%!PS-Adobe-2.0 %%Creator: dvipsk 5.519b Copyright 1986, 1993 Radical Eye Software %%Title: 3corpi.dvi %%Pages: 41 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips 3corpi.dvi -o 3corpi.ps %DVIPSSource: TeX output 1996.04.26:1036 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}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 dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put 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 /IE 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 IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /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 dup definefont setfont}B /ch-width{ch-data dup length 5 sub get} B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]{ ch-image}imagemask restore}B /D{/cc X dup type /stringtype ne{]}if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 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 dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore showpage userdict /eop-hook known{eop-hook}if}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 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 /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false} ifelse}{false}ifelse end{{gsave TR -.1 -.1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 -.1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail{dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail} B /c{-4 M}B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{ 3 M}B /k{4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{ 3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end %%EndProcSet TeXDict begin 40258431 52099146 1000 300 300 (3corpi.dvi) @start /Fa 10 118 df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b 1 35 df<000FF0007FFC01E01E03 0007040000080000100000100000100000100000100000087F000780800CFF0010000020 0000400000400000800000800000800000800000800010C000206000603C03801FFF0003 F800181C7E9A1C>34 D E /Fc 4 91 df<007F0401E1C406807C09002411001422000C42 000442000444000484000084000084000084000084000084000084000084000044000042 0000420000210002110004088008066070019FE0007F00171A7F9812>67 D78 D84 D<3FFFF02302202C0420500840600840401080401100002100004200 0044000084000108000108000210000220000420080840080880181080182100302100D0 420110440210FFFFF015187F9712>90 D E /Fd 3 106 df<3078F06005047C830D>46 D<01E007100C1018083810701070607F80E000E000E000E000E000E0086010602030C01F 000D127B9113>101 D<01800380010000000000000000000000000000001C0026004700 47008E008E000E001C001C001C0038003800710071007100720072003C00091C7C9B0D> 105 D E /Fe 37 124 df45 D<1C007F00FF80FF80FF80FF80FF807F001C0009097B8813>I<000E00001E00007E0007 FE00FFFE00FFFE00F8FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000 FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000 FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE007FFFFE7FFFFE7FFFFE17 277BA622>49 D<00FF800003FFF0000FFFFC003F03FF007C00FF807C007FC0FE007FC0FF 003FE0FF003FE0FF003FE0FF001FE07E001FE03C003FE000003FE000003FC000003FC000 007F8000007F800000FF000001FE000001FC000003F0000007E000000FC000001F000000 3E0000007C00E0007800E000F000E001E001C0038001C0070001C00FFFFFC01FFFFFC03F FFFFC07FFFFFC0FFFFFF80FFFFFF80FFFFFF801B277DA622>I<007F800003FFF00007FF FC001F81FE001F00FF003F80FF003F807F803FC07F803F807F803F807F801F007F800000 FF800000FF000000FF000001FE000003F8000007F00000FFC00000FFF0000001FC000000 FF0000007F8000007FC000003FC000003FE000003FE000003FE03C003FE07E003FE0FF00 3FE0FF003FE0FF003FC0FF007FC0FE007F807C00FF803F01FF001FFFFC0007FFF00000FF 80001B277DA622>I<00000E0000001E0000003E0000007E000000FE000000FE000001FE 000003FE0000077E00000E7E00000E7E00001C7E0000387E0000707E0000E07E0000E07E 0001C07E0003807E0007007E000E007E000E007E001C007E0038007E0070007E00E0007E 00FFFFFFF8FFFFFFF8FFFFFFF80000FE000000FE000000FE000000FE000000FE000000FE 000000FE000000FE00007FFFF8007FFFF8007FFFF81D277EA622>I<0C0003000F803F00 0FFFFE000FFFFE000FFFFC000FFFF8000FFFE0000FFFC0000FFE00000E0000000E000000 0E0000000E0000000E0000000E0000000E7FC0000FFFF8000F80FE000E007F000C003F80 00003F8000001FC000001FC000001FE000001FE018001FE07E001FE0FE001FE0FE001FE0 FE001FE0FE001FE0FE001FC078003FC078003F803C007F001F01FE000FFFFC0003FFF000 00FF80001B277DA622>I<1C007F00FF80FF80FF80FF80FF807F001C0000000000000000 00000000000000000000001C007F00FF80FF80FF80FF80FF807F001C00091B7B9A13>58 D<000003800000000007C00000000007C0000000000FE0000000000FE0000000000FE000 0000001FF0000000001FF0000000003FF8000000003FF8000000003FF80000000073FC00 00000073FC00000000F3FE00000000E1FE00000000E1FE00000001C0FF00000001C0FF00 000003C0FF80000003807F80000007807FC0000007003FC0000007003FC000000E003FE0 00000E001FE000001E001FF000001C000FF000001FFFFFF000003FFFFFF800003FFFFFF8 0000780007FC0000700003FC0000700003FC0000E00001FE0000E00001FE0001E00001FF 0001C00000FF0001C00000FF00FFFE001FFFFEFFFE001FFFFEFFFE001FFFFE2F297EA834 >65 D70 D75 D77 DI<007F806003FFF0E00FFFFFE01F807FE03F001FE0 7E0007E07E0003E07C0003E0FC0001E0FC0001E0FC0000E0FE0000E0FE0000E0FF000000 FFC000007FFE00007FFFE0003FFFFC003FFFFF001FFFFF8007FFFFC003FFFFE000FFFFF0 0007FFF000007FF000000FF8000007F8000003F8E00003F8E00001F8E00001F8E00001F8 F00001F8F00001F0F80003F0FC0003E0FF0007E0FFE01FC0FFFFFF00E0FFFE00C01FF000 1D297CA826>83 D<7FFFFFFFFFC07FFFFFFFFFC07FFFFFFFFFC07F803FC03FC07E003FC0 07C078003FC003C078003FC003C070003FC001C0F0003FC001E0F0003FC001E0E0003FC0 00E0E0003FC000E0E0003FC000E0E0003FC000E0E0003FC000E000003FC0000000003FC0 000000003FC0000000003FC0000000003FC0000000003FC0000000003FC0000000003FC0 000000003FC0000000003FC0000000003FC0000000003FC0000000003FC0000000003FC0 000000003FC0000000003FC0000000003FC0000000003FC0000000003FC0000000003FC0 000000003FC0000000003FC00000007FFFFFE000007FFFFFE000007FFFFFE0002B287EA7 30>I<01FF800007FFF0000F81FC001FC0FE001FC07F001FC07F001FC03F800F803F8000 003F8000003F8000003F80000FFF8000FFFF8007FC3F801FE03F803F803F807F803F807F 003F80FE003F80FE003F80FE003F80FE007F80FF007F807F00FFC03F83DFFC0FFF0FFC01 FC03FC1E1B7E9A21>97 DI<001FF80000FFFE0003F01F000FE03F801FC03F803F803F80 3F803F807F801F007F000000FF000000FF000000FF000000FF000000FF000000FF000000 FF000000FF000000FF0000007F0000007F8000003F8001C03FC001C01FC003C00FE00780 03F01F0000FFFC00001FE0001A1B7E9A1F>I<00003FF80000003FF80000003FF8000000 03F800000003F800000003F800000003F800000003F800000003F800000003F800000003 F800000003F800000003F800000003F800000003F800001FE3F80000FFFBF80003F03FF8 000FE00FF8001FC007F8003F8003F8003F8003F8007F8003F8007F0003F800FF0003F800 FF0003F800FF0003F800FF0003F800FF0003F800FF0003F800FF0003F800FF0003F800FF 0003F8007F0003F8007F0003F8003F8003F8003F8007F8001FC00FF8000FE01FF80003F0 3FFF8000FFF3FF80003FC3FF80212A7EA926>I<003FE00001FFF80003F07E000FE03F00 1FC01F803F800FC03F800FC07F000FC07F0007E0FF0007E0FF0007E0FF0007E0FFFFFFE0 FFFFFFE0FF000000FF000000FF000000FF0000007F0000007F8000003F8000E03F8001E0 1FC001C00FE003C003F81F8000FFFE00001FF0001B1B7E9A20>I<0007F0003FFC00FE3E 01FC7F03F87F03F87F07F07F07F03E07F00007F00007F00007F00007F00007F00007F000 FFFFC0FFFFC0FFFFC007F00007F00007F00007F00007F00007F00007F00007F00007F000 07F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F000 7FFF807FFF807FFF80182A7EA915>I<00FF81F003FFE7FC0FC1FE7C1F80FC7C3F80FE7C 3F007E107F007F007F007F007F007F007F007F007F007F007F007F003F007E003F80FE00 1F80FC000FC1F8001FFFE00018FF8000380000003C0000003C0000003E0000003FFFF800 3FFFFF001FFFFFC00FFFFFE007FFFFF01FFFFFF07E0007F87C0001F8F80001F8F80000F8 F80000F8F80000F8FC0001F87E0003F03F0007E00FC01F8003FFFE00007FF0001E287E9A 22>II<07001FC01FE03FE03FE03FE01FE01FC007000000000000000000000000000000 FFE0FFE0FFE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE0 0FE00FE00FE00FE00FE00FE0FFFEFFFEFFFE0F2B7DAA14>I108 DII<003FE00001FFFC0003F07E00 0FC01F801F800FC03F800FE03F0007E07F0007F07F0007F07F0007F0FF0007F8FF0007F8 FF0007F8FF0007F8FF0007F8FF0007F8FF0007F8FF0007F87F0007F07F0007F03F800FE0 3F800FE01F800FC00FC01F8007F07F0001FFFC00003FE0001D1B7E9A22>II114 D<03FE300FFFF03E03F07800F07000F0F00070F00070F80070FC0000FFE000FFFE007FFF C03FFFE01FFFF007FFF800FFFC0003FC0000FCE0007CE0003CF0003CF0003CF80078FC00 78FF01F0F7FFC0C1FF00161B7E9A1B>I<00700000700000700000700000F00000F00000 F00001F00003F00003F00007F0001FFFF0FFFFF0FFFFF007F00007F00007F00007F00007 F00007F00007F00007F00007F00007F00007F00007F00007F00007F03807F03807F03807 F03807F03807F03807F03803F87001F8F000FFE0001F8015267FA51B>III120 DI123 D E /Ff 1 50 df<18F818181818181818181818FF080D7D8C0E>49 D E /Fg 2 107 df0 D106 D E /Fh 4 111 df<1E01F807004005804005804008C080 086080086080083080101900101900100D00100700300600F80200150E7E8D16>78 D<0808000000007098B0303060646870060F7D8E0B>105 D<7800180018001800300030 C03360344078007E0063006320C340C1800B0E7E8D10>107 D<73809C40986030C030C0 30C03188619060E00D097D8812>110 D E /Fi 17 123 df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j 45 124 df<00000FE00000301800 00E01C0001C03C0001803C00038038000380000003800000070000000700000007000000 07000000070000000E000000FFFFE0000E00E0000E00E0000E01C0001C01C0001C01C000 1C01C0001C0380001C038000380380003803800038070000380700003807000070070800 700E1000700E1000700E1000700E2000E0062000E003C000E0000000E0000000C0000001 C0000001C0000071800000F1800000F3000000620000003C0000001E2D82A21B>12 D<00000FF3800000303F000000607F000000C07F000001C037000001800E000003800E00 0003800E000003800E000003801C000007001C000007001C000007001C00000700380000 FFFFF800000E003800000E003800000E007000000E007000000E007000001C007000001C 00E000001C00E000001C00E000001C00E000003801C000003801C000003801C000003801 C20000380384000070038400007003840000700388000070018800007000F00000E00000 0000E000000000E000000000C000000000C00000007180000000F180000000F100000000 62000000003C00000000212D82A21D>I<0C1E3F3F1D02020204040810204080080F75A2 0F>39 D<00008000010000020000040000080000100000300000600000C00000C0000180 000300000300000600000600000E00000C00001C00001800001800003800003000003000 00700000700000600000600000E00000E00000E00000C00000C00000C00000C00000C000 00C00000C00000C00000C00000C00000C00000C00000C000004000006000006000002000 00300000100000080000113278A414>I<00080000040000060000020000030000030000 010000018000018000018000018000018000018000018000018000018000018000018000 0180000180000380000380000380000300000300000700000700000600000600000E0000 0C00000C00001C0000180000380000300000300000600000600000C00001800001800003 0000060000040000080000100000200000400000800000113280A414>I<0E1E1E1E1E02 020404080810204080070F7D840F>44 DI<70F8F8F0E005 057A840F>I<0000000C0000001C00000018000000380000003000000070000000E00000 00C0000001C00000018000000380000003000000070000000E0000000C0000001C000000 18000000380000007000000060000000E0000000C0000001C00000038000000300000007 000000060000000E0000001C0000001800000038000000300000007000000060000000E0 000001C00000018000000380000003000000070000000E0000000C0000001C0000001800 0000380000007000000060000000E0000000C00000001E3180A419>I<00010003000300 06001E002E03CE001C001C001C001C0038003800380038007000700070007000E000E000 E000E001C001C001C001C003800380038003800780FFFC10217AA019>49 D<000FC000106000603800801800801C01001C02201E02101E04101E04101E04101E0820 3C08203C0840380840780880F00700E00001C000030000060000180000200000C0000100 000200000400100800301000202000605F80C063FFC040FF80807F00801E0017227CA019 >I<0000180000380000380000700000700000700000E00000E00000E00000C00001C000 0180000380000300000300000600000600000C00000C000018000010000031800061C000 43800083800183800303800207000407000807003FC700403E00800FF0000E00000E0000 1C00001C00001C00001C00003800003800003800003000152B7EA019>52 D<07000F800F800F000E00000000000000000000000000000000000000000000007000F8 00F800F000E00009157A940F>58 D<00E001F001F001E001C00000000000000000000000 00000000000000000000000E001E001E001E001E00020002000400040008000800100020 00400080000C1F7D940F>I<0000030000000300000007000000070000000F0000000F00 00001F0000002F0000002F0000004F0000004F8000008780000087800001078000020780 0002078000040780000407800008078000080780001007800030078000200780007FFF80 004007C0008007C0008003C0010003C0030003C0020003C0040003C0040003C00C0003C0 3C0007C0FF003FFC1E237DA224>65 D<00FFFFF000000F003C00000F000E00000F000700 001E000380001E000380001E0001C0001E0001C0003C0001C0003C0001E0003C0001E000 3C0001E000780001E000780001E000780001E000780001E000F00003C000F00003C000F0 0003C000F00003C001E000078001E000078001E000070001E0000F0003C0000E0003C000 1C0003C0003C0003C00038000780007000078000E000078001C00007800700000F801C00 00FFFFF0000023227DA125>68 D<00FFFFFF000F000F000F0003000F0003001E0003001E 0003001E0002001E0002003C0002003C0002003C0102003C010000780200007802000078 060000780E0000FFFC0000F00C0000F00C0000F00C0001E0080001E0080001E0080001E0 000003C0000003C0000003C0000003C00000078000000780000007800000078000000F80 0000FFFC000020227DA120>70 D<00FFFC00000F8000000F0000000F0000001E0000001E 0000001E0000001E0000003C0000003C0000003C0000003C000000780000007800000078 00000078000000F0000000F0000000F0000000F0000001E0000001E0000001E0002001E0 002003C0004003C0004003C0008003C0008007800180078001000780030007800F000F80 3E00FFFFFE001B227DA11F>76 D<1FFFFFF81E03C0381803C0183003C018200780182007 80184007801040078010400F0010800F0010800F0010000F0000001E0000001E0000001E 0000001E0000003C0000003C0000003C0000003C00000078000000780000007800000078 000000F0000000F0000000F0000000F0000001E0000001E0000001E0000001E0000003E0 0000FFFF00001D2277A123>84 D87 D<00F8C00185C00705C00E03800E03801C03803C03 80380700780700780700780700F00E00F00E00F00E00F00E10F01C20701C20703C20305C 40308C400F078014157B9419>97 D<03C03F8003800380038007000700070007000E000E 000E000E001C001CF81D0C1E0E3C0638073807380F700F700F700F700FE01EE01EE01EE0 3CE038E038607060E031C01F0010237BA216>I<007E0001C1000301800703800E07801C 07803C0000380000780000780000780000F00000F00000F00000F00000F0010070010070 0200300C001830000FC00011157B9416>I<00003C0003F8000038000038000038000070 0000700000700000700000E00000E00000E00000E00001C000F9C00185C00705C00E0380 0E03801C03803C0380380700780700780700780700F00E00F00E00F00E00F00E10F01C20 701C20703C20305C40308C400F078016237BA219>I<00F803840E021C023C0238027804 F018FFE0F000F000E000E000E000E000E002E0026004701830600F800F157A9416>I<00 003E0000470000CF00018F00018600038000038000038000070000070000070000070000 0700000E0000FFF0000E00000E00000E00001C00001C00001C00001C00001C0000380000 380000380000380000380000700000700000700000700000700000E00000E00000E00000 E00000C00001C00001C000718000F18000F300006200003C0000182D82A20F>I<001F18 0030B800E0B801C07001C0700380700780700700E00F00E00F00E00F00E01E01C01E01C0 1E01C01E01C01E03800E03800E0780060B8006170001E700000700000700000E00000E00 000E00701C00F01800F0300060E0003F8000151F7E9416>I<00F0000FE00000E00000E0 0000E00001C00001C00001C00001C000038000038000038000038000070000071F000721 8007C0C00F00E00F00E00E00E00E00E01C01C01C01C01C01C01C01C03803803803803803 80380704700708700E08700E10700610E006206003C016237DA219>I<00C001E001C001 C0000000000000000000000000000000001C002300430043008700870087000E000E001C 001C001C00380038003840708070807080710032001C000B217BA00F>I<0000E00001E0 0001E00000C0000000000000000000000000000000000000000000000000001E00002300 004380008380008380010380010380000700000700000700000700000E00000E00000E00 000E00001C00001C00001C00001C00003800003800003800003800007000007000007000 70E000F0C000F180006300003C0000132B82A00F>I<01E01FC001C001C001C003800380 0380038007000700070007000E000E000E000E001C001C001C001C003800380038003800 7000700070007100E200E200E200E200640038000B237CA20C>108 D<1C0F80F8002610C10C0047606606008780780700878078070087007007008700700700 0E00E00E000E00E00E000E00E00E000E00E00E001C01C01C001C01C01C001C01C01C001C 01C03820380380384038038070403803807080380380308070070031003003001E002315 7B9428>I<1C0F002631C04740C08780E08780E08700E08700E00E01C00E01C00E01C00E 01C01C03801C03801C03801C0704380708380E08380E103806107006203003C016157B94 1B>I<007E0001C3000381800701C00E01C01C01E03C01E03801E07801E07801E07801E0 F003C0F003C0F00380F00780700700700E00700C0030180018700007C00013157B9419> I<01C1F002621804741C08780C08700E08700E08701E00E01E00E01E00E01E00E01E01C0 3C01C03C01C03C01C07803807003807003C0E003C1C0072380071E000700000700000E00 000E00000E00000E00001C00001C00001C0000FFC000171F7F9419>I<00F8400184C007 05C00E03800E03801C03803C0380380700780700780700780700F00E00F00E00F00E00F0 0E00F01C00701C00703C00305C0030B8000F380000380000380000700000700000700000 700000E00000E00000E0000FFE00121F7B9416>I<1C1F002620804741C08783C08703C0 8701808700000E00000E00000E00000E00001C00001C00001C00001C0000380000380000 38000038000070000030000012157B9415>I<00FC000183000200800401800C03800C03 000C00000F00000FF00007FC0003FE00003E00000F00000700700700F00600F00600E004 004008002030001FC00011157D9414>I<00C001C001C001C001C0038003800380038007 00FFF8070007000E000E000E000E001C001C001C001C0038003800380038107020702070 40708031001E000D1F7C9E10>I<1E00602300E04380E04381C08381C08701C08701C007 03800E03800E03800E03801C07001C07001C07001C07081C0E10180E101C0E101C1E200C 262007C3C015157B941A>I<1E03802307C04387C04383C08381C08700C08700C0070080 0E00800E00800E00801C01001C01001C01001C02001C02001C04001C08001C08000C3000 03C00012157B9416>I<1E0060E02300E1F04380E1F04381C0F08381C0708701C0308701 C030070380200E0380200E0380200E0380201C0700401C0700401C0700401C0700801C07 00801C0701001C0F01000C0F020006138C0003E0F0001C157B9420>I<03C1E004621008 3470103CF02038F020386020380000700000700000700000700000E00000E00000E00000 E02061C040F1C040F1C080E2C100446200383C0014157D9416>I<1E0030230070438070 4380E08380E08700E08700E00701C00E01C00E01C00E01C01C03801C03801C03801C0380 1C07001C07001C07001C0F000C3E0003CE00000E00000E00001C00601C00F03800F03000 E0600080C0004380003E0000141F7B9418>I123 D E /Fk 35 122 df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l 10 57 df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m 2 91 df78 D<7FFE60448084010801080210022004200840088010801100210042024402841EFFFE0F 117F9044>90 D E /Fn 6 107 df0 D<0000C00003C0000F00003C0000F00003C0000F00003C0000F00000F000003C00000F00 0003C00000F000003C00000F000003C00000C0000000000000000000000000FFFFC0FFFF C012187D9018>20 D<081C1C3838383070706060C0C0060D7E8D09>48 D<01FE0FFE1E00380070006000E000C000FFFEFFFEC000E0006000700038001E000FFE01 FE0F127D8E15>50 D<0FFF003183E0418070C18030018038010018030018030018030018 0300100200300600200600400401800C06000838001FC00015117F9018>68 D106 D E /Fo 12 114 df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p 24 116 df12 D<0006000C001800300070006000C001C0018003800300070006000E000C00 1C001C0018003800380038003000700070007000700070007000E000E000E000E000E000 E000E000E000E000E000E000E000E000E000E000E000E000E00070007000700070007000 7000300038003800380018001C001C000C000E000600070003000380018001C000C00060 007000300018000C00060F4A788119>16 DI<00003000006000 00C0000180000300000700000E00000C0000180000380000300000700000E00000C00001 C0000180000380000380000300000700000600000E00000E00000C00001C00001C00001C 000018000038000038000038000038000070000070000070000070000070000070000070 0000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E0 0000E00000E00000E00000E00000E00000E00000E00000E00000E0000070000070000070 00007000007000007000007000003800003800003800003800001800001C00001C00001C 00000C00000E00000E000006000007000003000003800003800001800001C00000C00000 E000007000003000003800001800000C00000E000007000003000001800000C000006000 0030146377811F>II<0000180000300000600000 E00000C0000180000380000700000600000E00000C00001C000038000038000070000070 0000E00000E00001E00001C00001C0000380000380000380000780000700000700000F00 000E00000E00001E00001E00001E00001C00001C00003C00003C00003C00003C00003800 00780000780000780000780000780000780000780000780000700000F00000F00000F000 00F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F000 00F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F000007000 007800007800007800007800007800007800007800007800003800003C00003C00003C00 003C00001C00001C00001E00001E00001E00000E00000E00000F00000700000700000780 0003800003800003800001C00001C00001E00000E00000E0000070000070000038000038 00001C00000C00000E000006000007000003800001800000C00000E00000600000300000 18157C768121>32 DI<00001C00003C0000 F80001E00003C0000780000F00000E00001E00003C00003C00003C000078000078000078 000078000078000078000078000078000078000078000078000078000078000078000078 000078000078000078000078000078000078000078000078000078000078000078000078 0000780000780000780000780000780000780000780000780000780000780000780000F0 0000F00000F00001E00001E00003C0000380000700000E00001C0000780000E00000E000 007800001C00000E000007000003800003C00001E00001E00000F00000F00000F0000078 000078000078000078000078000078000078000078000078000078000078000078000078 000078000078000078000078000078000078000078000078000078000078000078000078 000078000078000078000078000078000078000078000078000078000078000078000078 00007800003C00003C00003C00001E00000E00000F000007800003C00001E00000F80000 3C00001C167C7B8121>40 D<0018007800F001E003C007800F001F001E003E003C007C00 7C007800F800F800F800F800F800F800F800F800F800F800F800F800F800F800F800F800 F800F800F800F800F800F800F8000D25707E25>56 D58 D<007C007C007C007C007C007C007C007C007C007C007C007C007C007C007C007C007C00 7C007C007C007C007C007C007C00F800F800F800F001F001E003E003C0078007000E001C 003800F000C000F00038001C000E000700078003C003E001E001F000F000F800F800F800 7C007C007C007C007C007C007C007C007C007C007C007C007C007C007C007C007C007C00 7C007C007C007C007C007C0E4D798025>60 D62 D80 DI88 D<00000003800000000660 0000000C700000000CF00000000CF00000001C6000000018000000003800000000380000 000038000000007000000000700000000070000000007000000000F000000000E0000000 00E000000000E000000001E000000001E000000001C000000001C000000003C000000003 C000000003C000000003C000000007800000000780000000078000000007800000000F80 0000000F800000000F000000000F000000001F000000001F000000001F000000001F0000 00001E000000003E000000003E000000003E000000003E000000003C000000007C000000 007C000000007C000000007C000000007800000000F800000000F800000000F800000000 F800000000F000000001F000000001F000000001F000000001F000000001E000000001E0 00000003E000000003E000000003C000000003C000000003C000000003C0000000078000 0000078000000007800000000780000000070000000007000000000F000000000F000000 000E000000000E000000000E000000001E000000001C000000001C000000001C00000000 1800000000380000000038000000003000000000700000006060000000F060000000F0C0 000000E18000000063000000001E00000000245C7E7F17>90 D<00300000FC0001CE0007 03801C00E0300030C0000C1607809E17>98 D<07C0041FE018307830601FE0800F801605 809D17>101 D104 DI<0000E00003E0000F80 001E00003C0000700000700000E00000E00000E00000E00000E00000E00000E00000E000 00E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E000 00E00000E00000E00001C00001C0000380000700000E00003C0000F00000F000003C0000 0E000007000003800001C00001C00000E00000E00000E00000E00000E00000E00000E000 00E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E000 00E00000E00000E00000E000007000007000003C00001E00000F800003E00000E0134A7C 811C>110 DI<00000000060000000006000000000E0000 00000C000000000C000000001C0000000018000000001800000000380000000030000000 003000000000700000000060000000006000000000E000000000C000000000C000000001 C00000000180000000018000000003800000000300000000030000000007000000000600 00000006000000000E000000000C000000000C000000001C000000001800000000180000 000038000000003000000000300000000070000000006000000000600008000060001C00 00E0003C0000C0007C0000C000DC0001C0008E000180000E000180000E00038000070003 0000070003000007000700000380060000038006000003800E000003800C000001C00C00 0001C01C000001C018000000E018000000E038000000E030000000E03000000070700000 007060000000706000000038E000000038C000000038C00000001DC00000001D80000000 1D800000001F800000000F000000000F000000000F00000000060000000006000000274B 7C812A>113 D<000000000600000000060000000006000000000E000000000C00000000 0C000000000C000000000C000000001C0000000018000000001800000000180000000018 000000003800000000300000000030000000003000000000300000000070000000006000 0000006000000000600000000060000000006000000000E000000000C000000000C00000 0000C000000000C000000001C00000000180000000018000000001800000000180000000 038000000003000000000300000000030000000003000000000700000000060000000006 00000000060000000006000000000E000000000C000000000C000000000C000000000C00 0000001C0000000018000000001800000000180000000018000000003800000000300000 000030000000003000000000300000000070000000006000000000600000000060000800 00600008000060001C0000E0001C0000C0003C0000C0003C0000C0005C0000C0005C0001 C0008E000180000E000180000E000180000E000180000E000380000E0003000007000300 000700030000070003000007000700000700060000038006000003800600000380060000 03800E000003800C000003800C000001C00C000001C00C000001C01C000001C018000001 C018000001C018000000E018000000E038000000E030000000E030000000E03000000070 3000000070700000007060000000706000000070600000007060000000386000000038E0 00000038C000000038C000000038C000000038C00000001DC00000001D800000001D8000 00001D800000001D800000000F800000000F000000000F000000000F000000000F000000 000F000000000600000000060000000006000000277D7C812A>115 D E /Fq 5 91 df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r 24 121 df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s 16 98 df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t 69 123 df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u 42 121 df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v 58 124 df<0000FC1E000007026180000E07C38000 180F878000380F838000700700000070070000007007000000E00E000000E00E000000E0 0E000000E00E000000E00E000000E00E00003FFFFFF00001C01C000001C01C000001C01C 000001C01C000001C01C0000038038000003803800000380380000038038000003803800 000380380000070070000007007000000700700000070070000007007000000700700000 0E00E000000F00F00000FFC7FF000021237FA21C>11 D<0000FC0000038300000E008000 1C03800038078000700780007003000070000000E0000000E0000000E0000000E0000000 E0000000E000003FFFFE0001C00E0001C00E0001C00E0001C00E0001C00E0003801C0003 801C0003801C0003801C0003801C0003801C000700380007003800070038000700380007 003800070038000E0070000F007800FFC3FF0019237FA21B>I<0000FE03F00003818E0C 000E00B802001C01F00E003803E01E007003C01E007001C00C007001C00000E003800000 E003800000E003800000E003800000E003800000E00380003FFFFFFFF801C007003801C0 07003801C007003801C007003801C007003803800E007003800E007003800E007003800E 007003800E007003800E007007001C00E007001C00E007001C00E007001C00E007001C00 E007001C00E00E003801C00F003C01E0FFE3FF9FFC27237FA229>14 D<60F07038181C0E0603080976A218>18 D<01C003C0078007000E00180030006000C000 80000A0A72A218>I<1C0E3E1F7E3F7E3F3A1D0201020104020402080408041008201040 208040100F7AA218>34 D<1C3E7E7E3A02020404080810204080070F78A20E>39 D<000100020004000800100020006000C00180018003000300060006000C000C001C0018 0018003800300030007000700070006000600060006000E000E000E000E000E000E00060 0060006000600060003000300030001000180008000C0004000200010010327BA413>I< 0080004000200030001000180008000C000C000C00060006000600060006000700070007 0007000700070006000600060006000E000E000E000C000C001C00180018003800300030 0060006000C000C0018001800300060004000800100020004000800010327FA413>I<1C 3E7E7E3A02020404080810204080070F7D840E>44 DI<3078F8 787005057C840E>I<060F1F1F0E00000000000000000000003078F8787008157C940E> 58 D<0000040000000006000000000E000000001E000000001E000000003E000000003F 000000004F000000004F000000008F000000008F000000010F0000000107800000020780 00000207800000040780000004078000000807C000000803C000001003C000001003C000 002003C000003FFFE000004001E000004001E000008001E000008001E000010001E00001 0000F000020000F000060000F000040000F0000C0000F0003E0001F800FF800FFF802123 7EA225>65 D<03FFFFE0003E0078003C003C003C001E003C001F003C000F003C000F0078 000F0078001F0078001F0078001F0078003E0078003C00F0007800F001F000F007C000FF FFC000F001F000F000F801E0007801E0007C01E0003C01E0003E01E0003E01E0003E03C0 003C03C0007C03C0007C03C000F803C000F003C001E0078003C007C00F80FFFFFC002022 7EA123>I<0001FC02000F0306003C008C0070005C00E0003C01C0003C0380001C070000 1C0F0000181E0000181E0000083E0000083C0000087C0000087C000000F8000000F80000 00F8000000F8000000F8000000F0000000F0000000F0000000F0000020F0000020F00000 40F80000407800004078000080380001003C0001001C0002000E000C000700100001C0E0 00003F00001F247AA223>I<03FFFFC000003E00F000003C003C00003C001E00003C000E 00003C000F00003C0007800078000780007800078000780007C000780003C000780003C0 00780007C000F00007C000F00007C000F00007C000F00007C000F00007C000F00007C001 E0000F8001E0000F8001E0000F8001E0000F0001E0001F0001E0001E0003C0001E0003C0 003C0003C000780003C000700003C000E00003C001C000078007800007C01E0000FFFFF8 000022227EA125>I<03FFFFFF003E001F003C000E003C0006003C0002003C0002003C00 0200780002007800020078000200780202007802000078060000F0040000F0040000F01C 0000FFFC0000F01C0000F00C0001E0080001E0080001E0080001E0080201E0000401E000 0403C0000403C0000803C0000803C0001803C0001003C00030078000F007C003E0FFFFFF E020227EA121>I<03FFFFFF003E001F003C000E003C0006003C0002003C0002003C0002 00780002007800020078000200780202007802000078060000F0040000F0040000F01C00 00FFFC0000F01C0000F00C0001E0080001E0080001E0080001E0080001E0000001E00000 03C0000003C0000003C0000003C0000003C0000003C0000007C0000007C00000FFFE0000 20227EA120>I<03FFF0003E00003C00003C00003C00003C00003C000078000078000078 0000780000780000780000F00000F00000F00000F00000F00000F00001E00001E00001E0 0001E00001E00001E00003C00003C00003C00003C00003C00003C00007800007C000FFFC 0014227EA112>73 D<03FFE00FF8003E0007E0003C000380003C000200003C000400003C 000800003C00100000780020000078008000007801000000780200000078040000007808 000000F018000000F038000000F07C000000F13C000000F23E000000F41E000001E81F00 0001F00F000001E00F000001E007800001E007800001E003C00003C003C00003C001E000 03C001E00003C000F00003C000F00003C000F800078000F80007C000FC00FFFC07FF8025 227EA126>75 D<03FFF000003E0000003C0000003C0000003C0000003C0000003C000000 780000007800000078000000780000007800000078000000F0000000F0000000F0000000 F0000000F0000000F0000001E0000001E0000001E0000001E0002001E0002001E0004003 C0004003C0004003C000C003C0008003C0018003C003800780078007C01F00FFFFFF001B 227EA11E>I<03FE00000FF8003E00001F80002F00001F00002F00002F00002F00002F00 002F00004F00002F00004F00004780009E00004780009E00004780011E00004780011E00 004780021E000043C0041E000083C0043C000083C0083C000083C0083C000083C0103C00 0081E0103C000081E0203C000101E02078000101E04078000101E04078000100F0807800 0100F08078000100F10078000200F100F0000200F200F00002007C00F00002007C00F000 02007800F00006007800F00006007001E0001F003001F000FFE0203FFF002D227EA12D> I<03FE001FF8003E0003C0003F000180002F000100002F80010000278001000027C00100 0043C002000043C002000041E002000041E002000040F002000040F00200008078040000 8078040000803C040000803C040000801E040000801E040001000F080001000F08000100 0F88000100078800010007C800010003C800020003D000020001F000020001F000020000 F000020000F000060000700006000060001F00002000FFE000200025227EA125>I<0001 FC00000E0700003801C0007000E001E000F003C00078078000780700003C0F00003C1E00 003C1E00003E3C00003E3C00003E7C00003E7C00003EF800003EF800003EF800003EF800 003EF800003EF800007CF000007CF000007CF00000F8F00000F8F80000F0F80001F07800 01E0780003C0780007803C0007001C000E000E001C000700780003C1E000007F00001F24 7AA226>I<03FFFFC0003E00F0003C0078003C003C003C003E003C001E003C003E007800 3E0078003E0078003E0078003E0078003C0078007C00F0007800F000F000F001E000F007 8000FFFE0000F0000001E0000001E0000001E0000001E0000001E0000001E0000003C000 0003C0000003C0000003C0000003C0000003C000000780000007C00000FFFC00001F227E A121>I<0001FC00000E0700003803C0007000E001E000F003C00078078000780780003C 0F00003C1E00003C1E00003E3E00003E3C00003E7C00003E7C00003EF800003EF800003E F800003EF800003EF800003EF000007CF000007CF000007CF0000078F00000F8F00000F0 F80001F0780001E0780E03C0781087803C2087001C204E000E405C000720780003E1E010 007F4010000060100000602000006060000070E000007FC000007FC000007F8000003F00 00001E001F2D7AA226>I<000FC0800030318000C00B0001800700038007000300030007 000300070003000E0002000E0002000F0002000F0000000F0000000F80000007E0000007 FE000003FFC00001FFE000007FF000000FF8000000F8000000780000003C0000003C0000 003C0020003C00400038004000380040003800600030006000700060006000F000C000E8 018000C607000081FC000019247DA21B>83 D<1FFFFFFE3C03E03E3803C00C3003C00420 03C0046003C0044003C00440078004C00780048007800480078004000780000007800000 0F0000000F0000000F0000000F0000000F0000000F0000001E0000001E0000001E000000 1E0000001E0000001E0000003C0000003C0000003C0000003C0000003C0000003C000000 7C000000FC00003FFFF0001F227AA123>I86 D<020104020804100820102010402040 2080408040B85CFC7EFC7E783C3018100F76A218>92 D<03FC000606000F03000F038006 01800001C0000380000380007F8003E3800F03801C0380380700780700F00708F00708F0 0F08F00F08F017107867A01F83C015157D9418>97 D<0780003F80000700000700000700 000700000700000700000E00000E00000E00000E00000E00000E00001C3F001CC1801D00 C01E00601C00701C00703800783800783800783800783800783800787000F07000F07000 E07001E07001C0700380E80700C61C0081F00015237BA21B>I<00FF000381C00603C00C 03C01C0180380000780000700000F00000F00000F00000F00000F00000E00000F00000F0 00807001007001003806001C180007E00012157C9416>I<00001E0000FE00001C00001C 00001C00001C00001C00001C00003800003800003800003800003800003800FC70038370 0700F00C00F01C00703800707800E07000E0F000E0F000E0F000E0F000E0E001C0E001C0 E001C0E001C07003C07003C0380F801C33C007C3F817237CA21B>I<00FE000383800701 C00C00E01C00E03800E07800E07000E0FFFFE0F00000F00000F00000F00000E00000E000 00F000407000803000801803000E0C0003F00013157D9416>I<0003E0000E30001C7000 38F000307000700000700000700000E00000E00000E00000E00000E00000E0003FFE0001 C00001C00001C00001C00001C00003800003800003800003800003800003800007000007 00000700000700000700000700000E00000F0000FFF00014237FA20F>I<00000780001F 88800070D18000E0E18001C0700003C0700003C070000780F0000780F0000780F0000780 E0000381E0000181C00002C30000027E00000400000004000000040000000600000007FF 800007FFE00007FFF0001C007800300018006000180060001800C0001800C0001800C000 3000600060003000C0001C07800003FC00001921809518>I<00780003F8000070000070 0000700000700000700000700000E00000E00000E00000E00000E00000E00001C3F001CC 1801D00C01E00E01E00E01C00E03C01C03801C03801C03801C03801C03801C0700380700 380700380700380700380700380E00700F0078FFE7FF18237FA21B>I<007000F001F000 F000E00000000000000000000000000000000001C00FC001C001C001C001C00380038003 800380038003800700070007000700070007000E000F00FFE00C227FA10E>I<00030000 0780000F80000F8000070000000000000000000000000000000000000000000000000000 0E0000FE00001E00000E00000E00000E00001C00001C00001C00001C00001C00001C0000 380000380000380000380000380000380000700000700000700000700000700000700000 E00000E00060E000F1C000F18000E300003E0000112C83A10F>I<0078000003F8000000 700000007000000070000000700000007000000070000000E0000000E0000000E0000000 E0000000E0000000E0000001C07FC001C03E0001C0380001C0300001C0400001C0800003 83000003840000038E000003BE000003CF00000387000007078000070380000701C00007 01C0000700E0000700F0000E00F0000F00F800FFE3FE001A237FA21A>I<007803F80070 0070007000700070007000E000E000E000E000E000E001C001C001C001C001C001C00380 038003800380038003800700070007000700070007000E000F00FFE00D237FA20E>I<01 C1F807E01FC60C183001D80E603801E007801C01E007801C01C007001C03C00F00380380 0E003803800E003803800E003803800E003803800E003807001C007007001C007007001C 007007001C007007001C007007001C00700E003800E00F003C00F0FFE3FF8FFE27157F94 2A>I<01C3F01FCC1801D00C01E00E01E00E01C00E03C01C03801C03801C03801C03801C 03801C0700380700380700380700380700380700380E00700F0078FFE7FF18157F941B> I<007E000383800600C00C00E01C0070380070780078700078F00078F00078F00078F000 78E000F0E000F0E000E0F001E07001C07003803807001C1C0007F00015157D9418>I<00 E1F8000FE60C0000E8060000F0070000E0038000E0038001C003C001C003C001C003C001 C003C001C003C001C003C003800780038007800380070003800F0003801E0003801C0007 4038000730E000070F80000700000007000000070000000E0000000E0000000E0000000E 0000000E0000001E000000FFC000001A1F80941B>I<00FC100382100701300E00F01C00 F03800F07800E07800E0F000E0F000E0F000E0F000E0F001C0F001C0F001C0F001C07003 C07005C0380B801C338007C3800003800003800003800007000007000007000007000007 00000F00007FE0141F7C941A>I<01C7C01FC8E001D1E001E1E001E0C001C00003C00003 80000380000380000380000380000700000700000700000700000700000700000E00000F 0000FFF00013157F9413>I<01F906070C0318031801180118021C001FE00FF807FC007E 000E4006400640066006600CE008D83087C010157E9413>I<0080008000800180010003 00030007000F001F00FFF80E000E000E000E000E001C001C001C001C001C001C00381038 10381038103810382038201C4007800D1F7C9E13>I<0E0070FE07F00E00F00E00700E00 700E00701C00E01C00E01C00E01C00E01C00E01C00E03801C03801C03801C03801C03803 C03805C0380B801C13C007E3F815157C941B>III<1FF83FC003E01E0001C0180000E0 100000E02000007040000078800000390000001E0000001C0000000E0000001F00000037 00000063800000C380000181C0000101E0000200E0000E00F0003E00F800FF03FF001A15 7F941A>I<0FFC0FE001E0078000E0030000E0020000E0040000E0040000700800007008 00007010000070300000782000003840000038400000388000003C8000001D0000001D00 00001E0000001C0000000C00000008000000080000001000000010000000200000004000 00F0400000F0800000F1000000C20000003C0000001B1F80941A>I<07FFF80780380600 700C00E00801C0080380080700100E00001C0000380000700000E00001C02003C0200380 200700600E00401C00C03801C0700380FFFF8015157F9416>II E /Fw 33 118 df<00180030006000C001C00380070007000E001E001C003C003C003C00 7800780078007800F800F000F000F000F000F000F000F000F000F000F000F000F8007800 7800780078003C003C003C001C001E000E0007000700038001C000C00060003000180D31 7BA416>40 DI<387CFEFEFE7C3807077C8610>46 D<00180000780001F800FFF800FFF80001F800 01F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F800 01F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F800 7FFFE07FFFE013207C9F1C>49 D<03FC000FFF003C1FC07007E07C07F0FE03F0FE03F8FE 03F8FE01F87C01F83803F80003F80003F00003F00007E00007C0000F80001F00003E0000 380000700000E01801C0180380180700180E00380FFFF01FFFF03FFFF07FFFF0FFFFF0FF FFF015207D9F1C>I<00FE0007FFC00F07E01E03F03F03F03F81F83F81F83F81F81F03F8 1F03F00003F00003E00007C0001F8001FE0001FF000007C00001F00001F80000FC0000FC 3C00FE7E00FEFF00FEFF00FEFF00FEFF00FC7E01FC7801F81E07F00FFFC001FE0017207E 9F1C>I<0000E00001E00003E00003E00007E0000FE0001FE0001FE00037E00077E000E7 E001C7E00187E00307E00707E00E07E00C07E01807E03807E07007E0E007E0FFFFFEFFFF FE0007E00007E00007E00007E00007E00007E00007E000FFFE00FFFE17207E9F1C>I<10 00201E01E01FFFC01FFF801FFF001FFE001FF8001BC00018000018000018000018000019 FC001FFF001E0FC01807E01803E00003F00003F00003F80003F83803F87C03F8FE03F8FE 03F8FC03F0FC03F07007E03007C01C1F800FFF0003F80015207D9F1C>I<001F8000FFE0 03F07007C0F00F01F81F01F83E01F83E01F87E00F07C00007C0000FC0800FC7FC0FCFFE0 FD80F0FF00F8FE007CFE007CFC007EFC007EFC007EFC007E7C007E7C007E7C007E3C007C 3E007C1E00F80F00F00783E003FFC000FF0017207E9F1C>I<6000007800007FFFFE7FFF FE7FFFFC7FFFF87FFFF87FFFF0E00060E000C0C00180C00300C00300000600000C00001C 0000180000380000780000780000F00000F00000F00001F00001F00001F00003F00003F0 0003F00003F00003F00003F00003F00001E00017227DA11C>I<00FE0003FFC00601E00C 00701800701800383800383C00383F00383F80783FE0701FF8E01FFFC00FFF8007FFC003 FFE007FFF01E7FF83C1FFC7807FC7801FEF000FEF0003EF0001EF0001EF0001CF8001C78 00383C00381F01F00FFFC001FF0017207E9F1C>I<000070000000007000000000F80000 0000F800000000F800000001FC00000001FC00000003FE00000003FE00000003FE000000 06FF000000067F0000000E7F8000000C3F8000000C3F800000183FC00000181FC0000038 1FE00000300FE00000300FE00000600FF000006007F00000E007F80000FFFFF80000FFFF F800018001FC00018001FC00038001FE00030000FE00030000FE000600007F000600007F 00FFE00FFFF8FFE00FFFF825227EA12A>65 D76 D<0007FC0000003FFF800000FC07E00003F0 01F80007E000FC000FC0007E001F80003F001F80003F003F00001F803F00001F807F0000 1FC07E00000FC07E00000FC0FE00000FE0FE00000FE0FE00000FE0FE00000FE0FE00000F E0FE00000FE0FE00000FE0FE00000FE0FE00000FE07E00000FC07F00001FC07F00001FC0 3F00001F803F80003F801F80003F000FC0007E0007E000FC0003F001F80000FC07E00000 3FFF80000007FC000023227DA12A>79 DI82 D<7FFFFFFF807FFFFFFF807E03F80F807803F807807003 F803806003F80180E003F801C0E003F801C0C003F800C0C003F800C0C003F800C0C003F8 00C00003F800000003F800000003F800000003F800000003F800000003F800000003F800 000003F800000003F800000003F800000003F800000003F800000003F800000003F80000 0003F800000003F800000003F800000003F800000003F800000003F8000003FFFFF80003 FFFFF80022227EA127>84 D<07FC001FFF803F07C03F03E03F01E03F01F01E01F00001F0 0001F0003FF003FDF01FC1F03F01F07E01F0FC01F0FC01F0FC01F0FC01F07E02F07E0CF8 1FF87F07E03F18167E951B>97 D<00FF8007FFE00F83F01F03F03E03F07E03F07C01E07C 0000FC0000FC0000FC0000FC0000FC0000FC00007C00007E00007E00003E00301F00600F C0E007FF8000FE0014167E9519>99 D<00FE0007FF800F87C01E01E03E01F07C00F07C00 F8FC00F8FC00F8FFFFF8FFFFF8FC0000FC0000FC00007C00007C00007E00003E00181F00 300FC07003FFC000FF0015167E951A>101 D<003F8000FFC001E3E003C7E007C7E00F87 E00F83C00F80000F80000F80000F80000F80000F8000FFFC00FFFC000F80000F80000F80 000F80000F80000F80000F80000F80000F80000F80000F80000F80000F80000F80000F80 000F80000F80000F80007FF8007FF80013237FA211>I104 D<1C003F007F007F007F003F001C000000000000000000000000000000FF00FF001F001F 001F001F001F001F001F001F001F001F001F001F001F001F001F001F001F001F00FFE0FF E00B247EA310>I107 DIII< 00FE0007FFC00F83E01E00F03E00F87C007C7C007C7C007CFC007EFC007EFC007EFC007E FC007EFC007EFC007E7C007C7C007C3E00F81F01F00F83E007FFC000FE0017167E951C> II114 D<0FF3003FFF00781F00600700E00300E00300F00300FC00007FE0007FF8003FFE000FFF 0001FF00000F80C00780C00380E00380E00380F00700FC0E00EFFC00C7F00011167E9516 >I<0180000180000180000180000380000380000780000780000F80003F8000FFFF00FF FF000F80000F80000F80000F80000F80000F80000F80000F80000F80000F80000F80000F 81800F81800F81800F81800F81800F830007C30003FE0000F80011207F9F16>II E /Fx 94 127 df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y 6 107 df0 D<40E04003037D880A>I<040004000400C460E4E03F800E003F80E4E0C4600400040004 000B0D7E8D11>3 D<03FC0FFC1C00380070006000E000C000C000FFFCFFFCC000C000E0 006000700038001C000FFC03FC0E147D9016>50 D<03FFC00FFFF030E1FC60C07CC0C01E 81C01E01C00E01C00E01C00E01800C01800C03801C030018030010070020060040060180 0C0E000FF8001FC00017147F931A>68 D106 D E /Fz 56 123 df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df<78FCFCFEFE7A0202040408083040070E7D850D>44 D<78FCFCFCFC7806067D850D>46 D<03F8000F1E001C07003C07803803807803C07803C0 7803C0F803E0F803E0F803E0F803E0F803E0F803E0F803E0F803E0F803E0F803E0F803E0 F803E07803C07803C03803803C07801C07000F1E0003F800131B7E9A18>48 D<00600001E0000FE000FFE000F3E00003E00003E00003E00003E00003E00003E00003E0 0003E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003E0 0003E0007FFF807FFF80111B7D9A18>I<07F8001FFE00383F80780FC0FC07C0FC07E0FC 03E0FC03E07803E00007E00007C00007C0000F80001F00001E0000380000700000E00001 80600300600600600800E01FFFC03FFFC07FFFC0FFFFC0FFFFC0131B7E9A18>I<03F800 1FFE003C1F003C0F807C07C07E07C07C07C03807C0000F80000F80001E00003C0003F800 001E00000F800007C00007C00007E03007E07807E0FC07E0FC07E0FC07C0780F80781F00 1FFE0007F800131B7E9A18>I<000180000380000780000F80001F80003F80006F8000CF 80008F80018F80030F80060F800C0F80180F80300F80600F80C00F80FFFFF8FFFFF8000F 80000F80000F80000F80000F80000F8001FFF801FFF8151B7F9A18>I<1801801FFF001F FE001FFC001FF8001FC00018000018000018000018000019F8001E0E00180F8010078000 07C00007E00007E00007E07807E0F807E0F807E0F807C0F007C0600F80381F001FFE0007 F000131B7E9A18>I<007E0003FF000781800F03C01E07C03C07C03C0380780000780000 F80000F8F800FB0E00FA0780FC0380FC03C0F803E0F803E0F803E0F803E07803E07803E0 7803C03C03C03C07801E0F0007FE0003F800131B7E9A18>I<03F8000FFF001C07803003 803001C07001C07801C07E01C07F83807FC3003FF6001FFC000FFE0007FF801DFF80387F C0781FE0F007E0F003E0F001E0F001E0F001E07801C07803803E07801FFE0003F800131B 7E9A18>56 D<00038000000380000007C0000007C0000007C000000FE000000FE000001F F000001BF000001BF0000031F8000031F8000061FC000060FC0000E0FE0000C07E0000C0 7E0001803F0001FFFF0003FFFF8003001F8003001F8006000FC006000FC00E000FE00C00 07E0FFC07FFEFFC07FFE1F1C7E9B24>65 D73 D75 D77 D 80 D82 D<07F8201FFEE03C07E07801E07000E0F000E0F00060F00060F80000FE00 00FFE0007FFE003FFF003FFF800FFFC007FFE0007FE00003F00001F00000F0C000F0C000 F0C000E0E000E0F001C0FC03C0EFFF0083FC00141C7D9B1B>I<7FFFFFE07FFFFFE0781F 81E0701F80E0601F8060E01F8070C01F8030C01F8030C01F8030C01F8030001F8000001F 8000001F8000001F8000001F8000001F8000001F8000001F8000001F8000001F8000001F 8000001F8000001F8000001F8000001F8000001F800007FFFE0007FFFE001C1C7E9B21> I<0FF8001C1E003E0F803E07803E07C01C07C00007C0007FC007E7C01F07C03C07C07C07 C0F807C0F807C0F807C0780BC03E13F80FE1F815127F9117>97 DI<03FC000E0E001C1F003C1F00781F00780E00F80000F800 00F80000F80000F80000F800007800007801803C01801C03000E0E0003F80011127E9115 >I<000FF0000FF00001F00001F00001F00001F00001F00001F00001F00001F00001F001 F9F00F07F01C03F03C01F07801F07801F0F801F0F801F0F801F0F801F0F801F0F801F078 01F07801F03C01F01C03F00F0FFE03F9FE171D7E9C1B>I<01FC000F07001C03803C01C0 7801C07801E0F801E0F801E0FFFFE0F80000F80000F800007800007C00603C00601E00C0 0F038001FC0013127F9116>I<007F0001E38003C7C00787C00F87C00F83800F80000F80 000F80000F80000F8000FFF800FFF8000F80000F80000F80000F80000F80000F80000F80 000F80000F80000F80000F80000F80000F80000F80007FF8007FF800121D809C0F>I104 D<1E003F003F003F003F001E0000 0000000000000000000000FF00FF001F001F001F001F001F001F001F001F001F001F001F 001F001F001F00FFE0FFE00B1E7F9D0E>I108 DII<01FC000F07801C01C0 3C01E07800F07800F0F800F8F800F8F800F8F800F8F800F8F800F87800F07800F03C01E0 1E03C00F078001FC0015127F9118>II114 D<1FD830786018E018E018F000FF807FE0 7FF01FF807FC007CC01CC01CE01CE018F830CFC00E127E9113>I<030003000300030007 0007000F000F003FFCFFFC1F001F001F001F001F001F001F001F001F001F0C1F0C1F0C1F 0C0F08079803F00E1A7F9913>II121 D123 D E /FB 35 124 df<1F003F807FC0FFE0FFE0FFF0FFF0FFF07FF03FF01F700070007000 F000E000E001E001C003C007800F001F003E003C0018000C197A8A17>44 D<1F003F807FC0FFE0FFE0FFE0FFE0FFE07FC03F801F000B0B7A8A17>46 D<0001E0000003E000000FE000007FE0001FFFE000FFFFE000FFBFE000E03FE000003FE0 00003FE000003FE000003FE000003FE000003FE000003FE000003FE000003FE000003FE0 00003FE000003FE000003FE000003FE000003FE000003FE000003FE000003FE000003FE0 00003FE000003FE000003FE000003FE000003FE000003FE000003FE000003FE000003FE0 00003FE000003FE000003FE000003FE000003FE000003FE000003FE0007FFFFFF07FFFFF F07FFFFFF01C2E7AAD29>49 D<003FF00001FFFE0007FFFF801FC07FE03F003FF07C001F F87F000FFCFF8007FEFFC007FEFFC003FFFFC003FFFFC003FFFFC003FF7F8003FF3F0003 FF000003FF000003FF000007FE000007FE000007FC00000FF800001FF800001FF000003F E000007F8000007F000000FE000001FC000003F0000007E000000FC007001F8007003E00 07007C000F00F8000E01F0000E01E0001E03FFFFFE07FFFFFE0FFFFFFE1FFFFFFE3FFFFF FE7FFFFFFCFFFFFFFCFFFFFFFCFFFFFFFC202E7CAD29>I<000FFC0000007FFF800001F0 1FE00003C00FF00007800FF8000FE007FC001FF007FE001FF807FE001FF807FE001FF807 FE001FF807FE001FF807FE000FF007FE0007E007FC0001800FFC0000000FF80000000FF8 0000001FF00000003FE00000007F8000001FFE0000001FFC0000001FFF800000001FF000 00000FF800000007FE00000003FF00000003FF00000001FF80000001FF800E0001FFC03F 8001FFC07FC001FFC0FFE001FFC0FFE001FFC0FFE001FFC0FFE001FFC0FFE001FF80FFE0 03FF807FC003FF007F0007FE003F000FFC001FE01FF80007FFFFE00001FFFF8000001FFC 0000222E7DAD29>I<0000007800000000F800000001F800000003F800000007F8000000 07F80000000FF80000001FF80000003FF80000007FF800000077F8000000F7F8000001E7 F8000003C7F800000787F800000707F800000F07F800001E07F800003C07F800007807F8 00007007F80000F007F80001E007F80003C007F800078007F8000F0007F8000F0007F800 1E0007F8003C0007F800780007F800F00007F800FFFFFFFFF0FFFFFFFFF0FFFFFFFFF000 000FF80000000FF80000000FF80000000FF80000000FF80000000FF80000000FF8000000 0FF80000000FF800000FFFFFF0000FFFFFF0000FFFFFF0242E7EAD29>I<0C0000380FC0 03F80FFFFFF80FFFFFF00FFFFFE00FFFFFC00FFFFF800FFFFF000FFFFC000FFFF0000FFF 00000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0FF8000F7F FF000FFFFFC00FF01FE00F800FF00F0007F80E0007FC000003FE000003FE000003FE0000 03FF000003FF1E0003FF7F8003FFFF8003FFFFC003FFFFC003FFFFC003FFFF8003FEFF80 03FE7F0007FC7C0007FC3C000FF81E001FF00FC07FE007FFFF8001FFFE00003FE000202E 7CAD29>I<0000007C0000000000007C000000000000FE000000000000FE000000000000 FE000000000001FF000000000001FF000000000003FF800000000003FF800000000007FF C00000000007FFC00000000007FFC0000000000FFFE0000000000F7FE0000000001F7FF0 000000001E3FF0000000001E3FF0000000003E3FF8000000003C1FF8000000007C1FFC00 000000780FFC00000000780FFC00000000F80FFE00000000F007FE00000001F007FF0000 0001E003FF00000001E003FF00000003E003FF80000003C001FF80000007C001FFC00000 078000FFC00000078000FFC000000FFFFFFFE000000FFFFFFFE000001FFFFFFFF000001E 00003FF000001E00003FF000003C00003FF800003C00001FF800007C00001FFC00007800 000FFC00007800000FFC0000F0000007FE0000F0000007FE0001F0000007FF0003F80000 03FF00FFFFC001FFFFFEFFFFC001FFFFFEFFFFC001FFFFFE37317DB03E>65 D<000003FF80038000003FFFF007800001FFFFFC0F800007FF007F1F80001FF8000FFF80 007FE00003FF8000FFC00001FF8001FF000000FF8003FE0000007F8007FE0000003F800F FC0000001F801FF80000001F801FF80000000F803FF80000000F803FF00000000F803FF0 00000007807FF000000007807FF000000007807FE00000000000FFE00000000000FFE000 00000000FFE00000000000FFE00000000000FFE00000000000FFE00000000000FFE00000 000000FFE00000000000FFE00000000000FFE00000000000FFE000000000007FE0000000 00007FF000000000007FF000000003803FF000000003803FF000000003803FF800000003 801FF800000007801FF800000007000FFC0000000F0007FE0000000E0003FF0000001E00 01FF0000003C0000FFC000007800007FE00001F000001FF80003E0000007FF003F800000 01FFFFFE000000003FFFF80000000003FF80000031317BB03C>67 D73 D75 D77 D80 D82 D<001FF0038000FFFF078003FFFFCF8007F0 0FFF801FC001FF801F80007F803F00003F807F00001F807E00000F807E00000F80FE0000 0780FE00000780FF00000380FF00000380FF80000380FFC0000000FFE0000000FFFC0000 007FFFE000007FFFFE00003FFFFFC0003FFFFFF0001FFFFFFC000FFFFFFE0007FFFFFF00 01FFFFFF00007FFFFF80001FFFFFC00000FFFFC0000007FFC0000000FFE00000007FE000 00003FE00000001FE0E000001FE0E000000FE0E000000FE0E000000FE0F000000FE0F000 000FC0F800000FC0F800001F80FC00001F80FF00003F00FFC0007E00FFFC01FC00F9FFFF F800F03FFFE000E007FF000023317BB02E>I<3FFFFFFFFFFF003FFFFFFFFFFF003FFFFF FFFFFF003FE00FFC01FF007F000FFC003F807E000FFC001F807C000FFC000F8078000FFC 00078078000FFC00078070000FFC00038070000FFC00038070000FFC00038070000FFC00 0380E0000FFC0001C0E0000FFC0001C0E0000FFC0001C0E0000FFC0001C000000FFC0000 0000000FFC00000000000FFC00000000000FFC00000000000FFC00000000000FFC000000 00000FFC00000000000FFC00000000000FFC00000000000FFC00000000000FFC00000000 000FFC00000000000FFC00000000000FFC00000000000FFC00000000000FFC0000000000 0FFC00000000000FFC00000000000FFC00000000000FFC00000000000FFC00000000000F FC00000000000FFC00000000000FFC00000000000FFC00000000000FFC00000000000FFC 00000000000FFC000000007FFFFFFF8000007FFFFFFF8000007FFFFFFF800032307DAF39 >I<007FF8000003FFFF00000FFFFFC0001FE01FF0001FF007F8001FF007FC001FF003FC 001FF001FE000FE001FE0007C001FE00010001FE00000001FE00000001FE000001FFFE00 003FFFFE0001FFF1FE0007FE01FE001FF001FE003FE001FE007FC001FE007F8001FE00FF 8001FE00FF0001FE00FF0001FE00FF0001FE00FF8003FE00FF8007FE007FC00FFE003FF0 3EFF001FFFF87FF807FFF03FF800FF800FF825207E9F28>97 D<01F8000000FFF8000000 FFF8000000FFF80000000FF800000007F800000007F800000007F800000007F800000007 F800000007F800000007F800000007F800000007F800000007F800000007F800000007F8 00000007F800000007F80FF00007F87FFE0007F9FFFF8007FFE03FE007FF000FF007FE00 07F807FC0007F807F80003FC07F80003FE07F80003FE07F80001FE07F80001FF07F80001 FF07F80001FF07F80001FF07F80001FF07F80001FF07F80001FF07F80001FF07F80001FF 07F80001FF07F80001FE07F80003FE07F80003FC07FC0003FC07FC0007F807FE000FF007 FF801FE007E7E07FC007C1FFFF8007807FFE0007001FE00028327EB12E>I<0007FF0000 7FFFE001FFFFF803FC03FC07F807FC0FF007FC1FE007FC3FC007FC3FC003F87FC001F07F 800040FF800000FF800000FF800000FF800000FF800000FF800000FF800000FF800000FF 800000FF8000007FC000007FC000003FC0000E3FE0000E1FE0001E0FF0003C07F8007803 FF01F001FFFFE0007FFF800007FC001F207D9F25>I<00000007E0000003FFE0000003FF E0000003FFE00000003FE00000001FE00000001FE00000001FE00000001FE00000001FE0 0000001FE00000001FE00000001FE00000001FE00000001FE00000001FE00000001FE000 00001FE0000FF81FE0007FFF1FE001FFFFDFE003FE03FFE00FF800FFE01FF0007FE01FE0 003FE03FC0001FE03FC0001FE07FC0001FE07F80001FE0FF80001FE0FF80001FE0FF8000 1FE0FF80001FE0FF80001FE0FF80001FE0FF80001FE0FF80001FE0FF80001FE0FF80001F E07F80001FE07FC0001FE07FC0001FE03FC0003FE01FE0007FE01FE000FFE00FF001FFE0 07FC07DFF001FFFF9FFF007FFE1FFF000FF01FFF28327DB12E>I<0007FC0000003FFF80 0000FFFFE00003FC07F00007F801F8000FE000FC001FE0007E003FC0007E003FC0003F00 7FC0003F007F80003F007F80003F80FF80003F80FF80003F80FFFFFFFF80FFFFFFFF80FF FFFFFF80FF80000000FF80000000FF800000007F800000007F800000003FC00000003FC0 0003801FC00003801FE00007800FF0000F0007F8001E0003FE00FC0000FFFFF800003FFF E0000003FF000021207E9F26>I<0000FF000007FFC0001FFFF0007FC7F000FF0FF801FF 0FF801FE0FF803FE0FF803FC0FF803FC07F003FC01C003FC000003FC000003FC000003FC 000003FC000003FC000003FC0000FFFFF800FFFFF800FFFFF80003FC000003FC000003FC 000003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC 000003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC 000003FC000003FC000003FC000003FC000003FC00007FFFF0007FFFF0007FFFF0001D32 7EB119>I<01F800000000FFF800000000FFF800000000FFF8000000000FF80000000007 F80000000007F80000000007F80000000007F80000000007F80000000007F80000000007 F80000000007F80000000007F80000000007F80000000007F80000000007F80000000007 F80000000007F807F8000007F83FFF000007F87FFF800007F9F03FC00007FBC01FE00007 FF801FE00007FF000FF00007FE000FF00007FC000FF00007FC000FF00007F8000FF00007 F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007 F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007 F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF000FF FFC1FFFF80FFFFC1FFFF80FFFFC1FFFF8029327DB12E>104 D<03C0000FF0000FF0001F F8001FF8001FFC001FF8001FF8000FF0000FF00003C00000000000000000000000000000 000000000000000000000001F800FFF800FFF800FFF8000FF80007F80007F80007F80007 F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007 F80007F80007F80007F80007F80007F80007F80007F80007F800FFFF80FFFF80FFFF8011 337DB217>I<01F800FFF800FFF800FFF8000FF80007F80007F80007F80007F80007F800 07F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F800 07F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F800 07F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F800 07F800FFFFC0FFFFC0FFFFC012327DB117>108 D<03F007F8000FF000FFF03FFF007FFE 00FFF07FFF80FFFF00FFF1F03FC3E07F800FF3C01FE7803FC007F7801FEF003FC007F700 0FFE001FE007FE000FFC001FE007FC000FF8001FE007FC000FF8001FE007F8000FF0001F E007F8000FF0001FE007F8000FF0001FE007F8000FF0001FE007F8000FF0001FE007F800 0FF0001FE007F8000FF0001FE007F8000FF0001FE007F8000FF0001FE007F8000FF0001F E007F8000FF0001FE007F8000FF0001FE007F8000FF0001FE007F8000FF0001FE007F800 0FF0001FE007F8000FF0001FE007F8000FF0001FE007F8000FF0001FE007F8000FF0001F E0FFFFC1FFFF83FFFFFFFFC1FFFF83FFFFFFFFC1FFFF83FFFF40207D9F47>I<03F007F8 0000FFF03FFF0000FFF07FFF8000FFF1F03FC0000FF3C01FE00007F7801FE00007F7000F F00007FE000FF00007FC000FF00007FC000FF00007F8000FF00007F8000FF00007F8000F F00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000F F00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000F F00007F8000FF00007F8000FF00007F8000FF00007F8000FF000FFFFC1FFFF80FFFFC1FF FF80FFFFC1FFFF8029207D9F2E>I<0007FE0000003FFFC00000FFFFF00003FC03FC0007 F000FE000FE0007F001FC0003F803FC0003FC03FC0003FC07F80001FE07F80001FE07F80 001FE0FF80001FF0FF80001FF0FF80001FF0FF80001FF0FF80001FF0FF80001FF0FF8000 1FF0FF80001FF07F80001FE07F80001FE07F80001FE03FC0003FC03FC0003FC01FE0007F 800FE0007F0007F801FE0003FE07FC0001FFFFF800003FFFC0000007FE000024207E9F29 >I<01F80FF000FFF87FFE00FFF9FFFF80FFFFE07FE00FFF001FF007FE000FF807FC0007 F807F80007FC07F80003FE07F80003FE07F80003FE07F80001FF07F80001FF07F80001FF 07F80001FF07F80001FF07F80001FF07F80001FF07F80001FF07F80001FF07F80003FF07 F80003FE07F80003FE07F80003FC07FC0007FC07FC000FF807FE000FF007FF801FE007FF E07FC007F9FFFF8007F87FFE0007F81FE00007F800000007F800000007F800000007F800 000007F800000007F800000007F800000007F800000007F800000007F800000007F80000 00FFFFC00000FFFFC00000FFFFC00000282E7E9F2E>I<03F03F00FFF0FFC0FFF1FFF0FF F3C7F00FF78FF807F70FF807FE0FF807FE0FF807FC07F007FC03E007FC008007FC000007 F8000007F8000007F8000007F8000007F8000007F8000007F8000007F8000007F8000007 F8000007F8000007F8000007F8000007F8000007F8000007F8000007F80000FFFFE000FF FFE000FFFFE0001D207E9F22>114 D<00FF870007FFEF001FFFFF003F007F007C001F00 7C000F00F8000F00F8000700FC000700FC000700FF000000FFF800007FFFC0007FFFF000 3FFFFC001FFFFE0007FFFF0001FFFF80001FFF800000FFC000001FC0E0000FC0E0000FC0 F00007C0F00007C0F80007C0FC000F80FE001F80FF803F00FFFFFE00F3FFF800E07FC000 1A207D9F21>I<00380000380000380000380000380000780000780000780000F80000F8 0001F80003F80007F8001FF800FFFFFEFFFFFEFFFFFE07F80007F80007F80007F80007F8 0007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F8 0707F80707F80707F80707F80707F80707F80707FC0F03FC0E03FE1E01FFFC007FF8000F E0182E7EAD20>I<01F80003F000FFF801FFF000FFF801FFF000FFF801FFF0000FF8001F F00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000F F00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000F F00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000FF00007F8000F F00007F8001FF00007F8001FF00007F8003FF00007F8007FF00003FC00FFF00003FE03EF F80001FFFFCFFF80007FFF0FFF80000FFC0FFF8029207D9F2E>I121 D123 D E /FC 2 107 df<000000C0000003C000000F0000003C000000F0000003C000000F00 00001C00000078000001E00000078000001E00000078000000E0000000780000001E0000 000780000001E0000000780000001C0000000F00000003C0000000F00000003C0000000F 00000003C0000000C0000000000000000000000000000000000000000000000000000000 00FFFFFFC0FFFFFFC01A247C9C23>20 D106 D E /FD 1 35 df<007C0001FF000703800C01001800001000001000001000001000000F F0000F6000108000200000400000400000800000800000C002004004007018003FF0000F C00011167F9415>34 D E /FE 32 118 df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df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df<7038F87CFC7EFC7E743A0402040204020804080410081008201040200F0E7E9F17> 34 D<0020004000800100020006000C000C001800180030003000300070006000600060 00E000E000E000E000E000E000E000E000E000E000E000E0006000600060007000300030 003000180018000C000C000600020001000080004000200B2E7DA112>40 D<800040002000100008000C00060006000300030001800180018001C000C000C000C000 E000E000E000E000E000E000E000E000E000E000E000E000C000C000C001C00180018001 8003000300060006000C00080010002000400080000B2E7DA112>I<70F8FCFC74040404 080810102040060E7C840D>44 DI<70F8F8F87005057C840D>I< 00030003000700060006000E000C000C001C0018001800380030003000700060006000E0 00C000C001C00180018001800380030003000700060006000E000C000C001C0018001800 380030003000700060006000E000C000C000102D7DA117>I<03F0000E1C001C0E001806 00380700700380700380700380700380F003C0F003C0F003C0F003C0F003C0F003C0F003 C0F003C0F003C0F003C0F003C0F003C0F003C07003807003807003807807803807001806 001C0E000E1C0003F000121F7E9D17>I<018003800F80F3800380038003800380038003 800380038003800380038003800380038003800380038003800380038003800380038003 8007C0FFFE0F1E7C9D17>I<000100000003800000038000000380000007C0000007C000 0007C0000009E0000009E0000009E0000010F0000010F0000010F0000020780000207800 0020780000403C0000403C0000403C0000801E0000801E0000FFFE0001000F0001000F00 01000F00020007800200078002000780040003C00E0003C01F0007E0FFC03FFE1F207F9F 22>65 D<000FC040007030C001C009C0038005C0070003C00E0001C01E0000C01C0000C0 3C0000C07C0000407C00004078000040F8000000F8000000F8000000F8000000F8000000 F8000000F8000000F8000000F8000000780000007C0000407C0000403C0000401C000040 1E0000800E000080070001000380020001C0040000703800000FC0001A217D9F21>67 D<0FFFC0007C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C 00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00203C 00F83C00F83C00F83C00F0380040780040700030E0000F800012207E9E17>74 D<07E0800C1980100780300380600180600180E00180E00080E00080E00080F00000F000 007800007F00003FF0001FFC000FFE0003FF00001F800007800003C00003C00001C08001 C08001C08001C08001C0C00180C00380E00300F00600CE0C0081F80012217D9F19>83 D<7FFFFFE0780F01E0600F0060400F0020400F0020C00F0030800F0010800F0010800F00 10800F0010000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F00 00000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F00 00000F0000000F0000001F800007FFFE001C1F7E9E21>I87 D<080410082010201040204020804080408040B85CFC7EFC7E7C3E381C 0F0E7B9F17>92 D<1FE000303000781800781C00300E00000E00000E00000E0000FE0007 8E001E0E00380E00780E00F00E10F00E10F00E10F01E10781E103867200F83C014147E93 17>97 D<0E0000FE00000E00000E00000E00000E00000E00000E00000E00000E00000E00 000E00000E3E000EC3800F01C00F00E00E00E00E00700E00700E00780E00780E00780E00 780E00780E00780E00700E00700E00E00F00E00D01C00CC300083E0015207F9F19>I<03 F80E0C1C1E381E380C70007000F000F000F000F000F000F00070007000380138011C020E 0C03F010147E9314>I<000380003F800003800003800003800003800003800003800003 8000038000038000038003E380061B801C0780380380380380700380700380F00380F003 80F00380F00380F00380F003807003807003803803803807801C07800E1B8003E3F81520 7E9F19>I<03F0000E1C001C0E00380700380700700700700380F00380F00380FFFF80F0 0000F00000F000007000007000003800801800800C010007060001F80011147F9314>I< 007C00C6018F038F07060700070007000700070007000700FFF007000700070007000700 07000700070007000700070007000700070007000700070007007FF01020809F0E>I<00 00E003E3300E3C301C1C30380E00780F00780F00780F00780F00780F00380E001C1C001E 380033E0002000002000003000003000003FFE001FFF800FFFC03001E0600070C00030C0 0030C00030C000306000603000C01C038003FC00141F7F9417>I<0E0000FE00000E0000 0E00000E00000E00000E00000E00000E00000E00000E00000E00000E3E000E43000E8180 0F01C00F01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C0 0E01C00E01C00E01C00E01C0FFE7FC16207F9F19>I<1C001E003E001E001C0000000000 00000000000000000E007E000E000E000E000E000E000E000E000E000E000E000E000E00 0E000E000E000E000E00FFC00A1F809E0C>I<0E00FE000E000E000E000E000E000E000E 000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E 000E000E000E000E00FFE00B20809F0C>108 D<0E1F01F000FE618618000E81C81C000F 00F00E000F00F00E000E00E00E000E00E00E000E00E00E000E00E00E000E00E00E000E00 E00E000E00E00E000E00E00E000E00E00E000E00E00E000E00E00E000E00E00E000E00E0 0E000E00E00E00FFE7FE7FE023147F9326>I<0E3E00FE43000E81800F01C00F01C00E01 C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01 C00E01C0FFE7FC16147F9319>I<01F800070E001C03803801C03801C07000E07000E0F0 00F0F000F0F000F0F000F0F000F0F000F07000E07000E03801C03801C01C0380070E0001 F80014147F9317>I<0E3E00FEC3800F01C00F00E00E00E00E00F00E00700E00780E0078 0E00780E00780E00780E00780E00700E00F00E00E00F01E00F01C00EC3000E3E000E0000 0E00000E00000E00000E00000E00000E00000E0000FFE000151D7F9319>I<03E0800619 801C05803C0780380380780380700380F00380F00380F00380F00380F00380F003807003 807803803803803807801C0B800E138003E3800003800003800003800003800003800003 80000380000380003FF8151D7E9318>I<0E78FE8C0F1E0F1E0F0C0E000E000E000E000E 000E000E000E000E000E000E000E000E000E00FFE00F147F9312>I<1F9030704030C010 C010C010E00078007F803FE00FF00070803880188018C018C018E030D0608F800D147E93 12>I<020002000200060006000E000E003E00FFF80E000E000E000E000E000E000E000E 000E000E000E000E080E080E080E080E080610031001E00D1C7F9B12>I<0E01C0FE1FC0 0E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C0 0E01C00E01C00E03C00603C0030DC001F1FC16147F9319>III<7F C3FC0F01E00701C007018003810001C20000E40000EC00007800003800003C00007C0000 4E000087000107000303800201C00601E01E01E0FF07FE1714809318>II123 D E /FH 7 117 df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df<007E1F0001C1B1800303E3C00703 C3C00E03C1800E01C0000E01C0000E01C0000E01C0000E01C0000E01C000FFFFFC000E01 C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01 C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0007F87FC001A1D809C18> 11 D<007E0001C1800301800703C00E03C00E01800E00000E00000E00000E00000E0000 FFFFC00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C0 0E01C00E01C00E01C00E01C00E01C07F87F8151D809C17>I<003F07E00001C09C180003 80F018000701F03C000E01E03C000E00E018000E00E000000E00E000000E00E000000E00 E000000E00E00000FFFFFFFC000E00E01C000E00E01C000E00E01C000E00E01C000E00E0 1C000E00E01C000E00E01C000E00E01C000E00E01C000E00E01C000E00E01C000E00E01C 000E00E01C000E00E01C000E00E01C000E00E01C007FC7FCFF80211D809C23>14 D18 D<60F0F0F0F0F0F0F0606060606060606060606060 60000000000060F0F060041E7C9D0C>33 D<6060F0F0F8F8686808080808080810101010 2020404080800D0C7F9C15>I<0003018000030180000301800007038000060300000603 0000060300000E0700000C0600000C0600000C0600000C0600001C0E007FFFFFF8FFFFFF FC0030180000301800003018000070380000603000006030000060300000603000FFFFFF FC7FFFFFF801C0E0000180C0000180C0000180C0000381C0000301800003018000030180 00070380000603000006030000060300001E257E9C23>I<60F0F8680808081010204080 050C7C9C0C>39 D<004000800100020006000C000C001800180030003000700060006000 6000E000E000E000E000E000E000E000E000E000E000E000E00060006000600070003000 3000180018000C000C00060002000100008000400A2A7D9E10>I<800040002000100018 000C000C000600060003000300038001800180018001C001C001C001C001C001C001C001 C001C001C001C001C0018001800180038003000300060006000C000C0018001000200040 0080000A2A7E9E10>I<0006000000060000000600000006000000060000000600000006 0000000600000006000000060000000600000006000000060000FFFFFFE0FFFFFFE00006 000000060000000600000006000000060000000600000006000000060000000600000006 00000006000000060000000600001B1C7E9720>43 D<60F0F0701010101020204080040C 7C830C>II<60F0F06004047C830C>I<00030003000700060006 000E000C001C0018001800380030003000700060006000E000C000C001C0018003800300 03000700060006000E000C000C001C001800180038003000700060006000E000C000C000 10297E9E15>I<03C00C301818300C300C700E60066006E007E007E007E007E007E007E0 07E007E007E007E007E007E00760066006700E300C300C18180C3007E0101D7E9B15>I< 030007003F00C70007000700070007000700070007000700070007000700070007000700 070007000700070007000700070007000F80FFF80D1C7C9B15>I<07C01830201C400C40 0EF00FF80FF807F8077007000F000E000E001C001C00380070006000C00180030006010C 01180110023FFE7FFEFFFE101C7E9B15>I<07E01830201C201C781E780E781E381E001C 001C00180030006007E00030001C001C000E000F000F700FF80FF80FF80FF00E401C201C 183007E0101D7E9B15>I<000C00000C00001C00003C00003C00005C0000DC00009C0001 1C00031C00021C00041C000C1C00081C00101C00301C00201C00401C00C01C00FFFFC000 1C00001C00001C00001C00001C00001C00001C0001FFC0121C7F9B15>I<300C3FF83FF0 3FC020002000200020002000200023E024302818301C200E000E000F000F000F600FF00F F00FF00F800E401E401C2038187007C0101D7E9B15>I<00F0030C06040C0E181E301E30 0C700070006000E3E0E430E818F00CF00EE006E007E007E007E007E00760076007700630 0E300C18180C3003E0101D7E9B15>I<4000007FFF807FFF007FFF004002008004008004 0080080000100000100000200000600000400000C00000C00001C0000180000180000380 00038000038000038000078000078000078000078000078000078000030000111D7E9B15 >I<03E00C301008200C20066006600660067006780C3E083FB01FE007F007F818FC307E 601E600FC007C003C003C003C00360026004300C1C1007E0101D7E9B15>I<03C00C3018 18300C700C600EE006E006E007E007E007E007E0076007700F300F18170C2707C7000600 06000E300C780C78187010203030C00F80101D7E9B15>I<60F0F0600000000000000000 000060F0F06004127C910C>I<60F0F0600000000000000000000060F0F0701010101020 204080041A7C910C>I<7FFFFFC0FFFFFFE0000000000000000000000000000000000000 0000000000000000000000000000FFFFFFE07FFFFFC01B0C7E8F20>61 D<003F800000C0600003001800040004000800020010000100201F00802070808040E040 4040C0384041C03840818038208380382083803820838038208380382083803820818038 2041C0382040C0384040E0784020709880201F0F00100000000800000004000000030001 E000C01F80003FF0001B1D7E9C20>64 D<000600000006000000060000000F0000000F00 00000F00000017800000178000001780000023C0000023C0000023C0000041E0000041E0 000041E0000080F0000080F0000180F8000100780001FFF80003007C0002003C0002003C 0006003E0004001E0004001E000C001F001E001F00FF80FFF01C1D7F9C1F>II<001F808000E0618001801980070007800E0003801C 0003801C00018038000180780000807800008070000080F0000000F0000000F0000000F0 000000F0000000F0000000F0000000F0000000700000807800008078000080380000801C 0001001C0001000E000200070004000180080000E03000001FC000191E7E9C1E>IIII72 DI<1FFF00F8007800780078007800780078007800780078 00780078007800780078007800780078007800787078F878F878F878F0F040E021C01F00 101D7F9B15>II III<003F800000E0E0000380380007001C000E000E001C 0007003C00078038000380780003C0780003C0700001C0F00001E0F00001E0F00001E0F0 0001E0F00001E0F00001E0F00001E0F00001E0700001C0780003C0780003C0380003803C 0007801C0007000E000E0007001C000380380000E0E000003F80001B1E7E9C20>II82 D<07E0801C1980300580700380600180E00180E00080E00080E00080F00000F800007C00 007FC0003FF8001FFE0007FF0000FF80000F800007C00003C00001C08001C08001C08001 C0C00180C00180E00300D00200CC0C0083F800121E7E9C17>I<7FFFFFC0700F01C0600F 00C0400F0040400F0040C00F0020800F0020800F0020800F0020000F0000000F0000000F 0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F 0000000F0000000F0000000F0000000F0000000F0000001F800003FFFC001B1C7F9B1E> II87 D<7FF0FFC00FC03E00078018 0003C0180003E0100001E0200001F0600000F0400000788000007D8000003D0000001E00 00001F0000000F0000000F8000000F80000013C0000023E0000021E0000041F00000C0F8 000080780001007C0003003C0002001E0006001F001F003F80FFC0FFF01C1C7F9B1F>I< FEFEC0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0 C0C0C0FEFE07297C9E0C>91 D<08081010202040404040808080808080B0B0F8F8787830 300D0C7A9C15>II<0C0012002100408080400A057B9B15>I< 1FC000307000783800781C00301C00001C00001C0001FC000F1C00381C00701C00601C00 E01C40E01C40E01C40603C40304E801F870012127E9115>97 DI<07E00C301878307870306000E000E000E000E000E000E000 60007004300418080C3007C00E127E9112>I<003F000007000007000007000007000007 0000070000070000070000070000070003E7000C1700180F00300700700700600700E007 00E00700E00700E00700E00700E00700600700700700300700180F000C370007C7E0131D 7E9C17>I<03E00C301818300C700E6006E006FFFEE000E000E000E00060007002300218 040C1803E00F127F9112>I<00F8018C071E061E0E0C0E000E000E000E000E000E00FFE0 0E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E007FE00F1D 809C0D>I<00038003C4C00C38C01C3880181800381C00381C00381C00381C001818001C 38000C300013C0001000003000001800001FF8001FFF001FFF803003806001C0C000C0C0 00C0C000C06001803003001C0E0007F800121C7F9215>II<18003C003C0018000000000000000000000000000000FC001C00 1C001C001C001C001C001C001C001C001C001C001C001C001C001C001C00FF80091D7F9C 0C>I<00C001E001E000C000000000000000000000000000000FE000E000E000E000E000 E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E060E0F0C0F1 C061803E000B25839C0D>IIIII<03F0000E1C00180600300300700380600180E001C0E001C0E001C0E001C0E0 01C0E001C06001807003803003001806000E1C0003F00012127F9115>II<03C1000C3300180B00300F00700700700700E00700E00700E00700E007 00E00700E00700600700700700300F00180F000C370007C7000007000007000007000007 00000700000700000700003FE0131A7E9116>II<1F9030704030C010C0 10E010F8007F803FE00FF000F880388018C018C018E010D0608FC00D127F9110>I<0400 0400040004000C000C001C003C00FFE01C001C001C001C001C001C001C001C001C001C10 1C101C101C101C100C100E2003C00C1A7F9910>IIII<7F8FF00F03800F030007020003840001C80001D80000F000007000007800 00F800009C00010E00020E000607000403801E07C0FF0FF81512809116>II<7FFC70386038407040F040E041C003C0038007000F040E041C043C0C 380870087038FFF80E127F9112>II<1C043F0843F080E00E047D9B 15>126 D E /FJ 20 118 df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df<00380000380000380000380000380000380060380CF8103E 7C107C1F11F00793C001D700007C00007C0001D7000793C01F11F07C107CF8103E60380C 003800003800003800003800003800003800171A7D9B1E>3 D E /FL 19 124 df<000001FF80000000001FFFF8000000007F00FE00000001F8001F800000 07F0000FE000000FC00003F000001F800001F800003F000000FC00007E0000007E0000FC 0000003F0001FC0000003F8003F80000001FC007F00000000FE007F00000000FE00FF000 00000FF00FE000000007F01FE000000007F81FC000000003F83FC000000003FC3FC00000 0003FC3FC000000003FC7FC000000003FE7F8000000001FE7F8000000001FE7F80000000 01FEFF8000000001FFFF8000000001FFFF8000000001FFFF8000000001FFFF8000000001 FFFF8000000001FFFF8000000001FFFF8000000001FFFF8000000001FFFF8000000001FF FF8000000001FF7F8000000001FE7FC000000003FE7FC000000003FE7FC000000003FE7F C000000003FE3FC000000003FC3FE000000007FC1FE000000007F81FE000000007F80FF0 0000000FF00FF00000000FF007F00000000FE007F80000001FE003F80000001FC001FC00 00003F8000FE0000007F00007F000000FE00003F000000FC00001F800001F800000FE000 07F0000007F0000FE0000001FC003F800000007F00FE000000001FFFF80000000001FF80 0000383D7BBB43>79 D<003FC0000001FFF8000007C07E00000E001F80001E000FC0003F 8007E0003FC007E0003FC003F0003FC003F0003FC001F8001F8001F8000F0001F8000000 01F800000001F800000001F800000001F80000001FF8000007FFF800003FF9F80000FF01 F80003F801F80007F001F8001FE001F8001FC001F8003F8001F8007F0001F8007F0001F8 00FF0001F80CFE0001F80CFE0001F80CFE0001F80CFE0003F80CFE0003F80CFF0007F80C 7F000EF80C3F800C7C181FC03C7E380FE0703FF003FFE01FE0007F800F8026287CA62B> 97 D<01F800000000FFF800000000FFF800000000FFF80000000007F80000000001F800 00000001F80000000001F80000000001F80000000001F80000000001F80000000001F800 00000001F80000000001F80000000001F80000000001F80000000001F80000000001F800 00000001F80000000001F80000000001F80000000001F80000000001F80FF0000001F83F FE000001F8F01F800001F9C007C00001FB8003F00001FF0001F80001FE0000FC0001FC00 00FC0001F800007E0001F800007F0001F800003F0001F800003F8001F800003F8001F800 001F8001F800001F8001F800001FC001F800001FC001F800001FC001F800001FC001F800 001FC001F800001FC001F800001FC001F800001FC001F800001FC001F800001F8001F800 003F8001F800003F8001F800003F0001F800003F0001F800007E0001FC00007E0001FC00 00FC0001FE0001F80001F70001F00001E38003E00001E1C00FC00001C0F03F000001807F FC000000000FE000002A3D7EBB30>I<0003FE00001FFFC0007E01F000F8003801F0003C 03E000FE07E001FE0FC001FE1F8001FE1F8001FE3F8000FC3F0000787F0000007F000000 7E000000FE000000FE000000FE000000FE000000FE000000FE000000FE000000FE000000 FE000000FE0000007E0000007F0000007F0000007F0000033F0000033F8000071F800006 0FC0000E0FC0000C07E0001C03F0003800F800F0007E03C0001FFF000003FC0020287DA6 26>I<00000007E000000003FFE000000003FFE000000003FFE0000000001FE000000000 07E00000000007E00000000007E00000000007E00000000007E00000000007E000000000 07E00000000007E00000000007E00000000007E00000000007E00000000007E000000000 07E00000000007E00000000007E00000000007E00000000007E0000001FC07E000000FFF 87E000003F03C7E00000FC00F7E00001F0003FE00003E0001FE00007E0001FE0000FC000 0FE0001F800007E0001F800007E0003F000007E0003F000007E0007F000007E0007F0000 07E0007E000007E000FE000007E000FE000007E000FE000007E000FE000007E000FE0000 07E000FE000007E000FE000007E000FE000007E000FE000007E0007E000007E0007E0000 07E0007F000007E0007F000007E0003F000007E0003F800007E0001F80000FE0000FC000 0FE0000FC0001FE00007E0003FE00003F00077F80000F800E7FFC0007E03C7FFC0001FFF 07FFC00003FC07E0002A3D7DBB30>I<0003FC0000001FFF0000007E07C00000F803F000 03F001F80007E000FC0007C0007C000FC0007E001F80003E003F80003F003F00001F003F 00001F007F00001F807F00001F807E00001F80FE00001F80FE00001F80FFFFFFFF80FFFF FFFF80FE00000000FE00000000FE00000000FE00000000FE00000000FE000000007E0000 00007F000000007F000000003F000001803F000001801F800003801F800003000FC00007 0007E0000E0003E0000C0001F0003C0000FC007000003F01E000000FFF80000001FE0000 21287EA626>I<00003F000001FFC00003E0F0000FC3F0001F87F8003F07F8003F07F800 7E07F8007E07F800FE03F000FC000000FC000000FC000000FC000000FC000000FC000000 FC000000FC000000FC000000FC000000FC000000FC000000FC000000FC0000FFFFFC00FF FFFC00FFFFFC0000FC000000FC000000FC000000FC000000FC000000FC000000FC000000 FC000000FC000000FC000000FC000000FC000000FC000000FC000000FC000000FC000000 FC000000FC000000FC000000FC000000FC000000FC000000FC000000FC000000FC000000 FC000000FC000000FC000000FC000000FC000001FE00007FFFFC007FFFFC007FFFFC001D 3D7FBC1A>I<01F800000000FFF800000000FFF800000000FFF80000000007F800000000 01F80000000001F80000000001F80000000001F80000000001F80000000001F800000000 01F80000000001F80000000001F80000000001F80000000001F80000000001F800000000 01F80000000001F80000000001F80000000001F80000000001F80000000001F807F80000 01F81FFE000001F8781F800001F8E00FC00001F9C007E00001FB8007E00001FF0007E000 01FE0003F00001FE0003F00001FC0003F00001FC0003F00001FC0003F00001F80003F000 01F80003F00001F80003F00001F80003F00001F80003F00001F80003F00001F80003F000 01F80003F00001F80003F00001F80003F00001F80003F00001F80003F00001F80003F000 01F80003F00001F80003F00001F80003F00001F80003F00001F80003F00001F80003F000 01F80003F00001F80003F00001F80003F00003FC0007F800FFFFF1FFFFE0FFFFF1FFFFE0 FFFFF1FFFFE02B3C7EBB30>104 D<01C00007F0000FF8000FF8000FF8000FF8000FF800 07F00001C000000000000000000000000000000000000000000000000000000000000000 00000001F8007FF8007FF8007FF80007F80001F80001F80001F80001F80001F80001F800 01F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F800 01F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80003FC00 FFFFE0FFFFE0FFFFE0133A7FB917>I<01F800FFF800FFF800FFF80007F80001F80001F8 0001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F8 0001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F8 0001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F8 0001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F8 0001F80003FC00FFFFF0FFFFF0FFFFF0143C7FBB17>108 D<01F803F80003F80000FFF8 1FFF001FFF0000FFF87C0FC07C0FC000FFF8E007E0E007E00007F9C003E1C003E00001FB 8003F38003F00001FF0003F70003F00001FE0001FE0001F80001FE0001FE0001F80001FC 0001FC0001F80001FC0001FC0001F80001FC0001FC0001F80001F80001F80001F80001F8 0001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F8 0001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F8 0001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F8 0001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F8 0001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F8 0001F80001F80003FC0003FC0003FC00FFFFF0FFFFF0FFFFF0FFFFF0FFFFF0FFFFF0FFFF F0FFFFF0FFFFF044267EA549>I<01F807F80000FFF81FFE0000FFF8781F8000FFF8E00F C00007F9C007E00001FB8007E00001FF0007E00001FE0003F00001FE0003F00001FC0003 F00001FC0003F00001FC0003F00001F80003F00001F80003F00001F80003F00001F80003 F00001F80003F00001F80003F00001F80003F00001F80003F00001F80003F00001F80003 F00001F80003F00001F80003F00001F80003F00001F80003F00001F80003F00001F80003 F00001F80003F00001F80003F00001F80003F00001F80003F00001F80003F00001F80003 F00003FC0007F800FFFFF1FFFFE0FFFFF1FFFFE0FFFFF1FFFFE02B267EA530>I<0001FE 0000000FFFC000003F03F00000F8007C0001F0003E0003E0001F0007C0000F800F800007 C00F800007C01F000003E03F000003F03F000003F03F000003F07E000001F87E000001F8 7E000001F8FE000001FCFE000001FCFE000001FCFE000001FCFE000001FCFE000001FCFE 000001FCFE000001FCFE000001FC7E000001F87E000001F87F000003F83F000003F03F00 0003F01F000003E01F800007E00F800007C007C0000F8003E0001F0001F0003E0000F800 7C00003F03F000000FFFC0000001FE000026287EA62B>I<01F80FF00000FFF83FFE0000 FFF8F01F8000FFF9C00FC00003FB8007F00001FF0003F80001FE0001FC0001FC0000FC00 01F80000FE0001F800007F0001F800007F0001F800003F8001F800003F8001F800003F80 01F800003F8001F800001FC001F800001FC001F800001FC001F800001FC001F800001FC0 01F800001FC001F800001FC001F800001FC001F800001FC001F800003F8001F800003F80 01F800003F8001F800003F0001F800007F0001F800007E0001FC0000FE0001FC0000FC00 01FE0001F80001FF0003F00001FB8007E00001F9C00FC00001F8F03F000001F87FFC0000 01F80FE0000001F80000000001F80000000001F80000000001F80000000001F800000000 01F80000000001F80000000001F80000000001F80000000001F80000000001F800000000 01F80000000003FC00000000FFFFF0000000FFFFF0000000FFFFF00000002A377EA530> I<03F00F80FFF03FE0FFF0F1F0FFF1C3F807F383F801F303F801F703F801F601F001FE00 4001FC000001FC000001FC000001FC000001F8000001F8000001F8000001F8000001F800 0001F8000001F8000001F8000001F8000001F8000001F8000001F8000001F8000001F800 0001F8000001F8000001F8000001F8000001F8000001F8000001F8000003FC0000FFFFF8 00FFFFF800FFFFF8001D267EA522>114 D<007F818003FFE3800F807F801E001F803C00 0F807800078070000380F0000380F0000380F0000180F8000180F8000180FC000180FE00 00007F8000007FFC00003FFFC0001FFFF0000FFFFC0007FFFE0001FFFF00001FFF800000 FFC000001FC0C0000FE0C00007E0C00003E0E00003E0E00001E0E00001E0F00001E0F000 01E0F80001C0F80003C0FC000380FE000780FF000F00F3C03E00E0FFF800C03FC0001B28 7DA622>I<00180000001800000018000000180000001800000038000000380000003800 0000380000007800000078000000F8000000F8000001F8000003F8000007F800001FFFFF 00FFFFFF00FFFFFF0001F8000001F8000001F8000001F8000001F8000001F8000001F800 0001F8000001F8000001F8000001F8000001F8000001F8000001F8000001F8000001F800 0001F8000001F8000001F8018001F8018001F8018001F8018001F8018001F8018001F801 8001F8018001F8018001F8018000FC038000FC0300007C0300007E0700003F0E00000FFC 000003F00019367EB421>I121 D123 D E end %%EndProlog %%BeginSetup %%Feature: *Resolution 300dpi TeXDict begin %%EndSetup %%Page: 1 1 1 0 bop 60 279 a FL(On)28 b(the)h(stabilit)o(y)j(of)d(realistic)j (three{b)r(o)r(dy)e(problems)1899 248 y FK(\003)286 427 y FJ(Alessandra)19 b(Celletti)274 502 y FI(Dipartimen)o(to)12 b(di)h(Matematica)337 577 y(Univ)o(ersit\022)-21 b(a)14 b(dell'Aquila)243 651 y(67100{Coppito,)d(L'Aquila)h(\(Italy\))175 726 y(\(In)o(ternet:)j(celletti@axscaq.aquila.infn.i)o(t\))1202 427 y FJ(Luigi)20 b(Chierc)n(hia)1142 502 y FI(Dipartimen)o(to)12 b(di)h(Matematica)1187 577 y(Univ)o(ersit\022)-21 b(a)14 b(di)g(Roma)d(T)m(re)1031 651 y(via)i(Corrado)h(Segre)h(2,)e(00146)g (Roma)e(\(Italy\))1022 726 y(\(In)o(ternet:)k(luigi@matrm)o(3.)o(m)o(a) o(t.unirom)n(a3.)o(it\))842 843 y FJ(April)k(1996)879 1080 y FH(Abstract)168 1165 y FG(W)l(e)k(deriv)o(e)g(a)g(three{b)q(o)q (dy)h(problem)f(for)f(the)h(Sun{Jupiter{Ceres)i(system)d(and,)h(after) 168 1221 y(substituting)e(the)g(mass)g(ratio)f(of)g(Jupiter/Sun)j (\(whic)o(h)e(is)h(appro)o(ximately)f(10)1617 1205 y FF(\000)p FE(3)1663 1221 y FG(\))g(with)168 1278 y(a)c(parameter)g FD(")p FG(,)h(w)o(e)f(pro)o(v)o(e)h(the)f(existence)j(of)d(stable)h (quasi{p)q(erio)q(dic)i(motions)e(with)g(fre-)168 1334 y(quencies)13 b(close)f(to)f(the)h(observ)o(ed)f(\(a)o(v)o(erage\))f (frequencies)j(rep)q(orted)f(in)g(\\The)g(Astronomical)168 1391 y(Almanac")j(for)g FC(j)p FD(")p FC(j)d(\024)h FG(10)604 1374 y FF(\000)p FE(6)650 1391 y FG(.)i(The)h(pro)q(of)e(is)i (\\computer{assisted".)46 1544 y FB(Con)n(ten)n(ts)46 1636 y FA(1)38 b(In)o(tro)q(duction)12 b(and)j(Theorem)g(1.1)1145 b(2)46 1727 y(2)38 b(Restricted,)13 b(circular,)i(planar)f(three{b)q(o) q(dy)g(problem)772 b(6)46 1818 y(3)38 b(A)16 b(mo)q(del)f(from)f(the)h (Solar)g(system)1122 b(8)46 1910 y(4)38 b(A)16 b(KAM)h(Theorem)1381 b(10)108 1960 y FI(4.1)42 b(Algebraic)13 b(Sc)o(heme)k Fz(:)j(:)g(:)g(:)h(:)f(:)g(:)h(:)f(:)g(:)g(:)h(:)f(:)g(:)h(:)f(:)g(:)g (:)h(:)f(:)g(:)h(:)f(:)g(:)g(:)h(:)f(:)g(:)h(:)f(:)g(:)g(:)h(:)f(:)g(:) h(:)f(:)g(:)g(:)h(:)58 b FI(11)108 2009 y(4.2)42 b(Analytic)13 b(T)m(o)q(ols)38 b Fz(:)21 b(:)f(:)g(:)g(:)h(:)f(:)g(:)h(:)f(:)g(:)g(:) h(:)f(:)g(:)h(:)f(:)g(:)g(:)h(:)f(:)g(:)h(:)f(:)g(:)g(:)h(:)f(:)g(:)h (:)f(:)g(:)g(:)h(:)f(:)g(:)h(:)f(:)g(:)g(:)h(:)58 b FI(14)108 2059 y(4.3)42 b(KAM)14 b(Algorithm)34 b Fz(:)20 b(:)g(:)g(:)h(:)f(:)g (:)h(:)f(:)g(:)g(:)h(:)f(:)g(:)h(:)f(:)g(:)g(:)h(:)f(:)g(:)h(:)f(:)g(:) g(:)h(:)f(:)g(:)h(:)f(:)g(:)g(:)h(:)f(:)g(:)h(:)f(:)g(:)g(:)h(:)58 b FI(17)108 2109 y(4.4)42 b(KAM)14 b(theorem)41 b Fz(:)21 b(:)f(:)g(:)g(:)h(:)f(:)g(:)h(:)f(:)g(:)g(:)h(:)f(:)g(:)h(:)f(:)g(:)g (:)h(:)f(:)g(:)h(:)f(:)g(:)g(:)h(:)f(:)g(:)h(:)f(:)g(:)g(:)h(:)f(:)g(:) h(:)f(:)g(:)g(:)h(:)58 b FI(21)46 2200 y FA(5)38 b(Pro)q(of)15 b(of)g(Theorem)g(1.1)1306 b(28)108 2250 y FI(5.1)42 b(Step)14 b(1:)f(F)m(ormal)f Fz(")p FI({expansion)i(and)f(initial)f(appro)o (ximate)g(solution)g Fz(:)21 b(:)f(:)g(:)h(:)f(:)g(:)g(:)h(:)f(:)g(:)h (:)f(:)g(:)g(:)h(:)58 b FI(28)108 2300 y(5.2)42 b(Step)14 b(2:)f(Norm)g(b)q(ounds)h(relativ)o(e)g(to)g(the)g(initial)e(appro)o (ximate)g(solution)33 b Fz(:)21 b(:)f(:)g(:)g(:)h(:)f(:)g(:)h(:)f(:)g (:)g(:)h(:)58 b FI(32)108 2350 y(5.3)42 b(Step)14 b(3:)f(Application)g (of)g(the)i(KAM)f(algorithm)d(and)j(of)f(the)i(KAM)f(Theorem)40 b Fz(:)20 b(:)h(:)f(:)g(:)h(:)f(:)g(:)g(:)h(:)58 b FI(36)46 2441 y FA(A)26 b(Some)15 b(computer{assist)o(ed)e(data)1138 b(38)p 46 2531 747 2 v 101 2562 a Fy(\003)121 2577 y FI(This)12 b(w)o(ork)h(is)g(part)g(of)g(the)g(researc)o(h)i(program)c (of)i(the)g(Europ)q(ean)h(Net)o(w)o(ork)f(on:)g(\\Stabilit)o(y)e(and)i (Univ)o(ersalit)o(y)46 2627 y(in)g(Classical)g(Mec)o(hanics",)i(#)e (ER{BCHRX)o(CT940460.)967 2863 y Fx(1)p eop %%Page: 2 2 2 1 bop 46 78 a FB(1)83 b(In)n(tro)r(duction)27 b(and)g(Theorem)e(1.1) 46 187 y Fw(1\))14 b Fx(Since)f(h)o(uman)h(kind)g(dev)o(elop)q(ed)f(a)i (de\014nite)e(mathematical)e(taste)k(\(whatev)o(er)f(this)g(means\))f (the)46 248 y(\\stabilit)o(y")k(of)h(planetary)g(motions)f(migh)o(t)f (b)q(e)i(considered)g(as)g(one)g(of)g(the)g(cen)o(tral)f(questions)g (in)46 308 y(mathematics.)f(No)o(w)o(ada)o(ys)j(consciousness)h(ab)q (out)h(new)e(phenomena)f(\(p)q(ollution,)h(just)h(to)f(name)46 368 y(one\))d(has)g(dra)o(wn)g(the)g(atten)o(tion)f(of)h(scien)o(tists) f(and)h(non)g(scien)o(tists)f(to)h(other)g(t)o(yp)q(es)f(of)h (stabilities)46 428 y(\(in)e(other)g(w)o(ords,)g(extinction)f(of)i (living)e(sp)q(ecies)g(will)h(dep)q(end)g(more)f(and)i(more)d(up)q(on)k (the)e(c)o(haotic)46 488 y(e\013ects)23 b(of)g(p)q(ollution)g(rather)g (than)h(on)f(\\the)h(sky)e(falling)h(on)g(our)h(heads"\).)f(Nev)o (ertheless)e(the)46 549 y(stabilit)o(y)11 b(problem)h(for)h(man)o(y{b)q (o)q(dy)g(systems)e(in)o(teracting)h(only)h(through)h(gra)o(vitation)f (still)e(stems)46 609 y(out)19 b(as)g(one)g(of)g(the)g(more)e(in)o (triguing)h(and)i(ric)o(h)d(problems)h(in)g(mathematics.)e(In)i(mo)q (dern)g(times)46 669 y(outstanding)i(con)o(tributions)e(came,)f(ab)q(o) o(v)o(e)i(all,)e(from)h(H.)g(P)o(oincar)o(\023)-23 b(e)18 b([13],)g(V.)g(I.)g(Arnold)g([2])g(and)46 729 y(J.)k(Moser)g([12].)f (In)h(particular)g(the)f(so{called)i(KAM)e(\(Kolmogoro)o(v,)g(Arnold,)g (Moser\))h(theory)46 789 y(\(see)17 b([1)o(])g(and)h(references)e (therein\))g(ga)o(v)o(e)g(a)i(\\p)q(ositiv)o(e")f(answ)o(er)g(to)h(the) f(ab)q(o)o(v)o(e)g(stabilit)o(y)f(problem)46 850 y(in)i(the)h(sense)f (that)h(it)f(pro)o(v)o(ed)g(\([2]\))g(the)h(p)q(ossibilit)o(y)e(of)i (the)g(existence)e(of)i(man)o(y{b)q(o)q(dy)f(systems)46 910 y(\(\\planetary)c(systems"\))f(whose)h(time)d(ev)o(olution)i(ma)o (y)f(b)q(e)i(describ)q(ed)f(b)o(y)g(a)h(linear)f(\015o)o(w)h(on)g(a)g (torus)46 970 y(\(\\quasi{p)q(erio)q(dic)j(motion",)f(here,)g(synon)o (ymous)h(of)g(\\stable)g(motion"\).)f(The)h(dra)o(wbac)o(k)g(of)g(this) 46 1030 y(b)q(eautiful)g(result)g(is)h(that,)f(taking)h(the)f (estimates)f(con)o(tained)h(in)h(it)f(seriously)l(,)f(it)i(turns)g(out) g(that)46 1090 y(the)12 b(mass)g(ratio)h(of)g(the)f(planets)h(of)g(suc) o(h)f(h)o(yp)q(othetical)g(planetary)g(system)f(with)i(their)e(star)j (should)46 1151 y(b)q(e)i(more)f(or)i(less)f(comparable)f(to)i(the)f (mass)f(ratio)i(of)f(a)h(proton)g(with)f(the)g(Sun)1553 1132 y FE(1)1573 1151 y Fx(.)140 1211 y(One)g(then)g(ma)o(y)f(p)q(ose)i (the)f(question)g(of)h(stabilit)o(y)e(of)h Fv(realistic)f(man)o(y{b)q (o)q(dy)h(systems)p Fx(.)140 1271 y(Of)c(course,)f(to)h(giv)o(e)e(a)i (mathematical)d(con)o(ten)o(t)h(to)i(the)f(w)o(ord)h(\\realistic")f(is) h(clearly)e(imp)q(ossible)46 1331 y(and)21 b(the)g(b)q(est)g(w)o(e)g (could)f(do)i(w)o(as)f(to)g(get)g(inspiration)g(from)f(our)h(o)o(wn)h (planetary)e(system.)f(W)l(e)46 1391 y(considered)f(one)i(of)f(the)g (most)g(p)q(opular)h(three{b)q(o)q(dy)g(problems)e(of)h(the)g(Solar)g (system)1721 1373 y FE(2)1759 1391 y Fx(namely)46 1452 y(Sun,)13 b(Jupiter)f(and)h(Ceres)g(\(one)g(of)g(the)f(ma)s(jor)g(b)q (o)q(dies)i(of)f(the)g(so{called)f(asteroid)h(b)q(elt\).)f(F)l(ollo)o (wing)46 1512 y(Delauna)o(y)20 b([6],)f(w)o(e)h(then)g(deriv)o(ed)f(a)i (Hamiltonian)d(mo)q(del.)h(Clearly)l(,)g(in)h(deriving)f(a)i(mo)q(del)e (one)46 1572 y(simpli\014es)9 b(quite)h(a)h(bit)g(Nature:)f(w)o(e)h (tried)f(to)i(motiv)m(ate)d(suc)o(h)i(simpli\014cations)e(in)i Fu(x)g Fx(3.)g(W)l(e)g(then)g(to)q(ok)46 1632 y(our)h(mo)q(del)f (seriously)h(\(from)f(the)h(mathematical)d(p)q(oin)o(t)j(of)g(view\))g (and)h(replacing)e(the)h(Jupiter/Sun)46 1692 y(mass)18 b(ratio,)g(whic)o(h)g(is)g(appro)o(ximately)e(10)864 1674 y FF(\000)p FE(3)912 1692 y Fx(,)i(with)g(a)h(\\p)q(erturbativ)o (e)f(parameter")g Ft(")p Fx(,)f(w)o(e)h(ask)o(ed)46 1752 y(for)e(ho)o(w)h(large)f(v)m(alues)g(of)g Ft(")g Fx(one)g(could)g (\014nd)h(quasi{p)q(erio)q(dic)f(motions)f(with)h(\\frequencies")f (close)46 1813 y(to)20 b(the)g(observ)o(ed)g(frequencies)f(of)h(the)g (Sun{Jupiter{Ceres)g(system)f(\(whic)o(h)h(ma)o(y)e(b)q(e)j(found)f(in) 46 1873 y(the)c(ephemeris)e([16]\).)140 1933 y(F)l(or)23 b(suc)o(h)f(a)h(mo)q(del)e(w)o(e)h(pro)o(v)o(ed)f(stabilit)o(y)h(for)g Fu(j)p Ft(")p Fu(j)i(\024)g Fx(10)1253 1915 y FF(\000)p FE(6)1301 1933 y Fx(,)e(b)q(eing)g(therefore)g(a)o(w)o(a)o(y)g(from)46 1993 y(\\realit)o(y")g(b)o(y)g(three)h(orders)g(of)g(magnitude.)f(W)l (e)g(lea)o(v)o(e)g(it)g(to)h(the)g(reader)g(to)g(judge)g(if)f(this)h (is)46 2053 y(realistic)15 b(or)i(not.)g(W)l(e)g(b)q(eliev)o(e,)d(ho)o (w)o(ev)o(er,)h(that)i(with)g(some)f(more)f(e\013orts)j(one)f(should)g (indeed)f(b)q(e)46 2114 y(able)g(to)g(pro)o(v)o(e)g(stabilit)o(y)f(up)h (to)h Ft(")d Fx(=)f(10)796 2096 y FF(\000)p FE(3)861 2114 y Fx(and)j(w)o(e)g(regard)h(our)g(result)e(as)i(a)g(\014rst)g (step)f(to)o(w)o(ards)h(a)46 2174 y(pro)q(of)g(of)g(the)f(mathematical) d(stabilit)o(y)i(of)h(realistic)f(man)o(y{b)q(o)q(dy)h(problems.)140 2234 y(Our)21 b(result)e(relies)g(basically)h(on)g(t)o(w)o(o)h(tec)o (hniques:)d(\(i\))i(a)g(\(new\))g(KAM)g(sc)o(heme)e(presen)o(ted)46 2294 y(in)k Fu(x)h Fx(4)h(\(for)f(the)f(exp)q(erts:)h(a)g(KAM)f(result) g(in)h(Hamiltonian)e(setting)h(in)h(the)g(st)o(yle)e(of)i(Moser,)46 2354 y(Salamon)11 b(and)h(Zehnder)e([15)q(])g(with)i(emphasis)e(on)i (analytical)e(dep)q(endence)h(up)q(on)h(parameters\);)e(\(ii\))46 2415 y(computer{assisted)k(\(rigorous\))i(estimates,)d(whic)o(h)h(are)h (needed)f(in)h(order)g(to)g(apply)g(\\e\013ectiv)o(ely")46 2475 y(the)h(KAM)f(sc)o(heme)f(to)j(our)g(three{b)q(o)q(dy)g(problem.)p 46 2518 747 2 v 102 2549 a Fs(1)121 2564 y FI(Mass)d(of)f(proton/mass)g (of)g(Sun=)i(1)p Fz(:)p FI(6724)7 b Fr(\001)i FI(10)866 2549 y Fy(\000)p Fs(21)926 2564 y Fz(=)p FI(8)p Fz(:)p FI(4078)f Fr(\001)h FI(10)1135 2549 y Fs(30)1181 2564 y Fr(')j FI(8)p Fz(:)p FI(4078)7 b Fr(\001)i FI(10)1412 2549 y Fy(\000)p Fs(52)1472 2564 y FI(;)14 b(compare)f([8)o(].)102 2599 y Fs(2)121 2614 y FI(F)m(or)g(a)h(recen)o(t)h(application)e(of)g (Arnold's)g(result)i(to)f(other)g(three-b)q(o)q(dy)h(problems)e(see)j ([14)o(].)967 2863 y Fx(2)p eop %%Page: 3 3 3 2 bop 140 78 a Fx(It)17 b(is)f(w)o(ell)f(kno)o(wn)i(that)g(computers) f(ma)o(y)f(b)q(e)i(used)f(to)i(pro)o(v)o(e)d(theorems)h(\(see,)f(e.g.,) h([7],)g([11],)46 138 y([10])i(or)g(think)f(to)i(the)e(famous)h (\\four{colour)h(theorem"\).)d(W)l(e)i(w)o(ere)f(not)h(really)f(en)o (th)o(usiastic)g(to)46 198 y(rely)i(on)h(mac)o(hines)e(to)i(pro)o(v)o (e)f(our)i(result)e(but)h(couldn't)f(get)h(a)o(w)o(a)o(y)g(without)g (it.)f(In)h(fact)g(w)o(e)f(are)46 259 y(prett)o(y)j(sure)h(that)h(with) f(more)e(re\014ned)i(tec)o(hniques)f(and/or)i(with)f(new)g(ideas)g(one) h(migh)o(t)d(get)46 319 y(b)q(etter)d(results)h(without)g(the)g (\(essen)o(tially)e(trivial)h(but\))h(length)o(y)f(computations)g(whic) o(h)g(are)h(the)46 379 y(only)d(reason)h(to)g(call)e(in)h(mac)o(hines.) 46 460 y Fw(2\))11 b Fx(W)l(e)h(giv)o(e)e(no)o(w)i(a)g(precise)f Fw(form)n(ulation)h(of)h(the)g(main)f(result)p Fx(.)e(Let)i(\()p Ft(`;)c(g)r Fx(\))14 b Fu(2)g Fq(T)1602 442 y FE(2)1627 460 y Fu(\021)8 b Fq(R)1713 442 y FE(2)1729 460 y Ft(=)p Fx(\(2)p Ft(\031)r Fq(Z)1863 442 y FE(2)1879 460 y Fx(\);)46 521 y(let)101 502 y FE(3)478 581 y Ft(L)511 588 y FE(0)539 581 y Fu(\021)g Fx(0)p Ft(:)p Fx(729305)19 b Ft(;)56 b(G)895 588 y FE(0)924 581 y Fu(\021)8 b Fx(0)p Ft(:)p Fx(727162)18 b Ft(;)57 b(r)1264 588 y FE(0)1292 581 y Fu(\021)8 b Fx(0)p Ft(:)p Fx(001)17 b Ft(;)332 b Fx(\(1.1\))46 662 y(and)17 b(let)257 760 y Ft(B)11 b Fu(\021)d(f)p Fx(\()p Ft(L;)g(G)p Fx(\))14 b Fu(2)g Fq(C)602 739 y FE(2)639 760 y Fx(:)29 b Fu(j)p Ft(L)11 b Fu(\000)g Ft(L)823 767 y FE(0)843 760 y Fu(j)j(\024)f Ft(r)945 767 y FE(0)981 760 y Ft(;)24 b Fu(j)p Ft(G)12 b Fu(\000)f Ft(G)1171 767 y FE(0)1191 760 y Fu(j)j(\024)f Ft(r)1293 767 y FE(0)1313 760 y Fu(g)p Ft(;)56 b(B)1445 767 y FE(0)1473 760 y Fu(\021)8 b Ft(B)14 b Fu(\\)d Fq(R)1654 739 y FE(2)1687 760 y Ft(:)111 b Fx(\(1.2\))46 857 y(On)14 b Fq(T)162 839 y FE(2)186 857 y Fu(\002)8 b Ft(B)270 864 y FE(0)289 857 y Fx(,)14 b(endo)o(w)o(ed)g(with)h(the)f(standard)i(symplectic)11 b(form)j Ft(d`)8 b Fu(^)g Ft(dL)g Fx(+)g Ft(dg)i Fu(^)e Ft(dG)p Fx(,)15 b(consider)f(the)46 917 y(one{parameter)h(family)f(of)j (Hamiltonian)d(functions)j(giv)o(en)e(b)o(y)297 1034 y Ft(H)t Fx(\()p Ft(`;)8 b(g)r(;)g(L;)g(G)p Fx(;)g Ft(")p Fx(\))59 b Fu(\021)e Fx(\()812 1000 y(1)p 785 1022 78 2 v 785 1068 a(2)p Ft(L)842 1054 y FE(2)878 1034 y Fu(\000)11 b Ft(G)p Fx(\))985 1013 y FE(2)1033 1034 y Fx(+)27 b(2)p Ft(")17 b Fx(\()1212 1000 y(1)p 1186 1022 V 1186 1068 a(2)p Ft(L)1243 1054 y FE(2)1279 1034 y Fu(\000)11 b Ft(G)p Fx(\))16 b Ft(R)p Fx(\()p Ft(`;)8 b(g)r(;)g(L;)g(G)p Fx(\))665 1179 y Fu(\021)41 b Ft(h)p Fx(\()p Ft(L;)8 b(G)p Fx(\))k(+)f Ft(")16 b(f)5 b Fx(\()p Ft(`;)j(g)r(;)g(L;)g(G)p Fx(\))17 b Ft(;)528 b Fx(\(1.3\))46 1276 y(with)16 b(the)g(\\p)q (erturbing)h(function")f Ft(R)h Fx(de\014ned)f(as)424 1374 y Ft(R)p Fx(\()p Ft(`;)8 b(g)r(;)g(L;)g(G)p Fx(\))31 b Fu(\021)870 1332 y Fp(X)863 1424 y Fo(n)p Fn(2)p Fm(Z)923 1417 y Fl(2)786 1453 y(0)p Fn(\024j)p Fo(n)853 1460 y Fl(1)871 1453 y Fn(j)p Fl(+)p Fn(j)p Fo(n)931 1460 y Fl(2)950 1453 y Fn(j\024)p Fl(10)1027 1374 y Ft(R)1064 1381 y Fk(n)1088 1374 y Fx(\()p Ft(L;)8 b(G)p Fx(\))g(cos)q(\()p Ft(n)1341 1381 y FE(1)1361 1374 y Ft(`)j Fx(+)g Ft(n)1470 1381 y FE(2)1490 1374 y Ft(g)r Fx(\))278 b(\(1.4\))46 1540 y(where)16 b Ft(R)224 1547 y Fk(n)256 1540 y Fu(\021)8 b Ft(R)340 1547 y Fk(n)361 1552 y Fl(1)379 1547 y Fk(n)400 1552 y Fl(2)436 1540 y Fx(v)m(anishes)16 b(unless)g(it)g(b)q(elongs)h (to)g(the)f(follo)o(wing)g(list:)342 1667 y Ft(R)379 1674 y FE(00)466 1667 y Fu(\021)560 1633 y Ft(L)593 1615 y FE(4)p 560 1655 53 2 v 574 1701 a Fx(4)618 1667 y(\(1)11 b(+)738 1633 y(9)p 726 1655 49 2 v 726 1701 a(16)780 1667 y Ft(L)813 1646 y FE(4)844 1667 y Fx(+)898 1633 y(3)p 898 1655 25 2 v 898 1701 a(2)927 1667 y Ft(e)950 1646 y FE(2)970 1667 y Fx(\))16 b Ft(;)73 b(R)1129 1674 y FE(10)1175 1667 y Fu(\021)19 b(\000)1287 1633 y Ft(L)1320 1615 y FE(4)1340 1633 y Ft(e)p 1287 1655 76 2 v 1313 1701 a Fx(2)1368 1667 y(\(1)11 b(+)1476 1633 y(9)p 1476 1655 25 2 v 1476 1701 a(8)1506 1667 y Ft(L)1539 1646 y FE(4)1559 1667 y Fx(\))16 b Ft(;)204 b Fx(\(1.5\))342 1791 y Ft(R)379 1798 y FE(11)466 1791 y Fu(\021)560 1757 y Fx(3)p 560 1779 V 560 1825 a(8)589 1791 y Ft(L)622 1770 y FE(6)642 1791 y Fx(\(1)12 b(+)751 1757 y(5)p 751 1779 V 751 1825 a(8)780 1791 y Ft(L)813 1770 y FE(4)833 1791 y Fx(\))k Ft(;)219 b(R)1138 1798 y FE(12)1184 1791 y Fu(\021)19 b(\000)1297 1757 y Ft(L)1330 1739 y FE(4)1350 1757 y Ft(e)p 1297 1779 76 2 v 1323 1825 a Fx(4)1378 1791 y(\(9)11 b(+)g(5)p Ft(L)1538 1770 y FE(4)1558 1791 y Fx(\))16 b Ft(;)205 b Fx(\(1.6\))342 1914 y Ft(R)379 1921 y FE(22)466 1914 y Fu(\021)560 1881 y Ft(L)593 1863 y FE(4)p 560 1903 53 2 v 574 1949 a Fx(4)618 1914 y(\(3)11 b(+)726 1881 y(5)p 726 1903 25 2 v 726 1949 a(4)756 1914 y Ft(L)789 1894 y FE(4)809 1914 y Fx(\))16 b Ft(;)251 b(R)1146 1921 y FE(32)1192 1914 y Fu(\021)1244 1881 y Fx(3)p 1244 1903 V 1244 1949 a(4)1274 1914 y Ft(L)1307 1894 y FE(4)1327 1914 y Ft(e)15 b(;)342 2031 y(R)379 2038 y FE(33)466 2031 y Fu(\021)560 1997 y Fx(5)p 560 2019 V 560 2065 a(8)589 2031 y Ft(L)622 2010 y FE(6)642 2031 y Fx(\(1)d(+)763 1997 y(7)p 751 2019 49 2 v 751 2065 a(16)804 2031 y Ft(L)837 2010 y FE(4)857 2031 y Fx(\))17 b Ft(;)202 b(R)1146 2038 y FE(44)1192 2031 y Fu(\021)1244 1997 y Fx(35)p 1244 2019 V 1244 2065 a(64)1298 2031 y Ft(L)1331 2010 y FE(8)1367 2031 y Ft(;)342 2148 y(R)379 2155 y FE(55)466 2148 y Fu(\021)572 2114 y Fx(63)p 560 2136 74 2 v 560 2182 a(128)638 2148 y Ft(L)671 2127 y FE(10)725 2148 y Ft(;)1073 b Fx(\(1.7\))46 2260 y(and)17 b(the)g(\\eccen)o(tricit)o(y")d Ft(e)p Fx(,)i(whic)o(h)g(is)h(a)g (function)g(of)g(the)f(\\action)i(v)m(ariables")f(\()p Ft(L;)8 b(G)p Fx(\),)17 b(is)f(de\014ned)46 2321 y(as)89 2302 y FE(4)723 2411 y Ft(e)8 b Fu(\021)g Ft(e)p Fx(\()p Ft(L;)g(G)p Fx(\))g Fu(\021)1010 2328 y Fp(s)p 1052 2328 154 2 v 83 x Fx(1)j Fu(\000)1142 2377 y Ft(G)1180 2363 y FE(2)p 1142 2399 59 2 v 1145 2445 a Ft(L)1178 2430 y FE(2)1221 2411 y Ft(:)p 46 2481 747 2 v 102 2512 a Fs(3)121 2527 y FI(All)i(the)h(follo)o(wing)d(n)o(um)o(b)q(ers,)i(quan) o(tities)h(and)g(functions)g(will)e(b)q(e)j(ph)o(ysically)d(motiv)n (ated)g(in)i Fr(x)g FI(2)g(and)f Fr(x)i FI(3.)102 2562 y Fs(4)121 2577 y FI(In)g(this)g(pap)q(er)h(the)g(letter)h Fz(e)e FI(will)f(alw)o(a)o(ys)g(refer)j(to)e(\\eccen)o(tricities")i (and)e(nev)o(er)h(to)f(the)h(Nep)q(er)h(n)o(um)o(b)q(er;)d(the)46 2627 y(exp)q(onen)o(tial)f(function)h(will)e(b)q(e)j(denoted)g(exp\()p Fz(x)p FI(\).)967 2863 y Fx(3)p eop %%Page: 4 4 4 3 bop 46 78 a Fx(Let)16 b Ft(\013)h Fx(b)q(e)f(the)g(golden)h(mean)e (\()p Ft(\013)8 b Fu(\021)726 31 y FF(p)p 756 31 18 2 v 756 59 a FE(5)o FF(\000)p FE(1)p 726 67 93 2 v 764 96 a(2)823 78 y Fx(\))17 b(and)g(let)230 211 y(\012)265 218 y FF(\000)309 211 y Fu(\021)366 177 y Fx(5)p 366 199 25 2 v 366 245 a(2)407 211 y(+)518 177 y(1)p 461 199 141 2 v 461 245 a(13)11 b(+)g Ft(\013)620 211 y Fx(=)i(2)p Ft(:)p Fx(573432)p Ft(:::)k(;)106 b Fx(\012)1067 218 y FE(+)1110 211 y Fu(\021)1168 177 y Fx(5)p 1168 199 25 2 v 1168 245 a(2)1208 211 y(+)1320 177 y(1)p 1262 199 141 2 v 1262 245 a(12)12 b(+)f Ft(\013)1421 211 y Fx(=)j(2)p Ft(:)p Fx(579251)p Ft(:::)j(:)46 336 y Fx(The)f(\\observ)o (ed)f(a)o(v)o(erage)h(frequency")f(of)h(Ceres)f(is)h(appro)o(ximately)d Fu(\000)p Fx(\012)1420 343 y Fk(c)1446 336 y Fu(\021)18 b(\000)10 b Fx(2)p Ft(:)p Fx(577107)18 b(\(see)e Fu(x)g Fx(3)46 396 y(b)q(elo)o(w\))g(so)h(that)g(\012)404 403 y FF(\000)447 396 y Ft(<)d Fx(\012)534 403 y Fk(c)565 396 y Ft(<)g Fx(\012)652 403 y FE(+)682 396 y Fx(.)i(Let)571 515 y Ft(L)604 522 y FF(\006)642 515 y Fu(\021)8 b Fx(\012)724 490 y FF(\000)p FE(1)p Fk(=)p FE(3)724 525 y FF(\006)823 515 y Ft(;)105 b(G)980 522 y FF(\006)1040 515 y Fu(\021)30 b Ft(L)1142 522 y FF(\006)1188 462 y Fp(q)p 1230 462 128 2 v 53 x Fx(1)11 b Fu(\000)g Ft(e)1338 498 y FE(2)1338 526 y(0)1373 515 y Ft(;)46 621 y Fx(where)19 b Ft(e)213 628 y FE(0)250 621 y Fx(=)g(0)p Ft(:)p Fx(0766)h(is)f(the)g(\\observ)o (ed")g(eccen)o(tricit)o(y)d(of)j(Ceres)g(as)h(found)f(in)g([16].)f (Finally)l(,)f(w)o(e)46 681 y(de\014ne)f(the)g Ft(h)p Fx(-frequencies)437 806 y Ft(!)469 786 y FE(\()p FF(\006)p FE(\))556 806 y Fu(\021)30 b Fx(\()p Ft(E)680 813 y FF(\006)709 806 y Fx(\012)744 813 y FF(\006)774 806 y Ft(;)24 b(E)848 813 y FF(\006)878 806 y Fx(\))16 b Ft(;)105 b(E)1068 813 y FF(\006)1112 806 y Fu(\021)13 b(\000)p Fx(2\()1283 773 y(1)p 1251 795 88 2 v 1251 841 a(2)p Ft(L)1308 824 y FE(2)1308 850 y FF(\006)1355 806 y Fu(\000)d Ft(G)1442 813 y FF(\006)1472 806 y Fx(\))17 b Ft(:)290 b Fx(\(1.8\))46 946 y(F)l(or)16 b(later)g(use)g(w)o(e)g(p)q(oin)o(t)g(out)h(that)g Ft(!)755 928 y FE(\()p FF(\006)p FE(\))828 946 y Fx(is)f(a)h(\\Diophan) o(tine)f(v)o(ector")g(satisfying)555 1052 y Fu(j)p Ft(!)601 1031 y FE(\()p FF(\006)p FE(\))669 1052 y Fu(\001)11 b Ft(n)p Fu(j)j(\025)f Fx(\()p Ft(\015)847 1059 y FF(\006)877 1052 y Fu(j)p Ft(n)p Fu(j)p Fx(\))953 1031 y FF(\000)p FE(1)1016 1052 y Ft(;)57 b Fu(8)15 b Ft(n)f Fu(2)g Fq(Z)1257 1031 y FE(2)1274 1052 y Fu(nf)p Fx(0)p Fu(g)i Ft(;)409 b Fx(\(1.9\))46 1158 y(with)16 b Ft(\015)182 1165 y FF(\006)228 1158 y Fx(giv)o(en)f(b)o(y)407 1140 y FE(5)696 1218 y Ft(\015)721 1225 y FF(\006)765 1218 y Fx(=)e(2)p Fu(j)p Ft(E)890 1225 y FF(\006)920 1218 y Fu(j)p Fx(\()953 1174 y Fu(p)p 995 1174 25 2 v 44 x Fx(5)e(+)g(24)h Fu(\006)f Fx(1\))16 b Ft(:)525 b Fx(\(1.10\))46 1303 y(The)16 b(result)g (discussed)g(in)g(1\))h(ab)q(o)o(v)o(e)f(can)g(no)o(w)h(b)q(e)f(form)o (ulated)f(as)i(follo)o(ws.)46 1412 y Fw(Theorem)f(1.1)24 b Fj(L)n(et)18 b Ft(H)k Fj(b)n(e)c(as)f(in)h Fx(\(1.3\))p Fu(\004)p Fx(\(1.5\))p Fj(;)f(let)i Ft(B)1086 1419 y FE(0)1123 1412 y Fj(b)n(e)f(as)g(in)g Fx(\(1.1\))p Fj(,)g Fx(\(1.2\))p Fj(;)g(let)g Ft(!)1672 1394 y FE(\()p FF(\006)p FE(\))1747 1412 y Fj(b)n(e)g(as)g(in)46 1473 y Fx(\(1.8\))p Fj(.)j(Then,)g(for)g(al)r(l)h Fx(0)f Fu(\024)f(j)p Ft(")p Fu(j)f(\024)h Fx(10)763 1455 y FF(\000)p FE(6)832 1473 y Fj(ther)n(e)h(exist)h(\(unique\))g(two{dimensional)h(analytic)e(tori) 46 1533 y Fu(S)76 1540 y Fk(")94 1533 y Fx(\()p Ft(!)145 1515 y FE(\()p FF(\006)p FE(\))202 1533 y Fx(\))14 b Fu(\032)g Ft(B)325 1540 y FE(0)356 1533 y Fu(\002)d Fq(T)442 1515 y FE(2)459 1533 y Fj(,)17 b(dep)n(ending)i(analytic)n(al)r(ly)g (also)f(on)g(the)g(p)n(ar)n(ameter)e Ft(")h Fj(\(for)g Fu(j)p Ft(")p Fu(j)d(\024)f Fx(10)1759 1515 y FF(\000)p FE(6)1807 1533 y Fj(\),)18 b(on)46 1593 y(which)g(the)g Ft(H)t Fj({\015ow)f(is)h(\(analytic)n(al)r(ly\))g(c)n(onjugate)n(d)g (to)g(the)g(line)n(ar)f(\015ow)h Ft(\022)d Fu(2)f Fq(T)1527 1575 y FE(2)1558 1593 y Fu(!)f Ft(\022)f Fu(\000)f Ft(!)1738 1575 y FE(\()p FF(\006)p FE(\))1795 1593 y Ft(t)p Fj(.)46 1702 y Fw(Remark)16 b(1.1)24 b Fx(In)f Fu(x)p Fx(3)g(w)o(e)f(giv)o(e)g (a)h(ph)o(ysical)f(motiv)m(ation)g(for)h(ha)o(ving)f(c)o(hosen)h(as)g (p)q(erturbating)46 1762 y(function)11 b(a)h(trigonometric)e(p)q (olynomial.)f(W)l(e)i(b)q(eliev)o(e,)e(ho)o(w)o(ev)o(er,)h(that)i (considering)f(p)q(erturbations)46 1823 y(with)19 b(an)h(in\014nite)f (n)o(um)o(b)q(er)f(of)i(non)g(v)m(anishing)g(F)l(ourier)f(co)q (e\016cien)o(ts)g(w)o(ould)g(lead)h(to)g(essen)o(tially)46 1883 y(the)15 b(same)f(results)h(making)e(only)i(tec)o(hnically)e(more) h(in)o(v)o(olv)o(ed)f(\(and)i(more)f(exp)q(ensiv)o(e\))g(the)g(pro)q (of.)46 2016 y Fw(3\))i(\(On)g(the)f(pro)r(of)5 b(\))14 b Fx(The)g(pro)q(of)h(of)f(Theorem)f(1.1)h(is)g(giv)o(en)f(in)h Fu(x)g Fx(5)g(and,)g(as)h(already)f(men)o(tioned,)46 2076 y(is)i(a)h(\\computer)e(assisted")i(application)f(of)g(the)g(KAM)g (sc)o(heme)1261 2058 y FE(6)1294 2076 y Fx(of)h Fu(x)f Fx(4.)140 2136 y(As)k(it)f(is)h(w)o(ell)e(kno)o(wn)i(KAM)f(sc)o(hemes)e (are)j(\\Newton)g(algorithms":)f(they)g(are)h(pro)q(cedures)46 2196 y(to)j(iterativ)o(ely)e(construct)i(solutions)h(for)f(certain)g (nonlinear)g(equations)g(\(with)g(\\loss)h(of)g(regu-)46 2256 y(larit)o(y"\),)19 b(starting)h(from)g(some)f(initial)f(\\appro)o (ximate)i(solution",)g(with)g(a)h(quadratical)f(rate)g(of)46 2317 y(con)o(v)o(ergence.)15 b Fv(Our)i(initial)e(appro)o(ximate)h (solution)h(is)g(a)g(suitable)g(truncation)g Fx(\(actually)f(a)i (\\\014fth)46 2377 y(order)h(truncation"\))h Fv(of)f(the)g(so{called)g (\\Lindstedt)g(series")g Fx(\(see)g([1])f(for)h(generalities\),)f(i.e.) f(,)i(of)p 46 2422 747 2 v 102 2453 a Fs(5)121 2468 y FI(In)c(general,)g(n)o(um)o(b)q(ers)g(of)g(the)h(form)d Fz(a)h FI(=)807 2449 y Fi(p)p 807 2458 18 2 v 807 2482 a(q)839 2468 y Fr(\006)911 2452 y Fs(1)p 887 2459 66 2 v 887 2482 a Fi(k)q Fs(+)p Fi(\013)972 2468 y FI(with)h Fz(p;)7 b(q)q(;)g(k)15 b FI(non)h(negativ)o(e)f(in)o(tegers,)g Fz(q)g(>)g FI(0)g(and)g Fz(k)g Fr(\025)f FI(2,)46 2527 y(satisfy)f Fr(j)p Fz(an)c FI(+)h Fz(m)p Fr(j)h(\025)h FI(\()p Fz(\015)r Fr(j)p Fz(n)p Fr(j)p FI(\))492 2512 y Fy(\000)p Fs(1)550 2527 y FI(for)i(an)o(y)f Fz(n;)7 b(m)12 b Fr(2)f Fq(Z)-11 b FI(,)10 b Fz(n)i Fr(6)p FI(=)g(0)h(with)h Fz(\015)g FI(=)e Fz(q)1190 2512 y Fs(2)1209 2527 y FI(\()p Fz(k)e FI(+)f Fz(\013)p FI(\))14 b(\(see,)h(e.g.,)e([3)o(]\).)102 2562 y Fs(6)121 2577 y FI(See)25 b([4)o(])f(and)g([5)o(])g(for)g (general)h(informations,)c(references)27 b(and)e(a)f(di\013eren)o(t)h (\\KAM)f(computer{assisted)46 2627 y(algorithm".)967 2863 y Fx(4)p eop %%Page: 5 5 5 4 bop 46 78 a Fx(the)22 b(formal)g Ft(")p Fx({p)q(o)o(w)o(er)h (series)f(solution)h(for)g(the)f(in)o(v)m(arian)o(t)g(torus)i(equation) e(asso)q(ciated)i(to)f(the)46 138 y(lo)q(ok)o(ed)14 b(for)h(quasi{p)q (erio)q(dic)f(solution.)g(T)l(o)h(this)f(initial)f(datum)h(w)o(e)g (apply)g(the)g(\\KAM)g(algorithm")46 198 y(presen)o(ted)j(in)h Fu(x)h Fx(4.)f(The)g(\\KAM)g(algorithm")g(is)g(based)h(on)f(an)h (algebraic)f(sc)o(heme)e(whic)o(h,)h(start-)46 259 y(ing)f(from)f(a)i (giv)o(en)e(\\appro)o(ximate)g(solution",)i(pro)q(duces)g(a)f(new)h (function)f(solving)g(the)g(in)o(v)m(arian)o(t)46 319 y(torus)k(equation)f(up)h(to)f(an)h(error)g(whic)o(h)e(is)h (\\quadratically)g(smaller")f(than)i(the)f(one)h(pro)q(duced)46 379 y(b)o(y)e(the)g(starting)h(appro)o(ximate)e(solution)i(\()p Fu(x)f Fx(4.1\).)g(This)h(algebraic)f(sc)o(heme)e(\(whic)o(h,)h(as)i (already)46 439 y(men)o(tioned,)13 b(is)i(new\))g(is)h(equipp)q(ed)f (with)g(a)h(set)f(of)h(\\accurate")g(estimates)e(\()p Fu(x)i Fx(4.3\).)f(The)g(algebraic)46 499 y(sc)o(heme)10 b(plus)j(the)g(set)f(of)h(estimates)f(is)g(what)i(w)o(e)e(call)g(\\the) h(KAM)f(algorithm".)g(W)l(e)g(then)h(w)o(ork)g(out)46 560 y(a)j(criterion)e(\(the)i(KAM)e(Theorem)h(of)g Fu(x)h Fx(4.4\))g(whic)o(h)f(guaran)o(tees)h(the)f(applicabilit)o(y)f(of)i (the)f(KAM)46 620 y(algorithm)k(an)i(in\014nite)f(n)o(um)o(b)q(er)f(of) h(times)f(yielding)g(a)i(solution)g(of)g(the)f(in)o(v)m(arian)o(t)g (torus)h(equa-)46 680 y(tion.)16 b(Suc)o(h)h(criterion)f(is)h(obtained) g(simplifying)d(the)j(estimates)e(and)j(getting)f(a)g(unique)f (stronger)46 740 y(condition)g(\(the)g(\\KAM)g(condition"\))g(ensuring) h(the)f(inde\014nite)f(applicabilit)o(y)f(of)i(the)g(sc)o(heme.)140 800 y(As)21 b(w)o(e)g(already)g(p)q(oin)o(ted)g(out)g(in)g(previous)g (pap)q(ers)h(\(see)f([5])f(and)i(references)e(therein\),)f(in)46 860 y(concrete)d(applications,)g(it)g(is)g(con)o(v)o(enien)o(t)f(to)i (iterate)f(a)h(few)f(times)f(the)i(KAM)e(algorithm)h Fv(b)q(efore)46 921 y Fx(trying)g(to)g(apply)h(the)f(KAM)f(theorem.)140 981 y(Both)g(the)f(computation)g(of)h(the)g(initial)e(appro)o(ximate)g (solution)i(and)g(the)g(application)f(of)h(the)46 1041 y(KAM)g(algorithm)g(are)i(computer{assisted.)140 1101 y(W)l(e)j(remark)e(that)i(it)g(is)f(quite)g(di\013eren)o(t)g(to)h (explicitly)d(compute)h(the)i Fv(initial)e Fx(appro)o(ximate)46 1161 y(solution)g(from)e(the)i(\\computation")g(of)g(the)f(new)h(appro) o(ximate)e(solutions)i(based)g(on)g(the)g(KAM)46 1222 y(algorithm:)h(the)h(calculation)g(of)h(the)f(truncated)g(Lindstedt)h (series)f(is)g(completely)d(explicit)h(\(w)o(e)46 1282 y(compute)g(n)o(um)o(b)q(ers!\))f(while)h(the)h(construction)g(of)h (the)f(sequence)f(of)h(quadratically)g(b)q(etter)f(and)46 1342 y(b)q(etter)j(appro)o(ximate)f(solutions)i(is)f(only)h Fv(implici)o(tly)c(describ)q(ed)j Fx(b)o(y)g(the)g(KAM)g(algorithm)f (and)46 1402 y(what)d(w)o(e)e(actually)h(compute)e(are)i(b)q(ounds)i (on)e(norms)g(relativ)o(e)e(to)i(suc)o(h)g(appro)o(ximate)f(solutions.) 46 1487 y Fw(4\))22 b(\(On)h(the)f(use)h(of)f(computers\))c Fx(Our)i(pro)q(of)h(is)e(\\computer{assisted")h(in)f(the)h(sense)g (that)46 1547 y(certain)c(form)o(ulae)f(\(deriv)o(ed)g(b)q(elo)o(w\))h (ha)o(v)o(e)g(b)q(een)h(implem)o(en)n(ted)d(on)j(a)g(computer)e(\(a)i (V)-5 b(AX\))15 b(k)o(eep-)46 1608 y(ing)24 b(rigorous)g(con)o(trol,)f (b)o(y)g(means)g(of)h(the)f(so{called)h(\\in)o(terv)m(al)f(arithmetic") e(\(see)j(b)q(elo)o(w\),)f(of)46 1668 y(the)d(n)o(umerical)e(errors)i (in)o(tro)q(duced)h(b)o(y)e(the)i(mac)o(hine.)c(W)l(e)k(rep)q(ort)f(in) h Fu(x)f Fx(5.2)h(and)g Fu(x)g Fx(5.3)f(all)g(the)46 1728 y(computer{aided)f(calculations)h(needed)f(to)i(pro)o(v)o(e)e (Theorem)g(1.1.)h(Instead)g(w)o(e)g(do)h(not)f(include)46 1788 y(the)12 b(computer)f(program)h(whic)o(h)g(an)o(yb)q(o)q(dy)h(can) g(write)f(b)o(y)g(her/himself)1377 1770 y FE(7)1394 1788 y Fx(.)g(W)l(e)h(are)f(ob)o(viously)g(a)o(w)o(are)46 1848 y(of)j(the)f(\(ph)o(yloso\014cal?\))g(problem)f(of)i(pro)q (ceeding)g(in)f(suc)o(h)g(a)h(w)o(a)o(y:)f(It)g(is)g(clear)g(that)h (writing)f(a)h(pro-)46 1909 y(gram)g(in)h(a)g(sligh)o(tly)e(di\013eren) o(t)h(w)o(a)o(y)h(or)g(using)g(di\013eren)o(t)f(mac)o(hines)f(migh)o(t) g(\(b)q(etter:)h(will\))g(pro)q(duce)46 1969 y(slighlt)o(y)d (di\013eren)o(t)g(outputs,)i(whic)o(h)e(in)h(our)h(case)f(are)h Fv(in)o(terv)m(als)e(of)h(rational)h(n)o(um)o(b)q(ers)e Fx(\(see)h(b)q(elo)o(w\).)46 2029 y(Ho)o(w)o(ev)o(er)j(w)o(e)h(regard)i (the)f(computer{impl)o(em)o(e)o(n)o(tation,)d Fv(once)j(it)f(is)h (clearly)e(settled)i(the)f(t)o(yp)q(e)h(of)46 2089 y(rigorous)13 b(metho)q(d)e(used)h(to)h(con)o(trol)e(the)h(propagation)i(of)e(n)o (umerical)e(errors)i Fx(\(here)f(\\in)o(terv)m(al)h(arith-)46 2149 y(metic"\))k(as)j(a)f(detail)g(at)g(the)g(same)f(lev)o(el,)f(sa)o (y)l(,)h(of)i(the)f(details)f(needed)h(to)g(w)o(ork)h(out)f (explicitely)46 2210 y(the)13 b(estimates)f(of)h(the)g(KAM)g(algorithm) f(of)i(sections)f(4.3)g(and)h(4.4.)f(Of)h(course,)e(w)o(e)h(shall)g(b)q (e)h(happ)o(y)46 2270 y(to)i(send)h(to)f(the)g(in)o(terested)f(readers) i(the)f(computer)e(programs)j(con)o(tained)f(in)f(this)i(pap)q(er.)140 2330 y(Let)f(us)g(no)o(w)g(brie\015y)f(discuss)h Fv(in)o(terv)m(al)e (arithmetic)f Fx(whic)o(h)i(is)h(the)f(tec)o(hnical)f(to)q(ol)j(w)o(e)e (used)h(to)46 2390 y(con)o(trol)10 b(the)h(n)o(umerical)d(errors)k(in)o (tro)q(duced)e(b)o(y)h(the)f(mac)o(hines.)f(Real)h(n)o(um)o(b)q(ers)g (are)h(represen)o(ted)f(b)o(y)p 46 2434 747 2 v 102 2464 a Fs(7)121 2480 y FI(The)15 b(\(11189{line\))d(computer)j(program)e(is) h(\\just")g(a)h(translation)f(in)g(computer)g(language)g(\(F)o(OR)m (TRAN)f(77\))46 2529 y(of)d(the)i(form)o(ulae)d(of)h Fr(x)i FI(5.1)e(and)g Fr(x)i FI(4.3)e(after)h(the)h(standard)f (arithmetic)f(\(basic)h(op)q(erations\))g(is)g(replaced)h(b)o(y)f(\\in) o(terv)n(al)46 2579 y(arithmetic")h(\(the)j(\\arithmetic)e(routines")h (ma)o(y)e(b)q(e)i(found,)f(e.g.,)g(at)h(pages)g(153{158)e(of)h([4)o (]\).)967 2863 y Fx(5)p eop %%Page: 6 6 6 5 bop 46 78 a Fx(computers)15 b(as)h(sign{exp)q(onen)o(t{fraction)h (quan)o(tities,)d(with)i(the)g(length)g(of)g(the)f(exp)q(onen)o(t)h (and)h(of)46 138 y(the)g(fraction)f(dep)q(ending)i(on)f(the)g(mac)o (hine.)d(An)o(y)i(result)h(among)f(elemen)o(tary)e(op)q(erations)19 b(\(sum,)46 198 y(subtraction,)i(m)o(ultiplication)e(and)j(division\))e (is)i(rounded)g(b)o(y)f(the)g(computer)f(up)i(to)g(a)g(certain)46 259 y(decimal)17 b(digit.)h(T)l(o)i(rigorously)g(implem)o(en)o(t)c(on)k (a)g(computer)e(a)h(certain)g(sequence)f(of)i(form)o(ulae,)46 319 y(one)i(\014rst)g(reduces)f(suc)o(h)h(form)o(ulae)e(to)i(a)g (sequence)e(of)j(elemen)o(t)o(ary)c(op)q(erations)1602 301 y FE(8)1623 319 y Fx(.)i(The)h(idea)g(of)46 379 y(the)c(\\in)o (terv)m(al)f(arithmetic")g(is)h(then)g(to)g(construct)h(an)f(in)o(terv) m(al)g(\(exactly)f(represen)o(table)g(on)h(the)46 439 y(computer\))j(con)o(taining)h(the)g Fv(exact)g Fx(result)g(of)h(an)g (elemen)o(tary)d(op)q(eration)j(and)g(to)g(replace)e(\(in)46 499 y(the)e(ob)o(vious)g(w)o(a)o(y\))g(algebra)g(on)h(n)o(um)o(b)q(ers) d(with)i(algebra)h(on)g(in)o(terv)m(als.)d(In)i(our)h(F)o(OR)l(TRAN)d (77)46 560 y(programs)c(w)o(e)h(de\014ne)f(quadruple)g(precision)g (\(H{\015oating\))h(v)m(ariables,)f(whic)o(h)g(are)g(allo)o(w)o(ed)g (to)h(v)m(ary)46 620 y(in)j(a)h(range)g(b)q(et)o(w)o(een)f(0)p Ft(:)p Fx(84)c Fu(\001)e Fx(10)642 602 y FF(\000)p FE(4392)761 620 y Fx(and)18 b(0)p Ft(:)p Fx(59)13 b Fu(\001)f Fx(10)1030 602 y FE(4392)1103 620 y Fx(.)17 b(The)h(binary)f(structure)g(of)h(a)g (quadruple)46 680 y(precision)f(datum)f(is)i(comp)q(osed)f(b)o(y)g(128) i(bits,)d(with)i(1)g(sign)g(bit,)e(15)j(bits)e(for)h(the)f(exp)q(onen)o (t)g(and)46 740 y(the)g(remaining)f(bits)i(for)f(the)h(fraction.)f(Tw)o (o)h(extra)f(hidden)g Fv(guard)i Fx(bits)e(are)h(used)g(to)g(guaran)o (tee)46 800 y(the)f(result)g(of)h(an)g(elemen)o(tary)d(op)q(eration)j (\\up)h(to)e(1/2)i(of)f(the)f(last)h(signi\014can)o(t)f(bit")h (\([17]\).)f(The)46 860 y(in)o(terv)m(al)c(con)o(taining)h(the)g (result)f(of)i(an)f(elemen)o(tary)e(op)q(eration)j(is)f(therefore)f (obtained)h(increasing)46 921 y(or)24 b(decreasing)g(b)o(y)g(one)h(bit) f(the)g(last)g(bit)g(of)g(the)g(man)o(tissa,)f(taking)i(ev)o(en)o (tually)d(care)i(of)g(the)46 981 y(propagation)g(of)f(the)f(carry)l(.)f (F)l(or)i(further)f(informations)f(and)i(for)g(the)f(necessary)g (routines)g(w)o(e)46 1041 y(refer)15 b(to)i([4])f(and)h([5)o(].)46 1126 y(W)l(e)g(\014nally)g(men)o(tion)f(that)i(in)f(the)g(App)q(endix)g (w)o(e)g(rep)q(ort)h(a)g(few)g(computer{assisted)f(data)h(with)46 1186 y(the)d(follo)o(wing)g(doublefold)g(aim.)e(F)l(rom)h(one)i(side)f (the)g(reader)g(repro)q(ducing)g(our)h(estimates)e(migh)o(t)46 1247 y(c)o(hec)o(k)f(her/his)i(results)f(with)h(ours;)g(from)f(the)g (other)h(side)g(the)f(reader)h(who)g(is)g(not)g(going)h(to)f(w)o(aste) 46 1307 y(time)f(p)q(erforming)h(the)g(computations)h(will)f(ha)o(v)o (e)g(an)h(idea)g(of)g(the)g(t)o(yp)q(e)f(of)i(outputs)f(one)h(needs)e (in)46 1367 y(this)h(pap)q(er.)46 1533 y FB(2)83 b(Restricted,)26 b(circular,)g(planar)h(three{b)r(o)r(dy)g(problem)46 1643 y Fx(Here)10 b(w)o(e)h(recall)f(the)h(Hamiltonian)f(form)o (ulation)g(of)h(the)g(\\restricted,)g(circular,)e(planar)j(three{b)q(o) q(dy)46 1703 y(problem")j(\(for)h(general)g(informations)g(see)g([6])f (or)i([1]\).)46 1788 y(Consider)d(\014rst)h(a)g(Keplerian)e(t)o(w)o (o{b)q(o)q(dy)i(problem)e(made)g(up)i(of)f(t)o(w)o(o)g(material)f(p)q (oin)o(ts)i(\(\\b)q(o)q(dies"\))46 1848 y Ft(P)77 1855 y FE(1)118 1848 y Fx(and)22 b Ft(P)249 1855 y FE(2)290 1848 y Fx(with)f(masses)f Ft(m)614 1855 y FE(1)654 1848 y Fx(and)i Ft(m)797 1855 y FE(2)837 1848 y Fx(and)g(let)e Ft(P)1043 1855 y FE(2)1084 1848 y Fx(rev)o(olv)o(e)f(on)j(a)f(circular) f(orbit)h(around)h Ft(P)1878 1855 y FE(1)1898 1848 y Fx(.)46 1908 y(Consider)f(no)o(w)h(a)f(third)g(b)q(o)q(dy)i Ft(A)d Fx(mo)o(ving)g(on)i(the)f(orbital)g(plane)g(of)g Ft(P)1445 1915 y FE(1)1487 1908 y Fx(and)g Ft(P)1617 1915 y FE(2)1659 1908 y Fx(and)h(sub)s(ject)46 1969 y(to)f(the)g(gra)o (vitational)f(attraction)i(of)f Ft(P)814 1976 y FE(1)855 1969 y Fx(and)g Ft(P)985 1976 y FE(2)1005 1969 y Fx(.)g(Let)g(the)f (mass)h Ft(m)1388 1976 y Fk(A)1437 1969 y Fx(of)g Ft(A)f Fx(b)q(e)h(m)o(uc)o(h)e(smaller)46 2029 y(than)e Ft(m)203 2036 y FE(1)223 2029 y Fx(,)f Ft(m)296 2036 y FE(2)332 2029 y Fx(and)h(assume)g(that)g(the)f(motion)g(of)h Ft(P)1040 2036 y FE(1)1077 2029 y Fx(and)g Ft(P)1203 2036 y FE(2)1240 2029 y Fx(is)g(not)g(a\013ected)g(b)o(y)f Ft(A)p Fx(.)g(The)h(study)46 2089 y(of)j(the)f(dynamics)f(asso)q(ciated)i(to)g(suc)o(h)f(a)h(mo)q (del)e(is)h(kno)o(wn)h(in)f(the)g(literature)f(as)i Fv(the)g(circular,) 46 2149 y(planar,)j(restricted)e(three{b)q(o)q(dy)j(problem)p Fx(.)d(In)h(particular,)h(w)o(e)f(shall)h(b)q(e)g(in)o(terested)e(in)i (phase)46 2209 y(space)15 b(regions)h(for)g(whic)o(h)e(the)h(resulting) g(motion)f(of)i Ft(A)f Fx(is)g(a)h(nearly)f(circular)f(orbit)h (\\around")i Ft(P)1878 2216 y FE(1)1898 2209 y Fx(.)p 46 2253 747 2 v 102 2284 a Fs(8)121 2299 y FI(Elemen)o(tary)e (functions)h(\(suc)o(h)g(as)g(ro)q(ots,)g(exp)q(onen)o(tials,)g (trigonometric)e(functions,)i(etc.\),)g(will)e(b)q(e)j(appro)o(xi-)46 2349 y(mated)d(b)o(y)h(a)g(\014nite)g(sequence)j(of)c(elemen)o(tary)h (op)q(erations)g(using)g(T)m(a)o(ylor)f(p)q(olynomial)o(s)f(k)o(eeping) i(trac)o(k)h(of)e(errors;)46 2398 y(see)h([5)o(])f Fr(x)p FI(8.3.)967 2863 y Fx(6)p eop %%Page: 7 7 7 6 bop 46 78 a Fx(A)16 b(con)o(v)o(enien)o(t)e(Hamiltonian)h(form)o (ulation)f(of)j(suc)o(h)f(a)h(three{b)q(o)q(dy)g(problem)e(is)h(based)h (up)q(on)g(the)46 138 y(classical)11 b(\\planar)i(Delauna)o(y)e(v)m (ariables")h([6].)f(Let)h Fq(T)s Fu(\021)6 b Fq(R)p Ft(=)p Fx(\(2)p Ft(\031)r Fq(Z)-14 b Fx(\))9 b(and)j(consider)g(the)f(phase)h (space)1880 120 y FE(9)367 248 y Fu(P)17 b Fx(=)d Fu(f)p Fx(\()p Ft(\025;)8 b(\015)s(;)g( )r Fx(\))14 b Fu(2)g Fq(T)765 228 y FE(3)782 248 y Fu(g)d(\002)f(f)p Fx(\(\003)p Ft(;)e Fx(\000)p Ft(;)g(E)s Fx(\))14 b Fu(2)g Fq(R)1177 228 y FE(3)1208 248 y Fx(:)f(\003)h Fu(6)p Fx(=)g(0)p Ft(;)8 b Fu(j)p Fx(\000)p Fu(j)14 b Ft(<)g Fu(j)p Fx(\003)p Fu(jg)46 358 y Fx(endo)o(w)o(ed)19 b(with)g(the)g(standard)h (symplectic)c(form)j Ft(d\025)13 b Fu(^)h Ft(d)p Fx(\003)f(+)g Ft(d\015)k Fu(^)c Ft(d)p Fx(\000)h(+)f Ft(d )i Fu(^)e Ft(dE)s Fx(.)19 b(Then,)g(the)46 418 y(dynamics)14 b(asso)q(ciated)i (with)f(the)h(circular,)d(planar)j(restricted)f(three{b)q(o)q(dy)h (problem)e(is)h(giv)o(en)g(b)o(y)46 479 y(the)h(Hamiltonian)e(\015o)o (w)j(generated)f(b)o(y)g(the)g(Hamiltonian)1152 460 y FE(10)351 608 y Ft(H)391 615 y FE(0)411 608 y Fx(\()p Ft(\025;)8 b(\015)s(;)g( )r(;)g Fx(\003)p Ft(;)g Fx(\000)p Ft(;)g(E)s Fx(\))30 b Fu(\021)882 574 y Fx(1)p 855 596 78 2 v 855 642 a(2\003)913 628 y FE(2)966 608 y Fx(+)d Ft(E)j Fx(+)e Ft(")16 b(R)1239 615 y FE(0)1259 608 y Fx(\()p Ft(\025;)8 b(\015)14 b Fu(\000)d Ft( )r(;)d Fx(\003)p Ft(;)g Fx(\000\))15 b Ft(;)205 b Fx(\(2.1\))46 734 y(where)14 b Ft(")8 b Fu(\021)h Ft(m)307 741 y FE(2)326 734 y Ft(=m)393 741 y FE(1)428 734 y Fx(and)15 b(the)g(\\p)q(erturbation)h(function")f Ft(R)1164 741 y FE(0)1198 734 y Fx(is)g(giv)o(en)f(as)i(follo)o(ws.)e (Let)h Ft(\027)i Fu(2)d Fq(T)7 b Fx(\(the)46 794 y(\\eccen)o(tric)14 b(anomaly"\))h(b)q(e)h(implicitl)o(y)d(de\014ned)i(for)h Fu(j)p Ft(e)p Fu(j)d Ft(<)h Fx(1)i(\()p Ft(e)e Fu(2)g Fq(R)p Fx(\))f(b)o(y)i(the)h(relation)f(\(\\Kepler's)46 854 y(equation"\))777 914 y Ft(\025)31 b Fx(=)f Ft(\027)g Fu(\000)d Ft(e)8 b Fx(sin)h Ft(\027)19 b Fx(;)46 1001 y(let)c Ft(')f Fu(2)g Fq(T)9 b Fx(\(the)16 b(\\true)g(anomaly"\))g(b)q (e)g(implicitl)o(y)d(de\014ned)j(\(again)h(for)g Fu(j)p Ft(e)p Fu(j)c Ft(<)h Fx(1\))i(b)o(y)g(the)g(relation)659 1133 y(tg)717 1100 y Ft(')11 b Fu(\000)f Ft(\015)p 717 1122 121 2 v 765 1168 a Fx(2)873 1133 y(=)941 1085 y Fp(\020)971 1100 y Fx(1)h(+)g Ft(e)p 970 1122 108 2 v 970 1168 a Fx(1)h Fu(\000)f Ft(e)1083 1085 y Fp(\021)1108 1097 y FE(1)p Fk(=)p FE(2)1188 1133 y Fx(tg)1245 1100 y Ft(\027)p 1245 1122 28 2 v 1246 1168 a Fx(2)1293 1133 y Ft(;)46 1260 y Fx(and)17 b(de\014ne)f(the)g(\\orbital)g(radius")h Ft(r)h Fx(as)528 1387 y Ft(r)10 b Fu(\021)686 1354 y Ft(a)p Fx(\(1)h Fu(\000)g Ft(e)839 1336 y FE(2)858 1354 y Fx(\))p 612 1376 340 2 v 612 1422 a(1)g(+)g Ft(e)d Fx(cos)q(\()p Ft(')j Fu(\000)f Ft(\015)s Fx(\))973 1387 y Ft(;)105 b Fx(where)48 b Ft(a)8 b Fu(\021)g Fx(\003)1380 1367 y FE(2)1416 1387 y Ft(:)46 1525 y Fx(The)16 b(function)g Ft(R)374 1532 y FE(0)410 1525 y Fx(in)g(\(2.1\))h(is)f(then)g(giv)o(en) f(b)o(y)270 1658 y Ft(R)307 1665 y FE(0)326 1658 y Fx(\()p Ft(\025;)8 b(\015)15 b Fu(\000)c Ft( )r(;)d Fx(\003)p Ft(;)g Fx(\000\))g Fu(\021)19 b(\000)761 1609 y Fp(\020)786 1658 y Ft(r)10 b Fx(cos\()p Ft(')h Fu(\000)g Ft( )r Fx(\))f Fu(\000)1359 1624 y Fx(1)p 1113 1646 516 2 v 1113 1654 a Fp(q)p 1155 1654 474 2 v 50 x Fx(1)h(+)g Ft(r)1262 1690 y FE(2)1293 1704 y Fu(\000)g Fx(2)p Ft(r)f Fx(cos\()p Ft(')h Fu(\000)g Ft( )r Fx(\))1634 1609 y Fp(\021)1675 1658 y Ft(;)46 1816 y Fx(where)16 b Ft(e)g Fx(is)g(de\014ned)g(as)830 1906 y Ft(e)8 b Fu(\021)908 1823 y Fp(s)p 949 1823 149 2 v 949 1906 a Fx(1)k Fu(\000)1041 1872 y Fx(\000)1071 1858 y FE(2)p 1040 1894 54 2 v 1040 1940 a Fx(\003)1074 1926 y FE(2)1114 1906 y Ft(:)46 2012 y Fx(W)l(e)24 b(recall)f(that)h(a) h(con)o(v)o(enien)o(t)d(represen)o(tation)h(of)i Ft(R)1114 2019 y FE(0)1158 2012 y Fx(is)e(obtained)i(b)o(y)e(means)h(of)g (Legendre)46 2072 y(p)q(olynomials)302 2054 y FE(11)338 2072 y Fx(:)16 b(if)g Ft(r)f(<)f Fx(1)i(\(whic)o(h)g(will)f(b)q(e)h (the)g(case)h(for)f(our)h(sp)q(eci\014c)e(mo)q(del\))g(one)i(\014nds) 646 2211 y Ft(R)683 2218 y FE(0)717 2211 y Fx(=)d(1)d(+)866 2157 y FF(1)854 2170 y Fp(X)853 2261 y Fk(j)r FE(=2)923 2211 y Ft(r)946 2190 y Fk(j)964 2211 y Ft(P)995 2218 y Fk(j)1014 2211 y Fx(\(cos\()p Ft(')g Fu(\000)g Ft( )r Fx(\)\))16 b Ft(:)p 46 2305 747 2 v 102 2336 a Fs(9)121 2351 y FI(The)11 b(Delauna)o(y)g(co)q(ordinates)h Fz(\025;)7 b(\015)r(;)g( )13 b FI(are)e(often)h(called,)f(resp)q(ectiv)o(ely)m(,)g (the)h(\\mean)e(anomaly",)e(the)k(\\argumen)o(t)46 2401 y(of)h(the)i(p)q(erihelion")e(and)h(the)g(\\longitude")f(\(of)g(the)i (\\planet")e Fz(P)1084 2407 y Fs(2)1103 2401 y FI(\).)85 2436 y Fs(10)121 2451 y FI(W)m(e)g(ha)o(v)o(e)g(c)o(hosen)h(the)h (units)e(of)g(measure)g(in)h(suc)o(h)g(a)f(w)o(a)o(y)g(that)g Fz(m)1187 2457 y Fs(1)1215 2451 y FI(+)c Fz(m)1292 2457 y Fi(A)1331 2451 y FI(=)i(1)j(and)f(that)g(the)i(p)q(erio)q(d)e(of)g Fz(P)1852 2457 y Fs(2)1884 2451 y FI(is)46 2500 y(2)p Fz(\031)q FI(.)85 2543 y Fs(11)121 2558 y Fz(P)148 2564 y Fs(0)166 2558 y FI(\()p Fz(x)p FI(\))e(=)h(1;)h Fz(P)350 2564 y Fs(1)369 2558 y FI(\()p Fz(x)p FI(\))e(=)h Fz(x)p FI(;)h Fz(P)556 2564 y Fi(k)q Fs(+1)618 2558 y FI(\()p Fz(x)p FI(\))f(=)735 2537 y Fs(\(2)p Fi(k)q Fs(+1\))p Fi(P)860 2541 y Fh(k)877 2537 y Fs(\()p Fi(x)p Fs(\))p Fi(x)p Fy(\000)p Fi(k)q(P)1007 2541 y Fh(k)p Fg(\000)p Ff(1)1060 2537 y Fs(\()p Fi(x)p Fs(\))p 735 2548 371 2 v 890 2572 a Fi(k)q Fs(+1)1110 2558 y FI(,)i(\(for)f Fz(k)g Fr(\025)e FI(1\).)967 2863 y Fx(7)p eop %%Page: 8 8 8 7 bop 46 78 a Fx(A)17 b(trivial)f(reduction)i(sho)o(ws)g(that)g(the)g (dynamics)e(generated)h(b)o(y)h(\(2.1\))f(ma)o(y)g(b)q(e)g(describ)q (ed)g(b)o(y)h(a)46 138 y(t)o(w)o(o{degree{of{freedom)c(Hamiltonian:)f (under)i(the)g(canonical)g(\(or)g(\\symplectic"\))e(transforma-)46 198 y(tion)388 259 y(\()p Ft(`;)8 b(g)r(;)g(\034)e Fx(\))14 b Fu(\021)f Fx(\()p Ft(\025;)8 b(\015)14 b Fu(\000)d Ft( )r(;)d( )r Fx(\))15 b Ft(;)106 b Fx(\()p Ft(L;)8 b(G;)g(T)f Fx(\))13 b Fu(\021)h Fx(\(\003)p Ft(;)8 b Fx(\000)p Ft(;)g Fx(\000)j(+)g Ft(E)s Fx(\))17 b Ft(;)46 344 y(H)86 351 y FE(0)122 344 y Fx(tak)o(es)f(the)g(form)460 467 y Ft(H)500 474 y FE(1)520 467 y Fx(\()p Ft(`;)8 b(g)r(;)g(L;)g(G)p Fx(\))31 b(=)871 434 y(1)p 844 456 78 2 v 844 502 a(2)p Ft(L)901 487 y FE(2)954 467 y Fu(\000)c Ft(G)h Fx(+)f Ft(")16 b(R)1227 474 y FE(0)1247 467 y Fx(\()p Ft(`;)8 b(g)r(;)g(L;)g(G)p Fx(\))17 b Ft(;)314 b Fx(\(2.2\))46 591 y(ha)o(ving)13 b(omitted)f(the)h(dumm)o(y)d(v)m(ariable)i Ft(T)7 b Fx(;)13 b(the)g(phase)g(space)h(is)f(no)o(w)g Fq(T)1396 573 y FE(2)1418 591 y Fu(\002)5 b(f)p Fx(\()p Ft(L;)j(G)p Fx(\))14 b Fu(2)g Fq(R)1718 573 y FE(2)1748 591 y Fx(:)f Ft(L)h Fu(6)p Fx(=)g(0,)46 651 y Fu(j)p Ft(G)p Fu(j)g Ft(<)g Fu(j)p Ft(L)p Fu(jg)p Fx(.)46 817 y FB(3)83 b(A)27 b(mo)r(del)f(from)f(the)j(Solar)f(system)46 927 y Fx(Let)17 b(us)h(no)o(w)f(fo)q(cus)h(on)g(the)f(case)g(in)g(whic) o(h)g Ft(P)914 934 y FE(1)951 927 y Fx(is)g(the)g(Sun,)g Ft(P)1229 934 y FE(2)1267 927 y Fx(is)g(Jupiter)f(and)i Ft(A)f Fx(is)g(Ceres)g(\(one)46 987 y(of)f(the)g(largest)h(b)q(o)q (dies)g(in)f(the)g(asteroid)g(b)q(elt\).)140 1047 y(As)j(the)f(reader)h (has)g(certainly)e(noticed,)h(the)g(Hamiltonian)f(in)i(\(1.3\))f(is)h (di\013eren)o(t)f(from)f(the)46 1107 y(Hamiltonian)c(in)h(\(2.2\):)g(b) q(esides)h(the)f(factor)h(\()938 1088 y FE(1)p 918 1096 59 2 v 918 1125 a(2)p Fk(L)960 1115 y Fl(2)989 1107 y Fu(\000)8 b Ft(G)p Fx(\))15 b(\(and)g(2)p Ft(")g Fx(in)f(place)g(of)h Ft(")p Fx(\),)e(the)i(main)e(di\013er-)46 1168 y(ence)i(is)h(that)g Ft(R)h Fx(is)f(a)g(trigonometric)e(p)q(olynomial)h(of)h(degree)g(10)g (while)g Ft(R)1442 1175 y FE(0)1477 1168 y Fx(con)o(tains)g(in\014nite) f(non)46 1228 y(v)m(anishing)d(F)l(ourier)g(harmonics.)f(The)h (\\selection)f(rule")h(whic)o(h)g(led)f(us)i(to)f(the)g(c)o(hoice)f(of) i(the)f(\\ph)o(ys-)46 1288 y(ically)e(relev)m(an)o(t")i(F)l(ourier)f (mo)q(des)h(is)g(based)g(on)h(the)f(follo)o(wing)f(trivial)g(observ)m (ation.)i(In)e(considering)46 1348 y(Ceres{Jupiter{Sun)20 b(as)g(a)g(mathematical)d(three{b)q(o)q(dy)k(problem,)c(w)o(e)j(are)f (neglecting)g(\(b)q(esides)46 1408 y(minor)14 b(forces)i(suc)o(h)f(as)i (dissipativ)o(e)d(ones\))j(the)e(gra)o(vitational)h(e\013ects)f(on)h (Ceres)g(of)g(asteroids)g(and)46 1469 y(planets)f(and)h(most)e(notably) i(the)f(attraction)g(excerted)f(b)o(y)h(Saturn)h(\(whic)o(h,)e(after)h (Jupiter,)f(is)h(the)46 1529 y(largest)g(planet)g(in)f(the)h(Solar)g (system)758 1511 y FE(12)794 1529 y Fx(\).)f Fv(Therefore,)g(after)h (ha)o(ving)g(de\014ned)g(a)g(\(rough\))h(measure,)46 1589 y Fu(G)76 1596 y FE(Sa)115 1589 y Fv(,)21 b(of)h(the)f (Ceres{Saturn)i(attraction,)e(w)o(e)g(disregard)h(in)g(the)f(F)l (ourier)g(expansion)h(of)g Ft(R)1803 1596 y FE(0)1844 1589 y Fv(the)46 1649 y(terms)15 b(exceeding)f Fu(G)432 1656 y FE(Sa)488 1649 y Fv(in)i(absolute)g(v)m(alue)p Fx(.)140 1709 y(In)f(order)f(to)h(de\014ne)f Fu(G)552 1716 y FE(Sa)605 1709 y Fx(w)o(e)g(\014rst)h(lo)q(ok)g(up)g(a)g(few)f (astronomical)g(data)h(in)f(the)g(ephemeris)e(\(see)46 1769 y(The)k(Astronomical)e(Almanac)g([16]\).)h(In)g(particular)g(w)o (e)h(w)o(an)o(t)f(to)h(de\014ne)g(the)f(\\reference)g(v)m(alues")46 1830 y(of)23 b Ft(L)141 1837 y FE(0)183 1830 y Fx(and)h Ft(G)323 1837 y FE(0)366 1830 y Fx(for)f(the)f(Sun{Jupiter{Ceres)h (system.)e(Observ)m(ations)i(of)g(the)g(true)f(motion)g(of)46 1890 y(Ceres,)14 b(as)i(found)f(in)g([16],)f(indicate)g(that)i(Ceres)e (mo)o(v)o(es)f(on)j(a)f(nearly)f(elliptical)f(orbit)i(of)g(\\a)o(v)o (erage)46 1950 y(eccen)o(tricit)o(y)o(")862 2010 y Ft(e)885 2017 y FE(0)913 2010 y Fu(\021)8 b Fx(0)p Ft(:)p Fx(0766)718 b(\(3.1\))46 2096 y(and)23 b(whose)h(a)o(v)o(erage)e(semima)s(jor)e (axis)j(is)f(appro)o(ximately)f(0)p Ft(:)p Fx(532.)i(Hence,)e(the)i (corresp)q(onding)46 2156 y(\\a)o(v)o(erage)16 b(frequency")f(of)i (Ceres,)e(computed)g(b)o(y)h(Kepler's)f(third)h(la)o(w,)g(yields)f(a)h (v)m(alue)g(of)768 2262 y Fu(\000)p Fx(\012)842 2269 y Fk(c)868 2262 y Fu(\021)j(\000)11 b Fx(2)p Ft(:)p Fx(577107)18 b Ft(:)46 2368 y Fx(Since)d Fu(\000)p Fx(\012)247 2375 y Fk(c)278 2368 y Fu(')f Ft(@)357 2375 y Fk(L)383 2368 y Ft(H)423 2375 y FE(1)443 2368 y Fu(j)457 2375 y Fk(")p FE(=0)534 2368 y Fx(=)g Fu(\000)p Ft(L)658 2350 y FF(\000)p FE(3)721 2368 y Fx(w)o(e)i(tak)o(e)g(as)h(\\reference)e Ft(L)p Fx({v)m(alue")h(the)g(quan)o(tit)o(y)768 2491 y Ft(L)801 2498 y FE(0)829 2491 y Fu(\021)8 b Fx(0)p Ft(:)p Fx(729305)17 b Fu(')c Fx(\012)1167 2447 y Fl(1)p 1167 2453 16 2 v 1167 2474 a(3)1162 2494 y Fk(c)p 46 2531 747 2 v 85 2562 a Fs(12)121 2577 y FI(Ev)o(en)19 b(though)h(the)g(orbit)f(of)g(Ceres)i(is)e(closer)h(to)f(the)i(orbit)e (of)f(Mars)i(than)g(to)f(the)h(orbit)f(of)g(Saturn,)g(the)46 2627 y(di\013erence)d(in)d(mass)g(mak)o(es)g(the)h(gra)o(vitational)e (attraction)i(of)f(Saturn)h(the)h(largest)f(one)g(after)g(that)g(of)f (Jupiter.)967 2863 y Fx(8)p eop %%Page: 9 9 9 8 bop 46 83 a Fx(and,)16 b(since)g Ft(G)312 90 y FE(0)346 83 y Fx(=)e Ft(L)431 90 y FE(0)450 31 y Fp(q)p 492 31 128 2 v 52 x Fx(1)d Fu(\000)g Ft(e)600 66 y FE(2)600 94 y(0)620 83 y Fx(,)k(w)o(e)h(tak)o(e,)f(as)i(\\reference)e Ft(G)p Fx({v)m(alue")j(the)e(quan)o(tit)o(y)815 183 y Ft(G)853 190 y FE(0)882 183 y Fu(\021)8 b Fx(0)p Ft(:)p Fx(727162)18 b Ft(:)46 283 y Fx(Suc)o(h)13 b(reference)f(v)m(alues)i (ha)o(v)o(e)f(b)q(een)h(tak)o(en)f(as)h(cen)o(ter)f(of)h(the)f Fv(analyticit)o(y)f(domain)h(for)h(the)g(\\action)46 343 y(v)m(ariables")p Fx(;)k(see)h(\(1.1\))g(and)g(\(1.2\).)f(Notice)g (that)h(with)f(our)i(c)o(hoice)d(of)i(the)f(analyticit)o(y)f(radius)i Ft(r)1892 350 y FE(0)46 403 y Fx(one)d(\014nds)h(that)g(the)f(function) g Ft(e)p Fx(\()p Ft(L;)8 b(G)p Fx(\))16 b(satis\014es)418 503 y(0)p Ft(:)p Fx(019799)g Ft(<)d Fu(j)p Ft(e)p Fx(\()p Ft(L;)8 b(G)p Fx(\))p Fu(j)14 b Ft(<)g Fx(0)p Ft(:)p Fx(106364)k Ft(;)106 b Fu(8)15 b Fx(\()p Ft(L;)8 b(G)p Fx(\))14 b Fu(2)g Ft(B)19 b(:)46 603 y Fx(Let)13 b(us)h(turn)f(no)o(w)h (to)f(the)g(de\014nition)g(of)h Fu(G)828 610 y FE(Sa)867 603 y Fx(.)e(In)h(general,)g(for)g(planets)h(whose)g(orbits)f(ha)o(v)o (e)f(a)i(larger)46 663 y(semima)s(jor)f(axis)j(than)g(that)g(of)h (Ceres,)e(the)g(\\secular)h(term")f(of)h Ft(H)1308 670 y FE(1)1344 663 y Fx(is)g(giv)o(en)f(b)o(y)1572 645 y FE(13)1624 663 y Ft(")h Fx(\()8 b Fu(\021)24 b Fx(mass)16 b(of)46 723 y(the)j(planet/mass)g(of)h(the)f(Sun\))g(times)f(the)h (term)e Ft(R)1055 730 y FE(00)1101 723 y Fu(\021)8 b Ft(R)1185 730 y FE(00)1223 723 y Fx(\()p Ft(L)p Fx(;)g Ft(e)p Fx(\))19 b(in)g(\(1.5\).)g(Keeping)g(in)g(mind)46 783 y(that,)f(in)g(the)f(in)o(tegrable)g(limit,)e Ft(L)j Fx(is)g(the)g(ratio)g(of)g(the)g(semima)s(jor)d(axis)j(of)h(Ceres)e (with)h(that)h(of)46 843 y(the)c(planet)h(w)o(e)f(de\014ne)g(\(for)h (planets)f(whose)h(orbits)g(ha)o(v)o(e)f(a)h(larger)f(semima)s(jor)e (axis)j(than)g(that)g(of)46 904 y(Ceres\))701 964 y Fu(G)731 971 y FE(P)765 964 y Fu(\021)8 b Ft(")p Fx(\()p Ft(P)f Fx(\))k Fu(\002)f Ft(R)1008 971 y FE(00)1046 964 y Fx(\()p Ft(L)p Fx(\()p Ft(P)d Fx(\);)h Ft(e)1219 971 y FE(0)1238 964 y Fx(\))555 b(\(3.2\))46 1047 y(where)16 b Ft(e)210 1054 y FE(0)246 1047 y Fx(is)h(the)g(observ)o(ed)f(\\a)o(v)o(erage)h (eccen)o(tricit)n(y")d(of)j(Ceres)g(\(3.1\),)f Ft(")p Fx(\()p Ft(P)7 b Fx(\))17 b(is)g(the)f(mass)h(ratio)g(of)46 1107 y(the)g(planet)g Ft(P)24 b Fx(and)18 b(of)g(the)e(Sun)i(and)g Ft(L)p Fx(\()p Ft(P)7 b Fx(\))17 b(is)g(the)g(ratio)h(of)f(the)g (semima)s(jor)e(axis)i(of)g(Ceres)g(with)46 1167 y(that)g(of)f(the)g (planet)g Ft(P)7 b Fx(.)16 b(Lo)q(oking)i(up)e(the)g(\\true")h(v)m (alues)f(in)g([16])g(one)h(\014nds)668 1267 y Fu(G)698 1274 y FE(Sa)745 1267 y Fu(\021)8 b(G)822 1274 y FE(Saturn)941 1267 y Fx(=)14 b(6)p Ft(:)p Fx(3778)f Fu(\001)e Fx(10)1213 1246 y FF(\000)p FE(6)1277 1267 y Ft(:)46 1367 y Fx(F)l(or)16 b(comparison)g(purp)q(oses)h(w)o(e)f(rep)q(ort)h(also)g(the)f(v)m(alue) g(for)g(Jupiter,)f(whic)o(h)h(is)727 1466 y Fu(G)757 1473 y FE(Jupiter)882 1466 y Fx(=)e(7)p Ft(:)p Fx(8850)f Fu(\001)e Fx(10)1154 1446 y FF(\000)p FE(5)1218 1466 y Ft(:)46 1566 y Fx(Neglecting)16 b(in)h(the)g(expansion)h(of)g Ft("R)775 1573 y FE(0)812 1566 y Fx(those)g(terms)e(whose)i(size)e(is)i (smaller)d(than)1633 1548 y FE(14)1689 1566 y Fu(G)1719 1573 y FE(Sa)1758 1566 y Fx(,)i(one)g(is)46 1626 y(led)e(to)i(consider) f(a)h(\\three{b)q(o)q(dy)g(problem")e(go)o(v)o(erned)h(b)o(y)g(the)g (Hamiltonian)462 1746 y Ft(H)502 1753 y FE(2)522 1746 y Fx(\()p Ft(`;)8 b(g)r(;)g(L;)g(G)p Fx(;)g Ft(")p Fx(\))31 b Fu(\021)918 1712 y Fx(1)p 892 1734 78 2 v 892 1780 a(2)p Ft(L)949 1765 y FE(2)1001 1746 y Fu(\000)c Ft(G)h Fx(+)f Ft(")16 b(R)p Fx(\()p Ft(`;)8 b(g)r(;)g(L;)g(G)p Fx(\))318 b(\(3.3\))46 1861 y(with)16 b Ft(R)h Fx(giv)o(en)e(in)h (\(1.4\))h(and)f(\(1.5\))h(of)f Fu(x)h Fx(1.)46 1943 y(The)d(\014nal)h(mo)q(di\014cation)f(of)h Ft(H)624 1950 y FE(2)659 1943 y Fx(whic)o(h)e(giv)o(es)h(the)h(Hamiltonian)d(in)j (\(1.3\),)f(\(1.4\))h(is)f(due)h(to)g(merely)46 2003 y(tec)o(hnical)f(reasons.)140 2064 y(As)j(men)o(tioned)d(ab)q(o)o(v)o (e,)i(our)h(results)f(are)h(based)g(on)g(computer{assisted)f(KAM)g (theory)l(,)f(and)46 2124 y(one)23 b(of)g(the)f(standard)i(h)o(yp)q (otheses)f(of)g(KAM)e(theorems)h(is)g(that)h(the)g(unp)q(erturb)q(ed)g (Hamilto-)46 2184 y(nian)c(\()p Ft(")f Fx(=)g(0\))h(is)g (non{degenerate,)g Fj(i.e.)h Fx(has)f(an)g(in)o(v)o(ertible)d(Hessian)j (matrix)e(on)i(its)g(domain)f(of)46 2244 y(analyticit)o(y)l(.)c(In)i (the)g(case)g(of)h(\(3.3\),)f(the)g(unp)q(erturb)q(ed)g(Hamiltonian)f (is)h(giv)o(en)f(b)o(y)716 2366 y Ft(h)744 2373 y FE(0)764 2366 y Fx(\()p Ft(L;)8 b(G)p Fx(\))30 b(=)1025 2332 y(1)p 998 2355 V 998 2400 a(2)p Ft(L)1055 2386 y FE(2)1108 2366 y Fu(\000)d Ft(G)17 b(;)p 46 2431 747 2 v 85 2462 a Fs(13)121 2477 y FI(The)f(\\secular)g(term")e(of)h Fz(H)574 2483 y Fs(1)608 2477 y FI(is)g(the)i(a)o(v)o(erage)e(o)o(v)o (er)h(the)g(angular)e(v)n(ariables)h Fz(`)h FI(and)f Fz(\015)k FI(of)c(the)h(\\p)q(erturbation")46 2527 y Fz("R)97 2533 y Fs(0)115 2527 y FI(;)g(the)i(computation)d(is)h (immediately)d(c)o(hec)o(k)o(ed)18 b(using,)e(e.g.,)f(the)j(ab)q(o)o(v) o(e)e(men)o(tioned)f(expansion)i(in)f(terms)h(of)46 2577 y(Legendre)e(P)o(olynomials.)85 2612 y Fs(14)121 2627 y FI(More)f(precisely)h(w)o(e)f(omit)e(all)g(the)j(terms)f(suc)o(h)g (that)g Fz(")p Fr(j)p Fz(R)1047 2633 y Fi(n)1069 2627 y FI(\()p Fz(L)1113 2633 y Fs(0)1132 2627 y Fz(;)7 b(G)1184 2633 y Fs(0)1202 2627 y FI(\))p Fr(j)k Fz(<)h Fr(G)1310 2633 y Fs(Sa)1347 2627 y FI(.)967 2863 y Fx(9)p eop %%Page: 10 10 10 9 bop 46 78 a Fx(whose)14 b(Hessian)g(matrix)e(is)h Fv(not)h Fx(in)o(v)o(ertible.)d(There)i(are)h(a)g(few)f(w)o(ell)f(kno)o (wn)i(metho)q(ds)g(to)g(o)o(v)o(ercome)46 138 y(this)20 b(minor)f(problem)465 120 y FE(15)521 138 y Fx(and)i(it)f(turns)h(out)g (that)g(for)f(our)h(purp)q(oses)h(the)e(most)g(con)o(v)o(enien)o(t)e (one)46 198 y(is)f(to)h(follo)o(w)g(P)o(oincar)o(\023)-23 b(e's)17 b(tric)o(k)f([13],)h(whic)o(h)g(consists)h(in)f(replacing)g (the)h(Hamiltonian)d Ft(H)1753 205 y FE(2)1791 198 y Fx(b)o(y)i(its)46 259 y(square)183 240 y FE(16)220 259 y Fx(.)f(Therefore)g(w)o(e)g(let)f Ft(H)653 266 y FE(3)682 259 y Fu(\021)8 b Fx(\()p Ft(H)788 266 y FE(2)808 259 y Fx(\))827 240 y FE(2)847 259 y Fx(:)298 391 y Ft(H)338 398 y FE(3)358 391 y Fx(\()p Ft(`;)g(g)r(;)g(L;)g(G)p Fx(;)g Ft(")p Fx(\))42 b(=)58 b(\()811 357 y(1)p 785 379 78 2 v 785 425 a(2)p Ft(L)842 411 y FE(2)878 391 y Fu(\000)11 b Ft(G)p Fx(\))985 370 y FE(2)1032 391 y Fx(+)27 b(2)p Ft(")17 b Fx(\()1211 357 y(1)p 1185 379 V 1185 425 a(2)p Ft(L)1242 411 y FE(2)1278 391 y Fu(\000)11 b Ft(G)p Fx(\))17 b Ft(R)p Fx(\()p Ft(`;)8 b(g)r(;)g(L;)g(G)p Fx(\))744 484 y(+)17 b Ft(")822 463 y FE(2)857 484 y Fx([)p Ft(R)p Fx(\()p Ft(`;)8 b(g)r(;)g(L;)g(G)p Fx(\)])1142 463 y FE(2)1179 484 y Ft(:)619 b Fx(\(3.4\))46 594 y(The)16 b(Hessian)g(of)h(the)f(unp)q(erturb)q(ed)g(Hamiltonian)f(\()p Ft(H)1085 601 y FE(3)1105 594 y Fu(j)1119 601 y Fk(")p FE(=0)1182 594 y Fx(\))i(is)f(equal)f(to)645 727 y Ft(A)8 b Fu(\021)g Ft(A)p Fx(\()p Ft(L;)g(G)p Fx(\))g Fu(\021)968 667 y Fp(\022)1024 678 y FE(5)p 1012 686 42 2 v 1012 714 a Fk(L)1036 705 y Fl(6)1069 697 y Fu(\000)1124 678 y FE(6)p Fk(G)p 1124 686 46 2 v 1126 714 a(L)1150 705 y Fl(4)1240 678 y FE(2)p 1228 686 42 2 v 1228 714 a Fk(L)1252 705 y Fl(3)1082 738 y FE(2)p 1070 746 V 1070 775 a Fk(L)1094 765 y Fl(3)1237 757 y Fx(2)1283 667 y Fp(\023)1812 727 y Fx(\(3.5\))46 861 y(and,)16 b(if)g(\()p Ft(L;)8 b(G)p Fx(\))14 b Fu(2)g Ft(B)s Fx(,)i(one)g(has)315 993 y Fu(j)8 b Fx(det)g Ft(A)p Fu(j)41 b Fx(=)585 944 y Fp(\014)585 969 y(\014)585 993 y(\014)606 960 y Fx(12)p 604 982 53 2 v 604 1028 a Ft(L)637 1013 y FE(4)678 993 y Fx(\()729 960 y(1)p 702 982 78 2 v 702 1028 a(2)p Ft(L)759 1013 y FE(2)795 993 y Fu(\000)11 b Ft(G)p Fx(\))902 944 y Fp(\014)902 969 y(\014)902 993 y(\014)505 1110 y Fu(\025)672 1076 y Fx(12)p 590 1098 213 2 v 590 1144 a(\()p Ft(L)642 1151 y FE(0)673 1144 y Fx(+)g Ft(r)744 1151 y FE(0)764 1144 y Fx(\))783 1130 y FE(4)807 1062 y Fp(\020)943 1076 y Fx(1)p 837 1098 237 2 v 837 1144 a(2\()p Ft(L)913 1151 y FE(0)945 1144 y Fx(+)g Ft(r)1016 1151 y FE(0)1035 1144 y Fx(\))1054 1130 y FE(2)1090 1110 y Fu(\000)g Fx(\()p Ft(G)1197 1117 y FE(0)1228 1110 y Fx(+)g Ft(r)1299 1117 y FE(0)1319 1110 y Fx(\))1338 1062 y Fp(\021)1376 1110 y Fu(\025)j Fx(8)p Ft(:)p Fx(830153)k Ft(:)46 1248 y Fx(T)l(o)d(b)q(e)f(consisten)o(t)g(with)g(the)g(criterion)f(that)i(led) f(us)g(to)h(the)f(Hamiltonian)e(\(3.3\),)i(w)o(e)g(ha)o(v)o(e)g(to)g (omit)46 1308 y(the)19 b(term)f(of)h(order)h Ft(")465 1290 y FE(2)504 1308 y Fx(in)f(\(3.4\))h(and)g(this)f(leads)g(us)h(to)g (the)f(Hamiltonian)f(\(1.3\))h(in)o(tro)q(duced)g(in)46 1368 y Fu(x)d Fx(1,)g(2\).)46 1535 y FB(4)83 b(A)27 b(KAM)h(Theorem)46 1644 y Fx(Here)16 b(w)o(e)h(pro)o(v)o(e)g(a)h(KAM)e(result,)h(in)g(the) g(st)o(yle)f(of)i([5],)e(whic)o(h)h(will)g(b)q(e)g(the)g(basis)h(of)g (the)f(pro)q(of)i(of)46 1704 y(Theorem)c(1.1.)46 1790 y(First)g(w)o(e)g(pro)o(vide)f(a)i(\\KAM)f(algorithm")g(\(in)g (Hamiltonian)e(con)o(text\),)h(whic)o(h)h(yields)f(a)i(sequence)46 1850 y(of)c(quadratically)f(b)q(etter)g(and)h(b)q(etter)f(appro)o (ximations)g(to)h(the)g(conjugacy)g(function)f(of)h(a)g(maximal)46 1910 y(in)o(v)m(arian)o(t)h(\(Diophan)o(tine\))h(torus,)g(and)g(then)g (w)o(e)f(form)o(ulate)f(a)j(criterion)d(ensuring)i(the)g(applicabil-)46 1970 y(it)o(y)h(of)i(the)f(algorithm)f(an)i(in\014nite)e(n)o(um)o(b)q (er)g(of)i(times)d(and)j(hence)e(the)i(existence)d(of)j(an)g(in)o(v)m (arian)o(t)46 2030 y(torus.)g(T)l(ec)o(hnically)l(,)c(the)k(algorithm,) e(whic)o(h)h(do)q(es)h(not)g(use)f(symplectic)e(transformations)i (\(used,)46 2091 y(instead,)h(in)g(the)g(original)g(w)o(orks)h(of)f (the)g(masters\),)f(ma)o(y)g(b)q(e)i(view)o(ed)d(as)j(a)g(Hamiltonian)e (v)o(ersion)46 2151 y(of)j(the)f(Lagrangian)i(approac)o(h)g(dev)o(elop) q(ed)d(in)h(the)g(eigh)o(ties)g(b)o(y)g(Moser,)g(Salamon)g(and)h (Zehnder)46 2211 y(\(see)d([5])f(and)i(references)e(therein\).)p 46 2257 747 2 v 85 2288 a Fs(15)121 2303 y FI(F)m(or)10 b(example,)g(one)h(can)g(replace)i(the)e(non{degeneracy)i(h)o(yp)q (othesis)e(with)g(a)g(\\iso{energetic)h(non{degeneracy")46 2353 y(\(see,)j(e.g.,)d([1]\),)h(whic)o(h)g(is)h(satis\014ed)h(b)o(y)e Fz(h)715 2359 y Fs(0)734 2353 y FI(.)85 2388 y Fs(16)121 2403 y FI(Note)h(that)g(the)h(dynamics)e(generated)i(b)o(y)f(a)g (Hamiltonian)d(function)j Fz(h)d FI(=)i Fz(h)p FI(\()p Fz(q)q(;)7 b(p)p FI(\))13 b(and)h(b)o(y)g Fz(h)1623 2388 y Fs(2)1656 2403 y FI(coincide)g(up)g(to)46 2453 y(a)h(time)g(scale:)h (if)f Fz(z)r FI(\()p Fz(t)p FI(\))g(=)h(\()p Fz(q)q FI(\()p Fz(t)p FI(\))p Fz(;)7 b(p)p FI(\()p Fz(t)p FI(\)\))16 b(is)f(an)h Fz(h)p FI({motion)d(than)j Fz(z)r FI(\(2)p Fz(E)r(t)p FI(\),)g(with)f Fz(E)i FI(=)f Fz(h)p FI(\()p Fz(z)r FI(\(0\)\))g(is)g(the)g(corresp)q(onding)46 2502 y Fz(h)70 2487 y Fs(2)88 2502 y FI({motion.)955 2863 y Fx(10)p eop %%Page: 11 11 11 10 bop 46 78 a Fe(4.1)69 b(Algebraic)22 b(Sc)n(heme)46 170 y Fx(Let)15 b(us)f(consider)h(a)g(smo)q(oth)f(\(later)g (real{analytic\))g(Hamiltonian)f Ft(h)p Fx(\()p Ft(x;)8 b(y)r Fx(\))13 b(where)i Ft(x)f Fx(v)m(aries)g(on)h(the)46 231 y(standard)h Ft(N)5 b Fx({torus)15 b Fq(T)476 212 y Fk(N)515 231 y Fu(\021)8 b Fq(R)601 212 y Fk(N)631 231 y Ft(=)p Fx(\(2)p Ft(\031)r Fq(Z)765 212 y Fk(N)795 231 y Fx(\))15 b(and)g Ft(y)h Fx(v)m(aries)e(in)g(some)g(op)q(en)h (ball)f Ft(B)1522 212 y Fk(N)1569 231 y Fu(\032)g Fq(R)1661 212 y Fk(N)1691 231 y Fx(;)g(\()p Ft(x;)8 b(y)r Fx(\))14 b(are)46 291 y(standard)j(symplectic)c(co)q(ordinates)731 273 y FE(17)770 291 y Fx(.)i(The)i(problem)d(is)i(to)h(construct)f(an)h (in)o(v)m(arian)o(t)f Ft(N)5 b Fx({torus)17 b Fu(S)46 351 y Fx(on)f(whic)o(h)f(the)g(\015o)o(w)i(is)e(conjugated)h(to)g(the)g (linear)f(\015o)o(w)h Ft(\022)f Fu(2)f Fq(T)1232 333 y Fk(N)1276 351 y Fu(!)g Ft(\022)d Fx(+)f Ft(!)r(t)15 b Fx(for)h(some)f(\\rationally)46 411 y(indep)q(enden)o(t")g(v)o(ector) 477 393 y FE(18)529 411 y Ft(!)h Fu(2)e Fq(R)661 393 y Fk(N)692 411 y Fx(.)h(The)h Fu(S)t Fx({em)o(b)q(edding)f(function)g Ft(\022)g Fu(2)f Fq(T)1438 393 y Fk(N)1482 411 y Fu(!)g Fx(\()p Ft(\022)d Fx(+)f Ft(u)p Fx(\()p Ft(\022)q Fx(\))p Ft(;)e(v)r Fx(\()p Ft(\022)q Fx(\)\))13 b Fu(2)46 471 y Fq(T)82 453 y Fk(N)125 471 y Fu(\002)e Ft(B)215 453 y Fk(N)266 471 y Fx(is)17 b(immediatel)o(y)d(seen)k(to)f(satisfy)h(the) f(follo)o(wing)h(quasi{linear,)e(degenerate)h(PDE)i(on)46 531 y Fq(T)82 513 y Fk(N)113 531 y Fx(:)653 641 y Ft(!)13 b Fx(+)e Ft(D)q(u)h Fu(\000)f Ft(h)904 648 y Fk(y)924 641 y Fx(\()p Ft(\022)i Fx(+)e Ft(u;)d(v)r Fx(\))40 b(=)h(0)17 b Ft(;)747 714 y(D)q(v)c Fx(+)e Ft(h)902 721 y Fk(x)924 714 y Fx(\()p Ft(\022)i Fx(+)e Ft(u;)d(v)r Fx(\))40 b(=)h(0)17 b Ft(;)515 b Fx(\(4.1\))46 824 y(where)16 b Ft(D)i Fx(denotes)e(the)g (deriv)m(ativ)o(es)f(in)h(the)g Ft(!)i Fx(direction:)733 959 y Ft(D)10 b Fu(\021)e Ft(!)13 b Fu(\001)e Ft(@)924 966 y Fk(\022)952 959 y Fu(\021)1021 905 y Fk(N)1007 917 y Fp(X)1009 1009 y Fk(i)p FE(=1)1075 959 y Ft(!)1105 966 y Fk(i)1143 925 y Ft(@)p 1125 947 66 2 v 1125 993 a(@)s(\022)1177 1000 y Fk(i)1211 959 y Ft(;)587 b Fx(\(4.2\))46 1101 y(and)17 b Ft(h)169 1108 y Fk(x)191 1101 y Fx(,)f Ft(h)249 1108 y Fk(y)286 1101 y Fx(denote)g(the)g(gradien)o(t)g(of)g Ft(h)h Fx(with)f(resp)q(ect)g(to)g Ft(x)p Fx(,)g Ft(y)r Fx(.)46 1186 y(As)e(usual,)h(w)o(e)f(assume)g(that)h Ft(!)h Fx(is)f(a)g Fv(Diophan)o(tine)f(v)o(ector)p Fx(:)f Fj(i.e.)j Fx(there)e(exist)g Ft(\015)j(>)d Fx(0)h(and)g(a)g(p)q(ositiv) o(e)46 1247 y(in)o(teger)g Ft(\034)22 b Fx(suc)o(h)16 b(that)431 1391 y Fu(j)p Ft(!)d Fu(\001)e Ft(n)p Fu(j)d(\021)612 1341 y Fp(\014)612 1366 y(\014)612 1391 y(\014)648 1337 y Fk(N)634 1349 y Fp(X)635 1441 y Fk(i)p FE(=1)702 1391 y Ft(!)732 1398 y Fk(i)746 1391 y Ft(n)775 1398 y Fk(i)790 1341 y Fp(\014)790 1366 y(\014)790 1391 y(\014)13 b Fu(\025)h Fx(\()p Ft(\015)s Fu(j)p Ft(n)p Fu(j)974 1370 y Fk(\034)995 1391 y Fx(\))1014 1370 y FF(\000)p FE(1)1078 1391 y Ft(;)105 b Fu(8)15 b Ft(n)f Fu(2)g Fq(Z)1367 1370 y Fk(N)1398 1391 y Fu(nf)p Fx(0)p Fu(g)i Ft(:)285 b Fx(\(4.3\))46 1536 y(The)17 b(starting)h(p)q(oin)o(t)g(of)f(a)h(KAM)f(algorithm)f(is) h(an)h Fv(appro)o(ximate)e Fx(solution)i(\()p Ft(u;)8 b(v)r Fx(\),)16 b(whic)o(h)h(solv)o(es)46 1596 y(\(4.1\))g(up)g(to)g (some)e(\\error".)i(In)g(order)f(to)h(form)o(ulate)e(a)i(precise)f (result)g(w)o(e)g(need)g(some)g(notations)46 1656 y(and)h(some)e (assumptions.)46 1742 y(Giv)o(en)g(a)i(function)f Ft(u)d Fx(:)h Fq(T)524 1723 y Fk(N)568 1742 y Fu(!)g Fq(R)671 1723 y Fk(N)718 1742 y Fx(w)o(e)i(denote)g(b)o(y)g Ft(u)1043 1749 y Fk(\022)1078 1742 y Fx(or)h(b)o(y)f Ft(@)1232 1749 y Fk(\022)1251 1742 y Ft(u)g Fx(its)g(Jacobian)h(matrix)838 1874 y(\()p Ft(u)885 1881 y Fk(\022)905 1874 y Fx(\))924 1881 y Fk(ij)962 1874 y Fu(\021)1014 1840 y Ft(@)s(u)1071 1847 y Fk(i)p 1014 1862 71 2 v 1014 1908 a Ft(@)s(\022)1066 1915 y Fk(j)1106 1874 y Fx(;)46 2010 y Ft(h)74 2017 y Fk(xy)131 2010 y Fx(denotes)f(the)g(matrix)f(with)h(en)o(tries)787 2103 y Fp(\020)812 2151 y Ft(h)840 2158 y Fk(xy)880 2103 y Fp(\021)905 2178 y Fk(ij)944 2151 y Fu(\021)1028 2118 y Ft(@)1057 2099 y FE(2)1076 2118 y Ft(h)p 996 2140 141 2 v 996 2185 a(@)s(x)1053 2192 y Fk(i)1066 2185 y Ft(@)s(y)1119 2192 y Fk(j)1158 2151 y Fx(;)46 2294 y(th)o(us)f Ft(h)180 2301 y Fk(y)q(x)235 2294 y Fx(is)g(the)f(transp)q(ose)i(of)f Ft(h)664 2301 y Fk(xy)705 2294 y Fx(,)f Fj(i.e.)j Ft(h)845 2301 y Fk(y)q(x)899 2294 y Fx(=)d Ft(h)979 2276 y Fk(T)979 2307 y(xy)1020 2294 y Fx(;)g(if)g Ft(A)h Fx(is)f(a)h(square)g(matrix,)e (w)o(e)h(denote)h(b)o(y)f Ft(A)1881 2276 y FE(#)46 2355 y Fx(the)i(an)o(tisymme)o(tric)d(part)k(of)f Ft(A)g Fx(times)e(t)o(w)o (o:)822 2465 y Ft(A)859 2444 y FE(#)898 2465 y Fu(\021)8 b Ft(A)j Fu(\000)f Ft(A)1079 2444 y Fk(T)1123 2465 y Fx(;)p 46 2508 747 2 v 85 2545 a Fs(17)121 2560 y FI(That)j(is,)h(the)g (symplectic)f(structure)j(is)e(giv)o(en)g(b)o(y)f(the)i(standard)f (2{form)1324 2529 y Fp(P)1368 2539 y Fi(N)1368 2572 y(i)p Fs(=1)1430 2560 y Fz(dx)1476 2566 y Fi(i)1499 2560 y Fr(^)9 b Fz(dy)1578 2566 y Fi(i)1591 2560 y FI(.)85 2595 y Fs(18)121 2610 y Fd(i.e.)14 b FI(if)f Fz(!)d Fr(\001)f Fz(n)j FI(=)f(0)j(for)f(some)g Fz(n)f Fr(2)f Fq(Z)682 2595 y Fi(N)725 2610 y FI(then)j Fz(n)g FI(m)o(ust)f(b)q(e)h(0.)955 2863 y Fx(11)p eop %%Page: 12 12 12 11 bop 46 78 a Fx(\014nally)l(,)15 b(if)i Ft(\022)e Fu(2)g Fq(T)371 60 y Fk(N)417 78 y Fu(!)f Ft(f)5 b Fx(\()p Ft(\022)q Fx(\))15 b Fu(2)g Fq(R)674 60 y Fk(s)706 78 y Fx(\()p Ft(s)f Fu(\025)h Fx(1\))i(is)f(a)h(smo)q(oth)g(function)g (with)f(v)m(anishing)h(mean)f(v)m(alue,)46 138 y Fj(i.e.)667 208 y Fu(h)q Ft(f)5 b Fu(i)k(\021)846 174 y Fx(1)p 796 196 126 2 v 796 242 a(\(2)p Ft(\031)r Fx(\))888 228 y Fk(N)926 150 y Fp(Z)943 244 y Fc(T)964 234 y Fo(N)997 208 y Ft(f)c Fx(\()p Ft(\022)q Fx(\))17 b Ft(d\022)32 b Fx(=)13 b(0)k Ft(;)46 325 y Fx(\(and)i(if)f Ft(D)i Fx(is)e(as)i(in)e(\(4.2\),)g(\(4.3\)\),)g(w)o(e)g(denote)h(b)o(y)f Ft(D)1070 307 y FF(\000)p FE(1)1118 325 y Ft(f)24 b Fx(the)18 b(unique)f(solution)i(with)g(v)m(anishing)46 386 y(mean)c(v)m(alue)h (of)g(the)g(equation)h Ft(D)q(g)f Fx(=)e Ft(f)5 b Fx(;)16 b(suc)o(h)g(a)h(solution)g(in)f(F)l(ourier)f(expansion)i(has)g(the)f (form)609 517 y Ft(D)650 497 y FF(\000)p FE(1)698 517 y Ft(f)j Fx(=)847 476 y Fp(X)793 572 y Fk(n)p FF(2)p Fc(Z)858 562 y Fo(N)890 572 y FF(nf)p FE(0)p FF(g)1007 484 y Ft(f)1031 491 y Fk(n)p 974 506 114 2 v 974 552 a Ft(i!)13 b Fu(\001)e Ft(n)1101 517 y Fx(exp)o(\()p Ft(in)g Fu(\001)g Ft(\022)q Fx(\))16 b Ft(;)46 681 y Fx(where)g Ft(f)211 688 y Fk(n)251 681 y Fx(denote)g(F)l(ourier)f(co)q (e\016cien)o(ts)g(and)i Ft(i)d Fx(=)1000 642 y Fu(p)p 1042 642 64 2 v 39 x(\000)p Fx(1.)46 795 y Fw(Assumption)k(4.1)24 b Fj(L)n(et)15 b Ft(\022)g Fu(2)f Fq(T)656 777 y Fk(N)701 795 y Fu(!)f Fx(\()p Ft(u;)8 b(v)r Fx(\))13 b Fu(2)h Fq(R)977 777 y Fk(N)1015 795 y Fu(\002)7 b Fq(R)1100 777 y Fk(N)1147 795 y Fj(b)n(e)16 b(a)f(smo)n(oth)g(function)i(and)f (let)h Fu(M)f Fj(and)46 855 y Ft(h)74 837 y FE(0)74 868 y Fk(y)q(y)131 855 y Fj(b)n(e)h(the)h(matric)n(es)450 837 y FE(19)478 978 y Fu(M)8 b(\021)g Ft(I)14 b Fx(+)d Ft(u)706 985 y Fk(\022)743 978 y Ft(;)107 b(h)892 957 y FE(0)892 990 y Fk(y)q(y)932 978 y Fx(\()p Ft(\022)q Fx(\))8 b Fu(\021)g Ft(h)1077 985 y Fk(y)q(y)1117 930 y Fp(\020)1141 978 y Ft(\022)13 b Fx(+)e Ft(u)p Fx(\()p Ft(\022)q Fx(\))p Ft(;)d(v)r Fx(\()p Ft(\022)q Fx(\))1426 930 y Fp(\021)1467 978 y Ft(:)331 b Fx(\(4.4\))46 1096 y Fj(Denoting)237 1078 y FE(20)771 1156 y Fu(T)21 b(\021)8 b(M)926 1136 y FF(\000)p FE(1)973 1156 y Ft(h)1001 1136 y FE(0)1001 1169 y Fk(y)q(y)1041 1156 y Fu(M)1101 1136 y FF(\000)p Fk(T)1173 1156 y Ft(;)625 b Fx(\(4.5\))46 1244 y Fj(we)18 b(assume)g(that,)f(for)g(any)h Ft(\022)c Fu(2)g Fq(T)696 1225 y Fk(N)726 1244 y Fj(,)k(the)g(matric)n(es)f Fu(M)p Fj(,)g Ft(h)1156 1225 y FE(0)1156 1256 y Fk(y)q(y)1213 1244 y Fj(and)g Fu(h)q(T)c(i)k Fj(ar)n(e)g(invertible.)46 1358 y Fw(Prop)r(osition)h(4.1)24 b Fj(L)n(et)17 b Ft(h)p Fj(,)g Ft(u)p Fj(,)g Ft(v)i Fj(and)f Ft(!)h Fj(satisfy,)e(r)n(esp)n(e)n (ctively,)h(Assumption)f(4.1)h(and)f Fx(\(4.3\))h Fj(and)46 1418 y(de\014ne)h Ft(f)k Fj(and)17 b Ft(g)j Fj(by)650 1528 y Ft(!)13 b Fx(+)e Ft(D)q(u)h Fu(\000)f Ft(h)901 1535 y Fk(y)921 1528 y Fx(\()p Ft(\022)h Fx(+)f Ft(u;)d(v)r Fx(\))41 b(=)g Ft(f)23 b(;)744 1601 y(D)q(v)13 b Fx(+)e Ft(h)899 1608 y Fk(x)921 1601 y Fx(\()p Ft(\022)h Fx(+)f Ft(u;)d(v)r Fx(\))41 b(=)g Ft(g)20 b(:)516 b Fx(\(4.6\))46 1711 y Fj(Then,)18 b(if)f(we)h(de\014ne)h(the)f(ve)n(ctor/matrix{value) n(d)h(functions)f Ft(b)p Fx(\()p Ft(\022)q Fx(\))f Fj(and)h Fu(B)r Fx(\()p Ft(\022)q Fx(\))f Fj(by)499 1837 y Ft(b)8 b Fu(\021)g Ft(v)601 1816 y Fk(T)599 1849 y(\022)628 1837 y Ft(f)16 b Fu(\000)11 b(M)778 1816 y Fk(T)806 1837 y Ft(g)19 b(;)108 b Fu(B)9 b(\021)1059 1789 y Fp(\020)1084 1837 y Fu(M)1144 1816 y Fk(T)1171 1837 y Ft(g)1194 1844 y Fk(\022)1225 1837 y Fu(\000)i Ft(v)1301 1816 y Fk(T)1299 1849 y(\022)1328 1837 y Ft(f)1352 1844 y Fk(\022)1372 1789 y Fp(\021)1396 1800 y FE(#)1445 1837 y Ft(;)46 1947 y Fj(we)18 b(have)738 2007 y Fu(h)p Ft(b)p Fu(i)c Fx(=)g(0)k Ft(;)107 b Fu(h)q(B)r(i)13 b Fx(=)h(0)k Ft(:)591 b Fx(\(4.7\))46 2094 y Fj(F)l(urthermor)n(e,)16 b(the)i(fol)r(lowing)i(e)n(quation)e (holds)931 2076 y FE(21)627 2204 y Ft(!)13 b Fx(+)e Ft(D)q(u)788 2184 y FF(0)811 2204 y Fu(\000)g Ft(h)889 2211 y Fk(y)910 2204 y Fx(\()p Ft(\022)h Fx(+)f Ft(u)1041 2184 y FF(0)1052 2204 y Ft(;)d(v)1100 2184 y FF(0)1111 2204 y Fx(\))42 b(=)f Ft(f)1280 2184 y FF(0)1309 2204 y Ft(;)721 2277 y(D)q(v)788 2256 y FF(0)811 2277 y Fx(+)11 b Ft(h)888 2284 y Fk(x)910 2277 y Fx(\()p Ft(\022)h Fx(+)f Ft(u)1041 2256 y FF(0)1052 2277 y Ft(;)d(v)1100 2256 y FF(0)1111 2277 y Fx(\))42 b(=)f Ft(g)1276 2256 y FF(0)1305 2277 y Ft(;)493 b Fx(\(4.8\))p 46 2323 747 2 v 85 2354 a Fs(19)121 2369 y Fz(I)14 b FI(is)e(the)g(iden)o(tit)o(y)e(matrix.)f(T)m(o)i(b)q (e)h(precise)h(w)o(e)e(should)h(replace,)f(in)g(\(4.4\),)f Fz(u)h FI(with)h Fz(p)t Fr(\016)t Fz(u)p FI(,)e Fz(p)h FI(b)q(eing)h(the)g(pro)r(jection)46 2419 y(of)h Fq(R)132 2404 y Fi(N)174 2419 y FI(on)o(to)h Fq(T)304 2404 y Fi(N)333 2419 y FI(;)f(ho)o(w)o(ev)o(er)h(w)o(e)g(shall)f(omit,)f(here)j(and)f (in)f(other)h(circumstances,)h(suc)o(h)f(pro)r(jection.)85 2454 y Fs(20)121 2469 y FI(The)g(sup)q(erscript)420 2454 y Fy(\000)p Fi(T)486 2469 y FI(denotes)h(the)f(transp)q(ose)i(of)d(the) h(in)o(v)o(erse:)h Fz(A)1193 2454 y Fy(\000)p Fi(T)1256 2469 y FI(=)d(\()p Fz(A)1347 2454 y Fy(\000)p Fs(1)1392 2469 y FI(\))1408 2454 y Fi(T)1434 2469 y FI(.)85 2503 y Fs(21)121 2518 y FI(Here)h(and)g(in)f(what)h(follo)o(ws,)d(the)j (prime)f(attac)o(hed)h(to)g(a)f(function)g(will)g(nev)o(er)h(denote)h (deriv)n(ates)f(but)g(just)g(new)46 2568 y(functions.)955 2863 y Fx(12)p eop %%Page: 13 13 13 12 bop 46 78 a Fj(wher)n(e)17 b Ft(u)211 60 y FF(0)223 78 y Ft(;)8 b(v)271 60 y FF(0)282 78 y Ft(;)g(f)333 60 y FF(0)345 78 y Ft(;)g(g)392 60 y FF(0)421 78 y Fj(ar)n(e)16 b(de\014ne)n(d)j(at)e(the)h(end)h(of)e(the)h(fol)r(lowing)h(list)f(of)g (de\014nitions)1602 60 y FE(22)278 187 y Ft(b)299 194 y FE(0)327 187 y Fu(\021)8 b(T)431 139 y Fp(\020)456 187 y Ft(D)497 167 y FF(\000)p FE(1)545 187 y Ft(b)j Fx(+)g Ft(c)647 194 y FE(1)667 139 y Fp(\021)703 187 y Fu(\000)f(M)812 167 y FF(\000)p FE(1)859 187 y Ft(f)23 b(;)108 b(c)1049 194 y FE(1)1077 187 y Fu(\021)8 b(hT)13 b(i)1202 163 y FF(\000)p FE(1)1249 139 y Fp(\020)1274 187 y Fu(h)q(M)1354 167 y FF(\000)p FE(1)1401 187 y Ft(f)5 b Fu(i)11 b(\000)g(h)q(T)h Ft(D)1610 167 y FF(\000)p FE(1)1658 187 y Ft(b)p Fu(i)1698 139 y Fp(\021)1741 187 y Ft(;)278 278 y(z)f Fu(\021)d(M)419 230 y Fp(\020)443 278 y Ft(D)484 258 y FF(\000)p FE(1)532 278 y Ft(b)553 285 y FE(0)584 278 y Fx(+)j Ft(c)654 285 y FE(2)673 230 y Fp(\021)716 278 y Ft(;)107 b(c)858 285 y FE(2)886 278 y Fu(\021)19 b(\000)11 b(hM)p Ft(D)1114 258 y FF(\000)p FE(1)1162 278 y Ft(b)1183 285 y FE(0)1203 278 y Fu(i)18 b Ft(;)278 378 y(w)10 b Fu(\021)370 330 y Fp(\020)395 378 y Ft(h)423 357 y FE(0)423 390 y Fk(y)q(y)462 330 y Fp(\021)487 341 y FF(\000)p FE(1)534 330 y Fp(\020)559 378 y Ft(D)q(z)k Fu(\000)c Ft(h)714 357 y FE(0)714 390 y Fk(y)q(x)755 378 y Ft(z)j Fx(+)e Ft(f)869 330 y Fp(\021)278 461 y Ft(q)300 468 y FE(1)328 461 y Fu(\021)d Ft(h)403 468 y Fk(x)425 461 y Fx(\()p Ft(\022)k Fx(+)f Ft(u)g Fx(+)g Ft(z)r(;)d(v)k Fx(+)f Ft(w)q Fx(\))g Fu(\000)g Ft(h)892 441 y FE(0)892 474 y Fk(x)925 461 y Fu(\000)g Ft(h)1003 441 y FE(0)1003 474 y Fk(xx)1045 461 y Ft(z)i Fu(\000)e Ft(h)1159 441 y FE(0)1159 474 y Fk(xy)1199 461 y Ft(w)19 b(;)278 539 y(q)300 546 y FE(2)328 539 y Fu(\021)8 b Ft(h)403 546 y Fk(y)424 539 y Fx(\()p Ft(\022)k Fx(+)f Ft(u)g Fx(+)g Ft(z)r(;)d(v)k Fx(+)f Ft(w)q Fx(\))g Fu(\000)g Ft(h)891 518 y FE(0)891 551 y Fk(y)923 539 y Fu(\000)g Ft(h)1001 518 y FE(0)1001 551 y Fk(y)q(x)1041 539 y Ft(z)i Fu(\000)e Ft(h)1155 518 y FE(0)1155 551 y Fk(y)q(y)1195 539 y Ft(w)18 b(;)278 617 y(q)300 624 y FE(3)328 617 y Fu(\021)8 b Ft(f)404 597 y Fk(T)399 630 y(\022)432 617 y Ft(w)k Fu(\000)f Ft(g)554 597 y Fk(T)552 630 y(\022)582 617 y Ft(z)i Fu(\000)d(M)727 597 y Fk(T)755 617 y Ft(q)777 624 y FE(1)814 617 y Ft(;)278 690 y(f)307 670 y FF(0)328 690 y Fu(\021)19 b(\000)11 b Ft(q)458 697 y FE(2)494 690 y Ft(;)278 763 y(g)303 742 y FF(0)323 763 y Fu(\021)d(M)430 742 y FF(\000)p Fk(T)485 715 y Fp(\020)510 763 y Ft(D)551 715 y Fp(h)572 763 y Fx(\()p Ft(D)632 742 y FF(\000)p FE(1)680 763 y Fu(B)r Fx(\))p Fu(M)794 742 y FF(\000)p FE(1)840 763 y Ft(z)13 b Fx(+)e Fu(M)985 742 y Fk(T)1012 763 y Fx(\()p Ft(h)1059 742 y FE(0)1059 775 y Fk(y)q(y)1098 763 y Fx(\))1117 742 y FF(\000)p FE(1)1164 763 y Ft(f)1188 770 y Fk(\022)1208 763 y Fu(M)1268 742 y FF(\000)p FE(1)1315 763 y Ft(z)1340 715 y Fp(i)1371 763 y Fu(\000)f Ft(q)1442 770 y FE(3)1462 715 y Fp(\021)1504 763 y Ft(;)278 843 y(u)306 823 y FF(0)326 843 y Fu(\021)e Ft(u)j Fx(+)g Ft(z)19 b(;)108 b(v)651 823 y FF(0)670 843 y Fu(\021)8 b Ft(v)13 b Fx(+)e Ft(w)18 b(:)46 956 y Fw(Remark)e(4.1)24 b Fx(\(i\))16 b(It)g(is)g(imme)o(diate) d(to)k(c)o(hec)o(k)d(that)j(if)e(w)o(e)h(replace)f Ft(f)22 b Fx(and)17 b Ft(g)h Fx(b)o(y)e Ft("f)21 b Fx(and)c Ft("g)h Fx(then)46 1017 y Ft(z)r(;)8 b(w)17 b Fx(=)g Ft(O)q Fx(\()p Ft(")p Fx(\))h(and)g Ft(f)442 999 y FF(0)454 1017 y Ft(;)8 b(g)501 999 y FF(0)530 1017 y Fx(=)16 b Ft(O)q Fx(\()p Ft(")664 999 y FE(2)684 1017 y Fx(\),)i Fj(i.e.)h Fv(the)e(errors)h (asso)q(ciated)h(to)f Ft(u)1369 999 y FF(0)1399 1017 y Fv(and)g Ft(v)1521 999 y FF(0)1550 1017 y Fv(are)g(quadratically)46 1077 y(smaller)13 b(than)j(the)f(errors)g(asso)q(ciated)i(to)e Ft(u)g Fv(and)h Ft(v)r Fx(.)e(W)l(e)h(shall)g(call)g(a)g(couple)g(\()p Ft(u;)8 b(v)r Fx(\))14 b(as)i(in)f(\(4.6\))h(an)46 1137 y Fv(appro)o(ximate)f(solution)h Fx(for)h(\(4.1\))f(and)h(the)f (relativ)o(e)e(couple)i(\()p Ft(f)s(;)8 b(g)r Fx(\))16 b(the)g Fv(error)g(function)p Fx(.)46 1222 y(\(ii\))i(Note)h(that)h (the)g(constan)o(ts)g Ft(c)688 1229 y FE(1)727 1222 y Fx(and)g Ft(c)846 1229 y FE(2)885 1222 y Fx(are)g(de\014ned)f(so)h (that)g(the)f(functions)g Ft(b)1634 1229 y FE(0)1673 1222 y Fx(and)h Ft(z)i Fx(ha)o(v)o(e)46 1282 y(v)m(anishing)17 b(mean)e(v)m(alue.)46 1420 y Fw(Pro)r(of)j Fx(T)l(o)f(c)o(hec)o(k)e (the)h(\014rst)g(of)h(\(4.7\),)f(observ)o(e)f(that)735 1529 y Ft(@)761 1536 y Fk(\022)780 1529 y Ft(h)808 1508 y FE(0)842 1529 y Fx(=)f Fu(M)954 1508 y Fk(T)981 1529 y Ft(h)1009 1508 y FE(0)1009 1541 y Fk(x)1042 1529 y Fx(+)d Ft(v)1117 1508 y Fk(T)1115 1541 y(\022)1144 1529 y Ft(h)1172 1508 y FE(0)1172 1541 y Fk(y)1209 1529 y Fx(;)46 1638 y(no)o(w,)17 b(m)o(ultiply)c(the)k(\014rst)g(of)g(\(4.6\)) g(b)o(y)f Ft(v)810 1620 y Fk(T)808 1650 y(\022)837 1638 y Fx(,)h(the)f(second)h(b)o(y)g Fu(\000M)1278 1620 y Fk(T)1305 1638 y Fx(,)f(add)h(them)f(together)h(and)g(use)46 1698 y(in)o(tegration)f(b)o(y)g(parts)g(to)h(get)f(rid)g(of)h(the)f (terms)f(con)o(taining)h Ft(\022)q Fx({deriv)m(ativ)o(es.)46 1758 y(T)l(o)h(c)o(hec)o(k)d(the)i(second)h(of)f(\(4.7\),)g(tak)o(e)g (the)g Ft(\022)q Fx({gradien)o(t)g(of)h(\(4.6\))f(to)h(obtain)708 1868 y Ft(D)q Fu(M)e Fx(=)f Ft(h)904 1847 y FE(0)904 1880 y Fk(y)q(x)944 1868 y Fu(M)d Fx(+)g Ft(h)1092 1847 y FE(0)1092 1880 y Fk(y)q(y)1131 1868 y Ft(v)1155 1875 y Fk(\022)1186 1868 y Fx(+)g Ft(f)1259 1875 y Fk(\022)1294 1868 y Ft(;)708 1945 y(D)q(v)773 1952 y Fk(\022)807 1945 y Fx(=)j Fu(\000)p Ft(h)926 1924 y FE(0)926 1957 y Fk(xx)967 1945 y Fu(M)d(\000)g Ft(h)1116 1924 y FE(0)1116 1957 y Fk(xy)1157 1945 y Ft(v)1181 1952 y Fk(\022)1211 1945 y Fx(+)g Ft(g)1283 1952 y Fk(\022)1319 1945 y Ft(:)479 b Fx(\(4.9\))46 2054 y(Let)811 2114 y Fu(A)8 b(\021)906 2066 y Fp(\020)931 2114 y Fu(M)991 2094 y Fk(T)1018 2114 y Ft(v)1042 2121 y Fk(\022)1061 2066 y Fp(\021)1086 2078 y FE(#)1134 2114 y Ft(;)639 b Fx(\(4.10\))46 2207 y(and)17 b(notice)e(\(in)o(tegration)h(b)o(y)g(parts\))h(that)880 2316 y Fu(hAi)d Fx(=)g(0)j Ft(:)708 b Fx(\(4.11\))46 2426 y(F)l(rom)15 b(\(4.9\),)h(it)g(follo)o(ws)g(that)h(the)f(matrix)e Fu(A)i Fx(satis\014es)h(the)f(equation)873 2535 y Ft(D)q Fu(A)f Fx(=)f Fu(B)j Ft(;)702 b Fx(\(4.12\))p 46 2581 747 2 v 85 2612 a Fs(22)121 2627 y FI(As)14 b(ab)q(o)o(v)o(e)g(if)f Fz(h)e FI(=)h Fz(h)p FI(\()p Fz(x;)7 b(y)q FI(\),)13 b Fz(h)586 2612 y Fs(0)605 2627 y FI(\()p Fz(\022)q FI(\))i(denotes)g (the)f(function)g Fz(h)p FI(\()p Fz(\022)d FI(+)e Fz(u)p FI(\()p Fz(\022)q FI(\))p Fz(;)e(v)q FI(\()p Fz(\022)q FI(\)\).)955 2863 y Fx(13)p eop %%Page: 14 14 14 13 bop 46 78 a Fx(from)15 b(whic)o(h)h(the)g(second)g(of)h(\(4.7\))f (follo)o(ws)g(at)h(once.)46 138 y(The)f(\014rst)h(of)f(\(4.8\))h(follo) o(ws)f(imme)o(diately)d(from)i(the)h(de\014nitions:)192 248 y Ft(!)d Fx(+)e Ft(D)q(u)353 228 y FF(0)376 248 y Fu(\000)g Ft(h)454 255 y Fk(y)475 248 y Fx(\()p Ft(\022)h Fx(+)f Ft(u)606 228 y FF(0)617 248 y Ft(;)d(v)665 228 y FF(0)676 248 y Fx(\))42 b(=)f Ft(!)13 b Fx(+)e Ft(D)q(u)h Fx(+)f Ft(D)q(z)j Fu(\000)d Ft(h)1194 255 y Fk(y)1214 248 y Fx(\()p Ft(\022)i Fx(+)e Ft(u)f Fx(+)h Ft(z)r(;)d(v)13 b Fx(+)e Ft(w)q Fx(\))737 321 y(=)41 b Ft(h)844 328 y Fk(y)865 321 y Fx(\()p Ft(\022)12 b Fx(+)f Ft(u;)d(v)r Fx(\))i Fu(\000)h Ft(h)1151 328 y Fk(y)1172 321 y Fx(\()p Ft(\022)h Fx(+)f Ft(u)g Fx(+)g Ft(z)r(;)d(v)k Fx(+)f Ft(w)q Fx(\))g(+)g Ft(D)q(z)j Fx(+)d Ft(f)737 393 y Fx(=)41 b Fu(\000)p Ft(q)877 400 y FE(2)907 393 y Fu(\000)11 b Ft(h)985 373 y FE(0)985 406 y Fk(y)q(x)1026 393 y Ft(z)i Fu(\000)e Ft(h)1140 373 y FE(0)1140 406 y Fk(y)q(y)1179 393 y Ft(w)h Fx(+)f Ft(D)q(z)j Fx(+)d Ft(f)737 466 y Fx(=)41 b Ft(f)845 446 y FF(0)873 466 y Ft(:)900 b Fx(\(4.13\))46 576 y(The)16 b(c)o(hec)o(k)e(of)i(the)f(second)h(of)g(\(4.8\))h(is)e (more)g(tric)o(ky)l(.)e(First)j(observ)o(e)f(that)h(last)g(iden)o(tit)o (y)e(in)h(\(4.13\))46 636 y(can)h(b)q(e)h(rewritten)e(as)724 696 y Ft(D)q(z)i Fx(=)c Ft(h)884 676 y FE(0)884 709 y Fk(y)q(x)925 696 y Ft(z)g Fx(+)e Ft(h)1038 676 y FE(0)1038 709 y Fk(y)q(y)1077 696 y Ft(w)i Fu(\000)e Ft(f)21 b(:)553 b Fx(\(4.14\))46 784 y(Next,)15 b(from)g(the)h(de\014nition)g(of)g Ft(z)r Fx(,)g(it)g(follo)o(ws)g(that)555 894 y Ft(D)596 846 y Fp(\020)622 894 y Fu(M)682 873 y Fk(T)709 894 y Fx(\()p Ft(h)756 873 y FE(0)756 906 y Fk(y)q(y)796 894 y Fx(\))815 873 y FF(\000)p FE(1)862 846 y Fp(h)881 894 y Fu(M)p Ft(D)q Fx(\()p Fu(M)1061 873 y FF(\000)p FE(1)1109 894 y Ft(z)r Fx(\))11 b(+)g Ft(f)1242 846 y Fp(i\021)1300 894 y Fx(=)j Ft(b)i(:)384 b Fx(\(4.15\))46 1012 y(Solving)14 b(for)g Ft(h)316 994 y FE(0)316 1024 y Fk(y)q(y)369 1012 y Fx(in)g(the)g(\014rst)g(of)g(\(4.9\))g(and)h(inserting)e(the)h (obtained)g(expression)g(in)g(the)f(de\014nition)46 1072 y(of)j Ft(w)q Fx(,)g(w)o(e)g(get)291 1182 y Ft(w)42 b Fx(=)g(\()p Ft(h)495 1162 y FE(0)495 1195 y Fk(y)q(y)534 1182 y Fx(\))553 1162 y FF(\000)p FE(1)600 1134 y Fp(\020)625 1182 y Ft(D)q(z)14 b Fu(\000)753 1134 y Fp(h)772 1182 y Fx(\()p Ft(D)q Fu(M)p Fx(\))p Fu(M)971 1162 y FF(\000)p FE(1)1030 1182 y Fu(\000)d Ft(h)1108 1162 y FE(0)1108 1195 y Fk(y)q(y)1147 1182 y Ft(v)1171 1189 y Fk(\022)1190 1182 y Fu(M)1250 1162 y FF(\000)p FE(1)1308 1182 y Fu(\000)g Ft(f)1382 1189 y Fk(\022)1402 1182 y Fu(M)1462 1162 y FF(\000)p FE(1)1509 1134 y Fp(i)1528 1182 y Ft(z)i Fx(+)e Ft(f)1642 1134 y Fp(\021)368 1274 y Fx(=)42 b Ft(v)472 1281 y Fk(\022)491 1274 y Fu(M)551 1253 y FF(\000)p FE(1)598 1274 y Ft(z)13 b Fx(+)e(\()p Ft(h)730 1253 y FE(0)730 1286 y Fk(y)q(y)769 1274 y Fx(\))788 1253 y FF(\000)p FE(1)835 1225 y Fp(\020)860 1274 y Ft(f)884 1281 y Fk(\022)904 1274 y Fu(M)964 1253 y FF(\000)p FE(1)1011 1274 y Ft(z)i Fx(+)e Fu(M)p Ft(D)q Fx(\()p Fu(M)1276 1253 y FF(\000)p FE(1)1323 1274 y Ft(f)5 b Fx(\))12 b(+)f Ft(f)1461 1225 y Fp(\021)1502 1274 y Ft(:)271 b Fx(\(4.16\))46 1390 y(F)l(rom)15 b(the)h(de\014nition)g(of)g Ft(g)555 1372 y FF(0)567 1390 y Fx(,)g(\(4.15\),)g(\(4.16\),)h(\(4.10\))p Fu(\004)p Fx(\(4.12\),)f(it)f(follo)o(ws)h(that)98 1506 y Fu(M)158 1485 y Fk(T)185 1506 y Ft(g)210 1485 y FF(0)233 1506 y Fx(+)11 b Ft(q)304 1513 y FE(3)334 1506 y Fx(+)g Ft(b)j Fx(=)g Ft(D)q Fx(\()p Fu(M)590 1485 y Fk(T)618 1506 y Ft(w)e Fu(\000)f Ft(v)741 1485 y Fk(T)739 1518 y(\022)768 1506 y Ft(z)r Fx(\))16 b Ft(;)56 b Fj(i)5 b Ft(:)p Fj(e)t Ft(:)66 b(g)1063 1485 y FF(0)1088 1506 y Fx(=)14 b Fu(M)1200 1485 y FF(\000)p Fk(T)1255 1458 y Fp(\020)1280 1506 y Ft(D)q Fx(\()p Fu(M)1400 1485 y Fk(T)1428 1506 y Ft(w)e Fu(\000)f Ft(v)1551 1485 y Fk(T)1549 1518 y(\022)1578 1506 y Ft(z)r Fx(\))g Fu(\000)g Ft(b)f Fu(\000)h Ft(q)1786 1513 y FE(3)1806 1458 y Fp(\021)1847 1506 y Ft(:)46 1622 y Fx(F)l(rom)18 b(this)g(iden)o(tit)o(y)l(,)f (recalling)g(the)i(de\014nitions)g(of)g Ft(b)g Fx(and)g Ft(q)1207 1629 y FE(3)1245 1622 y Fx(and)h(using)f(\(4.9\))g(to)g (eliminate)d Ft(f)1892 1629 y Fk(\022)46 1682 y Fx(and)h Ft(g)164 1689 y Fk(\022)184 1682 y Fx(,)e(one)i(obtains)146 1792 y Ft(g)171 1772 y FF(0)197 1792 y Fx(=)d Fu(M)309 1772 y FF(\000)p Fk(T)363 1792 y Ft(v)389 1772 y Fk(T)387 1805 y(\022)416 1744 y Fp(h)436 1792 y Ft(h)464 1772 y FE(0)464 1805 y Fk(y)q(x)504 1792 y Ft(z)f Fx(+)e Ft(h)617 1772 y FE(0)617 1805 y Fk(y)q(y)657 1792 y Ft(w)h Fu(\000)f Ft(f)783 1744 y Fp(i)814 1792 y Fx(+)863 1744 y Fp(h)883 1792 y Ft(g)i Fx(+)e Ft(h)996 1772 y FE(0)996 1805 y Fk(xx)1038 1792 y Ft(z)i Fx(+)e Ft(h)1151 1772 y FE(0)1151 1805 y Fk(xy)1191 1792 y Ft(w)i Fx(+)e Ft(q)1310 1799 y FE(1)1329 1744 y Fp(i)1360 1792 y Fx(+)g Ft(D)q(w)i Fu(\000)e(M)1608 1772 y FF(\000)p Fk(T)1662 1792 y Ft(v)1688 1772 y Fk(T)1686 1805 y(\022)1715 1792 y Ft(D)q(z)19 b(;)46 1909 y Fx(whic)o(h,)14 b(in)h(view)f(of)h(\(4.14\),)g(the)g (de\014nition)g(of)g Ft(q)947 1916 y FE(1)982 1909 y Fx(and)g(the)g(second)g(of)h(\(4.6\),)f(yields)f(the)g(second)i(of)46 1969 y(\(4.8\).)p 235 1969 21 42 v 46 2163 a Fe(4.2)69 b(Analytic)22 b(T)-6 b(o)r(ols)46 2255 y Fx(F)l(rom)13 b(no)o(w)j(on)f(w)o(e)f(shall)h(w)o(ork)g(in)f(the)h Fv(real{analytic)f(category)p Fx(;)g(in)h(this)g(section)f(w)o(e)g (review)g(some)46 2316 y(basic)i(tec)o(hnical)f(facts.)46 2401 y(W)l(e)k(shall)g(consider)h(the)f(Banac)o(h)g(space)h(of)g(p)q (erio)q(dic)f(functions)g Ft(f)25 b Fx(real{analytic)19 b(on)h(the)f(torus)46 2461 y Fq(T)82 2443 y Fk(N)113 2461 y Fx(,)c(admitting)g(\(for)i(some)e(pre\014xed)h Ft(\030)g(>)e Fx(0\))j(analytic)e(extension)h(on)h(the)f(closed)g (strip)494 2571 y(\001)535 2578 y Fk(\030)562 2571 y Fu(\021)8 b(f)p Ft(\022)15 b Fu(2)f Fq(C)752 2550 y Fk(N)802 2571 y Fx(:)30 b Fu(j)8 b Fx(Im)f Ft(\022)958 2578 y Fk(i)971 2571 y Fu(j)14 b(\024)g Ft(\030)19 b(;)24 b Fu(8)15 b Ft(i)e Fx(=)h(1)p Ft(;)8 b(:::;)g(N)d Fu(g)15 b Ft(;)955 2863 y Fx(14)p eop %%Page: 15 15 15 14 bop 46 78 a Fx(equipp)q(ed)16 b(with)g(the)g(\\F)l(ourier)g (norm")692 177 y Fu(k)p Ft(f)5 b Fu(k)771 184 y Fk(\030)799 177 y Fu(\021)873 136 y Fp(X)854 231 y Fk(n)p FF(2)p Fc(Z)919 222 y Fo(N)959 177 y Fu(j)p Ft(f)997 184 y Fk(n)1021 177 y Fu(j)j Fx(exp)o(\()p Fu(j)p Ft(n)p Fu(j)p Ft(\030)r Fx(\))17 b(;)521 b(\(4.17\))46 322 y(in)16 b Fq(C)136 304 y Fk(N)189 322 y Fx(\(and)g(its)g(subsets)h Fq(R)578 304 y Fk(N)625 322 y Fx(and)g Fq(Z)756 304 y Fk(N)787 322 y Fx(\))f(w)o(e)g(shall)g(use)g(the)g(1{norm,)775 458 y Fu(j)p Ft(y)r Fu(j)8 b(\021)f(j)p Ft(y)r Fu(j)937 465 y FE(1)964 458 y Fu(\021)1034 404 y Fk(N)1020 417 y Fp(X)1021 508 y Fk(i)p FE(=1)1088 458 y Fu(j)p Ft(y)1126 465 y Fk(i)1140 458 y Fu(j)16 b Ft(:)46 596 y Fx(If)22 b Ft(f)29 b Fx(:)24 b Fq(T)228 578 y Fk(N)283 596 y Fu(!)g Fq(C)390 578 y Fk(s)434 596 y Fx(is)e(a)g(v)o(ector)g(v)m(alued,)g (real{analytic)f(function,)h(analytic)g(on)g(\001)1672 603 y Fk(\030)1691 596 y Fx(,)g(its)g(norm)46 656 y(is)h(de\014ned)f (as)i Fu(k)p Ft(f)5 b Fu(k)422 663 y Fk(\030)449 656 y Fu(\021)505 623 y Fp(P)548 667 y Fk(i)571 656 y Fu(k)p Ft(f)620 663 y Fk(i)634 656 y Fu(k)659 663 y Fk(\030)678 656 y Fx(,)22 b(whic)o(h)g(coincides)g(with)h(\(4.17\))h(if)e Ft(f)1414 663 y Fk(n)1461 656 y Fx(denotes)h(the)g Ft(s)p Fx({v)o(ector)46 716 y(whose)c(comp)q(onen)o(ts)f(are)h(giv)o(en)e(b)o (y)h(the)h(F)l(ourier)f(co)q(e\016cien)o(ts)f(of)i(the)f(comp)q(onen)o (ts)g(of)h Ft(f)5 b Fx(.)18 b(These)46 777 y(de\014nitions)g(are)g(imm) o(ediatel)o(y)d(extended)i(to)i(matric/tensor{v)m(alued)e(functions)h (b)o(y)f(making)g(use)46 837 y(of)d(the)g(standard)h(\\op)q(erator)g (norm":)e Fj(e.g.)31 b Fx(if)13 b Ft(A)p Fx(\()p Ft(\022)q Fx(\))h(is)f(a)h(matrix{v)m(alued)f(p)q(erio)q(dic)g(functions)h(with) 46 897 y(analytic)h(extension)h(on)h(\001)556 904 y Fk(\030)574 897 y Fx(,)f(w)o(e)g(set)705 996 y Fu(k)p Ft(A)p Fu(k)792 1003 y Fk(\030)818 996 y Fu(\021)71 b Fx(sup)874 1039 y Fk(c)p FF(2)p Fc(C)935 1030 y Fo(N)966 1039 y FE(:)p FF(j)p Fk(c)p FF(j)p FE(=1)1097 996 y Fu(k)p Ft(Ac)p Fu(k)1205 1003 y Fk(\030)1240 996 y Fx(;)46 1135 y(or,)11 b(if)g Ft(@)183 1116 y FE(3)180 1147 y Fk(x)203 1135 y Ft(f)16 b Fx(is)c(the)f(tensor)h(of)g(order)g(three)e(of)i(the)g (deriv)m(ativ)o(es)e(of)i(a)g(p)q(erio)q(dic,)e(real{analytic)h (function)46 1195 y Ft(f)19 b Fx(:)13 b(\001)157 1202 y Fk(\030)190 1195 y Fu(!)g Fq(C)c Fx(,)19 b(its)d(F)l(ourier)g(norm)f (is)h(giv)o(en)g(b)o(y)506 1333 y Fu(k)p Ft(@)560 1312 y FE(3)557 1345 y Fk(x)579 1333 y Ft(f)5 b Fu(k)633 1340 y Fk(\030)661 1333 y Fu(\021)50 b Fx(sup)716 1374 y FF(j)p Fk(b)p FF(j)p FE(=)p FF(j)p Fk(c)p FF(j)p FE(=1)881 1279 y Fk(N)866 1291 y Fp(X)868 1382 y Fk(i)p FE(=1)935 1333 y Fu(k)996 1279 y Fk(N)982 1291 y Fp(X)968 1384 y Fk(j;k)q FE(=1)1142 1299 y Ft(@)1171 1281 y FE(3)1190 1299 y Ft(f)p 1070 1321 223 2 v 1070 1367 a(@)s(x)1127 1374 y Fk(i)1140 1367 y Ft(@)s(x)1197 1374 y Fk(j)1214 1367 y Ft(@)s(x)1271 1374 y Fk(k)1297 1333 y Ft(b)1318 1340 y Fk(k)1339 1333 y Ft(c)1360 1340 y Fk(j)1378 1333 y Fu(k)1403 1340 y Fk(\030)1438 1333 y Ft(:)46 1475 y Fx(Finally)l(,)22 b(w)o(e)h(shall)h(also)g(consider)f(functions)h(\(p)q(ossibly)g(v)o (ector/matrix/tensor{v)m(alued\))e Ft(h)27 b Fx(=)46 1535 y Ft(h)p Fx(\()p Ft(x;)8 b(y)r Fx(\))16 b(p)q(erio)q(dic)i(in)f Ft(x)g Fx(and)h(real{analytic)f(on)h(the)f(closed)g(domain)f(\001)1396 1542 y Fk(\030)1427 1535 y Fu(\002)1490 1519 y Fp(b)1478 1535 y Ft(B)1515 1542 y Fk(r)1534 1535 y Fx(\()p Ft(y)1577 1542 y FE(0)1596 1535 y Fx(\))h(where)1787 1519 y Fp(b)1775 1535 y Ft(B)1812 1542 y Fk(r)1831 1535 y Fx(\()p Ft(y)1874 1542 y FE(0)1893 1535 y Fx(\))54 1595 y Fu(\021)133 1579 y Fp(b)120 1595 y Ft(B)160 1577 y Fk(N)157 1607 y(r)193 1595 y Fx(\()p Ft(y)236 1602 y FE(0)256 1595 y Fx(\))i(denotes)g(the)g (closed)g(complex)e(ball)h(of)i(radius)f Ft(r)i Fx(around)f Ft(y)1441 1602 y FE(0)1479 1595 y Fu(2)f Fq(C)1564 1577 y Fk(N)1600 1595 y Fx(.)g(F)l(or)g(an)o(y)g(suc)o(h)46 1655 y(function,)c(whic)o(h)h(admits)f(the)h(expansion)844 1637 y FE(23)898 1655 y Fx(\(con)o(v)o(ergen)o(t)f(on)i(\001)1270 1662 y Fk(\030)1299 1655 y Fu(\002)1361 1639 y Fp(b)1349 1655 y Ft(B)1386 1662 y Fk(r)1405 1655 y Fx(\()p Ft(y)1448 1662 y FE(0)1468 1655 y Fx(\)\))551 1755 y Ft(h)p Fx(\()p Ft(x;)8 b(y)r Fx(\))13 b(=)775 1713 y Fp(X)762 1806 y Fo(n)p Fn(2)p Fm(Z)822 1799 y Fo(N)763 1838 y(k)q Fn(2)p Fm(N)822 1831 y Fo(N)861 1755 y Ft(h)889 1762 y Fk(n;k)950 1755 y Fx(exp\()p Ft(in)d Fu(\001)h Ft(x)p Fx(\))16 b(\()p Ft(y)d Fu(\000)e Ft(y)1318 1762 y FE(0)1337 1755 y Fx(\))1356 1734 y Fk(k)1394 1755 y Ft(;)379 b Fx(\(4.18\))46 1918 y(w)o(e)16 b(set)624 1978 y Fu(k)p Ft(h)p Fu(k)702 1985 y Fk(\030)q(;r)755 1978 y Fu(\021)828 1937 y Fp(X)816 2030 y Fo(n)p Fn(2)p Fm(Z)875 2023 y Fo(N)816 2061 y(k)q Fn(2)p Fm(N)875 2054 y Fo(N)914 1978 y Fu(j)p Ft(h)956 1985 y Fk(n;k)1009 1978 y Fu(j)8 b Fx(exp)o(\()p Fu(j)p Ft(n)p Fu(j)p Ft(\030)r Fx(\))17 b Ft(r)1263 1958 y FF(j)p Fk(k)q FF(j)1321 1978 y Ft(:)452 b Fx(\(4.19\))46 2129 y(The)13 b(elemen)o(tary)d(prop)q(erties)k(of)f(in)o(terest)f(in)h(the) g(presen)o(t)f(con)o(text)h(are)g(collected)f(in)g(the)h(follo)o(wing) 46 2231 y Fw(Lemm)o(a)j(4.1)24 b Fx(\(i\))15 b Fj(L)n(et)g Ft(f)k Fx(:)13 b Fq(T)595 2213 y Fk(N)640 2231 y Fu(!)h Fq(R)e Fj(have)j(an)h(analytic)g(extension)h(on)e Fx(\001)1435 2238 y Fk(\030)1469 2231 y Fj(\(for)f(some)h Ft(\030)i(>)c Fx(0)p Fj(\))j(and)46 2291 y(let)k Ft(!)g Fu(2)d Fq(R)255 2273 y Fk(N)305 2291 y Fj(b)n(e)j(a)f(r)n(ational)r(ly)g(indep)n (endent)i(ve)n(ctor.)f(Then,)g(for)f(al)r(l)h Fx(0)e Ft(<)f(\016)i Fu(\024)e Ft(\030)r Fj(,)j(for)e(any)i Ft(p)d Fu(2)h Fq(Z)46 2351 y Fj(and)g(for)e(any)i Ft(k)e Fu(2)e Fq(N)k Fj(or)f(any)g Ft(k)f Fu(2)e Fq(N)732 2333 y Fk(N)784 2351 y Fj(one)k(has)946 2333 y FE(24)683 2451 y Fu(k)p Ft(D)749 2430 y FF(\000)p Fk(p)797 2451 y Ft(@)826 2430 y Fk(k)823 2463 y(x)847 2451 y Ft(f)5 b Fu(k)901 2458 y Fk(\030)q FF(\000)p Fk(\016)978 2451 y Fu(\024)14 b(k)p Ft(f)5 b Fu(k)1110 2458 y Fk(\030)1146 2451 y Ft(\033)1174 2458 y Fk(pk)1213 2451 y Fx(\()p Ft(\016)r Fx(\))p 46 2495 747 2 v 85 2532 a Fs(23)121 2547 y FI(W)m(e)13 b(use)i(the)f (standard)h(notation:)d(\()p Fz(y)f Fr(\000)f Fz(y)795 2553 y Fs(0)814 2547 y FI(\))830 2532 y Fi(k)862 2547 y FI(=)906 2516 y Fp(Q)945 2526 y Fi(k)945 2560 y(i)p Fs(=1)1001 2547 y FI(\()p Fz(y)1037 2553 y Fi(i)1060 2547 y Fr(\000)g Fz(y)1122 2553 y Fs(0)p Fi(i)1153 2547 y FI(\))1169 2532 y Fi(k)1187 2536 y Fh(i)1202 2547 y FI(.)85 2594 y Fs(24)121 2609 y FI(If)j Fz(k)g Fr(2)e Fq(N)p FI(,)j Fz(@)321 2594 y Fi(k)319 2619 y(x)342 2609 y Fz(f)k FI(denotes)e(the)e Fz(k)q FI({tensor)h(of)e(the)h(deriv)n (ativ)o(es)g(of)f Fz(f)t FI(;)i(if)e Fz(k)f Fr(2)f Fq(N)1340 2594 y Fi(N)1373 2609 y FI(,)i Fz(@)1422 2594 y Fi(k)1420 2619 y(x)1443 2609 y Fz(f)j FI(=)1582 2590 y Fi(@)1602 2578 y Fg(j)p Fh(k)p Fg(j)1639 2590 y Fi(f)p 1528 2599 185 2 v 1528 2632 a(@)r(x)1567 2616 y Fh(k)1583 2622 y Ff(1)1567 2642 y(1)1600 2632 y Fy(\001\001\001)p Fi(@)r(x)1669 2615 y Fh(k)1685 2622 y(N)1669 2643 y(N)1718 2609 y FI(.)955 2863 y Fx(15)p eop %%Page: 16 16 16 15 bop 46 78 a Fj(wher)n(e,)17 b(if)h Ft(k)e Fx(=)d(0)p Fj(,)18 b Ft(f)23 b Fj(is)17 b(assume)n(d)g(to)h(have)g(vanishing)h(me) n(an)e(value,)i(and)52 194 y Ft(\033)80 201 y Fk(pk)119 194 y Fx(\()p Ft(\016)r Fx(\))8 b Fu(\021)127 b Fx(sup)244 237 y FF(f)p Fk(n)p FF(2)p Fc(Z)327 227 y Fo(N)359 237 y FF(nf)p FE(0)p FF(g)n FE(:)p Fk(f)456 241 y Fo(n)477 237 y FF(6)p FE(=0)p FF(g)583 146 y Fp(\020)608 194 y Ft(\031)636 201 y Fk(nk)696 194 y Fu(j)p Ft(!)13 b Fu(\001)e Ft(n)p Fu(j)821 173 y FF(\000)p Fk(p)894 194 y Fx(exp)o(\()p Fu(\000)p Ft(\016)r Fu(j)p Ft(n)p Fu(j)p Fx(\))1126 146 y Fp(\021)1168 194 y Ft(;)107 b(\031)1317 201 y Fk(nk)1368 194 y Fu(\021)1423 121 y Fp(\()1477 163 y Fu(j)p Ft(n)p Fu(j)1534 145 y Fk(k)1573 163 y Ft(;)58 b Fx(if)17 b Ft(k)f Fu(2)e Fq(N)19 b Ft(;)1477 223 y Fu(j)p Ft(n)1520 205 y Fk(k)1542 223 y Fu(j)e Ft(;)58 b Fx(if)17 b Ft(k)f Fu(2)e Fq(N)1814 205 y Fk(N)1866 223 y Ft(:)46 321 y Fx(\(ii\))i Fj(L)n(et)h Ft(f)s(;)8 b(g)16 b Fx(:)d Fq(T)364 303 y Fk(N)408 321 y Fu(!)h Fq(R)g Fj(have)k(an)g(analytic)g(extension) i(to)d Fx(\001)1206 328 y Fk(\030)1242 321 y Fj(then)773 409 y Fu(k)p Ft(f)5 b(g)r Fu(k)877 416 y Fk(\030)910 409 y Fu(\024)13 b(k)p Ft(f)5 b Fu(k)1041 416 y Fk(\030)1060 409 y Fu(k)p Ft(g)r Fu(k)1135 416 y Fk(\030)1172 409 y Ft(:)46 496 y Fx(\(iii\))16 b Fj(L)n(et)i Fx(0)c Ft(<)h(\030)i(<)413 483 y Fx(\026)408 496 y Ft(\030)s Fj(;)g(let)i Ft(h)c Fx(:)f Fq(T)640 478 y Fk(N)682 496 y Fu(\002)d(f)p Ft(y)16 b Fu(2)e Fq(R)883 478 y Fk(s)913 496 y Fx(:)g Fu(j)p Ft(y)f Fu(\000)e Ft(y)1066 503 y FE(0)1085 496 y Fu(j)j(\024)g Ft(r)q Fu(g)h(!)f Fq(R)h Fj(have)j(an)g(analytic)h(extension)46 557 y(on)e Fx(\001)160 559 y FE(\026)156 567 y Fk(\030)183 557 y Fu(\002)244 540 y Fp(b)231 557 y Ft(B)271 538 y Fk(s)268 569 y(r)289 557 y Fx(\()p Ft(y)332 564 y FE(0)352 557 y Fx(\))p Fj(,)f Ft(f)j Fx(:)14 b Fq(T)509 538 y Fk(N)553 557 y Fu(!)g Fq(R)656 538 y Fk(N)703 557 y Fj(and)j Ft(g)f Fx(:)d Fq(T)899 538 y Fk(N)944 557 y Fu(!)h Fq(R)1047 538 y Fk(s)1078 557 y Fj(have)k(analytic)f(extension)i(on)e Fx(\001)1698 564 y Fk(\030)1716 557 y Fj(.)g(Assume)46 617 y(that)g Fu(k)p Ft(f)5 b Fu(k)224 624 y Fk(\030)257 617 y Fu(\024)315 604 y Fx(\026)310 617 y Ft(\030)14 b Fu(\000)d Ft(\030)r Fj(,)18 b Fu(k)p Ft(g)13 b Fu(\000)e Ft(y)586 624 y FE(0)605 617 y Fu(k)630 624 y Fk(\030)663 617 y Fu(\024)i Ft(r)q Fj(.)18 b(Then,)g(denoting)h Ft(\036)p Fx(\()p Ft(\022)q Fx(\))8 b Fu(\021)g Fx(\()p Ft(\022)k Fx(+)f Ft(f)5 b Fx(\()p Ft(\022)q Fx(\))p Ft(;)j(g)r Fx(\()p Ft(\022)q Fx(\)\))p Fj(,)18 b(one)g(has)782 704 y Fu(k)p Ft(h)11 b Fu(\016)g Ft(\036)p Fu(k)936 711 y Fk(\030)969 704 y Fu(\024)i(k)p Ft(h)p Fu(k)1103 706 y FE(\026)1099 715 y Fk(\030)q(;r)1162 704 y Ft(:)46 811 y Fw(Pro)r(of)18 b Fx(\(i\):)e(The)g(claim)e(follo)o(ws)i(immedi)o (ately)d(b)o(y)j(expanding)g Ft(f)22 b Fx(in)16 b(F)l(ourier)f(series.) 46 890 y(\(ii\):)g(In)h(the)g(follo)o(wing)g(sums)f(the)h(indices)f Ft(n;)8 b(m)16 b Fx(run)h(o)o(v)o(er)e Fq(Z)1198 872 y Fk(N)1229 890 y Fx(.)332 983 y Fu(k)p Ft(f)5 b(g)r Fu(k)436 990 y Fk(\030)505 983 y Fu(\021)594 942 y Fp(X)613 1029 y Fk(n)662 983 y Fu(j)p Fx(\()p Ft(f)g(g)r Fx(\))768 990 y Fk(n)792 983 y Fu(j)j Fx(exp\()p Fu(j)p Ft(n)p Fu(j)p Ft(\030)r Fx(\))14 b(=)1073 942 y Fp(X)1092 1029 y Fk(n)1141 933 y Fp(\014)1141 958 y(\014)1141 983 y(\014)1163 942 y(X)1177 1029 y Fk(m)1231 983 y Ft(f)1255 990 y Fk(m)1289 983 y Ft(g)1312 990 y Fk(n)p FF(\000)p Fk(m)1394 933 y Fp(\014)1394 958 y(\014)1394 983 y(\014)25 b Fx(exp)o(\()p Fu(j)p Ft(n)p Fu(j)p Ft(\030)s Fx(\))505 1095 y Fu(\024)595 1054 y Fp(X)594 1141 y Fk(n;m)665 1095 y Fx(exp)7 b Fu(j)p Ft(m)p Fu(j)h Fx(exp)g Fu(j)p Ft(n)j Fu(\000)g Ft(m)p Fu(jj)p Ft(f)1108 1102 y Fk(m)1140 1095 y Fu(j)17 b(j)p Ft(g)1208 1102 y Fk(n)p FF(\000)p Fk(m)1290 1095 y Fu(j)506 1206 y Fx(=)50 b Fu(k)p Ft(f)5 b Fu(k)673 1213 y Fk(\030)692 1206 y Fu(k)p Ft(g)r Fu(k)767 1213 y Fk(\030)802 1206 y Ft(:)46 1293 y Fx(\(iii\):)15 b(In)i(the)h(follo)o(wing)f(sums)f(the) h(indices)g Ft(n;)8 b(m)17 b Fx(run)g(o)o(v)o(er)g Fq(Z)1223 1275 y Fk(N)1253 1293 y Fx(,)g(the)g(index)g Ft(k)i Fx(runs)f(o)o(v)o (er)f Fq(N)1797 1275 y Fk(s)1833 1293 y Fx(and)46 1353 y Ft(j)i Fx(o)o(v)o(er)c Fq(N)p Fx(.)i(Using)f(\(ii\),)f(one)i(gets)188 1465 y Fu(j)p Ft(h)11 b Fu(\016)g Ft(\036)p Fu(j)320 1472 y Fk(\030)380 1465 y Fx(=)460 1424 y Fp(X)479 1511 y Fk(n)528 1416 y Fp(\014)528 1440 y(\014)528 1465 y(\014)564 1424 y(X)550 1516 y Fk(m;k)q(;j)650 1432 y Ft(h)678 1439 y Fk(m;k)p 650 1454 91 2 v 677 1500 a Ft(j)s Fx(!)762 1465 y Ft(i)779 1445 y Fk(j)813 1417 y Fp(\020)838 1465 y Fx(\()p Ft(m)f Fu(\001)h Ft(f)5 b Fx(\))983 1445 y Fk(j)1002 1465 y Fx(\()p Ft(g)13 b Fu(\000)e Ft(y)1131 1472 y FE(0)1150 1465 y Fx(\))1169 1445 y Fk(k)1191 1417 y Fp(\021)1215 1492 y Fk(n)p FF(\000)p Fk(m)1298 1416 y Fp(\014)1298 1440 y(\014)1298 1465 y(\014)24 b Fx(exp)o(\()p Fu(j)p Ft(n)p Fu(j)p Ft(\030)r Fx(\))380 1614 y Fu(\024)489 1573 y Fp(X)460 1665 y Fk(n;m;k)q(;j)586 1614 y Fx(exp)o(\()p Fu(j)p Ft(m)p Fu(j)p Ft(\030)r Fx(\))797 1581 y Fu(j)p Ft(h)839 1588 y Fk(m;k)902 1581 y Fu(j)p 797 1603 119 2 v 838 1649 a Ft(j)s Fx(!)937 1565 y Fp(\014)937 1589 y(\014)937 1614 y(\014)950 1566 y(\020)975 1614 y Fx(\()p Ft(m)11 b Fu(\001)g Ft(f)5 b Fx(\))1121 1594 y Fk(j)1139 1614 y Fx(\()p Ft(g)14 b Fu(\000)c Ft(y)1268 1621 y FE(0)1288 1614 y Fx(\))1307 1594 y Fk(k)1328 1566 y Fp(\021)1353 1641 y Fk(n)p FF(\000)p Fk(m)1435 1565 y Fp(\014)1435 1589 y(\014)1435 1614 y(\014)25 b Fx(exp)o(\()p Fu(j)p Ft(n)11 b Fu(\000)g Ft(m)p Fu(j)p Ft(\030)r Fx(\))380 1763 y(=)473 1722 y Fp(X)460 1814 y Fk(m;k)q(;j)560 1730 y Fu(j)p Ft(h)602 1737 y Fk(m;k)664 1730 y Fu(j)p 560 1752 V 600 1798 a Ft(j)s Fx(!)699 1763 y Fu(k)p Fx(\()p Ft(m)g Fu(\001)f Ft(f)5 b Fx(\))869 1743 y Fk(j)888 1763 y Fx(\()p Ft(g)13 b Fu(\000)e Ft(y)1017 1770 y FE(0)1037 1763 y Fx(\))1056 1743 y Fk(k)1077 1763 y Fu(k)1102 1770 y Fk(\030)1129 1763 y Fx(exp)o(\()p Fu(j)p Ft(m)p Fu(j)p Ft(\030)r Fx(\))380 1912 y Fu(\024)473 1871 y Fp(X)460 1963 y Fk(m;k)q(;j)560 1879 y Fu(j)p Ft(h)602 1886 y Fk(m;k)664 1879 y Fu(j)p 560 1901 V 600 1947 a Ft(j)s Fx(!)699 1912 y Fu(k)p Ft(m)g Fu(\001)f Ft(f)5 b Fu(k)856 1889 y Fk(j)856 1925 y(\030)892 1912 y Fu(k)p Ft(g)13 b Fu(\000)e Ft(y)1027 1919 y FE(0)1046 1912 y Fu(k)1071 1887 y FF(j)p Fk(k)q FF(j)1071 1925 y Fk(\030)1137 1912 y Fx(exp)o(\()p Fu(j)p Ft(m)p Fu(j)p Ft(\030)r Fx(\))380 2061 y Fu(\024)473 2020 y Fp(X)460 2112 y Fk(m;k)q(;j)560 2028 y Fu(j)p Ft(h)602 2035 y Fk(m;k)664 2028 y Fu(j)p 560 2050 V 600 2096 a Ft(j)s Fx(!)699 2061 y Fu(j)p Ft(m)p Fu(j)770 2041 y Fk(j)804 2061 y Fx(\()828 2048 y(\026)823 2061 y Ft(\030)j Fu(\000)c Ft(\030)r Fx(\))949 2041 y Fk(j)985 2061 y Ft(r)1008 2041 y FF(j)p Fk(k)q FF(j)1057 2061 y Fx(exp\()p Fu(j)p Ft(m)p Fu(j)p Ft(\030)r Fx(\))380 2181 y(=)42 b Fu(k)p Ft(h)p Fu(k)542 2183 y FE(\026)538 2191 y Fk(\030)q(;r)600 2181 y Ft(:)p 672 2181 21 42 v 46 2269 a Fw(Remark)16 b(4.2)24 b Fx(\(i\))15 b(Note)g(that)h(the)f (result)f(migh)o(t)g(b)q(e)h(empt)o(y)e(if)i Ft(!)i Fx(is)e(\\to)q(o)i (w)o(ell)d(appro)o(ximable)g(b)o(y)46 2329 y(rationals)j(v)o(ectors".)e (If)h Ft(!)i Fx(satis\014es)f(\(4.3\))f(then)h(one)f(c)o(hec)o(ks)f (easily)g(that)143 2448 y Ft(\033)171 2455 y Fk(pk)209 2448 y Fx(\()p Ft(\016)r Fx(\))f Fu(\024)337 2375 y Fp(\()391 2417 y Ft(\015)419 2399 y Fk(p)439 2417 y Ft(\016)463 2399 y FF(\000)p FE(\()p FF(j)p Fk(k)q FF(j)p FE(+)p Fk(p\034)t FE(\))623 2417 y Fx(\()p Fu(j)p Ft(k)r Fu(j)d Fx(+)g Ft(p\034)6 b Fx(\)!)16 b Ft(;)85 b Fx(if)16 b Ft(p)e Fu(\025)g Fx(0)p Ft(;)391 2478 y Fx(\012)426 2460 y FF(j)p Fk(p)p FF(j)466 2478 y Ft(\016)490 2460 y FF(\000)p FE(\()p FF(j)p Fk(k)q FF(j)p FE(+)p FF(j)p Fk(p)p FF(j)p FE(\))650 2478 y Fx(\()p Fu(j)p Ft(k)r Fu(j)d Fx(+)g Fu(j)p Ft(p)p Fu(j)p Fx(\)!)16 b Ft(;)57 b Fx(if)16 b Ft(p)e(<)g Fx(0)j Ft(;)203 b Fx(\012)8 b Fu(\021)17 b Fx(max)23 b Fu(j)p Ft(!)1607 2485 y Fk(i)1621 2478 y Fu(j)16 b Ft(:)1787 2448 y Fx(\(4.20\))46 2566 y(\(ii\))f(It)h(is)g (easy)g(to)h(c)o(hec)o(k)d(that)j(\(i\))e(and)i(\(iii\))e(of)i(Lemma)d (4.1)i(holds)h(also)g(if)e Ft(f)5 b Fx(,)16 b(resp)q(ectiv)o(ely)l(,)e Ft(h)i Fx(are)46 2627 y(v)o(ector)f(v)m(alued.)955 2863 y(16)p eop %%Page: 17 17 17 16 bop 46 78 a Fe(4.3)69 b(KAM)23 b(Algorithm)46 170 y Fx(Here)15 b(w)o(e)h(describ)q(e)f(the)h(\\KAM)g(algorithm")g(asso)q (ciated)h(to)f(the)g(sc)o(heme)e(of)j Fu(x)f Fx(4.1,)g Fj(i.e.)i Fx(w)o(e)e(equip)46 231 y(the)d(algebraic)f(sc)o(heme)f (describ)q(ed)h(in)h(Prop)q(osition)h(4.1)f(with)g(\\accurate")g(and)h (detailed)e(estimates)46 291 y(so)17 b(as)h(to)f(end)g(up)g(with)f(a)i (map)e Fu(K)q Fx(,)g(whic)o(h)g(to)h(giv)o(en)f(b)q(ounds)i(on)g(norms) e(of)h(the)f(relev)m(an)o(t)g(ob)s(jects)46 351 y(relativ)o(e)k(to)j(a) g(certain)f(appro)o(ximate)f(solution)i(\()p Ft(u;)8 b(v)r Fx(\),)21 b(asso)q(ciates)i(corresp)q(onding)h(b)q(ounds)g(on)46 411 y(the)18 b(new)g(appro)o(ximation)f(\()p Ft(u)608 393 y FF(0)620 411 y Ft(;)8 b(v)668 393 y FF(0)679 411 y Fx(\).)18 b(More)g(precisely)l(,)e(let)h(us)i(start)g(b)o(y)e(making) h(quan)o(titativ)o(e)e(the)46 471 y(h)o(yp)q(otheses)g(form)o(ulated)f (in)h Fu(x)g Fx(4.1.)46 585 y Fw(Assumption)i(4.2)24 b Fj(L)n(et)18 b Fx(0)f Ft(<)f(\030)j(<)733 572 y Fx(\026)728 585 y Ft(\030)r Fj(,)g Ft(r)q Fj(,)p 842 546 40 2 v 19 w Ft(E)s Fj(,)g Ft(E)951 592 y Fk(p;q)1017 585 y Fj(\()p Ft(p;)8 b(q)18 b Fu(2)e Fq(N)p Fj(\))k(b)n(e)f(such)g(that)g Fx(\()p Ft(x;)8 b(y)r Fx(\))16 b Fu(!)g Ft(h)p Fx(\()p Ft(x;)8 b(y)r Fx(\))18 b Fj(is)46 655 y(r)n(e)n(al)f(analytic)h(on)g Fx(\001)438 657 y FE(\026)434 666 y Fk(\030)463 655 y Fu(\002)p 513 616 V 11 w Ft(B)553 626 y Fk(N)553 668 y(r)586 655 y Fx(\()p Ft(y)629 662 y FE(0)649 655 y Fx(\))f Fj(and)533 765 y Fu(k)p Fx(\()p Ft(h)605 772 y Fk(y)q(y)644 765 y Fx(\))663 745 y FF(\000)p FE(1)711 765 y Fu(k)740 767 y FE(\026)736 776 y Fk(\030;r)795 765 y Fu(\024)p 847 726 V 13 w Ft(E)k(;)107 b Fu(k)p Ft(@)1079 745 y Fk(p)1076 778 y(x)1099 765 y Ft(@)1128 745 y Fk(q)1125 778 y(y)1146 765 y Ft(h)p Fu(k)1203 767 y FE(\026)1199 776 y Fk(\030)q(;r)1258 765 y Fu(\024)14 b Ft(E)1347 772 y Fk(p;q)1411 765 y Fx(;)362 b(\(4.21\))46 880 y Fj(assume)17 b(that)g(the)g(appr)n(oximate)g(solution)g Fx(\()p Ft(u;)8 b(v)r Fx(\))16 b Fj(is)h(r)n(e)n(al)f(analytic)i(on)f Fx(\001)1422 887 y Fk(\030)1457 880 y Fj(and)g(let)h Ft(U)5 b Fj(,)17 b Ft(V)12 b Fj(,)k Ft(M)5 b Fj(,)p 1845 841 53 2 v 18 w Ft(M)g Fj(,)54 925 y Fp(e)46 941 y Ft(V)11 b Fj(,)18 b Ft(F)7 b Fj(,)16 b Ft(G)p Fj(,)269 925 y Fp(e)259 941 y Ft(T)24 b Fj(b)n(e)18 b(p)n(ositive)f(numb)n(ers)h(b)n (ounding)g(the)g(fol)r(lowing)i(norms)396 1051 y Fu(k)p Ft(u)p Fu(k)474 1058 y Fk(\030)506 1051 y Fu(\024)14 b Ft(U)22 b(;)58 b Fu(k)p Ft(v)r Fu(k)762 1058 y Fk(\030)794 1051 y Fu(\024)14 b Ft(V)28 b(;)58 b Fu(kMk)1085 1058 y Fk(\030)1117 1051 y Fu(\024)14 b Ft(M)23 b(;)57 b Fu(kM)1396 1030 y FF(\000)p FE(1)1443 1051 y Fu(k)1468 1058 y Fk(\030)1501 1051 y Fu(\024)p 1553 1012 V 13 w Ft(M)23 b(;)396 1123 y Fu(k)p Ft(v)445 1130 y Fk(\022)464 1123 y Fu(k)489 1130 y Fk(\030)521 1123 y Fu(\024)582 1107 y Fp(e)574 1123 y Ft(V)29 b(;)57 b Fu(k)p Ft(f)5 b Fu(k)781 1130 y Fk(\030)814 1123 y Fu(\024)14 b Ft(F)24 b(;)57 b Fu(k)p Ft(g)r Fu(k)1069 1130 y Fk(\030)1102 1123 y Fu(\024)14 b Ft(G)j(;)58 b Fu(jhT)13 b(i)1374 1099 y FF(\000)p FE(1)1422 1123 y Fu(j)g(\024)1512 1107 y Fp(e)1502 1123 y Ft(T)24 b(;)218 b Fx(\(4.22\))46 1233 y Fj(wher)n(e)16 b Fu(M)p Fj(,)g Fu(T)d Fj(,)i Ft(f)22 b Fj(and)16 b Ft(g)i Fj(ar)n(e,)d(r)n(esp) n(e)n(ctively,)i(as)e(in)i Fx(\(4.4\))p Fj(,)f Fx(\(4.5\))g Fj(and)g Fx(\(4.6\))p Fj(.)h(Final)r(ly)g(assume)f(that)604 1343 y Ft(U)j Fu(\024)713 1330 y Fx(\026)708 1343 y Ft(\030)14 b Fu(\000)d Ft(\030)20 b(;)107 b(\032)8 b Fu(\021)h(k)p Ft(v)j Fu(\000)f Ft(y)1171 1350 y FE(0)1191 1343 y Fu(k)1216 1350 y Fk(\030)1248 1343 y Ft(<)j(r)19 b(:)432 b Fx(\(4.23\))46 1482 y(No)o(w)16 b(let)f(0)g Ft(<)e(\016)j Fu(\024)d Ft(\030)19 b Fx(and)e(de\014ne)865 1543 y Ft(\030)888 1522 y FF(0)908 1543 y Fu(\021)8 b Ft(\030)14 b Fu(\000)d Ft(\016)18 b(:)693 b Fx(\(4.24\))46 1630 y(In)16 b(the)g(rest)g(of)g (this)h(section)e(w)o(e)h(shall)g(de\014ne)g(the)g Fv(KAM)g(map)p Fx(,)f(i.e.,)f(an)i(explicit)f(map)340 1740 y Fu(K)f Fx(:)g(\()p Ft(U;)8 b(V)s(;)g(M)r(;)p 618 1701 V 8 w(M)d(;)700 1724 y Fp(e)692 1740 y Ft(V)11 b(;)d(F)q(;)g(G;)879 1724 y Fp(e)868 1740 y Ft(T)f Fx(\))14 b Fu(!)f Fx(\()p Ft(U)1057 1719 y FF(0)1069 1740 y Ft(;)8 b(V)1130 1719 y FF(0)1142 1740 y Ft(;)g(M)1216 1719 y FF(0)1228 1740 y Ft(;)p 1250 1701 V 8 w(M)1302 1711 y FF(0)1314 1740 y Ft(;)1344 1724 y Fp(e)1336 1740 y Ft(V)1375 1719 y FF(0)1386 1740 y Ft(;)g(F)1447 1719 y FF(0)1458 1740 y Ft(;)g(G)1518 1719 y FF(0)1530 1740 y Ft(;)1562 1724 y Fp(e)1552 1740 y Ft(T)1588 1719 y FF(0)1599 1740 y Fx(\))169 b(\(4.25\))46 1850 y(where)10 b(\()p Ft(U)238 1832 y FF(0)250 1850 y Ft(;)e(:::;)g(G)374 1832 y FF(0)385 1850 y Ft(;)417 1834 y Fp(e)407 1850 y Ft(T)443 1832 y FF(0)454 1850 y Fx(\))i(are)h(b)q(ounds)i(on)e(the)f(norms)h Fu(k)p Ft(u)1059 1832 y FF(0)1070 1850 y Fu(k)1095 1857 y Fk(\030)1112 1848 y Fn(0)1125 1850 y Fx(,...,)e Fu(k)p Ft(g)1254 1832 y FF(0)1265 1850 y Fu(k)1290 1857 y Fk(\030)1307 1848 y Fn(0)1331 1850 y Fx(and)i(on)h(the)e(n)o(um)o(b)q(er)f Fu(jh)q(T)1806 1832 y FF(0)1818 1850 y Fu(i)1837 1826 y FF(\000)p FE(1)1885 1850 y Fu(j)p Fx(,)46 1910 y(with)15 b Ft(u)184 1892 y FF(0)195 1910 y Fx(,)f Ft(v)249 1892 y FF(0)260 1910 y Fx(,)h Ft(f)318 1892 y FF(0)330 1910 y Fx(,)f Ft(g)383 1892 y FF(0)410 1910 y Fx(de\014ned)g(in)h(Prop)q (osition)h(4.1,)e(while)g Fu(M)1169 1892 y FF(0)1195 1910 y Fx(and)i Fu(T)1329 1892 y FF(0)1355 1910 y Fx(are)f(\(ob)o (viously\))f(de\014ned)h(as)330 2025 y Fu(M)390 2004 y FF(0)410 2025 y Fu(\021)8 b Ft(I)14 b Fx(+)d Ft(u)570 2004 y FF(0)570 2037 y Fk(\022)598 2025 y Fu(\021)d(M)j Fx(+)g Ft(z)788 2032 y Fk(\022)823 2025 y Ft(;)105 b Fu(T)982 2004 y FF(0)1002 2025 y Fu(\021)8 b(M)1109 2004 y FF(0)1121 1999 y(\000)p FE(1)1168 2025 y Ft(h)1196 2032 y Fk(y)q(y)1235 2025 y Fx(\()p Ft(\022)k Fx(+)f Ft(u)1366 2004 y FF(0)1378 2025 y Ft(;)d(v)1426 2004 y FF(0)1437 2025 y Fx(\))16 b Fu(M)1532 2004 y FF(0)1543 1999 y(\000)p Fk(T)1615 2025 y Ft(:)46 2135 y Fx(W)l(e)g(start)h(no)o (w)f(to)h(w)o(ork)f(out)h(the)f(necessary)g(estimates.)e(By)h(Lemma)g (4.1)h(and)h(\(4.22\),)f(w)o(e)g(get)754 2252 y Fu(kT)c(k)843 2259 y Fk(\030)876 2252 y Fu(\024)h Ft(T)i Fu(\021)p 1019 2213 V 8 w Ft(M)1072 2223 y FE(2)1091 2252 y Ft(E)1127 2259 y FE(0)p Fk(;)p FE(2)1191 2252 y Ft(:)46 2362 y Fw(Remark)h(4.3)24 b Fx(In)g(principle,)e(the)i(b)q(ound)h(on)g Fu(kT)13 b(k)1061 2369 y Fk(\030)1104 2362 y Fx(could)24 b(b)q(e)g(impro)o(v)o(ed)e(replacing)h Ft(E)1789 2369 y FE(0)p Fk(;)p FE(2)1861 2362 y Fx(b)o(y)46 2422 y Fu(k)p Fx(\()p Ft(h)118 2404 y FE(0)118 2434 y Fk(y)q(y)157 2422 y Fx(\))176 2404 y FF(\000)p FE(1)223 2422 y Fu(k)248 2429 y Fk(\030)283 2422 y Fx(\(without)16 b(in)o(v)o(oking)e(p)q(oin)o (t)i(\(iii\))f(of)h(Lemma)d(4.1\);)j(in)f(practice,)f(ho)o(w)o(ev)o (er,)g(suc)o(h)i(norm)46 2482 y(is)g(di\016cult)f(to)h(ev)m(aluate)g (accurately)g(and)h(one)f(w)o(ould)g(not)h(get)f(signi\014can)o(t)g(b)q (etter)g(estimates.)955 2863 y(17)p eop %%Page: 18 18 18 17 bop 46 78 a Fx(All)13 b(other)i(estimates)f(are)h(imme)o(diatel)o (y)d(obtained)j(from)f(Lemma)f(4.1)i(\(and)h(from)e(the)g (de\014nitions)46 138 y(giv)o(en)h(in)h(Prop)q(osition)i(4.1\).)e(Here) f(is)h(the)g(complete)e(list:)571 241 y Fu(kT)635 220 y FF(\000)p FE(1)682 241 y Fu(k)707 248 y Fk(\030)768 241 y Fu(\024)p 848 202 36 2 v 41 w Ft(T)h Fu(\021)8 b Ft(M)991 220 y FE(2)p 1011 202 40 2 v 1011 241 a Ft(E)19 b Fx(;)637 313 y Fu(k)p Ft(b)p Fu(k)708 320 y Fk(\030)768 313 y Fu(\024)41 b Ft(F)880 320 y FF(\003)908 313 y Fu(\021)963 297 y Fp(e)955 313 y Ft(V)11 b(F)17 b Fx(+)11 b Ft(M)5 b(G)17 b Fx(;)571 412 y Fu(kB)r(k)656 426 y Fk(\030)q FF(\000)705 412 y Fo(\016)p 704 418 16 2 v 704 439 a Fl(2)768 412 y Fu(\024)41 b Ft(s)871 419 y FE(1)891 412 y Ft(F)923 419 y FF(\003)958 412 y Ft(;)106 b(s)1101 419 y FE(1)1129 412 y Fu(\021)8 b Fx(2)p Ft(\033)1228 419 y FE(01)1265 364 y Fp(\020)1295 379 y Ft(\016)p 1295 401 25 2 v 1295 446 a Fx(2)1324 364 y Fp(\021)1365 412 y Fx(;)658 513 y Fu(j)p Ft(c)693 520 y FE(1)712 513 y Fu(j)42 b(\024)858 497 y Fp(e)848 513 y Ft(T)883 465 y Fp(\020)p 908 474 53 2 v 48 x Ft(M)6 b(F)17 b Fx(+)11 b Ft(T)c(\033)1123 520 y FE(10)1159 513 y Fx(\()p Ft(\030)r Fx(\))p Ft(F)1252 520 y FF(\003)1272 465 y Fp(\021)1313 513 y Fx(;)46 622 y(\(note)15 b(that)g(in)g(the)f(last)i(estimate)d(w)o (e)h(ha)o(v)o(e)g(used)h(the)g(fact)g(that)g(the)g(suprem)o(um)d(norm)i (sup)1791 633 y Fc(T)1812 624 y Fo(N)1854 622 y Fu(j)8 b(\001)g(j)46 682 y Fx(is)16 b(dominated)f(b)o(y)h(the)g(0{F)l(ourier)g (norm)g Fu(k)11 b(\001)f(k)918 689 y FE(0)938 682 y Fx(\);)129 809 y Fu(kAk)219 816 y Fk(\030)236 807 y Fn(0)299 809 y Fu(\021)49 b(k)p Ft(D)453 788 y FF(\000)p FE(1)501 809 y Fu(B)r(k)561 816 y Fk(\030)578 807 y Fn(0)604 809 y Fu(\024)14 b Ft(s)680 816 y FE(2)700 809 y Ft(F)732 816 y FF(\003)767 809 y Ft(;)106 b(s)910 816 y FE(2)938 809 y Fu(\021)8 b Ft(\033)1013 816 y FE(10)1050 761 y Fp(\020)1080 775 y Ft(\016)p 1080 797 25 2 v 1080 843 a Fx(2)1109 761 y Fp(\021)1134 809 y Ft(s)1157 816 y FE(1)1193 809 y Fx(;)87 928 y Fu(k)p Ft(b)133 935 y FE(0)153 928 y Fu(k)178 942 y Fk(\030)q FF(\000)227 928 y Fo(\016)p 227 934 16 2 v 227 955 a Fl(2)299 928 y Fu(\024)49 b Ft(B)424 935 y FF(\003)453 928 y Fu(\021)8 b Ft(T)f(\033)564 935 y FE(10)600 880 y Fp(\020)630 894 y Ft(\016)p 630 916 25 2 v 630 962 a Fx(2)659 880 y Fp(\021)684 928 y Ft(F)716 935 y FF(\003)746 928 y Fx(+)k Ft(T)841 912 y Fp(e)831 928 y Ft(T)866 880 y Fp(\020)p 891 889 53 2 v 48 x Ft(M)5 b(F)18 b Fx(+)11 b Ft(T)c(F)1110 935 y FF(\003)1128 928 y Ft(\033)1156 935 y FE(10)1193 928 y Fx(\()p Ft(\030)r Fx(\))1254 880 y Fp(\021)1291 928 y Fx(+)p 1340 889 V 11 w Ft(M)e(F)23 b Fx(;)181 1047 y Fu(j)p Ft(c)216 1054 y FE(2)235 1047 y Fu(j)50 b(\024)f Ft(M)5 b Fu(k)p Ft(D)505 1027 y FF(\000)p FE(1)554 1047 y Ft(b)575 1054 y FE(0)594 1047 y Fu(k)619 1054 y FE(0)653 1047 y Fu(\024)13 b Ft(M)5 b(\033)785 1054 y FE(10)823 999 y Fp(\020)848 1047 y Ft(\030)13 b Fu(\000)938 1013 y Ft(\016)p 937 1036 25 2 v 937 1081 a Fx(2)966 999 y Fp(\021)991 1047 y Ft(B)1028 1054 y FF(\003)1064 1047 y Fx(;)144 1166 y Fu(k)p Ft(z)r Fu(k)219 1173 y Fk(\030)236 1164 y Fn(0)299 1166 y Fu(\024)49 b Ft(B)424 1173 y FF(\003)444 1166 y Ft(s)467 1173 y FE(3)503 1166 y Ft(;)106 b(s)646 1173 y FE(3)674 1166 y Fu(\021)8 b Ft(M)d(\033)801 1173 y FE(10)838 1118 y Fp(\020)868 1133 y Ft(\016)p 868 1155 V 868 1200 a Fx(2)897 1118 y Fp(\021)933 1166 y Fx(+)11 b Ft(M)1034 1146 y FE(2)1054 1166 y Ft(\033)1082 1173 y FE(10)1119 1118 y Fp(\020)1144 1166 y Ft(\030)j Fu(\000)1234 1133 y Ft(\016)p 1234 1155 V 1234 1200 a Fx(2)1263 1118 y Fp(\021)1304 1166 y Fx(;)130 1256 y Fu(k)p Ft(u)183 1235 y FF(0)194 1256 y Fu(k)219 1263 y Fk(\030)236 1254 y Fn(0)299 1256 y Fu(\021)49 b(k)p Ft(u)11 b Fx(+)g Ft(z)r Fu(k)550 1263 y Fk(\030)567 1254 y Fn(0)594 1256 y Fu(\024)i Ft(U)k Fx(+)11 b Fu(k)p Ft(z)r Fu(k)820 1263 y Fk(\030)837 1254 y Fn(0)299 1329 y Fu(\024)49 b Ft(U)425 1308 y FF(0)446 1329 y Fu(\021)8 b Ft(U)16 b Fx(+)11 b Ft(B)628 1336 y FF(\003)648 1329 y Ft(s)671 1336 y FE(3)707 1329 y Fx(;)102 1401 y Fu(k)p Ft(D)q(z)r Fu(k)218 1408 y Fk(\030)235 1399 y Fn(0)299 1401 y Fx(=)50 b Fu(k)p Fx(\()p Ft(D)q Fu(M)p Fx(\)\()p Ft(D)611 1381 y FF(\000)p FE(1)660 1401 y Ft(b)681 1408 y FE(0)711 1401 y Fx(+)12 b Ft(c)782 1408 y FE(2)801 1401 y Fx(\))f(+)g Fu(M)p Ft(b)961 1408 y FE(0)981 1401 y Fu(k)1006 1408 y Fk(\030)1023 1399 y Fn(0)299 1474 y Fu(\024)49 b Ft(B)424 1481 y FF(\003)444 1474 y Ft(s)467 1481 y FE(4)503 1474 y Ft(;)106 b(s)646 1481 y FE(4)674 1474 y Fu(\021)8 b Ft(s)744 1481 y FE(3)763 1474 y Ft(\033)791 1481 y FF(\000)p FE(10)856 1474 y Fx(\()p Ft(\016)r Fx(\))i(+)h Ft(M)22 b Fx(;)127 1547 y Fu(k)p Ft(z)175 1554 y Fk(\022)194 1547 y Fu(k)219 1554 y Fk(\030)236 1545 y Fn(0)299 1547 y Fx(=)50 b Fu(kM)472 1554 y Fk(\022)492 1547 y Fx(\()p Ft(D)552 1526 y FF(\000)p FE(1)600 1547 y Ft(b)621 1554 y FE(0)651 1547 y Fx(+)11 b Ft(c)721 1554 y FE(2)741 1547 y Fx(\))g(+)g Fu(M)p Ft(D)921 1526 y FF(\000)p FE(1)969 1547 y Ft(@)995 1554 y Fk(\022)1014 1547 y Ft(b)1035 1554 y FE(0)1055 1547 y Fu(k)1080 1554 y Fk(\030)1097 1545 y Fn(0)299 1645 y Fu(\024)49 b Ft(B)424 1652 y FF(\003)444 1645 y Ft(s)467 1652 y FE(5)503 1645 y Ft(;)106 b(s)646 1652 y FE(5)674 1645 y Fu(\021)8 b Ft(\033)749 1652 y FE(01)786 1645 y Fx(\()p Ft(\016)r Fx(\))p Ft(s)871 1652 y FE(3)901 1645 y Fx(+)j Ft(M)5 b(\033)1030 1652 y FE(11)1067 1597 y Fp(\020)1098 1612 y Ft(\016)p 1097 1634 V 1097 1680 a Fx(2)1127 1597 y Fp(\021)1168 1645 y Fx(;)85 1738 y Fu(k)p Ft(D)q(z)174 1745 y Fk(\022)194 1738 y Fu(k)219 1745 y Fk(\030)236 1736 y Fn(0)299 1738 y Fx(=)50 b Fu(k)p Fx(\()p Ft(D)q Fu(M)532 1745 y Fk(\022)553 1738 y Fx(\)\()p Ft(D)632 1718 y FF(\000)p FE(1)679 1738 y Ft(b)700 1745 y FE(0)731 1738 y Fx(+)11 b Ft(c)801 1745 y FE(2)821 1738 y Fx(\))g(+)g Ft(D)q Fu(M)p Ft(D)1042 1718 y FF(\000)p FE(1)1090 1738 y Ft(@)1116 1745 y Fk(\022)1136 1738 y Ft(b)1157 1745 y FE(0)1187 1738 y Fx(+)g Fu(M)1296 1745 y Fk(\022)1316 1738 y Ft(b)1337 1745 y FE(0)1367 1738 y Fx(+)g Fu(M)p Ft(@)1502 1745 y Fk(\022)1521 1738 y Ft(b)1542 1745 y FE(0)1562 1738 y Fu(k)1587 1745 y Fk(\030)1604 1736 y Fn(0)299 1837 y Fu(\024)49 b Ft(B)424 1844 y FF(\003)444 1837 y Ft(s)467 1844 y FE(6)503 1837 y Ft(;)57 b(s)597 1844 y FE(6)625 1837 y Fu(\021)8 b Ft(\033)700 1844 y FF(\000)p FE(11)764 1837 y Fx(\()p Ft(\016)r Fx(\))p Ft(s)849 1844 y FE(3)880 1837 y Fx(+)j Ft(M)981 1789 y Fp(h)1001 1837 y Ft(\033)1029 1844 y FF(\000)p FE(10)1093 1837 y Fx(\()p Ft(\016)r Fx(\))p Ft(\033)1183 1844 y FE(11)1220 1789 y Fp(\020)1250 1803 y Ft(\016)p 1249 1825 V 1249 1871 a Fx(2)1279 1789 y Fp(\021)1315 1837 y Fx(+)g Ft(\033)1392 1844 y FE(01)1429 1837 y Fx(\()p Ft(\016)r Fx(\))f(+)h Ft(\033)1578 1844 y FE(01)1615 1789 y Fp(\020)1645 1803 y Ft(\016)p 1645 1825 V 1645 1871 a Fx(2)1674 1789 y Fp(\021i)1735 1837 y Fx(;)133 1938 y Fu(k)p Ft(w)q Fu(k)219 1945 y Fk(\030)236 1936 y Fn(0)299 1938 y Fu(\024)p 387 1899 40 2 v 49 w Ft(E)427 1889 y Fp(\020)451 1938 y Fu(k)p Ft(D)q(z)r Fu(k)567 1945 y Fk(\030)584 1936 y Fn(0)609 1938 y Fx(+)g Ft(E)694 1945 y FE(1)p Fk(;)p FE(1)741 1938 y Fu(k)p Ft(z)r Fu(k)816 1945 y Fk(\030)833 1936 y Fn(0)857 1938 y Fx(+)g Ft(F)945 1889 y Fp(\021)299 2021 y Fu(\024)p 387 1982 V 49 w Ft(E)t Fx(\()p Ft(s)469 2028 y FE(7)488 2021 y Ft(B)525 2028 y FF(\003)556 2021 y Fx(+)g Ft(F)c Fx(\))16 b Ft(;)105 b(s)821 2028 y FE(7)849 2021 y Fu(\021)8 b Ft(s)919 2028 y FE(4)950 2021 y Fx(+)j Ft(E)1035 2028 y FE(1)p Fk(;)p FE(1)1082 2021 y Ft(s)1105 2028 y FE(3)1141 2021 y Fx(;)132 2094 y Fu(k)p Ft(v)183 2074 y FF(0)194 2094 y Fu(k)219 2101 y Fk(\030)236 2092 y Fn(0)299 2094 y Fu(\021)49 b(k)p Ft(v)13 b Fx(+)e Ft(w)q Fu(k)559 2101 y Fk(\030)576 2092 y Fn(0)603 2094 y Fu(\024)i Ft(V)23 b Fx(+)11 b Fu(k)p Ft(w)q Fu(k)841 2101 y Fk(\030)858 2092 y Fn(0)299 2167 y Fu(\024)49 b Ft(V)427 2146 y FF(0)447 2167 y Fu(\021)8 b Ft(V)22 b Fx(+)p 593 2128 V 11 w Ft(E)s Fx(\()p Ft(s)674 2174 y FE(7)694 2167 y Ft(B)731 2174 y FF(\003)762 2167 y Fx(+)11 b Ft(F)c Fx(\))15 b(;)115 2239 y Fu(k)p Ft(w)175 2246 y Fk(\022)194 2239 y Fu(k)219 2246 y Fk(\030)236 2237 y Fn(0)299 2239 y Fx(=)50 b Fu(k)p Ft(@)438 2246 y Fk(\022)458 2239 y Fx(\()p Ft(h)505 2219 y FE(0)505 2252 y Fk(y)q(y)544 2239 y Fx(\))563 2219 y FF(\000)p FE(1)610 2239 y Fx(\()p Ft(D)q(z)14 b Fu(\000)d Ft(h)785 2219 y FE(0)785 2252 y Fk(y)q(x)825 2239 y Ft(z)i Fx(+)e Ft(f)5 b Fx(\))12 b(+)f(\()p Ft(h)1066 2219 y FE(0)1066 2252 y Fk(y)q(y)1105 2239 y Fx(\))1124 2219 y FF(\000)p FE(1)1171 2239 y Fx(\()p Ft(D)q(z)1254 2246 y Fk(\022)1285 2239 y Fu(\000)g Ft(@)1361 2246 y Fk(\022)1380 2239 y Ft(h)1408 2219 y FE(0)1408 2252 y Fk(y)q(x)1449 2239 y Ft(z)i Fu(\000)e Ft(h)1563 2219 y FE(0)1563 2252 y Fk(y)q(x)1603 2239 y Ft(z)1626 2246 y Fk(\022)1657 2239 y Fx(+)g Ft(f)1730 2246 y Fk(\022)1749 2239 y Fx(\))p Fu(k)1793 2246 y Fk(\030)1810 2237 y Fn(0)299 2317 y Fu(\024)49 b Ft(\033)415 2324 y FE(01)453 2317 y Fx(\()p Ft(\016)r Fx(\))p 515 2278 V Ft(E)r Fx(\()p Fu(k)p Ft(D)q(z)r Fu(k)688 2324 y Fk(\030)705 2315 y Fn(0)730 2317 y Fx(+)11 b Ft(E)815 2324 y FE(1)p Fk(;)p FE(1)862 2317 y Fu(k)p Ft(z)r Fu(k)937 2324 y Fk(\030)954 2315 y Fn(0)978 2317 y Fx(+)g Ft(F)c Fx(\))387 2397 y(+)p 425 2358 V Ft(E)464 2349 y Fp(h)484 2397 y Fu(k)p Ft(D)q(z)573 2404 y Fk(\022)593 2397 y Fu(k)618 2404 y Fk(\030)635 2395 y Fn(0)659 2397 y Fx(+)k Ft(\033)736 2404 y FE(01)773 2397 y Fx(\()p Ft(\016)r Fx(\))p Ft(E)871 2404 y FE(1)p Fk(;)p FE(1)918 2397 y Fu(k)p Ft(z)r Fu(k)993 2404 y Fk(\030)1010 2395 y Fn(0)1034 2397 y Fx(+)g Ft(E)1119 2404 y FE(1)p Fk(;)p FE(1)1166 2397 y Fu(k)p Ft(z)1214 2404 y Fk(\022)1233 2397 y Fu(k)1258 2404 y Fk(\030)1275 2395 y Fn(0)1299 2397 y Fx(+)g Ft(\033)1376 2404 y FE(01)1413 2397 y Fx(\()p Ft(\016)r Fx(\))p Ft(F)1514 2349 y Fp(i)299 2481 y Fu(\024)p 387 2442 V 49 w Ft(E)t(B)464 2488 y FF(\003)483 2481 y Ft(s)506 2488 y FE(8)537 2481 y Fx(+)p 586 2442 V 11 w Ft(E)s(F)c(s)687 2488 y FE(9)722 2481 y Ft(;)387 2554 y(s)410 2561 y FE(8)438 2554 y Fu(\021)i Ft(\033)514 2561 y FE(01)551 2554 y Fx(\()p Ft(\016)r Fx(\))p Ft(s)636 2561 y FE(7)666 2554 y Fx(+)i Ft(s)738 2561 y FE(6)769 2554 y Fx(+)g Ft(\033)846 2561 y FE(01)883 2554 y Fx(\()p Ft(\016)r Fx(\))p Ft(E)981 2561 y FE(1)p Fk(;)p FE(1)1027 2554 y Ft(s)1050 2561 y FE(3)1081 2554 y Fx(+)g Ft(E)1166 2561 y FE(1)p Fk(;)p FE(1)1213 2554 y Ft(s)1236 2561 y FE(5)1272 2554 y Ft(;)387 2627 y(s)410 2634 y FE(9)438 2627 y Fu(\021)e Fx(2)p Ft(\033)538 2634 y FE(01)575 2627 y Fx(\()p Ft(\016)r Fx(\))16 b(;)955 2863 y(18)p eop %%Page: 19 19 19 18 bop 128 94 a Fu(k)p Ft(q)175 101 y FE(1)194 94 y Fu(k)219 101 y Fk(\030)236 92 y Fn(0)299 94 y Fu(\024)392 60 y Fx(1)p 392 82 25 2 v 392 128 a(2)422 94 y Fu(k)p Ft(h)475 101 y Fk(xxx)536 94 y Fu(k)565 96 y FE(\026)561 105 y Fk(\030)q(;r)607 94 y Fu(k)p Ft(z)r Fu(k)682 73 y FE(2)682 106 y Fk(\030)699 97 y Fn(0)723 94 y Fx(+)11 b Fu(k)p Ft(h)825 101 y Fk(xxy)885 94 y Fu(k)914 96 y FE(\026)910 105 y Fk(\030)q(;r)956 94 y Fu(k)p Ft(z)r Fu(k)1031 101 y Fk(\030)1048 92 y Fn(0)1060 94 y Fu(k)p Ft(w)q Fu(k)1146 101 y Fk(\030)1163 92 y Fn(0)1188 94 y Fx(+)1242 60 y(1)p 1242 82 V 1242 128 a(2)1271 94 y Fu(k)p Ft(h)1324 101 y Fk(xy)q(y)1383 94 y Fu(k)1412 96 y FE(\026)1408 105 y Fk(\030)q(;r)1454 94 y Fu(k)p Ft(w)q Fu(k)1540 73 y FE(2)1540 106 y Fk(\030)1557 97 y Fn(0)299 187 y Fu(\024)49 b Ft(s)410 194 y FE(10)448 187 y Ft(B)488 166 y FE(2)485 199 y FF(\003)518 187 y Fx(+)11 b Ft(s)590 194 y FE(11)628 187 y Ft(B)665 194 y FF(\003)684 187 y Ft(F)716 194 y FF(\003)747 187 y Fx(+)g Ft(s)819 194 y FE(12)856 187 y Ft(F)895 166 y FE(2)930 187 y Ft(;)387 281 y(s)410 288 y FE(10)456 281 y Fu(\021)508 248 y Fx(1)p 508 270 V 508 316 a(2)538 281 y Ft(E)574 288 y FE(3)p Fk(;)p FE(0)621 281 y Ft(s)644 261 y FE(2)644 294 y(3)675 281 y Fx(+)p 724 242 40 2 v 11 w Ft(E)s(E)799 288 y FE(2)p Fk(;)p FE(1)846 281 y Ft(s)869 288 y FE(3)889 281 y Ft(s)912 288 y FE(7)942 281 y Fx(+)996 248 y(1)p 996 270 25 2 v 996 316 a(2)1026 281 y Ft(E)1062 288 y FE(1)p Fk(;)p FE(2)p 1109 242 40 2 v 1109 281 a Ft(E)1148 252 y FE(2)1168 281 y Ft(s)1191 261 y FE(2)1191 294 y(7)1227 281 y Ft(;)387 383 y(s)410 390 y FE(11)456 383 y Fu(\021)d Ft(s)526 390 y FE(3)546 383 y Ft(E)582 390 y FE(2)p Fk(;)p FE(1)p 629 344 V 629 383 a Ft(E)14 b Fx(+)d Ft(s)751 390 y FE(7)771 383 y Ft(E)807 390 y FE(1)p Fk(;)p FE(2)p 854 344 V 854 383 a Ft(E)893 354 y FE(2)929 383 y Ft(;)387 479 y(s)410 486 y FE(12)456 479 y Fu(\021)508 445 y Fx(1)p 508 468 25 2 v 508 513 a(2)p 538 440 40 2 v 538 479 a Ft(E)577 450 y FE(2)596 479 y Ft(E)632 486 y FE(1)p Fk(;)p FE(2)696 479 y Fx(;)128 596 y Fu(k)p Ft(f)182 575 y FF(0)194 596 y Fu(k)219 603 y Fk(\030)236 594 y Fn(0)299 596 y Fu(\021)49 b(k)p Ft(q)434 603 y FE(2)454 596 y Fu(k)479 603 y Fk(\030)496 594 y Fn(0)523 596 y Fu(\024)580 562 y Fx(1)p 580 584 25 2 v 580 630 a(2)609 596 y Fu(k)p Ft(h)662 603 y Fk(y)q(xx)723 596 y Fu(k)752 598 y FE(\026)748 606 y Fk(\030)q(;r)793 596 y Fu(k)p Ft(z)r Fu(k)868 575 y FE(2)868 608 y Fk(\030)885 599 y Fn(0)909 596 y Fx(+)11 b Fu(k)p Ft(h)1011 603 y Fk(y)q(xy)1070 596 y Fu(k)1099 598 y FE(\026)1095 606 y Fk(\030)q(;r)1141 596 y Fu(k)p Ft(z)r Fu(k)1216 603 y Fk(\030)1233 594 y Fn(0)1246 596 y Fu(k)p Ft(w)q Fu(k)1332 603 y Fk(\030)1349 594 y Fn(0)1373 596 y Fx(+)1427 562 y(1)p 1427 584 V 1427 630 a(2)1456 596 y Fu(k)p Ft(h)1509 603 y Fk(y)q(y)q(y)1567 596 y Fu(k)1596 598 y FE(\026)1592 606 y Fk(\030)q(;r)1638 596 y Fu(k)p Ft(w)q Fu(k)1724 575 y FE(2)1724 608 y Fk(\030)1741 599 y Fn(0)299 688 y Fu(\024)49 b Ft(F)426 668 y FF(0)446 688 y Fu(\021)8 b Ft(s)516 668 y FF(0)516 701 y FE(10)553 688 y Ft(B)593 668 y FE(2)590 701 y FF(\003)624 688 y Fx(+)j Ft(s)696 668 y FF(0)696 701 y FE(11)733 688 y Ft(B)770 695 y FF(\003)790 688 y Ft(F)17 b Fx(+)11 b Ft(s)911 668 y FF(0)911 701 y FE(12)948 688 y Ft(F)987 668 y FE(2)1023 688 y Ft(;)387 783 y(s)410 763 y FF(0)410 795 y FE(10)456 783 y Fu(\021)508 749 y Fx(1)p 508 772 V 508 817 a(2)538 783 y Ft(E)574 790 y FE(2)p Fk(;)p FE(1)621 783 y Ft(s)644 763 y FE(2)644 795 y(3)675 783 y Fx(+)p 724 744 40 2 v 11 w Ft(E)s(E)799 790 y FE(1)p Fk(;)p FE(2)846 783 y Ft(s)869 790 y FE(3)889 783 y Ft(s)912 790 y FE(7)942 783 y Fx(+)996 749 y(1)p 996 772 25 2 v 996 817 a(2)1026 783 y Ft(E)1062 790 y FE(0)p Fk(;)p FE(3)p 1109 744 40 2 v 1109 783 a Ft(E)1148 754 y FE(2)1168 783 y Ft(s)1191 763 y FE(2)1191 795 y(7)1227 783 y Ft(;)387 885 y(s)410 864 y FF(0)410 897 y FE(11)456 885 y Fu(\021)d Ft(s)526 892 y FE(3)546 885 y Ft(E)582 892 y FE(1)p Fk(;)p FE(2)p 629 846 V 629 885 a Ft(E)14 b Fx(+)d Ft(s)751 892 y FE(7)771 885 y Ft(E)807 892 y FE(0)p Fk(;)p FE(3)p 854 846 V 854 885 a Ft(E)893 855 y FE(2)929 885 y Ft(;)387 981 y(s)410 960 y FF(0)410 993 y FE(12)456 981 y Fu(\021)508 947 y Fx(1)p 508 969 25 2 v 508 1015 a(2)p 538 942 40 2 v 538 981 a Ft(E)577 952 y FE(2)596 981 y Ft(E)632 988 y FE(0)p Fk(;)p FE(3)696 981 y Fx(;)128 1069 y Fu(k)p Ft(q)175 1076 y FE(3)194 1069 y Fu(k)219 1076 y Fk(\030)236 1067 y Fn(0)299 1069 y Fu(\024)49 b Ft(\033)415 1076 y FE(01)453 1069 y Fx(\()p Ft(\016)r Fx(\))p Ft(F)7 b Fu(k)p Ft(w)q Fu(k)640 1076 y Fk(\030)657 1067 y Fn(0)679 1069 y Fx(+)k Ft(\033)756 1076 y FE(01)794 1069 y Fx(\()p Ft(\016)r Fx(\))p Ft(G)p Fu(k)p Ft(z)r Fu(k)969 1076 y Fk(\030)986 1067 y Fn(0)1009 1069 y Fx(+)g Ft(M)5 b Fu(k)p Ft(q)1157 1076 y FE(1)1177 1069 y Fu(k)1202 1076 y Fk(\030)1219 1067 y Fn(0)299 1142 y Fu(\024)49 b Ft(s)410 1149 y FE(13)448 1142 y Ft(B)485 1149 y FF(\003)505 1142 y Ft(F)17 b Fx(+)11 b Ft(s)626 1149 y FE(14)663 1142 y Ft(B)700 1149 y FF(\003)720 1142 y Ft(G)h Fx(+)f Ft(s)842 1149 y FE(15)879 1142 y Ft(F)918 1121 y FE(2)948 1142 y Fx(+)g Ft(s)1020 1149 y FE(16)1057 1142 y Ft(B)1097 1121 y FE(2)1094 1154 y FF(\003)1133 1142 y Ft(;)387 1214 y(s)410 1221 y FE(13)456 1214 y Fu(\021)d Ft(\033)531 1221 y FE(01)568 1214 y Fx(\()p Ft(\016)r Fx(\))p 630 1175 V Ft(E)s(s)692 1221 y FE(7)722 1214 y Fx(+)j Ft(M)5 b(s)846 1221 y FE(11)900 1214 y Ft(;)106 b(s)1043 1221 y FE(14)1088 1214 y Fu(\021)9 b Ft(\033)1164 1221 y FE(01)1201 1214 y Fx(\()p Ft(\016)r Fx(\))p Ft(s)1286 1221 y FE(3)1321 1214 y Ft(;)387 1287 y(s)410 1294 y FE(15)456 1287 y Fu(\021)f Ft(\033)531 1294 y FE(01)568 1287 y Fx(\()p Ft(\016)r Fx(\))p 630 1248 V Ft(E)14 b Fx(+)d Ft(M)5 b(s)804 1294 y FE(12)858 1287 y Ft(;)105 b(s)1000 1294 y FE(16)1046 1287 y Fu(\021)8 b Ft(M)d(s)1168 1294 y FE(10)1222 1287 y Fx(;)132 1366 y Fu(k)p Ft(g)182 1345 y FF(0)194 1366 y Fu(k)219 1373 y Fk(\030)236 1364 y Fn(0)299 1366 y Fx(=)50 b Fu(kM)472 1345 y FF(\000)p Fk(T)527 1318 y Fp(h)547 1366 y Fu(B)r(M)642 1345 y FF(\000)p FE(1)688 1366 y Ft(z)13 b Fx(+)e Fu(A)p Fx(\()p Ft(D)q Fu(M)933 1345 y FF(\000)p FE(1)981 1366 y Fx(\))p Ft(z)h Fx(+)f Fu(AM)1184 1345 y FF(\000)p FE(1)1231 1366 y Ft(D)q(z)j Fx(+)d(\()p Ft(D)q Fu(M)1478 1345 y Fk(T)1506 1366 y Fx(\)\()p Ft(h)1572 1345 y FE(0)1572 1378 y Fk(y)q(y)1612 1366 y Fx(\))1631 1345 y FF(\000)p FE(1)1678 1366 y Ft(f)1702 1373 y Fk(\022)1721 1366 y Fu(M)1781 1345 y FF(\000)p FE(1)1828 1366 y Ft(z)1853 1318 y Fp(i)387 1449 y Fx(+)p Ft(D)q Fx(\()p Ft(h)513 1429 y FE(0)513 1462 y Fk(y)q(y)554 1449 y Fx(\))573 1429 y FF(\000)p FE(1)636 1449 y Ft(f)660 1456 y Fk(\022)680 1449 y Fu(M)740 1429 y FF(\000)p FE(1)787 1449 y Ft(z)i Fx(+)e(\()p Ft(h)919 1429 y FE(0)919 1462 y Fk(y)q(y)958 1449 y Fx(\))977 1429 y FF(\000)p FE(1)1024 1449 y Ft(D)q(f)1089 1456 y Fk(\022)1126 1449 y Fu(M)1186 1429 y FF(\000)p FE(1)387 1528 y Fx(+\()p Ft(h)472 1507 y FE(0)472 1540 y Fk(y)q(y)512 1528 y Fx(\))531 1507 y FF(\000)p FE(1)578 1528 y Ft(f)602 1535 y Fk(\022)621 1528 y Fx(\()p Ft(D)q Fu(M)741 1507 y FF(\000)p FE(1)789 1528 y Fx(\))17 b Ft(z)12 b Fx(+)f(\()p Ft(h)956 1507 y FE(0)956 1540 y Fk(y)q(y)996 1528 y Fx(\))1015 1507 y FF(\000)p FE(1)1062 1528 y Ft(f)1086 1535 y Fk(\022)1106 1528 y Fu(M)1166 1507 y FF(\000)p FE(1)1213 1528 y Ft(D)q(z)i Fu(\000)e(M)1400 1507 y FF(\000)p Fk(T)1455 1528 y Ft(q)1477 1535 y FE(3)1496 1528 y Fu(k)1521 1535 y Fk(\030)1538 1526 y Fn(0)299 1613 y Fu(\024)p 387 1574 53 2 v 49 w Ft(M)440 1565 y Fp(h)459 1613 y Ft(s)482 1620 y FE(1)502 1613 y Ft(F)534 1620 y FF(\003)p 553 1574 V 553 1613 a Ft(M)6 b(B)643 1620 y FF(\003)663 1613 y Ft(s)686 1620 y FE(3)716 1613 y Fx(+)11 b Ft(s)788 1620 y FE(2)808 1613 y Ft(F)840 1620 y FF(\003)859 1613 y Ft(\033)887 1620 y FF(\000)p FE(10)952 1613 y Fx(\(2)p Ft(\016)r Fx(\))p 1038 1574 V Ft(M)5 b(B)1127 1620 y FF(\003)1147 1613 y Ft(s)1170 1620 y FE(3)1201 1613 y Fx(+)11 b Ft(s)1273 1620 y FE(2)1292 1613 y Ft(F)1324 1620 y FF(\003)p 1344 1574 V 1344 1613 a Ft(M)5 b(B)1433 1620 y FF(\003)1453 1613 y Ft(s)1476 1620 y FE(4)387 1704 y Fx(+)p Ft(\033)453 1711 y FF(\000)p FE(10)518 1704 y Fx(\(2)p Ft(\016)r Fx(\))p 604 1665 40 2 v Ft(E)s(\033)671 1711 y FE(01)708 1704 y Fx(\(2)p Ft(\016)r Fx(\))p Ft(F)p 833 1665 53 2 v 7 w(M)t(B)921 1711 y FF(\003)941 1704 y Ft(s)964 1711 y FE(3)984 1656 y Fp(i)387 1788 y Fx(+)p Ft(\033)453 1795 y FF(\000)p FE(10)518 1788 y Fx(\()p Ft(\016)r Fx(\))p 580 1749 40 2 v Ft(E)r(\033)646 1795 y FE(01)683 1788 y Fx(\(2)p Ft(\016)r Fx(\))p Ft(F)p 808 1749 53 2 v 7 w(M)g(B)897 1795 y FF(\003)917 1788 y Ft(s)940 1795 y FE(3)971 1788 y Fx(+)p 1020 1749 40 2 v 11 w Ft(E)s(\033)1087 1795 y FF(\000)p FE(11)1151 1788 y Fx(\(2)p Ft(\016)r Fx(\))p Ft(F)p 1276 1749 53 2 v 7 w(M)g(B)1365 1795 y FF(\003)1384 1788 y Ft(s)1407 1795 y FE(3)387 1861 y Fx(+)p 425 1822 40 2 v Ft(E)s(\033)492 1868 y FE(01)530 1861 y Fx(\(2)p Ft(\016)r Fx(\))p Ft(F)i(\033)683 1868 y FF(\000)p FE(10)p 746 1822 53 2 v -7 x Ft(M)f(B)836 1868 y FF(\003)855 1861 y Ft(s)878 1868 y FE(3)909 1861 y Fx(+)p 958 1822 40 2 v 11 w Ft(E)s(\033)1025 1868 y FE(01)1062 1861 y Fx(\(2)p Ft(\016)r Fx(\))p Ft(F)p 1187 1822 53 2 v 7 w(M)f(B)1276 1868 y FF(\003)1296 1861 y Ft(s)1319 1868 y FE(4)387 1940 y Fx(+)p 425 1901 V Ft(M)478 1892 y Fp(\020)503 1940 y Ft(s)526 1947 y FE(13)563 1940 y Ft(B)600 1947 y FF(\003)620 1940 y Ft(F)17 b Fx(+)11 b Ft(s)741 1947 y FE(14)778 1940 y Ft(B)815 1947 y FF(\003)835 1940 y Ft(G)h Fx(+)f Ft(s)957 1947 y FE(15)994 1940 y Ft(F)1033 1919 y FE(2)1063 1940 y Fx(+)g Ft(s)1135 1947 y FE(16)1173 1940 y Ft(B)1213 1919 y FE(2)1210 1952 y FF(\003)1232 1892 y Fp(\021)299 2024 y Fu(\024)49 b Ft(G)425 2003 y FF(0)446 2024 y Fu(\021)8 b Ft(s)516 2031 y FE(17)553 2024 y Ft(F)585 2031 y FF(\003)605 2024 y Ft(B)642 2031 y FF(\003)672 2024 y Fx(+)j Ft(s)744 2031 y FE(18)p 782 1985 40 2 v 782 2024 a Ft(E)s(F)c(B)897 2031 y FF(\003)927 2024 y Fx(+)k Ft(s)999 2031 y FE(14)p 1036 1985 53 2 v 1036 2024 a Ft(M)6 b(B)1126 2031 y FF(\003)1145 2024 y Ft(G)12 b Fx(+)f Ft(s)1267 2031 y FE(15)p 1304 1985 V 1304 2024 a Ft(M)6 b(F)1396 2003 y FE(2)1426 2024 y Fx(+)11 b Ft(s)1498 2031 y FE(16)p 1535 1985 V 1535 2024 a Ft(M)6 b(B)1628 2003 y FE(2)1625 2036 y FF(\003)1663 2024 y Ft(;)387 2103 y(s)410 2110 y FE(17)456 2103 y Fu(\021)p 503 2064 V 8 w Ft(M)556 2074 y FE(2)575 2103 y Fx(\()p Ft(s)617 2110 y FE(1)637 2103 y Ft(s)660 2110 y FE(3)691 2103 y Fx(+)11 b Ft(\033)768 2110 y FF(\000)p FE(10)832 2103 y Fx(\(2)p Ft(\016)r Fx(\))p Ft(s)941 2110 y FE(2)961 2103 y Ft(s)984 2110 y FE(3)1014 2103 y Fx(+)g Ft(s)1086 2110 y FE(2)1106 2103 y Ft(s)1129 2110 y FE(4)1149 2103 y Fx(\))16 b Ft(;)387 2182 y(s)410 2189 y FE(18)456 2182 y Fu(\021)p 503 2143 V 8 w Ft(M)6 b(s)579 2189 y FE(3)598 2134 y Fp(h)p 618 2143 V 48 x Ft(M)f(\033)698 2189 y FF(\000)p FE(10)763 2182 y Fx(\(2)p Ft(\016)r Fx(\))p Ft(\033)877 2189 y FE(01)914 2182 y Fx(\(2)p Ft(\016)r Fx(\))11 b(+)g Ft(\033)1088 2189 y FF(\000)p FE(10)1152 2182 y Fx(\()p Ft(\016)r Fx(\))p Ft(\033)1242 2189 y FE(01)1279 2182 y Fx(\()p Ft(\016)r Fx(\))f(+)h Ft(\033)1428 2189 y FF(\000)p FE(11)1493 2182 y Fx(\(2)p Ft(\016)r Fx(\))485 2274 y(+)p Ft(\033)551 2281 y FE(01)588 2274 y Fx(\(2)p Ft(\016)r Fx(\))p Ft(\033)702 2281 y FF(\000)p FE(10)766 2274 y Fx(\(2)p Ft(\016)r Fx(\))852 2226 y Fp(i)883 2274 y Fx(+)p 932 2235 V 11 w Ft(M)5 b(\033)1012 2281 y FE(01)1049 2274 y Fx(\(2)p Ft(\016)r Fx(\))p Ft(s)1158 2281 y FE(4)1189 2274 y Fx(+)p 1238 2235 V 11 w Ft(M)p 1307 2235 40 2 v 22 w(E)1346 2245 y FF(\000)p FE(1)1393 2274 y Ft(s)1416 2281 y FE(13)1469 2274 y Fx(;)98 2355 y Fu(kM)183 2334 y FF(0)194 2355 y Fu(k)219 2362 y Fk(\030)236 2353 y Fn(0)299 2355 y Fu(\021)49 b(kM)11 b Fx(+)g Ft(z)555 2362 y Fk(\022)574 2355 y Fu(k)599 2362 y Fk(\030)616 2353 y Fn(0)643 2355 y Fu(\024)j Ft(M)i Fx(+)11 b Fu(k)p Ft(z)856 2362 y Fk(\022)875 2355 y Fu(k)900 2362 y Fk(\030)917 2353 y Fn(0)299 2427 y Fu(\024)49 b Ft(M)439 2407 y FF(0)460 2427 y Fu(\021)8 b Ft(M)16 b Fx(+)11 b Ft(B)656 2434 y FF(\003)676 2427 y Ft(s)699 2434 y FE(5)735 2427 y Ft(:)955 2863 y Fx(19)p eop %%Page: 20 20 20 19 bop 46 78 a Fx(T)l(o)15 b(de\014ne)p 255 39 53 2 v 14 w Ft(M)308 49 y FF(0)334 78 y Fx(w)o(e)f(ha)o(v)o(e)g(to)h (distinguish)g(t)o(w)o(o)f(cases)h(according)g(to)g(whether)p 1491 39 V 14 w Ft(M)5 b(B)1580 85 y FF(\003)1600 78 y Ft(s)1623 85 y FE(5)1658 78 y Fx(is)14 b(greater)h(or)46 138 y(smaller)f(than)j(1.)f(In)g(the)g(\014rst)h(case)f(w)o(e)g (de\014ne)p 944 99 V 16 w Ft(M)996 109 y FF(0)1024 138 y Fx(to)g(b)q(e)h(in\014nite:)p 640 185 V 640 224 a Ft(M)5 b(B)729 231 y FF(\003)749 224 y Ft(s)772 231 y FE(5)805 224 y Fu(\025)14 b Fx(1)128 b(=)-8 b Fu(\))p 1119 185 V 29 w Ft(M)1172 195 y FF(0)1192 224 y Fu(\021)8 b(1)16 b Ft(;)46 310 y Fx(while)f(if)p 858 331 V 858 370 a Ft(M)5 b(B)947 377 y FF(\003)967 370 y Ft(s)990 377 y FE(5)1024 370 y Ft(<)13 b Fx(1)688 b(\(4.26\))46 447 y(w)o(e)16 b(pro)q(ceed)g(as)h(follo)o(ws:)334 533 y Fu(kM)419 513 y FF(0)430 508 y(\000)p FE(1)477 533 y Fu(k)502 540 y Fk(\030)519 531 y Fn(0)574 533 y Fx(=)42 b Fu(k)p Fx(\()p Ft(I)14 b Fx(+)d Fu(M)843 513 y FF(\000)p FE(1)890 533 y Ft(z)913 540 y Fk(\022)932 533 y Fx(\))951 513 y FF(\000)p FE(1)999 533 y Fu(M)1059 513 y FF(\000)p FE(1)1106 533 y Fu(k)1131 540 y Fk(\030)1148 531 y Fn(0)1174 533 y Fu(\024)p 1227 494 V 14 w Ft(M)5 b Fx(\(1)12 b Fu(\000)p 1384 494 V 11 w Ft(M)5 b Fu(k)p Ft(z)1484 540 y Fk(\022)1503 533 y Fu(k)1528 540 y Fk(\030)1545 531 y Fn(0)1558 533 y Fx(\))1577 513 y FF(\000)p FE(1)574 612 y Fu(\024)p 654 573 V 41 w Ft(M)706 582 y FF(0)726 612 y Fu(\021)p 773 573 V 8 w Ft(M)h Fx(\(1)11 b Fu(\000)p 930 573 V 11 w Ft(M)5 b(B)1019 619 y FF(\003)1039 612 y Ft(s)1062 619 y FE(5)1082 612 y Fx(\))1101 591 y FF(\000)p FE(1)1164 612 y Ft(:)46 698 y Fx(Next,)571 784 y Fu(k)p Ft(v)622 763 y FF(0)620 796 y Fk(\022)639 784 y Fu(k)664 791 y Fk(\030)681 782 y Fn(0)744 784 y Fu(\021)50 b(k)p Ft(v)882 791 y Fk(\022)912 784 y Fx(+)11 b Ft(w)996 791 y Fk(\022)1015 784 y Fu(k)1040 791 y Fk(\030)1057 782 y Fn(0)1084 784 y Fu(\024)1145 767 y Fp(e)1136 784 y Ft(V)23 b Fx(+)11 b Fu(k)p Ft(w)1296 791 y Fk(\022)1315 784 y Fu(k)1340 791 y Fk(\030)1357 782 y Fn(0)744 856 y Fu(\024)841 840 y Fp(e)833 856 y Ft(V)872 836 y FF(0)892 856 y Fu(\021)947 840 y Fp(e)939 856 y Ft(V)22 b Fx(+)p 1038 817 40 2 v 11 w Ft(E)s(B)1114 863 y FF(\003)1134 856 y Ft(s)1157 863 y FE(8)1188 856 y Fx(+)p 1237 817 V 11 w Ft(E)s(F)7 b(s)1338 863 y FE(9)1373 856 y Ft(:)46 942 y Fx(Of)18 b(course,)g(as)h(it)e(is)h(clear)g(from)f(the)h(de\014nition)f(of)p 1043 903 53 2 v 19 w Ft(M)1095 913 y FF(0)1107 942 y Fx(,)h(the)g(KAM)f(algorithm)g(will)g(b)q(e)i(of)f(some)46 1003 y(use)h(only)g(if)g(\(4.26\))h(is)f(satis\014ed.)g(T)l(o)g (complete)e(the)i(computation)g(of)g Fu(K)q Fx(,)g(it)g(remains)e(to)j (b)q(ound)46 1063 y Fu(jhT)119 1045 y FF(0)131 1063 y Fu(i)150 1039 y FF(\000)p FE(1)197 1063 y Fu(j)p Fx(,)15 b Fj(i.e.)i Fx(to)f(compute)590 1047 y Fp(e)580 1063 y Ft(T)616 1045 y FF(0)627 1063 y Fx(.)f(If)p 704 1024 V 15 w Ft(M)756 1034 y FF(0)782 1063 y Fx(=)e Fu(1)p Fx(,)i(w)o(e)g(de\014ne)g(also)1231 1047 y Fp(e)1220 1063 y Ft(T)1256 1045 y FF(0)1276 1063 y Fu(\021)8 b(1)p Fx(,)15 b(otherwise)g(w)o(e)g(pro)q(ceed)g(as)46 1123 y(follo)o(ws.)h(A)f(bit)h(of)h(algebra)g(sho)o(ws)g(that,)f(if)g(w)o(e) f(de\014ne)551 1235 y Fu(C)44 b(\021)8 b Fx(\()p Ft(I)15 b Fx(+)c Fu(M)833 1214 y FF(\000)p FE(1)880 1235 y Ft(z)903 1242 y Fk(\022)922 1235 y Fx(\))941 1214 y FF(\000)p FE(1)999 1235 y Fu(\000)g Ft(I)17 b Fx(=)1154 1181 y FF(1)1142 1193 y Fp(X)1140 1286 y Fk(k)q FE(=1)1204 1235 y Fx(\()p Fu(\000M)1322 1214 y FF(\000)p FE(1)1369 1235 y Ft(z)1392 1242 y Fk(\022)1411 1235 y Fx(\))1430 1214 y Fk(k)1468 1235 y Ft(;)551 1348 y Fu(C)580 1328 y FF(0)633 1348 y Fu(\021)8 b(C)s(M)769 1328 y FF(\000)p FE(1)832 1348 y Ft(;)551 1421 y Fu(C)580 1401 y FF(00)610 1421 y Fu(\021)g Ft(h)685 1428 y Fk(y)q(y)724 1421 y Fx(\()p Ft(\022)13 b Fx(+)e Ft(u)856 1401 y FF(0)867 1421 y Ft(;)d(v)915 1401 y FF(0)926 1421 y Fx(\))j Fu(\000)g Ft(h)1034 1401 y FE(0)1034 1433 y Fk(y)q(y)1089 1421 y Ft(;)46 1507 y Fx(then)16 b(w)o(e)g(can)g(write)g Fu(T)483 1489 y FF(0)511 1507 y Fx(as)833 1567 y Fu(T)872 1547 y FF(0)898 1567 y Fx(=)e Fu(T)24 b Fx(+)11 b Fu(C)1076 1574 y FF(\003)1112 1567 y Ft(;)46 1644 y Fx(where)323 1730 y Fu(C)349 1737 y FF(\003)419 1730 y Fu(\021)49 b(M)567 1710 y FF(\000)p FE(1)614 1730 y Ft(h)642 1710 y FE(0)642 1743 y Fk(y)q(y)681 1730 y Fu(C)710 1710 y FF(0)722 1704 y Fk(T)761 1730 y Fx(+)11 b Fu(M)870 1710 y FF(\000)p FE(1)917 1730 y Fu(C)946 1710 y FF(00)967 1730 y Fu(M)1027 1710 y FF(\000)p FE(1)1085 1730 y Fx(+)g Fu(M)1194 1710 y FF(\000)p FE(1)1241 1730 y Fu(C)1270 1710 y FF(00)1292 1730 y Fu(C)1321 1710 y FF(0)1332 1704 y Fk(T)1371 1730 y Fx(+)g Fu(C)1449 1710 y FF(0)1461 1730 y Ft(h)1489 1710 y FE(0)1489 1743 y Fk(y)q(y)1528 1730 y Fu(M)1588 1710 y FF(\000)p FE(1)507 1814 y Fx(+)p Fu(C)574 1794 y FF(0)586 1814 y Ft(h)614 1794 y FE(0)614 1827 y Fk(y)q(y)653 1814 y Fu(C)682 1794 y FF(0)694 1788 y Fk(T)733 1814 y Fx(+)g Fu(C)811 1794 y FF(0)822 1814 y Fu(C)851 1794 y FF(00)873 1814 y Fu(M)933 1794 y FF(\000)p FE(1)991 1814 y Fx(+)g Fu(C)1069 1794 y FF(0)1081 1814 y Fu(C)1110 1794 y FF(00)1131 1814 y Fu(C)1160 1794 y FF(0)1172 1788 y Fk(T)1215 1814 y Ft(:)46 1900 y Fx(Th)o(us,)23 b(since)g(sup)390 1912 y Fc(T)410 1903 y Fo(N)453 1900 y Fu(j)p Ft(z)490 1907 y Fk(\022)509 1900 y Fu(j)i(\024)h(k)p Ft(z)661 1907 y Fk(\022)680 1900 y Fu(k)705 1907 y FE(0)748 1900 y Fx(\(and)e(a)g(b)q(ound)g(on)g Fu(k)p Ft(z)1197 1907 y Fk(\022)1216 1900 y Fu(k)1241 1907 y FE(0)1284 1900 y Fx(is)f(obtained)h(exactly)e(as)i(ab)q(o)o(v)o (e)46 1960 y(replacing)16 b Ft(\016)h Fx(with)f Ft(\030)r Fx(\),)h(w)o(e)f(obtain)472 2047 y(sup)483 2090 y Fc(T)503 2080 y Fo(N)554 2047 y Fu(jC)s(j)41 b(\024)p 732 2007 V 41 w Ft(M)6 b Fu(k)p Ft(z)833 2054 y Fk(\022)852 2047 y Fu(k)877 2054 y FE(0)896 2047 y Fx(\(1)12 b Fu(\000)p 1001 2007 V 11 w Ft(M)5 b Fu(k)p Ft(z)1101 2054 y Fk(\022)1120 2047 y Fu(k)1145 2054 y FE(0)1165 2047 y Fx(\))1184 2026 y FF(\000)p FE(1)652 2153 y Fu(\024)41 b Ft(C)12 b Fu(\021)p 826 2114 V 8 w Ft(M)5 b(B)918 2132 y FE(0)915 2165 y FF(\003)938 2153 y Ft(s)961 2132 y FE(0)961 2165 y(5)980 2153 y Fx(\(1)12 b Fu(\000)p 1085 2114 V 11 w Ft(M)5 b(B)1177 2132 y FE(0)1174 2165 y FF(\003)1196 2153 y Ft(s)1219 2132 y FE(0)1219 2165 y(5)1239 2153 y Fx(\))1258 2132 y FF(\000)p FE(1)1321 2153 y Fx(;)460 2225 y(sup)471 2268 y Fc(T)491 2259 y Fo(N)542 2225 y Fu(jC)585 2205 y FF(0)597 2225 y Fu(j)41 b(\024)g Ft(C)771 2205 y FF(0)791 2225 y Fu(\021)8 b Ft(C)p 877 2186 V 4 w(M)21 b Fx(;)451 2328 y(sup)461 2371 y Fc(T)482 2362 y Fo(N)532 2328 y Fu(jC)575 2308 y FF(00)597 2328 y Fu(j)41 b(\024)g Ft(E)768 2335 y FE(1)p Fk(;)p FE(2)815 2328 y Fu(k)p Ft(z)r Fu(k)890 2335 y FE(0)921 2328 y Fx(+)11 b Ft(E)1006 2335 y FE(0)p Fk(;)p FE(3)1053 2328 y Fu(k)p Ft(w)q Fu(k)1139 2335 y FE(0)652 2434 y Fu(\024)41 b Ft(C)771 2414 y FF(00)800 2434 y Fu(\021)8 b Ft(E)883 2441 y FE(1)p Fk(;)p FE(2)931 2434 y Ft(B)971 2414 y FE(0)968 2447 y FF(\003)990 2434 y Ft(s)1013 2414 y FE(0)1013 2447 y(3)1044 2434 y Fx(+)j Ft(E)1129 2441 y FE(0)p Fk(;)p FE(3)p 1176 2395 40 2 v 1176 2434 a Ft(E)s Fx(\()p Ft(s)1257 2414 y FE(0)1257 2447 y(7)1277 2434 y Ft(B)1317 2414 y FE(0)1314 2447 y FF(\003)1347 2434 y Fx(+)g Ft(F)c Fx(\))15 b(;)455 2515 y(sup)466 2559 y Fc(T)486 2549 y Fo(N)537 2515 y Fu(jC)577 2522 y FF(\003)597 2515 y Fu(j)41 b(\024)g Ft(C)767 2522 y FF(\003)795 2515 y Fu(\021)8 b Fx(2)p 866 2476 53 2 v Ft(M)22 b(E)971 2522 y FE(0)p Fk(;)p FE(2)1018 2515 y Ft(C)1057 2495 y FF(0)1080 2515 y Fx(+)p 1129 2476 V 11 w Ft(M)1181 2486 y FE(2)1201 2515 y Ft(C)1240 2495 y FF(00)1272 2515 y Fx(+)11 b(2)p 1345 2476 V Ft(M)5 b(C)1436 2495 y FF(00)1457 2515 y Ft(C)1496 2495 y FF(0)830 2627 y Fx(+)p Ft(C)907 2606 y FF(0)918 2601 y FE(2)938 2627 y Ft(E)974 2634 y FE(0)p Fk(;)p FE(2)1032 2627 y Fx(+)11 b Ft(C)1120 2606 y FF(0)1131 2601 y FE(2)1151 2627 y Ft(C)1190 2606 y FF(00)1227 2627 y Ft(;)546 b Fx(\(4.27\))955 2863 y(20)p eop %%Page: 21 21 21 20 bop 46 78 a Fv(where)16 b Ft(B)227 60 y FE(0)224 90 y FF(\003)246 78 y Ft(;)8 b(s)291 60 y FE(0)291 90 y(5)311 78 y Ft(;)g(s)356 60 y FE(0)356 90 y(3)375 78 y Ft(;)g(s)420 60 y FE(0)420 90 y(7)456 78 y Fv(are)16 b(de\014ned)g(as)h(ab)q(o)o(v)o(e)f(but)h(with)f Ft(\016)i Fv(replaced)d(b)o(y)h Ft(\030)r Fx(.)g(No)o(w,)g(if)676 172 y Fp(e)666 188 y Ft(T)6 b(C)736 195 y FF(\003)770 188 y Fu(\025)13 b Fx(1)144 b(=)-8 b Fu(\))1110 172 y Fp(e)1100 188 y Ft(T)1136 167 y FF(0)1161 188 y Fx(=)14 b Fu(1)i Ft(;)46 298 y Fx(otherwise,)f(if)884 342 y Fp(e)874 358 y Ft(T)6 b(C)944 365 y FF(\003)978 358 y Ft(<)14 b Fx(1)i Ft(;)703 b Fx(\(4.28\))46 445 y(w)o(e)16 b(ha)o(v)o(e)568 555 y Fu(jh)q(T)641 535 y FF(0)653 555 y Fu(i)672 531 y FF(\000)p FE(1)720 555 y Fu(j)41 b Fx(=)h Fu(j)p Fx(\()p Ft(I)14 b Fx(+)d Fu(h)q(T)i(i)1052 531 y FF(\000)p FE(1)1099 555 y Fu(h)q(C)1145 562 y FF(\003)1164 555 y Fu(i)q Fx(\))1203 535 y FF(\000)p FE(1)1250 555 y Fu(hT)g(i)1329 531 y FF(\000)p FE(1)1376 555 y Fu(j)775 628 y(\024)865 612 y Fp(e)855 628 y Ft(T)891 607 y FF(0)910 628 y Fu(\021)968 612 y Fp(e)957 628 y Ft(T)7 b Fx(\(1)k Fu(\000)1107 612 y Fp(e)1097 628 y Ft(T)6 b(C)1167 635 y FF(\003)1187 628 y Fx(\))1206 607 y FF(\000)p FE(1)1269 628 y Ft(:)504 b Fx(\(4.29\))46 738 y(The)16 b(computation)g(of)g(the)g(map)g Fu(K)h Fx(is)f(completed.)46 852 y Fw(Remark)g(4.4)24 b Fx(\()p Fv(\\KAM)h(algorithm")p Fx(\))g(If)g(\(4.23\))h(is)g (satis\014ed)f(when)h Ft(U;)8 b(v)r(;)g(\030)27 b Fx(are)f(replaced)f (b)o(y)46 912 y Ft(U)84 894 y FF(0)96 912 y Ft(;)8 b(v)144 894 y FF(0)155 912 y Ft(;)g(\030)200 894 y FF(0)233 912 y Fx(then)20 b(the)h(map)f Fu(K)h Fx(can)g(b)q(e)g(re{applied)g(and,)f (iterating)h(when)g(p)q(ossible,)f(one)h(obtains)46 973 y(a)d(sequence)f(\()p Ft(u)339 954 y FE(\()p Fk(j)r FE(\))384 973 y Ft(;)8 b(v)432 954 y FE(\()p Fk(j)r FE(\))477 973 y Fx(\))18 b(of)g(appro)o(ximate)e(solutions)i([with)f(relativ)o(e)f (error)i(functions)g(\()p Ft(f)1741 954 y FE(\()p Fk(j)r FE(\))1787 973 y Ft(;)8 b(g)1834 954 y FE(\()p Fk(j)r FE(\))1879 973 y Fx(\)])46 1033 y(and)19 b(corresp)q(onding)g(norm{b)q (ounds)g Ft(U)798 1040 y Fk(j)817 1033 y Fx(,)f Ft(V)877 1040 y Fk(j)895 1033 y Fx(,...,)973 1017 y Fp(e)965 1033 y Ft(T)994 1040 y Fk(j)1010 1033 y Fx(.)g(More)g(precisely)l(,)e(\014x) i(n)o(um)o(b)q(ers)f Ft(\016)1673 1040 y Fk(j)1708 1033 y Fu(&)g Fx(0)i(suc)o(h)46 1093 y(that)156 1060 y Fp(P)208 1093 y Ft(\016)230 1100 y Fk(j)269 1093 y Ft(<)h(\030)j Fx(\(where)d Ft(\016)557 1100 y FE(0)585 1093 y Fu(\021)8 b Ft(\016)22 b Fx(as)f(ab)q(o)o(v)o(e\);)f(let,)f(for)i Ft(j)i Fu(\025)e Fx(1,)f Ft(\030)1264 1100 y Fk(j)1291 1093 y Fu(\021)8 b Ft(\030)1359 1100 y Fk(j)r FF(\000)p FE(1)1437 1093 y Fu(\000)13 b Ft(\016)1511 1100 y Fk(j)r FF(\000)p FE(1)1574 1093 y Fx(;)20 b(call)g(the)g(ab)q(o)o(v)o(e)46 1153 y(\\\014rst)c(appro)o(ximate)e(solution")i(\()p Ft(u)706 1135 y FE(\(0\))753 1153 y Ft(;)8 b(v)801 1135 y FE(\(0\))847 1153 y Fx(\),)15 b(\()p Ft(f)943 1135 y FE(\(0\))991 1153 y Ft(;)8 b(g)1038 1135 y FE(\(0\))1085 1153 y Fx(\))15 b(the)h(relativ)o(e)d(error)j(function)f(and)h(attac)o (h)46 1213 y(to)f Fu(M)p Fx(,)f Fu(T)28 b Fx(and)15 b(to)g(their)g (norm)e(b)q(ound)j(an)g(index)e(0.)g(Then,)h(giv)o(en,)e(for)j Ft(j)g Fu(\025)e Fx(0,)h(\()p Ft(u)1579 1195 y FE(\()p Fk(j)r FE(\))1624 1213 y Ft(;)8 b(v)1672 1195 y FE(\()p Fk(j)r FE(\))1717 1213 y Fx(\))15 b(and)g(the)46 1273 y(relativ)o(e)j(norm)h(b)q(ounds)i(\()p Ft(U)579 1280 y Fk(j)598 1273 y Ft(;)8 b(V)648 1280 y Fk(j)666 1273 y Ft(;)g(:::;)761 1257 y Fp(e)751 1273 y Ft(T)780 1280 y Fk(j)797 1273 y Fx(\))20 b(w)o(e)g(let)f(\()p Ft(u)1033 1255 y FE(\()p Fk(j)r FE(+1\))1124 1273 y Ft(;)8 b(v)1172 1255 y FE(\()p Fk(j)r FE(+1\))1262 1273 y Fx(\))19 b(b)q(e)i(the)e (appro)o(ximate)g(solution)46 1334 y(constructed)i(in)355 1316 y FE(25)413 1334 y Fx(Prop)q(osition)h(4.1)f(and,)h(if)e (conditions)h(\(4.23\),)g(\(4.26\))h(and)g(\(4.28\))f(are)g(satis-)46 1394 y(\014ed)c(\(see,)g(again,)h(fo)q(otnote)h(25\),)f(then)g(\()p Ft(U)855 1401 y Fk(j)r FE(+1)919 1394 y Ft(;)8 b(:::;)1013 1378 y Fp(e)1003 1394 y Ft(T)1032 1401 y Fk(j)r FE(+1)1095 1394 y Fx(\))16 b(=)g Fu(K)q Fx(\()p Ft(U)1275 1401 y Fk(j)1294 1394 y Ft(;)8 b(:::;)1388 1378 y Fp(e)1378 1394 y Ft(T)1407 1401 y Fk(j)1425 1394 y Fx(\))17 b(are)h(the)g(norm)f (b)q(ounds)46 1454 y(con)o(trolling)e(the)h(new)h(appro)o(ximate)d (solution)j(\()p Ft(u)986 1436 y FE(\()p Fk(j)r FE(+1\))1076 1454 y Ft(;)8 b(v)1124 1436 y FE(\()p Fk(j)r FE(+1\))1215 1454 y Fx(\).)46 1598 y Fe(4.4)69 b(KAM)23 b(theorem)46 1691 y Fx(Here)15 b(w)o(e)h(pro)o(v)o(e)f(a)i(KAM)e(theorem)g(based)i (on)f(the)g(KAM)g(algorithm)f(describ)q(ed)h(ab)q(o)o(v)o(e)1713 1673 y FE(26)1750 1691 y Fx(.)46 1805 y Fw(Theorem)g(4.1)24 b Fj(L)n(et)17 b Ft(!)j Fj(satisfy)d Fx(\(4.3\))p Fj(,)h(let)g (Assumption)g(4.2)f(hold,)h(let)668 1953 y Fx(0)c Ft(<)763 1940 y Fx(^)758 1953 y Ft(\030)j(<)c(\030)20 b(;)1012 1940 y Fx(^)1010 1953 y Ft(\016)10 b Fu(\021)1094 1919 y Fx(1)p 1094 1941 25 2 v 1094 1987 a(2)1146 1919 y Ft(\030)j Fu(\000)1235 1906 y Fx(^)1230 1919 y Ft(\030)p 1146 1941 109 2 v 1188 1987 a(\030)1276 1953 y(;)46 2088 y Fj(let)18 b Ft(\021)i Fj(b)n(e)d(the)h(fol)r(lowing)i(norm)d(on)h Fx(\()p Ft(f)s(;)8 b(g)r Fx(\))329 2198 y Ft(\021)i Fu(\021)16 b Fx(max)o Fu(f)p 534 2159 40 2 v Ft(E)k(E)626 2205 y FE(0)p Fk(;)p FE(2)691 2198 y Ft(;)25 b(E)766 2205 y FE(0)p Fk(;)p FE(2)824 2182 y Fp(e)814 2198 y Ft(T)6 b Fu(g)26 b Fx(max)n Fu(f)p Ft(\015)s(F)e(;)h(\015)1166 2177 y FE(2)1186 2198 y Ft(E)1222 2205 y FE(0)p Fk(;)p FE(2)1278 2182 y Fp(e)1269 2198 y Ft(V)12 b(F)23 b(;)j(\015)1432 2177 y FE(2)1451 2198 y Ft(E)1487 2205 y FE(0)p Fk(;)p FE(2)1535 2198 y Ft(G)p Fu(g)18 b Ft(;)p 46 2246 747 2 v 85 2278 a Fs(25)131 2293 y FI(Of)10 b(course,)i(the)f(unprimed)f (quan)o(tities)g(in)g(Prop)q(osition)g(4.1)f(\(b)q(esides)p 1246 2260 34 2 v 13 w Fz(E)j FI(and)e Fz(E)1397 2299 y Fi(p;q)1453 2293 y FI(whic)o(h)g(remain)f(unc)o(hanged\))46 2343 y(corresp)q(ond)15 b(here)h(to)e(the)g(index)g Fz(j)j FI(while)c(the)i(primed)e(ones)h(corresp)q(ond)i(here)f(to)f(the)h (index)f Fz(j)d FI(+)f(1.)j(Analogously)m(,)46 2393 y(one)k(has)g(to)f (attac)o(h)h(in)g(the)g(ob)o(vious)f(w)o(a)o(y)g(an)h(index)f Fz(j)k FI(or)c Fz(j)e FI(+)d(1)17 b(in)f(the)i(form)o(ulae)c(de\014ned) k(in)e(this)h(section;)g(for)46 2443 y(example)p 208 2410 31 2 v 12 w Fz(T)238 2449 y Fi(j)267 2443 y FI(=)12 b Fz(M)356 2428 y Fs(2)351 2454 y Fi(j)p 375 2410 34 2 v 375 2443 a Fz(E)r FI(,)h Fz(F)460 2449 y Fy(\003)p Fi(j)506 2443 y FI(=)555 2432 y Fp(e)550 2443 y Fz(V)574 2449 y Fi(j)591 2443 y Fz(F)618 2449 y Fi(j)645 2443 y FI(+)c Fz(M)726 2449 y Fi(j)744 2443 y Fz(G)777 2449 y Fi(j)794 2443 y FI(,)14 b Fz(s)839 2449 y Fs(1)869 2443 y FI(=)e(2)p Fz(\033)958 2449 y Fs(01)993 2443 y FI(\()p Fz(\016)1027 2449 y Fi(j)1045 2443 y Fz(=)p FI(2\))h(etc.)85 2479 y Fs(26)121 2494 y FI(W)m(e)i(remind)g(that)g(in)h(order)g(to)g (apply)f(suc)o(h)h(a)g(theorem)f(in)g(an)h(\\e\013ectiv)o(e)h(w)o(a)o (y",)d(one)i(should)f(apply)g(it)h(only)46 2544 y(after)e(a)g(few)g (iteration)f(of)g(the)i(map)d Fr(K)q FI(:)h(see)i([5)o(])e(for)h(more)f (informations.)955 2863 y Fx(21)p eop %%Page: 22 22 22 21 bop 46 78 a Fj(de\014ne)18 b(the)f(fol)r(lowing)i(p)n(ar)n (ameters)c(r)n(elate)n(d)i(to)g Ft(!)h Fj(and)f(to)g(the)g(quantities)h (intr)n(o)n(duc)n(e)n(d)e(in)h(Assump-)46 138 y(tion)h(4.2:)337 248 y Fx(\012)71 b Fu(\021)29 b Fx(max)499 278 y FE(1)p FF(\024)p Fk(i)p FF(\024)p Fk(N)623 248 y Fu(j)p Ft(!)667 255 y Fk(i)681 248 y Fu(j)17 b Ft(;)76 b Fx(\012)837 255 y FE(1)865 248 y Fu(\021)16 b Fx(max)o Fu(f)p Fx(\012)i Ft(;)25 b(E)1164 255 y FE(1)p Fk(;)p FE(1)1211 248 y Fu(g)17 b Ft(;)327 355 y Fx(\012)362 362 y FF(\003)443 355 y Fu(\021)g Fx(max)n Fu(f)p Ft(E)650 362 y FE(2)p Fk(;)p FE(1)715 355 y Ft(;)p 754 316 40 2 v 25 w(E)s(E)829 362 y FE(1)p Fk(;)p FE(2)876 355 y Fx(\012)911 362 y FE(1)949 355 y Ft(;)25 b(E)1024 362 y FE(0)p Fk(;)p FE(3)p 1071 316 V 1071 355 a Ft(E)1110 326 y FE(2)1130 355 y Fx(\012)1165 334 y FE(2)1165 367 y(1)1185 355 y Fu(g)17 b Ft(;)325 436 y(H)365 443 y FF(\003)443 436 y Fu(\021)g Fx(max)n Fu(f)p Ft(E)650 443 y FE(3)p Fk(;)p FE(0)715 436 y Ft(;)p 754 397 V 25 w(E)s(E)829 443 y FE(2)p Fk(;)p FE(1)876 436 y Fx(\012)911 443 y FE(1)949 436 y Ft(;)25 b(E)1024 443 y FE(1)p Fk(;)p FE(2)p 1071 397 V 1071 436 a Ft(E)1110 407 y FE(2)1130 436 y Fx(\012)1165 415 y FE(2)1165 448 y(1)1185 436 y Fu(g)17 b Ft(;)333 509 y(H)377 490 y FF(0)373 521 y(\003)443 509 y Fu(\021)g Fx(max)n Fu(f)p Ft(H)654 516 y FF(\003)692 509 y Ft(;)p 731 469 V 25 w(E)s Fx(\012\012)840 516 y FE(1)860 509 y Fu(g)p Ft(;)25 b(H)968 488 y FF(00)964 521 y(\003)998 509 y Fu(\021)17 b Fx(max)n Fu(f)p Ft(H)1213 488 y FF(0)1209 521 y(\003)1247 509 y Ft(;)1294 492 y Fp(e)1286 509 y Ft(V)11 b Fx(\012)1360 516 y FF(\003)1380 509 y Fu(g)18 b Ft(;)329 581 y(\013)360 588 y FF(\003)443 581 y Fu(\021)f Fx(max)n Fu(f)p 614 542 V Ft(E)k(E)707 588 y FE(0)p Fk(;)p FE(2)771 581 y Ft(;)26 b(E)847 588 y FE(0)p Fk(;)p FE(2)904 565 y Fp(e)894 581 y Ft(T)6 b Fu(g)11 b(\001)g Fx(max)o Fu(f)p Ft(\015)1134 561 y FE(2)1154 581 y Ft(E)1190 588 y FE(0)p Fk(;)p FE(2)1237 581 y Ft(H)1281 561 y FF(00)1277 593 y(\003)1320 581 y Ft(;)25 b(\015)s Fx(\012)1422 588 y FF(\003)1442 581 y Fu(g)17 b Ft(;)343 659 y(\022)77 b Fu(\021)17 b Fx(max)n Fu(f)p Fx(\()p Ft(\030)d Fu(\000)723 646 y Fx(^)718 659 y Ft(\030)r Fx(\))760 639 y FE(2)p Fk(\034)817 659 y Ft(;)25 b Fx(\()p Ft(\030)14 b Fu(\000)965 646 y Fx(^)960 659 y Ft(\030)s Fx(\))1003 639 y FE(2)p Fk(\034)t FE(+2)1087 659 y Fu(g)j Ft(;)320 765 y(\013)351 772 y FF(\003\003)443 765 y Fu(\021)8 b Ft(\022)i Fx(max)n Fu(f)p Fx(1)18 b Ft(;)p 732 692 V 732 731 a(E)s Fx(\012)806 738 y FE(1)p 724 753 110 2 v 724 799 a Ft(r)13 b Fu(\000)e Ft(\032)856 765 y(;)p 900 692 40 2 v 900 731 a(E)s Fx(\012)974 738 y FE(1)p 900 753 94 2 v 936 788 a Fp(e)928 804 y Ft(V)1017 765 y(;)1066 748 y Fp(e)1056 765 y Ft(T)6 b(E)1127 772 y FE(0)p Fk(;)p FE(2)1192 765 y Ft(;)1242 748 y Fp(e)1231 765 y Ft(T)h(E)1303 772 y FE(1)p Fk(;)p FE(2)1368 765 y Ft(;)1417 748 y Fp(e)1407 765 y Ft(T)f(E)1478 772 y FE(0)p Fk(;)p FE(3)p 1525 726 40 2 v 1525 765 a Ft(E)s Fx(\012)1599 772 y FE(1)1619 765 y Fu(g)18 b Ft(;)339 862 y(\013)73 b Fu(\021)17 b Fx(max)n Fu(f)p Ft(\013)645 869 y FF(\003)682 862 y Ft(;)26 b(\013)753 869 y FF(\003\003)790 862 y Fu(g)17 b Ft(:)46 972 y Fj(Ther)n(e)g(exists)h(a)g(p)n(olynomial) f Ft(\027)k Fj(in)c Fx(\()p Ft(\030)r(;)778 959 y Fx(^)775 972 y Ft(\016)r Fx(\))h Fj(satisfying)1035 954 y FE(27)330 1071 y Fx(5)p 330 1093 25 2 v 330 1139 a(4)373 1105 y Fu(\024)13 b Ft(\027)s Fx(\()p Ft(\030)r(;)519 1091 y Fx(^)516 1105 y Ft(\016)s Fx(\))g Fu(\024)h Fx(21)e(+)f(88)d(max)p Fu(f)p Ft(\030)20 b(;)25 b(\030)1011 1084 y FE(6)1032 1105 y Fu(g)17 b Ft(;)58 b Fu(8)p Ft(\030)15 b(>)f Fx(0)j Ft(;)26 b Fu(8)16 b Fx(0)e Ft(<)1481 1091 y Fx(^)1479 1105 y Ft(\016)h(<)1573 1071 y Fx(1)p 1573 1093 V 1573 1139 a(2)1620 1105 y Ft(;)46 1227 y Fj(such)j(that,)f(if)479 1292 y Ft(\021)r(\013)h(M)606 1271 y FE(7)p 626 1253 53 2 v 626 1292 a Ft(M)678 1263 y FE(9)698 1292 y Fx(\()p Ft(\030)c Fu(\000)807 1279 y Fx(^)802 1292 y Ft(\030)r Fx(\))844 1271 y FF(\000)p FE(2\(2)p Fk(\034)t FE(+1\))1001 1292 y Fx(2)1025 1271 y FE(8)p Fk(\034)t FE(+13)1145 1292 y Ft(\034)6 b Fx(!)1186 1271 y FE(4)1222 1292 y Ft(\027)s Fx(\()p Ft(\030)r(;)1317 1279 y Fx(^)1313 1292 y Ft(\016)s Fx(\))14 b Fu(\024)f Fx(1)18 b Ft(;)308 b Fx(\(4.31\))46 1379 y Fj(then)21 b(ther)n(e)e(exists)i(a)f(solution)g Fx(\()s(~)-27 b Ft(u;)10 b Fx(~)-26 b Ft(v)q Fx(\))20 b Fj(of)g Fx(\(4.1\))p Fj(,)g(which)g(is)g(r)n(e)n(al{analytic)g(on)g Fx(\001)1573 1385 y FE(^)1569 1394 y Fk(\030)1607 1379 y Fj(and)g(is)g Ft(\021)r Fj({close)46 1439 y(to)g Fx(\()p Ft(u;)8 b(v)r Fx(\))p Fj(.)19 b(F)l(urthermor)n(e)g Fu(h)s Fx(~)-27 b Ft(u)p Fu(i)19 b Fx(=)f Fu(h)q Ft(u)p Fu(i)i Fj(and)g(the)g(solution)h Fx(\()s(~)-27 b Ft(u;)10 b Fx(~)-26 b Ft(v)q Fx(\))20 b Fj(is)g(the)g(unique)i(solution)f(in)f(an) g Ft(\021)46 1499 y Fj(neighb)n(ourho)n(o)n(d)d(of)g Fx(\()p Ft(u;)8 b(v)r Fx(\))16 b Fj(such)i(that)g Fu(h)s Fx(~)-27 b Ft(u)p Fu(i)14 b Fx(=)g Fu(h)p Ft(u)p Fu(i)p Fj(.)46 1613 y Fx(F)l(or)23 b(us)h(it)f(will)f(b)q(e)h(particularly)g (imp)q(ortan)o(t)f(to)i(in)o(v)o(estigate)e Fv(dep)q(endence)g(up)q(on) i(parameters)p Fx(:)46 1674 y(assume)e(that)h(the)g(Hamiltonian)e Ft(h)26 b Fx(=)f Ft(h)p Fx(\()p Ft(x;)8 b(y)r Fx(;)g Ft(\026)p Fx(\))22 b(dep)q(ends)h(analytically)f(also)i(up)q(on)g(a)f (set)g(of)46 1734 y(parameters)15 b(v)m(arying)i(in)e(some)h(complex)e (domain)838 1844 y Ft(\026)g Fu(2)g(D)i(\032)d Fq(C)1067 1823 y Fk(p)1106 1844 y Ft(:)46 1954 y Fx(Then)22 b(the)f(co)q (e\016cien)o(ts)f Ft(h)548 1961 y Fk(n;k)622 1954 y Fx(in)h(\(4.18\))i (will)d(b)q(e)i(analytic)f(functions)g(of)h Ft(\026)h Fu(2)h(D)f Fx(and)f(w)o(e)f(shall)46 2014 y Fv(rede\014ne)15 b(the)h(norm)g(in)g Fx(\(4.19\))h Fv(b)o(y)554 2130 y Fu(k)p Ft(h)p Fu(k)632 2137 y Fk(\030)q(;r)686 2130 y Fu(\021)759 2089 y Fp(X)746 2182 y Fo(n)p Fn(2)p Fm(Z)805 2175 y Fo(N)746 2213 y(k)q Fn(2)p Fm(N)805 2206 y Fo(N)845 2082 y Fp(\020)878 2130 y Fx(sup)878 2170 y Fk(\026)p FF(2D)959 2130 y Fu(j)p Ft(h)1001 2137 y Fk(n;k)1054 2130 y Fu(j)1068 2082 y Fp(\021)1101 2130 y Fx(exp)o(\()p Fu(j)p Ft(n)p Fu(j)p Ft(\030)r Fx(\))g Ft(r)1333 2110 y FF(j)p Fk(k)q FF(j)1391 2130 y Ft(:)382 b Fx(\(4.32\))p 46 2254 747 2 v 85 2284 a Fs(27)121 2299 y FI(The)13 b(p)q(olynomial)c Fz(\027)14 b FI(\(of)e(degree)i(12)e(and)g(with)h(p)q (ositiv)o(e)f(co)q(e\016cien)o(ts\))i(is)e(explicitly)f(giv)o(en)h(in)g (the)i(pro)q(of)e(b)q(elo)o(w.)46 2349 y(F)m(or)h(later)h(use)h(w)o(e)f (rep)q(ort)h(also)e(the)i(follo)o(wing)c(v)n(alues:)554 2458 y Fz(\027)s FI(\(1)p Fz(;)637 2430 y FI(1)p 637 2448 21 2 v 637 2486 a(4)663 2458 y(\))h(=)f(17)p Fz(:)p FI(4281)p Fz(:::)g(;)48 b(\027)s FI(\(0)p Fz(:)p FI(0025)p Fz(;)1157 2430 y FI(1)p 1157 2448 V 1157 2486 a(4)1183 2458 y(\))11 b(=)h(6)p Fz(:)p FI(5190)p Fz(:::)399 b FI(\(4.30\))955 2863 y Fx(22)p eop %%Page: 23 23 23 22 bop 46 78 a Fx(Analogously)l(,)14 b(for)g(a)g(function)g Ft(f)5 b Fx(\()p Ft(x)p Fx(;)j Ft(\026)p Fx(\))14 b(analytic)g(in)f (\001)1068 85 y Fk(\030)1093 78 y Fu(\002)6 b(D)16 b Fx(w)o(e)e(shall)g Fv(rede\014ne)f(the)h(F)l(ourier)f(norm)46 138 y Fx(\(4.17\))k Fv(b)o(y)622 205 y Fu(k)p Ft(f)5 b Fu(k)701 212 y Fk(\030)729 205 y Fu(\021)803 163 y Fp(X)784 259 y Fk(n)p FF(2)p Fc(Z)849 249 y Fo(N)890 156 y Fp(\020)923 205 y Fx(sup)923 244 y Fk(\026)p FF(2D)1004 205 y Fu(j)p Ft(f)1042 212 y Fk(n)1066 205 y Fu(j)1080 156 y Fp(\021)1113 205 y Fx(exp)o(\()p Fu(j)p Ft(n)p Fu(j)p Ft(\030)r Fx(\))17 b Ft(:)46 328 y Fx(Note)d(that)h(w)o(e)f(are) g(not)h(c)o(hanging)g(the)f(sym)o(b)q(ols)f(of)i(the)f(norms)g(since)f Fv(the)i(domain)e Fu(D)j Fv(will)e(remain)46 388 y(unc)o(hanged)i(in)g (the)g(pro)q(ofs)p Fx(.)46 471 y(Finally)l(,)e(Assumptions)i(4.1)g(and) h(4.2)g(will)e(b)q(e)h(mo)q(di\014ed)f(as)i(follo)o(ws:)46 578 y Fw(Assumption)h(4.3)24 b Fj(L)n(et)18 b Fx(\()p Ft(\022)q(;)8 b(\026)p Fx(\))16 b Fu(2)h Fq(T)753 560 y Fk(N)796 578 y Fu(\002)12 b(D)18 b(!)e Fx(\()p Ft(u;)8 b(v)r Fx(\))15 b Fu(2)h Fq(C)1180 560 y Fk(N)1229 578 y Fu(\002)c Fq(C)1313 560 y Fk(N)1368 578 y Fj(b)n(e)19 b(a)g(r)n(e)n(gular)f(function)i(of)f Ft(\022)46 639 y Fj(and)f(let)g Fu(M)f Fj(and)h Ft(h)409 620 y FE(0)409 651 y Fk(y)q(y)466 639 y Fj(b)n(e)g(the)f(matric)n(es)432 755 y Fu(M)8 b(\021)g Ft(I)14 b Fx(+)d Ft(u)660 762 y Fk(\022)697 755 y Ft(;)108 b(h)847 734 y FE(0)847 767 y Fk(y)q(y)894 755 y Fu(\021)8 b Ft(h)969 762 y Fk(y)q(y)1009 707 y Fp(\020)1034 755 y Ft(\022)k Fx(+)f Ft(u)p Fx(\()p Ft(\022)q Fx(;)d Ft(\026)p Fx(\))p Ft(;)g(v)r Fx(\()p Ft(\022)q Fx(;)g Ft(\026)p Fx(\);)g Ft(\026)1471 707 y Fp(\021)1513 755 y Ft(:)260 b Fx(\(4.33\))46 858 y Fj(We)17 b(assume)h(that,)g(if)f(we)h(denote)771 962 y Fu(T)j(\021)8 b(M)926 941 y FF(\000)p FE(1)973 962 y Ft(h)1001 941 y FE(0)1001 974 y Fk(y)q(y)1041 962 y Fu(M)1101 941 y FF(\000)p Fk(T)1173 962 y Ft(;)600 b Fx(\(4.34\))46 1066 y Fj(then)18 b(for)f(any)g Fx(\()p Ft(\022)q(;)8 b(\026)p Fx(\))14 b Fu(2)h Fq(T)536 1048 y Fk(N)577 1066 y Fu(\002)c(D)q Fj(,)18 b(the)g(matric)n(es)g Fu(M)p Fj(,)f Ft(h)1097 1048 y FE(0)1097 1078 y Fk(y)q(y)1154 1066 y Fj(and)g Fu(h)q(T)12 b(i)18 b Fj(ar)n(e)f(invertible.)46 1178 y Fw(Assumption)h(4.4)24 b Fj(L)n(et)15 b Fx(0)f Ft(<)g(\030)i(<)719 1164 y Fx(\026)714 1178 y Ft(\030)s Fj(,)g Ft(r)q Fj(,)p 823 1139 40 2 v 16 w Ft(E)s Fj(,)f Ft(E)928 1185 y Fk(p;q)991 1178 y Fj(\()p Ft(p;)8 b(q)15 b Fu(2)f Fq(N)p Fj(\))j(b)n(e)f(such)g(that)g Fx(\()p Ft(x;)8 b(y)r(;)g(\026)p Fx(\))13 b Fu(!)h Ft(h)p Fx(\()p Ft(x;)8 b(y)r Fx(;)g Ft(\026)p Fx(\))46 1248 y Fj(is)15 b(r)n(e)n(al)g(analytic)i(on)f Fx(\001)483 1250 y FE(\026)479 1258 y Fk(\030)504 1248 y Fu(\002)p 550 1209 V 7 w Ft(B)590 1218 y Fk(N)590 1260 y(r)623 1248 y Fx(\()p Ft(y)666 1255 y FE(0)686 1248 y Fx(\))7 b Fu(\002)g(D)17 b Fj(and)f(such)g(that) g Fx(\(4.21\))g Fj(holds;)g(assume)g(that)g Fx(\()p Ft(u;)8 b(v)r Fx(\))14 b Fj(is)i(r)n(e)n(al)46 1315 y(analytic)k(on)f Fx(\001)344 1322 y Fk(\030)375 1315 y Fu(\002)12 b(D)21 b Fj(and)e(let)h Ft(U;)8 b(V)s(;)g(M)r(;)p 830 1276 53 2 v 8 w(M)e(;)913 1299 y Fp(e)905 1315 y Ft(V)11 b(;)d(F)q(;)g(G;)1091 1299 y Fp(e)1081 1315 y Ft(T)25 b Fj(b)n(e)20 b(p)n(ositive)f(numb)n (ers)g(for)g(which)g Fx(\(4.22\))46 1375 y Fj(holds)153 1357 y FE(28)207 1375 y Fj(\(with)e Fu(M)p Fj(,)f Fu(T)d Fj(,)j Ft(f)22 b Fj(and)16 b Ft(g)j Fj(as)d(in)h Fx(\(4.33\))p Fj(,)g Fx(\(4.34\))g Fj(and)g Fx(\(4.6\))p Fj(\).)g(Final)r(ly,)g (assume)g(that)f Fx(\(4.23\))46 1435 y Fj(holds.)46 1542 y Fx(With)g(these)g(mo)q(di\014cations)f(one)i(obtains)46 1649 y Fw(Theorem)f(4.2)24 b Fj(L)n(et)g Ft(!)r Fj(,)g Ft(\030)r Fj(,)608 1636 y Fx(^)603 1649 y Ft(\030)s Fj(,)668 1636 y Fx(^)666 1649 y Ft(\016)q Fj(,)g Ft(\027)j Fj(b)n(e)d(as)g(in)g (The)n(or)n(em)f(4.1;)g(assume)h(that)g Ft(h)p Fj(,)g Fx(\()p Ft(f)s(;)8 b(g)r Fx(\))24 b Fj(satisfy)46 1709 y(Assumption)e(4.4.)f(L)n(et)g Ft(\013)h Fj(and)g Ft(\021)h Fj(b)n(e)f(de\014ne)n(d)g(as)g(in)g(The)n(or)n(em)e(4.1)h(and)h(let)h Fx(\(4.31\))f Fj(hold.)f(Then)46 1769 y(ther)n(e)f(exists)h(a)f (solution)h Fx(\()s(~)-27 b Ft(u;)10 b Fx(~)-26 b Ft(v)q Fx(\))21 b Fj(of)f Fx(\(4.1\))p Fj(,)g(which)h(is)f(r)n(e)n (al{analytic)g(on)h Fx(\001)1467 1775 y FE(^)1463 1784 y Fk(\030)1495 1769 y Fu(\002)13 b(D)22 b Fj(and)e(is)g Ft(\021)r Fj({close)46 1830 y(to)g Fx(\()p Ft(u;)8 b(v)r Fx(\))p Fj(.)19 b(F)l(urthermor)n(e)g Fu(h)s Fx(~)-27 b Ft(u)p Fu(i)19 b Fx(=)f Fu(h)q Ft(u)p Fu(i)i Fj(and)g(the)g(solution) h Fx(\()s(~)-27 b Ft(u;)10 b Fx(~)-26 b Ft(v)q Fx(\))20 b Fj(is)g(the)g(unique)i(solution)f(in)f(an)g Ft(\021)46 1890 y Fj(neighb)n(ourho)n(o)n(d)d(of)g Fx(\()p Ft(u;)8 b(v)r Fx(\))16 b Fj(such)i(that)g Fu(h)s Fx(~)-27 b Ft(u)p Fu(i)14 b Fx(=)g Fu(h)p Ft(u)p Fu(i)p Fj(.)46 1997 y Fw(Remark)i(4.5)24 b Fx(\(i\))g(W)l(e)g(shall)g(not)g(pro)o(v)o(e)g (here)f(the)h(lo)q(cal)g(uniqueness)g(\(under)g(the)g(constrain)46 2057 y Fu(h)s Fx(~)-27 b Ft(u)p Fu(i)14 b Fx(=)g Fu(h)p Ft(u)p Fu(i)p Fx(\))h(of)f(the)g(solution)g(\()s(~)-27 b Ft(u;)10 b Fx(~)-26 b Ft(v)q Fx(\))14 b(and)h(w)o(e)f(refer)f(the)h (reader)g(to)h([5])e(\(where)h(a)h(similar)d(statemen)o(t)46 2117 y(is)k(pro)o(v)o(en)f(in)h(full)g(detail\).)46 2177 y(\(ii\))h(Precise)g(estimates)g(concerning)g(the)h Ft(\021)i Fx(closeness)e(\(whic)o(h)g(is)f(of)i(course)f(in)g(the)g(sense)g(of)g (the)46 2237 y Fu(k)11 b(\001)g(k)136 2243 y FE(^)132 2252 y Fk(\030)167 2237 y Fx(norm\))k(can)i(b)q(e)f(easily)f(deduced)h (from)f(the)h(detailed)f(estimates)g(giv)o(en)g(in)h(the)g(pro)q(of.)46 2367 y Fw(Pro)r(of)i Fx(\(of)f(Theorem)e(4.1\))h(The)h(argumen)o(t)e (is)h(b)o(y)g(induction.)f(W)l(e)h(de\014ne,)f(for)i Ft(j)g Fu(\025)c Fx(0,)516 2509 y Ft(\030)537 2516 y Fk(j)563 2509 y Fu(\021)616 2496 y Fx(^)611 2509 y Ft(\030)g Fx(+)699 2475 y Ft(\030)h Fu(\000)789 2462 y Fx(^)784 2475 y Ft(\030)p 699 2497 109 2 v 732 2543 a Fx(2)756 2529 y Fk(j)829 2509 y Ft(;)105 b(\016)970 2516 y Fk(j)996 2509 y Fu(\021)8 b Ft(\030)1064 2516 y Fk(j)1094 2509 y Fu(\000)j Ft(\030)1165 2516 y Fk(j)r FE(+1)1242 2509 y Fx(=)1299 2475 y Ft(\030)j Fu(\000)1389 2462 y Fx(^)1384 2475 y Ft(\030)p 1299 2497 V 1309 2543 a Fx(2)1333 2529 y Fk(j)r FE(+1)1429 2509 y Ft(:)p 46 2575 747 2 v 85 2612 a Fs(28)121 2627 y FI(Last)g(inequalit)o(y)e(m)o(ust)h(b)q(e)i (rewritten)g(as)f(sup)859 2637 y Fi(\026)p Fy(2D)937 2627 y Fr(jhT)c(i)1014 2606 y Fy(\000)p Fs(1)1059 2627 y Fr(j)h(\024)1133 2616 y Fp(e)1126 2627 y Fz(T)6 b FI(.)955 2863 y Fx(23)p eop %%Page: 24 24 24 23 bop 46 78 a Fx(W)l(e)21 b(assume)f(\(b)o(y)h(induction\))g(to)g (ha)o(v)o(e)g(constructed)g(\(b)o(y)f(iterating)h Ft(j)k Fu(\025)d Fx(0)g(times)d(the)i(sc)o(heme)46 138 y(of)16 b(Prop)q(osition)h(4.1)g(starting)f(with)g(\()p Ft(u)781 120 y FE(\(0\))828 138 y Ft(;)8 b(v)876 120 y FE(\(0\))922 138 y Fx(\))g Fu(\021)h Fx(\()p Ft(u;)f(v)r Fx(\)\))15 b(the)g(appro)o(ximate)g(solution)h(\()p Ft(u)1741 120 y FE(\()p Fk(j)r FE(\))1787 138 y Ft(;)8 b(v)1835 120 y FE(\()p Fk(j)r FE(\))1879 138 y Fx(\);)46 198 y(denote)16 b(b)o(y)g(\()p Ft(U)323 205 y Fk(j)341 198 y Ft(;)8 b(:::;)436 182 y Fp(e)425 198 y Ft(T)454 205 y Fk(j)472 198 y Fx(\))17 b(the)f(relativ)o(e)e(norm{b)q(ounds)1053 180 y FE(29)1107 198 y Fx(and)j(assume)f(that,)g(for)h(0)d Fu(\024)f Ft(i)h Fu(\024)f Ft(j)s Fx(,)46 308 y Fu(k)p Ft(u)99 288 y FE(\()p Fk(i)p FE(\))140 308 y Fu(k)165 315 y Fk(\030)181 320 y Fo(i)210 308 y Fu(\024)267 295 y Fx(\026)262 308 y Ft(\030)6 b Fu(\000)s Ft(\030)352 315 y Fk(i)379 308 y Ft(;)56 b Fu(k)p Ft(v)500 288 y FE(\()p Fk(i)p FE(\))545 308 y Fu(\000)s Ft(y)611 315 y FE(0)630 308 y Fu(k)655 315 y Fk(\030)671 320 y Fo(i)699 308 y Fu(\024)14 b Ft(r)g(;)858 292 y Fp(e)850 308 y Ft(V)878 315 y Fk(i)907 308 y Fu(\024)f Fx(2)991 292 y Fp(e)983 308 y Ft(V)1011 315 y FE(0)1044 308 y Ft(;)48 b(M)1153 315 y Fk(i)1181 308 y Fu(\024)14 b Fx(2)p Ft(M)1305 315 y FE(0)1338 308 y Ft(;)p 1400 269 53 2 v 48 w(M)1453 315 y Fk(i)1480 308 y Fu(\024)g Fx(2)p 1557 269 V Ft(M)1610 315 y FE(0)1642 308 y Ft(;)1714 292 y Fp(e)1704 308 y Ft(T)1733 315 y Fk(i)1761 308 y Fu(\024)f Fx(2)1848 292 y Fp(e)1837 308 y Ft(T)1866 315 y FE(0)1898 308 y Ft(:)1787 369 y Fx(\(4.35\))46 429 y(W)l(e)18 b(w)o(an)o(t)h(to)h(sho)o(w)f(that)g(\(4.35\))h(is)f (satis\014ed)g(also)g(for)g Ft(i)f Fx(=)g Ft(j)e Fx(+)d(1)19 b(and)h(that)f(the)g(norms)f(of)h Ft(z)1857 411 y FE(\()p Fk(i)p FE(\))1898 429 y Fx(,)46 489 y Ft(w)82 471 y FE(\()p Fk(i)p FE(\))124 489 y Fx(,)d Ft(f)183 471 y FE(\()p Fk(i)p FE(\))242 489 y Fx(and)i Ft(g)363 471 y FE(\()p Fk(i)p FE(\))422 489 y Fx(deca)o(y)f(esp)q(onen)o(tially)f(fast)i(as)f Ft(i)g Fx(tends)h(to)f(in\014nit)o(y)f(so)i(that)g(the)f(claim)e(of)i (the)46 549 y(theorem)e(follo)o(ws.)46 634 y(W)l(e)k(no)o(w)h(list)f (estimates)f(on)i(v)m(arious)g(quan)o(tities)f(en)o(tering)f(the)i (de\014nition)f(of)h(the)f(map)g Fu(K)h Fx(\()p Fj(i.e.)46 694 y Fx(on)g(the)g(de\014nitions)g(of)g(the)g(parameters,)e(in)o(tro)q (duced)i(in)f(the)h(previous)g(section,)f(whic)o(h)g(b)q(ound)46 755 y(the)d(norms)f(of)i(the)f(relev)m(an)o(t)f(ob)s(jects)h(asso)q (ciated)h(to)1042 737 y FE(30)1096 755 y Ft(u)1124 737 y FE(\()p Fk(i)p FE(\))1166 755 y Ft(;)8 b(v)1214 737 y FE(\()p Fk(i)p FE(\))1254 755 y Fx(\);)16 b(the)g(estimates)f(are)h (completely)46 815 y(elemen)o(tary)g(and)j(w)o(e)g(shall)g(giv)o(e)f (details)g(only)h(for)g(some)f(of)h(the)g(\014rst)g(estimates)f(\(all)g (the)h(other)46 875 y(estimates)14 b(b)q(eing)j(obtained)f(in)g(a)h (complete)d(analogous)k(w)o(a)o(y\).)46 935 y(In)c(the)g(follo)o(wing)g (estimates)f(w)o(e)h(shall)h(mak)o(e)d(systematic)h(use)h(of)h (\(4.35\),)g(\(4.20\),)f(of)h(the)f(fact)h(that)46 995 y Ft(\034)k Fu(\025)14 b Fx(1)j(and)g(of)f(the)g(follo)o(wing)g(simple) e(observ)m(ations)1033 977 y FE(31)1071 995 y Fx(:)204 1127 y Ft(\015)s Fx(\012)g Fu(\025)g Fx(1)i Ft(;)49 b(M)484 1134 y FE(0)517 1127 y Fu(\025)14 b Fx(1)i Ft(;)p 681 1088 V 57 w(M)734 1134 y FE(0)767 1127 y Fu(\025)e Fx(1)i Ft(;)p 923 1088 40 2 v 49 w(E)s(E)998 1134 y FE(0)p Fk(;)p FE(2)1059 1127 y Fu(\025)d Fx(1)k Ft(;)1219 1094 y(\016)1241 1101 y Fk(j)p 1219 1116 V 1219 1161 a Ft(\030)1240 1168 y Fk(j)1278 1127 y Fx(=)d(1)d Fu(\000)1420 1094 y Ft(\030)1441 1101 y Fk(j)r FE(+1)p 1420 1116 85 2 v 1443 1161 a Ft(\030)1464 1168 y Fk(j)1523 1127 y Fu(\024)1578 1114 y Fx(^)1576 1127 y Ft(\016)18 b(;)157 b Fx(\(4.36\))46 1264 y(where)792 1336 y(^)790 1350 y Ft(\016)10 b Fu(\021)874 1316 y Ft(\016)896 1323 y FE(0)p 874 1338 42 2 v 874 1384 a Ft(\030)895 1391 y FE(0)934 1350 y Fx(=)991 1316 y(1)p 991 1338 25 2 v 991 1384 a(2)1025 1316 y Ft(\030)k Fu(\000)1115 1303 y Fx(^)1109 1316 y Ft(\030)p 1025 1338 109 2 v 1067 1384 a(\030)1154 1350 y(:)619 b Fx(\(4.37\))46 1459 y(Here)15 b(is)h(the)g(list:)312 1581 y Ft(\033)340 1588 y FE(01)377 1533 y Fp(\020)407 1548 y Ft(\016)429 1555 y Fk(i)p 407 1570 36 2 v 416 1616 a Ft(t)448 1533 y Fp(\021)486 1581 y Fu(\024)e Ft(t\016)581 1561 y FF(\000)p FE(1)579 1594 y Fk(i)643 1581 y Ft(;)25 b Fx(\()p Fu(8)14 b Ft(t)g(>)g Fx(0\))i(;)57 b Ft(\033)986 1588 y FE(10)1023 1581 y Fx(\()p Ft(\030)1063 1588 y Fk(i)1077 1581 y Fx(\))14 b Fu(\024)g Ft(\015)s(\030)1214 1561 y FF(\000)p Fk(\034)1212 1594 y(i)1280 1581 y Ft(\034)6 b Fx(!)15 b(;)312 1700 y Ft(\033)340 1707 y FE(10)377 1652 y Fp(\020)407 1667 y Ft(\016)429 1674 y Fk(i)p 407 1689 V 413 1735 a Fx(2)448 1652 y Fp(\021)486 1700 y Fu(\024)f Fx(2)563 1680 y Fk(\034)585 1700 y Ft(\015)s(\016)637 1680 y FF(\000)p Fk(\034)635 1713 y(i)685 1700 y Ft(\034)6 b Fx(!)16 b Ft(;)57 b(\033)841 1707 y FE(10)878 1652 y Fp(\020)902 1700 y Ft(\030)923 1707 y Fk(i)949 1700 y Fu(\000)1004 1667 y Ft(\016)1026 1674 y Fk(i)p 1004 1689 V 1009 1735 a Fx(2)1044 1652 y Fp(\021)1083 1700 y Fu(\024)14 b Ft(\033)1164 1707 y FE(10)1201 1700 y Fx(\()p Ft(\030)1241 1707 y Fk(i)1255 1700 y Fx(\))g Fu(\024)f Ft(\015)s(\030)1391 1680 y FF(\000)p Fk(\034)1389 1713 y(i)1441 1700 y Ft(\034)6 b Fx(!)16 b(;)312 1820 y Ft(\033)340 1827 y FF(\000)p FE(10)405 1771 y Fp(\020)435 1786 y Ft(\016)457 1793 y Fk(i)p 435 1808 V 444 1854 a Ft(t)475 1771 y Fp(\021)514 1820 y Fu(\024)d Ft(t)j Fx(\012)p Ft(\016)659 1799 y FF(\000)p FE(1)657 1832 y Fk(i)722 1820 y Ft(;)25 b Fx(\()p Fu(8)14 b Ft(t)g(>)g Fx(0\);)57 b Ft(\033)1049 1827 y FE(11)1086 1771 y Fp(\020)1115 1786 y Ft(\016)1137 1793 y Fk(i)p 1115 1808 V 1121 1854 a Fx(2)1156 1771 y Fp(\021)1195 1820 y Fu(\024)13 b Ft(\015)s Fx(2)1299 1799 y Fk(\034)t FE(+1)1366 1820 y Ft(\016)1390 1794 y FF(\000)p FE(\()p Fk(\034)t FE(+1\))1388 1831 y Fk(i)1511 1820 y Fx(\()p Ft(\034)k Fx(+)11 b(1\)!)16 b(;)312 1939 y Ft(\033)340 1946 y FF(\000)p FE(11)405 1890 y Fp(\020)435 1905 y Ft(\016)457 1912 y Fk(i)p 435 1927 V 444 1973 a Ft(t)475 1890 y Fp(\021)514 1939 y Fu(\024)d Fx(2)p Ft(t)608 1918 y FE(2)644 1939 y Fx(\012)p Ft(\016)703 1918 y FF(\000)p FE(2)701 1951 y Fk(i)767 1939 y Ft(;)24 b Fx(\()p Fu(8)15 b Ft(t)e(>)h Fx(0\))i(;)312 2040 y Ft(T)341 2047 y Fk(i)369 2040 y Fu(\024)d Fx(4)p 445 2001 53 2 v Ft(M)498 2011 y FE(2)498 2052 y(0)518 2040 y Ft(E)554 2047 y FE(0)p Fk(;)p FE(2)617 2040 y Fx(;)p 688 2001 36 2 v 57 w Ft(T)723 2047 y Fk(i)751 2040 y Fu(\024)h Fx(4)p Ft(M)880 2019 y FE(2)875 2052 y(0)p 900 2001 40 2 v 900 2040 a Ft(E)20 b(;)56 b(F)1058 2047 y FF(\003)p Fk(i)1103 2040 y Fu(\024)14 b Fx(2)1188 2024 y Fp(e)1180 2040 y Ft(V)1208 2047 y FE(0)1229 2040 y Ft(F)1261 2047 y Fk(i)1285 2040 y Fx(+)d(2)p Ft(M)1405 2047 y FE(0)1426 2040 y Ft(G)1464 2047 y Fk(i)1494 2040 y Fx(;)312 2121 y Ft(s)335 2128 y FE(1)369 2121 y Fu(\024)i Fx(4)p Ft(\016)469 2100 y FF(\000)p FE(1)467 2133 y Fk(i)533 2121 y Fx(;)56 b Ft(s)626 2128 y FE(2)660 2121 y Fu(\024)13 b Fx(2)736 2100 y FE(2+)p Fk(\034)803 2121 y Ft(\015)s(\016)855 2095 y FF(\000)p FE(\()p Fk(\034)t FE(+1\))853 2132 y Fk(i)976 2121 y Ft(\034)6 b Fx(!)16 b(;)p 46 2167 747 2 v 85 2198 a Fs(29)121 2213 y FI(See)c(previous)g (section.)g(Note)g(that,)f(with)g(these)i(de\014nitions,)e Fz(\030)1124 2219 y Fs(0)1155 2213 y FI(=)h Fz(\030)r FI(,)f Fz(U)1270 2219 y Fs(0)1300 2213 y FI(=)h Fz(U)5 b FI(,...,)p Fz(M)1477 2219 y Fs(0)1505 2213 y FI(=)12 b Fz(M)5 b FI(,)11 b(etc.)h(The)g(\\norm{)46 2263 y(parameters")g(are)h (de\014ned)g(in)f(the)h(previous)g(section)g(where)h(the)f(primed)e (quan)o(tities)h(corresp)q(ond)i(to)f Fz(u)1709 2248 y Fs(\()p Fi(i)p Fs(+1\))1790 2263 y Fz(;)7 b(v)1830 2248 y Fs(\()p Fi(i)p Fs(+1\))46 2313 y FI(while)13 b(the)i(unprimed)e (ones)h(corresp)q(ond)i(to)d Fz(u)791 2297 y Fs(\()p Fi(i)p Fs(\))831 2313 y Fz(;)7 b(v)871 2297 y Fs(\()p Fi(i)p Fs(\))911 2313 y FI(.)85 2347 y Fs(30)121 2362 y FI(Recall)i(that)i(the)g(primed)e(quan)o(tities)h(of)g(section)h(4.3) f(corresp)q(ond)i(here)g(to)e(the)h(index)f Fz(i)r FI(+)r(1)i(while)e (the)h(unprimed)46 2412 y(ones)16 b(to)f(the)i(index)e Fz(i)p FI(,)h(that)f(the)h(index)g Fz(i)f FI(=)f(0)i(corresp)q(onds)h (to)f(the)g(\\initial")d(appro)o(ximate)h(solution)g(namely)g(to)46 2462 y(the)g(quan)o(tities)g(de\014ned)h(in)e(Assumption)g(4.2)g(and,)g (\014nally)m(,)f(that)i Fz(E)1144 2468 y Fi(p;q)1203 2462 y FI(and)p 1284 2429 34 2 v 14 w Fz(E)i FI(are)e(indep)q(enden)o (t)h(of)e Fz(i)p FI(.)85 2499 y Fs(31)121 2514 y FI(The)f(\014rst)i (relation)e(in)f(\(4.36\))h(follo)o(ws)f(from)f(\(4.3\))i(letting)g Fz(n)g FI(=)f Fz(e)1163 2499 y Fs(\()p Fi(i)p Fs(\))1203 2514 y FI(,)h(where)i Fr(f)p Fz(e)1386 2499 y Fs(\()p Fi(i)p Fs(\))1426 2514 y Fr(g)e FI(is)g(the)h(standard)g(orthonor-)46 2564 y(mal)i(basis)i(of)g Fq(R)325 2548 y Fi(N)353 2564 y FI(;)g(the)h(second)g(and)f(third)h(relations)f(follo)o(w)e(b)o(y)i (observing)h(that)f Fz(e)1444 2548 y Fs(\(1\))1506 2564 y FI(is)g(an)g(eigen)o(v)o(ector)h(with)46 2613 y(eigen)o(v)n(alue)12 b(1)g(of)g Fr(M)372 2598 y Fi(T)398 2613 y FI(\()p Fz(\022)433 2619 y Fs(0)453 2613 y FI(\))g(and)h(of)f Fr(M)657 2598 y Fy(\000)p Fi(T)709 2613 y FI(\()p Fz(\022)744 2619 y Fs(0)763 2613 y FI(\),)g(if)g Fz(\022)859 2619 y Fs(0)891 2613 y FI(is)g(a)g(critical)h(p)q(oin)o(t)f(of)g Fz(u)1279 2619 y Fs(1)1297 2613 y FI(\()p Fz(\022)q FI(\);)h(last)g(t)o(w)o(o)f (relations)g(are)h(ob)o(vious.)955 2863 y Fx(24)p eop %%Page: 25 25 25 24 bop 312 96 a Ft(s)335 103 y FE(3)369 96 y Fu(\024)13 b Ft(M)468 103 y FE(0)488 96 y Fx(2)512 76 y Fk(\034)t FE(+1)579 96 y Ft(\015)s(\016)631 76 y FF(\000)p Fk(\034)629 109 y(i)680 96 y Ft(\034)6 b Fx(!)k(+)h(4)p Ft(M)856 76 y FE(2)851 109 y(0)877 96 y Ft(\015)s(\030)928 76 y FF(\000)p Fk(\034)926 109 y(i)977 96 y Ft(\034)6 b Fx(!)14 b(=)f Ft(M)1135 76 y FE(2)1130 109 y(0)1155 96 y Fx(2)1179 76 y Fk(\034)t FE(+1)1246 96 y Ft(\015)s(\016)1298 76 y FF(\000)p Fk(\034)1296 109 y(i)1347 96 y Ft(\034)6 b Fx(!)1388 48 y Fp(h)1433 63 y Fx(1)p 1412 85 67 2 v 1412 131 a Ft(M)1459 138 y FE(0)1495 96 y Fx(+)11 b(2)1568 48 y Fp(\020)1610 63 y Ft(\016)1632 70 y Fk(i)p 1598 85 60 2 v 1598 131 a Fx(2)p Ft(\030)1643 138 y Fk(i)1663 48 y Fp(\021)1687 60 y Fk(\034)1709 48 y Fp(i)369 207 y Fu(\024)i Ft(M)473 186 y FE(2)468 219 y(0)493 207 y Fx(2)517 186 y Fk(\034)t FE(+1)584 207 y Ft(\015)s(\016)636 186 y FF(\000)p Fk(\034)634 219 y(i)685 207 y Ft(\034)6 b Fx(!)726 159 y Fp(\020)750 207 y Fx(1)12 b(+)837 194 y(^)835 207 y Ft(\016)858 159 y Fp(\021)891 207 y Fu(\021)c Ft(M)990 186 y FE(2)985 219 y(0)1010 207 y Fx(2)1034 186 y Fk(\034)t FE(+1)1101 207 y Ft(\015)s(\016)1153 186 y FF(\000)p Fk(\034)1151 219 y(i)1202 207 y Ft(\034)e Fx(!)15 b Ft(\027)1282 214 y FE(1)1318 207 y Ft(;)25 b(\027)1381 214 y FE(1)1409 207 y Fu(\021)8 b Fx(1)j(+)1542 194 y(^)1540 207 y Ft(\016)18 b Fx(;)312 321 y Ft(s)335 328 y FE(4)369 321 y Fu(\024)13 b Fx(\()p Ft(\015)s Fx(\012\))p Ft(M)574 301 y FE(2)569 334 y(0)595 321 y Fx(2)619 301 y Fk(\034)t FE(+1)686 321 y Ft(\016)710 296 y FF(\000)p FE(\()p Fk(\034)t FE(+1\))708 333 y Fk(i)831 321 y Ft(\034)6 b Fx(!)15 b Ft(\027)911 328 y FE(2)947 321 y Ft(;)24 b(\027)1009 328 y FE(2)1038 321 y Fu(\021)8 b Ft(\027)1109 328 y FE(1)1139 321 y Fx(+)1193 288 y Ft(\016)1217 270 y FE(2)1215 300 y(0)p 1193 310 44 2 v 1203 356 a Fx(2)1258 321 y(;)312 427 y Ft(s)335 434 y FE(5)369 427 y Fu(\024)13 b Ft(\015)s(M)501 407 y FE(2)496 440 y(0)521 427 y Fx(2)545 407 y Fk(\034)t FE(+2)613 427 y Ft(\016)637 402 y FF(\000)p FE(\()p Fk(\034)t FE(+1\))635 439 y Fk(i)758 427 y Fx(\()p Ft(\034)j Fx(+)11 b(1\)!)16 b Ft(\027)960 434 y FE(3)997 427 y Ft(;)24 b(\027)1059 434 y FE(3)1087 427 y Fu(\021)8 b Fx(1)j(+)1223 394 y Ft(\027)1247 401 y FE(1)p 1223 416 V 1233 462 a Fx(4)1288 427 y(;)312 544 y Ft(s)335 551 y FE(6)369 544 y Fu(\024)i Fx(\()p Ft(\015)s Fx(\012\))p Ft(M)574 523 y FE(2)569 556 y(0)595 544 y Fx(2)619 523 y Fk(\034)t FE(+2)686 544 y Ft(\016)710 518 y FF(\000)p FE(\()p Fk(\034)t FE(+2\))708 555 y Fk(i)831 544 y Fx(\()p Ft(\034)j Fx(+)11 b(1\)!)17 b Ft(\027)1034 551 y FE(4)1070 544 y Ft(;)24 b(\027)1132 551 y FE(4)1160 544 y Fu(\021)8 b Fx(1)j(+)1296 510 y Ft(\027)1320 517 y FE(1)p 1296 532 V 1306 578 a Fx(2)1356 544 y(+)1422 510 y(3)p 1410 532 49 2 v 1410 578 a(16)1464 544 y Ft(\016)1488 523 y FE(2)1486 556 y(0)1523 544 y Fx(;)312 645 y Ft(s)335 652 y FE(7)369 645 y Fu(\024)i Fx(\()p Ft(\015)s Fx(\012)503 652 y FE(1)523 645 y Fx(\))p Ft(M)594 624 y FE(2)589 657 y(0)614 645 y Fx(2)638 624 y Fk(\034)t FE(+1)705 645 y Ft(\016)729 620 y FF(\000)p FE(\()p Fk(\034)t FE(+1\))727 656 y Fk(i)850 645 y Ft(\034)6 b Fx(!)16 b Ft(\027)931 652 y FE(5)967 645 y Ft(;)24 b(\027)1029 652 y FE(5)1057 645 y Fu(\021)8 b Ft(\027)1128 652 y FE(2)1159 645 y Fx(+)j Ft(\016)1230 652 y FE(0)1250 645 y Ft(\027)1274 652 y FE(1)1310 645 y Fx(;)312 742 y Ft(s)335 749 y FE(8)369 742 y Fu(\024)i Fx(\()p Ft(\015)s Fx(\012)503 749 y FE(1)523 742 y Fx(\))p Ft(M)594 722 y FE(2)589 755 y(0)614 742 y Fx(2)638 722 y Fk(\034)t FE(+2)705 742 y Ft(\016)729 717 y FF(\000)p FE(\()p Fk(\034)t FE(+2\))727 754 y Fk(i)850 742 y Fx(\()p Ft(\034)k Fx(+)11 b(1\)!)16 b Ft(\027)1053 749 y FE(6)1089 742 y Ft(;)25 b(\027)1152 749 y FE(6)1180 742 y Fu(\021)8 b Ft(\027)1251 749 y FE(4)1282 742 y Fx(+)1336 709 y Ft(\027)1360 716 y FE(5)p 1336 731 44 2 v 1345 777 a Fx(4)1395 742 y(+)1449 709 y Ft(\027)1473 716 y FE(1)1493 709 y Ft(\016)1515 716 y FE(0)p 1449 731 86 2 v 1480 777 a Fx(4)1551 742 y(+)j Ft(\027)1624 749 y FE(3)1643 742 y Ft(\016)1665 749 y FE(0)1701 742 y Fx(;)312 835 y Ft(s)335 842 y FE(9)369 835 y Fu(\024)i Fx(2)p Ft(\016)469 815 y FF(\000)p FE(1)467 847 y Fk(i)533 835 y Fx(;)312 914 y Ft(s)335 921 y FE(10)386 914 y Fu(\024)h Ft(\015)467 894 y FE(2)487 914 y Ft(H)527 921 y FF(\003)547 914 y Ft(M)599 894 y FE(4)594 927 y(0)619 914 y Fx(2)643 894 y FE(2)p Fk(\034)t FE(+1)728 914 y Ft(\016)752 889 y FF(\000)p FE(2\()p Fk(\034)t FE(+1\))750 926 y Fk(i)890 914 y Ft(\034)6 b Fx(!)931 894 y FE(2)967 914 y Ft(\027)991 921 y FE(7)1027 914 y Ft(;)24 b(\027)1089 921 y FE(7)1117 914 y Fu(\021)8 b Fx(\()p Ft(\027)1207 921 y FE(5)1238 914 y Fx(+)j Ft(\027)1311 921 y FE(1)1331 914 y Ft(\016)1353 921 y FE(0)1372 914 y Fx(\))1391 894 y FE(2)1427 914 y Fx(;)312 994 y Ft(s)335 1001 y FE(11)386 994 y Fu(\024)j Ft(\015)467 973 y FE(2)487 994 y Ft(H)527 1001 y FF(\003)547 994 y Ft(M)599 973 y FE(2)594 1006 y(0)619 994 y Fx(2)643 973 y Fk(\034)t FE(+1)710 994 y Ft(\016)734 968 y FF(\000)p FE(\()p Fk(\034)t FE(+1\))732 1005 y Fk(i)855 994 y Ft(\034)6 b Fx(!)16 b Ft(\027)936 1001 y FE(8)972 994 y Ft(;)24 b(\027)1034 1001 y FE(8)1062 994 y Fu(\021)8 b Ft(\027)1133 1001 y FE(5)1164 994 y Fx(+)j Ft(\027)1237 1001 y FE(1)1257 994 y Ft(\016)1279 1001 y FE(0)1314 994 y Fx(;)312 1096 y Ft(s)335 1103 y FE(12)386 1096 y Fu(\024)444 1062 y Ft(\015)472 1044 y FE(2)492 1062 y Ft(H)532 1069 y FF(\003)p 444 1084 108 2 v 486 1130 a Fx(2)573 1096 y(;)312 1197 y Ft(s)335 1177 y FF(0)335 1209 y FE(10)386 1197 y Fu(\024)j Ft(\015)467 1177 y FE(2)487 1197 y Fx(\012)522 1204 y FF(\003)542 1197 y Ft(M)594 1177 y FE(4)589 1209 y(0)614 1197 y Fx(2)638 1177 y FE(2)p Fk(\034)t FE(+1)722 1197 y Ft(\016)746 1172 y FF(\000)p FE(2\()p Fk(\034)t FE(+1\))744 1209 y Fk(i)885 1197 y Ft(\034)6 b Fx(!)926 1177 y FE(2)961 1197 y Ft(\027)985 1204 y FE(7)1021 1197 y Fx(;)312 1276 y Ft(s)335 1256 y FF(0)335 1289 y FE(11)386 1276 y Fu(\024)14 b Ft(\015)467 1256 y FE(2)487 1276 y Fx(\012)522 1283 y FF(\003)542 1276 y Ft(M)594 1256 y FE(2)589 1289 y(0)614 1276 y Fx(2)638 1256 y Fk(\034)t FE(+1)705 1276 y Ft(\016)729 1251 y FF(\000)p FE(\()p Fk(\034)t FE(+1\))727 1288 y Fk(i)850 1276 y Ft(\034)6 b Fx(!)15 b Ft(\027)930 1283 y FE(8)966 1276 y Fx(;)312 1378 y Ft(s)335 1358 y FF(0)335 1391 y FE(12)386 1378 y Fu(\024)444 1345 y Ft(\015)472 1327 y FE(2)492 1345 y Fx(\012)527 1352 y FF(\003)p 444 1367 103 2 v 483 1413 a Fx(2)568 1378 y(;)312 1480 y Ft(s)335 1487 y FE(13)386 1480 y Fu(\024)f Ft(\015)467 1459 y FE(2)487 1480 y Ft(H)531 1459 y FF(0)527 1492 y(\003)547 1480 y Ft(M)599 1459 y FE(3)594 1492 y(0)619 1480 y Fx(2)643 1459 y Fk(\034)t FE(+1)710 1480 y Ft(\016)734 1454 y FF(\000)p FE(\()p Fk(\034)t FE(+2\))732 1491 y Fk(i)855 1480 y Ft(\034)6 b Fx(!)16 b Ft(\027)936 1487 y FE(9)972 1480 y Ft(;)24 b(\027)1034 1487 y FE(9)1062 1480 y Fu(\021)8 b Ft(\027)1133 1487 y FE(5)1164 1480 y Fx(+)j(2)p Ft(\016)1259 1487 y FE(0)1279 1480 y Ft(\027)1303 1487 y FE(8)1339 1480 y Fx(;)312 1559 y Ft(s)335 1566 y FE(14)386 1559 y Fu(\024)j Ft(\015)s(M)519 1538 y FE(2)514 1571 y(0)539 1559 y Fx(2)563 1538 y Fk(\034)t FE(+1)630 1559 y Ft(\016)654 1533 y FF(\000)p FE(\()p Fk(\034)t FE(+1\))652 1570 y Fk(i)775 1559 y Ft(\034)6 b Fx(!)16 b Ft(\027)856 1566 y FE(1)892 1559 y Fx(;)312 1632 y Ft(s)335 1639 y FE(15)386 1632 y Fu(\024)e Ft(\015)467 1611 y FE(2)487 1632 y Ft(H)531 1611 y FF(0)527 1644 y(\003)547 1632 y Ft(M)594 1639 y FE(0)614 1632 y Ft(\016)638 1611 y FF(\000)p FE(1)636 1644 y Fk(i)701 1632 y Ft(\027)725 1639 y FE(10)778 1632 y Ft(;)24 b(\027)840 1639 y FE(10)886 1632 y Fu(\021)8 b Fx(1)k(+)f Ft(\016)1040 1639 y FE(0)1075 1632 y Fx(;)312 1711 y Ft(s)335 1718 y FE(16)386 1711 y Fu(\024)j Ft(\015)467 1690 y FE(2)487 1711 y Ft(H)527 1718 y FF(\003)547 1711 y Ft(M)599 1690 y FE(5)594 1723 y(0)619 1711 y Fx(2)643 1690 y FE(2)p Fk(\034)t FE(+2)728 1711 y Ft(\016)752 1685 y FF(\000)p FE(2\()p Fk(\034)t FE(+1\))750 1722 y Fk(i)890 1711 y Ft(\034)6 b Fx(!)931 1690 y FE(2)967 1711 y Ft(\027)991 1718 y FE(7)1027 1711 y Fx(;)312 1813 y Ft(s)335 1820 y FE(17)386 1813 y Fu(\024)14 b Ft(\015)s Fx(\()p Ft(\015)s Fx(\012\)\()p Ft(M)634 1820 y FE(0)p 654 1774 53 2 v 654 1813 a Ft(M)706 1820 y FE(0)726 1813 y Fx(\))745 1792 y FE(2)765 1813 y Fx(2)789 1792 y FE(2)p Fk(\034)t FE(+5)874 1813 y Ft(\016)898 1787 y FF(\000)p FE(2\()p Fk(\034)t FE(+1\))896 1824 y Fk(i)1036 1813 y Ft(\034)6 b Fx(!)1077 1792 y FE(2)1112 1813 y Ft(\027)1136 1820 y FE(11)1190 1813 y Ft(;)24 b(\027)1252 1820 y FE(11)1298 1813 y Fu(\021)8 b Ft(\027)1369 1820 y FE(2)1400 1813 y Fx(+)1454 1779 y Ft(\027)1478 1786 y FE(1)p 1454 1801 44 2 v 1464 1847 a Fx(2)1514 1813 y(+)1568 1779 y Ft(\027)1592 1786 y FE(1)1612 1779 y Ft(\016)1636 1761 y FE(2)1634 1792 y(0)p 1568 1801 88 2 v 1599 1847 a Fx(2)1676 1813 y(;)312 1929 y Ft(s)335 1936 y FE(18)386 1929 y Fu(\024)14 b Ft(\015)467 1909 y FE(2)p 487 1890 40 2 v 487 1929 a Ft(E)526 1900 y FF(\000)p FE(1)573 1929 y Ft(H)617 1909 y FF(0)613 1942 y(\003)633 1929 y Ft(M)685 1909 y FE(3)680 1942 y(0)p 705 1890 53 2 v 705 1929 a Ft(M)758 1900 y FE(2)758 1942 y(0)777 1929 y Ft(M)824 1936 y FE(0)844 1929 y Fx(2)868 1909 y Fk(\034)t FE(+3)935 1929 y Ft(\016)959 1904 y FF(\000)p FE(\()p Fk(\034)t FE(+2\))957 1941 y Fk(i)1080 1929 y Ft(\034)6 b Fx(!)16 b Ft(\027)1161 1936 y FE(12)1214 1929 y Ft(;)25 b(\027)1277 1936 y FE(12)1322 1929 y Fu(\021)1374 1896 y Fx(9)p 1374 1918 25 2 v 1374 1964 a(8)1404 1929 y Ft(\027)1428 1936 y FE(1)1459 1929 y Fx(+)1513 1896 y Ft(\027)1537 1903 y FE(9)p 1513 1918 44 2 v 1522 1964 a Fx(2)1572 1929 y(+)1626 1896 y Ft(\027)1650 1903 y FE(2)p 1626 1918 V 1636 1964 a Fx(4)1692 1929 y Ft(:)46 2030 y Fx(Let)18 b(no)o(w)g Ft(\021)262 2037 y Fk(i)294 2030 y Fx(b)q(e)g(as)h Ft(\021)g Fx(\(de\014ned)f(in)f(the)h(text)f(of) h(Theorem)f(4.1\))h(but)g(with)g Ft(F)1503 2037 y Fk(i)1516 2030 y Fx(,)g Ft(G)1586 2037 y Fk(i)1618 2030 y Fx(in)g(place)f(of)h Ft(F)7 b Fx(,)46 2090 y Ft(G)p Fx(.)16 b(Then,)202 2195 y Ft(\015)s(B)267 2202 y FF(\003)p Fk(i)312 2195 y Fu(\024)e Ft(\021)389 2202 y Fk(i)403 2195 y Ft(M)450 2202 y FE(0)p 470 2156 53 2 v 470 2195 a Ft(M)522 2166 y FE(4)522 2207 y(0)542 2195 y Fx(2)566 2174 y Fk(\034)t FE(+4)633 2195 y Ft(\016)657 2174 y FF(\000)p Fk(\034)655 2207 y(i)706 2195 y Ft(\034)6 b Fx(!)15 b Ft(\027)786 2202 y FE(13)840 2195 y Ft(;)24 b(\027)902 2202 y FE(13)948 2195 y Fu(\021)8 b Fx(1)j(+)g(4)1105 2182 y(^)1103 2195 y Ft(\016)i Fx(+)1204 2161 y(9)p 1192 2183 49 2 v 1192 2229 a(16)1246 2195 y Ft(\016)1268 2202 y FE(0)1287 2195 y Fx(;)202 2296 y Fu(k)p Ft(z)252 2276 y FE(\()p Fk(i)p FE(\))293 2296 y Fu(k)318 2303 y Fk(\030)334 2308 y Fo(i)p Fl(+1)401 2296 y Fu(\024)h Ft(\021)478 2303 y Fk(i)492 2296 y Ft(M)544 2276 y FE(3)539 2308 y(0)p 564 2257 53 2 v 564 2296 a Ft(M)616 2267 y FE(4)616 2308 y(0)636 2296 y Fx(2)660 2276 y FE(2)p Fk(\034)t FE(+5)745 2296 y Ft(\016)769 2276 y FF(\000)p FE(2)p Fk(\034)767 2308 y(i)835 2296 y Ft(\034)6 b Fx(!)876 2276 y FE(2)911 2296 y Ft(\027)935 2303 y FE(14)989 2296 y Ft(;)24 b(\027)1051 2303 y FE(14)1097 2296 y Fu(\021)8 b Ft(\027)1168 2303 y FE(1)1188 2296 y Ft(\027)1212 2303 y FE(13)1265 2296 y Fx(;)202 2379 y Fu(k)p Ft(z)252 2354 y FE(\()p Fk(i)p FE(\))250 2392 y Fk(\022)293 2379 y Fu(k)318 2386 y Fk(\030)334 2391 y Fo(i)p Fl(+1)401 2379 y Fu(\024)14 b Ft(\021)478 2386 y Fk(i)492 2379 y Ft(M)544 2359 y FE(3)539 2392 y(0)p 564 2340 V 564 2379 a Ft(M)616 2350 y FE(4)616 2392 y(0)636 2379 y Fx(2)660 2359 y FE(2)p Fk(\034)t FE(+6)745 2379 y Ft(\016)769 2354 y FF(\000)p FE(\(2)p Fk(\034)t FE(+1\))767 2391 y Fk(i)907 2379 y Ft(\034)6 b Fx(!)948 2359 y FE(2)967 2379 y Fx(\()p Ft(\034)17 b Fx(+)11 b(1\))17 b Ft(\027)1157 2386 y FE(15)1211 2379 y Ft(;)24 b(\027)1273 2386 y FE(15)1318 2379 y Fu(\021)8 b Ft(\027)1389 2386 y FE(3)1409 2379 y Ft(\027)1433 2386 y FE(13)1487 2379 y Fx(;)202 2485 y Fu(k)p Ft(w)263 2465 y FE(\()p Fk(i)p FE(\))304 2485 y Fu(k)329 2492 y Fk(\030)345 2497 y Fo(i)p Fl(+1)412 2485 y Fu(\024)14 b Ft(\021)489 2492 y Fk(i)503 2485 y Fx(\()p 522 2446 40 2 v Ft(E)s Fx(\012)596 2492 y FE(1)616 2485 y Fx(\))p Ft(M)687 2465 y FE(3)682 2497 y(0)p 707 2446 53 2 v 707 2485 a Ft(M)760 2456 y FE(4)760 2497 y(0)779 2485 y Fx(2)803 2465 y FE(2)p Fk(\034)t FE(+5)888 2485 y Ft(\016)912 2460 y FF(\000)p FE(\(2)p Fk(\034)t FE(+1\))910 2497 y Fk(i)1051 2485 y Ft(\034)6 b Fx(!)1092 2465 y FE(2)1127 2485 y Ft(\027)1151 2492 y FE(16)1205 2485 y Ft(;)24 b(\027)1267 2492 y FE(16)1313 2485 y Fu(\021)8 b Ft(\027)1384 2492 y FE(5)1403 2485 y Ft(\027)1427 2492 y FE(13)1476 2485 y Fx(+)1530 2451 y Ft(\016)1554 2433 y FE(3)1552 2464 y(0)p 1530 2473 45 2 v 1530 2519 a Fx(2)1554 2505 y FE(7)1595 2485 y Fx(;)202 2609 y Fu(k)p Ft(w)263 2584 y FE(\()p Fk(i)p FE(\))262 2622 y Fk(\022)304 2609 y Fu(k)329 2616 y Fk(\030)345 2621 y Fo(i)p Fl(+1)412 2609 y Fu(\024)14 b Ft(\021)489 2616 y Fk(i)503 2609 y Fx(\()p 522 2570 40 2 v Ft(E)s Fx(\012)596 2616 y FE(1)616 2609 y Fx(\))p Ft(M)687 2588 y FE(3)682 2621 y(0)p 707 2570 53 2 v 707 2609 a Ft(M)760 2580 y FE(4)760 2621 y(0)779 2609 y Fx(2)803 2588 y FE(2)p Fk(\034)t FE(+6)888 2609 y Ft(\016)912 2584 y FF(\000)p FE(2\()p Fk(\034)t FE(+1\))910 2620 y Fk(i)1051 2609 y Ft(\034)6 b Fx(!)1092 2588 y FE(2)1111 2609 y Fx(\()p Ft(\034)16 b Fx(+)11 b(1\))17 b Ft(\027)1300 2616 y FE(17)1354 2609 y Ft(;)24 b(\027)1416 2616 y FE(17)1462 2609 y Fu(\021)8 b Ft(\027)1533 2616 y FE(6)1553 2609 y Ft(\027)1577 2616 y FE(13)1625 2609 y Fx(+)1679 2575 y Ft(\016)1703 2557 y FE(3)1701 2588 y(0)p 1679 2597 45 2 v 1679 2643 a Fx(2)1703 2629 y FE(8)1744 2609 y Fx(;)955 2863 y(25)p eop %%Page: 26 26 26 25 bop 202 101 a Fu(k)p Ft(f)256 81 y FE(\()p Fk(i)p FE(+1\))343 101 y Fu(k)368 108 y Fk(\030)384 113 y Fo(i)p Fl(+1)451 101 y Fu(\024)13 b Fx(\012)538 108 y FF(\003)574 101 y Ft(\021)600 81 y FE(2)598 114 y Fk(i)620 101 y Ft(M)672 81 y FE(6)667 114 y(0)p 692 62 53 2 v 692 101 a Ft(M)745 72 y FE(8)745 114 y(0)764 101 y Fx(2)788 81 y FE(4)p Fk(\034)t FE(+9)873 101 y Ft(\016)897 76 y FF(\000)p FE(\(4)p Fk(\034)t FE(+2\))895 113 y Fk(i)1036 101 y Ft(\034)6 b Fx(!)1077 81 y FE(4)1112 101 y Ft(\027)1136 108 y FE(18)1190 101 y Ft(;)24 b(\027)1252 108 y FE(18)1297 101 y Fu(\021)9 b Ft(\027)1369 108 y FE(7)1388 101 y Ft(\027)1415 81 y FE(2)1412 114 y(13)1461 101 y Fx(+)1515 68 y Ft(\016)1539 50 y FE(3)1537 80 y(0)p 1515 90 45 2 v 1515 136 a Fx(2)1539 121 y FE(6)1564 101 y Ft(\027)1588 108 y FE(8)1608 101 y Ft(\027)1632 108 y FE(13)1680 101 y Fx(+)1744 68 y Ft(\016)1768 50 y FE(6)1766 80 y(0)p 1734 90 62 2 v 1734 136 a Fx(2)1758 121 y FE(14)1817 101 y Fx(;)202 203 y Fu(k)p Ft(g)252 182 y FE(\()p Fk(i)p FE(+1\))338 203 y Fu(k)363 210 y Fk(\030)379 215 y Fo(i)p Fl(+1)446 203 y Fu(\024)14 b Ft(H)543 182 y FF(0)539 215 y(\003)575 203 y Ft(\021)601 182 y FE(2)599 215 y Fk(i)621 203 y Ft(M)673 182 y FE(7)668 215 y(0)p 693 164 53 2 v 693 203 a Ft(M)746 174 y FE(9)746 215 y(0)765 203 y Fx(2)789 182 y FE(4)p Fk(\034)t FE(+12)892 203 y Ft(\016)916 177 y FF(\000)p FE(\(4)p Fk(\034)t FE(+2\))914 214 y Fk(i)1054 203 y Ft(\034)6 b Fx(!)1095 182 y FE(4)1130 203 y Ft(\027)1154 210 y FE(19)1208 203 y Ft(;)485 308 y(\027)509 315 y FE(19)555 308 y Fu(\021)i Ft(\027)626 315 y FE(7)646 308 y Ft(\027)673 288 y FE(2)670 321 y(13)718 308 y Fx(+)772 275 y Ft(\016)794 282 y FE(0)p 772 297 42 2 v 781 343 a Fx(2)819 308 y Ft(\027)843 315 y FE(11)880 308 y Ft(\027)904 315 y FE(13)953 308 y Fx(+)1007 275 y Ft(\016)1031 257 y FE(2)1029 287 y(0)p 1007 297 45 2 v 1007 343 a Fx(2)1031 328 y FE(6)1056 308 y Ft(\027)1080 315 y FE(12)1117 308 y Ft(\027)1141 315 y FE(13)1190 308 y Fx(+)1244 275 y Ft(\016)1268 257 y FE(3)1266 287 y(0)p 1244 297 V 1244 343 a Fx(2)1268 328 y FE(7)1293 308 y Ft(\027)1317 315 y FE(1)1337 308 y Ft(\027)1361 315 y FE(13)1409 308 y Fx(+)1472 275 y Ft(\016)1496 257 y FE(5)1494 287 y(0)p 1463 297 62 2 v 1463 343 a Fx(2)1487 328 y FE(14)1530 308 y Ft(\027)1554 315 y FE(10)1608 308 y Fx(;)165 b(\(4.38\))46 439 y(\(in)16 b(the)g(last)g(inequalit)o (y)e(w)o(e)i(used)g(also)h(the)f(fact)g(that)g(from)p 1205 400 40 2 v 15 w Ft(E)t(E)1281 446 y FE(0)p Fk(;)p FE(2)1342 439 y Fu(\025)d Fx(1)k(and)f Ft(\015)s Fx(\012)1592 446 y FE(1)1626 439 y Fu(\025)e Fx(1)i(it)g(follo)o(ws)46 499 y(that)h Ft(\015)s Fx(\012)8 b(max)o Fu(f)347 483 y Fp(e)339 499 y Ft(V)j(F)q(;)d(G)p Fu(g)14 b(\024)g Ft(H)607 481 y FF(0)603 511 y(\003)623 499 y Ft(\021)647 506 y Fk(i)661 499 y Fx(\).)i(W)l(e)g(let)f(no)o(w)756 631 y Ft(\027)d Fu(\021)c Ft(\027)s Fx(\()p Ft(\030)906 638 y FE(0)926 631 y Ft(;)950 618 y Fx(^)948 631 y Ft(\016)q Fx(\))g Fu(\021)h Ft(\027)1070 638 y FE(19)1118 631 y Fx(+)1172 598 y(1)p 1172 620 25 2 v 1172 666 a(4)1787 631 y(\(4.39\))46 756 y(and)24 b(observ)o(e)e(that)i(all)e(the)h Ft(\027)633 738 y FF(0)645 756 y Ft(s)g Fx(are)g(grater)h(or)f(equal)g (than)h(1.)f(Putting)g(together)g(the)g(ab)q(o)o(v)o(e)46 816 y(de\014nitions)16 b(of)g(the)g(v)m(arious)h Ft(\027)612 823 y Fk(i)627 816 y Fx('s)f(and)g(recalling)g(that)g Ft(\016)1093 823 y FE(0)1126 816 y Fx(=)e Ft(\030)1204 803 y Fx(^)1201 816 y Ft(\016)r Fx(,)i(one)h(\014nds)127 946 y Ft(\027)44 b Fx(=)280 912 y(5)p 280 934 V 280 980 a(4)320 946 y(+)11 b(10)437 933 y(^)434 946 y Ft(\016)i Fx(+)e(33)585 933 y(^)583 946 y Ft(\016)607 925 y FE(2)637 946 y Fx(+)g(40)753 933 y(^)751 946 y Ft(\016)775 925 y FE(3)805 946 y Fx(+)g(16)922 933 y(^)919 946 y Ft(\016)943 925 y FE(4)974 946 y Fx(+)1028 912 y(47)p 1028 934 49 2 v 1040 980 a(8)1100 933 y(^)1098 946 y Ft(\016)q(\030)j Fx(+)1210 912 y(101)p 1210 934 74 2 v 1234 980 a(2)1307 933 y(^)1304 946 y Ft(\016)1328 925 y FE(2)1348 946 y Ft(\030)f Fx(+)1436 912 y(1161)p 1436 934 98 2 v 1473 980 a(8)1557 933 y(^)1555 946 y Ft(\016)1579 925 y FE(3)1598 946 y Ft(\030)275 1062 y Fx(+)318 1029 y(329)p 318 1051 74 2 v 342 1096 a(2)414 1049 y(^)412 1062 y Ft(\016)436 1042 y FE(4)455 1062 y Ft(\030)h Fx(+)d(64)606 1049 y(^)604 1062 y Ft(\016)628 1042 y FE(5)647 1062 y Ft(\030)j Fx(+)736 1029 y(5257)p 736 1051 98 2 v 748 1096 a(512)857 1049 y(^)855 1062 y Ft(\016)879 1042 y FE(2)898 1062 y Ft(\030)921 1042 y FE(2)953 1062 y Fx(+)1007 1029 y(39527)p 1007 1051 122 2 v 1031 1096 a(512)1152 1049 y(^)1150 1062 y Ft(\016)1174 1042 y FE(3)1193 1062 y Ft(\030)1216 1042 y FE(2)1247 1062 y Fx(+)1301 1029 y(50415)p 1301 1051 V 1326 1096 a(256)1447 1049 y(^)1445 1062 y Ft(\016)1469 1042 y FE(4)1488 1062 y Ft(\030)1511 1042 y FE(2)1542 1062 y Fx(+)d(194)1683 1049 y(^)1681 1062 y Ft(\016)1705 1042 y FE(5)1724 1062 y Ft(\030)1747 1042 y FE(2)275 1179 y Fx(+64)380 1166 y(^)378 1179 y Ft(\016)402 1158 y FE(6)421 1179 y Ft(\030)444 1158 y FE(2)476 1179 y Fx(+)530 1145 y(77319)p 530 1167 V 542 1213 a(8192)675 1166 y(^)673 1179 y Ft(\016)697 1158 y FE(3)716 1179 y Ft(\030)739 1158 y FE(3)770 1179 y Fx(+)824 1145 y(455303)p 824 1167 147 2 v 849 1213 a(8192)994 1166 y(^)992 1179 y Ft(\016)1016 1158 y FE(4)1035 1179 y Ft(\030)1058 1158 y FE(3)1090 1179 y Fx(+)1144 1145 y(6105)p 1144 1167 98 2 v 1168 1213 a(64)1265 1166 y(^)1262 1179 y Ft(\016)1286 1158 y FE(5)1306 1179 y Ft(\030)1329 1158 y FE(3)1360 1179 y Fx(+)g(50)1476 1166 y(^)1474 1179 y Ft(\016)1498 1158 y FE(6)1517 1179 y Ft(\030)1540 1158 y FE(3)1572 1179 y Fx(+)1626 1145 y(1131)p 1626 1167 V 1638 1213 a(256)1747 1166 y(^)1745 1179 y Ft(\016)1769 1158 y FE(4)1788 1179 y Ft(\030)1811 1158 y FE(4)275 1295 y Fx(+)318 1262 y(8415)p 318 1284 V 330 1330 a(512)439 1282 y(^)437 1295 y Ft(\016)461 1275 y FE(5)480 1295 y Ft(\030)503 1275 y FE(4)534 1295 y Fx(+)588 1262 y(921)p 588 1284 74 2 v 600 1330 a(64)685 1282 y(^)683 1295 y Ft(\016)707 1275 y FE(6)726 1295 y Ft(\030)749 1275 y FE(4)780 1295 y Fx(+)834 1262 y(15447)p 834 1284 122 2 v 834 1330 a(16384)980 1282 y(^)977 1295 y Ft(\016)1001 1275 y FE(5)1021 1295 y Ft(\030)1044 1275 y FE(5)1075 1295 y Fx(+)1129 1262 y(925)p 1129 1284 74 2 v 1129 1330 a(512)1226 1282 y(^)1224 1295 y Ft(\016)1248 1275 y FE(6)1267 1295 y Ft(\030)1290 1275 y FE(5)1321 1295 y Fx(+)1387 1262 y(1369)p 1375 1284 122 2 v 1375 1330 a(16384)1521 1282 y(^)1518 1295 y Ft(\016)1542 1275 y FE(6)1562 1295 y Ft(\030)1585 1275 y FE(6)1621 1295 y Ft(:)46 1423 y Fx(W)l(e)16 b(note,)g(for)g(later)g (use,)g(that)627 1405 y FE(32)540 1542 y Ft(\027)h Fu(\025)c Fx(max)o Fu(f)770 1508 y Ft(\027)794 1515 y FE(18)p 770 1530 62 2 v 789 1576 a Fx(4)853 1542 y Ft(;)24 b(\027)915 1549 y FE(14)969 1542 y Ft(;)g(\027)1031 1549 y FE(15)1085 1542 y Ft(;)g(\027)1147 1549 y FE(16)1195 1542 y Fx(+)11 b Ft(\016)1266 1549 y FE(0)1286 1542 y Ft(\027)1310 1549 y FE(14)1363 1542 y Fu(g)17 b Ft(:)368 b Fx(\(4.40\))46 1667 y(Th)o(us,)16 b(\(using)h(the)f(inequalities)e Ft(\013)g Fu(\025)g Ft(\013)799 1674 y FF(\003)835 1667 y Fx(and)j Ft(\027)g Fu(\025)c Fx(max)o Fu(f)1144 1648 y Fk(\027)1161 1653 y Fl(18)p 1144 1656 50 2 v 1160 1684 a FE(4)1199 1667 y Ft(;)8 b(\027)1245 1674 y FE(19)1282 1667 y Fu(g)p Fx(\),)16 b(w)o(e)f(\014nd)831 1777 y Ft(\021)855 1784 y Fk(i)p FE(+1)928 1777 y Fu(\024)f Ft(\024\025)1037 1757 y Fk(i)1051 1777 y Ft(\021)1077 1757 y FE(2)1075 1790 y Fk(i)1113 1777 y Ft(;)46 1887 y Fx(where)171 1869 y FE(33)437 1947 y Ft(\024)8 b Fu(\021)h Ft(\013M)604 1927 y FE(7)599 1960 y(0)p 624 1908 53 2 v 624 1947 a Ft(M)676 1918 y FE(9)676 1960 y(0)696 1947 y Fx(2)720 1927 y FE(8)p Fk(\034)t FE(+13)823 1947 y Fx(\()p Ft(\030)k Fu(\000)931 1934 y Fx(^)926 1947 y Ft(\030)s Fx(\))969 1927 y FF(\000)p FE(\(4)p Fk(\034)t FE(+2\))1108 1947 y Ft(\034)6 b Fx(!)1149 1927 y FE(4)1184 1947 y Ft(\027)20 b(;)56 b(\025)8 b Fu(\021)h Fx(2)1406 1927 y FE(4)p Fk(\034)t FE(+2)1507 1947 y Ft(:)46 2035 y Fx(Iterating,)15 b(for)i(all)f(0)e Fu(\024)f Ft(i)h Fu(\024)f Ft(j)h Fx(+)d(1,)17 b(w)o(e)e(get)851 2017 y FE(34)830 2178 y Ft(\021)854 2185 y Fk(i)882 2178 y Fu(\024)940 2145 y Fx(\()p Ft(\021)r(\024\025)p Fx(\))1060 2127 y FE(2)1078 2115 y Fo(i)p 940 2167 154 2 v 958 2212 a Ft(\024\025)1014 2198 y Fk(i)p FE(+1)1114 2178 y Ft(;)p 46 2247 747 2 v 85 2277 a Fs(32)121 2292 y FI(As)d(it)g(is)h(imm)o (ediate)d(to)i(c)o(hec)o(k)i(after)e(ha)o(ving)f(written)i(do)o(wn)f (explicitly)g(the)h(de\014nition)f(of)f Fz(\027)k FI(\(whic)o(h,)d (recalling)46 2347 y(that)g Fz(\016)152 2353 y Fs(0)183 2347 y FI(=)g Fz(\030)245 2353 y Fs(0)265 2336 y FI(^)263 2347 y Fz(\016)r FI(,)g(turns)h(out)f(to)g(b)q(e)h(a)f(p)q(olynomial)d (of)i(degree)j(12)e(in)f Fz(\030)1127 2353 y Fs(0)1158 2347 y FI(and)1239 2336 y(^)1237 2347 y Fz(\016)k FI(with)c(p)q(ositiv) o(e)h(\(rational\))g(co)q(e\016cien)o(ts\))46 2396 y(and)i(of)f(the)h (other)h Fz(\027)375 2402 y Fi(k)395 2396 y FI('s.)85 2431 y Fs(33)121 2446 y FI(The)c(reason)h(for)e(ha)o(ving)g(in)g(this)h (form)o(ula)e Fz(\013)h FI(in)h(place)g(of)f Fz(\013)1051 2452 y Fy(\003)1081 2446 y FI(will)f(b)q(e)j(plain)e(when)h(w)o(e)g (shall)g(c)o(hec)o(k)g(the)h(inductiv)o(e)46 2496 y(assumptions)h (\(4.35\))g(for)g Fz(j)f FI(+)d(1.)85 2536 y Fs(34)121 2551 y FI(Recall)k(that)336 2520 y Fp(P)379 2530 y Fi(i)p Fy(\000)p Fs(1)379 2563 y Fi(k)q Fs(=0)449 2551 y FI(2)470 2536 y Fi(k)502 2551 y FI(=)e(2)566 2536 y Fi(i)589 2551 y Fr(\000)f FI(1,)677 2520 y Fp(P)721 2530 y Fi(i)p Fy(\000)p Fs(1)721 2563 y Fi(k)q Fs(=1)783 2551 y FI(\()p Fz(i)g Fr(\000)f Fz(k)q FI(\)2)924 2536 y Fi(k)q Fy(\000)p Fs(1)998 2551 y FI(=)j(2)1063 2536 y Fi(i)1086 2551 y Fr(\000)e Fz(i)f Fr(\000)h FI(1,)j(\(and)h(that)g Fz(\021)1447 2557 y Fs(0)1476 2551 y FI(=)e Fz(\021)q FI(\).)955 2863 y Fx(26)p eop %%Page: 27 27 27 26 bop 46 78 a Fx(whence,)16 b(in)h(view)f(of)h(the)g(\\KAM)g (condition")g(\(4.31\))h(\(whic)o(h)e(is)h(no)o(w)h(recognized)e(as)i (equiv)m(alen)o(t)46 138 y(to)e(require)f Ft(\021)r(\024\025)g Fu(\024)e Fx(1\),)849 221 y Ft(\021)873 228 y Fk(i)901 221 y Fu(\024)1004 187 y Fx(1)p 958 209 116 2 v 958 255 a Ft(\024\025)1014 240 y Fk(i)p FE(+1)1095 221 y Ft(:)678 b Fx(\(4.41\))46 329 y(This)18 b(b)q(ound)h(allo)o(ws)g(to)f(get)g (simple)e(estimates)h(on)h(the)g(norms)g(of)g Ft(z)1360 311 y FE(\()p Fk(i)p FE(\))1420 329 y Fx(and)g Ft(w)1552 311 y FE(\()p Fk(i)p FE(\))1612 329 y Fx(for)h(0)e Fu(\024)g Ft(i)g Fu(\024)f Ft(j)s Fx(.)46 389 y(Using)g(\(4.41\))h(and)g(the)f (facts)g(that)316 499 y Ft(\027)h Fu(\025)d Fx(max)n Fu(f)p Ft(\027)549 506 y FE(14)603 499 y Ft(;)24 b(\027)665 506 y FE(15)702 499 y Fu(g)49 b Fx(and)h Ft(\013)14 b Fu(\025)f Fx(max)o Fu(f)p Fx(\()p Ft(\030)h Fu(\000)1226 486 y Fx(^)1221 499 y Ft(\030)r Fx(\))1263 478 y FE(2)p Fk(\034)t FE(+2)1364 499 y Ft(;)24 b Fx(\()p Ft(\030)14 b Fu(\000)1510 486 y Fx(^)1505 499 y Ft(\030)s Fx(\))1548 478 y FE(2)p Fk(\034)1587 499 y Fu(g)i Ft(;)46 609 y Fx(from)f(\(4.38\))i(w)o(e)f(get)424 741 y(max)o Fu(fk)p Ft(z)590 721 y FE(\()p Fk(i)p FE(\))631 741 y Fu(k)656 748 y Fk(\030)672 753 y Fo(i)p Fl(+1)741 741 y Ft(;)25 b Fu(k)p Ft(z)830 716 y FE(\()p Fk(i)p FE(\))828 754 y Fk(\022)871 741 y Fu(k)896 748 y Fk(\030)912 753 y Fo(i)p Fl(+1)965 741 y Fu(g)14 b(\024)f Fx(min)o Fu(f)p Ft(\016)1185 748 y Fk(i)1215 741 y Ft(;)1282 708 y Fx(1)p 1258 730 73 2 v 1258 738 53 2 v 1258 777 a Ft(M)1310 784 y FE(0)1375 708 y Fx(1)p 1356 730 62 2 v 1356 775 a(2)1380 761 y FE(15)1451 708 y Fx(1)p 1444 730 39 2 v 1444 775 a(8)1468 761 y Fk(i)1487 741 y Fu(g)k Ft(:)244 b Fx(\(4.42\))46 873 y(Analogously)l(,)16 b(using)430 1002 y Ft(\027)h Fu(\025)c Ft(\027)547 1009 y FE(16)601 1002 y Ft(;)57 b(\013)14 b Fu(\025)p 769 963 40 2 v 13 w Ft(E)t Fx(\012)844 1009 y FE(1)880 1002 y Fx(\()p Ft(\030)f Fu(\000)988 989 y Fx(^)983 1002 y Ft(\030)s Fx(\))1026 982 y FE(2)p Fk(\034)t FE(+1)1135 1002 y Fx(max)o Fu(f)1298 969 y Fx(1)p 1256 991 110 2 v 1256 1037 a Ft(r)f Fu(\000)f Ft(\032)1386 1002 y(;)1437 969 y Fx(1)p 1429 991 40 2 v 1437 1026 a Fp(e)1429 1042 y Ft(V)1473 1002 y Fu(g)17 b Ft(;)46 1138 y Fx(from)e(\(4.38\))i(there)f(follo)o(ws)589 1270 y Fu(k)p Ft(w)650 1249 y FE(\()p Fk(i)p FE(\))692 1270 y Fu(k)717 1277 y Fk(\030)733 1282 y Fo(i)p Fl(+1)800 1270 y Fu(\024)d Fx(min)o Fu(f)p Ft(r)f Fu(\000)f Ft(\032)16 b(;)1130 1254 y Fp(e)1122 1270 y Ft(V)11 b Fu(g)1226 1236 y Fx(1)p 1208 1258 62 2 v 1208 1304 a(2)1232 1290 y FE(18)1303 1236 y Fx(1)p 1296 1258 39 2 v 1296 1304 a(8)1320 1290 y Fk(i)1355 1270 y Ft(:)418 b Fx(\(4.43\))46 1395 y(T)l(o)18 b(c)o(hec)o(k)e(the)h(inductiv)o(e)f(h)o(yp)q(otheses)h (\(4.35\))h(w)o(e)g(shall)f(also)h(need)f(simple)e(b)q(ounds)k(on)f (the)g(con-)46 1456 y(stan)o(ts)f Ft(C)224 1463 y Fk(i)238 1456 y Fx(,)e Ft(C)306 1437 y FF(0)302 1468 y Fk(i)318 1456 y Fx(,)g Ft(C)386 1437 y FF(00)382 1468 y Fk(i)407 1456 y Fx(,)h Ft(C)472 1463 y FF(\003)p Fk(i)520 1456 y Fx([recall)e(\(4.27\))j(and)g(\(4.29\)].)f(Using)684 1566 y Ft(\027)h Fu(\025)c Fx(max)o Fu(f)p Ft(\027)917 1573 y FE(15)971 1566 y Ft(;)24 b(\027)1033 1573 y FE(16)1081 1566 y Fx(+)11 b Ft(\016)1152 1573 y FE(0)1172 1566 y Ft(\027)1196 1573 y FE(14)1233 1566 y Fu(g)46 1676 y Fx(and)17 b(the)f(fact)g(that)399 1786 y Ft(\013)e Fu(\025)f Fx(\()p Ft(\030)h Fu(\000)605 1772 y Fx(^)600 1786 y Ft(\030)s Fx(\))643 1765 y FE(2)p Fk(\034)t FE(+1)751 1786 y Fx(max)o Fu(f)p Fx(1)j Ft(;)956 1769 y Fp(e)946 1786 y Ft(T)6 b(E)1017 1793 y FE(0)p Fk(;)p FE(2)1081 1786 y Ft(;)1129 1769 y Fp(e)1119 1786 y Ft(T)g(E)1190 1793 y FE(1)p Fk(;)p FE(2)1254 1786 y Ft(;)1302 1769 y Fp(e)1292 1786 y Ft(T)g(E)1363 1793 y FE(0)p Fk(;)p FE(3)p 1411 1747 40 2 v 1411 1786 a Ft(E)s Fx(\012)1485 1793 y FE(1)1505 1786 y Fu(g)16 b Ft(;)46 1896 y Fx(and)h(that)p 482 1926 53 2 v 482 1966 a Ft(M)534 1973 y Fk(i)549 1966 y Ft(B)586 1973 y FF(\003)p Fk(i)617 1966 y Ft(s)640 1973 y FE(5)674 1966 y Fu(\024)755 1932 y Fx(1)p 731 1954 73 2 v 731 1970 53 2 v 731 2009 a Ft(M)784 1980 y FE(2)784 2022 y(0)848 1932 y Fx(1)p 830 1954 62 2 v 830 2000 a(2)854 1985 y FE(14)925 1932 y Fx(1)p 918 1954 39 2 v 918 2000 a(8)942 1985 y Fk(i)969 1966 y Fx(min)o Fu(f)p Fx(1)f Ft(;)1206 1932 y Fx(1)p 1159 1954 119 2 v 1170 1989 a Fp(e)1159 2005 y Ft(T)7 b(E)1231 2012 y FE(0)p Fk(;)p FE(2)1283 1966 y Fu(g)14 b(\024)1398 1932 y Fx(1)p 1379 1954 62 2 v 1379 2000 a(2)1403 1985 y FE(14)1462 1966 y Ft(;)46 2088 y Fx(w)o(e)i(obtain)304 2207 y Ft(C)339 2214 y Fk(i)367 2207 y Fu(\024)d Fx(min)o Fu(f)p Fx(1)j Ft(;)656 2174 y Fx(1)p 609 2196 119 2 v 619 2231 a Fp(e)609 2247 y Ft(T)7 b(E)681 2254 y FE(0)p Fk(;)p FE(2)733 2207 y Fu(g)787 2174 y Fx(1)p 763 2196 73 2 v 763 2212 53 2 v 763 2251 a Ft(M)815 2222 y FE(2)815 2264 y(0)863 2174 y Fx(1)p 845 2196 62 2 v 845 2241 a(2)869 2227 y FE(13)940 2174 y Fx(1)p 933 2196 39 2 v 933 2241 a(8)957 2227 y Fk(i)992 2207 y Ft(;)304 2346 y(C)343 2325 y FF(0)339 2358 y Fk(i)368 2346 y Fu(\024)13 b Fx(min)o Fu(f)p Fx(1)j Ft(;)657 2312 y Fx(1)p 610 2334 119 2 v 621 2369 a Fp(e)610 2385 y Ft(T)7 b(E)682 2392 y FE(0)p Fk(;)p FE(2)734 2346 y Fu(g)788 2312 y Fx(1)p 764 2334 73 2 v 764 2343 53 2 v 764 2382 a Ft(M)816 2389 y FE(0)865 2312 y Fx(1)p 846 2334 62 2 v 846 2380 a(2)870 2366 y FE(12)941 2312 y Fx(1)p 934 2334 39 2 v 934 2380 a(8)958 2366 y Fk(i)994 2346 y Ft(;)304 2467 y(C)343 2446 y FF(00)339 2479 y Fk(i)377 2467 y Fu(\024)14 b Fx(max)o Fu(f)p Ft(E)582 2474 y FE(1)p Fk(;)p FE(2)645 2467 y Ft(;)24 b(E)719 2474 y FE(0)p Fk(;)p FE(3)p 767 2428 40 2 v 767 2467 a Ft(E)s Fx(\012)841 2474 y FE(1)860 2467 y Fu(g)17 b Ft(\021)926 2474 y Fk(i)956 2467 y Ft(M)1008 2446 y FE(3)1003 2479 y(0)p 1028 2428 53 2 v 1028 2467 a Ft(M)1081 2437 y FE(4)1081 2479 y(0)1117 2467 y Ft(\016)1141 2446 y FF(\000)p FE(2)p Fk(\034)t FE(+1)1139 2479 y Fk(i)1268 2467 y Fx(2)1292 2446 y FE(2)p Fk(\034)t FE(+5)1377 2467 y Ft(\034)6 b Fx(!)1418 2446 y FE(2)1437 2467 y Fx(\()p Ft(\027)1480 2474 y FE(16)1528 2467 y Fx(+)11 b Ft(\016)1599 2474 y FE(0)1619 2467 y Ft(\027)1643 2474 y FE(14)1680 2467 y Fx(\))16 b Ft(;)304 2563 y(C)339 2570 y FF(\003)p Fk(i)384 2563 y Fu(\024)447 2547 y Fp(e)437 2563 y Ft(T)473 2542 y FF(\000)p FE(1)466 2575 y(0)534 2529 y Fx(1)p 524 2551 45 2 v 524 2597 a(2)548 2583 y FE(8)602 2529 y Fx(1)p 595 2551 39 2 v 595 2597 a(8)619 2583 y Fk(i)654 2563 y Ft(:)1119 b Fx(\(4.44\))955 2863 y(27)p eop %%Page: 28 28 28 27 bop 46 78 a Fx(W)l(e)16 b(are)g(ready)g(to)h(c)o(hec)o(k)d (\(4.35\))j(for)g Ft(i)10 b Fx(+)h(1:)17 b(F)l(rom)e(\(4.35\))i(and)f (\(4.42\))h(w)o(e)f(\014nd)360 188 y Fu(k)p Ft(u)413 167 y FE(\()p Fk(i)p FE(+1\))499 188 y Fu(k)524 195 y Fk(\030)540 200 y Fo(i)p Fl(+1)607 188 y Fu(\024)e(k)p Ft(u)713 167 y FE(\()p Fk(i)p FE(\))754 188 y Fu(k)779 195 y Fk(\030)795 200 y Fo(i)821 188 y Fx(+)d Fu(k)p Ft(z)920 167 y FE(\()p Fk(i)p FE(\))961 188 y Fu(k)986 195 y Fk(\030)1002 200 y Fo(i)p Fl(+1)1069 188 y Fu(\024)1127 175 y Fx(\026)1122 188 y Ft(\030)i Fu(\000)e Ft(\030)1227 195 y Fk(i)1253 188 y Fx(+)g Ft(\016)1324 195 y Fk(i)1351 188 y Fx(=)1408 175 y(\026)1403 188 y Ft(\030)j Fu(\000)d Ft(\030)1509 195 y Fk(i)p FE(+1)1584 188 y Ft(:)46 298 y Fx(F)l(rom)k(\(4.23\))i(and)g(\(4.43\),)f(w)o(e)g(get)462 444 y Fu(k)p Ft(v)513 424 y FE(\()p Fk(i)p FE(+1\))610 444 y Fu(\000)11 b Ft(y)684 451 y FE(0)703 444 y Fu(k)728 451 y Fk(\030)744 456 y Fo(i)p Fl(+1)839 444 y Fu(\024)41 b(k)p Ft(v)970 424 y FE(\(0\))1028 444 y Fu(\000)11 b Ft(y)1102 451 y FE(0)1121 444 y Fu(k)1146 451 y Fk(\030)1176 444 y Fx(+)1250 391 y Fk(i)1226 403 y Fp(X)1225 494 y Fk(j)r FE(=0)1295 444 y Fu(k)p Ft(w)1356 424 y FE(\()p Fk(j)r FE(\))1401 444 y Fu(k)1426 451 y Fk(\030)1442 456 y Fo(i)p Fl(+1)839 594 y Fu(\024)41 b Ft(\032)12 b Fx(+)1010 560 y Ft(r)g Fu(\000)f Ft(\032)p 1010 582 110 2 v 1033 628 a Fx(2)1057 614 y FE(18)1145 540 y FF(1)1133 552 y Fp(X)1132 643 y Fk(j)r FE(=0)1216 560 y Fx(1)p 1207 582 43 2 v 1207 628 a(8)1231 614 y Fk(j)1268 594 y Fu(\024)j Ft(r)j(:)46 745 y Fx(Finally)l(,)10 b(using)i(\(4.43\),)g (\(4.42\),)g(\(4.44\))g(one)g(easily)f(obtains)h(the)f(remaining)f (inductiv)o(e)g(h)o(yp)q(otheses.)46 830 y(Observ)o(e)17 b(that)j(from)d(\(4.41\),)i(\(4.42\))g(and)g(\(4.43\))h(it)e(follo)o (ws)g(that)h(the)g(error)f(functions)h Ft(f)1773 812 y FE(\()p Fk(i)p FE(\))1833 830 y Fx(and)46 891 y Ft(g)71 873 y FE(\()p Fk(i)p FE(\))135 891 y Fx(go)k(to)g(zero)g(exp)q(onen)o (tially)e(fast,)h(while)g Ft(u)964 873 y FE(\()p Fk(i)p FE(\))1028 891 y Fx(and)h Ft(v)1155 873 y FE(\()p Fk(i)p FE(\))1218 891 y Fx(con)o(v)o(erge)f(\(exp)q(onen)o(tially)f(fast\))i (to)46 951 y(real{analytic)15 b(functions)20 b(~)-27 b Ft(u)16 b Fx(and)i(~)-25 b Ft(v)q Fx(.)46 1036 y(This)16 b(concludes)g(the)g(pro)q(of)h(of)g(Theorem)e(4.1.)p 1001 1036 21 42 v 46 1171 a Fw(Pro)r(of)i Fx(\(of)f(Theorem)e(4.2\))i (The)f(reader)g(will)g(ha)o(v)o(e)f(no)i(di\016cult)o(y)e(in)h(c)o(hec) o(king)e(that)j(the)f(previous)46 1231 y(pro)q(of)20 b(go)q(es)f(through)h(w)o(ord)f(b)o(y)f(w)o(ord)i(so)f(that)g(the)g (claim)d(follo)o(ws)j(from)e(uniformit)o(y)g(in)h Ft(\026)g Fu(2)h(D)q Fx(.)p 46 1291 V 46 1507 a FB(5)83 b(Pro)r(of)28 b(of)g(Theorem)e(1.1)46 1617 y Fx(The)19 b(pro)q(of)g(of)g(Theorem)f (1.1)h(will)e(b)q(e)i(divided)e(in)i(three)f(steps:)g(\(1\))h (construction)g(of)g(the)f(start-)46 1677 y(ing)k(appro)o(ximate)e (solution)i(\(Remark)e(4.1\))i(using)g Ft(")p Fx({expansions;)g(\(2\))g (b)q(ounds)h(\(4.22\))f(on)g(the)46 1737 y(norms)16 b(relativ)o(e)g(to) h(starting)h(appro)o(ximate)e(solution)h(with)g Fu(D)10 b(\021)e(f)p Ft(")15 b Fu(2)h Fq(C)27 b Fx(:)17 b Fu(j)p Ft(")p Fu(j)e(\024)g Ft(")1623 1744 y FE(0)1651 1737 y Fu(\021)8 b Fx(10)1746 1719 y FF(\000)p FE(6)1794 1737 y Fu(g)p Fx(,)16 b(see)46 1797 y(Assumption)f(4.4;)h(\(3\))h(iteration) f(of)g(the)g(map)f Fu(K)i Fx(\(4.25\))g(and)g(application)f(of)h (Theorem)e(4.2.)46 1942 y Fe(5.1)69 b(Step)22 b(1:)g(F)-6 b(ormal)20 b Fb(")p Fe({expansion)i(and)g(initial)d(appro)n(ximate)g (solu-)202 2017 y(tion)46 2109 y Fx(The)e(Hamiltonian)e Ft(H)21 b Fx(in)16 b(\(1.3\))h(con)o(tains)g Ft(")f Fx(as)i(a)f (parameter:)e Ft(")i Fx(corresp)q(onds)h(to)f(the)f(parameter)46 2169 y Ft(\026)j Fx(of)h(Theorem)d(4.2)j(and)g Fu(D)g Fx(corresp)q(onds)h(to)e(the)g(complex)e(ball)i Fu(f)p Ft(")f Fu(2)h Fq(C)30 b Fx(:)19 b Fu(j)p Ft(")p Fu(j)f(\024)g Ft(")1665 2176 y FE(0)1685 2169 y Fu(g)h Fx(for)g(some)46 2229 y Ft(")69 2236 y FE(0)102 2229 y Ft(>)14 b Fx(0)i(to)g(b)q(e)g (determined)d(b)q(elo)o(w)695 2211 y FE(35)733 2229 y Fx(.)i(As)g(w)o(ell)g(kno)o(wn)h(to)g(P)o(oincar)o(\023)-23 b(e,)15 b(Lindstedt)h(&)f(Co.,)h(one)g(ma)o(y)46 2290 y Fv(compute)11 b(formally)g Fx(the)i Ft(")p Fx({expansion)g(of)g (quasi{p)q(erio)q(dic)g Fv(formal)e Fx(solutions)i(\(\\Lindstedt)h (series";)46 2350 y(see)g([1]\).)g Fv(Our)g(starting)h(appro)o(ximate)e (solution)i(will)f(b)q(e)g(a)h(suitable)f(truncation)h(of)g(suc)o(h)f (a)h(formal)46 2410 y(expansion)p Fx(.)p 46 2454 747 2 v 85 2484 a Fs(35)121 2499 y FI(Clearly)f Fz(H)19 b FI(is)c(an)g(en)o(tire)h(function)f(of)g Fz(")g FI(and)h(the)g (restriction)g(on)f Fr(D)i FI(is)e(needed)i(in)e(order)h(to)f(meet)g (the)h(basic)46 2549 y(condition)g(\(4.31\);)f(the)i(c)o(hoice)f(of)g (the)h(\\b)q(est)h(v)n(alue)d(for)h Fz(")995 2555 y Fs(0)1014 2549 y FI(")h(\(i.e.,)e(the)i(largest)f(one\))h(has)g(b)q(een)g(done)g (simply)d(b)o(y)46 2599 y(\\trial)f(and)g(error".)955 2863 y Fx(28)p eop %%Page: 29 29 29 28 bop 46 78 a Fx(Here)13 b(w)o(e)h(deduce)g(a)g(few)h(elemen)o (tary)c(form)o(ulae)i(whic)o(h)g(allo)o(w)h(to)h(explicitly)d(compute)g (recursiv)o(ely)46 138 y(the)k(formal)f(solution.)46 220 y(Let)j Ft(N)j Fx(=)16 b(2,)h Ft(x)8 b Fu(\021)g Fx(\()p Ft(`;)g(g)r Fx(\),)18 b Ft(y)10 b Fu(\021)e Fx(\()p Ft(L;)g(G)p Fx(\))18 b(and)g(let)f(us)h(rewrite)e(explicitly)f(the)i (equation)h(\(4.1\))f(for)h(the)46 281 y(Hamiltonian)c Ft(H)t Fx(:)124 390 y Ft(!)154 397 y FE(1)186 390 y Fx(+)d Ft(D)q(u)304 397 y FE(1)324 390 y Fx(\()p Ft(\022)q Fx(\))41 b(=)h Fu(\000)p Fx(2\()616 357 y(1)p 594 379 70 2 v 594 424 a(2)p Ft(v)644 407 y FE(2)642 435 y(1)679 390 y Fu(\000)11 b Ft(v)753 397 y FE(2)773 390 y Fx(\))807 357 y(1)p 797 379 46 2 v 797 424 a Ft(v)823 407 y FE(3)821 435 y(1)858 390 y Fx(+)g(2)p Ft(")p Fx(\()1000 357 y(1)p 978 379 70 2 v 978 424 a(2)p Ft(v)1028 407 y FE(2)1026 435 y(1)1063 390 y Fu(\000)g Ft(v)1137 397 y FE(2)1156 390 y Fx(\))p Ft(R)1212 397 y Fk(L)1239 390 y Fx(\()p Ft(\022)h Fx(+)f Ft(u;)d(v)r Fx(\))i Fu(\000)h Fx(2)p Ft(")1559 357 y Fx(1)p 1549 379 46 2 v 1549 424 a Ft(v)1575 407 y FE(3)1573 435 y(1)1599 390 y Ft(R)p Fx(\()p Ft(\022)i Fx(+)e Ft(u;)d(v)r Fx(\))124 518 y Ft(!)154 525 y FE(2)186 518 y Fx(+)j Ft(D)q(u)304 525 y FE(2)324 518 y Fx(\()p Ft(\022)q Fx(\))41 b(=)h Fu(\000)p Fx(2\()616 484 y(1)p 594 506 70 2 v 594 552 a(2)p Ft(v)644 535 y FE(2)642 563 y(1)679 518 y Fu(\000)11 b Ft(v)753 525 y FE(2)773 518 y Fx(\))g(+)g(2)p Ft(")p Fx(\()945 484 y(1)p 923 506 V 923 552 a(2)p Ft(v)973 535 y FE(2)971 563 y(1)1008 518 y Fu(\000)g Ft(v)1082 525 y FE(2)1101 518 y Fx(\))p Ft(R)1157 525 y Fk(G)1187 518 y Fx(\()p Ft(\022)h Fx(+)f Ft(u;)d(v)r Fx(\))i Fu(\000)h Fx(2)p Ft("R)p Fx(\()p Ft(\022)i Fx(+)e Ft(u;)d(v)r Fx(\))239 645 y Ft(D)q(v)304 652 y FE(1)324 645 y Fx(\()p Ft(\022)q Fx(\))41 b(=)h Fu(\000)p Fx(2)p Ft(")p Fx(\()639 611 y(1)p 617 633 V 617 679 a(2)p Ft(v)667 662 y FE(2)665 690 y(1)702 645 y Fu(\000)11 b Ft(v)776 652 y FE(2)795 645 y Fx(\))p Ft(R)851 652 y Fk(`)868 645 y Fx(\()p Ft(\022)h Fx(+)f Ft(u;)d(v)r Fx(\))239 773 y Ft(D)q(v)304 780 y FE(2)324 773 y Fx(\()p Ft(\022)q Fx(\))41 b(=)h Fu(\000)p Fx(2)p Ft(")p Fx(\()639 739 y(1)p 617 761 V 617 807 a(2)p Ft(v)667 790 y FE(2)665 818 y(1)702 773 y Fu(\000)11 b Ft(v)776 780 y FE(2)795 773 y Fx(\))p Ft(R)851 780 y Fk(g)872 773 y Fx(\()p Ft(\022)h Fx(+)f Ft(u;)d(v)r Fx(\))15 b(;)713 b(\(5.1\))46 904 y(here)13 b Ft(!)j Fx(and)e Ft(D)h Fx(are)f(short)g(for)g Ft(!)647 886 y FE(\()p FF(\006)p FE(\))718 904 y Fx(and)g(for)g Ft(!)914 886 y FE(\()p FF(\006)p FE(\))977 904 y Fu(\001)6 b Ft(@)1023 911 y Fk(\022)1042 904 y Fx(,)13 b Ft(\022)i Fu(2)f Fq(T)1190 886 y FE(2)1221 904 y Fx(and)g(recall)e(\(1.8\),)i(\(1.9\))g(and)g (\(1.10\).)46 987 y(It)i(is)g(w)o(ell)f(kno)o(wn)h(that)h(\(5.1\))f (admits)f(a)i Fv(formal)e(solution)491 1115 y Fx(~)-27 b Ft(u)p Fx(\()p Ft(\022)q Fx(\))14 b Fu(\030)657 1061 y FF(1)645 1074 y Fp(X)644 1165 y Fk(j)r FE(=0)717 1115 y Fx(~)-27 b Ft(u)742 1095 y FE(\()p Fk(j)r FE(\))787 1115 y Fx(\()p Ft(\022)q Fx(\))17 b Ft(")889 1095 y Fk(j)923 1115 y Ft(;)107 b Fx(~)-26 b Ft(v)r Fx(\()p Ft(\022)q Fx(\))14 b Fu(\030)1209 1061 y FF(1)1197 1074 y Fp(X)1196 1165 y Fk(j)r FE(=0)1268 1115 y Fx(~)-26 b Ft(v)1292 1095 y FE(\()p Fk(j)r FE(\))1337 1115 y Fx(\()p Ft(\022)q Fx(\))p Ft(")1422 1095 y Fk(j)1456 1115 y Ft(;)342 b Fx(\(5.2\))46 1262 y(with)19 b(~)-27 b Ft(u)185 1244 y FE(\()p Fk(j)r FE(\))247 1262 y Fx(and)18 b(~)-25 b Ft(v)368 1244 y FE(\()p Fk(j)r FE(\))429 1262 y Fx(b)q(eing)16 b(v)o(ector{v)m(alued)g Fv(real{analytic)f(on)i Fq(T)1260 1244 y FE(2)1293 1262 y Fx(functions:)567 1362 y(~)-27 b Ft(u)592 1342 y FE(\()p Fk(j)r FE(\))646 1362 y Fu(\021)8 b Fx(\()s(~)-27 b Ft(u)740 1337 y FE(\()p Fk(j)r FE(\))740 1373 y(1)786 1362 y Ft(;)10 b Fx(~)-26 b Ft(u)836 1337 y FE(\()p Fk(j)r FE(\))836 1373 y(2)881 1362 y Fx(\))16 b Ft(;)107 b Fx(~)-26 b Ft(v)1061 1342 y FE(\()p Fk(j)r FE(\))1115 1362 y Fu(\021)8 b Fx(\()r(~)-26 b Ft(v)1207 1337 y FE(\()p Fk(j)r FE(\))1205 1373 y(1)1252 1362 y Ft(;)10 b Fx(~)-26 b Ft(v)1300 1337 y FE(\()p Fk(j)r FE(\))1298 1373 y(2)1345 1362 y Fx(\))16 b Ft(:)46 1462 y Fx(F)l(urthermore,)9 b(suc)o(h)i(a)g(formal)f(solution)h(is)g Fv(uniquely)f(determined)e Fx(b)o(y)j(requiring)f(that)h(the)g(a)o(v)o (erages)46 1522 y(of)16 b(the)j(~)-27 b Ft(u)213 1504 y FE(\()p Fk(j)r FE(\))259 1522 y Fx('s)16 b(v)m(anish:)697 1583 y Fu(h)t Fx(~)-27 b Ft(u)745 1562 y FE(\()p Fk(j)r FE(\))790 1583 y Fu(i)14 b Fx(=)g(0)j Ft(;)105 b Fx(\()p Fu(8)15 b Ft(j)i Fu(\025)c Fx(0\))k Ft(:)46 1665 y Fx(In)f(particular)g (this)g(implies)d(that)890 1726 y(~)-27 b Ft(u)915 1705 y FE(\(0\))970 1726 y Fu(\021)8 b Fx(0)17 b Ft(:)46 1808 y Fx(Instead)d(\(as)i(one)e(c)o(hec)o(ks)f(immediatel)o(y)e(b)o(y)j (inserting)h(\(5.2\))f(in)o(to)h(\(5.1\))f(and)i(lo)q(oking)f(at)g(the) f(order)46 1869 y(zero)i(in)g Ft(")p Fx(\))i(~)-26 b Ft(v)290 1851 y FE(\(0\))353 1869 y Fx(has)17 b(to)f(b)q(e)h(c)o(hosen) f(so)h(that)401 1975 y Ft(@)427 1982 y FE(\()p Fk(L;G)p FE(\))517 1975 y Ft(H)561 1925 y Fp(\014)561 1950 y(\014)561 1975 y(\014)575 2002 y Fk(")p FE(=0)639 1975 y Fx(\()r(~)-26 b Ft(v)684 1954 y FE(\(0\))730 1975 y Fx(\))14 b(=)g Ft(!)k(;)106 b Fj(i)5 b Ft(:)p Fj(e)t Ft(:)33 b(;)107 b Fx(~)-26 b Ft(v)1235 1954 y FE(\(0\))1290 1975 y Fu(\021)8 b Fx(\()p Ft(L)1389 1982 y FF(\006)1419 1975 y Ft(;)g(G)1479 1982 y FF(\006)1508 1975 y Fx(\))17 b Ft(:)46 2083 y Fw(Remark)f(5.1)24 b Fx(The)14 b(formal)f(solv)m(abilit)o(y)g(of)i (\(5.1\))f(implies)e(that)i(the)g(righ)o(t)g(hand)h(sides)f(of)g(the)g (last)46 2143 y(t)o(w)o(o)19 b(equations)g(of)g(\(5.1\))g Fv(ha)o(v)o(e)f(v)m(anishing)i(mean)d(v)m(alue)1115 2125 y FE(36)1171 2143 y Fv(o)o(v)o(er)h Fq(T)1314 2125 y FE(2)1331 2143 y Fx(.)h(The)g(a)o(v)o(erages)f(of)h(the)i(~)-26 b Ft(v)1834 2125 y FE(\()p Fk(j)r FE(\))1879 2143 y Fx('s)46 2203 y(ha)o(v)o(e)16 b(then)h(to)h(b)q(e)f(c)o(hosen)g(so)g(that)h(the) f(\014rst)g(t)o(w)o(o)g(equations)g(in)g(\(5.1\))h(are)f(solv)m(able.)f (This)h(lea)o(v)o(es)46 2263 y(free)e(the)h(a)o(v)o(erages)g(of)h(the)i (~)-27 b Ft(u)585 2245 y FE(\()p Fk(j)r FE(\))630 2263 y Fx('s,)16 b(whic)o(h,)f(as)i(already)f(said,)g(will)f(b)q(e)i(tak)o (en)e(to)i(b)q(e)f(zero.)46 2356 y Fv(As)g(initial)f(appro)o(ximate)g (solution)h(w)o(e)g(tak)o(e)406 2489 y Ft(u)434 2469 y FE(\(0\))481 2489 y Fx(\()p Ft(\022)q Fx(;)8 b Ft(")p Fx(\))g Fu(\021)673 2436 y FE(5)652 2448 y Fp(X)651 2539 y Fk(j)r FE(=0)724 2489 y Fx(~)-27 b Ft(u)749 2469 y FE(\()p Fk(j)r FE(\))794 2489 y Fx(\()p Ft(\022)q Fx(\))16 b Ft(")895 2469 y Fk(j)930 2489 y Ft(;)105 b(v)1075 2469 y FE(\(0\))1122 2489 y Fx(\()p Ft(\022)q Fx(;)8 b Ft(")p Fx(\))g Fu(\021)1314 2436 y FE(5)1293 2448 y Fp(X)1292 2539 y Fk(j)r FE(=0)1363 2489 y Fx(~)-25 b Ft(v)1388 2469 y FE(\()p Fk(j)r FE(\))1433 2489 y Fx(\()p Ft(\022)q Fx(\))16 b Ft(")1534 2469 y Fk(j)1812 2489 y Fx(\(5.3\))p 46 2581 747 2 v 85 2612 a Fs(36)121 2627 y FI(F)m(or)d(an)o(y)g(p)q (erio)q(dic)i(function)e Fz(f)t FI(\()p Fz(\022)q FI(\),)i Fz(\022)f Fr(2)d Fq(T)806 2612 y Fi(N)834 2627 y FI(,)j(the)g(in)o (tegral)f(o)o(v)o(er)h Fq(T)1208 2612 y Fi(N)1250 2627 y FI(of)f Fz(D)q(f)19 b FI(v)n(anishes.)955 2863 y Fx(29)p eop %%Page: 30 30 30 29 bop 46 78 a Fv(\(recall)15 b(that,)h(with)g(the)g(ab)q(o)o(v)o(e) g(con)o(v)o(en)o(tions,)h Fx(~)-26 b Ft(v)950 60 y FE(\(0\))1005 78 y Fu(\021)8 b Fx(\()p Ft(L)1104 85 y FF(\006)1134 78 y Ft(;)g(G)1194 85 y FF(\006)1223 78 y Fx(\))p Fv(,)19 b Fx(~)-27 b Ft(u)1300 60 y FE(\(0\))1355 78 y Fu(\021)9 b Fx(0)16 b Fv(and)h Fu(h)s Fx(~)-27 b Ft(u)1585 60 y FE(\()p Fk(j)r FE(\))1631 78 y Fu(i)14 b Fx(=)g(0)p Fv(\))p Fx(.)46 163 y(W)l(e)19 b(pro)q(ceed)h(b)o(y)f(writing)h(do)o(wn)g(the)g (explicit)d(form)o(ulae)h(whic)o(h,)h(implem)o(en)o(te)o(d)e(on)j(a)h (mac)o(hine,)46 223 y(allo)o(w)15 b(to)i(compute)414 205 y FE(37)465 223 y Fx(the)f(functions)j(~)-27 b Ft(u)787 205 y FE(\()p Fk(j)r FE(\))832 223 y Fx(,)17 b(~)-26 b Ft(v)887 205 y FE(\()p Fk(j)r FE(\))948 223 y Fx(or,)16 b(more)e(precisely)l(,)g Fv(allo)o(w)h(to)i(compute)d(in)o(terv)m(als) 46 283 y(of)i(real)g(n)o(um)o(b)q(ers)f(con)o(taining)h(the)g(F)l (ourier)g(co)q(e\016cien)o(ts)f(of)h(the)g Fx(\()s(~)-27 b Ft(u)1316 265 y FE(\()p Fk(j)r FE(\))1362 283 y Ft(;)9 b Fx(~)-25 b Ft(v)1410 265 y FE(\()p Fk(j)r FE(\))1455 283 y Fx(\))p Fv('s)p Fx(.)46 369 y(Recalling)18 b(the)i(explicit)e (form)g(of)j(the)e(function)h Ft(R)g Fx([see)f(\(1.4\),)h(\(1.5\)],)f (one)h(sees)g(that)g(the)f(righ)o(t)46 429 y(hand)e(sides)f(of)g (\(5.1\))h(ha)o(v)o(e)e(the)h(form)458 522 y Fk(M)447 534 y Fp(X)448 626 y Fk(i)p FE(=1)515 576 y Ft(r)537 583 y Fk(i)568 576 y Ft(")591 555 y Fk(s)607 560 y Fo(i)638 576 y Fx(\()p Ft(v)683 551 y FE(\(0\))681 587 y(1)730 576 y Fx(\))749 555 y Fk(p)767 560 y Fo(i)798 576 y Fx(\()p Ft(v)843 551 y FE(\(0\))841 587 y(2)889 576 y Fx(\))908 555 y Fk(q)924 560 y Fo(i)956 576 y Ft(e)979 555 y Fk(\033)999 560 y Fo(i)1014 576 y Fx(\()p Ft(v)1059 551 y FE(\(0\))1057 587 y(1)1105 576 y Ft(;)8 b(v)1153 551 y FE(\(0\))1151 587 y(2)1200 576 y Fx(\))16 b Ft(c)1256 583 y Fk(n)1277 588 y Fo(i)1292 576 y Fx(\()p Ft(\022)d Fx(+)e Ft(u)1424 555 y FE(\(0\))1470 576 y Fx(\))17 b Ft(;)292 b Fx(\(5.4\))46 718 y(where:)12 b Ft(M)19 b(<)14 b Fu(1)p Fx(;)f Ft(r)414 725 y Fk(i)441 718 y Fx(are)g(rational)h(n)o(um)o(b)q(ers;)d Ft(s)928 725 y Fk(i)942 718 y Fx(,)i Ft(p)993 725 y Fk(i)1007 718 y Fx(,)g Ft(q)1056 725 y Fk(i)1070 718 y Fx(,)g Ft(\033)1125 725 y Fk(i)1152 718 y Fx(are)g(in)o(tegers)f(ob)q(eying)i(the)f (constrain)o(ts)343 828 y(0)h Fu(\024)g Ft(s)457 835 y Fk(i)485 828 y Fu(\024)f Fx(1)k Ft(;)57 b Fu(\000)p Fx(5)13 b Fu(\024)h Ft(p)802 835 y Fk(i)830 828 y Fu(\024)g Fx(10)j Ft(;)57 b Fx(0)14 b Fu(\024)g Ft(q)1132 835 y Fk(i)1159 828 y Fu(\024)g Fx(1)i Ft(;)57 b Fu(\000)p Fx(1)14 b Fu(\024)f Ft(\033)1480 835 y Fk(i)1508 828 y Fu(\024)h Fx(1)i(;)46 938 y Ft(n)75 945 y Fk(i)103 938 y Fu(2)e Fq(Z)186 920 y FE(2)219 938 y Fx(with)h Fu(j)p Ft(n)372 945 y Fk(i)386 938 y Fu(j)f(\024)g Fx(10;)i(\014nally)f Ft(c)713 945 y Fk(n)736 938 y Fx(\()p Ft(x)p Fx(\))h(is)f(either)g(cos) 9 b Ft(n)h Fu(\001)g Ft(x)15 b Fx(or)h(sin)9 b Ft(n)h Fu(\001)g Ft(x)p Fx(.)15 b(W)l(e)g(shall)h(denote)g(b)o(y)f([)p Fu(\001)p Fx(])1895 945 y Fk(j)46 999 y Fx(the)f(op)q(erator)j(that)e (acts)g(on)g(a)g(\(formal\))f Ft(")p Fx({p)q(o)o(w)o(er)h(series,)1153 965 y Fp(P)1205 999 y Ft(")1228 980 y Fk(k)1249 999 y Ft(a)1275 980 y FE(\()p Fk(k)q FE(\))1324 999 y Fx(,)f(b)o(y)g(asso)q (ciating)i(to)g(it)e(the)h Ft(j)1877 980 y FE(th)46 1059 y Fx(co)q(e\016cien)o(t)f Ft(a)299 1041 y FE(\()p Fk(j)r FE(\))345 1059 y Fx(:)772 1071 y Fp(h)802 1077 y(X)800 1170 y Fk(k)q FF(\025)p FE(0)873 1119 y Ft(")896 1098 y Fk(k)917 1119 y Ft(a)943 1098 y FE(\()p Fk(k)q FE(\))991 1071 y Fp(i)1011 1146 y Fk(j)1037 1119 y Fu(\021)9 b Ft(a)1111 1098 y FE(\()p Fk(j)r FE(\))1172 1119 y Ft(:)46 1252 y Fx(Let)22 b Ft(p;)8 b(q)r(;)g(\033)25 b Fu(2)f Fq(Z)8 b Fx(with)22 b Ft(q)j(>)f Fx(0)f(and)f Fu(j)p Ft(\033)r Fu(j)h(\024)h Fx(1)e(and)h(let)e(us)i(compute)d(the)i Ft(j)1521 1234 y FE(th)1578 1252 y Ft(")p Fx({co)q(e\016cien)o(t)f(of,) 46 1312 y(resp)q(ectiv)o(ely)l(,)c Ft(a)351 1294 y Fk(p)371 1312 y Ft(b)392 1294 y Fk(q)410 1312 y Ft(e)433 1294 y Fk(\033)456 1312 y Fx(\()p Ft(a;)8 b(b)p Fx(\))20 b(and)h(of)f Ft(c)762 1319 y Fk(n)786 1312 y Fx(\()p Ft(\022)15 b Fx(+)e Ft(')p Fx(\))20 b(where)g Ft(a)p Fx(,)g Ft(b)g Fx(and)h Ft(')f Fx(are)g(formal)f Ft(")p Fx({p)q(o)o(w)o(er)i(series)46 1372 y(\(with)16 b(p)q(erio)q(dic)g(real{analytic)f(co)q(e\016cien)o (ts\))g(giv)o(en)h(b)o(y)499 1482 y Ft(a)e Fu(\030)594 1441 y Fp(X)592 1533 y Fk(k)q FF(\025)p FE(0)664 1482 y Ft(a)690 1489 y Fk(k)711 1482 y Fx(\()p Ft(\022)q Fx(\))p Ft(")796 1461 y Fk(k)834 1482 y Ft(;)56 b(b)14 b Fu(\030)993 1441 y Fp(X)991 1533 y Fk(k)q FF(\025)p FE(0)1064 1482 y Ft(b)1085 1489 y Fk(k)1106 1482 y Fx(\()p Ft(\022)q Fx(\))p Ft(")1191 1461 y Fk(k)1228 1482 y Ft(;)499 1609 y(')g Fu(\030)600 1568 y Fp(X)598 1660 y Fk(k)q FF(\025)p FE(0)670 1609 y Ft(')702 1589 y FE(\()p Fk(k)q FE(\))751 1609 y Fx(\()p Ft(\022)q Fx(\))p Ft(")836 1589 y Fk(k)866 1609 y Fu(\021)913 1561 y Fp(\020)948 1568 y(X)946 1660 y Fk(k)q FF(\025)p FE(0)1018 1609 y Ft(')1050 1584 y FE(\()p Fk(k)q FE(\))1050 1620 y(1)1099 1609 y Fx(\()p Ft(\022)q Fx(\))p Ft(")1184 1589 y Fk(k)1205 1609 y Ft(;)1229 1568 y Fp(X)1227 1660 y Fk(k)q FF(\025)p FE(0)1300 1609 y Ft(')1332 1584 y FE(\()p Fk(k)q FE(\))1332 1620 y(2)1381 1609 y Fx(\()p Ft(\022)q Fx(\))p Ft(")1466 1589 y Fk(k)1487 1561 y Fp(\021)1528 1609 y Ft(:)46 1767 y Fx(As)19 b(ab)q(o)o(v)o(e,)h (w)o(e)f(denote)g Ft(y)536 1774 y FE(0)564 1767 y Fu(\021)10 b Fx(~)-26 b Ft(v)637 1749 y FE(\(0\))703 1767 y Fx(and)21 b(write)e(the)g(expansions)h(of)g Ft(e)p Fx(\()p Ft(y)r Fx(\))f(and)i(of)e Ft(e)1611 1749 y FF(\000)p FE(1)1658 1767 y Fx(\()p Ft(y)r Fx(\))8 b Fu(\021)g Fx(1)p Ft(=e)p Fx(\()p Ft(y)r Fx(\))46 1827 y(as)89 1809 y FE(38)417 1887 y Ft(e)p Fx(\()p Ft(y)r Fx(\))g Fu(\021)580 1846 y Fp(X)567 1940 y Fk(h)p FF(2)p Fc(N)633 1931 y Fl(2)662 1887 y Ft(e)685 1894 y Fk(h)707 1887 y Fx(\()p Ft(y)k Fu(\000)f Ft(y)836 1894 y FE(0)856 1887 y Fx(\))875 1867 y Fk(h)913 1887 y Ft(;)57 b(e)1007 1867 y FF(\000)p FE(1)1054 1887 y Fx(\()p Ft(y)r Fx(\))8 b Fu(\021)1194 1846 y Fp(X)1181 1940 y Fk(h)p FF(2)p Fc(N)1247 1931 y Fl(2)1277 1887 y Fx(~)-26 b Ft(e)1298 1894 y Fk(h)1320 1887 y Fx(\()p Ft(y)13 b Fu(\000)e Ft(y)1450 1894 y FE(0)1469 1887 y Fx(\))1488 1867 y Fk(h)1527 1887 y Ft(:)46 2012 y Fx(Then,)16 b(for)g Ft(p)f Fu(\025)e Fx(0,)j(one)h(\014nds)432 2122 y([)p Ft(a)472 2102 y FF(\000)p Fk(p)519 2122 y Ft(b)540 2102 y Fk(q)558 2122 y Ft(e)581 2102 y Fk(\033)604 2122 y Fx(\()p Ft(a;)8 b(b)p Fx(\)])725 2129 y Fk(j)432 2228 y Fx(=)545 2186 y Fp(X)484 2280 y FE(\()p Fk(k)q(;h)p FE(\))p FF(2)p Fk(I)600 2284 y Fn(\000)p Fo(p;q)675 2155 y Fp( )718 2194 y Ft(p)707 2262 y(k)732 2269 y FE(1)752 2155 y Fp(! )829 2194 y Ft(q)818 2262 y(k)843 2269 y FE(2)863 2155 y Fp(!)896 2228 y Ft(y)922 2204 y Fk(k)940 2209 y Fl(1)957 2204 y FF(\000)p Fk(p)920 2239 y FE(01)1021 2228 y Ft(y)1047 2204 y Fk(q)q FF(\000)p Fk(k)1109 2209 y Fl(2)1045 2239 y FE(02)1145 2228 y Ft(e)1168 2235 y Fk(\033)o(;h)1235 2228 y Ft(a)1261 2202 y FE([)p Fk(h)1291 2207 y Fl(1)1308 2202 y FE(])1261 2240 y Fk(k)1279 2245 y Fl(3)1337 2228 y Fx(~)-24 b Ft(a)1363 2202 y FE([)p Fk(k)1391 2207 y Fl(1)1408 2202 y FE(])1363 2240 y Fk(k)1381 2245 y Fl(4)1436 2228 y Ft(b)1457 2202 y FE([)p Fk(h)1487 2207 y Fl(2)1504 2202 y FE(+)p Fk(k)1549 2207 y Fl(2)1567 2202 y FE(])1457 2240 y Fk(k)1475 2245 y Fl(5)1595 2228 y Ft(;)203 b Fx(\(5.5\))p 46 2328 747 2 v 85 2360 a Fs(37)121 2375 y FI(Notice)16 b(that)g(since)g(the)h(Hamiltonian)12 b Fz(H)19 b FI(is)d(a)f(trigonometric)g(p)q(olynomial)d(in)k Fz(x)p FI(,)f(the)h(functions)g(\()s(~)-24 b Fz(u)1769 2360 y Fs(\()p Fi(j)r Fs(\))1812 2375 y Fz(;)8 b FI(~)-22 b Fz(v)1852 2360 y Fs(\()p Fi(j)r Fs(\))1896 2375 y FI(\))46 2425 y(are)14 b(also)f(trigonometric)g(p)q(olynomials.)85 2470 y Fs(38)121 2485 y FI(W)m(e)k(use)i(standard)f(m)o(ultiindex)e (notation:)h(if)g Fz(n)h Fr(2)g Fq(N)1009 2470 y Fi(N)1042 2485 y FI(,)f Fr(j)p Fz(n)p Fr(j)h FI(=)1189 2454 y Fp(P)1232 2464 y Fi(N)1232 2497 y(i)p Fs(=1)1295 2485 y Fz(n)1320 2491 y Fi(i)1334 2485 y FI(;)f(if)h Fz(z)i Fr(2)e Fq(R)1530 2470 y Fi(N)1559 2485 y FI(,)f Fz(@)1612 2470 y Fi(n)1610 2495 y(z)1654 2485 y FI(=)1773 2468 y Fi(@)1793 2456 y Fg(j)p Fh(n)p Fg(j)p 1709 2475 187 2 v 1709 2503 a Fi(@)r(z)1746 2487 y Fh(n)1764 2493 y Ff(1)1745 2513 y(1)1782 2503 y Fy(\001\001\001)p Fi(@)r(z)1849 2486 y Fh(n)1867 2493 y(N)1848 2508 y(n)1900 2485 y FI(;)46 2552 y Fz(n)p FI(!)11 b(=)h Fz(n)163 2558 y Fs(1)181 2552 y FI(!)7 b Fr(\001)g(\001)g(\001)e Fz(n)280 2558 y Fi(N)311 2552 y FI(!;)13 b Fz(z)369 2537 y Fi(n)404 2552 y FI(=)e Fz(z)468 2533 y Fi(n)489 2537 y Ff(1)466 2563 y Fs(1)514 2552 y Fr(\001)c(\001)g(\001)f Fz(z)591 2533 y Fi(n)612 2537 y Fh(N)589 2564 y Fi(N)640 2552 y FI(.)955 2863 y Fx(30)p eop %%Page: 31 31 31 30 bop 46 78 a Fx(where:)162 179 y Ft(I)184 186 y FF(\000)p Fk(p;q)307 179 y Fu(\021)50 b(f)p Fx(\()p Ft(k)r(;)8 b(h)p Fx(\))14 b Fu(2)g Fq(N)632 159 y FE(5)663 179 y Fu(\002)d Fq(N)748 159 y FE(2)783 179 y Fx(:)29 b(0)15 b Fu(\024)e Ft(k)942 186 y FE(1)976 179 y Fu(\024)h Ft(p)i(;)24 b Fx(0)15 b Fu(\024)e Ft(k)1223 186 y FE(2)1257 179 y Fu(\024)h Ft(q)j(;)25 b(k)1413 186 y FE(3)1446 179 y Fu(\025)14 b Ft(h)1527 186 y FE(1)1563 179 y Ft(;)24 b(k)1626 186 y FE(4)1660 179 y Fu(\025)14 b Ft(k)1738 186 y FE(1)1774 179 y Ft(;)493 252 y(k)518 259 y FE(5)552 252 y Fu(\025)g Ft(k)630 259 y FE(2)661 252 y Fx(+)d Ft(h)738 259 y FE(2)774 252 y Ft(;)24 b(k)837 259 y FE(3)868 252 y Fx(+)11 b Ft(k)942 259 y FE(4)974 252 y Fx(+)g Ft(k)1048 259 y FE(5)1081 252 y Fx(=)j Ft(j)19 b Fu(g)e Fx(;)697 413 y Ft(e)720 420 y Fk(\033)o(;h)780 413 y Fu(\021)835 313 y Fp(8)835 350 y(>)835 363 y(<)835 437 y(>)835 450 y(:)895 352 y Fx(~)-26 b Ft(e)916 359 y Fk(h)1032 352 y Fx(if)16 b Ft(\033)f Fx(=)f Fu(\000)p Fx(1)893 412 y Ft(e)916 419 y Fk(h)1032 412 y Fx(if)i Ft(\033)f Fx(=)f(1)893 472 y Ft(\016)915 480 y FE(0)p FF(j)p Fk(h)p FF(j)1032 472 y Fx(if)i Ft(\033)f Fx(=)f(0)46 568 y Ft(c)67 550 y FE([)p Fk(k)q FE(])108 568 y Fx(\()p Ft(")p Fx(\))23 b(denotes)h(the)f Ft(k)494 550 y FE(th)553 568 y Fx(p)q(o)o(w)o(er)g (of)h(a)g(formal)e(p)q(o)o(w)o(er)h(series)g Ft(c)j Fu(\030)1372 535 y Fp(P)1416 579 y Fk(j)r FF(\025)p FE(0)1487 568 y Ft(c)1508 575 y Fk(j)1527 568 y Ft(")1550 550 y Fk(j)1591 568 y Fx(and)e Ft(c)1714 543 y FE([)p Fk(k)q FE(])1714 580 y Fk(j)1778 568 y Fx(its)g Ft(j)1877 550 y FE(th)46 628 y Ft(")p Fx({co)q(e\016cien)o(t;)14 b(\014nally)j(~)-25 b Ft(a)16 b Fx(is)g(the)g(p)q(o)o(w)o(er)g(series)g(de\014ned)g(b)o(y) 843 749 y(~)-25 b Ft(a)14 b Fu(\030)940 715 y Fx(1)p 939 738 26 2 v 939 783 a Ft(a)981 749 y Fu(\000)1046 715 y Fx(1)p 1036 738 46 2 v 1036 783 a Ft(a)1062 790 y FE(0)1102 749 y Ft(:)46 871 y Fx(Analogously)l(,)i(for)h Ft(p)d(>)g Fx(0,)i(one)g(gets)478 972 y([)p Ft(a)518 952 y Fk(p)537 972 y Ft(b)558 952 y Fk(q)577 972 y Ft(e)600 952 y Fk(\033)623 972 y Fx(\()p Ft(a;)8 b(b)p Fx(\)])744 979 y Fk(j)478 1077 y Fx(=)579 1036 y Fp(X)530 1130 y FE(\()p Fk(k)q(;h)p FE(\))p FF(2)p Fk(I)646 1134 y Fo(p;q)697 1004 y Fp( )740 1044 y Ft(p)730 1111 y(k)755 1118 y FE(1)775 1004 y Fp(! )851 1044 y Ft(q)841 1111 y(k)866 1118 y FE(2)886 1004 y Fp(!)918 1077 y Ft(y)944 1054 y Fk(p)p FF(\000)p Fk(k)1007 1059 y Fl(1)942 1088 y FE(01)1027 1077 y Ft(y)1053 1054 y Fk(q)q FF(\000)p Fk(k)1115 1059 y Fl(2)1051 1088 y FE(02)1135 1077 y Ft(e)1158 1084 y Fk(\033)o(;h)1225 1077 y Ft(a)1251 1052 y FE([)p Fk(h)1281 1057 y Fl(1)1298 1052 y FE(+)p Fk(k)1343 1057 y Fl(1)1361 1052 y FE(])1251 1090 y Fk(k)1269 1095 y Fl(3)1389 1077 y Ft(b)1410 1052 y FE([)p Fk(h)1440 1057 y Fl(2)1458 1052 y FE(+)p Fk(k)1503 1057 y Fl(2)1521 1052 y FE(])1410 1090 y Fk(k)1428 1095 y Fl(4)1549 1077 y Ft(;)249 b Fx(\(5.6\))46 1223 y(where:)69 1324 y Ft(I)91 1331 y Fk(p;q)187 1324 y Fu(\021)50 b(f)p Fx(\()p Ft(k)r(;)8 b(h)p Fx(\))13 b Fu(2)h Fq(N)511 1304 y FE(4)543 1324 y Fu(\002)d Fq(N)628 1304 y FE(2)662 1324 y Fx(:)30 b(0)14 b Fu(\024)g Ft(k)822 1331 y FE(1)856 1324 y Fu(\024)f Ft(p)k(;)24 b Fx(0)14 b Fu(\024)g Ft(k)1103 1331 y FE(2)1137 1324 y Fu(\024)g Ft(q)j(;)24 b(k)1292 1331 y FE(3)1326 1324 y Fu(\025)14 b Ft(k)1404 1331 y FE(1)1435 1324 y Fx(+)d Ft(h)1512 1331 y FE(1)1548 1324 y Ft(;)24 b(k)1611 1331 y FE(4)1645 1324 y Fu(\025)14 b Ft(k)1723 1331 y FE(2)1754 1324 y Fx(+)d Ft(h)1831 1331 y FE(2)1867 1324 y Ft(;)373 1397 y(k)398 1404 y FE(3)429 1397 y Fx(+)g Ft(k)503 1404 y FE(4)537 1397 y Fx(=)j Ft(j)19 b Fu(g)d Ft(:)1129 b Fx(\(5.7\))46 1498 y(Computing)16 b([)p Ft(c)336 1505 y Fk(n)359 1498 y Fx(\()p Ft(\022)c Fx(+)f Ft(')p Fx(\)])527 1505 y Fk(j)561 1498 y Fx(is)16 b(clearly)f(equiv)m(alen)o(t)f(to)j(ev)m(aluate)f([exp) o(\()p Ft(in)11 b Fu(\001)g Ft(')p Fx(\)])1502 1505 y Fk(j)1536 1498 y Fx(and)16 b(letting)628 1600 y Ft(E)667 1579 y FE(\(0\))664 1612 y Fk(n)722 1600 y Fu(\021)8 b Fx(1)17 b Ft(;)57 b Fu(8)15 b Ft(n)f Fu(2)g Fq(Z)1051 1579 y FE(2)1084 1600 y Ft(;)628 1710 y(E)667 1689 y FE(\()p Fk(k)q FE(\))664 1722 y Fk(n)716 1710 y Fx(\()p Ft(\022)q Fx(\))8 b Fu(\021)839 1676 y Fx(1)p 838 1698 28 2 v 838 1744 a Ft(k)899 1656 y Fk(k)878 1668 y Fp(X)879 1761 y Fk(`)p FE(=1)947 1710 y Ft(`)16 b(E)1022 1689 y FE(\()p Fk(k)q FF(\000)p Fk(`)p FE(\))1019 1722 y Fk(n)1129 1710 y Fx(\()p Ft(\022)q Fx(\))h Ft(n)11 b Fu(\001)g Ft(')1305 1717 y Fk(`)1321 1710 y Fx(\()p Ft(\022)q Fx(\))16 b Ft(;)46 1841 y Fx(one)g(gets)712 1853 y Fp(h)740 1901 y Fx(exp)o(\()p Ft(in)11 b Fu(\001)g Ft(')p Fx(\))966 1853 y Fp(i)986 1928 y Fk(j)1018 1901 y Fx(=)i Ft(E)1108 1880 y FE(\()p Fk(j)r FE(\))1105 1913 y Fk(n)1154 1901 y Fx(\()p Ft(\022)q Fx(\))j Ft(:)566 b Fx(\(5.8\))46 1999 y(Inserting)16 b(\(5.2\))g(in)g(\(5.1\))h(one)f(obtains)h (recursiv)o(e)e(relations)h(of)g(the)g(t)o(yp)q(e)567 2100 y Ft(D)t Fx(~)-27 b Ft(u)636 2080 y FE(\()p Fk(k)q FE(\))699 2100 y Fx(=)14 b Ft(A)r Fx(~)-26 b Ft(v)814 2080 y FE(\()p Fk(k)q FE(\))873 2100 y Fx(+)11 b(\010)957 2080 y FE(\()p Fk(k)q FE(\))1022 2100 y Ft(;)57 b(D)s Fx(~)-26 b Ft(v)1160 2080 y FE(\()p Fk(k)q FE(\))1223 2100 y Fx(=)13 b(\011)1312 2080 y FE(\()p Fk(k)q FE(\))1377 2100 y Ft(;)421 b Fx(\(5.9\))46 2202 y(where)24 b Ft(A)g Fx(is)g(the)f(\(constan)o(t\))i(matrix)e(giv)o(en)g(in)h(\(3.5\))g(ev)m (aluated)g(at)h(\()p Ft(L;)8 b(G)p Fx(\))g Fu(\021)h Fx(\()p Ft(L)1671 2209 y FF(\006)1700 2202 y Ft(;)f(G)1760 2209 y FF(\006)1790 2202 y Fx(\))24 b(and)46 2262 y(the)18 b(v)o(ectors)h(\010)336 2244 y FE(\()p Fk(k)q FE(\))403 2262 y Fx(and)h(\011)539 2244 y FE(\()p Fk(k)q FE(\))606 2262 y Fx(dep)q(end)f(on)k(~)-27 b Ft(u)877 2244 y FE(\(0\))924 2262 y Fx(,...,)s(~)g Ft(u)1022 2244 y FE(\()p Fk(k)q FF(\000)p FE(1\))1113 2262 y Fx(,)20 b(~)-26 b Ft(v)1171 2244 y FE(\(0\))1218 2262 y Fx(,...,)r(~)g Ft(v)1314 2244 y FE(\()p Fk(k)q FF(\000)p FE(1\))1424 2262 y Fx(\(and)19 b(on)h Ft(\022)q Fx(\))e(and)i(can)f(b)q(e)46 2322 y(explicitly)c (written)j(do)o(wn)h(b)o(y)f(using)g(the)g(remark)f(leading)h(to)h (\(5.4\))f(and)h(the)f(expansions)h(\(5.5\),)46 2382 y(\(5.6\))d(and)h(\(5.8\).)46 2465 y(Assume)10 b(no)o(w)i(that)h(\()s (~)-27 b Ft(u)468 2447 y FE(\()p Fk(j)r FE(\))513 2465 y Ft(;)10 b Fx(~)-26 b Ft(v)561 2447 y FE(\()p Fk(j)r FE(\))606 2465 y Fx(\),)11 b(for)h Ft(j)17 b Fx(=)d(0)p Ft(;)8 b(:::;)g(k)t Fu(\000)r Fx(1)i(are)i(kno)o(wn)g(and)h(let)e(us)h (determine)d(\()s(~)-27 b Ft(u)1735 2447 y FE(\()p Fk(k)q FE(\))1783 2465 y Ft(;)10 b Fx(~)-26 b Ft(v)1831 2447 y FE(\()p Fk(k)q FE(\))1879 2465 y Fx(\).)46 2525 y(In)o(v)o(erting)14 b(the)i(op)q(erator)i Ft(D)g Fx(in)e(the)g(second)g(of)h(\(5.9\))f (\(recalling)g(Remark)e(5.1\))j(w)o(e)e(let)746 2627 y(~)-26 b Ft(v)770 2606 y FE(\()p Fk(k)q FE(\))827 2627 y Fu(\021)10 b Fx(\026)-26 b Ft(v)900 2606 y FE(\()p Fk(k)q FE(\))959 2627 y Fx(+)11 b Ft(D)1049 2606 y FF(\000)p FE(1)1097 2627 y Fx(\011)1135 2606 y FE(\()p Fk(k)q FE(\))1200 2627 y Ft(;)955 2863 y Fx(31)p eop %%Page: 32 32 32 31 bop 46 78 a Fx(where)17 b(\026)-26 b Ft(v)212 60 y FE(\()p Fk(k)q FE(\))276 78 y Fx(denotes)16 b(the)f(a)o(v)o(erage)g (of)j(~)-26 b Ft(v)790 60 y FE(\()p Fk(k)q FE(\))854 78 y Fx(and)16 b(it)g(has)g(to)g(b)q(e)g(determined)d(so)j(that)g(the)g (equations)46 138 y(for)g Ft(D)t Fx(~)-27 b Ft(u)189 120 y FE(\()p Fk(k)q FE(\))255 138 y Fx(ha)o(v)o(e)15 b(a)i(righ)o(t)f(hand)h(side)f(with)g(v)m(anishing)h(mean)e(v)m(alue,)g (i.e.,)437 235 y(\026)-26 b Ft(v)461 214 y FE(\()p Fk(k)q FE(\))518 235 y Fu(\021)19 b(\000)11 b Ft(A)663 214 y FF(\000)p FE(1)709 235 y Fu(h)q Fx(\010)764 214 y FE(\()p Fk(k)q FE(\))813 235 y Fu(i)16 b Ft(;)109 b Fx(~)-27 b Ft(u)996 214 y FE(\()p Fk(k)q FE(\))1052 235 y Fu(\021)9 b Ft(D)1141 214 y FF(\000)p FE(1)1189 186 y Fp(\020)1213 235 y Ft(A)r Fx(~)-26 b Ft(v)1276 214 y FE(\()p Fk(k)q FE(\))1335 235 y Fx(+)11 b(\010)1419 214 y FE(\()p Fk(k)q FE(\))1468 186 y Fp(\021)1509 235 y Ft(:)46 343 y Fx(The)j(form)o(ulae) f(for)h(the)g(recursiv)o(e)e(computation)i(of)g(the)g(functions)g(\()s (~)-27 b Ft(u)1371 325 y FE(\()p Fk(j)r FE(\))1417 343 y Ft(;)10 b Fx(~)-26 b Ft(v)1465 325 y FE(\()p Fk(j)r FE(\))1510 343 y Fx(\))14 b(\(and)h(hence)e(of)i(our)46 403 y(c)o(hoice)i(of)i(the)f(initial)g(appro)o(ximate)f(solution\))h (are)h(complete.)d(In)i(App)q(endix)g(A)g(w)o(e)g(rep)q(ort)h(the)46 463 y(n)o(um)o(b)q(er)c(of)j(F)l(ourier)f(co)q(e\016cien)o(ts)f(of)i (the)f(functions)g(\()s(~)-27 b Ft(u)1099 445 y FE(\()p Fk(j)r FE(\))1144 463 y Ft(;)10 b Fx(~)-26 b Ft(v)1192 445 y FE(\()p Fk(j)r FE(\))1237 463 y Fx(\))18 b(for)f Ft(j)i Fu(\024)d Fx(5)h(and,)h(as)g(an)g(example,)46 523 y(the)e(list)f(of)i(in)o(terv)m(als)e(trapping)i(the)f(F)l(ourier)g (co)q(e\016cien)o(ts)f(of)h(the)g(\014rst)h(comp)q(onen)o(t)e(of)20 b(~)-27 b Ft(u)1733 505 y FE(\(1\))1780 523 y Fx(.)46 666 y Fe(5.2)69 b(Step)25 b(2:)h(Norm)c(b)r(ounds)27 b(relativ)n(e)c(to)i(the)f(initial)e(appro)n(ximate)202 740 y(solution)46 833 y Fx(Ha)o(ving)14 b(de\014ned)h(the)g(initial)f (appro)o(ximate)g(solution)h(as)h(the)f(\014fth)h(order)f(truncation)g (\(5.3\))h(of)f(the)46 893 y Ft(")p Fx({expansion)j(of)f(the)g(formal)f (solution)i(of)f(\(5.1\),)g(w)o(e)g(w)o(an)o(t)g(no)o(w)h(to)f (estimate)f(the)h(relativ)o(e)e(norm)46 953 y(parameters)g(as)i (de\014ned)f(in)g(\(4.22\).)46 1024 y(W)l(e)f(attac)o(h)g(an)h(index)e (0)i(to)f(the)g(quan)o(tities)f(related)h(to)g(our)h(starting)g(appro)o (ximate)d(solution)j(\(i.e.)46 1084 y(the)h(\014fth)g(order)g (truncation)g(\(5.3\))g(of)g(the)g Ft(")p Fx({expansion)g(of)g(the)g (formal)f(solution)h(of)g(\(5.1\)\);)g(th)o(us)46 1144 y(the)k(sym)o(b)q(ols)e Ft(\030)r(;)8 b(U;)g(V)k Fx(,...,)543 1128 y Fp(e)535 1144 y Ft(T)25 b Fx(of)c Fu(x)h Fx(4.3)f(corresp)q(ond) h(here)e(to)i Ft(\030)1225 1151 y FE(0)1245 1144 y Ft(;)8 b(U)1300 1151 y FE(0)1319 1144 y Ft(;)g(V)1369 1151 y FE(0)1389 1144 y Fx(,...,)1467 1128 y Fp(e)1459 1144 y Ft(T)1488 1151 y FE(0)1526 1144 y Fx(\(see)21 b(Remark)e(4.4\).)46 1204 y(Let)720 1265 y Ft(\030)741 1272 y FE(0)769 1265 y Fu(\021)8 b Fx(0)p Ft(:)p Fx(2)17 b Ft(;)105 b(")1037 1272 y FE(0)1065 1265 y Fu(\021)8 b Fx(10)1160 1244 y FF(\000)p FE(6)1224 1265 y Ft(;)46 1346 y Fx(and)21 b(recall)f(that)h (the)g(norm)e(in)i(\(4.32\))g(con)o(tains)g(also)g(a)h(suprem)o(um)c (tak)o(en)i(o)o(v)o(er)g(the)g(complex)46 1406 y(parameter)15 b(region)h Fu(D)723 1466 y(D)10 b(\021)e(f)p Ft(")14 b Fu(2)g Fq(C)26 b Fx(:)j Fu(j)p Ft(")p Fu(j)14 b(\024)f Ft(")1160 1473 y FE(0)1180 1466 y Fu(g)j Ft(:)46 1548 y Fx(The)f(ev)m(aluation)h(of)f Ft(U)466 1555 y FE(0)486 1548 y Fx(,)g Ft(V)543 1555 y FE(0)563 1548 y Fx(,)g Ft(M)639 1555 y FE(0)674 1548 y Fx(and)776 1532 y Fp(e)768 1548 y Ft(V)796 1555 y FE(0)831 1548 y Fx(are)g(easily)g(obtained)g(ha) o(ving)h(computed)d(the)i(\\explicit")46 1608 y(\(in)d(the)g(sense)h (of)g(in)o(terv)m(al)e(arithmetic\))f(form)h(of)i(the)f(appro)o(ximate) f(solution)i(\()p Ft(u;)8 b(v)r Fx(\).)j(Ha)o(ving)h(\()p Ft(u;)c(v)r Fx(\))46 1668 y(giv)o(en)15 b(as)221 1798 y Ft(u)249 1777 y FE(\(0\))310 1798 y Fx(=)383 1744 y FE(5)362 1756 y Fp(X)362 1847 y Fk(j)r FE(=1)431 1798 y Ft(")454 1777 y Fk(j)524 1756 y Fp(X)481 1850 y FE(0)p Fk(<)p FF(j)p Fk(n)p FF(j\024)p Fk(\027)611 1855 y Fo(j)639 1798 y Fx(~)-27 b Ft(u)664 1777 y FE(\()p Fk(j)r FE(\))664 1810 y Fk(n)718 1798 y Fx(exp)o(\()p Ft(in)11 b Fu(\001)g Ft(\022)q Fx(\))16 b Ft(;)106 b(v)1098 1777 y FE(\(0\))1158 1798 y Fx(=)1232 1744 y FE(5)1210 1756 y Fp(X)1210 1847 y Fk(j)r FE(=0)1279 1798 y Ft(")1302 1777 y Fk(j)1350 1756 y Fp(X)1329 1851 y FF(j)p Fk(n)p FF(j\024)p Fk(\027)1416 1839 y Fn(0)1414 1862 y Fo(j)1441 1798 y Fx(~)-26 b Ft(v)1465 1777 y FE(\()p Fk(j)r FE(\))1463 1810 y Fk(n)1519 1798 y Fx(exp)o(\()p Ft(in)11 b Fu(\001)g Ft(\022)q Fx(\))46 1949 y(\(where)16 b(the)h Ft(\027)315 1956 y Fk(j)333 1949 y Ft(;)8 b(\027)382 1931 y FF(0)379 1961 y Fk(j)414 1949 y Fx(are)17 b(the)g(trigonometric)e(degrees)i(listed)f(in)g(App)q (endix)g(A\),)g(w)o(e)h(let)f Ft(U)1732 1956 y FE(0)1768 1949 y Fx(and)i Ft(V)1892 1956 y FE(0)46 2009 y Fx(b)q(e)e(upp)q(er)h (b)q(ounds)g(on)346 2085 y FE(5)325 2097 y Fp(X)324 2188 y Fk(j)r FE(=1)394 2139 y Ft(")417 2115 y Fk(j)417 2150 y FE(0)488 2097 y Fp(X)445 2191 y FE(0)p Fk(<)p FF(j)p Fk(n)p FF(j\024)p Fk(\027)575 2196 y Fo(j)600 2139 y Fu(j)s Fx(~)-27 b Ft(u)642 2118 y FE(\()p Fk(j)r FE(\))642 2151 y Fk(n)688 2139 y Fu(j)8 b Fx(exp)o(\()p Fu(j)p Ft(n)p Fu(j)p Ft(\030)881 2146 y FE(0)901 2139 y Fx(\))16 b Ft(;)1077 2085 y FE(5)1056 2097 y Fp(X)1056 2188 y Fk(j)r FE(=0)1125 2139 y Ft(")1148 2115 y Fk(j)1148 2150 y FE(0)1197 2097 y Fp(X)1176 2192 y FF(j)p Fk(n)p FF(j\024)p Fk(\027)1263 2180 y Fn(0)1261 2203 y Fo(j)1286 2139 y Fu(j)r Fx(~)-26 b Ft(v)1326 2118 y FE(\()p Fk(j)r FE(\))1324 2151 y Fk(n)1371 2139 y Fu(j)8 b Fx(exp\()p Fu(j)p Ft(n)p Fu(j)p Ft(\030)1565 2146 y FE(0)1585 2139 y Fx(\))16 b Ft(:)46 2292 y Fx(The)g(computer{assisted)g(ev)m(aluations)g(of)h (suc)o(h)f(sums)f(yields)g(the)h(v)m(alues)1442 2274 y FE(39)439 2388 y Ft(U)477 2363 y FE(\(+\))472 2399 y(0)542 2388 y Fu(\021)8 b Fx(1)p Ft(:)p Fx(319112913)q(12)q(182)q(00)q (558)q(424)q(17)q(988)q(72)q(644)q(1)14 b Fu(\001)d Fx(10)1516 2368 y FF(\000)p FE(5)1580 2388 y Ft(;)p 46 2430 747 2 v 85 2462 a Fs(39)121 2477 y FI(Recall)h(that)h(the)h(plus)f(sign)g (corresp)q(onds)i(to)e(the)h(case)i(~)-23 b Fz(v)1033 2462 y Fs(\(0\))1090 2477 y FI(=)12 b(\()p Fz(L)1178 2483 y Fs(+)1206 2477 y Fz(;)7 b(G)1258 2483 y Fs(+)1285 2477 y FI(\))13 b(while)g(the)g(min)o(us)f(sign)h(corresp)q(onds)46 2527 y(to)i(~)-22 b Fz(v)118 2512 y Fs(\(0\))175 2527 y FI(=)12 b(\()p Fz(L)263 2533 y Fy(\000)292 2527 y Fz(;)7 b(G)344 2533 y Fy(\000)371 2527 y FI(\).)14 b(Note)g(also)g(that)g(in)o (terv)n(al)f(arithmetic)g(yields)h(\(as)h(the)f(name)f(sa)o(ys\))i(in)o (terv)n(als)e(with)h(rational)46 2577 y(endp)q(oin)o(ts)g(trapping)e (the)i(actual)f(quan)o(tit)o(y)g(one)g(is)g(computing;)f(but)h(since)h (w)o(e)g(need)g(\(usually\))f Fa(upp)q(er)h(b)q(ounds)g FI(w)o(e)46 2627 y(shall)f(rep)q(ort)i(only)e(the)h(righ)o(t)g(endp)q (oin)o(ts)g(of)f(the)i(computed)e(in)o(terv)n(als)h(whic)o(h)g(shall)f Fa(de\014ne)i FI(our)e(norm{b)q(ounds.)955 2863 y Fx(32)p eop %%Page: 33 33 33 32 bop 439 79 a Ft(U)477 53 y FE(\()p FF(\000)p FE(\))472 89 y(0)542 79 y Fu(\021)8 b Fx(1)p Ft(:)p Fx(309817593)q(02)q(981)q(79) q(879)q(143)q(98)q(304)q(54)q(021)q(5)14 b Fu(\001)d Fx(10)1516 58 y FF(\000)p FE(5)1580 79 y Ft(;)439 156 y(V)478 131 y FE(\(+\))467 167 y(0)543 156 y Fu(\021)d Fx(1)p Ft(:)p Fx(456220359)q(87)q(166)q(272)q(21)q(542)q(61)q(047)q(77) q(053)20 b Ft(;)439 234 y(V)478 209 y FE(\()p FF(\000)p FE(\))467 245 y(0)543 234 y Fu(\021)8 b Fx(1)p Ft(:)p Fx(457317243)q(81)q(047)q(957)q(87)q(786)q(95)q(386)q(47)q(097)20 b Ft(:)46 344 y Fx(It)c(is)g(easy)g(to)h(c)o(hec)o(k)d(that)j(these)f (b)q(ounds)h(imply)601 405 y Fp(\020)650 453 y Fx(sup)639 487 y Fo(\022)q Fn(2)p Fl(\001)701 496 y Fo(\030)715 503 y Fl(0)657 525 y Fo(")p Fn(2D)747 453 y Fu(j)p Ft(\022)12 b Fx(+)f Ft(u)873 433 y FE(\(0\))920 453 y Fu(j)16 b Ft(;)40 b Fx(sup)993 487 y Fo(\022)q Fn(2)p Fl(\001)1055 496 y Fo(\030)1069 503 y Fl(0)1011 525 y Fo(")p Fn(2D)1101 453 y Fu(j)p Ft(v)1141 433 y FE(\(0\))1188 453 y Fu(j)1202 405 y Fp(\021)1240 453 y Fu(2)14 b Ft(B)19 b(:)46 625 y Fx(Analogously)l(,)d(since)436 606 y FE(40)46 767 y Fu(M)8 b(\021)g Ft(I)i Fx(+)259 713 y FE(5)238 726 y Fp(X)237 817 y Fk(j)r FE(=1)307 767 y Ft(")330 747 y Fk(j)400 726 y Fp(X)356 820 y FE(0)p Fk(<)p FF(j)p Fk(n)p FF(j\024)p Fk(\027)486 825 y Fo(j)511 767 y Ft(i)528 719 y Fp(\020)556 767 y Fx(~)-27 b Ft(u)581 747 y FE(\()p Fk(j)r FE(\))581 779 y Fk(n)633 767 y Fu(\012)6 b Ft(n)707 719 y Fp(\021)740 767 y Fx(exp\()p Ft(in)g Fu(\001)g Ft(\022)q Fx(\))14 b Ft(;)105 b(v)1108 742 y FE(\(0\))1106 780 y Fk(\022)1168 767 y Fx(=)1242 713 y FE(5)1221 726 y Fp(X)1220 817 y Fk(j)r FE(=1)1290 767 y Ft(")1313 747 y Fk(j)1360 726 y Fp(X)1339 820 y FF(j)p Fk(n)p FF(j\024)p Fk(\027)1426 809 y Fn(0)1424 832 y Fo(j)1450 767 y Ft(i)1467 719 y Fp(\020)1493 767 y Fx(~)-26 b Ft(v)1517 747 y FE(\()p Fk(j)r FE(\))1515 779 y Fk(n)1568 767 y Fu(\012)6 b Ft(n)1642 719 y Fp(\021)1676 767 y Fx(exp)o(\()p Ft(in)g Fu(\001)g Ft(\022)q Fx(\))14 b Ft(;)46 929 y Fx(one)i(gets)h(the)f (computer{assisted)f(ev)m(aluations)438 1038 y Ft(M)490 1013 y FE(\(+\))485 1049 y(0)556 1038 y Fu(\021)8 b Fx(1)p Ft(:)p Fx(0000212664)q(65)q(664)q(701)q(23)q(867)q(71)q(708)q(75)q(501) 20 b Ft(;)438 1116 y(M)490 1091 y FE(\()p FF(\000)p FE(\))485 1127 y(0)556 1116 y Fu(\021)8 b Fx(1)p Ft(:)p Fx(0000214390)q(75)q(256) q(126)q(27)q(143)q(24)q(317)q(16)q(151)20 b Ft(;)446 1178 y Fp(e)438 1194 y Ft(V)478 1169 y FE(\(+\))466 1205 y(0)543 1194 y Fu(\021)8 b Fx(7)p Ft(:)p Fx(4535718585)q(179)q(57)q (815)q(59)q(273)q(044)q(38)q(744)q(32)15 b Fu(\001)c Fx(10)1517 1173 y FF(\000)p FE(7)1581 1194 y Ft(;)446 1256 y Fp(e)438 1272 y Ft(V)478 1246 y FE(\()p FF(\000)p FE(\))466 1283 y(0)543 1272 y Fu(\021)d Fx(7)p Ft(:)p Fx(5312705198)q(483)q(94)q(217)q(44)q(721)q(217)q(63)q(111)q(94)15 b Fu(\001)c Fx(10)1517 1251 y FF(\000)p FE(7)1581 1272 y Ft(:)192 b Fx(\(5.10\))46 1381 y(The)16 b(ev)m(aluation)h(of)p 436 1342 53 2 v 16 w Ft(M)488 1388 y FE(0)524 1381 y Fx(is)f(immediatel)o(y)d(obtained)k(using)f(\(5.10\):)387 1513 y Fu(kM)472 1492 y FF(\000)p FE(1)519 1513 y Fu(k)544 1520 y Fk(\030)560 1525 y Fl(0)628 1513 y Fu(\021)50 b(k)p Fx(\()p Ft(I)14 b Fx(+)d Ft(u)874 1520 y Fk(\022)894 1513 y Fx(\))913 1492 y FF(\000)p FE(1)960 1513 y Fu(k)985 1520 y Fk(\030)1001 1525 y Fl(0)1033 1513 y Fu(\024)1187 1479 y Fx(1)p 1091 1501 218 2 v 1091 1547 a(1)g Fu(\000)g(k)p Ft(u)1229 1554 y Fk(\022)1248 1547 y Fu(k)1273 1554 y Fk(\030)1289 1559 y Fl(0)1327 1513 y Fu(\024)1463 1479 y Fx(1)p 1385 1501 182 2 v 1385 1557 a(1)g Fu(\000)1477 1544 y Fx(~)1470 1557 y Ft(V)1509 1531 y FE(\()p FF(\006)p FE(\))1498 1568 y(0)628 1638 y Fu(\024)p 717 1599 53 2 v 50 w Ft(M)769 1609 y FE(\()p FF(\006)p FE(\))769 1651 y(0)843 1638 y Ft(;)46 1748 y Fx(whic)o(h)k(leads)i(to)p 502 1818 V 502 1857 a Ft(M)554 1828 y FE(\(+\))554 1870 y(0)619 1857 y Fu(\021)9 b Fx(1)p Ft(:)p Fx(00002126691)q(793)q(68)q (813)q(37)q(383)q(03)q(697)q(581)20 b Ft(;)p 502 1901 V 502 1940 a(M)554 1911 y FE(\()p FF(\000)p FE(\))554 1953 y(0)619 1940 y Fu(\021)9 b Fx(1)p Ft(:)p Fx(00002143953)q(489)q (99)q(284)q(47)q(344)q(04)q(412)q(283)q(1)20 b Ft(:)46 2050 y Fx(T)l(o)d(estimate)323 2034 y Fp(e)313 2050 y Ft(T)342 2057 y FE(0)378 2050 y Fx(write)e(the)h(matrix)f Fu(T)29 b Fx(as)841 2159 y Fu(T)21 b(\021)8 b Ft(A)973 2166 y FE(0)1004 2159 y Fx(+)j Fu(B)17 b Ft(;)p 46 2203 747 2 v 85 2233 a Fs(40)121 2249 y FI(If)12 b Fz(a;)7 b(b)k Fr(2)g Fq(C)303 2233 y Fi(N)337 2249 y FI(,)i(w)o(e)g(denote)h(b) o(y)e Fz(a)7 b Fr(\012)h Fz(b)k FI(the)i Fz(N)e Fr(\002)7 b Fz(N)18 b FI(matrix)11 b(with)h(en)o(tries)i(\()p Fz(a)7 b Fr(\012)h Fz(b)p FI(\))1394 2255 y Fi(ij)1434 2249 y FI(=)k Fz(a)1500 2255 y Fi(i)1514 2249 y Fz(b)1532 2255 y Fi(j)1549 2249 y FI(.)h(W)m(e)f(also)g(recall)h(that)46 2325 y(b)o(y)f(default)h(w)o(e)f(use)i(the)f(1{norm)e(on)h(v)o(ectors)i (so)f(that)g(the)g(\(\\op)q(erator"\))g(norm)e(on)h(a)h(2)7 b Fr(\002)g FI(2)12 b(matrix)f Fz(A)g FI(=)1752 2266 y Fp(\022)1789 2300 y Fz(a)44 b(b)1791 2350 y(c)g(d)1881 2266 y Fp(\023)46 2397 y FI(ma)o(y)12 b(b)q(e)i(b)q(ounded)h(as)f (follo)o(ws)202 2498 y Fr(k)p Fz(A)p Fr(k)d FI(=)29 b(sup)330 2535 y Fy(j)p Fi(x)p Fy(j)369 2539 y Ff(1)384 2535 y Fs(=1)433 2498 y Fr(j)p Fz(Ax)p Fr(j)11 b(\024)h FI(min)643 2452 y Fp(n)677 2498 y FI(max)o Fr(fj)p Fz(a)p Fr(j)p Fz(;)7 b Fr(j)p Fz(b)p Fr(jg)f FI(+)j(max)n Fr(fj)p Fz(c)p Fr(j)p Fz(;)e Fr(j)p Fz(d)p Fr(jg)k Fz(;)c FI(max)n Fr(fj)p Fz(a)p Fr(j)p Fz(;)g Fr(j)p Fz(c)p Fr(jg)f FI(+)j(max)o Fr(fj)p Fz(b)p Fr(j)p Fz(;)e Fr(j)p Fz(d)p Fr(j)o(g)1706 2452 y Fp(o)1745 2498 y Fz(:)955 2863 y Fx(33)p eop %%Page: 34 34 34 33 bop 46 78 a Fx(where)16 b Ft(A)224 85 y FE(0)251 78 y Fu(\021)8 b Ft(A)p Fx(\()p Ft(L)387 85 y FF(\006)417 78 y Ft(;)g(G)477 85 y FF(\006)506 78 y Fx(\))17 b([)p Ft(A)p Fx(\()p Ft(L;)8 b(G)p Fx(\))16 b(b)q(eing)g(the)g(in)o(v)o (ertible)d(matrix)i(de\014ned)h(in)g(\(3.5\)].)f(Hence)404 220 y Fu(khT)e(i)507 200 y FF(\000)p FE(1)554 220 y Fu(k)h Fx(=)g Fu(k)p Fx(\()p Ft(I)g Fx(+)d Ft(A)811 200 y FF(\000)p FE(1)811 232 y(0)858 220 y Fu(hB)r(i)p Fx(\))950 200 y FF(\000)p FE(1)997 220 y Ft(A)1034 200 y FF(\000)p FE(1)1034 232 y(0)1081 220 y Fu(k)j(\024)1281 186 y(k)p Ft(A)1343 166 y FF(\000)p FE(1)1343 197 y(0)1390 186 y Fu(k)p 1177 208 343 2 v 1177 254 a Fx(1)e Fu(\000)e(k)p Ft(A)1324 261 y FE(0)1344 254 y Fu(k)1369 240 y FF(\000)p FE(1)1416 254 y Fu(kB)r(k)1501 261 y FE(0)1541 220 y Ft(:)46 358 y Fx(T)l(o)15 b(estimate)d Fu(kB)r(k)394 365 y FE(0)427 358 y Fx(w)o(e)h(write)h(explicitly)d(the)j (de\014nition)g(of)g Fu(B)i Fx(and)e(use)g(the)g(follo)o(wing)g (computer{)46 418 y(assisted)j(b)q(ounds)458 528 y Fu(k)p Ft(R)p Fu(k)545 535 y FE(0)579 528 y Fu(\024)82 b Fx(1)p Ft(:)p Fx(437804901)q(53)q(865)q(71)q(516)q(681)q(21)q(552)q(90)q(155) 20 b Ft(;)432 601 y Fu(k)p Ft(R)494 608 y Fk(L)520 601 y Fu(k)545 608 y FE(0)579 601 y Fu(\024)82 b Fx(68)p Ft(:)p Fx(80715992)q(09)q(183)q(46)q(704)q(866)q(57)q(300)q(38)q(651)20 b Ft(;)429 674 y Fu(k)p Ft(R)491 681 y Fk(G)520 674 y Fu(k)545 681 y FE(0)579 674 y Fu(\024)82 b Fx(64)p Ft(:)p Fx(30605456)q(12)q(114)q(28)q(014)q(777)q(66)q(934)q(41)q(263)20 b Ft(;)408 746 y Fu(k)p Ft(R)470 753 y Fk(LL)520 746 y Fu(k)545 753 y FE(0)579 746 y Fu(\024)82 b Fx(56614)p Ft(:)p Fx(10721)q(16)q(429)q(90)q(416)q(531)q(77)q(102)q(56)q(315)20 b Ft(;)405 819 y Fu(k)p Ft(R)467 826 y Fk(LG)520 819 y Fu(k)545 826 y FE(0)579 819 y Fu(\024)82 b Fx(56755)p Ft(:)p Fx(02346)q(06)q(639)q(06)q(373)q(116)q(85)q(415)q(98)q(635)20 b Ft(;)401 891 y Fu(k)p Ft(R)463 898 y Fk(GG)520 891 y Fu(k)545 898 y FE(0)579 891 y Fu(\024)82 b Fx(28205)p Ft(:)p Fx(78724)q(21)q(132)q(38)q(403)q(228)q(51)q(337)q(72)q(315)20 b Ft(:)46 1001 y Fx(In)c(this)g(w)o(a)o(y)g(one)g(obtains)516 1095 y Fp(e)506 1111 y Ft(T)542 1086 y FE(\(+\))535 1122 y(0)607 1111 y Fu(\021)8 b Fx(8)p Ft(:)p Fx(5671005796)q(027)q(44)q (072)q(85)q(201)q(908)q(57)q(221)q(71)20 b Ft(;)516 1173 y Fp(e)506 1189 y Ft(T)542 1164 y FE(\()p FF(\000)p FE(\))535 1200 y(0)607 1189 y Fu(\021)8 b Fx(8)p Ft(:)p Fx(5596217784)q(728)q(45) q(438)q(20)q(966)q(702)q(13)q(687)q(91)20 b Ft(:)46 1299 y Fx(A)12 b(bit)h(more)e(delicate)h(is)h(the)f(ev)m(aluation)h(of)g Ft(F)903 1306 y FE(0)936 1299 y Fx(and)g Ft(G)1065 1306 y FE(0)1085 1299 y Fx(,)g Fj(i.e.)h Fx(of)f(the)g(norms)f(of)h(the)g (error)g(functions)46 1360 y Ft(f)22 b Fx(and)c Ft(g)r Fx(.)e(W)l(e)h(shall)g(follo)o(w)g(the)f(classical)h Fv(Cauc)o(h)o(y's)f(ma)s(joran)o(t)g(metho)q(d)g Fx(\(see)h([9],)f(c)o (hapter)g(5)i(for)46 1420 y(generalities\).)46 1505 y(W)l(e)e(recall)f (that)i(if)348 1615 y Ft(\013)p Fx(\()p Ft(y)r Fx(\))8 b Fu(\021)519 1573 y Fp(X)506 1667 y Fk(k)q FF(2)p Fc(N)571 1657 y Fo(p)601 1615 y Ft(\013)632 1622 y Fk(k)653 1615 y Fx(\()p Ft(y)k Fu(\000)f Ft(y)782 1622 y FE(0)802 1615 y Fx(\))821 1594 y Fk(k)940 1615 y Fx(and)103 b(\026)-29 b Ft(\013)p Fx(\()p Ft(y)r Fx(\))8 b Fu(\021)1287 1573 y Fp(X)1274 1667 y Fk(k)q FF(2)p Fc(N)1339 1657 y Fo(p)1373 1615 y Fx(\026)-28 b Ft(\013)1400 1622 y Fk(k)1421 1615 y Fx(\()p Ft(y)13 b Fu(\000)d Ft(y)1550 1622 y FE(0)1570 1615 y Fx(\))1589 1594 y Fk(k)46 1761 y Fx(are)16 b(smo)q(oth)g (functions)g(of)h Ft(y)g Fx(in)f(a)h(neigh)o(b)q(ourho)q(o)q(d)h(of)e Ft(y)1116 1768 y FE(0)1149 1761 y Fu(2)e Fq(R)1235 1743 y Fk(p)1252 1761 y Fx(,)21 b(\026)-29 b Ft(\013)16 b Fx(is)g(said)h(to)f Fv(ma)s(jorize)e Ft(\013)j Fx(and)f(is)46 1822 y(denoted)899 1882 y Ft(\013)f Fu(\036)j Fx(\026)-29 b Ft(\013)17 b(;)46 1969 y Fx(if)761 2029 y Fu(j)p Ft(\013)806 2036 y Fk(k)828 2029 y Fu(j)c(\024)19 b Fx(\026)-29 b Ft(\013)939 2036 y Fk(k)976 2029 y Ft(;)106 b Fu(8)15 b Ft(k)j(:)46 2116 y Fx(F)l(or)e(example,)e(for)i(an)o(y)g Ft(n)e Fu(2)g Fq(Z)631 2098 y Fk(p)648 2116 y Fx(,)79 2226 y(cos)9 b Ft(n)i Fu(\001)g Ft(x)i Fu(\036)h Fx(cosh\()p Fu(j)p Ft(n)466 2233 y FE(1)486 2226 y Fu(j)p Ft(x)528 2233 y FE(1)558 2226 y Fx(+)d Fu(\001)d(\001)g(\001)j Fx(+)g Fu(j)p Ft(n)768 2233 y Fk(p)788 2226 y Fu(j)p Ft(x)830 2233 y Fk(p)849 2226 y Fx(\))16 b Ft(;)57 b Fx(sin)8 b Ft(n)j Fu(\001)g Ft(x)i Fu(\036)h Fx(sinh\()p Fu(j)p Ft(n)1331 2233 y FE(1)1351 2226 y Fu(j)p Ft(x)1393 2233 y FE(1)1423 2226 y Fx(+)d Fu(\001)d(\001)g(\001)j Fx(+)g Fu(j)p Ft(n)1633 2233 y Fk(p)1652 2226 y Fu(j)p Ft(x)1694 2233 y Fk(p)1714 2226 y Fx(\))16 b Ft(:)24 b Fx(\(5.11\))46 2336 y(W)l(e)16 b(need)g(a)g(little)f(tec)o(hnical)f (Lemma.)46 2450 y Fw(Lemm)o(a)i(5.1)24 b Fj(L)n(et)18 b Ft(\027;)8 b(p)14 b Fu(2)g Fq(Z)592 2457 y FE(+)636 2450 y Fj(and)k(let)640 2560 y Ft(\013)p Fx(\()p Ft(x;)8 b(y)r Fx(\))g Fu(\021)911 2519 y Fp(X)849 2613 y Fk(n)p FF(2)p Fc(Z)914 2604 y Fo(p)935 2613 y FE(:)p FF(j)p Fk(n)p FF(j\024)p Fk(\027)1058 2560 y Ft(c)1079 2567 y Fk(n)1103 2560 y Fx(\()p Ft(x)p Fx(\))p Ft(\013)1200 2567 y Fk(n)1223 2560 y Fx(\()p Ft(y)r Fx(\))17 b Ft(;)955 2863 y Fx(34)p eop %%Page: 35 35 35 34 bop 46 78 a Fj(wher)n(e)17 b Ft(c)204 85 y Fk(n)228 78 y Fx(\()p Ft(x)p Fx(\))p Fj(,)f(for)h(e)n(ach)g Ft(n)p Fj(,)g(is)g(either)g Fx(cos)9 b Ft(n)h Fu(\001)g Ft(x)17 b Fj(or)f Fx(sin)8 b Ft(n)i Fu(\001)g Ft(x)17 b Fj(\()p Ft(x)d Fu(2)g Fq(T)1328 60 y Fk(p)1345 78 y Fj(\))j(and)g(the)g Ft(\013)1587 85 y Fk(n)1611 78 y Fj('s)g(ar)n(e)f(analytic)46 138 y(functions)g(on)f(a)g(c)n(omplex)h(b)n(al)r(l)g(ar)n(ound)f Ft(y)817 145 y FE(0)850 138 y Fu(2)f Fq(R)936 120 y Fk(p)953 138 y Fj(.)h(Fix)g Ft(M)k Fu(2)14 b Fq(N)p Fj(.)i(L)n(et)f Ft(b)1349 145 y FE(0)1376 138 y Fu(\021)8 b Ft(y)1447 145 y FE(0)1482 138 y Fj(and,)15 b(for)f Fx(1)h Fu(\024)e Ft(j)k Fu(\024)d Ft(M)5 b Fj(,)46 198 y(let)21 b Ft(a)143 180 y FE(\()p Fk(j)r FE(\))188 198 y Fx(\()p Ft(\022)q Fx(\))g Fj(and)f Ft(b)389 180 y FE(\()p Fk(j)r FE(\))435 198 y Fx(\()p Ft(\022)q Fx(\))g Fj(b)n(e)h(given)g Fq(R)748 180 y Fk(p)765 198 y Fj({value)n(d)g(functions)h(of)e Ft(\022)g Fu(2)f Fq(T)1348 180 y Fk(p)1385 198 y Fj(analytic)i(on)f Fx(\001)1685 205 y Fk(\030)1724 198 y Fj(for)g(some)46 259 y Ft(\030)c(>)e Fx(0)p Fj(.)k(L)n(et)k Fx(\026)-29 b Ft(\013)307 266 y Fk(n)331 259 y Fx(\()p Ft(y)r Fx(\))16 b Fj(b)n(e)i(analytic)h(functions)f(such)g(that)690 369 y Ft(\013)721 376 y Fk(n)758 369 y Fu(\036)g Fx(\026)-29 b Ft(\013)841 376 y Fk(n)882 369 y Ft(;)108 b Fx(\()p Fu(8)16 b(j)p Ft(n)p Fu(j)e(\024)f Ft(\027)s Fx(\))18 b Ft(:)46 479 y Fj(L)n(et)i Fx(\026)-25 b Ft(a)158 486 y FE(0)186 479 y Fu(\021)8 b Fx(\()p Ft(\030)r(;)g(:::;)g(\030)r Fx(\))p Fj(,)435 465 y Fx(\026)437 479 y Ft(b)458 486 y FE(0)486 479 y Fu(\021)g Fx(\()p Fu(j)p Ft(y)590 486 y FE(01)627 479 y Fu(j)p Ft(;)g(:::;)g Fu(j)p Ft(y)765 486 y FE(0)p Fk(p)800 479 y Fu(j)p Fx(\))19 b Fj(and,)h(for)f Fx(1)f Fu(\024)f Ft(j)j Fu(\024)e Ft(M)25 b Fj(let)c Fx(\026)-25 b Ft(a)1407 460 y FE(\()p Fk(j)r FE(\))1472 479 y Fj(and)1567 465 y Fx(\026)1569 479 y Ft(b)1590 460 y FE(\()p Fk(j)r FE(\))1654 479 y Fj(b)n(e)20 b Ft(p)p Fj({ve)n(ctors)46 539 y(with)e(nonne)n(gative)h(c)n(omp)n(onents)f (such)g(that)228 649 y Fu(k)p Ft(a)279 623 y FE(\()p Fk(j)r FE(\))279 660 y Fk(i)324 649 y Fu(k)349 656 y Fk(\030)381 649 y Fu(\024)d Fx(\026)-25 b Ft(a)460 623 y FE(\()p Fk(j)r FE(\))460 660 y Fk(i)523 649 y Ft(;)107 b Fu(k)p Ft(b)690 623 y FE(\()p Fk(j)r FE(\))690 660 y Fk(i)736 649 y Fu(k)761 656 y Fk(\030)793 649 y Fu(\024)844 636 y Fx(\026)846 649 y Ft(b)867 623 y FE(\()p Fk(j)r FE(\))867 660 y Fk(i)930 649 y Ft(;)g Fx(\()p Fu(8)16 b Fx(1)e Fu(\024)g Ft(j)j Fu(\024)c Ft(M)23 b(;)58 b Fu(8)16 b Fx(1)e Fu(\024)g Ft(i)f Fu(\024)h Ft(p)p Fx(\))k Ft(:)46 759 y Fj(Final)r(ly,)g(denote)384 897 y Fx(\026)-25 b Ft(c)404 904 y Fk(n)428 897 y Fx(\()p Ft(x)p Fx(\))8 b Fu(\021)557 824 y Fp(\()611 867 y Fx(cosh\()p Fu(j)p Ft(n)765 874 y FE(1)785 867 y Fu(j)p Ft(x)827 874 y FE(1)857 867 y Fx(+)j Fu(\001)d(\001)g(\001)k Fx(+)f Fu(j)p Ft(n)1068 874 y Fk(p)1087 867 y Fu(j)p Ft(x)1129 874 y Fk(p)1149 867 y Fx(\))59 b(if)17 b Ft(c)1294 874 y Fk(n)1331 867 y Fx(=)d(cos)8 b Ft(n)j Fu(\001)g Ft(x)611 927 y Fx(sinh\()p Fu(j)p Ft(n)760 934 y FE(1)780 927 y Fu(j)p Ft(x)822 934 y FE(1)852 927 y Fx(+)g Fu(\001)d(\001)g(\001)j Fx(+)g Fu(j)p Ft(n)1062 934 y Fk(p)1082 927 y Fu(j)p Ft(x)1124 934 y Fk(p)1143 927 y Fx(\))65 b(if)17 b Ft(c)1294 934 y Fk(n)1331 927 y Fx(=)d(sin)8 b Ft(n)j Fu(\001)g Ft(x)46 1035 y Fj(and)18 b(let)574 1170 y Fx(~)-29 b Ft(\013)p Fx(\()p Ft(";)8 b(\022)q Fx(\))g Fu(\021)g Ft(\013)793 1122 y Fp(\020)819 1170 y Ft(\022)k Fx(+)915 1116 y Fk(M)903 1129 y Fp(X)903 1220 y Fk(j)r FE(=1)972 1170 y Ft(")995 1150 y Fk(j)1013 1170 y Ft(a)1039 1150 y FE(\()p Fk(j)r FE(\))1085 1170 y Fx(\()p Ft(\022)q Fx(\))p Ft(;)1181 1116 y Fk(M)1169 1129 y Fp(X)1169 1220 y Fk(j)r FE(=0)1238 1170 y Ft(")1261 1150 y Fk(j)1279 1170 y Ft(b)1300 1150 y FE(\()p Fk(j)r FE(\))1346 1170 y Fx(\()p Ft(\022)q Fx(\))1408 1122 y Fp(\021)1450 1170 y Ft(;)569 1297 y(\014)s Fx(\()p Ft(";)c(\022)q Fx(\))g Fu(\021)803 1255 y Fp(X)770 1347 y Fk(j)r FF(\025)p Fk(M)t FE(+1)904 1297 y Ft(")927 1276 y Fk(j)945 1249 y Fp(h)970 1297 y Fx(~)-29 b Ft(\013)p Fx(\()p Fu(\001)p Ft(;)8 b(\022)q Fx(\))1094 1249 y Fp(i)1114 1324 y Fk(j)1149 1297 y Ft(;)569 1455 y(')p Fx(\()p Ft(")p Fx(\))g Fu(\021)739 1414 y Fp(X)725 1508 y FF(j)p Fk(n)p FF(j\024)p Fk(\027)823 1455 y Fx(\026)-25 b Ft(c)843 1462 y Fk(n)866 1407 y Fp(\020)911 1401 y Fk(M)900 1414 y Fp(X)899 1505 y Fk(j)r FE(=0)969 1455 y Ft(")992 1435 y Fk(j)1011 1455 y Fx(\026)g Ft(a)1036 1435 y FE(\()p Fk(j)r FE(\))1081 1407 y Fp(\021)1146 1455 y Fx(\026)c Ft(\013)1172 1462 y Fk(n)1195 1407 y Fp(\020)1240 1401 y Fk(M)1229 1414 y Fp(X)1229 1505 y Fk(j)r FE(=0)1298 1455 y Ft(")1321 1435 y Fk(j)1337 1442 y Fx(\026)1339 1455 y Ft(b)1360 1435 y FE(\()p Fk(j)r FE(\))1406 1407 y Fp(\021)1448 1455 y Ft(;)46 1612 y Fj(wher)n(e,)17 b(of)h(c)n(ourse,)f(the)h(op)n(er)n(ator)e Fx([)p Fu(\001)p Fx(])737 1619 y Fk(j)771 1612 y Fj(r)n(efers)h(to)h Ft(")p Fj({exp)n(ansions.)f(Then,)h(for)f(al)r(l)22 b Fx(\026)-27 b Ft(")14 b(>)f Fx(0)658 1722 y Fu(k)p Fx([)5 b(~)-29 b Ft(\013)p Fx(\()p Fu(\001)p Ft(;)8 b(\022)q Fx(\)])840 1729 y Fk(j)858 1722 y Fu(k)883 1729 y Fk(\030)915 1722 y Fu(\024)14 b Fx([)p Ft(')p Fx(])1028 1729 y Fk(j)1063 1722 y Ft(;)107 b Fu(8)16 b Ft(j)h Fu(\025)d Fx(0)k(;)662 1833 y(sup)658 1875 y FF(j)p Fk(")p FF(j\024)r FE(\026)-20 b Fk(")746 1833 y Fu(k)p Ft(\014)s Fu(k)827 1840 y Fk(\030)859 1833 y Fu(\024)14 b Ft(')p Fx(\()s(\026)-27 b Ft(")p Fx(\))11 b Fu(\000)1077 1779 y Fk(M)1066 1792 y Fp(X)1065 1883 y Fk(j)r FE(=0)1138 1833 y Fx(\026)-27 b Ft(")1158 1812 y Fk(j)1176 1833 y Fx([)p Ft(')p Fx(])1236 1840 y Fk(j)1271 1833 y Ft(:)502 b Fx(\(5.12\))46 2014 y Fw(Pro)r(of)18 b Fx(Observ)o(e)e(that,)g(b)o(y)g(\(5.11\),)g(for)h(an)o(y)f(\014xed)g Ft(\022)f Fu(2)f Fx(\001)1140 2021 y Fk(\030)1174 2014 y Fx(one)j(has)615 2162 y Ft(c)636 2169 y Fk(n)660 2114 y Fp(\020)705 2108 y Fk(M)694 2121 y Fp(X)693 2212 y Fk(j)r FE(=0)763 2162 y Ft(")786 2142 y Fk(j)804 2162 y Ft(a)830 2142 y FE(\()p Fk(j)r FE(\))875 2162 y Fx(\()p Ft(\022)q Fx(\))937 2114 y Fp(\021)976 2162 y Fu(\036)d Fx(\026)-25 b Ft(c)1049 2169 y Fk(n)1073 2114 y Fp(\020)1118 2108 y Fk(M)1106 2121 y Fp(X)1106 2212 y Fk(j)r FE(=0)1175 2162 y Ft(")1198 2142 y Fk(j)1217 2162 y Fx(\026)g Ft(a)1242 2142 y FE(\()p Fk(j)r FE(\))1288 2114 y Fp(\021)1329 2162 y Ft(;)46 2316 y Fx(and)17 b(also)g(that)328 2298 y FE(41)610 2401 y Ft(\013)641 2408 y Fk(n)665 2353 y Fp(\020)710 2347 y Fk(M)699 2359 y Fp(X)698 2451 y Fk(j)r FE(=0)767 2401 y Ft(")790 2380 y Fk(j)809 2401 y Ft(b)830 2380 y FE(\()p Fk(j)r FE(\))875 2401 y Fx(\()p Ft(\022)q Fx(\))937 2353 y Fp(\021)976 2401 y Fu(\036)h Fx(\026)-29 b Ft(\013)1059 2408 y Fk(n)1083 2353 y Fp(\020)1128 2347 y Fk(M)1117 2359 y Fp(X)1116 2451 y Fk(j)r FE(=0)1185 2401 y Ft(")1208 2380 y Fk(j)1225 2388 y Fx(\026)1227 2401 y Ft(b)1248 2380 y FE(\()p Fk(j)r FE(\))1293 2353 y Fp(\021)1334 2401 y Ft(:)p 46 2495 747 2 v 85 2526 a Fs(41)121 2541 y FI(Recall)16 b(the)h(basic)g(facts)g(of)g(ma)r (jorization)d(theory)k(\(see,)f(e.g.,)f([9)o(]\):)g(sum,)g(m)o (ultiplicati)o(on)e(and)j(comp)q(osition)46 2591 y(are)d(preserv)o(ed)i (b)o(y)e(ma)r(jorizations.)955 2863 y Fx(35)p eop %%Page: 36 36 36 35 bop 46 78 a Fx(These)15 b(relations)g(imply)e(that)j(for)f(all)g Ft(\022)g Fu(2)f Fx(\001)890 85 y Fk(\030)909 78 y Fx(,)19 b(~)-28 b Ft(\013)p Fx(\()p Fu(\001)p Ft(;)8 b(\022)q Fx(\))13 b Fu(\036)h Ft(')p Fx(,)h(whic)o(h)f(implies)f(immedi)o(ately) f(\(5.12\).)p 46 138 21 42 v 46 248 a(In)k(order)h(to)f(apply)h(the)f (Lemma)e(to)j(our)g(situations)g(w)o(e)f(need)g(the)g(follo)o(wing)h (ob)o(vious)f(ma)s(joriza-)46 308 y(tions.)i(Let)h Ft(z)g Fu(2)f Fq(C)30 b Fx(with)18 b Fu(j)p Ft(z)r Fu(j)f Ft(<)h Fx(1;)g(let)768 290 y FE(42)823 308 y Ft(y)847 315 y FE(10)884 308 y Ft(;)8 b(y)930 315 y FE(20)985 308 y Fx(b)q(e)19 b(p)q(ositiv)o(e)f(n)o(um)o(b)q(ers)e(and)j(let)f Fu(j)p Ft(y)1643 315 y FE(1)1675 308 y Fu(\000)12 b Ft(y)1750 315 y FE(10)1787 308 y Fu(j)18 b Ft(<)f Fx(1,)46 369 y Fu(j)p Ft(y)84 376 y FE(2)114 369 y Fu(\000)11 b Ft(y)188 376 y FE(20)225 369 y Fu(j)j Ft(<)g Fx(1;)i(\014nally)g(let)f Ft(s)h Fx(b)q(e)h(a)f(p)q(ositiv)o(e)g(in)o(teger.)f(Then)355 444 y Fu(p)p 396 444 111 2 v 396 485 a Fx(1)d Fu(\000)f Ft(z)k Fu(\036)f Fx(2)d Fu(\000)658 444 y(p)p 700 444 V 41 x Fx(1)g Fu(\000)g Ft(z)18 b(;)105 b Fx(\(1)12 b Fu(\000)f Ft(z)r Fx(\))1094 465 y FF(\000)1126 451 y Fl(1)p 1126 457 16 2 v 1126 478 a(2)1162 485 y Fu(\036)j Fx(\(1)d Fu(\000)g Ft(z)r Fx(\))1363 465 y FF(\000)1395 451 y Fl(1)p 1395 457 V 1395 478 a(2)1434 485 y Fx(;)360 539 y Ft(y)384 546 y FE(2)p 360 561 44 2 v 360 606 a Ft(y)384 613 y FE(1)422 572 y Fu(\036)553 539 y Ft(y)577 546 y FE(2)p 480 561 190 2 v 480 606 a Fx(2)p Ft(y)528 613 y FE(10)576 606 y Fu(\000)g Ft(y)650 613 y FE(1)691 572 y Ft(;)810 524 y Fp(\020)840 539 y Ft(y)864 546 y FE(2)p 840 561 44 2 v 840 606 a Ft(y)864 613 y FE(1)888 524 y Fp(\021)913 536 y FE(2)947 572 y Fu(\036)999 524 y Fp(\020)1102 539 y Ft(y)1126 546 y FE(2)p 1029 561 190 2 v 1029 606 a Fx(2)p Ft(y)1077 613 y FE(10)1126 606 y Fu(\000)g Ft(y)1200 613 y FE(1)1224 524 y Fp(\021)1249 536 y FE(2)1285 572 y Fx(;)355 708 y Ft(e)p Fx(\()p Ft(y)421 715 y FE(1)440 708 y Ft(;)d(y)486 715 y FE(2)505 708 y Fx(\))14 b Fu(\036)h Fx(\026)-25 b Ft(e)o Fx(\()p Ft(y)656 715 y FE(1)676 708 y Ft(;)8 b(y)722 715 y FE(2)741 708 y Fx(\))g Fu(\021)g Fx(2)k Fu(\000)901 636 y Fp(s)p 942 636 355 2 v 942 708 a Fx(1)g Fu(\000)1027 660 y Fp(\020)1130 674 y Ft(y)1154 681 y FE(2)p 1057 697 190 2 v 1057 742 a Fx(2)p Ft(y)1105 749 y FE(10)1154 742 y Fu(\000)f Ft(y)1228 749 y FE(1)1252 660 y Fp(\021)1277 671 y FE(2)1313 708 y Fx(;)355 862 y Ft(e)378 841 y FF(\000)p FE(1)424 862 y Fx(\()p Ft(y)467 869 y FE(1)487 862 y Ft(;)d(y)533 869 y FE(2)552 862 y Fx(\))14 b Fu(\036)638 789 y Fp( )671 862 y Fx(1)d Fu(\000)756 814 y Fp(\020)859 828 y Ft(y)883 835 y FE(2)p 786 850 V 786 896 a Fx(2)p Ft(y)834 903 y FE(10)882 896 y Fu(\000)g Ft(y)956 903 y FE(1)980 814 y Fp(\021)1005 825 y FE(2)1025 789 y Fp(!)1058 800 y FF(\000)p FE(1)p Fk(=)p FE(2)1165 862 y Fx(;)355 994 y Ft(y)381 974 y FF(\000)p Fk(s)379 1007 y FE(1)440 994 y Fu(\036)493 946 y Fp(\020)605 961 y Fx(1)p 522 983 V 522 1028 a(2)p Ft(y)570 1035 y FE(10)619 1028 y Fu(\000)g Ft(y)693 1035 y FE(1)717 946 y Fp(\021)742 958 y Fk(s)777 994 y Ft(;)105 b Fu(\000)p Fx(2)959 946 y Fp(\020)1012 961 y Fx(1)p 989 983 70 2 v 989 1028 a(2)p Ft(y)1039 1011 y FE(2)1037 1039 y(1)1075 994 y Fu(\000)11 b Ft(y)1149 1001 y FE(2)1168 946 y Fp(\021)1207 994 y Fu(\036)i Fx(2)p Ft(y)1307 1001 y FE(2)1338 994 y Fx(+)1387 946 y Fp(\020)1500 961 y Fx(1)p 1417 983 190 2 v 1417 1028 a(2)p Ft(y)1465 1035 y FE(10)1514 1028 y Fu(\000)d Ft(y)1587 1035 y FE(1)1612 946 y Fp(\021)1637 958 y FE(2)1673 994 y Ft(:)46 1128 y Fx(Using)15 b(the)h(ab)q(o)o(v)o(e)g(observ)m(ations,)g(one)g(gets)g (the)g(follo)o(wing)f(ev)m(aluations)h(of)g(the)g(norm)f(b)q(ounds)i Ft(F)1893 1135 y FE(0)46 1188 y Fx(and)g Ft(G)179 1195 y FE(0)199 1188 y Fx(:)430 1298 y Ft(F)469 1273 y FE(\(+\))462 1309 y(0)533 1298 y Fu(\021)9 b Fx(1)p Ft(:)p Fx(87553018275)q(383)q (71)q(921)q(26)q(197)q(35)q(567)q(217)q(9)14 b Fu(\001)d Fx(10)1507 1277 y FF(\000)p FE(23)1589 1298 y Ft(;)430 1376 y(F)469 1350 y FE(\()p FF(\000)p FE(\))462 1387 y(0)533 1376 y Fu(\021)e Fx(1)p Ft(:)p Fx(88144322864)q(402)q(60)q(730) q(96)q(600)q(12)q(365)q(262)q(4)14 b Fu(\001)d Fx(10)1507 1355 y FF(\000)p FE(23)1589 1376 y Ft(;)430 1454 y(G)468 1428 y FE(\(+\))468 1465 y(0)534 1454 y Fu(\021)d Fx(1)p Ft(:)p Fx(52434097388)q(63)q(086)q(26)q(744)q(645)q(63)q(989)q(60)q(68) 14 b Fu(\001)d Fx(10)1507 1433 y FF(\000)p FE(20)1589 1454 y Ft(;)430 1532 y(G)468 1506 y FE(\()p FF(\000)p FE(\))468 1543 y(0)534 1532 y Fu(\021)d Fx(1)p Ft(:)p Fx(57324136745)q(26)q(825)q(46)q(591)q(639)q(16)q(027)q(36)q(34)14 b Fu(\001)d Fx(10)1507 1511 y FF(\000)p FE(20)1589 1532 y Ft(:)46 1676 y Fe(5.3)69 b(Step)33 b(3:)h(Application)c(of)k(the)e (KAM)h(algorithm)d(and)k(of)g(the)202 1751 y(KAM)23 b(Theorem)46 1843 y Fx(W)l(e)17 b(pro)q(ceed)h(to)g(apply)g(the)f(KAM)g(algorithm)g (and)h(the)f(KAM)g(Theorem)g(w)o(ork)o(ed)g(out)h(in)f Fu(x)h Fx(4.3:)46 1903 y(recall)d(3\))i(of)f Fu(x)g Fx(1)h(and)g (Remark)e(4.4.)46 1988 y(Also)21 b(this)g(step)h(is)f (computer{assisted;)f(ho)o(w)o(ev)o(er)g(double)h(precision)g(\(rather) g(than)h(quadruple)46 2049 y(precision\))15 b(will)g(usually)h(b)q(e)h (go)q(o)q(d)h(enough.)46 2134 y(W)l(e)25 b(discuss)g(in)f(detail)g(the) h(case)g(with)g(initial)e(data)j(\()p Ft(L)1166 2141 y FE(+)1196 2134 y Ft(;)8 b(G)1256 2141 y FE(+)1286 2134 y Fx(\);)24 b(the)h(case)g(with)g(initial)e(data)46 2194 y(\()p Ft(L)98 2201 y FF(\000)128 2194 y Ft(;)8 b(G)188 2201 y FF(\000)217 2194 y Fx(\))23 b(is)f(completely)e(analogous)773 2176 y FE(43)835 2194 y Fx(and)j(w)o(e)f(spare)h(the)g(reader)f(more)g (data)h(whic)o(h)f(w)o(ould)46 2254 y(not)17 b(shed)f(m)o(uc)o(h)e (more)h(ligh)o(t.)46 2368 y Fw(Remark)h(5.2)24 b Fx(The)15 b(quan)o(tities)e(with)h(an)h(index)f(0)h(refer)f(\(as)h(in)f(the)g(ab) q(o)o(v)o(e)g(step)h(2\))g(to)f(our)h(c)o(hoice)46 2428 y(of)h(the)g Fv(initial)f Fx(appro)o(ximate)g(solution.)h(T)l(o)g(suc)o (h)g(initial)f(appro)o(ximate)g(solution)h(w)o(e)g(apply)g(a)g(few)p 46 2472 747 2 v 85 2503 a Fs(42)121 2518 y FI(T)m(o)d(a)o(v)o(oid)f (confusion)i(with)f(the)i(norm)d(parameter)i Fz(G)981 2524 y Fs(0)1013 2518 y FI(w)o(e)g(denote)h(here)g(the)f(action{v)n (ariables)f(b)o(y)h(\()p Fz(y)1767 2524 y Fs(1)1786 2518 y Fz(;)7 b(y)1825 2524 y Fs(2)1843 2518 y FI(\).)85 2553 y Fs(43)121 2568 y FI(Actually)m(,)14 b(the)i(\()p Fz(L)416 2574 y Fs(+)444 2568 y Fz(;)7 b(G)496 2574 y Fs(+)523 2568 y FI(\){case)17 b(\\con)o(v)o(erges")f(after)g(three)h(iterations) f(of)f Fr(K)h FI(while)f(the)i(\()p Fz(L)1599 2574 y Fy(\000)1627 2568 y Fz(;)7 b(G)1679 2574 y Fy(\000)1706 2568 y FI(\))16 b(case)h(after)46 2617 y(four.)955 2863 y Fx(36)p eop %%Page: 37 37 37 36 bop 46 78 a Fx(times)19 b([resp)q(ectiv)o(ely)f(three)i(times)f (for)i(the)f(case)h(with)g(initial)e(data)j(\()p Ft(L)1433 85 y FE(+)1463 78 y Ft(;)8 b(G)1523 85 y FE(+)1553 78 y Fx(\))20 b(and)i(four)f(times)46 138 y(for)16 b(the)g(case)g(with)g (initial)e(data)j(\()p Ft(L)721 145 y FF(\000)751 138 y Ft(;)8 b(G)811 145 y FF(\000)841 138 y Fx(\)])15 b(the)h(KAM)f (algorithm)g(obtaining)h(new)g(appro)o(ximate)46 198 y(solutions)k(\()p Ft(u)300 180 y FE(\(1\))347 198 y Ft(;)8 b(v)395 180 y FE(\(1\))441 198 y Fx(\),...,\()p Ft(u)577 180 y FE(\(3\))622 198 y Ft(;)g(v)670 180 y FE(\(3\))716 198 y Fx(\))20 b([for)f(the)g(case)h(with)f(initial)f (data)i(\()p Ft(L)1464 205 y FE(+)1494 198 y Ft(;)8 b(G)1554 205 y FE(+)1584 198 y Fx(\),)19 b(while)g(for)g(the)46 259 y(case)k(with)h(initial)e(data)i(\()p Ft(L)593 266 y FF(\000)623 259 y Ft(;)8 b(G)683 266 y FF(\000)713 259 y Fx(\))23 b(w)o(e)g(consider)g(also)h(\()p Ft(u)1183 240 y FE(\(4\))1230 259 y Ft(;)8 b(v)1278 240 y FE(\(4\))1325 259 y Fx(\)].)22 b(The)i Fv(input)f Fx(of)h(the)f(KAM)46 319 y(Theorem)d(will)g(b)q(e)h(resp)q(ectiv)o(ely)e(\()p Ft(u;)8 b(v)r Fx(\))g Fu(\021)f Fx(\()p Ft(u)911 301 y FE(\(3\))958 319 y Ft(;)h(v)1006 301 y FE(\(3\))1052 319 y Fx(\))21 b(\(with)g(the)g(relativ)o(e)e(norm)h(b)q(ounds)j(giv)o (en)46 379 y(b)o(y)c(the)g(KAM)f(algorithm\))g(for)i(the)f(case)g(with) g(initial)f(data)i(\()p Ft(L)1281 386 y FE(+)1311 379 y Ft(;)8 b(G)1371 386 y FE(+)1400 379 y Fx(\))20 b(and)f(\()p Ft(u;)8 b(v)r Fx(\))g Fu(\021)g Fx(\()p Ft(u)1752 361 y FE(\(4\))1799 379 y Ft(;)g(v)1847 361 y FE(\(4\))1893 379 y Fx(\))46 439 y(for)20 b(the)g(case)g(with)g(initial)f(data)i(\()p Ft(L)746 446 y FF(\000)776 439 y Ft(;)8 b(G)836 446 y FF(\000)866 439 y Fx(\).)19 b Fv(Quan)o(tities)h(referring)f(to)i(the)e (input)h(of)h(the)f(KAM)46 499 y(Theorem)15 b(will)g(carry)h(no)h (index)e Fx(\(and)i(b)q(ew)o(are)f(of)h(the)f(di\013erence)f(b)q(et)o (w)o(een)j(~)-27 b Ft(u)1522 481 y FE(\()p Fk(j)r FE(\))1584 499 y Fx(and)17 b Ft(u)1707 481 y FE(\()p Fk(j)r FE(\))1752 499 y Fx(\).)46 613 y(Since)f(w)o(e)h(shall)g(consider)g(only)g(the)h (case)f(with)g(initial)f(data)i(\()p Ft(L)1269 620 y FE(+)1299 613 y Ft(;)8 b(G)1359 620 y FE(+)1389 613 y Fx(\))17 b(w)o(e)g(shall)g(drop)h(from)e(the)46 674 y(notation)h(the)f (su\016x)454 656 y FE(\(+\))511 674 y Fx(.)46 759 y(T)l(o)e(apply)g (the)f(KAM)g(algorithm)g(w)o(e)h(ha)o(v)o(e)f(to)h(\014x)f(the)h(v)m (alues)g(of)g(the)f(\\angle")i(analyticit)o(y)d(widths,)46 819 y(namely)h(the)h(v)m(alues)h(of)499 806 y(\026)494 819 y Ft(\030)j Fx(and)d Ft(\030)647 826 y FE(0)682 819 y Fx([see)f(\(4.23\)])h(and)h(of)f Ft(\030)1095 826 y FE(1)1123 819 y Fu(\021)8 b Ft(\030)1193 801 y FF(0)1220 819 y Fx([see)14 b(\(4.24\)])h(and)1542 801 y FE(44)1595 819 y Ft(\030)1616 826 y FE(2)1645 819 y Fu(\021)8 b Ft(\030)1715 801 y FF(0)1713 831 y FE(1)1733 819 y Fx(,)14 b Ft(\030)1782 826 y FE(3)1810 819 y Fu(\021)8 b Ft(\030)1880 801 y FF(0)1878 831 y FE(2)1898 819 y Fx(.)46 879 y(W)l(e)16 b(c)o(ho)q(ose)268 861 y FE(45)661 976 y Fx(\026)656 989 y Ft(\030)11 b Fu(\021)d Fx(0)p Ft(:)p Fx(201)17 b Ft(;)57 b(\030)954 996 y FE(0)982 989 y Fu(\021)8 b Fx(0)p Ft(:)p Fx(2)17 b Ft(;)56 b(\030)1199 996 y FE(1)1233 989 y Fx(=)14 b(0)p Ft(:)p Fx(1)i Ft(;)656 1062 y(\030)677 1069 y FE(2)710 1062 y Fx(=)e(0)p Ft(:)p Fx(05)j Ft(;)57 b(\030)957 1069 y FE(3)985 1062 y Fu(\021)8 b Fx(0)p Ft(:)p Fx(0025)18 b Ft(:)46 1172 y Fx(W)l(e)e(also)h(\014x)f(the)g(v)m (alue)g Ft(r)h Fx(app)q(earing)h(in)e(\(4.23\))g(as)h Ft(r)10 b Fu(\021)e Fx(0)p Ft(:)p Fx(001.)46 1257 y(After)13 b(three)h(steps)h(of)g(the)f(KAM)g(algorithm)f(w)o(e)h(shall)h(apply)f (the)g(KAM)g(Theorem,)f(w)o(e)h(therefore)46 1317 y(let)576 1390 y Ft(\030)d Fu(\021)d Ft(\030)676 1397 y FE(3)705 1390 y Fu(\021)g Fx(0)p Ft(:)p Fx(0025)18 b Ft(;)1028 1376 y Fx(^)1023 1390 y Ft(\030)11 b Fu(\021)1107 1356 y Ft(\030)p 1107 1378 25 2 v 1107 1424 a Fx(2)1145 1390 y Fu(\021)d Fx(0)p Ft(:)p Fx(00125)18 b Ft(:)46 1492 y Fx(Next,)d(w)o(e)i(need)g(estimates)e(on)j(the)e(deriv)m(ativ)o(es)g (of)h(the)g(Hamiltonian)e Ft(h)p Fx(\()p Ft(x;)8 b(y)r Fx(\))17 b(app)q(earing)h(in)e(the)46 1552 y(de\014nitions)e(of)h (\012,...)p Ft(\021)f Fx(in)h(Theorem)e(1.1)h([recall)f(\(4.21\)].)h (These)h(estimates)e(\(whic)o(h)g(do)i(not)g(c)o(hange)46 1613 y(in)h(the)g(iteration\))g(are)g(straigh)o(tforw)o(ard)h(and)g (one)f(obtains:)p 439 1684 40 2 v 439 1723 a Ft(E)h Fu(\024)d Fx(0)p Ft(:)p Fx(83456917806)q(28)q(014)q(16)q(845)q(92)q(592)q(214)q (36)q(76)20 b Ft(;)439 1795 y(E)475 1802 y FE(0)p Fk(;)p FE(2)536 1795 y Fu(\024)14 b Fx(56)p Ft(:)p Fx(548643863)q(316)q(20)q (846)q(77)q(890)q(66)q(978)q(800)q(6)20 b Ft(;)439 1868 y(E)475 1875 y FE(1)p Fk(;)p FE(1)536 1868 y Fu(\024)14 b Fx(1)p Ft(:)p Fx(1866945834)q(052)q(33)q(167)q(66)q(697)q(74)q(467)q (302)q(61)h Fu(\001)c Fx(10)1516 1847 y FF(\000)p FE(3)1580 1868 y Ft(;)439 1941 y(E)475 1948 y FE(0)p Fk(;)p FE(3)536 1941 y Fu(\024)j Fx(24606)p Ft(:)p Fx(142191)q(612)q(73)q(592)q(68)q (759)q(35)q(854)q(661)q(2)20 b Ft(;)439 2013 y(E)475 2020 y FE(1)p Fk(;)p FE(2)536 2013 y Fu(\024)14 b Fx(3)p Ft(:)p Fx(9626581982)q(761)q(49)q(193)q(70)q(110)q(45)q(794)q(301)q(0) 20 b Ft(;)439 2086 y(E)475 2093 y FE(2)p Fk(;)p FE(1)536 2086 y Fu(\024)14 b Fx(2581)p Ft(:)p Fx(0532581)q(721)q(42)q(973)q(37)q (992)q(70)q(162)q(472)q(5)20 b Ft(;)439 2158 y(E)475 2165 y FE(3)p Fk(;)p FE(0)536 2158 y Fu(\024)14 b Fx(1)p Ft(:)p Fx(7479497240)q(057)q(94)q(044)q(98)q(723)q(22)q(169)q(573)q(51) h Fu(\001)c Fx(10)1516 2138 y FF(\000)p FE(4)1580 2158 y Ft(:)46 2341 y Fx(No)o(w)16 b(w)o(e)f(iterate)g(the)h(map)f Fu(K)i Fx(three)e(times:)f(conditions)i(\(4.23\),)g(\(4.26\))h(and)f (\(4.28\))h(are)f(satis\014ed)46 2401 y(for)i Ft(j)j Fx(=)c(0)p Ft(;)8 b Fx(1)p Ft(;)g Fx(2)p Ft(;)g Fx(3)20 b(\(as)f(it)e(is)i(easy)f(to)h(c)o(hec)o(k)d(using)j(the)f(v)m(alues)g (\()p Ft(U)1302 2408 y Fk(j)1321 2401 y Ft(;)8 b(:::;)1415 2385 y Fp(e)1405 2401 y Ft(T)1434 2408 y Fk(j)1452 2401 y Fx(\))18 b(rep)q(orted)h(here\).)e(The)p 46 2449 747 2 v 85 2480 a Fs(44)121 2495 y FI(Recall)c(that)h Fz(\030)354 2501 y Fi(j)r Fs(+1)425 2495 y FI(=)e Fz(\030)487 2501 y Fi(j)514 2495 y Fr(\000)d Fz(\016)573 2501 y Fi(j)605 2495 y FI(where)15 b(the)f Fz(\016)814 2501 y Fi(j)832 2495 y FI('s)g(are)g(suc)o(h)h(that)f Fz(\016)1145 2501 y Fs(0)1176 2495 y Fz(>)d(\016)1237 2501 y Fs(1)1268 2495 y Fz(>)h(\016)1330 2501 y Fs(2)1349 2495 y FI(...)g(and)1478 2464 y Fp(P)1529 2495 y Fz(\016)1547 2501 y Fi(j)1576 2495 y Fz(<)g(\030)r FI(.)85 2530 y Fs(45)121 2545 y FI(These)j(v)n(alues)e(ha)o(v)o(e)h(b)q(een)h(c)o(hosen)g(b)o(y)f (\\optimizing")c(\(trial)k(and)f(error\))i(the)g(KAM)f(algorithm.)955 2863 y Fx(37)p eop %%Page: 38 38 38 37 bop 46 78 a Fx(\(double)11 b(precision\))g(iterated)g(v)m(alues)h (of)f(the)h(map)e Fu(K)j Fx(are)e(the)h(follo)o(wing.)e(After)h(the)g (\014rst)h(iteration:)297 173 y Ft(U)330 180 y FE(1)364 173 y Fu(\024)i Fx(1)p Ft(:)p Fx(30981759332)q(26)q(89)g Fu(\001)d Fx(10)904 152 y FF(\000)p FE(5)969 173 y Ft(;)56 b(V)1067 180 y FE(1)1101 173 y Fu(\024)14 b Fx(1)p Ft(:)p Fx(4562203600)q(617)q(7)19 b Ft(;)297 245 y(M)344 252 y FE(1)378 245 y Fu(\024)14 b Fx(1)p Ft(:)p Fx(00002126646)q(58)q(2)19 b Ft(;)p 897 206 53 2 v 57 w(M)949 252 y FE(1)983 245 y Fu(\024)13 b Fx(1)p Ft(:)p Fx(000021266)q(91)q(809)20 b Ft(;)305 305 y Fx(~)297 318 y Ft(V)325 325 y FE(1)359 318 y Fu(\024)14 b Fx(7)p Ft(:)p Fx(4535834217)q(897)q(68)h Fu(\001)c Fx(10)900 297 y FF(\000)p FE(7)964 318 y Ft(;)56 b(F)1066 325 y FE(1)1100 318 y Fu(\024)13 b Fx(4)p Ft(:)p Fx(006354647)q(833)q(25)q(5)h Fu(\001)d Fx(10)1640 297 y FF(\000)p FE(24)1722 318 y Ft(;)297 391 y(G)335 398 y FE(1)369 391 y Fu(\024)j Fx(1)p Ft(:)p Fx(3686567169)q(49)q(379)h Fu(\001)c Fx(10)910 370 y FF(\000)p FE(25)991 391 y Ft(;)1072 378 y Fx(~)1062 391 y Ft(T)1091 398 y FE(1)1124 391 y Fu(\024)j Fx(0)p Ft(:)p Fx(8346046752)q(338)q(53)20 b Ft(:)46 485 y Fx(After)15 b(the)h(second)h(iteration:)297 580 y Ft(U)330 587 y FE(2)364 580 y Fu(\024)d Fx(1)p Ft(:)p Fx(30981759332)q(27)q(18)g Fu(\001)d Fx(10)904 559 y FF(\000)p FE(5)969 580 y Ft(;)56 b(V)1067 587 y FE(2)1101 580 y Fu(\024)14 b Fx(1)p Ft(:)p Fx(4562203600)q(617)q(7)19 b Ft(;)297 652 y(M)344 659 y FE(2)378 652 y Fu(\024)14 b Fx(1)p Ft(:)p Fx(00002126646)q(58)q(2)19 b Ft(;)p 897 613 V 57 w(M)949 659 y FE(2)983 652 y Fu(\024)13 b Fx(1)p Ft(:)p Fx(000021266)q(91)q(809)20 b Ft(;)305 712 y Fx(~)297 725 y Ft(V)325 732 y FE(2)359 725 y Fu(\024)14 b Fx(7)p Ft(:)p Fx(4535834260)q(514)q(43)h Fu(\001)c Fx(10)900 704 y FF(\000)p FE(7)964 725 y Ft(;)56 b(F)1066 732 y FE(2)1100 725 y Fu(\024)13 b Fx(1)p Ft(:)p Fx(209224397)q(282)q(49)q(1) h Fu(\001)d Fx(10)1640 704 y FF(\000)p FE(31)1722 725 y Ft(;)297 798 y(G)335 805 y FE(2)369 798 y Fu(\024)j Fx(2)p Ft(:)p Fx(3012337757)q(74)q(239)h Fu(\001)c Fx(10)910 777 y FF(\000)p FE(33)991 798 y Ft(;)1072 785 y Fx(~)1062 798 y Ft(T)1091 805 y FE(2)1124 798 y Fu(\024)j Fx(0)p Ft(:)p Fx(8346046752)q(338)q(62)20 b Ft(:)46 965 y Fx(After)15 b(the)h(third)g(iteration:)297 1059 y Ft(U)330 1066 y FE(3)364 1059 y Fu(\024)e Fx(1)p Ft(:)p Fx(30981759332)q(27)q(18)g Fu(\001)d Fx(10)904 1039 y FF(\000)p FE(5)969 1059 y Ft(;)56 b(V)1067 1066 y FE(3)1101 1059 y Fu(\024)14 b Fx(1)p Ft(:)p Fx(4562203600)q(617)q(7)19 b Ft(;)297 1132 y(M)344 1139 y FE(3)378 1132 y Fu(\024)14 b Fx(1)p Ft(:)p Fx(00002126646)q(58)q(2)19 b Ft(;)p 897 1093 V 57 w(M)949 1139 y FE(3)983 1132 y Fu(\024)13 b Fx(1)p Ft(:)p Fx(000021266)q(91)q (809)20 b Ft(;)305 1192 y Fx(~)297 1205 y Ft(V)325 1212 y FE(3)359 1205 y Fu(\024)14 b Fx(7)p Ft(:)p Fx(4535834260)q(514)q(48)h Fu(\001)c Fx(10)900 1184 y FF(\000)p FE(7)964 1205 y Ft(;)56 b(F)1066 1212 y FE(3)1100 1205 y Fu(\024)13 b Fx(8)p Ft(:)p Fx(855523608)q(162)q(04)q(2)h Fu(\001)d Fx(10)1640 1184 y FF(\000)p FE(44)1722 1205 y Ft(;)297 1277 y(G)335 1284 y FE(3)369 1277 y Fu(\024)j Fx(3)p Ft(:)p Fx(1747327327)q(16)q(713)h Fu(\001)c Fx(10)910 1257 y FF(\000)p FE(46)991 1277 y Ft(;)1072 1265 y Fx(~)1062 1277 y Ft(T)1091 1284 y FE(3)1124 1277 y Fu(\024)j Fx(0)p Ft(:)p Fx(8346046752)q(338)q(62)20 b Ft(:)46 1372 y Fx(The)d(quan)o (tities)e(\012)407 1379 y FE(1)427 1372 y Fx(,...,)p Ft(\021)i Fx(de\014ned)f(in)h(Theorem)e(4.2)i(\(recalling)f(Theorem)g (4.1\))h(are)g(imm)o(ediatel)o(y)46 1432 y(computed)e(using)h(the)g(ab) q(o)o(v)o(e)h(v)m(alues)f(and)h(one)f(obtains)359 1527 y Ft(\013)e Fu(\024)g Fx(8853099925)q(5)p Ft(:)p Fx(18)q(87)20 b Ft(;)56 b(\021)16 b Fu(\024)e Fx(1)p Ft(:)p Fx(18566843620)q(726)q(9) g Fu(\001)d Fx(10)1503 1506 y FF(\000)p FE(38)1585 1527 y Ft(:)46 1621 y Fx(With)16 b(suc)o(h)g(v)m(alues)g(condition)g (\(4.31\))h(is)f(satis\014ed,)g(in)g(fact)g(w)o(e)g(obtained)1441 1603 y FE(46)267 1723 y Ft(\021)r(\013)h(M)393 1702 y FE(7)p 413 1684 V 413 1723 a Ft(M)465 1694 y FE(9)485 1723 y Ft(\030)508 1702 y FF(\000)p FE(2\(2)p Fk(\034)t FE(+1\))665 1723 y Fx(2)689 1702 y FE(16)p Fk(\034)t FE(+23)826 1723 y Ft(\034)6 b Fx(!)867 1702 y FE(4)900 1723 y Fu(\024)13 b Fx(3)p Ft(:)p Fx(584973875)q(295)q(102)i Fu(\001)c Fx(10)1440 1702 y FF(\000)p FE(8)1502 1723 y Ft(<)i Fx(1)k Ft(;)46 1818 y Fx(so)g(that)f(Theorem)f(1.1)i(holds.)p 696 1818 21 42 v 46 2023 a FB(A)83 b(Some)25 b(computer{ass)o(isted)f (data)46 2132 y Fx(W)l(e)11 b(\014rst)g(rep)q(ort)g(the)g Fv(trigonometric)e(degrees)i Ft(\027)929 2139 y Fk(j)948 2132 y Fx(,)f Ft(\027)999 2114 y FF(0)996 2144 y Fk(j)1026 2132 y Fx(app)q(earing)i(in)e(the)h(F)l(ourier{T)l(a)o(ylor)g (expansion)46 2198 y(of)19 b(the)g(appro)o(ximate)e(solution)j(\()p Ft(u)710 2180 y FE(\(0\))757 2198 y Ft(;)8 b(v)805 2180 y FE(\(0\))851 2198 y Fx(\))19 b(\(obtained)g(as)h(the)e(\014fth)h Ft(")p Fx({order)h(truncation)f(of)g(the)46 2258 y(formal)c(solution\)) 213 2389 y Ft(u)241 2368 y FE(\(0\))302 2389 y Fx(=)376 2335 y FE(5)354 2347 y Fp(X)354 2438 y Fk(j)r FE(=1)423 2389 y Ft(")446 2368 y Fk(j)516 2347 y Fp(X)473 2441 y FE(0)p Fk(<)p FF(j)p Fk(n)p FF(j\024)p Fk(\027)603 2446 y Fo(j)631 2389 y Fx(~)-27 b Ft(u)656 2368 y FE(\()p Fk(j)r FE(\))656 2401 y Fk(n)710 2389 y Fx(exp)8 b Ft(in)j Fu(\001)g Ft(\022)17 b(;)105 b(v)1060 2368 y FE(\(0\))1121 2389 y Fx(=)1194 2335 y FE(5)1173 2347 y Fp(X)1172 2438 y Fk(j)r FE(=0)1242 2389 y Ft(")1265 2368 y Fk(j)1335 2347 y Fp(X)1291 2442 y FE(0)p Fk(<)p FF(j)p Fk(n)p FF(j\024)p Fk(\027)1423 2430 y Fn(0)1421 2453 y Fo(j)1449 2389 y Fx(~)-26 b Ft(v)1473 2368 y FE(\()p Fk(j)r FE(\))1471 2401 y Fk(n)1526 2389 y Fx(exp)8 b Ft(in)j Fu(\001)g Ft(\022)17 b(:)46 2540 y Fx(W)l(e)f(giv)o(e)f(the)h(result)g(b)o(y)g (comp)q(onen)o(ts:)f(for)k(~)-27 b Ft(u)902 2522 y FE(\()p Fk(j)r FE(\))964 2540 y Fx(w)o(e)16 b(found:)p 46 2579 747 2 v 85 2612 a Fs(46)121 2627 y FI(According)e(to)g(the)g(ab)q(o)o (v)o(e)g(con)o(v)o(en)o(tion)f Fz(M)19 b FI(and)p 905 2593 45 2 v 14 w Fz(M)f FI(are,)c(in)g(fact,)f(the)h(v)n(alues)g Fz(M)1421 2633 y Fs(3)1454 2627 y FI(and)p 1534 2593 V 13 w Fz(M)1579 2633 y Fs(3)1598 2627 y FI(.)955 2863 y Fx(38)p eop %%Page: 39 39 39 38 bop 46 79 a Fx(Order)16 b Ft(j)117 b Fx(F)l(ourier)15 b(deg.)h(of)k(~)-27 b Ft(u)679 53 y FE(\()p Fk(j)r FE(\))679 89 y(1)773 79 y Fx(F)l(ourier)16 b(deg.)g(of)j(~)-27 b Ft(u)1128 53 y FE(\()p Fk(j)r FE(\))1128 89 y(2)46 139 y Fx(1)254 b(5)425 b(5)46 199 y(2)254 b(15)401 b(15)46 259 y(3)254 b(25)401 b(25)46 319 y(4)254 b(35)401 b(35)46 379 y(5)254 b(45)401 b(45)46 546 y(F)l(or)18 b(~)-26 b Ft(v)159 528 y FE(\()p Fk(j)r FE(\))204 546 y Fx(:)46 661 y(Order)16 b Ft(j)117 b Fx(F)l(ourier)15 b(deg.)h(of)j(~)-26 b Ft(v)677 635 y FE(\()p Fk(j)r FE(\))675 671 y(1)771 661 y Fx(F)l(ourier)15 b(deg.)h(of)j(~)-26 b Ft(v)1124 635 y FE(\()p Fk(j)r FE(\))1122 671 y(2)46 721 y Fx(0)254 b(0)423 b(0)46 781 y(1)254 b(5)423 b(5)46 841 y(2)254 b(14)399 b(11)46 901 y(3)254 b(24)399 b(21)46 961 y(4)254 b(34)399 b(31)46 1022 y(5)254 b(44)399 b(41)46 1201 y(The)22 b Fv(n)o(um)o(b)q(er)f(of)i(F)l(ourier)e(comp)q(onen)o(ts)h Fx(of)h Ft(u)937 1183 y FE(\(0\))1006 1201 y Fx(and)g(of)g Ft(v)1195 1183 y FE(\(0\))1264 1201 y Fx(is)f(giv)o(en)f(for)i(eac)o(h) f(order)h(of)f(the)46 1261 y(T)l(a)o(ylor)17 b(series)h(expansion)g(in) f(p)q(o)o(w)o(ers)h(of)g(the)g(p)q(erturbing)g(parameter)e Ft(")p Fx(.)h(W)l(e)h(rep)q(ort)g(only)g(those)46 1321 y(comp)q(onen)o(ts)d(with)h(F)l(ourier)g(index)g(\()p Ft(n;)8 b(m)p Fx(\))15 b(with)i Ft(n)c(>)h Fx(0)j(or)f Ft(n)e Fx(=)g(0)j(and)g Ft(m)c Fu(\025)h Fx(0.)46 1407 y(F)l(or)i Ft(u)161 1388 y FE(\(0\))224 1407 y Fx(w)o(e)g(found:)46 1521 y(Order)g Ft(j)117 b Fx(F)l(ourier)15 b(co)q(e\013.)i(of)i(~)-27 b Ft(u)703 1495 y FE(\()p Fk(j)r FE(\))703 1532 y(1)797 1521 y Fx(F)l(ourier)16 b(co)q(e\013.)g(of)k(~)-27 b Ft(u)1177 1495 y FE(\()p Fk(j)r FE(\))1177 1532 y(2)46 1581 y Fx(0)254 b(0)449 b(0)46 1641 y(1)254 b(5)449 b(5)46 1701 y(2)254 b(35)425 b(35)46 1761 y(3)254 b(93)425 b(93)46 1822 y(4)254 b(179)401 b(179)46 1882 y(5)254 b(293)401 b(293)46 2086 y(F)l(or)16 b Ft(v)159 2068 y FE(\(0\))222 2086 y Fx(w)o(e)g(found:)46 2200 y(Order)g Ft(j)117 b Fx(F)l(ourier)15 b(co)q(e\013.)i(of)h(~)-26 b Ft(v)701 2175 y FE(\()p Fk(j)r FE(\))699 2211 y(1)795 2200 y Fx(F)l(ourier)16 b(co)q(e\013.)g(of)i(~)-25 b Ft(v)1173 2175 y FE(\()p Fk(j)r FE(\))1171 2211 y(2)46 2260 y Fx(0)254 b(6)447 b(5)46 2321 y(1)254 b(34)423 b(26)46 2381 y(2)254 b(92)423 b(76)46 2441 y(3)254 b(178)399 b(156)46 2501 y(4)254 b(292)399 b(264)46 2561 y(5)254 b(436)399 b(400)955 2863 y(39)p eop %%Page: 40 40 40 39 bop 46 79 a Fx(Finally)l(,)14 b(w)o(e)h(rep)q(ort)h(no)o(w)g(the) f(in)o(terv)m(als)g(con)o(taining)h(the)f(\014v)o(e)g(comp)q(onen)o(ts) g(of)h(the)f(function)k(~)-27 b Ft(u)1852 53 y FE(\(1\))1852 89 y(1)1898 79 y Fx(.)46 139 y(F)l(or)16 b(the)g(initial)f(data)i(\()p Ft(L)521 146 y FE(+)551 139 y Ft(;)8 b(G)611 146 y FE(+)641 139 y Fx(\))16 b(w)o(e)g(obtained)932 121 y FE(47)969 139 y Fx(:)407 245 y(0)p Ft(:)p Fx(6730562643)q(92)q(395)q(519)q(99)q (654)q(49)q(172)q(78)q(0)e(+)d([58)p Ft(;)d Fx(87])k Fu(\001)e Fx(10)1491 224 y FF(\000)p FE(33)1573 245 y Ft(;)407 317 y Fx(0)p Ft(:)p Fx(2979062447)q(30)q(211)q(719)q(15)q(265) q(26)q(718)q(57)q(1)k(+)d([77)p Ft(;)d Fx(85])k Fu(\001)e Fx(10)1491 297 y FF(\000)p FE(33)1573 317 y Ft(;)407 390 y Fx(1)p Ft(:)p Fx(4829465980)q(95)q(398)q(011)q(28)q(556)q(36)q (917)q(81)15 b(+)c([19)p Ft(;)d Fx(25])j Fu(\001)g Fx(10)1467 369 y FF(\000)p FE(32)1549 390 y Ft(;)407 462 y Fu(\000)p Fx(0)p Ft(:)p Fx(4589344174)q(885)q(537)q(36)q(667)q(97)q(797)q(22)q (512)k(+)c([53)p Ft(;)d Fx(41])j Fu(\001)g Fx(10)1530 442 y FF(\000)p FE(33)1612 462 y Ft(;)407 535 y Fu(\000)p Fx(0)p Ft(:)p Fx(2010745230)q(469)q(404)q(09)q(405)q(38)q(366)q(35)q (164)k(+)c([22)p Ft(;)d Fx(14])j Fu(\001)g Fx(10)1530 515 y FF(\000)p FE(33)1612 535 y Ft(:)46 641 y Fx(F)l(or)16 b(the)g(initial)f(data)i(\()p Ft(L)521 648 y FF(\000)551 641 y Ft(;)8 b(G)611 648 y FF(\000)641 641 y Fx(\))16 b(w)o(e)g(obtained:)407 747 y(0)p Ft(:)p Fx(6766043577)q(88)q(015)q (853)q(96)q(564)q(02)q(750)q(18)q(3)e(+)d([17)p Ft(;)d Fx(48])k Fu(\001)e Fx(10)1491 726 y FF(\000)p FE(33)1573 747 y Ft(;)407 819 y Fx(0)p Ft(:)p Fx(3061845611)q(99)q(587)q(897)q(36) q(747)q(29)q(226)q(74)q(6)k(+)d([45)p Ft(;)d Fx(54])k Fu(\001)e Fx(10)1491 799 y FF(\000)p FE(33)1573 819 y Ft(;)407 892 y Fx(1)p Ft(:)p Fx(4887986478)q(29)q(940)q(896)q(17)q(127) q(11)q(038)q(77)15 b(+)c([83)p Ft(;)d Fx(90])j Fu(\001)g Fx(10)1467 871 y FF(\000)p FE(32)1549 892 y Ft(;)407 965 y Fu(\000)p Fx(0)p Ft(:)p Fx(4659443843)q(426)q(364)q(96)q(164)q (61)q(137)q(72)q(143)k(+)c([22)p Ft(;)d Fx(11])j Fu(\001)g Fx(10)1530 944 y FF(\000)p FE(33)1612 965 y Ft(;)407 1037 y Fu(\000)p Fx(0)p Ft(:)p Fx(2018899370)q(701)q(671)q(72)q(193)q (31)q(533)q(71)q(788)k(+)c([70)p Ft(;)d Fx(62])j Fu(\001)g Fx(10)1530 1017 y FF(\000)p FE(33)1612 1037 y Ft(:)46 1203 y FB(References)46 1313 y Fx([1])24 b(Arnold)17 b(V.)g(I.)f(\(editor\):)h Fv(Encyclopaedia)g(of)h(Mathematical)e (Sciences,)g Fx(Dynamical)f(Systems)122 1373 y(I)q(I)q(I,)g(Springer{V) l(erlag)h Fw(3)g Fx(\(1988\).)46 1473 y([2])24 b(Arnold)c(V.)f(I.:)g Fv(Small)f(divisor)i(problems)e(in)i(classical)f(and)i(Celestial)e(Mec) o(hanics)p Fx(,)f(Russian)122 1534 y(Math.)e(Surv)o(ey)f Fw(18)p Fx(,)h(85{191)i(\(1963\))46 1634 y([3])24 b(Celletti)14 b(A.:)g Fv(Analysis)h(of)h(resonances)g(in)g(the)f(spin{orbit)i (problem)d(in)h(Celestial)g(Mec)o(hanics:)122 1694 y(The)f(sync)o (hronous)g(resonance)h(\(P)o(art)f(I\),)f Fx(J.)g(of)i(Applied)d(Math.) i(and)h(Ph)o(ys.)e(\(ZAMP\))g Fw(41)p Fx(,)h(174)122 1754 y(\(1990\))46 1855 y([4])24 b(Celletti)13 b(A.,)h(Chierc)o(hia)h (L.:)g Fv(Construction)h(of)f(Analytic)f(KAM)h(Surfaces)g(and)h (E\013ectiv)o(e)e(Sta-)122 1915 y(bilit)o(y)g(Bounds)p Fx(,)i(Comm)o(un.)e(Math.)i(Ph)o(ys.)g Fw(118)p Fx(,)f(119{161)k (\(1988\))46 2016 y([5])24 b(Celletti)13 b(A.,)h(Chierc)o(hia)g(L.:)g Fv(A)h(Constructiv)o(e)f(Theory)h(of)h(Lagrangian)h(T)l(ori)e(and)h (Computer{)122 2076 y(assisted)22 b(Applications)p Fx(,)e(Dynamics)g (Rep)q(orted)h(\(C.K.R.T.)f(Jones,)h(U.)g(Kirc)o(hgrab)q(er,)f(H.O.)122 2136 y(W)l(alther)c(Managing)h(Editors\),)f(V)l(ol.)f(4)i(\(New)f (Series\),)f(Springer{V)l(erlag)h(\(1995\))46 2237 y([6])24 b(Delauna)o(y)18 b(C.,)g Fv(Th)o(\023)-23 b(eorie)18 b(du)g(Mouv)o(emen)o(t)e(de)i(la)g(Lune)p Fx(,)h(M)o(\023)-23 b(emoires)16 b(de)i(l'Acad)o(\023)-23 b(emie)15 b(des)j(Sci-)122 2297 y(ences)d Fw(1)p Fx(,)h(T)l(ome)f(XXVI)q(I)q(I,)f(P)o(aris)j (\(1860\))p 46 2342 747 2 v 85 2373 a Fs(47)121 2388 y FI(The)d(notation)f Fz(x)c FI(+)g Fz(\016)r FI([)p Fz(a;)e(b)p FI(],)12 b(with)i Fz(\016)f(>)f FI(0)i(and)f Fz(a)f(<)g(b)p FI(,)h(means)g([)p Fz(x)8 b FI(+)i Fz(\016)r(a;)d(x)h FI(+)i Fz(\016)r(b)p FI(]:)i(for)i(example,)270 2476 y(0)p Fz(:)p FI(67305626439)o(2395)o(5199)o(9654)o(4917)o(2780)6 b(+)j([58)p Fz(;)e FI(87])g Fr(\001)i FI(10)1193 2459 y Fy(\000)p Fs(33)1265 2476 y FI(=)270 2538 y([0)p Fz(:)p FI(673056264)o(3923)o(9551)o(9996)o(5449)o(17278)o(058)p Fz(;)d FI(0)p Fz(:)o FI(6730)o(5626)o(4392)o(3955)o(1999)o(6544)o(9172) o(7808)o(7])11 b Fz(:)955 2863 y Fx(40)p eop %%Page: 41 41 41 40 bop 46 78 a Fx([7])24 b(Ec)o(kmann)18 b(J.{P)l(.,)h(Witt)o(w)o (er)f(P)l(.:)h Fv(Computer)g(metho)q(ds)g(and)i(Borel)d(sommabilit)o(y) e(applied)k(to)122 138 y(F)l(eigen)o(baum's)14 b(equation)p Fx(,)i(Springer)g(Lecture)f(Notes)i(in)e(Ph)o(ysics)h Fw(227)g Fx(\(1985\))46 240 y([8])24 b(H)o(\023)-23 b(enon)14 b(M.:)f Fv(Explorationes)h(n)o(um)o(\023)-23 b(erique)11 b(du)k(probl)o(\022)-23 b(eme)12 b(restrein)o(t)h(IV:)g(Masses)h (egales,)g(orbites)122 300 y(non)j(p)q(erio)q(dique)p Fx(,)e(Bullettin)f(Astronomique)h Fw(3)p Fx(,)h(no.)g(1,)g(fasc.)g(2,)g (49{66)i(\(1966\))46 402 y([9])24 b(John)17 b(F.:)e Fv(P)o(artial)h (Di\013eren)o(tial)f(Equations)p Fx(,)h(Springer{V)l(erlag)g(\(1982\)) 46 503 y([10])24 b(Ko)q(c)o(h)15 b(H.,)f(Sc)o(henk)o(er)g(A.,)f(Witt)o (w)o(er)h(P)l(.:)h Fv(Computer{Assisted)f(Pro)q(ofs)i(in)f(Analysis)f (and)i(Pro-)122 564 y(gramming)e(in)i(Logic:)g(A)g(Case)h(Study)p Fx(,)f(Preprin)o(t)f(Univ)o(ersit)o(\023)-23 b(e)14 b(de)i(Gen)o(\022) -23 b(ev)o(e)15 b(\(1994\))46 665 y([11])24 b(Mey)o(er)17 b(K.R.,)g(Sc)o(hmidt)g(D.)h(\(editors\):)g Fv(Computer)f(Aided)h(Pro)q (ofs)i(in)e(Analysis)p Fx(,)g(Springer{)122 726 y(V)l(erlag)d(\(1991\)) 46 827 y([12])24 b(Moser)16 b(J.:)f Fv(Stable)h(and)g(random)g(motions) f(in)h(dynamical)e(systems,)g(with)i(sp)q(ecial)f(emphasis)122 887 y(on)h(Celestial)g(Mec)o(hanics)p Fx(,)e(Ann.)i(Math.)g(Studies,)f Fw(77)h Fx(Princeton)g(Univ)o(ersit)o(y)e(Press)j(\(1973\))46 989 y([13])24 b(P)o(oincar)o(\022)-23 b(e)19 b(H.:)e Fv(Les)j(Metho)q(des)f(Nouv)o(elles)e(de)i(la)g(Mec)o(hanique)e (Celeste)p Fx(,)h(Gauthier)h(Villars,)122 1049 y(P)o(aris)d(\(1892\))46 1151 y([14])24 b(Robutel)c(P)l(.:)g Fv(Stabilit)o(y)e(of)j(the)f (planetary)g(three{b)q(o)q(dy)h(problem.)d(I)q(I.)i(KAM)f(Theory)h(and) 122 1211 y(existence)14 b(of)j(quasi{p)q(erio)q(dic)f(motions)p Fx(,)f(Celestial)g(Mec)o(hanics)g Fw(62)p Fx(,)h(219{261)j(\(1995\))46 1313 y([15])24 b(Salamon)f(D.,)g(Zehnder)f(E.:)h Fv(KAM)g(theory)g(in)g (con\014guration)h(space)p Fx(,)f(Commen)o(t.)d(Math.)122 1373 y(Helv)o(eticae)13 b Fw(64)j Fx(84)h(\(1989\))46 1475 y([16])24 b(\(no)13 b(author)g(listed\))e Fv(The)h(Astronomical)e (Almanac)p Fx(,)g(W)l(ashington:)j(U.S.)e(Go)o(v)o(ernmen)o(t)e(Prin)o (t-)122 1535 y(ing)16 b(O\016ce)f(\(1990\))46 1637 y([17])24 b(\(no)j(author)g(listed\))f Fv(V)l(ax)g(Arc)o(hitecture)e(handb)q(o)q (ok,)k Fx(Digital)e(Equipmen)o(t)e(Corp)q(oration)122 1697 y(\(1981\))955 2863 y(41)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF