This is a multi-part message in MIME format. ---------------0308220828855 Content-Type: text/plain; name="03-380.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-380.keywords" Navier-Stokes equations, Oseen vortices, stability, long-time asymptotics, self-similar variables ---------------0308220828855 Content-Type: application/postscript; name="global.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="global.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86d Copyright 1999 Radical Eye Software %%Title: global.dvi %%Pages: 35 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMR12 CMBX10 CMR10 CMBX12 CMMI12 CMSY10 CMR8 %%+ CMEX10 CMBX8 CMR6 CMSY8 CMTI12 CMMI8 CMMI6 CMMIB10 CMSY6 MSAM10 %%+ MSBM10 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips global -o global.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.08.21:1754 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro %! 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 17 /eta put dup 18 /theta put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 27 /sigma put dup 28 /tau put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 39 /phi1 put dup 44 /arrowhookleft put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 97 /a put dup 98 /b put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueXX 5087386 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 69 /E put dup 71 /G put dup 73 /I put dup 76 /L put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 90 /Z put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueXX 5000769 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5bfae5c1f476ee6f05eb1d2caeb 269958b194521197b312fcced4867f3c8fbd030bd715d8ffda1dcd454b174e7a 1a97b59fe770e67702519d9d9b23d61ac08424d555242a8ca08c49aef300945d 99b999a79ce74804ae6bfde623f4463371442f6523a5f6ce19c839a708c02513 2e22c696c8ccade45680e5197189d0f98e7f0d5f955e353970b392cf530a68cc 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMIB10 %!PS-AdobeFont-1.1: CMMIB10 1.100 %%CreationDate: 1996 Jul 23 07:54:00 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMIB10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMIB10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 17 /eta put dup 24 /xi put readonly def /FontBBox{-15 -250 1216 750}readonly def /UniqueXX 5087392 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d78041490e364babe4b85935b8c9c7758c2dbe253653c 892b1fc2c6a01103e0b8553f847bb38d8e4fcd563ebe51cb7d5f0877af29249d b61636618fb31eeb6cbb4ea4db687fd77ed1574da8c9f9d33b592ec0e38d798c 498a205cab252f7e2f1ddb82a38f331f917b8452af11b2b93e34c1ef55ee005c e8418dbcd0fb9d4d3fa8cb388ec91f16c8864913ad3fc8960d3c424650bdd12c 4db4049367faae97c660f00021f712e9eb6366b7ce74255ff62b72396ca43287 7c77fecdc53883427c90c1c3ff7bd46e472cea9bd72eef0e1be36abb3a50a7bf 6f5d460fa2fc81466014df9ee1e15365642c2de78706dfa740e035b4f28f290d 79b8ba16e5bca13fd9c3eb7eb54006ac2c1cf0bf363b3b69e923797a95b76515 073892c75cf733da7c07da8ed1a8d1ded902f784764c6df6fcd0fd28083d88c8 4c41aeaa6c2c2c1bb10219f4feaab75ee904f4314fc0b9de5e96054f2472ad86 42140315385400686cb07a0585d922ae4b1b25ae722edc6b4afe053560fc9ac9 20a1d2fb8d0921c52c9468d34dba9bed42e3a8308742d36186ed0a3b43f690a4 dca6bb2af8bd698fb303fd68df53edc5e01912cf007dd31ca7481ce844be1258 7002d4aafb204a8075fc6df86f22249d61dd840e7ef87362d72e35f4e8ece044 95ae41ee5c02d6aafaad37141935867a8770fbd9e9e894ea6cc5cf3770f35d98 09be73c56fb053ebb622cd4a3826bc95c192bc2248e1a30e641799994debd48f c5a0ca09be0b2831672271e6a087154d1541899171bc12c8ae00dac586f5c0dc a9c8f4bc7910fe68531c2adf25f29002b74151ec41cfe904b25dc975f97ca163 59d6ea5787a3b0529b3a7a7f8b5b8606cf11dfa1029106881e531349e8317d6d 256ecbe471ffe359862eb096355369e65f5d2da53e18801fcd24c7cf73c5462b 3ce96f74ad5fbc351b169fd8f482242e05c8681e7f97522bbb57e5e114cfbb8a c1759c130b14b6a195be7b7027a4c8ed3ae16c07a1a2973078348052a57c371c fa1a081b7fc2bf7c56af331d8f9ef6b042ed16b36e95f4487fbc785dc057b35a f3b9d9005a37a851b69e2a789262a524ccf94e8f123a35099bc9146bb1489bfd 8aa1c526745ed4fd73e97f86a6c426292a9f67786f216c47affdc88c42e54358 f6f31b5dafacd94b4a6bb328351429cacb759cc0a51889369da0222bcc793b1c c6e5a092cb5bd05e84f9054b0e3cf3e669e029369a67c01b01b4d6b98b60fd71 7643a9bb25585f868e1edcc0f3ee73eeff1e6a652bde12005d110e264d077e18 57443d39882fb3dcaddaa1276692a5000aafc65fcd992ffde06b36982d539930 947ff1b8868c67e784d2c6d6b27d00befb9bf389c2f18ef1354145e6d1ae66e5 42f03b0e3f3b2f05a9c7b5fedc4209f1856980d069686dea11a62acbf347b940 743e4da47a4a76e331dcf5317e7b28e6e0ad699f7e21f5e273226288c3ebdacd 2236b3aacb04632daf544e8370d6b1e8f651b02e1486f26c232480cffec7d1be 46c855233a55b53f03d24d85c4d4307482620157d411bb6b491e27215a37cc90 040f881754fc4efcc987ad273d463bc9004010083bd306689b7e800327da2723 aec12ae12a5bd1cf478b3c0f870a8a330968fee3e801015153ba38eb0c732ff2 a920c63a1111043d64ec64068554b73e0e37612166917806c11c06e846ec98d9 c565642c1398acb76eeecda42dd818222445c5a4dc3d4c8b7384dedf3571053f 94088a1f66d4171108dacf54fc2856a0bec695edd060754d3947291d256c0462 4a2dac0335d3c1d71837dc7816cac7f2c518de22f0a7e6aac0fbdce3adb5c172 7bd8cef65465add635700bd36edb0fbb49f0e25ba29069222431fb8572ba677a 6dcfe773d3b97062ac5df8cab947ae0c1b39b2e6c479f230 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 110 /n put dup 112 /p put dup 113 /q put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueXX 5087381 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a1e67ed7e41cc25acc04702f6 8ef703e3d45722c1a91fdef7100a48631a02a6f02a08c6b1f9b4df8310385b86 8632718fd87119a233f219d9411383b7fa9f3e4780d8c27e2e89e0cae883d664 c3eac57a3aef8988a2e9f0f8c7f53e0a80bdfc4620e21287d0390e1975398544 7f3ea66401024bea75e1b4c4437b7bb188f76f96b918ac7c6ad7e8ae7f21d8c2 790f08cccec904fe48ef39e597ed4d4237c1d1f596f5906b19ea308020f7a35c 168e327ec3246b1dfabe912f6b6daac09974876d3996e57d180261110db05f15 e3e8eebba3d90b5764c03df3033a1ed678ebc679569a2fb297378b25434c0f20 5313ecb8a952f07242d3ee731b0cdc086a4481178a3d65129c47c09b22e9c431 e11b3747b94c26a757c38d06001798c6a568303d541385244b967d3b1786edea f65bb53c4c2fe75e4b1b15c2c78d930b4296c80f08bad86012451edc8e9f0854 c3b390a16e27b11b3d45a9f72eff8baded2242dc928a61685d79e09681c97425 5b90a498614cf560fa5b1718981388268ba206a96989e6d0b5d485d9aca5594a e67dd7b34d8a369adb06647f8aff8814d6d9cdc04a4835918e557174c5bc0f3f bcea9907a04cf93c12727ec40db3f2f77596dca477862747435bdedacd9b2311 6cc97fa47ffdd7d897fb6bdd5572e35d34e7e1cb5e7273a4ffd86525323ace4a 84e1297028c2bd5469baa2e75d19360c2c9042139d5e7dd4390a6a3935424711 de21910126d750ae279916ceb71da3591d60dc62db333c5021e2c1cd61ade51e 93964fa67d27fae86cba2166fd5bcdea975949a3e065b95233f13db0b2d68e11 aea8c9cd57ad2eccdafb5768cf05b646c7fe218cb97a48367eb3aa0991cfa385 6404b2ba8486db5a412ae0945e96b37fbffffba1b8c9711a66bbe310bbd03c34 666b95810734f1ed3302ff7012de440963cc9f357780337fd6abe69c8ccc16f6 d70f15ab16b3a60d5ab51fab9e5ed31e7edf3513a635f78315aea8349d350ff0 970cfd6a9fb692f02df7a377996a6e69188a5e6527c251314939e83e088f7b41 04fe10c58c709b1a2d2b4bcb1ead58039cef5aa278942d52ba0795ba1cf7e4c8 26568f78f46f2ee3139d86a31a8d73e8ce14d553c189c5e41a1ceec93eacf46b 08cff263b9add2c31eff49452d8ab50da5760ec4c3730ffaefcd707bfe313b4b 7a31ecb929d988db76059a5c98434da4f3403c5e73f8bf5f42940e780a14c715 055a15a7deaa3606c9e4bcb4c248a53b2e457b6b4d18c0908627c6780283500f 88871cae28afbad5621570c1dbe13918195cfcd8d3c49ef037857a75d167c980 c13ac341b1f70d1c6f2ee0ece94aa383595c61281c435585de905e0cfc591891 331f0d85d6ec751296a7c07e46f573b3e7902ef7ffd7214eb0dadb1a065ec0b6 c2effb8b4dfd942b3a4ab1afd38394bd3430a9e424859ac38e6da6c4cd11de12 ad692f0143c7609057f167afcb094a19f87f527bbd0daf145014d854491d64fd d8f1df8958560075ac68207c494acccc38718436eceab8c0b15ab8dd8e98b051 d21f61b780d6d46af16a475a06ba96c20892c141e474874010c1c529f49528de c0901dd8071dbbd832692260e3be30bd5a8a57ea5ff6558f17e98b8e77190fa6 5ea6ba3d51b388eadca86a7da4d3ff28b67cb2b3564c45d4871a11a2b49d8e9b 01529f15be36fdd4b1e0b0aef421672b8306fd03176da5bb9482cda566501473 6372b7c2467e5aebbbacc7b35a319bf5834683f5aeca2e9e64285f998b574662 f6aab7416393d4951bac366327a2b20ec67dc19d5a3da060584abac27079d647 3d4251bf641fe89d187fa39afc7a53fd2f10fbcff8b7123c92cef380454da103 0697cab6eca6820b543b9080becc04cd4f556679da25f3f2b6cc7245f7642c96 75cb598dca7278766bb11b2dbe5d0931da11be7989b2c13548d14d77932435ae 90ba41dfbcc7a99dd90ab6f8e23d87d8426f0180b8998132fab96d85ee095eab 52b015dcb3fd037055bdf7a6debeec5f2031e759aec69813542f6823a1a1c0f9 35fb39fdd174c8fcc10453f87868b0f150b9a8d2bf3dc4e3ff81f4611561e760 f048f9a013fac107dc3982af7d809ab9bb31545d422493c387ea4b0553bf96c8 c3b53d361b56524a18428fc621b869567870a6d8362f0364d1b5a1cb6527381b d94afdcd71ad9cc246f44d186bd25d5bc66cee2e449658866182f9f386f4a510 f291ecb39e883424a3bcdae38987ce7ce82b3902623c0bae4d055bc7e8e509f8 9626c4c9c5a8fcf978c9 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 18 /theta put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 28 /tau put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 70 /F put dup 71 /G put dup 72 /H put dup 76 /L put dup 82 /R put dup 84 /T put dup 88 /X put dup 89 /Y put dup 98 /b put dup 102 /f put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueXX 5087383 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a196cf14c95a4e02f9cd2da8c db2546c6df52e524745992e18d9ff87aa25e4e1800bbe4ebb357c6ef55ed6d03 6d3a00c1ee8073266c21d2f0ac85d656abf61d7e5a4fa87da8ec3b5329e434d0 d2adab706b42a2e5331be5295399d803ccac03f631f01f39a022fcdf63486b68 7d15ef284a77def7fde4898543e7b5f7ec267756103e477f547cfb8d2311c4b0 09deff56085f5d419697af1846c8b88c1bbbae149f0f19ca3c8dafe19cec48fe 6b38357246d8b5cef80b53bab4cbbbe8bc8318cc462a4158258f456e697f6058 86010493f252073ee4e4c9531f23485415ad6bc901bbaea659ef701f825f4b6d 5c9bed0dce7b63810f906b33ea4955531ca1ed1fc120cc56dc1e277a33222cc6 c3a97552607b865783faf0895cbeccd0df54f57bd162a88494768435eda71a39 69718c0c5912d3167bb0971546d733e735aa9bcd89f5aa154656031ab3f6fe41 94ec225ad03ab8b67a9da43ab365c4df17c23e79f685c3b6fd5b654e4a6f2603 6cec3a8641b0b57f93eb95f9e238832aeac2061f3153e0f375513c974f1c38f7 b59edcc94716f2862cf71c9a2be8430985da1cd813b1e1c1b9310564a2e9bdb8 c536db7caeb9c37a0693a01df0045080f8236ece54dcfa7b743562771f452398 100dd95b678b6f190cdef25df0e2b10423aef1d27dfd5b022a0f895423aebb50 34ea52f46b077f2022e28ecfa6137fdd4b13935ac19769600b2e0dc86b0a2359 4675106e26eea6b5033525c1d79e046ee941c9ad410ad3284092f9fb725d366e e88a7a4ceaa6caf2ff6a2c80749780481db49bf09d5b7dd77cb83ed6c74fe223 0076eb41e5e84c5f714b9cd7ae6b36396bb68c992ac812a85b87db809bcac7b8 ab12df0c53ad4c89b7fa497020a34f8f8ffd8ee030d37b8668342950290d0919 1fe9b1f6e432446414671097b7866fbef89e9933e78faccdd4aa40aed452ad87 fd8e2c085cec089850cc55aaafaa636db647a92555268d133883d3ee5319c5f2 cc06d876e9ff5900f11cf6ab6ee369abda8277c432aa2ae460a6e332aec914a1 b90f0466ce591cb9ccca4720ab5419a2060b97360886ac95fa653ea31f2dde80 64d8d11b0761fdda48bd4d5e442e785a6be1e43b1d869b6a672f7c4d6becf546 27b0451682eb3af3ed0f3628182393dde35b1b95d6e1ed7b3f3afc64b5cc6293 abb34890b202bf2fcaebead98de17f184bdfca12e17a18f00490a7e1a62e2617 40847a689f40d9006f498218c6c9e2d0b6db8d825649ce4028701a15ae62bb47 eb2ff69cb3e43eeaa056dec181141b6d48117750b0a6ec439a3070f8fea5b8e4 15b4b4740e0cb42959dee3636b5dcb75173711eeddd7bc12fbc65525db614003 27db2e04e3a260dbd2774cabece3994efeb954c9a1de4a943b6f593c5abd9d39 1f57fb120eed30e246b1e5799f70393d4683d2b32ef0106738084ae36c1ce265 62e45ee808b2a0019548776a31361fde671d49801ab8183a76b925bf09c6ba62 24b532c68319da229be9e0400cb0207dae3954b5186317289de014e50cce6ff8 7d5eb5038f0be6fb4d539e4506420f417b9163775db28674d62c6ad76a9bc274 d8dd3349996a1623605b914feeb3f5d5964658e321bc3ecd6e3ff13d4bfd8154 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 19 /acute put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-36 -251 1103 750}readonly def /UniqueXX 5000829 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd803271dc8bb1656323089e9e2bb50f2b95088 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 2 /multiply put dup 3 /asteriskmath put dup 20 /lessequal put dup 21 /greaterequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 50 /element put dup 63 /perpendicular put dup 106 /bar put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueXX 5000818 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b66566ca5ba42b1911a 5d7f1bf343015eece988b7a93bce0c7aa61344d48aed9c92c8698d4b7c9951c8 7d103f2414b39e1437f9d2e50c4ee5f218f2e6716926a79ea978f13b1f855345 191dd7d31d8f82c2e3343c7a5894d95bdc492c28226834efcb5c12fea36ac5cc 430e0aa604961e34888adf6c1f3954cbc2498e225d953cf5685852162346f474 5a2a7087d5d7ad486de16d2ca8e15cee26e012671ba3bdc7d95cc8c98bb774f5 08625e968aee27ff7d1a06e63bcfb5aa4876c3f8f13b30ccccee73c3caf4e70d 98e6ed2f422dbb4950bf789680e064150995941a9f4dd68a575949847a7d012b b910bf03a4254c8f517e6ea430fac1155de17dc54e402e9f22beb39a64dbf0f4 ac415a28a2342100ebae2bbdd0e51225bd1825c5e0a2b0a5b4be4818bf1211e0 c4d56495f952a0a94dcc6ef1db7da90c9dd357202bfcee62fb05105a08c59872 884993c815ac627506c9f474f44c30acaf6c59a4ba7b1ca1a738c41823379039 cdbbcbdb11e581744824f66183ff293916f4320268968c55b5db380ad305ab05 7d11be177f36e26085f65d2ca44973d456f6c10d877d92f99928fd09f568ade3 103f61887e5b51bee994689a73d6888decedc1ce1278d591d8a7fa542d9cdd23 5e4f24a45c5486f510e31e3d9a784cf6addf3fe85da3537a7d148578459ec7b3 74f3a6386fab0529231e372328dce0c46e1036070981e7e770619ffb0b66691e b9dd52de25b70bc420b88611cb2baf177ee58c96bfa5b0285236968694bd4101 c74809a4a9de2c47c85515a8ef4a2e6168928777a3b823d281e386b801528896 d4c9a4a4765f0709d3da344ba4fea59d31aab84e705833f1b48126c1f7961548 ebc0a3771937f0525d7e3dea081263c437bd99776bde4af8ef1df88dad8a1627 6ce08e6d148db40bccf484a0bea9c01d46de53a825e2360418006cefd29ea19e f1d64b05c05cede044beb33fc2225179bccf7e3bc009e0f58ad25ca2b8cc779b dad7a05b29ccdc45144dc21a9367c7fc76afc4474c3d9be5a730f5984179646f d385a32f12f8ccb00b4c3a75331b5b4819d9916b6628185da38fedd7a5e9d2b8 e33b840ca2ead9c75ab578e175999cbda468065d4e6c473b4bbcd5760952eed7 9ea5f723d5d52817a3ef0eec97d1c1d6fcef261e5ab777c697a2df7a1ee51cbf e90e8e457d3b5a0f5f41588d3ae358a77c074da1c6d792e1b65d385b8c3481d4 e83a0ac9db72cf07e5b385cf202f61cd9bb1f0c2ff9b19f8d5e91b3ae40610e6 2d1debdd637fb29ddc13e9002082dbcce80aaddd7ae802fae166cef03445a47f d5150671b7c0efff1242d9e37a3c0e090d8413c6df6dc1329899200d16ec82aa eac96194f0af2d459d8d6d21b28002c6e7ecaa5b47ab8394e018215fcf59b1c9 dc9937eaf96d70caa3ae74c60b451575bf76b742afe32b13a5ed14bd94a3e448 f9f30ac2ddee43e3f63a343f22d847e3bf7378b4f7969ef1801f95cf873d0ccc b540aa218e04b20039809ebd8a14c44a0b6bc086e5f88dc3814da7147a258f51 c5a9d90a00b5bd90a3f921ba25e0f451fd019e2e0eccf252efe5fe2b71898d9f f6cfbb877b01b7dd1d77acd2ca1c9d0f74b4129ff88c1fe381cda26c6abcb17a f6f492a66c27c2fc77e815a1f47ed493b99eddd74e47bf21af06c05d8f8f01c0 872572ebc5143e990c4eb5383f11def469fe0224e0ec583a328b9022752e2ca5 03959d2e051224746d3ad6048acba9812247d50887e6d0f839dca5656211aa75 843eec5bd0e7f58a5c9a6c819660064cc4e2c37fa1bc3842ddae0b5d9fb77fc7 2bf6f686f1aa133a21fb2bb68a4921f8e2262289c6b5e59104e8c63ffc667a4f 5277a15fbb8de660482f3a87ddbf3cba57c38f9e4439fd3fa579cce35c360348 f513cd21fff105319be9e5a1b8a8656b71da65ec2eaeca3f223ae69159f766b4 d454ec53a4e6decce5fea52e20da43e1e9a931a46c11af1e4ca360030936c936 5d15009223d45c533152d89f3831c6977026df7b83206434d9aa632a01a386ea e15f079703f5fd980886ad87133d3876cbc0ff1a2cd67a805cbda0c2e3ec4459 185f860c1e97e086837ee87cc5eab829225aac64d56164c361b1183f965facb4 39f131bdb5ef96e893d42d1612a3d22a3754e1719a74d26314abaa9f678c53ff e7aba57cda462d4be833d469c80948d1aa0b33a01eca0c0287468768ad86cc8f ef9f25e708cc68560c4dabbf96a872397792076f1f890f0dd716f53404500e1e 0e85786f599054b5ffec6bb074514eaa6b688360de0868219c7a50d579625769 7a806cb9c37ffc65a1640f7e16d552c67191c355415313d16d475bc6345eb41a 0cf6d96e8aadd3cbb18c3509414a1e31e0ce5b583f810207c36f9f17807cce60 a6cee4d01f827cae392f19a4225067fdb6c4070776b83c149b63abc59608ac87 995b54053edefb14bbabc803de44cec434913c5fc68a2838750e9750c1cb24c0 d80e303ef66bed28d91f0d790a7450010c2f137c870b9da1b9534e5bea6f3d45 5bd6ff1bcff8b74d8ccb53302e973d2e687e903b35f4b9a2a2ec974756a31b63 223f6ffbe1623d931290cd76fa97f3c721a2611e3da077204966d8097535f32f 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueXX 5000789 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b031f3fc6c13d7da7277a94ba 018e9998b3dd888011a5d7c4204989f30f908b95533bda845746b673ab71ea57 65a0d14f4350707e47c8276305b28513cbe1bb0dbd269a53719bda46e536685d df78ca0146b6b93e760256b74d939d4e35b5e77238f04c92298dfdd188feea30 e053eefbcbb52f2011772b3aae39f5805597bbc1e8bb75a446ce014030f4f2f0 f49f9e962ee4a1024a746fa92a3628db5270732b54e43fe5ecfa524f127e5fcc 788e77e66098336ad67fe4cccaf0253272d5df79864bf4b734cb9a5859d557d8 bc11b8e00221ebc12e97de4b1f466ead83a4c894709363bca9040410a52d592e 34ee40cc7e5efa920546b981aa659513a24b1b85c221a1875b62d0b89e57a368 321b8043a5b094e0379760a443d632892b14ad6d19dacc8c78093243ad67e6a3 08e56e6b68412ee690b10dac6e17708754a00d51fc957b500eb80175716eef4b 2ca1ef867614659bee3f2b7319e97b6fdf1efc847bf3cee3156f72f21751da8e 5fb6898919e6799820d3de0642d756e09d6fae4ff08dd3deda3173bff4bb11f7 9109c97ddc05897af709ea199a90fcee8ce4c7a3c15b18170c41c04de2d3fba8 f34296a95b8e1e8de3739b17273f8f2c85e914615e8eac5e8bd2387ba3b1edf4 7968f06e2067d836d0f9f3e085cdfd2de06a62c81d786b304326f7002e83160a 36598589228b4dddddc43c85e1d126f8fe81b828028e26317af5894aaccf4f69 6301e1a9fc45935d8a414957f08febebbc3a72ada80f101e47447d019ade56e9 f4fab969bba2b44e47399fedf5caa1bcea216d7ba713d523d8c1e5c02115408a 277a0db93878d14fcdfef5999b0428be4c77940f21ba6b90ee390bc7f4276626 3bc4af384cb7c24e18d62a91cb21d46e9820f481bd7cb5dfbdc63ab8e563b429 d6be6ecf2c64f91a51e8c2d08d396c5a158a303fd73bc8bed6f75bb60b3e2836 1711b4be659b31729a7773c431048b9f457d2117fec6973c5ecf3764e1f16543 77fd41bd8fdf09c8b0ac1db016a9fef4bc29194f26f5b4de4d5a33cc520271ad 7dac397c0e79233251f31d85dd89c907b74a31520ea8331dbbbe0ccd4c9b3b85 44aef20b7aa6d6d2bcd26ec1cb76510de9e824d53bfe113b5a85e865401e9280 45f6fe64153940e36021f9d4c902d28c6a952ab769a114c2a34153092e3d791e 11792c25164d3537e3b80a03aab728e5ea285ba5c0eef10ba148712fe4220c91 47ce770dbd06226cc481d6f21cd6d2689acbcd1b5261791acde8d530df1e8eb8 5b87f058fc253051c17f69386178a23447eaa40c83e722d7b5c199fa9bce299b 1a94d0c92da7ac3ecf5cca495db6f9894e5ea16e695c0c28eda0a72ff937d731 c22e427854a2b0be8a9aa7afa394d0484317628cdb42aec4e8538a50764a886e bcd7708b54b5d189f061e099ddb5244115cb014e56967ace798205767954e14d a9851e6bb373472bd346ad40c3f6cb42750854748733268b6aa97051baece213 5552045900df4da8fa5980bf0227c89d876f1219ce77c703966c112bc6098d12 6348c5617ecc31c03bab5e8dbf3600d9e8b52a993e5fe2bcdb25d60e7798475a 9075f4f439411036b7648d6b3e3e53e4cf924f2694daf1bf6c51aef197eb9dd6 d3af7363c65b87ad4a61b9c9a41bc395d67cdb5010ce1c3f1f6553eb07abcec5 ab666444f00b0c35023defa2b60d9e0ff4ec42da1134b7ef498900b7071e8a14 ade16283c520aa506c7688e42a6b44824dcbeafc79741f68117dbbcc70763ed9 497d76acce5efd644474eac3eb7fc30aeb1ba00f8611485e9863a6276ae03fd0 dc4fa22e0f5de99fd19e7924d0232608d602a4eb0feabdfa2bb1b1ed0a8b958f 70ff55e8f52c1d3a3f327ea81449 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX8 %!PS-AdobeFont-1.1: CMBX8 1.0 %%CreationDate: 1991 Aug 20 16:36:07 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 82 /R put dup 90 /Z put readonly def /FontBBox{-59 -250 1235 750}readonly def /UniqueXX 5000766 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be1f91da738298311c1360c8158d61cb73427 3c21bce59b0f2e1d38c2fbc0984ce6a04f5d3bc84e2601397499a981b200b307 771141805f1f5cd7b12befe020f7121ec103dc2baf17f0aca1d0740c92a8dd61 12fa9c0d30cdae4f2af757b9f713c127d6ca02ef315205b81eff629104b21462 17d2b0f823133bccc80c9e400d6840db5fdf828f845313290fedee040d34f309 8f1b5a6b8941c81d1e607dcb310b14169cb433329994b4d1ac1b491e7422d518 2a8faf7ed3cbc2e23c58a2e310b56da54f51360556d27d58783694c51aaf62a3 a2db91ac8df80303c35cfd503341005020a5cf501fc0b64ff4f497951df6cd50 f00fafb04725d23bdd36616c7c06cd9052305682f6b6e521882922377b4979a2 e6f031b91b2a86d8631e81de17f90158ca05045c3b823da8b2d2aefd37a32728 45bace7594ceaafaecbf6140212789b3937a24cdd8dee327c934a9b0e328fd83 a1b535d9bbea6f0204ce5ee74cc30769734d4fa3dd0cac3986510c99ba062868 9285146cbe64caa89cc2f18a61d924e7f312fa189e0ae4deb913ed1cdfeae61b 4e0fe25f2775a8b2edabadebab4eb2ecf47258c1a77101c810cccd3b06633907 f765fc80bb1a2527f8a59c52760fffabbbac1d65a11708a3bea4557f24536773 d17dfd0eeee4e22f406e04765cf582d6ca4414f2441b6ff463c56018e44aa2fd c9af3f0480da99856aaa97e56b4bc28d9a74e6d74e5faf2016475d2991256254 6b9edd4874e8a3acaa1630376769be1149a2f5bab8c18a7d7d3a1bfe32aa1e53 05e5cdddc7ae6ea0b7945065d40d736808683107ea60c6a28e64b890097100f5 4da60fd7d785a9159e668cfcb1710f359ccdbfc25ecada02943aceeeae0ec38b 4bb1cbb26f4f5e99aec1e96415e1e2d58c726d576aa4a9cdf17420c396f4f87a c3e7e966f8d06c0bc29bcb3fc35578dbc1d6579e74da662c7d863adc0fe776de 58ad7df512fb17e7a67f96b365354ff588909b2208bb3228b7794afaabe2239d 565791ffa4640592c8d33bcb29f27f3496580420509b3dfa6ef6a53c17582cfe deb59b8ee85b3553d73d6c411e0ef02349a4da28f6391d756ad96c2f5a502948 fc5b2b4fa7d3d57659b9511cedc94874b489107cd8fd6048603fd94c691c534f ceddff1a8d6d20d810b952552931f5fcf8916c4dc390645c95f996492586042f 048a792e2127e2a7f3925ee8b69c177d64525345d263ed880b1872fdde3d077c 5e8c6dbf0ac7a4f542160f1f6f1d39805a275c6006966d9b13dad1c57b697d5c 74492b67137c53d8f07340d9b4c2bc83927d2693b2c99d06a425cf01f3678c6f 136f4138842a74931768e5ce0a76589789d46651b8f61a2b6737169c470350b0 a1a1e726e596f07f31c31ba6ba5ca4c05f72394c7fed28e42ef9535fa56062b1 7e452996778fc5cc7340b4d0e098e2f246dc5c517f0e38989fb60a294f005665 31b36ac62a5a26704a8af9dd24821b31b060070d2c09f4e92d60816b00a75f8f 04dbf9d1841adf9e1789b79da6da84159bd7f2644b665f0007ed1bc82886380b 7c75bc0b95812ad05103e3bcdb149f72cda75d536690b5cb125c40a6ffe1986c b6d224396f76884023438e0df98047a6246108620b8518ede510 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /vextendsingle put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 80 /summationtext put dup 82 /integraltext put dup 90 /integraldisplay put dup 110 /braceleftBig put dup 111 /bracerightBig put dup 112 /radicalbig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueXX 5000774 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d5649664203ab24acb938d58d246707ffe7 d62f04bec4b70c21ef75beb2b812622b3c74e969d72d3cd11bd7106294a99caf 0b1629bc7d4de6b96ca82930831d64575f23f4ad06a0e45e315b1d392411be8d 6d73c998789ff258a07a3c8c2057325784514c845500bfd1a971310cfc11d41c 1a167dbd5ff012c60add4e87325f6e5299032a839de65fb1473a166aae1876a4 414a434f22c1d241591fb36f857df6fa930608750ffc0c54f44994662b1f00f1 400bf752ea8d83ffc4cb77a290bc2d99981ae59a191748ba5c7ba1a9d2583fd2 1398452b6ff5d83a059f7eadcd2ef744e9dd22bdf9c79d049bf06835e878c32b 7765c69bdd8ef4deb4ea7cfff4cf9354a4ddffa689de961d16772491c7afbd7f ffde42400764c68e954ee5c455a5687959829bc3b319b2147deaab3628662c80 30c5e02fea09609abe4eaa12e217bc3af673f1bc36a7039eb13fcacb4218fe0f c5a3f9452d4edf46cc91db67b624d4f2d37502fb9f11af4da18ca40b61097f95 d44329375467ed13c5cb585ec53f62b83ef9502cc755af44bf32b87b8ae9f3f2 f8dbf72dab90acafbacd280db6aaffaefdff6d5eff26669bac56280a950560e3 d01714ba38659b0e4e28bf8cdc1f15ff7e3ea9d95a6afbbe3d8e6905aa6d531d 6b6dcab89e2036cc492c65f76ddcc3cfcff33fde3d9ffd10f7a90530fd5c7683 4329baa902e701ca42717476298c9ff511b1941244bb1dfd0e5cab92a43970cd 1bdb292eadd055c97972844fdcdb0aa06401e7b2632b9d134f96bce0064860f3 9859da7a9a82be0a840672a1331212752ae1b521b693b384a72dba8d37a86eaf a0341e7fc5f6ef7a8c566aacb91e8bf5005c18837894c4eacff41dba54e071ca 83fd0296a3417dccd62dfac26c11aa2febdecd723f34cfe7e46937bc49bf3db6 91fd8a440df8b01573f172ecda38fa01898e415a77a5ed24884e389e8b3c1d40 026efc776db7764f4c07eca7f9891bccc47c4157d520f98b531e246aa3c806fb cab036a40cdd6d75422e5cbf1e9ddf1b55cfc94096307ad24cbef2b93081de34 782bb35682d4ef9d729c8fccc64bdc493f0489581cb1d035e26d431178a75919 c4bf0c31d0fcd0b748d528a7092e3d62aa42faaa880871d68cdd44d065e88200 8ff4dcfdbadd5524bd792b18c60a43f79f0d76ba004af4f8c0bfc4039b8eb4e1 ca603a79d23916e62217d083fb7b947958452b1388c63caacce0ea9dfe22b83c c5b9e9ce5186225ed03fec99bda9da3758f498733a8f5663402cf8c14bcf3951 90a6c22fb8cd0f755b7b1aa666de2fabc0cdd75103022cfee777219288db2c66 e9cf6e150be75780f45489d510f353cf4f5d60e4dd3ac22b82244bb79edcf473 6acae0f5a08c484156fe6263f8cd2088cf38b9522196e488b83b50562faf918c 80a9ed22f5103f7751d8fee03be047f5a4c2962e26f50e752ae0591ed886ecb9 d1b42bba73385692b2793002dfffded760f178d8e76eeee3e72f02bf7dd5043b 32b3912423fec1ce3fc969464d949afd55c764499947bc883038bef67f6c0e88 b4e0d0349ce921662c050e0dfe3f7aa5c2e4651ff4a0accecb6ac47f9795f5f8 481ade0e361f812ae7fa15cca85d597c951a06542f927e5851fc153b50ea5a9e c2682485055ef59bc2a8dbc76a1901d98347c3c733569a498b209e676744af47 065463451e68c82feadc1c5d60233e72b2a4a5738506120611f53c58c61ddb33 90a503d65a56de0af87250e8982efeb80390483026e0d8d0b51682bff30bfe3c ae9a8cefc232cadea47575b96f7189ad261c2c013558b9005a84bd9bf3b5e912 e488657ae2dac3cd1ac377d354027dcc0f5c60b54e414618110a7d4b7d286ab5 264632cd7c4b9e6377d8b5c3c7f0bdfd6250e64aee1a3c9496c261cdc31f8ad4 cac5feb737c7b9196b797205d538db8641af1214a88b9326e2da3568a63e2cc9 988cb9a73d49344e8202b43c443c9a0bd85951f9936298703e08c10446e0d957 fc46734aee3b3b58ebdec493c71551f3427691a0ab074302f96afc2d3ec61582 03d1c6ce2e20a55e918bb885b29fd9270f5b1d2fee5fca398229f2dfd504eb2e fdad243a7496847515841b4bf51bd5a4b933bdc4ef8597ecd9deb8b7b0248ef5 503f7f1ecf5576b2697432ebe5bfd49046ba92e9a615c55edc93d0b36b866b21 fdec2fd4b685ba39f81457f24b5df9e2c28f4ad0c8f6756b23fe39214522d964 61cb60e7bf82f067f11a36e856aea24a6230ad622f8792593ffdd9a22253cde0 d0cb001ed5d70ceaecfd682917984a1fd54bf20c464d641878418667b2bb3ba9 2a76d13a5d946406244b18214db7902e548a23b5d211326290999a9e882e0e17 90785b214850e36134887bfa95f80c7aabe141bfa3705ee41efdd262e7c82929 2bc89c909f3170634bd72921163b2172bf7cb3910db40aaadf0e41cf93a53a46 c5586bfd4991a1474ea6497b68a4f2cc1bf1a48f45a9c3228fd19c77afd98f62 aa073576a26f2ff98ec91dc32b790deea0a660e55ed256c9b0df6a9f877a4839 4b6e351af2dafe8a318fb1718ea471c025cab4a113aa0dc24b697d85c870a25b b5526b05ce41e09cbe640502022006e64cc2b032ed0c2c707af7d3568f7fe38d e0cd93e6c3c84e4cab6fde92e224f540c0ebd80efe9e4f00d8aeb6ec0b745931 3ee27a0cefbda065d8ae3eeda88deb65e9964093dd89ed7f64797c0cabd5bd33 da1939a1332e63986333ee39e2e9ca06725acfd2586393546d566fd83095a1d9 e08f627e5196d2c787a20e63d0ac5f471b5b243bb6057548ba2fe9121b7d7187 64fef28fdeef177f0b6680199c9ea49d8520ce416581bf67bd0c1c48cddda38e ae94b2e330dbd0ac6b81d5bac2dec62426dcea926f8995d4b6470db24f6555a1 deb7294ffaa1f9d0d4b107cc5b616ee15bdab9aecf79979c3e99a9ca3cfe6a90 263f17ebccd1a416afe963e1638a8146d4e8e06734a837bd5f72bd20c2c85690 115e6778303d2e7f49876a2014ebb526455d578f199c02bd963cdd8d6afad12e 039a7ba061ef5f57ff50ee3677242fc70f07eca48332856c08b79394877f8f9b e5fd7b0ae18c2a8a814382297da6260b8e584442601e79394b66e28d513f9f5a 2f509ea54509236c5707175e270f0d21198c107785a9eb463f9dbe337c9c50a1 4d514581aea10c9b2cde74242f76faa81dc21d4cb3fbd98a6f17d975890ea034 5b6662b64b5af9df8c3a2a91a16dff97f369c62857653523c7dfed5c020e604b e5cc4b6b627d650923e1de193a09a9e3c37ca56cd8b02673b7b951e862098bfd ff5615cceae26fb4c9658aaaf915886dab950e7b963728214fb05268adb19c79 1eda1049be11c2ac65777879e46768070b79cd2c27de519b3514581cc14f7dba 71a3049c4486d926723b0f5d01e494957c4cddec5e559355cd64a6418bad8856 66673291ab75d69a1d65d823e75556311a2ff75d82678c979987298a5c6a85ff 50964af5490121ad23aa9058b06ddbb9bcfae4e6061b9c72fbec49dd6d2266c0 8bbc4f14f4d2dbc27557f85d1a3e5cb1efa7c8cbea0635c8d7131f022dd56ea4 547ea1adb6faed9e3d2b527e99fffc20b6b9e2863f06039c9a4a1ba7b9714dca 38c37336dfcd2edf89564f49c1d36e7217cbc584f7768a5e2b69a8ff37940976 6a40f42136e22f1aecbcca813dd96375e2294ee197a03a8a4412b6ce3a3293e8 5ff13bf5812a85213b056267700ee00604f4afb0a625cfa18b7394821e950c9e ccd32d423d096d4fae5b5e354a3e94ac5550ba213b5f9900ab6f0d9c9100a770 7042193ad87f6573dab9eab50eeb4427b6cf35051126570b72252c9bd325bec4 025fdfb6c5d1d2f5e686c619d0eea7d656e706b061eafc36d80c4fd335335229 0edc096668cef81d4cafdf265af4df410c881f34a272dae36910646744b027fb f0acf0301ddef4e990b87dfa05029c46c30c64895d424929b5ace76ee67c93b1 0fe4701381f9e6287c83bcb12cc434fe14c6e44f8236d5c008d663c11185daba 7c5302fdd5a967468e63d67cdcf183afa1f4a505417e51398b3ad654ea314dfc 413a2546d30659930ea8b62c39495c0e9472ecc2ecdbd8eff3c9939a98d1b5f8 85494ee9c0d47f01657b0ec051cda2d2aa92f39033329de48676afcd9ad2c36b 146819fb12a3862f39109b0e00f5fec38f261817c746c7aa9ff63d795b77923d 37654fecb0e61cd2387dd7da6280f1aad052664b35bb41adc755bcf1010742a2 86329d6322c0d5d52c29ccf0ee6f934f80818bea8e4eec4afa03fa7ddce9549e 927497fa8a432aaa1eb40eaf1b343d 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 84 /T put dup 91 /bracketleft put dup 93 /bracketright put dup 101 /e put dup 115 /s put dup 126 /tilde put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueXX 5000791 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b094541f345ec367ae85ca9bd 24568a01d3b9f8095f7420e6c423c414b3dcce6da48dd1c89a56d078e0d0e2f2 62a13640a06d17e44ee3866c3471fb58fedf5a3b77294517651c16bdd7267d39 a54e7171752dbde63ac19bb4b3021ce95eb5fe67390b09ae4d9ed4d704a67443 f55dce17acd996c1f5e023c9e5a18cbeecc3097f23763acb86cdd7cd13381ae7 4e48495ec7fa520539d87f8a8dcb3c826275469b6800876a457e7d1e5be867c7 b1ccad69742a8c9b0ad943482bf2a4ad0aed40baeb69a0233bad36b4ca2d2da7 322956c70375d152653500b2f22d2ab6990cadde2da14b4917f7515e64bc3d96 bf775258fc7dae4e42a4c9b6da8eddec4a800c8aadc8d75e48cae52137e05c03 677f5d6a82fa46d9f2fc7f56d62e5c605a1b7898b8d1401c2cac1a0122a2c8a7 aae09607f2c5f29293a09b9959399283be89051452898238b777db9830ff4318 a298b221c4a820c700ca964fc99e6b1d9eb0bafc39be9aa9dffa2fa326b2a466 1e67261bb3dbc85844e76f9313ab533ee16bcb478166de5502b68e6ec1ea6110 f53206cf1cfea419571b0243244b32ea38b828a01e62b03bcf476a32c2341cec b8a0fa2a3f3f339315d5f3aa7704408f6617d026ffee4925aa48537a4b2e556d 225be09392546ad561cd46cf766f0ba14c6854c5af1a528c3842b0c5aeefefcf 19f19cbce5d967153c4f88e51c776cfbe10d39328466f7a3b83d85601ef4d1ce 0cc136610fa6a74d9676e4e99291d565cad27b9eb2337b0eba4148efd0eb2866 ebf0a4209ba135ad1376cf9cc2d4110f36121055caa3f2e50a08c99d708d7f1b 7f2f2a9fa872f63f820b3b60dc8b518db5d16f8f0036dcfef0bfb148800fcc58 09f4e4b215a147dca09ee07fa968d2375fdacf271ed8980b6f619d95c8a4c44f 0bf94efc5813e10858db2d74719863f85d6e6c87a5cc2f9d56bb8db10a063a72 3013057bce04c467ad6b663f6ef724238de88c1571bdaea9c50dc0921e5f5677 c94176c4d7503ae0f0c58aa7f6de447c1db49698695bc19a13824fee13c360c5 bf1fb37da88a2848e548dd01c20b88096971af2fbada6a21557a9d2196d0295e 3a5cbb4776d8f4a12acbf7c30b99020425ec8f6aceb627f6b3d83210269a856e cfd77c4c2d2bc9ac74434fe31ed256a7f97bc8ac077dcb99cbc52b8d8b9b72c3 4610d9985597c8689c08d167cf869287c9946efd227b8966b49137449ba156ca 194f10724d90a85382b20c6bed2f71715b57a43df69debbe6e98a5d6f59e25e5 25ece314bd887ba5d83dce75f1952fc37c3eac2867aa0a8b717b6c0d511bab29 60f6f04990852419c5cda0e89ff193ca1beba1b52acad3b0aa4cfd50a39a6104 19e9be03d0bf9a1165c38711abcca073cfc30f87002622424b080f2a81b33441 9b19949ecc41bdab4754516ed1037dda6ead3cdb97ad5281e77cb5fb9a6d9d63 ba0d8721cf6e2410f4866d7ee0278983ca0635f6eacc45905b0105228dfb307b 711a20485ce9760ede4eae12f27fc2b2cfbf9034c1376b1ae88c77f3e6b02f67 8d14e42f281a2b1777b0ab622cffa61a193b33d4072c544a6582e794754adf15 feacf1452d52db362c1713477f21f6a93b8147a77aab12baa4827a94f86c74e0 2dcd131af5d8164f6e75d9439e31d7f150627e6b3565646307ad06266f0ddeb5 e2f5e032d1a6f71d0ed6f6a517429d2ba41384b1e1ba7d5b2b7454d9bce27659 9ee807f899972a8dc371a10052c0050c82490013134089d7502d05d2c992e3d3 6223abe5c069e363537fc40294fee0be581eba22614307c39c851c6f84a76956 7a6176077a5807a62a3c34e9cea1a2d974da5a8fdcd0860256093384a4e92cc8 66d569a73bdb1a62737d45768caf96af2b418f863f313957f8d38f8b157f494a f2b70fdbd93192536d2b0e699134a517274d8729d9c95ef50277dbc96dc4e0da 5e45c144a1753b62458316b9903565b1a0cdd505803c75ad37247567a37b497c e7c09de2a20bc50d7a9d0776fa636757d249f9b6872c95e4423b138eb7e55db2 cfa893b5afce3fbbbadefffb343bf860b7d048a21bf85aacb2eaeeaa418a9864 a56320c44f61340c1e1c8d844f7c3de26d5bed7b55263cc5049372663e893590 9428b218c4323e293996c724e5ceefa0c1f25a7d632f816c9b1a4c063c45c3c4 0abad3d9be5a1fd3e220d6727c5159775eaab3c876c531e58ac1a8607a0050b8 18be4130f8c8f341e6382281856a13bdc6f183be42a164e01f789b33ef5e770b 5ff7430df9dceb67ee756d0a1f0e9d8c86e3bbe52072b33dc83f327509595ff7 34e729f4a7009347c21474db430c546a13aba505f0df26cf0004cf40cbd908a9 88a44a5dbc15f9a6ac42397ae44e3d48fced2b9a416ef63d87c5b193c66c7e8a 9d322ad49181b51df5fd7d1be2df672003264c5d3ef8603d904868080a108a4d 9d0043f007b7a77dbb826a5e74b1888d302c22f5acc9f86bf9fc73107a55f0f2 9414f14370a79b5e140a33c72e74434bd1719ca7ccbcdac525a4eaca0aa04627 8db625de5739a8f29fd851fc9cf19c008bd93c324413e1dafc802a8da9d96b85 63c250fdd1bdf6e6132815da7cf4cfd2cb650dbf8ab7c84e2b962e12a880289f 65b1792882be2d256fdb54961b6504881f688a6b45e736ac0ee5d7b057f1ce5a 2a0c63790408d0a43aca5dae391e106ecedee13e06ea3f9a5754576dcf09aec1 ade1f814fda8511a157f059ed51476f04e523d06ddff5ac86959b6173d37e226 2a97b16f4af3c0c153c468f5d913d954e334ce14167a6bacb493c31ca275a550 1417fea7c6e77759c098fe8ab52c53341dca063ca1bd66448db08735d018bbe1 407651a9c1b63c9f9aeee6c348365f78f7fcf357d1ff3a435d6880f6014edb9b 6f3472442681da49cd84e0b4abdd49685100f75eab1ae36d84cf19dab00b2d79 1f89cae975fdada4c9226e16cf6992ccf0fda2317f9141f050da417650828dec b90afd54a5ddefbea6a2ea62b79f6e702a1472fe7bb7c9d890aefda47792475d 3af3377a7b6959000e0c3e27a264a6286ff8dff61f9ebc51446d1ff6982aa7a7 9b612a78f6b1ec57927b1dd22d470b9b9519292ae70890667340a16622708b75 3bcd317a0787b180eb8791aea028e68b7db36669f7433455438d4d1fc7514557 30a67caae33fe4ece02f32ede5d6fa68dc64781abdf61183a23d960abbec56e9 4d2ea0dbf5bbfc81a296b46da5cdf64b094d25b4a78b3b5fd1a626c715182dba c72ace70bd803a53ee9cdb5d1bd8af5b0392b3823a964fe8a90979d42426377d 5923c475a2be720d3c2e0818afe31023cea2fd8711317f78d2c77a9e069ca009 c094e6310cc66a309b9bd312676e50ba773c42df57ee07ef73538ac977c08cbc 295568a0f690ec9b8041c78c2bc4ab5a545126510a94c778f91790df49a88467 5bd50ea881da717ceb8a8d242e32f1d5b64e0a5c1fa83bd286793ae3d710ed53 6ff5730e8575206031bdc4e65ea0df554bd999f22c04c2d46a74969252d3a56e a4920bb2900a43ffcb32b4aa4d09bc3944e818714eacf7497e378e79a2c8c78e ece5119b589d36dd6a40e6e4edc75a7c83eb6ad295231a46ccf45687682753bb 58965efe82f78be8a170bbe2ee7f08a0fcad057f3c85a522e018636504 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 26 /propersubset put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 55 /mapsto put dup 65 /A put dup 66 /B put dup 68 /D put dup 75 /K put dup 76 /L put dup 79 /O put dup 83 /S put dup 84 /T put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 107 /bardbl put dup 112 /radical put dup 114 /nabla put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueXX 5000820 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8 3750d597b842d8d96c423ba1273ddd32f3a54a912a443fcd44f7c3a6fe3956b0 aa1e784aaec6fce08dae0c76da9d0a3eba57b98a6233d9e9f0c3f00fcc6b2c6a 9ba23af389e6dfff4efec3de05d6276c6be417703ce508377f25960ef4ed83b4 9b01b873f3a639ce00f356229b6477a081933fef3bb80e2b9dffa7f75567b1fa 4d739b772f8d674e567534c6c5bbf1cf615372be20b18472f7aa58be8c216dbd df81cc0a86b6d8318ca68fe22c8af13b54d7576fe4ca5a7af9005ea5cc4edb79 c0ab668e4fec4b7f5a9eb5f0e4c088cd818ecc4feb4b40ec8bd2981bf2336074 b64c4301da1e778dc34c37478847febbb8373a4470c67c9c7f4a4ee2bbec944c b6065f544f156c55c4b78d4209061602874835ff6d992c7260173bf08b452149 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 3 /Lambda put dup 6 /Sigma put dup 8 /Phi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 18 /grave put dup 19 /acute put dup 22 /macron put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 126 /tilde put dup 127 /dieresis put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueXX 5000794 def currentdict end currentfile eexec 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2373a5013a9ce40d63fb93ed205bc28b1477b1d1d36f402d2ecdfcd0ab6af0e8 c4ce27292b5faf59f7313bd84a4fa980ca1b83fa9f55b42fc63282dd095d809a b90df2eb2ba1939e2a34cf4e913ebcf5342ce6dbf4b80f76b65b24e527d8d1b8 f157311e2e31d009d9b4ca755a1d7b394a40017a773a931725618a4d5c23dbea d5274438070c870329aff97edf39048bc6846e795a647d72617b740544cfc4a4 73be3b65a496d17f72f5f0d754d758c44ac7f53a560f77353c62b0f943f50f3b c34ae72b51a6ab04f4eb157090dde585f5760e8a277769fc1b7addb918918aa1 3129d69abf892e78659018a6fe2ba7e462413fd9e2aefeef4f0a4eb125ff38e7 dd0719cbd03e4692b9af0d567640262112269a0a8f5bfc51868aa86b7da9a895 03cc2906053e3fb1d286097607588fcebfea1fdb68c16c31eb9e9f4ce0727bd9 0a114f222e842aaffa4974ba7e1426e9975649afac966dac4b99c4d15483ac61 36664fbb01fc22020848f8053a2f676a2b1f015c118309784fda82fd1e8a46f6 da9bf8f3ef9b55e3f827d7ed316d23b2f29a296bbd93a111c3cb1517fea1873c a22482ef83af9609f48ba87cb4facdb80f041742736ddb61a4d5c88669babe20 dbd7c00dcd65806fac8a84680d81bdcfe5a50284a2b95c4831a507c2cac7076a 8ce73fc504d1cb94a8a473aa3587d0163cdaea4bd453c6b40fcfce3b57d4df97 f54216cb66f0b170b44f37b78648f58d461e7a7839c6b331ef47bde11d13dcfc fbc2fd28dd36b2f36eadc66835f3fc5e5e900b0f70b636b2cdc6dbeebf693db2 b24cbc1c452675cbc5e96287936021eb417f86ea667d86c48fa0c1819f3b5691 c195db056ea52006af21d7f23cb6337ba4ce06bd13dad6ca6dbd4317cb5a271f bb89db30652bcf1b0e76fcb0add24e8632d429ef87d1d848858b3e1b7229ba2f be162853ebac1e2a2bc1bdeb55887d84349eac0e6934ac7ad2168d52f7dde934 344af5bd6ca36f524006ae58e19d12f39ff4c8c26c4f43f9c899aaba6a4179c8 2aa8353733b34b038dd2433420140cc896fa164ebd95adbee2d655dda9a9b16e 846480e66cc118ce9e1aeb8ed771691d5e0ac1f47c9fb22939e7c8f489d98667 eb50fc31235b36b2c7891904c59456e9f9b0198e97c4814e2881b407fd33a48d 01f91a32e490e8e4ce3154d9b9ccd984437dadba2178eab416be166faa718928 c9394879c3962a72fe63e36bcc723e0d7bffb1ec25c3e01828035ad6a21db6e3 425882631e45744ce9ad0af2543181cbd9dad49cbf62e725a3b91644e0dd6baa 12964283cf0052d7433a92ba2bbb8c75055619244b0d20da69c6a2de9e5ec8c8 7660a9cf2a8a385f992182d6af7b1096d6289c3d4f7f71d83073aa51531b14df e3b760add6a0b10d9d144324ddff5239a0734f822e1b33354c98818c56f71271 5ba6488c59c520e3a946786daeb2be34631efbefc7ac0f6d14c05b2301d48896 9c78042d394110f42bb10574e43e8ecee2e4404e228d6d4d835c39be236c0302 6df47bf86ddac4a694fdfc1b2a7ed1c812bd3e959c5757d61b6bf2eefa95066b 52c853320a30ad36b29c2471dac3777ac7c2cc611877fa7122827e51615d8204 320ee311e17f86c3548946a28488695a6dbb0715d3ddc179614f7567af26c526 6a48f7ecee8ed5e105ce59adf22cdf63eaeb0cc66ea962502e56f16516680ffb bcb9eac4aa172cf36001a5d7d9d86d31a642dccc7cbe422a4197b1c3fbd6ea37 0f089997ea90630f7698f151e13ff914356fc2f5a721f23cec8a9d06542fd7b8 96e6ae9527a3f9064376882555d00a534908c402231c090b1487f384b61c2796 5547896e0b44e2b5fb34f3351d3fc6069ffecef34da2206d2359c958d42d8734 b8b6d87fbf0e248c29e75734072fbb10c541d7246f1364cfd0a7de9c4be7e89b d2abd55a30c4e3566c9ed3fc7e1288a8dc15139e08c28ad26c5ba42d9aef2d67 fca3c99377f4f87d9ab3c376e6e33649e468159cfe0241416ef9744f8cf86f09 7fca6917e90841cab63217167b5fdbdf226739dbdcdd5105363069f4398b582e 9f17b6df978c3eb2ba27c85360ba647cbde76b7fcc574934081ad03fcebba760 4a8605f243b5d15dc04d44d717c1245a395533b83994ad63f6fd4633a650ea3c ca65b3198024fa10aaab7479bcff1d128abaf73a3152a8e8242e024c1900b7cc 843c011b84a8220e1cfb7fc8597634f55e657cf0116a887fbf1089ee2c71a0af 8222e904a4c9a5780685ba4432f85b9812a8054f61a4b53d779d357c8d3fbe5d 4d2c26ea10a6343c18fc0beb29efb9159a89f8d2a48e5afc9bb546a8fbfb2fb6 f36fcd9150af954dff7c30a2daa9883b6db151f5217eb063aad319a17937403d c252888ccdb040188058d86d234a4c1900f277cc9469f47ea0fd8f4b7fdfd5b1 dc9b8beb4af873d87b830142a164d262f35271a46775dab1ae94e6bc2ffd2bfe 32fa33ebb2cb09b85a0439fdc16c90b5840d783a24f95bbfb4b5825c4d5e8cf1 d9f4c0243a31a801933b3684c508b5f0508404977bf049874ab7dc0c0692bb97 b4a5d282e4403065d4c79d1265ec1f8c0aa06a5aa43b820a290ee1e12b9f1fbb f467ccf540c4df40c371604f148d2346fdd8d2749a060bffa9a6a3b4e6c13add e39faf16cd1c99ce3d4f245dd1 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 65 /A put dup 97 /a put dup 98 /b put dup 99 /c put dup 114 /r put dup 115 /s put dup 116 /t put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueXX 5000768 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306462c563d51ece66fcc9c831843ecae 1fefc1a232e2724f7baee428ae03aadb95c3035345c15e9922fe49e1f2cfc980 237316572dbc57064edac9b0db8913a5e2d45e97e19a91435ccf8adfc835b585 48e74b291446d689c7f2f8c4325e8356e974ae30c3e2977477baeaa33d141fb7 80f59351e84bfc88c87b3a4a1d25e168b9d134554f6a581378c7d2d6eca8ac09 045cc3a5e0ed86f147133f094029e3483bebc81cfde69942cf645ea20d0a2b64 5140ccc2f9cb4a3d969f4b0a79cd3329b440dfcff5285494406f3dd08d804c86 df191e57232f6f1f83df6430f214e03fa5412b1e13d83c60c52a698bc9a38159 19a64162af7b83a9468fae9507c6cb55ea80df8c696dfbc3296c663902d77e89 a181502627281181ac8624f65be9de5fb32f056cc7dd7cbc7366a91717b5892a 34ef6b441871c6993da3a81afc9bf70ff3842a21a29c0e49da1ccba37dd6b920 2702db510b8b1109acf4c27ee8d676bdbd16abe189f34a9b61532eb777457698 cee0561838eaab2d3939c5093afbb215bac433d1918c739b62566906bb2187f3 378521fb52e70c18886e1aa15f53a1f8a169528cf2ddd54c203613ff88e619f2 6c9c50c2c3b8b8c868a88afab1dc8472e7b6f3f992a161723a3c0904865a7c42 3281601eee91f35222c4a71f343739f102eb43145a47e4fac3bd666e4c114a90 5950618ecb1bdc29f90f6ebed1d8325fe6b52e14a8870e9b4bb088e585c71b91 493c12845a101937e3473ac5e9ae24f7c358f06c8b42dee8ef2a1a7a9f536037 57517f2fb6da0ea305d9d20cc74d922ec4b95b8b299459b71825f6f67abee3a6 f5267c6283c7c62aa8d36663071f18e1a3f0b35dcd88d739caf8dbc344630a28 7f583f037768d861f37d7929f248146b5f31ed9f61ef0a9659b50f47ec328516 9fd999b19687e50303557945476fe9845e5c4780045d84d7df253fe61bf69607 fd277a265296b02b4e4b35884f985d29521ee7327887d983bb92dd84ff646661 2b348b5e558d3bc65047afcb202750ea6e9c18a769a4e945e0954787ee65948e 7e8e443a9f2b9f5408e0febf523423b2200166ade223570bd17a6f7dd0386f3d 2246ebb40648548e0a3ff7bab6637a16137c24b042010a3cce2ba9afb9317fcf 4a5b893c4915859ff2b53a77ff0334966f7a234ea8a4349c3e048acd6876f5f3 17b9c238f37ed4231b5e8b661ba52c32821de47f7a0f6fa9c7e0f4bf09bf98a1 8008087aa44eaf37894e83891f552626f1f46563b5188ad949521e1a398eeba3 b9b8f4ac3c24785d50bdf01e5c9b7e4ea1d6b7b6c8019d2d5f353cacf216e30b 71f09d9b95805210964ccbf8f2e30aa04ac3a3a57bde635c9d2be1f5c8f64ad6 19764c72aca8152a51003bc396bce21a56c29306bccc36a37d816de2568e7794 a8f80d857132a3dbaf3ef75409cace85e1b85a8233ef2757a507158bbba8f5f6 d9c08ab4ee39f33a1c1a41ff2b8e81320050b6c4c167596b93918824a65fe1b0 6e7dcf5d7e128a77e5509f3a3a2af6f71de243cccfa5788088d277175e59bd61 a3b5cdc10cfd232cc14682d3b37d3aca68df524af6f38511e5627286034a7c08 1c7596bfc3adecc7d1868cb9d645f8cb61041bccad92249142542934977000c1 bfbbaf570bf1a3fad2368d1e6480ed485642acd1d44258a885562397035dc979 b08877eadba7c99963e7d866be89270b10629964b5c88439c08a7721e2fd6a7d 2a03d9cf2d7fa713411824db51a9141a7c3ca37feeb758ab37376d75b9ad8534 6fe75968a1e7f6b9b549b7bbef5da78116a7d029e6e95db78593abf789ab985e 301f5945c1628f591b4867079526cd9c779b05cd30de4d9dbb820ce964cb3acf cf693b39b15335996a208373a9a1cd362e93d740535d4fe06aaef81a77034010 883c098f8fe19617d6a3cc3ccaa63742d3b251b8baa94aba7ebb7f3cad53c3c5 2c626e3918191af1e343757e0fc5f9fa620ce7e655482aaab7303710167e3041 148e3e6aae44ebe904d96a7dea9281fdbdf93b03ccd08a315df83531214f1726 dd6eff4d5f494262983201ee2afd6cc77eac76155fc59d9e42a5c45735a37af6 c3c9c4aa7c535717838a1ce8ca3424188a3ddc2bb7f58483ef08fe9b6ea090c8 bd5a639bb56a54bed34a443f0f1624f13fffd6b4f24024573a6cdc110f2a85b3 f669c706b0bcf205ee9ff8342b38525e9cbf1f1b772a71c7718b30bd59f46490 dd92c121c9fd0b99edf33c57611ce19ca35fe17079c90ce99afdfd9227184562 f4e2dc14aea6d0989a8f01f17c0983ae121614d8f578d34ead9c05a88007bc4f 90e135d97abf57c1e32fd94ed63ceeddab6c1d42f3238e6cf85a063d1239a1d9 c5de6f9e7954b873ccef98d8b44f1761e66da28b9759a24f181c9f7ca8ca190d 4d5a52b1286ad43b5f8caebe64283da13e6a9e66536a821ebc1b50e98073d79e 335c244c2b8efcd7c0b223983db1608d6f1837ae6f73d970e567960e7e03d1a6 5f7dcaeecd4f9afd0266592ed051f800ee3d5a8954460f3f57a9fb9716f4dca2 616d5e77e19918f2b1c290fa9d049267df21a4b33a06e29c758e0ecac9746be0 b7988da1634f2e7162c2c483f1c8d78eaefc94e907d3bcc79cc389979ab0dc15 bf1b2371fe907e9c6f46e26260566e38ed55175ce116e4da7024db03b6657b44 1543e575e850440d7613b96baa8320e2cde5ab834d1f333a9a749c3ab079546c 3429c90c61bfda4b758861c2e351866a18c4d14d3bee4d863edce298231a71dc 9e4c701ac2d01fead65c2244a81865874b1c22125e9f11720450c10366eb6eec 694ba3acd45ea6f9cda354cc938222a523494fa570ba76d359afd28f6ca82c35 dc9890f6e09346d93734ef8c82d7b62344fb8103a1fe6f84fb56f55e1969c4bb f61c916f5fc7aaf919d620ed2c25d2bac648ede51f981afe1d4ecf8263830258 75069803a3032b0fba0522 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR17 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 45 /hyphen put dup 71 /G put dup 78 /N put dup 83 /S put dup 97 /a put dup 98 /b put dup 100 /d put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-33 -250 945 749}readonly def /UniqueXX 5000795 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be356c29c3aec91ef17f5d31183b1e489f1fa 559a6693fbdd04d4f7e99886d8cb7cca13d4aa9ceb34a708af22e69ccef0e504 c3997075243585edf60b6447c8d01cffb04b6e3ddbaa7da5891286861322a795 e9d990fd56af2f5c2b35801a0b07874e4ac3170481d232e150443253c92a8b05 513d4a4154a99d91f629ccd30376500f5d16aaa9b203bad6350e4b19424c0e33 75f2949bc1927a6f6221252f8a96618e646aa1b0810753c98becc26b37837775 24bca72b818d314074c1c91f4e1f4148d3d703d12d179ee6970bd55d04aa9a9d 97f63f2c7dbd129ec2ed3ba5f8080270ea5854ea165dbcc061e2f47f1aaf0997 2ac9a689a2a9620b91b53f6b5092884372f5f7afca495f331138d85a97c5bb83 d8072dc88010fc89108b60bc06d4f93c722e8908d7ce3c1ea3c70e9e65edd769 7db17ce0e012ad37d4030f33d30a73fd655b7a2e572f3378ac933dd75082f2f1 68702be19bc60cef6bd5f0ada180d21408c04b7f7db9b6dbaefe09e809a80ddf 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119.552 /CMMI12 rf /Ff 134[71 2[71 75 52 53 55 1[75 67 75 112 37 2[37 75 67 41 61 75 60 1[65 14[100 1[92 3[81 6[88 1[97 96 102 12[67 67 67 67 2[37 46[{}31 119.552 /CMBX12 rf /Fg 231[51 6[60 17[{}2 99.6264 /CMMIB10 rf /Fh 142[28 32 1[38 98[40 11[{}4 49.8132 /CMMI6 rf /Fi 133[33 35 40 51 1[41 25 33 32 32 36 1[43 62 1[37 29 24 2[34 3[30 8[41 58 3[41 1[53 5[48 3[58 55 45 7[55 35 55 20 30[31 2[41 31 35 42 3[33 3[31 36 40 45 11[{}39 66.4176 /CMMI8 rf /Fj 133[40 47 45 65 45 52 32 40 41 45 50 50 55 80 25 45 30 30 50 45 30 45 50 45 45 50 3[30 1[30 3[97 72 72 70 55 71 75 66 75 72 87 61 75 51 38 72 75 64 66 74 70 69 72 7[50 50 50 50 50 50 50 50 50 50 1[30 35 30 2[40 40 30 19[50 5[57 55 60 11[{}71 99.6264 /CMTI12 rf /Fk 149[20 42[55 12[47 71 19 14[71 11[55 55 16[35 55 1[55{}11 66.4176 /CMSY8 rf /Fl 204[30 30 30 30 48[{}4 49.8132 /CMR6 rf /Fm 165[50 7[61 82[{}2 66.4176 /CMBX8 rf /Fn 143[100 66 66 19[55 7[47 1[105 52[75 75 6[73 73 59 59 3[33 12[{}13 99.6264 /CMEX10 rf /Fo 129[35 10[28 13[31 7[20 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Fi(m)2082 5498 y Fp(\024)28 b Fr(K)40 b(;)114 b Fs(for)32 b(all)g Fr(n)c Fp(2)g Fq(N)k Fr(;)1865 5800 y Fs(8)p eop %%Page: 9 9 9 8 bop 0 100 a Fs(whic)m(h)34 b(implies)g(\(25\).)146 220 y(T)-8 b(o)38 b(pro)m(v)m(e)h(that)e(\(20\))g(has)h(a)f(unique)i (solution)f(in)f Fr(X)8 b Fs(,)39 b(w)m(e)g(pro)s(ceed)f(as)g(in)f (\([13],)i(Lemma)f(3.1\).)0 340 y(Giv)m(en)33 b Fr(w)352 355 y Fo(1)392 340 y Fr(;)17 b(w)506 355 y Fo(2)572 340 y Fp(2)28 b Fr(X)8 b Fs(,)33 b(w)m(e)g(de\014ne)762 606 y Fr(R)q Fs(\()p Fr(\034)11 b Fs(\))28 b(=)1098 470 y Fn(Z)1198 496 y Fi(\034)1153 696 y Fo(0)1257 606 y Fp(S)7 b Fs(\()p Fr(\034)35 b Fp(\000)22 b Fr(s)p Fs(\))p Fq(v)1681 621 y Fo(1)1721 606 y Fs(\()p Fr(s)p Fs(\))g Fp(\001)f(r)p Fr(w)2067 621 y Fo(2)2107 606 y Fs(\()p Fr(s)p Fs(\))c(d)p Fr(s)32 b(;)114 b Fs(0)27 b Fp(\024)i Fr(\034)39 b Fp(\024)28 b Fr(T)46 b(;)0 878 y Fs(where)f Fq(v)352 893 y Fo(1)392 878 y Fs(\()p Fr(s)p Fs(\))e(is)i(the)f(v)m(elo)s(cit)m(y)i(\014eld)e (obtained)h(from)e Fr(w)2164 893 y Fo(1)2203 878 y Fs(\()p Fr(s)p Fs(\))h(via)g(the)g(Biot-Sa)m(v)-5 b(art)43 b(la)m(w)i(\(14\).) 77 b(If)0 999 y Fr(q)31 b Fp(2)e Fs(\(1)p Fr(;)17 b Fs(2\),)31 b(w)m(e)j(kno)m(w)g(from)e([13])g(that)945 1269 y Fp(k)p Fr(R)q Fs(\()p Fr(\034)11 b Fs(\))p Fp(k)1249 1284 y Fi(m)1344 1269 y Fp(\024)28 b Fr(C)1526 1159 y Fn(\020)1585 1134 y(Z)1685 1160 y Fi(\034)1641 1359 y Fo(0)1871 1202 y Fs(1)p 1754 1246 282 4 v 1754 1338 a Fr(a)p Fs(\()p Fr(s)p Fs(\))1927 1309 y Fo(1)p Fi(=q)2063 1269 y Fs(d)p Fr(s)2163 1159 y Fn(\021)2222 1269 y Fp(k)p Fr(w)2342 1284 y Fo(1)2381 1269 y Fp(k)2431 1284 y Fi(X)2499 1269 y Fp(k)p Fr(w)2619 1284 y Fo(2)2657 1269 y Fp(k)2707 1284 y Fi(X)2807 1269 y Fr(:)0 1539 y Fs(It)33 b(remains)g(to)g (estimate)985 1784 y Fp(r)p Fr(R)q Fs(\()p Fr(\034)11 b Fs(\))29 b(=)1404 1649 y Fn(Z)1504 1675 y Fi(\034)1459 1874 y Fo(0)1563 1784 y Fp(rS)7 b Fs(\()p Fr(\034)35 b Fp(\000)22 b Fr(s)p Fs(\))p Fq(v)2070 1799 y Fo(1)2110 1784 y Fs(\()p Fr(s)p Fs(\))g Fp(\001)g(r)p Fr(w)2457 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(p)s(oin)m(t)g(argumen)m(t)g(that,)h(if)e Fr(T)53 b(>)39 b Fs(0)g(is)h(su\016cien)m(tly)i(small,)0 4516 y(then)33 b(\(20\))f(has)h(a)f(unique)i(solution)f(in)g Fr(X)41 b Fs(with)33 b(the)g(desired)h(prop)s(erties.)924 b Fb(\003)0 4686 y Fs(Next,)48 b(w)m(e)c(decomp)s(ose)i Fr(w)s Fs(\()p Fr(\030)5 b(;)17 b(\034)11 b Fs(\))45 b(=)i Fr(\013)q(G)p Fs(\()p Fr(\030)5 b Fs(\))28 b(+)i Fr(R)q Fs(\()p Fr(\030)5 b(;)17 b(\034)11 b Fs(\),)46 b(where)f Fr(G)p Fs(\()p Fr(\030)5 b Fs(\))43 b(is)h(giv)m(en)h(b)m(y)g(\(8\))e(and)h Fr(\013)k Fs(=)0 4727 y Fn(R)47 4841 y Fm(R)108 4822 y Fl(2)163 4807 y Fr(w)s Fs(\()p Fr(\030)5 b(;)17 b(\034)11 b Fs(\))17 b(d)p Fr(\030)37 b Fs(is)d(time-indep)s(enden)m(t)i(due)e (to)g(\(5\).)46 b(Then)34 b Fr(R)q Fs(\()p Fp(\001)p Fr(;)17 b(\034)11 b Fs(\))30 b Fp(2)g Fr(L)2710 4771 y Fo(2)2710 4831 y(0)2749 4807 y Fs(\()p Fr(m)p Fs(\))k(for)f(all)h Fr(\034)41 b Fp(\025)30 b Fs(0,)j(where)0 4927 y Fr(L)66 4891 y Fo(2)66 4952 y(0)106 4927 y Fs(\()p Fr(m)p Fs(\))g(is)g (de\014ned)h(in)e(\(15\).)43 b(Moreo)m(v)m(er,)35 b Fr(R)e Fs(satis\014es)h(the)f(ev)m(olution)h(equation)1277 5147 y Fr(@)1328 5162 y Fi(\034)1371 5147 y Fr(R)46 b Fs(=)e Fp(L)p Fr(R)23 b Fp(\000)f Fr(\013)q Fs(\003)p Fr(R)h Fp(\000)g Fr(N)10 b Fs(\()p Fr(R)q Fs(\))33 b Fr(;)1103 b Fs(\(28\))0 5367 y(where)866 5488 y(\003)p Fr(R)29 b Fs(=)e Fq(v)1201 5446 y Fi(G)1282 5488 y Fp(\001)22 b(r)p Fr(R)h Fs(+)f Fq(v)1671 5446 y Fi(R)1751 5488 y Fp(\001)g(r)p Fr(G)32 b(;)114 b(N)10 b Fs(\()p Fr(R)q Fs(\))29 b(=)e Fq(v)2566 5446 y Fi(R)2646 5488 y Fp(\001)22 b(r)p Fr(R)33 b(:)693 b Fs(\(29\))1865 5800 y(9)p eop %%Page: 10 10 10 9 bop 0 100 a Fs(Here)31 b Fq(v)289 63 y Fi(G)379 100 y Fs(is)g(giv)m(en)g(b)m(y)h(\(8\))e(and)h Fq(v)1265 63 y Fi(R)1353 100 y Fs(denotes)g(the)g(v)m(elo)s(cit)m(y)i(\014eld)e (asso)s(ciated)h(to)e(the)h(v)m(orticit)m(y)h Fr(R)g Fs(b)m(y)0 220 y(the)h(Biot-Sa)m(v)-5 b(art)32 b(la)m(w)h(\(14\).)43 b(The)33 b(corresp)s(onding)h(in)m(tegral)f(equation)g(is)405 491 y Fr(R)q Fs(\()p Fr(\034)11 b Fs(\))44 b(=)h Fp(S)7 b Fs(\()p Fr(\034)k Fs(\))p Fr(R)1044 506 y Fo(0)1107 491 y Fp(\000)22 b Fr(\013)1286 355 y Fn(Z)1385 381 y Fi(\034)1341 581 y Fo(0)1445 491 y Fp(S)7 b Fs(\()p Fr(\034)34 b Fp(\000)23 b Fr(s)p Fs(\)\003)p Fr(R)q Fs(\()p Fr(s)p Fs(\))17 b(d)p Fr(s)k Fp(\000)2313 355 y Fn(Z)2413 381 y Fi(\034)2368 581 y Fo(0)2472 491 y Fp(S)7 b Fs(\()p Fr(\034)35 b Fp(\000)22 b Fr(s)p Fs(\))p Fr(N)10 b Fs(\()p Fr(R)q Fs(\()p Fr(s)p Fs(\)\))17 b(d)p Fr(s)33 b(:)231 b Fs(\(30\))0 763 y(Equations)36 b(\(12\))f(and)g(\(28\))g(are)g (clearly)h(equiv)-5 b(alen)m(t.)54 b(In)35 b(particular,)h(Theorem)h (1.4)e(implies)i(that,)0 884 y(giv)m(en)d(an)m(y)f Fr(\013)28 b Fp(2)g Fq(R)p Fs(,)33 b(the)g(Cauc)m(h)m(y)h(problem)g(for)e(\(28\))g (is)h(globally)f(w)m(ell-p)s(osed)i(in)f Fr(L)3134 848 y Fo(2)3134 908 y(0)3174 884 y Fs(\()p Fr(m)p Fs(\))g(if)f Fr(m)c(>)g Fs(1.)146 1054 y(W)-8 b(e)33 b(no)m(w)h(pro)m(v)m(e)f(that)g (p)s(ositiv)m(e)h(tra)5 b(jectories)33 b(of)f(\(12\))g(in)h Fr(L)2362 1018 y Fo(2)2402 1054 y Fs(\()p Fr(m)p Fs(\))g(are)f(relativ) m(ely)j(compact.)0 1282 y Fq(Lemma)k(2.2)49 b Fj(L)-5 b(et)36 b Fr(w)828 1297 y Fo(0)897 1282 y Fp(2)29 b Fr(L)1058 1246 y Fo(2)1098 1282 y Fs(\()p Fr(m)p Fs(\))36 b Fj(with)f Fr(m)30 b(>)f Fs(1)p Fj(,)36 b(and)f(let)h Fr(w)31 b Fp(2)f Fr(C)2444 1246 y Fo(0)2483 1282 y Fs(\([0)p Fr(;)17 b Fp(1)p Fs(\))p Fr(;)g(L)2889 1246 y Fo(2)2928 1282 y Fs(\()p Fr(m)p Fs(\)\))36 b Fj(b)-5 b(e)35 b(the)h(solution)0 1403 y(of)28 b(\(12\))g(with)h(initial)f(data)g Fr(w)1082 1418 y Fo(0)1121 1403 y Fj(.)43 b(Then)28 b(the)g(tr)-5 b(aje)g(ctory)29 b Fp(f)p Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))p Fp(g)2331 1418 y Fi(\034)d Fk(\025)p Fo(0)2493 1403 y Fj(is)28 b(r)-5 b(elatively)29 b(c)-5 b(omp)g(act)27 b(in)i Fr(L)3549 1366 y Fo(2)3588 1403 y Fs(\()p Fr(m)p Fs(\))p Fj(.)0 1631 y Fq(Pro)s(of:)40 b Fs(Since)c Fr(w)e Fp(2)e Fr(C)879 1595 y Fo(0)919 1631 y Fs(\([0)p Fr(;)17 b Fp(1)p Fs(\))p Fr(;)g(L)1325 1595 y Fo(2)1363 1631 y Fs(\()p Fr(m)p Fs(\)\),)36 b(it)f(is)g(su\016cien)m(t)i(to)e(sho)m(w) h(that)f Fp(f)p Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))p Fp(g)3119 1646 y Fi(\034)d Fk(\025)p Fo(1)3286 1631 y Fs(is)36 b(relativ)m(ely)0 1751 y(compact)45 b(in)f Fr(L)593 1715 y Fo(2)633 1751 y Fs(\()p Fr(m)p Fs(\).)78 b(W)-8 b(e)44 b(decomp)s(ose)i Fr(w)s Fs(\()p Fr(\030)5 b(;)17 b(\034)11 b Fs(\))46 b(=)h Fr(\013)q(G)p Fs(\()p Fr(\030)5 b Fs(\))29 b(+)g Fr(R)q Fs(\()p Fr(\030)5 b(;)17 b(\034)11 b Fs(\))44 b(as)g(ab)s(o)m(v)m(e,)k(and)c(consider)0 1872 y(the)39 b(in)m(tegral)g(equation)g(\(30\))e(satis\014ed)j(b)m(y)f Fr(R)q Fs(.)61 b(Let)39 b Fr(R)2090 1887 y Fo(1)2130 1872 y Fs(\()p Fr(\034)11 b Fs(\))37 b(=)h Fr(R)q Fs(\()p Fr(\034)11 b Fs(\))26 b Fp(\000)h(S)7 b Fs(\()p Fr(\034)k Fs(\))p Fr(R)3014 1887 y Fo(0)3055 1872 y Fs(,)40 b(so)e(that)g Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))38 b(=)0 1992 y Fr(\013)q(G)27 b Fs(+)h Fp(S)7 b Fs(\()p Fr(\034)k Fs(\))p Fr(R)541 2007 y Fo(0)610 1992 y Fs(+)28 b Fr(R)788 2007 y Fo(1)827 1992 y Fs(\()p Fr(\034)11 b Fs(\).)69 b(Since)42 b Fr(R)1389 2007 y Fo(0)1471 1992 y Fp(2)g Fr(L)1645 1956 y Fo(2)1645 2017 y(0)1685 1992 y Fs(\()p Fr(m)p Fs(\),)h(it)e(follo)m(ws)h(from)e (\(22\))h(that)g Fp(S)7 b Fs(\()p Fr(\034)k Fs(\))p Fr(R)3294 2007 y Fo(0)3375 1992 y Fs(con)m(v)m(erges)0 2112 y(to)41 b(zero)i(in)f Fr(H)556 2076 y Fo(1)595 2112 y Fs(\()p Fr(m)p Fs(\))f(as)h Fr(\034)55 b Fp(!)43 b(1)p Fs(.)71 b(In)42 b(particular,)i(since)f Fp(f)p Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))p Fp(g)2533 2127 y Fi(\034)d Fk(\025)p Fo(1)2708 2112 y Fs(is)42 b(b)s(ounded)h(in)f Fr(H)3435 2076 y Fo(1)3474 2112 y Fs(\()p Fr(m)p Fs(\))g(b)m(y)0 2233 y(Lemma)33 b(2.1,)f(w)m(e)i(see)f(that)f Fp(f)p Fr(R)1168 2248 y Fo(1)1208 2233 y Fs(\()p Fr(\034)11 b Fs(\))p Fp(g)1387 2248 y Fi(\034)d Fk(\025)p Fo(1)1553 2233 y Fs(is)33 b(also)f(b)s(ounded)h(in)g Fr(H)2447 2197 y Fo(1)2486 2233 y Fs(\()p Fr(m)p Fs(\).)43 b(No)m(w,)34 b(w)m(e)f(shall)g(pro)m(v)m(e)h(that)0 2353 y Fp(f)p Fr(R)124 2368 y Fo(1)164 2353 y Fs(\()p Fr(\034)11 b Fs(\))p Fp(g)343 2368 y Fi(\034)d Fk(\025)p Fo(0)510 2353 y Fs(is)36 b(b)s(ounded)f(in)g Fr(L)1193 2317 y Fo(2)1232 2353 y Fs(\()p Fr(m)24 b Fs(+)g(1\).)49 b(By)35 b(Rellic)m(h's)h(criterion)f(\(see)h([24],)f(Theorem)h(XI)s(I)s (I.65\),)0 2474 y(this)h(implies)h(that)e Fp(f)p Fr(R)868 2489 y Fo(1)907 2474 y Fs(\()p Fr(\034)11 b Fs(\))p Fp(g)1086 2489 y Fi(\034)d Fk(\025)p Fo(1)1256 2474 y Fs(is)37 b(relativ)m(ely)h(compact)f(in)f Fr(L)2364 2437 y Fo(2)2404 2474 y Fs(\()p Fr(m)p Fs(\).)55 b(Since)37 b Fp(S)7 b Fs(\()p Fr(\034)k Fs(\))p Fr(R)3175 2489 y Fo(0)3252 2474 y Fs(con)m(v)m(erges)39 b(to)0 2594 y(zero)f(as)g Fr(\034)49 b Fp(!)36 b(1)p Fs(,)j(it)f(follo)m(ws)g(that)g Fp(f)p Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))p Fp(g)1676 2609 y Fi(\034)d Fk(\025)p Fo(1)1847 2594 y Fs(is)38 b(also)g(relativ)m(ely)i(compact)e(in)g Fr(L)3162 2558 y Fo(2)3202 2594 y Fs(\()p Fr(m)p Fs(\),)i(whic)m(h)f(is)0 2714 y(the)33 b(desired)h(result.)146 2835 y(T)-8 b(o)33 b(pro)m(v)m(e)h(the)f(claim,)g(w)m(e)h(decomp)s(ose)g Fr(R)1717 2850 y Fo(1)1756 2835 y Fs(\()p Fr(\034)11 b Fs(\))28 b(=)g Fr(R)2091 2850 y Fo(2)2131 2835 y Fs(\()p Fr(\034)11 b Fs(\))22 b(+)g Fr(R)2454 2850 y Fo(3)2494 2835 y Fs(\()p Fr(\034)11 b Fs(\))22 b(+)g Fr(R)2817 2850 y Fo(4)2857 2835 y Fs(\()p Fr(\034)11 b Fs(\))33 b(where)786 3105 y Fr(R)860 3120 y Fo(2)900 3105 y Fs(\()p Fr(\034)11 b Fs(\))83 b(=)g Fp(\000)p Fr(\013)1428 2970 y Fn(Z)1527 2996 y Fi(\034)1483 3195 y Fo(0)1587 3105 y Fr(e)1632 3064 y Fk(\000)1697 3037 y Fl(1)p 1697 3049 31 3 v 1697 3090 a(2)1737 3064 y Fo(\()p Fi(\034)8 b Fk(\000)p Fi(s)p Fo(\))1923 3105 y Fp(r)22 b(\001)g(S)7 b Fs(\()p Fr(\034)34 b Fp(\000)23 b Fr(s)p Fs(\))p Fq(v)2504 3064 y Fi(G)2563 3105 y Fr(R)q Fs(\()p Fr(s)p Fs(\))17 b(d)p Fr(s)32 b(;)786 3367 y(R)860 3382 y Fo(3)900 3367 y Fs(\()p Fr(\034)11 b Fs(\))83 b(=)g Fp(\000)p Fr(\013)1428 3231 y Fn(Z)1527 3257 y Fi(\034)1483 3456 y Fo(0)1587 3367 y Fr(e)1632 3325 y Fk(\000)1697 3298 y Fl(1)p 1697 3310 V 1697 3351 a(2)1737 3325 y Fo(\()p Fi(\034)8 b Fk(\000)p Fi(s)p Fo(\))1923 3367 y Fp(r)22 b(\001)g(S)7 b Fs(\()p Fr(\034)34 b Fp(\000)23 b Fr(s)p Fs(\))p Fq(v)2504 3325 y Fi(R)2561 3367 y Fs(\()p Fr(s)p Fs(\))p Fr(G)17 b Fs(d)p Fr(s)32 b(;)786 3628 y(R)860 3643 y Fo(4)900 3628 y Fs(\()p Fr(\034)11 b Fs(\))83 b(=)g Fp(\000)1365 3492 y Fn(Z)1465 3518 y Fi(\034)1420 3718 y Fo(0)1524 3628 y Fr(e)1569 3586 y Fk(\000)1634 3559 y Fl(1)p 1634 3571 V 1634 3613 a(2)1675 3586 y Fo(\()p Fi(\034)8 b Fk(\000)p Fi(s)p Fo(\))1860 3628 y Fp(r)22 b(\001)g(S)7 b Fs(\()p Fr(\034)35 b Fp(\000)22 b Fr(s)p Fs(\))p Fq(v)2441 3586 y Fi(R)2499 3628 y Fs(\()p Fr(s)p Fs(\))p Fr(R)q Fs(\()p Fr(s)p Fs(\))17 b(d)p Fr(s)32 b(:)0 3907 y Fs(T)-8 b(o)33 b(b)s(ound)f Fr(R)518 3922 y Fo(2)558 3907 y Fs(\()p Fr(\034)11 b Fs(\),)33 b(w)m(e)h(observ)m(e)g(that)e Fr(b)p Fq(v)1551 3871 y Fi(G)1638 3907 y Fp(2)d Fr(L)1799 3871 y Fk(1)1874 3907 y Fs(\()p Fq(R)1996 3871 y Fo(2)2035 3907 y Fs(\).)43 b(Using)33 b(\(21\),)f(w)m(e)i(th)m(us)f(\014nd)368 4178 y Fp(k)p Fr(R)492 4193 y Fo(2)532 4178 y Fs(\()p Fr(\034)11 b Fs(\))p Fp(k)711 4193 y Fi(m)p Fo(+1)952 4178 y Fs(=)83 b Fp(j)p Fr(\013)q Fp(j)1247 4063 y Fn(\014)1247 4123 y(\014)1247 4183 y(\014)1279 4178 y Fr(b)1320 4137 y Fi(m)p Fo(+1)1494 4042 y Fn(Z)1594 4068 y Fi(\034)1549 4268 y Fo(0)1653 4178 y Fr(e)1698 4137 y Fk(\000)1763 4109 y Fl(1)p 1763 4121 V 1763 4163 a(2)1804 4137 y Fo(\()p Fi(\034)8 b Fk(\000)p Fi(s)p Fo(\))1989 4178 y Fp(r)22 b(\001)g(S)7 b Fs(\()p Fr(\034)35 b Fp(\000)22 b Fr(s)p Fs(\))p Fq(v)2570 4137 y Fi(G)2629 4178 y Fr(R)q Fs(\()p Fr(s)p Fs(\))2826 4063 y Fn(\014)2826 4123 y(\014)2826 4183 y(\014)2859 4247 y Fo(2)2915 4178 y Fs(d)p Fr(s)951 4439 y Fp(\024)83 b Fr(C)7 b Fp(j)p Fr(\013)q Fp(j)1324 4303 y Fn(Z)1422 4330 y Fi(\034)1378 4529 y Fo(0)1482 4439 y Fr(e)1527 4398 y Fk(\000)1592 4371 y Fl(1)p 1592 4383 V 1592 4424 a(2)1633 4398 y Fo(\()p Fi(\034)h Fk(\000)p Fi(s)p Fo(\))2005 4371 y Fs(1)p 1828 4416 403 4 v 1828 4524 a Fr(a)p Fs(\()p Fr(\034)34 b Fp(\000)23 b Fr(s)p Fs(\))2187 4460 y Fl(1)p 2186 4472 31 3 v 2186 4513 a(2)2241 4439 y Fp(j)p Fr(b)2310 4398 y Fi(m)p Fo(+1)2467 4439 y Fq(v)2528 4398 y Fi(G)2587 4439 y Fr(R)q Fs(\()p Fr(s)p Fs(\))p Fp(j)2812 4454 y Fo(2)2867 4439 y Fs(d)p Fr(s)639 b Fs(\(31\))951 4720 y Fp(\024)83 b Fr(C)7 b Fp(j)p Fr(\013)q Fp(j)1324 4584 y Fn(Z)1422 4611 y Fi(\034)1378 4810 y Fo(0)1482 4720 y Fr(e)1527 4679 y Fk(\000)1592 4652 y Fl(1)p 1592 4664 V 1592 4705 a(2)1633 4679 y Fo(\()p Fi(\034)h Fk(\000)p Fi(s)p Fo(\))2005 4653 y Fs(1)p 1828 4697 403 4 v 1828 4805 a Fr(a)p Fs(\()p Fr(\034)34 b Fp(\000)23 b Fr(s)p Fs(\))2187 4741 y Fl(1)p 2186 4753 31 3 v 2186 4794 a(2)2241 4720 y Fp(j)p Fr(b)p Fq(v)2371 4679 y Fi(G)2430 4720 y Fp(j)2458 4735 y Fk(1)2532 4720 y Fp(k)p Fr(R)q Fs(\()p Fr(s)p Fs(\))p Fp(k)2829 4735 y Fi(m)2912 4720 y Fs(d)p Fr(s)28 b Fp(\024)g Fr(C)7 b(K)3312 4679 y Fo(2)3305 4745 y(1)3384 4720 y Fr(;)0 5013 y Fs(for)29 b(all)h Fr(\034)39 b Fp(\025)28 b Fs(0,)j(where)g Fr(K)934 5028 y Fo(1)1001 5013 y Fs(=)d(sup)1251 5036 y Fi(\034)8 b Fk(\025)p Fo(0)1401 5013 y Fp(k)p Fr(w)s Fs(\()p Fr(\034)j Fs(\))p Fp(k)1703 5028 y Fi(m)1769 5013 y Fs(.)43 b(T)-8 b(o)30 b(b)s(ound)g Fr(R)2352 5028 y Fo(3)2392 5013 y Fs(\()p Fr(\034)11 b Fs(\),)30 b(w)m(e)h(pro)s(ceed) g(in)f(the)h(same)f(w)m(a)m(y)-8 b(.)0 5142 y(Using)33 b(H\177)-49 b(older's)34 b(inequalit)m(y)-8 b(,)34 b(estimate)g(\(19\)) e(and)h(the)g(em)m(b)s(edding)h Fr(L)2642 5105 y Fo(2)2682 5142 y Fs(\()p Fr(m)p Fs(\))27 b Fr(,)-17 b Fp(!)28 b Fr(L)3074 5105 y Fo(4)p Fi(=)p Fo(3)3185 5142 y Fs(,)k(w)m(e)i(obtain) 807 5362 y Fp(j)p Fr(b)876 5320 y Fi(m)p Fo(+1)1033 5362 y Fq(v)1094 5320 y Fi(R)1151 5362 y Fr(G)p Fp(j)1256 5377 y Fo(2)1323 5362 y Fp(\024)28 b(j)p Fq(v)1517 5320 y Fi(R)1574 5362 y Fp(j)1602 5377 y Fo(4)1642 5362 y Fp(j)p Fr(b)1711 5320 y Fi(m)p Fo(+1)1867 5362 y Fr(G)p Fp(j)1972 5377 y Fo(4)2039 5362 y Fp(\024)g Fr(C)7 b Fp(j)p Fr(R)q Fp(j)2352 5377 y Fo(4)p Fi(=)p Fo(3)2489 5362 y Fp(\024)29 b Fr(C)7 b Fp(k)p Fr(R)q Fp(k)2847 5377 y Fi(m)2945 5362 y Fr(;)634 b Fs(\(32\))1841 5800 y(10)p eop %%Page: 11 11 11 10 bop 0 100 a Fs(so)35 b(that)f Fp(k)p Fr(R)459 115 y Fo(3)499 100 y Fs(\()p Fr(\034)11 b Fs(\))p Fp(k)678 115 y Fi(m)p Fo(+1)866 100 y Fp(\024)32 b Fr(C)7 b(K)1142 63 y Fo(2)1135 124 y(1)1216 100 y Fs(for)34 b(all)g Fr(\034)43 b Fp(\025)32 b Fs(0.)49 b(Finally)-8 b(,)36 b(to)e(b)s(ound)h Fr(R)2675 115 y Fo(4)2715 100 y Fs(\()p Fr(\034)11 b Fs(\))35 b(w)m(e)g(\014x)h Fr(q)f Fp(2)c Fs(\(1)p Fr(;)17 b Fs(2\))34 b(suc)m(h)0 220 y(that)e Fr(q)g Fp(\025)c Fs(2)p Fr(=m)p Fs(.)43 b(Com)m(bining)34 b(\(21\))e(and)h(\(23\),)f(w)m (e)h(obtain)588 489 y Fp(k)p Fr(R)712 504 y Fo(4)752 489 y Fs(\()p Fr(\034)11 b Fs(\))p Fp(k)931 504 y Fi(m)p Fo(+1)1115 489 y Fp(\024)28 b Fr(C)1314 353 y Fn(Z)1414 380 y Fi(\034)1369 579 y Fo(0)1473 489 y Fr(e)1518 448 y Fk(\000)1583 421 y Fl(1)p 1583 433 31 3 v 1583 474 a(2)1624 448 y Fo(\()p Fi(\034)8 b Fk(\000)p Fi(s)p Fo(\))2023 421 y Fs(1)p 1819 466 457 4 v 1819 557 a Fr(a)p Fs(\()p Fr(\034)34 b Fp(\000)23 b Fr(s)p Fs(\))2168 528 y Fo(1)p Fi(=q)2286 489 y Fp(j)p Fr(b)2355 448 y Fi(m)p Fo(+1)2512 489 y Fq(v)2573 448 y Fi(R)2631 489 y Fs(\()p Fr(s)p Fs(\))p Fr(R)q Fs(\()p Fr(s)p Fs(\))p Fp(j)2978 504 y Fi(q)3032 489 y Fs(d)p Fr(s)32 b(:)415 b Fs(\(33\))0 770 y(By)33 b(H\177)-49 b(older's)34 b(inequalit)m(y)-8 b(,)34 b Fp(j)p Fr(b)p Fq(v)1137 734 y Fi(R)1195 770 y Fr(b)1236 734 y Fi(m)1303 770 y Fr(R)q Fp(j)1406 785 y Fi(q)1472 770 y Fp(\024)28 b(j)p Fr(b)p Fq(v)1707 734 y Fi(R)1765 770 y Fp(j)1826 768 y Fl(2)p Fh(q)p 1802 783 109 3 v 1802 824 a Fl(2)p Fc(\000)p Fh(q)1925 770 y Fp(k)p Fr(R)q Fp(k)2100 785 y Fi(m)2166 770 y Fs(.)44 b(Applying)34 b(Prop)s(osition)f(B.1\(2\))f(in)h([13)o(],)0 924 y(w)m(e)38 b(also)g(obtain)f Fp(j)p Fr(b)p Fq(v)787 888 y Fi(R)845 924 y Fp(j)907 922 y Fl(2)p Fh(q)p 883 937 V 883 978 a Fl(2)p Fc(\000)p Fh(q)1041 924 y Fp(\024)g Fr(C)7 b Fp(k)p Fr(R)q Fp(k)1417 922 y Fl(2)p 1416 934 31 3 v 1416 975 a Fh(q)1497 924 y Fp(\024)36 b Fr(C)7 b Fp(k)p Fr(R)q Fp(k)1862 939 y Fi(m)1928 924 y Fs(.)58 b(Inserting)39 b(these)g(b)s(ounds)f(in)m(to)f(\(33\),)h(w)m(e)h (\014nd)0 1075 y(that)e Fp(k)p Fr(R)340 1090 y Fo(4)380 1075 y Fs(\()p Fr(\034)11 b Fs(\))p Fp(k)559 1090 y Fi(m)661 1075 y Fp(\024)36 b Fr(C)7 b(K)941 1039 y Fo(2)934 1100 y(1)1018 1075 y Fs(for)37 b(all)g Fr(\034)48 b Fp(\025)36 b Fs(0.)57 b(Summarizing,)40 b(w)m(e)e(ha)m(v)m(e)h(sho)m(wn)g(that)e Fp(k)p Fr(R)3291 1090 y Fo(1)3330 1075 y Fs(\()p Fr(\034)11 b Fs(\))p Fp(k)3509 1090 y Fi(m)p Fo(+1)3702 1075 y Fp(\024)0 1196 y Fr(C)c(K)167 1160 y Fo(2)160 1220 y(1)239 1196 y Fs(for)32 b(all)h Fr(\034)39 b Fp(\025)28 b Fs(0,)k(whic)m(h)i (concludes)h(the)e(pro)s(of.)1747 b Fb(\003)146 1365 y Fs(Finally)-8 b(,)51 b(w)m(e)c(sho)m(w)h(that)e(negativ)m(e)i(or)e (complete)i(tra)5 b(jectories)48 b(of)e(\(12\))g(in)h Fr(L)3177 1329 y Fo(2)3216 1365 y Fs(\()p Fr(m)p Fs(\))g(that)f(are)0 1486 y(b)s(ounded)33 b(in)g Fr(L)578 1450 y Fo(2)618 1486 y Fs(\()p Fr(m)p Fs(\))g(for)f(some)h Fr(m)28 b(>)g Fs(1)k(are)h(also)f(relativ)m(ely)j(compact.)0 1712 y Fq(Lemma)k(2.3)49 b Fj(Assume)35 b(that)g Fr(m)28 b(>)f Fs(1)35 b Fj(and)f(that)h Fr(w)30 b Fp(2)e Fr(C)2111 1676 y Fo(0)2150 1712 y Fs(\()p Fq(R)p Fr(;)17 b(L)2382 1676 y Fo(2)2421 1712 y Fs(\()p Fr(m)p Fs(\)\))35 b Fj(is)f(a)h (solution)f(of)g(\(12\))g(which)0 1832 y(is)h(b)-5 b(ounde)g(d)34 b(in)g Fr(L)661 1796 y Fo(2)701 1832 y Fs(\()p Fr(m)p Fs(\))p Fj(.)45 b(Then)34 b Fp(f)p Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))p Fp(g)1493 1847 y Fi(\034)d Fk(2)p Fm(R)1678 1832 y Fj(is)35 b(r)-5 b(elatively)35 b(c)-5 b(omp)g(act)34 b(in)g Fr(L)2759 1796 y Fo(2)2799 1832 y Fs(\()p Fr(m)p Fs(\))p Fj(.)0 2058 y Fq(Pro)s(of:)50 b Fs(By)43 b(assumption,)48 b(there)c(exists)h Fr(K)1696 2073 y Fo(1)1781 2058 y Fr(>)h Fs(0)d(suc)m(h)i(that)e Fp(k)p Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))p Fp(k)2750 2073 y Fi(m)2862 2058 y Fp(\024)46 b Fr(K)3068 2073 y Fo(1)3151 2058 y Fs(for)d(all)g Fr(\034)57 b Fp(2)47 b Fq(R)p Fs(.)0 2179 y(As)e(in)f(the)h(pro)s(of)e (of)h(Lemma)h(2.2,)i(w)m(e)e(decomp)s(ose)h Fr(w)s Fs(\()p Fr(\030)5 b(;)17 b(\034)11 b Fs(\))47 b(=)g Fr(\013)q(G)p Fs(\()p Fr(\030)5 b Fs(\))29 b(+)h Fr(R)q Fs(\()p Fr(\030)5 b(;)17 b(\034)11 b Fs(\),)47 b(where)e Fr(\013)k Fs(=)0 2219 y Fn(R)47 2334 y Fm(R)108 2315 y Fl(2)163 2299 y Fr(w)s Fs(\()p Fr(\030)5 b(;)17 b(\034)11 b Fs(\))17 b(d)p Fr(\030)5 b Fs(.)42 b(The)33 b(remainder)h Fr(R)q Fs(\()p Fr(\034)11 b Fs(\))33 b(satis\014es)h(the)f(in)m(tegral)g (equation)157 2578 y Fr(R)q Fs(\()p Fr(\034)11 b Fs(\))44 b(=)h Fp(S)7 b Fs(\()p Fr(\034)34 b Fp(\000)23 b Fr(\034)849 2593 y Fo(0)888 2578 y Fs(\))p Fr(R)q Fs(\()p Fr(\034)1081 2593 y Fo(0)1121 2578 y Fs(\))f Fp(\000)h Fr(\013)1360 2442 y Fn(Z)1460 2468 y Fi(\034)1415 2668 y(\034)1446 2677 y Fl(0)1519 2578 y Fp(S)7 b Fs(\()p Fr(\034)35 b Fp(\000)22 b Fr(s)p Fs(\)\003)p Fr(R)q Fs(\()p Fr(s)p Fs(\))17 b(d)p Fr(s)22 b Fp(\000)2387 2442 y Fn(Z)2487 2468 y Fi(\034)2443 2668 y(\034)2474 2677 y Fl(0)2547 2578 y Fp(S)7 b Fs(\()p Fr(\034)34 b Fp(\000)23 b Fr(s)p Fs(\))p Fr(N)10 b Fs(\()p Fr(R)q Fs(\()p Fr(s)p Fs(\)\))17 b(d)p Fr(s)32 b(;)157 b Fs(\(34\))0 2862 y(for)34 b(all)g Fr(\034)330 2877 y Fo(0)401 2862 y Fr(<)c(\034)11 b Fs(.)49 b(Since)36 b Fr(R)q Fs(\()p Fr(\034)1048 2877 y Fo(0)1088 2862 y Fs(\))30 b Fp(2)h Fr(L)1319 2826 y Fo(2)1319 2887 y(0)1359 2862 y Fs(\()p Fr(m)p Fs(\))k(and)f Fp(k)p Fr(R)q Fs(\()p Fr(\034)1951 2877 y Fo(0)1991 2862 y Fs(\))p Fp(k)2079 2877 y Fi(m)2176 2862 y Fp(\024)d Fr(C)7 b(K)2444 2877 y Fo(1)2517 2862 y Fs(for)34 b(all)g Fr(\034)2847 2877 y Fo(0)2918 2862 y Fp(2)d Fq(R)p Fs(,)j(it)h(follo)m(ws)g(from)0 2982 y(\(22\))f(that)g Fp(kS)7 b Fs(\()p Fr(\034)36 b Fp(\000)24 b Fr(\034)797 2997 y Fo(0)836 2982 y Fs(\))p Fr(R)q Fs(\()p Fr(\034)1029 2997 y Fo(0)1069 2982 y Fs(\))p Fp(k)1157 2997 y Fi(m)1254 2982 y Fp(!)31 b Fs(0)j(as)h Fr(\034)1632 2997 y Fo(0)1703 2982 y Fp(!)30 b(\0001)p Fs(.)50 b(Moreo)m(v)m(er,)36 b(pro)s(ceeding)g(as)f(in)f(the)h(pro)s (of)f(of)0 3103 y(Lemma)40 b(2.2)f(and)h(using)g(the)g(analogues)g(of)e (estimates)k(\(31\),)e(\(33\),)h(it)e(is)h(easy)h(to)e(see)h(that)g(b)s (oth)0 3223 y(in)m(tegrals)34 b(in)f(the)g(righ)m(t-hand)f(side)i(of)e (\(34\))g(ha)m(v)m(e)i(a)f(limit)g(in)g Fr(L)2388 3187 y Fo(2)2428 3223 y Fs(\()p Fr(m)p Fs(\))g(\(and)f(ev)m(en)j(in)e Fr(L)3252 3187 y Fo(2)3291 3223 y Fs(\()p Fr(m)23 b Fs(+)f(1\)\))32 b(as)0 3343 y Fr(\034)42 3358 y Fo(0)109 3343 y Fp(!)c(\0001)p Fs(.)43 b(Th)m(us,)35 b(w)m(e)e(ha)m(v)m(e)h(the)f(represen)m(tation) 237 3607 y Fr(R)q Fs(\()p Fr(\034)11 b Fs(\))45 b(=)f Fp(\000)p Fr(\013)763 3471 y Fn(Z)862 3498 y Fi(\034)818 3697 y Fk(\0001)964 3607 y Fp(S)7 b Fs(\()p Fr(\034)35 b Fp(\000)22 b Fr(s)p Fs(\)\003)p Fr(R)q Fs(\()p Fr(s)p Fs(\))17 b(d)p Fr(s)22 b Fp(\000)1832 3471 y Fn(Z)1932 3498 y Fi(\034)1888 3697 y Fk(\0001)2034 3607 y Fp(S)7 b Fs(\()p Fr(\034)34 b Fp(\000)23 b Fr(s)p Fs(\))p Fr(N)10 b Fs(\()p Fr(R)q Fs(\()p Fr(s)p Fs(\)\))17 b(d)p Fr(s)32 b(;)115 b(\034)39 b Fp(2)28 b Fq(R)k Fr(;)237 b Fs(\(35\))0 3889 y(whic)m(h)47 b(implies)f(that)f Fp(k)p Fr(R)q Fs(\()p Fr(\034)11 b Fs(\))p Fp(k)1163 3904 y Fi(m)p Fo(+1)1369 3889 y Fp(\024)49 b Fr(C)7 b(K)1662 3853 y Fo(2)1655 3913 y(1)1746 3889 y Fs(for)45 b(all)g Fr(\034)60 b Fp(2)50 b Fq(R)p Fs(.)80 b(This)46 b(sho)m(ws)h(that)e Fp(f)p Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))p Fp(g)3519 3904 y Fi(\034)d Fk(2)p Fm(R)3714 3889 y Fs(is)0 4009 y(b)s(ounded)33 b(in)f Fr(L)577 3973 y Fo(2)617 4009 y Fs(\()p Fr(m)21 b Fs(+)g(1\).)43 b(On)32 b(the)g(other)g(hand,)g(it)g(follo)m(ws)h (from)f(Lemma)g(2.1)g(that)g Fp(f)p Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))p Fp(g)3532 4024 y Fi(\034)d Fk(2)p Fm(R)3714 4009 y Fs(is)0 4129 y(b)s(ounded)27 b(in)g Fr(H)589 4093 y Fo(1)628 4129 y Fs(\()p Fr(m)p Fs(\),)g(hence)h Fp(f)p Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))p Fp(g)1410 4144 y Fi(\034)d Fk(2)p Fm(R)1587 4129 y Fs(is)27 b(relativ)m(ely)h(compact)f (in)f Fr(L)2655 4093 y Fo(2)2695 4129 y Fs(\()p Fr(m)p Fs(\))g(b)m(y)i(Rellic)m(h's)g(criterion.)3702 4250 y Fb(\003)0 4476 y Fq(Remark)38 b(2.4)49 b Fj(By)g(a)h(b)-5 b(o)g(otstr)g(ap)49 b(ar)-5 b(gument,)53 b(it)d(is)f(cle)-5 b(ar)49 b(fr)-5 b(om)49 b(the)h(pr)-5 b(o)g(of)49 b(of)g(L)-5 b(emma)49 b(2.3)g(that)0 4596 y Fp(f)p Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))p Fp(g)302 4611 y Fi(\034)d Fk(2)p Fm(R)497 4596 y Fj(is)43 b(b)-5 b(ounde)g(d)44 b(in)g Fr(H)1209 4560 y Fi(k)1251 4596 y Fs(\()p Fr(m)1374 4560 y Fk(0)1397 4596 y Fs(\))g Fj(for)g(al)5 b(l)44 b Fr(k)k Fp(2)d Fq(N)f Fj(and)g(al)5 b(l)43 b Fr(m)2567 4560 y Fk(0)2636 4596 y Fp(2)i Fq(N)p Fj(.)72 b(In)44 b(other)g(wor)-5 b(ds,)46 b(the)0 4717 y(tr)-5 b(aje)g(ctory)35 b Fp(f)p Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))p Fp(g)740 4732 y Fi(\034)d Fk(2)p Fm(R)926 4717 y Fj(is)34 b(b)-5 b(ounde)g(d)34 b(in)h(the)g(Schwartz)f(sp)-5 b(ac)g(e)34 b Fp(S)7 b Fs(\()p Fq(R)2538 4681 y Fo(2)2578 4717 y Fs(\))p Fj(.)0 5005 y Ff(2.2)135 b(Compactness)46 b(in)f Fe(L)1359 4962 y Fd(1)1404 5005 y Fw(\()p Ff(R)1550 4962 y Fd(2)1596 5005 y Fw(\))0 5190 y Fs(W)-8 b(e)40 b(no)m(w)h(study)g(the)g (compactness)h(prop)s(erties)f(of)e(the)i(solutions)g(of)e(\(12\))h(in) g Fr(L)3104 5154 y Fo(1)3144 5190 y Fs(\()p Fq(R)3266 5154 y Fo(2)3305 5190 y Fs(\).)66 b(W)-8 b(e)40 b(\014rst)0 5310 y(recall)30 b(t)m(w)m(o)g(imp)s(ortan)m(t)g(estimates)h(for)d(the) i(solutions)h(of)e(the)g(original)g(v)m(orticit)m(y)i(equation)g (\(2\),)e(and)0 5431 y(use)37 b(the)g(c)m(hange)g(of)f(v)-5 b(ariables)37 b(\(11\))f(to)g(obtain)g(the)h(corresp)s(onding)g(b)s (ounds)h(on)e(the)h(solutions)g(of)0 5551 y(\(12\).)1841 5800 y(11)p eop %%Page: 12 12 12 11 bop 146 100 a Fs(The)44 b(\014rst)f(is)g(a)g(smo)s(othing)g (estimate,)j(see)e(for)e(instance)i([19].)74 b(Under)43 b(the)g(assumptions)i(of)0 220 y(Theorem)34 b(1.1,)e(the)h(v)m(orticit) m(y)h Fr(!)t Fs(\()p Fr(x;)17 b(t)p Fs(\))32 b(satis\014es)j(for)d(all) g Fr(p)c Fp(2)g Fs(\(1)p Fr(;)17 b Fp(1)p Fs(])1278 502 y Fp(jr)p Fr(!)t Fs(\()p Fp(\001)p Fr(;)g(t)p Fs(\))p Fp(j)1665 517 y Fi(p)1731 502 y Fp(\024)1890 426 y Fr(C)1967 390 y Fk(0)1960 451 y Fi(p)p 1846 479 197 4 v 1846 597 a Fr(t)1891 523 y Fl(3)p 1891 535 31 3 v 1891 577 a(2)1932 551 y Fk(\000)1998 523 y Fl(1)p 1996 535 33 3 v 1996 577 a Fh(p)2085 502 y Fr(;)114 b(t)28 b(>)g Fs(0)k Fr(;)1105 b Fs(\(36\))0 780 y(where)37 b Fr(C)362 744 y Fk(0)355 804 y Fi(p)431 780 y Fs(dep)s(ends)h(only)e(on)g Fp(j)p Fr(!)1257 795 y Fo(0)1296 780 y Fp(j)1324 795 y Fo(1)1363 780 y Fs(.)55 b(The)37 b(second)g(is)g(a)f(nice)h(p)s(oin)m(t)m(wise)h (estimate)f(due)g(to)f(Carlen)0 900 y(and)i(Loss,)i(see)g(Theorem)f(3)f (in)g([9].)60 b(F)-8 b(or)38 b(an)m(y)h Fr(\014)j Fp(2)c Fs(\(0)p Fr(;)17 b Fs(1\),)39 b(there)f(exists)i Fr(C)2880 915 y Fi(\014)2964 900 y Fr(>)d Fs(0)h(\(dep)s(ending)i(on)0 1020 y Fp(j)p Fr(!)89 1035 y Fo(0)128 1020 y Fp(j)156 1035 y Fo(1)195 1020 y Fs(\))32 b(suc)m(h)i(that)425 1300 y Fp(j)p Fr(!)t Fs(\()p Fr(x;)17 b(t)p Fs(\))p Fp(j)27 b(\024)h Fr(C)958 1315 y Fi(\014)1021 1164 y Fn(Z)1077 1390 y Fm(R)1138 1371 y Fl(2)1203 1232 y Fs(1)p 1203 1277 49 4 v 1210 1368 a Fr(t)1278 1300 y Fs(exp)1427 1189 y Fn(\020)1487 1300 y Fp(\000)p Fr(\014)1634 1232 y Fp(j)p Fr(x)23 b Fp(\000)g Fr(y)t Fp(j)1920 1196 y Fo(2)p 1635 1277 324 4 v 1754 1368 a Fs(4)p Fr(t)1968 1189 y Fn(\021)2027 1300 y Fp(j)p Fr(!)2116 1315 y Fo(0)2155 1300 y Fs(\()p Fr(y)t Fs(\))p Fp(j)17 b Fs(d)p Fr(y)34 b(;)115 b(x)28 b Fp(2)g Fq(R)2867 1259 y Fo(2)2938 1300 y Fr(;)115 b(t)27 b(>)h Fs(0)k Fr(:)252 b Fs(\(37\))146 1576 y(Assume)30 b(no)m(w)f(that)f Fr(w)979 1591 y Fo(0)1046 1576 y Fp(2)g Fr(L)1206 1540 y Fo(1)1246 1576 y Fs(\()p Fq(R)1368 1540 y Fo(2)1407 1576 y Fs(\),)h(and)f(let)h Fr(w)h Fp(2)e Fr(C)2094 1540 y Fo(0)2133 1576 y Fs(\([0)p Fr(;)17 b Fp(1)p Fs(\))p Fr(;)g(L)2539 1540 y Fo(1)2578 1576 y Fs(\()p Fq(R)2700 1540 y Fo(2)2739 1576 y Fs(\)\))28 b(b)s(e)g(the)h(solution)f(of)g(\(12\))0 1697 y(with)33 b(initial)g(data)f Fr(w)796 1712 y Fo(0)835 1697 y Fs(.)43 b(Then)34 b(the)e(function)h Fr(!)t Fs(\()p Fr(x;)17 b(t)p Fs(\))32 b(de\014ned)i(b)m(y)f(\(11\))f(is)h(a)f(solution)h(of)f (\(2\))g(on)g(the)0 1817 y(time)d(in)m(terv)-5 b(al)30 b([1)p Fr(;)17 b Fp(1)p Fs(\))28 b(with)h(initial)h(data)e Fr(!)t Fs(\()p Fr(x;)17 b Fs(1\))27 b(=)g Fr(w)2057 1832 y Fo(0)2096 1817 y Fs(\()p Fr(x)p Fs(\).)43 b(Applying)30 b(\(36\))e(and)h(returning)h(to)e(the)0 1938 y(rescaled)34 b(v)-5 b(ariables,)33 b(w)m(e)h(obtain)1183 2214 y Fp(jr)p Fr(w)s Fs(\()p Fp(\001)p Fr(;)17 b(\034)11 b Fs(\))p Fp(j)1596 2229 y Fi(p)1662 2214 y Fp(\024)1894 2138 y Fr(C)1971 2102 y Fk(0)1964 2163 y Fi(p)p 1777 2191 342 4 v 1777 2309 a Fr(a)p Fs(\()p Fr(\034)g Fs(\))1967 2236 y Fl(3)p 1968 2248 31 3 v 1968 2289 a(2)2008 2263 y Fk(\000)2074 2236 y Fl(1)p 2073 2248 33 3 v 2073 2289 a Fh(p)2162 2214 y Fr(;)114 b(\034)39 b(>)28 b Fs(0)k Fr(;)1010 b Fs(\(38\))0 2518 y(where)34 b Fr(a)p Fs(\()p Fr(\034)11 b Fs(\))28 b(=)g(1)22 b Fp(\000)g Fr(e)809 2482 y Fk(\000)p Fi(\034)907 2518 y Fs(.)44 b(Similarly)-8 b(,)33 b(w)m(e)h(deduce)g (from)e(\(37\))g(that)143 2804 y Fp(j)p Fr(w)s Fs(\()p Fr(\030)5 b(;)17 b(\034)11 b Fs(\))p Fp(j)25 b(\024)k Fr(C)694 2819 y Fi(\014)757 2669 y Fn(Z)812 2894 y Fm(R)873 2875 y Fl(2)1004 2737 y Fs(1)p 939 2782 181 4 v 939 2873 a Fr(a)p Fs(\()p Fr(\034)11 b Fs(\))1162 2804 y(exp)1311 2694 y Fn(\020)1371 2804 y Fp(\000)p Fr(\014)1518 2737 y Fp(j)p Fr(\030)27 b Fp(\000)c Fr(y)t(e)1813 2701 y Fk(\000)p Fi(\034)8 b(=)p Fo(2)1981 2737 y Fp(j)2009 2701 y Fo(2)p 1519 2782 529 4 v 1669 2873 a Fs(4)p Fr(a)p Fs(\()p Fr(\034)j Fs(\))2058 2694 y Fn(\021)2117 2804 y Fp(j)p Fr(w)2215 2819 y Fo(0)2254 2804 y Fs(\()p Fr(y)t Fs(\))p Fp(j)17 b Fs(d)p Fr(y)34 b(;)115 b(\030)32 b Fp(2)c Fq(R)2958 2763 y Fo(2)3029 2804 y Fr(;)115 b(\034)39 b(>)27 b Fs(0)33 b Fr(:)142 b Fs(\(39\))0 3075 y(With)33 b(these)g(estimates)h(at)e(hand,)g(w)m(e)h(are)g(no)m(w)f(ready)h(to)f (pro)m(v)m(e)i(the)e(main)h(result)g(of)e(this)i(section.)0 3303 y Fq(Lemma)39 b(2.5)49 b Fj(L)-5 b(et)43 b Fr(w)835 3318 y Fo(0)916 3303 y Fp(2)e Fr(L)1089 3267 y Fo(1)1129 3303 y Fs(\()p Fq(R)1251 3267 y Fo(2)1290 3303 y Fs(\))p Fj(,)j(and)e(let)h Fr(w)g Fp(2)f Fr(C)2042 3267 y Fo(0)2082 3303 y Fs(\([0)p Fr(;)17 b Fp(1)p Fs(\))p Fr(;)g(L)2488 3267 y Fo(1)2526 3303 y Fs(\()p Fq(R)2648 3267 y Fo(2)2687 3303 y Fs(\)\))43 b Fj(b)-5 b(e)42 b(the)g(solution)g(of)g(\(12\))0 3423 y(with)35 b(initial)f(data)h Fr(w)785 3438 y Fo(0)824 3423 y Fj(.)45 b(Then)34 b Fp(f)p Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))p Fp(g)1455 3438 y Fi(\034)d Fk(\025)p Fo(0)1622 3423 y Fj(is)35 b(r)-5 b(elatively)35 b(c)-5 b(omp)g(act)34 b(in)g Fr(L)2703 3387 y Fo(1)2743 3423 y Fs(\()p Fq(R)2865 3387 y Fo(2)2904 3423 y Fs(\))p Fj(.)0 3652 y Fq(Pro)s(of:)42 b Fs(Again,)37 b(it)g(is)g(su\016cien)m(t)i(to)d(sho)m(w)h(that)g Fp(f)p Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))p Fp(g)2165 3667 y Fi(\034)d Fk(\025)p Fo(1)2334 3652 y Fs(is)37 b(relativ)m(ely)h(compact.)56 b(Fix)37 b Fr(R)e(>)g Fs(0.)0 3772 y(F)-8 b(rom)32 b(\(39\))g(one)h(has)g Fp(j)p Fr(w)s Fs(\()p Fr(\030)5 b(;)17 b(\034)11 b Fs(\))p Fp(j)26 b(\024)i Fr(w)1366 3787 y Fo(1)1405 3772 y Fs(\()p Fr(\030)5 b(;)17 b(\034)11 b Fs(\))22 b(+)g Fr(w)1816 3787 y Fo(2)1855 3772 y Fs(\()p Fr(\030)5 b(;)17 b(\034)11 b Fs(\))32 b(where)568 4058 y Fr(w)638 4073 y Fo(1)677 4058 y Fs(\()p Fr(\030)5 b(;)17 b(\034)11 b Fs(\))82 b(=)h Fr(C)1209 4073 y Fi(\014)1273 3923 y Fn(Z)1328 4148 y Fk(j)p Fi(y)r Fk(j\024)p Fi(R)1610 3991 y Fs(1)p 1544 4035 181 4 v 1544 4127 a Fr(a)p Fs(\()p Fr(\034)11 b Fs(\))1767 4058 y(exp)1916 3948 y Fn(\020)1976 4058 y Fp(\000)p Fr(\014)2123 3991 y Fp(j)p Fr(\030)27 b Fp(\000)c Fr(y)t(e)2418 3955 y Fk(\000)p Fi(\034)8 b(=)p Fo(2)2586 3991 y Fp(j)2614 3955 y Fo(2)p 2124 4035 529 4 v 2274 4127 a Fs(4)p Fr(a)p Fs(\()p Fr(\034)j Fs(\))2663 3948 y Fn(\021)2722 4058 y Fp(j)p Fr(w)2820 4073 y Fo(0)2859 4058 y Fs(\()p Fr(y)t Fs(\))p Fp(j)17 b Fs(d)p Fr(y)34 b(;)568 4352 y(w)638 4367 y Fo(2)677 4352 y Fs(\()p Fr(\030)5 b(;)17 b(\034)11 b Fs(\))82 b(=)h Fr(C)1209 4367 y Fi(\014)1273 4216 y Fn(Z)1328 4441 y Fk(j)p Fi(y)r Fk(j\025)p Fi(R)1610 4284 y Fs(1)p 1544 4329 181 4 v 1544 4420 a Fr(a)p Fs(\()p Fr(\034)11 b Fs(\))1767 4352 y(exp)1916 4241 y Fn(\020)1976 4352 y Fp(\000)p Fr(\014)2123 4284 y Fp(j)p Fr(\030)27 b Fp(\000)c Fr(y)t(e)2418 4248 y Fk(\000)p Fi(\034)8 b(=)p Fo(2)2586 4284 y Fp(j)2614 4248 y Fo(2)p 2124 4329 529 4 v 2274 4420 a Fs(4)p Fr(a)p Fs(\()p Fr(\034)j Fs(\))2663 4241 y Fn(\021)2722 4352 y Fp(j)p Fr(w)2820 4367 y Fo(0)2859 4352 y Fs(\()p Fr(y)t Fs(\))p Fp(j)17 b Fs(d)p Fr(y)34 b(:)0 4717 y Fs(If)g Fp(j)p Fr(\030)5 b Fp(j)30 b(\025)h Fs(2)p Fr(R)q Fs(,)k(then)g Fp(j)p Fr(\030)28 b Fp(\000)c Fr(y)t(e)1048 4681 y Fk(\000)p Fi(\034)8 b(=)p Fo(2)1216 4717 y Fp(j)30 b(\025)h(j)p Fr(\030)5 b Fp(j)23 b(\000)h(j)p Fr(y)t Fp(j)29 b(\025)i(j)p Fr(\030)5 b Fp(j)p Fr(=)p Fs(2)33 b(whenev)m(er)k Fp(j)p Fr(y)t Fp(j)29 b(\024)j Fr(R)j Fs(and)g Fr(\034)42 b Fp(\025)31 b Fs(0.)49 b(It)35 b(follo)m(ws)0 4837 y(that)724 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Fr(\014)2426 5304 y Fp(j)p Fr(z)t Fp(j)2531 5268 y Fo(2)p 2427 5348 145 4 v 2450 5440 a Fs(16)2581 5261 y Fn(\021)2657 5371 y Fs(d)p Fr(z)k Fs(=:)c Fr(")2999 5386 y Fo(1)3038 5371 y Fs(\()p Fr(R)q Fs(\))32 b Fr(:)1841 5800 y Fs(12)p eop %%Page: 13 13 13 12 bop 0 100 a Fs(Moreo)m(v)m(er,)34 b(using)f(F)-8 b(ubini's)34 b(theorem,)f(w)m(e)h(\014nd)789 212 y Fn(Z)845 437 y Fm(R)906 418 y Fl(2)961 347 y Fr(w)1031 362 y Fo(2)1070 347 y Fs(\()p Fr(\030)5 b(;)17 b(\034)11 b Fs(\))17 b(d)p Fr(\030)48 b Fp(\024)d Fr(C)1645 362 y Fi(\014)1702 280 y Fs(4)p Fr(\031)p 1702 324 108 4 v 1725 416 a(\014)1836 212 y Fn(Z)1891 437 y Fk(j)p Fi(y)r Fk(j\025)p Fi(R)2097 347 y Fp(j)p Fr(w)2195 362 y Fo(0)2234 347 y Fs(\()p Fr(\030)5 b Fs(\))p Fp(j)17 b Fs(d)p Fr(\030)47 b Fs(=:)e Fr(")2741 362 y Fo(2)2780 347 y Fs(\()p Fr(R)q Fs(\))32 b Fr(:)0 616 y Fs(Th)m(us)i(w)m(e)g(ha)m(v)m(e)g(sho)m(wn)644 844 y(sup)653 923 y Fi(\034)8 b(>)p Fo(0)808 709 y Fn(Z)863 934 y Fk(j)p Fi(\030)s Fk(j\025)p Fo(2)p Fi(R)1100 844 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Fp(j)p Fr(w)s Fs(\()p Fr(\030)c Fp(\000)c Fr(y)t(;)17 b(\034)11 b Fs(\))22 b Fp(\000)g Fr(w)s Fs(\()p Fr(\030)5 b(;)17 b(\034)11 b Fs(\))p Fp(j)17 b Fs(d)p Fr(\030)48 b Fp(\024)c Fr(\031)t(R)1902 2841 y Fo(2)1942 2882 y Fp(j)p Fr(y)t Fp(j)77 b Fs(sup)2067 2968 y Fk(j)p Fi(\030)s Fk(j\024)p Fi(R)p Fo(+1)2353 2882 y Fp(jr)p Fr(w)s Fs(\()p Fr(\030)5 b(;)17 b(\034)11 b Fs(\))p Fp(j)26 b(\024)i Fr(\031)t(R)3051 2841 y Fo(2)3090 2882 y Fr(\016)t(K)35 b Fp(\024)3372 2815 y Fr(")p 3370 2859 49 4 v 3370 2950 a Fs(3)3462 2882 y Fr(;)0 3167 y Fs(so)e(that)331 3087 y Fn(R)378 3202 y Fm(R)439 3183 y Fl(2)494 3167 y Fp(j)p Fr(w)s Fs(\()p Fr(\030)26 b Fp(\000)d Fr(y)t(;)17 b(\034)11 b Fs(\))21 b Fp(\000)i Fr(w)s Fs(\()p Fr(\030)5 b(;)17 b(\034)11 b Fs(\))p Fp(j)17 b Fs(d)p Fr(\030)30 b Fp(\024)e Fr(")p Fs(.)43 b(Th)m(us)34 b(w)m(e)g(ha)m(v)m(e)g(sho)m(wn)739 3430 y(sup)748 3511 y Fi(\034)8 b Fk(\025)p Fo(1)911 3430 y Fs(sup)902 3516 y Fk(j)p Fi(y)r Fk(j\024)p Fi(\016)1084 3295 y Fn(Z)1139 3520 y Fm(R)1200 3501 y Fl(2)1255 3430 y Fp(j)p Fr(w)s Fs(\()p Fr(\030)26 b Fp(\000)d Fr(y)t(;)17 b(\034)11 b Fs(\))21 b Fp(\000)i Fr(w)s Fs(\()p Fr(\030)5 b(;)17 b(\034)11 b Fs(\))p Fp(j)17 b Fs(d)p Fr(\030)30 b Fp(!)d Fs(0)98 b(as)32 b Fr(\016)g Fp(!)27 b Fs(0)33 b Fr(:)565 b Fs(\(41\))0 3710 y(By)34 b(the)g(Riesz)g(criterion)g (\([24],)f(Theorem)i(XI)s(I)s(I.66\),)e(it)h(follo)m(ws)g(from)f (\(40\),)g(\(41\))g(that)g Fp(f)p Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))p Fp(g)3647 3725 y Fi(\034)d Fk(\025)p Fo(1)0 3831 y Fs(is)33 b(relativ)m(ely)i(compact)e(in)g Fr(L)1094 3794 y Fo(1)1133 3831 y Fs(\()p Fq(R)1255 3794 y Fo(2)1294 3831 y Fs(\),)g(whic)m(h)h(is)f(the)g(desired)h(result.)1169 b Fb(\003)146 3996 y Fs(T)-8 b(o)36 b(conclude)i(this)e(subsection,)j (w)m(e)e(remark)f(that)g(Lemma)g(2.3)g(has)g(no)g(analogue)g(in)g Fr(L)3514 3960 y Fo(1)3554 3996 y Fs(\()p Fq(R)3676 3960 y Fo(2)3714 3996 y Fs(\),)0 4117 y(namely)k(negativ)m(e)f(tra)5 b(jectories)40 b(of)e(\(12\))g(that)g(are)g(b)s(ounded)h(in)g Fr(L)2558 4081 y Fo(1)2598 4117 y Fs(\()p Fq(R)2720 4081 y Fo(2)2759 4117 y Fs(\))f(need)i(not)e(b)s(e)h(relativ)m(ely)0 4237 y(compact)33 b(in)g(that)g(space.)45 b(T)-8 b(o)33 b(see)h(this,)f(let)g Fr(!)1736 4252 y Fo(0)1803 4237 y Fp(2)c Fr(L)1964 4201 y Fo(1)2004 4237 y Fs(\()p Fq(R)2126 4201 y Fo(2)2165 4237 y Fs(\))j(and)h(let)g Fr(!)t Fs(\()p Fr(x;)17 b(t)p Fs(\))32 b(b)s(e)h(the)g(solution)h(of)e(\(2\))0 4358 y(giv)m(en)c(b)m(y)g(Theorem)g(1.1.)42 b(If)26 b Fr(w)s Fs(\()p Fr(\030)5 b(;)17 b(\034)11 b Fs(\))27 b(=)h Fr(e)1540 4321 y Fi(\034)1583 4358 y Fr(!)t Fs(\()p Fr(\030)5 b(e)1779 4321 y Fi(\034)j(=)p Fo(2)1892 4358 y Fr(;)17 b(e)1981 4321 y Fi(\034)2024 4358 y Fs(\),)28 b(then)f Fp(f)p Fr(w)s Fs(\()p Fp(\001)p Fr(;)17 b(\034)11 b Fs(\))p Fp(g)2707 4373 y Fi(\034)d Fk(\024)p Fo(0)2866 4358 y Fs(is)28 b(a)f(b)s(ounded)g(negativ)m(e)0 4478 y(tra)5 b(jectory)32 b(in)g Fr(L)633 4442 y Fo(1)673 4478 y Fs(\()p Fq(R)795 4442 y Fo(2)834 4478 y Fs(\).)43 b(Ho)m(w)m(ev)m(er,)34 b Fr(w)s Fs(\()p Fp(\001)p Fr(;)17 b(\034)11 b Fs(\))31 b(is)h(\\ev)-5 b(anescen)m(t")34 b(as)e Fr(\034)39 b Fp(!)27 b(\0001)p Fs(,)32 b(hence)h Fp(f)p Fr(w)s Fs(\()p Fp(\001)p Fr(;)17 b(\034)11 b Fs(\))p Fp(g)3550 4493 y Fi(\034)d Fk(\024)p Fo(0)3714 4478 y Fs(is)0 4598 y(not)32 b(relativ)m(ely)j(compact)e(in)g Fr(L)1169 4562 y Fo(1)1209 4598 y Fs(\()p Fq(R)1331 4562 y Fo(2)1370 4598 y Fs(\))f(unless)i Fr(!)1789 4613 y Fo(0)1856 4598 y Fp(\021)28 b Fs(0.)0 4885 y Ff(2.3)135 b(Preserv)-7 b(ation)46 b(of)f(p)t(ositivit)l(y)0 5070 y Fs(A)26 b(more)g(qualitativ)m(e)h(prop)s(ert)m(y)f(that)f(will)i(b)s (e)e(essen)m(tial)j(for)d(our)h(analysis)h(is)f(the)g(fact)f(that)h (solutions)0 5190 y(of)41 b(\(2\),)i(and)f(hence)h(of)d(\(12\),)k (satisfy)e(a)f(maxim)m(um)i(principle.)72 b(W)-8 b(e)41 b(state)h(this)g(prop)s(ert)m(y)g(in)g(the)0 5310 y(original)f(v)-5 b(ariables,)44 b(and)e(for)e(a)h(generalized)i(v)m(ersion)g(of)d(\(2\)) 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y(\010)h(is)g(constan)m(t.)1841 5800 y(14)p eop %%Page: 15 15 15 14 bop 0 100 a Fq(Lemma)39 b(3.1)49 b Fj(L)-5 b(et)43 b Fr(w)835 115 y Fo(0)916 100 y Fp(2)e Fr(L)1089 63 y Fo(1)1129 100 y Fs(\()p Fq(R)1251 63 y Fo(2)1290 100 y Fs(\))p Fj(,)j(and)e(let)h Fr(w)g Fp(2)f Fr(C)2042 63 y Fo(0)2082 100 y Fs(\([0)p Fr(;)17 b Fp(1)p Fs(\))p Fr(;)g(L)2488 63 y Fo(1)2526 100 y Fs(\()p Fq(R)2648 63 y Fo(2)2687 100 y Fs(\)\))43 b Fj(b)-5 b(e)42 b(the)g(solution)g(of) g(\(12\))0 220 y(with)c(initial)g(data)g Fr(w)795 235 y Fo(0)834 220 y Fj(.)56 b(Then)37 b Fs(\010\()p Fr(w)s Fs(\()p Fr(\034)11 b Fs(\)\))34 b Fp(\024)h Fs(\010\()p Fr(w)1849 235 y Fo(0)1888 220 y Fs(\))k Fj(for)f(al)5 b(l)38 b Fr(\034)45 b Fp(\025)35 b Fs(0)p Fj(.)55 b(Mor)-5 b(e)g(over)38 b Fs(\010\()p Fr(w)s Fs(\()p Fr(\034)11 b Fs(\)\))34 b(=)g(\010\()p Fr(w)3702 235 y Fo(0)3742 220 y Fs(\))0 340 y Fj(for)h(al)5 b(l)34 b Fr(\034)39 b Fp(\025)28 b Fs(0)35 b Fj(if)g(and)f(only)h(if)f Fr(w)1225 355 y Fo(0)1292 340 y Fp(2)28 b 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Fs(log)18 b Fr(w)788 1166 y Fo(1)849 1151 y Fp(\000)23 b Fr(w)1019 1166 y Fo(2)1075 1151 y Fs(log)17 b Fr(w)1287 1166 y Fo(2)1326 1151 y Fp(j)44 b(\024)h Fr(C)1597 1041 y Fn(\020)1657 1151 y Fp(j)p Fr(w)1755 1166 y Fo(1)1816 1151 y Fp(\000)22 b Fr(w)1985 1166 y Fo(2)2024 1151 y Fp(j)2052 1110 y Fo(1)p Fi(=)p Fo(2)2184 1151 y Fs(+)g Fp(j)p Fr(w)2380 1166 y Fo(1)2441 1151 y Fp(\000)h Fr(w)2611 1166 y Fo(2)2650 1151 y Fp(j)p Fs(\()p Fr(w)2789 1100 y Fo(1)p Fi(=)p Fo(2)2786 1176 y(1)2920 1151 y Fs(+)f Fr(w)3091 1100 y Fo(1)p Fi(=)p Fo(2)3088 1176 y(2)3201 1151 y Fs(\))3239 1041 y Fn(\021)3331 1151 y Fr(;)0 1386 y Fs(one)32 b(v)m(eri\014es)j (that)c Fr(H)40 b Fs(is)33 b(con)m(tin)m(uous)g(on)f Fr(L)1621 1350 y Fo(2)1661 1386 y Fs(\()p Fr(m)p Fs(\))21 b Fp(\\)h Fs(\006)2001 1401 y Fo(+)2092 1386 y Fs(\(equipp)s(ed)34 b(with)f(the)f(top)s(ology)g(of)f Fr(L)3513 1350 y Fo(2)3553 1386 y Fs(\()p Fr(m)p Fs(\)\).)0 1507 y(W)-8 b(e)34 b(no)m(w)h(sho)m(w) g(that)f Fr(H)41 b Fs(is)35 b(indeed)g(a)f(Ly)m(apuno)m(v)h(function)f (for)g(the)g(semi\015o)m(w)i(de\014ned)f(b)m(y)g(\(12\))f(on)0 1627 y Fr(L)66 1591 y Fo(2)106 1627 y Fs(\()p Fr(m)p Fs(\))22 b Fp(\\)h Fs(\006)448 1642 y Fo(+)507 1627 y Fs(.)0 1837 y Fq(Lemma)39 b(3.2)49 b Fj(Assume)28 b(that)g Fr(w)1208 1852 y Fo(0)1275 1837 y Fp(2)g Fr(L)1435 1801 y Fo(2)1475 1837 y Fs(\()p Fr(m)p Fs(\))6 b Fp(\\)g Fs(\006)1784 1852 y Fo(+)1872 1837 y Fj(with)28 b Fr(m)g(>)f Fs(3)p Fj(,)i(and)e(let)h Fr(w)i Fp(2)e Fr(C)2984 1801 y Fo(0)3023 1837 y Fs(\([0)p Fr(;)17 b Fp(1)p Fs(\))p Fr(;)g(L)3429 1801 y Fo(2)3468 1837 y Fs(\()p Fr(m)p Fs(\)\))28 b Fj(b)-5 b(e)0 1957 y(the)32 b(solution)f(of)g(\(12\))g(with)g(initial)h(data)f Fr(w)1621 1972 y Fo(0)1660 1957 y Fj(.)44 b(Then)30 b Fr(H)8 b Fs(\()p Fr(w)s Fs(\()p Fr(\034)j Fs(\)\))27 b Fp(\024)h Fr(H)8 b Fs(\()p Fr(w)2680 1972 y Fo(0)2719 1957 y Fs(\))32 b Fj(for)f(al)5 b(l)31 b Fr(\034)39 b Fp(\025)28 b Fs(0)p Fj(.)44 b(Mor)-5 b(e)g(over)0 2077 y Fr(H)8 b Fs(\()p Fr(w)s Fs(\()p 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3192 y Fo(2)2411 3281 y Fs(d)p Fr(\030)32 b Fp(\025)c Fs(0)k Fr(:)853 b Fs(\(48\))0 3504 y(Remark)41 b(that)e Fr(I)8 b Fs(\()p Fr(w)s Fs(\))39 b(v)-5 b(anishes)42 b(if)e(and)g(only)g(if)g Fr(w)i Fs(is)f(prop)s (ortional)e(to)h Fr(G)p Fs(.)65 b(Th)m(us,)44 b(under)c(the)h(as-)0 3624 y(sumptions)c(ab)s(o)m(v)m(e,)g(it)e(is)h(clear)g(that)f Fr(I)8 b Fs(\()p Fr(w)s Fs(\()p Fr(\034)j Fs(\)\))32 b Fr(>)g Fs(0)j(at)g(least)h(for)f Fr(\034)44 b(>)32 b Fs(0)j(su\016cien)m(tly)j(small,)f(hence)0 3745 y Fr(H)8 b Fs(\()p Fr(w)s Fs(\()p Fr(\034)j Fs(\)\))27 b Fr(<)g(H)8 b Fs(\()p Fr(w)694 3760 y Fo(0)733 3745 y Fs(\))32 b(for)h(all)f Fr(\034)39 b(>)28 b Fs(0.)146 3865 y(Th)m(us,)34 b(all)e(that)f (remains)i(is)g(to)e(pro)m(v)m(e)i(\(47\).)43 b(Assume)33 b(\014rst)g(that)e Fr(w)2691 3880 y Fo(0)2762 3865 y Fs(b)s(elongs)i(to)e(the)h(Sc)m(h)m(w)m(artz)0 3985 y(space)h Fp(S)7 b Fs(\()p Fq(R)449 3949 y Fo(2)489 3985 y Fs(\).)43 b(Then)34 b Fr(w)c Fp(2)e Fr(C)1123 3949 y Fo(1)1162 3985 y Fs(\([0)p 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Fn(\021)2762 4884 y Fr(w)2832 4899 y Fo(0)2871 4884 y Fs(\()p Fr(y)t Fs(\))17 b(d)p Fr(y)35 b(;)426 b Fs(\(49\))0 5143 y(see)35 b(also)e(\(39\).)46 b(Using)34 b(these)h(prop)s(erties,)f(it)g(is)g(straigh)m(tforw)m(ard)g(to)f(v)m (erify)i(that)e Fr(\034)41 b Fp(7!)28 b Fr(H)8 b Fs(\()p Fr(w)s Fs(\()p Fr(\034)j Fs(\)\))33 b(is)0 5264 y(di\013eren)m(tiable)h (for)e Fr(\034)39 b(>)28 b Fs(0,)k(and)h(that)304 5444 y(d)p 278 5488 108 4 v 278 5579 a(d)p Fr(\034)395 5511 y(H)8 b Fs(\()p Fr(w)s Fs(\()p Fr(\034)j Fs(\)\))44 b(=)926 5375 y Fn(Z)981 5601 y Fm(R)1042 5582 y Fl(2)1081 5400 y Fn(\020)1140 5511 y Fs(1)22 b(+)g(log)1464 5444 y Fr(w)p 1462 5488 77 4 v 1462 5579 a(G)1549 5400 y Fn(\021)1608 5511 y Fr(@)1659 5526 y Fi(\034)1703 5511 y Fr(w)d Fs(d)p Fr(\030)32 b Fs(=)2025 5375 y Fn(Z)2080 5601 y Fm(R)2141 5582 y Fl(2)2180 5400 y Fn(\020)2239 5511 y Fs(1)22 b(+)g(log)2563 5444 y Fr(w)p 2561 5488 V 2561 5579 a(G)2648 5400 y Fn(\021)2707 5511 y Fs(\()p Fp(L)p Fr(w)i Fp(\000)f Fq(v)h Fp(\001)e(r)p Fr(w)s Fs(\))17 b(d)p Fr(\030)36 b(:)1841 5800 y Fs(16)p eop %%Page: 17 17 17 16 bop 0 100 a Fs(Next,)34 b(using)f(the)g(iden)m(tit)m(y)h Fp(L)p Fr(w)c Fs(=)e(div)r(\()p Fr(G)p Fp(r)p Fs(\()1703 60 y Fi(w)p 1702 77 55 4 v 1702 134 a(G)1766 100 y Fs(\)\))33 b(and)f(in)m(tegrating)h(b)m(y)h(parts,)f(w)m(e)g(obtain)508 233 y Fn(Z)563 458 y Fm(R)624 440 y Fl(2)663 258 y Fn(\020)722 368 y Fs(1)22 b(+)g(log)1046 301 y Fr(w)p 1044 346 77 4 v 1044 437 a(G)1131 258 y Fn(\021)1190 368 y Fs(\()p Fp(L)p Fr(w)s Fs(\))17 b(d)p Fr(\030)86 b Fs(=)d Fp(\000)1861 233 y Fn(Z)1917 458 y Fm(R)1978 440 y Fl(2)2033 368 y Fp(r)2116 258 y Fn(\020)2175 368 y Fs(log)2330 301 y Fr(w)p 2328 346 V 2328 437 a(G)2415 258 y Fn(\021)2497 368 y Fp(\001)2556 301 y Fr(G)p 2556 346 V 2558 437 a(w)2643 368 y Fp(r)2726 258 y Fn(\020)2798 301 y Fr(w)p 2796 346 V 2796 437 a(G)2883 258 y Fn(\021)2942 368 y Fr(w)19 b Fs(d)p Fr(\030)1608 627 y Fs(=)83 b Fp(\000)1861 492 y Fn(Z)1917 717 y Fm(R)1978 698 y Fl(2)2033 627 y Fr(w)2106 513 y Fn(\014)2106 572 y(\014)2106 632 y(\014)2139 627 y Fp(r)2222 517 y Fn(\020)2281 627 y Fs(log)2436 560 y Fr(w)p 2434 604 V 2434 696 a(G)2521 517 y Fn(\021)2580 513 y(\014)2580 572 y(\014)2580 632 y(\014)2613 539 y Fo(2)2669 627 y Fs(d)p Fr(\030)49 b Fs(=)44 b Fp(\000)p Fr(I)8 b Fs(\()p Fr(w)s Fs(\))32 b Fr(:)0 891 y Fs(On)h(the)g(other)f (hand,)h(using)h Fq(v)23 b Fp(\001)f(r)p Fr(w)30 b Fs(=)e(div)r(\()p Fq(v)r Fr(w)s Fs(\))j(and)i(in)m(tegrating)g(b)m(y)h(parts,)e(w)m(e)i (\014nd)70 1025 y Fn(Z)126 1251 y Fm(R)187 1232 y Fl(2)225 1050 y Fn(\020)285 1161 y Fs(1)22 b(+)g(log)609 1094 y Fr(w)p 606 1138 V 606 1229 a(G)693 1050 y Fn(\021)753 1161 y Fs(\()p Fq(v)h Fp(\001)f(r)p Fr(w)s Fs(\))17 b(d)p Fr(\030)87 b Fs(=)1477 1025 y Fn(Z)1532 1251 y Fm(R)1593 1232 y Fl(2)1632 1161 y Fs(\(1)22 b(+)g(log)q(\(4)p Fr(\031)t(w)s Fs(\)\)\()p Fq(v)g Fp(\001)g(r)p Fr(w)s Fs(\))17 b(d)p Fr(\030)26 b Fs(+)2861 1025 y Fn(Z)2916 1251 y Fm(R)2977 1232 y Fl(2)3042 1094 y Fp(j)p Fr(\030)5 b Fp(j)3146 1057 y Fo(2)p 3042 1138 143 4 v 3089 1229 a Fs(4)3194 1161 y(\()p Fq(v)24 b Fp(\001)e(r)p Fr(w)s Fs(\))17 b(d)p Fr(\030)1318 1420 y Fs(=)83 b Fp(\000)1571 1284 y Fn(Z)1626 1510 y Fm(R)1687 1491 y Fl(2)1742 1420 y Fq(v)24 b Fp(\001)e(r)p Fr(w)d Fs(d)p Fr(\030)27 b Fp(\000)2280 1352 y Fs(1)p 2280 1397 49 4 v 2280 1488 a(2)2356 1284 y Fn(Z)2411 1510 y Fm(R)2472 1491 y Fl(2)2511 1420 y Fs(\()p Fg(\030)d Fp(\001)e Fq(v)r Fs(\))p Fr(w)d Fs(d)p Fr(\030)5 b(:)0 1678 y Fs(W)-8 b(e)38 b(claim)g(that)f(b)s(oth)g(in)m(tegrals)h(in)g (the)g(last)f(expression)j(v)-5 b(anish.)58 b(This)39 b(is)f(ob)m(vious)g(for)f(the)h(\014rst)0 1798 y(one,)33 b(since)i Fq(v)24 b Fp(\001)e(r)p Fr(w)31 b Fs(=)d(div)r(\()p Fq(v)r Fr(w)s Fs(\).)44 b(As)33 b(for)g(the)g(second)h(one,)g(using)f (\(14\))g(and)g(F)-8 b(ubini's)34 b(theorem)f(w)m(e)0 1918 y(obtain)289 2031 y Fn(Z)344 2257 y Fm(R)405 2238 y Fl(2)444 2167 y Fs(\()p Fg(\030)24 b Fp(\001)e Fq(v)r Fs(\()p Fr(\030)5 b Fs(\)\))p Fr(w)s Fs(\()p Fr(\030)g Fs(\))17 b(d)p Fr(\030)85 b Fs(=)1424 2100 y(1)p 1395 2144 108 4 v 1395 2235 a(2)p Fr(\031)1529 2031 y Fn(Z)1585 2257 y Fm(R)1646 2238 y Fl(2)1680 2257 y Fk(\002)p Fm(R)1796 2238 y Fl(2)1851 2167 y Fg(\030)25 b Fp(\001)1986 2100 y Fs(\()p Fg(\030)g Fp(\000)e Fg(\021)s Fs(\))2301 2064 y Fk(?)p 1986 2144 374 4 v 2016 2235 a Fp(j)p Fr(\030)j Fp(\000)c Fr(\021)t Fp(j)2292 2207 y Fo(2)2370 2167 y Fr(w)s Fs(\()p Fr(\021)t Fs(\))p Fr(w)s Fs(\()p Fr(\030)5 b Fs(\))17 b(d)p Fr(\021)h Fs(d)p Fr(\030)1226 2447 y Fs(=)1424 2379 y(1)p 1395 2424 108 4 v 1395 2515 a(4)p Fr(\031)1529 2311 y Fn(Z)1585 2537 y Fm(R)1646 2518 y Fl(2)1680 2537 y Fk(\002)p Fm(R)1796 2518 y Fl(2)1834 2447 y Fs(\()p Fg(\030)25 b Fp(\000)e Fg(\021)s Fs(\))f Fp(\001)2231 2379 y Fs(\()p Fg(\030)j Fp(\000)d Fg(\021)t Fs(\))2546 2343 y Fk(?)p 2231 2424 374 4 v 2260 2515 a Fp(j)p Fr(\030)k Fp(\000)d Fr(\021)t Fp(j)2537 2486 y Fo(2)2615 2447 y Fr(w)s Fs(\()p Fr(\021)t Fs(\))p Fr(w)s Fs(\()p Fr(\030)5 b Fs(\))17 b(d)p Fr(\021)h Fs(d)p Fr(\030)32 b Fs(=)27 b(0)33 b Fr(:)0 2712 y Fs(Summarizing,)h(w)m(e)f(ha)m(v)m(e)h (sho)m(wn)g(that)972 2968 y Fr(H)8 b Fs(\()p Fr(w)s Fs(\()p Fr(\034)1252 2983 y Fo(1)1291 2968 y Fs(\)\))22 b Fp(\000)h Fr(H)8 b Fs(\()p Fr(w)s Fs(\()p Fr(\034)1769 2983 y Fo(0)1807 2968 y Fs(\)\))28 b(=)f Fp(\000)2108 2832 y Fn(Z)2208 2859 y Fi(\034)2239 2868 y Fl(1)2164 3058 y Fi(\034)2195 3067 y Fl(0)2295 2968 y Fr(I)8 b Fs(\()p Fr(w)s Fs(\()p Fr(\034)j Fs(\)\))17 b(d)p Fr(\034)43 b(;)799 b Fs(\(50\))0 3235 y(for)32 b(all)h Fr(\034)327 3250 y Fo(1)394 3235 y Fr(>)28 b(\034)540 3250 y Fo(0)607 3235 y Fr(>)g Fs(0.)146 3356 y(W)-8 b(e)37 b(no)m(w)g(return)g(to)f(the)g(general)h(case)g (where)h Fr(w)2027 3371 y Fo(0)2100 3356 y Fp(2)c Fr(L)2266 3319 y Fo(2)2306 3356 y Fs(\()p Fr(m)p Fs(\))25 b Fp(\\)g Fs(\006)2653 3371 y Fo(+)2712 3356 y Fs(.)55 b(Giv)m(en)37 b Fr(\034)3122 3371 y Fo(1)3196 3356 y Fr(>)c(\034)3347 3371 y Fo(0)3421 3356 y Fr(>)h Fs(0,)j(the)0 3476 y(solution)i Fr(w)i Fs(of)d(\(12\))g(satis\014es)j Fr(w)f Fp(2)f Fr(C)1473 3440 y Fo(0)1512 3476 y Fs(\([)p Fr(\034)1619 3491 y Fo(0)1659 3476 y Fr(;)17 b(\034)1745 3491 y Fo(1)1784 3476 y Fs(])p Fr(;)g(H)1944 3440 y Fi(k)1986 3476 y Fs(\()p Fr(m)p Fs(\)\))39 b(for)f(an)m(y)i Fr(k)h Fp(2)d Fq(N)p Fs(,)i(where)g Fr(H)3298 3440 y Fi(k)3340 3476 y Fs(\()p Fr(m)p Fs(\))f(is)g(the)0 3596 y(w)m(eigh)m(ted)34 b(Sob)s(olev)g (space)f(de\014ned)h(in)f(analogy)f(with)i(\(24\).)43 b(Moreo)m(v)m(er,)34 b(the)f(map)974 3807 y Fr(w)1044 3822 y Fo(0)1111 3807 y Fp(2)28 b Fr(L)1271 3766 y Fo(2)1310 3807 y Fs(\()p Fr(m)p Fs(\))126 b Fp(7!)f Fr(w)30 b Fp(2)e Fr(C)2093 3766 y Fo(0)2132 3807 y Fs(\([)p Fr(\034)2239 3822 y Fo(0)2279 3807 y Fr(;)17 b(\034)2365 3822 y Fo(1)2405 3807 y Fs(])p Fr(;)g(H)2565 3766 y Fi(k)2607 3807 y Fs(\()p Fr(m)p Fs(\)\))0 4018 y(is)39 b(con)m(tin)m(uous.)65 b(On)38 b(the)i(other)f(hand,)i(it)e(is)g(not)g(di\016cult)h(to)e(v)m (erify)i(that)f(the)g(quan)m(tit)m(y)i Fr(I)8 b Fs(\()p Fr(w)s Fs(\))38 b(is)0 4139 y(\014nite)j(for)e(an)m(y)i(p)s(ositiv)m(e) g Fr(w)i Fp(2)e Fr(H)1282 4103 y Fi(k)1324 4139 y Fs(\()p Fr(m)p Fs(\))f(if)g Fr(k)j Fs(is)e(su\016cien)m(tly)h(large)e(\(sa)m(y) -8 b(,)43 b Fr(k)h(>)c Fs(3\),)h(and)g(that)e Fr(I)8 b Fs(\()p Fr(w)s Fs(\))0 4259 y(dep)s(ends)34 b(con)m(tin)m(uously)g (on)e Fr(w)i Fs(in)e(that)g(top)s(ology)-8 b(.)43 b(Th)m(us)33 b(w)m(e)g(see)g(that)f(b)s(oth)g(sides)h(of)f(\(50\))f(dep)s(end)0 4379 y(con)m(tin)m(uously)36 b(on)e(the)g(initial)g(data)g Fr(w)1451 4394 y Fo(0)1524 4379 y Fs(in)g(the)g(top)s(ology)f(of)g Fr(L)2388 4343 y Fo(2)2428 4379 y Fs(\()p Fr(m)p Fs(\).)48 b(Since)35 b(\(50\))e(holds)h(for)g(all)f Fr(w)3740 4394 y Fo(0)0 4500 y Fs(in)f(the)g(dense)i(subset)f Fp(S)7 b Fs(\()p Fq(R)1033 4464 y Fo(2)1073 4500 y Fs(\))21 b Fp(\\)g Fs(\006)1289 4515 y Fo(+)1348 4500 y Fs(,)32 b(it)g(follo)m(ws)h(that)e(\(50\))h(is)g(v)-5 b(alid)32 b(for)f(all)h Fr(w)2925 4515 y Fo(0)2992 4500 y Fp(2)c Fr(L)3152 4464 y Fo(2)3192 4500 y Fs(\()p Fr(m)p Fs(\))21 b Fp(\\)g Fs(\006)3531 4515 y Fo(+)3622 4500 y Fs(and)0 4620 y(all)32 b Fr(\034)177 4635 y Fo(1)245 4620 y Fr(>)c(\034)391 4635 y Fo(0)458 4620 y Fr(>)g Fs(0.)43 b(This)33 b(concludes)i(the)e (pro)s(of.)1941 b Fb(\003)0 4908 y Ff(3.2)135 b(Con)l(v)l(ergence)47 b(in)d Fe(L)1331 4864 y Fd(2)1377 4908 y Fw(\()p Fe(m)p Fw(\))0 5092 y Fs(Using)e(the)h(compactness)h(prop)s(erties)e(of)g (Section)g(2)g(and)g(the)g(t)m(w)m(o)h(Ly)m(apuno)m(v)g(functions)g(of) e(the)0 5213 y(previous)34 b(subsection,)h(w)m(e)e(are)g(no)m(w)g(able) g(to)f(pro)m(v)m(e)i(Prop)s(osition)f(1.5.)0 5431 y Fq(Lemma)39 b(3.3)90 b Fj(Assume)41 b(that)g Fr(m)e(>)g Fs(1)h Fj(and)h(that)g Fp(f)p Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))p Fp(g)2236 5446 y Fi(\034)d Fk(2)p Fm(R)2427 5431 y Fj(is)41 b(a)f(c)-5 b(omplete)40 b(tr)-5 b(aje)g(ctory)42 b(of)e(\(12\))0 5551 y(which)34 b(is)h(b)-5 b(ounde)g(d)34 b(in)g Fr(L)935 5515 y Fo(2)975 5551 y Fs(\()p Fr(m)p Fs(\))p Fj(.)45 b(Then)34 b Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))27 b(=)h Fr(\013)q(G)35 b Fj(for)f(al)5 b(l)35 b Fr(\034)k Fp(2)28 b Fq(R)p Fj(,)34 b(wher)-5 b(e)34 b Fr(\013)29 b Fs(=)3060 5471 y Fn(R)3107 5586 y Fm(R)3168 5567 y Fl(2)3224 5551 y Fr(w)s Fs(\()p Fr(\030)5 b(;)17 b Fs(0\))g(d)p Fr(\030)5 b Fj(.)1841 5800 y Fs(17)p eop %%Page: 18 18 18 17 bop 0 100 a Fq(Pro)s(of:)29 b Fs(W)-8 b(e)26 b(kno)m(w)h(from)e (Lemma)h(2.3)f(that)h Fp(f)p Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))p Fp(g)1962 115 y Fi(\034)d Fk(2)p Fm(R)2138 100 y Fs(is)26 b(relativ)m(ely)h(compact)f(in)g Fr(L)3203 63 y Fo(2)3243 100 y Fs(\()p Fr(m)p Fs(\).)41 b(In)26 b(view)0 220 y(of)h(Remark)h(2.4,)h(w)m(e)f(can)g(assume)h(without)g (loss)f(of)f(generalit)m(y)i(that)e Fr(m)h(>)g Fs(3.)41 b(Let)28 b(\012)g(b)s(e)g(the)g Fr(!)t Fs(-limit)0 340 y(set)34 b(of)f(the)h(tra)5 b(jectory)34 b Fp(f)p Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))p Fp(g)1192 355 y Fi(\034)d Fk(2)p Fm(R)1342 340 y Fs(.)46 b(Since)35 b(b)m(y)f(Lemma)g(3.1)f(\010) h(is)g(a)f(Ly)m(apuno)m(v)i(function)f(whic)m(h)h(is)0 461 y(strictly)c(decreasing)h(except)g(on)e(\006,)h(it)f(follo)m(ws)h (from)f(LaSalle's)h(in)m(v)-5 b(ariance)31 b(principle)h(that)e(\012)e Fp(\032)g Fs(\006.)0 581 y(In)22 b(particular,)j(since)e(the)g(total)e (mass)i(is)f(conserv)m(ed,)27 b(an)m(y)43 b(\026)-69 b Fr(w)30 b Fp(2)e Fs(\012)22 b(satis\014es)i(\010\()c(\026)-69 b Fr(w)s Fs(\))27 b(=)h Fp(j)3172 501 y Fn(R)3218 616 y Fm(R)3279 597 y Fl(2)3354 581 y Fs(\026)-69 b Fr(w)s Fs(\()p Fr(\030)5 b Fs(\))17 b(d)p Fr(\030)5 b Fp(j)26 b Fs(=)0 702 y Fp(j)p Fr(\013)q Fp(j)p Fs(.)53 b(The)38 b(same)f(is)f(true)h(for)e(an)m(y)i(function)g Fr(w)p 1689 717 73 4 v 39 w Fs(in)f(the)h Fr(\013)q Fs(-limit)f(set)g Fp(A)p Fs(.)54 b(As)37 b Fr(\034)45 b Fp(7!)34 b Fs(\010\()p Fr(w)s Fs(\()p Fr(\034)11 b Fs(\)\))36 b(is)h(non-)0 822 y(increasing,)32 b(it)d(follo)m(ws)i(that)e(\010\()p Fr(w)s Fs(\()p Fr(\034)11 b Fs(\)\))28 b(=)g Fp(j)p Fr(\013)q Fp(j)g Fs(for)h(all)h Fr(\034)39 b Fp(2)28 b Fq(R)p Fs(.)42 b(By)31 b(Lemma)f(3.1)f(again,)h(w)m(e)h(conclude)0 942 y(that)36 b Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))35 b Fp(2)g Fs(\006)i(for)f(all)h Fr(\034)46 b Fp(2)35 b Fq(R)p Fs(.)56 b(Th)m(us,)39 b(up)s(on)e(replacing)g Fr(w)s Fs(\()p Fr(\030)2414 957 y Fo(1)2453 942 y Fr(;)17 b(\030)2540 957 y Fo(2)2579 942 y Fr(;)g(\034)11 b Fs(\))36 b(b)m(y)i Fp(\000)p Fr(w)s Fs(\()p Fr(\030)3121 957 y Fo(2)3160 942 y Fr(;)17 b(\030)3247 957 y Fo(1)3286 942 y Fr(;)g(\034)11 b Fs(\),)37 b(w)m(e)h(can)0 1063 y(assume)c(that)e Fp(f)p Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))p Fp(g)850 1078 y Fi(\034)d Fk(2)p Fm(R)1028 1063 y Fp(\032)29 b Fr(L)1200 1027 y Fo(2)1239 1063 y Fs(\()p Fr(m)p Fs(\))23 b Fp(\\)f Fs(\006)1581 1078 y Fo(+)1641 1063 y Fs(.)146 1183 y(W)-8 b(e)31 b(no)m(w)f(use)h (the)f(second)h(Ly)m(apuno)m(v)h(function)e Fr(H)8 b Fs(.)42 b(By)30 b(Lemma)h(3.2)e(and)h(LaSalle's)h(principle,)0 1303 y Fp(A)21 b Fs(and)h(\012)g(are)g(con)m(tained)h(in)f(the)g(line)h (of)e(equilibria)i Fp(f)p Fr(\013)2022 1267 y Fk(0)2045 1303 y Fr(G)p Fp(g)2172 1318 y Fi(\013)2217 1299 y Fc(0)2239 1318 y Fk(\025)p Fo(0)2334 1303 y Fs(.)40 b(Since)22 b(the)h(total)e(mass)i(is)f(conserv)m(ed,)0 1424 y(w)m(e)37 b(necessarily)h(ha)m(v)m(e)f Fp(A)c Fs(=)f(\012)i(=)f Fp(f)p Fr(\013)q(G)p Fp(g)p Fs(.)52 b(As)36 b Fr(H)43 b Fs(is)36 b(non-increasing,)i(it)e(follo)m(ws)g(that)f Fr(H)8 b Fs(\()p Fr(w)s Fs(\()p Fr(\034)j Fs(\)\))33 b(=)0 1544 y Fr(H)8 b Fs(\()p Fr(\013)q(G)p Fs(\))29 b(=)h Fr(\013)17 b Fs(log)q(\()p Fr(\013)q Fs(\))33 b(for)g(all)h Fr(\034)41 b Fp(2)31 b Fq(R)p Fs(.)47 b(By)34 b(Lemma)g(3.2)g(again,)g (w)m(e)g(conclude)i(that)d Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))30 b(=)g Fr(\013)q(G)j Fs(for)0 1665 y(all)f Fr(\034)40 b Fp(2)28 b Fq(R)p Fs(.)3280 b Fb(\003)0 1835 y Fq(Pro)s(of)26 b(of)h(Prop)s(osition)f(1.5:)39 b Fs(Let)23 b Fr(w)1457 1850 y Fo(0)1524 1835 y Fp(2)28 b Fr(L)1684 1799 y Fo(2)1724 1835 y Fs(\()p Fr(m)p Fs(\))23 b(with)h Fr(m)k(>)f Fs(1,)e(and)e(let)g Fr(w)30 b Fp(2)f Fr(C)3021 1799 y Fo(0)3060 1835 y Fs(\([0)p Fr(;)17 b Fs(+)p Fp(1)p Fs(\))p Fr(;)g(L)3542 1799 y Fo(2)3580 1835 y Fs(\()p Fr(m)p Fs(\)\))0 1955 y(b)s(e)36 b(the)g(solution)g(of)f(\(12\))h(with) g(initial)g(data)f Fr(w)1808 1970 y Fo(0)1847 1955 y Fs(.)53 b(F)-8 b(rom)35 b(Lemma)i(2.2,)f(w)m(e)g(kno)m(w)h(that)f Fp(f)p Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))p Fp(g)3647 1970 y Fi(\034)d Fk(\025)p Fo(0)0 2076 y Fs(is)35 b(relativ)m(ely)h (compact)f(in)f Fr(L)1100 2039 y Fo(2)1140 2076 y Fs(\()p Fr(m)p Fs(\).)48 b(Let)35 b(\012)30 b Fp(\032)h Fr(L)1827 2039 y Fo(2)1867 2076 y Fs(\()p Fr(m)p Fs(\))j(denote)h(the)g Fr(!)t Fs(-limit)e(set)i(of)f(this)h(tra)5 b(jectory)-8 b(.)0 2196 y(As)38 b(is)g(w)m(ell-kno)m(wn,)j(\012)d(is)g(non-empt)m(y) -8 b(,)40 b(compact,)g(fully)e(in)m(v)-5 b(arian)m(t)38 b(under)g(the)g(ev)m(olution)h(of)e(\(12\),)0 2316 y(and)e(\012)g (attracts)f Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))34 b(in)h Fr(L)1080 2280 y Fo(2)1120 2316 y Fs(\()p Fr(m)p Fs(\))g(as)f Fr(\034)43 b Fp(!)31 b Fs(+)p Fp(1)p Fs(.)48 b(If)55 b(\026)-69 b Fr(w)33 b Fp(2)f Fs(\012,)j(there)g(exists)i(a)d(complete) i(tra)5 b(jectory)0 2437 y Fp(f)20 b Fs(\026)-69 b Fr(w)r Fs(\()p Fr(\034)11 b Fs(\))p Fp(g)301 2452 y Fi(\034)d Fk(2)p Fm(R)494 2437 y Fs(of)41 b(\(12\))g(suc)m(h)i(that)62 b(\026)-69 b Fr(w)r Fs(\()p Fr(\034)11 b Fs(\))44 b Fp(2)g Fs(\012)e(for)f(all)g Fr(\034)55 b Fp(2)44 b Fq(R)d Fs(and)62 b(\026)-69 b Fr(w)r Fs(\(0\))43 b(=)63 b(\026)-69 b Fr(w)r Fs(.)71 b(By)42 b(Lemma)h(3.3,)20 2557 y(\026)-69 b Fr(w)r Fs(\()p Fr(\034)11 b Fs(\))29 b(=)f Fr(\013)q(G)33 b Fs(for)f(all)h Fr(\034)39 b Fp(2)29 b Fq(R)p Fs(,)k(where)h Fr(\013)29 b Fs(=)1589 2477 y Fn(R)1636 2592 y Fm(R)1697 2573 y Fl(2)1772 2557 y Fs(\026)-69 b Fr(w)r Fs(\()p Fr(\030)5 b Fs(\))17 b(d)p Fr(\030)32 b Fs(=)2198 2477 y Fn(R)2245 2592 y Fm(R)2306 2573 y Fl(2)2361 2557 y Fr(w)2431 2572 y Fo(0)2471 2557 y Fs(\()p Fr(\030)5 b Fs(\))17 b(d)p Fr(\030)5 b Fs(.)42 b(Th)m(us)35 b(\012)29 b(=)f Fp(f)p Fr(\013)q(G)p Fp(g)p Fs(,)k(whic)m(h)0 2677 y(is)h(the)g(desired)h(result.)2840 b Fb(\003)0 2966 y Ff(3.3)135 b(Con)l(v)l(ergence)47 b(in)d Fe(L)1331 2923 y Fd(1)1377 2966 y Fw(\()p Ff(R)1523 2923 y Fd(2)1568 2966 y Fw(\))0 3151 y Fs(W)-8 b(e)29 b(no)m(w)h(study)g(the)f(b)s(eha)m (vior)h(of)e(the)h(solutions)h(of)e(\(12\))g(in)h Fr(L)2348 3115 y Fo(1)2388 3151 y Fs(\()p Fq(R)2510 3115 y Fo(2)2549 3151 y Fs(\))g(and)g(pro)m(v)m(e)h(Theorem)g(1.2)e(and)0 3271 y(Prop)s(osition)33 b(1.3.)0 3500 y Fq(Prop)s(osition)38 b(3.4)49 b Fj(L)-5 b(et)43 b Fr(w)1044 3515 y Fo(0)1126 3500 y Fp(2)h Fr(L)1302 3464 y Fo(1)1342 3500 y Fs(\()p Fq(R)1464 3464 y Fo(2)1503 3500 y Fs(\))p Fj(,)h(and)d(let)i Fr(w)h Fp(2)f Fr(C)2261 3464 y Fo(0)2300 3500 y Fs(\([0)p Fr(;)17 b Fp(1)p Fs(\))p Fr(;)g(L)2706 3464 y Fo(1)2745 3500 y Fs(\()p Fq(R)2867 3464 y Fo(2)2906 3500 y Fs(\)\))43 b Fj(b)-5 b(e)43 b(the)g(solution)g(of)0 3620 y(\(12\))34 b(with)h(initial)f(data)h Fr(w)999 3635 y Fo(0)1038 3620 y Fj(.)45 b(Then)34 b Fp(j)p Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))22 b Fp(\000)g Fr(\013)q(G)p Fp(j)1886 3635 y Fo(1)1953 3620 y Fp(!)27 b Fs(0)35 b Fj(as)f Fr(\034)39 b Fp(!)28 b(1)p Fj(,)34 b(wher)-5 b(e)34 b Fr(\013)28 b Fs(=)3130 3540 y Fn(R)3177 3655 y Fm(R)3238 3636 y Fl(2)3293 3620 y Fr(w)3363 3635 y Fo(0)3402 3620 y Fs(\()p Fr(\030)5 b Fs(\))17 b(d)p Fr(\030)5 b Fj(.)0 3848 y Fq(Pro)s(of:)33 b Fs(W)-8 b(e)30 b(kno)m(w)g(from)f(Lemma)h(2.5)f(that) g Fp(f)p Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))p Fp(g)1988 3863 y Fi(\034)d Fk(\025)p Fo(0)2150 3848 y Fs(is)29 b(relativ)m(ely)i(compact)f(in)g Fr(L)3230 3812 y Fo(1)3269 3848 y Fs(\()p Fq(R)3391 3812 y Fo(2)3430 3848 y Fs(\).)43 b(Let)29 b(\012)0 3969 y(b)s(e)g(the)h Fr(!)t Fs(-limit)e(set)i(of)f (this)g(tra)5 b(jectory)-8 b(.)43 b(Then)31 b(\012)e(is)h(non-empt)m(y) -8 b(,)31 b(compact,)f(fully)g(in)m(v)-5 b(arian)m(t)30 b(under)0 4089 y(the)f(ev)m(olution)h(of)e(\(12\),)h(and)f(\012)h (attracts)g Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))28 b(in)h Fr(L)1970 4053 y Fo(1)2010 4089 y Fs(\()p Fq(R)2132 4053 y Fo(2)2170 4089 y Fs(\))g(as)g Fr(\034)39 b Fp(!)27 b Fs(+)p Fp(1)p Fs(.)42 b(If)48 b(\026)-69 b Fr(w)30 b Fp(2)e Fs(\012,)i(there)f(exists)i(a)0 4210 y(sequence)j Fr(\034)445 4225 y Fi(n)520 4210 y Fp(!)27 b(1)32 b Fs(suc)m(h)g(that)g Fp(j)p Fr(w)s Fs(\()p Fr(\034)1389 4225 y Fi(n)1435 4210 y Fs(\))20 b Fp(\000)40 b Fs(\026)-69 b Fr(w)s Fp(j)1691 4225 y Fo(1)1757 4210 y Fp(!)28 b Fs(0)j(and)g Fr(w)s Fs(\()p Fr(\030)5 b(;)17 b(\034)2398 4225 y Fi(n)2444 4210 y Fs(\))28 b Fp(!)47 b Fs(\026)-69 b Fr(w)r Fs(\()p Fr(\030)5 b Fs(\))31 b(for)g(almost)h(all)f Fr(\030)h Fp(2)c Fq(R)3713 4173 y Fo(2)3752 4210 y Fs(.)0 4330 y(Using)33 b(\(39\))f(and)h(Leb)s(esgue's)h(dominated)g(con)m(v)m (ergence)h(theorem,)e(w)m(e)h(obtain)746 4559 y Fp(j)19 b Fs(\026)-68 b Fr(w)r Fs(\()p Fr(\030)5 b Fs(\))p Fp(j)26 b Fs(=)52 b(lim)1128 4619 y Fi(n)p Fk(!1)1329 4559 y Fp(j)p Fr(w)s Fs(\()p Fr(\030)5 b(;)17 b(\034)1602 4574 y Fi(n)1647 4559 y Fs(\))p Fp(j)27 b(\024)h Fr(C)1915 4574 y Fi(\014)1962 4559 y Fp(j)p Fr(w)2060 4574 y Fo(0)2099 4559 y Fp(j)2127 4574 y Fo(1)2183 4559 y Fr(e)2228 4518 y Fk(\000)p Fi(\014)s Fk(j)p Fi(\030)s Fk(j)2400 4495 y Fl(2)2433 4518 y Fi(=)p Fo(4)2541 4559 y Fr(;)114 b(\030)32 b Fp(2)c Fq(R)2935 4518 y Fo(2)3007 4559 y Fr(;)572 b Fs(\(51\))0 4816 y(since)36 b Fr(a)p Fs(\()p Fr(\034)372 4831 y Fi(n)419 4816 y Fs(\))31 b Fp(!)f Fs(1)k(as)g Fr(n)d Fp(!)f(1)p Fs(.)48 b(In)34 b(particular,)h(this)g(sho)m(ws)h (that)e(\012)h(is)f(b)s(ounded)h(in)g Fr(L)3291 4780 y Fo(2)3331 4816 y Fs(\()p Fr(m)p Fs(\))f(for)g(all)0 4936 y Fr(m)28 b(>)g Fs(1.)146 5057 y(No)m(w,)h(since)f(\012)f(is)g(in) m(v)-5 b(arian)m(t)27 b(under)g(the)g(semi\015o)m(w)i(of)d(\(12\),)h (there)g(exists)i(a)d(complete)i(tra)5 b(jectory)0 5177 y Fp(f)20 b Fs(\026)-69 b Fr(w)r Fs(\()p Fr(\034)11 b Fs(\))p Fp(g)301 5192 y Fi(\034)d Fk(2)p Fm(R)498 5177 y Fs(suc)m(h)48 b(that)65 b(\026)-69 b Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))50 b Fp(2)h Fs(\012)c(for)e(all)h Fr(\034)62 b Fp(2)51 b Fq(R)45 b Fs(and)66 b(\026)-69 b Fr(w)r Fs(\(0\))50 b(=)71 b(\026)-69 b Fr(w)r Fs(.)84 b(As)46 b(w)m(e)h(just)f(observ)m(ed,)0 5298 y Fp(f)20 b Fs(\026)-69 b Fr(w)r Fs(\()p Fr(\034)11 b Fs(\))p Fp(g)301 5313 y Fi(\034)d Fk(2)p Fm(R)490 5298 y Fs(is)37 b(b)s(ounded)h(in)g Fr(L)1180 5261 y Fo(2)1219 5298 y Fs(\()p Fr(m)p Fs(\))g(for)e(all)h Fr(m)f(>)g Fs(1.)57 b(Applying)38 b(Lemma)g(3.3,)g(w)m(e)g(conclude)h (that)20 5418 y(\026)-69 b Fr(w)r Fs(\()p Fr(\034)11 b Fs(\))50 b(=)f Fr(\013)q(G)c Fs(for)f(all)h Fr(\034)61 b Fp(2)49 b Fq(R)p Fs(,)f(where)f Fr(\013)i Fs(=)1778 5338 y Fn(R)1825 5453 y Fm(R)1886 5434 y Fl(2)1961 5418 y Fs(\026)-69 b Fr(w)s Fs(\()p Fr(\030)5 b Fs(\))17 b(d)p Fr(\030)52 b Fs(=)2429 5338 y Fn(R)2476 5453 y Fm(R)2537 5434 y Fl(2)2593 5418 y Fr(w)2663 5433 y Fo(0)2702 5418 y Fs(\()p Fr(\030)5 b Fs(\))17 b(d)p Fr(\030)5 b Fs(.)79 b(This)47 b(pro)m(v)m(es)g(that)0 5538 y(\012)28 b(=)g Fp(f)p Fr(\013)q(G)p Fp(g)p Fs(,)k(whic)m(h)i(is)f(the)g(desired)h (result.)2060 b Fb(\003)1841 5800 y Fs(18)p eop %%Page: 19 19 19 18 bop 0 100 a Fq(Pro)s(of)40 b(of)h(Theorem)h(1.2:)49 b Fs(Let)36 b Fr(!)g Fp(2)d Fr(C)1596 63 y Fo(0)1636 100 y Fs(\([0)p Fr(;)17 b Fp(1)p Fs(\))p Fr(;)g(L)2042 63 y Fo(1)2080 100 y Fs(\()p Fq(R)2202 63 y Fo(2)2241 100 y Fs(\)\))35 b(b)s(e)h(the)g(solution)g(of)f(\(2\))g(with)h (initial)0 220 y(data)27 b Fr(!)278 235 y Fo(0)317 220 y Fs(.)42 b(If)27 b(w)m(e)h(set)g Fr(w)s Fs(\()p Fr(\030)5 b(;)17 b(\034)11 b Fs(\))26 b(=)i Fr(e)1232 184 y Fi(\034)1275 220 y Fr(!)t Fs(\()p Fr(\030)20 b(e)1486 184 y Fi(\034)8 b(=)p Fo(2)1600 220 y Fr(;)17 b(e)1689 184 y Fi(\034)1732 220 y Fp(\000)q Fs(1\),)28 b(then)g Fr(w)i Fp(2)e Fr(C)2440 184 y Fo(0)2479 220 y Fs(\([0)p Fr(;)17 b Fp(1)p Fs(\))p Fr(;)g(L)2885 184 y Fo(1)2924 220 y Fs(\()p Fq(R)3046 184 y Fo(2)3085 220 y Fs(\)\))27 b(is)g(the)h(solution)0 340 y(of)k(\(12\))f(with)i(initial)g(data)f Fr(w)1112 355 y Fo(0)1178 340 y Fs(=)c Fr(!)1343 355 y Fo(0)1382 340 y Fs(.)43 b(Applying)34 b(Prop)s(osition)e(3.4)g(and)g(returning)h (to)f(the)h(original)0 461 y(function)g Fr(!)t Fs(\()p Fr(x;)17 b(t)p Fs(\),)32 b(w)m(e)i(obtain)1137 672 y(lim)1122 732 y Fi(t)p Fk(!1)1305 558 y Fn(\014)1305 618 y(\014)1305 677 y(\014)1338 672 y Fr(!)t Fs(\()p Fp(\001)p Fr(;)17 b(t)p Fs(\))k Fp(\000)1765 605 y Fr(\013)p 1717 649 160 4 v 1717 741 a(t)p Fs(+1)1887 672 y Fr(G)1964 562 y Fn(\020)2141 605 y Fp(\001)p 2033 649 243 4 v 2033 669 a(p)p 2116 669 160 4 v 79 x Fr(t)p Fs(+1)2286 562 y Fn(\021)2345 558 y(\014)2345 618 y(\014)2345 677 y(\014)2379 741 y Fo(1)2446 672 y Fs(=)27 b(0)33 b Fr(;)0 912 y Fs(whic)m(h)39 b(is)f(equiv)-5 b(alen)m(t)39 b(to)e(\(9\))g(for)g Fr(p)e Fs(=)h(1.)58 b(Next,)39 b(in)m(terp)s(olating)f(b)s(et)m(w)m(een)h (\(9\))e(for)g Fr(p)f Fs(=)f(1)i(and)h(\(4\))0 1032 y(for)31 b Fr(p)c Fs(=)h Fp(1)p Fs(,)j(w)m(e)i(obtain)e(\(9\))g(for)g Fr(p)c Fp(2)h Fs(\(1)p Fr(;)17 b Fp(1)p Fs(\).)42 b(Then,)33 b(using)f(\(19\))f(if)g(2)c Fr(<)h(q)j(<)d Fp(1)j Fs(or)g(Lemma)h (2.1\(b\))0 1153 y(in)39 b([13)o(])f(if)h Fr(q)i Fs(=)c Fp(1)p Fs(,)i(w)m(e)h(arriv)m(e)f(at)f(\(10\).)60 b(Finally)-8 b(,)40 b(using)f(the)g(previous)g(results)h(and)e(the)h(in)m(tegral)0 1273 y(equation)32 b(satis\014ed)g(b)m(y)g Fr(!)t Fs(\()p Fr(x;)17 b(t)p Fs(\),)30 b(it)h(is)g(straigh)m(tforw)m(ard)h(to)e(sho)m (w)i(that)f(\(9\))f(also)h(holds)h(for)e Fr(p)d Fs(=)h Fp(1)p Fs(.)0 1393 y Fb(\003)146 1553 y Fs(As)k(w)m(e)h(already)f (observ)m(ed,)h(a)e(b)s(ounded)i(negativ)m(e)f(tra)5 b(jectory)32 b(of)f(\(12\))g(in)h Fr(L)3011 1517 y Fo(1)3050 1553 y Fs(\()p Fq(R)3172 1517 y Fo(2)3211 1553 y Fs(\))g(need)g(not)f (b)s(e)0 1674 y(relativ)m(ely)i(compact.)44 b(Ho)m(w)m(ev)m(er,)33 b(if)e(w)m(e)h(assume)h(that)e(the)g(tra)5 b(jectory)32 b(is)g(relativ)m(ely)g(compact)g(or)f(at)0 1794 y(least)g(that)g(its)g Fr(\013)q Fs(-limit)f(set)i(is)f(nonempt)m(y)-8 b(,)33 b(then)e(w)m(e)h(ha)m(v)m(e)g(the)f(follo)m(wing)g(result)h(whic)m(h)g (generalizes)0 1915 y(Lemma)h(3.3.)0 2102 y Fq(Prop)s(osition)38 b(3.5)49 b Fj(L)-5 b(et)32 b Fp(f)p Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))p Fp(g)1265 2117 y Fi(\034)d Fk(2)p Fm(R)1447 2102 y Fj(b)-5 b(e)32 b(a)g(c)-5 b(omplete)31 b(tr)-5 b(aje)g(ctory)33 b(of)e(\(12\))h(in)f Fr(L)2985 2066 y Fo(1)3025 2102 y Fs(\()p Fq(R)3147 2066 y Fo(2)3186 2102 y Fs(\))p Fj(,)i(and)e(assume)0 2222 y(that)k Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))35 b Fj(has)g(a)g(c)-5 b(onver)g(gent)34 b(subse)-5 b(quenc)g(e)34 b(in)h Fr(L)1897 2186 y Fo(1)1936 2222 y Fs(\()p Fq(R)2058 2186 y Fo(2)2097 2222 y Fs(\))g Fj(as)g Fr(\034)40 b Fp(!)27 b(\0001)p Fj(.)46 b(Then)34 b Fr(w)s Fs(\()p Fr(\034)11 b Fs(\))28 b(=)f Fr(\013)q(G)35 b Fj(for)g(al)5 b(l)0 2343 y Fr(\034)39 b Fp(2)28 b Fq(R)p Fj(,)35 b(wher)-5 b(e)34 b Fr(\013)28 b Fs(=)793 2262 y Fn(R)840 2377 y Fm(R)901 2358 y Fl(2)956 2343 y Fr(w)s Fs(\()p Fr(\030)5 b(;)17 b Fs(0\))g(d)p Fr(\030)5 b Fj(.)0 2530 y Fq(Pro)s(of:)47 b Fs(By)41 b(assumption,)k(there)c(exists)63 b(\026)-69 b Fr(w)43 b Fs(in)f Fr(L)1902 2494 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Fj(,)e(but)d(we)f(ar)-5 b(e)49 b(not)g(able)g(to)g(give)g(a)g(sharp)g(estimate)g(in)g(that)h(c) -5 b(ase.)88 b(Numeric)-5 b(al)0 4943 y(c)g(alculations)31 b(in)h([22)o(])g(indic)-5 b(ate)31 b(that)h(the)g(eigenvalues)f(that)h (ar)-5 b(e)31 b(not)h(fr)-5 b(ozen)31 b(by)h(symmetries)g(have)f(a)0 5063 y(r)-5 b(e)g(al)35 b(p)-5 b(art)35 b(that)g(c)-5 b(onver)g(ges)34 b(to)h Fp(\0001)g Fj(as)g Fp(j)p Fr(\013)q Fp(j)27 b(!)h(1)p Fj(,)34 b(ther)-5 b(eby)36 b(suggesting)e(that)h(a)g (fast)g(r)-5 b(otation)35 b(has)g(a)0 5184 y(stabilizing)g(e\013e)-5 b(ct)35 b(on)g(the)h(vortex.)46 b(Pr)-5 b(op)g(osition)35 b(4.1)g(shows)g(rigor)-5 b(ously)36 b(that)g(at)f(the)h(sp)-5 b(e)g(ctr)g(al)35 b(level)0 5304 y(p)-5 b(erturb)g(ations)33 b(of)g(the)g(vortex)f(solutions)h(de)-5 b(c)g(ay)33 b(at)g(le)-5 b(ast)33 b(as)f(fast)h(when)f Fr(\013)i Fj(is)f(lar)-5 b(ge)32 b(as)h(when)f Fr(\013)c Fs(=)g(0)p Fj(.)1841 5800 y Fs(23)p eop %%Page: 24 24 24 23 bop 146 100 a Fs(T)-8 b(o)32 b(pro)m(v)m(e)i(Prop)s(osition)e (4.1,)g(w)m(e)h(pro)s(ceed)g(in)g(three)f(steps.)45 b(First,)33 b(w)m(e)g(observ)m(e)h(that)e(the)g(linear)0 220 y(op)s(erators)k Fp(L)f Fs(and)h(\003)g(are)g(in)m(v)-5 b(arian)m(t)36 b(under)h(the)f(group)g(of)f(rotations)h Fr(S)6 b(O)s Fs(\(2\).)52 b(Th)m(us,)38 b(using)f(p)s(olar)0 340 y(co)s(ordinates)42 b(in)g Fq(R)734 304 y Fo(2)815 340 y Fs(and)f(expanding)i(the)f (angular)g(v)-5 b(ariable)41 b(in)h(F)-8 b(ourier)42 b(series,)j(w)m(e)e(reduce)g(the)0 461 y(eigen)m(v)-5 b(alue)30 b(equation)g(for)e(the)h(op)s(erator)f Fp(L)14 b(\000)g Fr(\013)q Fs(\003)29 b(to)f(a)g(\(nonlo)s(cal\))h(ordinary)g (di\013eren)m(tial)h(equation)0 581 y(in)44 b(the)g(radial)g(v)-5 b(ariable.)77 b(Next,)48 b(a)43 b(careful)i(study)g(of)e(this)i (di\013eren)m(tial)g(equation)f(rev)m(eals)i(that,)0 702 y(if)g Fr(\025)51 b Fp(2)g Fq(C)46 b Fs(is)h(an)f(isolated)h(eigen) m(v)-5 b(alue)48 b(of)e Fp(L)31 b(\000)h Fr(\013)q Fs(\003,)49 b(the)d(corresp)s(onding)i(eigenfunction)f(has)g(a)0 822 y(Gaussian)32 b(deca)m(y)h(at)d(in\014nit)m(y)-8 b(.)45 b(Finally)-8 b(,)32 b(w)m(e)g(pro)m(v)m(e)h(that)e(the)h(op)s (erators)f Fp(L)g Fs(and)g(\003)g(are)g(resp)s(ectiv)m(ely)0 942 y(self-adjoin)m(t)25 b(and)f(sk)m(ew-symmetric)k(in)c(a)g(w)m(eigh) m(ted)j Fr(L)2018 906 y Fo(2)2082 942 y Fs(space)e(with)g(appropriate)f (Gaussian)h(w)m(eigh)m(t,)0 1063 y(and)33 b(Prop)s(osition)g(4.1)f (then)h(follo)m(ws)g(from)g(elemen)m(tary)h(considerations.)0 1322 y Fq(4.1.1)112 b(P)m(olar)38 b(co)s(ordinates)0 1507 y Fs(Fix)33 b Fr(m)28 b(>)f Fs(0.)44 b(F)-8 b(or)31 b(an)m(y)j Fr(n)27 b Fp(2)i Fq(Z)p Fs(,)k(let)g Fr(P)1382 1522 y Fi(n)1461 1507 y Fs(b)s(e)g(the)g(orthogonal)e(pro)5 b(jection)34 b(in)f Fr(L)2893 1471 y Fo(2)2932 1507 y Fs(\()p Fr(m)p Fs(\))g(de\014ned)h(b)m(y)1063 1727 y(\()p Fr(P)1164 1742 y Fi(n)1211 1727 y Fr(w)s Fs(\)\()p Fr(r)18 b Fs(cos)f Fr(\022)s(;)g(r)j Fs(sin)d Fr(\022)s Fs(\))28 b(=)f Fr(!)2139 1742 y Fi(n)2186 1727 y Fs(\()p Fr(r)s Fs(\))p Fr(e)2354 1686 y Fi(in\022)2492 1727 y Fr(;)114 b Fs(where)1063 1937 y Fr(!)1124 1952 y Fi(n)1171 1937 y Fs(\()p Fr(r)s Fs(\))27 b(=)1464 1870 y(1)p 1434 1915 108 4 v 1434 2006 a(2)p Fr(\031)1569 1802 y Fn(Z)1668 1828 y Fo(2)p Fi(\031)1624 2027 y Fo(0)1767 1937 y Fr(w)s Fs(\()p Fr(r)19 b Fs(cos)e Fr(\022)s(;)g(r)i Fs(sin)f Fr(\022)s Fs(\))p Fr(e)2511 1896 y Fk(\000)p Fi(in\022)2721 1937 y Fs(d)p Fr(\022)36 b(:)723 b Fs(\(63\))0 2214 y(Clearly)-8 b(,)36 b Fr(P)423 2229 y Fi(n)470 2214 y Fr(P)533 2229 y Fi(n)576 2210 y Fc(0)632 2214 y Fs(=)30 b Fr(\016)781 2229 y Fi(nn)867 2210 y Fc(0)894 2214 y Fr(P)957 2229 y Fi(n)1038 2214 y Fs(and)1229 2140 y Fn(P)1334 2243 y Fi(n)p Fk(2)p Fm(Z)1495 2214 y Fr(P)1558 2229 y Fi(n)1635 2214 y Fs(=)g(1.)48 b(If)34 b Fr(w)f Fp(2)e Fr(L)2230 2178 y Fo(2)2270 2214 y Fs(\()p Fr(m)p Fs(\),)k(the)f(functions)h Fr(!)3145 2229 y Fi(n)3223 2214 y Fs(:)30 b Fq(R)3364 2229 y Fo(+)3453 2214 y Fp(!)g Fq(C)k Fs(in)0 2335 y(\(63\))e(b)s (elong)h(to)f(the)h(Hilb)s(ert)g(space)657 2605 y Fr(Z)7 b Fs(\()p Fr(m)p Fs(\))44 b(=)1057 2495 y Fn(n)1123 2605 y Fr(!)31 b Fs(:)d Fq(R)1354 2620 y Fo(+)1440 2605 y Fp(!)f Fq(C)1665 2491 y Fn(\014)1665 2551 y(\014)1665 2610 y(\014)1731 2470 y(Z)1831 2496 y Fk(1)1787 2695 y Fo(0)1922 2605 y Fr(r)s Fs(\(1)22 b(+)g Fr(r)2223 2564 y Fo(2)2262 2605 y Fs(\))2300 2564 y Fi(m)2366 2605 y Fp(j)p Fr(!)t Fs(\()p Fr(r)s Fs(\))p Fp(j)2610 2564 y Fo(2)2665 2605 y Fs(d)p Fr(r)30 b(<)e Fp(1)2997 2495 y Fn(o)3095 2605 y Fr(:)484 b Fs(\(64\))0 2878 y(F)-8 b(or)32 b(an)m(y)h Fr(n)28 b Fp(2)g Fq(Z)p Fs(,)33 b(let)g Fp(L)877 2893 y Fi(n)956 2878 y Fs(b)s(e)g(the)g(linear)g(op)s(erator)f (on)g Fr(Z)7 b Fs(\()p Fr(m)p Fs(\))33 b(de\014ned)h(b)m(y)986 3157 y Fp(L)1055 3172 y Fi(n)1102 3157 y Fr(!)48 b Fs(=)c Fr(@)1387 3116 y Fo(2)1382 3182 y Fi(r)1427 3157 y Fr(!)26 b Fs(+)1612 3047 y Fn(\020)1682 3090 y Fr(r)p 1681 3134 49 4 v 1681 3226 a Fs(2)1762 3157 y(+)1870 3090 y(1)p 1870 3134 V 1871 3226 a Fr(r)1929 3047 y Fn(\021)1988 3157 y Fr(@)2039 3172 y Fi(r)2078 3157 y Fr(!)f Fs(+)2262 3047 y Fn(\020)2322 3157 y Fs(1)d Fp(\000)2502 3090 y Fr(n)2560 3054 y Fo(2)p 2502 3134 98 4 v 2508 3226 a Fr(r)2555 3197 y Fo(2)2610 3047 y Fn(\021)2669 3157 y Fr(!)36 b(:)813 b Fs(\(65\))0 3408 y(Let)33 b(also)f(\003)438 3423 y Fi(n)518 3408 y Fs(b)s(e)g(the)h(\(b)s(ounded\))h(linear)f(op)s (erator)e(on)i Fr(Z)7 b Fs(\()p Fr(m)p Fs(\))33 b(de\014ned)h(b)m(y)f (\003)2899 3423 y Fo(0)2966 3408 y Fs(=)27 b(0)33 b(and)1213 3628 y(\003)1281 3643 y Fi(n)1328 3628 y Fr(!)47 b Fs(=)e Fr(in)p Fs(\()p Fr('!)25 b Fp(\000)e Fr(g)t Fs(\012\))32 b Fr(;)114 b(n)28 b Fp(6)p Fs(=)g(0)k Fr(;)1040 b Fs(\(66\))0 3848 y(where)884 3988 y Fr(')p Fs(\()p Fr(r)s Fs(\))43 b(=)1317 3921 y(1)p 1244 3965 194 4 v 1244 4056 a(2)p Fr(\031)t(r)1399 4028 y Fo(2)1448 3988 y Fs(\(1)22 b Fp(\000)h Fr(e)1702 3947 y Fk(\000)p Fi(r)1791 3924 y Fl(2)1825 3947 y Fi(=)p Fo(4)1900 3988 y Fs(\))32 b Fr(;)115 b(g)t Fs(\()p Fr(r)s Fs(\))43 b(=)2488 3921 y(1)p 2459 3965 108 4 v 2459 4056 a(4)p Fr(\031)2593 3988 y(e)2638 3947 y Fk(\000)p Fi(r)2727 3924 y Fl(2)2762 3947 y Fi(=)p Fo(4)2869 3988 y Fr(;)0 4188 y Fs(and)649 4341 y(\012\()p Fr(r)s Fs(\))h(=)1073 4273 y(1)p 1016 4318 163 4 v 1016 4409 a(4)p Fp(j)p Fr(n)p Fp(j)1205 4200 y Fn(\022)1279 4205 y(Z)1378 4231 y Fi(r)1334 4431 y Fo(0)1416 4230 y Fn(\020)1486 4273 y Fr(z)p 1486 4318 50 4 v 1488 4409 a(r)1545 4230 y Fn(\021)1605 4252 y Fk(j)p Fi(n)p Fk(j)1691 4341 y Fr(z)t(!)t Fs(\()p Fr(z)t Fs(\))17 b(d)p Fr(z)27 b Fs(+)2171 4205 y Fn(Z)2271 4231 y Fk(1)2227 4431 y Fi(r)2346 4230 y Fn(\020)2417 4273 y Fr(r)p 2415 4318 V 2415 4409 a(z)2475 4230 y Fn(\021)2535 4252 y Fk(j)p Fi(n)p Fk(j)2621 4341 y Fr(z)t(!)t Fs(\()p Fr(z)t Fs(\))17 b(d)p Fr(z)2980 4200 y Fn(\023)3103 4341 y Fr(:)476 b Fs(\(67\))0 4567 y(It)33 b(is)g(easy)g(to)g(see)g(that)g(the)g(op)s (erator)f(\003)1536 4582 y Fi(n)1615 4567 y Fs(is)h(indeed)h(w)m (ell-de\014ned:)0 4795 y Fq(Lemma)39 b(4.3)49 b Fj(If)38 b Fr(n)e Fp(2)f Fq(Z)958 4759 y Fk(\003)1037 4795 y Fj(and)j Fr(!)h Fp(2)c Fr(Z)7 b Fs(\()p Fr(m)p Fs(\))39 b Fj(for)g(some)f Fr(m)e(>)f Fs(0)p Fj(,)k(then)g(\(67\))f(de\014nes)g(a)h(c)-5 b(ontinuous)0 4915 y(function)42 b Fs(\012)f(:)f Fq(R)653 4930 y Fo(+)753 4915 y Fp(!)g Fq(C)p Fj(.)66 b(Mor)-5 b(e)g(over,)43 b Fs(\012\()p Fr(r)s Fs(\))p Fr(=r)h Fj(c)-5 b(onver)g(ges)41 b(to)h(zer)-5 b(o)41 b(as)h Fr(r)h Fp(!)d(1)i Fj(and)f(is)h(at)g(most)0 5036 y(lo)-5 b(garithmic)g(al)5 b(ly)34 b(diver)-5 b(gent)34 b(as)h Fr(r)30 b Fp(!)d Fs(0)p Fj(.)0 5264 y Fq(Pro)s(of:)37 b Fs(The)d(pro)s(of)e(is)h (straigh)m(tforw)m(ard)g(using)g(\(64\),)f(\(67\))g(and)h(H\177)-49 b(older's)33 b(inequalit)m(y)-8 b(.)419 b Fb(\003)146 5434 y Fs(W)-8 b(e)33 b(no)m(w)h(sho)m(w)f(that)g Fp(L)1040 5449 y Fi(n)1119 5434 y Fs(and)f(\003)1376 5449 y Fi(n)1456 5434 y Fs(are)g(the)h(expressions)j(of)c Fp(L)g Fs(and)h(\003)f(in)h(p) s(olar)f(co)s(ordinates:)1841 5800 y(24)p eop %%Page: 25 25 25 24 bop 0 100 a Fq(Lemma)39 b(4.4)49 b Fj(The)44 b(op)-5 b(er)g(ators)44 b Fp(L)g Fj(and)g Fs(\003)g Fj(c)-5 b(ommute)44 b(with)g(the)h(pr)-5 b(oje)g(ctions)43 b Fr(P)3035 115 y Fi(n)3082 100 y Fj(.)73 b(If)44 b Fr(n)h Fp(2)h Fq(Z)f Fj(and)0 220 y Fr(w)30 b Fp(2)e(D)271 235 y Fi(m)338 220 y Fs(\()p Fp(L)p Fs(\))34 b Fj(for)h(some)f Fr(m)28 b(>)f Fs(0)p Fj(,)35 b(then)967 437 y Fs(\()p Fp(L)p Fr(P)1137 452 y Fi(n)1183 437 y Fr(w)s Fs(\)\()p Fr(r)19 b Fs(cos)e Fr(\022)s(;)g(r)i Fs(sin)e Fr(\022)s Fs(\))84 b(=)e Fr(e)2206 396 y Fi(in\022)2313 437 y Fs(\()p Fp(L)2420 452 y Fi(n)2466 437 y Fr(!)2527 452 y Fi(n)2574 437 y Fs(\)\()p Fr(r)s Fs(\))34 b Fr(;)810 b Fs(\(68\))968 583 y(\(\003)p Fr(P)1137 598 y Fi(n)1183 583 y Fr(w)s Fs(\)\()p Fr(r)19 b Fs(cos)e Fr(\022)s(;)g(r)i Fs(sin)e Fr(\022)s Fs(\))84 b(=)e Fr(e)2206 542 y Fi(in\022)2313 583 y Fs(\(\003)2419 598 y Fi(n)2465 583 y Fr(!)2526 598 y Fi(n)2573 583 y Fs(\)\()p Fr(r)s Fs(\))34 b Fr(;)811 b Fs(\(69\))0 800 y Fj(wher)-5 b(e)34 b Fr(!)336 815 y Fi(n)383 800 y Fs(\()p Fr(r)s Fs(\))27 b(=)h Fr(e)682 764 y Fk(\000)p Fi(in\022)843 800 y Fs(\()p Fr(P)944 815 y Fi(n)990 800 y Fr(w)s Fs(\)\()p Fr(r)19 b Fs(cos)e Fr(\022)s(;)g(r)i Fs(sin)f Fr(\022)s Fs(\))p Fj(.)0 1025 y Fq(Pro)s(of:)39 b Fs(All)c(w)m(e)g(need)h(is)f(to)f(pro)m(v)m(e)h (\(68\))f(and)g(\(69\).)49 b(The)35 b(\014rst)g(relation)f(follo)m(ws)h (from)g(\(58\))e(b)m(y)j(an)0 1146 y(elemen)m(tary)29 b(calculation.)43 b(T)-8 b(o)27 b(pro)m(v)m(e)i(\(69\),)f(assume)g (that)g Fr(w)s Fs(\()p Fr(r)18 b Fs(cos)f Fr(\022)s(;)g(r)j Fs(sin)d Fr(\022)s Fs(\))28 b(=)f Fr(!)3076 1161 y Fi(n)3123 1146 y Fs(\()p Fr(r)s Fs(\))p Fr(e)3291 1110 y Fi(in\022)3424 1146 y Fs(for)g(some)0 1266 y Fr(n)h Fp(2)g Fq(Z)33 b Fs(and)g(some)g Fr(!)776 1281 y Fi(n)850 1266 y Fp(2)c Fr(Z)7 b Fs(\()p Fr(m)p Fs(\).)43 b(Then)867 1528 y Fq(v)928 1487 y Fi(G)1009 1528 y Fp(\001)22 b(r)p Fr(w)47 b Fs(=)1441 1461 y(1)p 1389 1505 155 4 v 1389 1596 a(2)p Fr(\031)t(r)1553 1528 y Fs(\(1)22 b Fp(\000)g Fr(e)1806 1487 y Fk(\000)p Fi(r)1895 1463 y Fl(2)1930 1487 y Fi(=)p Fo(4)2005 1528 y Fs(\))2053 1461 y(1)p 2053 1505 49 4 v 2054 1596 a Fr(r)2111 1528 y(@)2162 1543 y Fi(\022)2202 1528 y Fr(w)47 b Fs(=)d Fr(ine)2575 1487 y Fi(in\022)2681 1528 y Fr('!)2806 1543 y Fi(n)2885 1528 y Fr(:)0 1771 y Fs(On)f(the)h(other)g(hand,)i (the)e(v)m(elo)s(cit)m(y)h(\014eld)f Fr(v)j Fs(corresp)s(onding)d(to)f Fr(w)j Fs(satis\014es)f Fr(@)3072 1786 y Fo(1)3112 1771 y Fr(v)3159 1786 y Fo(2)3228 1771 y Fp(\000)30 b Fr(@)3386 1786 y Fo(2)3426 1771 y Fr(v)3473 1786 y Fo(1)3558 1771 y Fs(=)46 b Fr(w)s Fs(,)0 1891 y Fr(@)51 1906 y Fo(1)91 1891 y Fr(v)138 1906 y Fo(1)200 1891 y Fs(+)22 b Fr(@)349 1906 y Fo(2)389 1891 y Fr(v)436 1906 y Fo(2)503 1891 y Fs(=)27 b(0.)44 b(In)33 b(p)s(olar)f(co)s(ordinates,)h(these)h (relations)f(b)s(ecome)826 2080 y(1)p 826 2125 V 827 2216 a Fr(r)885 2147 y(@)936 2162 y Fi(r)974 2147 y Fs(\()p Fr(r)s(v)1106 2162 y Fi(\022)1145 2147 y Fs(\))22 b Fp(\000)1314 2080 y Fs(1)p 1314 2125 V 1315 2216 a Fr(r)1373 2147 y(@)1424 2162 y Fi(\022)1464 2147 y Fr(v)1511 2162 y Fi(r)1593 2147 y Fs(=)44 b Fr(w)35 b(;)1969 2080 y Fs(1)p 1969 2125 V 1970 2216 a Fr(r)2028 2147 y(@)2079 2162 y Fi(r)2118 2147 y Fs(\()p Fr(r)s(v)2250 2162 y Fi(r)2287 2147 y Fs(\))22 b(+)2455 2080 y(1)p 2455 2125 V 2456 2216 a Fr(r)2514 2147 y(@)2565 2162 y Fi(\022)2605 2147 y Fr(v)2652 2162 y Fi(\022)2735 2147 y Fs(=)44 b(0)33 b Fr(:)0 2403 y Fs(W)-8 b(e)33 b(lo)s(ok)f(for)g(a)h(solution)g(of)f (the)h(form)f Fr(v)1535 2418 y Fi(r)1601 2403 y Fs(=)f(\026)-53 b Fr(v)1751 2418 y Fi(r)1790 2403 y Fs(\()p Fr(r)s Fs(\))17 b Fr(e)1975 2367 y Fi(in\022)2080 2403 y Fs(,)32 b Fr(v)2186 2418 y Fi(\022)2253 2403 y Fs(=)g(\026)-53 b Fr(v)2404 2418 y Fi(\022)2443 2403 y Fs(\()p Fr(r)s Fs(\))17 b Fr(e)2628 2367 y Fi(in\022)2733 2403 y Fs(.)44 b(Then)977 2621 y(\()p Fr(r)6 b Fs(\026)-52 b Fr(v)1109 2636 y Fi(\022)1148 2621 y Fs(\))1186 2580 y Fk(0)1232 2621 y Fp(\000)22 b Fr(in)t Fs(\026)-53 b Fr(v)1469 2636 y Fi(r)1552 2621 y Fs(=)44 b Fr(r)s(!)1780 2636 y Fi(n)1859 2621 y Fr(;)114 b Fs(\()p Fr(r)6 b Fs(\026)-52 b Fr(v)2132 2636 y Fi(r)2170 2621 y Fs(\))2208 2580 y Fk(0)2254 2621 y Fs(+)22 b Fr(in)t Fs(\026)-53 b Fr(v)2490 2636 y Fi(\022)2574 2621 y Fs(=)44 b(0)32 b Fr(:)0 2838 y Fs(Eliminating)37 b(\026)-53 b Fr(v)579 2853 y Fi(\022)619 2838 y Fs(,)32 b(w)m(e)i(\014nd)f(the)g (follo)m(wing)g(ODE)f(for)g Fr(h)c Fs(=)g Fr(r)6 b Fs(\026)-52 b Fr(v)2277 2853 y Fi(r)2314 2838 y Fs(:)1310 3109 y(\()p Fr(r)s(h)1451 3068 y Fk(0)1474 3109 y Fs(\))1512 3068 y Fk(0)1557 3109 y Fp(\000)1667 3042 y Fr(n)1725 3006 y Fo(2)p 1667 3086 98 4 v 1692 3178 a Fr(r)1775 3109 y(h)22 b Fs(+)g Fr(inr)s(!)2150 3124 y Fi(n)2241 3109 y Fs(=)44 b(0)32 b Fr(:)0 3352 y Fs(The)i(general)f(solution)g(is:)267 3621 y Fr(h)p Fs(\()p Fr(r)s Fs(\))44 b(=)656 3554 y Fr(in)p 620 3598 163 4 v 620 3690 a Fs(2)p Fp(j)p Fr(n)p Fp(j)809 3481 y Fn(\022)883 3486 y(Z)982 3512 y Fi(r)938 3711 y Fo(0)1020 3511 y Fn(\020)1090 3554 y Fr(z)p 1090 3598 50 4 v 1092 3690 a(r)1149 3511 y 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Fr(\030)5 b Fs(\))17 b(d)p Fr(\030)47 b Fs(=)e(0)32 b(for)g Fr(j)i Fs(=)27 b(1)p Fr(;)17 b Fs(2)2564 5400 y Fn(o)2674 5511 y Fs(=)44 b Fr(X)30 b Fp(\\)23 b Fr(L)3060 5470 y Fo(2)3060 5536 y(1)3099 5511 y Fs(\()p Fr(m)p Fs(\))33 b Fr(:)1841 5800 y Fs(27)p eop %%Page: 28 28 28 27 bop 0 100 a Fs(According)38 b(to)f(Lemma)g(4.5,)h(if)f Fr(\026)g Fs(is)h(an)f(eigen)m(v)-5 b(alue)39 b(of)d Fp(L)25 b(\000)h Fr(\013)q Fs(\003)37 b(in)g Fr(L)2702 63 y Fo(2)2742 100 y Fs(\()p Fr(m)p Fs(\))g(with)h(Re)17 b(\()p Fr(\026)p Fs(\))35 b Fr(>)3590 60 y Fo(1)p Fk(\000)p Fi(m)p 3590 77 153 4 v 3649 134 a Fo(2)3752 100 y Fs(,)0 220 y(the)41 b(corresp)s(onding)h(eigenfunction)g(b)s(elongs)f(to)f Fr(X)8 b Fs(.)67 b(This)42 b(result)g(is)f(v)m(ery)h(useful)g(b)s (ecause)g(b)s(oth)0 340 y(op)s(erators)32 b Fp(L)h Fs(and)f(\003)h(ha)m (v)m(e)g(nice)h(prop)s(erties)f(in)g(this)g(space.)0 560 y Fq(Lemma)39 b(4.7)49 b Fj(The)29 b(line)-5 b(ar)29 b(op)-5 b(er)g(ator)29 b Fp(L)h Fj(is)f(self-adjoint)f(in)h Fr(X)8 b Fj(,)31 b(and)e Fp(L)e(\024)h Fs(0)p Fj(.)43 b(Mor)-5 b(evo)g(er,)30 b Fp(L)e(\024)g(\000)p Fs(1)p Fr(=)p Fs(2)0 680 y Fj(on)34 b Fr(X)220 695 y Fo(0)295 680 y Fj(and)g Fp(L)27 b(\024)i(\000)p Fs(1)35 b Fj(on)f Fr(X)1067 695 y Fo(1)1107 680 y Fj(.)0 900 y Fq(Pro)s(of:)j Fs(De\014ne)c Fr(L)28 b Fs(:)g Fp(D)s Fs(\()p Fr(L)p Fs(\))g Fp(!)f Fr(L)1234 864 y Fo(2)1274 900 y Fs(\()p Fq(R)1396 864 y Fo(2)1435 900 y Fs(\))32 b(b)m(y)i Fp(D)s Fs(\()p Fr(L)p Fs(\))27 b(=)h Fp(f)p Fr( )j Fp(2)d Fr(H)2321 864 y Fo(2)2360 900 y Fs(\()p Fq(R)2482 864 y Fo(2)2521 900 y Fs(\))17 b Fp(j)g(j)p Fr(\030)5 b Fp(j)2725 864 y Fo(2)2762 900 y Fr( )32 b Fp(2)c Fr(L)3017 864 y Fo(2)3057 900 y Fs(\()p Fq(R)3179 864 y Fo(2)3218 900 y Fs(\))p Fp(g)k Fs(and)1128 1172 y Fr(L)45 b Fs(=)f Fr(G)1436 1131 y Fk(\000)p Fo(1)p Fi(=)p Fo(2)1601 1172 y Fp(L)17 b Fr(G)1764 1131 y Fo(1)p Fi(=)p Fo(2)1917 1172 y Fs(=)45 b(\001)22 b Fp(\000)2251 1105 y(j)p Fr(\030)5 b Fp(j)2355 1068 y Fo(2)p 2251 1149 143 4 v 2273 1240 a Fs(16)2425 1172 y(+)2533 1105 y(1)p 2533 1149 49 4 v 2533 1240 a(2)2624 1172 y Fr(:)0 1415 y Fs(In)34 b(quan)m(tum)g(mec)m(hanics,)i(the)e(op)s (erator)e Fp(\000)p Fr(L)i Fs(is)g(\(up)f(to)g(n)m(umerical)i(constan)m (ts\))g(the)e(Hamiltonian)0 1536 y(of)c(the)h(harmonic)g(oscillator)g (in)f Fq(R)1307 1500 y Fo(2)1346 1536 y Fs(.)43 b(As)30 b(is)g(w)m(ell-kno)m(wn)h(\(see)g(for)e(example)i([17)o(]\),)f Fr(L)g Fs(is)g(self-adjoin)m(t)0 1656 y(in)39 b Fr(L)186 1620 y Fo(2)226 1656 y Fs(\()p Fq(R)348 1620 y Fo(2)387 1656 y Fs(\))g(and)g Fr(\033)t Fs(\()p Fr(L)p Fs(\))g(=)g Fp(f\000)p Fr(n=)p Fs(2)17 b Fp(j)g Fr(n)38 b Fp(2)h Fq(N)p Fp(g)p Fs(.)63 b(By)39 b(construction,)j(the)e(op)s(erator)e Fp(L)h Fs(:)f Fp(D)s Fs(\()p Fp(L)p Fs(\))g Fp(!)h Fr(X)0 1777 y Fs(with)33 b(domain)760 1916 y Fp(D)s Fs(\()p Fp(L)p Fs(\))27 b(=)1116 1806 y Fn(n)1182 1916 y Fr(w)j Fp(2)e Fr(X)1482 1802 y Fn(\014)1482 1862 y(\014)1482 1921 y(\014)1531 1916 y Fp(j)p Fr(\030)5 b Fp(j)1635 1875 y Fo(2)1674 1916 y Fr(w)30 b Fp(2)e Fr(X)40 b(;)49 b Fs(\001)p Fr(w)25 b Fs(+)2349 1849 y(1)p 2349 1894 V 2349 1985 a(2)2408 1916 y Fr(\030)h Fp(\001)c(r)p Fr(w)30 b Fp(2)e Fr(X)2893 1806 y Fn(o)2992 1916 y Fr(;)0 2113 y Fs(is)g(th)m(us)g(self-adjoin)m(t)g(in)g Fr(X)35 b Fs(with)28 b(the)f(same)i(sp)s(ectrum.)43 b(In)28 b(particular,)g Fp(L)g(\024)g Fs(0.)41 b(No)m(w,)29 b(observ)m(e)h(that)0 2233 y(0)39 b(is)h(a)f(simple)i(eigen)m(v)-5 b(alue)41 b(of)e Fp(L)g Fs(with)h(eigenfunction)g Fr(G)p Fs(,)h(and)f(that)f Fr(X)2766 2248 y Fo(0)2845 2233 y Fs(is)h(just)f(the)h(orthogonal)0 2354 y(complemen)m(t)e(of)e(the)h(eigenspace)h Fq(R)p Fr(G)e Fs(in)g Fr(X)8 b Fs(.)54 b(Th)m(us)38 b Fr(X)2132 2369 y Fo(0)2208 2354 y Fs(is)e(stable)h(under)h Fp(L)d Fs(and)i(the)f(restriction)0 2474 y(of)g Fp(L)g Fs(to)h Fr(X)425 2489 y Fo(0)500 2474 y Fs(is)h(a)e(self-adjoin)m(t)h(op)s (erator)f(satisfying)h Fp(L)d(\024)h(\000)p Fs(1)p Fr(=)p Fs(2.)56 b(Similarly)-8 b(,)38 b(one)f(can)g(sho)m(w)h(that)0 2595 y Fp(L)27 b(\024)i(\000)p Fs(1)j(on)h Fr(X)577 2610 y Fo(1)616 2595 y Fs(.)3059 b Fb(\003)0 2814 y Fq(Lemma)39 b(4.8)49 b Fj(The)34 b(line)-5 b(ar)35 b(op)-5 b(er)g(ator)34 b Fs(\003)h Fj(is)f(skew-symmetric)g(in)h Fr(X)8 b Fj(.)0 3034 y Fq(Pro)s(of:)49 b Fs(Since)44 b(\003)p Fr(G)h Fs(=)g(0)e(and)g(since)h(the)f(subspace)i Fr(X)2146 3049 y Fo(0)2228 3034 y Fs(is)f(stable)g(under)f(\003,)j(it)c(is)i (su\016cien)m(t)h(to)0 3154 y(sho)m(w)40 b(that)e(\003)g(is)i(sk)m (ew-symmetric)i(on)c Fr(X)1612 3169 y Fo(0)1652 3154 y Fs(.)61 b(Let)39 b Fr(w)s(;)d Fs(~)-68 b Fr(w)40 b Fp(2)e Fr(X)2333 3169 y Fo(0)2399 3154 y Fp(\\)26 b(D)s Fs(\()p Fp(L)p Fs(\),)40 b(and)e(denote)i(b)m(y)f Fq(v)r Fr(;)3551 3153 y Fs(~)3545 3154 y Fq(v)h Fs(the)0 3275 y(corresp)s(onding)30 b(v)m(elo)s(cit)m(y)h(\014elds.)44 b(Without)29 b(loss)h(of)f(generalit)m(y)-8 b(,)31 b(w)m(e)f(assume)g (that)f Fr(w)s(;)36 b Fs(~)-68 b Fr(w)31 b Fs(\(hence)f(also)0 3395 y Fq(v)r Fr(;)110 3394 y Fs(~)105 3395 y Fq(v)q 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Fi(G)1424 4220 y Fp(\001)22 b(r)p Fr(w)d Fs(d)p Fr(\030)32 b Fs(=)c Fp(\000)1973 4085 y Fn(Z)2028 4310 y Fm(R)2089 4291 y Fl(2)2168 4153 y Fs(1)p 2154 4197 V 2154 4289 a Fr(G)2258 4220 y(w)s Fq(v)2392 4179 y Fi(G)2472 4220 y Fp(\001)22 b(r)e Fs(~)-69 b Fr(w)19 b Fs(d)p Fr(\030)37 b(:)751 b Fs(\(73\))0 4480 y(On)33 b(the)g(other)f(hand,)h(the)g(follo)m(wing)g(iden)m(tit)m(y)i(is)e (easy)g(to)f(c)m(hec)m(k:)263 4692 y(~)-69 b Fr(w)s Fs(\()p Fq(v)23 b Fp(\001)f Fr(\030)5 b Fs(\))22 b(+)g Fr(w)s Fs(\()809 4691 y(~)803 4692 y Fq(v)h Fp(\001)f Fr(\030)5 b Fs(\))43 b(=)i(\()p Fr(\030)1266 4707 y Fo(1)1305 4692 y Fr(@)1356 4707 y Fo(1)1418 4692 y Fp(\000)22 b Fr(\030)1560 4707 y Fo(2)1599 4692 y Fr(@)1650 4707 y Fo(2)1690 4692 y Fs(\)\()p Fr(v)1813 4707 y Fo(1)1856 4692 y Fs(~)-52 b Fr(v)1900 4707 y Fo(2)1961 4692 y Fs(+)22 b Fr(v)2106 4707 y Fo(2)2150 4692 y Fs(~)-53 b Fr(v)2193 4707 y Fo(1)2233 4692 y Fs(\))22 b(+)g(\()p Fr(\030)2472 4707 y Fo(1)2511 4692 y Fr(@)2562 4707 y Fo(2)2624 4692 y Fs(+)g Fr(\030)2765 4707 y Fo(2)2804 4692 y Fr(@)2855 4707 y Fo(1)2895 4692 y Fs(\)\()p Fr(v)3018 4707 y Fo(2)3061 4692 y Fs(~)-53 b Fr(v)3104 4707 y Fo(2)3166 4692 y Fp(\000)23 b Fr(v)3313 4707 y Fo(1)3356 4692 y Fs(~)-53 b Fr(v)3399 4707 y Fo(1)3439 4692 y Fs(\))32 b Fr(:)0 4905 y Fs(Since)41 b Fr(w)s(;)36 b Fs(~)-68 b Fr(w)41 b Fs(ha)m(v)m(e)f(zero)g(mean,)i(it)d(follo)m(ws)i (from)e(\([13)o(],)j(App)s(endix)f(B\))e(that)g Fq(v)r Fr(;)3086 4904 y Fs(~)3080 4905 y Fq(v)i Fs(deca)m(y)g(at)e(least)0 5025 y(lik)m(e)34 b(1)p Fr(=)p Fp(j)p Fr(\030)5 b Fp(j)381 4989 y Fo(2)451 5025 y Fs(as)33 b Fr(\030)f Fp(!)27 b(1)p Fs(.)43 b(Th)m(us,)35 b(in)m(tegrating)d(b)s(oth)h(sides,)h(w)m(e)g (obtain)1190 5150 y Fn(Z)1246 5376 y Fm(R)1307 5357 y Fl(2)1345 5286 y Fs(\()20 b(~)-69 b Fr(w)s Fs(\()p Fq(v)23 b Fp(\001)f Fr(\030)5 b Fs(\))22 b(+)g Fr(w)s Fs(\()1949 5285 y(~)1943 5286 y Fq(v)h Fp(\001)f Fr(\030)5 b Fs(\)\))17 b(d)p Fr(\030)48 b Fs(=)c(0)32 b Fr(:)1017 b Fs(\(74\))0 5551 y(Com)m(bining)34 b(\(73\))e(and)g(\(74\),)h(w)m(e)g(see)h(that)e (\()20 b(~)-69 b Fr(w)s(;)17 b Fs(\003)p Fr(w)s Fs(\))1971 5566 y Fi(X)2059 5551 y Fs(+)22 b(\(\003)d(~)-68 b Fr(w)r(;)17 b(w)s Fs(\))2490 5566 y Fi(X)2584 5551 y Fs(=)28 b(0.)938 b Fb(\003)1841 5800 y Fs(28)p eop %%Page: 29 29 29 28 bop 0 100 a Fq(Pro)s(of)36 b(of)g(Prop)s(osition)g(4.1.)43 b Fs(Fix)31 b Fr(m)d(>)g Fs(1)j(and)g(assume)i(that)e Fr(\025)g Fs(is)g(an)g(eigen)m(v)-5 b(alue)33 b(of)e Fp(L)19 b(\000)h Fr(\013)q Fs(\003)30 b(in)0 220 y Fr(L)66 184 y Fo(2)66 245 y(0)106 220 y Fs(\()p Fr(m)p Fs(\))37 b(with)h(Re)17 b(\()p Fr(\025)p Fs(\))35 b Fr(>)953 181 y Fo(1)p Fk(\000)p Fi(m)p 953 197 153 4 v 1012 254 a Fo(2)1115 220 y Fs(.)58 b(By)38 b(Lemma)g(4.5,)g(there)g(exists)h(a)e (nonzero)h Fr(w)g Fp(2)e Fr(X)3174 235 y Fo(0)3239 220 y Fp(\\)26 b(D)s Fs(\()p Fp(L)p Fs(\))36 b(suc)m(h)0 340 y(that)c(\()p Fp(L)22 b(\000)h Fr(\013)q Fs(\003\))p Fr(w)29 b Fs(=)f Fr(\025w)s Fs(.)43 b(In)32 b(particular,)1091 543 y Fr(\025)p Fs(\()p Fr(w)s(;)17 b(w)s Fs(\))1414 558 y Fi(X)1524 543 y Fs(=)44 b(\()p Fr(w)s(;)17 b Fp(L)p Fr(w)s Fs(\))1979 558 y Fi(X)2067 543 y Fp(\000)23 b Fr(\013)q Fs(\()p Fr(w)s(;)17 b Fs(\003)p Fr(w)s Fs(\))2564 558 y Fi(X)2661 543 y Fr(;)0 745 y Fs(hence)957 885 y(Re)g(\()p Fr(\025)p Fs(\)\()p Fr(w)s(;)g(w)s Fs(\))1488 900 y Fi(X)1598 885 y Fs(=)44 b(\()p Fr(w)s(;)17 b Fp(L)p Fr(w)s Fs(\))2053 900 y Fi(X)2163 885 y Fp(\024)45 b(\000)2372 817 y Fs(1)p 2372 862 49 4 v 2372 953 a(2)2431 885 y(\()p Fr(w)s(;)17 b(w)s Fs(\))2697 900 y Fi(X)2795 885 y Fr(;)0 1082 y Fs(since)34 b(\003)e(is)h(sk)m(ew-symmetric)k(and)32 b Fp(L)c(\024)g(\000)p Fs(1)p Fr(=)p Fs(2)k(on)h Fr(X)2010 1097 y Fo(0)2049 1082 y Fs(.)44 b(Th)m(us,)34 b(Re)17 b(\()p Fr(\025)p Fs(\))27 b Fp(\024)h(\000)p Fs(1)p Fr(=)p Fs(2.)146 1203 y(Similarly)-8 b(,)42 b(if)d Fr(m)g(>)f Fs(2)h(and)g Fr(\025)g Fs(is)h(an)f(eigen)m(v)-5 b(alue)40 b(of)f Fp(L)26 b(\000)h Fr(\013)q Fs(\003)38 b(in)i Fr(L)2694 1167 y Fo(2)2694 1227 y(1)2733 1203 y Fs(\()p Fr(m)p Fs(\))g(with)f(Re)17 b(\()p Fr(\025)p Fs(\))38 b Fr(>)3590 1164 y Fo(1)p Fk(\000)p Fi(m)p 3590 1180 153 4 v 3649 1237 a Fo(2)3752 1203 y Fs(,)0 1323 y(there)d(exists)h(a)e(nonzero)h Fr(w)e Fp(2)e Fr(X)1251 1338 y Fo(1)1314 1323 y Fp(\\)23 b(D)s Fs(\()p Fp(L)p Fs(\))34 b(suc)m(h)h(that)f(\()p Fp(L)23 b(\000)h Fr(\013)q Fs(\003\))p Fr(w)33 b Fs(=)d Fr(\025w)s Fs(.)48 b(Pro)s(ceeding)35 b(as)g(ab)s(o)m(v)m(e)0 1444 y(and)e(using)g(the)g(fact)f(that)h Fp(L)27 b(\024)h(\000)p Fs(1)33 b(on)g Fr(X)1594 1459 y Fo(1)1633 1444 y Fs(,)g(w)m(e)g(obtain) g(Re)16 b(\()p Fr(\025)p Fs(\))28 b Fp(\024)g(\000)p Fs(1.)1012 b Fb(\003)0 1651 y Fq(Remark)38 b(4.9)49 b Fj(If)37 b Fr(m)d(>)f Fs(3)k Fj(and)h Fr(\025)g Fj(is)f(an)h (eigenvalue)e(of)i Fp(L)24 b(\000)h Fr(\013)q Fs(\003)37 b Fj(in)h Fr(L)2717 1615 y Fo(2)2717 1676 y(2)2756 1651 y Fs(\()p Fr(m)p Fs(\))h Fj(with)e Fs(Re)17 b(\()p Fr(\025)p Fs(\))33 b Fr(>)3587 1612 y Fo(1)p Fk(\000)p Fi(m)p 3587 1629 V 3646 1686 a Fo(2)3750 1651 y Fj(,)0 1772 y(ther)-5 b(e)30 b(exists)f(a)h(nonzer)-5 b(o)28 b Fr(w)i Fp(2)f(D)s Fs(\()p Fp(L)p Fs(\))11 b Fp(\\)g Fr(L)1512 1736 y Fo(2)1512 1796 y(2)1550 1772 y Fs(\()p Fr(m)p Fs(\))30 b Fj(such)g(that)g Fs(\()p Fp(L)11 b(\000)g Fr(\013)q Fs(\003\))p Fr(w)29 b Fs(=)e Fr(\025w)s Fj(.)43 b(The)29 b(ar)-5 b(gument)29 b(ab)-5 b(ove)0 1892 y(shows)34 b(that)i Fs(Re)16 b(\()p Fr(\025)p Fs(\))28 b Fp(\024)h(\000)p Fs(1)p Fj(,)35 b(and)g(that)g Fs(Re)17 b(\()p Fr(\025)p Fs(\))28 b(=)g Fp(\000)p Fs(1)35 b Fj(if)g(and)g(only)g(if)g Fp(L)p Fr(w)30 b Fs(=)e Fp(\000)p Fr(w)s Fj(.)45 b(But)36 b(this)f(implies)0 2013 y(that)k Fr(w)h Fj(is)f(a)f(line)-5 b(ar)37 b(c)-5 b(ombination)37 b(of)h Fs(\()p Fr(@)1554 1976 y Fo(2)1549 2037 y(1)1619 2013 y Fp(\000)25 b Fr(@)1777 1976 y Fo(2)1772 2037 y(2)1818 2013 y Fs(\))p Fr(G)38 b Fj(and)g Fr(@)2215 2028 y Fo(1)2255 2013 y Fr(@)2306 2028 y Fo(2)2346 2013 y Fr(G)p Fj(,)h(and)e(a)i(dir)-5 b(e)g(ct)38 b(c)-5 b(alculation)37 b(shows)0 2133 y(that)d(no)g(such)f Fr(w)k Fj(c)-5 b(an)33 b(b)-5 b(e)34 b(an)f(eigenfunction)g(of)h Fp(L)19 b(\000)i Fr(\013)q Fs(\003)33 b Fj(if)h Fr(\013)28 b Fp(6)p Fs(=)g(0)p Fj(.)44 b(Thus)33 b(any)h(eigenfunction)f Fr(\025)h Fj(of)0 2253 y Fp(L)22 b(\000)g Fr(\013)q Fs(\003)35 b Fj(in)f Fr(L)541 2217 y Fo(2)541 2278 y(2)581 2253 y Fs(\()p Fr(m)p Fs(\))h Fj(satis\014es)f Fs(Re)17 b(\()p Fr(\025)p Fs(\))27 b Fr(<)h Fp(\000)p Fs(1)35 b Fj(if)g Fr(\013)28 b Fp(6)p Fs(=)g(0)p Fj(.)0 2540 y Ff(4.2)135 b(Bounds)44 b(on)h(the)g(linear)h(ev)l(olution)0 2724 y Fs(In)28 b(this)g(subsection)h(w)m(e)g(pro)m(v)m(e)f(that)g(the)f(eigen)m(v)-5 b(alue)29 b(estimates)g(of)e(the)h(previous)h(subsection)g(imply)0 2845 y(analogous)38 b(b)s(ounds)i(on)e(the)h(linear)g(ev)m(olution.)63 b(Fix)39 b Fr(\013)f Fp(2)h Fq(R)f Fs(and)h(consider)h(the)f(linear)g (equation)0 2965 y Fr(@)51 2980 y Fi(\034)95 2965 y Fr(R)29 b Fs(=)e Fp(L)p Fr(R)c Fp(\000)g Fr(\013)q Fs(\003)p Fr(R)33 b Fs(whic)m(h)h(is)g(the)f(linearization)g(of)f(\(12\))g(a)h (the)g(v)m(ortex)h Fr(\013)q(G)p Fs(,)e(see)i(\(28\),)e(\(57\).)44 b(The)0 3085 y(corresp)s(onding)33 b(in)m(tegral)g(equation)h(reads)983 3333 y Fr(R)q Fs(\()p Fr(\034)11 b Fs(\))28 b(=)g Fp(S)7 b Fs(\()p Fr(\034)k Fs(\))p Fr(R)1589 3348 y Fo(0)1652 3333 y Fp(\000)23 b Fr(\013)1831 3197 y Fn(Z)1931 3224 y Fi(\034)1886 3423 y Fo(0)1990 3333 y Fp(S)7 b Fs(\()p Fr(\034)35 b Fp(\000)22 b Fr(s)p Fs(\)\003)p Fr(R)q Fs(\()p Fr(s)p Fs(\))17 b(d)p Fr(s)32 b(;)810 b Fs(\(75\))0 3588 y(where)37 b Fp(S)7 b Fs(\()p Fr(\034)k Fs(\))35 b(=)e(exp)q(\()p Fr(\034)11 b Fp(L)p Fs(\).)54 b(Pro)s(ceeding)37 b(as)f(in)g(Section)h (2.1,)f(it)g(is)h(straigh)m(tforw)m(ard)f(to)g(sho)m(w)h(b)m(y)g(a)0 3708 y(con)m(traction)26 b(mapping)g(argumen)m(t)g(that)f(this)i (equation)f(de\014nes)h(a)e(strongly)h(con)m(tin)m(uous)h(semigroup)0 3828 y Fp(T)54 3843 y Fi(\013)104 3828 y Fs(\()p Fr(\034)11 b Fs(\))37 b(in)g Fr(L)454 3792 y Fo(2)493 3828 y Fs(\()p Fr(m)p Fs(\))g(for)f(an)m(y)i Fr(m)d(>)f Fs(1,)k(namely)g Fr(R)q Fs(\()p Fr(\034)11 b Fs(\))35 b(=)f Fp(T)2123 3843 y Fi(\013)2173 3828 y Fs(\()p Fr(\034)11 b Fs(\))p Fr(R)2376 3843 y Fo(0)2416 3828 y Fs(.)56 b(W)-8 b(e)37 b(\014rst)g(pro)m(v)m(e)h(that)f Fp(T)3413 3843 y Fi(\013)3463 3828 y Fs(\()p Fr(\034)11 b Fs(\))37 b(is)g(a)0 3949 y(compact)c(p)s(erturbation)g(of)f Fp(S)7 b Fs(\()p Fr(\034)k Fs(\).)0 4157 y Fq(Lemma)39 b(4.10)49 b Fj(L)-5 b(et)30 b Fr(m)e(>)f Fs(1)p Fj(.)43 b(The)29 b(line)-5 b(ar)28 b(op)-5 b(er)g(ator)29 b Fp(K)2063 4172 y Fi(\013)2112 4157 y Fs(\()p Fr(\034)11 b Fs(\))28 b(=)g Fp(T)2427 4172 y Fi(\013)2477 4157 y Fs(\()p Fr(\034)11 b Fs(\))f Fp(\000)g(S)d Fs(\()p Fr(\034)k Fs(\))30 b Fj(is)f(c)-5 b(omp)g(act)29 b(in)g Fr(L)3579 4121 y Fo(2)3618 4157 y Fs(\()p Fr(m)p Fs(\))0 4277 y Fj(for)35 b(e)-5 b(ach)34 b Fr(\034)39 b(>)28 b Fs(0)p Fj(.)0 4485 y Fq(Pro)s(of:)j Fs(All)c(the)h(necessary)h(estimates)f(are)f(already)h(con)m(tained)g (in)f(Section)h(2.1.)41 b(Observ)m(e)29 b(that)e(the)0 4605 y(term)g Fp(K)303 4620 y Fi(\013)352 4605 y Fs(\()p Fr(\034)11 b Fs(\))p Fr(R)555 4620 y Fo(0)621 4605 y Fs(in)27 b(\(75\))e(is)i(precisely)i(what)d(w)m(e)h(called)g Fr(R)2131 4620 y Fo(2)2171 4605 y Fs(\()p Fr(\034)11 b Fs(\))e(+)g Fr(R)2468 4620 y Fo(3)2508 4605 y Fs(\()p Fr(\034)i Fs(\))27 b(in)f(the)h(pro)s(of)e(of)h(Lemma)h(2.2.)0 4726 y(Rep)s(eating)33 b(the)g(estimates)h(pro)m(v)m(ed)g(there,)f(w)m (e)h(obtain)1034 4928 y Fp(kK)1160 4943 y Fi(\013)1210 4928 y Fs(\()p Fr(\034)11 b Fs(\))p Fr(R)1413 4943 y Fo(0)1453 4928 y Fp(k)1503 4943 y Fi(m)p Fo(+1)1687 4928 y Fp(\024)28 b Fr(C)7 b Fs(\()p Fr(\034)k Fs(\))p Fp(k)p Fr(R)2122 4943 y Fo(0)2162 4928 y Fp(k)2212 4943 y Fi(m)2311 4928 y Fr(;)114 b(\034)39 b(>)28 b Fs(0)k Fr(;)0 5130 y Fs(for)44 b(some)i Fr(C)7 b Fs(\()p Fr(\034)k Fs(\))49 b Fr(>)f Fs(0.)80 b(Similarly)-8 b(,)49 b(the)c(result)h(of)e(Lemma)h (2.1)g(applies)h(to)e Fp(T)3042 5145 y Fi(\013)3092 5130 y Fs(\()p Fr(\034)11 b Fs(\))p Fr(R)3295 5145 y Fo(0)3335 5130 y Fs(,)48 b(hence)e(to)0 5251 y Fp(K)76 5266 y Fi(\013)125 5251 y Fs(\()p Fr(\034)11 b Fs(\))p Fr(R)328 5266 y Fo(0)368 5251 y Fs(,)33 b(and)g(yields)978 5508 y Fp(krK)1187 5523 y Fi(\013)1236 5508 y Fs(\()p Fr(\034)11 b Fs(\))p Fr(R)1439 5523 y Fo(0)1479 5508 y Fp(k)1529 5523 y Fi(m)1623 5508 y Fp(\024)1780 5440 y Fr(C)c Fs(\()p Fr(\034)k Fs(\))p 1738 5485 291 4 v 1738 5576 a Fr(a)p Fs(\()p Fr(\034)g Fs(\))1918 5547 y Fo(1)p Fi(=)p Fo(2)2055 5508 y Fp(k)p Fr(R)2179 5523 y Fo(0)2219 5508 y Fp(k)2269 5523 y Fi(m)2368 5508 y Fr(;)114 b(\034)39 b(>)28 b Fs(0)k Fr(;)1841 5800 y Fs(29)p eop %%Page: 30 30 30 29 bop 0 100 a Fs(where)34 b Fr(a)p Fs(\()p Fr(\034)11 b Fs(\))28 b(=)g(1)22 b Fp(\000)g Fr(e)809 63 y Fk(\000)p Fi(\034)907 100 y Fs(.)44 b(The)33 b(conclusion)h(then)g(follo)m(ws)f (from)f(Rellic)m(h's)j(criterion.)505 b Fb(\003)146 268 y Fs(W)-8 b(e)27 b(no)m(w)h(use)f(this)h(result)f(to)g(b)s(ound)g(the)g (essen)m(tial)h(sp)s(ectrum)g(of)f Fp(T)2651 283 y Fi(\013)2700 268 y Fs(\()p Fr(\034)11 b Fs(\).)42 b(W)-8 b(e)27 b(recall)g(that)g Fr(\025)g Fp(2)i Fq(C)0 388 y Fs(is)h(in)f(the)g(essen)m(tial)i(sp)s (ectrum)g(of)e(a)f(linear)i(op)s(erator)e Fr(L)i Fs(if)e Fr(\025)h Fs(is)h(not)f(a)g Fj(normal)h(p)-5 b(oint)29 b Fs(for)g Fr(L)p Fs(,)h(i.e.)43 b(if)29 b Fr(\025)0 508 y Fs(is)i(not)e(in)i(the)f(resolv)m(en)m(t)i(set)f(of)e Fr(L)i Fs(and)f(is)g(not)g(an)g(isolated)h(eigen)m(v)-5 b(alue)31 b(of)f Fr(L)g Fs(of)f(\014nite)i(m)m(ultiplicit)m(y)-8 b(,)0 629 y(see)31 b(e.g.)43 b([18].)g(F)-8 b(or)29 b(an)m(y)i Fr(r)g(>)c Fs(0,)k(w)m(e)g(denote)g(b)m(y)g Fp(B)s Fs(\()p Fr(r)s Fs(\))g(the)f(closed)i(disk)f(of)f(radius)h Fr(r)i Fs(cen)m(tered)f(at)e(the)0 749 y(origin)i(in)h Fq(C)p Fs(:)1333 870 y Fp(B)s Fs(\()p Fr(r)s Fs(\))45 b(=)f Fp(f)p Fr(z)32 b Fp(2)c Fq(C)17 b Fp(j)g(j)p Fr(z)t Fp(j)27 b(\024)h Fr(r)s Fp(g)k Fr(:)0 1089 y Fq(Lemma)39 b(4.11)49 b Fj(L)-5 b(et)46 b Fr(m)i(>)g Fs(1)p Fj(.)76 b(F)-7 b(or)45 b(any)h Fr(\034)59 b(>)47 b Fs(0)p Fj(,)h(the)e(essential)f(sp) -5 b(e)g(ctrum)45 b(of)g Fp(T)3157 1104 y Fi(\013)3207 1089 y Fs(\()p Fr(\034)11 b Fs(\))46 b Fj(in)g Fr(L)3579 1053 y Fo(2)3618 1089 y Fs(\()p Fr(m)p Fs(\))0 1210 y Fj(satis\014es)1292 1330 y Fr(\033)1351 1289 y Fo(ess)1347 1355 y Fi(m)1442 1330 y Fs(\()p Fp(T)1534 1345 y Fi(\013)1583 1330 y Fs(\()p Fr(\034)11 b Fs(\)\))45 b(=)f Fp(B)s Fs(\()p Fr(e)2066 1289 y Fi(\034)8 b Fo(\(1)p Fk(\000)p Fi(m)p Fo(\))p Fi(=)p Fo(2)2388 1330 y Fs(\))35 b Fr(:)1118 b Fs(\(76\))0 1550 y Fq(Pro)s(of:)41 b Fs(Fix)36 b Fr(\034)45 b(>)34 b Fs(0.)53 b(The)37 b(results)g(of)f(\([13)o(],)h(App)s(endix)h (A\))e(imply)h(that)f(the)g(essen)m(tial)i(sp)s(ectrum)0 1670 y(of)33 b Fp(S)7 b Fs(\()p Fr(\034)k Fs(\))34 b(in)g Fr(L)523 1634 y Fo(2)563 1670 y Fs(\()p Fr(m)p Fs(\))f(is)h(exactly)h Fp(B)s Fs(\()p Fr(e)1344 1634 y Fi(\034)8 b Fo(\(1)p Fk(\000)p Fi(m)p Fo(\))p Fi(=)p Fo(2)1666 1670 y Fs(\).)46 b(Since)34 b Fp(T)2086 1685 y Fi(\013)2136 1670 y Fs(\()p Fr(\034)11 b Fs(\))34 b(is)g(a)f(compact)h(p)s(erturbation)f(of)g Fp(S)7 b Fs(\()p Fr(\034)k Fs(\),)0 1791 y(it)35 b(follo)m(ws)h(from)e (\([18],)h(Theorem)i(A.1\))d(that)h(the)g(complemen)m(t)i(of)e(the)g Fp(B)s Fs(\()p Fr(e)2909 1755 y Fi(\034)8 b Fo(\(1)p Fk(\000)p Fi(m)p Fo(\))p Fi(=)p Fo(2)3231 1791 y Fs(\))35 b(in)g Fq(C)g Fs(either)0 1911 y(consists)42 b(en)m(tirely)g(of)e (eigen)m(v)-5 b(alues)42 b(of)e Fp(T)1531 1926 y Fi(\013)1581 1911 y Fs(\()p Fr(\034)11 b Fs(\),)43 b(or)d(en)m(tirely)i(of)e(normal) g(p)s(oin)m(ts)h(for)f Fp(T)3236 1926 y Fi(\013)3286 1911 y Fs(\()p Fr(\034)11 b Fs(\).)67 b(In)41 b(our)0 2032 y(case,)47 b(the)e(\014rst)f(p)s(ossibilit)m(y)h(is)f(excluded.)79 b(Indeed,)48 b(assume)e(that)d Fr(\025)j Fp(2)h Fq(C)d Fs(is)g(an)g(eigen)m(v)-5 b(alue)45 b(of)0 2152 y Fp(T)54 2167 y Fi(\013)104 2152 y Fs(\()p Fr(\034)11 b Fs(\))31 b(with)g Fp(j)p Fr(\025)p Fp(j)c Fr(>)h Fs(1.)43 b(By)31 b(the)g(sp)s(ectral)h(mapping)f(theorem,)h Fr(\025)c Fs(=)g Fr(e)2567 2116 y Fi(\027)2641 2152 y Fs(where)k Fr(\027)37 b Fs(is)32 b(an)f(eigen)m(v)-5 b(alue)32 b(of)0 2272 y Fp(L)14 b(\000)g Fr(\013)q Fs(\003)29 b(in)g Fr(L)510 2236 y Fo(2)549 2272 y Fs(\()p Fr(m)p Fs(\))g(with)h(Re)16 b(\()p Fr(\027)6 b Fs(\))28 b Fr(>)g Fs(0,)h(whic)m(h)h(con)m(tradicts) g(Prop)s(osition)f(4.1.)42 b(Th)m(us,)31 b Fr(\033)3274 2236 y Fo(ess)3270 2297 y Fi(m)3365 2272 y Fs(\()p Fp(T)3457 2287 y Fi(\013)3507 2272 y Fs(\()p Fr(\034)11 b Fs(\)\))28 b Fp(\032)0 2393 y(B)s Fs(\()p Fr(e)151 2357 y Fi(\034)8 b Fo(\(1)p Fk(\000)p Fi(m)p Fo(\))p Fi(=)p Fo(2)473 2393 y Fs(\),)34 b(and)h(since)g Fp(S)7 b Fs(\()p Fr(\034)k Fs(\))35 b(is)g(also)f(a)g(compact)h(p)s(erturbation)f(of)g Fp(T)2747 2408 y Fi(\013)2797 2393 y Fs(\()p Fr(\034)11 b Fs(\))34 b(the)h(same)g(argumen)m(t)0 2513 y(sho)m(ws)f(that)f Fr(\033)551 2477 y Fo(ess)547 2538 y Fi(m)642 2513 y Fs(\()p Fp(T)734 2528 y Fi(\013)783 2513 y Fs(\()p Fr(\034)11 b Fs(\)\))28 b(=)g Fp(B)s Fs(\()p Fr(e)1233 2477 y Fi(\034)8 b Fo(\(1)p Fk(\000)p Fi(m)p Fo(\))p Fi(=)p Fo(2)1555 2513 y Fs(\).)2082 b Fb(\003)146 2681 y Fs(By)43 b(construction,)i(the) d(sp)s(ectral)g(subspaces)j Fr(L)1980 2645 y Fo(2)1980 2706 y(0)2019 2681 y Fs(\()p Fr(m)p Fs(\))d(and)g Fr(L)2487 2645 y Fo(2)2487 2706 y(1)2527 2681 y Fs(\()p Fr(m)p Fs(\))g(are)f(left)h(in)m(v)-5 b(arian)m(t)42 b(b)m(y)h(the)0 2802 y(semigroup)c Fp(T)526 2817 y Fi(\013)576 2802 y Fs(\()p Fr(\034)11 b Fs(\).)60 b(Com)m(bining)39 b(Lemma)f(4.11)g(with) g(the)h(eigen)m(v)-5 b(alue)39 b(estimates)h(of)d(Section)i(4.1,)0 2922 y(w)m(e)34 b(obtain)e(precise)i(b)s(ounds)g(on)e(the)h(gro)m(wth)g (of)f Fp(T)1903 2937 y Fi(\013)1953 2922 y Fs(\()p Fr(\034)11 b Fs(\))33 b(in)g(these)g(subspaces:)0 3142 y Fq(Prop)s(osition)38 b(4.12)49 b Fj(Fix)34 b Fr(\013)28 b Fp(2)g Fq(R)p Fj(.)45 b(Assume)34 b(that)i(either)0 3262 y Fq(a\))f Fs(0)27 b Fr(<)h(\026)f Fp(\024)h Fs(1)p Fr(=)p Fs(2)35 b Fj(and)f Fr(R)949 3277 y Fo(0)1016 3262 y Fp(2)28 b Fr(L)1176 3226 y Fo(2)1176 3287 y(0)1216 3262 y Fs(\()p Fr(m)p Fs(\))35 b Fj(for)g(some)f Fr(m)28 b(>)g Fs(1)21 b(+)h(2)p Fr(\026)p Fj(,)35 b(or)0 3383 y Fq(b\))g Fs(1)p Fr(=)p Fs(2)27 b Fr(<)g(\026)h Fp(\024)g Fs(1)34 b Fj(and)h Fr(R)957 3398 y Fo(0)1024 3383 y Fp(2)28 b Fr(L)1184 3346 y Fo(2)1184 3407 y(1)1224 3383 y Fs(\()p Fr(m)p Fs(\))35 b Fj(for)g(some)f Fr(m)28 b(>)f Fs(1)22 b(+)g(2)p Fr(\026)p Fj(.)0 3503 y(Then)34 b(ther)-5 b(e)35 b(exists)f Fr(C)h(>)28 b Fs(0)34 b Fj(\(indep)-5 b(endent)34 b(of)g Fr(R)1824 3518 y Fo(0)1864 3503 y Fj(\))h(such)f(that)1050 3716 y Fp(kT)1154 3731 y Fi(\013)1203 3716 y Fs(\()p Fr(\034)11 b Fs(\))p Fr(R)1406 3731 y Fo(0)1446 3716 y Fp(k)1496 3731 y Fi(m)1590 3716 y Fp(\024)28 b Fr(C)c(e)1834 3674 y Fk(\000)p Fi(\026\034)1975 3716 y Fp(k)p Fr(R)2099 3731 y Fo(0)2138 3716 y Fp(k)2188 3731 y Fi(m)2289 3716 y Fr(;)117 b(\034)39 b Fp(\025)28 b Fs(0)35 b Fr(:)876 b Fs(\(77\))0 3935 y Fq(Pro)s(of:)38 b Fs(Assume)e(\014rst)e(that)g(0) 29 b Fr(<)g(\026)g(<)h Fs(1)p Fr(=)p Fs(2.)46 b(By)34 b(Lemma)g(4.11,)f(the)h(essen)m(tial)i(sp)s(ectrum)f(of)e Fp(T)3605 3950 y Fi(\013)3655 3935 y Fs(\(1\))0 4056 y(in)42 b Fr(L)189 4020 y Fo(2)189 4080 y(0)229 4056 y Fs(\()p Fr(m)p Fs(\))f(satis\014es)i Fr(\033)858 4020 y Fo(ess)854 4080 y Fi(m)950 4056 y Fs(\()p Fp(T)1042 4071 y Fi(\013)1091 4056 y Fs(\(1\)\))g(=)g Fp(B)s Fs(\()p Fr(e)1567 4020 y Fo(\(1)p Fk(\000)p Fi(m)p Fo(\))p Fi(=)p Fo(2)1849 4056 y Fs(\),)h(and)e Fr(e)2202 4020 y Fo(\(1)p Fk(\000)p Fi(m)p Fo(\))p Fi(=)p Fo(2)2527 4056 y Fr(<)h(e)2691 4020 y Fk(\000)p Fi(\026)2834 4056 y Fs(since)g Fr(m)g(>)g Fs(1)28 b(+)g(2)p Fr(\026)p Fs(.)70 b(If)0 4176 y Fr(\025)40 b Fp(2)h Fr(\033)t Fs(\()p Fp(T)355 4191 y Fi(\013)405 4176 y Fs(\(1\)\))f(satis\014es)h Fp(j)p Fr(\025)p Fp(j)f Fr(>)g(e)1288 4140 y Fo(\(1)p Fk(\000)p Fi(m)p Fo(\))p Fi(=)p Fo(2)1571 4176 y Fs(,)i(then)f Fr(\025)e Fs(is)i(an)f(isolated)h (eigen)m(v)-5 b(alue)41 b(of)f Fp(T)3227 4191 y Fi(\013)3277 4176 y Fs(\(1\))f(and)i(\(b)m(y)0 4297 y(the)30 b(sp)s(ectral)g (mapping)g(theorem\))g(there)g(exists)h(an)f(eigen)m(v)-5 b(alue)31 b Fr(\027)k Fs(of)29 b Fp(L)16 b(\000)g Fr(\013)q Fs(\003)29 b(in)g Fr(L)3153 4260 y Fo(2)3153 4321 y(0)3193 4297 y Fs(\()p Fr(m)p Fs(\))h(suc)m(h)h(that)0 4417 y Fr(e)45 4381 y Fi(\027)116 4417 y Fs(=)d Fr(\025)p Fs(.)41 b(Applying)29 b(Prop)s(osition)f(4.1,)g(w)m(e)h(obtain)e(Re)17 b(\()p Fr(\027)6 b Fs(\))28 b Fp(\024)g Fs(1)p Fr(=)p Fs(2,)g(hence)h Fp(j)p Fr(\025)p Fp(j)e(\024)h Fr(e)3053 4381 y Fk(\000)p Fo(1)p Fi(=)p Fo(2)3246 4417 y Fr(<)f(e)3394 4381 y Fk(\000)p Fi(\026)3496 4417 y Fs(.)42 b(Th)m(us)0 4537 y(the)37 b(sp)s(ectral)g(radius)g(of)f Fp(T)1008 4552 y Fi(\013)1058 4537 y Fs(\(1\))g(in)h Fr(L)1403 4501 y Fo(2)1403 4562 y(0)1442 4537 y Fs(\()p Fr(m)p Fs(\))g(is)g(strictly)h(less)f(than)g Fr(e)2540 4501 y Fk(\000)p Fi(\026)2642 4537 y Fs(,)g(and)g(\(77\))f(follo)m(ws)h (\(see)g(e.g.)0 4658 y([12],)28 b(Prop)s(osition)g(IV.2.2\).)42 b(A)27 b(similar)h(argumen)m(t)h(pro)m(v)m(es)g(\(77\))e(if)g Fr(R)2596 4673 y Fo(0)2663 4658 y Fp(2)h Fr(L)2823 4622 y Fo(2)2823 4682 y(1)2863 4658 y Fs(\()p Fr(m)p Fs(\))g(and)f(1)p Fr(=)p Fs(2)g Fr(<)h(\026)f(<)h Fs(1.)146 4778 y(No)m(w,)47 b(assume)e(that)e Fr(\026)j Fs(=)g(1)p Fr(=)p Fs(2)d(and)g Fr(m)j(>)g Fs(2.)76 b(An)m(y)44 b Fr(R)2294 4793 y Fo(0)2380 4778 y Fp(2)j Fr(L)2559 4742 y Fo(2)2559 4803 y(0)2599 4778 y Fs(\()p Fr(m)p Fs(\))c(can)h(b)s(e)f(decomp)s(osed)i(as)0 4898 y Fr(R)74 4913 y Fo(0)141 4898 y Fs(=)28 b Fr(\014)300 4913 y Fo(1)339 4898 y Fr(F)402 4913 y Fo(1)464 4898 y Fs(+)22 b Fr(\014)617 4913 y Fo(2)656 4898 y Fr(F)719 4913 y Fo(2)781 4898 y Fs(+)900 4873 y(~)879 4898 y Fr(R)953 4913 y Fo(0)993 4898 y Fs(,)32 b(where)i Fr(\014)1389 4913 y Fi(j)1453 4898 y Fs(=)1557 4818 y Fn(R)1604 4933 y Fm(R)1665 4914 y Fl(2)1720 4898 y Fr(\030)1763 4913 y Fi(j)1799 4898 y Fr(R)1873 4913 y Fo(0)1930 4898 y Fs(d)p Fr(\030)5 b Fs(,)32 b Fr(F)2154 4913 y Fi(j)2218 4898 y Fs(=)c Fp(\000)p Fr(@)2450 4913 y Fi(j)2487 4898 y Fr(G)p Fs(,)33 b(and)2835 4873 y(~)2813 4898 y Fr(R)2887 4913 y Fo(0)2955 4898 y Fp(2)28 b Fr(L)3115 4862 y Fo(2)3115 4923 y(1)3155 4898 y Fs(\()p Fr(m)p Fs(\).)43 b(It)33 b(follo)m(ws)0 5019 y(that)985 5139 y Fp(T)1039 5154 y Fi(\013)1089 5139 y Fs(\()p Fr(\034)11 b Fs(\))p Fr(R)1292 5154 y Fo(0)1360 5139 y Fs(=)27 b Fr(e)1508 5098 y Fk(\000)p Fi(\034)8 b(=)p Fo(2)1677 5139 y Fs(\()p Fr(\014)1770 5154 y Fo(1)1810 5139 y Fr(F)1873 5154 y Fo(1)1934 5139 y Fs(+)22 b Fr(\014)2087 5154 y Fo(2)2127 5139 y Fr(F)2190 5154 y Fo(2)2230 5139 y Fs(\))g(+)g Fp(T)2442 5154 y Fi(\013)2491 5139 y Fs(\()p Fr(\034)11 b Fs(\))2642 5114 y(~)2620 5139 y Fr(R)2694 5154 y Fo(0)2767 5139 y Fr(;)0 5310 y Fs(and)43 b(w)m(e)i(already)f(kno)m(w)g(that)f Fp(kT)1301 5325 y Fi(\013)1351 5310 y Fs(\()p Fr(\034)11 b Fs(\))1501 5285 y(~)1480 5310 y Fr(R)1554 5325 y Fo(0)1594 5310 y Fp(k)1644 5325 y Fi(m)1756 5310 y Fp(\024)47 b Fr(C)7 b(e)2002 5274 y Fk(\000)p Fi(\027)t(\034)2139 5310 y Fp(k)2210 5285 y Fs(~)2189 5310 y Fr(R)2263 5325 y Fo(0)2302 5310 y Fp(k)2352 5325 y Fi(m)2462 5310 y Fs(for)43 b(some)h Fr(\027)52 b(>)46 b Fs(1)p Fr(=)p Fs(2.)75 b(Th)m(us)45 b(\(77\))0 5431 y(holds)36 b(and)f(is)h(sharp)g (in)f(this)h(case.)52 b(A)36 b(similar)g(argumen)m(t)g(sho)m(ws)g(that) f(\(77\))g(holds)h(with)g Fr(\026)c Fs(=)g(1)j(if)0 5551 y Fr(R)74 5566 y Fo(0)141 5551 y Fp(2)28 b Fr(L)301 5515 y Fo(2)301 5576 y(1)341 5551 y Fs(\()p Fr(m)p Fs(\))33 b(for)f(some)h Fr(m)28 b(>)g Fs(3.)2481 b Fb(\003)1841 5800 y Fs(30)p eop %%Page: 31 31 31 30 bop 146 100 a Fs(Finally)-8 b(,)51 b(w)m(e)c(will)h(need)f(in)g (the)g(follo)m(wing)g(subsection)i Fr(L)2401 63 y Fi(p)2441 100 y Fs(-)p Fr(L)2540 63 y Fi(q)2624 100 y Fs(estimates)g(of)d Fp(T)3247 115 y Fi(\013)3296 100 y Fs(\()p Fr(\034)11 b Fs(\))47 b(and)g(its)0 220 y(deriv)-5 b(ativ)m(es,)35 b(in)d(the)h(spirit)h(of)e(\(23\).)0 431 y Fq(Prop)s(osition)38 b(4.13)49 b Fj(Under)30 b(the)h(assumptions)f(of)h(Pr)-5 b(op)g(osition)30 b(4.12,)h(if)g Fr(R)2885 446 y Fo(0)2956 431 y Fj(satis\014es)f(in)g(addition)0 551 y Fr(b)41 515 y Fi(m)108 551 y Fr(R)182 566 y Fo(0)249 551 y Fp(2)e Fr(L)409 515 y Fi(q)448 551 y Fs(\()p Fq(R)570 515 y Fo(2)609 551 y Fs(\))35 b Fj(for)f(some)g Fr(q)e Fp(2)c Fs(\(1)p Fr(;)17 b Fs(2\))p Fj(,)34 b(then)215 813 y Fp(kT)319 828 y Fi(\013)369 813 y Fs(\()p Fr(\034)11 b Fs(\))p Fr(R)572 828 y Fo(0)612 813 y Fp(k)662 828 y Fi(m)756 813 y Fp(\024)901 745 y Fr(C)24 b(e)1040 709 y Fk(\000)p Fi(\026\034)p 871 790 341 4 v 871 907 a Fr(a)p Fs(\()p Fr(\034)11 b Fs(\))1061 834 y Fl(1)p 1061 846 31 3 v 1061 887 a Fh(q)1102 861 y Fk(\000)1167 834 y Fl(1)p 1167 846 V 1167 887 a(2)1238 813 y Fp(j)p Fr(b)1307 772 y Fi(m)1373 813 y Fr(R)1447 828 y Fo(0)1487 813 y Fp(j)1515 828 y Fi(q)1588 813 y Fr(;)116 b Fp(krT)1918 828 y Fi(\013)1968 813 y Fs(\()p Fr(\034)11 b Fs(\))p Fr(R)2171 828 y Fo(0)2211 813 y Fp(k)2261 828 y Fi(m)2355 813 y Fp(\024)2470 745 y Fr(C)23 b(e)2608 709 y Fk(\000)p Fi(\026\034)p 2470 790 280 4 v 2492 907 a Fr(a)p Fs(\()p Fr(\034)11 b Fs(\))2682 834 y Fl(1)p 2682 846 31 3 v 2682 887 a Fh(q)2776 813 y Fp(j)p Fr(b)2845 772 y Fi(m)2911 813 y Fr(R)2985 828 y Fo(0)3025 813 y Fp(j)3053 828 y Fi(q)3126 813 y Fr(;)116 b(\034)39 b(>)28 b Fs(0)34 b Fr(:)0 1110 y Fq(Pro)s(of:)42 b Fs(Fix)37 b(1)e Fr(<)g(q)k(<)c Fs(2)i(and)g(assume)h(that)f Fr(b)1790 1074 y Fi(m)1857 1110 y Fr(R)1931 1125 y Fo(0)2006 1110 y Fp(2)e Fr(L)2173 1074 y Fi(q)2212 1110 y Fs(\()p Fq(R)2334 1074 y Fo(2)2373 1110 y Fs(\).)56 b(Giv)m(en)38 b Fr(T)49 b(>)35 b Fs(0,)j(w)m(e)g (consider)g(the)0 1231 y(function)33 b(space)792 1435 y Fr(Y)104 b Fs(=)83 b Fp(f)p Fr(R)29 b Fp(2)f Fr(C)1436 1394 y Fo(0)1475 1435 y Fs(\(\(0)p Fr(;)17 b(T)d Fs(])p Fr(;)j(H)1875 1394 y Fo(1)1913 1435 y Fs(\()p Fr(m)p Fs(\)\))g Fp(j)g(k)p Fr(R)q Fp(k)2349 1450 y Fi(Y)2436 1435 y Fr(<)28 b Fp(1g)k Fr(;)114 b Fs(where)635 1608 y Fp(k)p Fr(R)q Fp(k)810 1623 y Fi(Y)953 1608 y Fs(=)127 b(sup)1112 1690 y Fo(0)p Fi(<\034)8 b Fk(\024)p Fi(T)1347 1498 y Fn(n)1414 1608 y Fr(a)p Fs(\()p Fr(\034)j Fs(\))1604 1535 y Fl(1)p 1604 1547 V 1604 1588 a Fh(q)1644 1562 y Fk(\000)1709 1535 y Fl(1)p 1709 1547 V 1709 1588 a(2)1754 1608 y Fp(k)p Fr(R)q Fs(\()p Fr(\034)g Fs(\))p Fp(k)2058 1623 y Fi(m)2147 1608 y Fs(+)22 b Fr(a)p Fs(\()p Fr(\034)11 b Fs(\))2435 1535 y Fl(1)p 2435 1547 V 2435 1588 a Fh(q)2480 1608 y Fp(kr)p Fr(R)q Fs(\()p Fr(\034)g Fs(\))p Fp(k)2867 1623 y Fi(m)2933 1498 y Fn(o)3032 1608 y Fr(:)0 1878 y Fs(Using)25 b(\(23\))f(and)g(pro)s(ceeding)i(as)e(in)h(the)g(pro)s (of)e(of)h(Lemma)h(2.1,)h(it)e(is)h(straigh)m(tforw)m(ard)g(to)f(sho)m (w)i(that,)0 1999 y(if)g Fr(T)42 b(>)27 b Fs(0)f(is)h(su\016cien)m(tly) i(small,)g(there)e(exists)h Fr(C)35 b(>)27 b Fs(0)f(suc)m(h)i(that)e (\(75\))g(has)h(a)f(unique)i(solution)e(in)h(the)0 2119 y(ball)k(of)g(radius)g Fr(C)7 b Fp(j)p Fr(b)735 2083 y Fi(m)802 2119 y Fr(R)876 2134 y Fo(0)916 2119 y Fp(j)944 2134 y Fi(q)1013 2119 y Fs(cen)m(tered)33 b(at)d(the)i(origin)f(in)g Fr(Y)21 b Fs(.)43 b(This)32 b(pro)m(v)m(es)h(the)f(desired)g(estimates) h(for)0 2239 y(0)c Fr(<)h(\034)41 b Fp(\024)30 b Fr(T)14 b Fs(,)34 b(and)g(the)g(general)h(case)f(follo)m(ws)h(from)e(Prop)s (osition)h(4.12)f(if)h(one)g(uses)h(the)f(semigroup)0 2360 y(prop)s(ert)m(y)c(and)g(the)g(fact)g(that)f(the)h(subspaces)i Fr(L)1819 2324 y Fo(2)1819 2384 y(0)1859 2360 y Fs(\()p Fr(m)p Fs(\))e(and)g Fr(L)2303 2324 y Fo(2)2303 2384 y(1)2342 2360 y Fs(\()p Fr(m)p Fs(\))g(are)g(left)g(in)m(v)-5 b(arian)m(t)30 b(b)m(y)g Fp(T)3453 2375 y Fi(\013)3503 2360 y Fs(\()p Fr(\034)11 b Fs(\).)43 b Fb(\003)0 2646 y Ff(4.3)135 b(Bounds)44 b(on)h(the)g(nonlinear)h(ev)l(olution)0 2831 y Fs(In)37 b(this)g(subsection,)i(w)m(e)e(sho)m(w)h(that)e(our)g (estimates)i(on)e(the)h(linear)g(semigroup)g Fp(T)3152 2846 y Fi(\013)3202 2831 y Fs(\()p Fr(\034)11 b Fs(\))37 b(generated)0 2951 y(b)m(y)31 b Fp(L)17 b(\000)h Fr(\013)q Fs(\003)29 b(imply)i(similar)g(b)s(ounds)g(on)f(the)h(full)f(nonlinear) g(ev)m(olution)i(\(28\).)42 b(As)31 b(is)f(easy)h(to)f(v)m(erify)-8 b(,)0 3072 y(the)33 b(in)m(tegral)g(equation)g(\(30\))f(satis\014ed)i (b)m(y)g Fr(R)q Fs(\()p 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y Fk(\000)p Fi(\026\034)2037 3911 y Fs(\))35 b Fj(as)f Fr(\034)40 b Fp(!)27 b(1)p Fj(.)0 4122 y Fq(Pro)s(of:)i Fs(Let)d Fr(R)j Fp(2)f Fr(C)774 4085 y Fo(0)813 4122 y Fs(\([0)p Fr(;)17 b Fs(+)p Fp(1)p Fs(\))p Fr(;)g(L)1295 4085 y Fo(2)1295 4146 y(0)1334 4122 y Fs(\()p Fr(m)p Fs(\)\))25 b(b)s(e)h(the)g(solution)g(of)f (\(28\))g(with)h(initial)g(data)f Fr(R)3289 4137 y Fo(0)3328 4122 y Fs(.)41 b(W)-8 b(e)26 b(kno)m(w)0 4242 y(from)39 b(Prop)s(osition)g(1.5)f(that)h Fp(k)p Fr(R)q Fs(\()p Fr(\034)11 b Fs(\))p Fp(k)1453 4257 y Fi(m)1558 4242 y Fs(con)m(v)m(erges)42 b(to)c(zero)h(as)g Fr(\034)50 b Fp(!)38 b(1)p Fs(,)j(hence)f(w)m(e)g(can)f(assume)0 4362 y(without)33 b(loss)g(of)g(generalit)m(y)g(that)g Fp(k)p Fr(R)1445 4377 y Fo(0)1484 4362 y Fp(k)1534 4377 y Fi(m)1633 4362 y Fs(is)g(small.)44 b(Giv)m(en)34 b Fr(T)41 b(>)28 b Fs(0,)k(w)m(e)i(de\014ne)536 4595 y Fr(M)10 b Fs(\()p Fr(T)k Fs(\))28 b(=)g(max)1100 4484 y Fn(n)1227 4595 y Fs(sup)1183 4676 y Fo(0)p Fk(\024)p Fi(\034)8 b Fk(\024)p 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Fp(j)3301 5496 y Fi(q)3355 5481 y Fs(d)p Fr(s)33 b(;)1841 5800 y Fs(31)p eop %%Page: 32 32 32 31 bop 0 100 a Fs(where)34 b(1)27 b Fr(<)h(q)j(<)d Fs(2.)43 b(As)33 b(in)g(the)g(pro)s(of)f(of)g(Lemma)h(2.1,)f(w)m(e)i (can)f(b)s(ound)570 349 y Fp(j)p Fr(b)639 307 y Fi(m)705 349 y Fq(v)766 307 y Fi(R)824 349 y Fs(\()p Fr(s)p Fs(\))22 b Fp(\001)f(r)p Fr(R)q Fs(\()p Fr(s)p Fs(\))p Fp(j)1325 364 y Fi(q)1391 349 y Fp(\024)28 b Fr(C)7 b Fp(k)p Fr(R)q Fs(\()p Fr(s)p Fs(\))p Fp(k)1870 364 y Fi(m)1936 349 y Fp(kr)p Fr(R)q Fs(\()p Fr(s)p Fs(\))p Fp(k)2316 364 y Fi(m)2410 349 y Fp(\024)2525 281 y Fr(C)23 b(e)2663 245 y Fk(\000)p Fo(2)p Fi(\026s)p 2525 326 309 4 v 2565 434 a Fr(a)p Fs(\()p Fr(s)p Fs(\))2748 370 y Fl(1)p 2748 382 31 3 v 2748 423 a(2)2860 349 y Fr(M)10 b Fs(\()p Fr(T)k Fs(\))3111 307 y Fo(2)3183 349 y Fr(;)0 611 y Fs(hence)490 838 y Fp(k)p Fr(R)q Fs(\()p Fr(\034)d Fs(\))p Fp(k)794 853 y Fi(m)943 838 y Fp(\024)84 b Fr(e)1149 797 y Fk(\000)p Fi(\026\034)1290 727 y Fn(\020)1349 838 y Fr(C)7 b Fp(k)p Fr(R)1550 853 y Fo(0)1590 838 y Fp(k)1640 853 y Fi(m)1761 838 y Fs(+)22 b Fr(C)7 b(M)j Fs(\()p Fr(T)k Fs(\))2187 797 y Fo(2)2243 702 y Fn(Z)2343 728 y Fi(\034)2298 928 y Fo(0)2668 770 y Fr(e)2713 734 y Fk(\000)p Fi(\026s)p 2412 815 692 4 v 2412 932 a Fr(a)p Fs(\()p Fr(\034)d Fp(\000)q Fr(s)p Fs(\))2726 859 y Fl(1)p 2726 871 31 3 v 2726 912 a Fh(q)2767 886 y Fk(\000)2832 859 y Fl(1)p 2831 871 V 2831 912 a(2)2876 932 y Fr(a)p Fs(\()p Fr(s)p Fs(\))3059 868 y Fl(1)p 3059 880 V 3059 922 a(2)3114 727 y Fn(\021)3190 838 y Fs(d)p Fr(s)943 1105 y Fp(\024)84 b Fr(e)1149 1064 y Fk(\000)p Fi(\026\034)1290 994 y Fn(\020)1349 1105 y Fr(C)1419 1120 y Fo(1)1458 1105 y Fp(k)p Fr(R)1582 1120 y Fo(0)1622 1105 y Fp(k)1672 1120 y Fi(m)1793 1105 y Fs(+)22 b Fr(C)1961 1120 y Fo(2)2000 1105 y Fr(M)10 b Fs(\()p Fr(T)k Fs(\))2251 1064 y Fo(2)2291 994 y Fn(\021)2383 1105 y Fr(;)114 b Fs(0)28 b Fp(\024)g Fr(\034)39 b Fp(\024)28 b Fr(T)46 b(;)0 1322 y Fs(where)39 b Fr(C)357 1337 y Fo(1)396 1322 y Fr(;)17 b(C)510 1337 y Fo(2)585 1322 y Fr(>)36 b Fs(0)h(are)g(indep)s(enden)m(t)j(of)d Fr(T)14 b Fs(.)58 b(Without)38 b(loss)g(of)f(generalit)m(y)-8 b(,)40 b(w)m(e)e(assume)h(in)f(what)0 1443 y(follo)m(ws)33 b(that)g Fr(C)602 1458 y Fo(1)669 1443 y Fp(\025)28 b Fr(C)844 1458 y Fo(0)883 1443 y Fs(.)43 b(Di\013eren)m(tiating)33 b(\(78\))f(and)h(using)g(similar)h(estimates,)g(w)m(e)f(also)g(obtain) 603 1694 y Fp(kr)p Fr(R)q Fs(\()p Fr(\034)11 b Fs(\))p Fp(k)990 1709 y Fi(m)1085 1694 y Fp(\024)1225 1627 y Fr(e)1270 1591 y Fk(\000)p Fi(\026\034)p 1200 1671 236 4 v 1200 1779 a Fr(a)p Fs(\()p Fr(\034)g Fs(\))1390 1715 y Fl(1)p 1390 1727 31 3 v 1390 1769 a(2)1445 1584 y Fn(\020)1504 1694 y Fr(C)1574 1709 y Fo(1)1614 1694 y Fp(k)p Fr(R)1738 1709 y Fo(0)1777 1694 y Fp(k)1827 1709 y Fi(m)1948 1694 y Fs(+)22 b Fr(C)2116 1709 y Fo(2)2155 1694 y Fr(M)10 b Fs(\()p Fr(T)k Fs(\))2406 1653 y Fo(2)2446 1584 y Fn(\021)2538 1694 y Fr(;)114 b Fs(0)28 b Fr(<)f(\034)40 b Fp(\024)28 b Fr(T)46 b(:)0 1957 y Fs(Summarizing,)34 b(w)m(e)f(ha)m(v)m(e)h(sho)m (wn)1197 2152 y Fr(M)10 b Fs(\()p Fr(T)k Fs(\))28 b Fp(\024)g Fr(C)1651 2167 y Fo(1)1690 2152 y Fp(k)p Fr(R)1814 2167 y Fo(0)1854 2152 y Fp(k)1904 2167 y Fi(m)2025 2152 y Fs(+)22 b Fr(C)2193 2167 y Fo(2)2232 2152 y Fr(M)10 b Fs(\()p Fr(T)k Fs(\))2483 2111 y Fo(2)2555 2152 y Fr(:)1024 b Fs(\(79\))0 2347 y(No)m(w,)42 b(assume)f(that)e Fr(R)894 2362 y Fo(0)974 2347 y Fs(is)h(small)g(enough)g(so)g(that)f(4)p Fr(C)2148 2362 y Fo(1)2187 2347 y Fr(C)2257 2362 y Fo(2)2296 2347 y Fp(k)p Fr(R)2420 2362 y Fo(0)2460 2347 y Fp(k)2510 2362 y Fi(m)2616 2347 y Fr(<)g Fs(1.)64 b(As)40 b Fr(C)3092 2362 y Fo(1)3171 2347 y Fp(\025)g Fr(C)3358 2362 y Fo(0)3397 2347 y Fs(,)i(w)m(e)e(also)0 2467 y(ha)m(v)m(e)34 b Fr(C)295 2482 y Fo(0)334 2467 y Fp(k)p Fr(R)458 2482 y Fo(0)498 2467 y Fp(k)548 2482 y Fi(m)642 2467 y Fp(\024)783 2442 y Fs(\026)747 2467 y Fr(M)11 b Fs(,)32 b(where)1046 2682 y(\026)1010 2707 y Fr(M)38 b Fs(=)1310 2640 y(1)p 1256 2684 159 4 v 1256 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Fs(for)k(all)h Fr(\034)39 b Fp(\025)28 b Fs(0.)3155 b Fb(\003)0 3396 y Fq(Remark)38 b(4.15)49 b Fj(Similarly,)35 b(if)h Fr(R)1291 3411 y Fo(0)1360 3396 y Fp(2)30 b Fr(L)1522 3360 y Fo(2)1522 3420 y(2)1561 3396 y Fs(\()p Fr(m)p Fs(\))36 b Fj(for)g(some)f Fr(m)30 b(>)f Fs(3)p Fj(,)36 b(the)f(solution)h Fr(R)q Fs(\()p Fr(\034)11 b Fs(\))36 b Fj(of)g(\(28\))f(with)0 3516 y(initial)g(data)h Fr(R)579 3531 y Fo(0)654 3516 y Fj(satis\014es)f Fp(k)p Fr(R)q Fs(\()p Fr(\034)11 b Fs(\))p Fp(k)1325 3531 y Fi(m)1421 3516 y Fs(=)29 b Fp(O)s Fs(\()p Fr(e)1691 3480 y Fk(\000)p Fi(\026\034)1832 3516 y Fs(\))35 b Fj(as)g Fr(\034)41 b Fp(!)29 b(1)p Fj(,)35 b(for)h(some)f Fr(\026)28 b Fp(\025)i Fs(1)36 b Fj(dep)-5 b(ending)34 b(on)h Fr(\013)q Fj(.)0 3637 y(We)g(know)f(fr)-5 b(om)34 b(R)-5 b(emark)34 b(4.9)g(that)h Fr(\026)27 b(>)h Fs(1)34 b Fj(if)h Fr(\013)28 b Fp(6)p Fs(=)g(0)p Fj(,)34 b(but)h(we)g(have)f(no)g(sharp)g(r)-5 b(esult)35 b(in)f(that)h(c)-5 b(ase.)146 3837 y Fs(W)d(e)46 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Fs(=)g(\(0)p Fr(;)17 b Fs(0\),)32 b(the)i(righ)m(t-hand)f (side)h(of)f(\(80\))g(should)h(b)s(e)f(replaced)0 5551 y(b)m(y)g Fp(O)s Fs(\()p Fr(\034)11 b(e)353 5515 y Fk(\000)p Fi(\034)452 5551 y Fs(\).)1841 5800 y(32)p eop %%Page: 33 33 33 32 bop 0 100 a Fq(Remark)38 b(4.16)49 b Fj(Assume)35 b(that)g Fr(m)29 b(>)f Fs(3)p Fj(,)35 b Fr(\013)28 b Fp(6)p Fs(=)g(0)p Fj(,)35 b(and)2066 19 y Fn(R)2113 134 y Fm(R)2174 115 y Fl(2)2229 100 y Fp(j)p Fr(\030)5 b Fp(j)2333 63 y Fo(2)2371 100 y Fr(w)2441 115 y Fo(0)2480 100 y Fs(\()p Fr(\030)g Fs(\))17 b(d)p Fr(\030)32 b Fp(6)p Fs(=)c(0)p Fj(.)45 b(If)34 b(we)h(tr)-5 b(anslate)35 b(and)0 220 y(r)-5 b(esc)g(ale)27 b Fr(w)378 235 y Fo(0)445 220 y Fj(appr)-5 b(opriately,)28 b(we)f(c)-5 b(an)27 b(pr)-5 b(o)g(duc)g(e)28 b(a)f(new)g(initial)g(c)-5 b(ondition)47 b Fs(\026)-69 b Fr(w)2751 235 y Fo(0)2818 220 y Fj(with)47 b Fs(\026)-69 b Fr(w)3092 235 y Fo(0)3138 220 y Fp(\000)6 b Fr(\013)q(G)28 b Fp(2)g Fr(L)3549 184 y Fo(2)3549 245 y(2)3588 220 y Fs(\()p 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y(and)c([20].)0 1985 y Ft(References)49 2204 y Fs([1])49 b(A.)23 b(Arnold,)i(P)-8 b(.)23 b(Mark)m(o)m(wic)m(h,)k(G.)c(T)-8 b(oscani,)25 b(and)e(A.)g(Un)m(terreiter.)29 b(On)23 b(con)m(v)m(ex)i(Sob)s(olev)f(inequal-)201 2324 y(ities)40 b(and)g(the)g(rate)g(of)g(con)m(v)m(ergence)i(to)d(equilibrium)j(for)e (Fokk)m(er-Planc)m(k)h(t)m(yp)s(e)g(equations.)201 2445 y Fj(Comm.)33 b(Partial)i(Di\013er)-5 b(ential)34 b(Equations)p Fs(,)e(26\(1-2\):43{100,)e(2001.)49 2648 y([2])49 b(M.)41 b(Ben-Artzi.)68 b(Global)40 b(solutions)h(of)g(t)m(w)m(o-dimensional)h (Na)m(vier-Stok)m(es)h(and)e(Euler)g(equa-)201 2769 y(tions.)i Fj(A)n(r)-5 b(ch.)35 b(R)-5 b(ational)34 b(Me)-5 b(ch.)35 b(A)n(nal.)p Fs(,)d(128\(4\):329{358,)e(1994.)49 2972 y([3])49 b(A.)24 b(J.)h(Berno\013)g(and)g(J.)g(F.)f(Lingevitc)m(h.)33 b(Rapid)24 b(relaxation)i(of)e(an)g(axisymmetric)k(v)m(ortex.)k Fj(Phys.)201 3092 y(Fluids)p Fs(,)g(6\(11\):3717{3721,)d(No)m(v)m(em)m (b)s(er)35 b(1994.)49 3296 y([4])49 b(L.)43 b(Brandolese.)77 b Fj(L)-5 b(o)g(c)g(alisation,)47 b(Oscil)5 b(lations)43 b(et)j(Comp)-5 b(ortement)43 b(Asymptotique)j(p)-5 b(our)45 b(les)215 3391 y(\023)201 3416 y(Equations)29 b(de)h(Navier-Stokes)p Fs(.)k(PhD)28 b(thesis,)1948 3391 y(\023)1939 3416 y(Ecole)g(Normale)g (Sup)m(\023)-46 b(erieure)29 b(de)f(Cac)m(han,)h(2001.)49 3619 y([5])49 b(L.)41 b(Brandolese.)73 b(On)42 b(the)g(lo)s(calization) g(of)g(symmetric)i(and)e(asymmetric)i(solutions)f(of)e(the)201 3740 y(Na)m(vier-Stok)m(es)g(equations)g(in)f Fq(R)1472 3704 y Fi(n)1519 3740 y Fs(.)64 b Fj(C.)41 b(R.)g(de)g(l'A)-5 b(c)g(ad.)41 b(de)g(Scienc)-5 b(e)40 b(de)i(Paris)p Fs(,)f(t.)e(332\(S) m(\023)-46 b(erie)201 3860 y(1\):125{130,)30 b(2001.)49 4064 y([6])49 b(L.)40 b(Brandolese.)70 b(Space-time)42 b(deca)m(y)h(of)d(Na)m(vier-Stok)m(es)j(\015o)m(ws)f(in)m(v)-5 b(arian)m(t)42 b(under)g(rotations.)201 4184 y(Preprin)m(t,)33 b(April)g(2003.)43 b(arXiv:math.AP/0304436.)49 4387 y([7])49 b(H.)37 b(Brezis.)59 b(Remarks)38 b(on)g(the)f(preceding)i(pap)s(er)e (b)m(y)i(M.)f(Ben-Artzi:)53 b(\\Global)37 b(solutions)h(of)201 4508 y(t)m(w)m(o-dimensional)j(Na)m(vier-Stok)m(es)g(and)f(Euler)g (equations".)65 b Fj(A)n(r)-5 b(ch.)41 b(R)-5 b(ational)40 b(Me)-5 b(ch.)41 b(A)n(nal.)p Fs(,)201 4628 y(128\(4\):359{360,)30 b(1994.)49 4832 y([8])49 b(M.)37 b(Cannone)g(and)g(F.)f(Planc)m(hon.)57 b(Self-similar)37 b(solutions)h(for)e(Na)m(vier-Stok)m(es)i(equations)g (in)201 4952 y Fq(R)285 4916 y Fo(3)324 4952 y Fs(.)43 b Fj(Commun.)34 b(Partial)h(Di\013er.)e(Equations)p Fs(,)g (21\(1-2\):179{193,)d(1996.)49 5155 y([9])49 b(E.)42 b(A.)g(Carlen)h(and)f(M.)g(Loss.)72 b(Optimal)42 b(smo)s(othing)h(and)f (deca)m(y)h(estimates)h(for)d(viscously)201 5276 y(damp)s(ed)23 b(conserv)-5 b(ation)23 b(la)m(ws,)j(with)e(applications)f(to)g(the)g (2-D)e(Na)m(vier-Stok)m(es)k(equation.)j Fj(Duke)201 5396 y(Math.)35 b(J.)p Fs(,)d(81\(1\):135{157)f(\(1996\),)g(1995.)1841 5800 y(33)p eop %%Page: 34 34 34 33 bop 0 100 a Fs([10])49 b(A.)37 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