Content-Type: multipart/mixed; boundary="-------------0608260521117" This is a multi-part message in MIME format. ---------------0608260521117 Content-Type: text/plain; name="06-230.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-230.keywords" nonlinear wave equation, infinite dimensional hamiltonian systems, periodic solutions, Birkhoff Normal Form ---------------0608260521117 Content-Type: application/postscript; name="PhDthesis_digregorio.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="PhDthesis_digregorio.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: PhDthesis_digregorio.dvi %%CreationDate: Tue Mar 28 13:39:33 2006 %%Pages: 219 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMSL12 CMR12 CMBX12 CMMI12 CMMI8 CMR8 CMSY10 %%+ CMEX10 CMR7 CMR10 CMSY8 MSBM10 CMR6 CMMI6 CMTI12 MSBM7 CMMI10 %%+ CMCSC10 CMSY6 MSAM10 CMSL10 CMMI7 CMSY7 CMMI5 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips.exe -P pdf -G0 PhDthesis_digregorio %DVIPSParameters: dpi=8000, compressed %DVIPSSource: TeX output 2006.03.28:1339 %%BeginProcSet: tex.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/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]{Ci}imagemask restore}B/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: alt-rule.pro %! % Patch by TVZ % Makes dvips files draw rules with stroke rather than fill. % Makes narrow rules more predictable at low resolutions % after distilling to PDF. % May have unknown consequences for very thick rules. % Tested only with dvips 5.85(k). 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta 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 27 /sigma put dup 28 /tau put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 36 /pi1 put dup 38 /sigma1 put dup 39 /phi1 put dup 42 /arrowrighttophalf put dup 44 /arrowhookleft put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 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 93 /sharp put dup 96 /lscript 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 126 /vector put readonly def /FontBBox{-30 -250 1026 750}readonly def /UniqueXX 5087386 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 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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 99 /c 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 2b28c1eb3d82e4d3fe225223f4266d2b6147e344aad2d4884a9308aadc7801af a606838925e3e50ee7aaf73f090baefcfd4e34b795cd92d83c944d91a4940f2f c955d727ea5bd76f0c8057b12621eceb0947b0a5dd10dbcbc52957dd978d7db0 c794660afdd4e946b3d7fc1ba60907fb3f756aab0406c0e7bce5a2238fb3e12e 07e2c0a7f465e90057642d704434ae0b57bd46c8ea91ab2a386e24c6deaf9cb0 8cdaa8fc4ec3fadc06bf28659cb878453db16f4b03925b2127b83be8fc240f33 23ea950b51d70c8fe861e30c346efa0a0125137598caca5478991e35f5bbbc19 4c787488ba339830bca15f8333431902e82b66a514a41c0bff92719a6a0d9587 4cff62c622fc61a0308a69f9fb7ef51bb274ce9e059c04b64e817abdbbb39375 bb4d84ca40de5fa46b84b04f3056d74afe38378db988b6f5a88ad1118268e7c8 be173700f3c1185dab6b65909feb4002a6c9d800dce3c6c8996dc32962ae96fb 492e0ad5e18df6f58b28770cdebecb4e3674c7bf351d2b1d14f96aa19bd20742 4f6b8ab61f1fa190e8df0cc2a82e6054f3d59b293f8ac8655f37907e15398e56 d776b79a811d4ce9d003777ba747eee5d1ea854d0e1f241d4c8e183b5d71aa3b 67be8ab26b8b39445eab031bad299d870e907f02b79c26df3afe7951a2821a94 27dcb595517c9ff3911b44725d7f877fe158a9f15a40f76f45a54605dc4fb08a fb161426ed70d24fda9d4e4828995ad36545a1d640111bcfb4127c7e84eef35f 9761d8fa23b8bdcb092a45f897759787c8543e589c86b3887f3513fae52f 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 50 /element put dup 73 /I 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 4a93d9b8745f749460afc7ec123d154bb5b7dcdd7481a3ec469b5f3f30e93e2e b2988b10c6620829d2f4df0b112ff7a71932b7ace24aaaed1e933aeba8853b28 b59c2feae928001ea944aa96f99ec6485e6621a11b3f363248117ef3d8fa1ef5 c9b53fb8118bfdd1ed7e866df8fd8de82828af17e40f31cd1328b2631b8b8f61 f1f54c897f59812391f182f41c04b1c3cad54fc0e4c55bebe319f82e3d4ac61c 1aad4f27303a0b855bdfcfad7876f73ef25777ac549cd66f396ae29b1568051b 50bb8cce02c3c9ba7054d49dd1c425c204b551b86d2ab02ef90462083181624c 6a83105481919411e4c05f78913319fc17ff79348e9c8484ba17a0ac80439899 9bf898cc70936a5d6ce963d96ffbee9d2d30bc51e46f7840242a277411eae9dc a3d0e62e4922e159d02e195a3be0522f9db3cc64783746bfcdc3f351bcffac22 1078d34dfb55036beedfa0105f03317775c081e63fc97c4413c31df4fc12f055 c2273aee0413d49ab760c570c3a3d988507e4aa73cf22cc14adbc573f9f23694 3056bbebcb034ee859316492a2b18ffa8f1d260b9f8f8dee064218e97e5b0c52 dcf59fc0b3e0eab1410816782ca229457c6b0273aa376d2314a5f4dca64fe999 2620514362279c6dc772e1a8aa10501e5bda05de67405a2d4684556a98a9c72d d2141d630683cac73b25a54fa42a6ccedf1085b866ca8d5444d9cbc80d865ce1 1434e198afe5b43722b2e8bd08d2b19cdecd6ceebe6da365cc4bcdc9f3ea56e6 6c37a119903f80 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 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 67 /C put dup 73 /I put dup 78 /N put dup 81 /Q put dup 82 /R put dup 84 /T put dup 90 /Z put dup 106 /gimel put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueXX 5031982 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d56491bdecdcafdc988206c5a457a19270b 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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 36 /pi1 put dup 59 /comma put dup 78 /N put dup 97 /a put dup 105 /i put dup 107 /k put dup 115 /s put readonly def /FontBBox{0 -250 1171 750}readonly def /UniqueXX 5087382 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930f01bdaae649990604a33aa9eaffbe5b5489 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%%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 0 /parenleftbig put dup 1 /parenrightbig put dup 2 /bracketleftbig put dup 3 /bracketrightbig put dup 8 /braceleftbig put dup 9 /bracerightbig put dup 10 /angbracketleftbig put dup 11 /angbracketrightbig put dup 12 /vextendsingle put dup 13 /vextenddouble put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 20 /bracketleftbigg put dup 21 /bracketrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 34 /bracketleftBigg put dup 35 /bracketrightBigg put dup 40 /braceleftBigg put dup 41 /bracerightBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 66 /parenleftex put dup 67 /parenrightex put dup 80 /summationtext put dup 81 /producttext put dup 82 /integraltext put dup 83 /uniontext put dup 88 /summationdisplay put dup 89 /productdisplay put dup 90 /integraldisplay put dup 91 /uniondisplay put dup 92 /intersectiondisplay put dup 100 /hatwidest 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 dup 113 /radicalBig put dup 114 /radicalbigg put dup 115 /radicalBigg put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueXX 5000774 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 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6c046db132477ae6316f9263121d7ac9451ff757b6b9a81e91d13d09e552db90 5386df08fe5c0a776a9f61b055244eefcc51776d7923dea058cf960b0ae87dd9 668d7ad381958eabba55a95948f29b962e88f1432600954418197883ba75f23c d4dd6a41f9e3d5e2b1a92f385542ac77c36e467b8450111145b1c3b9cbec4f42 d16c0473c13650d8 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 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 82 /R put dup 90 /Z put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueXX 5032014 def currentdict end currentfile eexec b5e2457af4a6caaa4536453c498a618ecdce7b5f6f2d5944265309f833bd3d50 3df8bd14c071f404d2177500681a801856cb8c9c9cf0112c9bd17ccba9ab8944 1759a320b2078df0fafdd3e357994bd81740d1a7522ddfafb75d58f5def046f7 084d10fa2f702e570b09a7073d70efe37b48c355a7089772238b7b4538c4912f cf6703a14fa416057ea2e1b94455913dd271b42846ee10d78270a8118e7e4bcb 8deb048bb280c240d59409d47046c890e5f79e5dd4b0be65a28a77aeb53dd33e ac506a6da87bb55e329f433627e06c3c94b078838b908538d93f888e504e25e0 3a8ccf7d2c02122ab3d485525564295390d1154e15fee35edff4eb016fadd32c 26d1b4c09de87943780954cebc4965aa890fe1f80b03eb07103e13d468245c7b c26455f229e8859585c6eeafd9474acdb27290fd62950dc52f893c99a2a614d4 2a7e3618266f3fb0c68a01a30373e048385ffe7dd5844e208a55d1a9263ab2db cb9b19038724f95af00349840157be45cb337871b55a304b87ade64b10ff0eba 21aa0eaeef74aa01f049303a8b25de0acc26bb3a33fcf267f479a2a73f7dece3 2aa23083687322b417fe86e8f5afc2136d045ad6b392364811a17630a94ff097 038eedeedb8da4ebf8fa5a96ce27a2ebf771d14f4a11fb1f9f48cdd207209934 885ef9118463552984b0fd3cf15d55de967fbb051549dd65f60dfe07f299690e 58252f032bca78935b7d2df852ed688bd13dd171b91cf7bd46d5433aef9f50f1 41a2517c4912b837793f84ddcf1116f70f7e9b1e6ec752dbc9183d4791d92602 397bafe29ac6f879ceb72c673c8a1aa0135d6f92d0db7791f9a2f1a5b55347e0 3e63723efd258754f5f4a9ad35873a98d54b6c54b17a39892dc1d38660597448 fdba9258493a0db0d560c01d49f2d889a99eaa8a5b5919cbd33f6b3d97bf27b3 e0b94b4dcf3fe48b07398eec882c325567e28b9eb4d699111343b1910da9117d 74d6e9de75247ad908f8ce2cfc49e0faa22dbf4d1abe2c035b8f8eee2d1967e0 7cb9b5d8de5e0b1c3a86c7a08a580e51be75a8afdaba2cda08f87d352652a3c7 9bda6bcd32fd59e8079fee3fe7fa161ff626461e85ff8b9b60e2e6bc72884c6f dbaa5e8ad5912206cdd26b204cb34a5403c4fe9cc6d0a2e6805a98624c56baf0 3e673abaa926e941739ba23126dd78ca82499f376d15bfc7cc70f1586f5f10a1 fe99eed1a1f44ace181d76ee8aee83d31488053143530d7ded9f985e650b028e 29840a34637e40769c936bf028c06e9a0a3826c5d7a610747f5d2f57ca9a9771 ffe980db46681f8b5d6ad2578c044cdc53db5dd7d5d237e76f210f719743e052 3cd6aeb335abd84dea6f907e166a9ccad9fadde3f6cdd5f93833ba7ca40f0b64 0cfda47adb541b7214b5da377e37a906136a3e0880bc8d3599eca809261705e6 02db231e07275b1a5da961aff0988fa7e6298c4a43c38d75ad65f6337074dc2e dc80dcc6dfdc4f11600fc843a230160f771f7f21e411cd1b866a042270589f0f 97c07add4010471e6758d0a0063ba56370fbd6efd47111405d7f23fa09e5c1cc 713c8c77658da9e34bdc855afd6ca2f827fb1b5564b6f595f5b9ff4816f3b276 f275b3195ab110c87f0bfcc2c3eb489262331d84b56ab425cf1cdcca54dd3cee 7940564c830d75f498300e0c08917f3e6a3be956ed0ce87ed78c8b2c4a967371 c6a6d2b6dc4de08762bd7b3fd7b533168709b5e71df8efa4812600cb270824c0 34bdebffd9f8497ad058d7f3aac63e073403c8919807573572026ba7143089b3 44ece81f764c63b5347eaa42bc38a31a4c4c587970b6242be2c2a6eea8630749 91f4db3d178d9600ceec181280659d2fae7574c83edb661b2866679f2b843b49 930a07c045198779db19feef96160794a23a45ed5070937afde89e50291ca6d4 15e27ccf458bac123bee460dfe39e34575302ef59ea9f2b73217919359b8738a 1de44dc64b3aa5a38652af188cc6a010eea15dbf89abed3b63de4cbe88dfed69 69f3db8a9d10f91306c87813a39324caf527b1816843d3d83196f43796c1b89c 9af9506b3b4ed576bb7b0521e3f26e7bea71070cba88c1809e755c2f3e9cc001 9334d5704c656bd087ac98b33bffd3dd0e969855fe0e3b94b484d656e22320e8 c5524efc070fdc15726b9c83fae87760e51731caa1bcc664477588c8c3ade0b3 6ddd819a6cdefde5c2bda016cdcf12186268ff3c0fca77bbb8aba56a7c05a057 18a9054078e59bcc23ee27b6d4b91669ef53be002593418d767e19d4b67f423e a6351f5273874337fc6cca78bf17a641e83c5e1e8d3c7f3e8779b939ac93150c 8f8783ce37e4855aa5f400f746985d4455f6459095c062172ce51b86c8d8e7a5 3369ededd676d7dd14206878baffb8cf08a5fd1bc1e667b0e9bb98f0d6cb3a89 cd1bd4d368b3bd5a3cda2549c3d8f12ecd3a8abcd89af08845eaad73 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 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 3 /square put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueXX 5031981 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6f33457e3200f0935ea98f1a0cfaff49cfa2 d877ff805620cba84126dfd175a82e1b4c6ec58461fce97cdd749a012e2c4243 86b199a97c306b282350cbe5af90cf1d8e7371dcaedbd5fa3346dd3e558072c5 fd51a2edfee6890f62efd1b552699123f4f52c2817dadd5ece30b3bc178a3ef4 f45fd8ad0ee5bf30db5f9eee9ddefe249f3c349a886cf9130cba68ae559ad719 c439930ece1d52d38366116c390cbeabb8fdb36811cf3b8e2416825e8c880b24 6521ec6880ebe7f127c8741c756b29e1f89e3fd5d638a7bbade8c3a0fdbbf7ba ec37cd372afb2d9dd25d509726d20a05914e418f6b7e759c440beb61519df02a 06734ec1226890d17c51f25f775e6879fdc231bc4d7fcb4b33d04fa273bdb8e8 f124171714614f8a94e98a6c2d6bdc1254913999f9c181e42c0886556cdf48af 92a265c1eb06bb30fb2745840c17bd31c7b8bdfd37dfb9643a418a2eaae14094 6752e0a315207aef85c3c2630538907971ec1b05687208571a3504bb3369b5c8 9a43e5abe343d0599fd8d41a1bf01de49f5856bdd67d63a2b3ecea75d721a6ca 0c8751ab41e892e9cf674218dd44acfd40019bf59f4d69ea40d5cf9068a6ff96 9f8cc772b920baf17334833d732431a2969b9a30fe38bd44772cef653935a957 af61dd568f041847ccce3c587656cba7bbf46d8cefec936c6d34dbe51bc04929 0440354f3b29ed8e6b4f21ad53b5809779d67339f284563d54ed4d2a504b014d 974a31c0ceb57b364af519bad4e85c41d521a3ffc26dbac555b42dc191367152 1d60c30b95b5bbafb3c7d87a1c043ea5f0e6dab83a4de99a742e75f02837b923 c50eb478ac4131ee604e3d7b784bd8c93eb9430291d452c21be0495718546333 66a602a9f142a563fdae3c8d40de6a8aa581ff3172367ca325e6d23102c5c6a2 02cc99580287dbd76a05bb4cca591b12a9dd3d602d7957376ec30b16051a63c7 e26ddf009e21fdca5a31cc973f0b1f6967a66b30b973ddd9d9a3095b673ea8f8 5db8dbf3acb7bb2d85c04a0ea8853e4592e4f8e581adaadb78bc3ae58a2d4c80 3b09839eafc7bc3e7cdf784e852d6590ab1c4b7558e28505887c6858072f09dc 2caba91b6871920e5256b602a0de02b87e5eb456878d3634e8eb0e069fa2bb5c c10d5cd98401de1cf07ec9f7de62aaa20b583e2f7aad2fb7e727ebc43f712c74 e8eed1fa76133428ac71589b933cc37ad200c516045fc1a06d0086c541103b5d 4ec54dc77b 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 3 /asteriskmath put dup 6 /plusminus put dup 8 /circleplus put dup 14 /openbullet put dup 15 /bullet put dup 17 /equivalence put dup 18 /reflexsubset put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 25 /approxequal put dup 26 /propersubset put dup 27 /propersuperset put dup 28 /lessmuch put dup 33 /arrowright put dup 40 /arrowdblleft put dup 41 /arrowdblright put dup 49 /infinity put dup 50 /element put dup 51 /owner put dup 52 /triangle put dup 54 /negationslash put dup 55 /mapsto put dup 56 /universal put dup 57 /existential put dup 59 /emptyset put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 88 /X put dup 92 /intersection put dup 94 /logicaland put dup 95 /logicalor put dup 102 /braceleft put dup 103 /braceright put dup 104 /angbracketleft put dup 105 /angbracketright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put dup 112 /radical put dup 114 /nabla put dup 126 /heart put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueXX 5000820 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f67052788e6703bb5ee49533870bca1f113ad8 3750d597b842d8d96c423ba1273ddd32f3a54a912a443fcd44f7c3a6fe3956b0 aa1e784aaec6fce08dae0c76da9d0a3eba57b98a6233d9e9f0c3f00fcc6b2c6a 9ba23af389e6dfff4efec3de05d6276c6be417703ce508377f25960ef4ed83b4 9b01b873f3a639ce00f356229b6477a081933fef3bb80e2b9dffa7f75567b1fa 4d739b772f8d674e567534c6c5bbf1cf615372be20b18472f7aa58be8c216dbd df81cc0a86b6d8318ca68fe22c8af13b54d7576fe4ca5a7af9005ea5cc4edb79 c0ab668e4fec4b7f5a9eb5f0e4c088cd818ecc4feb4b40ec8bd2981bf2336074 b64c430528009c080fb967acbabd53399a40a6921f25ea92ca1bc17040162b8e 5416034f8b553aed8a223370503a3e04c94deb155d5822960c4516917d706a6c b6de5f98a9c99a32ba44b1940beaef727bfe3e1c7e5fb4243acdbf9129a14535 bdf8a1dcc0407058c77a7706c41c29d379d24d41346ea547178a397ab7f47440 eb8299c6839111e9bf5bc1ef02232a5beb7f5f84049656aa332df71fd07416bd 04f19f2025a71783db3c9bb1d3980752e7e195cbdae53f8b2dc9c78f702e8c2c f191cdb1d6785875c08ec57ede2f1026e47c1c6627a0f8e500719226768436b3 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSL10 %!PS-AdobeFont-1.1: CMSL10 1.0 %%CreationDate: 1991 Aug 20 16:40:20 % 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 (CMSL10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -9.46 def /isFixedPitch false def end readonly def /FontName /CMSL10 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 97 /a put dup 101 /e put dup 107 /k put dup 119 /w put readonly def /FontBBox{-62 -250 1123 750}readonly def /UniqueXX 5000798 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd800bcb325468c79ffacb9cd3036f2ed7bb65d c6f12e33dd6d45b61272314f98990d08c0f8274e999f4b46af4b5c5da923683c 960face50087915d2044d7211a18e0c22967ef89552662ba990e2dfb9267e09e 354f41e74d44bd80a2d905d2edd60d8829e820c6ab0fdf322964c1ecb6569818 a465d5266d3366d748b412446e602f0cc52f3cad2213ff6ad92c04f6ce8bba94 890fb44eadc41a0c42eda4fe9722e6d4b17a3f6d35cf755d0a1c90d0c2033b16 ba76c456137acee0112e35edfb7206a9cb18dd9274401c70c3039059a583940f 44cc19ffed32940fe8203292e103d2abcda173df85995aee98bf4a1f65ead55f b80bc0c737589fa10cf51999c634e05d80a7dffb61433bbdba17ea9a7b92f763 586105c8277d6a961d092617c64745450b7df4efa70884f6017d9b0331814cf7 02f1140425eb8bf4c269ec042fde895a7f625c4832c1a143a14abeb798e23b59 f88ebd1414a66520435a4c8f5dd7a548a7d6a2fa96f0995bc0fbcf91b1a1eb71 1026a99d2404b2103bb3626978385830a00a02b6aa820cf33785f7aa45e267e6 ebe7d737ded351c1d4e944733feb30a3e6f81c1a7b74ddc9148b46b02a525539 c54a7eb25dec18807557b6ea94c36db0c91121160f61bd37cff40bb2b11aceb7 5f146be38109daeb77803d8f580f9086bc460e5c96f764c1254e36feb77c48a4 419b9a9cb5ec916b73efcc4c5a698909d5c98ca014b7c8e2ebcde195f5ad44f5 75bd2dcebce29979618bee325c726f5d4967cf0e3f7d35058e04dbb7fbcf60f1 436fade30418ba3c7dde0b6aab803140e6aa7c9808ab4d3d31432f5d4cbc007d 32d7541ccb867f6ccba8bd18c3d18524a692ed2ad29db54d7ca84a618052d3af 3491bec27c918795bdbcd6cd338751b0b06f57e638a8bbeffc8bc885fa92e7dd 9f8c9f090cd35a401e8accc0d352203927b380de762b3b4ced6cc5dfe06fa0ce 8852ce20d526babb1529dee5696ce64e93b9c238d27c6eb66cc7462eaf591261 9eb44dd9beae6df572859ddfeb8397812c945d91a1d4bd4b1ae0872020c50cbb 0ac701d3b19f419ca776b80102fd1b762dcae498e464729f8d881e6ceb839b87 fd451b6efee0a1fab2c7324ead58e1aa66bec07eceb42d730a3162452c43484e 3d0d9e9d7bcaf7364c58d1910624136b55de597b87bfcef073f000f1cda57aea 47ff8aa225eb712d6fecdfd34134f22a6733cdcde0f30853cf12c73a48b0627b a35bb7f7c8443385821668984797daafe76d81a680296f636546377b6945b3d9 2a86013bb57479b64a268cd2e7b77f6dd982d7161f40f756506c6095a7c214d5 8dc2148966864f029762d0f49403246ad004d18f9b0481328f4ab30e73e08097 f905eb50b10de5be824a1f73fc97ab5c13a4cdccde2729ba7976a861ef2d7f3f 1cc44fa62b5f27852e7cef5f4ccf673cb17648e38fed3597a3aed05d45a9303a 419321d0c7c8dea5b1fe6e5f448a97fa03ba6816d9463b337e49499eca21dded 36763c733ecc7469a8638d2d67949d6f5679ae3795eb8b69616ea2cd102d4dfb 99382fc6a754eec485a5fd4851edff709c288f539c829cf2c31f71ab69e2d4f0 7e1de973d6061192c29a4e690bdeb95977126e7433a4a2aec787f461941abc34 57d4a8d7bc5a52bb9b7e956e868276c427bb09f98fb7205ae66433a3702da678 190d5450df8655302ab7f10fbcae4e00f69ea62923a39ebee1020e9918472548 63ef952c7298f13d61894cb7e7e5d10fd0d9a58c93373129041db948301306e5 77a75e01fd44553b5428691c87b0a2496a6082d61468669a75a36bff83721fc2 ed8ea0b3c8e2dad50e82b07aa21979c4ab576183865a1d650040fb12f37b5477 dee192a40bec37fabbd0e716ec3cc7e27218bd439003d7b4360b2b92a4802206 c823c6c471c1ccb64bb774862269dd23b85c7c26a2d9e054a06d87d3096d966e 7e489892d288bbd7da313d2a774976f537c548d56c5b1646af69f4b4659279ee e586e19506be9d539be93bdedba902212c6db89489ef9126d28c920a9017e0c5 b2294dd28dcfcee663478f6605bab789b8e2d9e5ffd0b50eeab0249897b7474d 9ad310b2882dd08d70b4356bf847b1c415bbbb6b03edcf61e2b14ae66182251f 2f6eb2e6073247be363a45d662867d1e38fb8ebff258872eceba8628fd685b80 ab3606df715799b4ab14a5ced3189803969580a84b84600e6cada0b0798912e0 cdf224aef60bb5cfaa4410bba607c014a9dcdc5d5bbe06375b667cc462adfc66 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 dup 2 /multiply put dup 3 /asteriskmath put dup 48 /prime put dup 49 /infinity put dup 63 /perpendicular put dup 73 /I put dup 76 /L 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 c43e28519dd651f01d5a9d75c77408441f594c3c1610cb694cdc3ec0bdfce4db 4c1df652d0d618f364a911ebc9fd6ddb118dfa8973cf9f90f3a333adfa839ed7 b03672d24f104bbb3a444738520440875ea13c9eea99a7e296607e56471b705b 5137adfddfd82ce1756a73b47c9889dbd256f24644d8ae1553edfecd6ee278c8 aaab53167af948eeafc3077c4b60ba0f5d1ee7c4defb06a0a1fc464476541d83 63c3df8de54a4d966abbce0f911381649161faf2e7c1782ea0faeb7fe3c38d38 85accdaaf48233c7a8907463192cd536d39fdc4274473231de8c698b8aadade9 f22ef9707a13ff091da8a3bc0d22f2bfe77d0fb744ca81316380c330eb7b6938 101c2603dc1620cda2a7229cfe78f16b6e54540a36073fd4e05e5bf0ae3c331a 006e7c60d5604a76a5cba3228ea7b55ed8d875ec9797ce91402d887660533f27 7a6ae06a5d8fcc21b3359538e72218b9498f72d27967309dfd383bebcd334e7f 39e841bc59fcc97e6daf968062585c2917c034af0cfc53071cf89905e5846f02 71918d8c807f273f76ae79b4d1c2269e9f228385c0786ae079510e767fd746b8 a8e5c4d0d143b9ab1db77425fb06c3c539d78e00005c83622e7efe294fc4d5aa 52f9a3d82a6847ec8faa635e57398ea0daee88a85ae1780894e335dfbb4e362b 159429cfe14aed4f06f7d8473c34bfd9119baef7f08d1a3a3136711f9dc25eba 2fa1df8dbe46d2f7e3af7f744c09db0ebd85b0ea5b96bbf00042577e9f776772 5200e0faf45bf41c71ccfd97213eec2f6983da0aabdd07d9ed665c57f9158723 618e365980f29ebf384869b6e6ebf285d1624db5d767d286bf0cfb8598357154 edc884a663417fcff183cc2c4897a1904df2ad84314e62e9aa9c365930f4e28b a0645787c5127f73c5ed767c663a2e826826fa246056e7e7be98d6105081a26a 9e76ba79cd4ffded80190b546d8fb2d1e95364270f2be3c7db8a8966b8cfdcd3 94c15128eabcf37fed2628d184e930a57f4afe5db0f366e160de82af0faaa04d e5ef1ffcec9b9ac55a00e43fb7f470cfec493cac8bf0c8ee8edc730bb2d628a6 38480e1b560c5916f2890e0f0e1c055d6681b0ac0ea50aa7293e075f7a9275e4 be9b172c6da03f7b4f1e7f4cc55fc5748685053fe4fb1f3526ff03f5b5782653 3f55d9f91a13f68637dde1c3758f823ce777fd5e8fa0434a226d7c9765c34b85 22f710f2a87c7a3f4f77e998b4cc4f5bcbaf144d3613fdecb0aa6b405f11f2ae 2f0b9c8d21232676434e1a10249ef04b001fd7d8760bfb1c6aa3cd2cb30e4bf7 d81387ade4e060bbd7c712acbda479fc03c647f84fb09280a96b70ba15e1232e 7eecc45fa5c348674f14bfa4720c3491dc4c920782300b36d834f8807ffa3671 91abbc586903c95219026a1e4bfd22b69b80d387ecfbbbbe51249200a8f912b8 fd4066f24ddd16a64383ee27e4eb6b22c71a5cc4a36dfbcfb42af65b8b4e993f b551e6119b4cc4eb6f5b7d16517ed5e48ea55fe42107a21b2272e35585a2c1fb f47ff3c1e62c278b9a54978dbde05992aa0b380c8dafbe3630cc6cfca2a429a0 88cd2d7beed0594843b33fdb3d75a43939bd47902f230d91077773a3971b8540 9fc69513bfd5db858ca018b9d91a99eb8c26c87160ba5e8a3ba17744b251c993 fbc7a7d7ad9c81570c8472101e8821a4071a13e43aecfe91aaa2f5c4245eb0a0 a54ce2655b578f04c954fde5fb41fdcc5ef90e1508c751f64f7b86984e2e75e4 efcaa015dc39d2168c72958eaf1c08ff148886aab301925ef59629626cd3b520 390f4769c6ee896efb7f01946354bd86874e20d0c5bc535a44805943a8c0f848 15efa6c69b9175e0ce815b2687cabdbc7cbae70bb583b701f2fba1374ad3d067 24abab3f40901685487426b54a6425b8510e1c8e47e5db9f4bd5ab74d2d835c8 fdc9fd9f0b13f0adc6de5e8dc3b88917f5379ebd4a01 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMCSC10 %!PS-AdobeFont-1.1: CMCSC10 1.0 %%CreationDate: 1991 Aug 18 17:46:49 % 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 (CMCSC10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMCSC10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 80 /P put dup 83 /S put dup 84 /T put dup 97 /a 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 121 /y put dup 122 /z put dup 123 /endash put readonly def /FontBBox{14 -250 1077 750}readonly def /UniqueXX 5000772 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd800bcb325468c79ffacb9cd3037a112066957 7889d67fff57549064b985d67ede2fb03272083723541051ebe936eff01a4ab3 cd845d565dfdf25d9ddd2792db17f52f894bea484480f06c832140fdd6c3af0e fef0584c6b78b5e2bbb9a752f10581fd1c48cb498a822afb18cddebba34695b9 b418010368fdfb7bf758d50f3af03360b81c776325c33c9c81130b1e1be840cf 68c71ffd091dccc0df7b507fe8953dc3836af0da08ff9ff3e9cf7288d9eb8baf ffac546a0fd987ae8e0bcf7b759f4408e13524f478de5896d816cb0ad2904271 cbec82a5ca50f0598502b434ecd94531b4a65181b5b3afcb650904b15157737a a4386f1a7f1252caf2adbd5c8162bf3b065c01ca471d520c884ebbe1b29f9549 134e285dea2fe158c15a08a744cdda00bb6c62158262e24702803d9e1dc1d3a3 1383085631154b077fc20488ee3aa03f47ed2766644330b94133f2f9389b99b8 96d3707d1c5bb227fd3bbdf3a6ff572bf07164107c7a1c0710a5c7cfb67282f9 c238100fa138b75c553d7cbef17e2292a320d3d86a6210e6e3a1e590b10dac6e 177094d5dd163db623d7b344478752f6f3d7d5b3fa92e6bc72dc6fc343d2f946 9153e3d14c5a6d6b1d9ca52947b789cb4cf6196a02973aff0b097333f8d615c0 a1726ab7e31c378eab77150faa28cce535afe3ae5798402ca51cbd18dedc2c43 5281d0781351df30f4d11761579676449ae4a47a10f677cdd31ca9120fc2606a 9025d3cc09d58856a9bbfb17669f706ff8c7efe37a3656c9b20609cccb58b309 0595ca5666833f2bbf884aac1cd632fce69b4e0ea2e0a25b9c4d471f0e322e09 4cefda321359fdbe0a82656d3ecaeb68b7e0c7e733d85066ed7474c22b67169a 1bf1bdf7dfd0ce0aaeef78ee393ce6ea65edc1fdcdf3f744f714005d8221a156 4e76923daea523536614df44b4057ca99603ebbd24724b58a82c48691bd1f278 16a987b0196888c1ce4e9c24ac4fd9e039a91880c644488d5d1db6efa810d3a9 57bbeaa93fe23c9706d9ecfeab860359d8f888d1f0faa258e0bd0d69c5211149 e01e817a0a98d66201a21cc2046f43c87ccdf13206b62c61151c11bdacfeee15 616fd7c92f50042eac689ff81a0caa5f2dfd5cda0708e2eb83423389446001d8 bc632ff3f97b86922846a1ffd4748e17ccd5a444bf2b6022489fb4a674bad5db 073c67574576fe6df24426ac13d3016c39f877904840f88ccff450f9b4cada6a a8fa31e77e9bbd507f8a60c5b9a31e9a44038cd1fed1b1e79123c3bb1c24dc19 1388b9346512db995c6147af41763b74506d8ded05c7903849c965af5054b983 5e390a2d71ae1fb14df3a3a6d10633632c5d115edd341df853d97cd759bc5086 aac0ae9907d1cfc83be401d4aa09d73b6529aacadbb5d4c322a16cd12ba6d239 c15c6da86b0402970a49be700dc2ed5b87aad044a7cd8d189c77e596ebed30f9 eb253164e0c343059a3e336731c59f0170fa34dca8c30eb67631e80bf168658c 60fd95e78dde9531182d460839390406d4054852b53de7a088d7639b8674b838 409ab84b26d9dc97ef323307e9ffc08a26beee7b8aa7d56d0b6cdac35b1a4b36 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206690d6b6405947e183c2e69a4abafdf0c501caebc8fd62d3c2b29d006eca2e 9e048b3bd47330ae0a36f8b96a5034bdba5f829aa9a3ea5714f315fceb856a14 8ee436f316348ab86f60ff8d95e0a73f 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 22 /mu put dup 28 /tau put dup 30 /phi put dup 34 /epsilon put dup 36 /pi1 put dup 58 /period put dup 59 /comma put dup 64 /partialdiff put dup 71 /G put dup 72 /H put dup 73 /I put dup 75 /K put dup 76 /L put dup 79 /O put dup 86 /V put dup 88 /X put dup 96 /lscript put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 107 /k put dup 110 /n put dup 116 /t put dup 117 /u put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-32 -250 1048 750}readonly def /UniqueXX 5087385 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 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f163f77efcabe951268624b36c8653df90471eb53d670eae0876863dbcb15b50 04a708b10bfed8d47c2e77f4379995d08b8804c4c72384dbe1ae4c6d2ce77469 27 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 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 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 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 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 127 /dieresis put readonly def /FontBBox{-36 -251 1103 750}readonly def /UniqueXX 5000829 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 22 /mu put dup 23 /nu put dup 27 /sigma put dup 28 /tau put dup 38 /sigma1 put dup 59 /comma put dup 61 /slash put dup 69 /E put dup 70 /F put dup 72 /H put dup 73 /I put dup 78 /N put dup 82 /R put dup 84 /T put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 115 /s put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueXX 5087381 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 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55e4ffe1ed217063e2936b82dec04437c25f7fb6ab84d4579b43ddb3891731ab 3d9862560b8132fdf313b664f46ec0f668d2db3f4ed442170ad808e770984691 1c0245ef31a5b84fa935ba98db8bb8e853ab4d92530b7cde3f94928e03d17f81 d4537c4ac046f408a54a7a879ec135c017902d2edb32f7961cfff6bdc361962e 84c7b1e781354aff9e387faa1226d53696b91c90283ec40157fdb1d0c1afd7c2 cd8a71b44cadcaed985901c1d8879f7f462015e462055607b35e3639ab610680 da73e4df9175c1aea7610abb45723e418a1eb75d7dbe55b13f68efc476df09ec 5850b647907fe51e14593354b67010b14fb92e6585cca3b7d960b046aa54a420 ddb5c468c866ac9fe547600abada590b508678a0ea504dd37c6d7be10babe82e 4ae8451bcb7e0a05d3ed2190269336a8fe33e064609a26aa763e88b2d94cdb8a 22d2f7aecd01da20408dbde060b90e534f8f0b1d52cfba05a55d51b323300ce3 6f053b2490080ebc5d7d5d70afaa5affa2a4d93165ab3d4e20320db363106ea4 8dc02e2e335cd264923f0e885e7a696387dc9f24c90aa73a834bea64d34f8afe 4ca205317093dc689e52af6593fa65c5a1e68dabfffb84967eb1c0dcf702eaf2 eea063df0ed043955a34e07e898b1a788e913f88817939afa48485827f90cfb0 4844b6a711c94dd475cd10bb335c77ce9e03621cedfd7e2fe11af42558ff787a 9739f5f345fd324cf05703fbd6233057b17103b36660758a2bd4b97e19fe5467 ab2307117519723b036289f4f4503c073d4e07dec63f8ff6bfd94bb51c76bb7c b86a2a808b54c2793e27e8403840bbd067f87270d017f6900af49d1d70b8ec63 087d0c988da5ebf971d6473f34e674d862e75b9e9e7af1b2e2df7ba9e4276013 fe23f6d7f2e6e7ad5de77323cdcfca0cec37f965beabc97cb38474de57eefd00 b147ed7f15500f26e6d5eb8cf2f9dd5177515b0bd931b4667f13c975daca1153 0e663c4eedaf2e308227f253860e8e26f1bafb00d033a4aa924e0c190e618da3 a1417667d8c794fc284b618d2858d3507689164955efea7c2cb73281e894e7a9 72405504383ebc5943fe701290f4c0fbb6049de67e7e27779d2093901431841b c7a061a55d5c98f9e13a5a216c1f5ec8718c4ae8a2f25d424bebc1330a808485 9eb6e459ecae51d0901508f9fac075f8298c05fc3bc9df485ff54e4bb298ec82 a5f73c06cd9fade0f4036d637f3900cef1292eb8b250dfcde0ebc7d75aba09c1 bb09ad1125a8dce750eb8f0f9765a974f7fda6d65c333e2eab6f1c22f77faaa2 b6675986161d0a314cf330d7a1c4acf56258cd80c7e054becfebd6c9c891e35a 8c7cff19a123bab4e1bb37d5ab87d5e9630cfc207af6e864f59fa3f9024887ae 1694364897851260a2ea1ec70804836542af3662e4b9b4057b94c5dbb0894375 f2592b98f62c8bf629be0a4cc641739266f93c61a73d29ceda0fbecc3f0ee92a 17b6664950d34aaada9be30eda8548b8b63d1cfcd9b323763f9f44b7fccfc616 ba9f7550bf1f009a71427ecc19cb4dc599b076a328ae253d28a3baa230d5151e 028218e57dd85d460697993e65d4e7466b7f4e1cb051fc3cf23e519b52678927 6c9900e120ffbfd06d8f6ce5eb0db6b0e1902c41d61ed77deb880549ef9db6d4 883b08a713cd5a59123e8e2215a1b74416dfb14fc9f36af4f715297d89f63a6f d0c21737f0db0059f7987cd1b3ccd959473a950fd25ec0503458427cca4abaa5 85fc90b04a8f2f4dc6a7b0c316a214718259e655271d63b1a3465c4a42048b18 88f6e490ba4822d143d9eadb01ebd36b1eb77c352c3352a1686ba72a64c89d7b fb56a9198211ca523184d26f71abb0d77f33d006f55b8d37a75891cc990b25a4 b69baac61b23619cafe1493926eb3c9a31edb8442c15f4a9dbf8d2fc6f4b0523 58856ce5eaff4a2c40967ec9463469372055865745d680babf4ddbca4502bb21 2318c8ffb50730ad15f1904d8bc7c1c2932c064cff5527cba27c260c2220bf2f 0afc3000a4febb548b4827ae60722722fab49d84d9b7ba8a105ebfd0ea8909a5 b06cc8576d5da524f50797960dd2741c514b1de30cbb36e213aff6560bc4905f d79cb189562092798e378f89cda5b0f55a3d6caf6f97c9049e9a7d5c3f884e06 96db555b2cecd3e190ef57ef91d9e77a082e5a1b715a51b9baad2777455dec6e 85f6e434c6d05a5e3325797e6ae02ad59d1c57fbd6c06baf729347a176728e6f 226546a06e0d79cfa5a6595f225702119740eb5f431d9afa88518ff35c28c5fa a130b7628c07814a2d9ea166b9412f45ca5bcea44df176a158c12bc3fd501826 9a75f1e9695a686dae10f37f5560e586a11d00df05779d1e9a0f5e108b3544ba b1adcf02a657f54bb50f133a3f7a80a8717c6db6d6ef3cd8a09642411cfe3af2 1f8686af6abeff17d761774196bd2344c733ada68c4f0b2f7d392f7739afb2d7 de2cef017e8379872aeb673880f107df7ea03db6ba9b91a1a2941139fcde023c e536f685910cdd36a62f6f39b872270ebc307311eb83b26440abab0e10c8b3a6 245c9b6be389745aa24d535ba93f85da8787f39ea7cea53805d386c28aa52160 f16ab603b0a3687935b84df224f90014c38ff8b02dc73c1cbf992c4d49ce5f64 c157428060bc7c3568600b4e60f884a5879d586b637894729240a26580d1047a 3ef78da4fd8843215604d40b6811b8fb0d1958bcd02231859369a5975c58258f 6b8f395ac6df0156af8243083a 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 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 56 /eight put dup 61 /equal 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 f4fab969bba2b44e47399fedf5caa1bcea216d7ba713d523dc2b1a89801ad4ff 2a26c7929cd309eeca94b69ec04056f1b1a21c014f5d1966b4921d36693771ef f6a3bb9981e9cd2d38070ca74c183820fd98593a70be07cfb97b2c96f8c8595e 60d4f7388fc866be90bbb73b0b284a6f11edd2ba662782c4a04b9b17ae7b48bd cce8463148c29a60e10a26084b263293cb04b7b1bfd7cda5528caf9126e362bb 989f6c307dd2fd2ff1b2c99d872cd44df0d73f3de2f1e0cc9193e40d9757509f 4953d748469802fa74c052385ae4a3f98cd5e05f26095f323747e6668a21e52d 70179cb6f89c7eb41523295588e704d736964199469d158414525bf3eb3d31aa a0d98aba1588898e2444647994a4b686ef685a72ace83475f3244dd4b6d630e7 6ce1bd8d8b4d9a45a18f2a28f79ad6662c90c9a81daded4afcd0828872d2a9bb 5584f70cc205a62d75ba07f30217dfd211ae85f9045adc280453ae046b6a0f23 cb6dc0810e5e48509c678fdf7f3353c4759aad2281f9ffd8f80ade96053459c6 ec9092e69d611e882e046c2694dbd1ac4a1de3bf62bab1e5c4197e6fb1da7c93 abf83327c2977dfcfa9ff50d4cceeaa3cdbbbfdd69259b4019107c8daf215efd 24af60db31a382b530dded686b435fa1bed7b4b5f241211b5d7831c4389f110d a92cf64867045c7966a97e61459b6c1089f1a8a41d6e5fc8358b9ca9d2f23594 9d07eb53b74b306586f72283ebc19f52da4873a6a237dc86436fc490f4865d51 00fd740ede9879b574c6bee9fc81ee24b73c95872a9d8ec55adaaa86cfabf68a 67fc290047f507d3f0cc8d53d0c8a0de0145d9871ac99536ed11da686afcfb38 d2a2800bc6ca27e5e8a359f81402a2182fcb6601c5026e73a691d04491cdf8c9 f4919c9865887d3c7e02a4e6ab10e720e2305830d94c2377d6387d303913f3ec 6a7e94cee186533395f2bbcca5101f3e120d3d36daee66620d013de5077147ab 6d9446677aa8ca598e57098d26cbaa1004ebf99ecbae2c67642011467e285766 a0e56cf30238e9c0d39a45e33cee4af8aaec5de0c38600149d95fdfc4ad9e9f3 a0ed8e2cea5cf720caa6468f77ebb2a48ff364321c7e047e48d44e4d8775b847 7bdf4d18cd48a38221768fba1102ac400933bbc388f531e33317fc5527972b96 03e8ba83b41fc36bee1c47c34d2550739488f1bc9d25f0ec10a90dc02db2f979 30558b7e878e776a818b39829accd08ad31a943e0d25dbe7c5a11127ff6bca14 66a9397aeac813fb1619bbdb970a13d7a4bf6923b778098b338a6a04d8878463 2475d885921b24ddbc5bdc80dc5bd50f3b5f3aad032004dd104357d4ec83d4d6 12d2e6d1d592e063b761b65193a07eb409f5b146873501f5583ba2a88d9fd11a 7966fd3fcd59740e6591ee56eeacca04ff9024f020bebe55e7d0f1021d45e4d6 f7ce70da4a6220e5f2e1fbb88445b70354b3641f59c9ec89d684a76e8e9f3b24 88fd08e3b2a161db00987ef1a5aeacd26bcb15a0f91110d423efbfba84677354 c346b4d65f3137e2a359d81b79d4c1dc9a232dc7a8e2568a72c8445e809af4ec 4e97f321f092079d7fd2262787c65b7be1c83f06a3c2e1f9aafd5e418eff2c04 8870b23fa4bcef0eabb174d3433286c886add08a6a369c88e298022c8e7e1dd7 21c456517128e73ae51fafdb62cfaa594a60e28e94908ccd0f0fc3a899fc054c 9fc1066bf2402b0a8f5497f0d13b84d3d49ecbe75a17a86e6a38358b32ea7ced f1e50fa0a9acee9e395f0c5474d7ab6f6811cd3df4db509bf7287c7ccc3f2555 36e63f9c36fa4e1bbe759b43826b69b9bdf8db5c2819d24d56622582732ad777 848f29a4a81bf65641f164e6f3ba70db456740694dd328b3a165b7eb26c9cc0b 0eb5f63a46e157e4b62bb0b86521d7e912610a5cf661b5dd39766843ecd604cf aded5de9f4f3aa0f1587333e66727afba0b7d432514b7578295193ce674ac970 6fc66490e0823989c7355200dcb79bf80f68c605e4269be057d4afddd2bb4612 8a021a56d962719ef38cbba23eb3d36aacdec0d9b2d1c4ee1ee1a182b8d9fd4c d78a6d9aa57effdd05dbc6f842d0b7c6accf5984645fa9f6c2c75276abe70ed9 5fbb17daa776d50a8d854b5d49852e21244264b544094c4cc3f727212b1b4680 6235484ae479 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 6 /plusminus put dup 14 /openbullet 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 69 /E 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 80 /P put dup 82 /R put dup 83 /S put dup 95 /logicalor put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put dup 112 /radical 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 b910bf03a681f0c4c84b3054a623f973471de50e2254753ada995f3e7ce33d8a 7d01b39f0e5a9675d7c24920d117dea104893f987390bb84e1ef04bca7664aa2 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444a2ebd5c4df2023fca3f237752f55e781a9b381102b24f419f8fe78f6d7b4b 822ef6fe4051cfc8110a2da4cd49a569a122be5e4ee6b25dbe400f1c02d0c04c fd6c0eeb5d3aefd74a0b4772653564495f3e3bbd61645ffc81fdf7ed8d82f8d5 09d9c3a0a1b67d99480e9c0a3c7a030ed4a1acdeb1576f98e58b6f39e3503362 f0f9b5387d63c679d4ac7a9680aacd70a17d26c0d20a3067b07416a9077a886f ccb598f9d246a0d5ffafd5f19f8c6c5aa5b139e07e259c03f3903acceb0bd2bb 814f367e782f840b2b94e5d232e1bdd5cc 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 10 /Omega put dup 12 /fi 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 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 58 /colon 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 76 /L put dup 78 /N put dup 80 /P put dup 83 /S put dup 84 /T put dup 87 /W put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 94 /circumflex put dup 95 /dotaccent 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 readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueXX 5000793 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306457c7e62c1d269f306bd3402e266de fc3b5e7d8a8d2f5bf0fe6ddd40d07391df4fad4a6018dce29a2b8f692b29f202 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 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 10 /Omega 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 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 51e80f0a88ab50fe8fdae4a5de93c1ae2fdca06150b37246140c0e87cb2325a6 0d2349162ae3ac93144eee1e665a1289105318fdfe86b6e76251cb25adc967d3 d0b97fe5e279e1161736ab22b4ca510b964342383a840defd38f96a7280e6ac1 34e48d740607ff2e7804164a16d47735864db847c97335e6d4215cb99911a1ec 015a3edaac1f28fedd56d2467130d07bae9416c15f0827d27c6c79f59054282a 418c12c157c91223a829947f47592f7cafd93ca182b25a73a9419127e3b12a9e 5167ac3963f2b019b338ac46d63880f94dda4b538835884d2a5538c85528d6bd 977f844d32b43b0e48caed5a4bdabcefe71695d69ca784d64d476eddddd04fb4 dd6833f4b71645d7f76cc408f4aab6476e7c58a62af2693158dee00e15147afb 1ad390821f90872f0ec9c322cc6823a6837c5d9c9506c8dc44144588283e02a6 db49cc3281594301a76a0573d9c95d986f108e988d3900ebb6edca7b8580d4dd 500c8d5ce83fa8d80cce38a5a3647ca9a87ce5e1558729e80e20e641cd833d27 f0f881ee92ed59dcc51d037d4d869189c996959b7a0d6c35f8612381133ebb4a 608587e67e0fdf52cfaaabee88cd286475815456a89f3885824c1d7ed2e6b7b8 aeb41c9df9493a8b534f8a8fd885ed7dbcf7450eba81b7737f134d25a87ab9d7 5149bc8fe8c74760e692a670f9ef5fd9c3675c117b93f516788de7f2b39b9fcc 0b97378b7281a470bff80c5a4fbe6875d7039cf7b137ab684cbc6ec90d036368 74bf006ac8e3d5f5f39e653959b5e242d1f21f8e4d3fd16b8f8bd5970e0059b8 ee0a30b1fc4ae04ffbbe21c2d022553be724a798bb2ff5def3255be08f1dc225 4c88e1b5b03138660723c1b7102a8b31395df987478779579a06a40b64b1393a ebafb6bbbafc6d0ce6ff41b71ce24ad0151e04bf3f0355eb751802e4202d7849 873652e3a03f37178e31b93e1f464b8d9d92ea501e24dee5c3bf19bb4f4a83e4 9e102c48e54bda9b985416954383d2cf5e229459eac9de818cbc8a4480b8cdc9 312d9b875d3e1db34ffbc94adf8aab1086d8bd3d8fa807c5bd6b3a98cf6c7c2c 422f36809f409561bc46908f1205f0d2519edb70cb905e8e0745fc777f418b43 27cce2dd107c85cbb72e258290bc6dc418289038fed9a3ddd8517b1ba4dec7f1 0e9622b4775551d99da549443490e4c16adaa4096b1b4cceb36ca7a1fa209d65 704e612367fe9f200813fe2e8e3f24d30c9cb7e46c24bd46d23dfb9c96c17525 c013529a0f3c7893d77492234be84022871210749df4baa7070a97d2c25a1f37 749d9b2810cf4f0493dbe296da53b7434eded84a4ec747e1a326f1f318e95d35 f764653cc578bb592909d3c94abcd0bfbaf22f3c0b57476c8ca0fa010d7c9dd7 d9a0752fdd1eb9f022808adc6510d71130608d6a8c8c679e94aeef8e7045ca8c a4602f968728717329073a5aa9c0c5b2839bb53379c33bbbdf7e2da664737244 088d60a58922b2c15fefac3156f855ad24d2ddc5ad60b4996924a6770c2423fa a490783110a58c95739d9184d7f101d4dab76687f4c51a39b42df7fae94561f5 6d740bd4bc7662ec6ce9d0e48f66d5edc677fe87b30d67ee6ea6f2d684606299 f959a2c3887c62334c16a61dce519d10007c80f420f0517e81827b7302feb379 6f0b8d5a48905666705c735bd0be513fa5b9e1e7fb655e3674ac1d1a713c525d 6b7aaa8e0ed907a527857fa9c25ef08deb741c21339c0a5e0f465e995fb25c10 5f9ab4f08cbd3196320434d0e30db15ba7963bb9a24324dc9214008c9d528440 8af7c9ce027c0ec21618ced55c5b13d7cf63843378697bd34e71501059145033 959cb72f21fc296bd506a453e540050ef6291d4777d6ac99fe8c5fea8d4be685 17f6fe7356603d47c7097c793a53b42520fb06b25918eb7e80116c78cdb38087 faabe176b782268a66bea308a5b47c72a789b6e9d2db4b44ac685e9111102119 7929aeed524b436ad0ac8873419a2d73ce46542d61cc0da69b81013411a237f3 523246000a3cd6805c156b46a37354d18efdf8dce218765a1433adbb863728be e1c4b73e5d2abdce64f0ff89039b3a378f81d6e9be86106ac4911daf27491bed d71ee7f246ebc99a1d2baf3a282e9cb2085f08f0e4a0ae99ccf429fa7475c66c d949215d46edc1a04c4463f306 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 0 /Gamma put dup 5 /Pi put dup 10 /Omega put dup 22 /macron put dup 33 /exclam 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 58 /colon put dup 61 /equal put dup 73 /I put dup 91 /bracketleft put dup 93 /bracketright put dup 94 /circumflex put dup 99 /c put dup 100 /d put dup 101 /e put dup 104 /h put dup 105 /i put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 116 /t put dup 122 /z put dup 126 /tilde put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueXX 5000791 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b094541f345ec367ae85ca9bd 24568a01d3b9f8095f7420e6c423c414b3dcce6da48dd1c89a56d078e0d0e2f2 62a13640a06d17e44ee3866c3471fb58fedf5a3b77294517651c16bdd7267d39 a54e7171752dbde63ac19bb4b3021ce95eb5fe67390b09ae4d9ed4d704a67443 f55dce17acd996c1f5e023c9e5a18cbeecc3097f23763acb86cdd7cd13381ae7 4e48495ec7fa520539d87f8a8dcb3c826275469b6800876a457e7d1e5be867c7 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta 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 27 /sigma put dup 28 /tau put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 36 /pi1 put dup 38 /sigma1 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 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 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 93 /sharp put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e 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 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 126 /vector put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueXX 5087383 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff 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90b4ecd8f1b447669e55837955c57346a07eb140c44c0e6711bad4a569a23c7f b8512914ab82fe435e812a67495901566a08519d1c6fc05cd3d5c582c749c202 66b4755f4e58f5151a4e37b78b1f040ef34d6720f1f0013609dd7f0a9c48fe58 8100aadaba97227e13e3355a8c2dc84602ff0b732dc4ef41d85f83e91629d9a4 86e01edffd82488b6bcd04bb2c5a1c99e98fd681eb2d4f0cab5f557c26cad10d d831605c5bd8f56377552ceeb611d2c1e2365c47e195bc16b4a26ea084ff808b c38fc5e024f8fd8649b1dc6d 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 34 /quotedblright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon 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 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 dup 123 /endash put dup 127 /dieresis put readonly def /FontBBox{-53 -251 1139 750}readonly def /UniqueXX 5000769 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSL12 %!PS-AdobeFont-1.1: CMSL12 1.0 %%CreationDate: 1991 Aug 20 16:40:41 % 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 (CMSL12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -9.46 def /isFixedPitch false def end readonly def /FontName /CMSL12 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 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 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 76 /L put dup 77 /M put dup 80 /P put dup 83 /S put dup 84 /T put dup 92 /quotedblleft put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put readonly def /FontBBox{-56 -251 1102 750}readonly def /UniqueXX 5000799 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd803271dc8bb1656323089e9e2bb50f2b95088 806eb7b8c434269c8a30d52dd3a6e85ebcf727aecf5607b54ac3942d00365274 9fef6277965f241846e12c5fc12f452046f9c7df8d1d31e228d6011de7485f19 4de778c725bd819dd7c91f2cfd0a9994b0ccdf95f200c0247f2c39f74a6ec77a 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3635d7974fdc1af0cf3878d82cb75ae6ce3dd2c440 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR17 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 58 /colon put dup 73 /I 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 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 121 /y put readonly def /FontBBox{-33 -250 945 749}readonly def /UniqueXX 5000795 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont TeXDict begin 39158280 55380996 1000 8000 8000 (PhDthesis_digregorio.dvi) @start /Fa 129[780 126[{}1 1594.02 /CMR12 rf /Fb 171[915 960 14[1296 68[{}3 1594.02 /CMMI12 rf /Fc 181[1035 41[1218 32[{}2 1912.83 /CMMI12 rf /Fd 156[359 99[{}1 553.48 /CMMI5 rf /Fe 182[501 22[597 49[692{}3 774.872 /CMSY7 rf /Ff 173[799 14[799 67[{}2 1106.96 /MSBM10 rf /Fg 140[418 7[471 1[313 7[480 18[701 18[263 22[736 36[{}7 774.872 /CMMI7 rf /Fh 150[444 444 23[993 80[{}3 885.568 /CMEX10 rf /Fi 173[480 3[480 78[{}2 664.176 /MSBM7 rf /Fj 173[523 1[1168 70[646 646 8[{}4 1106.96 /CMEX10 rf /Fk 252[861 3[{}1 1106.96 /MSAM10 rf /Fl 148[553 307 2[553 553 24[1329 26[738 28[861 19[307 861{}9 1106.96 /CMSY10 rf /Fm 136[799 11[584 5[492 3[553 97[{}4 1106.96 /CMSL10 rf /Fn 252[1033 3[{}1 1328.35 /MSAM10 rf /Fo 179[568 2[471 9[640 13[812 243 44[424 640 1[640{}8 664.176 /CMSY6 rf /Fp 132[734 667 815 1[1110 815 815 785 608 800 844 741 844 815 992 682 844 1[401 815 852 711 741 829 785 1[815 12[1044 812 2[985 1[1081 1314 908 2[539 1081 1[946 985 1101 1044 1[1081 7[734 734 734 734 734 734 734 734 734 734 1[424 502 45[{}49 1328.35 /CMCSC10 rf /Fq 134[543 633 792 1[634 400 5[664 2[576 1[381 638 1[542 515 4[461 7[917 1[646 6[844 2[753 940 1[487 920 870 6[588 4[307 307 21[917 1[516 3[660 1[484 5[667 22[{}28 1106.96 /CMMI10 rf /Fr 165[590 7[640 3[640 78[{}3 885.568 /MSBM7 rf /Fs 128[664 3[664 531 631 598 863 598 697 432 531 548 598 664 664 731 1063 332 598 399 399 664 598 399 598 664 598 598 664 3[399 664 399 3[1298 966 966 930 731 948 996 881 996 966 1165 815 999 1[501 966 1005 848 881 981 930 914 966 5[399 399 664 664 664 664 664 664 664 664 664 664 1[399 465 399 2[531 531 399 4[664 14[664 664 3[1146 1[731 797 11[{}77 1328.35 /CMTI12 rf /Ft 140[388 4[511 726 1[439 340 296 4[427 371 364 448 344 11[486 1[612 3[642 4[364 664 1[515 598 7[424 1[252 20[324 9[387 476 3[413 498 9[464 534 11[{}27 664.176 /CMMI6 rf /Fu 194[621 4[406 3[406 406 406 406 406 4[621 1[320 320 40[{}10 664.176 /CMR6 rf /Fv 149[590 15[886 5[886 1[959 1033 2[959 4[517 5[959 67[{}8 1328.35 /MSBM10 rf /Fw 143[784 1[470 2[470 261 2[470 470 6[627 11[570 803 1[658 1[770 1120 647 715 638 518 3[497 5[732 8[0 3[627 941 255 14[941 11[732 732 5[470 7[732 2[470 732 261 732{}31 885.568 /CMSY8 rf /Fx 134[584 584 799 584 615 430 437 434 584 615 553 615 922 307 584 338 307 615 553 338 492 615 492 615 553 1[307 553 307 553 307 3[1138 2[799 615 2[753 1[830 1[692 2[400 830 869 723 753 846 799 784 830 3[861 2[307 2[553 553 553 553 553 553 553 2[307 369 307 861 1[430 430 307 4[553 21[615 1[799 10[{}64 1106.96 /CMR10 rf /Fy 202[441 441 441 441 441 441 37[632 10[{}7 774.872 /CMR7 rf /Fz 140[1328 1328 1328 1328 886 886 4[627 627 2[738 1919 7[1476 1476 738 1697 1919 4[1107 627 1255 1402 12[1162 1162 1162 1162 1[1181 1[1181 1[1181 1[1181 6[1162 1162 6[1070 1070 4[775 775 1052 1052 4[996 996 4[701 701 978 978 793 793 2[738 443 627 627 775 775 4[553 553 609 609{}53 1328.35 /CMEX10 rf /FA 129[1033 11[1107 1[1107 1[664 2[664 369 517 517 664 664 6[886 886 1[886 3[948 1312 1[831 723 804 1126 1[924 1058 1090 1595 916 1012 900 723 1122 790 955 701 1025 699 872 1061 5[664 1[738 738 0 0 1[1181 886 886 1328 7[1328 1328 6[1328 4[1328 1033 1033 1033 1033 2[1033 1033 1[1033 1033 1[664 664 5[1033 1[1033 2[664 1033 369 1033{}64 1328.35 /CMSY10 rf /FB 129[470 3[418 5[366 1[367 1[523 470 523 784 3[261 523 2[418 523 418 4[470 261 1[261 17[339 11[732 2[261 470 470 470 470 470 470 470 470 470 470 4[732 1[366 366 6[261 10[470 11[680 4[705 4[588{}37 885.568 /CMR8 rf /FC 129[470 3[438 465 530 675 458 545 340 435 425 420 474 1[571 832 281 490 381 320 542 2[436 484 408 403 500 384 2[366 2[638 545 774 888 548 1[548 572 709 1[597 1[747 904 640 793 514 408 773 733 595 690 772 671 710 1[496 1[732 470 732 261 261 19[344 1[781 1[434 585 615 584 559 1[418 538 482 540 414 463 566 549 542 1[439 472 414 380 416 485 528 604 11[{}78 885.568 /CMMI8 rf /FD 129[650 3[604 634 739 930 629 740 470 613 586 577 653 625 776 1138 387 676 538 444 749 623 641 603 676 560 553 683 546 2[506 2[890 755 1079 1228 758 883 762 800 990 1030 838 994 1042 1258 885 1107 719 574 1078 1026 842 963 1080 931 988 975 687 1[1012 650 1012 361 361 13[361 1[1301 2[852 470 1[1075 1[609 810 846 815 770 1[565 739 671 738 568 643 783 759 749 1[606 642 568 525 576 675 734 827 11[{}88 1328.35 /CMMI12 rf /FE 132[747 664 789 789 1079 789 830 581 589 610 789 830 747 830 1245 415 789 1[415 830 747 457 682 830 664 830 726 3[415 772 415 1[1128 1[1544 2[1039 830 1115 1[1021 1[1168 1418 898 1170 1[557 1168 1175 939 1[1145 1079 1062 1128 6[415 747 747 747 747 747 747 747 747 747 747 1[415 1[415 2[581 581 5[772 21[830 872 1079 10[{}66 1328.35 /CMBX12 rf /FF 128[897 4[797 946 946 1295 946 996 697 707 732 946 996 897 996 1494 498 946 1[498 996 897 548 818 996 797 996 872 3[498 926 498 3[1852 2[1247 996 1338 1[1225 1[1402 1701 1077 1404 1[668 1402 1410 1127 1177 1374 1[1275 1354 7[897 897 897 897 897 897 897 897 897 897 1[498 598 10[926 22[1046 11[{}60 1594.02 /CMBX12 rf /FG 132[1549 1[1635 1[2238 1[1721 1205 1222 1265 1635 1721 1549 1721 2582 861 1635 1[861 1721 1549 947 1414 1721 1377 1721 1506 9[3200 3[1721 2[2117 1[2423 2939 1862 2[1154 2423 4[2238 2203 2340 6[861 4[1549 1549 1549 1549 1549 37[1807 11[{}41 2754.12 /CMBX12 rf /FH 128[1076 3[1076 956 1136 1[1554 1136 1196 837 849 879 1136 1196 1076 1196 1793 598 1136 1[598 1196 1076 658 982 1196 956 1196 1046 13[1196 1605 1616 1470 1616 1683 2041 1293 2[801 1683 1[1353 1412 1649 1[1530 1625 6[598 1076 1076 1076 1076 1076 1076 1076 1076 1076 1076 1[598 717 32[1196 1255 11[{}56 1912.83 /CMBX12 rf /FI 143[867 780 1[1301 434 2[434 7[780 31[1170 46[780 18[{}8 1594.02 /CMSL12 rf /FJ 128[650 650 2[650 578 686 686 939 686 723 506 513 506 686 723 650 723 1084 361 686 397 361 723 650 397 578 723 578 723 650 1[361 650 361 650 361 795 975 975 1336 975 975 939 723 957 1012 885 1012 975 1192 813 1011 668 469 975 1021 849 885 993 939 921 975 3[1012 1[361 361 650 650 650 650 650 650 650 650 650 650 1[361 434 361 1012 1[506 506 361 4[650 361 10[650 1[650 650 650 1[361 1[1084 723 723 759 939 1012 939 1012 939 975 867 903 1[1084 813{}101 1328.35 /CMR12 rf /FK 132[650 578 686 686 939 686 723 506 513 506 686 723 650 723 1084 361 686 397 361 723 650 397 578 723 578 723 650 2[650 1[650 7[939 723 2[885 2[1192 813 5[849 885 993 939 1[975 13[650 650 650 650 1[361 434 361 2[506 506 361 4[650 14[650 650 5[723 759 11[{}54 1328.35 /CMSL12 rf /FL 134[1335 1[1839 1335 1407 975 990 975 1335 1407 1263 1407 2126 688 2[688 1407 1263 1[1119 1407 1119 1407 1263 23[902 14[688 45[1375 12[{}24 2754.12 /CMR17 rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 8000dpi TeXDict begin %%BeginPaperSize: a4 a4 %%EndPaperSize %%EndSetup %%Page: 1 1 1 0 bop 5790 9661 a FL(In\014nite)832 b(dimensional)f(hamiltonian)g (systems)12210 12850 y(and)g(nonlinear)h(w)-72 b(a)g(v)g(e)831 b(equation:)5955 16039 y(p)72 b(erio)g(dic)831 b(orbits)h(with)f(long)g (minimal)g(p)72 b(erio)g(d)8250 49140 y FK(Candidate)29717 b(Sup)36 b(ervisors)8250 51690 y FJ(Dott.)579 b(Laura)433 b(Di)h(Gregorio)15961 b(Prof.)579 b(Luigi)433 b(Chierc)-36 b(hia)39928 54240 y(Dott.)578 b(Luca)434 b(Biasco)12346 68444 y FK(Thesis)g(submitted)e(for)i(the)f(degree)g(of)868 b(Do)36 b(ctor)435 b(Philosophiae)21031 70994 y FJ(Academic)f(Y)-108 b(ear)433 b(2004-2005)p eop %%Page: 2 2 2 1 bop eop %%Page: 3 3 3 2 bop 43457 21343 a FI(Al)520 b(mio)g(pap\022)-780 b(a)p eop %%Page: 4 4 4 3 bop eop %%Page: 5 5 5 4 bop 6299 4488 a FH(Ringraziamen)-60 b(ti)6299 7409 y FJ(Sem)-36 b(bra)483 b(incredibile)h(ma)g(questo)g(momen)-36 b(to)447 b(\022)-614 b(e)484 b(arriv)-72 b(ato...)730 b(Anzitutto)483 b(v)-36 b(oglio)486 b(ringra-)6299 9014 y(ziare)508 b(Luca.)801 b(T)-108 b(ra)508 b(p)36 b(o)g(co)508 b(div)-36 b(en)g(ter\022)-650 b(a)507 b(mio)h(marito)g(e)g(quindi)g(il) g(nostro)f(legame)i(v)-72 b(a)508 b(b)36 b(en)6299 10619 y(oltre)382 b(quan)-36 b(to)382 b(si)h(p)36 b(ossa)382 b(raccon)-36 b(tare)382 b(in)g(questa)g(pagina)h(\(e)e(p)36 b(oi)383 b(siamo)g(gelosissimi)g(delle)6299 12224 y(nostre)432 b(cose!\).)578 b(Quello)432 b(c)-36 b(he)432 b(desidero)g(far)g(sap)36 b(ere)432 b(a)h(quan)-36 b(ti)432 b(leggeranno)396 b(\022)-614 b(e)432 b(c)-36 b(he)432 b(non)f(so)6299 13829 y(se)445 b(sarei)g(riuscita)g(a)g(scriv)-36 b(ere)445 b(una)f(tesi)g(cos)-145 b(\022)-506 b(\020)447 b(b)36 b(ella)445 b(\(lo)g(dico)g(p)36 b(erc)-36 b(h)g(\023)-614 b(e)444 b(ho)g(ricevuto)h(tan)-36 b(ti)6299 15434 y(complimen)g(ti!\))960 b(senza)561 b(il)g(suo)g (aiuto.)960 b(La)-36 b(v)g(orare)561 b(insieme)525 b(\022)-614 b(e)561 b(stato)g(div)-36 b(erten)g(tissimo,)6299 17040 y(inclusi)299 b(gli)h(infuo)36 b(cati)300 b(litigi)g(p)36 b(er)299 b(l'inglese)h(e)f(le)h(notazioni.)534 b(T)-108 b(utto)298 b(ha)h(con)-36 b(tribuito)298 b(a)i(farci)6299 18645 y(innamorare)518 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FC(w)18923 46420 y FJ(\()p FA(B)40 b FJ(\))351 b FA(\002)g FD(T)23344 46619 y FC(w)24098 46420 y FJ(\()p FA(B)40 b FJ(\))508 b FA(\000)-221 b(!)507 b Fv(R)p FJ(,)536 b(where)514 b FD(T)35633 46619 y FC(w)36388 46420 y FJ(\()p FA(B)40 b FJ(\))515 b(is)g(the)f(tangen)-36 b(t)515 b(space)6299 48025 y(to)469 b FA(B)510 b FJ(in)469 b FD(w)36 b FJ(,)477 b(with)469 b(the)f(follo)-36 b(wing)471 b(prop)36 b(erties:)649 b(\(i\))469 b FD(d!)476 b FJ(=)429 b(0,)478 b(\(ii\))469 b FD(!)516 b FJ(is)470 b(non)e(degenerate)6299 49630 y(in)523 b(an)-36 b(y)524 b(p)36 b(oin)-36 b(t)523 b(of)h FA(B)40 b FJ(.)848 b(W)-108 b(e)524 b(note)f(that)g(if)h FA(B)564 b FJ(is)524 b(\014nite)e(dimensional,)547 b(then)522 b(it)h(has)h(ev)-36 b(en)6299 51236 y(dimension.)809 b(Giv)-36 b(en)510 b(a)g(hamiltonian)h(function)f FD(H)602 b FJ(:)721 b FA(B)541 b(!)500 b Fv(R)p FJ(,)530 b(w)-36 b(e)511 b(de\014ne)e(the)h(asso)36 b(ci-)6299 52841 y(ated)361 b(hamiltonian)g(v)-36 b(ector)361 b(\014eld)g FD(X)23924 53040 y 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27412 a(dt)19393 25424 y Fz(\000)20002 26500 y FD(p)20655 25952 y FB(2)21181 26500 y FJ(\()p FD(t)p FJ(\))294 b(+)h FD(q)24889 25952 y FB(2)25414 26500 y FJ(\()p FD(t)p FJ(\))26896 25424 y Fz(\001)27726 26500 y FD(dt)368 b FJ(=)30621 24692 y Fz(Z)31949 25043 y FC(T)31359 27700 y FB(0)32905 26500 y FJ(2)33555 25424 y Fz(\000)34164 26500 y FD(p)34817 25952 y FB(2)35342 26500 y FJ(\()p FD(t)p FJ(\))295 b(+)g FD(q)39051 25952 y FB(2)39576 26500 y FJ(\()p FD(t)p FJ(\))41058 25424 y Fz(\001)41887 26500 y FD(dt)221 b(;)6299 30428 y FJ(from)434 b(whic)-36 b(h)434 b(it)f(follo)-36 b(ws)436 b(that)21481 29352 y Fz(\000)22090 30428 y FD(p)p FJ(\()p FD(t)p FJ(\))p FD(;)221 b(q)48 b FJ(\()p FD(t)p FJ(\))26914 29352 y Fz(\001)27890 30428 y FA(\021)369 b FJ(0.)8250 33243 y(W)-108 b(e)607 b(ha)-36 b(v)g(e)607 b(just)f(said)h(that)f(the)g (study)h(of)g(the)f(oscillations)i(of)g(a)f(system)g(in)f(the)6299 34848 y(neigh)-36 b(b)36 b(orho)g(o)g(d)463 b(of)h(an)f(equilibrium)h (p)36 b(oin)-36 b(t)463 b(or)g(a)h(p)36 b(erio)g(dic)464 b(motion)f(usual)g(b)36 b(egins)464 b(with)6299 36453 y(linearization.)839 b(After)520 b(this,)541 b(the)520 b(main)g(prop)36 b(erties)519 b(of)i(the)e(oscillations)j(in)e(the)f (origi-)6299 38058 y(nal)457 b(system)g(can)g(frequen)-36 b(tly)457 b(b)36 b(e)456 b(determined)f(using)i(the)f(theory)h(of)g (normal)g(forms)g(of)6299 39663 y(P)-36 b(oincar)g(\023)-614 b(e-Birkho\013.)905 b(This)542 b(theory)g(is)h(an)f(analogue)h(of)g(p) 36 b(erturbation)541 b(theory)-108 b(.)904 b(Here)6299 41268 y(the)389 b(linearized)g(system)h(pla)-36 b(ys)390 b(the)e(role)i(of)g(the)f(unp)36 b(erturb)g(ed)386 b(system)k(with)f (resp)36 b(ect)389 b(to)6299 42873 y(the)433 b(original)i(one.)6299 44917 y(T)-108 b(o)395 b(\014x)f(ideas,)403 b(let)394 b(us)g(consider)g(a)h(nonlinear)f(p)36 b(erturbation)393 b(of)i(a)g(2)p FD(n)p FJ(-dimensional)g(in)-36 b(te-)6299 46522 y(grable)445 b(hamiltonian)g(system,)i(in)d(a)h(neigh)-36 b(b)36 b(orho)g(o)g(d)444 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FD(q)26319 57710 y FB(2)26271 58586 y(1)26844 58258 y FJ(\))g(+)g FD(:)221 b(:)g(:)297 b FJ(+)32214 57359 y(1)p 32214 57953 V 32214 59169 a(2)32997 58258 y FD(!)33807 58457 y FC(n)34433 58258 y FJ(\()p FD(p)35592 57710 y FB(2)35592 58586 y FC(n)36513 58258 y FJ(+)e FD(q)38445 57710 y FB(2)38397 58586 y FC(n)39023 58258 y FJ(\))221 b FD(:)6299 61650 y FJ(Cho)36 b(osing)495 b(co)36 b(ordinates)494 b FD(w)507 b FJ(=:)472 b(\()p FD(p;)221 b(q)48 b FJ(\))471 b(=)g(\()p FD(p)28274 61849 y FB(1)28799 61650 y FD(;)221 b(:)g(:)g(:)j(;)d(p)32365 61849 y FC(n)32992 61650 y FD(;)g(q)34151 61849 y FB(1)34677 61650 y FD(;)g(:)g(:)g(:)j(;)d(q)38167 61849 y FC(n)38793 61650 y FJ(\))494 b(in)g(the)f(phase)h(space)6299 63255 y Fv(R)7258 62773 y FC(n)8180 63255 y FA(\002)295 b Fv(R)10467 62773 y FC(n)11094 63255 y FJ(,)434 b(w)-36 b(e)434 b(can)f(write)h (the)f(Hamiltonian)i(in)e(the)g(follo)-36 b(wing)436 b(w)-36 b(a)g(y:)16304 67287 y FD(H)471 b FJ(=)19908 65626 y FC(n)19234 66025 y Fz(X)19376 68824 y 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b(though)e(this)e(is)h(not)g(neces-)0 74991 y(sary)-92 b(.)21397 78312 y FJ(34)p eop %%Page: 35 35 35 34 bop 6299 4488 a FJ(Diric)-36 b(hlet)517 b(b)36 b(oundary)516 b(conditions,)538 b(one)517 b(can)f(v)-36 b(erify)518 b(the)e(existence)h(of)h(a)f(m)-36 b(ulti{index)6299 6094 y FD(k)414 b FJ(=)369 b(\()p FD(:)221 b(:)g(:)i(;)e(k)12282 6293 y Fw(\000)p FB(1)13541 6094 y FD(;)g(k)14799 6293 y FB(0)15325 6094 y FD(;)g(k)16583 6293 y FB(1)17110 6094 y FD(;)g(:)g(:)g(:)s FJ(\))369 b FA(2)f Fv(Z)22235 5611 y Fw(1)23231 6094 y FJ(,)434 b(with)f FA(j)p FD(k)45 b FA(j)369 b FD(<)g FA(1)p FJ(,)434 b(suc)-36 b(h)433 b(that,)g(b)36 b(eing)434 b FD(!)417 b FJ(:=)368 b(\()p FD(!)46222 6293 y FB(1)46748 6094 y FD(;)221 b(:)g(:)g(:)r FJ(\),)22083 9302 y FA(h)p FD(!)48 b(;)221 b(k)45 b FA(i)368 b FJ(=)27027 8040 y Fz(X)27169 10846 y FC(j)51 b Fw(\025)p FB(1)29167 9302 y FD(!)29977 9501 y FC(j)30464 9302 y FD(k)31140 9501 y FC(j)31996 9302 y FJ(=)369 b(0)221 b FD(:)6299 13555 y FJ(In)495 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y(equation,)420 b(one)c(can)g(deal)g(with)f(p)36 b(erio)g(dic)416 b(b.c.,)k(adapting)c(the)f(de\014nition)g(of)h(the)f (phase)6299 74991 y(space)509 b(and)g(of)h(the)f(asso)36 b(ciated)510 b(Hamiltonian.)806 b(Let)508 b(us)h(rewrite)g(the)g (Hamiltonian)h(in)27696 78312 y(35)p eop %%Page: 36 36 36 35 bop 0 4488 a FJ(in\014nitely)338 b(man)-36 b(y)338 b(co)36 b(ordinates)338 b(in)g(the)f(Hilb)36 b(ert)338 b(space)g FD(`)27493 4006 y FB(2)28356 4488 y FJ(of)h(all)g(complex)f (v)-72 b(alued)338 b(square)0 6094 y(in)-36 b(tegrable)434 b(sequences)f FD(q)417 b FJ(:=)368 b(\()p FD(q)15755 6293 y FB(1)16281 6094 y FD(;)221 b(:)g(:)g(:)r FJ(\).)578 b(Mak)-36 b(e)434 b(the)f(ansatz)11324 9337 y FD(u)p FJ(\()p FD(t;)221 b(x)p FJ(\))369 b(=)f FA(S)100 b FD(q)417 b FJ(=)19894 8075 y Fz(X)20037 10881 y FC(j)51 b Fw(\025)p FB(1)22035 9337 y FD(q)22612 9536 y FC(j)23098 9337 y FJ(\()p FD(t)p FJ(\))p FD(\037)25395 9536 y FC(j)25881 9337 y FJ(\()p FD(x)p FJ(\))221 b FD(;)1523 b(j)444 b FA(\025)369 b 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33648 a(@)168 b FJ(\026)-746 b FD(q)21198 33847 y FC(j)22097 32737 y FD(;)1523 b(j)444 b FA(\025)369 b FJ(1)221 b FD(:)13174 b FJ(\(1.23\))0 36459 y(They)404 b(are)g(classical)h(hamiltonian)f(equations)g(of)g(motion)g(for)g(the)f (real)h(and)f(imaginary)0 38064 y(part)374 b(of)h FD(q)4758 38263 y FC(j)5614 38064 y FJ(=)368 b FD(x)7733 38263 y FC(j)8395 38064 y FJ(+)174 b(i)p FD(y)10576 38263 y FC(j)11063 38064 y FJ(.)558 b(In)375 b(order)f(to)g(pro)-36 b(v)g(e)374 b(that)g(the)g(origin)h(is)g(an)f(elliptic)h(equilibrium)0 39669 y(p)36 b(oin)-36 b(t)436 b(for)i(the)e(hamiltonian)i(system)f (\(1.22\),)i(one)d(can)h(decomp)36 b(ose)437 b FD(q)35205 39868 y FC(j)36129 39669 y FJ(in)g(its)g(real)g(and)0 41274 y(imaginary)e(part,)e(namely)h FD(q)14378 41473 y FC(j)15234 41274 y FJ(=)369 b FD(x)17354 41473 y FC(j)18136 41274 y FJ(+)295 b(i)p FD(y)20438 41473 y FC(j)20925 41274 y FJ(,)434 b(and)f(equation)h(\(1.23\))h(b)36 b(ecomes)11819 45100 y FD(@)12506 45299 y FC(t)13123 43227 y 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FD(:)14917 b FJ(\(2.31\))6299 32985 y(It)434 b(is)g(ob)-36 b(vious)434 b(b)-36 b(y)433 b(de\014nition)g(that,)g(for)i(an)-36 b(y)433 b(admissible)h(v)-72 b(ariation)435 b FD(\036)369 b FA(2)f FD(V)289 b FJ(,)18396 36281 y(0)369 b FA(\025)g FD(D)21897 36480 y FC(v)22442 36281 y FJ(\010\()49 b(\026)-699 b FD(v)48 b FJ(\)[)p FD(\036)p FJ(])369 b(=)28312 34473 y Fz(Z)29050 37480 y FB(\012)30006 36281 y FD(f)142 b FJ(\()49 b(\026)-699 b FD(v)342 b FJ(+)295 b FD(w)36 b FJ(\()49 b(\026)-699 b FD(v)47 b FJ(\)\))p FD(\036)433 b(:)8773 b FJ(\(2.32\))6299 40610 y(By)386 b(no)-36 b(w,)396 b(one)385 b(w)-36 b(an)g(ts)386 b(to)f(c)-36 b(ho)36 b(ose)386 b(a)g(suitable)f(admissible)h(v)-72 b(ariation)387 b(in)e(order)g(to)h(obtain)6299 42215 y(a)434 b FD(L)8268 41733 y Fw(1)9264 42215 y FJ({estimate)h(and)e(a)g FD(H)19918 41733 y FB(1)20444 42215 y FJ({estimate.)6299 43820 y(By)h(\(2.32\))h (one)e(has,)h(since)g FD(w)404 b FJ(=)369 b FD(O)36 b FJ(\()p FD(")p FJ(\))p FD(;)433 b FJ(that)22145 45382 y 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FA(j)p FD(q)48 b FJ(\()h(\026)-699 b FD(v)18750 24703 y FB(+)19537 24504 y FJ(\))p FA(j)294 b FJ(+)h FA(j)p FD(q)48 b FJ(\()h(\026)-699 b FD(v)24142 24703 y Fw(\000)24928 24504 y FJ(\))p FA(j)14968 b FJ(\(2.36\))0 28021 y(where)433 b(the)h(last)g(inequalit)-36 b(y)434 b(holds)g(b)36 b(ecause)433 b(of)h(\(2.34\).)580 b(So)6084 31600 y FD(\014)7113 29792 y Fz(Z)8712 31600 y FJ(\026)-699 b FD(v)47 b(\036)369 b FA(\024)11880 29674 y Fz(\014)11880 30471 y(\014)11880 31268 y(\014)11880 32065 y(\014)12544 29792 y(Z)13282 32799 y FB(\012)14238 31600 y FD(f)142 b FJ(\(0\))p FD(\036)17453 29674 y Fz(\014)17453 30471 y(\014)17453 31268 y(\014)17453 32065 y(\014)18265 31600 y FA(\024)369 b FD(C)20914 29792 y Fz(Z)21652 32799 y FB(\012)22609 31600 y FA(j)p FD(\036)p FA(j)f(\024)h FD(C)27135 29792 y Fz(Z)27873 32799 y FB(\012)28829 31600 y FA(j)p FD(q)48 b FJ(\()h(\026)-699 b FD(v)30958 31799 y FB(+)31744 31600 y FJ(\))p FA(j)295 b FJ(+)g FA(j)p FD(q)48 b FJ(\()h(\026)-699 b FD(v)36350 31799 y Fw(\000)37136 31600 y FJ(\))p FA(j)0 35179 y FJ(and)433 b(using)g(\(2.36\),)i(one)f (has,)g(if)g FA(j)p FD(")p FA(j)369 b(\024)g FD(")20036 35378 y FB(1)20561 35179 y FJ(\()p FD(R)11 b FJ(\),)19526 38638 y FD(M)508 b FA(\024)22827 37740 y FD(C)p 22827 38333 1026 54 v 22936 39550 a(\014)24207 38638 y(:)0 42230 y FJ(Cho)36 b(osing)435 b FD(M)508 b FJ(:=)369 b(\(1)p FD(=)p FJ(2\))221 b(sup)g FA(j)14969 42052 y FJ(^)14969 42230 y(\026)-699 b FD(v)48 b FA(j)433 b FJ(and)g(since)h (sup)220 b FA(j)49 b FJ(^)-699 b FD(v)48 b FA(j)369 b(\025)27613 41707 y FB(1)p 27613 41924 471 54 v 27613 42688 a(2)28438 42230 y FJ(sup)220 b FA(j)49 b FJ(\026)-699 b FD(v)48 b FA(j)p FJ(,)15716 45045 y FD(M)508 b FA(\031)370 b FJ(sup)220 b FA(j)49 b FJ(\026)-699 b FD(v)48 b FA(j)369 b FJ(=)f FA(k)49 b FJ(\026)-699 b FD(v)48 b FA(k)26233 45244 y FC(L)26873 44992 y Fo(1)28017 45045 y FD(:)0 47860 y FJ(No)-36 b(w)540 b(one)g(w)-36 b(an)g(ts)539 b(to)h(de\014ne)e(a)i(suitable)g(admissible)g(function)f FD(\036)g 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FD(=dx)48729 20331 y FB(2)49382 20813 y FJ(+)6299 22418 y FD(f)6940 22617 y FB(1)7466 22418 y FJ(\()p FD(x)p FJ(\),)434 b(w)-36 b(e)434 b(can)f(write)h(the)f(solutions)h(of)g(the)f (linearized)h(equation)g(of)h(\(2.4\),)f(that)f(is)21892 25156 y FD(v)22521 25355 y FC(tt)23551 25156 y FA(\000)295 b FD(v)25508 25355 y FC(xx)26918 25156 y FJ(+)f FD(f)28865 25355 y FB(1)29391 25156 y FJ(\()p FD(x)p FJ(\))p FD(v)417 b FJ(=)368 b(0)221 b FD(;)6299 27895 y FJ(in)434 b(the)f(form)18885 29500 y FD(v)48 b FJ(\()p FD(x;)221 b(t)p FJ(\))368 b(=)24114 28238 y Fz(X)24256 31044 y FC(j)51 b Fw(\025)p FB(1)26254 29500 y FD(r)26840 29699 y FC(j)27548 29500 y FJ(cos)q(\()p FD(!)30606 29699 y FC(j)31093 29500 y FD(t)294 b FJ(+)h FD(\030)33732 29699 y FC(j)34219 29500 y FJ(\))p FD( )35571 29699 y FC(j)36057 29500 y FJ(\()p FD(x)p FJ(\))6299 33018 y(whic)-36 b(h)434 b(are)f(parametrized)h(b)-36 b(y)433 b(the)g(amplitudes)g FD(r)31352 33217 y FC(j)32273 33018 y FJ(and)g(the)g(phases)g FD(\030)41743 33217 y 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FD(`)295 b FA(\000)g FD(!)23785 70847 y FC(j)24641 70648 y FA(6)p FJ(=)369 b(0)221 b FD(;)2823 b(`)369 b FA(\025)g FJ(1)p FD(;)221 b(j)445 b FA(6)p FJ(=)369 b(1)221 b FD(;)6299 73386 y FJ(then)463 b(there)h(exists)h(a)f(p)36 b(erio)g(dic)465 b(orbit)f(with)g(frequency)g(close)h(to)g FD(!)40098 73585 y FB(1)40623 73386 y FJ(.)670 b(The)464 b(result)g(con-)6299 74991 y(tained)409 b(in)g(this)g(pap)36 b(er)408 b(is)h(an)g(extension) h(of)f(the)g(Ly)-36 b(apuno)g(v)409 b(Cen)-36 b(ter)408 b(Theorem)h(to)g(some)27696 78312 y(53)p eop %%Page: 54 54 54 53 bop 0 4488 a FJ(partial)539 b(di\013eren)-36 b(tial)539 b(equations.)894 b(The)539 b(stated)f(theorem)h(is)g(not)f(really)i (new)f(but)f(the)0 6094 y(pro)36 b(of)373 b(is)g(so)h(simple)f(that)f (it)h(mak)-36 b(es)373 b(the)f(pap)36 b(er)373 b(v)-36 b(ery)373 b(in)-36 b(teresting.)558 b(Let)372 b(us)g(consider)h(the)0 7699 y(nonlinear)433 b(w)-36 b(a)g(v)g(e)435 b(equation)15811 10439 y FD(u)16551 10638 y FC(tt)17581 10439 y FA(\000)295 b FD(u)19649 10638 y FC(xx)21059 10439 y FJ(+)g FD(\026u)369 b FJ(=)g FD( )48 b FJ(\()p FD(u)p FJ(\))12486 b(\(2.42\))0 13179 y(where)519 b FD( )48 b FJ(\()p FD(u)p FJ(\))514 b(=)h FD( )9376 13378 y FB(0)9901 13179 y FD(u)10641 12697 y FC(r)11501 13179 y FJ(+)353 b FD( )13760 12697 y FB(\(1\))15017 13179 y FJ(\()p FD(u)p FJ(\),)541 b(with)519 b(some)h FD(r)551 b(>)515 b FJ(2,)541 b(satisfying)521 b(a)e(suitable)h(nonde-)0 14784 y(generacy)406 b(condition.)569 b(Moreo)-36 b(v)g(er)406 b FD( )18134 14302 y FB(\(1\))19796 14784 y FJ(admits)f(t)-36 b(w)g(o)406 b(Lipsc)-36 b(hitz)404 b(deriv)-72 b(ativ)-36 b(es,)412 b(suc)-36 b(h)405 b(that)0 16389 y FD( )894 15907 y FB(\(1\))2151 16389 y FJ(\(0\))369 b(=)5562 15313 y Fz(\000)6171 16389 y FD( )7065 15907 y FB(\(1\))8322 15313 y Fz(\001)8931 15611 y Fw(0)9242 16389 y FJ(\(0\))g(=)f(0)434 b(and)f(ful\014lls)h(the)f(inequalit)-36 b(y)7855 20026 y(sup)6031 21170 y Fw(j)p FC(x)p Fw(j)p FC(<)p FB(2)p FC(";)p Fw(j)p FC(y)32 b Fw(j)p FC(<)p FB(2)p 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b(quation)596 b(ab)-66 b(ove)597 b(p)-66 b(ossesses)600 b(a)e(lo)-66 b(c)g(al,)630 b(p)-66 b(ositive)596 b(me)-66 b(asur)g(e,)630 b FJ(2)p FD(b)p Fs({dimensional)0 27448 y(invariant)422 b(Cantor)i(manifold)f FA(E)16021 27647 y Fw(I)17080 27448 y Fs(given)f(by)i(a)h(Whitney)d (smo)-66 b(oth)424 b(emb)-66 b(e)g(dding)422 b FJ(\010)369 b(:)646 b FA(T)41960 27647 y Fw(I)42595 27448 y FJ([)p FA(C)77 b FJ(])0 29053 y FA(\000)-221 b(!)393 b(E)3234 29252 y Fw(I)3868 29053 y Fs(,)481 b(which)c(is)h(a)g(higher)f(or)-66 b(der)478 b(p)-66 b(erturb)g(ation)475 b(of)j(the)f(inclusion)g(map)h FJ(\010)38742 29252 y FB(0)39661 29053 y FJ(:)614 b FD(E)41599 29252 y Fw(I)42626 29053 y FD(,)-221 b FA(!)0 30658 y(P)109 b Fs(,)600 b(r)-66 b(estricte)g(d)571 b(to)j FA(T)10212 30857 y Fw(I)10846 30658 y FJ([)p FA(C)77 b FJ(])p Fs(.)924 b(Mor)-66 b(e)g(over,)599 b(the)573 b(Cantor)g(manifold)f FA(E)32914 30857 y Fw(I)34122 30658 y Fs(is)h(foliate)-66 b(d)572 b(by)h(r)-66 b(e)g(al)0 32263 y(analytic,)562 b(line)-66 b(arly)542 b(stable,)563 b FD(b)543 b Fs(dimensional)g (invariant)e(tori)i(c)-66 b(arrying)542 b(quasi{p)-66 b(erio)g(dic)0 33868 y(solutions.)0 36585 y FJ(The)383 b(pro)36 b(of)384 b(is)f(based)g(on)g(in\014nite)f(dimensional)h(KAM)g (theorem,)393 b(partial)384 b(normal)f(form)0 38190 y(and)527 b(scaling)h(skills,)553 b(follo)-36 b(wing)529 b([P96a)q(].)860 b(W)-108 b(e)527 b(note)h(that)e(this)h(result)g(can)h(b)36 b(e)527 b(applied)0 39795 y(also)435 b(to)e(the)g(nonlinear)h(b)36 b(eam)433 b(equation.)0 42734 y FE(P)-42 b(erio)42 b(dic)499 b(b)42 b(oundary)499 b(conditions)0 45672 y FA(~)520 b FE([Bou94)q(])g FJ(The)g(same)g(result)g(of)g(the)f(nonlinear)h(Sc) -36 b(hr\177)-650 b(odinger)519 b(equation,)542 b(see)520 b(pag.)0 47277 y(92)434 b(b)36 b(elo)-36 b(w,)435 b(but)d(with)i FD(g)48 b FJ(\()p FD(x)p FJ(\))368 b FA(6)p FJ(=)h(0.)0 50215 y FA(~)434 b FE([ChY00)q(])g FJ(Let)f(us)g(consider)h(the)f (equation)13732 53152 y FD(u)14472 53351 y FC(tt)15502 53152 y 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b(is)f(based)g(on)g(an)f(in\014nite)6299 36044 y(dimensional)434 b(KAM)g(theorem.)6299 39294 y FA(~)g FE([BePr05)q(])p FJ(:)578 b(Consider)434 b(the)f(completely)i (resonan)-36 b(t)433 b(forced)g(equation)21407 42417 y FD(u)22147 42616 y FC(tt)23178 42417 y FA(\000)295 b FD(u)25246 42616 y FC(xx)26656 42417 y FJ(+)g FD(f)142 b FJ(\()p FD(!)30062 42616 y FB(1)30587 42417 y FD(t;)221 b(u)p FJ(\))370 b(=)e(0)11785 b(\(3.11\))6299 45540 y(where)394 b(the)f(nonlinear)h(forcing)g(term)f FD(f)142 b FJ(\()p FD(!)27311 45739 y FB(1)27837 45540 y FD(t;)221 b(u)p FJ(\))394 b(is)g(2)p FD(\031)48 b FJ({p)36 b(erio)g(dic)394 b(in)g(time)f(and)h(v)-72 b(anishes)6299 47146 y(for)581 b FD(u)619 b FJ(=)g(0)581 b(with)f(a)h(zero)g(of)g(order)f(at)g(least)h (2.)1020 b(The)580 b(authors)g(deal)h(with)f(the)g(case)6299 48751 y FD(!)7109 48950 y FB(1)8004 48751 y FJ(=)369 b FD(n=m)g FA(2)f Fv(Q)434 b FJ(and)f FD(!)18378 48950 y FB(1)19273 48751 y FA(2)369 b Fv(R)295 b FA(n)g Fv(Q)p FJ(.)579 b(Assume)433 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FC(xx)15837 26662 y FJ(+)g FD(mu)g FJ(+)g FD(f)142 b FJ(\()p FA(j)p FD(u)p FA(j)23391 26114 y FB(2)23917 26662 y FJ(\))p FD(u)369 b FJ(=)g(0)221 b FD(;)1523 b(m)368 b FA(2)h Fv(R)7383 b FJ(\(3.12\))0 29835 y(with)532 b FD(f)673 b FJ(real)532 b(analytic)h(in)e(some)h(neigh)-36 b(b)36 b(orho)g(o)g(d)531 b(of)h(the)f(origin)h(in)f Fv(C)p FJ(.)873 b(Assume)531 b(that)0 31440 y FD(f)142 b FJ(\(0\))369 b(=)f(0)434 b(and)f FD(f)8590 30958 y Fw(0)8901 31440 y FJ(\(0\))369 b FA(6)p FJ(=)f(0)434 b(\(non)f(degeneracy)h (condition\).)0 34551 y FE(Theorem)386 b(3.2.7.)479 b Fs(F)-100 b(or)374 b(al)66 b(l)375 b FD(m)369 b FA(2)g Fv(R)p Fs(,)392 b(for)374 b(al)66 b(l)375 b FD(n)369 b FA(2)g Fv(N)374 b Fs(and)g(for)g FD(f)516 b Fs(r)-66 b(e)g(al)374 b(analytic)f(and)h(non-)0 36157 y(de)-66 b(gener)g(ate,)446 b(ther)-66 b(e)443 b(exists)g(a)i(ful)66 b(l)445 b(density)e(Cantor)h(manifold)f(of)h(r)-66 b(e)g(al)443 b(analytic)h(line)-66 b(arly)0 37762 y(stable,)464 b(diophantine)g FD(n)p 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1186 54 v 857 x FD(!)26846 68350 y FC(i)27576 65432 y Fz(Z)28904 65782 y FC(\031)28313 68439 y FB(0)29753 67240 y FD(f)142 b FJ(\()p FD(u)p FJ(\))p FD(\037)33103 67439 y FC(i)33699 67240 y FD(dx)221 b(;)11374 b FJ(\(4.10\))6299 70326 y(hence,)433 b(w)-36 b(e)434 b(ha)-36 b(v)g(e,)434 b(b)-36 b(y)434 b(\(4.9\))g(and)f(\(4.4\),)11888 73829 y FD(u)12628 74028 y FC(tt)14470 73829 y FJ(=)16589 72567 y Fz(X)16787 75373 y FC(i)p Fw(\025)p FB(1)18957 72931 y FJ(\177)-745 b FD(q)19439 73130 y FC(i)19814 72931 y FJ(\()p FD(t)p FJ(\))p 18862 73524 2434 54 v 18933 73883 a FA(p)p 20039 73883 1186 54 v 20039 74740 a FD(!)20849 74939 y FC(i)21650 73829 y FD(\037)22465 74028 y FC(i)22840 73829 y FJ(\()p FD(x)p FJ(\))27371 78312 y(103)p eop %%Page: 104 104 104 103 bop 8171 5086 a FJ(=)10290 3824 y Fz(X)10488 6630 y FC(i)p Fw(\025)p FB(1)13384 4188 y FJ(1)p 12563 4781 2293 54 v 12563 5140 a FA(p)p 13670 5140 1186 54 v 857 x FD(!)14480 6196 y FC(i)15210 3213 y Fz(\022)16187 5086 y FA(\000)p FD(!)18078 4538 y FB(2)18030 5415 y FC(i)18604 5086 y FD(q)19181 5285 y FC(i)19851 5086 y FA(\000)296 b FD(!)21990 5285 y FC(i)22498 4188 y FD(@)72 b(G)p 22498 4781 1785 54 v 22535 5997 a(@)g(q)23871 6196 y FC(i)24416 3213 y Fz(\023)25615 5086 y FD(\037)26430 5285 y FC(i)8171 9250 y FJ(=)1107 b FA(\000)11544 7988 y Fz(X)11742 10794 y FC(i)p Fw(\025)p FB(1)13817 8351 y FD(q)14394 8550 y FC(i)14770 8351 y FJ(\()p FD(t)p FJ(\))p 13817 8945 2434 54 v 13888 9304 a FA(p)p 14995 9304 1186 54 v 857 x FD(!)15805 10360 y FC(i)16827 9250 y FJ(\()o FD(\026)295 b FA(\000)g FD(@)20425 9449 y FC(xx)21540 9250 y FJ(\))221 b FD(\037)23082 9449 y FC(i)23458 9250 y FJ(\()p FD(x)p FJ(\))294 b FA(\000)26832 7988 y Fz(X)27030 10794 y FC(i)p Fw(\025)p FB(1)28972 9250 y FD(\037)29787 9449 y FC(i)30384 7442 y Fz(Z)31712 7793 y FC(\031)31122 10449 y FB(0)32561 9250 y FD(f)142 b FJ(\()p FD(u)p FJ(\))p FD(\037)35911 9449 y FC(i)36508 9250 y FD(dx)221 b(;)0 13465 y FJ(b)36 b(ecause)374 b FD(!)5611 12983 y FB(2)5563 13793 y FC(i)6511 13465 y FJ(are)g(the)g(eigen)-36 b(v)-72 b(alues)375 b(of)g(the)f(op)36 b(erator)375 b FD(\026)174 b FA(\000)g FD(@)29102 13664 y FC(xx)30216 13465 y FJ(.)559 b(Moreo)-36 b(v)g(er,)387 b(b)36 b(eing)374 b FD(\037)41418 13664 y FC(i)41793 13465 y FJ(,)387 b(for)0 15070 y FD(i)369 b FA(\025)g FJ(1,)434 b(a)g(complete)g(orthonormal)f(basis)h(for)g(the) f FD(L)25984 14588 y FB(2)26944 15070 y FJ(functions)g(on)g([0)p FD(;)221 b(\031)48 b FJ(],)435 b(w)-36 b(e)434 b(obtain)3840 18576 y FD(u)4580 18775 y FC(tt)5684 18576 y FJ(=)369 b FA(\000)p FJ(\()p FD(\026)294 b FA(\000)i FD(@)11697 18775 y FC(xx)12811 18576 y FJ(\))p FD(u)f FA(\000)15681 17314 y Fz(X)15879 20120 y FC(i)p Fw(\025)p FB(1)17821 18576 y FD(\037)18636 18775 y FC(i)19233 16768 y Fz(Z)20561 17119 y FC(\031)19971 19775 y FB(0)21410 18576 y FD(f)142 b FJ(\()p FD(u)p FJ(\))p FD(\037)24760 18775 y FC(i)25357 18576 y FD(dx)369 b FJ(=)f FA(\000)p FJ(\()p FD(\026)295 b FA(\000)g FD(@)33153 18775 y FC(xx)34267 18576 y FJ(\))p FD(u)h FA(\000)f FD(f)142 b FJ(\()p FD(u)p FJ(\))221 b FD(:)43061 22635 y Fn(\003)0 25509 y FJ(Let)530 b(us)g(note)h(that)f FD(\037)10911 25708 y FC(i)11286 25509 y FJ(,)555 b(for)531 b FD(i)j FA(\025)g FJ(1,)556 b(are)531 b(a)f(complete)h(orthonormal)g (basis)g(for)g(the)f FD(L)43569 25027 y FB(2)0 27114 y FJ(functions)572 b(on)f([0)p FD(;)221 b(\031)48 b FJ(],)608 b(but)571 b(not)g(for)i(all)f(analytic)h(function)f(on)f([0)p FD(;)221 b(\031)48 b FJ(].)995 b(T)-108 b(o)572 b(o)-36 b(v)g(ercome)0 28719 y(this)605 b(problem,)648 b(let)605 b(us)g(consider)g FD(`)18016 28237 y FB(2)18016 29065 y FC(b)19147 28719 y FJ(and)f FD(L)22732 28237 y FB(2)23258 28719 y FJ(,)649 b(resp)36 b(ectiv)-36 b(ely)-108 b(,)649 b(the)604 b(Hilb)36 b(ert)605 b(spaces)h(of)0 30324 y(all)599 b FD(bi)p FJ(-in\014nite,)638 b(square)598 b(in)-36 b(tegrable)598 b(sequences)g(with)f(complex)i(co)36 b(e\016cien)-36 b(ts)598 b(and)g(all)0 31929 y(square-in)-36 b(tegrable)581 b(complex)g(v)-72 b(alued)581 b(functions)g(on)g([)p FA(\000)p FD(\031)48 b(;)221 b(\031)48 b FJ(].)1020 b(T)-108 b(o)582 b(iden)-36 b(tify)581 b(the)f(t)-36 b(w)g(o)0 33534 y(spaces)434 b(w)-36 b(e)434 b(can)f(consider)g(the)h(in)-36 b(v)g(erse)433 b(discrete)g(F)-108 b(ourier)433 b(transform,)6371 36883 y FA(F)501 b FJ(:)591 b FD(`)9325 36334 y FB(2)9325 37211 y FC(b)10219 36883 y FA(\000)-221 b(!)369 b FD(L)13613 36334 y FB(2)14360 36883 y FD(;)4124 b(q)416 b FA(7\000)-221 b(!)22348 35807 y Fz(\002)22901 36883 y FA(F)132 b FD(q)24613 35807 y Fz(\003)25166 36883 y FJ(\()p FD(x)p FJ(\))368 b(:=)30106 35984 y(1)p 29160 36577 2543 54 v 29160 36843 a FA(p)p 30267 36843 1436 54 v 1099 x FJ(2)p FD(\031)32278 35621 y Fz(X)32469 38445 y FC(i)p Fw(2)p Fr(Z)34419 36883 y FD(q)34996 37082 y FC(i)35371 36883 y FD(e)35974 36334 y FB(i)p FC(ix)37362 36883 y FD(;)0 40892 y FJ(whic)-36 b(h)440 b(de\014nes)f(an)i(isometry)g(b)36 b(et)-36 b(w)g(een)439 b(the)h(t)-36 b(w)g(o)441 b(spaces.)598 b(Let)440 b FD(a)381 b FA(\025)f FJ(0)441 b(and)f FD(s)380 b FA(\025)g FJ(0.)599 b(The)0 42497 y(subspaces)433 b FD(`)6493 41867 y FC(a;s)6493 42867 y(b)8113 42497 y FA(\032)369 b FD(`)10061 42015 y FB(2)10061 42842 y FC(b)11021 42497 y FJ(con)-36 b(tain)433 b(all)i(bi-in\014nite)d(sequences)h(whose)h(norm)f(is)h(de\014ned)e(b) -36 b(y)14613 45597 y FA(k)p FD(q)48 b FA(k)16566 45049 y FB(2)16566 45925 y FC(a;s)18186 45597 y FJ(:=)19928 44335 y Fz(X)20119 47159 y FC(i)p Fw(2)p Fr(Z)22068 45597 y FA(j)p FD(q)23014 45796 y FC(i)23390 45597 y FA(j)23759 45049 y FB(2)24285 45597 y FA(j)p FD(i)p FA(j)25467 45049 y FB(2)p FC(s)25467 45925 y Fw(\003)26427 45597 y FD(e)27030 45049 y FB(2)p FC(a)p Fw(j)p FC(i)p Fw(j)29120 45597 y FD(;)0 49813 y FJ(where)489 b FA(j)p FD(i)p FA(j)4995 50012 y Fw(\003)5984 49813 y FJ(:=)463 b(max)10462 48737 y Fz(\010)11237 49813 y FA(j)p FD(i)p FA(j)p FD(;)221 b FJ(1)13651 48737 y Fz(\011)14427 49813 y FJ(.)744 b(In)489 b(this)g(w)-36 b(a)g(y)490 b(w)-36 b(e)489 b(obtain,)503 b(through)488 b(the)h(F)-108 b(ourier)488 b(trans-)0 51418 y(form)434 b FA(F)132 b FJ(,)434 b(the)f(subspaces)f FD(W)14548 50936 y FC(a;s)16169 51418 y FA(\032)369 b FD(L)18456 50936 y FB(2)19415 51418 y FJ(endo)-36 b(w)g(ed)433 b(with)h(the)f(norm)17133 54248 y FA(kF)132 b FD(q)48 b FA(k)20173 54447 y FC(a;s)21793 54248 y FJ(=)369 b FA(k)p FD(q)48 b FA(k)25127 54447 y FC(a;s)26600 54248 y FD(:)0 57078 y FJ(F)-108 b(or)510 b FD(a)499 b(>)g FJ(0,)530 b(the)509 b(subspaces)h FD(W)16390 56595 y FC(a;s)18151 57078 y FJ(consist)h(of)g(all)g(2)p FD(\031)48 b FJ(-p)36 b(erio)g(dic)509 b(functions)h(whic)-36 b(h)510 b(are)0 58683 y(analytic)462 b(and)f(b)36 b(ounded)459 b(in)i(the)g(complex)h(strip)e FA(j)p FJ(Im)p FD(z)59 b FA(j)416 b FD(<)g(a)461 b FJ(with)g(trace)g(functions)g(on)0 60288 y FA(j)p FJ(Im)p FD(z)59 b FA(j)369 b FJ(=)g FD(a)399 b FJ(b)36 b(elonging)401 b(to)e(the)g(standard)g(Sob)36 b(olev)400 b(space)g FD(H)30226 59806 y FC(s)30716 60288 y FJ(.)567 b(In)399 b(this)g(w)-36 b(a)g(y)-108 b(,)408 b(w)-36 b(e)400 b(obtain)0 61893 y(an)433 b(orthonormal)h(basis)g(for)g (all)h(analytic)f(functions)g(on)f([0)p FD(;)221 b(\031)48 b FJ(].)0 63498 y(The)388 b(follo)-36 b(wing)390 b(t)-36 b(w)g(o)389 b(results)f(w)-36 b(ere)388 b(pro)-36 b(v)g(ed)388 b(in)g([P96a)q(],)398 b(for)388 b(completeness)g(w)-36 b(e)389 b(giv)-36 b(e)389 b(here)0 65103 y(the)433 b(pro)36 b(ofs:)0 67726 y FE(Lemma)470 b(4.1.2.)535 b Fs(F)-100 b(or)443 b FD(a)369 b FA(\025)g FJ(0)442 b Fs(and)g FD(s)369 b(>)f FJ(1)p FD(=)p FJ(2)p Fs(,)447 b(the)442 b(sp)-66 b(ac)g(e)441 b FD(`)28941 67097 y FC(a;s)28941 68096 y(b)30635 67726 y Fs(is)g(a)i(Hilb)-66 b(ert)439 b(algebr)-66 b(a)442 b(with)0 69331 y(r)-66 b(esp)g(e)g(ct)463 b(to)i(c)-66 b(onvolution)463 b(of)i(the)g(se)-66 b(quenc)g(es,)463 b FJ(\()p FD(q)342 b FA(\003)296 b FD(p)p FJ(\))26143 69530 y FC(j)26998 69331 y FJ(:=)28740 68335 y Fz(P)30142 69719 y FC(k)30932 69331 y FD(q)31509 69530 y FC(j)51 b Fw(\000)p FC(k)33241 69331 y FD(p)33894 69530 y FC(k)34463 69331 y Fs(,)13460 72161 y FA(k)p FD(q)342 b FA(\003)296 b FD(p)221 b FA(k)17541 72360 y FC(a;s)19162 72161 y FA(\024)369 b FJ(const)221 b FA(k)p FD(q)48 b FA(k)25708 72360 y FC(a;s)27181 72161 y FA(k)p FD(p)221 b FA(k)29383 72360 y FC(a;s)0 74991 y Fs(with)465 b(a)g(c)-66 b(onstant)464 b(dep)-66 b(ending)463 b(only)h(on)h FD(s)p Fs(.)21072 78312 y FJ(104)p eop %%Page: 105 105 105 104 bop 6299 4488 a Fp(pr)-30 b(oof.)659 b FJ(Let)433 b FD(\015)14297 4687 y FC(j)51 b(k)15666 4488 y FJ(:=)17408 3412 y Fz(\000)18016 4488 y FA(j)p FD(j)370 b FA(\000)296 b FD(k)45 b FA(j)21712 4687 y Fw(\003)22238 4488 y FA(j)p FD(k)g FA(j)23697 4687 y Fw(\003)24223 4488 y FD(=)p FA(j)p FD(j)75 b FA(j)26224 4687 y Fw(\003)26749 3412 y Fz(\001)27358 4488 y FJ(.)579 b(By)434 b(the)f(Sc)-36 b(h)g(w)g(arz)433 b(inequalit)-36 b(y)-108 b(,)10385 6031 y Fz(\014)10385 6828 y(\014)10385 7625 y(\014)10385 8422 y(\014)10385 9219 y(\014)10828 7094 y(X)10921 9923 y FC(k)24 b Fw(2)p Fr(Z)12968 8356 y FD(x)13707 8555 y FC(k)14276 6031 y Fz(\014)14276 6828 y(\014)14276 7625 y(\014)14276 8422 y(\014)14276 9219 y(\014)14718 6382 y FB(2)15613 8356 y FJ(=)16994 6031 y Fz(\014)16994 6828 y(\014)16994 7625 y(\014)16994 8422 y(\014)16994 9219 y(\014)17437 7094 y(X)17530 9923 y FC(k)g Fw(2)p Fr(Z)19710 7320 y FD(\015)20457 6838 y FC(s)20385 7666 y(j)51 b(k)21385 7320 y FD(x)22124 7519 y FC(k)p 19710 8050 2984 54 v 20364 9267 a FD(\015)21111 8810 y FC(s)21039 9637 y(j)g(k)22826 6031 y Fz(\014)22826 6828 y(\014)22826 7625 y(\014)22826 8422 y(\014)22826 9219 y(\014)23268 6382 y FB(2)24163 8356 y FA(\024)25565 6084 y Fz( )26617 7094 y(X)26711 9923 y FC(k)24 b Fw(2)p Fr(Z)29419 7457 y FJ(1)p 28890 8050 1708 54 v 28890 9267 a FD(\015)29637 8810 y FB(2)p FC(s)29565 9637 y(j)51 b(k)30730 6084 y Fz(!)221 b( )33055 7094 y(X)33149 9923 y FC(k)24 b Fw(2)p Fr(Z)35195 8356 y FD(\015)35942 7808 y FB(2)p FC(s)35870 8684 y(j)51 b(k)36903 8356 y FA(j)p FD(x)38011 8555 y FC(k)38580 8356 y FA(j)38949 7808 y FB(2)39475 6084 y Fz(!)40969 8356 y FD(;)2823 b FA(8)221 b FD(j)297 b(:)6299 11892 y FJ(On)433 b(the)g(other)g(hand,)g(from)h(the)f(de\014nition,)17763 14004 y(1)p 17251 14597 1676 54 v 17251 15814 a FD(\015)17926 16013 y FC(j)51 b(k)19428 14903 y FA(\024)20963 14004 y(j)p FD(j)370 b FA(\000)295 b FD(k)45 b FA(j)24658 14203 y Fw(\003)25479 14004 y FJ(+)295 b FA(j)p FD(k)45 b FA(j)28245 14203 y Fw(\003)p 20963 14597 7809 54 v 21764 15814 a FA(j)p FD(j)370 b FA(\000)295 b FD(k)45 b FA(j)25459 16013 y Fw(\003)25985 15814 y FA(j)p FD(k)g FA(j)27444 16013 y Fw(\003)29272 14903 y FJ(=)32571 14004 y(1)p 30786 14597 4222 54 v 30786 15814 a FA(j)p FD(j)370 b FA(\000)295 b FD(k)45 b FA(j)34481 16013 y Fw(\003)35435 14903 y FJ(+)37542 14004 y(1)p 36875 14597 1985 54 v 36875 15814 a FA(j)p FD(k)g FA(j)38334 16013 y Fw(\003)39214 14903 y FD(;)6299 17966 y FJ(so)434 b(that)8576 19708 y Fz(X)8670 22537 y FC(k)24 b Fw(2)p Fr(Z)11378 20071 y FJ(1)p 10849 20664 1708 54 v 10849 21881 a FD(\015)11596 21424 y FB(2)p FC(s)11524 22251 y(j)51 b(k)13059 20970 y FA(\024)14461 19708 y Fz(X)14555 22537 y FC(k)24 b Fw(2)p Fr(Z)16601 19097 y Fz(\022)19497 20071 y FJ(1)p 17712 20664 4222 54 v 17712 21881 a FA(j)p FD(j)370 b FA(\000)295 b FD(k)45 b FA(j)21407 22080 y Fw(\003)22361 20970 y FJ(+)24468 20071 y(1)p 23800 20664 1985 54 v 23800 21881 a FA(j)p FD(k)g FA(j)25259 22080 y Fw(\003)25918 19097 y Fz(\023)26896 19394 y FB(2)p FC(s)28226 20970 y FA(\024)369 b FJ(4)30278 20422 y FC(s)30990 19708 y Fz(X)31084 22537 y FC(k)24 b Fw(2)p Fr(Z)34148 20071 y FJ(1)p 33263 20664 2420 54 v 33263 21881 a FA(j)p FD(k)45 b FA(j)34722 21497 y FB(2)p FC(s)34722 22210 y Fw(\003)36185 20970 y FJ(:=)369 b FD(C)38953 20422 y FB(2)39848 20970 y FD(<)f FA(1)221 b FD(;)2824 b FA(8)221 b FD(j)297 b(:)6299 24506 y FJ(In)434 b(conclusion,)20643 25095 y Fz(\014)20643 25892 y(\014)20643 26689 y(\014)20643 27486 y(\014)20643 28283 y(\014)21086 26158 y(X)21180 28987 y FC(k)24 b Fw(2)p Fr(Z)23226 27420 y FD(x)23965 27619 y FC(k)24534 25095 y Fz(\014)24534 25892 y(\014)24534 26689 y(\014)24534 27486 y(\014)24534 28283 y(\014)24977 25446 y FB(2)25872 27420 y FA(\024)369 b FD(C)28300 26871 y FB(2)29047 26158 y Fz(X)29141 28987 y FC(k)24 b Fw(2)p Fr(Z)31187 27420 y FD(\015)31934 26871 y FB(2)p FC(s)31862 27748 y(j)51 b(k)32895 27420 y FA(j)p FD(x)34003 27619 y FC(k)34572 27420 y FA(j)34941 26871 y FB(2)35688 27420 y FD(:)11021 b FJ(\(4.11\))6299 30679 y(Hence,)434 b(for)g(all)h FD(a)368 b FJ(=)h(0,)434 b(w)-36 b(e)434 b(obtain)10256 34340 y FA(k)p FD(q)343 b FA(\003)295 b FD(p)p FA(k)14116 33792 y FB(2)14116 34668 y FC(a;s)16486 34340 y FJ(=)18615 33078 y Fz(X)18750 35902 y FC(j)51 b Fw(2)p Fr(Z)20755 34340 y FA(j)p FD(j)75 b FA(j)22106 33792 y FB(2)p FC(s)22106 34668 y Fw(\003)23288 32015 y Fz(\014)23288 32812 y(\014)23288 33609 y(\014)23288 34406 y(\014)23288 35203 y(\014)23731 33078 y(X)23825 35907 y FC(k)24 b Fw(2)p Fr(Z)25871 34340 y FD(q)26448 34539 y FC(j)51 b Fw(\000)p FC(k)28180 34340 y FD(p)28833 34539 y FC(k)29402 32015 y Fz(\014)29402 32812 y(\014)29402 33609 y(\014)29402 34406 y(\014)29402 35203 y(\014)29844 32366 y FB(2)16475 38045 y FA(\024)1107 b FD(C)19641 37496 y FB(2)20388 36783 y Fz(X)20523 39607 y FC(j)51 b Fw(2)p Fr(Z)22528 38045 y FA(j)p FD(j)75 b FA(j)23879 37496 y FB(2)p FC(s)23879 38373 y Fw(\003)25061 36783 y Fz(X)25155 39612 y FC(k)24 b Fw(2)p Fr(Z)27201 38045 y FD(\015)27948 37496 y FB(2)p FC(s)27876 38373 y(j)51 b(k)28909 38045 y FA(j)p FD(q)29855 38244 y FC(j)g Fw(\000)p FC(k)31587 38045 y FD(p)32240 38244 y FC(k)32809 38045 y FA(j)33178 37496 y FB(2)16486 41749 y FJ(=)1117 b FD(C)19641 41201 y FB(2)20615 40487 y Fz(X)20388 43316 y FC(j)o(;k)24 b Fw(2)p Fr(Z)22981 41749 y FA(j)p FD(j)370 b FA(\000)296 b FD(k)45 b FA(j)26677 41201 y FB(2)p FC(s)26677 42078 y Fw(\003)27638 41749 y FA(j)p FD(q)28584 41948 y FC(j)51 b Fw(\000)p FC(k)30315 41749 y FA(j)30684 41201 y FB(2)31210 41749 y FA(j)p FD(k)45 b FA(j)32669 41201 y FB(2)p FC(s)32669 42078 y Fw(\003)33630 41749 y FA(j)p FD(p)34652 41948 y FC(k)35221 41749 y FA(j)35590 41201 y FB(2)36485 41749 y FJ(=)368 b FD(C)38891 41201 y FB(2)39417 41749 y FA(k)p FD(q)48 b FA(k)41370 41201 y FB(2)41370 42078 y FC(a;s)42621 41749 y FA(k)p FD(p)p FA(k)44602 41201 y FB(2)44602 42078 y FC(a;s)46075 41749 y FD(:)6299 45402 y FJ(The)434 b(case)g FD(a)369 b(>)f FJ(0)434 b(is)g(a)g(simple)g (v)-72 b(ariation)434 b(of)h(the)e(last)h(estimate.)49361 47007 y Fn(\003)6299 49221 y FE(Lemma)369 b(4.1.3.)467 b Fs(F)-100 b(or)361 b FD(a)369 b FA(\025)g FJ(0)361 b Fs(and)g FD(s)368 b(>)h FJ(0)p Fs(,)381 b(the)360 b(gr)-66 b(adient)359 b FD(G)35422 49420 y FC(q)36298 49221 y FJ(:=)38040 47746 y Fz(\020)39040 48698 y FC(@)52 b(G)p 38966 48915 1430 54 v 38966 49679 a(@)g(q)39934 49802 y Fu(1)40529 49221 y FD(;)41318 48698 y FC(@)g(G)p 41244 48915 V 41244 49679 a(@)g(q)42212 49802 y Fu(2)42806 49221 y FD(;)221 b(:)g(:)g(:)44915 47746 y Fz(\021)46069 49221 y Fs(is)361 b(a)g(r)-66 b(e)g(al)6299 51247 y(analytic)464 b(map)h(fr)-66 b(om)464 b(a)h(neighb)-66 b(orho)g(o)g(d)463 b(of)i(the)f(origin)g(in)g FD(`)35621 50765 y FC(a;s)37337 51247 y Fs(into)g FD(`)40573 50765 y FC(a;s)p FB(+1)43027 51247 y Fs(.)597 b(Mor)-66 b(e)g(over)21637 53600 y FA(k)p FD(G)23327 53799 y FC(q)23834 53600 y FA(k)24498 53799 y FC(a;s)p FB(+1)27321 53600 y FJ(=)369 b FD(O)36 b FJ(\()p FA(k)p FD(q)31527 53051 y FB(3)32052 53600 y FA(k)32716 53799 y FC(a;s)33968 53600 y FJ(\))221 b FD(:)6299 55952 y Fp(pr)-30 b(oof.)641 b FJ(Let)383 b(b)36 b(e)383 b FD(q)417 b FA(2)368 b FD(`)18068 55470 y FC(a;s)19320 55952 y FJ(.)561 b(F)-108 b(rom)384 b(\(4.2\),)394 b(w)-36 b(e)384 b(ha)-36 b(v)g(e)384 b FD(u)369 b FJ(=)g FA(S)100 b FD(q)416 b FA(2)369 b FD(`)38037 55470 y FC(a;s)p FB(+1)p FC(=)p FB(2)41815 55952 y FJ(on)384 b([)p FA(\000)p FD(\031)48 b(;)221 b(\031)48 b FJ(])384 b(with)6299 57557 y FA(k)p FD(u)p FA(k)8367 57764 y FC(a;s)p FB(+1)p FC(=)p FB(2)12132 57557 y FA(\024)369 b(k)p FD(q)48 b FA(k)15487 57756 y FC(a;s)17164 57557 y FJ(for)427 b FD(a)369 b FA(\025)g FJ(0.)576 b(By)427 b(the)f(algebra)h(prop)36 b(ert)-36 b(y)426 b(and)f(the)h(analit)-36 b(yticit)g(y)428 b(of)6299 59162 y FD(f)142 b FJ(,)434 b(the)f(function)g FD(f)142 b FJ(\()p FD(u)p FJ(\))434 b(also)g(b)36 b(elongs)434 b(to)g FD(`)27592 58680 y FC(a;s)p FB(+1)p FC(=)p FB(2)31420 59162 y FJ(with)19505 61514 y FA(k)p FD(f)142 b FJ(\()p FD(u)p FJ(\))p FA(k)23368 61721 y FC(a;s)p FB(+1)p FC(=)p FB(2)27132 61514 y FA(\024)369 b FJ(const)221 b FA(k)p FD(u)p FA(k)33793 60966 y FB(3)33793 61843 y FC(a;s)p FB(+1)p FC(=)p FB(2)6299 63866 y FJ(in)567 b(a)g(su\016cien)-36 b(tly)567 b(small)h(neigh)-36 b(b)36 b(orho)g(o)g(d)566 b(of)i(the)e(origin.)979 b(F)-108 b(rom)566 b(\(4.2\),)601 b(w)-36 b(e)567 b(ha)-36 b(v)g(e)567 b(the)6299 65472 y(estimate)11587 67824 y FA(k)p FD(G)13277 68023 y FC(q)13784 67824 y FA(k)14448 68023 y FC(a;s)p FB(+1)17271 67824 y FA(\024)369 b(k)p FD(f)142 b FJ(\()p FD(u)p FJ(\))p FA(k)22536 68030 y FC(a;s)p FB(+1)p FC(=)p FB(2)26300 67824 y FA(\024)369 b FJ(const)222 b FA(k)p FD(u)p FA(k)32962 67275 y FB(3)32962 68152 y FC(a;s)p FB(+1)p FC(=)p FB(2)36726 67824 y FA(\024)369 b FJ(const)221 b FA(k)p FD(q)48 b FA(k)43272 67275 y FB(3)43272 68152 y FC(a;s)44745 67824 y FD(;)6299 70176 y FJ(that)575 b(is)g FD(G)11734 70375 y FC(q)12850 70176 y FA(2)609 b FD(`)14891 69694 y FC(a;s)p FB(+1)17345 70176 y FJ(.)1002 b(The)574 b(regularit)-36 b(y)576 b(of)f FD(G)30237 70375 y FC(q)31319 70176 y FJ(follo)-36 b(ws)577 b(from)e(the)g(regularit)-36 b(y)575 b(of)h(its)6299 71781 y(comp)36 b(onen)-36 b(ts)581 b(and)h(its)g(lo)36 b(cal)583 b(b)36 b(oundedness,)617 b(as)582 b(it)g(is)g(pro)-36 b(v)g(ed)582 b(in)f(the)h(follo)-36 b(wing)583 b(t)-36 b(w)g(o)6299 73386 y(Lemmata,)434 b(see)g([PT].)49361 74991 y Fn(\003)27371 78312 y FJ(105)p eop %%Page: 106 106 106 105 bop 0 4488 a FE(Lemma)623 b(4.1.4.)609 b Fs(L)-66 b(et)563 b(b)-66 b(e)563 b FD(E)642 b Fs(and)564 b FD(F)745 b Fs(Banach)564 b(c)-66 b(omplex)563 b(sp)-66 b(ac)g(es.)894 b(L)-66 b(et)563 b(b)-66 b(e)563 b FD(U)703 b Fs(an)564 b(op)-66 b(en)0 6094 y(subset)465 b(of)g FD(E)78 b Fs(.)597 b(L)-66 b(et)464 b(b)-66 b(e)464 b FD(f)511 b FJ(:)369 b FD(U)508 b FA(\000)-221 b(!)369 b FD(H)103 b Fs(.)597 b(Then)464 b(ar)-66 b(e)464 b(e)-66 b(quivalent:)1140 8923 y(\(i\))651 b FD(f)607 b Fs(is)464 b(analytic)g(on)h FD(U)139 b Fs(,)465 b(namely)f(it)g(is)h(c)-66 b(ontinuously)464 b(di\013er)-66 b(entiable)462 b(on)j FD(U)139 b Fs(;)742 11753 y(\(ii\))650 b FD(f)655 b Fs(is)512 b(lo)-66 b(c)g(al)66 b(ly)513 b(b)-66 b(ounde)g(d)513 b(and)g(we)-66 b(akly)512 b(analytic)g(on)h FD(U)139 b Fs(,)525 b(namely)512 b(for)h(e)-66 b(ach)512 b FD(x)458 b FA(2)g FD(U)139 b Fs(,)3252 13358 y FD(h)368 b FA(2)h FD(E)543 b Fs(and)464 b FD(L)369 b FA(2)g FD(F)13185 12876 y Fw(\003)13710 13358 y Fs(,)465 b(the)f(function)18686 16362 y FD(z)428 b FA(\000)-221 b(!)369 b FD(Lf)142 b FJ(\()p FD(x)295 b FJ(+)g FD(z)59 b(h)p FJ(\))3252 19366 y Fs(is)445 b(analytic)g(in)g(some)h(neighb)-66 b(orho)g(o)g(d)444 b(of)i(the)f(origin)g(in)g Fv(C)h Fs(in)f(the)h(usual)h(sense)f(of)3252 20971 y(one)464 b(c)-66 b(omplex)464 b(variable;)343 23801 y(\(iii\))650 b FD(f)587 b Fs(is)446 b(in\014nitely)d(di\013er)-66 b(entiable)443 b(on)i FD(U)585 b Fs(and)445 b(is)h(r)-66 b(epr)g(esente)g(d)443 b(by)i(the)h(T)-100 b(aylor)446 b(series)3252 25406 y(in)464 b(a)h(neighb)-66 b(orho)g(o)g(d)463 b(of)i(e)-66 b(ach)464 b(p)-66 b(oint)464 b(in)g FD(U)139 b Fs(.)0 28236 y Fp(pr)-30 b(oof.)927 b FJ(\()p FD(i)p FJ(\))511 b(=)-221 b FA(\))511 b FJ(\()p FD(ii)p FJ(\))516 b(Supp)36 b(ose)516 b FD(f)660 b FJ(is)517 b(analytic.)831 b(Hence)517 b(it)g(is)h(a)f(di\013eren)-36 b(tiable)517 b(map)0 29841 y(and)397 b(it)i(implies)f(that)f(it)i(is)f(con)-36 b(tin)g(uous)397 b(and)g(lo)36 b(cally)400 b(b)36 b(ounded.)565 b(The)398 b(map)g FD(Lf)142 b FJ(\()p FD(x)222 b FJ(+)h FD(z)59 b(h)p FJ(\))0 31446 y(for)434 b(all)h FD(L)369 b FA(2)f FD(F)7325 30964 y Fw(\003)8284 31446 y FJ(is)434 b(con)-36 b(tin)g(uously)433 b(di\013eren)-36 b(tiable)433 b(in)h FD(z)59 b FJ(.)578 b(Hence,)434 b FD(f)576 b FJ(is)434 b(w)-36 b(eakly)435 b(analytic.)0 33075 y(\()p FD(ii)p FJ(\))390 b(=)-221 b FA(\))391 b FJ(\()p FD(i)p FJ(\))446 b(Supp)36 b(ose)445 b FD(f)589 b FJ(is)447 b(w)-36 b(eakly)448 b(analytic)g(and)e(lo)36 b(cally)449 b(b)36 b(ounded.)616 b(The)447 b(idea)f(is)h(to)0 34680 y(sho)-36 b(w)500 b(that)f FD(f)641 b FJ(is)500 b(con)-36 b(tin)g(uous)499 b(in)g(order)g(to)h(use)f(Cauc)-36 b(h)g(y's)500 b(form)-36 b(ula.)777 b(Fix)500 b FD(x)481 b FA(2)g FD(U)639 b FJ(and)0 36285 y(c)-36 b(ho)36 b(ose)434 b FD(r)405 b(>)369 b FJ(0)434 b(small)g(suc)-36 b(h)433 b(that)14558 39289 y(sup)14204 40433 y Fw(k)p FC(h)p Fw(k\024)p FC(r)17091 39289 y FA(k)p FD(f)142 b FJ(\()p FD(x)296 b FJ(+)f FD(h)p FJ(\))p FA(k)368 b FJ(=)h FD(M)508 b(<)368 b FA(1)p FD(:)0 43219 y FJ(By)434 b(a)g(c)-36 b(hain)433 b(rule)h(in)f(the)g(Cauc)-36 b(h)g(y's)434 b(form)-36 b(ula,)8095 46882 y FD(Lf)142 b FJ(\()p FD(x)295 b FJ(+)g FD(z)59 b(h)p FJ(\))295 b FA(\000)g FD(Lf)142 b FJ(\()p FD(x)p FJ(\))369 b(=)22067 45984 y(1)p 21452 46577 1880 54 v 21452 47793 a(2)p FD(\031)48 b(i)23686 45074 y Fz(Z)24424 48081 y Fw(j)p FC(\030)35 b Fw(j)p FB(=1)27007 45984 y FD(Lf)142 b FJ(\()p FD(x)295 b FJ(+)f FD(\030)64 b(h)p FJ(\))p 27007 46577 6402 54 v 28242 47793 a(\()p FD(\030)359 b FA(\000)296 b FD(z)59 b FJ(\))33674 45984 y FD(z)p 33674 46577 664 54 v 33690 47793 a(\030)34692 46882 y(d\030)4835 b FJ(\(4.12\))0 50809 y(uniformly)446 b(for)h FA(j)p FD(z)59 b FA(j)390 b FD(<)f FJ(1,)450 b FA(k)p FD(h)p FA(k)390 b FD(<)f(r)36 b FJ(.)615 b(Hence,)449 b(denoting)c(b)-36 b(y)446 b FA(k)p FD(L)p FA(k)g FJ(the)f(norm)h(op)36 b(erator)446 b(of)0 52414 y FD(L)369 b FA(2)f FD(F)3531 51932 y Fw(\003)4057 52414 y FJ(,)434 b(for)g FA(j)p FD(z)59 b FA(j)369 b FD(<)g FJ(1)p FD(=)p FJ(2,)435 b(w)-36 b(e)434 b(obtain)f(from)h (\(4.12\))12634 55252 y FA(j)p FD(Lf)142 b FJ(\()p FD(x)295 b FJ(+)f FD(z)59 b(h)p FJ(\))295 b FA(\000)h FD(Lf)142 b FJ(\()p FD(x)p FJ(\))p FA(j)p 12634 55845 12213 54 v 18039 57062 a(j)p FD(z)59 b FA(j)25348 56151 y(\024)369 b FJ(2)p FD(M)139 b FA(k)p FD(L)p FA(k)221 b FD(;)0 59900 y FJ(for)526 b FA(j)p FD(z)59 b FA(j)526 b FD(<)f FJ(1,)549 b FA(k)p FD(h)p FA(k)526 b FD(<)f(r)36 b FJ(.)855 b(It)525 b(follo)-36 b(ws)528 b(the)d(con)-36 b(tin)g(uit)g(y)525 b(of)h FD(f)142 b FJ(.)855 b(No)-36 b(w)526 b(w)-36 b(e)526 b(can)f(apply)h(the)0 61505 y(Cauc)-36 b(h)g(y's)434 b(form)-36 b(ula)11965 65168 y FD(f)142 b FJ(\()p FD(x)295 b FJ(+)g FD(z)59 b(h)p FJ(\))369 b(=)20010 64269 y(1)p 19396 64862 1880 54 v 19396 66079 a(2)p FD(\031)48 b(i)21629 63360 y Fz(Z)22367 66367 y Fw(j)p FC(\030)35 b Fw(j)p FB(=1)24950 64269 y FD(f)142 b FJ(\()p FD(x)295 b FJ(+)g FD(\030)64 b(h)p FJ(\))p 24950 64862 5517 54 v 26249 66079 a FD(\030)359 b FA(\000)296 b FD(z)30821 65168 y(d\030)0 69094 y FJ(for)458 b FA(j)p FD(z)59 b FA(j)410 b FD(<)g FJ(1)458 b(and)f FA(k)p FD(h)p FA(k)410 b FD(<)g(r)36 b FJ(.)650 b(It)458 b(follo)-36 b(ws)459 b(that)f FD(f)599 b FJ(has)458 b(directional)g(deriv)-72 b(ativ)-36 b(es)458 b(in)g(ev)-36 b(ery)0 70699 y(direction)433 b FD(h)p FJ(,)h(namely)11883 74394 y FD(\016)12459 74593 y FC(x)13044 74394 y FJ(\()p FD(h)p FJ(\))369 b(=)408 b(lim)16554 75191 y FC(z)37 b Fw(!)p FB(0)18802 73495 y FJ(1)p 18795 74088 664 54 v 18795 75305 a FD(z)20034 72521 y Fz(\024)20868 73495 y FD(f)142 b FJ(\()p FD(x)295 b FJ(+)g FD(z)59 b(h)p FJ(\))295 b FA(\000)g FD(f)142 b FJ(\()p FD(x)p FJ(\))p 20868 74088 9706 54 v 25389 75305 a FD(z)30706 72521 y Fz(\025)31850 74394 y FD(:)21072 78312 y FJ(106)p eop %%Page: 107 107 107 106 bop 6299 4488 a FJ(Since)8846 5880 y Fz(\015)8846 6677 y(\015)8846 7474 y(\015)8846 8271 y(\015)9724 6908 y FJ(1)p 9717 7501 664 54 v 9717 8718 a FD(z)10735 6730 y Fz(\002)11288 7806 y FD(f)142 b FJ(\()p FD(\030)360 b FJ(+)294 b FD(z)59 b(h)p FJ(\))296 b FA(\000)f FD(f)142 b FJ(\()p FD(\030)64 b FJ(\))20780 6730 y Fz(\003)21628 7806 y FA(\000)296 b FD(\016)23533 8005 y FC(\030)24036 7806 y FJ(\()p FD(h)p FJ(\))25797 5880 y Fz(\015)25797 6677 y(\015)25797 7474 y(\015)25797 8271 y(\015)26904 7806 y FJ(=)28284 5880 y Fz(\015)28284 6677 y(\015)28284 7474 y(\015)28284 8271 y(\015)29770 6908 y FJ(1)p 29155 7501 1880 54 v 29155 8718 a(2)p FD(\031)48 b(i)31389 5998 y Fz(Z)32127 9006 y Fw(j)p FC(\030)35 b Fw(j)p FB(=1)34710 6908 y FD(f)142 b FJ(\()p FD(x)295 b FJ(+)f FD(\030)64 b(h)p FJ(\))p 34710 7501 5517 54 v 34924 8718 a FD(\030)35556 8334 y FB(2)36082 8718 y FJ(\()p FD(\030)359 b FA(\000)295 b FD(z)59 b FJ(\))40580 7806 y FD(d\030)41888 5880 y Fz(\015)41888 6677 y(\015)41888 7474 y(\015)41888 8271 y(\015)42996 7806 y FA(\024)369 b FJ(2)p FD(M)139 b FA(j)p FD(z)59 b FA(j)6299 11615 y FJ(for)434 b FA(j)p FD(z)59 b FA(j)370 b FD(<)e FJ(1)p FD(=)p FJ(2,)435 b(the)e(limit)h(is)g (uniform)g(in)f FA(k)p FD(\030)360 b FA(\000)295 b FD(x)p FA(k)369 b FD(<)g(r)36 b(=)p FJ(2)435 b(and)e FA(k)p FD(h)p FA(k)369 b FD(<)f(r)36 b(=)p FJ(2,)435 b(th)-36 b(us)19286 15234 y FD(\016)19862 15433 y FC(x)20447 15234 y FJ(\()p FD(h)p FJ(\))369 b(=)24705 14335 y(1)p 24090 14928 1880 54 v 24090 16145 a(2)p FD(\031)48 b(i)26324 13426 y Fz(Z)27062 16433 y Fw(j)p FC(\030)35 b Fw(j)p FB(=1)29645 14335 y FD(f)142 b FJ(\()p FD(x)295 b FJ(+)g FD(\030)64 b(h)p FJ(\))p 29645 14928 5517 54 v 31824 16145 a FD(\030)32456 15761 y FB(2)35516 15234 y FD(d\030)285 b(:)6299 18969 y FJ(F)-108 b(rom)412 b(this)h(it)g(follo)-36 b(ws)414 b(that)f FD(f)554 b FJ(is)413 b(con)-36 b(tin)g(uously)413 b(di\013eren)-36 b(tiable)412 b(on)h FD(U)139 b FJ(,)417 b(hence)412 b(analytic.)49361 20574 y Fn(\003)6299 23245 y FE(Lemma)630 b(4.1.5.)612 b Fs(L)-66 b(et)569 b(b)-66 b(e)569 b FD(U)708 b Fs(an)569 b(op)-66 b(en)569 b(subset)h(of)f(a)h (Banach)g(c)-66 b(omplex)569 b(sp)-66 b(ac)g(e)569 b FD(E)647 b Fs(and)6299 24850 y FD(H)682 b Fs(a)580 b(Hilb)-66 b(ert)578 b(sp)-66 b(ac)g(e)579 b(with)g(orthonormal)h(b)-66 b(asis)579 b FD(e)31520 25049 y FC(n)32146 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40176 y(Ther)-66 b(e)459 b(exists)h(a)g(r)-66 b(e)g(al)459 b(analytic,)h(close)g(to)g(the)f (identity,)g(symple)-66 b(ctic)458 b(change)i(of)f(c)-66 b(o)g(or)g(di-)6299 41781 y(nates)465 b FJ(z)369 b(:=)g(\000\(z)14306 41980 y Fw(\003)14831 41781 y FJ(\))465 b Fs(de\014ne)-66 b(d)463 b(in)h FD(B)22767 41980 y FC(r)23643 41781 y FA(\032)369 b FD(`)25591 41299 y FC(a;s)27307 41781 y Fs(into)464 b FD(B)30985 41980 y FB(2)p FC(r)32331 41781 y FA(\032)369 b FD(`)34279 41299 y FC(a;s)35995 41781 y Fs(with)25201 44826 y FD(r)405 b FJ(:=)369 b FD(c)28494 45025 y FB(1)29020 43934 y FA(p)p 30127 43934 783 54 v 892 x FD(\026)221 b(;)15578 b FJ(\(4.16\))6299 47872 y Fs(verifying)19947 50425 y FA(k)p FJ(z)296 b FA(\000)f FJ(z)23391 50624 y Fw(\003)23917 50425 y FA(k)24581 50624 y FC(a;s)p FB(+1)27404 50425 y FJ(=)369 b FD(O)30036 48552 y Fz(\022)31147 49406 y FA(k)p FJ(z)32389 49605 y Fw(\003)32915 49406 y FA(k)33579 48924 y FB(3)33579 49734 y FC(a;s)p 31147 50119 3684 54 v 32598 51336 a 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(\(4.48\))e(clearly)h(app)36 b(ears)441 b(as)h(a)g(non)f(resonance)g (condition)g(b)36 b(et)-36 b(w)g(een)441 b(the)6299 74991 y(frequency)434 b(of)h(the)e(torus)g FD(!)481 b FJ(and)433 b(the)g(normal)h(ones)f FA(f)p FD(!)34080 75190 y FC(i)34456 74991 y FA(g)35120 75190 y FC(i)p Fw(2I)36645 74938 y Ft(c)37129 74991 y FJ(.)27371 78312 y(119)p eop %%Page: 120 120 120 119 bop 0 4488 a FF(4.2.2)1792 b(Small)599 b(divisors)g(estimate)0 6951 y FJ(In)568 b(order)h(to)g(estimate)f(the)h(quan)-36 b(tities)568 b(in)h(\(4.57\))h(w)-36 b(e)568 b(will)i(p)36 b(erform)569 b(the)f(expansion)0 8556 y FD(\034)148 b(!)1523 8755 y FC(i)2268 8556 y FJ(=)368 b FD(\034)4361 7405 y Fz(p)p 5690 7405 3355 54 v 5690 8556 a FD(i)6134 8173 y FB(2)6955 8556 y FJ(+)295 b FD(\026)433 b FJ(=)368 b FD(i\034)443 b FJ(+)13750 7973 y FC(\026\034)p 13750 8251 1086 54 v 13898 9015 a FB(2)p FC(i)15264 8556 y FJ(+)295 b FD(O)36 b FJ(\()p FD(\026)18890 8074 y FB(2)19415 8556 y FD(\034)148 b FJ(\))434 b(requiring)f(that)h 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b(it)569 b(is)g(not)g(ob)-36 b(vious)569 b(that)g(w)-36 b(e)569 b(can)g(mak)-36 b(e)0 20391 y FD(\026)783 19909 y FB(2)1308 20391 y FD(\034)592 b FJ(small)444 b(for)f(some)h(\014xed)f FD(\034)533 b FA(\025)386 b FJ(1)p FD(:)444 b FJ(Moreo)-36 b(v)g(er,)446 b(ev)-36 b(en)443 b(if)h(it)f(is)h(not)e(necessary)-108 b(,)446 b(w)-36 b(e)444 b(limit)0 21997 y(for)552 b(simplicit)-36 b(y)552 b(to)f(consider)g FD(\034)717 b FA(\030)569 b FD(\026)18712 21515 y Fw(\000)p FB(2)19969 21997 y FJ(,)581 b(namely)551 b FD(c)26104 22196 y FB(4)26630 21997 y FD(=)p FJ(2)p FD(\026)28713 21515 y FB(2)29808 21997 y FA(\024)569 b FD(\034)717 b FA(\024)569 b FD(c)34854 22196 y FB(4)35380 21997 y FD(=\026)36813 21515 y FB(2)37339 21997 y FJ(.)930 b(Again)552 b(for)0 23602 y(tec)-36 b(hnical)534 b(reasons,)560 b(w)-36 b(e)535 b(will)g(need)f(that)g FD(\034)682 b FJ(is)535 b(an)f FK(in)-36 b(teger)534 b FJ(and)g(that)g FD(\026\034)682 b FJ(is)535 b(far)f(a)-36 b(w)g(a)g(y)0 25207 y(from)434 b(ev)-36 b(en)434 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10058 y(@)20083 9205 y FJ(~)19772 9540 y FD(B)21048 6073 y Fz(0)21048 8411 y(B)21048 9208 y(B)21048 10058 y(@)23905 6650 y FJ(.)23905 7093 y(.)23905 7536 y(.)22897 8589 y FD(!)23707 8788 y FC(h)p 22897 9182 1408 54 v 23226 10399 a FD(h)24708 9488 y FJ(~)-699 b FD(\024)23905 11192 y FJ(.)23905 11635 y(.)23905 12078 y(.)25961 6073 y Fz(1)25961 8411 y(C)25961 9208 y(C)25961 10058 y(A)27123 6073 y(1)27123 8411 y(C)27123 9208 y(C)27123 10058 y(A)28286 12615 y FC(i)29030 9540 y FJ(=)30411 8278 y Fz(X)30496 11107 y FC(h)p Fw(2I)32684 8642 y FD(!)33494 8841 y FC(h)p 32684 9235 V 33013 10451 a FD(h)34495 9540 y FJ(~)g FD(\024)368 b FJ(=)418 b(~)-699 b FD(\024N)433 b FJ(+)295 b FD(O)36 b FJ(\()p FD(\026)p FJ(\))221 b FD(:)3188 b FJ(\(4.63\))6299 14906 y(No)-36 b(w)434 b(w)-36 b(e)434 b(need)f(the)g(follo)-36 b(wing)436 b(lemma)e(pro)-36 b(v)g(ed)433 b(on)h(page)f(122.)6299 17557 y FE(Lemma)447 b(4.2.4.)519 b Fs(L)-66 b(et)422 b FJ(0)369 b FD(<)g(\026)f(<)h FJ(1)p Fs(.)583 b(Then)423 b FA(8)221 b FD(h)369 b FA(2)f Fv(N)31948 17075 y FB(+)33159 17557 y Fs(ther)-66 b(e)422 b(exist)g FD(\016)39961 16878 y FB(\()p FC(k)24 b FB(\))39911 17927 y FC(h)41630 17557 y FD(>)369 b FJ(0)p FD(;)423 b(k)414 b FJ(=)369 b(1)p FD(;)221 b FJ(2)p FD(;)g FJ(3)p FD(;)6299 19163 y Fs(such)466 b(that)15309 22025 y FD(!)16119 22224 y FC(h)17824 22025 y FJ(=)1106 b FD(h)295 b FJ(+)g FD(\016)22919 21346 y FB(\(1\))22869 22395 y FC(h)24398 22025 y FD(;)10273 b(\016)35658 21346 y FB(\(1\))35608 22395 y FC(h)37285 22025 y FD(<)39106 21127 y(\026)p 38798 21720 1400 54 v 38798 22937 a FJ(2)p FD(h)47070 22025 y FJ(\(4.64\))17824 25331 y(=)1106 b FD(h)295 b FJ(+)22734 24432 y FD(\026)p 22426 25025 V 22426 26242 a FJ(2)p FD(h)24253 25331 y FA(\000)g FD(\016)26207 24652 y FB(\(2\))26157 25701 y FC(h)27686 25331 y FD(;)6930 b(\016)35603 24652 y FB(\(2\))35553 25701 y FC(h)37230 25331 y FD(<)39051 24432 y(\026)39834 23950 y FB(2)p 38743 25025 1925 54 v 38743 26242 a FJ(8)p FD(h)40142 25858 y FB(3)41022 25331 y FD(:)5687 b FJ(\(4.65\))6299 28628 y Fs(Mor)-66 b(e)g(over,)463 b(if)h FD(\034)517 b FA(2)369 b Fv(N)16970 28145 y FB(+)17757 28628 y Fs(,)465 b(ther)-66 b(e)464 b(exists)g FD(n)26188 28827 y FC(h)27155 28628 y FJ(:=)369 b FD(n)29673 28827 y FC(h)30271 28628 y FJ(\()p FD(\026;)221 b(\034)148 b FJ(\))369 b FA(2)g Fv(Z)464 b Fs(such)i(that)11070 30752 y Fz(\014)11070 31549 y(\014)11070 32346 y(\014)11513 32280 y FD(!)12323 32479 y FC(h)12921 32280 y FA(f)p FD(!)14395 32479 y FC(h)14992 32280 y FD(\034)148 b FA(g)296 b(\000)18126 31381 y FD(\026\034)p 18126 31974 1496 54 v 18549 33191 a FJ(2)20050 32280 y FA(\000)f FD(n)22154 32479 y FC(h)22752 30752 y Fz(\014)22752 31549 y(\014)22752 32346 y(\014)23564 32280 y FA(\024)25407 31381 y FD(\026)p 25099 31974 1400 54 v 25099 33191 a FJ(2)p FD(h)26926 32280 y FJ(+)28366 31381 y FD(\026)29149 30899 y FB(2)29674 31381 y FD(\034)p 28366 31974 2022 54 v 28414 33191 a FJ(8)p FD(h)29813 32807 y FB(2)30742 32280 y FD(;)5535 b FA(8)221 b FD(h)369 b FA(2)f Fv(N)40928 31731 y FB(+)41937 32280 y FD(:)4772 b FJ(\(4.66\))6299 35612 y(By)510 b(the)f(elemen)-36 b(tary)510 b(inequalit)-36 b(y)510 b(0)499 b FD(<)e FJ(1)347 b FA(\000)g FJ(\(1)g(+)g FD(x)p FJ(\))32707 35130 y Fw(\000)p FB(1)34462 35612 y FD(<)498 b(x;)510 b FA(8)221 b FD(x)499 b(>)e FJ(0)p FD(;)510 b FJ(w)-36 b(e)510 b(get,)529 b(using)6299 37217 y(\(4.64\),)14713 39984 y(0)369 b FD(<)17246 39086 y FJ(1)p 17246 39679 651 54 v 17349 40896 a FD(i)18324 39984 y FA(\000)20053 39086 y FJ(1)p 19785 39679 1186 54 v 19785 40896 a FD(!)20595 41095 y FC(i)21473 39984 y FJ(=)22986 39086 y(1)p 22986 39679 651 54 v 23089 40896 a FD(i)23991 37713 y Fz( )25042 39984 y FJ(1)296 b FA(\000)29739 39086 y FJ(1)p 27449 39679 5230 54 v 27449 41181 a(1)g(+)e FD(\016)30327 40503 y FB(\(1\))30277 41522 y FC(i)31585 41181 y FD(=i)32812 37713 y Fz(!)34232 39984 y FD(<)35746 39086 y(\016)36372 38407 y FB(\(1\))36322 39426 y FC(i)p 35746 39679 1884 54 v 36203 40896 a FD(i)36647 40512 y FB(2)38131 39984 y FD(<)40063 39086 y(\026)p 39645 39679 1620 54 v 39645 40896 a FJ(2)p FD(i)40739 40512 y FB(3)41619 39984 y FD(;)6299 43367 y FJ(namely)23438 44592 y(1)p 23170 45186 1186 54 v 23170 46402 a FD(!)23980 46601 y FC(i)24784 45491 y FA(\000)26245 44592 y FJ(1)p 26245 45186 651 54 v 26348 46402 a FD(i)27397 45491 y FJ(=)369 b FD(O)30029 44017 y Fz(\020)31049 44592 y FD(\026)p 30956 45186 970 54 v 30956 46402 a(i)31400 46018 y FB(3)32058 44017 y Fz(\021)33294 45491 y FD(:)13415 b FJ(\(4.67\))6299 48398 y(Since)433 b FA(jf)p FD(!)11538 48597 y FC(h)12136 48398 y FD(\034)148 b FA(gj)369 b(\024)h FJ(1)p FD(=)p FJ(2)p FD(;)434 b FJ(substituting)f(\(4.67\))h(in)g(\(4.62\),)g(w)-36 b(e)434 b(get)16324 52423 y(\()p FD(B)67 b(I)18459 52622 y FB(0)18984 52423 y FJ(\))19490 52622 y FC(i)20235 52423 y FJ(=)21983 51524 y(4)p 21748 52117 1120 54 v 21748 53334 a FD(di)23222 50151 y Fz( )24323 52423 y FJ(^)-700 b FD(\024)295 b FA(\000)26646 51161 y Fz(X)26731 53990 y FC(h)p Fw(2I)28786 52423 y FD(!)29596 52622 y FC(h)30193 52423 y FA(f)p FD(!)31667 52622 y FC(h)32265 52423 y FD(\034)148 b FA(g)33642 50151 y Fz(!)34989 52423 y FJ(+)295 b FD(O)37547 50948 y Fz(\020)38567 51524 y FD(\026)p 38474 52117 970 54 v 38474 53334 a(i)38918 52950 y FB(3)39576 50948 y Fz(\021)6299 56543 y FJ(hence,)433 b(b)-36 b(y)434 b(\(4.66\))g(and)f(\(4.63\))10900 60569 y(\()p FD(B)67 b(I)13035 60768 y FB(0)13561 60569 y FJ(\))14067 60768 y FC(i)14811 60569 y FJ(=)16559 59670 y(4)p 16324 60263 1120 54 v 16324 61480 a FD(di)17798 58297 y Fz( )18899 60569 y FJ(~)-699 b FD(\024N)433 b FA(\000)22402 59307 y Fz(X)22488 62136 y FC(h)p Fw(2I)24543 59094 y Fz(\020)25469 59670 y FD(\026\034)p 25469 60263 1496 54 v 25891 61480 a FJ(2)27392 60569 y(+)295 b FD(n)29475 60768 y FC(h)30073 59094 y Fz(\021)30867 58297 y(!)32213 60569 y FJ(+)g FD(O)34772 58696 y Fz(\022)35882 59670 y FD(\026)p 35882 60263 783 54 v 36051 61480 a(i)37093 60569 y FJ(+)38533 59670 y FD(\026)39316 59188 y FB(2)39841 59670 y FD(\034)p 38533 60263 2022 54 v 39322 61480 a(i)40687 58696 y Fz(\023)42108 60569 y FD(:)4601 b FJ(\(4.68\))6299 64689 y(Moreo)-36 b(v)g(er,)435 b(since)e(b)-36 b(y)434 b(\(4.65\))g(w)-36 b(e)434 b(get)18563 68341 y FD(\034)19276 67125 y Fz(p)p 20605 67125 3355 54 v 20605 68341 a FD(i)21049 67957 y FB(2)21870 68341 y FJ(+)295 b FD(\026)368 b FJ(=)h FD(\034)148 b(i)295 b FJ(+)28600 67443 y FD(\026\034)p 28600 68036 1496 54 v 28801 69252 a FJ(2)p FD(i)30524 68341 y FJ(+)g FD(O)33083 66468 y Fz(\022)34193 67443 y FD(\026)34976 66960 y FB(2)35502 67443 y FD(\034)p 34193 68036 2022 54 v 34719 69252 a(i)35163 68869 y FB(3)36348 66468 y Fz(\023)37768 68341 y FD(;)6299 71968 y FJ(using)434 b(\(4.68\),)g(w)-36 b(e)434 b(ha)-36 b(v)g(e)7776 74991 y FD(`)222 b FA(\000)f FD(\034)10511 73775 y Fz(p)p 11840 73775 3355 54 v 11840 74991 a FD(i)12284 74608 y FB(2)13105 74991 y FJ(+)294 b FD(\026)h FA(\000)16817 73915 y Fz(\000)17426 74991 y FD(B)67 b(I)19055 75190 y FB(0)19581 74991 y FJ(\()p FD(\034)148 b FJ(\))21306 73915 y Fz(\001)21915 75512 y FC(i)22659 74991 y FJ(=)27371 78312 y(121)p eop %%Page: 122 122 122 121 bop 4385 5485 a FJ(=)368 b FD(`)295 b FA(\000)g FD(\034)148 b(i)296 b FA(\000)10848 4586 y FD(\026\034)p 10848 5179 1496 54 v 11049 6396 a FJ(2)p FD(i)12772 5485 y FJ(+)14446 4586 y(4)p 14211 5179 1120 54 v 14211 6396 a FD(di)15685 3213 y Fz( )16737 4223 y(X)16822 7052 y FC(h)p Fw(2I)18877 4010 y Fz(\020)19803 4586 y FD(\026\034)p 19803 5179 1496 54 v 20226 6396 a FJ(2)21727 5485 y(+)f FD(n)23810 5684 y FC(h)24407 4010 y Fz(\021)25496 5485 y FA(\000)345 b FJ(~)-700 b FD(\024N)28754 3213 y Fz(!)30100 5485 y FJ(+)295 b FD(O)32659 3612 y Fz(\022)33769 4586 y FD(\026)p 33769 5179 783 54 v 33938 6396 a(i)34980 5485 y FJ(+)36419 4586 y FD(\026)37202 4104 y FB(2)37728 4586 y FD(\034)p 36419 5179 2022 54 v 37208 6396 a(i)38574 3612 y Fz(\023)39994 5485 y FD(:)416 b FJ(\(4.69\))0 9587 y(F)-108 b(rom)433 b(the)g(h)-36 b(yp)36 b(othesis)433 b FD(\034)518 b FA(2)368 b(T)15123 9786 y FC(\026)16179 9587 y 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FJ(\))295 b FA(\000)g FJ(4)49 b(~)-699 b FD(\024N)434 b FJ(+)294 b(4)26706 29127 y Fz(X)26792 31956 y FC(h)p Fw(2I)28847 30389 y FD(n)29623 30588 y FC(h)30221 28117 y Fz(!)31273 28064 y(\014)31273 28861 y(\014)31273 29658 y(\014)31273 30455 y(\014)31273 31252 y(\014)32085 30389 y FA(\025)33620 29490 y FJ(1)p 33620 30083 651 54 v 33620 31300 a(2)34624 30389 y FD(:)5786 b FJ(\(4.71\))0 34476 y(Since)12736 36912 y(2)13607 34640 y Fz( )14659 36912 y FD(id)p FJ(\()p FD(`)294 b FA(\000)i FD(\034)148 b(i)p FJ(\))295 b FA(\000)g FJ(4)49 b(~)-699 b FD(\024N)434 b FJ(+)294 b(4)26792 35650 y Fz(X)26878 38479 y FC(h)p Fw(2I)28933 36912 y FD(n)29709 37111 y FC(h)30307 34640 y Fz(!)0 40487 y FJ(is)434 b(an)f(ev)-36 b(en)434 b(in)-36 b(teger,)434 b(\(4.71\))g(follo)-36 b(ws)436 b(b)-36 b(y)433 b(h)-36 b(yp)36 b(othesis)434 b FD(\034)517 b FA(2)368 b(T)30041 40686 y FC(\026)30663 40487 y FJ(.)12037 b Fn(\003)0 43412 y Fp(pr)-30 b(oof)548 b(of)g(lemma)e(4.2.4.)1499 b FJ(Since)476 b FD(!)20551 43611 y FC(h)21589 43412 y FJ(=)440 b FD(h)23790 42364 y FA(p)p 24897 42364 2992 54 v 1048 x FJ(1)295 b(+)g FD(x)476 b FJ(with)g FD(x)441 b FJ(:=)g FD(\026=h)36545 42930 y FB(2)37070 43412 y FD(;)476 b FJ(0)441 b FD(<)g(x)f(<)h FJ(1)p FD(;)0 45018 y FJ(\(4.64\),)435 b(\(4.65\))f(directly)g(follo) -36 b(w)436 b(b)-36 b(y)433 b(the)g(elemen)-36 b(tary)434 b(inequalities)8947 48725 y(1)369 b FD(<)11347 47612 y FA(p)p 12454 47612 V 1113 x FJ(1)295 b(+)g FD(x)222 b(;)2823 b FA(\000)20017 47827 y FD(x)20756 47345 y FB(2)p 20017 48420 1265 54 v 20324 49637 a FJ(8)21784 48725 y FD(<)23164 47612 y FA(p)p 24271 47612 2992 54 v 1113 x FJ(1)296 b(+)e FD(x)i FA(\000)f FJ(1)h FA(\000)31293 47827 y FD(x)p 31293 48420 740 54 v 31337 49637 a FJ(2)32534 48725 y FD(<)368 b FJ(0)221 b FD(;)0 52057 y FJ(holding)434 b(for)g(an)-36 b(y)434 b(0)369 b FD(<)f(x)i(<)e FJ(1)p FD(:)434 b FJ(W)-108 b(e)434 b(no)-36 b(w)434 b(pro)-36 b(v)g(e)433 b(\(4.66\).)579 b(Being)434 b FA(jf)p FD(!)33629 52256 y FC(h)34227 52057 y FD(\034)148 b FA(gj)369 b(\024)h FJ(1,)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)4160 55284 y FA(j)p FD(!)5339 55483 y FC(h)5936 55284 y FA(f)p FD(!)7410 55483 y FC(h)8008 55284 y FD(\034)148 b FA(g)295 b(\000)h FD(h)p FA(f)p FD(!)13232 55483 y FC(h)13829 55284 y FD(\034)148 b FA(gj)369 b FJ(=)g FA(j)p FJ(\()p FD(!)19010 55483 y FC(h)19903 55284 y FA(\000)295 b FD(h)p FJ(\))p FA(f)p FD(!)23960 55483 y FC(h)24557 55284 y FD(\034)148 b FA(gj)369 b(\024)h(j)p FD(!)29254 55483 y FC(h)30146 55284 y FA(\000)296 b FD(h)p FA(j)368 b(\024)34805 54385 y FD(\026)p 34496 54978 1400 54 v 34496 56195 a FJ(2)p FD(h)36250 55284 y(;)4160 b FJ(\(4.72\))0 58616 y(where)433 b(in)h(the)f(last)h (inequalit)-36 b(y)435 b(w)-36 b(e)433 b(ha)-36 b(v)g(e)434 b(used)f(\(4.64\).)579 b(Moreo)-36 b(v)g(er)434 b(b)-36 b(y)434 b(\(4.65\))8232 61843 y FA(f)p FD(!)9706 62042 y FC(h)10303 61843 y FD(\034)148 b FA(g)370 b FJ(=)e FD(!)14240 62042 y FC(h)14838 61843 y FD(\034)443 b FA(\000)296 b FJ([)p FD(!)18346 62042 y FC(h)18943 61843 y FD(\034)148 b FJ(])370 b(=)f FD(h\034)443 b FJ(+)24964 60944 y FD(\026\034)p 24964 61537 1496 54 v 25012 62754 a FJ(2)p FD(h)26888 61843 y FA(\000)295 b FD(\016)28842 61164 y FB(\(2\))28792 62213 y FC(h)30100 61843 y FD(\034)443 b FA(\000)296 b FJ([)p FD(!)33608 62042 y FC(h)34205 61843 y FD(\034)148 b FJ(])221 b FD(;)0 65101 y FJ(from)434 b(whic)-36 b(h)433 b(w)-36 b(e)434 b(get)3984 67207 y Fz(\014)3984 68004 y(\014)3984 68801 y(\014)4427 68735 y FD(h)p FA(f)p FD(!)6650 68934 y FC(h)7248 68735 y FD(\034)148 b FA(g)295 b(\000)10381 67836 y FD(\026\034)p 10381 68429 V 10804 69646 a FJ(2)12305 68735 y FA(\000)g FD(n)14409 68934 y FC(h)15007 67207 y Fz(\014)15007 68004 y(\014)15007 68801 y(\014)15819 68735 y FA(\024)17354 67836 y FD(\026)18137 67354 y FB(2)18663 67836 y FD(\034)p 17354 68429 2022 54 v 17402 69646 a FJ(8)p FD(h)18801 69262 y FB(2)19730 68735 y FD(;)4124 b FJ(where)1300 b FD(n)29615 68934 y FC(h)30582 68735 y FJ(:=)369 b FD(h)33073 68186 y FB(2)33599 68735 y FD(\034)443 b FA(\000)295 b FD(h)p FJ([)p FD(!)37855 68934 y FC(h)38453 68735 y FD(\034)148 b FJ(])221 b FD(:)0 72066 y FJ(Then)433 b(the)g(pro)36 b(of)434 b(follo)-36 b(ws)436 b(b)-36 b(y)433 b(the)g(previous)h(inequalit)-36 b(y)435 b(and)e(\(4.72\).)8291 b Fn(\003)0 74991 y FJ(W)-108 b(e)434 b(no)-36 b(w)433 b(giv)-36 b(e)435 b(a)f(lo)-36 b(w)g(er)434 b(estimate)g(on)f(the)g (cardinalit)-36 b(y)435 b(of)f FA(T)30067 75190 y FC(\026)30689 74991 y FJ(.)21072 78312 y(122)p eop %%Page: 123 123 123 122 bop 6299 4488 a FE(Lemma)499 b(4.2.5.)555 b Fs(F)-100 b(or)465 b FD(\026)g Fs(smal)66 b(l)465 b(enough)24900 7075 y FD(])f FA(T)26593 7274 y FC(\026)27584 7075 y FA(\025)29556 6176 y FD(c)30116 6375 y FB(4)p 29119 6769 1959 54 v 29119 7986 a FJ(6)p FD(\026)30552 7602 y FB(2)31432 7075 y FD(:)6299 10025 y Fp(pr)-30 b(oof.)659 b FJ(W)-108 b(e)433 b(\014rst)g(claim)h(that)14580 12808 y FD(])15741 10935 y Fz(\032)16737 12808 y FD(\034)518 b FA(2)368 b Fv(N)221 b FD(;)1523 b FJ(s.t.)1301 b FD(\026\034)517 b FA(2)28300 10935 y Fz(\024)29001 12808 y FD(n)295 b FA(\000)31533 11910 y FJ(1)p 31533 12503 651 54 v 31533 13720 a(2)32317 12808 y FD(;)221 b(n)296 b FJ(+)35411 11910 y(1)p 35411 12503 V 35411 13720 a(2)36194 10935 y Fz(\025\033)38260 12808 y FA(\025)39662 10935 y Fz(\024)40562 11910 y FJ(1)p 40496 12503 783 54 v 40496 13720 a FD(\026)41411 10935 y Fz(\025)47070 12808 y FJ(\(4.73\))6299 15924 y(for)541 b(an)-36 b(y)541 b FD(n)g FJ(o)36 b(dd.)899 b(Indeed)539 b(if)j FD(\034)21864 16123 y FB(0)22940 15924 y FJ(:=)f(min)o FA(f)p FD(\034)700 b FA(2)550 b Fv(N)541 b FJ(s.t.)g FD(\026\034)699 b FA(\025)551 b FD(n)368 b FA(\000)h FJ(1)p FD(=)p FJ(2)p FA(g)p FJ(,)568 b(then)540 b FD(\026\034)48306 16123 y FB(0)49382 15924 y FD(<)6299 17529 y(\026)340 b FJ(+)f FD(n)h FA(\000)g FJ(1)p FD(=)p FJ(2)500 b(and)f(therefore)g FD(\026\034)23176 17728 y FB(0)24041 17529 y FJ(+)339 b FD(\026m)480 b FA(\024)g FD(n)340 b FJ(+)g(1)p FD(=)p FJ(2)500 b(for)f(an)-36 b(y)500 b(0)480 b FA(\024)h FD(m)f FA(\024)h(\000)p FJ(1)340 b(+)f(1)p FD(=\026)p FJ(,)6299 19135 y(pro)-36 b(ving)434 b(\(4.73\).)579 b(Since)11383 22250 y FD(])12544 20377 y Fz(\032)13974 22250 y FD(n)434 b FJ(o)36 b(dd)1300 b(s.t.)21965 21352 y FD(c)22525 21551 y FB(4)p 21791 21945 1433 54 v 21791 23162 a FJ(2)p FD(\026)23652 22250 y FJ(+)25091 21352 y(1)p 25091 21945 651 54 v 25091 23162 a(2)26244 22250 y FA(\024)369 b FD(n)g FA(\024)30326 21352 y FD(c)30886 21551 y FB(4)p 30326 21945 1086 54 v 30478 23162 a FD(\026)31840 22250 y FA(\000)33301 21352 y FJ(1)p 33301 21945 651 54 v 33301 23162 a(2)34518 20377 y Fz(\033)35883 22250 y FA(\025)37591 21352 y FD(c)38151 21551 y FB(4)p 37418 21945 1433 54 v 37418 23162 a FJ(4)p FD(\026)39279 22250 y FA(\000)295 b FJ(2)369 b FA(\025)43335 21352 y FD(c)43895 21551 y FB(4)p 43161 21945 V 43161 23162 a FJ(5)p FD(\026)44949 22250 y(;)6299 25293 y FJ(then)21845 27329 y FD(])433 b FA(T)23507 27528 y FC(\026)24498 27329 y FA(\025)26206 26431 y FD(c)26766 26630 y FB(4)p 26033 27024 V 26033 28241 a FJ(5)p FD(\026)27820 25456 y Fz(\024)28720 26431 y FJ(1)p 28654 27024 783 54 v 28654 28241 a FD(\026)29569 25456 y Fz(\025)30639 27329 y FA(\025)32611 26431 y FD(c)33171 26630 y FB(4)p 32174 27024 1959 54 v 32174 28241 a FJ(6)p FD(\026)33607 27857 y FB(2)34487 27329 y FD(;)6299 30095 y FJ(for)h FD(\026)f FJ(small)i(enough.)32047 b Fn(\003)6299 33860 y FF(4.2.3)1793 b(F)-149 b(unctional)598 b(setting)6299 36323 y FJ(Since)496 b(the)g(problem)f(is)i(hamiltonian,)512 b(an)-36 b(y)497 b FD(T)181 b FJ({p)36 b(erio)g(dic)497 b(solution)f(of)h(the)f(Hamilton's)6299 37928 y(equations)434 b(for)14456 37593 y Fz(e)14110 37928 y FA(H)447 b FJ(in)433 b(\(4.35\),)i(namely)6819 39118 y Fz(8)6819 40314 y(>)6819 40712 y(<)6819 43103 y(>)6819 43502 y(:)8856 40202 y FJ(_)8553 40538 y FD(I)1513 b FJ(=)1106 b FA(\000)p FD(\021)14481 40056 y FB(4)15007 40538 y FD(@)15694 40737 y FC(\036)16612 40202 y Fz(e)16308 40538 y FD(K)95 b FJ(\()p FD(I)104 b(;)221 b(\036;)301 b FJ(^)-730 b FD(z)60 b(;)21954 40187 y FJ(\026)21954 40538 y(^)-730 b FD(z)59 b FJ(\))8865 41964 y(_)8553 42315 y FD(\036)1317 b FJ(=)1106 b FD(!)343 b FJ(+)295 b FD(\021)15908 41833 y FB(2)16433 42315 y FD(AI)399 b FJ(+)295 b FD(\021)20378 41833 y FB(2)20903 42315 y FD(B)21958 41833 y FC(t)22354 42315 y FD(z)138 b FJ(\026)-729 b FD(z)355 b FJ(+)294 b FD(\021)25972 41833 y FB(4)26498 42315 y FD(@)27185 42514 y FC(I)28021 41979 y Fz(e)27717 42315 y FD(K)95 b FJ(\()p FD(I)104 b(;)221 b(\036;)300 b FJ(^)-729 b FD(z)60 b(;)33363 41964 y FJ(\026)33363 42315 y(^)-730 b FD(z)59 b FJ(\))8776 43741 y(_)8632 44091 y(^)-729 b FD(z)9157 44290 y FC(i)10640 44091 y FJ(=)1106 b(i\(\012)296 b(+)f FD(\021)16857 43609 y FB(2)17382 44091 y FD(B)67 b(I)104 b FJ(\))19621 44290 y FC(i)20297 44091 y FJ(^)-729 b FD(z)20822 44290 y FC(i)21493 44091 y FJ(+)295 b(i)p FD(\021)23851 43609 y FB(2)24376 44091 y FD(@)25118 44290 y FB(\026)-525 b FC(z)25501 44425 y Ft(i)26204 43755 y FJ(^)25907 44091 y FD(G)p FJ(\()79 b(^)-729 b FD(z)60 b(;)28764 43741 y FJ(\026)28764 44091 y(^)-729 b FD(z)59 b FJ(\))296 b(+)e(i)p FD(\021)32507 43609 y FB(4)33033 44091 y FD(@)33775 44290 y FB(\026)-525 b FC(z)34158 44425 y Ft(i)34868 43755 y Fz(e)34564 44091 y FD(K)95 b FJ(\()p FD(I)104 b(;)221 b(\036;)301 b FJ(^)-730 b FD(z)60 b(;)40210 43741 y FJ(\026)40210 44091 y(^)-730 b FD(z)59 b FJ(\))221 b FD(;)3373 b(i)369 b FA(2)f(I)48142 43609 y FC(c)48827 44091 y FD(:)47070 45804 y FJ(\(4.74\))6299 47409 y(is)434 b(a)g(critical)h(p)36 b(oin)-36 b(t)433 b(of)h(the)f(lagrangian)i(action)f(functional)13090 50850 y FD(S)77 b FJ(\()p FD(I)104 b(;)221 b(\036;)301 b FJ(^)-730 b FD(z)61 b FJ(\))368 b(=)20005 49042 y Fz(Z)21333 49392 y FC(T)20743 52049 y FB(0)22289 48578 y Fz( )23340 50850 y FD(I)399 b FA(\001)25290 50499 y FJ(_)24978 50850 y FD(\036)295 b FA(\000)g FJ(i)27971 49588 y Fz(X)27953 52407 y FC(i)p Fw(2I)29478 52155 y Ft(c)30128 50850 y FD(z)30732 51049 y FC(i)31331 50672 y FJ(_)31187 50850 y(\026)-729 b FD(z)31712 51049 y FC(i)32383 50850 y FA(\000)34057 50514 y Fz(e)33711 50850 y FA(H)13 b FJ(\()p FD(I)104 b(;)221 b(\036;)301 b FJ(^)-730 b FD(z)61 b(;)39290 50499 y FJ(\026)39290 50850 y(^)-729 b FD(z)59 b FJ(\))40380 48578 y Fz(!)41875 50850 y FD(dt)221 b(;)3467 b FJ(\(4.75\))6299 54583 y(in)434 b(the)f(space)g(of)i FD(T)181 b FJ({p)36 b(erio)g(dic,)434 b FA(P)22878 54782 y FC(a;s)24130 54583 y FJ({v)-72 b(alued)434 b(curv)-36 b(es)32844 53507 y Fz(\000)33453 54583 y FD(I)104 b FJ(\()p FD(t)p FJ(\))p FD(;)221 b(\036)p FJ(\()p FD(t)p FJ(\))p FD(;)300 b FJ(^)-729 b FD(z)59 b FJ(\()p FD(t)p FJ(\))41174 53507 y Fz(\001)41782 54583 y FD(:)8250 56453 y FJ(In)433 b(particular)g(w)-36 b(e)434 b(are)f(lo)36 b(oking)435 b(for)f(p)36 b(erio)g(dic)433 b(orbits)g(of)h(the)f(Hamiltonian)46715 56118 y Fz(e)46369 56453 y FA(H)446 b FJ(near)6299 58059 y(the)433 b(family)i FA(F)566 b FJ(de\014ned)432 b(in)h(\(4.47\),)i(namely)f(w)-36 b(e)434 b(seek)g(solutions)g(of)g(the)f(form)20297 59265 y Fz(8)20297 60460 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FJ(;)221 b FD(\036)22910 68852 y FB(0)23437 68653 y FJ(\))23943 67577 y Fz(\013)24939 68653 y FJ(=)368 b(0)0 71219 y(or)434 b(equiv)-72 b(alen)-36 b(tly)18069 72824 y FD(@)18756 73023 y FC(\036)19315 73146 y Fu(0)19832 72824 y FA(S)100 b FJ(\()p FD(\036)22012 73023 y FB(0)22537 72824 y FJ(\))369 b(=)f(0)221 b FD(:)14096 b FJ(\(4.107\))0 74991 y(Indeed)433 b(\(4.107\))i(is)e(a)h(corollary)i (of)e(\(4.35\),)h(\(4.82\))f(and)f(of)i(the)e(follo)-36 b(wing)21072 78312 y(130)p eop %%Page: 131 131 131 130 bop 6299 4488 a FE(Lemma)499 b(4.2.11.)555 b Fs(The)464 b(r)-66 b(e)g(duc)g(e)g(d)464 b(action)g(functional)f (satis\014es)i(the)f(fol)66 b(lowing)17053 7331 y FD(@)17740 7530 y FC(\036)18299 7653 y Fu(0)18815 7331 y FA(S)100 b FJ(\()p FD(\036)20995 7530 y FB(0)21520 7331 y FJ(\))369 b(=)g FA(\000)p FD(T)25752 6255 y Fz(\012)26379 7331 y FD(@)27066 7530 y FC(\036)28027 6996 y Fz(e)27681 7331 y FA(H)12 b FJ(\()p FD(I)29895 7530 y FC(\036)30454 7653 y Fu(0)30971 7331 y FD(;)221 b(\036)32323 7530 y FC(\036)32882 7653 y Fu(0)33399 7331 y FD(;)300 b FJ(^)-729 b FD(z)34585 7530 y FC(\036)35144 7653 y Fu(0)35662 7331 y FD(;)36323 6981 y FJ(\026)36323 7331 y(^)g FD(z)36848 7530 y FC(\036)37407 7653 y Fu(0)37925 7331 y FJ(\))38431 6255 y Fz(\013)39279 7331 y FD(:)6299 10174 y Fp(pr)-30 b(oof.)659 b FJ(W)-108 b(e)433 b(ha)-36 b(v)g(e)7595 13586 y FD(@)8282 13785 y FC(\036)8841 13908 y Fu(0)9357 13586 y FA(S)100 b FJ(\()p FD(\036)11537 13785 y FB(0)12063 13586 y FJ(\))1106 b(=)15794 11777 y Fz(Z)17122 12128 y FC(T)16532 14785 y FB(0)18077 12111 y Fz(h)18705 12510 y(\000)19626 13235 y FJ(_)19314 13586 y FD(\036)20084 13785 y FC(\036)20643 13908 y Fu(0)21454 13586 y FA(\000)296 b FD(@)23470 13785 y FC(I)24348 13250 y Fz(e)24002 13586 y FA(H)25137 12510 y Fz(\001)25745 13586 y FD(@)26432 13785 y FC(\036)26991 13908 y Fu(0)27508 13586 y FD(I)28082 13785 y FC(\036)28641 13908 y Fu(0)29453 13586 y FJ(+)e FD(I)31333 13785 y FC(\036)31892 13908 y Fu(0)32409 13586 y FD(@)33096 13785 y FC(\036)33655 13908 y Fu(0)34484 13235 y FJ(_)34172 13586 y FD(\036)34942 13785 y FC(\036)35501 13908 y Fu(0)36312 13586 y FA(\000)i FD(@)38328 13785 y FC(\036)39288 13250 y Fz(e)38942 13586 y FA(H)446 b FD(@)41197 13785 y FC(\036)41756 13908 y Fu(0)42273 13586 y FD(\036)43043 13785 y FC(\036)43602 13908 y Fu(0)18829 16755 y FA(\000)19862 15679 y Fz(\000)20471 16755 y FJ(i)21243 16577 y(_)21098 16755 y(\026)-916 b FD(w)21762 16954 y FC(\036)22321 17077 y Fu(0)23133 16755 y FJ(+)295 b FD(@)25182 16954 y FB(^)-525 b FC(z)26003 16419 y Fz(e)25657 16755 y FA(H)26791 15679 y Fz(\001)27400 16755 y FD(@)28087 16954 y FC(\036)28646 17077 y Fu(0)29163 16755 y FD(w)30093 16954 y FC(\036)30652 17077 y Fu(0)31464 16755 y FA(\000)295 b FJ(i)p FD(w)34083 16954 y FC(\036)34642 17077 y Fu(0)35159 16755 y FD(@)35846 16954 y FC(\036)36405 17077 y Fu(0)37332 16577 y FJ(_)37187 16755 y(\026)-916 b FD(w)37851 16954 y FC(\036)38410 17077 y Fu(0)39222 16755 y FA(\000)296 b FD(@)41293 16805 y FB(\026)41293 17039 y(^)-525 b FC(z)42114 16419 y Fz(e)41768 16755 y FA(H)446 b FD(@)44023 16954 y FC(\036)44582 17077 y Fu(0)45364 16755 y FJ(\026)-916 b FD(w)46028 16954 y FC(\036)46587 17077 y Fu(0)47104 15281 y Fz(i)47953 16755 y FD(dt)13675 20124 y FJ(=)1107 b FD(I)16368 20323 y FC(\036)16927 20446 y Fu(0)17443 20124 y FD(@)18130 20323 y FC(\036)18689 20446 y Fu(0)19206 20124 y FD(\036)19976 20323 y FC(\036)20535 20446 y Fu(0)21052 18596 y Fz(\014)21052 19393 y(\014)21052 20190 y(\014)21494 18947 y FC(T)21494 21043 y FB(0)22523 20124 y FA(\000)296 b FJ(i)p FD(w)25143 20323 y FC(\036)25702 20446 y Fu(0)26219 20124 y FD(@)26906 20323 y FC(\036)27465 20446 y Fu(0)28247 20124 y FJ(\026)-916 b FD(w)28911 20323 y FC(\036)29470 20446 y Fu(0)29987 18596 y Fz(\014)29987 19393 y(\014)29987 20190 y(\014)30430 18947 y FC(T)30430 21043 y FB(0)31459 20124 y FA(\000)32787 18316 y Fz(Z)34116 18667 y FC(T)33525 21323 y FB(0)35071 19048 y Fz(\012)35698 20124 y FD(@)36385 20323 y FC(\036)37345 19788 y Fz(e)36999 20124 y FA(H)38134 19048 y Fz(\013)38761 20124 y FD(@)39448 20323 y FC(\036)40007 20446 y Fu(0)40524 20124 y FD(\036)41294 20323 y FC(\036)41853 20446 y Fu(0)42803 20124 y FD(dt)221 b(;)6299 23668 y FJ(b)-36 b(y)494 b(an)f(in)-36 b(tegration)494 b(b)-36 b(y)494 b(parts)f(and)g(since)h FD(\020)28366 23867 y FC(\036)28925 23990 y Fu(0)29935 23668 y FJ(satis\014es)g(the)f(range)h(equation)g(\(4.100\).)6299 25475 y(Moreo)-36 b(v)g(er,)435 b(since)e FD(\020)16123 25674 y FC(\036)16682 25797 y Fu(0)17633 25475 y FJ(is)h(p)36 b(erio)g(dic)433 b(and)26456 24405 y Fz(R)27341 24755 y FC(T)27083 25936 y FB(0)28296 25475 y FD( )29142 25674 y FC(\036)29701 25797 y Fu(0)30587 25475 y FJ(=)368 b(0,)435 b(w)-36 b(e)433 b(get)10531 29067 y FD(I)11105 29266 y FC(\036)11664 29389 y Fu(0)12181 29067 y FD(@)12868 29266 y FC(\036)13427 29389 y Fu(0)13944 29067 y FD(\036)14714 29266 y FC(\036)15273 29389 y Fu(0)15789 27539 y Fz(\014)15789 28336 y(\014)15789 29133 y(\014)16232 27890 y FC(T)16232 29986 y FB(0)18073 29067 y FJ(=)1106 b FD(I)20765 29266 y FC(\036)21324 29389 y Fu(0)21841 29067 y FJ(\(0\))p FD(@)24190 29266 y FC(\036)24749 29389 y Fu(0)25265 27991 y Fz(\000)25874 29067 y FD(\036)26644 29266 y FC(\036)27203 29389 y Fu(0)27720 29067 y FJ(\()p FD(T)181 b FJ(\))295 b FA(\000)g FD(\036)32068 29266 y FC(\036)32627 29389 y Fu(0)33144 29067 y FJ(\(0\))34806 27991 y Fz(\001)35784 29067 y FJ(=)368 b FD(I)37738 29266 y FC(\036)38297 29389 y Fu(0)38814 29067 y FD(@)39501 29266 y FC(\036)40060 29389 y Fu(0)40680 29067 y FJ(~)-754 b FD(!)48 b(T)550 b FJ(=)368 b(0)221 b FD(;)10015 32211 y(w)10945 32410 y FC(\036)11504 32533 y Fu(0)12021 32211 y FD(@)12708 32410 y FC(\036)13267 32533 y Fu(0)14050 32211 y FJ(\026)-916 b FD(w)14714 32410 y FC(\036)15273 32533 y Fu(0)15789 30683 y Fz(\014)15789 31480 y(\014)15789 32277 y(\014)16232 31034 y FC(T)16232 33129 y FB(0)18073 32211 y FJ(=)1106 b FD(w)21121 32410 y FC(\036)21680 32533 y Fu(0)22197 32211 y FJ(\(0\))p FD(@)24546 32410 y FC(\036)25105 32533 y Fu(0)25622 31135 y Fz(\000)26496 32211 y FJ(\026)-916 b FD(w)27160 32410 y FC(\036)27719 32533 y Fu(0)28236 32211 y FJ(\()p FD(T)181 b FJ(\))295 b FA(\000)561 b FJ(\026)-916 b FD(w)32744 32410 y FC(\036)33303 32533 y Fu(0)33820 32211 y FJ(\(0\))35482 31135 y Fz(\001)36460 32211 y FJ(=)368 b(0)9988 33827 y Fz(Z)11317 34177 y FC(T)10726 36834 y FB(0)12272 35635 y FD(\036)13042 35834 y FC(\036)13601 35957 y Fu(0)14118 35635 y FJ(\()p FD(t)p FJ(\))221 b FD(dt)1106 b FJ(=)20191 33827 y Fz(Z)21520 34177 y FC(T)20929 36834 y FB(0)22475 34559 y Fz(\000)23084 35635 y FD(\036)23854 35834 y FB(0)24675 35635 y FJ(+)398 b(~)-754 b FD(!)48 b(t)27309 34559 y Fz(\001)28139 35635 y FD(dt)295 b FJ(+)30886 33827 y Fz(Z)32215 34177 y FC(T)31624 36834 y FB(0)33170 35635 y FD( )34016 35834 y FC(\036)34575 35957 y Fu(0)35091 35635 y FJ(\()p FD(t)p FJ(\))221 b FD(dt)369 b FJ(=)f FD(\036)40459 35834 y FB(0)40985 35635 y FD(T)476 b FJ(+)398 b(~)-753 b FD(!)44520 34736 y(T)45463 34254 y FB(2)p 44520 35329 1469 54 v 44930 36546 a FJ(2)46343 35635 y FD(:)6299 39105 y FJ(Finally)7060 42775 y FD(@)7747 42974 y FC(\036)8306 43097 y Fu(0)8823 42775 y FA(S)100 b FJ(\()p FD(\036)11003 42974 y FB(0)11528 42775 y FJ(\))369 b(=)f FA(\000)15037 40967 y Fz(Z)16366 41317 y FC(T)15776 43974 y FB(0)17322 41699 y Fz(\012)17949 42775 y FD(@)18636 42974 y FC(\036)19596 42439 y Fz(e)19250 42775 y FA(H)20385 41699 y Fz(\013)21012 42775 y FD(@)21699 42974 y FC(\036)22258 43097 y Fu(0)22775 42775 y FD(\036)23545 42974 y FC(\036)24104 43097 y Fu(0)25054 42775 y FD(dt)g FJ(=)h FA(\000)28982 41699 y Fz(\012)29609 42775 y FD(@)30296 42974 y FC(\036)31257 42439 y Fz(e)30911 42775 y FA(H)32045 41699 y Fz(\013)32673 42775 y FD(@)33360 42974 y FC(\036)33919 43097 y Fu(0)34656 40967 y Fz(Z)35985 41317 y FC(T)35394 43974 y FB(0)36940 42775 y FD(\036)37710 42974 y FC(\036)38269 43097 y Fu(0)38786 42775 y FJ(\()p FD(t)p FJ(\))221 b FD(dt)368 b FJ(=)h FA(\000)44417 41699 y Fz(\012)45044 42775 y FD(@)45731 42974 y FC(\036)46691 42439 y Fz(e)46345 42775 y FA(H)47480 41699 y Fz(\013)48107 42775 y FD(T)403 b(:)49361 46219 y Fn(\003)6299 50057 y FF(4.2.8)1793 b(Existence)6299 52520 y FJ(By)564 b(\(4.107\))h(ev)-36 b(ery)564 b(critical)h(p)36 b(oin)-36 b(t)563 b FD(\036)25332 52719 y FB(0)26448 52520 y FA(2)590 b Fv(T)28810 52038 y FC(N)30269 52520 y FJ(of)565 b(the)e(reduced)f(action)i(functional)g FA(S)6299 54125 y FJ(de\014ned)410 b(in)i(\(4.105\))h(solv)-36 b(es)413 b(the)e(\\reduced")g(k)-36 b(ernel)412 b(equation)g(\(4.101\)) h(and,)i(therefore,)6299 55730 y(the)492 b(curv)-36 b(e)12125 54654 y Fz(\000)12734 55730 y FD(I)13308 55929 y FC(\036)13867 56052 y Fu(0)14383 55730 y FJ(\()p FD(t)p FJ(\))p FD(;)221 b(\036)17217 55929 y FC(\036)17776 56052 y Fu(0)18293 55730 y FJ(\()p FD(t)p FJ(\))p FD(;)300 b FJ(^)-729 b FD(z)20961 55929 y FC(\036)21520 56052 y Fu(0)22037 55730 y FJ(\()p FD(t)p FJ(\))23519 54654 y Fz(\001)24620 55730 y FJ(de\014ned)490 b(in)i(\(4.106\))i(is)f(a)f(solution)h(of)g(the)e (Hamil-)6299 57335 y(ton's)584 b(equations)f(\(4.74\).)1029 b(In)583 b(particular)g(w)-36 b(e)583 b(exp)36 b(ect)584 b(the)f(existence)g(of)h(at)g(least)f FD(N)6299 58940 y FJ(geometrically)368 b(distinct)d FD(T)181 b FJ({p)36 b(erio)g(dic)367 b(solutions,)380 b(namely)366 b(solutions)g(not)f (obtained)h(one)6299 60545 y(from)434 b(eac)-36 b(h)434 b(other)f(simply)h(b)-36 b(y)434 b(time)f(translations.)8250 62151 y(Indeed)441 b(let)i(us)f(consider)f(the)h(restriction)g(of)h FA(S)542 b FJ(to)442 b(the)g(plane)g FD(E)461 b FJ(:=)384 b([)104 b(~)-754 b FD(!)48 b FJ(])44524 61669 y Fw(?)45311 62151 y FD(:)443 b FJ(The)f(set)6299 63756 y Fv(Z)7185 63274 y FC(N)8250 63756 y FA(\\)168 b FD(E)449 b FJ(is)371 b(a)h(lattice)g(of)g FD(E)78 b FJ(,)384 b(hence)371 b FA(S)471 b FJ(can)372 b(b)36 b(e)371 b(de\014ned)f(on)h(the)g(quotien) -36 b(t)372 b(space)f(\000)e(:=)g FD(E)78 b(=)6299 65361 y FJ(\()p Fv(Z)7691 64879 y FC(N)8901 65361 y FA(\\)313 b FD(E)78 b FJ(\))414 b FA(\030)h Fv(T)14395 64879 y FC(N)94 b Fw(\000)p FB(1)16493 65361 y FJ(.)659 b(Due)460 b(to)h(the)f(in)-36 b(v)-72 b(ariance)460 b(of)i FA(S)560 b FJ(with)460 b(resp)36 b(ect)460 b(to)h(the)f(time)g(shift,)6299 66966 y(an)-36 b(y)455 b(critical)g(p)36 b(oin)-36 b(t)454 b(of)h FA(S)18856 67172 y Fw(j)p FB(\000)19908 66966 y FJ(:)450 b(\000)404 b FA(\000)-221 b(!)405 b Fv(R)454 b FJ(is)h(also)g(a)g(critical)g(p)36 b(oin)-36 b(t)454 b(of)h FA(S)248 b FJ(:)450 b Fv(T)42981 66484 y FC(N)44281 66966 y FA(\000)-221 b(!)404 b Fv(R)p FJ(.)641 b(By)6299 68571 y(the)522 b(Lusternik-Sc)-36 b(hnirelman)519 b(category)524 b(theory)-108 b(,)544 b(since)522 b(cat)221 b(\000)519 b(=)j(cat)221 b Fv(T)42675 68089 y FC(N)94 b Fw(\000)p FB(1)45296 68571 y FJ(=)519 b FD(N)139 b FJ(,)543 b(w)-36 b(e)6299 70176 y(can)475 b(de\014ne)g(the)f FD(N)614 b FJ(min-max)475 b(critical)i(v)-72 b(alues)475 b(c)30829 70375 y FB(1)31795 70176 y FA(\024)441 b FJ(c)33847 70375 y FB(2)34812 70176 y FA(\024)g FD(:)221 b(:)g(:)441 b FA(\024)f FJ(c)40303 70375 y FC(N)41675 70176 y FJ(for)476 b(the)f(reduced)6299 71781 y(action)389 b(functional)g FA(S)17021 71988 y Fw(j)p FB(\000)17926 71781 y FJ(.)564 b(If)389 b(the)f(critical)i(lev)-36 b(els)389 b(c)30600 71980 y FC(i)31365 71781 y FJ(are)g(all)g(distinct,)398 b(the)388 b(corresp)36 b(onding)6299 73386 y FD(T)181 b FJ(-p)36 b(erio)g(dic)377 b(solutions)h(are)g(surely)f(geometrically) j(distinct,)388 b(since)378 b(their)f(actions)g(c)47906 73585 y FC(i)48659 73386 y FJ(are)6299 74991 y(di\013eren)-36 b(t.)556 b(On)369 b(the)h(other)f(hand,)383 b(if)370 b(some)h(min-max)f(critical)h(lev)-36 b(els)371 b(coincide,)383 b(then)369 b FA(S)49489 75198 y Fw(j)p FB(\000)27371 78312 y FJ(131)p eop %%Page: 132 132 132 131 bop 0 4488 a FJ(p)36 b(ossesses)503 b(in\014nitely)g(man)-36 b(y)503 b(critical)h(p)36 b(oin)-36 b(ts.)785 b(Although)502 b(not)h(all)h(the)e(corresp)36 b(onding)0 6094 y FD(T)181 b FJ({p)36 b(erio)g(dic)623 b(solutions)g(are)g(necessarily)g (geometrically)i(distinct,)670 b(since)622 b(a)h(p)36 b(erio)g(dic)0 7699 y(solution)349 b(can)h(cross)f(\000)g(at)g(most)g (a)h(\014nite)e(n)-36 b(um)g(b)36 b(er)348 b(of)i(times,)366 b(the)349 b(existence)h(of)f(in\014nitely)0 9304 y(man)-36 b(y)434 b(geometrically)h(distinct)e(orbits)h(follo)-36 b(ws)435 b(\(see)f([BeBiV04)q(])g(for)g(further)f(details\).)0 12235 y FE(Prop)42 b(osition)551 b(4.2.12.)577 b Fs(Supp)-66 b(ose)506 b FD(\034)593 b FA(2)444 b(T)21135 12434 y FC(\026)22263 12235 y Fs(and)505 b FD(\026)h Fs(smal)66 b(l)506 b(enough.)720 b(Then)505 b(the)h(system)0 13841 y FJ(\(4.74\))466 b Fs(p)-66 b(ossesses)466 b(\(at)f(le)-66 b(ast\))465 b FD(N)603 b Fs(ge)-66 b(ometric)g(al)66 b(ly)463 b(distinct)h FD(T)181 b Fs({p)-66 b(erio)g(dic)463 b(solutions)0 16061 y Fz(\000)609 17137 y FD(I)1183 17649 y FC(\036)1742 17178 y Fu(\()p Ft(j)39 b Fu(\))1742 17931 y(0)2870 17137 y FJ(\()p FD(t)p FJ(\))p FD(;)221 b(\036)5704 17649 y FC(\036)6263 17178 y Fu(\()p Ft(j)39 b Fu(\))6263 17931 y(0)7392 17137 y FJ(\()p 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b(then)542 b(e)-66 b(quation)542 b FJ(\(4.101\))i Fs(has)0 66200 y(solution)465 b FD(\036)5685 66399 y FB(0)6580 66200 y FJ(=)368 b(0)465 b Fs(by)g(symmetry.)0 70323 y FF(4.2.9)1792 b(Regularit)-50 b(y)0 73034 y FJ(The)632 b(solutions)8503 71958 y Fz(\000)9112 73034 y FD(I)9686 73233 y FB(0)10212 73034 y FJ(\()p FD(\034)148 b FJ(\))p FD(;)221 b(\036)13289 72355 y FB(\()p FC(j)51 b FB(\))13289 73359 y(0)14938 73034 y FJ(+)533 b(~)-754 b FD(!)48 b(t;)221 b FJ(0)18939 71958 y Fz(\001)20180 73034 y FJ(+)631 b FD(\020)22391 73545 y FC(\036)22950 73074 y Fu(\()p Ft(j)39 b Fu(\))22950 73828 y(0)24079 73034 y FJ(\()p FD(t)p FJ(\))631 b(of)h(the)f(system)h (\(4.74\))g(describ)36 b(ed)0 74991 y(in)565 b(\(4.108\))i(b)36 b(elong)565 b(to)h FD(H)13378 74509 y FB(1)13275 75327 y FC(T)78 b(;a;s)15979 74991 y FJ(and)564 b(v)-36 b(erify)567 b(the)e(estimate)g(\(4.109\).)975 b(With)565 b(the)g(same)21072 78312 y(132)p eop %%Page: 133 133 133 132 bop 6299 4488 a FJ(pro)36 b(cedure,)433 b(for)h(an)-36 b(y)434 b(\014xed)f FD(k)414 b FA(\025)370 b FJ(1,)434 b(w)-36 b(e)434 b(can)f(also)i(\014nd)d(solutions)i(in)f FD(H)41860 4006 y FC(k)41757 4824 y(T)78 b(;a;s)44330 4488 y FJ(v)-36 b(erifying)23314 8049 y FA(k)p FD(\020)24546 8561 y FC(\036)25105 8090 y Fu(\()p Ft(j)39 b Fu(\))25105 8843 y(0)26235 8049 y FA(k)26899 8358 y FC(H)27740 8053 y Ft(k)27672 8666 y(T)65 b(;a;s)30042 8049 y FA(\024)31729 7151 y FD(\026)p 31577 7744 1086 54 v 31577 8960 a(c)32137 9159 y FB(7)33017 8049 y FD(:)13041 b FJ(\(4.110\))6299 12180 y(Indeed)613 b(w)-36 b(e)614 b(can)g(solv)-36 b(e)615 b(the)f(range)g(equation)g(\(4.100\))h(in)p 36652 11113 1181 54 v 614 w FD(H)37832 11357 y FC(k)37832 12508 y(T)78 b(;a;s)40585 12180 y FJ(adapting)614 b(Lemma)6299 13785 y(4.2.8.)580 b(Ho)-36 b(w)g(ev)g(er,)435 b(in)e(this)h(case,)g(the)f (constan)-36 b(t)433 b FD(c)30726 13984 y FB(7)31685 13785 y FK(dep)36 b(ends)432 b(on)i FD(k)45 b FJ(.)6299 15438 y(On)334 b(the)h(other)f(hand,)355 b(giving)336 b(up)e(the)g FD(H)26211 14956 y FC(k)26108 15774 y(T)78 b(;a;s)28247 15438 y FJ({estimate)336 b(in)f(\(4.110\),)356 b(it)335 b(is)g(an)-36 b(yho)g(w)336 b(p)36 b(ossi-)6299 17043 y(ble)313 b(to)f(pro)-36 b(v)g(e,)338 b(b)-36 b(y)312 b(a)h(b)36 b(o)g(otstrap)313 b(argumen)-36 b(t,)336 b(that)312 b(the)g(solutions)h(\(4.108\))h(of)f(Prop)36 b(osition)6299 18648 y(4.2.12)449 b(actually)f(b)36 b(elong)447 b(to)g FD(H)21985 18166 y FC(k)21882 18984 y(T)78 b(;a;s)24468 18648 y FJ(for)447 b(an)-36 b(y)447 b FD(k)437 b FA(\025)391 b FJ(1.)619 b(In)446 b(particular)h(w)-36 b(e)447 b(sho)-36 b(w)447 b(that)g FD(\020)49318 18847 y FC(\036)49877 18970 y Fu(0)6299 20705 y FJ(of)435 b(Corollary)g(4.2.9)g(b)36 b(elongs)434 b(to)p 22890 19637 V 434 w FD(H)24070 19882 y FC(k)24070 21033 y(T)78 b(;a;s)26209 20705 y FJ(.)6299 22757 y(Indeed,)433 b FD(\020)11455 22956 y FC(\036)12014 23079 y Fu(0)12900 22757 y FA(2)p 14155 21690 V 369 w FD(H)15335 21934 y FB(1)15335 23086 y FC(T)78 b(;a;s)17908 22757 y FJ(solv)-36 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b(4.2.7.)6299 31238 y(Noting)523 b(that)p 13593 30171 V 521 w FD(H)14773 30415 y FC(k)14773 31566 y(T)78 b(;a;s)p Fw(\000)p FB(1)18634 31238 y FA(\032)p 20187 30171 V 520 w FD(H)21367 30415 y FC(k)21367 31566 y(T)g(;)15 b FB(~)-485 b FC(a)q(;s)24028 31238 y FJ(for)523 b(an)-36 b(y)522 b FD(k)565 b FA(\025)519 b FJ(1)j(and)g(0)e FD(<)535 b FJ(~)-666 b FD(a)519 b(<)g(a)p FJ(,)544 b(w)-36 b(e)523 b(ha)-36 b(v)g(e)522 b FD(\020)47913 31437 y FC(\036)48472 31560 y Fu(0)49508 31238 y FA(2)p 6299 32220 V 6299 33287 a FD(H)7480 32464 y FB(2)7480 33615 y FC(T)78 b(;)15 b FB(~)-485 b FC(a;s)9619 33287 y FJ(.)779 b(De\014ning)500 b FD(a)16693 33486 y FC(k)17745 33287 y FJ(:=)483 b FD(a)20505 32211 y Fz(\000)21247 32764 y FB(1)p 21247 32981 471 54 v 21247 33745 a(2)22145 33287 y FJ(+)23839 32764 y FB(1)p 23585 32981 979 54 v 23585 33748 a(2)24055 33496 y Ft(k)24696 32211 y Fz(\001)25305 33287 y FJ(,)517 b(w)-36 b(e)501 b(pro)-36 b(v)g(e)501 b(that)f FD(\020)35190 33486 y FC(\036)35749 33609 y Fu(0)36749 33287 y FA(2)p 38117 32220 1181 54 v 482 w FD(H)39298 32464 y FC(k)39298 33615 y(T)78 b(;a)40685 33771 y Ft(k)41193 33615 y FC(;s)42445 33287 y FJ(for)501 b(an)-36 b(y)501 b FD(k)528 b FA(\025)483 b FJ(1)6299 35336 y(and)449 b(\014nally)-108 b(,)453 b FD(\020)13622 35535 y FC(\036)14181 35658 y Fu(0)15093 35336 y FA(2)p 16373 34269 V 394 w FD(H)17553 34513 y FC(k)17553 35664 y(T)78 b(;a=)p FB(2)p FC(;s)21082 35336 y FJ(for)450 b(an)-36 b(y)449 b FD(k)440 b FA(\025)395 b FJ(1.)625 b(Ho)-36 b(w)g(ev)g(er,)453 b(in)c(this)g(fashion,)k(the)c FD(H)47708 34854 y FC(k)47605 35672 y(T)78 b(;a;s)49743 35336 y FJ({)6299 36941 y(estimates)434 b(deteriorate)g(while)g FD(k)479 b FJ(increases.)6299 38594 y(Summarizing,)392 b(b)-36 b(y)381 b(the)g(Sob)36 b(olev)382 b(immersions,)392 b(w)-36 b(e)381 b(ha)-36 b(v)g(e)382 b(that)e FD(\020)38494 38793 y FC(\036)39053 38916 y Fu(0)39939 38594 y FA(2)369 b FD(C)42220 38112 y FC(k)42789 38594 y FJ(\()p Fv(R)p FD(;)221 b FA(P)45760 38801 y FC(a=)p FB(2)p FC(;s)47953 38594 y FJ(\))381 b(for)6299 40199 y(an)-36 b(y)434 b FD(k)414 b FA(\025)369 b FJ(1)p FD(:)434 b FJ(W)-108 b(e)434 b(ha)-36 b(v)g(e)434 b(sho)-36 b(wn)433 b(the)g(follo)-36 b(wing:)6299 43150 y FE(Corollary)521 b(4.2.14.)554 b Fs(The)465 b(solutions)h FJ(\(4.108\))g Fs(of)e FJ(\(4.74\))i Fs(b)-66 b(elong)464 b(to)h FD(C)42110 42668 y Fw(1)43106 43150 y FJ(\()p Fv(R)p FD(;)221 b FA(P)46077 43357 y FC(a=)p FB(2)p FC(;s)48270 43150 y FJ(\))p Fs(.)6299 47296 y FF(4.2.10)1792 b(Minimal)599 b(p)50 b(erio)g(d)6299 49857 y FE(Lemma)499 b(4.2.15.)555 b Fs(L)-66 b(et)464 b FD(h;)221 b(k)414 b FA(2)369 b Fv(N)23260 49375 y FB(+)24047 49857 y Fs(,)465 b FD(h)368 b(<)h(k)45 b(:)465 b Fs(We)g(have)19513 53168 y FJ([)p FD(!)20684 53367 y FC(k)21254 53168 y FD(\034)443 b FA(\000)295 b FD(k)45 b(\034)148 b FJ(])370 b FA(\024)f FJ([)p FD(!)28328 53367 y FC(h)28926 53168 y FD(\034)443 b FA(\000)295 b FD(h\034)148 b FJ(])370 b FD(<)34968 52269 y(\026\034)p 34968 52863 1496 54 v 35016 54079 a FJ(2)p FD(h)36818 53168 y(:)9240 b FJ(\(4.111\))6299 56632 y Fp(pr)-30 b(oof.)659 b FJ(W)-108 b(e)433 b(\014rst)g(pro)-36 b(v)g(e)434 b(that)23429 59709 y FD(!)24239 59908 y FC(k)25103 59709 y FA(\000)295 b FD(k)414 b(<)369 b(!)29712 59908 y FC(h)30604 59709 y FA(\000)296 b FD(h)221 b(;)13155 b FJ(\(4.112\))6299 62786 y(whic)-36 b(h)351 b(implies)h FD(!)15081 62985 y FC(k)15650 62786 y FD(\034)276 b FA(\000)128 b FD(k)45 b(\034)517 b(<)368 b(!)21645 62985 y FC(h)22243 62786 y FD(\034)276 b FA(\000)128 b FD(h\034)499 b FJ(and)351 b(the)g(\014rst)f(inequalit)-36 b(y)353 b(in)e(\(4.111\).)552 b(Dividing)6299 64391 y(b)-36 b(y)434 b FD(k)45 b FJ(,)434 b(\(4.112\))g(is)g(equiv)-72 b(alen)-36 b(t)434 b(to)8947 67612 y FD(f)142 b FJ(\()p FD(x)p FJ(\))368 b(:=)13591 66412 y Fz(p)p 14919 66412 5547 54 v 1200 x FD(x)15658 67229 y FB(2)16480 67612 y FJ(+)294 b FD(\026=k)19940 67229 y FB(2)20761 67612 y FA(\000)i FD(x)f FA(\000)24452 66412 y Fz(p)p 25781 66412 4933 54 v 25781 67612 a FJ(1)g(+)g FD(\026=k)30187 67229 y FB(2)31008 67612 y FJ(+)g(1)369 b FD(>)g FJ(0)221 b FD(;)2823 b FJ(for)1302 b(0)369 b FD(<)g(x)g(<)f FJ(1)221 b FD(;)6299 70689 y FJ(where)551 b FD(x)568 b FJ(:=)h FD(h=k)45 b(:)551 b FJ(Since)f FD(f)142 b FJ(\(1\))569 b(=)f(0)551 b(and)f FD(f)29192 70207 y Fw(0)29502 70689 y FJ(\()p FD(x)p FJ(\))569 b(=)f FD(x)p FJ(\()p FD(x)35386 70207 y FB(2)36286 70689 y FJ(+)375 b FD(\026=k)39827 70207 y FB(2)40353 70689 y FJ(\))40859 70207 y Fw(\000)p FB(1)p FC(=)p FB(2)43432 70689 y FA(\000)g FJ(1)569 b FD(<)f FJ(0)551 b(for)6299 72294 y(0)370 b FD(<)e(x)h(<)g FJ(1,)406 b(w)-36 b(e)399 b(get)g FD(f)142 b FJ(\()p FD(x)p FJ(\))368 b FD(>)h FJ(0)399 b(and)f(\(4.112\))i(follo) -36 b(ws.)569 b(Then)398 b(the)g(second)g(inequalit)-36 b(y)400 b(in)6299 73899 y(\(4.111\))435 b(directly)f(follo)-36 b(ws)436 b(from)e(\(4.64\).)22895 b Fn(\003)27371 78312 y FJ(133)p eop %%Page: 134 134 134 133 bop 0 4488 a FE(Lemma)498 b(4.2.16.)554 b Fs(L)-66 b(et)464 b(b)-66 b(e)463 b FD(T)14860 4006 y FB(min)16948 4488 y Fs(the)h(minimal)f(p)-66 b(erio)g(d)463 b(of)h(a)g FD(T)181 b Fs(-p)-66 b(erio)g(dic)463 b(orbit)g FJ(\(4.108\))0 6094 y Fs(of)i(Pr)-66 b(op)g(osition)463 b(4.2.12.)597 b(If)464 b FD(N)508 b FA(\025)369 b FJ(2)465 b Fs(then)18911 9278 y FD(T)19854 8730 y FB(min)21847 9278 y FA(\025)23382 8379 y FD(c)23942 8578 y FB(9)p 23382 8972 1086 54 v 23533 10189 a FD(\026)24822 9278 y(:)14937 b FJ(\(4.113\))0 13038 y Fp(pr)-30 b(oof.)973 b FJ(Let)7735 11962 y Fz(\000)8344 13038 y FD(I)104 b FJ(\()p FD(t)p FJ(\))p FD(;)221 b(\036)p FJ(\()p FD(t)p FJ(\))p FD(;)300 b FJ(^)-729 b FD(z)59 b FJ(\()p FD(t)p FJ(\))16065 11962 y Fz(\001)17205 13038 y FJ(b)36 b(e)531 b(a)h FD(T)181 b FJ(-p)36 b(erio)g(dic)531 b(solution)h(of)g(Prop)36 b(osition)532 b(4.2.12.)0 14643 y(W)-108 b(e)415 b(kno)-36 b(w)415 b(that)f FD(\036)p FJ(\()p FD(T)181 b FJ(\))256 b FA(\000)h FD(\036)p FJ(\(0\))369 b(=)f(2)p FD(\031)48 b(k)460 b FJ(with)415 b FD(k)f FA(2)368 b Fv(Z)25595 14161 y FC(N)26906 14643 y FJ(de\014ned)413 b(in)i(\(4.54\).)573 b(Denoting)415 b(b)-36 b(y)0 16248 y FD(T)943 15766 y FB(min)762 16593 y FC(\036)2936 16248 y FA(\024)369 b FD(T)5281 15766 y FB(min)7310 16248 y FJ(the)405 b(minimal)h(p)36 b(erio)g(d)406 b(of)g FD(\036)p FJ(\()p FD(t)p FJ(\),)411 b(w)-36 b(e)406 b(ha)-36 b(v)g(e)406 b(that)f(there)g(exist)h FD(n)370 b FA(2)e Fv(N)40401 15766 y FB(+)41594 16248 y FJ(suc)-36 b(h)0 18108 y(that)365 b FD(nT)4469 17626 y FB(min)4288 18454 y FC(\036)6462 18108 y FJ(=)g FD(T)546 b FJ(and)11643 17758 y(~)11608 18108 y FD(k)414 b FA(2)368 b Fv(Z)14838 17626 y FC(N)16100 18108 y FJ(suc)-36 b(h)364 b(that)h FD(\036)p FJ(\()p FD(T)23934 17626 y FB(min)23753 18454 y FC(\036)25557 18108 y FJ(\))156 b FA(\000)g FD(\036)p FJ(\(0\))368 b(=)h(2)p FD(\031)33060 17758 y FJ(~)33025 18108 y FD(k)45 b FJ(,)379 b(v)-36 b(erifying)367 b FD(n)40576 17758 y FJ(~)40541 18108 y FD(k)415 b FJ(=)368 b FD(k)45 b(:)0 19713 y FJ(Hence)433 b(w)-36 b(e)434 b(deduce)f(that)g FD(n)h FJ(divides)g FD(g)416 b FJ(:=)369 b(gcd\()p FD(k)24437 19912 y FC(i)24757 20035 y Fu(1)25274 19713 y FD(;)221 b(:)g(:)g(:)j(;)d(k)28863 19912 y FC(i)29183 20061 y Ft(N)30009 19713 y FJ(\))434 b(and)f(w)-36 b(e)434 b(get)f(that)15335 23324 y FD(T)16278 22775 y FB(min)18270 23324 y FA(\025)369 b FD(T)20615 22775 y FB(min)20434 23652 y FC(\036)22608 23324 y FJ(=)24121 22425 y FD(T)p 24121 23018 943 54 v 24204 24235 a(n)25566 23324 y FA(\025)27101 22425 y FD(T)p 27101 23018 V 27237 24235 a(g)28398 23324 y(:)11361 b FJ(\(4.114\))0 26871 y(W)-108 b(e)434 b(claim)g(that)14885 28768 y(~)-696 b FD(g)416 b FJ(:=)369 b(gcd\()p FD(k)20753 28967 y FC(i)21073 29090 y Fu(1)21590 28768 y FD(;)221 b(k)22848 28967 y FC(i)23168 29090 y Fu(2)23687 28768 y FJ(\))368 b FD(<)26075 27869 y(\026\034)148 b(i)28015 28068 y FB(2)p 26075 28462 2466 54 v 26823 29679 a FD(i)27267 29878 y FB(1)28894 28768 y FD(:)10865 b FJ(\(4.115\))0 31724 y(Then)324 b(the)f(Lemma)h(follo)-36 b(ws)326 b(b)-36 b(y)324 b(\(4.114\))h(and)f(\(4.115\))h(noting)f(that)370 b(~)-696 b FD(g)416 b FA(\025)369 b FD(g)j FJ(and)323 b(recalling)0 33329 y(that)433 b FD(T)550 b FJ(=)369 b(2)p FD(\031)48 b(\034)581 b FJ(\(with)434 b FD(c)12122 33528 y FB(9)13016 33329 y FJ(=)369 b(2)p FD(\031)48 b(i)16277 33528 y FB(1)16802 33329 y FD(=i)17896 33528 y FB(2)18422 33329 y FJ(\).)0 34940 y(W)-108 b(e)347 b(no)-36 b(w)347 b(pro)-36 b(v)g(e)347 b(\(4.115\).)550 b(Since)347 b FD(\034)517 b FA(2)368 b Fv(N)347 b FJ(w)-36 b(e)348 b(ha)-36 b(v)g(e)347 b FD(k)25447 35139 y FC(i)26169 34940 y FJ(=)369 b([)p FD(!)28721 35139 y FC(i)29097 34940 y FD(\034)148 b FJ(])118 b(+)g FD(\024)32168 35139 y FC(i)32912 34940 y FJ(=)347 b FD(i\034)266 b FJ(+)118 b([)p FD(!)37847 35139 y FC(i)38222 34940 y FD(\034)266 b FA(\000)118 b FD(i\034)148 b FJ(])118 b(+)g FD(\024)43719 35139 y FC(i)0 36545 y FJ(for)434 b(all)h FD(i)368 b FA(2)h(I)532 b FJ(and)4791 39495 y FD(i)5235 39694 y FB(2)5760 39495 y FD(k)6436 39694 y FC(i)6756 39817 y Fu(1)7569 39495 y FA(\000)295 b FD(i)9341 39694 y FB(1)9867 39495 y FD(k)10543 39694 y FC(i)10863 39817 y Fu(2)12498 39495 y FJ(=)1117 b FD(i)15071 39694 y FB(2)15597 39495 y FJ([)p FD(!)16768 39694 y FC(i)17088 39817 y Fu(1)17605 39495 y FD(\034)443 b FA(\000)296 b FD(i)20386 39694 y FB(1)20911 39495 y FD(\034)148 b FJ(])296 b FA(\000)f FD(i)24053 39694 y FB(1)24579 39495 y FJ([)p FD(!)25750 39694 y FC(i)26070 39817 y Fu(2)26587 39495 y FD(\034)443 b FA(\000)296 b FD(i)29368 39694 y FB(2)29893 39495 y FD(\034)148 b FJ(])296 b(+)f FD(i)33014 39694 y FB(2)33539 39495 y FD(\024)34288 39694 y FC(i)34608 39817 y Fu(1)35420 39495 y FA(\000)g FD(i)37192 39694 y FB(1)37718 39495 y FD(\024)38467 39694 y FC(i)38787 39817 y Fu(2)12487 42196 y FA(\025)1107 b FJ(\()p FD(i)15577 42395 y FB(2)16398 42196 y FA(\000)295 b FD(i)18170 42395 y FB(1)18696 42196 y FJ(\)[)p FD(!)20373 42395 y FC(i)20693 42518 y Fu(1)21210 42196 y FD(\034)443 b FA(\000)295 b FD(i)23990 42395 y FB(1)24516 42196 y FD(\034)148 b FJ(])296 b(+)27192 40323 y Fz(\022)28303 41297 y FD(i)28747 41496 y FB(2)p 28303 41890 970 54 v 28303 43107 a FD(i)28747 43306 y FB(1)29701 42196 y FA(\000)31162 41297 y FD(i)31606 41496 y FB(1)p 31162 41890 V 31162 43107 a FD(i)31606 43306 y FB(2)32264 40323 y Fz(\023)33513 42196 y FJ(~)-700 b FD(\024)369 b(>)g FJ(0)3508 b(\(4.116\))0 45910 y(b)-36 b(y)518 b(\(4.111\))h(and)e(recalling)i(\(4.51\).)832 b(Moreo)-36 b(v)g(er,)540 b(since)564 b(~)-696 b FD(g)560 b FJ(=)512 b(gcd\()p FD(k)34085 46109 y FC(i)34405 46232 y Fu(1)34922 45910 y FD(;)221 b(k)36180 46109 y FC(i)36500 46232 y Fu(2)37018 45910 y FJ(\))518 b(there)f(exist)0 47515 y FD(h)749 47714 y FB(1)1275 47515 y FD(;)221 b(h)2606 47714 y FB(2)3712 47515 y FA(2)580 b Fv(N)558 b FJ(suc)-36 b(h)557 b(that)g FD(k)13371 47714 y FC(i)13691 47837 y Fu(1)14788 47515 y FJ(=)580 b FD(h)17129 47714 y FB(1)17701 47515 y FJ(~)-696 b FD(g)605 b FJ(and)557 b FD(k)22212 47714 y FC(i)22532 47837 y Fu(2)23630 47515 y FJ(=)579 b FD(h)25970 47714 y FB(2)26542 47515 y FJ(~)-696 b FD(g)47 b FJ(.)951 b(F)-108 b(rom)557 b(\(4.116\))i(w)-36 b(e)558 b(ha)-36 b(v)g(e)558 b(that)0 49120 y FD(i)444 49319 y FB(2)970 49120 y FD(k)1646 49319 y FC(i)1966 49442 y Fu(1)2778 49120 y FA(\000)295 b FD(i)4550 49319 y FB(1)5076 49120 y FD(k)5752 49319 y FC(i)6072 49442 y Fu(2)6958 49120 y FD(>)369 b FJ(0)434 b(and)f(therefore)12366 52070 y FD(i)12810 52269 y FB(2)13335 52070 y FD(k)14011 52269 y FC(i)14331 52392 y Fu(1)15144 52070 y FA(\000)295 b FD(i)16916 52269 y FB(1)17442 52070 y FD(k)18118 52269 y FC(i)18438 52392 y Fu(2)19324 52070 y FJ(=)20705 50994 y Fz(\000)21314 52070 y FD(i)21758 52269 y FB(2)22283 52070 y FD(h)23032 52269 y FB(1)23853 52070 y FA(\000)g FD(i)25625 52269 y FB(1)26151 52070 y FD(h)26900 52269 y FB(2)27425 50994 y Fz(\001)28080 52070 y FJ(~)-696 b FD(g)416 b FA(\025)g FJ(~)-696 b FD(g)268 b(:)8392 b FJ(\(4.117\))0 55020 y(Finally)435 b(b)-36 b(y)433 b(\(4.111\))4662 57970 y FD(i)5106 58169 y FB(2)5631 57970 y FD(k)6307 58169 y FC(i)6627 58292 y Fu(1)7440 57970 y FA(\000)295 b FD(i)9212 58169 y FB(1)9738 57970 y FD(k)10414 58169 y FC(i)10734 58292 y Fu(2)12358 57970 y FJ(=)1107 b FD(i)14921 58169 y FB(2)15446 57970 y FJ([)p FD(!)16617 58169 y FC(i)16937 58292 y Fu(1)17455 57970 y FD(\034)443 b FA(\000)295 b FD(i)20235 58169 y FB(1)20761 57970 y FD(\034)148 b FJ(])296 b FA(\000)f FD(i)23903 58169 y FB(1)24428 57970 y FJ([)p FD(!)25599 58169 y FC(i)25919 58292 y Fu(2)26437 57970 y FD(\034)443 b FA(\000)295 b FD(i)29217 58169 y FB(2)29743 57970 y FD(\034)148 b FJ(])295 b(+)g FD(i)32863 58169 y FB(2)33389 57970 y FD(\024)34138 58169 y FC(i)34458 58292 y Fu(1)35270 57970 y FA(\000)g FD(i)37042 58169 y FB(1)37568 57970 y FD(\024)38317 58169 y FC(i)38637 58292 y Fu(2)12358 60514 y FD(<)1107 b(i)14921 60713 y FB(2)15446 60514 y FJ([)p FD(!)16617 60713 y FC(i)16937 60836 y Fu(1)17455 60514 y FD(\034)443 b FA(\000)295 b FD(i)20235 60713 y FB(1)20761 60514 y FD(\034)148 b FJ(])296 b(+)23570 59616 y FD(i)24014 59815 y FB(2)p 23570 60209 V 23570 61425 a FD(i)24014 61624 y FB(1)24722 60514 y FJ(~)-699 b FD(\024)368 b FA(\024)27325 59616 y FD(i)27769 59815 y FB(2)p 27325 60209 V 27325 61425 a FD(i)27769 61624 y FB(1)28649 59040 y Fz(\020)29575 59616 y FD(\026\034)p 29575 60209 1496 54 v 29998 61425 a FJ(2)31499 60514 y(+)344 b(~)-699 b FD(\024)33554 59040 y Fz(\021)34717 60514 y FA(\024)36252 59616 y FD(i)36696 59815 y FB(2)37221 59616 y FD(\026\034)p 36252 60209 2466 54 v 37000 61425 a(i)37444 61624 y FB(1)39071 60514 y FD(:)688 b FJ(\(4.118\))0 64077 y(for)434 b FD(\026)f FJ(small)i(enough.)578 b(Then)433 b(\(4.115\))i(follo)-36 b(ws)435 b(from)f(\(4.117\))h(and)e(\(4.118\).)5052 b Fn(\003)0 67962 y FF(4.2.11)1792 b(Distinct)600 b(orbits)0 70436 y FJ(T)-108 b(ak)-36 b(e)387 b FD(\034)76 b(;)221 b(\034)5032 69954 y Fw(0)5712 70436 y FA(2)369 b(T)7690 70635 y FC(\026)8312 70436 y FJ(,)395 b(F)-108 b(or)386 b FD(\026)f FJ(small)i(enough,)395 b FD(\034)76 b(;)221 b(\034)22614 69954 y Fw(0)23312 70436 y FJ(satisfy)387 b(the)f(h)-36 b(yp)36 b(otheses)385 b(of)i(Prop)36 b(osition)0 72041 y(4.2.12.)580 b(Therefore,)435 b(let)e(b)36 b(e)9495 73915 y Fz(\000)10104 74991 y FD(I)104 b FJ(\()p FD(t)p FJ(\))p FD(;)221 b(\036)p FJ(\()p FD(t)p FJ(\))p FD(;)299 b FJ(^)-728 b FD(z)59 b FJ(\()p FD(t)p FJ(\))17825 73915 y Fz(\001)18802 74991 y FJ(=)20183 73915 y Fz(\000)20792 74991 y FD(I)21366 75190 y FB(0)21891 74991 y FJ(\()p FD(\034)148 b FJ(\))p FD(;)221 b(\036)24968 75190 y FB(0)25789 74991 y FJ(+)399 b(~)-754 b FD(!)48 b(t;)221 b FJ(0)29656 73915 y Fz(\001)30560 74991 y FJ(+)295 b FD(\020)100 b FJ(\()p FD(t)p FJ(\))221 b FD(;)21072 78312 y FJ(134)p eop %%Page: 135 135 135 134 bop 6299 4554 a FJ(with)434 b FD(\036)10032 4753 y FB(0)10926 4554 y FA(2)369 b Fv(T)13067 4072 y FC(N)13963 4554 y FJ(,)434 b FD(\020)469 b FA(2)p 17050 3486 1181 54 v 369 w FD(H)18230 3731 y FB(1)18230 4882 y FC(T)78 b(;a;s)20369 4554 y FJ(,)434 b FD(T)550 b FJ(=)369 b(2)p FD(\031)48 b(\034)148 b FJ(,)434 b(and)15018 6740 y Fz(\000)15626 7816 y FD(I)16304 7268 y Fw(0)16615 7816 y FJ(\()p FD(t)p FJ(\))p FD(;)221 b(\036)19449 7268 y Fw(0)19759 7816 y FJ(\()p FD(t)p FJ(\))p FD(;)300 b FJ(^)-729 b FD(z)22486 7268 y Fw(0)22797 7816 y FJ(\()p FD(t)p FJ(\))24279 6740 y Fz(\001)25257 7816 y FJ(=)26637 6740 y Fz(\000)27246 7816 y FD(I)27820 8015 y FB(0)28345 7816 y FJ(\()p FD(\034)29564 7268 y Fw(0)29875 7816 y FJ(\))p FD(;)221 b(\036)31733 7268 y Fw(0)31733 8145 y FB(0)32554 7816 y FJ(+)399 b(~)-754 b FD(!)48 b(t;)221 b FJ(0)36421 6740 y Fz(\001)37325 7816 y FJ(+)295 b FD(\020)39300 7268 y Fw(0)39611 7816 y FJ(\()p FD(t)p FJ(\))221 b FD(;)6299 11307 y FJ(with)440 b FD(\036)10038 10825 y Fw(0)10038 11636 y FB(0)10943 11307 y FA(2)380 b Fv(T)13095 10825 y FC(N)13991 11307 y FJ(,)441 b FD(\020)15461 10825 y Fw(0)16152 11307 y FA(2)p 17417 10240 V 379 w FD(H)18597 10485 y FB(1)18597 11636 y FC(T)19276 11384 y Fo(0)19574 11636 y FC(;a;s)21087 11307 y FJ(,)h FD(T)22833 10825 y Fw(0)23523 11307 y FJ(=)379 b(2)p FD(\031)48 b(\034)27063 10825 y Fw(0)27374 11307 y FJ(,)441 b(t)-36 b(w)g(o)441 b(solutions)f(of)g(\(4.74\))h (found)e(in)h(Prop)36 b(o-)6299 12913 y(sition)434 b(4.2.12;)i(w)-36 b(e)434 b(recall)g(that)f(b)-36 b(y)434 b(\(4.109\))16832 16348 y FA(k)p FD(\020)100 b FA(k)18828 16624 y FC(H)19669 16318 y Fu(1)19601 16932 y Ft(T)65 b(;a;s)21972 16348 y FA(\024)23659 15449 y FD(\026)p 23507 16042 1086 54 v 23507 17259 a(c)24067 17458 y FB(7)24947 16348 y FD(;)5425 b FA(k)p FD(\020)32065 15800 y Fw(0)32376 16348 y FA(k)33040 16624 y FC(H)33881 16318 y Fu(1)33813 17038 y Ft(T)34407 16847 y Fo(0)34704 17038 y Ft(;a;s)36525 16348 y FA(\024)38211 15449 y FD(\026)p 38060 16042 V 38060 17259 a(c)38620 17458 y FB(7)39499 16348 y FD(:)6559 b FJ(\(4.119\))6299 20014 y(Supp)36 b(ose)470 b(that)g(they)h(are)g(geometrically)h(the)f (same)g(solution,)480 b(namely)-108 b(,)481 b(up)470 b(to)h(a)g(time)6299 21619 y(translation,)12191 23670 y Fz(\000)12800 24746 y FD(I)104 b FJ(\()p FD(t)p FJ(\))p FD(;)221 b(\036)p FJ(\()p FD(t)p FJ(\))p FD(;)300 b FJ(^)-729 b FD(z)59 b FJ(\()p FD(t)p FJ(\))20521 23670 y Fz(\001)21498 24746 y FJ(=)22879 23670 y Fz(\000)23488 24746 y FD(I)24166 24198 y Fw(0)24476 24746 y FJ(\()p FD(t)p FJ(\))p FD(;)221 b(\036)27310 24198 y Fw(0)27621 24746 y FJ(\()p FD(t)p FJ(\))p FD(;)300 b FJ(^)-729 b FD(z)30348 24198 y Fw(0)30659 24746 y FJ(\()p FD(t)p FJ(\))32141 23670 y Fz(\001)32970 24746 y FD(;)2823 b FA(8)221 b FD(t)369 b FA(2)g Fv(R)p FD(:)5892 b FJ(\(4.120\))6299 27873 y(W)-108 b(e)434 b(claim)g(that)22160 29932 y FA(j)p FD(I)23103 30131 y FB(0)23628 29932 y FJ(\()p FD(\034)148 b FJ(\))295 b FA(\000)h FD(I)27551 30131 y FB(0)28076 29932 y FJ(\()p FD(\034)29295 29384 y Fw(0)29606 29932 y FJ(\))p FA(j)369 b(\024)32385 29034 y FJ(2)p FD(\026)p 32385 29627 1433 54 v 32559 30844 a(c)33119 31043 y FB(7)34172 29932 y FD(:)11886 b FJ(\(4.121\))6299 33095 y(Indeed,)433 b(using)h FD(I)104 b FJ(\()p FD(t)p FJ(\))368 b(=)h FD(I)18878 32613 y Fw(0)19188 33095 y FJ(\()p FD(t)p FJ(\),)7461 36223 y FA(j)p FD(I)8404 36422 y FB(0)8929 36223 y FJ(\()p FD(\034)148 b FJ(\))295 b FA(\000)h FD(I)12852 36422 y FB(0)13377 36223 y FJ(\()p FD(\034)14596 35674 y Fw(0)14907 36223 y FJ(\))p FA(j)1117 b FJ(=)g FA(j)p FD(J)123 b FJ(\()p FD(t)p FJ(\))295 b FA(\000)g FD(J)24186 35674 y Fw(0)24497 36223 y FJ(\()p FD(t)p FJ(\))p FA(j)368 b(\024)h(j)p FD(J)123 b FJ(\()p FD(t)p FJ(\))p FA(j)294 b FJ(+)h FA(j)p FD(J)33992 35674 y Fw(0)34303 36223 y FJ(\()p FD(t)p FJ(\))p FA(j)16888 38752 y(\024)1107 b(k)p FD(\020)100 b FJ(\()p FD(t)p FJ(\))p FA(k)22506 38951 y Fw(P)23164 39062 y Ft(a;s)24659 38752 y FJ(+)295 b FA(k)p FD(\020)27298 38204 y Fw(0)27609 38752 y FJ(\()p FD(t)p FJ(\))p FA(k)29755 38951 y Fw(P)30413 39062 y Ft(a;s)31981 38752 y FA(\024)369 b(k)p FD(\020)100 b FA(k)35379 39028 y FC(H)36220 38723 y Fu(1)36152 39336 y Ft(T)65 b(;a;s)38450 38752 y FJ(+)295 b FA(k)p FD(\020)41089 38204 y Fw(0)41400 38752 y FA(k)42064 39028 y FC(H)42905 38723 y Fu(1)42837 39442 y Ft(T)43431 39251 y Fo(0)43728 39442 y Ft(;a;s)45549 38752 y FA(\024)47084 37854 y FJ(2)p FD(\026)p 47084 38447 V 47258 39663 a(c)47818 39862 y FB(7)48871 38752 y FD(;)6299 42492 y FJ(recalling)435 b(\(4.78\))f(and)f(\(4.119\).)6299 44161 y(Moreo)-36 b(v)g(er)434 b(w)-36 b(e)434 b(claim)h(that)e(\(4.120\))h(implies)g (also)10669 47597 y FA(f)p FD(!)12143 47796 y FC(i)12463 47919 y Fu(1)12980 47597 y FJ(\()p FD(\034)14199 47048 y Fw(0)14805 47597 y FA(\000)295 b FD(\034)148 b FJ(\))p FA(g)370 b FD(<)20286 46698 y(\026)p 19899 47291 1557 54 v 19899 48508 a(c)20459 48707 y FB(10)25490 47597 y FJ(or)3902 b FA(f)p FD(!)32022 47796 y FC(i)32342 47919 y Fu(1)32859 47597 y FJ(\()p FD(\034)443 b FA(\000)296 b FD(\034)36415 47048 y Fw(0)36725 47597 y FJ(\))p FA(g)369 b FD(<)40165 46698 y(\026)p 39778 47291 V 39778 48508 a(c)40338 48707 y FB(10)41688 47597 y FD(;)4370 b FJ(\(4.122\))6299 51427 y(with)331 b FD(c)9719 51626 y FB(10)11084 51427 y FJ(:=)368 b FD(c)13385 51626 y FB(7)13911 51427 y FD(=)p FJ(2)p FA(k)p FD(A)p FA(k)p FD(:)332 b FJ(Indeed,)350 b(since)330 b FA(j)p FD(A)27139 50945 y Fw(\000)p FB(1)28397 51427 y FD(v)48 b FA(j)368 b(\025)i(k)p FD(A)p FA(k)33517 50945 y Fw(\000)p FB(1)34775 51427 y FA(j)p FD(v)48 b FA(j)329 b FJ(for)i(an)-36 b(y)331 b FD(v)416 b FA(2)369 b Fv(R)44016 50945 y FC(N)44912 51427 y FJ(,)352 b(recalling)6299 53032 y(\(4.53\))526 b(and)e(c)-36 b(ho)36 b(osing)525 b FD(v)571 b FJ(:=)524 b FA(f)p FD(!)48 b(\034)23473 52550 y Fw(0)23784 53032 y FA(g)357 b(\000)g(f)p FD(!)48 b(\034)148 b FA(g)p FJ(,)548 b(w)-36 b(e)525 b(ha)-36 b(v)g(e)525 b FD(I)35673 53231 y FB(0)36198 53032 y FJ(\()p FD(\034)148 b FJ(\))357 b FA(\000)g FD(I)40244 53231 y FB(0)40770 53032 y FJ(\()p FD(\034)41989 52550 y Fw(0)42299 53032 y FJ(\))524 b(=)f FD(A)45839 52550 y Fw(\000)p FB(1)47097 53032 y FD(v)572 b FJ(and)6299 54638 y(therefore)7316 57765 y FA(j)p FD(I)8259 57964 y FB(0)8784 57765 y FJ(\()p FD(\034)148 b FJ(\))295 b FA(\000)h FD(I)12707 57964 y FB(0)13232 57765 y FJ(\()p FD(\034)14451 57216 y Fw(0)14762 57765 y FJ(\))p FA(j)369 b(\025)g(k)p FD(A)p FA(k)19711 57216 y Fw(\000)p FB(1)20969 57765 y FA(jf)p FD(!)48 b(\034)23573 57216 y Fw(0)23883 57765 y FA(g)296 b(\000)f(f)p FD(!)48 b(\034)148 b FA(gj)369 b(\025)g(k)p FD(A)p FA(k)33513 57216 y Fw(\000)p FB(1)34771 57765 y FA(jf)p FD(!)36614 57964 y FC(i)36934 58087 y Fu(1)37451 57765 y FD(\034)38164 57216 y Fw(0)38475 57765 y FA(g)295 b(\000)h(f)p FD(!)42237 57964 y FC(i)42557 58087 y Fu(1)43074 57765 y FD(\034)148 b FA(gj)221 b FD(:)1017 b FJ(\(4.123\))6299 60892 y(Noting)434 b(that)12983 60410 y FB(3)6299 64019 y FA(jf)p FD(!)8142 64218 y FC(i)8462 64341 y Fu(1)8979 64019 y FD(\034)9692 63471 y Fw(0)10003 64019 y FA(g)153 b(\000)g(f)p FD(!)13480 64218 y FC(i)13800 64341 y Fu(1)14317 64019 y FD(\034)148 b FA(gj)369 b FJ(=)g FA(f)p FD(!)19287 64218 y FC(i)19607 64341 y Fu(1)20124 64019 y FJ(\()p FD(\034)21343 63471 y Fw(0)21806 64019 y FA(\000)153 b FD(\034)148 b FJ(\))p FA(g)2602 b FJ(or)g FA(jf)p FD(!)33078 64218 y FC(i)33398 64341 y Fu(1)33915 64019 y FD(\034)34628 63471 y Fw(0)34938 64019 y FA(g)153 b(\000)g(f)p FD(!)38415 64218 y FC(i)38735 64341 y Fu(1)39252 64019 y FD(\034)148 b FA(gj)370 b FJ(=)e FA(f)p FD(!)44222 64218 y FC(i)44542 64341 y Fu(1)45059 64019 y FJ(\()p FD(\034)301 b FA(\000)153 b FD(\034)48330 63471 y Fw(0)48641 64019 y FJ(\))p FA(g)221 b FD(;)6299 67146 y FJ(\(4.122\))435 b(follo)-36 b(ws)436 b(b)-36 b(y)433 b(\(4.121\))i(and)e(\(4.123\).)p 6299 68645 17638 45 v 7795 69461 a Fy(3)8292 69862 y Fx(Indeed,)369 b(if)h Fl(f)p Fq(x)p Fl(g)308 b(\025)f(f)p Fq(y)40 b Fl(g)370 b Fx(then)10680 72427 y Fl(jf)p Fq(x)p Fl(g)247 b(\000)f(f)p Fq(y)40 b Fl(gj)308 b Fx(=)g Fl(f)p Fq(x)p Fl(g)246 b(\000)g(f)p Fq(y)40 b Fl(g)308 b Fx(=)23810 71530 y Fj(\010)24456 72427 y Fl(f)p Fq(x)p Fl(g)247 b(\000)f(f)p Fq(y)40 b Fl(g)29238 71530 y Fj(\011)30191 72427 y Fx(=)31360 71530 y Fj(\010)32006 72427 y Fq(x)245 b Fl(\000)h Fq(y)286 b Fl(\000)246 b Fx([)p Fq(x)p Fx(])g(+)g([)p Fq(y)40 b Fx(])39724 71530 y Fj(\011)40678 72427 y Fx(=)307 b Fl(f)p Fq(x)246 b Fl(\000)g Fq(y)40 b Fl(g)184 b Fq(;)6299 74991 y Fx(on)370 b(the)f(other)h(hand,)f(if)h Fl(f)p Fq(y)40 b Fl(g)308 b(\025)g(f)p Fq(x)p Fl(g)369 b Fx(then)h Fl(jf)p Fq(x)p Fl(g)247 b(\000)f(f)p Fq(y)40 b Fl(gj)308 b Fx(=)f Fl(f)p Fq(y)286 b Fl(\000)246 b Fq(x)p Fl(g)p Fq(:)27371 78312 y FJ(135)p eop %%Page: 136 136 136 135 bop 0 4488 a FE(Lemma)499 b(4.2.17.)555 b Fs(L)-66 b(et)4607 7880 y FA(M)369 b FJ(:=)8313 6007 y Fz(\032)9774 7880 y FD(n)g FA(2)g Fv(Z)221 b FD(;)1550 b FA(j)p FD(n)p FA(j)369 b(\024)19047 6981 y FD(c)19607 7180 y FB(4)p 18610 7574 1959 54 v 18610 8791 a FJ(2)p FD(\026)20043 8407 y FB(2)20923 7880 y FD(;)1550 b FJ(s)p FD(:)p FJ(t)p FD(:)1329 b FA(f)p FD(!)27378 8079 y FC(i)27698 8202 y Fu(1)28215 7880 y FD(n)p FA(g)370 b(\024)31946 6981 y FD(\026)p 31559 7574 1557 54 v 31559 8791 a(c)32119 8990 y FB(10)33713 6007 y Fz(\033)35152 7880 y FD(:)4607 b FJ(\(4.124\))0 11530 y Fs(Then)18383 13422 y FD(])221 b FA(M)370 b(\024)22610 12523 y FJ(10)p FD(c)24470 12722 y FB(4)p 22610 13116 2387 54 v 22634 14333 a FD(c)23194 14532 y FB(10)24189 14333 y FD(\026)25350 13422 y(:)0 16775 y Fp(pr)-30 b(oof.)658 b FJ(W)-108 b(e)434 b(\014rst)f(note)g (that,)h(for)g FD(\026)f FJ(small)h(enough,)g(w)-36 b(e)433 b(get)9257 18552 y Fz(\024)10099 19526 y FJ(2)p FD(c)11309 19725 y FB(4)p 10091 20119 1753 54 v 10091 21336 a FD(i)10535 21535 y FB(1)11060 21336 y FD(\026)11976 18552 y Fz(\025)221 b(\024)13732 19526 y FD(i)14176 19725 y FB(1)p 13732 20119 970 54 v 13825 21336 a FD(\026)14835 18552 y Fz(\025)15905 20425 y FA(\025)17307 18552 y Fz(\022)18425 19526 y FJ(2)p FD(c)19635 19725 y FB(4)p 18417 20119 1753 54 v 18417 21336 a FD(i)18861 21535 y FB(1)19387 21336 y FD(\026)20597 20425 y FA(\000)296 b FJ(1)22576 18552 y Fz(\023)221 b(\022)24886 19526 y FD(i)25330 19725 y FB(1)p 24886 20119 970 54 v 24979 21336 a FD(\026)26284 20425 y FA(\000)295 b FJ(1)28262 18552 y Fz(\023)29609 20425 y FA(\025)31255 19526 y FD(c)31815 19725 y FB(4)p 31144 20119 1309 54 v 31144 21336 a FD(\026)31927 20952 y FB(2)32880 20425 y FJ(+)g(1)0 24075 y(and,)433 b(therefore,)1001 26250 y Fz(\024)1702 28123 y FA(\000)3304 27224 y FD(c)3864 27423 y FB(4)p 2868 27818 1959 54 v 2868 29034 a FJ(2)p FD(\026)4301 28651 y FB(2)4959 28123 y FD(;)6111 27224 y(c)6671 27423 y FB(4)p 5675 27818 V 5675 29034 a FJ(2)p FD(\026)7108 28651 y FB(2)7766 26250 y Fz(\025)8836 28123 y FA(\022)12089 26861 y Fz([)10239 30094 y FB(1)p Fw(\024)p FC(m)p Fw(\024)13005 29115 y Fh(h)13591 29613 y Fu(2)p Ft(c)14368 29812 y Fu(4)p 13582 29892 1256 40 v 13582 30441 a Ft(i)13878 30640 y Fu(1)14339 30441 y Ft(\026)14970 29115 y Fh(i)15636 25852 y Fz(")16632 26250 y(\024)17333 28123 y FA(\000)18936 27224 y FD(c)19496 27423 y FB(4)p 18499 27818 1959 54 v 18499 29034 a FJ(2)p FD(\026)19932 28651 y FB(2)20591 26250 y Fz(\025)21587 28123 y FJ(+)295 b(\()p FD(m)g FA(\000)g FJ(1\))27538 26250 y Fz(\024)28372 27224 y FD(i)28816 27423 y FB(1)p 28372 27818 970 54 v 28465 29034 a FD(\026)29475 26250 y Fz(\025)30397 28123 y FD(;)30979 26250 y Fz(\024)31681 28123 y FA(\000)33284 27224 y FD(c)33844 27423 y FB(4)p 32847 27818 1959 54 v 32847 29034 a FJ(2)p FD(\026)34280 28651 y FB(2)34939 26250 y Fz(\025)35935 28123 y FJ(+)g FD(m)38601 26250 y Fz(\024)39435 27224 y FD(i)39879 27423 y FB(1)p 39435 27818 970 54 v 39528 29034 a FD(\026)40537 26250 y Fz(\025)41459 25852 y(!)42732 28123 y FD(:)40120 31909 y FJ(\(4.125\))0 33514 y(No)-36 b(w)434 b(w)-36 b(e)434 b(claim)g(that)8161 36749 y FD(])8888 35274 y Fz(\020)9681 36749 y FA(M)296 b(\\)12752 35673 y Fz(\002)13369 36749 y FJ(\026)-713 b FD(n;)285 b FJ(\026)-714 b FD(n)297 b FJ(+)d([)q FD(i)17849 36948 y FB(1)18374 36749 y FD(=\026)p FJ(])20390 35673 y Fz(\001)20999 35274 y(\021)22161 36749 y FA(\024)23696 35850 y FJ(4)p FD(i)24790 36049 y FB(1)p 23696 36443 1620 54 v 23728 37660 a FD(c)24288 37859 y FB(10)25744 36749 y FJ(+)h(1)221 b FD(;)2823 b FA(8)285 b FJ(\026)-714 b FD(n)370 b FA(2)f Fv(Z)221 b FD(:)4186 b FJ(\(4.126\))0 40321 y(Then)433 b(from)h(\(4.125\))h(and)e(\(4.126\))i(w)-36 b(e)434 b(ha)-36 b(v)g(e)10084 44045 y FD(])221 b FA(M)370 b(\024)14178 42172 y Fz(\022)15289 43146 y FJ(4)p FD(i)16383 43345 y FB(1)p 15289 43739 V 15321 44956 a FD(c)15881 45155 y FB(10)17336 44045 y FJ(+)295 b(1)19293 42172 y Fz(\023)222 b(\024)21335 43146 y FJ(2)p FD(c)22545 43345 y FB(4)p 21327 43739 1753 54 v 21327 44956 a FD(i)21771 45155 y FB(1)22296 44956 y FD(\026)23212 42172 y Fz(\025)24282 44045 y FA(\024)25817 43146 y FJ(5)p FD(i)26911 43345 y FB(1)p 25817 43739 1620 54 v 25849 44956 a FD(c)26409 45155 y FB(10)27710 43146 y FJ(2)p FD(c)28920 43345 y FB(4)p 27702 43739 1753 54 v 27702 44956 a FD(i)28146 45155 y FB(1)28672 44956 y FD(\026)29956 44045 y FA(\024)31491 43146 y FJ(10)p FD(c)33351 43345 y FB(4)p 31491 43739 2387 54 v 31515 44956 a FD(c)32075 45155 y FB(10)33071 44956 y FD(\026)0 47695 y FJ(and)433 b(the)g(lemma)h(follo)-36 b(ws.)0 49308 y(Nothing)376 b(remains)h(but)e(to)i(pro)-36 b(v)g(e)376 b(\(4.126\).)561 b(If)377 b FA(M)179 b(\\)26040 48232 y Fz(\002)26655 49308 y FJ(\026)-713 b FD(n)q(;)285 b FJ(\026)-714 b FD(n)179 b FJ(+)g([)p FD(i)30902 49507 y FB(1)31427 49308 y FD(=\026)p FJ(])33443 48232 y Fz(\001)34421 49308 y FJ(=)368 b FA(;)p FJ(,)389 b(then)375 b(\(4.126\))0 50914 y(is)434 b(trivially)i(true.)577 b(Otherwise)433 b(let)12403 54165 y FD(n)13179 54364 y FB(0)14074 54165 y FJ(:=)369 b(min)18205 52690 y Fz(\020)18998 54165 y FA(M)296 b(\\)22070 53089 y Fz(\002)22686 54165 y FJ(\026)-713 b FD(n;)285 b FJ(\026)-714 b FD(n)297 b FJ(+)d([)q FD(i)27166 54364 y FB(1)27691 54165 y FD(=\026)p FJ(])29707 53089 y Fz(\001)30316 52690 y(\021)31330 54165 y FD(:)0 57490 y FJ(By)434 b(de\014nition)f FD(n)524 b(=)-805 b FA(2)369 b(M)434 b FJ(for)g(an)-36 b(y)496 b(\026)-713 b FD(n)370 b FA(\024)f FD(n)g(<)g(n)22544 57689 y FB(0)23070 57490 y FD(:)434 b FJ(Moreo)-36 b(v)g(er)434 b(b)-36 b(y)434 b(\(4.64\),\(4.65\))h(w)-36 b(e)434 b(ha)-36 b(v)g(e)9219 60758 y FA(f)p FD(!)10693 60957 y FC(i)11013 61080 y Fu(1)11530 60758 y FD(n)p FA(g)369 b FJ(=)g FA(f)p FD(\016)16010 60079 y FB(\(1\))15960 61099 y FC(i)16280 61222 y Fu(1)17268 60758 y FD(n)p FA(g)2602 b FJ(with)25691 59859 y FD(\026)p 25272 60453 1620 54 v 25272 61669 a FJ(4)p FD(i)26366 61868 y FB(1)27394 60758 y FD(<)368 b(\016)29400 60079 y FB(\(1\))29350 61099 y FC(i)29670 61222 y Fu(1)31027 60758 y FD(<)32959 59859 y(\026)p 32540 60453 V 32540 61669 a FJ(2)p FD(i)33634 61868 y FB(1)34514 60758 y FD(:)5245 b FJ(\(4.127\))0 64256 y(W)-108 b(e)434 b(no)-36 b(w)433 b(pro)-36 b(v)g(e)434 b(that)1364 67216 y FD(n)370 b FA(2)e(M)296 b(\\)6835 66140 y Fz(\002)7452 67216 y FJ(\026)-713 b FD(n;)285 b FJ(\026)-714 b FD(n)296 b FJ(+)f([)q FD(i)11932 67415 y FB(1)12457 67216 y FD(=\026)p FJ(])14473 66140 y Fz(\001)18052 67216 y FJ(=)-221 b FA(\))2970 b FD(n)369 b FJ(=)g FD(n)26443 67415 y FB(0)27264 67216 y FJ(+)295 b 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FJ(sin)221 b FD(ix)21072 78312 y FJ(138)p eop %%Page: 139 139 139 138 bop 6299 4488 a FJ(b)36 b(elongs)434 b(to)g FD(C)13583 4006 y Fw(1)14579 3412 y Fz(\000)15188 4488 y Fv(R)295 b FA(\002)h FJ([0)p FD(;)221 b(\031)48 b FJ(])p FD(;)221 b Fv(R)22052 3412 y Fz(\001)23096 4488 y FJ(and)433 b(is)h(a)g (solution)g(of)g(\(4\).)579 b(De\014ning)15300 8955 y(~)-731 b FD(u)p FJ(\()p FD(t;)221 b(x)p FJ(\))369 b(:=)g FD(\021)21784 7693 y Fz(X)21980 10512 y FC(i)p Fw(2I)23924 8955 y FJ(2)24574 6499 y Fz(s)p 25903 6499 2753 54 v 26036 8056 a FJ(\()p FD(I)27116 8255 y FB(0)27641 8056 y FJ(\))28147 8255 y FC(i)p 26036 8649 2487 54 v 26293 9866 a FD(\031)48 b(!)27889 10065 y FC(i)28876 8955 y FJ(cos)30839 7879 y Fz(\000)31552 8955 y FJ(~)-754 b FD(!)48 b(t)294 b FJ(+)h(\()p FD(\036)35653 9154 y FB(0)36179 8955 y FJ(\))36685 9154 y FC(i)37060 7879 y Fz(\001)37890 8955 y FJ(sin)222 b FD(ix)f(;)4945 b FJ(\(4.138\))6299 13136 y(w)-36 b(e)434 b(ha)-36 b(v)g(e,)434 b(for)g(an)-36 b(y)434 b FD(t)369 b 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FC(&)69 b Fw(\000)p FB(1)23834 56195 y FC(i)25505 55854 y FJ(,)606 b(for)572 b(all)g FD(i)p FJ(.)992 b(It)571 b(follo)-36 b(ws)574 b(that)d FD(x)41988 56053 y FC(i)42967 55854 y FJ(=)604 b FD(r)640 b FJ(=)603 b(const)6299 57459 y(and)594 b(then,)635 b(b)36 b(eing)594 b FD(g)48 b FJ(\()p FD(x)p FJ(\))643 b(=)g(1,)635 b(one)595 b(has)g FD(mr)29333 56977 y FC(&)30445 57459 y FJ(=)643 b(1,)635 b(namely)596 b FD(r)679 b FJ(=)42288 56936 y FB(1)p 41477 57154 2092 54 v 41477 57967 a FC(m)42309 57705 y Fu(1)p Ft(=&)43702 57459 y FJ(.)1061 b(It)595 b(results)6299 59206 y FD(f)7082 58130 y Fz(\000)8634 58683 y FB(1)p 7824 58901 V 7824 59713 a FC(m)8656 59452 y Fu(1)p Ft(=&)10048 59206 y FD(;)221 b(:)g(:)g(:)j(;)13905 58683 y FB(1)p 13095 58901 V 13095 59713 a FC(m)13927 59452 y Fu(1)p Ft(=&)15319 58130 y Fz(\001)16297 59206 y FJ(=)369 b FD(m)18816 58724 y FC(&)69 b Fw(\000)p FB(1)20487 59206 y FJ(.)49361 60811 y Fn(\003)6299 63511 y FE(Lemma)499 b(5.1.3.)555 b Fs(L)-66 b(et)463 b(b)-66 b(e)464 b FD(&)477 b(>)368 b FJ(1)p Fs(,)465 b FD(m)369 b FA(2)f Fv(N)p Fs(,)465 b FD(j)444 b FA(2)369 b Fv(Z)p Fs(.)596 b(Then)10848 65889 y Fz(X)9433 68718 y FC(k)9923 68841 y Fu(1)10384 68718 y FC(;:::)r(;k)12181 68829 y Ft(m)12963 68718 y Fw(2)p Fr(Z)14402 65278 y Fz(\022)23738 66252 y FA(j)p FD(j)75 b FA(j)25089 66451 y Fw(\003)p 15513 66846 18327 54 v 15513 68062 a FA(j)p FD(j)370 b FA(\000)295 b FD(k)18794 68261 y FB(1)19615 68062 y FA(\000)h FD(:)221 b(:)g(:)297 b FA(\000)e FD(k)24770 68261 y FC(m)25658 68062 y FA(j)26027 68261 y Fw(\003)26553 68062 y FA(j)p FD(k)27598 68261 y FB(1)28124 68062 y FA(j)28493 68261 y Fw(\003)29240 68062 y FA(\001)221 b(\001)g(\001)h(j)p FD(k)32056 68261 y FC(m)32945 68062 y FA(j)33314 68261 y Fw(\003)33972 65278 y Fz(\023)34950 65575 y FC(&)35788 67151 y FA(\024)369 b FJ(\()p FD(m)295 b FJ(+)g(1\))41592 66603 y FC(&)42061 67151 y FD(B)43116 66603 y FC(m)43049 67479 y(&)44225 67151 y FD(;)3134 b FJ(\(5.3\))6299 71370 y Fs(wher)-66 b(e)22537 73236 y FD(B)23525 73435 y FC(&)24363 73236 y FJ(:=)26105 71974 y Fz(X)26199 74803 y FC(k)24 b Fw(2)p Fr(Z)29045 72337 y FJ(1)p 28378 72930 1985 54 v 28378 74147 a FA(j)p FD(k)45 b FA(j)29837 73763 y FC(&)29837 74475 y Fw(\003)30864 73236 y FD(<)369 b FA(1)221 b FD(:)13565 b FJ(\(5.4\))27371 78312 y(145)p eop %%Page: 146 146 146 145 bop 0 4488 a Fp(pr)-30 b(oof.)658 b FJ(By)434 b(Lemma)g(5.1.2)h(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b(that)4507 7499 y FA(j)p FD(j)75 b FA(j)5858 6950 y FC(&)5858 7827 y Fw(\003)7490 7499 y FA(\024)9630 6423 y Fz(\000)10239 7499 y FA(j)p FD(j)370 b FA(\000)296 b FD(k)13521 7698 y FB(1)14342 7499 y FA(\000)f FD(:)221 b(:)g(:)297 b FA(\000)f FD(k)19497 7698 y FC(m)20385 7499 y FA(j)20754 7698 y Fw(\003)21575 7499 y FJ(+)e FA(j)p FD(k)23926 7698 y FB(1)24453 7499 y FA(j)24822 7698 y Fw(\003)25643 7499 y FJ(+)h FD(:)221 b(:)g(:)296 b FJ(+)f FA(j)p FD(k)31123 7698 y FC(m)32011 7499 y FA(j)32380 7698 y Fw(\003)32906 6423 y Fz(\001)33515 6720 y FC(&)7490 9436 y FA(\024)1107 b FJ(\()p FD(m)295 b FJ(+)g(1\))14032 8888 y FC(&)69 b Fw(\000)p FB(1)15703 8360 y Fz(\000)16312 9436 y FA(j)p FD(j)370 b FA(\000)295 b FD(k)19593 9635 y FB(1)20415 9436 y FA(\000)g FD(:)221 b(:)g(:)297 b FA(\000)e FD(k)25569 9635 y FC(m)26458 9436 y FA(j)26827 8888 y FC(&)26827 9764 y Fw(\003)27648 9436 y FJ(+)f FA(j)p FD(k)29999 9635 y FB(1)30526 9436 y FA(j)30895 8888 y FC(&)30895 9764 y Fw(\003)31716 9436 y FJ(+)g FD(:)221 b(:)g(:)297 b FJ(+)e FA(j)p FD(k)37196 9635 y FC(m)38084 9436 y FA(j)38453 8888 y FC(&)38453 9764 y Fw(\003)38979 8360 y Fz(\001)41421 9436 y FJ(\(5.5\))0 12446 y(and,)433 b(using)h(\(5.5\),)g(one)g(can)f(write)3285 14990 y Fz(X)1870 17818 y FC(k)2360 17941 y Fu(1)2822 17818 y FC(;:::)q(;k)4618 17929 y Ft(m)5400 17818 y Fw(2)p Fr(Z)8094 14378 y Fz(\022)17429 15353 y FA(j)p FD(j)75 b FA(j)18780 15552 y Fw(\003)p 9204 15946 18327 54 v 9204 17163 a FA(j)p FD(j)370 b FA(\000)296 b FD(k)12486 17362 y FB(1)13307 17163 y FA(\000)g FD(:)221 b(:)g(:)296 b FA(\000)g FD(k)18462 17362 y FC(m)19350 17163 y FA(j)19719 17362 y Fw(\003)20245 17163 y FA(j)p FD(k)21290 17362 y FB(1)21816 17163 y FA(j)22185 17362 y Fw(\003)22932 17163 y FA(\001)221 b(\001)g(\001)h(j)p FD(k)25748 17362 y FC(m)26636 17163 y FA(j)27005 17362 y Fw(\003)27664 14378 y Fz(\023)28642 14676 y FC(&)6839 20321 y FA(\024)1107 b FJ(\()p FD(m)295 b FJ(+)g(1\))13381 19772 y FC(&)69 b Fw(\000)p FB(1)16079 19059 y Fz(X)15274 21888 y FC(k)15764 22011 y Fu(2)16225 21888 y FC(;:::)q(;k)18021 21999 y Ft(m)22476 19422 y FJ(1)p 19158 20015 7287 54 v 19158 21232 a FA(j)p FD(k)20203 21431 y FB(2)20729 21232 y FA(j)21098 20848 y FC(&)21098 21561 y Fw(\003)21845 21232 y FA(\001)221 b(\001)g(\001)h(j)p FD(k)24661 21431 y FC(m)25549 21232 y FA(j)25918 20848 y FC(&)25918 21561 y Fw(\003)8979 24531 y FA(\001)9569 23269 y Fz(X)10053 26098 y FC(k)10543 26221 y Fu(1)11843 23632 y FA(j)p FD(j)370 b FA(\000)295 b FD(k)15124 23831 y FB(1)15945 23632 y FA(\000)h FD(:)221 b(:)g(:)297 b FA(\000)e FD(k)21100 23831 y FC(m)21988 23632 y FA(j)22357 23150 y FC(&)22357 23960 y Fw(\003)23178 23632 y FJ(+)g FA(j)p FD(k)25530 23831 y FB(1)26056 23632 y FA(j)26425 23150 y FC(&)26425 23960 y Fw(\003)27246 23632 y FJ(+)g FD(:)221 b(:)g(:)297 b FJ(+)e FA(j)p FD(k)32727 23831 y FC(m)33615 23632 y FA(j)33984 23150 y FC(&)33984 23960 y Fw(\003)p 11843 24225 22667 54 v 16423 25442 a FA(j)p FD(j)370 b FA(\000)295 b FD(k)19704 25641 y FB(1)20526 25442 y FA(\000)g FD(:)221 b(:)g(:)297 b FA(\000)e FD(k)25680 25641 y FC(m)26568 25442 y FA(j)26937 25058 y FC(&)26937 25770 y Fw(\003)27463 25442 y FA(j)p FD(k)28508 25641 y FB(1)29034 25442 y FA(j)29403 25058 y FC(&)29403 25770 y Fw(\003)6850 29121 y FJ(=)1117 b(\()p FD(m)295 b FJ(+)g(1\))13381 28572 y FC(&)69 b Fw(\000)p FB(1)16079 27859 y Fz(X)15274 30688 y FC(k)15764 30811 y Fu(2)16225 30688 y FC(;:::)q(;k)18021 30799 y Ft(m)22476 28222 y FJ(1)p 19158 28815 7287 54 v 19158 30032 a FA(j)p FD(k)20203 30231 y FB(2)20729 30032 y FA(j)21098 29648 y FC(&)21098 30360 y Fw(\003)21845 30032 y FA(\001)221 b(\001)g(\001)h(j)p FD(k)24661 30231 y FC(m)25549 30032 y FA(j)25918 29648 y FC(&)25918 30360 y Fw(\003)26872 29121 y FA(\001)27536 26849 y Fz( )28809 27859 y(X)29293 30688 y FC(k)29783 30811 y Fu(1)31990 28222 y FJ(1)p 31082 28815 2467 54 v 31082 30032 a FA(j)p FD(k)32127 30231 y FB(1)32653 30032 y FA(j)33022 29648 y FC(&)33022 30360 y Fw(\003)8979 33760 y FJ(+)10212 32498 y Fz(X)10696 35327 y FC(k)11186 35450 y Fu(1)17680 32862 y FJ(1)p 12485 33455 11041 54 v 12485 34671 a FA(j)p FD(j)370 b FA(\000)296 b FD(k)15767 34870 y FB(1)16588 34671 y FA(\000)f FD(:)221 b(:)g(:)297 b FA(\000)f FD(k)21743 34870 y FC(m)22631 34671 y FA(j)23000 34288 y FC(&)23000 35000 y Fw(\003)23954 33760 y FJ(+)25260 32498 y Fz(X)25744 35327 y FC(k)26234 35450 y Fu(1)29274 32862 y FA(j)p FD(k)30319 33061 y FB(2)30845 32862 y FA(j)31214 32380 y FC(&)31214 33190 y Fw(\003)32035 32862 y FJ(+)f FD(:)221 b(:)g(:)297 b FJ(+)e FA(j)p FD(k)37516 33061 y FC(m)38404 32862 y FA(j)38773 32380 y FC(&)38773 33190 y Fw(\003)p 27533 33455 13507 54 v 27533 34671 a FA(j)p FD(j)370 b FA(\000)296 b FD(k)30815 34870 y FB(1)31636 34671 y FA(\000)g FD(:)221 b(:)g(:)296 b FA(\000)g FD(k)36791 34870 y FC(m)37679 34671 y FA(j)38048 34288 y FC(&)38048 35000 y Fw(\003)38574 34671 y FA(j)p FD(k)39619 34870 y FB(1)40145 34671 y FA(j)40514 34288 y FC(&)40514 35000 y Fw(\003)41173 31489 y Fz(!)6839 37781 y FA(\024)1107 b FJ(\()p FD(m)295 b FJ(+)g(1\))13381 37232 y FC(&)69 b Fw(\000)p FB(1)16079 36519 y Fz(X)15274 39348 y FC(k)15764 39471 y Fu(2)16225 39348 y FC(;:::)q(;k)18021 39459 y Ft(m)22476 36882 y FJ(1)p 19158 37475 7287 54 v 19158 38692 a FA(j)p FD(k)20203 38891 y FB(2)20729 38692 y FA(j)21098 38308 y FC(&)21098 39020 y Fw(\003)21845 38692 y FA(\001)221 b(\001)g(\001)h(j)p FD(k)24661 38891 y FC(m)25549 38692 y FA(j)25918 38308 y FC(&)25918 39020 y Fw(\003)8979 42420 y FA(\001)9569 40148 y Fz( )10621 42420 y FJ(2)p FD(B)12259 42619 y FC(&)13024 42420 y FJ(+)14331 41158 y Fz(X)14814 43987 y FC(k)15304 44110 y Fu(1)18344 41521 y FA(j)p FD(k)19389 41720 y FB(2)19916 41521 y FA(j)20285 41039 y FC(&)20285 41850 y Fw(\003)21106 41521 y FJ(+)295 b FD(:)221 b(:)g(:)296 b FJ(+)f FA(j)p FD(k)26586 41720 y FC(m)27474 41521 y FA(j)27843 41039 y FC(&)27843 41850 y Fw(\003)p 16604 42114 13507 54 v 16604 43331 a FA(j)p FD(j)370 b FA(\000)295 b FD(k)19885 43530 y FB(1)20706 43331 y FA(\000)h FD(:)221 b(:)g(:)297 b FA(\000)e FD(k)25861 43530 y FC(m)26749 43331 y FA(j)27118 42947 y FC(&)27118 43660 y Fw(\003)27644 43331 y FA(j)p FD(k)28689 43530 y FB(1)29215 43331 y FA(j)29584 42947 y FC(&)29584 43660 y Fw(\003)30243 40148 y Fz(!)6839 47010 y FA(\024)1107 b FJ(\()p FD(m)295 b FJ(+)g(1\))13381 46462 y FC(&)69 b Fw(\000)p FB(1)15052 44739 y Fz( )16104 47010 y FJ(2)p FD(B)17809 46462 y FC(m)17742 47339 y(&)18992 47010 y FJ(+)20299 45748 y Fz(X)20783 48577 y FC(k)21273 48700 y Fu(1)23480 46112 y FJ(1)p 22572 46705 2467 54 v 22572 47921 a FA(j)p FD(k)23617 48120 y FB(1)24143 47921 y FA(j)24512 47538 y FC(&)24512 48250 y Fw(\003)8979 51600 y FA(\001)10375 50338 y Fz(X)9569 53167 y FC(k)10059 53290 y Fu(2)10521 53167 y FC(;:::)q(;k)12317 53278 y Ft(m)17605 50702 y FA(j)p FD(k)18650 50901 y FB(2)19176 50702 y FA(j)19545 50220 y FC(&)19545 51030 y Fw(\003)20366 50702 y FJ(+)295 b FD(:)221 b(:)g(:)297 b FJ(+)d FA(j)p FD(k)25846 50901 y FC(m)26735 50702 y FA(j)27104 50220 y FC(&)27104 51030 y Fw(\003)p 13454 51295 18327 54 v 13454 52512 a FA(j)p FD(j)370 b FA(\000)295 b FD(k)16735 52711 y FB(1)17556 52512 y FA(\000)h FD(:)221 b(:)g(:)297 b FA(\000)e FD(k)22711 52711 y FC(m)23599 52512 y FA(j)23968 52128 y FC(&)23968 52840 y Fw(\003)24494 52512 y FA(j)p FD(k)25539 52711 y FB(2)26065 52512 y FA(j)26434 52128 y FC(&)26434 52840 y Fw(\003)27181 52512 y FA(\001)221 b(\001)g(\001)h(j)p FD(k)29997 52711 y FC(m)30886 52512 y FA(j)31255 52128 y FC(&)31255 52840 y Fw(\003)31913 49329 y Fz(!)6850 56240 y FJ(=)1117 b(\()p FD(m)295 b FJ(+)g(1\))13381 55691 y FC(&)69 b Fw(\000)p FB(1)15052 53968 y Fz( )16104 56240 y FJ(2)p FD(B)17809 55691 y FC(m)17742 56568 y(&)18992 56240 y FJ(+)295 b(\()p FD(m)g FA(\000)g FJ(1\))24943 54978 y Fz(X)25427 57807 y FC(k)25917 57930 y Fu(1)28124 55341 y FJ(1)p 27216 55934 2467 54 v 27216 57151 a FA(j)p FD(k)28261 57350 y FB(1)28788 57151 y FA(j)29157 56767 y FC(&)29157 57479 y Fw(\003)8979 60830 y FA(\001)10375 59568 y Fz(X)9569 62397 y FC(k)10059 62520 y Fu(3)10521 62397 y FC(;:::)q(;k)12317 62508 y Ft(m)13321 59568 y Fz(X)13805 62397 y FC(k)14295 62520 y Fu(2)24432 59931 y FJ(1)p 15594 60525 18327 54 v 15594 61741 a FA(j)p FD(j)370 b FA(\000)295 b FD(k)18875 61940 y FB(1)19697 61741 y FA(\000)g FD(:)221 b(:)g(:)297 b FA(\000)e FD(k)24851 61940 y FC(m)25739 61741 y FA(j)26108 61357 y FC(&)26108 62070 y Fw(\003)26634 61741 y FA(j)p FD(k)27679 61940 y FB(3)28205 61741 y FA(j)28574 61357 y FC(&)28574 62070 y Fw(\003)29321 61741 y FA(\001)221 b(\001)g(\001)i(j)p FD(k)32138 61940 y FC(m)33026 61741 y FA(j)33395 61357 y FC(&)33395 62070 y Fw(\003)34053 58559 y Fz(!)6850 64274 y FJ(=)1117 b(\()p FD(m)295 b FJ(+)g(1\))13381 63725 y FC(&)69 b Fw(\000)p FB(1)15274 63198 y Fz(\002)15827 64274 y FJ(2)p FD(B)17532 63725 y FC(m)17465 64602 y(&)18715 64274 y FJ(+)295 b(\()p FD(m)g FA(\000)g FJ(1\))p FD(B)25500 63725 y FC(m)25433 64602 y(&)26388 63198 y Fz(\003)27311 64274 y FJ(=)368 b(\()p FD(m)295 b FJ(+)g(1\))33093 63725 y FC(&)33562 64274 y FD(B)34617 63725 y FC(m)34550 64602 y(&)35726 64274 y FD(;)0 67284 y FJ(getting)434 b(the)f(estimate)h(\(5.3\).)43061 68915 y Fn(\003)0 70546 y FJ(F)-108 b(rom)433 b(no)-36 b(w)434 b(on,)g(w)-36 b(e)433 b(will)i(denote)e(b)-36 b(y)434 b FA(k)516 b(\001)h(k)21391 70745 y FB(op)22873 70546 y FJ(the)433 b(standard)g(op)36 b(eratorial)435 b(norm.)0 73386 y FE(Lemma)364 b(5.1.4.)465 b Fs(L)-66 b(et)356 b(b)-66 b(e)356 b FD(L)13617 73585 y FC(n)14612 73386 y FJ(:)370 b FD(E)134 b FA(\002)56 b(\001)221 b(\001)g(\001)55 b(\002)h FD(E)78 b FJ(\()p FD(n)443 b(times)p 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Fz(\001)7931 36361 y FA(\001)8521 35099 y Fz(X)9265 37898 y FC(j)10661 34886 y Fz(\020)11455 36361 y FA(j)p FD(j)370 b FA(\000)295 b FD(k)14736 36560 y FB(1)15557 36361 y FA(\000)h FD(:)221 b(:)g(:)297 b FA(\000)e FD(k)20712 36560 y FC(n)p Fw(\000)p FB(1)22541 36361 y FA(j)22910 35812 y FB(2)p FC(\033)22910 36689 y Fw(\003)24005 36361 y FA(k)p FD(h)25418 35682 y FB(\()p FC(n)p FB(\))25418 36731 y FC(j)51 b Fw(\000)p FC(k)27072 36854 y Fu(1)27533 36731 y Fw(\000)p FC(:::)q Fw(\000)p FC(k)30271 36854 y Ft(n)p Fo(\000)p Fu(1)31938 36361 y FA(k)32602 35812 y FB(2)32602 36689 y FC(E)33396 34886 y Fz(\021)34411 36361 y FD(e)35014 35812 y FB(2)p FC(\013)p Fw(j)p FC(j)g Fw(j)5791 40971 y FA(\024)1107 b(k)p FD(L)9480 41170 y FC(n)10106 40971 y FA(k)10770 40422 y FB(2)10770 41299 y(op)11819 40971 y FD(n)12595 40422 y FB(2)p FC(\033)13691 40971 y FD(B)14746 40422 y FC(n)p Fw(\000)p FB(1)14679 41299 y(2)p FC(\033)16796 38699 y Fz( )17847 39709 y(X)18331 42538 y FC(k)18821 42661 y Fu(1)19987 40971 y FD(e)20590 40422 y FB(2)p FC(\013)p Fw(j)p FC(k)22415 40545 y Fu(1)22877 40422 y Fw(j)23194 40971 y FA(j)p FD(k)24239 41170 y FB(1)24765 40971 y FA(j)25134 40422 y FB(2)p FC(\033)25134 41299 y Fw(\003)26230 40971 y FA(k)p FD(h)27643 40292 y FB(\(1\))27643 41341 y FC(k)28133 41464 y Fu(1)28900 40971 y FA(k)29564 40422 y FB(2)29564 41299 y FC(E)30359 38699 y Fz(!)31632 40971 y FA(\001)221 b(\001)g(\001)7931 45959 y(\001)g(\001)g(\001)9702 43289 y Fz(0)9702 45680 y(@)10956 44697 y(X)10864 47526 y FC(k)11354 47649 y Ft(n)p Fo(\000)p Fu(1)13187 45959 y FD(e)13790 45411 y FB(2)p FC(\013)p Fw(j)p FC(k)15615 45534 y Ft(n)p Fo(\000)p Fu(1)17228 45411 y Fw(j)17545 45959 y FA(j)p FD(k)18590 46158 y FC(n)p Fw(\000)p FB(1)20419 45959 y FA(j)20788 45411 y FB(2)p FC(\033)20788 46288 y Fw(\003)21883 45959 y FA(k)p FD(h)23296 45280 y FB(\()p FC(n)p Fw(\000)p FB(1\))23296 46329 y FC(k)23786 46452 y Ft(n)p Fo(\000)p Fu(1)25857 45959 y FA(k)26521 45411 y FB(2)26521 46288 y FC(E)27315 43289 y Fz(1)27315 45680 y(A)7931 50786 y FA(\001)8521 48514 y Fz( )9573 49524 y(X)10316 52323 y FC(j)11713 50786 y FD(e)12316 50237 y FB(2)p FC(\013)p Fw(j)p FC(j)51 b Fw(\000)p FC(k)15305 50360 y Fu(1)15766 50237 y Fw(\000)p FC(:::)q Fw(\000)p FC(k)18504 50360 y Ft(n)p Fo(\000)p Fu(1)20116 50237 y Fw(j)20433 50786 y FA(j)p FD(j)370 b FA(\000)295 b FD(k)23714 50985 y FB(1)24536 50786 y FA(\000)g FD(:)221 b(:)g(:)297 b FA(\000)e FD(k)29690 50985 y FC(n)p Fw(\000)p FB(1)31519 50786 y FA(j)31888 50237 y FB(2)p FC(\033)31888 51114 y Fw(\003)32983 50786 y FA(k)p FD(h)34396 50107 y FB(\()p FC(n)p FB(\))34396 51156 y FC(j)51 b Fw(\000)p FC(k)36050 51279 y Fu(1)36511 51156 y Fw(\000)p FC(:::)q Fw(\000)p FC(k)39249 51279 y Ft(n)p Fo(\000)p Fu(1)40916 50786 y FA(k)41580 50237 y FB(2)41580 51114 y FC(E)42375 48514 y Fz(!)5801 54284 y FJ(=)1118 b FA(k)p FD(L)9480 54483 y FC(n)10106 54284 y FA(k)10770 53735 y FB(2)10770 54612 y(op)11819 54284 y FD(n)12595 53735 y FB(2)p FC(\033)13691 54284 y FD(B)14746 53735 y FC(n)p Fw(\000)p FB(1)14679 54612 y(2)p FC(\033)16574 54284 y FA(k)p FD(h)17987 53735 y FB(\(1\))19245 54284 y FA(k)19909 53735 y FB(2)19909 54668 y FC(H)20750 54234 y Ft(\013;\033)20682 54976 y(E)22368 54284 y FA(\001)221 b(\001)g(\001)i(k)p FD(h)25553 53735 y FB(\()p FC(n)p FB(\))26911 54284 y FA(k)27575 53735 y FB(2)27575 54668 y FC(H)28416 54234 y Ft(\013;\033)28348 54976 y(E)30034 54284 y FD(;)0 57496 y FJ(where)427 b(w)-36 b(e)427 b(used)f(that)h FD(e)12037 57014 y FB(2)p FC(\013)p Fw(j)p FC(j)51 b Fw(j)14490 57496 y FA(\024)369 b FD(e)16495 57014 y FB(2)p FC(\013)p Fw(j)p FC(k)18320 57137 y Fu(1)18782 57014 y Fw(j)19321 57496 y FA(\001)221 b(\001)g(\001)h FD(e)21695 57014 y FB(2)p FC(\013)p Fw(j)p FC(j)51 b Fw(\000)p FC(k)24684 57137 y Fu(1)25145 57014 y Fw(\000)p FC(:::)q Fw(\000)p FC(k)27883 57137 y Ft(n)p Fo(\000)p Fu(1)29495 57014 y Fw(j)30239 57496 y FJ(and,)428 b(in)f(the)f(last)h (equalit)-36 b(y)-108 b(,)0 59101 y(that)1754 61969 y FA(k)p FD(h)3167 61421 y FB(\()p FC(n)p FB(\))4525 61969 y FA(k)5189 61421 y FB(2)5189 62353 y FC(H)6030 61919 y Ft(\013;\033)5962 62662 y(E)7796 61969 y FJ(:=)9538 60707 y Fz(X)10282 63507 y FC(j)11678 61969 y FD(e)12281 61421 y FB(2)p FC(\013)p Fw(j)p FC(j)51 b Fw(\000)p FC(k)15270 61544 y Fu(1)15732 61421 y Fw(\000)p FC(:::)q Fw(\000)p FC(k)18470 61544 y Ft(n)p Fo(\000)p Fu(1)20082 61421 y Fw(j)20398 61969 y FA(j)p FD(j)370 b FA(\000)296 b FD(k)23680 62168 y FB(1)24501 61969 y FA(\000)g FD(:)221 b(:)g(:)296 b FA(\000)g FD(k)29656 62168 y FC(n)p Fw(\000)p FB(1)31485 61969 y FA(j)31854 61421 y FB(2)p FC(\033)31854 62298 y Fw(\003)32949 61969 y FA(k)p FD(h)34362 61290 y FB(\()p FC(n)p FB(\))34362 62339 y FC(j)51 b Fw(\000)p FC(k)36016 62462 y Fu(1)36477 62339 y Fw(\000)p FC(:::)p Fw(\000)p FC(k)39214 62462 y Ft(n)p Fo(\000)p Fu(1)40882 61969 y FA(k)41546 61421 y FB(2)41546 62298 y FC(E)0 66108 y FJ(b)-36 b(y)433 b(the)h(de\014nition)e(of)j(the)e(norm.)43061 67713 y Fn(\003)0 70369 y FE(Lemma)499 b(5.1.5.)554 b Fs(L)-66 b(et)464 b(b)-66 b(e)464 b FD(f)511 b FJ(:)591 b FD(E)446 b FA(\000)-221 b(!)370 b FD(F)645 b Fs(analytic)464 b(for)h FA(k)p FD(x)p FA(k)29740 70568 y FC(E)30904 70369 y FD(<)368 b(r)32870 70568 y FB(0)33396 70369 y Fs(,)13093 73829 y FD(f)142 b FJ(\()p FD(x)p FJ(\))369 b(=)17584 72567 y Fz(X)17376 75373 y FC(n)p Fw(\025)p FC(n)19250 75496 y Fu(0)20309 72931 y FJ(1)p 20066 73524 1138 54 v 20066 74740 a FD(n)p FJ(!)21557 73829 y FD(d)22233 73281 y FC(n)22860 73829 y FD(f)142 b FJ(\(0\)[)p FD(x;)221 b(:)g(:)g(:)j(;)d(x)p FJ(])g FD(:)10421 b FJ(\(5.9\))21072 78312 y(148)p eop %%Page: 149 149 149 148 bop 6299 4488 a Fs(Then)15299 6556 y FA(k)p FD(d)16639 6008 y FC(n)17265 6556 y FD(f)142 b FJ(\(0\))p FA(k)20374 6755 y FB(op)21792 6556 y FA(\024)369 b FD(M)24452 6755 y FC(r)24877 6878 y Fu(1)25615 4683 y Fz(\022)26894 5657 y FD(n)p 26726 6250 1112 54 v 26726 7467 a(r)27312 7666 y FB(1)27971 4683 y Fz(\023)28948 4980 y FC(n)30017 6556 y FD(;)2879 b FA(8)443 b FJ(0)369 b FD(<)g(r)37424 6755 y FB(1)38318 6556 y FD(<)g(r)40285 6755 y FB(0)41032 6556 y FD(;)6299 9458 y Fs(wher)-66 b(e)465 b FD(M)11227 9657 y FC(r)11652 9780 y Fu(1)12537 9458 y FJ(:=)369 b(max)16700 9665 y Fw(k)p FC(x)p Fw(k)18170 9814 y Ft(E)18859 9665 y Fw(\024)p FC(r)20016 9788 y Fu(1)20753 9458 y FA(k)p FD(f)142 b FJ(\()p FD(x)p FJ(\))p FA(k)24615 9657 y FC(F)25397 9458 y Fs(.)6299 11797 y Fp(pr)-30 b(oof.)659 b FJ(Consider)433 b(the)g(map)h FD(d)22525 11315 y FC(n)23151 11797 y FD(f)142 b FJ(\(0\))433 b(de\014ned)f(b)-36 b(y)31882 11315 y FB(2)6907 15028 y FD(d)7583 14479 y FC(n)8209 15028 y FD(f)142 b FJ(\(0\)[)p FD(h)11764 15227 y FB(1)12289 15028 y FD(;)221 b(:)g(:)g(:)j(;)d(h)15951 15227 y FC(n)16578 15028 y FJ(])369 b(=)20214 14129 y(1)p 18821 14722 3435 54 v 18821 15939 a(\(2)p FD(\031)48 b FJ(i\))21630 15555 y FC(n)22611 13220 y Fz(Z)23349 16227 y Fw(j)p FC(\020)24024 16350 y Fu(1)24485 16227 y Fw(j)p FB(=)p FC(")26189 15028 y FD(:)221 b(:)g(:)27937 13220 y Fz(Z)28675 16227 y Fw(j)p FC(\020)29350 16338 y Ft(n)29917 16227 y Fw(j)p FB(=)p FC(")31754 14129 y FD(f)142 b FJ(\()p FD(\020)33611 14328 y FB(1)34137 14129 y FD(h)34886 14328 y FB(1)35706 14129 y FJ(+)295 b FD(:)221 b(:)g(:)297 b FJ(+)e FD(\020)40710 14328 y FC(n)41336 14129 y FD(h)42085 14328 y FC(n)42711 14129 y FJ(\))p 31754 14722 11463 54 v 35295 15939 a FD(\020)35963 15481 y FB(2)35863 16265 y(1)36711 15939 y FA(\001)221 b(\001)g(\001)h FD(\020)39150 15555 y FB(2)39050 16267 y FC(n)43571 15028 y FD(d\020)44815 15227 y FB(1)45562 15028 y FA(\001)f(\001)g(\001)i FD(d\020)48578 15227 y FC(n)49425 15028 y FD(:)6299 18448 y FJ(De\014ning)434 b FD(")368 b FJ(:=)h FD(r)14789 18647 y FB(1)15315 18448 y FD(=n)p FJ(,)435 b(w)-36 b(e)434 b(ha)-36 b(v)g(e,)434 b(if)g FA(k)p FD(h)25417 18647 y FC(i)25792 18448 y FA(k)26456 18647 y FC(E)27620 18448 y FJ(=)369 b(1,)434 b(for)g(all)h(1)369 b FA(\024)g FD(i)g FA(\024)g FD(n)p FJ(,)10222 20947 y FA(k)p FD(\020)11454 21146 y FB(1)11980 20947 y FD(h)12729 21146 y FB(1)13550 20947 y FJ(+)294 b FD(:)221 b(:)g(:)297 b FJ(+)e FD(\020)18553 21146 y FC(n)19179 20947 y FD(h)19928 21146 y FC(n)20554 20947 y FA(k)21218 21146 y FC(E)22382 20947 y FA(\024)369 b FD(")p FJ(\()p FA(k)p FD(h)26312 21146 y FB(1)26838 20947 y FA(k)27502 21146 y FC(E)28591 20947 y FJ(+)295 b FD(:)221 b(:)g(:)297 b FJ(+)e FA(k)p FD(h)34440 21146 y FC(n)35066 20947 y FA(k)35730 21146 y FC(E)36525 20947 y FJ(\))369 b(=)f FD("n)h FJ(=)g FD(r)42501 21146 y FB(1)43396 20947 y FD(<)f(r)45362 21146 y FB(0)46110 20947 y FD(:)6299 23445 y FJ(Therefore)434 b(one)g(gets)13825 25943 y FA(k)p FD(d)15165 25395 y FC(n)15791 25943 y FD(f)142 b FJ(\(0\))p FA(k)18900 26142 y FB(op)21056 25943 y FJ(:=)3612 b(sup)23536 27087 y Fw(k)p FC(h)24548 27222 y Ft(i)24900 27087 y Fw(k)25370 27236 y Ft(E)26058 27087 y FB(=1)158 b FC(;)p FB(1)p Fw(\024)p FC(i)p Fw(\024)p FC(n)30726 25943 y FA(k)p FD(d)32066 25395 y FC(n)32692 25943 y FD(f)142 b FJ(\(0\)[)p FD(h)36247 26142 y FB(1)36773 25943 y FD(;)221 b(:)g(:)g(:)i(;)e(h)40434 26142 y FC(n)41061 25943 y FJ(])p FA(k)42086 26142 y FC(F)21226 29639 y FA(\024)23961 28740 y FJ(1)p 23668 29333 1235 54 v 23668 30550 a FD(")24277 30166 y FC(n)25258 29639 y FA(k)p FD(f)142 b FJ(\()p FD(\020)27779 29838 y FB(1)28304 29639 y FD(h)29053 29838 y FB(1)29874 29639 y FJ(+)295 b FD(:)221 b(:)g(:)297 b FJ(+)e FD(\020)34878 29838 y FC(n)35504 29639 y FD(h)36253 29838 y FC(n)36879 29639 y FJ(\))p FA(k)38049 29838 y FC(F)21226 32950 y FA(\024)23668 32051 y FD(M)24926 32250 y FC(r)25351 32373 y Fu(1)p 23668 32644 2200 54 v 24151 33861 a FD(")24760 33477 y FC(n)26370 32950 y FJ(=)368 b FD(M)29008 33149 y FC(r)29433 33272 y Fu(1)30393 31077 y Fz(\022)31671 32051 y FD(n)p 31503 32644 1112 54 v 31503 33861 a(r)32089 34060 y FB(1)32748 31077 y Fz(\023)33726 31374 y FC(n)34795 32950 y FD(:)49361 36112 y Fn(\003)6299 38451 y FE(Theorem)523 b(5.1.6.)565 b Fs(L)-66 b(et)483 b(b)-66 b(e)483 b FD(f)546 b FJ(:)626 b FD(E)482 b FA(\000)-221 b(!)404 b FD(F)181 b Fs(,)488 b FD(f)142 b FJ(\()p FD(x)p FJ(\))404 b(=)33010 37455 y Fz(P)34412 38839 y FC(n)p Fw(\025)p FC(n)36286 38962 y Fu(0)37337 37928 y FB(1)p 37156 38146 833 54 v 37156 38909 a FC(n)p FB(!)38121 38451 y FD(d)38797 37969 y FC(n)39423 38451 y FD(f)142 b FJ(\(0\)[)p FD(x;)221 b(:)g(:)g(:)k(;)c (x)p FJ(])p Fs(,)489 b(ana-)6299 40056 y(lytic)519 b(for)g FA(k)p FD(x)p FA(k)13407 40255 y FC(E)14673 40056 y FD(<)470 b(r)16741 40255 y FB(0)17267 40056 y Fs(,)533 b(such)520 b(that)g FD(f)142 b FJ(\(0\))470 b(=)g(0)p Fs(,)534 b(namely)519 b FD(n)35215 40255 y FB(0)36211 40056 y FA(\025)471 b FJ(1)p Fs(.)762 b(Ther)-66 b(e)519 b(exist)g FD(M)609 b(>)470 b FJ(0)6299 41661 y Fs(and)523 b FD(r)513 b(>)476 b FJ(0)523 b Fs(such)h(that)e(if)g FD(h)477 b FA(2)e FD(H)23457 41032 y FC(\013;\033)23354 42021 y(E)25470 41661 y Fs(with)523 b FD(\013)485 b(>)476 b FJ(0)p Fs(,)537 b FD(\033)525 b(>)476 b FJ(1)p FD(=)p FJ(2)524 b Fs(and)f FA(k)p FD(h)p FA(k)42622 41924 y FC(H)43463 41490 y Ft(\013;\033)43395 42232 y(E)45336 41661 y FD(<)476 b(r)559 b Fs(then)6299 43266 y FD(f)142 b FJ(\()p FD(h)p FJ(\))368 b FA(2)h FD(H)11647 42637 y FC(\013;\033)11544 43626 y(F)13602 43266 y Fs(and)18335 45853 y FA(k)p FD(f)142 b FJ(\()p FD(h)p FJ(\))p FA(k)22207 46116 y FC(H)23048 45681 y Ft(\013;\033)22980 46424 y(F)24814 45853 y FA(\024)369 b FD(M)28043 44591 y Fz(X)27835 47397 y FC(n)p Fw(\025)p FC(n)29709 47520 y Fu(0)30391 43980 y Fz(\022)31501 44764 y FA(k)p FD(h)p FA(k)33578 45027 y FC(H)34419 44593 y Ft(\013;\033)34351 45335 y(E)p 31501 45548 4316 54 v 33348 46764 a FD(r)35949 43980 y Fz(\023)36927 44118 y FC(n)37996 45853 y FD(;)6299 49319 y Fs(wher)-66 b(e)598 b FD(f)142 b FJ(\()p FD(h)p FJ(\))616 b(:)838 b Fv(R)617 b FA(\000)-221 b(!)617 b FD(F)181 b Fs(,)631 b FD(t)616 b FA(7\000)-221 b(!)617 b FD(f)142 b FJ(\()p FD(h)p FJ(\()p FD(t)p FJ(\)\))597 b Fs(and)i FJ([)p FD(f)142 b FJ(\()p FD(h)p FJ(\)]\()p FD(t)p FJ(\))615 b(:=)i FD(f)142 b FJ(\()p FD(h)p FJ(\()p FD(t)p FJ(\)\))p Fs(.)997 b(Mor)-66 b(e)g(over,)6299 50924 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Ft(\013;\033)18568 72778 y(F)20402 72207 y FA(\024)369 b FD(n)22580 71658 y FC(\033)23206 72207 y FD(B)24394 71029 y Ft(n)p Fo(\000)p Fu(1)p 24394 71201 1557 40 v 24969 71749 a(2)24194 72533 y FB(2)p FC(\033)26138 72207 y FA(k)p FD(d)27478 71658 y FC(n)28105 72207 y FD(f)142 b FJ(\(0\))p FA(k)31214 72406 y FB(op)32262 72207 y FA(k)p FD(h)p FA(k)34339 71657 y FC(n)p Fw(\000)p FB(1)34339 72682 y FC(H)35180 72248 y Ft(\013;\033)35112 72990 y(E)36577 72207 y FA(k)37254 71856 y FJ(~)37241 72207 y FD(h)p FA(k)38654 72470 y FC(H)39495 72035 y Ft(\013;\033)39427 72778 y(E)41114 72207 y FD(:)p 6299 73774 17638 45 v 7795 74590 a Fy(2)8292 74991 y Fx(See)369 b([PT)q(])g(pg.)493 b(136,)371 b(137.)27371 78312 y FJ(149)p eop %%Page: 150 150 150 149 bop 0 4488 a Fp(pr)-30 b(oof.)658 b FJ(F)-108 b(or)433 b FD(h)369 b FA(2)g FD(H)10876 3859 y FC(\013;\033)10773 4849 y(E)12366 4488 y FJ(,)433 b(b)-36 b(y)434 b(Lemma)f(5.1.4)i(with)f FD(L)26564 4687 y FC(n)27559 4488 y FJ(:=)369 b FD(d)29977 4006 y FC(n)30603 4488 y FD(f)142 b FJ(\(0\),)7640 7969 y FA(k)p FD(f)g FJ(\()p FD(h)p FJ(\))p FA(k)11512 8231 y FC(H)12353 7797 y Ft(\013;\033)12285 8540 y(F)14867 7969 y FJ(=)16997 6441 y Fz(\015)16997 7238 y(\015)16997 8035 y(\015)18164 6707 y(X)17956 9513 y FC(n)p Fw(\025)p FC(n)19830 9636 y Fu(0)20889 7070 y FJ(1)p 20645 7663 1138 54 v 20645 8880 a FD(n)p FJ(!)22137 7969 y FD(d)22813 7420 y FC(n)23439 7969 y FD(f)g FJ(\(0\)[)p FD(h;)221 b(:)g(:)g(:)j(;)d(h)p FJ(])31017 6441 y Fz(\015)31017 7238 y(\015)31017 8035 y(\015)31756 8231 y FC(H)32597 7797 y Ft(\013;\033)32529 8540 y(F)14857 11992 y FA(\024)17205 10730 y Fz(X)16997 13537 y FC(n)p Fw(\025)p FC(n)18871 13660 y Fu(0)19930 11094 y FJ(1)p 19686 11687 V 19686 12904 a FD(n)p FJ(!)20956 11992 y FA(k)p FD(d)22296 11444 y FC(n)22923 11992 y FD(f)142 b FJ(\(0\)[)p FD(h;)221 b(:)g(:)g(:)i(;)e(h)p FJ(])p FA(k)31164 12255 y FC(H)32005 11821 y Ft(\013;\033)31937 12563 y(F)14857 16016 y FA(\024)17205 14754 y Fz(X)16997 17560 y FC(n)p Fw(\025)p FC(n)18871 17683 y Fu(0)19930 15118 y FJ(1)p 19686 15711 V 19686 16927 a FD(n)p FJ(!)21178 16016 y FD(n)21954 15468 y FC(\033)22579 16016 y FD(B)23767 14838 y Ft(n)p Fo(\000)p Fu(1)p 23767 15010 1557 40 v 24342 15559 a(2)23567 16342 y FB(2)p FC(\033)25512 16016 y FA(k)p FD(d)26852 15468 y FC(n)27478 16016 y FD(f)142 b FJ(\(0\))p FA(k)30587 16215 y FB(op)31636 16016 y FA(k)p FD(h)p FA(k)33713 15468 y FC(n)33713 16400 y(H)34554 15966 y Ft(\013;\033)34486 16709 y(E)14857 20211 y FA(\024)17205 18949 y Fz(X)16997 21755 y FC(n)p Fw(\025)p FC(n)18871 21878 y Fu(0)19553 18338 y Fz(\022)20907 19313 y FJ(1)p 20664 19906 1138 54 v 20664 21122 a FD(n)p FJ(!)22156 20211 y FD(n)22932 19663 y FC(\033)23557 20211 y FD(B)24745 19033 y Ft(n)p Fo(\000)p Fu(1)p 24745 19205 1557 40 v 25320 19754 a(2)24545 20537 y FB(2)p FC(\033)26490 20211 y FD(M)27748 20410 y FC(r)28173 20533 y Fu(1)28822 19313 y FD(n)29598 18831 y FC(n)p 28822 19906 1403 54 v 28899 21122 a FD(r)29521 20665 y FC(n)29485 21448 y FB(1)30357 18338 y Fz(\023)31557 20211 y FA(k)p FD(h)p FA(k)33634 19663 y FC(n)33634 20595 y(H)34475 20161 y Ft(\013;\033)34407 20904 y(E)36093 20211 y FD(;)0 24325 y FJ(where,)441 b(in)e(the)g(last)h(inequalit)-36 b(y)-108 b(,)442 b(w)-36 b(e)439 b(used)g(Lemma)g(5.1.5.)597 b(Therefore,)442 b(w)-36 b(e)440 b(can)f(c)-36 b(ho)36 b(ose)0 25931 y FD(M)508 b FJ(:=)369 b FD(M)4766 26130 y FC(r)5191 26253 y Fu(1)6141 25931 y FJ(and)433 b FD(r)405 b(<)369 b(r)11628 26130 y FB(1)12587 25931 y FJ(suitably)434 b(small.)0 27536 y(No)-36 b(w,)434 b(let)g(us)f(pro)-36 b(v)g(e)434 b(that)f FD(f)13835 27735 y Fw(\003)14795 27536 y FJ(is)g(con)-36 b(tin)g(uously)434 b(di\013eren)-36 b(tiable.)578 b(W)-108 b(e)433 b(ha)-36 b(v)g(e)13261 30396 y FD(D)36 b(f)15018 30595 y Fw(\003)15913 30396 y FJ(:)369 b FD(H)17824 29766 y FC(\013;\033)17721 30756 y(E)19683 30396 y FA(\000)-221 b(!)369 b(L)23108 29320 y Fz(\000)23717 30396 y FD(H)24898 29766 y FC(\013;\033)24795 30756 y(E)26388 30396 y FD(;)221 b(H)28151 29766 y FC(\013;\033)28048 30756 y(F)29642 29320 y Fz(\001)30472 30396 y FD(:)0 33458 y FJ(Noting)369 b(that)e FD(f)7769 32976 y Fw(0)8449 33458 y FJ(:)i FD(E)447 b FA(\000)-222 b(!)370 b(L)p FJ(\()p FD(E)78 b(;)221 b(F)181 b FJ(\),)381 b(one)368 b(has)g FD(D)36 b(f)24744 33657 y Fw(\003)25270 33458 y FJ(\()p FD(h)p FJ(\))368 b(=)h FD(f)29563 32976 y Fw(0)30035 33458 y FA(\016)161 b FD(h)p FJ(,)382 b(namely)369 b(for)f(ev)-36 b(ery)42104 33108 y(~)42091 33458 y FD(h)369 b FA(2)0 35428 y FD(H)1181 34799 y FC(\013;\033)1078 35789 y(E)2671 35428 y FJ(,)3510 34353 y Fz(\000)4119 35428 y FD(D)36 b(f)5876 35627 y Fw(\003)6402 35428 y FJ(\()p FD(h)p FJ(\))8163 34353 y Fz(\001)8771 35428 y FJ([)9145 35078 y(~)9132 35428 y FD(h)p FJ(])429 b FA(2)g FD(H)13167 34799 y FC(\013;\033)13064 35789 y(F)14657 35428 y FJ(,)478 b(is)469 b(de\014ned)e(b)-36 b(y)23196 33954 y Fz(\020)23990 34353 y(\000)24598 35428 y FD(D)36 b(f)26355 35627 y Fw(\003)26881 35428 y FJ(\()p FD(h)p FJ(\))28642 34353 y Fz(\001)29251 35428 y FJ([)29625 35078 y(~)29612 35428 y FD(h)p FJ(])30722 33954 y Fz(\021)31515 35428 y FJ(\()p FD(t)p FJ(\))429 b(:=)35228 34353 y Fz(\000)35837 35428 y FD(f)36620 34946 y Fw(0)36930 35428 y FJ(\()p FD(h)p FJ(\()p FD(t)p FJ(\)\))40173 34353 y Fz(\001)40781 35428 y FJ([)41155 35078 y(~)41142 35428 y FD(h)p FJ(\()p FD(t)p FJ(\)].)0 37399 y(Recalling)435 b(\(5.9\),)f(w)-36 b(e)434 b(ha)-36 b(v)g(e)10549 39775 y Fz(\000)11158 40851 y FD(f)11941 40303 y Fw(0)12251 40851 y FJ(\()p FD(x)p FJ(\))14002 39775 y Fz(\001)14611 40851 y FJ([)81 b(~)-731 b FD(x)p FJ(])370 b(:=)18392 39589 y Fz(X)18184 42395 y FC(n)p Fw(\025)p FC(n)20058 42518 y Fu(0)21116 39953 y FJ(1)p 20873 40546 1138 54 v 20873 41762 a FD(n)p FJ(!)22365 40851 y FD(n)221 b(d)24038 40303 y FC(n)24664 40851 y FD(f)142 b FJ(\(0\)[)p FD(x;)221 b(:)g(:)g(:)k(;)c(x;)303 b FJ(~)-732 b FD(x)q FJ(])7226 b(\(5.10\))0 45039 y(and)433 b(hence,)g(using)h(the)f(symmetry)h(of)g FD(d)20381 44557 y FC(n)21007 45039 y FD(f)142 b FJ(\(0\),)2072 47482 y Fz(\000)2681 48558 y FD(f)3464 48009 y Fw(0)3774 48558 y FJ(\()p FD(x)p FJ(\))5525 47482 y Fz(\001)6134 48558 y FJ([)81 b(~)-731 b FD(x)p FJ(])370 b(=)9784 47659 y FD(d)p 9478 48252 1289 54 v 9478 49469 a(ds)10899 48395 y Fz(\014)10899 49192 y(\014)11342 50045 y Ft(s)p Fu(=0)12868 48558 y FD(f)142 b FJ(\()p FD(x)295 b FJ(+)g FD(s)81 b FJ(~)-731 b FD(x)p FJ(\))368 b(:=)20674 47296 y Fz(X)20466 50102 y FC(n)p Fw(\025)p FC(n)22340 50225 y Fu(0)25042 47659 y FJ(1)p 23155 48252 4424 54 v 23155 49469 a(\()p FD(n)296 b FA(\000)f FJ(1\)!)27933 48558 y FD(d)28609 48009 y FC(n)29235 48558 y FD(f)142 b FJ(\(0\)[)p FD(x;)221 b(:)g(:)g(:)j(;)d(x;)303 b FJ(~)-732 b FD(x)r FJ(])221 b FD(:)2073 b FJ(\(5.11\))0 52657 y(Denoting)12007 54262 y FD(g)12630 54461 y FC(n)13256 54262 y FJ(\()p FD(t)p FJ(\))369 b(:=)g FD(d)17525 53714 y FC(n)18151 54262 y FD(f)142 b FJ(\(0\)[)p FD(h)p FJ(\()p FD(t)p FJ(\))p FD(;)221 b(:)g(:)g(:)i(;)e(h)p FJ(\()p FD(t)p FJ(\))p FD(;)28926 53912 y FJ(~)28913 54262 y FD(h)p FJ(\()p FD(t)p FJ(\)])g FD(;)0 56555 y FJ(w)-36 b(e)434 b(obtain)f(that)g FD(D)36 b(f)10536 56754 y Fw(\003)11062 56555 y FJ(\()p FD(h)p FJ(\))433 b(is)h(b)36 b(ounded,)433 b(indeed)1014 58651 y Fz(\015)1014 59449 y(\015)1752 59781 y FD(D)36 b(f)3509 59980 y Fw(\003)4035 59781 y FJ(\()p FD(h)p FJ(\))5796 58651 y Fz(\015)5796 59449 y(\015)6534 60301 y Fw(L)p FB(\()p FC(H)8388 59867 y Ft(\013;\033)8320 60609 y(E)9729 60301 y FC(;H)10831 59867 y Ft(\013;\033)10763 60609 y(F)12173 60301 y FB(\))13712 59781 y FJ(=)2537 b(sup)15842 61088 y Fw(k)16322 60854 y FB(~)16312 61088 y FC(h)p Fw(k)17324 61427 y Ft(H)18038 61069 y(\013;\033)17988 61781 y(E)19436 61088 y FB(=1)20859 58253 y Fz(\015)20859 59050 y(\015)20859 59847 y(\015)21597 59781 y FJ(\()p FD(D)36 b(f)23860 59980 y Fw(\003)24386 59781 y FJ(\()p FD(h)p FJ(\)\)[)27027 59430 y(~)27014 59781 y FD(h)o FJ(])28123 58253 y Fz(\015)28123 59050 y(\015)28123 59847 y(\015)28862 60699 y FC(H)29703 60265 y Ft(\013;\033)29635 61008 y(F)13712 64111 y FJ(=)2537 b(sup)15842 65418 y Fw(k)16322 65185 y FB(~)16312 65418 y FC(h)p Fw(k)17324 65757 y Ft(H)18038 65399 y(\013;\033)17988 66111 y(E)19436 65418 y FB(=1)20859 62584 y Fz(\015)20859 63381 y(\015)20859 64178 y(\015)22027 62849 y(X)21819 65655 y FC(n)p Fw(\025)p FC(n)23693 65778 y Fu(0)26394 63212 y FJ(1)p 24508 63806 V 24508 65022 a(\()p FD(n)295 b FA(\000)h FJ(1\)!)29285 64111 y FD(g)29908 64310 y FC(n)30534 62584 y Fz(\015)30534 63381 y(\015)30534 64178 y(\015)31272 65030 y FC(H)32113 64596 y Ft(\013;\033)32045 65338 y(F)13702 68442 y FA(\024)2526 b FJ(sup)15842 69749 y Fw(k)16322 69515 y FB(~)16312 69749 y FC(h)p Fw(k)17324 70088 y Ft(H)18038 69730 y(\013;\033)17988 70442 y(E)19436 69749 y FB(=1)21068 67180 y Fz(X)20859 69986 y FC(n)p Fw(\025)p FC(n)22733 70109 y Fu(0)25435 67543 y FJ(1)p 23549 68136 V 23549 69353 a(\()p FD(n)295 b FA(\000)g FJ(1\)!)28326 68442 y FA(k)p FD(g)29613 68641 y FC(n)30239 68442 y FA(k)30903 68704 y FC(H)31744 68270 y Ft(\013;\033)31676 69012 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FB(i)p FC(k)24 b(t=\034)28102 6386 y FD(L)28987 6585 y FC(k)29556 6386 y Fs(,)710 b FD(L)31550 6585 y FC(k)32852 6386 y FA(2)733 b(L)p FJ(\()p FD(E)78 b(;)221 b(F)181 b FJ(\))p Fs(,)710 b(then)661 b(the)g(\\pr)-66 b(o)g(duct)6299 7991 y(op)g(er)g(ator",)464 b(namely)g(the)h(T)-100 b(o)-66 b(epliz)465 b(op)-66 b(er)g(ator,)463 b FD(L)29932 7509 y Fw(\003)30458 7991 y Fs(,)i(de\014ne)-66 b(d)463 b(as)14208 10075 y Fz(\000)14816 11151 y FD(L)15701 10603 y Fw(\003)16227 11151 y FJ([)p FD(x)p FJ(])17688 10075 y Fz(\001)18298 11151 y FJ(\()p FD(t)p FJ(\))368 b(:=)21890 10075 y Fz(\000)22499 11151 y FD(L)p FJ(\()p FD(t)p FJ(\))24866 10075 y Fz(\001)25474 11151 y FJ([)p FD(x)p FJ(\()p FD(t)p FJ(\)])h(=)30166 9889 y Fz(X)30869 12718 y FC(k)32307 11151 y FD(e)32910 10603 y FB(i)p FC(k)24 b(t=\034)35291 9889 y Fz(X)36058 12718 y FC(`)37431 11151 y FD(L)38316 11350 y FC(k)g Fw(\000)p FC(`)40001 11151 y FJ([)p FD(x)41101 11350 y FC(`)41541 11151 y FJ(])221 b FD(;)6299 15355 y Fs(b)-66 b(elongs)464 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y FJ(W)-108 b(e)417 b(p)36 b(erformed)417 b(a)g(partial)h (Birkho\013)f(Normal)h(F)-108 b(orm,)420 b(see)d(Prop)36 b(osition)417 b(4.1.6,)423 b(de\014ning)6299 43082 y(a)434 b(symplectic)g(c)-36 b(hange)434 b(of)g(co)36 b(ordinates)18331 46084 y(z)369 b(=)g(\000\(z)22556 46283 y Fw(\003)23081 46084 y FJ(\))g(:=)25698 45008 y Fz(\000)26307 46084 y FJ(\000)27120 46283 y FB(1)27645 46084 y FJ(\(z)28729 46283 y Fw(\003)29255 46084 y FD(;)185 b FJ(\026)-614 b(z)30415 46283 y Fw(\003)30941 46084 y FJ(\))p FD(;)221 b FJ(\000)32842 46283 y FB(2)33368 46084 y FJ(\(z)34452 46283 y Fw(\003)34978 46084 y FD(;)185 b FJ(\026)-614 b(z)36138 46283 y Fw(\003)36665 46084 y FJ(\))37171 45008 y Fz(\001)38001 46084 y FD(;)8708 b FJ(\(5.13\))6299 49086 y(suc)-36 b(h)452 b(that)g FD(H)410 b FA(\016)308 b FJ(\000)401 b(=)g FD(H)410 b FA(\016)309 b FD(X)20820 48604 y FC(t)20716 49421 y(F)21498 47957 y Fz(\014)21498 48754 y(\014)21940 49606 y FC(t)p FB(=1)23538 49086 y FJ(.)635 b(The)452 b(transformation)h(\000)f(is)h(obtained)f(as)h(the)f (map)6299 50691 y(at)609 b(time)f FD(t)666 b FJ(=)g(1)608 b(of)h(the)f(\015o)-36 b(w)609 b(of)g(the)e(hamiltonian)i(v)-36 b(ector\014eld)608 b FD(X)41240 50890 y FC(F)42022 50691 y FJ(,)652 b(giv)-36 b(en)609 b(b)-36 b(y)608 b(the)6299 52296 y(Hamiltonian)21144 53969 y FD(F)549 b FJ(=)24147 52707 y Fz(X)23916 55536 y FC(i;j)o(;k)24 b(;`)26297 52925 y Fw(0)26608 53969 y FD(F)27450 54168 y FC(ij)51 b(k)24 b(`)29376 53969 y FD(w)30306 54168 y FC(i)30682 53969 y FD(w)31612 54168 y FC(j)32099 53969 y FD(w)33029 54168 y FC(k)33597 53969 y FD(w)34527 54168 y FC(`)35188 53969 y FD(;)11521 b FJ(\(5.14\))6299 57748 y(with)583 b(co)36 b(e\016cien)-36 b(ts)583 b(\(4.20\),)620 b(where)583 b FD(w)25284 57947 y FC(i)26282 57748 y FJ(:=)622 b(z)28855 57947 y Fw(\003)p FC(i)30284 57748 y FJ(and)582 b FD(w)33892 57947 y Fw(\000)p FC(i)35622 57748 y FJ(:=)k(\026)-614 b(z)38195 57947 y Fw(\003)p FC(i)39041 57748 y FJ(.)1026 b(Therefore)583 b(\000\()p FD(t)p FJ(\))621 b(=)6299 58277 y Fz(\000)6908 59353 y FJ(\000)7721 59552 y FB(1)8247 59353 y FJ(\()p FD(t)p FJ(\))p FD(;)221 b FJ(\000)11124 59552 y FB(2)11649 59353 y FJ(\()p FD(t)p FJ(\))13131 58277 y Fz(\001)14173 59353 y FJ(satis\014es)434 b(the)f(equations) 21520 61483 y Fz(\032)23559 62207 y FJ(_)23070 62542 y(\000)23883 62741 y FB(1)24777 62542 y FJ(=)369 b FA(\000)p FJ(i)p FD(@)28213 62741 y FB(\026)-444 b(z)28657 62852 y Fo(\003)29192 62542 y FD(F)181 b FJ(\(\000)31534 62741 y FB(1)32059 62542 y FD(;)221 b FJ(\000)33454 62741 y FB(2)33981 62542 y FJ(\))23559 63905 y(_)23070 64241 y(\000)23883 64440 y FB(2)24777 64241 y FJ(=)369 b(i)p FD(@)27206 64440 y FB(z)27624 64551 y Fo(\003)28159 64241 y FD(F)181 b FJ(\(\000)30501 64440 y FB(1)31026 64241 y FD(;)221 b FJ(\000)32421 64440 y FB(2)32947 64241 y FJ(\))g FD(;)47070 63356 y FJ(\(5.15\))6299 67226 y(with)566 b(initial)h(datum)e(\000\(0\))594 b(=)g(\(z)23294 67425 y Fw(\003)23819 67226 y FD(;)185 b FJ(\026)-614 b(z)24979 67425 y Fw(\003)25506 67226 y FJ(\).)975 b(It)566 b(follo)-36 b(ws)568 b(that)d(\(z)p FD(;)185 b FJ(\026)-614 b(z)q(\))594 b(=)g(\(\000)42512 67425 y FB(1)43037 67226 y FJ(\(1\))p FD(;)221 b FJ(\000)46094 67425 y FB(2)46620 67226 y FJ(\(1\)\))594 b(=)6299 68831 y(\000\(z)8196 69030 y Fw(\003)8722 68831 y FD(;)185 b FJ(\026)-614 b(z)9882 69030 y Fw(\003)10408 68831 y FJ(\).)6299 71657 y FE(Lemma)499 b(5.2.1.)555 b Fs(L)-66 b(et)463 b(b)-66 b(e)464 b FD(F)646 b Fs(a)465 b(Hamiltonian)e(as)i(in)f FJ(\(5.14\))p Fs(.)599 b(Then)11646 74659 y FD(@)12307 74858 y FB(\026)-444 b(z)12751 74969 y Fo(\003)13286 74659 y FD(F)181 b FJ(\(z)15393 74858 y Fw(\003)15918 74659 y FD(;)k FJ(\026)-614 b(z)17078 74858 y Fw(\003)17605 74659 y FJ(\))369 b(=)f FD(@)20547 74858 y FB(z)20965 74969 y Fo(\003)21500 74659 y FD(F)181 b FJ(\(z)23607 74858 y Fw(\003)24132 74659 y FD(;)k FJ(\026)-614 b(z)25292 74858 y Fw(\003)25819 74659 y FJ(\))221 b FD(;)p 29786 73504 6465 54 v 2879 w(@)30473 74858 y FB(z)30891 74969 y Fo(\003)31425 74659 y FD(F)181 b FJ(\(z)33532 74858 y Fw(\003)34058 74659 y FD(;)k FJ(\026)-614 b(z)35218 74858 y Fw(\003)35744 74659 y FJ(\))369 b(=)g FD(@)38661 74858 y FB(\026)-444 b(z)39105 74969 y Fo(\003)39639 74659 y FD(F)181 b FJ(\(z)41746 74858 y Fw(\003)42272 74659 y FD(;)k FJ(\026)-614 b(z)43432 74858 y Fw(\003)43958 74659 y FJ(\))221 b FD(:)27371 78312 y FJ(155)p eop %%Page: 156 156 156 155 bop 0 4488 a Fp(pr)-30 b(oof.)658 b FJ(De\014ne,)434 b(recalling)h(\(4.20\),)13399 7395 y(~)13105 7731 y FD(F)13947 7930 y FC(i;j)o(;k)24 b(;`)16753 7731 y FJ(:=)23377 6832 y FD(G)24403 7031 y FC(ij)51 b(k)24 b(`)p 18628 7425 12230 54 v 18628 8642 a FJ(16\()p FD(!)21292 8185 y Fw(0)21244 8983 y FC(i)21915 8642 y FJ(+)295 b FD(!)24080 8185 y Fw(0)24032 8983 y FC(j)24814 8642 y FJ(+)g FD(!)26979 8185 y Fw(0)26931 9012 y FC(k)27794 8642 y FJ(+)g FD(!)29959 8185 y Fw(0)29911 9012 y FC(`)30351 8642 y FJ(\))40771 7731 y(\(5.16\))0 11476 y(suc)-36 b(h)405 b(that)5991 11141 y(~)5696 11476 y FD(F)6538 11675 y FC(i;j)o(;k)24 b(;`)9381 11476 y FJ(are)406 b(real)h(and)e(they)h(satis\014es)24481 11141 y(~)24186 11476 y FD(F)25028 11675 y FC(i;j)o(;k)24 b(;`)27834 11476 y FJ(=)369 b FA(\000)30542 11141 y FJ(~)30248 11476 y FD(F)31090 11675 y Fw(\000)p FC(i;)p Fw(\000)p FC(j)o(;)p Fw(\000)p FC(k)24 b(;)p Fw(\000)p FC(`)36454 11476 y FJ(.)569 b(F)-108 b(rom)406 b(\(5.14\))0 13081 y(and)433 b(\(4.20\),)i(w)-36 b(e)434 b(ha)-36 b(v)g(e)12171 16054 y FD(F)181 b FJ(\()-36 b(\026)-614 b(z)14278 16253 y Fw(\003)14804 16054 y FD(;)221 b FJ(z)15964 16253 y Fw(\003)16490 16054 y FJ(\))369 b(=)g FA(\000)p FJ(i)20372 14792 y Fz(X)20140 17621 y FC(i;j)o(;k)24 b(;`)22522 15010 y Fw(0)23127 15718 y FJ(~)22832 16054 y FD(F)23674 16253 y FC(i;j)o(;k)g(;`)26333 16054 y FD(w)27263 16253 y FC(i)27638 16054 y FD(w)28568 16253 y FC(j)29055 16054 y FD(w)29985 16253 y FC(k)30554 16054 y FD(w)31484 16253 y FC(`)40771 16054 y FJ(\(5.17\))0 19953 y(where)384 b FD(w)4638 20152 y FC(i)5383 19953 y FJ(=)368 b(z)7341 20152 y Fw(\003)p FC(i)8572 19953 y FJ(for)384 b FD(i)369 b FA(\025)g FJ(1)385 b(and)f FD(w)17169 20152 y FC(i)17913 19953 y FJ(=)332 b(\026)-613 b(z)19872 20152 y Fw(\003)p FC(i)21102 19953 y FJ(for)385 b FD(i)368 b FA(\024)i(\000)p FJ(1.)562 b(Therefore,)395 b(since)384 b FD(@)37870 20152 y FB(\026)-444 b(z)38314 20263 y Fo(\003)p Ft(n)39729 19953 y FJ(=)369 b FD(@)41797 20152 y FC(w)42472 20271 y Fo(\000)p Ft(n)43733 19953 y FJ(,)156 23792 y FD(@)817 23991 y FB(\026)-444 b(z)1261 24102 y Fo(\003)p Ft(n)2307 23792 y FD(F)181 b FJ(\(z)4414 23991 y Fw(\003)4939 23792 y FD(;)k FJ(\026)-614 b(z)6099 23991 y Fw(\003)6626 23792 y FJ(\))1106 b(=)h FD(@)11044 23991 y FC(w)11719 24110 y Fo(\000)p Ft(n)13202 21521 y Fz( )14253 23792 y FA(\000)p FJ(i)15879 22530 y Fz(X)15647 25359 y FC(i;j)o(;k)24 b(;`)18029 22748 y Fw(0)18635 23456 y FJ(~)18340 23792 y FD(F)19182 23991 y FC(i;j)o(;k)g(;`)21840 23792 y FD(w)22770 23991 y FC(i)23146 23792 y FD(w)24076 23991 y FC(j)24562 23792 y FD(w)25492 23991 y FC(k)26061 23792 y FD(w)26991 23991 y FC(`)27431 21521 y Fz(!)8238 27643 y FJ(=)1107 b FA(\000)p FJ(i)11983 26381 y Fz(X)11751 29210 y FC(i;j)o(;k)24 b(;`)14133 26599 y Fw(0)14738 27307 y FJ(~)14444 27643 y FD(F)15286 27842 y FC(i;j)o(;k)g(;`)17723 26169 y Fz(\020)18516 27643 y FD(\016)19092 27842 y Fw(\000)p FC(n;i)21253 27643 y FD(w)22183 27842 y FC(j)22670 27643 y FD(w)23600 27842 y FC(k)24169 27643 y FD(w)25099 27842 y FC(`)25833 27643 y FJ(+)295 b FD(:)221 b(:)g(:)297 b FJ(+)e FD(\016)30845 27842 y Fw(\000)p FC(n;`)33070 27643 y FD(w)34000 27842 y FC(i)34375 27643 y FD(w)35305 27842 y FC(j)35792 27643 y FD(w)36722 27842 y FC(k)37291 26169 y Fz(\021)8238 31494 y FJ(=)1107 b FA(\000)p FJ(i)11983 30232 y Fz(X)11751 33061 y FC(i;j)o(;k)24 b(;`)14133 30450 y Fw(0)14738 31158 y FJ(~)14444 31494 y FD(F)15286 31693 y Fw(\000)p FC(i;)p Fw(\000)p FC(j)o(;)p Fw(\000)p FC(k)g(;)p Fw(\000)p FC(`)20650 30020 y Fz(\020)21443 31494 y FD(\016)22019 31693 y FC(n;i)23449 31494 y FD(w)24379 31693 y Fw(\000)p FC(j)25597 31494 y FD(w)26527 31693 y Fw(\000)p FC(k)27828 31494 y FD(w)28758 31693 y Fw(\000)p FC(`)30224 31494 y FJ(+)295 b FD(:)221 b(:)g(:)297 b FJ(+)e FD(\016)35236 31693 y FC(n;`)36729 31494 y FD(w)37659 31693 y Fw(\000)p FC(i)38766 31494 y FD(w)39696 31693 y Fw(\000)p FC(j)40915 31494 y FD(w)41845 31693 y Fw(\000)p FC(k)43145 30020 y Fz(\021)8238 35345 y FJ(=)1107 b(i)10950 34083 y Fz(X)10718 36912 y FC(i;j)o(;k)24 b(;`)13100 34301 y Fw(0)13705 35009 y FJ(~)13411 35345 y FD(F)14253 35544 y FC(i;j)o(;k)g(;`)16689 33871 y Fz(\020)17483 35345 y FD(\016)18059 35544 y FC(n;i)19754 35345 y FJ(\026)-916 b FD(w)20418 35544 y FC(j)21171 35345 y FJ(\026)g FD(w)21835 35544 y FC(k)22670 35345 y FJ(\026)g FD(w)23334 35544 y FC(`)24068 35345 y FJ(+)295 b FD(:)221 b(:)g(:)297 b FJ(+)e FD(\016)29080 35544 y FC(n;`)30839 35345 y FJ(\026)-916 b FD(w)31503 35544 y FC(i)32145 35345 y FJ(\026)g FD(w)32809 35544 y FC(j)33561 35345 y FJ(\026)g FD(w)34225 35544 y FC(k)34794 33871 y Fz(\021)8238 39462 y FJ(=)p 10357 37766 25467 54 v 1107 w FA(\000)p FJ(i)11983 38200 y Fz(X)11751 41029 y FC(i;j)o(;k)24 b(;`)14133 38418 y Fw(0)14738 39126 y FJ(~)14444 39462 y FD(F)15286 39661 y FC(i;j)o(;k)g(;`)17944 39462 y FJ(\()p FD(\016)19026 39661 y FC(n;i)20234 39462 y FD(w)21164 39661 y FC(j)21651 39462 y FD(w)22581 39661 y FC(k)23149 39462 y FD(w)24079 39661 y FC(`)24814 39462 y FJ(+)295 b FD(:)221 b(:)g(:)297 b FJ(+)d FD(\016)29825 39661 y FC(n;`)31097 39462 y FD(w)32027 39661 y FC(i)32403 39462 y FD(w)33333 39661 y FC(j)33820 39462 y FD(w)34750 39661 y FC(k)35318 39462 y FJ(\))0 43360 y(where)350 b(w)-36 b(e)351 b(used)f FD(\016)8969 43559 y Fw(\000)p FC(n;)p Fw(\000)p FC(i)12009 43360 y FJ(=)369 b FD(\016)13966 43559 y FC(n;i)15524 43360 y FJ(and)350 b FD(w)18900 43559 y Fw(\000)p FC(i)20376 43360 y FJ(=)635 b(\026)-916 b FD(w)22687 43559 y FC(i)23062 43360 y FJ(.)551 b(In)350 b(the)g(same)h(w)-36 b(a)g(y)-108 b(,)368 b(one)350 b(has)g FD(@)38897 43559 y FB(z)39315 43670 y Fo(\003)p Ft(n)40730 43360 y FJ(=)369 b FD(@)42798 43559 y FC(w)43473 43670 y Ft(n)0 44966 y FJ(and)433 b(hence)4262 47564 y FD(@)4949 47763 y FB(z)5367 47874 y Fo(\003)p Ft(n)6413 47564 y FD(F)181 b FJ(\(z)8520 47763 y Fw(\003)9045 47564 y FD(;)k FJ(\026)-614 b(z)10205 47763 y Fw(\003)10732 47564 y FJ(\))368 b(=)h FA(\000)p FJ(i)14613 46302 y Fz(X)14381 49131 y FC(i;j)o(;k)24 b(;`)16763 46520 y Fw(0)17368 47228 y FJ(~)17074 47564 y FD(F)17916 47763 y FC(i;j)o(;k)g(;`)20353 46090 y Fz(\020)21146 47564 y FD(\016)21722 47763 y FC(n;i)23151 47564 y FD(w)24081 47763 y FC(j)24568 47564 y FD(w)25498 47763 y FC(k)26067 47564 y FD(w)26997 47763 y FC(`)27731 47564 y FJ(+)295 b FD(:)221 b(:)g(:)297 b FJ(+)e FD(\016)32743 47763 y FC(n;`)34236 47564 y FD(w)35166 47763 y FC(i)35542 47564 y FD(w)36472 47763 y FC(j)36958 47564 y FD(w)37888 47763 y FC(k)38457 46090 y Fz(\021)39472 47564 y FD(;)0 51463 y FJ(from)434 b(whic)-36 b(h)433 b(it)h(follo)-36 b(ws)436 b(that)p 13905 52906 6976 54 v 13905 54061 a FD(@)14566 54260 y FB(\026)-444 b(z)15010 54371 y Fo(\003)p Ft(n)16056 54061 y FD(F)181 b FJ(\(z)18163 54260 y Fw(\003)18689 54061 y FD(;)k FJ(\026)-614 b(z)19849 54260 y Fw(\003)20375 54061 y FJ(\))369 b(=)g FD(@)23318 54260 y FB(z)23736 54371 y Fo(\003)p Ft(n)24782 54061 y FD(F)181 b FJ(\(z)26889 54260 y Fw(\003)27414 54061 y FD(;)k FJ(\026)-614 b(z)28574 54260 y Fw(\003)29101 54061 y FJ(\))221 b FD(:)43061 56660 y Fn(\003)0 58265 y FJ(No)-36 b(w)434 b(w)-36 b(e)434 b(can)g(pro)-36 b(v)g(e)433 b(the)g(follo)-36 b(wing)0 60690 y FE(Lemma)591 b(5.2.2.)596 b Fs(L)-66 b(et)538 b(b)-66 b(e)538 b FJ(\000\()p FD(t)p FJ(\))504 b(=)17771 59614 y Fz(\000)18380 60690 y FJ(\000)19193 60889 y FB(1)19718 60690 y FJ(\()p FD(t)p FJ(\))p FD(;)221 b FJ(\000)22595 60889 y FB(2)23121 60690 y FJ(\()p FD(t)p FJ(\))24603 59614 y Fz(\001)25750 60690 y Fs(the)538 b(symple)-66 b(ctic)537 b(change)i(of)f(c)-66 b(o)g(or)g(di-)0 62295 y(nates)465 b(as)g(in)f FJ(\(5.13\))p Fs(,)h FJ(\000)369 b(=)g FD(X)14608 61813 y FC(t)14504 62631 y(F)15286 61166 y Fz(\014)15286 61963 y(\014)15728 62815 y FC(t)p FB(=1)17791 62295 y Fs(with)465 b FD(F)645 b Fs(as)466 b(in)e FJ(\(5.14\))p Fs(.)598 b(Then)p 15616 64026 2820 54 v 15616 65181 a FJ(\000)16429 65380 y FB(1)16955 65181 y FJ(\()p FD(t)p FJ(\))368 b(=)h(\000)20999 65380 y FB(2)21524 65181 y FJ(\()p FD(t)p FJ(\))221 b FD(;)2878 b FA(8)221 b FD(t)g(:)0 67780 y Fp(pr)-30 b(oof.)658 b FJ(It)434 b(results)f(that)p 13352 66624 3001 54 v 433 w(\000)14165 67979 y FB(1)14691 67780 y FJ(\(0\))369 b(=)332 b(\026)-614 b(z)18680 67979 y Fw(\003)19575 67780 y FJ(=)369 b(\000)21769 67979 y FB(2)22294 67780 y FJ(\(0\).)579 b(Moreo)-36 b(v)g(er,)434 b(b)-36 b(y)433 b(Lemma)h(5.2.1,)7961 70098 y(_)p 6732 69632 2820 54 v 6732 70788 a(\000)7545 70987 y FB(1)8070 70788 y FJ(\()p FD(t)p FJ(\))369 b(=)p 11301 69411 V 11527 70452 a(_)11301 70788 y(\000)12114 70987 y FB(1)12640 70788 y FJ(\()p FD(t)p FJ(\))f(=)h(i)p FD(@)16893 70987 y FB(\026)-444 b(z)17337 71098 y Fo(\003)17872 70788 y FD(F)181 b FJ(\(\000)20214 70987 y FB(1)20739 70788 y FD(;)221 b FJ(\000)22134 70987 y FB(2)22660 70788 y FJ(\))369 b(=)f(i)p FD(@)25963 70987 y FB(z)26381 71098 y Fo(\003)26917 70788 y FD(F)181 b FJ(\(\000)29259 70987 y FB(1)29783 70788 y FD(;)221 b FJ(\000)31178 70987 y FB(2)31705 70788 y FJ(\))368 b(=)34186 70452 y(_)33960 70788 y(\000)34773 70987 y FB(2)35299 70788 y FJ(\()p FD(t)p FJ(\))220 b FD(;)0 73386 y FJ(and)433 b(the)g(thesis)h(follo)-36 b(ws.)43061 74991 y Fn(\003)21072 78312 y FJ(156)p eop %%Page: 157 157 157 156 bop 6299 4488 a FE(Lemma)499 b(5.2.3.)555 b Fs(L)-66 b(et)463 b(b)-66 b(e)464 b FD(F)646 b Fs(as)465 b(in)f FJ(\(5.14\))p Fs(.)599 b(Then)10613 7373 y FD(@)11274 7572 y FB(\026)-444 b(z)11718 7683 y Fo(\003)12253 7373 y FD(F)181 b FJ(\()-36 b(\026)-614 b(z)14360 7572 y Fw(\003)14885 7373 y FD(;)221 b FJ(z)16045 7572 y Fw(\003)16572 7373 y FJ(\))368 b(=)h FA(\000)p FD(@)20547 7572 y FB(z)20965 7683 y Fo(\003)21500 7373 y FD(F)181 b FJ(\(z)23607 7572 y Fw(\003)24132 7373 y FD(;)k FJ(\026)-614 b(z)25292 7572 y Fw(\003)25819 7373 y FJ(\))221 b FD(;)2879 b(@)30473 7572 y FB(z)30891 7683 y Fo(\003)31425 7373 y FD(F)181 b FJ(\()-36 b(\026)-614 b(z)33532 7572 y Fw(\003)34058 7373 y FD(;)221 b FJ(z)35218 7572 y Fw(\003)35744 7373 y FJ(\))369 b(=)g FA(\000)p FD(@)39694 7572 y FB(\026)-444 b(z)40138 7683 y Fo(\003)40673 7373 y FD(F)181 b FJ(\(z)42780 7572 y Fw(\003)43305 7373 y FD(;)k FJ(\026)-614 b(z)44465 7572 y Fw(\003)44992 7373 y FJ(\))221 b FD(:)6299 10258 y Fp(pr)-30 b(oof.)659 b FJ(W)-108 b(e)433 b(claim)i(that)21839 13142 y FD(F)181 b FJ(\()-36 b(\026)-614 b(z)23946 13341 y Fw(\003)24471 13142 y FD(;)221 b FJ(z)25631 13341 y Fw(\003)26158 13142 y FJ(\))369 b(=)f FA(\000)p FD(F)181 b FJ(\(z)31553 13341 y Fw(\003)32079 13142 y FD(;)k FJ(\026)-614 b(z)33239 13341 y Fw(\003)33765 13142 y FJ(\))221 b FD(:)12217 b FJ(\(5.18\))6299 16027 y(Indeed,)433 b(from)h(\(5)p FD(:)p FJ(17\))h(and)e(recalling)25742 15691 y(~)25447 16027 y FD(F)26289 16226 y FC(i;j)o(;k)24 b(;`)29095 16027 y FJ(=)369 b FA(\000)31804 15691 y FJ(~)31509 16027 y FD(F)32351 16226 y Fw(\000)p FC(i;)p Fw(\000)p FC(j)o(;)p Fw(\000)p FC(k)24 b(;)p Fw(\000)p FC(`)37715 16027 y FJ(,)434 b(one)g(has)16269 19325 y FD(F)181 b FJ(\()-36 b(\026)-614 b(z)18376 19524 y Fw(\003)18901 19325 y FD(;)221 b FJ(z)20061 19524 y Fw(\003)20588 19325 y FJ(\))1107 b(=)f FA(\000)p FJ(i)25945 18063 y Fz(X)25713 20892 y FC(i;j)o(;k)24 b(;`)28095 18281 y Fw(0)28700 18989 y FJ(~)28406 19325 y FD(F)29248 19524 y FC(i;j)o(;k)g(;`)31906 19325 y FD(w)32836 19524 y Fw(\000)p FC(i)33944 19325 y FD(w)34874 19524 y Fw(\000)p FC(j)36092 19325 y FD(w)37022 19524 y Fw(\000)p FC(k)38323 19325 y FD(w)39253 19524 y Fw(\000)p FC(`)22201 23176 y FJ(=)1106 b FA(\000)p FJ(i)25945 21914 y Fz(X)25713 24743 y FC(i;j)o(;k)24 b(;`)28095 22132 y Fw(0)28700 22840 y FJ(~)28406 23176 y FD(F)29248 23375 y Fw(\000)p FC(i;)p Fw(\000)p FC(j)o(;)p Fw(\000)p FC(k)g(;)p Fw(\000)p FC(`)34834 23176 y FD(w)35764 23375 y FC(i)36139 23176 y FD(w)37069 23375 y FC(j)37556 23176 y FD(w)38486 23375 y FC(k)39054 23176 y FD(w)39984 23375 y FC(`)22201 27027 y FJ(=)1106 b(i)24912 25765 y Fz(X)24680 28594 y FC(i;j)o(;k)24 b(;`)27062 25983 y Fw(0)27667 26691 y FJ(~)27373 27027 y FD(F)28215 27226 y FC(i;j)o(;k)g(;`)30873 27027 y FD(w)31803 27226 y FC(i)32179 27027 y FD(w)33109 27226 y FC(j)33595 27027 y FD(w)34525 27226 y FC(k)35094 27027 y FD(w)36024 27226 y FC(`)22201 30338 y FJ(=)1106 b FA(\000)p FD(F)181 b FJ(\(z)27459 30537 y Fw(\003)27985 30338 y FD(;)k FJ(\026)-614 b(z)29145 30537 y Fw(\003)29671 30338 y FJ(\))221 b FD(:)6299 33223 y FJ(Therefore,)435 b(w)-36 b(e)433 b(get)12091 36693 y FD(@)12752 36892 y FB(\026)-444 b(z)13196 37003 y Fo(\003)13731 36693 y FD(F)181 b FJ(\()-36 b(\026)-614 b(z)15838 36892 y Fw(\003)16363 36693 y FD(;)221 b FJ(z)17523 36892 y Fw(\003)18050 36693 y FJ(\))368 b(=)20990 35794 y FD(d)p 20438 36387 1780 54 v 20438 37604 a(d)p FJ(z)21692 37803 y Fw(\003)22350 36693 y FD(F)181 b FJ(\()-36 b(\026)-614 b(z)24457 36892 y Fw(\003)24983 36693 y FD(;)221 b FJ(z)26143 36892 y Fw(\003)26669 36693 y FJ(\))369 b(=)29609 35794 y FD(d)p 29057 36387 V 29057 37604 a(d)p FJ(z)30311 37803 y Fw(\003)30970 36693 y FJ(\(z)32054 36892 y Fw(\003)32580 36693 y FD(;)185 b FJ(\026)-614 b(z)33740 36892 y Fw(\003)34266 36693 y FJ(\))369 b(=)g FA(\000)p FD(@)38242 36892 y FB(z)38660 37003 y Fo(\003)39195 36693 y FD(F)181 b FJ(\(z)41302 36892 y Fw(\003)41827 36693 y FD(;)k FJ(\026)-614 b(z)42987 36892 y Fw(\003)43514 36693 y FJ(\))221 b FD(:)6299 40116 y FJ(In)434 b(the)f(same)h(w)-36 b(a)g(y)-108 b(,)434 b(one)f(can)h(pro)-36 b(v)g(e)434 b(the)f(second)g(equalit)-36 b(y)435 b(of)f(the)f(thesis.)49361 41721 y Fn(\003)6299 44391 y FE(Lemma)592 b(5.2.4.)j Fs(L)-66 b(et)538 b(b)-66 b(e)538 b FJ(\000\()p FD(t)p FJ(\))505 b(=)24070 43315 y Fz(\000)24679 44391 y FJ(\000)25492 44590 y FB(1)26017 44391 y FJ(\()p FD(t)p FJ(\))p FD(;)221 b FJ(\000)28894 44590 y FB(2)29420 44391 y FJ(\()p FD(t)p FJ(\))30902 43315 y Fz(\001)32049 44391 y Fs(the)538 b(symple)-66 b(ctic)538 b(change)g(of)g(c)-66 b(o)g(or)g(di-)6299 45996 y(nates)465 b(as)g(in)f FJ(\(5.13\))p Fs(,)i FJ(\000)368 b(=)h FD(X)20907 45514 y FC(t)20803 46332 y(F)21585 44867 y Fz(\014)21585 45664 y(\014)22028 46517 y FC(t)p FB(=1)24090 45996 y Fs(with)465 b FD(F)645 b Fs(as)466 b(in)e FJ(\(5.14\))p Fs(.)598 b(Then)14294 48881 y FJ(\000)15107 49080 y FB(1)15632 48881 y FJ(\()-36 b(\026)-614 b(z)16716 49080 y Fw(\003)17242 48881 y FD(;)221 b FJ(z)18402 49080 y Fw(\003)18929 48881 y FJ(\))369 b(=)f(\000)21997 49080 y FB(2)22523 48881 y FJ(\(z)23607 49080 y Fw(\003)24132 48881 y FD(;)185 b FJ(\026)-614 b(z)25292 49080 y Fw(\003)25819 48881 y FJ(\))221 b FD(;)2879 b FJ(\000)30599 49080 y FB(2)31124 48881 y FJ(\()-36 b(\026)-614 b(z)32208 49080 y Fw(\003)32734 48881 y FD(;)221 b FJ(z)33894 49080 y Fw(\003)34420 48881 y FJ(\))369 b(=)g(\000)37489 49080 y FB(1)38014 48881 y FJ(\(z)39098 49080 y Fw(\003)39624 48881 y FD(;)185 b FJ(\026)-614 b(z)40784 49080 y Fw(\003)41310 48881 y FJ(\))221 b FD(:)4672 b FJ(\(5.19\))6299 51766 y Fp(pr)-30 b(oof.)1180 b FJ(De\014ne)596 b FD(\013)8 b(;)221 b(\014)671 b FJ(suc)-36 b(h)595 b(that)g(\000)25715 51965 y FB(1)26240 51766 y FJ(\()-36 b(\026)-614 b(z)27324 51965 y Fw(\003)27850 51766 y FD(;)221 b FJ(z)29010 51965 y Fw(\003)29537 51766 y FJ(\))644 b(=:)h FD(\013)8 b FJ(\(1\),)638 b(\000)37014 51965 y FB(2)37539 51766 y FJ(\()-36 b(\026)-614 b(z)38623 51965 y Fw(\003)39149 51766 y FD(;)221 b FJ(z)40309 51965 y Fw(\003)40835 51766 y FJ(\))645 b(=:)g FD(\014)74 b FJ(\(1\))596 b(where)6299 53371 y FD(\013)8 b FJ(\(0\))370 b(=)332 b(\026)-613 b(z)11125 53570 y Fw(\003)11650 53371 y FJ(,)434 b FD(\014)74 b FJ(\(0\))369 b(=)g(z)17243 53570 y Fw(\003)18202 53371 y FJ(and)22288 55212 y Fz(\032)24111 56217 y FJ(_)-634 b FD(\013)378 b FJ(=)368 b FA(\000)p FJ(i)p FD(@)28478 56416 y FB(\026)-444 b(z)28922 56527 y Fo(\003)29458 56217 y FD(F)181 b FJ(\()p FD(\013)8 b(;)221 b(\014)74 b FJ(\))24170 57580 y(_)23838 57930 y FD(\014)443 b FJ(=)369 b(i)p FD(@)27444 58129 y FB(z)27862 58240 y Fo(\003)28397 57930 y FD(F)181 b FJ(\()p FD(\013)8 b(;)221 b(\014)74 b FJ(\))221 b FD(:)47070 57085 y FJ(\(5.20\))6299 60931 y(By)358 b(Lemma)f(5.2.3,)374 b(w)-36 b(e)357 b(ha)-36 b(v)g(e)357 b(that)23739 59855 y Fz(\000)24348 60931 y FJ(\000)25161 61130 y FB(2)25686 60931 y FJ(\()p FD(t)p FJ(\))p FD(;)221 b FJ(\000)28563 61130 y FB(1)29089 60931 y FJ(\()p FD(t)p FJ(\))30571 59855 y Fz(\001)31536 60931 y FJ(satis\014es)357 b(the)g(system)g(\(5.20\),)373 b(indeed)16707 62945 y Fz(\032)18745 63668 y FJ(_)18256 64004 y(\000)19069 64203 y FB(2)19964 64004 y FJ(=)368 b(i)p FD(@)22366 64203 y FB(\026)-444 b(z)22810 64314 y Fo(\003)23346 64004 y FD(F)181 b FJ(\(\000)25688 64203 y FB(1)26212 64004 y FD(;)221 b FJ(\000)27607 64203 y FB(2)28134 64004 y FJ(\))368 b(=)h FA(\000)p FJ(i)p FD(@)32444 64203 y FB(\026)-444 b(z)32888 64314 y Fo(\003)33423 64004 y FD(F)181 b FJ(\(\000)35765 64203 y FB(2)36290 64004 y FD(;)221 b FJ(\000)37685 64203 y FB(1)38211 64004 y FJ(\))18745 65367 y(_)18256 65703 y(\000)19069 65902 y FB(1)19964 65703 y FJ(=)368 b FA(\000)p FJ(i)p FD(@)23399 65902 y FB(\026)-444 b(z)23843 66013 y Fo(\003)24379 65703 y FD(F)181 b FJ(\(\000)26721 65902 y FB(1)27246 65703 y FD(;)221 b FJ(\000)28641 65902 y FB(2)29167 65703 y FJ(\))369 b(=)f(i)p FD(@)32470 65902 y FB(z)32888 66013 y Fo(\003)33423 65703 y FD(F)181 b FJ(\(\000)35765 65902 y FB(2)36290 65703 y FD(;)221 b FJ(\000)37685 65902 y FB(1)38211 65703 y FJ(\))g FD(:)6299 68571 y FJ(Being)484 b FD(\013)8 b FJ(\(0\))456 b(=)418 b(\026)-614 b(z)15013 68770 y Fw(\003)15993 68571 y FJ(=)454 b(\000)18272 68770 y FB(2)18798 68571 y FJ(\(0\))484 b(and)f FD(\014)74 b FJ(\(0\))454 b(=)g(z)28491 68770 y Fw(\003)29472 68571 y FJ(=)g(\000)31751 68770 y FB(1)32276 68571 y FJ(\(0\),)497 b(it)484 b(follo)-36 b(ws)486 b(that)d FD(\013)8 b FJ(\()p FD(t)p FJ(\))455 b(=)f(\000)48387 68770 y FB(2)48912 68571 y FJ(\()p FD(t)p FJ(\))6299 70176 y(and)379 b FD(\014)74 b FJ(\()p FD(t)p FJ(\))368 b(=)h(\000)13626 70375 y FB(1)14151 70176 y FJ(\()p FD(t)p FJ(\))379 b(for)g(all)i FD(t)p FJ(.)559 b(Then,)390 b FD(\013)8 b FJ(\(1\))370 b(=)f(\000)29862 70375 y FB(2)30387 70176 y FJ(\(1\))g(=)g(\000)34612 70375 y FB(2)35137 70176 y FJ(\(z)36221 70375 y Fw(\003)36747 70176 y FD(;)185 b FJ(\026)-614 b(z)37907 70375 y Fw(\003)38434 70176 y FJ(\))378 b(and)h FD(\014)74 b FJ(\(1\))369 b(=)g(\000)46826 70375 y FB(1)47351 70176 y FJ(\(1\))g(=)6299 71781 y(\000)7112 71980 y FB(1)7638 71781 y FJ(\(z)8722 71980 y Fw(\003)9247 71781 y FD(;)185 b FJ(\026)-614 b(z)10407 71980 y Fw(\003)10934 71781 y FJ(\),)434 b(from)g(whic)-36 b(h)433 b(the)g(thesis)g(follo)-36 b(ws.)49361 73386 y Fn(\003)6299 74991 y FJ(No)g(w)434 b(w)-36 b(e)434 b(can)g(pro)-36 b(v)g(e)433 b(the)g(follo)-36 b(wing)436 b(theorem)27371 78312 y(157)p eop %%Page: 158 158 158 157 bop 0 4488 a FE(Theorem)591 b(5.2.5.)k Fs(L)-66 b(et)537 b(b)-66 b(e)537 b FJ(\000\()p FD(t)p FJ(\))503 b(=)18804 3412 y Fz(\000)19413 4488 y FJ(\000)20226 4687 y FB(1)20752 4488 y FJ(\()p FD(t)p FJ(\))p FD(;)221 b FJ(\000)23629 4687 y FB(2)24154 4488 y FJ(\()p FD(t)p FJ(\))25636 3412 y Fz(\001)26782 4488 y Fs(a)539 b(symple)-66 b(ctic)536 b(change)h(of)h(c)-66 b(o)g(or)g(di-)0 6094 y(nates)411 b(such)g(that)g FJ(z)369 b(:=)g(\000\(z)13457 6293 y Fw(\003)13983 6094 y FJ(\))411 b Fs(and)g(b)-66 b(e)410 b FA(H)382 b FJ(:=)368 b FD(H)278 b FA(\016)176 b FJ(\000)369 b(=)f FD(H)278 b FA(\016)176 b FD(X)30292 5611 y FC(t)30188 6429 y(F)30970 4964 y Fz(\014)30970 5761 y(\014)31413 6614 y FC(t)p FB(=1)33421 6094 y Fs(with)411 b FD(F)592 b Fs(as)411 b(in)g FJ(\(5.14\))0 7699 y Fs(and)465 b FD(H)567 b Fs(such)466 b(that)e FD(H)103 b FJ(\(z)p FD(;)185 b FJ(\026)-614 b(z\))369 b(=)g FD(H)103 b FJ(\()-36 b(\026)-614 b(z)o FD(;)221 b FJ(z\))p Fs(.)598 b(Then)15944 10348 y FA(H)13 b FJ(\()-36 b(\026)-614 b(z)18163 10547 y Fw(\003)18689 10348 y FD(;)221 b FJ(z)19849 10547 y Fw(\003)20375 10348 y FJ(\))369 b(=)g FA(H)13 b FJ(\(z)24850 10547 y Fw(\003)25375 10348 y FD(;)185 b FJ(\026)-614 b(z)26535 10547 y Fw(\003)27062 10348 y FJ(\))221 b FD(:)0 12998 y Fp(pr)-30 b(oof.)658 b FJ(W)-108 b(e)434 b(ha)-36 b(v)g(e,)434 b(b)-36 b(y)433 b(Lemma)h(5.2.2)h(and)e(Lemma)h(5.2.4,)195 15648 y FA(H)13 b FJ(\()-36 b(\026)-614 b(z)2414 15847 y Fw(\003)2939 15648 y FD(;)221 b FJ(z)4099 15847 y Fw(\003)4626 15648 y FJ(\))1107 b(=)f FD(H)9538 14572 y Fz(\000)10146 15648 y FJ(\000)10959 15847 y FB(1)11485 15648 y FJ(\()-36 b(\026)-614 b(z)12569 15847 y Fw(\003)13095 15648 y FD(;)221 b FJ(z)14255 15847 y Fw(\003)14781 15648 y FJ(\))p FD(;)g FJ(\000)16682 15847 y FB(2)17208 15648 y FJ(\()-36 b(\026)-614 b(z)18292 15847 y Fw(\003)18818 15648 y FD(;)221 b FJ(z)19978 15847 y Fw(\003)20504 15648 y FJ(\))21010 14572 y Fz(\001)21988 15648 y FJ(=)369 b FD(H)24550 14572 y Fz(\000)25158 15648 y FJ(\000)25971 15847 y FB(2)26496 15648 y FJ(\(z)27580 15847 y Fw(\003)28106 15648 y FD(;)185 b FJ(\026)-614 b(z)29266 15847 y Fw(\003)29793 15648 y FJ(\))p FD(;)221 b FJ(\000)31694 15847 y FB(1)32220 15648 y FJ(\(z)33304 15847 y Fw(\003)33829 15648 y FD(;)185 b FJ(\026)-614 b(z)34989 15847 y Fw(\003)35516 15648 y FJ(\))36022 14572 y Fz(\001)6239 17818 y FJ(=)1106 b FD(H)9538 16742 y Fz(\000)p 10146 16662 5141 54 v 10146 17818 a FJ(\000)10959 18017 y FB(1)11485 17818 y FJ(\(z)12569 18017 y Fw(\003)13095 17818 y FD(;)185 b FJ(\026)-614 b(z)14255 18017 y Fw(\003)14781 17818 y FJ(\))p FD(;)221 b FJ(\000)16682 18017 y FB(1)17208 17818 y FJ(\(z)18292 18017 y Fw(\003)18818 17818 y FD(;)185 b FJ(\026)-614 b(z)19978 18017 y Fw(\003)20504 17818 y FJ(\))21010 16742 y Fz(\001)21988 17818 y FJ(=)369 b FD(H)24550 16742 y Fz(\000)25158 17818 y FJ(\000)25971 18017 y FB(1)26496 17818 y FJ(\(z)27580 18017 y Fw(\003)28106 17818 y FD(;)185 b FJ(\026)-614 b(z)29266 18017 y Fw(\003)29793 17818 y FJ(\))p FD(;)p 30881 16662 V 221 w FJ(\000)31694 18017 y FB(1)32220 17818 y FJ(\(z)33304 18017 y Fw(\003)33829 17818 y FD(;)185 b FJ(\026)-614 b(z)34989 18017 y Fw(\003)35516 17818 y FJ(\))36022 16742 y Fz(\001)36999 17818 y FJ(=)369 b FA(H)13 b FJ(\(z)40599 18017 y Fw(\003)41124 17818 y FD(;)185 b FJ(\026)-614 b(z)42284 18017 y Fw(\003)42811 17818 y FJ(\))221 b FD(;)0 20467 y FJ(where)433 b(w)-36 b(e)434 b(used)f(also)i(the)e(symmetry)h(of)g(the)f(Hamiltonian)h FD(H)103 b FJ(.)43061 22073 y Fn(\003)0 23678 y FJ(W)-108 b(e)542 b(pro)-36 b(v)g(ed)541 b(that)h(after)g(the)f(Birkho\013)h (Normal)h(F)-108 b(orm,)568 b(the)542 b(Hamiltonian)g FA(H)555 b FJ(is)542 b(still)0 25283 y(symmetric,)340 b(namely)316 b FA(H)13 b FJ(\()-36 b(\026)-614 b(z)13214 25482 y Fw(\003)13739 25283 y FD(;)221 b FJ(z)14899 25482 y Fw(\003)15426 25283 y FJ(\))369 b(=)f FA(H)13 b FJ(\(z)19900 25482 y Fw(\003)20426 25283 y FD(;)185 b FJ(\026)-614 b(z)21586 25482 y Fw(\003)22112 25283 y FJ(\))221 b FD(:)316 b FJ(Let)f(us)h(in)-36 b(tro)36 b(duce)314 b(the)h(new)h(co)36 b(ordinates)0 26888 y(\(p)p FD(;)221 b FJ(q\))434 b(and)f(the)g (parameter)g FD(\021)417 b FA(2)368 b Fv(R)p FJ(,)434 b(b)-36 b(y)10020 29846 y(z)10598 30045 y Fw(\003)p FC(i)11813 29846 y FJ(:=)14222 28947 y FD(\021)p 13688 29540 1758 54 v 13688 29806 a FA(p)p 14795 29806 651 54 v 1099 x FJ(2)15578 29846 y(\(q)16770 30045 y FC(i)17441 29846 y FJ(+)295 b(ip)19832 30045 y FC(i)20207 29846 y FJ(\))221 b FD(;)1446 b FJ(\026)-614 b(z)23355 30045 y Fw(\003)p FC(i)24570 29846 y FJ(:=)26979 28947 y FD(\021)p 26445 29540 1758 54 v 26445 29806 a FA(p)p 27552 29806 651 54 v 1099 x FJ(2)28335 29846 y(\(q)29527 30045 y FC(i)30198 29846 y FA(\000)296 b FJ(ip)32611 30045 y FC(i)32986 29846 y FJ(\))221 b FD(:)6697 b FJ(\(5.21\))0 33442 y(In)598 b(the)g(v)-72 b(ariables)600 b(\(p)p FD(;)221 b FJ(q\),)640 b(de\014ning)598 b FD(H)20015 32960 y Fw(\003)20540 33442 y FJ(\(p)p FD(;)221 b FJ(q\))650 b(=)g FD(H)27036 32960 y Fw(\003)27561 33442 y FJ(\(p)p FD(;)221 b FJ(q;)g FD(\021)48 b FJ(\))650 b(:=)g FA(H)35644 32366 y Fz(\000)36252 33442 y FD(\021)48 b FJ(\(q)295 b(+)g(ip\))p FD(=)41976 32343 y FA(p)p 43083 32343 V 1099 x FJ(2)p FD(;)0 35223 y(\021)48 b FJ(\(q)295 b FA(\000)g FJ(ip\))p FD(=)5745 34124 y FA(p)p 6852 34124 V 1099 x FJ(2)7503 34147 y Fz(\001)8111 35223 y FJ(,)434 b(one)g(has)5083 38113 y FD(H)6264 37564 y Fw(\003)6789 38113 y FJ(\()p FA(\000)p FJ(p)p FD(;)221 b FJ(q\))1108 b(=)e FA(H)15186 37037 y Fz(\000)15795 38113 y FD(\021)48 b FJ(\(q)295 b FA(\000)g FJ(ip\))p FD(=)21540 36949 y FA(p)p 22647 36949 V 1164 x FJ(2)p FD(;)221 b(\021)48 b FJ(\(q)296 b(+)f(ip\))p FD(=)29604 36949 y FA(p)p 30711 36949 V 1164 x FJ(2)31361 37037 y Fz(\001)11933 40291 y FJ(=)1106 b FA(H)15186 39215 y Fz(\000)15795 40291 y FD(\021)48 b FJ(\(q)295 b(+)f(ip\))p FD(=)21518 39127 y FA(p)p 22625 39127 V 1164 x FJ(2)q FD(;)221 b(\021)48 b FJ(\(q)295 b FA(\000)h FJ(ip\))p FD(=)29604 39127 y FA(p)p 30711 39127 V 1164 x FJ(2)31361 39215 y Fz(\001)32339 40291 y FJ(=)368 b FD(H)34900 39742 y Fw(\003)35425 40291 y FJ(\(p)p FD(;)221 b FJ(q\))g FD(;)0 42940 y FJ(that)433 b(is)15655 44545 y FD(H)16836 43997 y Fw(\003)17361 44545 y FJ(\()p FA(\000)p FJ(p)p FD(;)221 b FJ(q\))370 b(=)e FD(H)24328 43997 y Fw(\003)24853 44545 y FJ(\(p)p FD(;)221 b FJ(q\))g FD(;)12333 b FJ(\(5.22\))0 46748 y(namely)409 b FD(H)5672 46266 y Fw(\003)6606 46748 y FJ(is)g(ev)-36 b(en)408 b(in)h(the)f(p-v)-72 b(ariable.)570 b(No)-36 b(w)409 b(w)-36 b(e)409 b(in)-36 b(tro)36 b(duce)408 b(action{angle)i(v)-72 b(ariables)0 48354 y(\()p FD(I)104 b(;)221 b(\036)p FJ(\))729 b FA(2)f Fv(R)6344 47872 y FC(N)6344 48682 y FB(+)7680 48354 y FA(\002)440 b Fv(T)10039 47872 y FC(N)11580 48354 y FJ(on)645 b(the)f FA(I)98 b FJ({mo)36 b(des)646 b(b)-36 b(y)645 b(the)f(follo)-36 b(wing)647 b(symplectic)f(c)-36 b(hange)645 b(of)0 49959 y(co)36 b(ordinates)6294 51220 y FA(p)p 7401 51220 V 1099 x FJ(2)p 6294 52912 1758 54 v 6828 54129 a FD(\021)8405 53218 y FJ(z)8983 53417 y Fw(\003)p FC(i)10936 53218 y FJ(=)1107 b(q)13741 53417 y FC(i)14412 53218 y FJ(+)295 b(ip)16803 53417 y FC(i)17548 53218 y FJ(:=)19290 51995 y Fz(p)p 20618 51995 950 54 v 1223 x FD(I)21192 53417 y FC(i)21567 53218 y FJ(\(cos)222 b FD(\036)24806 53417 y FC(i)25477 53218 y FJ(+)294 b(i)221 b(sin)h FD(\036)29954 53417 y FC(i)30330 53218 y FJ(\))f FD(;)2823 b(i)369 b FA(2)f(I)320 b FD(;)6294 54936 y FA(p)p 7401 54936 651 54 v 1099 x FJ(2)p 6294 56628 1758 54 v 6828 57845 a FD(\021)8369 56934 y FJ(\026)-614 b(z)8983 57133 y Fw(\003)p FC(i)10936 56934 y FJ(=)1107 b(q)13741 57133 y FC(i)14412 56934 y FA(\000)296 b FJ(ip)16825 57133 y FC(i)17569 56934 y FJ(:=)19311 55711 y Fz(p)p 20639 55711 950 54 v 1223 x FD(I)21213 57133 y FC(i)21589 56934 y FJ(\(cos)221 b FD(\036)24827 57133 y FC(i)25498 56934 y FA(\000)295 b FJ(i)221 b(sin)h FD(\036)29997 57133 y FC(i)30373 56934 y FJ(\))f FD(;)2823 b(i)369 b FA(2)f(I)320 b FD(:)0 60181 y FJ(W)-108 b(e)434 b(note)f(that)6358 61569 y Fz(X)6554 64388 y FC(i)p Fw(2I)8498 62831 y FD(dI)9748 63030 y FC(i)10419 62831 y FA(^)294 b FD(d\036)13045 63030 y FC(i)13790 62831 y FJ(=)368 b FA(\000)p FJ(i)16564 61357 y Fz(\020)17579 61569 y(X)17776 64388 y FC(i)p Fw(2I)19720 62831 y FD(d)p FJ(z)20974 63030 y Fw(\003)p FC(i)22115 62831 y FA(^)516 b FD(d)-36 b FJ(\026)-614 b(z)24771 63030 y Fw(\003)p FC(i)25617 61357 y Fz(\021)26410 62831 y FD(=\021)27750 62283 y FB(2)28645 62831 y FJ(=)30026 61569 y Fz(X)30222 64388 y FC(i)p Fw(2I)32166 62831 y FD(d)p FJ(p)33565 63030 y FC(i)34235 62831 y FA(^)295 b FD(d)p FJ(q)36778 63030 y FC(i)37375 62831 y FD(:)0 66729 y FJ(Denote)516 b(b)-36 b(y)515 b(\()110 b(^)-760 b FD(p;)317 b FJ(^)-746 b FD(q)48 b FJ(\))515 b(the)g(in\014nite)g(v) -36 b(ector)516 b(obtained)f(b)-36 b(y)515 b(excising)i(\(p)p FD(;)221 b FJ(q\))516 b(of)g(its)g FA(I)98 b FJ({com-)0 68334 y(p)36 b(onen)-36 b(ts,)439 b(namely)g(\()110 b(^)-760 b FD(p;)317 b FJ(^)-746 b FD(q)48 b FJ(\))377 b(=)g(\(p)15604 68533 y FC(i)15979 68334 y FD(;)221 b FJ(q)17247 68533 y FC(i)17624 68334 y FJ(\))18130 68533 y FC(i)p Fw(2I)19655 68281 y Ft(c)20139 68334 y FJ(.)593 b(Suc)-36 b(h)437 b(\()110 b(^)-760 b FD(p;)317 b FJ(^)-746 b FD(q)48 b FJ(\))438 b(ha)-36 b(v)g(e)439 b(not)f(to)h(b)36 b(e)438 b(confused)g(with)0 69939 y(the)433 b(\()p FD(p;)221 b(q)48 b FJ(\))433 b(used)g(in)g(c)-36 b(hapter)433 b(4.)579 b(With)433 b(a)h(little)g(abuse)g(of)g(notation,)g(w)-36 b(e)434 b(denote)4927 72253 y(~)4590 72589 y FD(H)102 b FJ(\()p FD(I)i(;)221 b(\036;)332 b FJ(^)-761 b FD(p)q(;)317 b FJ(^)-746 b FD(q)48 b FJ(\))1107 b(:=)15179 72253 y(~)14842 72589 y FD(H)102 b FJ(\()p FD(I)i(;)221 b(\036;)332 b FJ(^)-761 b FD(p)q(;)317 b FJ(^)-746 b FD(q)48 b FJ(;)221 b FD(\021)48 b FJ(\))12362 74659 y(:=)1107 b FD(H)16023 74111 y Fw(\003)16548 74659 y FJ(\()17054 73470 y FA(p)p 18161 73470 679 54 v 1189 x FD(I)325 b FJ(sin)222 b FD(\036;)22231 73470 y FA(p)p 23338 73470 V 1189 x FD(I)326 b FJ(cos)221 b FD(\036;)331 b FJ(^)-760 b FD(p)q(;)317 b FJ(^)-746 b FD(q)48 b FJ(;)221 b FD(\021)48 b FJ(\))369 b(=)f FD(H)34121 74111 y Fw(\003)34646 74659 y FJ(\(p)p FD(;)221 b FJ(q;)g FD(\021)48 b FJ(\))221 b FD(:)1268 b FJ(\(5.23\))21072 78312 y(158)p eop %%Page: 159 159 159 158 bop 6299 4488 a FF(5.2.1)1793 b(Space)598 b(of)f(the)i (solutions)6299 6951 y FJ(Consider)341 b FD(w)405 b FA(2)368 b FD(W)15450 7150 y FC(s;\013;\033)18006 6951 y FJ(:=)h FD(H)20929 6322 y FC(\013;\033)20826 7321 y(`)21210 7069 y Ft(a;s)22354 7321 y Fw(\002)p FC(`)23470 7069 y Ft(a;s)24670 6951 y FJ(,)359 b(with)342 b FD(a;)221 b(s;)g(\013)8 b(;)221 b(\033)419 b(>)368 b FJ(0)342 b(parameters,)359 b(endo)-36 b(w)g(ed)341 b(with)6299 8556 y(the)433 b(norm)18988 10162 y FA(k)p FD(w)36 b FA(k)21282 9613 y FB(2)21282 10490 y FC(s;\013;\033)23838 10162 y FJ(:=)25580 8900 y Fz(X)26282 11729 y FC(k)27720 10162 y FD(e)28323 9613 y FB(2)p FC(\013)p Fw(j)p FC(k)24 b Fw(j)30489 10162 y FA(j)p FD(k)45 b FA(j)31948 9613 y FB(2)p FC(\033)31948 10490 y Fw(\003)33044 10162 y FA(k)p FD(w)34638 10361 y 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y(\()p Fv(I)294 b FA(\000)i FD(M)18685 22661 y FB(1)19211 22462 y FJ(\))19938 20589 y Fz(\024)21569 21305 y FJ(^)21256 21641 y FD(J)21975 21840 y FC(p)21459 23045 y FJ(^)21193 23396 y FD( )22039 23595 y FC(p)23121 20589 y Fz(\025)24191 22462 y FJ(=)368 b FD(M)26829 22661 y FB(2)27356 22462 y FJ([)p FD(p)p FJ(])221 b FD(;)0 26008 y FJ(one)434 b(gets)14259 26678 y Fz(\022)16166 27395 y FJ(^)15854 27730 y FD(J)16573 27929 y FC(p)16056 29134 y FJ(^)15790 29485 y FD( )16636 29684 y FC(p)17718 26678 y Fz(\023)19065 28551 y FJ(=)20993 26891 y Fw(1)20504 27289 y Fz(X)20446 30079 y FC(m)p FB(=0)22702 28551 y FD(M)24099 28003 y FC(m)23960 28880 y FB(1)24986 27475 y Fz(\002)25540 28551 y FD(M)26798 28750 y FB(2)27324 28551 y FJ([)p FD(p)p FJ(])28699 27475 y Fz(\003)29474 28551 y FD(:)0 31911 y FJ(W)-108 b(e)434 b(need)e(t)-36 b(w)g(o)434 b(tec)-36 b(hnical)434 b(lemmata.)0 34515 y FE(Lemma)499 b(5.4.3.)554 b Fs(L)-66 b(et)464 b(b)-66 b(e)464 b FD(M)14430 34714 y FB(1)15421 34515 y Fs(as)i(in)e 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Fv(R)29471 44428 y FC(N)30368 44910 y FJ(\))509 b(and)g(w)-36 b(e)510 b(ha)-36 b(v)g(e)510 b(just)g(seen)0 46515 y(that)433 b FA(j)-221 b(j)g(j)p Fv(j)p FA(j)g(j)g(j)368 b(\024)h FJ(const)p FD(\034)582 b FJ(in)433 b(\(5.36\).)579 b(Moreo)-36 b(v)g(er,)435 b(b)36 b(eing)433 b(\(recall)h(\(5.32\)\))11576 48247 y Fz(\000)12185 49323 y FD(@)12872 49522 y FC(J)13837 48987 y FJ(~)13524 49323 y FD(J)123 b FJ(\()p FD(w)15802 49522 y FC(])16223 49323 y FJ(\))16729 48247 y Fz(\001)17338 49323 y FJ([)18012 48987 y(^)17699 49323 y FD(J)g FJ(]\()p FD(t)p FJ(\))368 b(=)h FD(\021)22823 48774 y FB(4)23348 49323 y FD(@)24107 48774 y FB(2)24035 49651 y FC(I)69 b(I)25392 48987 y FJ(~)25044 49323 y FD(K)95 b FJ(\()p FD(w)27682 49522 y FB(0)28208 49323 y FJ(\()p FD(t)p FJ(\)\))30507 48987 y(^)30196 49323 y FD(J)122 b FJ(\()p FD(t)p FJ(\))0 52131 y(a)478 b(pro)36 b(duct)477 b(op)36 b(erator)478 b(of)g(norm)f FA(j)-221 b(j)g(j)325 b(\001)g(j)-221 b(j)g(j)443 b(\024)h FJ(const)p FD(\021)24227 51649 y FB(4)24752 52131 y FJ(,)489 b(w)-36 b(e)478 b(obtain)g(that)f FA(j)-221 b(j)g(j)p FD(M)36437 52330 y FB(1)36963 52131 y FA(j)g(j)g(j)443 b(\024)h FJ(const)p FD(\021)43208 51649 y FB(2)43733 52131 y FJ(.)0 53736 y(Since)503 b FA(L)4382 53254 y FC(\013;\033)5873 53736 y FJ(\()p Fv(R)7338 53254 y FC(N)8578 53736 y FA(\002)343 b Fv(R)10913 53254 y FC(N)11810 53736 y FD(;)221 b Fv(R)13351 53254 y FC(N)14591 53736 y FA(\002)343 b Fv(R)16926 53254 y FC(N)17823 53736 y FJ(\))504 b(is)g(a)g(Banac)-36 b(h)504 b(space,)521 b(one)504 b(gets)g(that)38096 52739 y Fz(P)39498 53090 y Fw(1)39498 54123 y FC(m)p FB(=0)41810 53736 y FD(M)43207 53254 y FC(m)43068 54064 y FB(1)0 55341 y FJ(con)-36 b(v)g(erges.)43061 56946 y Fn(\003)0 59550 y FE(Lemma)499 b(5.4.4.)554 b Fs(L)-66 b(et)464 b(b)-66 b(e)464 b FD(M)14430 59749 y FB(2)15421 59550 y Fs(as)i(in)e FJ(\(5.56\))p Fs(.)598 b(Then)12323 62358 y FD(p)369 b FA(7\000)-221 b(!)369 b FD(M)17112 62557 y FB(2)17638 62358 y FJ([)p FD(p)p FJ(])h FA(2)e(L)21553 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169 169 169 168 bop 6299 4488 a Fp(pr)-30 b(oof.)659 b FJ(Let)433 b(us)g(estimate)8639 8413 y FA(k)p FD(h)p FA(k)10716 7865 y FB(2)10716 8742 y FC(s;\013)12016 8490 y Fo(0)12315 8742 y FC(;\033)13146 8490 y Fo(0)14617 8413 y FJ(=)17103 6753 y FC(L)17743 6864 y Ft(n)16747 7151 y Fz(X)16848 9980 y FC(k)24 b FB(=1)18887 8413 y FD(e)19490 7865 y FB(2)p FC(\013)20564 7552 y Fo(0)20863 7865 y FC(k)21431 8413 y FD(k)22152 7865 y FB(2)p FC(\033)23192 7552 y Fo(0)23546 8413 y FA(k)p FD(h)24959 8612 y FC(k)25528 8413 y FA(k)26192 7865 y FB(2)26192 8742 y FC(a;s)27813 8413 y FJ(=)29549 6753 y FC(L)30189 6864 y Ft(n)29193 7151 y Fz(X)29295 9980 y FC(k)g FB(=1)31333 8413 y FD(e)31936 7865 y FB(2)p FC(\013k)33580 8413 y FD(k)34301 7865 y FB(2)p FC(\033)35396 8413 y FA(k)p FD(h)36809 8612 y FC(k)37378 8413 y FA(k)38042 7865 y FB(2)38042 8742 y FC(a;s)39294 8413 y FD(e)39897 7865 y FB(2)p FC(k)g FB(\()p FC(\013)41851 7552 y Fo(0)42149 7865 y Fw(\000)p FC(\013)p FB(\))43906 8413 y FD(k)44627 7865 y 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Fw(\002)p FC(`)37787 74910 y Ft(a;s)27371 78312 y FJ(169)p eop %%Page: 170 170 170 169 bop 0 4488 a Fs(with)465 b FD(e)3426 4687 y FC(j)4377 4488 y Fs(the)g(standar)-66 b(d)465 b(orthonormal)g(b)-66 b(asis)464 b(of)h FD(`)24467 4006 y FC(a;s)26014 4488 y FA(\002)295 b FD(`)27888 4006 y FC(a;s)29139 4488 y Fs(.)598 b(De\014ne)463 b(the)i(op)-66 b(en)464 b(set)2405 8112 y FJ(\001)3489 7563 y FC(\027)3489 8440 y(n)4115 8112 y FJ(\()p FD(\030)5189 8311 y FB(0)5715 8112 y FJ(\))1107 b(:=)9807 6239 y Fz(\032)10804 8112 y FD($)404 b FA(2)369 b FJ(\(0)p FD(;)221 b($)16351 8311 y Fw(\003)16877 8112 y FJ(\))443 b(s)p FD(:)p FJ(t)p FD(:)20010 6584 y Fz(\014)20010 7381 y(\014)20010 8178 y(\014)20453 8112 y FD($)36 b(k)339 b FA(\000)24052 7776 y FJ(~)23908 8112 y(\012)24847 8311 y FC(j)25629 8112 y FJ(+)295 b FD(")27545 7036 y Fz(\000)28154 8112 y FA(M)29749 7563 y FB(22)29749 8440 y FC(k)30745 8112 y FJ(\()p FD(\030)31819 8311 y FB(0)32345 8112 y FJ(\))32851 7036 y Fz(\001)33459 8632 y FC(j)51 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30385 y FD(\013)17469 30584 y FC(n)18465 30385 y FJ(=)368 b FD(\013)20672 30584 y FC(n)p Fw(\000)p FB(1)22797 30385 y FA(\000)24821 29486 y FD(\013)25648 29685 y FB(0)p 24258 30079 V 24258 31296 a FJ(2)24908 30912 y FC(n)p FB(+1)27091 30385 y FD(:)0 33519 y FJ(It)434 b(results)f(that)15698 35994 y FD(\013)16525 36193 y FC(n)17521 35994 y FJ(=)369 b FD(\013)19729 36193 y FB(0)20476 33722 y Fz( )21528 35994 y FJ(1)295 b FA(\000)23874 34333 y FC(n)p FB(+1)23801 34732 y Fz(X)23944 37531 y FC(j)51 b FB(=2)26318 35095 y FJ(1)p 26074 35688 1138 54 v 26074 36905 a(2)26724 36521 y FC(j)27345 33722 y Fz(!)40771 35994 y FJ(\(5.70\))0 39604 y(so)434 b(that)f FD(\013)5242 39803 y FC(n)6238 39604 y FA(!)369 b FD(\013)8762 39803 y FB(0)9288 39604 y FD(=)p FJ(2.)0 41521 y(The)468 b(follo)-36 b(wing)470 b(Lemma)d(is)h(crucial)g(for)h(the)e(iterativ) -36 b(e)468 b(sc)-36 b(heme.)681 b(Its)467 b(pro)36 b(of)469 b(is)f(the)f(real)0 43126 y(core)434 b(of)g(the)f(issue)h(and)f(it)h (will)g(b)36 b(e)434 b(pro)-36 b(v)g(ed)433 b(in)g(the)g(follo)-36 b(wing)436 b(sections.)0 45719 y FE(Lemma)485 b(5.5.4)g(\(In)-42 b(v)g(ertibilit)g(y\).)546 b Fs(If)675 b FD(\030)518 b Fs(is)453 b(\\su\016ciently")g(smal)66 b(l)455 b(then)e(for)g(al)66 b(l)455 b FD(n)369 b FA(\025)g FJ(1)0 47505 y Fs(and)408 b FD($)c FA(2)369 b FJ(\001)6285 47023 y FC(\027)6285 47834 y(n)6911 47505 y FJ(\()p FD(\030)64 b FJ(\))408 b Fs(ther)-66 b(e)406 b(exist)h(op)-66 b(er)g(ators)408 b FA(G)21566 47023 y FB(\()p FC(n)p FB(\))23145 47505 y FJ(\()p FD($)36 b(;)221 b FA(\001)p FJ(\))369 b(:)408 b FD(W)28766 46826 y FB(\()p FC(n)p FB(\))28585 47642 y FC(s;\013)29885 47753 y Ft(n)30452 47642 y FC(;\033)31707 47505 y FA(\000)-221 b(!)370 b FD(W)35626 46826 y FB(\()p FC(n)p FB(\))35445 47642 y FC(s;\013)36745 47753 y Ft(n)37312 47642 y FC(;\033)38606 47505 y Fs(such)408 b(that,)0 49291 y(for)465 b(any)f FD(h)369 b FA(2)g FD(W)8348 48612 y FB(\()p FC(n)p FB(\))8167 49428 y FC(s;\013)9467 49539 y Ft(n)10034 49428 y FC(;\033)10920 49291 y Fs(,)464 b(it)h(r)-66 b(esults)2873 53291 y Fz(\015)2873 54088 y(\015)3611 54421 y FA(G)4480 53872 y FB(\()p FC(n)p FB(\))5838 54421 y FJ(\()p FD($)36 b(;)221 b(\030)64 b FJ(\)[)p FD(h)p FJ(])10646 53291 y Fz(\015)10646 54088 y(\015)11385 54941 y FC(\013)11989 55052 y Ft(n)12556 54941 y FC(;\033)13811 54421 y FA(\024)15213 50900 y Fz(8)15213 52096 y(>)15213 52494 y(>)15213 52893 y(<)15213 55284 y(>)15213 55682 y(>)15213 56081 y(:)16947 52119 y FD(C)17878 52318 y FB(0)18625 52119 y FA(k)p FD(h)p FA(k)20702 52318 y FC(\013)21306 52429 y Ft(n)21873 52318 y FC(;\033)22759 52119 y FD(;)6918 b FJ(if)757 b FD(L)32438 52318 y FC(n)33434 52119 y FA(\024)369 b FD(\034)148 b(=)p FJ(2)q(;)18082 55113 y FD(C)19013 55312 y FB(0)p 17080 55706 3460 54 v 17080 56923 a FD(\015)294 b(\034)18762 56539 y FC(\027)58 b Fw(\000)p FB(1)20894 56012 y FJ(\()p FD(L)22285 56211 y FC(n)22911 56012 y FJ(\))23417 55464 y FC(\027)24214 56012 y FA(k)p FD(h)p FA(k)26291 56211 y FC(\013)26895 56322 y Ft(n)27462 56211 y FC(;\033)28570 56012 y FD(;)1107 b FJ(if)757 b FD(L)32438 56211 y FC(n)33434 56012 y FD(>)368 b(\034)148 b(=)p FJ(2)q FD(:)40771 54421 y FJ(\(5.71\))0 59485 y Fs(Mor)-66 b(e)g(over)12495 61090 y FA(L)13411 60542 y FB(\()p FC(n)p FB(\))14769 61090 y FA(G)15638 60542 y FB(\()p FC(n)p FB(\))16996 61090 y FD(h)369 b FJ(=)f FD(h)221 b(;)2879 b FA(8)221 b FD(h)369 b FA(2)g FD(W)28445 60542 y FB(\()p FC(n)p FB(\))28264 61418 y FC(s;\013)29564 61529 y Ft(n)30131 61418 y FC(;\033)31238 61090 y FD(;)9172 b FJ(\(5.72\))0 63508 y Fs(wher)-66 b(e)465 b FA(L)4586 63026 y FB(\()p FC(n)p FB(\))6313 63508 y FJ(:=)369 b FD(P)8893 63707 y FC(n)9519 63508 y FA(L)p Fs(.)0 67339 y FF(5.5.2)1792 b(Iteration)0 69802 y FJ(F)-108 b(rom)433 b(\(5.70\),)i(w)-36 b(e)434 b(ha)-36 b(v)g(e)433 b(that)7240 73813 y FD(\013)8067 74012 y FC(n)p FB(+1)10191 73813 y FA(\000)295 b FD(\013)12346 74012 y FC(i)13829 73813 y FJ(=)1107 b FD(\013)16775 74012 y FB(0)17301 71940 y Fz(\022)18698 72153 y FC(i)p FB(+1)18500 72551 y Fz(X)18642 75350 y FC(j)51 b FB(=2)21016 72914 y FJ(1)p 20773 73508 1138 54 v 20773 74724 a(2)21423 74340 y FC(j)22338 73813 y FA(\000)23739 72153 y FC(n)p FB(+2)23666 72551 y Fz(X)23809 75350 y FC(j)g FB(=2)26183 72914 y FJ(1)p 25939 73508 V 25939 74724 a(2)26589 74340 y FC(j)27209 71940 y Fz(\023)28556 73813 y FJ(=)369 b FA(\000)p FD(\013)31797 74012 y FB(0)33001 72153 y FC(n)p FB(+2)32928 72551 y Fz(X)32545 75350 y FC(j)51 b FB(=)p FC(i)p FB(+2)35828 72914 y FJ(1)p 35585 73508 V 35585 74724 a(2)36235 74340 y FC(j)21072 78312 y FJ(170)p eop %%Page: 171 171 171 170 bop 20128 5539 a FJ(=)1107 b FA(\000)23850 4640 y FD(\013)24677 4839 y FB(0)p 23413 5233 2229 54 v 23413 6450 a FJ(2)24063 6066 y FC(i)p FB(+2)26143 3878 y FC(n)p Fw(\000)p FC(i)25995 4277 y Fz(X)26138 7076 y FC(j)51 b FB(=0)28512 4640 y FJ(1)p 28268 5233 1138 54 v 28268 6450 a(2)28918 6066 y FC(j)29907 5539 y FA(\024)369 b(\000)32913 4640 y FD(\013)33740 4839 y FB(0)p 32475 5233 2229 54 v 32475 6450 a FJ(2)33125 6066 y FC(i)p FB(+2)35058 5539 y FD(:)11651 b FJ(\(5.73\))6299 9618 y(Let)433 b FD(\030)9197 9817 y FC(n)10193 9618 y FJ(=)11573 8622 y Fz(P)12975 8972 y FC(n)12975 10006 y(i)p FB(=1)14775 9618 y FD(h)15524 9817 y FC(i)16333 9618 y FJ(for)h(certain)f FD(h)23404 9817 y FC(i)24149 9618 y FA(2)368 b FD(W)26812 8939 y FB(\()p FC(i)p FB(\))26631 9754 y FC(\013)27235 9889 y Ft(i)27587 9754 y FC(;\033)28907 9618 y FJ(satisfying)17846 12200 y FA(k)p FD(h)19259 12399 y FC(i)19635 12200 y FA(k)20299 12399 y FC(\013)20903 12534 y Ft(i)21255 12399 y FC(;\033)22510 12200 y FA(\024)h FD(c)24472 12399 y FB(0)24997 12200 y FD(\021)25687 11652 y FB(3)26434 12200 y FD(e)27037 11652 y Fw(\000)p FC(\037)28353 11339 y Ft(i)28704 11652 y FC(\013)29308 11775 y Fu(0)29770 11652 y FC(=)p FB(8)30987 12200 y FD(;)1523 b FJ(1)369 b FD(<)g(\037)g(<)f FJ(2)p FD(:)8224 b FJ(\(5.74\))6299 14589 y(Since,)434 b(b)-36 b(y)433 b(\(5.63\))i(and)e(\(5.74\),)7655 17989 y FA(k)p FD(\030)8887 18188 y FC(n)9513 17989 y FA(k)10177 18333 y FC(\013)10781 18456 y Ft(n)p Fu(+1)12376 18333 y FC(;\033)32 b FB(+)14072 17960 y Fu(2)p Ft(\027)45 b Fu(+1)p 14071 18132 1892 40 v 14274 18680 a Ft(\027)g Fu(+1)17257 17989 y FA(\024)20071 16329 y FC(n)19397 16727 y Fz(X)19595 19526 y FC(i)p FB(=1)21537 17989 y FA(k)p FD(h)22950 18188 y FC(i)23326 17989 y FA(k)23990 18333 y FC(\013)24594 18456 y Ft(n)p Fu(+1)26188 18333 y FC(;\033)32 b FB(+)27884 17960 y Fu(2)p Ft(\027)45 b Fu(+1)p 27884 18132 V 28087 18680 a Ft(\027)g Fu(+1)30332 17989 y FA(\024)369 b FD(c)p FJ(\()p FD(\027)12 b(;)221 b(\013)34864 18188 y FB(0)35390 17989 y FD(;)g(\033)48 b FJ(\))38160 16329 y FC(n)37486 16727 y Fz(X)37684 19526 y FC(i)p FB(=1)39626 17989 y FJ(2)40276 17407 y FB(\()p FC(i)p FB(+2\))42663 17033 y Fu(2)p Ft(\027)d Fu(+1)p 42664 17205 V 42867 17754 a Ft(\027)g Fu(+1)44743 17989 y FA(k)p FD(h)46156 18188 y FC(i)46532 17989 y FA(k)47196 18188 y FC(\013)47800 18323 y Ft(i)48152 18188 y FC(;\033)17257 22277 y FA(\024)1107 b FD(c)p FJ(\()p FD(\027)12 b(;)221 b(\013)22527 22476 y FB(0)23053 22277 y FD(;)g(\033)48 b FJ(\))p FD(c)25488 22476 y FB(0)26235 22277 y FD(\021)26925 21728 y FB(3)28346 20616 y FC(n)27672 21015 y Fz(X)27870 23814 y FC(i)p FB(=1)29812 21201 y Fz(\000)30421 22277 y FJ(2)31071 21728 y FC(i)p FB(+2)32649 21201 y Fz(\001)33391 21125 y Fu(2)p Ft(\027)d Fu(+1)p 33391 21297 V 33594 21845 a Ft(\027)g Fu(+1)35470 22277 y FD(e)36073 21728 y Fw(\000)p FC(\037)37389 21416 y Ft(i)37740 21728 y FC(\013)38344 21851 y Fu(0)38806 21728 y FC(=)p FB(8)40023 22277 y FD(;)6686 b FJ(\(5.75\))6299 25837 y(from)434 b(\(5.75\),)h(it)f(follo)-36 b(ws)435 b(that)17690 28225 y FA(k)p FD(\030)18922 28424 y FC(n)19549 28225 y FA(k)20213 28569 y FC(\013)20817 28692 y Ft(n)p Fu(+1)22411 28569 y FC(;\033)32 b FB(+)24107 28196 y Fu(2)p Ft(\027)45 b Fu(+1)p 24106 28368 V 24309 28916 a Ft(\027)g Fu(+1)26555 28225 y FA(\024)369 b FJ(const\()p FD(\027)12 b(;)221 b(\013)33497 28424 y FB(0)34022 28225 y FD(;)g(\033)48 b FJ(\))221 b FD(c)36678 28424 y FB(0)37205 28225 y FD(\021)37895 27677 y FB(3)38641 28225 y FD(:)8068 b FJ(\(5.76\))6299 30899 y(No)-36 b(w,)435 b(b)-36 b(y)433 b(Theorem)h(5.1.6,)h(if)g FA(k)p FD(\030)22814 31098 y FC(n)23440 30899 y FA(k)24104 31243 y FC(\013)24708 31366 y Ft(n)p Fu(+1)26302 31243 y FC(;\033)32 b FB(+)27998 30869 y Fu(2)p Ft(\027)45 b Fu(+1)p 27998 31041 V 28201 31590 a Ft(\027)g Fu(+1)30446 30899 y FA(\024)369 b FD(r)470 b FJ(then,)12216 33744 y Fz(\015)12216 34542 y(\015)12954 34874 y FA(N)196 b FJ(\()p FD(\030)15314 35073 y FC(n)15940 34874 y FJ(\))16446 33744 y Fz(\015)16446 34542 y(\015)17183 35394 y FC(s)p FB(+1)p FC(;\013)19685 35517 y Ft(n)p Fu(+1)21280 35394 y FC(;\033)32 b FB(+)22976 35020 y Fu(2)p Ft(\027)45 b Fu(+1)p 22976 35192 V 23179 35741 a Ft(\027)g Fu(+1)26162 34874 y FA(\024)1107 b FD(M)29978 33612 y Fz(X)29920 36418 y FC(m)p Fw(\025)p FB(0)32176 33001 y Fz(\022)33287 33572 y FA(k)p FD(\030)34519 33771 y FC(n)35145 33572 y FA(k)35809 33916 y FC(s;\013)37109 34039 y Ft(n)p Fu(+1)38704 33916 y FC(;\033)32 b FB(+)40400 33543 y Fu(2)p Ft(\027)45 b Fu(+1)p 40400 33715 V 40602 34263 a Ft(\027)g Fu(+1)p 33287 34568 9192 54 v 37572 35785 a FD(r)42612 33001 y Fz(\023)43589 33298 y FC(m)26162 38139 y FA(\024)1107 b FJ(const)221 b FD(;)15216 b FJ(\(5.77\))6299 40527 y(where)434 b(const)369 b(=)f(const\()p FD(a;)221 b(s;)g(\027)12 b(;)221 b(\013)22776 40726 y FB(0)23303 40527 y FD(;)g(\033)48 b FJ(\))434 b(and,)f(in)g(the)g(same)h(w)-36 b(a)g(y)-108 b(,)18694 41786 y Fz(\015)18694 42583 y(\015)19432 42915 y FD(D)36 b FA(N)196 b FJ(\()p FD(\030)22908 43114 y FC(n)23534 42915 y FJ(\))24040 41786 y Fz(\015)24040 42583 y(\015)24777 43436 y FB(op)26195 42915 y FA(\024)369 b FJ(const\()p FD(a;)221 b(s;)g(\027)12 b(;)221 b(\013)35597 43114 y FB(0)36124 42915 y FD(;)g(\033)48 b FJ(\))9071 b(\(5.78\))6299 45424 y(where)434 b(the)f(op)36 b(eratorial)434 b(norm)g(is)g(on)f(the)g(space)17684 47960 y FA(L)18600 47377 y FC(\013)19204 47500 y Ft(n)p Fu(+1)20798 47377 y FC(;\033)32 b FB(+)22494 47004 y Fu(2)p Ft(\027)45 b Fu(+1)p 22494 47176 1892 40 v 22697 47725 a Ft(\027)g Fu(+1)24573 47960 y FJ(\()p FD(`)25625 47412 y FC(a;s)27171 47960 y FA(\002)296 b FD(`)29046 47412 y FC(a;s)30297 47960 y FD(;)221 b(`)31425 47412 y FC(a;s)p FB(+1)34175 47960 y FA(\002)295 b FD(`)36049 47412 y FC(a;s)p FB(+1)38503 47960 y FJ(\))6299 50348 y(and)433 b(it)h(is)g(b)36 b(ounded)432 b(b)-36 b(y)433 b(Prop)36 b(osition)434 b(5.4.2)i(\(recall)e (\(5.48\)\).)6299 52593 y FE(Lemma)684 b(5.5.5)f(\(Iterativ)-42 b(e)684 b(Lemma\).)636 b Fs(L)-66 b(et)612 b(b)-66 b(e)612 b FD(\030)33508 52792 y FB(0)34676 52593 y FJ(=)642 b(0)p Fs(,)649 b FD(A)39003 52792 y FB(0)40172 52593 y FJ(:=)642 b(\(0)p FD(;)221 b($)45000 52792 y Fw(\003)45526 52593 y FJ(\))p Fs(.)1040 b(Then)6299 54198 y(ther)-66 b(e)473 b(exists)h(a)h(se)-66 b(quenc)g(e)472 b(of)i(op)-66 b(en)474 b(sets)g FD(A)27680 54397 y FC(n)28693 54198 y FA(\022)386 b FD(A)31087 54397 y FC(n)p Fw(\000)p FB(1)33302 54198 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FB(0)11947 29341 y FJ(w)-36 b(as)434 b(de\014ned)e(in)h(Lemma)h (5.5.4.)580 b(No)-36 b(w)434 b(\014x)f FD(n)34280 29540 y FB(0)35240 29341 y FJ(suc)-36 b(h)433 b(that)7924 32576 y(const)221 b FD(C)12046 32775 y FB(0)12572 32576 y FJ(\()p FD(L)13963 32775 y FC(n)14589 32576 y FJ(\))15095 32027 y FC(\027)15670 31101 y Fz(\020)16464 32576 y FD(e)17067 32027 y Fw(\000)p FC(\037)18383 31715 y Ft(n)18949 32027 y FC(\013)19553 32150 y Fu(0)20015 32027 y FC(=)p FB(4)21306 32576 y FJ(+)295 b FD(e)23216 32027 y Fw(\000)p FB(2)24418 31715 y Ft(n)24985 32027 y FC(\013)25589 32150 y Fu(0)26050 32027 y FC(=)p FB(4)27342 32576 y FJ(+)f FD(e)29251 32027 y FB(\()p Fw(\000)p FC(\037)30933 31715 y Ft(n)31500 32027 y Fw(\000)p FB(2)32702 31715 y Ft(n)p Fu(+1)34296 32027 y FB(\))p FC(\013)35266 32150 y Fu(0)35727 32027 y FC(=)p FB(8)36723 31101 y Fz(\021)37886 32576 y FA(\024)369 b FD(e)39891 32027 y Fw(\000)p FC(\037)41207 31715 y Ft(n)p Fu(+1)42800 32027 y FC(\013)43404 32150 y Fu(0)43866 32027 y FC(=)p FB(8)45084 32576 y FD(;)1625 b FJ(\(5.89\))6299 35870 y(for)637 b(all)g FD(n)715 b FA(\025)f FD(n)14512 36069 y FB(0)15038 35870 y FJ(.)1187 b(Suc)-36 b(h)635 b FD(n)20708 36069 y FB(0)21871 35870 y FJ(exists)i(since)f FD(\037)714 b(<)g FJ(2,)688 b FD(L)34893 36069 y FC(n)36233 35870 y FJ(=)714 b(4)38609 35388 y FC(n)39872 35870 y FJ(and)636 b(recalling)h(that)6299 37475 y(1)p FD(=)p FJ(\()p FD(\015)72 b(\034)9565 36993 y FC(\027)58 b Fw(\000)p FB(1)11344 37475 y FJ(\))772 b(=)h FD($)15518 36993 y FC(\027)58 b Fw(\000)p FB(1)p Fw(\000)p FC(\017)19180 37475 y FA(\034)774 b FJ(1.)1290 b(T)-108 b(aking)672 b FD($)706 b FJ(small)672 b(enough)e(\(equiv)-72 b(alen)-36 b(tly)672 b FD(\034)819 b FJ(large)6299 39080 y(enough\))433 b(suc)-36 b(h)433 b(that)g FD(L)17922 39279 y FC(n)18493 39402 y Fu(0)19378 39080 y FA(\024)369 b FD(\034)148 b(=)p FJ(2,)435 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b(that,)f(for)h(all)h FD(n)369 b FA(\025)g FD(n)38799 39279 y FB(0)39326 39080 y FJ(,)433 b(b)-36 b(y)434 b(\(5.89\),)10389 42727 y(const)14826 41828 y FD(C)15757 42027 y FB(0)p 13935 42421 3239 54 v 13935 43638 a FD(\015)72 b(\034)15395 43254 y FC(\027)58 b Fw(\000)p FB(1)17306 42727 y FJ(\()p FD(L)18697 42926 y FC(n)19323 42727 y FJ(\))19829 42178 y FC(\027)20626 41252 y Fz(\020)21419 42727 y FD(c)21979 42178 y FB(2)21979 43055 y(0)22726 42727 y FD("e)23938 42178 y Fw(\000)p FC(\037)25254 41866 y Ft(n)25820 42178 y FC(\013)26424 42301 y Fu(0)26886 42178 y FC(=)p FB(4)28177 42727 y FJ(+)295 b FD(e)30087 42178 y Fw(\000)p FB(2)31289 41866 y Ft(n)31856 42178 y FC(\013)32460 42301 y Fu(0)32921 42178 y FC(=)p FB(4)34213 42727 y FJ(+)g FD(c)36080 42926 y FB(0)36827 42727 y FD("e)38039 42178 y FB(\()p Fw(\000)p FC(\037)39721 41866 y Ft(n)40287 42178 y Fw(\000)p FB(2)41489 41866 y Ft(n)p Fu(+1)43082 42178 y FB(\))p FC(\013)44052 42301 y Fu(0)44514 42178 y FC(=)p FB(8)45510 41252 y Fz(\021)14688 45628 y FA(\024)369 b FD(c)16650 45827 y FB(0)17175 45628 y FD(e)17778 45080 y Fw(\000)p FC(\037)19094 44767 y Ft(n)p Fu(+1)20688 45080 y FC(\013)21292 45203 y Fu(0)21753 45080 y FC(=)p FB(8)22971 45628 y FD(;)23738 b FJ(\(5.90\))6299 48630 y(c)-36 b(ho)36 b(osing)609 b FD(c)12316 48829 y FB(0)13451 48630 y FJ(large)g(enough)f(and)h (subsequen)-36 b(tly)608 b FD($)644 b FJ(small)610 b(enough.)1103 b(Finally)-108 b(,)653 b(from)6299 50236 y(Lemma)434 b(5.5.4,)h(\(5.87\),)g(\(5.88\))f(and)f(\(5.90\),)i(it)f(follo)-36 b(ws)435 b(that)19930 53365 y FA(k)p FD(h)21343 53564 y FC(n)p FB(+1)23171 53365 y FA(k)23835 53564 y FC(\013)24439 53687 y Ft(n)p Fu(+1)26033 53564 y FC(;\033)27288 53365 y FA(\024)370 b FD(c)29251 53564 y FB(0)29998 53365 y FD("e)31210 52816 y Fw(\000)p FC(\037)32526 52504 y Ft(n)p Fu(+1)34119 52816 y FC(\013)34723 52939 y Fu(0)35184 52816 y FC(=)p FB(8)36402 53365 y FD(;)6299 56367 y FJ(pro)-36 b(ving)434 b(b)-36 b(y)434 b(induction)e(\(5.81\).)6299 58351 y(No)-36 b(w)386 b(w)-36 b(e)385 b(obtain)g(a)g(similar)h (estimate)g(for)f(the)g(deriv)-72 b(ativ)-36 b(e)386 b(of)f FD(h)37720 58550 y FC(n)38731 58351 y FJ(with)h(resp)36 b(ect)384 b(to)h FD(\021)48 b FJ(.)562 b(In)6299 59956 y(particular)434 b(w)-36 b(e)434 b(w)-36 b(an)g(t)433 b(to)h(pro)-36 b(v)g(e)17173 62958 y FA(k)p FD(@)18524 63157 y FC(\021)19083 62958 y FD(h)19832 63157 y FC(n)20458 62958 y FA(k)21122 63157 y FC(\013)21726 63268 y Ft(n)22718 62958 y FA(\024)369 b FD(c)24680 63157 y FB(0)25205 62958 y FD(e)25808 62410 y Fw(\000)109 b FB(~)-579 b FC(\037)27124 62097 y Ft(n)27691 62410 y FC(\013)28295 62533 y Fu(0)28756 62410 y FC(=)p FB(8)29974 62958 y FD(;)2823 b FJ(1)369 b FD(<)523 b FJ(~)-804 b FD(\037)369 b(<)f(\037)221 b(:)7551 b FJ(\(5.91\))6299 65984 y(Using)434 b(\(5.72\))h(in)e(\(5.84\))h(w)-36 b(e)434 b(get)8909 68986 y FA(\000L)10858 68438 y FB(\()p FC(n)p FB(+1\))13419 68986 y FD(h)14168 69185 y FC(n)p FB(+1)16365 68986 y FJ(=)369 b FD(")18355 67910 y Fz(\000)18963 68986 y FD(P)19982 68438 y FC(n)p FB(+1)19801 69315 y FC(n)21811 68986 y FA(N)196 b FJ(\()p FD(\030)24171 69185 y FC(n)p Fw(\000)p FB(1)25999 68986 y FJ(\))295 b(+)g FD(P)29126 68438 y FC(n)p FB(+1)28945 69315 y FC(n)30954 68986 y FD(D)36 b FA(N)196 b FJ(\()p FD(\030)34430 69185 y FC(n)p Fw(\000)p FB(1)36258 68986 y FJ(\)[)p FD(h)37874 69185 y FC(n)38500 68986 y FJ(])296 b(+)f FD(R)11 b FJ(\()p FD(h)42720 69185 y FC(n)43345 68986 y FJ(\))43851 67910 y Fz(\001)47070 68986 y FJ(\(5.92\))6299 71989 y(where)9629 74991 y FD(R)g FJ(\()p FD(h)11885 75190 y FC(n)12511 74991 y FJ(\))369 b(=)f FD(P)15604 75190 y FC(n)p FB(+1)17433 73915 y Fz(\000)18042 74991 y FA(N)196 b FJ(\()p FD(\030)20402 75190 y FC(n)21028 74991 y FJ(\))295 b FA(\000)g(N)196 b FJ(\()p FD(\030)25517 75190 y FC(n)p Fw(\000)p FB(1)27345 74991 y FJ(\))295 b FA(\000)h FD(D)36 b FA(N)196 b FJ(\()p FD(\030)32951 75190 y FC(n)p Fw(\000)p FB(1)34779 74991 y FJ(\)[)p FD(h)36395 75190 y FC(n)37021 74991 y FJ(])37382 73915 y Fz(\001)38360 74991 y FJ(=)369 b FD(O)36 b FJ(\()p FD(h)42026 74443 y FB(2)42026 75320 y FC(n)42651 74991 y FJ(\))221 b FD(:)3331 b FJ(\(5.93\))27371 78312 y(173)p eop %%Page: 174 174 174 173 bop 0 4488 a FJ(Deriving)435 b(\(5.93\))f(with)g(resp)36 b(ect)433 b(to)h FD(\021)48 b FJ(,)433 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)362 7812 y FD(@)1049 8011 y FC(\021)1608 6736 y Fz(\000)2217 7812 y FD(R)11 b FJ(\()p FD(h)4473 8011 y FC(n)5098 7812 y FJ(\))5604 6736 y Fz(\001)6877 7812 y FJ(=)664 b FD(P)9391 8011 y FC(n)p FB(+1)11220 6338 y Fz(\020)12013 7812 y FD(@)12700 8011 y FC(\021)13259 7812 y FD(N)139 b FJ(\()p FD(\030)15514 8011 y FC(n)16140 7812 y FJ(\))295 b(+)f FD(D)36 b FA(N)196 b FJ(\()p FD(\030)21723 8011 y FC(n)22349 7812 y FJ(\)[)p FD(@)23903 8011 y FC(\021)24462 7812 y FD(\030)25030 8011 y FC(n)25657 7812 y FJ(])295 b FA(\000)g FD(D)36 b FA(N)196 b FJ(\()p FD(\030)31117 8011 y FC(n)p Fw(\000)p FB(1)32946 7812 y FJ(\)[)p FD(@)34500 8011 y FC(\021)35059 7812 y FD(\030)35627 8011 y FC(n)p Fw(\000)p FB(1)37455 7812 y FJ(])8996 10237 y FA(\000)p FD(@)10716 10436 y FC(\021)11275 10237 y FA(N)g FJ(\()p FD(\030)13635 10436 y FC(n)p Fw(\000)p FB(1)15463 10237 y FJ(\))295 b FA(\000)g FD(@)18279 10436 y FC(\021)18838 10237 y FD(D)36 b FA(N)196 b FJ(\()p FD(\030)22314 10436 y FC(n)p Fw(\000)p FB(1)24142 10237 y FJ(\)[)p FD(h)25758 10436 y FC(n)26384 10237 y FJ(])296 b FA(\000)f FD(D)29485 9689 y FB(2)30011 10237 y FA(N)196 b FJ(\()p FD(\030)32371 10436 y FC(n)p Fw(\000)p FB(1)34199 10237 y FJ(\)[)p FD(@)35753 10436 y FC(\021)36312 10237 y FD(\030)36880 10436 y FC(n)p Fw(\000)p FB(1)38709 10237 y FD(;)221 b(h)40040 10436 y FC(n)40667 10237 y FJ(])8996 12579 y FA(\000)p FD(D)36 b FA(N)196 b FJ(\()p FD(\030)13505 12778 y FC(n)p Fw(\000)p FB(1)15333 12579 y FJ(\)[)p FD(@)16887 12778 y FC(\021)17446 12579 y FD(h)18195 12778 y FC(n)18821 12579 y FJ(])19182 11105 y Fz(\021)6877 15413 y FJ(=)664 b FD(P)9391 15612 y FC(n)p FB(+1)11220 13939 y Fz(\020)12013 15413 y FD(D)36 b FA(N)196 b FJ(\()p FD(\030)15489 15612 y FC(n)16115 15413 y FJ(\)[)p FD(@)17669 15612 y FC(\021)18228 15413 y FD(h)18977 15612 y FC(n)19603 15413 y FJ(])296 b FA(\000)f FD(D)36 b FA(N)196 b FJ(\()p FD(\030)25064 15612 y FC(n)p Fw(\000)p FB(1)26892 15413 y FJ(\)[)p FD(@)28446 15612 y FC(\021)29005 15413 y FD(h)29754 15612 y FC(n)30380 15413 y FJ(])8553 17838 y(+)p FD(D)36 b FA(N)196 b FJ(\()p FD(\030)13041 18037 y FC(n)13666 17838 y FJ(\)[)p FD(@)15220 18037 y FC(\021)15779 17838 y FD(\030)16347 18037 y FC(n)p Fw(\000)p FB(1)18176 17838 y FJ(])74 b FA(\000)g FD(D)36 b FA(N)196 b FJ(\()p FD(\030)23194 18037 y FC(n)p Fw(\000)p FB(1)25022 17838 y FJ(\)[)p FD(@)26576 18037 y FC(\021)27135 17838 y FD(\030)27703 18037 y FC(n)p Fw(\000)p FB(1)29532 17838 y FJ(])74 b FA(\000)g FD(D)32190 17290 y FB(2)32716 17838 y FA(N)196 b FJ(\()p FD(\030)35076 18037 y FC(n)p Fw(\000)p FB(1)36904 17838 y FJ(\)[)p FD(@)38458 18037 y FC(\021)39017 17838 y FD(\030)39585 18037 y FC(n)p Fw(\000)p FB(1)41414 17838 y FD(;)221 b(h)42745 18037 y FC(n)43371 17838 y FJ(])8553 20180 y(+)p FD(@)10252 20379 y FC(\021)10811 20180 y FA(N)196 b FJ(\()p FD(\030)13171 20379 y FC(n)13796 20180 y FJ(\))295 b FA(\000)h FD(@)16613 20379 y FC(\021)17172 20180 y FA(N)196 b FJ(\()p FD(\030)19532 20379 y FC(n)p Fw(\000)p FB(1)21360 20180 y FJ(\))295 b FA(\000)g FD(@)24176 20379 y FC(\021)24735 20180 y FD(D)36 b FA(N)196 b FJ(\()p FD(\030)28211 20379 y FC(n)p Fw(\000)p FB(1)30039 20180 y FJ(\)[)p FD(h)31655 20379 y FC(n)32281 20180 y FJ(])32642 18706 y Fz(\021)33657 20180 y FD(:)6753 b FJ(\(5.94\))0 23504 y(F)-108 b(rom)433 b(\(5.94\))h(w)-36 b(e)434 b(obtain)14 26462 y FA(k)p FD(@)1365 26661 y FC(\021)1924 25386 y Fz(\000)2533 26462 y FD(R)11 b FJ(\()p FD(h)4789 26661 y FC(n)5415 26462 y FJ(\))5921 25386 y Fz(\001)6529 26462 y FA(k)7193 26661 y FC(\013)7797 26784 y Ft(n)p Fu(+1)10554 26462 y FA(\024)1107 b FJ(const)p FA(k)p FD(h)17077 26661 y FC(n)17703 26462 y FA(k)18367 26661 y FC(\013)18971 26772 y Ft(n)19593 26462 y FA(k)p FD(@)20944 26661 y FC(\021)21503 26462 y FD(h)22252 26661 y FC(n)22878 26462 y FA(k)23542 26661 y FC(\013)24146 26772 y Ft(n)25064 26462 y FJ(+)295 b(const)221 b FA(k)p FD(h)30975 26661 y FC(n)31601 26462 y FA(k)32265 25914 y FB(2)32265 26791 y FC(\013)32869 26902 y Ft(n)33787 26462 y FJ(+)294 b(const)p FD(")38672 25914 y Fw(\000)p FB(1)40151 26462 y FA(k)p FD(h)41564 26661 y FC(n)42190 26462 y FA(k)42854 25914 y FB(2)42854 26791 y FC(\013)43458 26902 y Ft(n)10554 28583 y FA(\024)1107 b FJ(const)221 b FD(c)16445 28034 y FB(2)16445 28911 y(0)17192 28583 y FD("e)18404 28034 y Fw(\000)p FC(\037)19720 27722 y Ft(n)20286 28034 y FC(\013)20890 28157 y Fu(0)21352 28034 y FC(=)p FB(8)p Fw(\000)109 b FB(~)-579 b FC(\037)23608 27722 y Ft(n)24175 28034 y FC(\013)24779 28157 y Fu(0)25241 28034 y FC(=)p FB(8)26458 28583 y FD(:)13952 b FJ(\(5.95\))0 31541 y(Deriving)381 b(\(5.92\))g(with)f(resp)36 b(ect)379 b(to)h FD(\021)427 b FJ(and)380 b(recalling)h(that)e FD(\021)29933 31059 y FB(2)30827 31541 y FJ(=)369 b FD($)415 b FJ(and)380 b FD(\021)36864 31059 y FB(3)37758 31541 y FJ(=)369 b FD(")p FJ(,)390 b(w)-36 b(e)380 b(get)2510 34499 y FA(\000)p FD(@)4230 34698 y FC(\021)4789 34499 y FA(L)5705 33951 y FB(\()p FC(n)p FB(+1\))8266 34499 y FD(@)8953 34698 y FC(\021)9511 34499 y FD(h)10260 34698 y FC(n)p FB(+1)13196 34499 y FJ(=)15314 33423 y Fz(\000)15923 34499 y FD(@)16610 34698 y FC(\021)17169 34499 y FA(L)18085 33951 y FB(\()p FC(n)p FB(+1\))20646 33423 y Fz(\001)21254 34499 y FD(h)22003 34698 y FC(n)p FB(+1)15314 36934 y FJ(+)p FD(P)17345 36386 y FC(n)p FB(+1)17164 37263 y FC(n)19174 35460 y Fz(h)19801 36934 y FJ(3)p FD($)36 b FA(N)196 b FJ(\()p FD(\030)23922 37133 y FC(n)p Fw(\000)p FB(1)25750 36934 y FJ(\))295 b(+)g FD("@)29154 37133 y FC(\021)29712 36934 y FA(N)196 b FJ(\()p FD(\030)32072 37133 y FC(n)p Fw(\000)p FB(1)33900 36934 y FJ(\))15314 39237 y(+)p FD("D)36 b FA(N)196 b FJ(\()p FD(\030)20411 39436 y FC(n)p Fw(\000)p FB(1)22239 39237 y FJ(\)[)p FD(@)23793 39436 y FC(\021)24352 39237 y FD(\030)24920 39436 y FC(n)p Fw(\000)p FB(1)26748 39237 y FJ(])296 b(+)f(3)p FD($)36 b(D)g FA(N)196 b FJ(\()p FD(\030)33949 39436 y FC(n)p Fw(\000)p FB(1)35777 39237 y FJ(\)[)p FD(h)37393 39436 y FC(n)38019 39237 y FJ(])15314 41579 y(+)p FD("@)17622 41778 y FC(\021)18180 41579 y FD(D)36 b FA(N)196 b FJ(\()p FD(\030)21656 41778 y FC(n)p Fw(\000)p FB(1)23485 41579 y FJ(\)[)p FD(h)25101 41778 y FC(n)25727 41579 y FJ(])295 b(+)g FD("D)29415 41030 y FB(2)29941 41579 y FA(N)196 b FJ(\()p FD(\030)32301 41778 y FC(n)p Fw(\000)p FB(1)34129 41579 y FJ(\)[)p FD(@)35683 41778 y FC(\021)36242 41579 y FD(\030)36810 41778 y FC(n)p Fw(\000)p FB(1)38638 41579 y FD(;)221 b(h)39969 41778 y FC(n)40596 41579 y FJ(])40957 40104 y Fz(i)15314 44014 y FJ(+3)p FD($)36 b(R)11 b FJ(\()p FD(h)20343 44213 y FC(n)20969 44014 y FJ(\))295 b(+)f FD("@)24372 44213 y FC(\021)24931 42938 y Fz(\000)25540 44014 y FD(R)11 b FJ(\()p FD(h)27796 44213 y FC(n)28422 44014 y FJ(\))28928 42938 y Fz(\001)29758 44014 y FD(:)10652 b FJ(\(5.96\))0 46972 y(F)-108 b(rom)420 b(the)f(de\014nition)h(of)h (the)e(linear)i(op)36 b(erator)420 b FD(L)g FJ(in)h(\(5.43\),)j(using)c (\(4.55\))h(and)f(noting)0 48577 y(that)433 b(b)-36 b(y)434 b(\(4.53\))17430 50207 y FD(@)18117 50406 y FC(\021)18676 50207 y FD(I)19250 50406 y FB(0)20144 50207 y FJ(=)369 b FD(O)36 b FJ(\(1)p FD(=\021)25051 49659 y FB(3)25576 50207 y FJ(\))221 b FD(;)14107 b FJ(\(5.97\))0 52561 y(one)434 b(has)15633 54191 y FD(@)16320 54390 y FC(\021)16879 54191 y FD(L)369 b FJ(=)g FD(O)36 b FJ(\()p FD(\021)48 b FJ(\))294 b(+)h FD(O)36 b FJ(\(1)p FD(=\021)48 b FJ(\))221 b FD(;)0 56544 y FJ(hence)433 b(w)-36 b(e)434 b(obtain)14337 58174 y FD(@)15024 58373 y FC(\021)15583 58174 y FA(L)16499 57626 y FB(\()p FC(n)p FB(+1\))19429 58174 y FJ(=)368 b FD(O)36 b FJ(\()p FD(\021)48 b FJ(\))295 b(+)g FD(O)36 b FJ(\(1)p FD(=\021)48 b FJ(\))221 b FD(;)0 60528 y FJ(namely)9451 62158 y FA(k)10115 61082 y Fz(\000)10724 62158 y FD(@)11411 62357 y FC(\021)11970 62158 y FA(L)12886 61609 y FB(\()p FC(n)p FB(+1\))15446 61082 y Fz(\001)16055 62158 y 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y FA(\024)369 b FJ(const)221 b FD(")21669 36621 y FB(3)22563 37103 y FJ(=)369 b(const)p FD($)28025 36621 y FB(3)p FC(=)p FB(2)29491 37103 y Fs(.)6299 39298 y Fp(pr)-30 b(oof.)659 b FJ(By)434 b(\(5.81\))10200 42795 y FA(k)p FD(\030)11432 42994 y FC(n)p FB(+)p FC(m)13918 42795 y FA(\000)295 b FD(\030)15814 42994 y FC(n)16441 42795 y FA(k)17105 43001 y FC(\013)17709 43124 y Fu(0)18170 43001 y FC(=)p FB(2)p FC(;\033)21115 42795 y FJ(=)23244 40470 y Fz(\015)23244 41267 y(\015)23244 42064 y(\015)23244 42861 y(\015)23244 43658 y(\015)23982 41134 y FC(n)p FB(+)p FC(m)24091 41533 y Fz(X)24289 44332 y FC(i)p FB(=1)26339 42795 y FD(h)27088 42994 y FC(i)27758 42795 y FA(\000)29761 41134 y FC(n)29087 41533 y Fz(X)29285 44332 y FC(i)p FB(=1)31227 42795 y FD(h)31976 42994 y FC(i)32351 40470 y Fz(\015)32351 41267 y(\015)32351 42064 y(\015)32351 42861 y(\015)32351 43658 y(\015)33089 44520 y FC(\013)33693 44643 y Fu(0)34155 44520 y FC(=)p FB(2)p FC(;\033)36351 42795 y FJ(=)37732 40470 y Fz(\015)37732 41267 y(\015)37732 42064 y(\015)37732 42861 y(\015)37732 43658 y(\015)38815 41134 y FC(n)p FB(+)p FC(m)38923 41533 y Fz(X)38470 44332 y FC(i)p FB(=)p FC(n)p FB(+1)41516 42795 y FD(h)42265 42994 y FC(i)42641 40470 y Fz(\015)42641 41267 y(\015)42641 42064 y(\015)42641 42861 y(\015)42641 43658 y(\015)43379 44613 y FC(\013)43983 44736 y Fu(0)44444 44613 y FC(=)p FB(2)p FC(;\033)21104 47451 y FA(\024)24187 45791 y Fw(1)23698 46189 y Fz(X)23244 48989 y FC(i)p FB(=)p FC(n)p FB(+1)26291 47451 y FD(c)26851 47650 y FB(0)27377 47451 y FD("e)28589 46903 y Fw(\000)p FC(\037)29905 46590 y Ft(i)30256 46903 y FC(\013)30860 47026 y Fu(0)31321 46903 y FC(=)p FB(8)32539 47451 y FD(:)6299 51046 y FJ(It)434 b(follo)-36 b(ws)435 b(that)e FA(k)p FD(\030)16028 51245 y FC(n)16950 51046 y FA(\000)295 b FD(\030)64 b FA(k)19574 51253 y FC(\013)20178 51376 y Fu(0)20640 51253 y FC(=)p FB(2)p FC(;\033)22837 51046 y FA(!)369 b FJ(0.)579 b(F)-108 b(rom)433 b(\(5.85\))h(and)f (\(5.83\),)i(one)f(has)11207 53523 y FA(L)12123 52975 y FB(\()p FC(n)p FB(\))13481 53523 y FD(h)14230 53722 y FC(n)15225 53523 y FJ(=)369 b FA(\000)p FD(Lh)19273 53722 y FC(n)20194 53523 y FA(\000)295 b FD("P)22969 53722 y FC(n)23596 53523 y FD(D)36 b FA(N)196 b FJ(\()p FD(\030)27072 53722 y FC(n)p Fw(\000)p FB(1)28900 53523 y FJ(\)[)p FD(h)30516 53722 y FC(n)31142 53523 y FJ(])369 b(=)g FD(L\030)34706 53722 y FC(n)p Fw(\000)p FB(1)36829 53523 y FJ(+)295 b FD("P)39583 53722 y FC(n)40210 53523 y FA(N)196 b FJ(\()p FD(\030)42570 53722 y FC(n)p Fw(\000)p FB(1)44398 53523 y FJ(\))221 b FD(:)6299 56034 y FJ(T)-108 b(aking)435 b(the)e(limit)h(for)g FD(n)369 b FA(!)h(1)434 b FJ(in)f(the)g(space)h FD(W)31216 55356 y FB(\()p FC(n)p FB(\))31035 56454 y FC(s)p Fw(\000)p FB(1)p FC(;\013)33537 56577 y Fu(0)33999 56454 y FC(=)p FB(2)p FC(;\033)35826 56034 y FJ(,)g(one)g(gets)f(that)17462 58507 y FD(L\030)18915 58706 y FC(n)p Fw(\000)p FB(1)21039 58507 y FJ(+)295 b FD("P)23793 58706 y FC(n)24419 58507 y FA(N)196 b FJ(\()p FD(\030)26779 58706 y FC(n)p Fw(\000)p FB(1)28607 58507 y FJ(\))369 b FA(\000)-221 b(!)369 b FD(L\030)359 b FJ(+)295 b FD(")p FA(N)196 b FJ(\()p FD(\030)64 b FJ(\))221 b FD(:)6299 60838 y FJ(No)-36 b(w)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b(to)g(pro)-36 b(v)g(e)433 b(that)19134 63169 y FA(\000)p FD(Lh)21801 63368 y FC(n)22722 63169 y FA(\000)295 b FD("P)25497 63368 y FC(n)26123 63169 y FD(D)36 b FA(N)196 b FJ(\()p FD(\030)29599 63368 y FC(n)p Fw(\000)p FB(1)31428 63169 y FJ(\)[)p FD(h)33044 63368 y FC(n)33670 63169 y FJ(])369 b FA(\000)-221 b(!)369 b FJ(0)6299 65499 y(in)434 b(the)f(same)h(space.)578 b(Indeed,)15694 66701 y Fz(\015)15694 67498 y(\015)16432 67830 y FD(Lh)18066 68029 y FC(n)18692 66701 y Fz(\015)18692 67498 y(\015)19430 68350 y FC(s)p Fw(\000)p FB(1)p FC(;\013)21932 68473 y Fu(0)22394 68350 y FC(=)p FB(2)p FC(;\033)24590 67830 y FA(\024)370 b(k)p FD(L)p FA(k)28206 68029 y FC(s;s)p Fw(\000)p FB(1)30595 67830 y FA(k)p FD(h)32008 68029 y FC(n)32634 67830 y FA(k)33298 68036 y 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53843 y FJ(:)369 b FD(`)35545 53361 y FC(a;s)36911 53843 y FA(\002)114 b FD(`)38604 53361 y FC(a;s)40225 53843 y FA(\000)-221 b(!)369 b FD(`)43280 53361 y FC(a;s)p FB(+1)45848 53843 y FA(\002)114 b FD(`)47541 53361 y FC(a;s)p FB(+1)49995 53843 y Fs(,)6299 55448 y(as)466 b(in)e FJ(\(5.66\))p Fs(.)599 b(Then,)464 b(ther)-66 b(e)464 b(exists)g FD(")25061 55647 y FB(0)25956 55448 y FD(>)369 b FJ(0)p Fs(,)465 b(indep)-66 b(endent)463 b(of)h FD(k)45 b Fs(,)465 b(such)h(that,)e FA(8)221 b(j)p FD(")p FA(j)370 b(\024)f FD(")49868 55647 y FB(0)6299 57053 y Fs(and)492 b FA(8)221 b FD(k)45 b(;)221 b(j)75 b Fs(,)500 b(ther)-66 b(e)491 b(exist)g(eigenvalues)g FD(\025)26532 57252 y FC(k)24 b(j)27532 57053 y FJ(\()p FD(")p FJ(\))491 b Fs(and)h(asso)-66 b(ciate)g(d)492 b(eigenve)-66 b(ctors)489 b FD(')46468 57252 y FC(k)24 b(j)47469 57053 y FJ(\()p FD(")p FJ(\))418 b FA(2)6299 58658 y FD(`)6845 58176 y FC(a;s)8392 58658 y FA(\002)295 b FD(`)10266 58176 y FC(a;s)11982 58658 y Fs(with)465 b FA(j)p 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FJ(~)24560 6320 y(\012)25499 6519 y FC(j)25987 6320 y FD(;)221 b FJ(0\))27725 5244 y Fz(\003)28280 5541 y Fw(\000)p FB(1)29906 6320 y FA(2)369 b(L)p FJ(\()p FD(Z)23 b(;)221 b(Y)290 b FJ(\).)0 7925 y(Let)580 b(us)f(recall)i(that,)617 b(if)581 b FD(e)13212 8124 y FC(j)14279 7925 y FJ(is)f(a)h(basis)f(of)h FD(`)22478 7443 y FC(a;s)24124 7925 y FA(\002)395 b FD(`)26098 7443 y FC(a;s)27930 7925 y FJ(then)628 b(~)-699 b FD(e)31642 8124 y FC(j)32747 7925 y FJ(:=)619 b FD(j)75 b(e)35955 8124 y FC(j)37021 7925 y FJ(is)581 b(a)f(basis)h(of)0 9530 y FD(`)546 9048 y FC(a;s)p Fw(\000)p FB(1)3326 9530 y FA(\002)327 b FD(`)5232 9048 y FC(a;s)p Fw(\000)p FB(1)7686 9530 y FJ(.)718 b(Let)479 b(b)36 b(e)480 b FD(v)527 b FJ(a)481 b(v)-36 b(ector,)492 b(denote)479 b(b)-36 b(y)480 b FD(v)26285 9729 y FC(j)27220 9530 y FJ(:=)448 b FA(h)p FD(e)30161 9729 y FC(j)30647 9530 y FD(;)221 b(v)48 b FA(i)32423 9729 y FC(s)33393 9530 y FJ(the)480 b FD(j)36293 9048 y FC(th)37230 9530 y FJ({comp)36 b(onen)-36 b(t)0 11135 y(in)526 b(the)f(space)h FD(`)8056 10653 y FC(a;s)9665 11135 y FA(\002)358 b FD(`)11602 10653 y FC(a;s)13379 11135 y FJ(and)525 b(b)-36 b(y)575 b(~)-699 b FD(v)18528 11334 y FC(j)19541 11135 y FJ(:=)526 b FA(h)p FD(e)22560 11334 y FC(j)23046 11135 y FD(;)221 b(v)48 b FA(i)24822 11334 y FC(s)25838 11135 y FJ(the)525 b FD(j)28783 10653 y FC(th)29720 11135 y FJ({comp)36 b(onen)-36 b(t)526 b(in)f(the)g(space)0 12740 y FD(`)546 12258 y FC(a;s)p Fw(\000)p FB(1)3295 12740 y FA(\002)295 b FD(`)5169 12258 y FC(a;s)p Fw(\000)p FB(1)7623 12740 y FJ(.)0 14345 y(Compute)6033 17245 y FD(@)6720 17444 y FC(y)7272 17245 y FD(F)181 b FJ(\()p FD(e)9404 17444 y FC(j)9890 17245 y FD(;)10618 16909 y FJ(~)10472 17245 y(\012)11411 17444 y FC(j)11899 17245 y FD(;)221 b FJ(0\)[)p FD(w)36 b(;)221 b(\026)p FJ(])1108 b(=)19917 16169 y Fz(\000)20821 17245 y FA(\000)295 b FE(\012)p FD(w)331 b FJ(+)25941 16909 y(~)25796 17245 y(\012)26735 17444 y FC(j)27222 17245 y FD(w)g FJ(+)295 b FD(\026e)31176 17444 y FC(j)31662 17245 y FD(;)221 b FA(h)p FD(e)33364 17444 y FC(j)33851 17245 y FD(;)g(w)36 b FA(i)35916 17444 y FC(s)36406 16169 y Fz(\001)17798 19547 y FJ(=)19917 18471 y Fz(\000)20526 19547 y FD(\026e)21912 19746 y FC(j)22693 19547 y FJ(+)24000 18285 y Fz(X)24218 21114 y FC(i)p Fw(6)p FB(=)p FC(j)25919 19547 y FJ(\()26570 19211 y(~)26425 19547 y(\012)27364 19746 y FC(j)28146 19547 y FA(\000)29619 19211 y FJ(~)29475 19547 y(\012)30414 19746 y FC(i)30790 19547 y FJ(\))p FD(w)32226 19746 y FC(i)32601 19547 y FD(e)33204 19746 y FC(i)33579 19547 y FD(;)221 b(w)35091 19746 y FC(j)35579 18471 y Fz(\001)17798 24187 y FJ(=)19917 21915 y Fz( )21102 23288 y FD(\026)p 21102 23881 783 54 v 21187 25098 a(j)22287 24187 y FJ(~)-699 b FD(e)22841 24386 y FC(j)23623 24187 y FJ(+)24930 22925 y Fz(X)25148 25754 y FC(i)p Fw(6)p FB(=)p FC(j)27347 22952 y FJ(~)27203 23288 y(\012)28142 23487 y FC(j)28924 23288 y FA(\000)30397 22952 y FJ(~)30253 23288 y(\012)31192 23487 y FC(i)p 27203 23881 4365 54 v 29164 25098 a FD(i)31922 24187 y(w)32852 24386 y FC(i)33276 24187 y FJ(~)h FD(e)33831 24386 y FC(i)34206 24187 y FD(;)221 b(w)35718 24386 y FC(j)36205 21915 y Fz(!)37700 24187 y FD(;)0 28681 y FJ(and,)433 b(b)36 b(eing)434 b FD(T)476 b FA(\016)295 b FD(@)9279 28880 y FC(y)9831 28681 y FD(F)550 b FJ(=)368 b Fv(I)p FJ(,)434 b(denoting)f FD(@)19949 28880 y FC(y)20501 28681 y FD(F)181 b FJ(\()p FD(e)22633 28880 y FC(j)23119 28681 y FD(;)23846 28345 y FJ(~)23701 28681 y(\012)24640 28880 y FC(j)25128 28681 y FD(;)221 b FJ(0\)[)p FD(w)36 b(;)221 b(\026)p FJ(])370 b(=)f(\()266 b(~)-916 b FD(w)35 b(;)324 b FJ(~)-753 b FD(\026)p FJ(\),)434 b(one)g(obtains)11292 32782 y FD(T)181 b FJ([)266 b(~)-916 b FD(w)36 b(;)324 b FJ(~)-753 b FD(\026)p FJ(])369 b(=)17038 30511 y Fz( )18192 32782 y FJ(~)-752 b FD(\026)o(e)19475 32981 y FC(j)20257 32782 y FJ(+)21564 31520 y Fz(X)21781 34349 y FC(i)p Fw(6)p FB(=)p FC(j)24655 31884 y FD(i)266 b FJ(~)-916 b FD(w)26029 32083 y FC(i)26405 31884 y FD(e)27008 32083 y FC(i)p 23837 32477 V 23982 33518 a FJ(~)23837 33853 y(\012)24776 34052 y FC(j)25558 33853 y FA(\000)27031 33518 y FJ(~)26887 33853 y(\012)27826 34052 y FC(i)28335 32782 y FD(;)221 b(j)341 b FJ(~)-916 b FD(w)30460 32981 y FC(j)30947 30511 y Fz(!)32441 32782 y FD(:)7318 b FJ(\(5.119\))0 36982 y(Hence,)434 b(b)-36 b(y)433 b(Lemma)h(5.7.3,)4576 40970 y FA(k)p FD(T)181 b FA(k)6847 40421 y FB(2)6847 41298 y FC(s)p Fw(\000)p FB(1)p FC(;s)10355 40970 y FJ(=)3235 b(sup)12484 42129 y Fw(j)193 b FB(~)-663 b FC(w)24 b Fw(j)13705 42252 y Ft(s)p Fo(\000)p Fu(1)15195 42129 y FB(+)p Fw(j)74 b FB(~)-544 b FC(\026)p Fw(j)17015 41877 y Fu(2)17477 42129 y FB(=1)18900 38698 y Fz( )19952 39044 y(\014)19952 39841 y(\014)19952 40638 y(\014)19952 41435 y(\014)20497 40970 y FJ(~)-752 b FD(\026)o(e)21780 41169 y FC(j)22562 40970 y FJ(+)23869 39708 y Fz(X)24086 42537 y FC(i)p Fw(6)p FB(=)p FC(j)26960 40071 y FD(i)266 b FJ(~)-916 b FD(w)28334 40270 y FC(i)28710 40071 y FD(e)29313 40270 y FC(i)p 26142 40664 V 26287 41705 a FJ(~)26142 42041 y(\012)27081 42240 y FC(j)27863 42041 y FA(\000)29336 41705 y FJ(~)29192 42041 y(\012)30131 42240 y FC(i)30640 39044 y Fz(\014)30640 39841 y(\014)30640 40638 y(\014)30640 41435 y(\014)31082 39394 y FB(2)31082 42287 y FC(s)31903 40970 y FJ(+)295 b FA(j)p FD(j)341 b FJ(~)-916 b FD(w)35122 41169 y FC(j)35609 40970 y FA(j)35978 40421 y FB(2)36503 38698 y Fz(!)10355 45609 y FJ(=)3235 b(sup)12484 46769 y Fw(j)193 b FB(~)-663 b FC(w)24 b Fw(j)13705 46892 y Ft(s)p Fo(\000)p Fu(1)15195 46769 y FB(+)p Fw(j)74 b FB(~)-544 b FC(\026)p Fw(j)17015 46517 y Fu(2)17477 46769 y FB(=1)18900 43338 y Fz( )19952 45609 y FA(j)102 b FJ(~)-752 b FD(\026)o FA(j)21472 45061 y FB(2)22293 45609 y FJ(+)23600 44347 y Fz(X)23818 47176 y FC(i)p Fw(6)p FB(=)p FC(j)25740 43736 y Fz(\022)27670 44710 y FD(i)265 b FJ(~)-915 b FD(w)29044 44909 y FC(i)29419 44710 y FD(e)30022 44909 y FC(i)p 26851 45304 V 26996 46345 a FJ(~)26851 46680 y(\012)27790 46879 y FC(j)28572 46680 y FA(\000)30045 46345 y FJ(~)29901 46680 y(\012)30840 46879 y FC(i)31349 43736 y Fz(\023)32326 44033 y FB(2)33147 45609 y FJ(+)295 b FD(j)35067 45061 y FB(2)35859 45609 y FJ(~)-916 b FD(w)36559 45061 y FB(2)36523 45937 y FC(j)37084 43338 y Fz(!)10344 50248 y FA(\024)3225 b FJ(sup)12484 51408 y Fw(j)193 b FB(~)-663 b FC(w)24 b Fw(j)13705 51531 y Ft(s)p Fo(\000)p Fu(1)15195 51408 y FB(+)p Fw(j)74 b FB(~)-544 b FC(\026)p Fw(j)17015 51156 y Fu(2)17477 51408 y FB(=1)18900 47977 y Fz( )19952 50248 y FA(j)102 b FJ(~)-752 b FD(\026)o FA(j)21472 49700 y FB(2)22293 50248 y FJ(+)295 b(const)221 b FD(j)27404 49700 y FB(2)28151 48987 y Fz(X)28951 51786 y FC(i)30558 50248 y FJ(~)-916 b FD(w)31258 49700 y FB(2)31222 50577 y FC(i)31783 47977 y Fz(!)33203 50248 y FA(\024)370 b FJ(const)221 b FD(j)38410 49700 y FB(2)39157 50248 y FD(;)0 54377 y FJ(where)433 b(in)h(the)f(last)h(inequalit)-36 b(y)435 b(w)-36 b(e)433 b(used)g(that)23715 53380 y Fz(P)25118 54764 y FC(i)25981 54377 y FJ(~)-916 b FD(w)26681 53894 y FB(2)26645 54705 y FC(i)27575 54377 y FJ(=)369 b FA(j)266 b FJ(~)-916 b FD(w)35 b FA(j)30659 53894 y FB(2)30659 54705 y FC(s)p Fw(\000)p FB(1)32352 54377 y FJ(.)578 b(It)434 b(follo)-36 b(ws)435 b(that)16638 57276 y FA(k)p FD(T)181 b FA(k)18909 57475 y FC(s)p Fw(\000)p FB(1)p FC(;s)21668 57276 y FA(\024)369 b FJ(const)221 b FD(j)296 b(:)12664 b FJ(\(5.120\))0 60175 y(W)-108 b(e)434 b(ha)-36 b(v)g(e)433 b(to)h(c)-36 b(ho)36 b(ose)434 b(the)f(radius)g FD(\032)h FJ(suc)-36 b(h)432 b(that)13692 63309 y(sup)13475 64394 y FC(x)p Fw(2)p FC(X)15406 64517 y Fu(0)16089 63309 y FA(k)p FD(F)181 b FJ(\()p FD(y)18916 63508 y FB(0)19441 63309 y FD(;)221 b(x)p FJ(\))p FA(k)21932 63508 y FC(Z)23058 63309 y FA(\024)26913 62411 y FD(\032)p 24593 63004 5311 54 v 24593 64220 a FJ(2)p FA(k)p FD(T)181 b FA(k)27514 64419 y FC(s)p Fw(\000)p FB(1)p FC(;s)30258 63309 y FD(:)0 67272 y FJ(It)482 b(results)f FD(F)181 b FJ(\()p FD(y)7801 67471 y FB(0)8327 67272 y FD(;)221 b(x)p FJ(\))451 b(=)g FD(F)181 b FJ(\()p FD(e)14200 67471 y FC(j)14686 67272 y FD(;)15413 66937 y FJ(~)15268 67272 y(\012)16207 67471 y FC(j)16695 67272 y FJ(;)221 b FD(")p FJ(\))451 b(=)g(\()p FA(\000)p FD(")p FA(M)24049 67471 y FC(k)24618 67272 y FD(e)25221 67471 y FC(j)25707 67272 y FD(;)221 b FJ(0\))483 b(and)e FA(k)p FD(F)181 b FA(k)32856 67471 y FC(Z)34063 67272 y FJ(=)450 b FD(")p FA(jM)38098 67471 y FC(k)38667 67272 y FD(e)39270 67471 y FC(j)39757 67272 y FA(j)40126 67471 y FC(s)p Fw(\000)p FB(1)41819 67272 y FJ(.)723 b(In)0 68877 y(particular,)434 b(denoting)f(b)-36 b(y)433 b FD(M)509 b FJ(=)368 b(diag)19 b(\()p FD(j)75 b FJ(\),)433 b FD(j)444 b FA(\025)369 b FJ(1,)5127 71777 y FA(jM)7091 71976 y FC(k)7660 71777 y FD(e)8263 71976 y FC(j)8750 71777 y FA(j)9119 71976 y FC(s)p Fw(\000)p FB(1)11930 71777 y FJ(=)1117 b FA(jM)16023 71976 y FC(k)16592 71777 y FD(M)139 b(M)19386 71228 y Fw(\000)p FB(1)20644 71777 y FD(e)21247 71976 y FC(j)21733 71777 y FA(j)22102 71976 y FC(s)p Fw(\000)p FB(1)24164 71777 y FA(\024)369 b(kM)27825 71976 y FC(k)28395 71777 y FA(k)29059 71976 y FC(s)p Fw(\000)p FB(2)p FC(;s)p Fw(\000)p FB(1)32650 71777 y FA(k)p FD(M)139 b FA(k)35375 71976 y FC(s)p Fw(\000)p FB(1)p FC(;s)p Fw(\000)p FB(2)11919 74486 y FA(\024)1107 b(j)p FD(M)15825 73938 y Fw(\000)p FB(1)17083 74486 y FD(e)17686 74685 y FC(j)18172 74486 y FA(j)18541 74685 y FC(s)p Fw(\000)p FB(1)20603 74486 y FJ(=)22116 73587 y FA(kM)24375 73786 y FC(k)24945 73587 y FA(k)25609 73786 y FC(s)p Fw(\000)p FB(2)p FC(;s)p Fw(\000)p FB(1)p 22116 74180 7084 54 v 25089 75397 a FD(j)25702 75013 y FB(2)29555 74486 y FD(;)21072 78312 y FJ(180)p eop %%Page: 181 181 181 180 bop 6299 4488 a FJ(where)434 b(w)-36 b(e)434 b(used)e(that)i FA(j)p FD(M)19526 4006 y Fw(\000)p FB(1)20783 4488 y FD(e)21386 4687 y FC(j)21873 4488 y FA(j)22242 4687 y FC(s)p Fw(\000)p FB(1)24304 4488 y FJ(=)368 b FA(j)p FD(e)26656 4687 y FC(j)27143 4488 y FD(=j)75 b FA(j)28775 4687 y FC(s)p Fw(\000)p FB(1)30837 4488 y FJ(=)368 b FA(j)49 b FJ(~)-699 b FD(e)33189 4687 y FC(j)33676 4488 y FD(=j)34939 4006 y FB(2)35465 4488 y FA(j)35834 4687 y FC(s)p Fw(\000)p FB(1)37896 4488 y FJ(=)369 b(1)p FD(=j)41190 4006 y FB(2)41716 4488 y FJ(.)578 b(Hence)17963 8058 y FA(k)p FD(F)181 b FA(k)20314 8257 y FC(Z)21439 8058 y FA(\024)22974 7159 y FD(")p FA(kM)25842 7358 y FC(k)26411 7159 y FA(k)27075 7358 y FC(s)p Fw(\000)p FB(2)p FC(;s)p Fw(\000)p FB(1)p 22974 7752 7693 54 v 26251 8969 a FD(j)26864 8585 y FB(2)31168 8058 y FA(\024)35023 7159 y FD(\032)p 32703 7752 5311 54 v 32703 8969 a FJ(2)p FA(k)p FD(T)g FA(k)35624 9168 y FC(s)p Fw(\000)p FB(1)p FC(;s)38368 8058 y FD(;)6299 11640 y FJ(where)434 b(the)f(last)h (inequalit)-36 b(y)434 b(holds)g(if,)g(b)-36 b(y)434 b(\(5.120\),)24062 14746 y FD(\032)369 b FJ(:=)g(const)30737 13847 y FD(")p 30168 14440 1748 54 v 30168 15657 a(")30777 15856 y FB(0)31302 15657 y FD(j)32269 14746 y(;)13789 b FJ(\(5.121\))6299 18141 y(where)16097 20178 y FD(")16706 20377 y FB(0)17600 20178 y FJ(:=)369 b(const)221 b(min)24922 18305 y Fz(\032)28066 19279 y FJ(1)p 26051 19872 4680 54 v 26051 21089 a FA(kM)28310 21288 y FC(k)28880 21089 y FA(k)29544 21288 y FC(s;s)30864 20178 y FD(;)34796 19279 y FJ(1)p 31579 19872 7084 54 v 31579 21089 a FA(kM)33838 21288 y FC(k)34407 21089 y FA(k)35071 21288 y FC(s)p Fw(\000)p FB(2)p FC(;s)p Fw(\000)p FB(1)38796 18305 y Fz(\033)40235 20178 y FD(:)5823 b FJ(\(5.122\))6299 23229 y(W)-108 b(e)666 b(note)f(that)h FA(kM)17203 23428 y FC(k)17772 23229 y FA(k)18436 23428 y FC(s;s)20289 23229 y FJ(and)f FA(kM)25309 23428 y FC(k)25878 23229 y FA(k)26542 23428 y FC(s)p Fw(\000)p FB(2)p FC(;s)p Fw(\000)p FB(1)30800 23229 y FJ(are)h(b)36 b(ounded)664 b(b)-36 b(y)665 b(a)h(constan)-36 b(t)666 b FK(inde-)6299 24834 y(p)36 b(enden)-36 b(t)607 b FJ(of)578 b FD(k)623 b FJ(\(recall)578 b (\(5.59\),\(5.66\),\(5.67\)\).)1014 b(No)-36 b(w,)615 b(w)-36 b(e)578 b(ha)-36 b(v)g(e)578 b(to)g(compute)f(\()p FA(\000)p Fv(I)393 b FJ(+)6299 26439 y FD(T)181 b(@)7929 26638 y FC(y)8481 26439 y FD(F)g FJ(\)\()p FD(v)48 b(;)221 b(\025)p FJ(;)g FD(")p FJ(\)[)p FD(w)36 b(;)221 b(\026)p FJ(].)578 b(First,)434 b(compute)6403 29237 y FD(@)7090 29436 y FC(y)7642 29237 y FD(F)181 b FJ(\()p FD(v)48 b(;)221 b(\025)p FJ(;)g FD(")p FJ(\)[)p FD(w)36 b(;)221 b(\026)p FJ(])g(=)17393 28161 y Fz(\000)18297 29237 y FA(\000)19770 28901 y FJ(~)19626 29237 y(\012)p FD(w)331 b FJ(+)295 b FD(\025w)330 b FJ(+)295 b FD(\026v)342 b FA(\000)295 b FD(")p FA(M)31745 29436 y FC(k)32314 29237 y FD(w)36 b(;)221 b FA(h)p FD(v)48 b(;)221 b(w)36 b FA(i)37121 29436 y FC(s)37611 28161 y Fz(\001)16160 32469 y FJ(=)17172 30197 y Fz( )18224 31207 y(X)19023 34006 y FC(i)20364 30596 y Fz(\022)21474 31570 y FD(\025)295 b FA(\000)24001 31234 y FJ(~)23857 31570 y(\012)24796 31769 y FC(i)p 21474 32163 3698 54 v 23101 33380 a FD(i)25304 32469 y(w)26234 32668 y FC(i)26905 32469 y FJ(+)28345 31570 y FD(\026v)29757 31769 y FC(i)p 28345 32163 1788 54 v 29016 33380 a FD(i)30265 30596 y Fz(\023)31291 32469 y FJ(~)-699 b FD(e)31845 32668 y FC(i)32516 32469 y FA(\000)74 b FD(")34232 31207 y Fz(X)35031 34006 y FC(i)36151 32469 y FA(hM)38263 32668 y FC(k)38831 32469 y FD(w)36 b(;)270 b FJ(~)-699 b FD(e)40982 32668 y FC(i)41358 32469 y FA(i)41875 32668 y FC(s)p Fw(\000)p FB(1)43616 32469 y FJ(~)g FD(e)44170 32668 y FC(i)44546 32469 y FD(;)221 b FA(h)p FD(v)48 b(;)221 b(w)36 b FA(i)48387 32668 y FC(s)48877 30197 y Fz(!)49928 32469 y FD(:)6299 36365 y FJ(Hence,)434 b(w)-36 b(e)434 b(obtain)9456 39162 y FA(\000)443 b FJ(\()p FD(w)36 b(;)221 b(\026)p FJ(\))295 b(+)f FD(T)181 b(@)17506 39361 y FC(y)18058 39162 y FD(F)g FJ(\()p FD(v)48 b(;)221 b(\025)p FJ(;)g FD(")p FJ(\)[)p FD(w)36 b(;)221 b(\026)p FJ(])369 b(=)11596 42301 y(=)12977 40030 y Fz( )14324 42301 y FA(\000)295 b FD(w)331 b FJ(+)294 b FA(h)p FD(v)48 b(;)221 b(w)36 b FA(i)21478 42500 y FC(s)21968 42301 y FD(e)22571 42500 y FC(j)23353 42301 y FJ(+)24659 41039 y Fz(X)24877 43868 y FC(i)p Fw(6)p FB(=)p FC(j)26932 41403 y FJ(\()p FD(\025)295 b FA(\000)29965 41067 y FJ(~)29820 41403 y(\012)30759 41602 y FC(i)31135 41403 y FJ(\))p FD(w)32571 41602 y FC(i)33242 41403 y FJ(+)g FD(\026v)35961 41602 y FC(i)36631 41403 y FA(\000)g FD("i)p FA(hM)41124 41602 y FC(k)41692 41403 y FD(w)36 b(;)270 b FJ(~)-699 b FD(e)43843 41602 y FC(i)44219 41403 y FA(i)44736 41602 y FC(s)p Fw(\000)p FB(1)p 26932 41996 19497 54 v 34642 43037 a FJ(~)34498 43373 y(\012)35437 43572 y FC(j)36219 43373 y FA(\000)37692 43037 y FJ(~)37548 43373 y(\012)38487 43572 y FC(i)46783 42301 y FD(e)47386 42500 y FC(i)47761 42301 y FD(;)11596 46941 y FA(\000)p FD(\026)295 b FJ(+)g(\()p FD(\025)g FA(\000)18046 46605 y FJ(~)17902 46941 y(\012)18841 47140 y FC(j)19328 46941 y FJ(\))p FD(w)20764 47140 y FC(j)21546 46941 y FJ(+)g FD(\026v)24265 47140 y FC(j)25046 46941 y FA(\000)g FD("j)75 b FA(hM)29708 47140 y FC(k)30277 46941 y FD(w)36 b(;)269 b FJ(~)-698 b FD(e)32428 47140 y FC(j)32915 46941 y FA(i)33432 47140 y FC(s)p Fw(\000)p FB(1)35124 44669 y Fz(!)11596 51369 y FJ(=)12977 49097 y Fz( )14028 51369 y FA(h)p FJ(\()p FD(v)342 b FA(\000)296 b FD(e)17954 51568 y FC(j)18440 51369 y FJ(\))p FD(;)221 b(w)36 b FA(i)21011 51568 y FC(s)21502 51369 y FD(e)22105 51568 y FC(j)22886 51369 y FJ(+)24193 50107 y Fz(X)24411 52936 y FC(i)p Fw(6)p FB(=)p FC(j)26466 50470 y FJ(\()p FD(\025)295 b FA(\000)29499 50134 y FJ(~)29354 50470 y(\012)30293 50669 y FC(i)30669 50470 y FJ(\))p FD(w)32105 50669 y FC(i)32776 50470 y FJ(+)g FD(\026v)35495 50669 y FC(i)36165 50470 y FA(\000)g FD(")p FA(hM)40214 50669 y FC(k)40783 50470 y FD(w)36 b(;)221 b(e)42934 50669 y FC(i)43309 50470 y FA(i)43826 50669 y FC(s)p 26466 51063 17851 54 v 33353 52104 a FJ(~)33209 52440 y(\012)34148 52639 y FC(j)34930 52440 y FA(\000)36403 52104 y FJ(~)36259 52440 y(\012)37198 52639 y FC(i)44671 51369 y FD(e)45274 51568 y FC(i)45649 51369 y FD(;)11596 56008 y(\026)p FJ(\()p FD(v)13514 56207 y FC(j)14295 56008 y FA(\000)296 b FJ(1\))f(+)g(\()p FD(\025)g FA(\000)21414 55672 y FJ(~)21270 56008 y(\012)22209 56207 y FC(j)22696 56008 y FJ(\))p FD(w)24132 56207 y FC(j)24914 56008 y FA(\000)g FD(")p FA(hM)28963 56207 y FC(k)29532 56008 y FD(w)36 b(;)221 b(e)31683 56207 y FC(i)32058 56008 y FA(i)32575 56207 y FC(s)33065 53736 y Fz(!)34338 56008 y FD(;)6299 59968 y FJ(where)434 b(w)-36 b(e)434 b(used)e(that)i FA(hM)19872 60167 y FC(k)20440 59968 y FD(w)36 b(;)270 b FJ(~)-699 b FD(e)22591 60167 y FC(j)23078 59968 y FA(i)23595 60167 y FC(s)p Fw(\000)p FB(1)25656 59968 y FJ(=)369 b FA(hM)29149 60167 y FC(k)29718 59968 y FD(w)36 b(;)221 b(e)31869 60167 y FC(j)32356 59968 y FA(i)32873 60167 y FC(s)33363 59968 y FJ(.)578 b(W)-108 b(e)434 b(write)7064 62261 y Fz(\015)7064 63058 y(\015)7802 63390 y Fv(I)295 b FA(\000)g FD(T)181 b(@)11572 63589 y FC(y)12124 63390 y FD(F)g FJ(\()p FD(v)48 b(;)221 b(\025)p FJ(;)g FD(")p FJ(\))17368 62261 y Fz(\015)17368 63058 y(\015)18105 62612 y FB(2)18105 63911 y Fw(L)p FB(\()p FC(Y)52 b(;Y)211 b FB(\))21828 63390 y FJ(=)3014 b(sup)23736 64550 y Fw(j)193 b FB(~)-663 b FC(w)24 b Fw(j)24957 64673 y Ft(s)p Fo(\000)p 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FD(;)221 b(')27056 28833 y FC(k)24 b(i)28241 28634 y FA(\000)295 b FD(e)30172 28833 y FC(i)30548 28634 y FA(i)31065 28833 y FC(s)17409 31400 y FJ(=)1107 b FD(\016)20104 31599 y FC(ij)21206 31400 y FJ(+)295 b FD(O)23765 29926 y Fz(\020)24691 30501 y FD(")p 24691 31095 609 54 v 24773 32311 a(i)25432 29926 y Fz(\021)26521 31400 y FJ(+)g FD(O)29079 29527 y Fz(\022)30192 30501 y FD(")p 30190 31095 613 54 v 30190 32311 a(j)30935 29527 y Fz(\023)32208 31400 y FJ(+)g FD(O)34767 29527 y Fz(\022)35877 30501 y FD(")36486 30019 y FB(2)p 35877 31095 1135 54 v 35916 32311 a FD(ij)37145 29527 y Fz(\023)38492 31400 y FJ(=)368 b FD(\016)40448 31599 y FC(ij)41551 31400 y FJ(+)295 b FD(O)36 b FJ(\()p FD(")p FJ(\))221 b FD(;)6299 34933 y FJ(namely)23621 36539 y FD(d)24297 36738 y FC(ij)25473 36539 y FJ(=)369 b FD(\016)27430 36738 y FC(ij)28532 36539 y FJ(+)295 b FD(O)36 b FJ(\()p FD(")p FJ(\))221 b FD(:)13347 b FJ(\(5.130\))6299 39173 y FE(Lemma)499 b(5.7.8.)555 b Fs(The)464 b(op)-66 b(er)g(ator)465 b FD(B)24394 39372 y FC(k)24 b(j)25859 39173 y Fs(de\014ne)-66 b(d)464 b(in)g FJ(\(5.128\))i Fs(is)f(invertible)d(and)23433 42017 y FA(k)p FD(B)25085 42216 y FC(k)24 b(j)26085 42017 y FA(k)26749 42216 y FC(s;s)28306 42017 y FA(\024)369 b FJ(const)221 b FD(:)6299 44860 y Fp(pr)-30 b(oof.)929 b FJ(Let)517 b(b)36 b(e)518 b FD(b)f FJ(a)h(v)-36 b(ector)519 b(of)f FD(`)24167 44378 y FC(a;s)25771 44860 y FA(\002)352 b FD(`)27702 44378 y FC(a;s)28954 44860 y FJ(.)830 b(By)519 b(Remark)f(5.7.7)h(w)-36 b(e)518 b(can)g(write)g FD(b)512 b FJ(=)6299 45469 y Fz(P)7701 46853 y FC(i)8250 46115 y FJ(^)8298 46465 y FD(b)8851 46664 y FC(i)9227 46465 y FD(')10079 46664 y FC(k)24 b(i)11542 46465 y FJ(and)572 b(b)-36 b(y)573 b(the)g(de\014nition)f(of)i(\005)27052 46664 y FC(k)24 b(j)28053 46465 y FJ(,)608 b(w)-36 b(e)574 b(ha)-36 b(v)g(e)573 b(\005)35190 46664 y FC(k)24 b(j)36190 46465 y FD(b)607 b FJ(:=)g FD(b)390 b FA(\000)g(h)p FD(b;)221 b(')44200 46664 y FC(k)24 b(j)45201 46465 y FA(i)45718 46664 y FC(s)46208 46465 y FD(')47060 46664 y FC(k)g(j)48060 46465 y FJ(.)998 b(It)6299 48070 y(follo)-36 b(ws)436 b(that)15061 51346 y(\005)16036 51545 y FC(k)24 b(j)17036 49473 y Fz(\022)18235 50084 y(X)19035 52883 y FC(i)20327 50995 y FJ(^)20376 51346 y FD(b)20929 51545 y FC(i)21304 51346 y FD(')22156 51545 y FC(k)g(i)23046 49473 y Fz(\023)24392 51346 y FJ(=)25773 50084 y Fz(X)26572 52883 y FC(i)27864 50995 y FJ(^)27913 51346 y FD(b)28466 51545 y FC(i)28842 51346 y FD(')29694 51545 y FC(k)g(i)30878 51346 y FA(\000)32207 49473 y Fz(\022)33406 50084 y(X)33623 52913 y FC(i)p Fw(6)p FB(=)p FC(j)35497 50995 y FJ(^)35546 51346 y FD(b)36099 51545 y FC(i)36475 51346 y FD(d)37151 51545 y FC(i;j)38219 49473 y Fz(\023)39197 51346 y FD(')40049 51545 y FC(k)g(j)41271 51346 y FD(:)4787 b FJ(\(5.131\))6299 55489 y(Let)433 b(b)36 b(e)434 b FD(y)48 b(;)221 b(b)369 b FA(2)g FD(E)14804 55688 y FC(k)24 b(j)15804 55489 y FJ(,)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)433 b(to)h(in)-36 b(v)g(ert)22694 58332 y(\005)23669 58531 y FC(k)24 b(j)24669 58332 y FJ(\()p FD(S)25975 58531 y FC(k)26839 58332 y FA(\000)295 b FD(\025)28926 58531 y FC(k)24 b(j)29926 58332 y FJ(\))p FD(y)417 b FJ(=)369 b FD(b)221 b(:)12420 b FJ(\(5.132\))6299 61176 y(Since)19120 63213 y FD(y)417 b FJ(=)21551 61951 y Fz(X)22351 64750 y FC(i)23780 63213 y FJ(^)-738 b FD(y)24326 63412 y FC(i)24701 63213 y FD(')25553 63412 y FC(k)24 b(i)26738 63213 y FA(\000)28066 61340 y Fz(\022)29265 61951 y(X)29483 64780 y FC(i)p Fw(6)p FB(=)p FC(j)31494 63213 y FJ(^)-738 b FD(y)32040 63412 y FC(i)32415 63213 y FD(d)33091 63412 y FC(i;j)34160 61340 y Fz(\023)35138 63213 y FD(')35990 63412 y FC(k)24 b(j)37212 63213 y FD(;)6299 66799 y FJ(w)-36 b(e)434 b(obtain)10029 70074 y(\()p FD(S)11335 70273 y FC(k)12199 70074 y FA(\000)295 b FD(\025)14286 70273 y FC(k)24 b(j)15286 70074 y FJ(\))p FD(y)1155 b FJ(=)19699 68812 y Fz(X)20498 71611 y FC(i)21927 70074 y FJ(^)-738 b FD(y)22473 70273 y FC(i)22849 70074 y FJ(\()p FD(S)24155 70273 y FC(k)25019 70074 y FA(\000)295 b FD(\025)27106 70273 y FC(k)24 b(j)28107 70074 y FJ(\))p FD(')29465 70273 y FC(k)g(i)30649 70074 y FA(\000)31977 68201 y Fz(\022)33176 68812 y(X)33394 71641 y FC(i)p Fw(6)p FB(=)p FC(j)35404 70074 y FJ(^)-738 b FD(y)35950 70273 y FC(i)36326 70074 y FD(d)37002 70273 y FC(i;j)38071 68201 y Fz(\023)39049 70074 y FJ(\()p FD(S)40355 70273 y FC(k)41218 70074 y FA(\000)296 b FD(\025)43306 70273 y FC(k)24 b(j)44306 70074 y FJ(\))p FD(')45664 70273 y FC(k)g(j)17581 73783 y FJ(=)19699 72522 y Fz(X)19917 75350 y FC(i)p Fw(6)p FB(=)p FC(j)21927 73783 y FJ(^)-738 b FD(y)22473 73982 y FC(i)22849 73783 y FJ(\()p FD(\025)24114 73982 y FC(k)24 b(i)25298 73783 y FA(\000)296 b FD(\025)27386 73982 y FC(k)24 b(j)28386 73783 y FJ(\))p FD(')29744 73982 y FC(k)g(i)30854 73783 y FD(:)27371 78312 y FJ(183)p eop %%Page: 184 184 184 183 bop 0 4488 a FJ(Since)433 b FD(b)369 b FA(2)g FD(E)6536 4687 y FC(k)24 b(j)7536 4488 y FJ(,)434 b(it)f(is)h(of)h(the) e(form)h(\(5.131\).)579 b(Hence,)434 b(w)-36 b(e)434 b(get,)g(from)g(\(5.132\),)253 8291 y FD(B)1241 8490 y FC(k)24 b(j)2241 8291 y FD(y)1155 b FJ(=)1107 b(\005)7124 8490 y FC(k)24 b(j)8124 8291 y FJ(\()p FD(S)9430 8490 y FC(k)10294 8291 y FA(\000)295 b FD(\025)12381 8490 y FC(k)24 b(j)13381 8291 y FJ(\))p FD(y)417 b FJ(=)16319 7029 y Fz(X)17118 9828 y FC(i)18547 8291 y FJ(^)-738 b FD(y)19093 8490 y FC(i)19469 8291 y FJ(\()p FD(\025)20734 8490 y FC(k)24 b(i)21918 8291 y FA(\000)295 b FD(\025)24005 8490 y FC(k)24 b(j)25005 8291 y FJ(\))p FD(')26363 8490 y FC(k)g(i)27547 8291 y FA(\000)28876 6418 y Fz(\022)30075 7029 y(X)30292 9858 y FC(i)p Fw(6)p FB(=)p FC(j)32303 8291 y FJ(^)-738 b FD(y)32849 8490 y FC(i)33225 8291 y FJ(\()p FD(\025)34490 8490 y FC(k)24 b(i)35674 8291 y FA(\000)295 b FD(\025)37761 8490 y FC(k)24 b(j)38761 8291 y FJ(\))p FD(d)39943 8490 y FC(i;j)41012 6418 y Fz(\023)41989 8291 y FD(')42841 8490 y FC(k)g(j)4030 12532 y FJ(=)6149 11270 y Fz(X)6948 14069 y FC(i)8240 12181 y FJ(^)8289 12532 y FD(b)8842 12731 y FC(i)9217 12532 y FD(')10069 12731 y FC(k)g(i)11254 12532 y FA(\000)12582 10659 y Fz(\022)13781 11270 y(X)13999 14099 y FC(i)p Fw(6)p FB(=)p FC(j)15873 12181 y FJ(^)15922 12532 y FD(b)16475 12731 y FC(i)16850 12532 y FD(d)17526 12731 y FC(i;j)18595 10659 y Fz(\023)19573 12532 y FD(')20425 12731 y FC(k)g(j)21646 12532 y FD(:)0 16831 y FJ(Th)-36 b(us)4841 20508 y(^)-738 b FD(y)5387 20707 y FC(i)6132 20508 y FJ(=)9648 19259 y(^)9696 19609 y FD(b)10249 19808 y FC(i)p 7645 20203 5031 54 v 7645 21419 a FD(\025)8404 21618 y FC(k)24 b(i)9588 21419 y FA(\000)296 b FD(\025)11676 21618 y FC(k)24 b(j)13030 20508 y FD(;)2823 b FA(8)221 b FD(i)369 b FA(6)p FJ(=)g FD(j)2030 b FJ(and)1822 b(^)-738 b FD(y)26399 20707 y FC(j)27255 20508 y FJ(=)369 b FA(\000)29890 19246 y Fz(X)30108 22075 y FC(i)p Fw(6)p FB(=)p FC(j)32118 20508 y FJ(^)-738 b FD(y)32664 20707 y FC(i)33040 20508 y FD(d)33716 20707 y FC(i;j)35006 20508 y FD(:)4753 b FJ(\(5.133\))0 25158 y(F)-108 b(rom)465 b(\(5.133\),)474 b(it)465 b(follo)-36 b(ws)467 b(that)e FA(j)88 b FJ(^)-738 b FD(y)17741 25357 y FC(i)18117 25158 y FA(j)422 b(\024)h FJ(const)221 b FA(j)23875 24808 y FJ(^)23924 25158 y FD(b)24477 25357 y FC(i)24853 25158 y FA(j)465 b FJ(for)h(all)g FD(i)422 b FA(6)p FJ(=)g FD(j)75 b FJ(,)473 b(b)36 b(eing)465 b FA(j)p FD(\025)37955 25357 y FC(k)24 b(i)39161 25158 y FA(\000)317 b FD(\025)41270 25357 y FC(k)24 b(j)42270 25158 y FA(j)422 b(\025)0 26763 y FJ(const)f FD(>)f FJ(0,)472 b(as)465 b(it)f(follo)-36 b(ws)466 b(b)-36 b(y)464 b(\(5.123\).)671 b(Moreo)-36 b(v)g(er,)472 b(using)464 b(\(5.130\))h(and)f(the)f(Cauc) -36 b(h)g(y{)0 28368 y(Sc)g(h)g(w)g(arz)433 b(inequalit)-36 b(y)-108 b(,)435 b(w)-36 b(e)433 b(get)7072 32303 y FA(j)88 b FJ(^)-738 b FD(y)8075 32502 y FC(j)8562 32303 y FA(j)1107 b(\024)12178 31041 y Fz(X)12395 33870 y FC(i)p Fw(6)p FB(=)p FC(j)16502 31405 y FA(j)16822 31054 y FJ(^)16871 31405 y FD(b)17424 31604 y FC(i)17799 31405 y FA(j)p 14451 31998 5769 54 v 14451 33214 a(j)p FD(\025)15579 33413 y FC(k)24 b(i)16763 33214 y FA(\000)295 b FD(\025)18850 33413 y FC(k)24 b(j)19850 33214 y FA(j)20574 32303 y(j)p FD(d)21619 32502 y FC(i;j)22687 32303 y FA(j)369 b(\024)g FJ(const)221 b FD(")28848 31041 y Fz(X)29066 33870 y FC(i)p Fw(6)p FB(=)p FC(j)33172 31405 y FA(j)33492 31054 y FJ(^)33541 31405 y FD(b)34094 31604 y FC(i)34470 31405 y FA(j)p 31121 31998 V 31121 33214 a(j)p FD(\025)32249 33413 y FC(k)24 b(i)33434 33214 y FA(\000)295 b FD(\025)35521 33413 y FC(k)24 b(j)36521 33214 y FA(j)10038 36858 y(\024)1107 b FJ(const)221 b FD(")15978 34985 y Fz(\022)17177 35596 y(X)17394 38425 y FC(i)p Fw(6)p FB(=)p FC(j)22272 35959 y FJ(1)p 19450 36552 6295 54 v 19450 37769 a FA(j)p FD(\025)20578 37968 y FC(k)24 b(i)21762 37769 y FA(\000)295 b FD(\025)23849 37968 y FC(k)24 b(j)24849 37769 y FA(j)25218 37385 y FB(2)25877 34985 y Fz(\023)26855 35282 y FB(1)p FC(=)p FB(2)28322 35383 y Fz(\020)29336 35596 y(X)29554 38425 y FC(i)p Fw(6)p FB(=)p FC(j)31477 36858 y FA(j)31797 36507 y FJ(^)31846 36858 y FD(b)32399 37057 y FC(i)32774 36858 y FA(j)33143 36309 y FB(2)33669 35383 y Fz(\021)34462 35681 y FB(1)p FC(=)p FB(2)10038 41014 y FA(\024)1107 b FJ(const)221 b FD(")15978 39539 y Fz(\020)16992 39752 y(X)17210 42581 y FC(i)p Fw(6)p FB(=)p FC(j)19133 41014 y FA(j)19453 40663 y FJ(^)19502 41014 y FD(b)20055 41213 y FC(i)20430 41014 y FA(j)20799 40465 y FB(2)21325 39539 y Fz(\021)22118 39837 y FB(1)p FC(=)p FB(2)23807 41014 y FD(:)0 45511 y FJ(Therefore,)434 b(b)-36 b(y)434 b(Prop)36 b(osition)434 b(5.7.16,)i(de\014ning)c FD(B)25142 44962 y Fw(\000)p FB(1)25075 45881 y FC(k)24 b(j)26400 45511 y FD(b)369 b FJ(:=)g FD(y)417 b FJ(=)31495 44515 y Fz(P)32898 45899 y FC(i)33583 45511 y FJ(^)-738 b FD(y)34129 45710 y FC(i)34505 45511 y FD(')35357 45710 y FC(k)24 b(i)36246 45511 y FJ(,)434 b(one)f(obtains)383 49485 y FA(j)p FD(B)1807 48935 y Fw(\000)p FB(1)1740 49855 y FC(k)24 b(j)3065 49485 y FD(b)p FA(j)3987 49684 y FC(s)4846 49485 y FJ(=)369 b FA(j)p FD(y)48 b FA(j)7647 49684 y FC(s)8506 49485 y FJ(=)9887 47559 y Fz(\014)9887 48356 y(\014)9887 49153 y(\014)9887 49950 y(\014)10551 48223 y(X)11350 51022 y FC(i)12779 49485 y FJ(^)-738 b FD(y)13325 49684 y FC(i)13701 49485 y FD(')14553 49684 y FC(k)24 b(i)15442 47559 y Fz(\014)15442 48356 y(\014)15442 49153 y(\014)15442 49950 y(\014)16254 49485 y FA(\024)369 b FJ(2)18306 48010 y Fz(\020)19321 48223 y(X)20120 51022 y FC(i)21549 49485 y FJ(^)-738 b FD(y)22143 48936 y FB(2)22095 49813 y FC(i)22669 48010 y Fz(\021)23462 48308 y FB(1)p FC(=)p FB(2)25298 49485 y FA(\024)369 b FJ(const)29891 48010 y Fz(\020)30906 48223 y(X)31705 51022 y FC(i)32997 49134 y FJ(^)33046 49485 y FD(b)33599 48936 y FB(2)33599 49813 y FC(i)34125 48010 y Fz(\021)34918 48308 y FB(1)p FC(=)p FB(2)36754 49485 y FA(\024)g FJ(const)221 b FA(j)p FD(b)p FA(j)42638 49684 y FC(s)43350 49485 y FD(;)0 53582 y FJ(from)434 b(whic)-36 b(h)433 b(it)h(follo)-36 b(ws)436 b(that)13214 56582 y FA(k)p FD(B)14933 56032 y Fw(\000)p FB(1)14866 56952 y FC(k)24 b(j)16191 56582 y FA(k)16855 56781 y FC(s;s)p FB(=)19301 56582 y FJ(sup)18995 57726 y Fw(j)p FC(b)p Fw(j)19920 57837 y Ft(s)20364 57726 y FB(=1)21787 56582 y FA(j)p FD(B)23144 56781 y FC(k)g(j)24145 56582 y FD(b)p FA(j)25067 56781 y FC(s)25926 56582 y FA(\024)369 b FJ(const)222 b FD(:)43061 60482 y Fn(\003)0 63304 y FE(Lemma)713 b(5.7.9.)649 b Fs(L)-66 b(et)635 b(b)-66 b(e)635 b FA(M)15418 63503 y FC(k)16623 63304 y Fs(the)g(op)-66 b(er)g(ator)635 b(de\014ne)-66 b(d)635 b(in)g FJ(\(5.66\))i Fs(and)f(let)f(b)-66 b(e)635 b FD(e)41577 63503 y FC(j)42700 63304 y Fs(an)0 64909 y(orthonormal)465 b(b)-66 b(asis)464 b(of)h FD(`)12634 64427 y FC(a;s)14181 64909 y FA(\002)295 b FD(`)16055 64427 y FC(a;s)17306 64909 y Fs(,)465 b(then)13521 68388 y FA(jM)15485 68587 y FC(k)16054 68388 y FD(e)16657 68587 y FC(j)17144 68388 y FA(j)17513 68587 y FC(s)18372 68388 y FA(\024)19907 67489 y FJ(const)p 19907 68082 2970 54 v 21085 69299 a FD(j)23231 68388 y(;)2878 b(j)444 b FA(2)369 b(I)29528 67839 y FC(c)30212 68388 y FD(:)0 71992 y Fp(pr)-30 b(oof.)658 b FJ(Since)10040 74991 y FA(k)p FD(M)139 b FA(M)13696 75190 y FC(k)14266 74991 y FA(k)14930 75190 y FC(s;s)16486 74991 y FJ(=)368 b FA(kM)20125 75190 y FC(k)20695 74991 y FD(M)139 b FA(k)22756 75190 y FC(s;s)24312 74991 y FA(\024)369 b(kM)27973 75190 y FC(k)28542 74991 y FA(k)29206 75190 y FC(s;s)30762 74991 y FD(<)g FA(1)221 b FD(;)6067 b FJ(\(5.134\))21072 78312 y(184)p eop %%Page: 185 185 185 184 bop 6299 4488 a FJ(w)-36 b(e)434 b(get)13806 7254 y FA(jM)15770 7453 y FC(k)16339 7254 y FD(e)16942 7453 y FC(j)17429 7254 y FA(j)17798 7453 y FC(s)19395 7254 y FJ(=)1107 b FA(jM)23478 7453 y FC(k)24047 7254 y FD(M)139 b(M)26841 6705 y Fw(\000)p FB(1)28099 7254 y FD(e)28702 7453 y FC(j)29188 7254 y FA(j)29557 7453 y FC(s)30417 7254 y FA(\024)369 b(kM)34078 7453 y FC(k)34647 7254 y FD(M)139 b FA(k)36708 7453 y FC(s;s)37895 7254 y FA(j)p FD(M)39661 6705 y Fw(\000)p FB(1)40919 7254 y FD(e)41522 7453 y FC(j)42009 7254 y FA(j)42378 7453 y FC(s)19395 9963 y FJ(=)1107 b FA(kM)23773 10162 y FC(k)24342 9963 y FA(k)25006 10162 y FC(s;s)26193 9963 y FA(j)p FD(e)27165 10162 y FC(j)27873 9963 y FD(j)28486 9414 y Fw(\000)p FB(1)29744 9963 y FA(j)30113 10162 y FC(s)30972 9963 y FJ(=)32485 9064 y FA(kM)34744 9263 y FC(k)35314 9064 y FA(k)35978 9263 y FC(s;s)p 32485 9657 4680 54 v 34519 10874 a FD(j)37667 9963 y FA(\024)39202 9064 y FJ(const)p 39202 9657 2970 54 v 40380 10874 a FD(j)42526 9963 y(;)6299 13399 y FJ(with)434 b(constan)-36 b(t)433 b FK(indep)36 b(enden)-36 b(t)462 b FJ(of)434 b FD(k)478 b FJ(\(recall)435 b(\(5.59\),\(5.66\),\(5.67\)\).)49361 15004 y Fn(\003)6299 17572 y FE(Lemma)440 b(5.7.10.)515 b Fs(L)-66 b(et)416 b(b)-66 b(e)417 b FD(")g Fs(suitably)h(smal)66 b(l)418 b(such)g(that)g(The)-66 b(or)g(em)416 b(5.7.6)h(holds.)582 b(L)-66 b(et)417 b(b)-66 b(e)6299 19177 y FA(f)p FD(')7815 19376 y FC(k)24 b(j)8816 19177 y FA(g)9480 19376 y FC(j)51 b Fw(2I)11117 19124 y Ft(c)12065 19177 y Fs(the)464 b(eigenve)-66 b(ctors)463 b(of)h(the)h(op)-66 b(er)g(ator)464 b FD(S)31113 19376 y FC(k)31682 19177 y Fs(.)597 b(Then)20521 22535 y FA(j)p FD(')21742 21986 y Fw(0)21742 22863 y FC(k)24 b(j)22742 22535 y FA(j)23111 22734 y FC(s)23971 22535 y FA(\024)25505 21636 y FJ(const)p 25505 22229 V 26684 23446 a FD(j)28830 22535 y(;)2878 b(j)444 b FA(2)368 b(I)35126 21986 y FC(c)35811 22535 y FD(:)6299 25971 y Fp(pr)-30 b(oof.)659 b FJ(By)434 b(Lemma)f(5.7.8)i(and)e(from)h (\(5.127\),)h(it)f(follo)-36 b(ws)436 b(that)21585 28736 y FD(')22437 28188 y Fw(0)22437 29065 y FC(k)24 b(j)23807 28736 y FJ(=)368 b FA(\000)p FD(B)27275 28187 y Fw(\000)p FB(1)27208 29106 y FC(k)24 b(j)28533 28736 y FJ(\005)29508 28935 y FC(k)g(j)30509 28736 y FA(M)32104 28935 y FC(k)32673 28736 y FD(')33525 28935 y FC(k)g(j)34746 28736 y FD(:)11312 b FJ(\(5.135\))6299 31501 y(Since,)556 b(b)-36 b(y)531 b(h)-36 b(yp)36 b(othesis,)556 b(Theorem)532 b(5.7.6)h(holds,)556 b FD(')32510 31700 y FC(k)24 b(j)34041 31501 y FJ(satisfy)533 b(\(5.124\).)873 b(Hence,)556 b(from)6299 33107 y(\(5.135\),)435 b(using)f(Lemma)f(5.7.8,)j(w)-36 b(e)433 b(get)9529 35872 y FA(j)p FD(')10750 35323 y Fw(0)10750 36200 y FC(k)24 b(j)11750 35872 y FA(j)12119 36071 y FC(s)13717 35872 y FA(\024)1107 b(k)p FD(B)17576 35322 y Fw(\000)p FB(1)17509 36242 y FC(k)24 b(j)18834 35872 y FA(k)19498 36071 y FC(s;s)20685 35872 y FA(jM)22649 36071 y FC(k)23218 35872 y FD(')24070 36071 y FC(k)g(j)25070 35872 y FA(j)25439 36071 y FC(s)26299 35872 y FA(\024)369 b FJ(const)30892 34796 y Fz(\000)31501 35872 y FA(jM)33465 36071 y FC(k)34034 35872 y FD(e)34637 36071 y FC(j)35124 35872 y FA(j)35493 36071 y FC(s)36279 35872 y FJ(+)294 b FA(jM)39549 36071 y FC(k)40118 35872 y FJ(\()p FD(')41476 36071 y FC(k)24 b(j)42772 35872 y FA(\000)295 b FD(e)44703 36071 y FC(j)45190 35872 y FJ(\))p FA(j)46065 36071 y FC(s)46555 34796 y Fz(\001)13717 38783 y FA(\024)1107 b FJ(const)19269 36910 y Fz(\022)20380 37884 y FJ(1)p 20380 38477 651 54 v 20399 39694 a FD(j)21458 38783 y FJ(+)22900 37884 y FD(")p 22898 38477 613 54 v 22898 39694 a(j)23644 36910 y Fz(\023)24991 38783 y FA(\024)26526 37884 y FJ(const)p 26526 38477 2970 54 v 27704 39694 a FD(j)29850 38783 y(:)6299 42238 y FJ(where)434 b(in)f(the)g(last)h(inequalit)-36 b(y)435 b(w)-36 b(e)434 b(used)f(Lemma)g(5.7.9.)49361 43843 y Fn(\003)6299 45448 y FJ(Recall)301 b(\(5.126\),)329 b(w)-36 b(e)300 b(are)g(going)h(to)f(study)f(the)h(asymptotic)g(b)36 b(eha)-36 b(vior)301 b(of)g(the)e(eigen)-36 b(v)-72 b(alues)6299 47053 y FD(\025)7058 47252 y FC(k)24 b(j)8058 47053 y FJ(.)6299 49621 y FE(Prop)42 b(osition)570 b(5.7.11.)585 b Fs(L)-66 b(et)520 b(b)-66 b(e)520 b FD(")g Fs(suitably)g(smal)66 b(l)522 b(such)f(that)g(The)-66 b(or)g(em)519 b(5.7.6)h(holds.)6299 51226 y(L)-66 b(et)423 b(b)-66 b(e)423 b FA(f)p FD(\025)11477 51425 y FC(k)24 b(j)12477 51226 y FA(g)13141 51425 y FC(j)51 b Fw(2I)14778 51173 y Ft(c)15684 51226 y Fs(the)424 b(eigenvalues)e(asso)-66 b(ciate)g(d)423 b(to)h(the)f(eigenve)-66 b(ctors)421 b FA(f)p FD(')43005 51425 y FC(k)24 b(j)44006 51226 y FA(g)44670 51425 y FC(j)51 b Fw(2I)46307 51173 y Ft(c)47214 51226 y Fs(of)423 b(the)6299 52831 y(op)-66 b(er)g(ator)464 b FD(S)12213 53030 y FC(k)12782 52831 y Fs(.)597 b(Then)16133 54701 y Fz(\014)16133 55498 y(\014)16575 55831 y FD(\025)17334 55282 y Fw(0)17334 56159 y FC(k)24 b(j)18630 55831 y FA(\000)295 b(hM)22070 56030 y FC(k)22639 55831 y FD(e)23242 56030 y FC(j)23729 55831 y FD(;)221 b(e)24914 56030 y FC(j)25401 55831 y FA(i)25918 56030 y FC(s)26408 54701 y Fz(\014)26408 55498 y(\014)27220 55831 y FA(\024)369 b FJ(const)31990 54932 y FD(")p 31725 55525 1139 54 v 31725 56742 a(j)32338 56358 y FB(2)33217 55831 y FD(;)2879 b(j)444 b FA(2)368 b(I)39514 55282 y FC(c)40199 55831 y FD(;)5859 b FJ(\(5.136\))6299 59193 y Fs(with)465 b(c)-66 b(onstant)464 b(indep)-66 b(endent)462 b(of)j FD(k)45 b Fs(.)6299 61761 y Fp(pr)-30 b(oof.)1006 b FJ(Recall)543 b(the)e(expression)h(for)g FD(\025)27242 61279 y Fw(0)27242 62106 y FC(k)24 b(j)28784 61761 y FJ(in)542 b(\(5.126\).)903 b(Using)542 b(the)f(linearit)-36 b(y)543 b(of)f(the)6299 63366 y(scalar)435 b(pro)36 b(duct,)432 b(one)i(obtains)7126 66131 y FA(hM)9238 66330 y FC(k)9807 66131 y FD(')10659 66330 y FC(k)24 b(j)11659 66131 y FD(;)221 b(')13093 66330 y FC(k)24 b(j)14094 66131 y FA(i)14611 66330 y FC(s)16208 66131 y FJ(=)1107 b FA(hM)20439 66330 y FC(k)21008 66131 y FD(e)21611 66330 y FC(j)22097 66131 y FD(;)221 b(e)23282 66330 y FC(j)23770 66131 y FA(i)24287 66330 y FC(s)25072 66131 y FJ(+)295 b FA(hM)28491 66330 y FC(k)29060 66131 y FD(e)29663 66330 y FC(j)30149 66131 y FD(;)221 b(')31583 66330 y FC(k)24 b(j)32880 66131 y FA(\000)295 b FD(e)34811 66330 y FC(j)35298 66131 y FA(i)35815 66330 y FC(s)36600 66131 y FJ(+)37907 65055 y Fz(\012)38534 66131 y FA(M)40129 66330 y FC(k)40698 66131 y FJ(\()p FD(')42056 66330 y FC(k)24 b(j)43351 66131 y FA(\000)296 b FD(e)45283 66330 y FC(j)45770 66131 y FA(i)46287 66330 y FC(s)46777 66131 y FD(;)221 b(e)47962 66330 y FC(j)48449 65055 y Fz(\013)49076 66651 y FC(s)18327 68223 y FJ(+)19339 67147 y Fz(\012)19966 68223 y FA(M)21561 68422 y FC(k)22130 68223 y FJ(\()p FD(')23488 68422 y FC(k)24 b(j)24783 68223 y FA(\000)296 b FD(e)26715 68422 y FC(j)27202 68223 y FJ(\))p FD(;)221 b(')29142 68422 y FC(k)24 b(j)30438 68223 y FA(\000)295 b FD(e)32369 68422 y FC(j)32856 67147 y Fz(\013)33483 68744 y FC(s)33973 68223 y FD(:)12085 b FJ(\(5.137\))6299 70988 y(By)434 b(\(5.124\))h(and)e(Lemma)h(5.7.9,)h(w)-36 b(e)434 b(ha)-36 b(v)g(e)11736 72958 y Fz(\014)11736 73755 y(\014)11736 74552 y(\014)12179 73410 y(\012)12806 74486 y FA(M)14401 74685 y FC(k)14970 74486 y FD(e)15573 74685 y FC(j)16060 74486 y FD(;)221 b(')17494 74685 y FC(k)24 b(j)18790 74486 y FA(\000)295 b FD(e)20721 74685 y FC(j)21208 73410 y Fz(\013)21835 75006 y FC(s)22326 72958 y Fz(\014)22326 73755 y(\014)22326 74552 y(\014)23138 74486 y FA(\024)369 b(jM)26504 74685 y FC(k)27073 74486 y FD(e)27676 74685 y FC(j)28163 74486 y FA(j)28532 74685 y FC(s)29155 73587 y FJ(const)221 b FA(j)p FD(")p FA(j)p 29155 74180 4538 54 v 31117 75397 a FD(j)34194 74486 y FA(\024)35729 73587 y FJ(const)h FA(j)p FD(")p FA(j)p 35729 74180 V 37429 75397 a FD(j)38042 75013 y FB(2)40622 74486 y FD(:)5436 b FJ(\(5.138\))27371 78312 y(185)p eop %%Page: 186 186 186 185 bop 0 4488 a FJ(Recalling)435 b(that)e FD(M)508 b FJ(=)369 b(diag)18 b(\()p FD(j)75 b FJ(\),)434 b(b)-36 b(y)433 b(\(5.124\),)i(\(5.134\))g(and)e(Lemma)h(5.7.9,)h(w)-36 b(e)434 b(get)5980 6123 y Fz(\014)5980 6920 y(\014)5980 7717 y(\014)6423 6575 y(\012)7050 7651 y FA(M)8645 7850 y FC(k)9214 7651 y FJ(\()p FD(')10572 7850 y FC(k)24 b(j)11867 7651 y FA(\000)296 b FD(e)13799 7850 y FC(j)14285 7651 y FJ(\))p FD(;)221 b(e)15976 7850 y FC(j)16464 6575 y Fz(\013)17091 8171 y FC(s)17581 6123 y Fz(\014)17581 6920 y(\014)17581 7717 y(\014)19142 7651 y FJ(=)21271 6123 y Fz(\014)21271 6920 y(\014)21271 7717 y(\014)21714 6575 y(\012)22341 7651 y FD(M)23738 7102 y Fw(\000)p FB(1)24996 7651 y FD(M)139 b FA(M)27988 7850 y FC(k)28557 7651 y FJ(\()p FD(')29915 7850 y FC(k)24 b(j)31211 7651 y FA(\000)295 b FD(e)33142 7850 y FC(j)33629 7651 y FJ(\))p FD(;)221 b(e)35320 7850 y FC(j)35807 6575 y Fz(\013)36434 8171 y FC(s)36925 6123 y Fz(\014)36925 6920 y(\014)36925 7717 y(\014)19142 10485 y FJ(=)21271 8957 y Fz(\014)21271 9754 y(\014)21271 10551 y(\014)21714 9409 y(\012)22341 10485 y FD(M)139 b FA(M)25333 10684 y FC(k)25902 10485 y FJ(\()p FD(')27260 10684 y FC(k)24 b(j)28556 10485 y FA(\000)295 b FD(e)30487 10684 y FC(j)30974 10485 y FJ(\))p FD(;)221 b(M)33459 9936 y Fw(\000)p FB(1)34717 10485 y FD(e)35320 10684 y FC(j)35807 9409 y Fz(\013)36434 11005 y FC(s)36925 8957 y Fz(\014)36925 9754 y(\014)36925 10551 y(\014)19131 12910 y FA(\024)1107 b(k)p FD(M)139 b FA(M)24927 13109 y FC(k)25497 12910 y FA(k)26161 13109 y FC(s;s)27348 12910 y FA(j)p FD(')28569 13109 y FC(k)24 b(j)29864 12910 y FA(\000)296 b FD(e)31796 13109 y FC(j)32282 12910 y FA(j)32651 13109 y FC(s)33142 12910 y FA(j)p FD(M)34908 12361 y Fw(\000)p FB(1)36166 12910 y FD(e)36769 13109 y FC(j)37255 12910 y FA(j)37624 13109 y FC(s)19131 15619 y FA(\024)21404 14720 y FJ(const)221 b FA(j)p FD(")p FA(j)p 21404 15314 4538 54 v 23104 16530 a FD(j)23717 16146 y FB(2)26075 15619 y FD(:)13684 b FJ(\(5.139\))0 19014 y(In)433 b(the)g(same)h(w)-36 b(a)g(y)664 20998 y Fz(\014)664 21795 y(\014)664 22592 y(\014)1107 21450 y(\012)1734 22526 y FA(M)3329 22725 y FC(k)3898 22526 y FJ(\()p FD(')5256 22725 y FC(k)24 b(j)6338 22526 y FA(\000)81 b FD(e)8055 22725 y FC(j)8542 22526 y FJ(\))p FD(;)221 b(')10482 22725 y FC(k)24 b(j)11564 22526 y FA(\000)81 b FD(e)13281 22725 y FC(j)13768 21450 y Fz(\013)14395 23046 y FC(s)14886 20998 y Fz(\014)14886 21795 y(\014)14886 22592 y(\014)15697 22526 y FA(\024)369 b(kM)19358 22725 y FC(k)19928 22526 y FA(k)20592 22725 y FC(s;s)21779 22526 y FA(j)p FD(')23000 22725 y FC(k)24 b(j)24081 22526 y FA(\000)81 b FD(e)25798 22725 y FC(j)26285 22526 y FA(j)26654 22725 y FC(s)27145 22526 y FA(j)p FD(')28366 22725 y FC(k)24 b(j)29447 22526 y FA(\000)81 b FD(e)31164 22725 y FC(j)31651 22526 y FA(j)32020 22725 y FC(s)32880 22526 y FA(\024)34415 21627 y FJ(const)221 b FD(")38215 21145 y FB(2)p 34415 22220 4326 54 v 36008 23437 a FD(j)36621 23053 y FB(2)39095 22526 y FD(:)664 b FJ(\(5.140\))0 25994 y(Hence)433 b(b)-36 b(y)434 b(\(5.137\),)h(\(5.138\),)g (\(5.139\),)g(w)-36 b(e)434 b(get)10753 29524 y FA(hM)12865 29723 y FC(k)13434 29524 y FD(')14286 29723 y FC(k)24 b(j)15286 29524 y FD(;)221 b(')16720 29723 y FC(k)24 b(j)17721 29524 y FA(i)18238 29723 y FC(s)19097 29524 y FJ(=)369 b FA(hM)22590 29723 y FC(k)23159 29524 y FD(e)23762 29723 y FC(j)24248 29524 y FD(;)221 b(e)25433 29723 y FC(j)25921 29524 y FA(i)26438 29723 y FC(s)27223 29524 y FJ(+)295 b FD(O)29560 28049 y Fz(\020)30486 28625 y FA(j)p FD(")p FA(j)p 30486 29218 1347 54 v 30590 30435 a FD(j)31203 30051 y FB(2)31965 28049 y Fz(\021)32980 29524 y FD(:)6779 b FJ(\(5.141\))0 32992 y(By)434 b(\(5.134\),)h(Lemma) f(5.7.9)h(and)e(Lemma)g(5.7.10,)j(w)-36 b(e)434 b(obtain)1094 35025 y Fz(\014)1094 35822 y(\014)1537 36154 y FA(hM)3649 36353 y FC(k)4218 36154 y FD(')5070 35606 y Fw(0)5070 36483 y FC(k)24 b(j)6070 36154 y FD(;)221 b(')7504 36353 y FC(k)24 b(j)8505 36154 y FA(i)9022 36353 y FC(s)9512 35025 y Fz(\014)9512 35822 y(\014)11073 36154 y FJ(=)13202 34627 y Fz(\014)13202 35424 y(\014)13202 36221 y(\014)13645 35078 y(\012)14272 36154 y FD(M)15669 35606 y Fw(\000)p FB(1)16927 36154 y FD(M)139 b FA(M)19919 36353 y FC(k)20488 36154 y FD(')21340 35606 y Fw(0)21340 36483 y FC(k)24 b(j)22341 36154 y FD(;)221 b(')23775 36353 y FC(k)24 b(j)24776 35078 y Fz(\013)25403 36674 y FC(s)25893 34627 y Fz(\014)25893 35424 y(\014)25893 36221 y(\014)11073 38988 y FJ(=)13202 37460 y Fz(\014)13202 38257 y(\014)13202 39054 y(\014)13645 37912 y(\012)14272 38988 y FD(M)139 b FA(M)17264 39187 y FC(k)17833 38988 y FD(')18685 38440 y Fw(0)18685 39316 y FC(k)24 b(j)19686 38988 y FD(;)221 b(M)21665 38440 y Fw(\000)p FB(1)22923 38988 y FD(')23775 39187 y FC(k)24 b(j)24776 37912 y Fz(\013)25403 39508 y FC(s)25893 37460 y Fz(\014)25893 38257 y(\014)25893 39054 y(\014)26705 38988 y FA(\024)369 b(k)p FD(M)139 b FA(M)31763 39187 y FC(k)32333 38988 y FA(k)32997 39187 y FC(s;s)34184 38988 y FA(j)p FD(')35405 38440 y Fw(0)35405 39316 y FC(k)24 b(j)36405 38988 y FA(j)36774 39187 y FC(s)37265 38988 y FA(j)p FD(M)39031 38440 y Fw(\000)p FB(1)40288 38988 y FD(')41140 39187 y FC(k)g(j)42141 38988 y FA(j)42510 39187 y FC(s)11062 42010 y FA(\024)13335 41111 y FJ(const)p 13335 41704 2970 54 v 14514 42921 a FD(j)16438 40535 y Fz(\020)17231 42010 y FA(j)p FD(M)139 b(e)19600 42209 y FC(j)20087 42010 y FA(j)20456 42209 y FC(s)21241 42010 y FJ(+)295 b FA(j)p FD(M)24314 41461 y Fw(\000)p FB(1)25572 42010 y FJ(\()p FD(')26930 42209 y FC(k)24 b(j)28225 42010 y FA(\000)296 b FD(e)30157 42209 y FC(j)30643 42010 y FJ(\))p FA(j)31518 42209 y FC(s)32009 40535 y Fz(\021)11062 45548 y FA(\024)13335 44650 y FJ(const)p 13335 45243 V 14514 46459 a FD(j)16659 43675 y Fz(\022)17770 44650 y FJ(1)p 17770 45243 651 54 v 17789 46459 a FD(j)18848 45548 y FJ(+)20288 44650 y(const)221 b FA(j)p FD(")p FA(j)p 20288 45243 4538 54 v 22250 46459 a FD(j)24958 43675 y Fz(\023)26305 45548 y FA(\024)27840 44650 y FJ(const)p 27840 45243 2970 54 v 28756 46459 a FD(j)29369 46076 y FB(2)31164 45548 y FD(:)8595 b FJ(\(5.142\))0 49035 y(Analogously)-108 b(,)398 b(b)-36 b(y)387 b(Lemma)g(5.7.9,)398 b(Lemma)387 b(5.7.10)i(and)e(b)-36 b(y)387 b(the)f(linearit)-36 b(y)388 b(of)g(the)f(scalar)0 50640 y(pro)36 b(duct,)433 b(one)g(has)929 52308 y Fz(\014)929 53105 y(\014)1372 53437 y FA(hM)3484 53636 y FC(k)4053 53437 y FD(')4905 53636 y FC(k)24 b(j)5905 53437 y FD(;)221 b(')7339 52889 y Fw(0)7339 53766 y FC(k)24 b(j)8340 53437 y FA(i)8857 53636 y FC(s)9347 52308 y Fz(\014)9347 53105 y(\014)10908 53437 y FJ(=)1117 b FA(jM)15001 53636 y FC(k)15570 53437 y FD(')16422 53636 y FC(k)24 b(j)17423 53437 y FA(j)17792 53636 y FC(s)18282 53437 y FA(j)p FD(')19503 52889 y Fw(0)19503 53766 y FC(k)g(j)20504 53437 y FA(j)20873 53636 y FC(s)10897 56096 y FA(\024)13170 55198 y FJ(const)p 13170 55791 V 14349 57008 a FD(j)16273 56096 y FA(jM)18237 56295 y FC(k)18806 56096 y FD(')19658 56295 y FC(k)g(j)20658 56096 y FA(j)21027 56295 y FC(s)21887 56096 y FA(\024)23422 55198 y FJ(const)p 23422 55791 V 24600 57008 a FD(j)26524 54622 y Fz(\020)27318 56096 y FA(jM)29282 56295 y FC(k)29851 56096 y FD(e)30454 56295 y FC(j)30941 56096 y FA(j)31310 56295 y FC(s)32095 56096 y FJ(+)295 b FA(jM)35366 56295 y FC(k)35935 56096 y FJ(\()p FD(')37293 56295 y FC(k)24 b(j)38588 56096 y FA(\000)296 b FD(e)40520 56295 y FC(j)41007 56096 y FJ(\))p FA(j)41882 56295 y FC(s)42372 54622 y Fz(\021)10897 59635 y FA(\024)13170 58736 y FJ(const)p 13170 59329 V 14349 60546 a FD(j)16494 57762 y Fz(\022)17605 58736 y FJ(1)p 17605 59329 651 54 v 17624 60546 a FD(j)18683 59635 y FJ(+)20123 58736 y(const)221 b FA(j)p FD(")p FA(j)p 20123 59329 4538 54 v 22085 60546 a FD(j)24794 57762 y Fz(\023)26140 59635 y FA(\024)27675 58736 y FJ(const)p 27675 59329 2970 54 v 28591 60546 a FD(j)29204 60162 y FB(2)30999 59635 y FD(:)8760 b FJ(\(5.143\))0 63196 y(Hence)551 b(from)h(\(5.126\),)582 b(b)-36 b(y)551 b(\(5.141\),)582 b(\(5.142\),)g(\(5.143\))553 b(w)-36 b(e)551 b(reac)-36 b(h)551 b(the)g(\014nal)g(estimate)0 64801 y(\(5.136\).)43061 66406 y Fn(\003)0 68011 y FJ(F)-108 b(rom)433 b(\(5.136\),)i(the)e(asymptotic)h(estimate)g(for)g(the)f (eigen)-36 b(v)-72 b(alues)435 b(follo)-36 b(ws:)8032 70468 y Fz(\014)8032 71265 y(\014)8475 71597 y FD(\025)9234 71796 y FC(k)24 b(j)10529 71597 y FA(\000)12002 71261 y FJ(~)11857 71597 y(\012)12796 71796 y FC(j)13579 71597 y FA(\000)295 b FD(")p FA(hM)17628 71796 y FC(k)18197 71597 y FD(e)18800 71796 y FC(j)19286 71597 y FD(;)221 b(e)20471 71796 y FC(j)20959 71597 y FA(i)21476 71796 y FC(s)21966 70468 y Fz(\014)21966 71265 y(\014)22778 71597 y FA(\024)369 b FJ(const)27284 70698 y FD(")27893 70216 y FB(2)p 27282 71291 1139 54 v 27282 72508 a FD(j)27895 72124 y FB(2)28775 71597 y FD(;)2823 b(j)444 b FA(2)369 b(I)35017 71048 y FC(c)35701 71597 y FD(;)4058 b FJ(\(5.144\))0 74991 y(with)434 b(constan)-36 b(t)433 b(indep)36 b(enden)-36 b(t)431 b(of)j FD(k)45 b FJ(.)21072 78312 y(186)p eop %%Page: 187 187 187 186 bop 6299 4488 a FJ(In)405 b(order)g(to)h(pro)-36 b(v)g(e)405 b(the)g(in)-36 b(v)g(ertibilit)g(y)406 b(of)g(the)f(linear) h(op)36 b(erator)405 b FD(D)39107 4006 y FB(\()p FC(n)p FB(\))40465 4488 y FJ(,)412 b(nothing)405 b(remains)6299 6094 y(but)383 b(to)g(pro)-36 b(v)g(e)383 b(that)g(its)h(eigen)-36 b(v)-72 b(alues)384 b(are)g(di\013eren)-36 b(t)382 b(from)i(zero.)562 b(W)-108 b(e)383 b(reac)-36 b(h)383 b(an)g(estimate)6299 7699 y(from)327 b(b)36 b(elo)-36 b(w)326 b(for)h(the)e(eigen)-36 b(v)-72 b(alues,)348 b(in)326 b(particular)g(it)g(is)g(stated)g(in)g (the)f(follo)-36 b(wing)328 b(lemma.)6299 9304 y(Let)433 b(us)h(de\014ne)23793 11235 y FD(\026)24576 11434 y FC(k)24 b(;j)26206 11235 y FJ(:=)28081 10337 y 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187 bop 9998 5400 a FJ(=)1117 b(\(1)295 b(+)g(const)221 b FD(")p FJ(\))19191 3527 y Fz(\022)21648 3739 y FC(n)20974 4138 y Fz(X)20390 6937 y FC(i)p FB(=)p FC(m)p FB(+1)23698 5400 y FD(y)24380 4851 y FB(2)24332 5728 y FC(i)24906 3527 y Fz(\023)25884 3824 y FB(1)p FC(=)p FB(2)27720 5400 y FA(\000)-222 b(!)370 b FJ(0)2601 b(as)877 b FD(m)369 b FA(!)g(1)221 b FD(:)43061 9324 y Fn(\003)0 11887 y FE(Prop)42 b(osition)529 b(5.7.14.)567 b Fs(The)488 b(op)-66 b(er)g(ators)488 b FD(V)22010 12086 y FC(k)23067 11887 y Fs(in)g FJ(\(5.146\))h Fs(ar)-66 b(e)488 b(line)-66 b(ar)487 b(and)h(c)-66 b(ontinuous.)0 13492 y(Mor)g(e)g(over,)463 b FD(V)6871 13691 y FC(k)7905 13492 y Fs(ar)-66 b(e)464 b(invertible)f(and)16975 16252 y FA(k)p FD(V)18687 15702 y Fw(\000)p FB(1)18397 16622 y FC(k)19944 16252 y FA(k)20608 16451 y FC(s;s)22165 16252 y FA(\024)369 b FJ(const)221 b FD(:)0 19011 y Fp(pr)-30 b(oof.)903 b FJ(First)509 b(w)-36 b(e)510 b(pro)-36 b(v)g(e)510 b(the)f(b)36 b(oundness)508 b(of)i FD(V)25097 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FJ(1)296 b(+)e FD(")16480 30670 y Fz(\022)17679 31281 y(X)18478 34080 y FC(i)19819 32543 y FD(y)20501 31995 y FB(2)20453 32871 y FC(i)21027 30670 y Fz(\023)22005 30967 y FB(1)p FC(=)p FB(2)23472 30670 y Fz(\022)24671 31281 y(X)25470 34080 y FC(i)27103 31644 y FJ(1)p 26944 32237 970 54 v 26944 33454 a FD(i)27388 33070 y FB(2)28046 30670 y Fz(\023)29024 30967 y FB(1)p FC(=)p FB(2)30491 30271 y Fz(!)31911 32543 y FA(\024)369 b FJ(1)296 b(+)f(const)p FD(")368 b FA(\024)h FJ(2)221 b FD(;)0 36696 y FJ(where)564 b(w)-36 b(e)564 b(used)g(\(5.124\).)971 b(Moreo)-36 b(v)g(er)564 b FD(V)20935 36895 y FC(k)22068 36696 y FJ(are)g(in)-36 b(v)g(ertible.)970 b(Indeed,)596 b(de\014ning)41002 36360 y(~)40803 36696 y FD(V)41561 36895 y FC(k)42722 36696 y FJ(:=)0 38301 y FD(V)758 38500 y FC(k)1622 38301 y FA(\000)296 b Fv(I)p FJ(,)433 b(namely)h FD(V)9536 38500 y FC(k)10474 38301 y FJ(=)369 b Fv(I)295 b FJ(+)14172 37965 y(~)13974 38301 y FD(V)14732 38500 y FC(k)15301 38301 y FJ(,)434 b(then)104 41352 y FA(k)966 41016 y FJ(~)768 41352 y FD(V)1526 41551 y FC(k)2095 41352 y FA(k)2759 41551 y FC(s;s)5064 41352 y FJ(=)1471 b(sup)7193 42496 y Fw(j)p FC(y)32 b Fw(j)8212 42607 y Ft(s)8657 42496 y FB(=1)10080 39824 y Fz(\014)10080 40621 y(\014)10080 41418 y(\014)10744 40090 y(X)11544 42889 y FC(i)12885 41352 y FD(y)13519 41551 y FC(i)13894 41352 y FJ(\()p FD(')15252 41551 y FC(k)24 b(i)16437 41352 y FA(\000)295 b FD(e)18368 41551 y FC(i)18744 41352 y FJ(\))19250 39824 y Fz(\014)19250 40621 y(\014)19250 41418 y(\014)19692 42271 y FC(s)20552 41352 y FA(\024)722 b FJ(sup)21954 42496 y Fw(j)p FC(y)32 b Fw(j)22973 42607 y Ft(s)23417 42496 y FB(=1)24841 40090 y Fz(X)25640 42889 y FC(i)26981 41352 y FA(j)p FD(y)27984 41551 y FC(i)28360 41352 y FA(jj)p FD(')29950 41551 y FC(k)24 b(i)31134 41352 y FA(\000)295 b FD(e)33065 41551 y FC(i)33441 41352 y FA(j)33810 41551 y FC(s)5053 45705 y FA(\024)1107 b FJ(const)222 b FD(")574 b FJ(sup)11215 46849 y Fw(j)p FC(y)32 b Fw(j)12234 46960 y Ft(s)12678 46849 y FB(=1)14102 44443 y Fz(X)14901 47242 y FC(i)16374 44806 y FA(j)p FD(y)17377 45005 y FC(i)17753 44806 y FA(j)p 16374 45399 1748 54 v 17027 46616 a FD(i)18624 45705 y FA(\024)369 b FJ(const)222 b FD(")23827 43832 y Fz(\022)25025 44443 y(X)25825 47242 y FC(i)27166 45705 y FD(y)27848 45156 y FB(2)27800 46033 y FC(i)28373 43832 y Fz(\023)29351 44129 y FB(1)p FC(=)p FB(2)30818 43832 y Fz(\022)32017 44443 y(X)32816 47242 y FC(i)34450 44806 y FJ(1)p 34290 45399 970 54 v 34290 46616 a FD(i)34734 46232 y FB(2)35392 43832 y Fz(\023)36370 44129 y FB(1)p FC(=)p FB(2)38206 45705 y FA(\024)369 b FJ(const)221 b FD(")g(:)43061 49537 y Fn(\003)0 52100 y FE(Remark)697 b(5.7.15.)643 b Fs(F)-100 b(r)-66 b(om)624 b(R)-66 b(emark)623 b(5.7.7,)662 b(it)623 b(fol)66 b(lows)626 b(that)d FD(y)712 b FJ(=)663 b FD(V)35673 52299 y FC(k)36330 52100 y FJ(^)-738 b FD(y)672 b Fs(wher)-66 b(e)711 b FJ(^)-738 b FD(y)712 b FJ(:=)0 52709 y Fz(P)1402 54092 y FC(i)2087 53705 y FJ(^)-738 b FD(y)2633 53904 y FC(i)3009 53705 y FD(e)3612 53904 y FC(i)4453 53705 y Fs(and)552 b FJ(^)-738 b FD(y)417 b FJ(=)369 b FD(V)10455 53155 y Fw(\000)p FB(1)10166 54075 y FC(k)11713 53705 y FD(y)48 b Fs(.)0 56268 y FJ(The)434 b(follo)-36 b(wing)435 b(prop)36 b(osition)434 b(holds:)0 58831 y FE(Prop)42 b(osition)500 b(5.7.16.)554 b Fs(L)-66 b(et)464 b(b)-66 b(e)464 b FD(y)417 b FA(2)369 b FD(`)19550 58349 y FC(a;s)21096 58831 y FA(\002)296 b FD(`)22971 58349 y FC(a;s)24687 58831 y Fs(such)465 b(that)g FD(y)417 b FJ(=)32731 57835 y Fz(P)34133 59218 y FC(i)34818 58831 y FJ(^)-738 b FD(y)35364 59030 y FC(i)35740 58831 y FD(')36592 59030 y FC(k)24 b(i)37481 58831 y Fs(.)597 b(Then)11388 61825 y FJ(1)p 11388 62418 651 54 v 11388 63635 a(2)12171 60850 y Fz(\022)13370 61461 y(X)14169 64261 y FC(i)15598 62723 y FJ(^)-738 b FD(y)16192 62175 y FB(2)16144 63052 y FC(i)16718 60850 y Fz(\023)17696 61148 y FB(1)p FC(=)p FB(2)19531 62723 y FA(\024)370 b(j)p FD(y)48 b FA(j)22354 62922 y FC(s)23213 62723 y FA(\024)369 b FJ(2)25265 60850 y Fz(\022)26465 61461 y(X)27264 64261 y FC(i)28693 62723 y FJ(^)-738 b FD(y)29287 62175 y FB(2)29239 63052 y FC(i)29812 60850 y Fz(\023)30790 61148 y FB(1)p FC(=)p FB(2)32478 62723 y FD(:)0 66581 y Fp(pr)-30 b(oof.)658 b FJ(By)434 b(the)f(Cauc)-36 b(h)g(y{Sc)g(h)g(w)g(arz)434 b(inequalit)-36 b(y)-108 b(,)435 b(w)-36 b(e)433 b(ha)-36 b(v)g(e)6267 69632 y FA(j)p FD(y)48 b FA(j)7687 69831 y FC(s)9295 69632 y FJ(=)11424 68105 y Fz(\014)11424 68902 y(\014)11424 69699 y(\014)12088 68370 y(X)12887 71170 y FC(i)14316 69632 y FJ(^)-738 b FD(y)14862 69831 y FC(i)15238 69632 y FD(')16090 69831 y FC(k)24 b(i)16979 68105 y Fz(\014)16979 68902 y(\014)16979 69699 y(\014)17422 70551 y FC(s)18282 69632 y FA(\024)19684 68105 y Fz(\014)19684 68902 y(\014)19684 69699 y(\014)20348 68370 y(X)21147 71170 y FC(i)22576 69632 y FJ(^)-738 b FD(y)23122 69831 y FC(i)23498 69632 y FD(e)24101 69831 y FC(i)24476 68105 y Fz(\014)24476 68902 y(\014)24476 69699 y(\014)24919 70551 y FC(s)25705 69632 y FJ(+)27012 68105 y Fz(\014)27012 68902 y(\014)27012 69699 y(\014)27676 68370 y(X)28475 71170 y FC(i)29904 69632 y FJ(^)g FD(y)30450 69831 y FC(i)30826 69632 y FJ(\()p FD(')32184 69831 y FC(k)24 b(i)33368 69632 y FA(\000)295 b FD(e)35299 69831 y FC(i)35675 69632 y FJ(\))36181 68105 y Fz(\014)36181 68902 y(\014)36181 69699 y(\014)36624 70551 y FC(s)9284 73985 y FA(\024)11424 72112 y Fz(\022)12623 72723 y(X)13422 75523 y FC(i)14851 73985 y FJ(^)-738 b FD(y)15445 73437 y FB(2)15397 74314 y FC(i)15971 72112 y Fz(\023)16949 72410 y FB(1)p FC(=)p FB(2)18711 73985 y FJ(+)20017 72112 y Fz(\022)21217 72723 y(X)22016 75523 y FC(i)23445 73985 y FJ(^)g FD(y)24039 73437 y FB(2)23991 74314 y FC(i)24564 72112 y Fz(\023)25542 72410 y FB(1)p FC(=)p FB(2)27009 73985 y FJ(const)221 b FD(")30809 72112 y Fz(\022)32008 72723 y(X)32807 75523 y FC(i)34441 73087 y FJ(1)p 34281 73680 970 54 v 34281 74896 a FD(i)34725 74513 y FB(2)35383 72112 y Fz(\023)36361 72410 y FB(1)p FC(=)p FB(2)21072 78312 y FJ(188)p eop %%Page: 189 189 189 188 bop 15583 5400 a FA(\024)1107 b FJ(\(1)295 b(+)g(const)221 b FD(")p FJ(\))24787 3527 y Fz(\022)25986 4138 y(X)26785 6937 y FC(i)28214 5400 y FJ(^)-738 b FD(y)28808 4851 y FB(2)28760 5728 y FC(i)29334 3527 y Fz(\023)30312 3824 y FB(1)p FC(=)p FB(2)32147 5400 y FA(\024)370 b FJ(2)34200 3527 y Fz(\022)35399 4138 y(X)36198 6937 y FC(i)37627 5400 y FJ(^)-738 b FD(y)38221 4851 y FB(2)38173 5728 y FC(i)38747 3527 y Fz(\023)39725 3824 y FB(1)p FC(=)p FB(2)41413 5400 y FD(:)4645 b FJ(\(5.147\))6299 9486 y(Moreo)-36 b(v)g(er)7577 13234 y FA(j)p FD(y)48 b FA(j)8997 13433 y FC(s)9857 13234 y FA(\025)11259 11707 y Fz(\014)11259 12504 y(\014)11259 13301 y(\014)11923 11972 y(X)12722 14772 y FC(i)14151 13234 y FJ(^)-738 b FD(y)14697 13433 y FC(i)15073 13234 y FD(e)15676 13433 y FC(i)16051 11707 y Fz(\014)16051 12504 y(\014)16051 13301 y(\014)16494 14153 y FC(s)17280 13234 y FA(\000)18608 11707 y Fz(\014)18608 12504 y(\014)18608 13301 y(\014)19272 11972 y(X)20072 14772 y FC(i)21501 13234 y FJ(^)g FD(y)22047 13433 y FC(i)22422 13234 y FJ(\()p FD(')23780 13433 y FC(k)24 b(i)24965 13234 y FA(\000)295 b FD(e)26896 13433 y FC(i)27272 13234 y FJ(\))27778 11707 y Fz(\014)27778 12504 y(\014)27778 13301 y(\014)28220 14153 y FC(s)29080 13234 y FA(\025)369 b FJ(\(1)295 b FA(\000)g FJ(const)222 b FD(")p FJ(\))37568 11361 y Fz(\022)38766 11972 y(X)39565 14772 y FC(i)40994 13234 y FJ(^)-738 b FD(y)41588 12686 y FB(2)41540 13563 y FC(i)42114 11361 y Fz(\023)43092 11659 y FB(1)p FC(=)p FB(2)44780 13234 y FD(:)1278 b FJ(\(5.148\))6299 17409 y(The)434 b(thesis)f(follo)-36 b(ws)436 b(from)e(\(5.147\))h(and)e (\(5.148\).)49361 19037 y Fn(\003)6299 23031 y FF(5.7.3)1793 b(\\W)-149 b(eigh)-50 b(ted)599 b(asymmetric)h(diagonalization")e(of)f Fb(D)47018 22452 y Fx(\()p Fq(n)p Fx(\))6299 25541 y FJ(Let)433 b(us)h(de\014ne)e(the)h(op)36 b(erator)7069 31625 y FD(U)8091 31076 y FB(\()p FC(n)p FB(\))9818 31625 y FJ(:=)369 b(diag)13962 31938 y FC(`)p Fw(\024)p FC(L)15718 32049 y Ft(n)16562 27361 y Fz(0)16562 29698 y(B)16562 30495 y(B)16562 31293 y(B)16562 32090 y(B)16562 32940 y(@)18277 29632 y FJ(diag)20680 29946 y FC(j)21167 27361 y Fz( )22219 29632 y FJ(sign)24687 27759 y Fz(\022)25882 28733 y FD(`)p 25798 29327 714 54 v 25798 30543 a(\034)26939 29632 y FJ(+)295 b FD(\025)29005 29831 y FC(`j)29876 27759 y Fz(\023)31075 27361 y(!)39844 29632 y FJ(0)24877 33617 y(0)7707 b(diag)35636 33931 y FC(j)36123 31346 y Fz( )37175 33617 y FJ(sign)39643 31744 y Fz(\022)40838 32719 y FD(`)p 40754 33312 V 40754 34528 a(\034)41895 33617 y FA(\000)296 b FD(\025)43983 33816 y FC(`j)44854 31744 y Fz(\023)46053 31346 y(!)47658 27361 y(1)47658 29698 y(C)47658 30495 y(C)47658 31293 y(C)47658 32090 y(C)47658 32940 y(A)49263 31625 y FD(:)6299 38152 y FE(Lemma)737 b(5.7.17.)660 b Fs(We)655 b(have)g(that)g FD(U)26324 37670 y FB(\()p FC(n)p FB(\))28404 38152 y FJ(:)29865 37816 y(^)29486 38152 y FD(W)30895 37473 y FB(\()p FC(n)p FB(\))30714 38288 y FC(s;\013)32014 38399 y Ft(n)32582 38288 y FC(;\033)34189 38152 y FA(\000)-221 b(!)37430 37816 y FJ(^)37051 38152 y FD(W)38460 37473 y FB(\()p FC(n)p FB(\))38279 38288 y FC(s;\013)39579 38399 y Ft(n)40146 38288 y FC(;\033)41032 38152 y Fs(,)702 b FA(k)p FD(U)43819 37670 y FB(\()p FC(n)p FB(\))45178 38152 y FA(k)45842 38351 y FB(op)47612 38152 y FJ(=)721 b(1)p Fs(,)6299 39757 y(mor)-66 b(e)g(over)464 b FD(U)12934 39275 y FB(\()p FC(n)p FB(\))14757 39757 y Fs(is)g(invertible)f(with)i FJ(\()p FD(U)26198 39275 y FB(\()p FC(n)p FB(\))27556 39757 y FJ(\))28062 39275 y Fw(\000)p FB(1)29688 39757 y FJ(=)369 b FD(U)32091 39275 y FB(\()p FC(n)p FB(\))33449 39757 y Fs(.)6299 42585 y FJ(W)-108 b(e)434 b(no)-36 b(w)434 b(in)-36 b(tro)36 b(duce)432 b(the)h(parameter)h(1)p FD(=)p FJ(2)370 b FD(<)e(\016)419 b(<)369 b FJ(1.)6299 45412 y FE(Remark)513 b(5.7.18.)561 b Fs(Such)476 b(p)-66 b(ar)g(ameter)475 b(wil)66 b(l)476 b(b)-66 b(e)476 b(chosen)g(in)f(the) h(estimate)f(on)h(the)g(smal)66 b(l)6299 47017 y(divosors,)521 b(se)-66 b(e)509 b FJ(\(5.191\))p Fs(.)733 b(We)509 b(ne)-66 b(e)g(d)508 b(this)h(p)-66 b(ar)g(ameter)509 b FD(\016)559 b Fs(to)510 b(b)-66 b(alanc)g(e)508 b(the)h(asymmetry)g(of)6299 48622 y(the)465 b(op)-66 b(er)g(ator)464 b FD(D)14688 48140 y FB(\()p FC(n)p FB(\))16046 48622 y Fs(.)6299 51450 y FJ(De\014ne)14951 55150 y FA(D)15976 55349 y FC(k)16914 55150 y FJ(:=)18656 51683 y Fz(0)18656 54021 y(B)18656 54818 y(B)18656 55668 y(@)20371 53556 y FJ(diag)22774 53870 y FC(j)23482 51683 y Fz(\024)24316 52657 y FD(k)p 24316 53250 722 54 v 24320 54467 a(\034)25465 53556 y FJ(+)295 b FD(\025)27531 53755 y FC(k)24 b(j)28531 51683 y Fz(\025)34455 53556 y FJ(0)24477 56744 y(0)5212 b(diag)32742 57058 y FC(j)33450 54871 y Fz(\024)34284 55845 y FD(k)p 34284 56439 V 34288 57655 a(\034)35433 56744 y FA(\000)296 b FD(\025)37521 56943 y FC(k)24 b(j)38521 54871 y Fz(\025)39775 51683 y(1)39775 54021 y(C)39775 54818 y(C)39775 55668 y(A)41380 55150 y FD(;)6299 59839 y FJ(and)433 b(the)g(op)36 b(erators)8650 62842 y FD(Z)9635 62294 y FB(\()p FC(n)p FB(\))11362 62842 y FJ(:=)369 b(diag)15507 63156 y FC(k)24 b Fw(\024)p FC(L)17393 63267 y Ft(n)18014 62842 y FD(V)18772 63041 y FC(k)19341 62842 y FA(jD)20735 63041 y FC(k)21304 62842 y FA(j)21673 62294 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))25065 62842 y FD(;)28525 62506 y FJ(~)28249 62842 y FD(Z)29234 62294 y FB(\()p FC(n)p FB(\))30962 62842 y FJ(:=)368 b(diag)35106 63156 y FC(k)24 b Fw(\024)p FC(L)36992 63267 y Ft(n)37613 62842 y FA(jD)39007 63041 y FC(k)39576 62842 y FA(j)39945 62294 y Fw(\000)p FC(\016)41181 62842 y FD(V)42229 62292 y Fw(\000)p FB(1)41939 63212 y FC(k)43708 62842 y FD(;)2350 b FJ(\(5.149\))6299 65845 y(where)434 b FD(V)10815 66044 y FC(k)11817 65845 y FJ(are)g(de\014ned) e(in)i(\(5.146\))g(and,)g(for)g(an)-36 b(y)434 b FD(\024)368 b FA(2)h Fv(R)p FJ(,)9568 72002 y FA(jD)10962 72201 y FC(k)11530 72002 y FA(j)11899 71454 y Fw(\000)p FC(\024)13598 72002 y FJ(:=)15340 67738 y Fz(0)15340 70076 y(B)15340 70873 y(B)15340 71670 y(B)15340 72467 y(B)15340 73318 y(@)17055 70010 y FJ(diag)19458 70324 y FC(j)20166 67738 y Fz(")20941 68084 y(\014)20941 68881 y(\014)20941 69678 y(\014)20941 70475 y(\014)21517 69111 y FD(k)p 21517 69704 V 21521 70921 a(\034)22666 70010 y FJ(+)295 b FD(\025)24732 70209 y FC(k)24 b(j)25732 68084 y Fz(\014)25732 68881 y(\014)25732 69678 y(\014)25732 70475 y(\014)26175 68434 y Fw(\000)p FC(\024)27504 67738 y Fz(#)34683 70010 y FJ(0)22342 73995 y(0)6394 b(diag)31789 74309 y FC(j)32497 71724 y Fz(")33272 72069 y(\014)33272 72866 y(\014)33272 73663 y(\014)33272 74460 y(\014)33847 73096 y FD(k)p 33847 73690 V 33851 74906 a(\034)34996 73995 y FA(\000)296 b FD(\025)37084 74194 y FC(k)24 b(j)38084 72069 y Fz(\014)38084 72866 y(\014)38084 73663 y(\014)38084 74460 y(\014)38527 72419 y Fw(\000)p FC(\024)39856 71724 y Fz(#)41184 67738 y(1)41184 70076 y(C)41184 70873 y(C)41184 71670 y(C)41184 72467 y(C)41184 73318 y(A)42789 72002 y FD(:)3269 b FJ(\(5.150\))27371 78312 y(189)p eop %%Page: 190 190 190 189 bop 0 4488 a FJ(Since,)444 b(b)-36 b(y)442 b(\(5.145\),)1753 b(min)10363 5325 y FC(k)24 b Fw(\024)p FC(L)12249 5436 y Ft(n)12815 5325 y FC(;)434 b(j)51 b Fw(\025)p FB(1)15365 4488 y FA(j)p FD(k)45 b(=\034)450 b FA(\000)301 b FD(\025)20213 4687 y FC(k)24 b(j)21213 4488 y FA(j)384 b(\025)f FJ(const\()p FD(L)27743 4687 y FC(n)28369 4488 y FJ(\),)445 b(and,)f(b)-36 b(y)442 b(\(5.144\),)k FA(j)39682 3965 y FC(k)p 39679 4183 520 54 v 39679 4946 a(\034)40632 4488 y FA(\006)301 b FD(\025)42725 4687 y FC(k)24 b(j)43726 4488 y FA(j)0 6663 y(\030)369 b FD(j)509 b FJ(for)434 b FD(j)508 b FJ(large,)435 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)13448 9639 y FD(Z)14433 9091 y FB(\()p FC(n)p FB(\))16161 9639 y FJ(:)17270 9303 y(^)16891 9639 y FD(W)18300 8960 y FB(\()p FC(n)p FB(\))18119 10009 y FC(s)p Fw(\000)p FB(1+)p FC(\016)-19 b(;\013)21750 10120 y Ft(n)22318 10009 y FC(;\033)23573 9639 y FA(\000)-221 b(!)26461 9303 y FJ(^)26082 9639 y FD(W)27491 9091 y FB(\()p FC(n)p FB(\))27310 9967 y FC(s;\013)28610 10078 y Ft(n)29177 9967 y FC(;\033)30285 9639 y FD(;)9474 b FJ(\(5.151\))0 12615 y(and)14325 13927 y(~)14049 14263 y FD(Z)15034 13714 y FB(\()p FC(n)p FB(\))16762 14263 y FJ(:)17871 13927 y(^)17492 14263 y FD(W)18901 13714 y FB(\()p FC(n)p FB(\))18720 14591 y FC(s;\013)20020 14702 y Ft(n)20588 14591 y FC(;\033)21843 14263 y FA(\000)-221 b(!)24731 13927 y FJ(^)24352 14263 y FD(W)25761 13584 y FB(\()p FC(n)p FB(\))25580 14633 y FC(s)p FB(+)p FC(\016)-19 b(;\013)28009 14744 y Ft(n)28576 14633 y FC(;\033)29684 14263 y FD(:)10075 b FJ(\(5.152\))0 16637 y(Moreo)-36 b(v)g(er)434 b(w)-36 b(e)434 b(de\014ne)e(the)h(op)36 b(erators)1179 19613 y(\()p FD(Z)2670 19065 y FB(\()p FC(n)p FB(\))4028 19613 y FJ(\))4534 19065 y Fw(\000)p FB(1)6161 19613 y FJ(:=)369 b(diag)10305 19927 y FC(k)24 b Fw(\024)p FC(L)12191 20038 y Ft(n)12812 19613 y FA(jD)14206 19812 y FC(k)14775 19613 y FA(j)15144 19065 y FB(1)p Fw(\000)p FC(\016)16851 19613 y FD(V)17898 19064 y Fw(\000)p FB(1)17609 19983 y FC(k)19378 19613 y FD(;)2823 b FJ(\()23343 19278 y(~)23068 19613 y FD(Z)24053 19065 y FB(\()p FC(n)p FB(\))25411 19613 y FJ(\))25917 19065 y Fw(\000)p FB(1)27544 19613 y FJ(:=)368 b(diag)31688 19927 y FC(k)24 b Fw(\024)p FC(L)33574 20038 y Ft(n)34195 19613 y FD(V)34953 19812 y FC(k)35522 19613 y FA(jD)36916 19812 y FC(k)37485 19613 y FA(j)37854 19065 y FC(\016)38580 19613 y FD(:)1179 b FJ(\(5.153\))0 22589 y(W)-108 b(e)434 b(ha)-36 b(v)g(e)12605 24237 y(\()p FD(Z)14096 23689 y FB(\()p FC(n)p FB(\))15454 24237 y FJ(\))15960 23689 y Fw(\000)p FB(1)17587 24237 y FJ(:)18696 23901 y(^)18317 24237 y FD(W)19726 23689 y FB(\()p FC(n)p FB(\))19545 24566 y FC(s;\013)20845 24677 y Ft(n)21412 24566 y FC(;\033)22668 24237 y FA(\000)-222 b(!)25556 23901 y FJ(^)25177 24237 y FD(W)26586 23558 y FB(\()p FC(n)p FB(\))26405 24607 y FC(s)p Fw(\000)p FB(1+)p FC(\016)-19 b(;\013)30036 24718 y Ft(n)30603 24607 y FC(;\033)40120 24237 y FJ(\(5.154\))0 26612 y(and)12915 28259 y(\()13697 27924 y(~)13421 28259 y FD(Z)14406 27711 y 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38689 y FJ(.)0 41748 y FE(Lemma)499 b(5.7.19)g(\(\\Diagonalization"\).) 557 b FD(D)24535 41266 y FB(\()p FC(n)p FB(\))25893 41748 y FD(Z)26878 41266 y FB(\()p FC(n)p FB(\))28606 41748 y FA(j)29249 42036 y FB(^)28975 42259 y FC(W)29994 41788 y Fu(\()p Ft(n)p Fu(\))29863 42400 y Ft(s;\013)31037 42511 y(n)31604 42400 y(;\033)32467 41748 y FJ(=)464 b(\()34725 41412 y(~)34449 41748 y FD(Z)35434 41266 y FB(\()p FC(n)p FB(\))36793 41748 y FJ(\))37298 40881 y Fw(\000)p FB(1)38556 41748 y FD(U)39578 41266 y FB(\()p FC(n)p FB(\))40936 41748 y Fs(.)0 46272 y FH(5.8)2152 b(O\013-diagonal)716 b(term)0 49222 y FE(Remark)805 b(5.8.1.)691 b Fs(In)709 b(this)h(se)-66 b(ction)708 b(we)j(de)-66 b(al)709 b(with)h(the)g(op)-66 b(er)g(ator)709 b FD(T)36268 48740 y FB(\()p FC(n)p FB(\))38337 49222 y Fs(de\014ne)-66 b(d)708 b(in)0 50827 y FJ(\(5.116\))p Fs(.)802 b(We)533 b(p)-66 b(oint)531 b(out)j(that)e FD(T)16952 50345 y FB(\()p FC(n)p FB(\))18843 50827 y Fs(is)h(not)f(a)h(T)-100 b(o)-66 b(epliz)533 b(op)-66 b(er)g(ator)532 b(but)h(only)g(a)g (\\quasi{)0 52432 y(pr)-66 b(o)g(duct)464 b(op)-66 b(er)g(ator")465 b(ac)-66 b(c)g(or)g(ding)462 b(to)j(De\014nition)e(5.1.9.)0 55215 y FJ(De\014ne)16602 56527 y(~)16347 56863 y FD(T)17290 56315 y FB(\()p FC(n)p FB(\))19017 56863 y FJ(:=)21035 56527 y(~)20759 56863 y FD(Z)21744 56315 y FB(\()p FC(n)p FB(\))23103 56863 y FD(T)24046 56315 y FB(\()p FC(n)p FB(\))25404 56863 y FD(Z)26389 56315 y FB(\()p FC(n)p FB(\))40120 56863 y FJ(\(5.156\))0 59237 y(namely)11073 59809 y Fz(\000)11936 60549 y FJ(~)11682 60885 y FD(T)12625 60337 y FB(\()p FC(n)p FB(\))13983 59809 y Fz(\001)14592 61405 y FC(k)24 b(`)15914 60885 y FJ(:=)369 b FA(jD)19050 61084 y FC(k)19618 60885 y FA(j)19987 60337 y Fw(\000)p FC(\016)21224 60885 y FD(V)22271 60335 y Fw(\000)p FB(1)21982 61255 y FC(k)23655 60549 y FJ(^)23529 60885 y(\003)24432 61084 y FC(k)24 b(;`)25646 60885 y FD(V)26404 61084 y FC(`)27066 60885 y FA(jD)28460 61084 y FC(`)28899 60885 y FA(j)29268 60337 y Fw(\000)p 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y FJ(^)29359 70258 y FD(W)30768 69709 y FB(\()p FC(n)p FB(\))30587 70586 y FC(s;\013)31887 70697 y Ft(n)32454 70586 y FC(;\033)33561 70258 y FD(:)0 73386 y FJ(W)-108 b(e)371 b(w)-36 b(an)g(t)371 b(to)g(pro)-36 b(v)g(e)372 b(that)e(\004)13886 72904 y FB(\()p FC(n)p FB(\))15616 73386 y FJ(is)h(in)-36 b(v)g(ertible.)558 b(W)-108 b(e)371 b(need)f(the)h(follo)-36 b(wing)373 b FK(crucial)f FJ(lemma,)0 74991 y(that)433 b(will)i(b)36 b(e)433 b(pro)-36 b(v)g(ed)433 b(on)h(pg.)578 b(193.)21072 78312 y(190)p eop %%Page: 191 191 191 190 bop 6299 4488 a FE(Lemma)499 b(5.8.2.)555 b FA(k)16553 4153 y FJ(~)16299 4488 y FD(T)17242 4006 y FB(\()p FC(n)p FB(\))18600 4488 y FA(k)19264 4687 y FB(op)20682 4488 y FA(\024)369 b FJ(const)221 b FD(\015)26022 4006 y Fw(\000)p FB(1)27279 4488 y FD(\034)614 b Fs(with)464 b(c)-66 b(onstant)464 b(indep)-66 b(endent)462 b(of)j FD(n)p Fs(.)6299 7350 y FJ(By)575 b(Lemmata)e(5.7.17)j(and)d(5.8.2,)611 b(\004)25497 6868 y FB(\()p FC(n)p FB(\))27463 7350 y FJ(:)28811 7014 y(^)28432 7350 y FD(W)29841 6671 y FB(\()p FC(n)p FB(\))29660 7486 y FC(s;\013)30960 7597 y Ft(n)31528 7486 y FC(;\033)33021 7350 y FA(\000)-221 b(!)36148 7014 y FJ(^)35769 7350 y FD(W)37178 6671 y FB(\()p FC(n)p FB(\))36997 7486 y FC(s;\013)38297 7597 y Ft(n)38865 7486 y FC(;\033)40325 7350 y FJ(is)574 b(in)-36 b(v)g(ertible)574 b(for)g FD(")6299 8955 y FJ(small)435 b(enough)e(\(but)f(indep)36 b(enden)-36 b(t)431 b(of)k FD(n)p FJ(\).)578 b(So)434 b(w)-36 b(e)434 b(can)f(de\014ne)15582 11740 y(^)15325 12076 y FA(G)16194 11528 y FB(\()p FC(n)p FB(\))17921 12076 y FJ(:=)369 b FD(Z)20648 11528 y FB(\()p FC(n)p FB(\))22007 12076 y FJ(\(\004)23380 11528 y FB(\()p FC(n)p FB(\))24738 12076 y FJ(\))25244 11214 y Fw(\000)p FB(1)26777 11740 y FJ(~)26501 12076 y FD(Z)27486 11528 y FB(\()p FC(n)p FB(\))29214 12076 y FJ(:)30323 11740 y(^)29944 12076 y FD(W)31353 11528 y FB(\()p FC(n)p FB(\))31172 12404 y FC(s;\013)32472 12515 y Ft(n)33039 12404 y FC(;\033)34295 12076 y FA(\000)-222 b(!)37183 11740 y FJ(^)36804 12076 y FD(W)38213 11528 y FB(\()p FC(n)p FB(\))38032 12404 y FC(s;\013)39332 12515 y Ft(n)39899 12404 y FC(;\033)41007 12076 y FD(:)5051 b FJ(\(5.158\))6299 15191 y(It)451 b(results)g(that)14968 14856 y(^)14711 15191 y FA(G)15580 14709 y FB(\()p FC(n)p FB(\))17389 15191 y FJ(is)h(the)e(\(righ)-36 b(t\))450 b(in)-36 b(v)g(erse)451 b(of)31284 14856 y(^)30967 15191 y FA(L)31883 14709 y FB(\()p FC(n)p FB(\))33241 15191 y FJ(.)631 b(Namely)-108 b(,)456 b(ev)-36 b(en)451 b(if)43783 14856 y(^)43466 15191 y FA(L)44382 14709 y FB(\()p FC(n)p FB(\))45997 14856 y FJ(^)45740 15191 y FA(G)46609 14709 y FB(\()p FC(n)p FB(\))48418 15191 y FJ(is)g(a)6299 16977 y(priori)434 b(de\014ned)e(only)i(from)g FD(W)21654 16298 y FB(\()p FC(n)p FB(\))21473 17113 y FC(s;\013)22773 17224 y Ft(n)23341 17113 y FC(;\033)24660 16977 y FJ(in)-36 b(to)434 b FD(W)28707 16298 y FB(\()p FC(n)p FB(\))28526 17303 y FC(s)p Fw(\000)p FB(1)p FC(;\013)31028 17414 y Ft(n)31595 17303 y FC(;\033)32482 16977 y FJ(,)f(it)h(results)f (that)19239 19754 y(^)18922 20090 y FA(L)19838 19541 y FB(\()p FC(n)p FB(\))21453 19754 y FJ(^)21196 20090 y FA(G)22065 19541 y FB(\()p FC(n)p FB(\))23423 20090 y FD(y)417 b FJ(=)369 b FD(y)269 b(;)2823 b FA(8)221 b FD(y)417 b FA(2)369 b FD(W)34616 19541 y FB(\()p FC(n)p FB(\))34435 20418 y FC(s;\013)35735 20529 y Ft(n)36302 20418 y FC(;\033)37410 20090 y FD(:)8648 b FJ(\(5.159\))6299 23205 y(Recalling)435 b(that)15089 22869 y(^)14771 23205 y FA(L)15687 22723 y FB(\()p FC(n)p FB(\))17415 23205 y FJ(=)368 b FD(D)19911 22723 y FB(\()p FC(n)p FB(\))21565 23205 y FJ(+)294 b FD("T)24423 22723 y FB(\()p FC(n)p FB(\))25781 23205 y FJ(,)434 b(w)-36 b(e)434 b(get,)g(b)-36 b(y)433 b(Lemma)h(5.7.19)7559 26011 y(^)7242 26347 y FA(L)8158 25799 y FB(\()p FC(n)p FB(\))9773 26011 y FJ(^)9516 26347 y FA(G)10385 25799 y FB(\()p FC(n)p FB(\))12850 26347 y FJ(=)14969 24873 y Fz(\020)15762 26347 y FD(D)16878 25799 y FB(\()p FC(n)p FB(\))18532 26347 y FJ(+)294 b FD("T)21390 25799 y 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FJ(=)21467 49901 y Fz(\014)21467 50698 y(\014)21467 51495 y(\014)21910 51429 y FD(S)22710 51628 y FC(k)23278 51429 y FD(V)24036 51628 y FC(k)24827 50167 y Fz(X)25626 52967 y FC(i)26967 51429 y FD(y)27601 51628 y FC(i)27977 51429 y FD(e)28580 51628 y FC(i)29251 51429 y FA(\000)g FD(S)31379 51628 y FC(k)32169 50167 y Fz(X)32968 52967 y FC(i)34309 51429 y FD(y)34943 51628 y FC(i)35319 51429 y FD(e)35922 51628 y FC(i)36298 49901 y Fz(\014)36298 50698 y(\014)36298 51495 y(\014)36740 52348 y FC(s)9160 55070 y FJ(=)11289 53542 y Fz(\014)11289 54339 y(\014)11289 55136 y(\014)11732 55070 y FD(S)12532 55269 y FC(k)13322 53808 y Fz(X)14121 56607 y FC(i)15462 55070 y FD(y)16096 55269 y FC(i)16472 55070 y FD(')17324 55269 y FC(k)24 b(i)18509 55070 y FA(\000)295 b FD(S)20637 55269 y FC(k)21427 53808 y Fz(X)22226 56607 y FC(i)23567 55070 y FD(y)24201 55269 y FC(i)24577 55070 y FD(e)25180 55269 y FC(i)25556 53542 y Fz(\014)25556 54339 y(\014)25556 55136 y(\014)25998 55989 y FC(s)26858 55070 y FJ(=)28238 53542 y Fz(\014)28238 54339 y(\014)28238 55136 y(\014)28903 53808 y(X)29702 56607 y FC(i)31043 55070 y FD(y)31677 55269 y FC(i)32053 55070 y FD(\025)32812 55269 y FC(i)33187 55070 y FD(')34039 55269 y FC(k)24 b(i)35224 55070 y FA(\000)295 b FD(S)37352 55269 y FC(k)38142 53808 y Fz(X)38941 56607 y FC(i)40282 55070 y FD(y)40916 55269 y FC(i)41292 55070 y FD(e)41895 55269 y FC(i)42271 53542 y Fz(\014)42271 54339 y(\014)42271 55136 y(\014)42713 55989 y FC(s)9160 58710 y FJ(=)11289 57183 y Fz(\014)11289 57980 y(\014)11289 58777 y(\014)11953 57448 y(X)12753 60248 y FC(i)14093 58710 y FD(y)14727 58909 y FC(i)15103 58710 y FD(\025)15862 58909 y FC(i)16238 58710 y FD(e)16841 58909 y FC(i)17511 58710 y FJ(+)18818 57448 y Fz(X)19617 60248 y FC(i)20958 58710 y FD(y)21592 58909 y FC(i)21968 58710 y FD(\025)22727 58909 y FC(i)23103 58710 y FJ(\()p FD(')24461 58909 y FC(k)24 b(i)25645 58710 y FA(\000)295 b FD(e)27576 58909 y FC(i)27952 58710 y FJ(\))g FA(\000)g FD(S)30881 58909 y FC(k)31671 57448 y Fz(X)32471 60248 y FC(i)33812 58710 y FD(y)34446 58909 y FC(i)34821 58710 y FD(e)35424 58909 y FC(i)35800 57183 y Fz(\014)35800 57980 y(\014)35800 58777 y(\014)36243 59629 y FC(s)9160 62351 y FJ(=)11289 60823 y Fz(\014)11289 61620 y(\014)11289 62417 y(\014)11953 61089 y(X)12753 63889 y FC(i)14093 62351 y FD(y)14727 62550 y FC(i)15103 62351 y FD(\025)15862 62550 y FC(i)16238 62351 y FD(e)16841 62550 y FC(i)17511 62351 y FA(\000)18840 61089 y Fz(X)19639 63889 y FC(i)20980 62351 y FD(y)21614 62550 y FC(i)22134 62015 y FJ(~)21990 62351 y(\012)22929 62550 y FC(i)23305 62351 y FD(e)23908 62550 y FC(i)11289 65992 y FA(\000)p FD(")p FA(M)14526 66191 y FC(k)15316 64730 y Fz(X)16116 67529 y FC(i)17457 65992 y FD(y)18091 66191 y FC(i)18466 65992 y FD(e)19069 66191 y FC(i)19740 65992 y FJ(+)21047 64730 y Fz(X)21846 67529 y FC(i)23187 65992 y FD(y)23821 66191 y FC(i)24197 65992 y FD(\025)24956 66191 y FC(i)25331 65992 y FJ(\()p FD(')26689 66191 y FC(k)24 b(i)27874 65992 y FA(\000)295 b FD(e)29805 66191 y FC(i)30181 65992 y FJ(\))30687 64464 y Fz(\014)30687 65261 y(\014)30687 66058 y(\014)31129 66911 y FC(s)9160 69632 y FJ(=)11289 68105 y Fz(\014)11289 68902 y(\014)11289 69699 y(\014)11953 68370 y(X)12753 71170 y FC(i)14093 69632 y FD(y)14727 69831 y FC(i)15103 69632 y FJ(\()p FD(\025)16368 69831 y FC(i)17039 69632 y FA(\000)18512 69297 y FJ(~)18367 69632 y(\012)19306 69831 y FC(i)19682 69632 y FJ(\))p FD(e)20791 69831 y FC(i)21167 68105 y Fz(\014)21167 68902 y(\014)21167 69699 y(\014)21609 70551 y FC(s)22395 69632 y FA(\000)g FD(")24332 68503 y Fz(\014)24332 69300 y(\014)24775 69632 y FA(M)26370 69831 y FC(k)26939 69632 y FD(y)27621 68503 y Fz(\014)27621 69300 y(\014)28063 70153 y FC(s)28849 69632 y FJ(+)30156 68105 y Fz(\014)30156 68902 y(\014)30156 69699 y(\014)30820 68370 y(X)31619 71170 y FC(i)32960 69632 y FD(y)33594 69831 y FC(i)33970 69632 y FD(\025)34729 69831 y FC(i)35105 69632 y FJ(\()p FD(')36463 69831 y FC(k)24 b(i)37647 69632 y FA(\000)295 b FD(e)39578 69831 y FC(i)39954 69632 y FJ(\))40460 68105 y Fz(\014)40460 68902 y(\014)40460 69699 y(\014)40902 70551 y FC(s)9149 73985 y FA(\024)10846 72112 y Fz(\022)12046 72723 y(X)12845 75523 y FC(i)14186 73985 y FD(y)14868 73437 y FB(2)14820 74314 y FC(i)15393 73985 y FJ(\()p FD(\025)16658 74184 y FC(i)17329 73985 y FA(\000)18802 73649 y FJ(~)18657 73985 y(\012)19596 74184 y FC(i)19972 73985 y FJ(\))20478 73437 y FB(2)21004 72112 y Fz(\023)21982 72410 y FB(1)p FC(=)p FB(2)23744 73985 y FJ(+)f FD(")25659 72856 y Fz(\014)25659 73653 y(\014)26102 73985 y FA(M)27697 74184 y FC(k)28266 73985 y FD(y)28948 72856 y Fz(\014)28948 73653 y(\014)29391 74506 y FC(s)30176 73985 y FJ(+)h(ct)32567 72112 y Fz(\022)33766 72723 y(X)34565 75523 y FC(i)35906 73985 y FA(j)p FD(')37127 74184 y FC(k)24 b(i)38312 73985 y FA(\000)295 b FD(e)40243 74184 y FC(i)40619 73985 y FA(j)40988 73437 y FB(2)40988 74314 y FC(s)41513 72112 y Fz(\023)42491 72410 y FB(1)p FC(=)p FB(2)21072 78312 y FJ(192)p eop %%Page: 193 193 193 192 bop 15448 4488 a FA(\024)1107 b FJ(const)222 b FD(")f(;)6299 7433 y FJ(where)578 b(w)-36 b(e)577 b(used)g(that)18335 6437 y Fz(P)19737 7821 y FC(i)20334 7433 y FD(y)21016 6951 y FB(2)20968 7761 y FC(i)21542 7433 y FD(\025)22301 6951 y FB(2)22301 7761 y FC(i)23440 7433 y FA(\024)615 b FJ(const)o(.)1011 b(Therefore,)614 b(\(5.165\))579 b(is)e(true.)1010 b(The)577 b(last)6299 9038 y(estimate)434 b(w)-36 b(e)434 b(need)f(is)19582 10854 y Fz(\015)19582 11651 y(\015)20320 11983 y FD(S)21120 12182 y FC(k)21815 11647 y FJ(^)21689 11983 y(\003)22592 12182 y FC(k)24 b(;`)23806 10854 y Fz(\015)23806 11651 y(\015)24544 12503 y FC(s;s)26100 11983 y FA(\024)369 b FJ(const)221 b FA(k)31483 11647 y FJ(^)31357 11983 y(\003)32260 12182 y FC(k)24 b(;`)33475 11983 y FA(k)34139 12182 y FC(s;s)p FB(+1)36750 11983 y FD(;)9308 b FJ(\(5.166\))6299 15122 y(obtained)429 b(b)-36 b(y)429 b(\(5.117\).)578 b(Finally)-108 b(,)431 b(b)-36 b(y)429 b(\(5.161\),)i(\(5.162\),)h(\(5.163\),)f(\(5.165\),)h (\(5.166\))e(and)6299 16727 y(Lemma)k(5.7.14,)h(the)e(inequalit)-36 b(y)435 b(\(5.160\))g(in)e(the)g(thesis)h(follo)-36 b(ws.)49361 18336 y Fn(\003)6299 19945 y FJ(Lemma)523 b(5.8.3)i(will)f(b)36 b(e)523 b(used)g(in)g(Section)g(5.9)h(to)f(pro)-36 b(v)g(e)523 b(the)g(follo)-36 b(wing)525 b(fundamen)-36 b(tal)6299 21550 y(lemma)434 b(on)g(small)g(divisors.)6299 24280 y FE(Lemma)465 b(5.8.4)f(\(Small)h(divisors\).)533 b Fs(L)-66 b(et)436 b(b)-66 b(e)437 b FD(")368 b FJ(=)h FD(\021)33115 23798 y FB(3)33640 24280 y Fs(.)588 b(L)-66 b(et)436 b(b)-66 b(e)437 b FD(\021)416 b FJ(=)40848 23324 y FA(p)p 41955 23324 1111 54 v 956 x FD($)36 b Fs(,)442 b FD($)405 b FA(2)368 b FJ(\001)47725 23798 y FC(\027)47725 24609 y(n)48351 24280 y FJ(\()p FD(\030)64 b FJ(\))p Fs(.)6299 25886 y(Ther)-66 b(e)346 b(exists)h(a)g(c)-66 b(onstant)345 b FD(C)464 b(>)369 b FJ(0)347 b Fs(such)g(that,)370 b(for)347 b(al)66 b(l)347 b FD(`)369 b FA(6)p FJ(=)g FD(k)45 b Fs(,)370 b FJ(1)f FD(<)g(\027)452 b(<)369 b FJ(2)p Fs(,)h FD(\016)419 b FJ(=)369 b FD(\027)84 b(=)p FJ(\()p FD(\027)117 b FJ(+)33 b(1\))p Fs(,)7389 28009 y Fz(\015)7389 28806 y(\015)8127 29139 y FA(jD)9521 29338 y FC(`)9960 29139 y FA(j)10329 28590 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)g FB(\))13500 28009 y Fz(\015)13500 28806 y(\015)14238 29659 y FC(s;s)15425 28009 y Fz(\015)15425 28806 y(\015)16163 29139 y FD(M)17560 28590 y Fw(\000)p FB(1)18818 29139 y FA(jD)20212 29338 y FC(k)20780 29139 y FA(j)21149 28590 y Fw(\000)p FC(\016)22386 28009 y Fz(\015)22386 28806 y(\015)23124 29659 y FC(s;s)24680 29139 y FA(\024)369 b FD(C)27479 28240 y(\034)p 27462 28833 747 54 v 27462 30050 a(\015)28563 29139 y FA(j)p FD(`)295 b FA(\000)h FD(k)45 b FA(j)32192 28590 y FB(1)p Fw(\000)p FC(\027)58 b(\016)33 b FB(+\()p FC(\027)58 b Fw(\000)p FB(1\))p FC(=\014)39019 29139 y FJ(=)368 b FD(C)41796 28240 y(\034)p 41780 28833 V 41780 30050 a(\015)42881 29139 y FA(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)46642 28183 y Fu(2)p Ft(\027)g Fu(+1)p 46642 28355 1892 40 v 46845 28903 a Ft(\027)g Fu(+1)48943 29139 y FD(;)6299 32755 y Fs(wher)-66 b(e)465 b FD(\014)443 b FJ(=)368 b(\()p FD(\027)379 b FA(\000)295 b FJ(1\))p FD(=\027)84 b Fs(.)6299 35489 y Fp(pr)-30 b(oof.)659 b FJ(See)433 b(Section)g(5.9.)6299 37434 y(Using)h(Lemma)f(5.8.4)j(w) -36 b(e)433 b(giv)-36 b(e)435 b(the)e(follo)-36 b(wing)6299 39043 y Fp(Pr)-30 b(oof)504 b(of)e(Lemma)f(5.8.2.)580 b FJ(W)-108 b(e)433 b(ha)-36 b(v)g(e,)434 b(from)g(\(5.157\),)h (\(5.160\))g(and)e(Lemma)h(5.8.4,)7662 42094 y Fz(\015)7662 42891 y(\015)8400 42147 y(\000)9264 42888 y FJ(~)9009 43223 y FD(T)9952 42675 y FB(\()p FC(n)p FB(\))11310 43223 y FD(w)12276 42147 y Fz(\001)12884 43744 y FC(k)13453 42094 y Fz(\015)13453 42891 y(\015)14191 43744 y FC(a;s)16561 43223 y FJ(=)18690 41297 y Fz(\015)18690 42094 y(\015)18690 42891 y(\015)18690 43688 y(\015)20863 41563 y FC(L)21503 41674 y Ft(n)20507 41961 y Fz(X)19649 44790 y FC(`)p FB(=1)158 b FC(;`)p Fw(6)p FB(=)p FC(k)23505 43223 y FA(jD)24899 43422 y FC(k)25468 43223 y FA(j)25837 42675 y Fw(\000)p FC(\016)27073 43223 y FD(V)28121 42674 y Fw(\000)p FB(1)27831 43593 y FC(k)29505 42888 y FJ(^)29378 43223 y(\003)30281 43422 y FC(k)24 b(;`)31717 43223 y FD(V)32475 43422 y FC(`)32915 43223 y FA(jD)34309 43422 y FC(`)34970 43223 y FA(j)35339 42675 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))38510 43223 y FD(w)39440 43422 y FC(`)39879 41297 y Fz(\015)39879 42094 y(\015)39879 42891 y(\015)39879 43688 y(\015)40617 44541 y FC(s;s)16550 47937 y FA(\024)19904 46276 y FC(L)20544 46387 y Ft(n)19548 46675 y Fz(X)18690 49503 y FC(`)p FB(=1)157 b FC(;`)p Fw(6)p FB(=)p FC(k)22546 46807 y Fz(\015)22546 47604 y(\015)23284 47937 y FA(jD)24678 48136 y FC(k)25246 47937 y FA(j)25615 47388 y Fw(\000)p FC(\016)26852 47937 y FD(V)27899 47387 y Fw(\000)p FB(1)27610 48306 y FC(k)29283 47601 y FJ(^)29157 47937 y(\003)30060 48136 y FC(k)24 b(;`)31496 47937 y FD(V)32254 48136 y FC(`)32694 47937 y FA(jD)34088 48136 y FC(`)34527 47937 y FA(j)34896 47388 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))38067 46807 y Fz(\015)38067 47604 y(\015)38805 48457 y FC(a;s)40057 47937 y FA(k)p FD(w)41651 48136 y FC(`)42090 47937 y FA(k)42754 48136 y FC(a;s)16550 52650 y FA(\024)19904 50989 y FC(L)20544 51100 y Ft(n)19548 51388 y Fz(X)18690 54217 y FC(`)p FB(=1)157 b FC(;`)p Fw(6)p FB(=)p FC(k)22546 51521 y Fz(\015)22546 52318 y(\015)23284 52650 y FA(jD)24678 52849 y FC(k)25246 52650 y FA(j)25615 52101 y Fw(\000)p FC(\016)26852 52650 y FD(M)28249 52101 y Fw(\000)p FB(1)29507 51521 y Fz(\015)29507 52318 y(\015)30245 53170 y FC(s;s)31432 51521 y Fz(\015)31432 52318 y(\015)32170 52650 y FA(D)33195 52849 y FC(`)33634 52650 y FA(j)34003 52101 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))37174 51521 y Fz(\015)37174 52318 y(\015)37912 53170 y FC(s;s)39099 51521 y Fz(\015)39099 52318 y(\015)39963 52314 y FJ(^)39837 52650 y(\003)40740 52849 y FC(k)24 b(;`)41954 51521 y Fz(\015)41954 52318 y(\015)42692 53170 y FC(s;s)p FB(+1)45081 52650 y FA(k)p FD(w)46675 52849 y FC(`)47115 52650 y FA(k)47779 52849 y FC(a;s)16550 57363 y FA(\024)19904 55702 y FC(L)20544 55813 y Ft(n)19548 56101 y Fz(X)18690 58930 y FC(`)p FB(=1)157 b FC(;`)p Fw(6)p FB(=)p FC(k)22546 57363 y FD(\015)23293 56814 y Fw(\000)p FB(1)24550 57363 y FD(\034)148 b FA(j)p FD(`)296 b FA(\000)f FD(k)45 b FA(j)29025 56407 y Fu(2)p Ft(\027)g Fu(+1)p 29025 56579 V 29228 57128 a Ft(\027)g Fu(+1)31104 57363 y FA(k)31894 57027 y FJ(^)31768 57363 y(\003)32671 57562 y FC(k)24 b(;`)33886 57363 y FA(k)34550 57562 y FC(s;s)p FB(+1)36939 57363 y FA(k)p FD(w)38533 57562 y FC(`)38973 57363 y FA(k)39637 57562 y FC(a;s)41110 57363 y FD(;)6299 61608 y FJ(from)434 b(whic)-36 b(h)434 b(it)f(follo)-36 b(ws)436 b(that)10193 64327 y Fz(\015)10193 65124 y(\015)10931 64380 y(\000)11794 65121 y FJ(~)11540 65456 y FD(T)12483 64908 y FB(\()p FC(n)p FB(\))13841 65456 y FD(w)14807 64380 y Fz(\001)15415 65977 y FC(k)15984 64327 y Fz(\015)15984 65124 y(\015)16722 65977 y FC(a;s)18342 65456 y FA(\024)20959 63796 y FC(L)21599 63907 y Ft(n)20603 64194 y Fz(X)19745 67023 y FC(`)p FB(=1)157 b FC(;k)24 b Fw(6)p FB(=)p FC(`)23601 65456 y FD(\015)24348 64908 y Fw(\000)p FB(1)25605 65456 y FD(\034)148 b FA(j)p FD(`)295 b FA(\000)h FD(k)45 b FA(j)30080 64500 y Fu(2)p Ft(\027)g Fu(+1)p 30080 64672 V 30282 65221 a Ft(\027)g Fu(+1)32159 65456 y FA(k)32949 65121 y FJ(^)32823 65456 y(\003)33726 65655 y FC(k)24 b(;`)34940 65456 y FA(k)35604 65655 y FC(s;s)p FB(+1)37994 65456 y FA(k)p FD(w)39588 65655 y FC(`)40027 65456 y FA(k)40691 65655 y FC(a;s)42165 65456 y FD(:)3893 b FJ(\(5.167\))6299 69775 y(Hence,)434 b(using)f(\(4.11\))i(and)e(\(5.167\),)6378 72827 y Fz(\015)6378 73624 y(\015)7116 73956 y FD(T)8059 73407 y FB(\()p FC(n)p FB(\))9417 73956 y FD(w)10383 72827 y Fz(\015)10383 73624 y(\015)11121 73177 y FB(2)11121 74476 y FC(\013;\033)13729 73956 y FJ(=)16214 72295 y FC(L)16854 72406 y Ft(n)15858 72694 y Fz(X)15960 75523 y FC(k)24 b FB(=1)17998 73956 y FD(e)18601 73407 y FB(2)p FC(k)g(\013)20245 73956 y FD(k)20966 73407 y FB(2)p FC(\033)22061 72827 y Fz(\015)22061 73624 y(\015)22799 73956 y FJ(\()23560 73620 y(~)23305 73956 y FD(T)24248 73407 y FB(\()p FC(n)p FB(\))25606 73956 y FD(w)36 b FJ(\))27078 74155 y FC(k)27646 72827 y Fz(\015)27646 73624 y(\015)28384 73177 y FB(2)28384 74476 y FC(a;s)27371 78312 y FJ(193)p eop %%Page: 194 194 194 193 bop 7419 5558 a FA(\024)9709 4660 y FD(\034)10422 4178 y FB(2)p 9692 5253 1273 54 v 9692 6470 a FD(\015)10439 6086 y FB(2)11675 3898 y FC(L)12315 4009 y Ft(n)11319 4297 y Fz(X)11420 7125 y FC(k)24 b FB(=1)13459 5558 y FD(e)14062 5010 y FB(2)p FC(k)g(\013)15705 5558 y FD(k)16426 5010 y FB(2)p FC(\033)17522 3685 y Fz(\022)19935 3898 y FC(L)20575 4009 y Ft(n)19579 4297 y Fz(X)18721 7125 y FC(`)p FB(=1)157 b FC(;k)24 b Fw(6)p FB(=)p FC(`)22577 5558 y FA(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)26338 4603 y Fu(2)p Ft(\027)g Fu(+1)p 26338 4775 1892 40 v 26541 5323 a Ft(\027)g Fu(+1)28417 5558 y FA(k)29207 5223 y FJ(^)29081 5558 y(\003)29984 5757 y FC(k)24 b(;`)31199 5558 y FA(k)31863 5757 y FC(s;s)p FB(+1)34252 5558 y FA(k)p FD(w)35846 5757 y FC(`)36286 5558 y FA(k)36950 5757 y FC(a;s)38202 3685 y Fz(\023)39179 3983 y FB(2)7419 10272 y FA(\024)664 b FJ(ct)10571 9373 y FD(\034)11284 8891 y FB(2)p 10554 9966 1273 54 v 10554 11183 a FD(\015)11301 10799 y FB(2)12537 8611 y FC(L)13177 8722 y Ft(n)12181 9010 y Fz(X)12283 11839 y FC(k)24 b FB(=1)14100 10272 y FD(e)14703 9723 y FB(2)p FC(k)g(\013)16346 10272 y FD(k)17067 9723 y FB(2)p FC(\033)18934 8611 y(L)19574 8722 y Ft(n)18578 9010 y Fz(X)17720 11839 y FC(`)p FB(=1)158 b FC(;k)24 b Fw(6)p FB(=)p FC(`)21045 9373 y FA(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)24673 8891 y FB(2)p FC(\033)25768 9373 y FD(`)26314 8891 y FB(2)p FC(\033)p 21045 9966 6365 54 v 23319 11183 a FD(k)24040 10799 y FB(2)p FC(\033)27764 10272 y FA(j)p FD(k)340 b FA(\000)295 b FD(`)p FA(j)31392 9689 y FB(2)32152 9316 y Fu(2)p Ft(\027)45 b Fu(+1)p 32152 9488 1892 40 v 32355 10036 a Ft(\027)g Fu(+1)34231 10272 y FA(k)35021 9936 y FJ(^)34895 10272 y(\003)35798 10471 y FC(k)24 b(;`)37013 10272 y FA(k)37677 9723 y FB(2)37677 10600 y FC(s;s)p FB(+1)40066 10272 y FA(k)p FD(w)41660 10471 y FC(`)42100 10272 y FA(k)42764 9723 y FB(2)42764 10600 y FC(a;s)7419 14985 y FA(\024)664 b FJ(ct)10571 14086 y FD(\034)11284 13604 y FB(2)p 10554 14679 1273 54 v 10554 15896 a FD(\015)11301 15512 y FB(2)12537 13324 y FC(L)13177 13435 y Ft(n)12181 13723 y Fz(X)12347 16552 y FC(`)p FB(=1)14321 14985 y FD(e)14924 14436 y FB(2)p FC(`\013)16439 14985 y FD(`)16985 14436 y FB(2)p FC(\033)18080 14985 y FA(k)p FD(w)19674 15184 y FC(`)20113 14985 y FA(k)20777 14436 y FB(2)20777 15313 y FC(a;s)23086 13324 y(L)23726 13435 y Ft(n)22730 13723 y Fz(X)21808 16552 y FC(k)24 b FB(=1)156 b FC(;k)24 b Fw(6)p FB(=)p FC(`)25350 14985 y FA(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)28978 14377 y FB(2)29448 13301 y Fz(\000)30058 14377 y FC(\033)32 b FB(+)31493 14004 y Fu(2)p Ft(\027)45 b Fu(+1)p 31492 14176 1892 40 v 31695 14724 a Ft(\027)g Fu(+1)33516 13301 y Fz(\001)34180 14985 y FD(e)34783 14436 y FB(2)p Fw(j)p FC(`)p Fw(\000)p FC(k)24 b Fw(j)p FC(\013)38065 14985 y FA(k)38855 14649 y FJ(^)38729 14985 y(\003)39632 15184 y FC(k)g(;`)40847 14985 y FA(k)41511 14436 y FB(2)41511 15313 y FC(s;s)p FB(+1)0 19344 y FJ(where)433 b(in)h(the)f(last)h(inequalit)-36 b(y)435 b(w)-36 b(e)433 b(used)g FD(k)414 b FA(\024)370 b FD(`)295 b FJ(+)f FA(j)p FD(k)341 b FA(\000)295 b FD(`)p FA(j)p FJ(.)578 b(W)-108 b(e)434 b(can)f(estimate)5840 21300 y Fz(X)6543 24129 y FC(k)7980 22562 y FD(e)8583 22014 y FB(2)p Fw(j)p FC(`)p Fw(\000)p FC(k)24 b Fw(j)p FC(\013)11866 22562 y FA(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)15494 21980 y FB(2\()p FC(\033)32 b FB(+)17765 21606 y Fu(2)p Ft(\027)45 b Fu(+1)p 17765 21778 V 17968 22327 a Ft(\027)g Fu(+1)19789 21980 y FB(\))20210 22562 y FA(k)p FJ(\003)21777 21932 y FC(ij)21777 22932 y(`)p Fw(\000)p FC(k)23462 22562 y FA(k)24126 22014 y FB(2)24126 22890 y FC(s;s)p FB(+1)26884 22562 y FA(\024)369 b(k)p FJ(\003)29853 22014 y FC(ij)30661 22562 y FA(k)31325 22014 y FB(2)31325 23668 y FC(H)32166 22961 y Ft(\013;\033)24 b Fu(+)34206 22654 y(2)p Ft(\027)45 b Fu(+1)p 34206 22811 1892 33 v 34409 23271 a Ft(\027)g Fu(+1)32098 24152 y Fo(L)p Fu(\()p Ft(`)33330 23951 y(a;s)34473 24152 y(;`)35069 23951 y(a;s)p Fu(+1)37241 24152 y(\))37893 22562 y FD(:)0 26731 y FJ(In)433 b(the)g(same)h(w)-36 b(a)g(y)8080 28613 y Fz(X)8783 31442 y FC(k)10221 29875 y FD(e)10824 29327 y FB(2)p Fw(j)p FC(`)p Fw(\000)p FC(k)24 b Fw(j)p FC(\013)14475 29875 y FA(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)18103 29293 y FB(2\()p FC(\033)32 b FB(+)20374 28919 y Fu(2)p Ft(\027)45 b Fu(+1)p 20374 29091 1892 40 v 20577 29640 a Ft(\027)g Fu(+1)22398 29293 y FB(\))22819 29875 y FA(k)p FJ(\003)24386 29245 y FC(ij)24386 30245 y(k)24 b FB(+)p FC(`)26071 29875 y FA(k)26735 29327 y FB(2)26735 30204 y FC(s;s)p FB(+1)13220 33413 y FA(\024)15361 32151 y Fz(X)16063 34979 y FC(k)17501 33413 y FD(e)18104 32864 y FB(2)p Fw(j)p FC(k)g FB(+)p FC(`)p Fw(j)p FC(\013)21386 33413 y FA(j)p FD(k)340 b FJ(+)295 b FD(`)p FA(j)24993 32830 y FB(2\()p FC(\033)32 b FB(+)27264 32457 y Fu(2)p Ft(\027)45 b Fu(+1)p 27263 32629 V 27466 33177 a Ft(\027)g Fu(+1)29287 32830 y FB(\))29709 33413 y FA(k)p FJ(\003)31276 32783 y FC(ij)31276 33782 y(k)24 b FB(+)p FC(`)32961 33413 y FA(k)33625 32864 y FB(2)33625 33741 y FC(s;s)p FB(+1)13231 36689 y FJ(=)1118 b FA(k)p FJ(\003)16928 36141 y FC(ij)17735 36689 y FA(k)18399 36141 y FB(2)18399 37795 y FC(H)19240 37088 y Ft(\013;\033)24 b Fu(+)21280 36781 y(2)p Ft(\027)45 b Fu(+1)p 21280 36938 1892 33 v 21483 37398 a Ft(\027)g Fu(+1)19172 38278 y Fo(L)p Fu(\()p Ft(`)20404 38078 y(a;s)21548 38278 y(;`)22144 38078 y(a;s)p Fu(+1)24315 38278 y(\))24967 36689 y FD(:)0 40843 y FJ(Hence)8694 41828 y Fz(\015)8694 42625 y(\015)9432 42957 y FD(T)10375 42409 y FB(\()p FC(n)p FB(\))11733 42957 y FD(w)12699 41828 y Fz(\015)12699 42625 y(\015)13436 42179 y FB(2)13436 43478 y FC(\013;\033)15296 42957 y FA(\024)369 b FJ(const)20039 42059 y FD(\034)20752 41577 y FB(2)p 20022 42652 1273 54 v 20022 43868 a FD(\015)20769 43485 y FB(2)21649 42957 y FA(k)p FJ(\003)23216 42409 y FC(ij)24023 42957 y FA(k)24687 42409 y FB(2)24687 44064 y FC(H)25528 43356 y Ft(\013;\033)24 b Fu(+)27569 43049 y(2)p Ft(\027)45 b Fu(+1)p 27569 43206 1892 33 v 27771 43666 a Ft(\027)g Fu(+1)25460 44547 y Fo(L)p Fu(\()p Ft(`)26692 44346 y(a;s)27836 44547 y(;`)28432 44346 y(a;s)p Fu(+1)30603 44547 y(\))31034 42957 y FA(k)p FD(w)36 b FA(k)33328 42409 y FB(2)33328 43286 y FC(\013;\033)35040 42957 y FD(:)4719 b FJ(\(5.168\))0 46600 y(The)434 b(thesis)f(follo)-36 b(ws)436 b(from)e(estimate)f(\(5.76\))i(and)e(Prop)36 b(osition)434 b(5.4.2.)43061 48222 y Fn(\003)0 52766 y FH(5.9)2152 b(Small)716 b(divisors)0 55722 y FJ(Let)433 b(b)36 b(e)433 b FD(i)4544 55921 y FB(0)5504 55722 y FJ(and)g FD(j)8571 55921 y FB(0)9530 55722 y FJ(the)g(smallest)h(in)-36 b(tegers)434 b(suc)-36 b(h)433 b(that)11871 59397 y(min)12795 60204 y FC(i)14260 57471 y Fz(\014)14260 58268 y(\014)14260 59065 y(\014)14260 59862 y(\014)14836 58499 y FD(k)p 14836 59092 722 54 v 14840 60309 a(\034)15985 59397 y FA(\000)295 b FD(\025)18072 59596 y FC(k)24 b(i)18961 57471 y Fz(\014)18961 58268 y(\014)18961 59065 y(\014)18961 59862 y(\014)19773 59397 y FJ(=)21154 57471 y Fz(\014)21154 58268 y(\014)21154 59065 y(\014)21154 59862 y(\014)21729 58499 y FD(k)p 21729 59092 V 21733 60309 a(\034)22878 59397 y FA(\000)296 b FD(\025)24966 59596 y FC(k)24 b(i)25800 59719 y Fu(0)26316 57471 y Fz(\014)26316 58268 y(\014)26316 59065 y(\014)26316 59862 y(\014)27128 59397 y FJ(=)368 b FA(j)p FD(\026)29660 59596 y FC(k)24 b(;i)30755 59719 y Fu(0)31272 59397 y FA(j)221 b FD(;)7897 b FJ(\(5.169\))0 63073 y(and)11957 65162 y(min)12825 65968 y FC(j)14346 63236 y Fz(\014)14346 64033 y(\014)14346 64830 y(\014)14346 65627 y(\014)15005 64263 y FD(`)p 14922 64856 714 54 v 14922 66073 a(\034)16063 65162 y FA(\000)295 b FD(\025)18150 65361 y FC(`j)19021 63236 y Fz(\014)19021 64033 y(\014)19021 64830 y(\014)19021 65627 y(\014)19833 65162 y FJ(=)21214 63236 y Fz(\014)21214 64033 y(\014)21214 64830 y(\014)21214 65627 y(\014)21873 64263 y FD(`)p 21789 64856 V 21789 66073 a(\034)22931 65162 y FA(\000)g FD(\025)25018 65361 y FC(`j)25783 65484 y Fu(0)26299 63236 y Fz(\014)26299 64033 y(\014)26299 64830 y(\014)26299 65627 y(\014)27111 65162 y FJ(=)368 b FA(j)p FD(\026)29643 65361 y FC(`;j)30669 65484 y Fu(0)31186 65162 y FA(j)221 b FD(:)7983 b FJ(\(5.170\))0 68650 y FE(Remark)581 b(5.9.1.)591 b Fs(If)530 b FD(i)11699 68849 y FB(0)12716 68650 y FJ(=)490 b(1)531 b Fs(then)g FD(k)536 b(<)490 b FJ(3)p FD(\034)148 b Fs(.)796 b(Inde)-66 b(e)g(d,)545 b(by)531 b(c)-66 b(ontr)g(adiction,)544 b(if)530 b FD(k)536 b(>)490 b FJ(3)p FD(\034)148 b Fs(,)0 70255 y(then)475 b FA(j)p FD(k)45 b(=\034)452 b FA(\000)303 b FD(\025)7752 70454 y FC(k)24 b FB(1)8791 70255 y FA(j)389 b(\025)g FJ(2)475 b Fs(and,)j(for)d FD(i)17562 69773 y Fw(\003)18477 70255 y FA(\031)389 b FD(k)45 b(=\034)148 b Fs(,)478 b FA(j)p FD(k)45 b(=\034)452 b FA(\000)303 b FD(\025)27712 70454 y FC(k)24 b(i)28546 70203 y Fo(\003)29081 70255 y FA(j)388 b(\024)h FJ(1)p Fs(,)478 b(namely)d FD(i)37725 69773 y Fw(\003)38726 70255 y Fs(is)h(a)g(p)-66 b(oint)0 71861 y(in)464 b(which)h(it)f(takes)h(a)g(value)g(smal)66 b(ler)466 b(than)e(the)h(minimum.)595 b(If)464 b FD(i)32244 72060 y FB(0)33139 71861 y FD(>)368 b FJ(1)466 b Fs(then)e FD(i)38968 72060 y FB(0)39862 71861 y FA(\031)369 b FD(k)45 b(=\034)148 b Fs(.)0 74659 y FE(Remark)498 b(5.9.2.)555 b Fs(If)464 b FD(k)414 b(>)369 b(\034)148 b(=)p FJ(4)p Fs(,)465 b(then)f FD(i)19751 74858 y FB(0)20646 74659 y FA(\031)369 b FD(k)45 b(=\034)148 b Fs(.)21072 78312 y FJ(194)p eop %%Page: 195 195 195 194 bop 6299 4488 a FF(5.9.1)1793 b(Some)598 b(tec)-50 b(hnical)599 b(lemmata)6299 6951 y FE(Lemma)499 b(5.9.3.)555 b Fs(L)-66 b(et)463 b(b)-66 b(e)464 b FJ(1)370 b FD(<)e(\027)453 b(<)368 b FJ(2)p Fs(,)465 b FD($)405 b FA(2)368 b FJ(\001)29679 6469 y FC(\027)29679 7280 y(n)30770 6951 y Fs(and)465 b FD(`=\034)518 b FA(\025)369 b FJ(1)p FD(=)p FJ(4)p Fs(.)598 b(Then)20011 8601 y Fz(\014)20011 9398 y(\014)20011 10195 y(\014)20011 10992 y(\014)20671 9628 y FD(`)p 20587 10221 714 54 v 20587 11438 a(\034)21728 10527 y FA(\000)295 b FD(j)23669 8601 y Fz(\014)23669 9398 y(\014)23669 10195 y(\014)23669 10992 y(\014)24481 10527 y FA(\025)369 b FJ(cost)28263 9628 y FD(\034)28976 9146 y FC(\027)58 b Fw(\000)p FB(1)p 28263 10221 2492 54 v 28948 11438 a FD(`)29494 11054 y FC(\027)31109 10527 y FD(;)1616 b(j)444 b FA(6)p FJ(=)369 b(0)221 b FD(:)6299 14077 y Fp(pr)-30 b(oof.)659 b FJ(If)434 b FD(`)369 b(>)f(\034)148 b(=)p FJ(4,)435 b FD(j)444 b FA(\031)369 b FD(`=\034)582 b FJ(then,)433 b(for)h(the)f(Melnik)-36 b(o)g(v)434 b(condition)g(in)f (\(5.5.3\))15891 15726 y Fz(\014)15891 16523 y(\014)15891 17320 y(\014)15891 18117 y(\014)16551 16753 y FD(`)p 16467 17347 714 54 v 16467 18563 a(\034)17608 17652 y FA(\000)296 b FD(j)19550 15726 y Fz(\014)19550 16523 y(\014)19550 17320 y(\014)19550 18117 y(\014)20361 17652 y FA(\025)21896 16753 y FD(\015)p 21896 17347 747 54 v 21913 18563 a(\034)22776 17652 y(j)23389 17104 y FC(\027)24333 17652 y FA(\025)370 b FJ(const)221 b FD(\015)30079 16753 y(\034)30792 16271 y FC(\027)p 29807 17347 1835 54 v 29807 18563 a FD(\034)148 b(`)31066 18179 y FC(\027)32143 17652 y FJ(=)369 b(const)221 b FD(\015)37595 16753 y(\034)38308 16271 y FC(\027)58 b Fw(\000)p FB(1)p 37595 17347 2492 54 v 38280 18563 a FD(`)38826 18179 y FC(\027)40440 17652 y FD(:)49361 21102 y Fn(\003)6299 23688 y FE(Lemma)499 b(5.9.4.)555 b Fs(L)-66 b(et)463 b(b)-66 b(e)464 b FA(jD)20865 23887 y FC(`)21305 23688 y FA(j)21674 23206 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))25310 23688 y Fs(as)465 b(in)f FJ(\(5.150\))p Fs(.)599 b(Then)10444 27727 y Fz(\015)10444 28524 y(\015)11182 28857 y FA(jD)12576 29056 y FC(`)13015 28857 y FA(j)13384 28308 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))16555 27727 y Fz(\015)16555 28524 y(\015)17293 29377 y FC(s;s)18849 28857 y FA(\024)369 b FJ(max)22893 25336 y Fz(8)22893 26532 y(>)22893 26930 y(>)22893 27329 y(<)22893 29720 y(>)22893 30119 y(>)22893 30517 y(:)27404 26458 y FD(`)27950 25976 y FC(\027)58 b FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))p 24760 27052 8793 54 v 24760 28268 a FD(\015)25507 27885 y FB(\(1)p Fw(\000)p FC(\016)g FB(\))27946 28268 y FD(\034)28659 27885 y FB(\()p FC(\027)58 b Fw(\000)p FB(1\)\(1)p Fw(\000)p FC(\016)33 b FB(\))33686 27357 y FD(;)1550 b Fs(if)464 b FD(`)369 b(>)f(\034)148 b(=)p FJ(2)p Fs(;)24628 31255 y FJ(const)o FD(;)7639 b Fs(if)464 b FD(`)369 b FA(\024)g FD(\034)148 b(=)p FJ(2)p Fs(.)46419 28857 y FJ(\(5.171\))6299 34175 y Fp(pr)-30 b(oof.)1011 b FJ(F)-108 b(rom)543 b(the)g (de\014nition)f(of)i(the)e(op)36 b(erator)544 b FA(jD)34091 34374 y FC(`)34530 34175 y FA(j)34899 33693 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))38613 34175 y FJ(in)543 b(\(5.150\),)572 b(it)544 b(follo)-36 b(ws)6299 35780 y(that)18107 37385 y FA(kjD)20165 37584 y FC(`)20604 37385 y FA(j)20973 36837 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))24144 37385 y FA(k)24808 37584 y FC(s;s)26364 37385 y FA(\024)369 b FJ(max)28761 38192 y FC(j)30408 36309 y Fz(\010)31183 37385 y FA(j)p FD(\026)32335 37584 y FC(`;j)33467 37385 y FA(j)33836 36837 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))37007 36309 y Fz(\011)38225 37385 y FD(:)7833 b FJ(\(5.172\))6299 40260 y(Moreo)-36 b(v)g(er,)435 b(from)f(the)f FK(\014rst)f(Melnik)-36 b(o)g(v)435 b(condition)e FJ(in)h(\(5.5.3\),)h(w)-36 b(e)434 b(ha)-36 b(v)g(e)22633 43355 y FA(j)p FD(\026)23785 43554 y FC(`;j)24917 43355 y FA(j)369 b(\025)h(j)p FD(\026)28210 43554 y FC(`;j)29236 43677 y Fu(0)29752 43355 y FA(j)f(\025)32602 42456 y FD(\015)p 32025 43049 1902 54 v 32025 44266 a(\034)148 b(j)33351 43808 y FC(\027)33276 44592 y FB(0)6299 46880 y FJ(for)434 b FD(j)8824 47079 y FB(0)9719 46880 y FD(>)369 b FJ(1)434 b(and)f(using)g(that)g FD(j)21472 47079 y FB(0)22367 46880 y FA(\031)369 b FD(`=\034)148 b FJ(,)435 b(w)-36 b(e)433 b(get)h(the)f(estimate)19935 50548 y FA(j)p FD(\026)21087 50747 y FC(`;j)22219 50548 y FA(j)22588 50000 y FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))25396 50548 y FA(\025)29894 49650 y FD(\015)30641 49168 y FB(\(1)p Fw(\000)p FC(\016)g FB(\))p 26931 50243 9113 54 v 26931 51460 a FD(\034)27644 51076 y FB(\(1)p Fw(\000)p FC(\027)58 b FB(\)\(1)p Fw(\000)p FC(\016)33 b FB(\))32538 51460 y FD(`)33084 51076 y FC(\027)58 b FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))36397 50548 y FD(:)9661 b FJ(\(5.173\))6299 53748 y(In)434 b(particular)f(if)h FD(`)369 b FA(\024)g FD(\034)148 b(=)p FJ(2)435 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)20527 57416 y FA(j)p FD(\026)21679 57615 y FC(`;j)22811 57416 y FA(j)23180 56868 y FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))25988 57416 y FA(\025)27523 56518 y FD(\015)28270 56036 y FB(\(1)p Fw(\000)p FC(\016)g FB(\))p 27523 57111 3186 54 v 27540 58328 a FD(\034)28253 57944 y FB(\(1)p Fw(\000)p FC(\016)g FB(\))31211 57416 y FA(\025)369 b FJ(const)221 b FD(:)10254 b FJ(\(5.174\))6299 60616 y(F)-108 b(rom)433 b(\(5.173\))i(and)e(\(5.174\),)i(the)e(thesis)h(follo)-36 b(ws.)17878 b Fn(\003)6299 63202 y FE(Lemma)499 b(5.9.5.)555 b Fs(Supp)-66 b(ose)465 b(that)f FD(`)369 b(>)g(\034)148 b(=)p FJ(2)466 b Fs(and)17277 64851 y Fz(\014)17277 65648 y(\014)17277 66445 y(\014)17277 67242 y(\014)17937 65878 y FD(`)p 17853 66471 714 54 v 17853 67688 a(\034)18994 66777 y FA(\000)296 b FD(j)20861 66976 y FB(0)21386 64851 y Fz(\014)21386 65648 y(\014)21386 66445 y(\014)21386 67242 y(\014)22198 66777 y FA(\025)369 b FJ(const)26924 65878 y FD(\034)27637 65396 y FC(\027)58 b Fw(\000)p FB(1)p 26924 66471 2492 54 v 27323 67688 a FD(`)27869 67304 y FC(\027)g(\014)29770 66777 y FD(;)2878 b(\014)443 b(<)35700 65878 y(\027)378 b FA(\000)296 b FJ(1)p 35700 66471 3001 54 v 36837 67688 a FD(\027)39054 66777 y(:)6299 70253 y Fs(Then,)464 b(the)h(fol)66 b(lowing)464 b(estimate)g(holds) 16733 72831 y FJ(1)p 16733 73424 651 54 v 16733 74641 a(2)17737 71803 y Fz(\014)17737 72600 y(\014)17737 73397 y(\014)17737 74194 y(\014)18396 72831 y FD(`)p 18313 73424 714 54 v 18313 74641 a(\034)19454 73729 y FA(\000)295 b FD(j)21320 73928 y FB(0)21846 71803 y Fz(\014)21846 72600 y(\014)21846 73397 y(\014)21846 74194 y(\014)22658 73729 y FA(\025)24334 72831 y FD(\026)p 24193 73424 1064 54 v 24193 74641 a(j)24731 74840 y FB(0)25611 73729 y FD(;)5535 b FJ(if)1422 b FD(\014)443 b(<)36377 72831 y(\027)379 b FA(\000)295 b FJ(1)p 36377 73424 3001 54 v 37514 74641 a FD(\027)39732 73729 y(:)6326 b FJ(\(5.175\))27371 78312 y(195)p eop %%Page: 196 196 196 195 bop 0 4488 a Fp(pr)-30 b(oof.)658 b FJ(W)-108 b(e)434 b(w)-36 b(an)g(t)433 b(to)h(sho)-36 b(w)434 b(that)17231 7936 y(const)20556 7038 y FD(\034)21269 6556 y FC(\027)58 b Fw(\000)p FB(1)p 20556 7631 2492 54 v 20954 8848 a FD(`)21500 8464 y FC(\027)g(\014)23549 7936 y FA(\025)25224 7038 y FD(\026)p 25084 7631 1064 54 v 25084 8848 a(j)25622 9047 y FB(0)26502 7936 y FD(:)13257 b FJ(\(5.176\))0 11599 y(It)434 b(is)f(true,)h(since)f FD(\014)443 b(<)369 b FJ(\()p FD(\027)378 b FA(\000)296 b FJ(1\))p FD(=\027)84 b FJ(,)433 b FD(`)369 b FA(\025)g FD(\034)148 b(=)p FJ(4)435 b(and)e(b)-36 b(y)433 b(Remark)i(5.9.2.)579 b(Indeed)10653 14525 y FD(\034)11366 14042 y FC(\027)58 b Fw(\000)p FB(1)p 10653 15118 2492 54 v 11051 16334 a FD(`)11597 15951 y FC(\027)g(\014)13646 15423 y FA(\031)15181 14525 y FD(\034)15894 14042 y FC(\027)g(\014)p 15181 15118 1863 54 v 15231 16522 a FD(j)15844 15892 y FC(\027)g(\014)15769 16847 y FB(0)17545 15423 y FJ(=)369 b(const)22250 14525 y FD(\034)22963 14042 y FC(\027)58 b Fw(\000)p FB(1)p Fw(\000)p FC(\027)g(\014)p 22250 15118 4318 54 v 23527 16522 a FD(j)24140 15892 y FC(\027)g(\014)24065 16847 y FB(0)27069 15423 y FA(\025)369 b FJ(const)31936 14525 y FD(\026)p 31795 15118 1064 54 v 31795 16334 a(j)32333 16533 y FB(0)33213 15423 y FD(;)0 19267 y FJ(since)434 b FD(\027)84 b(\014)442 b(<)369 b(\027)378 b FA(\000)296 b FJ(1)433 b(implies)h FD(\027)84 b(\014)443 b(<)368 b FJ(1,)435 b(for)f(1)369 b FD(<)g(\027)452 b(<)369 b FJ(2.)43061 20891 y Fn(\003)0 23699 y FE(Lemma)499 b(5.9.6.)554 b Fs(The)465 b(fol)66 b(lowing)465 b(estimate)e(holds:)5272 29838 y FA(j)p FD(\026)6424 30037 y FC(k)24 b(;)p FB(1)7724 29838 y FA(j)8093 29290 y FC(\016)8967 29838 y FJ(=)10347 27912 y Fz(\014)10347 28709 y(\014)10347 29506 y(\014)10347 30303 y(\014)10923 28940 y FD(k)p 10923 29533 722 54 v 10927 30750 a(\034)12072 29838 y FA(\000)295 b FD(\025)14159 30037 y FC(k)24 b FB(1)15198 27912 y Fz(\014)15198 28709 y(\014)15198 29506 y(\014)15198 30303 y(\014)15641 28263 y FC(\016)16515 29838 y FA(\025)17917 25521 y Fz(8)17917 26717 y(>)17917 27115 y(>)17917 27514 y(>)17917 27912 y(>)17917 28311 y(<)17917 30702 y(>)17917 31100 y(>)17917 31499 y(>)17917 31897 y(>)17917 32296 y(:)19651 26617 y FD(k)20372 26069 y FC(\016)20877 26617 y FD(=\034)22240 26069 y FC(\016)22746 26617 y FD(;)1797 b Fs(if)464 b FD(k)414 b FA(\025)369 b FJ(2)p FD(\034)148 b Fs(;)19651 29867 y FD(\015)20398 29319 y FC(\016)20903 29867 y FD(=\034)22266 29319 y FC(\016)22772 29867 y FD(;)1771 b Fs(if)464 b FD(\034)148 b(=)p FJ(2)370 b FD(<)f(k)414 b(<)368 b FJ(2)p FD(\034)148 b Fs(;)19651 33077 y FJ(const)p FD(;)1922 b Fs(if)464 b FD(k)414 b FA(\024)369 b FD(\034)148 b(=)p FJ(2)p Fs(.)40120 29838 y FJ(\(5.177\))0 36071 y Fp(pr)-30 b(oof.)789 b FJ(W)-108 b(e)474 b(ha)-36 b(v)g(e)474 b(that,)484 b(for)475 b FD(k)483 b FA(\024)438 b FD(\034)148 b(=)p FJ(2,)486 b FD(k)45 b(=\034)622 b FJ(is)475 b(at)f(most)g(1)p FD(=)p FJ(2)h(hence)f FA(j)p FD(\026)37185 36270 y FC(k)24 b(;)p FB(1)38485 36071 y FA(j)438 b(\025)g FJ(const.)0 37676 y(F)-108 b(or)562 b FD(\034)148 b(=)p FJ(2)588 b FD(<)g(k)633 b(<)587 b FJ(2)p FD(\034)148 b FJ(,)595 b(w)-36 b(e)563 b(can)f(use)g(the)g(diophan)-36 b(tine)561 b(estimate)h(in)g(\(5.145\))i(and)d(w)-36 b(e)0 39281 y(obtain)487 b FA(j)p FD(\026)5252 39480 y FC(k)24 b(;)p FB(1)6552 39281 y FA(j)459 b(\025)g FD(\015)72 b(=\034)148 b FJ(.)738 b(Finally)-108 b(,)501 b(for)487 b FD(k)505 b(>)459 b FJ(2)p FD(\034)148 b FJ(,)500 b(b)36 b(eing)487 b FD(\025)28059 39480 y FC(k)24 b FB(1)29585 39281 y FJ(negligible)487 b(with)g(resp)36 b(ect)486 b(to)0 40886 y FD(k)45 b(=\034)148 b FJ(,)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b FA(j)p FD(\026)8945 41085 y FC(k)24 b(;)p FB(1)10245 40886 y FA(j)369 b(\025)g FJ(const)222 b FD(k)45 b(=\034)148 b FJ(.)579 b(Therefore)434 b(\(5.177\))h(follo)-36 b(ws.)43061 42511 y Fn(\003)0 44135 y FJ(Let)433 b(b)36 b(e)433 b FD(i)4544 44334 y FB(0)5504 44135 y FJ(suc)-36 b(h)432 b(that)h(\(5.169\))i(holds.)578 b(W)-108 b(e)434 b(ha)-36 b(v)g(e)434 b(that)9550 47890 y FA(k)p FD(M)11611 47341 y Fw(\000)p FB(1)12869 47890 y FA(jD)14263 48089 y FC(k)14831 47890 y FA(j)15200 47341 y Fw(\000)p FC(\016)16437 47890 y FA(k)17101 48089 y FC(s;s)18657 47890 y FA(\024)369 b FJ(max)22701 46017 y Fz(\032)25809 46991 y FJ(1)p 23830 47584 4607 54 v 23830 48801 a FD(i)24274 49000 y FB(0)24800 48801 y FA(j)p FD(\026)25952 49000 y FC(k)24 b(;i)27047 49123 y Fu(0)27563 48801 y FA(j)27932 48417 y FC(\016)28570 47890 y FD(;)30623 46991 y FJ(1)p 29285 47584 3327 54 v 29285 48801 a FA(j)p FD(\026)30437 49000 y FC(k)g(;)p FB(1)31738 48801 y FA(j)32107 48417 y FC(\016)32744 46017 y Fz(\033)34183 47890 y FD(:)5576 b FJ(\(5.178\))0 51684 y FE(Lemma)499 b(5.9.7.)554 b Fs(L)-66 b(et)464 b(b)-66 b(e)464 b FD(i)13616 51883 y FB(0)14607 51684 y Fs(such)h(that)g FJ(\(5.169\))h Fs(holds.)598 b(Then)4388 55649 y Fz(\015)4388 56446 y(\015)4388 57243 y(\015)5126 57177 y FD(M)6523 56628 y Fw(\000)p FB(1)7781 57177 y FA(jD)9175 57376 y FC(k)9743 57177 y FA(j)10112 56628 y Fw(\000)p FC(\016)11349 55649 y Fz(\015)11349 56446 y(\015)11349 57243 y(\015)12087 58095 y FC(s;s)13643 57177 y FA(\024)15045 53656 y Fz(8)15045 54852 y(>)15045 55250 y(>)15045 55649 y(<)15045 58040 y(>)15045 58439 y(>)15045 58837 y(:)16779 55704 y FJ(const)20120 54805 y FD(\034)20833 54323 y FC(\016)p 20103 55398 1252 54 v 20103 56615 a FD(\015)20850 56231 y FC(\016)21488 53831 y Fz(\022)22602 54805 y FD(\034)p 22599 55398 722 54 v 22599 56615 a(k)23452 53831 y Fz(\023)24430 54128 y FB(1)p Fw(\000)p FC(\027)58 b(\016)26879 55704 y FD(;)3321 b(k)414 b(>)369 b(\034)148 b(=)p FJ(2)16779 59694 y(const)221 b FD(;)6864 b(k)414 b FA(\024)369 b FD(\034)148 b(=)p FJ(2)221 b FD(:)40120 57177 y FJ(\(5.179\))0 62688 y Fp(pr)-30 b(oof.)1111 b FJ(If)575 b FD(k)653 b FA(\024)609 b FD(\034)148 b(=)p FJ(2)575 b(then)e FD(i)15992 62887 y FB(0)17126 62688 y FJ(=)608 b(1,)i(hence)574 b(from)g(\(5.177\))i (the)d(thesis)i(follo)-36 b(ws.)1002 b(If)0 64293 y FD(k)414 b(>)369 b(\034)148 b(=)p FJ(2,)434 b(from)g(\(5.178\),)h(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b(to)g(sho)-36 b(w)434 b(that)15606 67414 y(1)p 13628 68007 4607 54 v 13628 69223 a FD(i)14072 69422 y FB(0)14597 69223 y FA(j)p FD(\026)15749 69422 y FC(k)24 b(;i)16844 69545 y Fu(0)17360 69223 y FA(j)17729 68840 y FC(\016)18736 68312 y FA(\024)369 b FJ(const)23479 67414 y FD(\034)24192 66932 y FC(\016)p 23462 68007 1252 54 v 23462 69223 a FD(\015)24209 68840 y FC(\016)24847 66439 y Fz(\022)25962 67414 y FD(\034)p 25958 68007 722 54 v 25958 69223 a(k)26812 66439 y Fz(\023)27789 66737 y FB(1)p Fw(\000)p FC(\027)58 b(\016)30238 68312 y FD(;)9521 b FJ(\(5.180\))0 72014 y(and)15606 73474 y(1)p 14268 74067 3327 54 v 14268 75284 a FA(j)p FD(\026)15420 75483 y FC(k)24 b(;)p FB(1)16720 75284 y FA(j)17089 74900 y FC(\016)18096 74373 y FA(\024)369 b FJ(const)22839 73474 y FD(\034)23552 72992 y FC(\016)p 22822 74067 1252 54 v 22822 75284 a FD(\015)23569 74900 y FC(\016)24207 72500 y Fz(\022)25321 73474 y FD(\034)p 25317 74067 722 54 v 25317 75284 a(k)26171 72500 y Fz(\023)27149 72797 y FB(1)p Fw(\000)p FC(\027)58 b(\016)29598 74373 y FD(:)10161 b FJ(\(5.181\))21072 78312 y(196)p eop %%Page: 197 197 197 196 bop 6299 4488 a FJ(W)-108 b(e)560 b(note)f(that)f(\(5.180\))j (and)e(\(5.181\))h(coincide)g(if)g FD(i)33220 4687 y FB(0)34328 4488 y FJ(=)583 b(1.)956 b(In)559 b(the)g(case)h FD(i)45330 4687 y FB(0)46438 4488 y FJ(=)583 b(1,)591 b(if)6299 6094 y FD(\034)148 b(=)p FJ(2)563 b FA(\024)f FD(k)607 b FA(\024)563 b FJ(2)p FD(\034)148 b FJ(,)576 b(\(5.181\))548 b(directly)f(follo)-36 b(ws)549 b(from)f(\(5.177\).)920 b(F)-108 b(or)546 b FD(k)607 b(>)562 b FJ(2)p FD(\034)148 b FJ(,)576 b(w)-36 b(e)547 b(ha)-36 b(v)g(e,)6299 7699 y(using)434 b(that)f FD(\016)419 b(>)368 b FJ(1)p FD(=)p FJ(2)i FD(>)f FJ(1)295 b FA(\000)h FD(\027)84 b(\016)483 b FJ(and)433 b(from)h(\(5.177\),)19889 11671 y(const)23217 10772 y FD(\034)23930 10290 y FC(\016)p 23213 11365 1226 54 v 23213 12582 a FD(k)23934 12198 y FC(\016)24941 11671 y FA(\024)369 b FJ(const)29684 10772 y FD(\034)30397 10290 y FC(\016)p 29667 11365 1252 54 v 29667 12582 a FD(\015)30414 12198 y FC(\016)31051 9798 y Fz(\022)32166 10772 y FD(\034)p 32162 11365 722 54 v 32162 12582 a(k)33016 9798 y Fz(\023)33994 10095 y FB(1)p Fw(\000)p FC(\027)58 b(\016)36443 11671 y FD(;)6299 15378 y FJ(from)434 b(whic)-36 b(h)434 b(\(5.181\))g(follo)-36 b(ws.)6299 16987 y(If)434 b FD(i)8043 17186 y FB(0)8938 16987 y FD(>)369 b FJ(1,)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)433 b(that)g FD(i)19939 17186 y FB(0)20834 16987 y FA(\031)369 b FD(k)45 b(=\034)148 b FJ(,)435 b(therefore,)8960 20959 y FD(i)9404 21158 y FB(0)9930 20959 y FA(j)p FD(\026)11082 21158 y FC(k)24 b(;i)12177 21281 y Fu(0)12693 20959 y FA(j)13062 20410 y FC(\016)13936 20959 y FJ(=)369 b FD(i)15761 21158 y FB(0)16286 19033 y Fz(\014)16286 19830 y(\014)16286 20627 y(\014)16286 21424 y(\014)16862 20060 y FD(k)p 16862 20653 V 16866 21870 a(\034)18011 20959 y FA(\000)295 b FD(\025)20098 21158 y FC(k)24 b(i)20932 21281 y Fu(0)21448 19033 y Fz(\014)21448 19830 y(\014)21448 20627 y(\014)21448 21424 y(\014)21891 19383 y FC(\016)22765 20959 y FA(\025)369 b FD(i)24611 21158 y FB(0)25137 19086 y Fz(\022)26740 20060 y FD(\015)p 26247 20653 1733 54 v 26247 21870 a(\034)148 b(i)27404 21412 y FC(\027)27404 22196 y FB(0)28113 19086 y Fz(\023)29091 19383 y FC(\016)29964 20959 y FJ(=)32797 20060 y FD(\015)33544 19578 y FC(\016)p 31478 20653 3890 54 v 31478 21977 a FD(i)31922 21428 y FC(\027)58 b(\016)33 b Fw(\000)p FB(1)31922 22303 y(0)34149 21977 y FD(\034)34862 21594 y FC(\016)35869 20959 y FA(\025)369 b FJ(const)40595 20060 y FD(\015)41342 19578 y FC(\016)p 40595 20653 1252 54 v 40612 21870 a FD(\034)41325 21486 y FC(\016)41980 19086 y Fz(\022)43095 20060 y FD(\034)p 43091 20653 722 54 v 43091 21870 a(k)43945 19086 y Fz(\023)44922 19383 y FC(\027)58 b(\016)33 b Fw(\000)p FB(1)47371 20959 y FD(;)6299 24749 y FJ(that)433 b(is)h(just)g(\(5.180\).)49361 26357 y Fn(\003)6299 29087 y FE(Lemma)573 b(5.9.8.)587 b Fs(Be)524 b FD(`;)221 b(k)523 b FA(\025)479 b FJ(1)524 b Fs(such)g(that)g FA(j)p FD(`)338 b FA(\000)h FD(k)45 b FA(j)478 b FD(<)g FJ([max)222 b(\()p FD(`;)f(k)45 b FJ(\)])40428 28604 y FC(\014)41581 29087 y Fs(and)524 b FD(`;)221 b(k)524 b(>)477 b(\034)148 b(=)p FJ(2)p Fs(.)6299 30692 y(Then)23172 32307 y FA(j)p FD(`)295 b FA(\000)h FD(k)45 b FA(j)369 b(\024)g FJ(const)221 b FD(`)32309 31759 y FC(\014)33159 32307 y FD(:)6299 35036 y Fp(pr)-30 b(oof.)772 b FJ(If)470 b FD(k)k FA(\024)430 b FD(`)469 b FJ(then)f FA(j)p FD(`)319 b FA(\000)g FD(k)45 b FA(j)429 b FD(<)g(`)25459 34554 y FC(\014)26088 35036 y FJ(.)684 b(If)470 b FD(k)k(>)429 b(`)p FJ(,)478 b(then)468 b FA(j)p FD(`)319 b FA(\000)h FD(k)45 b FA(j)429 b FJ(=)g FD(k)364 b FA(\000)320 b FD(`)429 b FA(\024)g FD(k)46541 34554 y FC(\014)47170 35036 y FJ(,)478 b(that)6299 36641 y(is)446 b FD(k)348 b FA(\000)303 b FD(k)10700 36159 y FC(\014)11718 36641 y FA(\024)389 b FD(`)p FJ(.)613 b(Moreo)-36 b(v)g(er)445 b FD(k)349 b FA(\000)303 b FD(k)23461 36159 y FC(\014)24478 36641 y FA(\025)389 b FD(k)45 b(=)p FJ(2)446 b(if)g FD(k)434 b(>)388 b(\034)148 b(=)p FJ(2.)614 b(So)446 b(that)e FD(k)434 b FA(\024)389 b FJ(2)p FD(`)445 b FJ(de\014nitiv)-36 b(ely)-108 b(.)6299 38246 y(Then,)434 b(\(5.9.8\))g(directly)g(follo) -36 b(ws.)26257 b Fn(\003)6299 40976 y FE(Remark)499 b(5.9.9.)554 b Fs(Note)464 b(that,)g(for)h FD(`;)221 b(k)415 b FA(\025)369 b FJ(1)p Fs(,)464 b FD(`)369 b FA(6)p FJ(=)g FD(k)45 b Fs(,)464 b(the)h(fol)66 b(lowing)465 b(estimate)e(holds)19962 43680 y FD(`)p 19875 44273 V 19875 45490 a(k)21097 44578 y FA(\024)22632 43680 y(j)p FD(`)295 b FA(\000)h FD(k)45 b FA(j)295 b FJ(+)g FD(k)p 22632 44273 5952 54 v 25248 45490 a(k)29086 44578 y FA(\024)369 b FJ(1)295 b(+)g FA(j)p FD(`)g FA(\000)g FD(k)45 b FA(j)221 b FD(;)6299 47861 y Fs(henc)-66 b(e)24757 48905 y FD(`)p 24670 49498 722 54 v 24670 50715 a(k)25893 49803 y FA(\024)369 b FJ(2)p FA(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)221 b FD(:)14264 b FJ(\(5.182\))6299 53812 y FF(5.9.2)1793 b(Pro)50 b(of)597 b(of)g(Lemma)i(5.8.4)6299 56282 y FJ(Recall)435 b(\(5.179\))g(and)e(\(5.171\).)580 b(W)-108 b(e)433 b(distinguish)g (four)h(di\013eren)-36 b(t)432 b(cases.)6299 57891 y Fp(First)503 b(case)p FJ(:)578 b FA(j)p FD(`)295 b FA(\000)h FD(k)45 b FA(j)369 b(\025)g FJ([max)222 b(\()p FD(`;)f(k)45 b FJ(\)])25855 57409 y FC(\014)26701 57891 y FJ(.)6299 59499 y(If)434 b FD(`)369 b(>)g(\034)148 b(=)p FJ(2,)435 b(w)-36 b(e)433 b(ha)-36 b(v)g(e,)434 b(from)g(\(5.179\))h(and)e (\(5.171\),)10588 62197 y Fz(\015)10588 62994 y(\015)11326 63326 y FA(jD)12720 63525 y FC(`)13160 63326 y FA(j)13529 62777 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))16699 62197 y Fz(\015)16699 62994 y(\015)17437 63846 y FC(s;s)18624 62197 y Fz(\015)18624 62994 y(\015)19362 63326 y FD(M)20759 62777 y Fw(\000)p FB(1)22017 63326 y FA(jD)23411 63525 y FC(k)23980 63326 y FA(j)24349 62777 y Fw(\000)p FC(\016)25585 62197 y Fz(\015)25585 62994 y(\015)26323 63846 y FC(s;s)28617 63326 y FA(\024)34357 62427 y FD(`)34903 61945 y FC(\027)58 b FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))p 30890 63020 10440 54 v 30890 64237 a FD(\015)72 b(\034)32350 63853 y FB(\()p FC(\027)58 b Fw(\000)p FB(1\)\(1)p Fw(\000)p FC(\016)33 b FB(\))p Fw(\000)p FC(\016)g Fw(\000)p FB(1+)p FC(\027)58 b(\016)42744 62427 y FJ(1)p 41595 63020 2949 54 v 41595 64237 a FD(k)42316 63853 y FB(1)p Fw(\000)p FC(\027)g(\016)28628 66983 y FJ(=)30890 66084 y FD(`)31436 65602 y FC(\027)g FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))p 30890 66677 3506 54 v 31024 67894 a FD(\015)72 b(\034)32484 67510 y FC(\027)58 b Fw(\000)p FB(2)35810 66084 y FJ(1)p 34661 66677 2949 54 v 34661 67894 a FD(k)35382 67510 y FB(1)p Fw(\000)p FC(\027)g(\016)28628 70786 y FJ(=)31347 69887 y FD(`)31893 69405 y FC(\027)g Fw(\000)p FB(1)p 30890 70480 3239 54 v 30890 71697 a FD(\015)72 b(\034)32350 71313 y FC(\027)58 b Fw(\000)p FB(2)34261 68913 y Fz(\022)35459 69887 y FD(`)p 35372 70480 722 54 v 35372 71697 a(k)36226 68913 y Fz(\023)37203 69210 y FB(1)p Fw(\000)p FC(\027)g(\016)28617 74486 y FA(\024)30890 73587 y(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)34518 73105 y FB(1)p Fw(\000)p FC(\027)58 b(\016)p 30890 74181 5856 54 v 32199 75397 a FD(\015)72 b(\034)33659 75013 y FC(\027)58 b Fw(\000)p FB(2)37100 74486 y FA(j)p FD(`)295 b FA(\000)h FD(k)45 b FA(j)40729 73938 y FB(\()p FC(\027)58 b Fw(\000)p FB(1\))p FC(=\014)27371 78312 y FJ(197)p eop %%Page: 198 198 198 197 bop 22329 4915 a FJ(=)24639 4016 y(1)p 24591 4609 747 54 v 24591 5826 a FD(\015)25692 4915 y FA(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)29320 4366 y FB(\(1)p Fw(\000)p FC(\027)58 b(\016)33 b FB(\)+\()p FC(\027)58 b Fw(\000)p FB(1\))p FC(=\014)36732 4915 y FD(\034)37445 4366 y FB(2)p Fw(\000)p FC(\027)39444 4915 y FD(;)0 8641 y FJ(where)433 b(w)-36 b(e)434 b(used)f(\(5.182\))i(and)e(the)g(h)-36 b(yp)36 b(othesis)433 b FA(j)p FD(`)295 b FA(\000)h FD(k)45 b FA(j)369 b(\025)g FD(`)30174 8159 y FC(\014)30803 8641 y FJ(,)433 b(while,)i(if)f FD(`)369 b FA(\024)g FD(\034)148 b(=)p FJ(2,)3260 11299 y Fz(\015)3260 12096 y(\015)3998 12428 y FA(jD)5392 12627 y FC(`)5831 12428 y FA(j)6200 11880 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))9371 11299 y Fz(\015)9371 12096 y(\015)10109 12949 y FC(s;s)11296 11299 y Fz(\015)11296 12096 y(\015)12034 12428 y FD(M)13431 11880 y Fw(\000)p FB(1)14689 12428 y FA(jD)16083 12627 y FC(k)16651 12428 y FA(j)17020 11880 y Fw(\000)p FC(\016)18257 11299 y Fz(\015)18257 12096 y(\015)18995 12949 y FC(s;s)20551 12428 y FA(\024)22086 11530 y FJ(const)p 22086 12123 2970 54 v 22945 13340 a FD(\015)23692 12956 y FC(\016)25543 11530 y FD(\034)26256 11048 y FC(\016)g FB(+1)p Fw(\000)p FC(\027)58 b(\016)p 25543 12123 4123 54 v 26129 13340 a FD(k)26850 12956 y FB(1)p Fw(\000)p FC(\027)g(\016)30167 12428 y FA(\024)369 b FJ(const)35162 11530 y FD(\034)p 34893 12123 1252 54 v 34893 13340 a(\015)35640 12956 y FC(\016)36499 12428 y FD(:)3260 b FJ(\(5.183\))0 16161 y Fp(Second)502 b(case)p FJ(:)578 b(0)370 b FD(<)e FA(j)p FD(`)295 b FA(\000)h FD(k)45 b FA(j)369 b FD(<)f FJ([max)223 b(\()p FD(`;)e(k)45 b FJ(\)])23220 15679 y FC(\014)24282 16161 y FJ(and)434 b FD(`)368 b FA(\024)h FD(\034)148 b(=)p FJ(2)435 b(or)f FD(k)414 b FA(\024)369 b FD(\034)148 b(=)p FJ(2.)0 17773 y(Let)433 b(b)36 b(e)433 b(0)370 b FA(\024)f FD(`)g FA(\024)g FD(\034)148 b(=)p FJ(2.)579 b(If)434 b FD(`)369 b FJ(=)g(0,)434 b FD(\025)17592 17972 y FC(`;j)19093 17773 y FJ(=)369 b FD(\025)21233 17972 y FB(0)p FC(;j)22451 17773 y FJ(,)434 b(then)4200 20727 y(min)4467 21541 y FC(j)51 b Fw(\025)p FB(1)6589 20727 y FA(j)p FD(\025)7717 20926 y FB(0)p FC(;j)8935 20727 y FA(j)1107 b FJ(=)g(min)12797 21541 y FC(j)51 b Fw(\025)p FB(1)14919 20727 y FA(j)p FD(\025)16047 20926 y FC(j)16533 20727 y FA(j)10411 24215 y FJ(=)1107 b(min)12797 25028 y FC(j)51 b Fw(\025)p FB(1)14919 22289 y Fz(\014)14919 23086 y(\014)14919 23883 y(\014)14919 24680 y(\014)15362 22342 y(\024)16063 22998 y(p)p 17391 22998 3879 54 v 1217 x FD(j)18004 23831 y FB(2)18825 24215 y FJ(+)295 b FD(m)g FJ(+)f FD(\021)23561 23666 y FB(2)24087 24215 y FJ(\()p FD(B)67 b(I)26222 24414 y FB(0)26747 24215 y FJ(\))27253 24414 y FC(j)28035 24215 y FJ(+)295 b FD(O)30593 22342 y Fz(\022)31706 23316 y FD(")p 31704 23909 613 54 v 31704 25126 a(j)32450 22342 y Fz(\023)o(\025)34128 22289 y(\014)34128 23086 y(\014)34128 23883 y(\014)34128 24680 y(\014)34940 24215 y FA(\025)369 b FJ(const)222 b FD(:)0 27933 y FJ(Let)433 b(b)36 b(e)433 b(0)370 b FD(<)e(`)h FA(\024)g FD(\034)148 b(=)p FJ(2.)580 b(F)-108 b(rom)433 b(\(5.171\))h(and)f(from)h(\(5.179\),)i(w)-36 b(e)433 b(get)10814 29758 y Fz(\015)10814 30555 y(\015)11552 30887 y FA(jD)12946 31086 y FC(`)13385 30887 y FA(j)13754 30338 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))16925 29758 y Fz(\015)16925 30555 y(\015)17663 31407 y FC(s;s)19071 29758 y Fz(\015)19071 30555 y(\015)19809 30887 y FD(M)21206 30338 y Fw(\000)p FB(1)22464 30887 y FA(jD)23858 31086 y FC(k)24426 30887 y FA(j)24795 30338 y Fw(\000)p FC(\016)26032 29758 y Fz(\015)26032 30555 y(\015)26770 31407 y FC(s;s)28326 30887 y FA(\024)369 b FJ(const)221 b FD(;)6840 b FJ(\(5.184\))0 34035 y(if)434 b FD(k)414 b FA(\024)369 b FD(\034)148 b(=)p FJ(2)435 b(and,)e(if)i FD(k)414 b(>)368 b(\034)148 b(=)p FJ(2,)435 b(from)f(\(5.179\),)h(w)-36 b(e)434 b(get)10954 36694 y Fz(\015)10954 37491 y(\015)11692 37823 y FD(M)13089 37274 y Fw(\000)p FB(1)14347 37823 y FA(jD)15741 38022 y FC(k)16310 37823 y FA(j)16679 37274 y Fw(\000)p FC(\016)17915 36694 y Fz(\015)17915 37491 y(\015)18653 38343 y FC(s;s)20209 37823 y FA(\024)21783 36924 y FD(\034)22496 36442 y FB(1+)p FC(\016)33 b Fw(\000)p FC(\027)58 b(\016)p 21744 37517 4201 54 v 21744 38734 a FD(\015)22491 38350 y FC(\016)22996 38734 y FD(k)23717 38350 y FB(1)p Fw(\000)p FC(\027)g(\016)26447 37823 y FA(\024)369 b FJ(const)31190 36924 y FD(\034)31903 36442 y FC(\016)p 31173 37517 1252 54 v 31173 38734 a FD(\015)31920 38350 y FC(\016)32779 37823 y FD(;)0 41374 y FJ(therefore,)434 b(it)g(follo)-36 b(ws)435 b(that)9944 43959 y Fz(\015)9944 44756 y(\015)10682 45088 y FA(jD)12076 45287 y FC(`)12516 45088 y FA(j)12885 44539 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))16056 43959 y Fz(\015)16056 44756 y(\015)16793 45608 y FC(s;s)18202 43959 y Fz(\015)18202 44756 y(\015)18940 45088 y FD(M)20337 44539 y Fw(\000)p FB(1)21594 45088 y FA(jD)22988 45287 y FC(k)23557 45088 y FA(j)23926 44539 y Fw(\000)p FC(\016)25163 43959 y Fz(\015)25163 44756 y(\015)25901 45608 y FC(s;s)27457 45088 y FA(\024)369 b FJ(const)32200 44189 y FD(\034)32913 43707 y FC(\016)p 32183 44782 V 32183 45999 a FD(\015)32930 45615 y FC(\016)33789 45088 y FD(:)5970 b FJ(\(5.185\))0 48713 y(In)433 b(the)g(case)h FD(k)415 b FA(\024)369 b FD(\034)148 b(=)p FJ(2,)434 b(since,)g(from)g(\(5.179\),)14942 50538 y Fz(\015)14942 51335 y(\015)15680 51667 y FD(M)17077 51119 y Fw(\000)p FB(1)18335 51667 y FA(jD)19729 51866 y FC(k)20298 51667 y FA(j)20667 51119 y Fw(\000)p FC(\016)21903 50538 y Fz(\015)21903 51335 y(\015)22641 52188 y FC(s;s)24197 51667 y FA(\024)369 b FJ(const)222 b FD(;)0 54816 y FJ(from)434 b(\(5.171\))h(and)e(using)g(Remark)i(5.9.9,)g(w)-36 b(e)434 b(obtain)7585 57718 y Fz(\015)7585 58515 y(\015)8323 58847 y FA(jD)9717 59046 y FC(`)10156 58847 y FA(j)10525 58299 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))13696 57718 y Fz(\015)13696 58515 y(\015)14434 59368 y FC(s;s)16728 58847 y FA(\024)1107 b FJ(const)22060 56974 y Fz(\022)23258 57949 y FD(`)p 23170 58542 722 54 v 23170 59759 a(k)24024 56974 y Fz(\023)25002 57272 y FC(\027)58 b FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))30395 57949 y FD(k)31116 57467 y FC(\027)58 b FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))p 28094 58542 8283 54 v 28094 59759 a FD(\034)28807 59375 y FB(\()p FC(\027)58 b Fw(\000)p FB(1\)\(1)p Fw(\000)p FC(\016)33 b FB(\))33922 59759 y FD(\015)34669 59375 y FB(1)p Fw(\000)p FC(\016)16728 62597 y FA(\024)19001 61698 y(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)22629 61216 y FC(\027)58 b FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))p 19001 62291 6588 54 v 21068 63508 a FD(\015)21815 63124 y FB(1)p Fw(\000)p FC(\016)25943 62597 y FD(\034)26656 62049 y FB(1)p Fw(\000)p FC(\016)28585 62597 y FD(;)0 66149 y FJ(and)433 b(therefore)3968 68524 y Fz(\015)3968 69321 y(\015)4706 69653 y FA(jD)6100 69852 y FC(`)6540 69653 y FA(j)6909 69105 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))10079 68524 y Fz(\015)10079 69321 y(\015)10817 70173 y FC(s;s)12226 68524 y Fz(\015)12226 69321 y(\015)12964 69653 y FD(M)14361 69105 y Fw(\000)p FB(1)15618 69653 y FA(jD)17012 69852 y FC(k)17581 69653 y FA(j)17950 69105 y Fw(\000)p FC(\016)19187 68524 y Fz(\015)19187 69321 y(\015)19925 70173 y FC(s;s)21481 69653 y FA(\024)369 b FJ(const)26207 68755 y FA(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)29835 68272 y FC(\027)58 b FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))p 26207 69348 V 28274 70564 a FD(\015)29021 70181 y FB(1)p Fw(\000)p FC(\016)33149 69653 y FD(\034)33862 69105 y FB(1)p Fw(\000)p FC(\016)35791 69653 y FD(:)3968 b FJ(\(5.186\))0 73386 y Fp(Third)462 b(case)p FJ(:)559 b(0)369 b FD(<)g FA(j)p FD(`)217 b FA(\000)g FD(k)45 b FA(j)369 b FD(<)f FJ([max)223 b(\()p FD(`;)e(k)45 b FJ(\)])22067 72904 y FC(\014)22696 73386 y FJ(,)403 b FD(`;)221 b(k)415 b(>)368 b(\034)148 b(=)p FJ(2,)405 b(0)369 b FD(<)f FA(j)p FD(`)217 b FA(\000)g FD(k)45 b FA(j)369 b(\024)h FD(\034)148 b FA(j)p FD(j)39101 73585 y FB(0)39844 73386 y FA(\000)217 b FD(i)41538 73585 y FB(0)42064 73386 y FA(j)p FD(=)p FJ(4,)0 74991 y(where)433 b FD(i)4201 75190 y FB(0)5161 74991 y FJ(and)g FD(j)8228 75190 y FB(0)9187 74991 y FJ(are)h(suc)-36 b(h)433 b(that)g(\(5.169\))i (and)e(\(5.170\))h(hold.)21072 78312 y(198)p eop %%Page: 199 199 199 198 bop 6299 4488 a FJ(T)-108 b(o)434 b(reac)-36 b(h)433 b(the)g(\014nal)h(estimate,)g(let)g(us)f(consider)g(previously) 13225 7976 y FA(j)p FD(\026)14377 8175 y FC(`;j)15403 8298 y Fu(0)16215 7976 y FA(\000)295 b FD(\026)18326 8175 y FC(k)24 b(;i)19421 8298 y Fu(0)19937 7976 y FA(j)1118 b FJ(=)23553 6050 y Fz(\014)23553 6847 y(\014)23553 7644 y(\014)23553 8441 y(\014)24213 7078 y FD(`)p 24129 7671 714 54 v 24129 8888 a(\034)25270 7976 y FA(\000)296 b FD(\025)27358 8175 y FC(`j)28123 8298 y Fu(0)28934 7976 y FA(\000)30395 7078 y FD(k)p 30395 7671 722 54 v 30399 8888 a(\034)31544 7976 y FJ(+)f FD(\025)33610 8175 y FC(k)24 b(i)34444 8298 y Fu(0)34960 6050 y Fz(\014)34960 6847 y(\014)34960 7644 y(\014)34960 8441 y(\014)21413 11576 y FA(\025)1107 b(j)p FD(\025)24681 11775 y FC(k)24 b(i)25515 11898 y Fu(0)26327 11576 y FA(\000)295 b FD(\025)28414 11775 y FC(`j)29179 11898 y Fu(0)29695 11576 y FA(j)g(\000)31821 10677 y(j)p FD(`)g FA(\000)g FD(k)45 b FA(j)p 31821 11270 3629 54 v 33278 12487 a FD(\034)35951 11576 y FA(\025)37486 10677 y FJ(1)p 37486 11270 651 54 v 37486 12487 a(4)38269 11576 y FA(j)p FD(j)39176 11775 y FB(0)39997 11576 y FA(\000)295 b FD(i)41769 11775 y FB(0)42295 11576 y FA(j)221 b FD(;)6299 16311 y FJ(where,)699 b(in)645 b(the)g(last)h(inequalit)-36 b(y)-108 b(,)699 b(w)-36 b(e)646 b(used)e(that)h FA(j)p FD(\025)33557 16510 y FC(k)24 b(i)34885 16311 y FA(\000)440 b FD(\025)37117 16510 y FC(`j)37988 16311 y FA(j)730 b(\025)f(j)p FD(j)514 b FA(\000)440 b FD(i)p FA(j)p FD(=)p FJ(2.)1214 b(Since)6299 17916 y FA(j)p FD(j)7206 18115 y FB(0)8027 17916 y FA(\000)296 b FD(i)9800 18115 y FB(0)10325 17916 y FA(j)369 b(\025)g FJ(1,)434 b(w)-36 b(e)434 b(get)g(the)f (estimate)23171 21306 y FA(j)p FD(\026)24323 21505 y FC(`;j)25349 21628 y Fu(0)26160 21306 y FA(\000)296 b FD(\026)28272 21505 y FC(k)24 b(;i)29367 21628 y Fu(0)29883 21306 y FA(j)369 b(\025)32156 20408 y FJ(1)p 32156 21001 V 32156 22218 a(4)33161 21306 y FD(:)6299 24444 y FJ(It)434 b(follo)-36 b(ws)435 b(that)18542 26309 y FA(j)p FD(\026)19694 26508 y FC(`;j)20720 26631 y Fu(0)21237 26309 y FA(j)369 b(\025)23510 25410 y FJ(1)p 23510 26003 V 23510 27220 a(8)27443 26309 y(or)3149 b FA(j)p FD(\026)32900 26508 y FC(k)24 b(;i)33995 26631 y Fu(0)34512 26309 y FA(j)369 b(\025)36785 25410 y FJ(1)p 36785 26003 V 36785 27220 a(8)37789 26309 y FD(:)6299 28989 y FJ(In)434 b(the)f(case)h FA(j)p FD(\026)14070 29188 y FC(`;j)15096 29311 y Fu(0)15612 28989 y FA(j)369 b(\025)g FJ(1)p FD(=)p FJ(8,)435 b(from)f(\(5.179\),)h (w)-36 b(e)434 b(ha)-36 b(v)g(e,)434 b(b)36 b(eing)434 b FD(k)414 b(>)368 b(\034)148 b(=)p FJ(2,)9916 31725 y Fz(\015)9916 32522 y(\015)10654 32854 y FA(jD)12048 33053 y FC(`)12488 32854 y FA(j)12857 32306 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))16028 31725 y Fz(\015)16028 32522 y(\015)16766 33374 y FC(s;s)18174 31725 y Fz(\015)18174 32522 y(\015)18912 32854 y FD(M)20309 32306 y Fw(\000)p FB(1)21567 32854 y FA(jD)22961 33053 y FC(k)23529 32854 y FA(j)23898 32306 y Fw(\000)p FC(\016)25135 31725 y Fz(\015)25135 32522 y(\015)25873 33374 y FC(s;s)27429 32854 y FA(\024)369 b FJ(const)32172 31955 y FD(\034)32885 31473 y FC(\016)p 32155 32548 1252 54 v 32155 33765 a FD(\015)32902 33381 y FC(\016)33761 30981 y Fz(\022)34876 31955 y FD(\034)p 34872 32548 722 54 v 34872 33765 a(k)35726 30981 y Fz(\023)36703 31278 y FB(1)p Fw(\000)p FC(\027)58 b(\016)39300 32854 y FA(\024)40852 31955 y FD(\034)41565 31473 y FC(\016)p 40835 32548 1252 54 v 40835 33765 a FD(\015)41582 33381 y FC(\016)42441 32854 y FD(:)3617 b FJ(\(5.187\))6299 36416 y(In)434 b(the)f(case)h FA(j)p FD(\026)14070 36615 y FC(k)24 b(;i)15165 36738 y Fu(0)15681 36416 y FA(j)369 b(\025)g FJ(1)p FD(=)p FJ(8,)435 b(one)f(has,)f (generally)-108 b(,)11035 40281 y(min)11959 41088 y FC(i)13424 38408 y Fz(\032)14420 40281 y FD(i)14864 38355 y Fz(\014)14864 39152 y(\014)14864 39949 y(\014)14864 40746 y(\014)15440 39383 y FD(k)p 15440 39976 722 54 v 15444 41192 a(\034)16589 40281 y FA(\000)295 b FD(\025)18676 40480 y FC(k)24 b(i)19565 38355 y Fz(\014)19565 39152 y(\014)19565 39949 y(\014)19565 40746 y(\014)20008 38706 y FC(\016)20513 38408 y Fz(\033)21878 40281 y FJ(=)369 b(min)25648 38408 y Fz(\032)26644 38355 y(\014)26644 39152 y(\014)26644 39949 y(\014)26644 40746 y(\014)27220 39383 y FD(k)p 27220 39976 V 27224 41192 a(\034)28369 40281 y FA(\000)295 b FD(\025)30456 40480 y FC(k)24 b FB(1)31495 38355 y Fz(\014)31495 39152 y(\014)31495 39949 y(\014)31495 40746 y(\014)31938 38706 y FC(\016)32443 40281 y FD(;)221 b(i)33469 40480 y FB(0)33995 38355 y Fz(\014)33995 39152 y(\014)33995 39949 y(\014)33995 40746 y(\014)34571 39383 y FD(k)p 34571 39976 V 34575 41192 a(\034)35720 40281 y FA(\000)295 b FD(\025)37807 40480 y FC(k)24 b(i)38641 40603 y Fu(0)39157 38355 y Fz(\014)39157 39152 y(\014)39157 39949 y(\014)39157 40746 y(\014)39600 38706 y FC(\016)40105 38408 y Fz(\033)41322 40281 y FD(:)4736 b FJ(\(5.188\))6299 43843 y(and)433 b(in)h(particular,)g(w)-36 b(e)433 b(ha)-36 b(v)g(e)434 b FA(j)p FD(\026)22770 44042 y FC(k)24 b(;i)23865 44165 y Fu(0)24381 43843 y FA(j)369 b FJ(=)g FA(j)p FD(k)45 b(=\034)444 b FA(\000)295 b FD(\025)31336 44042 y FC(k)24 b(i)32170 44165 y Fu(0)32686 43843 y FA(j)369 b(\025)g FJ(1)p FD(=)p FJ(8;)435 b(hence)10123 48258 y(min)11046 49064 y FC(i)12512 46385 y Fz(\032)13508 48258 y FD(i)13952 46331 y Fz(\014)13952 47129 y(\014)13952 47926 y(\014)13952 48723 y(\014)14527 47359 y FD(k)p 14527 47952 V 14531 49169 a(\034)15676 48258 y FA(\000)296 b FD(\025)17764 48457 y FC(k)24 b(i)18653 46331 y Fz(\014)18653 47129 y(\014)18653 47926 y(\014)18653 48723 y(\014)19096 46682 y FC(\016)19600 46385 y Fz(\033)20966 48258 y FA(\025)22368 45534 y Fz(8)22368 46730 y(<)22368 49121 y(:)24102 46641 y FJ(const)221 b FD(k)28014 46159 y FC(\016)28519 46641 y FD(=\034)29882 46159 y FC(\016)30388 46641 y FD(;)1771 b FJ(if)434 b FD(k)414 b(>)369 b FJ(2)p FD(\034)148 b FJ(;)24102 49852 y FD(\015)24849 49370 y FC(\016)25354 49852 y FD(=\034)26717 49370 y FC(\016)27222 49852 y FD(;)4937 b FJ(if)434 b FD(\034)148 b(=)p FJ(2)370 b FD(<)f(k)414 b FA(\024)369 b FJ(2)p FD(\034)c FJ(.)46419 48258 y(\(5.189\))6299 52559 y(Indeed,)433 b(if)h FD(i)12523 52758 y FB(0)13418 52559 y FJ(=)369 b(1)433 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b(that)19044 56311 y(min)19968 57118 y FC(i)21433 54438 y Fz(\032)22429 56311 y FD(i)22873 54385 y Fz(\014)22873 55182 y(\014)22873 55979 y(\014)22873 56776 y(\014)23449 55413 y FD(k)p 23449 56006 V 23453 57223 a(\034)24598 56311 y FA(\000)295 b FD(\025)26685 56510 y FC(k)24 b(i)27574 54385 y Fz(\014)27574 55182 y(\014)27574 55979 y(\014)27574 56776 y(\014)28017 54736 y FC(\016)28522 54438 y Fz(\033)29887 56311 y FJ(=)31268 54385 y Fz(\014)31268 55182 y(\014)31268 55979 y(\014)31268 56776 y(\014)31843 55413 y FD(k)p 31843 56006 V 31847 57223 a(\034)32992 56311 y FA(\000)296 b FD(\025)35080 56510 y FC(k)24 b FB(1)36119 54385 y Fz(\014)36119 55182 y(\014)36119 55979 y(\014)36119 56776 y(\014)36561 54736 y FC(\016)37288 56311 y FD(;)6299 59873 y FJ(hence)493 b(\(5.189\))h(follo)-36 b(ws)496 b(from)d(\(5.177\).)759 b(On)492 b(the)h(other)g(hand,)508 b(if)494 b FD(i)40360 60072 y FB(0)41356 59873 y FD(>)470 b FJ(1,)509 b(then,)e(from)6299 61478 y(Remark)435 b(5.9.1,)g FD(i)15107 61677 y FB(0)16001 61478 y FA(\031)370 b FD(k)45 b(=\034)148 b FJ(,)434 b(hence)15921 65304 y FD(i)16365 65503 y FB(0)16890 63378 y Fz(\014)16890 64175 y(\014)16890 64972 y(\014)16890 65769 y(\014)17466 64406 y FD(k)p 17466 64999 V 17470 66216 a(\034)18615 65304 y FA(\000)295 b FD(\025)20702 65503 y FC(k)24 b(i)21536 65626 y Fu(0)22052 63378 y Fz(\014)22052 64175 y(\014)22052 64972 y(\014)22052 65769 y(\014)22495 63729 y FC(\016)23369 65304 y FA(\025)24997 64406 y FD(i)25441 64605 y FB(0)p 24904 64999 1156 54 v 24904 66216 a FJ(8)25554 65832 y FC(\016)26561 65304 y FA(\031)369 b FJ(const)31287 64406 y FD(k)p 31287 64999 722 54 v 31291 66216 a(\034)32510 65304 y FA(\025)g FJ(const)37104 63431 y Fz(\022)38214 64406 y FD(k)p 38214 64999 V 38218 66216 a(\034)39068 63431 y Fz(\023)40046 63729 y FC(\016)6299 68866 y FJ(where)466 b(w)-36 b(e)467 b(used)e(that)h FD(\016)475 b(<)424 b FJ(1.)677 b(Th)-36 b(us)466 b(\(5.189\))h(follo) -36 b(ws.)678 b(Therefore,)476 b(from)466 b(\(5.179\),)476 b(w)-36 b(e)6299 70471 y(get)11247 72451 y Fz(\015)11247 73248 y(\015)11985 73580 y FD(M)13382 73031 y Fw(\000)p FB(1)14639 73580 y FA(jD)16033 73779 y FC(k)16602 73580 y FA(j)16971 73031 y Fw(\000)p FC(\016)18207 72451 y Fz(\015)18207 73248 y(\015)18945 74100 y FC(s;s)20501 73580 y FA(\024)21904 70857 y Fz(8)21904 72052 y(<)21904 74443 y(:)23638 71964 y FJ(const)221 b FD(\034)27542 71482 y FC(\016)28047 71964 y FD(=k)29418 71482 y FC(\016)29923 71964 y FD(;)1329 b FJ(if)434 b FD(k)414 b(>)369 b FJ(2)p FD(\034)148 b FJ(;)23638 75174 y FD(\034)24351 74692 y FC(\016)24856 75174 y FD(=\015)26253 74692 y FC(\016)26758 75174 y FD(;)4494 b FJ(if)434 b FD(\034)148 b(=)p FJ(2)370 b FD(<)f(k)414 b FA(\024)369 b FJ(2)p FD(\034)148 b FJ(.)46419 73580 y(\(5.190\))27371 78312 y(199)p eop %%Page: 200 200 200 199 bop 0 4488 a FJ(Finally)-108 b(,)435 b(from)f(\(5.171\))g(and)f (\(5.190\),)j(w)-36 b(e)433 b(reac)-36 b(h)434 b(the)f(estimate)2354 10444 y FA(kjD)4412 10643 y FC(`)4852 10444 y FA(j)5221 9896 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))8392 10444 y FA(k)9056 10643 y FC(s;s)10243 10444 y FA(k)p FD(M)12304 9896 y Fw(\000)p FB(1)13562 10444 y FA(jD)14956 10643 y FC(k)15524 10444 y FA(j)15893 9896 y Fw(\000)p FC(\016)17130 10444 y FA(k)17794 10643 y FC(s;s)19350 10444 y FA(\024)20752 6127 y Fz(8)20752 7322 y(>)20752 7721 y(>)20752 8119 y(>)20752 8518 y(>)20752 8916 y(<)20752 11307 y(>)20752 11706 y(>)20752 12104 y(>)20752 12503 y(>)20752 12901 y(:)25621 7241 y FD(`)26167 6759 y FC(\027)58 b FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))p 22619 7834 9509 54 v 22619 9051 a FD(k)23340 8667 y FC(\016)24067 9051 y FD(\015)24814 8667 y FB(\(1)p Fw(\000)p FC(\016)g FB(\))27253 9051 y FD(\034)27966 8667 y FC(\027)58 b FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))p Fw(\000)p FB(1)32482 8140 y FD(;)2823 b FJ(if)434 b FD(k)414 b(>)369 b FJ(2)p FD(\034)148 b FJ(;)23678 12061 y FD(`)24224 11579 y FC(\027)58 b FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))p 22619 12654 5622 54 v 22619 13871 a FD(\015)72 b(\034)24079 13487 y FC(\027)58 b FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))p Fw(\000)p FB(1)28596 12959 y FD(;)2823 b FJ(if)434 b FD(\034)148 b(=)p FJ(2)370 b FD(<)e(k)414 b FA(\024)369 b FJ(2)p FD(\034)148 b FJ(.)0 16326 y(In)433 b(the)g(\014rst)g(case)h FD(k)414 b(>)369 b FJ(2)p FD(\034)148 b FJ(,)434 b(\014xing)17704 19095 y FD(\016)419 b FJ(:=)369 b FD(\027)84 b(=)p FJ(\()p FD(\027)378 b FJ(+)295 b(1\))221 b FD(;)13730 b FJ(\(5.191\))0 21864 y(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b FD(\027)84 b FJ(\(1)294 b FA(\000)i FD(\016)50 b FJ(\))368 b(=)h FD(\016)483 b FJ(and)434 b(therefore)934 25661 y FA(kjD)2992 25860 y FC(`)3432 25661 y FA(j)3801 25112 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))6972 25661 y FA(k)7636 25860 y FC(s;s)8823 25661 y FA(k)p FD(M)10884 25112 y Fw(\000)p FB(1)12142 25661 y FA(jD)13536 25860 y FC(k)14104 25661 y FA(j)14473 25112 y Fw(\000)p FC(\016)15710 25661 y FA(k)16374 25860 y FC(s;s)17930 25661 y FA(\024)19332 23788 y Fz(\022)20530 24762 y FD(`)p 20443 25355 722 54 v 20443 26572 a(k)21297 23788 y Fz(\023)22274 24085 y FC(\016)26252 24762 y FJ(1)p 22912 25355 7330 54 v 22912 26572 a FD(\015)23659 26188 y FB(1)p Fw(\000)p FC(\016)25366 26572 y FD(\034)26079 26188 y FC(\027)58 b FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))p Fw(\000)p FB(1)30743 25661 y FA(\024)32278 24762 y(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)35906 24280 y FC(\016)p 32278 25355 4134 54 v 33118 26572 a FD(\015)33865 26188 y FB(1)p Fw(\000)p FC(\016)36766 25661 y FD(\034)37479 25112 y FB(1)p Fw(\000)p FC(\016)40120 25661 y FJ(\(5.192\))0 29194 y(where)355 b(in)g(the)f(last)i(inequalit)-36 b(y)356 b(w)-36 b(e)355 b(used)f(\(5.182\).)554 b(In)354 b(the)h(second)f(case,)372 b FD(\034)148 b(=)p FJ(2)370 b FD(<)e(k)414 b FA(\024)370 b FJ(2)p FD(\034)148 b FJ(,)0 30799 y(w)-36 b(e)434 b(ha)-36 b(v)g(e)1497 34313 y FA(kjD)3555 34512 y FC(`)3995 34313 y FA(j)4364 33765 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))7535 34313 y FA(k)8199 34512 y FC(s;s)9386 34313 y FA(k)p FD(M)11447 33765 y Fw(\000)p FB(1)12705 34313 y FA(jD)14099 34512 y FC(k)14667 34313 y FA(j)15036 33765 y Fw(\000)p FC(\016)16273 34313 y FA(k)16937 34512 y FC(s;s)18493 34313 y FA(\024)20999 33414 y FD(k)21720 32932 y FC(\027)58 b FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))p 20028 34008 5622 54 v 20028 35224 a FD(\015)72 b(\034)21488 34840 y FC(\027)58 b FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))p Fw(\000)p FB(1)25783 32440 y Fz(\022)26981 33414 y FD(`)p 26894 34008 722 54 v 26894 35224 a(k)27747 32440 y Fz(\023)28725 32737 y FC(\027)58 b FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))32054 34313 y FA(\024)369 b FJ(const)36797 33414 y FD(\034)p 36780 34008 747 54 v 36780 35224 a(\015)37881 34313 y FA(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)41509 33765 y FC(\016)42236 34313 y FD(:)0 37772 y FJ(In)433 b(conclusion,)8278 40775 y FA(kjD)10336 40974 y FC(`)10775 40775 y FA(j)11144 40227 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))14315 40775 y FA(k)14979 40974 y FC(s;s)16166 40775 y FA(k)p FD(M)18227 40227 y Fw(\000)p FB(1)19485 40775 y FA(jD)20879 40974 y FC(k)21448 40775 y FA(j)21817 40227 y Fw(\000)p FC(\016)23053 40775 y FA(k)23717 40974 y FC(s;s)25273 40775 y FA(\024)370 b FJ(const)30017 39877 y FD(\034)p 30000 40470 V 30000 41687 a(\015)31101 40775 y FA(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)34729 40227 y FC(\016)35455 40775 y FD(:)4304 b FJ(\(5.193\))0 44316 y Fp(F)-39 b(our)-89 b(th)503 b(case)p FJ(:)578 b(0)369 b FD(<)g FA(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)369 b FD(<)g FJ([max)222 b(\()p FD(`;)f(k)45 b FJ(\)])23271 43834 y FC(\014)23901 44316 y FJ(,)433 b FD(`;)221 b(k)415 b(>)369 b(\034)148 b(=)p FJ(2,)434 b FA(j)p FD(`)295 b FA(\000)h FD(k)45 b FA(j)369 b FD(>)f(\034)148 b FA(j)p FD(j)38101 44515 y FB(0)38923 44316 y FA(\000)295 b FD(i)40695 44515 y FB(0)41220 44316 y FA(j)p FD(=)p FJ(4.)0 45922 y(Let)433 b(us)g(distinguish)g(the)g(t)-36 b(w)g(o)434 b(cases)g FD(i)18996 46121 y FB(0)19891 45922 y FJ(=)368 b FD(j)21809 46121 y FB(0)22769 45922 y FJ(and)433 b FD(i)25742 46121 y FB(0)26636 45922 y FA(6)p FJ(=)369 b FD(j)28555 46121 y FB(0)29081 45922 y FJ(.)0 47527 y(In)433 b(the)g(case)h FD(i)7062 47726 y FB(0)7957 47527 y FJ(=)369 b FD(j)9876 47726 y FB(0)10402 47527 y FJ(,)433 b(it)h(results)7208 50986 y FA(j)p FD(\026)8360 51185 y FC(`;j)9386 51308 y Fu(0)10198 50986 y FA(\000)295 b FD(\026)12309 51185 y FC(k)24 b(;i)13404 51308 y Fu(0)13920 50986 y FA(j)369 b FJ(=)16039 49059 y Fz(\014)16039 49856 y(\014)16039 50653 y(\014)16039 51451 y(\014)16698 50087 y FD(`)p 16615 50680 714 54 v 16615 51897 a(\034)17756 50986 y FA(\000)295 b FD(\025)19843 51185 y FC(`j)20608 51308 y Fu(0)21420 50986 y FA(\000)22881 50087 y FD(k)p 22881 50680 722 54 v 22885 51897 a(\034)24030 50986 y FJ(+)g FD(\025)26096 51185 y FC(k)24 b(i)26930 51308 y Fu(0)27446 49059 y Fz(\014)27446 49856 y(\014)27446 50653 y(\014)27446 51451 y(\014)28258 50986 y FA(\025)29792 50087 y(j)p FD(`)295 b FA(\000)h FD(k)45 b FA(j)p 29792 50680 3629 54 v 31250 51897 a FD(\034)33923 50986 y FA(\025)35489 50087 y FJ(1)p 35458 50680 714 54 v 35458 51897 a FD(\034)36525 50986 y(;)0 54518 y FJ(th)-36 b(us)432 b FA(j)p FD(\026)4013 54717 y FC(`;j)5039 54840 y Fu(0)5556 54518 y FA(j)369 b(\025)g FJ(1)p FD(=)p FJ(2)p FD(\034)583 b FJ(or)433 b FA(j)p FD(\026)13535 54717 y FC(k)24 b(;i)14630 54840 y Fu(0)15147 54518 y FA(j)369 b(\025)g FJ(1)p FD(=)p FJ(2)p FD(\034)148 b FJ(.)579 b(Let)434 b(b)36 b(e)433 b(\014rst)g FA(j)p FD(\026)28824 54717 y FC(`;j)29850 54840 y Fu(0)30366 54518 y FA(j)369 b(\025)g FJ(1)p FD(=)p FJ(2)p FD(\034)148 b FJ(.)580 b(Hence)13754 57287 y FA(kjD)15812 57486 y FC(`)16252 57287 y FA(j)16621 56739 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))19792 57287 y FA(k)20456 57486 y FC(s;s)22012 57287 y FA(\024)369 b FJ(const)221 b FD(\034)27318 56739 y FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))29979 57287 y FD(:)0 60056 y FJ(In)433 b(this)h(case)g(one)f (obtains,)h(from)g(\(5.179\),)h(using)f(that)f FD(k)414 b(>)368 b(\034)148 b(=)p FJ(2,)939 62724 y Fz(\015)939 63521 y(\015)1677 63853 y FA(jD)3071 64052 y FC(`)3510 63853 y FA(j)3879 63305 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))7050 62724 y Fz(\015)7050 63521 y(\015)7788 64374 y FC(s;s)8975 62724 y Fz(\015)8975 63521 y(\015)9713 63853 y FD(M)11110 63305 y Fw(\000)p FB(1)12368 63853 y FA(jD)13762 64052 y FC(k)14330 63853 y FA(j)14699 63305 y Fw(\000)p FC(\016)15936 62724 y Fz(\015)15936 63521 y(\015)16674 64374 y FC(s;s)18230 63853 y FA(\024)369 b FJ(const)22956 62955 y FD(\034)23669 62473 y FB(1)p Fw(\000)p FC(\016)25376 62955 y FD(\034)26089 62473 y FC(\016)p 22956 63548 3639 54 v 24149 64765 a FD(\015)24896 64381 y FC(\016)26949 61980 y Fz(\022)28063 62955 y FD(\034)p 28059 63548 722 54 v 28059 64765 a(k)28913 61980 y Fz(\023)29891 62278 y FB(1)p Fw(\000)p FC(\027)58 b(\016)32488 63853 y FA(\024)369 b FJ(const)37483 62955 y FD(\034)p 37214 63548 1252 54 v 37214 64765 a(\015)37961 64381 y FC(\016)38820 63853 y FD(:)939 b FJ(\(5.194\))0 67386 y(On)519 b(the)g(other)h(hand,) 541 b(if)520 b FA(j)p FD(\026)14214 67585 y FC(k)24 b(;i)15309 67708 y Fu(0)15826 67386 y FA(j)516 b(\025)g FJ(1)p FD(=)p FJ(2)p FD(\034)148 b FJ(,)543 b(w)-36 b(e)520 b(ha)-36 b(v)g(e)520 b(to)g(distinguish)f(the)h(case)g FD(i)40875 67585 y FB(0)41917 67386 y FJ(=)515 b(1)0 68991 y(from)434 b(the)f(case)h FD(i)8508 69190 y FB(0)9403 68991 y FD(>)368 b FJ(1.)0 70596 y(If)434 b FD(i)1744 70795 y FB(0)2639 70596 y FJ(=)368 b(1,)435 b(one)e(has,)h(from)g(\(5.177\),)3577 74394 y(min)4501 75200 y FC(i)5966 72521 y Fz(\032)6963 74394 y FD(i)7407 72467 y Fz(\014)7407 73264 y(\014)7407 74061 y(\014)7407 74858 y(\014)7982 73495 y FD(k)p 7982 74088 722 54 v 7986 75305 a(\034)9131 74394 y FA(\000)295 b FD(\025)11218 74593 y FC(k)24 b(i)12107 72467 y Fz(\014)12107 73264 y(\014)12107 74061 y(\014)12107 74858 y(\014)12550 72818 y FC(\016)13055 72521 y Fz(\033)14420 74394 y FA(\025)15822 72467 y Fz(\014)15822 73264 y(\014)15822 74061 y(\014)15822 74858 y(\014)16398 73495 y FD(k)p 16398 74088 V 16402 75305 a(\034)17547 74394 y FA(\000)295 b FD(\025)19634 74593 y FC(k)24 b(i)20468 74716 y Fu(0)20985 72467 y Fz(\014)20985 73264 y(\014)20985 74061 y(\014)20985 74858 y(\014)21427 72818 y FC(\016)22301 74394 y FJ(=)23682 72467 y Fz(\014)23682 73264 y(\014)23682 74061 y(\014)23682 74858 y(\014)24257 73495 y FD(k)p 24257 74088 V 24261 75305 a(\034)25406 74394 y FA(\000)296 b FD(\025)27494 74593 y FC(k)24 b FB(1)28533 72467 y Fz(\014)28533 73264 y(\014)28533 74061 y(\014)28533 74858 y(\014)28976 72818 y FC(\016)29850 74394 y FA(\025)369 b FJ(const)34576 73495 y FD(\015)35323 73013 y FC(\016)p 34576 74088 1252 54 v 34593 75305 a FD(\034)35306 74921 y FC(\016)36182 74394 y FD(;)3577 b FJ(\(5.195\))21072 78312 y(200)p eop %%Page: 201 201 201 200 bop 6299 4488 a FJ(therefore,)434 b(from)g(\(5.171\),)13542 7064 y Fz(\015)13542 7861 y(\015)14280 8193 y FA(jD)15674 8392 y FC(`)16113 8193 y FA(j)16482 7645 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))19653 7064 y Fz(\015)19653 7861 y(\015)20391 8713 y FC(s;s)21578 7064 y Fz(\015)21578 7861 y(\015)22316 8193 y FD(M)23713 7645 y Fw(\000)p FB(1)24970 8193 y FA(jD)26364 8392 y FC(k)26933 8193 y FA(j)27302 7645 y Fw(\000)p FC(\016)28539 7064 y Fz(\015)28539 7861 y(\015)29277 8713 y FC(s;s)30685 8193 y FA(\024)664 b FJ(const)35706 7295 y FD(\034)36419 6812 y FC(\016)p 35706 7888 1219 54 v 35942 9104 a FD(\015)38828 7295 y(`)39374 6812 y FB(1)p Fw(\000)p FC(\027)58 b(\016)p 37412 7888 5607 54 v 37412 9104 a FD(\034)38125 8721 y FB(\()p FC(\027)g Fw(\000)p FB(1\)\(1)p Fw(\000)p FC(\016)33 b FB(\))30696 11801 y FJ(=)674 b(const)37241 10902 y FD(`)37787 10420 y FB(1)p Fw(\000)p FC(\027)58 b(\016)p 35706 11496 5844 54 v 35706 12712 a FD(\015)294 b(\034)37388 12329 y FC(\027)58 b FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))p Fw(\000)p FB(1)41904 11801 y FD(:)6299 15344 y FJ(Hence)456 b(this)g(case)g(reduce)f(to)h(the)f(third)g(case)i(with) f FD(\034)148 b(=)p FJ(2)408 b FD(<)e(k)452 b FA(\024)408 b FJ(2)p FD(\034)148 b FJ(,)462 b(b)-36 b(y)456 b(using)f(\(5.182\)) 6299 16949 y(the)433 b(\014nal)h(estimate)f(is)14429 18691 y Fz(\015)14429 19488 y(\015)15167 19820 y FA(jD)16561 20019 y FC(`)17001 19820 y FA(j)17370 19272 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))20541 18691 y Fz(\015)20541 19488 y(\015)21279 20341 y FC(s;s)22465 18691 y Fz(\015)22465 19488 y(\015)23203 19820 y FD(M)24600 19272 y Fw(\000)p FB(1)25858 19820 y FA(jD)27252 20019 y FC(k)27821 19820 y FA(j)28190 19272 y Fw(\000)p FC(\016)29426 18691 y Fz(\015)29426 19488 y(\015)30164 20341 y FC(s;s)31720 19820 y FA(\024)369 b FJ(const)36463 18922 y FD(\034)p 36446 19515 747 54 v 36446 20731 a(\015)37548 19820 y FA(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)41176 19272 y FC(\016)41902 19820 y FD(:)6299 23363 y FJ(If)571 b FD(i)8180 23562 y FB(0)9307 23363 y FD(>)602 b FJ(1,)j(from)570 b(Remark)h(5.9.1,)607 b(one)570 b(has)g FD(i)29837 23562 y FB(0)30964 23363 y FA(\031)602 b FD(k)45 b(=\034)148 b FJ(.)989 b(In)570 b(this)g(case,)606 b(\(5.189\))571 b(and)6299 24968 y(\(5.188\))435 b(hold)e(with)h FD(i)17004 25167 y FB(0)17899 24968 y FD(>)368 b FJ(1.)579 b(Hence)433 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)10449 31045 y(min)11373 31851 y FC(i)12838 29172 y Fz(\032)13835 31045 y FD(i)14279 29118 y Fz(\014)14279 29916 y(\014)14279 30713 y(\014)14279 31510 y(\014)14854 30146 y FD(k)p 14854 30739 722 54 v 14858 31956 a(\034)16003 31045 y FA(\000)295 b FD(\025)18090 31244 y FC(k)24 b(i)18979 29118 y Fz(\014)18979 29916 y(\014)18979 30713 y(\014)18979 31510 y(\014)19422 29469 y FC(\016)19927 29172 y Fz(\033)21292 31045 y FA(\025)22694 26727 y Fz(8)22694 27923 y(>)22694 28321 y(>)22694 28720 y(>)22694 29118 y(>)22694 29517 y(<)22694 31908 y(>)22694 32307 y(>)22694 32705 y(>)22694 33104 y(>)22694 33502 y(:)24429 27823 y FD(k)25150 27341 y FC(\016)25654 27823 y FD(=\034)27017 27341 y FC(\016)27523 27823 y FD(;)2469 b FJ(if)434 b FD(k)414 b FA(\025)369 b FD(\034)34750 27341 y FB(1)p FC(=)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))38130 27823 y FJ(;)24429 31056 y FD(k)45 b(=\034)26513 30574 y FC(\016)33 b FB(+1)28220 31056 y FD(;)1772 b FJ(if)434 b(2)p FD(\034)518 b(<)368 b(k)414 b(<)369 b(\034)37842 30574 y FB(1)p FC(=)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))41222 31056 y FJ(;)24429 34266 y FD(\015)25176 33784 y FC(\016)25680 34266 y FD(=\034)27043 33784 y FC(\016)27549 34266 y FD(;)2443 b FJ(if)434 b FD(\034)148 b(=)p FJ(2)370 b FD(<)e(k)415 b FA(\024)369 b FJ(2)p FD(\034)148 b FJ(.)46419 31045 y(\(5.196\))6299 37271 y(In)434 b(the)f(case)h FD(k)414 b FA(\025)369 b FD(\034)16123 36789 y FB(1)p FC(=)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))19503 37271 y FJ(,)434 b(w)-36 b(e)434 b(obtain)9222 39822 y Fz(\015)9222 40619 y(\015)9960 40951 y FA(jD)11354 41150 y FC(`)11793 40951 y FA(j)12162 40403 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))15333 39822 y Fz(\015)15333 40619 y(\015)16071 41472 y FC(s;s)17258 39822 y Fz(\015)17258 40619 y(\015)17996 40951 y FD(M)19393 40403 y Fw(\000)p FB(1)20650 40951 y FA(jD)22044 41150 y FC(k)22613 40951 y FA(j)22982 40403 y Fw(\000)p FC(\016)24219 39822 y Fz(\015)24219 40619 y(\015)24957 41472 y FC(s;s)26513 40951 y FA(\024)369 b FJ(const)34689 40053 y(1)p 31239 40646 7551 54 v 31239 41862 a FD(\015)31986 41479 y FB(1)p Fw(\000)p FC(\016)33914 41862 y FD(\034)34627 41479 y FC(\027)58 b FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))p Fw(\000)p FB(1)39276 40053 y FD(`)39822 39571 y FC(\027)58 b FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))p 39276 40646 3506 54 v 40416 41862 a FD(k)41137 41479 y FC(\016)43136 40951 y FD(;)2922 b FJ(\(5.197\))6299 44420 y(th)-36 b(us,)433 b(this)g(case)h(reduces)f(to)h(the)f(third)g(one)g(with)h FD(k)414 b(>)368 b FJ(2)p FD(\034)148 b FJ(.)6299 46025 y(In)434 b(the)f(case)h(2)p FD(\034)517 b(<)369 b(k)414 b(<)368 b(\034)19214 45543 y FB(1)p FC(=)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))22595 46025 y FJ(,)434 b(one)f(gets,)h(using)g(that) f FD(\027)84 b FJ(\(1)294 b FA(\000)i FD(\016)50 b FJ(\))368 b(=)h FD(\016)50 b FJ(,)10396 48650 y Fz(\015)10396 49447 y(\015)11134 49779 y FA(jD)12528 49978 y FC(`)12968 49779 y FA(j)13337 49231 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))16507 48650 y Fz(\015)16507 49447 y(\015)17245 50299 y FC(s;s)18432 48650 y Fz(\015)18432 49447 y(\015)19170 49779 y FD(M)20567 49231 y Fw(\000)p FB(1)21825 49779 y FA(jD)23219 49978 y FC(k)23787 49779 y FA(j)24156 49231 y Fw(\000)p FC(\016)25393 48650 y Fz(\015)25393 49447 y(\015)26131 50299 y FC(s;s)28425 49779 y FA(\024)1107 b FJ(const)36820 48881 y FD(\034)37533 48399 y FB(1+)p FC(\016)p 33889 49474 8283 54 v 33889 50690 a FD(\015)34636 50307 y FB(1)p Fw(\000)p FC(\016)36565 50690 y FD(\034)37278 50307 y FB(\()p FC(\027)58 b Fw(\000)p FB(1\)\(1)p Fw(\000)p FC(\016)33 b FB(\))42659 48881 y FD(`)43205 48399 y FC(\027)58 b FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))p 42659 49474 3506 54 v 44051 50690 a FD(k)28436 53582 y FJ(=)1117 b(const)33889 52683 y FD(\034)34602 52201 y FB(2)p Fw(\000)p FC(\016)33 b Fw(\000)p FB(1+)p FC(\016)p 33889 53277 4805 54 v 35064 54493 a FD(\015)35811 54109 y FB(1)p Fw(\000)p FC(\016)39048 51709 y Fz(\022)40246 52683 y FD(`)p 40158 53277 722 54 v 40158 54493 a(k)41012 51709 y Fz(\023)41990 52006 y FC(\016)28425 56757 y FA(\024)1107 b FJ(const)34760 55859 y FD(\034)p 33889 56452 2455 54 v 33889 57669 a(\015)34636 57285 y FB(1)p Fw(\000)p FC(\016)36698 56757 y FA(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)40326 56209 y FC(\016)46419 56757 y FJ(\(5.198\))6299 60300 y(In)434 b(the)f(last)h(case,)g FD(\034)148 b(=)p FJ(2)370 b FD(<)e(k)415 b FA(\024)369 b FJ(2)p FD(\034)148 b FJ(,)434 b(from)g(\(5.171\),)7408 62925 y Fz(\015)7408 63722 y(\015)8146 64054 y FA(jD)9540 64253 y FC(`)9979 64054 y FA(j)10348 63506 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))13519 62925 y Fz(\015)13519 63722 y(\015)14257 64574 y FC(s;s)15444 62925 y Fz(\015)15444 63722 y(\015)16182 64054 y FD(M)17579 63506 y Fw(\000)p FB(1)18837 64054 y FA(jD)20231 64253 y FC(k)20799 64054 y FA(j)21168 63506 y Fw(\000)p FC(\016)22405 62925 y Fz(\015)22405 63722 y(\015)23143 64574 y FC(s;s)24699 64054 y FA(\024)369 b FJ(const)29425 63155 y FD(\034)30138 62673 y FC(\016)p 29425 63748 1219 54 v 29660 64965 a FD(\015)32181 63155 y(`)32727 62673 y FC(\027)58 b FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))p 31130 63748 5607 54 v 31130 64965 a FD(\034)31843 64581 y FB(\()p FC(\027)58 b Fw(\000)p FB(1\)\(1)p Fw(\000)p FC(\016)33 b FB(\))37239 64054 y FJ(=)39921 63155 y FD(`)40467 62673 y FC(\027)58 b FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))p 38752 63748 5844 54 v 38752 64965 a FD(\015)293 b(\034)40433 64581 y FC(\027)58 b FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))p Fw(\000)p FB(1)44950 64054 y FD(;)1108 b FJ(\(5.199\))6299 67597 y(th)-36 b(us,)433 b(this)g(case)h(reduces)f(to)h(the)f(third)g (one)g(with)h FD(\034)148 b(=)p FJ(2)370 b FD(<)e(k)414 b FA(\024)369 b FJ(2)p FD(\034)148 b FJ(.)6299 69202 y(On)401 b(the)g(other)h(hand,)407 b(in)402 b(the)f(case)i FD(i)24455 69401 y FB(0)25349 69202 y FA(6)p FJ(=)369 b FD(j)27268 69401 y FB(0)27794 69202 y FJ(,)408 b(it)402 b(results)f FA(j)p FD(`)230 b FA(\000)h FD(k)45 b FA(j)369 b FD(>)g(\034)148 b(=)p FJ(4,)409 b(therefore,)f(from)6299 70807 y(Lemma)434 b(5.9.3,)12062 72467 y Fz(\014)12062 73264 y(\014)12062 74061 y(\014)12062 74858 y(\014)12721 73495 y FD(`)p 12637 74088 714 54 v 12637 75305 a(\034)13778 74394 y FA(\000)296 b FD(j)15645 74593 y FB(0)16466 74394 y FA(\000)17927 73495 y FD(k)p 17927 74088 722 54 v 17931 75305 a(\034)19076 74394 y FJ(+)f FD(i)20827 74593 y FB(0)21352 72467 y Fz(\014)21352 73264 y(\014)21352 74061 y(\014)21352 74858 y(\014)22164 74394 y FJ(=)23545 72467 y Fz(\014)23545 73264 y(\014)23545 74061 y(\014)23545 74858 y(\014)24120 73495 y FA(j)p FD(`)g FA(\000)h FD(k)45 b FA(j)p 24120 74088 3629 54 v 25578 75305 a FD(\034)28177 74394 y FA(\000)295 b FJ(\()p FD(j)30549 74593 y FB(0)31370 74394 y FA(\000)g FD(i)33142 74593 y FB(0)33668 74394 y FJ(\))34174 72467 y Fz(\014)34174 73264 y(\014)34174 74061 y(\014)34174 74858 y(\014)34986 74394 y FA(\025)369 b FJ(const)40568 73495 y FD(\034)41281 73013 y FC(\027)58 b Fw(\000)p FB(1)p 39712 74088 4205 54 v 39712 75305 a FA(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)43340 74921 y FC(\027)44270 74394 y FD(:)27371 78312 y FJ(201)p eop %%Page: 202 202 202 201 bop 0 4488 a FJ(W)-108 b(e)434 b(ha)-36 b(v)g(e)433 b(t)-36 b(w)g(o)434 b(cases,)h(from)f(Lemma)f(5.9.8,)10904 6077 y Fz(\014)10904 6874 y(\014)10904 7671 y(\014)10904 8468 y(\014)11564 7104 y FD(`)p 11480 7697 714 54 v 11480 8914 a(\034)12621 8003 y FA(\000)295 b FD(j)14487 8202 y FB(0)15013 6077 y Fz(\014)15013 6874 y(\014)15013 7671 y(\014)15013 8468 y(\014)15825 8003 y FA(\025)369 b FJ(const)21408 7104 y FD(\034)22121 6622 y FC(\027)58 b Fw(\000)p FB(1)p 20551 7697 4205 54 v 20551 8914 a FA(j)p FD(`)295 b FA(\000)h FD(k)45 b FA(j)24180 8530 y FC(\027)25257 8003 y FA(\025)369 b FJ(const)29983 7104 y FD(\034)30696 6622 y FC(\027)58 b Fw(\000)p FB(1)p 29983 7697 2492 54 v 30382 8914 a FD(`)30928 8530 y FC(\027)g(\014)32829 8003 y FD(;)6930 b FJ(\(5.200\))0 11142 y(or)10947 11278 y Fz(\014)10947 12075 y(\014)10947 12872 y(\014)10947 13669 y(\014)11523 12305 y FD(k)p 11523 12898 722 54 v 11527 14115 a(\034)12672 13204 y FA(\000)296 b FD(i)14445 13403 y FB(0)14970 11278 y Fz(\014)14970 12075 y(\014)14970 12872 y(\014)14970 13669 y(\014)15782 13204 y FA(\025)369 b FJ(const)21364 12305 y FD(\034)22077 11823 y FC(\027)58 b Fw(\000)p FB(1)p 20508 12898 4205 54 v 20508 14115 a FA(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)24136 13731 y FC(\027)25214 13204 y FA(\025)369 b FJ(const)29940 12305 y FD(\034)30653 11823 y FC(\027)58 b Fw(\000)p FB(1)p 29940 12898 2492 54 v 30251 14115 a FD(k)30972 13731 y FC(\027)g(\014)32786 13204 y FD(:)6973 b FJ(\(5.201\))0 16291 y(Consider)446 b(at)f(\014rst)g(\(5.200\).)616 b(Since)445 b FD(")389 b FJ(=)g FD(\021)21180 15809 y FB(3)21706 16291 y FJ(,)449 b(from)d(the)f(asymptotic)h(estimates)g(for)g(the)0 17896 y(eigen)-36 b(v)-72 b(alues)434 b(it)g(results)f(that)4674 21386 y FA(j)p FD(\026)5826 21585 y FC(`;j)6852 21708 y Fu(0)7368 21386 y FA(j)1118 b FJ(=)10984 19460 y Fz(\014)10984 20257 y(\014)10984 21054 y(\014)10984 21851 y(\014)11643 20487 y FD(`)p 11560 21080 714 54 v 11560 22297 a(\034)12701 21386 y FA(\000)295 b FD(\025)14788 21585 y FC(`j)15553 21708 y Fu(0)16069 19460 y Fz(\014)16069 20257 y(\014)16069 21054 y(\014)16069 21851 y(\014)16881 21386 y FJ(=)18262 19460 y Fz(\014)18262 20257 y(\014)18262 21054 y(\014)18262 21851 y(\014)18921 20487 y FD(`)p 18837 21080 V 18837 22297 a(\034)19979 21386 y FA(\000)21307 19702 y Fz(q)p 22635 19702 3975 54 v 1684 x FD(j)23173 21585 y FB(0)23699 20865 y(2)24520 21386 y FJ(+)g FD(\026)g FA(\000)g FD(\021)28923 20837 y FB(2)29448 21386 y FJ(\()p FD(B)67 b(I)31583 21585 y FB(0)32109 21386 y FJ(\))32615 21585 y FC(j)32996 21708 y Fu(0)33807 21386 y FJ(+)295 b FD(O)36144 19513 y Fz(\022)37257 20487 y FD(")p 37255 21080 613 54 v 37255 22297 a(j)38000 19513 y Fz(\023)38978 19460 y(\014)38978 20257 y(\014)38978 21054 y(\014)38978 21851 y(\014)8855 25042 y FJ(=)10984 23116 y Fz(\014)10984 23913 y(\014)10984 24710 y(\014)10984 25507 y(\014)11643 24143 y FD(`)p 11560 24736 714 54 v 11560 25953 a(\034)12701 25042 y FA(\000)g FD(j)14567 25241 y FB(0)15388 25042 y FJ(+)g FD(O)17725 23169 y Fz(\022)18836 24143 y FD(\026)p 18836 24736 783 54 v 18921 25953 a(j)19751 23169 y Fz(\023)21024 25042 y FJ(+)g FD(O)23361 23169 y Fz(\022)24472 24143 y FD(\021)25162 23661 y FB(2)p 24472 24736 1216 54 v 24773 25953 a FD(j)25820 23169 y Fz(\023)27093 25042 y FJ(+)g FD(O)29430 23169 y Fz(\022)30543 24143 y FD(")p 30541 24736 613 54 v 30541 25953 a(j)31286 23169 y Fz(\023)32264 23116 y(\014)32264 23913 y(\014)32264 24710 y(\014)32264 25507 y(\014)8844 28673 y FA(\025)10984 26746 y Fz(\014)10984 27544 y(\014)10984 28341 y(\014)10984 29138 y(\014)11643 27774 y FD(`)p 11560 28367 714 54 v 11560 29584 a(\034)12701 28673 y FA(\000)g FD(j)14567 28872 y FB(0)15093 26746 y Fz(\014)15093 27544 y(\014)15093 28341 y(\014)15093 29138 y(\014)15831 28673 y FA(\000)17433 27774 y FD(\026)p 17292 28367 1064 54 v 17292 29584 a(j)17830 29783 y FB(0)18711 28673 y FD(:)21048 b FJ(\(5.202\))0 32162 y(Using)434 b(Lemma)f(5.9.5,)j(w)-36 b(e)434 b(obtain)5263 35676 y FA(j)p FD(\026)6415 35875 y FC(`;j)7441 35998 y Fu(0)7958 35676 y FA(j)369 b(\025)10098 33750 y Fz(\014)10098 34547 y(\014)10098 35344 y(\014)10098 36141 y(\014)10757 34778 y FD(`)p 10674 35371 714 54 v 10674 36588 a(\034)11815 35676 y FA(\000)295 b FD(j)13681 35875 y FB(0)14207 33750 y Fz(\014)14207 34547 y(\014)14207 35344 y(\014)14207 36141 y(\014)14945 35676 y FA(\000)16547 34778 y FD(\026)p 16406 35371 1064 54 v 16406 36588 a(j)16944 36787 y FB(0)17972 35676 y FA(\025)19507 34778 y FJ(1)p 19507 35371 651 54 v 19507 36588 a(2)20290 33750 y Fz(\014)20290 34547 y(\014)20290 35344 y(\014)20290 36141 y(\014)20949 34778 y FD(`)p 20866 35371 714 54 v 20866 36588 a(\034)22007 35676 y FA(\000)g FD(j)23873 35875 y FB(0)24399 33750 y Fz(\014)24399 34547 y(\014)24399 35344 y(\014)24399 36141 y(\014)25211 35676 y FA(\025)369 b FJ(const)30794 34778 y FD(\034)31507 34296 y FC(\027)58 b Fw(\000)p FB(1)p 29937 35371 4205 54 v 29937 36588 a FA(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)33565 36204 y FC(\027)34496 35676 y FD(:)5263 b FJ(\(5.203\))0 39166 y(Hence,)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)13426 40451 y Fz(\015)13426 41248 y(\015)14164 41580 y FA(jD)15558 41779 y FC(`)15997 41580 y FA(j)16366 41031 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))19537 40451 y Fz(\015)19537 41248 y(\015)20275 42100 y FC(s;s)21831 41580 y FA(\024)23366 40681 y(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)26994 40199 y FC(\027)58 b FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))p 23366 41274 6588 54 v 23856 42491 a FD(\034)24569 42107 y FB(\()p FC(\027)58 b Fw(\000)p FB(1\)\(1)p Fw(\000)p FC(\016)33 b FB(\))30308 41580 y FD(:)9451 b FJ(\(5.204\))0 44260 y(Therefore)434 b(it)g(follo)-36 b(ws)435 b(that,)f(using)f (\(5.179\))i(and)e FD(k)414 b(>)369 b(\034)148 b(=)p FJ(2,)434 b(b)-36 b(y)434 b(\(5.204\),)3408 46958 y Fz(\015)3408 47755 y(\015)4146 48087 y FA(jD)5540 48286 y FC(`)5979 48087 y FA(j)6348 47539 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))9519 46958 y Fz(\015)9519 47755 y(\015)10257 48608 y FC(s;s)11665 46958 y Fz(\015)11665 47755 y(\015)12403 48087 y FD(M)13800 47539 y Fw(\000)p FB(1)15058 48087 y FA(jD)16452 48286 y FC(k)17021 48087 y FA(j)17390 47539 y Fw(\000)p FC(\016)18626 46958 y Fz(\015)18626 47755 y(\015)19364 48608 y FC(s;s)21658 48087 y FA(\024)1107 b FJ(const)29186 47189 y FA(j)p FD(`)295 b FA(\000)h FD(k)45 b FA(j)32815 46707 y FC(\016)p 27122 47782 8262 54 v 27122 48999 a FD(\015)27869 48615 y FC(\016)28596 48999 y FD(\034)29309 48615 y FB(\()p FC(\027)58 b Fw(\000)p FB(1\)\(1)p Fw(\000)p FC(\016)33 b FB(\))p Fw(\000)p FC(\016)35517 46214 y Fz(\022)36631 47189 y FD(\034)p 36627 47782 722 54 v 36627 48999 a(k)37481 46214 y Fz(\023)38459 46512 y FB(1)p Fw(\000)p FC(\027)58 b(\016)21658 51788 y FA(\024)1107 b FJ(const)28230 50889 y FA(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)31858 50407 y FC(\016)p 27122 51482 6349 54 v 27122 52699 a FD(\015)27869 52315 y FC(\016)28596 52699 y FD(\034)29309 52315 y FC(\027)58 b FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))p Fw(\000)p FB(1)33825 51788 y FD(:)5934 b FJ(\(5.205\))0 55258 y(Let)433 b(us)g(consider,)h(no)-36 b(w,)434 b(\(5.201\).)0 56863 y(Using)g(Lemma)f(5.9.5,)j(w)-36 b(e)434 b(obtain)5363 60471 y FA(j)p FD(\026)6515 60670 y FC(k)24 b(;i)7610 60793 y Fu(0)8126 60471 y FA(j)369 b(\025)10266 58545 y Fz(\014)10266 59342 y(\014)10266 60139 y(\014)10266 60936 y(\014)10842 59573 y FD(k)p 10842 60166 722 54 v 10846 61383 a(\034)11991 60471 y FA(\000)295 b FD(i)13763 60670 y FB(0)14289 58545 y Fz(\014)14289 59342 y(\014)14289 60139 y(\014)14289 60936 y(\014)15027 60471 y FA(\000)16581 59573 y FD(\026)p 16488 60166 970 54 v 16488 61383 a(i)16932 61582 y FB(0)17959 60471 y FA(\025)19494 59573 y FJ(1)p 19494 60166 651 54 v 19494 61383 a(2)20277 58545 y Fz(\014)20277 59342 y(\014)20277 60139 y(\014)20277 60936 y(\014)20853 59573 y FD(k)p 20853 60166 722 54 v 20857 61383 a(\034)22002 60471 y FA(\000)g FD(i)23774 60670 y FB(0)24300 58545 y Fz(\014)24300 59342 y(\014)24300 60139 y(\014)24300 60936 y(\014)25112 60471 y FA(\025)369 b FJ(const)30328 59573 y FD(\034)31041 59091 y FB(\()p FC(\027)58 b Fw(\000)p FB(1\))p 29838 60166 4205 54 v 29838 61383 a FA(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)33466 60999 y FC(\027)34396 60471 y FD(:)5363 b FJ(\(5.206\))0 64035 y(Moreo)-36 b(v)g(er)588 b FD(i)6306 64234 y FB(0)7464 64035 y FA(\031)632 b FD(k)45 b(=\034)736 b FJ(for)589 b FD(i)14387 64234 y FB(0)15544 64035 y FA(\025)632 b FJ(1)589 b(b)-36 b(y)588 b(Remark)g(5.9.1)i(and)e(b)36 b(eing)588 b(3)p FD(\034)780 b(>)631 b(k)677 b FA(\025)632 b FD(\034)148 b(=)p FJ(2,)0 65640 y(therefore)12671 67941 y FD(i)13115 68140 y FB(0)13641 66015 y Fz(\014)13641 66812 y(\014)13641 67609 y(\014)13641 68406 y(\014)14216 67042 y FD(k)p 14216 67635 722 54 v 14220 68852 a(\034)15365 67941 y FA(\000)296 b FD(\025)17453 68140 y FC(k)24 b(i)18287 68263 y Fu(0)18803 66015 y Fz(\014)18803 66812 y(\014)18803 67609 y(\014)18803 68406 y(\014)19246 66365 y FC(\016)20120 67941 y FA(\025)21655 67042 y FD(k)p 21655 67635 V 21659 68852 a(\034)22730 67941 y FJ(const)26545 67042 y FD(\034)27258 66560 y FB(\()p FC(\027)58 b Fw(\000)p FB(1\))p FC(\016)p 26054 67635 4654 54 v 26054 68852 a FA(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)29682 68468 y FC(\027)58 b(\016)31062 67941 y FD(:)0 70898 y FJ(Th)-36 b(us)433 b(it)h(follo)-36 b(ws)435 b(that)12861 74394 y(min)13785 75200 y FC(i)15250 72521 y Fz(\032)16246 74394 y FD(i)16690 72467 y Fz(\014)16690 73264 y(\014)16690 74061 y(\014)16690 74858 y(\014)17266 73495 y FD(k)p 17266 74088 722 54 v 17270 75305 a(\034)18415 74394 y FA(\000)295 b FD(\025)20502 74593 y FC(k)24 b(i)21391 72467 y Fz(\014)21391 73264 y(\014)21391 74061 y(\014)21391 74858 y(\014)21834 72818 y FC(\016)22339 72521 y Fz(\033)23704 74394 y FA(\025)25239 73495 y FD(k)p 25239 74088 V 25243 75305 a(\034)26938 73495 y(\034)27651 73013 y FB(\()p FC(\027)58 b Fw(\000)p FB(1\))p FC(\016)p 26447 74088 4654 54 v 26447 75305 a FA(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)30075 74921 y FC(\027)58 b(\016)21072 78312 y FJ(202)p eop %%Page: 203 203 203 202 bop 6299 4488 a FJ(hence)19663 5532 y Fz(\015)19663 6330 y(\015)20401 6662 y FD(M)21798 6113 y Fw(\000)p FB(1)23056 6662 y FA(jD)24450 6861 y FC(k)25018 6662 y FA(j)25387 6113 y Fw(\000)p FC(\016)26624 5532 y Fz(\015)26624 6330 y(\015)27362 7182 y FC(s;s)28918 6662 y FA(\024)30457 5763 y FD(\034)p 30453 6356 722 54 v 30453 7573 a(k)31661 5763 y FA(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)35289 5281 y FC(\027)58 b(\016)p 31661 6356 4654 54 v 32151 7573 a FD(\034)32864 7189 y FB(\()p FC(\027)g Fw(\000)p FB(1\))p FC(\016)36669 6662 y FD(:)9389 b FJ(\(5.207\))6299 9477 y(F)-108 b(rom)433 b(\(5.207\))i(and)e(using)h(that)f FD(\027)84 b FJ(\(1)294 b FA(\000)i FD(\016)50 b FJ(\))368 b(=)h FD(\016)50 b FJ(,)434 b(it)f(follo)-36 b(ws)436 b(the)d(estimate)10825 13360 y FA(kjD)12883 13559 y FC(`)13323 13360 y FA(j)13692 12812 y Fw(\000)p FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))16862 13360 y FA(k)17526 13559 y FC(s;s)18714 13360 y FA(k)p FD(M)20775 12812 y Fw(\000)p FB(1)22032 13360 y FA(jD)23426 13559 y FC(k)23995 13360 y FA(j)24364 12812 y Fw(\000)p FC(\016)25601 13360 y FA(k)26265 13559 y FC(s;s)28559 13360 y FA(\024)30836 12461 y FD(\034)p 30832 13054 722 54 v 30832 14271 a(k)34428 12461 y(`)34974 11979 y FC(\027)58 b FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))p 32040 13054 8283 54 v 32040 14271 a FD(\015)32787 13887 y FB(1)p Fw(\000)p FC(\016)34715 14271 y FD(\034)35428 13887 y FB(\()p FC(\027)58 b Fw(\000)p FB(1\)\(1)p Fw(\000)p FC(\016)33 b FB(\))40809 12461 y FA(j)p FD(`)295 b FA(\000)h FD(k)45 b FA(j)44438 11979 y FC(\027)58 b(\016)p 40809 13054 4654 54 v 41300 14271 a FD(\034)42013 13887 y FB(\()p FC(\027)g Fw(\000)p FB(1\))p FC(\016)28559 17017 y FA(\024)30832 16118 y FD(`)31378 15636 y FC(\027)g FB(\(1)p Fw(\000)p FC(\016)33 b FB(\))34558 16118 y FA(j)p FD(`)295 b FA(\000)h FD(k)45 b FA(j)38187 15636 y FC(\027)58 b(\016)39433 16118 y FD(\034)40146 15636 y FB(\(2)p Fw(\000)p FC(\027)g FB(\))p 30832 16712 11825 54 v 33579 17928 a FD(\015)34326 17545 y FB(1)p Fw(\000)p FC(\016)36255 17928 y FD(k)36976 17545 y FB(1)p Fw(\000)p FC(\016)38683 17928 y FD(k)39404 17545 y FC(\016)28570 20820 y FJ(=)30699 18947 y Fz(\022)31897 19921 y FD(`)p 31810 20514 722 54 v 31810 21731 a(k)32663 18947 y Fz(\023)33641 19244 y FC(\016)34500 19921 y FA(j)p FD(`)295 b FA(\000)h FD(k)45 b FA(j)38129 19439 y FC(\027)58 b(\016)39375 19921 y FD(\034)40088 19439 y FB(\(2)p Fw(\000)p FC(\027)g FB(\))p 34500 20514 8099 54 v 35998 21731 a FD(\015)36745 21347 y FB(1)p Fw(\000)p FC(\016)38673 21731 y FD(k)39394 21347 y FB(1)p Fw(\000)p FC(\016)28559 24570 y FA(\024)30832 23671 y(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)34460 23189 y FB(\()p FC(\027)58 b FB(+1\))p FC(\016)37641 23671 y FD(\034)38354 23189 y FB(\(2)p Fw(\000)p FC(\027)g FB(\))p 30832 24264 10033 54 v 33296 25481 a FD(\015)34043 25097 y FB(1)p Fw(\000)p FC(\016)35972 25481 y FD(k)36693 25097 y FB(1)p Fw(\000)p FC(\016)28570 28372 y FJ(=)30832 27474 y FA(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)34460 26992 y FC(\027)35257 27474 y FD(\034)35970 26992 y FB(\(2)p Fw(\000)p FC(\027)58 b Fw(\000)p FB(1+)p FC(\016)33 b FB(\))p 30832 28067 V 34621 29284 a FD(\015)35368 28900 y FB(1)p Fw(\000)p FC(\016)41218 26499 y Fz(\022)42333 27474 y FD(\034)p 42329 28067 722 54 v 42329 29284 a(k)43183 26499 y Fz(\023)44161 26797 y FB(1)p Fw(\000)p FC(\016)28559 32073 y FA(\024)1107 b FJ(const)221 b FA(j)p FD(`)295 b FA(\000)g FD(k)45 b FA(j)37518 31524 y FC(\027)38448 31174 y FD(\034)39161 30692 y FB(1)p Fw(\000)p FC(\027)58 b FB(+)p FC(\016)p 38448 31767 3673 54 v 39058 32984 a FD(\015)39805 32600 y FB(1)p Fw(\000)p FC(\016)42475 32073 y FD(;)3583 b FJ(\(5.208\))6299 35744 y(where)434 b(w)-36 b(e)434 b(used)e(\(5.182\))j(and)e FD(k)414 b(>)369 b(\034)148 b(=)p FJ(2.)22638 b Fn(\003)6299 40319 y FH(5.10)2152 b(Measure)716 b(estimates)6299 43285 y FJ(In)478 b(this)g(section)g(w) -36 b(e)478 b(will)i(giv)-36 b(e)479 b(an)f(estimate)g(from)h(b)36 b(elo)-36 b(w)478 b(of)h(the)f(measure)f(of)i(the)f(set)6299 44890 y FA(C)e FJ(of)398 b(admissible)h(frequencies.)566 b(W)-108 b(e)398 b(\014rst)f(need)g(the)g(follo)-36 b(wing)400 b(standard)d(result)h(on)f(the)6299 46495 y(measure)434 b(of)g(Diophan)-36 b(tine)433 b(n)-36 b(um)g(b)36 b(ers.)6299 49318 y FE(Lemma)499 b(5.10.1.)555 b Fs(L)-66 b(et)8228 52775 y FD(D)405 b FJ(:=)11455 50902 y Fz(\032)12451 52775 y FD($)g(>)369 b FJ(0)443 b(s)p FD(:)p FJ(t)p FD(:)19032 51646 y Fz(\014)19032 52443 y(\014)19474 52775 y FD($)36 b(`)295 b FA(\000)g FD(j)23367 51646 y Fz(\014)23367 52443 y(\014)24179 52775 y FD(>)25692 51877 y($)26803 51395 y FB(1+)p FC(\017)p 25692 52470 2749 54 v 26472 53687 a FD(j)27085 53303 y FC(\027)28574 52775 y FD(;)443 b FA(8)221 b FD(`;)g(j)445 b FA(\025)369 b FJ(1)34500 50902 y Fz(\033)35939 52775 y FD(;)2082 b FJ(1)369 b FD(<)g(\027)452 b(<)369 b FJ(2)221 b FD(;)1929 b FJ(\(5.209\))6299 56466 y Fs(then)20430 58755 y FJ(lim)19668 59653 y FC($)20449 59776 y Fu(0)20910 59653 y Fw(!)p FB(0)22321 59401 y Fu(+)23353 57772 y FJ(meas)26399 56696 y Fz(\000)27008 57772 y FJ(\(0)p FD(;)221 b($)29821 57971 y FB(0)30347 57772 y FJ(])296 b FA(\\)f FD(D)33301 56696 y Fz(\001)p 23353 58450 10557 54 v 27831 59666 a FD($)28906 59865 y FB(0)34411 58755 y FJ(=)369 b(1)221 b FD(:)9395 b FJ(\(5.210\))6299 62160 y Fp(pr)-30 b(oof.)659 b FJ(Let)433 b(b)36 b(e)14760 65592 y FD(D)15840 65791 y FC(`j)17080 65592 y FJ(:=)18822 63719 y Fz(\032)19818 65592 y FD($)405 b(>)368 b FJ(0)443 b(s)p FD(:)p FJ(t)p FD(:)26398 64065 y Fz(\014)26398 64862 y(\014)26398 65659 y(\014)27474 64694 y FD(`)p 26974 65287 1547 54 v 26974 66504 a($)28085 66120 y FC(\017)28948 65592 y FA(\000)31477 64694 y FD(j)p 30409 65287 2749 54 v 30409 66504 a($)31520 66120 y FB(1+)p FC(\017)33291 64065 y Fz(\014)33291 64862 y(\014)33291 65659 y(\014)34103 65592 y FA(\024)35907 64694 y FJ(1)p 35638 65287 1189 54 v 35638 66504 a FD(j)36251 66120 y FC(\027)37328 65592 y FJ(=:)369 b FD(\016)39646 65791 y FC(j)40133 63719 y Fz(\033)41572 65592 y FD(:)6299 69356 y FJ(Let)377 b(b)36 b(e)378 b FD($)404 b FJ(:=)369 b FD($)14584 69555 y FC(`;j)15897 69356 y FJ(+)411 b(~)-880 b FD($)404 b FA(2)369 b FD(D)20904 69555 y FC(`;j)22414 69356 y FJ(with)377 b FD($)26395 69555 y FC(`;j)27896 69356 y FJ(:=)369 b FD(j)75 b(=`)378 b FJ(satisfying)h FD($)38639 69555 y FC(`;j)39771 69356 y FD(`)181 b FA(\000)g FD(j)443 b FJ(=)369 b(0.)560 b(W)-108 b(e)377 b(ha)-36 b(v)g(e)6299 70962 y(to)434 b(estimate)g(the)f(measure)g(of)i(the)e(translated)g (set)11748 74058 y(~)11443 74394 y FD(D)12523 74593 y FC(`j)13763 74394 y FJ(:=)15505 72521 y Fz(\032)16732 74394 y FJ(~)-881 b FD($)479 b FJ(s)p FD(:)p FJ(t)p FD(:)20682 72866 y Fz(\014)20682 73663 y(\014)20682 74460 y(\014)23969 73495 y FD(`)p 21257 74088 5969 54 v 21257 75305 a FJ(\()p FD(j)75 b(=`)295 b FJ(+)525 b(~)-880 b FD($)36 b FJ(\))26791 74921 y FC(\017)27654 74394 y FA(\000)32395 73495 y FD(j)p 29116 74088 7172 54 v 29116 75305 a FJ(\()p FD(j)75 b(=`)295 b FJ(+)525 b(~)-881 b FD($)36 b FJ(\))34649 74921 y FB(1+)p FC(\017)36420 72866 y Fz(\014)36420 73663 y(\014)36420 74460 y(\014)37231 74394 y FA(\024)370 b FD(\016)39210 74593 y FC(j)39697 72521 y Fz(\033)40914 74394 y FD(:)5144 b FJ(\(5.211\))27371 78312 y(203)p eop %%Page: 204 204 204 203 bop 0 4488 a FJ(De\014ne)7332 6648 y FD(f)142 b FJ(\()230 b(~)-880 b FD($)36 b FJ(\))368 b(:=)15193 5749 y FD(`)p 12481 6342 5969 54 v 12481 7559 a FJ(\()p FD(j)75 b(=`)295 b FJ(+)525 b(~)-880 b FD($)36 b FJ(\))18015 7175 y FC(\017)18878 6648 y FA(\000)23619 5749 y FD(j)p 20339 6342 7172 54 v 20339 7559 a FJ(\()p FD(j)75 b(=`)295 b FJ(+)525 b(~)-880 b FD($)36 b FJ(\))25873 7175 y FB(1+)p FC(\017)28012 6648 y FJ(=)31369 5749 y(~)-880 b FD($)36 b(`)32796 5267 y FB(2+)p FC(\017)p 29526 6342 6521 54 v 29526 7559 a FJ(\()p FD(j)370 b FJ(+)294 b FD(`)230 b FJ(~)-880 b FD($)36 b FJ(\))34409 7175 y FB(1+)p FC(\017)36401 6648 y FD(:)0 10052 y FJ(W)-108 b(e)434 b(ha)-36 b(v)g(e)433 b(that)g(meas)q(\()11693 9716 y(~)11388 10052 y FD(D)12468 10251 y FC(`j)13339 10052 y FJ(\))369 b FA(\030)g FD(\016)16192 10251 y FC(j)16679 10052 y FD(=)p FA(j)p FD(@)18553 10251 y FB(~)-638 b FC($)19247 10052 y FD(f)142 b FJ(\(0\))p FA(j)p FJ(.)579 b(Compute)2352 13911 y FD(@)3207 14110 y FB(~)-638 b FC($)3902 13911 y FD(f)142 b FJ(\()229 b(~)-879 b FD($)35 b FJ(\))369 b(=)8689 13012 y FD(`)9235 12530 y FB(2+)p FC(\017)10873 13012 y FJ(\()p FD(j)h FJ(+)295 b FD(`)230 b FJ(~)-880 b FD($)35 b FJ(\))15756 12530 y FB(1+)p FC(\017)17689 13012 y FA(\000)295 b FJ(\(1)h(+)f FD(\017)p FJ(\))230 b(~)-880 b FD($)35 b(`)24463 12530 y FB(3+)p FC(\017)26101 13012 y FJ(\()p FD(j)370 b FJ(+)295 b FD(`)230 b FJ(~)-880 b FD($)35 b FJ(\))30984 12530 y FC(\017)p 8689 13605 22731 54 v 16193 14822 a FJ(\()p FD(j)370 b FJ(+)294 b FD(`)230 b FJ(~)-880 b FD($)36 b FJ(\))21076 14438 y FB(2\(1+)p FC(\017)p FB(\))31921 13911 y FJ(=)33435 13012 y FD(`)33981 12530 y FB(2+)p FC(\017)35619 13012 y FJ(\()p FD(j)369 b FJ(+)295 b FD(\017`)230 b FJ(~)-880 b FD($)36 b FJ(\))p 33435 13605 7592 54 v 33970 14822 a(\()p FD(j)370 b FJ(+)295 b FD(`)230 b FJ(~)-880 b FD($)35 b FJ(\))38853 14438 y FB(2+)p FC(\017)41381 13911 y FD(;)0 17613 y FJ(th)-36 b(us)17917 19772 y FD(@)18772 19971 y FB(~)-638 b FC($)19466 19772 y FD(f)142 b FJ(\(0\))369 b(=)23827 18873 y FD(`)24373 18391 y FB(2+)p FC(\017)p 23794 19467 2251 54 v 23794 20683 a FD(j)24407 20300 y FB(1+)p FC(\017)0 22808 y FJ(and)8556 24967 y(meas\()p FD(D)12967 25166 y FC(`j)13838 24967 y FJ(\))g FA(\024)g FJ(meas\()19751 24631 y(~)19446 24967 y FD(D)20526 25166 y FC(`j)21397 24967 y FJ(\))g FA(\030)g FD(\016)24250 25166 y FC(j)24870 24068 y FD(j)25483 23586 y FB(1+)p FC(\017)p 24870 24661 V 24904 25878 a FD(`)25450 25494 y FB(2+)p FC(\017)27623 24967 y FJ(=)31655 24068 y(1)p 29136 24661 5687 54 v 29136 25878 a FD(`)29682 25494 y FB(2+)p FC(\017)31320 25878 y FD(j)31933 25494 y FC(\027)58 b Fw(\000)p FB(1)p Fw(\000)p FC(\017)35177 24967 y FD(:)4582 b FJ(\(5.212\))0 28076 y(Being,)434 b(b)-36 b(y)434 b(the)f (de\014nition)f(of)j FD(D)16417 28275 y FC(`j)17288 28076 y FJ(,)f FD(D)19163 28275 y FC(`j)20403 28076 y FJ(=)369 b FA(;)433 b FJ(if)i FD(`)368 b(<)h(j)75 b(=)p FJ(2)p FD($)29357 28275 y FB(0)29883 28076 y FJ(,)434 b(one)f(has)13330 31379 y(\(0)p FD(;)221 b($)16143 31578 y FB(0)16669 31379 y FJ(\))295 b FA(\\)g FD(D)19767 30830 y FC(c)20599 31379 y FJ(=)22059 30117 y Fz([)21980 32923 y FC(j)51 b Fw(\025)p FB(1)25405 30117 y Fz([)24278 32995 y FC(`)p Fw(\025)p FC(j)g(=)p FB(2)p FC($)27547 33118 y Fu(0)28230 31379 y FD(D)29310 31578 y FC(`j)30403 31379 y FD(:)0 35881 y FJ(Therefore)434 b(b)-36 b(y)434 b(\(5.212\),)4563 39589 y(meas)7388 38513 y Fz(\000)7997 39589 y FJ(\(0)p FD(;)221 b($)10810 39788 y FB(0)11336 39589 y FJ(\))295 b FA(\\)g FD(D)14434 39040 y FC(c)14897 38513 y Fz(\001)16613 39589 y FA(\024)1107 b FJ(const)21944 37780 y Fz(Z)23272 38131 y Fw(1)22682 40788 y FB(1)24490 37780 y Fz(Z)25818 38131 y Fw(1)25228 40788 y FC(j)51 b(=)p FB(2)p FC($)27381 40911 y Fu(0)30771 38690 y FJ(1)p 28253 39283 V 28253 40500 a FD(j)28866 40116 y FC(\027)58 b Fw(\000)p FB(1)p Fw(\000)p FC(\017)31756 40500 y FD(`)32302 40116 y FB(2+)p FC(\017)34294 39589 y FD(d`)221 b(d)-145 b(j)16624 43291 y FJ(=)1117 b(const)221 b FD($)23019 43490 y FB(0)23544 42742 y(1+)p FC(\017)25404 41483 y Fz(Z)26732 41833 y Fw(1)26142 44490 y FB(1)28352 42392 y FJ(1)p 28083 42985 1189 54 v 28083 44202 a FD(j)28696 43818 y FC(\027)29626 43291 y FD(d)-145 b(j)444 b FJ(=)368 b(const)222 b FD($)36786 43490 y FB(0)37311 42742 y(1+)p FC(\017)39170 43291 y FD(:)0 46949 y FJ(The)434 b(thesis)f(follo)-36 b(ws.)43061 48586 y Fn(\003)0 51460 y FE(Prop)42 b(osition)712 b(5.10.2.)648 b Fs(L)-66 b(et)634 b FD(A)16555 51659 y FC(n)17864 51460 y FJ(:=)19920 50384 y Fz(\010)20695 51460 y FD($)718 b FA(2)683 b FD(A)25032 51659 y FC(n)p Fw(\000)p FB(1)27768 51460 y Fs(s.t.)1040 b FD($)718 b FA(2)683 b FJ(\001)35015 50978 y FC(\027)35015 51788 y(n)35641 51460 y FJ(\()p FD(\030)36715 51659 y FC(n)p Fw(\000)p FB(1)38543 51460 y FJ(\))39049 50384 y Fz(\011)39824 51460 y Fs(,)676 b FD(A)41874 51659 y FB(0)43083 51460 y FJ(=)0 53065 y(\(0)p FD(;)221 b($)2813 53264 y Fw(\003)3339 53065 y FJ(\))p Fs(,)464 b(as)i(in)e FJ(\(5.79\))p Fs(.)598 b(De\014ne)18663 55000 y FA(C)447 b FJ(:=)21550 53738 y Fz(\\)22003 56527 y FC(n)23248 55000 y FD(A)24223 55199 y FC(n)25071 55000 y FD(:)0 58526 y Fs(Then)14228 60846 y FJ(lim)13466 61744 y FC($)14247 61867 y Fu(0)14709 61744 y Fw(!)p FB(0)16120 61492 y Fu(+)17151 59863 y FJ(meas)20197 58787 y Fz(\000)20806 59863 y FA(C)373 b(\\)295 b FJ(\(0)p FD(;)221 b($)25872 60062 y FB(0)26398 59863 y FJ(\))26904 58787 y Fz(\001)p 17151 60541 10363 54 v 21532 61757 a FD($)22607 61956 y FB(0)28015 60846 y FJ(=)368 b(1)221 b FD(:)9493 b FJ(\(5.213\))0 64302 y Fp(pr)-30 b(oof.)658 b FJ(F)-108 b(or)433 b(an)-36 b(y)434 b FD($)10854 64501 y FB(0)11749 64302 y FD(>)368 b FJ(0)434 b(\014xed)f(w)-36 b(e)434 b(note)g(that)16428 67333 y FA(C)373 b(\\)295 b FJ(\(0)p FD(;)221 b($)21494 67532 y FB(0)22020 67333 y FJ(\))369 b FA(\033)g FD(D)332 b FA(\\)27137 66997 y FJ(~)26889 67333 y FA(C)12532 b FJ(\(5.214\))0 70363 y(where)523 b FD(D)560 b FJ(is)523 b(de\014ned)f(in)i(\(5.209\))g(and)20426 70027 y(~)20178 70363 y FA(C)600 b FJ(=)23258 70027 y(~)23010 70363 y FA(C)78 b FJ(\()p FD($)25368 70562 y FB(0)25893 70363 y FJ(\))523 b FA(\032)f FJ(\(0)p FD(;)221 b($)31290 70562 y FB(0)31817 70363 y FJ(\))523 b(is)h(de\014ned)d(in)j(suc)-36 b(h)522 b(a)0 71968 y(w)-36 b(a)g(y)434 b(that)16216 73829 y(\(0)p FD(;)221 b($)19029 74028 y FB(0)19555 73829 y FJ(\))295 b FA(n)21563 73493 y FJ(~)21315 73829 y FA(C)447 b FJ(:=)24352 72567 y Fz([)24203 75373 y FC(n)p Fw(\025)p FB(0)26197 73829 y FD(B)27252 73281 y FC(n)40120 73829 y FJ(\(5.215\))21072 78312 y(204)p eop %%Page: 205 205 205 204 bop 6299 4488 a FJ(where)19839 6094 y FD(B)20894 5545 y FC(n)21889 6094 y FJ(=)369 b FD(B)24325 5545 y FC(n)24951 6094 y FJ(\()p FD($)26532 6293 y FB(0)27058 6094 y FJ(\))f(:=)31559 4832 y Fz([)29674 7660 y FC(j)51 b Fw(\025)p FB(1)p FC(;`)p Fw(\024)p FC(L)33325 7783 y Ft(n)p Fu(+1)35141 6094 y FD(F)36164 5545 y FC(n)35983 6422 y(`j)46419 6094 y FJ(\(5.216\))6299 9688 y(and)8516 12883 y FD(F)9539 12334 y FC(n)9358 13211 y(`j)11336 12883 y FJ(=)1107 b FD(F)14478 12334 y FC(n)14297 13211 y(`j)15168 12883 y FJ(\()p FD($)16749 13082 y FB(0)17274 12883 y FJ(\))369 b(:=)19891 11010 y Fz(\032)20887 12883 y FD($)405 b FA(2)368 b FJ(\(0)p FD(;)221 b($)26434 13082 y FB(0)26960 12883 y FJ(\))1301 b(s.t.)13455 14588 y Fz(\014)13455 15385 y(\014)13455 16182 y(\014)13455 16979 y(\014)14443 15615 y FD(k)p 14030 16208 1547 54 v 14030 17425 a($)15141 17041 y FC(\017)16005 16514 y FA(\000)18618 15615 y FD(i)p 17466 16208 2749 54 v 17466 17425 a($)18577 17041 y FB(1+)p FC(\017)20643 16514 y FJ(+)23152 15615 y FD(")p 22082 16208 V 22082 17425 a($)23193 17041 y FB(1+)p FC(\017)24964 15438 y Fz(\000)25573 16514 y FA(M)27168 15965 y FB(22)27168 16842 y FC(k)28164 16514 y FJ(\()p FD(\030)29238 16713 y FC(n)p Fw(\000)p FB(1)31067 16514 y FJ(\))31573 15438 y Fz(\001)32181 17034 y FC(ii)32877 14588 y Fz(\014)32877 15385 y(\014)32877 16182 y(\014)32877 16979 y(\014)33689 16514 y FD(>)35202 15615 y(\015)35877 15814 y FC(n)p 35202 16208 1301 54 v 35343 17425 a FD(i)35787 17041 y FC(\027)36858 16514 y FD(;)1522 b FA(8)221 b FD(i)369 b FA(\025)g FJ(1)p FD(;)444 b(k)414 b FA(\024)369 b FD(L)46747 16713 y FC(n)47594 16514 y FD(;)13455 18218 y Fz(\014)13455 19015 y(\014)13455 19812 y(\014)13455 20609 y(\014)14531 19246 y FD(`)p 14030 19839 1547 54 v 14030 21056 a($)15141 20672 y FC(\017)16005 20144 y FA(\000)18534 19246 y FD(j)p 17466 19839 2749 54 v 17466 21056 a($)18577 20672 y FB(1+)p FC(\017)20643 20144 y FJ(+)23152 19246 y FD(")p 22082 19839 V 22082 21056 a($)23193 20672 y FB(1+)p FC(\017)24964 19069 y Fz(\000)25573 20144 y FA(M)27168 19596 y FB(22)27168 20473 y FC(`)28164 20144 y FJ(\()p FD(\030)29238 20343 y FC(n)29864 20144 y FJ(\))30370 19069 y Fz(\001)30979 20665 y FC(j)51 b(j)31897 18218 y Fz(\014)31897 19015 y(\014)31897 19812 y(\014)31897 20609 y(\014)32709 20144 y FA(\024)34244 19246 y FD(\015)34919 19445 y FC(n)p FB(+1)p 34244 19839 2504 54 v 34901 21056 a FD(j)35514 20672 y FC(\027)36880 18271 y Fz(\033)38098 20144 y FD(:)7960 b FJ(\(5.217\))6299 23671 y(Using)434 b(\(5.210\))h(and)e(\(5.214\),)i(w)-36 b(e)434 b(note)f(that)g (\(5.213\))i(is)f(pro)-36 b(v)g(ed)433 b(once)h(w)-36 b(e)434 b(sho)-36 b(w)433 b(that)23188 27472 y(lim)22426 28370 y FC($)23207 28493 y Fu(0)23668 28370 y Fw(!)p FB(0)25079 28118 y Fu(+)26110 26489 y FJ(meas)29157 25413 y Fz(\000)30014 26153 y FJ(~)29766 26489 y FA(C)30543 25413 y Fz(\001)p 26110 27167 5042 54 v 27831 28383 a FD($)28906 28582 y FB(0)31653 27472 y FJ(=)369 b(1)221 b FD(;)6299 30817 y FJ(or,)434 b(equiv)-72 b(alen)-36 b(tly)-108 b(,)11205 34544 y(lim)10443 35442 y FC($)11224 35565 y Fu(0)11686 35442 y Fw(!)p FB(0)13097 35190 y Fu(+)14128 33561 y FJ(meas)17175 32485 y Fz(\000)17784 33561 y FJ(\(0)p FD(;)221 b($)20597 33760 y FB(0)21123 33561 y FJ(\))295 b FA(n)23131 33225 y FJ(~)22883 33561 y FA(C)23660 32485 y Fz(\001)p 14128 34238 10141 54 v 18398 35455 a FD($)19473 35654 y FB(0)24771 34544 y FJ(=)1130 b(lim)26151 35442 y FC($)26932 35565 y Fu(0)27394 35442 y Fw(!)p FB(0)28805 35190 y Fu(+)29836 33489 y FJ(meas)32882 32413 y Fz(\000)33713 32493 y(S)34820 33877 y FC(n)p Fw(\025)p FB(0)36870 33489 y FD(B)37925 33007 y FC(n)38551 32413 y Fz(\001)p 29836 34238 9325 54 v 33698 35455 a FD($)34773 35654 y FB(0)39662 34544 y FJ(=)368 b(0)221 b FD(:)4145 b FJ(\(5.218\))6299 37962 y(Nothing)434 b(remains)f(but)g (to)h(pro)-36 b(v)g(e)433 b(\(5.218\).)580 b(Since)9185 41091 y(meas)12231 39616 y Fz(\020)13395 39829 y([)13246 42635 y FC(n)p Fw(\025)p FB(0)15241 41091 y FD(B)16296 40542 y FC(n)16922 39616 y Fz(\021)18085 41091 y FA(\024)19487 39829 y Fz(X)19559 42635 y FC(n)p Fw(\025)p FB(0)21627 41091 y FJ(meas)24452 40015 y Fz(\000)25061 41091 y FD(B)26116 40542 y FC(n)26742 40015 y Fz(\001)27720 41091 y FA(\024)29122 39829 y Fz(X)29195 42635 y FC(n)p Fw(\025)p FB(0)31262 39829 y Fz(X)31405 42635 y FC(j)51 b Fw(\025)p FB(1)34118 39829 y Fz(X)33403 42658 y FC(`)p Fw(\024)p FC(L)35159 42781 y Ft(n)p Fu(+1)36974 41091 y FJ(meas)40020 40015 y Fz(\000)40629 41091 y FD(F)41652 40542 y FC(n)41471 41419 y(`j)42342 40015 y Fz(\001)43173 41091 y FD(;)2885 b FJ(\(5.219\))6299 45197 y(w)-36 b(e)434 b(are)g(going)g(to)g (estimate)g(the)f(measure)g(of)i FD(F)30460 44715 y FC(n)30279 45542 y(`j)31150 45197 y FJ(.)578 b(W)-108 b(e)434 b(immediately)g (observ)-36 b(e)434 b(that)23548 48909 y FD(F)24571 48360 y FC(n)24390 49237 y(`j)25631 48909 y FA(\032)27033 47036 y Fz(\022)28435 48010 y FD(j)p 28143 48603 1197 54 v 28143 49820 a FJ(2)p FD(`)29472 48909 y(;)30188 48010 y FJ(2)p FD(j)p 30188 48603 1264 54 v 30547 49820 a(`)31584 47036 y Fz(\023)32783 48909 y FD(;)13275 b FJ(\(5.220\))6299 52436 y(since,)434 b(for)g FD($)405 b FA(\024)369 b FD(j)75 b(=)p FJ(\(2)p FD(`)p FJ(\))433 b(or)h FD($)405 b FA(\025)369 b FJ(2)p FD(j)75 b(=`)p FJ(,)8850 54036 y Fz(\014)8850 54833 y(\014)8850 55630 y(\014)8850 56427 y(\014)9926 55064 y FD(`)p 9426 55657 1547 54 v 9426 56874 a($)10537 56490 y FC(\017)11400 55962 y FA(\000)13929 55064 y FD(j)p 12861 55657 2749 54 v 12861 56874 a($)13972 56490 y FB(1+)p FC(\017)16038 55962 y FJ(+)18548 55064 y FD(")p 17478 55657 V 17478 56874 a($)18589 56490 y FB(1+)p FC(\017)20359 54886 y Fz(\000)20968 55962 y FA(M)22563 55414 y FB(22)22563 56291 y FC(`)23559 55962 y FJ(\()p FD(\030)24633 56161 y FC(n)25260 55962 y FJ(\))25766 54886 y Fz(\001)26374 56483 y FC(j)51 b(j)27293 54036 y Fz(\014)27293 54833 y(\014)27293 55630 y(\014)27293 56427 y(\014)28842 55962 y FA(\025)31941 55064 y FD(`)p 31115 55657 2197 54 v 31115 56874 a FJ(2)p FD($)32876 56490 y FC(\017)33740 55962 y FA(\000)36271 55064 y FD(")p 35201 55657 2749 54 v 35201 56874 a($)36312 56490 y FB(1+)p FC(\017)38083 54435 y Fz(\014)38083 55232 y(\014)38083 56029 y(\014)38526 54886 y(\000)39134 55962 y FA(M)40729 55414 y FB(22)40729 56291 y FC(`)41726 55962 y FJ(\()p FD(\030)42800 56161 y FC(n)43426 55962 y FJ(\))43932 54886 y Fz(\001)44541 56483 y FC(j)g(j)45459 54435 y Fz(\014)45459 55232 y(\014)45459 56029 y(\014)28842 59488 y FA(\025)31941 58590 y FD(`)p 31115 59183 2197 54 v 31115 60399 a FJ(2)p FD($)32876 60016 y FC(\017)33740 59488 y FA(\000)295 b FJ(const)222 b FD($)39504 58578 y Fu(1)p 39503 58739 406 40 v 39503 59287 a(2)40042 58940 y Fw(\000)p FC(\017)41578 59488 y FA(\025)43939 58590 y FD(`)p 43113 59183 2197 54 v 43113 60399 a FJ(4)p FD($)44874 60016 y FC(\017)45812 59488 y FD(>)368 b FJ(2)28853 62313 y FD(>)31115 61414 y(\015)31790 61613 y FC(n)p FB(+1)p 31115 62007 2504 54 v 31773 63224 a FD(j)32386 62840 y FC(\027)33973 62313 y FD(;)6299 65673 y FJ(taking)435 b FD($)469 b FJ(small)434 b(enough.)578 b(Moreo)-36 b(v)g(er)434 b(w)-36 b(e)434 b(note)f(that)21051 68436 y FD(F)22074 67888 y FC(n)21893 68764 y(`j)23133 68436 y FJ(=)368 b FA(;)3903 b FJ(if)1394 b FD(`)368 b FA(\024)h FD(L)34433 68635 y FC(n)35281 68436 y FD(:)10777 b FJ(\(5.221\))6299 71199 y(Indeed)6356 72467 y Fz(\014)6356 73264 y(\014)6356 74061 y(\014)6356 74858 y(\014)7432 73495 y FD(`)p 6932 74088 1547 54 v 6932 75305 a($)8043 74921 y FC(\017)8906 74394 y FA(\000)11436 73495 y FD(j)p 10368 74088 2749 54 v 10368 75305 a($)11479 74921 y FB(1+)p FC(\017)13544 74394 y FJ(+)16054 73495 y FD(")p 14984 74088 V 14984 75305 a($)16095 74921 y FB(1+)p FC(\017)17865 73318 y Fz(\000)18474 74394 y FA(M)20069 73845 y FB(22)20069 74722 y FC(`)21066 74394 y FJ(\()p FD(\030)22140 74593 y FC(n)22766 74394 y FJ(\))23272 73318 y Fz(\001)23881 74914 y FC(j)51 b(j)24799 72467 y Fz(\014)24799 73264 y(\014)24799 74061 y(\014)24799 74858 y(\014)26349 74394 y FA(\025)28489 72467 y Fz(\014)28489 73264 y(\014)28489 74061 y(\014)28489 74858 y(\014)29565 73495 y FD(`)p 29065 74088 1547 54 v 29065 75305 a($)30176 74921 y FC(\017)31039 74394 y FA(\000)33568 73495 y FD(j)p 32500 74088 2749 54 v 32500 75305 a($)33611 74921 y FB(1+)p FC(\017)35677 74394 y FJ(+)38186 73495 y FD(")p 37116 74088 V 37116 75305 a($)38227 74921 y FB(1+)p FC(\017)39998 73318 y Fz(\000)40607 74394 y FA(M)42202 73845 y FB(22)42202 74722 y FC(`)43198 74394 y FJ(\()p FD(\030)44272 74593 y FC(n)p Fw(\000)p FB(1)46101 74394 y FJ(\))46607 73318 y Fz(\001)47215 74914 y FC(j)g(j)48134 72467 y Fz(\014)48134 73264 y(\014)48134 74061 y(\014)48134 74858 y(\014)27371 78312 y FJ(205)p eop %%Page: 206 206 206 205 bop 22190 5086 a FA(\000)24426 4188 y FD(")p 23356 4781 2749 54 v 23356 5997 a($)24467 5614 y FB(1+)p FC(\017)26459 3160 y Fz(\014)26459 3957 y(\014)26459 4754 y(\014)26459 5551 y(\014)26901 4010 y(\000)27510 5086 y FA(M)29105 4538 y FB(22)29105 5415 y FC(`)30102 5086 y FJ(\()p FD(\030)31176 5285 y FC(n)31802 5086 y FJ(\))32308 4010 y Fz(\001)32917 5606 y FC(j)51 b(j)34130 5086 y FA(\000)35459 4010 y Fz(\000)36067 5086 y FA(M)37662 4538 y FB(22)37662 5415 y FC(`)38659 5086 y FJ(\()p FD(\030)39733 5285 y FC(n)p Fw(\000)p FB(1)41561 5086 y FJ(\))42067 4010 y Fz(\001)42676 5606 y FC(j)g(j)43594 3160 y Fz(\014)43594 3957 y(\014)43594 4754 y(\014)43594 5551 y(\014)20050 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Fz(\020)31720 49140 y FD(j)p 31428 49733 1197 54 v 31428 50950 a FJ(2)p FD(`)32757 50039 y(;)33472 49140 y FJ(2)p FD(j)p 33472 49733 1264 54 v 33831 50950 a(`)34868 48564 y Fz(\021)35883 50039 y FD(;)12025 51744 y Fz(\014)12025 52541 y(\014)12025 53338 y(\014)12025 54135 y(\014)13101 52771 y FD(`)p 12601 53364 1547 54 v 12601 54581 a($)13712 54197 y FC(\017)14575 53670 y FA(\000)17104 52771 y FD(j)p 16036 53364 2749 54 v 16036 54581 a($)17147 54197 y FB(1+)p FC(\017)19213 53670 y FJ(+)21723 52771 y FD(")p 20653 53364 V 20653 54581 a($)21764 54197 y FB(1+)p FC(\017)23534 52594 y Fz(\000)24143 53670 y FA(M)25738 53121 y FB(22)25738 53998 y FC(`)26735 53670 y FJ(\()p FD(\030)27809 53869 y FC(n)28435 53670 y FJ(\))28941 52594 y Fz(\001)29549 54190 y FC(j)51 b(j)30468 51744 y Fz(\014)30468 52541 y(\014)30468 53338 y(\014)30468 54135 y(\014)31280 53670 y FA(\024)32815 52771 y FD(\015)33490 52970 y FC(n)p FB(+1)p 32815 53364 2504 54 v 33472 54581 a FD(j)34085 54197 y FC(\027)35451 51797 y Fz(\033)36668 53670 y FD(:)3091 b FJ(\(5.224\))0 57292 y(Let)433 b(us)g(in)-36 b(tro)36 b(duce)9883 60551 y FD(f)10666 60003 y FC(n)10524 60880 y(`j)11396 60551 y FJ(\()p FD($)g FJ(\))368 b(:=)16262 59653 y FD(`)p 15762 60246 1547 54 v 15762 61463 a($)16873 61079 y FC(\017)17736 60551 y FA(\000)20265 59653 y FD(j)p 19197 60246 2749 54 v 19197 61463 a($)20308 61079 y FB(1+)p FC(\017)22374 60551 y FJ(+)24883 59653 y FD(")p 23813 60246 V 23813 61463 a($)24924 61079 y FB(1+)p FC(\017)26695 59475 y Fz(\000)27304 60551 y FA(M)28899 60003 y FB(22)28899 60880 y FC(`)29895 60551 y FJ(\()p FD(\030)30969 60750 y FC(n)31595 60551 y FJ(\))32101 59475 y Fz(\001)32710 61072 y FC(j)51 b(j)33850 60551 y FD(:)0 64047 y FJ(Recalling)435 b(that)e FD(")369 b FJ(=)f FD($)11941 63565 y FB(3)p FC(=)p FB(2)13408 64047 y FJ(,)434 b FD(@)14890 64246 y FC($)15752 64047 y FD(")369 b FJ(=)f(3)18760 63090 y FA(p)p 19867 63090 1111 54 v 957 x FD($)36 b(=)p FJ(2,)435 b(w)-36 b(e)434 b(deriv)-36 b(e)433 b FD(f)29601 63565 y FC(n)29459 64392 y(`j)30764 64047 y FJ(with)h(resp)36 b(ect)433 b(to)h FD($)3426 67929 y(@)4113 68128 y FC($)4975 67929 y FD(f)5758 67380 y FC(n)5616 68257 y(`j)6488 67929 y FJ(\()p FD($)36 b FJ(\))1106 b(=)h FD($)12947 67380 y Fw(\000)p FB(2)p Fw(\000)p FC(\017)15316 66056 y Fz(\024)16017 67929 y FJ(\(1)295 b(+)g FD(\017)p FJ(\))p FD(j)370 b FA(\000)296 b FD("$)36 b(`)294 b FJ(+)26043 67030 y(3)p 26043 67623 651 54 v 26043 68840 a(2)26826 67929 y FD($)27937 67380 y FB(3)p FC(=)p FB(2)29403 66853 y Fz(\000)30012 67929 y FA(M)31607 67380 y FB(22)31607 68257 y FC(`)32603 67929 y FJ(\()p FD(\030)33677 68128 y FC(n)34304 67929 y FJ(\))34810 66853 y Fz(\001)35418 68449 y FC(j)51 b(j)11836 71161 y FJ(+)p FD($)13959 70613 y FB(5)p FC(=)p FB(2)15425 69687 y Fz(\020)16218 71161 y FD(D)17298 71360 y FC(\030)17801 70085 y Fz(\000)18410 71161 y FA(M)20005 70613 y FB(22)20005 71490 y FC(`)21002 70085 y Fz(\001)21610 71681 y FC(j)g(j)22529 69687 y Fz(\021)23322 71161 y FJ([)p FD(@)24370 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FB(22)33386 4834 y FC(`)34382 4488 y FJ(\()p FD(\030)35456 4687 y FC(n)36082 4488 y FJ(\))36588 3412 y Fz(\001)37197 5009 y FC(j)51 b(j)38509 4488 y FA(\031)393 b FJ(const)p FD(=$)44666 4006 y FB(2)45192 4488 y FJ(,)401 b(w)-36 b(e)394 b(ha)-36 b(v)g(e)6299 6214 y(that)19087 7940 y FD($)20198 7392 y FB(3)p FC(=)p FB(2)21664 6811 y Fz(\014)21664 7608 y(\014)22107 6864 y(\000)22716 7940 y FA(M)24311 7392 y FB(22)24311 8268 y FC(`)25307 7940 y FJ(\()p FD(\030)26381 8139 y FC(n)27008 7940 y FJ(\))27514 6864 y Fz(\001)28122 8460 y FC(j)51 b(j)29041 6811 y Fz(\014)29041 7608 y(\014)29852 7940 y FA(\024)370 b FJ(const)221 b FD($)35557 7392 y FB(3)p FC(=)p FB(2)37245 7940 y FD(;)8813 b FJ(\(5.226\))17342 11252 y FD($)18453 10704 y FB(5)p FC(=)p FB(2)19920 9725 y Fz(\014)19920 10522 y(\014)19920 11319 y(\014)20362 9778 y(\020)21156 11252 y FD(D)22236 11451 y FC(\030)22739 10176 y Fz(\000)23348 11252 y FA(M)24943 10704 y FB(22)24943 11581 y FC(`)25940 10176 y Fz(\001)26548 11773 y FC(j)51 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FA(\024)g FD(L)36070 37331 y FC(n)p FB(+1)38120 37132 y FD(;)1523 b FJ(1)369 b FA(\024)g FD(j)444 b FA(\024)369 b FJ(2)p FD($)46534 37331 y FB(0)47060 37132 y FD(L)47945 37331 y FC(n)p FB(+1)6299 40186 y FJ(Therefore,)435 b(b)-36 b(y)433 b(\(5.219\),)i(w)-36 b(e)434 b(ha)-36 b(v)g(e)9199 44574 y(meas)12245 43099 y Fz(\020)13409 43312 y([)13260 46118 y FC(n)p Fw(\025)p FB(0)15255 44574 y FD(B)16310 44025 y FC(n)16936 43099 y Fz(\021)18836 44574 y FA(\024)23008 43312 y Fz(X)20976 46131 y FC(n)p FB(:2)p FC($)23059 46254 y Fu(0)23522 46131 y FC(L)24162 46254 y Ft(n)p Fu(+1)25755 46131 y Fw(\025)p FB(1)27179 42816 y(2)p FC($)28430 42939 y Fu(0)28892 42816 y FC(L)29532 42939 y Ft(n)p Fu(+1)28193 43312 y Fz(X)28335 46118 y FC(j)51 b Fw(\025)p FB(1)31346 42816 y FC(L)31986 42939 y Ft(n)p Fu(+1)31504 43312 y Fz(X)31406 46141 y FC(`)p Fw(\025)p FC(M)33801 44574 y FJ(meas)q(\()p FD(F)37975 44773 y FC(`j)38846 44574 y FJ(\))18836 49417 y FA(\024)1107 b FJ(const)26420 48155 y Fz(X)24389 50974 y FC(n)p FB(:2)p FC($)26472 51097 y Fu(0)26934 50974 y FC(L)27574 51097 y Ft(n)p Fu(+1)29168 50974 y Fw(\025)p FB(1)30591 47660 y(2)p FC($)31842 47783 y Fu(0)32304 47660 y FC(L)32944 47783 y Ft(n)p Fu(+1)31605 48155 y Fz(X)31748 50961 y FC(j)51 b Fw(\025)p FB(1)34759 47660 y FC(L)35399 47783 y Ft(n)p Fu(+1)34916 48155 y Fz(X)34819 50984 y FC(`)p Fw(\025)p FC(M)39976 48519 y FJ(1)p 37347 49112 5909 54 v 37347 50329 a FD(j)37960 49945 y FC(\027)58 b Fw(\000)p FB(1)p Fw(\000)p FC(\017)41071 50329 y FD(`)41617 49945 y FB(2+)p FC(\017)18836 53828 y FA(\024)665 b FJ(const)25977 52566 y Fz(X)23946 55385 y FC(n)p FB(:2)p FC($)26029 55508 y Fu(0)26491 55385 y FC(L)27131 55508 y Ft(n)p Fu(+1)28725 55385 y Fw(\025)p FB(1)30149 52020 y Fz(Z)31477 52370 y FB(2)p FC($)32728 52493 y Fu(0)33190 52370 y FC(L)33830 52493 y Ft(n)p Fu(+1)30887 55027 y FB(1)37259 52929 y FJ(1)p 35833 53522 3504 54 v 35833 54739 a FD(j)36446 54355 y FC(\027)58 b Fw(\000)p FB(1)p Fw(\000)p FC(\017)39690 52020 y Fz(Z)41019 52370 y Fw(1)40428 55027 y FC(M)43136 52929 y FJ(1)p 42369 53522 2184 54 v 42369 54739 a FD(`)42915 54355 y FB(2+)p FC(\017)44907 53828 y FD(d`)221 b(d)-145 b(j)18836 58239 y FA(\024)1107 b FJ(const)26420 56977 y Fz(X)24389 59796 y FC(n)p FB(:2)p FC($)26472 59919 y Fu(0)26934 59796 y FC(L)27574 59919 y Ft(n)p Fu(+1)29168 59796 y Fw(\025)p FB(1)30370 56431 y Fz(Z)31698 56781 y FB(2)p FC($)32949 56904 y Fu(0)33411 56781 y FC(L)34051 56904 y Ft(n)p Fu(+1)31108 59438 y FB(1)38998 57340 y FJ(1)p 36054 57933 6539 54 v 36054 59150 a FD(M)37451 58766 y FB(1+)p FC(\017)39089 59150 y FD(j)39702 58766 y FC(\027)58 b Fw(\000)p FB(1)p Fw(\000)p FC(\017)42947 58239 y FD(d)-145 b(j)18847 62649 y FJ(=)1117 b(const)26524 61387 y Fz(X)24389 64391 y FC(n)p FB(:)25354 64029 y Fu(1)p 25354 64190 406 40 v 25354 64739 a(8)25893 64391 y Fw(\024)p FB(4)27095 64139 y Ft(n)27662 64391 y FC($)28443 64514 y Fu(0)28904 64391 y Fw(\0241)30577 60841 y Fz(Z)31905 61192 y FB(2)p FC($)33156 61315 y Fu(0)33618 61192 y FC(L)34258 61315 y Ft(n)p Fu(+1)31315 63849 y FB(1)39205 61751 y FJ(1)p 36261 62344 6539 54 v 36261 63560 a FD(M)37658 63177 y FB(1+)p FC(\017)39296 63560 y FD(j)39909 63177 y FC(\027)58 b Fw(\000)p FB(1)p Fw(\000)p FC(\017)43154 62649 y FD(d)-145 b(j)2196 b FJ(\(5.231\))6299 67428 y(recalling)435 b(that)e FD(L)15205 67627 y FC(n)16200 67428 y FJ(=)369 b(4)18231 66946 y FC(n)18857 67428 y FJ(.)579 b(W)-108 b(e)433 b(split)h(the)f(in)-36 b(tegral)434 b(\(5.231\))h(in)-36 b(to)18861 69574 y Fz(Z)20189 69925 y FB(2)p FC($)21440 70048 y Fu(0)21902 69925 y FC(L)22542 70048 y Ft(n)p Fu(+1)19599 72582 y FB(1)24560 71382 y FJ(=)25941 69574 y Fz(Z)27269 69925 y FB(2)p FC($)28520 70048 y Fu(0)28982 69925 y FC(L)29622 70036 y Ft(n)26678 72582 y FB(1)30465 71382 y FJ(+)31698 69574 y Fz(Z)33026 69925 y FB(2)p FC($)34277 70048 y Fu(0)34739 69925 y FC(L)35379 70048 y Ft(n)p Fu(+1)32436 72582 y FB(2)p FC($)33687 72705 y Fu(0)34149 72582 y FC(L)34789 72693 y Ft(n)37471 71382 y FD(:)p 6299 73774 17638 45 v 7795 74590 a Fy(4)8292 74991 y Fx(W)-92 b(e)368 b(recall)i(that,)h(b)-31 b(y)370 b(\(4.53\),)i Fq(@)21490 75157 y Fg($)22305 74991 y Fq(I)22792 75157 y Fy(0)23596 74991 y Fx(=)307 b Fq(O)31 b Fx(\()p Fq($)27017 74590 y Fe(\000)p Fy(2)28205 74991 y Fx(\),)370 b(since)g Fq(\034)432 b Fx(=)308 b Fq($)35058 74590 y Fe(\000)p Fy(1)36246 74991 y Fx(.)27371 78312 y FJ(207)p eop %%Page: 208 208 208 207 bop 0 4488 a FJ(F)-108 b(or)433 b(the)g(\014rst)g(term,)g (since)h(1)369 b FA(\024)h FD(j)443 b FA(\024)369 b FJ(2)p FD($)20436 4687 y FB(0)20962 4488 y FD(L)21847 4687 y FC(n)22473 4488 y FJ(,)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b(that)f FD(M)508 b FJ(=)369 b FD(L)35032 4687 y FC(n)35658 4488 y FJ(,)434 b(therefore)7149 6646 y Fz(Z)8478 6997 y FB(2)p FC($)9729 7120 y Fu(0)10191 6997 y FC(L)10831 7108 y Ft(n)7887 9653 y FB(1)14751 7556 y FJ(1)p 11807 8149 6539 54 v 11807 9365 a FD(j)12420 8982 y FC(\027)58 b Fw(\000)p FB(1)p Fw(\000)p FC(\017)15310 9365 y FD(M)16707 8982 y FB(1+)p FC(\017)18699 8454 y FD(d)-145 b(j)1193 b FJ(=)24159 7556 y(1)p 23223 8149 2523 54 v 23223 9365 a FD(L)24108 8982 y FB(1+)p FC(\017)24108 9694 y(n)26100 6646 y Fz(Z)27429 6997 y FB(2)p FC($)28680 7120 y Fu(0)29142 6997 y FC(L)29782 7108 y Ft(n)26838 9653 y FB(1)32184 7556 y FJ(1)p 30758 8149 3504 54 v 30758 9365 a FD(j)31371 8982 y FC(\027)58 b Fw(\000)p FB(1)p Fw(\000)p FC(\017)34615 8454 y FD(d)-145 b(j)20950 12283 y FA(\024)1107 b FJ(const)222 b FD($)27393 11734 y FB(1+)p FC(\017)27357 12611 y FB(0)29252 10410 y Fz(\022)31593 11384 y FJ(1)p 30362 11977 3112 54 v 30362 13194 a FD($)31437 13393 y FB(0)31963 13194 y FD(L)32848 13393 y FC(n)33607 10410 y Fz(\023)34584 10707 y FC(\027)58 b Fw(\000)p FB(1)36584 12283 y FJ(;)3175 b(\(5.232\))0 16186 y(for)434 b(the)f(second)g(term,)h(since)f FD(M)509 b FJ(=)368 b FD(j)75 b(=)p FJ(2)p FD($)21215 16385 y FB(0)21741 16186 y FJ(,)6636 18417 y Fz(Z)7964 18768 y FB(2)p FC($)9215 18891 y Fu(0)9677 18768 y FC(L)10317 18891 y Ft(n)p Fu(+1)7374 21425 y FB(2)p FC($)8625 21548 y Fu(0)9087 21425 y FC(L)9727 21536 y Ft(n)15264 19327 y FJ(1)p 12320 19920 6539 54 v 12320 21137 a FD(j)12933 20753 y FC(\027)58 b Fw(\000)p FB(1)p Fw(\000)p FC(\017)15823 21137 y FD(M)17220 20753 y FB(1+)p FC(\017)19213 20225 y FD(d)-145 b(j)1182 b FA(\024)1107 b FJ(const)221 b FD($)27906 19677 y FB(1+)p FC(\017)27870 20554 y FB(0)29765 18417 y Fz(Z)31093 18768 y Fw(1)30503 21425 y FB(2)p FC($)31754 21548 y Fu(0)32216 21425 y FC(L)32856 21536 y Ft(n)34101 19327 y FJ(1)p 33832 19920 1189 54 v 33832 21137 a FD(j)34445 20753 y FC(\027)35375 20225 y FD(d)-145 b(j)21464 24137 y FA(\024)1107 b FJ(const)221 b FD($)27906 23588 y FB(1+)p FC(\017)27870 24465 y FB(0)29765 22264 y Fz(\022)32106 23238 y FJ(1)p 30876 23831 3112 54 v 30876 25048 a FD($)31951 25247 y FB(0)32476 25048 y FD(L)33361 25247 y FC(n)34120 22264 y Fz(\023)35098 22561 y FC(\027)58 b Fw(\000)p FB(1)37097 24137 y FD(:)2662 b FJ(\(5.233\))0 28040 y(Therefore)434 b(b)-36 b(y)434 b(\(5.232\))g(and)f(\(5.233\),)j (w)-36 b(e)433 b(can)h(estimate)g(\(5.231\),)2770 30901 y Fz(X)635 33905 y FC(n)p FB(:)1600 33543 y Fu(1)p 1600 33704 406 40 v 1600 34252 a(8)2139 33905 y Fw(\024)p FB(4)3341 33653 y Ft(n)3908 33905 y FC($)4689 34028 y Fu(0)5150 33905 y Fw(\0241)6823 30355 y Fz(Z)8151 30705 y FB(2)p FC($)9402 30828 y Fu(0)9864 30705 y FC(L)10504 30828 y Ft(n)p Fu(+1)7561 33362 y FB(1)15451 31264 y FJ(1)p 12507 31857 6539 54 v 12507 33074 a FD(M)13904 32690 y FB(1+)p FC(\017)15542 33074 y FD(j)16155 32690 y FC(\027)58 b Fw(\000)p FB(1)p Fw(\000)p FC(\017)19400 32163 y FD(d)-145 b(j)1182 b FA(\024)1107 b FJ(const)221 b FD($)28093 31614 y FB(1+)p FC(\017)28057 32491 y FB(0)32308 30901 y Fz(X)30174 33905 y FC(n)p FB(:)31139 33543 y Fu(1)p 31139 33704 406 40 v 31139 34252 a(8)31678 33905 y Fw(\024)p FB(4)32880 33653 y Ft(n)33446 33905 y FC($)34227 34028 y Fu(0)34689 33905 y Fw(\0241)36583 30290 y Fz(\022)38807 31264 y FJ(1)p 37694 31857 2877 54 v 37694 33074 a FD($)38769 33273 y FB(0)39294 33074 y FJ(4)39944 32690 y FC(n)40703 30290 y Fz(\023)41681 30587 y FC(\027)58 b Fw(\000)p FB(1)21651 36667 y FA(\024)1107 b FJ(const)221 b FD($)28093 36118 y FB(1+)p FC(\017)28057 36995 y FB(0)29952 34859 y Fz(Z)31281 35209 y Fw(1)30823 37504 y Fu(1)p 30823 37665 406 40 v 30823 38213 a(8)32631 35768 y FD(dt)p 32631 36361 1146 54 v 32681 37578 a(t)33151 37194 y FC(\027)21651 39775 y FA(\024)1107 b FJ(const)221 b FD($)28093 39227 y FB(1+)p FC(\017)28057 40104 y FB(0)29952 39775 y FD(:)9807 b FJ(\(5.234\))0 42914 y(Therefore)434 b(b)-36 b(y)434 b(\(5.231\))g(and)f(\(5.234\))i(w)-36 b(e)434 b(ha)-36 b(v)g(e)13816 46419 y(meas)16862 44944 y Fz(\020)18026 45157 y([)17877 47963 y FC(n)p Fw(\025)p FB(0)19872 46419 y FD(B)20927 45870 y FC(n)21553 44944 y Fz(\021)22715 46419 y FA(\024)370 b FJ(const)221 b FD($)28420 45870 y FB(1+)p FC(\017)28384 46747 y FB(0)0 50886 y FJ(and)433 b(\(5.218\))i(follo)-36 b(ws.)43061 52560 y Fn(\003)0 57420 y FH(5.11)2152 b(Minimal)716 b(p)60 b(erio)g(d)0 60480 y FE(Lemma)579 b(5.11.1.)590 b Fs(L)-66 b(et)528 b FD(i)12949 60679 y FB(1)13962 60480 y FD(<)487 b(i)15905 60679 y FB(2)16917 60480 y FA(2)g Fv(N)19249 59998 y FB(+)20566 60480 y Fs(and)529 b FD(&)594 b(>)487 b FJ(1)p Fs(.)790 b(Then)528 b(ther)-66 b(e)528 b(exists)g FJ(\007)487 b FA(\032)h FJ(\(0)p FD(;)221 b FJ(1])p Fs(,)0 62085 y(with)465 b FJ(meas)5648 61009 y Fz(\000)6257 62085 y FJ(\007)7269 61009 y Fz(\001)8246 62085 y FJ(=)369 b(1)465 b Fs(such)h(that)f(for)f(al)66 b(l)466 b FD(\026)368 b FA(2)h FJ(\007)464 b Fs(the)h(ve)-66 b(ctor)14317 66121 y FJ(\026)-753 b FD(!)15024 66320 y FC(\026)16014 66121 y FJ(:=)17756 64248 y Fz(\022)18734 64503 y(q)p 20062 64503 3355 54 v 1618 x FD(i)20506 65663 y FB(2)20506 66447 y(1)21327 66121 y FJ(+)295 b FD(\026)o(;)23998 64503 y Fz(q)p 25327 64503 V 25327 66121 a FD(i)25771 65663 y FB(2)25771 66447 y(2)26592 66121 y FJ(+)g FD(\026)28903 64248 y Fz(\023)0 69950 y Fs(is)465 b(Diophantine,)e(namely)11176 72619 y Fz(\014)11176 73416 y(\014)11723 73748 y FJ(\026)-754 b FD(!)12429 73947 y FC(\026)13345 73748 y FA(\001)296 b FD(h)14759 72619 y Fz(\014)14759 73416 y(\014)15570 73748 y FD(>)17083 72849 y(c)p FJ(\()p FD(\026)p FJ(\))p 17083 73443 2354 54 v 17282 74659 a FA(j)p FD(h)p FA(j)18769 74275 y FC(&)19792 73748 y FD(;)4206 b FA(8)p FD(h)369 b FA(2)g Fv(Z)28356 73200 y FB(2)29176 73748 y FA(n)296 b(f)p FJ(0)p FA(g)443 b FD(:)21072 78312 y FJ(208)p eop %%Page: 209 209 209 208 bop 6299 4488 a Fp(pr)-30 b(oof.)659 b FJ(De\014ne,)433 b(forall)483 b(\026)-698 b FD(\015)441 b(>)369 b FJ(0)434 b(and)f FD(h)368 b FA(2)h Fv(Z)28409 4006 y FB(2)29229 4488 y FA(n)296 b(f)p FJ(0)p FA(g)p FJ(,)14973 7876 y FD(A)15981 7247 y FB(\026)-503 b FC(\015)15948 8246 y(h)16915 7876 y FJ(=)368 b FD(A)19303 7247 y FB(\026)-503 b FC(\015)19270 8246 y(h)19812 8369 y Fu(1)20274 8246 y FC(;h)21077 8369 y Fu(2)21963 7876 y FJ(:=)23705 6003 y Fz(\032)24701 7876 y FD(\026)369 b FA(2)f FJ(\(0)p FD(;)221 b FJ(1])444 b(s)p FD(:)p FJ(t)p FD(:)32485 6747 y Fz(\014)32485 7544 y(\014)33031 7876 y FJ(\026)-754 b FD(!)33737 8075 y FC(\026)34654 7876 y FA(\001)295 b FD(h)36067 6747 y Fz(\014)36067 7544 y(\014)36879 7876 y FA(\024)39066 6978 y FJ(\026)-698 b FD(\015)p 38414 7571 1956 54 v 38414 8788 a FA(j)p FD(h)p FA(j)39901 8404 y FC(&)40502 6003 y Fz(\033)6299 11190 y FJ(and)10284 14246 y(\007)11329 13698 y FB(\026)-503 b FC(\015)12257 14246 y FJ(:=)13999 12373 y Fz(\032)14995 14246 y FD(\026)369 b FA(2)f FJ(\(0)p FD(;)221 b FJ(1])445 b(s)p FD(:)p FJ(t)p FD(:)22779 13117 y Fz(\014)22779 13914 y(\014)23325 14246 y FJ(\026)-753 b FD(!)24032 14445 y FC(\026)24948 14246 y FA(\001)295 b FD(h)26361 13117 y Fz(\014)26361 13914 y(\014)27173 14246 y FD(>)29339 13348 y FJ(\026)-699 b FD(\015)p 28686 13941 V 28686 15157 a FA(j)p FD(h)p FA(j)30173 14774 y FC(&)30996 14246 y FD(;)1522 b FA(8)p FD(h)369 b FA(2)g Fv(Z)36876 13698 y FB(2)37696 14246 y FA(n)295 b(f)q FJ(0)p FA(g)40634 12373 y Fz(\033)42073 14246 y FD(:)3985 b FJ(\(5.235\))6299 17776 y(W)-108 b(e)434 b(note)f(that)g(if)i FD(A)16448 17146 y FB(\026)-503 b FC(\015)16415 18146 y(h)16957 18269 y Fu(1)17418 18146 y FC(;h)18221 18269 y Fu(2)19107 17776 y FA(6)p FJ(=)369 b FA(;)p FJ(,)434 b(then)459 b(~)-677 b FD(c)p FA(j)p FD(h)26587 17975 y FB(1)27113 17776 y FA(j)369 b(\024)g(j)p FD(h)30371 17975 y FB(2)30896 17776 y FA(j)g(\024)396 b FJ(\026)-677 b FD(c)p FA(j)p FD(h)34714 17975 y FB(1)35240 17776 y FA(j)p FJ(,)461 b(\026)-677 b FD(c)368 b(<)h FJ(1.)579 b(Moreo)-36 b(v)g(er)21427 20842 y(\007)22472 20294 y FB(\026)-503 b FC(\015)23400 20842 y FJ(=)369 b(\(0)p FD(;)221 b FJ(1])297 b FA(n)30099 19580 y Fz([)28786 22474 y FC(h)p Fw(2)p Fr(Z)30545 22222 y Fu(2)31007 22474 y Fw(nf)q FB(0)p Fw(g)33110 20842 y FD(A)34118 20213 y FB(\026)-503 b FC(\015)34085 21212 y(h)34904 20842 y FD(:)11154 b FJ(\(5.236\))6299 24881 y(W)-108 b(e)434 b(will)h(pro)-36 b(v)g(e)433 b(that)21929 26486 y(meas\(\007)26305 25937 y FB(\026)-503 b FC(\015)26864 26486 y FJ(\))368 b(=)h(1)296 b FA(\000)f FD(O)36 b FJ(\()48 b(\026)-698 b FD(\015)72 b FJ(\))221 b FD(;)11655 b FJ(\(5.237\))6299 28678 y(Note)434 b(that)f(meas\()p FD(A)16599 28048 y FB(\026)-503 b FC(\015)16566 29048 y(h)17164 28678 y FJ(\))369 b FA(\024)g FJ(const)23790 28094 y FB(\026)-503 b FC(\015)p 22765 28372 2522 54 v 22765 29136 a Fw(j)p FC(h)p Fw(j)23829 28884 y Ft(&)50 b Fu(+1)25419 28678 y FJ(.)578 b(In)434 b(fact,)g(w)-36 b(e)434 b(can)g(write)9242 32439 y FA(j)p FD(h)295 b FA(\001)399 b FJ(\026)-754 b FD(!)12129 32638 y FC(\026)12750 32439 y FA(j)369 b FJ(=)14869 30911 y Fz(\014)14869 31708 y(\014)14869 32505 y(\014)15312 32439 y FD(h)16061 32638 y FB(1)16586 30820 y Fz(q)p 17915 30820 3355 54 v 17915 32439 a FD(i)18359 31981 y FB(2)18359 32764 y(1)19179 32439 y FJ(+)295 b FD(\026)g FJ(+)g FD(h)23620 32638 y FB(2)24145 30820 y Fz(q)p 25474 30820 V 25474 32439 a FD(i)25918 31981 y FB(2)25918 32764 y(2)26738 32439 y FJ(+)g FD(\026)29049 30911 y Fz(\014)29049 31708 y(\014)29049 32505 y(\014)29861 32439 y FJ(=)32245 31540 y FA(j)p FD(h)33363 31058 y FB(2)33363 31868 y(1)33888 31540 y FJ(\()p FD(i)34838 31058 y FB(2)34838 31868 y(1)35659 31540 y FJ(+)f FD(\026)p FJ(\))h(+)g FD(h)40605 31058 y FB(2)40605 31868 y(2)41130 31540 y FJ(\()p FD(i)42080 31058 y FB(2)42080 31868 y(2)42901 31540 y FJ(+)g FD(\026)p FJ(\))p FA(j)p 31374 32133 15361 54 v 31374 33554 a(j)p FD(h)32492 33753 y FB(1)33018 33554 y FA(j)33387 32399 y Fz(p)p 34715 32399 3355 54 v 1155 x FD(i)35159 33096 y FB(2)35159 33879 y(1)35980 33554 y FJ(+)g FD(\026)g FJ(+)f FA(j)p FD(h)40789 33753 y FB(2)41315 33554 y FA(j)41684 32399 y Fz(p)p 43012 32399 V 1155 x FD(i)43456 33096 y FB(2)43456 33879 y(2)44277 33554 y FJ(+)h FD(\026)o FA(j)47090 32439 y FD(:)6299 36207 y FJ(Hence)434 b FD(\026)368 b FA(2)h FD(A)13579 35577 y FB(\026)-503 b FC(\015)13546 36577 y(h)14577 36207 y FJ(implies)434 b(that)13940 39356 y FA(j)p FD(f)142 b FJ(\()p FD(\026)p FJ(\))p FA(j)368 b FJ(:=)h FA(j)p FD(h)20484 38807 y FB(2)20484 39684 y(1)21009 39356 y FD(i)21453 38807 y FB(2)21453 39684 y(1)22274 39356 y FA(\000)295 b FD(h)24351 38807 y FB(2)24351 39684 y(2)24877 39356 y FD(i)25321 38807 y FB(2)25321 39684 y(2)26142 39356 y FJ(+)g FD(\026)p FJ(\()p FD(h)29487 38807 y FB(2)29487 39684 y(1)30307 39356 y FA(\000)g FD(h)32384 38807 y FB(2)32384 39684 y(2)32910 39356 y FJ(\))p FA(j)368 b(\024)h FJ(const)40133 38457 y(\026)-698 b FD(\015)p 38880 39050 3158 54 v 38880 40267 a FA(j)p FD(h)p FA(j)40367 39883 y FC(&)69 b Fw(\000)p FB(1)42392 39356 y FD(:)6299 42651 y FJ(Moreo)-36 b(v)g(er,)435 b(b)36 b(eing)18866 44256 y FA(j)p FD(f)20018 43708 y Fw(0)20328 44256 y FJ(\()p FD(\026)p FJ(\))p FA(j)368 b FJ(=)h FA(j)p FD(h)25359 43708 y FB(2)25359 44585 y(1)26180 44256 y FA(\000)295 b FD(h)28257 43708 y FB(2)28257 44585 y(2)28782 44256 y FA(j)369 b(\031)h(j)p FD(h)32041 44455 y FB(1)32566 44256 y FA(j)32935 43708 y FB(2)33830 44256 y FA(\031)f(j)p FD(h)p FA(j)36719 43708 y FB(2)37466 44256 y FD(;)6299 46448 y FJ(one)434 b(has)15736 48314 y(meas\()p FD(A)20075 47684 y FB(\026)-503 b FC(\015)20042 48684 y(h)20640 48314 y FJ(\))369 b FD(<)f FJ(const)27473 47415 y(\026)-698 b FD(\015)p 26219 48008 V 26219 49225 a FA(j)p FD(h)p FA(j)27706 48841 y FC(&)69 b Fw(\000)p FB(1)30545 47415 y FJ(1)p 29864 48008 2013 54 v 29864 49225 a FA(j)p FD(h)p FA(j)31351 48841 y FB(2)32379 48314 y FJ(=)368 b(const)38337 47415 y(\026)-698 b FD(\015)p 37083 48008 3158 54 v 37083 49225 a FA(j)p FD(h)p FA(j)38570 48841 y FC(&)69 b FB(+1)40595 48314 y FD(:)6299 51177 y FJ(Therefore)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b(that)10075 53801 y(meas)14435 52539 y Fz([)13122 55433 y FC(h)p Fw(2)p Fr(Z)14881 55181 y Fu(2)15343 55433 y Fw(nf)q FB(0)p Fw(g)17224 53801 y FJ(\()p FD(A)18738 53171 y FB(\026)-503 b FC(\015)18705 54171 y(h)19303 53801 y FJ(\))1107 b FA(\024)24148 52539 y Fz(X)23056 55433 y FC(h)p Fw(2)p Fr(Z)24815 55181 y Fu(2)25277 55433 y Fw(nf)q FB(0)p Fw(g)27380 53801 y FJ(meas\()p FD(A)31719 53171 y FB(\026)-503 b FC(\015)31686 54171 y(h)32284 53801 y FJ(\))20916 58303 y FA(\031)23056 56495 y Fz(Z)24384 56845 y Fw(1)23794 59502 y FB(1)25602 56495 y Fz(Z)26951 56845 y FB(\026)-491 b FC(c)157 b(h)28037 56968 y Fu(1)26361 59502 y FB(~)-491 b FC(c)156 b(h)27446 59625 y Fu(1)31999 57404 y FJ(\026)-698 b FD(\015)p 28908 57997 6834 54 v 28908 59214 a FJ(\()p FD(h)30163 59413 y FB(1)30983 59214 y FJ(+)295 b FD(h)33039 59413 y FB(2)33564 59214 y FJ(\))34070 58830 y FC(&)69 b FB(+1)36096 58303 y FD(dh)37521 58502 y FB(1)38046 58303 y FD(dh)39471 58502 y FB(2)20926 62033 y FJ(=)1166 b(\026)-698 b FD(\015)24024 60225 y Fz(Z)25352 60576 y Fw(1)24762 63232 y FB(1)26570 60225 y Fz(Z)27919 60576 y FB(\026)-491 b FC(c)27329 63232 y FB(~)g FC(c)32072 61135 y FD(h)32821 61334 y FB(1)p 28716 61728 7988 54 v 28716 63008 a FJ(\(1)295 b(+)g FD(\030)64 b FJ(\))32612 62624 y FC(&)69 b FB(+1)34283 63008 y FD(h)35032 62458 y FC(&)g FB(+1)35032 63333 y(1)37057 62033 y FD(d\030)64 b(dh)39790 62232 y FB(1)20926 65748 y FJ(=)1166 b(\026)-698 b FD(\015)24024 63940 y Fz(Z)25352 64290 y Fw(1)24762 66947 y FB(1)27015 64849 y FJ(1)p 26703 65442 1275 54 v 26703 66659 a FD(h)27452 66115 y FC(&)27452 66985 y FB(1)28332 65748 y FD(dh)29757 65947 y FB(1)30503 63940 y Fz(Z)31853 64290 y FB(\026)-491 b FC(c)31262 66947 y FB(~)g FC(c)35108 64849 y FJ(1)p 32649 65442 5568 54 v 32649 66659 a(\(1)295 b(+)g FD(\030)64 b FJ(\))36545 66275 y FC(&)69 b FB(+1)38571 65748 y FD(d\030)433 b FJ(=)368 b(const)44598 65947 y FC(&)45337 65748 y FJ(\026)-698 b FD(\015)293 b(:)6299 69117 y FJ(Then)433 b(\(5.237\))i(follo)-36 b(ws)436 b(b)-36 b(y)433 b(\(5.236\).)580 b(Cho)36 b(osing)434 b(\007)369 b(:=)33896 68121 y Fz(S)35036 69505 y FB(\026)-503 b FC(\015)52 b(>)p FB(0)37019 69117 y FJ(\007)38064 68635 y FB(\026)-503 b FC(\015)38623 69117 y FJ(,)434 b(w)-36 b(e)433 b(get,)h(b)-36 b(y)434 b(\(5.237\),)19751 71863 y(meas\(\007\))368 b(=)440 b(lim)26382 72660 y FB(\026)-503 b FC(\015)52 b Fw(!)p FB(0)28519 71863 y FJ(meas\(\007)32895 71314 y FB(\026)-503 b FC(\015)33454 71863 y FJ(\))368 b(=)h(1)221 b FD(:)49361 74991 y Fn(\003)27371 78312 y FJ(209)p eop %%Page: 210 210 210 209 bop 0 4488 a FE(Lemma)410 b(5.11.2.)495 b Fs(Fix)394 b FD(\032)368 b(<)h FJ(1)p FD(=)p FJ(2)p Fs(.)575 b(Ther)-66 b(e)393 b(exists)g FJ(\007)369 b FA(\032)g FJ(\(0)p FD(;)221 b FJ(1])p Fs(,)409 b(with)393 b FJ(meas)36581 3412 y Fz(\000)37190 4488 y FJ(\007)38202 3412 y Fz(\001)39179 4488 y FJ(=)369 b(1)394 b Fs(such)0 6094 y(that,)462 b(if)e FD(\026)369 b FA(2)f FJ(\007)p Fs(,)462 b FD(N)507 b FA(\025)369 b FJ(2)462 b Fs(and)g FD(T)16117 5611 y FB(min)18202 6094 y Fs(is)f(the)g(minimal)f(p)-66 b(erio)g(d)461 b(of)g(a)h FD(T)181 b Fs(-p)-66 b(erio)g(dic)459 b(solution)0 7699 y(of)465 b FJ(\(5.26\))p Fs(,)g(then)17251 9304 y FD(T)18194 8755 y FB(min)20187 9304 y FA(\025)369 b FJ(const)221 b FD(T)25723 8755 y FC(\032)26482 9304 y FD(;)13277 b FJ(\(5.238\))0 11603 y Fs(with)465 b FJ(const)369 b(=)f(const\()p FD(\026;)221 b FA(I)98 b FJ(\))p Fs(.)0 14263 y Fp(pr)-30 b(oof.)1001 b FJ(Let)7772 13187 y Fz(\000)8381 14263 y FD(I)104 b FJ(\()p FD(t)p FJ(\))p FD(;)221 b(\036)p FJ(\()p FD(t)p FJ(\))p FD(;)330 b FJ(^)-759 b FD(p)o FJ(\()p FD(t)p FJ(\))p FD(;)317 b FJ(^)-746 b FD(q)48 b FJ(\()p FD(t)p FJ(\))18780 13187 y Fz(\001)19928 14263 y FJ(a)540 b FD(T)181 b FJ(-p)36 b(erio)g(dic)540 b(solution)h(of)g (\(5.26\).)899 b(W)-108 b(e)540 b(kno)-36 b(w)0 15868 y(that)418 b FD(\036)p FJ(\()p FD(T)181 b FJ(\))368 b(=)h(2)p FD(\031)48 b(k)463 b FJ(with)419 b FD(k)414 b FA(2)369 b Fv(Z)16031 15386 y FC(N)17345 15868 y FJ(de\014ned)417 b(in)i(\(4.54\).)574 b(Denoting)419 b(b)-36 b(y)418 b FD(T)35908 15386 y FB(min)35727 16214 y FC(\036)37901 15868 y FA(\024)369 b FD(T)40246 15386 y FB(min)42288 15868 y FJ(the)0 17473 y(minimal)426 b(p)36 b(erio)g(d)426 b(of)g FD(\036)p FJ(\()p FD(t)p FJ(\),)g(w)-36 b(e)426 b(ha)-36 b(v)g(e)426 b(that)f(there)g(exist)h FD(n)369 b FA(2)g Fv(N)31054 16991 y FB(+)32267 17473 y FJ(suc)-36 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FJ(\))368 b(=)472 b(~)-754 b FD(!)48 b(T)476 b FJ(+)295 b FD(O)36 b FJ(\()p FD(\021)20001 28649 y FB(2)20526 29197 y FJ(\))369 b(=)g FD(!)48 b(T)475 b FJ(+)295 b FD(O)36 b FJ(\()p FD(\021)28410 28649 y FB(2)28935 29197 y FD(T)181 b FJ(\))369 b(=)g FD(!)48 b(T)475 b FJ(+)295 b FD(O)36 b FJ(\(1\))221 b FD(:)1310 b FJ(\(5.240\))0 32070 y(Consider)488 b(the)f(v)-36 b(ector)592 b(\026)-753 b FD(!)12542 32269 y FC(\026)13624 32070 y FJ(:=)462 b(\()p FD(!)16775 32269 y FC(i)17095 32392 y Fu(1)17611 32070 y FD(;)221 b(!)19003 32269 y FC(i)19323 32392 y Fu(2)19841 32070 y FJ(\).)741 b(One)487 b(has)h(2)p FD(\031)48 b(n)p FJ(\()29376 31720 y(~)29341 32070 y FD(k)30017 32269 y FB(1)30543 32070 y FD(;)31161 31720 y FJ(~)31125 32070 y FD(k)31801 32269 y FB(2)32328 32070 y FJ(\))462 b(=)564 b(\026)-753 b FD(!)35579 32269 y FC(\026)36200 32070 y FD(T)513 b FJ(+)332 b FD(O)36 b FJ(\(1\))488 b(and)0 33676 y(taking)434 b FD(h)369 b FJ(=)g(\()p FD(h)7764 33875 y FB(1)8289 33676 y FD(;)221 b(h)9620 33875 y FB(2)10146 33676 y FJ(\))369 b(=)g(\()12943 33325 y(~)12908 33676 y FD(k)13584 33875 y FB(2)14110 33676 y FD(;)221 b FA(\000)15760 33325 y FJ(~)15725 33676 y FD(k)16401 33875 y FB(1)16928 33676 y FJ(\),)434 b(one)f(gets)11115 36549 y(0)369 b(=)g(2)p FD(\031)48 b(n)p FJ(\()16268 36198 y(~)16233 36549 y FD(k)16909 36748 y FB(1)17435 36549 y FD(;)18053 36198 y FJ(~)18017 36549 y FD(k)18693 36748 y FB(2)19220 36549 y FJ(\))295 b FA(\001)h FD(h)368 b FJ(=)472 b(\026)-753 b FD(!)23994 36748 y FC(\026)24910 36549 y FA(\001)295 b FD(hT)476 b FJ(+)295 b FD(O)36 b FJ(\()p FA(j)p FD(h)p FA(j)p FJ(\))221 b FD(:)7141 b FJ(\(5.241\))0 39423 y(F)-108 b(rom)433 b(Lemma)g(5.11.1,)j(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b(that,)f(except)h(for)g FD(\026)f FJ(in)h(a)f(zero)h(measure)g(set,)13780 42955 y FA(j)104 b FJ(\026)-754 b FD(!)14959 43154 y FC(\026)15876 42955 y FA(\001)295 b FD(h)p FA(j)369 b(\025)19562 42056 y FD(c)p FJ(\()p FD(\026)p FJ(\))p 19562 42649 2354 54 v 19761 43866 a FA(j)p FD(h)p FA(j)21248 43482 y FC(&)22270 42955 y FD(;)4123 b(&)477 b(>)368 b FJ(1)221 b FD(;)0 46500 y FJ(from)434 b(whic)-36 b(h)433 b(it)h(follo)-36 b(ws)436 b(that)14331 49774 y FD(O)36 b FJ(\()p FA(j)p FD(h)p FA(j)p FJ(\))369 b(=)f FA(j)104 b FJ(\026)-754 b FD(!)20788 49973 y FC(\026)21705 49774 y FA(\001)295 b FD(h)p FA(j)p FD(T)550 b FA(\025)26334 48875 y FD(c)p FJ(\()p FD(\026)p FJ(\))p FD(T)p 26334 49468 3297 54 v 27004 50685 a FA(j)p FD(h)p FA(j)28491 50301 y FC(&)0 53449 y FJ(and)433 b(hence)g FA(j)p FD(h)p FA(j)7629 52967 y FC(&)69 b FB(+1)9669 53449 y FA(\025)369 b FJ(const)221 b FD(T)181 b FJ(,)434 b(namely)13469 56322 y FA(j)p FD(h)p FA(j)368 b(\025)i FJ(const)221 b FD(T)20861 55774 y FB(1)p FC(=)p FB(\()p FC(&)69 b FB(+1\))24574 56322 y FJ(=)368 b(const)222 b FD(T)30089 55774 y FC(\032)40120 56322 y FJ(\(5.242\))0 59196 y(where)18817 61061 y FD(\032)369 b FJ(:=)22822 60162 y(1)p 21732 60755 2830 54 v 21732 61972 a(1)295 b(+)g FD(&)24916 61061 y(:)0 64138 y FJ(F)-108 b(rom)433 b(\(5.240\),)i(from)f(\(5.242\))h(and)e(b)36 b(eing)434 b FA(j)p FD(h)p FA(j)368 b FJ(=)h FA(j)25336 63787 y FJ(~)25301 64138 y FD(k)45 b FA(j)p FJ(,)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)6655 67980 y FD(T)7598 67431 y FB(min)10329 67980 y FA(\025)12602 67081 y FD(T)p 12602 67674 943 54 v 12685 68891 a(n)14046 67980 y FA(\025)370 b FJ(2)p FD(\031)17086 67081 y FA(j)17490 66731 y FJ(~)17455 67081 y FD(k)45 b FA(j)p 17017 67674 1596 54 v 17017 68891 a(j)p FD(!)j FA(j)19041 67980 y(\000)295 b FD(O)21399 66505 y Fz(\020)23186 67081 y FJ(1)p 22325 67674 2372 54 v 22325 68891 a FD(n)p FA(j)p FD(!)48 b FA(j)24830 66505 y Fz(\021)25993 67980 y FJ(=)27506 67081 y(2)p FD(\031)g FA(j)p FD(h)p FA(j)p 27506 67674 2923 54 v 28169 68891 a(j)p FD(!)g FA(j)30857 67980 y(\000)295 b FD(O)33215 66505 y Fz(\020)35002 67081 y FJ(1)p 34141 67674 2372 54 v 34141 68891 a FD(n)p FA(j)p FD(!)48 b FA(j)36646 66505 y Fz(\021)10329 71472 y FA(\025)1107 b FJ(const)15851 70574 y FD(T)16794 70092 y FC(\032)p 15793 71167 1596 54 v 15793 72384 a FA(j)p FD(!)48 b FA(j)17816 71472 y(\000)296 b FD(O)20175 69998 y Fz(\020)21962 70574 y FJ(1)p 21101 71167 2372 54 v 21101 72384 a FD(n)p FA(j)p FD(!)48 b FA(j)23606 69998 y Fz(\021)24768 71472 y FA(\025)369 b FJ(const)222 b FD(T)30305 70924 y FC(\032)31063 71472 y FD(:)43061 74991 y Fn(\003)21072 78312 y FJ(210)p eop %%Page: 211 211 211 210 bop eop %%Page: 212 212 212 211 bop eop %%Page: 213 213 213 212 bop 6299 13344 a FG(Bibliograph)-86 b(y)6299 19377 y FJ([AmPr])651 b(A.)332 b(Am)-36 b(brosetti,)353 b(G.)332 b(Pro)36 b(di:)528 b Fs(A)371 b(Primer)g(of)h(Nonline)-66 b(ar)369 b(A)-33 b(nalysis)p FJ(,)353 b(Cam)-36 b(bridge)8973 20982 y(Univ)g(ersit)g(y)434 b(Press,)g(1993.)6299 23557 y([A])977 b(V.I.)475 b(Arnold:)659 b Fs(Mathematic)-66 b(al)500 b(Metho)-66 b(ds)502 b(of)g(Classic)-66 b(al)502 b(Me)-66 b(chanics)p FJ(,)483 b(Second)473 b(Edi-)8973 25162 y(tion,)434 b(Springer-V)-108 b(erlag,)433 b(1989.)6299 27738 y([A88])652 b(V.I.)587 b(Arnold:)884 b 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b(preprin)-36 b(t)432 b(2005.)6299 22848 y([LinSh88])651 b(B.V.)332 b(Lindskij,)353 b(E.I.)333 b(Sh)-36 b(ulman:)526 b Fs(Perio)-66 b(dic)370 b(solutions)i(of)f(the)h(e)-66 b(quation)370 b FD(u)48538 23047 y FC(tt)49361 22848 y FA(\000)8973 24453 y FD(u)9713 24652 y FC(xx)11123 24453 y FJ(+)295 b FD(u)13170 23971 y FB(3)14065 24453 y FJ(=)368 b(0,)435 b(F)-108 b(unct.)432 b(Anal.)i(Appl.)f(22)i(\(1988\),)f(332{333.)6299 27037 y([Lo69])652 b(H.)447 b(Lo)-36 b(vicaro)g(v)-72 b(a:)607 b Fs(Perio)-66 b(dic)476 b(solutions)h(of)g(a)g(we)-66 b(akly)477 b(nonline)-66 b(ar)476 b(wave)h(e)-66 b(quation)8973 28642 y(in)464 b(one)g(dimension)p FJ(,)433 b(Czec)-36 b(h.)433 b(Math.)h(J.)g(19)g(\(1969\),)h(324{342.)6299 31225 y([Ly07])652 b(A.M.)439 b(Ly)-36 b(apuno)g(v:)588 b Fs(Pr)-66 b(obl)-33 b(\022)-631 b(eme)469 b(g)-33 b(\023)-631 b(en)-33 b(\023)-631 b(er)-66 b(al)468 b(de)h(la)h(stabilit)-33 b(\023)-631 b(e)468 b(du)j(mouvement)p FJ(,)438 b(Ann.)8973 32830 y(Sc.)479 b(F)-108 b(ac.)480 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b(9,)f(\(1968\),)h(882{886.)6299 50211 y([Mo67])652 b(J.)618 b(Moser:)946 b Fs(Conver)-66 b(gent)631 b(series)i(exp)-66 b(ansions)633 b(for)h(quasi{p)-66 b(erio)g(dic)632 b(motions)p FJ(,)8973 51817 y(Math.)433 b(Ann.)g(169,)i(\(1967\),)g(136{176.)6299 54400 y([Mo73])652 b(J.)305 b(Moser:)514 b Fs(Stable)346 b(and)g(r)-66 b(andom)346 b(motions)g(in)g(dynamic)-66 b(al)345 b(systems)p FJ(,)331 b(Princeton:)8973 56005 y(Princeton)433 b(Univ.)h(Press,)g(1973.)6299 58589 y([Mo76])652 b(J.)480 b(Moser:)672 b Fs(Perio)-66 b(dic)506 b(orbits)i(ne)-66 b(ar)506 b(an)i(e)-66 b(quilibrium)505 b(and)j(a)g(the)-66 b(or)g(em)506 b(by)i(A)-33 b(lan)8973 60194 y(Weinstein)p FJ(,)431 b(Comm.)j(Pure)f(Appl.)h(Math.)f(29,)i (\(1976\),)f(727{747.)6299 62777 y([Mo77])652 b(J.)435 b(Moser:)580 b Fs(Pr)-66 b(o)g(of)465 b(of)h(a)g(gener)-66 b(alize)g(d)464 b(form)h(of)g(a)h(\014xe)-66 b(d)465 b(p)-66 b(oint)464 b(The)-66 b(or)g(em)465 b(due)h(to)8973 64382 y(G.D.)501 b(Birkho\013)p FJ(,)485 b(Geometry)475 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(9,)g(1059{1065)j(\(1989\).)27371 78312 y(217)p eop %%Page: 218 218 218 217 bop 0 4488 a FJ([P)-36 b(o])651 b(H.)479 b(P)-36 b(oincar)g(\023)-614 b(e:)667 b Fs(L)-66 b(es)506 b(M)-33 b(\023)-631 b(etho)-66 b(des)505 b(nouvel)66 b(les)507 b(de)e(la)i(M)-33 b(\023)-631 b(ec)-66 b(anique)504 b(C)-33 b(\023)-631 b(eleste)p FJ(,)488 b(Gauthier)2674 6094 y(Villars,)435 b(P)-36 b(aris,)434 b(1892.)0 8951 y([P82])652 b(J.)379 b(P\177)-650 b(osc)-36 b(hel:)551 b Fs(Inte)-66 b(gr)g(ability)411 b(of)j(Hamiltonian)f(systems)i(on)f(Cantor)g(sets)p FJ(,)391 b(Comm.)2674 10556 y(Pure)433 b(Appl.)g(Math.)g(35)i (\(1982\),)f(653{695.)0 13414 y([P89])652 b(J.)509 b(P\177)-650 b(osc)-36 b(hel:)729 b Fs(On)534 b(el)66 b(liptic)533 b(lower)h(dimensional)f(tori)g(in)h(hamiltonian)e(systems)p FJ(,)2674 15019 y(Math.)433 b(Z.)h(202)g(\(1989\),)h(559{608.)0 17876 y([P90])652 b(J.)454 b(P\177)-650 b(osc)-36 b(hel:)619 b Fs(Smal)66 b(l)483 b(divisors)h(with)f(sp)-66 b(atial)483 b(structur)-66 b(e)483 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b(P\177)-650 b(osc)-36 b(hel:)971 b Fs(A)644 b(KAM)g(the)-66 b(or)g(em)644 b(for)h(some)g(nonline)-66 b(ar)643 b(p)-66 b(artial)645 b(di\013er)-66 b(ential)2674 36080 y(e)g(quations)p FJ(,)432 b(Ann.)i(Scuola)f(Norm.)h(Sup.)f(Pisa)h (Cl.)h(Sci.)e(23)i(\(1996\),)f(119{148.)0 38937 y([P98])652 b(J.)693 b(P\177)-650 b(osc)-36 b(hel:)1097 b Fs(Nonline)-66 b(ar)702 b(p)-66 b(artial)703 b(di\013er)-66 b(ential)701 b(e)-66 b(quations,)761 b(Birkho\013)704 b(normal)2674 40542 y(forms)697 b(and)h(KAM)e(the)-66 b(ory)p FJ(,)750 b(Pro)36 b(ceedings)687 b(of)h(the)e(ECM-2,)750 b(Budap)36 b(est,)750 b(1996.)2674 42147 y(Progress)434 b(in)f(Mathematics)h(169)g (\(1998\),)h(167{186.)0 45005 y([P02])652 b(J.)462 b(P\177)-650 b(osc)-36 b(hel:)636 b Fs(On)491 b(the)g(c)-66 b(onstruction)490 b(of)h(almost)h(p)-66 b(erio)g(dic)489 b(solutions)j(for)f(a)g(non-) 2674 46610 y(line)-66 b(ar)488 b(Schr\177)-664 b(oe)-66 b(dinger)488 b(e)-66 b(quation)p FJ(,)466 b(Ergo)36 b(d.)461 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b(the)j(normal)g(forms)8973 37919 y(of)464 b(line)-66 b(ar)464 b(dynamic)-66 b(al)464 b(systems)p FJ(,)434 b(Am.)g(J.)g(Math.)f(58,)h(No.)h(1,)f(\(1936\),)h(141{163.) 27371 78312 y(219)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0608260521117--