b(Then,)1532 4434 y FE(\006)1602 4449 y Fq(\036)1644 4457 y Ft(n)1719 4434 y Fx(\024)1834 4367 y Fw(C)1904 4382 y Fq(K)p 1834 4411 139 4 v 1851 4503 a FE(2)p Fw(h)2004 4434 y FE(+)2112 4367 y Fw(h)17 b(C)2255 4382 y Fq(U)p 2112 4411 202 4 v 2189 4503 a FE(2)2341 4434 y Fw(;)456 5196 y FD(for)34 b(any)h Fw(n)28 b Fx(\025)g FE(1)p FD(.)p eop %%Page: 16 16 16 15 bop 456 251 a FC(16)666 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)456 450 y FD(Pr)k(o)g(of.)41 b FE(Fix)33 b Fw(w)f Fx(2)e Fn(R)1194 414 y Fq(N)1267 450 y FE(.)47 b(By)34 b(using)g(\(9\))f(and)h(the)g(Cauc)m(h)m(y-Sc)m(h)m(w)m(arz)j (Inequalit)m(y)-8 b(,)456 578 y(w)m(e)33 b(get)g(that)577 777 y Fw(\036)635 792 y Fq(n)p Fl(+1)772 777 y FE(\()p Fw(x)p FE(\))84 b Fx(\025)f(\000)p Fw(h)17 b(U)10 b FE(\()p Fw(x)p FE(\))1147 978 y Fx(\000)p Fw(\017)17 b FE(ln)1378 867 y Fv(\020)1455 842 y(Z)1510 1068 y Fu(R)1558 1049 y Ft(N)1636 978 y Fw(e)1681 937 y Fs(\000)1757 910 y Ft(h)p 1746 922 57 3 v 1746 963 a Fr(2)p Ft(\017)1813 937 y Fl(\()p Fq(K)5 b Fl(\()p Fq(v)r Fl(+)p Fq(w)r Fl(\)+)p Fq(K)g Fl(\()p Fq(v)r Fs(\000)p Fq(w)r Fl(\))p Fs(\000)p Fq(C)2526 948 y Ft(K)2585 937 y Fs(j)p Fq(w)r Fs(j)2678 913 y Fr(2)2712 937 y Fl(\))2743 978 y Fw(e)2788 937 y Fs(\000)2853 903 y Ft(\036)2890 911 y(n)2933 903 y Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p 2853 922 278 3 v 2979 963 a Ft(\017)3162 978 y Fw(dv)3264 867 y Fv(\021)987 1245 y FE(=)84 b Fx(\000)p Fw(h)17 b(U)10 b FE(\()p Fw(x)p FE(\))23 b Fx(\000)1637 1177 y Fw(h)17 b(C)1780 1192 y Fq(K)p 1637 1222 211 4 v 1718 1313 a FE(2)1858 1245 y Fx(j)p Fw(w)s Fx(j)1987 1204 y Fl(2)1147 1489 y Fx(\000)p Fw(\017)g FE(ln)1378 1378 y Fv(\020)1455 1353 y(Z)1510 1579 y Fu(R)1558 1560 y Ft(N)1636 1489 y Fw(e)1681 1448 y Fs(\000)1746 1414 y Ft(h)11 b(K)s Fr(\()p Ft(v)r Fr(+)p Ft(w)q Fr(\)+)p Ft(\036)2104 1422 y(n)2147 1414 y Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p 1746 1433 600 3 v 2018 1474 a(2)p Ft(\017)2359 1489 y Fw(e)2404 1448 y Fs(\000)2469 1414 y Ft(h)g(K)s Fr(\()p Ft(v)r Fm(\000)p Ft(w)q Fr(\)+)p Ft(\036)2828 1422 y(n)2872 1414 y Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p 2470 1433 601 3 v 2742 1474 a(2)p Ft(\017)3101 1489 y Fw(dv)3203 1378 y Fv(\021)987 1756 y Fx(\025)83 b(\000)p Fw(h)17 b(U)10 b FE(\()p Fw(x)p FE(\))23 b Fx(\000)1637 1688 y Fw(h)17 b(C)1780 1703 y Fq(K)p 1637 1733 211 4 v 1718 1824 a FE(2)1858 1756 y Fx(j)p Fw(w)s Fx(j)1987 1715 y Fl(2)1147 2000 y Fx(\000)1239 1933 y Fw(\017)p 1234 1977 49 4 v 1234 2068 a FE(2)1310 2000 y(ln)1408 1889 y Fv(\020)1484 1864 y(Z)1539 2090 y Fu(R)1587 2071 y Ft(N)1666 2000 y Fw(e)1711 1959 y Fs(\000)1776 1925 y Ft(h)10 b(K)s Fr(\()p Ft(v)r Fr(+)p Ft(w)q Fr(\)+)p Ft(\036)2133 1933 y(n)2176 1925 y Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p 1776 1944 600 3 v 2062 1985 a Ft(\017)2405 2000 y Fw(dv)1147 2266 y Fx(\001)1192 2130 y Fv(Z)1247 2356 y Fu(R)1295 2337 y Ft(N)1373 2266 y Fw(e)1418 2225 y Fs(\000)1483 2191 y Ft(h)h(K)s Fr(\()p Ft(v)r Fm(\000)p Ft(w)q Fr(\)+)p Ft(\036)1842 2199 y(n)1885 2191 y Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p 1483 2210 601 3 v 1770 2251 a Ft(\017)2114 2266 y Fw(dv)2216 2155 y Fv(\021)987 2533 y FE(=)84 b Fx(\000)p Fw(h)17 b(U)10 b FE(\()p Fw(x)p FE(\))23 b Fx(\000)1637 2465 y Fw(h)17 b(C)1780 2480 y Fq(K)p 1637 2510 211 4 v 1718 2601 a FE(2)1858 2533 y Fx(j)p Fw(w)s Fx(j)1987 2491 y Fl(2)1147 2777 y Fx(\000)1239 2709 y Fw(\017)p 1234 2754 49 4 v 1234 2845 a FE(2)1310 2777 y(ln)1408 2666 y Fv(\020)1484 2641 y(Z)1539 2867 y Fu(R)1587 2848 y Ft(N)1666 2777 y Fw(e)1711 2736 y Fs(\000)1776 2702 y Ft(h)10 b(K)s Fr(\()p Ft(v)r Fr(+)p Ft(w)q Fr(\)+)p Ft(\036)2133 2710 y(n)2176 2702 y Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p 1776 2721 600 3 v 2062 2762 a Ft(\017)2405 2777 y Fw(dv)2507 2666 y Fv(\021)1147 3043 y Fx(\000)1239 2975 y Fw(\017)p 1234 3020 49 4 v 1234 3111 a FE(2)1310 3043 y(ln)1408 2932 y Fv(\020)1484 2907 y(Z)1539 3133 y Fu(R)1587 3114 y Ft(N)1666 3043 y Fw(e)1711 3002 y Fs(\000)1776 2968 y Ft(h)g(K)s Fr(\()p Ft(v)r Fm(\000)p Ft(w)q Fr(\)+)p Ft(\036)2134 2976 y(n)2178 2968 y Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p 1776 2986 601 3 v 2063 3028 a Ft(\017)2386 3002 y Fl(\))2434 3043 y Fw(dv)2536 2932 y Fv(\021)2612 3043 y Fw(:)456 3282 y FE(Th)m(us,)34 b(c)m(hanging)e(v)-5 b(ariables)31 b(of)h(in)m (tegration,)551 3517 y Fw(\036)609 3532 y Fq(n)p Fl(+1)746 3517 y FE(\()p Fw(x)p FE(\))83 b Fx(\025)h(\000)p Fw(h)17 b(U)10 b FE(\()p Fw(x)p FE(\))23 b Fx(\000)1611 3450 y Fw(h)17 b(C)1754 3465 y Fq(K)p 1611 3494 211 4 v 1692 3586 a FE(2)1831 3517 y Fx(j)p Fw(w)s Fx(j)1960 3476 y Fl(2)1121 3761 y Fx(\000)1213 3694 y Fw(\017)p 1208 3739 49 4 v 1208 3830 a FE(2)1283 3761 y(ln)1381 3651 y Fv(\020)1457 3626 y(Z)1513 3851 y Fu(R)1561 3832 y Ft(N)1639 3761 y Fw(e)1684 3720 y Fs(\000)1749 3687 y Ft(h)11 b(K)s Fr(\()s(~)-33 b Ft(v)s Fr(\)+)p Ft(\036)2016 3695 y(n)2058 3687 y Fr(\()p Ft(x)p Fm(\000)p Ft(hw)q Fr(+)p Ft(h)s Fr(~)g Ft(v)r Fr(\))p 1749 3705 637 3 v 2055 3746 a Ft(\017)2417 3761 y Fw(d)t FE(~)-53 b Fw(v)2518 3651 y Fv(\021)1121 4027 y Fx(\000)1213 3960 y Fw(\017)p 1208 4004 49 4 v 1208 4096 a FE(2)1283 4027 y(ln)1381 3917 y Fv(\020)1457 3892 y(Z)1513 4117 y Fu(R)1561 4098 y Ft(N)1639 4027 y Fw(e)1684 3986 y Fs(\000)1749 3953 y Ft(h)11 b(K)s Fr(\()s(~)-33 b Ft(v)s Fr(\)+)p Ft(\036)2016 3961 y(n)2058 3953 y Fr(\()p Ft(x)p Fr(+)p Ft(hw)q Fr(+)p Ft(h)s Fr(~)g Ft(v)q Fr(\))p 1749 3971 636 3 v 2054 4012 a Ft(\017)2415 4027 y Fw(d)t FE(~)-53 b Fw(v)2517 3917 y Fv(\021)961 4294 y FE(=)84 b Fx(\000)1208 4227 y Fw(h)17 b(C)1351 4242 y Fq(K)p 1208 4271 211 4 v 1289 4362 a FE(2)1429 4294 y Fx(j)p Fw(w)s Fx(j)1558 4253 y Fl(2)1618 4294 y Fx(\000)23 b Fw(h)17 b(U)10 b FE(\()p Fw(x)p FE(\))23 b(+)2129 4227 y Fw(h)p 2129 4271 57 4 v 2133 4362 a FE(2)2195 4183 y Fv(\020)2254 4294 y Fw(U)10 b FE(\()p Fw(x)24 b FE(+)e Fw(hw)s FE(\))f(+)h Fw(U)10 b FE(\()p Fw(x)23 b Fx(\000)g Fw(hw)s FE(\))3290 4183 y Fv(\021)1121 4534 y FE(+)1207 4467 y(1)p 1207 4511 49 4 v 1207 4603 a(2)1265 4424 y Fv(\020)1325 4534 y Fw(\036)1383 4549 y Fq(n)p Fl(+1)1520 4534 y FE(\()p Fw(x)f FE(+)g Fw(hw)s FE(\))g(+)g Fw(\036)2078 4549 y Fq(n)p Fl(+1)2215 4534 y FE(\()p Fw(x)g Fx(\000)h Fw(hw)s FE(\))2597 4424 y Fv(\021)2672 4534 y Fw(;)456 4757 y FE(th)m(us,)33 b(from)f(\(8\),)510 4998 y Fw(\036)568 5013 y Fq(n)p Fl(+1)705 4998 y FE(\()p Fw(x)23 b FE(+)f Fw(hw)s FE(\))f(+)h Fw(\036)1263 5013 y Fq(n)p Fl(+1)1400 4998 y FE(\()p Fw(x)h Fx(\000)f Fw(hw)s FE(\))g Fx(\000)h FE(2)p Fw(\036)2011 5013 y Fq(n)p Fl(+1)2147 4998 y FE(\()p Fw(x)p FE(\))28 b Fx(\024)2411 4888 y Fv(\020)2481 4931 y Fw(C)2551 4946 y Fq(K)p 2481 4975 139 4 v 2498 5067 a FE(2)p Fw(h)2651 4998 y FE(+)2759 4931 y Fw(h)17 b(C)2902 4946 y Fq(U)p 2759 4975 202 4 v 2836 5067 a FE(2)2971 4888 y Fv(\021)3047 4998 y FE(\()p Fw(h)p Fx(j)p Fw(w)s Fx(j)p FE(\))3308 4957 y Fl(2)3363 4998 y Fw(;)456 5216 y FE(from)31 b(whic)m(h)i(the)g(desired)g(claim)d (follo)m(ws.)1327 b Fj(\003)p eop %%Page: 17 17 17 16 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)420 b(17)555 450 y FE(Notice)46 b(that)h(the)g(b)s(ound)f(pro)m(v)m(en)i(in)e(Lemma)f (11)h(do)s(es)h(not)f(really)g(lo)s(ok)456 578 y(satisfactory)32 b(for)g(small)e Fw(h)p FE(.)44 b(W)-8 b(e)33 b(no)m(w)g(sho)m(w)h(ho)m (w)f(to)f(impro)m(v)m(e)g(it.)456 732 y FI(Lemma)37 b(12)g FE(\(Semiconca)m(vit)m(y)c(impro)m(v)m(emen)m(t\))p FI(.)41 b FD(Fix)34 b Fw(n)28 b Fx(2)g Fn(N)9 b FD(.)51 b(Assume)35 b(that)456 917 y FE(\(37\))1123 b(\006)1823 932 y Fq(\036)1865 940 y Ft(n)1939 917 y Fx(\024)29 b Fw(\033)20 b(:)456 1102 y FD(Then,)706 1063 y Fl(4)456 1322 y FE(\(38\))756 b(\006)1456 1337 y Fq(\036)1498 1346 y Ft(n)p Fr(+1)1650 1322 y Fx(\024)28 b Fw(h)17 b(C)1898 1337 y Fq(U)1979 1322 y FE(+)2167 1255 y Fw(C)2237 1270 y Fq(K)2321 1255 y Fw(\033)p 2087 1299 374 4 v 2087 1391 a(C)2157 1406 y Fq(K)2247 1391 y FE(+)22 b Fw(h\033)2487 1322 y(:)456 1559 y FD(Pr)-5 b(o)g(of.)41 b FE(Fix)35 b Fw(y)i Fx(2)d Fn(R)1183 1523 y Fq(N)1293 1559 y FE(and)i Fw(\022)h Fx(2)d FE(\(0)p Fw(;)17 b FE(1\))35 b(and)i(set)g Fw(t)d FE(:=)f(1)25 b Fx(\000)g Fw(\022)s FE(.)54 b(Then,)39 b(c)m(hanging)456 1687 y(v)-5 b(ariable)30 b(in)i(the)h(in)m (tegration,)e(w)m(e)j(gather)e(that)636 1872 y Fw(\036)694 1887 y Fq(n)p Fl(+1)831 1872 y FE(\()p Fw(x)22 b Fx(\006)h Fw(hy)t FE(\))82 b(=)h Fx(\000)p Fw(h)17 b(U)10 b FE(\()p Fw(x)23 b Fx(\006)g Fw(hy)t FE(\))1433 2073 y Fx(\000)p Fw(\017)17 b FE(ln)1664 1962 y Fv(\020)1741 1937 y(Z)1796 2163 y Fu(R)1844 2144 y Ft(N)1922 2073 y Fw(e)1967 2032 y Fs(\000)2032 2005 y Ft(h)p 2032 2017 37 3 v 2037 2058 a(\017)2079 2032 y Fq(K)5 b Fl(\()p Fq(w)r Fs(\007)p Fq(\022)r(y)r Fl(\))2381 2073 y Fw(e)2426 2032 y Fs(\000)2491 2005 y Fr(1)p 2491 2017 31 3 v 2493 2058 a Ft(\017)2532 2032 y Fq(\036)2574 2040 y Ft(n)2616 2032 y Fl(\()p Fq(x)p Fl(+)p Fq(hw)r Fs(\006)p Fq(thy)r Fl(\))3038 2073 y Fw(dw)3162 1962 y Fv(\021)3237 2073 y Fw(;)456 2303 y FE(so)32 b(that,)h(b)m(y)g (means)f(of)g(\(8\))g(and)h(using)f(the)h(Cauc)m(h)m(y-Sc)m(h)m(w)m (arz)j(Inequalit)m(y)-8 b(,)32 b(w)m(e)456 2431 y(infer)g(that)1035 2616 y Fw(\036)1093 2631 y Fq(n)p Fl(+1)1230 2616 y FE(\()p Fw(x)23 b FE(+)f Fw(hy)t FE(\))f(+)h Fw(\036)1767 2631 y Fq(n)p Fl(+1)1904 2616 y FE(\()p Fw(x)g Fx(\000)h Fw(hy)t FE(\))e Fx(\000)i FE(2)p Fw(\036)2493 2631 y Fq(n)p Fl(+1)2630 2616 y FE(\()p Fw(x)p FE(\))875 2817 y Fx(\024)83 b Fw(h)17 b(C)1178 2832 y Fq(U)1237 2817 y FE(\()p Fw(h)p Fx(j)p Fw(y)t Fx(j)p FE(\))1477 2776 y Fl(2)1537 2817 y FE(+)22 b(2)p Fw(\017)17 b FE(ln)1838 2706 y Fv(\020)1914 2681 y(Z)1969 2907 y Fu(R)2017 2888 y Ft(N)2095 2817 y Fw(e)2140 2776 y Fs(\000)2205 2749 y Ft(h)p 2206 2761 37 3 v 2211 2802 a(\017)2252 2776 y Fq(K)5 b Fl(\()p Fq(w)r Fl(\))2427 2817 y Fw(e)2472 2776 y Fs(\000)2537 2749 y Fr(1)p 2537 2761 31 3 v 2539 2802 a Ft(\017)2578 2776 y Fq(\036)2620 2784 y Ft(n)2662 2776 y Fl(\()p Fq(x)p Fl(+)p Fq(hw)r Fl(\))2926 2817 y Fw(dw)3050 2706 y Fv(\021)875 3083 y Fx(\000)83 b Fw(\017)17 b FE(ln)1189 2972 y Fv(\020)1265 2947 y(Z)1321 3173 y Fu(R)1369 3154 y Ft(N)1447 3083 y Fw(e)1492 3042 y Fs(\000)1557 3014 y Ft(h)p 1557 3026 37 3 v 1562 3068 a(\017)1603 3042 y Fq(K)5 b Fl(\()p Fq(w)r Fs(\000)p Fq(\022)r(y)r Fl(\))1906 3083 y Fw(e)1951 3042 y Fs(\000)2016 3014 y Fr(1)p 2016 3026 31 3 v 2018 3068 a Ft(\017)2056 3042 y Fq(\036)2098 3050 y Ft(n)2141 3042 y Fl(\()p Fq(x)p Fl(+)p Fq(hw)r Fl(+)p Fq(thy)r Fl(\))2563 3083 y Fw(dw)1035 3348 y Fx(\001)1080 3213 y Fv(Z)1135 3438 y Fu(R)1183 3419 y Ft(N)1261 3348 y Fw(e)1306 3307 y Fs(\000)1371 3280 y Ft(h)p 1371 3292 37 3 v 1376 3333 a(\017)1418 3307 y Fq(K)g Fl(\()p Fq(w)r Fl(+)p Fq(\022)r(y)r Fl(\))1720 3348 y Fw(e)1765 3307 y Fs(\000)1830 3280 y Fr(1)p 1830 3292 31 3 v 1832 3333 a Ft(\017)1871 3307 y Fq(\036)1913 3315 y Ft(n)1955 3307 y Fl(\()p Fq(x)p Fl(+)p Fq(hw)r Fs(\000)p Fq(thy)r Fl(\))2377 3348 y Fw(dw)2501 3238 y Fv(\021)875 3614 y Fx(\024)83 b Fw(h)17 b(C)1178 3629 y Fq(U)1237 3614 y FE(\()p Fw(h)p Fx(j)p Fw(y)t Fx(j)p FE(\))1477 3573 y Fl(2)1537 3614 y FE(+)22 b(2)p Fw(\017)17 b FE(ln)1838 3504 y Fv(\020)1914 3479 y(Z)1969 3704 y Fu(R)2017 3685 y Ft(N)2095 3614 y Fw(e)2140 3573 y Fs(\000)2205 3546 y Ft(h)p 2206 3558 37 3 v 2211 3599 a(\017)2252 3573 y Fq(K)5 b Fl(\()p Fq(w)r Fl(\))2427 3614 y Fw(e)2472 3573 y Fs(\000)2537 3546 y Fr(1)p 2537 3558 31 3 v 2539 3599 a Ft(\017)2578 3573 y Fq(\036)2620 3581 y Ft(n)2662 3573 y Fl(\()p Fq(x)p Fl(+)p Fq(hw)r Fl(\))2926 3614 y Fw(dw)3050 3504 y Fv(\021)875 3880 y Fx(\000)83 b FE(2)p Fw(\017)17 b FE(ln)1238 3769 y Fv(\020)1314 3744 y(Z)1369 3970 y Fu(R)1417 3951 y Ft(N)1496 3880 y Fw(e)1541 3839 y Fs(\000)1616 3812 y Ft(h)p 1606 3824 57 3 v 1606 3865 a Fr(2)p Ft(\017)1672 3839 y Fl(\()p Fq(K)5 b Fl(\()p Fq(w)r Fs(\000)p Fq(\022)r(y)r Fl(\)+)p Fq(K)g Fl(\()p Fq(w)r Fl(+)p Fq(\022)r(y)r Fl(\)\))1035 4125 y Fx(\001)p Fw(e)1108 4084 y Fs(\000)1186 4057 y Fr(1)p 1173 4069 V 1173 4110 a(2)p Ft(\017)1240 4084 y Fl(\()p Fq(\036)1309 4092 y Ft(n)1351 4084 y Fl(\()p Fq(x)p Fl(+)p Fq(hw)r Fl(+)p Fq(thy)r Fl(\)+)p Fq(\036)1849 4092 y Ft(n)1892 4084 y Fl(\()p Fq(x)p Fl(+)p Fq(hw)r Fs(\000)p Fq(hty)r Fl(\)\))2341 4125 y Fw(dw)2465 4014 y Fv(\021)2540 4125 y Fw(:)p 456 4240 499 4 v 555 4307 a Fl(4)591 4339 y FH(Note)27 b(that,)h(if)1268 4481 y Fp(\033)e FF(\025)1429 4388 y Fo(\020)1479 4481 y Fp(h)14 b(C)1600 4493 y Fi(U)1674 4481 y FH(+)1757 4381 y Fo(q)p 1840 4381 623 4 v 100 x Fp(h)1888 4457 y Fy(2)1939 4481 y Fp(C)2004 4452 y Fy(2)1998 4505 y Fi(U)2072 4481 y FH(+)k(4)c Fp(C)2270 4493 y Fi(U)2339 4481 y Fp(C)2398 4493 y Fi(K)2463 4388 y Fo(\021)2512 4481 y Fp(=)p FH(2)g Fp(;)456 4635 y FH(then)25 b(the)h(righ)n(t)f(hand)g(side)h(of)f (\(38\))g(is)g(less)g(than)h Fp(\033)s FH(,)g(th)n(us)f(w)n(e)g(impro)n (v)n(ed)f(the)i(semiconca)n(vit)n(y)456 4745 y(b)r(ound)i(from)f(the)h Fp(n)p FH(th)g(step)f(to)h(the)g(\()p Fp(n)18 b FH(+)h(1\)th)27 b(step.)37 b(If,)28 b(on)g(the)g(other)f(hand,)1268 4929 y Fp(\033)f FF(\024)1429 4837 y Fo(\020)1479 4929 y Fp(h)14 b(C)1600 4941 y Fi(U)1674 4929 y FH(+)1757 4829 y Fo(q)p 1840 4829 V 100 x Fp(h)1888 4905 y Fy(2)1939 4929 y Fp(C)2004 4901 y Fy(2)1998 4954 y Fi(U)2072 4929 y FH(+)k(4)c Fp(C)2270 4941 y Fi(U)2339 4929 y Fp(C)2398 4941 y Fi(K)2463 4837 y Fo(\021)2512 4929 y Fp(=)p FH(2)g Fp(;)456 5103 y FH(then)26 b(the)g(righ)n(t)e(hand)i(side)g(of)f(\(38\))g(pro)n(vides)f(a)i (uniform)f(b)r(ound)h(for)f(\006)2760 5115 y Fi(\036)2800 5123 y Fh(n)p Fg(+1)2916 5103 y FH(.)36 b(Observ)-5 b(ations)456 5216 y(of)27 b(this)h(t)n(yp)r(e)g(will)g(pla)n(y)f(a)g(r^)-42 b(ole)26 b(in)i(the)g(pro)r(of)f(of)h(Theorem)f(13)f(here)h(b)r(elo)n (w.)p eop %%Page: 18 18 18 17 bop 456 251 a FC(18)666 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)456 450 y FE(Therefore,)33 b(b)m(y)h(\(9\))e(and)h(\(37\),) 1209 628 y Fw(\036)1267 643 y Fq(n)p Fl(+1)1404 628 y FE(\()p Fw(x)22 b FE(+)g Fw(hy)t FE(\))g(+)g Fw(\036)1941 643 y Fq(n)p Fl(+1)2078 628 y FE(\()p Fw(x)g Fx(\000)h Fw(hy)t FE(\))e Fx(\000)i FE(2)p Fw(\036)2667 643 y Fq(n)p Fl(+1)2803 628 y FE(\()p Fw(x)p FE(\))1048 788 y Fx(\024)84 b Fw(h)17 b(C)1352 803 y Fq(U)1427 788 y FE(\()p Fw(h)p Fx(j)p Fw(y)t Fx(j)p FE(\))1667 747 y Fl(2)1727 788 y FE(+)22 b Fw(h)17 b(C)1968 803 y Fq(K)2053 788 y FE(\()p Fw(\022)s Fx(j)p Fw(y)t Fx(j)p FE(\))2285 747 y Fl(2)2345 788 y FE(+)22 b Fw(\033)e FE(\()p Fw(ht)p Fx(j)p Fw(y)t Fx(j)p FE(\))2793 747 y Fl(2)2848 788 y Fw(;)456 966 y FE(hence,)34 b(since)f Fw(t)28 b FE(=)f(1)22 b Fx(\000)h Fw(\022)s FE(,)1195 1180 y(\006)1265 1195 y Fq(\036)1307 1204 y Ft(n)p Fr(+1)1459 1180 y Fx(\024)28 b Fw(h)17 b(C)1707 1195 y Fq(U)1788 1180 y FE(+)1896 1112 y Fw(C)1966 1127 y Fq(K)p 1896 1157 139 4 v 1937 1248 a Fw(h)2060 1180 y(\022)2108 1139 y Fl(2)2170 1180 y FE(+)22 b Fw(\033)t FE(\(1)g Fx(\000)h Fw(\022)s FE(\))2622 1139 y Fl(2)2678 1180 y Fw(:)456 1381 y FE(The)33 b(desired)g(result)g(follo)m(ws)e(no)m (w)i(b)m(y)h(c)m(ho)s(osing)e Fw(\022)f FE(:=)c Fw(\033)t(h=)p FE(\()p Fw(C)2792 1396 y Fq(K)2883 1381 y FE(+)22 b Fw(\033)t(h)p FE(\).)206 b Fj(\003)555 1565 y FE(W)-8 b(e)43 b(no)m(w)f(get)g(the)g (uniform)f(con)m(trol)g(on)h(\006)2222 1580 y Fq(\036)2264 1588 y Ft(n)2352 1565 y FE(b)m(y)h(a)f(suitable)f(iteration)f(of)456 1693 y(Lemma)31 b(12)h(\(th)m(us)h(impro)m(ving)e(Lemma)g(11)h(in)g (the)h(case)h(of)e(small)e Fw(h)p FE(\):)456 1844 y FI(Theorem)37 b(13)g FE(\(Uniform)31 b(semiconca)m(vit)m(y\))p FI(.)42 b FD(L)-5 b(et)1226 2090 y Fw(\033)1281 2105 y Fq(?)1348 2090 y FE(:=)1489 2021 y Fw(h)17 b(C)1632 2036 y Fq(U)1713 2021 y FE(+)1811 1935 y Fv(p)p 1910 1935 710 4 v 1910 2021 a Fw(h)1966 1992 y Fl(2)2023 2021 y Fw(C)2100 1986 y Fl(2)2093 2048 y Fq(U)2173 2021 y FE(+)23 b(4)17 b Fw(C)2408 2036 y Fq(U)2482 2021 y Fw(C)2552 2036 y Fq(K)p 1489 2067 1132 4 v 2030 2158 a FE(2)2647 2090 y Fw(:)456 2286 y FD(Then,)34 b(for)g(any)h Fw(n)28 b Fx(2)g Fn(N)9 b FD(,)1255 2464 y FE(\006)1325 2479 y Fq(\036)1367 2488 y Ft(n)p Fr(+1)1518 2464 y Fx(\024)29 b FE(max)1822 2384 y Fv(\010)1880 2464 y FE(\006)1950 2479 y Fq(\036)1992 2487 y Ft(n)2056 2464 y Fw(;)44 b(h)17 b(C)2270 2479 y Fq(U)2351 2464 y FE(+)22 b Fw(\033)2504 2479 y Fq(?)2543 2384 y Fv(\011)2618 2464 y Fw(:)456 2642 y FD(In)34 b(p)-5 b(articular,)1297 2776 y FE(\006)1367 2791 y Fq(\036)1409 2799 y Ft(n)1484 2776 y Fx(\024)28 b FE(max)1787 2695 y Fv(\010)1845 2776 y FE(\006)1915 2791 y Fq(\036)1957 2800 y Fr(0)2013 2776 y Fw(;)44 b(h)17 b(C)2227 2791 y Fq(U)2308 2776 y FE(+)22 b Fw(\033)2461 2791 y Fq(?)2501 2695 y Fv(\011)2576 2776 y Fw(:)456 2960 y FD(Pr)-5 b(o)g(of.)41 b FE(The)c(second)g(claim)d(follo)m(ws)h(from)g(the)i(\014rst)g(one,)g (b)m(y)g(a)f(simple)f(itera-)456 3088 y(tion.)k(T)-8 b(o)25 b(pro)m(v)m(e)g(the)g(\014rst)f(claim,)g(w)m(e)h(distinguish)e (t)m(w)m(o)i(cases:)41 b(either)24 b(\006)3128 3103 y Fq(\036)3170 3111 y Ft(n)3244 3088 y Fx(\025)29 b Fw(\033)3405 3103 y Fq(?)456 3215 y FE(or)37 b(\006)650 3230 y Fq(\036)692 3238 y Ft(n)776 3215 y Fx(\024)g Fw(\033)945 3230 y Fq(?)985 3215 y FE(.)59 b(Let)38 b(us)g(deal)f(\014rst)i(with)e(the)h(case)h (\006)2481 3230 y Fq(\036)2523 3238 y Ft(n)2607 3215 y Fx(\025)e Fw(\033)2776 3230 y Fq(?)2815 3215 y FE(.)60 b(Note)38 b(that,)g(if)456 3343 y Fw(\033)31 b Fx(\025)d Fw(\033)702 3358 y Fq(?)742 3343 y FE(,)33 b(then)1475 3512 y Fw(h)17 b(C)1618 3527 y Fq(U)1698 3512 y FE(+)1886 3445 y Fw(C)1956 3460 y Fq(K)2041 3445 y Fw(\033)p 1806 3489 374 4 v 1806 3580 a(C)1876 3595 y Fq(K)1967 3580 y FE(+)22 b Fw(h\033)2217 3512 y Fx(\024)29 b Fw(\033)20 b(:)456 3701 y FE(This)32 b(observ)-5 b(ation)32 b(and)h(Lemma)e(12)h (imply)f(that)1190 3914 y(\006)1260 3929 y Fq(\036)1302 3938 y Ft(n)p Fr(+1)1454 3914 y Fx(\024)d Fw(h)17 b(C)1702 3929 y Fq(U)1783 3914 y FE(+)1971 3847 y Fw(C)2041 3862 y Fq(K)2125 3847 y FE(\006)2195 3862 y Fq(\036)2237 3870 y Ft(n)p 1891 3891 474 4 v 1891 3983 a Fw(C)1961 3998 y Fq(K)2051 3983 y FE(+)22 b Fw(h)p FE(\006)2275 3998 y Fq(\036)2317 4006 y Ft(n)2402 3914 y Fx(\024)28 b FE(\006)2577 3929 y Fq(\036)2619 3937 y Ft(n)2683 3914 y Fw(;)456 4138 y FE(pro)m(ving)47 b(the)h(desired)g(result)f(when)i(\006)1978 4153 y Fq(\036)2020 4161 y Ft(n)2120 4138 y Fx(\025)k Fw(\033)2305 4153 y Fq(?)2345 4138 y FE(.)88 b(If,)51 b(on)c(the)h(other)g(hand,)456 4266 y(\006)526 4281 y Fq(\036)568 4289 y Ft(n)642 4266 y Fx(\024)28 b Fw(\033)802 4281 y Fq(?)842 4266 y FE(,)33 b(Lemma)e(12)h(implies)e(that)1303 4488 y(\006)1373 4503 y Fq(\036)1415 4512 y Ft(n)p Fr(+1)1622 4488 y Fx(\024)83 b Fw(h)17 b(C)1925 4503 y Fq(U)2006 4488 y FE(+)2194 4421 y Fw(C)2264 4436 y Fq(K)2348 4421 y FE(\006)2418 4436 y Fq(\036)2460 4444 y Ft(n)p 2114 4465 V 2114 4556 a Fw(C)2184 4571 y Fq(K)2274 4556 y FE(+)22 b Fw(h)p FE(\006)2498 4571 y Fq(\036)2540 4579 y Ft(n)1622 4694 y Fx(\024)83 b Fw(h)17 b(C)1925 4709 y Fq(U)2006 4694 y FE(+)22 b(\006)2174 4709 y Fq(\036)2216 4717 y Ft(n)1622 4854 y Fx(\024)83 b Fw(h)17 b(C)1925 4869 y Fq(U)2006 4854 y FE(+)22 b Fw(\033)2159 4869 y Fq(?)2215 4854 y Fw(;)456 5032 y FE(whic)m(h)33 b(pro)m(v)m(es)h(the)f (desired)g(result.)1566 b Fj(\003)555 5216 y FE(W)-8 b(e)33 b(are)f(no)m(w)g(in)g(p)s(osition)e(to)i(deduce)h(a)f(uniform)e (Lipsc)m(hitz)i(b)s(ound)h(on)f Fw(\036)3371 5231 y Fq(n)3417 5216 y FE(.)p eop %%Page: 19 19 19 18 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)420 b(19)456 450 y FI(Theorem)35 b(14)h FE(\(Uniform)30 b(Lipsc)m(hitz)h(b)s(ound\))p FI(.)41 b FD(L)-5 b(et)34 b Fw(h)28 b Fx(2)g FE(\(0)p Fw(;)17 b FE(1])p FD(.)43 b(Assume)34 b(that)456 578 y Fw(\036)514 593 y Fl(0)595 578 y FD(is)42 b Fn(Z)776 542 y Fq(N)840 578 y FD(-p)-5 b(erio)g(dic)41 b(and)h(semic)-5 b(onc)g(ave.)64 b(Then,)43 b(ther)-5 b(e)42 b(exists)g Fw(C)7 b FD(,)44 b(dep)-5 b(ending)456 706 y(only)34 b(on)h FE(\006)877 721 y Fq(\036)919 730 y Fr(0)958 706 y FD(,)g Fw(N)10 b FD(,)35 b Fw(C)1246 721 y Fq(U)1340 706 y FD(and)f Fw(C)1599 721 y Fq(K)1702 706 y FD(so)g(that)719 895 y Fx(j)p Fw(\036)805 910 y Fq(n)851 895 y FE(\()p Fw(x)p FE(\))22 b Fx(\000)h Fw(\036)1162 910 y Fq(n)1209 895 y FE(\()p Fw(y)t FE(\))p Fx(j)j(\024)i Fw(C)c Fx(j)p Fw(x)e Fx(\000)h Fw(y)t Fx(j)17 b Fw(;)214 b Fx(8)p Fw(x;)17 b(y)31 b Fx(2)d Fn(R)2526 854 y Fq(N)2616 895 y Fw(;)216 b Fx(8)p Fw(n)28 b Fx(\025)h FE(1)p Fw(:)456 1090 y FD(Pr)-5 b(o)g(of.)41 b FE(By)c(Prop)s(osition)d(5)j(and)f(Theorem)h(13,)g Fw(\036)2354 1105 y Fq(n)2437 1090 y FE(is)f Fn(Z)2608 1054 y Fq(N)2673 1090 y FE(-p)s(erio)s(dic)e(and)i(uni-)456 1218 y(formly)31 b(semiconca)m(v)m(e.)44 b(Then,)34 b(the)f(result)f (follo)m(ws)g(from)f(Theorem)i(44.)205 b Fj(\003)1182 1470 y FE(6.)48 b Fz(Some)38 b(pr)n(oper)-7 b(ties)36 b(of)i Fx(L)g Fz(and)2674 1445 y FE(^)2650 1470 y Fx(L)555 1656 y FE(In)30 b(analogy)f(with)g(the)h(results)g(in)f Fx(x)h FE(5,)g(w)m(e)g(no)m(w)h(p)s(oin)m(t)d(out)i(some)f(prop)s (erties)456 1784 y(of)34 b(the)i(linear)d(op)s(erators)i Fx(L)g FE(and)1767 1758 y(^)1743 1784 y Fx(L)p FE(.)50 b(The)36 b(follo)m(wing)d(ones)i(are)h(quite)f(easy)h(to)456 1912 y(c)m(hec)m(k:)456 2067 y FI(Prop)s(osition)31 b(15.)38 b FD(If)31 b Fw( )1400 2082 y Fl(0)1471 2067 y FD(is)g Fn(Z)1641 2031 y Fq(N)1706 2067 y FD(-p)-5 b(erio)g(dic,)31 b(then)g(so)g(is)g Fw( )2631 2082 y Fq(n)2706 2067 y FE(:=)c Fx(L)2905 2031 y Fq(n)2952 2067 y FE([)p Fw( )3042 2082 y Fl(0)3082 2067 y FE(])k FD(for)g(any)456 2196 y Fw(n)p FD(.)44 b(If)710 2170 y FE(^)690 2196 y Fw( )753 2211 y Fl(0)827 2196 y FD(is)34 b Fn(Z)1000 2160 y Fq(N)1065 2196 y FD(-p)-5 b(erio)g(dic,)33 b(then)h(so)g(is)1958 2170 y FE(^)1938 2196 y Fw( )2001 2211 y Fq(n)2076 2196 y FE(:=)2230 2171 y(^)2206 2196 y Fx(L)p FE([)2322 2170 y(^)2302 2196 y Fw( )2365 2211 y Fl(0)2405 2196 y FE(])g FD(for)g(any)h Fw(n)p FD(.)44 b(If)34 b Fw(U)45 b FD(and)34 b Fw( )3405 2211 y Fl(0)456 2324 y FD(b)-5 b(elongs)27 b(to)h Fw(C)970 2288 y Fl(1)1010 2324 y FE(\()p Fn(R)1114 2288 y Fq(N)1187 2324 y FE(\))p FD(,)h(then)f(so)g(do)-5 b(es)28 b Fw( )1883 2339 y Fq(n)1959 2324 y FD(for)g(any)g Fw(n)p FD(.)43 b(If)2534 2298 y FE(^)2514 2324 y Fw( )2577 2339 y Fl(0)2645 2324 y FD(b)-5 b(elongs)28 b(to)g Fw(C)3160 2288 y Fl(1)3199 2324 y FE(\()p Fn(R)3303 2288 y Fq(N)3377 2324 y FE(\))p FD(,)456 2453 y(then)34 b(so)h(do)-5 b(es)1031 2427 y FE(^)1011 2453 y Fw( )1074 2468 y Fq(n)1156 2453 y FD(for)35 b(any)g Fw(n)p FD(.)456 2609 y FI(Prop)s(osition)46 b(16)i FE(\(Monotonicit)m(y\))p FI(.)d FD(If)e Fw( )48 b Fx(\025)43 b Fw( )2356 2624 y Fs(\003)2396 2609 y FD(,)j(then)d Fx(L)p FE([)p Fw( )t FE(])g Fx(\025)h(L)p FE([)p Fw( )3210 2624 y Fs(\003)3250 2609 y FE(])p FD(.)70 b(If)476 2711 y FE(^)456 2737 y Fw( )31 b Fx(\025)675 2711 y FE(^)655 2737 y Fw( )718 2752 y Fs(\003)758 2737 y FD(,)k(then)1064 2712 y FE(^)1040 2737 y Fx(L)o FE([)1155 2711 y(^)1135 2737 y Fw( )5 b FE(])27 b Fx(\025)1386 2712 y FE(^)1362 2737 y Fx(L)p FE([)1478 2711 y(^)1458 2737 y Fw( )1521 2752 y Fs(\003)1561 2737 y FE(])p FD(.)456 2906 y FI(Prop)s(osition)34 b(17.)41 b Fx(L)p FE([)p Fw( )24 b FE(+)c Fw(r)s FE(])27 b(=)h Fx(L)p FE([)p Fw( )t FE(])20 b(+)g Fw(r)2097 2826 y Fv(R)2144 2941 y Fu(R)2192 2922 y Ft(N)2270 2906 y Fw(e)2315 2869 y Fs(\000)2380 2836 y Ft(h)11 b(L)p Fr(\()p Ft(x;v)r Fr(\))p 2380 2854 224 3 v 2479 2895 a Ft(\017)2635 2906 y Fw(dv)t FD(,)33 b(and)3012 2881 y FE(^)2988 2906 y Fx(L)p FE([)3104 2880 y(^)3084 2906 y Fw( )24 b FE(+)c Fw(r)s FE(])28 b(=)479 3019 y(^)456 3044 y Fx(L)o FE([)571 3018 y(^)551 3044 y Fw( )t FE(])23 b(+)f Fw(r)s FD(,)34 b(for)h(any)g Fw(r)30 b Fx(2)e Fn(R)5 b FD(.)555 3200 y FE(Also,)27 b(it)d(is)i(easy)g(to)g(see)g(that)1678 3175 y(^)1654 3200 y Fx(L)f FE(is)g(a)h(\(non-strict\))e(con)m (traction)i(on)f Fw(L)3127 3164 y Fs(1)3202 3200 y FE(\()p Fn(R)3306 3164 y Fq(N)3379 3200 y FE(\):)456 3356 y FI(Prop)s(osition) 35 b(18.)42 b FD(F)-7 b(or)34 b(any)1635 3330 y FE(^)1615 3356 y Fw( )t(;)1745 3330 y FE(^)1726 3356 y Fw( )1789 3371 y Fs(\003)1856 3356 y Fx(2)28 b Fw(L)2016 3320 y Fs(1)2091 3356 y FE(\()p Fn(R)2195 3320 y Fq(N)2268 3356 y FE(\))p FD(,)1070 3546 y Fx(k)1143 3521 y FE(^)1120 3546 y Fx(L)o FE([)1235 3520 y(^)1215 3546 y Fw( )t FE(])23 b Fx(\000)1455 3521 y FE(^)1431 3546 y Fx(L)p FE([)1547 3520 y(^)1527 3546 y Fw( )1590 3561 y Fs(\003)1630 3546 y FE(])p Fx(k)1707 3565 y Fq(L)1755 3546 y Fm(1)1820 3565 y Fl(\()p Fu(R)1895 3546 y Ft(N)1953 3565 y Fl(\))2012 3546 y Fx(\024)28 b(k)2187 3520 y FE(^)2167 3546 y Fw( )e Fx(\000)2376 3520 y FE(^)2356 3546 y Fw( )2419 3561 y Fs(\003)2459 3546 y Fx(k)2509 3565 y Fq(L)2557 3546 y Fm(1)2622 3565 y Fl(\()p Fu(R)2697 3546 y Ft(N)2755 3565 y Fl(\))2803 3546 y Fw(:)555 3735 y FE(In)33 b(particular:)456 3891 y FI(Prop)s(osition)i(19)j FE(\(Uniform)30 b Fw(L)1699 3854 y Fs(1)1774 3891 y FE(-b)s(ound\))p FI(.)1356 4080 y Fx(k)1430 4055 y FE(^)1406 4080 y Fx(L)o FE([)1521 4054 y(^)1501 4080 y Fw( )5 b FE(])p Fx(k)1646 4099 y Fq(L)1694 4080 y Fm(1)1758 4099 y Fl(\()p Fu(R)1833 4080 y Ft(N)1892 4099 y Fl(\))1951 4080 y Fx(\024)28 b(k)2126 4054 y FE(^)2106 4080 y Fw( )t Fx(k)2223 4099 y Fq(L)2271 4080 y Fm(1)2336 4099 y Fl(\()p Fu(R)2411 4080 y Ft(N)2469 4099 y Fl(\))2517 4080 y Fw(:)555 4269 y FE(Analogously)-8 b(,)456 4425 y FI(Prop)s(osition)27 b(20)j FE(\(Uniform)24 b Fw(C)1688 4389 y Fq(k)1756 4425 y FE(and)i Fw(W)2045 4389 y Fq(k)r(;)p Fs(1)2177 4425 y FE(-b)s(ound\))p FI(.)36 b FD(If)2699 4399 y FE(^)2679 4425 y Fw( )c Fx(2)c Fw(C)2945 4389 y Fq(k)2988 4425 y FE(\()p Fn(R)3092 4389 y Fq(N)3165 4425 y FE(\))p FD(,)i(then)479 4528 y FE(^)456 4554 y Fx(L)o FE([)571 4527 y(^)551 4554 y Fw( )t FE(])e Fx(2)g Fw(C)844 4517 y Fq(k)887 4554 y FE(\()p Fn(R)991 4517 y Fq(N)1064 4554 y FE(\))35 b FD(and)1376 4743 y Fx(k)1449 4718 y FE(^)1426 4743 y Fx(L)o FE([)1541 4717 y(^)1521 4743 y Fw( )t FE(])p Fx(k)1665 4762 y Fq(C)1720 4744 y Ft(k)1758 4762 y Fl(\()p Fu(R)1833 4744 y Ft(N)1892 4762 y Fl(\))1951 4743 y Fx(\024)28 b(k)2126 4717 y FE(^)2106 4743 y Fw( )t Fx(k)2223 4762 y Fq(C)2278 4744 y Ft(k)2316 4762 y Fl(\()p Fu(R)2391 4744 y Ft(N)2449 4762 y Fl(\))2497 4743 y Fw(:)456 4932 y FD(A)n(n)34 b(analo)-5 b(gous)34 b(statement)h(holds)f(by)h(r)-5 b(eplacing)34 b Fw(C)2375 4896 y Fq(k)2452 4932 y FD(with)h Fw(W)2770 4896 y Fq(k)r(;)p Fs(1)2902 4932 y FD(.)555 5088 y FE(W)-8 b(e)31 b(no)m(w)g(obtain)e(a)h (\(non-uniform)e(in)i Fw(h)g FE(and)g Fw(\017)p FE(\))h Fw(C)2463 5052 y Fl(1)2502 5088 y FE(-b)s(ound)f(of)g Fx(L)p FE([)p Fw( )t FE(])g(for)g(the)456 5216 y(dep)s(endence)35 b(of)d(the)h Fw(L)1324 5179 y Fs(1)1399 5216 y FE(-norm)e(of)h Fw( )t FE(:)p eop %%Page: 20 20 20 19 bop 456 251 a FC(20)666 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)456 450 y FI(Prop)s(osition)35 b(21)j FE(\(Impro)m(v)m(emen) m(t)33 b(of)f(regularit)m(y\))p FI(.)40 b FD(Assume)35 b(that)456 690 y FE(\(39\))1283 554 y Fv(Z)1338 780 y Fu(R)1386 761 y Ft(N)1464 690 y Fw(e)1509 649 y Fs(\000)1574 615 y Ft(hK)s Fr(\()p Ft(v)r Fr(\))p 1574 633 172 3 v 1647 675 a Ft(\017)1760 690 y Fx(j)p Fw(D)s(K)7 b FE(\()p Fw(v)t FE(\))p Fx(j)17 b Fw(dv)46 b(<)e FE(+)p Fx(1)17 b Fw(:)456 928 y FD(Then,)764 1093 y Fv(\015)764 1153 y(\015)764 1212 y(\015)820 1207 y Fw(D)s FE(\()965 1182 y(^)942 1207 y Fx(L)o FE([)p Fw( )t FE(]\))1169 1093 y Fv(\015)1169 1153 y(\015)1169 1212 y(\015)1224 1276 y Fq(L)1272 1257 y Fm(1)1337 1276 y Fl(\()p Fu(R)1412 1257 y Ft(N)1471 1276 y Fl(\))1546 1207 y Fx(\024)45 b(k)p Fw( )t Fx(k)1835 1226 y Fq(L)1883 1207 y Fm(1)1948 1226 y Fl(\()p Fu(R)2023 1207 y Ft(N)2081 1226 y Fl(\))2139 1054 y Fv(R)2186 1168 y Fu(R)2234 1150 y Ft(N)2313 1134 y Fw(e)2358 1097 y Fs(\000)2423 1064 y Ft(hK)s Fr(\()p Ft(v)r Fr(\))p 2423 1082 V 2495 1123 a Ft(\017)2608 1134 y Fx(j)p Fw(D)s(K)7 b FE(\()p Fw(v)t FE(\))p Fx(j)17 b Fw(dv)p 2139 1184 944 4 v 2281 1304 a(\017)2353 1224 y Fv(R)2400 1339 y Fu(R)2448 1320 y Ft(N)2527 1304 y Fw(e)2572 1267 y Fs(\000)2637 1234 y Ft(hK)s Fr(\()p Ft(v)r Fr(\))p 2637 1252 172 3 v 2710 1293 a Ft(\017)2839 1304 y Fw(dv)3109 1207 y(;)456 1499 y FD(for)34 b(any)h Fw( )d Fx(2)c Fw(L)1053 1463 y Fs(1)1128 1499 y FE(\()p Fn(R)1232 1463 y Fq(N)1305 1499 y FE(\))p FD(.)456 1659 y Fz(Remark.)86 b FE(F)-8 b(or)32 b Fw(K)7 b FE(\()p Fw(v)t FE(\))28 b(=)f Fx(j)p Fw(v)t Fx(j)1598 1623 y Fl(2)1637 1659 y FE(,)32 b(the)h(result)g(in)f(Prop)s(osition)e(21)j (yields)f(that)1116 1797 y Fv(\015)1116 1857 y(\015)1116 1917 y(\015)1171 1912 y Fw(D)s FE(\()1317 1887 y(^)1293 1912 y Fx(L)p FE([)p Fw( )t FE(]\))1521 1797 y Fv(\015)1521 1857 y(\015)1521 1917 y(\015)1576 1981 y Fq(L)1624 1962 y Fm(1)1689 1981 y Fl(\()p Fu(R)1764 1962 y Ft(N)1822 1981 y Fl(\))1898 1912 y Fx(\024)2047 1838 y FE(const)17 b Fx(k)p Fw( )t Fx(k)2453 1857 y Fq(L)2501 1838 y Fm(1)2566 1857 y Fl(\()p Fu(R)2641 1838 y Ft(N)2699 1857 y Fl(\))p 2030 1889 701 4 v 2291 1909 a Fx(p)p 2374 1909 96 4 v 85 x Fw(\017h)2757 1912 y(:)456 2161 y FD(Pr)-5 b(o)g(of.)41 b FE(By)33 b(c)m(hanging)f(v)-5 b(ariable,)31 b(w)m(e)j(ha)m(v)m(e)f (that)1156 2406 y(^)1132 2431 y Fx(L)p FE([)p Fw( )t FE(]\()p Fw(x)p FE(\))45 b(=)1628 2277 y Fv(R)1675 2392 y Fu(R)1723 2373 y Ft(N)1802 2358 y Fw(e)1847 2321 y Fs(\000)1912 2294 y Ft(h)p 1912 2306 37 3 v 1917 2347 a(\017)1958 2321 y Fq(K)5 b Fl(\(\()p Fq(w)r Fs(\000)p Fq(x)p Fl(\))p Fq(=h)p Fl(\))2359 2358 y Fw( )t FE(\()p Fw(w)s FE(\))17 b Fw(dw)p 1628 2408 1086 4 v 1799 2528 a(h)1855 2499 y Fq(N)1956 2448 y Fv(R)2003 2563 y Fu(R)2051 2544 y Ft(N)2129 2528 y Fw(e)2174 2491 y Fs(\000)2239 2458 y Ft(hK)s Fr(\()p Ft(v)r Fr(\))p 2239 2476 172 3 v 2312 2517 a Ft(\017)2441 2528 y Fw(dv)2740 2431 y(:)456 2711 y FE(Therefore,)643 2867 y Fv(\014)643 2926 y(\014)643 2986 y(\014)676 2981 y Fw(D)757 2996 y Fq(x)801 2981 y FE(\()863 2956 y(^)839 2981 y Fx(L)p FE([)p Fw( )t FE(]\))1067 2867 y Fv(\014)1067 2926 y(\014)1067 2986 y(\014)1183 2981 y Fx(\024)1354 2827 y Fv(R)1401 2942 y Fu(R)1449 2923 y Ft(N)1527 2908 y Fw(e)1572 2871 y Fs(\000)1637 2844 y Ft(h)p 1637 2856 37 3 v 1642 2897 a(\017)1683 2871 y Fq(K)5 b Fl(\(\()p Fq(w)r Fs(\000)p Fq(x)p Fl(\))p Fq(=h)p Fl(\))2084 2908 y Fx(j)p Fw(D)s(K)i FE(\(\()p Fw(w)24 b Fx(\000)f Fw(x)p FE(\))p Fw(=h)p FE(\))p Fx(j)17 b(j)p Fw( )t FE(\()p Fw(w)s FE(\))p Fx(j)g Fw(dw)p 1354 2958 1894 4 v 1909 3078 a(\017h)2004 3049 y Fq(N)2105 2998 y Fv(R)2152 3112 y Fu(R)2200 3093 y Ft(N)2278 3078 y Fw(e)2323 3041 y Fs(\000)2388 3007 y Ft(hK)s Fr(\()p Ft(v)r Fr(\))p 2388 3026 172 3 v 2461 3067 a Ft(\017)2590 3078 y Fw(dv)1184 3325 y FE(=)1354 3171 y Fv(R)1401 3286 y Fu(R)1449 3267 y Ft(N)1527 3251 y Fw(e)1572 3214 y Fs(\000)1637 3187 y Ft(h)p 1637 3199 37 3 v 1642 3240 a(\017)1683 3214 y Fq(K)5 b Fl(\()p Fq(v)r Fl(\))1843 3251 y Fx(j)p Fw(D)s(K)i FE(\()p Fw(v)t FE(\))p Fx(j)17 b(j)p Fw( )t FE(\()p Fw(x)k FE(+)h Fw(hv)t FE(\))p Fx(j)17 b Fw(dv)p 1354 3302 1461 4 v 1754 3421 a(\017)1827 3341 y Fv(R)1874 3456 y Fu(R)1922 3437 y Ft(N)2000 3421 y Fw(e)2045 3385 y Fs(\000)2110 3351 y Ft(hK)s Fr(\()p Ft(v)r Fr(\))p 2110 3369 172 3 v 2183 3411 a Ft(\017)2313 3421 y Fw(dv)2841 3325 y(;)456 3604 y FE(whic)m(h)33 b(implies)d(the)j(desired)g(result.)1536 b Fj(\003)555 3811 y FE(F)-8 b(urther)36 b(regularit)m(y)e(results)i (can)g(b)s(e)g(also)e(deduced)k(in)c(an)i(analogous)e(w)m(a)m(y)456 3939 y(from)44 b(Prop)s(osition)f(21.)81 b(Also,)48 b(it)d(is)f (straigh)m(tforw)m(ard)h(to)g(see)i(that)e(that)3399 3913 y(^)3376 3939 y Fx(L)456 4066 y FE(preserv)m(es)35 b(the)e(a)m(v)m(erage)h(o)m(v)m(er)f Fn(T)1663 4030 y Fq(N)1766 4066 y FE(of)f Fn(Z)1946 4030 y Fq(N)2011 4066 y FE(-p)s(erio)s(dic)f(function,)h(namely:)456 4226 y FI(Prop)s(osition)j(22.)42 b FD(If)34 b Fw( )e Fx(2)c Fw(L)1603 4190 y Fl(1)1643 4226 y FE(\()p Fn(T)1744 4190 y Fq(N)1814 4226 y FE(\))p FD(,)35 b(then)1539 4330 y Fv(Z)1595 4556 y Fu(T)1645 4537 y Ft(N)1741 4441 y FE(^)1717 4466 y Fx(L)p FE([)p Fw( )t FE(])45 b(=)2072 4330 y Fv(Z)2127 4556 y Fu(T)2177 4537 y Ft(N)2250 4466 y Fw( )20 b(:)555 4704 y FE(W)-8 b(e)31 b(remark)g(that)g(the)g(ab)s(o)m(v)m(e)g(b)s (ounds)h(and)f(the)g(Theorem)g(of)g(Ascoli)e(imply)456 4832 y(the)j(uniform)e(con)m(v)m(ergence,)35 b(up)d(to)g(a)g (subsequence,)k(of)31 b(the)i(iteration)3168 4807 y(^)3145 4832 y Fx(L)3214 4796 y Fq(n)3260 4832 y FE([)p Fw( )3350 4847 y Fl(0)3390 4832 y FE(].)456 4960 y(In)26 b(particular,)f(if)g Fw( )1188 4975 y Fl(0)1253 4960 y FE(is)g Fn(Z)1413 4924 y Fq(N)1478 4960 y FE(-p)s(erio)s(dic,)g(Prop)s(ositions)g(20)g(\(or)g (21\))g(and)h(22)f(imply)456 5088 y(that)689 5063 y(^)665 5088 y Fx(L)734 5052 y Fq(n)781 5088 y FE([)p Fw( )871 5103 y Fl(0)911 5088 y FE(])31 b(con)m(v)m(erges,)i(when)f Fw(n)c Fx(!)f FE(+)p Fx(1)p FE(,)k(up)g(to)g(a)f(subsequence,)35 b(to)c(a)f(\014xed)456 5216 y(p)s(oin)m(t)h(of)845 5190 y(^)821 5216 y Fx(L)i FE(with)f(the)h(same)f Fn(T)1620 5179 y Fq(N)1691 5216 y FE(-a)m(v)m(erage)h(of)f Fw( )2247 5231 y Fl(0)2287 5216 y FE(.)p eop %%Page: 21 21 21 20 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)420 b(21)1363 450 y FE(7.)49 b Fz(F)m(ormal)38 b(Asymptotics)555 636 y FE(In)46 b(this)f(section,)k(w)m(e)e(discuss)g(the)f(formal)d (asymptotic)i(b)s(eha)m(vior)g(of)g(the)456 764 y(sc)m(hemes)i(corresp) s(onding)f(to)1638 739 y(^)1614 764 y Fx(L)p FE(,)k Fx(L)45 b FE(and)i Fx(G)6 b FE(.)84 b(These)48 b(computations)d(pla)m(y)h(a)456 892 y(crucial)31 b(r^)-49 b(ole)32 b(in)f(the)i(rigorous)f(con)m(v)m (ergence)j(results)e(giv)m(en)g(in)f Fx(x)g FE(9.)555 1020 y(W)-8 b(e)39 b(deal)e(with)h(di\013eren)m(t)g(c)m(hoices)h(of)e (the)i(parameters)f Fw(\017)g FE(and)g Fw(h)p FE(.)60 b(Elliptic,)456 1147 y(parab)s(olic)40 b(and)k(Hamilton-Jacobi-t)m(yp)s (e)39 b(equations)44 b(will)c(sho)m(w)k(up)g(in)e(these)456 1275 y(asymptotics.)456 1428 y FI(Prop)s(osition)35 b(23)j FE(\(Asymptotics)32 b(for)1986 1403 y(^)1962 1428 y Fx(L)p FE(\))p FI(.)42 b FD(L)-5 b(et)805 1697 y Fw(b)846 1712 y Fq(i)903 1697 y FE(:=)1043 1543 y Fv(R)1090 1658 y Fu(R)1138 1639 y Ft(N)1217 1624 y Fw(e)1262 1587 y Fs(\000)1327 1553 y Ft(hK)s Fr(\()p Ft(v)r Fr(\))p 1327 1572 172 3 v 1399 1613 a Ft(\017)1512 1624 y Fw(v)1559 1639 y Fq(i)1588 1624 y Fw(dv)p 1043 1674 646 4 v 1081 1714 a Fv(R)1128 1828 y Fu(R)1176 1810 y Ft(N)1254 1794 y Fw(e)1299 1757 y Fs(\000)1364 1724 y Ft(hK)s Fr(\()p Ft(v)r Fr(\))p 1364 1742 172 3 v 1437 1783 a Ft(\017)1550 1794 y Fw(dv)1716 1697 y(;)216 b(a)2010 1712 y Fq(ij)2098 1697 y FE(:=)2239 1624 y Fw(h)2312 1543 y Fv(R)2358 1658 y Fu(R)2406 1639 y Ft(N)2485 1624 y Fw(e)2530 1587 y Fs(\000)2595 1553 y Ft(hK)s Fr(\()p Ft(v)r Fr(\))p 2595 1572 V 2668 1613 a Ft(\017)2781 1624 y Fw(v)2828 1639 y Fq(i)2856 1624 y Fw(v)2903 1639 y Fq(j)2940 1624 y Fw(dv)p 2239 1674 803 4 v 2322 1794 a FE(2)2388 1714 y Fv(R)2434 1828 y Fu(R)2482 1810 y Ft(N)2561 1794 y Fw(e)2606 1757 y Fs(\000)2671 1724 y Ft(hK)s Fr(\()p Ft(v)r Fr(\))p 2671 1742 172 3 v 2744 1783 a Ft(\017)2856 1794 y Fw(dv)3068 1697 y(:)456 1966 y FD(Supp)-5 b(ose)34 b(that)h Fx(j)p Fw(b)1095 1981 y Fq(i)1123 1966 y Fx(j)22 b FE(+)g Fx(j)p Fw(a)1350 1981 y Fq(ij)1411 1966 y Fx(j)27 b Fw(<)g FE(+)p Fx(1)p FD(.)45 b(L)-5 b(et)1666 2126 y FE(^)1646 2153 y Fw( )1709 2168 y Fq(n)p Fl(+1)1874 2153 y FE(=)2001 2127 y(^)1977 2153 y Fx(L)p FE([)2093 2126 y(^)2073 2153 y Fw( )2136 2168 y Fq(n)2184 2153 y FE(])17 b Fw(:)456 2335 y FD(Then,)456 2581 y FE(\(40\))915 2487 y(^)895 2513 y Fw( )958 2528 y Fq(n)p Fl(+1)1117 2513 y Fx(\000)1237 2487 y FE(^)1217 2513 y Fw( )1280 2528 y Fq(n)p 895 2558 433 4 v 1083 2649 a Fw(h)1365 2581 y FE(=)28 b Fw(b)1510 2596 y Fq(i)1538 2581 y Fw(D)1619 2596 y Fq(x)1659 2606 y Ft(i)1709 2554 y FE(^)1690 2581 y Fw( )1753 2596 y Fq(n)1822 2581 y FE(+)22 b Fw(a)1971 2596 y Fq(ij)2032 2581 y Fw(D)2116 2539 y Fl(2)2113 2605 y Fq(x)2153 2615 y Ft(i)2179 2605 y Fq(x)2219 2615 y Ft(j)2275 2554 y FE(^)2255 2581 y Fw( )2318 2596 y Fq(n)2388 2581 y FE(+)g FD(err)-5 b(or)35 b(terms)16 b Fw(:)456 2781 y FD(The)34 b(err)-5 b(or)35 b(terms)f(c)-5 b(an)35 b(b)-5 b(e)34 b(estimate)-5 b(d)35 b(by:)1149 3041 y FE(const)17 b Fw(h)1444 2999 y Fl(2)1484 3041 y Fx(k)p Fw(D)1618 2999 y Fl(3)1677 3014 y FE(^)1657 3041 y Fw( )1720 3056 y Fq(n)1767 3041 y Fx(k)1817 3059 y Fq(L)1865 3040 y Fm(1)1930 3059 y Fl(\()p Fu(R)2005 3040 y Ft(N)2063 3059 y Fl(\))2105 2891 y Fv(R)2188 2971 y Fx(j)p Fw(v)t Fx(j)2295 2935 y Fl(3)2333 2971 y Fw(e)2378 2934 y Fs(\000)2443 2901 y Ft(hK)s Fr(\()p Ft(v)r Fr(\))p 2443 2919 172 3 v 2516 2961 a Ft(\017)2629 2971 y Fw(dv)p 2105 3018 626 4 v 2178 3057 a Fv(R)2261 3137 y Fw(e)2306 3101 y Fs(\000)2371 3067 y Ft(hK)s Fr(\()p Ft(v)r Fr(\))p 2371 3085 172 3 v 2443 3127 a Ft(\017)2556 3137 y Fw(dv)2740 3041 y(:)456 3305 y Fz(Remark.)118 b FE(The)40 b(left)f(hand)i(side)f (of)f(\(40\))g(ma)m(y)h(b)s(e)g(though)m(t)g(as)g(a)g(discrete)456 3434 y(appro)m(ximation)e(of)i Fw(@)1284 3449 y Fq(t)1334 3408 y FE(^)1314 3434 y Fw( )k FE(as)d Fw(h)g Fx(!)g FE(0.)67 b(Though)41 b(the)g(co)s(e\016cien)m(ts)h(in)d(\(40\))h(de-) 456 3562 y(p)s(end)46 b(on)g(the)g(sp)s(eci\014c)h(form)d(of)i(the)g (kinetic)f(energy)i(and)f(on)g(the)g(relativ)m(e)456 3690 y(scale)40 b(of)h Fw(\017)g FE(and)g Fw(h)p FE(,)i(it)d(ma)m(y)h (b)s(e)g(useful)g(to)f(w)m(ork)i(out)f(the)g(case)h Fw(K)7 b FE(\()p Fw(v)t FE(\))41 b(=)h Fx(j)p Fw(v)t Fx(j)3406 3654 y Fl(2)456 3818 y FE(someho)m(w)47 b(explicitly)-8 b(.)85 b(In)48 b(this)e(case,)52 b(the)47 b(expressions)i(in)e(Prop)s (osition)e(23)456 3946 y(b)s(ecome)g Fw(b)859 3961 y Fq(i)938 3946 y FE(=)50 b(0,)e Fw(a)1239 3961 y Fq(ij)1350 3946 y FE(=)67 b(const)17 b Fw(\017)g(\016)1831 3961 y Fq(ij)1937 3946 y FE(and)46 b(the)g(error)g(term)f(is)g(b)s(ounded)h (b)m(y)456 4073 y(const)17 b Fw(h)751 4037 y Fl(1)p Fq(=)p Fl(2)861 4073 y Fw(\017)900 4037 y Fl(3)p Fq(=)p Fl(2)1011 4073 y Fx(k)p Fw(D)1145 4037 y Fl(3)1183 4073 y Fw( )1246 4088 y Fq(n)1294 4073 y Fx(k)1344 4088 y Fq(L)1392 4069 y Fm(1)1461 4073 y FE(.)456 4262 y FD(Pr)-5 b(o)g(of.)41 b FE(W)-8 b(e)33 b(ha)m(v)m(e)486 4469 y(^)466 4495 y Fw( )529 4510 y Fq(n)p Fl(+1)688 4495 y Fx(\000)808 4469 y FE(^)788 4495 y Fw( )851 4510 y Fq(n)p 466 4540 433 4 v 654 4631 a Fw(h)936 4563 y FE(=)1049 4379 y Fv(R)1132 4459 y Fw(e)1177 4422 y Fs(\000)1242 4389 y Ft(hK)s Fr(\()p Ft(v)r Fr(\))p 1243 4407 172 3 v 1315 4449 a Ft(\017)1445 4349 y Fv(h)1512 4433 y FE(^)1492 4459 y Fw( )1555 4474 y Fq(n)1602 4459 y FE(\()p Fw(x)23 b FE(+)f Fw(hv)t FE(\))g Fx(\000)2102 4433 y FE(^)2082 4459 y Fw( )2145 4474 y Fq(n)2193 4459 y FE(\()p Fw(x)p FE(\))2324 4349 y Fv(i)2388 4459 y Fw(dv)p 1049 4540 1440 4 v 1493 4660 a(h)1566 4579 y Fv(R)1649 4660 y Fw(e)1694 4623 y Fs(\000)1759 4589 y Ft(hK)s Fr(\()p Ft(v)r Fr(\))p 1759 4608 172 3 v 1831 4649 a Ft(\017)1944 4660 y Fw(dv)936 4875 y FE(=)27 b Fw(c)1081 4890 y Fq(\017;h)1191 4734 y Fv(\024)1244 4739 y(Z)1360 4875 y Fw(e)1405 4834 y Fs(\000)1470 4800 y Ft(hK)s Fr(\()p Ft(v)r Fr(\))p 1470 4819 V 1543 4860 a Ft(\017)1656 4875 y Fw(v)1703 4890 y Fq(i)1731 4875 y Fw(D)1812 4890 y Fq(x)1852 4900 y Ft(i)1902 4848 y FE(^)1882 4875 y Fw( )1945 4890 y Fq(n)1993 4875 y FE(\()p Fw(x)p FE(\))p Fw(dv)e FE(+)2355 4807 y Fw(h)p 2355 4852 57 4 v 2359 4943 a FE(2)2438 4739 y Fv(Z)2554 4875 y Fw(e)2599 4834 y Fs(\000)2664 4800 y Ft(hK)s Fr(\()p Ft(v)r Fr(\))p 2664 4819 172 3 v 2737 4860 a Ft(\017)2850 4875 y Fw(v)2897 4890 y Fq(i)2925 4875 y Fw(v)2972 4890 y Fq(j)3009 4875 y Fw(D)3093 4834 y Fl(2)3090 4899 y Fq(x)3130 4909 y Ft(i)3156 4899 y Fq(x)3196 4909 y Ft(j)3252 4848 y FE(^)3233 4875 y Fw( )3296 4890 y Fq(n)3343 4875 y FE(\()p Fw(x)p FE(\))p Fw(dv)3576 4734 y Fv(\025)1028 5154 y FE(+)d Fw(O)1220 5014 y Fv(\022)1293 5154 y Fw(h)1349 5113 y Fl(2)1389 5154 y Fw(c)1431 5169 y Fq(\017;h)1524 5154 y Fx(k)p Fw(D)1658 5113 y Fl(3)1697 5154 y Fw( )1760 5169 y Fq(n)1807 5154 y Fx(k)1857 5169 y Fs(1)1948 5019 y Fv(Z)2064 5154 y Fx(j)p Fw(v)t Fx(j)2171 5113 y Fl(3)2210 5154 y Fw(e)2255 5113 y Fs(\000)2320 5079 y Ft(hK)s Fr(\()p Ft(v)r Fr(\))p 2320 5098 V 2393 5139 a Ft(\017)2506 5154 y Fw(dv)2608 5014 y Fv(\023)2697 5154 y Fw(;)p eop %%Page: 22 22 22 21 bop 456 251 a FC(22)666 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)456 450 y FE(with)1439 658 y Fw(c)1481 673 y Fq(\017;h)1601 658 y FE(=)1705 518 y Fv(\024)1757 523 y(Z)1874 658 y Fw(e)1919 617 y Fs(\000)1984 584 y Ft(hK)s Fr(\()p Ft(v)r Fr(\))p 1984 602 172 3 v 2056 643 a Ft(\017)2169 658 y Fw(dv)2271 518 y Fv(\025)2323 540 y Fs(\000)p Fl(1)2434 658 y Fw(:)3367 890 y Fj(\003)555 1111 y FE(As)34 b(an)f(application)d (of)j(this)g(theorem,)g(w)m(e)h(study)g(the)g(\014xed)g(p)s(oin)m(ts) 3129 1072 y Fl(5)3197 1111 y FE(of)f(the)456 1239 y(sc)m(heme)h (\(15\).)44 b(Assume,)34 b(for)e(the)i(momen)m(t,)e(that)g Fw(h)d FE(=)f Fw(\017)p FE(.)45 b(Observ)m(e)35 b(that)e(if)e(w)m(e)456 1367 y(de\014ne)1505 1550 y Fw(a)1556 1565 y Fq(ij)1644 1550 y FE(=)1758 1482 y(1)p 1758 1527 49 4 v 1758 1618 a(2)1833 1414 y Fv(Z)1949 1550 y Fw(e)1994 1509 y Fs(\000)p Fq(K)5 b Fl(\()p Fq(v)r Fl(\))2209 1550 y Fw(v)2256 1565 y Fq(i)2285 1550 y Fw(v)2332 1565 y Fq(j)2368 1550 y Fw(;)456 1774 y FE(the)33 b(op)s(erator)1810 1941 y Fw(a)1861 1956 y Fq(ij)1922 1941 y Fw(@)1973 1956 y Fq(i)2002 1941 y Fw(@)2053 1956 y Fq(j)456 2130 y FE(is)f(\(p)s(ossibly)g (degenerate\))h(elliptic,)d(as)835 2383 y Fw(a)886 2398 y Fq(ij)947 2383 y Fw(\030)990 2398 y Fq(i)1018 2383 y Fw(\030)1061 2398 y Fq(j)1124 2383 y FE(=)1238 2316 y(1)p 1238 2360 V 1238 2451 a(2)1313 2247 y Fv(Z)1429 2383 y Fw(e)1474 2342 y Fs(\000)p Fq(K)5 b Fl(\()p Fq(v)r Fl(\))1690 2383 y Fw(v)1737 2398 y Fq(i)1765 2383 y Fw(\030)1808 2398 y Fq(i)1836 2383 y Fw(v)1883 2398 y Fq(j)1919 2383 y Fw(\030)1962 2398 y Fq(j)2026 2383 y FE(=)2140 2316 y(1)p 2140 2360 V 2140 2451 a(2)2215 2247 y Fv(Z)2331 2383 y Fw(e)2376 2342 y Fs(\000)p Fq(K)g Fl(\()p Fq(v)r Fl(\))2591 2383 y Fx(j)p Fw(v)26 b Fx(\001)c Fw(\030)5 b Fx(j)2818 2342 y Fl(2)2884 2383 y Fx(\025)28 b FE(0)p Fw(:)456 2630 y FE(F)-8 b(urthermore,)31 b(if)f Fw(K)38 b FE(is)31 b(ev)m(en,)i(then)f Fw(b)1863 2645 y Fq(i)1920 2630 y FE(=)27 b(0.)43 b(In)32 b(this)f(case,)h(the)g(\014xed)g(p)s (oin)m(ts)f Fw( )456 2758 y FE(of)h(the)h(sc)m(heme)h(\(15\))d(satisfy) 1428 2969 y Fw(a)1479 2984 y Fq(ij)1540 2969 y Fw(@)1591 2984 y Fq(i)1620 2969 y Fw(@)1671 2984 y Fq(j)1708 2969 y Fw( )g FE(=)d Fw(O)s FE(\()p Fw(h)p Fx(k)p Fw(D)2212 2928 y Fl(3)2250 2969 y Fw( )t Fx(k)2367 2928 y Fl(2)2407 2969 y FE(\))p Fw(;)456 3181 y FE(as)k(it)g(follo)m(ws)f(from)h(the)h (estimate)e(of)h(Prop)s(osition)f(23.)456 3347 y FI(Prop)s(osition)i (24)i FE(\(Asymptotics)c(for)f Fx(L)p FE(\))p FI(.)40 b FD(L)-5 b(et)34 b Fw(b)2340 3362 y Fq(i)2402 3347 y FD(and)e Fw(a)2640 3362 y Fq(ij)2734 3347 y FD(b)-5 b(e)33 b(as)g(in)g(Pr)-5 b(op)g(osi-)456 3475 y(tion)34 b(23.)45 b(Supp)-5 b(ose)34 b(that)1646 3642 y Fw( )1709 3657 y Fq(n)p Fl(+1)1874 3642 y FE(=)27 b Fx(L)p FE([)p Fw( )2136 3657 y Fq(n)2184 3642 y FE(])17 b Fw(:)p 456 3779 499 4 v 555 3850 a Fl(5)591 3882 y FH(Note)27 b(that)i(an)n(y)e(constan)n (t)g(is)h(a)f(\014xed)h(p)r(oin)n(t)g(for)2182 3861 y(^)2163 3882 y FF(L)p FH(.)38 b(Moreo)n(v)n(er,)25 b(if)k Fp(K)k FH(is)28 b(ev)n(en)f(in)h(an)n(y)g(of)456 3991 y(eac)n(h)e(v)-5 b(ariables,)27 b(giv)n(en)g(a)g(quadratic)f(p)r(olynomial)1487 4173 y Fp(p)p FH(\()p Fp(x)p FH(\))e(=)e Fp(b)c FH(+)g Fp(`)g FF(\001)h Fp(x)g FH(+)f Fp(M)9 b(x)18 b FF(\001)h Fp(x)14 b(;)456 4354 y FH(with)29 b Fp(b)d FF(2)g Ff(R)p FH(,)36 b Fp(`)25 b FF(2)h Ff(R)1097 4324 y Fi(N)1196 4354 y FH(and)j Fp(M)34 b FF(2)40 b FH(Mat\()p Fp(N)29 b FF(\002)19 b Fp(N)9 b FH(\),)30 b(one)e(easily)h(sees)f(that)i Fp(p)f FH(is)g(a)g(\014xed)g(p)r(oin)n(t)456 4464 y(for)602 4443 y(^)583 4464 y FF(L)f FH(if)g(and)f(only)g(if)1738 4607 y Fp(a)1782 4619 y Fi(j)1817 4607 y Fp(@)1866 4573 y Fy(2)1861 4627 y Fi(j)1903 4607 y Fp(p)37 b FH(=)f(0)14 b Fp(;)456 4769 y FH(where)1490 4966 y Fp(a)1534 4978 y Fi(j)1606 4966 y FH(:=)1740 4834 y Fo(R)1780 4931 y Fe(R)1827 4914 y Fh(N)1893 4901 y Fp(e)1932 4871 y Fd(\000)1993 4842 y Fh(hK)s Fg(\()p Fh(v)q Fg(\))p 1993 4858 161 3 v 2061 4891 a Fh(\017)2182 4901 y Fp(v)2225 4871 y Fy(2)2222 4923 y Fi(j)2277 4901 y Fp(dv)p 1740 4947 623 4 v 1788 4980 a Fo(R)1827 5076 y Fe(R)1874 5059 y Fh(N)1940 5046 y Fp(e)1979 5018 y Fd(\000)2041 4988 y Fh(hK)s Fg(\()p Fh(v)q Fg(\))p 2041 5004 161 3 v 2108 5038 a Fh(\017)2229 5046 y Fp(dv)2387 4966 y(:)456 5216 y FH(In)27 b(particular,)g(for)g Fp(K)6 b FH(\()p Fp(v)s FH(\))23 b(=)g FF(j)p Fp(v)s FF(j)1477 5185 y Fy(2)1514 5216 y FH(,)28 b Fp(p)g FH(is)f(a)g(\014xed) h(p)r(oin)n(t)g(of)2319 5195 y(^)2299 5216 y FF(L)g FH(if)g(and)g(only) f(if)h(it)g(is)f(harmonic.)p eop %%Page: 23 23 23 22 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)420 b(23)456 450 y FD(Then,)853 611 y Fw( )916 626 y Fq(n)p Fl(+1)1076 611 y Fx(\000)22 b Fw( )1238 626 y Fq(n)p 853 655 433 4 v 1041 746 a Fw(h)1379 678 y FE(=)83 b Fx(\000)1625 611 y Fw(U)p 1625 655 77 4 v 1644 746 a(\017)1712 678 y( )1775 693 y Fq(n)1379 931 y FE(+)1538 791 y Fv(\022)1611 931 y FE(1)22 b Fx(\000)1792 864 y Fw(hU)p 1792 908 133 4 v 1838 999 a(\017)1934 791 y Fv(\023)2024 821 y(\020)2084 931 y Fw(b)2125 946 y Fq(i)2153 931 y Fw(D)2234 946 y Fq(x)2274 956 y Ft(i)2305 931 y Fw( )2368 946 y Fq(n)2437 931 y FE(+)g Fw(a)2586 946 y Fq(ij)2647 931 y Fw(D)2731 890 y Fl(2)2728 956 y Fq(x)2768 966 y Ft(i)2794 956 y Fq(x)2834 966 y Ft(j)2870 931 y Fw( )2933 946 y Fq(n)2981 821 y Fv(\021)1379 1127 y FE(+)83 b FD(err)-5 b(or)35 b(terms)16 b Fw(:)456 1319 y FD(The)34 b(err)-5 b(or)35 b(terms)f(c)-5 b(an)35 b(b)-5 b(e)34 b(estimate)-5 b(d)35 b(by)1026 1555 y FE(const)1282 1414 y Fv(\024)1335 1410 y(\015)1335 1470 y(\015)1335 1530 y(\015)1335 1590 y(\015)1390 1555 y Fw(e)1435 1514 y Fs(\000)1500 1487 y Ft(hU)p 1500 1499 84 3 v 1529 1540 a(\017)1620 1555 y Fx(\000)22 b FE(1)g(+)1898 1487 y Fw(h)p 1898 1532 57 4 v 1906 1623 a(\017)1964 1555 y(U)2040 1410 y Fv(\015)2040 1470 y(\015)2040 1530 y(\015)2040 1590 y(\015)2096 1654 y Fq(L)2144 1635 y Fm(1)2247 1555 y Fx(k)2320 1530 y FE(^)2297 1555 y Fx(L)o FE([)p Fw( )2455 1570 y Fq(n)2503 1555 y FE(])p Fx(k)2580 1570 y Fq(L)2628 1551 y Fm(1)1036 1871 y FE(+)p Fw(h)1168 1830 y Fl(2)1241 1726 y Fv(\015)1241 1786 y(\015)1241 1846 y(\015)1241 1906 y(\015)1296 1871 y FE(1)g Fx(\000)1477 1803 y Fw(hU)p 1477 1848 133 4 v 1524 1939 a(\017)1620 1726 y Fv(\015)1620 1786 y(\015)1620 1846 y(\015)1620 1906 y(\015)1675 1970 y Fq(L)1723 1951 y Fm(1)1825 1871 y Fx(k)p Fw(D)1959 1830 y Fl(3)2018 1845 y FE(^)1998 1871 y Fw( )2061 1886 y Fq(n)2109 1871 y Fx(k)2159 1886 y Fq(L)2207 1867 y Fm(1)2286 1721 y Fv(R)2369 1802 y Fx(j)p Fw(v)t Fx(j)2476 1765 y Fl(3)2514 1802 y Fw(e)2559 1765 y Fs(\000)2624 1731 y Ft(hK)s Fr(\()p Ft(v)r Fr(\))p 2624 1750 172 3 v 2697 1791 a Ft(\017)2810 1802 y Fw(dv)p 2286 1848 626 4 v 2358 1887 a Fv(R)2441 1968 y Fw(e)2486 1931 y Fs(\000)2551 1897 y Ft(hK)s Fr(\()p Ft(v)r Fr(\))p 2551 1916 172 3 v 2624 1957 a Ft(\017)2737 1968 y Fw(dv)2921 1700 y Fv(#)3013 1871 y Fw(:)456 2149 y Fz(Remark.)142 b FE(If)45 b Fw(K)7 b FE(\()p Fw(v)t FE(\))49 b(=)h Fx(j)p Fw(v)t Fx(j)1633 2113 y Fl(2)1671 2149 y FE(,)f(formally)-8 b(,)46 b(this)f(means)h(that,)i(in)d(the)h(limit)456 2277 y Fw(h)27 b Fx(!)h FE(0,)1413 2459 y Fw( )1476 2474 y Fq(t)1534 2459 y Fx(')g(\000)1726 2392 y Fw(U)p 1726 2436 77 4 v 1745 2527 a(\017)1813 2459 y( )e FE(+)38 b(const)18 b Fw(\017)f FE(\001)p Fw( )t(:)456 2676 y FD(Pr)-5 b(o)g(of.)41 b FE(W)-8 b(e)33 b(ha)m(v)m(e)1318 2834 y Fw( )1381 2849 y Fq(n)p Fl(+1)1546 2834 y FE(=)28 b Fx(L)p FE([)p Fw( )1809 2849 y Fq(n)1856 2834 y FE(])g(=)g Fw(e)2060 2793 y Fs(\000)2125 2760 y Ft(hU)5 b Fr(\()p Ft(x)p Fr(\))p 2125 2778 167 3 v 2195 2819 a Ft(\017)2329 2809 y FE(^)2305 2834 y Fx(L)o FE([)p Fw( )2463 2849 y Fq(n)2511 2834 y FE(])17 b Fw(:)456 3004 y FE(Therefore,)876 3240 y Fw( )939 3255 y Fq(n)p Fl(+1)1099 3240 y Fx(\000)23 b Fw( )1262 3255 y Fq(n)1392 3240 y FE(=)1551 3099 y Fv(\022)1624 3240 y FE(1)f Fx(\000)1805 3172 y Fw(hU)p 1805 3217 133 4 v 1852 3308 a(\017)1948 3099 y Fv(\023)2054 3129 y(\020)2138 3214 y FE(^)2114 3240 y Fx(L)o FE([)p Fw( )2272 3255 y Fq(n)2320 3240 y FE(])g Fx(\000)h Fw( )2532 3255 y Fq(n)2579 3129 y Fv(\021)2661 3240 y Fx(\000)2770 3172 y Fw(hU)p 2770 3217 V 2817 3308 a(\017)2913 3240 y( )2976 3255 y Fq(n)1551 3519 y Fx(\000)1645 3378 y Fv(\022)1718 3519 y FE(1)f Fx(\000)h Fw(e)1934 3478 y Fs(\000)1999 3451 y Ft(hU)p 1999 3463 84 3 v 2028 3504 a(\017)2119 3519 y Fx(\000)2228 3452 y Fw(hU)p 2228 3496 133 4 v 2275 3587 a(\017)2371 3378 y Fv(\023)2501 3494 y FE(^)2478 3519 y Fx(L)o FE([)p Fw( )2636 3534 y Fq(n)2684 3519 y FE(])17 b Fw(:)456 3755 y FE(The)33 b(desired)g(result)g(th)m (us)g(follo)m(ws)f(from)f(Prop)s(osition)g(23.)693 b Fj(\003)456 3954 y FI(Prop)s(osition)37 b(25)j FE(\(Asymptotics)34 b(for)g Fx(G)6 b FE(\))p FI(.)43 b FD(L)-5 b(et)37 b Fw(\036)31 b Fx(2)g Fw(C)2580 3917 y Fl(3)2619 3954 y FE(\()p Fn(R)2723 3917 y Fq(N)2797 3954 y FE(\))p FD(.)49 b(Assume)37 b(that)456 4081 y Fw(K)7 b FE(\()p Fw(v)t FE(\))27 b(=)h Fx(j)p Fw(v)t Fx(j)911 4045 y Fl(2)984 4081 y FD(and)34 b(that)456 4273 y FE(\(41\))239 b Fw(h)17 b Fx(k)p Fw(D)1076 4237 y Fl(2)1115 4273 y Fw(\036)p Fx(k)1223 4292 y Fq(L)1271 4273 y Fm(1)1336 4292 y Fl(\()p Fu(R)1411 4273 y Ft(N)1469 4292 y Fl(\))1535 4273 y FD(is)35 b(smal)5 b(ler)34 b(than)g(a)h(suitably)g(c)-5 b(onstant.)456 4465 y(L)g(et)456 4702 y FE(\(42\))803 b Fw(a)45 b FE(:=)1686 4548 y Fv(R)1733 4663 y Fu(R)1781 4644 y Ft(N)1859 4628 y Fw(e)1904 4592 y Fs(\000j)p Fq(w)r Fs(j)2052 4569 y Fr(2)2106 4628 y Fx(j)p Fw(w)s Fx(j)2235 4592 y Fl(2)2290 4628 y Fw(dw)p 1686 4679 728 4 v 1737 4771 a FE(2)1819 4691 y Fv(R)1866 4806 y Fu(R)1914 4787 y Ft(N)1992 4771 y Fw(e)2037 4742 y Fs(\000j)p Fq(w)r Fs(j)2185 4723 y Fr(2)2239 4771 y Fw(dw)2440 4702 y(:)456 4947 y FD(Then,)456 5180 y FE(\(43\))735 5113 y Fx(G)6 b FE([)p Fw(\036)p FE(])22 b Fx(\000)h Fw(\036)p 735 5158 357 4 v 885 5249 a(h)1129 5180 y FE(=)28 b Fx(\000)p Fw(H)8 b FE(\()p Fw(D)s(\036)p FE(\()p Fw(x)p FE(\))p Fw(;)17 b(x)p FE(\))22 b(+)g Fw(a\017)p FE(\001)p Fw(\036)h Fx(\000)2328 5113 y Fw(\017N)p 2328 5158 128 4 v 2339 5249 a FE(2)p Fw(h)2483 5180 y FE(ln)2590 5113 y Fw(\017\031)p 2590 5158 99 4 v 2612 5249 a(h)2721 5180 y FE(+)f FD(err)-5 b(or)35 b(terms)16 b Fw(:)p eop %%Page: 24 24 24 23 bop 456 251 a FC(24)666 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)456 450 y FD(The)34 b(err)-5 b(or)35 b(terms)f(c)-5 b(an)35 b(b)-5 b(e)34 b(estimate)-5 b(d)35 b(by)1573 651 y Fw(C)1643 666 y Fq(?)1699 651 y FE(\()p Fw(h)22 b FE(+)g Fw(\017)1952 610 y Fl(3)p Fq(=)p Fl(2)2079 651 y Fw(h)2135 610 y Fl(1)p Fq(=)p Fl(2)2245 651 y FE(\))17 b Fw(;)456 851 y FD(wher)-5 b(e)29 b Fw(C)796 866 y Fq(?)864 851 y FD(is)h(a)f(suitable)g(p)-5 b(ositive)29 b(quantity)i(dep)-5 b(ending)28 b(only)h(on)g Fx(k)p Fw(D)3033 815 y Fq(j)3069 851 y Fw(\036)p Fx(k)3177 866 y Fq(L)3225 847 y Fm(1)3324 851 y FD(for)456 979 y FE(1)e Fx(\024)h Fw(j)34 b Fx(\024)28 b FE(3)p FD(.)456 1140 y Fz(Remark.)96 b FE(There)36 b(are)f(sev)m(eral)g(heuristic)f(w)m(a)m(ys)j(of)d(in)m(terpreting)g (the)h(result)456 1268 y(in)41 b(Prop)s(osition)g(25)h(according)g(to)g (the)h(sc)m(heme)h Fw(\036)2412 1283 y Fq(n)p Fl(+1)2593 1268 y FE(:=)h Fx(G)6 b FE([)p Fw(\036)2891 1283 y Fq(n)2938 1268 y FE(].)73 b(The)44 b(\014rst)456 1396 y(consists)36 b(in)f(taking)g Fw(h)e Fx(!)f FE(0)k(and)f Fw(\017=h)e Fx(!)g FE(0,)j(so)g(that,)g(formally)-8 b(,)33 b(\(43\))i(go)s(es)h(to) 456 1524 y(the)d(Hamilton-Jacobi)28 b(equation)1449 1724 y Fw(@)1500 1739 y Fq(t)1530 1724 y Fw(\036)44 b FE(=)g Fx(\000)p Fw(H)8 b FE(\()p Fw(D)2037 1739 y Fq(x)2081 1724 y Fw(\036)p FE(\()p Fw(x)p FE(\))p Fw(;)17 b(x)p FE(\))g Fw(:)456 1925 y FE(Another)43 b(p)s(ossibilit)m(y)e(is)h (taking)g Fw(\017)k(>)f FE(0)e(to)g(b)s(e)g(\014xed)h(and)f(sending)g Fw(h)j Fx(!)f FE(0.)456 2053 y(Then,)33 b(if)f(w)m(e)h(de\014ne)1494 2244 y Fw(\036)1552 2203 y Fs(\003)1552 2269 y Fq(n)1626 2244 y FE(:=)28 b Fw(\036)1815 2259 y Fq(n)1884 2244 y FE(+)1992 2177 y Fw(\017N)p 1992 2221 128 4 v 2031 2313 a FE(2)2163 2244 y(ln)2271 2177 y Fw(\017\031)p 2271 2221 99 4 v 2292 2313 a(h)2379 2244 y(;)456 2446 y FE(it)j(follo)m(ws)g(that)i(\(43\))f(go)s(es)g(formally)e(to)1215 2647 y Fw(@)1266 2662 y Fq(t)1296 2647 y Fw(\036)1354 2606 y Fs(\003)1438 2647 y FE(=)44 b Fx(\000)p Fw(H)8 b FE(\()p Fw(D)1843 2662 y Fq(x)1887 2647 y Fw(\036)1945 2606 y Fs(\003)1984 2647 y FE(\()p Fw(x)p FE(\))p Fw(;)17 b(x)p FE(\))22 b(+)g Fw(a\017)p FE(\001)p Fw(\036)2601 2606 y Fs(\003)2658 2647 y Fw(;)456 2847 y FE(that)29 b(is,)h(a)g(Hamilton-Jacobi)c(equation)j(of)g(\\v)-5 b(anishing)29 b(viscosit)m(y")h(t)m(yp)s(e.)43 b(The)456 2975 y(reader)33 b(ma)m(y)f(compare)g(this)g(case)i(with)e(the)h (example)f(discussed)j(in)c(\(35\).)555 3103 y(This)h(observ)-5 b(ation)31 b(ma)m(y)g(also)f(b)s(e)i(rephrased)h(as)e(follo)m(ws.)42 b(A)32 b(direct)f(compu-)456 3231 y(tation)g(sho)m(ws)j(that)1291 3448 y Fx(G)6 b FE([0]\()p Fw(x)p FE(\))28 b(=)g Fx(\000)p Fw(hU)10 b FE(\()p Fw(x)p FE(\))23 b Fx(\000)2195 3380 y Fw(\017N)p 2195 3425 128 4 v 2234 3516 a FE(2)2349 3448 y(ln)2457 3380 y Fw(\017\031)p 2457 3425 99 4 v 2478 3516 a(h)2582 3448 y(:)456 3666 y FE(Therefore,)33 b(if)f(w)m(e)h(de\014ne)456 3867 y(\(44\))1530 3842 y(^)1510 3867 y Fx(G)7 b FE([)p Fw(\036)p FE(])44 b(:=)g Fx(G)6 b FE([)p Fw(\036)p FE(])22 b Fx(\000)h(G)6 b FE([0])17 b Fw(;)456 4068 y FE(then)33 b(Prop)s(osition)e(25)h(giv)m(es)h(that) 456 4330 y(\(45\))973 4237 y(^)953 4262 y Fx(G)6 b FE([)p Fw(\036)p FE(])22 b Fx(\000)h Fw(\036)p 953 4307 357 4 v 1103 4398 a(h)1347 4330 y FE(=)28 b Fx(\000)p Fw(H)8 b FE(\()p Fw(D)s(\036)p FE(\()p Fw(x)p FE(\)\))22 b(+)g Fw(a\017)p FE(\001)p Fw(\036)h FE(+)f(error)32 b(terms)17 b Fw(:)456 4548 y FE(The)26 b(latter)e(in)m(terpretation)g(is)g (related)h(with)g(Prop)s(osition)e(8,)j(whic)m(h)g(giv)m(es)g(that)456 4749 y(\(46\))737 b Fx(k)1436 4724 y FE(^)1417 4749 y Fx(G)6 b FE([)p Fw(\036)p FE(])p Fx(k)1644 4768 y Fq(L)1692 4749 y Fm(1)1757 4768 y Fl(\()p Fu(R)1832 4749 y Ft(N)1890 4768 y Fl(\))1949 4749 y Fx(\024)28 b(k)p Fw(\036)p Fx(k)2212 4768 y Fq(L)2260 4749 y Fm(1)2325 4768 y Fl(\()p Fu(R)2400 4749 y Ft(N)2458 4768 y Fl(\))2506 4749 y Fw(;)456 4960 y FE(that)k(is,)h(a)g(uniform)e(b)s(ound)i(on)g(the)1865 4935 y(^)1846 4960 y Fx(G)6 b FE(-iterations.)43 b(W)-8 b(e)33 b(will)e(use)j(these)g(obser-)456 5088 y(v)-5 b(ations)30 b(in)h(Theorem)g(30)g(b)s(elo)m(w,)g(where)i(a)e(formal)e (pro)s(of)h(of)h(the)h(con)m(v)m(ergence)456 5216 y(of)g(the)754 5190 y(^)735 5216 y Fx(G)6 b FE(-sc)m(heme)33 b(for)f Fw(h)c Fx(!)f FE(0)33 b(will)d(b)s(e)j(pro)m(vided.)p eop %%Page: 25 25 25 24 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)420 b(25)456 450 y FD(Pr)-5 b(o)g(of.)41 b FE(W)-8 b(e)33 b(ha)m(v)m(e)1091 666 y Fx(G)6 b FE([)p Fw(\036)p FE(])28 b(=)f Fx(\000)p Fw(\017)17 b FE(ln)1631 530 y Fv(Z)1747 666 y Fw(e)1792 624 y Fs(\000)1857 591 y Ft(hL)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p 1857 609 495 3 v 2091 651 a Ft(\017)2366 666 y Fw(dv)-2008 b FE(\(47\))1296 927 y(=)27 b Fx(\000)p Fw(hU)10 b FE(\()p Fw(x)p FE(\))24 b Fx(\000)f Fw(\017)17 b FE(ln)2017 792 y Fv(Z)2133 927 y Fw(e)2178 886 y Fs(\000)2243 853 y Ft(h)p Fm(j)p Ft(v)r Fm(j)2350 832 y Fr(2)2384 853 y(+)p Ft(\036)p Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p 2243 871 424 3 v 2442 912 a Ft(\017)2680 927 y Fw(dv)t(:)456 1148 y FE(De\014ne)33 b Fw(v)809 1112 y Fs(\003)876 1148 y FE(=)27 b Fw(v)1030 1112 y Fs(\003)1069 1148 y FE(\()p Fw(x;)17 b(h)p FE(\))33 b(to)f(b)s(e)h(a)f(minimizer)d(of)k(the)g (function)1551 1328 y Fw(h)17 b Fx(j)p Fw(v)t Fx(j)1731 1287 y Fl(2)1791 1328 y FE(+)22 b Fw(\036)p FE(\()p Fw(x)h FE(+)f Fw(hv)t FE(\))17 b Fw(:)456 1509 y FE(Note)32 b(that)h(the)g(minimizing)28 b(prop)s(ert)m(y)33 b(of)f Fw(v)2135 1472 y Fs(\003)2207 1509 y FE(yields)g(that)1372 1689 y Fw(h)17 b Fx(j)p Fw(v)1524 1647 y Fs(\003)1563 1689 y Fx(j)1591 1647 y Fl(2)1652 1689 y FE(+)22 b Fw(\036)p FE(\()p Fw(x)g FE(+)g Fw(hv)2128 1647 y Fs(\003)2168 1689 y FE(\))27 b Fx(\024)i Fw(\036)p FE(\()p Fw(x)p FE(\))456 1869 y(and)j(so)1445 2004 y Fw(h)17 b Fx(j)p Fw(v)1597 1963 y Fs(\003)1635 2004 y Fx(j)1663 1963 y Fl(2)1730 2004 y Fx(\024)28 b Fw(h)17 b Fx(k)p Fw(D)s(\036)p Fx(k)2150 2019 y Fq(L)2198 2000 y Fm(1)2266 2004 y Fx(j)p Fw(v)2345 1963 y Fs(\003)2384 2004 y Fx(j)g Fw(;)456 2162 y FE(that)32 b(is)456 2342 y(\(48\))963 b Fx(j)p Fw(v)1672 2301 y Fs(\003)1711 2342 y Fx(j)44 b(\024)h(k)p Fw(D)s(\036)p Fx(k)2147 2357 y Fq(L)2195 2338 y Fm(1)2280 2342 y Fw(:)456 2522 y FE(By)33 b(the)g(minimizing)28 b(prop)s(ert)m(y)33 b(of)f Fw(v)1841 2486 y Fs(\003)1880 2522 y FE(,)h(it)e(also)h(follo)m(ws)f(that)456 2702 y(\(49\))810 b(2)p Fw(v)1540 2661 y Fs(\003)1601 2702 y FE(+)22 b Fw(D)1780 2717 y Fq(x)1824 2702 y Fw(\036)p FE(\()p Fw(x)g FE(+)g Fw(hv)2202 2661 y Fs(\003)2242 2702 y FE(\))27 b(=)h(0)456 2882 y(and)k(th)m(us)1016 3022 y Fw(h)p Fx(j)p Fw(v)t Fx(j)1179 2985 y Fl(2)1240 3022 y FE(+)22 b Fw(\036)p FE(\()p Fw(x)g FE(+)g Fw(hv)t FE(\))p 1016 3066 738 4 v 1366 3157 a Fw(\017)456 3089 y FE(\(50\))931 3344 y(=)1044 3277 y([)p Fw(h)p Fx(j)p Fw(v)1206 3240 y Fs(\003)1245 3277 y Fx(j)1273 3240 y Fl(2)1334 3277 y FE(+)g Fw(\036)p FE(\()p Fw(x)h FE(+)f Fw(hv)1811 3240 y Fs(\003)1850 3277 y FE(\)])g(+)g Fw(h)17 b Fx(j)p Fw(v)26 b Fx(\000)c Fw(v)2359 3240 y Fs(\003)2398 3277 y Fx(j)2426 3240 y Fl(2)p 1044 3321 1422 4 v 1735 3412 a Fw(\017)2498 3344 y FE(+)g(\004)2661 3359 y Fl(1)2700 3344 y FE(\()p Fw(x;)17 b(v)t FE(\))g Fw(;)456 3542 y FE(with)456 3670 y(\(51\))568 3845 y(\004)633 3860 y Fl(1)672 3845 y FE(\()p Fw(x;)g(v)t FE(\))44 b(:=)1100 3778 y Fw(\036)p FE(\()p Fw(x)22 b FE(+)g Fw(hv)t FE(\))g Fx(\000)g Fw(\036)p FE(\()p Fw(x)h FE(+)f Fw(hv)2016 3742 y Fs(\003)2055 3778 y FE(\))g Fx(\000)h Fw(h)17 b(D)2369 3793 y Fq(x)2412 3778 y Fw(\036)p FE(\()p Fw(x)23 b FE(+)f Fw(hv)2791 3742 y Fs(\003)2830 3778 y FE(\))g Fx(\001)g FE(\()p Fw(v)k Fx(\000)c Fw(v)3201 3742 y Fs(\003)3240 3778 y FE(\))p 1100 3822 2179 4 v 2169 3914 a Fw(\017)3305 3845 y(:)456 4043 y FE(Note)32 b(that,)h(b)m(y)g(construction,)456 4270 y(\(52\))670 b Fx(j)p FE(\004)1393 4285 y Fl(1)1432 4270 y Fx(j)27 b(\024)h Fw(C)1679 4202 y(h)1735 4166 y Fl(2)1775 4202 y Fx(k)p Fw(D)1909 4166 y Fl(2)1948 4202 y Fw(\036)p Fx(k)2056 4217 y Fq(L)2104 4198 y Fm(1)p 1679 4247 494 4 v 1906 4338 a Fw(\017)2199 4270 y Fx(j)p Fw(v)e Fx(\000)c Fw(v)2450 4228 y Fs(\003)2489 4270 y Fx(j)2517 4228 y Fl(2)2573 4270 y Fw(;)456 4468 y FE(where)42 b Fw(C)7 b FE(,)44 b(here)e(and)g(in)f(the)h(remainder)e(of)h(the)h (pro)s(of,)h(w)m(e)g(will)c(denote)j(an)456 4595 y(appropriate)27 b(univ)m(ersal)h(constan)m(t)i(\(p)s(ossibly)d(taking)h(a)g(di\013eren) m(t)h(v)-5 b(alue)28 b(at)g(dif-)456 4723 y(feren)m(t)35 b(steps)i(of)d(the)h(computation\).)49 b(W)-8 b(e)35 b(will)e(also)h(use)i(the)f(follo)m(wing)d(short)456 4851 y(hand)g(notation:)1211 4973 y Fv(Z)1227 5108 y Fx(\000)17 b Fw(f)11 b FE(\()p Fw(v)t FE(\))17 b Fw(dv)30 b FE(:=)1793 4955 y Fv(R)1840 5070 y Fu(R)1888 5051 y Ft(N)1966 5035 y Fw(e)2011 4998 y Fs(\000)2076 4965 y Ft(h)p Fm(j)p Ft(v)r Fm(\000)p Ft(v)2245 4944 y Fm(\003)2281 4965 y(j)2300 4944 y Fr(2)p 2076 4983 259 3 v 2193 5024 a Ft(\017)2349 5035 y Fw(f)11 b FE(\()p Fw(v)t FE(\))17 b Fw(dv)p 1793 5086 860 4 v 1944 5134 a Fv(R)1991 5249 y Fu(R)2039 5230 y Ft(N)2118 5214 y Fw(e)2163 5177 y Fs(\000)2228 5144 y Ft(h)p Fm(j)p Ft(v)r Fm(j)2335 5130 y Fr(2)p 2228 5162 142 3 v 2285 5203 a Ft(\017)2400 5214 y Fw(dv)2679 5108 y(:)p eop %%Page: 26 26 26 25 bop 456 251 a FC(26)666 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)456 450 y FE(Let)32 b(also)456 666 y(\(53\))692 b(\004)1387 681 y Fl(2)1427 666 y FE(\()p Fw(x)p FE(\))44 b(:=)1766 530 y Fv(Z)1783 666 y Fx(\000)17 b FE(\()p Fw(e)1960 625 y Fs(\000)p Fl(\004)2062 634 y Fr(1)2123 666 y Fx(\000)23 b FE(1)e(+)i(\004)2457 681 y Fl(1)2496 666 y FE(\))17 b Fw(:)456 907 y FE(Then,)33 b(b)m(y)h(\(52\))e(and)h (\(41\),)f(w)m(e)h(get)g(that)827 1175 y Fx(j)p FE(\004)920 1190 y Fl(2)959 1175 y Fx(j)83 b(\024)1247 1040 y Fv(Z)1263 1175 y Fx(\000)1411 1051 y Fs(1)1374 1081 y Fv(X)1382 1293 y Fq(k)r Fl(=2)1545 1108 y Fx(j)p FE(\004)1638 1123 y Fl(1)1677 1108 y Fx(j)1705 1072 y Fq(k)p 1545 1152 203 4 v 1605 1244 a Fw(k)s FE(!)1070 1506 y Fx(\024)1247 1370 y Fv(Z)1263 1506 y Fx(\000)17 b(j)p FE(\004)1450 1521 y Fl(1)1490 1506 y Fx(j)1518 1465 y Fl(2)1610 1382 y Fs(1)1573 1411 y Fv(X)1581 1624 y Fq(k)r Fl(=0)1744 1439 y Fx(j)p FE(\004)1837 1454 y Fl(1)1876 1439 y Fx(j)1904 1403 y Fq(k)p 1744 1483 V 1805 1574 a Fw(k)s FE(!)1071 1809 y(=)1247 1674 y Fv(Z)1263 1809 y Fx(\000)g(j)p FE(\004)1450 1824 y Fl(1)1490 1809 y Fx(j)1518 1768 y Fl(2)1557 1809 y Fw(e)1602 1768 y Fl(\004)1649 1777 y Fr(1)456 1809 y FE(\(54\))1070 2082 y Fx(\024)1240 2014 y Fw(C)24 b(h)1390 1978 y Fl(4)1446 2014 y Fx(k)p Fw(D)1580 1978 y Fl(2)1619 2014 y Fw(\036)p Fx(k)1727 1978 y Fl(2)1727 2040 y Fq(L)1775 2021 y Fm(1)p 1240 2059 604 4 v 1502 2150 a Fw(\017)1541 2121 y Fl(2)1887 1946 y Fv(Z)1903 2082 y Fx(\000)17 b(j)p Fw(v)26 b Fx(\000)d Fw(v)2249 2041 y Fs(\003)2288 2082 y Fx(j)2316 2041 y Fl(4)2355 2082 y Fw(e)2410 2001 y Ft(C)14 b(h)2504 1980 y Fr(2)2539 2001 y Fm(k)p Ft(D)2623 1980 y Fr(2)2657 2001 y Ft(\036)p Fm(k)2726 2020 y Ft(L)2768 2006 y Fm(1)2837 2001 y(j)p Ft(v)r Fm(\000)p Ft(v)2970 1980 y Fm(\003)3006 2001 y(j)3025 1980 y Fr(2)p 2410 2026 649 3 v 2722 2067 a Ft(\017)1070 2405 y Fx(\024)83 b Fw(C)24 b(h)1380 2364 y Fl(2)1436 2405 y Fx(k)p Fw(D)1570 2364 y Fl(2)1609 2405 y Fw(\036)p Fx(k)1717 2364 y Fl(2)1717 2429 y Fq(L)1765 2410 y Fm(1)1860 2251 y Fv(R)1907 2366 y Fu(R)1955 2347 y Ft(N)2034 2331 y Fw(e)2079 2294 y Fs(\000)2144 2261 y Fm(j)p Ft(w)q Fm(j)2227 2240 y Fr(2)p 2144 2279 119 3 v 2188 2320 a(8)2293 2331 y Fx(j)p Fw(w)s Fx(j)2422 2295 y Fl(4)2477 2331 y Fw(dw)p 1860 2382 740 4 v 1958 2394 a Fv(R)2005 2509 y Fu(R)2053 2490 y Ft(N)2132 2474 y Fw(e)2177 2445 y Fs(\000j)p Fq(w)r Fs(j)2325 2426 y Fr(2)2379 2474 y Fw(dw)1070 2633 y Fx(\024)83 b Fw(C)24 b(h)1380 2592 y Fl(2)1436 2633 y Fx(k)p Fw(D)1570 2592 y Fl(2)1609 2633 y Fw(\036)p Fx(k)1717 2592 y Fl(2)1717 2657 y Fq(L)1765 2638 y Fm(1)1850 2633 y Fw(:)456 2833 y FE(W)-8 b(e)33 b(also)e(de\014ne)456 3033 y(\(55\))393 b(\002)1099 3048 y Fl(1)1138 3033 y FE(\()p Fw(x;)17 b(v)t FE(\))44 b(:=)g Fx(\000)p Fw(hU)10 b FE(\()p Fw(x)p FE(\))24 b(+)e Fw(h)p Fx(j)p Fw(v)2152 2991 y Fs(\003)2191 3033 y Fx(j)2219 2991 y Fl(2)2280 3033 y FE(+)g Fw(\036)p FE(\()p Fw(x)g FE(+)g Fw(hv)2756 2991 y Fs(\003)2796 3033 y FE(\))17 b Fw(:)456 3232 y FE(Using)32 b(this)g(de\014nition,)g (\(47\),)g(\(50\))g(and)g(\(53\),)g(w)m(e)i(obtain)d(that)678 3482 y Fx(G)6 b FE([)p Fw(\036)p FE(])84 b(=)e(\002)1173 3497 y Fl(1)1213 3482 y FE(\()p Fw(x;)17 b(v)t FE(\))22 b Fx(\000)g Fw(\017)17 b FE(ln)1714 3342 y Fv(\022)1787 3347 y(Z)1843 3572 y Fu(R)1891 3553 y Ft(N)1969 3482 y Fw(e)2014 3441 y Fs(\000)2079 3407 y Ft(h)11 b Fm(j)p Ft(v)r Fm(\000)p Ft(v)2259 3387 y Fm(\003)2295 3407 y(j)2314 3387 y Fr(2)p 2079 3426 269 3 v 2201 3467 a Ft(\017)2379 3482 y Fw(e)2424 3441 y Fs(\000)p Fl(\004)2526 3450 y Fr(1)2561 3441 y Fl(\()p Fq(x;v)r Fl(\))2732 3482 y Fw(dv)2834 3342 y Fv(\023)939 3762 y FE(=)82 b(\002)1173 3777 y Fl(1)1235 3762 y Fx(\000)22 b Fw(\017)17 b FE(ln)1488 3621 y Fv(\022)1578 3626 y(Z)1595 3762 y Fx(\000)g Fw(e)1734 3720 y Fs(\000)p Fl(\004)1836 3729 y Fr(1)1871 3720 y Fl(\()p Fq(x;v)r Fl(\))2042 3762 y Fw(dv)2144 3621 y Fv(\023)2239 3762 y Fx(\000)23 b Fw(\017)17 b FE(ln)2493 3621 y Fv(\022)2566 3626 y(Z)2622 3851 y Fu(R)2670 3833 y Ft(N)2748 3762 y Fw(e)2793 3720 y Fs(\000)2858 3687 y Ft(h)p Fm(j)p Ft(v)r Fm(j)2965 3666 y Fr(2)p 2858 3705 142 3 v 2916 3746 a Ft(\017)3030 3762 y Fw(dv)3132 3621 y Fv(\023)456 3762 y FE(\(56\))939 4041 y(=)82 b(\002)1173 4056 y Fl(1)1235 4041 y Fx(\000)22 b Fw(\017)17 b FE(ln)1488 3900 y Fv(\022)1578 3905 y(Z)1595 4041 y Fx(\000)g FE(\(1)22 b Fx(\000)g FE(\004)1962 4056 y Fl(1)2002 4041 y FE(\))17 b Fw(dv)25 b FE(+)d(\004)2343 4056 y Fl(2)2383 4041 y FE(\()p Fw(x)p FE(\))2514 3900 y Fv(\023)2609 4041 y Fx(\000)2719 3973 y Fw(\017N)p 2719 4018 128 4 v 2758 4109 a FE(2)2890 4041 y(ln)2998 3973 y Fw(\017\031)p 2998 4018 99 4 v 3019 4109 a(h)939 4320 y FE(=)82 b(\002)1173 4335 y Fl(1)1235 4320 y Fx(\000)22 b Fw(\017)17 b FE(ln)1488 4180 y Fv(\022)1562 4320 y FE(1)22 b Fx(\000)1749 4185 y Fv(Z)1765 4320 y Fx(\000)17 b FE(\004)1924 4335 y Fl(1)1981 4320 y Fw(dv)25 b FE(+)d(\004)2267 4335 y Fl(2)2307 4320 y FE(\()p Fw(x)p FE(\))2438 4180 y Fv(\023)2533 4320 y Fx(\000)2643 4253 y Fw(\017N)p 2643 4297 128 4 v 2683 4389 a FE(2)2798 4320 y(ln)2905 4253 y Fw(\017\031)p 2905 4297 99 4 v 2926 4389 a(h)3030 4320 y(:)456 4564 y FE(W)-8 b(e)33 b(no)m(w)g(de\014ne)456 4780 y(\(57\))618 b(\004)1313 4795 y Fl(3)1352 4780 y FE(\()p Fw(x)p FE(\))45 b(:=)f(\004)1740 4795 y Fl(2)1780 4780 y FE(\()p Fw(x)p FE(\))22 b Fx(\000)2049 4645 y Fv(Z)2066 4780 y Fx(\000)17 b FE(\004)2225 4795 y Fl(1)2265 4780 y FE(\()p Fw(x;)g(v)t FE(\))g Fw(dv)i(:)456 5016 y FE(Note)32 b(that)456 5216 y(\(58\))853 b Fx(j)p FE(\004)1576 5231 y Fl(3)1615 5216 y Fx(j)44 b(\024)h Fw(C)24 b(h)17 b Fx(k)p Fw(D)2110 5174 y Fl(2)2148 5216 y Fw(\036)p Fx(k)2256 5231 y Fq(L)2304 5212 y Fm(1)2390 5216 y Fw(;)p eop %%Page: 27 27 27 26 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)420 b(27)456 450 y FE(thanks)33 b(to)f(\(52\),)g(\(54\))g(and)h(\(41\).)43 b(Let)33 b(also)456 628 y(\(59\))822 b(\004)1517 643 y Fl(4)1601 628 y FE(:=)44 b(ln)o(\(1)22 b(+)g(\004)2101 643 y Fl(3)2140 628 y FE(\))g Fx(\000)h FE(\004)2365 643 y Fl(3)2421 628 y Fw(:)456 805 y FE(Then,)456 983 y(\(60\))850 b Fx(j)p FE(\004)1573 998 y Fl(4)1612 983 y Fx(j)28 b(\024)g Fw(C)23 b(h)1922 942 y Fl(2)1979 983 y Fx(k)p Fw(D)2113 942 y Fl(2)2151 983 y Fw(\036)p Fx(k)2259 942 y Fl(2)2259 1007 y Fq(L)2307 988 y Fm(1)2393 983 y Fw(;)456 1160 y FE(due)33 b(to)f(\(58\).)43 b(Also,)32 b(using)g(\(56\),)g(\(57\))g(and)h(\(59\),)f(w)m(e)i(conclude)f(that) 986 1379 y Fx(G)6 b FE([)p Fw(\036)p FE(])28 b(=)g(\002)1371 1394 y Fl(1)1432 1379 y Fx(\000)1542 1311 y Fw(\017N)p 1542 1356 128 4 v 1581 1447 a FE(2)1713 1379 y(ln)1821 1311 y Fw(\017\031)p 1821 1356 99 4 v 1842 1447 a(h)1951 1379 y Fx(\000)23 b Fw(\017)17 b FE(\004)2172 1394 y Fl(3)2233 1379 y Fx(\000)23 b Fw(\017)17 b FE(\004)2454 1394 y Fl(4)456 1379 y FE(\(61\))1191 1623 y(=)28 b(\002)1371 1638 y Fl(1)1432 1623 y Fx(\000)1542 1555 y Fw(\017N)p 1542 1600 128 4 v 1581 1691 a FE(2)1696 1623 y(ln)1804 1555 y Fw(\017\031)p 1804 1600 99 4 v 1825 1691 a(h)1935 1623 y FE(+)22 b Fw(\017)2105 1487 y Fv(Z)2122 1623 y Fx(\000)17 b FE(\004)2281 1638 y Fl(1)2342 1623 y Fx(\000)23 b Fw(\017)p FE(\(\004)2584 1638 y Fl(2)2646 1623 y FE(+)f(\004)2809 1638 y Fl(4)2849 1623 y FE(\))p Fw(:)456 1836 y FE(Since)32 b(\004)775 1851 y Fl(2)846 1836 y FE(and)g(\004)1100 1851 y Fl(4)1172 1836 y FE(ha)m(v)m(e)h(the)g(same)f(order)g(of)f (magnitude,)g(it)g(is)h(con)m(v)m(enien)m(t)i(to)456 1964 y(de\014ne)1624 2096 y(\004)1689 2111 y Fl(5)1773 2096 y FE(:=)27 b(\004)1968 2111 y Fl(2)2030 2096 y FE(+)22 b(\004)2193 2111 y Fl(4)2249 2096 y Fw(;)456 2252 y FE(so)32 b(that)h(\(54\))f(and)g(\(60\))g(giv)m(e)456 2429 y(\(62\))858 b Fx(j)p FE(\004)1581 2444 y Fl(5)1621 2429 y Fx(j)27 b(\024)h Fw(C)c(h)1931 2388 y Fl(2)1987 2429 y Fx(k)p Fw(D)2121 2388 y Fl(2)2160 2429 y Fw(\036)p Fx(k)2268 2388 y Fl(2)2268 2454 y Fq(L)2316 2435 y Fm(1)2384 2429 y Fw(;)456 2607 y FE(and)32 b(w)m(e)i(obtain)d(from)h(\(61\))g(that)456 2825 y(\(63\))423 b Fx(G)6 b FE([)p Fw(\036)p FE(])28 b(=)f(\002)1437 2840 y Fl(1)1499 2825 y Fx(\000)1608 2758 y Fw(\017N)p 1608 2802 128 4 v 1648 2893 a FE(2)1779 2825 y(ln)1887 2758 y Fw(\017\031)p 1887 2802 99 4 v 1908 2893 a(h)2017 2825 y FE(+)22 b Fw(\017)2188 2689 y Fv(Z)2205 2825 y Fx(\000)17 b FE(\004)2364 2840 y Fl(1)2420 2825 y Fw(dv)25 b Fx(\000)e Fw(\017)17 b FE(\004)2764 2840 y Fl(5)2820 2825 y Fw(:)456 3038 y FE(W)-8 b(e)33 b(no)m(w)g(de\014ne)651 3242 y(\004)716 3257 y Fl(6)755 3242 y FE(\()p Fw(x;)17 b(v)t FE(\))83 b(:=)g(\004)1315 3257 y Fl(1)1354 3242 y FE(\()p Fw(x;)17 b(v)t FE(\))22 b Fx(\000)1712 3175 y Fw(h)1768 3139 y Fl(2)p 1712 3219 96 4 v 1716 3311 a FE(2)p Fw(\017)1817 3242 y(@)1868 3257 y Fq(ij)1929 3242 y Fw(\036)p FE(\()p Fw(x)p FE(\))17 b(\()p Fw(v)2220 3257 y Fq(i)2270 3242 y Fx(\000)23 b Fw(v)2421 3201 y Fs(\003)2417 3267 y Fq(i)2460 3242 y FE(\)\()p Fw(v)2583 3257 y Fq(j)2642 3242 y Fx(\000)f Fw(v)2792 3201 y Fs(\003)2788 3267 y Fq(j)2832 3242 y FE(\))222 b(and)745 3487 y(\004)810 3502 y Fl(7)849 3487 y FE(\()p Fw(x)p FE(\))84 b(:=)1266 3351 y Fv(Z)1283 3487 y Fx(\000)17 b FE(\004)1442 3502 y Fl(6)1482 3487 y FE(\()p Fw(x;)g(v)t FE(\))g Fw(dv)i(:)-1413 b FE(\(64\))456 3705 y(Note)32 b(that,)h(from)e(\(51\))h(and)h(\(48\),)813 3934 y Fx(j)p FE(\004)906 3949 y Fl(6)946 3934 y Fx(j)82 b(\024)1217 3820 y Fv(\014)1217 3880 y(\014)1217 3939 y(\014)1250 3934 y FE(\004)1315 3949 y Fl(1)1377 3934 y Fx(\000)1486 3867 y Fw(h)1542 3831 y Fl(2)p 1486 3911 V 1490 4003 a FE(2)p Fw(\017)1592 3934 y(@)1643 3949 y Fq(ij)1704 3934 y Fw(\036)p FE(\()p Fw(x)22 b FE(+)g Fw(hv)2082 3893 y Fs(\003)2121 3934 y FE(\))17 b(\()p Fw(v)2261 3949 y Fq(i)2311 3934 y Fx(\000)23 b Fw(v)2462 3893 y Fs(\003)2458 3959 y Fq(i)2501 3934 y FE(\)\()p Fw(v)2624 3949 y Fq(j)2683 3934 y Fx(\000)f Fw(v)2833 3893 y Fs(\003)2829 3959 y Fq(j)2873 3934 y FE(\))2911 3820 y Fv(\014)2911 3880 y(\014)2911 3939 y(\014)1217 4189 y FE(+)1303 4122 y Fw(C)7 b(h)1436 4086 y Fl(3)p 1303 4166 173 4 v 1369 4258 a Fw(\017)1502 4189 y Fx(j)p Fw(v)1581 4148 y Fs(\003)1620 4189 y Fx(j)17 b(k)p Fw(D)1799 4148 y Fl(3)1837 4189 y Fw(\036)p Fx(k)1945 4204 y Fq(L)1993 4185 y Fm(1)2062 4189 y Fx(j)p Fw(v)25 b Fx(\000)e Fw(v)2313 4148 y Fs(\003)2352 4189 y Fx(j)2380 4148 y Fl(2)1056 4444 y Fx(\024)1227 4377 y Fw(C)7 b(h)1360 4341 y Fl(3)p 1227 4421 V 1293 4513 a Fw(\017)1426 4444 y Fx(k)p Fw(D)1560 4403 y Fl(3)1599 4444 y Fw(\036)p Fx(k)1707 4459 y Fq(L)1755 4440 y Fm(1)1840 4444 y FE(\()p Fx(j)p Fw(v)26 b Fx(\000)c Fw(v)2129 4403 y Fs(\003)2168 4444 y Fx(j)g FE(+)g Fx(k)p Fw(D)s(\036)p Fx(k)2558 4459 y Fq(L)2606 4440 y Fm(1)2675 4444 y FE(\))17 b Fx(j)p Fw(v)25 b Fx(\000)d Fw(v)2980 4403 y Fs(\003)3019 4444 y Fx(j)3047 4403 y Fl(2)456 4640 y FE(and)32 b(so)456 4817 y(\(65\))426 b Fx(j)p FE(\004)1149 4832 y Fl(7)1188 4817 y Fx(j)44 b(\024)h Fw(C)24 b(h)1532 4776 y Fl(2)1588 4817 y Fx(k)p Fw(D)1722 4776 y Fl(3)1761 4817 y Fw(\036)p Fx(k)1869 4832 y Fq(L)1917 4813 y Fm(1)2002 4817 y FE(\()2040 4727 y Fv(p)p 2140 4727 145 4 v 90 x Fw(\017=h)e FE(+)g Fx(k)p Fw(D)s(\036)p Fx(k)2646 4832 y Fq(L)2694 4813 y Fm(1)2762 4817 y FE(\))17 b Fw(:)456 4995 y FE(Moreo)m(v)m(er,)1452 5027 y Fv(Z)1469 5163 y Fx(\000)g FE(\004)1628 5178 y Fl(1)1684 5163 y Fw(dv)48 b FE(=)c Fw(ah)p FE(\001)p Fw(\036)22 b FE(+)g(\004)2381 5178 y Fl(7)2437 5163 y Fw(;)p eop %%Page: 28 28 28 27 bop 456 251 a FC(28)666 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)456 450 y FE(due)22 b(to)f(\(64\),)j(\(42\))d(and)h(a)f (parit)m(y)h(argumen)m(t.)39 b(Therefore,)25 b(w)m(e)e(deduce)g(from)e (\(63\))456 578 y(that)972 723 y Fx(G)6 b FE([)p Fw(\036)p FE(])28 b(=)g(\002)1357 738 y Fl(1)1418 723 y Fx(\000)1528 656 y Fw(\017N)p 1528 700 128 4 v 1567 791 a FE(2)1699 723 y(ln)1807 656 y Fw(\017\031)p 1807 700 99 4 v 1828 791 a(h)1937 723 y FE(+)22 b Fw(a\017h)p FE(\001)p Fw(\036)h FE(+)f Fw(\017)17 b FE(\004)2562 738 y Fl(7)2624 723 y Fx(\000)22 b Fw(\017)17 b FE(\004)2844 738 y Fl(5)2900 723 y Fw(:)456 892 y FE(Accordingly)-8 b(,)31 b(if)h(\004)1171 907 y Fl(8)1238 892 y FE(:=)c(\004)1434 907 y Fl(7)1495 892 y Fx(\000)23 b FE(\004)1660 907 y Fl(5)1699 892 y FE(,)33 b(w)m(e)h(ha)m(v)m(e)g(that)456 1102 y(\(66\))484 b Fx(G)6 b FE([)p Fw(\036)p FE(])27 b(=)h(\002)1498 1117 y Fl(1)1559 1102 y Fx(\000)1669 1034 y Fw(\017N)p 1669 1079 128 4 v 1709 1170 a FE(2)1840 1102 y(ln)1948 1034 y Fw(\017\031)p 1948 1079 99 4 v 1969 1170 a(h)2078 1102 y FE(+)22 b Fw(a\017h)p FE(\001)p Fw(\036)h FE(+)f Fw(\017)17 b FE(\004)2703 1117 y Fl(8)2759 1102 y Fw(:)456 1299 y FE(Also,)32 b(b)m(y)h(\(62\))f(and)h(\(65\),)456 1493 y(\(67\))223 b Fx(j)p FE(\004)946 1508 y Fl(8)986 1493 y Fx(j)44 b(\024)g Fw(C)7 b(h)1312 1452 y Fl(2)1368 1382 y Fv(h)1416 1493 y Fx(k)p Fw(D)1550 1452 y Fl(2)1588 1493 y Fw(\036)p Fx(k)1696 1508 y Fq(L)1744 1489 y Fm(1)1835 1493 y FE(+)22 b Fx(k)p Fw(D)2067 1452 y Fl(3)2106 1493 y Fw(\036)p Fx(k)2214 1508 y Fq(L)2262 1489 y Fm(1)2331 1493 y FE(\()2369 1403 y Fv(p)p 2469 1403 145 4 v 90 x Fw(\017=h)g FE(+)g Fx(k)p Fw(D)s(\036)p Fx(k)2975 1508 y Fq(L)3023 1489 y Fm(1)3091 1493 y FE(\))3129 1382 y Fv(i)3193 1493 y Fw(:)456 1681 y FE(W)-8 b(e)33 b(no)m(w)g(de\014ne)628 1872 y(\002)704 1887 y Fl(2)743 1872 y FE(\()p Fw(x;)17 b(v)931 1831 y Fs(\003)971 1872 y FE(\))83 b(=)1260 1804 y Fw(h)p 1260 1849 57 4 v 1264 1940 a FE(4)1343 1761 y Fv(\020)1403 1872 y Fx(j)p Fw(D)s(\036)p FE(\()p Fw(x)p FE(\))p Fx(j)1732 1831 y Fl(2)1792 1872 y Fx(\000)23 b(j)p Fw(D)s(\036)p FE(\()p Fw(x)f FE(+)g Fw(hv)2382 1831 y Fs(\003)2421 1872 y FE(\))p Fx(j)2487 1831 y Fl(2)2526 1761 y Fv(\021)628 2117 y FE(\002)704 2132 y Fl(3)743 2117 y FE(\()p Fw(x;)17 b(v)931 2076 y Fs(\003)971 2117 y FE(\))83 b(=)g Fw(\036)p FE(\()p Fw(x)p FE(\))22 b Fx(\000)1571 2050 y Fw(h)p 1571 2094 V 1575 2185 a FE(4)1654 2117 y Fx(j)p Fw(D)s(\036)p FE(\()p Fw(x)g FE(+)g Fw(hv)2144 2076 y Fs(\003)2183 2117 y FE(\))p Fx(j)2249 2076 y Fl(2)2310 2117 y Fx(\000)h Fw(h)p Fx(j)p Fw(v)2545 2076 y Fs(\003)2584 2117 y Fx(j)2612 2076 y Fl(2)2673 2117 y Fx(\000)g Fw(\036)p FE(\()p Fw(x)f FE(+)g Fw(hv)3151 2076 y Fs(\003)3190 2117 y FE(\))17 b Fw(;)456 2321 y FE(so)32 b(that)754 2282 y Fl(6)789 2321 y FE(,)h(b)m(y)h(\(55\),)456 2495 y(\(68\))449 b(\002)1155 2510 y Fl(1)1222 2495 y FE(=)28 b Fx(\000)p Fw(hH)8 b FE(\()p Fw(D)s(\036)p FE(\()p Fw(x)p FE(\))p Fw(;)17 b(x)p FE(\))22 b(+)g Fw(\036)p FE(\()p Fw(x)p FE(\))g(+)g(\002)2501 2510 y Fl(2)2562 2495 y Fx(\000)h FE(\002)2738 2510 y Fl(3)2794 2495 y Fw(:)456 2669 y FE(Note)32 b(that)1003 2842 y Fx(j)p FE(\002)1107 2857 y Fl(2)1146 2842 y Fx(j)83 b(\024)g FE(2)p Fw(h)17 b Fx(k)p Fw(D)s(\036)p Fx(k)1781 2857 y Fq(L)1829 2839 y Fm(1)1897 2728 y Fv(\014)1897 2788 y(\014)1897 2847 y(\014)1930 2842 y Fw(D)s(\036)p FE(\()p Fw(x)22 b FE(+)g Fw(hv)2392 2801 y Fs(\003)2431 2842 y FE(\))g Fx(\000)h Fw(D)s(\036)p FE(\()p Fw(x)p FE(\))2864 2728 y Fv(\014)2864 2788 y(\014)2864 2847 y(\014)1257 3031 y Fx(\024)83 b Fw(C)24 b(h)1567 2990 y Fl(2)1623 3031 y Fx(k)p Fw(D)s(\036)p Fx(k)1865 3046 y Fq(L)1913 3027 y Fm(1)1998 3031 y Fx(k)p Fw(D)2132 2990 y Fl(2)2171 3031 y Fw(\036)p Fx(k)2279 3046 y Fq(L)2327 3027 y Fm(1)2412 3031 y Fx(j)p Fw(v)2491 2990 y Fs(\003)2530 3031 y Fx(j)1257 3191 y(\024)83 b Fw(C)24 b(h)1567 3150 y Fl(2)1623 3191 y Fx(k)p Fw(D)s(\036)p Fx(k)1865 3150 y Fl(2)1865 3216 y Fq(L)1913 3197 y Fm(1)1998 3191 y Fx(k)p Fw(D)2132 3150 y Fl(2)2171 3191 y Fw(\036)p Fx(k)2279 3206 y Fq(L)2327 3187 y Fm(1)2412 3191 y Fw(;)456 3365 y FE(thanks)33 b(to)f(\(48\).)43 b(F)-8 b(urthermore,)32 b(b)m(y)i(using)e(\(48\))g(and)h(\(49\),)f(one)h(has)g(that)721 3539 y Fx(j)p FE(\002)825 3554 y Fl(3)864 3539 y Fx(j)82 b(\024)i(j)p Fw(\036)p FE(\()p Fw(x)p FE(\))22 b Fx(\000)g Fw(\036)p FE(\()p Fw(x)h FE(+)f Fw(hv)1852 3498 y Fs(\003)1891 3539 y FE(\))g(+)g Fw(h)17 b(D)s(\036)p FE(\()p Fw(x)22 b FE(+)g Fw(hv)2584 3498 y Fs(\003)2623 3539 y FE(\))g Fx(\001)g Fw(v)2784 3498 y Fs(\003)2823 3539 y Fx(j)1135 3736 y FE(+)p Fw(h)1284 3621 y Fv(\014)1284 3681 y(\014)1284 3741 y(\014)1327 3668 y FE(1)p 1327 3713 49 4 v 1327 3804 a(4)1385 3736 y Fx(j)p Fw(D)s(\036)p FE(\()p Fw(x)g FE(+)g Fw(hv)1875 3695 y Fs(\003)1914 3736 y FE(\))p Fx(j)1980 3695 y Fl(2)2042 3736 y FE(+)g Fx(j)p Fw(v)2219 3695 y Fs(\003)2257 3736 y Fx(j)2285 3695 y Fl(2)2347 3736 y FE(+)g Fw(D)s(\036)p FE(\()p Fw(x)g FE(+)g Fw(hv)2907 3695 y Fs(\003)2946 3736 y FE(\))g Fx(\001)g Fw(v)3107 3695 y Fs(\003)3146 3621 y Fv(\014)3146 3681 y(\014)3146 3741 y(\014)974 3928 y Fx(\024)84 b Fw(C)23 b(h)1284 3887 y Fl(2)1341 3928 y Fx(k)p Fw(D)1475 3887 y Fl(2)1514 3928 y Fw(\036)p Fx(k)1622 3943 y Fq(L)1670 3924 y Fm(1)1755 3928 y Fx(j)p Fw(v)1834 3887 y Fs(\003)1873 3928 y Fx(j)1901 3887 y Fl(2)1979 3928 y FE(+)38 b(0)974 4088 y Fx(\024)84 b Fw(C)23 b(h)1284 4047 y Fl(2)1341 4088 y Fx(k)p Fw(D)s(\036)p Fx(k)1583 4047 y Fl(2)1583 4113 y Fq(L)1631 4094 y Fm(1)1716 4088 y Fx(k)p Fw(D)1850 4047 y Fl(2)1888 4088 y Fw(\036)p Fx(k)1996 4103 y Fq(L)2044 4084 y Fm(1)2130 4088 y Fw(:)456 4262 y FE(Therefore,)33 b(setting)456 4436 y(\(69\))828 b(\002)1534 4451 y Fs(\003)1618 4436 y FE(:=)44 b(\002)1841 4451 y Fl(2)1903 4436 y Fx(\000)22 b FE(\002)2078 4451 y Fl(3)2140 4436 y FE(+)g Fw(\017)17 b FE(\004)2359 4451 y Fl(8)2415 4436 y Fw(;)456 4610 y FE(the)33 b(estimates)f(ab)s(o)m(v)m (e)h(and)g(\(67\))f(yield)g(that)726 4803 y Fx(j)p FE(\002)830 4818 y Fs(\003)869 4803 y Fx(j)83 b(\024)h Fw(C)23 b(h)1290 4762 y Fl(2)1346 4693 y Fv(n)1413 4803 y Fx(k)p Fw(D)s(\036)p Fx(k)1655 4762 y Fl(2)1655 4828 y Fq(L)1703 4809 y Fm(1)1788 4803 y Fx(k)p Fw(D)1922 4762 y Fl(2)1960 4803 y Fw(\036)p Fx(k)2068 4818 y Fq(L)2116 4799 y Fm(1)456 4803 y FE(\(70\))1141 5023 y(+)p Fw(\017)1256 4912 y Fv(h)1303 5023 y Fx(k)p Fw(D)1437 4982 y Fl(2)1476 5023 y Fw(\036)p Fx(k)1584 5038 y Fq(L)1632 5019 y Fm(1)1723 5023 y FE(+)f Fx(k)p Fw(D)1955 4982 y Fl(3)1994 5023 y Fw(\036)p Fx(k)2102 5038 y Fq(L)2150 5019 y Fm(1)2218 5023 y FE(\()2256 4933 y Fv(p)p 2356 4933 145 4 v 90 x Fw(\017=h)g FE(+)g Fx(k)p Fw(D)s(\036)p Fx(k)2862 5038 y Fq(L)2910 5019 y Fm(1)2979 5023 y FE(\))3017 4912 y Fv(io)3147 5023 y Fw(:)p 456 5115 499 4 v 555 5183 a Fl(6)591 5216 y FH(Of)27 b(course,)g(if)h Fp(K)6 b FH(\()p Fp(v)s FH(\))23 b(=)g FF(j)p Fp(v)s FF(j)1446 5185 y Fy(2)1483 5216 y FH(,)28 b(one)f(has)g(that)h Fp(H)7 b FH(\()p Fp(p;)14 b(x)p FH(\))24 b(=)e FF(j)p Fp(p)p FF(j)2479 5185 y Fy(2)2516 5216 y Fp(=)p FH(4)c(+)g Fp(U)9 b FH(\()p Fp(x)p FH(\).)p eop %%Page: 29 29 29 28 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)420 b(29)456 450 y FE(Also,)32 b(b)m(y)h(collecting)e(the)i(iden)m(tities)e(in)h (\(66\),)g(\(68\))g(and)h(\(69\),)f(w)m(e)h(ha)m(v)m(e)h(that)704 673 y Fx(G)6 b FE([)p Fw(\036)p FE(])28 b(=)g Fx(\000)p Fw(hH)8 b FE(\()p Fw(D)s(\036)p FE(\()p Fw(x)p FE(\))p Fw(;)17 b(x)p FE(\))22 b(+)g Fw(\036)p FE(\()p Fw(x)p FE(\))g Fx(\000)2123 606 y Fw(\017N)p 2123 650 128 4 v 2163 741 a FE(2)2295 673 y(ln)2402 606 y Fw(\017\031)p 2402 650 99 4 v 2423 741 a(h)2533 673 y FE(+)g Fw(a\017h)p FE(\001)p Fw(\036)h FE(+)f(\002)3113 688 y Fs(\003)3169 673 y Fw(:)456 879 y FE(This)32 b(and)h(\(70\))f(yield)g(the)h(desired) g(result.)1292 b Fj(\003)555 1067 y FE(The)34 b(previous)f(result)f (immediately)e(applies)h(to)i(a)f(\014xed)i(p)s(oin)m(t)2956 1028 y Fl(7)3023 1067 y FE(problem:)456 1219 y FI(Corollary)j(26.)42 b FD(Supp)-5 b(ose)34 b(that)456 1402 y FE(\(71\))1007 b Fx(G)6 b FE([)p Fw(\036)p FE(])28 b(=)f Fw(\036)22 b FE(+)g Fw(\025)2180 1417 y Fq(?)2236 1402 y Fw(:)456 1584 y FD(Then,)34 b(we)g(have)846 1802 y Fw(\017a)p FE(\001)p Fw(\036)23 b Fx(\000)f Fw(H)8 b FE(\()p Fw(D)1405 1817 y Fq(x)1449 1802 y Fw(\036;)17 b(x)p FE(\))27 b(=)1785 1735 y Fw(\025)1842 1750 y Fq(?)p 1785 1779 97 4 v 1805 1871 a Fw(h)1913 1802 y FE(+)2021 1735 y Fw(\017N)p 2021 1779 128 4 v 2032 1871 a FE(2)p Fw(h)2175 1802 y FE(ln)2283 1735 y Fw(\017N)p 2283 1779 V 2319 1871 a(h)2443 1802 y FE(+)22 b FD(err)-5 b(or)35 b(terms)p Fw(:)456 2002 y FD(The)d(c)-5 b(o)g(e\016cient)33 b Fw(a)g FD(and)g(the)g(err)-5 b(or)33 b(terms)g(have)g(the)g(same)f(form)h(as)g(in)g(Pr)-5 b(op)g(o-)456 2130 y(sition)34 b(25.)456 2283 y Fz(Remark.)99 b FE(Heuristically)-8 b(,)35 b(as)g Fw(h)e Fx(!)g FE(0)i(and)h Fw(\017=h)d Fx(!)f FE(0,)k(the)g(result)g(in)f(Corol-)456 2410 y(lary)f(26)h(means)g(that)g Fw(\025)1361 2425 y Fq(?)1400 2410 y Fw(=h)g FE(pla)m(ys)g(the)h(r^)-49 b(ole)34 b(of)h(the)g(\\e\013ectiv)m(e)h(Hamiltonian")p 456 2458 89 4 v 456 2538 a Fw(H)k FE(in)31 b(the)i(equation)1519 2676 y Fw(H)8 b FE(\()p Fw(D)1727 2691 y Fq(x)1770 2676 y Fw(\036;)17 b(x)p FE(\))22 b(+)p 2085 2596 V 22 w Fw(H)36 b FE(=)27 b(0)p Fw(:)456 2835 y FE(A)32 b(formal)e(justi\014cation)i (for)g(this)g(will)e(b)s(e)j(giv)m(en)g(in)f(Theorem)g(38.)1345 3069 y(8.)49 b Fz(Fixed)37 b(point)h(pr)n(oblems)555 3255 y FE(W)-8 b(e)34 b(no)m(w)g(deriv)m(e)f(from)f(the)i(ab)s(o)m(v)m (e)g(estimates)f(a)g(\014xed)h(p)s(oin)m(t)e(result)h(on)g(our)456 3383 y(iteration)d(sc)m(heme)k Fx(G)6 b FE(.)44 b(Namely)-8 b(,)32 b(an)m(y)h(iteration)e(on)h(the)h(sc)m(heme)h Fx(G)39 b FE(con)m(v)m(erges,)456 3511 y(in)g(the)h Fx(k)28 b(\001)e(k)934 3526 y Fq(])1006 3511 y FE(de\014ned)41 b(in)f(\(17\),)h(to)f(a)g(function)f(whic)m(h)i(solv)m(es)g(a)f (\014xed)h(p)s(oin)m(t)456 3639 y(t)m(yp)s(e)33 b(problem,)f(up)g(to)h (additiv)m(e)f(constan)m(ts.)456 3791 y FI(Theorem)d(27.)36 b FD(Ther)-5 b(e)28 b(exists)g Fw(\025)1693 3806 y Fq(?)1760 3791 y Fx(2)g Fn(R)1920 3755 y Fq(N)2021 3791 y FD(and)g(a)g Fn(Z)2351 3755 y Fq(N)2416 3791 y FD(-p)-5 b(erio)g(dic)27 b(Lipschitz)h(semi-)456 3919 y(c)-5 b(onc)g(ave)33 b(function)i Fw(\036)1256 3934 y Fq(?)1330 3919 y FD(so)f(that)456 4101 y FE(\(72\))967 b Fx(G)6 b FE([)p Fw(\036)1747 4116 y Fq(?)1787 4101 y FE(])28 b(=)f Fw(\036)2003 4116 y Fq(?)2065 4101 y FE(+)22 b Fw(\025)2220 4116 y Fq(?)2275 4101 y Fw(:)456 4284 y FD(F)-7 b(urthermor)i(e,)32 b(given)h(any)g Fn(Z)1554 4247 y Fq(N)1619 4284 y FD(-p)-5 b(erio)g(dic)32 b(semic)-5 b(onc)g(ave)32 b(function)h Fw(\036)3009 4299 y Fl(0)3048 4284 y FD(,)h(we)f(have)456 4411 y(that)456 4593 y FE(\(73\))809 b(lim)1387 4653 y Fq(n)p Fs(!)p Fl(+)p Fs(1)1642 4593 y Fx(kG)1757 4552 y Fq(n)1804 4593 y FE([)p Fw(\036)1889 4608 y Fl(0)1929 4593 y FE(])22 b Fx(\000)h Fw(\036)2136 4608 y Fq(?)2175 4593 y Fx(k)2225 4608 y Fq(])2300 4593 y FE(=)45 b(0)17 b Fw(:)456 4802 y FD(The)24 b(Lipschitz)g(c)-5 b(onstant)25 b(and)f(the)h(semic)-5 b(onc)g(avity)24 b(mo)-5 b(dulus)24 b(of)h Fw(\036)2896 4817 y Fq(?)2960 4802 y FD(ar)-5 b(e)25 b(b)-5 b(ounde)g(d)456 4930 y(by)35 b(a)f(c)-5 b(onstant)35 b(dep)-5 b(ending)33 b(only)i(on)f Fw(N)10 b FD(,)36 b Fw(C)2084 4945 y Fq(U)2177 4930 y FD(and)f Fw(C)2437 4945 y Fq(K)2505 4930 y FD(.)p 456 5007 499 4 v 555 5074 a Fl(7)591 5106 y FH(F)-7 b(or)28 b(the)h(existence)f(of)h(functions)g(satisfying)g(\(71\),)f(see)h (Theorem)f(27.)40 b(Note)28 b(also)g(that,)456 5216 y(if)g Fp(\036)g FH(satis\014es)f(\(71\),)g(so)g(do)r(es)g FF(G)1456 5185 y Fi(n)1502 5216 y FH([)p Fp(\036)p FH(],)h(for)f(an)n(y)g Fp(n)c FF(2)g Ff(N)t FH(,)34 b(due)28 b(to)f(Prop)r(osition)f(7.)p eop %%Page: 30 30 30 29 bop 456 251 a FC(30)666 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)555 450 y FD(A)n(lso,)42 b Fw(\036)870 465 y Fq(?)950 450 y FD(and)e Fw(\025)1202 465 y Fq(?)1282 450 y FD(enjoy)h(the)f(fol)5 b(lowing)40 b(uniqueness)g(pr)-5 b(op)g(erties:)56 b(if)40 b(ther)-5 b(e)456 578 y(exist)41 b Fw(\025)g Fx(2)g Fn(R)960 542 y Fq(N)1075 578 y FD(and)g(a)h (Lipschitz)f Fn(Z)1851 542 y Fq(N)1916 578 y FD(-p)-5 b(erio)g(dic)41 b(function)g Fw(\036)h FD(so)f(that)i Fx(G)6 b FE([)p Fw(\036)p FE(])41 b(=)456 706 y Fw(\036)21 b FE(+)h Fw(\025)p FD(,)35 b(then)g Fw(\036)27 b FE(=)h Fw(\036)1219 721 y Fq(?)1280 706 y FE(+)22 b Fw(c)35 b FD(for)g(some)f Fw(c)27 b Fx(2)h Fn(R)46 b FD(and)35 b Fw(\025)27 b FE(=)h Fw(\025)2565 721 y Fq(?)2604 706 y FD(.)555 834 y(Final)5 b(ly,)34 b(if)456 1061 y FE(\(74\))1283 926 y Fv(Z)1338 1151 y Fu(R)1386 1132 y Ft(N)1464 1061 y Fw(e)1509 1020 y Fs(\000)1574 986 y Ft(hK)s Fr(\()p Ft(v)r Fr(\))p 1574 1005 172 3 v 1647 1046 a Ft(\017)1760 1061 y Fx(j)p Fw(D)s(K)7 b FE(\()p Fw(v)t FE(\))p Fx(j)17 b Fw(dv)46 b(<)e FE(+)p Fx(1)17 b Fw(;)456 1293 y FD(then)28 b(the)g(ab)-5 b(ove)27 b(uniqueness)h(pr)-5 b(op)g(erty)28 b(holds)f(even)h(if)g(we)g(r)-5 b(eplac)g(e)27 b(the)h(assump-)456 1421 y(tion)34 b(that)i Fw(\036)e FD(is)h(Lipschitz)f(with)h(the)g(one) f(that)h(it)g(is)g(b)-5 b(ounde)g(d.)456 1620 y(Pr)g(o)g(of.)41 b FE(Let)k(us)h(consider)f(the)g(quotien)m(t)h(space)g Fw(Y)70 b FE(:=)49 b Fw(C)2678 1584 y Fl(0)2717 1620 y FE(\()p Fn(R)2821 1584 y Fq(N)2894 1620 y FE(\))p Fw(=)p Fn(R)5 b FE(,)54 b(that)45 b(is,)456 1748 y(let)34 b(us)h(iden)m(tify)g (con)m(tin)m(uous)g(functions)g(if)f(and)h(only)g(if)e(they)j(di\013er) f(b)m(y)h(a)e(real)456 1876 y(n)m(um)m(b)s(er.)42 b(Then,)29 b Fw(Y)47 b FE(is)27 b(a)f(Banac)m(h)h(space,)i(with)e(norm)e Fx(k)10 b(\001)g(k)2634 1891 y Fq(])2692 1876 y FE(as)27 b(de\014ned)h(in)e(\(17\):)456 2003 y(see,)i(e.g.,)f(Theorem)f(3.14-A)e (on)i(page)f(105)g(of)g([T)-8 b(a)m(y58)q(])25 b(for)g(a)h(pro)s(of)e (of)h(this)g(fact.)456 2131 y(Note)39 b(that)f(the)i(semiconca)m(vit)m (y)f(de\014nition)e(in)i(\(36\))f(passes)i(to)f(the)g(quotien)m(t)456 2259 y(in)46 b Fw(Y)21 b FE(:)72 b(th)m(us)48 b(w)m(e)g(ma)m(y)f(tak)m (e)h Fw(S)53 b FE(to)46 b(b)s(e)i(the)f(\(class)g(of)g(the\))g (functions)g(in)f Fw(Y)456 2387 y FE(with)34 b(semiconca)m(vit)m(y)h (mo)s(dulus)e(b)s(ounded)j(b)m(y)f(an)g(appropriately)e(large)h(\006.) 50 b(It)456 2515 y(is)c(easily)g(seen)i(that)f Fw(S)53 b FE(is)46 b(closed)h(in)g Fw(Y)21 b FE(.)87 b(The)47 b(results)h(in)e(Prop)s(ositions)g(5)456 2643 y(and)39 b(7)g(also)g(sho)m(w)h(that)g Fx(G)45 b FE(is)39 b(w)m(ell)g(de\014ned) i(on)e Fw(Y)61 b FE(and,)41 b(b)m(y)g(Theorem)e(13,)i Fx(G)456 2771 y FE(sends)i Fw(S)k FE(in)m(to)41 b(itself.)69 b(F)-8 b(unctions)41 b(in)g Fw(S)47 b FE(are)42 b(also)e(uniformly)g (Lipsc)m(hitz)h(\(see)456 2898 y(Theorem)32 b(44\),)h(due)g(to)f(their) g(semiconca)m(vit)m(y)h(prop)s(erties.)555 3026 y(Therefore,)k(b)m(y)f (Theorem)g(9)f(and)g(Theorem)g(41,)h(w)m(e)g(deduce)h(that)e Fx(G)41 b FE(has)36 b(a)456 3154 y(unique)f(\014xed)i(p)s(oin)m(t)d(in) h Fw(S)j Fx(\022)33 b Fw(Y)56 b FE(and)36 b(that)f(\(73\))g(holds.)51 b(Unfolding)34 b(bac)m(k)i(the)456 3282 y(quotien)m(t)c(space)i Fw(Y)54 b FE(to)32 b(the)h(co)m(v)m(er)h(space)g Fw(X)8 b FE(,)32 b(w)m(e)i(get)e(\(72\).)555 3410 y(Let)46 b(us)f(no)m(w)h (deal)f(with)g(the)g(uniqueness)i(prop)s(ert)m(y)-8 b(.)82 b(Observ)m(e)47 b(that,)h(b)m(y)456 3538 y(c)m(hanging)32 b(v)-5 b(ariable)31 b(of)h(in)m(tegration)f(in)g(\(6\),)i(w)m(e)g(ha)m (v)m(e)h(that)1024 3780 y Fx(G)6 b FE([)p Fw(\036)p FE(])28 b(=)f Fx(\000)p Fw(\017)17 b FE(ln)1564 3639 y Fv(\024)1663 3712 y FE(1)p 1626 3757 124 4 v 1626 3848 a Fw(h)1682 3819 y Fq(N)1776 3644 y Fv(Z)1831 3869 y Fu(R)1879 3851 y Ft(N)1958 3780 y Fw(e)2003 3738 y Fs(\000)2068 3705 y Ft(hL)p Fr(\()p Ft(x;)p Fr(\()p Ft(w)q Fm(\000)p Ft(x)p Fr(\))p Ft(=h)p Fr(\)+)p Ft(\036)p Fr(\()p Ft(w)q Fr(\))p 2068 3723 602 3 v 2356 3764 a Ft(\017)2700 3780 y Fw(dw)2824 3639 y Fv(\025)456 4021 y FE(and)31 b(therefore,)i(if)d Fx(G)6 b FE([)p Fw(\036)p FE(])28 b(=)g Fw(\036)20 b FE(+)g Fw(\025)p FE(,)31 b(it)g(follo)m(ws)f(that)i Fw(\036)f FE(is)g(Lipsc)m(hitz,)h(since)g(so)g(is)456 4149 y Fw(K)39 b FE(\(recall)31 b(also)h(\(74\)\).)43 b(Therefore,)34 b(b)m(y)f(Theorem)g(9,)977 4341 y Fx(k)p Fw(\036)22 b Fx(\000)g Fw(\036)1264 4356 y Fq(?)1303 4341 y Fx(k)1353 4356 y Fq(])1469 4341 y FE(=)83 b Fx(kG)6 b FE([)p Fw(\036)p FE(])22 b Fx(\000)h(G)6 b FE([)p Fw(\036)2127 4356 y Fq(?)2166 4341 y FE(])23 b Fx(\000)f Fw(\025)g FE(+)g Fw(\025)2549 4356 y Fq(?)2589 4341 y Fx(k)2639 4356 y Fq(])1469 4501 y FE(=)83 b Fx(kG)6 b FE([)p Fw(\036)p FE(])22 b Fx(\000)h(G)6 b FE([)p Fw(\036)2127 4516 y Fq(?)2166 4501 y FE(])p Fx(k)2243 4516 y Fq(])1468 4684 y Fx(\024)1628 4573 y Fv(\020)1688 4684 y FE(1)22 b Fx(\000)g Fw(C)i Fx(k)p Fw(\036)e Fx(\000)g Fw(\036)2239 4699 y Fq(?)2278 4684 y Fx(k)2328 4643 y Fq(N)2328 4708 y(])2395 4573 y Fv(\021)2472 4684 y Fx(k)p Fw(\036)f Fx(\000)i Fw(\036)2759 4699 y Fq(?)2798 4684 y Fx(k)2848 4699 y Fq(])2896 4684 y Fw(;)456 4896 y FE(sho)m(wing)32 b(that)h Fx(k)p Fw(\036)21 b Fx(\000)i Fw(\036)1326 4911 y Fq(?)1365 4896 y Fx(k)1415 4911 y Fq(])1474 4896 y FE(=)28 b(0)k(and)h(th)m(us)g Fw(\036)28 b FE(=)f Fw(\036)2310 4911 y Fq(?)2372 4896 y FE(+)22 b Fw(c)p FE(.)555 5024 y(But)33 b(then,)g(b)m(y)h(Prop)s (osition)d(7,)1179 5216 y Fw(\025)22 b Fx(\000)h Fw(\025)1415 5231 y Fq(?)1482 5216 y FE(=)k Fx(G)6 b FE([)p Fw(\036)p FE(])22 b Fx(\000)h Fw(\036)f Fx(\000)h(G)6 b FE([)p Fw(\036)2214 5231 y Fq(?)2253 5216 y FE(])22 b(+)g Fw(\036)2458 5231 y Fq(?)2525 5216 y FE(=)28 b(0)17 b Fw(:)p eop %%Page: 31 31 31 30 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)420 b(31)3367 450 y Fj(\003)555 634 y FE(W)-8 b(e)28 b(remark)g(that)g(a)f(\014xed)i(p)s (oin)m(t)e(result)h(ma)m(y)f(also)g(b)s(e)h(pro)m(v)m(ed)h(applying)e (the)456 762 y(Sc)m(hauder)34 b(Fixed)e(P)m(oin)m(t)g(Theorem)h(\(see,) h(e.g.,)f Fx(x)g FE(9.2.2)f(in)g([Ev)-5 b(a98)o(]\))33 b(to)1562 940 y Fx(G)6 b FE([)p Fw(\036)p FE(]\()p Fw(x)p FE(\))23 b Fx(\000)f(G)6 b FE([)p Fw(\036)p FE(]\(0\))17 b Fw(;)456 1118 y FE(seen)29 b(as)f(acting)f(on)h(the)h(space)g(of)e (functions)h(with)g(suitably)f(b)s(ounded)i Fw(L)3184 1081 y Fs(1)3287 1118 y FE(and)456 1245 y(Lipsc)m(hitz)d(norm)f(\(and)h (suc)m(h)i(space)f(is)e(con)m(v)m(ex)k(and)d(compact)g(b)m(y)h(the)f (Theorem)456 1373 y(of)37 b(Ascoli\).)59 b(The)39 b(estimates)f(needed) h(for)f(making)e(suc)m(h)k(an)e(argumen)m(t)f(w)m(ork)456 1501 y(are)d(giv)m(en)g(b)m(y)i(Prop)s(osition)c(5)i(and)h(Theorem)g (14.)48 b(This)35 b(approac)m(h,)g(ho)m(w)m(ev)m(er,)456 1629 y(only)24 b(giv)m(es)h(the)g(\014xed)h(p)s(oin)m(t)e(prop)s(ert)m (y)i(\(72\))e(but)h(it)f(do)s(es)h(not)g(giv)m(e)g(information)456 1757 y(either)k(on)g(the)g(con)m(v)m(ergence)j(of)c(the)i(iterations)e (of)g Fx(G)35 b FE(or)29 b(on)g(the)h(uniqueness)h(of)456 1885 y(the)i(\014xed)g(p)s(oin)m(ts,)g(while)e(Theorems)j(27)e(and)g (41)h(do.)456 2035 y FI(Corollary)39 b(28.)k FD(Fix)35 b Fw(x)c Fx(2)g Fn(R)1566 1999 y Fq(N)1640 2035 y FD(.)49 b(Under)36 b(the)h(assumptions)e(and)h(the)h(notation)456 2163 y(of)d(The)-5 b(or)g(em)34 b(27,)g(let)h(the)g(me)-5 b(asur)g(e)35 b Fw(\026)1875 2178 y Fq(?)1949 2163 y FD(b)-5 b(e)34 b(de\014ne)-5 b(d)34 b(by)456 2353 y FE(\(75\))591 b Fw(d\026)1331 2368 y Fq(?)1397 2353 y FE(:=)28 b Fw(e)1573 2312 y Fs(\000)1638 2278 y Ft(h)11 b(L)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft(\036)1946 2286 y(?)1981 2278 y Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p Fm(\000)p Ft(\036)2265 2286 y(?)2300 2278 y Fr(\()p Ft(x)p Fr(\))p Fm(\000)p Ft(\025)2468 2286 y(?)p 1638 2297 866 3 v 2058 2338 a(\017)2534 2353 y Fw(dv)20 b(:)456 2531 y FD(Then,)33 b Fw(\026)798 2546 y Fq(?)871 2531 y FD(is)g(a)h(pr)-5 b(ob)g(ability)33 b(me)-5 b(asur)g(e)34 b(on)f Fn(R)2107 2495 y Fq(N)2180 2531 y FD(.)45 b(F)-7 b(urthermor)i(e,)33 b(if)g Fw( )3003 2546 y Fq(?)3071 2531 y FE(:=)28 b Fx(E)9 b FE([)p Fw(\036)3349 2546 y Fq(?)3387 2531 y FE(])p FD(,)456 2659 y(then)1534 2792 y Fw( )1597 2807 y Fq(?)1637 2792 y FE(\()p Fw(x)p FE(\))28 b(=)g Fw(e)1945 2751 y Fs(\000)2010 2721 y Ft(\025)2047 2729 y(?)p 2010 2735 73 3 v 2033 2777 a(\017)2096 2792 y Fx(L)p FE([)p Fw( )2255 2807 y Fq(?)2295 2792 y FE(])17 b Fw(:)456 2976 y FD(Pr)-5 b(o)g(of.)41 b FE(Straigh)m(tforw)m(ard)32 b(from)f(\(72\).)1480 b Fj(\003)555 3159 y FE(Other)43 b(features)g(of)f(the)h(\014xed)h(p)s(oin)m(ts)e(for)g Fx(G)49 b FE(will)40 b(b)s(e)j(discussed)h(in)e Fx(x)h FE(10,)456 3287 y(where)e(w)m(e)g(will)c(relate)j(the)g(term)f Fw(\025)1838 3302 y Fq(?)1918 3287 y FE(in)g(\(72\))g(and)h(the)h (measure)f Fw(\026)3071 3302 y Fq(?)3150 3287 y FE(in)f(\(75\))456 3415 y(with)32 b(an)g(en)m(trop)m(y)i(p)s(enalized)e(Mather)h(measure.) 1391 3638 y(9.)49 b Fz(Conver)n(gence)36 b(issues)555 3824 y FE(In)25 b(this)e(section,)j(w)m(e)f(use)g(a)f(v)-5 b(ariation)22 b(of)h(the)i(results)f(b)m(y)h([Sou85])f(and)g([BS91])456 3952 y(to)i(pro)m(v)m(e)h(the)g FD(c)-5 b(onver)g(genc)g(e)28 b(of)g(the)i(entr)-5 b(opy)29 b(p)-5 b(enalize)g(d)28 b(scheme)d FE(for)h(small)e(time)456 4080 y(step)44 b Fw(h)p FE(.)76 b(W)-8 b(e)44 b(will)d(pro)m(v)m(e)j(a)f(rigorous)f (relation)g(b)s(et)m(w)m(een)j(our)f(linear)d(\(resp.,)456 4208 y(non-linear\))32 b(sc)m(heme)j(and)g(the)f(heat)h(equation)f (\(resp.,)i(the)e(Hamilton-Jacobi)456 4335 y(equation\).)555 4463 y(W)-8 b(e)35 b(\014rst)h(deal)e(with)g(the)h(con)m(v)m(ergence)i (of)e(the)2380 4438 y(^)2356 4463 y Fx(L)p FE(-sc)m(heme)g(as)g Fw(h)c Fx(!)g FE(0,)k(for)g(a)456 4591 y(\014xed)658 4553 y Fl(8)721 4591 y Fw(\017)28 b(>)f FE(0.)41 b(F)-8 b(or)26 b(this,)h(w)m(e)h(stress)g(that)2017 4566 y(^)1993 4591 y Fx(L)f FE(dep)s(ends)h(on)e Fw(h)h FE(b)m(y)g(writing)3155 4566 y(^)3131 4591 y Fx(L)3200 4606 y Fq(h)3271 4591 y FE(\(the)456 4719 y(dep)s(endence)f(on)e Fw(\017)g FE(is)g(not)f(explicitly)f(written\).)41 b(Consider)30 b(\026)-55 b Fw(u)27 b Fx(2)i Fw(W)2925 4683 y Fl(1)p Fq(;)p Fs(1)3054 4719 y FE(\()p Fn(T)3155 4683 y Fq(N)3225 4719 y FE(\))24 b(and)p 456 4788 499 4 v 555 4855 a Fl(8)591 4887 y FH(The)31 b(case)f(when)h Fp(\017)e FF(!)g FH(0)i(to)r(o)g(is)g (actually)f(easier)g(and)h(can)g(b)r(e)g(dealt)g(with)h(b)n(y)f(a)g(mo) r(di-)456 4996 y(\014cation)d(of)h(the)h(pro)r(of)e(of)h(Theorem)g(29.) 40 b(In)30 b(this)f(circumstance,)g(Theorem)f(29)g(holds)h(with)456 5106 y Fp(\017)34 b FH(=)g(0)h(in)g(\(77\),)h(namely)e Fp(u)p FH(\()p Fp(x;)14 b(t)p FH(\))35 b(=)40 b(\026)-48 b Fp(u)p FH(\()p Fp(x)p FH(\).)59 b(W)-7 b(e)35 b(omit)g(the)g(details) f(of)h(the)g(pro)r(of,)h(since)e(it)456 5216 y(closely)26 b(follo)n(ws)h(the)h(one)f(of)h(Theorem)e(29.)p eop %%Page: 32 32 32 31 bop 456 251 a FC(32)666 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)456 450 y FE(assume,)31 b(for)e(simplicit)m(y)-8 b(,)1398 411 y Fl(9)1462 450 y FE(that)30 b(1)p Fw(=h)d Fx(2)h Fn(N)9 b FE(.)49 b(W)-8 b(e)30 b(de\014ne)i Fw(u)2589 465 y Fq(h)2661 450 y FE(:)c Fn(T)2779 414 y Fq(N)2866 450 y Fx(\002)18 b FE([0)p Fw(;)f FE(1])27 b Fx(\000)-16 b(!)27 b Fn(R)456 578 y FE(in)k(the)i(follo)m(wing)d(w)m(a)m(y)-8 b(.)45 b(Let)1599 760 y Fw(u)1655 775 y Fq(h)1699 760 y FE(\()p Fw(x;)17 b FE(0\))44 b(:=)50 b(\026)-55 b Fw(u)p FE(\()p Fw(x)p FE(\))456 941 y(and)32 b(then,)i(iterativ)m(ely)-8 b(,)1345 1123 y Fw(u)1401 1138 y Fq(h)1446 1123 y FE(\()p Fw(x;)17 b(t)p FE(\))28 b(:=)1838 1098 y(^)1814 1123 y Fx(L)1883 1138 y Fq(t)p Fs(\000)p Fq(ih)2032 1123 y FE([)p Fw(u)2115 1138 y Fq(h)2160 1123 y FE(\()p Fx(\001)p Fw(;)17 b(ih)p FE(\)]\()p Fw(x)p FE(\))456 1305 y(if)24 b Fw(t)k Fx(2)g FE(\()p Fw(ih;)17 b FE(\()p Fw(i)9 b FE(+)g(1\))p Fw(h)p FE(],)28 b(for)d Fw(i)j FE(=)g(0)p Fw(;)17 b(:)g(:)g(:)e(;)i FE(\(1)p Fw(=h)p FE(\))9 b Fx(\000)g FE(1.)41 b(Then,)28 b(w)m(e)f(ha)m(v)m(e)h(the)e(follo)m (wing)456 1433 y(con)m(v)m(ergence)35 b(result:)456 1585 y FI(Theorem)i(29.)k FD(L)-5 b(et)36 b Fw(K)7 b FE(\()p Fw(v)t FE(\))27 b(=)h Fx(j)p Fw(v)t Fx(j)1739 1549 y Fl(2)1812 1585 y FD(and)456 1837 y FE(\(76\))858 b Fw(a)27 b FE(:=)1707 1683 y Fv(R)1754 1798 y Fu(R)1802 1779 y Ft(N)1880 1764 y Fw(e)1925 1728 y Fs(\000j)p Fq(v)r Fs(j)2056 1704 y Fr(2)2095 1764 y Fx(j)p Fw(v)t Fx(j)2202 1728 y Fl(2)2257 1764 y Fw(dv)p 1707 1814 652 4 v 1747 1906 a FE(2)1813 1826 y Fv(R)1860 1941 y Fu(R)1908 1922 y Ft(N)1986 1906 y Fw(e)2031 1878 y Fs(\000j)p Fq(v)r Fs(j)2162 1859 y Fr(2)2217 1906 y Fw(dv)2385 1837 y(:)456 2078 y FD(L)-5 b(et)35 b Fw(u)27 b FE(=)h Fw(u)p FE(\()p Fw(x;)17 b(t)p FE(\))34 b FD(b)-5 b(e)35 b(the)g(unique)f(\(visc)-5 b(osity\))35 b(solution)f(of)456 2260 y FE(\(77\))840 b Fw(@)1521 2275 y Fq(t)1551 2260 y Fw(u)p FE(\()p Fw(x;)17 b(t)p FE(\))28 b(=)f Fw(\017a)p FE(\001)2119 2275 y Fq(x)2164 2260 y Fw(u)p FE(\()p Fw(x;)17 b(t)p FE(\))456 2441 y FD(in)31 b Fn(T)635 2405 y Fq(N)720 2441 y Fx(\002)15 b FE(\(0)p Fw(;)i FE(1])p FD(,)32 b(with)f Fw(u)p FE(\()p Fw(x;)17 b FE(0\))27 b(=)33 b(\026)-55 b Fw(u)p FE(\()p Fw(x)p FE(\))p FD(.)44 b(Then,)31 b Fw(u)2297 2456 y Fq(h)2373 2441 y FD(c)-5 b(onver)g(ges)30 b(uniformly)h(to)h Fw(u)p FD(.)456 2629 y(Pr)-5 b(o)g(of.)41 b FE(The)22 b(pro)s(of)f(is)h(a)f(minor)f(v)-5 b(ariation)19 b(of)j(the)g(one)g(in) f(Theorem)h(2.1)f(of)g([BS91])456 2757 y(\(see)36 b(also)f([Sou85]\).) 53 b(Since,)37 b(in)e(our)h(case,)i(the)e(argumen)m(ts)g(in)m(v)m(olv)m (ed)g(are)g(ele-)456 2885 y(men)m(tary)-8 b(,)32 b(w)m(e)i(pro)m(vide)f (full)e(details,)g(for)h(the)h(reader's)h(con)m(v)m(enience.)555 3013 y(W)-8 b(e)33 b(note)g(that,)f(if)g Fw(t)c Fx(2)g FE(\()p Fw(ih;)17 b FE(\()p Fw(i)22 b FE(+)g(1\))p Fw(h)p FE(])33 b(and)f Fw(x)d Fx(2)f Fn(T)2420 2977 y Fq(N)1374 3194 y Fx(j)p Fw(u)1458 3209 y Fq(h)1502 3194 y FE(\()p Fw(x;)17 b(t)p FE(\))p Fx(j)28 b(\024)49 b FE(sup)1873 3283 y Fq(x)p Fs(2)p Fu(T)2010 3264 y Ft(N)2078 3194 y Fx(j)p Fw(u)2162 3209 y Fq(h)2206 3194 y FE(\()p Fw(x;)17 b(ih)p FE(\))p Fx(j)p Fw(;)456 3429 y FE(and)32 b(so)703 3611 y(sup)456 3699 y Fq(x)p Fs(2)p Fu(T)593 3681 y Ft(N)644 3699 y Fq(;)20 b(t)p Fs(2)p Fl(\()p Fq(ih;)p Fl(\()p Fq(i)p Fl(+1\))p Fq(h)p Fl(])1114 3611 y Fx(j)p Fw(u)1198 3626 y Fq(h)1242 3611 y FE(\()p Fw(x;)d(t)p FE(\))p Fx(j)27 b(\024)276 b FE(sup)1613 3699 y Fq(x)p Fs(2)p Fu(T)1750 3681 y Ft(N)1801 3699 y Fq(;)19 b(t)p Fs(2)p Fl(\(\()p Fq(i)p Fs(\000)p Fl(1\))p Fq(h;ih)p Fl(])2271 3611 y Fx(j)p Fw(u)2355 3626 y Fq(h)2399 3611 y FE(\()p Fw(x;)e(t)p FE(\))p Fx(j)27 b(\024)h(\001)17 b(\001)g(\001)26 b(\024)j(k)6 b FE(\026)-55 b Fw(u)o Fx(k)3174 3629 y Fq(L)3222 3610 y Fm(1)3287 3629 y Fl(\()p Fu(T)3364 3610 y Ft(N)3416 3629 y Fl(\))3464 3611 y Fw(;)456 3857 y FE(that)32 b(is)1301 3994 y Fx(k)p Fw(u)1407 4009 y Fq(h)1452 3994 y Fx(k)1502 4013 y Fq(L)1550 3994 y Fm(1)1614 4013 y Fl(\()p Fu(T)1691 3994 y Ft(N)1744 4013 y Fs(\002)p Fl([0)p Fq(;)p Fl(1]\))2004 3994 y Fx(\024)45 b(k)6 b FE(\026)-55 b Fw(u)o Fx(k)2281 4013 y Fq(L)2329 3994 y Fm(1)2394 4013 y Fl(\()p Fu(T)2471 3994 y Ft(N)2523 4013 y Fl(\))2572 3994 y Fw(:)456 4154 y FE(Therefore,)33 b(w)m(e)h(ma)m(y)e(de\014ne)456 4335 y(\(78\))624 b Fw(u)1310 4350 y Fs(\000)1368 4335 y FE(\()p Fw(x;)17 b(t)p FE(\))28 b(:=)159 b(lim)17 b(inf)1737 4402 y Fl(\()p Fq(y)r(;s)p Fl(\))p Fs(!)p Fl(\()p Fq(x;t)p Fl(\))p Fq(;)12 b(h)p Fs(!)p Fl(0)2286 4335 y Fw(u)2342 4350 y Fq(h)2386 4335 y FE(\()p Fw(y)t(;)17 b(s)p FE(\))g Fw(:)456 4565 y FE(W)-8 b(e)46 b(sho)m(w)i(that)e Fw(u)1174 4580 y Fs(\000)1278 4565 y FE(is)g(a)g(viscosit)m(y)h(sup)s(ersolution) e(of)h(\(77\).)84 b(F)-8 b(or)45 b(this,)50 b(\014x)456 4693 y(\()p Fw(x)549 4708 y Fl(0)588 4693 y Fw(;)17 b(t)667 4708 y Fl(0)707 4693 y FE(\))34 b Fx(2)g Fn(T)942 4657 y Fq(N)1037 4693 y Fx(\002)25 b FE(\(0)p Fw(;)17 b FE(1\))35 b(and)i(let)e Fw(\036)h FE(b)s(e)g(a)g(smo)s(oth)f(function,)i(so)f (that)g Fw(u)3201 4708 y Fs(\000)3285 4693 y Fx(\000)25 b Fw(\036)456 4821 y FE(has)f(a)f(strict)g(minim)m(um)e(at)i(\()p Fw(x)1570 4836 y Fl(0)1610 4821 y Fw(;)17 b(t)1689 4836 y Fl(0)1728 4821 y FE(\).)41 b(Let)24 b(\()p Fw(x)2093 4836 y Fq(h)2138 4821 y Fw(;)17 b(t)2217 4836 y Fq(h)2261 4821 y FE(\))28 b Fx(2)g Fn(T)2484 4785 y Fq(N)2559 4821 y Fx(\002)t FE([0)p Fw(;)17 b FE(1])23 b(b)s(e)h(minimizers)p 456 4898 499 4 v 555 4964 a Fl(9)591 4996 y FH(The)34 b(reader)e(ma)n(y)i(con)n(vince)f(herself)h(that,)i(with)f(minor)e(mo)r (di\014cations,)j(it)e(w)n(ould)g(b)r(e)456 5106 y(p)r(ossible)c(to)h (consider)e(more)h(general)f(partitions)h(of)h(an)f(in)n(terv)-5 b(al,)31 b(instead)g(of)f(the)i(uniform)456 5216 y Fp(h)p FH(-mesh)27 b(that,)h(for)f(simplicit)n(y)-7 b(,)28 b(w)n(e)f(deal)g (with.)p eop %%Page: 33 33 33 32 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)420 b(33)456 450 y FE(for)23 b(the)h(function)g Fw(u)1185 465 y Fq(h)1233 450 y Fx(\000)t Fw(\036)p FE(.)41 b(W)-8 b(e)25 b(sho)m(w)g(that)e(w)m (e)i(ma)m(y)f(c)m(ho)s(ose)h(a)e(sequence)k Fw(h)3198 465 y Fq(k)3268 450 y Fx(!)h FE(0)456 578 y(suc)m(h)34 b(that)456 773 y(\(79\))816 b(lim)1396 836 y Fq(k)r Fs(!)p Fl(+)p Fs(1)1631 773 y FE(\()p Fw(x)1724 788 y Fq(h)1765 800 y Ft(k)1807 773 y Fw(;)17 b(t)1886 788 y Fq(h)1927 800 y Ft(k)1969 773 y FE(\))44 b(=)g(\()p Fw(x)2264 788 y Fl(0)2304 773 y Fw(;)17 b(t)2383 788 y Fl(0)2422 773 y FE(\))g Fw(:)456 1003 y FE(F)-8 b(or)31 b(this,)i(let)f(\()p Fw(y)1075 1018 y Fq(k)1117 1003 y Fw(;)17 b(s)1207 1018 y Fq(k)1249 1003 y FE(\))28 b Fx(!)f FE(\()p Fw(x)1535 1018 y Fl(0)1575 1003 y Fw(;)17 b(t)1654 1018 y Fl(0)1693 1003 y FE(\))33 b(and)f Fw(h)2009 1018 y Fq(k)2080 1003 y Fx(!)27 b FE(0)32 b(b)s(e)h(a)f(sequence)k(so)c(that)1328 1198 y Fw(u)1384 1213 y Fs(\000)1442 1198 y FE(\()p Fw(x)1535 1213 y Fl(0)1575 1198 y Fw(;)17 b(t)1654 1213 y Fl(0)1693 1198 y FE(\))44 b(=)67 b(lim)1896 1260 y Fq(k)r Fs(!1)2092 1198 y Fw(u)2148 1213 y Fq(h)2189 1225 y Ft(k)2230 1198 y FE(\()p Fw(y)2316 1213 y Fq(k)2359 1198 y Fw(;)17 b(s)2449 1213 y Fq(k)2491 1198 y FE(\))g Fw(;)456 1421 y FE(according)47 b(to)h(\(78\).)88 b(W)-8 b(e)49 b(ma)m(y)e(also)g(assume,)53 b(up)48 b(to)g(subsequences,)56 b(that)456 1548 y(\()p Fw(x)549 1563 y Fq(h)590 1575 y Ft(k)632 1548 y Fw(;)17 b(t)711 1563 y Fq(h)752 1575 y Ft(k)794 1548 y FE(\))27 b Fx(!)h FE(\()6 b(\026)-55 b Fw(x;)1125 1530 y FE(\026)1124 1548 y Fw(t)p FE(\))28 b Fx(2)g Fn(T)1382 1512 y Fq(N)1468 1548 y Fx(\002)16 b FE([0)p Fw(;)h FE(1])29 b(as)h Fw(k)g Fx(!)e FE(+)p Fx(1)p FE(.)42 b(Then,)30 b(\(79\))f(is)g(pro)m(v)m(ed)i (if)d(w)m(e)456 1676 y(sho)m(w)33 b(that)456 1871 y(\(80\))966 b(\()6 b(\026)-55 b Fw(x)q(;)1735 1853 y FE(\026)1734 1871 y Fw(t)p FE(\))44 b(=)g(\()p Fw(x)2064 1886 y Fl(0)2104 1871 y Fw(;)17 b(t)2183 1886 y Fl(0)2222 1871 y FE(\))g Fw(:)456 2066 y FE(F)-8 b(or)31 b(this,)i(let)f(us)h(observ)m(e)h (that,)f(b)m(y)g(construction,)1024 2261 y(\()p Fw(u)1118 2276 y Fs(\000)1199 2261 y Fx(\000)22 b Fw(\036)p FE(\)\()p Fw(x)1487 2276 y Fl(0)1527 2261 y Fw(;)17 b(t)1606 2276 y Fl(0)1645 2261 y FE(\))84 b(=)106 b(lim)1927 2324 y Fq(k)r Fs(!1)2106 2261 y FE(\()p Fw(u)2200 2276 y Fq(h)2241 2288 y Ft(k)2305 2261 y Fx(\000)22 b Fw(\036)p FE(\)\()p Fw(y)2586 2276 y Fq(k)2628 2261 y Fw(;)17 b(s)2718 2276 y Fq(k)2760 2261 y FE(\))1766 2449 y Fx(\025)106 b FE(lim)1927 2512 y Fq(k)r Fs(!1)2106 2449 y FE(\()p Fw(u)2200 2464 y Fq(h)2241 2476 y Ft(k)2305 2449 y Fx(\000)22 b Fw(\036)p FE(\)\()p Fw(x)2593 2464 y Fq(h)2634 2476 y Ft(k)2676 2449 y Fw(;)17 b(t)2755 2464 y Fq(h)2796 2476 y Ft(k)2838 2449 y FE(\))1766 2637 y Fx(\025)84 b FE(\()p Fw(u)2021 2652 y Fs(\000)2101 2637 y Fx(\000)23 b Fw(\036)p FE(\)\()6 b(\026)-55 b Fw(x;)2435 2618 y FE(\026)2434 2637 y Fw(t)p FE(\))17 b Fw(:)456 2832 y FE(Therefore,)27 b(since)f(\()p Fw(x)1243 2847 y Fl(0)1282 2832 y Fw(;)17 b(t)1361 2847 y Fl(0)1401 2832 y FE(\))24 b(w)m(as)i(assumed)g(to)e(b)s(e)i(a)e (strict)h(minim)m(um)c(for)k Fw(u)3211 2847 y Fs(\000)3276 2832 y Fx(\000)6 b Fw(\036)p FE(,)456 2959 y(the)33 b(ab)s(o)m(v)m(e)g (estimate)f(pro)m(v)m(es)i(\(80\))e(and,)h(thence,)h(\(79\).)555 3087 y(W)-8 b(e)26 b(no)m(w)g(consider)g(the)g(sequence)i Fw(h)1899 3102 y Fq(k)1967 3087 y FE(in)d(\(79\),)h(and)g(w)m(e)g (denote)g Fw(h)i FE(:=)g Fw(h)3198 3102 y Fq(k)3268 3087 y Fx(!)g FE(0)456 3215 y(for)k(short.)43 b(The)34 b(fact)e(that)1282 3410 y Fw(u)1338 3425 y Fq(h)1405 3410 y Fx(\000)22 b Fw(u)1560 3425 y Fq(h)1604 3410 y FE(\()p Fw(x)1697 3425 y Fq(h)1743 3410 y Fw(;)17 b(t)1822 3425 y Fq(h)1866 3410 y FE(\))28 b Fx(\025)g Fw(\036)22 b Fx(\000)h Fw(\036)p FE(\()p Fw(x)2368 3425 y Fq(h)2413 3410 y Fw(;)17 b(t)2492 3425 y Fq(h)2536 3410 y FE(\))g Fw(;)456 3605 y FE(together)32 b(with)h(Prop)s(ositions)e(16)h(and)h(17,)f(implies)e(that)851 3775 y(^)827 3800 y Fx(L)896 3815 y Fq(\032)936 3800 y FE([)p Fw(u)1019 3815 y Fq(h)1063 3800 y FE(\()p Fx(\001)p Fw(;)17 b(t)p FE(\)]\()p Fw(x)p FE(\))22 b Fx(\000)h Fw(u)1582 3815 y Fq(h)1626 3800 y FE(\()p Fw(x)1719 3815 y Fq(h)1765 3800 y Fw(;)17 b(t)1844 3815 y Fq(h)1888 3800 y FE(\))28 b Fx(\025)2083 3775 y FE(^)2059 3800 y Fx(L)2128 3815 y Fq(\032)2168 3800 y FE([)p Fw(\036)p FE(\()p Fx(\001)p Fw(;)17 b(t)p FE(\)]\()p Fw(x)p FE(\))22 b Fx(\000)g Fw(\036)p FE(\()p Fw(x)2866 3815 y Fq(h)2911 3800 y Fw(;)17 b(t)2990 3815 y Fq(h)3035 3800 y FE(\))456 3995 y(for)48 b(an)m(y)j Fw(x)56 b Fx(2)h Fn(T)1120 3959 y Fq(N)1191 3995 y FE(,)c(an)m(y)d Fw(t)57 b Fx(2)g FE([0)p Fw(;)17 b FE(1])49 b(and)g(an)m(y)h Fw(\032)57 b(>)f FE(0.)94 b(Thence,)55 b(if)49 b Fw(i)3277 4010 y Fq(h)3378 3995 y Fx(2)456 4123 y(f)p FE(0)p Fw(;)17 b(:)g(:)g(:)e(;)i FE(\(1)p Fw(=h)p FE(\))k Fx(\000)i FE(1)p Fx(g)32 b FE(is)g(so)h(that)f Fw(t)1719 4138 y Fq(h)1792 4123 y Fx(2)c FE(\()p Fw(i)1957 4138 y Fq(h)2002 4123 y Fw(h;)17 b FE(\()p Fw(i)2173 4138 y Fq(h)2240 4123 y FE(+)22 b(1\))p Fw(h)p FE(])33 b(and)f(w)m(e)i(set)1617 4318 y Fw(\032)1667 4333 y Fq(h)1756 4318 y FE(:=)44 b Fw(t)1938 4333 y Fq(h)2006 4318 y Fx(\000)22 b Fw(i)2138 4333 y Fq(h)2183 4318 y Fw(h)17 b(;)456 4513 y FE(w)m(e)33 b(ha)m(v)m(e)h(that)1720 4663 y(lim)1714 4726 y Fq(h)p Fs(!)p Fl(0)1877 4663 y Fw(\032)1927 4678 y Fq(h)2017 4663 y FE(=)44 b(0)456 4858 y(and)1081 5053 y(0)83 b(=)1397 5028 y(^)1373 5053 y Fx(L)1442 5068 y Fq(\032)1478 5080 y Ft(h)1522 5053 y FE([)p Fw(u)1605 5068 y Fq(h)1650 5053 y FE(\()p Fx(\001)p Fw(;)17 b(t)1795 5068 y Fq(h)1861 5053 y Fx(\000)23 b Fw(\032)2011 5068 y Fq(h)2056 5053 y FE(\)]\()p Fw(x)2214 5068 y Fq(h)2259 5053 y FE(\))f Fx(\000)h Fw(u)2475 5068 y Fq(h)2519 5053 y FE(\()p Fw(x)2612 5068 y Fq(h)2658 5053 y Fw(;)17 b(t)2737 5068 y Fq(h)2781 5053 y FE(\))1213 5216 y Fx(\025)1397 5190 y FE(^)1373 5216 y Fx(L)1442 5231 y Fq(\032)1478 5243 y Ft(h)1522 5216 y FE([)p Fw(\036)p FE(\()p Fx(\001)p Fw(;)g(t)1752 5231 y Fq(h)1819 5216 y Fx(\000)22 b Fw(\032)1968 5231 y Fq(h)2013 5216 y FE(\)]\()p Fw(x)2171 5231 y Fq(h)2217 5216 y FE(\))g Fx(\000)g Fw(\036)p FE(\()p Fw(x)2527 5231 y Fq(h)2572 5216 y Fw(;)17 b(t)2651 5231 y Fq(h)2696 5216 y FE(\))g Fw(:)-2322 b FE(\(81\))p eop %%Page: 34 34 34 33 bop 456 251 a FC(34)666 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)456 450 y FE(Also,)32 b(since)h Fw(\036)f FE(is)g(smo)s(oth,)g(w)m(e)h(infer)f(from)g(Prop)s(osition)f(23)h(that) 456 578 y(\(82\))523 773 y(lim)518 836 y Fq(h)p Fs(!)p Fl(0)715 680 y FE(^)691 706 y Fx(L)760 721 y Fq(\032)796 733 y Ft(h)840 706 y FE([)p Fw(\036)p FE(\()p Fx(\001)p Fw(;)17 b(t)1070 721 y Fq(h)1137 706 y Fx(\000)22 b Fw(\032)1286 721 y Fq(h)1331 706 y FE(\)]\()p Fw(x)1489 721 y Fq(h)1534 706 y FE(\))h Fx(\000)f Fw(\036)p FE(\()p Fw(x)1845 721 y Fq(h)1890 706 y Fw(;)17 b(t)1969 721 y Fq(h)2014 706 y FE(\))p 691 750 1361 4 v 1324 841 a Fw(\032)1374 856 y Fq(h)2106 773 y FE(=)44 b Fw(a\017)p FE(\001)2397 788 y Fq(x)2442 773 y Fw(\036)p FE(\()p Fw(x)2593 788 y Fl(0)2632 773 y Fw(;)17 b(t)2711 788 y Fl(0)2751 773 y FE(\))22 b Fx(\000)h Fw(@)2962 788 y Fq(t)2992 773 y Fw(\036)p FE(\()p Fw(x)3143 788 y Fl(0)3182 773 y Fw(;)17 b(t)3261 788 y Fl(0)3301 773 y FE(\))g Fw(:)456 1016 y FE(By)33 b(collecting)d(\(81\))i(and)h(\(82\),)f(w)m(e)i(get)e(that)1303 1215 y Fw(a\017)p FE(\001)1474 1230 y Fq(x)1519 1215 y Fw(\036)p FE(\()p Fw(x)1670 1230 y Fl(0)1709 1215 y Fw(;)17 b(t)1788 1230 y Fl(0)1828 1215 y FE(\))22 b Fx(\000)g Fw(@)2038 1230 y Fq(t)2069 1215 y Fw(\036)p FE(\()p Fw(x)2220 1230 y Fl(0)2259 1215 y Fw(;)17 b(t)2338 1230 y Fl(0)2377 1215 y FE(\))28 b Fx(\024)g FE(0)456 1415 y(if)h Fw(u)599 1430 y Fs(\000)675 1415 y Fx(\000)18 b Fw(\036)29 b FE(has)i(a)f(minim) m(um)d(at)j(\()p Fw(x)1752 1430 y Fl(0)1792 1415 y Fw(;)17 b(t)1871 1430 y Fl(0)1910 1415 y FE(\).)43 b(That)30 b(is,)h Fw(u)2439 1430 y Fs(\000)2527 1415 y FE(is)f(a)g(viscosit)m(y)h (sup)s(erso-)456 1543 y(lution)f(of)j(\(77\).)555 1671 y(Analogously)-8 b(,)31 b(if)h(w)m(e)i(de\014ne)456 1870 y(\(83\))624 b Fw(u)1310 1885 y Fl(+)1368 1870 y FE(\()p Fw(x;)17 b(t)p FE(\))28 b(:=)144 b(lim)17 b(sup)1737 1956 y Fl(\()p Fq(y)r(;s)p Fl(\))p Fs(!)p Fl(\()p Fq(x;t)p Fl(\))p Fq(;)12 b(h)p Fs(!)p Fl(0)2286 1870 y Fw(u)2342 1885 y Fq(h)2386 1870 y FE(\()p Fw(y)t(;)17 b(s)p FE(\))g Fw(;)456 2138 y FE(w)m(e)33 b(ha)m(v)m(e)h(that)e Fw(u)1091 2153 y Fl(+)1183 2138 y FE(is)g(a)g(viscosit)m(y)h(subsolution)f(of)g (\(77\).)555 2266 y(Also,)g(if)g Fw(s)27 b Fx(2)h FE(\(0)p Fw(;)17 b(h)p FE(\),)476 2465 y Fx(j)p Fw(u)560 2480 y Fq(h)604 2465 y FE(\()p Fw(y)t(;)g(s)p FE(\))k Fx(\000)29 b FE(\026)-55 b Fw(u)o FE(\()p Fw(x)p FE(\))p Fx(j)84 b FE(=)g Fx(j)1452 2440 y FE(^)1429 2465 y Fx(L)1498 2480 y Fq(s)1534 2465 y FE([)6 b(\026)-55 b Fw(u)p FE(]\()p Fw(y)t FE(\))21 b Fx(\000)29 b FE(\026)-56 b Fw(u)p FE(\()p Fw(x)p FE(\))p Fx(j)1240 2727 y(\024)1411 2574 y Fv(R)1458 2688 y Fu(R)1506 2669 y Ft(N)1584 2654 y Fw(e)1629 2617 y Fs(\000)1694 2583 y Ft(s)11 b Fm(j)p Ft(v)r Fm(j)1805 2563 y Fr(2)p 1694 2602 146 3 v 1754 2643 a Ft(\017)1870 2654 y Fx(j)6 b FE(\026)-55 b Fw(u)o FE(\()p Fw(y)25 b FE(+)d Fw(sv)t FE(\))g Fx(\000)29 b FE(\026)-55 b Fw(u)o FE(\()p Fw(y)t FE(\))p Fx(j)17 b Fw(dv)p 1411 2704 1337 4 v 1799 2753 a Fv(R)1846 2868 y Fu(R)1894 2849 y Ft(N)1972 2833 y Fw(e)2017 2796 y Fs(\000)2082 2763 y Ft(s)11 b Fm(j)p Ft(v)r Fm(j)2193 2749 y Fr(2)p 2082 2781 146 3 v 2142 2822 a Ft(\017)2258 2833 y Fw(dv)2779 2727 y FE(+)22 b Fx(j)6 b FE(\026)-55 b Fw(u)o FE(\()p Fw(y)t FE(\))22 b Fx(\000)28 b FE(\026)-55 b Fw(u)p FE(\()p Fw(x)p FE(\))p Fx(j)1240 3022 y(\024)84 b(k)p Fw(D)9 b FE(\026)-55 b Fw(u)o Fx(k)1640 3037 y Fq(L)1688 3018 y Fm(1)1757 2912 y Fv(\020)1816 2946 y Fx(p)p 1899 2946 86 4 v 76 x Fw(\017s)39 b FE(+)g Fx(j)p Fw(x)22 b Fx(\000)h Fw(y)t Fx(j)2423 2912 y Fv(\021)2498 3022 y Fw(;)456 3237 y FE(and)32 b(so)1365 3392 y Fw(u)1421 3407 y Fs(\000)1480 3392 y FE(\()p Fw(x;)17 b FE(0\))27 b(=)h Fw(u)1891 3407 y Fl(+)1949 3392 y FE(\()p Fw(x;)17 b FE(0\))28 b(=)33 b(\026)-55 b Fw(u)p FE(\()p Fw(x)p FE(\))17 b Fw(:)456 3569 y FE(W)-8 b(e)33 b(claim)d(that)2282 3769 y Fw(u)2338 3784 y Fs(\000)2429 3769 y FE(\(resp.,)j Fw(u)2783 3784 y Fl(+)2842 3769 y FE(\))f(is)-2522 b(\(84\))756 3928 y(lo)m(w)m(er)33 b(semicon)m(tin)m(uous)f(\(resp.,)i(upp)s(er)f(semicon)m(tin)m(uous\).) 456 4128 y(F)-8 b(or)34 b(pro)m(ving)h(this,)h(\014x)g Fw(z)h FE(:=)32 b(\()p Fw(x;)17 b(y)t FE(\))31 b Fx(2)i Fn(T)1987 4092 y Fq(N)2082 4128 y Fx(\002)24 b FE([0)p Fw(;)17 b FE(1])35 b(and)g(consider)h(a)f(sequence)456 4256 y Fw(z)501 4271 y Fq(j)569 4256 y Fx(2)d Fn(T)730 4220 y Fq(N)824 4256 y Fx(\002)24 b FE([0)p Fw(;)17 b FE(1],)35 b(so)g(that)f Fw(z)1563 4271 y Fq(j)1632 4256 y Fx(!)d Fw(z)39 b FE(as)c Fw(j)i Fx(!)32 b FE(+)p Fx(1)p FE(.)49 b(F)-8 b(or)34 b(eac)m(h)i Fw(j)h Fx(2)32 b Fn(N)9 b FE(,)41 b(let)34 b(also)456 4384 y Fw(h)512 4399 y Fq(j;k)630 4384 y Fw(>)28 b FE(0)k(and)h Fw(\020)1048 4399 y Fq(j;k)1166 4384 y Fx(2)28 b Fn(T)1323 4348 y Fq(N)1416 4384 y Fx(\002)22 b FE([0)p Fw(;)17 b FE(1])32 b(b)s(e)h(so)g(that)997 4584 y(lim)948 4646 y Fq(k)r Fs(!)p Fl(+)p Fs(1)1199 4584 y Fw(h)1255 4599 y Fq(j;k)1373 4584 y FE(=)28 b(0)17 b Fw(;)288 b FE(lim)1809 4646 y Fq(k)r Fs(!)p Fl(+)p Fs(1)2060 4584 y Fw(\020)2103 4599 y Fq(j;k)2221 4584 y FE(=)27 b Fw(z)2369 4599 y Fq(j)2629 4584 y FE(and)1696 4778 y Fw(u)1752 4793 y Fs(\000)1811 4778 y FE(\()p Fw(z)1894 4793 y Fq(j)1931 4778 y FE(\))g(=)77 b(lim)2100 4841 y Fq(k)r Fs(!)p Fl(+)p Fs(1)2351 4778 y Fw(u)2407 4793 y Fq(h)2448 4805 y Ft(j;k)2533 4778 y FE(\()p Fw(\020)2614 4793 y Fq(j;k)2705 4778 y FE(\))17 b Fw(;)456 5013 y FE(for)32 b(an)m(y)h(\014xed)h Fw(j)f Fx(2)28 b Fn(N)9 b FE(.)50 b(Fix)32 b(no)m(w)h Fw(\021)e(>)d FE(0.)43 b(Let)33 b Fw(k)2241 5028 y Fl(0)2280 5013 y FE(\()p Fw(\021)t(;)17 b(j)6 b FE(\))32 b(b)s(e)g(so)h(that)1437 5212 y Fx(j)p Fw(u)1521 5227 y Fq(h)1562 5239 y Ft(j;k)1647 5212 y FE(\()p Fw(\020)1728 5227 y Fq(j;k)1819 5212 y FE(\))22 b Fx(\000)g Fw(u)2034 5227 y Fs(\000)2093 5212 y FE(\()p Fw(z)2176 5227 y Fq(j)2213 5212 y FE(\))p Fx(j)27 b(\024)h Fw(\021)p eop %%Page: 35 35 35 34 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)420 b(35)456 450 y FE(for)32 b(an)m(y)h Fw(k)e Fx(\025)d Fw(k)1027 465 y Fl(0)1066 450 y FE(\()p Fw(\021)t(;)17 b(j)6 b FE(\).)43 b(Let)32 b(also)g Fw(k)1775 465 y Fl(1)1814 450 y FE(\()p Fw(j)6 b FE(\))33 b(suc)m(h)h(that)1470 706 y Fx(j)p Fw(h)1554 721 y Fq(j;k)1645 706 y Fx(j)21 b FE(+)h Fx(j)p Fw(\020)1863 721 y Fq(j;k)1976 706 y Fx(\000)g Fw(z)2120 721 y Fq(j)2157 706 y Fx(j)27 b(\024)2327 639 y FE(1)p 2327 683 49 4 v 2328 774 a Fw(j)2403 706 y(;)456 968 y FE(for)32 b(an)m(y)h Fw(k)e Fx(\025)d Fw(k)1027 983 y Fl(1)1066 968 y FE(\()p Fw(j)6 b FE(\).)43 b(W)-8 b(e)33 b(de\014ne)1224 1206 y Fw(k)1275 1221 y Fq(?)1315 1206 y FE(\()p Fw(\021)t(;)17 b(j)6 b FE(\))82 b(:=)h(max)1999 1096 y Fv(n)2065 1206 y Fw(k)2116 1221 y Fl(0)2156 1206 y FE(\()p Fw(\021)t(;)17 b(j)6 b FE(\))17 b Fw(;)59 b(k)2528 1221 y Fl(1)2568 1206 y FE(\()p Fw(j)6 b FE(\))2690 1096 y Fv(o)2772 1206 y Fw(;)1439 1381 y(h)1495 1396 y Fq(j)1615 1381 y FE(:=)83 b Fw(h)1857 1396 y Fq(j;k)1943 1404 y Ft(?)1978 1396 y Fl(\()p Fq(\021)r(;j)t Fl(\))1084 1540 y FE(and)212 b Fw(\020)1496 1555 y Fq(j)1615 1540 y FE(:=)83 b Fw(\020)1844 1556 y Fq(j;k)1930 1564 y Ft(?)1964 1556 y Fl(\()p Fq(\021)r(;j)t Fl(\))2130 1540 y Fw(:)456 1759 y FE(Then,)33 b(b)m(y)h(construction,)f Fw(h)1515 1774 y Fq(j)1579 1759 y Fx(!)27 b FE(0)33 b(and)f Fw(\020)2020 1774 y Fq(j)2084 1759 y Fx(!)27 b Fw(z)38 b FE(as)32 b Fw(j)i Fx(!)27 b FE(+)p Fx(1)p FE(.)43 b(Therefore,)1064 1978 y Fw(u)1120 1993 y Fs(\000)1179 1978 y FE(\()p Fw(z)t FE(\))28 b Fx(\024)75 b FE(lim)1437 2039 y Fq(j)t Fs(!)p Fl(+)p Fs(1)1682 1978 y Fw(u)1738 1993 y Fq(h)1779 2003 y Ft(j)1815 1978 y FE(\()p Fw(\020)1896 1993 y Fq(j)1932 1978 y FE(\))28 b Fx(\024)74 b FE(lim)2103 2039 y Fq(j)t Fs(!)p Fl(+)p Fs(1)2348 1978 y Fw(u)2404 1993 y Fs(\000)2463 1978 y FE(\()p Fw(z)2546 1993 y Fq(j)2582 1978 y FE(\))22 b(+)g Fw(\021)f(:)456 2230 y FE(Since)32 b Fw(\021)37 b FE(is)32 b(arbitrary)-8 b(,)31 b(w)m(e)j(conclude)f(that)1483 2449 y Fw(u)1539 2464 y Fs(\000)1598 2449 y FE(\()p Fw(z)t FE(\))28 b Fx(\024)74 b FE(lim)1856 2509 y Fq(j)t Fs(!)p Fl(+)p Fs(1)2101 2449 y Fw(u)2157 2464 y Fs(\000)2216 2449 y FE(\()p Fw(z)2299 2464 y Fq(j)2335 2449 y FE(\))17 b Fw(;)456 2701 y FE(for)32 b(an)m(y)h(sequence)i Fw(z)1238 2716 y Fq(j)1303 2701 y Fx(!)27 b Fw(z)t FE(,)33 b(and)g(th)m(us)h Fw(u)2000 2716 y Fs(\000)2091 2701 y FE(is)e(lo)m(w)m(er)g(semicon)m (tin)m(uous.)44 b(Analo-)456 2829 y(gously)-8 b(,)28 b(one)g(sees)i(that)d Fw(u)1400 2844 y Fl(+)1487 2829 y FE(is)g(upp)s(er)i(semicon)m(tin)m(uous,)f(th)m(us)h(con\014rming)e (\(84\).)555 2956 y(Thanks)44 b(to)f(\(84\),)h(the)f(Comparison)f (Principle)f(\(see,)46 b(e.g.,)g(Theorem)d(8.2)456 3084 y(in)31 b([CIL92)q(]\))h(yields)g(that)g Fw(u)1499 3099 y Fl(+)1586 3084 y Fx(\024)c Fw(u)1747 3099 y Fs(\000)1805 3084 y FE(.)44 b(Since,)32 b(b)m(y)i(\(78\))e(and)g(\(83\),)g(the)h (opp)s(osite)456 3212 y(inequalit)m(y)e(also)h(holds,)g(w)m(e)i(obtain) d(that)1630 3431 y Fw(u)1686 3446 y Fs(\000)1789 3431 y FE(=)44 b Fw(u)1965 3446 y Fl(+)2068 3431 y FE(=:)g Fw(u)456 3650 y FE(and)32 b(th)m(us)i Fw(u)e FE(is)g(the)h(unique)1498 3611 y Fl(10)1602 3650 y FE(\(con)m(tin)m(uous)g(viscosit)m(y\))g (solution)e(of)h(\(77\).)555 3778 y(T)-8 b(o)43 b(complete)f(the)i(pro) s(of)e(of)g(the)h(desired)h(result,)h(w)m(e)f(sho)m(w)g(the)f(uniform) 456 3906 y(con)m(v)m(ergence)j(of)e Fw(u)1185 3921 y Fq(h)1274 3906 y FE(to)g Fw(u)p FE(,)i(b)m(y)g(arguing)c(as)j(follo)m (ws.)77 b(If,)47 b(b)m(y)e(con)m(tradiction,)456 4033 y Fw(u)512 4048 y Fq(h)604 4033 y FE(did)i(not)g(con)m(v)m(erge)i (uniformly)d(to)h Fw(u)p FE(,)k(there)d(w)m(ould)g(exist)g Fw(\021)57 b(>)c FE(0,)e(and)456 4161 y Fw(z)501 4176 y Fq(h)573 4161 y Fx(2)29 b Fn(T)731 4125 y Fq(N)823 4161 y Fx(\002)23 b FE([0)p Fw(;)17 b FE(1])32 b(so)h(that)1525 4380 y Fx(j)p Fw(u)1609 4395 y Fq(h)1653 4380 y FE(\()p Fw(z)1736 4395 y Fq(h)1781 4380 y FE(\))22 b Fx(\000)h Fw(u)p FE(\()p Fw(z)2080 4395 y Fq(h)2125 4380 y FE(\))p Fx(j)k(\025)h Fw(\021)456 4599 y FE(for)k(in\014nitely)g(man)m(y)h Fw(h)c Fx(!)f FE(0.)44 b(Then,)35 b(either)e Fw(u)2230 4614 y Fq(h)2274 4599 y FE(\()p Fw(z)2357 4614 y Fq(h)2402 4599 y FE(\))22 b Fx(\000)h Fw(u)p FE(\()p Fw(z)2701 4614 y Fq(h)2746 4599 y FE(\))28 b Fx(\025)h Fw(\021)37 b FE(or)c Fw(u)p FE(\()p Fw(z)3262 4614 y Fq(h)3306 4599 y FE(\))23 b Fx(\000)456 4727 y Fw(u)512 4742 y Fq(h)556 4727 y FE(\()p Fw(z)639 4742 y Fq(h)684 4727 y FE(\))37 b Fx(\025)h Fw(\021)k FE(for)c(in\014nitely)f(man)m(y)h Fw(h)g Fx(!)f FE(0.)60 b(Let)38 b(us)h(assume)g(the)g(latter)e(\(the) 456 4855 y(other)30 b(case)i(b)s(eing)d(analogous\))h(and)g(assume)h (also)f(that)g Fw(z)2630 4870 y Fq(h)2703 4855 y Fx(!)d Fw(z)33 b Fx(2)28 b Fn(T)3065 4819 y Fq(N)3154 4855 y Fx(\002)18 b FE([0)p Fw(;)f FE(1])p 456 5007 499 4 v 555 5074 a Fl(10)626 5106 y FH(The)29 b(results)g(w)n(e)g(use)h(ab)r (out)f(viscosit)n(y)f(sub/sup)r(er/solutions)g(ma)n(y)g(b)r(e)i(found,) h(for)e(in-)456 5216 y(stance,)e(in)h([CIL92)o(].)p eop %%Page: 36 36 36 35 bop 456 251 a FC(36)666 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)456 450 y FE(for)32 b(this)g(set)h(of)f Fw(h)p FE('s.)44 b(Then,)1523 623 y Fw(u)p FE(\()p Fw(z)t FE(\))84 b(=)g Fw(u)2004 638 y Fs(\000)2062 623 y FE(\()p Fw(z)t FE(\))1787 783 y Fx(\024)89 b FE(lim)1948 846 y Fq(h)p Fs(!)p Fl(0)2111 783 y Fw(u)2167 798 y Fq(h)2211 783 y FE(\()p Fw(z)2294 798 y Fq(h)2339 783 y FE(\))1787 971 y Fx(\024)84 b Fw(u)p FE(\()p Fw(z)t FE(\))22 b Fx(\000)h Fw(\021)d(;)456 1144 y FE(whic)m(h)33 b(is)f(a)g(con)m(tradiction.)1864 b Fj(\003)555 1323 y FE(W)-8 b(e)33 b(no)m(w)g(deal)f(with)h(the)g(sc)m (heme)g Fx(G)39 b FE(as)33 b Fw(h)28 b Fx(!)f FE(0.)43 b(W)-8 b(e)33 b(de\014ne)2873 1298 y(^)2854 1323 y Fx(G)39 b FE(as)32 b(in)g(\(44\).)555 1451 y(W)-8 b(e)41 b(will)e(\014x)i Fw(\017)h(>)g FE(0)e(and)h(tak)m(e)h Fw(h)f Fx(!)h FE(0.)68 b(W)-8 b(e)41 b(will)d(explicilt)m(y)h(write)3232 1425 y(^)3213 1451 y Fx(G)3272 1466 y Fq(h)3358 1451 y FE(to)456 1579 y(stress)34 b(the)f(dep)s(endence)i(on)d Fw(h)h FE(in)1771 1553 y(^)1752 1579 y Fx(G)6 b FE(.)555 1706 y(W)-8 b(e)37 b(\014x)57 b(\026)-68 b Fw(w)37 b Fx(2)e Fw(C)1155 1670 y Fl(3)1194 1706 y FE(\()p Fn(T)1295 1670 y Fq(N)1366 1706 y FE(\),)i(w)m(e)h(assume)1920 1668 y Fl(11)2028 1706 y FE(that)e(1)p Fw(=h)e Fx(2)h Fn(N)52 b FE(and)37 b(w)m(e)g(de\014ne)h Fw(w)3338 1721 y Fq(h)3417 1706 y FE(:)456 1834 y Fn(T)519 1798 y Fq(N)611 1834 y Fx(\002)23 b FE([0)p Fw(;)17 b FE(1])27 b Fx(\000)-16 b(!)27 b Fn(R)44 b FE(as)33 b(follo)m(ws:)42 b(\014rst,)33 b(w)m(e)h(set)1600 2008 y Fw(w)1670 2023 y Fq(h)1714 2008 y FE(\()p Fw(x;)17 b FE(0\))28 b(:=)47 b(\026)-69 b Fw(w)s FE(\()p Fw(x)p FE(\))456 2181 y(and)32 b(then,)i(recursiv)m (ely)-8 b(,)1336 2354 y Fw(w)1406 2369 y Fq(h)1450 2354 y FE(\()p Fw(x;)17 b(t)p FE(\))28 b(:=)1838 2329 y(^)1819 2354 y Fx(G)1878 2369 y Fq(t)p Fs(\000)p Fq(ih)2027 2354 y FE([)p Fw(w)2124 2369 y Fq(h)2169 2354 y FE(\()p Fx(\001)p Fw(;)17 b(ih)p FE(\)]\()p Fw(x)p FE(\))456 2528 y(if)46 b Fw(t)53 b Fx(2)h FE(\()p Fw(ih;)17 b FE(\()p Fw(i)33 b FE(+)f(1\))p Fw(h)p FE(],)51 b(for)c Fw(i)53 b FE(=)g(0)p Fw(;)17 b(:)g(:)g(:)f(;)h FE(\(1)p Fw(=h)p FE(\))32 b Fx(\000)g FE(1.)89 b(Then,)52 b(the)c(follo)m(wing)456 2656 y(con)m(v)m(ergence)35 b(result)d(holds:)456 2804 y FI(Theorem)49 b(30.)e FD(L)-5 b(et)45 b Fw(a)g FD(b)-5 b(e)44 b(as)g(in)g(\(76\).)74 b(Then,)46 b(as)e Fw(h)i Fx(!)f FE(0)p FD(,)i Fw(w)2956 2819 y Fq(h)3045 2804 y FD(c)-5 b(onver)g(ges)456 2932 y(uniformly)34 b(to)h Fw(w)s FD(,)f(wher)-5 b(e)35 b Fw(w)i FD(satis\014es)456 3105 y FE(\(85\))460 b Fw(@)1141 3120 y Fq(t)1171 3105 y Fw(w)s FE(\()p Fw(x;)17 b(t)p FE(\))22 b(+)g Fw(K)7 b FE(\()p Fw(D)1783 3120 y Fq(x)1827 3105 y Fw(w)s FE(\()p Fw(x;)17 b(t)p FE(\)\))44 b(=)g Fw(a\017)p FE(\001)2483 3120 y Fq(x)2528 3105 y Fw(w)s FE(\()p Fw(x;)17 b(t)p FE(\))456 3279 y FD(in)34 b(the)h(visc)-5 b(osity)35 b(sense,)f(with)g Fw(w)s FE(\()p Fw(x;)17 b FE(0\))27 b(=)48 b(\026)-69 b Fw(w)r FE(\()p Fw(x)p FE(\))p FD(.)456 3457 y(Pr)-5 b(o)g(of.)41 b FE(As)k(in)e(Theorem)i(29,)h(the)f(pro)s (of)e(is)h(a)g(v)-5 b(ariation)42 b(of)i(the)g(argumen)m(ts)456 3585 y(in)31 b([Sou85])i(and)g([BS91)o(].)44 b(First,)32 b(w)m(e)h(observ)m(e)i(that)1286 3758 y Fx(k)p Fw(w)1406 3773 y Fq(h)1450 3758 y Fx(k)1500 3777 y Fq(L)1548 3758 y Fm(1)1613 3777 y Fl(\()p Fu(T)1690 3758 y Ft(N)1742 3777 y Fs(\002)p Fl([0)p Fq(;)p Fl(1]\))2003 3758 y Fx(\024)44 b(k)20 b FE(\026)-69 b Fw(w)s Fx(k)2297 3777 y Fq(L)2345 3758 y Fm(1)2409 3777 y Fl(\()p Fu(T)2486 3758 y Ft(N)2539 3777 y Fl(\))2587 3758 y Fw(;)456 3932 y FE(due)33 b(to)f(\(46\).)43 b(Th)m(us,)34 b(w)m(e)g(ma)m(y)e(de\014ne)1239 4105 y Fw(w)1309 4120 y Fs(\000)1368 4105 y FE(\()p Fw(x;)17 b(t)p FE(\))28 b(:=)159 b(lim)17 b(inf)1737 4172 y Fl(\()p Fq(y)r(;s)p Fl(\))p Fs(!)p Fl(\()p Fq(x;t)p Fl(\))p Fq(;)12 b(h)p Fs(!)p Fl(0)2286 4105 y Fw(w)2356 4120 y Fq(h)2400 4105 y FE(\()p Fw(y)t(;)17 b(s)p FE(\))g Fw(:)456 4327 y FE(W)-8 b(e)45 b(sho)m(w)i(that)e Fw(w)1185 4342 y Fs(\000)1289 4327 y FE(is)f(a)h(viscosit)m(y)h(sup)s(ersolution)f(of)f (\(85\).)81 b(F)-8 b(or)45 b(this,)j(\014x)456 4454 y(\()p Fw(x)549 4469 y Fl(0)588 4454 y Fw(;)17 b(t)667 4469 y Fl(0)707 4454 y FE(\))32 b Fx(2)h Fn(T)939 4418 y Fq(N)1033 4454 y Fx(\002)25 b FE(\(0)p Fw(;)17 b FE(1\))34 b(and)i(let)e Fw(\036)h FE(b)s(e)h(a)f(smo)s(oth)f(function,)i(so)f(that)g Fw(w)3202 4469 y Fs(\000)3285 4454 y Fx(\000)25 b Fw(\036)456 4582 y FE(has)f(a)f(strict)g(minim)m(um)e(at)i(\()p Fw(x)1570 4597 y Fl(0)1610 4582 y Fw(;)17 b(t)1689 4597 y Fl(0)1728 4582 y FE(\).)41 b(Let)24 b(\()p Fw(x)2093 4597 y Fq(h)2138 4582 y Fw(;)17 b(t)2217 4597 y Fq(h)2261 4582 y FE(\))28 b Fx(2)g Fn(T)2484 4546 y Fq(N)2559 4582 y Fx(\002)t FE([0)p Fw(;)17 b FE(1])23 b(b)s(e)h(minimizers)456 4710 y(for)31 b(the)h(function)f Fw(w)1222 4725 y Fq(h)1287 4710 y Fx(\000)21 b Fw(\036)p FE(.)43 b(As)32 b(sho)m(wn)h(in)e (\(79\),)g(w)m(e)i(ma)m(y)e(and)h(do)g(assume,)g(up)456 4838 y(to)g(a)g(subsequence,)k(that)d(\()p Fw(x)1539 4853 y Fq(h)1584 4838 y Fw(;)17 b(t)1663 4853 y Fq(h)1708 4838 y FE(\))27 b Fx(!)h FE(\()p Fw(x)1994 4853 y Fl(0)2033 4838 y Fw(;)17 b(t)2112 4853 y Fl(0)2152 4838 y FE(\).)p 456 4898 499 4 v 555 4964 a Fl(11)626 4996 y FH(F)-7 b(or)34 b(the)i(sak)n(e)d(of)i(simplicit)n(y)-7 b(,)37 b(w)n(e)e(assumed)52 b(\026)-59 b Fp(w)37 b FH(to)e(b)r(e)h(smo)r(oth)e (enough,)j(in)e(order)f(to)456 5106 y(use)26 b(\(45\))g(in)g(estimate)h (\(86\))f(here)g(b)r(elo)n(w.)36 b(W)-7 b(e)27 b(remark)d(that,)j(b)n (y)g(applying)e(the)i(lo)n(w)n(er)e(order)456 5216 y(argumen)n(ts)h(in) i(Prop)r(osition)e(25)g(directly)i(to)44 b(\026)-59 b Fp(w)s FH(,)28 b(less)f(smo)r(othness)g(ma)n(y)f(b)r(e)i(required.)p eop %%Page: 37 37 37 36 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)420 b(37)555 450 y FE(Since)34 b Fw(w)881 465 y Fq(h)949 450 y Fx(\000)24 b Fw(w)1120 465 y Fq(h)1164 450 y FE(\()p Fw(x)1257 465 y Fq(h)1302 450 y Fw(;)17 b(t)1381 465 y Fq(h)1426 450 y FE(\))30 b Fx(\025)h Fw(\036)23 b Fx(\000)g Fw(\036)p FE(\()p Fw(x)1934 465 y Fq(h)1979 450 y Fw(;)17 b(t)2058 465 y Fq(h)2103 450 y FE(\),)35 b(b)m(y)f(exploiting)e(\(45\),)i(w)m(e) h(gather)456 578 y(that)841 766 y(^)822 791 y Fx(G)881 806 y Fq(\032)922 791 y FE([)p Fw(w)1019 806 y Fq(h)1063 791 y FE(\()p Fx(\001)p Fw(;)17 b(t)p FE(\)]\()p Fw(x)p FE(\))22 b Fx(\000)h Fw(w)1596 806 y Fq(h)1641 791 y FE(\()p Fw(x)1734 806 y Fq(h)1779 791 y Fw(;)17 b(t)1858 806 y Fq(h)1902 791 y FE(\))28 b Fx(\025)2093 766 y FE(^)2073 791 y Fx(G)2132 806 y Fq(\032)2173 791 y FE([)p Fw(\036)p FE(\()p Fx(\001)p Fw(;)17 b(t)p FE(\)]\()p Fw(x)p FE(\))22 b Fx(\000)g Fw(\036)p FE(\()p Fw(x)2871 806 y Fq(h)2916 791 y Fw(;)17 b(t)2995 806 y Fq(h)3040 791 y FE(\))456 1004 y(for)48 b(an)m(y)j Fw(x)56 b Fx(2)h Fn(T)1120 967 y Fq(N)1191 1004 y FE(,)c(an)m(y)d Fw(t)57 b Fx(2)g FE([0)p Fw(;)17 b FE(1])49 b(and)g(an)m(y)h Fw(\032)57 b(>)f FE(0.)94 b(Thence,)55 b(if)49 b Fw(i)3277 1019 y Fq(h)3378 1004 y Fx(2)456 1131 y(f)p FE(0)p Fw(;)17 b(:)g(:)g(:)e(;)i FE(\(1)p Fw(=h)p FE(\))29 b Fx(\000)h FE(1)p Fx(g)43 b FE(is)h(so)f(that)h Fw(t)1779 1146 y Fq(h)1870 1131 y Fx(2)j FE(\()p Fw(i)2054 1146 y Fq(h)2100 1131 y Fw(h;)17 b FE(\()p Fw(i)2271 1146 y Fq(h)2345 1131 y FE(+)30 b(1\))p Fw(h)p FE(])43 b(and)h(w)m(e)h(set)f Fw(\032)3233 1146 y Fq(h)3341 1131 y FE(:=)456 1259 y Fw(t)491 1274 y Fq(h)558 1259 y Fx(\000)22 b Fw(i)690 1274 y Fq(h)736 1259 y Fw(h)p FE(,)1071 1472 y(0)84 b(=)1383 1447 y(^)1364 1472 y Fx(G)1423 1487 y Fq(\032)1459 1499 y Ft(h)1503 1472 y FE([)p Fw(w)1600 1487 y Fq(h)1645 1472 y FE(\()p Fx(\001)p Fw(;)17 b(t)1790 1487 y Fq(h)1857 1472 y Fx(\000)22 b Fw(\032)2006 1487 y Fq(h)2051 1472 y FE(\)]\()p Fw(x)2209 1487 y Fq(h)2255 1472 y FE(\))g Fx(\000)g Fw(w)2484 1487 y Fq(h)2529 1472 y FE(\()p Fw(x)2622 1487 y Fq(h)2667 1472 y Fw(;)17 b(t)2746 1487 y Fq(h)2791 1472 y FE(\))1203 1635 y Fx(\025)1383 1609 y FE(^)1364 1635 y Fx(G)1423 1650 y Fq(\032)1459 1662 y Ft(h)1503 1635 y FE([)p Fw(\036)p FE(\()p Fx(\001)p Fw(;)g(t)1733 1650 y Fq(h)1800 1635 y Fx(\000)22 b Fw(\032)1949 1650 y Fq(h)1995 1635 y FE(\)]\()p Fw(x)2153 1650 y Fq(h)2198 1635 y FE(\))g Fx(\000)g Fw(\036)p FE(\()p Fw(x)2508 1650 y Fq(h)2553 1635 y Fw(;)17 b(t)2632 1650 y Fq(h)2677 1635 y FE(\))g Fw(:)456 1847 y FE(Then,)33 b(b)m(y)h(means)f(of)f (\(45\),)g(w)m(e)h(ha)m(v)m(e)h(that)1394 2127 y(lim)1388 2190 y Fq(h)p Fs(!)p Fl(0)1581 2035 y FE(^)1561 2060 y Fx(G)1620 2075 y Fq(\032)1656 2087 y Ft(h)1701 2060 y FE([)p Fw(\036)p FE(\()p Fx(\001)p Fw(;)17 b(t)1931 2075 y Fq(h)1998 2060 y Fx(\000)22 b Fw(\032)2147 2075 y Fq(h)2192 2060 y FE(\)]\()p Fw(x)2350 2075 y Fq(h)2395 2060 y FE(\))h Fx(\000)f Fw(\036)p FE(\()p Fw(x)2706 2075 y Fq(h)2751 2060 y Fw(;)17 b(t)2830 2075 y Fq(h)2875 2060 y FE(\))p 1561 2104 1352 4 v 2189 2196 a Fw(\032)2239 2211 y Fq(h)811 2330 y FE(=)44 b Fx(\000)p Fw(K)7 b FE(\()p Fw(D)1217 2345 y Fq(x)1262 2330 y Fw(\036)p FE(\()p Fw(x)1413 2345 y Fl(0)1452 2330 y Fw(;)17 b(t)1531 2345 y Fl(0)1571 2330 y FE(\)\))22 b(+)g Fw(a\017)p FE(\001)1938 2345 y Fq(x)1982 2330 y Fw(\036)p FE(\()p Fw(x)2133 2345 y Fl(0)2173 2330 y Fw(;)17 b(t)2252 2345 y Fl(0)2291 2330 y FE(\))22 b Fx(\000)h Fw(@)2502 2345 y Fq(t)2532 2330 y Fw(\036)p FE(\()p Fw(x)2683 2345 y Fl(0)2723 2330 y Fw(;)17 b(t)2802 2345 y Fl(0)2841 2330 y FE(\))g Fw(:)456 2543 y FE(Therefore,)1009 2756 y Fw(a\017)p FE(\001)1180 2771 y Fq(x)1224 2756 y Fw(\036)p FE(\()p Fw(x)1375 2771 y Fl(0)1415 2756 y Fw(;)g(t)1494 2771 y Fl(0)1533 2756 y FE(\))22 b Fx(\000)h Fw(@)1744 2771 y Fq(t)1774 2756 y Fw(\036)p FE(\()p Fw(x)1925 2771 y Fl(0)1965 2756 y Fw(;)17 b(t)2044 2771 y Fl(0)2083 2756 y FE(\))28 b Fx(\024)g Fw(K)7 b FE(\()p Fw(D)2463 2771 y Fq(x)2507 2756 y Fw(\036)p FE(\()p Fw(x)2658 2771 y Fl(0)2697 2756 y Fw(;)17 b(t)2776 2771 y Fl(0)2816 2756 y FE(\)\))456 2968 y(if)32 b Fw(w)616 2983 y Fs(\000)698 2968 y Fx(\000)24 b Fw(\036)33 b FE(has)h(a)g(minim) m(um)c(at)k(\()p Fw(x)1799 2983 y Fl(0)1839 2968 y Fw(;)17 b(t)1918 2983 y Fl(0)1957 2968 y FE(\).)47 b(That)34 b(is,)g Fw(w)2511 2983 y Fs(\000)2604 2968 y FE(is)f(a)h(viscosit)m(y)g (sup)s(er-)456 3096 y(solution)d(of)h(\(85\).)555 3224 y(Analogously)-8 b(,)31 b(if)h(w)m(e)i(de\014ne)1239 3437 y Fw(w)1309 3452 y Fl(+)1368 3437 y FE(\()p Fw(x;)17 b(t)p FE(\))28 b(:=)144 b(lim)17 b(sup)1737 3523 y Fl(\()p Fq(y)r(;s)p Fl(\))p Fs(!)p Fl(\()p Fq(x;t)p Fl(\))p Fq(;)12 b(h)p Fs(!)p Fl(0)2286 3437 y Fw(w)2356 3452 y Fq(h)2400 3437 y FE(\()p Fw(y)t(;)17 b(s)p FE(\))g Fw(;)456 3717 y FE(w)m(e)33 b(ha)m(v)m(e)h(that)e Fw(w)1105 3732 y Fl(+)1197 3717 y FE(is)g(a)g(viscosit)m(y)h(subsolution)f(of)g(\(85\).) 43 b(Also,)32 b(if)g Fw(s)27 b Fx(2)h FE(\(0)p Fw(;)17 b(h)p FE(\),)456 3930 y(\(86\))926 b Fx(j)1603 3905 y FE(^)1584 3930 y Fx(G)1643 3945 y Fq(s)1680 3930 y FE([)20 b(\026)-69 b Fw(w)r FE(])22 b Fx(\000)43 b FE(\026)-69 b Fw(w)r Fx(j)28 b(\024)g Fw(C)23 b(s)17 b(;)456 4143 y FE(thanks)33 b(to)f(\(45\),)g(where)i Fw(C)40 b FE(dep)s(ends)34 b(only)e(on)g Fx(k)20 b FE(\026)-69 b Fw(w)s Fx(k)2416 4161 y Fq(C)2471 4143 y Fr(3)2505 4161 y Fl(\()p Fu(T)2582 4143 y Ft(N)2635 4161 y Fl(\))2666 4143 y FE(,)33 b(and)f(so)759 4366 y Fx(j)p Fw(w)857 4381 y Fq(h)901 4366 y FE(\()p Fw(y)t(;)17 b(s)p FE(\))k Fx(\000)43 b FE(\026)-69 b Fw(w)r FE(\()p Fw(x)p FE(\))p Fx(j)84 b FE(=)f Fx(j)1761 4341 y FE(^)1742 4366 y Fx(G)1801 4381 y Fq(s)1838 4366 y FE([)20 b(\026)-69 b Fw(w)s FE(]\()p Fw(y)t FE(\))21 b Fx(\000)43 b FE(\026)-69 b Fw(w)r FE(\()p Fw(x)p FE(\))p Fx(j)1554 4526 y(\024)83 b(j)1761 4500 y FE(^)1742 4526 y Fx(G)1801 4541 y Fq(s)1838 4526 y FE([)20 b(\026)-69 b Fw(w)s FE(]\()p Fw(y)t FE(\))21 b Fx(\000)43 b FE(\026)-69 b Fw(w)r FE(\()p Fw(y)t FE(\))p Fx(j)21 b FE(+)h Fx(j)d FE(\026)-68 b Fw(w)r FE(\()p Fw(y)t FE(\))21 b Fx(\000)43 b FE(\026)-69 b Fw(w)r FE(\()p Fw(x)p FE(\))p Fx(j)1554 4705 y(\024)83 b Fw(C)1808 4595 y Fv(\020)1868 4705 y Fw(s)38 b FE(+)h Fx(j)p Fw(x)22 b Fx(\000)h Fw(y)t Fx(j)2352 4595 y Fv(\021)2427 4705 y Fw(:)456 4938 y FE(This)44 b(implies)d(that)j Fw(w)1325 4953 y Fs(\000)1383 4938 y FE(\()p Fw(x;)17 b FE(0\))47 b(=)g Fw(w)1847 4953 y Fl(+)1906 4938 y FE(\()p Fw(x;)17 b FE(0\))46 b(=)67 b(\026)-69 b Fw(w)s FE(\()p Fw(x)p FE(\).)77 b(Th)m(us,)49 b(b)m(y)44 b(using)g(the)456 5066 y(Comparison)35 b(Principle)g(and)i (arguing)e(as)h(in)g(the)h(pro)s(of)f(of)g(Theorem)g(29,)h(w)m(e)456 5194 y(obtain)31 b(that)h Fw(w)1040 5209 y Fq(h)1118 5194 y FE(con)m(v)m(erges)i(uniformly)d(to)h Fw(w)2188 5209 y Fs(\000)2274 5194 y FE(=)c Fw(w)2448 5209 y Fl(+)2507 5194 y FE(.)833 b Fj(\003)p eop %%Page: 38 38 38 37 bop 456 251 a FC(38)666 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)926 450 y FE(10.)48 b Fz(Entr)n(opy)38 b(penalized)f(Ma)-7 b(ther)37 b(measures)555 636 y FE(In)44 b(Mather's)h(theory)g(\(see,)i (e.g.,)d([MF94])g(and)g(references)h(therein\),)i(one)456 764 y(lo)s(oks)29 b(for)g(probabilit)m(y)f(measures)j Fw(\026)e FE(on)h Fn(T)2045 728 y Fq(N)2132 764 y Fx(\002)18 b Fn(R)2293 728 y Fq(N)2396 764 y FE(that)29 b(minimize)e(the)j(action) 456 1006 y(\(87\))1467 870 y Fv(Z)1522 1096 y Fu(T)1572 1077 y Ft(N)1624 1096 y Fs(\002)p Fu(R)1727 1077 y Ft(N)1806 1006 y Fw(L)p FE(\()p Fw(x;)17 b(v)t FE(\))p Fw(d\026)p FE(\()p Fw(x;)g(v)t FE(\))456 1265 y(and)32 b(satisfy)h(the)g (\\holonom)m(y")e(\(or)h(\\\015o)m(w)h(in)m(v)-5 b(ariancy"\))31 b(constrain)m(t)456 1512 y(\(88\))1096 1377 y Fv(Z)1152 1602 y Fu(T)1202 1583 y Ft(N)1254 1602 y Fs(\002)p Fu(R)1357 1583 y Ft(N)1435 1512 y FE([)p Fw(')p FE(\()p Fw(x)22 b FE(+)g Fw(hv)t FE(\))g Fx(\000)h Fw(')p FE(\()p Fw(x)p FE(\)])17 b Fw(d\026)p FE(\()p Fw(x;)g(v)t FE(\))27 b(=)g(0)17 b Fw(;)456 1778 y FE(for)40 b(all)f Fw(')j Fx(2)g Fw(C)7 b FE(\()p Fn(R)1152 1742 y Fq(N)1225 1778 y FE(\).)69 b(In)41 b(this)g(section,)i(w)m(e)g(discuss)f(an)f(en)m(trop)m(y)h(p)s (enalized)456 1906 y(v)m(ersion)j(of)g(Mather's)i(problem,)g(and)f(w)m (e)g(presen)m(t)h(a)e(solution)f(in)h(terms)g(of)456 2034 y(\014xed)31 b(p)s(oin)m(ts)e(of)h(the)g(op)s(erator)f Fx(G)6 b FE(.)43 b(The)31 b(en)m(trop)m(y)g(p)s(enalized)e(Mather)i (problem)456 2162 y(consists)i(in)f(minimizing)456 2404 y(\(89\))1266 2268 y Fv(Z)1321 2494 y Fu(T)1371 2475 y Ft(N)1423 2494 y Fs(\002)p Fu(R)1526 2475 y Ft(N)1605 2404 y Fw(L)p FE(\()p Fw(x;)17 b(v)t FE(\))p Fw(d\026)p FE(\()p Fw(x;)g(v)t FE(\))k(+)2370 2336 y Fw(\017)p 2362 2381 57 4 v 2362 2472 a(h)2428 2404 y(S)6 b FE([)p Fw(\026)p FE(])p Fw(;)456 2657 y FE(in)31 b(whic)m(h)1011 2841 y Fw(S)6 b FE([)p Fw(\026)p FE(])28 b(=)1321 2706 y Fv(Z)1376 2931 y Fu(T)1426 2912 y Ft(N)1478 2931 y Fs(\002)p Fu(R)1581 2912 y Ft(N)1660 2841 y Fw(\026)p FE(\()p Fw(x;)17 b(v)t FE(\))g(ln)2228 2774 y Fw(\026)p FE(\()p Fw(x;)g(v)t FE(\))p 2069 2818 603 4 v 2069 2830 a Fv(R)2116 2945 y Fu(R)2164 2926 y Ft(N)2242 2911 y Fw(\026)p FE(\()p Fw(x;)g(w)s FE(\))p Fw(dw)2681 2841 y(dxdv)456 3085 y FE(is)30 b(the)h(\\en)m(trop)m(y)h(term".)42 b(The)31 b(minimization)26 b(in)k(\(89\))g(is)g(p)s(erformed)g(o)m(v)m(er)i(the) 456 3213 y(space)h(of)f(probabilit)m(y)f(densities)i(on)f Fn(T)1921 3176 y Fq(N)2014 3213 y Fx(\002)23 b Fn(R)2180 3176 y Fq(N)456 3463 y FE(\(90\))877 3323 y Fv(\032)952 3463 y Fw(\026)k Fx(2)i Fw(L)1199 3422 y Fl(1)1238 3463 y FE(\()p Fn(T)1339 3422 y Fq(N)1432 3463 y Fx(\002)23 b Fn(R)1598 3422 y Fq(N)1671 3463 y FE(\))17 b Fw(;)44 b(\026)27 b Fx(\025)h FE(0)44 b Fw(a:e:)17 b(;)2320 3328 y Fv(Z)2375 3553 y Fu(T)2425 3535 y Ft(N)2477 3553 y Fs(\002)p Fu(R)2580 3535 y Ft(N)2659 3463 y Fw(d\026)27 b FE(=)g(1)2948 3323 y Fv(\033)456 3729 y FE(that)37 b(satisfy)944 3690 y Fl(12)1053 3729 y FE(the)h(constrain)m(t)g (\(88\).)58 b(In)39 b(this)e(setting,)i(w)m(e)g(study)g(the)f(mini-)456 3857 y(mizers)g(of)g(the)h(functional)e(in)g(\(89\))h(in)g(the)h(space) h(giv)m(en)e(b)m(y)i(\(90\))e(under)h(the)456 3985 y(constrain)m(t)k (in)h(\(88\),)i(that)d(is)h(the)g(\\en)m(trop)m(y)h(p)s(enalized)e (Mather)i(problem".)456 4112 y(W)-8 b(e)33 b(will)e(sho)m(w)j(that)f (these)h(p)s(enalized)f(minimal)c(measures)34 b(alw)m(a)m(ys)f FD(exist)g FE(\(see)456 4240 y(Theorem)38 b(32)f(here)h(b)s(elo)m(w\))g (and)g(that)f(they)i(are)e FD(unique)h FE(\(see)h(Theorem)f(36\).)456 4368 y(An)27 b FD(explicit)i(formula)d FE(for)h(this)g(measure)g(will)e (b)s(e)i(pro)m(vided)g(b)m(y)h(using)f(the)g(\014xed)456 4496 y(p)s(oin)m(t)36 b(structure)i(of)f Fx(G)43 b FE(\(see)38 b(form)m(ula)d(\(105\)\).)56 b(In)38 b(pro)m(ving)e(these)j(results,)f (an)456 4624 y(imp)s(ortan)m(t)g(r^)-49 b(ole)40 b(is)g(pla)m(y)m(ed)i (b)m(y)f(the)h Fn(T)1926 4588 y Fq(N)1996 4624 y FE(-pro)5 b(jection)40 b(of)h(p)s(enalized)f(measures)456 4752 y(and)32 b(on)h(their)f(analytic)f(b)s(ounds)i(\(see)h(Prop)s(ositions) d(34)h(and)h(35\).)555 4880 y(F)-8 b(or)32 b(this,)g(w)m(e)i(need)g (the)f(follo)m(wing)c(preliminary)h(result:)p 456 5007 499 4 v 555 5074 a Fl(12)626 5106 y FH(As)24 b(customary)-7 b(,)24 b(with)h(a)f(sligh)n(t)f(abuse)h(of)g(notation,)h(giv)n(en)e(a)h (function)h Fp(\026)e FF(2)h Fp(L)3117 5076 y Fy(1)3153 5106 y FH(\()p Ff(T)12 b FF(\002)g Ff(R)p FH(\),)456 5216 y(w)n(e)27 b(denote)g Fp(d\026)p FH(\()p Fp(x;)14 b(v)s FH(\))25 b(=)d Fp(\026)p FH(\()p Fp(x;)14 b(v)s FH(\))g Fp(dx)g(dv)s FH(.)p eop %%Page: 39 39 39 38 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)420 b(39)456 450 y FI(Lemma)37 b(31.)k FD(L)-5 b(et)36 b Fw(\036)27 b Fx(2)h Fw(W)1491 414 y Fl(1)p Fq(;)p Fs(1)1620 450 y FE(\()p Fn(T)1721 414 y Fq(N)1792 450 y FE(\))35 b FD(satisfy)456 629 y FE(\(91\))1035 b Fx(G)6 b FE([)p Fw(\036)p FE(])28 b(=)f Fw(\036)22 b FE(+)g Fw(\025:)456 808 y FD(Then,)34 b(ther)-5 b(e)34 b(exists)h Fw(\022)c Fx(2)d Fw(L)1485 771 y Fl(1)1525 808 y FE(\()p Fn(T)1626 771 y Fq(N)1696 808 y FE(\))35 b FD(satisfying)456 986 y FE(\(92\))1167 b Fw(\022)47 b Fx(\025)e FE(0)17 b Fw(;)456 1257 y FE(\(93\))929 1121 y Fv(Z)985 1347 y Fu(R)1033 1328 y Ft(N)1111 1257 y Fw(\022)s FE(\()p Fw(x)23 b Fx(\000)f Fw(hv)t FE(\))p Fw(e)1564 1216 y Fs(\000)1629 1182 y Ft(hL)p Fr(\()p Ft(x)p Fm(\000)p Ft(hv)r(;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(x)p Fr(\))p Fm(\000)p Ft(\036)p Fr(\()p Ft(x)p Fm(\000)p Ft(hv)r Fr(\))p Fm(\000)p Ft(\025)p 1629 1201 866 3 v 2049 1242 a(\017)2525 1257 y Fw(dv)48 b FE(=)c Fw(\022)s FE(\()p Fw(x)p FE(\))456 1475 y FD(and)456 1670 y FE(\(94\))1581 1534 y Fv(Z)1637 1760 y Fu(T)1687 1741 y Ft(N)1759 1670 y Fw(\022)s FE(\()p Fw(x)p FE(\))17 b Fw(dx)45 b FE(=)f(1)17 b Fw(:)456 1894 y FD(Pr)-5 b(o)g(of.)41 b FE(The)f(follo)m(wing)c (elemen)m(tary)j(observ)-5 b(ation)38 b(will)f(b)s(e)i(used)h (throughout)456 2022 y(this)32 b(pro)s(of:)42 b(if)32 b Fw(\021)f Fx(2)d Fw(L)1267 1986 y Fl(1)1307 2022 y FE(\()p Fn(T)1408 1986 y Fq(N)1479 2022 y FE(\))k(and)h Fw(\030)f Fx(2)c Fn(R)1974 1986 y Fq(N)2047 2022 y FE(,)33 b(then)1405 2106 y Fv(Z)1461 2332 y Fu(T)1511 2313 y Ft(N)1583 2242 y Fw(\021)48 b FE(=)1800 2106 y Fv(Z)1855 2332 y Fu(T)1905 2313 y Ft(N)1978 2242 y Fw(\021)t FE(\()p Fw(x)22 b FE(+)g Fw(\030)5 b FE(\))17 b Fw(dx)g(:)456 2460 y FE(F)-8 b(urther,)32 b(w)m(e)i(note)f(that)456 2674 y(\(95\))1236 2539 y Fv(Z)1291 2764 y Fu(R)1339 2745 y Ft(N)1418 2674 y Fw(e)1463 2633 y Fs(\000)1528 2599 y Ft(hL)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p Fm(\000)p Ft(\036)p Fr(\()p Ft(x)p Fr(\))p Fm(\000)p Ft(\025)p 1528 2618 748 3 v 1888 2659 a(\017)2306 2674 y Fw(dv)47 b FE(=)e(1)17 b Fw(;)456 2905 y FE(for)32 b(an)m(y)h Fw(x)28 b Fx(2)g Fn(T)1029 2869 y Fq(N)1100 2905 y FE(,)k(due)i(to)e(\(91\).)43 b(No)m(w,)33 b(de\014ne)h(for)e Fw(#)c Fx(2)g Fw(L)2632 2869 y Fl(1)2672 2905 y FE(\()p Fn(T)2773 2869 y Fq(N)2843 2905 y FE(\),)456 3124 y(\(96\))187 b Fx(F)10 b FE([)p Fw(#)p FE(]\()p Fw(x)p FE(\))44 b(:=)1333 2989 y Fv(Z)1388 3214 y Fu(R)1436 3196 y Ft(N)1514 3124 y Fw(#)p FE(\()p Fw(x)23 b Fx(\000)g Fw(hv)t FE(\))p Fw(e)1977 3083 y Fs(\000)2042 3050 y Ft(hL)p Fr(\()p Ft(x)p Fm(\000)p Ft(hv)r(;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(x)p Fr(\))p Fm(\000)p Ft(\036)p Fr(\()p Ft(x)p Fm(\000)p Ft(hv)r Fr(\))p Fm(\000)p Ft(\025)p 2042 3068 866 3 v 2462 3109 a(\017)2938 3124 y Fw(dv)d(:)456 3348 y FE(Let)32 b(us)i(observ)m(e)g (that)e(\(95\))g(and)h(\(96\))f(imply)f(that)456 3568 y(\(97\))1543 3432 y Fv(Z)1598 3658 y Fu(T)1648 3639 y Ft(N)1721 3568 y Fx(F)10 b FE([)p Fw(#)p FE(])44 b(=)2078 3432 y Fv(Z)2134 3658 y Fu(T)2184 3639 y Ft(N)2256 3568 y Fw(#)17 b(:)456 3786 y FE(Let)32 b(no)m(w)i Fw(\022)879 3801 y Fl(0)946 3786 y FE(:=)28 b(1)k(and,)h(recursiv)m(ely)-8 b(,)456 3965 y(\(98\))1014 b Fw(\022)1689 3980 y Fq(n)p Fl(+1)1854 3965 y FE(:=)28 b Fx(F)10 b FE([)p Fw(\022)2139 3980 y Fq(n)2186 3965 y FE(])17 b Fw(;)456 4144 y FE(for)39 b(an)m(y)i Fw(n)f Fx(2)h Fn(N)9 b FE(.)72 b(Note)40 b(that)g Fw(\022)1681 4159 y Fq(n)1768 4144 y FE(is)g Fn(Z)1943 4108 y Fq(N)2008 4144 y FE(-p)s(erio)s(dic)d(for)j(an)m(y)h Fw(n)f Fx(2)h Fn(N)9 b FE(,)48 b(b)s(ecause)456 4272 y(so)40 b(are)h Fw(\022)799 4287 y Fl(0)839 4272 y FE(,)h Fw(U)51 b FE(and)41 b Fw(\036)p FE(.)67 b(Analogously)-8 b(,)41 b Fw(\022)2002 4287 y Fq(n)2090 4272 y Fx(\025)h FE(0,)g(b)s(ecause)g(so)e(is)g Fw(\022)2974 4287 y Fl(0)3014 4272 y FE(.)67 b(Thence,)456 4400 y(b)m(y)40 b(\(97\),)h(w)m(e)f(ha)m (v)m(e)h(that)f Fw(d\022)1537 4415 y Fq(n)1584 4400 y FE(\()p Fw(x)p FE(\))f(=)h Fw(\022)1915 4415 y Fq(n)1962 4400 y FE(\()p Fw(x)p FE(\))17 b Fw(dx)40 b FE(is)f(a)g(sequence)j(of)d (probabilit)m(y)456 4527 y(mesures)33 b(on)g Fn(T)1024 4491 y Fq(N)1094 4527 y FE(,)g(that)f(is)456 4742 y(\(99\))1661 4606 y Fv(Z)1716 4831 y Fu(T)1766 4813 y Ft(N)1839 4742 y Fw(d\022)1935 4757 y Fq(n)2026 4742 y FE(=)45 b(1)17 b Fw(;)456 4960 y FE(for)41 b(an)m(y)h Fw(n)h Fx(2)h Fn(N)9 b FE(.)76 b(W)-8 b(e)42 b(ma)m(y)f(th)m(us)i(supp)s(ose)g(that,) h(up)e(to)f(subsequence,)47 b Fw(d\022)3397 4975 y Fq(n)456 5088 y FE(w)m(eakly)e(con)m(v)m(erges)i(to)e(a)f(Radon)g(measure)h Fw(d\022)j FE(on)d Fn(T)2533 5052 y Fq(N)2648 5088 y FE(\(see,)k(e.g.,)f(page)d(55)456 5216 y(of)e([EG92)o(]\).)77 b(Also,)46 b Fw(d\022)h FE(is)c(a)g(probabilit)m(y)f(measure,)47 b(since)d Fn(T)2863 5179 y Fq(N)2977 5216 y FE(is)g(compact)p eop %%Page: 40 40 40 39 bop 456 251 a FC(40)666 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)456 450 y FE(\(see,)34 b(e.g.,)h(Theorem)e(1-\(ii\))e(on)j (page)f(54)g(of)g([EG92]\).)46 b(Our)33 b(ob)5 b(jectiv)m(e)35 b(no)m(w)f(is)456 578 y(to)e(sho)m(w)i(that)e Fw(d\022)k FE(is)c(absolutely)g(con)m(tin)m(uous)i(with)e(resp)s(ect)i(to)f(the)g (Leb)s(esgue)456 706 y(measure.)88 b(F)-8 b(or)46 b(this,)51 b(let)c Fw(\021)56 b Fx(2)e Fw(C)7 b FE(\()p Fn(T)1901 670 y Fq(N)1971 706 y FE(\))47 b(b)s(e)h(so)g(that)f(0)52 b Fx(\024)i Fw(\021)i Fx(\024)e FE(1.)87 b(Then,)456 834 y(exploiting)30 b(\(98\),)i(w)m(e)i(get)e(that)920 936 y Fv(Z)975 1162 y Fu(T)1025 1143 y Ft(N)1098 1072 y Fw(\021)t FE(\()p Fw(x)p FE(\))17 b Fw(d\022)1394 1087 y Fq(n)p Fl(+1)1531 1072 y FE(\()p Fw(x)p FE(\))g Fw(dx)761 1338 y FE(=)920 1202 y Fv(Z)975 1428 y Fu(T)1025 1409 y Ft(N)1098 1338 y Fw(\021)t FE(\()p Fw(x)p FE(\))g Fw(\022)1343 1353 y Fq(n)p Fl(+1)1480 1338 y FE(\()p Fw(x)p FE(\))g Fw(dx)761 1604 y FE(=)920 1468 y Fv(Z)975 1693 y Fu(T)1025 1675 y Ft(N)1098 1604 y Fw(\021)t FE(\()p Fw(x)p FE(\))g Fx(F)10 b FE([)p Fw(\022)1452 1619 y Fq(n)1498 1604 y FE(]\()p Fw(x)p FE(\))17 b Fw(dx)-1323 b FE(\(100\))761 1869 y(=)920 1734 y Fv(Z)975 1959 y Fu(T)1025 1940 y Ft(N)1098 1734 y Fv(Z)1153 1959 y Fu(R)1201 1940 y Ft(N)1279 1869 y Fw(\021)t FE(\()p Fw(x)p FE(\))17 b Fw(\022)1524 1884 y Fq(n)1572 1869 y FE(\()p Fw(x)22 b Fx(\000)h Fw(hv)t FE(\))17 b Fw(e)1994 1828 y Fs(\000)2059 1795 y Ft(hL)p Fr(\()p Ft(x)p Fm(\000)p Ft(hv)r(;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(x)p Fr(\))p Fm(\000)p Ft(\036)p Fr(\()p Ft(x)p Fm(\000)p Ft(hv)r Fr(\))p Fm(\000)p Ft(\025)p 2058 1813 866 3 v 2478 1854 a(\017)2954 1869 y Fw(dv)j(dx)d(:)456 2106 y FE(W)-8 b(e)36 b(no)m(w)g(denote)h(b)m(y)g Fw(\024)1346 2121 y Fq(i)1410 2106 y FE(some)e(appropriate)g(p)s(ositiv)m(e)g(quan)m (tities,)i(whic)m(h)f(de-)456 2234 y(p)s(end)i(only)g(on)f Fw(N)10 b FE(,)40 b Fw(\036)p FE(,)f Fw(L)p FE(,)h Fw(\025)p FE(,)f Fw(\017)g FE(and)f Fw(h)g FE(\(but)g(not)g(on)g Fw(\021)t FE(\).)59 b(Then,)40 b(b)m(y)f(the)g(fact)456 2362 y(that)32 b Fw(K)40 b FE(is)32 b(sup)s(erlinear)g(and)g(\(99\),)g (w)m(e)i(deduce)g(that)1388 2600 y Fw(h)1444 2559 y Fq(N)1528 2464 y Fv(Z)1583 2690 y Fu(R)1631 2671 y Ft(N)1709 2600 y Fw(\022)1754 2615 y Fq(n)1802 2600 y FE(\()p Fw(x)22 b Fx(\000)h Fw(hv)t FE(\))17 b Fw(e)2224 2559 y Fs(\000)2289 2525 y Ft(hK)s Fr(\()p Ft(v)r Fr(\))p 2288 2544 172 3 v 2361 2585 a Ft(\017)2491 2600 y Fw(dv)1228 2866 y FE(=)1388 2730 y Fv(Z)1443 2956 y Fu(R)1491 2937 y Ft(N)1569 2866 y Fw(\022)1614 2881 y Fq(n)1662 2866 y FE(\()p Fw(w)s FE(\))g Fw(e)1873 2825 y Fs(\000)1938 2791 y Ft(hK)s Fr(\(\()p Ft(x)p Fm(\000)p Ft(w)q Fr(\))p Ft(=h)p Fr(\))p 1937 2810 384 3 v 2115 2851 a Ft(\017)2351 2866 y Fw(dw)1228 3132 y FE(=)1409 3037 y Fv(X)1388 3256 y Fq(k)r Fs(2)p Fu(Z)1523 3237 y Ft(N)1592 2996 y Fv(Z)1647 3222 y Fu(T)1697 3203 y Ft(N)1770 3132 y Fw(\022)1815 3147 y Fq(n)1862 3132 y FE(\()p Fw(w)s FE(\))g Fw(e)2073 3091 y Fs(\000)2138 3057 y Ft(hK)s Fr(\(\()p Ft(x)p Fm(\000)p Ft(w)q Fm(\000)p Ft(k)q Fr(\))p Ft(=h)p Fr(\))p 2137 3075 466 3 v 2357 3117 a Ft(\017)2633 3132 y Fw(dw)1227 3446 y Fx(\024)84 b Fw(\024)1444 3461 y Fl(1)1538 3351 y Fv(X)1516 3570 y Fq(k)r Fs(2)p Fu(Z)1652 3552 y Ft(N)1720 3310 y Fv(Z)1776 3536 y Fu(T)1826 3517 y Ft(N)1899 3446 y Fw(\022)1944 3461 y Fq(n)1991 3446 y FE(\()p Fw(w)s FE(\))17 b Fw(e)2202 3405 y Fs(\000)p Fq(\024)2298 3414 y Fr(2)2331 3405 y Fs(j)p Fq(k)r Fs(j)2430 3446 y Fw(dw)1227 3760 y Fx(\024)84 b Fw(\024)1444 3775 y Fl(3)1500 3625 y Fv(Z)1555 3850 y Fu(T)1605 3831 y Ft(N)1678 3760 y Fw(\022)1723 3775 y Fq(n)1770 3760 y FE(\()p Fw(w)s FE(\))17 b Fw(dw)1228 3934 y FE(=)84 b Fw(\024)1444 3949 y Fl(3)1500 3934 y Fw(:)456 4131 y FE(This)32 b(estimate,)g(together)h(with)f(\(100\),)g (giv)m(es)h(that)1168 4233 y Fv(Z)1223 4459 y Fu(T)1273 4440 y Ft(N)1346 4369 y Fw(\021)t FE(\()p Fw(x)p FE(\))17 b Fw(d\022)1642 4384 y Fq(n)p Fl(+1)1779 4369 y FE(\()p Fw(x)p FE(\))44 b Fx(\024)h Fw(\024)2132 4384 y Fl(4)2205 4233 y Fv(Z)2260 4459 y Fu(T)2310 4440 y Ft(N)2383 4369 y Fw(\021)t FE(\()p Fw(x)p FE(\))17 b Fw(dx)g(:)456 4606 y FE(T)-8 b(aking)35 b(no)m(w)i(the)g(limit)c(as)j Fw(n)f Fx(!)e FE(+)p Fx(1)p FE(,)k(using)f(the)h(w)m(eak)g(con)m(v)m(ergence)i (of)d Fw(\022)3370 4621 y Fq(n)3417 4606 y FE(,)456 4734 y(w)m(e)d(gather)g(that)456 4966 y(\(101\))1243 4831 y Fv(Z)1299 5056 y Fu(T)1349 5037 y Ft(N)1421 4966 y Fw(\021)t FE(\()p Fw(x)p FE(\))17 b Fw(d\022)s FE(\()p Fw(x)p FE(\))44 b Fx(\024)h Fw(\024)2073 4981 y Fl(4)2129 4831 y Fv(Z)2185 5056 y Fu(T)2235 5037 y Ft(N)2307 4966 y Fw(\021)t FE(\()p Fw(x)p FE(\))17 b Fw(dx)g(;)456 5216 y FE(for)32 b(an)m(y)h Fw(\021)e Fx(2)d Fw(C)7 b FE(\()p Fn(T)1140 5179 y Fq(N)1211 5216 y FE(\).)p eop %%Page: 41 41 41 40 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)420 b(41)555 450 y FE(T)-8 b(ak)m(e)26 b(no)m(w)f(an)m(y)h(measurable)e(set)h Fw(A)j Fx(\032)g Fn(T)2070 414 y Fq(N)2141 450 y FE(,)e(with)f(small)d (Leb)s(esgue)k(measure,)456 578 y(sa)m(y)31 b Fx(j)p Fw(A)p Fx(j)c(\024)h Fw(\017)p FE(,)j(for)f Fw(\017)e(>)g FE(0.)42 b(W)-8 b(e)31 b(also)e(denote)i(the)g Fw(\022)s FE(-measure)g(of)e Fw(A)i FE(b)m(y)g Fw(\022)s FE(\()p Fw(A)p FE(\).)43 b(By)456 706 y(standard)34 b(results)g(on)g(the)h (Radon)e(measure)i(appro)m(ximation)c(with)j(op)s(en)g(and)456 834 y(compact)d(sets)i(\(see,)g(e.g.,)g(Theorem)f(4)f(on)h(page)g(8)g (of)f([EG92)o(]\),)h(w)m(e)h(ha)m(v)m(e)h(that)456 962 y(there)f(exist)g(a)f(compact)g(set)i Fw(K)39 b FE(and)33 b(an)f(op)s(en)h(set)g Fw(U)44 b FE(so)32 b(that)1642 1294 y Fw(K)52 b Fx(\022)44 b Fw(A)h Fx(\022)g Fw(U)27 b(;)1051 1787 y(\022)s FE(\()p Fw(A)p FE(\))g Fx(\024)i Fw(\022)s FE(\()p Fw(K)7 b FE(\))22 b(+)g Fw(\017)61 b FE(and)f Fx(j)p Fw(U)10 b Fx(j)28 b(\024)g(j)p Fw(A)p Fx(j)21 b FE(+)h Fw(\017)28 b Fx(\024)h FE(2)p Fw(\017)17 b(:)456 2098 y FE(Exploiting)45 b(the)j(classical)e(Urysohn's)j(Lemma)d (\(see,)52 b(e.g.,)g(Theorem)47 b(2)g(on)456 2225 y(page)37 b(15)f(of)g([DS58]\),)i(w)m(e)g(see)g(that)e(there)i(exists)g(a)e (function)h Fw(\021)h Fx(2)e Fw(C)7 b FE(\()p Fn(T)3212 2189 y Fq(N)3282 2225 y FE(\))37 b(so)456 2353 y(that)h(0)g Fx(\024)h Fw(\021)j Fx(\024)d FE(1,)i Fw(\021)t FE(\()p Fw(x)p FE(\))d(=)g(0)h(for)f(an)m(y)i Fw(x)e Fx(2)h Fn(T)2229 2317 y Fq(N)2326 2353 y Fx(n)27 b Fw(U)49 b FE(and)39 b Fw(\021)t FE(\()p Fw(x)p FE(\))f(=)h(1)f(for)h(an)m(y)456 2481 y Fw(x)28 b Fx(2)g Fw(K)7 b FE(.)555 2609 y(Accordingly)-8 b(,)1327 2942 y Fw(\022)s FE(\()p Fw(A)p FE(\))83 b Fx(\024)h Fw(\022)s FE(\()p Fw(K)7 b FE(\))39 b(+)f Fw(\017)1607 3142 y Fx(\024)1768 3007 y Fv(Z)1823 3232 y Fu(T)1873 3214 y Ft(N)1946 3142 y Fw(\021)t FE(\()p Fw(x)p FE(\))17 b Fw(d\022)s FE(\()p Fw(x)p FE(\))38 b(+)h Fw(\017)1607 3408 y Fx(\024)84 b Fw(\024)1824 3423 y Fl(4)1896 3273 y Fv(Z)1952 3498 y Fu(T)2002 3479 y Ft(N)2075 3408 y Fw(\021)t FE(\()p Fw(x)p FE(\))17 b Fw(dx)38 b FE(+)h Fw(\017)1607 3613 y Fx(\024)84 b Fw(\024)1824 3628 y Fl(4)1880 3613 y Fx(j)p Fw(U)10 b Fx(j)39 b FE(+)f Fw(\017)1607 3773 y Fx(\024)84 b FE(\(2)p Fw(\024)1911 3788 y Fl(4)1972 3773 y FE(+)22 b(1\))17 b Fw(\017)g(;)456 4106 y FE(b)m(y)33 b(means)g(of)f(\(101\).)555 4234 y(Thence,)49 b Fw(d\022)e FE(is)d(absolutely)f(con)m(tin)m(uous)i(with)e(resp)s(ect)j(to)e(the)g (Leb)s(esgue)456 4361 y(measure.)555 4489 y(Th)m(us,)34 b(w)m(e)g(write)1596 4777 y Fw(d\022)s FE(\()p Fw(x)p FE(\))28 b(=)g Fw(\022)s FE(\()p Fw(x)p FE(\))17 b Fw(dx)g(;)456 5088 y FE(with)22 b Fw(\022)s FE(\()p Fw(x)p FE(\))28 b Fx(2)g Fw(L)1035 5052 y Fl(1)1075 5088 y FE(\()p Fn(T)1176 5052 y Fq(N)1246 5088 y FE(\).)40 b(Since)23 b Fw(d\022)i FE(is,)f(b)m(y)g(construction,)g(a)f(probabilit)m(y)d(measure)456 5216 y(on)32 b Fn(T)654 5179 y Fq(N)725 5216 y FE(,)g(w)m(e)i(ha)m(v)m (e)g(that)e Fw(\022)k FE(satis\014es)d(\(92\))f(and)h(\(94\).)p eop %%Page: 42 42 42 41 bop 456 251 a FC(42)666 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)555 450 y FE(Moreo)m(v)m(er,)30 b(giv)m(en)e(an)m(y)h Fw(\021)i Fx(2)d Fw(C)7 b FE(\()p Fn(T)1787 414 y Fq(N)1858 450 y FE(\),)28 b(w)m(e)h(deduce)g(from)d(\(100\))h(and)h(the)g(w)m (eak)456 578 y(con)m(v)m(ergence)35 b(of)d Fw(d\022)1202 593 y Fq(n)1282 578 y FE(that)814 665 y Fv(Z)869 891 y Fu(T)919 872 y Ft(N)992 801 y Fw(\021)t FE(\()p Fw(x)p FE(\))17 b Fw(\022)s FE(\()p Fw(x)p FE(\))g Fw(dx)655 1067 y FE(=)814 931 y Fv(Z)869 1157 y Fu(T)919 1138 y Ft(N)992 1067 y Fw(\021)t FE(\()p Fw(x)p FE(\))g Fw(d\022)s FE(\()p Fw(x)p FE(\))655 1333 y(=)134 b(lim)814 1392 y Fq(n)p Fs(!)p Fl(+)p Fs(1)1069 1197 y Fv(Z)1125 1423 y Fu(T)1175 1404 y Ft(N)1247 1333 y Fw(\021)t FE(\()p Fw(x)p FE(\))17 b Fw(d\022)1543 1348 y Fq(n)p Fl(+1)1680 1333 y FE(\()p Fw(x)p FE(\))655 1598 y(=)134 b(lim)814 1658 y Fq(n)p Fs(!)p Fl(+)p Fs(1)1069 1463 y Fv(Z)1125 1688 y Fu(T)1175 1669 y Ft(N)1247 1463 y Fv(Z)1303 1688 y Fu(R)1351 1669 y Ft(N)1429 1598 y Fw(\021)t FE(\()p Fw(x)p FE(\))17 b Fw(\022)1674 1613 y Fq(n)1721 1598 y FE(\()p Fw(x)23 b Fx(\000)f Fw(hv)t FE(\))17 b Fw(e)2143 1557 y Fs(\000)2208 1524 y Ft(hL)p Fr(\()p Ft(x)p Fm(\000)p Ft(hv)r(;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(x)p Fr(\))p Fm(\000)p Ft(\036)p Fr(\()p Ft(x)p Fm(\000)p Ft(hv)r Fr(\))p Fm(\000)p Ft(\025)p 2208 1542 866 3 v 2627 1583 a(\017)3104 1598 y Fw(dv)j(dx)655 1864 y FE(=)134 b(lim)814 1924 y Fq(n)p Fs(!)p Fl(+)p Fs(1)1069 1729 y Fv(Z)1125 1954 y Fu(R)1173 1935 y Ft(N)1251 1729 y Fv(Z)1306 1954 y Fu(T)1356 1935 y Ft(N)1429 1864 y Fw(\021)t FE(\()p Fw(y)25 b FE(+)d Fw(hv)t FE(\))17 b Fw(\022)1897 1879 y Fq(n)1944 1864 y FE(\()p Fw(y)t FE(\))g Fw(e)2134 1823 y Fs(\000)2199 1790 y Ft(hL)p Fr(\()p Ft(y)r(;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(y)r Fr(+)p Ft(hv)r Fr(\))p Fm(\000)p Ft(\036)p Fr(\()p Ft(y)r Fr(\))p Fm(\000)p Ft(\025)p 2198 1808 743 3 v 2556 1849 a(\017)2971 1864 y Fw(dy)i(dv)655 2130 y FE(=)134 b(lim)814 2190 y Fq(n)p Fs(!)p Fl(+)p Fs(1)1069 1994 y Fv(Z)1125 2220 y Fu(R)1173 2201 y Ft(N)1251 1994 y Fv(Z)1306 2220 y Fu(T)1356 2201 y Ft(N)1429 2130 y Fw(\021)t FE(\()p Fw(y)25 b FE(+)d Fw(hv)t FE(\))17 b Fw(e)1897 2089 y Fs(\000)1962 2055 y Ft(hL)p Fr(\()p Ft(y)r(;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(y)r Fr(+)p Ft(hv)r Fr(\))p Fm(\000)p Ft(\036)p Fr(\()p Ft(y)r Fr(\))p Fm(\000)p Ft(\025)p 1961 2074 V 2320 2115 a(\017)2735 2130 y Fw(d\022)2831 2145 y Fq(n)2878 2130 y FE(\()p Fw(y)t FE(\))g Fw(dv)655 2396 y FE(=)814 2260 y Fv(Z)869 2486 y Fu(R)917 2467 y Ft(N)996 2260 y Fv(Z)1051 2486 y Fu(T)1101 2467 y Ft(N)1174 2396 y Fw(\021)t FE(\()p Fw(y)25 b FE(+)d Fw(hv)t FE(\))17 b Fw(e)1642 2355 y Fs(\000)1707 2321 y Ft(hL)p Fr(\()p Ft(y)r(;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(y)r Fr(+)p Ft(hv)r Fr(\))p Fm(\000)p Ft(\036)p Fr(\()p Ft(y)r Fr(\))p Fm(\000)p Ft(\025)p 1706 2340 V 2064 2381 a(\017)2480 2396 y Fw(d\022)s FE(\()p Fw(y)t FE(\))g Fw(dv)655 2662 y FE(=)814 2526 y Fv(Z)869 2752 y Fu(R)917 2733 y Ft(N)996 2526 y Fv(Z)1051 2752 y Fu(T)1101 2733 y Ft(N)1174 2662 y Fw(\021)t FE(\()p Fw(y)25 b FE(+)d Fw(hv)t FE(\))17 b Fw(\022)s FE(\()p Fw(y)t FE(\))g Fw(e)1835 2620 y Fs(\000)1900 2587 y Ft(hL)p Fr(\()p Ft(y)r(;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(y)r Fr(+)p Ft(hv)r Fr(\))p Fm(\000)p Ft(\036)p Fr(\()p Ft(y)r Fr(\))p Fm(\000)p Ft(\025)p 1898 2605 V 2256 2647 a(\017)2671 2662 y Fw(dy)j(dv)655 2927 y FE(=)814 2792 y Fv(Z)869 3017 y Fu(R)917 2998 y Ft(N)996 2792 y Fv(Z)1051 3017 y Fu(T)1101 2998 y Ft(N)1174 2927 y Fw(\021)t FE(\()p Fw(x)p FE(\))d Fw(\022)s FE(\()p Fw(x)22 b Fx(\000)h Fw(hv)t FE(\))17 b Fw(e)1844 2886 y Fs(\000)1909 2853 y Ft(hL)p Fr(\()p Ft(x)p Fm(\000)p Ft(hv)r(;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(x)p Fr(\))p Fm(\000)p Ft(\036)p Fr(\()p Ft(x)p Fm(\000)p Ft(hv)r Fr(\))p Fm(\000)p Ft(\025)p 1908 2871 866 3 v 2328 2912 a(\017)2804 2927 y Fw(dx)g(dv)j(:)456 3160 y FE(Since)32 b Fw(\021)k FE(here)d(ab)s(o)m(v)m(e)g(is)e (arbitrary)-8 b(,)31 b(w)m(e)j(get)e(that)g Fw(\022)j FE(satis\014es)e(\(93\),)e(as)i(desired.)3367 3288 y Fj(\003)555 3482 y FE(Suc)m(h)h Fw(\022)i FE(pla)m(ys)c(a)h(decisiv)m (e)g(r^)-49 b(ole)31 b(in)h(the)h(construction)g(of)f(p)s(enalized)g (Mather)456 3610 y(measures,)i(as)g(w)m(e)h(are)f(no)m(w)g(going)e(to)h (sho)m(w.)48 b(W)-8 b(e)34 b(also)f(remark)g(that)h(the)g(reg-)456 3738 y(ularit)m(y)d(and)h(uniqueness)j(of)c(the)i(function)f Fw(\022)k FE(will)30 b(b)s(e)i(dealt)g(with)g(in)g(Prop)s(osi-)456 3866 y(tions)g(34)g(and)g(35)g(b)s(elo)m(w.)555 3994 y(Let)h(us)g(no)m(w)g(deal)f(with)h(the)g(en)m(trop)m(y)h(p)s(enalized) d(Mather)i(measures:)456 4149 y FI(Theorem)k(32.)k FD(L)-5 b(et)36 b Fw(\036)27 b Fx(2)h Fw(W)1569 4113 y Fl(1)p Fq(;)p Fs(1)1698 4149 y FE(\()p Fn(T)1799 4113 y Fq(N)1870 4149 y FE(\))35 b FD(satisfy)456 4336 y FE(\(102\))986 b Fx(G)6 b FE([)p Fw(\036)p FE(])28 b(=)f Fw(\036)22 b FE(+)g Fw(\025:)456 4524 y FD(L)-5 b(et)35 b Fw(\022)c Fx(2)d Fw(L)860 4488 y Fl(1)900 4524 y FE(\()p Fn(T)1001 4488 y Fq(N)1071 4524 y FE(\))35 b FD(b)-5 b(e)35 b(so)f(that)1552 4485 y Fl(13)1658 4524 y Fw(\022)d Fx(\025)d FE(0)p FD(,)456 4752 y FE(\(103\))946 4617 y Fv(Z)1001 4842 y Fu(R)1049 4823 y Ft(N)1128 4752 y Fw(\022)s FE(\()p Fw(x)22 b Fx(\000)h Fw(hv)t FE(\))p Fw(e)1581 4711 y Fs(\000)1646 4678 y Ft(hL)p Fr(\()p Ft(x)p Fm(\000)p Ft(hv)r(;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(x)p Fr(\))p Fm(\000)p Ft(\036)p Fr(\()p Ft(x)p Fm(\000)p Ft(hv)r Fr(\))p Fm(\000)p Ft(\025)p 1646 4696 V 2066 4737 a(\017)2542 4752 y Fw(dv)31 b FE(=)d Fw(\022)s FE(\()p Fw(x)p FE(\))p 456 4898 499 4 v 555 4964 a Fl(13)626 4996 y FH(The)i(existence)h(of)f(a)h Fp(\022)i FH(satisfying)d(\(103\))f(and)i(\(104\))f(is)g(assured)g(b)n (y)g(Lemma)g(31)g(here)456 5106 y(ab)r(o)n(v)n(e.)72 b(The)40 b(existence)f(\(and)h(essen)n(tial)f(uniqueness\))h(of)f Fp(\036)i FH(and)e Fp(\025)i FH(satisfying)e(\(102\))g(is)456 5216 y(assured)26 b(b)n(y)h(Theorem)g(27.)p eop %%Page: 43 43 43 42 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)420 b(43)456 450 y FD(and)456 659 y FE(\(104\))1581 523 y Fv(Z)1637 749 y Fu(T)1687 730 y Ft(N)1759 659 y Fw(\022)s FE(\()p Fw(x)p FE(\))17 b Fw(dx)45 b FE(=)f(1)17 b Fw(:)456 891 y FD(L)-5 b(et)456 1083 y FE(\(105\))486 b Fw(\026)p FE(\()p Fw(x;)17 b(v)t FE(\))44 b(:=)g Fw(\022)s FE(\()p Fw(x)p FE(\))p Fw(e)1865 1042 y Fs(\000)1930 1009 y Ft(hL)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p Fm(\000)p Ft(\036)p Fr(\()p Ft(x)p Fr(\))p Fm(\000)p Ft(\025)p 1930 1027 748 3 v 2291 1068 a(\017)2708 1083 y Fw(:)456 1276 y FD(Then,)d Fw(\026)f FD(minimizes)f(the)h(functional) g(\(89\))g(over)g(the)h(sp)-5 b(ac)g(e)39 b(\(90\))h(under)g(the)456 1404 y(c)-5 b(onstr)g(aint)34 b(\(88\).)456 1562 y Fz(Remark.)86 b FE(The)34 b(result)e(in)g(Theorem)h(32)f(here)i(ab)s(o)m(v)m(e)f(th)m (us)h(sa)m(ys)g(that)1071 1767 y Fw(d\026)p FE(\()p Fw(x;)17 b(v)t FE(\))27 b(=)g Fw(\022)s FE(\()p Fw(x)p FE(\))p Fw(e)1761 1725 y Fs(\000)1826 1692 y Ft(hL)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p Fm(\000)p Ft(\036)p Fr(\()p Ft(x)p Fr(\))p Fm(\000)p Ft(\025)p 1827 1710 V 2187 1751 a(\017)2605 1767 y Fw(dx)17 b(dv)456 1959 y FE(is)47 b(an)h(\\en)m(trop)m(y)i(p)s(enalized)d (Mather)i(measure".)90 b(The)49 b(uniqueness)h(of)e(the)456 2087 y(measure)33 b Fw(\026)f FE(will)e(b)s(e)j(pro)m(v)m(ed)h(in)e (Theorem)g(36.)456 2288 y FD(Pr)-5 b(o)g(of.)41 b FE(Exploiting)30 b(\(103\))i(and)h(\(95\),)f(w)m(e)h(deduce)h(that)968 2386 y Fv(Z)1023 2611 y Fu(T)1073 2593 y Ft(N)1125 2611 y Fs(\002)p Fu(R)1228 2593 y Ft(N)1307 2522 y Fw(')p FE(\()p Fw(x)22 b FE(+)g Fw(hv)t FE(\))17 b Fw(d\026)p FE(\()p Fw(x;)g(v)t FE(\))809 2794 y(=)968 2659 y Fv(Z)1023 2884 y Fu(T)1073 2865 y Ft(N)1125 2884 y Fs(\002)p Fu(R)1228 2865 y Ft(N)1307 2794 y Fw(')p FE(\()p Fw(x)22 b FE(+)g Fw(hv)t FE(\))17 b Fw(\022)s FE(\()p Fw(x)p FE(\))g Fw(e)1987 2753 y Fs(\000)2052 2720 y Ft(hL)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p Fm(\000)p Ft(\036)p Fr(\()p Ft(x)p Fr(\))p Fm(\000)p Ft(\025)p 2052 2738 V 2412 2779 a(\017)2830 2794 y Fw(dx)g(dv)809 3067 y FE(=)968 2931 y Fv(Z)1023 3157 y Fu(T)1073 3138 y Ft(N)1125 3157 y Fs(\002)p Fu(R)1228 3138 y Ft(N)1307 3067 y Fw(')p FE(\()p Fw(x)p FE(\))g Fw(\022)s FE(\()p Fw(x)22 b Fx(\000)h Fw(hv)t FE(\))17 b Fw(e)1989 3026 y Fs(\000)2054 2992 y Ft(hL)p Fr(\()p Ft(x)p Fm(\000)p Ft(hv)r(;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(x)p Fr(\))p Fm(\000)p Ft(\036)p Fr(\()p Ft(x)p Fm(\000)p Ft(hv)r Fr(\))p Fm(\000)p Ft(\025)p 2053 3011 866 3 v 2473 3052 a(\017)2950 3067 y Fw(dx)g(dv)809 3340 y FE(=)968 3204 y Fv(Z)1023 3430 y Fu(T)1073 3411 y Ft(N)1146 3340 y Fw(')p FE(\()p Fw(x)p FE(\))g Fw(\022)s FE(\()p Fw(x)p FE(\))g Fw(dx)809 3606 y FE(=)968 3470 y Fv(Z)1023 3695 y Fu(T)1073 3677 y Ft(N)1125 3695 y Fs(\002)p Fu(R)1228 3677 y Ft(N)1307 3606 y Fw(')p FE(\()p Fw(x)p FE(\))g Fw(\022)s FE(\()p Fw(x)p FE(\))g Fw(e)1760 3564 y Fs(\000)1825 3531 y Ft(hL)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p Fm(\000)p Ft(\036)p Fr(\()p Ft(x)p Fr(\))p Fm(\000)p Ft(\025)p 1825 3549 748 3 v 2185 3590 a(\017)2603 3606 y Fw(dx)g(dv)809 3878 y FE(=)968 3743 y Fv(Z)1023 3968 y Fu(T)1073 3949 y Ft(N)1125 3968 y Fs(\002)p Fu(R)1228 3949 y Ft(N)1307 3878 y Fw(')p FE(\()p Fw(x)p FE(\))g Fw(d\026)p FE(\()p Fw(x;)g(v)t FE(\))g Fw(;)456 4130 y FE(for)32 b(an)m(y)i(function)f Fw(')c Fx(2)g Fw(L)1427 4094 y Fs(1)1502 4130 y FE(\()p Fn(T)1603 4094 y Fq(N)1673 4130 y FE(\),)34 b(whic)m(h)g(sho)m(ws)g (that)f(the)h(constrain)m(t)f(\(88\))g(is)456 4258 y(satis\014ed)g(b)m (y)g Fw(\026)p FE(.)43 b(In)33 b(addition,)e(from)g(\(105\))h(and)h (\(95\),)f(w)m(e)h(get)g(that)456 4492 y(\(106\))1464 4356 y Fv(Z)1519 4582 y Fu(R)1567 4563 y Ft(N)1646 4492 y Fw(\026)p FE(\()p Fw(x;)17 b(v)t FE(\))g Fw(dv)47 b FE(=)d Fw(\022)s FE(\()p Fw(x)p FE(\))17 b Fw(;)456 4737 y FE(for)32 b(an)m(y)h Fw(x)28 b Fx(2)g Fn(T)1029 4701 y Fq(N)1100 4737 y FE(.)43 b(Therefore,)34 b(b)m(y)f(\(104\),)1187 4835 y Fv(Z)1243 5061 y Fu(T)1293 5042 y Ft(N)1345 5061 y Fs(\002)p Fu(R)1448 5042 y Ft(N)1542 4971 y Fw(d\026)p FE(\()p Fw(x;)17 b(v)t FE(\))27 b(=)2009 4835 y Fv(Z)2064 5061 y Fu(T)2114 5042 y Ft(N)2187 4971 y Fw(\022)s FE(\()p Fw(x)p FE(\))17 b Fw(dx)28 b FE(=)f(1)17 b Fw(;)456 5216 y FE(and)32 b(so)h Fw(\026)f FE(b)s(elongs)g(to)h(the)g(space)g(giv)m (en)g(b)m(y)g(\(90\).)p eop %%Page: 44 44 44 43 bop 456 251 a FC(44)666 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)555 450 y FE(T)h(o)41 b(pro)m(v)m(e)g(the)g(minimizing)35 b(prop)s(ert)m(y)41 b(of)f Fw(\026)g FE(it)f(su\016ces)k(to)c(sho)m(w)j (that)e(for)456 578 y(an)m(y)34 b(other)h(densit)m(y)42 b(~)-56 b Fw(\026)33 b FE(in)h(\(90\))f(that)h(satis\014es)h(\(88\))e (and)i(an)m(y)f(0)c Fx(\024)h Fw(\034)42 b Fx(\024)31 b FE(1,)j(one)456 706 y(has)f(that)f(the)h(function)456 1118 y(\(107\))625 b Fw(I)8 b FE([)p Fw(\034)j FE(])28 b(=)1594 982 y Fv(Z)1649 1208 y Fu(T)1699 1189 y Ft(N)1751 1208 y Fs(\002)p Fu(R)1854 1189 y Ft(N)1932 1118 y Fw(Ld\026)2108 1133 y Fq(\034)2173 1118 y FE(+)2290 1050 y Fw(\017)p 2281 1095 57 4 v 2281 1186 a(h)2347 1118 y(S)6 b FE([)p Fw(\026)2499 1133 y Fq(\034)2542 1118 y FE(])p Fw(;)456 1565 y FE(with)40 b Fw(\026)745 1580 y Fq(\034)831 1565 y FE(:=)i(\(1)28 b Fx(\000)h Fw(\034)11 b FE(\))p Fw(\026)28 b FE(+)g Fw(\034)19 b FE(~)-57 b Fw(\026)p FE(,)43 b(is)e(con)m(v)m(ex) i(and)f(that)f Fw(I)2566 1529 y Fs(0)2589 1565 y FE([0])h(=)h(0.)69 b(Denote)41 b(b)m(y)474 1693 y(_)-45 b Fw(\026)515 1708 y Fq(\034)585 1693 y FE(:=)35 b(~)-56 b Fw(\026)22 b Fx(\000)g Fw(\026)32 b FE(Then,)913 2133 y Fw(I)964 2092 y Fs(0)987 2133 y FE([)p Fw(\034)11 b FE(])83 b(=)1336 1998 y Fv(Z)1392 2223 y Fu(T)1442 2204 y Ft(N)1494 2223 y Fs(\002)p Fu(R)1597 2204 y Ft(N)1675 1993 y Fv(\024)1728 2133 y Fw(Ld)18 b FE(_)-45 b Fw(\026)1904 2148 y Fq(\034)1969 2133 y FE(+)2085 2066 y Fw(\017)p 2077 2110 V 2077 2202 a(h)2165 2133 y Fx(\001)2215 1993 y Fv(\022)2288 2133 y FE(ln)2499 2066 y Fw(\026)2558 2081 y Fq(\034)p 2396 2110 308 4 v 2396 2122 a Fv(R)2479 2203 y Fw(\026)2538 2218 y Fq(\034)2581 2203 y Fw(dw)2714 1993 y Fv(\023)2809 2133 y Fx(\001)41 b FE(_)-46 b Fw(\026)2918 2148 y Fq(\034)1346 2418 y Fx(\000)1442 2350 y Fw(\017)p 1433 2395 57 4 v 1433 2486 a(h)1522 2418 y Fx(\001)1685 2350 y Fw(\026)1744 2365 y Fq(\034)p 1582 2395 308 4 v 1582 2407 a Fv(R)1665 2487 y Fw(\026)1724 2502 y Fq(\034)1767 2487 y Fw(dw)1916 2277 y Fv(\022)1990 2282 y(Z)2124 2418 y FE(_)h Fw(\026)2165 2433 y Fq(\034)2208 2418 y Fw(dw)2332 2277 y Fv(\023)o(\025)2490 2418 y Fw(;)-2061 b FE(\(108\))456 2863 y(where)33 b(w)m(e)h(ha)m(v)m (e)g(used)g(the)f(fact)f(that)1065 3139 y Fv(Z)1121 3365 y Fu(T)1171 3346 y Ft(N)1223 3365 y Fs(\002)p Fu(R)1326 3346 y Ft(N)1422 3275 y FE(_)-45 b Fw(\026)1463 3290 y Fq(t)1520 3275 y FE(=)1623 3139 y Fv(Z)1679 3365 y Fu(T)1729 3346 y Ft(N)1781 3365 y Fs(\002)p Fu(R)1884 3346 y Ft(N)1945 3275 y FE(\()8 b(~)-57 b Fw(\026)22 b Fx(\000)h Fw(\026)p FE(\))k(=)h(1)22 b Fx(\000)g FE(1)28 b(=)f(0)17 b Fw(:)456 3717 y FE(F)-8 b(urther,)32 b(notice)h(that,)f (since)h(b)s(oth)f Fw(\026)h FE(and)40 b(~)-57 b Fw(\026)32 b FE(are)h(probabilit)m(y)e(measures,)1202 4022 y Fv(Z)1257 4247 y Fu(T)1307 4228 y Ft(N)1359 4247 y Fs(\002)p Fu(R)1462 4228 y Ft(N)1699 4090 y Fw(\026)p 1551 4134 356 4 v 1551 4146 a Fv(R)1598 4261 y Fu(R)1646 4242 y Ft(N)1724 4227 y Fw(\026dw)1933 4017 y Fv(\022)2006 4022 y(Z)2061 4247 y Fu(R)2109 4228 y Ft(N)2171 4157 y FE(\()8 b(~)-57 b Fw(\026)22 b Fx(\000)g Fw(\026)p FE(\))p Fw(dw)2610 4017 y Fv(\023)2716 4157 y Fw(dx)17 b(dv)1043 4455 y FE(=)1202 4320 y Fv(Z)1257 4545 y Fu(T)1307 4526 y Ft(N)1401 4301 y Fv(R)1448 4416 y Fu(R)1496 4397 y Ft(N)1574 4382 y Fw(\026dv)p 1390 4432 V 1390 4444 a Fv(R)1437 4559 y Fu(R)1485 4540 y Ft(N)1564 4524 y Fw(\026dw)1789 4315 y Fv(\022)1862 4320 y(Z)1917 4545 y Fu(R)1965 4526 y Ft(N)2027 4455 y FE(\()8 b(~)-57 b Fw(\026)22 b Fx(\000)g Fw(\026)p FE(\))p Fw(dw)2466 4315 y Fv(\023)2572 4455 y Fw(dx)1043 4744 y FE(=)1202 4608 y Fv(Z)1257 4834 y Fu(T)1307 4815 y Ft(N)1380 4603 y Fv(\022)1454 4608 y(Z)1509 4834 y Fu(R)1557 4815 y Ft(N)1619 4744 y FE(\()8 b(~)-57 b Fw(\026)21 b Fx(\000)i Fw(\026)p FE(\))p Fw(dw)2058 4603 y Fv(\023)2163 4744 y Fw(dx)-1813 b FE(\(109\))1043 5014 y(=)1202 4879 y Fv(Z)1257 5104 y Fu(T)1307 5085 y Ft(N)1359 5104 y Fs(\002)p Fu(R)1462 5085 y Ft(N)1524 5014 y FE(\()8 b(~)-57 b Fw(\026)22 b Fx(\000)h Fw(\026)p FE(\))1043 5216 y(=)83 b(1)22 b Fx(\000)h FE(1)44 b(=)g(0)17 b Fw(:)p eop %%Page: 45 45 45 44 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)420 b(45)456 450 y FE(Moreo)m(v)m(er,)34 b(making)d(use)i(of)g(\(106\))e(and)i(\(105\),) f(w)m(e)h(deduce)i(that)1189 537 y Fv(Z)1245 763 y Fu(T)1295 744 y Ft(N)1347 763 y Fs(\002)p Fu(R)1450 744 y Ft(N)1528 673 y Fw(hL)17 b(d)p FE(\()8 b(~)-57 b Fw(\026)21 b Fx(\000)i Fw(\026)p FE(\))1189 945 y(+)p Fw(\017)1321 810 y Fv(Z)1376 1035 y Fu(T)1426 1016 y Ft(N)1478 1035 y Fs(\002)p Fu(R)1581 1016 y Ft(N)1660 945 y FE(ln)1916 878 y Fw(\026)p 1768 922 356 4 v 1768 934 a Fv(R)1815 1049 y Fu(R)1863 1030 y Ft(N)1941 1015 y Fw(\026dw)2133 945 y FE(\()8 b(~)-57 b Fw(\026)22 b Fx(\000)g Fw(\026)p FE(\))17 b Fw(dx)g(dv)1030 1234 y FE(=)1189 1098 y Fv(Z)1245 1324 y Fu(T)1295 1305 y Ft(N)1347 1324 y Fs(\002)p Fu(R)1450 1305 y Ft(N)1528 1094 y Fv(\022)1601 1234 y Fw(hL)p FE(\()p Fw(x;)g(v)t FE(\))23 b(+)f Fw(\017)17 b FE(ln)2401 1167 y Fw(\026)p FE(\()p Fw(x;)g(v)t FE(\))p 2233 1211 620 4 v 2233 1223 a Fv(R)2281 1338 y Fu(R)2329 1319 y Ft(N)2407 1303 y Fw(\026)p FE(\()p Fw(x;)g(w)s FE(\))g Fw(dw)2863 1094 y Fv(\023)1189 1453 y Fx(\001)p FE(\()8 b(~)-57 b Fw(\026)p FE(\()p Fw(x;)17 b(v)t FE(\))k Fx(\000)i Fw(\026)p FE(\()p Fw(x;)17 b(v)t FE(\)\))g Fw(dx)g(dv)1030 1663 y FE(=)1189 1527 y Fv(Z)1245 1752 y Fu(T)1295 1734 y Ft(N)1347 1752 y Fs(\002)p Fu(R)1450 1734 y Ft(N)1528 1522 y Fv(\022)1601 1663 y Fw(hL)p FE(\()p Fw(x;)g(v)t FE(\))23 b(+)f Fw(\017)17 b FE(ln)2233 1595 y Fw(\026)p FE(\()p Fw(x;)g(v)t FE(\))p 2233 1640 285 4 v 2286 1731 a Fw(\022)s FE(\()p Fw(x)p FE(\))2528 1522 y Fv(\023)1189 1874 y Fx(\001)p FE(\()8 b(~)-57 b Fw(\026)p FE(\()p Fw(x;)17 b(v)t FE(\))k Fx(\000)i Fw(\026)p FE(\()p Fw(x;)17 b(v)t FE(\)\))g Fw(dx)g(dv)-1766 b FE(\(110\))1030 2075 y(=)1189 1940 y Fv(Z)1245 2165 y Fu(T)1295 2146 y Ft(N)1347 2165 y Fs(\002)p Fu(R)1450 2146 y Ft(N)1528 1965 y Fv(\020)1610 2075 y Fx(\000)22 b Fw(\036)p FE(\()p Fw(x)g FE(+)g Fw(hv)t FE(\))g(+)g Fw(\036)p FE(\()p Fw(x)p FE(\))h(+)f Fw(\025)2612 1965 y Fv(\021)1189 2287 y Fx(\001)p FE(\()8 b(~)-57 b Fw(\026)p FE(\()p Fw(x;)17 b(v)t FE(\))k Fx(\000)i Fw(\026)p FE(\()p Fw(x;)17 b(v)t FE(\)\))g Fw(dx)g(dv)1030 2488 y FE(=)1189 2352 y Fv(Z)1305 2377 y(\020)1365 2488 y Fw(\036)p FE(\()p Fw(x)p FE(\))22 b Fx(\000)h Fw(\036)p FE(\()p Fw(x)f FE(+)g Fw(hv)t FE(\))2092 2377 y Fv(\021)2168 2488 y Fw(d)8 b FE(~)-57 b Fw(\026)1189 2749 y FE(+)1282 2614 y Fv(Z)1398 2639 y(\020)1457 2749 y Fw(\036)p FE(\()p Fw(x)23 b FE(+)f Fw(hv)t FE(\))g Fx(\000)g Fw(\036)p FE(\()p Fw(x)p FE(\))2184 2639 y Fv(\021)2260 2749 y Fw(d\026)1189 3020 y FE(+)p Fw(\025)1339 2879 y Fv(\022)1412 2884 y(Z)1528 3020 y Fw(d)8 b FE(~)-57 b Fw(\026)22 b Fx(\000)1759 2884 y Fv(Z)1876 3020 y Fw(d\026)1986 2879 y Fv(\023)1030 3230 y FE(=)83 b(0)22 b(+)g(0)g(+)g Fw(\025)p FE(\(1)g Fx(\000)g FE(1\))44 b(=)h(0)17 b Fw(;)456 3411 y FE(where)32 b(w)m(e)g(ha)m(v)m(e)g(also)e(used)i(again)e(that)h(b)s (oth)f Fw(\026)h FE(and)39 b(~)-57 b Fw(\026)31 b FE(are)f(probabilit)m (y)f(mea-)456 3539 y(sures)34 b(satisfying)d(\(88\).)555 3667 y(Collecting)g(\(108\),)h(\(109\))f(and)i(\(110\),)f(it)f(follo)m (ws)h(that)1758 3848 y Fw(I)1809 3807 y Fs(0)1832 3848 y FE([0])c(=)f(0)p Fw(:)456 4030 y FE(Th)m(us,)34 b(to)e(pro)m(v)m(e)i (that)e Fw(\026)g FE(is)h(minimal,)28 b(it)k(is)g(su\016cien)m(t)h(to)g (sho)m(w)g(that)1745 4212 y Fw(I)1796 4171 y Fs(00)1838 4212 y FE([)p Fw(\034)11 b FE(])29 b Fx(\025)f FE(0)p Fw(:)456 4393 y FE(T)-8 b(o)32 b(c)m(hec)m(k)j(this,)d(w)m(e)i(tak)m(e) f(a)g(further)g(deriv)-5 b(ativ)m(e)32 b(in)f(\(108\),)h(th)m(us)i (obtaining)963 4588 y Fw(h)p 963 4632 57 4 v 971 4723 a(\017)1046 4655 y(I)1097 4614 y Fs(00)1139 4655 y FE([)p Fw(\034)11 b FE(])28 b(=)1378 4520 y Fv(Z)1433 4745 y Fu(T)1483 4726 y Ft(N)1535 4745 y Fs(\002)p Fu(R)1638 4726 y Ft(N)1745 4588 y FE(_)-45 b Fw(\026)1786 4552 y Fl(2)1786 4612 y Fq(\034)p 1727 4632 102 4 v 1727 4723 a Fw(\026)1786 4738 y Fq(\034)1861 4655 y Fx(\000)22 b FE(2)2054 4586 y(_)-45 b Fw(\026)2095 4601 y Fq(\034)2154 4506 y Fv(R)2255 4586 y FE(_)g Fw(\026)2296 4601 y Fq(\034)2339 4586 y Fw(dw)p 2036 4632 427 4 v 2095 4644 a Fv(R)2178 4724 y Fw(\026)2237 4739 y Fq(\034)2280 4724 y Fw(dw)2494 4655 y FE(+)2602 4581 y Fw(\026)2661 4596 y Fq(\034)2720 4501 y Fv(\000R)2868 4581 y FE(_)f Fw(\026)2908 4596 y Fq(\034)2951 4581 y Fw(dw)3075 4501 y Fv(\001)3120 4523 y Fl(2)p 2602 4632 558 4 v 2661 4665 a Fv(\000R)2790 4745 y Fw(\026)2849 4760 y Fq(\034)2892 4745 y Fw(dw)3016 4665 y Fv(\001)3060 4687 y Fl(2)456 4655 y FE(\(111\))1274 4987 y(=)1378 4851 y Fv(Z)1433 5076 y Fu(T)1483 5058 y Ft(N)1535 5076 y Fs(\002)p Fu(R)1638 5058 y Ft(N)1717 4987 y Fw(\026)1776 5002 y Fq(\034)1835 4846 y Fv(\022)1937 4919 y FE(_)g Fw(\026)1977 4934 y Fq(\034)p 1918 4964 102 4 v 1918 5055 a Fw(\026)1977 5070 y Fq(\034)2052 4987 y Fx(\000)2162 4837 y Fv(R)2263 4917 y FE(_)h Fw(\026)2304 4932 y Fq(\034)2347 4917 y Fw(dw)p 2162 4964 308 4 v 2162 4976 a Fv(R)2245 5056 y Fw(\026)2304 5071 y Fq(\034)2347 5056 y Fw(dw)2480 4846 y Fv(\023)2553 4863 y Fl(2)2620 4987 y Fx(\025)28 b FE(0)17 b Fw(:)3367 5216 y Fj(\003)p eop %%Page: 46 46 46 45 bop 456 251 a FC(46)666 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)555 450 y FE(As)24 b(a)f(consequence)k(of)c(Theorem)g(32,)i (w)m(e)g(pro)m(vide)e(the)h(follo)m(wing)d(v)-5 b(ariational)456 578 y(c)m(haracterization)31 b(for)h(the)h(\014xed)h(p)s(oin)m(t)e (problem)f(of)h(Theorem)h(27.)456 730 y FI(Corollary)45 b(33.)g FD(L)-5 b(et)42 b Fw(\036)e Fx(2)g Fw(W)1636 694 y Fl(1)p Fq(;)p Fs(1)1765 730 y FE(\()p Fn(T)1866 694 y Fq(N)1936 730 y FE(\))i FD(and)f Fw(\025)e Fx(2)h Fn(R)52 b FD(satisfy)42 b Fx(G)6 b FE([)p Fw(\036)p FE(])40 b(=)f Fw(\036)27 b FE(+)g Fw(\025)p FD(.)456 858 y(Then,)34 b(the)g(in\014mum)g(of)1253 939 y Fv(Z)1309 1165 y Fu(T)1359 1146 y Ft(N)1411 1165 y Fs(\002)p Fu(R)1514 1146 y Ft(N)1592 1075 y Fw(h)17 b(L)p FE(\()p Fw(x;)g(v)t FE(\))g Fw(d\026)p FE(\()p Fw(x;)g(v)t FE(\))k(+)h Fw(\017S)6 b FE([)p Fw(\026)p FE(])456 1309 y FD(when)36 b Fw(\026)h FD(is)g(in)g(the)g(sp)-5 b(ac)g(e)36 b(given)g(by)i(\(90\))e(and)h(satis\014es)f(the)h(c)-5 b(onstr)g(aint)37 b(\(88\),)456 1436 y(is)d(e)-5 b(qual)35 b(to)g Fw(\025)g FD(\(which,)e(in)i(turn,)g(is)g(e)-5 b(qual)34 b(to)h Fx(G)6 b FE([)p Fw(\036)p FE(])23 b Fx(\000)f Fw(\036)p FD(\).)456 1624 y(Pr)-5 b(o)g(of.)41 b FE(Theorem)26 b(32)g(sa)m(ys)h(that)f(the)g(ab)s(o)m(v)m(e)h(in\014m) m(um)e(is)g(attained)g(at)h(the)g(mea-)456 1752 y(sure)34 b Fw(\026)g FE(giv)m(en)f(b)m(y)i(\(105\),)e(for)g(whic)m(h,)i(b)m(y)f (a)g(direct)f(computation)g(\(and)g(recall-)456 1880 y(ing)e(\(88\)\),)h(w)m(e)i(ha)m(v)m(e:)1214 1967 y Fv(Z)1270 2192 y Fu(T)1320 2173 y Ft(N)1372 2192 y Fs(\002)p Fu(R)1475 2173 y Ft(N)1553 2102 y Fw(h)17 b(L)p FE(\()p Fw(x;)g(v)t FE(\))g Fw(d\026)p FE(\()p Fw(x;)g(v)t FE(\))k(+)h Fw(\017S)6 b FE([)p Fw(\026)p FE(])1055 2375 y(=)1214 2239 y Fv(Z)1270 2465 y Fu(T)1320 2446 y Ft(N)1372 2465 y Fs(\002)p Fu(R)1475 2446 y Ft(N)1553 2264 y Fv(\020)1612 2375 y Fw(\036)p FE(\()p Fw(x)p FE(\))23 b Fx(\000)f Fw(\036)p FE(\()p Fw(x)g FE(+)h Fw(hv)t FE(\))e(+)h Fw(\025)2516 2264 y Fv(\021)2592 2375 y Fw(d\026)p FE(\()p Fw(x;)17 b(v)t FE(\))1055 2648 y(=)1214 2512 y Fv(Z)1270 2738 y Fu(T)1320 2719 y Ft(N)1372 2738 y Fs(\002)p Fu(R)1475 2719 y Ft(N)1553 2537 y Fv(\020)1612 2648 y Fw(\036)p FE(\()p Fw(x)p FE(\))23 b Fx(\000)f Fw(\036)p FE(\()p Fw(x)g FE(+)h Fw(hv)t FE(\))2340 2537 y Fv(\021)2415 2648 y Fw(d\026)p FE(\()p Fw(x;)17 b(v)t FE(\))22 b(+)g Fw(\025)1055 2854 y FE(=)83 b Fw(\025)17 b(:)3367 3035 y Fj(\003)555 3223 y FE(W)-8 b(e)32 b(no)m(w)g(further)f (in)m(v)m(estigate)g(some)g(prop)s(erties)g(of)g(the)h(function)e Fw(\022)s FE(,)i(whic)m(h)456 3351 y(w)m(as)f(in)m(tro)s(duced)g(in)f (Lemma)g(31)g(and)h(pla)m(y)m(ed)h(an)e(imp)s(ortan)m(t)f(r^)-49 b(ole)30 b(in)g(the)i(con-)456 3479 y(struction)j(of)g(the)i(p)s (enalized)e(Mather)h(measure)g Fw(\026)f FE(in)g(Theorem)h(32.)53 b(First)35 b(of)456 3607 y(all,)30 b(w)m(e)k(ha)m(v)m(e)g(the)f(follo)m (wing)d(regularit)m(y)h(result:)456 3759 y FI(Prop)s(osition)23 b(34.)33 b FD(Consider)24 b(the)h(setting)g(of)f(L)-5 b(emma)24 b(31.)41 b(Then,)26 b Fw(\022)31 b Fx(2)d Fw(L)3208 3723 y Fs(1)3283 3759 y FE(\()p Fn(T)3384 3723 y Fq(N)3455 3759 y FE(\))456 3887 y FD(and)34 b(ther)-5 b(e)35 b(exist)f(c)-5 b(onstants)35 b Fw(\024)1599 3851 y Fs(\003)1599 3911 y Fl(1)1673 3887 y FD(and)g Fw(\024)1919 3851 y Fs(\003)1919 3911 y Fl(2)1993 3887 y FD(such)g(that)456 4068 y FE(\(112\))863 b(0)27 b Fw(<)h(\024)1778 4027 y Fs(\003)1778 4093 y Fl(1)1845 4068 y Fx(\024)g Fw(\022)s FE(\()p Fw(x)p FE(\))g Fx(\024)h Fw(\024)2319 4027 y Fs(\003)2319 4093 y Fl(2)456 4250 y FD(for)34 b(any)h Fw(x)28 b Fx(2)g Fn(T)1038 4214 y Fq(N)1109 4250 y FD(.)555 4378 y(Mor)-5 b(e)g(over,)35 b(if)456 4594 y FE(\(113\))1269 4459 y Fv(Z)1324 4684 y Fu(R)1372 4666 y Ft(N)1451 4594 y Fw(e)1496 4553 y Fs(\000)1561 4520 y Ft(h)10 b(K)s Fr(\()p Ft(v)r Fr(\))p 1561 4538 183 3 v 1639 4579 a Ft(\017)1774 4594 y Fx(j)p Fw(D)s(K)d FE(\()p Fw(v)t FE(\))p Fx(j)17 b Fw(dv)46 b(<)e FE(+)p Fx(1)17 b Fw(;)456 4828 y FD(then,)34 b Fw(\022)d Fx(2)d Fw(W)978 4792 y Fl(1)p Fq(;)p Fs(1)1107 4828 y FE(\()p Fn(T)1208 4792 y Fq(N)1279 4828 y FE(\))35 b FD(and)f(ther)-5 b(e)35 b(exists)f(a)h(c)-5 b(onstant)35 b Fw(\024)2580 4792 y Fs(\003)2580 4853 y Fl(3)2654 4828 y FD(such)g(that)456 5009 y FE(\(114\))885 b Fx(k)p Fw(\022)s Fx(k)1712 5028 y Fq(W)1789 5009 y Fr(1)p Ft(;)p Fm(1)1903 5028 y Fl(\()p Fu(T)1980 5009 y Ft(N)2032 5028 y Fl(\))2091 5009 y Fx(\024)29 b Fw(\024)2253 4968 y Fs(\003)2253 5034 y Fl(3)2309 5009 y Fw(:)456 5191 y FD(The)34 b(c)-5 b(onstants)34 b Fw(\024)1139 5155 y Fs(\003)1139 5216 y Fq(i)1179 5191 y FD(,)h(for)f FE(1)28 b Fx(\024)g Fw(i)g Fx(\024)g FE(3)p FD(,)35 b(dep)-5 b(end)33 b(only)i(on)g Fw(N)10 b FD(,)35 b Fw(\036)p FD(,)f Fw(L)p FD(,)h Fw(\025)p FD(,)g Fw(\017)g FD(and)g Fw(h)p FD(.)p eop %%Page: 47 47 47 46 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)420 b(47)456 450 y FD(Pr)-5 b(o)g(of.)41 b FE(In)30 b(the)h(course)g(of)f(this)g(pro)s (of)f(w)m(e)j(denote)f(b)m(y)g Fw(\024)f FE(a)g(con)m(v)m(enien)m(t)i (p)s(ositiv)m(e)456 578 y(quan)m(tit)m(y)-8 b(,)38 b(p)s(ossibly)f(dep) s(ending)g(only)g(on)g Fw(N)10 b FE(,)38 b Fw(\036)p FE(,)g Fw(L)p FE(,)h Fw(\025)p FE(,)f Fw(\017)f FE(and)g Fw(h)p FE(,)i(and)e(whic)m(h)456 706 y(ma)m(y)32 b(b)s(e)h(di\013eren)m (t)g(at)f(di\013eren)m(t)h(steps)g(of)g(the)g(computation.)555 834 y(By)g(c)m(hanging)f(v)-5 b(ariable)31 b(in)h(\(93\),)g(w)m(e)i(ha) m(v)m(e)g(that)456 1055 y(\(115\))235 b Fw(\022)s FE(\()p Fw(x)p FE(\))45 b(=)1305 987 y(1)p 1268 1032 124 4 v 1268 1123 a Fw(h)1324 1094 y Fq(N)1435 919 y Fv(Z)1490 1145 y Fu(R)1538 1126 y Ft(N)1616 1055 y Fw(\022)s FE(\()p Fw(w)s FE(\))p Fw(e)1858 1014 y Fs(\000)1923 980 y Ft(hL)p Fr(\()p Ft(w)q(;)p Fr(\()p Ft(x)p Fm(\000)p Ft(w)q Fr(\))p Ft(=h)p Fr(\)+)p Ft(\036)p Fr(\()p Ft(x)p Fr(\))p Fm(\000)p Ft(\036)p Fr(\()p Ft(w)q Fr(\))p Fm(\000)p Ft(\025)p 1923 999 866 3 v 2343 1040 a(\017)2819 1055 y Fw(dw)19 b(:)456 1280 y FE(Therefore,)33 b(recalling)e(\(92\))h(and)g(\(94\),) 898 1501 y Fw(\022)s FE(\()p Fw(x)p FE(\))84 b Fx(\025)1369 1434 y FE(1)p 1331 1478 124 4 v 1331 1570 a Fw(h)1387 1541 y Fq(N)1498 1366 y Fv(Z)1553 1591 y Fu(T)1603 1572 y Ft(N)1676 1501 y Fw(\022)s FE(\()p Fw(w)s FE(\))p Fw(e)1918 1460 y Fs(\000)1983 1427 y Ft(hL)p Fr(\()p Ft(w)q(;)p Fr(\()p Ft(x)p Fm(\000)p Ft(w)q Fr(\))p Ft(=h)p Fr(\)+)p Ft(\036)p Fr(\()p Ft(x)p Fr(\))p Fm(\000)p Ft(\036)p Fr(\()p Ft(w)q Fr(\))p Fm(\000)p Ft(\025)p 1983 1445 866 3 v 2402 1486 a(\017)2878 1501 y Fw(dw)1161 1767 y Fx(\025)f Fw(\024)1411 1631 y Fv(Z)1466 1857 y Fu(T)1516 1838 y Ft(N)1589 1767 y Fw(\022)s FE(\()p Fw(w)s FE(\))17 b Fw(dw)-1468 b FE(\(116\))1162 1940 y(=)83 b Fw(\024)17 b(:)456 2120 y FE(On)k(the)i(other)e(hand,)k(since)d Fw(L)g FE(is)f(sup)s(erlinear)g(in)g Fw(v)t FE(,)j(one)e(also)f (deduces)i(from)e(\(115\))456 2248 y(and)32 b(\(94\))g(that)1215 2469 y Fw(\022)s FE(\()p Fw(x)p FE(\))84 b Fx(\024)f Fw(\024)1749 2375 y Fv(X)1727 2594 y Fq(k)r Fs(2)p Fu(Z)1863 2575 y Ft(N)1932 2334 y Fv(Z)1987 2559 y Fu(T)2037 2540 y Ft(N)2110 2469 y Fw(\022)s FE(\()p Fw(w)s FE(\))17 b Fw(e)2369 2428 y Fs(\000)p Fq(\024)p Fs(j)p Fq(k)r Fs(j)2562 2469 y Fw(dw)1478 2784 y Fx(\024)83 b Fw(\024)1727 2648 y Fv(Z)1783 2874 y Fu(T)1833 2855 y Ft(N)1905 2784 y Fw(\022)s FE(\()p Fw(w)s FE(\))17 b Fw(dw)1478 2957 y FE(=)84 b Fw(\024)17 b(:)456 3137 y FE(This)32 b(and)h(\(116\))f (yield)f(\(112\).)555 3265 y(F)-8 b(urthermore,)32 b(b)m(y)i (di\013eren)m(tiating)c(\(115\),)i(w)m(e)i(get)e(that)699 3445 y Fx(j)p Fw(D)s(\022)s FE(\()p Fw(x)p FE(\))p Fx(j)539 3646 y(\024)83 b Fw(\024)788 3510 y Fv(Z)844 3736 y Fu(R)892 3717 y Ft(N)970 3646 y Fw(\022)s FE(\()p Fw(w)s FE(\))p Fw(e)1212 3605 y Fs(\000)1277 3571 y Ft(hL)p Fr(\()p Ft(w)q(;)p Fr(\()p Ft(x)p Fm(\000)p Ft(w)q Fr(\))p Ft(=h)p Fr(\)+)p Ft(\036)p Fr(\()p Ft(x)p Fr(\))p Fm(\000)p Ft(\036)p Fr(\()p Ft(w)q Fr(\))p Fm(\000)p Ft(\025)p 1277 3590 V 1696 3631 a(\017)2173 3535 y Fv(h)2220 3531 y(\014)2220 3591 y(\014)2220 3651 y(\014)2253 3646 y Fw(D)s(K)2427 3535 y Fv(\020)2496 3579 y Fw(x)23 b Fx(\000)f Fw(w)p 2496 3623 250 4 v 2593 3714 a(h)2756 3535 y Fv(\021)2815 3531 y(\014)2815 3591 y(\014)2815 3651 y(\014)2871 3646 y FE(+)g Fx(j)p Fw(D)s(\036)p FE(\()p Fw(x)p FE(\))p Fx(j)3298 3535 y Fv(i)3360 3646 y Fw(dw)539 3912 y Fx(\024)83 b Fw(\024)788 3776 y Fv(Z)844 4002 y Fu(R)892 3983 y Ft(N)970 3912 y Fw(e)1015 3871 y Fs(\000)1080 3837 y Ft(h)11 b(K)s Fr(\()p Ft(v)r Fr(\))p 1080 3855 183 3 v 1158 3897 a Ft(\017)1293 3801 y Fv(h)1340 3912 y Fx(j)p Fw(D)s(K)c FE(\()p Fw(v)t FE(\))p Fx(j)21 b FE(+)h(1)1865 3801 y Fv(i)1929 3912 y Fw(dv)539 4105 y Fx(\024)83 b Fw(\024)17 b(;)456 4285 y FE(thanks)33 b(to)f(\(112\))g(and)h(\(113\).) 42 b(This)33 b(and)g(\(112\))f(yield)f(\(114\).)585 b Fj(\003)555 4472 y FE(W)-8 b(e)46 b(no)m(w)f(apply)g(the)h(Hopf-Cole)e (transform)g(metho)s(d)g(to)h Fw(\022)s FE(,)k(that)44 b(is,)k(w)m(e)456 4599 y(de\014ne)456 4780 y(\(117\))861 b Fw(&)8 b FE(\()p Fw(x)p FE(\))45 b(:=)f Fx(\000)p Fw(\017)17 b FE(ln)f Fw(\022)s FE(\()p Fw(x)p FE(\))h Fw(:)456 4960 y FE(Note)27 b(that,)h(in)f(the)g(setting)g(of)g(\(13\),)h(w)m(e)g(ha)m (v)m(e)h(that)e Fw(\022)k FE(=)c Fx(E)9 b FE([)p Fw(&)f FE(].)42 b(Also,)26 b(\(117\))h(is)g(a)456 5088 y(b)s(ona)32 b(\014de)i(de\014nition,)f(since)h Fw(\022)e(>)c FE(0)33 b(b)m(y)i(\(112\).)44 b(Moreo)m(v)m(er,)35 b Fw(&)42 b FE(is)33 b Fn(Z)2979 5052 y Fq(N)3043 5088 y FE(-p)s(erio)s(dic,)456 5216 y(since)g(so)f(is)g Fw(\022)s FE(.)p eop %%Page: 48 48 48 47 bop 456 251 a FC(48)666 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)555 450 y FE(W)h(e)35 b(no)m(w)g(sho)m(w)g(that)f Fw(&)43 b FE(is)34 b(the)g(solution)f(of)h(an)g(en)m(trop)m(y)i(p)s (enalized)d(sc)m(heme)456 578 y(analogous)e(to)h(\(6\).)43 b(Namely)-8 b(,)32 b(let)p 887 785 67 4 v 887 865 a Fw(L)p FE(\()p Fw(x;)17 b(v)t FE(\))44 b(:=)h Fw(L)p FE(\()p Fw(x)23 b FE(+)f Fw(hv)t(;)17 b Fx(\000)p Fw(v)t FE(\))k(+)2097 798 y Fw(\036)p FE(\()p Fw(x)p FE(\))i Fx(\000)f Fw(\036)p FE(\()p Fw(x)g FE(+)g Fw(hv)t FE(\))g Fx(\000)h Fw(\025)p 2097 842 906 4 v 2522 933 a(h)456 1123 y FE(and,)32 b(for)g(an)m(y)i (function)e Fw(')p FE(,)g(consider)456 1407 y(\(118\))p 1029 1326 66 4 v 350 w Fx(G)6 b FE([)p Fw(')p FE(]\()p Fw(x)p FE(\))45 b(:=)f Fx(\000)p Fw(\017)34 b FE(ln)1783 1266 y Fv(\024)1836 1271 y(Z)1891 1496 y Fu(R)1939 1478 y Ft(N)2018 1407 y Fw(e)2063 1365 y Fs(\000)2128 1332 y Ft(h)p 2164 1291 43 3 v(L)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft(')p Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p 2128 1350 499 3 v 2364 1391 a Ft(\017)2657 1407 y Fw(dv)2759 1266 y Fv(\025)2844 1407 y Fw(:)456 1696 y FE(The)27 b(reader)g(ma)m(y)f (compare)f(\(6\))h(and)h(\(118\))e(and)h(note)h(that)f(the)g(function)g Fw(&)18 b FE(+)9 b Fw(\036)456 1824 y FE(is)27 b(a)g(solution)f(to)h(a) g(time-rev)m(ersed)h Fx(G)33 b FE(sc)m(heme)c(with)e(Lagrangian)e Fw(L)p FE(\()p Fw(x)12 b FE(+)g Fw(hv)t(;)17 b(v)t FE(\),)456 1952 y(that)30 b(is)g Fw(&)39 b FE(is)29 b(the)i(di\013erence)h(b)s(et) m(w)m(een)g(a)e(forw)m(ard)h(and)f(bac)m(kw)m(ard)i(\014xed)g(p)s(oin)m (t.)555 2080 y(In)h(this)f(framew)m(ork,)h(the)g(follo)m(wing)d(result) i(holds:)456 2258 y FI(Prop)s(osition)46 b(35.)g FD(Consider)c(the)i (setting)f(of)g(L)-5 b(emma)43 b(31)g(and)g(let)g Fw(&)52 b FD(b)-5 b(e)43 b(as)456 2386 y(in)28 b(\(117\).)42 b(Then,)29 b Fw(&)36 b Fx(2)28 b Fw(L)1381 2350 y Fs(1)1456 2386 y FE(\()p Fn(T)1557 2350 y Fq(N)1627 2386 y FE(\))h FD(and)f(its)h Fw(L)2074 2350 y Fs(1)2149 2386 y FD(-norm)f(is)g(b)-5 b(ounde)g(d)28 b(by)h(a)g(quantity)456 2514 y(dep)-5 b(ending)33 b(only)i(on)f Fw(N)10 b FD(,)35 b Fw(\036)p FD(,)g Fw(L)p FD(,)g Fw(\025)p FD(,)g Fw(\017)g FD(and)f Fw(h)p FD(.)555 2642 y(A)n(lso,)p 804 2562 66 4 v 34 w Fx(G)7 b FE([)p Fw(&)h FE(])28 b(=)f Fw(&)8 b FD(.)555 2770 y(Mor)-5 b(e)g(over,)51 b(if)d(\(113\))f(holds,)k(then)d Fw(&)60 b Fx(2)53 b Fw(W)2261 2733 y Fl(1)p Fq(;)p Fs(1)2390 2770 y FE(\()p Fn(T)2491 2733 y Fq(N)2562 2770 y FE(\))p FD(,)e(its)d Fw(W)2937 2733 y Fl(1)p Fq(;)p Fs(1)3066 2770 y FD(-norm)g(is)456 2897 y(b)-5 b(ounde)g(d)34 b(by)h(a)f (quantity)i(dep)-5 b(ending)33 b(only)i(on)f Fw(N)10 b FD(,)36 b Fw(\036)p FD(,)e Fw(L)p FD(,)h Fw(\025)p FD(,)g Fw(\017)g FD(and)g Fw(h)p FD(.)555 3025 y(A)n(lso,)f(in)h(this)g (c)-5 b(ase,)34 b(the)h(function)f Fw(\022)k FD(in)d(L)-5 b(emma)34 b(31)h(is)f(unique.)456 3278 y(Pr)-5 b(o)g(of.)41 b FE(The)22 b Fw(L)1009 3241 y Fs(1)1085 3278 y FE(-b)s(ound)f (\(resp.,)k(the)d Fw(W)1960 3241 y Fl(1)p Fq(;)p Fs(1)2089 3278 y FE(-b)s(ound\))g(plainly)d(follo)m(ws)i(from)f(\(117\))456 3405 y(and)33 b(\(112\))g(\(resp.,)i(and)f(\(114\)\).)46 b(The)34 b(fact)g(that)f Fw(&)42 b FE(is)33 b(a)h(\014xed)h(p)s(oin)m (t)d(of)p 3198 3325 V 34 w Fx(G)39 b FE(is)34 b(a)456 3533 y(consequence)h(of)d(\(93\).)555 3662 y(Let)f(us)h(no)m(w)g(sho)m (w)g(the)f(uniqueness)i(of)d Fw(\022)s FE(.)43 b(Assume)32 b(that)2755 3636 y(^)2747 3662 y Fw(\022)i FE(also)c(ful\014lls)f(the) 456 3790 y(thesis)e(of)g(Lemma)f(31)g(and)i(let)j(^)-54 b Fw(&)35 b FE(b)s(e)28 b(so)f(that)2141 3763 y(^)2133 3790 y Fw(\022)k FE(=)d Fx(E)9 b FE([)c(^)-54 b Fw(&)7 b FE(].)42 b(By)28 b(what)f(w)m(e)h(ha)m(v)m(e)h(just)456 3918 y(pro)m(v)m(ed,)44 b(w)m(e)e(ha)m(v)m(e)g(that)f(b)s(oth)f Fw(&)49 b FE(and)e(^)-55 b Fw(&)49 b FE(are)41 b(uniformly)e(Lipsc)m (hitz)i(and)g(that)456 4045 y(they)35 b(are)g(b)s(oth)g(\014xed)h(p)s (oin)m(ts)f(of)p 1722 3965 V 34 w Fx(G)6 b FE(.)51 b(Then,)37 b(using)d(Theorem)i(9)e(\(applied)g(to)3374 4007 y Fl(14)p 456 4093 V 456 4173 a Fx(G)k FE(instead)33 b(of)f Fx(G)6 b FE(\),)33 b(w)m(e)g(ha)m(v)m(e)h(that)1119 4413 y Fx(k)p Fw(&)c Fx(\000)e FE(^)-54 b Fw(&)8 b Fx(k)1427 4428 y Fq(])1542 4413 y FE(=)84 b Fx(k)p 1752 4333 V(G)6 b FE([)p Fw(&)i FE(])22 b Fx(\000)p 2036 4333 V 23 w(G)6 b FE([)f(^)-54 b Fw(&)9 b FE(])p Fx(k)2249 4428 y Fq(])1541 4573 y Fx(\024)84 b FE(\(1)22 b Fx(\000)g Fw(C)i Fx(k)p Fw(&)30 b Fx(\000)e FE(^)-54 b Fw(&)8 b Fx(k)2312 4588 y Fq(])2343 4573 y FE(\))17 b Fx(k)p Fw(&)30 b Fx(\000)e FE(^)-54 b Fw(&)8 b Fx(k)2706 4588 y Fq(])2754 4573 y Fw(;)456 4812 y FE(for)32 b(some)g Fw(C)j(>)27 b FE(0.)43 b(This)33 b(sho)m(ws)h(that)f Fw(&)j FE(=)d(^)-55 b Fw(&)31 b FE(+)22 b Fw(c)p FE(,)32 b(for)g(some)h Fw(c)27 b Fx(2)h Fn(R)5 b FE(.)p 456 5007 499 4 v 555 5074 a Fl(14)626 5106 y FH(Note)27 b(that)p 1006 5039 57 4 v 28 w Fp(L)g FH(is)g(sup)r(erlinear,)f(since)h(so)g(is) g Fp(L)p FH(.)36 b(Also,)p 2337 5039 V 27 w Fp(L)23 b FF(2)g Fp(W)2585 5076 y Fy(1)p Fi(;)p Fd(1)2708 5106 y FH(\()p Ff(T)2796 5076 y Fi(N)2877 5106 y FF(\002)17 b Ff(R)3013 5076 y Fi(N)3082 5106 y FH(\),)28 b(b)r(ecause)456 5216 y Fp(\036)23 b FF(2)h Fp(W)697 5185 y Fy(1)p Fi(;)p Fd(1)820 5216 y FH(\()p Ff(T)908 5185 y Fi(N)970 5216 y FH(\).)37 b(No)28 b(further)f(regularit)n(y)f(of)i Fp(L)f FH(w)n(as)f(used)i(in)g(the)g(pro)r(of)f(of)g(Theorem)g(9.)p eop %%Page: 49 49 49 48 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)420 b(49)555 450 y FE(But)33 b(then,)g(recalling)d(\(94\),)1418 543 y Fv(Z)1473 769 y Fu(T)1523 750 y Ft(N)1596 679 y Fw(e)1641 638 y Fs(\000)1706 611 y Ft(&)p 1706 623 29 3 v 1707 664 a(\017)1776 679 y FE(=)1879 543 y Fv(Z)1934 769 y Fu(T)1984 750 y Ft(N)2057 679 y Fw(\022)h FE(=)d(1)1259 945 y(=)1418 809 y Fv(Z)1473 1035 y Fu(T)1523 1016 y Ft(N)1603 919 y FE(^)1596 945 y Fw(\022)j FE(=)1775 809 y Fv(Z)1830 1035 y Fu(T)1880 1016 y Ft(N)1953 945 y Fw(e)1998 904 y Fs(\000)2067 877 y Fr(^)-34 b Ft(&)p 2063 889 V 2064 930 a(\017)2133 945 y FE(=)28 b Fw(e)2292 877 y Ft(c)p 2292 889 28 3 v 2293 930 a(\017)2350 809 y Fv(Z)2406 1035 y Fu(T)2456 1016 y Ft(N)2528 945 y Fw(e)2573 904 y Fs(\000)2638 877 y Ft(&)p 2638 889 29 3 v 2639 930 a(\017)2697 945 y Fw(;)456 1199 y FE(thence)34 b Fw(c)27 b FE(=)h(0)k(and,)h(therefore,)g Fw(&)j FE(=)d(^)-54 b Fw(&)8 b FE(.)43 b(Consequen)m(tly)-8 b(,)35 b Fw(\022)c FE(=)2771 1172 y(^)2763 1199 y Fw(\022)t FE(.)528 b Fj(\003)555 1393 y FE(W)-8 b(e)47 b(are)f(no)m(w)h(in)f(p)s(osition)e(to)i(pro)m(v) m(e)i(the)e(uniqueness)j(of)c(the)i(p)s(enalized)456 1521 y(Mather)33 b(measure)g(constructed)h(in)e(Theorem)h(32:)456 1676 y FI(Theorem)53 b(36.)d FD(Assume)e(\(113\).)84 b(Then,)51 b(the)d(me)-5 b(asur)g(e)48 b Fw(\026)f FD(c)-5 b(onstructe)g(d)49 b(in)456 1804 y(The)-5 b(or)g(em)33 b(32)h(is)g(the)g(unique)h(minimizer)e(of)h(the)g(functional)g(in)g (\(89\))f(over)h(the)456 1932 y(sp)-5 b(ac)g(e)34 b(given)g(by)h (\(90\))f(under)h(the)g(c)-5 b(onstr)g(aint)34 b(\(88\).)456 2127 y(Pr)-5 b(o)g(of.)41 b FE(Assume)31 b(that)37 b(~)-57 b Fw(\026)30 b FE(is)f(also)g(a)h(minimizer.)39 b(Let)30 b Fw(\026)2500 2142 y Fq(\034)2571 2127 y FE(:=)d(\(1)17 b Fx(\000)g Fw(\034)11 b FE(\))p Fw(\026)17 b FE(+)f Fw(\034)j FE(~)-57 b Fw(\026)p FE(,)31 b(for)456 2255 y Fw(\034)39 b Fx(2)28 b FE([0)p Fw(;)17 b FE(1].)43 b(Let)33 b(also)50 b(_)-46 b Fw(\026)1326 2270 y Fq(\034)1397 2255 y FE(:=)35 b(~)-57 b Fw(\026)22 b Fx(\000)h Fw(\026)32 b FE(and)456 2484 y(\(119\))1467 2457 y(~)1459 2484 y Fw(\022)s FE(\()p Fw(x)p FE(\))45 b(:=)1813 2348 y Fv(Z)1869 2574 y Fu(R)1917 2555 y Ft(N)2003 2484 y FE(~)-57 b Fw(\026)p FE(\()p Fw(x;)17 b(v)t FE(\))g Fw(dv)i(:)456 2717 y FE(These)34 b(de\014nitions)e(and)h(\(106\))e(giv)m(e)i(that)1530 2946 y(_)-46 b Fw(\026)1570 2961 y Fq(\034)1630 2810 y Fv(Z)1685 3036 y Fu(R)1733 3017 y Ft(N)1812 2946 y Fw(\026)17 b(dw)23 b Fx(\000)g Fw(\026)2208 2810 y Fv(Z)2262 3036 y Fu(R)2310 3017 y Ft(N)2407 2946 y FE(_)-45 b Fw(\026)2448 2961 y Fq(\034)2507 2946 y Fw(dw)1353 3168 y FE(=)82 b Fw(\022)10 b FE(~)-56 b Fw(\026)22 b Fx(\000)1748 3142 y FE(~)1740 3168 y Fw(\022)s(\026)17 b(:)-1435 b FE(\(120\))456 3356 y(Let)37 b(no)m(w)g Fw(I)44 b FE(b)s(e)37 b(as)g(in)f(\(107\).)55 b(Since)37 b(b)s(oth)g Fw(\026)f FE(and)44 b(~)-56 b Fw(\026)36 b FE(are)h(minimizers)d(for)i(the)456 3484 y(functional)j(in)i(\(89\),)i(w)m(e)f(ha)m(v)m(e)h(that)e(b)s(oth)g Fw(\034)54 b FE(=)42 b(0)f(and)g Fw(\034)54 b FE(=)42 b(1)f(minimize)d Fw(I)8 b FE(.)456 3612 y(On)37 b(the)h(other)f(hand,)i (w)m(e)f(kno)m(w)h(from)d(the)i(pro)s(of)e(of)h(Theorem)h(32)f(that)g Fw(I)45 b FE(is)456 3740 y(con)m(v)m(ex.)f(Therefore,)31 b Fw(I)37 b FE(m)m(ust)29 b(b)s(e)g(constan)m(t)h(for)f(an)m(y)h Fw(\034)39 b Fx(2)28 b FE([0)p Fw(;)17 b FE(1].)42 b(In)29 b(particular,)456 3868 y Fw(I)507 3832 y Fs(00)549 3868 y FE([0])f(=)f(0.)43 b(By)33 b(\(111\),)f(this)g(sa)m(ys)i(that)1012 4196 y(0)44 b(=)1225 4060 y Fv(Z)1280 4286 y Fu(T)1330 4267 y Ft(N)1382 4286 y Fs(\002)p Fu(R)1485 4267 y Ft(N)1574 3982 y Fv(\020)1652 4092 y FE(_)-46 b Fw(\026)1692 4107 y Fq(\034)1752 4012 y Fv(R)1799 4127 y Fu(R)1847 4108 y Ft(N)1925 4092 y Fw(\026)17 b(dw)24 b Fx(\000)e Fw(\026)2321 4012 y Fv(R)2368 4127 y Fu(R)2416 4108 y Ft(N)2513 4092 y FE(_)-46 b Fw(\026)2553 4107 y Fq(\034)2612 4092 y Fw(dw)2736 3982 y Fv(\021)2795 4004 y Fl(2)p 1574 4173 1261 4 v 1865 4316 a Fw(\026)22 b Fx(\001)1996 4205 y Fv(\020)2072 4236 y(R)2119 4350 y Fu(R)2167 4332 y Ft(N)2246 4316 y Fw(\026)17 b(dw)2446 4205 y Fv(\021)2503 4228 y Fl(2)2861 4196 y Fw(:)456 4524 y FE(This)32 b(and)h(\(120\))f(yield)f (that)456 4712 y(\(121\))1065 b Fw(\022)28 b FE(~)-57 b Fw(\026)44 b FE(=)2040 4685 y(~)2032 4712 y Fw(\022)20 b(\026)456 4900 y FE(almost)46 b(ev)m(erywhere)k(in)d Fn(T)1499 4864 y Fq(N)1602 4900 y Fx(\002)33 b Fn(R)1778 4864 y Fq(N)1851 4900 y FE(.)89 b(Thence,)53 b(in)47 b(the)h(ligh)m(t)e(of)i(\(105\),)i(w)m(e)456 5028 y(conclude)33 b(that)1214 5189 y(~)1206 5216 y Fw(\022)s FE(\()p Fw(x)p FE(\))45 b(=)51 b(~)-56 b Fw(\026)o FE(\()p Fw(x;)17 b(v)t FE(\))g Fw(e)1906 5141 y Ft(hL)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p Fm(\000)p Ft(\036)p Fr(\()p Ft(x)p Fr(\))p Fm(\000)p Ft(\025)p 1906 5159 748 3 v 2266 5200 a(\017)2667 5216 y Fw(;)p eop %%Page: 50 50 50 49 bop 456 251 a FC(50)666 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)456 450 y FE(and)32 b(therefore)456 623 y(\(122\))917 597 y(~)910 623 y Fw(\022)s FE(\()p Fw(x)22 b Fx(\000)h Fw(hv)t FE(\))44 b(=)52 b(~)-57 b Fw(\026)p FE(\()p Fw(x)22 b Fx(\000)h Fw(hv)t(;)17 b(v)t FE(\))g Fw(e)2068 548 y Ft(hL)p Fr(\()p Ft(x)p Fm(\000)p Ft(hv)r(;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(x)p Fr(\))p Fm(\000)p Ft(\036)p Fr(\()p Ft(x)p Fm(\000)p Ft(hv)r Fr(\))p Fm(\000)p Ft(\025)p 2067 567 866 3 v 2487 608 a(\017)2963 623 y Fw(:)456 796 y FE(Th)m(us,)30 b(giv)m(en)f(an)m(y)g Fw(')e Fx(2)h Fw(C)7 b FE(\()p Fn(T)1520 759 y Fq(N)1591 796 y FE(\),)29 b(w)m(e)g(gather)g(from)e(\(88\),)h(\(119\))g(and)g(\(122\))g(that)966 874 y Fv(Z)1021 1099 y Fu(T)1071 1080 y Ft(N)1144 874 y Fv(Z)1199 1099 y Fu(R)1247 1080 y Ft(N)1326 1009 y Fw(')p FE(\()p Fw(x)p FE(\))1545 983 y(~)1538 1009 y Fw(\022)s FE(\()p Fw(x)22 b Fx(\000)h Fw(hv)t FE(\))p Fw(e)1991 968 y Fs(\000)2056 935 y Ft(hL)p Fr(\()p Ft(x)p Fm(\000)p Ft(hv)r(;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(x)p Fr(\))p Fm(\000)p Ft(\036)p Fr(\()p Ft(x)p Fm(\000)p Ft(hv)r Fr(\))p Fm(\000)p Ft(\025)p 2056 953 V 2475 994 a(\017)2952 1009 y Fw(dv)d(dx)807 1275 y FE(=)966 1139 y Fv(Z)1021 1365 y Fu(T)1071 1346 y Ft(N)1144 1139 y Fv(Z)1199 1365 y Fu(R)1247 1346 y Ft(N)1326 1275 y Fw(')p FE(\()p Fw(x)p FE(\))k(~)-56 b Fw(\026)o FE(\()p Fw(x)23 b Fx(\000)f Fw(hv)t(;)17 b(v)t FE(\))g Fw(dv)i(dx)807 1541 y FE(=)966 1405 y Fv(Z)1021 1631 y Fu(R)1069 1612 y Ft(N)1148 1405 y Fv(Z)1203 1631 y Fu(T)1253 1612 y Ft(N)1326 1541 y Fw(')p FE(\()p Fw(y)25 b FE(+)d Fw(hv)t FE(\))i(~)-56 b Fw(\026)o FE(\()p Fw(y)t(;)17 b(v)t FE(\))g Fw(dy)g(dv)807 1807 y FE(=)966 1671 y Fv(Z)1021 1897 y Fu(T)1071 1878 y Ft(N)1123 1897 y Fs(\002)p Fu(R)1226 1878 y Ft(N)1304 1807 y Fw(')p FE(\()p Fw(y)26 b FE(+)c Fw(hv)t FE(\))17 b Fw(d)8 b FE(~)-57 b Fw(\026)n FE(\()p Fw(y)t(;)17 b(v)t FE(\))807 2079 y(=)966 1944 y Fv(Z)1021 2169 y Fu(T)1071 2150 y Ft(N)1123 2169 y Fs(\002)p Fu(R)1226 2150 y Ft(N)1304 2079 y Fw(')p FE(\()p Fw(y)t FE(\))g Fw(d)8 b FE(~)-57 b Fw(\026)n FE(\()p Fw(y)t(;)17 b(v)t FE(\))807 2352 y(=)966 2217 y Fv(Z)1021 2442 y Fu(T)1071 2423 y Ft(N)1144 2217 y Fv(Z)1199 2442 y Fu(R)1247 2423 y Ft(N)1326 2352 y Fw(')p FE(\()p Fw(y)t FE(\))23 b(~)-55 b Fw(\026)n FE(\()p Fw(y)t(;)17 b(v)t FE(\))g Fw(dv)h(dy)807 2618 y FE(=)966 2482 y Fv(Z)1021 2708 y Fu(T)1071 2689 y Ft(N)1144 2618 y Fw(')p FE(\()p Fw(y)t FE(\))1359 2592 y(~)1353 2618 y Fw(\022)r FE(\()p Fw(y)t FE(\))p Fw(dy)g(:)456 2854 y FE(Since)27 b Fw(')h FE(here)g(ab)s(o)m(v)m(e)g(is)f(arbitrary) -8 b(,)27 b(this)h(means)f(that)2489 2827 y(~)2481 2854 y Fw(\022)k FE(satis\014es)d(\(93\).)41 b(Also,)3404 2827 y(~)3396 2854 y Fw(\022)456 2981 y FE(satis\014es)29 b(\(92\))f(and)g(\(94\),)h(due)h(to)e(\(119\))f(and)i(the)g(fact)f (that)36 b(~)-56 b Fw(\026)28 b FE(is)g(a)g(probabilit)m(y)456 3109 y(measure.)43 b(Then,)33 b(b)m(y)f(the)g(uniqueness)i(result)d(in) g(Prop)s(osition)f(35,)h(w)m(e)i(deduce)456 3237 y(that)675 3211 y(~)667 3237 y Fw(\022)e FE(=)c Fw(\022)s FE(.)44 b(Then,)34 b(\(121\))d(yields)i(that)40 b(~)-57 b Fw(\026)27 b FE(=)h Fw(\026)p FE(.)1104 b Fj(\003)986 3447 y FE(11.)48 b Fz(Conver)n(gence)37 b(to)h(Ma)-7 b(ther)37 b(measures)555 3633 y FE(In)d(this)g(section,)g(w)m(e)h(use)g(dualit)m(y)e(metho)s(ds) h(to)f(study)i(the)f(en)m(trop)m(y)i(p)s(enal-)456 3760 y(ized)26 b(Mather)h(problem)e(and)i(its)f(con)m(v)m(ergence)j(as)e Fw(\017=h)h Fx(!)f FE(0)g(\(that)f(is,)h(when)h(the)456 3888 y(en)m(trop)m(y)41 b(v)-5 b(anishes\).)65 b(W)-8 b(e)40 b(will)d(pro)m(v)m(e)k(a)e FD(variational)i(identity)e FE(whic)m(h)h(relates)456 4016 y(the)k(action)e(functional)g(with)h (the)h(\014xed)h(p)s(oin)m(ts)e(of)g(the)h(non-linear)e(sc)m(heme)456 4144 y(\(see)30 b(Theorem)g(37\).)42 b(This)30 b(iden)m(tit)m(y)g (turns)g(out)g(to)f(b)s(e)h(useful)g(in)f(sho)m(wing)h(that)456 4272 y FD(the)g(minimal)g(action)g(value,)h(the)g(\014xe)-5 b(d)30 b(p)-5 b(oint)31 b(of)f(the)h(scheme)e(and)i(the)f(entr)-5 b(opy)456 4400 y(p)g(enalize)g(d)26 b(Mather)j(me)-5 b(asur)g(e)27 b(c)-5 b(onver)g(ge)25 b FE(to)f(the)i(analogous)e (features)i(in)f(Mather)456 4528 y(theory)33 b(as)g(the)g(en)m(trop)m (y)h(v)-5 b(anishes)33 b(\(see)g(Theorems)h(38,)e(39)g(and)h(40\).)456 4676 y FI(Theorem)k(37.)k FD(We)35 b(have)g(the)f(fol)5 b(lowing)34 b(identity:)1120 4884 y FE(inf)1158 4944 y Fq(\026)1255 4748 y Fv(Z)1371 4884 y Fw(hLd\026)22 b FE(+)g Fw(\017S)6 b FE([)p Fw(\026)p FE(])-1485 b(\(123\))961 5146 y(=)83 b Fx(\000)112 b FE(inf)1214 5215 y Fq( )r Fs(2)p Fq(C)5 b Fl(\()p Fu(T)1441 5196 y Ft(N)1494 5215 y Fl(\))1559 5146 y FE(sup)1538 5235 y Fq(x)p Fs(2)p Fu(T)1675 5216 y Ft(N)1743 5146 y Fw(\017)17 b FE(ln)1897 5010 y Fv(Z)1953 5236 y Fu(R)2001 5217 y Ft(N)2079 5146 y Fw(e)2124 5105 y Fs(\000)2189 5071 y Ft(hL)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft( )r Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p Fm(\000)p Ft( )r Fr(\()p Ft(x)p Fr(\))p 2189 5089 674 3 v 2513 5131 a Ft(\017)2877 5146 y Fw(dv)j(;)p eop %%Page: 51 51 51 50 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)420 b(51)456 450 y FD(wher)-5 b(e)32 b(the)h(\014rst)g(in\014mum)f(is)h(taken)g(over)f (al)5 b(l)33 b(me)-5 b(asur)g(es)32 b Fw(\026)h FD(in)g(the)g(sp)-5 b(ac)g(e)32 b(given)456 578 y(by)j(\(90\))f(under)h(the)f(c)-5 b(onstr)g(aint)35 b(in)g(\(88\).)555 706 y(F)-7 b(urthermor)i(e,)42 b(if)f Fw(\036)e Fx(2)h Fw(W)1566 670 y Fl(1)p Fq(;)p Fs(1)1696 706 y FE(\()p Fn(T)1797 670 y Fq(N)1867 706 y FE(\))h FD(and)g Fw(\025)e Fx(2)h Fn(R)52 b FD(satisfy)2726 667 y Fl(15)2838 706 y Fx(G)6 b FE([)p Fw(\036)p FE(])40 b(=)f Fw(\036)27 b FE(+)g Fw(\025)p FD(,)456 834 y(then)33 b Fw(\036)g FD(is)g(an)h(optimal)e(function)i(for)f(the)h (right-hand-side)d(of)j(\(123\))e(and)h Fw(\025)h FD(is)456 962 y(e)-5 b(qual)34 b(to)h(the)g(quantity)h(given)e(in)g(\(123\).)456 1173 y(Pr)-5 b(o)g(of.)41 b FE(Clearly)-8 b(,)1080 1134 y Fl(16)1284 1411 y FE(inf)1190 1481 y Fq( )r Fs(2)p Fq(C)5 b Fl(\()p Fu(T)1417 1462 y Ft(N)1470 1481 y Fl(\))1535 1411 y FE(sup)1514 1500 y Fq(x)p Fs(2)p Fu(T)1651 1481 y Ft(N)1719 1411 y Fw(\017)17 b FE(ln)1873 1276 y Fv(Z)1928 1501 y Fu(R)1976 1482 y Ft(N)2055 1411 y Fw(e)2100 1370 y Fs(\000)2165 1337 y Ft(hL)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft( )r Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p Fm(\000)p Ft( )r Fr(\()p Ft(x)p Fr(\))p 2165 1355 674 3 v 2489 1396 a Ft(\017)2853 1411 y Fw(dv)-2495 b FE(\(124\))1029 1690 y Fx(\024)105 b FE(sup)1190 1779 y Fq(x)p Fs(2)p Fu(T)1327 1760 y Ft(N)1395 1690 y Fw(\017)17 b FE(ln)1549 1555 y Fv(Z)1604 1780 y Fu(R)1652 1761 y Ft(N)1731 1690 y Fw(e)1776 1649 y Fs(\000)1841 1616 y Ft(hL)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft(\036)p Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p Fm(\000)p Ft(\036)p Fr(\()p Ft(x)p Fr(\))p 1841 1634 663 3 v 2159 1675 a Ft(\017)2517 1690 y Fw(dv)31 b FE(=)d Fx(\000)p Fw(\025)17 b(:)456 1952 y FE(On)32 b(the)h(other)g(hand,)g(from)e(\(6\),)1162 2196 y(inf)1068 2265 y Fq( )r Fs(2)p Fq(C)5 b Fl(\()p Fu(T)1295 2246 y Ft(N)1348 2265 y Fl(\))1413 2196 y FE(sup)1392 2284 y Fq(x)p Fs(2)p Fu(T)1529 2266 y Ft(N)1597 2196 y Fw(\017)17 b FE(ln)1751 2060 y Fv(Z)1806 2285 y Fu(R)1854 2267 y Ft(N)1933 2196 y Fw(e)1978 2154 y Fs(\000)2043 2121 y Ft(hL)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft( )r Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p Fm(\000)p Ft( )r Fr(\()p Ft(x)p Fr(\))p 2043 2139 674 3 v 2367 2180 a Ft(\017)2731 2196 y Fw(dv)1764 2416 y FE(=)139 b(inf)1884 2485 y Fq( )r Fs(2)p Fq(C)5 b Fl(\()p Fu(T)2111 2467 y Ft(N)2164 2485 y Fl(\))2229 2416 y FE(sup)2208 2505 y Fq(x)p Fs(2)p Fu(T)2345 2486 y Ft(N)2413 2416 y Fw( )27 b Fx(\000)22 b(G)6 b FE([)p Fw( )t FE(])17 b Fw(:)456 2314 y FE(\(125\))456 2684 y(Th)m(us,)43 b(tak)m(e)d(an)m(y)h(function)e Fw( )44 b Fx(2)d Fw(C)7 b FE(\()p Fn(T)1930 2648 y Fq(N)2000 2684 y FE(\))40 b(and)g(c)m(ho)s(ose)h(a)e(p)s(oin)m(t)g Fw(x)2997 2699 y Fl(0)3077 2684 y FE(in)g(whic)m(h)456 2813 y Fw( )28 b Fx(\000)d Fw(\036)36 b FE(ha)m(v)m(e)i(a)e(maxim)m (um.)52 b(Let)1752 2786 y(^)1740 2813 y Fw(\036)33 b FE(:=)h Fw(\036)25 b Fx(\000)g Fw(\036)p FE(\()p Fw(x)2304 2828 y Fl(0)2343 2813 y FE(\))g(+)g Fw( )t FE(\()p Fw(x)2667 2828 y Fl(0)2706 2813 y FE(\),)38 b(so)e(that)3160 2786 y(^)3147 2813 y Fw(\036)e Fx(\025)g Fw( )t FE(.)456 2941 y(Therefore,)f(in)f(the)h(ligh)m(t)e(of)h(Prop)s(ositions)g(6)g(and)h (7,)1172 3143 y Fx(G)6 b FE([)p Fw(\036)p FE(])22 b Fx(\000)h Fw(\036)p FE(\()p Fw(x)1622 3158 y Fl(0)1661 3143 y FE(\))f(+)g Fw( )t FE(\()p Fw(x)1979 3158 y Fl(0)2019 3143 y FE(\))28 b(=)f Fx(G)6 b FE([)2293 3117 y(^)2280 3143 y Fw(\036)p FE(])28 b Fx(\025)g(G)6 b FE([)p Fw( )t FE(])17 b Fw(:)456 3346 y FE(Consequen)m(tly)-8 b(,)686 3549 y(sup)665 3638 y Fq(x)p Fs(2)p Fu(T)802 3619 y Ft(N)870 3549 y Fw( )26 b Fx(\000)d(G)6 b FE([)p Fw( )t FE(])28 b Fx(\025)g Fw( )t FE(\()p Fw(x)1538 3564 y Fl(0)1578 3549 y FE(\))22 b Fx(\000)h(G)6 b FE([)p Fw( )t FE(]\()p Fw(x)2017 3564 y Fl(0)2057 3549 y FE(\))28 b Fx(\025)g Fw(\036)p FE(\()p Fw(x)2379 3564 y Fl(0)2418 3549 y FE(\))22 b Fx(\000)h(G)6 b FE([)p Fw(\036)p FE(]\()p Fw(x)2848 3564 y Fl(0)2888 3549 y FE(\))27 b(=)h Fx(\000)p Fw(\025)17 b(:)456 3818 y FE(Th)m(us,)33 b(b)m(y)g(taking)e(the)h(in\014m)m(um)e(o)m(v)m(er)j (all)d(functions)i Fw( )f Fx(2)d Fw(C)7 b FE(\()p Fn(T)2841 3781 y Fq(N)2912 3818 y FE(\),)32 b(w)m(e)h(deduce)456 3945 y(from)e(the)i(latter)e(estimate,)h(\(125\))g(and)h(\(124\))e (that)905 4189 y Fx(\000)112 b FE(inf)999 4259 y Fq( )r Fs(2)p Fq(C)5 b Fl(\()p Fu(T)1226 4240 y Ft(N)1279 4259 y Fl(\))1344 4189 y FE(sup)1323 4278 y Fq(x)p Fs(2)p Fu(T)1460 4259 y Ft(N)1528 4189 y Fw(\017)17 b FE(ln)1682 4054 y Fv(Z)1738 4279 y Fu(R)1786 4260 y Ft(N)1864 4189 y Fw(e)1909 4148 y Fs(\000)1974 4115 y Ft(hL)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft( )r Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p Fm(\000)p Ft( )r Fr(\()p Ft(x)p Fr(\))p 1974 4133 V 2298 4174 a Ft(\017)2662 4189 y Fw(dv)31 b FE(=)d Fw(\025)17 b(:)456 4451 y FE(Then,)33 b(the)g(claim)e(in)g(\(123\))h (follo)m(ws)f(from)h(Corollary)f(33.)726 b Fj(\003)p 456 4571 499 4 v 555 4637 a Fl(15)626 4669 y FH(Again,)32 b(w)n(e)g(recall)e(that)i(the)h(existence)e(of)h Fp(\036)g FH(and)g Fp(\025)g FH(as)f(desired)g(follo)n(ws)g(from)g(Theo-)456 4779 y(rem)c(27.)555 4855 y Fl(16)626 4887 y FH(Note)32 b(that,)i(in)f(general,)f(a)g(pro)r(of)g(of)g(statemen)n(ts)g(as)g(the) h(ones)f(in)g(Theorem)g(37)f(ma)n(y)456 4996 y(require)e(the)i(use)f (of)g(the)h(Legendre-F)-7 b(enc)n(hel)29 b(Dualit)n(y)i(Theorem)e(or)h (some)g(adv)-5 b(anced)30 b(tec)n(h-)456 5106 y(nology)-7 b(.)35 b(Ho)n(w)n(ev)n(er,)25 b(in)i(our)f(case,)g(as)h(w)n(e)f(kno)n (w)g(the)i(optimal)e(measure)g(b)n(y)h(Theorem)f(32,)g(w)n(e)456 5216 y(manage)g(to)h(giv)n(e)g(an)g(elemen)n(tary)g(pro)r(of.)p eop %%Page: 52 52 52 51 bop 456 251 a FC(52)666 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)555 450 y FE(A)m(t)44 b(this)f(p)s(oin)m(t,)i(it)e(is)f(con) m(v)m(enien)m(t)k(to)d(recall)f(that)h(the)h(Mather)f(problem,)456 578 y(whic)m(h)h(consists)g(in)f(minimizing)c(\(87\))k(under)i(the)f (constrain)m(t)f(in)g(\(88\),)j(also)456 706 y(has)33 b(a)f(dual)g(problem.)42 b(Namely)-8 b(,)32 b(if)p 1811 626 89 4 v 31 w Fw(H)1899 721 y Fl(0)1971 706 y FE(is)h(the)g(in\014m)m (um)e(of)1671 862 y Fv(Z)1726 1087 y Fu(T)1776 1069 y Ft(N)1828 1087 y Fs(\002)p Fu(R)1931 1069 y Ft(N)2010 998 y Fw(Ld\026)17 b(;)456 1313 y FE(o)m(v)m(er)31 b(all)d(the)j (probabilit)m(y)d(measures)j(on)f Fn(T)2069 1277 y Fq(N)2157 1313 y Fx(\002)17 b Fn(R)2317 1277 y Fq(N)2421 1313 y FE(satisfying)29 b(the)h(constrain)m(t)456 1441 y(in)h(\(88\),)h(one)h (has)g(that)456 1697 y(\(126\))177 b(inf)761 1766 y Fq( )r Fs(2)p Fq(C)5 b Fl(\()p Fu(T)988 1748 y Ft(N)1041 1766 y Fl(\))1242 1697 y FE(sup)1085 1786 y Fl(\()p Fq(x;v)r Fl(\))p Fs(2)p Fu(T)1332 1767 y Ft(N)1385 1786 y Fs(\002)p Fu(R)1488 1767 y Ft(N)1562 1697 y Fx(\000)17 b FE([)q Fw(hL)p FE(\()p Fw(x;)g(v)t FE(\))22 b(+)g Fw( )t FE(\()p Fw(x)g FE(+)g Fw(hv)t FE(\))g Fx(\000)h Fw( )t FE(\()p Fw(x)p FE(\)])28 b(=)g Fx(\000)p Fw(h)p 3189 1617 V(H)3278 1712 y Fl(0)3334 1697 y Fw(:)456 2024 y FE(W)-8 b(e)23 b(refer)h(to)f([Gom05)n(])g(for)g(further)h(details)e(ab)s(out)h(this)g (fact.)40 b(W)-8 b(e)24 b(just)f(men)m(tion)456 2152 y(here)39 b(that)p 889 2072 V 37 w Fw(H)978 2167 y Fl(0)1056 2152 y FE(\(p)s(ossibly)e(up)i(to)e(a)h(sign)g(con)m(v)m(en)m(tion\))h (is)f(what)g(is)g(called)f(the)456 2280 y(\\e\013ectiv)m(e)32 b(Hamiltonian")27 b(in)k(the)h(literature.)42 b(F)-8 b(urthermore,)31 b(there)h(exists)g(an)456 2407 y(optimal)d Fw(\036)871 2422 y Fl(0)943 2407 y FE(suc)m(h)34 b(that)456 2664 y(\(127\))269 b(inf)912 2733 y Fq(v)r Fs(2)p Fu(R)1043 2714 y Ft(N)1118 2583 y Fv(\002)1159 2664 y Fw(hL)p FE(\()p Fw(x;)17 b(v)t FE(\))23 b(+)f Fw(\036)1686 2679 y Fl(0)1725 2664 y FE(\()p Fw(x)h FE(+)f Fw(hv)t FE(\))g Fx(\000)g Fw(\036)2263 2679 y Fl(0)2302 2664 y FE(\()p Fw(x)p FE(\))h Fx(\000)f Fw(h)p 2611 2584 V(H)2700 2679 y Fl(0)2740 2583 y Fv(\003)2809 2664 y FE(=)27 b(0)p Fw(;)456 2966 y FE(for)e(all)e Fw(x)28 b Fx(2)g Fn(T)966 2930 y Fq(N)1063 2966 y FE(\(see,)g(e.g.,)f([Gom05)n(]\).)42 b(F)-8 b(or)24 b(con)m(v)m(enience,)30 b(w)m(e)c(de\014ne)h Fw(\025)3136 2981 y Fq(\017;h)3257 2966 y FE(:=)g Fw(\025)456 3094 y FE(and)32 b Fw(\036)703 3109 y Fq(\017;h)824 3094 y FE(:=)27 b Fw(\036)p FE(,)33 b(where)g Fw(\036)g FE(and)f Fw(\025)h FE(are)f(as)h(in)f(Theorem)h(37,)f(and)h(w)m(e)g(set)p 1341 3314 V 1341 3394 a Fw(H)1430 3409 y Fq(\017=h)1583 3394 y FE(:=)44 b(inf)1768 3454 y Fq(\026)1865 3258 y Fv(Z)1981 3394 y Fw(Ld\026)22 b FE(+)2295 3326 y Fw(\017)p 2287 3371 57 4 v 2287 3462 a(h)2353 3394 y(S)6 b FE([)p Fw(\026)p FE(])p Fw(;)456 3686 y FE(where)32 b(the)f(in\014m)m(um)f(is) h(tak)m(en)h(o)m(v)m(er)g(all)d(measures)j Fw(\026)f FE(b)s(elonging)e(to)i(the)g(space)456 3813 y(giv)m(en)48 b(b)m(y)i(\(90\))e(and)h(that)f(satisfy)h(the)g(constrain)m(t)g (\(88\).)91 b(W)-8 b(e)49 b(will)d(see)k(in)456 3941 y(Theorem)33 b(38)g(b)s(elo)m(w)h(that)p 1489 3861 89 4 v 33 w Fw(H)1578 3957 y Fq(\017=h)1720 3941 y FE(ma)m(y)f(b)s(e)h (seen)h(as)e(an)h(appro)m(ximate)e(e\013ectiv)m(e)456 4070 y(Hamiltonian,)d(whic)m(h)k(will)d(tend)j(to)p 1848 3990 V 32 w Fw(H)1937 4085 y Fl(0)2009 4070 y FE(when)h Fw(\017=h)28 b Fx(!)f FE(0.)555 4198 y(It)33 b(follo)m(ws)e(from)g (Theorem)i(37)f(that)456 4471 y(\(128\))p 1662 4391 V 983 w Fw(H)1751 4486 y Fq(\017=h)1904 4471 y FE(=)2034 4403 y Fw(\025)2091 4418 y Fq(\017;h)p 2034 4448 150 4 v 2081 4539 a Fw(h)2210 4471 y(:)456 4745 y FI(Theorem)37 b(38.)k FD(We)35 b(have)p 1696 4877 89 4 v 1696 4957 a Fw(H)1785 4972 y Fq(\017=h)1921 4957 y Fx(!)p 2049 4877 V 28 w Fw(H)2137 4972 y Fl(0)2176 4957 y Fw(;)456 5191 y FD(as)f Fw(\017=h)28 b Fx(!)f FE(0)p FD(.)p eop %%Page: 53 53 53 52 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)420 b(53)456 450 y FD(Pr)-5 b(o)g(of.)41 b FE(W)-8 b(e)47 b(will)e(mak)m(e)i(use)h(of)e (the)i(follo)m(wing)c(elemen)m(tary)2787 411 y Fl(17)2904 450 y FE(fact:)72 b(if)46 b Fw(f)63 b Fx(2)456 578 y Fw(C)7 b FE(\()p Fn(R)636 542 y Fq(N)710 578 y FE(\))32 b(is)g(sup)s(erlinear,)g(then)456 834 y(\(129\))555 b(lim)1209 902 y Fq(\017)p Fs(!)p Fl(0)1344 883 y Fr(+)1411 694 y Fv(\022)1484 699 y(Z)1539 924 y Fu(R)1587 905 y Ft(N)1666 834 y Fw(e)1711 793 y Fs(\000)1776 760 y Ft(f)6 b Fr(\()p Ft(v)r Fr(\))p 1776 778 117 3 v 1821 819 a Ft(\017)1923 834 y Fw(dv)2025 694 y Fv(\023)2098 716 y Fq(\017)2175 834 y FE(=)44 b Fw(e)2340 793 y Fs(\000)12 b Fl(inf)2492 819 y Fk(R)2534 805 y Ft(N)2602 793 y Fq(f)2664 834 y Fw(:)456 1083 y FE(Also,)32 b(since)1264 1242 y(lim)1201 1310 y Fq(\017=h)p Fs(!)p Fl(0)1412 1291 y Fr(+)p 1479 1162 89 4 v 1479 1242 a Fw(H)1568 1258 y Fq(\017=h)1704 1242 y FE(=)52 b(lim)1807 1310 y Fq(\017)p Fs(!)p Fl(0)1942 1291 y Fr(+)p 2009 1162 V 2009 1242 a Fw(H)2098 1257 y Fq(\017)2158 1242 y FE(=)g(lim)2261 1310 y Fq(\017)p Fs(!)p Fl(0)2396 1291 y Fr(+)p 2463 1162 V 2463 1242 a Fw(H)2552 1258 y Fq(\017=)p Fl(1)2672 1242 y Fw(;)456 1468 y FE(w)m(e)33 b(ma)m(y)g(and)f(do)h(\014x)g Fw(h)28 b FE(:=)f(1)33 b(in)f(this)g(pro)s(of.)555 1596 y(Let)h Fw(\036)788 1611 y Fl(0)860 1596 y FE(b)s(e)f(as)h(in)f(\(127\).)43 b(By)33 b(\(128\))e(and)i(Theorem)g(37,)f(w)m(e)i(ha)m(v)m(e)g(that) 1049 1841 y Fx(\000)p 1126 1761 V Fw(H)1215 1856 y Fq(\017)1276 1841 y Fx(\024)49 b FE(sup)1381 1930 y Fq(x)p Fs(2)p Fu(T)1518 1911 y Ft(N)1586 1841 y Fw(\017)17 b FE(ln)1740 1705 y Fv(Z)1796 1931 y Fu(R)1844 1912 y Ft(N)1922 1841 y Fw(e)1967 1800 y Fs(\000)2032 1764 y Ft(L)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft(\036)2293 1779 y Fr(0)2327 1764 y(\()p Ft(x)p Fr(+)p Ft(v)r Fr(\))p Fm(\000)p Ft(\036)2575 1779 y Fr(0)2609 1764 y(\()p Ft(x)p Fr(\))p 2032 1785 660 3 v 2349 1826 a Ft(\017)2705 1841 y Fw(dv)k(:)456 2104 y FE(Therefore,)33 b(using)g(\(129\))e(and)i(\(127\),)456 2308 y(\(130\))872 b Fx(\000)41 b FE(lim)1645 2375 y Fq(\017)p Fs(!)p Fl(0)1780 2356 y Fr(+)p 1846 2228 89 4 v 1846 2308 a Fw(H)1935 2323 y Fq(\017)1995 2308 y Fx(\024)28 b(\000)p 2177 2228 V Fw(H)2266 2323 y Fl(0)2322 2308 y Fw(:)555 2539 y FE(T)-8 b(o)37 b(pro)m(v)m(e)h(the)f(rev)m(erse) i(inequalit)m(y)-8 b(,)36 b(w)m(e)i(observ)m(e)h(that)d Fw(\036)2721 2554 y Fq(\017;)p Fl(1)2845 2539 y FE(is)g(a)h(uniformly) 456 2667 y(Lipsc)m(hitz)29 b(function,)g(thanks)i(to)e(Theorem)h(27,)f (and,)h(therefore,)h(w)m(e)f(ma)m(y)g(tak)m(e)456 2795 y(the)g(in\014m)m(um)e(on)h(the)h(righ)m(t)f(hand)h(side)g(of)f (\(123\))g(o)m(v)m(er)h(the)h(space)f(of)f(Lipsc)m(hitz)456 2923 y(functions)24 b(with)g(Lipsc)m(hitz)h(constan)m(t)g(b)s(ounded)h (b)m(y)f(some)g(suitably)e(large)h(n)m(um-)456 3051 y(b)s(er)32 b(\003.)44 b(In)33 b(particular,)e(b)m(y)j(Theorem)f(37,)f(\014xed)i (an)m(y)g Fw(\016)e(>)27 b FE(0,)33 b(there)g(exists)h(a)e Fw( )456 3178 y FE(in)f(suc)m(h)j(a)f(Lipsc)m(hitz)f(space,)i(so)f (that)456 3418 y(\(131\))158 b Fx(\000)p 914 3338 V Fw(H)1003 3433 y Fq(\017)1058 3418 y FE(+)22 b Fw(\016)32 b FE(=)27 b Fx(\000)p Fw(\025)1468 3433 y Fq(\017;)p Fl(1)1579 3418 y FE(+)22 b Fw(\016)31 b Fx(\025)49 b FE(sup)1856 3507 y Fq(x)p Fs(2)p Fu(T)1993 3488 y Ft(N)2062 3418 y Fw(\017)17 b FE(ln)2216 3283 y Fv(Z)2271 3508 y Fu(R)2319 3489 y Ft(N)2397 3418 y Fw(e)2442 3377 y Fs(\000)2507 3343 y Ft(L)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft( )r Fr(\()p Ft(x)p Fr(+)p Ft(v)r Fr(\))p Fm(\000)p Ft( )r Fr(\()p Ft(x)p Fr(\))p 2507 3362 602 3 v 2795 3403 a Ft(\017)3140 3418 y Fw(dv)j(:)p 456 3611 499 4 v 555 3677 a Fl(17)626 3709 y FH(F)-7 b(or)33 b(completeness,)i(w)n(e)f(giv)n(e)f(a)g(sk)n (etc)n(h)g(of)h(the)g(pro)r(of)g(of)g(\(129\).)55 b(Let)34 b Fp(f)9 b FH(\()p Fp(v)s FH(\))34 b FF(\025)f Fp(c)p FF(j)p Fp(v)s FF(j)h FH(if)456 3819 y FF(j)p Fp(v)s FF(j)23 b(\025)g Fp(R)719 3831 y Fy(0)756 3819 y FH(,)28 b(for)f Fp(c)c(>)f FH(0.)37 b(T)-7 b(ak)n(e)26 b Fp(R)e FF(\025)f Fp(R)1620 3831 y Fy(0)1657 3819 y FH(,)28 b(then)647 3914 y Fo(Z)693 4102 y Fe(R)740 4086 y Fh(N)806 4027 y Fp(e)845 3992 y Fd(\000)907 3966 y Fh(f)p 907 3980 34 3 v 911 4012 a(\017)977 4027 y FF(\024)1065 3914 y Fo(Z)1111 4102 y Fd(j)p Fi(v)r Fd(j\024)p Fi(R)1306 4027 y Fp(e)1345 3992 y Fd(\000)1406 3966 y Fg(inf)14 b Fh(f)p 1406 3980 114 3 v 1450 4012 a(\017)1552 4027 y FH(+)1635 3914 y Fo(Z)1681 4102 y Fd(j)p Fi(v)r Fd(j\025)p Fi(R)1876 4027 y Fp(e)1915 3992 y Fd(\000)1976 3963 y Fh(c)p Fc(j)p Fh(v)q Fc(j)p 1976 3979 97 3 v 2012 4012 a Fh(\017)2110 4027 y FF(\024)23 b FH(const)13 b(\()p Fp(R)2497 3992 y Fi(N)2560 4027 y Fp(e)2599 3992 y Fd(\000)2661 3966 y Fg(inf)g Fh(f)p 2661 3980 114 3 v 2704 4012 a(\017)2806 4027 y FH(+)18 b Fp(\017)2923 3992 y Fi(N)3000 4027 y Fp(e)3039 3992 y Fd(\000)3100 3970 y Fh(cR)p 3100 3979 70 3 v 3108 4012 a Fg(2)p Fh(\017)3184 4027 y FH(\))c Fp(:)456 4253 y FH(In)27 b(particular,)g(if)h Fp(R)c FH(:=)e(1)p Fp(=\017)27 b FH(and)g Fp(\017)g FH(is)h(small)f(enough,) 1433 4347 y Fo(Z)1479 4536 y Fe(R)1526 4520 y Fh(N)1592 4460 y Fp(e)1631 4426 y Fd(\000)1693 4399 y Fh(f)p 1693 4413 34 3 v 1697 4446 a(\017)1763 4460 y FF(\024)c FH(const)13 b Fp(\017)2088 4426 y Fd(\000)p Fi(N)2203 4460 y Fp(e)2242 4426 y Fd(\000)2303 4399 y Fg(inf)h Fh(f)p 2303 4413 114 3 v 2347 4446 a(\017)2444 4460 y Fp(;)456 4672 y FH(whic)n(h)24 b(giv)n(es)e(one)i(inequalit)n(y)g(in)g(\(129\).)35 b(F)-7 b(or)23 b(the)i(other)e(one,)i(\014x)f Fp(\016)i(>)d FH(0.)35 b(Then,)25 b(there)f(exists)456 4782 y(a)j(ball)g Fp(B)32 b FH(so)27 b(that)h Fp(f)9 b FH(\()p Fp(v)s FH(\))23 b FF(\024)g FH(inf)e Fp(f)27 b FH(+)18 b Fp(\016)31 b FH(for)c(an)n(y)f Fp(v)h FF(2)c Fp(B)t FH(.)37 b(Then,)1322 4876 y Fo(Z)1368 5065 y Fe(R)1415 5048 y Fh(N)1481 4989 y Fp(e)1520 4955 y Fd(\000)1582 4928 y Fh(f)p 1582 4942 34 3 v 1586 4975 a(\017)1652 4989 y FF(\025)1740 4876 y Fo(Z)1786 5065 y Fi(B)1857 4989 y Fp(e)1896 4955 y Fd(\000)1957 4928 y Fh(f)p 1957 4942 V 1961 4975 a(\017)2028 4989 y FF(\025)22 b Fp(e)2154 4955 y Fd(\000)2216 4928 y Fg(inf)14 b Fh(f)5 b Fg(+)p Fh(\016)p 2216 4942 185 3 v 2295 4975 a(\017)2428 4989 y FF(j)p Fp(B)t FF(j)14 b Fp(;)456 5216 y FH(then)28 b(tak)n(e)f(the)h Fp(\017)p FH(-p)r(o)n(w)n(er)d(to)j(b)r(oth)g(sides,)f(then)h(send)g Fp(\017)23 b FF(!)g FH(0)k(and)g(then)h Fp(\016)e FF(!)e FH(0.)p eop %%Page: 54 54 54 53 bop 456 251 a FC(54)666 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)456 450 y FE(Giv)m(en)32 b(suc)m(h)i(a)e Fw( )t FE(,)h(w)m(e)h(de\014ne)f(\()p Fw(x)1684 465 y Fq( )1737 450 y Fw(;)17 b(v)1828 465 y Fq( )1881 450 y FE(\))32 b(to)h(b)s(e)f(suc)m(h)i(that)561 655 y Fx(\000)22 b FE([)p Fw(L)p FE(\()p Fw(x)846 670 y Fq( )900 655 y Fw(;)17 b(v)991 670 y Fq( )1043 655 y FE(\))22 b(+)g Fw( )t FE(\()p Fw(x)1361 670 y Fq( )1437 655 y FE(+)g Fw(v)1582 670 y Fq( )1634 655 y FE(\))h Fx(\000)f Fw( )t FE(\()p Fw(x)1954 670 y Fq( )2007 655 y FE(\)])1471 815 y(=)184 b(sup)1574 904 y Fl(\()p Fq(x;v)r Fl(\))p Fs(2)p Fu(T)1821 885 y Ft(N)1874 904 y Fs(\002)p Fu(R)1977 885 y Ft(N)2051 815 y Fx(\000)17 b FE([)p Fw(L)p FE(\()p Fw(x;)g(v)t FE(\))23 b(+)f Fw( )t FE(\()p Fw(x)g FE(+)g Fw(v)t FE(\))g Fx(\000)h Fw( )t FE(\()p Fw(x)p FE(\)])33 b Fw(:)456 1053 y FE(W)-8 b(e)33 b(denote)g(b)m(y)g Fw(\033)1128 1068 y Fq( )1214 1053 y FE(the)g(ab)s(o)m(v)m(e)g(suprem)m(um.)555 1181 y(Let)c(us)g(no)m(w)f(lo)s(calize)e Fw(v)1432 1196 y Fq( )1485 1181 y FE(:)41 b(since)29 b Fw( )j FE(is)c(Lipsc)m(hitz,)h (the)g(de\014nition)e(of)h(\()p Fw(x)3210 1196 y Fq( )3263 1181 y Fw(;)17 b(v)3354 1196 y Fq( )3406 1181 y FE(\))456 1309 y(implies)30 b(that)1464 1438 y Fw(K)7 b FE(\()p Fw(v)1639 1453 y Fq( )1691 1438 y FE(\))28 b Fx(\024)g Fw(K)7 b FE(\(0\))22 b(+)g(\003)2265 1348 y Fx(p)p 2348 1348 89 4 v 90 x Fw(N)456 1591 y FE(and)38 b(so,)j(since)e Fw(K)46 b FE(is)38 b(sup)s(erlinear,)i(w)m(e)f(ha)m(v)m(e)h(that)f Fx(j)p Fw(v)2494 1606 y Fq( )2546 1591 y Fx(j)f(\024)2737 1566 y FE(^)2728 1591 y(\003)o(,)j(for)d(a)g(suitably)456 1722 y(large)31 b(univ)m(ersal)h(n)m(um)m(b)s(er)1471 1696 y(^)1461 1722 y(\003.)44 b(Let)32 b(no)m(w)1525 1870 y(~)1516 1895 y(\003)27 b(:=)89 b(sup)1742 1994 y Fs(j)p Fq(v)r Fs(j\024)1880 1977 y Fl(^)1873 1994 y(\003+2)2028 1895 y Fx(j)p Fw(D)s(K)7 b FE(\()p Fw(v)t FE(\))p Fx(j)456 2143 y FE(and)32 b(consider)h(the)g(univ)m(ersal)g(quan)m(tit)m(y)1651 2316 y(\003)1719 2331 y Fs(\003)1802 2316 y FE(:=)45 b(\003)22 b(+)2147 2291 y(~)2138 2316 y(\003)16 b Fw(:)456 2490 y FE(Then,)33 b(if)f Fx(j)p Fw(v)25 b Fx(\000)e Fw(v)1074 2505 y Fq( )1127 2490 y Fx(j)k(\024)h Fw(\017)g(<)g FE(1,)k(w)m(e)i(ha)m(v)m(e)g(that)e Fx(j)p Fw(v)t Fx(j)27 b(\024)2395 2464 y FE(^)2385 2490 y(\003)22 b(+)g(1)32 b(and)h(so)1147 2663 y Fx(j)p Fw(\033)1230 2678 y Fq( )1305 2663 y FE(+)22 b Fw(L)p FE(\()p Fw(x)1562 2678 y Fq( )1615 2663 y Fw(;)17 b(v)t FE(\))22 b(+)g Fw( )t FE(\()p Fw(x)2028 2678 y Fq( )2103 2663 y FE(+)g Fw(v)t FE(\))g Fx(\000)g Fw( )t FE(\()p Fw(x)2571 2678 y Fq( )2624 2663 y FE(\))p Fx(j)986 2823 y(\024)84 b(j)p Fw(K)7 b FE(\()p Fw(v)1350 2838 y Fq( )1402 2823 y FE(\))23 b Fx(\000)f Fw(K)7 b FE(\()p Fw(v)t FE(\))p Fx(j)22 b FE(+)g Fx(j)p Fw( )t FE(\()p Fw(x)2115 2838 y Fq( )2190 2823 y FE(+)g Fw(v)t FE(\))f Fx(\000)i Fw( )t FE(\()p Fw(x)2658 2838 y Fq( )2733 2823 y FE(+)f Fw(v)2878 2838 y Fq( )2931 2823 y FE(\))p Fx(j)-2541 b FE(\(132\))986 2982 y Fx(\024)84 b FE(\003)1215 2997 y Fs(\003)1271 2982 y Fw(\017)17 b(;)456 3156 y FE(for)32 b(an)m(y)h Fw(v)e Fx(2)d Fw(B)1035 3171 y Fq(\017)1068 3156 y FE(\()p Fw(v)1153 3171 y Fq( )1206 3156 y FE(\).)555 3284 y(By)33 b(collecting)e(the)i(estimates)f(in)g(\(131\))g(and)g (\(132\),)g(w)m(e)i(obtain)896 3498 y Fx(\000)p 973 3418 V Fw(H)1062 3513 y Fq(\017)1117 3498 y FE(+)22 b Fw(\016)87 b Fx(\025)104 b FE(sup)1505 3587 y Fq(x)p Fs(2)p Fu(T)1642 3568 y Ft(N)1711 3498 y Fw(\017)17 b FE(ln)1865 3362 y Fv(Z)1920 3588 y Fq(B)1973 3596 y Ft(\017)2003 3588 y Fl(\()p Fq(v)2064 3600 y Ft( )2112 3588 y Fl(\))2160 3498 y Fw(e)2205 3457 y Fs(\000)2270 3423 y Ft(L)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft( )r Fr(\()p Ft(x)p Fr(+)p Ft(v)r Fr(\))p Fm(\000)p Ft( )r Fr(\()p Ft(x)p Fr(\))p 2270 3441 602 3 v 2558 3483 a Ft(\017)2902 3498 y Fw(dv)1345 3764 y Fx(\025)83 b Fw(\017)17 b FE(ln)1659 3653 y Fv(\020)1719 3764 y Fx(j)p Fw(B)1821 3779 y Fq(\017)1853 3764 y FE(\()p Fw(v)1938 3779 y Fq( )1991 3764 y FE(\))p Fx(j)g Fw(e)2129 3675 y Ft(\033)2165 3692 y( )p 2128 3708 83 3 v 2157 3749 a(\017)2221 3723 y Fs(\000)p Fl(\003)2325 3731 y Fm(\003)2365 3653 y Fv(\021)1346 3944 y FE(=)83 b Fw(\033)1560 3959 y Fq( )1636 3944 y FE(+)38 b(const)18 b Fw(\017)33 b FE(ln\()17 b(const)f Fw(\017)p FE(\))h Fw(:)456 4117 y FE(Since,)32 b(b)m(y)i(\(126\))e(and)g(the)h(de\014nition)f(of)g Fw(\033)2085 4132 y Fq( )2138 4117 y FE(,)h(w)m(e)g(ha)m(v)m(e)h(that) 1705 4290 y Fx(\000)p 1782 4210 89 4 v Fw(H)1871 4305 y Fl(0)1938 4290 y Fx(\024)28 b Fw(\033)2098 4305 y Fq( )2168 4290 y Fw(;)456 4463 y FE(it)j(follo)m(ws)g(from)h(the)h(ab)s(o)m(v)m (e)g(computation)e(that)1513 4637 y Fw(\016)26 b Fx(\000)48 b FE(lim)1682 4704 y Fq(\017)p Fs(!)p Fl(0)1817 4685 y Fr(+)p 1884 4556 V 1884 4637 a Fw(H)1972 4652 y Fq(\017)2032 4637 y Fx(\025)29 b(\000)p 2215 4556 V Fw(H)2304 4652 y Fl(0)2360 4637 y Fw(:)456 4837 y FE(Thence,)34 b(since)f Fw(\016)k FE(is)32 b(arbitrary)-8 b(,)1551 5010 y Fx(\000)41 b FE(lim)1645 5077 y Fq(\017)p Fs(!)p Fl(0)1780 5058 y Fr(+)p 1846 4930 V 1846 5010 a Fw(H)1935 5025 y Fq(\017)1995 5010 y Fx(\025)28 b(\000)p 2177 4930 V Fw(H)2266 5025 y Fl(0)2322 5010 y Fw(:)456 5216 y FE(This)k(and)h(\(130\))f(complete)g (the)h(pro)s(of)e(of)h(the)h(desired)h(result.)533 b Fj(\003)p eop %%Page: 55 55 55 54 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)420 b(55)555 450 y FE(Ha)m(ving)23 b(completed)f(the)i(pro)s(of)e(of)g(the)h(con)m(v)m (ergence)j(of)c(the)i(e\013ectiv)m(e)f(Hamil-)456 578 y(tonian,)31 b(w)m(e)j(no)m(w)f(address)h(the)f(con)m(v)m(ergence)i(of) d(the)h(\\normalized")d(function)456 765 y(\(133\))711 b Fw(\036)1448 780 y Fq(\017)1481 765 y FE(\()p Fw(x)p FE(\))44 b(:=)h Fw(\036)1862 780 y Fq(\017;h)1954 765 y FE(\()p Fw(x)p FE(\))23 b Fx(\000)f Fw(\036)2265 780 y Fq(\017;h)2358 765 y FE(\(0\))p Fw(;)456 951 y FE(as)32 b Fw(\017)c Fx(!)g FE(0,)k(for)g(a)g(\014xed)i Fw(h)28 b(>)g FE(0:)456 1106 y FI(Theorem)i(39.)37 b Fw(\036)1162 1121 y Fq(\017)1222 1106 y Fx(!)28 b Fw(\036)1408 1121 y Fl(0)1477 1106 y FD(uniformly)h(as)g Fw(\017)f Fx(!)f FE(0)2278 1070 y Fl(+)2337 1106 y FD(,)k(in)e(which)g Fw(\036)2839 1121 y Fl(0)2906 1106 y Fx(2)f Fw(W)3106 1070 y Fl(1)p Fq(;)p Fs(1)3235 1106 y FE(\()p Fn(T)3336 1070 y Fq(N)3406 1106 y FE(\))456 1234 y FD(is)j(a)g(solution)g(to)g (\(127\).)43 b(A)n(lso,)31 b Fw(\036)1723 1249 y Fl(0)1794 1234 y FD(is)g(semic)-5 b(onc)g(ave.)41 b(The)31 b(Lipschitz)g(c)-5 b(onstant)456 1361 y(and)28 b(the)h(semic)-5 b(onc)g(avity)27 b(mo)-5 b(dulus)29 b(of)g Fw(\036)1946 1376 y Fl(0)2014 1361 y FD(ar)-5 b(e)28 b(b)-5 b(ounde)g(d)28 b(by)h(a)g(c)-5 b(onstant)29 b(dep)-5 b(end-)456 1489 y(ing)34 b(only)h(on)f Fw(N)10 b FD(,)35 b Fw(C)1194 1504 y Fq(U)1288 1489 y FD(and)f Fw(C)1547 1504 y Fq(K)1616 1489 y FD(.)456 1682 y(Pr)-5 b(o)g(of.)41 b FE(Along)34 b(this)h(pro)s(of,)g(w)m(e)h(will)d (mak)m(e)j(explicit)e(the)h(dep)s(endence)j(on)d Fw(\017)h FE(of)456 1810 y Fx(G)i FE(b)m(y)c(denoting)e(it)f(b)m(y)j Fx(G)1382 1825 y Fq(\017)1447 1810 y FE(and)f(w)m(e)h(de\014ne)f Fx(G)2121 1825 y Fl(0)2194 1810 y FE(to)f(b)s(e)h(the)g(op)s(erator) 1151 1997 y Fx(G)1210 2012 y Fl(0)1250 1997 y FE([)p Fw( )t FE(]\()p Fw(x)p FE(\))44 b(:=)h(inf)1734 2057 y Fq(v)1828 1997 y Fw(hL)p FE(\()p Fw(x;)17 b(v)t FE(\))23 b(+)f Fw( )t FE(\()p Fw(x)g FE(+)g Fw(hv)t FE(\))p Fw(:)456 2208 y FE(Notice)32 b(that,)g(from)g(\(128\))f(and)i(Theorem)g(38,)456 2395 y(\(134\))916 b(lim)1570 2462 y Fq(\017)p Fs(!)p Fl(0)1705 2443 y Fr(+)1772 2395 y Fw(\025)1829 2410 y Fq(\017;h)1966 2395 y FE(=)44 b Fw(h)p 2159 2315 89 4 v 17 w(H)2247 2410 y Fl(0)2303 2395 y Fw(:)456 2608 y FE(Moreo)m(v)m(er,)34 b(due)f(to)g(Theorem)g(27,)f(w)m(e)i(kno)m(w)g (that)f Fw(\036)2453 2623 y Fq(\017)2518 2608 y FE(is)f(uniformly)e (Lipsc)m(hitz)456 2736 y(con)m(tin)m(uous)37 b(and)f(semiconca)m(v)m(e) h(and)g(so,)g(b)m(y)g(\(133\),)f Fw(\036)2523 2751 y Fq(\017)2592 2736 y FE(con)m(v)m(erges)j(uniformly)456 2864 y(to)30 b(a)g(suitable)f Fw(\036)1071 2879 y Fl(0)1138 2864 y Fx(2)f Fw(W)1338 2828 y Fl(1)p Fq(;)p Fs(1)1467 2864 y FE(\()p Fn(T)1568 2828 y Fq(N)1639 2864 y FE(\),)j(and)f(b)s (oth)g(the)h(Lipsc)m(hitz)f(constan)m(t)h(and)g(the)456 2992 y(semiconca)m(vit)m(y)k(mo)s(dulus)f(of)h Fw(\036)1650 3007 y Fl(0)1725 2992 y FE(are)g(univ)m(ersally)g(b)s(ounded.)53 b(What)35 b(is)g(more,)456 3120 y(b)m(y)e(Prop)s(osition)e(8,)456 3307 y(\(135\))198 b(lim)852 3374 y Fq(\017)p Fs(!)p Fl(0)987 3355 y Fr(+)1054 3307 y Fx(kG)1163 3322 y Fq(\017)1195 3307 y FE([)p Fw(\036)1280 3322 y Fq(\017)1313 3307 y FE(])22 b Fx(\000)h(G)1521 3322 y Fq(\017)1554 3307 y FE([)p Fw(\036)1639 3322 y Fl(0)1678 3307 y FE(])p Fx(k)1755 3325 y Fq(L)1803 3306 y Fm(1)1868 3325 y Fl(\()p Fu(T)1945 3306 y Ft(N)1997 3325 y Fl(\))2057 3307 y Fx(\024)53 b FE(lim)2162 3374 y Fq(\017)p Fs(!)p Fl(0)2297 3355 y Fr(+)2364 3307 y Fx(k)p Fw(\036)2472 3322 y Fq(\017)2526 3307 y Fx(\000)23 b Fw(\036)2684 3322 y Fl(0)2723 3307 y Fx(k)2773 3325 y Fq(L)2821 3306 y Fm(1)2886 3325 y Fl(\()p Fu(T)2963 3306 y Ft(N)3015 3325 y Fl(\))3074 3307 y FE(=)28 b(0)17 b Fw(:)456 3526 y FE(Notice)32 b(also)f(that,)i(b)m(y)g(\(129\))f(and)h(the)g(con)m(tin)m(uit)m(y)f (of)g Fw(\036)2573 3541 y Fl(0)2613 3526 y FE(,)g(w)m(e)i(ha)m(v)m(e)g (that)456 3712 y(\(136\))713 b(lim)1367 3779 y Fq(\017)p Fs(!)p Fl(0)1502 3761 y Fr(+)1569 3712 y Fx(G)1628 3727 y Fq(\017)1661 3712 y FE([)p Fw(\036)1746 3727 y Fl(0)1785 3712 y FE(]\()p Fw(x)p FE(\))45 b(=)f Fx(G)2167 3727 y Fl(0)2207 3712 y FE([)p Fw(\036)2292 3727 y Fl(0)2331 3712 y FE(]\()p Fw(x)p FE(\))17 b Fw(;)456 3938 y FE(for)32 b(an)m(y)h Fw(x)28 b Fx(2)g Fn(T)1029 3902 y Fq(N)1100 3938 y FE(.)555 4066 y(Therefore,)34 b(w)m(e)g(use)f(\(134\),)f (\(135\))g(and)g(\(136\))g(to)g(deduce)i(that)702 4252 y Fw(h)p 775 4172 V 17 w(H)863 4267 y Fl(0)986 4252 y FE(=)108 b(lim)1145 4320 y Fq(\017)p Fs(!)p Fl(0)1280 4301 y Fr(+)1346 4252 y Fw(\025)1403 4267 y Fq(\017;h)986 4445 y FE(=)g(lim)1145 4512 y Fq(\017)p Fs(!)p Fl(0)1280 4493 y Fr(+)1346 4445 y Fx(G)1405 4460 y Fq(\017)1438 4445 y FE([)p Fw(\036)1523 4460 y Fq(\017)1556 4445 y FE(])22 b Fx(\000)h Fw(\036)1763 4460 y Fq(\017)986 4677 y FE(=)108 b(lim)1145 4744 y Fq(\017)p Fs(!)p Fl(0)1280 4725 y Fr(+)1346 4677 y Fx(G)1405 4692 y Fq(\017)1438 4677 y FE([)p Fw(\036)1523 4692 y Fl(0)1563 4677 y FE(])22 b(+)1710 4566 y Fv(\020)1769 4677 y Fx(G)1828 4692 y Fq(\017)1861 4677 y FE([)p Fw(\036)1946 4692 y Fq(\017)1979 4677 y FE(])g Fx(\000)h(G)2187 4692 y Fq(\017)2220 4677 y FE([)p Fw(\036)2305 4692 y Fl(0)2344 4677 y FE(])2371 4566 y Fv(\021)2453 4677 y Fx(\000)f Fw(\036)2610 4692 y Fl(0)2672 4677 y FE(+)2770 4566 y Fv(\020)2829 4677 y Fw(\036)2887 4692 y Fl(0)2948 4677 y Fx(\000)h Fw(\036)3106 4692 y Fq(\017)3138 4566 y Fv(\021)986 4869 y FE(=)83 b Fx(G)1204 4884 y Fl(0)1243 4869 y FE([)p Fw(\036)1328 4884 y Fl(0)1368 4869 y FE(])22 b(+)g(0)g Fx(\000)g Fw(\036)1743 4884 y Fl(0)1805 4869 y FE(+)g(0)986 5029 y(=)83 b Fx(G)1204 5044 y Fl(0)1243 5029 y FE([)p Fw(\036)1328 5044 y Fl(0)1368 5029 y FE(])22 b Fx(\000)h Fw(\036)1575 5044 y Fl(0)1630 5029 y Fw(;)456 5216 y FE(and)32 b(so)h Fw(\036)823 5231 y Fl(0)895 5216 y FE(is)f(a)g(solution)f(of)h(\(127\).)1563 b Fj(\003)p eop %%Page: 56 56 56 55 bop 456 251 a FC(56)666 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)555 450 y FE(W)h(e)48 b(no)m(w)f(consider)g(the)h(p)s (enalized)e(Mather)i(measures)f Fw(\026)2827 465 y Fq(\017)2912 450 y FE(:=)52 b Fw(\026)3126 465 y Fq(\017;h)3266 450 y FE(con-)456 578 y(structed)34 b(in)e(Theorem)h(32)f(and)h(w)m(e)g (pro)m(v)m(e)h(its)f(con)m(v)m(ergence)i(as)e Fw(\017)28 b Fx(!)g FE(0)3128 542 y Fl(+)3187 578 y FE(,)k(for)h(a)456 706 y(\014xed)g Fw(h)28 b(>)g FE(0:)456 865 y FI(Theorem)42 b(40.)j FD(As)39 b Fw(\017)e Fx(!)f FE(0)p FD(,)k Fw(\026)1665 880 y Fq(\017)1737 865 y FD(c)-5 b(onver)g(ges)38 b(we)-5 b(akly)39 b(to)g(a)g(\(discr)-5 b(ete\))39 b(Mather)456 993 y(me)-5 b(asur)g(e,)34 b(that)h(is,)g(a)f(pr)-5 b(ob)g(ability)35 b(me)-5 b(asur)g(e)34 b(on)h Fn(T)2331 957 y Fq(N)2423 993 y Fx(\002)23 b Fn(R)2589 957 y Fq(N)2697 993 y FD(that)35 b(minimizes)1679 1090 y Fv(Z)1735 1316 y Fu(T)1785 1297 y Ft(N)1837 1316 y Fs(\002)p Fu(R)1940 1297 y Ft(N)2018 1226 y Fw(Ld\026;)456 1475 y FD(under)f(the)h(c)-5 b(onstr)g(aint)35 b(in)f(\(88\).)555 1603 y(Mor)-5 b(e)g(over,)35 b(the)g(supp)-5 b(ort)35 b(of)f Fw(\026)1694 1618 y Fl(0)1768 1603 y FD(is)h(c)-5 b(omp)g(act.)456 1808 y(Pr)g(o)g(of.)41 b FE(W)-8 b(e)41 b(will)e(drop)j(the)g Fw(h)f FE(dep)s(endence)j(in)c (some)h(of)g(the)h(indices)f(of)g(this)456 1936 y(pro)s(of,)31 b(since)i(no)g(confusion)f(can)h(arise.)43 b(W)-8 b(e)33 b(recall)e(that,)i(b)m(y)g(\(105\),)1253 2150 y Fw(\026)1312 2165 y Fq(\017)1372 2150 y FE(=)28 b Fw(\022)1521 2165 y Fq(\017)1554 2150 y FE(\()p Fw(x)p FE(\))p Fw(e)1730 2109 y Fs(\000)1795 2075 y Ft(hL)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft(\036)2092 2083 y(\017)2122 2075 y Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p Fm(\000)p Ft(\036)2406 2083 y(\017)2435 2075 y Fr(\()p Ft(x)p Fr(\))p Fm(\000)p Ft(\025)2603 2083 y(\017)p 1795 2094 838 3 v 2201 2135 a(\017)456 2347 y FE(with)1449 2380 y Fv(Z)1505 2606 y Fu(R)1553 2587 y Ft(N)1631 2516 y Fw(\026)1690 2531 y Fq(\017)1723 2516 y FE(\()p Fw(x;)17 b(v)t FE(\))g Fw(dv)30 b FE(=)d Fw(\022)2242 2531 y Fq(\017)2276 2516 y FE(\()p Fw(x)p FE(\))17 b Fw(:)456 2736 y FE(F)-8 b(urthermore,)41 b Fw(\036)1115 2751 y Fq(\017)1187 2736 y Fx(!)e Fw(\036)1384 2751 y Fl(0)1463 2736 y FE(uniformly)-8 b(,)39 b(where)i Fw(\036)2282 2751 y Fl(0)2361 2736 y FE(satis\014es)g(\(127\),)g(and)e Fw(\025)3272 2751 y Fq(\017)3345 2736 y Fx(!)456 2863 y Fw(h)p 512 2783 89 4 v(H)600 2878 y Fl(0)640 2863 y FE(,)32 b(according)g(to)h(Theorems)g(38)f(and)h(39.)555 2991 y(In)24 b(particular,)g Fx(k)p Fw(\036)1246 3006 y Fq(\017)1278 2991 y Fx(k)1328 3006 y Fq(L)1376 2987 y Fm(1)1469 2991 y FE(and)f Fx(j)p Fw(\025)1734 3006 y Fq(\017)1767 2991 y Fx(j)g FE(are)h(b)s(ounded)g(uniformly)d(in)i Fw(\017)p FE(.)41 b(Therefore,)456 3119 y(from)31 b(the)i(fact)f(that)h Fw(K)39 b FE(is)32 b(sup)s(erlinear,)456 3352 y Fw(\026)515 3367 y Fq(\017)547 3241 y Fv(\020)606 3352 y Fn(T)669 3311 y Fq(N)762 3352 y Fx(\002)23 b FE(\()p Fn(R)966 3311 y Fq(N)1061 3352 y Fx(n)f Fw(B)1207 3367 y Fq(R)1265 3352 y FE(\))1303 3241 y Fv(\021)1446 3352 y FE(=)1606 3216 y Fv(Z)1661 3442 y Fu(T)1711 3423 y Ft(N)1763 3442 y Fs(\002)p Fl(\()p Fu(R)1893 3423 y Ft(N)1951 3442 y Fs(n)p Fq(B)2039 3453 y Ft(R)2090 3442 y Fl(\))2138 3352 y Fw(\022)2183 3367 y Fq(\017)2216 3352 y FE(\()p Fw(x)p FE(\))17 b Fw(e)2409 3311 y Fs(\000)2474 3277 y Ft(hK)s Fr(\()p Ft(v)r Fr(\))p Fm(\000)p Ft(hU)5 b Fr(\()p Ft(x)p Fr(\)+)p Ft(\036)2943 3285 y(\017)2974 3277 y Fr(\()p Ft(x)p Fr(+)p Ft(hv)r Fr(\))p Fm(\000)p Ft(\036)3258 3285 y(\017)3287 3277 y Fr(\()p Ft(x)p Fr(\))p Fm(\000)p Ft(\025)3455 3285 y(\017)p 2474 3296 1011 3 v 2967 3337 a(\017)1445 3634 y Fx(\024)84 b Fw(e)1661 3557 y Ft(c)1689 3572 y Fr(1)1724 3557 y Fm(\000)p Ft(c)1800 3572 y Fr(2)1834 3557 y Ft(hR)p 1661 3578 256 3 v 1776 3619 a(\017)1947 3499 y Fv(Z)2002 3724 y Fu(T)2052 3705 y Ft(N)2104 3724 y Fs(\002)p Fl(\()p Fu(R)2234 3705 y Ft(N)2292 3724 y Fs(n)p Fq(B)2380 3735 y Ft(R)2431 3724 y Fl(\))2479 3634 y Fw(\022)2524 3649 y Fq(\017)2557 3634 y FE(\()p Fw(x)p FE(\))p Fw(e)2733 3593 y Fs(\000)2798 3557 y Ft(c)2826 3572 y Fr(2)2861 3557 y Ft(h)p Fm(j)p Ft(v)r Fm(j)p 2799 3578 170 3 v 2870 3619 a Ft(\017)2999 3634 y Fw(dx)17 b(dv)-2710 b FE(\(137\))1445 3917 y Fx(\024)84 b Fw(c)1648 3932 y Fl(3)1687 3917 y Fw(\017)1726 3875 y Fq(N)1794 3917 y Fw(e)1839 3875 y Fs(\000)1904 3839 y Ft(c)1932 3854 y Fr(1)p 1904 3860 63 3 v 1922 3902 a Ft(\017)1997 3781 y Fv(Z)2053 4007 y Fu(T)2103 3988 y Ft(N)2175 3917 y Fw(\022)2220 3932 y Fq(\017)2253 3917 y FE(\()p Fw(x)p FE(\))17 b Fw(dx)1446 4145 y FE(=)84 b Fw(c)1648 4160 y Fl(3)1687 4145 y Fw(\017)1726 4104 y Fq(N)1794 4145 y Fw(e)1839 4104 y Fs(\000)1904 4068 y Ft(c)1932 4083 y Fr(1)p 1904 4089 V 1922 4130 a Ft(\017)1997 4145 y Fw(;)456 4343 y FE(pro)m(vided)29 b(that)g Fw(R)g Fx(\025)f FE(4)p Fw(c)1360 4358 y Fl(1)1399 4343 y Fw(=)p FE(\()p Fw(c)1528 4358 y Fl(2)1567 4343 y Fw(h)p FE(\),)i(where)h(the)e Fw(c)2203 4358 y Fq(i)2232 4343 y FE('s)g(are)g(suitable)g(p)s(ositiv)m (e)f(quan-)456 4470 y(tities)j(indep)s(enden)m(t)j(of)e Fw(\017)p FE(.)555 4598 y(Also,)40 b(since)g Fw(\026)1115 4613 y Fq(\017)1186 4598 y FE(is)e(a)h(probabilit)m(y)e(measure,)k(w)m (e)f(ha)m(v)m(e)g(that)e Fw(\026)2955 4613 y Fq(\017)3026 4598 y Fw(*)g(\026)3221 4613 y Fl(0)3260 4598 y FE(,)j(for)456 4726 y(some)34 b(measure)h Fw(\026)1143 4741 y Fl(0)1217 4726 y FE(on)f Fn(T)1417 4690 y Fq(N)1511 4726 y Fx(\002)24 b Fn(R)1678 4690 y Fq(N)1786 4726 y FE(\(see,)36 b(e.g.,)g(page)e(55)g (of)h([EG92)o(]\).)50 b(W)-8 b(e)35 b(no)m(w)456 4854 y(sho)m(w)25 b(that)f Fw(\026)951 4869 y Fl(0)1014 4854 y FE(is)g(a)g(probabilit)m(y)e(measure,)27 b(and)d(that)g(it)f(has)i (compact)f(supp)s(ort.)456 4982 y(It)32 b(is)g(standard)h(\(see,)h (e.g.,)f(Theorem)g(1-\(ii\))d(on)i(page)h(54)f(of)g([EG92]\))g(that) 1063 5179 y Fw(\026)1122 5194 y Fl(0)1161 5179 y FE(\()p Fn(T)1262 5138 y Fq(N)1355 5179 y Fx(\002)23 b Fn(R)1521 5138 y Fq(N)1594 5179 y FE(\))k Fx(\024)i FE(lim)17 b(inf)1832 5239 y Fq(\017)p Fs(!)p Fl(0)2051 5179 y Fw(\026)2110 5194 y Fq(\017)2143 5179 y FE(\()p Fn(T)2244 5138 y Fq(N)2336 5179 y Fx(\002)23 b Fn(R)2502 5138 y Fq(N)2575 5179 y FE(\))28 b(=)f(1)17 b Fw(:)p eop %%Page: 57 57 57 56 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)420 b(57)456 450 y FE(On)32 b(the)h(other)g(hand)g(\(see)g(again)f(Theorem)g(1-\(ii\))e (on)j(page)f(54)g(of)h([EG92)o(]\),)500 665 y Fw(\026)559 680 y Fl(0)598 665 y FE(\()p Fn(T)699 623 y Fq(N)792 665 y Fx(\002)22 b Fw(B)965 680 y Fq(R)1023 665 y FE(\))28 b Fx(\025)g FE(lim)17 b(sup)1276 744 y Fq(\017)p Fs(!)p Fl(0)1509 665 y Fw(\026)1568 680 y Fq(\017)1601 665 y FE(\()p Fn(T)1702 623 y Fq(N)1795 665 y Fx(\002)22 b Fw(B)1968 680 y Fq(R)2026 665 y FE(\))28 b Fx(\025)g FE(lim)17 b(sup)2279 744 y Fq(\017)p Fs(!)p Fl(0)2512 554 y Fv(\020)2572 665 y FE(1)22 b Fx(\000)g Fw(c)2784 680 y Fl(3)2824 665 y Fw(\017)2863 623 y Fq(N)2930 665 y Fw(e)2975 623 y Fs(\000)3040 587 y Ft(c)3068 602 y Fr(1)p 3040 608 63 3 v 3059 649 a Ft(\017)3117 554 y Fv(\021)3204 665 y FE(=)28 b(1)17 b Fw(;)456 903 y FE(pro)m(vided)39 b(that)g Fw(R)h FE(is)e(large)g(enough,)k(thanks)e(to)e(\(137\).)62 b(Therefore,)42 b Fw(\026)3213 918 y Fl(0)3291 903 y FE(is)d(a)456 1031 y(probabilit)m(y)30 b(measure)j(with)f(compact)g (supp)s(ort.)555 1159 y(Also,)g Fw(\026)861 1174 y Fl(0)933 1159 y FE(satis\014es)h(the)g(constrain)m(t)g(in)f(\(88\),)g(since)h Fw(\026)2564 1174 y Fq(\017)2628 1159 y FE(do)s(es.)555 1287 y(W)-8 b(e)33 b(no)m(w)g(pro)m(v)m(e)h(that)1037 1481 y(if)d(\()p Fw(x;)17 b(v)t FE(\))32 b(is)g(a)h(p)s(oin)m(t)e(in)h (the)h(supp)s(ort)g(of)f Fw(\026)2631 1496 y Fl(0)2670 1481 y FE(,)1129 1641 y(then)h Fw(v)j FE(is)c(a)h(minimizer)c(of)j(the) h(function)-2242 b(\(138\))1157 1801 y Fw(v)48 b Fx(7!)c Fw(hL)p FE(\()p Fw(x;)17 b(v)t FE(\))23 b(+)f Fw(\036)1923 1816 y Fl(0)1962 1801 y FE(\()p Fw(x)g FE(+)g Fw(hv)t FE(\))g Fx(\000)h Fw(\036)2500 1816 y Fl(0)2539 1801 y FE(\()p Fw(x)p FE(\))p Fw(:)456 1996 y FE(T)-8 b(o)26 b(pro)m(v)m(e)i(this,)g(tak)m(e)g(a)e(p)s(oin)m(t)g(\()p Fw(x)1686 2011 y Fl(1)1726 1996 y Fw(;)17 b(v)1817 2011 y Fl(1)1856 1996 y FE(\))28 b Fx(2)g Fn(T)2079 1959 y Fq(N)2160 1996 y Fx(\002)10 b Fn(R)2313 1959 y Fq(N)2413 1996 y FE(whic)m(h)27 b(is)g(not)f(a)h(minimizer)456 2123 y(of)g(the)h(ab)s(o)m(v)m(e)h(function.)41 b(Cho)s(ose)29 b(a)e(small)f(neigh)m(b)s(orho)s(o)s(d)g Fw(V)2741 2138 y Fl(1)2808 2123 y FE(of)i Fw(x)2970 2138 y Fl(1)3037 2123 y FE(in)f Fn(T)3209 2087 y Fq(n)3287 2123 y FE(and)456 2251 y(and)44 b(a)g(small)e(ball)h Fw(B)1293 2266 y Fl(1)1380 2251 y Fx(\032)48 b Fn(R)1571 2215 y Fq(n)1668 2251 y FE(cen)m(tered)f(in)c Fw(v)2240 2266 y Fl(1)2280 2251 y FE(.)79 b(Then,)48 b(b)m(y)d(\(127\),)i(if)c(these)456 2379 y(neigh)m(b)s(orho)s(o)s(d)f(are)i(small)d(enough,)47 b(there)e(exists)f(a)g(suitably)f(small)e Fw(\016)51 b(>)c FE(0)456 2507 y(suc)m(h)34 b(that)456 2701 y(\(139\))392 b Fw(hL)p FE(\()p Fw(x;)17 b(v)t FE(\))22 b(+)g Fw(\036)1597 2716 y Fl(0)1636 2701 y FE(\()p Fw(x)h FE(+)f Fw(hv)t FE(\))g Fx(\000)h Fw(\036)2175 2716 y Fl(0)2214 2701 y FE(\()p Fw(x)p FE(\))28 b Fx(\025)g Fw(h)p 2534 2621 89 4 v(H)2623 2716 y Fl(0)2684 2701 y FE(+)22 b Fw(\016)456 2896 y FE(for)34 b(an)m(y)h(\()p Fw(x;)17 b(v)t FE(\))32 b Fx(2)g Fw(V)1206 2911 y Fl(1)1269 2896 y Fx(\002)24 b Fw(B)1444 2911 y Fl(1)1484 2896 y FE(.)50 b(No)m(w)35 b(c)m(ho)s(ose)h(a)f(p)s(oin)m(t)f Fw(v)2485 2911 y Fl(0)2559 2896 y FE(so)h(to)g(minimizing)30 b(the)456 3024 y(function)1112 3174 y Fw(v)48 b Fx(7!)c Fw(hL)p FE(\()p Fw(x)1566 3189 y Fl(1)1607 3174 y Fw(;)17 b(v)t FE(\))k(+)h Fw(\036)1917 3189 y Fl(0)1956 3174 y FE(\()p Fw(x)2049 3189 y Fl(1)2111 3174 y FE(+)g Fw(hv)t FE(\))g Fx(\000)h Fw(\036)2534 3189 y Fl(0)2573 3174 y FE(\()p Fw(x)2666 3189 y Fl(1)2706 3174 y FE(\))17 b Fw(:)456 3346 y FE(Thanks)30 b(to)f(\(127\),)g(w)m(e) h(ma)m(y)f(no)m(w)h(tak)m(e)g(a)f(small)e(ball)g Fw(B)2543 3361 y Fl(0)2610 3346 y Fx(\032)h Fn(R)2781 3310 y Fq(n)2863 3346 y FE(cen)m(tered)j(in)e Fw(v)3405 3361 y Fl(0)456 3474 y FE(and)j(a)h(small)d(neigh)m(b)s(orho)s(o)s(d)h Fw(V)1651 3489 y Fl(0)1723 3474 y FE(of)h Fw(x)1889 3489 y Fl(1)1929 3474 y FE(,)h(in)e(suc)m(h)j(a)f(w)m(a)m(y)h(that)456 3704 y(\(140\))381 b Fw(hL)p FE(\()p Fw(x;)17 b(v)t FE(\))22 b(+)g Fw(\036)1586 3719 y Fl(0)1626 3704 y FE(\()p Fw(x)g FE(+)g Fw(hv)t FE(\))g Fx(\000)h Fw(\036)2164 3719 y Fl(0)2203 3704 y FE(\()p Fw(x)p FE(\))28 b Fx(\024)g Fw(h)p 2523 3624 V(H)2612 3719 y Fl(0)2673 3704 y FE(+)2782 3637 y Fw(\016)p 2781 3682 49 4 v 2781 3773 a FE(4)456 3922 y(for)d(an)m(y)i(\()p Fw(x;)17 b(v)t FE(\))27 b Fx(2)i Fw(V)1181 3937 y Fl(0)1229 3922 y Fx(\002)9 b Fw(B)1389 3937 y Fl(0)1429 3922 y FE(.)41 b(W)-8 b(e)27 b(no)m(w)g(consider)f(the)h(neigh)m(b)s(orho)s(o)s(d)e(of)g Fw(x)3156 3937 y Fl(1)3222 3922 y FE(giv)m(en)456 4050 y(b)m(y)1641 4200 y Fw(V)66 b FE(:=)45 b Fw(V)1969 4215 y Fl(0)2030 4200 y Fx(\\)23 b Fw(V)2176 4215 y Fl(1)2231 4200 y Fw(:)456 4372 y FE(By)36 b(the)g(uniform)d(con)m(v)m(ergence)38 b(of)d Fw(\036)1865 4387 y Fq(\017)1898 4372 y FE(,)g(\(139\))g(and)h (\(140\),)f(w)m(e)i(ha)m(v)m(e)g(that,)f(if)e Fw(\017)456 4500 y FE(is)e(small)e(enough,)1032 4726 y Fw(hL)p FE(\()p Fw(x;)17 b(v)t FE(\))23 b(+)f Fw(\036)1559 4741 y Fq(\017)1591 4726 y FE(\()p Fw(x)h FE(+)f Fw(hv)t FE(\))g Fx(\000)g Fw(\036)2129 4741 y Fq(\017)2162 4726 y FE(\()p Fw(x)p FE(\))28 b Fx(\025)g Fw(h)p 2482 4646 89 4 v(H)2571 4741 y Fl(0)2632 4726 y FE(+)2740 4659 y(3)p 2740 4703 49 4 v 2740 4794 a(4)2799 4726 y Fw(\016)n(;)456 4944 y FE(for)k(an)m(y)h(\()p Fw(x;)17 b(v)t FE(\))27 b Fx(2)h Fw(V)44 b Fx(\002)23 b Fw(B)1411 4959 y Fl(1)1483 4944 y FE(and)1053 5180 y Fw(hL)p FE(\()p Fw(x;)17 b(v)t FE(\))23 b(+)f Fw(\036)1580 5195 y Fq(\017)1612 5180 y FE(\()p Fw(x)h FE(+)f Fw(hv)t FE(\))g Fx(\000)g Fw(\036)2150 5195 y Fq(\017)2183 5180 y FE(\()p Fw(x)p FE(\))28 b Fx(\024)g Fw(h)p 2503 5100 89 4 v(H)2591 5195 y Fl(0)2653 5180 y FE(+)2762 5113 y Fw(\016)p 2761 5158 49 4 v 2761 5249 a FE(2)2820 5180 y Fw(:)p eop %%Page: 58 58 58 57 bop 456 251 a FC(58)666 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)456 450 y FE(for)32 b(an)m(y)h(\()p Fw(x;)17 b(v)t FE(\))27 b Fx(2)h Fw(V)44 b Fx(\002)23 b Fw(B)1411 465 y Fl(0)1450 450 y FE(.)44 b(Thence,)472 677 y Fw(\026)531 692 y Fq(\017)563 677 y FE(\()p Fw(V)g Fx(\002)23 b Fw(B)876 692 y Fl(1)915 677 y FE(\))28 b(=)1084 541 y Fv(Z)1140 767 y Fq(V)16 b Fs(\002)p Fq(B)1305 776 y Fr(1)1360 677 y Fw(\022)1405 692 y Fq(\017)1438 677 y FE(\()p Fw(x)p FE(\))h Fw(e)1631 636 y Fs(\000)1696 602 y Ft(hL)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft(\036)1993 610 y(\017)2023 602 y Fr(\()p Ft(x)p Fr(+)p Ft(v)r Fr(\))p Fm(\000)p Ft(\036)2271 610 y(\017)2300 602 y Fr(\(0\))p Fm(\000)p Ft(\025)2463 610 y(\017)p 1696 621 798 3 v 2082 662 a(\017)2535 677 y Fx(\024)28 b Fw(\022)2685 692 y Fq(\017)2718 677 y FE(\()p Fw(V)22 b FE(\))p Fw(e)2918 636 y Fs(\000)2983 591 y Ft(h)p Fr(\()p 3043 550 54 3 v Ft(H)3097 606 y Fr(0)3131 591 y Fm(\000)p 3179 550 V Ft(H)3233 599 y(\017)3263 591 y Fr(\)+)3343 572 y(3)p Ft(\016)p 3343 580 61 3 v 3358 615 a Fr(4)p 2983 621 431 3 v 3185 662 a Ft(\017)456 907 y FE(and)487 1109 y Fw(\026)546 1124 y Fq(\017)578 1109 y FE(\()p Fw(V)44 b Fx(\002)23 b Fw(B)891 1124 y Fl(0)930 1109 y FE(\))28 b(=)1099 974 y Fv(Z)1155 1199 y Fq(V)15 b Fs(\002)p Fq(B)1319 1208 y Fr(0)1375 1109 y Fw(\022)1420 1124 y Fq(\017)1453 1109 y FE(\()p Fw(x)p FE(\))i Fw(e)1646 1068 y Fs(\000)1711 1035 y Ft(hL)p Fr(\()p Ft(x;v)r Fr(\)+)p Ft(\036)2008 1043 y(\017)2038 1035 y Fr(\()p Ft(x)p Fr(+)p Ft(v)r Fr(\))p Fm(\000)p Ft(\036)2286 1043 y(\017)2315 1035 y Fr(\(0\))p Fm(\000)p Ft(\025)2478 1043 y(\017)p 1711 1053 798 3 v 2097 1094 a(\017)2550 1109 y Fx(\025)28 b Fw(\022)2700 1124 y Fq(\017)2733 1109 y FE(\()p Fw(V)22 b FE(\))p Fw(e)2933 1068 y Fs(\000)2998 1024 y Ft(h)p Fr(\()p 3058 982 54 3 v Ft(H)3112 1039 y Fr(0)3146 1024 y Fm(\000)p 3194 982 V Ft(H)3248 1032 y(\017)3278 1024 y Fr(\)+)3359 1004 y Ft(\016)p 3358 1012 31 3 v 3358 1047 a Fr(2)p 2998 1053 401 3 v 3185 1094 a Ft(\017)456 1340 y FE(Th)m(us,)1437 1496 y Fw(\026)1496 1511 y Fq(\017)1529 1496 y FE(\()p Fw(V)43 b Fx(\002)23 b Fw(B)1841 1511 y Fl(1)1880 1496 y FE(\))p 1437 1541 482 4 v 1437 1632 a Fw(\026)1496 1647 y Fq(\017)1529 1632 y FE(\()p Fw(V)43 b Fx(\002)23 b Fw(B)1841 1647 y Fl(0)1880 1632 y FE(\))1956 1564 y Fx(\024)28 b Fw(e)2106 1522 y Fs(\000)2185 1495 y Ft(\016)p 2171 1507 57 3 v 2171 1548 a Fr(4)p Ft(\017)2270 1564 y Fx(!)f FE(0)p Fw(;)456 1794 y FE(as)44 b Fw(\017)49 b Fx(!)e FE(0.)79 b(In)45 b(particular,)h Fw(\026)1662 1809 y Fq(\017)1694 1794 y FE(\()p Fw(V)52 b Fx(\002)31 b Fw(B)2023 1809 y Fl(1)2063 1794 y FE(\))47 b Fx(!)h FE(0)c(as)h Fw(\017)j Fx(!)g FE(0.)79 b(This)45 b(implies)456 1922 y(that)34 b Fw(\026)728 1937 y Fl(0)767 1922 y FE(\()p Fw(V)45 b Fx(\002)24 b Fw(B)1082 1937 y Fl(1)1121 1922 y FE(\))31 b(=)g(0)j(\(see,)i(e.g.,)f(Theorem)g(1-\(ii\))d(on)i(page)h(54)f(of)g ([EG92)o(]\).)456 2049 y(Therefore,)f(the)g(pro)s(of)f(of)g(\(138\))g (is)g(complete.)555 2177 y(It)h(follo)m(ws)e(from)g(\(138\))h(and)h (\(127\))f(that)1198 2359 y Fw(L)p FE(\()p Fw(x;)17 b(v)t FE(\))22 b(+)g Fw(\036)1668 2374 y Fl(0)1707 2359 y FE(\()p Fw(x)h FE(+)f Fw(hv)t FE(\))g Fx(\000)g Fw(\036)2245 2374 y Fl(0)2285 2359 y FE(\()p Fw(x)p FE(\))28 b(=)p 2547 2279 89 4 v 27 w Fw(H)2636 2374 y Fl(0)2675 2359 y Fw(;)456 2541 y FE(for)40 b(an)m(y)i(\()p Fw(x;)17 b(v)t FE(\))41 b(in)g(the)h(supp)s(ort)f(of)g Fw(\026)1921 2556 y Fl(0)1960 2541 y FE(.)70 b(Th)m(us,)45 b(in)m(tegrating)40 b(and)h(using)g(the)456 2669 y(constrain)m(t)32 b(in)g(\(88\),)g(w)m(e) i(obtain)d(that)903 2756 y Fv(Z)1019 2891 y Fw(Ld\026)1195 2906 y Fl(0)1262 2891 y FE(=)1366 2756 y Fv(Z)1482 2891 y FE([)p Fw(L)23 b FE(+)f Fw(\036)1754 2906 y Fl(0)1793 2891 y FE(\()p Fw(x)g FE(+)g Fw(hv)t FE(\))g Fx(\000)h Fw(\036)2331 2906 y Fl(0)2370 2891 y FE(\()p Fw(x)p FE(\)])17 b Fw(d\026)2655 2906 y Fl(0)2722 2891 y FE(=)p 2825 2811 V 27 w Fw(H)2914 2906 y Fl(0)2970 2891 y Fw(:)456 3127 y FE(This)35 b(completes)g(the)h(pro)s(of)f(of)g(the)h(desired)g (result,)g(b)m(y)g(the)g(de\014niton)f(of)p 3316 3047 V 35 w Fw(H)3405 3142 y Fl(0)456 3255 y FE(giv)m(en)d(on)h(page)f(52.) 2169 b Fj(\003)679 3488 y Fz(Appendix)37 b FE(A.)49 b Fz(A)38 b(Bana)n(ch-Ca)n(ccioppoli-type)e(Theorem)555 3674 y FE(W)-8 b(e)34 b(state)g(and)g(pro)m(v)m(e)h(a)e(v)-5 b(ariation)31 b(of)i(the)h(standard)g(Banac)m(h-Cacciopp)s(oli)456 3802 y(Theorem)k(ab)s(out)g(the)h(existence)i(and)d(uniqueness)j(of)d (\014xed)h(p)s(oin)m(ts)g(of)f(strict)456 3930 y(con)m(tractions.)48 b(The)35 b(result)f(w)m(e)h(presen)m(t)h(is)e(giv)m(en)g(in)f(a)h(form) f(whic)m(h)h(is)g(con)m(v)m(e-)456 4058 y(nien)m(t)e(for)g(the)h(pro)s (of)f(of)g(Theorem)h(27.)456 4210 y FI(Theorem)46 b(41.)g FD(L)-5 b(et)43 b Fw(S)49 b FD(b)-5 b(e)42 b(a)h(close)-5 b(d)41 b(subset)i(of)f(a)h(Banach)e(sp)-5 b(ac)g(e,)44 b(endowe)-5 b(d)456 4338 y(with)34 b(a)h(norm)f Fx(k)22 b(\001)g(k)p FD(.)45 b(Assume)34 b(that)i Fw(g)31 b FE(:)c Fw(S)34 b Fx(!)27 b Fw(S)41 b FD(is)34 b(so)h(that)456 4520 y FE(\(141\))362 b Fx(k)p Fw(g)t FE(\()p Fw(x)p FE(\))21 b Fx(\000)i Fw(g)t FE(\()p Fw(y)t FE(\))p Fx(k)42 b(\024)j FE(\(1)22 b Fx(\000)g Fw(\013)q Fx(k)p Fw(x)g Fx(\000)h Fw(y)t Fx(k)2387 4479 y Fq(\014)2433 4520 y FE(\))17 b Fx(k)p Fw(x)22 b Fx(\000)h Fw(y)t Fx(k)17 b Fw(;)456 4701 y FD(for)42 b(any)i Fw(x;)60 b(y)46 b Fx(2)d Fw(S)49 b FD(and)42 b(some)h(given)f(c)-5 b(onstants)42 b Fw(\013)q(;)60 b(\014)48 b(>)43 b FE(0)p FD(.)69 b(Then,)44 b(ther)-5 b(e)456 4829 y(exists)43 b(a)g(unique)h Fw(x)1202 4793 y Fq(?)1285 4829 y Fx(2)g Fw(S)50 b FD(so)43 b(that)h Fw(g)t FE(\()p Fw(x)1990 4793 y Fq(?)2029 4829 y FE(\))g(=)f Fw(x)2285 4793 y Fq(?)2325 4829 y FD(.)71 b(F)-7 b(urthermor)i(e,)45 b(given)d(any)456 4957 y Fw(x)511 4972 y Fl(0)578 4957 y Fx(2)28 b Fw(S)6 b FD(,)35 b(we)f(have)g(that)456 5139 y FE(\(142\))874 b Fw(x)1608 5098 y Fq(?)1676 5139 y FE(=)79 b(lim)1779 5199 y Fq(n)p Fs(!)p Fl(+)p Fs(1)2035 5139 y Fw(g)2086 5098 y Fq(n)2132 5139 y FE(\()p Fw(x)2225 5154 y Fl(0)2265 5139 y FE(\))17 b Fw(:)p eop %%Page: 59 59 59 58 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)420 b(59)456 450 y FD(Pr)-5 b(o)g(of.)41 b FE(Let)36 b Fw(x)987 465 y Fl(0)1061 450 y Fx(2)f Fw(S)6 b FE(.)54 b(F)-8 b(or)36 b(an)m(y)h Fw(n)d Fx(2)h Fn(N)9 b FE(,)43 b(w)m(e)38 b(de\014ne)f Fw(x)2493 465 y Fq(n)2575 450 y FE(:=)d Fw(g)2763 414 y Fq(n)2809 450 y FE(\()p Fw(x)2902 465 y Fl(0)2942 450 y FE(\))i(and)h Fw(d)3261 465 y Fq(n)3341 450 y FE(:=)456 578 y Fx(k)p Fw(x)561 593 y Fq(n)p Fl(+1)711 578 y Fx(\000)13 b Fw(x)856 593 y Fq(n)905 578 y Fx(k)p FE(.)42 b(Note)28 b(that)g Fw(x)1517 593 y Fq(n)1592 578 y Fx(2)g Fw(S)35 b FE(for)27 b(an)m(y)i Fw(n)f Fx(2)g Fn(N)9 b FE(,)36 b(b)s(ecause)29 b Fw(g)t FE(\()p Fw(S)6 b FE(\))27 b Fx(\022)h Fw(S)6 b FE(.)42 b(Also,)456 706 y(b)m(y)33 b(\(141\),)456 890 y(\(143\))712 b Fx(k)p Fw(g)t FE(\()p Fw(x)p FE(\))22 b Fx(\000)g Fw(g)t FE(\()p Fw(y)t FE(\))p Fx(k)43 b(\024)h(k)p Fw(x)23 b Fx(\000)f Fw(y)t Fx(k)17 b Fw(;)456 1074 y FE(for)32 b(an)m(y)h Fw(x;)45 b(y)31 b Fx(2)d Fw(S)38 b FE(and)33 b(therefore)1395 1259 y Fw(d)1446 1274 y Fq(n)1577 1259 y FE(=)83 b Fx(k)p Fw(g)t FE(\()p Fw(x)1930 1274 y Fq(n)1977 1259 y FE(\))22 b Fx(\000)g Fw(g)t FE(\()p Fw(x)2280 1274 y Fq(n)p Fs(\000)p Fl(1)2417 1259 y FE(\))p Fx(k)1576 1418 y(\024)83 b(k)p Fw(x)1841 1433 y Fq(n)1911 1418 y Fx(\000)22 b Fw(x)2065 1433 y Fq(n)p Fs(\000)p Fl(1)2203 1418 y Fx(k)1577 1578 y FE(=)83 b Fw(d)1787 1593 y Fq(n)p Fs(\000)p Fl(1)1941 1578 y Fw(;)456 1763 y FE(for)41 b(an)m(y)i Fw(n)g Fx(\025)i FE(1,)f(that)d(is)h Fw(d)1530 1778 y Fq(n)1618 1763 y FE(is)g(a)g(non-increasing)e(sequence.)75 b(Th)m(us,)46 b(there)456 1890 y(exists)33 b Fw(d)27 b Fx(\025)h FE(0)33 b(so)g(that)1661 2030 y Fw(d)27 b FE(=)79 b(lim)1843 2090 y Fq(n)p Fs(!)p Fl(+)p Fs(1)2098 2030 y Fw(d)2149 2045 y Fq(n)2212 2030 y Fw(:)456 2218 y FE(W)-8 b(e)28 b(no)m(w)g(sho)m(w)h(that)f Fw(d)f FE(=)h(0.)42 b(Indeed,)30 b(let)d(us)i(assume,)g(b)m(y)g(con)m(tradiction,)f(that)456 2402 y(\(144\))1134 b Fw(d)27 b(>)h FE(0)17 b Fw(:)456 2587 y FE(Then,)33 b(using)g(\(141\),)e(one)i(has)g(that)981 2771 y Fw(d)83 b FE(=)136 b(lim)1275 2831 y Fq(n)p Fs(!)p Fl(+)p Fs(1)1530 2771 y Fx(k)p Fw(x)1635 2786 y Fq(n)p Fl(+1)1795 2771 y Fx(\000)22 b Fw(x)1949 2786 y Fq(n)1997 2771 y Fx(k)1115 2963 y FE(=)136 b(lim)1275 3022 y Fq(n)p Fs(!)p Fl(+)p Fs(1)1530 2963 y Fx(k)p Fw(g)t FE(\()p Fw(x)1724 2978 y Fq(n)1771 2963 y FE(\))22 b Fx(\000)g Fw(g)t FE(\()p Fw(x)2074 2978 y Fq(n)p Fs(\000)p Fl(1)2211 2963 y FE(\))p Fx(k)1114 3167 y(\024)136 b FE(lim)1275 3227 y Fq(n)p Fs(!)p Fl(+)p Fs(1)1514 3167 y FE(\(1)22 b Fx(\000)g Fw(\013)q Fx(k)p Fw(x)1890 3182 y Fq(n)1959 3167 y Fx(\000)h Fw(x)2114 3182 y Fq(n)p Fs(\000)p Fl(1)2251 3167 y Fx(k)2301 3126 y Fq(\014)2348 3167 y FE(\))17 b Fx(k)p Fw(x)2508 3182 y Fq(n)2577 3167 y Fx(\000)23 b Fw(x)2732 3182 y Fq(n)p Fs(\000)p Fl(1)2870 3167 y Fx(k)1115 3377 y FE(=)136 b(lim)1275 3437 y Fq(n)p Fs(!)p Fl(+)p Fs(1)1514 3377 y FE(\(1)22 b Fx(\000)g Fw(\013)q(d)1836 3330 y Fq(\014)1836 3402 y(n)p Fs(\000)p Fl(1)1973 3377 y FE(\))p Fw(d)2062 3392 y Fq(n)p Fs(\000)p Fl(1)1115 3581 y FE(=)84 b(\(1)22 b Fx(\000)g Fw(\013)q(d)1597 3540 y Fq(\014)1644 3581 y FE(\))p Fw(d)17 b(;)456 3766 y FE(that)32 b(is,)g(b)m(y)i(\(144\),)1655 3905 y(1)44 b Fx(\024)h FE(1)22 b Fx(\000)h Fw(\013)q(d)2155 3864 y Fq(\014)2218 3905 y Fw(;)456 4067 y FE(whic)m(h)44 b(giv)m(es)g Fw(d)i Fx(\024)i FE(0,)e(con)m(tradicting)c(\(144\).)77 b(This)43 b(sho)m(ws)j(that)d Fw(d)k FE(=)f(0)e(or,)456 4195 y(equiv)-5 b(alen)m(tly)32 b(that)456 4379 y(\(145\))790 b(lim)1418 4439 y Fq(n)p Fs(!)p Fl(+)p Fs(1)1673 4379 y Fx(k)p Fw(g)t FE(\()p Fw(x)1867 4394 y Fq(n)1914 4379 y FE(\))22 b Fx(\000)g Fw(x)2128 4394 y Fq(n)2176 4379 y Fx(k)44 b FE(=)g(0)17 b Fw(:)456 4590 y FE(W)-8 b(e)33 b(sho)m(w)g(that)g Fw(x)1132 4605 y Fq(n)1212 4590 y FE(is)f(a)g(Cauc)m(h)m(y)i(sequence)i(in)c Fw(S)6 b FE(.)43 b(Indeed,)34 b(\014x)f Fw(\017)28 b(>)g FE(0)k(and)h(let)456 4823 y(\(146\))1006 b Fw(\021)32 b FE(:=)1905 4756 y Fw(\013)17 b(\017)2023 4720 y Fq(\014)s Fl(+1)p 1905 4801 257 4 v 2009 4892 a FE(2)2188 4823 y Fw(:)456 5031 y FE(Thanks)34 b(to)e(\(145\),)g(w)m(e)h(ma)m(y)g(assumne)g(that)1582 5216 y Fx(k)p Fw(g)t FE(\()p Fw(x)1776 5231 y Fq(n)1822 5216 y FE(\))22 b Fx(\000)h Fw(x)2037 5231 y Fq(n)2084 5216 y Fx(k)28 b(\024)g Fw(\021)p eop %%Page: 60 60 60 59 bop 456 251 a FC(60)666 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)456 450 y FE(pro)m(vided)30 b Fw(n)e Fx(\025)g Fw(n)1103 465 y Fl(0)1142 450 y FE(,)j(for)e(a)g(suitable)g Fw(n)1843 465 y Fl(0)1910 450 y FE(=)e Fw(n)2071 465 y Fl(0)2111 450 y FE(\()p Fw(\017)p FE(\).)43 b(Com)m(bining)28 b(this)h(with)g(\(141\),)456 578 y(w)m(e)k(gather)g(that)1031 770 y Fx(k)p Fw(x)1136 785 y Fq(n)1205 770 y Fx(\000)23 b Fw(x)1360 785 y Fq(m)1427 770 y Fx(k)870 930 y(\024)84 b(k)p Fw(x)1136 945 y Fq(n)1205 930 y Fx(\000)23 b Fw(g)t FE(\()p Fw(x)1449 945 y Fq(n)1495 930 y FE(\))p Fx(k)f FE(+)g Fx(k)p Fw(g)t FE(\()p Fw(x)1897 945 y Fq(n)1944 930 y FE(\))g Fx(\000)g Fw(g)t FE(\()p Fw(x)2247 945 y Fq(m)2313 930 y FE(\))p Fx(k)g FE(+)g Fx(k)p Fw(g)t FE(\()p Fw(x)2715 945 y Fq(m)2781 930 y FE(\))g Fx(\000)h Fw(x)2996 945 y Fq(m)3063 930 y Fx(k)870 1089 y(\024)84 b FE(2)p Fw(\021)25 b FE(+)d(\(1)g Fx(\000)h Fw(\013)q Fx(k)p Fw(x)1628 1104 y Fq(n)1697 1089 y Fx(\000)f Fw(x)1851 1104 y Fq(m)1918 1089 y Fx(k)1968 1048 y Fq(\014)2015 1089 y FE(\))17 b Fx(k)p Fw(x)2175 1104 y Fq(n)2244 1089 y Fx(\000)23 b Fw(x)2399 1104 y Fq(m)2466 1089 y Fx(k)871 1249 y FE(=)84 b(2)p Fw(\021)25 b Fx(\000)e Fw(\013)q Fx(k)p Fw(x)1421 1264 y Fq(n)1490 1249 y Fx(\000)g Fw(x)1645 1264 y Fq(m)1712 1249 y Fx(k)1762 1208 y Fq(\014)s Fl(+1)1921 1249 y FE(+)f Fx(k)p Fw(x)2124 1264 y Fq(n)2193 1249 y Fx(\000)h Fw(x)2348 1264 y Fq(m)2415 1249 y Fx(k)456 1441 y FE(pro)m(vided)33 b(that)f Fw(n;)45 b(m)28 b Fx(\025)g Fw(n)1474 1456 y Fl(0)1513 1441 y FE(.)44 b(That)33 b(is,)1425 1633 y(0)28 b Fx(\024)g FE(2)p Fw(\021)d Fx(\000)e Fw(\013)q Fx(k)p Fw(x)1997 1648 y Fq(n)2066 1633 y Fx(\000)g Fw(x)2221 1648 y Fq(m)2288 1633 y Fx(k)2338 1592 y Fq(\014)s Fl(+1)456 1825 y FE(and)32 b(so)1641 1972 y Fx(k)p Fw(x)1746 1987 y Fq(n)1815 1972 y Fx(\000)23 b Fw(x)1970 1987 y Fq(m)2037 1972 y Fx(k)k(\024)i Fw(\017)456 2141 y FE(due)42 b(to)f(\(146\).)70 b(This)42 b(sho)m(ws)h(that)e Fw(x)1893 2156 y Fq(n)1982 2141 y FE(is)g(a)h(Cauc)m(h)m(y)h(sequence.)74 b(Since)41 b Fw(S)48 b FE(is)456 2269 y(complete,)32 b(w)m(e)h(ha)m(v)m(e)h(that)e (there)i(exists)f Fw(x)2048 2233 y Fq(?)2116 2269 y Fx(2)28 b Fw(S)38 b FE(so)33 b(that)456 2461 y(\(147\))992 b(lim)1620 2521 y Fq(n)p Fs(!)p Fl(+)p Fs(1)1875 2461 y Fw(x)1930 2476 y Fq(n)2022 2461 y FE(=)44 b Fw(x)2197 2420 y Fq(?)2253 2461 y Fw(;)456 2685 y FE(pro)m(ving)32 b(\(142\).)43 b(Also,)32 b(com)m(bining)e(\(143\))i(and)h(\(147\),)1562 2876 y(lim)1510 2936 y Fq(n)p Fs(!)p Fl(+)p Fs(1)1765 2876 y Fw(g)t FE(\()p Fw(x)1909 2891 y Fq(n)1956 2876 y FE(\))28 b(=)f Fw(g)t FE(\()p Fw(x)2269 2835 y Fq(?)2308 2876 y FE(\))17 b Fw(:)456 3100 y FE(This,)32 b(together)h(with)f (\(147\))g(and)h(\(145\),)f(giv)m(es)h(that)1156 3292 y Fx(k)p Fw(g)t FE(\()p Fw(x)1350 3251 y Fq(?)1389 3292 y FE(\))22 b Fx(\000)h Fw(x)1604 3251 y Fq(?)1644 3292 y Fx(k)83 b FE(=)134 b(lim)1936 3352 y Fq(n)p Fs(!)p Fl(+)p Fs(1)2191 3292 y Fx(k)p Fw(g)t FE(\()p Fw(x)2385 3307 y Fq(n)2432 3292 y FE(\))22 b Fx(\000)g Fw(x)2646 3307 y Fq(n)2694 3292 y Fx(k)1777 3473 y FE(=)83 b(0)17 b Fw(;)456 3665 y FE(thence)34 b Fw(x)820 3629 y Fq(?)892 3665 y FE(is)e(the)h(desired)g(\014xed)h(p)s(oin)m(t.)555 3793 y(W)-8 b(e)37 b(no)m(w)g(sho)m(w)h(that)f(suc)m(h)h(\014xed)f(p)s (oin)m(t)f(is)g(unique.)56 b(If,)38 b(b)m(y)f(con)m(tradiction,)456 3921 y(there)42 b(existed)g Fw(y)1103 3884 y Fq(?)1184 3921 y Fx(2)h Fw(S)48 b FE(with)41 b Fx(k)p Fw(x)1737 3884 y Fq(?)1805 3921 y Fx(\000)28 b Fw(y)1962 3884 y Fq(?)2001 3921 y Fx(k)42 b(6)p FE(=)h(0)e(and)h Fw(g)t FE(\()p Fw(y)2642 3884 y Fq(?)2679 3921 y FE(\))h(=)g Fw(y)2931 3884 y Fq(?)2969 3921 y FE(,)h(w)m(e)e(w)m(ould)456 4048 y(deriv)m(e)33 b(from)e(\(141\))h(that)1457 4240 y Fx(k)p Fw(x)1562 4199 y Fq(?)1623 4240 y Fx(\000)23 b Fw(y)1775 4199 y Fq(?)1814 4240 y Fx(k)1297 4400 y FE(=)84 b Fx(k)p Fw(g)t FE(\()p Fw(x)1651 4359 y Fq(?)1690 4400 y FE(\))22 b Fx(\000)g Fw(g)t FE(\()p Fw(y)1990 4359 y Fq(?)2028 4400 y FE(\))p Fx(k)1296 4560 y(\024)84 b FE(\(1)21 b Fx(\000)i Fw(\013)q Fx(k)p Fw(x)1833 4519 y Fq(?)1895 4560 y Fx(\000)f Fw(y)2046 4519 y Fq(?)2085 4560 y Fx(k)2135 4519 y Fq(\014)2182 4560 y FE(\))17 b Fx(k)p Fw(x)2342 4519 y Fq(?)2403 4560 y Fx(\000)23 b Fw(y)2555 4519 y Fq(?)2593 4560 y Fx(k)17 b Fw(;)456 4752 y FE(and)32 b(th)m(us)i(that)1494 4899 y(1)27 b Fx(\024)h FE(1)22 b Fx(\000)h Fw(\013)q Fx(k)p Fw(x)2014 4858 y Fq(?)2075 4899 y Fx(\000)g Fw(y)2227 4858 y Fq(?)2266 4899 y Fx(k)2316 4858 y Fq(\014)2379 4899 y Fw(:)456 5068 y FE(This)43 b(w)m(ould)g(giv)m(e)g(that)g Fw(x)1472 5032 y Fq(?)1557 5068 y FE(=)j Fw(y)1731 5032 y Fq(?)1769 5068 y FE(,)g(con)m(tradicting)c(our)h(assumptions.)75 b(This)456 5196 y(sho)m(ws)34 b(the)f(uniqueness)h(of)e(the)h(\014xed)h (p)s(oin)m(t)e Fw(x)2220 5160 y Fq(?)2260 5196 y FE(.)1080 b Fj(\003)p eop %%Page: 61 61 61 60 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)420 b(61)723 450 y Fz(Appendix)37 b FE(B.)49 b Fz(Some)38 b(element)-7 b(ar)g(y)37 b(pr)n(oper)-7 b(ties)36 b(of)j(the)1698 578 y(semiconca)-9 b(ve)38 b(functions)555 764 y FE(The)e(purp)s(ose)g (of)f(this)g(section)g(is)g(to)g(recall)f(some)h(standard)h(prop)s (erties)f(of)456 892 y(the)g(semiconca)m(v)m(e)h(functions,)g(and)g(to) e(express)k(suc)m(h)e(prop)s(erties)f(in)g(a)g(frame-)456 1020 y(w)m(ork)e(con)m(v)m(enien)m(t)i(for)d(our)h(pro)s(ofs.)44 b(First)32 b(of)g(all,)f(w)m(e)j(remark)e(that)h Fw(\036)f FE(is)h(semi-)456 1147 y(conca)m(v)m(e)h(if)d(and)i(only)f(if)456 1370 y(\(148\))49 b Fw(\025\036)p FE(\()p Fw(x)p FE(\))8 b(+)g(\(1)g Fx(\000)g Fw(\025)p FE(\))17 b Fw(\036)p FE(\()p Fw(y)t FE(\))8 b Fx(\000)g Fw(\036)1695 1260 y Fv(\020)1753 1370 y Fw(\025x)g FE(+)g(\(1)g Fx(\000)g Fw(\025)p FE(\))p Fw(y)2284 1260 y Fv(\021)2388 1370 y Fx(\024)2519 1303 y FE(\006)2589 1318 y Fq(\036)2652 1303 y Fw(\025)17 b FE(\(1)22 b Fx(\000)g Fw(\025)p FE(\))p 2519 1348 510 4 v 2750 1439 a(2)3056 1370 y Fx(j)p Fw(x)8 b Fx(\000)g Fw(y)t Fx(j)3312 1329 y Fl(2)3367 1370 y Fw(;)456 1575 y FE(for)38 b(an)m(y)h Fw(\025)f Fx(2)h FE([0)p Fw(;)17 b FE(1].)61 b(Indeed,)41 b(\(148\))d(is)g(due)h(to)g (\(36\))f(and)h(the)g(con)m(tin)m(uit)m(y)g(of)456 1703 y Fw(\036)p FE(;)45 b(see,)f(e.g.,)f(Prop)s(osition)d(1.1.3)g(in)g ([CS04)q(].)2198 1664 y Fl(18)2337 1703 y FE(W)-8 b(e)42 b(also)e(recall)f(the)j(follo)m(w-)456 1831 y(ing)35 b(standard)j(result,)f(the)h(pro)s(of)e(of)g(whic)m(h)h(ma)m(y)g(b)s(e) g(found,)h(for)e(instance)h(in)456 1959 y(Theorem)32 b(2.1.7)h(and)f(Remark)g(2.1.8)g(of)g([CS04)q(]:)456 2111 y FI(Lemma)49 b(42.)e FD(A)e(b)-5 b(ounde)g(d)44 b(semic)-5 b(onc)g(ave)43 b(function)h Fw(\036)h FD(is)f(Lipschitz)g(c) -5 b(ontin-)456 2238 y(uous.)71 b(Mor)-5 b(e)g(over,)45 b(its)f(Lipschitz)f(c)-5 b(onstants)43 b(dep)-5 b(ends)43 b(only)g(on)h Fw(N)10 b FD(,)46 b FE(\006)3200 2253 y Fq(\036)3290 2238 y FD(and)456 2366 y Fx(k)p Fw(\036)p Fx(k)614 2385 y Fq(L)662 2366 y Fm(1)726 2385 y Fl(\()p Fu(R)801 2366 y Ft(N)859 2385 y Fl(\))891 2366 y FD(.)555 2518 y FE(Also,)30 b(b)m(y)h(mo)s(difying)c(the)j(pro)s(of)f(of)g (Theorem)h(2.1.7)g(in)f([CS04],)h(the)h(follo)m(w-)456 2646 y(ing)g(result)i(follo)m(ws:)456 2798 y FI(Lemma)24 b(43.)34 b FD(L)-5 b(et)25 b Fw(\036)g FD(b)-5 b(e)24 b(semic)-5 b(onc)g(ave)23 b(and)h Fn(Z)2158 2762 y Fq(N)2223 2798 y FD(-p)-5 b(erio)g(dic.)40 b(Then,)26 b Fw(\036)i Fx(2)g Fw(L)3181 2762 y Fs(1)3256 2798 y FE(\()p Fn(R)3360 2762 y Fq(N)3433 2798 y FE(\))456 2926 y FD(and)1594 3062 y FE(sup)1571 3151 y Fq(x)p Fs(2)p Fu(R)1706 3132 y Ft(N)1780 3062 y Fx(j)p Fw(\036)p FE(\()p Fw(x)p FE(\))22 b Fx(\000)h Fw(\036)p FE(\(0\))p Fx(j)456 3274 y FD(is)34 b(b)-5 b(ounde)g(d)34 b(by)h(a)g(c)-5 b(onstant)35 b(which)f(dep)-5 b(ends)33 b(only)i(on)f Fw(N)46 b FD(and)34 b FE(\006)2903 3289 y Fq(\036)2950 3274 y FD(.)456 3461 y(Pr)-5 b(o)g(of.)41 b FE(Let)33 b(us)g(de\014ne)1523 3597 y Fw(u)p FE(\()p Fw(x)p FE(\))27 b(=)h Fw(\036)p FE(\()p Fw(x)p FE(\))22 b Fx(\000)g Fw(\036)p FE(\(0\))17 b Fw(;)456 3756 y FE(so)34 b(that)f Fw(u)h FE(is)f Fn(Z)1047 3719 y Fq(N)1112 3756 y FE(-p)s(erio)s(dic)f(and)i(semiconca)m(v)m(e,)h(with)f(\006)2584 3771 y Fq(u)2659 3756 y FE(=)c(\006)2835 3771 y Fq(\036)2916 3756 y FE(b)m(y)k(\(36\),)g(and)456 3883 y Fw(u)p FE(\(0\))g(=)g(0.)56 b(Let)37 b(no)m(w)h Fw(x)1355 3898 y Fl(1)1431 3883 y FE(and)f Fw(x)1680 3898 y Fl(2)1757 3883 y FE(b)s(e)g(t)m(w)m(o)g (consecutiv)m(e)i(v)m(ertices)f(of)e(the)i(cub)s(e)456 4011 y([0)p Fw(;)17 b FE(1])652 3975 y Fq(N)718 4011 y FE(.)44 b(Then,)1412 4148 y Fw(u)p FE(\()p Fw(x)1561 4163 y Fl(1)1601 4148 y FE(\))27 b(=)h Fw(u)p FE(\()p Fw(x)1919 4163 y Fl(2)1958 4148 y FE(\))g(=)f Fw(u)p FE(\(0\))g(=)h(0)456 4306 y(b)m(y)33 b(the)g(p)s(erio)s(dicit)m(y)e(of) h Fw(u)g FE(and)1614 4487 y Fx(j)p Fw(x)1697 4502 y Fl(1)1759 4487 y Fx(\000)23 b Fw(x)1914 4502 y Fl(2)1954 4487 y Fx(j)k(\024)2114 4398 y(p)p 2197 4398 89 4 v 89 x Fw(N)456 4668 y FE(b)m(y)33 b(construction.)44 b(Th)m(us,)34 b(in)e(the)h(ligh)m (t)e(of)h(\(148\),)g(w)m(e)h(ha)m(v)m(e)h(that)698 4897 y Fw(u)754 4786 y Fv(\020)813 4897 y Fw(\025x)925 4912 y Fl(1)987 4897 y FE(+)22 b(\(1)g Fx(\000)h Fw(\025)p FE(\))p Fw(x)1444 4912 y Fl(2)1483 4786 y Fv(\021)1571 4897 y Fx(\025)28 b(\000)1763 4829 y FE(\006)1833 4844 y Fq(\036)1896 4829 y Fw(\025)17 b FE(\(1)22 b Fx(\000)g Fw(\025)p FE(\))p 1763 4874 510 4 v 1994 4965 a(2)2300 4897 y Fx(j)p Fw(x)2383 4912 y Fl(1)2444 4897 y Fx(\000)h Fw(x)2599 4912 y Fl(2)2639 4897 y Fx(j)2667 4856 y Fl(2)2734 4897 y Fx(\025)28 b(\000)2926 4829 y FE(\006)2996 4844 y Fq(\036)3060 4829 y Fw(N)p 2926 4874 222 4 v 3013 4965 a FE(8)3175 4897 y Fw(;)p 456 5007 499 4 v 555 5074 a Fl(18)626 5106 y FH(As)c(a)f(notation)h(remark,)f(w)n(e)h(note)f(that)i (what)f(w)n(e)f(call)h(here)f(\\semiconca)n(v)n(e)f(functions")456 5216 y(are)k(called)h(\\semiconca)n(v)n(e)f(functions)h(with)i(linear)d (mo)r(dulus")i(in)g([CS04)o(].)p eop %%Page: 62 62 62 61 bop 456 251 a FC(62)666 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)456 450 y FE(for)34 b(an)m(y)i Fw(\025)d Fx(2)f FE([0)p Fw(;)17 b FE(1].)52 b(That)35 b(is,)h(if)e Fw(x)i FE(b)s(elongs)e(to)h(an)m(y)h(of)f(the)h(one-dimensional)456 578 y(faces)d(of)f(the)h(cub)s(e)g([0)p Fw(;)17 b FE(1])1397 542 y Fq(N)1464 578 y FE(,)33 b(w)m(e)g(ha)m(v)m(e)h(that)1689 763 y Fw(u)p FE(\()p Fw(x)p FE(\))28 b Fx(\025)g(\000)p Fw(b)2127 778 y Fl(1)2184 763 y Fw(;)456 949 y FE(with)1688 1105 y Fw(b)1729 1120 y Fl(1)1796 1105 y FE(:=)1937 1038 y(\006)2007 1053 y Fq(\036)2070 1038 y Fw(N)p 1937 1082 222 4 v 2023 1173 a FE(8)2185 1105 y Fw(:)456 1286 y FE(This)f(pro)s(cedure)h(can)f(b)s(e)g(iterated.)41 b(Namely)-8 b(,)27 b(assume)h(that)e(for)h(an)m(y)g Fw(x)h FE(b)s(elong-)456 1414 y(ing)j(to)h(an)m(y)i(of)e(the)h(\()p Fw(n)22 b Fx(\000)h FE(1\)-dimensional)29 b(faces)k(of)f([0)p Fw(;)17 b FE(1])2618 1378 y Fq(N)2685 1414 y FE(,)33 b(w)m(e)g(ha)m(v)m(e)h (that)456 1599 y(\(149\))961 b Fw(u)p FE(\()p Fw(x)p FE(\))28 b Fx(\025)g(\000)p Fw(b)2078 1614 y Fq(n)p Fs(\000)p Fl(1)2233 1599 y Fw(;)456 1785 y FE(for)35 b(a)i(suitable)e Fw(b)1102 1800 y Fq(n)p Fs(\000)p Fl(1)1274 1785 y Fx(\025)g FE(0.)54 b(Then,)39 b(tak)m(e)e(an)m(y)g Fw(n)p FE(-dimensional)d(face) j(of)f([0)p Fw(;)17 b FE(1])3351 1748 y Fq(N)3417 1785 y FE(,)456 1912 y(and)32 b(an)m(y)h Fw(x)f FE(on)g(this)g(face.)43 b(Let)33 b Fw(x)1706 1927 y Fl(1)1778 1912 y FE(and)f Fw(x)2022 1927 y Fl(2)2094 1912 y FE(b)s(elong)f(to)h(a)g(\()p Fw(n)21 b Fx(\000)h FE(1\)-dimensional)456 2040 y(face)32 b(of)h([0)p Fw(;)17 b FE(1])961 2004 y Fq(N)1060 2040 y FE(so)32 b(that)1500 2226 y Fw(x)c FE(=)f Fw(\025x)1798 2241 y Fl(1)1860 2226 y FE(+)22 b(\(1)g Fx(\000)h Fw(\025)p FE(\))p Fw(x)2317 2241 y Fl(2)2373 2226 y Fw(;)456 2411 y FE(with)32 b Fw(\025)27 b Fx(2)h FE([0)p Fw(;)17 b FE(1].)43 b(Then,)34 b(b)m(y)g(using)e(\(148\))g(and)g(\(149\),)g(w)m (e)i(conclude)f(that)723 2616 y Fw(u)p FE(\()p Fw(x)p FE(\))83 b(=)h Fw(u)1209 2506 y Fv(\020)1268 2616 y Fw(\025x)1380 2631 y Fl(1)1442 2616 y FE(+)22 b(\(1)g Fx(\000)h Fw(\025)p FE(\))p Fw(x)1899 2631 y Fl(2)1938 2506 y Fv(\021)993 2863 y Fx(\025)83 b Fw(\025u)p FE(\()p Fw(x)1359 2878 y Fl(1)1398 2863 y FE(\))22 b(+)g(\(1)g Fx(\000)h Fw(\025)p FE(\))17 b Fw(u)p FE(\()p Fw(x)2026 2878 y Fl(2)2065 2863 y FE(\))22 b Fx(\000)2234 2796 y FE(\006)2304 2811 y Fq(\036)2368 2796 y Fw(\025)17 b FE(\(1)k Fx(\000)i Fw(\025)p FE(\))p 2234 2840 510 4 v 2465 2932 a(2)2771 2863 y Fx(j)p Fw(x)2854 2878 y Fl(1)2916 2863 y Fx(\000)f Fw(x)3070 2878 y Fl(2)3110 2863 y Fx(j)3138 2822 y Fl(2)993 3107 y Fx(\025)83 b(\000)p Fw(\025b)1328 3122 y Fq(n)p Fs(\000)p Fl(1)1488 3107 y Fx(\000)23 b FE(\(1)f Fx(\000)g Fw(\025)p FE(\))p Fw(b)1932 3122 y Fq(n)p Fs(\000)p Fl(1)2092 3107 y Fx(\000)2202 3040 y FE(\006)2272 3055 y Fq(\036)2335 3040 y Fw(N)p 2202 3085 222 4 v 2288 3176 a FE(8)2450 3107 y Fw(;)456 3311 y FE(that)32 b(is)1685 3451 y Fw(u)p FE(\()p Fw(x)p FE(\))c Fx(\025)g(\000)p Fw(b)2123 3466 y Fq(n)2188 3451 y Fw(;)456 3614 y FE(with)1535 3771 y Fw(b)1576 3786 y Fq(n)1651 3771 y FE(:=)f Fw(b)1822 3786 y Fq(n)p Fs(\000)p Fl(1)1982 3771 y FE(+)2090 3703 y(\006)2160 3718 y Fq(\036)2223 3703 y Fw(N)p 2090 3748 V 2177 3839 a FE(8)2338 3771 y Fw(:)456 3952 y FE(Note)32 b(that)h(the)g(iteration)d(sho)m(ws)k(that)1682 4153 y Fw(b)1723 4168 y Fq(n)1798 4153 y FE(=)1912 4086 y(\006)1982 4101 y Fq(\036)2045 4086 y Fw(N)27 b(n)p 1912 4130 297 4 v 2035 4221 a FE(8)456 4356 y(and)32 b(so,)h(taking)f Fw(n)c FE(=)f Fw(N)10 b FE(,)34 b(w)m(e)f(deduce)h(that)456 4542 y(\(150\))1030 b Fw(u)p FE(\()p Fw(x)p FE(\))27 b Fx(\025)h(\000)p Fw(b)17 b(;)456 4727 y FE(with)1688 4894 y Fw(b)28 b FE(:=)1897 4827 y(\006)1967 4842 y Fq(\036)2031 4827 y Fw(N)2119 4791 y Fl(2)p 1897 4872 262 4 v 2004 4963 a FE(8)2185 4894 y Fw(;)456 5088 y FE(for)k(an)m(y)i Fw(x)c Fx(2)f FE([0)p Fw(;)17 b FE(1])1166 5052 y Fq(N)1266 5088 y FE(and)33 b(th)m(us,)i(b)m(y)f(p)s(erio)s(dicit)m(y)-8 b(,)32 b(for)h(an)m(y)h Fw(x)29 b Fx(2)g Fn(R)2923 5052 y Fq(N)2997 5088 y FE(.)45 b(This)34 b(will)456 5216 y(b)s(e)e(a)h(b)s(ound)f(on)h Fw(\036)f FE(from)g(b)s(elo)m(w.)43 b(W)-8 b(e)33 b(no)m(w)g(p)s(erform)f(a)g(b)s(ound)h(from)e(ab)s(o)m(v) m(e.)p eop %%Page: 63 63 63 62 bop 954 251 a FC(ENTR)n(OPY)32 b(PENALIZA)-6 b(TION)31 b(F)n(OR)i(HAMIL)-6 b(TON-JA)n(COBI)420 b(63)555 450 y FE(F)-8 b(or)33 b(this,)h(w)m(e)h(tak)m(e)g Fw(x)30 b Fx(2)h FE([0)p Fw(;)17 b FE(1])1686 414 y Fq(N)1752 450 y FE(,)35 b(with)e Fw(x)e Fx(6)p FE(=)f(0)j(\(recall)g(that)g Fw(u)p FE(\(0\))c(=)h(0\),)k(and)456 578 y(w)m(e)f(apply)f(again)g (\(148\),)f(as)i(follo)m(ws:)1105 752 y Fx(j)p Fw(x)p Fx(j)p 1020 796 280 4 v 1020 887 a FE(1)22 b(+)g Fx(j)p Fw(x)p Fx(j)1310 819 y Fw(u)1366 709 y Fv(\020)1447 819 y Fx(\000)1584 752 y Fw(x)p 1557 796 111 4 v 1557 887 a Fx(j)p Fw(x)p Fx(j)1677 709 y Fv(\021)1759 819 y FE(+)1983 752 y(1)p 1867 796 280 4 v 1867 887 a(1)g(+)g Fx(j)p Fw(x)p Fx(j)2157 819 y Fw(u)p FE(\()p Fw(x)p FE(\))27 b Fx(\024)2486 752 y(j)p Fw(x)p Fx(j)p FE(\006)2667 767 y Fq(\036)p 2486 796 228 4 v 2576 887 a FE(2)456 1061 y(and)32 b(therefore,)i(b)m(y)f(means)g(of)f(\(150\),)456 1255 y(\(151\))835 b Fw(u)p FE(\()p Fw(x)p FE(\))28 b Fx(\024)g FE(2)p Fw(N)f FE(\(\006)2096 1270 y Fq(\036)2165 1255 y FE(+)22 b Fw(b)p FE(\))17 b Fw(;)456 1449 y FE(for)33 b(an)m(y)h Fw(x)c Fx(2)g FE([0)p Fw(;)17 b FE(1])1168 1413 y Fq(N)1234 1449 y FE(.)47 b(By)34 b(p)s(erio)s(dicit)m(y)-8 b(,)32 b(\(151\))g(holds)i(for)f(an)m(y)h Fw(x)c Fx(2)g Fn(R)3064 1413 y Fq(N)3137 1449 y FE(,)k(whic)m(h)456 1577 y(is)e(the)h(desired)g(upp)s(er)g(b)s(ound.)555 1704 y(Collecting)e(the)i(estimates)f(in)g(\(150\))g(and)g(\(151\),)g (w)m(e)i(obtain)d(that)1441 1898 y Fx(j)p Fw(\036)p FE(\()p Fw(x)p FE(\))22 b Fx(\000)g Fw(\036)p FE(\(0\))p Fx(j)27 b(\024)h FE(4)p Fw(N)2259 1857 y Fl(3)2299 1898 y FE(\006)2369 1913 y Fq(\036)2432 1898 y Fw(:)3367 2092 y Fj(\003)555 2293 y FE(Putting)k(together)h(the)g(results)g(in)f(Lemmata)f(42)h(and) g(43)h(w)m(e)g(th)m(us)h(obtain:)456 2451 y FI(Theorem)47 b(44.)f FD(If)d Fw(\036)f FD(is)h(semic)-5 b(onc)g(ave)41 b(and)i Fn(Z)2282 2415 y Fq(N)2346 2451 y FD(-p)-5 b(erio)g(dic,)44 b(then)f(it)g(is)g(Lips-)456 2579 y(chitz)37 b(c)-5 b(ontinuous.)51 b(Mor)-5 b(e)g(over,)38 b(its)f(Lipschitz)g(c)-5 b(onstant)37 b(dep)-5 b(ends)36 b(only)h(on)g Fw(N)456 2706 y FD(and)d FE(\006)715 2721 y Fq(\036)761 2706 y FD(.)456 2908 y(Pr)-5 b(o)g(of.)41 b FE(P)m(ossibly)49 b(replacing)e Fw(\036)i FE(with)f Fw(\036)33 b Fx(\000)h Fw(\036)p FE(\(0\),)53 b(w)m(e)c(ma)m(y)g(and)g(do)g(assume)456 3035 y(that)33 b Fw(\036)p FE(\(0\))c(=)h(0.)47 b(Then,)36 b(b)m(y)e(Lemma)f(43,)h Fx(k)p Fw(\036)p Fx(k)2196 3054 y Fq(L)2244 3035 y Fm(1)2308 3054 y Fl(\()p Fu(R)2383 3035 y Ft(N)2441 3054 y Fl(\))2506 3035 y FE(is)g(uniformly)d(b)s(ounded.)456 3163 y(Therefore,)48 b(thanks)e(to)e(Lemma)f(43,)k(the)e(Lipsc)m(hitz)g(constan)m(t)g(of)f Fw(\036)g FE(is)g(also)456 3291 y(uniformly)30 b(b)s(ounded,)j(as)g (desired.)1597 b Fj(\003)1732 3733 y Fb(References)456 3900 y FH([Ana04])66 b(Nalini)41 b(Anan)n(tharaman.)f(On)h(the)g (zero-temp)r(erature)e(or)h(v)-5 b(anishing)41 b(viscosit)n(y)802 4010 y(limit)31 b(for)e(certain)g(Mark)n(o)n(v)f(pro)r(cesses)g (arising)g(from)i(Lagrangian)d(dynamics.)i Fa(J.)802 4120 y(Eur.)h(Math.)h(So)l(c.)f(\(JEMS\))p FH(,)e(6\(2\):207{276,)c (2004.)456 4229 y([BS91])111 b(G.)20 b(Barles)e(and)h(P)-7 b(.)19 b(E.)h(Souganidis.)e(Con)n(v)n(ergence)f(of)j(appro)n(ximation)d (sc)n(hemes)i(for)802 4339 y(fully)34 b(nonlinear)e(second)h(order)f (equations.)g Fa(Asymptotic)k(A)n(nal.)p FH(,)f(4\(3\):271{283,)802 4448 y(1991.)456 4558 y([CIL92])74 b(Mic)n(hael)31 b(G.)h(Crandall,)g (Hitoshi)g(Ishii,)h(and)f(Pierre-Louis)d(Lions.)j(User's)f(guide)802 4668 y(to)f(viscosit)n(y)g(solutions)f(of)i(second)f(order)f(partial)g (di\013eren)n(tial)i(equations.)e Fa(Bul)t(l.)802 4777 y(A)n(mer.)h(Math.)h(So)l(c.)f(\(N.S.\))p FH(,)e(27\(1\):1{67,)d(1992.) 456 4887 y([CS04])110 b(Piermarco)67 b(Cannarsa)h(and)i(Carlo)e (Sinestrari.)h Fa(Semic)l(onc)l(ave)g(functions,)802 4996 y(Hamilton-Jac)l(obi)43 b(e)l(quations,)i(and)e(optimal)g(c)l (ontr)l(ol)p FH(.)e(Progress)c(in)k(Nonlinear)802 5106 y(Di\013eren)n(tial)22 b(Equations)f(and)h(their)g(Applications,)h(58.) f(Birkh\177)-42 b(auser)20 b(Boston)h(Inc.,)802 5216 y(Boston,)27 b(MA,)h(2004.)p eop %%Page: 64 64 64 63 bop 456 251 a FC(64)666 b(DIOGO)24 b(A.)i(GOMES,)g(ENRICO)f(V)-9 b(ALDINOCI)456 450 y FH([DS58])107 b(Nelson)41 b(Dunford)g(and)g(Jacob) e(T.)j(Sc)n(h)n(w)n(artz.)d Fa(Line)l(ar)k(Op)l(er)l(ators.)f(I.)h (Gener)l(al)802 560 y(The)l(ory)p FH(.)32 b(With)g(the)f(assistance)e (of)i(W.)g(G.)g(Bade)f(and)h(R.)g(G.)g(Bartle.)f(Pure)g(and)802 669 y(Applied)40 b(Mathematics,)h(V)-7 b(ol.)39 b(7.)g(In)n(terscience) f(Publishers,)j(Inc.,)h(New)d(Y)-7 b(ork,)802 779 y(1958.)456 888 y([EG92])94 b(La)n(wrence)21 b(C.)i(Ev)-5 b(ans)21 b(and)i(Ronald)f(F.)h(Gariep)n(y)-7 b(.)21 b Fa(Me)l(asur)l(e)26 b(the)l(ory)f(and)h(\014ne)f(pr)l(op-)802 998 y(erties)32 b(of)g(functions)p FH(.)d(Studies)h(in)g(Adv)-5 b(anced)29 b(Mathematics.)g(CR)n(C)g(Press,)f(Bo)r(ca)802 1108 y(Raton,)f(FL,)h (1992.)456 1217 y([Ev)-5 b(a98])78 b(La)n(wrence)25 b(C.)h(Ev)-5 b(ans.)25 b Fa(Partial)30 b(di\013er)l(ential)g(e)l(quations)p FH(,)d(v)n(olume)e(19)h(of)g Fa(Gr)l(aduate)802 1327 y(Studies)f(in)h(Mathematics)p FH(.)e(American)e(Mathematical)h(So)r (ciet)n(y)-7 b(,)23 b(Pro)n(vidence,)f(RI,)802 1436 y(1998.)456 1546 y([Ev)-5 b(a04])78 b(La)n(wrence)38 b(C.)i(Ev)-5 b(ans.)39 b(A)h(surv)n(ey)f(of)g(en)n(trop)n(y)g(metho)r(ds)h(for)f (partial)g(di\013eren-)802 1656 y(tial)25 b(equations.)e Fa(Bul)t(l.)28 b(A)n(mer.)f(Math.)h(So)l(c.)g(\(N.S.\))p FH(,)d(41\(4\):409{438)c(\(electronic\),)802 1765 y(2004.)456 1875 y([FS86])116 b(W.)39 b(H.)f(Fleming)h(and)f(P)-7 b(.)38 b(E.)f(Souganidis.)h(A)g(PDE)g(approac)n(h)e(to)i(some)g(large) 802 1984 y(deviations)22 b(problems.)f(In)i Fa(Nonline)l(ar)i(systems)g (of)h(p)l(artial)g(di\013er)l(ential)g(e)l(quations)802 2094 y(in)39 b(applie)l(d)j(mathematics,)h(Part)d(1)f(\(Santa)g(F)-6 b(e,)43 b(N.M.,)g(1984\))p FH(,)f(v)n(olume)37 b(23)h(of)802 2204 y Fa(L)l(e)l(ctur)l(es)33 b(in)h(Appl.)h(Math.)p FH(,)g(pages)c(441{447.)e(Amer.)j(Math.)g(So)r(c.,)h(Pro)n(vidence,)802 2313 y(RI,)28 b(1986.)456 2423 y([Gom05])40 b(Diogo)h(A.)g(Gomes.)g (Viscosit)n(y)g(solution)g(metho)r(ds)g(and)g(the)h(discrete)f(Aubry-) 802 2532 y(Mather)27 b(problem.)g Fa(Discr)l(ete)j(Contin.)g(Dyn.)g (Syst.)p FH(,)e(13\(1\):103{116,)c(2005.)456 2642 y([Mat91])66 b(John)29 b(N.)g(Mather.)g(Action)h(minimizing)f(in)n(v)-5 b(arian)n(t)28 b(measures)g(for)h(p)r(ositiv)n(e)g(de\014-)802 2751 y(nite)f(Lagrangian)d(systems.)i Fa(Math.)k(Z.)p FH(,)d(207\(2\):169{207,)23 b(1991.)456 2861 y([MF94])86 b(John)27 b(N.)h(Mather)f(and)h(Gio)n(v)-5 b(anni)27 b(F)-7 b(orni.)27 b(Action)h(minimizing)g(orbits)f(in)h(Hamil-)802 2971 y(tonian)19 b(systems.)g(In)g Fa(T)-6 b(r)l(ansition)22 b(to)g(chaos)h(in)f(classic)l(al)i(and)e(quantum)f(me)l(chanics)802 3080 y(\(Monte)l(c)l(atini)k(T)-6 b(erme,)26 b(1991\))p FH(,)f(v)n(olume)c(1589)f(of)i Fa(L)l(e)l(ctur)l(e)h(Notes)i(in)f (Math.)p FH(,)h(pages)802 3190 y(92{186.)g(Springer,)i(Berlin,)g(1994.) 456 3299 y([Sou85])82 b(P)n(anagiotis)28 b(E.)i(Souganidis.)f(Appro)n (ximation)g(sc)n(hemes)h(for)f(viscosit)n(y)g(solutions)802 3409 y(of)f(Hamilton-Jacobi)e(equations.)h Fa(J.)i(Di\013er.)i (Equations)p FH(,)d(59:1{43,)d(1985.)456 3519 y([T)-7 b(a)n(y58])79 b(Angus)27 b(E.)f(T)-7 b(a)n(ylor.)25 b Fa(Intr)l(o)l(duction)k(to)g(functional)h(analysis)p FH(.)e(John)e(Wiley)h(&)g(Sons)802 3628 y(Inc.,)h(New)g(Y)-7 b(ork,)27 b(1958.)456 3738 y([Vil03])108 b(C)n(\023)-39 b(edric)30 b(Villani.)i Fa(T)-6 b(opics)35 b(in)f(optimal)h(tr)l(ansp)l (ortation)p FH(,)d(v)n(olume)f(58)g(of)h Fa(Gr)l(aduate)802 3847 y(Studies)25 b(in)h(Mathematics)p FH(.)e(American)e(Mathematical)h (So)r(ciet)n(y)-7 b(,)23 b(Pro)n(vidence,)f(RI,)802 3957 y(2003.)1674 4455 y FC(Diogo)27 b(A.)e(Gomes)507 4555 y(Departamen)n(to)f(de)i(Matem\023)-38 b(atica,)27 b(Instituto)e(Sup)r (erior)h(T)n(\023)-36 b(ecnico,)27 b(Lisb)r(oa,)g(1049-001,)h(P)n (ortugal)1423 4656 y(e-mail:)34 b(dgomes@math.ist.utl.pt)1671 4886 y(Enrico)26 b(V)-6 b(aldino)r(ci)531 4987 y(Dipartimen)n(to)25 b(di)h(Matematica,)h(Univ)n(ersit\022)-38 b(a)25 b(di)h(Roma)f(T)-6 b(or)26 b(V)-6 b(ergata,)26 b(Roma,)g(I-00133,)h(Italy)1370 5087 y(e-mail:)34 b(v)l(aldino)r(ci@mat.uniroma2.it)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0511070752145--