Content-Type: multipart/mixed; boundary="-------------0303250103788" This is a multi-part message in MIME format. ---------------0303250103788 Content-Type: text/plain; name="03-135.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-135.keywords" Mode conversion, polarization, Born-Oppenheimer approximation, eigenvalues crossing, pseudo-differential systems, semi-classical analysis, normal form ---------------0303250103788 Content-Type: application/postscript; name="cross2.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="cross2.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86d Copyright 1999 Radical Eye Software %%Title: cross2_beta.dvi %%Pages: 17 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMR17 CMR12 CMR8 CMBX12 CMMI12 CMTI12 CMR7 CMTI10 CMR10 %%+ CMSY10 CMEX10 MSBM10 CMMI8 CMSY8 CMMI6 CMR6 MSAM10 CMSY6 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips cross2_beta.dvi -o cross2.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.03.25:0752 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro %! 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 6 /plusminus put dup 7 /minusplus put dup 10 /circlemultiply put dup 14 /openbullet put dup 15 /bullet put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 26 /propersubset put dup 33 /arrowright put dup 49 /infinity put dup 50 /element put dup 54 /negationslash put dup 88 /X put dup 92 /intersection put dup 94 /logicaland put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put readonly def /FontBBox{-29 -960 1116 775}readonly def /UniqueXX 5000820 def currentdict end currentfile eexec 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%%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR17 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 45 /hyphen put dup 46 /period put dup 50 /two put dup 65 /A put dup 66 /B put dup 67 /C put dup 69 /E put dup 71 /G put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 89 /Y put dup 97 /a put dup 99 /c put dup 101 /e put dup 104 /h put dup 105 /i put dup 109 /m put dup 110 /n put dup 114 /r put dup 115 /s put dup 116 /t put readonly def /FontBBox{-33 -250 945 749}readonly def /UniqueXX 5000795 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSAM10 %!PS-AdobeFont-1.1: MSAM10 2.1 %%CreationDate: 1993 Sep 17 09:05:00 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSAM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSAM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /square put readonly def /FontBBox{8 -463 1331 1003}readonly def /UniqueXX 5031981 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6f33457e3200f0935ea98f1a0cfaff49cfa2 d877ff805620cba84126dfd175a82e1b4c6ec58461fce97cdd749a012e2c4243 86b199a97c306b282350cbe5af90cf1d8e7371dcaedbd5fa3346dd3e558072c5 fd51a2edfee6890f62efd1b552699123f4f52c2817dadd5ece30b3bc178a3ef4 f45fd8ad0ee5bf30db5f9eee9ddefe249f3c349a886cf9130cba68ae559ad719 c439930ece1d52d38366116c390cbeabb8fdb36811cf3b8e2416825e8c880b24 6521ec6880ebe7f127c8741c756b29e1f89e3fd5d638a7bbade8c3a0fdbbf7ba ec37cd372afb2d9dd25d509726d20a05914e418f6b7e759c440beb61519df02a 06734ec1226890d17c51f25f775e6879fdc231bc4d7fcb4b33d04fa273bdb8e8 f124171714614f8a94e98a6c2d6bdc1254913999f9c181e42c0886556cdf48af 92a265c1eb06bb30fb2745840c17bd31c7b8bdfd37dfb9643a418a2eaae14094 6752e0a315207aef85c3c2630538907971ec1b05687208571a3504bb3369b5c8 9a43e5abe343d0599fd8d41a1bf01de49f5856bdd67d63a2b3ecea75d721a6ca 0c8751ab41e892e9cf674218dd44acfd40019bf59f4d69ea40d5cf9068a6ff96 9f8cc772b920baf17334833d732431a2969b9a30fe38bd44772cef653935a957 af61dd568f041847ccce3c587656cba7bbf46d8cefec936c6d34dbe51bc04929 0440354f3b29ed8e6b4f21ad53b5809779d67339f284563d54ed4d2a504b014d 974a31c0ceb57b364af519bad4e85c41d521a3ffc26dbac555b42dc191367152 1d60c30b95b5bbafb3c7d87a1c043ea5f0e6dab83a4de99a742e75f02837b923 c50eb478ac4131ee604e3d7b784bd8c93eb9430291d452c21be0495718546333 66a602a9f142a563fdae3c8d40de6a8aa581ff3172367ca325e6d23102c5c6a2 02cc99580287dbd76a05bb4cca591b12a9dd3d602d7957376ec30b16051a63c7 e26ddf009e21fdca5a31cc973f0b1f6967a66b30b973ddd9d9a3095b673ea8f8 5db8dbf3acb7bb2d85c04a0ea8853e4592e4f8e581adaadb78bc3ae58a2d4c80 3b09839eafc7bc3e7cdf784e852d6590ab1c4b7558e28505887c6858072f09dc 2caba91b6871920e5256b602a0de02b87e5eb456878d3634e8eb0e069fa2bb5c c10d5cd98401de1cf07ec9f7de62aaa20b583e2f7aad2fb7e727ebc43f712c74 e8eed1fa76133428ac71589b933cc37ad200c516045fc1a06d0086c541103b5d 4ec54dc77b 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueXX 5000789 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b031f3fc6c13d7da7277a94ba 018e9998b3dd888011a5d7c4204989f30f908b95533bda845746b673ab71ea57 65a0d14f4350707e47c8276305b28513cbe1bb0dbd269a53719bda46e536685d df78ca0146b6b93e760256b74d939d4e35b5e77238f04c92298dfdd188feea30 e053eefbcbb52f2011772b3aae39f5805597bbc1e8bb75a446ce014030f4f2f0 f49f9e962ee4a1024a746fa92a3628db5270732b54e43fe5ecfa524f127e5fcc 788e77e66098336ad67fe4cccaf0253272d5df79864bf4b734cb9a5859d557d8 bc11b8e00221ebc12e97de4b1f466ead83a4c894709363bca9040410a52d592e 34ee40cc7e5efa920546b981aa659513a24b1b85c221a1875b62d0b89e57a368 321b8043a5b094e0379760a443d632892b14ad6d19dacc8c78093243ad67e6a3 08e56e6b68412ee690b10dac6e17708754a00d51fc957b500eb80175716eef4b 2ca1ef867614659bee3f2b7319e97b6fdf1efc847bf3cee3156f72f21751da8e 5fb6898919e6799820d3de0642d756e09d6fae4ff08dd3deda3173bff4bb11f7 9109c97ddc05897af709ea199a90fcee8ce4c7a3c15b18170c41c04de2d3fba8 f34296a95b8e1e8de3739b17273f8f2c85e914615e8eac5e8bd2387ba3b1edf4 7968f06e2067d836d0f9f3e085cdfd2de06a62c81d786b304326f7002e83160a 36598589228b4dddddc43c85e1d126f8fe81b828028e26317af5894aaccf4f69 6301e1a9fc45935d8a414957f08febebbc3a72ada80f101e47447d019ade56e9 f4fab969bba2b44e47399fedf5caa1bcea216d7ba713d523d8c1e5c02115408a 277a0db93878d14fcdfef5999b0428be4c77940f21ba6b90ee390bc7f4276626 3bc4af384cb7c24e18d62a91cb21d46e9820f481bd7cb5dfbdc63ab8e563b429 d6be6ecf2c64f91a51e8c2d08d396c5a158a303fd73bc8bed6f75bb60b3e2836 1711b4be659b31729a7773c431048b9f457d2117fec6973c5ecf3764e1f16543 77fd41bd8fdf09c8b0ac1db016a9fef4bc29194f26f5b4de4d5a33cc520271ad 7dac397c0e79233251f31d85dd89c907b74a31520ea8331dbbbe0ccd4c9b3b85 44aef20b7aa6d6d2bcd26ec1cb76510de9e824d53bfe113b5a85e865401e9280 45f6fe64153940e36021f9d4c902d28c6a952ab769a114c2a34153092e3d791e 11792c25164d3537e3b80a03aab728e5ea285ba5c0eef10ba148712fe4220c91 47ce770dbd06226cc481d6f21cd6d2689acbcd1b5261791acde8d530df1e8eb8 5b87f058fc253051c17f69386178a23447eaa40c83e722d7b5c199fa9bce299b 1a94d0c92da7ac3ecf5cca495db6f9894e5ea16e695c0c28eda0a72ff937d731 c22e427854a2b0be8a9aa7afa394d0484317628cdb42aec4e8538a50764a886e bcd7708b54b5d189f061e099ddb5244115cb014e56967ace798205767954e14d a9851e6bb373472bd346ad40c3f6cb42750854748733268b6aa97051baece213 5552045900df4da8fa5980bf0227c89d876f1219ce77c703966c112bc6098d12 6348c5617ecc31c03bab5e8dbf3600d9e8b52a993e5fe2bcdb25d60e7798475a 9075f4f439411036b7648d6b3e3e53e4cf924f2694daf1bf6c51aef197eb9dd6 d3af7363c65b87ad4a61b9c9a41bc395d67cdb5010ce1c3f1f6553eb07abcec5 ab666444f00b0c35023defa2b60d9e0ff4ec42da1134b7ef498900b7071e8a14 ade16283c520aa506c7688e42a6b44824dcbeafc79741f68117dbbcc70763ed9 497d76acce5efd644474eac3eb7fc30aeb1ba00f8611485e9863a6276ae03fd0 dc4fa22e0f5de99fd19e7924d0232608d602a4eb0feabdfa2bb1b1ed0a8b958f 70ff55e8f52c1d3a3f327ea81449 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 34 /epsilon put dup 59 /comma put dup 78 /N put dup 104 /h put dup 106 /j put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueXX 5087381 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a1e67ed7e41cc25acc04702f6 8ef703e3d45722c1a91fdef7100a48631a02a6f02a08c6b1f9b4df8310385b86 8632718fd87119a233f219d9411383b7fa9f3e4780d8c27e2e89e0cae883d664 c3eac57a3aef8988a2e9f0f8c7f53e0a80bdfc4620e21287d0390e1975398544 7f3ea66401024bea75e1b4c4437b7bb188f76f96b918ac7c6ad7e8ae7f21d8c2 790f08cccec904fe48ef39e597ed4d4237c1d1f596f5906b19ea308020f7a35c 168e327ec3246b1dfabe912f6b6daac09974876d3996e57d180261110db05f15 e3e8eebba3d90b5764c03df3033a1ed678ebc679569a2fb297378b25434c0f20 5313ecb8a952f07242d3ee731b0cdc086a4481178a3d65129c47c09b22e9c431 e11b3747b94c26a757c38d06001798c6a568303d541385244b967d3b1786edea f65bb53c4c2fe75e4b1b15c2c78d930b4296c80f08bad86012451edc8e9f0854 c3b390a16e27b11b3d45a9f72eff8baded2242dc928a61685d79e09681c97425 5b90a498614cf560fa5b1718981388268ba206a96989e6d0b5d485d9aca5594a e67dd7b34d8a369adb06647f8aff8814d6d9cdc04a4835918e557174c5bc0f3f 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 6 /plusminus put dup 48 /prime put dup 49 /infinity put dup 106 /bar put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueXX 5000818 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bac8ced9b09a275ab231194ecf829352 05826f4e975dcecec72b2cf3a18899ccde1fd935d09d813b096cc6b83cdf4f23 b9a60db41f9976ac333263c908dcefcdbd4c8402ed00a36e7487634d089fd45a f4a38a56a4412c3b0baffaeb717bf0de9ffb7a8460bf475a6718b0c73c571145 d026957276530530a2fbefc6c8f059084178f5ab59e11b66566ca5ba42b1911a 5d7f1bf343015eece988b7a93bce0c7aa61344d48aed9c92c8698d4b7c9951c8 7d103f2414b39e1437f9d2e50c4ee5f218f2e6716926a79ea978f13b1f855345 191dd7d31d8f82c2e3343c7a5894d95bdc492c28226834efcb5c12fea36ac5cc 430e0aa604961e34888adf6c1f3954cbc2498e225d953cf5685852162346f474 5a2a7087d5d7ad486de16d2ca8e15cee26e012671ba3bdc7d95cc8c98bb774f5 08625e968aee27ff7d1a06e63bcfb5aa4876c3f8f13b30ccccee73c3caf4e70d 98e6ed2f422dbb4950bf789680e064150995941a9f4dd68a575949847a7d012b b910bf03a368f97cc4b552b0cb393586ef85adfbe8b725cd15e9c696e58e1b11 21c368ba02f172045741e4815d8cca8af35469573ec88979a32605484af842c7 c8113b9e8ed45959f6509328d38df9339718d0330e7678d6f8dd1eda9c753d8e ca84e0ec721ca748f66909e9d28baff19be1c356f7240cb81f058c2b15e4169f 61fba2c3d19e803f68526090cba235753238d5a7d593b10b60e96fcc899a9266 a13ec5e5d8a256ba1831fae44cac90a56bf0e96b941fe4005e93505caa64a511 b40b304c7c1440f553ed4cb0f67fa70b7e40a4d61d9cd8b81cc11d6e4b414589 708517e8ddb987559aa183511726983c4fcdd1c75ec98d0088e97e1aa4c07e7f 471d325b8a8c4fdb9a4ba03ab9c33de118a056bb7cb4028a02c00a925c994bdb b3e460b5f28b218be690024eb021ea62681acf4cbf035166c4e1e050ccc4f716 69fd2a9a254d769c3c226291948cf0f7cfeb085708da607e1fb3eed28efaaf07 471b27226357e9fa3b9ad314213e9d7636c7eb6a00020650ece5d983e95d1a4c 3e5386a93062049236c8ab2a2c0c7882b3d5baa9b05927254ca4423de151b54f 6a815c6227de5018301fbd2521c7b37036063c21110ad82b9af1521e53560763 339d9b569db7563ad47465a19dca1d5147bc60e7d5e23ac3643ec2dea521eab3 1ab0a6b5b371c608ee0c1e98581c8974a75000f864d481d8843934cb7088532a 2275587abbeb9f02932f7208576954d8c8103df338d2691116898313d6125ace bc2e9456e5fba1e0bab4ef24904a4ec453ecc199a7703e5e77388c0a4ca505ee 5b7b5590b520b2a391d517d0703d6541f7572857d30d144b79e9468772985491 0f1f86139bc19f90acdc4b4205a22427cfef24dd174394b4a507e8dbe09dcc65 ea2f405a14b25a59939477e6f425d8aeae2f83234afd52934c05eb5ca74bc5bd a8ce6def64bbe2eb851e1b00cee512d0c21220ba7b1de0d9d0f50efeb549b0c6 98ca65fa 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 21 /lambda put dup 24 /xi put dup 34 /epsilon put dup 59 /comma put dup 63 /star put dup 64 /partialdiff put dup 69 /E put dup 78 /N put dup 81 /Q put dup 83 /S put dup 84 /T put dup 87 /W put dup 88 /X put dup 89 /Y put dup 100 /d put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 110 /n put dup 112 /p put dup 120 /x put readonly def /FontBBox{-24 -250 1110 750}readonly def /UniqueXX 5087383 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d725e830d1c76fba12e12486e989c98 74c2b527f0925722787027f44470d484262c360cdfdddf3657533a57bb16f730 48bfbbfcb73a650484015441fdc837add94ac8fbd2022e3ec8f115d4b4bb7b7f 15388f22cc6198efe768bd9fceb3446ee4a8dc27d6cd152485384ef5f59381ff da43f2d20c8fb08aa27ab2015b774db10dacfdcd33e60f178c461553146ab427 bdd7da12534ba078ad3d780414930e72218b3075925ce1192f11fc8530fcd5e3 038e3a6a6db2dcfbae3b4653e7e02730314e02b54a1e296d2bef8a79411d9225 dad7b4e6d6f9cf0688b69ba21193bf1495807e7a196cf14c95a4e02f9cd2da8c db2546c6df52e524745992e18d9ff87aa25e4e1800bbe4ebb357c6ef55ed6d03 6d3a00c1ee8073266c21d2f0ac85d656abf61d7e5a4fa87da8ec3b5329e434d0 d2adab706b42a2e5331be5295399d803ccac03f631f01f39a022fcdf63486b68 7d15ef284a77def7fde4898543e7b5f7ec267756103e477f547cfb8d2311c4b0 09deff56085f5d419697af1846c8b88c1bbbae149f0f19ca3c8dafe19cec48fe 6b38357246d8b5cef80b53bab4cbbbe8bc8318cc462a4158258f456e697f6058 86010493f252073ee4e4c9531f23485415ad6bc901bbaea659ef701f825f4b6d 5c9bed0dce7b63810f906b33ea4955531ca1ed1fc120cc56dc1e277a33222cc6 c3a97552607b865783faf0895cbeccd0df54f57bd162a88494768435eda71a39 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4fdc99d684940cc28d0e25e8c663c6ffb41769ac23feb3ecea567b95e28a88d8 02c8f7cac6e8160d08d50772d2db127b3333894eb4203db093f079d83b9c82be aad1ae165c6f33fb61625edda65422cf883c273a89dabb091c621dbf0c55efcc 41d0f536d3c871040b4513aa5042ff7d940d6053d415b84b077605a5687073a4 a429f99fe8eb67a22fef34f83e0bde339f8dc98d5171ea825d72d4e8636d2d94 d7b4a6a460292dd007b86b24929c37b94eac17c779ea66747d954d4270a36f22 cb094fec0e41789cee233c548668988216e2f70147ce36d6441c20e6a97e60d7 b2f1b0b9113928f18c896720e510ef4a98485188cb65b1b78c2746ba754812b0 9fa2007b0aae53febbba740e2bbe7ca0c77f377baf9ee77c36cd974bd3c5d141 65fffe01befe17326b47559b39a386071a62ae1bb0912f7666a9deb6c8a60102 eeb8a66bf87113920ad832db372d390cb0ced2595ff1f7897130c985750d120a 366cd72f76d6cdd0d4b50c609fb56dc300ffc57e294bcab10097aa2a2e871b55 9af7749fb64d766b1ca85091e1f7a613cc73928f090c04be5a32b73ca23c7b89 baf756f1e0 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 78 /N put dup 82 /R put dup 91 /hatwide put dup 92 /hatwider put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueXX 5031982 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d56491bdecdcafdc988206c5a457a19270b 37d0ab776e03eaa7eb568eeab6b5e79dec03b0dcbf923a2aa8e4f4deda2cb043 858d8b430efef0ed914a80ffc9818aba0fb30dae3694e5b31df3855892d59b89 7f82ff79fce8444c7926ffc4ea008e63bc518936c9098afc2d1c14030a2c15bc c9285c1b57e80570d2f6301009e63d9ddeba44e3251c75ca9616a2c5de9ade66 1a15db37b40b67c1bd71f77343912c62f47dd24be0e03dd833ea60d2eb020407 eaf042fed30f0c2f01956c83d46d982b50dd979c2026ed5f71979dd98c9c1eee 25054000a5eed5ce9498b2cb59a7ee55e0538ab47d86e69b2ac79229c8c6bfc9 3a24e822a2f1c9964fe630344579a25f66646113f73d1315c447f070c575de76 266c6773b01137e4a039df5e4c2f199e8e3396ef9fd001eebe090e04384c4089 ec9f13a07ad8733c6af775f432f154ef50ec4477e08760c03c006e4d1eb1c7cd acb8955ca4b0d62c7de046ff58d431bb9883d28586552c51809e3ced4dfc2198 b97ad9e52ea33c91b9e335a80d9b77d3430e0487930f992459664abca5dd9177 150b0fb502fbb04ccaa94ffe2697ed07906f95fd98e7b325bfbf10ea6b39fd76 c1e5295022cb39e9cd471b7b0b3f9d72da99deb0615d35cc22f42dbf5cfb70dc 6279fd520d99f32201abf25e621cfe1d37961d9c0ad36ef3d9264c547638152a be30812b1188a10f6327c9e8714449de2ae636420f56a44cebd774f0b131c629 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37647e19e15007f0bc7767d35080fe17a0dfc6a13431eff43bc94a0c19f51f85 0d4f0a98bca6abaa7867b534357b01bc94c66ef35462fc7b5c171187df6b3923 c65e9764bef4f1212e5736470b811b9c30512f4eb9a1ff43d31ed92235b82152 1cc809d24170588bc48d8128f50936cf63834586f4a3b81d47a800ef8c9dfe5a 87a9ce66455a4682352a5f6bde283ec09f2acb2b967bbb47041c0e091bc75b97 c01d1cdcf0ab3c97fcc0e4715c9fe0b494424a7412f6f22d80faa440b7b7d8d9 84e8fb10ff30177a15a65ae81c1e56a9158c6f53d25531ceec392797e0856938 e01fa5a14b33bdbce82a7b95888f0f9db687ad7d250f741ae828a4b99241b461 290f20da92608b2a076ca5e6607ff0dd06361a559f9e926dbe4875456d840646 9da7e4d5fdcf879e1d778eae5a4610649f550304168951663a1318538d42e2f9 71cb5b1d5d642dfef43fa662e3595eca733280a17293da9c5966ee9b0be5d0dd 2a008e05f81f2300713ffa5db678a10ce478b8606acde6cd177509a5f4b71780 355587f74f791b931c2e08e7cbc5e55ee1c720fba00673e234dc1ad065513adc 9c207e0bc00749d510504524a6f185e277ef34f17f22ebd86b67c0b5003f666a c8fbb87a7777a20ab39d335ba2bc9a501518ff5fac1ce34a3359c63e38f4934f 761b954cd4267a3c1dd91927596823d7356304745738bb66c1bbdd913babd9c7 2ec71e9515276f90d360ee268b2fbcf4d2eb1aeee33880eccdc0719acc81524c b17d3680787623845e51d963a6be85a311cff7ac8cbea91d267d024a2f3a2bd2 79f9448578a97f0ed78238431ce290b89f4354a46042f5863eeb2b2124896df2 ff2a4e5017df7b4e830d5066fde526cca5d3f7a6ff495a056920e44d698023b3 c2dab352747266f0be8c8de61ae91710777974d770de13b975c4342f85d4dccf 74990f7020898f1b494c32a89ee5dd16b00fc418f7e53d1b3d01033308fb9070 4b9b45a818bcb977f28ca484dc90cfbe57a5546e61f0689a21de64c36f453ebe 1e32eee091a70a9ecf8d9b83414a1cb7186dc1d447023d6be183f81495cb2b7c 45352397648321d6516802cee2fdc18f30f993b1d82e366d2246a4af309ba287 33a53d3cfa7ea6d383a278a33e7adcaf1add06c7 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 26 /braceleftbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 40 /braceleftBigg put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 80 /summationtext put dup 88 /summationdisplay put dup 98 /hatwide put dup 112 /radicalbig put readonly def /FontBBox{-24 -2960 1454 772}readonly def /UniqueXX 5000774 def currentdict end currentfile eexec 80347982ab3942d930e069a70d0d48311d7190fa2d133a583138f76695558e7a e9348d37cac6651806d08527c1bb4a062a4835ac37784cc39ad8841404e438b4 d52d3901e47a1de4f7924e0fb3daf442499175bab1226edf692a4956739f8828 e80592f450c5d5c22ac88bcfbe9748f61d18243a16f4a4467f084e8e2be46ef4 7fc51c3a8199e3cda62ff9c4fb73956dab8b6683d2156377808cb35026073e80 523f59a30d195fcf9b9fce4ffafc6d5649664203ab24acb938d58d246707ffe7 d62f04bec4b70c21ef75beb2b812622b3c74e969d72d3cd11bd7106294a99caf 0b1629bc7d4de6b96ca82930831d64575f23f4ad06a0e45e315b1d392411be8d 6d73c998789ff258a07a3c8c2057325784514c845500bfd1a971310cfc11d41c 1a167dbd5ff012c60add4e87325f6e5299032a839de65fb1473a166aae1876a4 414a434f22c1d241591fb36f857df6fa930608750ffc0c54f44994662b1f00f1 400bf752ea8d83ffc4cb77a290bc2d99981ae59a191748ba5c7ba1a9d2583fd2 1398452b6ff5d83a059f7eadcd2ef744e9dd22bdf9c79d049bf06835e878c32b 7765c69bdd8ef4deb4ea7cfff4cf9354a4ddffa689de961d16772491c7afbd7f ffde42400764c68e954ee5c455a5687959829bc3b319b2147deaab3628662c80 30c5e02fea09609abe4eaa12e217bc3af673f1bc36a7039eb13fcacb4218fe0f c5a3f9452d4edf46cc91db67b624d4f2d37502fb9f11af4da18ca40b61097f95 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 44 /comma put dup 45 /hyphen put dup 48 /zero put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 74 /J put dup 75 /K put dup 77 /M put dup 79 /O put dup 81 /Q put dup 83 /S put dup 97 /a put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-251 -250 1009 969}readonly def /UniqueXX 5000793 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3fca78881823c5537fe6c3dda8ee5b8 97e17cb027f5c73fdbb56b0a7c25fc3512b55fe8f3acfbffcc7f4a382d8299cc 8fd37d3cea49dabdca92847af0560b404ef71134b0f3d99934fc9d0b4e602011 b9cfb856c23f958f3c5a2fbe0ef8587d1f5774879c324e51fcb22888b74f2415 50d7401eb990d4f3a7af635198422283cac1b6cd446ddbcbd915db9bff88844e 784c6bf7389803d9450b0c21756a017306457c7e62c1d269f306bd3402e266de fc3b5e7d8a8d2f5bf0fe6ddd40d07391df4fad4a6018dce29a2b8f692b29f202 3a7c0e66de8ed85c14f1f8492167357f51a7e84cc5d92e0fee4d81cf7fbc8de5 2d2e7bb57142033993f9c08c315abade8dbc4a732e84e142d3bee51557910e12 cd8aa37c459a5e6b7f5269f59078aba3be4641a11ac48d0b625c8325b38ec08e 4c9e5e7fed976a5650d99d82114f449b9ca14c4ec957702295a39a93ef93f618 99b8ea06b092c3c1e503e6e436e0a9fa22576c8930ab3dc8c20f5d82b69cddf8 ff4dacfa9c54bed5a3aa3ea5b129fe96be632843b9b6bc91b615581a985db56b 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 48 /zero put dup 50 /two put dup 58 /colon put dup 65 /A put dup 67 /C put dup 75 /K put dup 77 /M put dup 83 /S put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 121 /y put readonly def /FontBBox{-163 -250 1146 969}readonly def /UniqueXX 5000828 def currentdict end currentfile eexec 8053514d28ec28da1630165fab262882d3ffd20326947b1065649b533eb5e9e3 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0249d1edf892b29d07cd624598fe88e6342eb9c76fceedc46d3713ef7f22a4dd 82b338421eb52d1fd710987144f6c7db2c1c2629c91c36eec45e49c7c38078a6 f1628b14dee1f1e90a9cce970fb82fa20691b4271ce624cb471e3d823df05271 9316495ce6a5e98483df484befd814b02e8e2699cfa178a8339e44c9f340c3c1 5131959058635430bab6dee97c130705b871f29b3281ea2539e9c42a218a681a 10aaf6009336e852a521f785fdbd7bf773eb1f9711cdaabd87de7872e1c175ee 93d38341eb502b7719ea72ef8e4021b1fdc311eb529886cdcbd9bbb998f8116c 646140f932686d19aa0638080707978e76a5fcfbfe34c29a5e699767f43daf7b 86a612e8a8ef72a9ec08ae5a6c045a3a7c6d6cb268f8e42cd45406b900f2cd55 b8e81ce06a867e1a5a388240dc0546daff346ce57b1e5c017100693d1fef3fcc 81a0606a141bd763b1d779379c2d68d4175ac5a3c68e422576812744af7f46c5 b657be86423a494becf8638a378c526416327ed6f81befe3d85eeb23176b2a42 5c17da3eefdda3ba8f49e35767ac1cbc0f2d087a66b3acde27fbca0dd7d5af1e f5e77cb4915ea5968ffa7ee3dd9d67e85f6281590b793453efb26ddaab1df27b 8e4cad0094fa8f8c059ad0944ff01c640951f7109a3438df173b5e8dc5ecc12b 9fba4f3c3c20535bc563456ee565e977d6dc1a5657e38b80ac0d718016b268fa 290b15c7bb8dc87f235789b5b0626c8571b6306373f0f02c2f618858fa87a70b 94a2addf7e0786fd60e3492dd4de48f1edb9f94e03ee68cdb900bc76b7d9ea8d 199413d20a65f7fe95ade17da2a7ac3129f11d4eb2c385b84b04a93dcf7d3486 fe32290974db614bb50adb5e40ecba55b52857661e6724a3055c7c5d264055ae 3bd40b73a3c1e46502674dde7d04394b2b6284e43a7ec6981101aa2b23db4e24 43372cb2158d1497d852585dab03bf88ba650464edc951b21040bf8bdcbe92ae 424c14682b00e8e6ed2a6ef3ce4d691b8e2e55ddef95337581a12212bf9733b8 ae66b3a47b93b2e04e30a8b4b7e586441bbd470750e58cdbe44c39cec76fdc43 f6c30d3a58f887ac2bc9163c21c84df5eb2bb229f813814b9597ea00a2d73810 b2f6de3dfc40ecaf2592b622e917795ec1a338ccd85df2e8224abeba6c957d75 05984b81feadfd4392558cad760cb8ab73b90697607f7f37d87d46d3bc3b307f ead665f2e320d770de0591a569e1f79826cb4a 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 49 /one put dup 50 /two put dup 51 /three put readonly def /FontBBox{-27 -250 1122 750}readonly def /UniqueXX 5000790 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b448685940729e6e22f4be0e08c6505cba868f7950 93f556b64b808dd710eb936d3ac83e5783b5f7e56d363a59a026a5619a357d21 c4e77bea121eb24b09027d1aad93158edf912940e91cd2f17922b35885894644 7773730bb19af9e70453e1b78d5edc123b273a979891d6d5e624f4a388f3897e 13b045e34b4ee2f5e4fc5e996d3549640010c671de14d18b0b3cd300e57dac03 22e59bfbf7b29422230870f5897fcfaad4b50c7c1c58edcd119218163d6f6e56 25ccb882db7b6d09a1586508e8cb642a11c29a46915e4a96e282079cb21922c1 c2e360b487a45302fd22ec8c5fab63e54b5e844d4b17ca2fff37c69c366dd72a d02922c14c0932f65ed03e4219c117962edbad2dcdeaa9c10ce8af38a4ae52e2 b377245b0be19a77d6c936e7530cc4d0b78d0cc4a92698fa2870fa54f2d8503e 2d17b3d52fb2febb09f2b2af0c2a1892039ebe19a690098799a858e3d39631bd 6925a154d161df3918074ada6bd52baddd0adc3f07e2d9f15e27cbf7fe8b98c4 07205c811121fa91e059f2f99322fed63f359ac9da97aec383f067f23e5de331 51e80f0a88ab50fe8fdae4a5de93c1ae2fdca06150b37246140c0e87cb2325a6 0d2349162ae3ac93144eee1e665a1289105318fdfe86b6e76251cb25adc967d3 d0b97fe5e279e1161736ab22b4ca510b964342383a840defd38f96a7280e6ac1 34e48d740607ff2e7804164a16d47735864db847c97335e6d4215cb99911a1ec 015a3edaac1f28fedd56d2467130d07bae9416c15f0827d27c6c79f59054282a 418c12c157c91223a829947f47592f7cafd93ca182b25a73a9419127e3b12a9e 5167ac3963f2b019b338ac46d63880f94dda4b538835884d2a5538c85528d6bd 977f844d32b43b0e48caed5a4bdabcefe71695d69ca784da87b5d0b13f35180e 58efc5cf12de78792a88665f1d5e60d4569e5c614792303616a6d0bc41d20457 507b75a7973223073a715fbc7fe4b888df49c5b1841fb540bdb8dd0128f1626a 801f8d3871c2b4d5e5d8bccca9cb92d282289128d69c0f75f990aece52ea6ab5 cfd2d5dd01ce4ff38d21301e6014664c0c937b57a7af2be7e8aae980929cd77e 66694fbd58c24e5531988dfde0d8ed096c25719fddaa11dd9354bb2e379cc716 b80194087a8d4822d289871d6510625aaf06726d6833cbff90afccf2af6d3def 95fcfdb1aa0e890467164da5107dbf2243e48acc60c1b304f73ab4a3209fb636 060f9c812fe5b50da3682923ee15290c5aa0a9895bb2e7836549c403770d825a 4f4092ca19ca195e16fbbaab55c6868efc2e341b1f90ba3da1379c374194ad97 570658336de5a24600f4dafed3d142e7c94d9e959dcfc983a65a6217132bd1dd e2873796c347663955f778f8f887a6d6426a88f82a01129a12f9f0973bce51ac 1ff4aebce6f82f86218b5dd3f4a69957fd9cbfe75420e3e9e435f8c88fb1eec0 29c19d1fdee97387ad365f6b1d2f271ccc612444eb45b3a687626fc3b0507649 bdde0c6fc6aa18807d3d792b3d20bd263e4608dd80082dfdc97a83713ec740b3 b98281ee6e2ebf13735f48c2d41a70b84db608c3a0fbb6687419e864012f5742 73b91877dfdc859488f6f80bb8116e89d022825d409c5e6670870c38fad9c3e0 884c8ee63f1abfa5874b88524b2f09c3375a0dbb7f0296839b31443c40eef3d7 c0417fe2a23156aae7c2350b99c930e346c40da420c0737dd71aa62eb30614da fc309f7cc5668984ad2bf770e5bdc8439cfe623e905a116b0eb1 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI12 %!PS-AdobeFont-1.1: CMTI12 1.0 %%CreationDate: 1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 19 /acute put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 58 /colon put dup 65 /A put dup 67 /C put dup 69 /E put dup 70 /F put dup 72 /H put dup 73 /I put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 87 /W put dup 91 /bracketleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put readonly def /FontBBox{-36 -251 1103 750}readonly def /UniqueXX 5000829 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacee583a5c939393e012707b47e0c1f a47d284a1edc9d01a497d772bca8c543388e6dc0d1e2c4944740470e0914f65e fb0737b0851b2ba713a9a00b36d07da6bcb52920b9b59efe587734027a3c5e65 66aad332fe6fbcced1417802822a3b81d6187875263d6bbda04bbcf6e4870fee ad60f104bb3c5a766610dd11aea64a6b107b0b04439fa2888b8cc39232bb83f7 695aba81f0260cd5248b9e649cd803271dc8bb1656323089e9e2bb50f2b95088 87d31dc36e555668d9578b338402f8c259f6813b0b71e6cbd95e19c20ced8495 2559e47577e612a4d94de3c1b4c6c41a6a3fa9e2313457242de74c7e4c0d722b 3e224d18baa726369542682e9459b951b2ac638494880c42be4e2175a3c067e5 945bfc1a05d3de8c3e8ab97f80ab6000e1f78fc258b85d3c84b038d7d0f5be85 0d43fe03e26cdfbb41c36d14cb552e8447ca678efea1d275e85503123a2263b1 4077587683de69a8f5a88d666172668411d0d0fed3e037ee468627207505dbb3 59cfdc5d7f526fca5416d741073284f5dd2471e09f5916f4bff1a90eea4d1283 7731f6a36e5ea695f4efc3bf5669b7f60ab606df5819149c0c8a4817f06dc172 0543467645daeffe4457293e0c0407c4ec7857558d0452de2ac773f141783edf 81b30215ae8d2d062aeb3cb69eb0d62bf04780997c9c027a22896f456b784378 7f2ccdd4804375321e7946d3d254521249a4d28a360f4d3163912a688c16e807 15d0440bd27bb78fcbc89bc28028aab2f594f8b641e45b9c599180b69663eef0 03465a2425aa4f8b29dade493002fb335b23eb145108ef3ab828755d3753ea8d f0683265497b1cb5cb484c13f7a4e003fa6ab3ed5e24fcc6125cee4e33936122 3cc907af3c64e5aaad882c356e673e6ca2509a71920e99549d8797e9af709e51 4ebefe61aa0478a60f3a2b7b3d47542fbd126303e31718e6b9c711d202dff7af eb7c19098c9e2a52aa57844eb07e1bfe9067dbe7d0bcd87c86f4ea099deb2373 edf149ab7955979e26ad415a5c29c1dfd02b5f0678dc863d3d6e537ad3b40da7 6529df395597aad7f22d53ed7e16573ce620e179f10fc8e66d1eef855b5427e2 9a7aee650c79f4ee340b1bcc4aef263bba31c829effbd9df68ce46b8fd90d280 0c6578962691d753df719c78e9b1af6378adcc3f9d77d25a4058c2121e048628 cfd039df8adffa81a01872a0ff96d524af0850c0f1977e938d99f1a6432f47c8 e0ca1ce8b1afdbac35d30996b2b4a92f595cdadaadb93d5355bde82069c60c0a 67f148c357643f61c1cdbb721eaf926dcd3cd303dd526b0e836b95c572b03f30 29ecd71d2923c5e97ad392654b75278fed325f9e39799425b525080339f029eb d540b8456f429c3711f4d220c58e242fb29d7a138dfb5fe5edac1a6f24a2e7a7 bc1918014e672ad6e4591d6570f556d1544a1634b7bd1d7275cb90bcc21b5603 15835347712efca6a6ca5a2cb6f9d97158eb80ea51ba4f20631fefc592441402 dddf963c0cbb1e01852ce7b6ba40e9e7b2aea7c7d763692a745dba443d76f23e 4a6b28c07837b906db6886583c7b7fa47b577c203222f417405809ffda371e6d f061b686510526541ae0e7610a995ff774f6dd68a85ff741abdfdb4c3ff83a53 891de85ff4af172c3aa24d812a4c92b0a965d941b58cafc3a68dc8e286e3569a 7a81cf261eb100bf527d6957ef7837da6afe3aa3a3bf14bcaa78e2205db3b1af 06e999483412d1e2d1ceed4461dc8d5296a52f6266b5ccf50752e6f723df4576 dcac31a8079c60b05c1a723ef59da7a950bca1ac7603f3f5a844a6204752a89e 4951921a06a36b226412c22a23aa33e3d2b46bc966b200a974fc86fba30bf44c 2a29913a84971320a075c0b801e4a18887ce1f2b3a542d2e6e8b579396e5ce01 98f01ba7379560588c87a3cc0de160e812bc13ee674046e2806ef8a4cf55a54b eb80ddab53c72fa436c0a36cf123ae47647a62747fdb6b588d0d587339ad1622 974586f42539d6d68e794d465ec82bf90733c70ca0e76b9ebc7b3c89186b2f08 d5c8b271fdcc0cdd2d46829a04c9db23a1a21e7acca0731cbbf94bde429cf3c2 cbea3835fda6e2b3b65e847effd342231dc31f31cfaacd6ff384023bb943c413 d232adc5c2e5370bb8695e202195c59b5988ad058fc059ec683568ed336a2ec8 fa78a7c393a8d04692f41272b1facf770643254269ffdc618eb0eb900f86c051 205a80935fcdc54660936207a068981b8b321fe493548e358a4d50ba962e1262 255bdffa2ba2c1036b3e7b5d30a239af10efea0d0dfee7227cddc2beb78a199e 0483b16ec366be9188cc14cc97557a18a53d487944a93931745a8657914b35e3 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 13 /gamma put dup 14 /delta put dup 21 /lambda put dup 24 /xi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 31 /chi put dup 33 /omega put dup 34 /epsilon put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 63 /star put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 5 /Pi put dup 6 /Sigma put dup 9 /Psi put dup 10 /Omega put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 18 /grave put dup 19 /acute put dup 22 /macron put dup 33 /exclam put dup 34 /quotedblright put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-34 -251 988 750}readonly def /UniqueXX 5000794 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 6 /Sigma put dup 18 /grave put dup 19 /acute put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 55 /seven put dup 56 /eight put dup 58 /colon put dup 61 /equal put dup 64 /at put dup 66 /B put dup 67 /C put dup 70 /F put dup 72 /H put dup 73 /I put dup 74 /J put dup 77 /M put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 85 /U put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 126 /tilde put readonly def /FontBBox{-36 -250 1070 750}readonly def /UniqueXX 5000791 def currentdict end currentfile eexec 9b9c1569015f2c1d2bf560f4c0d52257bacdd6500abda5ed9835f6a016cfc8f0 0b6c052ed76a87856b50f4d80dfaeb508c97f8281f3f88b17e4d3b90c0f65ec3 79791aacdc162a66cbbc5be2f53aad8de72dd113b55a022fbfee658cb95f5bb3 2ba0357b5e050fddf264a07470bef1c52119b6fbd5c77ebed964ac5a2bbec9d8 b3e48ae5bb003a63d545774b922b9d5ff6b0066ece43645a131879b032137d6d 823385fe55f3402d557fd3b4486858b2a4b5a0cc2e1bf4e2a4a0e748483c3bcf 5de47cc5260a3a967cac70a7a35b88b54315191d0423b4065c7a432987938c6b edad3b72ad63c2918b6e5a2017457e0d4ebc204b094541f345ec367ae85ca9bd 24568a01d3b9f8095f7420e6c423c414b3dcce6da48dd1c89a56d078e0d0e2f2 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b(h)m(yp)s(erb)s(olic)g(corank)f(2)f(in)h(Born-Opp)s(enheimer) h(appro)m(ximation,)h([11)o(],)f([9])f(and)364 527 y([8].)43 b(In)33 b([7],)g(it)f(is)h(the)g(case)h(where)g Fr(E)6 b(:B)33 b Fs(=)27 b(0)33 b(and)f Fm(j)p Fr(E)6 b Fm(j)27 b Fr(>)h Fm(j)p Fr(B)5 b Fm(j)p Fs(.)p Black 239 728 a(3.)p Black 49 w(The)36 b(case)f(of)f(one)h(degree)h(of)e(freedom)i (with)f(parameters)h(is)f(studied)h(in)f([3])g(\(adia-)364 848 y(batic)24 b(limit,)h(h)m(yp)s(erb)s(olic)g(case\))g(\(see)f(also)g ([12)o(])g(and)f([15]\))g(and)h(in)g([5)o(])g(and)f([6])h(\(elliptic) 364 968 y(case:)44 b(band)33 b(crossings\).)120 1300 y Ft(2)161 b(The)53 b(general)g(strategy)120 1519 y Fs(W)-8 b(e)33 b(will)g(pro)s(ceed)h(for)e(eac)m(h)h(case)h(along)e(the)h(same) g(lines:)p Black 239 1713 a(1.)p Black 49 w(Reduction)i(to)f(a)f(2)23 b Fm(\002)h Fs(2)34 b(system.)50 b(This)35 b(part)f(is)h(alw)m(a)m(ys)g (the)g(same)g(and)f(is)h(recalled)364 1834 y(in)e(section)g(3)p Black 239 2034 a(2.)p Black 49 w(Finding)g(a)f(normal)g(form)h(for)f (the)h(disp)s(ersion)h(relation:)43 b(this)34 b(part)e(w)m(orks)i(b)m (y)p Black 480 2235 a Fm(\017)p Black 48 w Fs(Finding)f(a)g (\\Birkho\013)f(normal)h(form")f(along)g(the)h(singular)g(manifold)g (\006)p Black 480 2394 a Fm(\017)p Black 48 w Fs(Using)i(Stern)m(b)s (erg's)h(Theorem)g(in)e(order)h(to)e(get)i(a)f(normal)g(form)g(for)g (the)h(h)m(y-)578 2514 y(p)s(erb)s(olic)e(part)p Black 239 2715 a(3.)p Black 49 w(Using)40 b(a)e(general)i(result)g(stated)f (in)h(section)g(4)f(,)h(w)m(e)g(pass,)i(using)e(a)e(gauge)h(trans-)364 2835 y(form)32 b Fr(J)9 b Fs(,)33 b(from)g(a)f(normal)h(for)f(the)h (disp)s(ersion)h(relation)f(to)f(a)g(normal)h(form)f(for)h(the)364 2956 y(disp)s(ersion)h(matrix)p Black 239 3156 a(4.)p Black 49 w(In)24 b(order)g(to)f(get)h(the)g(semi-classical)i(normal)e (form,)h(w)m(e)g(need)g(to)e(solv)m(e)i(the)g(follo)m(wing)364 3276 y(t)m(yp)s(e)33 b(of)f(homological)h(equation)1350 3486 y Fm(f)p Fr(S;)17 b(H)1585 3501 y Fu(0)1624 3486 y Fm(g)22 b Fs(+)g Fr(C)1871 3445 y Fj(?)1910 3486 y Fr(H)1991 3501 y Fu(0)2053 3486 y Fs(+)g Fr(H)2232 3501 y Fu(0)2271 3486 y Fr(C)35 b Fs(=)27 b Fr(R)364 3696 y Fs(where)42 b Fr(H)735 3711 y Fu(0)815 3696 y Fs(is)f(the)g (classical)h(normal)f(form,)i Fr(R)f Fs(is)f(giv)m(en,)j Fr(S)i Fs(is)c(an)e(unkno)m(wn)j(real)364 3817 y(v)-5 b(alued)40 b(function)g(and)g Fr(C)46 b Fs(an)40 b(unkno)m(wn)h(matrix) f(v)-5 b(alued)40 b(function.)65 b(F)-8 b(ortunately)g(,)364 3937 y(this)40 b(equation)f(is)h(the)f(linearization)h(of)e(the)i (classical)g(normal)f(form,)i(so)e(that)g(w)m(e)364 4057 y(can)33 b(solv)m(e)h(it)e(for)g(free!)266 4252 y(The)j(realization)f (of)f(this)h(program)f(is)h(more)g(di\016cult)g(than)g(in)g([2)o(],)g (esp)s(ecially)i(in)e(the)120 4372 y(corank)f(2)f(case)i(whic)m(h)g(is) f(not)f(structurally)i(stable.)120 4704 y Ft(3)161 b(Reduction)52 b(to)h(a)h Fd(2)32 b Fc(\002)g Fd(2)53 b Ft(system)120 4923 y Fs(It)33 b(is)g(w)m(ell)h(kno)m(wn)f(\(see)h([1])f(or)f([2]\))g (that)h(near)f(a)h(p)s(oin)m(t)f Fr(z)2263 4938 y Fu(0)2331 4923 y Fm(2)c Fr(T)2496 4887 y Fj(?)2535 4923 y Fk(R)2607 4887 y Fj(n)2686 4923 y Fs(suc)m(h)35 b(that)1354 5133 y(dim)17 b(k)m(er)g Fr(H)1760 5148 y Fu(class)1906 5133 y Fs(\()p Fr(z)1989 5148 y Fu(0)2029 5133 y Fs(\))28 b(=)f(2)32 b Fr(;)120 5342 y Fs(w)m(e)c(can)f(split)g(microlo)s(cally)h (the)f(system)i(in)m(to)d(a)h(direct)g(sum)h(of)e(a)h(\()p Fr(d)10 b Fm(\000)g Fs(2\))g Fm(\002)g Fs(\()p Fr(d)g Fm(\000)g Fs(2\))28 b Fq(el)5 b(liptic)120 5463 y(blo)-5 b(ck)32 b Fs(and)g(a)h(2)22 b Fm(\002)g Fs(2)33 b(blo)s(c)m(k)g(whose)h (principal)f(sym)m(b)s(ol)h(v)-5 b(anishes)34 b(at)e Fr(z)2712 5478 y Fu(0)2752 5463 y Fs(.)p Black 1806 5712 a(4)p Black eop %%Page: 5 5 5 4 bop Black Black 266 407 a Fs(The)32 b(disp)s(ersion)g(relations)f (of)f(the)g(initial)h(system)h(and)f(the)g(small)g(one)g(are)f(the)h (same.)120 527 y(In)44 b(what)g(follo)m(ws,)k(w)m(e)c(will)h(alw)m(a)m (ys)g(assume)g(that)f(this)g(splitting)h(has)f(b)s(een)h(done)f(and)120 648 y(therefore)33 b(w)m(e)h(ha)m(v)m(e)g(a)e(2)22 b Fm(\002)h Fs(2)32 b(system)i(to)f(study)-8 b(.)266 768 y(F)g(or)48 b(con)m(v)m(enience,)55 b(the)49 b(transv)m(ersalit)m(y)i (h)m(yp)s(othesis)f(\()p Fr(?)p Fs(\))e(has)h(b)s(een)g(form)m(ulated)g (in)120 888 y(section)34 b(1)e(for)g(the)h(big)g(system.)120 1221 y Ft(4)161 b(A)54 b(lemma)g(ab)t(out)g(gauge)f(transforms)120 1440 y Fs(The)34 b(follo)m(wing)e(Lemma)i(will)f(b)s(e)g(used)g(sev)m (eral)i(times:)p Black 120 1668 a Fh(Lemma)k(2)p Black 49 w Fq(L)-5 b(et)35 b Fr(H)g Fs(:)28 b Fk(R)1033 1632 y Fu(4)1033 1694 y Fj(X)1123 1668 y Fm(\002)22 b Fk(R)1294 1632 y Fj(N)1294 1694 y(\025)1389 1668 y Fm(!)27 b Fs(Herm)q(\(2\))35 b Fq(b)-5 b(e)34 b(a)h(smo)-5 b(oth)34 b(map)g(such)h(that)1046 1888 y Fs(det)q(\()p Fr(H)8 b Fs(\()p Fr(X)r(;)17 b(\025)p Fs(\)\))26 b(=)i Fr(X)1818 1903 y Fu(1)1857 1888 y Fr(X)1938 1903 y Fu(2)2000 1888 y Fm(\000)23 b Fs(\()p Fr(X)2227 1847 y Fu(2)2219 1913 y(3)2288 1888 y Fs(+)f Fr(X)2475 1847 y Fu(2)2467 1913 y(4)2514 1888 y Fs(\))35 b Fr(:)120 2108 y Fq(Ther)-5 b(e)42 b(exist)g(uniquely)h(de\014ne)-5 b(d)42 b Fr(")f Fs(=)h Fm(\006)p Fs(1)p Fr(;)60 b(\013)42 b Fs(=)g Fm(\006)p Fs(1)h Fq(and)f(a)g(smo)-5 b(oth)42 b(germ)h(of)f(map)g Fr(J)51 b Fs(:)120 2229 y Fk(R)192 2193 y Fu(4)254 2229 y Fm(\002)22 b Fk(R)425 2193 y Fj(N)520 2229 y Fm(!)27 b Fr(GL)p Fs(\(2)p Fr(;)17 b Fk(C)p Fs(\))35 b Fq(such)f(that)910 2507 y Fr(J)973 2466 y Fj(?)1013 2507 y Fr(H)8 b Fs(\()p Fr(X)r(;)17 b(\025)p Fs(\))p Fr(J)36 b Fs(=)1556 2367 y Fl(\022)1800 2446 y Fr(\013)q(X)1944 2461 y Fu(1)2195 2446 y Fr(X)2276 2461 y Fu(3)2338 2446 y Fs(+)22 b Fr(i"X)2596 2461 y Fu(4)1670 2566 y Fr(X)1751 2581 y Fu(3)1813 2566 y Fm(\000)h Fr(i"X)2073 2581 y Fu(4)2323 2566 y Fr(\013)q(X)2467 2581 y Fu(2)2676 2367 y Fl(\023)266 2793 y Fs(The)34 b(pro)s(of)e(follo)m(ws)h(exactly)h(the) f(same)g(lines)h(as)f(the)g(pro)s(of)e(of)i(Lemma)g(5)f(in)h([2].)120 3126 y Ft(5)161 b(The)53 b(symplectic)g(case)120 3374 y Fb(5.1)135 b(The)45 b(normal)g(form)h(for)f(the)g(disp)t(ersion)g (relation)p Black 120 3559 a Fh(Prop)s(osition)38 b(2)p Black 48 w Fq(Assuming)i Fs(\()p Fr(?)p Fs(\))f Fq(and)h Fs(\(1)p Fr(:)p Fs(\))g Fq(\(we)f(ar)-5 b(e)40 b(in)f(the)h(symple)-5 b(ctic)39 b(c)-5 b(ase)40 b(and)f(b)-5 b(oth)120 3679 y(p)g(airs)31 b(of)h(eigenvalues)f(do)g(not)h(vanish\),)f(ne)-5 b(ar)32 b(any)g(p)-5 b(oint)31 b Fr(z)2338 3694 y Fu(0)2410 3679 y Fq(of)h(the)g(singular)f(set)h Fs(\006)p Fq(,)h(ther)-5 b(e)120 3800 y(exists)35 b(a)h(c)-5 b(anonic)g(al)34 b(tr)-5 b(ansformation)35 b Fr(\037)h Fq(and)f(two)h(invertible)f(p)-5 b(ositive)35 b Fs(\()p Fr(>)29 b Fs(0\))35 b Fq(germs)g Fr(e)p Fs(\()p Fr(z)t Fs(\))120 3920 y Fq(and)f Fr(b)p Fs(\()p Fr(\034)6 b(;)17 b(z)529 3884 y Fi(0)554 3920 y Fs(\))34 b Fq(so)h(that:)487 4140 y Fs(det\()p Fr(H)741 4155 y Fu(class)887 4140 y Fs(\))22 b Fm(\016)g Fr(\037)28 b Fs(=)g Fr(e)p Fs(\()p Fr(z)t Fs(\))1399 4059 y Fl(\000)1445 4140 y Fr(x)1500 4155 y Fu(1)1540 4140 y Fr(\030)1583 4155 y Fu(1)1644 4140 y Fm(\000)22 b Fr(b)1784 4099 y Fu(2)1824 4140 y Fs(\()p Fr(x)1917 4099 y Fu(2)1917 4165 y(2)1979 4140 y Fs(+)g Fr(\030)2125 4099 y Fu(2)2120 4165 y(2)2164 4140 y Fr(;)17 b(z)2257 4099 y Fi(0)2281 4140 y Fs(\)\()p Fr(x)2412 4099 y Fu(2)2412 4165 y(2)2474 4140 y Fs(+)22 b Fr(\030)2620 4099 y Fu(2)2615 4165 y(2)2659 4140 y Fs(\))2697 4059 y Fl(\001)2764 4140 y Fs(+)g Fr(O)2937 4155 y Fj(Y)2998 4140 y Fs(\()p Fm(1)p Fs(\))120 4360 y Fq(wher)-5 b(e)39 b Fr(Y)57 b Fs(=)36 b Fm(f)p Fr(x)731 4375 y Fu(2)807 4360 y Fs(=)h Fr(\030)963 4375 y Fu(2)1038 4360 y Fs(=)f(0)p Fm(g)p Fq(,)k Fr(z)h Fs(=)36 b(\()p Fr(x)1610 4375 y Fu(1)1650 4360 y Fr(;)17 b(\030)1737 4375 y Fu(1)1776 4360 y Fr(;)g(x)1875 4375 y Fu(2)1914 4360 y Fr(;)g(\030)2001 4375 y Fu(2)2040 4360 y Fr(;)g(z)2133 4324 y Fi(0)2193 4360 y Fs(=)36 b(\()p Fr(x)2398 4324 y Fi(0)2422 4360 y Fr(;)17 b(\030)2514 4324 y Fi(0)2536 4360 y Fs(\)\))40 b Fq(ar)-5 b(e)39 b(c)-5 b(anonic)g(al)38 b(c)-5 b(o)g(or)g(di-)120 4481 y(nates)35 b(ne)-5 b(ar)34 b Fs(0)27 b Fm(2)i Fr(T)839 4444 y Fj(?)878 4481 y Fk(R)950 4444 y Fj(n)997 4481 y Fq(.)120 4709 y(Pr)-5 b(o)g(of.{)p Black Black 388 4912 a Fs(W)d(e)24 b(start)g(using)g(the)h(same)g(kind) f(of)g(argumen)m(ts)h(as)f(in)g(the)g(pro)s(of)f(of)h(Theorem)364 5033 y(2)47 b(in)h([2].)90 b(W)-8 b(e)48 b(get)f(then)i(a)e(\(formal\)) h(Birkho\013)g(normal)g(form)f(along)h(the)364 5153 y(singular)33 b(set)g(\006)g(of)f(the)h(form:)1193 5373 y Fr(A)p Fs(\()p Fr(x)1359 5388 y Fu(1)1399 5373 y Fr(\030)1442 5388 y Fu(1)1481 5373 y Fr(;)17 b(x)1580 5332 y Fu(2)1580 5398 y(2)1642 5373 y Fs(+)22 b Fr(\030)1788 5332 y Fu(2)1783 5398 y(2)1827 5373 y Fr(;)17 b(z)1920 5332 y Fi(0)1944 5373 y Fs(\))22 b(+)g Fr(O)2177 5388 y Fu(\006)2231 5373 y Fs(\()p Fm(1)p Fs(\))32 b Fr(;)p Black 1806 5712 a Fs(5)p Black eop %%Page: 6 6 6 5 bop Black Black 364 407 a Fs(with)33 b Fr(A)g Fs(a)f(smo)s(oth)h (function)g(whic)m(h)h(satis\014es)846 646 y Fr(A)p Fs(\()p Fr(\034)999 661 y Fu(1)1039 646 y Fr(;)17 b(\034)1125 661 y Fu(2)1165 646 y Fr(;)g(z)1258 605 y Fi(0)1281 646 y Fs(\))28 b(=)g Fr(\025)p Fs(\()p Fr(z)1595 605 y Fi(0)1618 646 y Fs(\))p Fr(\034)1698 661 y Fu(1)1760 646 y Fm(\000)1870 579 y Fr(!)t Fs(\()p Fr(z)2022 543 y Fi(0)2045 579 y Fs(\))p 1870 623 214 4 v 1952 715 a(2)2093 646 y Fr(\034)2135 661 y Fu(2)2197 646 y Fs(+)22 b Fr(O)s Fs(\()p Fr(\034)2464 605 y Fu(2)2453 671 y(1)2525 646 y Fs(+)g Fr(\034)2676 605 y Fu(2)2665 671 y(2)2716 646 y Fs(\))33 b Fr(:)364 881 y Fs(There)28 b(is)f(a)f(min)m(us)i(sign)f(in)f(fron)m(t)h(of)f (the)h Fr(\034)1930 896 y Fu(2)1996 881 y Fs(term)g(b)s(ecause)h(it)e (is)h(the)g(only)g(w)m(a)m(y)364 1001 y(to)38 b(get)g(the)h (appropriate)g(signature)g(\(+)p Fr(;)17 b Fm(\000)p Fr(;)g Fm(\000)p Fr(;)g Fm(\000)p Fs(\))38 b(for)g Fr(p)2549 965 y Fi(00)2630 1001 y Fs(along)g(\006.)61 b(Using)364 1122 y(T)-8 b(a)m(ylor)33 b(form)m(ula,)g(w)m(e)g(can)g(rewrite)h Fr(A)f Fs(as)f(follo)m(ws)901 1331 y Fr(A)p Fs(\()p Fr(\034)1054 1346 y Fu(1)1094 1331 y Fr(;)17 b(\034)1180 1346 y Fu(2)1220 1331 y Fr(;)g(z)1313 1290 y Fi(0)1337 1331 y Fs(\))27 b(=)h Fr(F)14 b Fs(\()p Fr(\034)1663 1346 y Fu(1)1702 1331 y Fr(;)j(\034)1788 1346 y Fu(2)1828 1331 y Fr(;)g(z)1921 1290 y Fi(0)1944 1331 y Fs(\)\()p Fr(\034)2062 1346 y Fu(1)2124 1331 y Fm(\000)23 b Fr(\034)2266 1346 y Fu(2)2306 1331 y Fr(b)2347 1290 y Fu(2)2387 1331 y Fs(\()p Fr(\034)2467 1346 y Fu(2)2506 1331 y Fr(;)17 b(z)2599 1290 y Fi(0)2623 1331 y Fs(\)\))32 b Fr(:)364 1540 y Fs(Using)41 b(Stern)m(b)s(erg's)h (linearization)f(as)g(in)g([2],)h(w)m(e)g(get)f(the)g(result.)68 b(W)-8 b(e)41 b(use)364 1661 y(the)24 b(follo)m(wing)g(v)m(ersion)h(of) f(Stern)m(b)s(erg's)h(Theorem)g(whose)g(pro)s(of)e(can)h(b)s(e)g(giv)m (en)364 1781 y(using)33 b(the)g(same)h(argumen)m(ts)f(as)g(in)g (Nelson's)h(b)s(o)s(ok)e([14]:)364 1902 y Fh(Theorem)j(\(Stern)m(b)s (erg\))30 b Fq(L)-5 b(et)33 b Fr(X)40 b Fq(b)-5 b(e)33 b(a)f(smo)-5 b(oth)32 b(ve)-5 b(ctor)33 b(\014eld)f(on)g Fr(T)2951 1865 y Fj(?)2990 1902 y Fk(R)3062 1865 y Fj(n)3142 1902 y Fq(and)364 2022 y Fs(\006)c(=)g Fm(f)p Fr(x)671 2037 y Fu(1)738 2022 y Fs(=)g Fr(\030)885 2037 y Fu(1)951 2022 y Fs(=)g Fr(x)1110 2037 y Fu(2)1177 2022 y Fs(=)g Fr(\030)1324 2037 y Fu(2)1391 2022 y Fs(=)f(0)p Fm(g)p Fq(.)45 b(L)-5 b(et)35 b(us)g(assume)f(that)h Fr(X)h Fs(=)27 b Fr(X)2804 2037 y Fu(0)2866 2022 y Fs(+)22 b Fr(X)3045 2037 y Fu(1)3119 2022 y Fq(with)364 2142 y Fr(X)445 2157 y Fu(1)519 2142 y Fq(is)35 b(c)-5 b(omp)g(actly)34 b(supp)-5 b(orte)g(d)35 b(and)f Fr(X)1777 2157 y Fu(1)1844 2142 y Fs(=)28 b Fr(O)2023 2157 y Fu(\006)2077 2142 y Fs(\()p Fm(1)p Fs(\))p Fq(.)44 b(We)36 b(assume)1084 2352 y Fr(X)1165 2367 y Fu(0)1232 2352 y Fs(=)27 b Fr(x)1390 2367 y Fu(1)1430 2352 y Fr(@)1481 2367 y Fj(x)1521 2376 y Ff(1)1582 2352 y Fm(\000)c Fr(\030)1725 2367 y Fu(1)1764 2352 y Fr(@)1815 2367 y Fj(\030)1846 2376 y Ff(1)1907 2352 y Fs(+)f Fr(Y)2062 2367 y Fu(0)2101 2352 y Fs(\()p Fr(x)2194 2367 y Fu(2)2234 2352 y Fr(;)17 b(\030)2321 2367 y Fu(2)2360 2352 y Fr(;)g(z)2453 2311 y Fi(0)2477 2352 y Fs(\))34 b Fr(;)364 2561 y Fq(with)f Fr(Y)631 2576 y Fu(0)704 2561 y Fq(tangent)g(to)g(al)5 b(l)33 b(c)-5 b(o)g(dimension)32 b Fs(2)h Fq(subsp)-5 b(ac)g(es)32 b Fr(x)2422 2576 y Fu(1)2490 2561 y Fs(=)c Fr(a;)50 b(\030)2765 2576 y Fu(1)2832 2561 y Fs(=)27 b Fr(b)p Fq(.)45 b(Ther)-5 b(e)364 2681 y(exists)32 b(a)g(di\013e)-5 b(omorphism)30 b Fr(\037)i Fq(which)f(is)h(tangent)g(of)g(or)-5 b(der)32 b Fm(1)g Fq(to)h(the)f(identity)364 2802 y(along)i Fs(\006)h Fq(such)g(that)1189 3011 y Fr(\037)1250 2970 y Fj(?)1290 3011 y Fs(\()p Fr(X)1409 3026 y Fu(0)1470 3011 y Fs(+)22 b Fr(X)1649 3026 y Fu(1)1688 3011 y Fs(\))28 b(=)g Fr(X)1939 3026 y Fu(0)2000 3011 y Fs(+)22 b Fr(O)2173 3026 y Fj(Y)2233 3011 y Fs(\()p Fm(1)p Fs(\))35 b Fr(:)510 3221 y Fq(Mor)-5 b(e)g(over,)47 b(if)d Fr(X)52 b Fq(and)44 b Fr(X)1498 3236 y Fu(0)1581 3221 y Fq(ar)-5 b(e)45 b(Hamiltonian)e(ve)-5 b(ctor)44 b(\014elds,)i Fr(\037)f Fq(c)-5 b(an)43 b(b)-5 b(e)364 3341 y(cho)g(osen)34 b(to)h(b)-5 b(e)34 b(symple)-5 b(ctic.)3463 3535 y Fe(\003)p Black 120 3729 a Fh(Remark)38 b(1)p Black 49 w Fq(The)i(normal)g(form)h(is)g(c)-5 b(onver)g(gent)40 b(in)h(the)g(c)-5 b(ase)40 b(of)h(2)g(de)-5 b(gr)g(e)g(es)40 b(of)h(fr)-5 b(e)g(e)g(dom)120 3850 y(and)34 b(analytic)h(data.)44 b(This)35 b(is)f(implie)-5 b(d)34 b(by)h(a)g(r)-5 b(esult)35 b(of)f(Moser)h([13].)120 4137 y Fb(5.2)135 b(The)45 b(gauge)h (transform)120 4322 y Fs(Using)40 b(Lemma)g(2)f(with)h Fr(X)1154 4337 y Fu(1)1233 4322 y Fs(=)f Fr(x)1403 4337 y Fu(1)1443 4322 y Fr(;)56 b(X)1607 4337 y Fu(2)1686 4322 y Fs(=)39 b Fr(\030)1844 4337 y Fu(1)1883 4322 y Fr(;)56 b(X)2047 4337 y Fu(3)2126 4322 y Fs(=)39 b Fr(bx)2337 4337 y Fu(2)2378 4322 y Fr(;)17 b(X)2503 4337 y Fu(4)2581 4322 y Fs(=)39 b Fr(b\030)2780 4337 y Fu(2)2820 4322 y Fs(,)i(w)m(e)g(get)e(a)g(gauge)120 4442 y(transform.)k(The)30 b(v)-5 b(alue)30 b(of)f Fr(\013)h Fs(can)g(b)s(e)g(c)m(hanged)h(to)e (+1)g(using)h(the)g(canonical)h(transforma-)120 4563 y(tion)i(\()p Fr(x)414 4578 y Fu(1)453 4563 y Fr(;)17 b(\030)540 4578 y Fu(1)579 4563 y Fs(\))28 b Fm(!)f Fs(\()p Fm(\000)p Fr(x)942 4578 y Fu(1)982 4563 y Fr(;)17 b Fm(\000)p Fr(\030)1146 4578 y Fu(1)1186 4563 y Fs(\).)266 4683 y(Both)28 b(signs)g(of)f Fr(")g Fs(in)g(the)h(classical)h(normal)e (form)g(giv)m(e)h(non)g(equiv)-5 b(alen)m(t)29 b(Hamiltonians.)120 4803 y(Using)k(the)g(notations)g(of)f(Equation)h(\(1\))f(in)h(Section)h (1,)e(w)m(e)i(ha)m(v)m(e:)1473 5013 y Fr(")27 b Fs(=)h(sign\(\005\()p Fr(z)2012 5028 y Fu(0)2052 5013 y Fs(\)\))k Fr(:)120 5222 y Fs(So)37 b Fr(")d Fs(=)g(1)i(if)h(the)g(orien)m(tations)g(of)f (the)h(normal)g(bundle)h(to)e(\006)h(giv)m(en)h(b)m(y)f Fr(dp)3013 5237 y Fu(1)3077 5222 y Fm(^)26 b(\001)17 b(\001)g(\001)23 b(^)i Fr(dp)3501 5237 y Fu(4)120 5342 y Fs(and)35 b(\012)24 b Fm(^)g Fs(\012)36 b(are)f(the)g(same)h(and)f Fr(")c Fs(=)h Fm(\000)p Fs(1)j(if)g(they)h(are)f(not)g(the)g(same.)52 b(It)35 b(is)g(clear)h(that)e Fr(")120 5463 y Fs(is)f(in)m(v)-5 b(arian)m(t)33 b(b)m(y)h(gauge)e(transform,)h(the)g(group)f Fr(GL)p Fs(\(2)p Fr(;)17 b Fk(C)p Fs(\))32 b(b)s(eing)h(connected.)p Black 1806 5712 a(6)p Black eop %%Page: 7 7 7 6 bop Black Black Black 120 407 a Fh(Remark)38 b(2)p Black 49 w Fq(We)31 b(c)-5 b(an)30 b(se)-5 b(e)31 b(that)g(in)g(a)g (mor)-5 b(e)30 b(top)-5 b(olo)g(gic)g(al)30 b(way:)43 b(let)31 b(us)g(denote)f(by)h Fr(\025)3232 422 y Fi(\000)3319 407 y Fm(\024)d Fr(\025)3481 422 y Fu(+)120 527 y Fq(the)35 b(eigenvalues)e(close)h(to)h Fs(0)f Fq(of)g(the)h(disp)-5 b(ersion)33 b(matrix.)45 b(The)34 b(op)-5 b(en)34 b(c)-5 b(ones)33 b Fr(C)3080 542 y Fi(\006)3167 527 y Fm(\032)28 b Fr(p)3321 491 y Fi(\000)p Fu(1)3415 527 y Fs(\(0\))120 648 y Fq(which)d(c)-5 b(orr)g(esp)g(ond)25 b(r)-5 b(esp)g(e)g(ctively) 25 b(to)h Fr(\025)1527 663 y Fi(\000)1614 648 y Fs(=)h(0)h Fr(<)f(\025)1954 663 y Fu(+)2039 648 y Fq(\()p Fr(\025)2136 663 y Fi(\000)2223 648 y Fr(<)g(\025)2383 663 y Fu(+)2470 648 y Fs(=)g(0)p Fq(\))f(ar)-5 b(e)25 b(wel)5 b(l)26 b(de\014ne)-5 b(d)24 b(ne)-5 b(ar)120 768 y Fs(\006)p Fq(:)41 b(Morse)25 b(indic)-5 b(es)25 b(di\013ers)g(by)g Fs(1)h Fq(on)f(those)g(c)-5 b(ones.)41 b(Mor)-5 b(e)g(over,)27 b(b)-5 b(oth)26 b(c)-5 b(ones)24 b(ar)-5 b(e)26 b(oriente)-5 b(d)25 b(by)120 888 y Fr(p)j(>)f Fs(0)p Fq(.)44 b(The)31 b(sp)-5 b(ac)g(es)31 b Fm(f)p Fr(z)1009 852 y Fi(0)1061 888 y Fs(=)c Fr(constant)p Fm(g)33 b Fq(ar)-5 b(e)32 b(c)-5 b(o-oriente)g(d)31 b(by)h(the)g Fr(z)2613 852 y Fi(0)2669 888 y Fq(symple)-5 b(ctic)31 b(structur)-5 b(e,)120 1009 y(henc)g(e)37 b(oriente)-5 b(d.)51 b(It)38 b(fol)5 b(lows)36 b(that)i(the)f(b)-5 b(asis)37 b(of)g(the)h(c)-5 b(one)36 b Fr(C)2431 1024 y Fu(+)2514 1009 y Fm(\\)25 b(f)p Fr(z)2704 973 y Fi(0)2760 1009 y Fs(=)32 b Fr(constant)p Fm(g)38 b Fq(\(a)f(2-)120 1129 y(spher)-5 b(e\))27 b(is)h(a)h(wel)5 b(l)27 b(de\014ne)-5 b(d)27 b(homolo)-5 b(gy)28 b(class)f(of)h(the)h (germ)e(of)h Fr(C)2471 1144 y Fu(+)2530 1129 y Fq(.)43 b(Henc)-5 b(e)28 b(the)g(p)-5 b(olarization)120 1249 y(bund)5 b(le)32 b(have)g(a)h(wel)5 b(l)32 b(de\014ne)-5 b(d)32 b(\014rst)h(Chern)f(class)g(on)h Fr(C)2192 1264 y Fu(+)2284 1249 y Fq(and)f(b)-5 b(oth)33 b(signes)e(in)i(the)g(normal) 120 1370 y(form)h(gives)g(b)-5 b(oth)35 b(signes)f(in)h(the)g(Chern)f (class.)120 1659 y Fb(5.3)135 b(The)45 b(classical)h(normal)g(form)120 1843 y Fs(Using)33 b(the)g(previous)h(results,)g(w)m(e)g(get:)p Black 120 2047 a Fh(Theorem)k(1)p Black 49 w Fq(We)43 b(assume)e(that)i Fr(H)1521 2062 y Fu(class)1710 2047 y Fq(satis\014es)f Fs(\()p Fr(?)p Fs(\))g Fq(and)g(1.)67 b Fs(\(the)42 b(symplectic)h(case\))p Fq(.)120 2167 y(Then)52 b(ther)-5 b(e)53 b(exists)f(a)h(c)-5 b(anonic)g(al)52 b(tr)-5 b(ansformation)52 b Fr(\037)h Fq(and)f(a)h(gauge)f(tr)-5 b(ansform)52 b Fr(J)71 b Fm(2)120 2288 y Fr(GL)p Fs(\(2)p Fr(;)17 b Fk(C)p Fs(\))34 b Fq(such)h(that:)1083 2508 y Fr(J)1146 2466 y Fj(?)1202 2508 y Fs(\()p Fr(H)1321 2523 y Fu(class)1489 2508 y Fm(\016)22 b Fr(\037)p Fs(\))17 b Fr(J)36 b Fs(:=)28 b Fr(H)1979 2523 y Fu(symp)2168 2508 y Fs(+)22 b Fr(O)2341 2523 y Fj(Y)2402 2508 y Fs(\()p Fm(1)p Fs(\))120 2728 y Fq(wher)-5 b(e)540 2981 y Fr(H)621 2996 y Fu(symp)816 2981 y Fs(=)919 2840 y Fl(\022)1465 2920 y Fr(\030)1508 2935 y Fu(1)2061 2920 y Fr(b)p Fs(\()p Fr(x)2195 2884 y Fu(2)2195 2944 y(2)2258 2920 y Fs(+)22 b Fr(\030)2404 2884 y Fu(2)2399 2944 y(2)2442 2920 y Fr(;)17 b(z)2535 2884 y Fi(0)2559 2920 y Fs(\)\()p Fr(x)2690 2935 y Fu(2)2752 2920 y Fm(\006)23 b Fr(i\030)2928 2935 y Fu(2)2967 2920 y Fs(\))1034 3040 y Fr(b)p Fs(\()p Fr(x)1168 3004 y Fu(2)1168 3065 y(2)1231 3040 y Fs(+)f Fr(\030)1377 3004 y Fu(2)1372 3065 y(2)1416 3040 y Fr(;)17 b(z)1509 3004 y Fi(0)1532 3040 y Fs(\)\()p Fr(x)1663 3055 y Fu(2)1725 3040 y Fm(\007)23 b Fr(i\030)1901 3055 y Fu(2)1940 3040 y Fs(\))508 b Fr(x)2541 3055 y Fu(1)3046 2840 y Fl(\023)120 3263 y Fq(and)34 b Fr(b)28 b Fs(=)g Fr(b)p Fs(\()p Fr(x)616 3227 y Fu(2)616 3288 y(2)678 3263 y Fs(+)22 b Fr(\030)824 3227 y Fu(2)819 3288 y(2)863 3263 y Fr(;)17 b(z)956 3227 y Fi(0)980 3263 y Fs(\))28 b Fr(>)f Fs(0)35 b Fq(is)f(smo)-5 b(oth.)120 3552 y Fb(5.4)135 b(The)45 b(semi-classical)i(normal)e(form) p Black 120 3737 a Fh(Theorem)38 b(2)p Black 49 w Fq(We)43 b(assume)e(that)i Fr(H)1521 3752 y Fu(class)1710 3737 y Fq(satis\014es)f Fs(\()p Fr(?)p Fs(\))g Fq(and)g(1.)67 b Fs(\(the)42 b(symplectic)h(case\))p Fq(.)120 3857 y(Using)35 b(FIO)e(and)i(gauge)f(tr)-5 b(ansform,)34 b(we)h(get)g(the)g(fol)5 b(lowing)33 b(micr)-5 b(olo)g(c)g(al)34 b(normal)g(form:)1313 4134 y Fl(b)1291 4159 y Fr(H)h Fs(=)1511 3988 y Fl( )1708 4072 y(b)1695 4098 y Fr(\030)1738 4113 y Fu(1)1943 4073 y Fl(b)1923 4098 y Fr(B)5 b(a)1631 4232 y(a)1682 4195 y Fj(?)1742 4206 y Fl(b)1722 4232 y Fr(B)1801 4195 y Fj(?)1941 4232 y Fr(x)1996 4247 y Fu(1)2095 3988 y Fl(!)2196 4159 y Fs(+)22 b Fr(R)120 4460 y Fq(wher)-5 b(e)p Black 265 4664 a Fm(\017)p Black 384 4639 a Fl(b)364 4664 y Fr(B)35 b Fq(is)29 b(an)g(el)5 b(liptic)29 b Fs(\011)p Fr(D)s(O)j Fq(whose)c(total)i(symb)-5 b(ol)29 b(is)h Fr(>)d Fs(0)i Fq(and)h(dep)-5 b(ends)28 b(only)h(on)h Fr(x)3317 4628 y Fu(2)3317 4688 y(2)3367 4664 y Fs(+)10 b Fr(\030)3501 4628 y Fu(2)3496 4688 y(2)364 4784 y Fq(and)34 b Fr(z)602 4748 y Fi(0)p Black 265 4988 a Fm(\017)p Black 49 w Fr(a)28 b Fs(=)596 4965 y Fk(\\)546 4988 y Fr(x)601 5003 y Fu(2)663 4988 y Fm(\006)23 b Fr(i\030)839 5003 y Fu(2)p Black 265 5191 a Fm(\017)p Black 49 w Fq(The)34 b(ful)5 b(l)35 b(symb)-5 b(ol)34 b(of)h Fr(R)h Fq(is)e(\015at)i(on)e Fr(Y)120 5419 y Fq(Pr)-5 b(o)g(of.{)p Black 1806 5712 a Fs(7)p Black eop %%Page: 8 8 8 7 bop Black Black Black Black 510 527 a Fs(Using)24 b(the)g(same)g(metho)s(d)g(as)g(in)f(the)h(pro)s(of)f(of)g(Theorem)h(3) g(in)f([2])h(\(Lemma)364 648 y(4\),)32 b(w)m(e)i(need)f(to)g(solv)m(e)h (the)f(follo)m(wing)g(homological)f(equation:)1036 824 y Fm(f)p Fr(S;)17 b(H)1271 839 y Fu(symp)1438 824 y Fm(g)22 b Fs(+)g Fr(C)1685 782 y Fj(?)1724 824 y Fr(H)1805 839 y Fu(symp)1995 824 y Fs(+)g Fr(H)2174 839 y Fu(symp)2341 824 y Fr(C)34 b Fs(=)28 b Fr(R)364 999 y Fs(where)36 b Fr(R)g Fs(is)f(giv)m(en)h(and)f Fr(S)40 b Fs(\(real)35 b(v)-5 b(alued\))35 b(and)g Fr(C)42 b Fs(\(matrix)35 b(v)-5 b(alued\))35 b(are)g(un-)364 1120 y(kno)m(wn)43 b(functions.)73 b(This)43 b(can)f(b)s(e)g(done)h(directly)g(b)m(y)g (solving)g(the)f(normal)364 1240 y(form)33 b(problem)h(for)f Fr(H)1206 1255 y Fu(symp)1395 1240 y Fs(+)23 b Fr("R)34 b Fs(whic)m(h)h(satisfy)f(our)f(basic)h(h)m(yp)s(othesis)i(\(\()p Fr(?)p Fs(\))364 1361 y(and)31 b(2.\))43 b(for)31 b(all)g(small)h Fr(")f Fs(\(this)g(case)i(is)e Fq(structur)-5 b(al)5 b(ly)35 b(stable)p Fs(\))c(and)g(taking)h(the)364 1481 y(\014rst)h(order)g(term)g(in)f Fr(")p Fs(.)3463 1647 y Fe(\003)120 1929 y Fb(5.5)135 b(Microlo)t(cal)46 b(solutions)120 2114 y Fs(W)-8 b(e)32 b(will)h(study)g(the)f(+)f(case)i(of)e(the)i (normal)e(form.)43 b(The)33 b Fm(\000)f Fs(case)h(is)f(similar.)44 b(W)-8 b(e)32 b(will)h(see)120 2234 y(ho)m(w)g(to)f(extend)i(the)f (metho)s(d)g(of)g([7)o(])g(in)g(order)g(to)f(solv)m(e)i(the)f(lo)s(cal) f(mo)s(del.)44 b(W)-8 b(e)33 b(de\014ne)800 2486 y Fr(u)27 b Fs(=)1023 2361 y Fi(1)987 2391 y Fl(X)997 2601 y Fj(j)t Fu(=0)1147 2486 y Fr(a)1198 2501 y Fj(j)1235 2486 y Fs(\()p Fr(x)1328 2501 y Fu(1)1368 2486 y Fr(;)17 b(x)1467 2445 y Fi(0)1490 2486 y Fs(\))p Fr(')1592 2501 y Fj(j)1628 2486 y Fs(\()p Fr(x)1721 2501 y Fu(2)1761 2486 y Fs(\))p Fr(;)49 b(v)32 b Fs(=)2094 2361 y Fi(1)2057 2391 y Fl(X)2068 2601 y Fj(j)t Fu(=0)2218 2486 y Fr(b)2259 2501 y Fj(j)2296 2486 y Fs(\()p Fr(x)2389 2501 y Fu(1)2429 2486 y Fr(;)17 b(x)2528 2445 y Fi(0)2551 2486 y Fs(\))p Fr(')2653 2501 y Fj(j)2689 2486 y Fs(\()p Fr(x)2782 2501 y Fu(2)2822 2486 y Fs(\))120 2768 y(where)37 b Fr(')469 2783 y Fj(j)541 2768 y Fs(is)f(the)g(usual)h Fr(L)1138 2732 y Fu(2)1177 2768 y Fs(\()p Fk(R)p Fr(;)17 b(dx)1437 2783 y Fu(2)1476 2768 y Fs(\))36 b(orthonormal)f(basis)i(suc)m(h)g(that)f Fr(')2856 2783 y Fj(j)2925 2768 y Fs(=)d Fr(c)3076 2783 y Fj(j)3112 2768 y Fr(\013)3175 2732 y Fj(j)3211 2768 y Fr(')3275 2783 y Fu(0)3350 2768 y Fs(with)120 2888 y Fr(c)162 2903 y Fj(j)226 2888 y Fr(>)28 b Fs(0,)k Fr(')502 2903 y Fu(0)569 2888 y Fs(=)c Fr(c)715 2903 y Fu(0)754 2888 y Fs(exp)q(\()p Fm(\000)p Fr(x)1073 2852 y Fu(2)1073 2913 y(2)1113 2888 y Fr(=)p Fs(2)p Fr(h)p Fs(\))33 b(and)f Fr(\013)d Fs(=)e Fm(\000)p Fr(h@)1905 2903 y Fj(x)1945 2912 y Ff(2)2007 2888 y Fs(+)22 b Fr(x)2160 2903 y Fu(2)2232 2888 y Fs(is)33 b(the)g(creation)g(op)s(erator.)266 3035 y(W)-8 b(e)35 b(get)g(the)f(follo)m(wing)h(systems,)j(where)1866 3007 y Fk([)1859 3035 y Fr(B)5 b Fs(\()p Fr(j)h Fs(\))35 b(is)f(the)h(pseudo-di\013eren)m(tial)i(op)s(erator)120 3156 y(in)26 b(the)h Fr(x)444 3120 y Fi(0)494 3156 y Fs(v)-5 b(ariable)26 b(obtained)g(b)m(y)h(W)-8 b(eyl-quan)m(tizing)28 b(the)f(\()p Fr(>)g Fs(0\))f(sym)m(b)s(ol)h(of)f Fr(B)31 b Fs(at)26 b(the)h(v)-5 b(alue)120 3276 y Fr(x)175 3240 y Fu(2)175 3301 y(2)237 3276 y Fs(+)22 b Fr(\030)383 3240 y Fu(2)378 3301 y(2)450 3276 y Fs(=)27 b(\(2)p Fr(j)h Fs(+)22 b(1\))p Fr(h)p Fs(:)675 3369 y Fl( )1058 3439 y Fj(h)p 1058 3455 41 4 v 1066 3513 a(i)1108 3478 y Fr(@)1159 3493 y Fj(x)1199 3502 y Ff(1)1574 3393 y Fl(p)p 1674 3393 396 4 v 85 x Fs(2\()p Fr(j)28 b Fs(+)22 b(1\))p Fr(h)2076 3450 y Fk([)2070 3478 y Fr(B)5 b Fs(\()p Fr(j)h Fs(\))795 3540 y Fl(p)p 895 3540 V 85 x Fs(2\()p Fr(j)27 b Fs(+)22 b(1\))p Fr(h)1297 3596 y Fk([)1290 3625 y Fr(B)5 b Fs(\()p Fr(j)h Fs(\))384 b Fr(x)1930 3640 y Fu(1)2312 3369 y Fl(!)2408 3399 y(\022)2563 3478 y Fr(a)2614 3493 y Fj(j)2522 3599 y Fr(b)2563 3614 y Fj(j)t Fu(+1)2732 3399 y Fl(\023)2833 3539 y Fs(=)28 b(0)120 3797 y(and)33 b Fr(b)351 3812 y Fu(0)418 3797 y Fs(=)28 b(0.)120 4123 y Ft(6)161 b(The)53 b(corank)h(2)f(case)120 4371 y Fb(6.1)135 b(Singular)46 b(p)t(erturbations)f(and)g(homological)h(equations)120 4556 y Fh(6.1.1)112 b(In)m(tro)s(duction)120 4741 y Fs(The)30 b(corank)g(2)e(case)i(is)g(more)f(di\016cult,)i(b)s(ecause)g(it)e(is)g (not)g Fq(structur)-5 b(al)5 b(ly)33 b(stable)p Fs(:)41 b(a)29 b(generic)120 4861 y(p)s(erturbation)48 b(of)f(the)h(disp)s (ersion)i(matrix)e(will)g(b)s(e)g(in)g(the)g(symplectic)j(case.)89 b(In)49 b(the)120 4981 y(subsection)28 b(6.1.3,)f(w)m(e)g(will)g (\014nd)g(the)g(space)g(of)f(in\014nitesimal)h(deformations)g(of)f(the) g(corank)120 5102 y(2)39 b(case.)63 b(In)39 b(the)g(subsection)i (6.1.4,)f(w)m(e)g(will)g(lo)s(ok)e(at)h(a)f(Birkho\013)i(normal)f(form) f(for)h(the)120 5222 y(T)-8 b(a)m(ylor)37 b(expansion)h(along)f(\006)g (of)f(the)h(disp)s(ersion)h(relation.)56 b(In)37 b(the)g(subsection)i (6.1.5)d(w)m(e)120 5342 y(will)28 b(lo)s(ok)f(at)f(the)i(homological)f (equation)g(needed)i(for)e(\014nding)g(the)h(semi-classical)h(normal) 120 5463 y(form.)p Black 1806 5712 a(8)p Black eop %%Page: 9 9 9 8 bop Black Black 120 407 a Fh(6.1.2)112 b(Singular)39 b(deformations)g(of)f Fs(\006)p Black 120 592 a Fh(De\014nition)g(1)p Black 49 w Fq(L)-5 b(et)48 b Fs(\006)k(=)g Fm(f)p Fr(x)1284 607 y Fu(1)1375 592 y Fs(=)f Fr(\030)1545 607 y Fu(1)1636 592 y Fs(=)h Fr(x)1819 607 y Fu(2)1910 592 y Fs(=)g Fr(x)2093 607 y Fu(3)2184 592 y Fs(=)f(0)p Fm(g)h(\032)g Fs(\()p Fr(T)2700 555 y Fj(?)2739 592 y Fk(R)2811 555 y Fj(n)2858 592 y Fr(;)17 b Fs(\012\))p Fq(.)83 b(A)49 b(smo)-5 b(oth)120 712 y(deformation)34 b Fs(\006)737 727 y Fj(")809 712 y Fq(of)g Fs(\006)i Fq(is)e(c)-5 b(al)5 b(le)-5 b(d)34 b Fs(singular)i Fq(if)e(the)h(c)-5 b(or)g(ank)34 b(of)h Fs(\012)2525 727 y Fi(j)p Fu(\006)2596 735 y Fg(")2668 712 y Fq(is)g(c)-5 b(onstant)34 b(\()p Fm(\021)28 b Fs(2)p Fq(\).)266 832 y(A)36 b(deformation)d Fs(\()p Fr(F)s(;)17 b(G;)g(A;)g(B)5 b Fs(\))34 b Fq(given)g(by)409 1052 y Fs(\006)479 1067 y Fj(")544 1052 y Fs(=)27 b Fm(f)p Fr(x)752 1067 y Fu(1)820 1052 y Fs(=)g Fr("F)14 b Fs(\()p Fr(\033)t Fs(\))p Fr(;)51 b(\030)1302 1067 y Fu(1)1369 1052 y Fs(=)27 b Fr("G)p Fs(\()p Fr(\033)t Fs(\))p Fr(;)51 b(x)1863 1067 y Fu(2)1931 1052 y Fs(=)28 b Fr("A)p Fs(\()p Fr(\033)t Fs(\))p Fr(;)51 b(x)2422 1067 y Fu(3)2489 1052 y Fs(=)28 b Fr("B)5 b Fs(\()p Fr(\033)t Fs(\))34 b Fm(j)h Fr(\033)d Fm(2)c Fs(\006)p Fm(g)120 1272 y Fq(is)38 b(c)-5 b(al)5 b(le)-5 b(d)37 b Fs(in\014nitesimally)h(singular)h Fq(if)f(it)g(c)-5 b(an)38 b(b)-5 b(e)38 b(mo)-5 b(di\014e)g(d)36 b(by)j Fr(O)s Fs(\()p Fr(")2693 1236 y Fu(2)2731 1272 y Fs(\))f Fq(terms)g(so)g(that)g(the)120 1393 y(new)c(deformation)g(is)h (singular.)p Black 120 1621 a Fh(Lemma)k(3)p Black 49 w Fq(The)34 b(sp)-5 b(ac)g(e)34 b(of)h(in\014nitesimal)5 b(ly)33 b(singular)i(deformations)e(is)i(the)g(sp)-5 b(ac)g(e)1496 1885 y Fs(\()p Fr(F)s(;)17 b(G;)1780 1817 y(@)5 b(T)p 1774 1862 139 4 v 1774 1953 a(@)g(\030)1873 1968 y Fu(2)1923 1885 y Fr(;)1983 1817 y(@)g(T)p 1977 1862 V 1977 1953 a(@)g(\030)2076 1968 y Fu(3)2126 1885 y Fs(\))120 2150 y Fq(wher)-5 b(e)34 b Fr(F)s(;)52 b(G;)f(T)e Fq(ar)-5 b(e)34 b(arbitr)-5 b(ary)36 b(functions)e(on)h Fs(\006)p Fq(.)120 2378 y(Pr)-5 b(o)g(of.{)p Black Black 403 2581 a Fs(Let)38 b(us)i(start)e(with)i(a)e(singular)h(deformation)g (whose)h(in\014nitesimal)h(defor-)364 2702 y(mation)31 b(is)h(giv)m(en)h(b)m(y)g(\()p Fr(F)s(;)17 b(G;)g(A;)g(B)5 b Fs(\).)42 b(W)-8 b(e)32 b(see)g(that)g(the)f(pull-bac)m(k)i(of)e (\012)3053 2717 y Fi(j)p Fu(\006)3124 2725 y Fg(")3193 2702 y Fs(on)364 2822 y(\006)i(is)g(giv)m(en)h(b)m(y:)769 3042 y(\012)839 3057 y Fj(")904 3042 y Fs(=)27 b Fr(d\030)1106 3001 y Fi(0)1151 3042 y Fm(^)c Fr(dx)1346 3001 y Fi(0)1391 3042 y Fm(\000)g Fr(")p Fs(\()p Fr(dA)e Fm(^)i Fr(d\030)1903 3057 y Fu(2)1964 3042 y Fs(+)f Fr(dB)27 b Fm(^)22 b Fr(d\030)2396 3057 y Fu(3)2435 3042 y Fs(\))g(+)g Fr(O)s Fs(\()p Fr(")2755 3001 y Fu(2)2793 3042 y Fs(\))33 b Fr(:)364 3262 y Fs(W)-8 b(e)33 b(ha)m(v)m(e)457 3482 y(\012)527 3441 y Fj(n)p Fi(\000)p Fu(1)527 3507 y Fj(")692 3482 y Fs(=)27 b Fm(\000)p Fs(\()p Fr(n)c Fm(\000)g Fs(1\))p Fr(")p Fs(\()p Fr(d\030)1361 3441 y Fi(0)1405 3482 y Fm(^)f Fr(dx)1599 3441 y Fi(0)1623 3482 y Fs(\))1661 3441 y Fj(n)p Fi(\000)p Fu(2)1820 3482 y Fm(^)g Fs(\()p Fr(dA)g Fm(^)h Fr(d\030)2275 3497 y Fu(2)2336 3482 y Fs(+)f Fr(dB)27 b Fm(^)22 b Fr(d\030)2768 3497 y Fu(3)2807 3482 y Fs(\))g(+)g Fr(O)s Fs(\()p Fr(")3127 3441 y Fu(2)3165 3482 y Fs(\))364 3702 y(whose)34 b(v)-5 b(anishing)33 b(implies)h(there)f(exists)i Fr(T)46 b Fs(suc)m(h)34 b(that)1396 3966 y Fr(A)28 b Fs(=)1616 3899 y Fr(@)5 b(T)p 1610 3943 V 1610 4034 a(@)g(\030)1709 4049 y Fu(2)1759 3966 y Fr(;)49 b(B)33 b Fs(=)2061 3899 y Fr(@)5 b(T)p 2056 3943 V 2056 4034 a(@)g(\030)2155 4049 y Fu(3)2237 3966 y Fr(:)510 4231 y Fs(Con)m(v)m(ersely)-8 b(,)45 b(let)c(us)g(start)f(with)h(the)g(in\014nitesimal)h(deformation) e(giv)m(en)364 4351 y(b)m(y)j(\()p Fr(F)s(;)17 b(G;)g(T)d Fs(\).)72 b(Let)43 b Fr(S)51 b Fs(=)44 b Fr(T)f Fs(+)29 b Fr(\030)1650 4366 y Fu(1)1689 4351 y Fr(F)42 b Fm(\000)30 b Fr(x)1956 4366 y Fu(1)1996 4351 y Fr(G)p Fs(.)73 b(Let)43 b Fr(\037)2419 4366 y Fj(")2498 4351 y Fs(the)g(\015o)m(w)g(at)f(time)h Fr(")364 4472 y Fs(of)32 b(the)i(Hamiltonian)f(v)m(ector)h(\014eld)g Fm(X)1781 4487 y Fj(S)1865 4472 y Fs(generated)g(b)m(y)g Fr(S)6 b Fs(.)44 b(The)34 b(deformation)364 4592 y(\006)434 4607 y Fj(")522 4592 y Fs(=)51 b Fr(\037)710 4607 y Fj(")747 4592 y Fs(\(\006\))46 b(is)h(singular.)85 b(It)47 b(is)g(easy)g(to)f(c) m(hec)m(k)i(that)f(the)f(in\014nitesimal)364 4712 y(deformation)33 b(asso)s(ciated)g(to)f(\006)1562 4727 y Fj(")1632 4712 y Fs(is)h(\()p Fr(F)s(;)17 b(G;)2015 4673 y Fj(@)t(T)p 2008 4690 107 4 v 2008 4747 a(@)t(\030)2080 4756 y Ff(2)2125 4712 y Fr(;)2186 4673 y Fj(@)t(T)p 2178 4690 V 2178 4747 a(@)t(\030)2250 4756 y Ff(3)2295 4712 y Fs(\).)3463 4916 y Fe(\003)p Black 1806 5712 a Fs(9)p Black eop %%Page: 10 10 10 9 bop Black Black 120 407 a Fh(6.1.3)112 b(Singular)39 b(p)s(erturbations)f(of)g(the)f(disp)s(ersion)i(matrix)120 592 y Fs(Let)33 b(us)g(denote)g(b)m(y)h Fr(H)951 607 y Fu(h)n(yp)1101 592 y Fs(\(resp.)44 b Fr(H)1464 607 y Fu(ell)1539 592 y Fs(\))32 b(some)i(disp)s(ersion)g(matrices)f(giv)m (en)h(b)m(y)961 868 y Fr(H)1042 883 y Fu(h)n(yp)1188 868 y Fs(=)1291 727 y Fl(\022)1537 807 y Fr(\030)1580 822 y Fu(1)1834 807 y Fr(x)1889 822 y Fu(2)1951 807 y Fs(+)22 b Fr(ix)2137 822 y Fu(3)1406 927 y Fr(x)1461 942 y Fu(2)1523 927 y Fm(\000)h Fr(ix)1711 942 y Fu(3)1958 927 y Fr(x)2013 942 y Fu(1)2218 727 y Fl(\023)2314 868 y Fs(+)f Fr(O)2487 883 y Fu(\006)2542 868 y Fs(\(2\))827 1212 y(\(resp)p Fr(:)34 b(H)1180 1227 y Fu(ell)1282 1212 y Fs(=)1386 1071 y Fl(\022)1619 1151 y Fr(x)1674 1166 y Fu(2)1915 1151 y Fr(x)1970 1166 y Fu(1)2032 1151 y Fs(+)22 b Fr(i\030)2206 1166 y Fu(1)1500 1271 y Fr(x)1555 1286 y Fu(1)1617 1271 y Fm(\000)h Fr(i\030)1793 1286 y Fu(1)2033 1271 y Fr(x)2088 1286 y Fu(3)2287 1071 y Fl(\023)2383 1212 y Fs(+)f Fr(O)2556 1227 y Fu(\006)2610 1212 y Fs(\(2\))33 b(\))p Fr(:)p Black 120 1487 a Fh(De\014nition)38 b(2)p Black Black 194 w Fm(\017)p Black 49 w Fq(We)d(say)g(that)g(a)g (smo)-5 b(oth)34 b(deformation)1154 1702 y Fr(H)1235 1717 y Fj(")1299 1702 y Fs(=)28 b Fr(H)1484 1717 y Fu(h)n(yp)1636 1702 y Fs(\(resp)p Fr(:)36 b(H)1991 1717 y Fu(ell)2066 1702 y Fs(\))22 b(+)g Fr("K)2353 1717 y Fu(0)2414 1702 y Fs(+)g Fr(O)s Fs(\()p Fr(")2674 1661 y Fu(2)2712 1702 y Fs(\))665 b(\(2\))364 1917 y Fq(is)37 b Fs(singular)g Fq(if)g(it)g(satis\014es)g(the)g(hyp)-5 b(othesis)36 b(2)h(\(r)-5 b(esp.)51 b(3\))36 b(of)h(se)-5 b(ction)36 b(1)h(for)g Fr(")g Fq(smal)5 b(l)364 2038 y(enough.)p Black 265 2240 a Fm(\017)p Black 49 w Fq(A)n(n)44 b Fs(in\014nitesimal) g(deformation)g Fr(K)1735 2255 y Fu(0)1819 2240 y Fq(is)g Fs(singular)h Fq(if)f(it)g(c)-5 b(an)44 b(b)-5 b(e)43 b(emb)-5 b(e)g(dde)g(d)43 b(into)h(a)364 2360 y(smo)-5 b(oth)34 b(singular)g(deformation)p Black 120 2583 a Fh(Lemma)39 b(4)p Black Black 194 w Fm(\017)p Black 49 w Fq(A)n(n)33 b Fs(in\014nitesimal)f(deformation)h Fr(K)2203 2598 y Fu(0)2276 2583 y Fq(is)g Fs(singular)h Fq(if)f(and)f(only)h(if)g (ther)-5 b(e)364 2703 y(exists)34 b Fr(T)42 b Fs(:)28 b(\006)g Fm(!)f Fk(R)35 b Fq(so)f(that)1398 2943 y Fs(\(\()p Fr(K)1557 2958 y Fu(0)1596 2943 y Fs(\))1634 2958 y Fu(1)p Fj(;)p Fu(2)1729 2943 y Fs(\))1767 2975 y Fi(j)p Fu(\006)1869 2943 y Fs(=)1988 2876 y Fr(@)5 b(T)p 1983 2920 139 4 v 1983 3011 a(@)g(\030)2082 3026 y Fu(2)2154 2943 y Fs(+)22 b Fr(i)2301 2876 y(@)5 b(T)p 2295 2920 V 2295 3011 a(@)g(\030)2394 3026 y Fu(3)2479 2943 y Fr(:)p Black 265 3244 a Fm(\017)p Black 49 w Fq(The)28 b(same)f(r)-5 b(esult)29 b(holds)f(in)g(the)h Fs(elliptic)h Fq(c)-5 b(ase)27 b(by)i(r)-5 b(eplacing)27 b(the)i(pr)-5 b(evious)28 b(c)-5 b(ondition)364 3364 y(by:)1515 3396 y Fl(\()1637 3493 y Fs(\()o(\()p Fr(K)1795 3508 y Fu(0)1835 3493 y Fs(\))1873 3508 y Fu(1)p Fj(;)p Fu(1)1967 3493 y Fs(\))2005 3525 y Fi(j)p Fu(\006)2107 3493 y Fs(=)2228 3453 y Fj(@)t(T)p 2221 3470 107 4 v 2221 3527 a(@)t(\030)2293 3536 y Ff(2)1637 3627 y Fs(\()o(\()p Fr(K)1795 3642 y Fu(0)1835 3627 y Fs(\))1873 3642 y Fu(2)p Fj(;)p Fu(2)1967 3627 y Fs(\))2005 3659 y Fi(j)p Fu(\006)2107 3627 y Fs(=)2228 3587 y Fj(@)t(T)p 2221 3604 V 2221 3661 a(@)t(\030)2293 3670 y Ff(3)120 3871 y Fq(Pr)g(o)g(of.{)p Black Black 396 4070 a Fs(Lemma)34 b(4)e(is)h(an)f(easy)i(consequence)i (of)c(Lemma)h(3.)3463 4269 y Fe(\003)120 4528 y Fh(6.1.4)112 b(Homological)39 b(equation)f(to)f(high)h(order)120 4713 y Fs(W)-8 b(e)33 b(will)g(giv)m(e)h(a)e(Lemma)h(in)g(the)g(h)m(yp)s (erb)s(olic)h(case,)g(the)f(elliptic)g(case)h(w)m(orks)g(similarly:)p Black 120 4912 a Fh(Lemma)39 b(5)p Black 49 w Fq(L)-5 b(et)44 b Fr(H)880 4927 y Fj(N)991 4912 y Fq(b)-5 b(e)44 b(the)g(sp)-5 b(ac)g(e)43 b(of)h(function)f(homo)-5 b(gene)g(ous)43 b(of)g(de)-5 b(gr)g(e)g(e)44 b Fr(N)54 b Fq(w.r.)72 b(to)120 5033 y Fs(\()p Fr(x)213 5048 y Fu(1)253 5033 y Fr(;)17 b(\030)340 5048 y Fu(1)379 5033 y Fr(;)g(x)478 5048 y Fu(2)517 5033 y Fr(;)g(x)616 5048 y Fu(3)656 5033 y Fs(\))34 b Fq(and)h Fr(p)967 5048 y Fu(0)1034 5033 y Fs(=)27 b Fr(x)1192 5048 y Fu(1)1232 5033 y Fr(\030)1275 5048 y Fu(1)1336 5033 y Fm(\000)c Fs(\()p Fr(x)1529 4996 y Fu(2)1529 5057 y(2)1591 5033 y Fs(+)f Fr(x)1744 4996 y Fu(2)1744 5057 y(3)1784 5033 y Fs(\))p Fq(.)45 b(We)35 b(c)-5 b(an)34 b(solve)g(the)h(fol)5 b(lowing)33 b(e)-5 b(quation:)830 5248 y Fm(f)p Fr(U)32 b Fs(+)22 b Fr(W)m(;)17 b(p)1258 5263 y Fu(0)1298 5248 y Fm(g)22 b Fs(+)g Fr(V)f(p)1595 5263 y Fu(0)1657 5248 y Fs(+)h Fr(x)1810 5207 y Fj(N)1810 5272 y Fu(3)1877 5248 y Fr(\034)11 b Fs(\()p Fr(\033)t Fs(\))28 b(=)g Fr(\032)22 b Fs(+)g Fr(O)2442 5263 y Fu(\006)2497 5248 y Fs(\()p Fr(N)33 b Fs(+)22 b(1\))584 b(\(3\))120 5463 y Fq(wher)-5 b(e)34 b Fr(\032)28 b Fm(2)g Fr(H)648 5478 y Fj(N)750 5463 y Fq(is)35 b(given.)44 b(The)34 b(unknowns)g(functions)g(ar)-5 b(e:)p Black 1781 5712 a Fs(10)p Black eop %%Page: 11 11 11 10 bop Black Black Black 265 407 a Fm(\017)p Black 49 w Fr(U)38 b Fm(2)28 b Fr(H)643 422 y Fj(N)p Black 265 609 a Fm(\017)p Black 49 w Fr(W)41 b Fm(2)28 b Fr(H)672 624 y Fj(N)7 b Fi(\000)p Fu(1)857 609 y Fq(an)28 b(homo)-5 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bop Black Black Black 265 407 a Fm(\017)p Black 49 w Fq(In)34 b(the)h(hyp)-5 b(erb)g(olic)34 b(c)-5 b(ase)34 b Fr(H)1395 422 y Fu(0)1462 407 y Fs(=)28 b Fr(H)1647 422 y Fu(h)n(yp)1764 407 y Fq(,)35 b(e)-5 b(quation)35 b(\(4\))f(c)-5 b(an)34 b(b)-5 b(e)35 b(solve)-5 b(d)34 b(with)1057 688 y Fr(T)41 b Fs(=)28 b Fr(i)1309 547 y Fl(\022)1725 627 y Fs(0)384 b Fr(t)p Fs(\()p Fr(x)2286 642 y Fu(3)2326 627 y Fr(;)17 b(\030)2413 642 y Fu(2)2452 627 y Fr(;)g(\030)2539 642 y Fu(3)2578 627 y Fr(;)g(z)2671 591 y Fi(0)2694 627 y Fs(\))1423 747 y Fm(\000)p Fr(t)p Fs(\()p Fr(x)1628 762 y Fu(3)1669 747 y Fr(;)g(\030)1756 762 y Fu(2)1795 747 y Fr(;)g(\030)1882 762 y Fu(3)1920 747 y Fr(;)g(z)2013 711 y Fi(0)2037 747 y Fs(\))346 b(0)2774 547 y Fl(\023)364 960 y Fq(with)35 b Fr(t)g Fq(r)-5 b(e)g(al)34 b(value)-5 b(d.)p Black 265 1164 a Fm(\017)p Black 49 w Fq(In)34 b(the)h(el)5 b(liptic)34 b(c)-5 b(ase)34 b Fr(H)1264 1179 y Fu(0)1331 1164 y Fs(=)28 b Fr(H)1516 1179 y Fu(ell)1591 1164 y Fq(,)34 b(e)-5 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y(v)2503 428 y Fl(\023)2604 569 y Fs(=)28 b(0)364 821 y(where)34 b Fr(Q)e Fs(and)h Fr(R)h Fs(comm)m(ute)g(with)f Fr(x)1750 836 y Fu(1)1822 821 y Fs(and)2059 781 y Fj(@)p 2022 798 V 2022 855 a(@)t(x)2103 864 y Ff(1)2180 821 y Fs(in)g(the)g(elliptic)g(case.)120 1153 y Ft(7)161 b(One)53 b(dimensional)g(systems)g(with)f(parameters)120 1401 y Fb(7.1)135 b(Normal)46 b(forms)120 1586 y Fs(In)33 b(this)g(section)h(w)m(e)f(will)h(consider)g(the)f(case)g(of)f(a)g (system)1608 1796 y Fk([)1591 1824 y Fr(H)8 b Fs(\()p Fr(\025)p Fs(\))1821 1799 y Fr(~)1812 1824 y(U)39 b Fs(=)27 b(0)120 2068 y(where)434 2039 y Fk([)418 2068 y Fr(H)8 b Fs(\()p Fr(\025)p Fs(\))47 b(is)i(a)g Fr(d)32 b Fm(\002)i Fr(d)48 b Fs(self-adjoin)m(t)g(system)j(in)d(one)h(v)-5 b(ariable)49 b Fr(x)2810 2083 y Fu(1)2898 2068 y Fs(and)g(dep)s(ending) 120 2188 y(smo)s(othly)40 b(of)g(an)g(external)g(parameter)h Fr(\025)f Fm(2)g Fk(R)1944 2152 y Fj(N)2011 2188 y Fr(;)57 b(N)50 b Fm(\025)41 b Fs(2.)65 b(Usually)41 b Fr(\025)e 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y(critical)33 b(p)s(oin)m(t.)44 b(W)-8 b(e)33 b(ha)m(v)m(e)h(t)m(w)m(o)f(cases)h(the)f Fq(el)5 b(liptic)32 b Fs(one)h(and)f(the)h Fq(hyp)-5 b(erb)g(olic)32 b Fs(one.)266 3280 y(The)j(h)m(yp)s(erb)s(olic)f(case)h (is)f(strongly)g(related)g(to)f([3])g(\(see)i(also)e([15]\))g(while)i (the)f(elliptic)120 3400 y(normal)k(form)f(has)h(b)s(een)g(in)m(tro)s (duced)h(as)e(a)h Fq(mo)-5 b(del)36 b Fs(in)i([5])f(and)h([6].)58 b(Using)39 b(the)e(previous)120 3521 y(metho)s(ds,)d(one)e(can)h(sho)m (w)h(the)f(follo)m(wing)p Black 120 3724 a Fh(Theorem)38 b(5)p Black Black 194 w Fm(\017)p Black 49 w Fs(Elliptic)c(case:)46 b Fq(ne)-5 b(ar)34 b Fs(\(0)p Fr(;)17 b Fs(0)p Fr(;)g(\025)2034 3739 y Fu(0)2073 3724 y Fs(\))p Fq(,)35 b(one)f(c)-5 b(an)35 b(r)-5 b(e)g(duc)g(e)34 b(the)h(system)g(using)364 3845 y(a)g Fr(\025)p Fq(-dep)-5 b(endent)33 b(gauge)i(tr)-5 b(ansform)34 b(and)g(FIO's)g(to)1322 3976 y Fl( )1494 4085 y Fr(a)1545 4100 y Fj(h)1590 4085 y Fs(\()p Fr(\025)p Fs(\))135 b Fr(x)1913 4100 y Fu(1)1975 4085 y Fs(+)22 b Fr(i)2119 4059 y Fl(b)2106 4085 y Fr(\030)2149 4100 y Fu(1)1443 4220 y Fr(x)1498 4235 y Fu(1)1560 4220 y Fm(\000)g Fr(i)1705 4193 y Fl(b)1692 4220 y Fr(\030)1735 4235 y Fu(1)1913 4220 y Fr(b)1954 4235 y Fj(h)2000 4220 y Fs(\()p Fr(\025)p Fs(\))2229 3976 y Fl(!)2334 4121 y Fr(~)2325 4146 y(U)38 b Fs(=)28 b(0)p Black 265 4495 a Fm(\017)p Black 49 w Fs(Hyp)s(erb)s(olic)23 b(case:)40 b Fq(ne)-5 b(ar)25 b Fs(\(0)p Fr(;)17 b Fs(0)p Fr(;)g(\025)1584 4510 y Fu(0)1622 4495 y Fs(\))p Fq(,)27 b(one)d(c)-5 b(an)25 b(r)-5 b(e)g(duc)g(e)24 b(the)h(system)g(using)g Fr(\025)p Fq(-dep)-5 b(endent)364 4615 y(gauge)34 b(tr)-5 b(ansform)35 b(and)f(FIO's)f(to)1430 4754 y Fl(\022)1632 4815 y(b)1618 4841 y Fr(\030)1661 4856 y Fu(1)1857 4841 y Fr(a)1908 4856 y Fj(h)1953 4841 y Fs(\()p Fr(\025)p Fs(\))1569 4961 y(\026)-73 b Fr(a)1596 4976 y Fj(h)1641 4961 y Fs(\()p Fr(\025)p Fs(\))150 b Fr(x)1979 4976 y Fu(1)2127 4754 y Fl(\023)2226 4870 y Fr(~)2217 4895 y(U)38 b Fs(=)28 b(0)266 5180 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