%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: Wave9.dvi %%Pages: 29 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: CMBX12 CMR12 CMBX10 CMR10 CMMI10 CMMI7 CMR7 CMSY10 %%+ CMEX10 CMSY7 CMTI10 CMR6 CMR8 CMBX7 CMMI8 CMMI6 CMSY6 CMSY8 CMBX8 %%+ CMSY5 CMMI5 CMR5 CMCSC10 CMMI12 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o wave9.ps Wave9.dvi %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2002.11.18:1035 %%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 %! TeXDict begin/rf{findfont dup length 1 add dict begin{1 index/FID ne 2 index/UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]/Metrics exch def dict begin Encoding{exch dup type/integertype ne{pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def} ifelse}forall Metrics/Metrics currentdict end def[2 index currentdict end definefont 3 -1 roll makefont/setfont cvx]cvx def}def/ObliqueSlant{ dup sin S cos div neg}B/SlantFont{4 index mul add}def/ExtendFont{3 -1 roll mul exch}def/ReEncodeFont{CharStrings rcheck{/Encoding false def dup[exch{dup CharStrings exch known not{pop/.notdef/Encoding true def} if}forall Encoding{]exch pop}{cleartomark}ifelse}if/Encoding exch def} def end %%EndProcSet %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. 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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 46 /period 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 58 /colon put dup 65 /A put dup 67 /C put dup 73 /I put dup 80 /P put dup 82 /R put dup 84 /T 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 108 /l 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 readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueXX 5000769 def currentdict end 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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37fe3431e38a1ac9a9c161b627b8121f209f1dd309ab6d5be5722756bda098ef c85437a26f35e398855e55dfc1652293ab051c73bf1ec881944d9d0815ecf6f9 37ff928b31a690e76617cff8d0415c20b5112a38f35654c01ae0c056984451ff 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % 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 (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 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 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put readonly def /FontBBox{-341 -250 1304 965}readonly def /UniqueXX 5000788 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306457c7e62c1d269f306bd3402e266de fc3b5e7d8a8d2f5bf0fe6ddd40d07391df4fad4a6018dce29a2b8f692b29f202 3a7c0e66de8ed85c14f1f8492167357f51a7e84cc5d92e0fee4a71813d2ffb26 445026f89b4787516ecd1afc78f8bd19e91e9ccc9402e8c36d2449c1ff850a8a f61165aac3fe931332dd28e261b91b05edd18f44ea7d58a8f35fe88493b64aad 6bfac3a0136215fc2f4ca8e91d70c5010e6f4013e6d63b44f6fb00afdbd7cdb4 5ec5b1d9736f45cefc8a0124b815987cebd81bbe0d44dedb2d5ef37923b3d551 abb6a1cefca7868fc7bf3814ef7d7b6b1ae6e869cb77aa29e3d90b12b0dc3ff6 ec945922b5899bbf2f12e92731486d2ef1230c528bf8d7e0ae09ad7632a38966 5963de49d1ef3d65bc483e4a577b927c940f5e121169ba52f6576c85793e5fd7 7f5863c488e55bdefd5d8b2514795533aabbc12e7f51816c7e1484f1c441aa9a 66a5dc77158e79cd6692d299b95e8058b35e771a6972d2b5eb1cd6d2bb8e835d 361a6eac0c90c906aea1cd75c4412a1339dc1439f93e59917311bf20af4f2bc8 bba5a4626f75f28a47d4595c37f4019df480d9385975159142618e22e964f019 d88125210e2ef21d65baf500adf802297b24eebbf4e93c617a613f856bd0d45f e17a9480980ef83950bdb8ebf208c895ebc4a7132f0765b7c19a332ebbf8a3c9 ab65063a42e19ca9d43dec3de7aba46b9a5d24fb93d615c0a1726ab7e31c378e ab77150858535b91362ab0e2b968dee1c539964126d937f9d4980ccefa92bc81 b054996a65a13995c9c9d819ff05511cfc89cbfe67447c940405001d61be633b b771eff383fc6977ec06af30f981504260de7b9ec97359974734f495ac073935 fb30e3700679092ef5d971b6b2da0e15960f2391fa6b63c33e7f56b4a9b86951 1d3686a5e8e613a69a0add8b8c73b127ddcd397a48d785079e786d24b755e4d7 27a8feff23826aabdb372dec537221d5970606b25d431cb0cf0e009ca00f59c6 c38fb65c47526d3474e19dc40df792ea6957e771dbb5a63d340a23ee230a0290 f77cfed663d8d97e066872202c9ab0c25092810e0dfaeea6b7e9a59c21ae3dd0 975d54e2efed43cb9e79f3a159c353476f5c173bc9d7b61950076a0cf39d3b70 728e1ae5759d43dd641368cadc778ed239410e9b11e489b020143b95a03f1a2e b3bb94d7665bd365a49843b4a66df8035c66cd09d4fb3224f192e1654ed96072 91ef68cfaaebdd41a72a37e9cc44628cb07f5766a982ea9b7fe0d46d49a4073a 508af260460b7b72632f064dbae5498bf06090ede2f9cfcd5e620dd310b8f1ca c63450aff1424c4e31874d317a195653a4fcea2f81384f5258bcc4ee3550e0a3 7b208a24c6c194e733be4a255f324d9b522b5b259cedbd348a9561a1011af73d 5f9970a1805488747e3ccfbf3c89d685e515114192f202cc434ffb3735900bff 026c7ce9bac2a8629d78823dc866c57737b67b7e98dae4e47a91e30a7d156ef6 e2f812da875f63dbaa04aeac27ae2af8b7d0071351791fd10920d00420ab3593 41106f901a0175b8e4a2af391ec6fff9c7b11af467b912392106fd122563ab6f 477eb47db7a487d113019a42c67bd477b8b420945485314d5eefa3eb33b0e904 981b2ffcf971f94cf2309beabc79da5b020d547b4f3830d007f1e284f1f64b0b e35f15d43c19ce3d19b80f422722887cf23e7a88c64675d4eb44442677308f21 85bac7e29a43d3d22b8ec780c99f5b28c69ca61e594710eef6f8e46859ceec3c 4bdd5880361b6fb8b2ce8e5779e6af42b07b553e5b8239e3b92e2cf58bcf2136 9bddc8e1b63b670ff444e838b51bd58978fba5f32ee2c4863238fc8b3bf790bf baee897908921a3bbb094a6fa5bab1a9bb29e59549b8ae1f26bd2355fd9753a4 60d705983f52e5e24577018acd25ad5e26ee10c1cad6ec0e5d5ea25703eea833 4248e75287366e192df7906e1bd83e800228f49033a088135496724fb5445b94 a83f51f31ec3d4696f59c6de8af165fc82fba565adbe880b8e6a27eb447af470 32a9e9fca646385f28b939a29f9e2948f554562f4822af83444dc2910682979f d11eeb0f02cee1755bc9e7fe79822c7b2aca5deaa7b585674a1d524bc8d9fd60 6aa5b5e9d51cd153a644824842612fab5e60eca455ed62e3485b01d2b54395af 10972524d307c0efe2f9db6e4d3e1547cea7f6de8555 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % 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 (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 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 59 /comma put dup 65 /A put dup 97 /a put dup 105 /i put dup 107 /k put dup 110 /n put dup 113 /q put dup 114 /r put dup 115 /s put dup 121 /y put readonly def /FontBBox{37 -250 1349 750}readonly def /UniqueXX 5087380 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311e252234d51741e18db3a68e8ad10242 29e5817a10e796a78d2c7f7c1f50961b9a57aa604c9f821dbf5a9295197bc666 31678d7d2c7e1f8f2151ce0c29efce46270570f4301c5dad1b38884732e53dad 05dfc36b020e726ca575f71f04abf2b49e1f1d6d2a08a477658ca9a48f1c8cce 14382f0a42201de56e2821ee2b0e91a818b1b753d810ec2428dae5153b66af66 9442192024becf743a8f6264a58a0b3aa823510a4957d1bab3e1b83135362888 3e392f49926471a8021fdb48637baf0abcf31204b1b214f48c99f44af7782323 69d772ae4f49be53a1bf584747bf29b94f530d59a5be5370b8dd0691e32a21ba 9d0d592cf3bb50a99a3ecd18d6c4114cb4cfecc49705da90d7edb10e3a28f9eb a49347a09d31eec896f2a28e9607d99d42914aea16545c16e136bdd0b75fe486 61e24c7a9e9b233a76754c760822d0a88c575e08ae3e3f081d7bfb80ec4c03ac a08b65fb6474f71fc95c7c2462f2eb3754e647c29498288f2ced0bd3a2bca5c3 2f80a61fa667802d376cacd34de1f9572d1b3ab09d6f352750db4a768f869638 a87457889aa502850427b40e1294c5ba33e6b17532b2d13b5f50e71582f93daa 231e86597872a2a968127afb0f36257a76ce1eb24e6c4f29fce2aaf6532fb5cf df84da03f03a45b915209f67d398c7c4f7a3b0623750259087a478dca684b29e a8b7f52060bae9394369ed1cecc645d113cca3772a8dc36c6d916b5c5e65454c 0a188a33395fd3da67ad0a31ddab6a8434dd80f0bd0ea1a2a62eecc331d88b3e 369552daa1c575a98fe84383548626b65be7c24ae7eb61159f149872cfd87cf7 6c1a7c155c7186c7600e58cb61b8cb0c5dea98268a39e928bf6fc53985eaa763 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cf4ed8277024a94d3388dd4ebeacd961baf812c2f5e1e1808a0a4a07144e3fae 1af8c1932940221553de94e4d58d6c7780d30aa27e044027d126f09500772936 f92cb12acaf71b70e86469e18a5a577843fe58b596d70a50cc48345b0e7bc243 afa36a53762ccabe81e3a29c70375b4c746d8aec624eb58665ea20e255aa7cac c12f064a3bd1428b04ac0d20b978ffc1ac69f035ee0e004610b3f03a2a6956bd 91aedd513706eabc7e13e34d102375cd008ef00fe3e0625650d7119b88e2fa25 0dd848dd4f6a4d9eeacb96b26b59a15879a04bc393c3001ba02151898738524f 45bdcb045893d7be84885ea919152a40cb14bd2c7619759294357983731127dc ab7b01e9969e145a5ebfa0900bfa2a1c5ced65daae965ad15153f82ada5c0063 2095228ce3b199607cda3d4053309c8482bd166d0c42b5dc2114cb42ab800581 87505a6746e066fd0a025744ea73bdefe514dc2ee847d9b5eb0c7dafa80c94a9 af152e59e97af8ecb4f0a68669c3c669b4041293ff30afd9c92c3dc2cbde699d fe5c1a2065d8b25d84ea2e1147eb84b56e4e389421e9d22e45d66554ea735ade 15b349a27727edf4bcbe01da5e6820fab6c292fbdb3fffa7c70cd15f88856f36 2f5fa01ae7bb538ced9366462075dc23a6f2770ef9de4b42636943a815f95b72 2f599b80f1cdaa321e9f6b1df50d518c970437d50fb6d9c538193b4e9333a9d1 0b2803a8aa65420f472508497bc41ce8235abcf9f44f52aff88a9ebf5779f1b7 c3d11f609ed26827796a86f0ad66ca19314e93913d92931a4014a190a1674478 062c8996476573a0cb546630385384ee7768 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % 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 (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 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 48 /prime put readonly def /FontBBox{21 -944 1448 791}readonly def /UniqueXX 5000815 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6784decd2fb36dec8f 7478da853762e00bc9c52ba86ef50d222ff142abc18e4fb95b86b3e57d9390ca 86e8de6347bf7f48f93322a372c5d7830521b14e8fa0771961b027c81365426d d95d16b950e216a90f28ce806244ff28dd969cb512a23e05045e7b2bcaf7d6f0 ca8b51494f2ec0681bdfde60420628d9bce98a0ca1ecca9140f452a88db19a30 ed46ae881b098c843a51d06f9a153489357fd24de6a0c36cb5a9eb9b3b64dd75 18f83018ae7cefa1c12fd205df6880a705e28457af7b869a6dd28f6b37d82cc3 e51cba348dcdb34efb119b8ce5a996f15492b0f5d34538850a2c1b6a509fe5fa 015eceaf28b6f9385526e98445876548d0f2a09cfea14b84be3721b51afa2185 90c7b9ca643c7f4fe67cd8985f27855c40ae7ea592fadfbba29301071039d18a 8a6880cc0bff22fc8af50fd2802aff5a96381fb4bef240dd26b34a5461188ca0 c85e4b3f803f94dce14e8ddb2284e22abfe148560aff784967ed795332cc0995 80b762f864b2c15bc640a4cd7f6927c7dd95aef777fd226254b8e6e1b8ec6483 2750d9929627c4cecf9f5c8feb57a2c7ee5d1315593f3fa16c7c7292a594ce14 2c396d9b787665569dbd0f710822ae3155a3aa53d6f055cb19e897b235b65bce 4d507a4d41edc6a09dff1f737f2711478e6e998218b7f3e7c1aaf3938f68e9cf 9ea46e4a2904c459281cc156f5ce80d6356e357011cd0bc383318129bc8ce38a 758c05d1205192a457f4b1b31df97baec7611905cac0 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 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 c3e7e966f8d06c0bc29bcb3fc35578dbc1d6579e74da662c7d863adc0fe777ec 7c6391367f7a5a088e0315257051ed96b06ef9eac732e922616d6953372da9eb 182d27755483646061a012b995176d97530d0e7f003c6cc068486a776a164dcd 11fce16209f9e0933384592e058e65db48464ca93bcacafab64f5433f1689d04 003d1f96f7b5f1029a7b691fe177fcdce772ad3e53a3e42d15255814dcaf6018 456d61fa4965b544bb48ccbcb2de322c4087ed18482b353d54d5fa662fac6fa8 b5e263551700d19905d5eb9121efc13c16c4a62f2b17aca05dd4e1c4abfafeae b900a643283c409f80414d7755794a68d82a365afa12eb148d28297fbfc5f90a b043f8d9005e761988ca043d405e000100411469f44eb9999de3c538c8f5ff0f ff59addbd3d90b410ccef2b4bfc2fbba8376cfa56f69393a18f822032084cd5a 7b8879c8ef56430186566024514645f3a531458d785048cc835e4781f1599c63 40e90e358d58a9b21c2dc6c7e435dd1089a411a30a654fd3efa26e58b5311452 755662f9ce51684676b3 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 50 /element put dup 102 /braceleft put dup 103 /braceright 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 b910bf03a0fd0ff582d7bf32eef7d0676dea7d3546c46d4ebed5a1376fbb1951 cc390a70dcac49db843da51e5d86171dd97b36b4443863888375a775276425d6 d4a31dda6e7d08ce93049bd4daa8583300b0ec2139e1b8a761afb89bcd67d168 a5551eb9f7117f255c3c0b8522ccb1bdb20d51ff6ae88b7130b7d1f9d8c559a9 01425fee7457b3bf76f161e51d407f7f1229b8f650497f9b7b25855c60cbc31d b3ea5f99cc8cf2eaf00e8be72a211f866189c0afdc5738c350e62040ea8d3ce4 d9aad5064e2f95f812dccd2586e17f3cad42c49a4716fbde99d224938fbfc8a7 8f0032a117ef1709d02168306e886cd7eddb524202f926450fadb9beb76c7a7e a8be82db17b06471b73efc0635a4d5c9a7b75404fd91205dd08792559384fd50 78288e9ab82fcfc35c1fb8685e2a95256b4a881519be340843b0c357baadc3ca 01eaaf1ff661c639248265131d3d096df7019ab53c4efcbd6e0f54faa67b5c5f b65bc72aa8279d65b129b384ee6b6548e7a0935e3d761a9c467c2d4da3fc69a4 746332eebc29222415a0334c88f46673b67e7125112fb71ef5cc1fcfdb344cac 5609eaf952e8ee9de61309c7c1904e2e08858787869881d2b55d818404f00db3 e2c9f10cc95a59c77647fedeb25e9bcdc5b74fb366cd852d0691e357f143939b f6f64c8bd4c75a8c975540f0bc4901038e89fe5888b5ab65f182fcadf1f51aa5 10fa4bd977463acafdf4f4440579c2c745cfe5fff05b6c7e03b92d893fd2a0f0 0656409d5f0d73dd9b71c78dbbbe326a174768830b857516ab2f4b472c01f0f8 3d9edd0642ab894b1da16c6a6912484f1912b330f7545fabb9c4b57be07f08cc bad4faa28c387cdc145539b6ae3c402708445cd1defb6efecb6017d097fd0450 80f84b7a12cdd7641c59543d368b1cf3f672d0082aeec07dd5d5348e48a78573 1a4e0437b9550b1f8d63b4771fcfce0e324e24e6eb80600466a7fa81c73f2f7a fa62859ad1af6dce3907af327b1bfc985ac7d204e8ca9093c0432d2448f21ebb 9f5b81c401457ee524fe33e8f8002664 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % 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 (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 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 readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueXX 5000816 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b6a18979f258b8c6ed3 ccafbc21aca420c9c83eea371adc20e038b4d7b8ac303004b0aa205f04135140 76407216032fdd22e6219da8f16b28ca12524deb7bca073cc5eba65c102a5e85 fd48e6d062cd4283ee570a7774597e5bf0e3400b6be72db0115f3cb12db70ce0 83722870cddfadee715f10f1fcaf20e06f3c54afe5ca238539bfe2b596116e83 f5371ff18fa5003d8543226cfd4025f9940365b392a858d27f078d3abcffe4a1 54e78c7692d1a32bf935967c64f01b24788ff8325d61145e2d4a489fd986fb77 38e6b254522c77ca2797a504a9ce4676a77ebacb026eca94dde5922c936f8e90 c43e28519671e8def84a1526a8b89450ef2bd624857da91e7a4a832f89e4f9cc 36f6027a64831992e98b1707ddc56e562de9ad0278d89f3416dfab55ea0e5b25 9cfdc4787442890a5a8986bd394cfd87942f66b3712745aa18e52c6bd1d44d48 3f75ca3a5d39391f2bb9441314fbb25ed2b6813e66bbdec80f12a88c2f94f601 fdb6b48613c26c3ac738dcd4ab05a16c34aac07eb789901b095230281fc71f04 5ac83d60c6b82255c6276398134634a8135902a31a52e84b55aece4be46e9d93 72e8c2738a3ae29ce61298d5c2e792bc098a143f3f08614489 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 110 /n 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 9391472f46a58f4d47b46f95c41dbec5e4518446480981fc63a076168ed65f9a 9e472afa6a1c3ea322a64bf8c4d57f39421f2d847c0ab350d74a876418b68a29 34063e0931fbb89cd89433ef9fd3d740212b27e1e05c5da6cade43c00bc4f126 36deae33c75277c9f1e6da5788c01cf9c50ae4b09baf30691b48358f0447316a 7db6dc4bf0cb3b6a16a06cf590c3a9e11be32c7efca7c6d19b73f1e33621f13e 17fd0131a72f31cb08012afa59be090f8286789a89f705a9b5aac1110bbf27e3 aa59a5ec50aa211db20004f7202fe623d16b54472ece5b7623aed8fb549681dd d1b52548b64eee12c82685f8d34e28d693d9cd4026a97e5415058c393704414b 09f367e4f6a2931106f6cedbbbfe39176a9e53ebd402815c5b54f2e0f05e18f9 a82d82dd0f1f4aae0a4bc00458002ad1144c852fff1f34bd5bba3084f33e3327 2534c57bdfae5d4da4c32993c5c6a151b79f7d27cdf4647d58f32771ba8eb334 0e9475ecf5f74ea2481b54e0a277182138f81a8124ff284b009abab899664e3e 7fe939c8f72d02290af45eac 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 58 /period put dup 83 /S put dup 120 /x 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 b59edcc94716f2862cf71c9a2be8430985da1cd813b1e1c1b9310564a2eb1aff 52c190f50fc981dccb64bb2c1f7a62c216f211c162c2ed1c554ef49fbaf12b31 aa3b2a80fee259dc0a2a4242869ddec7c29e2941792b28bba7f2c6f77ef69638 12b07d6843f33b51e01c8626b26edc34e1e0fffadede5fb468e8cd5de7e5a64c 366967578f415484d122834de4a647541d66989f7d8b25d1712d8d16729bd9a6 472b144607560dfbab8831de2d060b0908ef550060e703333a5a820fb66e5699 77fa0925bdaa792232e2b60cb7702f4c2527a8482e00ad81f2db4de99cfca065 d3f6487fb1e279c27faa9b1decb8ff14d0a5c44627d5ea65fc909ab008519979 9d3d9a2cb0b11d2ac14bc6dc70d3cba6fc56664b0883f92411d858697b96c196 44c841ce814f2b57cc451c62cb520d92e18c235d9ebe80ea4de07d9dd4bedbff 35b9df7d20fb28e4efa3e94c6060ce5f2b713818e5aafa8726ac8adcbd2b3138 194c3224968c3488ce12b8f7cdea86dc9d8033011eee12c851a9cb22046f7dd2 dc2fb8890d94821d01dc336d2448311b56bc29b478b3d708e3a12fe8b738b563 f478953ae3bc073c56ffed282491d22a72cd6baa0c6829016865caf14fa2a9b1 a83f0ec589f5208c509614d23442a166def2148ec37a3bc687b0f6193e82c12f 19a425d89b65abc132ce32b6dacff4aff920dc62199058043f2456fec1d7d783 325d312e8e31202c224ccb53649aa965248aacc3284a0afae1b786ae776ef3f9 7765adbc3de3c65522e1e32d6cd7813b5122c41968df224e032a62af057b04b1 5c73611855fb2061d7ec9af277e8da74a335709848b4b1babca9b172c5be2a94 5a559eb898963b68bebbf32db78503f1ec4dacf28b66f3e3915cf4eb01c0911a bc50de632bc630f490ae56daae816c87bedc69d24ec35d0449d69fe87fe2780a f054e150cda1d15dac4d03c0783a38571d14d0e5536682406462147daaa5ead1 1aeda5f7a54b79101226700bd718610aed2f3689486d3f25ac584515a1363a04 288e4c5e94d62861467e30350016fe00397ba1f18855b7c6bf78d5db33ab828e f8e3a20e96b51b13fa7259707698ddcf6d7151e96296f5da007facc876c034d4 c65f7187f3bfa5673e33711783db10b4776828da89c282062c8de92632936e5c 95ba0e3beeae1a416fb5a1f181b723d345028d2d4a81fa1efd2bc7efae0bf373 444b77e54e1b1691788e3d91186eba6a104e 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX7 %!PS-AdobeFont-1.1: CMBX7 1.0 %%CreationDate: 1991 Aug 20 16:35:49 % 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 (CMBX7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX7 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 78 /N put dup 84 /T put dup 90 /Z put readonly def /FontBBox{-55 -250 1289 751}readonly def /UniqueXX 5000765 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486be79011d1f5bfae5c1f476ee6f05eb1d2caeb 269958b194521197b1ee2d9de5f5346241e154574f14a4532b35968596ef7a20 102dad629475cf5d654585d4caac68ba7b2ff21b103a98af3e9ec01fae180804 26b73d62e93f0328ce7775e81898de587dd0fbcfe98934a58afb1aa4dd8c6eec d19373ddd360a5e85cdd4b752b2f2baa13b48a282214a05632d28e915ec90824 21ff8f2166e940c7122c583f235bfd098f8aac0f1dbb9a625b02bd889869664e f1856608899ba509d6eb8ece665afc34a459dd5b54dc7711f6f04a288214efe4 53095e239bf19cd36b7028d46c07a2b8c17d452d8563ea88a3bc5e5ea29a9ea9 a7e38481679f893f9e6a983ccb5512bed4aca5888793526701796b3fcac0fb62 f69c66d0688eede0fefd590fbce954b2cdd2abc9017390077a2cf6f6fa7d0db2 dcb4b0963e3dfbf02be1dca9cad104f3d29495d19337fbffdd745e1729e7c0a6 41fdd3bf42c30d715670b3ade44f481d795071f1b24f4f03dff59d91bf9d678b 4560da27ad826c9fc8f8cd9a333248cb2007b4ac0a9e2d5c044774124f1ee23c 0e3e005a4a16e1d0cba15797fc42fa75cb45b24fcf8a8cf41de8e06388b5f995 04454646c3246074f495564cfc8ad75a949cc477ecf03f0511b90b6a7456b730 cf09773b5544f6ec17a9a179714e47e2eb1142d8005bd228affaa375eed7fb1b 075b3023be6386f3805aee05b47c7f370e8ad410110429d2ee8801d0d2da9e68 a453a323c147716ca588fbf563175e5f47d1086a21b04102cba338a3580225ce 00dfb1784fbcad0fcd11434afcc2ec13337ee24775c2952be02e20572aaab17f d21287775e4a4d9fbf877a487b252b5c880463541489f5578c7472d39fc27320 2e4ec86f9935ab2ded14ab9ea36770d79a1a72161102412bc57b891c4202dbd5 0d65e4fab544133726161946c3f021404d2acc1079af9fda5a0f0ebe67705f80 8a2cbcccee8576f6dec4e5e87a337cce18471c737d0ff5bb2a17e67df0a2af5f a1099c1a869eebf8e28e65aee114477d1d203e9d89fd2de634a6fdd5d0ecd5e7 702484ba7bc600e38366cdc662c69be2961c2bb922cff235b003d84c12e35d13 dc934febd246cd4064807b8de87dcaed293b7befff7fcab3c52fd3c67b8d3912 1d5faa7610f926f4bab7b97bd9584bc11c5d9f3934ff78ecccf5b5fb9e1c1872 400cd94c3dbcb2cc506a3905f09152eabf3a07ecc0a29c6f41f0f6ee5e4ea11a b5be4f65511f13a0fca7289d7600c255ff4fa0f0c6999c4ea837500f5aab0242 8e0ccb557e9e026fcf1f425dad56039a35dd621fef846de5e79c2a0b44b051dd 5b57fbab94d48860db06a4d91fb347ca5af496c7fe0918adac92192505ad4c8f 44b5dd688b15764aad349da738676c1a757994ed6acbda8739882d218478cd58 2dcbb5b273661b86758bdc40b144ce0dd884b7cdfb935f5ce9b6e25627c4eb6e 1a63202b0241bfdadc1897a40949467885693565f2c002488b74052a2514a958 282ac22768a7608e7f6b9d15cef0324dcf6d9482f36f366719ff4cbb5d956a18 96ac3ba6bcfbfed2ff03d24ed3ba92df65cb422fe2457a6793c19c1eb1a39b85 cd4df3ad0249d2e7a7821d41de8517e5da3a2a3621a76eb5a343a708266fb734 54b750ce92aade715672026f47c077acd589fd228331688ed509d1b7eae09bc4 399dd3bb57234a5fda2200248817259ad5a8706ce074318e6ec9728031 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 11 /ff put dup 46 /period put dup 49 /one put dup 58 /colon put dup 61 /equal put dup 68 /D put dup 69 /E put dup 72 /H put dup 77 /M put dup 78 /N put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 97 /a put dup 98 /b put dup 100 /d put dup 101 /e put dup 104 /h put dup 105 /i put dup 108 /l 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 121 /y 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 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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 49 /one put dup 50 /two 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 f4fab969bba2b44e47399fedf5caa1bcea216d7ba713d523de97a57cbed59a39 c235a135d115ebd4abbd5c9600b26694d650c64477d24d3e668cf2f77c682069 8a49e801bc835101d0380ff234ffc28ff2f6fe6f1ad59d35142a82e252b03506 4eefbe506595eee4c52ed8c9e252fe3a9894b250b5a8765bd0a3ccec511fc398 f294b9ef281f87d76802eced7414297358d29f12e1d246c0f39f9ec4536c3f4f 0549474956b5667672941778cd589fde416abb9915c030ac96e1b70221c9b072 04f75663bcb1dc96dbafd3ddde72bd6af74b96324b707d58c89ca136d07c38f1 bd08201d21d46b9067a4c1ec2857c203487a9237cfbf1dbef5af4c59fc1e726d f49161e746feae8bab48ec220e62fc4437ae29ddff8582dfeb1afa3531a3f86a 22606ab4a62e2107e4569716d1b12297b276035d91ee77babb02b1a5a9428528 b13475a731a58f734f2640f57946f21fd852f775db7637444b84f2dcf1a530e5 964667948960b901dcee0d45e7fbe71331faf54efa7462dd3d6c0e88334556a5 b1418fab410355d9856668d76d6943ca25da0ea62be33599f490997674a9eb45 5f94c730ca6580cdc3a7bd5f362d0ecffecb99b24d7301bca4a0fe45 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % 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 (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 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 14 /ffi put dup 18 /grave put dup 19 /acute put dup 34 /quotedblright 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 55 /seven put dup 56 /eight put dup 58 /colon put dup 59 /semicolon 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 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 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 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{-163 -250 1146 969}readonly def /UniqueXX 5000828 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3ffd20326947b1065649b533eb5e9e3 a88a87fe987918a687950b26f647d6e08bf14d983ed1d4a8ca1e2da8d985f944 2eb916f5b6ed0697ac7c33e1e36a3bf460d34ce45f1631871097cb04f18e3889 4cf4ac1538eb19481311d24fe3be7beaa4a3730e8b4831fe59d6d9ce2e46116b 629c7ba2f9ce3ecba2f43bc162a5a077ca1b2882a42afdcec3f4b75b5d63e0bc 8e5dc95257766d8ea467ad9cbaf47be60f797580cced6884b3a68f70c91f4fdf 80fd00ed9139e7f480dc3a76af72ad9b434187730bdfaefe4cbfe5c7edcaaf24 9204fd703011932e5a3c27be468b7dab69daa18dbbb6335ab8ddfdc607961c7c 02582763fa069d43563ba17704029945cf42fcb19cf78c51df0ec4d851086d43 c1ab38e865ef36865fff3a08b01cbcb070ba4893f7482dc7819ad03d337e520e 7d8cd83bb8ac7aff4df36751f4f12aaaea2b6d7260c09a26e987c904003e0723 7dbcc4a8e4e85cd3259f40d3caf55c742cbb40ad0db1044c20a4f122a63f7b81 1d945bfa69662189ce4d089464d2ead9f6ed60b581b04114d5d45e97be27a576 35154e9b8464f2a0df18da855eb51b8f64d3054b0b2e84621f025a0776d16fed b988a43d2d64c32768dd106b9f541ae8e68ac0bb7399e16f4d4ba074b2152d0a 4e9b5bfc21bfa0d531e28165e518ffc054dd4ed0d74eca079d20e79670419f92 e03bcbfeecf3554d661596f6f29e12170804d15a250d917ffe7a8b6e8fa53047 1ed97ba56481589a65ca13619129f1e26affe6c22f0c38540f8ce37efb6e9e9b 60b51a5a9cb4454fd7c7f92f2f9437159687ab590f35b946a54491a7739c9269 e0f1fb48b129811c82edb3171cbd5105fe65906974db8a7da92f4f43dc7f6b36 6981146d86d23613a0b38d9d6191db9fea0172e982cb1cc50e382f69da1b377c 1be9407f88a2a761b56f398467a057f35d18c01cc377e006182c891ec80272e2 2e3a5c7123cd27eccdc87392a9fdf63bb40f02b7649cfc53ef933c6518744cb8 9cbbd437830ef5b1d58b00b67bbcf4426dc1e6ef583b0b4ad3aebd741e3ff0f7 29ae271f032019b1e7268043fd215be590f9cf24da14a90f1683afcba8006cef 5d5632c2481fbc1bd8224e8952debb7c4596a237e5505ac143180956ca1f603e ec70cdb4a33e1c4ca174436a38b06fd9d2b9196f02114b77a9a5f66595727faa ba323668573ae328f1f84e5a3e0e7634d559479c134207aad7a949da589970c8 da77d71a20a8ffe72beb0643b81044daf53e79809e17a14cce790765c281b4c5 cd8c35a9f87a34fe5981c199ffae7fe41503e38d5f437d0134d3a0a7bbda0350 4bbe59c8585052ac7c18ce6b421449d73b16961298c14548b796d7a2198080f3 45be1b247c5a2e454e47886dbaa56f166a4b875a9cc7b735bd0686a3570da5a5 ae911fcd2159fef835f73b4721f47d4734828e4c7e30997976d6c457941ec11a e59c8de8fdcd954c78ea7937031d7fc78b28014875117e712d2e301e06a5aad5 8d1a7e3988ea54ca49f07a96456dc6e706026f2eb17a7173296dc0af329002a0 16f74ba820800c72fc1100acd6632eec4c67cef2c2ec5fea8c2d2f6da716705c 467f758cb956bab927f05bee8281c4e322fa3707356ca55fbe12bcce6286f88d 6b4e08232dd37aa580b4808808c2e6d1b6957c4b7c0fae009f566eedd88185c4 9cbfd959770eccec2314669b24b4e0411afb1818fdc04ceedd6605210feae5e1 b108fe642035f3e8a6de01a4e171ca75dd67f5d7f8a3026ee1636b8c79d1adc3 dff8d88f65b382e18b036d15a25725ef5fe4f3b6b43a417984f954ba168edbea ea5b1ec6f70d2877124be228fdc9763c07b77a284ee45470735322c155007aa9 50825ba2ba46ad6cbde6efcae9222e40a79227a4deab4b0983f02c43aeeb81ee 87e3bb08d17360f6e8ea7139bfcc53e1519b96171a3ce928fbfc3e86f9c4125e 3e2a3c348266d2b13d2fbc43f8ba7a03c079f4ce68feb2146041c64ae986097a a97e46def77b65b899c38abd8bc5361da3494c2b7053083fb710fa05cef53995 052741705365e8f6f6bd9d4cce5cc0852adfcf388aa69ae9f0c67f4edcde712d aec3d070951425ebd8d2065bf8038e4c7bb9d7cc18e978299c0158419861edc2 c95b17903e24537e5e865bde2e052442df8efac8abec5cc79e78936e26d2639d afb22d449f38ef77d78a53a12eaff4d80445b67dd6529dc9b53826ef5823907f 8caa740909dd03d8c5c251ad2c72f467d6e1ba691049d7ed540067a7697b56f9 16755296b3fdf465e357e061f1be3ee36dcb52011d913398b2e863186a1c22b3 5dc2b6f634d8b690aa6b05bd529d3bb0c4ee4559205f6657420ebf6a81be6263 d8dda0d60d984ff2b05301e1d8b21159b734d59734bebcebfb26162d3ec007c4 626f414c970f0032b4946219b1ea67e3a53f9fa4f5571158e18ec672508547a1 bf7cda63e69ab0b65baca6e89dd5a3be0104e231f09b821f159eafc4cc7f1efd 708634d41a14119d5d431b6832082f67799bca2d8f03b9c236d0db0b0bacfe2d c954c79320e272a0327f74a74c145c3ef54202fc06035d031fab6296859b63c0 d903850b259205cd4eb9cccab94c738cd9cc47d09afd04c4e61db3551bb1cadb 0445b0404e89697c1035f0ac95c5e02447d8c546d6b6fe4583bfe04c57206430 8302fa589e26de18831de1834abbd3d90d9c81cc6afdff1bf03eea996197322f c61c9199c60555a84e62ce528184721376dc8e2ade57623dbfe5dc90a29f5fbb 405bc744fa4c487c23d459883553e182005dafe5b3eeddaf9cfaf6a45d3f31d5 74fa7d8ef4db57c9ce66bc4d4d261a9edfb70af8f23368551b03ff83e4c62447 918760ed339aad18eed0be2a62ad9b9f43df3f6c7167990cd41c54d9e6337a37 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950428b8f1dc44e9895b2d7ec42788b8669ed30f7406f4ad810201d780153cf1 ff7a4cedbcafeca40e50fd3c413927b865166857f94ee43254838ad043f255ef 4285e41bf979926befa00fc5bfe7ce86669c916447f8cbdeb37a6c28331d1b50 1fb2c4021313d255482c9175a9bb921c93a08bf104dfb28ae7f162436ddc907d f190e038bf0059cd95f7106d0ef6dc1aba8c3a51c860a69f7c1cc0ea3a2178f7 c30a74d0e4c4cfe090394cc18aae7621063e2a4a5f028b121cbbfe376f4eec27 337e0c44fea339b5447c60dabf9eac9de8eb2064afa8bc645ecf8a9627c8d52c 178826f9dcb799060d74d0f294ea106f2cdcab168666e59be55d9606a4787dbb 5bb4d83294dfa3dfcd82076f5837d45aa93d9b565f895259ab85087a23532a5b 963686b2968c97c7e9a50587b6bb774ee441c3a04aa1a30be14d23038c88db45 d871ffac68b9e55a6a293f80f75ec439fa0496642c3b97afdbac707111c6b613 bb6f8d54cc5f81fece5f53a914b0333b523a3f4ff442f469023c2ea149cd5a96 e240411051a0cdbbbdde86046491c7744e709787eccece457650c11dc4929d8c 15410fcaaea231c805c39647225908c887acddac0a60657a982e73730a9dd467 f7ed365969d8404c50746e4ffb90ba8d4293bae1fc07faac6fe82d1cac34569d 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4d4e9c81f2d90bc76aa07bba393aa7fc0f7a9288ceba15294af872409031ba36 2543ec87868868d5a1be36e3993cd9b5d6578c25fef6874a812fd0dec94bc8a3 a8d7497cac77daa13197b1f8df09a3f6ef1a34b5ae29e59aaafa020d5c768d33 6516eb67c98f8a5bc1ff5d330d6b9d783b 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % 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 (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 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 54 /negationslash put dup 63 /perpendicular put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 110 /backslash put readonly def /FontBBox{-15 -951 1252 782}readonly def /UniqueXX 5000817 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f73e9b697ddc0351763500e23c4d170ea8a2d2 75ad0b6b23097f02fa803c1f46f9ac10fe0e527c431b11b1b9a50e874896cca6 17fe3e255960846bc319ca68b9b930a45d5d0fc36b352c3443ab6f7793b38d2c 7fb9c409ab7980ead55e9f8f6b83c70088d1fa6d7f2214da8928e40b96334ad2 72d1b5db9d1355a579b26a8dac21f021e393d7d837a95deb8de92ed818a8b80b 98ee9cdd8a3c57ad7d6eb7a82f43eaa6ec520eefa7302e750edc4b495f70bba2 24a2182eec37f1a1c5dc8e20973007150237daa6bf03d498826777eeebe09014 ebfe05a8731e1a421cfc513d47d673e3d94d805f5d0668ffa265e0962cb5a75a 4a93d9b871227522d95bfffec89bec308982e8aef933fcd407778ff1c9ce31ac 527b4567d954b095e8b1a9b07be61182c30c20a629c806df85f259578e4672c1 f4cba401ed2f2c5870aa59001999c842c5b85441933acf4d6c506131fc2b18cc dcc43234c753eb0b4e154a85a9c6d9bbfc380b2c7e58234f7a6a47acc7b70c2d 1a22eb0a91f348fee5495a2623c98d803696ac835960b4b5a8053698d999cf61 25369cda51a6dc1a327da57755e855ca676e3a399ab02dca1798de879049a9c9 8146ce6d38167914b16d08d6b63ceea52dab2a48eef0c5f1267bd1bc779eab1b d3c8f974bed5361acf8f359a21f9bd7af050ef371158ac932eea0de1e8aa785b aab7419797bc66da41dd0806da9dd9840866d881f1319fa522397ee42aaf776a bc8cb6f9890893bc7825aef5015f14539c4a3f4e20d226b79df871c556ea9ada 6a08767720101f0765134cabc333035e7862c4bac0dfdc3d6333df1fcea6ed19 ae61b2158b434d455e8c95ff8c6a77bf93c768e4a66b1ab333c69d1d489c4b88 fe7edf4e5b3a850a04154616af4648b607f5c6b63625a2a82a4850723074853e c1301d4138a60a32422b604a8adc3b569b2865f8f814d4e0cc83acbbb7a68e2f 2d04078a946a89da821d16431f730518a200fb4544c2defea7b27443bf366375 388dc365a56a543e9422ea49884ef0a5a3b45e1cbb46a053de0f97640a8be545 f43acc4ab82f2fac6d02481ffb7d6ff045d5ada9fccce73b65e3cf63c870c810 99c61503a4f32be3970e70a4f21383cec5483300fe93283c6605ea359dd6f5ee 9ffa364c8c39d069b18f831bb3517f63f85b2d0b7369a6725697871818f29ce6 99c271dc6f3afe51461c6aa34cbf2cc4bd50de69f2e267c8bc678062431321d4 140b854f005ae52a2c58d919755b1d22525c7042e7dbdea2f8ef1ad88277bccb 33bd509b64633316763a404216e22e637fb0452bf9f5b23783de87ab8196fe5a 1d8f8e62ae3cdbedd4cc9db40df28c1f7cf6853bb40e0a3903b1f4af663241f6 55283f283155dee791ef480a223654c1addfba2ad043f5b851e9cc5649fcd5f7 100bc7f74a33ea2cadf82ccc359201498a9bcb83d4a88f513b1c43a2d5a9fa4d 382c580aa900a8b486919265feeeb47c17193f0f723dbb6321c94774e8163d68 42391b6e0f54b2dafa9d7ad33e9aaa315401013bec88e153eb9241d9f4fd4d2a 7c1b70eb70ccbb9f387c190128eeee9cf81f670ed990d8467077a5730b030cf1 4841fc80e8d3abb8c38899dd5cbebf9be158ba36601fd11c3c2001c23fdb7eea 2d971fc4ac4b63ba5575b84dcb6e6ae70f3ae468fbcc08da3ce38a31938be23d 237f5e605646a269162bab20c402b69699bf82c80ef704ee38d7c8207ffb19f7 ca895272866148abca72ad138807e94ba4f1332771f7d1e8af69862ab66bccfc 58d97816d8316ffa19ce2b9c59c6bdbdcf8aed1208a138005305e401362fe53a 676fda8f9f561c6683c2e0bb346165c2ad7f8dd6608adfc45957cfc2f9ed9356 c73063a99f1f847a9b841ebcb0f065412c74b60e0a7123a5d05ef4ed8e5cb0aa 655b90b6a66cc34db03f4a1cca78071b229f59f94498b9dc89427231781fe142 e28df9215ddf956e16d7a022a0a06191d558a8ec624dc8f0fa9791fbfa776e68 168fd7568980848c83ac15cb91c85bd9f0f92042228df93ee3b466562397b416 0a6ad342f1f3f5bd3fa9a08604a141a6d7e490e9c501e0aea383e5bd00ac2330 cfd68e25e58150ee812386bb1d44719050000db6ee31f5e314f6a3748c9b9d8e aafecb3b88f334195647f1246d394f8f83d77a35023fc9205b567a89f27ba79b 26bc4c3965f017a32e532a22755876c5582d775cd43c1203b217eca93f7ae077 1e71d52c6a55da6285bfddc37603012da7558062e80ba880f1ef16155bdb2ac9 8cd7b6fb61251ffc190e7f27b63e1e61edd4f5dd10b479a1f05a5556b9438a83 9bb3fb38a6e7c5ba18bc6ef3be59284d2a63cbbd05d054997c52fb4a69562b4e d0234c7756d04f7d625e070ce46ae4e193958d1d4bf23f78e32f3d61d442683c 127a0b780c4242eabb307bd25a78ef4741b628e3e2ef12a009a96e57bec260a7 23faebcf81bdc10e8cd717514ce55b23184449c6e3216e906ac97cbb76b1203f 35ab602d2157bae1c78ddb3add8a841e41a06999fc82352e503c5bbf6dc58957 b25e33149ef35f1b2ef1a2b4872a0c0e2c51893db6ec32ebe6040142afa312c9 5343c729e7b031f3cabb4e3d5ef879da7e3b36aff72ee4f3a0dca566f36a7b12 0847cdc23d9c4f609d33ddbbd87340bcea9e60e9f8f86c6c738277330cc0f1f5 fcfc1f7b3698e4418895b4b3a3dd481f5a576b2417e9148173ac74d9351351fa 3f35a437a815b636f1ac5ad0e28d8cb1d4087b6bf0e6e3c255e3a3fe6205e89a 094e5baa15bceb5ffeb84106559bff286aca85edd364047f202b4bea79b75910 caecca093d6ebe4221c2a8bd051c49e4e6aa8a7c4c3fe5d502aad0679e179bfe 0727cd4c737ba951d3bc7c17c1600805709d54c96c9231b9af673cd57cc4496e a0fc2da09349a46699ba14c876f112958b54c2f47cd8351ff2cb2a0554c1f30d 79f306e88d6c845f76434376aef304783e7fe73bf8139065da8a7661c06be13b aa95c34f181ac6d77781ded7e1412478edb75411b14d123c4858f54a32e0a0aa a90bbb4bf73e8abfde776c31ca19d730c7a26a077b127e90d3b0eb8a7b1cac98 c46ed0006e5ec7a8d426f5bfb38892478c1719c8e7f7d9d177fbdec808a48ac5 46b4e295287aaeff2f239f62a59a79a599c594c96ed7f609cf2bf73ef41e1362 671673724eb0b8cf5b5a2f734607ecb69508c1c5041c346444ec5657ff083a67 21c72696de46cafdb85c86547c5a562474e354b05ba4286367e152a290b62ec3 e227b7da34af2c932e892a75c5f6ce9f0f9afd24d0938c468fa27fef4ec6c57f 65580f5625628091dcfd7dc37f5732ef29e41511c72c49778a63354170732932 bd881704cdcd8d8a40fd03358646307ad79d2201f45df51ab99feffec6cefcbf 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 26 /braceleftbigg put dup 80 /summationtext put dup 82 /integraltext put dup 88 /summationdisplay put dup 90 /integraldisplay put dup 101 /tildewide put dup 104 /bracketleftBig put dup 105 /bracketrightBig 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 83fd03a516d8b118b5f7f9f9b7cb6c7a14f7048737ab827e9a1f4d496f7e6b06 c671f756837d04d3d4f800cf9cfe8a9130669983d7bf613cb047d726c3aca830 8a553df43c4fe2a9bc91ed0f2d71fcfb27813fbfea58e9ae1c79c98a90ab136b e9934f8c4f97f99273f9ae6e19d75c63532ee83f80859b060bba3f06f768c7dc 52313f7e818b2ad66d66fbf03c026835ba6e411ff135f6578171d884219ebb8e 6e4c7c0883b5c7033f203f6a325cd7076fb3c8bf3ae6cae4fad853850f43e558 de73539ef315e3eecbe937c3a6ec3bb0e0b06e14d73d407f3cd1e7afcaa039ee 89da47a1094d37506c73c040b93389d54ecb11fc46e4c96977a62eb03921cbdf ca470ba4e7bc4188714b2cf79311f25ba7bf90e3fb4dcb33a947e25f1fdfa2e4 3a00ddda1534c976708813721ee882a66bd99c737c932c78abb3015b02cd904e fb45f49a9e74c94c7f314e6ae62ac594508edfeffb9a1070d02f40183240b93a 370da41c3707b6eeb18666e8216f2c22eff50da37a30463f20bdebb54e6f128b 7c70c1c86967dbaf059c5cd4ee1718641649008fbda19d7a881318760901e3ed c06eafc4f4873c51b0cb1a26394048091093f076ff2280fee13262a8f1e1d1f3 238b477dc80e7e6857a543aa617daed800b50683ea1778a1c0d3f33ad108a454 0815fba01a5912ec255c903085ac0eed8b7e2c610f1a1ff7133c02431de92494 287aa3e76bf51d83f6fe0f1f6ad5cc6d1ec0b46c5889a42be2a5b67f9ffd3cf6 d1bb6f012ac1bc2706d3de9d3a412e32b74bebeace58e65861f4d7ee92a12202 382506a9ffef6c971494c4b7ba8b4b185430382355750e2e4038d580574eedc3 350548e71760068b0315b02e889af407be595fbae9e71538b1dacbb0bdecf21e 30b3ffaf0a88edbe1b4ac87caddfb4302ac6fb2de5caf1432898d39f57089088 31744400f177f456613961f98aba9ee874a057d1939583a3cd776989d64117ff f9d8057baa0aca93423b8fa88c487be2b15ef6a4a68ddd27460c1167f72d2eb2 61ffb45bf83ee55015dc4eb3495b754a79484882735630dd60f60969aed20ef8 cb68b7d09116b5245f7a2049e961c4b09fd6d1b926cce809eadb13bbcc24b6f7 794bb4b409a40e026af02bcaf1c97ed2b66f5f1379e5e75958dd237634da3787 3d87c7f6674f83ae556c820dac379a1cc9dc40b16a515529457573f756538027 3971cbcabbde11390ffddf7bfdd57278f8cb4360db4f57a9f9562736aa7de1a0 8dac4f68cef404c20bb9324fb2889e6100dfc35ab34a00c10e8cc6d7e618e135 ac9b111166e4b00dd560d3ac51d429963bef2d2f17056ab6691fed6d25a423a0 75a2eaaa0621c1ac05a121626dfe306cc0c996c006f2001c0370f80782a0934e 4b3ad1160ec35160f05937567f3bd9c1eb2fbb5dc8e1cfa5af79bbfd3341ae8d 0d6637e85001fa6fa6ac48e30ca570f2bfc9df64173ff6c36258ce8531d2d2c1 0f5f13f49182b2f85cb5e94b7c46cbebbd00ac496807dda208553db01c09f6ab 886f681932eb5f3f14b3eb70d6f2233c9194c3e9dc1366f7b14390c865e2820a 6aeba70b6e2a912ccfdbc155ce837da3d1859b061e0e525ecf285e2c28cb16a4 749fbb279bfed958fab53aa960165e26d2f3a095a45ed968f062b08e66136702 50bb29764c636bd63389a8c49d4d33ae5b6ceb52ab96f176a96a471dc6d5a2f1 d71dec8bf33f58b6151ff4b965d91b7c8ab0a67e435259b0603d09e0866eec6a 627c054d317b5b902be88cbedde46146541f8af7cd2f31fed08f6ecb906a16fa 872d2f9836d222902f82df27f2f1b2333e212b1913f1c1540894ba6319ff73ce 8e413635c49e6a5c32c504b41b130e2f8bb2db6bbb548794bf5f28de41c5129e a02f4ceff2f65a9ae26a45c5e657e4dd2e5824814214b7e6010caf2eee744447 ca8c04bd6afeeaa694a6a004815161e6a099c3c0fe2189f2f6e0ce9d91854cfd c2f3e5d0d8d22af336fc5ded036bfb2de080ce3ca58bb688d72bbae21fd94ab8 1eb387d7a90762033fce8d97406ca7ca56977f985f2e6f71ed0257473d155097 05a5c62007293182bcf65c07af42207b30e05324466a3015e8ba6f488e297afc fc49da2b5352b6e1852d2b8dcb696e84314bd327036b9ca93e8a532ab770d317 707e3bd1b5dcd270d545020d382cef163036758ca4a830326a46d72d78649683 b84d30c2d8f21e1217749412da9598169aecd7b4f52b1d91e07242810fc45232 e4938f8b30e5dd69652753c9496149b201bef7023213bebcb6fbb9285eb77bf5 d298308c88c0fda0310e974c66c9af3a1830497953928fc2d707cddc975344f4 14e163b65c2e680f3a81f69e95198410027938a7debca8c08700a0b1608506cc 2004c09532b11ca4cabb8159bbf7ac5c996427d235eb8a85d7f242f16f2d048e 9dbf4bb1c9d286a17dbe7b0aacab272caec684fe6a906e777379dfa1b05947d8 1aca866b1c7821bbc86dae8fd4dc3c58a4815782475b52b116a6032dd01facf9 6d3f367c96e1dce9e9703e42fe2b92ae56bfa35a4021fb8e196a1c1554470268 8eb9bb28a7402329e9ab20574436250b09cb089eee84d8312884ca1ea7c6272e f42d34718f0929f0aad97dd4145264833d8299879d2ae00d70d5afcbfa233dc5 ea214bfb790c5613649b453ffa525de148bcbee8a319a1264d77353de2614ece 61da49410975b8e5a2f1f237e770c5a12249435b5233a24f28143a79593cb5e8 cc9cb1b31ff6c6615ab3805e4931883557840795421fe04e252eb0cdecd29701 8112378ea8bfce837a22414afa597059923c543caf8236af9c9d8d7fad01b309 49b90ac05f44eedc1b7abf19987a57f977fb7b20345e0a707e6d3ac8043e1a52 7d8964710bbaf6872ef4b67321b4eb09bec5161920c8eb7edc584f314d8e02c0 49a83a20badddfe07645da00cbb8f1902a547638081d8d3a193a6e7ae51ba9bb ce694bdf56fdf1014cbbbd1f54dcd2b4131221a53876be85f0889a96a6325a67 cce9bf838b6cb4cfe381029ac161fc36b52e82ce8b66e696616d02d55ac8a514 d09dcc5e04bf286714a10e9c47c57885c78c1c23cc53a4bd0ee82f963246b478 7faf1188597b2ab0cde4a4c64524983dc87d581401981be9a489a06bb1a7b881 51c2596242bc90269b042efb4c1fa83bd286793ae3d710ed536ff574dd090fac fef1cc4b82174c1d060c23df559c4c27fd6010f200a7308c8014ffd0fdb0c16e bfacb6a5035fdd2acb7619d4bcd8433e7214acffcde635c1c3c630c6025db899 06f67db0889a8e17a01c34501b6e46cd25aaa48a4190a50ec68f5a2bf7147337 ec1c3b294f12de4970e30059c9406323f7b78a 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 6 /plusminus put dup 8 /circleplus put dup 14 /openbullet put dup 15 /bullet put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 25 /approxequal put dup 26 /propersubset put dup 33 /arrowright put dup 41 /arrowdblright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 59 /emptyset put dup 67 /C put dup 68 /D put dup 69 /E put dup 71 /G put dup 72 /H put dup 73 /I put dup 76 /L put dup 87 /W put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 110 /backslash put dup 112 /radical 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 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % 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 (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 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 10 /Omega put 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 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 61 /equal put dup 91 /bracketleft put dup 93 /bracketright put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueXX 5000790 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b448685940729e6e22f4be0e08c6505cba868f7950 93f556b64b808dd710eb936d3ac83e5783b5f7e56d363a59a026a5619a357d21 c4e77bea121eb24b09027d1aad93158edf912940e91cd2f17922b35885894644 7773730bb19af9e70453e1b78d5edc123b273a979891d6d5e624f4a388f3897e 13b045e34b4ee2f5e4fc5e996d3549640010c671de14d18b0b3cd300e57dac03 22e59bfbf7b29422230870f5897fcfaad4b50c7c1c58edcd119218163d6f6e56 25ccb882db7b6d09a1586508e8cb642a11c29a46915e4a96e282079cb21922c1 c2e360b487a45302fd22ec8c5fab63e54b5e844d4b17ca2fff37c69c366dd72a d02922c14c0932f65ed03e4219c117962edbad2dcdeaa9c10ce8af38a4ae52e2 b377245b0be19a77d6c936e7530cc4d0b78d0cc4a92698fa2870fa54f2d8503e 2d17b3d52fb2febb09f2b2af0c2a1892039ebe19a690098799a858e3d39631bd 6925a154d161df3918074ada6bd52baddd0adc3f07e2d9f15e27cbf7fe8b98c4 07205c811121fa91e059f2f99322fed63f359ac9da97aec383f067f23e5de331 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68192016d97b7b787e78a7d1f0ce5011745f97692180fe0822027593056b600b e8b3153c4667bc99a7674591f38781740162eb1c89eecc7c586cc92285201233 f851b7f9975593addec3a99e978ffc47cbbcc20d58b897d66e818d922fef6bc7 d0f626bee2e6e717d8a5328eec6f36208bb79824201bf46394df9d2afb843a8c ab5a067580b0266f1a69b743b6f19077aab0ca9bde3c207241b034e386f36661 56b567d23fec2ff7b8b8e6cd8d4e698ba1f5538ff1b52c436d31648f3511aa05 3e417abe6ca576ae8d506f6a88602b426e1c5fb307d4a04d160c154b23e73089 4ae593053ea2da3b1d367287b6024c965d6c991a115bc7c622f35b4cf3d674b9 be8193ae34aefa9e986bf065de573200415d9591d8415825bebc483a4fbe6b9c b27059d024365417213bf176750a9b8902638d2480bc87d1a252ce3636f277c9 736675f7c294316fd00a6b38ea8fa14cace72b122cbda75fb9f2245b9f1f8fe8 3fa734ebe754718fd27894bbba8989ff50c812ef956863a45ea1c7c4edea073f b8e26b6c9ef11b3b70454572c81030ed7ec4fb182e6cb88cb84521826284561a 5976ee96a0b00a6bd727dffa8f1263fd5a6d39efaf0d5871f234bed13c0b8bfa 00a5309e9447296ddd56a663d813758a70e705249a5fa7bbbfc3abca4db5d020 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9e97976c3f47d33add7e9ff2f8bd1a3bf7f9312f96685fd42cdb309997d353e4 83c05c93612d7674450467cc80a9a6811eafe7f547f3186a8c9a8f6fa2120651 0cb4c78a8b6b5f8fed93af611c271bdefba5a30e89f0def464ccc61740ed5bfa 16517939898c674961fc121962d4346028063c5fed9296fd38301590e8a27cc6 de7c8f46c3d7bd682dde1264d42044cb4d256bd277e22d93c287274886c80791 0fc363634315bbc4b2fb3a4102122347ac02cb99930c5025951ecf5e5c21254e 2b88eeeb4afe02bfce41eb20f5a91d3febebdf2e26ee11ffeedc97e867857d58 35a9ec5b5d34eb735dfda95a98263053546b2fe2908db39cf0d0b6a2061ea179 4805f1165c5d1635ba9be7529780eb4781f18653570c5aa727540944cccef398 3ba3d2d64fdb8093e713afc3329fda2c8da00665c4af56c980dc69082ea70a1d 6f8e376da3da09e7c0cccf6e63e583a606c7fb83e0f553a7806dd34993663ddd cea686fdc3a206d7e860d1316666ac55c7e5b9478acef1a16d84138a90bc84f3 f3a95f6a9d847653df95f5b217d0d4156d2aa318a0f0f07417a45415f24906ee 9c03f26b24e33d0393c178a464c767d883ae64ad70b3f1d18158e3e94e228202 292cfe1bf326159702949dbc7cf6a58e2eabec5540c330566e25367c80af9186 8da8c80369283f17c74b1850ce1c590ef48f5fa30af2add056d2f883e467de63 7350bd872c0c168fe414f4c27e8327c0c0d6de743b03cae4f500d7b45e0b7254 e93333f50e00474ab70c9ade9066b6efc2692fc571bc56ceac3e3faf24ffb5f0 433ce69ed89b81d861f202d030d7149c 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % 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 (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 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 /beta put dup 13 /gamma put dup 14 /delta put dup 17 /eta put dup 23 /nu put dup 25 /pi put dup 26 /rho put dup 28 /tau put dup 33 /omega put dup 34 /epsilon put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 65 /A put dup 66 /B put dup 69 /E put dup 71 /G put dup 72 /H put dup 76 /L put dup 78 /N put dup 82 /R put dup 83 /S put dup 86 /V put dup 87 /W put dup 97 /a put dup 100 /d put dup 101 /e put dup 102 /f 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 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueXX 5087382 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930f01bdaae649990604a33aa9eaffbe5b5489 e5c7c9ff9d9be01b08220832c41caac64816b53bbc087ae4b621d2b44b21ae5e 3f7fe4ddb05c675abfe30510eee4a7c874bb57b2ffe521a6875eddfdfd18c781 25bfca5a097aad361dd94df46f14026c25ea988194899b95001209cb7009cead 4b4ea03473ea120e7a7e341d4e3926796347fdb3a124c29660ac09f4648b313c b58e824884382f24ce08d2edc686923acace1201c16a07a42803288cd0b946f7 546510e95b1fa1cc96f0bf72aa823d8e83d7c68c752c82a54c752eed2b1e699d 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6490901c0f5858f3be47418869ed27dad9b1bb1cef6e1de9329d2c0f28f6ef73 d73415f9c1b5342ffc97c0 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % 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 (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 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 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 28 /tau put dup 30 /phi put dup 33 /omega put dup 34 /epsilon put dup 39 /phi1 put dup 42 /arrowrighttophalf 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 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H 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 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y 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{-32 -250 1048 750}readonly def /UniqueXX 5087385 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % 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 (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 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 5 /Pi put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 19 /acute put dup 33 /exclam 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 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 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 88 /X put dup 89 /Y put dup 90 /Z 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 127 /dieresis put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueXX 5000793 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 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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 46 /period 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 76 /L put dup 78 /N put dup 80 /P put dup 82 /R put dup 84 /T put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e 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 119 /w put readonly def /FontBBox{-301 -250 1164 946}readonly def /UniqueXX 5000768 def 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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 44 /comma put dup 66 /B put dup 77 /M put dup 80 /P put dup 97 /a put dup 101 /e put dup 104 /h put dup 105 /i 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 readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueXX 5000794 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 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/Fk 135[40 36[43 24[20 58[{}3 66.4176 /CMMI8 rf /Fl 165[46 5[52 5[58 78[{}3 58.1154 /CMBX7 rf /Fm 134[37 3[39 27 28 28 37 39 35 39 1[20 2[20 39 2[31 39 1[39 35 10[53 53 51 39 52 3[53 65 4[53 2[48 54 6[55 2[20 8[35 2[20 34[41 11[{}31 66.4176 /CMR8 rf /Fn 205[30 30 49[{}2 49.8132 /CMR6 rf /Fo 133[34 40 39 55 38 45 28 34 35 38 42 42 47 68 21 38 1[25 42 38 25 38 42 38 38 42 3[25 43 25 1[62 1[83 62 62 59 47 61 64 56 64 62 74 52 64 1[32 62 64 54 56 63 59 58 62 5[25 25 1[42 42 2[42 42 42 42 42 1[25 30 25 2[34 34 25 4[43 14[42 42 3[73 1[47 51 11[{}72 83.022 /CMTI10 rf /Fp 145[34 3[20 2[34 34 38[52 8[0 3[45 66 19 14[66 11[52 52 16[34 52 1[52{}15 58.1154 /CMSY7 rf /Fq 143[83 55 55 4[39 39 2[46 10[46 1[120 5[39 1[88 53[62 8[50 50 3[28 12[{}14 83.022 /CMEX10 rf /Fr 134[71 71 97 71 75 52 53 55 71 75 67 75 1[37 2[37 75 67 41 61 75 60 75 65 12[94 1[100 8[50 7[102 10[67 67 67 67 67 67 49[{}32 119.552 /CMBX12 rf /Fs 143[69 1[42 3[23 32 32 42 42 9[55 55 3[82 10[57 2[45 70 49 1[44 64 44 7[42 1[46 46 0 0 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b(at)i(0)f(a)h (non-degenerate)d(minim)n(um,)36 b(then,)g(on)e(eac)n(h)e(small)i (energy)e(lev)n(el,)j(there)236 671 y(exist)i(at)g(least)g Fv(n)g Fw(p)r(erio)r(dic)g(orbits)f(geometrically)g(distinct.)66 b(This)37 b(Theorem)f(is)h(pro)n(v)n(ed)f(through)g(a)h(Ly)n(apuno)n (v-)236 770 y(Sc)n(hmidt)31 b(reduction:)42 b(one)30 b(\014rst)h(partially)e(solv)n(es)g(the)i(ab)r(o)n(v)n(e)e(problem)h(b) n(y)g(the)h(tec)n(hniques)f(of)g(bifurcation)h(theory)236 870 y(and)41 b(then)h(restricts)f(the)g(action)g(functional)h(to)f(the) g(pseudo-solutions)f(so)h(obtained.)78 b(It)41 b(turns)h(out)f(that)g (the)236 969 y(restricted)30 b(functional)h(is)g(de\014ned)g(on)f(a)h (manifold)g(homeomorphic)e(to)2551 939 y Ft(2)2619 969 y Fv(S)2675 939 y Ft(2)p Fu(n)p Fp(\000)p Ft(1)2838 969 y Fv(=S)2936 939 y Ft(1)2972 969 y Fw(,)j(and)f(admits,)g(for)f(top)r (ological)236 1069 y(reasons,)c(at)i(least)f Fv(n)h Fw(distinct)g (critical)g(p)r(oin)n(ts)f(giving)g(rise)g(to)h(p)r(erio)r(dic)g 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b(is)g(natural)f(to)h(try)g(to)g(extend)g(suc)n(h)g(t)n(yp)r(e)g(of)g (results)f(to)h(Hamiltonian)g(PDE's)f(as)h(the)g(w)n(a)n(v)n(e)e (equation,)236 1790 y(the)f(plate)f(equation,)h(the)g(nonlinear)e(Sc)n (hr\177)-42 b(odinger)35 b(equation,)k(etc...)65 b(.)g(The)37 b(main)g(di\016cult)n(y)g(in)h(carrying)d(o)n(v)n(er)236 1890 y(this)i(program)d(is)i(the)g(fact)g(that,)j(in)d(in\014nite)h (dimensional)f(systems,)i(\\small)d(denominator)g(problems")g(app)r (ear,)236 1989 y(whereas)f(in)g(\014nite)i(dimensional)e(systems)g (\\small)g(denominators")f(app)r(ear)g(only)i(in)f(the)h(construction)f (of)h(quasi-)236 2089 y(p)r(erio)r(dic)25 b(solutions.)35 b(This)26 b(is)f(simply)g(explained)f(as)h(follo)n(ws.)35 b(The)25 b(frequencies)g Fs(f)p Fv(!)2919 2101 y Fu(j)2953 2089 y Fs(g)2995 2101 y Fu(j)s Fp(2)p Fl(N)3157 2089 y Fw(of)g(the)h(linear)e(oscillations)236 2189 y(close)34 b(to)g(the)h(equilibrium)f(t)n(ypically)g(gro)n(w)f(as)g Fv(!)1861 2201 y Fu(j)1930 2189 y Fs(\031)h Fv(j)2068 2158 y Fu(d)2141 2189 y Fw(for)g(some)g Fv(d)g(>)g Fw(0)g(and)g(then)h (the)g(sequence)f Fs(f)p Fv(!)3701 2201 y Fu(i)3728 2189 y Fv(l)24 b Fs(\000)f Fv(!)3917 2201 y Fu(j)3951 2189 y Fs(g)p Fw(,)236 2288 y Fv(l)i Fs(2)e Fx(N)p Fv(;)14 b(j)28 b Fs(2)23 b Fx(N)p Fv(;)14 b(j)28 b Fs(6)p Fw(=)22 b Fv(i)p Fw(,)28 b(accum)n(ulates)e(to)i(0.)361 2388 y(Despite)35 b(this)g(di\016cult)n(y)-7 b(,)38 b(Kuksin)c([20)o(])h (and)g(W)-7 b(a)n(yne)35 b([31)o(],)i(w)n(ere)d(able)g(to)h(\014nd,)j (extending)c(in)i(a)e(suitable)h(w)n(a)n(y)236 2487 y(KAM)26 b(tec)n(hniques,)g(p)r(erio)r(dic)f(\(and)h(ev)n(en)f(quasi-p)r(erio)r (dic\))g(solutions)f(in)i(some)f(Hamiltonian)h(PDE's)e(in)i(one)f (spatial)236 2587 y(dimension)39 b(under)g(Diric)n(hlet)g(b)r(oundary)g (conditions.)71 b(As)39 b(usual)g(in)g(KAM-t)n(yp)r(e)g(results,)j(the) d(p)r(erio)r(ds)g(of)g(suc)n(h)236 2687 y(p)r(ersisten)n(t)23 b(solutions)g(satisfy)f(a)h(strong)f(irrationalit)n(y)f(condition,)j 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b(than)f(attempting)236 3384 y(to)31 b(mak)n(e)e(a)h(series)g(of)g (canonical)g(transformations)e(whic)n(h)j(bring)f(the)g(Hamiltonian)h (in)n(to)f(some)g(normal)f(form,)i(the)236 3484 y(solution)24 b(is)g(constructed)f(directly)-7 b(.)36 b(Making)23 b(the)h(ansatz)f (that)h(a)g(p)r(erio)r(dic)g(solution)f(exists)h(one)g(writes)f(this)h (solution)236 3583 y(as)33 b(a)f(F)-7 b(ourier)33 b(series)f(and)h (substitutes)g(that)h(series)e(in)n(to)h(the)g(partial)g(di\013eren)n (tial)g(equation.)52 b(In)34 b(this)f(w)n(a)n(y)f(one)h(is)236 3683 y(reduced)g(to)g(solv)n(e)f(t)n(w)n(o)h(equations:)47 b(the)34 b(so)e(called)h(\()p Fv(P)12 b Fw(\))34 b(equation,)g(whic)n (h)f(is)g(in\014nite)h(dimensional,)g(where)f(small)236 3783 y(denominators)25 b(app)r(ear,)h(and)g(the)h(\014nite)g (dimensional)e(\()p Fv(Q)p Fw(\))i(equation,)f(whic)n(h)g(corresp)r (onds)e(to)j(resonances.)34 b(Due)27 b(to)236 3882 y(the)33 b(presence)e(of)h(small)g(divisors)f(the)i(\()p Fv(P)12 b Fw(\))33 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4580 y(Con)n(traction)k(Mapping)i(Theorem.)35 b(Apart)27 b(its)g(simplicit)n(y)-7 b(,)27 b([6)o(])g(has)f(also)f(the) i(adv)-5 b(an)n(tage)26 b(of)g(a)n(v)n(oiding)f(some)h(further)236 4679 y(condition)k(presen)n(t)g(in)h([17)o(].)45 b(Other)30 b(related)f(results)h(on)g(the)h(existence)f(of)g(p)r(erio)r(dic)g (solutions)g(in)h(nonlinear)e(plate)236 4779 y(equations)e(in)h(higher) f(dimension)g(ha)n(v)n(e)g(b)r(een)h(giv)n(en)e(in)i([8].)361 4878 y(W)-7 b(e)26 b(remark)e(that)j(in)f(all)f(the)i(previous)e(cases) f(the)j(\()p Fv(Q)p Fw(\))f(equation)f(is)h(still)g(\014nite)h (dimensional)e(since)h(one)f(handles)236 4978 y(cases)i(with)h(only)f (a)g(\014nite)h(n)n(um)n(b)r(er)g(of)f(resonances)f(among)g(the)i (linear)f(frequencies.)361 5078 y(The)i(\014rst)g(results)f(on)h(the)h (existence)f(of)g(small)f(amplitude)i(p)r(erio)r(dic)f(solutions)f(for) h(some)g Fo(c)l(ompletely)j(r)l(esonant)236 5177 y Fw(PDE's)21 b(as)f(\(1.1\))h(ha)n(v)n(e)f(b)r(een)h(giv)n(en)g(in)g([23)o(],)i(for) 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y Ft(1)2945 1261 y Fw(\()p Fx(T)p Fw(\))p Fv(;)43 b(\021)s Fw(\()p Fs(\001)p Fw(\))28 b(o)r(dd)3436 1168 y Fq(o)3491 1261 y Fv(:)361 1462 y Fw(W)-7 b(e)28 b(next)g(consider)e(the)i(Lagrangian)d(action)i(functional)h(\011)22 b(:)i Fv(X)29 b Fs(!)23 b Fx(R)k Fw(de\014ned)h(b)n(y)1346 1698 y(\011\()p Fv(u)p Fw(\))23 b(:=)1656 1585 y Fq(Z)1739 1605 y Ft(2)p Fu(\031)1703 1774 y Ft(0)1831 1698 y Fv(dt)1918 1585 y Fq(Z)2001 1605 y Fu(\031)1965 1774 y Ft(0)2060 1698 y Fv(dx)2189 1642 y(!)2244 1612 y Ft(2)p 2189 1679 V 2214 1755 a Fw(2)2291 1698 y Fv(u)2339 1664 y Ft(2)2339 1718 y Fu(t)2394 1698 y Fs(\000)2487 1642 y Fw(1)p 2487 1679 42 4 v 2487 1755 a(2)2538 1698 y Fv(u)2586 1664 y Ft(2)2586 1718 y Fu(x)2646 1698 y Fs(\000)18 b Fv(F)12 b Fw(\()p Fv(u)p Fw(\))939 b(\(2.7\))236 1935 y(where)36 b Fv(F)12 b Fw(\()p Fs(\001)p Fw(\))37 b(is)f(the)h(primitiv)n(e)f(of)h Fv(f)45 b Fw(suc)n(h)36 b(that)g Fv(F)12 b Fw(\(0\))38 b(=)g(0,)g(i.e.)63 b Fv(F)12 b Fw(\()p Fv(u)p Fw(\))38 b(=)2757 1868 y Fq(R)2813 1888 y Fu(u)2796 1964 y Ft(0)2870 1935 y Fv(f)9 b Fw(\()p Fv(s)p Fw(\))36 b Fv(ds)p Fw(.)64 b(It)37 b(is)f(easy)f(to)i(see)f(that)236 2034 y(\011)23 b Fs(2)g Fv(C)467 2004 y Ft(1)505 2034 y Fw(\()p Fv(X)r(;)14 b Fx(R)p Fw(\))27 b(and)h(that)1016 2275 y Fv(D)r Fw(\011\()p Fv(u)p Fw(\)[)p Fv(h)p Fw(])22 b(=)1468 2162 y Fq(Z)1551 2182 y Ft(2)p Fu(\031)1514 2351 y Ft(0)1643 2275 y Fv(dt)1730 2162 y Fq(Z)1813 2182 y Fu(\031)1776 2351 y Ft(0)1872 2275 y Fv(dx)28 b(!)2045 2241 y Ft(2)2082 2275 y Fv(u)2130 2287 y Fu(t)2159 2275 y Fv(h)2207 2287 y Fu(t)2254 2275 y Fs(\000)18 b Fv(u)2385 2287 y Fu(x)2427 2275 y Fv(h)2475 2287 y Fu(x)2535 2275 y Fs(\000)g Fv(f)9 b Fw(\()p Fv(u)p Fw(\))p Fv(h;)96 b Fs(8)p Fv(h)23 b Fs(2)h Fv(X)r(:)608 b Fw(\(2.8\))236 2503 y(Critical)27 b(p)r(oin)n(ts)h(of)f(\011)h(are)e (w)n(eak)h(solutions)g(of)g(\(2.1\).)361 2626 y(In)h(order)e(to)i (\014nd)g(critical)f(p)r(oin)n(ts)h(of)g(\011)f(w)n(e)h(p)r(erform)f(a) g(Ly)n(apuno)n(v-Sc)n(hmidt)g(reduction.)37 b(The)28 b(space)f Fv(X)34 b Fw(can)27 b(b)r(e)236 2726 y(decomp)r(osed)g(as)g Fv(X)j Fw(=)22 b Fv(V)37 b Fs(\010)18 b Fv(W)40 b Fw(where)456 2927 y Fv(W)35 b Fw(:=)680 2835 y Fq(n)735 2927 y Fv(w)26 b Fs(2)d Fv(X)1001 2832 y Fq(\014)1001 2881 y(\014)1001 2931 y(\014)1056 2927 y Fw(\()p Fv(w)r(;)14 b(v)s Fw(\))1261 2942 y Fu(L)1307 2925 y Fd(2)1368 2927 y Fw(=)22 b(0)p Fv(;)42 b Fs(8)27 b Fv(v)g Fs(2)c Fv(V)1847 2835 y Fq(o)1925 2927 y Fw(=)2013 2835 y Fq(n)2143 2848 y(X)2082 3027 y Fu(l)p Fp(\025)p Ft(0)p Fu(;j)s Fp(\025)p Ft(1)2338 2927 y Fv(w)2397 2939 y Fu(lj)2467 2927 y Fw(cos\()p Fv(l)r(t)p Fw(\))14 b(sin)f Fv(j)5 b(x)24 b Fs(2)f Fv(X)3120 2832 y Fq(\014)3120 2881 y(\014)3120 2931 y(\014)3175 2927 y Fv(w)3234 2939 y Fu(j)s(j)3324 2927 y Fw(=)f(0)27 b Fs(8)p Fv(j)i Fs(\025)22 b Fw(1)3718 2835 y Fq(o)3773 2927 y Fv(:)236 3204 y Fw(In)28 b(fact,)g(if)f Fv(u)c Fw(=)761 3142 y Fq(P)862 3204 y Fv(u)910 3216 y Fu(lj)980 3204 y Fw(cos)o(\()p Fv(l)r(t)p Fw(\))14 b(sin\()p Fv(j)5 b(x)p Fw(\))24 b Fs(2)f Fv(X)7 b Fw(,)27 b(then)h(the)g Fv(H)2114 3174 y Ft(1)2178 3204 y Fw(norm)f(of)h(\005)2552 3216 y Fu(V)2610 3204 y Fw(\()p Fv(u)p Fw(\))23 b(:=)2855 3142 y Fq(P)2957 3204 y Fv(u)3005 3216 y Fu(j)s(j)3084 3204 y Fw(cos)o(\()p Fv(j)5 b(t)p Fw(\))14 b(sin)q(\()p Fv(j)5 b(x)p Fw(\))28 b(is)f(\014nite,)h(so)236 3304 y(that)k(\005)482 3316 y Fu(V)540 3304 y Fw(\()p Fv(u)p Fw(\))d Fs(2)g Fv(X)7 b Fw(.)47 b(As)31 b(a)g(consequence)f(\005)1644 3316 y Fu(W)1720 3304 y Fw(\()p Fv(u)p Fw(\))f(:=)g Fv(u)20 b Fs(\000)g Fw(\005)2193 3316 y Fu(V)2251 3304 y Fw(\()p Fv(u)p Fw(\))29 b Fs(2)h Fv(X)7 b Fw(.)47 b(Moreo)n(v)n(er)28 b(the)j(pro)5 b(jectors)30 b(\005)3594 3316 y Fu(V)3681 3304 y Fw(:)f Fv(X)35 b Fs(!)29 b Fv(V)236 3403 y Fw(and)i(\005)463 3415 y Fu(W)566 3403 y Fw(:)d Fv(X)34 b Fs(!)28 b Fv(W)42 b Fw(are)30 b(con)n(tin)n(uous.)44 b Fv(W)f Fw(is)30 b(also)g(the)g Fv(H)2147 3373 y Ft(1)2184 3403 y Fw(-orthogonal)e(of)j Fv(V)49 b Fw(in)31 b Fv(X)7 b Fw(.)45 b(Note)30 b(that)h(if)g Fv(v)g Fw(=)c(\005)3759 3415 y Fu(V)3817 3403 y Fv(u)j Fw(has)236 3503 y(minimal)e(p)r(erio)r(d)g(w.r.t)f(time)h(2)p Fv(\031)j Fw(then)d(also)e Fv(u)i Fw(has)f(minimal)g(p)r(erio)r(d)h (w.r.t.)37 b(time)28 b(2)p Fv(\031)s Fw(.)361 3626 y(Setting)23 b Fv(u)g Fw(:=)g Fv(v)13 b Fw(+)d Fv(w)25 b Fw(with)f Fv(v)i Fs(2)d Fv(V)43 b Fw(and)23 b Fv(w)i Fs(2)f Fv(W)12 b Fw(,)24 b(\(2.1\))f(is)g(equiv)-5 b(alen)n(t)23 b(to)g(the)h(follo)n (wing)e(system)h(of)g(t)n(w)n(o)g(equations)236 3726 y(\(called)28 b(resp.)36 b(the)28 b(\()p Fv(Q)p Fw(\))g(and)f(the)h(\() p Fv(P)12 b Fw(\))28 b(equations\))1436 3907 y(\()p Fv(Q)p Fw(\))185 b Fs(\000)18 b Fv(!)1889 3873 y Ft(2)1926 3907 y Fv(v)1966 3919 y Fu(tt)2039 3907 y Fw(+)g Fv(v)2162 3919 y Fu(xx)2265 3907 y Fw(=)k(\005)2414 3919 y Fu(V)2472 3907 y Fv(f)9 b Fw(\()p Fv(v)22 b Fw(+)c Fv(w)r Fw(\))p Fv(;)1408 4088 y Fw(\()p Fv(P)12 b Fw(\))185 b Fs(\000)18 b Fv(!)1860 4054 y Ft(2)1897 4088 y Fv(w)1956 4100 y Fu(tt)2030 4088 y Fw(+)g Fv(w)2172 4100 y Fu(xx)2274 4088 y Fw(=)23 b(\005)2424 4100 y Fu(W)2500 4088 y Fv(f)9 b Fw(\()p Fv(v)21 b Fw(+)d Fv(w)r Fw(\))p Fv(:)236 4237 y Fw(Note)27 b(that)g Fv(w)j Fw(solv)n(es)25 b(the)i(\()p Fv(P)12 b Fw(\))28 b(equation)e(\(in)h(a)g(w)n(eak)f(sense\))g(i\013)i (it)f(is)g(a)f(critical)g(p)r(oin)n(t)h(of)g(the)h(restricted)e (functional)236 4336 y Fv(w)g Fs(!)d Fw(\011\()p Fv(v)f Fw(+)c Fv(w)r Fw(\))24 b Fs(2)f Fx(R)p Fw(,)k Fo(i.e.)39 b Fw(if)28 b(and)f(only)h(if)766 4577 y Fv(D)r Fw(\011\()p Fv(v)22 b Fw(+)c Fv(w)r Fw(\)[)p Fv(h)p Fw(])24 b(=)1377 4464 y Fq(Z)1460 4484 y Ft(2)p Fu(\031)1423 4652 y Ft(0)1552 4577 y Fv(dt)1639 4464 y Fq(Z)1722 4484 y Fu(\031)1685 4652 y Ft(0)1781 4577 y Fv(dx)k(!)1954 4542 y Ft(2)1991 4577 y Fv(w)2050 4589 y Fu(t)2080 4577 y Fv(h)2128 4589 y Fu(t)2176 4577 y Fs(\000)18 b Fv(w)2318 4589 y Fu(x)2360 4577 y Fv(h)2408 4589 y Fu(x)2468 4577 y Fs(\000)g Fv(f)9 b Fw(\()p Fv(v)21 b Fw(+)d Fv(w)r Fw(\))p Fv(h)24 b Fw(=)f(0)p Fv(;)96 b Fs(8)p Fv(h)23 b Fs(2)h Fv(W)n(:)359 b Fw(\(2.9\))236 4804 y(W)-7 b(e)29 b(will)f(\014rst)g(solv)n(e)f(the)i(\()p Fv(P)12 b Fw(\))29 b(equation)e(in)i(lemma)f(2.3)f(through)h(the)h (standard)e(Con)n(traction)g(Mapping)g(Principle,)236 4904 y(assuming)j(that)h Fv(!)g Fs(2)e(W)35 b Fw(:=)28 b Fs([)1238 4916 y Fu(\015)t(>)p Ft(0)1366 4904 y Fv(W)1444 4916 y Fu(\015)1518 4904 y Fw(where)i Fv(W)1839 4916 y Fu(\015)1913 4904 y Fw(is)g(the)h(set)g(of)g(strongly)e(non-resonan)n (t)g(frequencies)h(in)n(tro)r(duced)236 5004 y(in)e([7])1382 5128 y Fv(W)1460 5140 y Fu(\015)1526 5128 y Fw(:=)1637 5036 y Fq(n)1693 5128 y Fv(!)d Fs(2)f Fx(R)1948 5033 y Fq(\014)1948 5083 y(\014)1948 5132 y(\014)2003 5128 y Fs(j)p Fv(!)s(l)c Fs(\000)e Fv(j)5 b Fs(j)23 b(\025)2392 5072 y Fv(\015)p 2392 5109 48 4 v 2403 5185 a(l)2449 5128 y(;)97 b Fs(8)p Fv(j)28 b Fs(6)p Fw(=)23 b Fv(l)2792 5036 y Fq(o)2847 5128 y Fv(:)934 b Fw(\(2.10\))236 5305 y(As)21 b(pro)n(v)n(ed)f(in)h(remark)f([2.4])g(of)h([7],)i(for)d Fv(\015)28 b(<)22 b Fw(1)p Fv(=)p Fw(3,)f(the)h(set)f Fv(W)2165 5317 y Fu(\015)2229 5305 y Fw(is)g(uncoun)n(table,)h(has)f (zero)f(measure)g(and)h(accum)n(ulates)236 5404 y(to)28 b Fv(!)d Fw(=)e(1)k(b)r(oth)h(from)g(the)f(left)i(and)e(from)g(the)h (righ)n(t.)37 b(It)28 b(is)f(also)g(easy)f(to)i(sho)n(w)e(that)i Fv(W)3074 5416 y Fu(\015)3140 5404 y Fw(=)23 b Fs(;)k Fw(for)g Fv(\015)h Fs(\025)22 b Fw(1.)361 5504 y(The)27 b(pro)r(ofs)g(of)h(the)g(next)g(lemmas)f(2.2)g(and)g(2.3)g(are)f(giv)n (en)h(in)h(the)g(App)r(endix.)2105 6059 y(6)p eop %%Page: 7 7 7 6 bop 236 272 a Fx(Lemma)32 b(2.2)40 b Fo(F)-6 b(or)30 b Fv(!)c Fs(2)d Fv(W)1127 284 y Fu(\015)1170 272 y Fo(,)31 b(let)e(us)g(de\014ne)1455 497 y Fv(")23 b Fw(=)1614 441 y Fv(!)1669 411 y Ft(2)1725 441 y Fs(\000)18 b Fw(1)p 1614 478 235 4 v 1711 554 a(2)1944 497 y(\()p Fv(")23 b Fs(\030)g Fv(!)e Fs(\000)d Fw(1)59 b(as)f Fv(!)26 b Fs(!)d Fw(1\))p Fv(:)236 699 y Fo(Then)32 b(the)f(op)l(er)l(ator)h Fv(L)978 711 y Fu(!)1051 699 y Fw(:=)25 b Fs(\000)p Fv(!)1284 669 y Ft(2)1321 699 y Fv(@)1365 711 y Fu(tt)1439 699 y Fw(+)18 b Fv(@)1566 711 y Fu(xx)1671 699 y Fw(:)26 b Fv(D)r Fw(\()p Fv(L)1880 711 y Fu(!)1928 699 y Fw(\))g Fs(\032)f Fv(W)37 b Fs(!)25 b Fv(W)43 b Fo(has)32 b(a)g(b)l(ounde)l(d)f (inverse)h Fv(L)3305 669 y Fp(\000)p Ft(1)3305 719 y Fu(!)3419 699 y Fw(:)25 b Fv(W)38 b Fs(!)25 b Fv(W)43 b Fo(which)236 799 y(writes)1308 921 y Fv(L)1365 886 y Fp(\000)p Ft(1)1365 941 y Fu(!)1454 921 y Fv(h)22 b Fw(:=)1758 842 y Fq(X)1635 1021 y Fu(l)p Fp(\025)p Ft(0)p Fu(;j)s Fp(\025)p Ft(1)p Fu(;j)s Fp(6)p Ft(=)p Fu(l)2170 865 y Fv(h)2218 877 y Fu(lj)p 2024 902 398 4 v 2024 978 a Fw(\()p Fv(!)2111 954 y Ft(2)2148 978 y Fv(l)2175 954 y Ft(2)2230 978 y Fs(\000)c Fv(j)2352 954 y Ft(2)2389 978 y Fw(\))2445 921 y(cos)o(\()p Fv(l)r(t)p Fw(\))c(sin\()p Fv(j)5 b(x)p Fw(\))861 b(\(2.11\))236 1158 y Fo(and)34 b(satis\014es,)g(for)g(a)g(p)l(ositive)g(c)l(onstant)f Fv(C)1650 1170 y Ft(3)1720 1158 y Fo(indep)l(endent)h(of)g Fv(\015)k Fo(and)c Fv(!)s Fo(,)g Fs(j)p Fv(L)2725 1127 y Fp(\000)p Ft(1)2725 1178 y Fu(!)2813 1158 y Fw(\005)2875 1170 y Fu(W)2951 1158 y Fv(u)p Fs(j)3022 1170 y Fu(!)3099 1158 y Fs(\024)29 b Fw(\()p Fv(C)3284 1170 y Ft(3)3322 1158 y Fv(=\015)5 b Fw(\))p Fs(j)p Fv(u)p Fs(j)3538 1170 y Fu(!)3618 1158 y Fo(for)34 b Fv(u)28 b Fs(2)i Fv(X)7 b Fo(.)236 1257 y(L)l(et)30 b Fv(L)437 1227 y Fp(\000)p Ft(1)548 1257 y Fw(:)23 b Fv(W)35 b Fs(!)23 b Fv(W)42 b Fo(b)l(e)30 b(the)g(inverse)g(op)l(er)l(ator)h(of)g Fs(\000)p Fv(@)1992 1269 y Fu(tt)2064 1257 y Fw(+)18 b Fv(@)2191 1269 y Fu(xx)2270 1257 y Fo(.)39 b(Ther)l(e)31 b(exists)e Fv(C)2859 1269 y Ft(4)2920 1257 y Fv(>)22 b Fw(0)30 b Fo(such)f(that)h Fs(8)p Fv(r)n(;)14 b(s)23 b Fs(2)h Fv(X)1076 1368 y Fq(\014)1076 1418 y(\014)1076 1468 y(\014)1118 1351 y(Z)1164 1539 y Ft(\012)1229 1464 y Fv(r)r Fw(\()p Fv(t;)14 b(x)p Fw(\))1446 1372 y Fq(\020)1497 1464 y Fv(L)1554 1430 y Fp(\000)p Ft(1)1554 1484 y Fu(!)1662 1464 y Fs(\000)k Fv(L)1802 1430 y Fp(\000)p Ft(1)1890 1372 y Fq(\021)1940 1464 y Fw(\(\005)2034 1476 y Fu(W)2110 1464 y Fv(s)p Fw(\()p Fv(t;)c(x)p Fw(\)\))31 b Fv(dt)f(dx)2583 1368 y Fq(\014)2583 1418 y(\014)2583 1468 y(\014)2634 1464 y Fs(\024)23 b Fv(C)2781 1476 y Ft(4)2833 1408 y Fv(")p 2828 1445 48 4 v 2828 1521 a(\015)2886 1464 y Fs(j)p Fv(r)r Fs(j)2971 1476 y Fu(!)3020 1464 y Fs(j)p Fv(s)p Fs(j)3105 1476 y Fu(!)3153 1464 y Fv(:)628 b Fw(\(2.12\))236 1675 y Fo(Mor)l(e)l(over,)32 b Fs(8)p Fv(r)n(;)14 b(s)23 b Fs(2)g Fv(X)1128 1717 y Fq(\014)1128 1766 y(\014)1128 1816 y(\014)1170 1699 y(Z)1216 1888 y Ft(\012)1281 1812 y Fv(r)r Fw(\()p Fv(t;)14 b(x)p Fw(\))p Fv(L)1555 1778 y Fp(\000)p Ft(1)1555 1833 y Fu(!)1646 1812 y Fw(\(\005)1740 1824 y Fu(W)1816 1812 y Fv(s)p Fw(\()p Fv(t;)g(x)p Fw(\)\))30 b Fv(dt)g(dx)2288 1717 y Fq(\014)2288 1766 y(\014)2288 1816 y(\014)2340 1812 y Fs(\024)22 b Fv(C)2486 1824 y Ft(4)2524 1720 y Fq(\020)2574 1812 y Fw(1)17 b(+)2731 1756 y Fv(")p 2726 1793 V 2726 1869 a(\015)2784 1720 y Fq(\021)2834 1812 y Fs(j)p Fv(r)r Fs(j)2919 1824 y Fu(!)2968 1812 y Fs(j)p Fv(s)p Fs(j)3053 1824 y Fu(!)3101 1812 y Fv(;)680 b Fw(\(2.13\))236 1995 y Fo(and)30 b(\014nal)t(ly,)h Fs(8)p Fv(s)24 b Fs(2)f Fv(X)7 b Fo(,)1534 2142 y Fs(j)p Fv(L)1614 2108 y Fp(\000)p Ft(1)1614 2163 y Fu(!)1703 2142 y Fw(\005)1765 2154 y Fu(W)1840 2142 y Fv(s)p Fs(j)1902 2157 y Fu(L)1948 2140 y Fd(2)2008 2142 y Fs(\024)22 b Fv(C)2154 2154 y Ft(4)2192 2050 y Fq(\020)2241 2142 y Fw(1)c(+)2394 2026 y Fs(p)p 2464 2026 39 4 v 2464 2086 a Fv(")p 2394 2123 108 4 v 2425 2199 a(\015)2512 2050 y Fq(\021)2562 2142 y Fs(j)p Fv(s)p Fs(j)2647 2154 y Fu(!)2695 2142 y Fv(:)1086 b Fw(\(2.14\))236 2360 y Fx(Lemma)32 b(2.3)40 b Fo(Ther)l(e)d(exists)e Fv(\032)e(>)g Fw(0)i Fo(such)h(that,)h Fs(8)p Fv(v)g Fs(2)d(D)2140 2372 y Fu(\032)2212 2360 y Fw(:=)f Fs(f)p Fv(v)j Fs(2)e Fv(V)55 b Fs(j)35 b(j)p Fv(v)s Fs(j)2790 2330 y Fu(p)p Fp(\000)p Ft(1)2790 2381 y Fu(!)2914 2360 y Fv(=\015)j(<)33 b(\032)p Fs(g)h Fo(ther)l(e)i(exists)f(a)h(unique)236 2460 y Fv(w)r Fw(\()p Fv(v)s Fw(\))e Fs(2)f Fv(W)47 b Fo(with)36 b Fs(j)p Fv(w)r Fw(\()p Fv(v)s Fw(\))p Fs(j)1051 2472 y Fu(!)1133 2460 y Fv(<)c Fs(j)p Fv(v)s Fs(j)1319 2472 y Fu(!)1402 2460 y Fo(solving)k(the)f Fw(\()p Fv(P)12 b Fw(\))36 b Fo(e)l(quation.)54 b(Mor)l(e)l(over,)38 b(for)e(some)g(p)l(ositive)g(c)l(onstant)e Fv(C)3823 2472 y Ft(5)3861 2460 y Fo(,)i(we)236 2560 y(have)361 2715 y Fs(\017)41 b Fv(i)p Fw(\))29 b Fs(j)p Fv(w)r Fw(\()p Fv(v)s Fw(\))p Fs(j)748 2727 y Fu(!)821 2715 y Fs(\024)23 b Fv(C)968 2727 y Ft(5)1005 2715 y Fs(j)p Fv(v)s Fs(j)1094 2684 y Fu(p)1094 2735 y(!)1143 2715 y Fv(=\015)5 b Fo(;)59 b Fs(j)p Fv(w)r Fw(\()p Fv(v)s Fw(\))p Fs(j)1531 2730 y Fu(L)1577 2713 y Fd(2)1637 2715 y Fs(\024)23 b Fv(C)1784 2727 y Ft(5)1822 2715 y Fw(\(1)18 b(+)1997 2655 y Fs(p)p 2066 2655 39 4 v 60 x Fv("=\015)5 b Fw(\))p Fs(j)p Fv(v)s Fs(j)2316 2684 y Fu(p)2316 2735 y(!)2364 2715 y Fo(;)361 2875 y Fs(\017)41 b Fv(ii)p Fw(\))59 b Fo(for)30 b(al)t(l)h Fv(r)26 b Fs(2)d Fv(X)7 b Fo(,)1229 2991 y Fq(\014)1229 3041 y(\014)1229 3090 y(\014)1271 2973 y(Z)1317 3162 y Ft(\012)1382 2994 y Fq(\020)1432 3086 y Fv(w)r Fw(\()p Fv(v)s Fw(\))20 b Fs(\000)e Fv(L)1760 3052 y Fp(\000)p Ft(1)1760 3107 y Fu(!)1849 3086 y Fw(\005)1911 3098 y Fu(W)1986 3086 y Fv(f)9 b Fw(\()p Fv(v)s Fw(\))2143 2994 y Fq(\021)2223 3086 y Fv(r)2262 2991 y Fq(\014)2262 3041 y(\014)2262 3090 y(\014)2314 3086 y Fs(\024)2411 3030 y Fv(C)2470 3042 y Ft(5)p 2411 3067 97 4 v 2436 3143 a Fv(\015)2518 2994 y Fq(\020)2567 3086 y Fw(1)18 b(+)2725 3030 y Fv(")p 2720 3067 48 4 v 2720 3143 a(\015)2778 2994 y Fq(\021)2827 3086 y Fs(j)p Fv(v)s Fs(j)2916 3052 y Ft(2)p Fu(p)p Fp(\000)p Ft(1)2916 3107 y Fu(!)3073 3086 y Fs(j)p Fv(r)r Fs(j)3158 3098 y Fu(!)3207 3086 y Fw(;)361 3332 y Fs(\017)41 b Fv(iii)p Fw(\))29 b Fv(w)r Fw(\()p Fs(\000)p Fv(v)s Fw(\)\()p Fv(t;)14 b(x)p Fw(\))25 b(=)d Fv(w)r Fw(\()p Fv(v)s Fw(\)\()p Fv(t)e Fw(+)e Fv(\031)s(;)c(\031) 22 b Fs(\000)c Fv(x)p Fw(\))p Fo(;)361 3493 y Fs(\017)41 b Fv(iv)s Fw(\))30 b Fv(v)c Fs(2)d Fv(V)770 3505 y Fu(n)869 3493 y Fs(\))52 b Fv(w)r Fw(\()p Fv(v)s Fw(\))32 b Fo(is)e Fw(2)p Fv(\031)s(=n)f Fo(p)l(erio)l(dic)i(w.r.t.)40 b Fv(t)p Fo(;)361 3653 y Fs(\017)h Fv(v)s Fw(\))60 b Fo(the)30 b(map)g Fv(v)c Fs(!)d Fv(w)r Fw(\()p Fv(v)s Fw(\))32 b Fo(is)e(in)g Fv(C)1527 3623 y Ft(1)1564 3653 y Fw(\()p Fv(V)5 b(;)14 b(W)e Fw(\))p Fo(.)361 3822 y Fw(In)24 b(order)f(to)g(solv)n(e)g(the)i(\()p Fv(Q)p Fw(\))f(equation)f(w)n(e)h (consider)f(the)h(reduced)g(Lagrangian)d(action)j(functional)g(\010) 3608 3834 y Fu(")3666 3822 y Fw(:)f Fs(D)3776 3834 y Fu(\032)3838 3822 y Fs(!)g Fx(R)236 3922 y Fw(de\014ned)28 b(b)n(y)343 4145 y(\010)403 4157 y Fu(")439 4145 y Fw(\()p Fv(v)s Fw(\))c(:=)e(\011\()p Fv(v)g Fw(+)c Fv(w)r Fw(\()p Fv(v)s Fw(\)\))25 b(=)1234 4032 y Fq(Z)1317 4053 y Ft(2)p Fu(\031)1280 4221 y Ft(0)1409 4145 y Fv(dt)1496 4032 y Fq(Z)1579 4053 y Fu(\031)1542 4221 y Ft(0)1638 4145 y Fv(dx)1766 4089 y(!)1821 4059 y Ft(2)p 1766 4126 92 4 v 1792 4202 a Fw(2)1868 4145 y(\()p Fv(v)1940 4157 y Fu(t)1989 4145 y Fw(+)18 b(\()p Fv(w)r Fw(\()p Fv(v)s Fw(\)\))2304 4157 y Fu(t)2335 4145 y Fw(\))2367 4111 y Ft(2)2423 4145 y Fs(\000)2516 4089 y Fw(1)p 2516 4126 42 4 v 2516 4202 a(2)2567 4145 y(\()p Fv(v)2639 4157 y Fu(x)2700 4145 y Fw(+)g(\()p Fv(w)r Fw(\()p Fv(v)s Fw(\)\))3015 4157 y Fu(x)3059 4145 y Fw(\))3091 4111 y Ft(2)3147 4145 y Fs(\000)g Fv(F)12 b Fw(\()p Fv(v)22 b Fw(+)c Fv(w)r Fw(\()p Fv(v)s Fw(\)\))p Fv(:)109 b Fw(\(2.15\))236 4360 y(By)28 b(\(2.15\))e(and)i(form)n(ula)e(\(2.9\))i(with)g Fv(h)23 b Fw(=)f Fv(w)r Fw(\()p Fv(v)s Fw(\))30 b(w)n(e)d(can)g(write)g (the)h(reduced)g(action)f(functional)g(as)928 4582 y(\010)988 4594 y Fu(")1023 4582 y Fw(\()p Fv(v)s Fw(\))d(=)1241 4469 y Fq(Z)1288 4658 y Ft(\012)1381 4582 y Fv(dt)k(dx)1610 4526 y(!)1665 4496 y Ft(2)p 1610 4563 92 4 v 1635 4639 a Fw(2)1712 4582 y Fv(v)1755 4548 y Ft(2)1752 4603 y Fu(t)1811 4582 y Fs(\000)1904 4526 y Fv(v)1947 4496 y Ft(2)1944 4547 y Fu(x)p 1904 4563 83 4 v 1924 4639 a Fw(2)2014 4582 y Fs(\000)18 b Fv(F)12 b Fw(\()p Fv(v)22 b Fw(+)c Fv(w)r Fw(\()p Fv(v)s Fw(\)\))i(+)2652 4526 y(1)p 2652 4563 42 4 v 2652 4639 a(2)2704 4582 y Fv(f)9 b Fw(\()p Fv(v)21 b Fw(+)d Fv(w)r Fw(\()p Fv(v)s Fw(\)\))p Fv(w)r Fw(\()p Fv(v)s Fw(\))p Fv(;)483 b Fw(\(2.16\))236 4803 y(Since)28 b Fs(j)p Fv(v)516 4815 y Fu(t)546 4803 y Fs(j)569 4773 y Ft(2)569 4830 y Fu(L)615 4813 y Fd(2)674 4803 y Fw(=)22 b Fs(j)p Fv(v)824 4815 y Fu(x)866 4803 y Fs(j)889 4773 y Ft(2)889 4830 y Fu(L)935 4813 y Fd(2)995 4803 y Fw(=)g Fs(jj)p Fv(v)s Fs(jj)1217 4773 y Ft(2)1217 4830 y Fu(H)1275 4813 y Fd(1)1313 4803 y Fv(=)p Fw(2,)27 b(w)n(e)g(ha)n(v)n(e)1008 5020 y(\010)1068 5032 y Fu(")1104 5020 y Fw(\()p Fv(v)s Fw(\))d(=)1334 4964 y Fv(")p 1332 5001 V 1332 5077 a Fw(2)1384 5020 y Fs(jj)p Fv(v)s Fs(jj)1519 4986 y Ft(2)1575 5020 y Fw(+)1658 4907 y Fq(Z)1704 5096 y Ft(\012)1797 5020 y Fv(dt)k(dx)2026 4964 y Fw(1)p 2026 5001 V 2026 5077 a(2)2078 5020 y Fv(f)9 b Fw(\()p Fv(v)21 b Fw(+)d Fv(w)r Fw(\()p Fv(v)s Fw(\)\))p Fv(w)r Fw(\()p Fv(v)s Fw(\))k Fs(\000)c Fv(F)12 b Fw(\()p Fv(v)22 b Fw(+)c Fv(w)r Fw(\()p Fv(v)s Fw(\)\))p Fv(:)562 b Fw(\(2.17\))236 5236 y(De\014ning)28 b(the)f(linear)f(op)r(erator)g Fs(I)j Fw(:)23 b Fv(X)29 b Fs(!)24 b Fv(X)33 b Fw(b)n(y)27 b(\()p Fs(I)6 b Fv(u)p Fw(\)\()p Fv(t;)14 b(x)p Fw(\))24 b(=)f Fv(u)p Fw(\()p Fv(t)17 b Fw(+)g Fv(\031)s(;)d(\031)21 b Fs(\000)16 b Fv(x)p Fw(\))28 b(it)g(results)e(that)i(\011)16 b Fs(\016)h(I)30 b Fw(=)22 b(\011.)37 b(Hence,)236 5335 y(b)n(y)28 b(lemma)f(2.3-)p Fv(iii)p Fw(\))f(and)h(since)g Fs(\000)p Fv(v)f Fw(=)d Fs(I)6 b Fv(v)s Fw(,)28 b(functional)g(\010) 2077 5347 y Fu(")2140 5335 y Fw(is)g(ev)n(en:)516 5504 y(\010)576 5516 y Fu(")611 5504 y Fw(\()p Fs(\000)p Fv(v)s Fw(\))c(=)e(\011\()p Fs(\000)p Fv(v)f Fw(+)d Fv(w)r Fw(\()p Fs(\000)p Fv(v)s Fw(\)\))25 b(=)d(\011\()p Fs(I)6 b Fw(\()p Fv(v)23 b Fw(+)18 b Fv(w)r Fw(\()p Fv(v)s Fw(\)\)\))25 b(=)d(\011\()p Fv(v)g Fw(+)c Fv(w)r Fw(\()p Fv(v)s Fw(\)\))25 b(=)d(\010)2861 5516 y Fu(")2897 5504 y Fw(\()p Fv(v)s Fw(\))p Fv(;)181 b Fs(8)p Fv(v)26 b Fs(2)d(D)3462 5516 y Fu(\032)3501 5504 y Fv(;)280 b Fw(\(2.18\))2105 6059 y(7)p eop %%Page: 8 8 8 7 bop 236 272 a Fw(By)28 b(lemma)f(2.3-)p Fv(v)s Fw(\))g(functional)h (\010)1321 284 y Fu(")1384 272 y Fw(is)f(in)h Fv(C)1629 242 y Ft(1)1667 272 y Fw(\()p Fv(V)5 b(;)14 b Fx(R)p Fw(\).)37 b(Arguing)27 b(as)g(in)g([1]-[2)o(]-[19)o(])h(w)n(e)f(\014nd) h(that,)g Fs(8)p Fv(h)23 b Fs(2)g Fv(V)236 495 y(D)r Fw(\010)367 507 y Fu(")403 495 y Fw(\()p Fv(v)s Fw(\)[)p Fv(h)p Fw(])h(=)e Fv(D)r Fw(\011\()p Fv(v)13 b Fw(+)d Fv(w)r Fw(\()p Fv(v)s Fw(\)\)[)p Fv(h)p Fw(])26 b(=)1418 382 y Fq(Z)1464 571 y Ft(\012)1530 495 y Fv(dt)e(dx)g(!)1796 461 y Ft(2)1833 495 y Fv(v)1873 507 y Fu(t)1902 495 y Fv(h)1950 507 y Fu(t)1990 495 y Fs(\000)10 b Fv(v)2105 507 y Fu(x)2147 495 y Fv(h)2195 507 y Fu(x)2247 495 y Fs(\000)g Fv(f)f Fw(\()p Fv(v)k Fw(+)d Fv(w)r Fw(\()p Fv(v)s Fw(\)\))p Fv(h)25 b Fw(=)e Fv(")p Fw(\()p Fv(v)s(;)14 b(h)p Fw(\))c Fs(\000)3209 382 y Fq(Z)3256 571 y Ft(\012)3321 495 y Fv(dt)24 b(dx)g(f)9 b Fw(\()p Fv(v)k Fw(+)d Fv(w)r Fw(\()p Fv(v)s Fw(\)\))p Fv(h:)3804 641 y Fw(\(2.19\))236 741 y(Hence)28 b(the)g(follo)n(wing)f(lemma)g(holds)236 921 y Fx(Lemma)32 b(2.4)40 b Fo(If)29 b Fv(v)d Fs(2)e Fv(V)47 b Fo(is)28 b(a)h(critic)l(al)g(p)l(oint)g(of)g(the)f(r)l(e)l (duc)l(e)l(d)h(action)g(functional)g Fw(\010)2931 933 y Fu(")2989 921 y Fw(:)23 b Fs(D)3099 933 y Fu(\032)3161 921 y Fs(!)g Fx(R)28 b Fo(then)g Fv(u)22 b Fw(=)h Fv(v)c Fw(+)14 b Fv(w)r Fw(\()p Fv(v)s Fw(\))236 1021 y Fo(is)30 b(a)g(critic)l(al)h(p)l(oint)g(of)f Fw(\010)1044 1033 y Fu(")1103 1021 y Fw(:)23 b Fv(X)29 b Fs(!)23 b Fx(R)p Fo(.)236 1201 y Fx(Remark)32 b(2.1)41 b Fo(On)30 b(the)h(other)h(hand)g (it)g(is)f(p)l(ossible)i(to)e(pr)l(ove)h(that)f(any)h(critic)l(al)h(p)l (oint)e Fv(u)g Fo(of)h Fw(\010)3329 1213 y Fu(")3364 1201 y Fo(,)g(su\016ciently)g(close)236 1301 y(to)e Fw(0)p Fo(,)g(c)l(an)f(b)l(e)h(written)g(as)g Fv(u)22 b Fw(=)h Fv(v)f Fw(+)c Fv(w)32 b Fo(with)e Fv(w)c Fw(=)d Fv(w)r Fw(\()p Fv(v)s Fw(\))h Fs(2)g Fv(W)n(;)14 b(v)26 b Fs(2)d Fv(V)49 b Fo(and)30 b Fv(v)j Fo(is)d(a)g(critic)l(al)h(p)l(oint)f(of)h Fw(\010)3493 1313 y Fu(")3528 1301 y Fo(.)236 1575 y Fr(3)135 b(An)44 b(abstract)i(result)236 1757 y Fw(Let)24 b Fv(E)29 b Fw(b)r(e)24 b(a)f(Hilb)r(ert)h(space)f(with)i(scalar)d(pro) r(duct)h(\()p Fs(\001)p Fv(;)14 b Fs(\001)p Fw(\))25 b(and)e(norm)g Fs(jj)11 b(\001)g(jj)p Fw(.)35 b(W)-7 b(e)24 b(will)g(denote)g Fv(B)j Fw(:=)c Fs(f)p Fv(v)j Fs(2)d Fv(E)29 b Fs(j)24 b(jj)p Fv(v)s Fs(jj)f(\024)g Fw(1)p Fs(g)236 1857 y Fw(the)28 b(closed)f(unit)h(ball)g(of)f Fv(E)33 b Fw(and)28 b Fv(S)f Fw(:=)c Fs(f)p Fv(v)j Fs(2)d Fv(E)33 b Fs(j)28 b(jj)p Fv(v)s Fs(jj)23 b Fw(=)g(1)p Fs(g)k Fw(the)h(unit)g(sphere.)361 1980 y(Let)d(us)h(consider)f(a)g Fv(C)1066 1950 y Ft(1)1129 1980 y Fw(functional)h(\010)d(:)g Fv(B)1709 1992 y Fu(r)1740 2000 y Fd(0)1799 1980 y Fs(\032)g Fv(E)28 b Fs(!)23 b Fx(R)p Fw(,)j(de\014ned)g(on)f(the)h(ball)f Fv(B)2963 1992 y Fu(r)2994 2000 y Fd(0)3054 1980 y Fw(:=)d Fs(f)p Fv(v)k Fs(2)e Fv(E)30 b Fs(j)c(jj)p Fv(v)s Fs(jj)49 b Fv(<)23 b(r)3800 1992 y Ft(0)3837 1980 y Fs(g)p Fw(,)j(b)n(y)1592 2182 y(\010\()p Fv(v)s Fw(\))e(=)1880 2126 y Fv(\026)p 1880 2163 51 4 v 1884 2239 a Fw(2)1940 2182 y Fs(jj)p Fv(v)s Fs(jj)2075 2148 y Ft(2)2131 2182 y Fs(\000)18 b Fv(G)p Fw(\()p Fv(v)s Fw(\))i(+)e Fv(R)q Fw(\()p Fv(v)s Fw(\))1185 b(\(3.1\))236 2390 y(where)27 b Fv(\026)c Fs(2)h Fw(\(0)p Fv(;)14 b Fw(+)p Fs(1)p Fw(\))27 b(and)361 2555 y Fs(\017)41 b Fw(\()p Fv(H)7 b Fw(1\))57 b Fv(G)24 b Fs(2)h Fv(C)917 2524 y Ft(1)954 2555 y Fw(\()p Fv(E)5 b(;)14 b Fx(R)p Fw(\))29 b(is)f(p)r(ositiv)n(ely)g(homogeneous)f(of)h (degree)f Fv(q)22 b Fw(+)d(1)28 b(with)h Fv(q)e(>)d Fw(1,)k(i.e.)40 b Fv(G)p Fw(\()p Fv(\025v)s Fw(\))26 b(=)e Fv(\025)3722 2524 y Fu(q)r Ft(+1)3843 2555 y Fv(G)p Fw(\()p Fv(v)s Fw(\))444 2654 y Fs(8)p Fv(\025)f Fs(2)h Fx(R)712 2666 y Ft(+)767 2654 y Fw(,)j(and)h Fv(D)r(G)23 b Fw(:)g Fv(V)42 b Fs(!)23 b Fv(V)1447 2624 y Fp(\003)1513 2654 y Fw(is)k(compact;)361 2819 y Fs(\017)41 b Fw(\()p Fv(H)7 b Fw(2\))55 b Fv(R)24 b Fs(2)f Fv(C)911 2789 y Ft(1)949 2819 y Fw(\()p Fv(B)1044 2831 y Fu(r)1075 2839 y Fd(0)1111 2819 y Fv(;)14 b Fx(R)p Fw(\))28 b(and)f Fv(D)r(R)d Fw(:)f Fv(B)1708 2831 y Fu(r)1739 2839 y Fd(0)1798 2819 y Fs(!)h Fv(V)1971 2789 y Fp(\003)2037 2819 y Fw(is)k(compact;)361 2985 y Fs(\017)41 b Fw(\()p Fv(H)7 b Fw(3\))55 b Fv(R)q Fw(\(0\))23 b(=)f(0)28 b(and)f Fs(9)p Fv(\013)d(>)e Fw(0)27 b(suc)n(h)h(that)g Fs(j)p Fv(D)r(R)q Fw(\()p Fv(v)s Fw(\)[)p Fv(v)s Fw(])p Fs(j)c(\024)e Fv(\013)28 b Fs(jj)p Fv(v)s Fs(jj)2543 2954 y Fu(q)r Ft(+1)2665 2985 y Fv(;)f Fs(8)p Fv(v)g Fs(2)c Fv(B)2969 2997 y Fu(r)3000 3005 y Fd(0)3036 2985 y Fv(:)236 3149 y Fw(By)32 b(the)h(compactness)e(of)h Fv(D)r(G)f Fw(:)g Fv(V)50 b Fs(!)30 b Fv(V)1602 3119 y Fp(\003)1641 3149 y Fw(,)j Fv(G)g Fw(maps)e(the)i(unit)g(sphere)f Fv(S)37 b Fw(in)n(to)32 b(a)f(b)r(ounded)i(subset)f(of)g Fx(R)g Fw(\(indeed)236 3262 y Fv(G)p Fw(\()p 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Fv(U)9 b Fw(\()p Fv(v)s Fw(\))24 b Fv(>)f Fw(0)p Fv(:)1300 b Fw(\(3.2\))236 4125 y(The)28 b(follo)n(wing)e(lemma,)i(quite)g(standard,)f(ensures)f (that)i Fv(U)37 b Fw(attains)27 b(its)h(maxim)n(um)f(on)g Fv(S)5 b Fw(.)236 4305 y Fx(Lemma)32 b(3.1)40 b Fv(K)809 4317 y Ft(0)869 4305 y Fw(:=)23 b Fs(f)p Fv(v)j Fs(2)d Fv(S)34 b Fs(j)c Fv(G)p Fw(\()p Fv(v)s Fw(\))24 b(=)f Fv(m)p Fs(g)f Fw(=)h Fs(f)p Fv(v)j Fs(2)d Fv(S)35 b Fs(j)29 b Fv(U)9 b Fw(\()p Fv(v)s Fw(\))24 b(=)f Fv(m)p Fs(g)29 b Fo(is)h(not)f(empty)h(and)h(c)l(omp)l(act.)236 4486 y Fc(Pr)n(oof)p Fw(.)78 b(Since)28 b Fv(D)r(G)p Fw(\()p Fv(B)t Fw(\))h(is)e(compact,)h Fs(8)p Fv(\016)e(>)c Fw(0)28 b(there)f(is)g(a)h(\014nite)g(dimensional)f(subspace)g Fv(F)3249 4498 y Fu(\016)3313 4486 y Fw(of)h Fv(E)33 b Fw(suc)n(h)27 b(that)1434 4666 y Fs(8)p Fv(v)g Fs(2)c Fv(B)60 b Fs(8)p Fv(h)23 b Fs(2)g Fv(F)2008 4632 y Fp(?)1996 4687 y Fu(\016)2119 4666 y Fs(j)p Fv(D)r(G)p Fw(\()p Fv(v)s Fw(\)[)p Fv(h)p Fw(])p Fs(j)i(\024)d Fv(\016)s Fs(jj)p 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b(can)f(deriv)n(e)g(from)g(this)h(prop)r(ert)n(y)f (that)h Fv(G)1794 5320 y Fp(j)p Fu(B)1905 5305 y Fw(is)g(con)n(tin)n (uous)f(for)g(the)h(w)n(eak)f(top)r(ology)f(in)i Fv(E)5 b Fw(.)56 b(Indeed)34 b Fv(G)3849 5320 y Fp(j)p Fu(B)3960 5305 y Fw(is)236 5404 y(the)j(uniform)f(limit,)j(as)c Fv(\016)40 b Fs(!)d Fw(0,)h(of)e Fv(G)24 b Fs(\016)g Fv(P)1661 5416 y Fu(\016)1734 5404 y Fw(and)36 b(eac)n(h)f(function)i Fv(G)24 b Fs(\016)g Fv(P)2641 5416 y Fu(\016)2713 5404 y Fw(is)36 b(con)n(tin)n(uous)f(in)i(the)f(w)n(eak)f(top)r(ology)236 5504 y(since)f Fv(P)499 5516 y Fu(\016)569 5504 y Fw(is)f(a)g(linear)g (compact)g(op)r(erator.)53 b(Since)34 b Fv(B)t Fw(,)h(the)f(unit)g (ball)f(of)h Fv(E)5 b Fw(,)35 b(is)f(compact)f(for)g(the)g(w)n(eak)g (top)r(ology)-7 b(,)2105 6059 y(8)p eop %%Page: 9 9 9 8 bop 236 272 a Fv(G)36 b Fw(attains)e(its)i(maxim)n(um)f(on)f Fv(B)40 b Fw(at)35 b(some)f(p)r(oin)n(t)i Fv(v)1954 284 y Ft(0)2027 272 y Fs(2)g Fv(B)t Fw(.)59 b(W)-7 b(e)36 b(ha)n(v)n(e)e Fv(G)p Fw(\()p Fv(v)2754 284 y Ft(0)2792 272 y Fw(\))i(=)f Fv(m)g(>)h Fw(0,)g(so)f(that)g Fv(v)3607 284 y Ft(0)3680 272 y Fs(6)p Fw(=)g(0,)i(and)236 372 y Fv(U)9 b Fw(\()p Fv(v)374 384 y Ft(0)412 372 y Fw(\))23 b(=)g Fv(m=)p Fs(jj)p Fv(v)756 384 y Ft(0)793 372 y Fs(jj)839 341 y Fu(q)r Ft(+1)983 372 y Fs(\025)f Fv(m)p Fw(.)37 b(Hence,)28 b(b)n(y)f(the)h(de\014nition)g(of)g Fv(m)p Fw(,)f Fv(U)9 b Fw(\()p Fv(v)2456 384 y Ft(0)2494 372 y Fw(\))23 b(=)g Fv(m)k Fw(and)h Fv(v)2939 384 y Ft(0)2999 372 y Fs(2)c Fv(S)5 b Fw(,)27 b Fo(i.e.)39 b Fv(v)3376 384 y Ft(0)3436 372 y Fs(2)24 b Fv(K)3586 384 y Ft(0)3622 372 y Fw(.)361 471 y(T)-7 b(o)26 b(pro)n(v)n(e)f(that)i Fv(K)955 483 y Ft(0)1018 471 y Fw(is)f(compact,)h(consider)e(a)h (sequence)g(\()p Fv(v)2260 483 y Fu(k)2302 471 y Fw(\))g(with)h Fv(v)2588 483 y Fu(k)2653 471 y Fs(2)c Fv(K)2802 483 y Ft(0)2865 471 y Fw(for)j(all)h Fv(k)s Fw(.)36 b(Since)27 b(\()p Fv(v)3499 483 y Fu(k)3540 471 y Fw(\))g(is)f(b)r(ounded,)236 571 y(w)n(e)i(ma)n(y)f(assume)h(that)g(\(up)h(to)f(a)g(subsequence\))g (\()p Fv(v)1906 583 y Fu(k)1947 571 y Fw(\))g(con)n(v)n(erges)e(to)i (some)f Fv(v)h Fs(2)c Fv(E)33 b Fw(for)28 b(the)g(w)n(eak)f(top)r (ology)-7 b(.)38 b(F)-7 b(or)27 b(all)236 671 y Fv(k)s Fw(,)k Fv(v)376 683 y Fu(k)443 671 y Fs(2)c Fv(B)34 b Fw(hence,)d(since)f Fv(B)k Fw(is)29 b(closed)g(for)h(the)g(w)n(eak)f (top)r(ology)-7 b(,)29 b Fv(v)h Fs(2)d Fv(B)t Fw(.)44 b(Moreo)n(v)n(er,)27 b(since)j Fv(G)3306 686 y Fp(j)p Fu(B)3413 671 y Fw(is)g(con)n(tin)n(uous)e(for)236 770 y(the)j(w)n(eak)f(top)r(ology)-7 b(,)30 b Fv(G)p Fw(\()p Fv(v)s Fw(\))f(=)f Fv(m)p Fw(.)46 b(By)30 b(the)h(same)f(argumen)n(t)g (as)g(ab)r(o)n(v)n(e,)g(w)n(e)g(get)g(that)h Fs(jj)p Fv(v)s Fs(jj)e Fw(=)e(1.)46 b(Hence)31 b(\()p Fv(v)3760 782 y Fu(k)3801 770 y Fw(\))d Fv(*)h(v)236 870 y Fw(and)f(\()p Fs(jj)p Fv(v)516 882 y Fu(k)557 870 y Fs(jj)p Fw(\))c Fs(!)f(jj)p Fv(v)s Fs(jj)28 b Fw(and)f(w)n(e)h(can)f(conclude)g(that)h (\()p Fv(v)1957 882 y Fu(k)1998 870 y Fw(\))c Fs(!)f Fv(v)31 b Fw(strongly)-7 b(.)p 2577 870 48 48 v 361 1019 a(The)26 b(main)f(result)h(of)g(this)g(section)f(-whic)n(h)h(will)g (allo)n(w)e(us)i(in)g(the)h(next)f(section)f(to)h(get)f(our)g(m)n (ultiplicit)n(y)i(results-)236 1119 y(is)h(the)g(follo)n(wing)e(Prop)r (osition.)236 1301 y Fx(Prop)s(osition)k(3.1)41 b Fo(Assume)24 b(that)g(the)h(functional)h Fw(\010)d Fs(2)g Fv(C)2130 1271 y Ft(1)2168 1301 y Fw(\()p Fv(E)5 b(;)14 b Fx(R)p Fw(\))24 b Fo(satis\014es)h Fw(\()p Fv(H)7 b Fw(1\))p Fo(,)26 b Fw(\()p Fv(H)7 b Fw(2\))p Fo(,)26 b Fw(\()p Fv(H)7 b Fw(3\))24 b Fo(and)i(\(3.2\).)38 b(Ther)l(e)236 1401 y(exists)30 b(a)g(smal)t(l)g(p)l(ositive)i(c)l(onstant)c Fv(C)1445 1413 y Ft(0)1483 1401 y Fo(,)i(dep)l(ending)h(only)g(on)f Fv(q)s Fo(,)g(such)g(that,)g(if)1532 1576 y Fv(\013)p 1523 1613 73 4 v 1523 1690 a(m)1628 1633 y Fs(\024)23 b Fv(C)1775 1645 y Ft(0)1898 1633 y Fw(and)2116 1540 y Fq(\020)2187 1576 y Fv(\026)p 2176 1613 V 2176 1690 a(m)2259 1540 y Fq(\021)2355 1535 y Fd(1)p 2318 1544 103 4 v 2318 1578 a Fe(q)q Ff(\000)p Fd(1)2457 1633 y Fs(\024)g Fv(C)2604 1645 y Ft(0)2642 1633 y Fv(r)2679 1645 y Ft(0)2716 1633 y Fv(;)1106 b Fw(\(3.3\))236 1839 y Fo(functional)31 b Fw(\010)e Fo(has)i(a)f(critic)l(al)h(p)l(oint)f Fv(v)c Fs(2)e Fv(B)1634 1851 y Fu(r)1665 1859 y Fd(0)1731 1839 y Fo(on)30 b(a)g(critic)l(al)h(level)1381 2081 y Fv(c)23 b Fw(=)1537 2025 y Fv(q)f Fs(\000)c Fw(1)p 1537 2062 184 4 v 1608 2138 a(2)1730 2081 y Fv(m)1803 1989 y Fq(\020)1998 2025 y Fv(\026)p 1863 2062 321 4 v 1863 2138 a Fw(\()p Fv(q)k Fw(+)c(1\))p Fv(m)2193 1989 y Fq(\021)2254 1978 y Fe(q)q Fd(+1)p 2253 1993 103 4 v 2253 2026 a Fe(q)q Ff(\000)p Fd(1)2369 1989 y Fq(h)2408 2081 y Fw(1)g(+)g Fv(O)2616 1989 y Fq(\020)2686 2025 y Fv(\013)p 2676 2062 73 4 v 2676 2138 a(m)2759 1989 y Fq(\021i)2848 2081 y Fv(:)974 b Fw(\(3.4\))236 2307 y Fo(Mor)l(e)l(over)1125 2439 y Fv(v)26 b Fw(=)1279 2347 y Fq(\020)1474 2383 y Fv(\026)p 1339 2420 321 4 v 1339 2496 a(m)p Fw(\()p Fv(q)21 b 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3019 y(c)l(ontinuing)d(them)h(thr)l(ough)g(the)f(Implicit)i(F)-6 b(unction)36 b(The)l(or)l(em)h(as)g(critic)l(al)h(p)l(oints)f(of)g Fw(\010)p Fo(,)i(we)d(\014nd)h(critic)l(al)g(p)l(oints)236 3118 y(of)c Fw(\010)f Fo(showing)i(that)e(the)g(nonline)l(ar)h(p)l (erturb)l(ation)f(term)g Fv(R)h Fo(do)l(es)g(not)f(a\013e)l(ct)g(the)g (mountain)g(p)l(ass)h(ge)l(ometry)f(of)i(the)236 3218 y(functional)f Fw(\()p Fv(\026=)p Fw(2\))p Fs(jj)p Fv(v)s Fs(jj)962 3188 y Ft(2)1019 3218 y Fs(\000)20 b Fv(G)p Fw(\()p Fv(v)s Fw(\))p Fo(.)46 b(A)l(ctual)t(ly,)33 b(in)f([7)q(],)h (non)f(de)l(gener)l(ate)g(critic)l(al)i(p)l(oints)e(of)g Fv(G)h Fo(c)l(onstr)l(aine)l(d)f(on)g Fv(S)k Fo(ar)l(e)236 3318 y(c)l(ontinue)l(d.)51 b(This)34 b(is)h(e)l(quivalent)f(to)f(what)i (said)g(b)l(efor)l(e)f(sinc)l(e)g(any)g(critic)l(al)h(p)l(oint)g Fq(e)-47 b Fv(v)37 b Fo(of)e Fv(G)f Fo(c)l(onstr)l(aine)l(d)g(to)f Fv(S)39 b Fo(gives)236 3417 y(rise,)31 b(by)f(homo)l(geneity,)i(to)e(a) g(critic)l(al)h(p)l(oint)f Fw(\()p 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b(and)f(\(3.8\))h(stem)g(from)f(\(3.3\))g(if)i Fv(C)1538 5064 y Ft(0)1605 5052 y Fw(is)e(c)n(hosen)g(appropriately)-7 b(.)42 b(W)-7 b(e)30 b(shall)f(pro)n(v)n(e)f(that,)j(for)e Fv(\021)3448 5064 y Ft(1)3515 5052 y Fw(small)g(enough,)236 5152 y Fv(')f Fw(has)f(a)g(critical)g(p)r(oin)n(t)h Fv(y)j Fw(on)c(a)g(critical)g(lev)n(el)g Fv(d)h Fw(suc)n(h)f(that)1512 5369 y Fv(d)c Fw(=)1666 5277 y Fq(\020)1725 5313 y Fw(1)p 1725 5350 42 4 v 1725 5426 a(2)1795 5369 y Fs(\000)1959 5313 y Fw(1)p 1888 5350 184 4 v 1888 5426 a Fv(q)e Fw(+)e(1)2081 5277 y Fq(\021)2131 5369 y Fw(\(1)f(+)g Fv(O)r Fw(\()p Fv(\021)s Fw(\()p Fv(\013;)c(m)p Fw(\)\)\))1107 b(\(3.9\))2105 6059 y(9)p eop %%Page: 10 10 10 9 bop 236 272 a Fw(and)1757 372 y(dist\()p Fv(y)s(;)14 b(K)2075 384 y Ft(0)2112 372 y Fw(\))23 b Fs(\024)g Fv(h)2303 279 y Fq(\020)2372 315 y Fv(\013)p 2362 353 73 4 v 2362 429 a(m)2445 279 y Fq(\021)3804 372 y Fw(\(3.10\))236 542 y(for)33 b(some)g(function)h Fv(h)f Fw(suc)n(h)g(that)h(lim)1489 554 y Fu(s)p Fp(!)p Ft(0)1638 542 y Fv(h)p Fw(\()p Fv(s)p Fw(\))f(=)f(0.)54 b(\(3.9\))33 b(and)g(\(3.10\))g(imply)g(resp.)54 b(\(3.4\))33 b(and)g(\(3.5\))g(since,)i(b)n(y)236 690 y(the)h(de\014nition)g(of)g Fv(')p Fw(,)i Fv(v)h Fw(:=)1186 598 y Fq(\020)1380 634 y Fv(\026)p 1245 671 321 4 v 1245 747 a Fw(\()p Fv(q)22 b Fw(+)c(1\))p Fv(m)1576 598 y Fq(\021)1625 615 y Ft(1)p Fu(=q)r Fp(\000)p Ft(1)1814 690 y Fv(y)38 b Fw(will)e(then)g(b)r(e)g(a)f(critical)g(p)r(oin)n(t)h (of)g(\010)f(at)h(lev)n(el)f Fv(c)h Fw(=)g Fv(d)p Fw(\()p Fv(q)28 b Fw(+)23 b(1\))p Fv(m)236 802 y Fq(\020)431 838 y Fv(\026)p 296 875 V 296 952 a Fw(\()p Fv(q)f Fw(+)c(1\))p Fv(m)626 802 y Fq(\021)676 820 y Fu(q)r Ft(+1)p Fu(=q)r Fp(\000)p Ft(1)948 895 y Fv(:)28 b Fw(W)-7 b(e)28 b(can)f(write)1528 1176 y Fv(')p Fw(\()p Fv(y)s Fw(\))c(=)1811 1120 y Fs(jj)p Fv(y)s Fs(jj)1947 1090 y Ft(2)p 1811 1157 174 4 v 1877 1233 a Fw(2)2012 1176 y Fs(\000)2179 1120 y Fv(G)p Fw(\()p Fv(y)s Fw(\))p 2105 1157 321 4 v 2105 1233 a(\()p Fv(q)f Fw(+)c(1\))p Fv(m)2454 1176 y Fw(+)g Fv(S)5 b Fw(\()p Fv(y)s Fw(\))p Fv(;)1080 b Fw(\(3.11\))236 1392 y(where,)27 b(b)n(y)h(the)g(prop)r(erties)e(of)i Fv(R)q Fw(,)1195 1564 y Fs(8)p Fv(y)e Fs(2)d Fv(B)1449 1576 y Fu(\016)1486 1564 y Fv(;)69 b Fs(j)p Fv(D)r(S)5 b Fw(\()p Fv(y)s Fw(\)[)p Fv(y)s Fw(])p Fs(j)23 b(\024)f Fv(\021)s Fw(\()p Fv(\013;)14 b(m)p Fw(\))p Fs(jj)p Fv(y)s Fs(jj)2466 1530 y Fu(q)r Ft(+1)2611 1564 y Fs(\024)23 b Fv(\021)2740 1576 y Ft(1)2777 1564 y Fs(jj)p Fv(y)s Fs(jj)2913 1530 y Fu(q)r Ft(+1)3034 1564 y Fv(:)747 b Fw(\(3.12\))236 1749 y(As)28 b(a)f(consequence,)g (since)g Fv(S)5 b Fw(\()p Fv(y)s Fw(\))23 b(=)1398 1682 y Fq(R)1453 1703 y Ft(1)1437 1779 y(0)1504 1749 y Fv(dsD)r(S)5 b Fw(\()p Fv(sy)s Fw(\)[)p Fv(y)s Fw(],)1367 1978 y Fs(j)p Fv(S)g Fw(\()p Fv(y)s Fw(\))p Fs(j)23 b(\024)1698 1921 y Fv(\021)s Fw(\()p Fv(\013;)14 b(m)p Fw(\))p 1698 1958 272 4 v 1742 2035 a Fv(q)22 b Fw(+)c(1)2007 1978 y Fs(jj)p Fv(y)s Fs(jj)2143 1943 y Fu(q)r Ft(+1)2287 1978 y Fs(\024)2437 1921 y Fv(\021)2478 1933 y Ft(1)p 2385 1958 184 4 v 2385 2035 a Fv(q)j Fw(+)d(1)2605 1978 y Fs(jj)p Fv(y)s Fs(jj)2741 1943 y Fu(q)r Ft(+1)2862 1978 y Fv(:)919 b Fw(\(3.13\))361 2189 y(W)-7 b(e)28 b(need)f(the)h(follo)n(wing)f(tec)n(hnical)g(lemma:) 236 2361 y Fx(Lemma)32 b(3.2)40 b Fo(L)l(et)30 b Fv(W)41 b Fo(b)l(e)30 b(some)h(\(smal)t(l\))f(neighb)l(orho)l(o)l(d)j(of)d Fw(0)g Fo(in)f Fx(R)p Fo(.)39 b(Ther)l(e)31 b(ar)l(e)f(c)l(onstants)f Fv(C)3308 2373 y Ft(6)3346 2361 y Fv(;)14 b(C)3442 2373 y Ft(7)3502 2361 y Fv(>)23 b Fw(0)29 b Fo(such)h(that,)236 2460 y Fs(8)p Fv(a)23 b Fs(2)h Fw([1)p Fv(=)p Fw(2)p Fv(;)14 b Fw(1])p Fo(,)24 b Fs(8)h Fw(0)e Fv(<)f(\030)28 b Fs(\024)22 b Fv(C)1161 2472 y Ft(6)1199 2460 y Fv(;)14 b Fs(8)25 b Fw(0)d Fv(<)h(\021)j Fs(\024)d Fv(C)1673 2472 y Ft(6)1710 2460 y Fo(,)k(for)f(al)t(l)g Fv(C)2069 2430 y Ft(1)2131 2460 y Fo(function)f Fv(r)h Fw(:)d([0)p Fv(;)14 b(\016)s Fw(])23 b Fs(!)g Fx(R)h Fo(with)i Fw(sup)3253 2480 y Fu(t)p Fp(2)p Ft([0)p Fu(;\016)r Ft(])3463 2460 y Fs(j)p Fv(r)r Fw(\()p Fv(t)p Fw(\))p Fs(j)8 b Fw(+)g Fs(j)p Fv(r)3785 2430 y Fp(0)3810 2460 y Fw(\()p Fv(t)p Fw(\))p Fs(j)24 b(\024)236 2571 y Fv(\030)t Fo(,)31 b(the)e(map)p Fv(f)662 2583 y Fu(a;r)778 2571 y Fw(:)24 b([0)p Fv(;)14 b(\016)s Fw(])22 b Fs(!)h Fx(R)30 b Fo(de\014ne)l(d)g(by)g Fv(f)1649 2583 y Fu(a;r)1741 2571 y Fw(\()p Fv(t)p Fw(\))24 b(:=)e(\()p Fv(t)2031 2541 y Ft(2)2069 2571 y Fv(=)p Fw(2\))c Fs(\000)g Fv(a)p Fw(\()p Fv(t)2392 2541 y Fu(q)r Ft(+1)2512 2571 y Fv(=q)j Fw(+)d(1\))h(+)f Fv(r)r Fw(\()p Fv(t)p Fw(\))p Fo(,)31 b(satis\014es)1088 2753 y Fv(t)23 b Fs(2)h Fw([0)p Fv(;)14 b(\016)s Fw(])p Fs(n)p Fv(W)41 b(;)i Fs(j)p Fv(f)1685 2719 y Fp(0)1676 2774 y Fu(a;r)1768 2753 y Fw(\()p Fv(t)p Fw(\))p Fs(j)24 b(\024)f Fv(\021)j Fw(=)-14 b Fs(\))23 b(j)p Fv(t)18 b Fs(\000)g Fv(a)2419 2719 y Fp(\000)p Ft(1)p Fu(=q)r Fp(\000)p Ft(1)2660 2753 y Fs(j)23 b(\024)g Fv(C)2853 2765 y Ft(7)2890 2753 y Fw(\()p Fv(\030)g Fw(+)18 b Fv(\021)s Fw(\))p Fv(;)641 b Fw(\(3.14\))236 2925 y Fo(and)1255 3052 y Fw(inf)992 3109 y Fu(t)p Fp(2)p Ft([0)p Fu(;\016)r Ft(])p Fp(n)p Fu(W)n(;)p Fp(j)p Fu(f)1358 3092 y Ff(0)1351 3126 y Fe(a;r)1435 3109 y Ft(\()p Fu(t)p Ft(\))p Fp(j\024)p Fu(\021)1634 3052 y Fv(f)1675 3064 y Fu(a;r)1767 3052 y Fw(\()p Fv(t)p Fw(\))24 b Fs(\025)1972 2959 y Fq(\020)2032 2995 y Fw(1)p 2032 3033 42 4 v 2032 3109 a(2)2102 3052 y Fs(\000)2265 2995 y Fw(1)p 2195 3033 184 4 v 2195 3109 a Fv(q)d Fw(+)d(1)2388 2959 y Fq(\021)o(\020)2498 2995 y Fw(1)p 2497 3033 44 4 v 2497 3109 a Fv(a)2551 2959 y Fq(\021)2600 2977 y Ft(2)p Fu(=q)r Fp(\000)p Ft(1)2807 3052 y Fs(\000)g Fv(C)2949 3064 y Ft(7)2987 3052 y Fw(\()p Fv(\030)23 b Fw(+)18 b Fv(\021)s Fw(\))p Fv(:)544 b Fw(\(3.15\))236 3309 y Fc(Pr)n(oof)p Fw(.)76 b(Consider)19 b(\014rst)h(the)h(function)f Fv(f)1592 3321 y Fu(a)1632 3309 y Fw(\()p Fv(t)p Fw(\))k(:=)1870 3276 y Fu(t)1895 3251 y Fd(2)p 1870 3290 58 4 v 1883 3337 a Ft(2)1941 3309 y Fs(\000)t Fv(a)2064 3276 y Fu(t)2089 3251 y Fe(q)q Fd(+1)p 2063 3290 129 4 v 2069 3337 a Fu(q)r Ft(+1)2202 3309 y Fw(.)35 b Fv(f)2310 3278 y Fp(0)2301 3329 y Fu(a)2361 3309 y Fw(p)r(ossesses)19 b(outside)h Fv(W)32 b Fw(a)19 b(unique)i(zero)e Fv(t)3620 3321 y Fu(a)3683 3309 y Fw(:=)k Fv(a)3838 3272 y Fp(\000)3936 3249 y Fd(1)p 3900 3258 103 4 v 3900 3292 a Fe(q)q Ff(\000)p Fd(1)236 3408 y Fw(at)28 b(the)g(lev)n(el)1574 3541 y Fv(f)1615 3553 y Fu(a)1655 3541 y Fw(\()p Fv(t)1717 3553 y Fu(a)1758 3541 y Fw(\))23 b(=)1900 3448 y Fq(\020)1960 3484 y Fw(1)p 1960 3521 42 4 v 1960 3597 a(2)2030 3541 y Fs(\000)2194 3484 y Fw(1)p 2123 3521 184 4 v 2123 3597 a Fv(q)e Fw(+)d(1)2316 3448 y Fq(\021\020)2426 3484 y Fw(1)p 2425 3521 44 4 v 2425 3597 a Fv(a)2479 3448 y Fq(\021)2575 3443 y Fd(2)p 2538 3452 103 4 v 2538 3486 a Fe(q)q Ff(\000)p Fd(1)2655 3541 y Fv(:)236 3728 y(f)286 3697 y Fp(00)277 3748 y Fu(a)328 3728 y Fw(\()p Fv(t)390 3740 y Fu(a)431 3728 y Fw(\))23 b(=)g(1)18 b Fs(\000)g Fv(q)26 b(<)c Fw(0,)28 b(and)f(there)g(are)g Fv(\027)h(>)23 b Fw(0,)p 1721 3682 47 4 v 27 w Fv(\027)29 b(>)22 b Fw(0)27 b(suc)n(h)h(that,)g Fs(8)p Fv(a)23 b Fs(2)g Fw([1)p Fv(=)p Fw(2)p Fv(;)14 b Fw(1],)450 3899 y Fs(8)p Fv(t)23 b Fs(2)h Fw([)p Fv(t)681 3911 y Fu(a)739 3899 y Fs(\000)18 b Fv(\027)q(;)c(t)931 3911 y Fu(a)990 3899 y Fw(+)k Fv(\027)5 b Fw(])55 b Fv(f)1247 3865 y Fp(00)1238 3920 y Fu(a)1289 3899 y Fw(\()p Fv(t)p Fw(\))24 b Fs(\024)e(\000)p Fv(\027)5 b(=)p Fw(2)82 b(and)h Fs(8)p Fv(t)23 b Fs(2)h Fw([0)p Fv(;)14 b(\016)s Fw(])p Fs(n)p Fw(\()p Fv(W)30 b Fs([)18 b Fw([)p Fv(t)2639 3911 y Fu(a)2698 3899 y Fs(\000)g Fv(\027)q(;)c(t)2890 3911 y Fu(a)2948 3899 y Fw(+)k Fv(\027)5 b Fw(]\))56 b Fs(j)p Fv(f)3261 3865 y Fp(0)3252 3920 y Fu(a)3292 3899 y Fw(\()p Fv(t)p Fw(\))p Fs(j)24 b(\025)p 3520 3854 V 22 w Fv(\027)6 b(:)214 b Fw(\(3.16\))236 4071 y(Cho)r(ose)29 b Fv(C)586 4083 y Ft(6)650 4071 y Fv(>)d Fw(0)k(suc)n(h)f(that)h(2)p Fv(C)1285 4083 y Ft(6)1349 4071 y Fv(<)p 1440 4025 V 26 w(\027)5 b Fw(.)43 b(If)30 b(sup)1762 4091 y Fu(t)p Fp(2)p Ft([0)p Fu(;\016)r Ft(])1973 4071 y Fs(j)p Fv(r)r Fw(\()p Fv(t)p Fw(\))p Fs(j)21 b Fw(+)e Fs(j)p Fv(r)2319 4041 y Fp(0)2344 4071 y Fw(\()p Fv(t)p Fw(\))p Fs(j)27 b(\024)f Fv(\030)31 b Fs(\024)26 b Fv(C)2796 4083 y Ft(6)2863 4071 y Fw(and)k Fs(j)p Fv(f)3100 4041 y Fp(0)3091 4092 y Fu(a;r)3183 4071 y Fw(\()p Fv(t)p Fw(\))p Fs(j)d(\024)f Fv(\021)k Fs(\024)c Fv(C)3639 4083 y Ft(6)3706 4071 y Fw(for)j(some)236 4171 y Fv(t)41 b(=)-52 b Fs(2)32 b Fv(W)12 b Fw(,)34 b(then)f Fs(j)p Fv(f)798 4141 y Fp(0)789 4191 y Fu(a)829 4171 y Fw(\()p Fv(t)p Fw(\))p Fs(j)f(\024)e Fv(\030)c Fw(+)21 b Fv(\021)35 b(<)p 1393 4125 V 31 w(\027)j Fw(and)32 b(therefore,)h(b)n(y)f(the)h(second)f(inequalit)n(y)g(in)h (\(3.16\),)g Fv(t)e Fs(2)g Fw([)p Fv(t)3518 4183 y Fu(a)3580 4171 y Fs(\000)22 b Fv(\027)q(;)14 b(t)3776 4183 y Fu(a)3837 4171 y Fw(+)21 b Fv(\027)5 b Fw(].)236 4270 y(Moreo)n(v)n(er)26 b(b)n(y)j(\(3)p Fv(:)p Fw(16\))f(again,)h Fs(j)p Fv(f)1280 4240 y Fp(0)1271 4291 y Fu(a)1311 4270 y Fw(\()p Fv(t)p Fw(\))p Fs(j)d(\025)f Fw(\()p Fv(\027)5 b(=)p Fw(2\))p Fs(j)p Fv(t)19 b Fs(\000)g Fv(t)1924 4282 y Fu(a)1964 4270 y Fs(j)29 b Fw(for)g Fv(t)c Fs(2)h Fw([)p Fv(t)2334 4282 y Fu(a)2393 4270 y Fs(\000)19 b Fv(\027)q(;)14 b(t)2586 4282 y Fu(a)2645 4270 y Fw(+)19 b Fv(\027)5 b Fw(],)30 b(and)f(hence)g Fs(j)p Fv(t)20 b Fs(\000)e Fv(t)3432 4282 y Fu(a)3473 4270 y Fs(j)25 b(\024)g Fw(2\()p Fv(\030)e Fw(+)c Fv(\021)s Fw(\))p Fv(=\027)5 b Fw(,)236 4370 y(whic)n(h)40 b(pro)n(v)n(es)d(estimate)j(\(3)p Fv(:)p Fw(14\).)72 b(Finally)-7 b(,)42 b(b)n(y)d(the)h(T)-7 b(a)n(ylor)38 b(form)n(ula)h Fv(f)2633 4382 y Fu(a)2672 4370 y Fw(\()p Fv(t)p Fw(\))44 b(=)e Fv(f)2958 4382 y Fu(a)2998 4370 y Fw(\()p Fv(t)3060 4382 y Fu(a)3101 4370 y Fw(\))26 b(+)g Fv(O)r Fw(\(\()p Fv(\030)32 b Fw(+)26 b Fv(\021)s Fw(\))3614 4340 y Ft(2)3652 4370 y Fw(\),)43 b(so)c(that)236 4470 y Fv(f)277 4482 y Fu(a;r)369 4470 y Fw(\()p Fv(t)p Fw(\))24 b(=)f Fv(f)616 4482 y Fu(a)655 4470 y Fw(\()p Fv(t)717 4482 y Fu(a)758 4470 y Fw(\))c(+)f Fv(O)r Fw(\()p Fv(\030)t Fw(\))h(+)f Fv(O)r Fw(\(\()p Fv(\030)24 b Fw(+)18 b Fv(\021)s Fw(\))1511 4440 y Ft(2)1549 4470 y Fw(\))28 b(and)f(estimate)h(\(3.15\))f(follo)n(ws)f(straigh)n(tforw)n(ardly)-7 b(.)p 3292 4470 48 48 v 361 4615 a(Let)28 b(us)f(de\014ne)h(the)g(op)r (en)g(set)1529 4735 y Fv(A)23 b Fw(:=)1725 4643 y Fq(n)1780 4735 y Fv(v)j Fs(2)e Fv(E)5 b Fs(nf)p Fw(0)p Fs(g)2212 4639 y Fq(\014)2212 4689 y(\014)2212 4739 y(\014)2268 4735 y Fv(U)k Fw(\()p Fv(v)s Fw(\))23 b Fv(>)2562 4679 y(m)p 2562 4716 73 4 v 2578 4792 a Fw(2)2645 4643 y Fq(o)2700 4735 y Fv(;)236 4901 y Fw(the)28 b(min-max)f(class)g(of)h(path)915 5093 y(\000)23 b(=)1078 5001 y Fq(n)1133 5093 y Fv(\015)28 b Fs(2)23 b Fv(C)6 b Fw(\([0)p Fv(;)14 b Fw(1])p Fv(;)g(A)19 b Fs([)f(f)p Fw(0)p Fs(g)p Fw(\))1949 4998 y Fq(\014)1949 5047 y(\014)1949 5097 y(\014)2033 5093 y Fv(\015)5 b Fw(\(0\))22 b(=)h(0)p Fv(;)41 b Fs(jj)p Fv(\015)5 b Fw(\()p Fv(s)p Fw(\))p Fs(jj)23 b(\024)g Fv(\016)o(;)69 b Fs(jj)p Fv(\015)5 b Fw(\(1\))p Fs(jj)23 b Fw(=)f Fv(\016)3281 5001 y Fq(o)236 5285 y Fw(and)28 b(the)g(\\restricted-to-)p Fv(A)p Fw(")d(moun)n(tain)i(pass)g(lev)n(el)g(of)h Fv(')1715 5457 y(d)c Fw(:=)34 b(inf)1892 5510 y Fu(\015)t Fp(2)p Ft(\000)2053 5457 y Fw(max)2030 5514 y Fu(s)p Fp(2)p Ft([0)p Fu(;)p Ft(1])2244 5457 y Fv(')p Fw(\()p Fv(\015)5 b Fw(\()p Fv(s)p Fw(\)\))p Fv(:)2084 6059 y Fw(10)p eop %%Page: 11 11 11 10 bop 236 272 a Fw(Before)33 b(pro)n(ving)g(that)i Fv(d)f Fw(is)g(a)g(critical)f(lev)n(el)h(for)g Fv(')p Fw(,)i(w)n(e)e(shall)f(c)n(hec)n(k)h(that,)i(pro)n(vided)d Fv(\021)3125 284 y Ft(1)3197 272 y Fw(has)g(b)r(een)i(c)n(hosen)e (small)236 372 y(enough,)c(it)h(satis\014es)e(\(3)p Fv(:)p Fw(9\).)42 b(F)-7 b(or)28 b Fv(y)h Fs(2)d Fv(S)1527 384 y Fu(A)1607 372 y Fw(:=)f Fv(S)f Fs(\\)c Fv(A)30 b Fw(w)n(e)f(de\014ne) g Fv(f)2368 384 y Fu(y)2433 372 y Fw(:)d([0)p Fv(;)14 b(\016)s Fw(])26 b Fs(!)f Fx(R)k Fw(b)n(y)g Fv(f)3040 384 y Fu(y)3080 372 y Fw(\()p Fv(t)p Fw(\))d(=)f Fv(')p Fw(\()p Fv(ty)s 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Fv(:)539 b Fw(\(3.18\))236 1256 y(W)-7 b(e)31 b(\014rst)g(pro)n(v)n (e)e(an)h(upp)r(er)h(b)r(ound)g(for)f Fv(d)p Fw(.)47 b(F)-7 b(or)30 b Fv(y)1838 1268 y Ft(0)1906 1256 y Fw(b)r(elonging)g (to)g Fv(K)2460 1268 y Ft(0)2528 1256 y Fw(w)n(e)g(clearly)g(ha)n(v)n (e)f Fv(d)g Fs(\024)f Fw(max)3435 1271 y Fu(t)p Fp(2)p Ft([0)p Fu(;\016)r Ft(])3646 1256 y Fv(f)3687 1268 y Fu(y)3721 1276 y Fd(0)3757 1256 y Fw(\()p Fv(t)p Fw(\))j(and)236 1356 y(hence,)d(from)f(\(3.17\))g(and)g(\(3.18\),)847 1590 y Fv(d)c Fs(\024)42 b Fw(max)1001 1648 y Fu(t)p Fp(2)p Ft([0)p Fu(;\016)r Ft(])1207 1498 y Fq(\020)1267 1534 y Fv(t)1297 1504 y Ft(2)p 1267 1571 68 4 v 1280 1647 a Fw(2)1362 1590 y Fs(\000)1472 1534 y Fv(t)1502 1504 y Fu(q)r Ft(+1)p 1455 1571 184 4 v 1455 1647 a Fv(q)22 b Fw(+)c(1)1648 1498 y Fq(\021)1716 1590 y Fw(+)1809 1534 y Fv(\021)s Fw(\()p Fv(\013;)c(m)p Fw(\))p 1809 1571 272 4 v 1854 1647 a Fv(q)21 b Fw(+)d(1)2091 1590 y Fv(\016)2131 1556 y Fu(q)r Ft(+1)2275 1590 y Fw(=)2363 1498 y Fq(\020)2422 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Fu(\027)2905 3992 y Fw(\()p Fv(X)36 b Fw(is)31 b(called)e(a)h(pseudo)g(gradien)n(t)236 4091 y(\014eld)e(for)f Fv(')p Fw(;)h(for)f(a)g(pro)r(of)g(of)h (existence,)f(see)h(for)f(example)g([30)o(]\).)361 4191 y(Let)h Fv(J)j Fw(:)23 b Fx(R)f Fs(!)h Fw([0)p Fv(;)14 b Fw(1])27 b(b)r(e)h(a)f(smo)r(oth)h(function)g(suc)n(h)f(that)h Fv(J)8 b Fw(\()p Fv(s)p Fw(\))24 b(=)e(0)27 b(if)i Fs(j)p Fv(s)p Fs(j)23 b(\025)f Fv(\027)5 b(=)p Fw(2)27 b(and)h Fv(J)8 b Fw(\()p Fv(s)p Fw(\))23 b(=)g(1)k(if)h Fs(j)p Fv(s)p Fs(j)23 b(\024)g Fv(\027)5 b(=)p Fw(4.)361 4291 y(Let)32 b Fv(\014)561 4303 y Ft(1)630 4291 y Fw(and)g Fv(\014)843 4303 y Ft(2)911 4291 y Fv(<)e(\016)35 b Fw(b)r(e)d(suc)n(h) g(that)g Fs(8)p Fv(v)i Fs(2)c Fv(A)j Fw(with)f Fs(jj)p Fv(v)s Fs(jj)f(\024)f Fv(\014)2372 4303 y Ft(1)2441 4291 y Fw(or)i Fs(jj)p Fv(v)s Fs(jj)e(\025)g Fv(\014)2855 4303 y Ft(2)2893 4291 y Fw(,)j Fv(')p Fw(\()p Fv(v)s Fw(\))e Fv(<)f(d)22 b Fs(\000)f Fv(\027)5 b Fw(.)50 b Fv(\014)3553 4303 y Ft(2)3623 4291 y Fw(do)r(es)31 b(exists)236 4390 y(b)r(ecause)h Fs(8)p Fv(y)h Fs(2)e Fv(S)805 4402 y Fu(A)859 4390 y Fw(,)j(b)n(y)e(\(3.17\)-\(3.18\),)f Fv(')p Fw(\()p Fv(\016)s(y)s Fw(\))g Fs(\024)h Fw(\()p Fv(\016)1946 4360 y Ft(2)1984 4390 y Fv(=)p Fw(2\))20 b Fs(\000)i Fw(\()p Fv(\016)2279 4360 y Fu(q)r Ft(+1)2400 4390 y Fv(=)p Fw(2\()p Fv(q)h Fw(+)e(1\)\)+)32 b Fv(O)r Fw(\()p Fv(\021)s Fw(\()p Fv(\013;)14 b(m)p Fw(\))p Fv(\016)3273 4360 y Fu(q)r Ft(+1)3396 4390 y Fw(\))32 b(=)f Fv(\016)3596 4360 y Ft(2)3665 4390 y Fw(\()p Fs(\000)p Fw(\(1)p Fv(=)p Fw(2\)+)236 4490 y Fv(O)r Fw(\()p Fv(\021)s Fw(\()p Fv(\013;)14 b(m)p Fw(\)\)\))37 b Fs(\024)c Fw(0,)j(b)n(y)e(the)g(c)n(hoice)g(of)g Fv(\016)j Fw(pro)n(vided)c Fv(\021)1993 4502 y Ft(1)2065 4490 y Fw(is)h(small)g(enough.)56 b(Let)34 b Fv(K)39 b Fw(:)c Fx(R)e Fs(!)h Fw([0)p Fv(;)14 b Fw(1])33 b(b)r(e)i(smo)r(oth)f (and)236 4590 y(satisfy)27 b Fv(K)6 b Fw(\()p Fv(s)p Fw(\))23 b(=)g(0)k(if)h Fs(j)p Fv(s)p Fs(j)23 b(\024)g Fv(\014)1178 4559 y Ft(2)1174 4610 y(1)1216 4590 y Fv(=)p Fw(4)j(or)h Fs(j)p Fv(s)p Fs(j)c(\025)g Fw(\()p Fv(\014)1703 4602 y Ft(2)1759 4590 y Fw(+)18 b Fv(\016)s Fw(\))1914 4559 y Ft(2)1951 4590 y Fv(=)p Fw(4)27 b(and)g Fv(K)6 b Fw(\()p Fv(s)p Fw(\))23 b(=)g(1)k(if)h Fv(\014)2710 4559 y Ft(2)2706 4610 y(1)2771 4590 y Fs(\024)23 b(j)p Fv(s)p Fs(j)g(\024)f Fv(\014)3105 4559 y Ft(2)3101 4610 y(2)3143 4590 y Fw(.)37 b(W)-7 b(e)28 b(de\014ne)485 4796 y Fv(Y)19 b Fw(\()p Fv(v)s Fw(\))24 b(=)e Fv(K)6 b Fw(\()p Fs(jj)p Fv(v)s Fs(jj)1014 4762 y Ft(2)1052 4796 y Fw(\))p Fv(J)i Fw(\()p Fv(')p Fw(\()p Fv(v)s Fw(\))20 b Fs(\000)e Fv(d)p Fw(\))1509 4704 y Fq(h)1567 4796 y Fs(\000)1650 4704 y Fq(\020)1700 4796 y Fw(1)g Fs(\000)g Fv(J)1897 4704 y Fq(\020)1946 4796 y Fv(U)9 b Fw(\()p Fv(v)s Fw(\))19 b Fs(\000)2231 4740 y Fv(m)p 2231 4777 V 2247 4853 a Fw(2)2314 4704 y Fq(\021\021)2413 4796 y Fv(X)7 b Fw(\()p Fv(v)s Fw(\))19 b Fs(\000)f Fv(J)2752 4704 y Fq(\020)2801 4796 y Fv(U)9 b Fw(\()p Fv(v)s Fw(\))19 b Fs(\000)3086 4740 y Fv(m)p 3086 4777 V 3102 4853 a Fw(2)3169 4704 y Fq(\021)3247 4796 y Fw(\()p Fv(D)r(')p Fw(\()p Fv(v)s Fw(\)[)p Fv(v)s Fw(]\))29 b Fv(v)3704 4704 y Fq(i)3744 4796 y Fv(:)236 5006 y Fw(The)k(v)n(ector\014eld)f Fv(Y)51 b Fw(is)33 b(w)n(ell)g(de\014ned)g(on)f Fv(A)1654 4976 y Fp(0)1710 5006 y Fw(:=)f Fv(A)22 b Fs([)h Fv(B)2054 5018 y Fu(\014)2092 5026 y Fd(1)2161 5006 y Fw(and)32 b(v)-5 b(anishes)33 b(ev)n(erywhere)e(in)i Fv(B)3261 5021 y Fu(\014)3299 5029 y Fd(1)3331 5021 y Fu(=)p Ft(2)3435 5006 y Fw(and)g(outside)f Fv(B)3956 5018 y Fu(\016)3993 5006 y Fw(.)236 5105 y(It)j(is)g(lo)r(cally)f(Lipsc)n(hitz)h(con)n(tin) n(uous.)58 b(Moreo)n(v)n(er)32 b(it)j(is)g(b)r(ounded,)i(and)e(it)g(is) g(easy)f(to)h(see)f(that)i(its)f(\015o)n(w)f Fv(\036)3780 5117 y Fu(r)3817 5105 y Fw(\()p Fv(v)s Fw(\))i(is)236 5205 y(de\014ned,)31 b(for)d(all)i(time)f Fv(r)r Fw(,)i(in)f Fv(A)1238 5175 y Fp(0)1291 5205 y Fw(\(notice)g(that)f(in)h(a)f(neigh)n (b)r(ourho)r(o)r(d)f(of)i Fv(@)5 b(A)p Fw(,)30 b Fv(Y)18 b Fw(\()p Fv(v)s Fw(\))27 b Fs(2)g Fx(R)p Fv(v)s Fw(,)i(so)g(that)h (the)g(\015o)n(w)e(lea)n(v)n(es)236 5305 y Fv(U)9 b Fw(\()p Fv(v)s Fw(\))27 b(in)n(v)-5 b(arian)n(t,)26 b(and)h(one)f(do)r(es)g (not)h(reac)n(h)e Fv(@)5 b(A)p Fw(\);)28 b Fv(r)d Fs(7!)f Fv(\036)2082 5317 y Fu(r)2119 5305 y Fw(\()p Fv(v)s Fw(\))j(is)g(the)g (unique)g(solution)f(of)h(\()p Fv(d=dr)r Fw(\))p Fv(\036)3434 5317 y Fu(r)3472 5305 y Fw(\()p Fv(v)s Fw(\))d(=)f Fv(Y)18 b Fw(\()p Fv(\036)3838 5317 y Fu(r)3876 5305 y Fw(\()p Fv(v)s Fw(\)\))236 5404 y(with)29 b Fv(\036)475 5416 y Ft(0)512 5404 y Fw(\()p Fv(v)s Fw(\))c(=)e Fv(v)s Fw(.)38 b(W)-7 b(e)28 b(observ)n(e)f(also)f(that)j(if)f Fv(\015)g Fs(2)c Fw(\000)k(then)g Fv(\015)2161 5416 y Fu(r)2226 5404 y Fw(de\014ned)g(b)n(y)g Fv(\015)2671 5416 y Fu(r)2708 5404 y Fw(\()p Fv(s)p Fw(\))c(=)f Fv(\036)2972 5416 y Fu(r)3009 5404 y Fw(\()p Fv(\015)5 b Fw(\()p Fv(s)p Fw(\)\))29 b(b)r(elongs)e(to)h(\000)g(for)f(all)h Fv(r)236 5504 y Fw(b)r(ecause)f Fv(\036)592 5516 y Fu(r)630 5504 y Fw(\()p Fv(y)s Fw(\))c(=)g Fv(y)30 b Fw(for)d Fs(jj)p Fv(y)s Fs(jj)c(\024)g Fv(\014)1341 5516 y Ft(1)1378 5504 y Fv(=)p Fw(2)k(and)g Fs(jj)p Fv(y)s Fs(jj)c(\025)g Fv(\016)s Fw(.)2084 6059 y(11)p eop %%Page: 12 12 12 11 bop 361 272 a Fw(Note)26 b(that)g(if)h Fv(J)8 b Fw(\()p Fv(')p Fw(\()p Fv(v)s Fw(\))17 b Fs(\000)e Fv(d)p Fw(\))23 b Fs(6)p Fw(=)g(0)j(then)g Fs(j)p Fv(')p Fw(\()p Fv(v)s Fw(\))17 b Fs(\000)e Fv(d)p Fs(j)23 b Fv(<)g(\027)5 b(=)p Fw(2.)36 b(In)26 b(that)g(case,)g(either)g Fs(j)p Fv(U)9 b Fw(\()p Fv(v)s Fw(\))16 b Fs(\000)f Fv(m=)p Fw(2)p Fs(j)22 b Fv(<)h(\027)5 b(=)p Fw(2)25 b(and)h(then)236 372 y(b)n(y)j(\(3.20\))g Fs(j)p Fv(D)r(')p Fw(\()p Fv(v)s Fw(\)[)p Fv(v)s Fw(])p Fs(j)f(\025)e Fv(\027)5 b Fw(,)31 b(or)d Fs(j)p Fv(U)9 b Fw(\()p Fv(v)s Fw(\))21 b Fs(\000)e Fv(m=)p Fw(2)p Fs(j)25 b(\025)h Fv(\027)5 b(=)p Fw(2)29 b(and)h(then)g Fv(Y)18 b Fw(\()p Fv(v)s Fw(\))27 b(=)f Fs(\000)p Fv(K)6 b Fw(\()p Fs(jj)p Fv(v)s Fs(jj)2995 341 y Ft(2)3032 372 y Fw(\))p Fv(J)i Fw(\()p Fv(')p Fw(\()p Fv(v)s Fw(\))22 b Fs(\000)d Fv(d)p Fw(\))p Fv(X)7 b Fw(\()p Fv(v)s Fw(\).)44 b(In)29 b(b)r(oth)236 471 y(cases,)e(b)n(y)g(\(3.19\)) g(and)g(the)h(prop)r(erties)f(of)h Fv(X)7 b Fw(\()p Fv(v)s Fw(\),)1313 654 y Fv(D)r(')p Fw(\()p Fv(v)s Fw(\)[)p Fv(Y)19 b Fw(\()p Fv(v)s Fw(\)])24 b Fs(\024)f(\000)p Fv(K)6 b Fw(\()p Fs(jj)p Fv(v)s Fs(jj)2186 620 y Ft(2)2223 654 y Fw(\))p Fv(J)i Fw(\()p Fv(')p Fw(\()p Fv(v)s Fw(\))20 b Fs(\000)e Fv(d)p Fw(\))p Fv(\027)2726 620 y Ft(2)2787 654 y Fs(\024)23 b Fw(0)p Fv(:)864 b Fw(\(3.21\))236 837 y(No)n(w,)32 b(if)g Fs(j)p Fv(')p Fw(\()p Fv(v)s Fw(\))22 b Fs(\000)e Fv(d)p Fs(j)29 b Fv(<)g(\027)5 b(=)p Fw(4)31 b(then)g Fv(J)8 b Fw(\()p Fv(')p Fw(\()p Fv(v)s Fw(\))22 b Fs(\000)f Fv(d)p Fw(\))29 b(=)g(1)i(and)g(b)n(y)g(the)g (de\014nition)h(of)f Fv(\014)2940 849 y Ft(1)2977 837 y Fv(;)14 b(\014)3061 849 y Ft(2)3098 837 y Fw(,)33 b Fv(\014)3201 849 y Ft(1)3267 837 y Fs(\024)28 b(jj)p Fv(v)s Fs(jj)i(\024)f Fv(\014)3666 849 y Ft(2)3703 837 y Fw(,)j(so)f(that)236 936 y Fv(K)6 b Fw(\()p Fs(jj)p Fv(v)s Fs(jj)480 906 y Ft(2)518 936 y Fw(\))23 b(=)g(1.)36 b(Therefore,)27 b(b)n(y)g(\(3.21\),)g(for)g(all)g Fv(v)f Fs(2)e Fv(A)1989 906 y Fp(0)2031 936 y Fs(\\)p 2105 869 68 4 v 19 w Fv(B)2172 948 y Fu(\016)2208 936 y Fw(,)1275 1162 y Fs(j)p Fv(')p Fw(\()p Fv(\036)1433 1174 y Fu(r)1471 1162 y Fw(\()p Fv(v)s Fw(\)\))c Fs(\000)e Fv(d)p Fs(j)23 b Fv(<)1900 1106 y(\027)p 1900 1143 47 4 v 1902 1219 a Fw(4)2062 1162 y(=)-14 b Fs(\))2332 1106 y Fv(d)p 2312 1143 83 4 v 2312 1219 a(dr)2405 1162 y(')p Fw(\()p Fv(\036)2540 1174 y Fu(r)2578 1162 y Fw(\()p Fv(v)s Fw(\)\))24 b Fs(\024)e(\000)p Fv(\027)2939 1128 y Ft(2)3804 1162 y Fw(\(3.22\))236 1373 y(Pic)n(k)34 b(some)h Fv(\015)40 b Fs(2)c Fw(\000)f(whic)n(h)g (satis\014es)f(max)1619 1388 y Fu(s)p Fp(2)p Ft([0)p Fu(;)p Ft(1])1836 1373 y Fv(')p Fw(\()p Fv(\015)5 b Fw(\()p Fv(s)p Fw(\)\))37 b Fs(\024)e Fv(d)23 b Fw(+)g Fv(\027)5 b(=)p Fw(4.)59 b(By)35 b(\(3)p Fv(:)p Fw(22\),)h(max)3173 1388 y Fu(s)p Fp(2)p Ft([0)p Fu(;)p Ft(1])3390 1373 y Fv(')p Fw(\()p Fv(\036)3525 1388 y Ft(1)p Fu(=)p Ft(2)p Fu(\027)3668 1373 y Fw(\()p Fv(\015)5 b Fw(\()p Fv(s)p Fw(\)\)\))36 b Fs(\024)236 1472 y Fv(d)19 b Fs(\000)f Fv(\027)5 b(=)p Fw(4,)27 b(whic)n(h)g(con)n(tradicts)g(the)h (de\014nition)g(of)f Fv(d)p Fw(.)p 1936 1472 48 48 v 361 1622 a(W)-7 b(e)20 b(can)f(no)n(w)g(pro)n(v)n(e)f(that)h Fv(d)h Fw(is)g(a)f(critical)g(lev)n(el.)33 b(Let)20 b Fv(y)2061 1634 y Fu(k)2121 1622 y Fw(b)r(e)g(the)g(sequence)f(giv)n(en) g(b)n(y)g(lemma)g(3.3.)34 b(If)20 b Fv(D)r(')p Fw(\()p Fv(y)3709 1634 y Fu(k)3750 1622 y Fw(\)[)p Fv(y)3846 1634 y Fu(k)3887 1622 y Fw(])j Fs(!)236 1721 y Fw(0)31 b(and)g Fv(U)9 b Fw(\()p Fv(y)613 1733 y Fu(k)654 1721 y Fw(\))30 b Fs(!)f Fv(m=)p Fw(2)i(then,)i(writing)e Fv(y)1564 1733 y Fu(k)1634 1721 y Fw(=)e Fv(t)1758 1733 y Fu(k)1798 1721 y Fv(z)1837 1733 y Fu(k)1909 1721 y Fw(with)j Fv(z)2141 1733 y Fu(k)2211 1721 y Fs(2)e Fv(S)5 b Fw(,)32 b(w)n(e)f(ha)n(v)n(e)f(lim)2844 1733 y Fu(k)2898 1721 y Fv(f)2948 1691 y Fp(0)2939 1742 y Fu(z)2971 1751 y Fe(k)3011 1721 y Fw(\()p Fv(t)3073 1733 y Fu(k)3114 1721 y Fw(\))g(=)f(0)i(and)g Fv(U)9 b Fw(\()p Fv(z)3645 1733 y Fu(k)3686 1721 y Fw(\))29 b Fs(!)h Fv(m=)p Fw(2)236 1821 y(\()p Fs(jj)p Fv(y)355 1833 y Fu(k)396 1821 y Fs(jj)f(\025)f Fv(\032)607 1833 y Ft(0)673 1821 y Fv(>)h Fw(0)h(since)h Fv(')p Fw(\()p Fv(y)1173 1833 y Fu(k)1214 1821 y Fw(\))e Fs(!)g Fv(d)g(>)f Fw(0\).)47 b(Therefore)30 b(in)h(this)h(case,)f(b)n (y)g(\(3.17\),)g(\(3.18\))f(and)h(lemma)g(3.2,)g(if)g Fv(\021)3891 1833 y Ft(1)3960 1821 y Fw(is)236 1921 y(small)c(enough,) 1269 2055 y(lim)14 b(inf)1430 2109 y Fu(k)1513 2055 y Fv(')p Fw(\()p Fv(y)1640 2067 y Fu(k)1681 2055 y Fw(\))23 b Fs(\025)1824 1963 y Fq(\020)1883 1999 y Fw(1)p 1883 2036 42 4 v 1883 2112 a(2)1953 2055 y Fs(\000)2117 1999 y Fw(1)p 2046 2036 184 4 v 2046 2112 a Fv(q)f Fw(+)c(1)2239 1963 y Fq(\021)2289 2055 y Fw(2)2331 2020 y Ft(2)p Fu(=q)r Fp(\000)p Ft(1)2538 2055 y Fw(+)g Fv(O)r Fw(\()p Fv(\021)2759 2067 y Ft(1)2797 2055 y Fw(\))23 b Fv(>)g(d)236 2248 y Fw(b)n(y)i(estimate)f(\(3.9\).)36 b(It)25 b(follo)n(ws)e(that)i Fv(D)r(')p Fw(\()p Fv(y)1639 2260 y Fu(k)1681 2248 y Fw(\))e Fs(!)g Fw(0)h(and)h(it)g(is)f(standard)g(to)g(pro)n(v)n(e)f (\(b)n(y)i(the)g(compactness)f(prop)r(erties)236 2348 y(of)j Fv(D)r(G)h Fw(and)e Fv(D)r(S)5 b Fw(\))28 b(that,)f(up)g(to)g(a) g(subsequence,)g(\()p Fv(y)1897 2360 y Fu(k)1937 2348 y Fw(\))h(con)n(v)n(erges)c(strongly)i(to)h(a)f(critical)h(p)r(oin)n(t) g Fv(y)j Fw(of)d Fv(')g Fw(on)g(the)g(lev)n(el)236 2447 y Fv(d)h Fw(\(P)n(alais)d(Smale)i(prop)r(ert)n(y)-7 b(,)26 b(see)h(e.g.)36 b([30)o(]\).)h(Note)28 b(that,)f(since)g Fv(')p Fw(\()p Fv(y)2430 2459 y Fu(k)2471 2447 y Fw(\))p Fv(tod)h Fw(and)f Fv(y)2846 2459 y Fu(k)2909 2447 y Fs(2)d Fv(A)p Fw(,)j Fv(y)3141 2459 y Fu(k)3205 2447 y Fs(2)c Fv(B)3346 2459 y Fu(\014)3384 2467 y Fd(2)3448 2447 y Fw(for)j(all)h Fv(k)j Fw(so)d(that)236 2547 y Fv(y)f Fs(2)d Fv(B)444 2559 y Fu(\016)481 2547 y Fw(.)361 2647 y(Let)34 b(us)f(no)n(w)g(pro)n(v)n(e)f(the)j(last)e(claim)g(of)h(Prop)r (osition)e(3.1.)55 b(W)-7 b(riting)33 b Fv(y)j Fw(=)d Fv(tz)k Fw(with)d Fv(z)j Fs(2)d Fv(S)k Fw(and)c Fv(f)3574 2617 y Fp(0)3565 2667 y Fu(z)3603 2647 y Fw(\()p Fv(t)p Fw(\))f(=)g(0,)i(b)n(y)236 2746 y(lemma)28 b(3.2)e(w)n(e)i(get)984 2972 y Fv(d)23 b Fs(\025)1138 2880 y Fq(\020)1197 2916 y Fw(1)p 1197 2953 42 4 v 1197 3029 a(2)1267 2972 y Fs(\000)1431 2916 y Fw(1)p 1360 2953 184 4 v 1360 3029 a Fv(q)f Fw(+)c(1)1553 2880 y Fq(\021\020)1712 2916 y Fv(m)p 1662 2953 173 4 v 1662 3029 a(G)p Fw(\()p Fv(z)t Fw(\))1844 2880 y Fq(\021)1894 2897 y Ft(2)p Fu(=q)r Fp(\000)p Ft(1)2101 2972 y Fs(\000)g Fv(C)2243 2984 y Ft(7)2281 2972 y Fv(\021)s Fw(\()p Fv(\013;)c(m)p Fw(\)\()p Fv(\016)2624 2938 y Fu(q)2680 2972 y Fw(+)k Fv(\016)2803 2938 y Fu(q)r Ft(+1)2924 2972 y Fv(=)p Fw(\()p Fv(q)j Fw(+)d(1\)\))p Fv(;)236 3204 y Fw(and)28 b(hence,)f(b)n(y)h(the) g(upp)r(er)f(estimate)h(on)f Fv(d)p Fw(,)h Fv(G)p Fw(\()p Fv(z)t Fw(\))23 b(=)g Fv(m)18 b Fw(+)g Fv(O)r Fw(\()p Fv(\021)s Fw(\()p Fv(\013;)c(m)p Fw(\)\).)40 b(Moreo)n(v)n(er)24 b(b)n(y)k(\(3)p Fv(:)p Fw(14\))f(w)n(e)g(ha)n(v)n(e)1224 3434 y Fv(t)c Fw(=)1365 3342 y Fq(\020)1474 3378 y Fv(m)p 1425 3415 V 1425 3491 a(G)p Fw(\()p Fv(z)t Fw(\))1607 3342 y Fq(\021)1656 3359 y Ft(1)p Fu(=q)r Fp(\000)p Ft(1)1863 3434 y Fw(+)18 b Fv(O)r Fw(\()p Fv(\021)s Fw(\()p Fv(\013;)c(m)p Fw(\)\))26 b(=)c(1)c(+)g Fv(O)r Fw(\()p Fv(\021)s Fw(\()p Fv(\013;)c(m)p Fw(\)\))p Fv(:)236 3665 y Fw(Hence)28 b(dist\()p Fv(y)s(;)14 b(K)801 3677 y Ft(0)838 3665 y Fw(\))23 b Fs(\024)g Fv(g)s Fw(\()p Fv(C)1115 3677 y Ft(8)1152 3665 y Fv(\021)s Fw(\()p Fv(\013;)14 b(m)p Fw(\)\))20 b(+)e Fv(C)1617 3677 y Ft(8)1655 3665 y Fv(\021)s Fw(\()p Fv(\013;)c(m)p Fw(\))28 b(for)f(some)g(p)r(ositiv)n(e)h (constan)n(t)e Fv(C)2990 3677 y Ft(8)3028 3665 y Fw(,)i(where)1233 3848 y Fv(g)s Fw(\()p Fv(s)p Fw(\))23 b(:=)197 b(sup)1513 3922 y Fu(v)r Fp(2)p Fu(S;G)p Ft(\()p Fu(v)r Ft(\))p Fp(\025)p Fu(m)p Fp(\000)p 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b Fv(m)23 b Fw(:)g Fx(R)1096 4906 y Ft(2)1156 4936 y Fs(!)g Fx(R)29 b Fo(b)l(e)h(lo)l(c)l(al)t(ly)h(inte)l(gr)l(able) g(and)f Fw(2)p Fv(\031)s Fo(-p)l(erio)l(dic)h(w.r.t.)40 b(b)l(oth)30 b(variables.)40 b(Then)998 5061 y Fq(Z)1081 5081 y Ft(2)p Fu(\031)1044 5249 y Ft(0)1173 5061 y Fq(Z)1256 5081 y Fu(\031)1219 5249 y Ft(0)1315 5174 y Fv(m)p Fw(\()p Fv(t)18 b Fw(+)g Fv(x;)c(t)19 b Fs(\000)f Fv(x)p Fw(\))31 b Fv(dt)f(dx)23 b Fw(=)2191 5118 y(1)p 2191 5155 42 4 v 2191 5231 a(2)2256 5061 y Fq(Z)2339 5081 y Ft(2)p Fu(\031)2302 5249 y Ft(0)2431 5061 y Fq(Z)2514 5081 y Ft(2)p Fu(\031)2477 5249 y Ft(0)2606 5174 y Fv(m)p Fw(\()p Fv(s)2750 5186 y Ft(1)2788 5174 y Fv(;)14 b(s)2864 5186 y Ft(2)2901 5174 y Fw(\))30 b Fv(ds)3045 5186 y Ft(1)3112 5174 y Fv(ds)3194 5186 y Ft(2)3231 5174 y Fv(:)2084 6059 y Fw(12)p eop %%Page: 13 13 13 12 bop 236 272 a Fc(Pr)n(oof)p Fw(.)78 b(Using)28 b(new)f(v)-5 b(ariables)27 b(\()p Fv(s)1417 284 y Ft(2)1454 272 y Fv(;)14 b(x)p Fw(\))24 b(=)f(\()p Fv(t)18 b Fs(\000)g 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Fw(\()p Fv(\021)1297 3977 y Fp(0)1321 4011 y Fw(\()p Fv(t)18 b Fw(+)g Fv(x)p Fw(\))i Fs(\000)e Fv(\021)1710 3977 y Fp(0)1733 4011 y Fw(\()p Fv(t)h Fs(\000)f Fv(x)p Fw(\)\))2008 3977 y Ft(2)2074 4011 y Fv(dt)28 b(dx)23 b Fw(=)2386 3955 y(1)p 2386 3992 42 4 v 2386 4068 a(2)2451 3898 y Fq(Z)2534 3919 y Ft(2)p Fu(\031)2497 4087 y Ft(0)2626 3898 y Fq(Z)2709 3919 y Ft(2)p Fu(\031)2672 4087 y Ft(0)2787 4011 y Fw(\()p Fv(\021)2863 3977 y Fp(0)2887 4011 y Fw(\()p Fv(s)2958 4023 y Ft(1)2996 4011 y Fw(\))18 b Fs(\000)g Fv(\021)3173 3977 y Fp(0)3197 4011 y Fw(\()p Fv(s)3268 4023 y Ft(2)3305 4011 y Fw(\)\))3369 3977 y Ft(2)3435 4011 y Fv(ds)3517 4023 y Ft(1)3582 4011 y Fv(ds)3664 4023 y Ft(2)770 4250 y Fw(=)928 4194 y(1)p 928 4231 V 928 4307 a(2)979 4158 y Fq(h)1018 4250 y Fw(2)p Fv(\031)1124 4137 y Fq(Z)1207 4158 y Ft(2)p Fu(\031)1170 4326 y Ft(0)1299 4250 y Fv(\021)1343 4216 y Fp(0)1367 4250 y Fw(\()p Fv(s)1438 4262 y Ft(1)1475 4250 y Fw(\))1507 4216 y Ft(2)1572 4250 y Fv(ds)1654 4262 y Ft(1)1710 4250 y Fs(\000)g Fw(2)1849 4137 y Fq(Z)1931 4158 y Ft(2)p Fu(\031)1894 4326 y Ft(0)2023 4250 y Fv(\021)2067 4216 y Fp(0)2091 4250 y Fw(\()p Fv(s)2162 4262 y Ft(1)2199 4250 y Fw(\))28 b Fv(ds)2341 4262 y Ft(1)2392 4137 y Fq(Z)2476 4158 y Ft(2)p Fu(\031)2439 4326 y Ft(0)2567 4250 y Fv(\021)2611 4216 y Fp(0)2635 4250 y Fw(\()p Fv(s)2706 4262 y Ft(2)2743 4250 y Fw(\))g Fv(ds)2885 4262 y Ft(2)2941 4250 y Fw(+)18 b(2)p Fv(\031)3130 4137 y Fq(Z)3213 4158 y Ft(2)p Fu(\031)3176 4326 y Ft(0)3305 4250 y Fv(\021)3349 4216 y Fp(0)3372 4250 y Fw(\()p Fv(s)3443 4262 y Ft(2)3481 4250 y Fw(\))3513 4216 y Ft(2)3578 4250 y Fv(ds)3660 4262 y Ft(2)3697 4158 y Fq(i)770 4489 y Fw(=)83 b(2)p Fv(\031)1023 4376 y Fq(Z)1106 4396 y Ft(2)p Fu(\031)1069 4565 y Ft(0)1198 4489 y Fv(\021)1242 4455 y Fp(0)1266 4489 y Fw(\()p Fv(s)p Fw(\))1369 4455 y Ft(2)1434 4489 y Fv(ds:)236 4740 y Fw(Hence)28 b Fs(jjL)586 4752 y Fu(n)632 4740 y Fv(z)t Fs(jj)721 4710 y Ft(2)780 4740 y Fw(=)f(2)p Fv(\031)978 4673 y Fq(R)1033 4693 y Ft(2)p Fu(\031)1017 4769 y Ft(0)1125 4740 y Fv(n)1175 4710 y Ft(2)1213 4740 y Fv(\021)1257 4710 y Fp(0)1280 4740 y Fw(\()p Fv(ns)p Fw(\))1433 4710 y Ft(2)1498 4740 y Fv(ds)c Fw(=)g Fv(n)1741 4710 y Ft(2)1778 4740 y Fs(jj)p Fv(z)t Fs(jj)1913 4710 y Ft(2)1950 4740 y Fv(;)28 b Fw(b)r(ecause)f Fv(\021)k Fw(is)c(2)p Fv(\031)s Fw(-p)r(erio)r(dic.)36 b(As)28 b(a)f(consequence)g(\(for)g Fv(z)g Fs(6)p Fw(=)22 b(0\),)1312 4979 y Fv(U)9 b Fw(\()p Fs(L)1467 4991 y Fu(n)1513 4979 y Fv(z)t Fw(\))22 b(=)1749 4923 y Fv(G)p Fw(\()p Fs(L)1903 4935 y Fu(n)1949 4923 y Fv(z)t Fw(\))p 1708 4960 358 4 v 1708 5036 a Fs(jjL)1811 5048 y Fu(n)1856 5036 y Fv(z)t Fs(jj)1945 5012 y Fu(q)r Ft(+1)2098 4979 y Fw(=)2273 4923 y Fv(G)p Fw(\()p Fs(L)2427 4935 y Fu(n)2473 4923 y Fv(z)t Fw(\))p 2196 4960 428 4 v 2196 5036 a Fv(n)2246 5012 y Fu(p)p Ft(+1)2368 5036 y Fs(jj)p Fv(z)t Fs(jj)2503 5012 y Fu(q)r Ft(+1)2657 4979 y Fv(<)g(U)9 b Fw(\()p Fv(z)t Fw(\))p Fv(;)236 5210 y Fw(b)n(y)28 b(\(4.1\).)36 b(Hence)28 b(for)f(all)g Fv(v)g Fs(2)c Fv(V)1264 5222 y Fu(n)1310 5210 y Fs(nf)p Fw(0)p Fs(g)p Fw(,)i Fv(n)e Fs(\025)g Fw(2,)k Fv(U)9 b Fw(\()p Fv(v)s Fw(\))24 b Fv(<)e Fw(max)2218 5225 y Fu(z)r Fp(2)p Fu(V)14 b Fp(nf)p Ft(0)p Fp(g)2503 5210 y Fv(U)9 b Fw(\()p Fv(z)t Fw(\))23 b(=)g Fv(m)k Fw(and)h(therefore)e Fv(V)3445 5222 y Fu(n)3509 5210 y Fs(\\)19 b Fv(K)3654 5222 y Ft(0)3714 5210 y Fw(=)k Fs(;)p Fw(.)p 3903 5210 48 48 v 361 5360 a(The)k(next)h(lemma)g(pro)n (vides)e(a)h(suitable)h(appro)n(ximation)d(for)i(the)h(functional)g (\010)2966 5372 y Fu(")3029 5360 y Fw(de\014ned)g(in)g(\(2.17\).)2084 6059 y(13)p eop %%Page: 14 14 14 13 bop 236 272 a Fx(Lemma)32 b(4.4)40 b Fo(The)31 b(functional)f Fw(\010)1358 284 y Fu(")1417 272 y Fw(:)23 b Fs(D)1527 284 y Fu(\032)1589 272 y Fs(!)g Fx(R)29 b Fo(de\014ne)l(d)h(in)g(\(2.17\))h(c)l(an)f(b)l(e)g(develop)l(e)l(d)i (as)1018 500 y Fw(\010)1078 512 y Fu(")1113 500 y Fw(\()p Fv(v)s Fw(\))24 b(=)1343 444 y Fv(")p 1342 481 42 4 v 1342 557 a Fw(2)1393 500 y Fs(jj)p Fv(v)s Fs(jj)1528 466 y Ft(2)1584 500 y Fs(\000)1667 387 y Fq(Z)1713 576 y Ft(\012)1779 500 y Fv(F)12 b Fw(\()p Fv(v)s Fw(\))19 b Fs(\000)2063 444 y Fw(1)p 2063 481 V 2063 557 a(2)2128 387 y Fq(Z)2174 576 y Ft(\012)2240 500 y Fv(f)9 b Fw(\()p Fv(v)s Fw(\))p Fv(L)2454 466 y Fp(\000)p Ft(1)2543 500 y Fw(\005)2605 512 y Fu(W)2681 500 y Fv(f)g Fw(\()p Fv(v)s Fw(\))18 b(+)g Fv(R)3002 512 y Fu(!)r(;)p Ft(1)3103 500 y Fw(\()p Fv(v)s Fw(\))p Fv(;)236 725 y Fo(with)1463 857 y Fv(D)r(R)1597 869 y Fu(!)r(;)p Ft(1)1698 857 y Fw(\()p Fv(v)s Fw(\)[)p Fv(v)s Fw(])24 b(=)f Fv(O)2071 765 y Fq(\020)2135 801 y Fv(")p 2131 838 48 4 v 2131 914 a(\015)2188 857 y Fs(j)p Fv(v)s Fs(j)2277 822 y Ft(2)p Fu(p)2277 877 y(!)2368 857 y Fw(+)2461 801 y Fs(j)p Fv(v)s Fs(j)2550 770 y Ft(3)p Fu(p)p Fp(\000)p Ft(1)2550 821 y Fu(!)p 2461 838 246 4 v 2560 914 a Fv(\015)2716 765 y Fq(\021)2766 857 y Fv(:)236 1089 y Fc(Pr)n(oof)p Fw(.)78 b(W)-7 b(e)28 b(ha)n(v)n(e,)f(since)1158 1022 y Fq(R)1197 1118 y Ft(\012)1263 1089 y Fv(f)9 b Fw(\()p Fv(v)s Fw(\))p Fv(L)1477 1059 y Fp(\000)p Ft(1)1566 1089 y Fw(\005)1628 1101 y Fu(W)1704 1089 y Fv(f)1754 1059 y Fp(0)1777 1089 y Fw(\()p Fv(v)s Fw(\))p Fv(v)27 b Fw(=)2038 1022 y Fq(R)2077 1118 y Ft(\012)2143 1089 y Fv(f)2193 1059 y Fp(0)2216 1089 y Fw(\()p Fv(v)s Fw(\))p Fv(v)s(L)2423 1059 y Fp(\000)p Ft(1)2512 1089 y Fw(\005)2574 1101 y Fu(W)2650 1089 y Fv(f)9 b Fw(\()p Fv(v)s Fw(\),)321 1323 y Fv(D)r(R)455 1335 y Fu(!)r(;)p Ft(1)556 1323 y Fw(\()p Fv(v)s Fw(\)[)p Fv(v)s Fw(])84 b(=)983 1210 y Fq(Z)1029 1398 y Ft(\012)1095 1230 y Fq(h)1134 1323 y Fv(f)9 b Fw(\()p Fv(v)s Fw(\))19 b(+)f Fv(f)1443 1288 y Fp(0)1466 1323 y Fw(\()p Fv(v)s Fw(\))p Fv(L)1630 1288 y Fp(\000)p Ft(1)1719 1323 y Fw(\005)1781 1335 y Fu(W)1857 1323 y Fv(f)9 b Fw(\()p Fv(v)s Fw(\))19 b Fs(\000)f Fv(f)9 b Fw(\()p Fv(v)21 b Fw(+)d Fv(w)r Fw(\()p Fv(v)s Fw(\)\))2542 1230 y Fq(i)2583 1323 y Fv(v)836 1544 y Fw(=)983 1431 y Fq(Z)1029 1620 y Ft(\012)1095 1452 y Fq(h)1134 1544 y Fv(f)9 b Fw(\()p Fv(v)s Fw(\))19 b(+)f Fv(f)1443 1510 y Fp(0)1466 1544 y Fw(\()p Fv(v)s Fw(\))p Fv(w)r Fw(\()p Fv(v)s Fw(\))i Fs(\000)e Fv(f)9 b Fw(\()p Fv(v)22 b Fw(+)c Fv(w)r Fw(\()p Fv(v)s Fw(\)\))2271 1452 y Fq(i)2312 1544 y Fv(v)k Fw(+)2457 1431 y Fq(Z)2503 1620 y Ft(\012)2568 1452 y Fq(h)2607 1544 y Fw(\()p Fv(L)2696 1510 y Fp(\000)p Ft(1)2804 1544 y Fs(\000)c Fv(L)2944 1510 y Fp(\000)p Ft(1)2944 1565 y Fu(!)3032 1544 y Fw(\)\005)3126 1556 y Fu(W)3203 1544 y Fv(f)9 b Fw(\()p Fv(v)s Fw(\))3360 1452 y Fq(i)3399 1544 y Fv(f)3449 1510 y Fp(0)3472 1544 y Fw(\()p Fv(v)s Fw(\))p Fv(v)836 1766 y Fw(+)983 1653 y Fq(Z)1029 1842 y Ft(\012)1095 1674 y Fq(h)1134 1766 y Fv(L)1191 1732 y Fp(\000)p Ft(1)1191 1787 y Fu(!)1280 1766 y Fw(\005)1342 1778 y Fu(W)1417 1766 y Fv(f)g Fw(\()p Fv(v)s Fw(\))19 b Fs(\000)f Fv(w)r Fw(\()p Fv(v)s Fw(\))1844 1674 y Fq(i)1885 1766 y Fv(f)1935 1732 y Fp(0)1958 1766 y Fw(\()p Fv(v)s Fw(\))p Fv(v)836 1999 y Fw(=)82 b Fv(O)1048 1907 y Fq(\020)1099 1999 y Fs(j)p Fv(v)s Fs(j)1188 1965 y Fu(p)p Fp(\000)p Ft(1)1188 2019 y Fu(!)1311 1999 y Fs(j)p Fv(w)r Fw(\()p Fv(v)s Fw(\))p Fs(j)1525 1965 y Ft(2)1525 2021 y Fu(L)1571 2005 y Fd(2)1609 1907 y Fq(\021)1677 1999 y Fw(+)18 b Fv(O)1825 1907 y Fq(\020)1890 1943 y Fv(")p 1885 1980 48 4 v 1885 2056 a(\015)1943 1999 y Fs(j)p Fv(v)s Fs(j)2032 1965 y Fu(p)2032 2019 y(!)2080 1999 y Fs(j)p Fv(v)s Fs(j)2169 1965 y Fu(p)2169 2019 y(!)2218 1907 y Fq(\021)2286 1999 y Fw(+)g Fv(O)2434 1907 y Fq(\020)2494 1943 y Fw(\()p Fv(")g Fw(+)g Fv(\015)5 b Fw(\))p Fs(j)p Fv(v)s Fs(j)2835 1913 y Ft(2)p Fu(p)p Fp(\000)p Ft(1)2835 1963 y Fu(!)p 2494 1980 499 4 v 2700 2056 a Fv(\015)2748 2032 y Ft(2)3002 1999 y Fs(j)p Fv(v)s Fs(j)3091 1965 y Fu(p)3091 2019 y(!)3139 1907 y Fq(\021)836 2229 y Fw(=)82 b Fv(O)1048 2137 y Fq(\020)1132 2173 y Fv(")p 1109 2210 85 4 v 1109 2286 a(\015)1157 2262 y Ft(2)1203 2229 y Fs(j)p Fv(v)s Fs(j)1292 2195 y Ft(3)p Fu(p)p Fp(\000)p Ft(1)1292 2249 y Fu(!)1468 2229 y Fw(+)18 b Fs(j)p Fv(v)s Fs(j)1640 2195 y Ft(3)p Fu(p)p Fp(\000)p Ft(1)1640 2249 y Fu(!)1796 2137 y Fq(\021)1864 2229 y Fw(+)g Fv(O)2012 2137 y Fq(\020)2077 2173 y Fv(")p 2073 2210 48 4 v 2073 2286 a(\015)2130 2229 y Fs(j)p Fv(v)s Fs(j)2219 2195 y Ft(2)p Fu(p)2219 2249 y(!)2291 2137 y Fq(\021)2359 2229 y Fw(+)g Fv(O)2507 2137 y Fq(\020)2567 2173 y Fv(\015)23 b Fw(+)18 b Fv(")p 2567 2210 188 4 v 2619 2286 a(\015)2667 2262 y Ft(2)2765 2229 y Fs(j)p Fv(v)s Fs(j)2854 2195 y Ft(3)p Fu(p)p Fp(\000)p Ft(1)2854 2249 y Fu(!)3011 2137 y Fq(\021)3083 2229 y Fw(=)23 b Fv(O)3236 2137 y Fq(\020)3301 2173 y Fv(")p 3296 2210 48 4 v 3296 2286 a(\015)3354 2229 y Fs(j)p Fv(v)s Fs(j)3443 2195 y Ft(2)p Fu(p)3443 2249 y(!)3533 2229 y Fw(+)3626 2173 y Fs(j)p Fv(v)s Fs(j)3715 2143 y Ft(3)p Fu(p)p Fp(\000)p Ft(1)3715 2193 y Fu(!)p 3626 2210 246 4 v 3725 2286 a Fv(\015)3882 2137 y Fq(\021)236 2461 y Fw(b)n(y)28 b(lemmas)f(2.1,)g (2.3,)f(form)n(ula)h(\(2.12\))g(and)g(b)r(ecause)h Fs(j)p Fv(v)s Fs(j)2068 2431 y Fu(p)p Fp(\000)p Ft(1)2068 2481 y Fu(!)2191 2461 y Fv(=\015)f Fs(\024)c Fv(\032)p Fw(.)p 2494 2461 48 48 v 361 2610 a(Critical)k(p)r(oin)n(ts)g(of)h(the)g (restriction)e(of)i(\010)1700 2622 y Fu(")1763 2610 y Fw(to)g(the)g(subspace)e Fv(V)2402 2622 y Fu(n)2471 2610 y Fs(\032)d Fv(V)46 b Fw(are)27 b(critical)g(p)r(oin)n(ts)g(of)h(\010) 3473 2622 y Fu(")3508 2610 y Fw(.)236 2793 y Fx(Lemma)k(4.5)40 b Fo(A)30 b(critic)l(al)h(p)l(oint)f Fv(v)1359 2805 y Fu(n)1434 2793 y Fo(of)g Fw(\010)1591 2808 y Fu(")p Fp(j)p Fu(V)1681 2816 y Fe(n)1750 2793 y Fw(:)23 b Fv(V)1844 2805 y Fu(n)1908 2793 y Fs(\\)c(D)2046 2805 y Fu(\032)2107 2793 y Fs(!)k Fx(R)30 b Fo(is)g(a)g(critic)l(al)h(p)l(oint)f(of)h Fw(\010)3123 2805 y Fu(")3181 2793 y Fw(:)23 b Fv(V)42 b Fs(!)23 b Fx(R)p Fo(.)236 2976 y Fc(Pr)n(oof)p Fw(.)97 b(W)-7 b(e)35 b(kno)n(w)e(that)h Fv(D)r Fw(\010)1306 2988 y Fu(")1341 2976 y Fw(\()p Fv(v)1413 2988 y Fu(n)1459 2976 y Fw(\)[)p Fv(h)p Fw(])g(=)f(0,)i Fs(8)p Fv(h)f Fs(2)f Fv(V)2081 2988 y Fu(n)2161 2976 y Fw(and)h(w)n(e)f(w)n(an)n(t)g (to)h(pro)n(v)n(e)e(that)j Fv(D)r Fw(\010)3322 2988 y Fu(")3357 2976 y Fw(\()p Fv(v)3429 2988 y Fu(n)3475 2976 y Fw(\)[)p Fv(h)p Fw(])f(=)f(0,)i Fs(8)p Fv(h)e Fs(2)236 3075 y Fv(V)303 3045 y Fp(?)284 3096 y Fu(n)378 3075 y Fs(\\)18 b Fv(V)h Fw(.)37 b(By)28 b(\(2.19\))e(it)i(is)g(su\016cien)n (t)g(to)f(pro)n(v)n(e)f(that)i Fs(8)p Fv(h)23 b Fs(2)g Fv(V)2231 3045 y Fp(?)2212 3096 y Fu(n)2306 3075 y Fs(\\)18 b Fv(V)1243 3204 y Fq(Z)1326 3225 y Fu(\031)1289 3393 y Ft(0)1385 3317 y Fv(dx)1517 3204 y Fq(Z)1600 3225 y Ft(2)p Fu(\031)1563 3393 y Ft(0)1720 3317 y Fv(dt)28 b(f)9 b Fw(\()p Fv(v)s Fw(\()p Fv(t;)14 b(x)p Fw(\))19 b(+)f Fv(w)r Fw(\()p Fv(v)s Fw(\)\()p Fv(t;)c(x)p Fw(\)\))p Fv(h)p Fw(\()p Fv(t;)g(x)p Fw(\))27 b(=)22 b(0)p Fv(:)236 3547 y Fw(This)f(holds)g(true)g(b)r(ecause,)h(b)n(y)f(lemma)g(2.3-)p Fv(iv)s Fw(\),)h Fv(f)9 b Fw(\()p Fv(v)s Fw(\()p Fv(t;)14 b(x)p Fw(\))6 b(+)g Fv(w)r Fw(\()p Fv(v)s Fw(\)\()p Fv(t;)14 b(x)p Fw(\)\))24 b(is)d(2)p Fv(\031)s(=n)f Fw(p)r(erio)r(dic)h(w.r.t)g Fv(t)g Fw(and)g Fv(t)j Fs(!)f Fv(h)p Fw(\()p Fv(t;)14 b(x)p Fw(\))23 b Fs(2)236 3646 y Fv(V)303 3616 y Fp(?)284 3667 y Fu(n)378 3646 y Fs(\\)18 b Fv(V)47 b Fw(do)r(es)27 b(not)h(con)n(tain)f(an)n(y)f(harmonic)h(with)h(a)f(frequency)h(m)n (ultiple)g(of)f Fv(n)p Fw(.)p 2859 3646 V 361 3796 a(Let)h(us)f (de\014ne)h(\010)916 3808 y Fu(";n)1035 3796 y Fw(:)23 b Fs(f)p Fv(v)j Fs(2)e Fv(V)46 b Fs(j)28 b(L)1470 3808 y Fu(n)1515 3796 y Fv(v)f Fs(2)c(D)1724 3808 y Fu(\032)1763 3796 y Fs(g)f(!)h Fx(R)28 b Fw(b)n(y)1763 3978 y(\010)1823 3990 y Fu(";n)1919 3978 y Fw(\()p Fv(v)s Fw(\))c(:=)e(\010)2220 3990 y Fu(")2256 3978 y Fw(\()p Fs(L)2345 3990 y Fu(n)2391 3978 y Fv(v)s Fw(\))p Fv(:)236 4161 y Fw(By)30 b(lemma)h(4.5,)f(if)h Fv(v)j Fw(is)d(a)f(critical)g(p)r(oin)n(t)g(of)h(\010)1769 4173 y Fu(";n)1896 4161 y Fw(\(of)f(minimal)h(p)r(erio)r(d)g(2)p Fv(\031)s Fw(\),)g(then)g Fv(v)3023 4173 y Fu(n)3096 4161 y Fw(:=)d Fs(L)3269 4173 y Fu(n)3314 4161 y Fv(v)34 b Fw(is)d(a)f(critical)g(p)r(oin)n(t)236 4261 y(of)g(\010)393 4273 y Fu(")459 4261 y Fw(\(of)g(minimal)g(p)r(erio)r(d)g(2)p Fv(\031)s(=n)p Fw(\).)44 b(Hence,)30 b(b)n(y)g(lemma)g(2.4,)g Fv(u)c Fw(=)h Fv(v)2483 4273 y Fu(n)2548 4261 y Fw(+)20 b Fv(w)r Fw(\()p Fv(v)2766 4273 y Fu(n)2812 4261 y Fw(\))31 b(is)e(a)h(solution)g(of)f(\(2.1\))h(and)g(it)g(has)236 4360 y(minimal)e(p)r(erio)r(d)g(with)g(resp)r(ect)f(to)h(time)g(2)p Fv(\031)s(=n)p Fw(.)361 4460 y(In)g(the)g(sequel)f(w)n(e)g(shall)g(use) h(the)g(norm)1478 4642 y Fs(j)p Fv(v)s Fs(j)1567 4654 y Fu(!)r(;n)1700 4642 y Fw(:=)22 b Fs(jL)1890 4654 y Fu(n)1936 4642 y Fv(v)s Fs(j)2002 4654 y Fu(!)2073 4642 y Fw(=)h Fs(j)p Fv(v)s Fs(j)2250 4654 y Fp(1)2339 4642 y Fw(+)18 b Fv(")2461 4608 y Ft(1)p Fu(=)p Ft(2)2565 4642 y Fv(n)p Fs(jj)p Fv(v)s Fs(jj)p Fv(:)1072 b Fw(\(4.2\))236 4825 y(Note)24 b(that)g(if)g Fs(j)p Fv("n)793 4795 y Ft(2)830 4825 y Fs(j)f Fv(<)f Fw(1,)i(whic)n(h)g(w)n(e)f(shall)g(alw)n (a)n(ys)f(assume)h(in)h(the)g(sequel,)g(then)g Fs(j)p Fv(v)s Fs(j)2912 4837 y Fu(!)r(;n)3044 4825 y Fs(\024)f Fv(C)6 b Fs(jj)p Fv(v)s Fs(jj)p Fw(.)36 b(As)24 b(a)f(consequence)236 4925 y(there)28 b(is)f Fv(\032)575 4937 y Ft(0)635 4925 y Fv(>)c Fw(0)k(suc)n(h)g(that)h Fs(D)e Fw(:=)c Fs(f)p Fv(v)k Fs(2)e Fv(V)46 b Fs(j)28 b(jj)p Fv(v)s Fs(jj)1826 4895 y Fu(p)p Fp(\000)p Ft(1)1950 4925 y Fv(=\015)f Fs(\024)c Fv(\032)2193 4937 y Ft(0)2230 4925 y Fs(g)k Fw(is)g(included)h(in)g (the)g(domain)f(of)h(de\014nition)g(of)g(\010)3868 4937 y Fu(";n)3964 4925 y Fw(.)361 5124 y(No)n(w)38 b(w)n(e)h(m)n(ust)f (distinguish)h(b)r(et)n(w)n(een)g(di\013eren)n(t)g(cases)f(dep)r (ending)h(whether)g(the)g(\014rst)f(term)h(in)g(the)h(T)-7 b(a)n(ylor)236 5224 y(expansion)26 b(of)i(of)f(the)g(nonlinearit)n(y)f Fv(f)9 b Fw(\()p Fv(u)p Fw(\))27 b(at)h(0)e(is)h(an)g(o)r(dd)h(p)r(o)n (w)n(er)e(of)h Fv(u)g Fw(\(lik)n(e)g(e.g.)36 b Fv(f)9 b Fw(\()p Fv(u)p Fw(\))23 b(=)g Fv(au)3255 5193 y Ft(3)3309 5224 y Fw(+)17 b Fv(h:o:t:)p Fw(\))28 b(or)e(an)h(ev)n(en)236 5323 y(p)r(o)n(w)n(er)i(of)h Fv(u)g Fw(\(lik)n(e)f(e.g.)44 b Fv(f)9 b Fw(\()p Fv(u)p Fw(\))27 b(=)g Fv(au)1384 5293 y Ft(2)1440 5323 y Fw(+)20 b Fv(h:o:t:)p Fw(\).)45 b(W)-7 b(e)30 b(consider)f(\014rst)h(the)g(easiest)f(case)h(of)f(an)h(o)r(dd)g (dominan)n(t)g(p)r(o)n(w)n(er)236 5423 y(for)d Fv(f)9 b Fw(\()p Fv(u)p Fw(\).)2084 6059 y(14)p eop %%Page: 15 15 15 14 bop 236 272 a Fb(4.1)113 b(Case)38 b(I:)f Fa(p)g Fb(o)s(dd)236 425 y Fw(In)27 b(this)f(case,)g(b)n(y)g(lemma)g(4.4,)g (the)g(reduced)g(action)g(functional)g(\010)2378 437 y Fu(")2440 425 y Fw(de\014ned)g(in)h(\(2.17\))e(can)h(b)r(e)h(written) f(in)h(the)f(form)236 525 y(\(3.1\))i(with)931 684 y Fv(\026)23 b Fw(=)g Fv(")f Fw(=)1251 628 y Fv(!)1306 598 y Ft(2)1361 628 y Fs(\000)c Fw(1)p 1251 665 235 4 v 1348 741 a(2)1496 684 y Fv(;)180 b(G)p Fw(\()p Fv(v)s Fw(\))24 b(:=)f Fv(a)2064 571 y Fq(Z)2109 759 y Ft(\012)2194 628 y Fv(v)2237 598 y Fu(p)p Ft(+1)p 2185 665 185 4 v 2185 741 a Fv(p)18 b Fw(+)g(1)2379 684 y Fv(;)346 b(a)23 b Fw(=)2913 628 y Fv(f)2963 598 y Ft(\()p Fu(p)p Ft(\))3053 628 y Fw(\(0\))p 2913 665 246 4 v 3004 741 a Fv(p)p Fw(!)3192 684 y Fs(6)p Fw(=)g(0)523 b(\(4.3\))236 879 y(\()p Fv(G)28 b Fw(is)g(homogeneous)e(of)h(degree)g Fv(p)18 b Fw(+)g(1\))28 b(and)977 1115 y Fv(R)q Fw(\()p Fv(v)s Fw(\))23 b(:=)g Fv(R)1345 1127 y Fu(!)r(;)p Ft(1)1446 1115 y Fw(\()p Fv(v)s Fw(\))c Fs(\000)1665 1059 y Fw(1)p 1665 1096 42 4 v 1665 1172 a(2)1731 1002 y Fq(Z)1777 1190 y Ft(\012)1842 1115 y Fv(f)9 b Fw(\()p Fv(v)s Fw(\))p Fv(L)2056 1080 y Fp(\000)p Ft(1)2145 1115 y Fw(\005)2207 1127 y Fu(W)2283 1115 y Fv(f)g Fw(\()p Fv(v)s Fw(\))19 b Fs(\000)2542 1002 y Fq(Z)2588 1190 y Ft(\012)2653 1023 y Fq(\020)2703 1115 y Fv(F)12 b Fw(\()p Fv(v)s Fw(\))19 b Fs(\000)f Fv(a)3040 1059 y(v)3083 1028 y Fu(p)p Ft(+1)p 3031 1096 185 4 v 3031 1172 a Fv(p)g Fw(+)g(1)3226 1023 y Fq(\021)236 1339 y Fw(satis\014es)1334 1471 y Fs(j)p Fv(D)r(R)q Fw(\()p Fv(v)s Fw(\)[)p Fv(v)s Fw(])p Fs(j)24 b(\024)e Fv(C)1887 1379 y Fq(\020)1937 1471 y Fs(j)p Fv(v)s Fs(j)2026 1437 y Fu(p)p Ft(+3)2026 1491 y Fu(!)2167 1471 y Fw(+)2265 1415 y Fv(")p 2260 1452 48 4 v 2260 1528 a(\015)2318 1471 y Fs(j)p Fv(v)s Fs(j)2407 1437 y Ft(2)p Fu(p)2407 1491 y(!)2497 1471 y Fw(+)2590 1415 y Fs(j)p Fv(v)s Fs(j)2679 1385 y Ft(3)p Fu(p)p Fp(\000)p Ft(1)2679 1435 y Fu(!)p 2590 1452 246 4 v 2689 1528 a Fv(\015)2846 1379 y Fq(\021)2895 1471 y Fv(:)927 b Fw(\(4.4\))236 1669 y(In)30 b(fact)509 1602 y Fq(R)578 1669 y Fv(f)9 b Fw(\()p Fv(v)s Fw(\))p Fv(v)23 b Fs(\000)d Fv(av)970 1639 y Fu(p)p Ft(+1)1120 1669 y Fw(=)26 b Fv(O)r Fw(\()p Fs(j)p Fv(v)s Fs(j)1397 1639 y Fu(p)p Ft(+3)1397 1690 y Fu(!)1521 1669 y Fw(\))31 b(since,)f(b)n(y)g(lemma)g(4.2,)2361 1602 y Fq(R)2430 1669 y Fv(v)2473 1639 y Fu(p)p Ft(+2)2622 1669 y Fw(=)d(0.)44 b(Moreo)n(v)n(er)3190 1602 y Fq(R)3259 1669 y Fv(f)3309 1639 y Fp(0)3332 1669 y Fw(\()p Fv(v)s Fw(\))p Fv(v)s(L)3539 1639 y Fp(\000)p Ft(1)3629 1669 y Fw(\005)3691 1681 y Fu(W)3767 1669 y Fv(f)9 b Fw(\()p Fv(v)s Fw(\))27 b(=)236 1769 y Fv(O)r Fw(\()p Fs(j)p Fv(f)406 1739 y Fp(0)430 1769 y Fw(\()p Fv(v)s Fw(\))p Fv(v)s Fs(j)603 1784 y Fu(L)649 1767 y Fd(2)686 1769 y Fs(j)p Fv(f)9 b Fw(\()p Fv(v)s Fw(\))p Fs(j)889 1784 y Fu(L)935 1767 y Fd(2)972 1769 y Fw(\))23 b(=)g Fv(O)r Fw(\()p Fs(j)p Fv(v)s Fs(j)1301 1739 y Ft(2)p Fu(p)1301 1789 y(!)1374 1769 y Fw(\))g(=)g Fv(O)r Fw(\()p Fs(j)p Fv(v)s Fs(j)1703 1739 y Fu(p)p Ft(+3)1703 1789 y Fu(!)1826 1769 y Fw(\))28 b(since)g Fv(p)22 b Fs(\025)h Fw(3.)361 1892 y(If)33 b Fv(a)f(>)g Fw(0,)j(the)e(functional)g Fv(G)p Fw(\()p Fs(\001)p Fw(\))h(de\014ned)g (in)f(\(4.3\))g(is)g(p)r(ositiv)n(e)g(\()p Fv(p)22 b Fw(+)g(1)33 b(is)g(ev)n(en\))g(and)g(then)g(condition)g(\(3.2\))g(is) 236 1992 y(v)n(eri\014ed.)51 b(On)32 b(the)h(con)n(trary)-7 b(,)32 b(if)h Fv(a)f(<)e Fw(0,)k(w)n(e)e(will)h(lo)r(ok)f(for)g (critical)g(p)r(oin)n(ts)g(of)h Fs(\000)p Fw(\010)2938 2004 y Fu(")3005 1992 y Fw(and)g(w)n(e)f(will)h(tak)n(e)e Fv(")g(<)g Fw(0,)j(i.e.)236 2091 y Fv(!)26 b(<)d Fw(1.)36 b(F)-7 b(or)27 b(de\014niteness)h(w)n(e)f(will)h(assume)f(in)h(the)g (sequel)f(that)h Fv(a)23 b(>)f Fw(0)27 b(and)h(so)f Fv(")c(>)f Fw(0.)361 2215 y(W)-7 b(e)28 b(can)f(easily)g(pro)n(v)n(e)f(that)236 2380 y Fx(Lemma)32 b(4.6)40 b Fv(D)r(G)24 b Fw(:)f Fv(V)42 b Fs(!)23 b Fv(V)1207 2350 y Fp(\003)1275 2380 y Fo(and)30 b Fv(D)r(R)24 b Fw(:)f Fs(D)i(!)e Fv(V)1902 2350 y Fp(\003)1970 2380 y Fo(ar)l(e)30 b(c)l(omp)l(act)g(maps.)236 2546 y Fc(Pr)n(oof)p Fw(.)87 b(Let)31 b Fs(jj)p Fv(v)844 2558 y Fu(k)885 2546 y Fs(jj)g Fw(b)r(e)g(b)r(ounded.)46 b(Then,)31 b(up)g(to)f(a)g(subsequence,)h Fv(v)2536 2558 y Fu(k)2605 2546 y Fv(*)p 2716 2500 44 4 v 28 w(v)g Fs(2)d Fv(V)50 b Fw(w)n(eakly)29 b(in)i Fv(H)3421 2516 y Ft(1)3489 2546 y Fw(and)f Fv(v)3693 2558 y Fu(k)3762 2546 y Fs(!)p 3873 2500 V 28 w Fv(v)k Fw(in)236 2645 y Fs(j)20 b(\001)h(j)346 2657 y Fp(1)416 2645 y Fw(.)45 b(Moreo)n(v)n(er)27 b(since)j Fv(w)1116 2657 y Fu(k)1184 2645 y Fw(:=)d Fv(w)r Fw(\()p Fv(v)1432 2657 y Fu(k)1474 2645 y Fw(\))k(to)r(o)f(is)g(b)r(ounded)g(w) n(e)g(can)g(also)f(assume)g(that)i Fv(w)3092 2657 y Fu(k)3160 2645 y Fv(*)p 3271 2600 62 4 v 28 w(w)h Fw(w)n(eakly)d(in)i Fv(H)3815 2615 y Ft(1)3882 2645 y Fw(and)236 2745 y Fv(w)295 2757 y Fu(k)360 2745 y Fs(!)p 466 2699 V 23 w Fv(w)e Fw(in)d Fs(j)17 b(\001)f(j)751 2757 y Fu(L)797 2741 y Fe(q)860 2745 y Fw(norm)26 b(for)g(all)g Fv(q)g(<)c Fs(1)p Fw(.)37 b(Since)27 b Fv(D)r(G)p Fw(\()p Fv(v)s Fw(\)[)p Fv(h)p Fw(])d(=)2273 2678 y Fq(R)2313 2775 y Ft(\012)2378 2745 y Fv(av)2465 2715 y Fu(p)2503 2745 y Fv(h)p Fw(,)j(it)g(follo)n (ws)e(easily)h(that)h Fv(D)r(G)p Fw(\()p Fv(v)3569 2757 y Fu(k)3610 2745 y Fw(\))d Fs(!)f Fv(D)r(G)p Fw(\()p 3940 2699 44 4 v Fv(v)s Fw(\))236 2856 y(in)37 b Fv(V)409 2826 y Fp(\003)447 2856 y Fw(.)65 b(W)-7 b(e)37 b(claim)g(that)g Fv(D)r(R)q Fw(\()p Fv(v)1314 2868 y Fu(k)1355 2856 y Fw(\))i Fs(!)p 1548 2789 64 4 v 39 w Fv(R)e Fw(where)p 1898 2789 V 37 w Fv(R)q Fw([)p Fv(h)p Fw(])h(=)2197 2789 y Fq(R)2236 2886 y Ft(\012)2302 2856 y Fv(f)9 b Fw(\()p 2384 2810 44 4 v Fv(v)27 b Fw(+)p 2541 2810 62 4 v 25 w Fv(w)r Fw(\))p Fv(h)e Fs(\000)f Fv(a)p 2840 2810 44 4 v(v)2883 2824 y Fu(p)2922 2856 y Fv(h)38 b Fw(=)g Fv(D)r(R)q Fw(\()p 3278 2810 V Fv(v)s Fw(\)[)p Fv(h)p Fw(].)66 b(Indeed,)39 b(since)236 2956 y Fv(w)295 2968 y Fu(k)360 2956 y Fs(!)p 466 2910 62 4 v 23 w Fv(w)26 b Fw(in)e Fs(j)10 b(\001)g(j)733 2968 y Fu(L)779 2952 y Fe(q)816 2956 y Fw(,)24 b(it)g(con)n(v)n(erges)d (\(up)j(to)g(a)f(subsequence\))g(also)f Fv(a:e)p Fw(.)35 b(W)-7 b(e)24 b(can)f(deduce,)i(b)n(y)e(the)h(Leb)r(esgue)f(dominated) 236 3055 y(con)n(v)n(ergence)d(theorem,)j(that)f Fv(f)9 b Fw(\()p Fv(v)1328 3067 y Fu(k)1376 3055 y Fw(+)e Fv(w)1507 3067 y Fu(k)1549 3055 y Fw(\))23 b Fs(!)g Fv(f)9 b Fw(\()p 1792 3010 44 4 v Fv(v)i Fw(+)p 1915 3010 62 4 v 7 w Fv(w)r Fw(\))23 b(in)f Fv(L)2179 3025 y Ft(2)2238 3055 y Fw(since)g Fv(f)9 b Fw(\()p Fv(v)2558 3067 y Fu(k)2606 3055 y Fw(+)e Fv(w)2737 3067 y Fu(k)2779 3055 y Fw(\))23 b Fs(!)g Fv(f)9 b Fw(\()p 3022 3010 44 4 v Fv(v)i Fw(+)p 3145 3010 62 4 v 7 w Fv(w)r Fw(\))23 b(a.e.)34 b(and)22 b(\()p Fv(f)9 b Fw(\()p Fv(v)3730 3067 y Fu(k)3779 3055 y Fw(+)e Fv(w)3910 3067 y Fu(k)3951 3055 y Fw(\)\))236 3155 y(is)28 b(b)r(ounded)g(in)g Fv(L)813 3125 y Fp(1)882 3155 y Fw(.)37 b(Hence,)28 b(since)g Fv(D)r(R)q Fw(\()p Fv(v)1623 3167 y Fu(k)1664 3155 y Fw(\)[)p Fv(h)p Fw(])23 b(=)1901 3088 y Fq(R)1940 3185 y Ft(\012)2005 3155 y Fv(f)9 b Fw(\()p Fv(v)2127 3167 y Fu(k)2187 3155 y Fw(+)18 b Fv(w)2329 3167 y Fu(k)2370 3155 y Fw(\))p Fv(h)h Fs(\000)f Fv(av)2639 3115 y Fu(p)2636 3180 y(k)2677 3155 y Fv(h)p Fw(,)28 b Fv(D)r(R)q Fw(\()p Fv(v)2983 3167 y Fu(k)3024 3155 y Fw(\))23 b Fs(!)p 3185 3088 64 4 v 23 w Fv(R)q Fw(.)p 3309 3155 48 48 v 361 3304 a Fv(G)p Fw(\()p Fv(v)s Fw(\))28 b(is)g(nonnegativ)n(e)e(for)h (all)h Fv(v)e Fs(2)d Fv(V)47 b Fw(and)27 b(it)h(is)g(easy)e(to)i(see)f (that)1475 3486 y Fv(G)p Fw(\()p Fs(L)1629 3498 y Fu(n)1675 3486 y Fw(\()p Fv(v)s Fw(\)\))d(=)f Fv(G)p Fw(\()p Fv(v)s Fw(\))p Fv(;)98 b Fs(8)p Fv(v)27 b Fs(2)c Fv(V)5 b(;)42 b(n)22 b Fs(2)i Fx(N)p Fv(:)1068 b Fw(\(4.5\))236 3668 y(Hence)1034 3800 y(\010)1094 3812 y Fu(";n)1190 3800 y Fw(\()p Fv(v)s Fw(\))24 b(:=)e(\010)1491 3812 y Fu(")1527 3800 y Fw(\()p Fs(L)1616 3812 y Fu(n)1662 3800 y Fv(v)s Fw(\))h(=)1858 3744 y Fv("n)1947 3714 y Ft(2)p 1858 3781 126 4 v 1900 3857 a Fw(2)1994 3800 y Fs(jj)p Fv(v)s Fs(jj)2129 3766 y Ft(2)2185 3800 y Fs(\000)18 b Fv(G)p Fw(\()p Fv(v)s Fw(\))h(+)f Fv(R)2605 3812 y Fu(n)2650 3800 y Fw(\()p Fv(v)s Fw(\))p Fv(;)181 b Fs(8)p Fv(v)27 b Fs(2)c(D)236 3977 y Fw(where)k Fv(R)539 3989 y Fu(n)585 3977 y Fw(\()p Fv(v)s Fw(\))c(:=)g Fv(R)q Fw(\()p Fs(L)979 3989 y Fu(n)1024 3977 y Fv(v)s Fw(\).)38 b(Of)28 b(course)e Fv(D)r(R)1666 3989 y Fu(n)1739 3977 y Fw(\(as)h Fv(D)r(R)q Fw(\))h(is)f(compact.)37 b(Moreo)n(v)n(er,)25 b(b)n(y)i(\(4.4\))g(and)h(\(4.2\))285 4221 y Fs(j)p Fv(D)r(R)442 4233 y Fu(n)487 4221 y Fw(\()p Fv(v)s Fw(\)[)p Fv(v)s Fw(])p Fs(j)c Fw(=)f Fs(j)p Fv(D)r(R)q Fw(\()p Fs(L)1065 4233 y Fu(n)1110 4221 y Fv(v)s Fw(\)[)p Fs(L)1265 4233 y Fu(n)1311 4221 y Fv(v)s Fw(])p Fs(j)h(\024)e Fv(C)1576 4129 y Fq(\020)1626 4221 y Fs(j)p Fv(v)s Fs(j)1715 4186 y Fu(p)p Ft(+3)1715 4241 y Fu(!)r(;n)1857 4221 y Fw(+)1954 4165 y Fv(")p 1950 4202 48 4 v 1950 4278 a(\015)2007 4221 y Fs(j)p Fv(v)s Fs(j)2096 4186 y Ft(2)p Fu(p)2096 4241 y(!)r(;n)2224 4221 y Fw(+)2317 4156 y Fs(j)p Fv(v)s Fs(j)2406 4126 y Ft(3)p Fu(p)p Fp(\000)p Ft(1)2406 4177 y Fu(!)r(;n)p 2317 4202 246 4 v 2416 4278 a Fv(\015)2572 4129 y Fq(\021)2645 4221 y Fs(\024)h Fv(C)2798 4129 y Fq(\020)2848 4221 y Fs(jj)p Fv(v)s Fs(jj)2983 4186 y Fu(p)p Ft(+3)3124 4221 y Fw(+)3221 4165 y Fv(")p 3217 4202 48 4 v 3217 4278 a(\015)3274 4221 y Fs(jj)p Fv(v)s Fs(jj)3409 4186 y Ft(2)p Fu(p)3500 4221 y Fw(+)3593 4165 y Fs(jj)p Fv(v)s Fs(jj)3728 4134 y Ft(3)p Fu(p)p Fp(\000)p Ft(1)p 3593 4202 292 4 v 3715 4278 a Fv(\015)3895 4129 y Fq(\021)3944 4221 y Fv(:)3845 4364 y Fw(\(4.6\))361 4464 y(W)-7 b(e)22 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242 y Ft(2)1117 272 y Fw(\).)37 b(The)28 b(second)f(condition)g(in)h(\(3.3\))f(is)h (trivially)f(satis\014ed)g(b)n(y)g(the)h(c)n(hoice)f(of)g Fv(r)3660 284 y Ft(0)3698 272 y Fw(.)361 372 y(Let)i(us)g(de\014ne,)g (for)g Fv(!)f Fs(2)d(W)32 b Fw(:=)25 b Fs([)1454 384 y Fu(\015)t(>)p Ft(0)1582 372 y Fv(W)1660 384 y Fu(\015)1703 372 y Fw(,)k Fv(\015)1798 384 y Fu(!)1872 372 y Fw(:=)c(max)o Fs(f)p Fv(\015)33 b Fs(j)c Fv(!)f Fs(2)e Fv(W)2548 384 y Fu(\015)2591 372 y Fs(g)p Fw(.)40 b(By)29 b(\(4.5\),)g(condition)g (\(4.1\))f(is)h(satis\014ed.)236 471 y(As)g(a)f(conclusion,)f(using)h (Prop)r(osition)f(3.1)h(and)g(lemma)g(4.3)g(w)n(e)g(obtain)g(the)h (follo)n(wing)e(result)h(\(w)n(e)g(obtain)g(pairs)g(of)236 571 y(solutions)f(b)r(ecause)g(\010)950 583 y Fu(")1013 571 y Fw(is)h(ev)n(en;)f(if)h Fv(u)p Fw(\()p Fv(t;)14 b(x)p Fw(\))28 b(is)g(a)f(solution)g(of)h(\(1.1\))f(then)h(so)f(is)g Fv(u)p Fw(\()p Fv(t)19 b Fw(+)f Fv(\031)s(;)c(\031)22 b Fs(\000)c Fv(x)p Fw(\))28 b(\))236 746 y 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y Fo(solutions)f(of)g(minimal)h(p)l(erio)l(d)g Fw(2)p Fv(\031)s(=)p Fw(\()p Fv(n!)s Fw(\))p Fo(.)361 1569 y Fw(The)26 b(regularit)n(y)f(of)h(the)h(solutions)f Fv(u)1537 1581 y Fu(n)1608 1569 y Fw(is)h(pro)n(v)n(ed)e(in)i(section)f(5.)36 b(By)26 b(condition)g(\(4.7\))h(w)n(e)f(can)g(\014nd)h Fv(N)3627 1581 y Fu(!)3698 1569 y Fs(\031)22 b Fv(O)r Fw(\()p Fv(\015)3925 1581 y Fu(!)3974 1569 y Fv(=)236 1598 y Fq(p)p 319 1598 244 4 v 71 x Fs(j)p Fv(!)f Fs(\000)d Fw(1)p Fs(j)p Fw(\))23 b(p)r(erio)r(dic)g(solutions)g(of)g(\(1.1\))f (with)i(p)r(erio)r(d)f(2)p Fv(\031)s(=!)s Fw(.)34 b(F)-7 b(or)23 b(a)f(discussion)h(on)f(the)i(optimalit)n(y)e(of)h(the)h (estimate)236 1768 y(\(4.7\))k(see)f(remark)f(4.5.)236 1944 y Fx(Remark)32 b(4.1)41 b Fo(The)25 b(unique)g(smal)t(l)g (amplitude)h Fw(2)p Fv(\031)i Fo(p)l(erio)l(dic)f(in)d(time)h(solution) g(of)h(\(1.1\))g(is)f(the)g(e)l(quilibrium)g(solution)236 2043 y Fv(u)g Fs(\021)f Fw(0)p Fo(.)42 b(Inde)l(e)l(d)31 b(it)g(must)f(b)l(e)h(a)g(critic)l(al)h(p)l(oint)f(of)h(the)f (functional)g Fw(\010)2414 2055 y Ft(0)2452 2043 y Fw(\()p Fv(v)s Fw(\))26 b(=)e(\()p Fv(a=p)19 b Fw(+)g(1\))3025 1976 y Fq(R)3094 2043 y Fv(v)3137 2013 y Fu(p)p Ft(+1)3278 2043 y Fw(+)g Fv(R)q Fw(\()p Fv(v)s Fw(\))p Fo(.)43 b(Using)31 b(\(4.4\))236 2143 y(we)f(se)l(e)g(that)g(its)f(unique)h(critic)l(al)h (p)l(oint)f(is)g Fv(v)c Fw(=)d(0)p Fo(.)236 2374 y Fb(4.2)113 b(Case)38 b(I)s(I:)f Fa(p)g Fb(ev)m(en)236 2527 y Fw(The)d(case)f(in)h (whic)n(h)f Fv(f)43 b Fw(satis\014es)33 b(\()p Fv(F)12 b Fw(1\))33 b(for)h(some)f(ev)n(en)g Fv(p)g Fw(is)h(more)f(di\016cult)h (to)g(deal)f(with)i(since,)g(b)n(y)e(lemma)h(4.2,)236 2560 y Fq(R)275 2656 y Ft(\012)341 2627 y Fv(v)384 2597 y Fu(p)p Ft(+1)530 2627 y Fw(=)22 b(0,)j Fs(8)p Fv(v)i Fs(2)c Fv(V)c Fw(.)36 b(Then)25 b(one)g(m)n(ust)g(lo)r(ok)f(more)h (carefully)f(for)g(the)i(dominan)n(t)e(nonlinear)h(term)f(in)i(the)f (reduced)236 2726 y(functional)j(\010)686 2738 y Fu(")721 2726 y 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Fv(v)s Fw(\))f(:=)236 4676 y(\()p Fv(b=d)18 b Fw(+)g(1\))592 4609 y Fq(R)631 4706 y Ft(\012)696 4676 y Fv(v)739 4646 y Fu(d)p Ft(+1)862 4676 y Fw(,)28 b(where)f Fv(b)c Fw(:=)g Fv(f)1373 4646 y Ft(\()p Fu(d)p Ft(\))1463 4676 y Fw(\(0\))p Fv(=d)p Fw(!,)k(and)h(b)n(y)f(lemma)h(4.4)e(\()i(since)2669 4609 y Fq(R)2708 4706 y Ft(\012)2773 4676 y Fv(v)2816 4646 y Fu(d)p Ft(+2)2962 4676 y Fw(=)23 b(0\))1333 4914 y Fs(j)p Fv(D)r(R)q Fw(\()p Fv(v)s Fw(\)[)p Fv(v)s Fw(])p Fs(j)h(\024)f Fv(C)1887 4821 y Fq(\020)1937 4914 y Fs(j)p Fv(v)s Fs(j)2026 4879 y Fu(d)p Ft(+3)2026 4934 y Fu(!)2167 4914 y Fw(+)2265 4857 y Fv(")p 2260 4894 48 4 v 2260 4970 a(\015)2318 4914 y Fs(j)p Fv(v)s Fs(j)2407 4879 y Ft(2)p Fu(p)2407 4934 y(!)2497 4914 y Fw(+)2590 4857 y Fs(j)p Fv(v)s Fs(j)2679 4827 y Ft(3)p Fu(p)p Fp(\000)p Ft(1)2679 4878 y Fu(!)p 2590 4894 246 4 v 2689 4970 a Fv(\015)2846 4821 y Fq(\021)2896 4914 y Fv(:)236 5133 y Fw(As)28 b(b)r(efore)g Fv(G)c Fs(\025)f Fw(0)k(and)h Fv(G)p Fw(\()p Fs(L)1171 5145 y Fu(n)1217 5133 y Fv(v)s Fw(\))c(=)g Fv(G)p Fw(\()p Fv(v)s Fw(\).)39 b(Applying)28 b(the)g(same)f(argumen)n(ts)g(as)g(in)i(the)f(previous)f(subsection,)g (with)236 5233 y Fv(q)f Fw(=)d Fv(d)p Fw(,)28 b Fv(r)518 5245 y Ft(0)579 5233 y Fw(=)22 b(\(1)p Fv(=C)841 5245 y Ft(0)878 5233 y Fw(\)\()p Fs(j)p Fv(")p Fs(j)p Fv(n)1077 5202 y Ft(2)1115 5233 y Fv(=m)p Fw(\))1262 5202 y Ft(1)p Fu(=d)p Fp(\000)p Ft(1)1452 5233 y Fw(,)28 b(and)1413 5464 y Fv(\013)23 b Fw(=)g Fv(O)1642 5372 y Fq(\020)1692 5464 y Fv(r)1729 5476 y Ft(0)1767 5430 y(2)1822 5464 y Fw(+)1915 5408 y Fs(j)p Fv(")p Fs(j)p 1915 5445 85 4 v 1934 5521 a Fv(\015)2010 5464 y(r)2047 5476 y Ft(0)2085 5430 y(2)p Fu(p)p Fp(\000)p Ft(1)p Fp(\000)p Fu(d)2346 5464 y Fw(+)2439 5408 y Fv(r)2476 5420 y Ft(0)2514 5378 y(3)p Fu(p)p Fp(\000)p Ft(2)p Fp(\000)p Fu(d)p 2439 5445 318 4 v 2574 5521 a Fv(\015)2767 5372 y Fq(\021)2816 5464 y Fv(;)2084 6059 y Fw(16)p eop %%Page: 17 17 17 16 bop 236 272 a Fw(w)n(e)31 b(obtain)g(that)h(condition)f(\(4.8\))g (ensures)g(that)h Fv(B)1921 284 y Fu(r)1952 292 y Fd(0)2017 272 y Fs(\032)d(D)34 b Fw(and)d(that)h(conditions)f(\(3.3\))g(hold,)h (whic)n(h)g(implies)f(the)236 372 y(existence)d(of)f(a)g(pair)g(of)h (critical)f(p)r(oin)n(ts)g(of)h(\010)1700 384 y Fu(";n)1824 372 y Fw(\(observ)n(e)e(that)i(2)p Fv(p)17 b Fs(\000)h Fw(1)g Fs(\000)g Fv(d)24 b Fs(\025)e Fw(2\).)p 2947 372 48 48 v 361 520 a(When)32 b(\()p Fv(N)9 b Fw(1\))32 b(is)f(not)h (satis\014ed)f(the)h(situation)g(is)f(more)g(delicate.)49 b(W)-7 b(e)33 b(shall)e(\014rst)g(consider)g(the)h(case)f(in)h(whic)n (h)236 620 y(\()p Fv(N)9 b Fw(2\))28 b(holds.)36 b(W)-7 b(e)28 b(can)f(write,)h(since)f(\005)1491 632 y Fu(W)1567 620 y Fv(v)1610 589 y Fu(p)1672 620 y Fw(=)22 b Fv(v)1802 589 y Fu(p)1841 620 y Fw(,)1433 786 y(1)p 1433 823 42 4 v 1433 899 a(2)1499 729 y Fq(Z)1545 918 y Ft(\012)1610 842 y Fv(f)9 b Fw(\()p Fv(v)s Fw(\))p Fv(L)1824 808 y Fp(\000)p Ft(1)1913 842 y Fw(\005)1975 854 y Fu(W)2051 842 y Fv(f)g Fw(\()p Fv(v)s Fw(\))23 b(=)g Fs(\000)p Fv(G)p Fw(\()p Fv(v)s Fw(\))c(+)f Fv(r)r Fw(\()p Fv(v)s Fw(\))p Fv(;)236 1064 y Fw(with)1054 1196 y Fv(G)p Fw(\()p Fv(v)s Fw(\))25 b(=)d Fs(\000)1413 1140 y Fv(a)1457 1110 y Ft(2)p 1413 1177 82 4 v 1433 1253 a Fw(2)1518 1083 y Fq(Z)1564 1272 y Ft(\012)1629 1196 y Fv(v)1672 1162 y Fu(p)1711 1196 y Fv(L)1768 1162 y Fp(\000)p Ft(1)1856 1196 y Fv(v)1899 1162 y Fu(p)2104 1196 y Fw(and)166 b Fv(D)r(r)r Fw(\()p Fv(v)s Fw(\)[)p Fv(v)s Fw(])25 b(=)d Fv(O)r Fw(\()p Fs(j)p Fv(v)s Fs(j)3008 1162 y Ft(2)p Fu(p)p Ft(+2)3008 1216 y Fu(!)3165 1196 y Fw(\))648 b(\(4.9\))236 1402 y(\(recall)24 b(that)i Fv(f)715 1372 y Ft(\()p Fu(p)p Ft(+1\))888 1402 y Fw(\(0\))e(=)e(0)j(b)n(y)f(\()p Fv(N)9 b Fw(2\)\).)36 b(Hence,)26 b(b)n(y)f(lemma)f(4.4,)h(w)n(e)f(ha)n(v)n(e) g Fs(\000)p Fw(\010)2790 1414 y Fu(")2825 1402 y Fw(\()p Fv(v)s Fw(\))g(=)f(\()p Fs(\000)p Fv("=)p Fw(2\))p Fs(jj)p Fv(v)s Fs(jj)3431 1372 y Ft(2)3480 1402 y Fs(\000)13 b Fv(G)p Fw(\()p Fv(v)s Fw(\))g(+)g Fv(R)q Fw(\()p Fv(v)s Fw(\))p Fv(;)236 1501 y Fw(with)1340 1633 y Fv(D)r(R)q Fw(\()p Fv(v)s Fw(\)[)p Fv(v)s Fw(])24 b(=)e Fv(O)1847 1541 y Fq(\020)1898 1633 y Fs(j)p Fv(v)s Fs(j)1987 1599 y Ft(2)p Fu(p)p Ft(+2)1987 1654 y Fu(!)2161 1633 y Fw(+)2258 1577 y Fv(")p 2254 1614 48 4 v 2254 1690 a(\015)2311 1633 y Fs(j)p Fv(v)s Fs(j)2400 1599 y Ft(2)p Fu(p)2400 1654 y(!)2491 1633 y Fw(+)2584 1577 y Fs(j)p Fv(v)s Fs(j)2673 1547 y Ft(3)p Fu(p)p Fp(\000)p Ft(1)2673 1598 y Fu(!)p 2584 1614 246 4 v 2683 1690 a Fv(\015)2839 1541 y Fq(\021)2889 1633 y Fv(:)892 b Fw(\(4.10\))236 1821 y Fv(D)r(G;)14 b(D)r(R)24 b Fw(:)f Fv(V)42 b Fs(!)23 b Fv(V)876 1791 y Fp(\003)942 1821 y Fw(are)k(still)h(compact)f(op)r(erators)e(and)j (no)n(w)f Fv(G)h Fw(is)f(homogeneous)f(of)h(degree)g Fv(q)21 b Fw(+)e(1)j(:=)h(2)p Fv(p)p Fw(.)361 1944 y(A)g(di\016cult)n (y)h(arises)e(for)g(pro)n(ving)g(that)h Fv(G)h Fw(satis\014es)e (\(4.1\).)35 b(The)24 b(follo)n(wing)e(lemmas)h(will)g(enable)g(to)g (get)g(a)g(suitable)236 2044 y(expression)j(of)i Fv(G)p Fw(\()p Fv(v)s Fw(\).)236 2224 y Fx(Lemma)k(4.7)40 b Fo(L)l(et)29 b Fv(w)r Fw(\()p Fv(t;)14 b(x)p Fw(\))25 b Fs(2)e Fv(W)41 b Fo(b)l(e)30 b(of)g(the)g(form)g Fv(w)r Fw(\()p Fv(t;)14 b(x)p Fw(\))24 b(=)f Fv(m)p Fw(\()p Fv(t)18 b Fw(+)f Fv(x;)d(t)k Fs(\000)f Fv(x)p Fw(\))30 b Fo(for)g(some)g Fv(m)23 b Fw(:)g Fx(R)3346 2193 y Ft(2)3406 2224 y Fs(!)g Fx(R)29 b Fw(2)p Fv(\031)s Fo(-p)l(erio)l(dic)236 2323 y(with)i(r)l(esp)l(e)l(ct)e(to)h(b)l(oth)g(variables.)41 b(Then)30 b(ther)l(e)g(exists)43 b Fq(e)-60 b Fv(m)p Fo(,)30 b Fv(a)g Fw(2)18 b Fs(\000)g Fv(\031)33 b Fo(p)l(erio)l(dic)f (such)d(that)h Fv(m)g Fo(c)l(an)g(b)l(e)f(de)l(c)l(omp)l(ose)l(d)j(as) 1369 2503 y Fv(m)p Fw(\()p Fv(s)1513 2515 y Ft(1)1550 2503 y Fv(;)14 b(s)1626 2515 y Ft(2)1663 2503 y Fw(\))24 b(=)36 b Fq(e)-60 b Fv(m)p Fw(\()p Fv(s)1950 2515 y Ft(1)1988 2503 y Fv(;)14 b(s)2064 2515 y Ft(2)2101 2503 y Fw(\))k(+)g Fv(a)p Fw(\()p Fv(s)2349 2515 y Ft(1)2387 2503 y Fw(\))g(+)h Fv(a)p Fw(\()p Fv(s)2636 2515 y Ft(2)2673 2503 y Fw(\))g(+)f Fv(\013;)921 b Fw(\(4.11\))236 2683 y Fo(wher)l(e)31 b Fv(\013)23 b Fs(2)h Fx(R)29 b Fo(\()p Fv(\013)23 b Fw(=)g Fs(h)p Fv(m)p Fs(i)p Fo(\),)30 b Fs(h)p Fv(a)p Fs(i)24 b Fw(=)f(0)p Fo(,)29 b Fs(h)13 b Fq(e)-59 b Fv(m)q Fs(i)1605 2695 y Fu(s)1636 2703 y Fd(1)1673 2683 y Fw(\()p Fv(s)1744 2695 y Ft(2)1781 2683 y Fw(\))23 b(=)g(0)p Fo(,)30 b Fs(h)13 b Fq(e)-59 b Fv(m)p Fs(i)2158 2695 y Fu(s)2189 2703 y Fd(2)2226 2683 y Fw(\()p Fv(s)2297 2695 y Ft(1)2335 2683 y Fw(\))23 b(=)g(0)p Fo(.)38 b(Mor)l(e)l(over)390 2809 y Fq(Z)473 2830 y Ft(2)p Fu(\031)436 2998 y Ft(0)565 2809 y Fq(Z)648 2830 y Fu(\031)611 2998 y Ft(0)707 2922 y Fv(L)764 2888 y Fp(\000)p Ft(1)852 2922 y Fw(\()p Fv(w)r Fw(\))p Fv(w)c(dt)c(dx)83 b Fw(=)g Fs(\000)1569 2866 y Fw(1)p 1569 2903 42 4 v 1569 2979 a(2)1634 2809 y Fq(Z)1717 2830 y Ft(2)p Fu(\031)1680 2998 y Ft(0)1809 2809 y Fq(Z)1892 2830 y Ft(2)p Fu(\031)1855 2998 y Ft(0)1984 2922 y Fv(M)9 b Fw(\()p Fv(s)2145 2934 y Ft(1)2182 2922 y Fv(;)14 b(s)2258 2934 y Ft(2)2295 2922 y Fw(\))f Fq(e)-59 b Fv(m)p Fw(\()p Fv(s)2471 2934 y Ft(1)2509 2922 y Fv(;)14 b(s)2585 2934 y Ft(2)2622 2922 y Fw(\))30 b Fv(ds)2766 2934 y Ft(1)2833 2922 y Fv(ds)2915 2934 y Ft(2)2971 2922 y Fw(+)18 b(2)p Fv(\031)3159 2809 y Fq(Z)3242 2830 y Ft(2)p Fu(\031)3205 2998 y Ft(0)3334 2922 y Fv(M)9 b Fw(\()p Fv(s;)14 b(s)p Fw(\))p Fv(a)p Fw(\()p Fv(s)p Fw(\))30 b Fv(ds)1346 3161 y Fw(+)83 b(2)p Fv(\031)s(\013)1653 3048 y Fq(Z)1736 3069 y Ft(2)p Fu(\031)1699 3237 y Ft(0)1828 3161 y Fv(M)9 b Fw(\()p Fv(s;)14 b(s)p Fw(\))30 b Fv(ds)18 b Fs(\000)g Fw(8)p Fv(\031)2416 3048 y Fq(Z)2499 3069 y Ft(2)p Fu(\031)2462 3237 y Ft(0)2591 3161 y Fv(A)p Fw(\()p Fv(s)p Fw(\))2756 3127 y Ft(2)2824 3161 y Fv(ds)g Fs(\000)3017 3105 y Fv(\013)3070 3075 y Ft(2)3108 3105 y Fv(\031)3158 3075 y Ft(4)p 3017 3142 179 4 v 3086 3218 a Fw(6)3205 3161 y Fv(;)236 3383 y Fo(wher)l(e)31 b Fv(M)38 b Fo(and)30 b Fv(A)g Fo(ar)l(e)g(the)g Fw(2)p Fv(\031)s Fo(-p)l(erio)l(dic)i(functions)d(de\014ne)l(d)h(by)374 3597 y Fv(A)436 3563 y Fp(0)459 3597 y Fw(\()p Fv(s)p Fw(\))24 b(=)684 3541 y(1)p 684 3578 42 4 v 684 3654 a(4)735 3597 y Fv(a)p Fw(\()p Fv(s)p Fw(\))p Fv(;)74 b Fs(h)p Fv(A)p Fs(i)24 b Fw(=)e(0)p Fv(;)73 b(@)1398 3609 y Fu(s)1429 3617 y Fd(1)1466 3597 y Fv(@)1510 3609 y Fu(s)1541 3617 y Fd(2)1578 3597 y Fv(M)9 b Fw(\()p Fv(s)1739 3609 y Ft(1)1776 3597 y Fv(;)14 b(s)1852 3609 y Ft(2)1889 3597 y Fw(\))23 b(=)2042 3541 y(1)p 2042 3578 V 2042 3654 a(4)2107 3597 y Fq(e)-59 b Fv(m)o Fw(\()p Fv(s)2237 3609 y Ft(1)2275 3597 y Fv(;)14 b(s)2351 3609 y Ft(2)2388 3597 y Fw(\))p Fv(;)74 b Fs(h)p Fv(M)9 b Fs(i)2671 3609 y Fu(s)2702 3617 y Fd(1)2739 3597 y Fw(\()p Fv(s)2810 3609 y Ft(2)2847 3597 y Fw(\))23 b(=)g(0)p Fv(;)73 b Fs(h)p Fv(M)9 b Fs(i)3282 3609 y Fu(s)3313 3617 y Fd(2)3350 3597 y Fw(\()p Fv(s)3421 3609 y Ft(1)3458 3597 y Fw(\))24 b(=)e(0)p Fv(:)138 b Fw(\(4.12\))236 3801 y Fc(Pr)n(oof)p Fw(.)78 b(The)28 b(pro)r(of)f(is)h(in)f(the)h(App) r(endix.)p 1721 3801 48 48 v 236 4029 a Fx(Lemma)k(4.8)40 b Fo(L)l(et)f Fv(m)h Fw(:)g Fx(R)1139 3999 y Ft(2)1216 4029 y Fs(!)h Fx(R)d Fo(b)l(e)i Fw(2)p Fv(\031)s Fo(-p)l(erio)l(dic)h (with)e(r)l(esp)l(e)l(ct)g(to)g(b)l(oth)h(variables)h(and)f(such)f (that)g Fv(w)r Fw(\()p Fv(t;)14 b(x)p Fw(\))42 b(=)236 4129 y Fv(m)p Fw(\()p Fv(t)19 b Fw(+)f Fv(x;)c(t)19 b Fs(\000)f Fv(x)p Fw(\))23 b Fs(2)h Fv(W)12 b Fo(.)38 b(F)-6 b(or)30 b Fv(n)23 b Fs(2)h Fx(N)29 b Fo(de\014ne)h Fw(\()p Fs(L)1765 4141 y Fu(n)1811 4129 y Fv(w)r Fw(\)\()p Fv(t;)14 b(x)p Fw(\))25 b(:=)d Fv(m)p Fw(\()p Fv(n)p Fw(\()p Fv(t)d Fw(+)f Fv(x)p Fw(\))p Fv(;)c(n)p Fw(\()p Fv(t)19 b Fs(\000)f Fv(x)p Fw(\)\))p Fo(.)40 b(Then)673 4255 y Fq(Z)756 4276 y Ft(2)p Fu(\031)719 4444 y Ft(0)848 4255 y Fq(Z)931 4276 y Fu(\031)894 4444 y Ft(0)990 4368 y Fv(L)1047 4334 y Fp(\000)p Ft(1)1136 4368 y Fw(\()p Fs(L)1225 4380 y Fu(n)1271 4368 y Fv(w)r Fw(\))p Fs(L)1421 4380 y Fu(n)1467 4368 y Fv(w)32 b(dx)f(dt)23 b Fw(=)g Fs(\000)1938 4312 y Fv(\031)1988 4282 y Ft(4)p 1937 4349 88 4 v 1960 4425 a Fw(6)2035 4368 y Fv(\013)2088 4334 y Ft(2)2144 4368 y Fw(+)2260 4312 y(1)p 2237 4349 V 2237 4425 a Fv(n)2287 4401 y Ft(2)2334 4276 y Fq(\020)2397 4255 y(Z)2480 4276 y Ft(2)p Fu(\031)2443 4444 y Ft(0)2572 4255 y Fq(Z)2655 4276 y Fu(\031)2618 4444 y Ft(0)2714 4368 y Fv(w)r(L)2832 4334 y Fp(\000)p Ft(1)2921 4368 y Fv(w)33 b(dx)d(dt)19 b Fw(+)3318 4312 y Fv(\031)3368 4282 y Ft(4)p 3318 4349 V 3341 4425 a Fw(6)3415 4368 y Fv(\013)3468 4334 y Ft(2)3506 4276 y Fq(\021)3556 4368 y Fv(:)225 b Fw(\(4.13\))236 4595 y Fc(Pr)n(oof)p Fw(.)77 b(W)-7 b(e)23 b(ha)n(v)n(e)e Fs(L)977 4607 y Fu(n)1022 4595 y Fv(w)r Fw(\()p Fv(t;)14 b(x)p Fw(\))25 b(=)d Fv(m)1446 4607 y Fu(n)1492 4595 y Fw(\()p Fv(t)8 b Fw(+)g Fv(x;)14 b(t)8 b Fs(\000)g Fv(x)p Fw(\),)24 b(with)f Fv(m)2213 4607 y Fu(n)2258 4595 y Fw(\()p Fv(s)2329 4607 y Ft(1)2367 4595 y Fv(;)14 b(s)2443 4607 y Ft(2)2480 4595 y Fw(\))23 b(:=)g Fv(m)p Fw(\()p Fv(ns)2840 4607 y Ft(1)2877 4595 y Fv(;)14 b(ns)3003 4607 y Ft(2)3040 4595 y Fw(\).)35 b(Using)23 b(the)f(decomp)r(osition)236 4694 y(giv)n(en)f(in)h(the)g (latter)f(lemma,)i(w)n(e)f(can)f(write)g Fv(m)1723 4706 y Fu(n)1769 4694 y Fw(\()p Fv(s)1840 4706 y Ft(1)1877 4694 y Fv(;)14 b(s)1953 4706 y Ft(2)1990 4694 y Fw(\))23 b(=)36 b Fq(e)-59 b Fv(m)2206 4706 y Fu(n)2251 4694 y Fw(\()p Fv(s)2322 4706 y Ft(1)2360 4694 y Fv(;)14 b(s)2436 4706 y Ft(2)2473 4694 y Fw(\))7 b(+)g Fv(a)2628 4706 y Fu(n)2672 4694 y Fw(\()p Fv(s)2743 4706 y Ft(1)2780 4694 y Fw(\))g(+)g Fv(a)2935 4706 y Fu(n)2980 4694 y Fw(\()p Fv(s)3051 4706 y Ft(2)3088 4694 y Fw(\))g(+)g Fv(\013)p Fw(,)23 b(where)34 b Fq(e)-59 b Fv(m)3605 4706 y Fu(n)3673 4694 y Fw(:=)23 b Fv(m)p Fw(\()p Fv(n)e Fs(\001)p Fw(\))236 4794 y(and)28 b Fv(a)442 4806 y Fu(n)510 4794 y Fw(:=)22 b Fv(a)p Fw(\()p Fv(n)28 b Fs(\001)p Fw(\).)37 b(Therefore)27 b(\(using)g(the)h(abbreviation)f Fv(w)2200 4806 y Fu(n)2268 4794 y Fw(:=)c Fs(L)2436 4806 y Fu(n)2481 4794 y Fv(w)r Fw(\))270 4920 y Fq(Z)353 4941 y Ft(2)p Fu(\031)317 5109 y Ft(0)445 4920 y Fq(Z)528 4941 y Fu(\031)492 5109 y Ft(0)587 5033 y Fv(L)644 4999 y Fp(\000)p Ft(1)733 5033 y Fw(\()p Fv(w)824 5045 y Fu(n)870 5033 y Fw(\))p Fv(w)961 5045 y Fu(n)1034 5033 y Fv(dx)29 b(dt)83 b Fw(=)g Fs(\000)1532 4977 y Fw(1)p 1532 5014 42 4 v 1532 5090 a(2)1596 4920 y Fq(Z)1680 4941 y Ft(2)p Fu(\031)1643 5109 y Ft(0)1771 4920 y Fq(Z)1855 4941 y Ft(2)p Fu(\031)1818 5109 y Ft(0)1947 5033 y Fv(M)2028 5045 y Fu(n)2072 5033 y Fw(\()p Fv(s)2143 5045 y Ft(1)2181 5033 y Fv(;)14 b(s)2257 5045 y Ft(2)2294 5033 y Fw(\))f Fq(e)-59 b Fv(m)2399 5045 y Fu(n)2444 5033 y Fw(\()p Fv(s)2515 5045 y Ft(1)2553 5033 y Fv(;)14 b(s)2629 5045 y Ft(2)2666 5033 y Fw(\))28 b Fv(ds)2808 5045 y Ft(1)2873 5033 y Fv(ds)2955 5045 y Ft(2)3010 5033 y Fw(+)18 b(2)p Fv(\031)3199 4920 y Fq(Z)3282 4941 y Ft(2)p Fu(\031)3245 5109 y Ft(0)3374 5033 y Fv(M)3455 5045 y Fu(n)3500 5033 y Fw(\()p Fv(s;)c(s)p Fw(\))p Fv(a)3723 5045 y Fu(n)3768 5033 y Fw(\()p Fv(s)p Fw(\))28 b Fv(ds)1309 5272 y Fw(+)83 b(2)p Fv(\031)s(\013)1616 5159 y Fq(Z)1699 5180 y Ft(2)p Fu(\031)1662 5348 y Ft(0)1791 5272 y Fv(M)1872 5284 y Fu(n)1916 5272 y Fw(\()p Fv(s;)14 b(s)p Fw(\))28 b Fv(ds)19 b Fs(\000)f Fw(8)p Fv(\031)2413 5159 y Fq(Z)2496 5180 y Ft(2)p Fu(\031)2459 5348 y Ft(0)2588 5272 y Fv(A)2650 5284 y Fu(n)2695 5272 y Fw(\()p Fv(s)p Fw(\))2798 5238 y Ft(2)2864 5272 y Fv(ds)g Fs(\000)3057 5216 y Fv(\013)3110 5186 y Ft(2)3148 5216 y Fv(\031)3198 5186 y Ft(4)p 3057 5253 179 4 v 3125 5329 a Fw(6)3245 5272 y Fv(;)236 5504 y Fw(where)27 b Fv(M)557 5516 y Fu(n)602 5504 y Fw(\()p Fv(s)673 5516 y Ft(1)710 5504 y Fv(;)14 b(s)786 5516 y Ft(2)824 5504 y Fw(\))23 b(:=)g Fv(M)9 b Fw(\()p Fv(ns)1201 5516 y Ft(1)1237 5504 y Fv(;)14 b(ns)1363 5516 y Ft(2)1400 5504 y Fw(\))p Fv(=n)1524 5474 y Ft(2)1589 5504 y Fw(and)27 b Fv(A)1812 5516 y Fu(n)1858 5504 y Fw(\()p Fv(s)p Fw(\))c(:=)g Fv(A)p Fw(\()p Fv(ns)p Fw(\))p Fv(=n)p Fw(.)37 b(\(4.13\))27 b(follo)n(ws)f(straigh)n (tforw)n(ardly)-7 b(.)p 3651 5504 48 48 v 2084 6059 a(17)p eop %%Page: 18 18 18 17 bop 361 272 a Fw(Lemma)31 b(4.7)f(allo)n(ws)g(to)i(obtain)f(a)g (handable)g(expression)e(for)i Fv(G)p Fw(\()p Fv(v)s Fw(\))g(=)e Fs(\000)p Fw(\()p Fv(a)2821 242 y Ft(2)2858 272 y Fv(=)p Fw(2\))2988 205 y Fq(R)3056 272 y Fv(v)3099 242 y Fu(p)3137 272 y Fv(L)3194 242 y Fp(\000)p Ft(1)3283 272 y Fv(v)3326 242 y Fu(p)3396 272 y Fw(when)j Fv(p)d Fw(=)g(2.)47 b(F)-7 b(or)236 372 y Fv(v)s Fw(\()p Fv(t;)14 b(x)p Fw(\))24 b(=)f Fv(\021)s Fw(\()p Fv(t)13 b Fw(+)g Fv(x)p Fw(\))g Fs(\000)g Fv(\021)s Fw(\()p Fv(t)g Fs(\000)g Fv(x)p Fw(\))22 b Fs(2)i Fv(V)43 b Fw(w)n(e)25 b(ha)n(v)n(e)e Fv(v)1755 341 y Ft(2)1793 372 y Fw(\()p Fv(t;)14 b(x)p Fw(\))24 b(=)e Fv(m)p Fw(\()p Fv(t)13 b Fw(+)g Fv(x;)h(t)f Fs(\000)g Fv(x)p Fw(\))22 b Fs(2)i Fv(W)36 b Fw(with)26 b Fv(m)p Fw(\()p Fv(s)3138 384 y Ft(1)3175 372 y Fv(;)14 b(s)3251 384 y Ft(2)3288 372 y Fw(\))23 b(=)g(\()p Fv(\021)s Fw(\()p Fv(s)3578 384 y Ft(1)3616 372 y Fw(\))13 b Fs(\000)g Fv(\021)s Fw(\()p Fv(s)3854 384 y Ft(2)3891 372 y Fw(\)\))3955 341 y Ft(2)3993 372 y Fw(.)236 471 y(W)-7 b(e)24 b(can)f(then)g(decomp) r(ose)g Fv(m)g Fw(as)g(in)g(\(4.11\))f(with)i Fv(\013)g Fw(=)e(2)p Fs(h)p Fv(\021)2109 441 y Ft(2)2147 471 y Fs(i)p Fw(,)i Fv(a)p Fw(\()p Fv(s)p Fw(\))g(=)e Fv(\021)2528 441 y Ft(2)2566 471 y Fw(\()p Fv(s)p Fw(\))10 b Fs(\000)g(h)p Fv(\021)2830 441 y Ft(2)2867 471 y Fs(i)23 b Fw(and)37 b Fq(e)-60 b Fv(m)p Fw(\()p Fv(s)3223 483 y Ft(1)3261 471 y Fv(;)14 b(s)3337 483 y Ft(2)3374 471 y Fw(\))23 b(=)g Fs(\000)p Fw(2)p Fv(\021)s Fw(\()p Fv(s)3739 483 y Ft(1)3775 471 y Fw(\))p Fv(\021)s Fw(\()p Fv(s)3922 483 y Ft(2)3960 471 y Fw(\).)236 571 y(Let)34 b(us)f(call)g Fv(P)714 583 y Ft(1)752 571 y Fw(,)i(resp.)54 b Fv(P)1089 583 y Ft(2)1127 571 y Fw(,)35 b(the)f(primitiv)n(e)f(of)g Fv(\021)s Fw(,)j(resp.)54 b Fv(\021)2168 541 y Ft(2)2228 571 y Fs(\000)22 b(h)p Fv(\021)2391 541 y Ft(2)2428 571 y Fs(i)p Fw(,)36 b(of)d(zero)g(mean-v)-5 b(alue.)54 b(The)33 b(functions)h Fv(A;)14 b(M)236 671 y Fw(de\014ned)27 b(in)g(\(4.12\))f(are)g Fv(A)p Fw(\()p Fv(s)p Fw(\))d(:=)g Fv(P)1346 683 y Ft(2)1384 671 y Fw(\()p Fv(s)p Fw(\))p Fv(=)p Fw(4)j(and)g Fv(M)9 b Fw(\()p Fv(s)1918 683 y Ft(1)1955 671 y Fv(;)14 b(s)2031 683 y Ft(2)2068 671 y Fw(\))24 b(=)e Fs(\000)p Fv(P)2329 683 y Ft(1)2367 671 y Fw(\()p Fv(s)2438 683 y Ft(1)2475 671 y Fw(\))p Fv(P)2560 683 y Ft(1)2598 671 y Fw(\()p Fv(s)2669 683 y Ft(2)2706 671 y Fw(\))p Fv(=)p Fw(2.)36 b(Hence,)27 b(b)n(y)g(lemma)f(4.7)g(w)n(e)g(obtain)236 770 y(the)i(follo)n(wing)f (expression)f(for)h Fv(G)p Fw(\()p Fv(v)s Fw(\))d(=)f Fs(\000)p Fw(\()p Fv(a)1682 740 y Ft(2)1719 770 y Fv(=)p Fw(2\))1849 703 y Fq(R)1917 770 y Fv(v)1960 740 y Ft(2)1997 770 y Fv(L)2054 740 y Fp(\000)p Ft(1)2143 770 y Fv(v)2186 740 y Ft(2)2251 770 y Fw(:)759 1013 y Fv(G)p Fw(\()p Fv(v)s Fw(\))h(=)1053 957 y Fv(a)1097 927 y Ft(2)p 1053 994 82 4 v 1073 1070 a Fw(2)1144 921 y Fq(h)1183 1013 y Fv(\031)1247 900 y Fq(Z)1330 921 y Ft(2)p Fu(\031)1294 1089 y Ft(0)1422 1013 y Fv(P)1475 1025 y Ft(1)1513 1013 y Fw(\()p Fv(s)p Fw(\))1616 979 y Ft(2)1654 921 y Fq(\020)1703 1013 y Fv(\021)s Fw(\()p Fv(s)p Fw(\))1850 979 y Ft(2)1907 1013 y Fw(+)18 b Fs(h)p Fv(\021)2066 979 y Ft(2)2103 1013 y Fs(i)2135 921 y Fq(\021)2213 1013 y Fv(ds)h Fw(+)2407 957 y Fv(\031)p 2407 994 51 4 v 2411 1070 a Fw(2)2481 900 y Fq(Z)2564 921 y Ft(2)p Fu(\031)2527 1089 y Ft(0)2656 1013 y Fv(P)2709 1025 y Ft(2)2746 1013 y Fw(\()p Fv(s)p Fw(\))2849 979 y Ft(2)2915 1013 y Fv(ds)f Fw(+)3108 957 y(2)p Fv(\031)3200 927 y Ft(4)p 3108 994 130 4 v 3152 1070 a Fw(3)3247 1013 y Fs(h)p Fv(\021)3323 979 y Ft(2)3361 1013 y Fs(i)3393 979 y Ft(2)3430 921 y Fq(i)3470 1013 y Fv(:)311 b Fw(\(4.14\))236 1239 y(F)-7 b(rom)27 b(\(4.14\))g(w)n(e)g (see)h(that)1053 1418 y Fs(8)p Fv(v)e Fs(2)d Fv(V)5 b(;)42 b(G)p Fw(\()p Fv(v)s Fw(\))24 b Fv(>)f Fw(0)165 b(and)h(0)23 b Fs(\024)f Fv(G)p Fw(\()p Fs(L)2458 1430 y Fu(n)2504 1418 y Fv(v)s Fw(\))i Fs(\024)f Fv(G)p Fw(\()p Fv(v)s Fw(\))p Fv(;)42 b Fs(8)p Fv(n)23 b Fs(\025)g Fw(2)p Fv(:)604 b Fw(\(4.15\))236 1598 y(In)28 b(general)e(w)n(e)h(can)h(pro)n(v)n(e)e (that)236 1761 y Fx(Lemma)32 b(4.9)40 b Fo(F)-6 b(or)30 b(any)g(even)g Fv(p)p Fo(,)g Fv(G)p Fw(\()p Fv(v)s Fw(\))24 b(:=)f Fs(\000)p Fw(\()p Fv(a)1788 1731 y Ft(2)1825 1761 y Fv(=)p Fw(2\))1955 1694 y Fq(R)1993 1790 y Ft(\012)2059 1761 y Fv(v)2102 1731 y Fu(p)2140 1761 y Fv(L)2197 1731 y Fp(\000)p Ft(1)2286 1761 y Fv(v)2329 1731 y Fu(p)2397 1761 y Fo(is)30 b(non)g(ne)l(gative.)236 1924 y Fc(Pr)n(oof)p Fw(.)83 b(W)-7 b(riting)30 b Fv(v)952 1894 y Fu(p)990 1924 y Fw(\()p Fv(t;)14 b(x)p Fw(\))27 b(=)e Fv(m)p Fw(\()p Fv(t)20 b Fw(+)f Fv(x;)14 b(t)20 b Fs(\000)f Fv(x)p Fw(\))30 b(with)g Fv(m)p Fw(\()p Fv(s)2186 1936 y Ft(1)2223 1924 y Fv(;)14 b(s)2299 1936 y Ft(2)2336 1924 y Fw(\))27 b(=)e(\()p Fv(\021)s Fw(\()p Fv(s)2632 1936 y Ft(1)2670 1924 y Fw(\))20 b Fs(\000)f Fv(\021)s Fw(\()p Fv(s)2921 1936 y Ft(2)2959 1924 y Fw(\)\))3023 1894 y Fu(p)3061 1924 y Fw(,)30 b(w)n(e)f(claim)g (that)h Fv(L)3700 1894 y Fp(\000)p Ft(1)3788 1924 y Fw(\()p Fv(v)3863 1894 y Fu(p)3902 1924 y Fw(\))c Fs(2)236 2024 y(\000)p Fw(\(1)p Fv(=)p Fw(8\))p Fv(M)9 b Fw(\()p Fv(t)17 b Fw(+)h Fv(x;)c(t)19 b Fs(\000)f Fv(x)p Fw(\))h(+)f Fv(V)h Fw(,)28 b(where)1505 2246 y Fv(M)9 b Fw(\()p Fv(s)1666 2258 y Ft(1)1703 2246 y Fv(;)14 b(s)1779 2258 y Ft(2)1816 2246 y Fw(\))23 b(:=)1982 2133 y Fq(Z)2028 2321 y Fu(R)2078 2329 y Fe(s)2106 2341 y Fd(1)2138 2329 y Fe(;s)2185 2341 y Fd(2)2240 2246 y Fv(m)p Fw(\()p Fv(\030)t(;)14 b(\027)5 b Fw(\))28 b Fv(d\030)k(d\027)q(;)1057 b Fw(\(4.16\))236 2497 y(and)28 b Fv(R)461 2509 y Fu(s)492 2517 y Fd(1)524 2509 y Fu(;s)575 2517 y Fd(2)635 2497 y Fw(:=)f Fs(f)p Fw(\()p Fv(\030)t(;)14 b(\027)5 b Fw(\))24 b Fs(2)f Fx(R)1153 2467 y Ft(2)1218 2497 y Fs(j)28 b Fv(s)1308 2509 y Ft(1)1368 2497 y Fs(\024)22 b Fv(\030)28 b Fs(\024)f Fv(s)1650 2509 y Ft(2)1705 2497 y Fw(+)18 b(2)p Fv(\031)s(;)42 b(s)1984 2509 y Ft(2)2044 2497 y Fs(\024)23 b Fv(\027)28 b Fs(\024)23 b Fv(s)2328 2509 y Ft(1)2365 2497 y Fs(g)p Fv(:)k Fw(In)h(fact)1148 2739 y Fv(@)1192 2751 y Fu(s)1223 2759 y Fd(1)1260 2739 y Fv(M)9 b Fw(\()p Fv(s)1421 2751 y Ft(1)1458 2739 y Fv(;)14 b(s)1534 2751 y Ft(2)1571 2739 y Fw(\))23 b(=)1714 2626 y Fq(Z)1797 2647 y Fu(s)1828 2655 y Fd(2)1861 2647 y Ft(+2)p Fu(\031)1760 2815 y(s)1791 2823 y Fd(1)2004 2739 y Fv(m)p Fw(\()p Fv(\030)t(;)14 b(s)2225 2751 y Ft(1)2262 2739 y Fw(\))28 b Fv(d\030)23 b Fs(\000)2507 2626 y Fq(Z)2590 2647 y Fu(s)2621 2655 y Fd(1)2553 2815 y Fu(s)2584 2823 y Fd(2)2672 2739 y Fv(m)p Fw(\()p Fv(s)2816 2751 y Ft(1)2853 2739 y Fv(;)14 b(\027)5 b Fw(\))28 b Fv(d\027)q(:)236 2973 y Fw(Hence,)g(since)f Fv(m)p Fw(\()p Fv(s)853 2985 y Ft(1)891 2973 y Fv(;)14 b(s)967 2985 y Ft(2)1004 2973 y Fw(\))23 b(=)g(\()p Fv(\021)s Fw(\()p Fv(s)1294 2985 y Ft(1)1332 2973 y Fw(\))c Fs(\000)f Fv(\021)s Fw(\()p Fv(s)1581 2985 y Ft(2)1618 2973 y Fw(\)\))1682 2943 y Fu(p)1749 2973 y Fw(with)28 b(ev)n(en)f Fv(p)p Fw(,)h(and)f Fv(\021)k Fw(is)c(2)p Fv(\031)s Fw(-p)r(erio)r(dic,)1189 3152 y Fv(@)1233 3164 y Fu(s)1264 3172 y Fd(2)1301 3152 y Fv(@)1345 3164 y Fu(s)1376 3172 y Fd(1)1413 3152 y Fv(M)k Fw(=)23 b Fv(m)p Fw(\()p Fv(s)1757 3164 y Ft(2)1813 3152 y Fw(+)18 b(2)p Fv(\031)s(;)c(s)2064 3164 y Ft(1)2101 3152 y Fw(\))k(+)g Fv(m)p Fw(\()p Fv(s)2378 3164 y Ft(1)2416 3152 y Fv(;)c(s)2492 3164 y Ft(2)2529 3152 y Fw(\))23 b(=)g(2)p Fv(m)p Fw(\()p Fv(s)2858 3164 y Ft(1)2895 3152 y Fv(;)14 b(s)2971 3164 y Ft(2)3008 3152 y Fw(\))p Fv(:)236 3332 y Fw(Moreo)n(v)n(er,)26 b(b)n(y)i(the)g(de\014nition)h(of)f Fv(M)9 b Fw(,)28 b Fv(M)9 b Fw(\()p Fv(s)1651 3344 y Ft(1)1688 3332 y Fv(;)14 b(s)1764 3344 y Ft(1)1801 3332 y Fw(\))25 b(=)e Fv(M)9 b Fw(\()p Fv(s)2107 3344 y Ft(1)2144 3332 y Fv(;)14 b(s)2220 3344 y Ft(1)2276 3332 y Fs(\000)19 b Fw(2)p Fv(\031)s Fw(\))24 b(=)f(0)28 b(and)g Fv(M)9 b Fw(\()p Fv(s)2989 3344 y Ft(1)3045 3332 y Fw(+)18 b(2)p Fv(\031)s(;)c(s)3296 3344 y Ft(2)3352 3332 y Fw(+)19 b(2)p Fv(\031)s Fw(\))24 b(=)f Fv(M)9 b Fw(\()p Fv(s)3833 3344 y Ft(1)3870 3332 y Fv(;)14 b(s)3946 3344 y Ft(2)3983 3332 y Fw(\))236 3431 y(for)28 b(all)f(\()p Fv(s)550 3443 y Ft(1)588 3431 y Fv(;)14 b(s)664 3443 y Ft(2)701 3431 y Fw(\))24 b Fs(2)f Fx(R)907 3401 y Ft(2)944 3431 y Fw(,)28 b(so)g(that)g(\()p Fv(t;)14 b(x)p Fw(\))24 b Fs(7!)g Fv(M)9 b Fw(\()p Fv(t)18 b Fw(+)h Fv(x;)14 b(t)19 b Fs(\000)f Fv(x)p Fw(\))24 b Fs(2)g Fv(X)7 b Fw(.)37 b(Hence)28 b Fv(L)p Fw([)p Fv(M)9 b Fw(\()p Fv(t)18 b Fw(+)g Fv(x;)c(t)19 b Fs(\000)g Fv(x)p Fw(\)])24 b(=)f Fs(\000)p Fw(4\()p Fv(@)3569 3443 y Fu(s)3600 3451 y Fd(2)3636 3431 y Fv(@)3680 3443 y Fu(s)3711 3451 y Fd(1)3748 3431 y Fv(M)9 b Fw(\)\()p Fv(t)19 b Fw(+)236 3531 y Fv(x;)14 b(t)20 b Fs(\000)e Fv(x)p Fw(\))26 b(=)e Fs(\000)p Fw(8)p Fv(v)797 3501 y Fu(p)835 3531 y Fw(\()p Fv(t;)14 b(x)p Fw(\))29 b(and)g(our)f(claim)g(follo)n(ws.)39 b(F)-7 b(rom)28 b(\(4.16\))g(w)n(e)g(see)g(that)h Fv(M)37 b Fw(is)29 b(non)f(negativ)n(e)g(and)g(as)g(a)g(result)236 3631 y Fs(\000)p Fv(a)345 3600 y Ft(2)396 3564 y Fq(R)465 3631 y Fv(L)522 3600 y Fp(\000)p Ft(1)611 3631 y Fw(\()p Fv(v)686 3600 y Fu(p)724 3631 y Fw(\))p Fv(v)799 3600 y Fu(p)861 3631 y Fw(=)23 b(\()p Fv(a)1025 3600 y Ft(2)1062 3631 y Fv(=)p Fw(8\))1192 3564 y Fq(R)1261 3631 y Fv(M)9 b(v)1394 3600 y Fu(p)1455 3631 y Fs(\025)23 b Fw(0.)p 1644 3631 48 48 v 361 3779 a(W)-7 b(e)25 b(do)g(not)g(kno)n(w)f (whether)h(\(4.15\))g(still)g(holds)g(true)g(for)f Fv(G)p Fw(\()p Fv(v)s Fw(\))g(=)f Fs(\000)p Fw(\()p Fv(a)2623 3748 y Ft(2)2660 3779 y Fv(=)p Fw(2\))2790 3712 y Fq(R)2858 3779 y Fv(v)2901 3748 y Fu(p)2940 3779 y Fv(L)2997 3748 y Fp(\000)p Ft(1)3085 3779 y Fv(v)3128 3748 y Fu(p)3167 3779 y Fw(,)j Fv(p)d Fs(\025)f Fw(4.)36 b(In)25 b(an)n(y)f(case,)h(w)n (e)236 3878 y(can)i(deduce)h(from)f(lemma)h(4.9,)f(that)h Fv(G)1531 3890 y Fu(n)1576 3878 y Fw(\()p Fv(v)s Fw(\))c(:=)f Fv(G)p Fw(\()p Fs(L)1972 3890 y Fu(n)2018 3878 y Fv(v)s Fw(\))28 b(satis\014es)f(\(4.1\))g(for)g Fv(n)c Fs(\025)g Fw(2.)36 b(Indeed,)28 b(b)n(y)f(\(4.13\),)1295 4111 y Fv(G)1360 4123 y Fu(n)1406 4111 y Fw(\()p Fv(v)s Fw(\))d(=)e Fv(G)p Fw(\()p Fs(L)1778 4123 y Fu(n)1824 4111 y Fv(v)s Fw(\))i(=)2021 4054 y Fv(G)p Fw(\()p Fv(v)s Fw(\))p 2021 4092 174 4 v 2064 4168 a Fv(n)2114 4144 y Ft(2)2222 4111 y Fw(+)2315 4054 y Fv(a)2359 4024 y Ft(2)2396 4054 y Fv(\031)2446 4024 y Ft(4)2484 4054 y Fv(\013)2537 4024 y Ft(2)p 2315 4092 260 4 v 2403 4168 a Fw(12)2584 4018 y Fq(\020)2634 4111 y Fw(1)18 b Fs(\000)2810 4054 y Fw(1)p 2787 4092 88 4 v 2787 4168 a Fv(n)2837 4144 y Ft(2)2884 4018 y Fq(\021)2933 4111 y Fv(;)848 b Fw(\(4.17\))236 4323 y(with)28 b Fv(\013)c Fw(:=)e(\(1)p Fv(=)p Fw(2)p Fv(\031)820 4293 y Ft(2)857 4323 y Fw(\))903 4256 y Fq(R)942 4353 y Ft(\012)1008 4323 y Fv(v)1051 4293 y Fu(p)1089 4323 y Fw(.)37 b(Hence,)28 b(since)f Fv(G)p Fw(\()p Fv(v)s Fw(\))d Fs(\025)f Fw(0,)k Fs(8)p Fv(n;)14 b(m)23 b Fs(\025)g Fw(2,)k(for)g(all)g Fv(v)g Fs(2)c Fv(V)c Fs(nf)p Fw(0)p Fs(g)p Fw(,)426 4564 y Fv(G)491 4576 y Fu(n)537 4564 y Fw(\()p Fs(L)626 4576 y Fu(m)689 4564 y Fv(v)s Fw(\))24 b(=)f 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b(=)e Fv(\021)s Fw(\()p Fv(t)d Fw(+)f Fv(x)p Fw(\))h Fs(\000)g Fv(\021)s Fw(\()p Fv(t)f Fs(\000)g Fv(x)p Fw(\))24 b Fs(2)g Fv(V)1385 5025 y Fs(\000)p Fw(\010)1510 5037 y Fu(";n)1605 5025 y Fw(\()p Fv(v)s Fw(\))g(=)1834 4968 y Fs(\000)p Fv("n)1988 4938 y Ft(2)p 1834 5005 191 4 v 1908 5081 a Fw(2)2034 5025 y Fs(jj)p Fv(v)s Fs(jj)2169 4990 y Ft(2)2225 5025 y Fs(\000)18 b Fv(G)2373 5037 y Fu(n)2419 5025 y Fw(\()p Fv(v)s Fw(\))h(+)f Fv(R)2691 5037 y Fu(n)2736 5025 y Fw(\()p Fv(v)s Fw(\))p Fv(;)236 5232 y Fw(where)27 b Fv(G)541 5244 y Fu(n)587 5232 y Fw(\()p Fv(v)s Fw(\))d(=)e Fv(G)p Fw(\()p Fs(L)959 5244 y Fu(n)1005 5232 y Fv(v)s Fw(\),)29 b Fv(R)1195 5244 y Fu(n)1240 5232 y Fw(\()p Fv(v)s Fw(\))24 b(=)e Fv(R)q Fw(\()p Fs(L)1611 5244 y Fu(n)1657 5232 y Fv(v)s Fw(\))28 b(satis\014es,)f(b)n(y)g(\(4.10\),)925 5464 y Fv(D)r(R)1059 5476 y Fu(n)1104 5464 y Fw(\()p Fv(v)s Fw(\)[)p Fv(v)s Fw(])d(=)f Fv(D)r(R)q Fw(\()p Fs(L)1636 5476 y Fu(n)1681 5464 y Fv(v)s Fw(\)[)p Fs(L)1836 5476 y Fu(n)1882 5464 y Fv(v)s Fw(])h(=)e Fv(O)2124 5372 y Fq(\020)2174 5464 y Fs(jj)p Fv(v)s Fs(jj)2309 5430 y Ft(2)p Fu(p)p Ft(+2)2484 5464 y Fw(+)2581 5408 y Fv(")p 2577 5445 48 4 v 2577 5521 a(\015)2634 5464 y Fs(jj)p Fv(v)s Fs(jj)2769 5430 y Ft(2)p Fu(p)2860 5464 y Fw(+)2953 5408 y Fs(jj)p Fv(v)s Fs(jj)3088 5378 y Ft(3)p Fu(p)p Fp(\000)p Ft(1)p 2953 5445 292 4 v 3075 5521 a Fv(\015)3255 5372 y Fq(\021)3304 5464 y Fv(:)477 b Fw(\(4.19\))2084 6059 y(18)p eop %%Page: 19 19 19 18 bop 236 272 a Fw(W)-7 b(e)34 b(shall)g(consider)f(frequencies)g Fv(!)j(<)d Fw(1)h(so)f(that)h Fs(\000)p Fv(")f(>)g Fw(0.)55 b(By)34 b(\(4.18\),)h(for)e Fv(n)g Fs(\025)h Fw(2,)h Fv(G)3146 284 y Fu(n)3191 272 y Fw(\()p Fv(v)s Fw(\))g(satis\014es)e (\(4.1\).)55 b(F)-7 b(or)236 372 y(\()p Fs(j)p Fv(")p Fs(j)p Fv(n)403 341 y Ft(2)440 372 y Fw(\))472 341 y Ft(1)p Fu(=)p Ft(2)577 372 y Fv(=\015)33 b Fw(small)c(enough)g(w)n(e)g (shall)g(apply)g(Prop)r(osition)e(3.1)i(and)g(lemma)g(4.3)f(to)h Fs(\000)p Fw(\010)3131 384 y Fu(";n)3253 372 y Fw(:)d Fv(B)3365 384 y Fu(r)3396 392 y Fd(0)3458 372 y Fs(!)g Fx(R)i Fw(as)h(b)r(efore,)236 471 y(with)f Fv(q)e Fw(=)d(2)p Fv(p)18 b Fs(\000)g Fw(1,)27 b Fv(\026)c Fw(=)g Fs(\000)p Fv("n)1168 441 y Ft(2)1204 471 y Fw(,)926 687 y Fv(r)963 699 y Ft(0)1024 687 y Fw(=)1149 631 y(1)p 1121 668 97 4 v 1121 744 a Fv(C)1180 756 y Ft(0)1228 595 y Fq(\020)1326 631 y Fs(j)p Fv(")p Fs(j)p Fv(n)1461 601 y Ft(2)p 1288 668 248 4 v 1288 744 a Fv(m)p Fw(\()p Fv(G)1458 756 y Fu(n)1503 744 y Fw(\))1545 595 y Fq(\021)1595 612 y Ft(1)p Fu(=)p Ft(\(2)p Fu(p)p Fp(\000)p Ft(2\))2036 687 y Fw(and)166 b Fv(\013)24 b Fw(=)e Fv(O)2565 595 y Fq(\020)2616 687 y Fv(r)2653 699 y Ft(0)2690 653 y(2)2746 687 y Fw(+)2839 631 y Fs(j)p Fv(")p Fs(j)p 2839 668 85 4 v 2858 744 a Fv(\015)2952 687 y Fw(+)3045 631 y Fv(r)3082 643 y Ft(0)3120 601 y Fu(p)p Fp(\000)p Ft(1)p 3045 668 199 4 v 3120 744 a Fv(\015)3253 595 y Fq(\021)3303 687 y Fv(:)236 915 y Fw(Note)40 b(that)g Fv(m)p Fw(\()p Fv(G)811 927 y Fu(n)856 915 y Fw(\))k Fs(!)f Fv(m)p Fw(\()p 1163 848 66 4 v Fv(G)q Fw(\))d(as)f Fv(n)k Fs(!)g Fw(+)p Fs(1)p Fw(,)f(where)p 2099 848 V 39 w Fv(G)q Fw(\()p Fv(v)s Fw(\))h(:=)g(\()p Fv(a)2522 885 y Ft(2)2560 915 y Fv(\031)2610 885 y Ft(4)2647 915 y Fv(=)p Fw(12\))p Fs(h)p Fv(v)2880 885 y Fu(p)2918 915 y Fs(i)2950 885 y Ft(2)3031 915 y Fw(=)f(\()p Fv(a)3214 885 y Ft(2)3252 915 y Fv(=)p Fw(48\)\()3442 848 y Fq(R)3510 915 y Fv(v)3553 885 y Fu(p)3592 915 y Fw(\))3624 885 y Ft(2)3661 915 y Fw(,)h(and)c(the)236 1014 y(sequence)e(\()p Fv(m)p Fw(\()p Fv(G)792 1026 y Fu(n)838 1014 y Fw(\)\))h(is)f(b)r (ounded)h(from)f(b)r(elo)n(w)g(b)n(y)g(a)g(p)r(ositiv)n(e)f(constan)n (t.)66 b(W)-7 b(e)37 b(deriv)n(e)g(the)g(follo)n(wing)g(Theorem,)236 1114 y(where)27 b(the)h(condition)g Fv(r)1023 1074 y Fu(p)p Fp(\000)p Ft(1)1021 1136 y(0)1147 1114 y Fv(=\015)k Fw(small,)27 b(i.e.)37 b(\()p Fs(j)p Fv(")p Fs(j)p Fv(n)1814 1084 y Ft(2)1851 1114 y Fw(\))1883 1084 y Ft(1)p Fu(=)p Ft(2)1988 1114 y Fv(=\015)31 b Fw(small,)d(is)f(required.)236 1282 y Fx(Theorem)32 b(4.3)40 b Fo(Assume)29 b(that)h Fv(f)38 b Fo(satis\014es)30 b Fw(\()p Fv(F)12 b Fw(1\))30 b Fo(and)h Fw(\()p Fv(N)9 b Fw(2\))p Fo(.)38 b(Then)31 b(ther)l(e)f(is)g(a)g(c)l(onstant)f Fv(C)3259 1294 y Ft(12)3360 1282 y Fo(dep)l(ending)i(only)g(on)236 1381 y Fv(f)39 b Fo(such)29 b(that,)i Fs(8)p Fv(!)25 b Fs(2)f(W)7 b Fo(,)30 b Fv(!)25 b(<)e Fw(1)p Fo(,)30 b Fs(8)p Fv(n)23 b Fs(\025)f Fw(2)30 b Fo(such)f(that)1756 1547 y Fw(\()p Fs(j)p Fv(!)21 b Fs(\000)d Fw(1)p Fs(j)p Fv(n)2082 1517 y Ft(2)2119 1547 y Fw(\))2151 1517 y Ft(1)p Fu(=)p Ft(2)p 1756 1584 500 4 v 1960 1660 a Fv(\015)2003 1672 y Fu(!)2289 1604 y Fs(\024)k Fv(C)2435 1616 y Ft(12)3804 1604 y Fw(\(4.20\))236 1820 y Fo(e)l(quation)37 b(\(1.1\))h(p)l(ossesses)g(at)e(le)l(ast)h (one)g(p)l(air)h(of)g(even)f(p)l(erio)l(dic)i(\(in)d(time\))h(classic)l (al)i Fv(C)3185 1790 y Ft(2)3259 1820 y Fo(solutions)e(of)h(minimal)236 1920 y(p)l(erio)l(d)31 b Fw(2)p Fv(\031)s(=)p Fw(\()p Fv(n!)s Fw(\))p Fo(.)38 b(If)31 b Fv(p)22 b Fw(=)h(2)29 b Fo(the)h(existenc)l(e)g(r)l(esult)f(holds)i(true)e(for)h Fv(n)23 b Fw(=)g(1)29 b Fo(as)h(wel)t(l.)236 2088 y Fx(Remark)i(4.2)41 b Fo(We)29 b(think)h(that)g(the)g(r)l(estriction)g Fv(n)23 b Fs(\025)g Fw(2)29 b Fo(in)h(the)g(c)l(ase)g Fv(p)23 b Fs(6)p Fw(=)f(2)29 b Fo(even)h(is)h(of)f(pur)l(ely)g(te)l(chnic)l(al) h(natur)l(e.)361 2255 y Fw(Finally)c(w)n(e)g(consider)g(the)h(case)f (in)h(whic)n(h)f(\()p Fv(N)9 b Fw(3\))28 b(holds.)36 b(W)-7 b(e)28 b(\014rst)g(assume)e(that)i Fv(b)23 b(<)g Fw(0.)36 b(W)-7 b(e)28 b(ha)n(v)n(e)1503 2457 y Fs(\000)p Fw(\010)1628 2469 y Fu(")1664 2457 y Fw(\()p Fv(v)s Fw(\))23 b(=)1892 2401 y Fs(\000)p Fv(")p 1892 2438 104 4 v 1923 2514 a Fw(2)2005 2457 y Fs(jj)p Fv(v)s Fs(jj)2140 2423 y Ft(2)2196 2457 y Fs(\000)18 b Fv(G)p Fw(\()p Fv(v)s Fw(\))i(+)e Fv(R)q Fw(\()p Fv(v)s Fw(\))p Fv(;)236 2648 y Fw(where)1374 2780 y Fv(G)p Fw(\()p Fv(v)s Fw(\))24 b(:=)f Fs(\000)1779 2724 y Fv(b)p 1756 2761 84 4 v 1756 2837 a Fw(2)p Fv(p)1863 2667 y Fq(Z)1909 2856 y Ft(\012)1974 2780 y Fv(v)2017 2746 y Ft(2)p Fu(p)2107 2780 y Fs(\000)2200 2724 y Fv(a)2244 2694 y Ft(2)p 2200 2761 82 4 v 2220 2837 a Fw(2)2305 2667 y Fq(Z)2351 2856 y Ft(\012)2417 2780 y Fv(v)2460 2746 y Fu(p)2498 2780 y Fv(L)2555 2746 y Fp(\000)p Ft(1)2644 2780 y Fv(v)2687 2746 y Fu(p)2749 2780 y Fv(>)f Fw(0)926 b(\(4.21\))236 2967 y(for)37 b(all)g Fv(v)42 b Fs(6)p Fw(=)d(0)e(and)g Fv(R)q Fw(\()p Fv(v)s Fw(\))h(satis\014es)f(estimate)g(\(4.10\).)65 b(Moreo)n(v)n(er,)37 b(de\014ning)h Fv(G)2891 2979 y Fu(n)2936 2967 y Fw(\()p Fv(v)s Fw(\))i(:=)f Fv(G)p Fw(\()p Fs(L)3364 2979 y Fu(n)3410 2967 y Fv(v)s Fw(\),)h(w)n(e)d(still)h(ha)n(v)n(e)236 3067 y Fv(G)301 3079 y Fu(n)347 3067 y Fw(\()p Fs(L)436 3079 y Fu(m)499 3067 y Fv(v)s Fw(\))31 b Fv(<)f(m)773 3036 y Ft(2)p Fu(p)844 3067 y Fv(G)909 3079 y Fu(n)954 3067 y Fw(\()p Fv(v)s Fw(\))j(for)f(all)f Fv(n;)14 b(m)30 b Fs(\025)g Fw(2)h(and)h Fv(v)h Fs(6)p Fw(=)d(0.)49 b(F)-7 b(or)31 b Fv(")f(<)g Fw(0)h(and)h Fs(j)p Fv(")p Fs(j)p Fv(n)2842 3036 y Ft(2)2879 3067 y Fv(=\015)k Fw(small)31 b(enough)g(w)n(e)h(obtain)f(the)236 3166 y(same)c(existence)h(result)f (as)g(in)h(Theorem)e(4.3.)361 3337 y(No)n(w)h(assume)g(that)h Fv(b)22 b(>)h Fw(0.)36 b(As)28 b(a)f(consequence)g(of)g(lemma)h(4.8)f (w)n(e)g(ha)n(v)n(e)710 3443 y Fq(Z)756 3631 y Ft(\012)822 3556 y Fs(L)879 3568 y Fu(n)924 3556 y Fv(w)r(L)1042 3521 y Fp(\000)p Ft(1)1132 3556 y Fw(\()p Fs(L)1221 3568 y Fu(n)1266 3556 y Fv(w)r Fw(\))d(=)f Fs(\000)1546 3499 y Fv(\031)1596 3469 y Ft(4)p 1545 3536 88 4 v 1568 3612 a Fw(6)1643 3556 y Fs(h)p Fv(m)p Fs(i)1780 3521 y Ft(2)1836 3556 y Fw(+)18 b Fv(O)1984 3463 y Fq(\020)2044 3497 y Fs(j)p Fv(m)p Fs(j)2163 3467 y Ft(2)2163 3523 y Fu(L)2209 3507 y Fd(2)2264 3497 y Fw(+)g Fs(j)p Fv(w)r Fs(j)2454 3467 y Ft(2)2454 3523 y Fu(L)2500 3507 y Fd(2)p 2044 3536 493 4 v 2247 3612 a Fv(n)2297 3588 y Ft(2)2547 3463 y Fq(\021)2619 3556 y Fw(=)23 b Fs(\000)2782 3499 y Fv(\031)2832 3469 y Ft(4)p 2782 3536 88 4 v 2805 3612 a Fw(6)2879 3556 y Fs(h)p Fv(w)r Fs(i)3004 3521 y Ft(2)3061 3556 y Fw(+)18 b Fv(O)3209 3463 y Fq(\020)3269 3497 y Fs(j)p Fv(w)r Fs(j)3376 3467 y Ft(2)3376 3523 y Fu(L)3422 3507 y Fd(2)p 3269 3536 190 4 v 3321 3612 a Fv(n)3371 3588 y Ft(2)3469 3463 y Fq(\021)3519 3556 y Fv(:)262 b Fw(\(4.22\))236 3770 y(By)28 b(\(4.22\),)e(w)n(e)i(ma)n(y)f(write)g(for)g Fv(v)s Fw(\()p Fv(t;)14 b(x)p Fw(\))24 b(=)f Fv(\021)s Fw(\()p Fv(t)c Fw(+)f Fv(x)p Fw(\))h Fs(\000)f Fv(\021)s Fw(\()p Fv(t)h Fs(\000)f Fv(x)p Fw(\))24 b Fs(2)f Fv(V)1472 3991 y Fw(\010)1532 4003 y Fu(";n)1628 3991 y Fw(\()p Fv(v)s Fw(\))h(=)1857 3934 y Fv("n)1946 3904 y Ft(2)p 1857 3971 126 4 v 1899 4047 a Fw(2)1992 3991 y Fs(jj)p Fv(v)s Fs(jj)2127 3956 y Ft(2)2183 3991 y Fs(\000)2283 3970 y Fq(e)2266 3991 y Fv(G)q Fw(\()p Fv(v)s Fw(\))19 b(+)f Fv(R)2604 4003 y Fu(n)2649 3991 y Fw(\()p Fv(v)s Fw(\))p Fv(;)236 4181 y Fw(with)493 4360 y Fq(e)477 4381 y Fv(G)p Fw(\()p Fv(v)s Fw(\))24 b(=)794 4325 y Fv(b)p 770 4362 84 4 v 770 4438 a Fw(2)p Fv(p)878 4268 y Fq(Z)924 4457 y Ft(\012)989 4381 y Fv(v)1032 4347 y Ft(2)p Fu(p)1122 4381 y Fs(\000)1216 4325 y Fv(a)1260 4295 y Ft(2)p 1215 4362 V 1215 4438 a Fw(48)1308 4289 y Fq(\020)1372 4268 y(Z)1418 4457 y Ft(\012)1483 4381 y Fv(v)1526 4347 y Fu(p)1565 4289 y Fq(\021)1614 4306 y Ft(2)1675 4381 y Fw(=)1796 4325 y Fv(b)p 1772 4362 V 1772 4438 a Fw(4)p Fv(p)1879 4268 y Fq(Z)1925 4457 y Fl(T)1977 4440 y Fd(2)2028 4289 y Fq(\020)2078 4381 y Fv(\021)s Fw(\()p Fv(s)2193 4393 y Ft(1)2230 4381 y Fw(\))19 b Fs(\000)f Fv(\021)s Fw(\()p Fv(s)2479 4393 y Ft(2)2517 4381 y Fw(\))2549 4289 y Fq(\021)2598 4306 y Ft(2)p Fu(p)2688 4381 y Fs(\000)2803 4325 y Fv(a)2847 4295 y Ft(2)p 2781 4362 125 4 v 2781 4438 a Fw(192)2916 4289 y Fq(\020)2979 4268 y(Z)3025 4457 y Fl(T)3077 4440 y Fd(2)3114 4381 y Fw(\()p Fv(\021)s Fw(\()p Fv(s)3261 4393 y Ft(1)3299 4381 y Fw(\))h Fs(\000)f Fv(\021)s Fw(\()p Fv(s)3548 4393 y Ft(2)3585 4381 y Fw(\)\))3649 4347 y Fu(p)3688 4289 y Fq(\021)3738 4306 y Ft(2)236 4591 y Fw(and)412 4799 y Fv(D)r(R)546 4811 y Fu(n)591 4799 y Fw(\()p Fv(v)s Fw(\)[)p Fv(v)s Fw(])25 b(=)d Fv(D)r(R)q Fw(\()p Fs(L)1123 4811 y Fu(n)1169 4799 y Fv(v)s Fw(\)[)p Fs(L)1324 4811 y Fu(n)1370 4799 y Fv(v)s Fw(])d(+)f Fv(O)1603 4707 y Fq(\020)1663 4735 y Fs(j)p Fv(v)s Fs(j)1752 4705 y Ft(2)p Fu(p)1752 4755 y(!)r(;n)p 1663 4780 199 4 v 1718 4856 a Fv(n)1768 4832 y Ft(2)1871 4707 y Fq(\021)1944 4799 y Fw(=)k Fv(O)2096 4707 y Fq(\020)2146 4799 y Fs(jj)p Fv(v)s Fs(jj)2281 4765 y Ft(2)p Fu(p)p Ft(+2)2456 4799 y Fw(+)2553 4743 y Fv(")p 2549 4780 48 4 v 2549 4856 a(\015)2606 4799 y Fs(jj)p Fv(v)s Fs(jj)2741 4765 y Ft(2)p Fu(p)2832 4799 y Fw(+)2925 4743 y Fs(jj)p Fv(v)s Fs(jj)3060 4713 y Ft(3)p Fu(p)p Fp(\000)p Ft(1)p 2925 4780 292 4 v 3047 4856 a Fv(\015)3245 4799 y Fw(+)3338 4743 y Fs(jj)p Fv(v)s Fs(jj)3473 4713 y Ft(2)p Fu(p)p 3338 4780 207 4 v 3398 4856 a Fv(n)3448 4832 y Ft(2)3555 4707 y Fq(\021)3605 4799 y Fv(:)176 b Fw(\(4.23\))236 5011 y(W)-7 b(e)29 b(are)f(no)h(longer)e(able)i(to)g(pro)n(v)n(e)e(that)i(the)g(condition) g(of)g(lemma)g(4.1)f(is)g(satis\014ed)h(b)n(y)f Fv(G)i Fw(de\014ned)f(in)g(\(4.21\))f(or)g(b)n(y)236 5121 y Fs(\000)p Fv(G)p Fw(.)49 b(That)32 b(is)f(wh)n(y)g(w)n(e)h(in)n(tro)r (duce)1432 5100 y Fq(e)1416 5121 y Fv(G)p Fw(.)49 b(Clearly)1862 5100 y Fq(e)1846 5121 y Fv(G)p Fw(\()p Fs(L)2000 5133 y Fu(n)2046 5121 y Fv(v)s Fw(\))30 b(=)2262 5100 y Fq(e)2245 5121 y Fv(G)p Fw(\()p Fv(v)s Fw(\))j(for)e(all)g Fv(v)i Fs(2)d Fv(V)51 b Fw(and)31 b Fv(n)f Fs(2)g Fx(N)p Fs(nf)p Fw(0)p Fs(g)p Fw(.)47 b(W)-7 b(e)32 b(ha)n(v)n(e)e(to)236 5231 y(sp)r(ecify)e(the)g(sign)f(of)936 5210 y Fq(e)920 5231 y Fv(G)p Fw(\()p Fv(v)s Fw(\).)38 b(W)-7 b(e)28 b(need)g(to)f(in)n(tro)r(duce)236 5441 y Fv(\024)p Fw(\()p Fv(p)p Fw(\))c(:=)95 b(sup)524 5514 y Fu(v)r Fp(2)p Fu(V)15 b Fp(nf)p Ft(0)p Fp(g)807 5349 y Fq(\020)870 5328 y(Z)916 5516 y Ft(\012)982 5441 y Fv(v)1025 5406 y Fu(p)1063 5349 y Fq(\021)1113 5366 y Ft(2)1150 5349 y Fq(\020)1213 5328 y(Z)1260 5516 y Ft(\012)1325 5441 y Fv(v)1368 5406 y Ft(2)p Fu(p)1440 5349 y Fq(\021)1489 5366 y Fp(\000)p Ft(1)1601 5441 y Fw(=)191 b(sup)1689 5514 y Fu(\021)r Fp(2)p Fu(H)1828 5498 y Fd(1)1861 5514 y Ft(\()p Fl(T)p Ft(\))p Fu(;\021)30 b(odd)2165 5349 y Fq(\020)2228 5328 y(Z)2274 5516 y Fl(T)2326 5500 y Fd(2)2363 5441 y Fw(\()p Fv(\021)s Fw(\()p Fv(s)2510 5453 y Ft(1)2548 5441 y Fw(\))14 b Fs(\000)g Fv(\021)s Fw(\()p Fv(s)2788 5453 y Ft(2)2826 5441 y Fw(\)\))2890 5406 y Fu(p)2929 5349 y Fq(\021)2978 5366 y Ft(2)3015 5349 y Fq(\020)3065 5441 y Fw(2)3121 5328 y Fq(Z)3167 5516 y Fl(T)3219 5500 y Fd(2)3255 5441 y Fw(\()p Fv(\021)s Fw(\()p Fv(s)3402 5453 y Ft(1)3440 5441 y Fw(\))h Fs(\000)e Fv(\021)s Fw(\()p Fv(s)3680 5453 y Ft(2)3718 5441 y Fw(\)\))3782 5406 y Ft(2)p Fu(p)3854 5349 y Fq(\021)3904 5366 y Fp(\000)p Ft(1)3993 5441 y Fv(:)2084 6059 y Fw(19)p eop %%Page: 20 20 20 19 bop 236 272 a Fw(W)-7 b(e)28 b(can)f(pro)n(v)n(e)f(that)i Fv(\024)p Fw(\()p Fv(p)p Fw(\))23 b(=)g Fv(\031)1250 242 y Ft(2)1288 272 y Fw(.)37 b(Indeed,)27 b(using)h(that)g Fv(\021)i Fw(is)e(o)r(dd,)g(w)n(e)f(get)534 423 y Fq(Z)580 612 y Fl(T)p Fp(\002)p Fl(T)740 536 y Fw(\()p Fv(\021)s Fw(\()p Fv(s)887 548 y Ft(1)925 536 y Fw(\))19 b Fs(\000)f Fv(\021)s Fw(\()p Fv(s)1174 548 y Ft(2)1211 536 y Fw(\)\))1275 502 y Fu(p)1342 536 y Fv(ds)1424 548 y Ft(1)1489 536 y Fv(ds)1571 548 y Ft(2)1631 536 y Fw(=)1729 425 y Fu(p=)p Ft(2)1720 457 y Fq(X)1719 636 y Fu(k)q Ft(=0)1854 536 y Fv(C)1919 502 y Ft(2)p Fu(k)1913 557 y(p)2007 423 y Fq(Z)2053 612 y Fl(T)2123 536 y Fv(\021)2167 502 y Ft(2)p Fu(k)2269 536 y Fv(ds)2365 423 y Fq(Z)2411 612 y Fl(T)2481 536 y Fv(\021)2525 502 y Fu(p)p Fp(\000)p Ft(2)p Fu(k)2713 536 y Fv(ds)23 b Fs(\024)2915 425 y Fu(p=)p Ft(2)2906 457 y Fq(X)2906 636 y Fu(k)q Ft(=0)3040 536 y Fv(C)3105 502 y Ft(2)p Fu(k)3099 557 y(p)3198 536 y Fs(\002)18 b Fw(2)p Fv(\031)3387 423 y Fq(Z)3433 612 y Fl(T)3503 536 y Fv(\021)3547 502 y Fu(p)3613 536 y Fv(ds;)236 782 y Fw(b)n(y)28 b(H\177)-42 b(older)27 b(inequalit)n(y)-7 b(.)36 b(Hence)1295 879 y Fq(Z)1341 1068 y Fl(T)p Fp(\002)p Fl(T)1502 992 y Fw(\()p Fv(\021)s Fw(\()p Fv(s)1649 1004 y Ft(1)1687 992 y Fw(\))18 b Fs(\000)h Fv(\021)s Fw(\()p Fv(s)1936 1004 y Ft(2)1973 992 y Fw(\)\))2037 958 y Fu(p)2104 992 y Fv(ds)2186 1004 y Ft(1)2251 992 y Fv(ds)2333 1004 y Ft(2)2393 992 y Fs(\024)k Fw(2)2523 958 y Fu(p)2561 992 y Fv(\031)2625 879 y Fq(Z)2671 1068 y Fl(T)2741 992 y Fv(\021)2785 958 y Fu(p)2851 992 y Fv(ds:)236 1213 y Fw(No)n(w,)k(b)n(y)h(Cauc)n(h)n(y-Sc)n(h)n(w)n(arz)c(inequalit)n(y)-7 b(,)27 b(for)g(all)h(0)22 b Fs(\024)h Fv(k)j Fs(\024)c Fv(p)p Fw(,)1221 1330 y Fq(\020)1284 1309 y(Z)1330 1498 y Fl(T)1400 1422 y Fv(\021)1444 1388 y Fu(p)1483 1330 y Fq(\021)1533 1347 y Ft(2)1593 1422 y Fw(=)1681 1330 y Fq(\020)1744 1309 y(Z)1790 1498 y Fl(T)1860 1422 y Fs(j)p Fv(\021)s Fs(j)1950 1388 y Fu(k)1991 1422 y Fs(j)p Fv(\021)s Fs(j)2081 1388 y Fu(p)p Fp(\000)p Fu(k)2209 1330 y Fq(\021)2258 1347 y Ft(2)2319 1422 y Fs(\024)2406 1309 y Fq(Z)2452 1498 y Fl(T)2523 1422 y Fv(\021)2567 1388 y Ft(2)p Fu(k)2654 1309 y Fq(Z)2701 1498 y Fl(T)2771 1422 y Fv(\021)2815 1388 y Ft(2)p Fu(p)p Fp(\000)p Ft(2)p Fu(k)3008 1422 y Fv(:)236 1635 y Fw(Hence)622 1672 y Fq(Z)668 1860 y Fl(T)p Fp(\002)p Fl(T)828 1785 y Fw(\()p Fv(\021)s Fw(\()p Fv(s)975 1797 y Ft(1)1013 1785 y Fw(\))d Fs(\000)f Fv(\021)s Fw(\()p Fv(s)1262 1797 y Ft(2)1300 1785 y Fw(\)\))1364 1750 y Ft(2)p Fu(p)1463 1785 y Fv(ds)1545 1797 y Ft(1)1610 1785 y Fv(ds)1692 1797 y Ft(2)1753 1785 y Fw(=)1884 1677 y Fu(p)1841 1706 y Fq(X)1840 1884 y Fu(k)q Ft(=0)1975 1785 y Fv(C)2040 1750 y Ft(2)p Fu(k)2034 1805 y Ft(2)p Fu(p)2128 1672 y Fq(Z)2174 1860 y Fl(T)2244 1785 y Fv(\021)2288 1750 y Ft(2)p Fu(k)2390 1785 y Fv(ds)2486 1672 y Fq(Z)2532 1860 y Fl(T)2602 1785 y Fv(\021)2646 1750 y Ft(2)p Fu(p)p Fp(\000)p Ft(2)p Fu(k)2867 1785 y Fv(ds)24 b Fs(\025)e Fw(2)3102 1750 y Ft(2)p Fu(p)p Fp(\000)p Ft(1)3258 1692 y Fq(\020)3322 1672 y(Z)3368 1860 y Fl(T)3438 1785 y Fv(\021)3482 1750 y Fu(p)3520 1692 y Fq(\021)3570 1710 y Ft(2)3607 1785 y Fv(:)236 2012 y Fw(As)34 b(a)g(result)f(w)n(e)h(obtain)f Fv(\024)p Fw(\()p Fv(p)p Fw(\))h Fs(\024)f Fv(\031)1407 1982 y Ft(2)1445 2012 y Fw(.)56 b(Cho)r(osing)33 b(for)g Fv(\021)k(H)2180 1982 y Ft(1)2217 2012 y Fw(-appro)n(ximations)32 b(of)h(the)i(2)p Fv(\031)s Fw(-p)r(erio)r(dic)e(function)i(that)236 2112 y(tak)n(es)23 b(v)-5 b(alue)23 b Fs(\000)p Fw(1)g(on)g(\()p Fs(\000)p Fv(\031)s(;)14 b Fw(0])23 b(and)g(v)-5 b(alue)23 b(1)g(on)g(\(0)p Fv(;)14 b(\031)s Fw(],)25 b(w)n(e)e(obtain)g(the)h (con)n(v)n(erse)d(inequalit)n(y)-7 b(.)35 b(Hence)24 b Fv(\024)p Fw(\()p Fv(p)p Fw(\))f(=)g Fv(\031)3747 2081 y Ft(2)3784 2112 y Fw(.)36 b(Note)236 2211 y(also)27 b(that)1595 2332 y(inf)1511 2390 y Fu(v)r Fp(2)p Fu(V)15 b Fp(nf)p Ft(0)p Fp(g)1794 2240 y Fq(\020)1857 2219 y(Z)1903 2408 y Ft(\012)1969 2332 y Fv(v)2012 2298 y Fu(p)2050 2240 y Fq(\021)2100 2257 y Ft(2)2137 2240 y Fq(\020)2201 2219 y(Z)2247 2408 y Ft(\012)2312 2332 y Fv(v)2355 2298 y Ft(2)p Fu(p)2427 2240 y Fq(\021)2476 2257 y Fp(\000)p Ft(1)2589 2332 y Fw(=)22 b(0)p Fv(:)236 2548 y Fw(In)32 b(fact,)g(for)f(all)g Fv(\016)i(>)c Fw(0)i(there)g(is)h(an)f(o)r(dd)g (smo)r(oth)g(2)p Fv(\031)s Fw(-p)r(erio)r(dic)g Fv(\021)2392 2560 y Fu(\016)2460 2548 y Fw(suc)n(h)h(that)2835 2481 y Fq(R)2891 2502 y Ft(2)p Fu(\031)2874 2578 y Ft(0)2983 2548 y Fv(\021)3027 2508 y Ft(2)p Fu(p)3024 2573 y(\016)3128 2548 y Fw(=)d(1)i(and)3460 2481 y Fq(R)3515 2502 y Ft(2)p Fu(\031)3499 2578 y Ft(0)3607 2548 y Fv(\021)3651 2508 y Ft(2)p Fu(q)3648 2573 y(\016)3751 2548 y Fs(\024)e Fv(\016)34 b Fw(for)236 2648 y(1)d Fs(\024)f Fv(q)k(<)c(p)p Fw(.)51 b(One)32 b(can)g(see)g(easily)f(that)i(for)e Fv(v)s Fw(\()p Fv(t;)14 b(x)p Fw(\))32 b(=)f Fv(\021)2101 2660 y Fu(\016)2137 2648 y Fw(\()p Fv(t)22 b Fw(+)f Fv(x)p Fw(\))i Fs(\000)e Fv(\021)2536 2660 y Fu(\016)2573 2648 y Fw(\()p Fv(t)g Fs(\000)g Fv(x)p Fw(\),)35 b(\()2911 2581 y Fq(R)2950 2677 y Ft(\012)3016 2648 y Fv(v)3059 2618 y Fu(p)3097 2648 y Fw(\))3129 2618 y Ft(2)3167 2648 y Fw(\()3199 2581 y Fq(R)3238 2677 y Ft(\012)3304 2648 y Fv(v)3347 2618 y Ft(2)p Fu(p)3418 2648 y Fw(\))3450 2618 y Fp(\000)p Ft(1)3571 2648 y Fs(\024)30 b Fv(\014)t Fw(\()p Fv(\016)s Fw(\))j(with)236 2747 y(lim)352 2759 y Fu(\016)r Fp(!)p Ft(0)501 2747 y Fv(\014)t Fw(\()p Fv(\016)s Fw(\))24 b(=)f(0.)361 2857 y(If)33 b Fv(b)d Fs(\025)h Fv(p\024)p Fw(\()p Fv(p)p Fw(\))p Fv(a)851 2827 y Ft(2)888 2857 y Fv(=)p Fw(24)e(=)i Fv(p\031)1231 2827 y Ft(2)1268 2857 y Fv(a)1312 2827 y Ft(2)1349 2857 y Fv(=)p Fw(24)g(then)i(for)f(all)g Fv(v)i Fs(2)d Fv(V)19 b Fw(,)2252 2836 y Fq(e)2236 2857 y Fv(G)p Fw(\()p Fv(v)s Fw(\))32 b Fs(\025)e Fw(0)i(and)g Fv(G)f Fs(6\021)g Fw(0.)51 b(In)32 b(this)h(case)e(w)n(e)h(obtain)g(an)236 2957 y(existence)c(result)f(for)g Fv(!)f(>)c Fw(1.)361 3057 y(If)27 b(0)c Fv(<)g(b)f(<)h(p\024)p Fw(\()p Fv(p)p Fw(\))p Fv(a)982 3026 y Ft(2)1019 3057 y Fv(=)p Fw(24)f(=)g Fv(p\031)1346 3026 y Ft(2)1384 3057 y Fv(a)1428 3026 y Ft(2)1465 3057 y Fv(=)p Fw(24)k(then)h(there)g(is)g Fv(v)2140 3069 y Ft(1)2201 3057 y Fs(2)c Fv(V)47 b Fw(suc)n(h)26 b(that)2756 3036 y Fq(e)2740 3057 y Fv(G)p Fw(\()p Fv(v)2877 3069 y Ft(1)2915 3057 y Fw(\))d Fv(<)g Fw(0)k(and)g(there)g(is)g Fv(v)3623 3069 y Ft(2)3684 3057 y Fs(2)c Fv(V)46 b Fw(suc)n(h)236 3167 y(that)433 3146 y Fq(e)416 3167 y Fv(G)p Fw(\()p Fv(v)553 3179 y Ft(2)591 3167 y Fw(\))23 b Fv(>)g Fw(0.)37 b(In)27 b(this)h(case)f(w)n(e)g(obtain)h(an)f(existence)g(result)g(b)r (oth)h(for)g Fv(!)d(<)e Fw(1)k(and)h Fv(!)d(>)e Fw(1.)361 3266 y(In)i(b)r(oth)g(cases)e(w)n(e)i(apply)f(the)h(same)f(argumen)n (ts)g(as)g(b)r(efore,)h(for)f(\()p Fs(j)p Fv(")p Fs(j)p Fv(n)2612 3236 y Ft(2)2649 3266 y Fw(\))2681 3236 y Ft(1)p Fu(=)p Ft(2)2786 3266 y Fv(=\015)29 b Fw(small)24 b(and)h Fv(n)f Fw(large)g(enough,)g(with)236 3366 y Fv(q)i Fw(=)d(2)p Fv(p)18 b Fs(\000)g Fw(1,)27 b Fv(\026)c Fw(=)g Fs(j)p Fv(")p Fs(j)p Fv(n)960 3336 y Ft(2)997 3366 y Fw(,)831 3591 y Fv(r)868 3603 y Ft(0)929 3591 y Fw(=)1054 3535 y(1)p 1026 3572 97 4 v 1026 3648 a Fv(C)1085 3660 y Ft(0)1133 3499 y Fq(\020)1240 3535 y Fs(j)p Fv(")p Fs(j)p Fv(n)1375 3505 y Ft(2)p 1192 3572 268 4 v 1192 3663 a Fv(m)p Fw(\()p Fs(\006)1378 3642 y Fq(e)1362 3663 y Fv(G)p Fw(\))1469 3499 y Fq(\021)1519 3516 y Ft(1)p Fu(=)p Ft(\(2)p Fu(p)p Fp(\000)p Ft(2\))1961 3591 y Fw(and)165 b Fv(\013)24 b Fw(=)e Fv(O)2489 3499 y Fq(\020)2540 3591 y Fv(r)2577 3603 y Ft(0)2614 3557 y(2)2670 3591 y Fw(+)2768 3535 y Fv(")p 2763 3572 48 4 v 2763 3648 a(\015)2839 3591 y Fw(+)2932 3535 y Fv(r)2969 3547 y Ft(0)3007 3505 y Fu(p)p Fp(\000)p Ft(1)p 2932 3572 199 4 v 3007 3648 a Fv(\015)3159 3591 y Fw(+)3274 3535 y(1)p 3252 3572 88 4 v 3252 3648 a Fv(n)3302 3624 y Ft(2)3349 3499 y Fq(\021)3398 3591 y Fv(:)236 3845 y(m)p Fw(\()p Fs(\000)422 3824 y Fq(e)406 3845 y Fv(G)p Fw(\))28 b(app)r(ears)f(only)g(in)h(the)g (second)f(case)f(when)i Fv(!)e(<)c Fw(1.)37 b(As)28 b(a)f(conclusion)g (w)n(e)g(ha)n(v)n(e)236 4017 y Fx(Theorem)32 b(4.4)40 b Fo(Assume)31 b(that)g Fv(f)40 b Fo(satis\014es)31 b Fw(\()p Fv(F)12 b Fw(1\))31 b Fo(and)h Fw(\()p Fv(N)9 b Fw(3\))p Fo(.)43 b(De\014ne)31 b Fv(J)j Fs(\032)25 b Fx(R)31 b Fo(as)h Fv(J)i Fw(=)25 b(\(0)p Fv(;)14 b Fw(1\))31 b Fo(if)h Fv(b)26 b(<)f Fw(0)p Fo(,)32 b Fv(J)i Fw(=)25 b(\(1)p Fv(;)14 b Fw(2\))236 4116 y Fo(if)30 b Fv(b)23 b Fs(\025)g Fv(p\031)555 4086 y Ft(2)592 4116 y Fv(a)636 4086 y Ft(2)673 4116 y Fv(=)p Fw(24)28 b Fo(and)i Fv(J)h Fw(=)23 b(\(0)p Fv(;)14 b Fw(1\))j Fs([)h Fw(\(1)p Fv(;)c Fw(2\))29 b Fo(if)h Fw(0)22 b Fv(<)h(b)g(<)f(p\031)2112 4086 y Ft(2)2150 4116 y Fv(a)2194 4086 y Ft(2)2231 4116 y Fv(=)p Fw(24)p Fo(.)37 b(Then)30 b(ther)l(e)f(is)h(a)g(c)l(onstant)e Fv(C)3391 4128 y Ft(13)3491 4116 y Fo(and)i(an)g(inte)l(ger)236 4216 y Fv(N)303 4228 y Ft(0)370 4216 y Fs(\025)g Fw(1)j Fo(dep)l(ending)i(only)f(on)g Fv(f)42 b Fo(\(with)34 b Fv(N)1605 4228 y Ft(0)1672 4216 y Fw(=)29 b(2)k Fo(if)i Fv(b)29 b(<)h Fw(0)j Fo(and)h Fv(p)c Fs(\025)f Fw(4)p Fo(,)35 b Fv(N)2661 4228 y Ft(0)2728 4216 y Fw(=)29 b(1)34 b Fo(if)g Fv(b)c(<)f Fw(0)k Fo(and)h Fv(p)c Fw(=)g(2)p Fo(\))j(such)g(that,)236 4316 y Fs(8)p Fv(!)26 b Fs(2)d(W)j(\\)19 b Fv(J)8 b Fo(,)30 b Fs(8)p Fv(n)23 b Fs(\025)f Fv(N)1002 4328 y Ft(0)1069 4316 y Fo(such)30 b(that)1756 4414 y Fw(\()p Fs(j)p Fv(!)21 b Fs(\000)d Fw(1)p Fs(j)p Fv(n)2082 4384 y Ft(2)2119 4414 y Fw(\))2151 4384 y Ft(1)p Fu(=)p Ft(2)p 1756 4451 500 4 v 1960 4527 a Fv(\015)2003 4539 y Fu(!)2289 4470 y Fs(\024)k Fv(C)2435 4482 y Ft(13)3804 4470 y Fw(\(4.24\))236 4662 y Fo(e)l(quation)37 b(\(1.1\))h(p)l (ossesses)g(at)e(le)l(ast)h(one)g(p)l(air)h(of)g(even)f(p)l(erio)l(dic) i(\(in)d(time\))h(classic)l(al)i Fv(C)3185 4632 y Ft(2)3259 4662 y Fo(solutions)e(of)h(minimal)236 4762 y(p)l(erio)l(d)31 b Fw(2)p Fv(\031)s(=)p Fw(\()p Fv(n!)s Fw(\))p Fo(.)361 4934 y Fw(In)c(the)h(next)g(remarks)e(w)n(e)h(sp)r(ecify)g(the)h(norms) f(and)g(the)h(energies)e(of)h(the)h(solutions)f Fv(u)3150 4946 y Fu(n)3222 4934 y Fw(obtained)g(in)h(Theorems)236 5033 y(4.1,)f(4.2,)g(4.3,)g(4.4.)236 5205 y Fx(Remark)32 b(4.3)41 b Fo(It)e(is)h(r)l(e)l(adily)h(se)l(en)e(by)h(the)g(choic)l(e) h(of)g Fv(r)2100 5217 y Ft(0)2177 5205 y Fo(in)f(the)g(di\013er)l(ent)f (c)l(ases)i(and)f(the)f(fact)i(that)e Fs(jL)3750 5217 y Fu(n)3796 5205 y Fv(v)s Fs(j)3862 5217 y Fu(!)3951 5205 y Fw(=)236 5305 y Fv(O)r Fw(\()p Fs(jj)p Fv(v)s Fs(jj)p Fw(\))47 b(=)42 b Fv(O)r Fw(\()p Fv(r)788 5317 y Ft(0)827 5305 y Fw(\))p Fo(,)k(for)d Fs(j)p Fv(")p Fs(j)p Fv(n)1210 5275 y Ft(2)1292 5305 y Fv(<)i Fw(1)p Fo(,)h(that)c(the)g(critic)l(al)h(p)l(oints)g Fv(v)2435 5317 y Fu(";n)2574 5305 y Fo(of)g Fw(\010)2744 5317 y Fu(")2821 5305 y Fo(that)f(we)h(obtain)g(satisfy)g Fs(j)p Fv(v)3738 5317 y Fu(";n)3834 5305 y Fs(j)3857 5317 y Fu(!)3951 5305 y Fw(=)236 5404 y Fv(O)r Fw(\(\()p Fs(j)p Fv(")p Fs(j)p Fv(n)500 5374 y Ft(2)538 5404 y Fw(\))570 5374 y Ft(1)p Fu(=)p Ft(\()p Fu(q)r Fp(\000)p Ft(1\))811 5404 y Fw(\))32 b Fo(with)h Fv(q)d Fw(=)c Fv(p)31 b Fo(if)i Fv(p)f Fo(is)g(o)l(dd)h(and)f Fv(q)e Fw(=)c Fv(d)32 b Fo(in)g(c)l(ase)g Fw(\()p Fv(N)9 b Fw(1\))p Fo(,)33 b Fv(q)d Fw(=)c(2)p Fv(p)19 b Fs(\000)h Fw(1)31 b Fo(in)h(c)l(ases)g Fw(\()p Fv(N)9 b Fw(2\))p Fo(-)p Fw(\()p Fv(N)g Fw(3\))p Fo(,)32 b(if)h Fv(p)e Fo(is)236 5504 y(even.)2084 6059 y Fw(20)p eop %%Page: 21 21 21 20 bop 236 272 a Fx(Remark)32 b(4.4)41 b Fo(L)l(et)24 b Fv(u)949 284 y Fu(!)r(;n)1080 272 y Fw(:=)f Fv(u)1239 284 y Fu(n)1308 272 y Fo(b)l(e)i(the)g(se)l(quenc)l(e)g(of)g(solutions) g(\()p Fv(u)2388 284 y Fu(!)r(;n)2521 272 y Fo(has)h(minimal)g(p)l (erio)l(d)g Fw(2)p Fv(\031)s(=n!)s Fo(\))d(obtaine)l(d)j(fr)l(om)236 372 y(the)k(ab)l(ove)g(The)l(or)l(ems.)39 b(L)l(et)29 b Fs(E)1197 384 y Fu(!)r(;n)1329 372 y Fw(:=)1439 305 y Fq(R)1495 325 y Fu(\031)1479 401 y Ft(0)1540 372 y Fw(\()p Fv(u)1620 384 y Fu(!)r(;n)1728 372 y Fw(\))1760 341 y Ft(2)1760 392 y Fu(t)1798 372 y Fv(=)p Fw(2)16 b(+)h(\()p Fv(u)2060 384 y Fu(!)r(;n)2168 372 y Fw(\))2200 341 y Ft(2)2200 392 y Fu(x)2243 372 y Fv(=)p Fw(2)f(+)g Fv(F)c Fw(\()p Fv(u)p Fw(\))29 b Fv(dx)h Fo(b)l(e)f(the)h(ener)l(gy)f (of)h Fv(u)3395 384 y Fu(!)r(;n)3504 372 y Fo(.)38 b(It)29 b(is)g(p)l(ossible)236 471 y(to)h(pr)l(ove,)h(using)f(Pr)l(op)l (osition)h(\(3.1\),)g(the)f(fol)t(lowing)j(estimates:)905 713 y Fs(jj)p Fv(u)999 725 y Fu(!)r(;n)1107 713 y Fs(jj)1153 728 y Fu(H)1211 711 y Fd(1)1272 713 y Fw(=)23 b Fv(n)1410 621 y Fq(\020)1486 657 y Fs(j)p Fv(!)f Fs(\000)c Fw(1)p Fs(j)p Fv(n)1781 627 y Ft(2)p 1469 694 366 4 v 1469 770 a Fv(m)1542 782 y Fu(n)1587 770 y Fw(\()p Fv(q)k Fw(+)c(1\))1845 621 y Fq(\021)1894 638 y Ft(1)p Fu(=)p Ft(\()p Fu(q)r Fp(\000)p Ft(1\))2135 621 y Fq(\020)2184 713 y Fw(1)g(+)g Fv(g)2367 725 y Ft(1)2404 621 y Fq(\020)2464 657 y Fw(\()p Fs(j)p Fv(!)j Fs(\000)d Fw(1)p Fs(j)p Fv(n)2790 627 y Ft(2)2827 657 y Fw(\))2859 627 y Ft(\()p Fu(p)p Fp(\000)p Ft(1\))p Fu(=)p Ft(\()p Fu(q)r Fp(\000)p Ft(1\))p 2464 694 775 4 v 2805 770 a Fv(\015)2848 782 y Fu(!)3248 621 y Fq(\021)o(\021)3804 713 y Fw(\(4.25\))970 1012 y Fs(E)1014 1024 y Fu(!)r(;n)1145 1012 y Fw(=)1245 956 y Fv(n)1295 926 y Ft(2)p 1243 993 92 4 v 1243 1069 a Fw(2)p Fv(\031)1345 920 y Fq(\020)1422 956 y Fs(j)p Fv(!)j Fs(\000)d Fw(1)p Fs(j)p Fv(n)1716 926 y Ft(2)p 1404 993 366 4 v 1404 1069 a Fv(m)1477 1081 y Fu(n)1522 1069 y Fw(\()p Fv(q)k Fw(+)c(1\))1780 920 y Fq(\021)1830 937 y Ft(2)p Fu(=)p Ft(\()p Fu(q)r Fp(\000)p Ft(1\))2070 920 y Fq(\020)2120 1012 y Fw(1)g(+)g Fv(g)2303 1024 y Ft(2)2340 920 y Fq(\020)2399 956 y Fw(\()p Fs(j)p Fv(!)k Fs(\000)c Fw(1)p Fs(j)p Fv(n)2726 926 y Ft(2)2762 956 y Fw(\))2794 926 y Ft(\()p Fu(p)p Fp(\000)p Ft(1\))p Fu(=)p Ft(\()p Fu(q)r Fp(\000)p Ft(1\))p 2399 993 775 4 v 2741 1069 a Fv(\015)2784 1081 y Fu(!)3183 920 y Fq(\021\021)236 1206 y Fo(wher)l(e)27 b Fv(q)f Fw(=)d Fv(p)i Fo(if)i Fv(p)f Fo(is)g(o)l(dd,)i Fv(q)e Fw(=)d Fv(d)j Fo(if)h Fv(p)e Fo(is)i(even)f(and)g Fv(d)d(<)g Fw(2)p Fv(p)10 b Fs(\000)g Fw(1)24 b Fo(is)j(the)f(smal)t(lest)g(or)l (der)h(of)g(the)f(o)l(dd)h(terms,)f Fv(q)h Fw(=)22 b(2)p Fv(p)10 b Fs(\000)g Fw(1)236 1305 y Fo(if)37 b Fv(p)f Fo(is)g(even)h(and)f(ther)l(e)g(is)h(no)f(o)l(dd)h(term)f(of)h(or)l (der)f Fv(<)e Fw(2)p Fv(p)23 b Fs(\000)f Fw(1)36 b Fo(and)g Fw(lim)2641 1317 y Fu(s)p Fp(!)p Ft(0)2789 1305 y Fv(g)2829 1317 y Fu(i)2857 1305 y Fw(\()p Fv(s)p Fw(\))f(=)f(0)p Fo(;)39 b Fv(m)3273 1317 y Fu(n)3352 1305 y Fw(:=)34 b Fv(m)p Fw(\()p Fv(G)3644 1317 y Fu(n)3690 1305 y Fw(\))h(=)f Fv(m)i Fo(if)236 1405 y Fv(p)f Fo(is)g(o)l(dd)h(and)g Fv(m)805 1417 y Fu(n)883 1405 y Fw(=)c Fv(m)22 b Fw(+)g Fv(O)r Fw(\(1)p Fv(=n)1393 1375 y Ft(2)1430 1405 y Fw(\))35 b Fo(if)h Fv(p)f Fo(is)g(even,)i(wher)l(e)f Fv(m)c(>)h Fw(0)h Fo(dep)l(ends)i(only)g(on)f Fv(f)9 b Fo(.)54 b(In)34 b(the)i(c)l(ase)f(when)h(the)236 1504 y(smal)t(lest)27 b(or)l(der)h(of)f(the)g(o)l(dd)g(terms)g(is)f Fw(2)p Fv(p)12 b Fs(\000)g Fw(1)24 b Fo(and)j Fw(0)c Fv(<)g(b)f(<)h(p\031)2250 1474 y Ft(2)2287 1504 y Fv(a)2331 1474 y Ft(2)2368 1504 y Fv(=)p Fw(24)p Fo(,)k(we)f(have)i(two)f(se)l(quenc)l(es)f(of)h (solutions)g(which)236 1604 y(c)l(orr)l(esp)l(ond)k(to)f(two)g(\(p)l (ossibly\))h(di\013er)l(ent)f(values)g(of)h Fv(m)p Fo(.)361 1787 y Fw(W)-7 b(e)28 b(\014nally)f(discuss)g(the)h(optimalit)n(y)f(of) h(the)g(estimate)g(on)f(the)h(n)n(um)n(b)r(er)f(of)h(solutions)f(that)g (w)n(e)h(\014nd.)236 1969 y Fx(Remark)k(4.5)41 b Fo(By)28 b(c)l(onditions)h(\(4.7\),)h(\(4.8\),)g(\(4.20\),)g(\(4.24\))g(in)e (The)l(or)l(ems)g(4.1,)i(4.2,)g(4.3,)g(4.4)g(we)e(c)l(an)g(\014nd)f Fv(N)3880 1981 y Fu(!)3951 1969 y Fs(\031)236 2069 y Fv(O)r Fw(\()p Fv(\015)381 2039 y Fu(\034)376 2090 y(!)425 2069 y Fv(=)467 1998 y Fq(p)p 550 1998 244 4 v 71 x Fs(j)p Fv(!)21 b Fs(\000)d Fw(1)p Fs(j)o Fw(\))38 b Fo(with)g Fw(1)f Fs(\024)f Fv(\034)47 b Fs(\024)36 b Fw(2)p Fo(,)j(p)l(erio)l (dic)h(solutions)d(of)i(\(1.1\))f(with)g(p)l(erio)l(d)h Fw(2)p Fv(\031)s(=!)s Fo(.)60 b(They)39 b(have)g(de)l(cr)l(e)l(asing) 236 2169 y(minimal)f(p)l(erio)l(ds)g Fw(2)p Fv(\031)s(=!)s(;)14 b(\031)s(=!)s(;)g(:)g(:)g(:)e(;)i Fw(2)p Fv(\031)s(=N)1615 2181 y Fu(!)1662 2169 y Fv(!)40 b Fo(and)d(incr)l(e)l(asing)h(ener)l (gies)f(in)g Fv(n)p Fo(,)i(se)l(e)e(r)l(emark)h(4.4.)61 b(We)37 b(exp)l(e)l(ct)g(the)236 2268 y(estimate)31 b(on)h(the)f(numb)l (er)f Fv(N)1192 2280 y Fu(!)1271 2268 y Fo(of)h(solutions)h(found,)g (with)g(smal)t(l)f Fs(j)20 b(\001)f(j)2488 2238 y Fu(p)p Fp(\000)p Ft(1)2488 2289 y Fu(!)2612 2268 y Fv(=\015)2697 2280 y Fu(!)2744 2268 y Fo(,)32 b(to)f(b)l(e)g(optimal)h(sinc)l(e)g(we) f(c)l(ould)h(have)236 2379 y(p)l(erforme)l(d)d(a)f(\014nite)f (dimensional)i(Lyapunov-Schmidt)g(r)l(e)l(duction)e Fv(u)c Fw(:=)g Fv(v)2642 2391 y Fu(N)2718 2379 y Fw(+)13 b Fv(w)30 b Fo(with)e Fv(v)3103 2391 y Fu(N)3189 2379 y Fw(:=)3300 2317 y Fq(P)3388 2337 y Fu(N)3388 2404 y(j)s Ft(=1)3521 2379 y Fv(\030)3557 2391 y Fu(j)3606 2379 y Fw(cos)13 b Fv(j)5 b(t)14 b Fw(sin)f Fv(j)5 b(x)236 2491 y Fo(and)32 b Fv(N)475 2461 y Ft(2)512 2491 y Fs(j)p Fv(!)23 b Fs(\000)d Fw(1)p Fs(j)26 b Fw(=)g Fv(O)r Fw(\()p Fv(\015)1017 2503 y Fu(!)1066 2491 y Fw(\))p Fo(.)45 b(Inde)l(e)l(d,)33 b(in)f(lemma)g(2.3,)i(the)e(\\smal)t(l)g(denominators")h(ar)l(e)f Fs(j)p Fv(!)3190 2461 y Ft(2)3227 2491 y Fv(l)3254 2461 y Ft(2)3311 2491 y Fs(\000)19 b Fv(j)3434 2461 y Ft(2)3471 2491 y Fs(j)27 b(\025)f Fv(\015)3655 2503 y Fu(!)3703 2491 y Fo(,)33 b Fs(8)p Fv(l)28 b Fs(6)p Fw(=)e Fv(j)5 b Fo(,)236 2591 y(and)30 b(for)h Fv(j)d Fw(=)22 b Fv(l)j Fs(\025)d Fv(N)39 b Fo(ar)l(e)29 b Fs(j)p Fv(!)1140 2560 y Ft(2)1177 2591 y Fv(l)1204 2560 y Ft(2)1259 2591 y Fs(\000)17 b Fv(j)1380 2560 y Ft(2)1418 2591 y Fs(j)23 b Fw(=)f Fs(j)p Fv(!)1629 2560 y Ft(2)1684 2591 y Fs(\000)c Fw(1)p Fs(j)p Fv(j)1871 2560 y Ft(2)1931 2591 y Fs(\025)k Fw(\()p Fs(j)p Fv(!)f Fs(\000)d Fw(1)p Fs(j)p Fv(=)p Fw(2\))p Fv(N)2486 2560 y Ft(2)2545 2591 y Fw(=)k Fv(O)r Fw(\()p Fv(\015)2772 2603 y Fu(!)2821 2591 y Fw(\))p Fo(.)39 b(In)29 b(this)h(way)g(we)g(r)l(e)l(duc)l(e)f(to)h(study)236 2690 y(a)k(even)f(functional)g(like)h Fw(\010)1115 2702 y Fu(")1184 2690 y 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Fw(\()p Fv(l)2348 1376 y Ft(6)2403 1410 y Fw(+)18 b(\(3)p Fv(l)r Fw(\))2619 1376 y Ft(6)2656 1410 y Fw(\))23 b Fs(\024)2818 1354 y Fv(C)p 2809 1391 85 4 v 2809 1467 a(\015)2857 1443 y Ft(2)2903 1410 y Fs(j)p Fv(u)2974 1422 y Fu(ttt)3053 1410 y Fs(j)3076 1376 y Ft(2)3076 1433 y Fu(L)3122 1416 y Fd(2)3182 1410 y Fs(\024)3279 1354 y Fv(C)3344 1324 y Fp(0)p 3279 1391 89 4 v 3281 1467 a Fv(\015)3329 1443 y Ft(2)3378 1410 y Fs(jj)p Fv(u)p Fs(jj)3518 1376 y Ft(2)3518 1433 y Fu(H)3576 1416 y Fd(3)3613 1410 y Fv(:)209 b Fw(\(5.4\))236 1689 y(By)28 b(\(5.2\),)f(\(5.3\))g (and)h(\(5.4\))f(w)n(e)g(get)1407 1626 y Fq(P)1494 1714 y Fu(l)p Fp(\025)p Ft(0)p Fu(;j)s Fp(\025)p Ft(1)p Fu(;j)s Fp(6)p Ft(=)p Fu(l)1877 1689 y Fs(j)p Fv(w)1959 1701 y Fu(lj)2016 1689 y Fs(j)2039 1658 y Ft(2)2076 1689 y Fw(\()p Fv(l)2135 1658 y Ft(6)2190 1689 y Fw(+)18 b Fv(j)2312 1658 y Ft(6)2350 1689 y Fw(\))23 b Fs(\024)g Fw(\()p Fv(C)t(=\015)5 b Fw(\))2710 1658 y Ft(2)2747 1689 y Fs(jj)p Fv(u)p Fs(jj)2887 1658 y Ft(2)2887 1715 y Fu(H)2945 1699 y Fd(3)2982 1689 y Fw(.)361 1788 y(No)n(w)27 b(w)n(e)g(pro)n(v)n(e)f (that)1717 1821 y Fq(X)1851 1900 y Fs(j)p Fv(w)1933 1912 y Fu(lj)1989 1900 y Fs(j)p Fw(\()p Fv(l)2071 1866 y Ft(2)2127 1900 y Fw(+)18 b Fv(j)2249 1866 y Ft(2)2286 1900 y Fw(\))23 b Fv(<)g Fs(1)p Fv(;)1310 b Fw(\(5.5\))236 2077 y(whic)n(h)28 b(implies)g(that)f Fv(w)f Fs(2)e Fv(C)1164 2047 y Ft(2)1201 2077 y Fw(\()p 1233 2010 60 4 v(\012\).)38 b(W)-7 b(e)28 b(ha)n(v)n(e)1034 2229 y Fq(X)911 2407 y Fu(l)p Fp(\025)p Ft(0)p Fu(;j)s Fp(\025)p Ft(1)p Fu(;j)s Fp(6)p Ft(=)p Fu(l)1290 2307 y Fs(j)p Fv(w)1372 2319 y Fu(lj)1429 2307 y Fs(j)p Fw(\()p Fv(l)1511 2273 y Ft(2)1566 2307 y Fw(+)18 b Fv(j)1688 2273 y Ft(2)1725 2307 y Fw(\))24 b(=)1991 2229 y Fq(X)1868 2407 y Fu(l)p Fp(\025)p Ft(0)p Fu(;j)s Fp(\025)p Ft(1)p Fu(;j)s Fp(6)p Ft(=)p Fu(l)2372 2251 y Fs(j)p Fv(u)2443 2263 y Fu(lj)2498 2251 y Fs(j)p 2257 2288 380 4 v 2257 2364 a(j)p Fv(!)2335 2340 y Ft(2)2372 2364 y Fv(l)2399 2340 y Ft(2)2454 2364 y Fs(\000)18 b Fv(j)2576 2340 y Ft(2)2613 2364 y Fs(j)2646 2307 y Fw(\()p Fv(l)2705 2273 y Ft(2)2761 2307 y Fw(+)g Fv(j)2883 2273 y Ft(2)2920 2307 y Fw(\))23 b(=)g Fv(S)3119 2273 y Fp(0)3114 2328 y Ft(1)3169 2307 y Fw(+)18 b Fv(S)3308 2273 y Fp(0)3303 2328 y Ft(2)3845 2307 y Fw(\(5.6\))236 2576 y(where)698 2703 y Fv(S)754 2668 y Fp(0)749 2723 y Ft(1)809 2703 y Fw(:=)951 2624 y Fq(X)920 2803 y Fu(j)s Fp(\025)p Ft(2)p Fu(!)r(l)1240 2646 y Fs(j)p Fv(u)1311 2658 y Fu(lj)1367 2646 y Fs(j)p 1125 2684 V 1125 2760 a(j)p Fv(!)1203 2736 y Ft(2)1240 2760 y Fv(l)1267 2736 y Ft(2)1322 2760 y Fs(\000)g Fv(j)1444 2736 y Ft(2)1481 2760 y Fs(j)1514 2703 y Fw(\()p Fv(l)1573 2668 y Ft(2)1629 2703 y Fw(+)g Fv(j)1751 2668 y Ft(2)1788 2703 y Fw(\))166 b(and)g Fv(S)2342 2668 y Fp(0)2337 2723 y Ft(2)2397 2703 y Fw(:=)2600 2624 y Fq(X)2508 2803 y Fu(j)s(<)p Ft(2)p Fu(!)r(l;j)s Fp(6)p Ft(=)p Fu(l)2950 2646 y Fs(j)p Fv(u)3021 2658 y Fu(lj)3077 2646 y Fs(j)p 2836 2684 V 2836 2760 a(j)p Fv(!)2914 2736 y Ft(2)2951 2760 y Fv(l)2978 2736 y Ft(2)3033 2760 y Fs(\000)18 b Fv(j)3155 2736 y Ft(2)3192 2760 y Fs(j)3225 2703 y Fw(\()p Fv(l)3284 2668 y Ft(2)3339 2703 y Fw(+)g Fv(j)3461 2668 y Ft(2)3498 2703 y Fw(\))p Fv(:)236 2948 y Fw(F)-7 b(or)27 b Fv(j)h Fs(\025)23 b Fw(2)p Fv(!)s(l)r Fw(,)j Fs(j)p Fv(!)786 2918 y Ft(2)823 2948 y Fv(l)850 2918 y Ft(2)905 2948 y Fs(\000)18 b Fv(j)1027 2918 y Ft(2)1064 2948 y Fs(j)24 b(\025)e Fw(3)p Fv(j)1279 2918 y Ft(2)1316 2948 y Fv(=)p Fw(4)27 b(and)g Fv(l)d Fs(\024)f Fv(j)5 b(=)p Fw(2)p Fv(!)25 b Fs(\024)d Fv(j)5 b Fw(,)28 b(hence)578 3178 y Fv(S)634 3144 y Fp(0)629 3199 y Ft(1)689 3178 y Fs(\024)23 b Fv(C)887 3100 y Fq(X)856 3278 y Fu(j)s Fp(\025)p Ft(2)p Fu(!)r(l)1051 3178 y Fs(j)p Fv(u)1122 3190 y Fu(lj)1178 3178 y Fs(j)g(\024)g Fv(C)1377 3086 y Fq(\020)1501 3100 y(X)1440 3278 y Fu(l)p Fp(\025)p Ft(0)p Fu(;j)s Fp(\025)p Ft(1)1696 3178 y Fs(j)p Fv(u)1767 3190 y Fu(lj)1823 3178 y Fs(j)1846 3144 y Ft(2)1883 3178 y Fw(\()p Fv(l)1942 3144 y Ft(2)1998 3178 y Fw(+)18 b Fv(j)2120 3144 y Ft(2)2157 3178 y Fw(\))2189 3144 y Ft(2)2226 3086 y Fq(\021)2276 3103 y Ft(1)p Fu(=)p Ft(2)2380 3086 y Fq(\020)2505 3100 y(X)2444 3278 y Fu(l)p Fp(\025)p Ft(0)p Fu(;j)s Fp(\025)p Ft(1)2860 3122 y Fw(1)p 2709 3159 344 4 v 2709 3235 a(\()p Fv(l)2768 3211 y Ft(2)2824 3235 y Fw(+)g Fv(j)2946 3211 y Ft(2)2983 3235 y Fw(\))3015 3211 y Ft(2)3062 3086 y Fq(\021)3112 3103 y Ft(1)p Fu(=)p Ft(2)3239 3178 y Fs(\024)23 b Fv(C)3392 3144 y Fp(0)3416 3178 y Fs(jj)p Fv(u)p Fs(jj)3556 3193 y Fu(H)3614 3177 y Fd(2)3651 3178 y Fv(:)171 b Fw(\(5.7\))236 3457 y(Note)28 b(that)617 3395 y Fq(P)704 3482 y Fu(l)p Fp(\025)p Ft(0)p Fu(;j)s Fp(\025)p Ft(1)964 3457 y Fs(j)p Fv(u)1035 3469 y Fu(lj)1091 3457 y Fs(j)1114 3427 y Ft(2)1151 3457 y Fw(\()p Fv(l)1210 3427 y Ft(2)1266 3457 y Fw(+)18 b Fv(j)1388 3427 y Ft(2)1425 3457 y Fw(\))1457 3427 y Ft(2)1518 3457 y Fw(=)k Fv(O)r Fw(\()p Fs(jj)p Fv(u)p Fs(jj)1842 3427 y Ft(2)1842 3483 y Fu(H)1900 3467 y Fd(2)1938 3457 y Fw(\).)38 b(W)-7 b(e)28 b(claim)f(that)1495 3708 y Fv(S)1551 3674 y Fp(0)1546 3729 y Ft(2)1606 3708 y Fs(\024)1786 3629 y Fq(X)1694 3808 y Fu(j)s(<)p Ft(2)p Fu(!)r(l;j)s Fp(6)p Ft(=)p Fu(l)2011 3708 y Fv(C)2169 3652 y Fs(j)p Fv(u)2240 3664 y Fu(lj)2296 3652 y Fs(j)p Fv(l)2346 3622 y Ft(2)p 2087 3689 380 4 v 2087 3765 a Fs(j)p Fv(!)2165 3741 y Ft(2)2202 3765 y Fv(l)2229 3741 y Ft(2)2284 3765 y Fs(\000)18 b Fv(j)2406 3741 y Ft(2)2443 3765 y Fs(j)2499 3708 y Fv(<)23 b Fw(+)p Fs(1)p Fv(:)1087 b Fw(\(5.8\))236 3976 y(Indeed)28 b(w)n(e)f(kno)n(w)g(that)1759 3997 y Fq(X)1728 4176 y Fu(j)s(<)p Ft(2)p Fu(!)r(l)1909 4076 y Fw(\()p Fv(u)1989 4088 y Fu(lj)2045 4076 y Fv(l)2072 4042 y Ft(2)2109 4076 y Fw(\))2141 4042 y Ft(2)2179 4076 y Fv(l)2206 4042 y Ft(2)2265 4076 y Fv(<)c Fw(+)p Fs(1)p Fv(:)1321 b Fw(\(5.9\))236 4316 y(As)27 b(in)g(the)g(pro)r(of)f(of)g (\(6.2\))g(in)h(lemma)g(2.2,)f(\(5.9\))g(implies)h(that)f Fv(S)2333 4286 y Fp(0)2328 4337 y Ft(2)2392 4316 y Fw(is)h(\014nite.)37 b(By)26 b(and)g(\(5.6\),)h(\(5.7\))f(and)h(\(5.8\))f(w)n(e)g(get)236 4416 y(\(5.5\).)37 b(The)27 b(lemma)h(is)f(pro)n(v)n(ed.)p 1291 4416 48 48 v 361 4565 a(By)i(lemma)h(5.1)f Fs(jj)p Fv(L)1002 4535 y Fp(\000)p Ft(1)1002 4586 y Fu(!)1091 4565 y Fw(\005)1153 4577 y Fu(W)1229 4565 y Fs(jj)1275 4577 y Fu(!)r(;)p Ft(3)1403 4565 y Fs(\024)d Fv(C)t(=\015)5 b Fw(.)43 b(Hence,)31 b(b)n(y)f(\(5.1\),)g(the)h(map)e Fs(G)2709 4577 y Fu(v)2776 4565 y Fw(:)e Fv(W)39 b Fs(!)27 b Fv(W)12 b Fw(,)31 b(de\014ned)f(in)g(the)h(pro)r(of)e(of)236 4665 y(lemma)f(2.3)e(b)n(y)i Fs(G)802 4677 y Fu(v)842 4665 y Fw(\()p Fv(w)r Fw(\))c(:=)f Fv(L)1159 4635 y Fp(\000)p Ft(1)1159 4685 y Fu(!)1247 4665 y Fw(\005)1309 4677 y Fu(W)1385 4665 y Fv(f)9 b Fw(\()p Fv(v)22 b Fw(+)c Fv(w)r Fw(\),)28 b(satis\014es)f(\(if)i Fv(v)s(;)14 b(w)25 b Fs(2)f Fv(H)2490 4635 y Ft(3)2527 4665 y Fw(\(\012\)\))1536 4914 y Fs(jG)1608 4926 y Fu(v)1648 4914 y Fw(\()p Fv(w)r Fw(\))p Fs(j)1796 4926 y Fu(!)r(;)p Ft(3)1921 4914 y Fs(\024)2035 4837 y Fq(e)2018 4858 y Fv(C)p 2018 4895 66 4 v 2027 4971 a(\015)2094 4822 y Fq(\020)2143 4914 y Fs(j)p Fv(v)s Fs(j)2232 4874 y Fu(p)2232 4936 y(!)r(;)p Ft(3)2352 4914 y Fw(+)18 b Fs(j)p Fv(w)r Fs(j)2542 4874 y Fu(p)2542 4936 y(!)r(;)p Ft(3)2644 4822 y Fq(\021)2693 4914 y Fv(;)1088 b Fw(\(5.10\))236 5160 y(where)499 5139 y Fq(e)482 5160 y Fv(C)40 b Fw(is)33 b(some)g(p)r(ositiv)n(e)g(constan) n(t.)54 b(De\014ne)1847 5138 y Fq(e)1845 5160 y Fv(\016)36 b Fw(:=)d([)p Fv(\015)5 b(=)p Fw(\(2)2241 5139 y Fq(e)2226 5160 y Fv(C)g Fw(\)])2345 5130 y Ft(1)p Fu(=)p Ft(\()p Fu(p)p Fp(\000)p Ft(1\))2621 5160 y Fw(and)34 b(assume)e(that)i Fs(j)p Fv(v)s Fs(j)3356 5172 y Fu(!)r(;)p Ft(3)3490 5160 y Fs(\024)3589 5138 y Fq(e)3588 5160 y Fv(\016)s Fw(.)54 b(Then)34 b(b)n(y)236 5273 y(\(5.10\),)27 b Fs(G)548 5285 y Fu(v)588 5273 y Fw(\()638 5252 y Fq(e)620 5273 y Fv(B)680 5305 y Fq(e)683 5310 y Fu(\016)720 5273 y Fw(\))c Fs(\032)880 5252 y Fq(e)863 5273 y Fv(B)923 5305 y Fq(e)926 5310 y Fu(\016)962 5273 y Fw(,)28 b(where)1447 5392 y Fq(e)1430 5413 y Fv(B)1490 5444 y Fq(e)1493 5449 y Fu(\016)1552 5413 y Fw(:=)1663 5321 y Fq(n)1718 5413 y Fv(w)e Fs(2)e Fv(W)30 b Fs(\\)19 b Fv(H)2140 5379 y Ft(3)2177 5413 y Fw(\(\012\))2329 5318 y Fq(\014)2329 5367 y(\014)2329 5417 y(\014)2384 5413 y Fs(j)p Fv(w)r Fs(j)2491 5425 y Fu(!)r(;)p Ft(3)2616 5413 y Fs(\024)2705 5391 y Fq(e)2704 5413 y Fv(\016)2744 5321 y Fq(o)2799 5413 y Fv(:)2084 6059 y Fw(22)p eop %%Page: 23 23 23 22 bop 236 272 a Fw(As)31 b(a)e(result,)i(if)g Fs(j)p Fv(v)s Fs(j)859 284 y Fu(!)r(;)p Ft(3)988 272 y Fs(\024)1081 250 y Fq(e)1080 272 y Fv(\016)i Fw(then)e Fv(w)1401 284 y Fu(k)1470 272 y Fw(:=)c Fs(G)1639 242 y Fu(k)1634 293 y(v)1680 272 y Fw(\(0\))h Fs(2)1914 251 y Fq(e)1897 272 y Fv(B)1957 303 y Fq(e)1960 308 y Fu(\016)2026 272 y Fw(for)i(all)g Fv(k)g Fs(\025)e Fw(1.)44 b(It)31 b(follo)n(ws)e(that)i (\()p Fv(w)3191 284 y Fu(k)3232 272 y Fw(\))g(con)n(v)n(erges)c(w)n (eakly)j(in)236 397 y Fv(H)312 367 y Ft(3)349 397 y Fw(\(\012\))e(to)g (some)42 b Fq(e)-61 b Fv(w)26 b Fs(2)991 376 y Fq(e)974 397 y Fv(B)1034 429 y Fq(e)1037 434 y Fu(\016)1073 397 y Fw(.)361 497 y(No)n(w,)32 b(recall)e(that)i Fv(w)r Fw(\()p Fv(v)s Fw(\))h(is)e(the)h(unique)f(\014xed)h(p)r(oin)n(t)f(of)h Fs(G)2268 509 y Fu(v)2339 497 y Fw(in)g(a)f(small)g(ball)g(around)f(0)h (for)g(the)h(norm)f Fs(j)21 b(\001)g(j)3867 509 y Fu(!)3947 497 y Fw(in)236 596 y Fv(W)41 b Fw(and)28 b(that,)i(b)n(y)e(the)h(Con)n (traction)e(Mapping)i(Principle,)f Fv(w)r Fw(\()p Fv(v)s Fw(\))i(is)f(the)g Fv(H)2671 566 y Ft(1)2737 596 y Fw(limit)g(of)g(\()p Fv(w)3123 608 y Fu(k)3164 596 y Fw(\).)41 b(Hence)29 b Fv(w)r Fw(\()p Fv(v)s Fw(\))d(=)39 b Fq(e)-60 b Fv(w)31 b Fw(and)236 707 y Fv(w)r Fw(\()p Fv(v)s Fw(\))25 b Fs(2)e Fv(H)583 677 y Ft(3)620 707 y Fw(\(\012\).)36 b(By)23 b(form)n(ula)f(\(5.1\))h(and)g(lemma)g(5.1)f(w)n(e)h(deriv)n(e)f(from)h (the)h(ab)r(o)n(v)n(e)d(considerations)h(that)h(if)h Fs(j)p Fv(v)s Fs(j)3764 719 y Fu(!)r(;)p Ft(3)3888 707 y Fs(\024)3977 685 y Fq(e)3976 707 y Fv(\016)236 807 y Fw(then)k Fv(w)r Fw(\()p Fv(v)s Fw(\))d Fs(2)e Fv(C)761 777 y Ft(2)799 807 y Fw(\()p 831 740 60 4 v(\012\),)28 b(and)f Fv(v)22 b Fw(+)c Fv(w)r Fw(\()p Fv(v)s Fw(\))25 b Fs(2)e Fv(C)1616 777 y Ft(2)1654 807 y Fw(\()p 1686 740 V(\012\))28 b(\(b)r(ecause)f Fv(V)38 b Fs(\\)18 b Fv(H)2380 777 y Ft(3)2417 807 y Fw(\(\012\))24 b Fs(\032)f Fv(C)2718 777 y Ft(2)2755 807 y Fw(\()p 2787 740 V(\012\)\).)361 907 y(There)30 b(remains)h(to)g(c)n(hec)n(k)f(that)h(when)h(condition)e (\(4.7\))h(or)g(\(4.8\))f(or)h(\(4.20\))f(or)g(\(4.24\))h(is)g (satis\014ed)f(\(according)236 1006 y(to)d(the)g(di\013eren)n(t)g (cases\))f(the)h(critical)f(p)r(oin)n(ts)h Fv(v)1753 1018 y Fu(";n)1876 1006 y Fw(of)g(\010)2030 1018 y Fu(")2092 1006 y Fw(that)g(w)n(e)f(ha)n(v)n(e)g(obtained)g(in)h(section)f(4)h (satisfy)f Fs(j)p Fv(v)3685 1018 y Fu(";n)3782 1006 y Fs(j)3805 966 y Fu(p)p Fp(\000)p Ft(1)3805 1028 y Fu(!)r(;)p Ft(3)3951 1006 y Fs(\024)236 1129 y Fv(\015)5 b(=)p Fw(\(2)416 1108 y Fq(e)400 1129 y Fv(C)g Fw(\).)37 b(Since)25 b(the)g(critical)f (p)r(oin)n(ts)g Fv(v)j Fw(:=)22 b Fv(v)1647 1141 y Fu(";n)1769 1129 y Fw(of)i(\010)1920 1141 y Fu(")1956 1129 y Fw(,)h(obtained)g(in)f (Theorems)g(4.1,)h(4.2,)f(4.3,)h(4.4)e(satisfy)i Fv(v)3708 1141 y Fu(tt)3785 1129 y Fw(=)e Fv(v)3913 1141 y Fu(xx)3993 1129 y Fw(,)236 1229 y(2)p Fv("v)357 1241 y Fu(tt)434 1229 y Fw(=)g(\005)584 1241 y Fu(V)642 1229 y Fv(f)9 b Fw(\()p Fv(v)21 b Fw(+)d Fv(w)r Fw(\()p Fv(v)s Fw(\)\),)459 1428 y Fs(jj)p Fv(v)s Fs(jj)594 1443 y Fu(H)652 1426 y Fd(3)713 1428 y Fw(=)23 b Fv(O)866 1336 y Fq(\020)926 1372 y Fw(1)p 926 1409 42 4 v 927 1485 a Fv(")978 1428 y Fs(jj)p Fw(\005)1086 1440 y Fu(V)1144 1428 y Fv(f)9 b Fw(\()p Fv(v)21 b Fw(+)d Fv(w)r Fw(\()p Fv(v)s Fw(\)\))p Fs(jj)1616 1336 y Fq(\021)1691 1428 y Fw(=)k Fv(O)1843 1336 y Fq(\020)1904 1372 y Fw(1)p 1904 1409 V 1905 1485 a Fv(")1955 1428 y Fs(jj)p Fv(f)9 b Fw(\()p Fv(v)22 b Fw(+)c Fv(w)r Fw(\()p Fv(v)s Fw(\)\))p Fs(jj)2474 1336 y Fq(\021)2525 1428 y Fv(;)97 b Fs(jj)p Fv(v)s Fs(jj)2780 1443 y Fu(H)2838 1426 y Fd(2)2899 1428 y Fw(=)23 b Fv(O)3052 1336 y Fq(\020)3112 1372 y Fw(1)p 3112 1409 V 3113 1485 a Fv(")3163 1428 y Fs(j)p Fv(f)9 b Fw(\()p Fv(v)22 b Fw(+)c Fv(w)r Fw(\()p Fv(v)s Fw(\)\))p Fs(j)3636 1443 y Fu(L)3682 1426 y Fd(2)3720 1336 y Fq(\021)3769 1428 y Fv(:)236 1611 y Fw(Therefore)27 b(b)n(y)g(lemmas)g(2.1)g(and)g(2.2) 760 1820 y Fs(j)p Fv(v)s Fs(j)849 1832 y Fu(!)r(;)p Ft(3)973 1820 y Fw(=)c Fv(O)1126 1728 y Fq(\020)1176 1820 y Fs(j)p Fv(v)s Fs(j)1265 1832 y Fu(!)1332 1820 y Fw(+)18 b Fs(j)p Fv(f)9 b Fw(\()p Fv(v)22 b Fw(+)c Fv(w)r Fw(\()p Fv(v)s Fw(\)\))p Fs(j)1888 1832 y Fu(!)1938 1728 y Fq(\021)2010 1820 y Fw(=)23 b Fv(O)2163 1728 y Fq(\020)2213 1820 y Fs(j)p Fv(v)s Fs(j)2302 1832 y Fu(!)2369 1820 y Fw(+)18 b Fs(j)p Fv(v)s Fs(j)2541 1785 y Fu(p)2541 1840 y(!)2589 1728 y Fq(\020)2639 1820 y Fw(1)g(+)2792 1764 y Fs(j)p Fv(v)s Fs(j)2881 1733 y Fu(p)p Fp(\000)p Ft(1)2881 1784 y Fu(!)p 2792 1801 213 4 v 2874 1877 a Fv(\015)3014 1728 y Fq(\021\021)3137 1820 y Fw(=)k Fv(O)r Fw(\()p Fs(j)p Fv(v)s Fs(j)3410 1832 y Fu(!)3460 1820 y Fw(\))312 b(\(5.11\))236 2042 y(b)r(ecause)27 b Fv(v)f Fs(2)e(D)752 2054 y Fu(\032)790 2042 y Fw(.)37 b(Hence,)28 b(b)n(y)e(remark)g(4.3,)h Fs(j)p Fv(v)s Fs(j)1764 2003 y Fu(p)p Fp(\000)p Ft(1)1764 2065 y Fu(!)r(;)p Ft(3)1887 2042 y Fv(=\015)g Fs(\024)c Fw(1)p Fv(=)p Fw(2)2228 2021 y Fq(e)2213 2042 y Fv(C)32 b Fw(pro)n(vided)26 b(the)i(constan)n(ts)e Fv(C)3214 2054 y Ft(10)3285 2042 y Fw(,)h Fv(C)3394 2054 y Ft(11)3465 2042 y Fw(,)g Fv(C)3574 2054 y Ft(12)3645 2042 y Fw(,)g Fv(C)3754 2054 y Ft(13)3852 2042 y Fw(ha)n(v)n(e)236 2142 y(b)r(een)h(c)n(hosen)f(small)g(enough.)236 2413 y Fr(6)135 b(App)t(endix)236 2595 y Fc(Pr)n(oof)72 b(of)35 b(lemma)c Fw(2.1.)46 b(By)31 b(standard)e(results,)j(if)f Fv(u)d Fs(2)h Fv(L)2242 2564 y Fp(1)2332 2595 y Fs(\\)21 b Fv(H)2484 2564 y Ft(1)2521 2595 y Fw(\()p Fx(T)h Fs(\002)e Fw(\(0)p Fv(;)14 b(\031)s Fw(\)\),)32 b(then)g Fv(f)9 b Fw(\()p Fv(u)p Fw(\))28 b Fs(2)h Fv(H)3549 2564 y Ft(1)3586 2595 y Fw(\()p Fx(T)21 b Fs(\002)f Fw(\(0)p Fv(;)14 b(\031)s Fw(\)\))236 2694 y(and)28 b(\()p Fv(f)9 b Fw(\()p Fv(u)p Fw(\)\))624 2706 y Fu(x)689 2694 y Fs(\021)22 b Fv(f)826 2664 y Fp(0)849 2694 y Fw(\()p Fv(u)p Fw(\))p Fv(u)1009 2706 y Fu(x)1051 2694 y Fw(,)27 b(\()p Fv(f)9 b Fw(\()p Fv(u)p Fw(\)\))1327 2706 y Fu(t)1380 2694 y Fs(\021)23 b Fv(f)1518 2664 y Fp(0)1540 2694 y Fw(\()p Fv(u)p Fw(\))p Fv(u)1700 2706 y Fu(t)1729 2694 y Fw(.)37 b(W)-7 b(e)28 b(get)369 2902 y Fs(j)p Fw(\()p Fv(f)9 b Fw(\()p Fv(u)p Fw(\)\))618 2914 y Fu(x)660 2902 y Fs(j)683 2917 y Fu(L)729 2900 y Fd(2)788 2902 y Fw(=)876 2810 y Fq(\020)939 2789 y(Z)986 2978 y Ft(\012)1051 2902 y Fs(j)p Fw(\()p Fv(f)g Fw(\()p Fv(u)p Fw(\)\))1300 2914 y Fu(x)1342 2902 y Fs(j)1365 2868 y Ft(2)1402 2810 y Fq(\021)1452 2827 y Ft(1)p Fu(=)p Ft(2)1579 2902 y Fw(=)1667 2810 y Fq(\020)1730 2789 y(Z)1777 2978 y Ft(\012)1842 2902 y Fs(j)p Fv(f)1915 2868 y Fp(0)1938 2902 y Fw(\()p Fv(u)p Fw(\))p Fs(j)2073 2868 y Ft(2)2110 2902 y Fs(j)p Fv(u)2181 2914 y Fu(x)2223 2902 y Fs(j)2246 2868 y Ft(2)2283 2810 y Fq(\021)2333 2827 y Ft(1)p Fu(=)p Ft(2)2460 2902 y Fs(\024)23 b(j)p Fv(f)2621 2868 y Fp(0)2644 2902 y Fw(\()p Fv(u)p Fw(\))p Fs(j)2779 2914 y Fp(1)2849 2902 y Fs(jj)p Fv(u)p Fs(jj)2989 2917 y Fu(H)3047 2900 y Fd(1)3107 2902 y Fs(\024)g Fv(C)6 b Fs(j)p Fv(u)p Fs(j)3354 2868 y Fu(p)p Fp(\000)p Ft(1)3354 2923 y Fp(1)3477 2902 y Fs(jj)p Fv(u)p Fs(jj)3617 2917 y Fu(H)3675 2900 y Fd(1)3845 2902 y Fw(\(6.1\))236 3109 y(for)33 b Fs(j)p Fv(u)p Fs(j)463 3121 y Fp(1)566 3109 y Fw(small)g(enough.)54 b(An)33 b(analogous)e(estimate)j(holds)f(for)f Fs(j)p Fw(\()p Fv(f)9 b Fw(\()p Fv(u)p Fw(\)\))2598 3121 y Fu(t)2628 3109 y Fs(j)2651 3124 y Fu(L)2697 3107 y Fd(2)2733 3109 y Fw(.)54 b(Since)34 b Fs(j)p Fv(f)9 b Fw(\()p Fv(u)p Fw(\))p Fs(j)32 b(\024)g Fv(C)6 b Fs(j)p Fv(u)p Fs(j)3529 3079 y Fu(p)3529 3129 y Fp(1)3599 3109 y Fw(,)35 b(w)n(e)e(obtain)236 3208 y(\(2.4\).)361 3308 y(W)-7 b(e)40 b(no)n(w)f(pro)n(v)n(e)f(that)i Fv(u)j Fs(!)h Fv(f)9 b Fw(\()p Fv(u)p Fw(\))39 b(is)h Fv(C)1709 3278 y Ft(1)1747 3308 y Fw(\()p Fv(X)r(;)14 b(X)7 b Fw(\).)73 b(It)40 b(is)f(easy)g(to)h(sho)n(w)f(that)h(it)g(is)g (Gateaux)f(di\013eren)n(tiable)236 3408 y(and)c(that)h Fv(D)662 3420 y Fu(G)717 3408 y Fv(f)9 b Fw(\()p Fv(u)p Fw(\)[)p Fv(h)p Fw(])35 b(=)g Fv(f)1158 3378 y Fp(0)1181 3408 y Fw(\()p Fv(u)p Fw(\))p Fv(h)p Fw(.)60 b(Moreo)n(v)n(er)32 b(with)k(estimates)e(similar)h(as)f(b)r(efore)h(w)n(e)g(ha)n(v)n(e)e (that)j Fv(u)f Fs(!)g Fv(D)3798 3420 y Fu(G)3854 3408 y Fv(f)9 b Fw(\()p Fv(u)p Fw(\))236 3507 y(is)38 b(con)n(tin)n(uous.)68 b(Hence)39 b(the)f(Nemitski)h(op)r(erator)d Fv(u)41 b Fs(!)g Fv(f)9 b Fw(\()p Fv(u)p Fw(\))38 b(is)g(in)g Fv(C)2607 3477 y Ft(1)2645 3507 y Fw(\()p Fv(X)r(;)14 b(X)7 b Fw(\))38 b(and)g(its)g(F)-7 b(rec)n(het)38 b(di\013eren)n(tial)g(is)236 3607 y Fv(h)23 b Fs(!)g Fv(D)r(f)9 b Fw(\()p Fv(u)p Fw(\)[)p Fv(h)p Fw(])23 b(=)g Fv(D)920 3619 y Fu(G)975 3607 y Fv(f)9 b Fw(\()p Fv(u)p Fw(\)[)p Fv(h)p Fw(])23 b(=)f Fv(f)1391 3577 y Fp(0)1414 3607 y Fw(\()p Fv(u)p Fw(\))p Fv(h)p Fw(.)37 b(Estimate)27 b(\(2.5\))h(can)f(b)r(e)h(obtained)f(in)h (the)g(same)f(w)n(a)n(y)f(as)h(\(2.4\).)p 3740 3607 48 48 v 236 3747 a Fc(Pr)n(oof)69 b(of)32 b(lemma)27 b Fw(2.2.)36 b(W)-7 b(riting)28 b Fv(w)r Fw(\()p Fv(t;)14 b(x)p Fw(\))25 b(=)1807 3685 y Fq(P)1895 3772 y Fu(l)p Fp(\025)p Ft(0)p Fu(;j)s Fp(\025)p Ft(1)p Fu(;j)s Fp(6)p Ft(=)p Fu(l)2277 3747 y Fv(w)2336 3759 y Fu(l;j)2427 3747 y Fw(cos)o(\()p Fv(l)r(t)p Fw(\))14 b(sin\()p Fv(j)5 b(x)p Fw(\))24 b Fs(2)f Fv(X)34 b Fw(w)n(e)27 b(ha)n(v)n(e)g(that)1124 3944 y Fv(L)1181 3910 y Fp(\000)p Ft(1)1181 3965 y Fu(!)1270 3944 y Fv(w)r Fw(\()p Fv(t;)14 b(x)p Fw(\))25 b(=)1744 3865 y Fq(X)1621 4044 y Fu(l)p Fp(\025)p Ft(0)p Fu(;j)s Fp(\025)p Ft(1)p Fu(;j)s Fp(6)p Ft(=)p Fu(l)2228 3888 y Fv(w)2287 3900 y Fu(l;j)p 2010 3925 573 4 v 2010 4001 a Fw(\()p Fv(!)s(l)20 b Fs(\000)e Fv(j)5 b Fw(\)\()p Fv(!)s(l)20 b Fw(+)e Fv(j)5 b Fw(\))2606 3944 y(cos)o(\()p Fv(l)r(t)p Fw(\))14 b(sin\()p Fv(j)5 b(x)p Fw(\))p Fv(:)236 4200 y Fw(Since)28 b Fv(!)e Fs(2)d Fv(W)687 4212 y Fu(\015)730 4200 y Fw(,)28 b(it)g(is)f(clear)g(that)h Fs(jj)p Fv(L)1429 4170 y Fp(\000)p Ft(1)1429 4221 y Fu(!)1518 4200 y Fv(w)r Fs(jj)23 b Fw(=)g Fv(O)r Fw(\()p Fs(jj)p Fv(w)r Fs(jj)p Fv(=\015)5 b Fw(\).)361 4300 y(W)-7 b(e)28 b(no)n(w)f(pro)n(v)n(e)f (that)i(for)f(all)g Fv(w)f Fs(2)d Fv(W)952 4514 y(S)28 b Fw(:=)1264 4436 y Fq(X)1141 4614 y Fu(l)p Fp(\025)p Ft(0)p Fu(;j)s Fp(\025)p Ft(1)p Fu(;j)s Fp(6)p Ft(=)p Fu(l)1716 4458 y Fs(j)p Fv(w)1798 4470 y Fu(l;j)1874 4458 y Fs(j)p 1530 4495 554 4 v 1530 4571 a(j)p Fv(!)s(l)19 b Fs(\000)f Fv(j)5 b Fs(j)p Fw(\()p Fv(!)s(l)20 b Fw(+)e Fv(j)5 b Fw(\))2116 4514 y Fs(\024)23 b Fv(C)2269 4422 y Fq(\020)2319 4514 y Fs(j)p Fv(w)r Fs(j)2426 4529 y Fu(L)2472 4513 y Fd(2)2527 4514 y Fw(+)2620 4458 y Fs(j)p Fv(!)e Fs(\000)d Fw(1)p Fs(j)2864 4428 y Ft(1)p Fu(=)p Ft(2)p 2620 4495 349 4 v 2770 4571 a 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Fv(!)i Fw(is)e(not)f(rational,)h Fv(e)p Fw(\()p Fv(l)r Fw(\))f(is)g(the)236 5203 y(only)c(in)n(teger)g Fv(e)g Fw(suc)n(h)h(that)g Fs(j)p Fv(e)18 b Fs(\000)g Fv(!)s(l)r Fs(j)k Fv(<)h Fw(1)p Fv(=)p Fw(2.)35 b(Let)28 b(us)f(de\014ne)1447 5411 y Fv(S)1498 5423 y Ft(1)1559 5411 y Fw(=)1872 5332 y Fq(X)1646 5514 y Fu(l)p Fp(\025)p Ft(0)p Fu(;j)s Fp(\025)p Ft(1)p Fu(;j)s Fp(6)p Ft(=)p Fu(l;j)s Fp(6)p Ft(=)p Fu(e)p Ft(\()p Fu(l)p Ft(\))2427 5355 y Fs(j)p Fv(w)2509 5367 y Fu(l;j)2586 5355 y Fs(j)p 2241 5392 554 4 v 2241 5468 a(j)p Fv(!)s(l)20 b Fs(\000)e Fv(j)5 b Fs(j)p Fw(\()p Fv(!)s(l)19 b Fw(+)f Fv(j)5 b Fw(\))2084 6059 y(23)p eop %%Page: 24 24 24 23 bop 236 272 a Fw(and)1479 404 y Fv(S)1530 416 y Ft(2)1590 404 y Fw(=)1770 326 y Fq(X)1678 508 y Fu(l)p Fp(\025)p Ft(0)p Fu(;e)p Ft(\()p Fu(l)p Ft(\))p Fp(6)p Ft(=)p Fu(l)2245 343 y Fs(j)p Fv(w)2327 358 y Fu(l;e)p Ft(\()p Fu(l)p Ft(\))2477 343 y Fs(j)p 2005 385 736 4 v 2005 461 a(j)p Fv(!)s(l)19 b Fs(\000)f Fv(e)p Fw(\()p Fv(l)r Fw(\))p Fs(j)p Fw(\()p Fv(!)s(l)i Fw(+)e Fv(e)p Fw(\()p Fv(l)r Fw(\)\))2750 404 y Fv(:)236 648 y Fw(F)-7 b(or)27 b Fv(j)h Fs(6)p Fw(=)23 b Fv(e)p Fw(\()p Fv(l)r Fw(\))k(w)n(e)g(ha)n(v)n(e)g(that)1126 874 y Fs(j)p Fv(!)s(l)20 b Fs(\000)e Fv(j)5 b Fs(j)23 b(\025)1515 818 y(j)p Fv(j)g Fs(\000)18 b Fv(e)p Fw(\()p Fv(l)r Fw(\))p Fs(j)p 1515 855 317 4 v 1652 931 a Fw(2)2007 874 y(and)166 b Fv(!)s(l)19 b Fw(+)f Fv(j)28 b Fs(\025)2649 818 y(j)p Fv(j)23 b Fs(\000)18 b Fv(e)p Fw(\()p Fv(l)r Fw(\))p Fs(j)g Fw(+)g Fv(l)p 2649 855 445 4 v 2850 931 a Fw(4)3103 874 y Fv(:)719 b Fw(\(6.4\))236 1080 y(Indeed,)36 b(for)e Fv(j)k Fs(6)p Fw(=)c Fv(e)p Fw(\()p Fv(l)r Fw(\))f(w)n(e)h(ha)n(v)n(e)f(that)68 b Fs(j)p Fv(j)28 b Fs(\000)22 b Fv(!)s(l)r Fs(j)33 b(\025)g(j)p Fv(j)28 b Fs(\000)22 b Fv(e)p Fw(\()p Fv(l)r Fw(\))p Fs(j)h(\000)f(j)p Fv(e)p Fw(\()p Fv(l)r Fw(\))g Fs(\000)h Fv(!)s(l)r Fs(j)32 b(\025)i(j)p Fv(j)27 b Fs(\000)c Fv(e)p Fw(\()p Fv(l)r Fw(\))p Fs(j)f(\000)g Fw(1)p Fv(=)p Fw(2)33 b Fs(\025)g(j)p Fv(j)28 b Fs(\000)22 b Fv(e)p Fw(\()p Fv(l)r Fw(\))p Fs(j)p Fv(=)p Fw(2.)236 1180 y(Moreo)n(v)n(er,)h(since)i Fs(j)p Fv(e)p Fw(\()p Fv(l)r Fw(\))13 b Fs(\000)g Fv(!)s(l)r Fs(j)23 b Fv(<)f Fw(1)p Fv(=)p Fw(2,)i(it)i(is)f(easy)f(to)h(see)g (\(remem)n(b)r(er)g(that)g Fv(!)h Fs(\025)c Fw(1)p Fv(=)p Fw(2\))i(that)i Fv(e)p Fw(\()p Fv(l)r Fw(\))13 b(+)g Fv(l)24 b Fs(\024)f Fw(4)p Fv(!)s(l)j Fw(and)f(hence)236 1279 y Fs(j)p Fv(j)f Fs(\000)18 b Fv(e)p Fw(\()p Fv(l)r Fw(\))p Fs(j)g Fw(+)g Fv(l)24 b Fs(\024)f Fv(j)g Fw(+)18 b Fv(e)p Fw(\()p Fv(l)r Fw(\))g(+)g Fv(l)25 b Fs(\024)e Fw(4\()p Fv(j)g Fw(+)18 b Fv(!)s(l)r Fw(\).)36 b(De\014ning)28 b Fv(w)2077 1291 y Fu(l;j)2181 1279 y Fw(b)n(y)f Fv(w)2355 1291 y Fu(l;j)2455 1279 y Fw(=)c(0)k(if)h Fv(j)g Fs(\024)23 b Fw(0)k(or)f Fv(j)i Fw(=)23 b Fv(l)r Fw(,)k(w)n(e)g(get)h(from)f (\(6.4\))1359 1508 y Fv(S)1410 1520 y Ft(1)1470 1508 y Fs(\024)1725 1429 y Fq(X)1558 1611 y Fu(l)p Fp(\025)p Ft(0)p Fu(;j)s Fp(2)p Fl(Z)p Fu(;j)s Fp(6)p Ft(=)p Fu(e)p Ft(\()p Fu(l)p Ft(\))2336 1452 y Fw(8)p Fs(j)p Fv(w)2460 1464 y Fu(l;j)2536 1452 y Fs(j)p 2035 1489 825 4 v 2035 1565 a(j)p Fv(j)d Fs(\000)18 b Fv(e)p Fw(\()p Fv(l)r Fw(\))p Fs(j)p Fw(\()p Fs(j)p Fv(j)23 b Fs(\000)18 b Fv(e)p Fw(\()p Fv(l)r Fw(\))p Fs(j)g Fw(+)g Fv(l)r Fw(\))2870 1508 y Fv(:)236 1783 y Fw(Hence,)28 b(b)n(y)f(the)h(Cauc)n(h)n(y-Sc)n (h)n(w)n(arz)d(inequalit)n(y)-7 b(,)27 b Fv(S)1835 1795 y Ft(1)1895 1783 y Fs(\024)c Fw(8)p Fv(R)2088 1795 y Ft(1)2125 1783 y Fs(j)p Fv(w)r Fs(j)2232 1798 y Fu(L)2278 1781 y Fd(2)2314 1783 y Fw(,)28 b(where)313 2000 y Fv(R)377 1966 y Ft(2)376 2021 y(1)437 2000 y Fw(=)692 1922 y Fq(X)525 2103 y Fu(l)p Fp(\025)p Ft(0)p Fu(;j)s Fp(2)p Fl(Z)p Fu(;j)s Fp(6)p Ft(=)p Fu(e)p Ft(\()p Fu(l)p Ft(\))1441 1944 y Fw(1)p 1003 1981 918 4 v 1003 2057 a(\()p Fv(j)23 b Fs(\000)18 b Fv(e)p Fw(\()p Fv(l)r Fw(\)\))1337 2033 y Ft(2)1374 2057 y Fw(\()p Fs(j)p Fv(j)24 b Fs(\000)18 b Fv(e)p Fw(\()p Fv(l)r Fw(\))p Fs(j)g Fw(+)g Fv(l)r Fw(\))1883 2033 y Ft(2)1953 2000 y Fw(=)2172 1922 y Fq(X)2041 2100 y Fu(l)p Fp(\025)p Ft(0)p Fu(;j)s Fp(2)p Fl(Z)p Fu(;j)s Fp(6)p Ft(=0)2622 1944 y Fw(1)p 2447 1981 391 4 v 2447 2057 a Fv(j)2486 2033 y Ft(2)2523 2057 y Fw(\()p Fs(j)p Fv(j)5 b Fs(j)19 b Fw(+)f Fv(l)r Fw(\))2801 2033 y Ft(2)2871 2000 y Fs(\024)3090 1922 y Fq(X)2958 2100 y Fu(l)p Fp(\025)p Ft(0)p Fu(;j)s Fp(2)p Fl(Z)p Fu(;j)s Fp(6)p Ft(=0)3517 1944 y Fw(1)p 3364 1981 348 4 v 3364 2057 a Fv(j)3403 2033 y Ft(2)3441 2057 y Fw(\(1)g(+)g Fv(l)r Fw(\))3675 2033 y Ft(2)3745 2000 y Fv(<)k Fs(1)p Fv(:)236 2275 y Fw(In)28 b(order)f(to)h(\014nd)g(an)g(upp)r(er)g(b)r (ound)g(for)g Fv(S)1615 2287 y Ft(2)1652 2275 y Fw(,)g(w)n(e)g(observ)n (e)e(that)i(if)h Fv(l)c(<)j Fw(1)p Fv(=)p Fw(\(2)p Fs(j)p Fv(!)20 b Fs(\000)e Fw(1)p Fs(j)p Fw(\))28 b(then)g Fs(j)p Fv(!)s(l)20 b Fs(\000)f Fv(l)r Fs(j)k Fv(<)g Fw(1)p Fv(=)p Fw(2)k(and)g(then)236 2375 y Fv(e)p Fw(\()p Fv(l)r Fw(\))c(=)g Fv(l)r Fw(.)36 b(Hence)28 b(w)n(e)f(can)g(write)1361 2607 y Fv(S)1412 2619 y Ft(2)1473 2607 y Fw(=)1770 2528 y Fq(X)1560 2710 y Fu(l)p Fp(\025)p Ft(1)p Fu(=)p Ft(2)p Fp(j)p Fu(!)r Fp(\000)p Ft(1)p Fp(j)p Fu(;e)p Ft(\()p Fu(l)p Ft(\))p Fp(6)p Ft(=)p Fu(l)2362 2545 y Fs(j)p Fv(w)2444 2560 y Fu(l;e)p Ft(\()p Fu(l)p Ft(\))2595 2545 y Fs(j)p 2123 2588 736 4 v 2123 2664 a(j)p Fv(!)s(l)19 b Fs(\000)f Fv(e)p Fw(\()p Fv(l)r Fw(\))p Fs(j)p Fw(\()p Fv(!)s(l)i Fw(+)e Fv(e)p Fw(\()p Fv(l)r Fw(\)\))2868 2607 y Fv(:)236 2887 y Fw(By)27 b(the)h(prop)r(erties)e(of)h Fv(!)f Fs(2)d Fv(W)1227 2899 y Fu(\015)1270 2887 y Fw(,)28 b(for)f Fv(l)d Fs(6)p Fw(=)f Fv(e)p Fw(\()p Fv(l)r Fw(\),)k Fs(j)p Fv(!)s(l)19 b Fs(\000)e Fv(e)p Fw(\()p Fv(l)r Fw(\))p Fs(jj)p Fv(!)s(l)i Fw(+)e Fv(e)p Fw(\()p Fv(l)r Fw(\))p Fs(j)23 b(\025)k Fv(\015)5 b Fw(\()p Fv(!)s(l)18 b Fw(+)g Fv(e)p Fw(\()p Fv(l)r Fw(\)\))p Fv(l)3044 2857 y Fp(\000)p Ft(1)3155 2887 y Fs(\025)27 b Fv(\015)5 b Fw(.)36 b(Hence,)28 b(still)f(b)n(y)g(the)236 2987 y(Cauc)n(h)n(y-Sc)n (h)n(w)n(arz)e(inequalit)n(y)-7 b(,)1310 3202 y Fv(S)1361 3214 y Ft(2)1421 3202 y Fs(\024)1521 3146 y Fw(1)p 1518 3183 48 4 v 1518 3259 a Fv(\015)1799 3124 y Fq(X)1590 3305 y Fu(l)p Fp(\025)p Ft(1)p Fu(=)p Ft(2)p Fp(j)p Fu(!)r Fp(\000)p Ft(1)p Fp(j)p Fu(;e)p Ft(\()p Fu(l)p Ft(\))p Fp(6)p Ft(=)p Fu(l)2142 3202 y Fs(j)p Fv(w)2224 3217 y Fu(l;e)p Ft(\()p Fu(l)p Ft(\))2375 3202 y Fs(j)23 b(\024)2518 3146 y Fw(2)p Fv(R)2623 3158 y Ft(2)p 2518 3183 142 4 v 2565 3259 a Fv(\015)2670 3202 y Fs(jj)p Fv(w)r Fs(jj)2823 3217 y Fu(H)2881 3201 y Fd(1)2919 3202 y Fv(;)236 3473 y Fw(where)1591 3591 y Fv(R)1655 3556 y Ft(2)1654 3611 y(2)1715 3591 y Fw(=)1914 3512 y Fq(X)1803 3694 y Fu(l)p Fp(\025)p Ft(1)p Fu(=)p Ft(2)p Fp(j)p Fu(!)r Fp(\000)p Ft(1)p Fp(j)2179 3535 y Fw(1)p 2168 3572 64 4 v 2168 3648 a Fv(l)2195 3624 y Ft(2)2265 3591 y Fs(\024)g Fw(2)p Fs(j)p Fv(!)d Fs(\000)e Fw(1)p Fs(j)p Fv(:)236 3840 y Fw(The)30 b(estimates)g Fs(j)p Fv(L)857 3809 y Fp(\000)p Ft(1)857 3860 y Fu(!)946 3840 y Fv(w)r Fs(j)1030 3852 y Fu(!)1106 3840 y Fs(\024)d Fw(\()p Fv(C)t(=\015)5 b Fw(\))p Fs(j)p Fv(w)r Fs(j)1522 3852 y Fu(!)1601 3840 y Fs(8)p Fv(w)30 b Fs(2)e Fv(W)42 b Fw(and)30 b Fs(j)p Fv(L)2183 3809 y Fp(\000)p Ft(1)2183 3860 y Fu(!)2271 3840 y Fw(\005)2333 3852 y Fu(W)2409 3840 y Fv(u)p Fs(j)2480 3852 y Fu(!)2555 3840 y Fs(\024)d Fw(\()p Fv(C)2738 3852 y Ft(3)2776 3840 y Fv(=\015)5 b Fw(\))p Fs(j)p Fv(u)p Fs(j)2992 3852 y Fu(!)3069 3840 y Fs(8)p Fv(u)27 b Fs(2)h Fv(X)37 b Fw(are)29 b(an)h(immediate)236 3939 y(consequence)37 b(of)h(the)g(b)r(ound)g(of)f Fs(jj)p Fv(L)1446 3909 y Fp(\000)p Ft(1)1446 3960 y Fu(!)1535 3939 y Fv(w)r Fs(jj)h Fw(and)g(\(6.3\),)i(since)d Fs(j)p Fw(\005)2384 3951 y Fu(W)2460 3939 y Fv(u)p Fs(j)2531 3951 y Fu(L)2577 3959 y Fd(2)2653 3939 y Fs(\024)i(j)p Fv(u)p Fs(j)2851 3951 y Fu(L)2897 3959 y Fd(2)2971 3939 y Fw(and)e Fs(jj)p Fw(\005)3250 3951 y Fu(W)3326 3939 y Fv(u)p Fs(jj)j(\024)f(jj)p Fv(u)p Fs(jj)e Fw(F)-7 b(or)37 b(the)236 4039 y(estimate)28 b(of)f Fs(j)p Fv(L)743 4009 y Fp(\000)p Ft(1)743 4059 y Fu(!)832 4039 y Fv(w)r Fs(j)916 4054 y Fu(L)962 4037 y Fd(2)1027 4039 y Fw(\()p Fv(w)f Fs(2)d Fv(W)12 b Fw(\),)28 b(w)n(e)f(use)h(the)g(decomp)r(osition)f Fs(j)p Fv(L)2431 4009 y Fp(\000)p Ft(1)2431 4059 y Fu(!)2519 4039 y Fv(w)r Fs(j)2603 4009 y Ft(2)2603 4065 y Fu(L)2649 4049 y Fd(2)2709 4039 y Fw(=)c Fv(S)2853 4009 y Fp(0)2848 4059 y Ft(1)2903 4039 y Fw(+)18 b Fv(S)3042 4009 y Fp(0)3037 4059 y Ft(2)3075 4039 y Fw(,)27 b(where)465 4293 y Fv(S)521 4259 y Fp(0)516 4314 y Ft(1)577 4293 y Fw(:=)913 4215 y Fq(X)687 4396 y Fu(j)s(>)p Ft(0)p Fu(;l)p Fp(\025)p Ft(0)p Fu(;j)s Fp(6)p Ft(=)p Fu(l;j)s Fp(6)p Ft(=)p Fu(e)p Ft(\()p Fu(l)p Ft(\))1538 4226 y Fv(w)1599 4196 y Ft(2)1597 4250 y Fu(l;j)p 1282 4274 647 4 v 1282 4350 a Fw(\()p Fv(!)s(l)20 b Fs(\000)e Fv(j)5 b Fw(\))1568 4326 y Ft(2)1605 4350 y Fw(\()p Fv(!)s(l)20 b Fw(+)e Fv(j)5 b Fw(\))1891 4326 y Ft(2)1939 4293 y Fv(;)179 b(S)2197 4259 y Fp(0)2192 4314 y Ft(2)2253 4293 y Fw(:=)2573 4215 y Fq(X)2363 4396 y Fu(l)p Fp(\025)p Ft(1)p Fu(=)p Ft(2)p Fp(j)p Fu(!)r Fp(\000)p Ft(1)p Fp(j)p Fu(;e)p Ft(\()p Fu(l)p Ft(\))p Fp(6)p Ft(=)p Fu(l)3235 4220 y Fv(w)3296 4190 y Ft(2)3294 4246 y Fu(l;e)p Ft(\()p Fu(l)p Ft(\))p 2926 4274 828 4 v 2926 4350 a Fw(\()p Fv(!)s(l)19 b Fs(\000)f Fv(e)p Fw(\()p Fv(l)r Fw(\)\))3302 4326 y Ft(2)3340 4350 y Fw(\()p Fv(!)s(l)h Fw(+)f Fv(e)p Fw(\()p Fv(l)r Fw(\)\))3716 4326 y Ft(2)3764 4293 y Fv(:)236 4574 y Fw(Recall)27 b(that)h(if)g Fv(l)d(<)e Fw(1)p Fv(=)p Fw(2)p Fs(j)p Fv(!)c Fs(\000)f Fw(1)p Fs(j)27 b Fw(then)i Fv(e)p Fw(\()p Fv(l)r Fw(\))22 b(=)h Fv(l)r Fw(.)36 b(Arguing)27 b(as)g(b)r(efore)g(it)h(is)g(ob)n(vious)e(that)i Fv(S)3166 4544 y Fp(0)3161 4594 y Ft(1)3221 4574 y Fs(\024)23 b Fv(C)6 b Fs(j)p Fv(w)r Fs(j)3481 4544 y Ft(2)3481 4600 y Fu(L)3527 4584 y Fd(2)3564 4574 y Fw(.)37 b(W)-7 b(e)28 b(ha)n(v)n(e)386 4828 y Fv(S)442 4794 y Fp(0)437 4849 y Ft(2)497 4828 y Fs(\024)794 4749 y Fq(X)585 4931 y Fu(l)p Fp(\025)p Ft(1)p Fu(=)p Ft(2)p Fp(j)p Fu(!)r Fp(\000)p Ft(1)p Fp(j)p Fu(;e)p Ft(\()p Fu(l)p Ft(\))p Fp(6)p Ft(=)p Fu(l)1147 4755 y Fv(w)1208 4724 y Ft(2)1206 4781 y Fu(l;e)p Ft(\()p Fu(l)p Ft(\))p 1147 4809 210 4 v 1209 4885 a Fv(\015)1257 4861 y Ft(2)1390 4828 y Fs(\024)1487 4772 y Fw(4\()p Fv(!)21 b Fs(\000)d Fw(1\))1791 4742 y Ft(2)p 1487 4809 341 4 v 1615 4885 a Fv(\015)1663 4861 y Ft(2)2061 4749 y Fq(X)1852 4931 y Fu(l)p Fp(\025)p Ft(1)p Fu(=)p Ft(2)p Fp(j)p Fu(!)r Fp(\000)p Ft(1)p Fp(j)p Fu(;e)p Ft(\()p Fu(l)p Ft(\))p Fp(6)p Ft(=)p Fu(l)2404 4828 y Fv(l)2431 4794 y Ft(2)2468 4828 y Fv(w)2529 4794 y Ft(2)2527 4851 y Fu(l;e)p Ft(\()p Fu(l)p Ft(\))2700 4828 y Fw(=)23 b Fv(O)2853 4736 y Fq(\020)2918 4772 y Fv(")2957 4742 y Ft(2)p 2913 4809 85 4 v 2913 4885 a Fv(\015)2961 4861 y Ft(2)3008 4828 y Fs(jj)p Fv(w)r Fs(jj)3161 4794 y Ft(2)3161 4851 y Fu(H)3219 4834 y Fd(1)3257 4736 y Fq(\021)3330 4828 y Fw(=)f Fv(O)3482 4736 y Fq(\020)3566 4772 y Fv(")p 3543 4809 V 3543 4885 a(\015)3591 4861 y Ft(2)3637 4828 y Fs(j)p Fv(w)r Fs(j)3744 4794 y Ft(2)3744 4849 y Fu(!)3793 4736 y Fq(\021)3843 4828 y Fv(;)236 5103 y Fw(whic)n(h)d(yields)g (estimate)g(\(2.14\).)34 b(W)-7 b(e)19 b(no)n(w)f(pro)n(v)n(e)g (\(2.12\).)33 b(W)-7 b(riting)19 b Fv(r)r Fw(\()p Fv(t;)14 b(x)p Fw(\))25 b(=)2693 5041 y Fq(P)2780 5128 y Fu(l)p Fp(\025)p Ft(0)p Fu(;j)s Fp(\025)p Ft(1)3040 5103 y Fv(r)3077 5115 y Fu(l;j)3168 5103 y Fw(cos)o(\()p Fv(l)r(t)p Fw(\))14 b(sin)o(\()p Fv(j)5 b(x)p Fw(\),)22 b Fv(s)p Fw(\()p Fv(t;)14 b(x)p Fw(\))24 b(=)236 5141 y Fq(P)324 5228 y Fu(l)p Fp(\025)p Ft(0)p Fu(;j)s Fp(\025)p Ft(1)584 5203 y Fv(s)623 5215 y Fu(l;j)712 5203 y Fw(cos\()p Fv(l)r(t)p Fw(\))14 b(sin)o(\()p Fv(j)5 b(x)p Fw(\))29 b(w)n(e)e(get)624 5418 y Fv(L)681 5382 y Fp(\000)p Ft(1)681 5440 y(1)769 5418 y Fw(\005)831 5430 y Fu(W)907 5418 y Fv(s)c Fw(=)1057 5339 y Fq(X)1065 5518 y Fu(j)s Fp(6)p Ft(=)p Fu(l)1264 5362 y Fv(s)1303 5374 y Fu(l;j)p 1200 5399 242 4 v 1200 5475 a Fv(l)1227 5451 y Ft(2)1283 5475 y Fs(\000)18 b Fv(j)1405 5451 y Ft(2)1466 5418 y Fw(cos)o(\()p Fv(l)r(t)p Fw(\))c(sin)o(\()p Fv(j)5 b(x)p Fw(\))p Fv(;)347 b(L)2390 5383 y Fp(\000)p Ft(1)2390 5438 y Fu(!)2479 5418 y Fv(s)23 b Fw(=)2629 5339 y Fq(X)2637 5518 y Fu(j)s Fp(6)p Ft(=)p Fu(l)2882 5362 y Fv(s)2921 5374 y Fu(l;j)p 2772 5399 334 4 v 2772 5475 a Fv(!)2827 5451 y Ft(2)2864 5475 y Fv(l)2891 5451 y Ft(2)2946 5475 y Fs(\000)18 b Fv(j)3068 5451 y Ft(2)3129 5418 y Fw(cos\()p Fv(l)r(t)p Fw(\))c(sin)o(\()p Fv(j)5 b(x)p Fw(\))2084 6059 y(24)p eop %%Page: 25 25 25 24 bop 236 272 a Fw(and)688 404 y Fv(S)27 b Fw(:=)877 291 y Fq(Z)974 404 y Fv(r)r Fw(\()p Fv(L)1102 370 y Fp(\000)p Ft(1)1102 425 y Fu(!)1210 404 y Fs(\000)18 b Fv(L)1350 370 y Fp(\000)p Ft(1)1439 404 y Fw(\)\(\005)1565 416 y Fu(W)1641 404 y Fv(s)p Fw(\))28 b Fv(dt)g(dx)23 b Fw(=)g Fv(\031)2092 370 y Ft(2)2143 325 y Fq(X)2152 504 y Fu(j)s Fp(6)p Ft(=)p Fu(l)2460 348 y Fv(s)2499 360 y Fu(l;j)2575 348 y Fv(r)2612 360 y Fu(l;j)2716 348 y Fv("l)2782 318 y Ft(2)p 2287 385 704 4 v 2287 461 a Fw(\()p Fv(!)2374 437 y Ft(2)2411 461 y Fv(l)2438 437 y Ft(2)2493 461 y Fs(\000)18 b Fv(j)2615 437 y Ft(2)2653 461 y Fw(\)\()p Fv(l)2744 437 y Ft(2)2799 461 y Fs(\000)g Fv(j)2921 437 y Ft(2)2959 461 y Fw(\))3024 404 y(=)k Fv(\031)3161 370 y Ft(2)3199 404 y Fw(\()p Fv(S)3282 416 y Ft(1)3338 404 y Fw(+)c Fv(S)3472 416 y Ft(2)3509 404 y Fw(\))p Fv(;)236 639 y Fw(where)298 860 y Fv(S)349 872 y Ft(1)409 860 y Fw(:=)614 781 y Fq(X)520 963 y Fu(j)s Fp(6)p Ft(=)p Fu(l;j)s Fp(6)p Ft(=)p Fu(e)p Ft(\()p Fu(l)p Ft(\))1191 804 y Fv(s)1230 816 y Fu(l;j)1306 804 y Fv(r)1343 816 y Fu(l;j)1447 804 y Fv("l)1513 774 y Ft(2)p 853 841 1035 4 v 853 917 a Fw(\()p Fv(!)s(l)i Fs(\000)e Fv(j)5 b Fw(\)\()p Fv(!)s(l)20 b Fw(+)e Fv(j)5 b Fw(\)\()p Fv(l)20 b Fs(\000)e Fv(j)5 b Fw(\)\()p Fv(l)21 b Fw(+)d Fv(j)5 b Fw(\))1898 860 y Fv(;)180 b(S)2152 872 y Ft(2)2212 860 y Fw(:=)2351 781 y Fq(X)2323 963 y Fu(l)p Fp(6)p Ft(=)p Fu(e)p Ft(\()p Fu(l)p Ft(\))2969 798 y Fv(s)3008 813 y Fu(l;e)p Ft(\()p Fu(l)p Ft(\))3158 798 y Fv(r)3195 813 y Fu(l;e)p Ft(\()p Fu(l)p Ft(\))3373 798 y Fv("l)3439 768 y Ft(2)p 2523 841 1398 4 v 2523 917 a Fw(\()p Fv(!)s(l)20 b Fs(\000)e Fv(e)p Fw(\()p Fv(l)r Fw(\)\)\()p Fv(!)s(l)i Fw(+)e Fv(e)p Fw(\()p Fv(l)r Fw(\)\)\()p Fv(l)i Fs(\000)e Fv(e)p Fw(\()p Fv(l)r Fw(\)\)\()p Fv(l)i Fw(+)e Fv(e)p Fw(\()p Fv(l)r Fw(\)\))3931 860 y Fv(:)236 1140 y Fw(F)-7 b(or)28 b Fv(j)h Fs(6)p Fw(=)24 b Fv(e)p Fw(\()p Fv(l)r Fw(\))k(and)h Fv(j)g Fs(6)p Fw(=)24 b Fv(l)30 b Fw(w)n(e)e(ha)n(v)n(e)f(that)i Fs(j)p Fv(!)s(l)20 b Fs(\000)e Fv(j)5 b Fs(j)25 b(\025)f(j)p Fv(j)g Fs(\000)18 b Fv(e)p Fw(\()p Fv(l)r Fw(\))p Fs(j)p Fv(=)p Fw(2)23 b Fs(\025)h Fw(1)p Fv(=)p Fw(2)j(hence)i Fs(j)p Fw(\()p Fv(!)s(l)20 b Fs(\000)f Fv(j)5 b Fw(\)\()p Fv(!)s(l)20 b Fw(+)f Fv(j)5 b Fw(\)\()p Fv(l)21 b Fs(\000)d Fv(j)5 b Fw(\)\()p Fv(l)21 b Fw(+)e Fv(j)5 b Fw(\))p Fs(j)24 b(\025)236 1239 y Fw(\(1)p Fv(=)p Fw(2\))p Fv(!)s(l)508 1209 y Ft(2)544 1239 y Fw(.)37 b(W)-7 b(e)28 b(obtain)1031 1461 y Fs(j)p Fv(S)1105 1473 y Ft(1)1142 1461 y Fs(j)23 b(\024)1286 1405 y Fw(2)p Fv(")p 1286 1442 81 4 v 1299 1518 a(!)1390 1382 y Fq(X)1398 1561 y Fu(j)s Fp(6)p Ft(=)p Fu(l)1524 1461 y Fs(j)p Fv(r)1584 1473 y Fu(l;j)1660 1461 y Fs(jj)p Fv(s)1745 1473 y Fu(l;j)1821 1461 y Fs(j)g Fw(=)g Fv(O)2020 1369 y Fq(\020)2080 1405 y Fw(2)p Fv(")p 2080 1442 V 2093 1518 a(!)2170 1461 y Fs(j)p Fv(r)r Fs(j)2255 1476 y Fu(L)2301 1459 y Fd(2)2338 1461 y Fs(j)p Fv(s)p Fs(j)2423 1476 y Fu(L)2469 1459 y Fd(2)2506 1369 y Fq(\021)2578 1461 y Fw(=)g Fv(O)2731 1369 y Fq(\020)2791 1405 y Fw(2)p Fv(")p 2791 1442 V 2804 1518 a(!)2881 1461 y Fs(j)p Fv(r)r Fs(j)2966 1473 y Fu(!)3015 1461 y Fs(j)p Fv(s)p Fs(j)3100 1473 y Fu(!)3148 1369 y Fq(\021)3198 1461 y Fv(:)236 1734 y Fw(As)e(b)r(efore,)h(for)e Fv(!)25 b Fs(2)f Fv(W)973 1746 y Fu(\015)1036 1734 y Fw(and)d Fv(l)j Fs(6)p Fw(=)f Fv(e)p Fw(\()p Fv(l)r Fw(\),)f(w)n(e)e(ha)n(v)n(e)f Fs(j)p Fv(!)s(l)6 b Fs(\000)t Fv(e)p Fw(\()p Fv(l)r Fw(\))p Fs(j)23 b(\025)f Fv(\015)5 b(=l)21 b Fw(and)g(so)f Fs(j)p Fw(\()p Fv(!)s(l)6 b Fs(\000)t Fv(e)p Fw(\()p Fv(l)r Fw(\)\)\()p Fv(!)s(l)g Fw(+)t Fv(e)p Fw(\()p Fv(l)r Fw(\)\)\()p Fv(l)g Fs(\000)t Fv(e)p Fw(\()p Fv(l)r Fw(\)\)\()p Fv(l)g Fw(+)t Fv(e)p Fw(\()p Fv(l)r Fw(\))p Fs(j)22 b(\025)236 1834 y Fv(\015)5 b(l)r Fw(.)36 b(Moreo)n(v)n(er)25 b Fv(l)f Fs(6)p Fw(=)f Fv(e)p Fw(\()p Fv(l)r Fw(\))k(implies)h Fv(l)d Fs(\025)d Fw(1)p Fv(=)p Fw(\(2)p Fs(j)p Fv(!)e Fs(\000)e Fw(1)p Fs(j)p Fw(\).)37 b(Hence)1091 2069 y Fs(j)p Fv(S)1165 2081 y Ft(2)1202 2069 y Fs(j)23 b(\024)f Fv(C)1411 2013 y(")1450 1983 y Ft(2)p 1411 2050 76 4 v 1425 2126 a Fv(\015)1579 1991 y Fq(X)1510 2173 y Fu(l)p Fp(\025)p Ft(1)p Fu(=)p Ft(\(2)p Fu(")p Ft(\))1781 2069 y Fs(j)p Fv(r)1841 2084 y Fu(l;e)p Ft(\()p Fu(l)p Ft(\))1991 2069 y Fs(jj)p Fv(s)2076 2084 y Fu(l;e)p Ft(\()p Fu(l)p Ft(\))2226 2069 y Fs(j)p Fv(l)2276 2035 y Ft(2)2336 2069 y Fw(=)h Fv(O)2489 1977 y Fq(\020)2549 2013 y Fv(")2588 1983 y Ft(2)p 2549 2050 V 2563 2126 a Fv(\015)2635 2069 y Fs(jj)p Fv(r)r Fs(jj)2766 2084 y Fu(H)2824 2068 y Fd(1)2862 2069 y Fs(jj)p Fv(s)p Fs(jj)2993 2084 y Fu(H)3051 2068 y Fd(1)3089 1977 y Fq(\021)3138 2069 y Fv(:)236 2349 y Fw(Estimate)34 b(\(2.12\))f(follo)n(ws)g(straigh)n(tforw)n(ardly)e (from)j(the)g(ab)r(o)n(v)n(e)f(estimates)h(of)g Fv(S)2896 2361 y Ft(1)2967 2349 y Fw(and)g Fv(S)3186 2361 y Ft(2)3223 2349 y Fw(.)57 b(Finally)34 b(\(2.13\))f(is)h(an)236 2449 y(immediate)28 b(consequence)f(of)g(\(2.12\))g(since,)h(for)f(all) g(\()p Fv(r)n(;)14 b(s)p Fw(\))24 b Fs(2)f Fv(X)i Fs(\002)18 b Fv(X)7 b Fw(,)1303 2561 y Fq(Z)1349 2749 y Ft(\012)1415 2674 y Fv(r)r(L)1511 2639 y Fp(\000)p Ft(1)1600 2674 y Fw(\(\005)1694 2686 y Fu(W)1770 2674 y Fv(s)p Fw(\))24 b(=)e Fv(O)r Fw(\()p Fs(j)p Fv(r)r Fs(j)2134 2689 y Fu(L)2180 2672 y Fd(2)2218 2674 y Fs(j)p Fv(s)p Fs(j)2303 2689 y Fu(L)2349 2672 y Fd(2)2385 2674 y Fw(\))i(=)e Fv(O)r Fw(\()p Fs(j)p Fv(r)r Fs(j)2710 2686 y Fu(!)2760 2674 y Fs(j)p Fv(s)p Fs(j)2845 2686 y Fu(!)2893 2674 y Fw(\))p Fv(:)p 236 2888 48 48 v 236 3037 a Fc(Pr)n(oof)78 b(of)41 b(lemma)36 b Fw(2.3.)63 b(It)37 b(is)f(su\016cien)n(t)h(to)f(sho)n(w)f (that)i(the)g(op)r(erator)e Fs(G)43 b Fw(:)38 b Fv(W)49 b Fs(!)38 b Fv(W)49 b Fw(de\014ned)37 b(b)n(y)f Fs(G)5 b Fw(\()p Fv(w)r Fw(\))39 b(:=)236 3137 y Fv(L)293 3107 y Fp(\000)p Ft(1)293 3157 y Fu(!)382 3137 y Fw(\005)444 3149 y Fu(W)520 3137 y Fv(f)9 b Fw(\()p Fv(v)25 b Fw(+)c Fv(w)r Fw(\))35 b(is)e(a)f(con)n(traction)g(on)h(a)f(ball)h Fv(B)1911 3149 y Fu(\016)1980 3137 y Fs(\032)e Fv(W)45 b Fw(for)33 b(some)f Fv(\016)37 b Fw(if)c Fs(j)p Fv(v)s Fs(j)2789 3107 y Fu(p)p Fp(\000)p Ft(1)2789 3157 y Fu(!)2913 3137 y Fv(=\015)k Fw(is)c(small)g(enough.)52 b(Since,)35 b(b)n(y)236 3237 y(lemma)28 b(2.2,)f Fs(jj)p Fv(L)764 3206 y Fp(\000)p Ft(1)764 3257 y Fu(!)852 3237 y Fw(\005)914 3249 y Fu(W)990 3237 y Fs(jj)1036 3249 y Fu(!)1107 3237 y Fs(\024)c Fv(C)1254 3249 y Ft(3)1292 3237 y Fv(=\015)31 b Fw(and)d(b)n(y)f(\(2.4\),)g(w)n(e)h(ha)n(v)n(e)e(that,)i(for)f(all)g Fv(w)f Fs(2)e Fv(B)2892 3249 y Fu(\016)796 3478 y Fs(jG)5 b Fw(\()p Fv(w)r Fw(\))p Fs(j)1021 3490 y Fu(!)1093 3478 y Fs(\024)1191 3422 y Fv(C)1250 3434 y Ft(3)p 1191 3459 97 4 v 1215 3535 a Fv(\015)1297 3478 y Fs(j)p Fv(f)k Fw(\()p Fv(v)22 b Fw(+)c Fv(w)r Fw(\))p Fs(j)1663 3490 y Fu(!)1735 3478 y Fs(\024)1833 3422 y Fv(C)p 1833 3459 66 4 v 1842 3535 a(\015)1908 3478 y Fs(j)p Fv(v)k Fw(+)c Fv(w)r Fs(j)2160 3444 y Fu(p)2160 3499 y(!)2232 3478 y Fs(\024)2329 3422 y Fv(C)p 2329 3459 V 2338 3535 a(\015)2405 3386 y Fq(\020)2454 3478 y Fs(j)p Fv(v)s Fs(j)2543 3490 y Fu(!)2610 3478 y Fw(+)g Fv(\016)2733 3386 y Fq(\021)2783 3403 y Fu(p)2844 3478 y Fs(\024)p 2942 3355 V 2942 3422 a Fv(C)p 2942 3459 V 2951 3535 a(\015)3017 3386 y Fq(\020)3066 3478 y Fs(j)p Fv(v)s Fs(j)3155 3444 y Fu(p)3155 3499 y(!)3222 3478 y Fw(+)g Fv(\016)3345 3444 y Fu(p)3384 3386 y Fq(\021)3433 3478 y Fv(:)389 b Fw(\(6.5\))236 3716 y(for)26 b(some)p 568 3649 V 25 w Fv(C)k(>)22 b Fw(0.)36 b(Moreo)n(v)n(er)24 b(b)n(y)i(lemma)g(2.1)f(the)h(op)r(erator) f Fs(G)31 b Fw(is)26 b(di\013eren)n(tiable)g(and)g Fv(D)r Fs(G)5 b Fw(\()p Fv(w)r Fw(\)[)p Fv(h)p Fw(])25 b(=)e Fv(L)3529 3686 y Fp(\000)p Ft(1)3529 3736 y Fu(!)3617 3716 y Fw(\005)3679 3728 y Fu(W)3755 3716 y Fw(\()p Fv(f)3837 3686 y Fp(0)3860 3716 y Fw(\()p Fv(v)c Fw(+)236 3816 y Fv(w)r Fw(\))p Fv(h)p Fw(\).)38 b(Hence)28 b(b)n(y)f(\(6.3\),)735 4065 y Fs(j)p Fv(D)r Fs(G)5 b Fw(\()p Fv(w)r Fw(\)[)p Fv(h)p Fw(])p Fs(j)1125 4077 y Fu(!)1198 4065 y Fs(\024)1296 4009 y Fv(C)1355 4021 y Ft(3)p 1296 4046 97 4 v 1320 4122 a Fv(\015)1402 4065 y Fs(j)p Fv(f)1475 4030 y Fp(0)1498 4065 y Fw(\()p Fv(v)22 b Fw(+)c Fv(w)r Fw(\))p Fv(h)p Fs(j)1839 4077 y Fu(!)1911 4065 y Fs(\024)2009 4009 y Fv(C)p 2009 4046 66 4 v 2018 4122 a(\015)2084 4065 y Fs(j)p Fv(v)k Fw(+)c Fv(w)r Fs(j)2336 4030 y Fu(p)p Fp(\000)p Ft(1)2336 4085 y Fu(!)2460 4065 y Fs(j)p Fv(h)p Fs(j)2554 4077 y Fu(!)2625 4065 y Fs(\024)2739 3988 y Fq(e)2723 4009 y Fv(C)p 2723 4046 V 2732 4122 a(\015)2798 3973 y Fq(\020)2847 4065 y Fs(j)p Fv(v)s Fs(j)2936 4030 y Fu(p)p Fp(\000)p Ft(1)2936 4085 y Fu(!)3079 4065 y Fw(+)g Fv(\016)3202 4030 y Fu(p)p Fp(\000)p Ft(1)3325 3973 y Fq(\021)3375 4065 y Fs(j)p Fv(h)p Fs(j)3469 4077 y Fu(!)3845 4065 y Fw(\(6.6\))236 4310 y(for)28 b(some)590 4289 y Fq(e)573 4310 y Fv(C)j(>)25 b Fw(0.)39 b(Let)29 b(us)g(c)n(ho)r(ose)e Fv(\016)h Fw(:=)c Fs(j)p Fv(v)s Fs(j)1645 4322 y Fu(!)1722 4310 y Fw(and)29 b Fv(\032)c Fw(:=)f(\(4)14 b(max)o(\()p 2339 4243 V Fv(C)7 b(;)2458 4289 y Fq(e)2442 4310 y Fv(C)f Fw(\)\))2571 4280 y Fp(\000)p Ft(1)2661 4310 y Fw(.)40 b(F)-7 b(or)28 b Fv(v)g Fs(2)d(D)3086 4322 y Fu(\032)3124 4310 y Fw(,)k(\()p 3208 4243 V Fv(C)7 b(=\015)e Fw(\)\()p Fs(j)p Fv(v)s Fs(j)3517 4280 y Fu(p)3517 4331 y(!)3584 4310 y Fw(+)19 b Fv(\016)3708 4280 y Fu(p)3746 4310 y Fw(\))25 b Fs(\024)g Fv(\016)s(=)p Fw(2)236 4423 y(and)i(\()446 4402 y Fq(e)429 4423 y Fv(C)6 b(=\015)f Fw(\)\()p Fs(j)p Fv(v)s Fs(j)737 4393 y Fu(p)p Fp(\000)p Ft(1)737 4444 y Fu(!)878 4423 y Fw(+)16 b Fv(\016)999 4393 y Fu(p)p Fp(\000)p Ft(1)1123 4423 y Fw(\))23 b Fs(\024)f Fw(1)p Fv(=)p Fw(2.)36 b(Hence)27 b Fs(G)32 b Fw(maps)26 b Fv(B)2056 4435 y Fu(\016)2120 4423 y Fw(in)n(to)g(the)h(closed)g(ball)p 2834 4356 68 4 v 26 w Fv(B)2901 4438 y Fu(\016)r(=)p Ft(2)3032 4423 y Fw(and)f(is)h(a)f(con)n(traction)g(on)g Fv(B)3956 4435 y Fu(\016)3993 4423 y Fw(.)236 4523 y(By)h(the)g(Con)n (traction)f(Mapping)g(Theorem)g(there)h(is)g(a)f(unique)i Fv(w)r Fw(\()p Fv(v)s Fw(\))g(in)f Fv(B)2652 4535 y Fu(\016)2716 4523 y Fw(suc)n(h)f(that)h Fs(G)5 b Fw(\()p Fv(w)r Fw(\()p Fv(v)s Fw(\)\))26 b(=)c Fv(w)r Fw(\()p Fv(v)s Fw(\).)39 b(W)-7 b(e)27 b(ha)n(v)n(e)1560 4764 y Fs(j)p Fv(w)r Fw(\()p Fv(v)s Fw(\))p Fs(j)1774 4776 y Fu(!)1847 4764 y Fw(=)22 b Fs(jG)5 b Fw(\()p Fv(w)r Fw(\()p Fv(v)s Fw(\)\))p Fs(j)2266 4776 y Fu(!)2340 4764 y Fs(\024)2438 4708 y Fw(2)p 2480 4641 66 4 v Fv(C)p 2438 4745 107 4 v 2467 4821 a(\015)2554 4764 y Fs(j)p Fv(v)s Fs(j)2643 4730 y Fu(p)2643 4785 y(!)3845 4764 y Fw(\(6.7\))236 4991 y(b)n(y)28 b(\(6.5\))f(and)g(the)h(c)n(hoice)f(of)h Fv(\016)s Fw(.)37 b(Moreo)n(v)n(er)668 5226 y Fs(j)p Fv(w)r Fw(\()p Fv(v)s Fw(\))p Fs(j)882 5241 y Fu(L)928 5224 y Fd(2)989 5226 y Fw(=)23 b Fs(j)p Fv(L)1157 5191 y Fp(\000)p Ft(1)1157 5246 y Fu(!)1245 5226 y Fw(\005)1307 5238 y Fu(W)1383 5226 y Fv(f)9 b Fw(\()p Fv(v)22 b Fw(+)c Fv(w)r Fw(\()p Fv(v)s Fw(\)\))p Fs(j)1833 5241 y Fu(L)1879 5224 y Fd(2)1940 5226 y Fs(\024)k Fv(C)2092 5134 y Fq(\020)2142 5226 y Fw(1)c(+)2295 5110 y Fs(p)p 2364 5110 39 4 v 60 x Fv(")p 2295 5207 108 4 v 2325 5283 a(\015)2413 5134 y Fq(\021)2463 5226 y Fs(j)p Fv(f)9 b Fw(\()p Fv(v)21 b Fw(+)d Fv(w)r Fw(\))p Fs(j)2828 5238 y Fu(!)2900 5226 y Fs(\024)23 b Fv(C)3053 5134 y Fq(\020)3103 5226 y Fw(1)18 b(+)3256 5110 y Fs(p)p 3325 5110 39 4 v 60 x Fv(")p 3256 5207 108 4 v 3286 5283 a(\015)3374 5134 y Fq(\021)3423 5226 y Fs(j)p Fv(v)s Fs(j)3512 5191 y Fu(p)3512 5246 y(!)3561 5226 y Fv(;)2084 6059 y Fw(25)p eop %%Page: 26 26 26 25 bop 236 272 a Fw(b)n(y)28 b(\(2.14\))e(and)i(\(6.7\).)36 b(In)28 b(addition)834 400 y Fq(\014)834 449 y(\014)834 499 y(\014)876 382 y(Z)922 571 y Ft(\012)987 495 y Fv(r)1026 403 y Fq(\020)1077 495 y Fv(w)r Fw(\()p Fv(v)s Fw(\))19 b Fs(\000)g Fv(L)1405 461 y Fp(\000)p Ft(1)1405 516 y Fu(!)1493 495 y Fw(\005)1555 507 y Fu(W)1631 495 y Fv(f)9 b Fw(\()p Fv(v)s Fw(\))1788 403 y Fq(\021)1838 400 y(\014)1838 449 y(\014)1838 499 y(\014)1949 495 y Fw(=)2096 400 y Fq(\014)2096 449 y(\014)2096 499 y(\014)2138 382 y(Z)2184 571 y Ft(\012)2249 495 y Fv(r)31 b(L)2374 461 y Fp(\000)p Ft(1)2374 516 y Fu(!)2462 495 y Fw(\005)2524 507 y Fu(W)2600 495 y Fw(\()p Fv(f)9 b Fw(\()p Fv(v)22 b Fw(+)c Fv(w)r Fw(\()p Fv(v)s Fw(\)\))i Fs(\000)e Fv(f)9 b Fw(\()p Fv(v)s Fw(\)\))3351 400 y Fq(\014)3351 449 y(\014)3351 499 y(\014)1949 699 y Fs(\024)82 b Fv(C)2155 711 y Ft(4)2193 607 y Fq(\020)2242 699 y Fw(1)18 b(+)2400 643 y Fv(")p 2395 680 48 4 v 2395 756 a(\015)2453 607 y Fq(\021)2502 699 y Fs(j)p Fv(f)9 b Fw(\()p Fv(v)22 b Fw(+)c Fv(w)r Fw(\()p Fv(v)s Fw(\)\))i Fs(\000)e Fv(f)9 b Fw(\()p Fv(v)s Fw(\))p Fs(j)3235 711 y Fu(!)3284 699 y Fs(j)p Fv(r)r Fs(j)3369 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b Fv(X)29 b Fs(!)23 b Fv(X)33 b Fw(the)27 b(linear)f(op)r (erator)f(de\014ned)j(b)n(y)e Fs(I)6 b Fv(u)p Fw(\()p Fv(t;)14 b(x)p Fw(\))24 b(=)e Fv(u)p Fw(\()p Fv(t)17 b Fw(+)f Fv(\031)s(;)e(\031)21 b Fs(\000)16 b Fv(x)p Fw(\).)236 1544 y(It)28 b(is)g(easy)e(to)i(see)f(that)h Fs(I)34 b Fw(and)27 b Fv(L)1305 1514 y Fp(\000)p Ft(1)1305 1565 y Fu(!)1394 1544 y Fw(\005)1456 1556 y Fu(W)1559 1544 y Fw(comm)n(ute.)37 b(Hence,)28 b(since)f Fs(\000)p Fv(v)f Fw(=)d Fs(I)6 b Fv(v)s Fw(,)510 1725 y Fv(w)r Fw(\()p Fv(v)s Fw(\))24 b(=)f Fv(L)847 1691 y Fp(\000)p Ft(1)847 1746 y Fu(!)936 1725 y Fw(\005)998 1737 y Fu(W)1073 1725 y Fv(f)9 b Fw(\()p Fv(v)22 b Fw(+)c Fv(w)r Fw(\()p Fv(v)s Fw(\)\))52 b Fs(\))f(I)6 b Fv(w)r Fw(\()p Fv(v)s Fw(\))25 b(=)e Fv(L)2075 1691 y Fp(\000)p Ft(1)2075 1746 y Fu(!)2163 1725 y Fw(\005)2225 1737 y Fu(W)2301 1725 y Fs(I)6 b Fv(f)j Fw(\()p Fv(v)22 b Fw(+)c Fv(w)r Fw(\()p Fv(v)s Fw(\)\))25 b(=)d Fv(L)2948 1691 y Fp(\000)p Ft(1)2948 1746 y Fu(!)3037 1725 y Fw(\005)3099 1737 y Fu(W)3175 1725 y Fv(f)9 b Fw(\()p Fs(\000)p Fv(v)21 b Fw(+)d Fs(I)6 b Fv(w)r Fw(\()p Fv(v)s Fw(\)\))p Fv(:)236 1906 y Fw(This)28 b(implies,)g(b)n(y)g(the)h(uniqueness)e(of)h(the)g(solution)g(of)g Fv(w)e Fw(=)d Fv(L)2287 1876 y Fp(\000)p Ft(1)2287 1926 y Fu(!)2376 1906 y Fw(\005)2438 1918 y Fu(W)2514 1906 y Fv(f)9 b Fw(\()p Fs(\000)p Fv(v)21 b Fw(+)e Fv(w)r Fw(\))29 b(in)f(the)g(ball)g(of)g(radius)f Fs(j)p Fv(v)s Fs(j)3764 1918 y Fu(!)3812 1906 y Fw(,)i(that)236 2005 y Fv(w)r Fw(\()p Fv(v)s Fw(\)\()p Fv(t)20 b Fw(+)e Fv(\031)s(;)c(\031) 22 b Fs(\000)c Fv(x)p Fw(\))24 b(=)f Fv(w)r Fw(\()p Fs(\000)p Fv(v)s Fw(\)\()p Fv(t;)14 b(x)p Fw(\).)361 2105 y(F)-7 b(or)35 b Fv(iv)s Fw(\),)i(w)n(e)e(remark)f(that)i(if)g Fv(v)j Fs(2)e Fv(V)1595 2117 y Fu(n)1641 2105 y Fw(,)g(then)f Fv(w)r Fw(\()p Fv(v)s Fw(\))h(is)f(the)f(unique)h(solution)f(in)h(an)f (appropriate)f(ball)h(of)g(the)236 2205 y(equation)26 b Fs(G)5 b Fw(\()p Fv(w)r Fw(\))25 b(=)e Fv(w)r Fw(,)k(where)g(the)g (map)f Fs(G)32 b Fw(satis\014es)26 b Fs(G)5 b Fw(\()p Fv(W)2092 2217 y Fu(n)2139 2205 y Fw(\))23 b Fs(\032)g Fv(W)2360 2217 y Fu(n)2432 2205 y Fw(\()p Fv(W)2542 2217 y Fu(n)2615 2205 y Fw(is)j(the)h(closed)f(subspace)g(of)h Fv(W)39 b Fw(con)n(taining)236 2304 y Fv(w)26 b Fs(2)d Fv(W)40 b Fw(whic)n(h)28 b(are)e(2)p Fv(\031)s(=n)h Fw(p)r(erio)r(dic)g (in)h(time\).)38 b(Since)28 b Fs(G)2044 2274 y Fu(k)2085 2304 y Fw(\(0\))23 b Fs(!)g Fv(w)r Fw(\()p Fv(v)s Fw(\))29 b(in)f Fv(W)12 b Fw(,)28 b(this)g(implies)g(that)f Fv(w)r Fw(\()p Fv(v)s Fw(\))e Fs(2)e Fv(W)3727 2316 y Fu(n)3773 2304 y Fw(.)361 2404 y(Finally)-7 b(,)36 b(the)f(map)g(\()p Fv(v)s(;)14 b(w)r Fw(\))36 b Fs(!)g(G)5 b Fw(\()p Fv(v)s Fw(;)14 b Fv(w)r Fw(\))36 b(:=)f Fv(w)26 b Fs(\000)c Fv(L)2017 2374 y Fp(\000)p Ft(1)2017 2424 y Fu(!)2106 2404 y Fw(\005)2168 2416 y Fu(W)2244 2404 y Fv(f)9 b Fw(\()p Fv(v)26 b Fw(+)d Fv(w)r Fw(\))36 b(is)e(of)h(class)f Fv(C)3068 2374 y Ft(1)3140 2404 y Fw(and)h(its)g(di\013eren)n(tial)f (with)236 2504 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b(o)r(dd)g(and)f(ev)n(en,)h Fv(a)1014 4422 y Ft(1)1074 4410 y Fw(=)22 b Fv(a)1205 4422 y Ft(2)1243 4410 y Fw(.)361 4509 y(W)-7 b(e)28 b(no)n(w)f(turn)g(to)h(the)g(expression)e(of)1601 4442 y Fq(R)1670 4509 y Fv(w)r(L)1788 4479 y Fp(\000)p Ft(1)1877 4509 y Fw(\()p Fv(w)r Fw(\).)38 b(Let)28 b(us)g(de\014ne)f Fv(\014)2605 4521 y Ft(1)2643 4509 y Fw(,)g Fv(\014)2740 4521 y Ft(2)2778 4509 y Fw(,)g Fv(\014)2875 4521 y Ft(3)2940 4509 y Fw(on)g([0)p Fv(;)14 b Fw(2)p Fv(\031)s Fw(])3272 4479 y Ft(2)3337 4509 y Fw(b)n(y)1111 4694 y Fv(\014)1158 4706 y Ft(1)1195 4694 y Fw(\()p Fv(s)1266 4706 y Ft(1)1304 4694 y Fv(;)g(s)1380 4706 y Ft(2)1417 4694 y Fw(\))83 b(:=)g Fs(\000)p Fv(M)9 b Fw(\()p Fv(s)1929 4706 y Ft(1)1965 4694 y Fv(;)14 b(s)2041 4706 y Ft(2)2078 4694 y Fw(\))19 b(+)2222 4661 y Ft(1)p 2222 4675 34 4 v 2222 4723 a(2)2265 4694 y Fw(\()p Fv(M)9 b Fw(\()p Fv(s)2458 4706 y Ft(1)2495 4694 y Fv(;)14 b(s)2571 4706 y Ft(1)2608 4694 y Fw(\))19 b(+)f Fv(M)9 b Fw(\()p Fv(s)2903 4706 y Ft(2)2940 4694 y Fv(;)14 b(s)3016 4706 y Ft(2)3053 4694 y Fw(\)\))p Fv(;)1111 4794 y(\014)1158 4806 y Ft(2)1195 4794 y Fw(\()p Fv(s)1266 4806 y Ft(1)1304 4794 y Fv(;)g(s)1380 4806 y Ft(2)1417 4794 y Fw(\))83 b(:=)g(\()p Fv(s)1774 4806 y Ft(1)1830 4794 y Fs(\000)18 b Fv(s)1952 4806 y Ft(2)1989 4794 y Fw(\)\()p Fv(A)p Fw(\()p Fv(s)2186 4806 y Ft(1)2224 4794 y Fw(\))h Fs(\000)f Fv(A)p Fw(\()p Fv(s)2491 4806 y Ft(2)2528 4794 y Fw(\)\))p Fv(;)1111 4915 y(\014)1158 4927 y Ft(3)1195 4915 y Fw(\()p Fv(s)1266 4927 y Ft(1)1304 4915 y Fv(;)c(s)1380 4927 y Ft(2)1417 4915 y Fw(\))83 b(:=)g Fs(\000)1778 4859 y Fv(\013)p 1778 4896 54 4 v 1784 4972 a Fw(8)1841 4915 y(\()p Fv(s)1912 4927 y Ft(1)1967 4915 y Fs(\000)18 b Fv(s)2089 4927 y Ft(2)2127 4915 y Fw(\)\(2)p Fv(\031)j Fs(\000)e Fw(\()p Fv(s)2456 4927 y Ft(1)2511 4915 y Fs(\000)f Fv(s)2633 4927 y Ft(2)2671 4915 y Fw(\)\))p Fv(;)236 5124 y Fw(and)36 b Fv(B)469 5136 y Ft(1)506 5124 y Fv(;)14 b(B)606 5136 y Ft(2)643 5124 y Fv(;)g(B)743 5136 y Ft(3)817 5124 y Fw(on)36 b([0)p Fv(;)14 b Fw(2)p Fv(\031)s Fw(])23 b Fs(\002)h Fw([0)p Fv(;)14 b(\031)s Fw(])36 b(b)n(y)g Fv(B)1668 5136 y Fu(i)1695 5124 y Fw(\()p Fv(t;)14 b(x)p Fw(\))38 b(:=)f Fv(\014)2083 5136 y Fu(i)2111 5124 y Fw(\()p Fv(t)24 b Fw(+)g Fv(x;)14 b(t)24 b Fs(\000)g Fv(x)p Fw(\).)63 b(As)36 b(an)g(immediate)g (consequence)g(of)g(its)236 5224 y(de\014nition,)44 b Fv(B)707 5236 y Fu(i)775 5224 y Fw(is)d(2)p Fv(\031)s Fw(-p)r(erio)r(dic)f(w.r.t.)75 b Fv(t)41 b Fw(and)f(for)g Fv(x)45 b Fs(2)g(f)p Fw(0)p Fv(;)14 b(\031)s Fs(g)p Fw(,)42 b Fv(B)2510 5236 y Fu(i)2538 5224 y Fw(\()p Fv(t;)14 b(x)p Fw(\))45 b(=)f(0.)75 b(Hence)41 b Fv(B)3333 5236 y Fu(i)3405 5224 y Fs(2)k Fv(X)7 b Fw(.)75 b(Moreo)n(v)n(er)236 5323 y Fs(\000)p Fv(@)345 5335 y Fu(s)376 5343 y Fd(1)413 5323 y Fv(@)457 5335 y Fu(s)488 5343 y Fd(2)525 5323 y Fv(\014)572 5335 y Fu(i)622 5323 y Fw(=)23 b(\(1)p Fv(=)p Fw(4\))p Fv(\025)948 5335 y Fu(i)975 5323 y Fw(,)28 b(where)776 5504 y Fv(\025)824 5516 y Ft(1)862 5504 y Fw(\()p Fv(s)933 5516 y Ft(1)970 5504 y Fv(;)14 b(s)1046 5516 y Ft(2)1083 5504 y Fw(\))24 b(:=)36 b Fq(e)-60 b Fv(m)p Fw(\()p Fv(s)1393 5516 y Ft(1)1431 5504 y Fv(;)14 b(s)1507 5516 y Ft(2)1544 5504 y Fw(\))p Fv(;)180 b(\025)1827 5516 y Ft(2)1865 5504 y Fw(\()p Fv(s)1936 5516 y Ft(1)1973 5504 y Fv(;)14 b(s)2049 5516 y Ft(2)2086 5504 y Fw(\))24 b(:=)e Fv(a)p Fw(\()p Fv(s)2367 5516 y Ft(1)2405 5504 y Fw(\))c(+)g Fv(a)p Fw(\()p Fv(s)2653 5516 y Ft(2)2691 5504 y Fw(\))p Fv(;)180 b(\025)2974 5516 y Ft(3)3012 5504 y Fw(\()p Fv(s)3083 5516 y Ft(1)3120 5504 y Fv(;)14 b(s)3196 5516 y Ft(2)3233 5504 y Fw(\))24 b(:=)e Fv(\013:)2084 6059 y Fw(26)p eop %%Page: 27 27 27 26 bop 236 272 a Fw(This)28 b(implies)h(that)f([)p Fs(\000)p Fw(\()p Fv(@)1053 284 y Fu(t)1082 272 y Fw(\))1114 242 y Ft(2)1171 272 y Fw(+)18 b(\()p Fv(@)1330 284 y Fu(x)1372 272 y Fw(\))1404 242 y Ft(2)1442 272 y Fw(])p Fv(B)1528 284 y Fu(i)1579 272 y Fw(=)24 b Fv(L)1725 284 y Fu(i)1780 272 y Fw(where)k(w)n(e)g(ha)n(v)n(e)f(de\014ned)h Fv(L)2679 284 y Fu(i)2706 272 y Fw(\()p Fv(t;)14 b(x)p Fw(\))25 b(:=)f Fv(\025)3069 284 y Fu(i)3097 272 y Fw(\()p Fv(t)19 b Fw(+)f Fv(x;)c(t)19 b Fs(\000)g Fv(x)p Fw(\).)39 b(As)28 b(a)g(result)236 372 y Fv(B)299 384 y Ft(1)355 372 y Fw(+)18 b Fv(B)501 384 y Ft(2)557 372 y Fw(+)g Fv(B)703 384 y Ft(3)763 372 y Fs(2)23 b Fv(L)898 341 y Fp(\000)p Ft(1)987 372 y Fv(w)e Fw(+)d Fv(V)47 b Fw(and)352 463 y Fq(Z)398 652 y Ft([0)p Fu(;)p Ft(2)p Fu(\031)r Ft(])p Fp(\002)p Ft([0)p Fu(;\031)r Ft(])764 576 y Fv(w)r Fw(\()p Fv(t;)14 b(x)p Fw(\))p Fv(L)1060 542 y Fp(\000)p Ft(1)1150 576 y Fw(\()p Fv(w)r Fw(\)\()p Fv(t;)g(x)p Fw(\))30 b Fv(dt)e(dx)23 b Fw(=)1785 463 y Fq(Z)1831 652 y Ft([0)p Fu(;)p Ft(2)p Fu(\031)r Ft(])p Fp(\002)p Ft([0)p Fu(;\031)r Ft(])2183 576 y Fw(\()p Fv(B)2278 588 y Ft(1)2334 576 y Fw(+)18 b Fv(B)2480 588 y Ft(2)2536 576 y Fw(+)g Fv(B)2682 588 y Ft(3)2719 576 y Fw(\)\()p Fv(t;)c(x)p Fw(\)\()p Fv(L)3018 588 y Ft(1)3074 576 y Fw(+)k Fv(L)3214 588 y Ft(2)3270 576 y Fw(+)g Fv(L)3410 588 y Ft(3)3446 576 y Fw(\)\()p Fv(t;)c(x)p Fw(\))29 b Fv(dt)f(dx:)236 803 y Fw(W)-7 b(e)28 b(observ)n(e)e(that,)i(since)f Fv(B)1144 815 y Fu(i)1200 803 y Fw(v)-5 b(anishes)27 b(for)g Fv(x)c Fw(=)g(0)k(and)h Fv(x)23 b Fw(=)g Fv(\031)s Fw(,)28 b(for)f(all)g Fv(i;)14 b(j)5 b Fw(,)388 898 y Fq(Z)485 1011 y Fv(B)548 1023 y Fu(i)575 1011 y Fv(L)632 1023 y Fu(j)694 1011 y Fv(dt)28 b(dx)c Fw(=)997 898 y Fq(Z)1094 1011 y Fv(B)1157 1023 y Fu(i)1184 919 y Fq(\020)1234 1011 y Fw(\()p Fv(B)1329 1023 y Fu(j)1364 1011 y Fw(\))1396 1023 y Fu(tt)1469 1011 y Fs(\000)18 b Fw(\()p Fv(B)1647 1023 y Fu(j)1682 1011 y Fw(\))1714 1023 y Fu(xx)1794 919 y Fq(\021)1871 1011 y Fv(dt)28 b(dx)c Fw(=)2173 898 y Fq(Z)2270 919 y(\020)2320 1011 y Fw(\()p Fv(B)2415 1023 y Fu(i)2443 1011 y Fw(\))2475 1023 y Fu(tt)2548 1011 y Fs(\000)18 b Fw(\()p Fv(B)2726 1023 y Fu(i)2754 1011 y Fw(\))2786 1023 y Fu(xx)2866 919 y Fq(\021)2915 1011 y Fv(B)2978 1023 y Fu(j)3041 1011 y Fv(dt)28 b(dx)23 b Fw(=)3343 898 y Fq(Z)3440 1011 y Fv(L)3497 1023 y Fu(i)3524 1011 y Fv(B)3587 1023 y Fu(j)3650 1011 y Fv(dt)k(dx:)236 1218 y Fw(Hence)510 1297 y Fq(Z)606 1410 y Fv(w)r(L)724 1376 y Fp(\000)p Ft(1)814 1410 y Fv(w)e Fw(=)986 1297 y Fq(Z)1032 1486 y Ft([0)p Fu(;)p Ft(2)p Fu(\031)r Ft(])p Fp(\002)p Ft([0)p Fu(;\031)r Ft(])1398 1410 y Fv(B)1461 1422 y Ft(1)1498 1410 y Fv(L)1555 1422 y Ft(1)1610 1410 y Fw(+)18 b(2)1749 1297 y Fq(Z)1845 1410 y Fv(B)1908 1422 y Ft(1)1946 1410 y Fv(L)2003 1422 y Ft(2)2058 1410 y Fw(+)g(2)2197 1297 y Fq(Z)2293 1410 y Fv(B)2356 1422 y Ft(1)2393 1410 y Fv(L)2450 1422 y Ft(3)2506 1410 y Fw(+)2589 1297 y Fq(Z)2685 1410 y Fv(B)2748 1422 y Ft(2)2786 1410 y Fv(L)2843 1422 y Ft(2)2898 1410 y Fw(+)g(2)3037 1297 y Fq(Z)3133 1410 y Fv(B)3196 1422 y Ft(2)3233 1410 y Fv(L)3290 1422 y Ft(3)3345 1410 y Fw(+)3428 1297 y Fq(Z)3525 1410 y Fv(B)3588 1422 y Ft(3)3626 1410 y Fv(L)3683 1422 y Ft(3)3719 1410 y Fv(:)236 1637 y Fw(W)-7 b(e)28 b(ha)n(v)n(e)236 1716 y Fq(Z)333 1829 y Fv(B)396 1841 y Ft(1)433 1829 y Fv(L)490 1841 y Ft(1)550 1829 y Fw(=)648 1773 y(1)p 648 1810 42 4 v 648 1886 a(2)713 1716 y Fq(Z)759 1904 y Fl(T)811 1888 y Fd(2)848 1829 y Fw(\()p Fs(\000)p Fv(M)9 b Fw(\()p Fv(s)1106 1841 y Ft(1)1143 1829 y Fv(;)14 b(s)1219 1841 y Ft(2)1256 1829 y Fw(\)+)1363 1773 y(1)p 1363 1810 V 1363 1886 a(2)1414 1829 y Fv(M)9 b Fw(\()p Fv(s)1575 1841 y Ft(1)1612 1829 y Fv(;)14 b(s)1688 1841 y Ft(1)1725 1829 y Fw(\)+)1832 1773 y(1)p 1832 1810 V 1832 1886 a(2)1883 1829 y Fv(M)9 b Fw(\()p Fv(s)2044 1841 y Ft(2)2082 1829 y Fv(;)14 b(s)2158 1841 y Ft(2)2195 1829 y Fw(\)\))f Fq(e)-59 b Fv(m)p Fw(\()p Fv(s)2403 1841 y Ft(1)2441 1829 y Fv(;)14 b(s)2517 1841 y Ft(2)2554 1829 y Fw(\))k Fv(ds)2686 1841 y Ft(1)2742 1829 y Fv(ds)2824 1841 y Ft(2)2885 1829 y Fw(=)k Fs(\000)3047 1773 y Fw(1)p 3047 1810 V 3047 1886 a(2)3112 1716 y Fq(Z)3158 1904 y Fl(T)3210 1888 y Fd(2)3261 1829 y Fv(M)9 b Fw(\()p Fv(s)3422 1841 y Ft(1)3459 1829 y Fv(;)14 b(s)3535 1841 y Ft(2)3572 1829 y Fw(\))f Fq(e)-59 b Fv(m)p Fw(\()p Fv(s)3748 1841 y Ft(1)3785 1829 y Fv(;)14 b(s)3861 1841 y Ft(2)3898 1829 y Fw(\))19 b Fv(ds)4031 1841 y Ft(1)4087 1829 y Fv(ds)4169 1841 y Ft(2)236 2046 y Fw(b)r(ecause)27 b Fs(h)13 b Fq(e)-59 b Fv(m)q Fs(i)681 2058 y Fu(s)712 2066 y Fe(i)766 2046 y Fw(=)22 b(0.)254 2145 y Fq(Z)351 2258 y Fv(B)414 2270 y Ft(1)451 2258 y Fv(L)508 2270 y Ft(2)568 2258 y Fw(=)666 2202 y(1)p 666 2239 V 666 2315 a(2)731 2145 y Fq(Z)777 2334 y Fl(T)829 2317 y Fd(2)866 2258 y Fw(\()p Fs(\000)p Fv(M)9 b Fw(\()p Fv(s)1124 2270 y Ft(1)1161 2258 y Fv(;)14 b(s)1237 2270 y Ft(2)1274 2258 y Fw(\))19 b(+)1418 2202 y(1)p 1418 2239 V 1418 2315 a(2)1469 2258 y Fv(M)9 b Fw(\()p Fv(s)1630 2270 y Ft(1)1667 2258 y Fv(;)14 b(s)1743 2270 y Ft(1)1780 2258 y Fw(\))19 b(+)1924 2202 y(1)p 1924 2239 V 1924 2315 a(2)1975 2258 y Fv(M)9 b Fw(\()p Fv(s)2136 2270 y Ft(2)2173 2258 y Fv(;)14 b(s)2249 2270 y Ft(2)2287 2258 y Fw(\)\)\()p Fv(a)p Fw(\()p Fv(s)2498 2270 y Ft(1)2536 2258 y Fw(\))k(+)g Fv(a)p Fw(\()p Fv(s)2784 2270 y Ft(2)2822 2258 y Fw(\)\))28 b Fv(ds)2996 2270 y Ft(1)3061 2258 y Fv(ds)3143 2270 y Ft(2)3204 2258 y Fw(=)22 b Fv(\031)3355 2145 y Fq(Z)3401 2334 y Fl(T)3472 2258 y Fv(M)9 b Fw(\()p Fv(s;)14 b(s)p Fw(\))p Fv(a)p Fw(\()p Fv(s)p Fw(\))27 b Fv(ds)236 2475 y Fw(b)r(ecause)g Fs(h)p Fv(M)9 b Fs(i)697 2487 y Fu(s)728 2495 y Fe(i)782 2475 y Fw(=)23 b(0)k(and)h Fs(h)p Fv(a)p Fs(i)23 b Fw(=)g(0.)635 2575 y Fq(Z)732 2688 y Fv(B)795 2700 y Ft(1)832 2688 y Fv(L)889 2700 y Ft(3)949 2688 y Fw(=)1047 2632 y Fv(\013)p 1047 2669 54 4 v 1053 2745 a Fw(2)1124 2575 y Fq(Z)1170 2764 y Fl(T)1222 2747 y Fd(2)1272 2596 y Fq(\020)1340 2688 y Fs(\000)c Fv(M)9 b Fw(\()p Fv(s)1585 2700 y Ft(1)1622 2688 y Fv(;)14 b(s)1698 2700 y Ft(2)1735 2688 y Fw(\))k(+)1878 2632 y(1)p 1878 2669 42 4 v 1878 2745 a(2)1930 2688 y Fv(M)9 b Fw(\()p Fv(s)2091 2700 y Ft(1)2128 2688 y Fv(;)14 b(s)2204 2700 y Ft(1)2241 2688 y Fw(\))19 b(+)2385 2632 y(1)p 2385 2669 V 2385 2745 a(2)2436 2688 y Fv(M)9 b Fw(\()p Fv(s)2597 2700 y Ft(2)2634 2688 y Fv(;)14 b(s)2710 2700 y Ft(2)2747 2688 y Fw(\))2779 2596 y Fq(\021)2852 2688 y Fw(=)23 b Fs(\000)p Fv(\031)s(\013)3122 2575 y Fq(Z)3168 2764 y Fl(T)3238 2688 y Fv(M)9 b Fw(\()p Fv(s;)14 b(s)p Fw(\))28 b Fv(ds)236 2905 y Fw(b)r(ecause)f Fs(h)p Fv(M)9 b Fs(i)24 b Fw(=)e(0.)432 3022 y Fq(Z)528 3135 y Fv(B)591 3147 y Ft(2)629 3135 y Fv(L)686 3147 y Ft(2)805 3135 y Fw(=)953 3022 y Fq(Z)1036 3042 y Ft(2)p Fu(\031)999 3210 y Ft(0)1128 3022 y Fq(Z)1211 3042 y Fu(\031)1174 3210 y Ft(0)1270 3135 y Fw(2)p Fv(x)1359 3042 y Fq(\020)1408 3135 y Fv(A)p Fw(\()p Fv(t)d Fw(+)g Fv(x)p Fw(\))g Fs(\000)f Fv(A)p Fw(\()p Fv(t)h Fs(\000)f Fv(x)p Fw(\))2121 3042 y Fq(\021\020)2221 3135 y Fv(a)p Fw(\()p Fv(t)g Fw(+)g Fv(x)p Fw(\))h(+)f Fv(a)p Fw(\()p Fv(t)h Fs(\000)f Fv(x)p Fw(\))2896 3042 y Fq(\021)2974 3135 y Fv(dt)28 b(dx)805 3373 y Fw(=)953 3260 y Fq(Z)1036 3281 y Fu(\031)999 3449 y Ft(0)1095 3373 y Fv(dx)g Fw(8)p Fv(x)1316 3260 y Fq(Z)1399 3281 y Ft(2)p Fu(\031)1362 3449 y Ft(0)1491 3281 y Fq(\020)1541 3373 y Fv(A)p Fw(\()p Fv(t)19 b Fw(+)f Fv(x)p Fw(\))h Fs(\000)f Fv(A)p Fw(\()p Fv(t)h Fs(\000)f Fv(x)p Fw(\))2253 3281 y Fq(\021\020)2353 3373 y Fv(A)2415 3339 y Fp(0)2438 3373 y Fw(\()p Fv(t)h Fw(+)f Fv(x)p Fw(\))h(+)f Fv(A)2845 3339 y Fp(0)2869 3373 y Fw(\()p Fv(t)h Fs(\000)f Fv(x)p Fw(\))3112 3281 y Fq(\021)3190 3373 y Fv(dt)805 3612 y Fw(=)953 3499 y Fq(Z)1036 3520 y Fu(\031)999 3688 y Ft(0)1095 3612 y Fv(dx)28 b Fw(8)p Fv(x)1316 3499 y Fq(Z)1399 3520 y Ft(2)p Fu(\031)1362 3688 y Ft(0)1516 3556 y Fv(d)p 1501 3593 74 4 v 1501 3669 a(dt)1584 3520 y Fq(h)1633 3556 y Fw(\()p Fv(A)p Fw(\()p Fv(t)20 b Fw(+)e Fv(x)p Fw(\))h Fs(\000)f Fv(A)p Fw(\()p Fv(t)h Fs(\000)f Fv(x)p Fw(\)\))2410 3526 y Ft(2)p 1633 3593 815 4 v 2020 3669 a Fw(2)2477 3612 y Fs(\000)g Fv(A)p Fw(\()p Fv(t)h Fs(\000)f Fv(x)p Fw(\))2865 3578 y Ft(2)2903 3520 y Fq(i)2960 3612 y Fw(+)g(2)p Fv(A)p Fw(\()p Fv(t)h Fw(+)f Fv(x)p Fw(\))p Fv(A)3452 3578 y Fp(0)3476 3612 y Fw(\()p Fv(t)h Fs(\000)f Fv(x)p Fw(\))28 b Fv(dt)805 3851 y Fw(=)953 3738 y Fq(Z)1036 3758 y Fu(\031)999 3926 y Ft(0)1095 3851 y Fv(dx)g Fw(16)p Fv(x)1358 3738 y Fq(Z)1441 3758 y Ft(2)p Fu(\031)1404 3926 y Ft(0)1533 3851 y Fv(A)p Fw(\()p Fv(t)19 b Fw(+)f Fv(x)p Fw(\))p Fv(A)1900 3817 y Fp(0)1924 3851 y Fw(\()p Fv(t)h Fs(\000)f Fv(x)p Fw(\))28 b Fv(dt)23 b Fw(=)2379 3738 y Fq(Z)2462 3758 y Fu(\031)2425 3926 y Ft(0)2521 3851 y Fv(dx)28 b Fw(16)p Fv(x)2784 3738 y Fq(Z)2867 3758 y Ft(2)p Fu(\031)2830 3926 y Ft(0)2959 3851 y Fv(A)p Fw(\()p Fv(s)p Fw(\))p Fv(A)3186 3817 y Fp(0)3210 3851 y Fw(\()p Fv(s)18 b Fs(\000)g Fw(2)p Fv(x)p Fw(\))28 b Fv(ds)805 4090 y Fw(=)953 3977 y Fq(Z)1036 3997 y Ft(2)p Fu(\031)999 4165 y Ft(0)1128 4090 y Fv(ds)g(A)p Fw(\()p Fv(s)p Fw(\))1417 3977 y Fq(Z)1500 3997 y Fu(\031)1464 4165 y Ft(0)1559 4090 y Fw(16)p Fv(xA)1752 4055 y Fp(0)1775 4090 y Fw(\()p Fv(s)19 b Fs(\000)f Fw(2)p Fv(x)p Fw(\))28 b Fv(dx)c Fw(=)2298 3977 y Fq(Z)2381 3997 y Ft(2)p Fu(\031)2344 4165 y Ft(0)2473 4090 y Fv(ds)k(A)p Fw(\()p Fv(s)p Fw(\))2748 3997 y Fq(h)2788 4090 y Fw(8)p Fv(xA)p Fw(\()p Fv(s)19 b Fs(\000)f Fw(2)p Fv(x)p Fw(\))3233 3997 y Fq(i)3272 4015 y Fu(x)p Ft(=0)3272 4148 y Fu(x)p Ft(=)p Fu(\031)805 4328 y Fw(=)83 b Fs(\000)p Fw(8)p Fv(\031)1123 4215 y Fq(Z)1206 4236 y Ft(2)p Fu(\031)1169 4404 y Ft(0)1298 4328 y Fv(A)p Fw(\()p Fv(s)p Fw(\))1463 4294 y Ft(2)1529 4328 y Fv(ds:)236 4545 y Fw(In)28 b(the)g(fore-last)e(line,)i(w)n(e)f (ha)n(v)n(e)g(in)n(tegrated)f(b)n(y)i(parts)f(\(w.r.t.)37 b Fv(x)p Fw(\))28 b(and)f(used)h(the)g(fact)g(that)g Fs(h)p Fv(A)p Fs(i)c Fw(=)e(0.)1103 4662 y Fq(Z)1200 4775 y Fv(B)1263 4787 y Ft(2)1300 4775 y Fv(L)1357 4787 y Ft(3)1417 4775 y Fw(=)g Fv(\013)1571 4662 y Fq(Z)1654 4682 y Fu(\031)1618 4851 y Ft(0)1713 4775 y Fv(dx)29 b Fw(2)p Fv(x)1935 4662 y Fq(Z)2018 4682 y Ft(2)p Fu(\031)1981 4851 y Ft(0)2096 4775 y Fw(\()p Fv(A)p Fw(\()p Fv(t)19 b Fw(+)f Fv(x)p Fw(\))h Fs(\000)f Fv(A)p Fw(\()p Fv(t)h Fs(\000)f Fv(x)p Fw(\)\))29 b Fv(dt)23 b Fw(=)g(0)p Fv(;)236 4992 y Fw(still)28 b(b)r(ecause)f Fs(h)p Fv(A)p Fs(i)d Fw(=)f(0.)36 b(A)n(t)28 b(last)1177 5108 y Fq(Z)1274 5221 y Fv(B)1337 5233 y Ft(3)1374 5221 y Fv(L)1431 5233 y Ft(3)1491 5221 y Fw(=)22 b Fs(\000)1653 5165 y Fv(\013)p 1653 5202 54 4 v 1659 5278 a Fw(8)1730 5108 y Fq(Z)1813 5129 y Ft(2)p Fu(\031)1776 5297 y Ft(0)1905 5108 y Fq(Z)1988 5129 y Fu(\031)1951 5297 y Ft(0)2047 5221 y Fw(2)p Fv(x)p Fw(\(2)p Fv(\031)f Fs(\000)d Fw(2)p Fv(x)p Fw(\))28 b Fv(dt)g(dx)c Fw(=)f Fs(\000)2888 5165 y Fv(\013)2941 5135 y Ft(2)2978 5165 y Fv(\031)3028 5135 y Ft(4)p 2887 5202 179 4 v 2955 5278 a Fw(6)236 5438 y(Summing)28 b(up,)g(w)n(e)f (get)h(\(4.12\).)p 1281 5438 48 48 v 2084 6059 a(27)p eop %%Page: 28 28 28 27 bop 236 272 a Fr(References)278 454 y Fw([1])41 b(A.)26 b(Am)n(brosetti,)f(M.)h(Badiale,)f Fo(Homo)l(clinics:)39 b(Poinc)l(ar)n(\023)-40 b(e-Melnikov)31 b(typ)l(e)d(r)l(esults)e(via)j (a)f(variational)i(appr)l(o)l(ach)p Fw(,)407 553 y(Annales)d(I.)h(H.)g (P)-7 b(.)27 b(-)h(Analyse)f(nonlin.,)h(v)n(ol.)e(15,)h(n.2,)h(pp.)g (233-252,)c(1998.)278 720 y([2])41 b(A.)29 b(Am)n(brosetti,)g(V.)g (Coti-Zelati,)g(I.)g(Ek)n(eland,)f Fo(Symmetry)j(br)l(e)l(aking)h(in)f (Hamiltonian)h(systems)p Fw(,)d(Journ.)f(Di\013.)407 819 y(Eq.)f(67,)g(165-184,)d(1987.)278 985 y([3])41 b(A.)31 b(Am)n(brosetti,)h(A.)f(Malc)n(hio)r(di,)h Fo(A)h(multiplicity)h(r)l (esult)e(for)i(the)f(Y)-6 b(amab)l(e)33 b(pr)l(oblem)h(on)f Fv(S)3360 955 y Fu(n)3405 985 y Fw(,)f(J.)f(F)-7 b(unct.)32 b(Anal.)407 1085 y(168,)26 b(529-561,)f(1999.)278 1251 y([4])41 b(A.)29 b(Am)n(brosetti,)g(G.)h(Pro)r(di,)e Fo(A)j(primer)h(of)g(nonline)l(ar)f(analysis)p Fw(,)g(Cam)n(bridge)d (Univ)n(ersit)n(y)g(Press,)g(Cam)n(bridge,)407 1350 y(1993.)278 1517 y([5])41 b(A.)24 b(Am)n(brosetti,)h(P)-7 b(.)23 b(Rabino)n(witz:)35 b Fo(Dual)26 b(V)-6 b(ariational)28 b(Metho)l(ds)g(in)e(Critic)l(al)i(Point)f(The)l(ory)h(and)f(Applic)l (ations)p Fw(,)407 1616 y(Journ.)g(F)-7 b(unc.)28 b(Anal,)f(14,)g (349-381,)e(1973.)278 1782 y([6])41 b(D.)24 b(Bam)n(busi:)33 b Fo(Lyapunov)27 b(Center)f(The)l(or)l(ems)g(for)h(some)f(nonline)l(ar) g(PDE's:)37 b(a)26 b(simple)h(pr)l(o)l(of)p Fw(,)e(Ann.)f(Sc.)f(Norm.) 407 1882 y(Sup.)28 b(di)g(Pisa,)e(Ser.)i(IV,)g(v)n(ol.)e(XXIX,)j(fasc.) e(4,)h(2000.)278 2048 y([7])41 b(D.)29 b(Bam)n(busi,)f(S.)h(P)n (aleari,)e Fo(F)-6 b(amilies)32 b(of)f(p)l(erio)l(dic)i(solutions)e(of) g(r)l(esonant)g(PDE's)p Fw(,)e(J.)g(Nonlinear)e(Sci.,)i(11,)f(pp.)407 2147 y(69-87,)d(2001.)278 2314 y([8])41 b(D.)33 b(Bam)n(busi,)h(S.)g(P) n(aleari,)e Fo(F)-6 b(amilies)36 b(of)g(p)l(erio)l(dic)g(orbits)g(for)f (some)h(PDE's)f(in)g(higher)h(dimensions)p Fw(,)g(Comm.)407 2413 y(Pure)27 b(and)g(Appl.)h(Analysis,)f(V)-7 b(ol.)28 b(1,)f(n.4,)h(2002.)278 2579 y([9])41 b(T.)28 b(Bartsc)n(h,)f Fo(A)i(gener)l(alization)j(of)f(the)f(Weinstein-Moser)h(The)l(or)l(ems) g(on)f(p)l(erio)l(dic)i(orbits)e(of)h(a)g(Hamiltonian)407 2679 y(system)e(ne)l(ar)h(an)g(e)l(quilibrium)p Fw(,)f(Ann.)f(Inst.)g (Henri)f(P)n(oincar)n(\023)-39 b(e,)25 b(Anal.)j(Non)f(Lin)n(\023)-39 b(eaire)26 b(14,)g(No.6,)i(691-718,)c(1997.)236 2845 y([10])41 b(M.)22 b(Berti,)h(A.)f(Malc)n(hio)r(di,)h Fo(Multiplicity)j(r)l(esults)e(and)h(multibump)f(solutions)h(for)g(the) g(Y)-6 b(amab)l(e)25 b(pr)l(oblem)g(on)g Fv(S)3905 2815 y Fu(n)3950 2845 y Fo(")p Fw(,)407 2944 y(J.)i(F)-7 b(unct.)29 b(Anal.,)e(180,)g(1,)g(2001.)236 3111 y([11])41 b(G.D.)34 b(Birkho)r(of,)g(D.C.)f(Lewis,)i Fo(On)f(the)h(p)l(erio)l(dic)i (motions)e(ne)l(ar)g(a)g(given)h(p)l(erio)l(dic)h(motion)e(of)h(a)f (dynamic)l(al)407 3210 y(system)p Fw(,)27 b(Ann.)i(Mat.)e(12,)g (117-133,)e(1933.)236 3376 y([12])41 b(J.)26 b(Bourgain,)f Fo(Construction)j(of)h(Quasi-Perio)l(dic)i(solutions)d(for)i (Hamiltonian)f(p)l(erturb)l(ations)f(of)i(line)l(ar)f(e)l(qua-)407 3476 y(tions)h(and)g(applic)l(ations)i(to)e(Nonline)l(ar)g(PDE)p Fw(,)e(In)n(t.)g(Math.)g(Res.)f(Not,)h(n.11,)f(1994.)236 3642 y([13])41 b(J.)36 b(Bourgain,)g Fo(Construction)h(of)i(p)l(erio)l (dic)g(solutions)f(of)g(nonline)l(ar)g(wave)g(e)l(quations)g(in)f (higher)j(dimension)p Fw(,)407 3742 y(Geom.)27 b(and)h(F)-7 b(unc.)28 b(Anal.,)g(v)n(ol.)f(5,)g(n.4,)g(1995.)236 3908 y([14])41 b(J.)31 b(Bourgain,)g Fo(Quasi-p)l(erio)l(dic)k (solutions)f(of)g(Hamiltonian)h(p)l(erturb)l(ations)e(of)h Fw(2)p Fv(D)h Fo(line)l(ar)f(Schr)l(o)l(dinger)h(e)l(qua-)407 4007 y(tions)p Fw(,)28 b(Ann.)g(of)g(Math.,)f(148,)g(363-439,)d(1998.) 236 4173 y([15])41 b(H.)26 b(Brezis,)f(J.)g(M.)h(Coron,)f(L.)h(Niren)n (b)r(erg,)f Fo(F)-6 b(r)l(e)l(e)28 b(vibr)l(ations)h(for)f(a)h (non-line)l(ar)f(wave)h(e)l(quation)f(and)g(a)h(The)l(or)l(em)407 4273 y(of)h(P.)h(R)l(abinowitz)p Fw(,)d(Comm.)g(Pure)f(and)g(Appl.)h (Math.)g(31,)f(1-30,)f(1978.)236 4439 y([16])41 b(J.)27 b(M.)g(Coron,)f Fo(Perio)l(dic)32 b(solutions)d(of)h(a)g(Non-line)l(ar) g(wave)g(e)l(quation)f(without)h(assumption)f(of)i(monotonicity)p Fw(,)407 4539 y(Math.)d(Ann.)g(262,)e(273-285,)f(1985.)236 4705 y([17])41 b(W.)36 b(Craig,)h(E.W)-7 b(a)n(yne,)38 b Fo(Newton)-8 b('s)37 b(metho)l(d)h(and)h(p)l(erio)l(dic)g(solutions)f (of)h(nonline)l(ar)f(wave)g(e)l(quation)p Fw(,)h(Comm.)407 4804 y(Pure)27 b(and)g(Appl.)h(Math,)g(v)n(ol.)f(XL)-9 b(VI,)28 b(1409-1498,)23 b(1993.)236 4970 y([18])41 b(L.)f(DeSimon,)j (G.)d(T)-7 b(orelli,)41 b Fo(Soluzioni)h(p)l(erio)l(diche)i(di)d(e)l (quazioni)h(al)t(le)g(derivate)g(p)l(arziali)h(di)f(tip)l(o)f(ip)l(erb) l(olic)l(o)407 5070 y(nonline)l(ari)p Fw(,)29 b(Rend.)f(Sem.)g(Mat.)f (Univ.)h(P)n(ado)n(v)-5 b(a,)26 b(40,)g(380-401,)f(1968.)236 5236 y([19])41 b(E.)28 b(R.)g(F)-7 b(adell,)28 b(P)-7 b(.)28 b(Rabino)n(witz,)g Fo(Gener)l(alize)l(d)k(c)l(ohomolo)l(gic)l (al)h(index)e(the)l(ories)g(for)g(the)f(gr)l(oup)h(actions)g(with)g(an) 407 5336 y(applic)l(ation)h(to)e(bifur)l(c)l(ations)g(question)g(for)h (Hamiltonian)f(systems)p Fw(,)e(In)n(v.)f(Math.)h(45,)f(139-174,)d (1978.)2084 6059 y(28)p eop %%Page: 29 29 29 28 bop 236 272 a Fw([20])41 b(S.)34 b(B.)h(Kuksin,)g Fo(Perturb)l(ation)h(of)h(c)l(onditional)t(ly)i(p)l(erio)l(dic)f (solutions)e(of)h(in\014nite-dimensional)g(Hamiltonian)407 372 y(systems)p Fw(,)27 b(Izv.)h(Ak)-5 b(ad.)28 b(Nauk)f(SSSR,)h(Ser.)g (Mat.)f(52,)g(No.1,)g(41-63,)f(1988.)236 538 y([21])41 b(S.)28 b(B.)g(Kuksin,)g Fo(A)n(nalysis)i(of)h(Hamiltonian)g(PDE's)e Fw(Oxford)e(Lecture)g(Series)h(in)g(Mathematics)g(and)f(its)h(Appli-) 407 637 y(cations.)f(19.)g(Oxford)f(Univ)n(ersit)n(y)h(Press,)f(2000.) 236 803 y([22])41 b(A.)24 b(M.)f(Ly)n(apuno)n(v,)g Fo(Pr)l(obl)n(\022) -40 b(eme)27 b(g)n(\023)-40 b(en)n(\023)g(er)l(al)26 b(de)h(la)f(stabilit)n(\023)-40 b(e)27 b(du)f(mouvement)p Fw(,)e(Ann.)g(Sc.)g(F)-7 b(ac.)23 b(T)-7 b(oulouse,)23 b(2,)h(203-474,)407 903 y(1907.)236 1069 y([23])41 b(B.)33 b(V.)h(Lidskij,)i(E.)d(I.)h(Sh)n(ulman,)h Fo(Perio)l(dic)j(solutions)d (of)h(the)g(e)l(quation)f Fv(u)2839 1081 y Fu(tt)2915 1069 y Fs(\000)22 b Fv(u)3050 1081 y Fu(xx)3152 1069 y Fw(+)g Fv(u)3287 1039 y Ft(3)3357 1069 y Fw(=)32 b(0,)j(F)-7 b(unct.)35 b(Anal.)407 1169 y(Appl.,)28 b(22,)f(332-333,)d(1988.)236 1335 y([24])41 b(J.)26 b(Moser,)f Fo(Pr)l(o)l(of)30 b(of)g(a)f(gener)l (alize)l(d)h(form)f(of)g(a)g(\014xe)l(d)f(p)l(oint)h(the)l(or)l(em)g (due)g(to)f(G.)h(D.)g(Birkho)l(of)p Fw(,)g(Geometry)d(and)407 1434 y(T)-7 b(op)r(ology)g(,)26 b(Lectures)h(Notes)g(in)h(Math.)g(597,) e(464-494,)f(1976.)236 1600 y([25])41 b(J.)30 b(Moser,)h Fo(Perio)l(dic)k(orbits)e(ne)l(ar)f(an)h(Equilibrium)h(and)f(a)g(The)l (or)l(em)g(by)g(A)n(lan)g(Weinstein)p Fw(,)f(Comm.)e(on)h(Pure)407 1700 y(and)c(Appl.)i(Math.,)e(v)n(ol.)g(XXIX,)i(1976.)236 1866 y([26])41 b(H.)28 b(P)n(oincar)n(\023)-39 b(e,)24 b Fo(L)l(es)30 b(m)n(\023)-40 b(etho)l(des)31 b(nouvel)t(les)f(de)g(la) h(m)n(\023)-40 b(ec)l(anique)30 b(c)n(\023)-40 b(eleste)p Fw(,)28 b(P)n(aris,)e(1899.)236 2032 y([27])41 b(P)-7 b(.)21 b(Rabino)n(witz,)h Fo(Time)j(p)l(erio)l(dic)h(solutions)f(of)g (nonline)l(ar)f(wave)h(e)l(quations)p Fw(,)e(Man)n(usc.)e(Math.)h(5,)g (165-194,)e(1971.)236 2198 y([28])41 b(P)-7 b(.)26 b(Rabino)n(witz,)g Fo(Perio)l(dic)31 b(solutions)e(of)h(nonline)l(ar)f(hyp)l(erb)l(olic)i (p)l(artial)f(di\013er)l(ential)g(e)l(quations)p Fw(,)d(Comm.)g(Pure) 407 2298 y(Appl.)h(Math.,)g(20,)f(145-205,)d(1967.)236 2464 y([29])41 b(P)-7 b(.)25 b(Rabino)n(witz,)g Fo(F)-6 b(r)l(e)l(e)27 b(vibr)l(ation)h(of)g(a)g(seminile)l(ar)h(wave)f(e)l (quation)p Fw(,)e(Comm.)f(on)g(Pure)f(Appl.)i(Math,)f(31,)g(31-68,)407 2563 y(1978.)236 2729 y([30])41 b(M.)22 b(Stru)n(w)n(e,)g Fo(V)-6 b(ariational)25 b(metho)l(ds.)g(Applic)l(ations)h(to)e(nonline) l(ar)h(Partial)g(Di\013er)l(ential)g(Equations)g(and)f(Hamil-)407 2829 y(tonian)30 b(Systems)p Fw(,)d(Springer)g(V)-7 b(erlag,)26 b(1990.)236 2995 y([31])41 b(E.)25 b(W)-7 b(a)n(yne,)25 b Fo(Perio)l(dic)k(and)f(quasi-p)l(erio)l(dic)i(solutions)e(of)g (nonline)l(ar)g(wave)g(e)l(quations)g(via)h(KAM)e(the)l(ory)p Fw(,)f(Com-)407 3095 y(m)n(un.)i(Math.)f(Ph)n(ys.)g(127,)f(No.3,)h (479-528,)e(1990.)236 3261 y([32])41 b(A.)28 b(W)-7 b(einstein,)28 b Fo(Normal)j(mo)l(des)f(for)h(Nonline)l(ar)f(Hamiltonian)h(Systems)p Fw(,)c(In)n(v.)g(Math,)h(20,)f(47-57,)e(1973.)2084 6059 y(29)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF