%!PS-Adobe-2.0 %%Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %%Title: CMP.DVI %%Pages: 24 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: "DVIPS CMP.DVI/PRINTER=LASER_PS1" %DVIPSParameters: dpi=300, comments removed %DVIPSSource: TeX output 1997.03.03:1436 %%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 round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /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 sub]{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 userdict /eop-hook known{eop-hook}if showpage}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /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 newpath 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 39158280 55380996 1200 300 300 (VMDIPD$DKA200:[COMUNE]CMP.DVI) @start /Fa 3 49 df0 DI<3C3C3C3C7878787870F0F0E0E0060D7E8D0B>48 D E /Fb 12 115 df23 D<03F007F80E381C1C1C3C3C38383838787CF07FE07F807000F000F000E000E0000E 107E8A12>26 D<3FFE7FFEFFFCC380030007000700070007000E000E000F0B7E8A11>28 D<18038038038070038060E38060E380E1E300E1C700F3CF00FFFE007FFC003EF800110B 7F8A16>33 D<00C000C001C0018001800380030003000700060006000E000C001C001800 1800380030003000700060006000E000C000C0000A197D9212>61 D<0FE0FF0FE0FF03F03803F030037830037830073C70073C60061E60061E600E0FE00E0F C00C07C00C07C01C03C07F0380FF018018117E901B>78 D<0FFF800FFFE003C0F003C070 0380700380700781E007FF8007FF800707800F03C00F03800E03800E03801E039C7F83F8 FF81F016117E901A>82 D<007C007C003C003C003800380FF81FF838F0707070F0F0F0E0 E0E1ECF3FC7FF83EF80E117E9012>100 D<070F06000000003C7EFEDC1C3C387B777E3C 08127E910D>105 D<3E003E001E001E001C001C003C783DF83BF83F387F387F8073C071 CCF1DCF1F8E0F00E117E9013>107 D<7C7C3C3C383878787070F0F0E0ECFCF8F806117E 900C>I<7BE0FFF0DE70DC703C703C00380038007800780070000C0B7E8A11>114 D E /Fc 3 91 df<003FE06001FFFCE007FFFFE00FF01FE01FC007E03F0003E07F0001E0 7E0001E0FE0000E0FC0000E0FC000000FC000000FC000000FC000000FC0000E0FE0000E0 7E0001E07F0001E03F0003C01FC007C00FF01F8007FFFF0001FFFC00003FF0001B187E97 20>67 D<7FFFFF807FFFFF807FFFFF80783F0780703F0380F03F03C0E03F01C0E03F01C0 E03F01C0003F0000003F0000003F0000003F0000003F0000003F0000003F0000003F0000 003F0000003F0000003F0000003F000007FFF80007FFF80007FFF8001A187E971F>84 D<7FFFFE7FFFFE7FFFFE7F01FC7C03F87807F8700FF0701FE0701FC0003F80007F0000FF 0001FE0001FC0E03F80E07F00E0FF01E1FE01E3FC03C3F807C7F01FCFFFFFCFFFFFCFFFF FC17187E971C>90 D E /Fd 8 62 df<0E1E1C3838707060E0E0E0E0E0E0E0E0E0E06070 7038381C1E0E071A7D920D>40 DI<00600000600000600000600000600000600000600000600000 6000006000FFFFF0FFFFF000600000600000600000600000600000600000600000600000 600000600014167D901A>43 D<06003E00FE00CE000E000E000E000E000E000E000E000E 000E000E00FFE0FFE00B107D8F11>49 D<3FC07FF0E0F0E078E03800380078007000E003 C007801E183C187FF0FFF0FFF00D107E8F11>I<1FC07FE070F07070207001F00FE00FC0 00F000780038E038E078E0F0FFE03FC00D107E8F11>I<00E001E003E007E00EE01CE038 E070E0E0E0FFF8FFF800E000E000E007F807F80D107E8F11>I61 D E /Fe 14 107 df0 D<70F8F8F87005057C8B0C>I<03000380030063 18F33C7FF81FE007801FE07FF8F33C63180300038003000E0F7D8F14>3 D<00000600001E00007C0001F00007C0001F00007C0001F00007C0001F00007C0000F000 00F000007C00001F000007C00001F000007C00001F000007C00001F000007C00001E0000 06000000000000000000000000000000000000FFFFFEFFFFFE17207C971F>20 DI<0F1F1F1F 3F3E3E3E3C7C78787870F0F0E06008127E920B>48 D<07F001F8001FFC07FE003FFF1F8F 00783FBE0380E01FF801C0E00FF001C0C007E000C0C003F000C0C001F800C0E003FC01C0 E007FE01C0701F7F07803C7E3FFF001FF80FFE0007E003F800220F7D8E28>I<00FFE007 FFE00F80001C0000380000300000700000600000E00000C00000C00000FFFFE0FFFFE0C0 0000C00000E000006000007000003000003800001C00000F800007FFE000FFE013187C94 1B>I<0303F00F8FF81F3FFC3F787C0FE03C0FC03C0FC0380F80F00F03E00F1F801F7FE0 1EFFF01E03F01E00F81C00783C00783C00783800707800F07800E07703C0FFFF80EFFE00 C7F00016187D971A>66 D<000FE0007FF001FFF003C0F00700F00E00E01C01C03C038038 0200780000700020F000E0F001E0F001E0F003C0F003C0F007C0F80F807C3F807FFF803F E7001F8F00000E00001C003FF8007FF0007FC000141B7D9718>71 D<000E00000200001E00000600001E00000E00001E00001E00001F00003E00003F00007E 00003F0000FC00003F8001FC0000378003FC0000378007BC000077800F3C000067C01E3C 000063C03C380000E3C038780000C3E070780000C1E1E0780001C1F3E078000180F7C078 000380FF00780003007E007800E7007C007800FE0038007F80FE0010007F00FC0000003C 0078000000000029197D972F>77 D<00FFFC0007FFFF001FFFFF803C381F80707807C0F0 7803C0E07803C0C07803C000780380007807800070070000F00E0000F01C0000F0780000 EFF00001FFC00001FE000001C0000003C0000003C000000380000007800000070000000F 0000000E000000080000001A1A7E971C>80 D<00700000700000F80000D80001DC00018C 00038E000306000707000603000E03800C01801C01C01800C03800E03000607000706000 30E00038C0001815147D931B>94 D106 D E /Ff 31 119 df<000800100020004000C0018003000300060006000C000C001C00180038003800 300070007000700070006000E000E000E000E000E000E000E000E000E000E000E000E000 E000E000E0006000700070007000700030003800380018001C000C000C00060006000300 0300018000C000400020001000080D3B798117>0 D<800040002000100018000C000600 0600030003000180018001C000C000E000E0006000700070007000700030003800380038 003800380038003800380038003800380038003800380038003000700070007000700060 00E000E000C001C00180018003000300060006000C00180010002000400080000D3B7E81 17>II I<0003000C0038007000E000C001C001C001C001C001C001C001C001C001C001C001C001 C001C001C001C001C001C001C00180038007000E003800C00038000E0007000380018001 C001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C001C000 C000E000700038000C0003103B7A811D>8 DI< C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C0C002207980 11>12 DI<0000600000C0000180000300000600000E00001C00001800003800007000006000 00E00001C00001C0000380000380000780000700000700000F00000E00001E00001E0000 1C00001C00003C00003C0000380000380000780000780000780000780000780000700000 F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000 F00000F00000F00000F00000F00000F00000F00000700000780000780000780000780000 7800003800003800003C00003C00001C00001C00001E00001E00000E00000F0000070000 07000007800003800003800001C00001C00000E000006000007000003800001800001C00 000E000006000003000001800000C0000060135977811E>16 DI<00000180000003000000060000000C000000180000003000 000060000000E0000001C000000180000003800000070000000E0000000E0000001C0000 001C000000380000007800000070000000F0000000E0000001E0000001C0000001C00000 03C00000038000000780000007800000070000000F0000000F0000000E0000001E000000 1E0000001E0000001C0000003C0000003C0000003C0000003C0000003800000078000000 78000000780000007800000078000000780000007800000070000000F0000000F0000000 F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000 F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000 F00000007000000078000000780000007800000078000000780000007800000078000000 380000003C0000003C0000003C0000003C0000001C0000001E0000001E0000001E000000 0E0000000F0000000F0000000700000007800000078000000380000003C0000001C00000 01C0000001E0000000E0000000F00000007000000078000000380000001C0000001C0000 000E0000000E00000007000000038000000180000001C0000000E0000000600000003000 0000180000000C000000060000000300000001801977768125>II<00000700001F00003C0000F80001E00003C0000780000780000F00001E00 001E00001E00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00 003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00 003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00 003C00003C0000780000780000700000F00000E00001C00003C0000780000E00003C0000 780000E000007800003C00000E000007800003C00001C00000E00000F000007000007800 007800003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00 003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00 003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00003C00 003C00001E00001E00001E00000F000007800007800003C00001E00000F800003C00001F 00000718777A8125>26 DI34 DI<0000003000000060000000E0000001C000000180 000003800000070000000E0000000E0000001C0000003C00000038000000700000007000 0000E0000001E0000001C0000003C0000003C0000007800000078000000F0000000F0000 001E0000001E0000001E0000003C0000003C0000007C0000007800000078000000F80000 00F0000000F0000001F0000001F0000001E0000003E0000003E0000003E0000007C00000 07C0000007C0000007C000000FC000000F8000000F8000000F8000000F8000001F800000 1F8000001F0000001F0000001F0000003F0000003F0000003F0000003F0000003E000000 3E0000003E0000003E0000007E0000007E0000007E0000007E0000007E0000007E000000 7E0000007E0000007C0000007C0000007C000000FC000000FC000000FC000000FC000000 FC000000FC000000FC000000FC000000FC000000FC000000FC000000FC000000FC000000 FC000000FC000000FC000000FC0000001C5A72802C>48 DI64 D<000003F0000003F0000003F0000003F0000003F0000003F0000003F0000003F000 0003F0000003F0000003F0000003F0000003F0000003F0000003F0000003F0000003F000 0003E0000003E0000003E0000007E0000007E0000007E0000007E0000007E0000007E000 0007E0000007E0000007C0000007C0000007C0000007C000000FC000000FC000000FC000 000FC000000F8000000F8000000F8000001F8000001F8000001F0000001F0000001F0000 001F0000003F0000003E0000003E0000003E0000003E0000007C0000007C0000007C0000 0078000000F8000000F8000000F0000000F0000001F0000001E0000001E0000003E00000 03C0000003C000000780000007800000078000000F0000000F0000001E0000001E000000 3C0000003C000000380000007800000070000000E0000000E0000001C0000003C0000003 80000007000000070000000E0000001C00000018000000380000007000000060000000C0 0000001C5A7F822C>I80 D88 D<000000000F00000000 0018800000000039C00000000033E00000000073E00000000063E000000000E1C0000000 00E00000000000E00000000001C00000000001C00000000001C00000000003C000000000 03C000000000038000000000078000000000078000000000078000000000070000000000 0F00000000000F00000000000F00000000000F00000000001F00000000001E0000000000 1E00000000001E00000000003E00000000003E00000000003C00000000003C0000000000 7C00000000007C00000000007C00000000007C0000000000FC0000000000F80000000000 F80000000000F80000000001F80000000001F80000000001F80000000001F00000000001 F00000000003F00000000003F00000000003F00000000003F00000000007E00000000007 E00000000007E00000000007E00000000007E0000000000FC0000000000FC0000000000F C0000000000FC0000000000FC0000000001F80000000001F80000000001F80000000001F 80000000001F80000000003F00000000003F00000000003F00000000003F00000000003E 00000000003E00000000007E00000000007E00000000007E00000000007C00000000007C 00000000007C0000000000FC0000000000F80000000000F80000000000F80000000000F8 0000000000F00000000000F00000000001F00000000001F00000000001E00000000001E0 0000000001E00000000003E00000000003C00000000003C00000000003C00000000003C0 000000000380000000000780000000000780000000000780000000000700000000000F00 000000000F00000000000E00000000000E00000000001E00000000001C00000000001C00 000000703800000000F83800000000F87000000000F8600000000070C000000000218000 0000001F00000000002B6F7D7F1C>90 D<600000000006F0000000000FF0000000000FF0 000000000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF0 000000000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF0 000000000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF0 000000000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF0 000000000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF0 000000000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF0 000000000FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF0 000000000FF0000000000FF0000000000FF0000000000FF0000000000FF8000000001F78 000000001E78000000001E78000000001E7C000000003E3C000000003C3E000000007C1F 00000000F80F00000000F00F80000001F007C0000003E003E0000007C001F800001F8000 FC00003F00007F0000FE00003FF00FFC00000FFFFFF0000007FFFFE0000001FFFF800000 001FF8000030467D7F37>I104 DI<00000000000400000000000C 000000000018000000000018000000000030000000000030000000000060000000000060 0000000000C00000000000C0000000000180000000000180000000000300000000000300 000000000600000000000600000000000600000000000C00000000000C00000000001800 00000000180000000000300000000000300000000000600000000000600000000000C000 00000000C000000000018000000000018000000000030000020000030000070000060000 0F00000600003780000C00006780000C000087800018000003C00018000003C000300000 01E00030000001E00060000000F00060000000F000C0000000F000C00000007801800000 007801800000003C01800000003C03000000001E03000000001E06000000001E06000000 000F0C000000000F0C0000000007980000000007980000000003F00000000003F0000000 0003E00000000001E00000000001C00000000000C00000002E3C7B8132>112 D<00000000000400000000000C00000000000C0000000000180000000000180000000000 180000000000180000000000300000000000300000000000300000000000300000000000 300000000000600000000000600000000000600000000000600000000000C00000000000 C00000000000C00000000000C00000000001800000000001800000000001800000000001 800000000003000000000003000000000003000000000003000000000006000000000006 00000000000600000000000600000000000600000000000C00000000000C00000000000C 00000000000C000000000018000000000018000000000018000000000018000000000030 000000000030000000000030000000000030000000000060000000000060000000000060 0000000000600000000000C00000000000C00000000000C00000000000C00000000000C0 000000000180000000000180000000000180000000000180000000000300000000000300 000000000300000200000300000700000600000700000600000F00000600001F00000600 001F00000C00002780000C00002780000C00004780000C00008780000C00000780001800 0003C00018000003C00018000003C00018000003C00030000001E00030000001E0003000 0001E00030000001E00060000001E00060000000F00060000000F00060000000F000C000 0000F000C00000007800C00000007800C000000078018000000078018000000078018000 00003C01800000003C01800000003C03000000003C03000000003C03000000001E030000 00001E06000000001E06000000001E06000000000F06000000000F0C000000000F0C0000 00000F0C000000000F0C0000000007980000000007980000000007980000000007980000 000003F00000000003F00000000003F00000000003F00000000003F00000000001E00000 000001E00000000001E00000000001E00000000001C00000000000C00000000000C00000 002E787B8132>114 D<0000000600000006000000060000000600000006000000060000 000600000006000000060000000600000006030000060300000607000006078000060F80 00060F8000061780000613C0000623C0000623C0000641E0000641E0000681E0000600F0 000600F0000600F00006007800060078000600780006003C0006003C0006003C0006001E 0006001E0006001E0006000F0006000F0006000F00060007800600078006000780060007 80060003C0060003C0060003C0060001E0060001E0060001E0060000F0060000F0060000 F00600007806000078060000780600003C0600003C0600003C0600001E0600001E060000 1E0600000F0600000F0600000F0600000786000007860000078600000786000003C60000 03C6000003C6000001E6000001E6000001E6000000F6000000F6000000F60000007E0000 007E0000007E0000003E0000003E0000003E0000001E0000001E0000001E0000000E0000 000E0000000E00000006000000041F5B7B8035>116 DI<7FFFF0FFFFF0C00000C0 0000C00000C00000C00000C00000C00000C00000C00000C00000C00000C00000C00000C0 0000C00000C00000C00000C00000C00000C00000C00000C00000C00000C00000C00000C0 0000C00000C00000C00000141F5E8135>I E /Fg 70 128 df<00001FC0F800007FF3FE 0000F1F7BE0001E1F73E0003C1FF3E0003C1EF3C0003C01F000007C01E000007801E0000 07801E000007801E000007803E00000F803C00000F003C0001FFFFFFE001FFFFFFE0000F 007C00001F007800001E007800001E007800001E007800001E00F800003E00F000003C00 F000003C00F000003C00F000003C01F000007C01E000007801E000007801E000007801E0 00007803E00000F803C00000F003C00000F003C00000F003C00000F007800001E0078000 01E007800079EF0F000079CF0F0000FBDF1E0000FB9F3C00007F0FF800003E07E0000027 2D82A21F>11 D<00000FF000007FFC0000F81E0001E03E0003C03E0003C03C0003C03C00 078000000780000007800000078000000F8000000F0000000F000001FFFFF001FFFFF000 1F01F0001E01E0001E01E0001E01E0001E03E0003E03C0003C03C0003C03C0003C07C000 3C0780007C07800078078C00780F9C00780F1800780F1800F80F3800F00F3000F007F000 F003E000F0000001E0000001E0000001E0000079C0000079C00000FB800000FF8000007F 0000003E0000001F2D82A21C>I<00001FE03FF000007FF8FFFC0000F8F9E01E0001E0FB C03E0001E0FF803E0003C0F7803C0003C00F003C0003C00F00000007C00F00000007801F 00000007801E00000007801E00000007801E0000000F801E000001FFFFFFFFF001FFFFFF FFF0000F003C01F0000F003C01E0001F003C01E0001E003C01E0001E007C03E0001E0078 03C0001E007803C0003E007803C0003C007807C0003C00F80780003C00F00780003C00F0 078C007C00F00F8C007800F00F18007801F00F18007801E00F38007801E00F3000F801E0 07F000F001E003E000F003C0000000F003C0000000F003C0000001E00380000079E78780 000079C787000000FBCF8F000000FB8F9E0000007F07FC0000003E03F00000002F2D82A2 2C>14 D<1F07C03F0FC03F8FE03F8FE03F8FE03F0FC00300C00300C00701C00E03800E03 801C0700380E00701C00E03800C03000131078A21A>34 D<1F003F003F803F803F803F00 0300030007000E000E001C0038007000E000C000091076A20F>39 D<0000C00003C0000780000F00000E00001C0000380000780000700000F00001E00001C0 0003C00003C0000780000780000700000F00000F00001E00001E00001E00001C00003C00 003C00003C0000380000780000780000780000700000700000700000F00000F00000F000 00F00000F00000E00000E00000F000007000007000007000007000003000003800003800 001C00000C000012327AA414>I<000C00000E0000070000070000030000038000038000 03800003800003C00001C00001C00003C00003C00003C00003C00003C000038000038000 0380000780000780000780000700000F00000F00000F00000E00001E00001E00001E0000 3C00003C0000380000780000780000F00000F00000E00001E00003C00003800007800007 00000E00001C00003C0000780000F00000C00000123280A414>I<1E3E3F3F3E3E060E0C 1C18387070E0C008107D850F>44 D<7FF8FFF8FFF8FFF00D047D8B12>I<78FCFCFCF8F0 06067B850F>I<000F80003FE000F0F001E07003C0780380780780780F80780F00781F00 781F00781E00F83E00F83E00F83E00F87C01F07C01F07C01F07C01F0F803E0F803E0F803 E0F803C0F007C0F007C0F00780F00F80F00F00F00E00F01E00F03C007878003FE0000F80 0015227BA019>48 D<000080000180000380000780000F00003F0003FF0003FF00001E00 001E00003E00003E00003C00003C00007C00007C0000780000780000F80000F80000F000 00F00001F00001F00001E00001E00003E00003E00003C00003C00007C000FFFC00FFFC00 11217BA019>I<000FE0003FF000787800F03C01C03C01C03E03983E03183E07183E0618 3E06387E0E307C0C307C0C70FC0CE0F80FC1F80781F00003E0000780000F00003E000078 0001F00003C0000780000E00181C00383800703F80F07FFFE073FFE0E0FFC0C07F80C03F 0017227DA019>I<000FE0003FF000787800E03C01C03C03801C03303C07303C06303C06 303C06707C07E07803C0F80000F00003E0007FC0007F00000F800007C00003C00003C000 03C00007C01007C07C07C07C07C07C0FC0F80F80E00F80601F00703E0078FC003FF0001F C00016227CA019>I<0000780000780000F80000F00000F00000F00001F00001E00001E0 0003E00003C00003C0000780000780000700000F00000E00001E00001C00003C00003800 00738000F7C000E78001C780038780078F800F0F001E0F003F8F007FFF00F0FE60C03FE0 001FC0003E00003C00003C00003C00007C00007800007800007800007000152B7EA019> I<00C01800F07801FFF001FFE001FFC001FF000380000300000300000300000700000600 00067E0007FF000FC7800F07800E03C00C03C00007C00007C00007C00007C0000FC0780F C0F80F80F80F80F81F00F01F00C03E00C03C00E07800F1F0007FE0003F800015227BA019 >I<0007E0001FF0007C7000F07001E1F003E1F003C1F00781E00F80000F00001F00001F 10003EFE003FFF003F8F007F07807E07807C07807C0F80FC0F80F80F80F80F80F81F80F8 1F00F01F00F01F00F03E00F03E00E03C00F07C00F0F80079F0007FC0001F800014227AA0 19>I<067C0607FC0E0FFE1C0FFE3C1FFE781F8FF83E03F03C00703800E07001E06001C0 E003C0000380000780000780000F00000F00001F00001E00003E00003E00007C00007C00 00FC0000F80000F80001F80001F80001F00003F00003F00003F00003E00003C00017227A A019>I<000FC0003FE000F8F001E07001C0780380380380780780700700700780F007C0 E007E1E007F3C007FF8003FE0003FE0001FF0007FF800F7F801E1F803C0F807807C07007 C0F00380E00380E00380E00780E00700E00F00E00E00F03C007878003FF0001FC0001522 7BA019>I<000F80003FE000F8E001F0F003E07003C07007C0F007C0F00F80F00F80F00F 81F01F81F01F01F01F01F01F03F01F03E01E03E01E07E01E0FE00F1FC00FFFC007F7C000 8F80000F80000F00001F00781E00F83E00F83C00F87800E0F000E3E0007FC0003F000014 227BA019>I<07800FC01FC01FC01F800F00000000000000000000000000000000000000 7800FC00FC00FC00F800F0000A157B940F>I<01E003F007F007F007E003C00000000000 000000000000000000000000001E003E003F003F003E003E0006000E000C001C00180038 0070007000E000C0000C1F7D940F>I<00000180000003800000038000000780000007C0 00000FC000000FC000001FC000003FC0000037C0000067C0000067C00000C7C00000C7C0 000187C0000187C0000307C0000707C0000607C0000C07C0000C07C0001807E0001807E0 003FFFE0003FFFE0006003E000E003E000C003E0018003E0018003E0030003E0070003E0 1F8007E07FE03FFEFFE03FFE1F237EA225>65 D<00FFFFE000FFFFF8000F807C000F803E 001F801E001F001F001F001F001F001F003F001E003E003E003E003E003E007C007E0078 007C00F0007C03E0007FFFC000FFFFC000F803E000F801F000F800F801F8007801F00078 01F0007C01F0007C03F0007803E000F803E000F803E001F007E001F007C003E007C007C0 0FC01F80FFFFFE00FFFFF00020227DA123>I<0000FF01800007FFC300000FC1E700003F 007F00007C003F0001F8003E0003E0001E0003E0001E0007C0001E000F80001C001F0000 1C001F00001C003E00001C003E000018007C000018007C000000007C00000000FC000000 00F800000000F800000000F800000000F800000000F000006000F000006000F00000E000 F00000C000F80001C000F80001800078000380007C000700007C000E00003E001C00001F 003800000FC1F0000007FFC0000000FF000000212479A224>I<00FFFFF00000FFFFFC00 000F803E00000F800F00001F800780001F000780001F0003C0001F0003C0003F0003C000 3E0003E0003E0003E0003E0003E0007E0003E0007C0003E0007C0003E0007C0003E000FC 0003C000F80007C000F80007C000F80007C001F8000F8001F0000F8001F0000F0001F000 1F0003F0001E0003E0003C0003E0007C0003E000780007E000F00007C001E00007C00780 000FC01F0000FFFFFC0000FFFFF0000023227DA126>I<00FFFFFF8000FFFFFF80000F80 1F80000F800780001F800380001F000380001F000300001F000300003F000300003E0303 00003E030300003E070300007E060000007C0E0000007C1E0000007FFE000000FFFC0000 00F81C000000F81C000000F81C000001F818000001F0180C0001F0181C0001F000180003 F000180003E000380003E000300003E000700007E000E00007C001E00007C003E0000FC0 0FC000FFFFFFC000FFFFFF800021227DA122>I<00FFFFFF00FFFFFF000F803F000F800F 001F8007001F0007001F0006001F0006003F0006003E0006003E0306003E0706007E0600 007C0600007C0E00007C1E0000FFFC0000FFFC0000F83C0000F81C0001F8180001F01800 01F0180001F0180003F0000003E0000003E0000003E0000007E0000007C0000007C00000 0FC00000FFFE0000FFFE000020227DA121>I<00FFFCFFFC00FFFCFFFC000F800F80000F 801F80001F801F80001F001F00001F001F00001F003F00003F003F00003E003E00003E00 3E00003E007E00007E007E00007C007C00007C007C00007FFFFC0000FFFFFC0000F800F8 0000F800F80000F801F80001F801F80001F001F00001F001F00001F003F00003F003F000 03E003E00003E003E00003E007E00007E007E00007C007C00007C007C0000FC00FC000FF FCFFFC00FFFCFFFC0026227DA125>72 D<00FFFC00FFF8000F80000F80001F80001F0000 1F00001F00003F00003E00003E00003E00007E00007C00007C00007C0000FC0000F80000 F80000F80001F80001F00001F00001F00003F00003E00003E00003E00007E00007C00007 C0000FC000FFFC00FFF80016227EA113>I<00FFFC0FFC00FFFC0FFC000F8007E0000F80 0780001F800E00001F001C00001F003800001F007000003F00E000003E01C000003E0780 00003E0E0000007E1C0000007C3C0000007C7C0000007CFE000000FDFE000000FBBE0000 00FF3F000000FC3F000001F83F000001F01F800001F01F800001F01F800003F00F800003 E00FC00003E00FC00003E007C00007E007E00007C007E00007C003E0000FC007F000FFFC 1FFE00FFFC1FFE0026227DA126>75 D<00FFFE0000FFFE00000FC000000F8000001F8000 001F0000001F0000001F0000003F0000003E0000003E0000003E0000007E0000007C0000 007C0000007C000000FC000000F8000000F8000000F8000001F8000001F000C001F000C0 01F001C003F0018003E0038003E0038003E0070007E0070007C00F0007C03E000FC0FE00 FFFFFE00FFFFFC001A227DA11F>I<00FFC0001FFC00FFC0001FFC000FC0003F80000FC0 007F80001FC0007F80001BC000DF00001BC000DF00001BC001BF00003BC001BF000033C0 033E000033C0063E000033C0067E000073C00C7E000063C00C7C000061E0187C000061E0 18FC0000E1E030FC0000C1E060F80000C1E060F80000C1E0C1F80001C1E0C1F8000181E1 81F0000181E181F0000181E303F0000381E603F0000301E603E0000301EC03E0000300FC 07E0000700F807E0000600F807C0000E00F007C0001F00E00FC000FFE0E0FFFC00FFE0C0 FFFC002E227DA12D>I<00FFC01FFC00FFC01FFC000FC007E0000FE00380001FE0038000 1BE00300001BF00300001BF007000039F007000031F806000031F806000030F80E000070 FC0E000060FC0C0000607C0C0000607E1C0000E07E1C0000C03E180000C03F180000C03F 380001C01F380001801FB00001801FB00001800FF00003800FF00003000FE000030007E0 00030007E000070007E000060003C0000E0003C0001F0003C000FFE001C000FFE0018000 26227DA125>I<0000FE000007FF80000F03E0003C01F0007800F800F0007801E0003C03 C0003C0780003C0F00003E0F00003E1E00003E3E00003E3E00003E7C00003E7C00003E7C 00003EF800007CF800007CF800007CF8000078F80000F8F00000F8F00001F0F00001F0F0 0003E0F00003C0F80007C0F8000F8078000F007C001E003C007C001E00F8000F83E00007 FF800001FC00001F2479A226>I<00FFFFE000FFFFF8000F807C001F801E001F801F001F 000F001F000F003F000F003F001F003E001F003E001F007E001E007E003E007C003C007C 007800FC00F000FC03E000FFFF8000FFFE0001F8000001F8000001F0000001F0000003F0 000003F0000003E0000003E0000007E0000007E0000007C0000007C000000FC00000FFFC 0000FFFC000020227DA122>I<0000FE000007FF80000F03E0003C01F0007800F800F000 7801E0007C03C0003C0780003C0F80003E0F00003E1F00003E3E00003E3E00003E7C0000 3E7C00003E7C00003EF800007CF800007CF800007CF8000078F80000F8F00000F8F00001 F0F00001F0F00003E0F00003C0F01E07C0F87F078078730F0078E19E003CC1BC001EC1F8 000FC3E00007FF804001FD8040000180C000018080000181800001C7800003FF000003FF 000001FE000001FC000000F8001F2D79A226>I<00FFFFC000FFFFF0000F80F8000F803C 001F803E001F001E001F001E001F001E003F003E003E003E003E003E003E007C007E0078 007C00F8007C01E0007C07C000FFFF0000FFFF0000F81F8000F80F8001F807C001F007C0 01F007C001F00FC003F00FC003E00F8003E00F8003E01F8007E01F8007C01F0607C01F06 0FC01F0EFFFC0F9CFFFC07F8000003F01F237DA124>I<0003F860000FFCE0001F1FC000 3C07C0007803C000F003C001F0038001E0038001E0038003E0038003E0030003E0000003 E0000003F0000003FE000003FFE00001FFF00000FFF800007FFC00001FFC000001FE0000 007E0000003E0000003E0000003E0030003C0030003C0030003C0030007C007000780078 00F0007801F0007C03E000EF07C000E7FF0000C1FC00001B247DA21C>I<1FFFFFF81FFF FFF81F03E0F83C03E0383807E0383007C0387007C0306007C030E00FC030C00F8030C00F 8030C00F8030001F8000001F0000001F0000001F0000003F0000003E0000003E0000003E 0000007E0000007C0000007C0000007C000000FC000000F8000000F8000000F8000001F8 000001F0000001F0000003F00000FFFFC000FFFFC0001D2277A124>I<3FFF07FF3FFF07 FF03E001F807E000E007E000E007C000C007C000C00FC001C00FC001800F8001800F8001 801F8003801F8003001F0003001F0003003F0007003F0006003E0006003E0006007E000E 007E000C007C000C007C000C007C001C00FC001800F8001800F8003800F8003000780070 007800E0007C01C0003C0380001F0F00000FFC000003F00000202377A125>I87 D<0300C00781E00F03C01C0700180600380E00701C00601800601800E03800FC3F00FE3F 80FE3F80FE3F80FC3F00781E00131075A21A>92 D<00FDE003FFE007CFE00F87C01F03C0 1E03C03E07C07E07807C07807C07807C0F80FC0F00F80F00F80F18F81F38F81E30783E30 787E707CFE603FEFE00F87C015157C9419>97 D<03E0003FE0003FC00007C00007C00007 C0000780000780000780000F80000F00000F00000F00001F00001EFC001FFE001F8F003F 0F003E07803C07803C0F807C0F80780F80780F80781F80F81F00F01F00F01F00F03E00F0 3E00F07C0070780079F0003FE0001F800011237BA217>I<007F0001FFC003E3C00F87C0 0F07C01F07C03E07803E00007C00007C00007C0000FC0000F80000F80000F80000F80180 7803807C07003E1E001FFC0007F00012157C9417>I<00003E0003FE0003FC00007C0000 7C00007C0000780000780000F80000F80000F00000F00001F00001F000FDE003FFE007CF E00F87E01F03C01E03C03E07C07E07C07C07807C07807C0F80FC0F80F80F00F80F18F81F 38F81F30783E30787E707CFE603FEFE00F87C017237CA219>I<00FC03FF0FC71F033E03 7C037C07FC1EFFFCFFE0F800F800F000F000F000F003F007780E7C3C3FF80FE010157B94 17>I<00007C0001FE0001FE0003FE0007BE0007BC000780000780000F80000F00000F00 000F00000F00001F0001FFF001FFF0001E00001E00003E00003C00003C00003C00003C00 007C0000780000780000780000780000F80000F00000F00000F00001F00001E00001E000 01E00001E00003C00003C0007BC0007B8000FB8000FF0000FE00007C0000172D82A20F> I<001FBC007FFC00F9FC01F0FC03E07803C07807C0F80FC0F80F80F00F80F00F81F01F81 F01F01E01F01E01F03E01F03E00F07C00F0FC00F9FC007FFC001F780000780000F80000F 80000F00780F00781E00F83E00F87C00FFF8003FC000161F7F9417>I<00F8000FF8000F F00001F00001F00001F00001E00001E00001E00003E00003C00003C00003C00007C00007 9F8007FFE007F1E00FC0F00F80F00F80F00F01F01F01E01E01E01E01E01E03E03E03C03C 03C03C07C63C078E7C078C780F8C780F1C780738F807F07003E017237EA219>I<007000 F800F800F000F0000000000000000000000000000000000F801FC039C033C073C063C0E7 C0078007800F800F000F001F001E301E703E603C603CE03DC03F801F000D227DA10F>I< 00F8000FF8000FF00001F00001F00001F00001E00001E00001E00003E00003C00003C000 03C00007C0000781E00787F0078F300F9CF00F39F00F71F00FE1F01FC1E01F80001FF000 1FF8003E78003C7C003C3C303C3C707C7C607878607878E0783DC0F83FC0700F0014237D A217>107 D<01F00FF01FE003E003E003E003C003C007C007C0078007800F800F800F00 0F001F001F001E001E003E003E003C003C007C007C00780078C0F9C0F980F180F380F300 7F003E000C237DA20D>I<1F03F03F003F8FFCFFC033DE3DE3C073F81F81E063F01F01E0 63E01F01E0E3C03E03E007C03C03C007803C03C007803C03C007807C07C00F807C07800F 007807800F00780F8C0F00F80F1C1F00F80F181E00F01F181E00F01E381E01F00E703E01 F00FE01C00E007C026157D9429>I<1F07E0003F9FF8003BFC780073F03C0063E03C0063 E03C00E3C07C0007C0780007807800078078000780F8000F80F0000F00F0000F01F1800F 01E3801F01E3001E03E3001E03C7001E01CE003E01FC001C00F80019157D941C>I<007F 0001FF8003E3C00F81E01F01F01F01F03E01F03E01F07C01F07C01F07C03F0FC03E0F803 E0F803E0F807C0F807C0780F80781F003E7E001FF8000FE00014157C9419>I<01E1F803 F3FC077F1E0E7E1E0C7C0F0C780F1CF81F00F81F00F01F00F01F01F03F01F03E01E03E01 E03E03E07C03E07C03E0F803E0F007F3E007FFC0079F000780000F80000F80000F00000F 00001F00001F00001E0000FFE000FFE000181F809419>I<00FC6003FFE007CFE00F87E0 1F03C01E03C03E03C07E07C07C07807C07807C0780FC0F80F80F00F80F00F80F00F81F00 783E00787E007CFE003FFE000FBC00003C00003C00007C0000780000780000780000F800 00F0000FFF000FFF00131F7C9417>I<0F1F803FBFC03BF9E073E1E063E3E063C3E0E3C3 E007C3C00780000780000780000F80000F00000F00000F00001F00001E00001E00001E00 003E00001C000013157D9415>I<00FC03FF0787070F0E0F0E0F0F060FC00FF80FFC0FFC 03FE007E381E781EF81CF81CF83CF0787FF01FC010157D9414>I<00E001E001E001E003 E003E003C003C007C007C0FFF8FFF80F800F800F000F001F001F001E001E003E003E003C 003C187C387C30787078E079E03FC01F000D1F7C9E11>I<0F803C1FC03C39E07C31E078 71E07863E078E3C0F803C0F007C0F00780F00781F00F81E00F01E00F01E30F03E70F03C6 0F03C60F07CE0F9FCC07FDFC03F0F818157D941B>I<0F81E01FC3F039E3F031E3F071E1 F063E0F0E3C07003C06007C0600780600780E00F80C00F00C00F01C00F01800F03800F03 000F07000F8E0007FC0001F80014157D9417>I<0F8078781FC078FC39E0F8FC31E0F0FC 71E0F0FC63E0F03CE3C1F01C03C1E01807C1E0180781E0180783E0380F83E0300F03C030 0F03C0300F03C0700F03C0600F07C0E00F07C1C00F8FE38007FDFF8001F87E001E157D94 21>I<07C3C01FF7E01CFE70387CF07079F06079F0E0F9F000F9E000F00000F00001F000 01E00001E00079E0307BE070FBE060FBC0E0F7C1C0E7E3C07EFF807C7E0014157D9417> I<0F80781FC07839E0F831E0F071E0F063E0F0E3C1F003C1E007C1E00781E00783E00F83 C00F03C00F03C00F07C00F07800F07800F0F800F9F8007FF0003FF00000F00001F00101E 007C1E007C3C007C7C0078780071F0007FE0001F8000151F7D9418>I<01F06003F8E007 FDC00FFFC00E0F800C0700000E00001C0000380000700000E00001C0000380000700C00E 01C01C01803F07803FFF007BFF00E1FE00C0780013157E9414>II<3C3C7C7CFCFCFCFCFCFC78780E0674A119>127 D E /Fh 26 128 df<3C7C7EFE7C7807067C850E>46 D<000006000000000E000000000E00000000 1E000000001F000000003F000000003F000000007F000000007F00000000DF00000001CF 800000018F800000038F800000030F800000070F800000060FC000000E07C000000C07C0 00001C07C000001807C000003807C000003007E000007003E000007FFFE00000FFFFE000 00C003E00001C003E000018003F000038001F000030001F000070001F000070001F0001F 0001F800FFE01FFF80FFC01FFF8021237EA225>65 D<03FFFFE00003FFFFFC00003E007E 00003E001F00003E001F80003E001F80003E000F80007E000F80007C000F80007C001F80 007C001F80007C003F00007C003E0000FC007C0000F801F80000FFFFF00000FFFFE00000 F801F80000F800FC0001F8007E0001F0007E0001F0003E0001F0003F0001F0003F0001F0 003F0003F0003E0003E0007E0003E0007E0003E000FC0003E001F80003E003F00007E007 E000FFFFFFC000FFFFFE000021227FA123>I<03FFFFF00007FFFFFC00003E007E00003E 001F00003E000F80003E0007C0003E0003C0007E0003E0007C0003E0007C0001F0007C00 01F0007C0001F0007C0001F000FC0003F000F80003F000F80003F000F80003F000F80003 F000F80003F001F80003E001F00007E001F00007E001F00007C001F0000FC001F0000F80 03F0000F8003E0001F0003E0003E0003E0007C0003E000F80003E001F00007E007E000FF FFFF8000FFFFFC000024227FA126>68 D<0001FC00000FFF80003F07E0007C01F000F000 F803E0007807C0007C07C0003E0F80003E1F00003E1F00003F3E00003F3E00003F7E0000 3F7E00003FFC00003FFC00003FFC00003FFC00003FFC00003FFC00007EF800007EF80000 7EF80000FCF80000FCFC0000F8FC0001F87C0001F07C0003E03E0007C03E000F801F001F 000F803E0007E0F80001FFE000007F800020247AA227>79 D<000FE0C0007FF9C000F87F C001E01F8003C00F8003C0078007800380078003800F8003800F0003000F8003000F8000 000FC000000FE000000FFC000007FFC00007FFF00003FFF80000FFFC00003FFC000003FC 000000FE0000007E0000003E0000003E0030003E0070003C0060003C0070003C00700078 00700078007800F000FC01E000FF83C000E7FF8000C1FE00001A247DA21C>83 D<03FF0007FFC00F83E00F81E00F81F00F01F00001F00003E001FFE007FFE01FC1E03F03 E07E03E0FC03C0F803C6F803C6F807C6FC0FC67E3FDC7FFBF81FE1E017157E9419>97 D<03C0003FC0003FC0000780000780000780000780000780000F80000F00000F00000F00 000F00000F00001F3F801EFFE01FE1F01F80F81E00781E007C3E007C3C007C3C007C3C00 7C3C007C3C00FC7C00F87800F87800F87801F07803E07C07C0FF0F80E7FF00C1FC001623 7BA21C>I<00FF0003FFC00F87C01F07C03E07C07C07807C0000FC0000F80000F80000F8 0000F80000F80000F80000F80000F80180F803807C07003F1E001FFC0007F00012157C94 16>I<00000F0000FF0000FF00001E00001E00001E00001E00003E00003E00003C00003C 00003C00003C00007C00FE7C07FFF80FC3F81F00F83E00F87C00F87C00F8FC00F0F800F0 F800F0F800F0F801F0F801F0F001E0F001E0F001E0F803E07807E07E1FE03FFBFC0FE3FC 18237CA21C>I<00FE0003FF800FC7C01F03E03E01E03C01E07C01E0FFFFE0FFFFE0F800 00F80000F80000F80000F00000F80000F800C07801C07C03803F0F001FFE0007F8001315 7D9416>I<0001F8000FFC001F3C003C7C00787C00F87C00F00000F00001F00001E00001 E00001E00001E00001E0003FFE003FFE0003C00003C00003C00003C00007C00007800007 80000780000780000780000F80000F00000F00000F00000F00000F00001F0000FFF000FF F00016237FA20F>I<000007C0003F8FC000FFFDC001F1F9C003E0F9C007C0780007C0F8 000FC0F8000F80F8000F80F8000F80F8000781F00007C3E00007FFC0000EFE00000C0000 000E0000000E0000000FFFC0000FFFF0000FFFF8001FFFFC003C00FC0078003C0070001C 00F0001C00E0001C00F0003C00700078007800F8003E03E0001FFFC00003FE00001A2180 9519>I<003C000003FC000003FC00000078000000780000007800000078000000780000 00F8000000F0000000F0000000F0000000F0000000F0000001F3FC0001EFFE0001FE3E00 01F81F0001F00F0001E01F0003E01F0003C01E0003C01E0003C01E0003C01E0003C03E00 07C03E0007803C0007803C0007803C0007803C000F807C000F807C00FFF3FF80FFF3FF80 19237FA21C>I<007800F801F801F801F800F0000000000000000000000000000000E00F E00FE003E001E001E003E003C003C003C003C003C007C0078007800780078007800F807F F0FFE00D2280A10D>I<003C03FC03FC0078007800780078007800F800F000F000F000F0 00F001F001E001E001E001E001E003E003C003C003C003C003C007C00780078007800780 0F800F807FF0FFF00E2380A20D>108 D<01E1FC07F01FEFFF3FFC1FFE1F787C03F80FE0 3E01F007C01E01E007801E03E00F803E03C00F003C03C00F003C03C00F003C03C00F003C 03C00F003C07C01F007C07801E007807801E007807801E007807801E007807801E00780F 803E00F8FFF3FFCFFFFFF3FFCFFF28157F942B>I<00E3FC001FEFFE001FFE3E0003F81F 0001F00F0001E01F0003E01F0003C01E0003C01E0003C01E0003C01E0003C03E0007C03E 0007803C0007803C0007803C0007803C0007807C000F807C00FFF3FF80FFF3FF8019157F 941C>I<007F0001FFC00781E00F00F01E00783C00787C007C78007CF8007CF8007CF800 7CF8007CF000F8F000F8F000F0F801F07801E07803C03E0F800FFE0003F00016157D9419 >I<01FC3007FE300FC7701F83F03F01F07E01F07C01F0FC01E0F801E0F801E0F801E0F8 03E0F803E0F803C0F803C0F803C0F807C07C0FC07E3FC03FF7800FC78000078000078000 0F80000F00000F00000F00000F00000F0000FFE000FFE0141F7C941A>113 D<00E7E01FEFF01FFDF003F9F001F1F001E1E003E00003E00003C00003C00003C00003C0 0007C0000780000780000780000780000780000F8000FFF800FFF80014157F9414>I<01 FD8007FF800F07801E03801C01801C01803E00003F80001FF8001FFE000FFE0003FF0000 3F00600F00600700700700700F00780E00FC3C00FFF800C7E00011157E9414>I<018001 800180038003000300070007000F001F00FFFCFFFC1E001E001E001E003E003C003C003C 003C003C007C30783078307830787078607CE03FC01F000E1F7C9E13>I<070038FF07F8 FF07F81F00F80F00780F00F81F00F81E00F01E00F01E00F01E00F01E01F03E01F03C01E0 3C01E03C03E03C07E03C0FE03E1FE01FFBFC0FE3FC16157C941C>I<0FFE1FF00FFE1FE0 01F00F8000F0070000F00E0000F00C0000F01C0000781800007830000078300000786000 007C6000003CC000003CC000003D8000003F8000003F0000001E0000001E0000001C0000 001C00000018000000180000003000000030000078600000F8C00000F9C00000F3800000 7F0000003C0000001C1F80941A>121 D<3C1E7C3FFE3FFE3F7C3E783E100677A119>127 D E /Fi 6 110 df45 D<0FF8003FFF003FFF803FFFC03E0FE01C03E00001E001FFE00FFFE03FFFE07FFFE0FF01 E0F801E0F001E0F003E0F807E0FE0FE07FFFFE3FFFFE1FFFFE07F07E17157E941A>97 D<01FC0007FF001FFF803FFFC07F07E07E03E0FC01F0F801F0FFFFF0FFFFF0FFFFF0FFFF F0F80000F800F07C00F07E01F03F83F03FFFE00FFFC007FF8001FC0014157D941A>101 D<01E00003F00003F00003F00003F00001E0000000000000000000000000007FF0007FF0 007FF0007FF00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F0 0000F00000F00000F000FFFFE0FFFFE0FFFFE0FFFFE0131F7D9E1A>105 D108 DI E /Fj 35 115 df0 D<70F8F8F87005057C8E0E>I<0000800000018000000180000001800000 018000000180000001800000018000000180000001800000018000000180000001800000 0180000001800000018000FFFFFFFFFFFFFFFF0001800000018000000180000001800000 018000000180000001800000018000000180000001800000018000000180000001800000 018000FFFFFFFFFFFFFFFF20227DA027>6 D<03F0000FFC001C0E003003006001806001 80C000C0C000C0C000C0C000C0C000C0C000C0C000C0C000C06001806001803003001C0E 000FFC0003F00012147D9519>14 D17 D<0000000C0000003C000000F0000003C000000F0000003C000000F0000003C00000 0F0000003C000000F0000003C000000F0000003C000000F0000000F00000003C0000000F 00000003C0000000F00000003C0000000F00000003C0000000F00000003C0000000F0000 0003C0000000F00000003C0000000C000000000000000000000000000000000000000000 00000000000000000000007FFFFFF8FFFFFFFC1E287C9F27>20 DI<000000040000000C0000000C00000018000000180000 00300000003000000060000000C0000003800000070000003C000001F800003FC000FFFE 0000FFFE0000003FC0000001F80000003C000000070000000380000000C0000000600000 00300000003000000018000000180000000C0000000C0000000400000000000000000000 00000000000000000000000000000000000000000000FFFFFFFCFFFFFFFC1E287C9F27> I<000FFFFC007FFFFC01F0000003800000060000000C0000001800000030000000300000 006000000060000000C0000000C0000000C0000000C0000000C0000000C0000000C00000 00C000000060000000600000003000000030000000180000000C00000006000000038000 0001E00000007FFFFC001FFFFC1E1E7C9A27>26 DI<00000001800000000001800000000001800000000001800000000000C0000000 0000C000000000006000000000003000000000003000000000001C00000000000E000000 00000700FFFFFFFFFFE0FFFFFFFFFFE0000000000700000000000E00000000001C000000 0000300000000000300000000000600000000000C00000000000C0000000000180000000 000180000000000180000000000180002B1A7D9832>33 D<000000600000000000300000 0000003000000000001800000000001800000000000C0000000000060000000000030000 FFFFFFFF8000FFFFFFFFE000000000007000000000001C00000000000F800000000003E0 000000000780000000001E00000000003800000000006000FFFFFFFFC000FFFFFFFF8000 0000000300000000000600000000000C0000000000180000000000180000000000300000 0000003000000000006000002B1C7D9932>41 D<03F80001F80007FE000FFE001E3F801C 03003807C03001802003E06000804001F0C000404000F9800040C0007F00002080003F00 002080003E00002080001F00002080000F80002080001F80002080001FC00060400033E0 0040400061F000402000C0F800803001807C03801807003F8F000FFE000FFC0003F00003 F8002B157D9432>49 D<001FFF007FFF01E0000380000600000C00001800003000003000 00600000600000600000C00000C00000FFFFFFFFFFFFC00000C000006000006000006000 003000003000001800000C000006000003800001E000007FFF001FFF181E7C9A21>I<7F F8007FFE000007800001C000006000003000001800000C00000C00000600000600000600 0003000003FFFFFFFFFFFF00000300000300000600000600000600000C00000C00001800 00300000600001C00007807FFE007FF800181E7C9A21>I<000003000003000006000006 00000C00000C0000180000180000300000300000600000600000C00000C00000C0000180 000180000300000300000600000600000C00000C00001800001800003000003000006000 00600000C00000C0000180000180000300000300000300000600000600000C00000C0000 180000180000300000300000600000600000C00000400000183079A300>54 DI<40000010C00000306000 00606000006060000060300000C0300000C0300000C018000180180001800C0003000C00 03000C00030007FFFE0007FFFE0003000C0003000C0003000C0001801800018018000180 180000C0300000C030000060600000606000006060000030C0000030C000001980000019 800000198000000F0000000F0000000F000000060000000600001C2480A21D>I<001007 E000701FF001F07FF803F0C0FC04F3007C00F7003C00FE003C00EC003800F8003801F800 7001F0006001F000C001E0018001E00E0003E07C0003C0FE0003C3FF8003C7FFC003800F E0078003E0078001F0078001F0070000F00F0000F00F0000F00E0000F00E0000E01E0000 E01C0001C01C0001803C8003003BC006003FF0380073FFF00061FFC000807E00001E247F A221>66 D<00007F000003FFC0000FFFE0001C07E0006001E000C001C0018001C0030003 80060003000E0007001C000C001C00080038000000380000007800000070000000700000 00F0000180F0000380F0000780F0000700F0000F00F0000F00F0001F00F8001E00F8003E 007C007E007E00DC003F839C001FFE3C000FFC380003E0380000007000000070000000E0 000000C000180180003E030000FFFC00003FF800000FE000001B297EA21E>71 D<0000180000001000007800000030000078000000600000F8000000600000F8000000E0 0000F8000001E00000FC000003E00000FC000007C00000BC00000FC000013C00000FC000 013C00001FC000013C00003BC000013E000073C000023E0000E3C000021E0001E3800002 1E0003C38000041E0003878000041F0007078000041F000E078000080F001C078000080F 0038078000080F8078070000100F80F0070000100781E00F0000100783C00F00002007C7 800F00002007CF000F00004003FE000F00004003FC000F0000C001F8000F00008001F000 0F00618000E0000F007F0000C0000F00FF000000000FF0FE000000000FE0FE0000000007 803C00000000000034257EA23C>77 D<00000000001E0000000000FE0000000001FC0000 000003FC0000600003F80001E00003C00001E00006000001F00004000003F00008000003 F00008000003F00008000002F80008000002F80010000002780010000002780010000006 7C00200000047C00200000043C00200000043C00200000043E00400000083E0040000008 1E00400000081F00400000081F00800000100F00800000100F00800000100F8080000020 0F810000002007810000002007C10000004007C10000004003E20000004003E200000080 01F20000018001F20000630000FE00007F0000FC0000FF00007C0000FE0000780000FC00 003000003800000000002F2980A629>I<000FFFE000007FFFFC0001FFFFFF00038780FF 800C07801FC018078007C038070003E070070003E0600F0001E0C00F0001E0000F0001E0 000F0001C0000E0001C0000E000380001E000380001E000700001E000600001C000C0000 1C001000003C006000003803800000387F00000079FC0000007BE0000000700000000070 00000000F000000000E000000000E000000001E000000001C000000001C0000000038000 000003800000000700000000060000000023247FA123>80 D<000FFFF800007FFFFF0001 FFFFFF800387803FC00C078007E018078003E038070001E070070001E0600F0001E0C00F 0001C0000F0001C0000F000380000E000300000E000600001E000C00001E001800001E00 E000001C1FC000001C3F0000003C7F000000381F000000380F800000780F8000007007C0 00007007C00000F003E00000E003E00000E001F00001E001F00301C000F80601C000F80C 03C0007C180380007FE00300003FC00400001F0028237FA12A>82 D<0000000002000000000C003FFFFFF800FFFFFFF003FFFFFFC004001C00001C003C0000 3C003C000078007C000078007800007000780000C0007800000000F800000000F0000000 00F000000000F000000001F000000001E000000001E000000001E000000003E000000003 C000000003C000000003C000000007C00000000780000000078000000007800000000F80 0000000F000000000F000000000F000000001E000000001E000000001C000000003C0000 0000380000000070000000006000000027277FA31B>84 D<00007FC0200001FFFFC00007 FFFF80001807FF0000380006000070000E0000E0001C0000C00038000100003000000000 6000000000C0000000018000000003000000000600000007FFC000000FFFE000000031C0 0000006080000000C00000000180000000030000000006000000000C0000000018000000 00300003000060000E0000C0001E000100001C000600003C000C000038001FFF8070003F FFFFC0007FFFFF8000800FFE000023227EA124>90 D<0003C0001E0000380000700000E0 0000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E0 0000E00000E00000E00000E00001C0000380000F0000F800000F000003800001C00000E0 0000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E0 0000E00000E00000E00000E000007000003800001E000003C012317DA419>102 DI<0020006000C000C000C0018001800180030003000600060006 000C000C000C0018001800300030003000600060006000C000C000600060006000300030 003000180018000C000C000C000600060006000300030001800180018000C000C000C000 6000200B327CA413>III<8010C030C030C0 30C030C030C030C030C030C030C030C030C030C030C030C030C030C030C030C030C030C0 30C030C030C030C030C030C030C030C030C030C030C030C030C030C030C030C030C030C0 30C030C030C030C030C030C030C030C030C03080100C327AA419>I110 D<0000000001000000000300000000060000000006000000000C000000000C0000 0000180000000018000000003000000000300000000060000000006000000000C0000000 00C00000000180000000018000000003000000000300000000060000000006000000000C 000000000C0000000018000000001800000000300006000030001E000060002F00006000 4F0000C000878000C000078001800003C001800003C003000003C003000001E006000001 E006000000F00C000000F00C000000781800000078180000003C300000003C300000001E 600000001E600000000FC00000000FC00000000780000000078000000003000000000300 000028327D812A>112 D114 D E /Fk 37 121 df<0060006000600060007E01FE03 FE07800F001E001C003800780070007000F000E000E000E000E000E000F000F8007F003F C01FF003F0007000700C700FE007C00F207D9812>16 D<0603000E07000E07000E07001E 0F001E0F001C0E001C0E003C1E003C1E60381C603C3CE07C7CC07FFFC07FCF80700000F0 0000F00000E00000E00000E00000C0000013167D8E18>22 D<7E037E071C071C073C0F3C 1E381C383C787878F071E077C0FF00FE00F000100F7D8E14>I<1FFFF83FFFF87FFFF0F1 8E00C18C00038C00038C00030C00070C00070C000E0E000E0E001E0F001C0F001C060015 0F7E8E17>25 D<00FC0001FF0007CF000707800E07801E07801C07801C07801C0F803C0F 00380F003C1E003E3C007FF80077E000700000700000F00000E00000E00000E00000C000 0011167D8E15>I<03FFF00FFFF01FFFF03C1E00780E00F00E00E00E00E00E00E01E00E0 1C00E03C00E07800F0F0007FE0001F8000140F7E8E17>I<1FFFC03FFFC07FFFC0F1C000 C180000180000380000380000380000700000700000700000700000E00000E0000120F7E 8E12>I<000C00000C00000C00001C00001C0000180000180000380000380003FF000FFF 801E73C03C71E07860E0F060E0E0E0E0E0E0E0E0C1E0E0C1C0E1C3C0E1C780718F007FFE 001FF800038000030000030000070000070000060000060000131F7D9718>30 D<0000C00000C00000C00000C00001C00001800001800001800003800F030E3F831F7BC3 1F73C71FE3860F078606070606070E06070C0E0F0C1C0F0C38071C7007D9E003FFC000FF 00003800003000003000003000007000006000006000181F7F971A>32 D<18003838007C30007C70007C60303CE07018C07018C07018C07038C0E070E1F0F0F7FB E0FFFFC07F7F803E3E00160F7E8E19>I<70F8F8F87005057C840C>58 D<70F8F8F87818183830706060050C7C840C>I<00000600001E00007C0001F00007C000 1F00007C0001F00007C0001F00007C0000F00000F000007C00001F000007C00001F00000 7C00001F000007C00001F000007C00001E00000617187C941F>I<000C000C001C001800 180038003000700060006000E000C000C001C001800180038003000700060006000E000C 000C001C001800180038003000700060006000E000C000C0000E237D9914>I<00003000 000070000000F0000000F8000001F8000001F800000378000006780000067800000C7800 001C78000018780000307C0000303C0000603C0000FFFC0000FFFC0001803C0003803C00 03003C0006003E001E003E007F81FFC0FF81FFC01A187E971E>65 D<07FFFE0007FFFF8000F807C000F003C000F001E001F001E001F001C001E003C001E007 8003E00F0003E0FE0003FFF80003FFFE0007C00F0007C0078007800780078007800F8007 800F800F000F000F000F003E001F007C00FFFFF000FFFFC0001B187E971E>I<000FE0C0 007FF9C001F83F8003E00F800F8007801F0007803E0003003C0003007C00030078000300 F8000000F0000000F0000000F0000000F0000C00F0000C00F0001C00F0001800F0003800 780070003C00E0003F03C0000FFF000003FC00001A187D971D>I<07FF07FF07FF07FF00 F800F800F000F000F000F001F001F001F001E001E001E001E001E003E003E003E003C003 FFFFC003FFFFC007C007C007C0078007800780078007800F800F800F800F000F000F000F 000F001F001F00FFE0FFE0FFE0FFE020187E9721>72 D<07FF01FF07FF01FF00F800F800 F001C000F0078001F00E0001F03C0001E0780001E0E00003E3C00003E7C00003DFE00003 FFE00007F1F00007E1F00007C0F8000780F8000F807C000F807C000F003E000F003E001F 003F00FFE0FFE0FFE0FFC020187E9721>75 D<07FC0003FC07FC0007FC00FC000FE000DE 000FC000DE001BC001DE0037C001DE003780019E006780019E00C780038F00CF80038F01 8F00030F030F00030F030F00070F061F00070F0C1E0006078C1E000607981E000E07B03E 000E07B03C000C07E03C000C07C03C001C03C07C00FF8383FF80FF8303FF8026187E9726 >77 D<07F801FF07FC01FF00FE003800FE003000DF003001DF007001CF8060018F806001 87C0600387C0E00383E0C00303E0C00301F0C00700F1C00700F9800600798006007D800E 003F800E003F000C001F000C001F001C000F00FF800E00FF80060020187E9720>I<07FF F80007FFFE0000F80F0000F0078000F0038001F003C001F0078001E0078001E0070003E0 0F0003E03C0003FFF80003FFF80007C0FC0007C07C0007803C0007803C000F807C000F80 7C000F0078000F0078601F007CE0FFE07FC0FFE01F801B187E971E>82 D<1FFFFFC03FFFFFC03C0F83C0380F01C0700F0180601F0180E01F0180C01E0180C01E01 80003E0000003E0000003C0000003C0000007C0000007C0000007800000078000000F800 0000F8000000F0000000F0000001F000003FFF00003FFF00001A187E9718>84 D<0180018061806180618061C063C07FC07FC0FF80FF80F180E180618061806180618061 806180618061C063C07FC07FC0FF80FF80F180E1806180600060000A1F7D9710>93 D<7E007E001E001C001C003C003C00380038007FE07FF07C787878F078F078E078E078E0 F8E0F0C0F0E1E0F3C07F803E000D187D9712>98 D<000F80001FC0003FC0003BC0003BC0 00780000700000700000700007FF0007FF0000E00000E00000E00000E00000E00001E000 01C00001C00001C00001C00003C000038000038000038000038000770000F70000FE0000 FE00007C0000121F7D9713>102 D<1F80001F80000780000700000700000700000F0000 0E00000E00000EF8001FFC001F9E001E0E001C1E003C1C00381C00381C00383C007838C0 7038C07079C0707380F07F00E03E0012187D9717>104 D<01E001E001E001C000000000 0000000000001F003F8073806380E780070007000F000E001E601C601CE01DC01FC00F00 0B187F970E>I<001C003C003C003C0000000000000000000001F007F807380E381C7800 780070007000F000F000E000E001E001E001C001C003C07380F780FF00FE007C000E1F7F 9710>I<1F801F8007800700070007000F000E000E000E1E1E3F1CFF1DCF1F8F3F0E3F80 3FC039E078E370E370E770E6F0FEE07C10187D9715>I<3F3F0F0E0E1E1E1C1C3C3C3838 78787070F0F6E6E6EEFC7808187D970D>I<1F1F803FBFC077F1E063E0E0E7C1E00781C0 0701C00701C00F03C00F038C0E038C0E079C1E07381E07F81C03E0160F7F8E19>110 D<1F3E003FFF0077E78063C780E787800787000700000700000F00000F00000E00000E00 001E00001E00001C0000110F7F8E12>114 D<07F00FF81E781C781C781F001FE01FF007 F070F0F070F070F1E0FFC03F800D0F7D8E13>I<01C001C003C003C0038003807FF0FFF0 070007000F000F000E000E001E001E301C301C701CE01FC00F800C157F940F>I<0F0180 3F83807B8380738380E38780078780070700070700070F000F0F300E0E300E1E700F3E60 07FFE003E7C0140F7F8E18>I<0F9F003FFFC079F3C070E3C0E1E3C001E38001C00001C0 0003C000738180F38180F38380E7C700FFFE007CFC00120F7E8E17>120 D E /Fl 17 94 df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m 64 123 df<003F800000FFE00003F1F07007C078600F807C60 1F003CE03E003CE03E003CC07C003DC07C003D807C003F80FC003F00F8003E00F8003E00 F8003E0078007E007800FE007C01FE303E0F9E701FFF0FE007F807C01C157F9420>11 D<00003F800000FFC00003E1E0000780E0000E00F0001C00F0003800F0003800F0007001 F0006001F000E001E000C001E000C003C001C003C00181FF800183FF000383FF800303FF C0030003C0030003E0070003E0060003E0060003E0060003E00E0007E00E0007C00C0007 C00C0007C01E000F801E000F801F001F001F003E003BC0FC0031FFF00030FFC000300400 0070000000600000006000000060000000E0000000C0000000C0000000C0000000C00000 001C2D80A21C>I<01F800C007FE01C00FFF01801FFF03803FFF8300780F83007003C700 E001C600C000CE000000CC000000EC0000007C0000007800000078000000700000007000 0000700000006000000060000000E0000000E0000000E0000000E0000001C0000001C000 0001C0000001C00000038000000380000003800000038000000300001A2080941A>I<00 3F80007FF000F3F000E1F000C0F000C00000C00000E00000E00000F00000F00000780000 7800007C00003E00007E0001FF0003FF00079F800F0F801E0F803C0F803C078078078078 0780780780F80700F00F00F00F00F00F00F00E00701E00781C007C38003FF8000FE00001 000014257EA316>I<007F03FF0FC01F003E007E007C00FC00FFFCFFFCF800F800F000F0 00F000F80078007C033E0F1FFE07F810157E9414>I<000400000600000E00000C00000C 00000FF8000FF8001FF8003BF000700000E00001C0000380000780000700000E00000E00 001C00001C0000380000380000380000780000700000700000700000700000F00000F000 00F000007800007800007E00007F80003FF0001FFC000FFF0001FF00003F80000F800007 8000078000070003CF0003FE0000FC00152E7EA316>I<01E00F0001E00F0003E01F0003 E01E0003C01E0003C01E0007C03E0007C03C0007803C0007803C000F807C000F8078000F 0078000F0078601F00F8E01F00F0C01F01F0C01F03F1C03F87FB803FFE7F003DFC3E003C 0000007C0000007C0000007800000078000000F8000000F8000000F0000000F0000000F0 000000E00000001B207F941E>22 D<0F800F7F800F7F801F0F801E0F001E0F003E1F003C 1F007C1E00781E00F03E01F03E03E03C07C03C0F807C1F007C3E0078F8007BF000FFC000 FF0000F8000018157E9419>I<07FFFFE01FFFFFE03FFFFFE07FFFFFC07870E000E060C0 00C060C00000E0C00000E1C00000C1C00001C1C00001C1C00003C1C0000381C0000781E0 000781E0000781E0000F01F0000F01F0000F01F0000E00E0001B157F941C>25 D<0007E0001FF8007C7C00F03C01E03E01E03E03C03E03C03E07803E07803E0F807E0F80 7C0F007C0F007C1F00F81F00F81F01F01F03E03FC7C03FFF003CFC003C00007C00007C00 00780000780000F80000F80000F00000F00000F00000E0000017207F941A>I<007FFFE0 01FFFFE003FFFFE007FFFFC00F83E0001F01F0003E00F0003C00F0007800F0007800F000 F801F000F801E000F001E000F001E000F003C000F007C00070078000780F00003C3E0000 1FF800000FE000001B157F941C>I<07FFFF1FFFFF3FFFFF7FFFFE781800E01800C03800 00380000380000700000700000700000F00000F00000E00001E00001E00001E00003E000 03E00001C00018157F9416>I<0000300000003000000070000000700000006000000060 000000E0000000E0000000C0000000C0000001C0000001C0000001800000018000003FF8 0000FFFE0003E31F00078307800F0707801E0703C03E0603C03C0603C0780E03C0780E03 C0780C07C0F80C0780F01C0780F01C0F80F0180F0078181E0078383C003C3878003E31F0 001FFFE00007FF000000700000006000000060000000E0000000E0000000C0000000C000 0001C0000001C00000018000001A2D7EA21E>30 D<0F8000303FE000703BF000E071F001 C060F8038000F80700007C0E00007C1E00007C1C00003E3800003E7000003EE000001FC0 00001F8000001F0000000F0000001F0000001F8000003F80000077800000E7C00001C7C0 000383C0000703E0000E03E0001E01F0001C01F0E03801F8C07000FDC0E0007F80C0003F 001C1F7F941F>I<0000060000000600000006000000060000000E0000000C0000000C00 00000C0000001C00000018000000180000001800000038000000300007C030781FE030F8 3CF070FC38F060FC70F060FC61F0603CE1E0E03C01E0C01803E0C01803C0C01803C1C038 07C180300781807007818060078380E0078301C007C3038003E30F0003F73E0000FFF800 003FE00000060000000E0000000C0000000C0000000C0000001C00000018000000180000 00180000003800001E2D80A220>I<060000F00E0001F80C0001F81C0001F8380001F838 0000783003803870078030600780306007803060078070E00F0060E00F00E0E00F01E0F0 1E03C0F83F07807FFFFF807FFFFF007FF7FE003FE7FC001F83F0001D1580941F>I<78FC FCFCFC7806067C850E>58 D<78FCFEFEFE7E0606060E0C1C1838706007107C850E>I<00 00001C0000007C000001F0000007C000001F0000007C000001F0000007C000001F000000 7C000001F0000007C000001F0000007C000000F0000000F00000007C0000001F00000007 C0000001F00000007C0000001F00000007C0000001F00000007C0000001F00000007C000 0001F00000007C0000001C1E1E7C9A27>I<0000C00000C00001C0000180000180000380 000300000300000700000600000600000E00000C00000C00001C00001800001800003800 00300000300000700000600000600000E00000C00001C000018000018000038000030000 0300000700000600000600000E00000C00000C00001C0000180000180000380000300000 300000700000600000600000E00000C00000C0000012317DA419>II<000FE000003FF80000783E0000E01E0001C00F0001C0078003 E0078003E0078003E003C003E003C0000003C0000003C0000203C0007FC7C000FFE7C003 E077C007C03FC00F801FC01F001F801F001F803E001F803E001F807E001F007C001F007C 001F007C003E00FC003E00F8003C00F8007C00F80078007800F8007800F0007801E0003C 07C0001E0F80000FFE000003F800001A257EA31A>64 D<000001C000000001C000000003 C000000003C000000007C00000000FC00000000FC00000001FC00000001FE000000037E0 00000033E000000063E0000000C3E0000000C3E000000183E000000183E000000303E000 000703E000000603E000000C03E000000C03F000001801F000001801F000003FFFF00000 7FFFF000006001F00000C001F00000C001F000018001F000038001F000070001F0000700 01F8001F8001F800FFE01FFF80FFE01FFF8021237EA225>I<00FFFFFC0000FFFFFF0000 07C00FC00007C003E0000FC003E0000F8001E0000F8001F0000F8001F0001F8001E0001F 0003E0001F0003E0001F0007C0003F000F80003E001F00003E007E00003FFFF800007FFF F000007C00FC00007C003E00007C001F0000FC001F0000F8000F0000F8000F8000F8000F 8001F8000F0001F0001F0001F0001F0001F0003E0003F0003C0003E0007C0003E001F800 07E007E000FFFFFFC000FFFFFE000024227FA126>I<00007F80C00003FFE180000FE0F3 80003F003780007C003F8000F8001F0001F0000F0003E0000F0007C0000F000F80000E00 1F00000E001F00000E003E00000E003E00000C007C00000C007C000000007C00000000FC 00000000F800000000F800000000F800000000F800000000F000003000F000003000F000 007000F000006000F80000E000F80000C000780001C0007C000380003C000700003E000E 00001F803C00000FE0F8000003FFE0000000FF00000022247DA224>I<00FFFFFC0000FF FFFF800007C00FC00007C003E0000FC001F0000F8000F0000F800078000F800078001F80 0078001F00007C001F00007C001F00007C003F00007C003E00007C003E00007C003E0000 7C007E000078007C0000F8007C0000F8007C0000F800FC0001F000F80001F000F80001E0 00F80003E001F80003C001F000078001F0000F8001F0000F0003F0001E0003E0007C0003 E000F00007E007E000FFFFFF8000FFFFFC000026227FA129>I<00007F80600003FFE0C0 000FE071C0003F003BC0007C001FC000F8000F8001F0000F8003E000078007C00007800F 800007001F800007001F000007003E000007003E000006007E000006007C000000007C00 000000FC00000000F800000000F800000000F800000000F8003FFF80F8003FFF80F00001 F800F00001F800F80001F000F80001F000F80001F000780003F0007C0003E0003E0003E0 003F0007E0001F801FE00007E07CC00003FFF04000007F80000023247DA227>71 D<00FFFE1FFFC000FFFE1FFFC00007C000F8000007C001F800000FC001F800000F8001F0 00000F8001F000000F8003F000001F8003F000001F0003E000001F0003E000001F0007E0 00003F0007E000003E0007C000003E0007C000003FFFFFC000007FFFFFC000007C000F80 00007C000F8000007C001F800000FC001F800000F8001F000000F8001F000000F8003F00 0001F8003F000001F0003E000001F0003E000001F0007E000003F0007E000003E0007C00 0003E0007C000007E000FC0000FFFE1FFFC000FFFE1FFFC0002A227FA129>I<00FFFE00 FFFE0007C00007C0000FC0000F80000F80000F80001F80001F00001F00001F00003F0000 3E00003E00003E00007E00007C00007C00007C0000FC0000F80000F80000F80001F80001 F00001F00001F00003F00003E00003E00007E000FFFE00FFFE0017227FA116>I<0003FF F80003FFF800001F8000001F8000001F0000001F0000001F0000003F0000003E0000003E 0000003E0000007E0000007C0000007C0000007C000000FC000000F8000000F8000000F8 000001F8000001F8000001F0000001F0000003F0000003F0001803E0007C03E0007E07E0 00FC07C000FC0FC000F80F8000E01F0000787E00003FFC00000FE000001D237EA11C>I< 00FFFE01FFC000FFFE01FFC00007C000FE000007C000F800000FC001E000000F8003C000 000F80070000000F800E0000001F803C0000001F00780000001F00E00000001F01C00000 003F07800000003E0F000000003E1F800000003E3F800000007EFF800000007DEFC00000 007F8FC00000007F07E0000000FE07E0000000FC03F0000000F803F0000000F803F00000 01F801F8000001F001F8000001F000FC000001F000FC000003F000FC000003E0007E0000 03E0007E000007E0007F0000FFFE03FFE000FFFE03FFE0002A227FA12A>I<00FFFF0000 FFFF000007E0000007C000000FC000000F8000000F8000000F8000001F8000001F000000 1F0000001F0000003F0000003E0000003E0000003E0000007E0000007C0000007C000000 7C000000FC000000F8000C00F8000C00F8001C01F8001801F0003801F0003801F0007003 F000F003E001F003E003E007E00FE0FFFFFFE0FFFFFFC01E227FA122>I<00FFE00003FF 8000FFE00003FF800007E00007F0000007E0000FF000000FE0000FF000000DE0001BE000 000DE0001BE000000DE00037E000001DE00067E0000019E00067C0000018F000C7C00000 18F000CFC0000038F0018FC0000030F0030F80000030F0030F80000030F0061F80000070 F0061F80000060F00C1F00000060F0181F0000006078183F000000E078303F000000C078 303E000000C078603E000000C078C07E000001C078C07E0000018079807C000001807980 7C000001807B00FC000003807E00FC000003003E00F8000007003C00F800000F803C01F8 0000FFF0383FFF8000FFF0303FFF800031227EA130>I<00FFE003FFC000FFE003FFC000 07F0007E000007F0003800000FF8003800000DF8003000000CF8003000000CFC00700000 1CFC00700000187E00600000187E00600000183E00E00000383F00E00000301F00C00000 301F80C00000301F81C00000700FC1C00000600FC18000006007C18000006007E3800000 E007E3800000C003F3000000C003F3000000C001F7000001C001FF0000018000FE000001 8000FE0000018000FE00000380007E00000300007C00000700003C00000F80003C0000FF F0003C0000FFF0001800002A227FA128>I<00007F00000003FFE000000F81F000003E00 78000078003C0000F0001E0001E0001E0003C0000F000780000F000F80000F000F00000F 801F00000F803E00000F803E00000F807C00000F807C00000F807C00000F80F800001F00 F800001F00F800001F00F800003E00F800003E00F800003E00F000007C00F000007C00F8 0000F800F80000F000F80001F000780003E000780007C0003C000F80003E001E00001F00 7C00000FC1F8000003FFE0000000FF00000021247DA226>I<00FFFFF00000FFFFFC0000 07C03F000007C00F80000FC007C0000F8003C0000F8003C0000F8003C0001F8007C0001F 0007C0001F0007C0001F000F80003F000F00003E001E00003E003C00003E01F800007FFF E000007FFFC000007C07F000007C01F00000FC01F80000F800F80000F800F80000F801F8 0001F801F80001F001F00001F001F00001F003F00003F003F00003E003E06003E003E060 07E003F0E0FFFE01F1C0FFFE00FF800000007F0023237EA126>82 D<1FFFFFFF1FFFFFFF1F01F01F3C01F00F3803F0073003E0077003E0066003E006E007E0 06E007C006C007C006C007C006000FC000000F8000000F8000000F8000001F8000001F00 00001F0000001F0000003F0000003E0000003E0000003E0000007E0000007C0000007C00 00007C000000FC000000F8000000F8000001F80000FFFFE000FFFFE00020227EA11D>84 D86 D<007FFE0FFF007FFE0FFF0007F003F00003E001C00003F003800003F007000001 F00E000001F80C000000F818000000FC30000000FC600000007CC00000007F800000007F 800000003F000000003F000000001F000000001F800000003F800000006F80000000CFC0 0000018FC000000387C000000707E000000E03E000000C03F000001803F000003001F000 006001F80000C001F80003C000FC000FE001FC00FFF00FFFC0FFF00FFFC028227FA129> 88 DI< 007FFFFE007FFFFE007F007C00F800FC00F001F800E003F001C007E001C007C003800FC0 03801F8003003F0003007E0000007C000000FC000001F8000003F0000007E000000FC000 000F8000001F8000003F0060007E006000FC006000F800E001F800C003F001C007E001C0 0FC003801F8007801F000F803F001F007E00FF00FFFFFF00FFFFFE001F227DA122>I<00 7E7801FF7807E3F80F81F01F01F01F00F03E01F03E01E07C01E07C01E07C03E0FC03C0F8 03C0F803C3F807C7F80786780F86781F8E3C7FDC1FF3F80FC1F018157F941A>97 D<03E0003FE0003FC00007C00007C00007C0000780000780000780000F80000F00000F00 000F00001F00001E7E001FFF801FC7C03F03C03E03E03C03E03C03E07C03E07803E07803 E07807E0F807C0F007C0F007C0F00F80F00F80701F00783E003C7C003FF0000FC0001323 7EA215>I<003F8000FFE003F1E007C3E00F83E01F03E03E03C03E00007C00007C00007C 0000FC0000F80000F80000F800007800607800E07C01C03E0F801FFF0007F80013157F94 16>I<00000F800000FF800000FF0000001F0000001F0000001F0000001E0000001E0000 003E0000003E0000003C0000003C0000007C0000007C00007E780001FF780007E3F8000F 81F8001F01F0001F00F0003E01F0003E01F0007C01E0007C01E0007C03E000FC03E000F8 03C000F803C300F807C700F807C600780F8600781F8E003C7FDC001FF3F8000FC1F00019 237FA21A>I<007F0003FF800FC1C01F00C03E00C07E00C07C01C0FC0F80FFFF00FFF800 F80000F80000F00000F00000F00000F000C0F801C07803803E1F001FFE0007F00012157E 9417>I<00003F000000FF800001EF800001CF800003CF800003CF000003C0000007C000 00078000000780000007800000078000000F8000000F000001FFFC0001FFFC00000F0000 001F0000001E0000001E0000001E0000001E0000001E0000003E0000003C0000003C0000 003C0000003C0000007C00000078000000780000007800000078000000F8000000F00000 00F0000000F0000000E0000001E0000079E0000079C00000FBC00000FB8000007F000000 3E000000192D7EA218>I<001F9E007FDE01F8FE03E07E07C07C07C03C0F807C0F807C1F 00781F00781F00F83F00F83E00F03E00F03E01F03E01F01E03E01E07E00F1FE007FFE003 F3C00003C00007C00007C0000780780F80780F00F81F00F87E00FFF8003FE000171F8094 18>I<00F800000FF800000FF0000001F0000001F0000001F0000001E0000001E0000001 E0000003E0000003C0000003C0000003C0000007C00000078FC00007BFF00007F8F0000F E078000FC078000F8078000F00F8001F00F0001E00F0001E00F0001E01F0003E01E0003C 01E0003C03E1803C03C3807C03C3007807C700780786007807DE00F803FC007001F00019 237EA21D>I<0038007C007C007800780000000000000000000000000000000007C01FE0 3DE031E071E061E0E3E003C003C007C0078007800F800F0C1F1C1E181E181E381E701FE0 07C00E2280A111>I<0000F00001F00001F00001F00001E0000000000000000000000000 000000000000000000000000001F00007F8000E3C001C3C00383C00303C00707C00007C0 000780000780000F80000F80000F00000F00001F00001F00001E00001E00003E00003E00 003C00003C00007C00007C0000780078F80078F000F9F000FBE000FF80007E0000142C81 A115>I<00F8000FF8000FF00001F00001F00001F00001E00001E00001E00003E00003C0 0003C00003C00007C0000780F80783FC07879C0F8E3C0F1C7C0F387C0F707C1FE0781FE0 001FF8001EFE003E3E003C1F003C0F0C3C0F1C7C1F18781E18781E38780F70F80FE07007 C016237EA21A>I<01F01FF01FE003E003E003E003C003C007C007C0078007800F800F80 0F000F001F001F001E001E003E003E003C003C007C007C00780078C0F8C0F9C0F180F380 FB007F003E000C237EA20F>I<0F81F8001FC7FE0039EF1E0031FC0F0061F80F0061F00F 00E1E01F0003E01E0003C01E0003C01E0003C03E0007C03C0007803C0007807C30078078 700F8078600F00F8E00F00F0C00F00FBC01F007F800E003E001C1580941E>110 D<03E0FC0007F3FF000E7F8F800C7E0780187C07C0187807C038F807C000F807C000F007 C000F007C001F00FC001F00F8001E00F8001E00F8003E01F0003E01F0003E03E0003F07C 0007F8F80007FFE000079F8000078000000F8000000F8000000F0000000F0000001F0000 001F0000001E000000FFE00000FFE000001A1F819419>112 D<007E1801FF7807E3F80F 81F81F01F01F00F03E00F03E01F07C01E07C01E07C01E0FC03E0F803C0F803C0F803C0F8 07C0780F80781F803C7F801FFF800FCF00000F00000F00001F00001E00001E00001E0000 3E00003C0003FFC003FFC0151F7F9416>I<0F87C01FCFE039FC7031F8F071F1F061E1F0 E1E1F003E1E003C00003C00003C00007C0000780000780000780000F80000F00000F0000 0F00001F00000E00001415809416>I<007E0001FF8003C3800387800707800707800783 0007E00007FC0007FF0007FF0001FF80001F80380780780780F80700F80700F80E00F03E 007FF8001FE00011157E9417>I<00E001E001E001E003E003E003C003C007C007C0FFFC FFFC0F800F800F000F001F001F001E001E003E003E003C003C0C7C1C7C18783878707CE0 3FC01F800E1F7F9E12>I<07C01E001FE01E003CF03E0038F03C0070F03C0061F03C00E1 E07C0001E0780003E0780003C0780003C0F80007C0F0000780F0000780F0C00781F1C007 81E1800781E1800783E38007CFF70003FEFE0001F87C001A1580941D>I<07C0781FE0FC 3CF0FC38F0FC70F0FC61F03CE1E03C01E01803E01803C01803C03807C030078030078070 0780600780E00781C007838007C78003FF0000FC001615809418>I<07E1F00FF3F81C7F 1C383E3C703C7C603C7CE07C7C007C7800780000780000F80000F00000F00078F00C79F0 1CF9F018F9E038F3F070E7F0E07F7FC03C1F8016157E941C>120 D<07C00F1FE00F3CF01F38F01E70F01E61F01EE1E03E01E03C03E03C03C03C03C07C07C0 780780780780780780F80780F00781F00781F007C7F003FFE000FDE00001E00003E00803 C03E07C03E07803E0F003C1E00387C003FF8000FE000181F809418>I<01F03003F87007 FEE00FFFC00E0FC00C0380000700000E00001C0000380000700001E0000380000700600E 00E01C01C01F03C03FFF807BFF80E1FE00C07C0014157E9417>I E /Fn 93 128 df5 D<03FFFE0003FFFE00000F8000000F8000000F8000000F8000000F8000003F E00001FFFC0007EFBF001F8F8FC03E0F83E03E0F83E07C0F81F0FC0F81F8FC0F81F8FC0F 81F8FC0F81F8FC0F81F8FC0F81F87C0F81F03E0F83E03E0F83E01F8F8FC007EFBF0001FF FC00003FE000000F8000000F8000000F8000000F8000000F800003FFFE0003FFFE001D22 7DA124>8 D<00FFFF8000FFFF800003E0000003E0000003E0000003E000FE03E07FFE03 E07F7F03E0FE3F03E0FC1F03E0F81F03E0F81F03E0F81F03E0F81F03E0F81F03E0F81F03 E0F81F03E0F80F83E1F80F83E1F00F83E1F007C3E3E007E3E7E003F3EFC001FFFF80007F FE00000FF8000003E0000003E0000003E0000003E0000003E00000FFFF8000FFFF802022 7DA127>I<001FF00000FFFE0003F83F8007E00FC00FC007E01F8003F01F0001F03F0001 F83F0001F87E0000FC7E0000FC7E0000FC7E0000FC7E0000FC7E0000FC7E0000FC3E0000 F83F0001F83F0001F81F0001F01F0001F00F8003E00F8003E0078003C0078003C003C007 80C3C00786C1C0070661C0070C60E00E0C60E00E0C7FE00FFC7FE00FFC3FE00FF83FE00F F81F237EA224>I<000FF0FC003FFBFE00F83F9F01F07F1F03E07E1F07C07E1F07803C00 07803C0007803C0007803C0007803C0007803C0007803C0007803C00FFFFFFE0FFFFFFE0 07803C0007803C0007803C0007803C0007803C0007803C0007803C0007803C0007803C00 07803C0007803C0007803C0007803C0007803C0007803C0007803C0007803C007FF1FFE0 7FF1FFE0202380A21D>I<000FE000003FF80000F8380001F07C0003E07C0007C07C0007 807C0007803800078000000780000007800000078000000780000007800000FFFFFC00FF FFFC0007807C0007803C0007803C0007803C0007803C0007803C0007803C0007803C0007 803C0007803C0007803C0007803C0007803C0007803C0007803C0007803C0007803C007F F1FFC07FF1FFC01A2380A21C>I<000FFC00007FFC0000F87C0001F07C0003E07C0007C0 7C0007803C0007803C0007803C0007803C0007803C0007803C0007803C0007803C00FFFF FC00FFFFFC0007803C0007803C0007803C0007803C0007803C0007803C0007803C000780 3C0007803C0007803C0007803C0007803C0007803C0007803C0007803C0007803C000780 3C007FF1FFC07FF1FFC01A2380A21C>I<0007F03F80003FFCFFE000FC3FE0E001F03FC1 F003E03F81F007C03F01F007803E01F007801E00E007801E000007801E000007801E0000 07801E000007801E000007801E0000FFFFFFFFF0FFFFFFFFF007801E01F007801E00F007 801E00F007801E00F007801E00F007801E00F007801E00F007801E00F007801E00F00780 1E00F007801E00F007801E00F007801E00F007801E00F007801E00F007801E00F007801E 00F07FF0FFC7FF7FF0FFC7FF282380A22A>I18 D<038007800F801F001E003C007800F000E0004000090A76A219 >I22 D<781E00FC3F00FE3F80FE3F80FE3F807E1F800601 800601800601800E03800C03001C0700180600380E00701C0060180011107EA219>34 D<78FCFEFEFE7E0606060E0C1C1838706007107CA20E>39 D<00E001E001C00380070007 000E001E001C001C003C00380038007800780070007000F000F000F000F000F000F000F0 00F000F000F000F000F000F000F000F000F0007000700078007800380038003C001C001C 001E000E0007000700038001C001E000E00B327CA413>II<00C00000E00000C00000C00000 C000E0C1C0F8C7C07CCF801FFE0007F80001E00007F8001FFE007CCF80F8C7C0E0C1C000 C00000C00000C00000E00000C00012157DA419>I<000180000001800000018000000180 000001800000018000000180000001800000018000000180000001800000018000000180 00000180000001800000018000FFFFFFFFFFFFFFFF000180000001800000018000000180 000001800000018000000180000001800000018000000180000001800000018000000180 0000018000000180000001800020227D9C27>I<78FCFEFEFE7E0606060E0C1C18387060 07107C850E>II<78FCFCFCFC7806067C850E>I<0000C000 00C00001C0000180000180000380000300000300000700000600000600000E00000C0000 0C00001C0000180000180000380000300000300000700000600000600000E00000C00001 C0000180000180000380000300000300000700000600000600000E00000C00000C00001C 0000180000180000380000300000300000700000600000600000E00000C00000C0000012 317DA419>I<01F80007FE000F0F001E07803C03C03C03C07801E07801E07801E07801E0 F801F0F801F0F801F0F801F0F801F0F801F0F801F0F801F0F801F0F801F0F801F0F801F0 F801F0F801F07801E07801E07801E07801E03C03C03C03C01E07800F0F0007FE0001F800 14227EA019>I<00C001C007C0FFC0FBC003C003C003C003C003C003C003C003C003C003 C003C003C003C003C003C003C003C003C003C003C003C003C003C003C003C003C0FFFFFF FF10217CA019>I<03F8000FFF003C3F80380FC07007E0E007E0F803F0FC03F0FC03F0FC 01F0FC01F03003F00003F00003E00007E00007C0000FC0000F80001F00003E00007C0000 F80001E00001C0000380300700300E00301C00703800607FFFE0FFFFE0FFFFE0FFFFE014 217EA019>I<03F8000FFF001E1F80380FC07807E07C07E07E07E07E03E07C07E03807E0 0007C00007C0000F80001F80003F0003FC0003FC00001F00000F800007C00007E00003E0 0003F03003F0FC03F0FC03F0FC03F0FC03F0F807E06007E0780FC03E1F801FFF0003F800 14227EA019>I<000380000380000780000F80000F80001F80003F8000378000678000E7 8000C7800187800387800307800607800E07800C0780180780380780300780600780E007 80FFFFFCFFFFFC00078000078000078000078000078000078000078000FFFC00FFFC1621 7FA019>I<1800C01E03C01FFFC01FFF801FFF001FFC001FF00018000018000018000018 000018000019FC001FFF001F0F801E07C01C03C01803E00001E00001F00001F00001F000 01F0F801F0F801F0F801F0F801F0F803E0E003E07007C03807803E1F000FFE0003F80014 227EA019>I<007F0001FF8003E1C00780E00F03E01E03E03E03E03C03E03C00007C0000 780000782000F9FE00FBFF00FF0780FE03C0FC03E0FC01E0FC01F0F801F0F801F0F801F0 F801F0F801F07801F07801F07801F07C01E03C03E03C03C01E07C00F0F8007FE0001FC00 14227EA019>I<6000007000007FFFF07FFFF07FFFF07FFFE06000C0E001C0C00380C003 00C00700000E00001C0000180000380000300000700000700000E00000E00001E00001E0 0001C00003C00003C00003C00003C00007C00007C00007C00007C00007C00007C00007C0 0003800014237DA119>I<01FC0007FF001F0F803C03C03801C07001E07000E07000E078 00E07C01E07E01C07F83C03FC7801FFF000FFE0007FE0003FF000FFF801E7FC03C1FE078 0FE07003F0F001F0E000F0E00070E00070E00070F000707000E07800E03C03C01F0F800F FF0003FC0014227EA019>I<01F80007FE001F0F001E07803C03C07C03C07801E0F801E0 F801E0F801E0F801F0F801F0F801F0F801F0F801F0F803F07803F07C03F03C07F01E0FF0 0FFDF007F9F00041E00001E00001E00003E07C03C07C03C07C07807C0F00701F003C3E00 1FF8000FE00014227EA019>I<78FCFCFCFC7800000000000000000078FCFCFCFC780615 7C940E>I<78FCFCFCFC7800000000000000000078F8FCFCFC7C0C0C0C1C181838707060 061F7C940E>I61 D<000FF000007FFE0001 F81F8003C003C0078001E00E0000701C0000381C07E038383FF81C307C3C0C70F80E0E60 F007C661E007C6E1E003C7C3E003C3C3C003C3C3C003C3C3C003C3C3C003C3C3C003C3C3 E003C3E1E003C361E007C360F007C770F80FC6307C3FEE383FF9FC1C07E0F81C0000000E 0000000780000003C0001F01F803FE007FFFF0000FFE0020237DA227>64 D<000180000003C0000003C0000003C0000007E0000007E0000007E000000FF000000DF0 00000DF000001DF8000018F8000018F8000038FC0000307C0000307C0000707E0000603E 0000603E0000C03F0000C01F0000C01F0001801F8001FFFF8001FFFF8003000FC0030007 C0030007C0060003E0060003E0060003E00E0001F01F0003F0FFC01FFFFFC01FFF20237E A225>II<000FF018003FFC1800FE0F3801F003F807E001F80FC000F80F8000F81F0000783F 0000783E0000387E0000387E0000387C000018FC000018FC000018FC000000FC000000FC 000000FC000000FC000000FC000000FC000000FC0000187C0000187E0000187E0000183E 0000383F0000301F0000700F8000600FC000E007E001C001F0038000FE0F00003FFE0000 0FF8001D247DA224>IIII<0007F80C00003FFE0C0000FE0F9C0001F803FC0003E000FC0007C0007C000F 80007C001F80003C003F00003C003E00001C007E00001C007E00001C007E00000C00FC00 000C00FC00000C00FC00000000FC00000000FC00000000FC00000000FC00000000FC0000 0000FC001FFF80FC001FFF807E00007C007E00007C007E00007C003F00007C003F00007C 001F80007C000F80007C0007C0007C0003E000FC0001F801FC0000FE079C00003FFF0C00 0007F8000021247DA227>III<01FFFC01FFFC0007C00007C00007C00007C00007C00007C0 0007C00007C00007C00007C00007C00007C00007C00007C00007C00007C00007C00007C0 0007C00007C00007C00007C00007C03007C0FC07C0FC07C0FC07C0FC0FC0F80F80701F80 7C3F001FFC0007F00016237FA11A>IIIII<000FF000003FFC0000F81F0001E0078007C003E0078001E00F8001F01F0000F8 3E00007C3E00007C7E00007E7E00007E7C00003E7C00003EFC00003FFC00003FFC00003F FC00003FFC00003FFC00003FFC00003FFC00003FFC00003F7C00003E7E00007E7E00007E 3E00007C3F0000FC1F0000F80F8001F00F8001F007C003E003F00FC000F81F00003FFC00 000FF00020247DA227>II82 D<03F8300FFE701E0FF03C03F07801F07800F0F00070F000 70F00030F00030F80030F80000FC00007E00007FE0003FFE003FFF801FFFC007FFE000FF F0001FF00003F00001F80000F80000F8C00078C00078C00078E00078E00070F000F0F800 F0FC01E0FF07C0E7FF80C1FE0015247DA21C>I<7FFFFFFC7FFFFFFC7C07C07C7007C01C 6007C00C6007C00CE007C00EE007C00EC007C006C007C006C007C006C007C0060007C000 0007C0000007C0000007C0000007C0000007C0000007C0000007C0000007C0000007C000 0007C0000007C0000007C0000007C0000007C0000007C0000007C0000007C0000007C000 0007C00003FFFF8003FFFF801F227EA124>IIII<7FFE0FFF007FFE0FFF0007F0 03F00003F001C00001F001800001F803800000FC030000007C060000007E0E0000003E0C 0000003F180000001FB80000000FB00000000FE000000007E000000003E000000003F000 000001F000000003F800000007FC000000067C0000000C7E0000001C3F000000181F0000 00301F800000700F800000600FC00000C007E00001C003E000018003F000038001F8000F C003FC00FFF00FFFC0FFF00FFFC022227FA125>I<7FFFFE7FFFFE7F007E7C00FC7000F8 7001F8E001F0E003F0C007E0C007E0C00FC0C00FC0001F80001F00003F00007E00007E00 00FC0000FC0001F80001F00303F00307E00307E0030FC0030FC0071F80071F00063F000E 3E000E7E003EFC00FEFFFFFEFFFFFE18227DA11E>90 DI<0C03001C0700380E00300C00701C00601800E03800C03000C03000C030 00FC3F00FE3F80FE3F80FE3F807E1F803C0F0011107BA219>II<06000F001F8039C070E0E070C0300C077AA119>I<7C7CFEFE7C7C 07067DA10E>I<0FF8003FFF003E1F803E07803E03C01C03C00003C00007C003FFC00FFF C03F83C07E03C07C03C0FC03C0F803C6F803C6F807C6FC0FC67E1FEE3FF9FC0FE0F01715 7F9419>97 D<0F0000FF0000FF00001F00000F00000F00000F00000F00000F00000F0000 0F00000F00000F00000F00000F1FC00F7FF00FE0F80FC07C0F003E0F003E0F001E0F001F 0F001F0F001F0F001F0F001F0F001F0F001F0F001E0F003E0F803C0FC07C0EE1F80E7FF0 0C1FC018237FA21C>I<03FE000FFF801F0F803E0F803C0F807C0700780000F80000F800 00F80000F80000F80000F80000F800007C00007C00C03E01C03F03801F87800FFE0003FC 0012157E9416>I<0000F0000FF0000FF00001F00000F00000F00000F00000F00000F000 00F00000F00000F00000F00000F003F8F00FFFF01F87F03E01F07C00F07C00F07800F0F8 00F0F800F0F800F0F800F0F800F0F800F0F800F07800F07C00F07C01F03E03F01F0FF80F FEFF03F8FF18237EA21C>I<01FC0007FF001F0F803E07C03C03C07C03E07801E0FFFFE0 FFFFE0F80000F80000F80000F80000F800007C00007C00603E00E03F01C01FC3C007FF00 01FC0013157F9416>I<003F0000FF8003E7C007C7C00787C00F87C00F00000F00000F00 000F00000F00000F00000F00000F0000FFF800FFF8000F00000F00000F00000F00000F00 000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00 00FFF000FFF000122380A20F>I<00007803F9FC0FFFDC1F1FBC3E0F9C7C07C07C07C07C 07C07C07C07C07C07C07C03E0F801F1F003FFE003BF8003000003800003C00003FFF803F FFE01FFFF03FFFF87801F8F0003CE0003CE0001CE0001CF0003C7000387C00F83F03F00F FFC003FF0016217F9519>I<0F000000FF000000FF0000001F0000000F0000000F000000 0F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F1FC000 0F7FF0000FF1F0000FC0F8000F8078000F8078000F0078000F0078000F0078000F007800 0F0078000F0078000F0078000F0078000F0078000F0078000F0078000F0078000F007800 FFF3FF80FFF3FF8019237FA21C>I<1E003F003F003F003F001E00000000000000000000 00000000000F007F007F001F000F000F000F000F000F000F000F000F000F000F000F000F 000F000F000F00FFE0FFE00B2280A10D>I<007800FC00FC00FC00FC0078000000000000 0000000000000000003C03FC03FC007C003C003C003C003C003C003C003C003C003C003C 003C003C003C003C003C003C003C003C003C003C003CF83CF87CF878F8F07FE03F800E2C 83A10F>I<0F0000FF0000FF00001F00000F00000F00000F00000F00000F00000F00000F 00000F00000F00000F00000F0FFC0F0FFC0F07E00F07C00F07000F0E000F1C000F38000F 7C000FFC000FFE000FBF000F1F000F0F800F0FC00F07C00F03E00F03E00F03F0FFE7FEFF E7FE17237FA21A>I<0F00FF00FF001F000F000F000F000F000F000F000F000F000F000F 000F000F000F000F000F000F000F000F000F000F000F000F000F000F000F000F000F000F 000F00FFF0FFF00C2380A20D>I<0F1FE07F80FF7FF9FFE0FFF0FBC3E01FC07F01F00F80 3E00F00F803E00F00F003C00F00F003C00F00F003C00F00F003C00F00F003C00F00F003C 00F00F003C00F00F003C00F00F003C00F00F003C00F00F003C00F00F003C00F00F003C00 F0FFF3FFCFFFFFF3FFCFFF28157F942B>I<0F1FC000FF7FF000FFF1F0001FC0F8000F80 78000F8078000F0078000F0078000F0078000F0078000F0078000F0078000F0078000F00 78000F0078000F0078000F0078000F0078000F007800FFF3FF80FFF3FF8019157F941C> I<00FC0007FF800F03C01E01E03C00F0780078780078F8007CF8007CF8007CF8007CF800 7CF8007CF8007C7800787C00F83C00F01E01E00F03C007FF8000FC0016157F9419>I<0F 1FC0FF7FF0FFE1F80FC0FC0F007E0F003E0F003E0F001F0F001F0F001F0F001F0F001F0F 001F0F001F0F003E0F003E0F807C0FC0FC0FE1F80F7FF00F1FC00F00000F00000F00000F 00000F00000F00000F00000F0000FFF000FFF000181F7F941C>I<03F8600FFCE01F8EE0 3F07E07E03E07C01E07C01E0F801E0F801E0F801E0F801E0F801E0F801E0F801E07C01E0 7C01E07E03E03E07E01F0FE00FFDE003F1E00001E00001E00001E00001E00001E00001E0 0001E00001E0001FFE001FFE171F7E941A>I<0F3E00FFFF80FFEF801FCF800F8F800F87 000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00 000F0000FFF800FFF80011157F9414>I<0FEC3FFC783CF01CE00CE00CF000FE00FFE07F F83FFC0FFC007EC01EC00EE00EE00EF01EFC3CFFF8CFE00F157E9414>I<060006000600 060006000E000E000E001E003E00FFFCFFFC1E001E001E001E001E001E001E001E001E00 1E001E0C1E0C1E0C1E0C1E0C1F1C0F380FF803E00E1F7F9E13>I<0F007800FF07F800FF 07F8001F00F8000F0078000F0078000F0078000F0078000F0078000F0078000F0078000F 0078000F0078000F0078000F0078000F00F8000F00F8000F01F8000783FC0007FF7F8001 FC7F8019157F941C>III<7FE3FF007FE3FF000FC1F80003C1E00003C1C00001E18000 00F3000000FF0000007E0000003C0000001E0000003E0000007F00000067800000C7C000 01C3C0000381E0000780F0001F81F800FFC3FF80FFC3FF80191580941A>II<3FFFC03FFFC03C0FC0380F8030 1F00703E00607E00607C0060F80001F00003F00003E0C007C0C00F80C01F80C01F01C03E 01807C0380FC0780FFFF80FFFF8012157F9416>II<1E023F EF7FFEF7FC407810057CA019>126 D<781E7C3EFC3FFC3F7C3E781E10067CA119>I E /Fo 59 128 df<000FF800007FFE0001FC1F0003F03F0007E03F000FE03F000FC07F80 0FC03F000FC03F000FC00C000FC000000FC000000FC0FF80FFFFFF80FFFFFF80FFFFFF80 0FC01F800FC01F800FC01F800FC01F800FC01F800FC01F800FC01F800FC01F800FC01F80 0FC01F800FC01F800FC01F800FC01F800FC01F800FC01F800FC01F807FF8FFF07FF8FFF0 7FF8FFF01C237FA220>12 D<0038007800F001E003C003C007800F800F001F001E003E00 3E003C007C007C007C007C00FC00F800F800F800F800F800F800F800F800F800F800F800 FC007C007C007C007C003C003E003E001E001F000F000F80078003C003C001E000F00078 00380D317BA416>40 DI<3C7EFFFFFFFF7E3C08087C8710>46 D<00FF0003FFC00FC3F01F81F83F 00FC3E007C7E007E7E007E7E007E7E007EFE007FFE007FFE007FFE007FFE007FFE007FFE 007FFE007FFE007FFE007FFE007FFE007FFE007F7E007E7E007E7E007E3E007C3F00FC1F 81F80FC3F003FFC000FF0018207E9F1D>48 D<001C00007C0001FC00FFFC00FFFC00FFFC 0001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC 0001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC0001FC007FFF F07FFFF07FFFF014207C9F1D>I<03FE001FFFC03FFFE07E3FF0FF0FF8FF07F8FF03FCFF 03FCFF01FC7E01FC3C03FC0003FC0003F80007F80007F0000FE0001FC0001F80003E0000 7C0000F81C01E01C03C01C07801C0F003C1FFFFC3FFFF87FFFF8FFFFF8FFFFF8FFFFF8FF FFF816207D9F1D>I<01FF0007FFE00FFFF01F87F83F83F83FC1FC3FC1FC3FC1FC3FC1FC 3FC3FC1F83F80007F80007F0001FE001FF8001FF800007F00001FC0001FE0000FE3E00FF 7F00FFFF80FFFF80FFFF80FFFF80FFFF81FE7F01FE7F07FC3FFFF80FFFE001FF0018207E 9F1D>I<0000F00000F00001F00003F00007F0000FF0001FF0001FF0003BF00073F000E3 F001C3F00383F00783F00703F00E03F01C03F03803F07003F0F003F0FFFFFFFFFFFFFFFF FF0007F00007F00007F00007F00007F00007F000FFFF00FFFF00FFFF18207E9F1D>I<30 00303E01F03FFFF03FFFE03FFFC03FFF803FFE003FF80038000038000038000038000039 FE003FFFC03F0FE03C07F03803F80003F80003FC0003FC3C03FC7E03FCFF03FCFF03FCFF 03FCFF03F8FE07F87C07F07E1FF03FFFC01FFF8007FC0016207D9F1D>I<001FC000FFF0 03FFF807F0780FC0FC1F81FC3F01FC3F01FC7F01FC7E00F87E0000FE0C00FE7FE0FEFFF0 FFC1FCFF80FCFF007EFF007EFE007FFE007FFE007FFE007F7E007F7E007F7E007F7E007F 3F007E3F80FC1FC1FC0FFFF803FFE000FF8018207E9F1D>I<7000007C00007FFFFF7FFF FF7FFFFF7FFFFE7FFFFC7FFFF8FFFFF8F000F0E001E0E003C0E00780E00F00000E00001E 00003E00003C00007C00007C0000FC0000F80001F80001F80001F80003F80003F80003F8 0003F80003F80003F80003F80003F80001F00018227DA11D>I<00FF8003FFE00FFFF01F 83F81F00FC3E00FC3E007C3F007C3FC07C3FE0FC3FF8F83FFFF81FFFE01FFFC00FFFF007 FFF80FFFFC1FFFFE3E3FFE7C0FFFFC03FFF801FFF8007FF8003FF8001FF8001FFC001E7E 003E7F80FC3FFFF80FFFE001FF8018207E9F1D>I<01FF0007FFE01FFFF03F83F83F01FC 7E00FC7E007EFE007EFE007EFE007FFE007FFE007FFE007FFE007F7E00FF7E00FF3F01FF 3F83FF0FFF7F07FE7F00307F00007E1F007E3F80FE3F80FC3F80FC3F81F83F03F01F0FF0 1FFFC00FFF8003FC0018207E9F1D>I<000078000000007800000000FC00000000FC0000 0000FC00000001FE00000001FE00000003FF00000003FF00000003FF000000077F800000 077F8000000F7FC000000E3FC000000E3FC000001C1FE000001C1FE000003C1FF0000038 0FF00000380FF000007007F800007007F80000F007FC0000FFFFFC0000FFFFFC0001FFFF FE0001C001FE0003C001FF00038000FF00038000FF000700007F80FFF00FFFFCFFF00FFF FCFFF00FFFFC26227EA12B>65 DI<0003FF00C0003FFFE1C000FFFFF7C001FF80FFC007FC003FC0 0FF8001FC01FE0000FC01FC00007C03FC00003C07F800003C07F800003C07F800001C0FF 000001C0FF00000000FF00000000FF00000000FF00000000FF00000000FF00000000FF00 000000FF00000000FF000000007F800001C07F800001C07F800001C03FC00003C01FC000 03801FE00007800FF8000F0007FC003E0001FF80FC0000FFFFF800003FFFE0000003FF00 0022227DA129>III I<0003FF8060001FFFE0E000FFFFFBE001FF80FFE007FC001FE00FF8000FE01FE00007E0 1FE00003E03FC00001E03F800001E07F800001E07F800000E0FF000000E0FF00000000FF 00000000FF00000000FF00000000FF00000000FF00000000FF000FFFFEFF000FFFFEFF00 0FFFFE7F80001FE07F80001FE03F80001FE03FC0001FE01FE0001FE01FE0001FE00FF800 1FE007FC003FE001FF807FE000FFFFFFE0001FFFF3E00003FF806027227DA12D>III76 DII<0007FE0000003FFFC0 0000FE07F00003F801FC0007F000FE000FE0007F001FC0003F801F80001F803F80001FC0 3F80001FC07F00000FE07F00000FE07F00000FE0FF00000FF0FF00000FF0FF00000FF0FF 00000FF0FF00000FF0FF00000FF0FF00000FF0FF00000FF0FF00000FF07F00000FE07F80 001FE07F80001FE03F80001FC01FC0003F801FC0003F800FE0007F0007F000FE0003F801 FC0000FE07F000003FFFC0000007FE000024227DA12B>II<0007FE0000003FFFC00000FE07F000 03F801FC0007F000FE000FE0007F001FC0003F801FC0003F803F80001FC03F80001FC07F 80001FE07F00000FE07F00000FE0FF00000FF0FF00000FF0FF00000FF0FF00000FF0FF00 000FF0FF00000FF0FF00000FF0FF00000FF0FF00000FF07F00000FE07F80001FE07F8000 1FE03F80001FC01F81F81F801FC3FC3F800FE70E7F0007F607FE0003FE07FC0000FF07F0 00003FFFC0000007FFC030000003E030000001F0F0000001FFF0000001FFF0000001FFE0 000000FFE0000000FFE00000007FC00000007F800000001F00242C7DA12B>II<01FF06000FFFCE 001FFFFE003F83FE007E007E007E003E00FC001E00FC001E00FC000E00FE000E00FF0000 00FFC00000FFFE00007FFFE0007FFFF8003FFFFC003FFFFE001FFFFE0007FFFF0001FFFF 00000FFF800000FF8000007F8000003F80E0001F80E0001F80E0001F80F0001F00F8003F 00FC003F00FF80FE00FFFFFC00E7FFF000C0FFC00019227DA120>I<7FFFFFFFC07FFFFF FFC07FFFFFFFC07E03FC0FC07C03FC03C07803FC03C07003FC01C0F003FC01E0F003FC01 E0E003FC00E0E003FC00E0E003FC00E0E003FC00E00003FC00000003FC00000003FC0000 0003FC00000003FC00000003FC00000003FC00000003FC00000003FC00000003FC000000 03FC00000003FC00000003FC00000003FC00000003FC00000003FC00000003FC000001FF FFF80001FFFFF80001FFFFF80023217EA028>II89 D<3FFFFFF03FFFFFF03FFFFFF03FC03FE03F003FE03E 007FC03C00FF807800FF807801FF007803FE007003FE007007FC00700FF800000FF80000 1FF000003FF000003FE000007FC00000FFC00000FF807001FF007001FF007003FE007007 FC007007FC00F00FF800F01FF000E01FF001E03FE003E07FC007E07FC03FE0FFFFFFE0FF FFFFE0FFFFFFE01C227DA123>I<0FFE00003FFFC0007F0FE0007F03F0007F03F8007F01 F8007F01F8003E01F8000001F80000FFF8000FFFF8001FE1F8007F81F8007E01F800FE01 F800FC01F800FC01F800FC03F800FE07F8007F0FFF803FFCFF800FF03F8019167E951C> 97 DI<01FF8007FFE01FC7F03F87F03F07F07F07F07E07F0FE03E0FE0000FE0000 FE0000FE0000FE0000FE0000FE00007F00007F00003F80383F80781FE1F007FFE001FF80 15167E9519>I<0001FF000001FF000001FF0000003F0000003F0000003F0000003F0000 003F0000003F0000003F0000003F0000003F0000003F0001FF3F0007FFFF001FE1FF003F 807F003F003F007F003F007E003F00FE003F00FE003F00FE003F00FE003F00FE003F00FE 003F00FE003F00FE003F007E003F007E003F003F007F003F80FF001FC3FFE007FFFFE001 FE3FE01B237EA220>I<01FE0007FF800FC7E01F03E03F01F07E01F07E01F8FE00F8FE00 F8FFFFF8FFFFF8FE0000FE0000FE0000FE00007E00007F00383F00381F80700FC1F007FF C000FF0015167E951A>I<001FE000FFF003F9F807F3F807E3F80FC3F80FC3F80FC1F00F C0000FC0000FC0000FC0000FC000FFFE00FFFE00FFFE000FC0000FC0000FC0000FC0000F C0000FC0000FC0000FC0000FC0000FC0000FC0000FC0000FC0000FC0000FC0000FC0007F FC007FFC007FFC0015237EA212>I<01FE1F800FFFFFC01F87F1C03F03F7C03E01F7C07E 01F9007E01F8007E01F8007E01F8007E01F8003E01F0003F03F0001F87E0001FFFC00039 FE000038000000380000003C0000003FFFF0003FFFFE001FFFFF001FFFFF803FFFFF807C 003FC0FC000FC0F80007C0F80007C0F80007C0FC000FC07E001F803F807F000FFFFC0001 FFE0001A217F951D>II<1E003F007F807F807F807F803F001E0000000000000000 0000000000FF80FF80FF801F801F801F801F801F801F801F801F801F801F801F801F801F 801F801F801F80FFF0FFF0FFF00C247EA30F>I108 DII<00FF0007FFE00FC3F01F00F83F00FC7E007E7E007E7E007EFE007F FE007FFE007FFE007FFE007FFE007FFE007F7E007E7E007E3F00FC1F00F80FC3F007FFE0 00FF0018167E951D>I I<01FE070007FF8F001FE3FF003FC0FF003F807F007F003F007F003F00FE003F00FE003F 00FE003F00FE003F00FE003F00FE003F00FE003F00FE003F007F003F007F003F003F807F 003F80FF001FE3FF0007FFBF0001FE3F0000003F0000003F0000003F0000003F0000003F 0000003F0000003F000001FFE00001FFE00001FFE01B207E951E>II<0FFB803FFF807C 0F80780780F00380F00380F80000FF8000FFF8007FFE007FFF001FFF8007FFC0003FC0E0 0FC0E007C0F003C0F003C0F80780FE0F80FFFF00C7FC0012167E9517>I<038000038000 0380000380000380000780000780000780000F80001F80003FFF00FFFF00FFFF001F8000 1F80001F80001F80001F80001F80001F80001F80001F80001F80001F83801F83801F8380 1F83801F83801FC7800FCF0007FE0001FC0011207F9F16>III120 DI<7FFFF07FFFF07C0FF0780FE0701FC0F03F80E03F80 E07F00E0FE0000FE0001FC0003F80007F07007F0700FE0701FC0F01FC0F03F80E07F01E0 FF07E0FFFFE0FFFFE014167E9519>I<3C0F007E1F80FE1FC0FE1FC0FE1FC07E1F803C0F 0012077BA31D>127 D E end %%EndProlog %%BeginSetup %%Feature: *Resolution 300dpi TeXDict begin %%PaperSize: A4 %%EndSetup %%Page: 1 1 1 0 bop 40 149 a Fo(LONG)19 b(TIME)g(ST)-5 b(ABILITY)19 b(OF)g(SOME)g(SMALL)i(AMPLITUDE)f(SOLUTIONS)359 209 y(IN)f(NONLINEAR)f (SCHR)989 197 y(\177)982 209 y(ODINGER)h(EQUA)-5 b(TIONS)792 468 y Fn(Dario)17 b(BAMBUSI*)514 528 y(Dipartimen)o(to)f(di)h (Matematica)g(dell'Univ)o(ersit\022)-25 b(a)577 588 y(Via)17 b(Saldini)h(50,)f(20133)f(MILANO,)h(Italy)l(.)0 904 y Fo(Abstract.)50 b Fn(W)l(e)13 b(consider)g(small)f(p)q(erturbations)g (of)h(the)h(Zakharo)o(v-Shabat)d(nonlinear)i(Sc)o(hr\177)-25 b(odinger)0 963 y(equation)17 b(on)g([0)p Fm(;)8 b(\031)r Fn(])16 b(with)i(v)m(anishing)f(or)f(p)q(erio)q(dic)i(b)q(oundary)d (conditions;)i(w)o(e)g(pro)o(v)o(e)e(a)i(Nekhoroshev)0 1023 y(t)o(yp)q(e)e(result)h(for)e(solutions)i(starting)f(in)h(the)f (neigh)o(b)q(ourho)q(o)q(d)f(\(in)h(the)h Fm(H)1398 1005 y Fl(1)1435 1023 y Fn(top)q(ology\))h(of)e(the)g(ma)s(jorit)o(y)0 1083 y(of)h(small)f(amplitude)h(\014nite)g(dimensional)f(in)o(v)m (arian)o(t)h(tori)h(of)f(the)f(linearized)j(system.)i(More)15 b(precisely)0 1143 y(w)o(e)g(will)j(pro)o(v)o(e)c(that)i(along)f(the)h (considered)f(solutions)g(all)i(the)f(actions)g(of)f(the)h(linearized)h (system)d(are)0 1202 y(appro)o(ximativ)o(ely)19 b(constan)o(t)f(up)g (to)h(times)f(gro)o(wing)g(exp)q(onen)o(tially)j(with)f(the)f(in)o(v)o (erse)f(of)h(a)f(suitable)0 1262 y(small)e(parameter.)0 1578 y Fo(1.)57 b(In)n(tro)r(duction)100 1713 y Fn(W)l(e)16 b(consider)h(the)f(equation)607 1859 y Fm(iu)653 1866 y Fk(t)681 1859 y Fn(+)11 b Fm(u)760 1866 y Fk(xx)818 1859 y Fj(\006)g Fm(\031)r(u)p Fj(j)p Fm(u)p Fj(j)984 1838 y Fl(2)1019 1859 y Fn(=)j Fm(\017)1098 1825 y(@)s(H)t Fn(\()p Fm(x;)8 b(u;)k Fn(\026)-29 b Fm(u)q Fn(\))p 1098 1848 244 2 v 1191 1893 a Fm(@)6 b Fn(\026)-28 b Fm(u)1364 1859 y(;)504 b Fn(\(1)p Fm(:)p Fn(1\))0 1996 y(with)21 b(v)m(anishing)f(or)f(p)q(erio)q(dic)i(b)q(oundary)d(conditions)j(on)e ([0)p Fm(;)8 b(\031)r Fn(].)31 b(Here)20 b Fm(H)k Fn(is)c(a)f (su\016cien)o(tly)i(smo)q(oth)0 2056 y(real)d(v)m(alued)g(function)h (\(the)f(co)q(e\016cien)o(t)g Fm(\031)i Fn(in)e(fron)o(t)f(of)h(the)g (third)g(order)e(term)g(has)h(b)q(een)h(in)o(tro)q(duced)0 2116 y(for)e(future)g(con)o(v)o(enience\).)100 2176 y(When)c Fm(\017)i Fn(=)g(0)e(eq.)h(\(1.1\))g(reduces)f(to)h(the)f(Zakharo)o (v-Shabat)f(\(ZS\))j(equation)f(whic)o(h)g(is)g(w)o(ell)h(kno)o(wn)0 2236 y(to)i(b)q(e)g(in)o(tegrable.)23 b(So,)15 b(it)i(has)e(an)h (in\014nite)h(sequence)e(of)h(constan)o(t)f(of)h(motion)f(and,)h(for)f (an)o(y)g(\014nite)i Fm(N)5 b Fn(,)0 2295 y(there)16 b(exist)h(families)f(of)g Fm(N)21 b Fn(dimensional)16 b(in)o(v)m(arian)o(t)g(tori)g(\(\014nite)h(gap)e(solutions\))i(whic)o (h,)e(as)h Fm(N)j Fj(!)14 b(1)p Fn(,)0 2355 y(\014ll)19 b(densely)f(the)g(phase)f(space.)25 b(In)18 b(the)g(case)f Fm(\017)g Fj(6)p Fn(=)e(0,)j(it)h(is)f(kno)o(wn)f(that)h(there)g (exists)h(a)f(sequence)f Fm(\017)1946 2362 y Fk(N)0 2415 y Fn(suc)o(h)f(that,)i(if)g Fj(j)p Fm(\017)p Fj(j)e(\024)e Fm(\017)419 2422 y Fk(N)457 2415 y Fn(,)k(then)f(the)h(ma)s(jorit)o(y)e (of)i(the)f(ab)q(o)o(v)o(e)g Fm(N)23 b Fn(dimensional)17 b(tori)h(can)f(b)q(e)h(con)o(tin)o(ued)0 2475 y(to)f(in)o(v)m(arian)o (t)g(tori)h(of)f(\(1.1\))524 2457 y Fl([10)p Fk(;)p Fl(14)p Fk(;)p Fl(11)p Fk(;)p Fl(6])741 2475 y Fn(\(see)g(also)g([7]\).)23 b(Ho)o(w)o(ev)o(er)16 b Fm(\017)1271 2482 y Fk(N)1323 2475 y Fj(!)e Fn(0)j(as)f Fm(N)k Fj(!)14 b(1)p Fn(,)j(and)f(therefore)0 2534 y(the)j(ab)q(o)o(v)o(e)g(results)g(giv)o(e)h(a)f(go)q(o)q(d)g (insigh)o(t)h(on)f(the)g(dynamics)f(of)h(\(1.1\))h(only)g(in)f(compact) f(subsets)g(of)0 2594 y(the)h(phase)e(space.)27 b(On)18 b(the)h(con)o(trary)l(,)e(v)o(ery)h(littl)q(e)j(is)e(kno)o(wn)1203 2576 y Fl([5])1265 2594 y Fn(on)g(the)f(dynamics)f(in)i(op)q(en)f (subsets)0 2654 y(of)f(the)g(phase)e(space)h(\(a)h(detailed)h (discussion)f(of)f(the)h(pap)q(er)f([5])g(will)j(b)q(e)e(giv)o(en)g(in) g(sect.)g(4\).)p 0 2714 600 2 v 50 2770 a(*)25 b Fi(e-mail)p Fn(:)e(BAMBUSI@MA)l(T.UNIMI.IT)p eop %%Page: 2 2 2 1 bop 0 30 a Fn(2)851 b Fh(D.Bam)o(busi)100 149 y Fn(In)22 b(the)i(presen)o(t)e(pap)q(er)g(w)o(e)h(giv)o(e)h(a)f(b)q(ound)f(on)g (the)i(long)f(time)g(di\013usion)h(of)f(the)g(actions)h(for)0 209 y(initial)18 b(data)d(in)g(op)q(en)f(subsets)g(of)h(the)g(phase)e (space,)i(in)g(the)g(spirit)g(of)g(Nekhoroshev's)f(theorem)1856 191 y Fl([12)p Fk(;)p Fl(13])1970 209 y Fn(.)0 269 y(Actually)l(,)h (our)c(result)h(restricts)g(to)g(solutions)h(whic)o(h)f(\(i\))h(are)f (of)g(small)f(amplitude,)i(and)e(\(ii\))j(corresp)q(ond)0 329 y(to)23 b(initial)j(data)c(close)i(to)f(the)g(ma)s(jorit)o(y)e(of)i (\014nite)g(dimensional)g(in)o(v)m(arian)o(t)g(tori)g(of)g(the)g (linearized)0 389 y(system,)15 b(namely)h(those)g(ful\014lling)j(a)e (suitable)h(\(generic\))f(nonresonance)e(condition.)100 450 y(The)c(reasons)f(of)i(the)g(limitations)h(\(i\))h(and)d(\(ii\))j (are)d(as)g(follo)o(ws:)21 b(due)11 b(to)h(\(i\))h(w)o(e)f(a)o(v)o(oid) f(all)j(di\016culties)0 509 y(related)22 b(to)g(the)g(actual)g(in)o (tegration)h(of)f(the)f(ZS)h(equation.)38 b(Indeed,)22 b(as)f(p)q(oin)o(ted)h(out)g(in)g([11],)g(the)0 569 y(ZS)g(equation)g (can)f(b)q(e)h(appro)o(ximated,)e(in)i(a)g(neigh)o(b)q(ourho)q(o)q(d)e (of)i(the)f(origin,)j(with)e(the)g(in)o(tegrable)0 629 y(nonlinear)g(system)f(giv)o(en)i(b)o(y)e(the)i(\014rst)e(order)g (Birkho\013)i(normal)d(form)g(of)j(the)f(ZS)g(equation)h(itself)0 689 y(\(a)o(v)o(eraged)15 b(system\).)21 b(The)16 b(adv)m(an)o(tage)g (is)g(that)h(the)f(Hamiltonian)h(function)f(of)h(the)f(a)o(v)o(eraged)f (system)0 749 y(can)g(b)q(e)h(explicitl)q(y)j(calculated)e(in)f(terms)e (of)i(the)g(action)g(v)m(ariables.)23 b(In)15 b(particular)h(this)g (allo)o(ws)h(one)e(to)0 808 y(study)c(the)g(dep)q(endence)g(of)h(the)f (frequency)g(on)g(the)h(initial)i(datum,)d(and)f(to)i(sho)o(w)e(that)i (the)f(ma)s(jorit)o(y)f(\(in)0 868 y(a)h(measure)f(sense\))h(of)g (\014nite)i(dimensional)e(in)o(v)m(arian)o(t)h(tori)g(ha)o(v)o(e)f (some)e(go)q(o)q(d)j(nonresonance)e(prop)q(erties.)0 928 y(Our)20 b(result)g(concerns)g(precisely)i(the)f(dynamics)e(in)i(a) g(neigh)o(b)q(ourho)q(o)q(d)e(of)i(suc)o(h)e(nonresonan)o(t)g(tori.)0 988 y(It)i(turns)f(out)h(that)g(the)g(condition)h(of)f(b)q(eing)h (close)f(to)g(suc)o(h)f(tori)i(can)e(b)q(e)h(form)o(ulated)f(in)h (terms)e(of)0 1047 y(in)o(v)m(arian)o(t)e(tori)g(of)g(the)g(linearized) h(system;)d(this)i(giv)o(es)g(rise)g(to)g(condition)h(\(ii\).)100 1139 y(The)g(pap)q(er)g(is)i(organized)e(as)h(follo)o(ws.)29 b(In)19 b(sect.)g(2)g(w)o(e)f(giv)o(e)i(a)f(precise)g(statemen)o(t)e (of)i(our)f(main)0 1198 y(result)h(\(theorem)f(2.1\),)h(and)g(in)g (sect.)g(3)g(w)o(e)g(presen)o(t)f(the)h(sc)o(heme)e(of)i(its)h(pro)q (of.)28 b(Sect.)20 b(4)f(con)o(tains)f(a)0 1258 y(discussion)11 b(on)h(the)f(relation)i(of)f(the)g(presen)o(t)e(result)i(with)g(some)e (recen)o(t)h(w)o(orks)g(on)g(long)h(time)g(b)q(eha)o(viour)0 1318 y(in)j(nonlinear)h(PDE's.)k(Sections)c(5,)f(6,)g(7)f(constitute)i (the)g(tec)o(hnical)g(part)e(of)h(the)g(pap)q(er.)21 b(In)15 b(sect.)g(5)g(w)o(e)0 1378 y(p)q(erform)h(a)i(transformation)e (putting)i(the)h(system)d(in)j(\014rst)e(order)g(Birkho\013)h(normal)f (form,)f(and)h(giv)o(e)0 1437 y(the)i(Hamiltonian)g(a)f(form)e(adapted) i(for)g(the)h(subsequen)o(t)d(steps)i(of)h(the)f(pro)q(of.)27 b(In)18 b(sect.)h(6)f(w)o(e)g(state)0 1497 y(the)f(abstract)g(normal)e (form)g(theorem)g(whic)o(h)i(constitutes)h(the)f(main)f(tec)o(hnical)j (result)e(of)g(the)g(pap)q(er)0 1557 y(as)h(w)o(ell)i(as)d(the)i(main)e (step)h(for)g(the)h(pro)q(of)f(of)g(theorem)f(2.1;)h(in)h(our)e (opinion)i(this)g(could)g(ha)o(v)o(e)e(some)0 1617 y(indep)q(enden)o(t) f(in)o(terest.)22 b(In)15 b(that)h(section)h(w)o(e)e(also)i(giv)o(e)f (the)g(original)i(elemen)o(ts)d(leading)h(to)h(the)f(pro)q(of)0 1677 y(of)k(the)g(normal)d(form)h(result.)31 b(In)20 b(sect.)f(7)h(w)o(e)f(apply)h(the)g(normal)e(form)g(theorem)f(to)j(the) g(nonlinear)0 1736 y(Sc)o(hr\177)-25 b(odinger)30 b(equation,)17 b(and)e(complete)h(the)g(pro)q(of)g(of)g(theorem)e(2.1.)22 b(W)l(e)16 b(also)g(giv)o(e)h(a)f(sligh)o(tly)i(more)0 1796 y(precise)f(description)g(of)f(the)h(dynamics.)0 1887 y Fg(A)m(cknow)s(le)m(dgements)24 b Fn(I)f(w)o(ould)g(lik)o(e)i (to)f(thank)f(An)o(tonio)h(Giorgilli)j(for)c(man)o(y)e(fruitful)k (discussions.)0 1947 y(This)h(w)o(ork)e(has)h(b)q(een)g(dev)o(elop)q (ed)h(with)h(the)e(partial)i(supp)q(ort)d(of)i(the)f(gran)o(t)g(EC)g (con)o(tract)g(ER-)0 2007 y(BCHRX)o(CT940460)14 b(for)g(the)h(pro)s (ject)f(\\Stabilit)o(y)j(and)d(univ)o(ersalit)o(y)i(in)f(classical)h (mec)o(hanics",)d(and)h(of)0 2066 y(the)j(gran)o(t)e(CE)i(n.)22 b(CHRX-CT96-0330/DG.)0 2450 y Fo(2.)57 b(Statemen)n(t)19 b(of)f(the)i(main)e(result)100 2586 y Fn(W)l(e)i(come)f(no)o(w)g(to)i (a)f(precise)g(statemen)o(t)f(of)h(our)g(result.)33 b(F)l(or)19 b(de\014niteness)g(w)o(e)h(\014rst)g(consider)0 2646 y(eq.)14 b(\(1.1\))g(with)h(v)m(anishing)f(b)q(oundary)f(conditions)h (on)g([0)p Fm(;)8 b(\031)r Fn(];)14 b(concerning)g(the)g(function)g Fm(H)k Fn(w)o(e)c(assume)0 2706 y(that)j(the)g(application)738 2770 y Fo(C)779 2749 y Fl(2)815 2770 y Fj(3)e Fn(\()p Fm(y)r(;)8 b(z)r Fn(\))15 b Fj(7!)e Fm(H)t Fn(\()p Fm(:;)8 b(y)r(;)g(z)r Fn(\))p eop %%Page: 3 3 3 2 bop 353 30 a Fh(On)16 b(long)h(time)f(stabilit)o(y)j(in)f (nonlinear)e(Sc)o(hr\177)-25 b(odinger)15 b(equation)354 b Fg(3)0 149 y Fn(tak)o(es)18 b(v)m(alues)h(in)f Fm(H)383 131 y Fl(1)379 162 y(0)406 149 y Fn(\([0)p Fm(;)8 b(\031)r Fn(])p Fm(;)g Fo(C)p Fn(\))19 b(\(Sob)q(olev)g(space)f(of)g Fm(L)1064 131 y Fl(2)1104 149 y Fn(functions)g(ha)o(ving)g Fm(L)1516 131 y Fl(2)1556 149 y Fn(deriv)m(ativ)o(es)i(and)d(sat-)0 209 y(isfying)k Fm(u)p Fn(\(0\))d(=)g Fm(u)p Fn(\()p Fm(\031)r Fn(\))h(=)f(0\))i(and)f(moreo)o(v)o(er)d(is)k(analytic)h(in)f (a)f(neigh)o(b)q(ourho)q(o)q(d)g(of)g(the)h(origin)g(as)f(an)0 269 y(application)d(from)c Fo(C)411 251 y Fl(2)448 269 y Fn(to)i Fm(H)551 251 y Fl(1)547 281 y(0)574 269 y Fn(\([0)p Fm(;)8 b(\031)r Fn(])p Fm(;)g Fo(C)p Fn(\).)22 b(W)l(e)15 b(recall)g(that)g(this)f(simply)g(means)e(that,)j(considering)f(the)0 329 y(T)l(a)o(ylor)i(expansion)h(of)g Fm(H)k Fn(in)c Fm(y)i Fn(and)c Fm(z)r Fn(,)i(namely)f Fm(H)t Fn(\()p Fm(x;)8 b(y)r(;)g(z)r Fn(\))17 b(=)1208 291 y Ff(P)1261 344 y Fk(k)q(;l)p Fe(\025)p Fl(0)1369 329 y Fm(H)1410 336 y Fk(k)q(l)1448 329 y Fn(\()p Fm(x)p Fn(\))p Fm(y)1540 311 y Fk(k)1566 329 y Fm(z)1591 311 y Fk(l)1606 329 y Fn(,)g(the)g(series)755 412 y Ff(X)742 519 y Fk(k)q(;l)p Fe(\025)p Fl(0)848 459 y Fj(k)p Fm(H)914 466 y Fk(k)q(l)952 459 y Fn(\()p Fm(:)p Fn(\))p Fj(k)1030 474 y Fk(H)1066 464 y Fd(1)1095 459 y Fj(j)p Fm(y)r Fj(j)1149 439 y Fk(l)1164 459 y Fj(j)p Fm(z)r Fj(j)1217 439 y Fk(k)1882 459 y Fn(\(2)p Fm(:)p Fn(1\))0 622 y(is)g(con)o(v)o(ergen)o(t)e(in)i(a)g(neigh)o(b)q (ourho)q(o)q(d)e(of)i(the)g(origin.)100 681 y(Consider)f(no)o(w)f(the)i (F)l(ourier)f(expansion)g(of)h Fm(u)p Fn(,)f(namely)716 830 y Fm(u)p Fn(\()p Fm(x)p Fn(\))e(=)878 782 y Ff(X)878 890 y Fk(k)q Fe(\025)p Fl(1)959 830 y Fm(u)988 837 y Fk(k)1012 751 y Ff(r)p 1062 751 43 2 v 1071 796 a Fn(2)p 1068 818 31 2 v 1068 864 a Fm(\031)1113 830 y Fn(sin)8 b Fm(k)r(x)17 b(;)613 b Fn(\(2)p Fm(:)p Fn(2\))0 998 y(and)17 b(de\014ne)h(the)g(action)h(v)m(ariables)g Fm(Z)724 1005 y Fk(k)764 998 y Fn(:=)d Fj(j)p Fm(u)876 1005 y Fk(k)900 998 y Fj(j)914 980 y Fl(2)936 998 y Fm(=)p Fn(2)i(of)g(the)g (linearized)i(system.)25 b(Fix)19 b(an)e(arbitrary)h Fm(N)0 1058 y Fn(dimensional)i(diophan)o(tine)f(v)o(ector)h(\()p Fm(!)756 1065 y Fl(1)779 1058 y Fm(;)8 b(:::;)g(!)896 1065 y Fk(N)934 1058 y Fn(\))19 b Fj(2)28 b Fn(I)-9 b(R)1079 1036 y Fk(N)1117 1058 y Fn(,)21 b(namely)e(a)h(v)o(ector)f(suc)o(h)g (that)h(there)g(exist)0 1117 y Fm(\034)f Fj(\025)14 b Fn(0)i(and)g Fm(\014)h(>)c Fn(0)k(with)g(the)g(prop)q(ert)o(y)f(that) 593 1275 y Fj(j)633 1212 y Fk(N)615 1227 y Ff(X)619 1333 y Fk(i)p Fl(=1)695 1275 y Fm(!)726 1282 y Fk(i)743 1275 y Fm(k)769 1282 y Fk(i)785 1275 y Fj(j)e(\025)896 1241 y Fm(\014)p 871 1263 81 2 v 871 1309 a Fj(j)p Fm(k)r Fj(j)927 1294 y Fk(\034)974 1275 y Fm(;)58 b Fj(8)5 b Fm(k)14 b Fj(2)g Fo(Z)1202 1254 y Fk(N)1245 1275 y Fj(n)8 b(f)p Fn(0)p Fj(g)25 b Fn(;)490 b(\(2)p Fm(:)p Fn(3\))0 1432 y(w)o(e)17 b(also)g(assume)d Fm(!)377 1439 y Fk(i)408 1432 y Fm(>)g Fn(0,)j Fm(i)d Fn(=)g(1)p Fm(;)8 b(:::;)g(N)d Fn(.)23 b(A)17 b(rescaling)h(of)f(the)g(v)o(ector)g Fm(!)f Fn(:=)d(\()p Fm(!)1510 1439 y Fl(1)1533 1432 y Fm(;)8 b(:::;)g(!)1650 1439 y Fk(N)1688 1432 y Fm(;)g Fn(0)p Fm(;)g Fn(0)p Fm(;)g Fn(0)p Fm(;)g Fn(0)p Fm(:::)p Fn(\))14 b Fj(2)9 1492 y Fn(I)-9 b(R)55 1470 y Fe(1)97 1492 y Fn(,)18 b(will)j(b)q(e)d(used)g(as)g(reference)f(v)m(alue)j(of)e(the)h (actions.)27 b(As)18 b(an)o(ticipated)i(ab)q(o)o(v)o(e,)e(our)f(main)g (result)0 1551 y(deals)f(with)h(small)f(amplitude)f(solutions.)23 b(In)15 b(the)i(statemen)o(t)d(of)j(the)f(forthcoming)e(theorem)h(the)h (role)0 1611 y(of)h(order)e(of)i(magnitude)e(of)i(the)g(solution)g (will)i(b)q(e)e(pla)o(y)o(ed)f(b)o(y)h(the)f(parameter)f Fm(R)p Fn(.)0 1691 y Fo(Theorem)78 b(2.1.)17 b Fg(Under)i(the)g(ab)m (ove)g(assumptions)h(ther)m(e)f(exist)g(p)m(ositive)h(c)m(onstants)g Fm(\026)1738 1698 y Fe(\003)1760 1691 y Fm(;)8 b(C)1818 1698 y Fl(1)1841 1691 y Fm(;)g(:::;)g(C)1963 1698 y Fl(4)0 1751 y Fg(with)18 b(the)g(fol)s(lowing)g(pr)m(op)m(erty:)25 b(\014x)18 b Fm(R)c(>)g Fn(0)k Fg(and)g(assume)h(that)776 1888 y Fm(\026)14 b Fn(:=)f Fm(R)e Fn(+)997 1854 y Fj(j)p Fm(\017)p Fj(j)p 991 1876 61 2 v 991 1922 a Fm(R)1029 1908 y Fl(5)1071 1888 y Fj(\024)j Fm(\026)1154 1895 y Fe(\003)1195 1888 y Fm(;)0 2019 y Fg(then)k(pr)m(ovide)m(d)i(the)e (initial)h(value)e(of)i(the)e(actions)i(is)g(su\016ciently)e(close)h (to)g Fm(R)1503 2001 y Fl(2)1526 2019 y Fm(!)r Fg(,)g(pr)m(e)m(cisely)h (satis\014es)576 2090 y Ff(X)575 2197 y Fk(k)q Fe(\025)p Fl(1)657 2137 y Fm(k)685 2117 y Fl(2)715 2095 y Ff(\014)715 2125 y(\014)731 2137 y Fm(Z)765 2144 y Fk(k)790 2137 y Fn(\(0\))12 b Fj(\000)f Fm(R)953 2117 y Fl(2)975 2137 y Fm(!)1006 2144 y Fk(k)1031 2095 y Ff(\014)1031 2125 y(\014)1061 2137 y Fj(\024)j Fm(C)1150 2144 y Fl(1)1172 2137 y Fm(\026)1202 2117 y Fl(1)p Fk(=)p Fl(\()p Fk(\034)t Fl(+1\))1348 2137 y Fm(R)1386 2117 y Fl(3)1882 2137 y Fn(\(2)p Fm(:)p Fn(4\))0 2305 y Fg(wher)m(e)20 b Fm(!)173 2312 y Fk(i)206 2305 y Fn(:=)c(0)j Fg(for)g Fm(i)e Fj(\025)f Fm(N)h Fn(+)12 b(1)p Fg(,)20 b(then)f(the)g(value)f(of)i(the)f(actions) i(r)m(emains)f(close)f(to)h Fm(R)1712 2287 y Fl(2)1734 2305 y Fm(!)r Fg(,)g(pr)m(e)m(cisely)0 2365 y(satis\014es)579 2385 y Ff(X)579 2492 y Fk(k)q Fe(\025)p Fl(1)660 2432 y Fm(k)688 2412 y Fl(2)718 2390 y Ff(\014)718 2420 y(\014)735 2432 y Fm(Z)769 2439 y Fk(k)793 2432 y Fn(\()p Fm(t)p Fn(\))12 b Fj(\000)f Fm(R)949 2412 y Fl(2)972 2432 y Fm(!)1003 2439 y Fk(k)1027 2390 y Ff(\014)1027 2420 y(\014)1058 2432 y Fj(\024)i Fm(C)1146 2439 y Fl(2)1168 2432 y Fm(\026)1198 2412 y Fl(1)p Fk(=)p Fl(\()p Fk(\034)t Fl(+1\))1345 2432 y Fm(R)1383 2412 y Fl(3)1882 2432 y Fn(\(2)p Fm(:)p Fn(5\))0 2574 y Fg(for)19 b(exp)m(onential)s(ly)f(long)g(times,)g(namely)g(for)g (al)s(l)g(times)f Fm(t)h Fg(satisfying)644 2727 y Fj(j)p Fm(t)p Fj(j)c(\024)779 2694 y Fm(C)815 2701 y Fl(3)p 763 2716 91 2 v 763 2761 a Fm(\026R)831 2747 y Fl(2)867 2727 y Fn(exp)952 2642 y Ff(")981 2657 y(\022)1023 2694 y Fm(C)1059 2701 y Fl(4)p 1023 2716 58 2 v 1037 2761 a Fm(\026)1087 2657 y Ff(\023)1124 2667 y Fl(1)p Fk(=)p Fl(\()p Fk(\034)t Fl(+1\))1271 2642 y Ff(#)1326 2727 y Fm(:)542 b Fn(\(2)p Fm(:)p Fn(6\))p eop %%Page: 4 4 4 3 bop 0 30 a Fn(4)851 b Fh(D.Bam)o(busi)p 1945 113 39 2 v 1945 147 2 35 v 1982 147 V 1945 149 39 2 v 100 259 a Fn(Consider)16 b(no)o(w)f(the)i Fm(N)5 b Fn(-dimensional)16 b(torus)26 395 y Fj(M)86 402 y Fk(R;!)170 395 y Fn(:=)237 325 y Ff(\032)274 395 y Fm(u)e Fj(2)g Fm(H)409 374 y Fl(1)405 407 y(0)431 395 y Fn(\([0)p Fm(;)8 b(\031)r Fn(])p Fm(;)g Fo(C)p Fn(\))32 b(:)719 361 y Fj(j)p Fm(u)762 368 y Fk(k)786 361 y Fj(j)800 343 y Fl(2)p 719 384 104 2 v 758 429 a Fn(2)842 395 y(=)14 b Fm(R)933 374 y Fl(2)955 395 y Fm(!)986 402 y Fk(k)1027 395 y Fm(;)8 b(k)16 b Fn(=)d(1)p Fm(;)8 b(:::;)g(N)22 b Fn(;)j Fm(u)1384 402 y Fk(k)1422 395 y Fn(=)13 b(0)k Fm(;)25 b(k)15 b Fj(\025)f Fm(N)i Fn(+)11 b(1)1780 325 y Ff(\033)1842 395 y Fm(;)26 b Fn(\(2)p Fm(:)p Fn(7\))0 545 y(and)16 b(denote)g(b)o(y)h Fm(d)353 554 y Fk(H)389 544 y Fd(1)410 545 y Fn(\()p Fm(:;)8 b(:)p Fn(\))17 b(the)g(distance)g(in)g(the)g Fm(H)984 527 y Fl(1)1023 545 y Fn(norm,)d(then)i(the)h(follo)o(wing)i (corollary)e(holds)0 624 y Fo(Corollary)69 b(2.2.)16 b Fg(Pr)m(ovide)m(d)i Fm(\026)c Fj(\024)g Fm(\026)740 631 y Fe(\003)779 624 y Fg(ther)m(e)i(exist)h(p)m(ositive)h(c)m (onstants)f Fm(C)1447 631 y Fl(5)1469 624 y Fm(;)8 b(C)1527 631 y Fl(6)1566 624 y Fg(such)16 b(that,)h(solutions)0 684 y(of)i(\(1.1\))g(c)m(orr)m(esp)m(ond)q(ing)i(to)d(initial)g(data)h Fm(u)844 691 y Fl(0)883 684 y Fg(satisfying)633 801 y Fm(d)659 810 y Fk(H)695 800 y Fd(1)716 801 y Fn(\()p Fm(u)764 808 y Fl(0)786 801 y Fm(;)8 b Fj(M)868 808 y Fk(R;!)939 801 y Fn(\))14 b Fj(\024)g Fm(C)1061 808 y Fl(5)1083 801 y Fm(R)1121 780 y Fl(2)1143 801 y Fm(\026)1173 780 y Fl(1)p Fk(=)p Fl(\()p Fk(\034)t Fl(+1\))1338 801 y Fm(;)0 917 y Fg(r)m(emain)19 b(close)f(to)g Fj(M)411 924 y Fk(R;!)481 917 y Fg(,)g(namely)g(satisfy)615 1040 y Fm(d)641 1049 y Fk(H)677 1039 y Fd(1)699 1040 y Fn(\()p Fm(u)p Fn(\()p Fm(t)p Fn(\))p Fm(;)8 b Fj(M)885 1047 y Fk(R;!)956 1040 y Fn(\))14 b Fj(\024)g Fm(C)1078 1047 y Fl(6)1100 1040 y Fm(R)1138 1020 y Fl(2)1160 1040 y Fm(\026)1190 1020 y Fl(1)p Fk(=)p Fl(\()p Fk(\034)t Fl(+1\))1355 1040 y Fm(;)0 1157 y Fg(for)19 b(exp)m(onential)s(ly)f(long)g(times,)g (namely)g(for)g(times)g(satisfying)i(\(2.6\).)p 1945 1120 39 2 v 1945 1155 2 35 v 1982 1155 V 1945 1157 39 2 v 100 1266 a Fn(A)d(particularly)h(in)o(teresting)f(case)f(of)h (\(1.1\))g(is)758 1388 y Fm(iu)804 1395 y Fk(t)832 1388 y Fn(+)11 b Fm(u)911 1395 y Fk(xx)972 1388 y Fn(=)j Fm(ug)r Fn(\()p Fj(j)o Fm(u)p Fj(j)1154 1363 y Fl(2)1177 1388 y Fn(\))j Fm(;)655 b Fn(\(2)p Fm(:)p Fn(8\))0 1505 y(where)21 b Fm(g)k Fn(:)31 b(I)-9 b(R)22 b Fj(!)31 b Fn(I)-9 b(R)22 b(is)h(analytic)h(in)e(a)g(neigh)o(b)q(ourho)q(o)q(d)f(of)h(the)g (origin,)h(and)f(satis\014es)f Fm(g)r Fn(\(0\))i(=)g(0,)0 1564 y Fm(g)26 1546 y Fe(0)40 1564 y Fn(\(0\))14 b Fj(6)p Fn(=)g(0.)21 b(F)l(or)16 b(suc)o(h)f(an)h(equation)i(w)o(e)e(ha)o(v)o (e)g(the)h(follo)o(wing)0 1644 y Fo(Theorem)106 b(2.3.)24 b Fg(Consider)h(e)m(q.)g(\(2.8\);)k(ther)m(e)c(exist)f(p)m(ositive)i(c) m(onstants)f Fm(R)1598 1651 y Fe(\003)1621 1644 y Fm(;)8 b(C)1679 1651 y Fl(7)1701 1644 y Fm(;)g(:::;)g(C)1823 1651 y Fl(10)1889 1644 y Fg(such)0 1704 y(that)20 b(for)h(any)f(p)m (ositive)i Fm(R)c(<)f(R)615 1711 y Fe(\003)658 1704 y Fg(the)i(fol)s(lowing)i(holds)g(true:)27 b(if)20 b(the)g(initial)h (value)e(of)i(the)f(actions)h(is)0 1764 y(su\016ciently)c(close)i(to)e Fm(R)471 1746 y Fl(2)494 1764 y Fm(!)r Fg(,)h(namely)g(satis\014es)569 1840 y Ff(X)573 1947 y Fk(i)p Fe(\025)p Fl(1)649 1887 y Fm(i)666 1867 y Fl(2)697 1845 y Ff(\014)697 1875 y(\014)714 1887 y Fm(Z)748 1894 y Fk(i)764 1887 y Fn(\(0\))12 b Fj(\000)f Fm(R)927 1867 y Fl(2)949 1887 y Fm(!)980 1894 y Fk(i)997 1845 y Ff(\014)997 1875 y(\014)1027 1887 y Fj(\024)j Fm(C)1116 1894 y Fl(7)1138 1887 y Fm(R)1176 1867 y Fl(3)1199 1887 y Fm(R)1237 1867 y Fl(1)p Fk(=)p Fl(\()p Fk(\034)t Fl(+1\))1401 1887 y Fm(;)467 b Fn(\(2)p Fm(:)p Fn(9\))0 2055 y Fg(then)18 b(one)g(has)573 2067 y Ff(X)576 2174 y Fk(i)p Fe(\025)p Fl(1)653 2115 y Fm(i)670 2094 y Fl(2)701 2072 y Ff(\014)701 2102 y(\014)717 2115 y Fm(Z)751 2122 y Fk(i)768 2115 y Fn(\()p Fm(t)p Fn(\))12 b Fj(\000)e Fm(R)923 2094 y Fl(2)946 2115 y Fm(!)977 2122 y Fk(i)994 2072 y Ff(\014)994 2102 y(\014)1024 2115 y Fj(\024)k Fm(C)1113 2122 y Fl(8)1135 2115 y Fm(R)1173 2094 y Fl(3)1195 2115 y Fm(R)1233 2094 y Fl(1)p Fk(=)p Fl(\()p Fk(\034)t Fl(+1\))1398 2115 y Fm(;)445 b Fn(\(2)p Fm(:)p Fn(10\))0 2259 y Fg(for)19 b(exp)m(onential)s(ly)f(long)g (times,)g(namely)g(for)650 2417 y Fj(j)p Fm(t)p Fj(j)13 b(\024)769 2384 y Fm(C)805 2391 y Fl(9)p 768 2406 61 2 v 768 2452 a Fm(R)806 2437 y Fl(3)842 2417 y Fn(exp)927 2332 y Ff(")956 2347 y(\022)999 2384 y Fm(C)1035 2391 y Fl(10)p 999 2406 78 2 v 1018 2452 a Fm(R)1082 2347 y Ff(\023)1119 2358 y Fl(1)p Fk(=)p Fl(\()p Fk(\034)t Fl(+1\))1266 2332 y Ff(#)1321 2417 y Fm(:)522 b Fn(\(2)p Fm(:)p Fn(11\))p 1945 2543 39 2 v 1945 2578 2 35 v 1982 2578 V 1945 2580 39 2 v 0 2710 a Fg(R)m(emark)71 b(2.4.)18 b Fn(Theorem)d(2.3,)h(exactly)i(in)f(the)g(same)d(form,)h(holds)h(for)g (eq.)h(\(1.1\))f(with)i Fm(\017)c Fn(=)f(1,)k(if)g(the)0 2770 y(series)g(\(2.1\))g(starts)f(with)h(terms)e(of)i(the)f(6)811 2752 y Fk(th)869 2770 y Fn(order)f(\()p Fm(l)d Fn(+)f Fm(k)16 b Fj(\025)d Fn(6\).)p 1945 2733 V 1945 2768 2 35 v 1982 2768 V 1945 2770 39 2 v eop %%Page: 5 5 5 4 bop 354 30 a Fh(On)16 b(long)h(time)f(stabilit)o(y)j(in)e (nonlinear)g(Sc)o(hr\177)-25 b(odinger)15 b(equation)355 b Fn(5)0 149 y Fg(R)m(emark)64 b(2.5.)16 b Fn(Strictly)h(sp)q(eaking)e (theorem)d(2.3)i(is)h(not)g(a)f(corollary)h(of)g(theorem)e(2.1,)h (indeed)h(in)g(this)0 209 y(case)h(one)h(has)f Fm(\017)e Fn(:=)f Fm(R)424 191 y Fl(6)447 209 y Fn(,)j(and)601 352 y Fm(H)t Fn(\()p Fm(u;)c Fn(\026)-29 b Fm(u)p Fn(\))15 b(=)855 318 y(1)p 837 340 61 2 v 837 386 a Fm(R)875 371 y Fl(6)904 352 y Fn([)p Fm(G)p Fn(\()p Fj(j)p Fm(u)p Fj(j)1032 327 y Fl(2)1055 352 y Fn(\))c Fj(\000)g Fm(G)1174 331 y Fe(0)1188 352 y Fn(\(0\))d Fj(j)q Fm(u)p Fj(j)1316 327 y Fl(2)1339 352 y Fn(])16 b Fm(;)0 490 y Fn(where)g Fm(G)g Fn(is)h(a)g(primitiv)o(e)g(of)f Fm(g)i Fn(with)g Fm(G)p Fn(\(0\))c(=)g(0,)i(and)g(therefore)g Fm(H)21 b Fn(dep)q(ends)15 b(also)i(on)f Fm(\017)p Fn(.)22 b(Ho)o(w)o(ev)o(er,) 16 b(as)0 550 y(it)i(will)h(b)q(e)e(clear,)g(the)f(pro)q(of)h(of)f (theorem)f(5.1)i(holds)f(also)h(in)g(the)g(case)f(of)h(eq.)g(\(2.8\).)p 1945 513 39 2 v 1945 548 2 35 v 1982 548 V 1945 550 39 2 v 100 630 a(In)k(the)g(case)h(of)f(p)q(erio)q(dic)i(b)q(oundary)e (conditions)h(theorem)d(2.1)j(\(and)f(also)h(corollary)g(2.2)f(and)0 690 y(theorem)10 b(2.3\))i(hold)h(essen)o(tially)h(unc)o(hanged,)d(the) h(only)h(di\013erence)f(b)q(eing)h(that)f(the)g(F)l(ourier)g(expansion) 0 750 y(has)18 b(to)h(b)q(e)g(done)g(on)f(the)h(basis)g Fm(e)653 732 y Fk(ik)q(x)714 750 y Fn(,)h Fm(k)f Fj(2)f Fo(Z)p Fn(,)h(so)g(that)g(the)g(index)h(in)f(\(2.4\))g(and)g(\(2.5\))g (runs)e(o)o(v)o(er)h Fo(Z)p Fn(,)0 810 y(and)e(the)h(co)q(e\016cien)o (t)g(of)472 767 y Ff(\014)472 797 y(\014)488 810 y Fm(Z)522 817 y Fl(0)556 810 y Fj(\000)11 b Fm(R)644 792 y Fl(2)666 810 y Fm(!)697 817 y Fl(0)720 767 y Ff(\014)720 797 y(\014)753 810 y Fn(is)17 b(1;)f(moreo)o(v)o(er,)e(the)j(v)o(ector)f Fm(!)j Fn(has)c(to)i(satisfy)h Fm(!)1708 817 y Fl(0)1744 810 y Fm(>)c Fn(0.)100 870 y(The)i(pro)q(of)h(of)g(theorem)e(2.1)h (will)j(b)q(e)e(obtained)g(using)g(a)g(metho)q(d)e(whic)o(h)i(w)o(e)g (exp)q(ect)h(to)f(w)o(ork)f(in)0 930 y(quite)i(general)f(situations)g (\(see)g(sect.)g(3)f(for)h(more)d(details\).)100 991 y(F)l(or)e(de\014niteness)h(in)h(all)h(the)f(rest)f(of)h(the)f(pap)q (er)g(w)o(e)g(will)j(consider)d(only)i(eq.)e(\(1.1\))h(with)h(v)m (anishing)0 1050 y(b)q(oundary)g(conditions.)0 1394 y Fo(3.)57 b(Sc)n(heme)19 b(of)f(the)h(pro)r(of)100 1529 y Fn(As)12 b(an)o(ticipated)i(ab)q(o)o(v)o(e)e(w)o(e)g(\014rst)g (consider)g(equation)h(\(1.1\))g(as)f(a)h(p)q(erturbation)f(of)h(the)f (completely)0 1589 y(resonan)o(t)i(linearized)j(system)d Fm(i)9 b Fn(_)-23 b Fm(u)9 b Fn(+)g Fm(u)719 1596 y Fk(xx)780 1589 y Fn(=)k(0,)j(and,)f(follo)o(wing)i([11])e(\(see)h(also)f([6]\),)h (w)o(e)f(p)q(erform)f(a)h(\014rst)0 1649 y(order)h(a)o(v)o(eraging.)24 b(Then)16 b(w)o(e)i(use)e(the)i(nonlinearit)o(y)h(of)e(the)h(a)o(v)o (eraged)e(system)g(to)h(tune)g(suitably)i(the)0 1709 y(frequency)j(of)h(the)f(unp)q(erturb)q(ed)f(motions)g(around)g(whic)o (h)h(w)o(e)g(study)g(the)g(dynamics)f(of)h(equation)0 1768 y(\(1.1\).)28 b(So,)19 b(w)o(e)f(consider)g(the)h(in)o(v)m(arian)o (t)g(torus)e(of)i(the)g(a)o(v)o(eraged)e(system)g(on)h(whic)o(h)h(the)g (actions)g Fm(Z)1960 1775 y Fk(k)0 1828 y Fn(are)d(equal)h(to)f Fm(R)312 1810 y Fl(2)335 1828 y Fm(!)i Fn(\()p Fm(!)g Fn(as)e(in)h(theorem)d(2.1\).)22 b(Expanding)16 b(the)g(Hamiltonian)h (in)g(a)f(neigh)o(b)q(ourho)q(o)q(d)f(of)0 1888 y(this)i(torus)f(w)o(e) g(obtain)439 2046 y Fm(H)i Fn(=)569 1984 y Fk(N)551 1999 y Ff(X)551 2106 y Fk(k)q Fl(=1)634 2046 y Fn(~)-27 b Fm(\027)657 2053 y Fk(k)681 2046 y Fm(I)703 2053 y Fk(k)738 2046 y Fn(+)821 1999 y Ff(X)788 2106 y Fk(k)q Fe(\025)p Fk(N)t Fl(+1)939 2046 y Fn(~)e Fm(!)966 2053 y Fk(k)991 2046 y Fm(Z)1025 2053 y Fk(k)1060 2046 y Fn(+)11 b(higher)16 b(order)g(terms)f Fm(;)336 b Fn(\(3)p Fm(:)p Fn(1\))0 2219 y(where)24 b(~)-28 b Fm(\027)174 2226 y Fk(k)219 2219 y Fn(are)21 b(the)g(frequencies)h(of)f(the)h(motions)e(on)h(the)g (torus)g(and)j(~)-29 b Fm(!)1431 2226 y Fk(k)1477 2219 y Fn(are)21 b(the)g(frequencies)h(the)0 2279 y(small)15 b(oscillati)q(ons)j(around)c(it)j(\(and)e Fm(I)738 2286 y Fk(k)776 2279 y Fn(:=)f Fm(Z)877 2286 y Fk(k)910 2279 y Fj(\000)9 b Fm(R)996 2261 y Fl(2)1019 2279 y Fm(!)1050 2286 y Fk(k)1074 2279 y Fn(\);)17 b(in)f(terms)e(of)i Fm(!)h Fn(suc)o(h)d(frequencies)j(are)e(giv)o(en)0 2338 y(b)o(y)390 2446 y(~)-27 b Fm(\027)413 2453 y Fk(k)450 2446 y Fn(:=)14 b Fm(k)545 2426 y Fl(2)578 2446 y Fn(+)d Fm(R)666 2426 y Fl(2)715 2384 y Fk(N)696 2399 y Ff(X)700 2505 y Fk(i)p Fl(=1)777 2446 y Fm(!)808 2453 y Fk(i)824 2446 y Fn(\(4)h Fj(\000)e Fm(\016)951 2453 y Fk(ik)990 2446 y Fn(\))17 b Fm(;)62 b Fn(~)-29 b Fm(!)1129 2453 y Fk(k)1168 2446 y Fn(:=)13 b Fm(k)1262 2426 y Fl(2)1295 2446 y Fn(+)e Fm(R)1383 2426 y Fl(2)1432 2384 y Fk(N)1414 2399 y Ff(X)1417 2505 y Fk(i)p Fl(=1)1494 2446 y Fn(4)p Fm(!)1550 2453 y Fk(i)1583 2446 y Fm(:)285 b Fn(\(3)p Fm(:)p Fn(2\))100 2590 y(Subsequen)o(tly)23 b(w)o(e)h(dev)o(elop)g (normal)f(form)f(theory)i(for)f(system)g(\(3.1\).)44 b(F)l(rom)22 b(the)i(tec)o(hnical)0 2650 y(p)q(oin)o(t)15 b(of)g(view)h(this)g(is)f(the)g(most)f(di\016cult)i(step,)e(the)i(main) d(di\016cult)o(y)j(b)q(eing)g(related)f(to)g(the)g(fact)h(that)4 2710 y(~)-29 b Fm(!)31 2717 y Fk(k)69 2710 y Fj(!)14 b(1)e Fn(when)f Fm(k)k Fj(!)f(1)p Fn(.)20 b(T)l(o)11 b(illustrate)j(this)e(p)q(oin)o(t)h(remark)c(that)j(\(at)h(least)f (formally\))g(it)h(is)g(p)q(ossible)f(to)0 2770 y(in)o(tro)q(duce)i (suitable)h(action)g(angle)g(v)m(ariables)g(\()p Fm(J)928 2777 y Fl(1)950 2770 y Fm(;)8 b(:::;)g(J)1064 2777 y Fe(1)1106 2770 y Fn(;)g Fm( )1160 2777 y Fl(1)1183 2770 y Fm(;)g(:::;)g( )1301 2777 y Fe(1)1343 2770 y Fn(\))14 b(for)g(the)g(main)f(part)h(of)g(\(3.1\))h(b)o(y)p eop %%Page: 6 6 6 5 bop 0 30 a Fn(6)851 b Fh(D.Bam)o(busi)0 149 y Fn(de\014ning)11 b Fm(J)211 156 y Fk(k)249 149 y Fn(:=)i Fj(\000)p Fm(I)376 156 y Fk(k)401 149 y Fn(+)q(4\()485 112 y Ff(P)537 124 y Fk(N)537 164 y(i)p Fl(=1)613 149 y Fm(I)635 156 y Fk(i)652 149 y Fn(+)692 112 y Ff(P)744 164 y Fk(i)p Fe(\025)p Fk(N)t Fl(+1)885 149 y Fm(Z)919 156 y Fk(i)936 149 y Fn(\),)g Fm(k)i Fn(=)e(1)p Fm(;)8 b(:::;)g(N)d Fn(,)13 b Fm(J)1286 156 y Fk(N)t Fl(+1)1388 149 y Fn(:=)1454 112 y Ff(P)1507 124 y Fk(N)1507 164 y(i)p Fl(=1)1582 149 y Fm(i)1599 131 y Fl(2)1621 149 y Fm(I)1643 156 y Fk(i)1660 149 y Fn(+)1700 112 y Ff(P)1752 164 y Fk(i)p Fe(\025)p Fk(N)t Fl(+1)1894 149 y Fm(i)1911 131 y Fl(2)1934 149 y Fm(Z)1968 156 y Fk(i)0 209 y Fn(and)j(completing)g(to)h(a)g (canonical)g(transformation.)j(In)d(suc)o(h)e(a)i(w)o(a)o(y)f(system)f (\(3.1\))i(tak)o(es)f(the)h(form)527 392 y Fm(R)565 371 y Fl(2)614 329 y Fk(N)596 344 y Ff(X)596 452 y Fk(k)q Fl(=1)677 392 y Fm(!)708 399 y Fk(k)732 392 y Fm(J)760 399 y Fk(k)795 392 y Fn(+)11 b Fm(J)873 399 y Fk(N)t Fl(+1)972 392 y Fn(+)g(higher)16 b(order)g(terms)e Fm(:)425 b Fn(\(3)p Fm(:)p Fn(3\))100 577 y(Pro)o(vided)20 b(\()p Fm(!)365 584 y Fl(1)388 577 y Fm(;)8 b(:::;)g(!)505 584 y Fk(N)542 577 y Fn(\))22 b(is)f(diophan)o(tine)g(w)o(e)g(are)g(th)o (us)f(reduced)g(to)h(a)g(p)q(erturbation)f(of)i(an)e(in-)0 637 y(tegrable)i(iso)q(c)o(hronous)d(system)h(with)i(a)f(\014nite)h(n)o (um)o(b)q(er)c(of)j(nonresonan)o(t)e(frequencies)1706 619 y Fl(1)1729 637 y Fn(.)36 b(Th)o(us)20 b(one)0 697 y(exp)q(ects)d(that)g(it)g(should)f(b)q(e)g(p)q(ossible)i(to)e(put)g (the)h(system)e(in)h(normal)f(form)g(up)g(to)i(an)f(exp)q(onen)o (tially)0 757 y(small)g(remainder)e(in)i(order)f(to)h(conclude)h(that)f Fm(J)952 764 y Fl(1)974 757 y Fm(;)8 b(:::;)g(J)1088 764 y Fk(N)t Fl(+1)1192 757 y Fn(are)15 b(appro)o(ximate)g(constan)o(t) g(of)h(motion.)0 817 y(Ho)o(w)o(ev)o(er,)f(it)i(is)g(immediate)e(to)h (realize)h(that,)f(since)h(the)f(\015o)o(w)f(generated)g(b)o(y)h Fm(J)1524 824 y Fk(N)t Fl(+1)1628 817 y Fn(is)g(essen)o(tially)i(the)0 876 y(\015o)o(w)f(of)h(the)g(linear)h(Sc)o(hr\177)-25 b(odinger)33 b(equation,)19 b(the)f(angle)g Fm( )1150 883 y Fk(N)t Fl(+1)1257 876 y Fn(is)g Fg(a)i(c)m(ontinuous)f(non)h (di\013er)m(entiable)0 936 y Fn(function)c(of)f(the)g(phase)f(p)q(oin)o (t.)22 b(As)15 b(a)g(consequence)g(the)g(p)q(erturbation,)g(when)f (expressed)g(in)i(terms)d(of)0 996 y(these)h(action)g(angle)h(v)m (ariables)f(dep)q(ends)f(only)h(con)o(tin)o(uously)g(on)f(the)h(angle)h Fm( )1494 1003 y Fk(N)t Fl(+1)1582 996 y Fn(.)21 b(In)14 b(particular)g(this)0 1056 y(has)k(the)h(consequence)g(that)g(one)g (cannot)f(use)h(F)l(ourier)f(series)h(to)g(solv)o(e)g(the)g (homological)h(equation,)0 1115 y(and)c(to)h(dev)o(elop)g(normal)e (form)f(theory)l(.)100 1180 y(W)l(e)24 b(solv)o(e)h(the)f(ab)q(o)o(v)o (e)g(di\016cult)o(y)h(b)o(y)e(in)o(tro)q(ducing)i(a)f(new)g(metho)q(d)f (for)g(the)i(solution)g(of)f(the)0 1240 y(homological)15 b(equation)g(whic)o(h)g(a)o(v)o(oids)f(the)g(in)o(tro)q(duction)h(of)g (the)g(angle)g Fm( )1428 1247 y Fk(N)t Fl(+1)1516 1240 y Fn(,)g(and)f(ev)o(en)g(an)g(implicit)0 1299 y(use)21 b(of)g(the)h(F)l(ourier)e(expansion)h(in)h(suc)o(h)e(an)h(angle)h (\(see)f(lemma)e(6.4)i(b)q(elo)o(w\).)37 b(Suc)o(h)21 b(a)g(metho)q(d)f(is)0 1359 y(based)c(on)g(the)h(fact)g(that)g(there)f (is)h(only)h Fg(one)f Fn(bad)e(angle.)100 1424 y(W)l(e)22 b(p)q(oin)o(t)h(out)g(that)g(one)f(could)g(pro)q(ceed)h(also)f(in)h(a)g (di\013eren)o(t)f(w)o(a)o(y)l(,)h(namely)f(exploiting)j(the)0 1483 y(fact)19 b(that)f(the)g(dynamics)f(generated)h(b)o(y)g Fm(J)846 1490 y Fk(N)t Fl(+1)951 1483 y Fn(is)h(m)o(uc)o(h)c(faster)j (than)g(the)g(dynamics)f(generated)h(b)o(y)0 1549 y Fm(R)38 1531 y Fl(2)69 1512 y Ff(P)121 1524 y Fk(N)121 1564 y(k)q Fl(=1)205 1549 y Fm(!)236 1556 y Fk(k)260 1549 y Fm(J)288 1556 y Fk(k)312 1549 y Fn(.)38 b(W)l(e)23 b(think)g(that)f(theorem)e (6.2)i(giving)h(a)f(normal)e(form)h(result)h(for)f(Hamiltonian)0 1609 y(systems)e(of)h(the)h(form)d(\(3.3\))j(\(without)h(making)d(use)h (of)h(the)f(di\013eren)o(t)h(time)f(scales)h(quoted)f(ab)q(o)o(v)o(e\)) 0 1668 y(could)d(b)q(e)g(in)o(teresting)g(in)g(itself.)100 1733 y(Besides)i(the)g(ab)q(o)o(v)o(e)f(di\016cult)o(y)h(there)g(is)g (the)g(w)o(ell)h(kno)o(wn)e(problem)f(that)i(action)g(v)m(ariables)h (are)0 1793 y(singular)d(at)h(the)g(origin,)g(but)f(suc)o(h)f(a)h (problem)f(can)h(b)q(e)g(solv)o(ed)h(b)o(y)f(the)g(same)f(metho)q(d)g (dev)o(elop)q(ed)i(in)0 1853 y([3].)31 b(So,)20 b(w)o(e)g(are)f(able)i (to)f(pro)o(v)o(e)e(a)i(normal)e(form)g(result)i(in)g(the)g(vicinit)o (y)j(of)d(a)g(nonresonan)o(t)d(torus.)0 1912 y(This)d(allo)o(ws)g(to)f (conclude)h(that)g(the)f(functions)h Fm(J)942 1919 y Fl(1)964 1912 y Fm(;)8 b(:::;)g(J)1078 1919 y Fk(N)t Fl(+1)1179 1912 y Fn(are)13 b(small)g(for)g(exp)q(onen)o(tially)j(long) e(times)0 1972 y(pro)o(vided)i(the)h(solution)g(do)q(es)g(not)f(lea)o (v)o(e)h(the)g(set)g(where)f(the)h(normal)e(form)f(holds.)100 2037 y(W)l(e)24 b(p)q(oin)o(t)h(out)g(that)g(the)f(ab)q(o)o(v)o(e)g (pro)q(cedure)g(can)g(b)q(e)h(used)e(also)i(to)g(study)f(p)q (erturbation)g(of)0 2096 y(in)o(tegrable)13 b(hamiltonian)g(PDE's)g (whic)o(h)f(\(i\))j(are)d(nonlinear)h(and)g(\(ii\))i(ha)o(v)o(e)d(an)g (equilibrium)i(p)q(oin)o(t)f(with)0 2156 y(the)k(prop)q(ert)o(y)g(that) g(the)h(sp)q(ectrum)d(of)i(small)g(oscillations)j(is)e(an)e(in\014nite) j(sequence)e(of)g(real)h(n)o(um)o(b)q(ers)0 2216 y(whic)o(h)k(are)h (linear)g(com)o(bination)f(with)h(in)o(teger)g(co)q(e\016cien)o(ts)g (of)f(a)h Fg(\014nite)f Fn(n)o(um)o(b)q(er)d(of)k(fundamen)o(tal)0 2276 y(frequencies.)100 2340 y(The)e(\014nal)g(step)g(to)h(pro)o(v)o(e) e(theorem)f(2.1)i(is)h(to)g(use)e(the)i(functions)f Fm(J)1469 2347 y Fl(1)1491 2340 y Fm(;)8 b(:::;)g(J)1605 2347 y Fk(N)t Fl(+1)1714 2340 y Fn(as)21 b(Ly)o(apunof)0 2400 y(functions)e(in)h(order)d(to)i(pro)o(v)o(e)f(that)h Fg(al)s(l)g Fn(the)g(actions)g(are)g(appro)o(ximate)e(constan)o(ts)h (of)h(motions)f(\(see)0 2460 y(lemma)j(7.4\))i(and)f(the)h(solution)h (do)q(es)f(not)g(lea)o(v)o(e)g(the)h(set)f(where)f(the)h(normal)f(form) f(holds)i(up)f(to)0 2519 y(exp)q(onen)o(tially)d(long)e(times.)p 0 2595 600 2 v 52 2632 a Fl(1)100 2650 y Fn(Actually)k(there)f(could)f (b)q(e)h(some)e(resonances)g(b)q(et)o(w)o(een)h Fm(R)1252 2632 y Fl(2)1274 2650 y Fm(!)1305 2657 y Fk(i)1341 2650 y Fn(on)g(the)h(one)f(side)h(and)f(1)g(on)g(the)0 2710 y(other)f(side,)g(ho)o(w)o(ev)o(er)f(w)o(e)g(will)k(pro)o(v)o(e)c(that) h(due)g(to)g(the)g Fm(R)1138 2692 y Fl(2)1179 2710 y Fn(factor)g(suc)o(h)e(resonances)h(are)g(of)i(so)e(high)0 2770 y(order)e(that)i(they)g(do)g(not)f(matter.)p eop %%Page: 7 7 7 6 bop 354 30 a Fh(On)16 b(long)h(time)f(stabilit)o(y)j(in)e (nonlinear)g(Sc)o(hr\177)-25 b(odinger)15 b(equation)355 b Fn(7)0 149 y Fo(4.)57 b(Discussion)100 284 y Fn(W)l(e)13 b(compare)e(here)i(our)f(result)i(with)g(some)e(recen)o(t)g(w)o(orks)h (on)g(long)g(time)g(b)q(eha)o(viour)g(in)h(nonlinear)0 344 y(PDE's.)21 b(In)c(particular)g(w)o(e)f(will)j(consider)d([5])g (and)g([4].)100 403 y(In)i([5])g(\(see)h(also)g([6]\))g(Bourgain)f(pro) o(v)o(ed)f(a)i(general)g(theorem)e(on)h(the)h(long)g(time)f(b)q(eha)o (viour)h(in)0 463 y(the)e(nonlinear)g(Sc)o(hr\177)-25 b(odinger)31 b(equation)424 608 y Fm(i)9 b Fn(_)-23 b Fm(u)10 b Fn(+)h Fm(u)559 615 y Fk(xx)617 608 y Fj(\000)g Fm(V)g Fn(\()p Fm(x)p Fn(\))p Fm(u)k Fn(=)f Fm(\017)896 574 y(@)s(H)t Fn(\()p Fm(u;)e Fn(\026)-29 b Fm(u)p Fn(\))p 896 596 193 2 v 963 642 a Fm(@)6 b Fn(\026)-28 b Fm(u)1111 608 y(;)58 b(u)p Fn(\(0\))14 b(=)f Fm(u)p Fn(\()p Fm(\031)r Fn(\))i(=)e(0)k Fm(;)321 b Fn(\(4)p Fm(:)p Fn(1\))0 743 y(where)20 b Fm(V)32 b Fn(is)21 b(a)f(t)o(ypical)i(smo)q(oth)d(p)q (oten)o(tial.)36 b(Bourgain's)20 b(theorem)e(applies)k(to)f Fm(H)1628 725 y Fk(s)1670 743 y Fn(\()p Fm(s)g Fj(\025)f Fn(1\))h(initial)0 803 y(data)d(suc)o(h)g(that)g(\\the)h(norm)d(stored) i(in)h(the)f(high)h(frequency)f(mo)q(des")f(is)i(v)o(ery)f(small,)h (namely)f(suc)o(h)0 863 y(that)f(the)g(relation)697 872 y Ff(v)697 900 y(u)697 929 y(u)697 959 y(t)p 749 872 369 2 v 779 917 a(X)749 1024 y Fk(i)p Fe(\025)p Fk(N)t Fl(+1)889 964 y Fm(i)906 950 y Fl(2)p Fk(s)953 930 y Fj(j)p Fm(u)996 937 y Fk(i)1012 930 y Fn(\(0\))p Fj(j)1089 916 y Fl(2)p 953 953 159 2 v 1020 998 a Fn(2)1131 964 y Fj(\024)d Fm(C)t(\017)1244 943 y Fk(M)1882 964 y Fn(\(4)p Fm(:)p Fn(2\))0 1126 y(holds)j(for)g(some)f(p)q(ositiv)o(e)j(\(large\)) f Fm(M)23 b Fn(\(here)17 b Fm(u)908 1133 y Fk(i)924 1126 y Fn(\(0\))h(is)g(the)g(comp)q(onen)o(t)d(of)j(the)f(initial)k(datum)15 b(on)i(the)0 1186 y Fm(i)p Fj(\000)p Fn(th)h(eigen)o(v)o(ector)h(of)f Fj(\000)p Fm(@)505 1193 y Fk(xx)565 1186 y Fn(+)12 b Fm(V)f Fn(\).)28 b(The)18 b(conclusion)g(is)h(that)g(the)f(solution)h (is)g Fm(O)q Fn(\()p Fm(\017)1632 1167 y Fk(M)1677 1186 y Fn(\))g(close)g(\(in)g(the)0 1245 y Fm(H)45 1227 y Fk(s)82 1245 y Fn(top)q(ology\))d(to)g(a)f(quasip)q(erio)q(dic)i (motion)d(up)g(to)h(times)g(of)g(order)f Fm(\017)1338 1227 y Fe(\000)p Fk(M)1414 1245 y Fn(.)21 b(Bourgain's)14 b(theorem)g(do)q(es)0 1305 y(not)i(apply)g(directly)i(to)e(eq.)g (\(1.1\),)g(but)g(w)o(e)f(think)i(that)f(it)h(should)e(apply)i(to)f(a)f (neigh)o(b)q(ourho)q(o)q(d)g(of)h(the)0 1365 y(manifold)g Fj(M)266 1372 y Fk(R;!)337 1365 y Fn(,)g(at)h(least)h(in)f(the)g(case)f (of)h(v)m(anishing)g(b)q(oundary)e(conditions.)100 1425 y(The)d(tec)o(hnique)h(of)g(the)g(presen)o(t)f(pap)q(er)g(allo)o(ws)h (to)g(impro)o(v)o(e)e(Bourgain's)h(result)h(in)g(the)g(case)g(of)f Fm(H)1961 1407 y Fl(1)0 1484 y Fn(initial)21 b(data;)e(the)f(impro)o(v) o(emen)n(t)e(b)q(eing)i(essen)o(tially)i(that)f(condition)g(\(4.2\))f (on)g(the)g(initial)k(datum)16 b(is)0 1544 y(remo)o(v)o(ed,)c(and)h (that)i(the)f(time)h(scales)f(on)g(whic)o(h)g(the)h(dynamics)e(is)h (con)o(trolled)h(are)f(m)o(uc)o(h)d(longer)k(than)0 1604 y(those)g(considered)g(b)o(y)h(Bourgain.)21 b(Moreo)o(v)o(er)14 b(the)i(presen)o(t)e(metho)q(d)h(allo)o(ws)h(to)g(deal)g(with)h(the)f (case)f(of)0 1664 y(p)q(erio)q(dic)g(b)q(oundary)d(conditions,)j(whic)o (h)e(seems)f(di\016cult)i(to)g(b)q(e)f(treated)h(with)g(Bourgain's)f (tec)o(hnique.)0 1724 y(Ho)o(w)o(ev)o(er,)19 b(in)h(the)g(general)g (case)f(of)h(p)q(erturbations)f(of)h(linear)g(systems)e(\(lik)o(e)k (eq.)e(\(4.1\)\))g(Bourgain's)0 1783 y(metho)q(d)15 b(is)i(m)o(uc)o(h)d (more)h(general)i(\(in)g(a)g(measure)d(sense\))i(than)g(that)h(of)g (the)g(presen)o(t)e(pap)q(er.)100 1873 y(W)l(e)j(come)f(no)o(w)g(to)i (the)f(w)o(ork)g([4],)g(where)f(the)h(p)q(erturbation)g(of)h(a)f (completely)h(resonan)o(t)d(linear)0 1933 y(system)h(w)o(as)g(studied.) 27 b(In)18 b(that)h(pap)q(er)e(a)h(metho)q(d)f(w)o(as)h(giv)o(en)g(for) g(constructing)g(a)h(family)g(of)f(closed)0 1993 y(phase)d(curv)o(es)g (whic)o(h)h(are)f(appro)o(ximate)g(solutions)h(of)g(the)g(considered)f (system,)g(and)g(are)h(stable)g(o)o(v)o(er)0 2052 y(exp)q(onen)o (tially)k(long)e(times.)25 b(Suc)o(h)16 b(a)i(metho)q(d)e(w)o(as)h (also)h(applied)g(to)g(the)g(nonlinear)g(w)o(a)o(v)o(e)f(equation.)0 2112 y(W)l(e)e(exp)q(ect)h(that)g(the)f(metho)q(d)e(of)j([4])e(applies) i(also)f(to)g(eq.)h(\(1.1\),)f(allo)o(wing)i(to)e(pro)o(v)o(e)f(the)h (existence)h(of)0 2172 y(a)i(family)h(of)g(closed)f(curv)o(es)g(with)h (the)f(prop)q(ert)o(y)g(that)h(solutions)f(starting)h Fm(O)q Fn(\()p Fm(R)1570 2154 y Fk(b)1591 2172 y Fn(\))g(close)g(\(in)g (the)f Fm(H)1961 2154 y Fl(1)0 2232 y Fn(top)q(ology\))j(to)g(one)e(of) h(these)g(curv)o(es)f(remain)g Fm(O)q Fn(\()p Fm(R)999 2214 y Fk(b=)p Fl(2)1060 2232 y Fn(\))h(close)h(to)f(it)h(\(in)g(the)f (same)e(top)q(ology\))k(up)d(to)0 2291 y(times)d(of)h(order)e Fm(R)358 2273 y Fk(b)387 2291 y Fn(exp\()p Fm(c=R)567 2273 y Fl(2)589 2291 y Fn(\).)100 2351 y(The)g(result)i(of)f(the)g (presen)o(t)f(pap)q(er)g(is)i(m)o(uc)o(h)c(stronger)h(since)j(it)g (applies)g(to)f(initial)j(data)d(close)h(to)0 2411 y(the)h(ma)s(jorit)o (y)e(of)h(\014nite)h(dimensional)f(tori,)h(instead)g(of)f(initial)k (data)c(close)h(to)g(the)g(particular)f(curv)o(es)0 2471 y(selected)e(b)o(y)e(the)h(metho)q(d)f(of)h([4].)20 b(On)13 b(the)h(other)g(hand)e(w)o(e)i(p)q(oin)o(t)g(out)g(that)g(the)g(metho)q (d)e(of)i([4])g(applies)0 2531 y(to)g(generic)g(p)q(erturbations)f(of)g (the)h(linear)h(Sc)o(hr\177)-25 b(odinger)25 b(equation,)15 b(while)g(the)f(metho)q(d)e(of)i(the)f(presen)o(t)0 2590 y(pap)q(er)j(strongly)h(rely)g(on)f(the)h(in)o(tegrabilit)o(y)i(and)d (on)g(the)h(nonlinearit)o(y)h(of)e(the)h(ZS)g(equation.)100 2650 y(W)l(e)23 b(also)g(recall)h(the)f(w)o(orks)f([1],[2],[3];)i(the)f (tec)o(hnique)h(of)f(the)g(presen)o(t)f(pap)q(er)g(is)h(essen)o(tially) 0 2710 y(a)f(generalization)h(of)f(the)g(tec)o(hniques)g(dev)o(elop)q (ed)g(in)g(those)g(pap)q(ers;)h(it)g(turns)e(out)g(that)h(the)g(main)0 2770 y(abstract)16 b(theorems)f(of)h([1])h(and)e([3])i(are)f(con)o (tained)g(in)h(theorem)e(6.2)i(of)f(the)h(presen)o(t)e(pap)q(er.)p eop %%Page: 8 8 8 7 bop 0 30 a Fg(8)850 b Fh(D.Bam)o(busi)0 149 y Fo(5.)57 b(First)19 b(Normalization)100 284 y Fn(It)c(is)h(w)o(ell)g(kno)o(wn)e (that)i(eq.)f(\(1.1\))h(with)g(v)m(anishing)f(b)q(oundary)f(conditions) i(is)g(Hamiltonian,)g(e.g.,)0 344 y(on)i(the)h(phase)f(space)g Fj(P)i Fn(=)d Fm(H)591 326 y Fl(1)587 356 y(0)614 344 y Fn(\([0)p Fm(;)8 b(\031)r Fn(])p Fm(;)g Fo(C)p Fn(\))20 b(endo)o(w)o(ed)d(with)i(the)g(scalar)f(pro)q(duct)g Fj(h)p Fm(:;)8 b(:)p Fj(i)p Fn(,)20 b(and)e(with)h(the)0 403 y(symplectic)e(form)e(\012\()p Fm(:;)8 b(:)p Fn(\))17 b(de\014ned)f(resp)q(ectiv)o(ely)i(b)o(y)425 545 y Fj(h)q Fm(u;)8 b(v)r Fj(i)13 b Fn(:=)h(Re)688 477 y Ff(Z)738 489 y Fk(\031)716 590 y Fl(0)776 545 y Fn(\026)-28 b Fm(u)802 552 y Fk(x)827 545 y Fm(v)851 552 y Fk(x)876 545 y Fm(dx)17 b(;)58 b Fn(\012\()p Fm(u;)8 b(v)r Fn(\))14 b(:=)f(Re)1317 477 y Ff(Z)1367 489 y Fk(\031)1345 590 y Fl(0)1403 545 y Fm(ui)r Fn(\026)-27 b Fm(v)q(dx)17 b(:)323 b Fn(\(5)p Fm(:)p Fn(1\))0 690 y(W)l(e)18 b(recall)g(that)g (the)f(Hamiltonian)i(v)o(ector)e(\014eld)h Fj(r)1004 672 y Fl(\012)1034 690 y Fm(f)23 b Fn(corresp)q(onding)16 b(to)i(a)f(di\013eren)o(tiable)i(function)0 750 y Fm(f)h Fn(:)13 b Fj(P)18 b(!)k Fn(I)-9 b(R)17 b(is)g(de\014ned)e(b)o(y)575 863 y(\012\()p Fj(r)672 843 y Fl(\012)703 863 y Fm(f)5 b Fn(\()p Fm(u)p Fn(\))p Fm(;)j(X)t Fn(\))15 b(=)f Fm(d)-8 b(f)5 b Fn(\()p Fm(u)p Fn(\))p Fm(X)21 b(;)58 b Fj(8)5 b Fm(X)17 b Fj(2)d(P)20 b Fm(;)473 b Fn(\(5)p Fm(:)p Fn(2\))0 977 y(so)16 b(that)h(the)g(Hamiltonian)g(function)g(corresp)q (onding)f(to)g(\(1.1\))h(is)h(giv)o(en)f(b)o(y)381 1084 y(1)p 381 1106 25 2 v 381 1152 a(2)420 1050 y Ff(Z)470 1062 y Fk(\031)448 1163 y Fl(0)506 1118 y Fj(j)o Fm(u)548 1125 y Fk(x)573 1118 y Fj(j)587 1093 y Fl(2)617 1118 y Fm(dx)12 b Fn(+)739 1084 y Fm(\031)p 739 1106 31 2 v 742 1152 a Fn(4)783 1050 y Ff(Z)833 1062 y Fk(\031)811 1163 y Fl(0)868 1118 y Fj(j)p Fm(u)p Fj(j)925 1097 y Fl(4)947 1118 y Fm(dx)f Fn(+)g Fm(\017)1090 1050 y Ff(Z)1140 1062 y Fk(\031)1118 1163 y Fl(0)1176 1118 y Fm(H)t Fn(\()p Fm(x;)d(u)p Fn(\()p Fm(x)p Fn(\))p Fm(;)13 b Fn(\026)-30 b Fm(u)r Fn(\()p Fm(x)p Fn(\)\))p Fm(dx)18 b(:)273 b Fn(\(5)p Fm(:)p Fn(3\))0 1256 y(W)l(e)17 b(will)i(also)e(denote)729 1332 y Fm(h)758 1339 y Fl(0)780 1332 y Fn(\()p Fm(u)p Fn(\))d(:=)934 1298 y(1)p 934 1320 25 2 v 934 1366 a(2)973 1264 y Ff(Z)1023 1276 y Fk(\031)1001 1377 y Fl(0)1058 1332 y Fj(j)p Fm(u)1101 1339 y Fk(x)1126 1332 y Fj(j)1139 1307 y Fl(2)1170 1332 y Fm(dx)j(:)100 1451 y Fn(It)g(is)g(useful)g (\(and)f(p)q(ossible\))i(to)f(in)o(tro)q(duce)g(canonical)g(co)q (ordinates.)23 b(T)l(o)16 b(this)i(end)e(consider)g(the)0 1511 y(F)l(ourier)g(expansion)g(of)h Fm(u)f Fn(\(cf.)h(eq.)g(\(2.2\)\)) g(and)f(de\014ne)685 1624 y Fm(q)707 1631 y Fk(k)745 1624 y Fn(:=)e(Im)8 b Fm(u)910 1631 y Fk(k)951 1624 y Fm(;)58 b(p)1048 1631 y Fk(k)1086 1624 y Fn(:=)13 b(Re)5 b Fm(u)1245 1631 y Fk(k)1286 1624 y Fm(;)0 1738 y Fn(then)19 b(\()p Fm(p)160 1745 y Fk(k)185 1738 y Fm(;)8 b(q)229 1745 y Fk(k)254 1738 y Fn(\))20 b(are)f(the)h(canonical)h(conjugated)e (co)q(ordinates)g(\(whic)o(h)h(ob)o(viously)h(are)e(not)g(orthonor-)0 1797 y(mal\).)j(Remark)14 b(that)j(in)g(terms)e(of)i(these)f(co)q (ordinates)h(one)f(has)561 1942 y Fm(h)590 1949 y Fl(0)613 1942 y Fn(\()p Fm(u)p Fn(\))e(=)752 1909 y(1)p 752 1931 V 752 1977 a(2)792 1895 y Ff(X)791 2002 y Fk(k)q Fe(\025)p Fl(1)873 1942 y Fm(k)901 1922 y Fl(2)923 1942 y Fj(j)p Fm(u)966 1949 y Fk(k)989 1942 y Fj(j)1003 1922 y Fl(2)1039 1942 y Fn(=)1093 1895 y Ff(X)1092 2002 y Fk(k)q Fe(\025)p Fl(1)1173 1942 y Fm(k)1201 1922 y Fl(2)1229 1909 y Fm(p)1254 1891 y Fl(2)1254 1923 y Fk(k)1290 1909 y Fn(+)d Fm(q)1364 1891 y Fl(2)1362 1923 y Fk(k)p 1229 1931 158 2 v 1295 1977 a Fn(2)1409 1942 y Fm(:)459 b Fn(\(5)p Fm(:)p Fn(4\))100 2113 y(F)l(ollo)o(wing)17 b([11])f(w)o(e)h(no)o(w)e(put)i(\(5.3\))g(in) g(normal)e(form)f(up)i(to)h(quan)o(tities)h(of)f(order)e Fj(j)p Fm(u)p Fj(j)1756 2095 y Fl(6)1789 2113 y Fn(+)c Fj(j)p Fm(\017)p Fj(j)p Fn(.)100 2172 y(So,)22 b(consider)f(the)h (complexi\014cation)h Fj(P)893 2154 y Fc(C)949 2172 y Fn(of)f Fj(P)t Fn(,)g(i.e.)38 b(allo)o(w)23 b Fm(p)1346 2179 y Fk(k)1392 2172 y Fn(and)e Fm(q)1516 2179 y Fk(k)1563 2172 y Fn(to)h(assume)d(complex)0 2232 y(v)m(alues,)i(and)e(denote)h(b) o(y)f Fm(B)540 2239 y Fk(R)592 2232 y Fj(\032)g(P)689 2214 y Fc(C)744 2232 y Fn(the)h(\(complex\))g(ball)h(of)f(radius)f Fm(R)h Fn(and)f(cen)o(ter)h(at)g(the)g(origin.)0 2292 y Fg(We)c(wil)s(l)g(also)i(use)e(the)h(notation)g Fm(a)d Fj(\026)g Fm(b)j Fg(to)g(me)m(an)g(\\ther)m(e)f(exists)h(a)g(p)m (ositive)i(c)m(onstant)e Fm(C)j Fg(indep)m(endent)0 2352 y(of)f Fm(R)f Fg(and)g Fm(\017)g Fg(such)h(that)f Fm(a)c Fj(\024)f Fm(C)t(b)p Fg(")p Fn(.)0 2431 y Fo(Lemma)77 b(5.1.)18 b Fg(Consider)h(the)g(Hamiltonian)g(system)g(\(5.3\))h(in)f (the)f(domain)i Fm(B)1606 2438 y Fk(R)1638 2431 y Fg(.)25 b(Ther)m(e)19 b(exists)g(a)0 2491 y(p)m(ositive)f Fm(\026)208 2498 y Fk(])244 2491 y Fg(such)f(that,)g(if)h Fj(j)p Fm(\017)p Fj(j)9 b Fn(+)f Fm(R)662 2473 y Fl(6)699 2491 y Fj(\024)13 b Fm(\026)781 2498 y Fk(])800 2491 y Fg(,)k(then)g(the)f (fol)s(lowing)i(holds)g(true:)k(ther)m(e)17 b(exists)g(an)h(analytic)0 2551 y(c)m(anonic)m(al)i(tr)m(ansformation)g Fj(T)586 2558 y Fl(1)626 2551 y Fg(such)e(that)33 2611 y(1.)27 b Fj(T)127 2618 y Fl(1)167 2611 y Fg(is)18 b(de\014ne)m(d)h(on)g Fm(B)498 2620 y Fl(3)p Fk(R=)p Fl(4)590 2611 y Fg(,)f(one)h(has)g Fm(B)845 2620 y Fk(R=)p Fl(2)931 2611 y Fj(\032)14 b(T)1011 2618 y Fl(1)1033 2611 y Fn(\()p Fm(B)1090 2620 y Fl(3)p Fk(R=)p Fl(4)1183 2611 y Fn(\))g Fj(\032)g Fm(B)1307 2618 y Fk(R)1357 2611 y Fg(and)773 2730 y Fn(sup)756 2772 y Fk(B)786 2779 y Fd(3)p Fb(R=)p Fd(4)874 2730 y Fj(k)p Fm(u)d Fj(\000)g(T)1016 2737 y Fl(1)1038 2730 y Fn(\()p Fm(u)p Fn(\))p Fj(k)1130 2745 y Fk(H)1166 2735 y Fd(1)1201 2730 y Fj(\026)j Fm(R)1292 2709 y Fl(3)1315 2730 y Fm(;)553 b Fn(\(5)p Fm(:)p Fn(5\))p eop %%Page: 9 9 9 8 bop 354 30 a Fh(On)16 b(long)h(time)f(stabilit)o(y)j(in)e (nonlinear)g(Sc)o(hr\177)-25 b(odinger)15 b(equation)355 b Fn(9)33 149 y Fg(2.)802 213 y Fm(H)16 b Fj(\016)11 b(T)922 220 y Fl(1)958 213 y Fn(=)j Fm(h)1040 220 y Fl(0)1073 213 y Fn(+)d Fj(h)p Fm(f)5 b Fj(i)12 b Fn(+)1263 200 y(~)1252 213 y Fm(f)606 b Fn(\(5)p Fm(:)p Fn(6\))100 311 y Fg(wher)m(e)613 387 y Fj(h)p Fm(f)5 b Fj(i)p Fn(\()p Fm(u)p Fn(\))16 b(:=)835 353 y(1)p 835 375 25 2 v 835 421 a(2)866 332 y Ff(\020)905 340 y(X)904 447 y Fk(k)q Fe(\025)p Fl(1)985 387 y Fj(j)p Fm(u)1028 394 y Fk(k)1052 387 y Fj(j)1066 362 y Fl(2)1097 332 y Ff(\021)1126 342 y Fl(2)1160 387 y Fj(\000)1216 353 y Fn(1)p 1216 375 V 1216 421 a(8)1255 340 y Ff(X)1255 447 y Fk(k)q Fe(\025)p Fl(1)1336 387 y Fj(j)p Fm(u)1379 394 y Fk(k)1403 387 y Fj(j)1417 366 y Fl(4)1457 387 y Fm(;)411 b Fn(\(5)p Fm(:)p Fn(7\))110 535 y(~)100 548 y Fm(f)23 b Fg(is)c(analytic)f(on)g Fm(B)498 557 y Fk(R=)p Fl(2)589 548 y Fg(and)h(satis\014es)g(the)f (estimates)662 671 y Fn(1)p 656 694 39 2 v 656 739 a Fm(R)743 705 y Fn(sup)708 747 y Fk(u)p Fe(2)p Fk(B)789 754 y Fb(R=)p Fd(2)860 648 y Ff(\015)860 678 y(\015)860 708 y(\015)888 705 y Fj(r)930 685 y Fl(\012)971 692 y Fn(~)960 705 y Fm(f)6 b Fn(\()p Fm(u)p Fn(\))1057 648 y Ff(\015)1057 678 y(\015)1057 708 y(\015)1085 740 y Fk(H)1121 730 y Fd(1)1156 705 y Fj(\026)14 b Fm(R)1247 685 y Fl(4)1280 705 y Fn(+)1342 671 y Fj(j)p Fm(\017)p Fj(j)p 1336 694 61 2 v 1336 739 a Fm(R)1374 725 y Fl(2)1420 705 y Fm(;)894 842 y Fn(sup)860 884 y Fk(u)p Fe(2)p Fk(B)941 891 y Fb(R=)p Fd(2)1012 785 y Ff(\014)1012 815 y(\014)1012 844 y(\014)1039 829 y Fn(~)1029 842 y Fm(f)5 b Fn(\()p Fm(u)p Fn(\))1125 785 y Ff(\014)1125 815 y(\014)1125 844 y(\014)1156 842 y Fj(\026)14 b Fm(R)1247 821 y Fl(6)1280 842 y Fn(+)d Fj(j)p Fm(\017)p Fj(j)1882 780 y Fn(\(5)p Fm(:)p Fn(8\))33 1006 y Fg(3.)27 b(F)l(or)18 b(any)g(analytic)g (function)h Fm(g)c Fn(:)f Fm(B)778 1013 y Fk(R)824 1006 y Fj(!)g(B)r Fg(,)j(with)h Fj(B)h Fg(a)f(Banach)i(sp)m(ac)m(e,)g(one)e (has)561 1129 y Fn(sup)527 1171 y Fk(u)p Fe(2)p Fk(B)608 1178 y Fb(R=)p Fd(2)679 1129 y Fj(k)p Fm(g)r Fn(\()p Fm(u)p Fn(\))11 b Fj(\000)f Fm(g)r Fn(\()p Fj(T)929 1136 y Fl(1)952 1129 y Fn(\()p Fm(u)p Fn(\)\))p Fj(k)1063 1144 y Fe(B)1107 1129 y Fj(\026)j Fm(R)1197 1108 y Fl(2)1245 1129 y Fn(sup)1228 1171 y Fk(u)p Fe(2)p Fk(B)1309 1176 y Fb(R)1345 1129 y Fj(k)p Fm(g)r Fn(\()p Fm(u)p Fn(\))p Fj(k)1488 1144 y Fe(B)1544 1129 y Fm(;)324 b Fn(\(5)p Fm(:)p Fn(9\))100 1301 y Fg(and)18 b(the)g(same)g(estimate)g(holds)h (for)g Fj(T)852 1279 y Fe(\000)p Fl(1)839 1314 y(1)905 1301 y Fg(.)p 1945 1264 39 2 v 1945 1299 2 35 v 1982 1299 V 1945 1301 39 2 v 100 1381 a Fn(Suc)o(h)14 b(a)h(lemma)f(is)i (just)f(a)g(reform)o(ulation)g(of)g(lemma)f(4)h(of)h([11].)21 b(W)l(e)15 b(ga)o(v)o(e)g(it)i(a)e(form)f(suitable)j(for)0 1441 y(our)f(purp)q(ose.)0 1502 y Fg(Pr)m(o)m(of.)22 b Fn(The)e(pro)q(of)f(can)h(b)q(e)g(obtained)g(b)o(y)f(standard)f(p)q (erturbation)i(theory)l(.)31 b(F)l(or)19 b(example)g(one)h(can)0 1562 y(de\014ne)13 b(the)h(transformation)e Fj(T)588 1569 y Fl(1)624 1562 y Fn(as)h(the)h(time)g(one)f(\015o)o(w)g(of)h(a)g (Hamiltonian)g(system)f(with)h(Hamiltonian)0 1622 y(function)j Fm(\037)g Fn(giv)o(en)g(b)o(y)518 1778 y Fm(\037)p Fn(\()p Fm(u)p Fn(\))d(:=)718 1744 y(1)p 702 1766 56 2 v 702 1812 a(2)p Fm(\031)772 1710 y Ff(Z)822 1722 y Fl(2)p Fk(\031)799 1823 y Fl(0)877 1778 y Fm(t)903 1737 y Ff(\000)926 1778 y Fm(f)5 b Fn(\()p Fm(e)997 1757 y Fk(At)1046 1778 y Fm(u)p Fn(\))11 b Fj(\000)g(h)p Fm(f)5 b Fj(i)p Fn(\()p Fm(e)1264 1757 y Fk(At)1313 1778 y Fm(u)p Fn(\))1361 1737 y Ff(\001)1392 1778 y Fm(dt)17 b(;)390 b Fn(\(5)p Fm(:)p Fn(10\))0 1933 y(where)16 b Fm(e)167 1915 y Fk(At)231 1933 y Fn(is)h(the)g(\015o)o(w)f(generated)g(b)o(y)g(the)h(linearized)h (system)e(\()p Fm(A)e Fn(=)g Fj(\000)p Fm(@)1431 1940 y Fk(xx)1479 1933 y Fn(\),)i(and)373 2092 y Fm(f)5 b Fn(\()p Fm(u)p Fn(\))14 b(:=)556 2058 y Fm(\031)p 556 2080 31 2 v 559 2126 a Fn(4)600 2024 y Ff(Z)650 2036 y Fk(\031)628 2137 y Fl(0)685 2092 y Fj(j)p Fm(u)p Fj(j)742 2071 y Fl(4)764 2092 y Fm(dx)34 b(;)107 b Fj(h)p Fm(f)5 b Fj(i)p Fn(\()p Fm(u)p Fn(\))16 b(:=)1210 2058 y(1)p 1195 2080 56 2 v 1195 2126 a(2)p Fm(\031)1265 2024 y Ff(Z)1314 2036 y Fl(2)p Fk(\031)1292 2137 y Fl(0)1370 2092 y Fm(f)5 b Fn(\()p Fm(e)1441 2071 y Fk(At)1489 2092 y Fm(u)p Fn(\))p Fm(dt)17 b(;)245 b Fn(\(5)p Fm(:)p Fn(11\))0 2240 y(are)19 b(resp)q(ectiv)o(ely)i(the)f(Hamiltonian)g(of)g(the)f (nonlinear)h(part)f(of)g(the)h(ZS)f(equation)h(and)f(its)h(a)o(v)o (erage)0 2300 y(with)h(resp)q(ect)g(to)f(the)h(unp)q(erturb)q(ed)e (\015o)o(w.)32 b(The)21 b(estimate)f(\(5.8\))h(can)f(b)q(e)h(obtained)f (exploiting)j(the)0 2360 y(relations)401 2446 y Fj(r)443 2426 y Fl(\012)473 2446 y Fm(\037)14 b Fn(=)592 2413 y(1)p 577 2435 V 577 2480 a(2)p Fm(\031)646 2378 y Ff(Z)696 2391 y Fl(2)p Fk(\031)674 2492 y Fl(0)751 2446 y Fm(te)792 2426 y Fe(\000)p Fk(At)879 2406 y Ff(\002)900 2446 y Fj(r)942 2426 y Fl(\012)973 2446 y Fm(f)5 b Fn(\()p Fm(e)1044 2426 y Fk(At)1092 2446 y Fm(u)p Fn(\))11 b Fj(\000)g(r)1243 2426 y Fl(\012)1274 2446 y Fj(h)p Fm(f)5 b Fj(i)q Fn(\()p Fm(e)1384 2426 y Fk(At)1432 2446 y Fm(u)p Fn(\))1480 2406 y Ff(\003)1509 2446 y Fm(dt)17 b(;)273 b Fn(\(5)p Fm(:)p Fn(12\))0 2577 y(\(whic)o(h)17 b(follo)o(ws)g(from)e(the)i(fact) g(that)g Fm(e)758 2559 y Fk(At)822 2577 y Fn(is)h(a)e(canonical)i (transformation\))d(and)396 2730 y(sup)359 2772 y Fk(B)389 2779 y Fb(R)p Fd(\(1)p Fa(\000)p Fb(d)p Fd(\))515 2688 y Ff(\015)515 2717 y(\015)543 2730 y Fj(r)585 2709 y Fl(\012)615 2730 y Fj(f)p Fm(g)r(;)8 b(g)712 2737 y Fl(1)734 2730 y Fj(g)759 2688 y Ff(\015)759 2717 y(\015)800 2730 y Fj(\024)879 2696 y Fn(1)p 859 2718 65 2 v 859 2764 a Fm(Rd)937 2660 y Ff(\022)974 2730 y Fn(sup)986 2772 y Fk(B)1016 2777 y Fb(r)1057 2688 y Ff(\015)1057 2717 y(\015)1085 2730 y Fj(r)1127 2709 y Fl(\012)1157 2730 y Fm(g)1183 2688 y Ff(\015)1183 2717 y(\015)1211 2660 y(\023)g(\022)1292 2730 y Fn(sup)1305 2772 y Fk(B)1335 2777 y Fb(r)1376 2688 y Ff(\015)1376 2717 y(\015)1403 2730 y Fj(r)1445 2709 y Fl(\012)1476 2730 y Fm(g)1500 2737 y Fl(1)1522 2688 y Ff(\015)1522 2717 y(\015)1550 2660 y(\023)1611 2730 y Fm(;)232 b Fn(\(5)p Fm(:)p Fn(13\))p eop %%Page: 10 10 10 9 bop 0 30 a Fn(10)838 b Fh(D.Bam)o(busi)0 149 y Fn(whic)o(h)21 b(holds)f(for)h(an)o(y)f(t)o(w)o(o)h(functions)g(whose)f(Hamiltonian)i (v)o(ector)f(\014eld)g(is)g(analytic.)37 b(Eq.)21 b(\(5.13\))0 209 y(follo)o(ws)c(from)d Fj(r)322 191 y Fl(\012)353 209 y Fj(f)p Fm(g)r(;)8 b(g)450 216 y Fl(1)471 209 y Fj(g)14 b Fn(=)g([)p Fj(r)619 191 y Fl(\012)649 209 y Fm(g)r(;)8 b Fj(r)739 191 y Fl(\012)769 209 y Fm(g)793 216 y Fl(1)815 209 y Fn(])16 b(\(for)g(the)h(complete)e(pro)q(of)h(of)g (\(5.13\))h(see)f([1],)f(lemma)f(5.2\).)0 269 y(F)l(or)h(more)g (details)j(on)e(the)h(estimate)g(\(5.8\))g(see)f([11])g(or)g([1].)100 329 y(One)h(has)g(also)i(to)f(pro)o(v)o(e)f(that)h Fj(h)p Fm(f)5 b Fj(i)p Fn(,)20 b(as)d(de\014ned)g(b)o(y)h(\(5.11\))g (coincides)h(with)g(the)g(function)f(\(5.7\).)0 389 y(This)g(can)g(b)q (e)g(obtained)g(b)o(y)f(explicit)k(calculation)f(exactly)g(as)d(in)h ([11])g(\(see)g(in)g(particular)g(lemma)e(5\);)0 448 y(the)h(details)h(are)e(omitted.)p 1945 411 39 2 v 1945 446 2 35 v 1982 446 V 1945 448 39 2 v 0 528 a Fg(R)m(emark)89 b(5.2.)23 b Fn(The)e(pro)q(of)g(has)f(b)q(een)h(obtained)h(b)o(y)e(p)q (erforming)g(a)h(resonan)o(t)f(normal)f(form,)i(i.e.)0 588 y(b)o(y)e(a)o(v)o(eraging)g(the)h(\014rst)f(order)f(part)i(of)f (the)h(p)q(erturbation)f(with)i(resp)q(ect)f(to)g(the)g(linear)g(\015o) o(w.)30 b(The)0 648 y(resonan)o(t)22 b(normal)f(form)g(th)o(us)h (obtained)i(turns)e(out)h(to)g(coincide)i(with)f(the)f(nonresonan)o(t)e (normal)0 707 y(form)12 b(obtained)j(in)f([11].)21 b(This)14 b(is)h(not)f(surprising)f(since)h(along)h(a)f(p)q(erio)q(d)g(of)g(the)h (\015o)o(w)e(of)h(the)h(linearized)0 767 y(system)g(all)j(the)f(angles) g(conjugated)f(to)h(the)g(linear)g(actions)g(v)m(ary)g(b)o(y)f(an)h(in) o(teger)f(m)o(ultiple)h(of)g(2)p Fm(\031)r Fn(.)p 1945 730 V 1945 765 2 35 v 1982 765 V 1945 767 39 2 v 0 867 a Fg(R)m(emark)72 b(5.3.)18 b Fn(P)o(articular)f(useful)g(cases)f(of)h (\(5.9\))g(are)658 985 y(sup)648 1027 y Fk(B)678 1034 y Fb(R=)p Fd(2)750 985 y Fj(j)p Fm(Z)798 992 y Fk(k)823 985 y Fn(\()p Fm(u)p Fn(\))11 b Fj(\000)g Fm(Z)985 992 y Fk(k)1009 985 y Fn(\()p Fj(T)1055 992 y Fl(1)1078 985 y Fn(\()p Fm(u)p Fn(\)\))p Fj(j)j(\026)g Fm(R)1283 965 y Fl(4)1322 985 y Fm(;)521 b Fn(\(5)p Fm(:)p Fn(14\))458 1211 y(sup)449 1253 y Fk(B)479 1260 y Fb(R=)p Fd(2)550 1109 y Ff(\014)550 1139 y(\014)550 1168 y(\014)550 1198 y(\014)550 1228 y(\014)550 1258 y(\014)600 1163 y(X)567 1271 y Fk(k)q Fe(\025)p Fk(N)t Fl(+1)714 1211 y Fm(k)742 1190 y Fl(2)764 1211 y Fm(Z)798 1218 y Fk(k)823 1211 y Fn(\()p Fm(u)p Fn(\))11 b Fj(\000)984 1163 y Ff(X)951 1271 y Fk(k)q Fe(\025)p Fk(N)t Fl(+1)1098 1211 y Fm(k)1126 1190 y Fl(2)1148 1211 y Fm(Z)1182 1218 y Fk(k)1206 1211 y Fn(\()p Fj(T)1252 1218 y Fl(1)1275 1211 y Fn(\()p Fm(u)p Fn(\)\))1361 1109 y Ff(\014)1361 1139 y(\014)1361 1168 y(\014)1361 1198 y(\014)1361 1228 y(\014)1361 1258 y(\014)1392 1211 y Fj(\026)j Fm(R)1483 1190 y Fl(4)1522 1211 y Fm(;)321 b Fn(\(5)p Fm(:)p Fn(15\))p 1945 1329 V 1945 1364 2 35 v 1982 1364 V 1945 1366 39 2 v 100 1446 a(W)l(e)14 b(concen)o(trate)g (no)o(w)f(on)h(system)f(\(5.6\).)22 b(Remark)12 b(that)j(it)g(is)g(a)f (p)q(erturbation)g(of)g(the)h(in)o(tegrable)0 1506 y(system)g Fm(h)194 1513 y Fl(0)228 1506 y Fn(+)10 b Fj(h)p Fm(f)5 b Fj(i)p Fn(,)18 b(for)f(whic)o(h)f(action)i(v)m(ariables)g(are)e Fm(Z)1069 1513 y Fk(k)1107 1506 y Fn(=)e(\()p Fm(p)1204 1488 y Fl(2)1204 1520 y Fk(k)1240 1506 y Fn(+)d Fm(q)1314 1488 y Fl(2)1312 1520 y Fk(k)1337 1506 y Fn(\))p Fm(=)p Fn(2.)22 b(W)l(e)17 b(will)i(use)e(p)q(erturbation)0 1565 y(theory)h(to)g(construct)f(a)h(lo)q(cal)i(normal)d(form)f(around) g(a)i(nonresonan)o(t)e(\014nite)i(dimensional)g(in)o(v)m(arian)o(t)0 1625 y(torus)e(of)g(suc)o(h)g(an)g(in)o(tegrable)h(system.)100 1685 y(W)l(e)c(giv)o(e)h(no)o(w)e(a)h(suitable)h(form)d(to)i(our)f (Hamiltonian.)22 b(Fix)14 b(a)f(p)q(ositiv)o(e)h(in)o(teger)g Fm(N)k Fn(and)12 b(in)o(tro)q(duce)0 1745 y(the)17 b(\014rst)f Fm(N)22 b Fn(action)17 b(angle)g(v)m(ariables)h(for)e(our)g(system,)f (namely)h(de\014ne)g(\()p Fm(Z)1451 1752 y Fk(i)1468 1745 y Fm(;)8 b(\036)1520 1752 y Fk(i)1536 1745 y Fn(\))17 b(b)o(y)410 1871 y Fm(p)435 1878 y Fk(i)465 1871 y Fn(=)518 1825 y Ff(p)p 568 1825 76 2 v 46 x Fn(2)p Fm(Z)627 1878 y Fk(i)651 1871 y Fn(cos)8 b Fm(\036)756 1878 y Fk(i)789 1871 y Fm(;)58 b(q)883 1878 y Fk(i)914 1871 y Fn(=)966 1825 y Ff(p)p 1016 1825 V 46 x Fn(2)p Fm(Z)1075 1878 y Fk(i)1100 1871 y Fn(sin)9 b Fm(\036)1200 1878 y Fk(i)1232 1871 y Fm(;)58 b(i)14 b Fn(=)g(1)p Fm(;)8 b(:::;)g(N)22 b(:)0 2009 y Fg(R)m(emark)73 b(5.4.)18 b Fn(Outside)f(the)f(set)h Fm(Z)720 2016 y Fk(i)751 2009 y Fn(=)d(0)i(for)h(some)e Fm(i)p Fn(,)h(this)i(is)f(an)f(analytic)j(c)o(hange)c(of)i(co)q (ordinates,)0 2069 y(and)f(therefore)317 2056 y(~)306 2069 y Fm(f)22 b Fn(is)c(analytic)g(in)f(terms)e(of)i(these)f(co)q (ordinates.)p 1945 2032 39 2 v 1945 2067 2 35 v 1982 2067 V 1945 2069 39 2 v 100 2149 a(As)e(in)i(the)f(in)o(tro)q(duction)g (w)o(e)g(\014x)g(an)f Fm(N)5 b Fn(-dimensional)15 b(v)o(ector)g(\()p Fm(!)1336 2156 y Fl(1)1358 2149 y Fm(;)8 b(:::;)g(!)1475 2156 y Fk(N)1513 2149 y Fn(\))15 b(with)h(p)q(ositiv)o(e)g(en)o(tries)0 2208 y(\()p Fm(!)50 2215 y Fk(i)86 2208 y Fm(>)i Fn(0,)j Fj(8)5 b Fm(i)17 b Fn(=)i(1)p Fm(;)8 b(:::;)g(N)d Fn(\),)21 b(whic)o(h)e(w)o(e)h(assume)d(to)j(b)q(e)g(diophan)o(tine,)g(precisely) h(to)f(satisfy)g(\(2.3\);)i(w)o(e)0 2275 y(also)17 b(assume)273 2237 y Ff(P)325 2250 y Fk(N)325 2290 y(i)p Fl(=1)400 2275 y Fm(i)417 2257 y Fl(2)440 2275 y Fm(!)471 2282 y Fk(i)502 2275 y Fj(\024)d Fn(1)p Fm(=)p Fn(8,)i(the)h(general)g(case) g(in)g(whic)o(h)g(suc)o(h)f(a)g(quan)o(tit)o(y)i(is)f(arbitrary)f(can)h (b)q(e)0 2334 y(dealt)h(with)f(minor)e(mo)q(di\014cations.)100 2394 y(In)h(what)g(follo)o(ws)i(w)o(e)e(will)j(denote)808 2513 y Fm(I)830 2520 y Fk(i)860 2513 y Fn(:=)13 b Fm(Z)960 2520 y Fk(i)988 2513 y Fj(\000)e Fm(R)1076 2492 y Fl(2)1098 2513 y Fm(!)1129 2520 y Fk(i)1163 2513 y Fm(:)0 2651 y Fg(R)m(emark)72 b(5.5.)18 b Fn(By)f(our)f(assumption)f(on)h Fm(!)r Fn(,)g(the)h(torus)262 2770 y Fj(M)c Fn(:=)h Fj(f)p Fn(\()p Fm(I)t(;)8 b(\036;)g(p;)g(q)r Fn(\))14 b Fj(2)g(P)35 b Fn(:)30 b Fm(I)833 2777 y Fk(k)871 2770 y Fn(=)14 b Fm(p)949 2777 y Fk(l)978 2770 y Fn(=)f Fm(q)1052 2777 y Fk(l)1082 2770 y Fn(=)g(0)p Fm(;)25 b(k)15 b Fn(=)f(1)p Fm(;)8 b(:::;)g(N)s(;)24 b(l)15 b Fj(\025)e Fm(N)k Fn(+)11 b(1)p Fj(g)134 b Fn(\(5)p Fm(:)p Fn(16\))p eop %%Page: 11 11 11 10 bop 340 30 a Fh(On)16 b(long)h(time)g(stabilit)o(y)i(in)e (nonlinear)g(Sc)o(hr\177)-25 b(odinger)14 b(equation)342 b Fg(11)0 149 y Fn(is)17 b(in)g Fm(B)146 158 y Fk(R=)p Fl(4)219 149 y Fn(,)f(therefore)h(it)g(is)g(in)h(the)e(domain)g(of)g (de\014nition)i(of)e Fj(T)1241 156 y Fl(1)1264 149 y Fn(;)g(b)o(y)g(\(5.5\))h(it)h(is)f Fm(O)q Fn(\()p Fm(R)1678 131 y Fl(3)1702 149 y Fn(\))g(close)g(in)g(the)0 209 y Fm(H)45 191 y Fl(1)84 209 y Fn(norm)e(to)h(the)h(torus)f Fj(M)549 216 y Fk(R;!)636 209 y Fn(de\014ned)g(b)o(y)g(\(2.7\).)p 1945 172 39 2 v 1945 207 2 35 v 1982 207 V 1945 209 39 2 v 100 289 a(W)l(e)g(also)h(in)o(tro)q(duce)g(the)g(notations)462 401 y Fm(r)f Fn(:=)d Fm(R)604 381 y Fl(1)p Fk(:)p Fl(5)675 401 y Fm(;)108 b(\020)17 b Fn(=)d(\()p Fj(f)p Fm(I)955 408 y Fk(i)972 401 y Fm(;)8 b(\036)1024 408 y Fk(i)1040 401 y Fj(g)1065 376 y Fk(N)1065 416 y(i)p Fl(=1)1140 401 y Fm(;)g Fj(f)p Fm(p)1212 408 y Fk(k)1237 401 y Fm(;)g(q)1281 408 y Fk(k)1306 401 y Fj(g)1331 416 y Fk(k)q Fe(\025)p Fk(N)t Fl(+1)1472 401 y Fn(\))17 b Fm(:)335 b Fn(\(5)p Fm(:)p Fn(17\))0 514 y(In)16 b(what)h(follo)o(ws)g(the)g(role)g(of)g(p) q(erturbativ)o(e)g(parameter)d(will)19 b(b)q(e)e(pla)o(y)o(ed)f(b)o(y)g Fm(r)q Fn(.)100 573 y(W)l(e)g(in)o(tro)q(duce)h(no)o(w)f(a)g(new)h Fm(r)q Fj(\000)p Fn(dep)q(enden)o(t)f(norm)e(in)j Fj(P)t Fn(,)g(b)o(y)364 737 y Fj(k)p Fm(\020)t Fj(k)d Fn(:=)538 675 y Fk(N)520 690 y Ff(X)524 796 y Fk(i)p Fl(=1)606 703 y Fm(i)623 685 y Fl(2)646 703 y Fj(j)p Fm(I)682 710 y Fk(i)698 703 y Fj(j)p 606 725 106 2 v 647 771 a Fm(r)729 737 y Fn(+)53 b(sup)779 779 y Fk(i)p Fl(=1)p Fk(;:::;N)946 737 y Fm(r)q Fj(j)p Fm(\036)1013 744 y Fk(i)1030 737 y Fj(j)11 b Fn(+)1105 645 y Ff(v)1105 672 y(u)1105 702 y(u)1105 732 y(t)p 1157 645 433 2 v 1187 690 a(X)1157 796 y Fk(i)p Fe(\025)p Fk(N)t Fl(+1)1297 737 y Fm(i)1314 722 y Fl(2)1342 703 y Fj(j)p Fm(p)1381 710 y Fk(i)1397 703 y Fj(j)1411 689 y Fl(2)1445 703 y Fn(+)g Fj(j)p Fm(q)1531 710 y Fk(i)1547 703 y Fj(j)1561 689 y Fl(2)p 1342 725 242 2 v 1450 771 a Fn(2)1606 737 y Fm(;)237 b Fn(\(5)p Fm(:)p Fn(18\))0 920 y(\(recall)21 b(that)f Fm(r)g Fn(=)f Fm(R)404 902 y Fl(1)p Fk(:)p Fl(5)458 920 y Fn(\))h(whic)o(h)f(is)h (clearly)h(equiv)m(alen)o(t)h(to)e(the)f Fm(H)1294 902 y Fl(1)1337 920 y Fn(norm)e(used)h(b)q(efore.)31 b Fg(F)l(r)m(om)21 b(now)0 980 y(on)e(we)f(wil)s(l)f(use)h(only)h(this)f(new)h(norm,)f (exc)m(ept)h(when)f(explicitly)g(sp)m(e)m(ci\014e)m(d)p Fn(.)26 b(In)16 b(what)h(follo)o(ws)h(w)o(e)f(will)0 1040 y(consider)f(domains)f(of)i(the)f(form)716 1152 y Fj(G)746 1159 y Fk(\032)783 1152 y Fn(:=)879 1105 y Ff([)849 1215 y Fk(\036)p Fe(2)p Fc(T)931 1205 y Fb(N)972 1152 y Fj(B)1005 1159 y Fk(\032)1028 1152 y Fn(\()p Fm(I)t(;)8 b(\036;)g(p;)g(q)r Fn(\))17 b Fm(;)0 1318 y Fn(where)436 1378 y Fj(B)469 1385 y Fk(\032)492 1378 y Fn(\()p Fm(I)t(;)8 b(\036;)g(p;)g(q)r Fn(\))15 b(=)e Fj(B)801 1385 y Fk(\032)824 1378 y Fn(\()p Fm(\020)t Fn(\))h(:=)g Fj(f)o Fm(\020)1015 1385 y Fl(1)1051 1378 y Fj(2)g(P)35 b Fn(:)30 b Fj(k)p Fm(\020)14 b Fj(\000)d Fm(\020)1345 1385 y Fl(1)1367 1378 y Fj(k)j(\024)g Fm(\032)p Fj(g)24 b Fm(:)309 b Fn(\(5)p Fm(:)p Fn(19\))100 1470 y(The)21 b(norm)e(\(5.18\))j(is)g(useful)g (since)g(in)h(the)e(domain)g Fj(G)1188 1477 y Fk(r)1231 1470 y Fn(the)h(actions)g Fm(I)j Fn(and)c(the)h(co)q(ordinates)0 1529 y(\()p Fm(p;)8 b(q)r Fn(\))18 b(are)e(small,)g(but)h(the)f(angles) h Fm(\036)g Fn(can)f(c)o(hange)g(b)o(y)g(quan)o(tities)i(of)f(order)e (one)i(also)g(in)g(the)g(complex)0 1589 y(directions.)0 1669 y Fg(R)m(emark)72 b(5.6.)18 b Fn(F)l(or)e Fm(R)g Fn(\(and)h(therefore)f Fm(r)q Fn(\))i(small)e(enough)g(w)o(e)g(ha)o(v)o (e)853 1781 y Fj(G)883 1788 y Fl(4)p Fk(r)938 1781 y Fj(\032)e Fm(B)1029 1790 y Fk(R=)p Fl(2)1118 1781 y Fm(;)725 b Fn(\(5)p Fm(:)p Fn(20\))0 1893 y(and)11 b(therefore)g Fj(G)326 1900 y Fl(4)p Fk(r)379 1893 y Fn(\(whic)o(h)h(will)i(b)q(e)d (of)h(particular)g(in)o(terest\))g(is)g(con)o(tained)f(in)h(the)g (domain)e(of)h(de\014nition)0 1953 y(of)17 b Fj(T)84 1960 y Fl(1)106 1953 y Fn(.)p 1945 1916 39 2 v 1945 1951 2 35 v 1982 1951 V 1945 1953 39 2 v 0 2053 a Fg(R)m(emark)76 b(5.7.)19 b Fn(Let)g Fm(u)c Fj(2)h Fm(H)570 2035 y Fl(1)566 2065 y(0)610 2053 y Fn(b)q(e)i(suc)o(h)f(that)h(its)g(co)q(ordinates)g Fm(\020)j Fn(are)d(real)g(and)f(b)q(elong)h(to)g Fj(G)1814 2060 y Fl(4)p Fk(r)1856 2053 y Fn(,)f(then)0 2113 y(one)f(has)g(\(for)h Fm(R)g Fn(small)f(enough\))780 2225 y Fm(d)806 2234 y Fk(H)842 2224 y Fd(1)863 2225 y Fn(\()p Fm(u;)8 b Fj(M)p Fn(\))14 b Fj(\026)g Fm(d)1105 2232 y Fe(P)1139 2225 y Fn(\()p Fm(\020)t(;)8 b Fj(M)p Fn(\))700 2330 y Fm(d)726 2337 y Fe(P)759 2330 y Fn(\()p Fm(\020)t(;)g Fj(M)p Fn(\))14 b Fj(\026)978 2296 y Fm(R)p 978 2319 V 985 2364 a(r)1022 2330 y(d)1048 2339 y Fk(H)1084 2329 y Fd(1)1105 2330 y Fn(\()p Fm(u;)8 b Fj(M)p Fn(\))17 b Fm(;)0 2461 y Fn(where)f Fm(d)170 2468 y Fe(P)204 2461 y Fn(\()p Fm(:;)8 b(:)p Fn(\))17 b(is)g(the)g(distance)g(in)g(the)g(norm)d(\(5.18\).)p 1945 2424 V 1945 2459 2 35 v 1982 2459 V 1945 2461 39 2 v 0 2560 a Fo(Prop)r(osition)102 b(5.8.)22 b Fg(In)h(terms)g(of)h (the)f(variables)h Fn(\()p Fm(I)t(;)8 b(\036;)g(p;)g(q)r Fn(\))p Fg(,)25 b(the)e(Hamiltonian)h(\(5.6\))h(c)m(an)f(b)m(e)0 2620 y(written)17 b(in)h(the)g(form)705 2726 y Fm(H)t Fn(\()p Fm(\020)t Fn(\))c(=)881 2664 y Fk(N)t Fl(+1)888 2678 y Ff(X)892 2785 y Fk(i)p Fl(=1)975 2726 y Fm(\027)1000 2733 y Fk(i)1016 2726 y Fm(J)1044 2733 y Fk(i)1060 2726 y Fn(\()p Fm(\020)t Fn(\))e(+)1196 2713 y(\026)1185 2726 y Fm(f)6 b Fn(\()p Fm(\020)t Fn(\))578 b(\(5)p Fm(:)p Fn(21\))p eop %%Page: 12 12 12 11 bop 0 30 a Fn(12)838 b Fh(D.Bam)o(busi)0 149 y Fg(wher)m(e)341 257 y Fm(J)369 264 y Fk(i)386 257 y Fn(\()p Fm(\020)t Fn(\))14 b(:=)f Fj(\000)p Fm(I)591 264 y Fk(i)618 257 y Fn(+)e(4)701 156 y Ff(0)701 246 y(@)764 194 y Fk(N)745 209 y Ff(X)745 316 y Fk(k)q Fl(=1)826 257 y Fm(I)848 264 y Fk(k)883 257 y Fn(+)967 209 y Ff(X)933 316 y Fk(k)q Fe(\025)p Fk(N)t Fl(+1)1086 223 y Fm(p)1111 205 y Fl(2)1111 237 y Fk(k)1147 223 y Fn(+)g Fm(q)1221 205 y Fl(2)1219 237 y Fk(k)p 1086 245 158 2 v 1152 291 a Fn(2)1249 156 y Ff(1)1249 246 y(A)1319 257 y Fm(;)110 b(i)14 b Fn(=)g(1)p Fm(;)8 b(:::;)g(N)23 b(;)270 445 y(J)298 452 y Fk(N)t Fl(+1)386 445 y Fn(\()p Fm(\020)t Fn(\))14 b(:=)548 382 y Fk(N)530 397 y Ff(X)534 504 y Fk(i)p Fl(=1)610 445 y Fm(i)627 424 y Fl(2)650 445 y Fm(I)672 452 y Fk(i)699 445 y Fn(+)779 397 y Ff(X)749 504 y Fk(i)p Fe(\025)p Fk(N)t Fl(+1)888 445 y Fm(i)905 424 y Fl(2)934 411 y Fm(p)959 393 y Fl(2)959 424 y Fk(i)992 411 y Fn(+)d Fm(q)1066 393 y Fl(2)1064 424 y Fk(i)p 934 433 155 2 v 999 479 a Fn(2)1112 445 y Fm(;)409 579 y(\027)434 586 y Fk(i)464 579 y Fn(:=)i Fm(R)568 558 y Fl(2)591 579 y Fm(!)622 586 y Fk(i)638 579 y Fm(;)59 b(i)14 b Fn(=)g(1)p Fm(;)8 b(:::;)g(N)23 b Fn(;)110 b Fm(\027)1118 586 y Fk(N)t Fl(+1)1219 579 y Fn(:=)14 b(1)k Fm(:)1857 388 y Fn(\(5)p Fm(:)p Fn(22\))0 661 y Fg(Mor)m(e)m(over)h(ther)m(e)f(exists)h Fm(r)502 668 y Fe(\003)543 661 y Fg(such)f(that,)g(for)h Fm(r)c Fj(\024)f Fm(r)965 668 y Fe(\003)1006 661 y Fg(the)k(estimate) 738 751 y Fn(1)p 738 773 25 2 v 739 819 a Fm(r)778 785 y Fn(sup)785 826 y Fe(G)809 831 y Fd(2)p Fb(r)861 742 y Ff(\015)861 772 y(\015)889 785 y Fj(r)931 764 y Fl(\012)972 772 y Fn(\026)961 785 y Fm(f)991 742 y Ff(\015)991 772 y(\015)1032 785 y Fj(\026)c Fm(r)1108 764 y Fl(2)1142 785 y Fn(+)1198 751 y Fj(j)p Fm(\017)p Fj(j)p 1198 773 48 2 v 1199 819 a Fm(r)1222 804 y Fl(2)1857 785 y Fn(\(5)p Fm(:)p Fn(23\))0 921 y Fg(holds,)20 b(and)250 908 y Fn(\026)239 921 y Fm(f)k Fg(is)18 b(analytic)h(in)f Fj(G)620 928 y Fl(2)p Fk(r)661 921 y Fg(.)p 1945 884 39 2 v 1945 919 2 35 v 1982 919 V 1945 921 39 2 v 0 1050 a(R)m(emark)72 b(5.9.)18 b Fn(The)f(\015o)o(w)e(of)i Fm(\027)624 1057 y Fk(i)640 1050 y Fm(J)668 1057 y Fk(i)684 1050 y Fn(,)g(is)g(p)q(erio) q(dic)h(with)g(p)q(erio)q(d)f Fm(T)1254 1057 y Fk(i)1284 1050 y Fn(:=)d(2)p Fm(\031)r(=\027)1456 1057 y Fk(i)1472 1050 y Fn(.)p 1945 1014 V 1945 1048 2 35 v 1982 1048 V 1945 1050 39 2 v 0 1137 a Fg(Pr)m(o)m(of.)19 b Fn(A)d(simple)g (calculation)j(sho)o(ws)c(that)i(\(apart)f(from)e(a)j(constan)o(t\))f Fj(h)p Fm(f)5 b Fj(i)15 b Fn(=)1525 1099 y Ff(P)1577 1112 y Fk(N)1577 1152 y(i)p Fl(=1)1652 1137 y Fm(\027)1677 1144 y Fk(i)1694 1137 y Fm(J)1722 1144 y Fk(i)1748 1137 y Fn(+)1798 1123 y(^)1798 1137 y Fm(h)p Fn(,)h(where)0 1196 y Fj(h)p Fm(f)5 b Fj(i)18 b Fn(is)f(the)g(function)g(\(5.7\))g (expressed)f(in)h(terms)d(of)j(the)g(new)f(v)m(ariables,)i(and)384 1341 y(^)383 1354 y Fm(h)c Fn(:=)f(2\()555 1292 y Fk(N)536 1307 y Ff(X)536 1414 y Fk(k)q Fl(=1)617 1354 y Fm(I)639 1361 y Fk(k)675 1354 y Fn(+)758 1307 y Ff(X)725 1414 y Fk(k)q Fe(\025)p Fk(N)t Fl(+1)872 1354 y Fm(Z)906 1361 y Fk(k)930 1354 y Fn(\))949 1333 y Fl(2)983 1354 y Fj(\000)1039 1320 y Fn(1)p 1039 1342 25 2 v 1039 1388 a(2)1078 1254 y Ff(0)1078 1343 y(@)1140 1292 y Fk(N)1122 1307 y Ff(X)1122 1414 y Fk(k)q Fl(=1)1202 1354 y Fm(I)1228 1333 y Fl(2)1224 1366 y Fk(k)1262 1354 y Fn(+)1345 1307 y Ff(X)1311 1414 y Fk(k)q Fe(\025)p Fk(N)t Fl(+1)1459 1354 y Fm(Z)1497 1333 y Fl(2)1493 1366 y Fk(k)1519 1254 y Ff(1)1519 1343 y(A)1587 1354 y Fm(:)256 b Fn(\(5)p Fm(:)p Fn(24\))0 1522 y(De\014ne)165 1509 y(\026)154 1522 y Fm(f)20 b Fn(:=)265 1509 y(^)264 1522 y Fm(h)11 b Fn(+)365 1509 y(~)354 1522 y Fm(f)6 b Fn(.)22 b(W)l(e)16 b(estimate)h(no)o(w)f Fj(r)851 1504 y Fl(\012)892 1509 y Fn(~)881 1522 y Fm(f)22 b Fn(in)c(the)e(new)h(norm.)i(One)e(has)28 1578 y Ff(v)28 1606 y(u)28 1636 y(u)28 1666 y(u)28 1696 y(t)p 81 1578 656 2 v 114 1651 a(X)81 1758 y Fk(k)q Fe(\025)p Fk(N)t Fl(+1)234 1665 y Fm(k)262 1650 y Fl(2)p 234 1687 50 2 v 247 1732 a Fn(2)298 1598 y Ff(0)298 1688 y(@)342 1611 y(\014)342 1641 y(\014)342 1671 y(\014)342 1701 y(\014)342 1731 y(\014)374 1665 y Fm(@)414 1651 y Fn(~)403 1665 y Fm(f)p 364 1687 79 2 v 364 1732 a(@)s(p)418 1739 y Fk(k)449 1611 y Ff(\014)449 1641 y(\014)449 1671 y(\014)449 1701 y(\014)449 1731 y(\014)466 1623 y Fl(2)499 1698 y Fn(+)549 1611 y Ff(\014)549 1641 y(\014)549 1671 y(\014)549 1701 y(\014)549 1731 y(\014)580 1665 y Fm(@)620 1651 y Fn(~)609 1665 y Fm(f)p 572 1687 76 2 v 572 1732 a(@)s(q)623 1739 y Fk(k)654 1611 y Ff(\014)654 1641 y(\014)654 1671 y(\014)654 1701 y(\014)654 1731 y(\014)670 1623 y Fl(2)693 1598 y Ff(1)693 1688 y(A)750 1699 y Fj(\024)803 1578 y Ff(v)803 1606 y(u)803 1636 y(u)803 1666 y(u)803 1696 y(t)p 855 1578 590 2 v -45 x(X)855 1758 y Fk(k)q Fe(\025)p Fl(1)942 1665 y Fm(k)970 1650 y Fl(2)p 942 1687 50 2 v 955 1732 a Fn(2)1007 1598 y Ff(0)1007 1688 y(@)1050 1611 y(\014)1050 1641 y(\014)1050 1671 y(\014)1050 1701 y(\014)1050 1731 y(\014)1083 1665 y Fm(@)1123 1651 y Fn(~)1112 1665 y Fm(f)p 1073 1687 79 2 v 1073 1732 a(@)s(p)1127 1739 y Fk(k)1157 1611 y Ff(\014)1157 1641 y(\014)1157 1671 y(\014)1157 1701 y(\014)1157 1731 y(\014)1174 1623 y Fl(2)1207 1698 y Fn(+)1257 1611 y Ff(\014)1257 1641 y(\014)1257 1671 y(\014)1257 1701 y(\014)1257 1731 y(\014)1288 1665 y Fm(@)1328 1651 y Fn(~)1317 1665 y Fm(f)p 1280 1687 76 2 v 1280 1732 a(@)s(q)1331 1739 y Fk(k)1362 1611 y Ff(\014)1362 1641 y(\014)1362 1671 y(\014)1362 1701 y(\014)1362 1731 y(\014)1378 1623 y Fl(2)1401 1598 y Ff(1)1401 1688 y(A)1458 1699 y Fj(\026)d Fm(R)1549 1678 y Fl(5)1582 1699 y Fn(+)1638 1665 y Fj(j)p Fm(\017)p Fj(j)p 1638 1687 48 2 v 1643 1732 a Fm(R)1706 1699 y(<)f(r)1781 1678 y Fl(3)1816 1699 y Fn(+)1872 1665 y Fj(j)p Fm(\017)p Fj(j)p 1872 1687 V 1884 1732 a Fm(r)1942 1699 y(:)0 1863 y Fn(Consider)j(no)o(w)720 1897 y Fk(N)702 1912 y Ff(X)706 2018 y Fk(i)p Fl(=1)788 1926 y Fm(i)805 1908 y Fl(2)p 788 1948 40 2 v 796 1993 a Fm(r)842 1872 y Ff(\014)842 1902 y(\014)842 1932 y(\014)842 1962 y(\014)842 1992 y(\014)873 1926 y Fm(@)913 1912 y Fn(~)902 1926 y Fm(f)p 865 1948 76 2 v 865 1993 a(@)s(\036)924 2000 y Fk(i)946 1872 y Ff(\014)946 1902 y(\014)946 1932 y(\014)946 1962 y(\014)946 1992 y(\014)974 1959 y Fn(+)11 b(sup)1054 2000 y Fk(i)1107 1872 y Ff(\014)1107 1902 y(\014)1107 1932 y(\014)1107 1962 y(\014)1107 1992 y(\014)1134 1926 y Fm(@)1174 1912 y Fn(~)1163 1926 y Fm(f)p 1129 1948 68 2 v 1129 1993 a(@)s(I)1180 2000 y Fk(i)1203 1872 y Ff(\014)1203 1902 y(\014)1203 1932 y(\014)1203 1962 y(\014)1203 1992 y(\014)1228 1959 y Fm(r)18 b(:)0 2100 y Fn(Notice)h(that,)e(due)f (to)h(\(5.20\))g(and)g(\(5.18\))p Fm(;)836 2087 y Fn(~)825 2100 y Fm(f)6 b Fn(,)16 b(as)h(a)g(function)g(of)g Fm(\036)1271 2107 y Fk(i)1304 2100 y Fn(is)h(analytic)g(in)g(the)f(complex)f(strip)0 2159 y Fj(j)p Fn(Im)8 b Fm(\036)113 2166 y Fk(i)130 2159 y Fj(j)13 b(\026)h Fn(1.)22 b(It)17 b(follo)o(ws,)g(b)o(y)f(Cauc)o(h)o (y)g(inequalit)o(y)l(,)j(that)543 2302 y(sup)550 2344 y Fe(G)574 2349 y Fd(2)p Fb(r)626 2215 y Ff(\014)626 2245 y(\014)626 2275 y(\014)626 2305 y(\014)626 2334 y(\014)657 2268 y Fm(@)697 2255 y Fn(~)686 2268 y Fm(f)p 649 2291 76 2 v 649 2336 a(@)s(\036)708 2343 y Fk(i)730 2215 y Ff(\014)730 2245 y(\014)730 2275 y(\014)730 2305 y(\014)730 2334 y(\014)761 2302 y Fj(\026)13 b Fn(sup)820 2344 y Fe(G)844 2349 y Fd(3)p Fb(r)896 2302 y Fj(j)921 2289 y Fn(~)910 2302 y Fm(f)6 b Fj(j)14 b(\026)f Fm(R)1058 2281 y Fl(6)1092 2302 y Fn(+)e Fj(j)p Fm(\017)p Fj(j)i Fn(=)h Fm(r)1279 2281 y Fl(4)1313 2302 y Fn(+)d Fj(j)p Fm(\017)p Fj(j)17 b Fm(:)0 2442 y Fn(Analogously)625 2544 y(sup)632 2586 y Fe(G)656 2591 y Fd(2)p Fb(r)709 2457 y Ff(\014)709 2486 y(\014)709 2516 y(\014)709 2546 y(\014)709 2576 y(\014)736 2510 y Fm(@)776 2497 y Fn(~)765 2510 y Fm(f)p 731 2532 68 2 v 731 2578 a(@)s(I)782 2585 y Fk(i)805 2457 y Ff(\014)805 2486 y(\014)805 2516 y(\014)805 2546 y(\014)805 2576 y(\014)835 2544 y Fj(\026)905 2510 y Fn(1)p 894 2532 47 2 v 894 2578 a Fm(r)917 2563 y Fl(2)954 2544 y Fn(sup)961 2586 y Fe(G)985 2591 y Fd(3)p Fb(r)1038 2544 y Fj(j)1062 2531 y Fn(~)1052 2544 y Fm(f)5 b Fj(j)14 b(\026)g Fm(r)1185 2523 y Fl(2)1219 2544 y Fn(+)1275 2510 y Fj(j)p Fm(\017)p Fj(j)p 1275 2532 48 2 v 1276 2578 a Fm(r)1299 2563 y Fl(2)1345 2544 y Fm(;)0 2693 y Fn(and)i(therefore)307 2636 y Ff(\015)307 2666 y(\015)307 2696 y(\015)334 2693 y Fj(r)376 2675 y Fl(\012)418 2680 y Fn(~)407 2693 y Fm(f)437 2636 y Ff(\015)437 2666 y(\015)437 2696 y(\015)479 2693 y Fj(\026)e Fm(r)555 2675 y Fl(3)589 2693 y Fn(+)d Fj(j)p Fm(\017)p Fj(j)p Fm(=r)q Fn(.)24 b(The)16 b(estimate)h(of)1132 2636 y Ff(\015)1132 2666 y(\015)1132 2696 y(\015)1160 2693 y Fj(r)1202 2675 y Fl(\012)1233 2680 y Fn(^)1232 2693 y Fm(h)1261 2636 y Ff(\015)1261 2666 y(\015)1261 2696 y(\015)1306 2693 y Fn(is)g(a)g(straigh)o(tforw)o(ard)e(calculation)0 2770 y(that)i(w)o(e)f(omit.)p 1945 2733 39 2 v 1945 2768 2 35 v 1982 2768 V 1945 2770 39 2 v eop %%Page: 13 13 13 12 bop 341 30 a Fh(On)16 b(long)h(time)g(stabilit)o(y)i(in)e (nonlinear)g(Sc)o(hr\177)-25 b(odinger)15 b(equation)342 b Fn(13)0 149 y Fo(6.)57 b(P)n(erturbation)19 b(theory)g(for)f (Hamiltonians)h(of)g(the)g(form)e(\(5.21\))100 284 y Fn(System)c(\(5.21\))j(has)e(the)h(form)e(of)i(a)g(small)g(p)q (erturbation)g(of)g(an)f(in)o(tegrable)i(iso)q(c)o(hronous)e(system)0 344 y(whose)h(dynamics)g(is)h(con)o(tin)o(uous)f(but)g(not)h (di\013eren)o(tiable)h(and)f(quasip)q(erio)q(dic)h(with)g(a)f(\014nite) g(n)o(um)o(b)q(er)0 403 y(of)j(frequencies;)i(w)o(e)e(will)j(\014nd)c (a)h(canonical)i(transformation)c(putting)j(it)g(in)g(resonan)o(t)e (normal)f(form)0 463 y(up)g(to)g(an)h(exp)q(onen)o(tially)i(small)d (remainder.)22 b(W)l(e)c(will)i(pro)o(v)o(e)c(a)h(result)h(for)f (abstract)g(systems)f(of)h(the)0 523 y(form)f(\(5.21\))h(without)i (taking)f(in)o(to)g(accoun)o(t)f(the)g(fact)h(that)g Fm(\027)i Fn(is)e(of)g(order)e Fm(R)1508 505 y Fl(2)1530 523 y Fn(;)i(in)g(our)f(opinion)g(suc)o(h)0 583 y(a)g(result)f(could)h (ha)o(v)o(e)f(some)f(indep)q(enden)o(t)h(in)o(terest.)100 643 y(So,)c(consider)e(a)i(w)o(eakly)g(symplectic)798 625 y Fl([8])855 643 y Fn(Banac)o(h)f(space)g Fj(P)k Fn(endo)o(w)o(ed)10 b(b)o(y)h(the)g(symplectic)h(structure)0 702 y(\012.)22 b(Let)17 b Fj(P)200 684 y Fc(C)250 702 y Fn(b)q(e)g(the)f(complexi\014cation)h(of)f Fj(P)t Fn(,)g(and)f(let)j Fj(G)e(\032)e(P)20 b Fn(b)q(e)c(a)g(domain.)k(F)l(or)15 b(eac)o(h)h Fm(\032)d(>)h Fn(0)i(de\014ne)0 762 y(the)h(complex)f (extension)h(of)g Fj(G)i Fn(b)o(y)807 829 y Fj(G)837 836 y Fk(\032)874 829 y Fn(:=)949 782 y Ff([)940 889 y Fk(\020)r Fe(2G)1022 829 y Fm(B)1060 836 y Fk(\032)1083 829 y Fn(\()p Fm(\020)t Fn(\))d Fm(;)0 976 y Fn(where)e Fm(B)180 983 y Fk(\032)203 976 y Fn(\()p Fm(\020)t Fn(\))g(is)h(the)f (\(complex\))h(ball)h(of)e(radius)g Fm(\032)g Fn(and)f(cen)o(ter)h Fm(\020)t Fn(.)21 b(W)l(e)14 b(\014x)h(once)f(for)g(all)i(a)e(real)g(n) o(um)o(b)q(er)0 1036 y Fm(r)i(>)d Fn(0)k(and)f(consider,)g(in)h Fj(G)525 1043 y Fl(2)p Fk(r)583 1036 y Fn(an)f(analytic)j(Hamiltonian)e (function)g(of)g(the)g(form)862 1135 y Fk(N)t Fl(+1)869 1150 y Ff(X)873 1256 y Fk(i)p Fl(=1)956 1197 y Fm(h)985 1177 y Fk(i)1012 1197 y Fn(+)11 b Fm(f)22 b(:)760 b Fn(\(6)p Fm(:)p Fn(1\))0 1363 y(Essen)o(tially)15 b(w)o(e)f(will)i(assume)11 b(that)j(the)g Fm(h)795 1344 y Fk(i)811 1363 y Fn('s)f(comm)o(u)o(te)e (and)i(that)h(eac)o(h)f(of)h(them)e(generates)h(a)h(p)q(erio)q(dic)0 1422 y(\015o)o(w)g(\010)138 1404 y Fk(i)154 1422 y Fn(,)h Fm(i)f Fn(=)f(1)p Fm(;)8 b(:::;)g(N)k Fn(+)7 b(1;)14 b(\014nally)i(w)o(e)e(assume)e(that)j(the)f(\015o)o(ws)f(\010)1264 1404 y Fk(i)1295 1422 y Fm(i)h Fn(=)g(1)p Fm(;)8 b(:::;)g(N)19 b Fn(are)14 b(analytic)j(in)e(time,)0 1482 y(while)j(\010)166 1464 y Fk(N)t Fl(+1)271 1482 y Fg(is)g(only)h(c)m(ontinuous)p Fn(.)100 1542 y(The)d(precise)h(assumptions)d(w)o(e)i(need)g(are)30 1602 y(1\))26 b(F)l(or)19 b Fm(i)i Fn(=)f(1)p Fm(;)8 b(:::;)g(N)26 b Fn(the)21 b(symplectic)g(gradien)o(t)f Fj(r)1047 1584 y Fl(\012)1078 1602 y Fm(h)1107 1584 y Fk(i)1143 1602 y Fn(\(de\014ned)g(b)o(y)h(\(5.2\)\))g(exists)h(in)f Fj(G)1788 1609 y Fl(2)p Fk(r)1850 1602 y Fn(and)f(is)100 1661 y(analytic,)e(moreo)o(v)o(er)c(there)i(exists)i(a)e(domain)f Fm(D)k Fn(dense)c(in)j Fj(G)1285 1668 y Fl(2)p Fk(r)1343 1661 y Fn(where)e Fj(r)1529 1643 y Fl(\012)1559 1661 y Fm(h)1588 1643 y Fk(N)t Fl(+1)1693 1661 y Fn(exists)30 1721 y(2\))26 b(there)19 b(exist)j(p)q(ositiv)o(e)f(constan)o(ts)e Fm(T)792 1728 y Fk(i)809 1721 y Fn(,)i Fm(i)e Fn(=)h(1)p Fm(;)8 b(:::;)g(N)18 b Fn(+)13 b(1)20 b(suc)o(h)f(that)h(the)g(\015o)o (w)g(\010)1665 1703 y Fk(i)1701 1721 y Fn(generated)g(b)o(y)100 1781 y Fj(r)142 1763 y Fl(\012)172 1781 y Fm(h)201 1763 y Fk(i)234 1781 y Fn(is)d(p)q(erio)q(dic)h(with)g(p)q(erio)q(d)f Fm(T)773 1788 y Fk(i)789 1781 y Fn(,)g(namely)698 1896 y(\010)734 1876 y Fk(i)734 1909 y(t)p Fl(+)p Fk(T)804 1914 y Fb(i)835 1896 y Fn(=)d(\010)924 1876 y Fk(i)924 1909 y(t)958 1896 y Fm(;)58 b(i)14 b Fn(=)f(1)p Fm(;)8 b(:::;)g(N)17 b Fn(+)11 b(1)16 b Fm(;)100 2012 y Fn(moreo)o(v)o(er)d (\010)350 1993 y Fk(i)350 2024 y(t)368 2012 y Fn(\()p Fj(G)417 2019 y Fk(\032)440 2012 y Fn(\))h Fj(\032)g(G)556 2019 y Fk(\032)595 2012 y Fj(8)5 b Fm(\032)12 b(<)i Fn(2)p Fm(r)q Fn(.)30 2071 y(3\))26 b(The)16 b(functions)h Fm(h)446 2053 y Fk(i)479 2071 y Fn(comm)n(ute)d(eac)o(h)i(other,)g(namely)g(one) h(has)566 2146 y Ff(\010)595 2187 y Fm(h)624 2166 y Fk(i)640 2187 y Fm(;)8 b(h)691 2166 y Fk(j)712 2146 y Ff(\011)755 2187 y Fj(\021)14 b Fn(0)i Fm(;)58 b(i;)8 b(j)17 b Fn(=)c(1)p Fm(;)8 b(:::;)g(N)16 b Fn(+)11 b(1)17 b Fm(;)58 b(i)14 b(<)f(j)19 b(:)364 b Fn(\(6)p Fm(:)p Fn(2\))0 2352 y Fg(R)m(emark)84 b(6.1.)21 b Fn(Using)g(the)f(\015o)o(ws)e(\010)746 2334 y Fk(i)763 2352 y Fn(,)i Fm(i)g Fn(=)f(1)p Fm(;)8 b(:::;)g(N)25 b Fn(one)19 b(can)h(de\014ne)f(an)g(action)i(\011)f(of)g Fo(T)1788 2334 y Fk(N)1845 2352 y Fn(on)g Fj(G)1948 2359 y Fk(\032)1970 2352 y Fn(,)0 2412 y(namely)450 2471 y(\011)489 2478 y Fk( )515 2483 y Fd(1)535 2478 y Fk(;:::; )621 2483 y Fb(N)655 2471 y Fn(\()p Fm(\020)t Fn(\))14 b(:=)f(\010)835 2451 y Fl(1)835 2485 y Fk( )861 2490 y Fd(1)881 2485 y Fk(=\027)921 2490 y Fd(1)954 2471 y Fj(\016)e Fn(\010)1026 2451 y Fl(2)1026 2485 y Fk( )1052 2490 y Fd(2)1072 2485 y Fk(=\027)1112 2490 y Fd(2)1146 2471 y Fj(\016)g Fm(:::)f Fj(\016)h Fn(\010)1306 2451 y Fk(N)1306 2485 y( )1332 2490 y Fb(N)1364 2485 y Fk(=\027)1404 2490 y Fb(N)1439 2471 y Fn(\()p Fm(\020)t Fn(\))17 b Fm(;)0 2570 y Fn(where)f Fm(\027)169 2577 y Fk(i)199 2570 y Fn(=)e(2)p Fm(\031)r(=T)361 2577 y Fk(i)377 2570 y Fn(,)j Fm(i)d Fn(=)f(1)p Fm(;)8 b(:::;)g(N)d Fn(.)p 1945 2534 39 2 v 1945 2568 2 35 v 1982 2568 V 1945 2570 39 2 v 100 2650 a(W)l(e)16 b(assume)30 2710 y(4\))26 b(denote)14 b(b)o(y)f Fo(T)364 2692 y Fk(N)408 2710 y Fn(+)6 b Fm(i\033)16 b Fn(the)e(set)g(of)g(the)g Fm( )r Fn('s)g(b)q(elonging)h(to)f(the)h(complexi\014ed)e Fm(N)5 b Fn(-dimensional)14 b(torus)100 2770 y(suc)o(h)g(that)i Fj(j)p Fn(Im)9 b Fm( )434 2777 y Fk(i)451 2770 y Fj(j)k(\024)h Fm(\033)r Fn(,)i(then)g(w)o(e)f(assume)e(that)k(there)e(exists)i(a)f Fm(\033)g(>)d Fn(0)j(suc)o(h)e(that)j Fj(8)5 b Fm(\020)16 b Fj(2)e(G)1864 2777 y Fk(r)1885 2770 y Fn(,)i(the)p eop %%Page: 14 14 14 13 bop 0 30 a Fg(14)838 b Fh(D.Bam)o(busi)100 149 y Fn(function)18 b Fo(T)336 131 y Fk(N)389 149 y Fj(3)e Fm( )i Fj(7!)e Fn(\011)593 156 y Fk( )623 149 y Fn(\()p Fm(\020)t Fn(\))g Fj(2)g(G)782 156 y Fk(r)821 149 y Fn(can)h(b)q(e)h (extended)g(to)g(a)g(complex)f(analytic)j(function)e(from)100 209 y Fo(T)140 191 y Fk(n)178 209 y Fn(+)10 b Fm(i\033)19 b Fn(to)e Fj(G)382 216 y Fl(2)p Fk(r)424 209 y Fn(.)30 269 y(5\))26 b(\010)136 248 y Fk(N)t Fl(+1)136 281 y Fk(t)224 269 y Fn(\()p Fm(D)q Fn(\))17 b Fj(\032)e Fm(D)q Fn(,)j(the)g(map)e(\010)684 248 y Fk(N)t Fl(+1)684 281 y Fk(t)789 269 y Fn(lea)o(v)o(es)i(in)o(v)m(arian)o(t)g Fj(G)1173 276 y Fk(\032)1214 269 y Fn(for)f(eac)o(h)g Fm(\032)e Fj(\024)g Fn(2)p Fm(r)20 b Fn(and)d(is)h(analytic)h(\(for)100 329 y(eac)o(h)d(\014xed)i Fm(t)p Fn(\);)f(moreo)o(v)o(er)e(it)j(is)g(a) f(canonical)i(transformation,)c(i.e.)25 b(its)18 b(di\013eren)o(tial)h (\(\010)1798 307 y Fk(N)t Fl(+1)1798 341 y Fk(t)1887 329 y Fn(\))1906 311 y Fe(0)1920 329 y Fn(\()p Fm(\020)t Fn(\))100 389 y(satisfy)550 460 y(\012)594 405 y Ff(h)617 420 y(\000)640 460 y Fn(\010)676 439 y Fk(N)t Fl(+1)676 473 y Fk(t)764 420 y Ff(\001)787 428 y Fe(0)810 460 y Fn(\()p Fm(\020)t Fn(\))p Fm(X)q(;)938 420 y Ff(\000)961 460 y Fn(\010)997 439 y Fk(N)t Fl(+1)997 473 y Fk(t)1085 420 y Ff(\001)1108 428 y Fe(0)1130 460 y Fn(\()p Fm(\020)t Fn(\))p Fm(Y)1235 405 y Ff(i)1272 460 y Fn(=)14 b(\012\()p Fm(X)q(;)8 b(Y)j Fn(\))17 b Fm(;)100 558 y Fn(and)e(satis\014es)i(also) g(the)g(estimate)745 657 y(sup)737 699 y Fk(\020)r Fe(2G)808 704 y Fb(r)837 600 y Ff(\015)837 630 y(\015)837 660 y(\015)864 617 y(\000)887 657 y Fn(\010)923 636 y Fk(N)t Fl(+1)923 669 y Fk(t)1011 617 y Ff(\001)1034 624 y Fe(0)1057 657 y Fn(\()p Fm(\020)t Fn(\))1121 600 y Ff(\015)1121 630 y(\015)1121 660 y(\015)1148 692 y Fe(P)s Fk(;)p Fe(P)1239 657 y Fj(\024)c Fn(1)k Fm(:)30 792 y Fn(6\))26 b(The)14 b(di\013eren)o(t)g(p)q(erio)q(dic)i(motions)e(ha)o(v)o(e)f(frequencies) i(whic)o(h)g(form)d(a)j(v)o(ector)f(close)h(to)g(a)f(diophan-)100 852 y(tine)j(one,)f(precisely)i(there)f(exist)h Fm(\015)e(>)e Fn(0,)i Fm(\034)j Fj(\025)14 b Fn(0)i(and)g Fm(M)k(>)13 b Fn(0)k(suc)o(h)e(that)191 910 y Ff(\014)191 940 y(\014)191 970 y(\014)191 1000 y(\014)191 1029 y(\014)226 935 y Fk(N)208 950 y Ff(X)212 1056 y Fk(i)p Fl(=1)288 997 y Fm(\027)313 1004 y Fk(i)329 997 y Fm(k)355 1004 y Fk(i)383 997 y Fn(+)c Fm(!)r(n)496 910 y Ff(\014)496 940 y(\014)496 970 y(\014)496 1000 y(\014)496 1029 y(\014)526 997 y Fj(\025)610 963 y Fm(\015)p 584 986 81 2 v 584 1031 a Fj(j)p Fm(k)r Fj(j)640 1017 y Fk(\034)687 997 y Fm(;)58 b Fj(8)5 b Fn(\()p Fm(k)r(;)j(n)p Fn(\))13 b Fj(2)h Fo(Z)1006 976 y Fk(N)t Fl(+1)1099 997 y Fj(n)8 b(f)p Fn(0)p Fj(g)25 b Fm(;)108 b Fj(j)p Fm(k)r Fj(j)13 b Fn(:=)1507 935 y Fk(N)1489 950 y Ff(X)1493 1056 y Fk(i)p Fl(=1)1569 997 y Fj(j)p Fm(k)1609 1004 y Fk(i)1626 997 y Fj(j)g(\024)h Fm(M)22 b(;)92 b Fn(\(6)p Fm(:)p Fn(3\))100 1142 y(here)16 b Fm(!)f Fn(:=)f(2)p Fm(\031)r(=T)430 1149 y Fk(N)t Fl(+1)518 1142 y Fn(,)j Fm(\027)574 1149 y Fk(i)604 1142 y Fn(=)c(2)p Fm(\031)r(=T)765 1149 y Fk(i)782 1142 y Fn(,)j Fm(i)e Fn(=)g(1)p Fm(;)8 b(:::;)g(N)d Fn(.)0 1222 y Fo(Theorem)77 b(6.2.)41 b Fg(Consider)20 b(the)e(Hamiltonian)h(system)f(\(6.1\),)j (assume)e(that)f(it)g(is)h(analytic)g(over)0 1282 y Fj(G)30 1289 y Fl(2)p Fk(r)72 1282 y Fg(,)h(that)g(it)g(satis\014es)h(\(1{6\))h (ab)m(ove;)g(assume)f(also)g(that)f Fj(r)1169 1264 y Fl(\012)1199 1282 y Fm(f)26 b Fg(is)20 b(analytic)h(on)f Fj(G)1598 1289 y Fl(2)p Fk(r)1640 1282 y Fg(,)g(and)h(that)f(ther)m(e)0 1341 y(exist)e(c)m(onstants)h Fm(!)366 1348 y Fk(f)392 1341 y Fg(,)f Fm(c)447 1348 y Fk(T)478 1341 y Fg(,)g(such)g(that)609 1432 y Fn(1)p 609 1454 25 2 v 610 1500 a Fm(r)665 1466 y Fn(sup)648 1508 y Fk(\020)r Fe(2G)719 1513 y Fd(2)p Fb(r)765 1423 y Ff(\015)765 1453 y(\015)792 1466 y Fj(r)834 1445 y Fl(\012)865 1466 y Fm(f)894 1423 y Ff(\015)894 1453 y(\015)936 1466 y Fj(\024)c Fm(!)1020 1473 y Fk(f)1063 1466 y Fm(;)566 1584 y Fn(sup)557 1626 y Fk(\020)r Fe(2G)628 1631 y Fb(r)657 1542 y Ff(\015)657 1571 y(\015)685 1584 y Fn(\011)724 1563 y Fe(0)724 1596 y Fk( )754 1584 y Fn(\()p Fm(\020)t Fn(\))818 1542 y Ff(\015)818 1571 y(\015)846 1606 y Fe(P)s Fk(;)p Fe(P)936 1584 y Fj(\024)g Fm(c)1011 1591 y Fk(T)1059 1584 y Fm(;)8 b Fj(8)d Fm( )15 b Fj(2)f Fo(T)1249 1563 y Fk(N)1298 1584 y Fn(+)d Fm(i\033)20 b(:)0 1717 y Fg(De\014ne)d(the)h(dimensionless)i(p)m(ar)m(ameter)749 1837 y Fm(\026)14 b Fn(:=)f(24\()p Fm(K)970 1844 y Fl(1)993 1837 y Fn(\))1012 1817 y Fk(\034)1037 1837 y Fm(e)1060 1817 y Fl(2)p Fk(\034)1111 1804 y Fm(\031)6 b Fn(~)-29 b Fm(!)1172 1811 y Fk(f)p 1111 1826 88 2 v 1140 1871 a Fm(\015)1222 1837 y(;)0 1960 y Fg(wher)m(e)308 2054 y Fn(~)g Fm(!)335 2061 y Fk(f)375 2054 y Fn(:=)13 b Fm(c)463 2061 y Fk(T)503 1984 y Ff(\022)545 2021 y Fn(1)e(+)g Fm(e)654 2002 y Fe(\000)p Fk(\033)q(=)p Fl(2)p 545 2043 208 2 v 545 2089 a Fn(1)g Fj(\000)g Fm(e)654 2075 y Fe(\000)p Fk(\033)q(=)p Fl(2)758 1984 y Ff(\023)795 1989 y Fk(N)841 2054 y Fm(!)872 2061 y Fk(f)916 2054 y Fm(;)59 b(K)1031 2061 y Fl(1)1067 2054 y Fn(:=)14 b(max)1235 1984 y Ff(\032)1272 2054 y Fn(1)p Fm(;)1325 2021 y(\034)j Fn(+)10 b(1)h(+)g(log)f(2)p 1325 2043 272 2 v 1421 2088 a Fm(\033)r(=)p Fn(2)1603 1984 y Ff(\033)1666 2054 y Fm(:)0 2167 y Fg(Then)18 b(the)g(fol)s (lowing)g(holds)h(true:)k(if)18 b Fm(\026)c(<)f Fn(1)18 b Fg(and)822 2288 y Fm(M)h Fj(\025)1004 2255 y Fm(K)1046 2262 y Fl(1)p 948 2277 177 2 v 948 2323 a Fm(\026)978 2309 y Fl(1)p Fk(=)p Fl(\()p Fk(\034)t Fl(+1\))1149 2288 y Fm(;)719 b Fn(\(6)p Fm(:)p Fn(4\))0 2415 y Fg(then)19 b(ther)m(e)h(exists)g(an)h(analytic)m(al)f(c)m(anonic)m(al)i(tr)m (ansformation)g Fj(T)29 b Fn(:)17 b Fj(G)1355 2424 y Fk(r)q(=)p Fl(2)1434 2415 y Fj(!)g(G)1531 2424 y Fl(3)p Fk(r)q(=)p Fl(4)1613 2415 y Fg(,)j(with)g Fj(T)13 b Fn(\()p Fj(G)1847 2424 y Fk(r)q(=)p Fl(2)1909 2415 y Fn(\))18 b Fj(\033)0 2474 y(G)30 2483 y Fk(r)q(=)p Fl(4)92 2474 y Fg(,)g(such)g(that)g Fm(H)e Fj(\016)11 b(T)30 b Fg(turns)18 b(out)f(to)h(b)m(e)g(of)h(the)f(form)582 2627 y Fm(H)t Fn(\()p Fj(T)c Fn(\()p Fm(\020)t Fn(\)\))g(=)837 2565 y Fk(N)t Fl(+1)844 2580 y Ff(X)847 2686 y Fk(i)p Fl(=1)931 2627 y Fm(h)960 2607 y Fk(i)976 2627 y Fn(\()p Fm(\020)t Fn(\))d(+)g Fm(Z)t Fn(\()p Fm(\020)t Fn(\))g(+)g Fj(R)p Fn(\()p Fm(\020)t Fn(\))18 b Fm(;)480 b Fn(\(6)p Fm(:)p Fn(5\))0 2770 y Fg(wher)m(e)p eop %%Page: 15 15 15 14 bop 341 30 a Fh(On)16 b(long)h(time)g(stabilit)o(y)i(in)e (nonlinear)g(Sc)o(hr\177)-25 b(odinger)15 b(equation)342 b Fn(15)-8 149 y Fg(R1\))27 b Fm(Z)d Fg(is)d(analytic)h(to)m(gether)e (with)i Fj(r)751 131 y Fl(\012)781 149 y Fm(Z)t Fg(;)g(it)f(is)g(in)g (normal)g(form,)h(namely)f(satis\014es)1707 109 y Ff(\010)1736 149 y Fm(Z)q(;)8 b(h)1822 131 y Fk(i)1838 109 y Ff(\011)1886 149 y Fj(\021)19 b Fn(0)p Fg(,)100 209 y Fm(i)14 b Fn(=)f(1)p Fm(;)8 b(:::;)g(N)16 b Fn(+)11 b(1)p Fg(,)18 b(and)h(mor)m(e)m(over)g (satis\014es)g(the)f(estimate)692 313 y Fn(1)p 692 335 25 2 v 693 381 a Fm(r)757 347 y Fn(sup)731 389 y Fk(\020)r Fe(2G)802 396 y Fb(r)q(=)p Fd(2)866 304 y Ff(\015)866 334 y(\015)894 347 y Fj(r)936 326 y Fl(\012)966 347 y Fm(Z)t Fn(\()p Fm(\020)t Fn(\))1068 304 y Ff(\015)1068 334 y(\015)1109 347 y Fj(\024)g Fn(~)-29 b Fm(!)1193 354 y Fk(f)1219 347 y Fn(\()p Fm(e)11 b Fn(+)g(1\))18 b(;)484 b(\(6)p Fm(:)p Fn(6\))-8 506 y Fg(R2\))27 b Fj(R)18 b Fg(is)g(an)g(exp)m(onential)s(ly)h(smal)s(l)e(r)m(emainder,)j(namely) e(one)g(has)228 634 y Fn(1)p 228 656 V 229 702 a Fm(r)289 668 y Fn(Sup)268 710 y Fk(\020)r Fe(2G)339 717 y Fb(r)q(=)p Fd(2)403 625 y Ff(\015)403 655 y(\015)430 668 y Fj(r)472 647 y Fl(\012)503 668 y Fj(R)p Fn(\()p Fm(\020)t Fn(\))609 625 y Ff(\015)609 655 y(\015)637 687 y Fe(P)684 668 y Fj(\024)c Fm(e)760 647 y Fk(\034)t Fl(+2)839 668 y Fn(~)-29 b Fm(!)866 675 y Fk(f)892 668 y Fm(\026)922 647 y Fl(1)p Fk(=)p Fl(\()p Fk(\034)t Fl(+1\))1077 668 y Fn(exp)1162 582 y Ff(")1191 668 y Fj(\000)p Fn(\()p Fm(\034)16 b Fn(+)11 b(1)g Fj(\000)g Fn(log)f(2\))1548 597 y Ff(\022)1593 634 y Fn(1)p 1591 656 31 2 v 1591 702 a Fm(\026)1627 597 y Ff(\023)1663 608 y Fl(1)p Fk(=)p Fl(\()p Fk(\034)t Fl(+1\))1810 582 y Ff(#)1848 668 y Fn(;)-8 829 y Fg(R3\))27 b(for)18 b(any)h(analytic)f(function)g Fm(g)d Fn(:)f Fj(G)756 836 y Fk(r)792 829 y Fj(!)f(B)r Fg(,)18 b(wher)m(e)g Fj(B)h Fg(is)g(a)f(Banach)h(sp)m(ac)m(e,)h(one)f(has)517 946 y Fn(Sup)495 988 y Fk(\020)r Fe(2G)566 995 y Fb(r)q(=)p Fd(2)630 946 y Fj(k)p Fm(g)r Fn(\()p Fm(\020)t Fn(\))11 b Fj(\000)g Fm(g)r Fn(\()p Fj(T)h Fn(\()p Fm(\020)t Fn(\)\))p Fj(k)999 961 y Fe(B)1042 946 y Fj(\024)i Fm(\026)1125 925 y Fl(1)p Fk(=)p Fl(\()p Fk(\034)t Fl(+1\))1284 946 y Fn(Sup)1280 988 y Fk(\020)r Fe(2G)1351 993 y Fb(r)1380 946 y Fj(k)o Fm(g)r Fn(\()p Fm(\020)t Fn(\))p Fj(k)1519 961 y Fe(B)1575 946 y Fm(:)293 b Fn(\(6)p Fm(:)p Fn(7\))p 1945 1070 39 2 v 1945 1105 2 35 v 1982 1105 V 1945 1107 39 2 v 100 1216 a(The)16 b(remainder)e(of)j(this)g(section)g(is)g(dev)o (oted)f(to)g(the)h(sc)o(heme)d(of)j(the)f(pro)q(of)g(of)h(this)g (theorem.)j(It)0 1276 y(is)f(almost)f(iden)o(tical)j(to)e(the)g(pro)q (of)f(of)h(theorem)e(4.1)h(of)h([3],)f(the)h(only)h(v)m(arian)o(t)f(b)q (eing)g(the)g(tec)o(hnique)0 1336 y(used)d(to)h(solv)o(e)g(the)f (homological)h(equation.)100 1396 y(First)c(w)o(e)f(in)o(tro)q(duce)h (a)g(suitable)i(F)l(ourier)d(expansion)g(for)h(functions:)20 b(related)14 b(to)g(the)f(action)h(\011)f(w)o(e)0 1455 y(de\014ne)k(the)g(\\)p Fm(k)r Fj(\000)p Fn(th)g(F)l(ourier)f(co)q (e\016cien)o(t")k(^)-27 b Fm(g)842 1462 y Fk(k)867 1455 y Fn(\()p Fm(\020)t Fn(\))17 b(of)h(a)f(function)h Fm(g)f Fn(:)e Fj(P)1353 1437 y Fc(C)1403 1455 y Fj(!)g(B)r Fn(,)h(where)h Fj(B)i Fn(is)f(a)f(Banac)o(h)0 1515 y(space,)f(b)o(y)549 1598 y(^)-27 b Fm(g)571 1605 y Fk(k)595 1598 y Fn(\()p Fm(\020)t Fn(\))14 b(:=)799 1564 y(1)p 746 1586 132 2 v 746 1632 a(\(2)p Fm(\031)r Fn(\))839 1617 y Fk(N)892 1530 y Ff(Z)919 1643 y Fc(T)950 1633 y Fb(N)994 1598 y Fm(g)9 b Fn(\()q(\011)1086 1605 y Fk( )1116 1598 y Fn(\()p Fm(\020)t Fn(\)\))f Fm(e)1230 1577 y Fk(ik)q Fe(\001)p Fk( )1309 1598 y Fm(d)1335 1577 y Fk(N)1372 1598 y Fm( )19 b(:)445 b Fn(\(6)p Fm(:)p Fn(8\))0 1717 y(It)17 b(is)g(easy)g(to)g(see)f(that,)h(for)f(smo)q(oth)f(functions)i (one)f(has)664 1841 y Fm(g)r Fn(\(\011)748 1848 y Fk( )778 1841 y Fn(\()p Fm(\020)t Fn(\)\))e(=)946 1794 y Ff(X)928 1904 y Fk(k)q Fe(2)p Fc(Z)1005 1894 y Fb(N)1045 1841 y Fm(e)1068 1821 y Fe(\000)p Fk(ik)q Fe(\001)p Fk( )1179 1841 y Fn(^)-27 b Fm(g)1201 1848 y Fk(k)1225 1841 y Fn(\()p Fm(\020)t Fn(\))17 b Fm(;)0 2008 y Fn(so)f(that,)h(in)g(particular)784 2075 y Fm(g)r Fn(\()p Fm(\020)t Fn(\))d(=)959 2028 y Ff(X)940 2138 y Fk(k)q Fe(2)p Fc(Z)1017 2128 y Fb(N)1059 2075 y Fn(^)-27 b Fm(g)1081 2082 y Fk(k)1105 2075 y Fn(\()p Fm(\020)t Fn(\))17 b Fm(:)0 2241 y Fg(R)m(emark)83 b(6.3.)21 b Fn(Using)f(standard)e(argumen)o(ts)f(and)i(h)o(yp)q(othesis)g(4\))h (of)g(the)g(theorem)e(it)i(is)h(p)q(ossible)0 2301 y(to)16 b(pro)o(v)o(e)f(exp)q(onen)o(tial)j(deca)o(y)e(of)g(the)g(F)l(ourier)f (co)q(e\016cien)o(ts)i(of)f(an)f(analytic)j(function;)f(moreo)o(v)o (er,)c(if)k(a)0 2361 y(function)g Fm(g)h Fn(has)e(symplectic)h(gradien) o(t)g(whic)o(h)f(is)h(analytic)i(then)d(the)h(follo)o(wing)h(estimate)f (holds)617 2478 y(sup)632 2519 y Fe(G)656 2524 y Fb(r)700 2435 y Ff(\015)700 2465 y(\015)728 2478 y Fj(r)770 2457 y Fl(\012)802 2478 y Fn(^)-27 b Fm(g)824 2485 y Fk(k)849 2435 y Ff(\015)849 2465 y(\015)890 2478 y Fj(\024)14 b Fm(c)965 2485 y Fk(T)996 2478 y Fm(e)1019 2457 y Fe(\000)p Fk(\033)q Fe(j)p Fk(k)q Fe(j)1131 2478 y Fn(sup)1138 2519 y Fe(G)1162 2524 y Fd(2)p Fb(r)1214 2435 y Ff(\015)1214 2465 y(\015)1242 2478 y Fj(r)1284 2457 y Fl(\012)1314 2478 y Fm(g)1340 2435 y Ff(\015)1340 2465 y(\015)1882 2478 y Fn(\(6)p Fm(:)p Fn(9\))0 2630 y(\(for)j(the)f(pro)q(of)h(see)f ([3])g(Lemma)e(7.4\))p 1945 2593 39 2 v 1945 2628 2 35 v 1982 2628 V 1945 2630 39 2 v 100 2710 a(Then)h(w)o(e)h(mak)o(e)f(the) h(so)g(called)h(ultra)o(violet)i(cuto\013.)j(So)15 b(w)o(e)h(\014x)g Fm(K)i Fj(2)c Fo(N)1481 2692 y Fl(+)1514 2710 y Fn(,)i(and)g(corresp)q (ondingly)0 2770 y(w)o(e)g(giv)o(e)i(the)f(follo)o(wing)p eop %%Page: 16 16 16 15 bop 0 30 a Fg(16)838 b Fh(D.Bam)o(busi)0 149 y Fo(De\014nition)18 b Fh(A)f(function)g Fm(g)e Fn(:)f Fj(G)611 156 y Fk(r)647 149 y Fj(!)f Fo(C)k Fh(is)g(said)g(to)g(b)q(e)g (of)f(class)h Fn(\005)1260 156 y Fk(l)1293 149 y Fh(if)g(one)f(has)776 278 y Fm(g)r Fn(\()p Fm(\020)t Fn(\))d(=)959 231 y Ff(X)932 340 y Fe(j)p Fk(k)q Fe(j)p Fk()h Fn(0)k Fg(c)m(onsider)h(r)m(e)m(al)g Fn(\()p Fm(p;)8 b(q)r(;)g(I)t(;)g(\036)p Fn(\))19 b Fg(and)g(assume)654 1591 y Fj(j)p Fm(J)696 1598 y Fk(i)712 1591 y Fn(\()p Fm(\020)t Fn(\))p Fj(j)14 b(\024)f Fm(\032)18 b(;)59 b(i)14 b Fn(=)g(1)p Fm(;)8 b(:::;)g(N)16 b Fn(+)11 b(1)17 b(;)552 b(\(7)p Fm(:)p Fn(7\))0 1704 y Fg(then)18 b(ther)m(e)g(exists)g(a)h(c)m(onstant)f Fm(C)649 1711 y Fl(11)709 1704 y Fg(such)g(that)419 1797 y Ff(X)390 1904 y Fk(i)p Fe(\025)p Fk(N)t Fl(+1)529 1844 y Fm(i)546 1824 y Fl(2)574 1811 y Fm(p)599 1793 y Fl(2)599 1825 y Fk(k)635 1811 y Fn(+)11 b Fm(q)709 1793 y Fl(2)707 1825 y Fk(k)p 574 1833 158 2 v 641 1879 a Fn(2)751 1844 y Fj(\024)j Fm(C)840 1851 y Fl(11)882 1844 y Fm(\032)j(;)59 b Fj(j)p Fm(I)1034 1851 y Fk(k)1059 1844 y Fj(j)13 b(\024)h Fm(C)1175 1851 y Fl(11)1217 1844 y Fm(\032)j(;)59 b(k)16 b Fn(=)d(1)p Fm(;)8 b(:::;)g(N)s(:)287 b Fn(\(7)p Fm(:)p Fn(8\))p 1945 1976 39 2 v 1945 2011 2 35 v 1982 2011 V 1945 2013 39 2 v 0 2092 a Fg(Pr)m(o)m(of.)19 b Fn(De\014ne)567 2198 y Fm(A)c Fn(:=)e(4)737 2136 y Fk(N)718 2151 y Ff(X)723 2258 y Fk(l)p Fl(=1)798 2198 y Fm(I)820 2205 y Fk(l)852 2198 y Fm(;)58 b(J)18 b Fn(:=)c(4)1104 2151 y Ff(X)1070 2258 y Fk(k)q Fe(\025)p Fk(N)t Fl(+1)1223 2165 y Fm(p)1248 2146 y Fl(2)1248 2179 y Fk(k)1284 2165 y Fn(+)d Fm(q)1358 2146 y Fl(2)1356 2179 y Fk(k)p 1223 2187 158 2 v 1289 2232 a Fn(2)1403 2198 y Fm(;)0 2344 y Fn(so)k(that,)h(one)g(has)f Fm(J)388 2351 y Fk(i)418 2344 y Fn(=)e Fj(\000)p Fm(I)531 2351 y Fk(l)555 2344 y Fn(+)c Fm(A)h Fn(+)f Fm(J)c Fn(,)15 b Fm(i)f Fn(=)g(1)p Fm(;)8 b(:::;)g(N)d Fn(.)21 b(The)16 b(\014rst)f Fm(N)21 b Fn(of)16 b(\(7.7\))g(are)f(therefore)g(equiv)m (alen)o(t)0 2404 y(to)675 2464 y Fm(A)c Fn(+)g Fm(J)k Fj(\000)c Fm(\032)j Fj(\024)f Fm(I)980 2471 y Fk(l)1009 2464 y Fj(\024)h Fm(A)d Fn(+)g Fm(J)16 b Fn(+)10 b Fm(\032)17 b(;)0 2555 y Fn(inserting)f(these)g(inequaliti)q(es)i(in)f(the)f (de\014nition)g(of)g Fm(A)h Fn(solving)g(with)f(resp)q(ect)g(to)g Fm(A)g Fn(and)g(substituting)0 2615 y(again)h(in)g(the)g(ab)q(o)o(v)o (e)f(inequalities)k(w)o(e)c(get)398 2751 y Fj(\000)508 2718 y Fn(1)p 443 2740 157 2 v 443 2785 a(4)p Fm(N)g Fj(\000)11 b Fn(1)605 2751 y Fm(J)k Fj(\000)704 2718 y Fn(8)p Fm(N)h Fj(\000)11 b Fn(1)p 704 2740 V 704 2785 a(4)p Fm(N)16 b Fj(\000)11 b Fn(1)866 2751 y Fm(\032)j Fj(\024)f Fm(I)980 2758 y Fk(l)1009 2751 y Fj(\024)h(\000)1172 2718 y Fn(1)p 1107 2740 V 1107 2785 a(4)p Fm(N)i Fj(\000)11 b Fn(1)1269 2751 y Fm(J)k Fn(+)1368 2718 y(8)p Fm(N)h Fj(\000)11 b Fn(1)p 1368 2740 V 1368 2785 a(4)p Fm(N)16 b Fj(\000)11 b Fn(1)1530 2751 y Fm(\032)17 b(:)p eop %%Page: 21 21 21 20 bop 341 30 a Fh(On)16 b(long)h(time)g(stabilit)o(y)i(in)e (nonlinear)g(Sc)o(hr\177)-25 b(odinger)15 b(equation)342 b Fn(21)0 149 y(Inserting)16 b(in)h(the)g(last)h(of)e(\(7.7\),)h(and)f (remem)n(b)q(ering)e(the)j(de\014nition)g(of)g Fm(J)k Fn(w)o(e)16 b(get)506 252 y Ff(X)477 359 y Fk(i)p Fe(\025)p Fk(N)t Fl(+1)616 259 y Ff(\002)637 299 y Fm(i)654 279 y Fl(2)687 299 y Fj(\000)11 b Fm(\013)p Fn(\()p Fm(N)5 b Fn(\))852 259 y Ff(\003)882 299 y Fm(Z)916 306 y Fk(i)947 299 y Fj(\024)999 229 y Ff(\022)1036 299 y Fn(1)11 b(+)1128 265 y(8)p Fm(N)16 b Fj(\000)11 b Fn(1)p 1128 288 157 2 v 1194 333 a(4)1290 299 y Fm(\013)p Fn(\()p Fm(N)5 b Fn(\))1405 229 y Ff(\023)1451 299 y Fm(\032)17 b(;)0 470 y Fn(where)778 558 y Fm(\013)p Fn(\()p Fm(N)5 b Fn(\))15 b(:=)e(4)1005 486 y Ff(P)1058 499 y Fk(N)1058 539 y(l)p Fl(=1)1132 524 y Fm(l)1148 506 y Fl(2)p 1005 547 165 2 v 1009 592 a Fn(4)p Fm(N)k Fj(\000)11 b Fn(1)1193 558 y Fm(:)0 671 y Fn(Recalling)19 b(that)687 715 y Fk(N)669 730 y Ff(X)673 837 y Fk(l)p Fl(=1)749 777 y Fm(l)765 757 y Fl(2)801 777 y Fn(=)14 b Fm(N)907 707 y Ff(\022)950 743 y Fm(N)995 725 y Fl(2)p 950 766 68 2 v 972 811 a Fn(3)1035 777 y(+)1091 743 y Fm(N)p 1091 766 46 2 v 1101 811 a Fn(2)1153 777 y(+)1209 743 y(1)p 1209 766 25 2 v 1209 811 a(6)1240 707 y Ff(\023)1302 777 y Fm(;)0 923 y Fn(it)20 b(is)g(easy)f(to)h(see)f(that)h(\()p Fm(N)e Fn(+)13 b(1\))648 905 y Fl(2)683 923 y Fj(\000)g Fm(\013)p Fn(\()p Fm(N)5 b Fn(\))20 b(is)g(p)q(ositiv)o(e,)h(so)e(that)h(there)f (exists)h(a)f(p)q(ositiv)o(e)i(constan)o(t)0 982 y(suc)o(h)15 b(that)739 1062 y(inf)703 1094 y Fk(i)p Fe(\025)p Fk(N)t Fl(+1)849 1028 y Fm(i)866 1010 y Fl(2)899 1028 y Fj(\000)c Fm(\013)p Fn(\()p Fm(N)5 b Fn(\))p 849 1050 217 2 v 937 1096 a Fm(i)954 1082 y Fl(2)1085 1062 y Fj(\025)14 b Fm(c)f(>)h Fn(0)i Fm(:)0 1185 y Fn(F)l(rom)e(this)j(the)g(thesis)g (follo)o(ws.)p 1945 1148 39 2 v 1945 1183 2 35 v 1982 1183 V 1945 1185 39 2 v 0 1264 a Fg(R)m(emark)82 b(7.5.)21 b Fn(The)e(ab)q(o)o(v)o(e)g(lemma)e(has)i(an)g(in)o(teresting)h (geometrical)g(in)o(terpretation.)30 b(Indeed)19 b(it)0 1324 y(can)e(b)q(e)g(sho)o(wn)e(to)i(b)q(e)g(equiv)m(alen)o(t)i(to)e (the)g(statemen)o(t)f(that)h Fm(J)1192 1331 y Fk(N)t Fl(+1)1297 1324 y Fn(restricted)g(to)g(the)g(surface)f Fm(J)1862 1331 y Fk(i)1893 1324 y Fn(=)e(0,)0 1384 y Fm(i)g Fn(=)g(1)p Fm(;)8 b(:::;)g(N)21 b Fn(has)16 b(a)h(minim)o(um)12 b(at)17 b(zero.)p 1945 1347 V 1945 1382 2 35 v 1982 1382 V 1945 1384 39 2 v 0 1484 a Fg(R)m(emark)76 b(7.6.)20 b Fn(By)e(lemma)e(7.4)h(one)h(has)f(that)h(there)f(exists)i Fm(C)1259 1491 y Fl(12)1301 1484 y Fn(,)f(suc)o(h)e(that,)j(pro)o (vided)e Fm(\032)g Fn(is)i(small)0 1543 y(enough)517 1616 y Fj(j)p Fm(J)559 1623 y Fk(i)575 1616 y Fn(\()p Fm(\020)t Fn(\))p Fj(j)14 b(\024)g Fm(\032)30 b Fn(=)-8 b Fj(\))13 b Fm(d)8 b Fn(\()q Fm(\020)t(;)g Fj(M)p Fn(\))13 b Fj(\024)h Fm(C)1153 1623 y Fl(12)1203 1560 y Ff(\020)1239 1582 y Fm(\032)p 1239 1604 26 2 v 1240 1650 a(r)1281 1616 y Fn(+)1331 1582 y Fj(p)p 1373 1582 V 34 x Fm(\032)1399 1560 y Ff(\021)1453 1616 y Fm(;)415 b Fn(\(7)p Fm(:)p Fn(9\))0 1729 y(where)16 b Fj(M)g Fn(w)o(as)g(de\014ned)g(b)o(y)g (\(5.16\).)p 1945 1692 39 2 v 1945 1727 2 35 v 1982 1727 V 1945 1729 39 2 v 100 1808 a(W)l(e)g(ha)o(v)o(e)g(no)o(w)g(the)h (follo)o(wing)0 1888 y Fo(Theorem)97 b(7.7.)21 b Fg(Consider)i(the)g (Hamiltonian)g(system)f(\(5.21\))j(with)d(r)m(e)m(al)i(initial)e(data.) 38 b(Ther)m(e)0 1948 y(exists)18 b Fm(\026)166 1955 y Fe(\003)189 1948 y Fg(,)g Fm(C)258 1955 y Fl(4)280 1948 y Fg(,)g(and)h Fm(C)446 1955 y Fl(13)506 1948 y Fg(such)f(that,)g(if)h Fm(\026)13 b Fj(\024)h Fm(\026)911 1955 y Fe(\003)952 1948 y Fg(then)k(c)m(orr)m(esp)m(onding)j(to)d(initial)g(data)h(such)f (that)454 2051 y Fk(N)435 2066 y Ff(X)439 2172 y Fk(i)p Fl(=1)516 2114 y Fm(i)533 2093 y Fl(2)555 2114 y Fj(j)p Fm(I)591 2121 y Fk(i)608 2114 y Fn(\()p Fm(\020)t Fn(\(0\)\))p Fj(j)11 b Fn(+)840 2066 y Ff(X)810 2173 y Fk(i)p Fe(\025)p Fk(N)t Fl(+1)949 2114 y Fm(i)966 2093 y Fl(2)989 2114 y Fm(Z)1023 2121 y Fk(i)1039 2114 y Fn(\()p Fm(\020)t Fn(\(0\)\))k Fj(\024)f Fm(C)1270 2121 y Fl(13)1312 2114 y Fm(R)1350 2093 y Fl(3)1372 2114 y Fm(\026)1402 2093 y Fl(1)p Fk(=)p Fl(\()p Fk(\034)t Fl(+1\))1857 2114 y Fn(\(7)p Fm(:)p Fn(10\))0 2284 y Fg(one)k(has)500 2318 y Fk(N)481 2333 y Ff(X)485 2439 y Fk(i)p Fl(=1)562 2381 y Fm(i)579 2360 y Fl(2)601 2381 y Fj(j)p Fm(I)637 2388 y Fk(i)653 2381 y Fn(\()p Fm(\020)t Fn(\()p Fm(t)p Fn(\)\))p Fj(j)12 b Fn(+)878 2333 y Ff(X)849 2440 y Fk(i)p Fe(\025)p Fk(N)t Fl(+1)988 2381 y Fm(i)1005 2360 y Fl(2)1028 2381 y Fm(Z)1062 2388 y Fk(i)1078 2381 y Fn(\()p Fm(\020)t Fn(\()p Fm(t)p Fn(\)\))j Fj(\026)f Fm(R)1304 2360 y Fl(3)1326 2381 y Fm(\026)1356 2360 y Fl(1)p Fk(=)p Fl(\()p Fk(\034)t Fl(+1\))0 2526 y Fg(for)19 b(al)s(l)e(times)h(satisfying)644 2628 y Fj(j)p Fm(t)p Fj(j)c(\026)795 2595 y Fn(1)p 763 2617 91 2 v 763 2663 a Fm(R)801 2648 y Fl(2)823 2663 y Fm(\026)867 2628 y Fn(exp)952 2543 y Ff(")981 2558 y(\022)1023 2595 y Fm(C)1059 2602 y Fl(4)p 1023 2617 58 2 v 1037 2663 a Fm(\026)1087 2558 y Ff(\023)1124 2569 y Fl(1)p Fk(=)p Fl(\()p Fk(\034)t Fl(+1\))1271 2543 y Ff(#)1326 2628 y Fm(:)517 b Fn(\(7)p Fm(:)p Fn(11\))p 1945 2733 39 2 v 1945 2768 2 35 v 1982 2768 V 1945 2770 39 2 v eop %%Page: 22 22 22 21 bop 0 30 a Fg(22)838 b Fh(D.Bam)o(busi)0 149 y Fg(Pr)m(o)m(of.)21 b Fn(Eq.)d(\(7.10\))h(implies)g Fj(j)p Fm(J)601 156 y Fk(i)617 149 y Fn(\()p Fm(\020)t Fn(\(0\)\))p Fj(j)e(\026)g Fm(R)869 131 y Fl(3)891 149 y Fm(\026)921 131 y Fl(1)p Fk(=)p Fl(\()p Fk(\034)t Fl(+1\))1086 149 y Fj(8)5 b Fm(i)16 b Fn(=)g(1)p Fm(;)8 b(:::;)g(N)18 b Fn(+)12 b(1,)18 b(and)g Fm(\020)t Fn(\(0\))f Fj(2)g(G)1770 158 y Fk(r)q(=)p Fl(2)1832 149 y Fn(,)i(so)f(w)o(e)0 209 y(can)e(apply)h(lemma)e(7.1)h(implying)h(\(7.5\),)g(from)e(whic)o (h)718 312 y Fj(j)p Fm(J)760 319 y Fk(i)776 312 y Fn(\()p Fm(\020)821 292 y Fe(0)835 312 y Fn(\(0\)\))p Fj(j)g(\026)e Fm(R)1036 292 y Fl(3)1059 312 y Fm(\026)1089 292 y Fl(1)p Fk(=)p Fl(\()p Fk(\034)t Fl(+1\))1252 312 y Fm(;)591 b Fn(\(7)p Fm(:)p Fn(12\))0 416 y(and)16 b(moreo)o(v)o(er)e(\(7.6\))j (holds.)k(It)c(implies)613 519 y Fj(j)p Fm(J)655 526 y Fk(i)671 519 y Fn(\()p Fm(\020)716 498 y Fe(0)730 519 y Fn(\()p Fm(t)p Fn(\)\))12 b Fj(\000)f Fm(J)895 526 y Fk(i)911 519 y Fn(\()p Fm(\020)956 498 y Fe(0)970 519 y Fn(\(0\)\))p Fj(j)k(\026)f Fm(R)1172 498 y Fl(3)1194 519 y Fm(\026)1224 498 y Fl(1)p Fk(=)p Fl(\()p Fk(\034)t Fl(+1\))1857 519 y Fn(\(7)p Fm(:)p Fn(13\))0 622 y(pro)o(vided)f Fm(t)h Fn(satis\014es)g(\(7.11\))h(and)e Fm(\020)677 604 y Fe(0)690 622 y Fn(\()p Fm(t)p Fn(\))i Fj(2)f(G)838 631 y Fk(r)q(=)p Fl(2)900 622 y Fm(:)g Fn(W)l(e)g(pro)o(v)o(e)f(suc)o (h)g(an)g(inclusion.)23 b(By)14 b(\(7.12\))h(and)e(\(7.13\))0 682 y(it)18 b(follo)o(ws)722 741 y Fj(j)p Fm(J)764 748 y Fk(i)780 741 y Fn(\()p Fm(\020)825 721 y Fe(0)839 741 y Fn(\()p Fm(t)p Fn(\)\))p Fj(j)c(\026)g Fm(R)1033 721 y Fl(3)1055 741 y Fm(\026)1085 721 y Fl(1)p Fk(=)p Fl(\()p Fk(\034)t Fl(+1\))1249 741 y Fm(;)0 829 y Fn(whic)o(h)h(b)o(y)f (\(7.9\))h(implies)g Fm(\020)518 811 y Fe(0)531 829 y Fn(\()p Fm(t)p Fn(\))g Fj(2)f(G)679 840 y Fk(C)r(R\026)764 830 y Fd(1)p Fb(=)p Fd(2\()p Fb(\034)s Fd(+1\))911 829 y Fn(,)h(whic)o(h)f(pro)o(vided)g Fm(\026)h Fn(is)g(small)f(enough)g (implies)h Fm(\020)1867 811 y Fe(0)1880 829 y Fn(\()p Fm(t)p Fn(\))g Fj(2)0 889 y(G)30 898 y Fk(r)q(=)p Fl(2)92 889 y Fm(:)h Fn(Using)h(again)g(\(7.5\))g(w)o(e)g(get)g(the)f(thesis.)p 1945 852 39 2 v 1945 887 2 35 v 1982 887 V 1945 889 39 2 v 100 948 a(Theorem)e(2.1)i(is)h(a)g(simple)f(corollary)i(of)e (theorem)f(7.7.)22 b(W)l(e)17 b(ha)o(v)o(e)0 1038 y Fo(Pro)r(of)j(of)h (theorem)f(2.1)p Fn(.)29 b(First)19 b(w)o(e)g(pro)o(v)o(e)e(that)j (\(2.4\))f(implies)h(\(7.10\).)29 b(Indeed,)19 b(denoting)g(b)o(y)g Fm(u)1971 1020 y Fe(0)0 1098 y Fn(the)e(function)g(whose)f(co)q (ordinates)h(are)f Fm(\020)t Fn(,)g(and)f(b)o(y)i Fm(u)c Fn(:=)g Fj(T)1133 1105 y Fl(1)1156 1098 y Fn(\()p Fm(u)1204 1080 y Fe(0)1218 1098 y Fn(\))k(w)o(e)f(ha)o(v)o(e)301 1187 y Fk(N)282 1202 y Ff(X)286 1308 y Fk(i)p Fl(=1)363 1250 y Fm(i)380 1229 y Fl(2)402 1250 y Fj(j)p Fm(I)438 1257 y Fk(i)454 1250 y Fj(j)11 b Fn(+)559 1202 y Ff(X)529 1309 y Fk(i)p Fe(\025)p Fk(N)t Fl(+1)668 1250 y Fm(i)685 1229 y Fl(2)714 1216 y Fm(p)739 1198 y Fl(2)739 1229 y Fk(i)772 1216 y Fn(+)g Fm(q)846 1198 y Fl(2)844 1229 y Fk(i)p 714 1238 155 2 v 779 1284 a Fn(2)888 1250 y(=)959 1187 y Fk(N)941 1202 y Ff(X)945 1308 y Fk(i)p Fl(=1)1021 1250 y Fm(i)1038 1229 y Fl(2)1069 1250 y Fj(j)p Fm(Z)1117 1257 y Fk(i)1133 1250 y Fn(\()p Fm(u)1181 1229 y Fe(0)1195 1250 y Fn(\))h Fj(\000)e Fm(\027)1300 1257 y Fk(i)1317 1250 y Fj(j)h Fn(+)1421 1202 y Ff(X)1391 1309 y Fk(i)p Fe(\025)p Fk(N)t Fl(+1)1531 1250 y Fm(i)1548 1229 y Fl(2)1570 1250 y Fm(Z)1604 1257 y Fk(i)1621 1250 y Fn(\()p Fm(u)1669 1229 y Fe(0)1683 1250 y Fn(\))747 1432 y Fj(\024)817 1369 y Fk(N)799 1384 y Ff(X)803 1490 y Fk(i)p Fl(=1)879 1432 y Fm(i)896 1411 y Fl(2)927 1432 y Fj(j)p Fm(Z)975 1439 y Fk(i)992 1432 y Fn(\()p Fm(u)p Fn(\))g Fj(\000)g Fm(Z)1154 1439 y Fk(i)1170 1432 y Fn(\()p Fm(u)1218 1411 y Fe(0)1232 1432 y Fn(\))p Fj(j)h Fn(+)1345 1369 y Fk(N)1326 1384 y Ff(X)1330 1490 y Fk(i)p Fl(=1)1407 1432 y Fm(i)1424 1411 y Fl(2)1454 1432 y Fj(j)p Fm(Z)1502 1439 y Fk(i)1519 1432 y Fn(\()p Fm(u)p Fn(\))f Fj(\000)g Fm(\027)1672 1439 y Fk(i)1688 1432 y Fj(j)697 1618 y Fn(+)744 1515 y Ff(\014)744 1545 y(\014)744 1575 y(\014)744 1605 y(\014)744 1635 y(\014)744 1665 y(\014)791 1570 y(X)761 1677 y Fk(i)p Fe(\025)p Fk(N)t Fl(+1)900 1618 y Fm(i)917 1597 y Fl(2)948 1618 y Fn(\()q Fm(Z)1002 1625 y Fk(i)1018 1618 y Fn(\()p Fm(u)1066 1597 y Fe(0)1080 1618 y Fn(\))g Fj(\000)g Fm(Z)1194 1625 y Fk(i)1211 1618 y Fn(\()p Fm(u)p Fn(\)\))1297 1515 y Ff(\014)1297 1545 y(\014)1297 1575 y(\014)1297 1605 y(\014)1297 1635 y(\014)1297 1665 y(\014)1325 1618 y Fn(+)1404 1570 y Ff(X)1375 1677 y Fk(i)p Fe(\025)p Fk(N)t Fl(+1)1514 1618 y Fm(i)1531 1597 y Fl(2)1554 1618 y Fm(Z)1588 1625 y Fk(i)1604 1618 y Fn(\()p Fm(u)p Fn(\))17 b Fm(:)0 1780 y Fn(Using)g(\(5.14\))g(and)f(\(5.15\))h(w)o(e)f(obtain)437 1870 y Fk(N)419 1885 y Ff(X)423 1991 y Fk(i)p Fl(=1)499 1932 y Fm(i)516 1912 y Fl(2)539 1932 y Fj(j)p Fm(I)575 1939 y Fk(i)591 1932 y Fj(j)11 b Fn(+)695 1885 y Ff(X)666 1992 y Fk(i)p Fe(\025)p Fk(N)t Fl(+1)805 1932 y Fm(i)822 1912 y Fl(2)850 1899 y Fm(p)875 1881 y Fl(2)875 1912 y Fk(i)909 1899 y Fn(+)g Fm(q)983 1881 y Fl(2)981 1912 y Fk(i)p 850 1921 V 915 1967 a Fn(2)1025 1932 y Fj(\026)i Fm(R)1115 1912 y Fl(4)1149 1932 y Fn(+)1199 1885 y Ff(X)1202 1992 y Fk(i)p Fe(\025)p Fl(1)1279 1932 y Fj(j)p Fm(Z)1327 1939 y Fk(i)1344 1932 y Fn(\()p Fm(u)p Fn(\))e Fj(\000)g Fm(\027)1497 1939 y Fk(i)1513 1932 y Fj(j)25 b Fm(;)0 2089 y Fn(whic)o(h)18 b(sho)o(ws)f(that)h(indeed)g(\(2.4\))h(implies)g (\(7.10\).)27 b(Using)18 b(the)g(analogous)g(of)g(\(5.14\))h(and)e (\(5.15\))i(for)0 2149 y Fj(T)40 2128 y Fe(\000)p Fl(1)27 2162 y(1)108 2149 y Fn(it)e(is)f(clear)f(that)h(also)f(the)h(con)o(v)o (erse)e(estimate)h(holds,)g(and)f(therefore)h(that)g(theorem)f(7.7)h (implies)0 2209 y(theorem)g(2.1.)p 1945 2172 39 2 v 1945 2207 2 35 v 1982 2207 V 1945 2209 39 2 v 100 2268 a(A)i(sligh)o(tly)h (di\013eren)o(t)f(description)g(of)g(the)f(dynamics)g(is)h(giv)o(en)g (b)o(y)f(the)h(follo)o(wing)0 2348 y Fo(Theorem)76 b(7.8.)17 b Fg(Ther)m(e)h(exist)h Fm(\026)699 2355 y Fe(\003)740 2348 y Fg(and)g Fm(C)873 2355 y Fl(14)932 2348 y Fg(such)g(that,)f(if)h Fm(\026)14 b Fj(\024)g Fm(\026)1339 2355 y Fe(\003)1380 2348 y Fg(then)k(ther)m(e)h(exists)f(a)h(manifold)0 2408 y Fj(N)i(\032)14 b(P)21 b Fg(isomorphic)f(to)e(an)h Fm(N)k Fg(dimensional)c(torus,)g(with)f(the)f(fol)s(lowing)i(pr)m(op)m(erties) 28 2468 y(1\))27 b(It)17 b(is)i(stable)e(over)i(exp)m(onential)s(ly)f (long)g(times,)g(namely)236 2571 y Fm(d)p Fn(\()p Fm(\020)t Fn(\(0\))p Fm(;)8 b Fj(N)f Fn(\))15 b Fj(\024)f Fm(C)563 2578 y Fl(14)604 2571 y Fm(r)104 b Fn(implies)243 2717 y Fm(d)p Fn(\()p Fm(\020)t Fn(\()p Fm(t)p Fn(\))p Fm(;)8 b Fj(N)f Fn(\))15 b Fj(\026)527 2612 y Ff(v)527 2640 y(u)527 2670 y(u)527 2700 y(t)p 579 2612 1110 2 v 579 2717 a Fm(r)q(d)p Fn(\()p Fm(\020)t Fn(\(0\))p Fm(;)8 b Fj(N)f Fn(\))13 b(+)e Fj(j)p Fm(t)p Fj(j)p Fm(\026R)1002 2703 y Fl(5)1024 2717 y Fm(\026)1054 2703 y Fl(1)p Fk(=)p Fl(\()p Fk(\034)t Fl(+1\))1209 2717 y Fn(exp)1294 2632 y Ff(")1323 2717 y Fj(\000)1370 2647 y Ff(\022)1412 2683 y Fm(C)1448 2690 y Fl(4)p 1412 2706 58 2 v 1426 2751 a Fm(\026)1476 2647 y Ff(\023)1513 2657 y Fl(1)p Fk(=)p Fl(\()p Fk(\034)t Fl(+1\))1660 2632 y Ff(#)1707 2717 y Fm(;)1857 2671 y Fn(\(7)p Fm(:)p Fn(14\))p eop %%Page: 23 23 23 22 bop 341 30 a Fh(On)16 b(long)h(time)g(stabilit)o(y)i(in)e (nonlinear)g(Sc)o(hr\177)-25 b(odinger)15 b(equation)342 b Fn(23)100 149 y Fg(for)18 b(the)g(times)g(satisfying)h(\(7.11\).)28 209 y(2\))27 b Fj(N)e Fg(is)19 b(close)g(to)g(the)f(manifold)i Fj(M)e Fg(de\014ne)m(d)i(by)f(\(5.16\))i(\(and)e(ther)m(efor)m(e)h (also)g(to)e Fj(M)1705 216 y Fk(R;!)1776 209 y Fg(\),)h(namely)100 269 y(one)f(has)761 329 y Fn(sup)751 371 y Fk(\020)r Fe(2M)854 329 y Fm(d)p Fn(\()p Fm(\020)t(;)8 b Fj(N)f Fn(\))14 b Fj(\026)g Fm(r)q(\026)1134 308 y Fl(1)p Fk(=)p Fl(2\()p Fk(\034)t Fl(+1\))1319 329 y Fm(:)524 b Fn(\(7)p Fm(:)p Fn(15\))p 1945 425 39 2 v 1945 459 2 35 v 1982 459 V 1945 461 39 2 v 0 571 a Fg(Pr)m(o)m(of.)17 b Fj(N)22 b Fn(is)16 b(de\014ned)e(as)g(the)i(set)f(suc)o(h)f(that)h Fm(I)891 553 y Fe(0)887 585 y Fk(k)925 571 y Fn(=)f Fm(p)1003 553 y Fe(0)1003 585 y Fk(l)1032 571 y Fn(=)f Fm(q)1108 553 y Fe(0)1106 585 y Fk(l)1136 571 y Fn(=)h(0,)h Fj(8)5 b Fm(k)14 b Fn(=)f(1)p Fm(;)8 b(:::;)g(N)d Fn(,)16 b Fj(8)5 b Fm(l)13 b Fj(\025)h Fm(N)f Fn(+)8 b(1.)21 b(Denote)0 631 y(for)16 b(simplicit)o(y)812 691 y Fm(\032)d Fn(:=)h Fm(d)p Fn(\()p Fm(\020)t Fn(\(0\))p Fm(;)8 b Fj(N)f Fn(\))18 b Fm(;)0 785 y Fn(then,)e(since)h(the)g(Lipsc)o(hitz)h(constan)o(t)e (of)g Fj(T)30 b Fn(is)17 b(b)q(ounded)e(w)o(e)i(ha)o(v)o(e)841 905 y Fm(d)p Fn(\()p Fm(\020)912 884 y Fe(0)908 917 y Fl(0)931 905 y Fm(;)8 b Fj(N)f Fn(\))14 b Fj(\026)g Fm(\032)i(;)0 1024 y Fn(with)h Fm(\020)139 1006 y Fe(0)135 1037 y Fl(0)171 1024 y Fn(:=)d Fj(T)278 1006 y Fe(\000)p Fl(1)331 1024 y Fn(\()p Fm(\020)t Fn(\(0\)\);)j(it)h(follo)o(ws,)g(b)o(y)e(the)h(v)o (ery)f(de\014nition)h(of)g(the)g(norm)d(and)i(of)h Fj(N)721 1144 y(j)p Fm(J)763 1151 y Fk(i)779 1144 y Fn(\()p Fm(\020)824 1123 y Fe(0)820 1156 y Fl(0)842 1144 y Fn(\))p Fj(j)d(\026)g Fn(\()p Fm(r)f Fn(+)e Fm(\032)p Fn(\))p Fm(\032)j Fj(\026)f Fm(r)q(\032)18 b(;)0 1263 y Fn(and,)e(b)o(y)g(\(7.6\),)380 1428 y Fj(j)p Fm(J)422 1435 y Fk(i)439 1428 y Fn(\()p Fm(\020)484 1407 y Fe(0)497 1428 y Fn(\()p Fm(t)p Fn(\)\))p Fj(j)f(\026)f Fm(r)q(\032)d Fn(+)g Fj(j)p Fm(t)p Fj(j)p Fm(\026R)878 1407 y Fl(5)900 1428 y Fm(\026)930 1407 y Fl(1)p Fk(=)p Fl(\()p Fk(\034)t Fl(+1\))1085 1428 y Fn(exp)1170 1343 y Ff(")1199 1428 y Fj(\000)1246 1358 y Ff(\022)1289 1394 y Fm(C)1325 1401 y Fl(4)p 1289 1416 58 2 v 1303 1462 a Fm(\026)1353 1358 y Ff(\023)1389 1368 y Fl(1)p Fk(=)p Fl(\()p Fk(\034)t Fl(+1\))1536 1343 y Ff(#)1590 1428 y Fn(;)0 1592 y(using)16 b(\(7.9\))h(w)o(e)g(get)361 1778 y Fm(d)p Fn(\()p Fm(\020)432 1758 y Fe(0)446 1778 y Fn(\()p Fm(t)p Fn(\))p Fm(;)8 b Fj(N)f Fn(\))15 b Fj(\026)659 1673 y Ff(v)659 1701 y(u)659 1731 y(u)659 1761 y(t)p 711 1673 882 2 v 711 1778 a Fm(r)q(\032)d Fn(+)f Fj(j)p Fm(t)p Fj(j)p Fm(R)906 1764 y Fl(5)928 1778 y Fm(\026)958 1764 y Fl(1)p Fk(=)p Fl(\()p Fk(\034)t Fl(+1\))1113 1778 y Fn(exp)1198 1693 y Ff(")1227 1778 y Fj(\000)1274 1708 y Ff(\022)1316 1744 y Fm(C)1352 1751 y Fl(4)p 1316 1767 58 2 v 1330 1812 a Fm(\026)1380 1708 y Ff(\023)1417 1718 y Fl(1)p Fk(=)p Fl(\()p Fk(\034)t Fl(+1\))1564 1693 y Ff(#)1609 1778 y Fm(;)0 1959 y Fn(and,)16 b(b)o(y)g(the)h(b)q (oundedness)d(of)j(the)g(Lipsc)o(hitz)g(constan)o(t)f(of)h Fj(T)1211 1938 y Fe(\000)p Fl(1)1198 1973 y(1)1281 1959 y Fn(w)o(e)f(get)h(\(7.14\).)100 2019 y(T)l(o)f(pro)o(v)o(e)f(\(7.15\)) i(use)f(\(7.3\))h(whic)o(h)g(giv)o(es)178 2177 y Fm(I)200 2184 y Fk(k)238 2177 y Fn(=)c Fm(p)315 2184 y Fk(l)345 2177 y Fn(=)g Fm(q)419 2184 y Fk(l)448 2177 y Fn(=)h(0)g(=)-8 b Fj(\))13 b(j)p Fm(I)674 2156 y Fe(0)670 2189 y Fk(k)694 2177 y Fj(j)h(\026)f Fm(\026)804 2156 y Fl(1)p Fk(=)p Fl(\()p Fk(\034)t Fl(+1\))951 2177 y Fm(R)989 2156 y Fl(3)1028 2177 y Fm(;)1130 2130 y Ff(X)1100 2236 y Fk(i)p Fe(\025)p Fk(N)t Fl(+1)1240 2177 y Fm(i)1257 2156 y Fl(2)1285 2143 y Fm(p)1310 2110 y Fa(0)1324 2125 y Fl(2)1310 2156 y Fk(i)1357 2143 y Fn(+)e Fm(q)1431 2110 y Fa(0)1444 2125 y Fl(2)1429 2156 y Fk(i)p 1285 2165 182 2 v 1363 2211 a Fn(2)1486 2177 y Fj(\026)j Fm(\026)1569 2156 y Fl(1)p Fk(=)p Fl(\()p Fk(\034)t Fl(+1\))1716 2177 y Fm(R)1754 2156 y Fl(3)1793 2177 y Fm(;)0 2352 y Fn(from)h(whic)o(h)h(the)h (thesis)g(immediately)g(follo)o(ws.)p 1945 2315 39 2 v 1945 2350 2 35 v 1982 2350 V 1945 2352 39 2 v eop %%Page: 24 24 24 23 bop 0 30 a Fn(24)838 b Fh(D.Bam)o(busi)858 149 y Fo(References)22 239 y Fn([1])25 b(D.)15 b(Bam)o(busi,)e(A.)i (Giorgilli)q(:)24 b Fg(Exp)m(onential)17 b(Stability)g(of)g(States)f (Close)h(to)f(R)m(esonanc)m(e)j(in)d(In\014nite)100 299 y(Dimensional)i(Hamiltonian)h(Systems.)46 b Fn(Jour.)20 b(Stat.)j(Ph)o(ys.)e Fo(71)p Fn(,)15 b(p.)h(569-606)g(\(1993\).)22 379 y([2])25 b(D.)12 b(Bam)o(busi:)19 b Fg(A)14 b(Nekhor)m(oshev{T)l (yp)n(e)j(The)m(or)m(em)f(for)f(the)f(Pauli{Fierz)h(Mo)m(del)h(of)f (Classic)m(al)h(Ele)m(c-)100 439 y(tr)m(o)m(dynamics.)24 b Fn(Ann.)e(Inst.)g(Henri)17 b(P)o(oincar)o(\023)-24 b(e,)16 b(Ph)o(ysique)h(th)o(\023)-24 b(eorique)17 b Fo(60)f Fn(339-371)f(\(1994\).)22 520 y([3])25 b(D.)15 b(Bam)o(busi:)k Fg(Exp)m(onential)f(Stability)f(of)g(Br)m(e)m(athers)i (in)e(Hamiltonian)g(Chains)h(of)g(We)m(akly)f(Cou-)100 579 y(ple)m(d)h(Oscil)s(lators)p Fn(.)23 b(Nonlinearit)o(y)c Fo(9)d Fn(433-457)f(\(1996\).)22 660 y([4])25 b(D.)18 b(Bam)o(busi,)e(N.N.)j(Nekhoroshev:)25 b Fg(A)19 b(pr)m(op)m(erty)i(of) f(exp)m(onential)h(stability)f(in)f(nonline)m(ar)i(wave)100 720 y(e)m(quation)c(ne)m(ar)h(the)f(fundamental)g(line)m(ar)h(mo)m(de)p Fn(.)k(Preprin)o(t)15 b(16/1996)g(Dipartimen)o(to)g(di)h(Matem-)100 780 y(atica,)h(Univ)o(ersit\022)-25 b(a)18 b(di)f(Milano.)22 860 y([5])25 b(J.)16 b(Bourgain:)24 b Fg(Construction)18 b(of)i(appr)m(oximative)h(and)e(almost)g(p)m(erio)m(dic)i(solutions)f (of)f(p)m(erturb)m(e)m(d)100 920 y(line)m(ar)f(Schr\177)-25 b(odinger)20 b(and)e(wave)g(e)m(quation)g Fn(GAF)-6 b(A)17 b Fo(6)g Fn(201{230)e(\(1995\).)22 1000 y([6])25 b(J.)16 b(Bourgain:)21 b(Nonlinear)d(Sc)o(hr\177)-25 b(odinger)15 b(equations.)22 b(Preprin)o(t)16 b(1995)22 1081 y([7])25 b(W.)20 b(Craig,)i(C.E.)e(W)l(a)o(yne:)30 b Fg(Newton)-5 b('s)22 b(Metho)m(d)h(and)g(Perio)m(dic)g(Solutions)f(of)g(Nonline)m (ar)h(Wave)100 1141 y(Equations)17 b Fn(Comm.)i(Pure)c(and)h(Appl.)23 b(Math.)e Fo(46)16 b Fn(1409-1501)g(\(1993\).)22 1221 y([8])25 b(M.P)l(.)14 b(Cherno\013,)g(J.E.)h(Marsden:)k(Prop)q(erties)d (of)f(In\014nite)h(Dimensional)f(Hamiltonian)h(Systems.)100 1281 y(Lect.)22 b(Notes)17 b(Math.)22 b(425,)16 b(Springer)g(V)l(erlag) h(1974.)22 1362 y([9])25 b(A.)18 b(Giorgilli)q(,)k(E.)c(Zehnder:)25 b Fg(Exp)m(onential)c(stability)e(for)i(time)e(dep)m(endent)h(p)m (otentials.)30 b Fn(J.)18 b(Appl.)100 1421 y(Math.)j(Ph)o(ys.)g (\(ZAMP\),)d Fo(43)e Fn(827{855,)g(\(1992\).)-3 1502 y([10])25 b(S.B.)14 b(Kuksin:)21 b(Nearly)15 b(In)o(tegrable)g (In\014nite-Dimensional)f(Hamiltonian)i(Systems,)d(Lect.)22 b(Notes)100 1562 y(Math.)f(1556)16 b(\(Springer)g(1994\))-3 1642 y([11])25 b(S.B.)18 b(Kuksin,)i(J.)e(P\177)-25 b(osc)o(hel:)27 b Fg(Invariant)20 b(Cantor)h(manifolds)g(of)g(quasi-p)m(erio)m(dic)i (oscil)s(lations)f(for)100 1702 y(a)c(nonline)m(ar)h(Schr\177)-25 b(odinger)19 b(e)m(quation)f Fn(Ann.)j(of)c(Math.)22 b Fo(142)15 b Fn(149{179)h(\(1995\).)-3 1782 y([12])25 b(N.N.)14 b(Nekhoroshev:)20 b Fg(Behaviour)d(of)f(Hamiltonian)g (systems)g(close)f(to)g(inte)m(gr)m(able.)23 b Fn(F)l(unct.)e(Anal.)100 1842 y(and)15 b(Appl.)23 b Fo(5)16 b Fn(338{339)g(\(1971\).)-3 1923 y([13])25 b(N.N.)16 b(Nekhoroshev:)21 b Fg(Exp)m(onential)e (estimate)e(of)h(the)f(stability)h(time)f(of)h(ne)m(ar)g(inte)m(gr)m (able)g(Hamil-)100 1983 y(tonian)g(systems.)24 b Fn(Russ.)c(Math.)i (Surv)o(eys.)f Fo(32)16 b Fn(\(6\))h(1-65)f(\(1977\))-3 2063 y([14])25 b(C.E.)17 b(W)l(a)o(yne:)51 b Fg(Perio)m(dic)21 b(and)f(Quasi-p)m(erio)m(d)q(ic)i(Solutions)e(of)g(Nonline)m(ar)h(Wave) e(Equation)h(via)100 2123 y(KAM)d(The)m(ory.)24 b Fn(Comm)o(u)o(n.)19 b(Math.)i(Ph)o(ys.)h Fo(127)p Fn(,)15 b(479-528)h(\(1990\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF